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--- author: - Attila Pór - 'David R. Wood' title: | The Big-Line-Big-Clique Conjecture\ is False for Infinite Point Sets --- Let $P$ be a finite set of points in the plane. Two distinct points $v$ and $w$ in the plane are *visible* with respect to $P$ if no point in $P$ is in the open line segment $\overline{vw}$. @KPW-DCG05 made the following Ramsey-theoretic conjecture, which has recently received considerable attention [@KPW-DCG05; @Luigi; @EmptyPentagon-GC; @ABBCLPW; @PorWood-JoCG; @Matousek09]. [**Big-Line-Big-Clique Conjecture**]{} [@KPW-DCG05] For all $k\geq2$ and $\ell\geq2$ there is an integer $n$ such that every finite set of at least $n$ points in the plane: - contains $\ell$ collinear points, or - contains $k$ pairwise visible points. This conjecture is true for $k\leq 5$ or $\ell\leq3$ [@KPW-DCG05; @Luigi; @EmptyPentagon-GC], and is open for $k=6$ or $\ell=4$. Note that the natural approach for attacking this conjecture using extremal graph theory fails [@PorWood-JoCG]. Another natural approach for attacking the Big-Line-Big-Clique Conjecture is to follow an infinitary compactness argument (which can be used to establish many results in Ramsey theory). The purpose of this note is to show that this conjecture is false for infinite point sets, which suggests that an infinitary compactness argument cannot work. There is a countably infinite point set with no 4 collinear points and no 3 pairwise visible points. Let $x_1,x_2,x_3$ be three non-collinear points in the plane. Given points $x_1,\dots,x_{n-1}$, define $x_n$ as follows. By the Sylvester-Gallai theorem, there is a line through exactly two of $x_1,\dots,x_{n-1}$. Choose such a line $\overleftrightarrow{x_ix_j}$ with $i<j$ such that $j$ is minimum and then $i$ is minimum. Insert $x_n$ on $\overline{x_ix_j}$, such that $\{x_i,x_n,x_j\}$ is the only collinear triple that contains $x_n$. This is possible, since there are only finitely many ($\leq\binom{n-3}{2}$) excluded locations for $x_n$. Repeat this step to obtain a point set $\{x_i:i\in\mathbb{N}\}$, which by construction, contains no 4 collinear points. Moreover, if $x_i$ and $x_k$ are visible with $i<k$, then $x_i$ and $x_k$ are collinear with some other point $x_{i'}$ (otherwise some point would be added at a later stage in $\overline{x_ix_k}$). Since $i<k$ we have $x_k\in\overline{x_ix_{i'}}$ and $i'<k$. Suppose on the contrary that three points $x_i,x_j,x_k$ are pairwise visible, where $i<j<k$. As proved above, $x_k\in\overline{x_ix_{i'}}$ and $x_k\in\overline{x_jx_{j'}}$, where $i',j'<k$. Since $x_k$ is in only one collinear triple amongst $x_1,\dots,x_k$, we have $i'=j$ and $j'=i$. Thus $x_i,x_k,x_j$ are collinear, and $x_i$ and $x_j$ are not visible. This contradiction proves that no 3 points are pairwise visible. \#1[0=7=0 7 by-1ext\#1\#1T\#1 \#1 d\#1\#1 D\#1\#1 l\#1\#1 L\#1\#1\#1]{} [6]{} \[1\][\#1]{} \[1\][`#1`]{} urlstyle \[1\][doi:\#1]{} <span style="font-variant:small-caps;">Zachary Abel, Brad Ballinger, Prosenjit Bose, Sébastien Collette, Vida Dujmović, Ferran Hurtado, Scott D. Kominers, Stefan Langerman, Attila Pór, and David R. Wood</span>. Every large point set contains many collinear points or an empty pentagon. *Proc. 21st Canadian Conference on Computational Geometry* (CCCG ’09), pp. 99–102, 2009. *Graphs and Combinatorics*, to appear. <http://arxiv.org/abs/0904.0262>. <span style="font-variant:small-caps;">Louigi Addario-Berry, Cristina Fernandes, Yoshiharu Kohayakawa, Jos Coelho de Pina, and Yoshiko Wakabayashi</span>. On a geometric [R]{}amsey-style problem, 2007. <http://crm.umontreal.ca/cal/en/mois200708.html>. <span style="font-variant:small-caps;">Greg Aloupis, Brad Ballinger, Prosenjit Bose, Sébastien Collette, Stefan Langerman, Attila Pór, and David R. Wood</span>. Blocking coloured point sets. In *Proc. 26th European Workshop on Computational Geometry* (EuroCG ’10), pp. 29–32. 2010. <http://arxiv.org/abs/1002.0190>. <span style="font-variant:small-caps;">Jan K[á]{}ra, Attila P[ó]{}r, and David R. Wood</span>. On the chromatic number of the visibility graph of a set of points in the plane. *Discrete Comput. Geom.*, 34(3):497–506, 2005. <http://dx.doi.org/10.1007/s00454-005-1177-z>. <span style="font-variant:small-caps;">Ji[ř]{}[í]{} Matou[š]{}ek</span>. Blocking visibility for points in general position. *Discrete Comput. Geom.*, 42(2):219–223, 2009. <http://dx.doi.org/10.1007/s00454-009-9185-z>. <span style="font-variant:small-caps;">Attila Pór and David R. Wood</span>. On visibility and blockers. *J. Comput. Geom.*, 1(1):29–40, 2010. <http://www.jocg.org/index.php/jocg/article/view/24>.

--- abstract: 'This article classifies the real forms of Lie Superalgebra by Vogan diagrams, developing Borel de Seibenthal theorem of semisimple Lie algebras for Lie superalgebras. A Vogan diagram is a Dynkin diagram of triplet $(\mathfrak{g}_{C},\mathfrak{h_{\overline{0}}},\triangle^{+})$, where $\mathfrak{g}_{C}$ is a real Lie superalgebra, $\mathfrak{h_{\overline{0}}}$ cartan subalgebra, $\triangle^{+}$ positive root system. Although the classification of real forms of contragradient Lie superalgebras already furnished but this article’s method is a quicker one to classify.' author: - 'B Ransingh$^{*}$ and K C Pati' title: A quick proof of classification of real forms of Basic Lie superalgebras by Vogan diagrams --- Department of Mathematics\ National Institute of Technology\ Rourkela (India) Email$^{*}$- bransingh@gmail.com 2010 AMS Subject Classification : 17B05, 17B22, 17B40\ Keywords : Lie superalgebras, Vogan diagrams Introduction ============ For a complex semi-simple Lie algebra $\mathfrak{g}$, it is well known that the conjugacy classes of real forms of $\mathfrak{g}$ are in one to one correspondence with the conjugacy classes of involutions of $\mathfrak{g}$, if we associate to a real form $\mathfrak{g}_{\mathbb{R}}$ one of its Cartan involutions $\theta$. Using a suited pair $(\mathfrak{h}, \prod)$ of a Cartan subalgebra $\mathfrak{h}$ and a basis $\prod$ of the associated root system, an involution is described by a “Vogan diagram”. Knapp [@Knapp:Lie; @groups] brought Vogan diagrams of simple Lie algebras into the light to represent the real forms of the complex simple Lie algebras. Batra [@batra:affine; @batra:vogan] developed a corresponding theory of Vogan diagrams for almost compact real forms of indecomposable nontwisted affine Kac-Moody Lie algebras. Tanushree [@tanushree:twisted] developed the theory of Vogan diagrams for almost compact real forms of indecomposable twisted affine Kac-Moody Lie algebras. Similar theory is also developed to find out the Vogan diagrams of hyperbolic Kac-Moody algebras[@ransingh:pati]. A Vogan diagram is a Dynkin diagram with some additional information as follows the 2-elements orbits under $\theta$ (Cartan involution) are exhibited by joining the corresponding simple roots by a double arrow and the 1-element orbit is painted in black (respectively, not painted), if the corresponding imaginary simple root is noncompact (respectively compact). The real form is defined as a real Lie superalgebra such that its complexification is the original complex Lie superalgebra. It can be seen easily that every standard real form is naturally associated to an antilinear involutive automorphism of the complex Lie superalgebra. The classification of real semisimple Lie algebras will use maximally compact and split Cartan subalgebras, The Vogan diagram is based on the classification of maximally compact Cartan subalgebras. Recently similar work has been done using Vogan superdiagrams to classify the real forms of contragradient Lie superalgebras[@chuah:vsuper], where the extended Dynkin diagrams of Lie superalgebra is used. But our method uses the ordinary Dynkin diagram as done by Knapp[@Knapp:Lie; @groups]. In this article we construct all the real forms of Lie superalgebras by Vogan diagrams. Preliminary =========== The general and special linear Lie superalgebras. ------------------------------------------------- Let $V=V_{\overline{0}}\oplus V_{\overline{1}}$ be a vector superspace, so that End$(V)$ is an associative superalgebra. The End$(V)$ with the supercummutator forms a Lie superalgebra, called the general linear Lie superalgebra and is denoted by $\mathfrak{gl}(m|n)$, where $V=\mathbb{C}^{m|n}$. With respect to an suitable ordered basis End$(V)$ and $\mathfrak{gl}(m|n)$ can be realized as $(m+n)\times(m+n)$ complex matrices of the block form. $\left(\begin{array}{cc} a & b\\ c & d \end{array}\right)$ where $a$ , $b$, $c$ and $d$ are respectivily $m\times m$, $m\times n$, $n\times m$ and $n\times n$ matrices. The even subalgebra of $\mathfrak{gl}(m|n)$ is $\mathfrak{gl}(m)\oplus\mathfrak{gl}(n)$ , which consists of matrices of the form $\left(\begin{array}{cc} a & 0\\ 0 & d \end{array}\right)$, While the odd subspace consists of $\left(\begin{array}{cc} 0 & b\\ c & 0 \end{array}\right)$ A Lie superalgebras $\mathfrak{g}$ is an algebra graded over $\mathbb{Z}_{2}$ , i.e., $\mathfrak{g}$ is a direct sum of vector spaces $\mathfrak{g}=\mathfrak{g}_{\overline{0}}\oplus \mathfrak{g}_{\overline{1}}$, and such that the bracket satisfies 1. $[\mathfrak{g}_{i}, \mathfrak{g}_{j}]\subset\mathfrak{g} _{i+j(mod2)}$, 2. $[x,y]=-(-1)^{|x||y|}[y,x]$, (Skew supersymmetry) $\forall$ homogenous $x,y,z\in \mathfrak{g}$ (Super Jacobi identity) 3. $[x,[y,z]]=[[x,y],z]+\left(-1\right)^{|x||y|}[y,[x,z]]\forall z\in \mathfrak{g}$ A bilinear form $(.,.):\mathfrak{g}\times\mathfrak{g}\rightarrow\mathbb{C}$ on a Lie superalgebra is called **invariant** if $([x,y],z)=(x,[y,z])$, for all $x,y,z\in \mathfrak{g}$ The Lie superalgebra $\mathfrak{g}$ has a root space decomposition with respect to $\mathfrak{h}$ $$\mathfrak{g}=\mathfrak{h}\oplus\bigoplus_{\alpha\in\triangle}\mathfrak{g}_\alpha$$ A root $\alpha$ is even if $\mathfrak{g}_{\alpha}\subset\mathfrak{g}_{\overline{0}}$ and it is odd if $\mathfrak{g}_{\alpha}\subset\mathfrak{g}_{\overline{1}}$ A *Cartan subalgebra* $\mathfrak{h}$ of diagonal matrices of $\mathfrak{g}$ is defined to be a Cartan subalgebra of the even subalgebra $\mathfrak{g}_{\overline{0}}$. Since every inner automorphism of $\mathfrak{g}_{\overline{0}}$ extends to one of Lie superalgebra $\mathfrak{g}$ and Cartan subalgebras of $\mathfrak{g}_{\overline{0}}$ are conjugate under inner automorphisms. So the Cartan subalgebras of $\mathfrak{g}$ are conjugate under inner automorphism. Real forms of Basic Lie superalgebras -------------------------------------- \[[@Parker:classification] Proposition 1.4\] Let $\mathcal{\mathfrak{g}}$ be a complex classical Lie superalgebra and let $C$ be an involutive semimorphism of $\mathcal{\mathfrak{g}}$. Then $\mathcal{\mathfrak{g}}_{C}=\left\{ x+Cx|x\in\mathfrak{g}\right\} $ is a real classical Lie superalgebra. \[[@Parker:classification] Proposition 1.5\] If $\mathcal{\mathfrak{g}_C}$ is a real classical Lie superalgebra, its complexification $\mathfrak{g}=\mathfrak{g}_{C}\otimes\mathbb{C}$ is a Lie superalgebra which is either classical or direct sum of two isomorphic ideals which are classical \[[@Parker:classification] Theorem 4\] Up to isomorphism, the real forms of the classical Lie superalgebras are uniquely determined by the real form $\mathcal{\mathfrak{g}}_{\overline{0}C}$ of the Lie subalgebra $\mathcal{\mathfrak{g}}_{\overline{0}}$. The real form is said to standard (graded) when the real structure is standard (graded). Let $\mathfrak{g}_{C}$ be a real form of $\mathfrak{g}$ and let $\omega$ be the corresponding complex conjugation. Then $\omega|_{\mathfrak{g}_{\overline{0}}}$ is an antilinear involution of the Lie algebra $\mathfrak{g}_{\overline{0}}$. Hence there is a corresponding Cartan decomposition $\mathfrak{g}_{\overline{0}}=t_{\overline{0}}\oplus p_{\overline{0}}$, with Cartan involution $C$. The following table gives a list of all the real forms associated with basic classical Lie superalgebras. ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- $\mathfrak{g}$ $\mathfrak{g}_{\overline{0}}$ $\mathfrak{g}_C$ $\mathfrak{g}_{\overline{0}C}$ ------------------- ------------------------------------------------------------------ ------------------------------------------ --------------------------------------------------------------------------------------------------- $A(m,n)$ $\mathfrak{sl}(m)\oplus \mathfrak{sl}(n)\oplus U(1)$ $\begin{array}{l} $\begin{array}{l} \mathfrak{sl}(m|n;\mathbb{R})\\ \mathfrak{sl}(m,\mathbb{R})\oplus \mathfrak{sl}(n,\mathbb{R})\oplus\mathbb{R}\\ \mathfrak{sl}(m|n;\mathbb{H})\\ \mathfrak{su}^{*}(m)\oplus \mathfrak{su}^{*}(n)\oplus\mathbb{R}\\ \mathfrak{su}(p,m-p|q,n-q) \mathfrak{su}(p,m-p)\oplus \mathfrak{su}(q,n-q)\oplus i\mathbb{R} \end{array}$ \end{array}$ $A(n,n)$ $\mathfrak{sl}(n)\oplus \mathfrak{sl}(n)$ $\begin{array}{l} $\begin{array}{l} \mathfrak{psl}(n|n;\mathbb{R})\\ \mathfrak{sl}(n,\mathbb{R})\oplus \mathfrak{sl}(n,\mathbb{R})\\ \mathfrak{psl}(n|n;\mathbb{H})\\ \mathfrak{su}^{*}(n)\oplus \mathfrak{su}^{*}(n)\\ \mathfrak{su}(p.n-p|q,n-q) \mathfrak{su}(p,n-p)\oplus \mathfrak{su}(q,n-q) \end{array}$ \end{array}$ $\begin{array}{l} $\begin{array}{l} $\begin{array}{l} $\begin{array}{l} B(m,n)\\ \mathfrak{so}(2m+1)\oplus \mathfrak{sp}(2n)\\ \mathfrak{osp}(p,2m+1-p|2n;\mathbb{R})\\ \mathfrak{so}(p,2m+1-p)\oplus \mathfrak{sp}(2n;\mathbb{R})\\ B(0,n) \mathfrak{sp}(2n) \mathfrak{osp}(1|2n;\mathbb{R}) \mathfrak{sp}(2n,\mathbb{R}) \end{array}$ \end{array}$ \end{array}$ \end{array}$ $C(n+1)$ $\mathfrak{so}(2)\oplus \mathfrak{sp}(2n)$ $\begin{array}{l} $\begin{array}{l} \mathfrak{osp}(p|2n;\mathbb{R})\\ \mathfrak{so}^{*}(2)\oplus \mathfrak{sp}(2n;\mathbb{R})\\ \mathfrak{osp}(2|2q,2n-2q;\mathbb{H}) \mathfrak{so}^{*}(2)\oplus \mathfrak{sp}(2q,2n-2q) \end{array}$ \end{array}$ $D(m,n)$ $\mathfrak{so}(2m)\oplus \mathfrak{sp}(2n)$ $\begin{array}{l} $\begin{array}{l} \mathfrak{osp}(p,2m-p|2n;\mathbb{R})\\ \mathfrak{so}(p,2m-p)\oplus \mathfrak{sp}(2n;\mathbb{R})\\ \mathfrak{osp}(2m|2q,2n-2q;\mathbb{H}) \mathfrak{so}^{*}(2m)\oplus \mathfrak{sp}(2q,2n-2q)\\ \end{array}$ \mathfrak{sp}(n)\oplus \mathfrak{so}^{*}(2m)\\ \mathfrak{so}(2m)\oplus \mathfrak{so}^{*}(2n,\mathbb{R}) \end{array}$ $F(4)$ $\mathfrak{sl}(2)\oplus \mathfrak{so}(7)$ $\begin{array}{l} $\begin{array}{l} F(4;0)\\ \mathfrak{sl}(2,\mathbb{R})\oplus \mathfrak{so}(7)\\ F(4;3)\\ \mathfrak{sl}(2,\mathbb{R})\oplus \mathfrak{so}(1,6)\\ F(4;2)\\ \mathfrak{sl}(2,\mathbb{R})\oplus \mathfrak{so}(2,5)\\ F(4;1) \mathfrak{sl}(2,\mathbb{R})\oplus \mathfrak{so}(3,4) \end{array}$ \end{array}$ $G(3)$ $\mathfrak{sl}(2)\oplus G_{2}$ $\begin{array}{l} $\begin{array}{l} G(3,0)\\ \mathfrak{sl}(2,\mathbb{R})\oplus G_{2,0}\\ G(3,1) \mathfrak{sl}(2,\mathbb{R})\oplus G_{2,2} \end{array}$ \end{array}$ $D(2,1;\alpha)$ $\mathfrak{sl}(2)\oplus \mathfrak{sl}(2)\oplus \mathfrak{sl}(2)$ $\begin{array}{l} $\begin{array}{l} D(2,1;\alpha;0)\\ \mathfrak{sl}(2,\mathbb{R})\oplus \mathfrak{sl}(2,\mathbb{R})\oplus \mathfrak{sl}(2,\mathbb{R})\\ D(2,1;\alpha;1)\\ \mathfrak{sl}(u)\oplus \mathfrak{sl}(u)\oplus \mathfrak{sl}(u)\\ D(2,1;\alpha;2) \mathfrak{sl}(2,\mathbb{C})\oplus \mathfrak{sl}(2,\mathbb{R}) \end{array}$ \end{array}$ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Vogan diagrams of Basic Lie Superalgebras ========================================= The following Lemma shows the existence of \[Lemma 4.7 [@kac:denominator]\]There exists $C\in Aut(\mathfrak{g})$ such that $C|_{g_{0}}$ is a nontrivial automorphism if and only if $\mathfrak{g}$ is of type $A(m,n)$, $D(m,n)$, $D(2,1;\alpha)$ ,$\alpha\in\{1,(-2)^{\pm 1}\}$ and $C|_{g_{0}}$is as follows. 1. If $\mathfrak{g}$ is of type $A(m,n)$ with $n\ne m$, then $C|_{\mathfrak{g}_0}$ restricts to the nontrivial diagram automorphism of both $A_n$ and $A_m$. 2. If $\mathfrak{g}$ is of type $A(n,n)$, then $C|_{\mathfrak{g}_{0}}$ is either the nontrivial diagram automorphism of both $A_n$ components, or it is the flip automorphism between the two $A_n$ components, or the composition of these two automorphisms. 3. If $\mathfrak{g}$ is of type $D(m,n),\,m>2$, then $C|_{\mathfrak{g}_0}$ is the unique diagram automorphism of $\mathfrak{g}_0$. 4. If $\mathfrak{g}$ is of type $D(2,1)\cong D(2,1,a)$, $a\in{1, (-2)^{\pm1} }$, then $C$ is the unique diagram automorphism of the diagram. The Vogan diagram of Lie superalgebras is the Vogan diagram of the even part of Lie superalgerbas. In addition to that - The vertices fixed by the Cartan involution of the even part is painted (or unpainted) depending whether the the root is noncompact (or compact). - Label the 2- elements orbit by the diagram automorphism indicated with two sided arrow. - The odd root remain unchanged. An abstract Vogan diagram is an abstract Dynkin diagram with two pieces of additional structure , one is an automorphism of order 1 or 2 of the diagram, which is to be indicated by labeling the 2 element orbits. The other is the subset of the 1 element orbits which is to be indicated by painting the vertices corresponding to the members of the subset of noncompact roots. Every Vogan diagram is of course an abstract Vogan diagram of Lie superalgebra. If an abstract Vogan diagram is given, then there exist a real Lie superalgebra $\mathfrak{g}_{C}$, a Cartan involution $\theta$, a maximally compact $\theta$ stable Cartan subalgebra and a positive system $\triangle_{\overline{0}}^{+}$ for $\triangle=\triangle(\mathfrak{g,h})$ that takes $it_{\overline{0}}$ before $a_{\overline{0}}$ such that the given diagram is the Vogan diagram of $(\mathfrak{g}_{C},\mathfrak{h}_{\overline{0}},\triangle_{0}^{+})$. Briefly the theorem says that any abstract Vogan diagram comes from some $\mathcal{\mathfrak{g}}_{\overline{0}}$. Thus the theorem is an analog for real semisimple Lie algebras of the existence theorem for complex semisimple Lie algebras. We will modify the Borel and de Siebenthal Theorem for Lie superalgebra. Let $\mathfrak{g}_{C}$ be a non complex real Lie superalgebra and Let the Vogan diagram of $\mathfrak{g}_{C}$ be given that correponding to the triple $(\mathfrak{g}_{C},\mathfrak{h}_{0},\triangle^{+})$. Then $\exists$ a simple system $\prod'$ for $\triangle=\triangle(\mathcal{\mathfrak{g}},\mathfrak{h})$, with corresponding positive system $\triangle^{+}$, such that $(\mathcal{\mathfrak{g}}_ {C},\mathfrak{h}_{\overline{0}},\triangle^{+})$ is a triple and there is at most two painted simple root in its Vogan diagrams of $A(m,n),D(m,n)$ and at most three painted vertices in $D(2,1;\alpha)$. Furthermore suppose the automorphism associated with the Vogan diagram is the identity,that $\prod'={\alpha_{1},\cdots,\alpha_{l}}$ and that ${\omega_{1},\cdots,\omega_{l}}$ is the dual basis for each even part such that $\left\langle\omega_{j},\alpha_{k}\right\rangle=\delta_{jk}/\epsilon_{kk} $, where $\epsilon_{kk}$ is the diagonal entries to make cartan matrix symmetric. The the double painted simple root of even parts may be chosen so that there is no $i'$ with $\left\langle \omega_{i}-\omega_{i'},\omega_{i'}\right\rangle>0$ for each even part. We know $\mathfrak{g}=\mathcal{\mathfrak{g}}_{\bar{0}}\oplus\mathcal{\mathfrak{g}}_{\bar{1}}$. The positive even root system $\triangle_{0}^{+}$ can be written as $$\triangle_{0}^{+}=\triangle_{01}^{+}\cup\triangle_{02}^{+}$$ where $\triangle_{01}^{+}$ are the even positive root system for simple root system formed by $e_{i}$ basis and $\triangle_{02}^{+}$ are for $\delta_{j}$ basis. For the even part, we take $<\omega_{i},\alpha_{j}>=\delta_{ij}/\epsilon_{kk}$ . This makes the Cartan matrix symmetric and so that we can get the inverse of cartan matrix of $A_{m}$ and $A_n$ for $A(m,n)$ Lie superalgebra. Similarly construction will follows for other Lie superalgebras. Each inverse for even part is associted with the dual basis $\omega$. For the odd part the condition is $<\omega_{i},\alpha_{j}>=\delta_{ij}$ and donot get any painted vertices. The Symmetrizable condition of Kac-Cartan matrix gives $S=\epsilon_{kk} A$, where $S$ is the symmetric cartan matrix. The below table gives the values of $\epsilon_{kk}$ for different superalgebras. Lie superalgebra $\epsilon_{kk}$ ------------------ ------------------------------------- $A(m,n)$ $(1,\cdots,1,-1,\cdots,-1)$ $B(m,n)$ $(1,\cdots,1,-1\cdots,-1,-2)$ $B(0,n)$ $1,\cdots,1,2$ $C(n)$ $(-1,1,\cdots,1,\frac{1}{2})$ $D(m,n)$ $(1,\cdots,1,-1-1,\cdots,-1,-1,-1)$ $D(2,1;\alpha)$ $(1,-1,\frac{1}{a})$ $F(4)$ $ (-1,1,\frac{1}{2}) $ $G(3)$ $(-\frac{1}{2},1,\frac{1}{3},)$ Taking suitable normalization condition for each type of Lie superalgebras and from the two Lemmas 6.97 and 6.98 [@Knapp:Lie; @groups] we get redudancy test for each even part. So now the Vogan diagram of Lie superalgebras becomes two painted vertices Vogan diagram. Let $\bigtriangleup$ be an irreducible abstract reduced root system in a real vector superspace $V$, let $\Pi_{01}$ and $\Pi_{02}$ be the two simple simple root system for even parts $e_{i}$ and $\delta_{j}$ basis respectivily and let $\omega$ and $\omega'$ be nonzero members of $V$ that are domiant relative to $\Pi_{i}'$s. Then $\left\langle \omega,\omega'\right\rangle >0$. Using the suitable normalisations of $e_i$ and $\delta_j$ we get the proof of the Lemma. Let $\mathcal{\mathfrak{g}}_{0C}$ be a noncomplex simple real Lie superalgebra and let the Vogan diagram of $\mathcal{\mathfrak{g}}_{0C}$ be given that corresponding to the triple $(\mathfrak{g}_{0},\mathfrak{h}_{0},\triangle^{+})$. Write $\mathfrak{h}_{01}=\mathfrak{t}_{01}\oplus\mathfrak{a}_{01}$ and $\mathfrak{h}_{02}=\mathfrak{t}_{02}\oplus\mathfrak{a}_{02}$ for two even parts. . Let $V$ be the span of simple roots that are imaginary, let $\triangle_{0}$ be the root system $\triangle\cap V$, let $\mathcal{H}$ be the subset of $it_{0}$ paired with $V$ and let $\Lambda$ be the subset of $\mathcal{H}$ where all roots of $\triangle_{0}$ take integer values and all noncompact roots of $\triangle_{0}$ take odd integer values. Then $\Lambda$ is nonempty. In fact if $\alpha_{1},\cdots,\alpha_{m}$ is any simple system for $\triangle_{0}$ and if $\omega_{1},\cdots,\omega_{m}$ in $V$ are defined by $\left\langle \omega,\alpha_{k}\right\rangle =\delta_{jk}$, then the element $$\omega=\underset{i\mbox{ with }\alpha_{i}\mbox{ noncompact}}{\sum}\omega_{i}$$ Fix a simple system $\alpha_{1},\cdots,\alpha_{m}$ for $\bigtriangleup_{\overline{0}}$ and let $\bigtriangleup^{+}_{\overline{0}}$ be the set of positive roots of $\bigtriangleup_{\overline{0}}$. Define $\omega_{1},\cdots,\omega_{m}$ by $\left\langle \omega,\alpha_{k}\right\rangle =\delta_{jk}$. If $\alpha=\sum_{i=1}^{m} n_{i}\alpha_{i}$ is a positive root of $\bigtriangleup_{\overline{0}}$, then $\left\langle \omega,\alpha\right\rangle $ is the sum of the $n_{i}$ for which $\alpha_{i}$ is noncompact. Using induction of the Lemma 6.98 [@Knapp:Lie; @groups] for even part of root system the above Lemma will be proved and each even roots satisfy compact root + compact root = compact root compact root+ noncompact root = noncompact root noncompact root + noncompact root = noncompact root 1. $A(m,n)$ The Vogan diagrams and real foms of Lie superalgebras $A(m,n)$ are as follows. $$\begin{picture}(60,10) \thicklines \put(-125,0){\circle{9}} \put(-97,0){\circle{9}}\put(-70,0){\circle{9}} \put(-43,0){\circle{9}} \put(-20,-2.5){$\bigotimes$}\put(12,0){\circle{9}} \put(39,0){\circle{9}}\put(66,0){\circle{9}}\put(93,0){\circle{9}} \put(-120,0){\line(1,0){19}} \put(-92,0){\dottedline{4}(1,0)(17,0)}\put(-65,0){\line(1,0){17}}\put(-38,0){\line(1,0){19}} \put(-10,0){\line(1,0){17}} \put(16,0){\line(1,0){18}} \put(43,0){\dottedline{4}(1,0)(18,0)}\put(70,0){\line(1,0){19}} \put(-15,-25){\makebox(0,0){$\mathfrak{sl}(n,\mathbb{C})$}} \end{picture}$$ $$\begin{picture}(60,10) \thicklines \put(-125,0){\circle{9}} \put(-97,0){\circle*{9}}\put(-70,0){\circle{9}} \put(-43,0){\circle{9}} \put(-20,-2.5){$\bigotimes$}\put(12,0){\circle{9}} \put(39,0){\circle{9}}\put(66,0){\circle*{9}}\put(93,0){\circle{9}} \put(-120,0){\dottedline{4}(1,0)(19,0)} \put(-92,0){\dottedline{4}(1,0)(17,0)}\put(-65,0){\line(1,0){17}}\put(-38,0){\line(1,0){19}} \put(-10,0){\line(1,0){17}} \put(16,0){\line(1,0){18}} \put(43,0){\dottedline{4}(1,0)(18,0)}\put(70,0){\dottedline{4}(1,0)(19,0)} \put(-65,-25){\makebox(0,0){$\mathfrak{su}(p,m-p)$}}\put(65,-25){\makebox(0,0){$\mathfrak{su}(r,n-r)$}} \end{picture}$$ $$\begin{picture}(60,10) \thicklines \put(-125,0){\circle{9}} \put(-97,0){\circle{9}}\put(-70,0){\circle{9}} \put(-43,0){\circle{9}} \put(-20,-2.5){$\bigotimes$}\put(12,0){\circle{9}} \put(39,0){\circle{9}}\put(66,0){\circle{9}}\put(93,0){\circle{9}} \put(-120,0){\line(1,0){19}} \put(-92,0){\dottedline{4}(1,0)(17,0)}\put(-65,0){\line(1,0){17}}\put(-38,0){\line(1,0){19}} \put(-10,0){\line(1,0){17}} \put(16,0){\line(1,0){18}} \put(43,0){\dottedline{4}(1,0)(18,0)}\put(70,0){\line(1,0){19}} \qbezier(-97,-6)(-81.5,-14)(-69, -6) \put(-97,-6){\vector( -2, 1){0}}\put(-69,-6){\vector( 2, 1){0}} \qbezier(-125,-6)(-81,-24)(-43, -6) \put(-125,-6){\vector( -2, 1){0}}\put(-43,-6){\vector( 2, 1){0}} \qbezier(39,-6)(52.5,-14)(66, -6) \put(39,-6){\vector( -2, 1){0}}\put(66,-6){\vector( 2, 1){0}} \qbezier(12,-6)(52.5,-24)(93, -6) \put(12,-6){\vector( -2, 1){0}}\put(93,-6){\vector( 2, 1){0}} \put(-75,-25){\makebox(0,0){$\mathfrak{sl}(m,\mathbb{R})$}}\put(55,-25){\makebox(0,0){$\mathfrak{sl}(n,\mathbb{R})$}} \end{picture}$$ $$\begin{picture}(60,10) \thicklines \put(-152,0){\circle{9}}\put(-125,0){\circle{9}} \put(-97,0){\circle*{9}}\put(-70,0){\circle{9}} \put(-43,0){\circle{9}} \put(-20,-2.5){$\bigotimes$}\put(12,0){\circle{9}} \put(39,0){\circle{9}}\put(66,0){\circle*{9}}\put(93,0){\circle{9}}\put(120,0){\circle{9}} \put(-147,0){\line(1,0){18}}\put(-120,0){\dottedline{4}(1,0)(19,0)} \put(-92,0){\dottedline{4}(1,0)(17,0)} \put(-65,0){\line(1,0){17}}\put(-38,0){\line(1,0){19}} \put(-10,0){\line(1,0){17}} \put(16,0){\line(1,0){18}} \put(43,0){\dottedline{4}(1,0)(18,0)}\put(70,0){\dottedline{4}(1,0)(19,0)} \put(98,0){\line(1,0){18}} \qbezier(-125,-6)(-97.5,-16)(-70, -6) \put(-125,-6){\vector( -3, 1){0}}\put(-70,-6){\vector( 3, 1){0}} \qbezier(-152,-6)(-97.5,-32)(-43, -6) \put(-152,-6){\vector( -2, 1){0}}\put(-43,-6){\vector(2, 1){0}} \qbezier(39,-6)(66,-16)(93, -6) \put(39,-6){\vector( -3, 1){0}}\put(93,-6){\vector( 3, 1){0}} \qbezier(12,-6)(66,-32)(120, -6) \put(12,-6){\vector( -2, 1){0}}\put(120,-6){\vector(2, 1){0}} \put(-95,-28){\makebox(0,0){$\mathfrak{su}^{*}(m)$}}\put(65,-28){\makebox(0,0){$\mathfrak{su}^{*}(n)$}} \end{picture}$$ 2. $B(m,n)$ The Lie subalgebra of $\mathfrak{g}_{0}$ is $C_{m}\oplus B_{n}$ The only trivial automophism of even part of Vogan diagram of $B(m,n)$ is shown below and the real form is $\mathfrak{sp}(2n,\mathbb{R})\oplus \mathfrak{so}(p,2m+1-p)$ $$\begin{picture}(60,10) \thicklines \put(-97,0){\circle{9}}\put(-70,0){\circle{9}} \put(-43,0){\circle{9}} \put(-20,-2.5){$\bigotimes$}\put(12,0){\circle*{9}} \put(39,0){\circle{9}}\put(66,0){\circle{9}} \put(-92,0){\line(1,0){19}}\put(-65,0){\dottedline{4}(1,0)(17,0)}\put(-38,0){\line(1,0){19}}\put(-11,0){\dottedline{4}(1,0)(18,0)} \put(16,0){\dottedline{4}(1,0)(18,0)}\put(37,0){{\begin{picture}(20,20) \put(6,-2){\line(1,0){20}}\put(6,2){\line(1,0){20}} \put(20,0){\line(-1,1){10}}\put(20,0){\line(-1,-1){10}} \end{picture}}} \put(45,-25){\makebox(0,0){$so(p,2m+1-p)$}} \end{picture}$$ Because of missing of real form of the first even part, we need and additional $C_{n}$ Dynkin diagram superimposed Vogan diagrams below. $$\begin{picture}(60,10) \thicklines \put(-97,0){\circle{9}}\put(-70,0){\circle{9}} \put(-43,0){\circle{9}} \put(-20,-2.5){$\bigotimes$}\put(12,0){\circle*{9}} \put(39,0){\circle{9}}\put(66,0){\circle{9}} \put(-65,0){\dottedline{4}(1,0)(17,0)}\put(-38,0){\line(1,0){19}}\put(-11,0){\dottedline{4}(1,0)(18,0)} \put(16,0){\dottedline{4}(1,0)(18,0)}\put(37,0){{\begin{picture}(20,20) \put(6,-2){\line(1,0){20}}\put(6,2){\line(1,0){20}} \put(20,0){\line(-1,1){10}}\put(20,0){\line(-1,-1){10}} \end{picture}}} \put(-65,-25){\makebox(0,0){$sp(2n,\mathbb{R})$}}\put(45,-25){\makebox(0,0){$so(p,2m+1-p)$}} \put(-100,0){\begin{picture}(20,20) \put(6,-2){\line(1,0){20}}\put(6,2){\line(1,0){20}} \put(20,0){\line(-1,1){10}}\put(20,0){\line(-1,-1){10}} \end{picture}}\end{picture}$$ The another real form of Lie superalgebra $B(m,n)$ is $sp(2n,\mathbb{R})\oplus so(2m+1)$ and the corresponding Vogan diagram is drawn below. $$\begin{picture}(60,10) \thicklines \put(-97,0){\circle{9}}\put(-70,0){\circle{9}} \put(-43,0){\circle{9}} \put(-20,-2.5){$\bigotimes$}\put(12,0){\circle{9}} \put(39,0){\circle{9}}\put(66,0){\circle{9}} \put(-65,0){\dottedline{4}(1,0)(17,0)}\put(-38,0){\line(1,0){19}}\put(-11,0){\dottedline{4}(1,0)(18,0)} \put(16,0){\dottedline{4}(1,0)(18,0)}\put(37,0){{\begin{picture}(20,20) \put(6,-2){\line(1,0){20}}\put(6,2){\line(1,0){20}} \put(20,0){\line(-1,1){10}}\put(20,0){\line(-1,-1){10}} \end{picture}}} \put(-65,-25){\makebox(0,0){$sp(2n,\mathbb{R})$}}\put(45,-25){\makebox(0,0){$so(2m+1)$}} \put(-100,0){\begin{picture}(20,20) \put(6,-2){\line(1,0){20}}\put(6,2){\line(1,0){20}} \put(20,0){\line(-1,1){10}}\put(20,0){\line(-1,-1){10}} \end{picture}}\end{picture}$$ 3. Case $B(0,n)$ The Vogan diagram below is a unpainted diagram but it consists of its own painted vertices on the extreme right.\ $$\begin{picture}(60,20) \thicklines \put(-43,0){\circle{9}} \put(-15,0){\circle{9}}\put(12,0){\circle{9}} \put(39,0){\circle{9}}\put(66,0){\circle*{9}} \put(-38,0){\line(1,0){19}}\put(-12,0){\dottedline{4}(1,0)(19,0)}\put(16,0){\line(1,0){19}}\put(37,0){{\begin{picture}(20,20) \put(6,-2){\line(1,0){20}}\put(6,2){\line(1,0){20}} \put(20,0){\line(-1,1){10}}\put(20,0){\line(-1,-1){10}} \end{picture}}} \put(25,-25){\makebox(0,0){$sp(2n,\mathbb{R})$}} \end{picture}$$ 4. Case $C(n+1)$ The unpainted Vogan diagram of $C(n+1)$ creats the real form $so^{*}(2)\oplus sp(2n,\mathbb{R})$\ $$\begin{picture}(60,20) \thicklines \put(-48,-3){$\bigotimes$} \put(-15,0){\circle{9}}\put(12,0){\circle{9}} \put(39,0){\circle{9}}\put(66,0){\circle{9}} \put(-38,0){\line(1,0){19}}\put(-12,0){\dottedline{4}(1,0)(19,0)}\put(16,0){\dottedline{4}(1,0)(19,0)}\put(37,0){{\begin{picture}(20,20) \put(6,-2){\line(1,0){20}}\put(6,2){\line(1,0){20}} \put(15,0){\line(1,-1){10}}\put(15,0){\line(1,1){10}} \end{picture}}} \end{picture}$$ The trivial automophism of the even part of $C(n+1)$ makes the Vogan diagram below and the real form is $so^{*}(2)\oplus sp(2q,2n-2q)$\ $$\begin{picture}(60,20) \thicklines \put(-48,-3){$\bigotimes$} \put(-15,0){\circle{9}}\put(12,0){\circle*{9}} \put(39,0){\circle{9}}\put(66,0){\circle{9}} \put(-38,0){\line(1,0){19}}\put(-12,0){\dottedline{4}(1,0)(19,0)}\put(16,0){\dottedline{4}(1,0)(19,0)}\put(37,0){{\begin{picture}(20,20) \put(6,-2){\line(1,0){20}}\put(6,2){\line(1,0){20}} \put(15,0){\line(1,-1){10}}\put(15,0){\line(1,1){10}} \end{picture}}} \end{picture}$$ 5. Case $D(m,n)$. The Lie subalgebra of $\mathfrak{g}_{0}$ is $C_{m}\oplus D_{n}$ . The compact real form of $C_{m}$ is $\mathfrak{sp}(m)$. The real form $sp(2n,\mathbb{R})\oplus so(2m)$ of the abstract Vogan diagram of $D(m,n)$ for the above subalgebras is followed below. $$\begin{picture}(-20,20)\thicklines \put(-137,0){\circle{9}}\put(-109,0){\circle{9}}\put(-81,0){\circle{9}}\put(-56,-2.5){$\bigotimes$} \put(-22,0){\circle{9}} \put(5,0){\circle{9}}\put(30,16){\circle{9}} \put(30,-16){\circle{9}} \put(-132,0){\dottedline{4}(1,0)(19,0)} \put(-105,0){\dottedline{4}(1,0)(19,0)} \put(-76,0){\line(1,0){20}} \put(-46,0){\dottedline{4}(1,0)(19,0)}\put(-18,0){\dottedline{4}(1,0)(19,0)} \put(10,2){\line(3,2){16}} \put(10,-2){\line(3,-2){16}} \put(5,-25){\makebox(0,0){$\mathfrak{so}(2m)$}} \end{picture}$$ Since from the diagram we get only $\mathfrak{so}(2m)$ part of real form, so for the $\mathfrak{sp}(2n)$ part we need the addition $C_{n}$ diagram in the diagram above. Subsequently the Vogan diagram becomes $$\begin{picture}(-20,20)\thicklines \put(-137,0){\circle{9}}\put(-109,0){\circle{9}}\put(-81,0){\circle{9}}\put(-56,-2.5){$\bigotimes$} \put(-22,0){\circle{9}} \put(5,0){\circle{9}}\put(30,16){\circle{9}} \put(30,-16){\circle{9}} \put(-105,0){\dottedline{4}(1,0)(19,0)} \put(-76,0){\line(1,0){20}} \put(-46,0){\dottedline{4}(1,0)(19,0)}\put(-18,0){\dottedline{4}(1,0)(19,0)} \put(10,2){\line(3,2){16}}\put(10,-2){\line(3,-2){16}} \put(-105,-25){\makebox(0,0){$\mathfrak{sp}(2n,\mathbb{R})$}}\put(5,-25){\makebox(0,0){$\mathfrak{so}(2m)$}} \put(-139,0){\begin{picture}(20,20) \put(6,-2){\line(1,0){20}}\put(6,2){\line(1,0){20}} \put(20,0){\line(-1,1){10}}\put(20,0){\line(-1,-1){10}} \end{picture}}\end{picture}$$ The first trivial involution for one of the even part for the Vogan diagram of $D(m,n)$ is given below and the real form of this diagram is $\mathfrak{sp}(2n,\mathbb{R})\oplus \mathfrak{so}(p,2m-p)$. $$\begin{picture}(-20,20)\thicklines \put(-137,0){\circle{9}}\put(-109,0){\circle{9}}\put(-81,0){\circle{9}}\put(-56,-2.5){$\bigotimes$} \put(-22,0){\circle*{9}} \put(5,0){\circle{9}}\put(30,16){\circle{9}} \put(30,-16){\circle{9}} \put(-105,0){\dottedline{4}(1,0)(19,0)} \put(-76,0){\line(1,0){20}} \put(-46,0){\dottedline{4}(1,0)(19,0)}\put(-18,0){\dottedline{4}(1,0)(19,0)} \put(10,2){\line(3,2){16}}\put(10,-2){\line(3,-2){16}} \put(-105,-25){\makebox(0,0){$\mathfrak{sp}(2n,\mathbb{R})$}}\put(5,-25){\makebox(0,0){$\mathfrak{so}(p,2m-p)$}} \put(-139,0){\begin{picture}(20,20) \put(6,-2){\line(1,0){20}}\put(6,2){\line(1,0){20}} \put(20,0){\line(-1,1){10}}\put(20,0){\line(-1,-1){10}} \end{picture}}\end{picture}$$ The nontrivial involution for the Vogan diagram of $D(m,n)$ is given below and the real form of this diagram is $sp(2q,2n-2q)\oplus so^{*}(2m)$. $$\begin{picture}(-20,20)\thicklines \put(-165,0){\circle{9}} \put(-137,0){\circle{9}} \put(-137,0){\circle{9}}\put(-109,0){\circle*{9}}\put(-81,0){\circle{9}}\put(-56,-2.5){$\bigotimes$} \put(-22,0){\circle{9}} \put(5,0){\circle{9}}\put(30,16){\circle{9}} \put(30,-16){\circle{9}} \put(-132,0){\dottedline{4}(1,0)(19,0)} \put(-105,0){\dottedline{4}(1,0)(19,0)} \put(-76,0){\line(1,0){20}} \put(-46,0){\dottedline{4}(1,0)(19,0)}\put(-18,0){\dottedline{4}(1,0)(19,0)} \put(10,2){\line(3,2){16}}\put(10,-2){\line(3,-2){16}} \qbezier(35,16)(51,1)(35, -16) \put(35,16){\vector( -1, 1){0}}\put(35,-16){\vector( -1, -1){0}} \put(-105,-25){\makebox(0,0){$\mathfrak{sp}(2q,2n-2q)$}}\put(5,-25){\makebox(0,0){$\mathfrak{so}^{*}(2m)$}} \put(-167,0){\begin{picture}(20,20) \put(6,-2){\line(1,0){20}}\put(6,2){\line(1,0){20}} \put(20,0){\line(-1,1){10}}\put(20,0){\line(-1,-1){10}} \end{picture}}\end{picture}$$ The below Vogan diagram is formed by the nontrivial involution . The real form for this superlagebra is $\mathfrak{sp}(n)\oplus \mathfrak{so}^{*}(2m)$. $$\begin{picture}(-20,20)\thicklines \put(-137,0){\circle{9}}\put(-109,0){\circle{9}}\put(-81,0){\circle{9}}\put(-56,-2.5){$\bigotimes$} \put(-22,0){\circle{9}} \put(5,0){\circle{9}}\put(30,16){\circle{9}} \put(30,-16){\circle{9}} \put(-105,0){\dottedline{4}(1,0)(19,0)} \put(-76,0){\line(1,0){20}} \put(-46,0){\dottedline{4}(1,0)(19,0)}\put(-18,0){\dottedline{4}(1,0)(19,0)} \put(10,2){\line(3,2){16}}\put(10,-2){\line(3,-2){16}} \qbezier(35,16)(51,1)(35, -16) \put(35,16){\vector( -1, 1){0}}\put(35,-16){\vector( -1, -1){0}} \put(-105,-25){\makebox(0,0){$\mathfrak{sp}(n)$}}\put(5,-25){\makebox(0,0){$\mathfrak{so}^{*}(2m)$}} \put(-139,0){\begin{picture}(20,20) \put(6,-2){\line(1,0){20}}\put(6,2){\line(1,0){20}} \put(20,0){\line(-1,1){10}}\put(20,0){\line(-1,-1){10}} \end{picture}}\end{picture}$$ 6. Case $D(2,1;\alpha)$ The unpainted and no two element orbit Vogan diagram is given below with the real form. Since we can get only the real form $\mathfrak{su}(2)\oplus \mathfrak{su}(2)$ from ordinary Vogan diagram $$\begin{picture}(40,20)\thicklines \put(0,-2.5){$\bigotimes$}\put(30,16){\circle{9}} \put(30,-16){\circle{9}} \put(10,2){\line(3,2){16}}\put(10,-2){\line(3,-2){16}} \put(100,0){\makebox(0,0){$\mathfrak{su}(2)\oplus \mathfrak{su}(2)$}} \end{picture}$$ So our requisite Vogan diagrams for the suitable real forms are $$\begin{picture}(40,20)\thicklines \put(-22,0){\circle{9}} \put(0,-2.5){$\bigotimes$}\put(30,16){\circle{9}} \put(30,-16){\circle{9}} \put(-18,0){\line(1,0){19}} \put(10,2){\line(3,2){16}}\put(10,-2){\line(3,-2){16}} \put(100,0){\makebox(0,0){$\mathfrak{su}(2)\oplus \mathfrak{su}(2)\oplus \mathfrak{su}(2)$}} \end{picture}$$ $$\begin{picture}(40,20)\thicklines \put(-22,0){\circle*{9}} \put(0,-2.5){$\bigotimes$}\put(30,16){\circle*{9}} \put(30,-16){\circle*{9}} \put(-18,0){\line(1,0){19}} \put(10,2){\line(3,2){16}}\put(10,-2){\line(3,-2){16}} \put(105,0){\makebox(0,0){$\mathfrak{sl}(2,\mathbb {R})\oplus\mathfrak{ sl}(2,\mathbb {R})\oplus \mathfrak{sl}(2,\mathbb {R})$}} \end{picture}$$ The nontrivial involution of the Dynkin diagram of $D(2,1;\alpha)$ makes the following Vogan diagram as shown below. $$\begin{picture}(40,20)\thicklines \put(-22,0){\circle*{9}} \put(0,-2.5){$\bigotimes$}\put(30,16){\circle{9}} \put(30,-16){\circle{9}} \put(-18,0){\line(1,0){19}} \put(10,2){\line(3,2){16}}\put(10,-2){\line(3,-2){16}} \qbezier(35,16)(51,1)(35, -16) \put(35,16){\vector( -1, 1){0}}\put(35,-16){\vector( -1, -1){0}} \put(120,0){\makebox(0,0){$\mathfrak{sl}(2,\mathbb{C})\oplus \mathfrak{sl}(2,\mathbb{R})$}} \end{picture}$$ 7. Case F(4) From the Dynkin diagram we can get only the real form $so(7)$ and the Vogan diagram $$\begin{picture}(60,20) \thicklines \put(-48,-3){$\bigotimes$} \put(-15,0){\circle{9}}\put(12,0){\circle{9}} \put(39,0){\circle{9}} \put(-38,0){\line(1,0){19}}\put(-18,0){{\begin{picture}(20,20) \put(6,-2){\line(1,0){20}}\put(6,2){\line(1,0){20}} \put(15,0){\line(1,-1){10}}\put(15,0){\line(1,1){10}} \end{picture}}}\put(16,0){\line(1,0){19}} \put(0,-32){\makebox(0,0){$\mathfrak{sl}(2,\mathbb{R})\oplus \mathfrak{so}(7)$}} \end{picture}$$ we add the extra even part root to get the desired real forms and Vogan diagrams. $$\begin{picture}(60,20) \thicklines \put(-75,0){\circle*{9}}\put(-48,-3){$\bigotimes$} \put(-15,0){\circle{9}}\put(12,0){\circle{9}} \put(39,0){\circle{9}} \put(-71,0){\line(1,0){24}} \put(-38,0){\line(1,0){19}}\put(-18,0){{\begin{picture}(20,20) \put(6,-2){\line(1,0){20}}\put(6,2){\line(1,0){20}} \put(15,0){\line(1,-1){10}}\put(15,0){\line(1,1){10}} \end{picture}}}\put(16,0){\line(1,0){19}} \put(0,-32){\makebox(0,0){$\mathfrak{sl}(2,\mathbb{R})\oplus \mathfrak{so}(7)$}} \end{picture}$$ $$\begin{picture}(60,20) \thicklines \put(-75,0){\circle{9}}\put(-48,-2.5){$\bigotimes$} \put(-15,0){\circle*{9}}\put(12,0){\circle{9}} \put(39,0){\circle{9}} \put(-71,0){\line(1,0){24}} \put(-38,0){\line(1,0){19}}\put(-18,0){{\begin{picture}(20,20) \put(6,-2){\line(1,0){20}}\put(6,2){\line(1,0){20}} \put(15,0){\line(1,-1){10}}\put(15,0){\line(1,1){10}} \end{picture}}}\put(16,0){\line(1,0){19}} \put(0,-32){\makebox(0,0){$\mathfrak{su}(2)\oplus \mathfrak{so}(1,6)$}} \end{picture}$$ $$\begin{picture}(60,20) \thicklines \put(-75,0){\circle{9}}\put(-48,-2.5){$\bigotimes$} \put(-15,0){\circle{9}}\put(12,0){\circle*{9}} \put(39,0){\circle{9}} \put(-71,0){\line(1,0){24}}\put(-38,0){\line(1,0){19}}\put(-18,0){{\begin{picture}(20,20) \put(6,-2){\line(1,0){20}}\put(6,2){\line(1,0){20}} \put(15,0){\line(1,-1){10}}\put(15,0){\line(1,1){10}} \end{picture}}}\put(16,0){\line(1,0){19}} \put(0,-32){\makebox(0,0){$\mathfrak{su}(2)\oplus \mathfrak{so}(2,5)$}} \end{picture}$$ $$\begin{picture}(60,20) \thicklines \put(-75,0){\circle{9}} \put(-48,-2.5){$\bigotimes$} \put(-15,0){\circle{9}}\put(12,0){\circle{9}} \put(39,0){\circle*{9}} \put(-71,0){\line(1,0){24}} \put(-38,0){\line(1,0){19}}\put(-18,0){{\begin{picture}(20,20) \put(6,-2){\line(1,0){20}}\put(6,2){\line(1,0){20}} \put(15,0){\line(1,-1){10}}\put(15,0){\line(1,1){10}} \end{picture}}}\put(16,0){\line(1,0){19}} \put(0,-32){\makebox(0,0){$\mathfrak{su}(2)\oplus \mathfrak{so}(3,4)$}} \end{picture}$$ 8. Case $G(3)$ The Vogan diagram of $G(3)$ with real form $\mathfrak{sl}(2,\mathbb{R})\oplus G_{2,0}$ and $\mathfrak{sl}(2,\mathbb{R})\oplus G_{2,2}$ are $$\begin{picture}(60,20) \thicklines \put(-34,0){\circle*{9}}\put(-7,-2.5){$\bigotimes$}\put(26,0){\circle{9}}\put(54,0){\circle{9}} \put(-30,0){\line(1,0){24}}\put(3,0){\line(1,0){19}}\put(23,0){{\begin{picture}(20,20) \put(7,0){\line(1,0){20}} \put(6,-3){\line(1,0){22}}\put(6,3){\line(1,0){22}} \put(15,0){\line(1,1){10}}\put(15,0){\line(1,-1){10}} \end{picture}}} \end{picture}$$ $$\begin{picture}(60,20) \thicklines \put(-34,0){\circle*{9}}\put(-7,-2.5){$\bigotimes$}\put(26,0){\circle{9}}\put(54,0){\circle*{9}} \put(-30,0){\line(1,0){24}}\put(3,0){\line(1,0){19}}\put(23,0){{\begin{picture}(20,20) \put(7,0){\line(1,0){20}} \put(6,-3){\line(1,0){22}}\put(6,3){\line(1,0){22}} \put(15,0){\line(1,1){10}}\put(15,0){\line(1,-1){10}} \end{picture}}} \end{picture}$$ #### Acknowledgement: The authors thank National Board of Higher Mathematics, India (Project Grant No. 48/3/2008-R&DII/196-R) for financial support. 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--- abstract: 'We prove that every finite set of homothetic copies of a given compact and convex body in the plane can be colored with four colors so that any point covered by at least two copies is covered by two copies with distinct colors. This generalizes a previous result from Smorodinsky (SIAM J. Disc. Math. 2007). Then we show that for any $k\geq 2$, every three-dimensional hypergraph can be colored with $6(k-1)$ colors so that every hyperedge $e$ contains $\min\{ |e|,k \}$ vertices with mutually distinct colors. This refines a previous result from Aloupis [*et al.*]{} (Disc. & Comp. Geom. 2009). As corollaries, we improve on previous results for conflict-free coloring, $k$-strong conflict-free coloring, and choosability. Proofs of the upper bounds are constructive and yield simple, polynomial-time algorithms.' author: - Jean Cardinal - Matias Korman bibliography: - 'fourCol.bib' title: 'Coloring Planar Homothets and Three-Dimensional Hypergraphs' --- Introduction ============ The well-known graph coloring problem has several natural generalizations to hypergraphs. A rich literature exists on these topics; in particular, the two-colorability of hypergraphs (also known as property B), has been studied since the sixties. In this paper, we concentrate on coloring geometric hypergraphs, defined by simple objects in the plane. Those hypergraphs serve as models for wireless sensor networks, and associated coloring problems have been investigated recently. This includes conflict-free colorings [@shakharcf; @HPS05], and covering decomposition problems [@pachtoth; @pachindecomp; @GV09]. Smorodinsky [@Smo07] investigated the chromatic number of geometric hypergraphs, defined as the minimum number of colors required to make every hyperedge non-monochromatic. He considered hypergraphs induced by a collection $S$ of regions in the plane, whose vertex set is $S$, and the hyperedges are all subsets $S'\subseteq S$ for which there exists a point $p$ such that $S'= \{ R \in S: p\in R \}$. He proved the following result. \[thm\_4col\] - Any hypergraph induced by a family of n simple Jordan regions in the plane such that the union complexity of any $m$ of them is given by $u(m)$ and $u(m)/m$ is non-decreasing is $O(u(n)/n)$-colorable so that no hyperedge is monochromatic. In particular, any finite family of pseudodisks can be colored with $O(1)$ colors. - Any hypergraph induced by a finite family of disks is 4-colorable Later, Aloupis, [*et al.*]{} [@ACCLS09] considered the quantity $c(k)$, defined as the minimum number of colors required to color a given hypergraph, such that every hyperedge of size $r$ has at least $\min \{ r, k\}$ vertices with distinct colors. For hypergraphs induced by a collection of regions in the plane, such that no point is covered more than $k$ times (a $k$-fold packing), this number corresponds to the minimum number of (1-fold) packings into which we can decompose this collection. It generalizes the usual chromatic number, equal to $c(2)$. They proved the following. \[thm\_ck\] Any finite family of pseudodisks in the plane can be colored with $24k+1$ colors in a way that any point covered by $r$ pseudodisks is covered by $\min \{r, k\}$ pseudodisks with distinct colors. #### Our results. We show in Section \[sec\_dual\] that the second statement of Theorem \[thm\_4col\] actually holds for homothets of any compact and convex body in the plane. The proof uses a lifting transformation that allows us to identify a planar graph, such that every hyperedge of the initial hypergraph contains an edge of the graph. The result then follows from the Four Color Theorem. We actually give two definitions of this graph: one is based on a weighted Voronoi diagram construction, while the other relates to Schnyder’s characterization of planar graphs. Schnyder showed that a graph is planar if and only if its vertex-edge incidence poset has dimension at most $3$ [@schnyder]. In Section \[sec\_3D\], we show that the chromatic number $c(k)$ of three-dimensional hypergraphs is at most $6(k-1)$. This improves the constant of Theorem \[thm\_ck\] for this special case, which includes hypergraphs induced by homothets of a triangle. In Section \[sec\_lb\], we give a lower bound for all the above problems. Finally, in Section \[sec\_appl\], we give some corollaries of these results involving other types of colorings, namely conflict-free and $k$-strong conflict-free colorings, and choosability. #### Definitions. We consider hypergraphs defined by [*ranges*]{}, which are compact and convex bodies ${Q}\subset {\mathbb{R}}^d$ containing the origin. The [*scaling*]{} of ${Q}$ by a factor $ \lambda\in{\mathbb{R}}$ is the set $\{\lambda x : x\in {Q}\}$. Note that, the scaling of ${Q}$ with $\lambda=-1$ is the reflection of ${Q}$ around the origin. The [*translate*]{} of ${Q}$ by a vector $t\in{\mathbb{R}}^d$ is the set $\{x+t : x\in {Q}\}$. The [*homothet*]{} of ${Q}$ of [*center*]{} $t$ and [*scaling*]{} $\lambda$ is the set $\{\lambda x + t : x\in {Q}\}$ and is denoted by ${Q}(t,\lambda)$. Given a finite collection $S$ of points in ${\mathbb{R}}^d$, the [*primal hypergraph*]{} defined by these points and a range ${Q}$ has $S$ as vertex set, and $\{ S\cap {Q}' : {Q}'\mathrm{\ homothet\ of\ }{Q}\}$ as hyperedge set. Similarly, the [*dual hypergraph*]{} defined by a finite set $S$ of homothets of ${Q}$ has $S$ as vertex set, and the hyperedges are all subsets $S'\subseteq S$ for which there exists a point $p\in{\mathbb{R}}^d$ such that $S'= \{ R \in S: p\in R \}$ (i.e., the set of ranges of $S$ that contain $p$). While we give these definitions for an arbitrary dimension $d$, we will be mostly concerned by the case $d=2$. For a given range ${Q}$, the chromatic number $c_{{Q}} (k)$ is the minimum number $c$ such that every primal hypergraph (induced by a set of points) can be colored with $c$ colors, so that every hyperedge of size $r$ contains $\min \{r, k\}$ vertices with mutually distinct colors. Similarly, the chromatic number $\bar{c}_{{Q}} (k)$ is the smallest number $c$ such that every dual hypergraph (induced by a set of homothets of ${Q}$) can be $c$-colored so that every hyperedge of size $r$ contains $\min \{r, k\}$ vertices with mutually distinct colors. In what follows, we refer to these two coloring problems as [*primal*]{} and [*dual*]{}, respectively. Such colorings are called [*polychromatic*]{}[^1]. Coloring Primal Hypergraphs {#sec_primal} =========================== As a warm-up, we consider the primal version of the problem for $k=2$. Given a set of points $S\subset\mathbb{R}^2$ and a two-dimensional range ${Q}$, the [*Delaunay graph*]{} of $S$ induced by ${Q}$ is the graph $G_Q(S)=(S,E)$ with $S$ as vertex set [@fortune]. For any two points $p,q\in S$, $pq\in E$ if and only if there exists a homothet ${Q'}$ of ${Q}$ such that ${Q'}\cap S =\{p,q\}$. Note that, the Delaunay graph induced by disks in the plane corresponds to the ordinary Delaunay triangulation, which is planar. In fact, planarity holds for many ranges.  [@BCCS08c; @sarioz]\[lem\_planar\] For any convex range ${Q}\subseteq {\mathbb{R}}^2$ and set of points $S$, $G_Q(S)$ is planar. Previously published versions of this result required that the points of $S$ are in general position (that is, no four points of $S$ are on the boundary of a range). The generalization to any point set was done by Bose [*et al.*]{} [@BCCS08c]. Whenever a homothet ${Q'}$ contains more than $3$ points on its boundary, the edges $uv,uw$ and $vw$ are added to $G_Q(S)$, where $u,v$, and $w$ are the three lexicographically smallest points of $S\cap {Q'}$. With this definition, they showed that planarity holds for any compact and convex range. The compactness requirement was afterwards removed by Sarioz [@sarioz]. ![Proof of Lemma \[lem\_edge\]: given a homothet ${Q'}$ (light grey), we shrink it until further shrinking will have less than two points (dark grey). Afterwards we keep shrinking while remaining tangent to a point $q$ on the boundary until the point in the interior of the range (if any) reaches the boundary. The resulting range ${Q''}$ is depicted in white. []{data-label="fig_shrink"}](fig_shrink){width="40.00000%"} \[lem\_edge\] For any homothet ${Q'}$ containing two or more points of $S$, there exist $p,q\in S \cap {Q'}$ such that $pq\in E$. Let ${Q'}$ be a homothet of center $c_0$ and scaling $\lambda_0$ that contains two or more points of $S$. We shrink it continuously keeping the same center; let $\lambda_{\min}$ be the smallest scaling such that ${Q}(c_0,\lambda_{\min})$ has two (or more) points of $S$. If ${Q}(c_0, \lambda_{\min})$ contains exactly two points $p,q\in S$, we have $pq\in E$ by definition of $G_Q(S)$. However, we might have some kind of degeneracy in which ${Q}(c_0,\lambda_{\min})$ contains three (or more) points of $S$. Observe that this can only happen if there are two or more points on the boundary and possibly an interior point. First consider the case in which there exists a point $p\in S$ in the interior of $\lambda_{\min}$. Pick any point $q\in S$ on the boundary of ${Q}(c_0,\lambda_{\min})$ and shrink the homothet remaining tangent to $q$ until $p$ reaches the boundary (see Figure \[fig\_shrink\]). After this shrinking process, both $p$ and $q$ will be on the boundary of the new homothet. Moreover, any other point that was previously in ${Q}(c_0,\lambda_{\min})$ either remains on the boundary or is not in the range anymore. That is, we can always shrink a range ${Q'}$ to another range ${Q''}\subseteq {Q'}$ that contains two or more points on its boundary and none in the interior. Hence, by the result of [@BCCS08c] we know that there will be an edge between the two lexicographically smallest points of $S \cap {Q''}$. \[theo\_primal\] For any two-dimensional range ${Q}$, we have $c_{Q}(2)\leq4$. By Lemma \[lem\_planar\], $G_Q(S)$ is planar, hence 4-colorable. By Lemma \[lem\_edge\], any homothet ${Q'}$ containing two or more points of $S$ must contain $p,q\in S\cap {Q'}$ such that $pq\in E$. In particular, these points cannot have the same color assigned, hence ${Q'}$ cannot be monochromatic. The proof yields an $O(n^2)$-time algorithm. The bound of Theorem \[theo\_primal\] is tight for a wide class of ranges (see Section \[sec\_lb\]). Coloring Dual Hypergraphs {#sec_dual} ========================= In this section we describe a similar approach for the dual variant of the problem in the plane. Recall that in the dual problem, we are given a set $S$ of compact and convex homothets of ${Q}$, and we are interested in coloring the elements of $S$ so that any point of the plane covered at least twice is covered by two homothets of different colors. For simplicity, we first suppose that no range of $S$ is contained in another; we will show how to remove this assumption afterwards. In order to solve this problem, we lift the two-dimensional ranges to a three dimensional space. We map the homothet of ${Q'}$ of center $(x,y)$ and scaling $\lambda$ to the three dimensional point $\rho ({Q'})=(x, y, \lambda)\in {\mathbb{R}}^3$. Given a set $S$ of homothets of ${Q}$, we define $\rho (S) = \{\rho ({Q'}) : {Q'}\in S\}$ as the set containing the images of the ranges in $S$. For any point $p=(x,y,d)$, we associate the three dimensional range $\pi(p)$ as the cone with apex at $(x,y,d)$ such that the intersection with the horizontal plane of height $z$ is $Q(0, z-d)$ (if $z\geq d$) or empty (if $z<d$), where ${Q}^*={Q}(0,-1)$ is the reflection of ${Q}$ about its center. Note that the cone $\pi (p)$ so defined is convex (see Figure \[fig\_trans\]). We define the [*downward cone*]{} $\pi^* (p)$ as the centrally symmetric image of $\pi(p)$ through $p$. By symmetry, we observe the following: [cc]{} {width="\textwidth"} {width="\textwidth"} \[lem\_dual\] For any $p,q\in{\mathbb{R}}^3$, we have $q\in\pi (p)\Leftrightarrow p\in\pi^*(q)$. Moreover, for any point $(x, y)\in {\mathbb{R}}^2$ and range ${Q'}$, $(x, y)\in {Q'}\Leftrightarrow \rho({Q'}) \in \pi((x, y, 0))$. It follows that any coloring of $\rho(S)$ with respect to the conic ranges $\pi$ is a valid coloring of $S$. Let $G_\pi(\rho(S))$ be the Delaunay graph in ${\mathbb{R}}^3$ with cones as ranges. That is, the vertex set of $G_\pi(\rho(S))$ is $S$ and two ranges ${Q'},{Q''}$ of $S$ are adjacent if and only if there exists a point $p\in{\mathbb{R}}^3$ such that $\pi (p) \cap \rho (S) = \{ \rho ({Q'}), \rho ({Q''})\}$. We claim that $G_\pi(\rho(S))$ satisfies properties similar to those of Lemmas \[lem\_planar\] and \[lem\_edge\]. In order to prove so, we first introduce some inclusion properties. \[lem\_inclus\] For any $p\in{\mathbb{R}}^3$, $q\in \pi (p)$ and $m$ on the line segment $pq$, we have $\pi (q) \subseteq \pi (p)$ and $q \in \pi (m)$. Observe that the projections of the cones $\pi (p)$ and $\pi (q)$ on any vertical plane (i.e., any plane of equation $ax+by+c=0$) are two-dimensional cones; that is, the set of points above two halflines with a common origin. Moreover, the slope of the halflines only depends on $a,b$ and ${Q}$. Let $z_q$ be the $z$-coordinate of $q$, and consider the intersections of the cones $\pi (p)$ and $\pi (q)$ with a horizontal plane $\Pi$ of $z$-coordinate $t \geq z_q$. We get two homothets of ${Q}^*$, say ${Q}^*_p$ and ${Q}^*_q$. We have to show that ${Q}^*_q \subseteq {Q}^*_p$ for any $t$. Suppose otherwise. There exists a vertical plane $\Pi'$ for which the projection of ${Q}^*_q$ on $\Pi'$ is not included in the projection of ${Q}^*_p$. To see this, we can find a common tangent to ${Q}^*_p$ and ${Q}^*_q$ in $\Pi$, slightly rotate it so that it is tangent to ${Q}^*_q$ only, then pick a plane that is orthogonal to that line. But the projections of $\pi (p)$ and $\pi (q)$ on $\Pi'$ are two cones with parallel bounding halflines, thus the projection of the apex of $\pi (q)$ cannot be in that of $\pi (p)$, a contradiction. The proof of the second claim is similar. We know that $q\in \pi (p)$, hence from Lemma \[lem\_dual\], $p\in\pi^* (q)$. Now from the convexity of $\pi^* (q)$, we have that $m\in \pi^*(q)$. Using again Lemma \[lem\_dual\], we obtain $q\in\pi (m)$. \[lem\_dualplanar\] The graph $G_\pi(\rho(S))$ is planar. By definition of $E(S)$, we know that for every edge ${Q'}{Q''}\in E$ there exists $p\in{\mathbb{R}}^3$ such that $\pi (p) \cap \rho (S) = \{ \rho ({Q'}), \rho ({Q''})\}$. We draw the edge ${Q'}{Q''}$ as the projection (on the horizontal plane $z=0$) of the two line segments connecting respectively $\rho ({Q'})$ and $\rho ({Q''})$ to $p$. First note that crossings involving two edges with a common endpoint can be eliminated by rerouting the two polygonal lines at their intersection point. Thus it suffices to show that this embedding has no crossing involving vertex-disjoint edges. Consider two edges $uu'$ and $vv'$, and their corresponding witness cones $\pi_1\ni u, u'$ and $\pi_2\ni v, v'$. By definition of witness, each cone must contain exactly two points. In particular, we have $u\not\in\pi_2$ and $v\not\in\pi_1$. Suppose that the projections of the segments connecting $u$ with the apex of $\pi_1$ and $v$ with the apex of $\pi_2$ cross at a point $x$ (other than the endpoints). Consider the vertical line $\ell$ that passes through $x$: by construction, this line must intersect with both segments at points $a$ and $b$, respectively. Without loss of generality we assume that $a$ has lower $z$ coordinate than $b$ (see Figure \[fig\_planar\]). From the convexity of $\pi_1$, we have $a\in \pi_1$. From Lemma \[lem\_inclus\], we have $v\in \pi (b)$, $b\in \pi (a)$, and $\pi (a)\subseteq \pi (b)$. In particular, we have $v\in \pi (b) \subseteq \pi (a) \subseteq\pi_1$, which contradicts $v\not\in \pi_1$. Alternative construction via weighted Voronoi diagrams {#sec:wVD} ------------------------------------------------------ We introduce an alternative definition of $G_\pi(\rho(S))$ so as to prove its planarity. For any point $p$, we define its distance to $q$ as $d(p,q)=\min\{\lambda \geq 0 | q\in {Q}(p,\lambda)\}$. That is, the smallest possible scaling so that a range of center $p$ contains $q$. This distance is called the [*convex distance function*]{} from $p$ to $q$ (with respect to ${Q}$). Given a set $S=\{{Q}_1, \ldots, {Q}_n\}$ of homothets of ${Q}$, we construct an additively weighted Voronoi diagram $V_{Q}(S)$ with respect to the convex distance function [@fortune]. Let $c_i$ and $\lambda_i$ be the center and scaling of ${Q}_i$. Then $V_{Q}(S)$ has $\{c_1,\ldots c_n\}$ as the set of sites, and each site $c_i$ is given the weight $-\lambda_i$. The additively weighted Voronoi diagram for this set of sites has a cell for each site $c_i$, defined as the locus of the points $p$ of the plane whose weighted distance $d(p,c_i)-\lambda_i$ to $c_i$ is the smallest among all sites. The dual graph for this Voronoi diagram has an edge between any two sites whose cells share a boundary. In the following we show that the dual of $V_{Q}(S)$ is $G_\pi(\rho(S))$. Let $p = (x,y)\in{\mathbb{R}}^2$ be any point covered by one or more ranges of $S$. We denote by $(p,z)$ the point of ${\mathbb{R}}^3$ of coordinates $(x,y,z)$, for any $z\in{\mathbb{R}}$. From Lemma \[lem\_dual\], the points of $\rho (S)$ contained in $\pi(p,0)$ are exactly the ranges of $S$ that contain $p$. We translate the cone $\pi (p,0)$ vertically upward; in this lifting process, the points of $\rho(S)\cap \pi(p,0)$ will leave the cone. For any homothet ${Q'}\in S$ such that $p\in {Q'}$, we define $z_{{Q'}}(p)$ as the height in which $\rho({Q'})$ is on the boundary of the cone $\pi(p,z_{{Q'}}(p))$. [cc]{} {width="\textwidth"} {width="\textwidth"} \[lem\_height\] For any homothet ${Q'}$ of center $c$ and scaling $\lambda$ and $p\in{Q'}$, $z_{{Q'}}(p)=\lambda - d(c,p)$. Consider the cone $\pi(p,z_{{Q'}}(p))$ and the halfplane $z=z_{{Q'}}(p)$. By Lemma \[lem\_dual\], the fact that $\rho({Q'})$ is on the boundary of $\pi(p,z_{{Q'}}(p))$ is equivalent to $(p,z_{{Q'}}(p))$ being on the boundary of the downwards cone $\pi^*(\rho({Q'}))=(c,\lambda)$. Observe that the distance between points $(c,z_{{Q'}}(p))$ and $(p,z_{{Q'}}(p))$ is exactly $d(c,p)$. Consider the plane that passes through the points $(c,0)$, $(p,0)$, and $(c,\lambda)$: we have two similar triangles whose bases have lengths $\lambda$ and $d(c,p)$, respectively. Since the height of the large triangle is $\lambda$, we conclude that the height of the smaller one must be $d(c,p)$ (see Figure \[fig\_vorono\]). That is, the difference in the $z$ coordinates between the points $(c,z_{{Q'}}(p))$ and $(c, \lambda)$ is $d(c,p)$. Since, by definition of $\rho$, the difference in $z$ coordinates between $(c,0)$ and $\rho({Q'})$ is exactly $\lambda$, we obtain the equality $\lambda=z_{{Q'}}(p)+d(c,p)$ and the Lemma follows. This shows the duality between the weighted Voronoi diagram and the graph $G_\pi(\rho(S))$: let $p\in {\mathbb{R}}^2$ be any point in the plane covered by at least one range of ${Q'}$. Consider the cone $\pi(p,0)$ and lift it continuously upward. The last point of $\rho(S)\cap \pi(p,0)$ to leave the cone will be one with highest $z_{{Q'}}(p)$. By Lemma \[lem\_height\], it will be the homothet ${Q'}$ of center $c'$ and scaling $\lambda'$ that has the [*smallest*]{} $d(c',p)-\lambda'$. Observe that this is exactly how we defined the weights of the sites, hence ${Q'}$ being the last range in the cone is equivalent to $c'$ being the [*closest*]{} site of $p$ in $V_{Q'}(S)$. This can be interpreted as shrinking simultaneously all ranges until $p$ is only covered by its closest homothet ${Q'}$. This shrinking process is simulated in our construction through the $z$ coordinate. \[lem\_dualVorDel\] The dual graph of $V_{Q'}(S)$ is $G_\pi(\rho(S))$. Let $p=(x,y)\in {\mathbb{R}}^2$ be a point on a bisector of $V_{Q}(S)$ between sites $c_1$ and $c_2$ (corresponding to ranges ${Q}_1$ and ${Q}_2$ of scaling $\lambda_1$ and $\lambda_2$, respectively). By definition, we have that $d(c_1,p)-\lambda_1=d(c_2,p)-\lambda_2$ and $d(c',p)-\lambda' > d(c_1,p)-\lambda_1$ for all other homothets ${Q'}\in S$ of center $c'$ and scaling $\lambda'$. Let $w_{\min}=d(c_1,p)-\lambda_1$ and consider the cone $\pi(x,y,-w_{\min})$: by Lemma \[lem\_height\], both $\rho({Q}_1)$ and $\rho({Q}_2)$ are on the boundary. Moreover, any other homothet ${Q'}\in S$ will satisfy ${Q'}\not\in \pi(x,y, -w_{\min})$ (since other ranges have larger weighted distance, which is equivalent to having smaller $z_{{Q'}}(p)$). That is, the cone $\pi(x,y,-w({Q}_1))$ contains exactly points $\rho({Q}_1)$ and $\rho({Q}_2)$. Moreover, no other point of $\rho(S)$ will be in the cone, hence ${Q}_1{Q}_2\in E$. The other inclusion is shown analogously: let ${Q}_1{Q}_2$ be two ranges such that ${Q}_1{Q}_2\in E$. Let $(x,y,z)\in {\mathbb{R}}^3$ be the apex of the minimal cone (with respect to inclusions) such that $\pi(x,y,z)\cap \rho(S)=\{\rho({Q}_1),\rho({Q}_2)\}$. Since $\pi(x,y,z)$ is minimal, both $\rho({Q}_1)$ and $\rho({Q}_2)$ must be on the boundary of the cone. In particular, $z_{{Q}_1}(x,y)=z_{{Q}_2}(x,y)$ and other ranges satisfy $z_{{Q'}}(x,y)<z_{{Q}_1}(x,y)$ (for all other ${Q'}\in S$). Using again Lemma \[lem\_height\], this is equivalent to the fact that $p=(x,y)$ is equidistant to sites $c_1$, $c_2$, and all other sites have strictly larger distance. #### Coloring. As an application of the above construction, we show how to solve the dual coloring problem. By Lemma \[lem\_dualplanar\], we already know that $G_\pi(\rho(S))$ is 4-colorable. For any point $p\in{\mathbb{R}}^2$, let $S_p$ be the set of ranges containing $p$ (i.e., $S_p=\{{Q'}\in S : p\in {Q'}\}$). \[lem\_enough\] For any $p\in {\mathbb{R}}^2$ such that $|S_p| \geq 2$, there exist ${Q}_1,{Q}_2 \in S_p$ such that ${Q}_1{Q}_2\in E(S)$. From the second property of Lemma \[lem\_dual\], the number of points of $\rho (S)$ contained in the cone $\pi(p,0)$ is the number of ranges of $S$ containing $p$. The proof is now analogous to Lemma \[lem\_edge\], where the shrinking operation is replaced by a vertical lifting of the cone. Let $z_0\geq 0$ be the largest value such that the cone $\pi(p,z_0)$ has two or more points of $\rho(S)$. If $\pi(p,z_0)$ contains exactly two points we are done, hence it only remains to treat the degeneracies. Remember that in such a case, there must be at least two points on the boundary of $\pi(p,z_0)$ (and possibly a point in its interior). If there is a point $\rho({Q}_1)$ in the interior, we select a second point $\rho({Q}_2)$ on the boundary of $\pi(p,z_0)$ and translate the apex of the cone towards $\rho({Q}_2)$. Note that when the apex is located at $\rho({Q}_2)$, the point $\rho({Q}_1)$ cannot be in the cone (since it would imply that ${Q}_1 \subseteq {Q}_2$, and we assumed otherwise). Thus, at some point in the translation $\rho({Q}_1)$ reaches the boundary of the cone. During this translation process, points that were on the boundary of $\pi(p,z_0)$ have either remained on the boundary or have left the cone. In either case, we obtain a cone $\pi(p',z')\subseteq \pi(z,0)$ with two (or more) vertices of $\rho(S)$ on its boundary and none in its interior. If it contains exactly two points of $\rho(S)$ we are done. Otherwise, by duality of Lemma \[lem\_dualVorDel\], $p'$ is a vertex of the weighted Voronoi diagram. We pick a point $p''$ in an edge $e$ of the Voronoi diagram incident to $p'$. Since $p'$ is in an edge of the Voronoi diagram, it is equidistant to two ranges ${Q}_1,{Q}_2 \in S_p$. Hence, by Lemma \[lem\_dualVorDel\], when we do the lifting operation on $p'$ we will obtain a cone that exactly contains ${Q}_1,{Q}_2 \in S_p$ on its boundary and no other point in its interior. It suffices to show the claim for points $p\in{\mathbb{R}}^2$ that are $3$ or more deep. We translate the cone vertically upward; points of $\rho(S)$ will be leaving the cone. If at some height exactly two points $\rho({Q}_1),\rho({Q}_2)$ remain, we have ${Q}_1{Q}_2\in E(S)$. Otherwise, we have that, at some height $z_0\geq 0$, the cone $\pi(p,z_0)$ has three or more points of $\rho(S)$ and any vertical lifting (even by an arbitrarily small amount) gives a cone with less than two points. If there is a point of $\rho({Q}_1)\in \rho(S)$ in the interior of $\pi(p,z_0)$ we select a second point $\rho({Q}_2)$ on the boundary of $\pi(p,z_0)$ and translate the apex of the cone towards $\rho({Q}_2)$. During this translation process, point $\rho({Q}_2)$ will always stay in the cone. Moreover, for a sufficiently small translation $\rho({Q}_1)$ will also be in the interior of the cone. Observe that the translated cone is be tangent to $\pi(p,z_0)$ only at the line that ray emanating from $(p,z_0)$ and passing through $\rho({Q}_2)$. Hence, no other point $\rho({Q'})$ can be in the ray (since then we would have a point of $\rho(S)$ dominating another, which is equivalent to a range being included in another range). That is, after a small translation, we have that only ${Q}_1$ and ${Q}_2$ will be in the cone. We now study the case in which there is no interior point. That is, the cone $\pi(p,z_0)$ has three or more points of $\rho(S)\cap \pi(p,z_0)$ on its boundary and none in the interior. By Lemma \[lem\_dualVorDel\], this case corresponds to the case in which $p$ is a vertex of the weighted Voronoi diagram. Hence, $p$ is equidistant to ranges ${Q}_1, {Q}_2, \ldots {Q}_k$ (for some $k\geq 3$). Let $e$ be any edge of the Voronoi diagram incident to $p$, and let $p'$ be any point of $e$. Without loss of generality, we can assume that $e$ is the bisector between ranges ${Q}_1$ and ${Q}_2$. By Lemma \[lem\_dualVorDel\], when we do the lifting operation on $p'$ we will obtain a cone that only contains $\rho({Q}_1)$ and $\rho({Q}_2)$ on its boundary and no other point of $\rho(S)$. As a result, we have ${Q}_1{Q}_2\in E$ completing the proof of the Lemma. \[theo\_dual\] $\bar{c}_{Q}(2) \leq 4$. Let $I\subset S$ be the set of homothets included in other homothets of $S$. Recall that we initially assumed that $I$ was empty. Thus, to finish the proof it only remains to study the $I\neq \emptyset$ case. First we color $S\setminus I$ with 4 colors, using a 4-coloring of $G_\pi(\rho(S))$. This is possible from Lemma \[lem\_dualplanar\]. Then, for each homothet ${Q'}$ of $I$ there exists one homothet ${Q''}$ in $S \setminus I$ that contains it. We assign to ${Q'}$ any color different than the one assigned to ${Q''}$. Any point $p\in {Q'}$ will also satisfy $p\in{Q''}$, since ${Q'}\subseteq{Q''}$, hence $p$ will be covered by two ranges of different colors. Coloring Three-dimensional Hypergraphs {#sec_3D} ====================================== Lemma \[lem\_dualplanar\] actually generalizes the “easy" direction of Schnyder’s characterization of planar graphs. We first give a brief overview of this fundamental result. The [*vertex-edge incidence poset*]{} of a hypergraph $G=(V,E)$ is a bipartite poset $P=(V\cup E,\preceq_P)$, such that $e\preceq_P v$ if and only if $e\in E$, $v\in V$, and $v\in e$. The [*dimension*]{} of a poset $P=(S,\preceq_P)$ is the smallest $d$ such that there exists an injective mapping $f:S\to{\mathbb{R}}^d$, such that $u\preceq_P v$ if and only if $f(u)\leq f(v)$, where the order $\leq$ is the componentwise partial order on $d$-dimensional vectors. When $P$ is the vertex-edge incidence poset of a hypergraph $G$, we will refer to this mapping as a [*realizer*]{} of $G$, and to $d$ as its [*dimension*]{}. There exists a relation between the dimension of a graph and its chromatic number. For example, the graphs of dimension $2$ or less are subgraphs of the path, hence are 2-colorable. Schnyder pointed out that all 4-colorable graphs have dimension at most $4$ [@schnyder], and completely characterized the graphs whose incidence poset has dimension $3$: A graph is planar if and only if its dimension is at most three. The “easy" direction of Schnyder’s theorem states that every graph of dimension at most three is planar. The non-crossing drawing that is considered in one of the proofs is similar to ours, and simply consists of, for every edge $e=uv$, projecting the two line segments $f(e)f(u)$, and $f(e)f(v)$ onto the plane $x+y+z=0$ [@trotter; @BD81]. It is easy to see that octants in ${\mathbb{R}}^3$ satisfy the equivalent of our Lemma \[lem\_enough\] (by translating the apex of the octant with vector $(1,1,1)$ for example). Combining this result with the Four Color Theorem gives the following result. \[lem\_3dim\] Every hypergraph of dimension at most three is 4-colorable. Upper bounds for three-dimensional hypergraphs {#sec_3dim} ---------------------------------------------- We now adapt the above result for higher values of $k$. That is, we are given a three-dimensional hypergraph $G=(V,H)$ and a constant $k\geq 2$. We would like to color the vertices of $G$ such that any hyperedge $e\in H$ contains at least $\min\{|e|,k\}$ vertices with different colors. We denote by $c_3(k)$ the minimum number of colors so that any three-dimensional hypergraph can be suitably colored. Note that the problem is self-dual: any instance of the dual coloring problem can be transformed into a primal coloring problem by symmetry. For simplicity, we assume that no two vertices of $V$ in the realizer share an $x$, $y$ or $z$ coordinate. This can be obtained by making a symbolic perturbation of the point set in ${\mathbb{R}}^3$. Recall that, from the definition of the realizer, the point $q_e$ dominates $u\in S$ if and only if $u\in e$. For any hyperedge $e\in H$, there exist many points in ${\mathbb{R}}^3$ that dominate the points of $e$. We also assume that hyperedge $e$ is mapped to the minimal point $q_e\in{\mathbb{R}}^3$, obtained by translating $q_e$ in each of the three coordinates until a point of hits the boundary of the upper octant whose apex is $q_e$. For any hyperedge $e\in H$, we define the $x$-extreme of $e$ as the point $x(e)\in e$ whose image has smallest $x$-coordinate. Analogously we define the $y$ and $z$-extremes and denote them $y(e)$ and $z(e)$, respectively. We say that a hyperedge $e$ is [*extreme*]{} if two extremes of $e$ are equal. \[lem\_boundeg\] For any $k\geq 2$, $G$ has up to $3n$ extreme hyperedges of size exactly $k$. We charge any extreme hyperedge to the point that is repeated. By the pigeonhole principle, if a point is charged more than three times, there exist two extreme hyperedges $e_1,e_2$ of size exactly $k$ that charge on the same coordinates. Without loss of generality, we have $x(e_1)=x(e_2)$ and $y(e_1)=y(e_2)$. Let $q_1$ and $q_2$ be the mappings of $e_1$ and $e_2$, respectively. By hypothesis, the $x$ and $y$ coordinates of $q_1$ and $q_2$ are equal. Without loss of generality, we assume that $q_1$ has higher $z$ coordinate than $q_2$. In particular, we have $q_1\subset q_2$. Since both have size $k$, we obtain $e_1=e_2$. Let $S$ be the the 3-dimensional realizer of the vertices of $G$. For simplicity, we assume that $G$ is maximal. That is, for any $e\subseteq S$, we have $e\in H$ if and only if there exists a point $q_e\in {\mathbb{R}}^3$ dominating exactly $e$. Since we are only adding hyperedges to $G$, any coloring of this graph is a valid coloring of $G$. For any $2\leq k\leq n$, we define the graph $G_k(S)=(S,E_k)$, where for any $u,v\in S$ we have $uv\in E_k$ if and only if there exists a point $q\in {\mathbb{R}}^3$ that dominates $u,v$ and at most $k-2$ other points of $S$ (that is, we replace hyperedges of $G$ whose size is at most $k$ by cliques). The main property of this graph is that any proper coloring of $G_k(S)$ induces a polychromatic coloring of $G$. Using Lemma \[lem\_boundeg\], we can bound the number of edges of $G_k(S)$. \[lem\_edges\] For any set $S$ of points and $2\leq k\leq n$, graph $G_k(S)$ has at most $3(k-1)n-6$ edges. The claim is true for $k=2$ from Schnyder’s characterization. Notice that $E_{k-1}\subseteq E_k$, thus it suffices to bound the total number of edges $uv\in E_k\setminus E_{k-1}$. By definition of $G_k$ and $G_{k-1}$, there must exist a hyperedge $e$ of size exactly $k$ such that $u,v\in e$. In the three-dimensional realizer, this corresponds to a point $q_e\in {\mathbb{R}}^3$ that dominates $u,v$ and $k-2>0$ other points of $S$. We translate the point $q_e$ upward on the $x$ coordinate until it dominates only $k-1$ points. By definition, the first point to leave must be the $x$-extreme point $x(e)$. After this translation we obtain point $q'_e$ that dominates $k-1$ points. All these points will form a clique in $E_{k-1}$. Since $uv\not\in E_{k-1}$, we either have $u=x(e)$ or $v=x(e)$. We repeat the same reasoning translating in the $y$ and $z$ coordinates instead and, combined with the fact that a point cannot be extreme in the three directions, either $uv\in E_{k-1}$ or $u$ and $v$ are the only two extremes of $e$. In particular, the hyperedge $e$ is extreme. From Lemma \[lem\_boundeg\] we know that this case can occur at most $3n$ times, hence we obtain the recurrence $|E_k|\leq |E_{k-1}|+3n$. \[theo\_colz\] For any $k\geq 2$, we have $c_3(k)\leq 6(k-1)$. From Lemma \[lem\_edges\] and the handshake lemma, the average degree of $G$ is strictly smaller than $6(k-1)$. In particular, there must exist a vertex whose degree is at most $6(k-1)-1$. Moreover, this property is also satisfied by any induced subgraph, as any edge $(u,v)\in E_k$ is an edge of $G_k(S\setminus\{w\})$, $\forall w\neq u,v$. Hence, for any $S'\subseteq S$, the induced subgraph $G_k(S) \setminus S'$ is a subgraph of $G_k(S\setminus\{S'\})$. In particular, the graph $G_k(S)$ is $(6(k-1)-1)$-degenerate, and can therefore be colored with $6(k-1)$ colors. Note that dual hypergraphs induced by collections of homothetic triangles have dimension at most 3, so our result directly applies. \[cor\_triangle\] For any $k\geq 3$, any set $S$ of homothets of a triangle can be colored with $6(k-1)$ colors so that any point $p\in{\mathbb{R}}^2$ covered by $r$ homothets is covered by $\min\{r,k\}$ homothets with distinct colors. Lower Bounds {#sec_lb} ============ We now give a lower bound on $c_{Q}(k)$. The normal vector of ${Q}$ at the boundary point $p$ is the unique unit vector orthogonal to the halfplane tangent to ${Q}$ at $p$, if it is well-defined. We say that a range has $m$ distinct directions if there exist $m$ different points with defined, pairwise linearly independent normal vectors. \[lem\_lowerprim\] Any range ${Q}$ with at least three distinct directions satisfies $c_{Q}(k)\geq 4\lfloor k/2 \rfloor$ and $\bar{c}_{Q}(k)\geq 4\lfloor k/2 \rfloor$. We first show that $c_{Q}(2)\geq 4$. Scale ${Q}$ by a large enough value so that it essentially becomes a halfplane. By hypothesis, we can obtain halfplane ranges with three different orientations. By making an affine transformation to the problem instance, we can assume that the halfplanes are of the form $x\geq c$, $y\geq \sqrt{3}x+c$ or $y\leq \sqrt{3}x+c$ for any constant $c\in {\mathbb{R}}$ (i.e. the directions of the equilateral triangle). Let $\Delta$ be the largest equilateral triangle with a side parallel to the abscissa that can be circumscribed in ${Q}$. Let $p_1,p_2, p_3$ and $p_4$ be the vertices and the incenter of $\Delta$, respectively (see Figure \[fig\_ranges\]). Note that any two points of $\{p_1,p_2,p_3,p_4\}$ can be selected with the appropriate halfplane range, hence any valid coloring must assign different colors to the four points. The proof of the dual bound is analogous: it suffices to consider the ranges that contain exactly two points of $\{p_1,p_2,p_3,p_4\}$. For higher values of $k$ it suffices to replace each point $p_i$ for a cluster of $\lfloor k/2 \rfloor$ points. That is, we have $4\lfloor k/2 \rfloor$ points clustered into four groups so that any two groups can be covered by one range. By the pigeonhole principle, any coloring that uses strictly less than $4\lfloor k/2 \rfloor$ colors must have two points with the same color. The range containing them (and any other $k-2$ points) will have at most $k-1$ colors, hence will not be polychromatic. Observe that parallelograms are the only ranges that do not have three or more distinct normal directions (in this case, we can show a weaker $3\lfloor k/2 \rfloor$ lower bound). In particular, the results of Sections \[sec\_primal\] and \[sec\_dual\] are tight for any range other than a parallelogram. Also notice that, since triangle containment posets are 3-dimensional, the lower bound also applies to $c_3(k)$. Applications to other coloring problems {#sec_appl} ======================================= #### Conflict-free colorings. A coloring of a hypergraph is said to be [*conflict-free*]{} if, for every hyperedge $e$ there is a vertex $v\in e$ whose color is distinct from all other vertices of $e$. Even [*et al.*]{} [@shakharcf] gave an algorithm for finding such a coloring. Their method repeatedly colors (in the polychromatic sense) the input hypergraph with few colors, and removes the largest color class. By repeating this process iteratively a conflict-free coloring is obtained. The number of colors is at most $\log_{\frac c{c-1}} n$, where $n$ is the number of vertices, and $c$ is the maximum number of colors used at each iteration. Our 4-colorability proof of Theorem \[theo\_dual\] is constructive and can be computed in $O(n^2)$ time. Hence, by combining both results we obtain the following corollary. Any dual hypergraph induced by a finite set of $n$ homothets of a compact and convex body in the plane has a conflict-free coloring using at most $\log_{4/3} n \leq 2.41\log_2 n$ colors. Furthermore, such a coloring can be found in $O(n^2\log n )$ time. #### $k$-strong conflict-free colorings. Abellanas [*et al.*]{} [@ABGHNR09] introduced the notion of $k$-strong conflict free colorings, in which every hyperedge $e$ has $\min\{|e|,k\}$ vertices with a unique color. Conflict-free colorings are $k$-strong conflict-free colorings for $k=1$. Recently, Horev, Krakovski, and Smorodinsky [@HKS10] showed how to find $k$-strong conflict-free colorings by iteratively removing the largest color class of a polychromatic coloring with $c(k)$ colors. Again, combining this result with Theorem \[theo\_colz\] yields the following corollary. Any dual hypergraph induced by a finite set of $n$ homothets of a compact and convex body in the plane has a $k$-strong conflict-free coloring using at most $\log_{(1+\frac{1}{6(k-1)})}n$ colors. #### Choosability. Cheilaris and Smorodinsky [@CS10] introduced the notion of choosability in geometric hypergraphs. A hypergraph with vertex set $V$ is said to be $k$-choosable whenever for any collection $\{L_v\}_{v\in V}$ of subsets of positive integers of size at least $k$, the hypergraph admits a proper coloring, where the color of vertex $v$ is chosen from $L_v$. Our construction of Section \[sec\_dual\] provides a planar graph, and planar graphs are known to be 5-choosable. This directly yields the following result. Any dual hypergraph induced by a finite set of homothets of a convex body in the plane is 5-choosable. [^1]: The term [*$k$-colorful*]{} is also used in the literature [@Smo07].

--- abstract: 'This paper considers the maximum generalized empirical likelihood (GEL) estimation and inference on parameters identified by high dimensional moment restrictions with weakly dependent data when the dimensions of the moment restrictions and the parameters diverge along with the sample size. The consistency with rates and the asymptotic normality of the GEL estimator are obtained by properly restricting the growth rates of the dimensions of the parameters and the moment restrictions, as well as the degree of data dependence. It is shown that even in the high dimensional time series setting, the GEL ratio can still behave like a chi-square random variable asymptotically. A consistent test for the over-identification is proposed. A penalized GEL method is also provided for estimation under sparsity setting.' author: - |           [Jinyuan Chang[^1]                     Song Xi Chen[^2]                Xiaohong Chen[^3]          ]{}\ \ [The University of Melbourne         Peking University             Yale University        ]{} date: 'First version: May 2013      This version: October 2014' title: High Dimensional Generalized Empirical Likelihood for Moment Restrictions with Dependent Data --- **JEL classification**: C14; C30; C40 **Key words**: Generalized empirical likelihood; High dimensionality; Penalized likelihood; Variable selection; Over-identification test; Weak dependence. Introduction ============ In economic, financial and statistical applications, econometric models defined with a growing number of parameters and moment restrictions are increasingly employed. Vector autoregressive models, dynamic asset pricing models, dynamic panel data models and high dimensional dynamic factor models are specific examples; see, e.g., [@BaiNg_2002_Ecma], [StockWatson\_2010]{} and [@FanLiao_2014]. Due to the desire to better capture large scale dynamic fundamental relations, these models with large number of unknown parameters of interest are typically used to for time series data of high dimension due to a large number of variables (relative to the sample size). The unconditional moment restriction models are the inferential settings of the Generalized Method of Moment (GMM) of [@Hansen_1982_Ecma], which is perhaps the most popular econometric method for semiparametric statistical inference. There are two dimensions that play essential roles in this method: the dimension of the moment restrictions and the dimension of the unknown parameters of interest. When both dimensions are fixed and finite, there is a huge established literature on inferential procedures, which include but not restrict to [@Rothenberg_1973] for the minimum distance, [@Hansen_1982_Ecma] and [@HansenSingleton_1982_Ecma] for the GMM, [@Owen_1988_Biometrika], [@QinLawless_1994_AOS] and [Kitamura\_1997\_AOS]{} for the empirical likelihood (EL), [@Smith_1997], [@NeweySmith_2004_Ecma] and [[@Anatolyev_2005] for the]{} generalized empirical likelihood (GEL). Among these methods, some members of the GEL (especially the EL) have the attractive properties of the Wilks’ theorem [@Owen_1988_Biometrika; @Owen_1990_AOS; @QinLawless_1994_AOS], Bartlett correction [@ChenCui_2006_Biometrika; @ChenCui_2007_JOE], and a smaller second order bias [@NeweySmith_2004_Ecma; @Anatolyev_2005]. See [Owen\_2001]{}, [@Kitamura_2007] and [@ChenVanKeilegom_2009_Test] for reviews. This paper investigates high dimensional GEL estimation and testing for weakly dependent observations when the dimensions of both the moment restrictions and the unknown parameters of interest may grow with the sample size $n$. Let $p$ and $r$ denote the dimension of the unknown parameters and the number of moment restrictions, respectively. When $r\geq p$, we investigate the impacts of $p$ and $r$ on the consistency, the rate of convergence and the asymptotic normality of the GEL estimator, the limiting behavior of the GEL ratio statistics as well as the overidentification test. To accommodate the potential serial dependence in the estimating functions induced by the original time series data, the blocking technique is employed. This paper establishes the consistency (with rate) and the asymptotic normality of the GEL estimator under either (fixed) finite or diverging block size $M$ under some suitable restrictions on $r$, $p$, $M$ and $n$. It is demonstrated that in general the blocking technique with a diverging block size delivers the estimation efficiency. We also discuss the impact of the smallest eigenvalue of the covariance matrix of the averaged estimating function on the consistency and the asymptotic normality of the GEL estimator. We show that, even in high dimensional nonlinear time series setting (with diverging $M$), the GEL ratio still behaves like a chi-square random variable asymptotically, which echoes a similar result by [FanZhangZhang\_2001]{} for nonparametric regression with iid data. A GEL based over-identification specification test is also presented for high dimensional time series models, which extends that of [DonaldImbensNewey\_2003\_JOE]{} for iid data from increasing dimension of moments ($r$) but fixed finite dimension of parameters ($p)$ to both dimensions are allowed to diverge (as long as $r-p>0$). Finally, when the parameter space is sparse, a penalized GEL method is proposed to allow for $p>r$, and is shown to attain the oracle property in the selection consistency as well as the asymptotic normality of the estimated non-zero parameters. There are some studies on the EL and its related methods under high dimensionality of both the moment restrictions and the parameters of interest. [@ChenPengQin_2009_Biometrika] and [HjortMcKeagueVanKeilegom\_2009\_AOS]{} evaluated the EL ratio statistic for the mean under high dimensional setting. [@TangLeng_2010_Biometrika] and [@LengTang_2011_Biometrika] evaluated a penalized EL when the underlying parameter is sparse in the context of the mean parameters and the general estimating equations, respectively. [@FanLiao_2014] considered penalized GMM estimation under high dimensionality and sparsity assumption. These papers assume independent data. Recently, by allowing for dependent data but losing the self-standardization property of the EL, [@Lahiri_2012] proposed a modified EL method by adding a penalty term to the original EL criterion for estimating the high-dimensional mean parameters with $r=p>n$. [@Lahiri_2012] did not implement data blocking in their modified (penalized) EL for the means despite the moment equations are serially dependent. The EL ratio statistic based on their modified EL method is no longer asymptotically pivotal. As a result, any inference based on this modified EL has to use data blocking or other HAC long-run variance estimation. The rationale in our paper is to preserve the attractive self-standardization property of the GEL in high dimensional time series setting; doing so makes our allowed dimensionality smaller than that in [Lahiri\_2012]{} but maintains simple GEL inference. The rest of the paper is organized as follows. Section 2 introduces the high dimensional model framework and the basic regularity conditions. Sections 3 and 4 establish the consistency, the rate of convergence and the asymptotic normality of the GEL estimator. Sections 5 and 6 derive the asymptotic properties of the GEL ratio statistic and the overidentification specification test respectively. [Section 7 presents a penalized GEL approach for parameter estimation and variable selection when the unknown parameter is sparse]{}. Section 8 reports some simulation results and Section 9 briefly concludes. Technical lemmas and all the proofs are given in Appendix. Preliminaries ============= Empirical Likelihood and Generalization --------------------------------------- Let $\{X_{t}\}_{t=1}^{n}$ be a sample of size $n$ from an $\mathbb{R}^{d}-$valued strictly stationary stochastic process, where $d$ denotes the dimension of $X_{t}$, and $\boldsymbol{\theta }=(\theta _{1},\ldots ,\theta _{p})^{\prime }$ be a $p$-dimensional parameter taking values in a parameter space $\Theta $. Consider a sequence of $r$-dimensional estimating equation $$g(X_{t},\boldsymbol{\theta })=(g_{1}(X_{t},\boldsymbol{\theta }),\ldots ,g_{r}(X_{t},\boldsymbol{\theta }))^{\prime }$$for $r\geq p$. The model information regarding the data and the parameter is summarized by moment restrictions $${E}\{g(X_{t},\boldsymbol{\theta }_{0})\}=\boldsymbol{\mathbf{0}} \label{eq:1}$$where $\boldsymbol{\theta }_{0}\in \Theta $ is the true parameter. As argued in [@HjortMcKeagueVanKeilegom_2009_AOS], the moment restrictions ([eq:1]{}) can be viewed as a triangular array where $r,d,X_{t},\boldsymbol{\theta }$ and $g(x,\boldsymbol{\theta })$ may all depend on the sample size $n$. We will explicitly allow $r$ and/or $p$ grow with $n$ while considering inference for $\boldsymbol{\theta }_{0}$ identified by (\[eq:1\]). Although there is often a connection between $d$ and $r$ which is dictated by the context of an econometrical or statistical analysis, the theoretical results established in this paper are written directly on the growth rates of $r$ and $p$ relative to $n$. Hence, we will not impose explicit conditions on $d$ which can be either growing or fixed. Certainly, when $d$ diverges, it would indirectly affect the underlying assumptions made in Section 2.3, for instance the moment condition and the rate of the mixing coefficients. We assume the dependence in the time series $\{X_{t}\}$ satisfies the $\alpha $-mixing condition [@Doukhan_1994]. Specifically, let $\mathscr{F}_{u}^{v}=\sigma (X_{t}:u\leq t\leq v)$ be the $\sigma $-field generated by the data from a time $u$ to a time $v$ for $v\geq u$. Then, the $\alpha $-mixing coefficients are defined as $$\alpha _{X}(k)=\sup_{d}\sup_{A\in \mathscr{F}_{-\infty }^{0},B\in \mathscr{F}_{k}^{\infty }}|{P}(A\cap B)-{P}(A){P}(B)|~~\text{for each}~k\geq 1.$$The $\alpha $-mixing condition means that $\alpha _{X}(k)\rightarrow 0$ as $k\rightarrow \infty $. When $\{X_{t}\}$ are independent, $\alpha _{X}(k)=0$ for all $k\geq 1$. We employ the blocking technique [@Hall_1985; @Carlstein_1986_AOS; @Kunsch_1989_AOS] to preserve the dependence among the underlying data. Let $M $ and $L$ be two integers denoting the block length and separation between adjacent blocks, respectively. Then, the total number of blocks is $Q=\lfloor (n-M)/L\rfloor +1$, where $\lfloor \cdot \rfloor $ is the integer truncation operator. For each $q=1,\ldots ,Q$, the $q$-th data block $B_{q}=(X_{(q-1)L+1},\ldots ,X_{(q-1)L+M})$. The average of the estimating equation over the $q$-th block is $$\phi _{M}\left( B_{q},\boldsymbol{\theta }\right) =\frac{1}{M}\sum\limits_{m=1}^{M}g(X_{(q-1)L+m},\boldsymbol{\theta }). \label{eq:phi}$$Clearly, ${E}\{\phi _{M}(B_{q},\boldsymbol{\theta }_{0})\}=\boldsymbol{\mathbf{0}}$. For any $n$ and $\boldsymbol{\theta }\in \Theta $, $\{\phi _{M}(B_{q},\boldsymbol{\theta })\}_{q=1}^{Q}$ is a new stationary sequence. The blockwise EL [@Kitamura_1997_AOS] is defined as $$\mathcal{L}(\boldsymbol{\theta })=\sup \bigg\{\prod\limits_{q=1}^{Q}\pi _{q}\bigg{|}\pi _{q}>0,\sum\limits_{q=1}^{Q}\pi _{q}=1,\sum\limits_{q=1}^{Q}\pi _{q}\phi _{M}(B_{q},\boldsymbol{\theta })=\boldsymbol{\mathbf{0}}\bigg\}. \label{eq:l}$$Employing the routine optimization procedure for the blockwise EL leads to $$\mathcal{L}(\boldsymbol{\theta })=\prod\limits_{q=1}^{Q}\bigg\{\frac{1}{Q}\frac{1}{1+\widehat{\lambda }(\boldsymbol{\theta })^{\prime }\phi _{M}(B_{q},\boldsymbol{\theta })}\bigg\}, \label{eq:L}$$where $\widehat{\lambda }(\boldsymbol{\theta })$ is a stationary point of the function $q(\lambda )=-\sum_{q=1}^{Q}\log \{1+\lambda ^{\prime }\phi _{M}(B_{q},\boldsymbol{\theta })\}$. The EL estimator for $\boldsymbol{\theta}_0$ is $\widehat{\boldsymbol{\theta}}_{EL}=\arg\max_{\boldsymbol{\theta}\in\Theta}\log\mathcal{\ L}(\boldsymbol{\theta}).$ The maximization in (\[eq:l\]) can be carried out more efficiently by solving the corresponding dual problem, which implies that $\widehat{\boldsymbol{\theta}}_{EL}$ can be obtained as $$\widehat{\boldsymbol{\theta}}_{EL}=\arg\min_{\boldsymbol{\theta}\in\Theta}\max_{\lambda\in\widehat{\Lambda}_n(\boldsymbol{\theta})}\sum_{q=1}^Q\log\{1+\lambda^{\prime }\phi_M(B_q,\boldsymbol{\theta})\}, \label{eq:el}$$ where $\widehat{\Lambda}_{n}(\boldsymbol{\theta} )=\left\{ \lambda \in \mathbb{R}^{r}:\lambda ^{\prime }\phi _{M}(B_q,\boldsymbol{\theta} )\in \mathcal{V},q=1,\ldots ,Q\right\} $ for any $\boldsymbol{\theta} \in \Theta $ and $\mathcal{V}$ is an open interval containing zero. The link function $\log (1+v)$ in (\[eq:el\]) can be replaced by a general concave function $\rho (v)$ [@Smith_1997]. The domain of $\rho (\cdot )$ contains $0$ as an interior point, and $\rho (\cdot )$ satisfies $\rho _{v}(0)\neq 0$ and $\rho _{vv}(0)<0$ where $\rho _{v}(v)=\partial \rho (v)/\partial v$ and $\rho _{vv}(v)=\partial ^{2}\rho(v) /\partial v^{2}$. The GEL estimator [@Smith_1997; @NeweySmith_2004_Ecma] is $$\widehat{\boldsymbol{\theta }}_{n}=\arg \min_{\boldsymbol{\theta }\in \Theta }\max_{\lambda \in \widehat{\Lambda }_{n}(\boldsymbol{\theta })}\sum_{q=1}^{Q}\rho (\lambda ^{\prime }\phi _{M}(B_{q},\boldsymbol{\theta })), \label{eq:gel}$$which includes the EL estimator $\widehat{\boldsymbol{\theta }}_{EL}$ of [@Owen_1988_Biometrika], the exponential tilting (ET) estimator of [KitamuraStutzer\_1997]{} and [@ImbensSpadyJonson_1998] (with $\rho (v)=-\exp (v)$), the continuous updating (CU) GMM estimator of [HansenHeatonYaron\_1996]{} (with a quadratic $\rho (v)$), and many others as special cases. Define $$\widehat{S}_{n}(\boldsymbol{\theta },\lambda )=\frac{1}{Q}\sum\limits_{q=1}^{Q}\rho (\lambda ^{\prime }\phi _{q}(\boldsymbol{\theta })). \label{eq:sn}$$Then $\widehat{\boldsymbol{\theta }}_{n}$ and its Lagrange multiplier $\widehat{\lambda }$ satisfy the score equation $$\nabla _{\lambda }\widehat{S}_{n}(\widehat{\boldsymbol{\theta }}_{n},\widehat{\lambda })=\boldsymbol{\mathbf{0}}.$$By the implicit function theorem \[Theorem 9.28 of [@Rudin_1976]\], for all $\boldsymbol{\theta }$ in a $\Vert \cdot \Vert _{2}$-neighborhood of $\widehat{\boldsymbol{\theta }}_{n}$, there is a $\widehat{\lambda }(\boldsymbol{\theta })$ such that $\nabla _{\lambda }\widehat{S}_{n}(\boldsymbol{\theta },\widehat{\lambda }(\boldsymbol{\theta }))=\boldsymbol{\mathbf{0}}$ and $\widehat{\lambda }(\boldsymbol{\theta })$ is continuously differentiable in $\boldsymbol{\theta }$. By the concavity of $\widehat{S}_{n}(\boldsymbol{\theta },\lambda )$ with respect to $\lambda $, $\widehat{S}_{n}(\boldsymbol{\theta },\widehat{\lambda }(\boldsymbol{\theta }))=\max_{\lambda \in \widehat{\Lambda }_{n}(\boldsymbol{\theta })}\widehat{S}_{n}(\boldsymbol{\theta },\lambda )$. From the envelope theorem, $$\boldsymbol{\mathbf{0}}=\nabla _{\boldsymbol{\theta }}\widehat{S}_{n}(\boldsymbol{\theta },\widehat{\lambda }(\boldsymbol{\theta }))\big|_{\boldsymbol{\theta }=\widehat{\boldsymbol{\theta }}_{n}}=\frac{1}{Q}\sum_{q=1}^{Q}\rho _{v}(\widehat{\lambda }(\widehat{\boldsymbol{\theta }}_{n})^{\prime }\phi _{q}(\widehat{\boldsymbol{\theta }}_{n}))\{\nabla _{\boldsymbol{\theta }}\phi _{q}(\widehat{\boldsymbol{\theta }}_{n})\}^{\prime }\widehat{\lambda }(\widehat{\boldsymbol{\theta }}_{n}). \label{eq:0.1}$$The role of the block size $M$ played in the consistency and the asymptotic normality of the GEL estimator $\widehat{\boldsymbol{\theta }}_{n}$ will be discussed in Sections 3 and 4, respectively. Examples -------- We illustrate the model setting of high dimensional moment restrictions framework through three examples. **Example 1** (High dimensional means): Suppose $\{X_{t}\}_{t=1}^{n}$ is a stationary sequence of observations, where $X_{t}\in \mathbb{R}^{d}$ and $\boldsymbol{\theta }_{0}={E}(X_{t})$. For high dimensional data, $d$ diverges and $g(X_{t},\boldsymbol{\theta })=X_{t}-\boldsymbol{\theta }$ constitutes the simplest high dimensional moment equation, which implies the dimension of observation $d$, the number of moment restrictions $r$ and the number of parameters $p$ all are the same. Under this setting and for independent data, [@ChenPengQin_2009_Biometrika] and [HjortMcKeagueVanKeilegom\_2009\_AOS]{} considered the asymptotic normality of the EL ratio, that mirrors the Wilks’ theorem for finite dimensional case. This framework can be used in other inference problems. For instance checking if two univariate stationary time series $\{Y_{t}\}$ and $\{Z_{t}\}$ have identical marginal distribution. Let $$f_{Y}(s)=E(e^{\mathbf{i}sY_{t}})~~\text{and}~~f_{Z}(s)=E(e^{\mathbf{i}sZ_{t}})$$denote the characteristic functions of the two series, respectively. Suppose all the moments of $Y_{t}$ and $Z_{t}$ exist, then the characteristic functions can be expressed as $$f_{Y}(s)=1+\sum_{k=1}^{\infty }\frac{(\mathbf{i}s)^{k}}{k!}E(Y_{t}^{k})~~\text{and}~~f_{Z}(s)=1+\sum_{k=1}^{\infty }\frac{(\mathbf{i}s)^{k}}{k!}E(Z_{t}^{k}).$$Let $X_{t}=(Y_{t},Z_{t})$ and $g(X_{t},\boldsymbol{\theta })=(a_{1}(Y_{t}-Z_{t}-\theta _{1}),\ldots ,a_{r}(Y_{t}^{r}-Z_{t}^{r}-\theta _{r}))^{\prime }$ for some nonzero constants $a_{1},\ldots ,a_{r}$. Here $\theta _{l}$ measures $E(Y_{t}^{l})-E(Z_{t}^{l})$ for $l=1,\ldots ,r$, and the $a_{i}$’s are used to account for the potential diverging moments case, i.e, either $E(Y_{t}^{l})$ or $E(Z_{t}^{l})$ may diverge as $l\rightarrow \infty $. Then, the test for whether $Y_{t}$ and $Z_{t}$ having the same marginal distribution can be conducted by testing if $\boldsymbol{\theta }_{0}=\mathbf{0}$ via the growing dimensional moment restrictions $E\{g(X_{t},\boldsymbol{\theta }_{0})\}=\boldsymbol{\mathbf{0}}$ by letting $r\rightarrow \infty $. **Example 2** (Time series regression): We assume a structural model for $s$-dimensional time series $Y_t$ which involve unknown parameter $\boldsymbol{\theta}\in\mathbb{R}^p$ of interest as well as time innovations with unknown distributional form. Specifically, assume $$h(Y_t,\ldots,Y_{t-m};\boldsymbol{\theta}_0)=\boldsymbol{\varepsilon}_t\in\mathbb{R}^r \label{eq:hd}$$ where $m\geq1$ is some constant. In this model, we can view $X_t=(Y_t^{\prime },\ldots,Y_{t-m}^{\prime })^{\prime }\in\mathbb{R}^d$ with $d=sm$ and $g(X_t,\boldsymbol{\theta})=h(Y_t,\ldots,Y_{t-m};\boldsymbol{\theta})$. If $E(\boldsymbol{\varepsilon}_t)=\boldsymbol{\mathbf{0}}$, it implies $$E\{g(X_t,\boldsymbol{\theta}_0)\}=\boldsymbol{\mathbf{0}}.$$ For conventional vector autoregressive models $$Y_t=\mathbf{A}_1Y_{t-1}+\cdots+\mathbf{A}_mY_{t-m}+\eta_t \label{eq:var}$$ where $\mathbf{A}_1,\ldots,\mathbf{A}_m$ are some coefficient matrices needed to be estimated and $\eta_t$ is the white noise series. This model is the special case of (\[eq:hd\]) with $$h(Y_t,\ldots,Y_{t-m};\boldsymbol{\theta}_0)=(Y_t-\mathbf{A}_1Y_{t-1}-\cdots-\mathbf{A}_mY_{t-m})\otimes(Y_t^{\prime },\ldots,Y_{t-m}^{\prime })^{\prime }. \label{eq:h}$$ In modern high dimensional time series analysis, we always assume the dimensionality of $Y_t$ is large in relation to sample size, i.e., $s\rightarrow\infty$ as $n\rightarrow\infty$. Under such background, the numbers of estimating equation and unknown parameters are both $s^2m$. If we replace $(Y_t^{\prime },\ldots,Y_{t-m}^{\prime })^{\prime }$ by $(Y_t^{\prime },\ldots,Y_{t-m-l}^{\prime })^{\prime }$ for some fixed $l\geq1$, the model will be over-identified. The phenomenon of over-parametrization in such model is well known [@Lutkepohl_(2006).]. [@Davis2012] considered the estimation of (\[eq:var\]) under the sparsity assumption on $\mathbf{A}_i$’s. Under the sparsity, the penalized method proposed in Section 7 can be applied. Some other models share the form (\[eq:hd\]) can be found in Section 3.1 of [@NordmanLahiri_2013]. **Example 3** (Conditional moment restrictions): Let $\{X_{t}=(Y_{t}^{\prime },Z_{t}^{\prime })^{\prime }\}_{t=1}^{n}$ be a set of observations, and $\rho (y,z,\boldsymbol{\theta })$ be a known $J$-dimensional vector of generalized residual function. The parameter $\boldsymbol{\theta }_{0}$ is uniquely defined via the following conditional moment restrictions $${E}\{\rho (Y_{t},Z_{t},\boldsymbol{\theta }_{0})|Y_{t}\}=\boldsymbol{\mathbf{0}}~~\hbox{almost surely}. \label{eq:condm}$$By different choices of the functional forms of the generalized residual function $\rho (y,z,\boldsymbol{\theta })$, the conditional moment restrictions (\[eq:condm\]) include many existing models in statistics and econometrics as special cases. The popular generalized linear models are special cases of (\[eq:condm\]). To appreciate this point, let $\mu (y)={E}(Z|Y=y)$ and $h(\mu (y))=y^{\prime }\boldsymbol{\theta }_{0}$ for an increasing link function $h(\cdot )$. Then the generalized linear models are special cases of (\[eq:condm\]) with $\rho (y,z,\boldsymbol{\theta }_{0})=z-h^{-1}(y^{\prime }\boldsymbol{\theta }_{0})$. Let $q^{K}(y)=(q_{1K}(y),\ldots ,q_{KK}(y))^{\prime }$ denote a $K\times 1$ vector of known basis functions that can approximate any square integrable functions of $Y$ well as $K\rightarrow \infty $, such as polynomial splines, B-splines, power series, Fourier series, wavelets, Hermite polynomials and others; see, e.g., [@AiChen_2003_Ecma] and [DonaldImbensNewey\_2003\_JOE]{}. Then, (\[eq:condm\]) implies $${E}\{\rho (Y_{t},Z_{t},\boldsymbol{\theta }_{0})\otimes q^{K}(Y_{t})\}=\boldsymbol{\mathbf{0}}. \label{eq:uncond}$$Moreover, the unknown parameter $\boldsymbol{\theta }_{0}$ is a solution to this set of increasing dimensional ($r=JK$) unconditional moment restrictions (\[eq:uncond\]). The dimension $K$ will increase with $n$ to guarantee the consistency of the estimator for $\boldsymbol{\theta }_{0}$ and its asymptotic efficiency. Define $g(X_{t},\boldsymbol{\theta })=\rho (Y_{t},Z_{t},\boldsymbol{\theta })\otimes q^{K}(Y_{t})$, then ([eq:uncond]{}) is a special case of (\[eq:1\]). The number of moment restrictions $r=JK$ increases as $K$ does. For this model with iid data, [@DonaldImbensNewey_2003_JOE] apply the GEL method to the increasing number of the unconditional moment restrictions (\[eq:uncond\]) to obtain efficient estimation for finite fixed dimensional $\boldsymbol{\theta }_{0}$. They find that the diverging rate of the moment restrictions $r=JK$ depends on the choice of the basis functions $q^{K}(y)$. For example, if $q^{K}(y)$ is a spline basis then $r=JK$ could grow at the rate of $K=o(n^{1/3})$. Notations and Technical Conditions ---------------------------------- Throughout the paper, we use $C$s, with different subscripts, to denote positive finite constants which does not depend on the sample size $n$. For a matrix $A$, we use $\Vert A\Vert _{F}$ and $\Vert A\Vert _{2}$ to denote its Frobenius-norm and operator-norm respectively, i.e., $\Vert A\Vert _{F}=\left\{ \text{tr}\left( A^{\prime }A\right) \right\} ^{1/2}$ and $\Vert A\Vert _{2}=\left\{ \lambda _{\text{max}}\left( A^{\prime }A\right) \right\} ^{1/2}$. If $a$ is a vector, $\Vert a\Vert _{2}$ denotes its $L_{2}$-norm. Without causing much confusion, we denote [the $i$-th component of $g(x,\boldsymbol{\theta})$ by $g_i(x,\boldsymbol{\theta})$; and simplify $g(X_t,\boldsymbol{\theta})$ and $\phi_M(B_q,\boldsymbol{\theta})$ by $g_t(\boldsymbol{\theta})$ and $\phi_q(\boldsymbol{\theta})$, respectively, where $\phi_M(B_q,\boldsymbol{\theta})$ is defined in ([eq:phi]{}). Furthermore, we use $g_{t,j}(\boldsymbol{\theta})$ and $\phi_{q,j}(\boldsymbol{\theta})$ to denote the $j$-th component of $g_t(\boldsymbol{\theta})$ and $\phi_q(\boldsymbol{\theta})$ respectively. Let $\bar{g}(\boldsymbol{\theta})={n}^{-1}\sum_{t=1}^ng_t(\boldsymbol{\theta})$ and $\bar{\phi}(\boldsymbol{\theta})={Q}^{-1}\sum_{q=1}^Q\phi_q(\boldsymbol{\theta})$. Additionally, define $$V_M=\text{Var}\{M^{1/2}{\phi}_q(\boldsymbol{\theta}_0)\}~~\hbox{and}~~V_n=\text{Var}\{n^{1/2}\bar{g}(\boldsymbol{\theta}_0)\}$$ which are the covariance of the averaged estimating functions over a block and the entire sample respectively. Clearly $V_M = V_n$ if $M=n$. The following regularity conditions are needed in our analysis. ]{} (A.1) (i) $\{X_t\}$ is strictly stationary and there exists $\gamma >2$ such that $\sum_{k=1}^{\infty }k\alpha _{X}(k)^{1-2/\gamma }<\infty$; (ii) $M\geq L$ and $M/L\rightarrow c\geq 1$; (iii) ${E}\{g_t(\boldsymbol{\theta} _{0})\}=\boldsymbol{\mathbf{0}}$ and there are positive functions $\Delta _{1}(r,p)$ and $\Delta _{2}(\varepsilon )$ such that for any $\varepsilon >0$, $$\inf_{\{\boldsymbol{\theta} \in \Theta:\Vert \boldsymbol{\theta} -\boldsymbol{\theta} _{0}\Vert _{2}\geq \varepsilon \}}\Vert {E}\{g_t(\boldsymbol{\theta})\}\Vert _{2}\geq \Delta _{1}(r,p)\Delta _{2}(\varepsilon )>0,$$where $\liminf_{r,p\rightarrow \infty }\Delta _{1}(r,p)>0$; (iv) $\sup_{\boldsymbol{\theta} \in \Theta}\Vert \bar{g}(\boldsymbol{\theta} )-{E}\{g_t(\boldsymbol{\theta} )\}\Vert _{2}=o_{p}\{\Delta _{1}(r,p)\}$. (A.2) (i) $\boldsymbol{\theta} _{0}\in \text{int}(\Theta)$ and $\Theta$ contains a small $\Vert \cdot \Vert _{2}$-neighborhood of $\boldsymbol{\theta} _{0}$ in which $g(x,\boldsymbol{\theta} )$ is continuously differentiable with respect to $\boldsymbol{\theta} $ for any $x\in \mathcal{X}$, the domain of $X_t$, and $$\bigg|\frac{\partial g_{i}(x,\boldsymbol{\theta} )}{\partial \theta _{j}}\bigg|\leq T_{n,ij}(x)~~(i=1,\ldots,r;j=1,\ldots,p)$$ for some functions $T_{n,ij}(x)$ with $E\{T_{n,ij}^2(X_t)\}\leq C$ for any $i,j$; (ii) $\sup_{\boldsymbol{\theta} \in \Theta}\Vert g(x,\boldsymbol{\theta} )\Vert _{2}\leq r^{1/2}B_{n}(x)$, where ${E}\{B_{n}^{\gamma }(X_{t})\}\leq C$ for $\gamma $ given in (A.1)(i); (iii) ${E}\{|g_{t,j}(\boldsymbol{\theta} _{0})|^{2\gamma }\}\leq C$ for all $j=1,\ldots ,r$; (iv) the eigenvalues of $[E\{\nabla _{\boldsymbol{\theta} }{g}_t(\boldsymbol{\theta} )\}]^{\prime }[E\{\nabla _{\boldsymbol{\theta} }{g}_t(\boldsymbol{\theta} )\}]$ in a $\|\cdot\|_2$-neighborhood of $\boldsymbol{\theta}_0$ are uniformly bounded away from zero and infinity; $\sup_{\boldsymbol{\theta} \in \Theta}\lambda _{\text{max}}\{n^{-1}\sum_{t=1}^{n}g_{t}(\boldsymbol{\theta} )g_{t}(\boldsymbol{\theta} )^{\prime }\}\leq C$ with probability approaching to 1. (A.3) In a $\|\cdot\|_2$-neighborhood of $\boldsymbol{\theta}_{0}$, $g(x,\boldsymbol{\theta})$ is twice continuously differentiable with respect to $\boldsymbol{\theta}$ for any $x\in\mathcal{X}$, and for some functions $K_{n,ijk}(x)$ with $E\{K_{n,ijk}^2(X_t)\}\leq C$ for any $i,j,k$, $$\bigg{|}\frac{\partial ^{2}g_{i}(x,\boldsymbol{\theta} )}{\partial \theta _{j}\partial \theta _{k}}\bigg{|}\leq K_{n,ijk}(x)~~(i=1,\ldots,r;j,k=1,\ldots,p).$$ Condition (A.1)(i) specifies the rate of decay for the mixing coefficients via a tuning parameter $\gamma $ as commonly assumed in the analysis of weakly dependent data. When the data are independent, $\alpha _{X}(k)=0$ for all $k\geq 1$ and this condition is automatically satisfied for any $\gamma >2$. [@Kitamura_1997_AOS] assumed $\sum_{k=1}^{\infty }\alpha _{X}(k)^{1-2/\gamma }<\infty $ for fixed finite dimensional EL, which implies $M^{-1}\sum_{k=1}^{M}k\alpha _{X}(k)^{1-2/\gamma }\rightarrow 0$ by Kronecker’s lemma. In the current high dimensional setting, we need stronger condition on the mixing coefficients in order to control remainder terms when analyzing the asymptotic properties of the GEL estimator and the GEL ratio. If $\{X_{t}\}$ is exponentially strong mixing [@FanYao_2003] so that $\alpha _{X}(k)\sim \varrho ^{k}$ for some $\varrho \in (0,1)$, then (A.1)(i) is automatically valid for any $\gamma >2$. (A.1)(ii) imposes a condition regarding the two blocking quantities $M$ and $L$, which is commonly assumed in the works of block bootstrap and blockwise EL. (A.1)(iii) is the population identification condition for the case of diverging parameter space. A similar assumption can be found in [Chen\_2007]{} and [@ChenPouzo_2012_Ecma]. The last part of (A.1) is an extension of the uniform convergence. If $p$ is fixed, under the assumption of the compactness of $\Theta $ and some other regularity conditions, following [@Newey_1991], $\sup_{\boldsymbol{\theta }\in \Theta }\Vert \bar{g}(\boldsymbol{\theta })-{E}\{g_{t}(\boldsymbol{\theta })\}\Vert _{2}=o_{p}(1)$ which is a special case of (A.1)(iv) with $\Delta _{1}(r,p)$ being a constant. As conditions (A.1)(iii) and (iv) are rather abstractive, we illustrate them via the examples given in Section 2.2. For Example 1, we can choose $\Delta_1(r,p)=1$ and $\Delta_2(\varepsilon)=\varepsilon$. For the conditional moment restrictions model (Example 3), a common assumption in the literature is that for any $a(Y_t)$ with $E\{a^2(Y_t)\}<\infty$ there exists a $K\times1$ vector $\gamma_K$ such that $E[\{a(Y_t)-\gamma_K^{\prime K}(Y_t)\}^2]\rightarrow0$ as $K\rightarrow\infty$. For any $\boldsymbol{\theta}\in\{\boldsymbol{\theta}:\|\boldsymbol{\theta}-\boldsymbol{\theta}_0\|_2\geq\varepsilon\}$, let $\Gamma_K(\boldsymbol{\theta})$ satisfy $E[\|E\{\rho(Y_t,Z_t,\boldsymbol{\theta})|Y_t\}-\Gamma_K(\boldsymbol{\theta})q^K(Y_t)\|_2^2]\rightarrow0$ as $K\rightarrow\infty$. If $\sup_{y}\|E\{\rho(Y_t,Z_t,\boldsymbol{\theta})|Y_t=y\}-\Gamma_K(\boldsymbol{\theta})q^K(y)\|_2=O(K^{-\lambda})$ for some $\lambda>1/2$, then $$\begin{split} \|E\{g_t(\boldsymbol{\theta})\}\|_2\geq&~ \big\|E[\{\Gamma_K(\boldsymbol{\theta}) q^K(Y_t)\}\otimes q^K(Y_t)]\big\|_2-\big\|E[\{\rho(Y_t,Z_t,\boldsymbol{\theta})-\Gamma_K(\boldsymbol{\theta})q^K(Y_t)\}\otimes q^K(Y_t)]\big\|_2 \\ \geq&~\lambda_{\min}\big(E\{q^K(Y_t)q^K(Y_t)^{\prime }\}\big)\|\Gamma_K(\boldsymbol{\theta})\|_F-O(K^{-\lambda})\big(\text{tr}[E\{q^K(Y_t)q^K(Y_t)^{\prime }\}]\big)^{1/2}. \end{split}$$ Under the assumption that the eigenvalues of $E\{q^K(Y_t)q^K(Y_t)^{\prime }\} $ are uniformly bounded away from zero and infinity, we have $$\begin{split} \inf_{\{\boldsymbol{\theta} \in \Theta:\Vert \boldsymbol{\theta} -\boldsymbol{\theta} _{0}\Vert _{2}\geq \varepsilon \}}\|E\{g_t(\boldsymbol{\theta})\}\|_2\geq&~ C\bigg[\inf_{\{\boldsymbol{\theta} \in \Theta:\Vert \boldsymbol{\theta} -\boldsymbol{\theta} _{0}\Vert _{2}\geq \varepsilon \}}\|\Gamma_K(\boldsymbol{\theta})\|_F-K^{1/2-\lambda}\bigg] \\ \geq&~C\bigg[\inf_{\{\boldsymbol{\theta} \in \Theta:\Vert \boldsymbol{\theta} -\boldsymbol{\theta} _{0}\Vert _{2}\geq \varepsilon \}}E\big[\big\|E\{\rho(Y_t,Z_t,\boldsymbol{\theta})|Y_t\}\big\|_2\big]-K^{1/2-\lambda}\bigg]. \\ \end{split}$$ Hence, as $\boldsymbol{\theta}_0$ is the unique root of $E\{\rho(Y_t,Z_t,\boldsymbol{\theta})|Y_t\}=\mathbf{0}$, (A.1)(iii) holds provided that the lower bound in the above inequality is greater than or equal to $\Delta _{1}(r,p)\Delta _{2}(\varepsilon )$. In addition, if the generalized residual function $\rho(y,z,\boldsymbol{\theta})$ is continuously differentiable with respect to $\boldsymbol{\theta}$. Then, $$\big\|E\{\rho(Y_t,Z_t,\boldsymbol{\theta})|Y_t\}\big\|_2\geq\|\boldsymbol{\theta}-\boldsymbol{\theta}_0\|_2\lambda_{\min}^{1/2}\big(E\big[\{\nabla_{\boldsymbol{\theta}} \rho(Y_t,Z_t,\boldsymbol{\theta}^*)\}^{\prime }\{\nabla_{\boldsymbol{\theta}} \rho(Y_t,Z_t,\boldsymbol{\theta}^*)\}|Y_t\big]\big)$$ where $\boldsymbol{\theta}^*$ is on the line joining $\boldsymbol{\theta}_0$ and $\boldsymbol{\theta}$. If the eigenvalues of $E[\{\nabla_{\boldsymbol{\theta}} \rho(Y_t,Z_t,\boldsymbol{\theta})\}^{\prime }\{\nabla_{\boldsymbol{\theta}} \rho(Y_t,Z_t,\boldsymbol{\theta})\}|Y_t]$ are uniformly bounded away from zero, $\Delta_1(r,p)$ and $\Delta_2(\varepsilon)$ can be chosen as some constant $C$ and $\varepsilon$, respectively. Condition (A.2)(i) assumes that the first derivatives of $g_{i}(x,\boldsymbol{\theta })$ near $\boldsymbol{\theta }_{0}$ are uniformly bounded by some functions which have bounded second moments. (A.2)(ii) generalizes the moment conditions on $g(x,\boldsymbol{\theta })$ for fixed dimensional case [@QinLawless_1994_AOS; @Kitamura_1997_AOS; @NeweySmith_2004_Ecma]. More generally, we can replace the factor $r^{1/2}$ by some function $\zeta (r)>0$. We let $\zeta (r)=r^{1/2}$ to simplify the presentation. (A.2)(iii) is the moment assumption on each $g_{t,j}(\boldsymbol{\theta }_{0})$. The first part of (A.2)(iv) is an extension of that assumed in the EL or the GEL in the fixed dimensional case [@QinLawless_1994_AOS; @Kitamura_1997_AOS; @NeweySmith_2004_Ecma]. The second one of (A.2)(iv) is to bound $\sup_{\boldsymbol{\theta }\in \Theta }\lambda _{\max }\{Q^{-1}\sum_{q=1}^{Q}\phi _{q}(\boldsymbol{\theta })\phi _{q}(\boldsymbol{\theta })\}$. Our proofs in Appendix can be easily extended to allow for $\sup_{\boldsymbol{\theta }\in \Theta }\lambda _{\max }\{Q^{-1}\sum_{q=1}^{Q}\phi _{q}(\boldsymbol{\theta })\phi _{q}(\boldsymbol{\theta })^{\prime }\}$ diverging in probability. Note that we do not assume the eigenvalues of $V_{M}$ or $V_{n}$ being bounded away from zero and infinity, but rather leave it open for specific treatments in Sections 3 and 4 for the consistency and the asymptotic normality of the GEL estimator. Our subsequent analysis shows that, to obtain the main results of the paper, $\lambda _{\min }(V_{M})$ is allowed to decay to zero at certain rates by properly restricting the diverging rates of $r$ and $p$. Condition (A.3) ensures the second derivatives of $g_{i}(x,\boldsymbol{\theta })$ near $\boldsymbol{\theta }_{0}$ are uniformly bounded by functions which have bounded second moments. Consistency and Convergence Rates ================================= To study the consistency of the GEL estimator $\widehat{\boldsymbol{\theta }}_{n}$ defined by (\[eq:gel\]), we need the following conditions regarding the dimensionality $r$, the block size $M$ and the sample size $n$:$$r^{2}M^{2-2/\gamma }n^{2/\gamma -1}=o(1)~~\text{and}~~r^{2}M^{3}n^{-1}=o(1). \label{eq:cond1}$$ Assume conditions , and that the eigenvalues of $V_M$ are uniformly bounded away from zero and infinity. Then, if holds, $\|\widehat{\boldsymbol{\theta}}_{n}-\boldsymbol{\theta}_{0}\|_2\xrightarrow{p}0. $ If in addition, $r^2pM^2n^{-1}=o(1)$, then $\|\widehat{\boldsymbol{\theta}}_{n}-\boldsymbol{\theta}_{0}\|_2=O_p(r^{1/2}n^{-1/2})$ and $\|\widehat{\lambda}(\widehat{\boldsymbol{\theta}}_{n})\|_2=O_p(r^{1/2}Mn^{-1/2})$. This theorem provides the consistency of the GEL estimator $\widehat{\boldsymbol{\theta }}_{n}$ for both independent and dependent data when the blocking size $M$ is either finite or diverging. For independent data, $V_{M}=E\{g_{1}(\boldsymbol{\theta }_{0})g_{1}(\boldsymbol{\theta }_{0})^{\prime }\}$ for any $M\geq 1$. Thus, to make $r$ have a faster diverging rate, we select the block size $M=1$. For dependent data, $$V_{M}=E\{g_{1}(\boldsymbol{\theta }_{0})g_{1}(\boldsymbol{\theta }_{0})^{\prime }\}+\sum_{k=1}^{M-1}\bigg(1-\frac{k}{M}\bigg)\big[E\{g_{1}(\boldsymbol{\theta }_{0})g_{1+k}(\boldsymbol{\theta }_{0})^{\prime }\}+E\{g_{1+k}(\boldsymbol{\theta }_{0})g_{1}(\boldsymbol{\theta }_{0})^{\prime }\}\big].$$However, if $\{g_{t}(\boldsymbol{\theta }_{0})\}_{t=1}^{n}$ is a martingale difference sequence, then $V_{M}\equiv E\{g_{1}(\boldsymbol{\theta }_{0})g_{1}(\boldsymbol{\theta }_{0})^{\prime }\}$ for any $M\geq 1$ and $M=1$ should be used to make $r$ have a faster diverging rate. Furthermore, if the eigenvalues of $V_{M}$ are uniformly bounded away from zero and infinity for some fixed $M$, (\[eq:cond1\]) is simplified to $$r^{2}n^{2/\gamma -1}=o(1).$$Here $\gamma $ determines the number of moments of the estimating equation as specified in (A.2)(ii) and (A.2)(iii). Then, $r=o(n^{1/2-1/\gamma })$ ensures the consistency of the GEL estimator $\widehat{\boldsymbol{\theta }}_{n}$. For large enough $\gamma $, $r$ will be made close to $o(n^{1/2})$, which is the best rate we can established. Theorem 1 encompasses the existing consistency results for the GEL estimator in the literature. Indeed, if $r$ is fixed and the data are independent, Theorem 1 implies that both $\Vert \widehat{\boldsymbol{\theta }}_{n}-\boldsymbol{\theta }_{0}\Vert _{2}$ and $\Vert \widehat{\lambda }(\widehat{\boldsymbol{\theta }}_{n})\Vert _{2}$ are $O_{p}(n^{-1/2})$, which are the same as the rates obtained in [@QinLawless_1994_AOS] for the EL and [NeweySmith\_2004\_Ecma]{} for the GEL. If $r$ is fixed but the data are dependent, Theorem 1 means that $\Vert \widehat{\boldsymbol{\theta }}_{n}-\boldsymbol{\theta }_{0}\Vert _{2}=O_{p}(n^{-1/2})$ and $\Vert \widehat{\lambda }(\widehat{\boldsymbol{\theta }}_{n})\Vert _{2}=O_{p}(Mn^{-1/2})$, which coincides with the result of [@Kitamura_1997_AOS] for the EL estimator. If $r$ is diverging and the data are independent, both $\Vert \widehat{\boldsymbol{\theta }}_{n}-\boldsymbol{\theta }_{0}\Vert _{2}$ and $\Vert \widehat{\lambda }(\widehat{\boldsymbol{\theta }}_{n})\Vert _{2}$ are $O_{p}(r^{1/2}n^{-1/2})$, which retain the results in [DonaldImbensNewey\_2003\_JOE]{} and [@LengTang_2011_Biometrika]. The following is an extension of Theorem 1 by allowing $V_{M}$ to be asymptotically singular, namely $\lambda _{\min }(V_{M})\rightarrow 0$ as $r\rightarrow \infty $, with $M$ being either fixed or diverging. \[cy1\] Assume conditions , , and that $\lambda_{\min}(V_M) \asymp r^{-\iota_1}$ for some $\iota_1>0$ and $\lambda_{\max}(V_M)$ is uniformly bounded away from zero and infinity. Then $\|\widehat{\boldsymbol{\theta}}_n-\boldsymbol{\theta}_0\|_2=O_p(r^{(1+\iota_1)/2}n^{-1/2})$ and $\|\widehat{\lambda}(\widehat{\boldsymbol{\theta}}_n)\|_2=O_p(r^{(1+3\iota_1)/2}Mn^{-1/2})$ provided that $r^{2+3\iota_1}M^{2-2/\gamma}n^{2/\gamma-1}=o(1)$, $r^{2+2\iota_1}M^3n^{-1}=o(1)$, $r^{2+3\iota_1}pMn^{-1}=o(1)$ and $r^{2+\iota_1}pM^2n^{-1}=o(1)$. This corollary shows that when the smallest eigenvalue of $V_{M}$ is not bounded away from zero, the convergence rates for $\widehat{\boldsymbol{\theta }}_{n}$ and the Lagrange multiplier $\widehat{\lambda }(\widehat{\boldsymbol{\theta }}_{n})$ become slower. Theorem 1 can be viewed as a special case of Corollary 1 with $\iota _{1}=0$. The convergence rate of $\Vert \widehat{\boldsymbol{\theta }}_{n}-\boldsymbol{\theta }_{0}\Vert _{2}$ attained in Theorem 1 is dictated by $r$, the number of the moment restrictions, rather than by $p$, the dimension of $\boldsymbol{\theta }$. Under slightly stronger conditions the next proposition improves the convergence rate to $O_{p}(p^{1/2}n^{-1/2})$. Under conditions , assume that the eigenvalues of $V_{M}$ and $V_{n}$ are uniformly bounded away from zero and infinity. Then $\Vert \widehat{\boldsymbol{\theta }}_{n}-\boldsymbol{\theta }_{0}\Vert _{2}=O_{p}(p^{1/2}n^{-1/2})$ provided that $r^{3}M^{2-2/\gamma }n^{2/\gamma -1}=o(1)$, $r^{3}M^{3}n^{-1}=o(1)$, $r^{3}pMn^{-1}=o(1)$ and $r^{3}p^{2}n^{-1}=o(1)$. Asymptotic Normality ==================== We now turn to the asymptotic normality of the GEL estimator $\widehat{\boldsymbol{\theta}}_{n}$. We are in particularly interested in the effect of the block size $M$ on the estimation efficiency. Based on the consistency of $\widehat{\boldsymbol{\theta}}_{n}$ and $\widehat{\lambda}(\widehat{\boldsymbol{\theta}}_{n})$ given in Theorem 1, expanding $\nabla _{\lambda }\widehat{S}_{n}(\widehat{\boldsymbol{\theta}}_{n},\widehat{\lambda}(\widehat{\boldsymbol{\theta}}_{n}))=\boldsymbol{\mathbf{0}}$ for $\widehat{\lambda}(\widehat{\boldsymbol{\theta}}_{n})$ around $\lambda =\boldsymbol{\mathbf{0}}$ gives $$\boldsymbol{\mathbf{0}}=\frac{1}{Q}\sum_{q=1}^{Q}\rho_v(0)\phi _{q}(\widehat{\boldsymbol{\theta}}_{n})+\frac{1}{Q}\sum_{q=1}^{Q}\rho_{vv}(\widetilde{\lambda}^{\prime }\phi_q(\widehat{\boldsymbol{\theta}}_n))\phi _{q}(\widehat{\boldsymbol{\theta}}_{n})\phi _{q}(\widehat{\boldsymbol{\theta}}_{n})^{\prime }\widehat{\lambda}(\widehat{\boldsymbol{\theta}}_{n}), \label{eq:0.2}$$where $\widetilde{\lambda}$ is on the line joining $\boldsymbol{\mathbf{0}}$ and $\widehat{\lambda}(\widehat{\boldsymbol{\theta}}_{n})$. From ([eq:0.1]{}) and (\[eq:0.2\]), it yields $$\bigg[\frac{1}{Q}\sum_{q=1}^{Q}\rho_v(\widehat{\lambda}(\widehat{\boldsymbol{\theta}}_n)^{\prime }\phi_q(\widehat{\boldsymbol{\theta}}_n))\{\nabla _{\boldsymbol{\theta} }\phi _{q}(\widehat{\boldsymbol{\theta}}_{n})\}^{\prime }\bigg]\bigg\{\frac{1}{Q}\sum_{q=1}^{Q}\rho_{vv}(\widetilde{\lambda}^{\prime }\phi_q(\widehat{\boldsymbol{\theta}}_n))\phi _{q}(\widehat{\boldsymbol{\theta}}_{n})\phi _{q}(\widehat{\boldsymbol{\theta}}_{n})^{\prime }\bigg\}^{-1}\bar{\phi} (\widehat{\boldsymbol{\theta}}_{n})=\boldsymbol{\mathbf{0}}. \label{eq:key}$$Based on (\[eq:key\]), we can establish the following proposition which is the starting point in our study of the asymptotic normality of $\widehat{\boldsymbol{\theta}}_{n}$. Under conditions , assume that the eigenvalues of $V_M$ and $V_n$ are uniformly bounded away from zero and infinity. If $r^2pM^2n^{-1}=o(1)$ and holds, then for any vector $\boldsymbol{\alpha}_{n}\in\mathbb{R}^p$ with unit $L_2$-norm, $$\begin{split} &\sqrt{n}\boldsymbol{\alpha}_{n}^{\prime }([{E}\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}]^{\prime }V_M^{-1}V_nV_M^{-1}[{E}\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}])^{-1/2}[E\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}]^{\prime }V_M^{-1}[{E}\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}](\widehat{\boldsymbol{\theta}}_{n}-\boldsymbol{\theta}_{0}) \\ =&-\sqrt{n}\boldsymbol{\alpha}_{n}^{\prime }([{E}\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}]^{\prime }V_M^{-1}V_nV_M^{-1}[{E}\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}])^{-1/2}[{E}\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}]^{\prime }V_M^{-1}\bar{g}(\boldsymbol{\theta}_{0}) \\ &+O_p(r^{3/2}p^{1/2}M^{1/2}n^{-1/2})+O_p(r^{3/2}pn^{-1/2})+O_p(r^{3/2}M^{1-1/\gamma}n^{1/\gamma-1/2})+O_p(r^{3/2}M^{3/2}n^{-1/2}). \end{split}$$ Proposition 2 covers both the finite and diverging $M$ cases. When $M$ is diverging, $\Vert V_{M}-V_{n}\Vert _{2}\rightarrow 0$ provided that $r=o(M)$. We can replace $V_{M}^{-1}V_{n}V_{M}^{-1}$ on the left-hand side of above asymptotic expansion by $V_{n}^{-1}$ via adding an extra high order term on the right-hand side of the asymptotic expansion. Let $$\boldsymbol{\beta }_{n}=-V_{M}^{-1}[{E}\{\nabla _{\boldsymbol{\theta }}g_{t}(\boldsymbol{\theta }_{0})\}]([{E}\{\nabla _{\boldsymbol{\theta }}g_{t}(\boldsymbol{\theta }_{0})\}]^{\prime }V_{M}^{-1}V_{n}V_{M}^{-1}[{E}\{\nabla _{\boldsymbol{\theta }}g_{t}(\boldsymbol{\theta }_{0})\}])^{-1/2}\boldsymbol{\alpha }_{n} \label{eq:betan}$$and $U_{n,t}=n^{-1/2}\boldsymbol{\beta }_{n}^{\prime }g_{t}(\boldsymbol{\theta }_{0})$ for $t=1,\ldots ,n$. From Proposition 2, a major point of interest is under what conditions $\sum_{t=1}^{n}U_{n,t}$ is asymptotically normal. Let us first consider the easier case where the observations $\{X_t\}_{t=1}^n $ are independent. From Lindeberg-Feller theorem [@Durrett_2010], to attain the asymptotic normality of $\sum_{t=1}^nU_{n,t}$, it suffices to verify the following two conditions,$$\text{(i)}~\sum_{t=1}^n{E}(U_{n,t}^2)\rightarrow1~~\text{and}~~\text{(ii)}~\sum_{t=1}^n{E}\{U_{n,t}^21_{(|U_{n,t}|>\varepsilon)}\}\rightarrow0 ~~\text{as}~~ n\rightarrow\infty~~\text{for any}~~\varepsilon>0.$$ Actually, $V_n=V_M={E}\{ g_t(\boldsymbol{\theta}_{0})g_t(\boldsymbol{\theta}_{0})^{\prime }\}$ in this case. Hence, $\sum_{t=1}^n{E}(U_{n,t}^2)=1$. Note that $\|\boldsymbol{\beta}_n\|_2\leq \lambda_{\text{min}}^{-1/2}(V_n)$ which is uniformly bounded away from infinity if $\lambda_{\min}(V_n)$ is uniformly bounded away from zero. Hence, by (A.2)(ii), $$(n^{1/2}\varepsilon)^{\gamma-2}{E}[|\boldsymbol{\beta}_n^{\prime }g_t(\boldsymbol{\theta}_{0})|^21_{\{|\boldsymbol{\beta}_n^{\prime }g_t(\boldsymbol{\theta}_{0})|>n^{1/2}\varepsilon\}}]\leq{E}\{|\boldsymbol{\beta}_n^{\prime }g_t(\boldsymbol{\theta}_{0})|^{\gamma}\}\leq Cr^{\gamma/2},$$ which implies that part (ii) holds if $rn^{2/\gamma-1}=o(1)$. Therefore, $\sum_{t=1}^nU_{n,t}\xrightarrow{d}N(0,1)$ provided that $rn^{2/\gamma-1}=o(1)$ for any selection of $\boldsymbol{\alpha}_{n} \in \mathbb{R}^p$ with unit $L_2$-norm. For dependent data, we need to assume $\sup_{n}{E}\{|\boldsymbol{\beta }_{n}^{\prime }g_{t}(\boldsymbol{\theta }_{0})|^{2+v}\}<\infty $ for some $v>0 $, namely $|\boldsymbol{\beta }_{n}^{\prime }g_{t}(\boldsymbol{\theta }_{0})| $ has a higher than two uniformly bounded moment. This is required in the central limit theorem for dependent processes as carried out in [PeligradUtev\_1997\_AOP]{} and [@FrancqZakoian_2005_ET]. It is used to guarantee the limit of $\text{Var}\{n^{-1/2}\sum_{t=1}^{n}\boldsymbol{\beta }_{n}^{\prime }g_{t}(\boldsymbol{\theta }_{0})\}$ can be well defined as $n\rightarrow \infty $. More specifically, notice that $$\text{Var}\bigg\{\frac{1}{n^{1/2}}\sum_{t=1}^{n}\boldsymbol{\beta }_{n}^{\prime }g_{t}(\boldsymbol{\theta }_{0})\bigg\}=E\{|\boldsymbol{\beta }_{n}^{\prime }g_{1}(\boldsymbol{\theta }_{0})|^{2}\}+2\sum_{k=1}^{n-1}\bigg(1-\frac{k}{n}\bigg)E\{\boldsymbol{\beta }_{n}^{\prime }g_{1}(\boldsymbol{\theta }_{0})g_{1+k}(\boldsymbol{\theta }_{0})^{\prime }\boldsymbol{\beta }_{n}\},$$to define the limit of above sum of series, we need that $\text{Var}\{n^{-1/2}\sum_{t=1}^{n}\boldsymbol{\beta }_{n}^{\prime }g_{t}(\boldsymbol{\theta }_{0})\}$ is absolutely convergent, i.e., $$\lim_{n\rightarrow \infty }\bigg[E\{|\boldsymbol{\beta }_{n}^{\prime }g_{1}(\boldsymbol{\theta }_{0})|^{2}\}+2\sum_{k=1}^{n-1}\bigg(1-\frac{k}{n}\bigg)|E\{\boldsymbol{\beta }_{n}^{\prime }g_{1}(\boldsymbol{\theta }_{0})g_{1+k}(\boldsymbol{\theta }_{0})^{\prime }\boldsymbol{\beta }_{n}\}|\bigg]<\infty .$$By Davydov inequality [@Davydov_1968_TPIA; @Rio_1993_AIHP], the absolute convergence of $\text{Var}\{n^{-1/2}\sum_{t=1}^{n}\boldsymbol{\beta }_{n}^{\prime }g_{t}(\boldsymbol{\theta }_{0})\}$ will hold by requiring $\sup_{n}E\{|\boldsymbol{\beta }_{n}^{\prime }g_{t}(\boldsymbol{\theta }_{0})|^{2+v}\}<\infty $ for some suitable $v$. For high dimensional moment equation $g(x,\theta)$ with diverging $r$, we need $$\sup_n{E}\{|\boldsymbol{\beta}_n^{\prime }g_t(\boldsymbol{\theta}_{0})|^{\gamma}\}<\infty \label{eq:moment-cond}$$ for $\boldsymbol{\beta}_n$ defined via (\[eq:betan\]) and $\gamma > 2$ specified in (A.1)(i). A sufficient condition for (\[eq:moment-cond\]) is to restrict$$\boldsymbol{\beta}_n\in\mathcal{D}(K):=\bigg\{(v_1,v_2,\ldots)\in\mathbb{R}^\infty:\sum_{k=1}^{\infty}|v_k|\leq K\bigg\},$$ where $K$ is a given finite constant. To appreciate this, write $\boldsymbol{\beta}_n=(\beta_{n,1},\ldots,\beta_{n,r})^{\prime }$ and let $\kappa_r=\sum_{j=1}^r|\beta_{n,j}|$. Then, $${E}\{|\boldsymbol{\beta}_n^{\prime }g_t(\boldsymbol{\theta}_{0})|^{\gamma}\}=\kappa_r^{\gamma}{E}\bigg\{\bigg|\sum_{j=1}^r\frac{|\beta_{n,j}|}{\kappa_r}\text{sign}(\beta_{n,j})g_{t,j}(\boldsymbol{\theta}_{0})\bigg|^{\gamma}\bigg\}\leq K^{\gamma} C,$$ where the last step is based on the Jensen’s inequality and (A.2)(iii). If $\sum_{j=1}^r|\beta_{n,j}|\rightarrow\infty$ as $n\rightarrow\infty$, we can construct a counter-example such that $\sup_n{E}\{|\boldsymbol{\beta}_n^{\prime }g_t(\boldsymbol{\theta}_{0})|^{2+v}\}\rightarrow\infty$ for any $v>0$. The following theorem establishes the asymptotic normality of $\widehat{\boldsymbol{\theta}}_{n}$. Under conditions , assume that the eigenvalues of $V_M$ and $V_n$ are uniformly bounded away from zero and infinity. For dependent data, if $$r^3M^{2-2/\gamma}n^{2/\gamma-1}=o(1),~r^3M^3n^{-1}=o(1),~r^{3}pMn^{-1}=o(1)~~\text{and}~~r^3p^2n^{-1}=o(1), \label{eq:condasynor}$$ then for any $\boldsymbol{\alpha}_{n}\in\mathbb{R}^{p}$ with unit $L_2$-norm such that holds, $$\sqrt{n}\boldsymbol{\alpha}_{n}^{\prime }([{E}\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}]^{\prime }V_M^{-1}V_nV_M^{-1}[{E}\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}])^{-1/2}[E\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}]^{\prime }V_M^{-1}[{E}\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}](\widehat{\boldsymbol{\theta}}_{n}-\boldsymbol{\theta}_{0})$$ converges to $N(0,1)$ as $n\rightarrow\infty$. For finite block size $M$, the above asymptotic distribution holds provided that $$r^3n^{2/\gamma-1}=o(1) \quad \hbox{and} \quad r^3p^2n^{-1}=o(1).$$ Since $$\begin{split} &([E\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}]^{\prime }V_M^{-1}[{E}\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}])^{-1}([{E}\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}]^{\prime }V_M^{-1}V_nV_M^{-1}[{E}\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}]) \\ &~~~~~~~~~~~~~~\times([E\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}]^{\prime }V_M^{-1}[{E}\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}])^{-1}\geq([E\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}]^{\prime }V_n^{-1}[{E}\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}])^{-1}, \end{split}$$ the GEL estimator is asymptotically efficient if $\|V_M-V_n\|_2\rightarrow 0$, which implies $V_M^{-1}V_nV_M^{-1}$ is asymptotically equivalent to $V_n^{-1}$. This means that if $\{g_t(\boldsymbol{\theta}_0)\}_{t=1}^n$ is a martingale difference sequence, as $V_M=V_n=E\{g_1(\boldsymbol{\theta}_0)g_1(\boldsymbol{\theta}_0)^{\prime }\}$ for any $M\geq1$, selecting $M=1$ will lead to the efficient GEL estimation. In a general case where the nature of the dependence in the estimating function is unknown, letting $M\rightarrow\infty$ at some suitable diverging rate, so that ([eq:condasynor]{}) is satisfied, will lead to the efficient estimation. Specifically, as $$\|V_M-V_n\|_2\leq CrM^{-1}\sum_{k=1}^Mk\alpha_X(k)^{1-2/\gamma},$$ under (A.1)(i) and (A.2)(ii), choosing $M \to \infty$ such that $r=o(M)$ produces the asymptotically efficient GEL estimator $\widehat{\boldsymbol{\theta}}_n$. According to (\[eq:condasynor\]) and $r=o(M)$, the divergence rate in $M$ is $M=O(n^{\{(\gamma-2)/(5\gamma-2)\}\wedge{1/6}})$ while $r=o(n^{\{(\gamma-2)/(5\gamma-2)\}\wedge {1/6}})$, regardless $p $ being fixed or diverging. Under such setting, the best growth rate for $r$ is $r=o(n^{1/6}) $ when $\gamma\geq10$. In comparison with the case of finite $M$, while letting $M\rightarrow\infty$ can guarantee the efficiency, it does slows the divergence of $r$. If the smallest eigenvalues of $V_M$ and $V_n$ decay to zero as $r\rightarrow\infty$, we assume $\lambda_{\min}(V_M)\asymp r^{-\iota_1}$ and $\lambda_{\min}(V_n)\asymp r^{-\iota_2}$ for some positive $\iota_1$ and $\iota_2$. Based on Corollary \[cy1\], by repeating the proof of Proposition 2 in Appendix, it can be shown that the leading term in the asymptotic expansion of Proposition 2 remains while the four remainder terms become $$\begin{split} &~O_p(r^{(3+6\iota_1+\iota_2)/2}p^{1/2}M^{1/2}n^{-1/2})+O_p(r^{(3+4\iota_1+\iota_2)/2}pn^{-1/2}) \\ +&~O_p(r^{(3+5\iota_1+\iota_2)/2}M^{1-1/\gamma}n^{1/\gamma-1/2})+O_p(r^{(3+5\iota_1+\iota_2)/2}M^{3/2}n^{-1/2}) \end{split}$$ provided that the conditions governing $r, p, M$ and $n$ assumed in Corollary \[cy1\] hold. By the central limit theorem established in [@FrancqZakoian_2005_ET], the leading term in the asymptotic expansion of Proposition 2 converges to $N(0,1)$ regardless $\lambda_{\min}(V_M)\rightarrow0$ and $\lambda_{\min}(V_n)\rightarrow0$ or not. Hence, the asymptotic normality of the GEL estimator $\widehat{\boldsymbol{\theta}}_n$ is valid free of the statue of the eigenvalues of $V_M$ and $V_n$. The difference is that when $\lambda_{\min}(V_M)\rightarrow0$ and $\lambda_{\min}(V_n)\rightarrow0$, the growth rate of $r$ and/or $M$ are reduced. To put the growth rate of $r$ into perspectives and to highlight the impacts of data dependence, we consider the independent analogue of Theorem 2 in the following, whose proof is obtained by assigning $\alpha_X(k)=0$ and $M=1$ in Proposition 2. Under conditions , assume that the eigenvalues of $E\{g_1(\boldsymbol{\theta}_0)g_1(\boldsymbol{\theta}_0)^{\prime }\}$ are uniformly bounded away from zero and infinity. For independent data, if $r^3p^2n^{-1}=o(1)$ and $r^3n^{2/\gamma-1}=o(1)$, then for any $\boldsymbol{\alpha}_{n}\in\mathbb{R}^{p}$ with unit $L_2$-norm, $$\sqrt{n}\boldsymbol{\alpha}_{n}^{\prime }([{E}\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}]^{\prime }V_n^{-1}[{E}\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}])^{1/2}(\widehat{\boldsymbol{\theta}}_{n}-\boldsymbol{\theta}_{0})\xrightarrow[]{d}N(0,1)~~\text{as}~n\rightarrow\infty.$$ The above corollary shows that under independence, the growth rate for $r$ is $o(n^{1/3-2/(3\gamma)})$ if $p$ is fixed. If $\gamma$ is sufficiently large, the rate of $r$ can be close to $o(n^{1/3})$. If $p$ grows with $r$ and $p/r\rightarrow y\in(0,1]$, then $r=o(n^{\{1/3-2/(3\gamma)\}\wedge1/5})$. In particular, if $\gamma\geq5$, $r=o(n^{1/5})$ which retains Theorem 2 in [@LengTang_2011_Biometrika] for the EL estimator. Comparing the growth rates for $r$ under the dependent and independent settings, when $M$ is diverging, we see a slowing down in the rate under dependence from $o(n^{1/5})$ to $o(n^{1/6})$ if the best moment conditions hold. If $p$, the dimension of $\boldsymbol{\theta }$, is fixed, as in a case of conditional moment restrictions in Example 3, the asymptotic normality of $\widehat{\boldsymbol{\theta }}_{n}$ can be attained with some ease. It can be shown that $\boldsymbol{\beta }_{n}$ is automatically in $\mathcal{D}(K)$ for a large enough $K$, which implies the condition (\[eq:moment-cond\]) holds for any $\boldsymbol{\alpha }_{n}\in \mathbb{R}^{p}$ with unit $L_{2}$-norm. This is summarized in the following corollary. Under conditions , assume that the eigenvalues of $V_M$ and $V_n$ are uniformly bounded away from zero and infinity. For dependent data, if $p$ is fixed, then for any $\boldsymbol{\alpha}_{n}\in\mathbb{R}^p$ with unit $L_2$-norm, $$\sqrt{n}\boldsymbol{\alpha}_{n}^{\prime }([{E}\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}]^{\prime }V_M^{-1}V_nV_M^{-1}[{E}\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}])^{-1/2}[E\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}]^{\prime }V_M^{-1}[{E}\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}](\widehat{\boldsymbol{\theta}}_{n}-\boldsymbol{\theta}_{0})$$ converges to $N(0,1)$ as $n\rightarrow\infty$, provided that $r^3M^{2-2/\gamma}n^{2/\gamma-1}=o(1)$ and $r^3M^3n^{-1}=o(1)$. This Corollary with $M=1$ recovers that in [@DonaldImbensNewey_2003_JOE] for iid data. Generalized Empirical Likelihood Ratios ======================================= The EL ratio $w_{n}(\boldsymbol{\theta })=-2\log \{Q^{Q}\mathcal{L}(\boldsymbol{\theta })\}$ for $\mathcal{L}(\boldsymbol{\theta })$ defined in (\[eq:L\]) plays an important role in the statistical inference. A prominent result for fixed dimensional EL is its resembling the parameter likelihood by have a limiting chi-square distribution under a wide range of situations, as demonstrated in [@Owen_1988_Biometrika], [ChenCui\_2003]{}, [@QinLawless_1994_AOS] and [ChenVanKeilegom\_2009\_Test]{} for independent data, and [Kitamura\_1997\_AOS]{} for dependent data. For GEL, we define the GEL ratio as $$w_n(\boldsymbol{\theta})=\frac{2\rho_{vv}(0)}{\rho_v^2(0)}\bigg\{Q\rho(0)-\max_{\lambda\in\widehat{\Lambda}_n(\boldsymbol{\theta})}\sum_{q=1}^Q\rho(\lambda^{\prime }\phi_q(\boldsymbol{\theta}))\bigg\} \label{eq:gelratio}$$ which is the extension of the EL ratio in the GEL framework. We consider the asymptotic distribution of the GEL ratio $w_{n}(\boldsymbol{\theta }_{0})$ when both $r$ and $p$ are diverging. Under such setting, a natural form of the Wilks’ theorem is $$(2r)^{-1/2}\{w_{n}(\boldsymbol{\theta }_{0})-r\}\xrightarrow {d}N(0,1)~~\text{as}~r\rightarrow \infty . \label{eq:anchisq}$$For the case of means where $g_{t}(\boldsymbol{\theta })=X_{t}-\boldsymbol{\theta }$ with independent observations, [@ChenPengQin_2009_Biometrika] and [@HjortMcKeagueVanKeilegom_2009_AOS] evaluated the impact of the dimensionality on the asymptotic distribution (\[eq:anchisq\]) for the EL ratio by providing various diverging rates for $r$. For parameters defined by general moment restrictions, establishing the limiting distribution of the GEL ratio is far more challenging. We need the following stronger version of (A.1)(i): (A.1)’(i) There is some $\eta>8$ such that $\alpha_X(k)^{1-2/\gamma}\asymp k^{-\eta}$ where $\gamma$ is given in (A.2). Condition (A.1)’(i) is used to guarantee the leading order term of ([eq:gelratio]{}) has the similar probabilistic behavior as the chi-square distribution. It is automatically satisfied with $\eta =\infty $ if $X_{t}$ is exponentially strong mixing or independent. We also need the following conditions: $$r^{3}M^{2-2/\gamma }n^{2/\gamma -1}=o(1),~~r^{3}M^{3}n^{-1}=o(1)~~\text{and}~~r^{3/2}M^{-1}\sum_{k=1}^{M}k\alpha _{X}(k)^{1-2/\gamma }=o(1). \label{eq:cond-ratio}$$Define $$\xi =\frac{\eta -8}{4\eta +4}1_{\{8<\eta <32\}}+\frac{2}{11}1_{\{32\leq \eta \leq \infty \}}+1_{\{\text{indenpendent data}\}}. \label{eq:xi}$$The next theorem establishes the asymptotic distribution of $w_{n}(\boldsymbol{\theta }_{0})$. Under conditions , and , assume that the eigenvalues of $V_n$ are uniformly bounded away from zero and infinity. If holds and $r=o(n^\xi)$ where $\xi$ is defined in , then $$(2r)^{-1/2}\{w_n(\boldsymbol{\theta}_{0})-r\}\xrightarrow[]{d}N(0,1)~~ \mbox{as $r\rightarrow\infty$.}$$ This theorem is new for dependent data and includes some established results for independent data as special cases. For independent data, this theorem implies that the asymptotic normality of the GEL ratio is valid if $r=o(n^{1/3-2/(3\gamma )})$, which is the same as that in [HjortMcKeagueVanKeilegom\_2009\_AOS]{} for the EL ratio with independent data. Our result is more general than theirs since we allow for GEL ratio and for dependent data. For dependent data, the block size is $M=O(n^{(\gamma -2)/(4\gamma -2)})$ if $2<\gamma <8$ and $M=O(n^{1/5})$ otherwise, and hence the asymptotic distribution (\[eq:anchisq\]) holds if $r=o(n^{\delta })$ with $$\delta =\min \left( \frac{\eta -8}{4\eta +4}1_{\{8<\eta <32\}}+\frac{2}{11}1_{\{32\leq \eta \leq \infty \}},\frac{\gamma -2}{6\gamma -3}1_{\{2<\gamma <8\}}+\frac{2}{15}1_{\{\gamma \geq 8\}}\right) .$$When $\eta $ and $\gamma $ are sufficiently large, the best diverging rate is $r=o(n^{2/15})$ for the dependent case, which is slower than the rate of $r=o(n^{1/3-2/(3\gamma )})$ for the independence case. Test for Over-identification ============================ [ For moment restrictions, it is important to check on the validity of the model by testing the following hypotheses $$H_{0}:{E}\{g(X_{t},\boldsymbol{\theta }_{0})\}=\boldsymbol{\mathbf{0}}~~\text{for some}~\boldsymbol{\theta }_{0}\in \Theta ~~\text{v.s.}~~H_{1}:{E}\{g(X_{t},\boldsymbol{\theta })\}\neq \boldsymbol{\mathbf{0}}~~\text{for any}~\boldsymbol{\theta }\in \Theta .$$We consider testing the above hypothesis when $r>p$, namely the moment equation overly identify the parameter $\boldsymbol{\theta }$.]{} We formulate the test statistic as the GEL ratio $w_{n}(\widehat{\boldsymbol{\theta }}_{n})$. For the EL ratio, it has been demonstrated in the fixed dimensional case by [@QinLawless_1994_AOS] and [@Kitamura_1997_AOS] that $$w_{n}(\widehat{\boldsymbol{\theta }}_{EL})\xrightarrow{d}\chi _{r-p}^{2}~~~\text{as}~n\rightarrow \infty$$under $H_{0}$. This mirrors the J-test of [@Hansen_1982_Ecma]’s GMM with fixed and finite dimensions $r$ and $p$. To formulate the GEL specification test allowing for increasing dimensions $r $ and $p$, we are to study the asymptotic distribution of $w_{n}(\widehat{\boldsymbol{\theta }}_{n})$ under $H_{0}$ first. We only need to consider its leading order $n\bar{g}(\widehat{\boldsymbol{\theta }}_{n})^{\prime }\{M\widehat{\Omega }(\widehat{\boldsymbol{\theta }}_{n})\}^{-1}\bar{g}(\widehat{\boldsymbol{\theta }}_{n})$ as the remainder terms in the asymptotic expansion of $w_{n}(\widehat{\boldsymbol{\theta }}_{n})$ can be shown to be of a smaller order. Since $\widehat{\boldsymbol{\theta }}_{n}$ is consistent for $\boldsymbol{\theta }_{0}$ under $H_{0}$, Lemma \[la19\] in Appendix establishes the relationship between the asymptotic distributions of $n\bar{g}(\widehat{\boldsymbol{\theta }}_{n})^{\prime }\{M\widehat{\Omega }(\widehat{\boldsymbol{\theta }}_{n})\}^{-1}\bar{g}(\widehat{\boldsymbol{\theta }}_{n})$ and $n\bar{g}(\boldsymbol{\theta }_{0})^{\prime }V_{n}^{-1}\bar{g}(\boldsymbol{\theta }_{0})$ under $H_{0}$. We need the following conditions: $$\label{eq:8} \begin{split} & ~~~~~~~~r^{3}pn^{-1}=o(1),~~pr^{-1/2}=o(1),~~r^{3}M^{3}n^{-1}=o(1), \\ & r^{3}M^{2-2/\gamma }n^{2/\gamma -1}=o(1)~~\text{and}~~r^{3/2}M^{-1}\sum_{k=1}^{M}k\alpha _{X}(k)^{1-2/\gamma }=o(1). \end{split}$$Compared with the conditions for the asymptotic distribution of $w_{n}(\boldsymbol{\theta }_{0})$ in (\[eq:cond-ratio\]), the first two restrictions in (\[eq:8\]) are the extra ones used to control the remainder terms. Under conditions , , , , and , assume that the eigenvalues of $V_n$ are bounded away from zero and infinity. If holds and $r=o(n^\xi)$, where $\xi$ is defined in , then $$\{2(r-p)\}^{-1/2}\{w_n(\widehat{\boldsymbol{\theta}} _{n})-(r-p)\}\xrightarrow{d}N(0,1) \quad \mbox{as $r\rightarrow\infty$.}$$ The asymptotic normality can be used to derive the over-identification test under high dimensionality and dependence. Specifically, $H_0$ is rejected if $$\{2(r-p)\}^{-1/2}\{w_n(\widehat{\boldsymbol{\theta}}_{n})-(r-p)\} > z_{1-\alpha}$$ where $z_{1-\alpha}$ is the $1-\alpha$ quantile of $N(0,1)$. [To show the above GEL test for over-identification is consistent, we assume that under the alternative hypothesis $H_{1}$, $$\inf_{\boldsymbol{\theta }\in \Theta }\Vert {E}\{g(X_{t},\boldsymbol{\theta })\}\Vert _{2}\geq \varsigma . \label{eq:cond61}$$The following theorem describes the behavior of $\{2(r-p)\}^{-1/2}\{w_{n}(\widehat{\boldsymbol{\theta }}_{n})-(r-p)\}$ under $H_{1}$.]{} Under conditions , , , and , if there is a positive constant $\epsilon$ such that $r^2M^{1-2/\gamma}n^{2/\gamma-1}(\log n)^\epsilon\varsigma^{-2}=O(1)$, $r^{1/2}Mn^{-1}\varsigma^{-1}=o(1)$ and $\Delta_1(r,p)\varsigma^{-1}=O(1)$, then $$\{2(r-p)\}^{-1/2}\{w_n(\widehat{\boldsymbol{\theta}}_n)-(r-p)\}\xrightarrow{p}\infty \quad \mbox{as $r\rightarrow\infty$}.$$ This theorem shows that the GEL test is consistent. Unlike Theorem 4, this theorem does not require the block size $M\rightarrow \infty $ and assumes the weaker condition (A.1)(i) (instead of the more restrictive one (A.1)’(i)). From the proof given in the Appendix which follows the technique developed in [@ChangTangWu_2013], the test statistic $\{2(r-p)\}^{-1/2}\{w_{n}(\widehat{\boldsymbol{\theta }}_{n})-(r-p)\}$ diverges to infinity at least at the rate of $O(r^{1/2})$ under $H_{1}$. Penalized Generalized Empirical Likelihood ========================================== In high dimensional data analysis, when the dimension of parameters is large, i.e., $p\rightarrow\infty$, a more reasonable assumption is that only a subset of the parameters are nonzero. Write $\boldsymbol{\theta}_0=(\theta_{01},\ldots,\theta_{0p})^{\prime }\in\mathbb{R}^p$ and define $\mathcal{A}=\{j:\theta_{0j}\neq0\}$ with its cardinality $s=|\mathcal{A}|$. Without loss of generality, let $\boldsymbol{\theta}=(\boldsymbol{\theta}^{(1)^{\prime }},\boldsymbol{\theta}^{(2)^{\prime }})^{\prime }$, where $\boldsymbol{\theta}^{(1)}\in\mathbb{R}^s$ and $\boldsymbol{\theta}^{(2)}\in\mathbb{R}^{p-s}$ correspond to the nonzero and zero components respectively, i.e., $\boldsymbol{\theta}_0=(\boldsymbol{\theta}_{0}^{(1)^{\prime }},\boldsymbol{\mathbf{0}}^{\prime })^{\prime }$. Under such sparsity, we can allow the number of parameters is larger than the number of estimating equations, i.e., $p>r$. However, we still need to assume $s\leq r$, which means that the “real" parameters can be uniquely identified by the moment restrictions (\[eq:1\]). To carry out the statistical inference on $\boldsymbol{\theta}$ under the sparsity assumption, we add a penalty term in (\[eq:gel\]) and the penalized GEL estimator is defined as $$\widehat{\boldsymbol{\theta}}_n^{(\mathrm{pe})}=\arg\min_{\boldsymbol{\theta}\in\Theta}\max_{\lambda\in\widehat{\Lambda}_n(\boldsymbol{\theta})}\bigg\{\sum_{q=1}^Q\rho(\lambda^{\prime }\phi_M(B_q,\boldsymbol{\theta}))+Q\sum_{j=1}^p p_\tau(|\theta_j|)\bigg\}$$ where $p_{\tau}(\cdot)$ is some penalty function with a tuning parameter $\tau$. The following conditions are imposed on the penalty function $p_{\tau}(\cdot)$ and the tuning parameter $\tau$. (A.4) $\liminf_{\tau\rightarrow0}\liminf_{\theta\rightarrow0+}p^{\prime }_{\tau}(\theta)/\tau>0. $ (A.5) There exists a positive constant $C$ such that $\max_{j\in\mathcal{A}}p_\tau(|\theta_{0j}|)\leq C\tau$. Conditions (A.4) and (A.5) hold for many penalty functions such as the one in [@FanLi_2001] and the minimax concave penalty of [@Zhang_2010]. Define $$\begin{split} {\mathbf{S}}(\boldsymbol{\theta}_0)=&~\big(\lbrack E\{\nabla_{\boldsymbol{\theta}}g_t(\boldsymbol{\theta}_0)\}]^{\prime }V_M^{-1}[E\{\nabla_{\boldsymbol{\theta}}g_t(\boldsymbol{\theta}_0)\}]\big)^{-1}\big(\lbrack E\{\nabla_{\boldsymbol{\theta}}g_t(\boldsymbol{\theta}_0)\}]^{\prime }V_M^{-1}V_nV_M^{-1}[E\{\nabla_{\boldsymbol{\theta}}g_t(\boldsymbol{\theta}_0)\}]\big) \\ &~~~~~~~~~~~~~~~~~\times\big(\lbrack E\{\nabla_{\boldsymbol{\theta}}g_t(\boldsymbol{\theta}_0)\}]^{\prime }V_M^{-1}[E\{\nabla_{\boldsymbol{\theta}}g_t(\boldsymbol{\theta}_0)\}]\big)^{-1}. \end{split}$$ We correspondingly decompose ${\mathbf{S}}(\boldsymbol{\theta}_0)$ as $$\label{eq:block} {\mathbf{S}}(\boldsymbol{\theta}_0)=\left( \begin{array}{cc} {\mathbf{S}}_{11}(\boldsymbol{\theta}_0) & {\mathbf{S}}_{12}(\boldsymbol{\theta}_0) \\ {\mathbf{S}}_{21}(\boldsymbol{\theta}_0) & {\mathbf{S}}_{22}(\boldsymbol{\theta}_0) \\ \end{array} \right)$$ where ${\mathbf{S}}_{11}(\boldsymbol{\theta}_0)$ and ${\mathbf{S}}_{22}(\boldsymbol{\theta}_0)$ are $s\times s$ and $(p-s)\times(p-s)$ matrices, respectively. The following restrictions are needed $$s\tau r^{-1}nM^{-1} =O(1)~~\text{and}~~ \tau(r^{-1}n)^{1/2}M^{-1}\rightarrow\infty. \label{eq:cond2}$$ Write the penalized GEL estimator $\widehat{\boldsymbol{\theta}}_n^{(\mathrm{pe})}=(\widehat{\boldsymbol{\theta}}_n^{(1)^{\prime }},\widehat{\boldsymbol{\theta}}_n^{(2)^{\prime }})^{\prime }$ and define $${\mathbf{S}}_p(\boldsymbol{\theta}_0)={\mathbf{S}}_{11}(\boldsymbol{\theta}_0)-{\mathbf{S}}_{12} (\boldsymbol{\theta}_0){\mathbf{S}}_{22}^{-1}(\boldsymbol{\theta}_0) {\mathbf{S}}_{21}(\boldsymbol{\theta}_0).$$ The following theorem describes the basic properties of the penalized GEL estimator. \[tm:pen\] Under conditions , assume that the eigenvalues of $V_M$ are uniformly bounded away from zero and infinity. If $\max_{j\in\mathcal{A}}p_{\tau}^{\prime }(|\theta_{0j}|)=o(r^{-1/2}n^{-1/2})$, $\min_{j\in\mathcal{A}}|\theta_{0j}|/\tau\rightarrow\infty$ and holds, the following results hold. 1. $P\{\widehat{\boldsymbol{\theta}}_n^{(2)}=\mathbf{0}\}\rightarrow1$ as $n\rightarrow\infty$, provided that holds and $r^2pM^2n^{-1}=o(1)$; 2. In addition, if the eigenvalues of $V_n$ are uniformly bounded away from zero and infinity, then for any $\boldsymbol{\alpha}_n\in\mathbb{R}^s$ with unit $L_2$-norm, then $$\sqrt{n}\boldsymbol{\alpha}_n^{\prime }{\mathbf{S}}_p^{-1/2}(\boldsymbol{\theta}_0)(\widehat{\boldsymbol{\theta}}_n^{(1)}-\boldsymbol{\theta}_{0}^{(1)})\xrightarrow {d} N(0,1)~~\text{as}~n\rightarrow\infty,$$ provided that 1. for independent data, $r^3p^2n^{-1}=o(1)$ and $r^3n^{2/\gamma-1}=o(1)$; 2. for dependent data, holds and $\boldsymbol{\alpha}_n$ satisfies with $$\begin{split} \boldsymbol{\beta}_n=&~-V_M^{-1}[E\{\nabla_{\boldsymbol{\theta}}g(\boldsymbol{\theta}_0)\}]\big(\lbrack E\{\nabla_{\boldsymbol{\theta}}g_t(\boldsymbol{\theta}_0)\}]^{\prime }V_M^{-1}V_nV_M^{-1}[E\{\nabla_{\boldsymbol{\theta}}g_t(\boldsymbol{\theta}_0)\}]\big)^{-1} \\ &~~~\times[E\{\nabla_{\boldsymbol{\theta}}g_t(\boldsymbol{\theta}_0)\}]^{\prime }V_M^{-1}[E\{\nabla_{\boldsymbol{\theta}^{(1)}}g_t(\boldsymbol{\theta}_0)\}]\{{\mathbf{S}}_{11}(\boldsymbol{\theta}_0)-{\mathbf{S}}_{12}(\boldsymbol{\theta}_0){\mathbf{S}}_{22}^{-1}(\boldsymbol{\theta}_0){\mathbf{S}}_{21}(\boldsymbol{\theta}_0)\}^{1/2}\boldsymbol{\alpha}_n. \end{split}$$ Similar to the consistency of GEL estimator, if the eigenvalues of $E\{g_{t}(\boldsymbol{\theta }_{0})g_{t}(\boldsymbol{\theta }_{0})^{\prime }\}$ are uniformly bounded away from zero and infinity, result (i) still holds without blocking technique if $r^{2}n^{2/\gamma -1}=o(1)$ and $r^{2}pn^{-1}=o(1)$ are satisfied. Comparing Theorem 6 with Theorem 2 and Corollary 2, since ${\mathbf{S}}_{p}(\boldsymbol{\theta }_{0})\leq {\mathbf{S}}_{11}(\boldsymbol{\theta }_{0})$, the penalized GEL estimator is more efficient in estimating the nonzero components. [LengTang\_2011\_Biometrika]{} considered the theoretical results of the penalized EL estimator for independent data by assuming $p/r\rightarrow c\in (0,1)$. Our results extend theirs to penalized GEL estimator for weakly dependent data without requiring $p/r\rightarrow c\in (0,1)$. Simulation Results ================== In this section, we present simulation results to compare the finite sample performance of the GEL estimators with the GMM estimator in the high dimensional time series setting. Three versions of the GEL estimators were considered in the simulations: the EL, the ET and the CU estimators. We experimented two forms of the moment restrictions: one was linear, and the other was nonlinear. The penalized GEL estimator was also considered in the non-linear case. We first conducted simulation for the linear moment restrictions with $g(X_{t},\boldsymbol{\theta })=X_{t}-\boldsymbol{\theta }$. The observations $\{X_{t}\}_{t=1}^{n}$ were generated according to the vector autoregressive (VAR) model of order $1$: $X_{t}=\psi X_{t-1}+\varepsilon _{t}$ where $\varepsilon _{t}\sim N(\boldsymbol{\mathbf{0}},\Sigma _{\varepsilon })$, $\Sigma _{\varepsilon }=(\sigma _{i,j})_{p\times p}$, $\sigma _{i,i}=1-\psi ^{2}$, $\sigma _{i,i\pm 1}=0.5(1-\psi ^{2})$ and $\sigma _{i,j}=0$ for $|i-j|>1$. The stationary distribution of $X_{t}$ is $N(\boldsymbol{\mathbf{0}},\Sigma _{x})$ where $\Sigma _{x}=(\tilde{\sigma}_{i,j})_{p\times p}$ and $\tilde{\sigma}_{i,i}=1$, $\tilde{\sigma}_{i,i\pm 1}=0.5$ and $\tilde{\sigma}_{i,j}=0$ for $|i-j|>1$. In this model, $p=r$ and the true parameter $\boldsymbol{\theta }_{0}=\mathbf{0}\in \mathbb{R}^{p}$. The second simulation model was the generalized linear model. The covariates $\{Z_{t}\}_{t=1}^{n}$ were generated with the same VAR(1) process as the $\{X_{t}\}_{t=1}^{n}$ in the first model setting. The response variables $\{Y_{t}\}_{t=1}^{n}$ were generated from the Bernoulli distribution such that $P(Y_{t}=1|Z_{t})=\exp (1+Z_{t}^{\prime }\boldsymbol{\theta }_{0})/\{1+\exp (1+Z_{t}^{\prime }\boldsymbol{\theta }_{0})\}$ with the true parameter $\boldsymbol{\theta }_{0}=(0.8,0.2,0,\ldots ,0)^{\prime }\in \mathbb{R}^{p}$. Then$$E\bigg\{Y_{t}-\frac{\exp (1+Z_{t}^{\prime }\boldsymbol{\theta }_{0})}{1+\exp (1+Z_{t}^{\prime }\boldsymbol{\theta }_{0})}\bigg|Z_{t}\bigg\}=0.$$In this setting, we have nonlinear moment restrictions $$g(X_{t},\boldsymbol{\theta })=\left( \begin{array}{c} Z_{t} \\ W_{t} \\ \end{array}\right) \bigg\{Y_{t}-\frac{\exp (1+Z_{t}^{\prime }\boldsymbol{\theta })}{1+\exp (1+Z_{t}^{\prime }\boldsymbol{\theta })}\bigg\},$$where $W_{t}=(Z_{1,t}^{2},\ldots ,Z_{p,t}^{2})^{\prime }$ for $Z_{t}=(Z_{1,t},\ldots ,Z_{p,t})^{\prime }$. This model is over-identified. We considered both non-penalized and penalized estimators under this model setting. In both simulation models, we chose $n=500$, $1000$ and $2000$, respectively. The parameter $\psi $ in the VAR(1) process capturing the serial dependence was set to be $0.1,0.3$ and $0.5$, respectively. The dimension $p$ was pegged to the sample size $n$ such that $p=\lfloor cn^{2/15}\rfloor $, where $c=10$ and $12$ in the first model setting, and $c=5$ and $6$ in the second model setting, respectively. Simulations results were based on $200$ repetitions. For each repetition of each model setting, we obtained the parameter estimates $\widehat{\boldsymbol{\theta }}$’s based on the four considered estimation methods: EL, GMM, ET and CU under five regimes regarding the blocking parameters $L$ and $M$: $$\begin{split} & \text{Regime~(i)}.~~L=M=1;~~~~~~~~~~~~\text{Regime~(ii)}.~~M=\lfloor n^{1/5}\rfloor ~\text{and}~L=\lfloor 0.5M\rfloor ; \\ & \text{Regime~(iii)}.~~L=M=\lfloor n^{1/5}\rfloor ;~~~~\text{Regime~(iv)}.~~M=\lfloor 3n^{1/5}\rfloor ~\text{and}~L=\lfloor 0.5M\rfloor ; \\ & \text{Regime~(v)}.~~L=M=\lfloor 3n^{1/5}\rfloor ~. \end{split}$$Regime (i) means no blocking. Regimes (ii) and (iv) assigned the block size $M$ to be twice of the block separation parameter $L$; and Regimes (iii) and (v) prescribed $M=L$. For each repetition of the second model setting, we additionally considered the parameter estimates $\widehat{\boldsymbol{\theta }}$’s based on the penalized GEL estimation methods. The penalty function $p_{\tau }(u)$ used in the simulation satisfied: $$p_{\tau }^{\prime }(u)=\tau \bigg\{I(u\leq \tau )+\frac{(a\tau -u)_{+}}{(a-1)\tau }I(u>\tau )\bigg\}$$for $u>0$, where $a=3.7$, and $s_{+}=s$ for $s>0$ and $0$ otherwise. This penalty function is given in [@FanLi_2001]. We applied the method given in [@LengTang_2011_Biometrika] to determine the penalty parameter $\tau $. In each simulation replication, we calculated the $L_{2}$ distance between $\widehat{\boldsymbol{\theta }}$ and $\boldsymbol{\theta }_{0}$ as $\Vert \widehat{\boldsymbol{\theta }}-\boldsymbol{\theta }_{0}\Vert _{2}=\{(\widehat{\boldsymbol{\theta }}-\boldsymbol{\theta }_{0})^{\prime }(\widehat{\boldsymbol{\theta }}-\boldsymbol{\theta }_{0})\}^{1/2}$. Tables 1 and 2 report empirical medians of the squared estimation errors for the EL, ET, CU and GMM estimators in the first simulation model with $c=10$ and $c=12$, respectively. And Tables 3 and 4 summarize the empirical median for the second simulation model with the extra penalized GEL estimators. We had also collected the average of the squared estimation errors, which exhibited similar patterns as the empirical median. Hence, we only report the median of squared estimation errors per the suggestion of one referee. It is noted that the performance of each estimator at each given blocking regime was improved when the sample size was increased, which confirms the convergence of these estimators. For the second nonlinear model, we observed that the performance of three GEL estimators and their penalized analogues were improved under the blocking regimes (ii)-(v) which were bona fide blocking since $L,M>1$. This was not that surprising since dependence was presence in both simulated models, and applying the blocking can improve the efficiency of the estimation. However, the performance of the GMM estimator were largely similar regardless of the blocking regimes used. The empirical medians of the squared estimation errors of the GMM estimator were much larger than those of the GEL estimators, which confirmed the existing research on GMM versus GEL for finite fixed dimensional settings [@NeweySmith_2004_Ecma; @Anatolyev_2005]. Among the three GEL estimators, we observed that while they were largely similar under the first simulation model, the EL and the ET estimators performed better than the CU estimator for the logistic regression model. This might be due to the multivariate asymmetry in the moment conditions, which makes the bias term of the CU estimator more pronounced, as shown in [@NeweySmith_2004_Ecma] and [@Anatolyev_2005]. We note that the estimation efficiency among the GEL estimator with respect to the different regimes of the blocking width selection was largely comparable to each other for the simple mean models. However, in the case of the generalized linear model, the regimes (iv) and (v), with the block width $M=\lfloor 3n^{1/5}\rfloor $, led to the best performance. We also observed that under the second model setting where the parameter is sparse, the penalized GEL estimators were much more efficient than their non-penalized counterparts, which confirmed our Theorem 6. Conclusion ========== In this paper, we have investigated the asymptotic properties of the GEL estimator, the GEL ratio statistic and the over-identification specification test for high dimensional moment restriction models with increasing number of parameters and weakly dependent data. We have also investigated a penalized GEL approach that is designed for the high dimensional sparse parameter situation with $p>r$, although the true but unknown number of non-zero parameters is not larger than $r$. We establish the oracle property of the penalized GEL estimator. Both theoretical and simulation studies find the penalization leads to efficiency gain for the GEL estimators even for dependent data. We establish the consistency and the asymptotic normality of the high-dimensional GEL and the penalized GEL estimators allowing for fixed block size $M$ for time series data. However, when the unconditional moment functions $\{g(X_{t},\boldsymbol{\theta }_{0})\}_{t=1}^{n}$ are autocorrelated, the simple limiting distributions of the GEL ratio statistic and the over-identification specification test are established when the block size $M$ diverges with the sample size $n$. How to practically select $M$ is a quite challenging problem. As indicated in [HallHorowitzJing\_1995]{} and [@Lahari_2003], although there has been much research in determining the order of magnitude of $M$, there is in general a lack of research for selecting the tuning parameter, the coefficient of $M$ for general nonlinear time series models. The simulation study reported in Section 8 shows that $M=\lfloor 3n^{1/5}\rfloor $ led to satisfactory performance. Instead of blocking, one could also perform local smoothing of the unconditional moment functions $\{g(X_{t},\boldsymbol{\theta }_{0})\}_{t=1}^{n}$ to reduce temproal dependence [@Smith_1997; @Anatolyev_2005; @Kitamura_2007], which introduces an alternative tuning parameter, however. We leave it to future research about the performance of this local smoothing GEL approach for high dimensinal time series models. Acknowledgements {#acknowledgements .unnumbered} ================ The authors are grateful to the co-editor Jianqing Fan, an Associate Editor, two anonymous referees and Qiwei Yao for constructive comments. Some results of this work is the Chapter 2 of Jinyuan Chang’s PhD thesis at Peking University. We thank Jean-Michel Zakoïan for sharing a longer version of his paper, and Chenlei Leng and Cheng Yong Tang for sharing their R code. Jinyuan Chang and Song Xi Chen were supported by National Natural Science Foundation of China Grants 11131002 and G0113, LMEQF and Center for Statistical Science at Peking University. Appendix {#appendix .unnumbered} ======== Throughout the Appendix, $C$ denotes a generic positive finite constant that may be different in different uses. For any $q=1,\ldots ,Q$ and $k=1,\ldots ,M$, let $\beta _{1}(q,k)=\#\{j<q:X_{(q-1)L+k}\in B_{j}\}$ and $\beta _{2}(q,k)=\#\{j>q:X_{(q-1)L+k}\in B_{j}\}$. These two quantities denote the times of the $k$-th element of the $q$-th block occurs in the blocks before and after the $q$-th block, respectively. Let $\bar{g}(\boldsymbol{\theta })={n}^{-1}\sum_{t=1}^{n}g_{t}(\boldsymbol{\theta })$, $\bar{\phi}(\boldsymbol{\theta })={Q}^{-1}\sum_{q=1}^{Q}\phi _{q}(\boldsymbol{\theta })$, $V_{M}=\text{Var}\{M^{1/2}{\phi }_{q}(\boldsymbol{\theta }_{0})\},$ $\widehat{\Omega }(\boldsymbol{\theta })={Q}^{-1}\sum_{q=1}^{Q}\phi _{q}(\boldsymbol{\theta })\phi _{q}(\boldsymbol{\theta })^{\prime }$ and $\Omega (\boldsymbol{\theta })={E}\{\phi _{q}(\boldsymbol{\theta })\phi _{q}(\boldsymbol{\theta })^{\prime }\}$. Some Lemmas I {#some-lemmas-i .unnumbered} ------------- The lemmas proposed in this subsection are used to prove Theorem 1. \[la1\] $\beta_1(q,k)=(q-1)\wedge\lfloor(M-k)/L\rfloor$ and $\beta_2(q,k)=(Q-q)\wedge\lfloor(k-1)/L\rfloor $ . <span style="font-variant:small-caps;">Proof</span>: For $t=(q-1)L+k$, suppose $X_t\in B_{\bar{q}}$ where $\bar{q}<q$. Then there exists a positive integer $\bar{k}\in [1,M]$ such that $(q-1)L+k=(\bar{q}-1)L+\bar{k}. $ It means $\bar{q}=q-(\bar{k}-k)/L $. From this, we can get $\bar{k}=k+iL$ for some $i\in\{1,\ldots,q-1\}$. Note that $\bar{k}\in[1,M]$, then $i\leq\lfloor(M-k)/L\rfloor$. Hence, $\beta_1(q,k)=(q-1)\wedge\lfloor(M-k)/L\rfloor $. By the same argument, $\beta_2(q,k)=(Q-q)\wedge\lfloor(k-1)/L\rfloor $. $\hfill \square$ \[la2\] Under conditions and , $\sup_{\boldsymbol{\theta}\in\Theta}\|\bar{\phi}(\boldsymbol{\theta})-\bar{g}(\boldsymbol{\theta})\|_2=O_p(r^{1/2}Mn^{-1}). $ <span style="font-variant:small-caps;">Proof</span>: By Jensen’s inequality, $${E}\bigg\{\sup_{\boldsymbol{\theta}\in\Theta}\|\bar{\phi}(\boldsymbol{\theta})-\bar{g}(\boldsymbol{\theta})\|_2\bigg\}\leq \frac{1}{MQ}\bigg\{n-(Q-1)L-M+\sum_{\beta_1(q,k)=0}\beta_2(q,k)+n-MQ\bigg\}\cdot{E}\bigg\{\sup_{\boldsymbol{\theta}\in\Theta}\| g_t(\boldsymbol{\theta}) \|_2\bigg\}.$$ From Lemma 1 and (A.1)(ii), $\sum_{\beta_1(q,k)=0}\beta_2(q,k)\leq (Q-1)(M-L) $ for sufficiently large $n$. Noting that $Q=\lfloor(n-M)/L\rfloor+1$, then for sufficiently large $n$ $${E}\bigg\{\sup_{\boldsymbol{\theta}\in\Theta}\|\bar{\phi}(\boldsymbol{\theta})-\bar{g}(\boldsymbol{\theta})\|_2\bigg\}\leq2LM^{-1}Q^{-1}\cdot{E}\bigg\{\sup_{\boldsymbol{\theta}\in\Theta}\|g_t(\boldsymbol{\theta})\|_2\bigg\}.$$ Hence, (A.1)(ii) and (A.2)(ii) lead to the conclusion. $\hfill \square$ \[la3\] Under conditions and , $\|\widehat{\Omega}(\boldsymbol{\theta}_{0})-\Omega(\boldsymbol{\theta}_{0})\|_F=O_p(rM^{1/2}n^{-1/2})$ . <span style="font-variant:small-caps;">Proof</span>: Note that $$\begin{split} {E}\{\|\widehat{\Omega}(\boldsymbol{\theta}_{0})-\Omega(\boldsymbol{\theta}_{0})\|_F^2\}&=Q^{-1}{E}(\text{tr} \{[\phi_q(\boldsymbol{\theta}_{0})\phi_q(\boldsymbol{\theta}_{0})^{\prime }-\Omega(\boldsymbol{\theta}_{0})]^2\}) \\ &~~~+Q^{-2}\sum_{q_1\neq q_2}{E}(\text{tr} \{[\phi_{q_1}(\boldsymbol{\theta}_{0})\phi_{q_1}(\boldsymbol{\theta}_{0})^{\prime }-\Omega(\boldsymbol{\theta}_{0})][\phi_{q_2}(\boldsymbol{\theta}_{0})\phi_{q_2}(\boldsymbol{\theta}_{0})^{\prime }-\Omega(\boldsymbol{\theta}_{0})]\}) \\ &=:A_1+A_2. \end{split}$$ As $A_1\leq Q^{-1}{E}\{\|\phi_q(\boldsymbol{\theta}_{0})\|_2^4\}, $ by Jensen’s inequality and (A.2)(iii), $A_1=O\left(r^2Mn^{-1}\right).$ At the same time, $$\begin{split} A_2&=Q^{-2}\sum_{u,v=1}^r\sum_{q_1\neq q_2} {E}\{ [ \phi_{q_1,u}(\boldsymbol{\theta}_{0})\phi_{q_1,v}(\boldsymbol{\theta}_{0})-\Omega_{u,v}(\boldsymbol{\theta}_{0}) ][ \phi_{q_2,v}(\boldsymbol{\theta}_{0})\phi_{q_2,u}(\boldsymbol{\theta}_{0})-\Omega_{v,u}(\boldsymbol{\theta}_{0}) ]\}, \\ \end{split}$$ where $\Omega_{u,v}(\boldsymbol{\theta}_{0})$ denotes the $(u,v)$-element of $\Omega(\boldsymbol{\theta}_{0})$. By Davydov inequality and (A.2)(iii), $|A_2|\leq Cr^{2}Q^{-2}\sum_{q_1\neq q_2}\alpha_{\phi}\{|q_1-q_2|\}^{1-2/\gamma}.$ Hence, by (A.1)(i), $A_2=O(r^{2}Mn^{-1}).$ From Markov inequality, $\| \widehat{\Omega}(\boldsymbol{\theta}_{0})-\Omega(\boldsymbol{\theta}_{0}) \|_F=O_p(rM^{1/2}n^{-1/2})$. $\hfill \square$ \[la4\] Under conditions , and , then $\sup_{\boldsymbol{\theta}\in\Theta}\lambda_{\max}\{\widehat{ \Omega}(\boldsymbol{\theta})\}=O_p(1)$ provided that $rMn^{-1}=o(1)$. <span style="font-variant:small-caps;">Proof</span>: Using the same approach as in the proof of Lemma \[la2\], $$\sup_{\boldsymbol{\theta} \in \Theta }\sup_{\Vert x\Vert _{2}=1}\bigg\{\bigg|\frac{1}{MQ}\sum_{q=1}^{Q}\sum_{t\in B_{q}}x^{\prime }g_{t}(\boldsymbol{\theta} )g_{t}(\boldsymbol{\theta} )^{\prime }x-\frac{1}{n}\sum_{t=1}^{n}x^{\prime }g_{t}(\boldsymbol{\theta} )g_{t}(\boldsymbol{\theta} )^{\prime }x\bigg|\bigg\}=O_{p}(rMn^{-1}).$$By Jensen’s inequality, for any $\|x\|_2=1$, $$\frac{1}{Q}\sum_{q=1}^{Q}x^{\prime }\phi _{q}(\boldsymbol{\theta} )\phi _{q}(\boldsymbol{\theta} )^{\prime }x\leq \frac{1}{MQ}\sum_{q=1}^{Q}\sum_{t\in B_{q}}x^{\prime }g_{t}(\boldsymbol{\theta} )g_{t}(\boldsymbol{\theta} )^{\prime }x.$$Then $\sup_{\boldsymbol{\theta} \in \Theta }\lambda _{\text{max}}\{\widehat{\Omega}(\boldsymbol{\theta} )\}\leq \sup_{\boldsymbol{\theta} \in \Theta }\lambda _{\text{max}}\{n^{-1}\sum_{i=1}^{n}g_{t}(\boldsymbol{\theta} )g_{t}(\boldsymbol{\theta} )^{\prime }\}+o_{p}(1)$. The result can be implied by (A.2)(iv). $\hfill \square $ \[la5\] Under condition , define $\delta_n=o(r^{-1/2}Q^{-1/\gamma})$ and $\Lambda_n=\{\lambda\in\mathbb{R}^{r}:\|\lambda\|_2\leq\delta_n\}$, we have $\sup_{1\leq q\leq Q, \boldsymbol{\theta}\in\Theta, \lambda\in\Lambda_n}|\lambda^{\prime }\phi_q(\boldsymbol{\theta})|\xrightarrow[]{p}0. $ Also w.p.a.1, $\Lambda_n\subset\widehat{\Lambda}_n(\boldsymbol{\theta})$ for all $\boldsymbol{\theta}\in\Theta$. <span style="font-variant:small-caps;">Proof</span>: From (A.2)(ii) and Markov inequality, $\sup_{1\leq q\leq Q, \boldsymbol{\theta}\in\Theta}\|\phi_q(\boldsymbol{\theta})\|_2=O_p(r^{1/2}Q^{1/\gamma}). $ Then, $$\sup_{1\leq q\leq Q, \boldsymbol{\theta}\in\Theta, \lambda\in\Lambda_n}|\lambda^{\prime }\phi_q(\boldsymbol{\theta})|\leq\delta_n\cdot\sup_{1\leq q\leq Q, \boldsymbol{\theta}\in\Theta}\|\phi_q(\boldsymbol{\theta})\|_2\xrightarrow[]{p}0.$$ It also implies w.p.a.1 $\lambda^{\prime }\phi_q(\boldsymbol{\theta})\in\mathcal{V}$ for all $\boldsymbol{\theta}\in\Theta$ and $\|\lambda\|_2\leq\delta_n$. $\hfill \square$ \[la7\] Under conditions , and , assume that $\lambda_{\max}(V_M)$ is uniformly bounded away from infinity. If $r^2M^3n^{-1}=o(1)$, $\|\boldsymbol{\theta}-\boldsymbol{\theta}_{0}\|_2=O_p(\tau_n)$ and $rpM\tau_n^2=o(1)$, then $\|\widehat{\Omega}(\boldsymbol{\theta})-\widehat{\Omega}(\boldsymbol{\theta}_{0})\|_2=O_p( r^{1/2}p^{1/2}M^{-1/2}\tau_n)$. <span style="font-variant:small-caps;">Proof</span>: Choose $x\in \mathbb{R}^{r}$ with unit $L_{2}$-norm such that $\lambda _{\text{max}}\{\widehat{\Omega}(\boldsymbol{\theta} )-\widehat{\Omega}(\boldsymbol{\theta} _{0})\}=x^{\prime }\{\widehat{\Omega}(\boldsymbol{\theta} )-\widehat{\Omega}(\boldsymbol{\theta} _{0})\}x$. Then, $$\begin{split} &|\lambda _{\text{max}}\{\widehat{\Omega}(\boldsymbol{\theta} )-\widehat{\Omega}(\boldsymbol{\theta} _{0})\}| \\ &~~~~~~\leq \frac{1}{Q}\sum_{q=1}^{Q}|x^{\prime }\phi _{q}(\boldsymbol{\theta} )-x^{\prime }\phi _{q}(\boldsymbol{\theta} _{0})|\cdot \Vert \phi _{q}(\boldsymbol{\theta} )-\phi _{q}(\boldsymbol{\theta} _{0})\Vert _{2}+\frac{2}{Q}\sum_{q=1}^{Q}|x^{\prime }\phi _{q}(\boldsymbol{\theta} _{0})|\cdot \Vert \phi _{q}(\boldsymbol{\theta} )-\phi _{q}(\boldsymbol{\theta} _{0})\Vert _{2} \\ &~~~~~~ \leq \frac{1}{Q}\sum_{q=1}^{Q}\Vert \phi _{q}(\boldsymbol{\theta} )-\phi _{q}(\boldsymbol{\theta} _{0})\Vert _{2}^{2}+2[ \lambda _{\text{max}}\{\widehat{\Omega}(\boldsymbol{\theta} _{0})\}] ^{1/2}\bigg\{\frac{1}{Q}\sum_{q=1}^{Q}\Vert \phi _{q}(\boldsymbol{\theta} )-\phi _{q}(\boldsymbol{\theta} _{0})\Vert _{2}^{2}\bigg\}^{1/2}. \\ \end{split}$$Note that $r^{2}M^{3}n^{-1}=o(1)$, by Lemmas \[la3\] and $\lambda_{\max}(V_M)$ is uniformly bounded away from infinity, $\lambda _{\text{max}}\{\widehat{\Omega}(\boldsymbol{\theta} _{0})\}=O_{p}(M^{-1})$. From (A.2)(i), $Q^{-1}\sum_{q=1}^{Q}\Vert \phi _{q}(\boldsymbol{\theta} )-\phi _{q}(\boldsymbol{\theta} _{0})\Vert _{2}^{2}=rp\cdot O_{p}(\Vert \boldsymbol{\theta} -\boldsymbol{\theta} _{0}\Vert _{2}^{2})$. If $rpM\tau _{n}^{2}=o(1)$, then $|\lambda _{\text{max}}\{\widehat{\Omega}(\boldsymbol{\theta} )-\widehat{\Omega}(\boldsymbol{\theta} _{0})\}|=O_{p}(r^{1/2}p^{1/2}M^{-1/2}\tau _{n})$. Using the same argument, $|\lambda _{\text{min}}\{\widehat{\Omega}(\boldsymbol{\theta} )-\widehat{\Omega}(\boldsymbol{\theta} _{0})\}|=O_{p}(r^{1/2}p^{1/2}M^{-1/2}\tau _{n})$. This completes the proof.$\hfill \square $ \[la9\] Under conditions , , , and , assume that the eigenvalues of $V_M$ are uniformly bounded away from zero and infinity. If $r^2M^{2-2/\gamma}n^{2/\gamma-1}=o(1)$, $r^2M^{3}n^{-1}=o(1)$, $\|\widetilde{\boldsymbol{\theta}}-\boldsymbol{\theta}_{0}\|_2=O_p(\tau_n)$, $rpM\tau_n^2=o(1)$ and $\|\bar{g}(\widetilde{\boldsymbol{\theta}})\|_2=O_p(r^{1/2}n^{-1/2})$, then $\widehat{\lambda}(\widetilde{\boldsymbol{\theta}})=\arg\max_{\lambda\in\widehat{\Lambda}_n(\widetilde{\boldsymbol{\theta}})}\widehat{S}_n(\widetilde{\boldsymbol{\theta}},\lambda)$ exists w.p.a.1, $\sup_{\lambda\in\widehat{\Lambda}_n(\widetilde{\boldsymbol{\theta}})}\widehat{S}_n(\widetilde{\boldsymbol{\theta}},\lambda)=\rho(0)+O_p(rMn^{-1})$ and $\|\widehat{\lambda}(\widetilde{\boldsymbol{\theta}})\|_2=O_p(r^{1/2}Mn^{-1/2})$ where $\widehat{S}_n(\boldsymbol{\theta},\lambda)$ is defined in . <span style="font-variant:small-caps;">Proof</span>: Pick $\delta_n=o(r^{-1/2}Q^{-1/\gamma})$ and $r^{1/2}Mn^{-1/2}=o(\delta_n)$, which is guaranteed by $r^2M^{2-2/\gamma}n^{2/\gamma-1}=o(1)$. From Lemma \[la2\] and Triangle inequality, then $\|\bar{\phi}(\widetilde{\boldsymbol{\theta}})\|_2\leq \|\bar{g}(\widetilde{\boldsymbol{\theta}}) \|_2+O_p(r^{1/2}Mn^{-1})$ which implies $\|\bar{\phi}(\widetilde{\boldsymbol{\theta}})\|_2=O_p(r^{1/2}n^{-1/2})$. Let $\bar{\lambda}=\arg\max_{\lambda\in\Lambda_n}\widehat{S}_n(\widetilde{\boldsymbol{\theta}},\lambda)$, where $\Lambda_n$ is defined in Lemma [la5]{}. By Lemmas \[la3\], \[la5\] and \[la7\], noting $\rho_{vv}(0)<0$, $$\begin{split} \rho(0)=\widehat{S}_n(\widetilde{\boldsymbol{\theta}},0)&\leq\widehat{S}_n(\widetilde{\boldsymbol{\theta}},\bar{\lambda})=\rho(0)+\rho_v(0)\bar{\lambda}^{\prime }\bar{\phi}(\widetilde{\boldsymbol{\theta}})+\frac{1}{2}\bar{\lambda}^{\prime }\bigg\{\frac{1}{Q}\sum\limits_{q=1}^Q\rho_{vv}(\dot{\lambda}^{\prime }\phi_q(\widetilde{\boldsymbol{\theta}}))\phi_q(\widetilde{\boldsymbol{\theta}})\phi_q(\widetilde{\boldsymbol{\theta}})^{\prime }\bigg\}\bar{\lambda} \\ &\leq\rho(0)+|\rho_v(0)|\cdot\|\bar{\lambda}\|_2\cdot\|\bar{\phi}(\widetilde{\boldsymbol{\theta}})\|_2-C\|\bar{\lambda}\|_2^2\cdot\{M^{-1}+o_p(M^{-1})\}. \end{split}$$ where $\dot{\lambda}$ lies on the jointing line between $\boldsymbol{\mathbf{0}}$ and $\bar{\lambda}$. Hence, $\|\bar{\lambda}\|_2\leq C\cdot M\cdot\|\bar{\phi}(\widetilde{\boldsymbol{\theta}})\|_2\cdot\{1+o_p(1)\}=O_p(r^{1/2}Mn^{-1/2})=o_p(\delta_n). $ Thus $\bar{\lambda}\in\text{int}(\Lambda_n)$ w.p.a.1. Since $\Lambda_n\subset\widehat{\Lambda}_n(\widetilde{\boldsymbol{\theta}})$ w.p.a.1, $\widehat{\lambda}(\widetilde{\boldsymbol{\theta}})=\bar{\lambda}$ w.p.a.1 by the concavity of $\widehat{S}_n(\widetilde{\boldsymbol{\theta}},\lambda)$ and $\widehat{\Lambda}_n(\widetilde{\boldsymbol{\theta}})$. Then, $$\widehat{S}_n(\widetilde{\boldsymbol{\theta}},\widehat{\lambda}(\widetilde{\boldsymbol{\theta}}))\leq\rho(0)+|\rho_v(0)|\cdot\|\widehat{\lambda}(\widetilde{\boldsymbol{\theta}})\|_2\cdot\|\bar{\phi}(\widetilde{\boldsymbol{\theta}})\|_2-C\cdot M^{-1}\cdot\|\widehat{\lambda}(\widetilde{\boldsymbol{\theta}})\|_2^2\cdot\{1+o_p(1)\}$$ leads to $\sup_{\lambda\in\widehat{\Lambda}_n(\widetilde{\boldsymbol{\theta}})}\widehat{S}_n(\widetilde{\boldsymbol{\theta}},\lambda)=\rho(0)+O_p(rMn^{-1})$. $\hfill \square$ Proof of Theorem 1 {#proof-of-theorem-1 .unnumbered} ------------------ Choose $\delta _{n}=o(r^{-1/2}Q^{-1/\gamma })$ and $r^{1/2}Mn^{-1/2}=o(\delta _{n})$. Let $\bar{\lambda}=\text{sign}\{\rho_v(0)\}\cdot\delta _{n}\bar{\phi}(\widehat{\boldsymbol{\theta}}_{n})/\Vert \bar{\phi}(\widehat{\boldsymbol{\theta}}_{n})\Vert _{2}$, then $\bar{\lambda}\in \Lambda _{n}$. By Taylor expansion, Lemmas \[la4\] and \[la5\], noting $\rho_{vv}(0)<0$, $$\begin{split} \widehat{S}_n(\widehat{\boldsymbol{\theta}}_{n},\bar{\lambda})& =\rho(0)+\rho_v(0)\bar{\lambda}^{\prime }\bar{\phi}(\widehat{\boldsymbol{\theta}}_{n})+\frac{1}{2}\bar{\lambda}^{\prime }\bigg\{\frac{1}{Q}\sum\limits_{q=1}^{Q}\rho_{vv}(\dot{\lambda}^{\prime }\phi_q(\widehat{\boldsymbol{\theta}}_n))\phi_{q}(\widehat{\boldsymbol{\theta}}_{n})\phi_{q}(\widehat{\boldsymbol{\theta}}_{n})^{\prime }\bigg\}\bar{\lambda} \\ & \geq \rho(0)+|\rho_v(0)|\cdot\delta _{n}\cdot \Vert \bar{\phi}(\widehat{\boldsymbol{\theta}}_{n})\Vert _{2}-C\cdot O_{p}(1)\cdot \Vert \bar{\lambda}\Vert _{2}^{2}. \end{split}$$Meanwhile, by the same way in the proof of Lemma \[la3\], $\Vert \bar{g}(\boldsymbol{\theta} _{0})-{E}\{g_{t}(\boldsymbol{\theta} _{0})\}\Vert _{2}=O_{p}(r^{1/2}n^{-1/2})$. Since ${E}\{g_{t}(\boldsymbol{\theta} _{0})\}=\boldsymbol{\mathbf{0}}$, $\Vert \bar{g}(\boldsymbol{\theta} _{0})\Vert _{2}=O_{p}(r^{1/2}n^{-1/2})$. Then, from Lemma \[la9\], $$\widehat{S}_n(\widehat{\boldsymbol{\theta}}_{n},\bar{\lambda})\leq \sup\limits_{\lambda \in \widehat{\Lambda}_{n}(\widehat{\boldsymbol{\theta}}_{n})}\widehat{S}_n(\widehat{\boldsymbol{\theta}}_{n},\lambda )\leq \sup\limits_{\lambda \in \widehat{\Lambda}_{n}(\boldsymbol{\theta} _{0})}\widehat{S}_n(\boldsymbol{\theta} _{0},\lambda )=\rho(0)+O_{p}(rMn^{-1}).$$Hence, $\Vert \bar{\phi}(\widehat{\boldsymbol{\theta}}_{n})\Vert _{2}=O_{p}(\delta _{n}).$ Consider any $\varepsilon _{n}\rightarrow 0$ and let $\tilde{\lambda}=\text{sign}\{\rho_v(0)\}\cdot\varepsilon _{n}\bar{\phi}(\widehat{\boldsymbol{\theta}}_{n})$, then $\Vert \widetilde{\lambda}\Vert _{2}=o_{p}(\delta _{n})$. Using the same way above, we can obtain $$|\rho_v(0)|\cdot\varepsilon _{n}\cdot \Vert \bar{\phi}(\widehat{\boldsymbol{\theta}}_{n})\Vert _{2}^{2}-C\cdot O_{p}(1)\cdot \varepsilon _{n}^{2}\cdot \Vert \bar{\phi}(\widehat{\boldsymbol{\theta}}_{n})\Vert _{2}^{2}=O_{p}(rMn^{-1}).$$Then, $\varepsilon _{n}\Vert \bar{\phi}(\widehat{\boldsymbol{\theta}}_{n})\Vert _{2}^{2}=O_{p}(rMn^{-1})$. Thus, $\Vert \bar{\phi}(\widehat{\boldsymbol{\theta}}_{n})\Vert _{2}^{2}=O_{p}(rMn^{-1})$. From Lemma [la2]{}, $\Vert \bar{g}(\widehat{\boldsymbol{\theta}}_{n})\Vert _{2}=O_{p}(r^{1/2}M^{1/2}n^{-1/2})$. If $\Vert \widehat{\boldsymbol{\theta}}_{n}-\boldsymbol{\theta} _{0}\Vert _{2}$ does not converge to zero in probability, then there exists a subsequence $\{(n_{\ast},M_{\ast},r_{\ast},p_{\ast })\}$ such that $\Vert \widehat{\boldsymbol{\theta}}_{n_{\ast }}-\boldsymbol{\theta} _{0}\Vert _{2}\geq \varepsilon$ a.s. for some positive constant $\varepsilon$. By (A.1)(iv), $\Vert {E}\{g_{t}(\widehat{\boldsymbol{\theta}}_{n_{\ast }})\}\Vert _{2}=o_{p}\{\Delta _{1}(r_{\ast },p_{\ast })\}+O_{p}(r_{\ast}^{1/2}M_{\ast}^{1/2}n_{\ast}^{-1/2})$. On the other hand, from (A.1)(iii), $\Vert {E}\{g_{t}(\widehat{\boldsymbol{\theta}}_{n_{\ast }})\}\Vert _{2}\geq \Delta _{1}(r_{\ast },p_{\ast })\Delta _{2}(\varepsilon )$. As $\liminf_{r,p\rightarrow\infty}\Delta_1(r,p)>0$, it is a contradiction. Hence, $\Vert \widehat{\boldsymbol{\theta}}_{n}-\boldsymbol{\theta} _{0}\Vert _{2}\xrightarrow[]{p}0$. By (A.2)(iv), $\|\bar{g}(\widehat{\boldsymbol{\theta}}_{n})-\bar{g}(\boldsymbol{\theta}_{0})\|_2\geq C\|\widehat{\boldsymbol{\theta}}_{n}-\boldsymbol{\theta}_{0}\|_2$ w.p.a.1. Then, $\|\widehat{\boldsymbol{\theta}}_{n}-\boldsymbol{\theta}_{0}\|_2=O_p(r^{1/2}M^{1/2}n^{-1/2})$. In addition, if $r^2pM^{2}n^{-1}=o(1)$, from Lemmas \[la3\] and \[la7\], $\lambda _{\text{max}}\{\widehat{\Omega}(\widehat{\boldsymbol{\theta}}_{n})\}\leq CM^{-1}$ w.p.a.1. By repeating the above arguments, we can obtain $\Vert \bar{\phi}(\widehat{\boldsymbol{\theta}}_{n})\Vert _{2}=O_{p}(r^{1/2}n^{-1/2})$ and $\Vert \widehat{\boldsymbol{\theta}}_{n}-\boldsymbol{\theta} _{0}\Vert _{2}=O_{p}(r^{1/2}n^{-1/2})$. From Lemma \[la9\], $\|\widehat{\lambda}(\widehat{\boldsymbol{\theta}}_{n})\|_2=O_p(r^{1/2}Mn^{-1/2})$. Therefore, we complete the proof of Theorem 1. $\hfill \square $ Proof of Corollary \[cy1\] {#proof-of-corollary-cy1 .unnumbered} -------------------------- To construct Corollary \[cy1\], we need the analogue of Lemma \[la9\] listed below. \[newla1\] Under conditions , , , and , assume that $\lambda_{\min}(V_M)\asymp r^{-\iota_1}$ for some $\iota_1>0$. 1. If $r^{2+3\iota_1}M^{2-2/\gamma}n^{2/\gamma-1}=o(1)$, $r^{2+2\iota_1}M^{3}n^{-1}=o(1)$, $\|\widetilde{\boldsymbol{\theta}}-\boldsymbol{\theta}_{0}\|_2=O_p(\tau_n)$, $r^{1+2\iota_1}pM\tau_n^2=o(1)$ and $\|\bar{g}(\widetilde{\boldsymbol{\theta}})\|_2=O_p(r^{(1+\iota_1)/2}n^{-1/2})$, then $\widehat{\lambda}(\widetilde{\boldsymbol{\theta}})=\arg\max_{\lambda\in\widehat{\Lambda}_n(\widetilde{\boldsymbol{\theta}})}\widehat{S}_n(\widetilde{\boldsymbol{\theta}},\lambda)$ exists w.p.a.1, $\sup_{\lambda\in\widehat{\Lambda}_n(\widetilde{\boldsymbol{\theta}})}\widehat{S}_n(\widetilde{\boldsymbol{\theta}},\lambda)=\rho(0)+O_p(r^{1+2\iota_1}Mn^{-1})$ and $\|\widehat{\lambda}(\widetilde{\boldsymbol{\theta}})\|_2=O_p(r^{(1+3\iota_1)/2}Mn^{-1/2})$ where $\widehat{S}_n(\boldsymbol{\theta},\lambda)$ is defined in . 2. If $r^{2+2\iota_1}M^{2-2/\gamma}n^{2/\gamma-1}=o(1)$ and $r^{2+2\iota_1}M^{3}n^{-1}=o(1)$, then $\widehat{\lambda}({\boldsymbol{\theta}}_0)=\arg\max_{\lambda\in\widehat{\Lambda}_n({\boldsymbol{\theta}}_0)}\widehat{S}_n({\boldsymbol{\theta}}_0 ,\lambda)$ exists w.p.a.1, $\sup_{\lambda\in\widehat{\Lambda}_n({\boldsymbol{\theta}}_0)}\widehat{S}_n({\boldsymbol{\theta}}_0,\lambda)=\rho(0)+O_p(r^{1+\iota_1}Mn^{-1})$. <span style="font-variant:small-caps;">Proof</span>: We first prove (a). Pick $\delta_n=o(r^{-1/2}Q^{-1/\gamma})$ and $r^{(1+3\iota_1)/2}Mn^{-1/2}=o(\delta_n)$, which is guaranteed by $r^{2+3\iota_1}M^{2-2/\gamma}n^{2/\gamma-1}=o(1)$. From Lemma \[la2\] and Triangle inequality, then $\|\bar{\phi}(\widetilde{\boldsymbol{\theta}})\|_2\leq \|\bar{g}(\widetilde{\boldsymbol{\theta}}) \|_2+O_p(r^{1/2}Mn^{-1}) $ which implies $\|\bar{\phi}(\widetilde{\boldsymbol{\theta}})\|_2=O_p(r^{(1+\iota_1)/2}n^{-1/2})$. Let $\bar{\lambda}=\arg\max_{\lambda\in\Lambda_n}\widehat{S}_n(\widetilde{\boldsymbol{\theta}},\lambda)$, where $\Lambda_n$ is defined in Lemma \[la5\]. By Lemmas [la3]{}, \[la5\] and \[la7\], noting $\rho_{vv}(0)<0$, $$\begin{split} \rho(0)=\widehat{S}_n(\widetilde{\boldsymbol{\theta}},0)&\leq\widehat{S}_n(\widetilde{\boldsymbol{\theta}},\bar{\lambda})=\rho(0)+\rho_v(0)\bar{\lambda}^{\prime }\bar{\phi}(\widetilde{\boldsymbol{\theta}})+\frac{1}{2}\bar{\lambda}^{\prime }\bigg\{\frac{1}{Q}\sum\limits_{q=1}^Q\rho_{vv}(\dot{\lambda}^{\prime }\phi_q(\widetilde{\boldsymbol{\theta}}))\phi_q(\widetilde{\boldsymbol{\theta}})\phi_q(\widetilde{\boldsymbol{\theta}})^{\prime }\bigg\}\bar{\lambda} \\ &\leq\rho(0)+|\rho_v(0)|\cdot\|\bar{\lambda}\|_2\cdot\|\bar{\phi}(\widetilde{\boldsymbol{\theta}})\|_2-C\|\bar{\lambda}\|_2^2\cdot\{M^{-1}r^{-\iota_1}+o_p(M^{-1}r^{-\iota_1})\}. \end{split}$$ where $\dot{\lambda}$ lies on the jointing line between $\boldsymbol{\mathbf{0}}$ and $\bar{\lambda}$. Therefore, $\|\bar{\lambda}\|_2\leq C\cdot Mr^{\iota_1}\cdot\|\bar{\phi}(\widetilde{\boldsymbol{\theta}})\|_2\cdot\{1+o_p(1)\}=O_p(r^{(1+3\iota_1)/2}Mn^{-1/2})=o_p(\delta_n). $ Thus $\bar{\lambda}\in\text{int}(\Lambda_n)$ w.p.a.1. Since $\Lambda_n\subset\widehat{\Lambda}_n(\widetilde{\boldsymbol{\theta}})$ w.p.a.1, $\widehat{\lambda}(\widetilde{\boldsymbol{\theta}})=\bar{\lambda}$ w.p.a.1 by the concavity of $\widehat{S}_n(\widetilde{\boldsymbol{\theta}},\lambda)$ and $\widehat{\Lambda}_n(\widetilde{\boldsymbol{\theta}})$. Then, $$\widehat{S}_n(\widetilde{\boldsymbol{\theta}},\widehat{\lambda}(\widetilde{\boldsymbol{\theta}}))\leq\rho(0)+|\rho_v(0)|\cdot\|\widehat{\lambda}(\widetilde{\boldsymbol{\theta}})\|_2\cdot\|\bar{\phi}(\widetilde{\boldsymbol{\theta}})\|_2-C\cdot M^{-1}r^{-\iota_1}\cdot\|\widehat{\lambda}(\widetilde{\boldsymbol{\theta}})\|_2^2\cdot\{1+o_p(1)\}$$ leads to $\sup_{\lambda\in\widehat{\Lambda}_n(\widetilde{\boldsymbol{\theta}})}\widehat{S}_n(\widetilde{\boldsymbol{\theta}},\lambda)=\rho(0)+O_p(r^{1+2\iota_1}Mn^{-1})$. The proof of (b) is similar to that of (a). $\hfill \square$ Here, we begin to prove Corollary \[cy1\]. Choose $\delta _{n}=o(r^{-1/2}Q^{-1/\gamma })$ and $r^{(1+3\iota_1)/2}Mn^{-1/2}=o(\delta _{n})$. Let $\bar{\lambda}=\text{sign}\{\rho_v(0)\}\cdot\delta _{n}\bar{\phi}(\widehat{\boldsymbol{\theta}}_{n})/\Vert \bar{\phi}(\widehat{\boldsymbol{\theta}}_{n})\Vert _{2}$, then $\bar{\lambda}\in \Lambda _{n}$. By Taylor expansion, Lemmas \[la4\] and \[la5\], noting $\rho_{vv}(0)<0$, $$\begin{split} \widehat{S}_n(\widehat{\boldsymbol{\theta}}_{n},\bar{\lambda})& =\rho(0)+\rho_v(0)\bar{\lambda}^{\prime }\bar{\phi}(\widehat{\boldsymbol{\theta}}_{n})+\frac{1}{2}\bar{\lambda}^{\prime }\bigg\{\frac{1}{Q}\sum\limits_{q=1}^{Q}\rho_{vv}(\dot{\lambda}^{\prime }\phi_q(\widehat{\boldsymbol{\theta}}_n))\phi_{q}(\widehat{\boldsymbol{\theta}}_{n})\phi_{q}(\widehat{\boldsymbol{\theta}}_{n})^{\prime }\bigg\}\bar{\lambda} \\ & \geq \rho(0)+|\rho_v(0)|\cdot\delta _{n}\cdot \Vert \bar{\phi}(\widehat{\boldsymbol{\theta}}_{n})\Vert _{2}-C\cdot O_{p}(1)\cdot \Vert \bar{\lambda}\Vert _{2}^{2}. \end{split}$$Meanwhile, by the same way in the proof of Lemma \[la3\], $\Vert \bar{g}(\boldsymbol{\theta} _{0})-{E}\{g_{t}(\boldsymbol{\theta} _{0})\}\Vert _{2}=O_{p}(r^{1/2}n^{-1/2})$. Since ${E}\{g_{t}(\boldsymbol{\theta} _{0})\}=\boldsymbol{\mathbf{0}}$, $\Vert \bar{g}(\boldsymbol{\theta} _{0})\Vert _{2}=O_{p}(r^{1/2}n^{-1/2})$. Then, from Lemma \[newla1\](b), $$\widehat{S}_n(\widehat{\boldsymbol{\theta}}_{n},\bar{\lambda})\leq \sup\limits_{\lambda \in \widehat{\Lambda}_{n}(\widehat{\boldsymbol{\theta}}_{n})}\widehat{S}_n(\widehat{\boldsymbol{\theta}}_{n},\lambda )\leq \sup\limits_{\lambda \in \widehat{\Lambda}_{n}(\boldsymbol{\theta} _{0})}\widehat{S}_n(\boldsymbol{\theta} _{0},\lambda )=\rho(0)+O_{p}(r^{1+\iota_1}Mn^{-1}).$$Hence, $\Vert \bar{\phi}(\widehat{\boldsymbol{\theta}}_{n})\Vert _{2}=O_{p}(\delta _{n}).$ Consider any $\varepsilon _{n}\rightarrow 0$ and let $\tilde{\lambda}=\text{sign}\{\rho_v(0)\}\cdot\varepsilon _{n}\bar{\phi}(\widehat{\boldsymbol{\theta}}_{n})$, then $\Vert \widetilde{\lambda}\Vert _{2}=o_{p}(\delta _{n})$. Using the same way above, we can obtain $$|\rho_v(0)|\cdot\varepsilon _{n}\cdot \Vert \bar{\phi}(\widehat{\boldsymbol{\theta}}_{n})\Vert _{2}^{2}-C\cdot O_{p}(1)\cdot \varepsilon _{n}^{2}\cdot \Vert \bar{\phi}(\widehat{\boldsymbol{\theta}}_{n})\Vert _{2}^{2}=O_{p}(r^{1+\iota_1}Mn^{-1}).$$Then, $\varepsilon _{n}\Vert \bar{\phi}(\widehat{\boldsymbol{\theta}}_{n})\Vert _{2}^{2}=O_{p}(r^{1+\iota_1}Mn^{-1})$. Thus, $\Vert \bar{\phi}(\widehat{\boldsymbol{\theta}}_{n})\Vert _{2}^{2}=O_{p}(r^{1+\iota_1}Mn^{-1})$. From Lemma \[la2\], $\Vert \bar{g}(\widehat{\boldsymbol{\theta}}_{n})\Vert _{2}=O_{p}(r^{(1+\iota_1)/2}M^{1/2}n^{-1/2})$. If $\Vert \widehat{\boldsymbol{\theta}}_{n}-\boldsymbol{\theta} _{0}\Vert _{2}$ does not converge to zero in probability, then there exists a subsequence $\{(n_{\ast},M_{\ast},r_{\ast},p_{\ast })\}$ such that $\Vert \widehat{\boldsymbol{\theta}}_{n_{\ast }}-\boldsymbol{\theta} _{0}\Vert _{2}\geq \varepsilon$ a.s. for some positive constant $\varepsilon$. By (A.1)(iv), $\Vert {E}\{g_{t}(\widehat{\boldsymbol{\theta}}_{n_{\ast }})\}\Vert _{2}=o_{p}\{\Delta _{1}(r_{\ast },p_{\ast })\}+O_{p}(r_{\ast}^{(1+\iota_1)/2}M_{\ast}^{1/2}n_{\ast}^{-1/2})$. On the other hand, from (A.1)(iii), $\Vert {E}\{g_{t}(\widehat{\boldsymbol{\theta}}_{n_{\ast }})\}\Vert _{2}\geq \Delta _{1}(r_{\ast },p_{\ast })\Delta _{2}(\varepsilon )$. As $\liminf_{r,p\rightarrow\infty}\Delta_1(r,p)>0$, it is a contradiction. Hence, $\Vert \widehat{\boldsymbol{\theta}}_{n}-\boldsymbol{\theta} _{0}\Vert _{2}\xrightarrow[]{p}0$. By (A.2)(iv), $\|\bar{g}(\widehat{\boldsymbol{\theta}}_{n})-\bar{g}(\boldsymbol{\theta}_{0})\|_2\geq C\|\widehat{\boldsymbol{\theta}}_{n}-\boldsymbol{\theta}_{0}\|_2$ w.p.a.1. Then, $\|\widehat{\boldsymbol{\theta}}_{n}-\boldsymbol{\theta}_{0}\|_2=O_p(r^{(1+\iota_1)/2}M^{1/2}n^{-1/2})$. In addition, if $r^{2+\iota_1}pM^{2}n^{-1}=o(1)$, from Lemmas \[la3\] and \[la7\], $\lambda _{\text{max}}\{\widehat{\Omega}(\widehat{\boldsymbol{\theta}}_{n})\}\leq CM^{-1}$ w.p.a.1. By repeating the above arguments, we can obtain $\Vert \bar{\phi}(\widehat{\boldsymbol{\theta}}_{n})\Vert _{2}=O_{p}(r^{(1+\iota_1)/2}n^{-1/2})$ and $\Vert \widehat{\boldsymbol{\theta}}_{n}-\boldsymbol{\theta} _{0}\Vert _{2}=O_{p}(r^{(1+\iota_1)/2}n^{-1/2})$. From Lemma \[newla1\], $\|\widehat{\lambda}(\widehat{\boldsymbol{\theta}}_{n})\|_2=O_p(r^{(1+3\iota_1)/2}Mn^{-1/2})$. Therefore, we complete the proof of Corollary \[cy1\]. $\hfill \square $ Some Lemmas II {#some-lemmas-ii .unnumbered} -------------- The lemmas proposed in this subsection are used to establish Proposition 1, Proposition 2 and Theorem 2. The proof of Proposition 1 is based on the asymptotic expansion given in Proposition 2, so we will first construct the proof of Proposition 2 later. \[la10\] Under conditions , assume that $\lambda_{\max}(V_M)$ is uniformly bounded away from infinity. If $r^2M^{2-2/\gamma}n^{2/\gamma-1}=o(1)$, $r^2pM^2n^{-1}=o(1)$ and $r^2M^3n^{-1}=o(1)$, then for any $x\in\mathbb{R}^p$, $y,z\in\mathbb{R}^r$, $$\begin{split} &~~\bigg\|\frac{1}{Q}\sum_{q=1}^Q\rho_v(\widehat{\lambda}(\widehat{\boldsymbol{\theta}}_n)^{\prime }\phi_q(\widehat{\boldsymbol{\theta}}_n))\nabla_{\boldsymbol{\theta}}\phi_q(\widehat{\boldsymbol{\theta}}_{n}) x-\frac{1}{Q}\sum_{q=1}^Q\rho_v(0)\nabla_{\boldsymbol{\theta}}\phi_q(\widehat{\boldsymbol{\theta}}_{n})x\bigg\|_2=O_p(rp^{1/2}M^{1/2}n^{-1/2})\cdot\|x\|_2, \\ &~~~~~~~~~~~\bigg|\frac{M}{Q}\sum_{q=1}^Qy^{\prime }\rho_{vv}(\widetilde{\lambda}^{\prime }\phi_q(\widehat{\boldsymbol{\theta}}_{n}))\phi_q(\widehat{\boldsymbol{\theta}}_{n})\phi_q(\widehat{\boldsymbol{\theta}}_{n})^{\prime }z-\frac{M}{Q}\sum_{q=1}^Q\rho_{vv}(0)y^{\prime }\phi_q(\widehat{\boldsymbol{\theta}}_{n})\phi_q(\widehat{\boldsymbol{\theta}}_{n})^{\prime }z\bigg| \\ &~~~~~~~~~~~~~~~~~~~~~=O_p(rM^{1-1/\gamma}n^{1/\gamma-1/2})\cdot\|y\|_2\cdot\|z\|_2, \end{split}$$ where $\widehat{\lambda}(\widehat{\boldsymbol{\theta}}_{n})$ and $\widetilde{\lambda}$ are defined in . <span style="font-variant:small-caps;">Proof</span>: From Theorem 1, both $\widehat{\lambda}(\widehat{\boldsymbol{\theta}}_{n})$ and $\widetilde{\lambda}$ are $O_p(r^{1/2}Mn^{-1/2})=o_p(\delta_n)$ where $\delta_n$ is defined in Lemma \[la5\]. By Taylor expansion and Cauchy-Schwarz inequality, $$\begin{split} &\bigg\|\frac{1}{Q}\sum_{q=1}^Q\rho_v(\widehat{\lambda}(\widehat{\boldsymbol{\theta}}_n)^{\prime }\phi_q(\widehat{\boldsymbol{\theta}}_n))\nabla_{\boldsymbol{\theta}}\phi_q(\widehat{\boldsymbol{\theta}}_{n}) x-\frac{1}{Q}\sum_{q=1}^Q\rho_v(0)\nabla_{\boldsymbol{\theta}}\phi_q(\widehat{\boldsymbol{\theta}}_{n})x\bigg\|_2^2 \\ &~~~~~~~~~~\leq\bigg[\frac{1}{Q}\sum_{q=1}^Q\rho_{vv}^2(\dot{\lambda}^{\prime }\phi_q(\widehat{\boldsymbol{\theta}}_{n}))\{\widehat{\lambda}(\widehat{\boldsymbol{\theta}}_{n})^{\prime }\phi_q(\widehat{\boldsymbol{\theta}}_{n})\}^2\bigg]\bigg[\frac{1}{Q}\sum_{q=1}^Qx^{\prime }\{\nabla_{\boldsymbol{\theta}}\phi_q(\widehat{\boldsymbol{\theta}}_{n})\}^{\prime }\{\nabla_{\boldsymbol{\theta}}\phi_q(\widehat{\boldsymbol{\theta}}_{n})\}x \bigg] \end{split}$$ where $\dot{\lambda}$ lies on the jointing line between $\boldsymbol{\mathbf{0}}$ and $\widehat{\lambda}(\widehat{\boldsymbol{\theta}}_n)$. From Lemma \[la5\] and $\lambda_{\text{max}}\{\widehat{\Omega}(\widehat{\boldsymbol{\theta}}_{n})\}=O_p(M^{-1})$ which is provided by Lemmas \[la3\] and [la7]{}, we obtain $$\sum_{q=1}^Q\rho_{vv}^2(\dot{\lambda}^{\prime }\phi_q(\widehat{\boldsymbol{\theta}}_{n}))\big\{\widehat{\lambda}(\widehat{\boldsymbol{\theta}}_{n})^{\prime }\phi_q(\widehat{\boldsymbol{\theta}}_{n})\big\}^2\leq C\sum_{q=1}^Q\big\{\widehat{\lambda}(\widehat{\boldsymbol{\theta}}_{n})^{\prime }\phi_q(\widehat{\boldsymbol{\theta}}_{n})\big\}^2\cdot\{1+o_p(1)\}=O_p(rMn^{-1}).$$ On the other hand, $$\frac{1}{Q}\sum_{q=1}^Qx^{\prime }\big\{\nabla_{\boldsymbol{\theta}}\phi_q(\widehat{\boldsymbol{\theta}}_{n})\big\}^{\prime }\big\{\nabla_{\boldsymbol{\theta}}\phi_q(\widehat{\boldsymbol{\theta}}_{n})\big\}x\leq \frac{1}{MQ}\sum_{q=1}^Q\sum_{t\in B_q}\big\|\nabla_{\boldsymbol{\theta}} g_t(\widehat{\boldsymbol{\theta}}_{n})\cdot x\big\|_2^2=O_p(rp)\cdot\|x\|_2^2.$$ Using the same argument, we can obtain the other result. $\hfill \square$ \[la11\] Under conditions , and , then $\|\nabla_{\boldsymbol{\theta}}\bar{\phi}(\boldsymbol{\theta})-\nabla_{\boldsymbol{\theta}}\bar{\phi}(\boldsymbol{\theta}^*)\|_F=O_p(r^{1/2}p\cdot\|\boldsymbol{\theta}-\boldsymbol{\theta}^*\|_2)$ for any $\boldsymbol{\theta}$, $\boldsymbol{\theta}^*$ in a neighborhood of $\boldsymbol{\theta}_{0}$, and $\|\nabla_{\boldsymbol{\theta}}\bar{\phi}(\boldsymbol{\theta}_{0})-{E}\{ \nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}\|_F=O_p(r^{1/2}p^{1/2}n^{-1/2})$ provided that $M=o(n^{1/2})$. <span style="font-variant:small-caps;">Proof</span>: Using Taylor expansion and noting (A.3), the first conclusion holds. Using the same method in the proof of Lemma \[la3\], $\|\nabla_{\boldsymbol{\theta}} \bar{g}(\boldsymbol{\theta}_{0})-{E}\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}\|_F=O_p(r^{1/2}p^{1/2}n^{-1/2}). $ By the same way in the proof of Lemma \[la2\], $\|\nabla_{\boldsymbol{\theta}}\bar{\phi}(\boldsymbol{\theta}_{0})-\nabla_{\boldsymbol{\theta}} \bar{g}(\boldsymbol{\theta}_{0})\|_F=O_p(r^{1/2}p^{1/2}Mn^{-1}). $ Hence, by Triangle inequality, we can obtain $\|\nabla_{\boldsymbol{\theta}}\bar{\phi}(\boldsymbol{\theta}_{0})-{E}\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}\|_F=O_p(r^{1/2}p^{1/2}n^{-1/2}). $ $\hfill \square$ Proof of Proposition 2 {#proof-of-proposition-2 .unnumbered} ---------------------- Define $$\boldsymbol{\beta}=([{E}\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}]^{\prime }V_M^{-1}V_nV_M^{-1}[{E}\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}])^{-1/2} \boldsymbol{\alpha}_{n},$$ then $$\begin{split} \|{E}\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}\cdot\boldsymbol{\beta}\|_2^2&=\boldsymbol{\alpha}_{n}^{\prime }(U^{\prime}U)^{ -1/2}U^{\prime }V_n^{-1/2}V_M^2V_n^{-1/2}U(U^{\prime}U)^{ -1/2}\boldsymbol{\alpha}_{n} \\ &\leq \lambda_{\text{max}}(V_n^{-1/2}V_M^2V_n^{-1/2})\cdot\|U(U^{\prime}U)^{ -1/2}\boldsymbol{\alpha}_{n}\|_2^2= \lambda_{\text{max}}^2(V_M)\lambda_{\min}^{-1}(V_n), \end{split}$$ where $U=V_n^{1/2}V_M^{-1}[{E}\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}]$. Therefore, $\|{E}\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}\cdot\boldsymbol{\beta}\|_2=O(1)$. Meanwhile, $$\begin{split} \|\boldsymbol{\beta}\|_2^2\leq&~ \lambda_{\text{min}}^{-1}([{E}\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}]^{\prime }V_M^{-1}V_nV_M^{-1}[{E}\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}]) \\ \leq&~\lambda_{\text{max}}^2(V_M)\lambda_{\text{min}}^{-1}([{E}\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}]^{\prime }[{E}\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}])\lambda_{\min}^{-1}(V_n). \end{split}$$ Hence, $\|\boldsymbol{\beta}\|_2\leq C$. From Lemma \[la5\], $$\frac{M}{Q}\sum_{q=1}^Q\rho_{vv}(\widetilde{\lambda}^{\prime }\phi_q(\widehat{\boldsymbol{\theta}}_{n}))\phi_q(\widehat{\boldsymbol{\theta}}_{n})\phi_q(\widehat{\boldsymbol{\theta}}_{n})^{\prime }=\rho_{vv}(0)M\widehat{\Omega}(\widehat{\boldsymbol{\theta}}_{n})\cdot\{1+o_p(1)\}.$$ Noting Lemmas \[la3\] and \[la7\], we know the eigenvalues of $M\widehat{\Omega}(\widehat{\boldsymbol{\theta}}_{n})$ are uniformly bounded away from zero and infinity w.p.a.1. Hence, the eigenvalues of $MQ^{-1}\sum_{q=1}^Q\rho_{vv}(\widetilde{\lambda}^{\prime }\phi_q(\widehat{\boldsymbol{\theta}}_{n}))\phi_q(\widehat{\boldsymbol{\theta}}_{n})\phi_q(\widehat{\boldsymbol{\theta}}_{n})^{\prime }$ are uniformly bounded away from zero and infinity w.p.a.1. By Lemma \[la10\] and (\[eq:key\]), $$\boldsymbol{\beta}^{\prime }\{\nabla_{\boldsymbol{\theta}}\bar{\phi}(\widehat{\boldsymbol{\theta}}_{n})\}^{\prime }\bigg\{\frac{M}{Q}\sum_{q=1}^Q\rho_{vv}(\widetilde{\lambda}^{\prime }\phi_q(\widehat{\boldsymbol{\theta}}_{n}))\phi _q(\widehat{\boldsymbol{\theta}}_{n})\phi_q(\widehat{\boldsymbol{\theta}}_{n})^{\prime } \bigg\}^{-1}\bar{\phi}(\widehat{\boldsymbol{\theta}}_n)=O_p(r^{3/2}p^{1/2}M^{1/2}n^{-1}).$$ From Lemmas \[la11\] and \[la10\], $$\begin{split} &~\boldsymbol{\beta}^{\prime }[{E}\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}]^{\prime }\{M\widehat{\Omega}(\widehat{\boldsymbol{\theta}}_{n})\}^{-1}\bar{\phi}(\widehat{\boldsymbol{\theta}}_n) \\ =&~O_p( r^{3/2}p^{1/2}M^{1/2}n^{-1})+O_p(r^{3/2}M^{1-1/\gamma}n^{1/\gamma-1})+O_p(r^{3/2}pn^{-1}). \end{split}$$ Note that Lemmas \[la3\] and \[la7\], $$\begin{split} &~\boldsymbol{\beta}^{\prime }[{E}\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}]^{\prime }V_M^{-1}\bar{\phi}(\widehat{\boldsymbol{\theta}}_n) \\ =&~O_p(r^{3/2}p^{1/2}M^{1/2}n^{-1})+O_p(r^{3/2}M^{1-1/\gamma}n^{1/\gamma-1})+O_p(r^{3/2}M^{3/2}n^{-1})+O_p(r^{3/2}pn^{-1}). \end{split}$$ Expanding $\bar{\phi}(\widehat{\boldsymbol{\theta}}_n)$ around $\boldsymbol{\theta}=\boldsymbol{\theta}_0$, by Lemmas \[la11\] and \[la2\], $$\begin{split} &~\boldsymbol{\beta}^{\prime }[{E}\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}]^{\prime }V_M^{-1}[{E}\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}](\widehat{\boldsymbol{\theta}}_{n}-\boldsymbol{\theta}_{0}) \\ =&-\boldsymbol{\beta}^{\prime }[{E}\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}]^{\prime }V_M^{-1}\bar{g}(\boldsymbol{\theta}_{0})+O_p( r^{3/2}p^{1/2}M^{1/2}n^{-1})+O_p(r^{3/2}M^{1-1/\gamma}n^{1/\gamma-1}) \\ &+O_p(r^{3/2}pn^{-1})+O_p(r^{3/2}M^{3/2}n^{-1}). \end{split}$$ Hence, we obtain Proposition 2.$\hfill \square$ Proof of Proposition 1 {#proof-of-proposition-1 .unnumbered} ---------------------- From Proposition 2, if we pick $\boldsymbol{\alpha}_{n}=(\widehat{\boldsymbol{\theta}}_{n}-\boldsymbol{\theta}_{0})/\|\widehat{\boldsymbol{\theta}}_{n}-\boldsymbol{\theta}_{0}\|_2$, we can obtain that $$\begin{split} &~\sqrt{n}\cdot\lambda_{\max}^{-1/2}\{[{E}\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}]^{\prime }V_M^{-1}V_nV_M^{-1}[{E}\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}]\} \\ &~~~~~~~~~~~~~~~~~~~~~~\cdot\lambda_{\text{min}}\{[{E}\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}]^{\prime }V_M^{-1}[{E}\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}]\}\cdot\|\widehat{\boldsymbol{\theta}}_{n}-\boldsymbol{\theta}_{0}\|_2 \\ =&~O_p\{\|\sqrt{n}\boldsymbol{\alpha}_{n}^{\prime }([{E}\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}]^{\prime }V_M^{-1}V_nV_M^{-1}[{E}\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}])^{-1/2}[{E}\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}]^{\prime }V_M^{-1}\bar{g}(\boldsymbol{\theta}_{0})\|_2\} \\ &+O_p(r^{3/2}p^{1/2}M^{1/2}n^{-1/2})+O_p(r^{3/2}pn^{-1/2})+O_p(r^{3/2}M^{1-1/\gamma}n^{1/\gamma-1/2})+O_p(r^{3/2}M^{3/2}n^{-1/2}). \label{eq:theta} \end{split}$$ Note that $$\lambda_{\max}^{-1/2}\{[{E}\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}]^{\prime }V_M^{-1}V_nV_M^{-1}[{E}\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}]\}\lambda_{\text{min}}\{[{E}\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}]^{\prime }V_M^{-1}[{E}\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}]\}>C,$$ which is assumed in (A.2)(iv) and the eigenvalues of $V_M$ and $V_n$ are uniformly bounded away from zero and infinity. Therefore, by $${E}(\|\sqrt{n}\boldsymbol{\alpha}_{n}^{\prime }([{E}\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}]^{\prime }V_M^{-1}V_nV_M^{-1}[{E}\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}])^{-1/2}[{E}\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}]^{\prime }V_M^{-1}\bar{g}(\boldsymbol{\theta}_{0})\|_2^2)=p,$$ we complete the proof of Proposition 1.$\hfill \square$ Proof of Theorem 2 {#proof-of-theorem-2 .unnumbered} ------------------ From Proposition 2, it is only need to show $$S_n:=-\sqrt{n}\boldsymbol{\alpha}_{n}^{\prime }([{E}\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}]^{\prime }V_M^{-1}V_nV_M^{-1}[{E}\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}])^{-1/2}[{E}\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}]^{\prime }V_M^{-1}\bar{g}(\boldsymbol{\theta}_{0})\xrightarrow{d}N(0,1).$$ Let $$x_{n,t}=-\boldsymbol{\alpha}_{n}^{\prime }([{E}\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}]^{\prime }V_M^{-1}V_nV_M^{-1}[{E}\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}])^{-1/2}[{E}\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}]^{\prime }V_M^{-1}\bar{g}(\boldsymbol{\theta}_{0})=:\boldsymbol{\beta}_n^{\prime }g_t(\boldsymbol{\theta}_{0}),$$ then $S_n=n^{-1/2}\sum_{t=1}^nx_{n,t}$. As restriction (\[eq:moment-cond\]) holds, $\sup_{n}\sup_{1\leq t\leq n}{E}(|x_{n,t}|^\gamma)<\infty$. On the other hand, $\text{Var}(S_n)=1$. Note that (A.1)(i), then by the central limit theorem proposed in Francq and Zakoïan (2005), we have $S_n\xrightarrow{d}N(0,1)$.$\hfill \square$ Some Lemmas III {#some-lemmas-iii .unnumbered} --------------- To prove Theorem 3, we employ the blocking technique by splitting the observations to big blocks of length $h$ and small blocks of length $b$. Suppose that $\tilde{B}_i=(X_{(i-1)(h+b)+1},\ldots,X_{i(h+b)})=(\tilde{B}_{i1},\tilde{B}_{i2})$, where $\tilde{B}_{i1}=(X_{(i-1)(h+b)+1},\ldots,X_{(i-1)(h+b)+h})$ , $\tilde{B}_{i2}=(X_{(i-1)(h+b)+(h+1)},\ldots,X_{i(h+b)})$ and $b<h$. Then $n=T(h+b)+m$, where $m<h+b$. Later, we will discuss the selection of $b$ and $h$. By a similar argument to those in finding the order of $\|\bar{g}(\boldsymbol{\theta}_{0})\|_2$ and the proof of Lemma \[la2\], we can obtain $\|\bar{g}(\boldsymbol{\theta}_{0})-(Th)^{-1}\sum_{i=1}^T\sum_{t\in\tilde{B}_{i1}}g_t(\boldsymbol{\theta}_{0})\|_2=O_p(r^{1/2}T^{1/2}b^{1/2}n^{-1}) $. Furthermore, define $\widetilde{V}_n=\text{Var}\{h^{-1/2}\sum_{t=1}^hg_t(\boldsymbol{\theta}_{0})\}$, $Z_{T,i}=h^{-1/2}\widetilde{V}_n^{-1/2}\sum_{t\in\tilde{B}_{i1}}g_t(\boldsymbol{\theta}_{0})$ and $G_{T,k}=\sum_{i=1}^kZ_{T,i}$, then ${E}(Z_{T,i})=\boldsymbol{\mathbf{0}}$ and ${E}( Z_{T,i}Z_{T,i}^{\prime })=I_r $. It can be shown that $|x^{\prime }(V_n-\widetilde{V}_n)y|\leq Crh^{-1}\sum_{k=1}^hk\alpha_{X}(k)^{1-2/\gamma}\cdot\|x\|_2\|y\|_2 $. Define $\mathscr{G}_{T,0}=\{\varnothing,\Omega\}$, $\mathscr{G}_{T,k}=\sigma(Z_{T,1},\ldots,Z_{T,k})$, $k=1,\ldots,T$, and $S_{T,k}=T^{-1}(2r)^{-1/2}\{(\sum_{i=1}^kZ_{T,i}^{\prime })(\sum_{i=1}^kZ_{T,i})-kr\}$. ${E}_{T,k}(\cdot)$ denote the conditional expectation given $\mathscr{G}_{T,k}$. Let $D_{T,k}=S_{T,k}-S_{T,k-1}=T^{-1}(2r)^{-1/2}(2Z_{T,k}^{\prime }G_{T,k-1}+\|Z_{T,k}\|_2^2-r)$. The following lemmas are used to establish Theorem 3. \[la13\] Under conditions and , assume that the eigenvalues of $V_n$ are uniformly bounded away from zero and infinity. Then $$\begin{split} {E}\{(\|Z_{T,k}\|_2^2-r)^2\}&\leq Cr^2h^2, \\ {E}\{|{E}_{T,k-1}(Z_{T,k}^{\prime }G_{T,k-1})|\}&\leq Crk^{1/2}\bigg\{\sum_{l=1}^h\alpha_X(b+l)^{1-2/\gamma}\bigg\}^{1/2}, \\ {E}\{{E}_{T,k-1}^2(\|Z_{T,k}\|_2^2-r)\}&\leq Cr^2h^2\alpha_X(b+1)^{1-2/\gamma}, \end{split}$$ and for any $i\neq j$, $$|{E}\{(\|Z_{T,i}\|_2^2-r)(\|Z_{T,j}\|_2^2-r)\}|\leq Cr^2h^2\alpha_X\{(b+h)|i-j|-h+1\}^{1-2/\gamma}, \\$$ provided that $rh^{-1}\sum_{k=1}^hk\alpha_X(k)^{1-2/\gamma}=o(1)$. <span style="font-variant:small-caps;">Proof</span>: As $rh^{-1}\sum_{k=1}^hk\alpha_X(k)^{1-2/\gamma}=o(1)$, $\lambda_{\text{max}}(\widetilde{V}_n^{-1})\leq C$. Then, ${E}(\|Z_{T,k}\|_2^4)\leq Ch^{-2}{E}\{ \|\sum_{t=1}^hg_t(\boldsymbol{\theta}_{0})\|_2^4\}$. By Triangle and Jensen’s inequalities, ${E}\{\|\sum_{t=1}^hg_t(\boldsymbol{\theta}_{0})\|_2^4\}\leq Cr^2h^4$. Hence, ${E}\{(\|Z_{T,k}\|_2^2-r)^2\}\leq Cr^2h^2$. By Cauchy-Schwarz inequality, $${E}\{|{E}_{T,k-1}(Z_{T,k}^{\prime }G_{T,k-1}) |\}\leq\{{E}(\|G_{T,k-1}\|_2^2)\}^{1/2}[{E}\{ \|{E}_{T,k-1}(Z_{T,k})\|_2^2\}]^{1/2}.$$ Using the same method in the proof of Lemma \[la3\], ${E}( \|G_{T,k-1}\|_2^2)\leq Crk$. On the other hand, note that $\|{E}_{T,k-1}(Z_{T,k})\|_2^2\leq C\sum_{t\in\tilde{B}_{k1}}\|{E}_{T,k-1}\{g_t(\boldsymbol{\theta}_{0})\}\|_2^2$. Hence, $$\begin{split} {E}\{\|{E}_{T,k-1}(Z_{T,k})\|_2^2\}&~\leq C\sum_{t\in\tilde{B}_{k1}}\sum_{j=1}^r{E}[{E}^2_{T,k-1}\{g_{tj}(\boldsymbol{\theta}_{0})\}] \\ &~\leq C\sum_{t\in\tilde{B}_{k1}}\sum_{j=1}^r\left[{E}\{|g_{tj}(\boldsymbol{\theta}_{0})|^\gamma\}\right]^{2/\gamma}\alpha_X\{t+b-(k-1)(h+b)\}^{1-2/\gamma} \\ &~= Cr\sum_{l=1}^h\alpha_X(b+l)^{1-2/\gamma}. \label{eq:4} \end{split}$$ This is based on the fact that if ${E}(X)=0$, then $({E}[\{{E}(X|\mathcal{F})\}^2])^{1/2}=\sup\{{E}(XY):Y\in\mathcal{F},{E}(Y^2)=1\}$ for any $\sigma$-field $\mathcal{F}$ (details can be found in Durrett (2010)), and Davydov inequality. Then, ${E}\{|{E}_{T,k-1}(Z_{T,k}^{\prime }G_{T,k-1})|\}\leq Crk^{1/2}\{\sum_{l=1}^h\alpha_X(b+l)^{1-2/\gamma}\}^{1/2}$. Using the same argument above, we can obtain ${E}(\|Z_{T,k}\|_2^{2\gamma})\leq Cr^{\gamma}h^{\gamma}$. Then, by the same argument of (\[eq:4\]), $${E}\{{E}_{T,k-1}^2(\|Z_{T,k}\|_2^2-r)\}\leq Cr^2h^2\alpha_X(b+1)^{1-2/\gamma}. \label{eq:t}$$ For any $i\neq j$, by Davydov inequality, $$|{E}\{(\|Z_{T,i}\|_2^2-r)(\|Z_{T,j}\|_2^2-r)\}|\leq C\{{E}(|\|Z_{T,i}\|_2^2-r|^{\gamma})\}^{2/\gamma}\alpha_X\{(b+h)|i-j|-h+1\}^{1-2/\gamma}.$$ Hence, we complete the proof of this lemma. $\hfill \square$ \[la14\] Under conditions and , assume that the eigenvalues of $V_n$ are uniformly bounded away from zero and infinity. Then $$\begin{split} {E}\{\|{E}_{T,k-1}(Z_{T,k}Z_{T,k}^{\prime })-I_r\|_F^2\}&\leq Cr^2h^2\alpha_X(b+1)^{1-2/\gamma}, \\ {E}[\{G_{T,k-1}^{\prime }{E}_{T,k-1}(Z_{T,k})\}^2]&\leq Cr^2h^2k^2\alpha_X(b+1)^{1/2-1/\gamma}. \end{split}$$ provided that $rh^{-1}\sum_{k=1}^hk\alpha_X(k)^{1-2/\gamma}=o(1)$. <span style="font-variant:small-caps;">Proof</span>: Note that $$\begin{split} &~{E}\{\|{E}_{T,k-1}(Z_{T,k}Z_{T,k}^{\prime })-I_r\|_F^2\} \\ \leq&~C{E}\bigg(\bigg\|{E}_{T,k-1}\bigg\{\bigg[h^{-1/2}\sum_{t\in\tilde{B}_{k1}}g_t(\boldsymbol{\theta}_{0})\bigg]\bigg[h^{-1/2}\sum_{t\in\tilde{B}_{k1}}g_t(\boldsymbol{\theta}_{0})\bigg]^{\prime }\bigg\}-\widetilde{V}_n\bigg\|_F^2\bigg) \\ \leq&~C\sum_{u,v=1}^r{E}\bigg(\bigg|{E}_{T,k-1}\bigg\{\bigg[h^{-1/2}\sum_{t\in\tilde{B}_{k1}}g_{tu}(\boldsymbol{\theta}_{0})\bigg]\bigg[h^{-1/2}\sum_{t\in\tilde{B}_{k1}}g_{tv}(\boldsymbol{\theta}_{0})\bigg]\bigg\}-\widetilde{V}_n(u,v)\bigg|^2\bigg) \\ \leq&~Cr^2h^2\alpha_X(b+1)^{1-2/\gamma}, \end{split}$$ where $\widetilde{V}_n(u,v)$ denotes the $(u,v)$-element of $\widetilde{V}_n$. The last step is similar to (\[eq:t\]). Then we obtain the first conclusion. As $$\begin{split} \|{E}_{T,k-1}(Z_{T,k})\|_2^4\leq&~\{{E}_{T,k-1}(Z_{T,k})\}^{\prime }\{{E}_{T,k-1}(Z_{T,k}Z_{T,k}^{\prime })\}\{{E}_{T,k-1}(Z_{T,k})\} \\ \leq&~ \|{E}_{T,k-1}(Z_{T,k})\|_2^2\|{E}_{T,k-1}(Z_{T,k}Z_{T,k}^{\prime })-I_r\|_F+\|{E}_{T,k-1}(Z_{T,k})\|_2^2, \end{split}$$ then by Cauchy-Schwarz inequality, $$\begin{split} &~{E}\{\|{E}_{T,k-1}(Z_{T,k})\|_2^4\} \\ \leq&~[{E}\{\|{E}_{T,k-1}(Z_{T,k})\|_2^4\}]^{1/2}[{E}\{\|{E}_{T,k-1}(Z_{T,k}Z_{T,k}^{\prime })-I_r\|_F^2\}]^{1/2}+{E}\{\|{E}_{T,k-1}(Z_{T,k})\|_2^2\}. \end{split}$$ Let $U=[{E}\{\|{E}_{T,k-1}(Z_{T,k})\|_2^4\}]^{1/2}$, from (\[eq:4\]) and the first result in this lemma, $$U^2\leq CUrh\alpha_{X}(b+1)^{1/2-1/\gamma}+Crh\alpha_X(b+1)^{1-2/\gamma}.$$ Then, $U\leq Crh\alpha_X(b+1)^{1/2-1/\gamma}$. Hence, ${E}\{\|{E}_{T,k-1}(Z_{T,k})\|_2^4\}\leq Cr^2h^2\alpha_X(b+1)^{1-2/\gamma}$. Also, $${E}[\{G_{T,k-1}^{\prime }{E}_{T,k-1}(Z_{T,k})\}^2]\leq \{{E}(\|G_{T,k-1}\|_2^4)\}^{1/2}[{E}\{\|{E}_{T,k-1}(Z_{T,k})\|_2^4\} ]^{1/2}\leq Cr^2h^2k^2\alpha_X(b+1)^{1/2-1/\gamma}.$$ We complete the proof.$\hfill \square$ \[la15\] Under conditions and , assume that the eigenvalues of $V_n$ are uniformly bounded away from zero and infinity. Then $r^{-1}T^{-2}\sum_{j=2}^T(T-j)G_{T,j-1}^{\prime }Z_{T,j}=o_p(1) $ provided that $rh^{-1}\sum_{k=1}^hk\alpha_X(k)^{1-2/\gamma}=o(1)$ and $n^2h^2\alpha_X(b+1)^{1-2/\gamma}=o(1)$. <span style="font-variant:small-caps;">Proof</span>: Note that $$\begin{split} &{E}\bigg(\bigg[\frac{1}{rT^2}\sum_{j=2}^T(T-j)\{G_{T,j-1}^{\prime }Z_{T,j}-{E}_{T,j-1}(G_{T,j-1}^{\prime }Z_{T,j})\}\bigg]^2\bigg) \\ &~~~~~~~~~~~~ =\frac{1}{r^2T^4}\sum_{j=2}^T(T-j)^2{E}[\{G_{T,j-1}^{\prime }Z_{T,j}-{E}_{T,j-1}(G_{T,j-1}^{\prime }Z_{T,j})\}^2]. \end{split}$$ By the first result of Lemma \[la14\], $$\begin{split} {E}\{(G_{T,j-1}^{\prime }Z_{T,j})^2\} &={E}(\|G_{T,j-1}\|_2^2)+{E}[G_{T,j-1}^{\prime }\{{E}_{T,j-1}(Z_{T,j}Z_{T,j}^{\prime })-I_r\}G_{T,j-1}] \\ &\leq Crj+\{{E}(\|G_{T,j-1}\|_2^4)\}^{1/2}[{E}\{\|{E}_{T,j-1}(Z_{T,j}Z_{T,j}^{\prime })-I_r\|_F^2\}]^{1/2} \\ &\leq Crj+Cr^2h^2j^2\alpha_X(b+1)^{1/2-1/\gamma}. \end{split}$$ Using Cauchy-Schwarz inequality and the fact ${E}\{|{E}_{T,j-1}(G_{T,j-1}^{\prime }Z_{T,j})|^2\}\leq {E}\{(G_{T,j-1}^{\prime }Z_{T,j})^2\}$, $${E}\bigg[\frac{1}{r^2T^4}\sum_{j=2}^T(T-j)^2\{G_{T,j-1}^{\prime }Z_{T,j}-{E}_{T,j-1}(G_{T,j-1}^{\prime }Z_{T,j})\}^2\bigg]\leq Cr^{-1}+Cnh\alpha_X(b+1)^{1/2-1/\gamma}\rightarrow0.$$ Then, $r^{-1}T^{-2}\sum_{j=2}^T(T-j)\{G_{T,j-1}^{\prime }Z_{T,j}-{E}_{T,j-1}(G_{T,j-1}^{\prime }Z_{T,j})\}= o_p(1)$. From Lemma \[la13\], we have $r^{-1}T^{-2}\sum_{j=2}^T(T-j){E}_{T,j-1}(G_{T,j-1}^{\prime }Z_{T,j})= o_p(1)$. Hence, we complete the proof. $\hfill \square$ \[la16\] Under conditions and , assume that the eigenvalues of $V_n$ are uniformly bounded away from zero and infinity. If $rh^{-1}\sum_{k=1}^hk\alpha_X(k)^{1-2/\gamma}=o(1)$, $rT\sum_{l=1}^h\alpha_X(b+l)^{1-2/\gamma}=o(1)$ and $rh^2\alpha_X(b+1)^{1-2/\gamma}=o(1)$, then $\sum_{k=1}^T{E}_{T,k-1}(D_{T,k})= o_p(1)$. <span style="font-variant:small-caps;">Proof</span>: Note that $$\begin{split} \sum_{k=1}^T{E}_{T,k-1}(D_{T,k})&=\frac{2}{(2r)^{1/2}T}\sum_{k=1}^T{E}_{T,k-1}(Z_{T,k}^{\prime }G_{T,k-1})+\frac{1}{(2r)^{1/2}T}\sum_{k=1}^T{E}_{T,k-1}(\|Z_{T,k}\|_2^2-r) \\ &=:I_1+I_2. \end{split}$$ From Lemma \[la13\], $${E}(|I_1|)\leq\frac{2}{(2r)^{1/2}T}\sum_{k=1}^T{E}\{|{E}_{T,k-1}(Z_{T,k}^{\prime }G_{T,k-1})|\}\leq C r^{1/2}T^{1/2}\bigg\{\sum_{l=1}^h\alpha_X(b+l)^{1-2/\gamma}\bigg\}^{1/2}\rightarrow0$$ and $${E}(|I_2|)\leq\frac{1}{(2r)^{1/2}T}\sum_{k=1}^T[{E}\{{E}^2_{T,k-1}(\|Z_{T,k}\|_2^2-r)\}]^{1/2}\leq Cr^{1/2}h\alpha_X(b+1)^{1/2-1/\gamma}\rightarrow0.$$ Then, we complete the proof of this lemma. $\hfill \square$ \[la17\] Under conditions and , assume that the eigenvalues of $V_n$ are uniformly bounded away from zero and infinity. If $rh^{-1}\sum_{k=1}^hk\alpha_X(k)^{1-2/\gamma}=o(1)$, $r^2n^{2}h^2\alpha_X(b+1)^{1-2/\gamma}=o(1)$ and $rh^3n^{-1}=o(1)$, then $S_{T,T}\xrightarrow[]{d}N(0,1)$. <span style="font-variant:small-caps;">Proof</span>: We will use the martingale central limit theorem to show $S_{T,T}\xrightarrow[]dN(0,1)$. Note that $$S_{T,T}=\sum_{k=1}^TD_{T,k}=\sum_{k=1}^T\{D_{T,k}-{E}_{T,k-1}(D_{T,k})\}+\sum_{k=1}^T{E}_{T,k-1}(D_{T,k}).$$ The first part on the right hand of above equation are the sum of a sequence of martingale difference with respect to $\{\mathscr{G}_{T,k}\}_{k=0}^T$. From Lemma \[la16\], $S_{T,T}=\sum_{k=1}^T\{D_{T,k}-{E}_{T,k-1}(D_{T,k})\}+o_p(1). $ By the martingale central limit theorem (Billingsley, 1995), in order to show the conclusion, it is sufficient to show that, letting $\sigma_{T,k}^2={E}_{T,k-1}[\{D_{T,k}-{E}_{T,k-1}(D_{T,k})\}^2]$, as $T\rightarrow\infty$, $$V_{T,T}:=\sum_{k=1}^T\sigma_{T,k}^2\xrightarrow[]p1~~~~\text{and}~~~~\sum_{k=1}^T{E}\{D_{T,k}-{E}_{T,k-1}(D_{T,k})\}^4\rightarrow0.$$ For the first part, $$\begin{split} V_{T,T}=&~\frac{2}{rT^2}\sum_{k=1}^T(G_{T,k-1}^{\prime }[{E}_{T,k-1}(Z_{T,k}Z_{T,k}^{\prime })-\{{E}_{T,k-1}(Z_{T,k})\}\{{E}_{T,k-1}(Z_{T,k}^{\prime })\}]G_{T,k-1}) \\ &+\frac{2}{rT^2}\sum_{k=1}^TG_{T,k-1}^{\prime }[{E}_{T,k-1}\{Z_{T,k}(\|Z_{T,k}\|_2^2-r)\}-{E}_{T,k-1}(Z_{T,k})\cdot{E}_{T,k-1}(\|Z_{T,k}\|_2^2-r)] \\ &+\frac{1}{2rT^2}\sum_{k=1}^T[{E}_{T,k-1}\{(\|Z_{T,k}\|_2^2-r)^2\}-{E}_{T,k-1}^2(\|Z_{T,k}\|_2^2-r)] \\ =:&~I_1+I_2+I_3. \end{split}$$ We will show that $I_1\xrightarrow[]p1, I_2\xrightarrow[]p0$ and $I_3\xrightarrow[]p0$. Note that $0\leq I_3 \leq (2r)^{-1}T^{-2}\sum_{k=1}^T{E}_{T,k-1}\{(\|Z_{T,k}\|_2^2-r)^2\} $ and ${E}\{(\|Z_{T,k}\|_2^2-r)^2\}\leq Cr^2h^2$, then $I_3\xrightarrow[]p0$. Using Cauchy-Schwarz, Triangle and Jensen’s inequalities, $$\begin{split} &|{E}_{T,k-1}\{G_{T,k-1}^{\prime }Z_k(\|Z_{T,k}\|_2^2-r)\}| \\ &~~~~~~~~\leq\{{E}_{T,k-1}(G_{T,k-1}^{\prime }Z_{T,k}Z_{T,k}^{\prime }G_{T,k-1})\}^{1/2}[{E}_{T,k-1}\{(\|Z_{T,k}\|_2^2-r)^2\}]^{1/2} \\ &~~~~~~~~\leq\{\|G_{T,k-1}\|_2^2+\|G_{T,k-1}\|_2^2\cdot\|{E}_{T,k-1}(Z_{T,k}Z_{T,k}^{\prime })-I_r\|_F\}^{1/2}[{E}_{T,k-1}\{(\|Z_{T,k}\|_2^2-r)^2\}]^{1/2} \\ &~~~~~~~~\leq C\|G_{T,k-1}\|_2[{E}_{T,k-1}\{(\|Z_{T,k}\|_2^2-r)^2\}]^{1/2}\{1+\|{E}_{T,k-1}(Z_{T,k}Z_{T,k}^{\prime })-I_r\|_F^{1/2}\}. \end{split}$$ Then, by Cauchy-Schwarz inequality and Lemmas \[la13\] and \[la14\], $$\begin{split} &{E}[|{E}_{T,k-1}\{G_{T,k-1}^{\prime }Z_k(\|Z_{T,k}\|_2^2-r)\}|] \\ &~~~~~~~~~~~~~\leq C\{{E}(\|G_{T,k-1}\|_2^2)\}^{1/2}[{E}\{(\|Z_{T,k}\|_2^2-r)^2\}]^{1/2} \\ &~~~~~~~~~~~~~~~~~+C\{{E}(\|G_{T,k-1}\|_2^2)\}^{1/2}[{E}\{(\|Z_{T,k}\|_2^2-r)^4\}]^{1/4}[{E}\{\|{E}_{T,k-1}(Z_{T,k}Z_{T,k}^{\prime })-I_r\|_F^2\}]^{1/4} \\ &~~~~~~~~~~~~~\leq Cr^{3/2}hk^{1/2}+Cr^2h^{3/2}k^{1/2}\alpha_X(b+1)^{1/4-1/(2\gamma)}. \end{split}$$ Hence, $r^{-1}T^{-2}\sum_{k=1}^T{E}[|G_{T,k-1}^{\prime }{E} _{T,k-1}\{Z_{T,k}(\|Z_{T,k}\|_2^2-r)\}|]\rightarrow0. $ By the same argument, we can obtain $r^{-1}T^{-2}\sum_{k=1}^T{E}\{|{E}_{T,k-1}(G_{T,k-1}^{\prime }Z_{T,k})\cdot{E}_{T,k-1}(\|Z_{T,k}\|_2^2-r)|\}\rightarrow0$. Thus, $I_2\xrightarrow[]p0$. Note that $$\begin{split} I_1=&~\frac{2}{rT^2}\sum_{k=1}^T\|G_{T,k-1}\|_2^2 \\ &+\frac{2}{rT^2}\sum_{k=1}^TG_{T,k-1}^{\prime }\{{E}_{T,k-1}(Z_{T,k}Z_{T,k}^{\prime })-{E}_{T,k-1}(Z_{T,k})\cdot{E}_{T,k-1}(Z_{T,k}^{\prime })-I_r\}G_{T,k-1}. \end{split}$$ By Triangle inequality, $$\begin{split} &\bigg|\frac{2}{rT^2}\sum_{k=1}^TG_{T,k-1}^{\prime }\{{E}_{T,k-1}(Z_{T,k}Z_{T,k}^{\prime })-{E}_{T,k-1}(Z_{T,k})\cdot{E}_{T,k-1}(Z_{T,k}^{\prime })-I_r\}G_{T,k-1}\bigg| \\ &~~~~~~~~\leq\frac{2}{rT^2}\sum_{k=1}^T\|G_{T,k-1}\|_2^2\|{E}_{T,k-1}(Z_{T,k}Z_{T,k}^{\prime })-I_r\|_F+\frac{2}{rT^2}\sum_{k=1}^T\{G_{T,k-1}^{\prime }{E}_{T,k-1}(Z_{T,k})\}^2. \end{split}$$ By Cauchy-Schwarz inequality and Lemma \[la14\], $$\begin{split} &~{E}\{\|G_{T,k-1}\|_2^2\|{E}_{T,k-1}(Z_{T,k}Z_{T,k}^{\prime })-I_r\|_F\} \\ \leq&~\{{E}(\|G_{T,k-1}\|_2^4)\}^{1/2}[{E}\{\|{E}_{T,k-1}(Z_{T,k}Z_{T,k}^{\prime })-I_r\|_F^2\}]^{1/2} \\ \leq&~ Cr^2h^2k^2\alpha_X(b+1)^{1/2-1/\gamma}. \end{split}$$ Then, $${E}\bigg\{\frac{2}{rT^2}\sum_{k=1}^T\|G_{T,k-1}\|_2^2\|{E}_{T,k-1}(Z_{T,k}Z_{T,k}^{\prime })-E_r\|_F\bigg\}\leq Crnh\alpha_X(b+1)^{1/2-1/\gamma}\rightarrow0.$$ On the other hand, by Lemma \[la14\], $${E}\bigg[\frac{2}{rT^2}\sum_{k=1}^T\{G_{T,k-1}^{\prime }{E}_{T,k-1}(Z_{T,k})\}^2\bigg]\leq Crnh\alpha_X(b+1)^{1/2-1/\gamma}\rightarrow0.$$ Then, $I_1=2r^{-1}T^{-2}\sum_{k=1}^T\|G_{T,k-1}\|_2^2+o_p(1). $ From Lemma \[la15\], $$\begin{split} \frac{2}{rT^2}\sum_{k=1}^T\|G_{T,k-1}\|_2^2&=\frac{2}{rT^2}\sum_{i=1}^T(T-i)\|Z_{T,i}\|_2^2+\frac{4}{rT^2}\sum_{j=2}^T(T-j)G_{T,j-1}^{\prime }Z_{T,j} \\ &=\frac{2}{rT^2}\sum_{i=1}^T(T-i)\|Z_{T,i}\|_2^2+o_p(1). \end{split}$$ In order to prove $I_1\xrightarrow[]p1$, it is only need to show $2r^{-1}T^{-2}\sum_{i=1}^T(T-i)(\|Z_{T,i}\|_2^2-r)\xrightarrow[]p0. $ Note that $${E}\bigg\{\frac{2}{rT^2}\sum_{i=1}^T(T-i)(\|Z_{T,i}\|_2^2-r)\bigg\}=0,$$ it is sufficient to show $$\frac{4}{r^2T^4}\bigg[\sum_{i=1}^T(T-i)^2{E}\{(\|Z_{T,i}\|_2^2-r)^2\}+\sum_{i\neq j}(T-i)(T-j){E}\{(\|Z_{T,i}\|_2^2-r)(\|Z_{T,j}\|_2^2-r)\}\bigg]\rightarrow0,$$ which can be derived from Lemma \[la13\]. Hence, $I_1\xrightarrow[]p1$. For the second part, we only need to prove $\sum_{k=1}^T{E}(D_{T,k}^4)\rightarrow0$. Note that $$D_{T,k}^4\leq Cr^{-2}T^{-4}\{(G_{T,k-1}^{\prime }Z_{T,k})^4+(\|Z_{T,k}\|_2^2-r)^4\}$$ and $$(G_{T,k-1}^{\prime }Z_{T,k})^4=\sum_{i_1,\ldots,i_4=1}^{k-1}\sum_{j_1,\ldots,j_4=1}^rZ_{T,i_1,j_1}Z_{T,i_2,j_2}Z_{T,i_3,j_3}Z_{T,i_4,j_4}Z_{T,k,j_1}Z_{T,k,j_2}Z_{T,k,j_3}Z_{T,k,j_4},$$ where $Z_{T,i,j}$ denotes the $j$th component of $Z_{T,i}$. By the same way of the Lemma 15 in Francq and Zakoïan (2007), $r^{-2}h^{-2}{E}\{(G_{T,k-1}^{\prime }Z_{T,k})^4\}\leq Ck^2. $ Then, $\sum_{k=1}^T{E}(D_{T,k}^4)\leq Ch^2T^{-1}+Cr^2h^{4}T^{-3}\rightarrow0. $ Hence, we complete the proof. $\hfill \square$ \[la18\] Under conditions and , assume that the eigenvalues of $V_n$ are uniformly bounded away from zero and infinity. Then $(2r)^{-1/2}\{n\bar{g}(\boldsymbol{\theta}_{0})^{\prime }V_n^{-1}\bar{g}(\boldsymbol{\theta}_{0})-r\}\xrightarrow[]{d}N(0,1)$ provided that $r^{3/2}h^{-1}\sum_{k=1}^hk\alpha_X(k)^{1-2/\gamma}=o(1)$, $rbh^{-1}=o(1)$, $r^2n^{2}h^2\alpha_X(b+1)^{1-2/\gamma}=o(1)$ and $rh^3n^{-1}=o(1)$. <span style="font-variant:small-caps;">Proof</span>: Note that $$\begin{split} &~(2r)^{-1/2}\{n\bar{g}(\boldsymbol{\theta}_{0})^{\prime }V_n^{-1}\bar{g}(\boldsymbol{\theta}_{0})-r\} \\ =&~\frac{n}{Th}S_{T,T}+O_p\bigg\{r^{3/2}h^{-1}\sum_{k=1}^hk\alpha_X(k)^{1-2/\gamma}\bigg\}+O_p(r^{1/2}b^{1/2}h^{-1/2}). \end{split}$$ Then, by Lemma \[la17\], we have $(2r)^{-1/2}\{n\bar{g}(\boldsymbol{\theta}_{0})^{\prime }V_n^{-1}\bar{g}(\boldsymbol{\theta}_{0})-r\}\xrightarrow[]{d}N(0,1)$. $\hfill \square$ Proof of Theorem 3 {#proof-of-theorem-3 .unnumbered} ------------------ Let $\widehat{\lambda}(\boldsymbol{\theta}_0)=\arg\max_{\lambda\in\widehat{\Lambda}_n(\boldsymbol{\theta}_0)}\sum_{q=1}^Q\rho(\lambda^{\prime }\phi_q(\boldsymbol{\theta}_0))$. From Lemma \[la2\], $\|\bar{\phi}(\boldsymbol{\theta}_{0})-\bar{g}(\boldsymbol{\theta}_{0})\|_2=O_p(r^{1/2}Mn^{-1})$. Hence, $\|\bar{\phi}(\boldsymbol{\theta}_{0})\|_2=O_p(r^{1/2}n^{-1/2})$. Then, by Lemma \[la9\], $\|\widehat{\lambda}(\boldsymbol{\theta}_{0})\|_2=O_p(r^{1/2}Mn^{-1/2})$. Meanwhile, $\sup_{1\leq q\leq Q}|\widehat{\lambda}(\boldsymbol{\theta}_{0})^{\prime }\phi_q(\boldsymbol{\theta}_{0})|=o_p(1)$. Expanding $\max_{\lambda\in\widehat{\Lambda}_n(\boldsymbol{\theta}_0)}\sum_{q=1}^Q\rho(\lambda^{\prime }\phi_q(\boldsymbol{\theta}_0))$ around $\lambda=\boldsymbol{\mathbf{0}}$, $$\begin{split} \max_{\lambda\in\widehat{\Lambda}_n(\boldsymbol{\theta}_0)}\sum_{q=1}^Q\rho(\lambda^{\prime }\phi_q(\boldsymbol{\theta}_0)) =&~\sum_{q=1}^Q\bigg[\rho(0)+\rho_v(0)\widehat{\lambda}(\boldsymbol{\theta}_0)^{\prime }\phi_q(\boldsymbol{\theta}_0)+\frac{1}{2}\rho_{vv}(\dot{\lambda}^{\prime }\phi_q(\boldsymbol{\theta}_0))\{\widehat{\lambda}(\boldsymbol{\theta}_0)^{\prime }\phi_q(\boldsymbol{\theta}_0)\}^2\bigg] \end{split}$$ where $\dot{\lambda}$ lies on the line joining $\widehat{\lambda}(\boldsymbol{\theta}_0)$ and $\boldsymbol{\mathbf{0}}$. On the other hand, from $\nabla_\lambda\widehat{S}_n(\boldsymbol{\theta}_0,\widehat{\lambda}(\boldsymbol{\theta}_0))=\boldsymbol{\mathbf{0}}$, we have $$\widehat{\lambda}(\boldsymbol{\theta}_0)=-\bigg\{\frac{1}{Q}\sum_{q=1}^Q\rho_{vv}(\ddot{\lambda}^{\prime }\phi_q(\boldsymbol{\theta}_0))\phi_q(\boldsymbol{\theta}_0)\phi_q(\boldsymbol{\theta}_0)^{\prime }\bigg\}^{-1}\bigg\{\frac{1}{Q}\sum_{q=1}^Q\rho_v(0)\phi_q(\boldsymbol{\theta}_0)\bigg\}$$ for some $\ddot{\lambda}$ lies on the line joining $\widehat{\lambda}(\boldsymbol{\theta}_0)$ and $\boldsymbol{\mathbf{0}}$. Hence, $$\begin{split} \max_{\lambda\in\widehat{\Lambda}_n(\boldsymbol{\theta}_0)}\sum_{q=1}^Q\rho(\lambda^{\prime }\phi_q(\boldsymbol{\theta}_0))=Q\rho(0)-Q\rho^2_v(0)\bar{\phi}(\boldsymbol{\theta}_0)^{\prime }\ddot{\Omega}^{-1}\bar{\phi}(\boldsymbol{\theta}_0)+\frac{1}{2}Q\rho_v^2(0)\bar{\phi}(\boldsymbol{\theta}_0)^{\prime }\ddot{\Omega}^{-1}\dot{\Omega}\ddot{\Omega}^{-1}\bar{\phi}(\boldsymbol{\theta}_0) \\ \end{split}$$ where $\dot{\Omega}=Q^{-1}\sum_{q=1}^Q\rho_{vv}(\dot{\lambda}\phi_q(\boldsymbol{\theta}_0))\phi_q(\boldsymbol{\theta}_0)\phi_q(\boldsymbol{\theta}_0)^{\prime }$ and $\ddot{\Omega}=Q^{-1}\sum_{q=1}^Q\rho_{vv}(\ddot{\lambda}\phi_q(\boldsymbol{\theta}_0))\phi_q(\boldsymbol{\theta}_0)\phi_q(\boldsymbol{\theta}_0)^{\prime }$. Then, the generalized empirical likelihood ratio can be written as $$\begin{split} w_n(\boldsymbol{\theta}_0)=&~2Q\rho_{vv}(0)\bar{\phi}(\boldsymbol{\theta}_0)^{\prime }\ddot{\Omega}^{-1}\bar{\phi}(\boldsymbol{\theta}_0)-Q\rho_{vv}(0)\bar{\phi}(\boldsymbol{\theta}_0)^{\prime }\ddot{\Omega}^{-1}\dot{\Omega}\ddot{\Omega}^{-1}\bar{\phi}(\boldsymbol{\theta}_0) \\ =&~Q\bar{\phi}(\boldsymbol{\theta}_0)^{\prime }\widehat{\Omega}^{-1}(\boldsymbol{\theta}_0)\bar{\phi}(\boldsymbol{\theta}_0)+Q\bar{\phi}(\boldsymbol{\theta}_0)^{\prime }\{2\rho_{vv}(0)\ddot{\Omega}^{-1}-\widehat{\Omega}^{-1}(\boldsymbol{\theta}_0)-\rho_{vv}(0)\ddot{\Omega}^{-1}\dot{\Omega}\ddot{\Omega}^{-1}\}\bar{\phi}(\boldsymbol{\theta}_0). \end{split}$$ By the same argument of Lemma \[la10\], $$\|\dot{\Omega}-\rho_{vv}(0)\widehat{\Omega}(\boldsymbol{\theta}_0)\|_2=O_p(rM^{-1/\gamma}n^{1/\gamma-1/2})=\|\ddot{\Omega}-\rho_{vv}(0)\widehat{\Omega}(\boldsymbol{\theta}_0)\|_2.$$ From Lemmas \[la3\] and $\|M\Omega(\boldsymbol{\theta}_0)-V_n\|_2=O(rM^{-1})$, we know the eigenvalues of $M\widehat{\Omega}(\boldsymbol{\theta}_0)$ are uniformly bounded away from zero and infinity. Hence, $$w_n(\boldsymbol{\theta}_0)=Q\bar{\phi}(\boldsymbol{\theta}_0)^{\prime }\widehat{\Omega}^{-1}(\boldsymbol{\theta}_0)\bar{\phi}(\boldsymbol{\theta}_0)+O_p(r^2M^{1-1/\gamma}n^{1/\gamma-1/2}).$$ By Lemmas \[la2\] and \[la3\], we have $$\begin{split} (2r)^{-1/2}\{w_n(\boldsymbol{\theta}_{0})-r\}=&~(2r)^{-1/2}\{n\bar{g}(\boldsymbol{\theta}_{0})^{\prime }V_n^{-1}\bar{g}(\boldsymbol{\theta}_{0})-r\}+O_p(r^{5/2}M^{2-2/\gamma}n^{2/\gamma-1}) \\ &+O_p(r^{3/2}M^{1-1/\gamma}n^{1/\gamma-1/2})+O_p(r^{3/2}M^{3/2}n^{-1/2}) \\ &+O_p\bigg\{r^{3/2}M^{-1}\sum_{k=1}^Mk\alpha_X(k)^{1-2/\gamma}\bigg\}. \label{eq:wn1} \end{split}$$ The key step is to show $(2r)^{-1/2}\{n\bar{g}(\boldsymbol{\theta}_{0})^{\prime }V_n^{-1}\bar{g}(\boldsymbol{\theta}_{0})-r\}\xrightarrow{d}N(0,1)$. In the independent case, the requirements in Lemma \[la18\] can be simplified as $rbh^{-1}=o(1)$ and $rh^3n^{-1}=o(1)$. We can pick $b=0$ and $h=1$, then $r=o(n)$. In this case, we can regard $\eta=\infty$. In the dependent case with $\eta<\infty$, suppose $b\asymp n^{\kappa_1}$ and $h\asymp n^{\kappa_2}$, where $0<\kappa_1<\kappa_2<1$. Note that (A.1)’(i), the requirements in Lemma \[la18\] turn to $$r=o(n^{2\kappa_2/3}),~~r=o(n^{\kappa_2-\kappa_1}),~~r=o(n^{(\eta\kappa_1-2-2\kappa_2)/2})~~\text{and}~~r=o(n^{1-3\kappa_2}),$$ where $\eta\kappa_1-2-2\kappa_2>0$ and $1-3\kappa_2>0$. In the following, we will consider the selection of $(\kappa_1,\kappa_2)$ to satisfy these inequalities. From $2\kappa_2+2-\eta\kappa_1< 0$, $3\kappa_2-1<0$ and $\kappa_1<\kappa_2$, we can get $\frac{2\kappa_2+2}{\eta}<\kappa_1<\kappa_2<\frac{1}{3}$. In order to guarantee there exists the solution for above inequalities in $(0,1)^2$, it is necessary to require $\eta>8$. If $8<\eta<\infty$, $$\begin{split} \xi:=&\sup_{\substack{ \frac{2}{\eta-2}<\kappa_2<\frac{1}{3} \\ \frac{2\kappa_2+2}{\eta}<\kappa_1<\kappa_2}}\min\bigg(\frac{2\kappa_2}{3},\kappa_2-\kappa_1, \frac{\eta\kappa_1-2-2\kappa_2}{2},1-3\kappa_2\bigg) \\ =&\sup_{\frac{2}{\eta-2}<\kappa_2<\frac{1}{3}}\min\bigg\{\frac{2\kappa_2}{3},\frac{(\eta-2)\kappa_2-2}{\eta+2},1-3\kappa_2\bigg\} \\ =&~\frac{\eta-8}{4\eta+4}1_{(8<\eta<32)}+\frac{2}{11}1_{(32\leq\eta<\infty)}. \end{split}$$ In the dependent case with $\eta=\infty$ where $X_t$ is exponentially strong mixing. The requirements in Lemma \[la18\] turn to $r^{3/2}h^{-1}=o(1)$, $rbh^{-1}=o(1)$ and $rh^3n^{-1}=o(1)$. Then, $$r=o(n^{2\kappa_2/3}),~~r=o(n^{\kappa_2-\kappa_1})~~\text{and}~~r=o(n^{1-3\kappa_2}).$$ In this setting, $$\xi:=\sup_{\substack{ 0<\kappa_2<\frac{1}{3} \\ 0<\kappa_1<\kappa_2}}\min\bigg(\frac{2\kappa_2}{3},\kappa_2-\kappa_1,1-3\kappa_2\bigg)=\frac{2}{11}.$$ Define $$\xi=\frac{\eta-8}{4\eta+4}1_{\{8<\eta<32\}}+\frac{2}{11}1_{\{32\leq\eta\leq\infty\}}+1_{\{\text{indenpendent data}\}}.$$ Hence, if $r=o(n^{\xi})$, then $(2r)^{-1/2}\{n\bar{g}(\boldsymbol{\theta}_{0})^{\prime }V_n^{-1}\bar{g}(\boldsymbol{\theta}_{0})-r\}\xrightarrow[]{d}N(0,1)$. If (\[eq:cond-ratio\]) holds, the other terms in (\[eq:wn1\]) are $o_p(1)$. We complete the proof of Theorem 3. $\hfill \square$ Proof of Theorem 4 {#proof-of-theorem-4 .unnumbered} ------------------ In order to establish Theorem 4, we need the following lemma. \[la19\] For any $\widetilde{\boldsymbol{\theta}}\in\Theta$ and $r\times r$ matrix $\widehat{V}_n$ such that $\|\widetilde{\boldsymbol{\theta}}-\boldsymbol{\theta}_{0}\|_2=O_p(p^{1/2}n^{-1/2})$ and $\|\widehat{V}_n-V_n\|_2=o_p(r^{-1/2})$, if $\|\nabla_{\boldsymbol{\theta}} \bar{g}(\dot{\boldsymbol{\theta}})-{E}\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}\|_2=o_p(p^{-1/2})$ for any $\dot{\boldsymbol{\theta}}$ with $\|\dot{\boldsymbol{\theta}}-\boldsymbol{\theta}_{0}\|_2\leq \|\widetilde{\boldsymbol{\theta}}-\boldsymbol{\theta}_{0}\|_2$, and the eigenvalues of $[{E}\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}]^{\prime }[{E}\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}]$ and $V_n$ are uniformly bounded away from zero and infinity, then $(2r)^{-1/2}\{n\bar{g}(\widetilde{\boldsymbol{\theta}})^{\prime }\widehat{V}_n^{-1}\bar{g}(\widetilde{\boldsymbol{\theta}})-n\bar{g}(\boldsymbol{\theta}_{0})^{\prime }V_n^{-1}\bar{g}(\boldsymbol{\theta}_{0})\}\xrightarrow{p}0 $ provided that $p=o(r^{1/2})$. <span style="font-variant:small-caps;">Proof</span>: Note that $$\begin{split} &~~~~(2r)^{-1/2}|n\bar{g}(\widetilde{\boldsymbol{\theta}})^{\prime }\widehat{V}_n^{-1}\bar{g}(\widetilde{\boldsymbol{\theta}})-n\bar{g}(\boldsymbol{\theta}_{0})^{\prime }V_n^{-1}\bar{g}(\boldsymbol{\theta}_{0})| \\ &=(2r)^{-1/2}|n\bar{g}(\widetilde{\boldsymbol{\theta}})^{\prime }\widehat{V}_n^{-1}\bar{g}(\widetilde{\boldsymbol{\theta}})-n\bar{g}(\boldsymbol{\theta}_{0})^{\prime }\widehat{V}_n^{-1}\bar{g}(\boldsymbol{\theta}_{0})|+(2r)^{-1}|n\bar{g}(\boldsymbol{\theta}_{0})^{\prime }\widehat{V}_n^{-1}\bar{g}(\boldsymbol{\theta}_{0})-n\bar{g}(\boldsymbol{\theta}_{0})^{\prime }V_n^{-1}\bar{g}(\boldsymbol{\theta}_{0})| \\ &=:I_1+I_2. \end{split}$$ We only need to show $I_1\xrightarrow{p}0$ and $I_2\xrightarrow{p}0$. For $I_1$, by Taylor expansion, $\bar{g}(\widetilde{\boldsymbol{\theta}})=\bar{g}(\boldsymbol{\theta}_{0})+\nabla_{\boldsymbol{\theta}} \bar{g}(\dot{\boldsymbol{\theta}})\cdot(\widetilde{\boldsymbol{\theta}}-\boldsymbol{\theta}_{0})$. Then, $$I_1\leq (2r)^{-1/2}|2n(\widetilde{\boldsymbol{\theta}}-\boldsymbol{\theta}_{0})^{\prime }\{\nabla_{\boldsymbol{\theta}} \bar{g}(\dot{\boldsymbol{\theta}})\}^{\prime }\widehat{V}_n^{-1}\bar{g}(\boldsymbol{\theta}_{0})|+(2r)^{-1/2}|n(\widetilde{\boldsymbol{\theta}}-\boldsymbol{\theta}_{0})^{\prime }\{\nabla_{\boldsymbol{\theta}} \bar{g}(\dot{\boldsymbol{\theta}})\}^{\prime }\widehat{V}_n^{-1}\{\nabla_{\boldsymbol{\theta}}\bar{g}(\dot{\boldsymbol{\theta}})\}(\widetilde{\boldsymbol{\theta}}-\boldsymbol{\theta}_{0})|.$$ As the eigenvalues of $V_n$ are uniformly bounded away from zero and infinity, and $\|\widehat{V}_n-V_n\|_2=o_p(r^{-1/2})$, then the eigenvalues of $\widehat{V}_n$ are uniformly bounded away from zero and infinity w.p.a.1. Hence, $$\begin{split} &~\|\{\nabla_{\boldsymbol{\theta}} \bar{g}(\dot{\boldsymbol{\theta}})\}^{\prime }\widehat{V}_n^{-1}-[{E}\{\nabla_{\boldsymbol{\theta}} g(\boldsymbol{\theta}_{0})\}]^{\prime }V_n^{-1}\|_2 \\ \leq&~\|(\{\nabla_{\boldsymbol{\theta}}\bar{g}(\dot{\boldsymbol{\theta}})\}^{\prime }-[{E}\{\nabla_{\boldsymbol{\theta}} g(\boldsymbol{\theta}_{0})\}])\widehat{V}_n^{-1}\|_2+\|[{E}\{\nabla_{\boldsymbol{\theta}} g(\boldsymbol{\theta}_{0})\}]^{\prime }(\widehat{V}_n^{-1}-V_n^{-1})\|_2 \\ =&~o_p(p^{-1/2})+o_p(r^{-1/2})=o_p(p^{-1/2}). \end{split}$$ On the other hand, $$\begin{split} {E}(\|[{E}\{\nabla_{\boldsymbol{\theta}} g(\boldsymbol{\theta}_{0})\}]^{\prime }V_n^{-1}\bar{g}(\boldsymbol{\theta}_{0})\|_2^2)=&~{E}\{\text{tr}(V_n^{-1}[{E}\{\nabla_{\boldsymbol{\theta}} g(\boldsymbol{\theta}_{0})\}][{E}\{\nabla_{\boldsymbol{\theta}} g(\boldsymbol{\theta}_{0})\}]^{\prime }V_n^{-1}\bar{g}(\boldsymbol{\theta}_{0})\bar{g}(\boldsymbol{\theta}_{0}))^{\prime }\} \\ =&~n^{-1}\text{tr}([{E}\{\nabla_{\boldsymbol{\theta}} g(\boldsymbol{\theta}_{0})\}]^{\prime }V_n^{-1}[{E}\{\nabla_{\boldsymbol{\theta}} g(\boldsymbol{\theta}_{0})\}]) \\ \leq&~Cpn^{-1}. \end{split}$$ Then, $$\|\{\nabla_{\boldsymbol{\theta}}\bar{g}(\dot{\boldsymbol{\theta}})\}^{\prime }\widehat{V}_n^{-1}\bar{g}(\boldsymbol{\theta}_{0})\|_2=O_p(p^{1/2}n^{-1/2})+o_p(r^{1/2}p^{-1/2}n^{-1/2}).$$ Therefore, $$I_1\leq O_p(pr^{-1/2})+o_p(1)\xrightarrow{p}0$$ provided that $p=o(r^{1/2})$. For $I_2$, $$I_2=(2r)^{-1/2}|n\bar{g}(\boldsymbol{\theta}_{0})^{\prime }(\widehat{V}_n^{-1}-V_n^{-1})\bar{g}(\boldsymbol{\theta}_{0})|=O(r^{-1/2}n)o_p(r^{-1/2})O_p(rn^{-1})=o_p(1).$$ Hence, we complete the proof of this lemma. $\hfill \square$ **Remark**: This lemma is similar to the Lemma 6.1 of Donald, Imbens and Newey (2003). However, we work on the operator-norm in establishing the consistency results, whereas Donald, Imbens and Newey (2003) employed the Frobenius-norm. The matrix $\widehat{V}_n$ and $\widetilde{\boldsymbol{\theta}}$ are the consistency estimators of $V_n$ and $\boldsymbol{\theta}_{0}$ respectively. Here, we begin to establish Theorem 4. From Proposition 1, we know $\|\widehat{\boldsymbol{\theta}}_{n}-\boldsymbol{\theta}_{0}\|_2=O_p(p^{1/2}n^{-1/2})$. By the same argument of the proof of Theorem 3, we have $$w_n(\widehat{\boldsymbol{\theta}}_n)=Q\bar{\phi}(\widehat{\boldsymbol{\theta}}_n)^{\prime }\widehat{\Omega}^{-1}(\widehat{\boldsymbol{\theta}}_n)\bar{\phi}(\widehat{\boldsymbol{\theta}}_n)+O_p(r^2M^{1-1/\gamma}n^{1/\gamma-1/2}).$$ Note Lemma \[la2\], $$\begin{split} w_n(\widehat{\boldsymbol{\theta}}_{n})=&~n\bar{g}(\widehat{\boldsymbol{\theta}}_{n})^{\prime }\{M\widehat{\Omega}(\widehat{\boldsymbol{\theta}}_{n})\}^{-1}\bar{g}(\widehat{\boldsymbol{\theta}}_{n})+O_p(r^{2}M^{1-1/\gamma}n^{1/\gamma-1/2})+O_p(rMn^{-1/2})+O_p(rM^2n^{-1}). \end{split}$$ By Lemmas \[la3\] and \[la7\], it yields that $$\begin{split} \big\|M\widehat{\Omega}(\widehat{\boldsymbol{\theta}}_{n})-V_n\big\|_2&=O_p(r^{1/2}pM^{1/2}n^{-1/2}+rM^{3/2}n^{-1/2})+O_p\bigg\{rM^{-1}\sum_{k=1}^Mk\alpha_X(k)^{1-2/\gamma}\bigg\} \\ &=o_p(r^{-1/2}). \end{split}$$ Noting Lemma \[la11\], for any $\dot{\boldsymbol{\theta}}$ such that $\|\dot{\boldsymbol{\theta}}-\boldsymbol{\theta}_{0}\|_2\leq\|\widehat{\boldsymbol{\theta}}_{n}-\boldsymbol{\theta}_{0}\|_2=O_p(p^{1/2}n^{-1/2})$, $$\|\nabla_{\boldsymbol{\theta}} \bar{g}(\dot{\boldsymbol{\theta}})-{E}\{\nabla_{\boldsymbol{\theta}} g_t(\boldsymbol{\theta}_{0})\}\|_F=O_p(r^{1/2}p^{3/2}n^{-1/2})=o_p(p^{-1/2}).$$ By Lemma \[la19\], we can get $n\bar{g}(\widehat{\boldsymbol{\theta}}_{n})^{\prime }\{M\widehat{\Omega}(\widehat{\boldsymbol{\theta}}_{n})\}^{-1}\bar{g}(\widehat{\boldsymbol{\theta}}_{n})-n\bar{g}(\boldsymbol{\theta}_{0})^{\prime }V_n^{-1}\bar{g}(\boldsymbol{\theta}_{0})=o_p(r^{1/2})$. Then, by Lemma \[la18\], we complete the proof of Theorem 4. $\hfill \square$ Proof of Theorem 5 {#proof-of-theorem-5 .unnumbered} ------------------ We only need to prove that for some $c>1$, $P\{w_n(\widehat{\boldsymbol{\theta}}_n)> cr\}\rightarrow1$. To prove this, we use the technique for the proof of Theorem 1 in [@ChangTangWu_2013]. Let $$\widetilde{\lambda}=\frac{-\rho_v(0)}{2\rho_{vv}(0)Q^\omega}\frac{e}{\max_{1\leq q\leq Q}\|\phi_q(\widehat{\boldsymbol{\theta}}_n)\|_2}$$ where $e$ is a $r$-dimensional vector with unit $L_2$-norm, and $\omega>0$ will be determined later. Then, $\widetilde{\lambda}\in\widehat{\Lambda}_n(\widehat{\boldsymbol{\theta}}_n)$ when $Q$ is sufficiently large. Note that $\rho_{vv}(0)<0$, by Taylor expansion, we have $$\begin{split} w_n(\widehat{\boldsymbol{\theta}}_n)=&~\frac{2\rho_{vv}(0)}{\rho_v^2(0)}\bigg\{Q\rho(0)-\max_{\lambda\in\widehat{\Lambda}_n(\widehat{\boldsymbol{\theta}}_n)}\sum_{q=1}^Q\rho(\lambda^{\prime }\phi_q(\widehat{\boldsymbol{\theta}}_n))\bigg\} \\ \geq&~\frac{1}{Q^\omega}\sum_{q=1}^Q\frac{e^{\prime }\phi_q(\widehat{\boldsymbol{\theta}}_n)}{\max_{1\leq q\leq Q}\|\phi_q(\widehat{\boldsymbol{\theta}}_n)\|_2} \\ &~~~~~~~~~~~~~~~~~~~~~-\frac{1}{4Q^{2\omega}\rho_{vv}(0)}\sum_{q=1}^Q\frac{\rho_{vv}(\dot{\lambda}\phi_q(\widehat{\boldsymbol{\theta}}_n))e^{\prime }\phi_q(\widehat{\boldsymbol{\theta}}_n)\phi_q(\widehat{\boldsymbol{\theta}}_n)^{\prime }e}{\max_{1\leq q\leq Q}\|\phi_q(\widehat{\boldsymbol{\theta}}_n)\|_2^2} \end{split}$$ where $\dot{\lambda}$ lies on the jointing line between $\widetilde{\lambda}$ and $\boldsymbol{\mathbf{0}}$. By the definition of $\widetilde{\lambda}$, we have $$\frac{1}{4Q^{2\omega}\rho_{vv}(0)}\sum_{q=1}^Q\frac{\rho_{vv}(\dot{\lambda}\phi_q(\widehat{\boldsymbol{\theta}}_n))e^{\prime }\phi_q(\widehat{\boldsymbol{\theta}}_n)\phi_q(\widehat{\boldsymbol{\theta}}_n)^{\prime }e}{\max_{1\leq q\leq Q}\|\phi_q(\widehat{\boldsymbol{\theta}}_n)\|_2^2}\leq \frac{1}{2}Q^{1-2\omega}~~\text{w.p.a. 1}.$$ Hence, for any $c>1$, $$P\{w_n(\widehat{\boldsymbol{\theta}}_n)\leq cr\}\leq P\bigg\{\sum_{q=1}^Q\frac{e^{\prime }\phi_q(\widehat{\boldsymbol{\theta}}_n)}{\max_{1\leq q\leq Q}\|\phi_q(\widehat{\boldsymbol{\theta}}_n)\|_2}\leq crQ^{\omega}+\frac{1}{2}Q^{1-\omega}\bigg\}+o(1).$$ From (A.2)(ii), we have $\|\phi_q(\widehat{\boldsymbol{\theta}}_n)\|_2\leq r^{1/2}M^{-1}\sum_{t\in B_q}B_n(X_t)$. Then, by Markov inequality, $$P\bigg\{\max_{1\leq q\leq Q}\|\phi_q(\widehat{\boldsymbol{\theta}}_n)\|_2>(cK)^{-1}r^{1/2}Q^{1/\gamma}(\log Q)^{\epsilon/2}\bigg\}\rightarrow0$$ for each fixed $K>0$, which implies that $$P\{w_n(\widehat{\boldsymbol{\theta}}_n)\leq cr\}\leq P\bigg\{\sum_{q=1}^Qe^{\prime }\phi_q(\widehat{\boldsymbol{\theta}}_n)\leq K^{-1}(rQ^{\omega}+Q^{1-\omega})r^{1/2}Q^{1/\gamma}(\log Q)^{\epsilon/2}\bigg\}+o(1).$$ Let $rQ^{\omega}=Q^{1-\omega}$, i.e., $Q^{\omega}=Q^{1/2}r^{-1/2}$, then $$P\{w_n(\widehat{\boldsymbol{\theta}}_n)\leq cr\}\leq P\bigg\{\sum_{q=1}^Qe^{\prime }\phi_q(\widehat{\boldsymbol{\theta}}_n)\leq 2K^{-1}rQ^{1/\gamma+1/2}(\log Q)^{\epsilon/2}\bigg\}+o(1).$$ On the other hand, by Lemma \[la2\] and (A.1)(iv), $$\sum_{q=1}^Qe^{\prime }\phi_q(\widehat{\boldsymbol{\theta}}_n)=Qe^{\prime }\bar{g}(\widehat{\boldsymbol{\theta}}_n)+O_p(r^{1/2})=Qe^{\prime }E\{g_t(\widehat{\boldsymbol{\theta}}_n)\}+O_p(r^{1/2})+o_p\{Q\Delta_1(r,p)\}.$$ Select $e=E\{g_t(\widehat{\boldsymbol{\theta}}_n)\}/\|E\{g_t(\widehat{\boldsymbol{\theta}}_n)\}\|_2$. Then, $$\begin{split} &~P\{w_n(\widehat{\boldsymbol{\theta}}_n)\leq cr\} \\ \leq&~ P\big[\|E\{g_t(\widehat{\boldsymbol{\theta}}_n)\}\|_2\leq 2K^{-1}rQ^{1/\gamma-1/2}(\log Q)^{\epsilon/2}+O_p(r^{1/2}Q^{-1})+o_p\{\Delta_1(r,p)\}\big]+o(1) \\ \leq&~P\big[\varsigma\leq 2K^{-1}rQ^{1/\gamma-1/2}(\log Q)^{\epsilon/2}+O_p(r^{1/2}Q^{-1})+o_p\{\Delta_1(r,p)\}\big]+o(1). \\ \end{split}$$ As $r^2M^{1-2/\gamma}n^{2/\gamma-1}(\log n)^{\epsilon}\varsigma^{-2}=O(1)$, $r^{1/2}Mn^{-1}\varsigma^{-1}=o(1)$ and $\Delta_1(r,p)\varsigma^{-1}=O(1)$, we can choose sufficiently large $K$ to guarantee $$P\big[\varsigma\leq 2K^{-1}rQ^{1/\gamma-1/2}(\log Q)^{\epsilon/2}+O_p(r^{1/2}Q^{-1})+o_p\{\Delta_1(r,p)\}\big]\rightarrow0,$$ which leads to $P\{w_n(\widehat{\boldsymbol{\theta}}_n)\leq cr\}\rightarrow0$ for any $c>1$. Hence, we complete the proof.$\hfill \square$ Proof of Theorem \[tm:pen\] {#proof-of-theorem-tmpen .unnumbered} =========================== Let $$\widehat{S}_n^{(\mathrm{pe})}(\boldsymbol{\theta},\lambda)=\frac{1}{Q}\sum_{q=1}^Q\rho(\lambda^{\prime }\phi_q(\boldsymbol{\theta}))+\sum_{j=1}^pp_{\tau}(|\theta_j|)~~\mbox{for any ${\boldsymbol{\theta}}\in\Theta$ and $\lambda\in\widehat{\Lambda}_n({\boldsymbol{\theta}})$}.$$ Then, $$\widehat{\boldsymbol{\theta}}_n^{(\mathrm{pe})}=\arg\min_{\boldsymbol{\theta}\in\Theta}\max_{\lambda\in\widehat{\Lambda}_n(\boldsymbol{\theta})}\widehat{S}_n^{(\mathrm{pe})}(\boldsymbol{\theta},\lambda)~~\text{and}~~\widehat{\boldsymbol{\theta}}_n=\arg\min_{\boldsymbol{\theta}\in\Theta}\max_{\lambda\in\widehat{\Lambda}_n(\boldsymbol{\theta})}\widehat{S}_n(\boldsymbol{\theta},\lambda).$$ The following lemma will be used to construct Theorem \[tm:pen\]. Under conditions , and , assume that the eigenvalues of $V_M$ are uniformly bounded away from zero and infinity. If holds, $r^2pM^2n^{-1}=o(1)$ and $s\tau r^{-1}M^{-1} n=O(1)$, then $\|\widehat{\boldsymbol{\theta}}_n^{(\mathrm{pe})}-\boldsymbol{\theta}_0\|_2=O_p(r^{1/2}n^{-1/2})$. **Proof**: Choose $\delta_n=o(r^{-1/2}Q^{-1/\gamma})$ and $r^{1/2}Mn^{-1/2}=o(\delta_n)$. Let $\bar{\lambda}=\text{sign}\{\rho_v(0)\}\delta_n\bar{\phi}(\widehat{\boldsymbol{\theta}}_n^{(\mathrm{pe})})/\|\bar{\phi}(\widehat{\boldsymbol{\theta}}_n^{(\mathrm{pe})})\|_2$, then $\bar{\lambda}\in\Lambda_n$ where $\Lambda_n$ is defined in Lemma \[la5\]. By Taylor expansion, Lemmas \[la4\] and \[la5\], noting $\rho_{vv}(0)<0$, we have $$\begin{split} \widehat{S}_n(\widehat{\boldsymbol{\theta}}_n^{(\mathrm{pe})},\bar{\lambda})=&~\rho(0)+\rho_v(0)\bar{\lambda}^{\prime }\bar{\phi}(\widehat{\boldsymbol{\theta}}_n^{(\mathrm{pe})})+\frac{1}{2}\bar{\lambda}^{\prime }\bigg\{\frac{1}{Q}\sum_{q=1}^Q\rho_{vv}(\dot{\lambda}^{\prime }\phi_q(\widehat{\boldsymbol{\theta}}_n^{(\mathrm{pe})}))\phi_q(\widehat{\boldsymbol{\theta}}_n^{(\mathrm{pe})})\phi_q(\widehat{\boldsymbol{\theta}}_n^{(\mathrm{pe})})^{\prime }\bigg\}\bar{\lambda} \\ \geq&~ \rho(0)+|\rho_v(0)|\delta_n\|\bar{\phi}(\widehat{\boldsymbol{\theta}}_n^{(\mathrm{pe})})\|_2-C\|\bar{\lambda}\|_2^2\cdot O_p(1). \end{split}$$ On the other hand, $$\widehat{S}_n^{(\mathrm{pe})}(\widehat{\boldsymbol{\theta}}_n^{(\mathrm{pe})},\bar{\lambda})\leq \sup_{\lambda\in\widehat{\Lambda}_n(\widehat{\boldsymbol{\theta}}_n^{(\mathrm{pe})})}\widehat{S}_n^{(\mathrm{pe})}(\widehat{\boldsymbol{\theta}}_n^{(\mathrm{pe})},\lambda)\leq\sup_{\lambda\in\widehat{\Lambda}_n(\boldsymbol{\theta}_0)}\widehat{S}_n^{(\mathrm{pe})}(\boldsymbol{\theta}_0,\lambda).$$ By Lemma 7 and (A.5), as $sr^{-1}\tau n=O(1)$, $$\begin{split} \sup_{\lambda\in\widehat{\Lambda}_n(\boldsymbol{\theta}_0)}\widehat{S}_n^{(\mathrm{pe})}(\boldsymbol{\theta}_0,\lambda)=&~\sup_{\lambda\in\widehat{\Lambda}_n(\boldsymbol{\theta}_0)}\widehat{S}_n(\boldsymbol{\theta}_0,\lambda)+\sum_{j=1}^pp_{\tau}(|\theta_{0j}|) \\ =&~\rho(0)+O_p(rMn^{-1}+s\tau)=\rho(0)+O_p(rMn^{-1}). \end{split}$$ Note that $\widehat{S}_n^{(\mathrm{pe})}(\boldsymbol{\theta},\lambda)\geq\widehat{S}_n(\boldsymbol{\theta},\lambda)$ for any $\boldsymbol{\theta}\in\Theta$ and $\lambda\in\widehat{\Lambda}_n(\boldsymbol{\theta})$, it yields $\|\bar{\phi}(\widehat{\boldsymbol{\theta}}_n^{(\mathrm{pe})})\|_2=O_p(\delta_n)$. Consider any $\varepsilon_n\rightarrow0$ and let $\widetilde{\lambda}=\text{sign}\{\rho_v(0)\}\varepsilon_n\bar{\phi}(\widehat{\boldsymbol{\theta}}_n^{(\mathrm{pe})})$, then $\|\widetilde{\lambda}\|_2=o_p(\delta_n)$. Using the same way above, we can obtain $$|\rho_v(0)|\cdot\varepsilon_n\|\bar{\phi}(\widehat{\boldsymbol{\theta}}_n^{(\mathrm{pe})})\|_2^2-O_p(1)\cdot\varepsilon_n^2\|\bar{\phi}(\widehat{\boldsymbol{\theta}}_n^{(\mathrm{pe})})\|_2^2=O_p(rMn^{-1}).$$ Then, $\varepsilon_n\|\bar{\phi}(\widehat{\boldsymbol{\theta}}_n^{(\mathrm{pe})})\|_2^2=O_p(rMn^{-1})$. Thus, $\|\bar{\phi}(\widehat{\boldsymbol{\theta}}_n^{(\mathrm{pe})})\|_2=O_p(r^{1/2}M^{1/2}n^{-1/2})$. Following the same arguments given in the proof of Theorem 1, we can obtain $\|\widehat{\boldsymbol{\theta}}_n^{(\mathrm{pe})}-\boldsymbol{\theta}_0\|_2=O_p(r^{1/2}n^{-1/2})$. $\hfill\Box$ Here, we begin to prove Theorem \[tm:pen\]. $\widehat{\boldsymbol{\theta}}_n^{(\mathrm{pe})}$ and its Lagrange multiplier $\widehat{\lambda}^{(\mathrm{pe})}$ satisfy the score equation $$\boldsymbol{\mathbf{0}}=\nabla_\lambda \widehat{S}_n^{(\mathrm{pe})}(\widehat{\boldsymbol{\theta}}_n^{(\mathrm{pe})},\widehat{\lambda}^{(\mathrm{pe})})=\nabla_\lambda \widehat{S}_n(\widehat{\boldsymbol{\theta}}_n^{(\mathrm{pe})},\widehat{\lambda}^{(\mathrm{pe})}).$$ By the implicit theorem (Theorem 9.28 of Rudin, 1976), for all $\boldsymbol{\theta}$ in a $\|\cdot\|_2$-neighborhood of $\widehat{\boldsymbol{\theta}}_n^{(\mathrm{pe})}$, there is a $\widehat{\lambda}(\theta)$ such that $\nabla_\lambda \widehat{S}_n^{(\mathrm{pe})}(\boldsymbol{\theta},\widehat{\lambda}(\boldsymbol{\theta}))=\boldsymbol{\mathbf{0}}$ and $\widehat{\lambda}(\boldsymbol{\theta})$ is continuously differentiable in $\boldsymbol{\theta}$. By the concavity of $\widehat{S}_n^{(\mathrm{pe})}(\boldsymbol{\theta},\lambda)$ with respect to $\lambda$, $\widehat{S}_n^{(\mathrm{pe})}(\boldsymbol{\theta},\widehat{\lambda}(\boldsymbol{\theta}))=\max_{\lambda\in\widehat{\Lambda}_n(\boldsymbol{\theta})}\widehat{S}_n(\boldsymbol{\theta},\lambda)$. From the envelope theorem, $$\label{eq:enve} \begin{split} \boldsymbol{\mathbf{0}}=&~\nabla_{\boldsymbol{\theta}}\widehat{S}_n^{(\mathrm{pe})}(\boldsymbol{\theta},\widehat{\lambda}(\boldsymbol{\theta}))\big|_{\boldsymbol{\theta}=\widehat{\boldsymbol{\theta}}_n^{(\mathrm{pe})}} \\ =&~\frac{1}{Q}\sum_{q=1}^Q\rho_v(\widehat{\lambda}(\widehat{\boldsymbol{\theta}}_n^{(\mathrm{pe})})^{\prime }\phi_q(\widehat{\boldsymbol{\theta}}_n^{(\mathrm{pe})}))\big\{\nabla_{\boldsymbol{\theta}}\phi_q(\widehat{\boldsymbol{\theta}}_n^{(\mathrm{pe})})\big\}^{\prime }\widehat{\lambda}(\widehat{\boldsymbol{\theta}}_n^{(\mathrm{pe})})+\sum_{q=1}^Q\nabla_{\boldsymbol{\theta}}p_{\tau}(|\theta_j|)\big|_{\boldsymbol{\theta}=\widehat{\boldsymbol{\theta}}_n^{(\mathrm{pe})}}. \\ \end{split}$$ For any $\boldsymbol{\theta}$ such that $\|\boldsymbol{\theta}-\boldsymbol{\theta}_0\|_2=O_p(r^{1/2}n^{-1/2})$ and $\|\bar{g}(\boldsymbol{\theta})\|_2=O_p(r^{1/2}n^{-1/2})$, define $$\mathbf{h}(\boldsymbol{\theta})=\frac{1}{Q}\sum_{q=1}^Q\rho_v(\widehat{\lambda}(\boldsymbol{\theta})^{\prime }\phi_q({\boldsymbol{\theta}}))\big\{\nabla_{\boldsymbol{\theta}}\phi_q({\boldsymbol{\theta}})\big\}^{\prime }\widehat{\lambda}({\boldsymbol{\theta}})+\sum_{q=1}^Q\nabla_{\boldsymbol{\theta}}p_{\tau}(|\theta_j|).$$ Write $\mathbf{h}(\boldsymbol{\theta})=(h_1(\boldsymbol{\theta}),\ldots,h_p(\boldsymbol{\theta}))^{\prime }$. From Lemma 7, it yields that $\|\widehat{\lambda}(\boldsymbol{\theta})\|_2=O_p(r^{1/2}Mn^{-1/2})$ which implies $\sup_{1\leq q\leq Q}|\widehat{\lambda}(\boldsymbol{\theta})^{\prime }\phi_q(\boldsymbol{\theta})|=o_p(1)$. For each $j=1,\ldots,p$, $$\begin{split} h_j(\boldsymbol{\theta})=&~\frac{1}{Q}\sum_{q=1}^Q\rho_v(0)\widehat{\lambda}(\boldsymbol{\theta}_0)^{\prime }\frac{\partial\phi_q(\boldsymbol{\theta}_0)}{\partial\theta_j}+\frac{1}{Q}\sum_{q=1}^Q\rho_v(0)\widehat{\lambda}(\boldsymbol{\theta}_0)^{\prime }\frac{\partial^2 \phi_q(\boldsymbol{\theta}_0)}{\partial\theta_j\partial \boldsymbol{\theta}^{\prime }}(\boldsymbol{\theta}-\boldsymbol{\theta}_0)+p_{\tau}^{\prime }(|\theta_j|)\text{sign}(\theta_j) \\ &+\text{higher order terms}. \end{split}$$ From (A.4), there exists a positive constant $C$ such that $p_{\tau}^{\prime }(|\theta_j|)\geq C\tau$. On the other hand, as $\tau (r^{-1}n)^{1/2}M^{-1}\rightarrow\infty$, $$\max_{j\notin\mathcal{A}}\bigg|\frac{1}{Q}\sum_{q=1}^Q\rho_v(0)\widehat{\lambda}(\boldsymbol{\theta}_0)^{\prime }\frac{\partial\phi_q(\boldsymbol{\theta}_0)}{\partial\theta_j}\bigg|= O_p(r^{1/2}Mn^{-1/2})=o_p(\tau).$$ Similarly, we can show $$\max_{j\notin\mathcal{A}}\bigg|\frac{1}{Q}\sum_{q=1}^Q\rho_v(0)\widehat{\lambda}(\boldsymbol{\theta}_0)^{\prime }\frac{\partial^2 \phi_q(\boldsymbol{\theta}_0)}{\partial\theta_j\partial \boldsymbol{\theta}^{\prime }}(\boldsymbol{\theta}-\boldsymbol{\theta}_0)\bigg|=o_p(\tau).$$ Hence, $p_{\tau}^{\prime }(|\theta_j|)\text{sign}(\theta_j)$ dominates the sign of $h_j(\boldsymbol{\theta})$ uniformly for all $j\notin\mathcal{A}$. If $\widehat{\boldsymbol{\theta}}_n^{(2)}\neq \boldsymbol{\mathbf{0}}$, there exists some $j\notin\mathcal{A}$ such that $\widehat{\theta}_{n,j}\neq0 $. Under our above arguments, we can find $$P\big\{h_j(\widehat{\boldsymbol{\theta}}_n^{(\mathrm{pe})})\neq 0\big\}\rightarrow1.$$ It is a contradiction. Hence, $\widehat{\boldsymbol{\theta}}_n^{(2)}=\boldsymbol{\mathbf{0}}$. Nextly, we consider the second result. From (\[eq:block\]), it yields $$\begin{split} &[E\{\nabla_{\boldsymbol{\theta}}g_t(\boldsymbol{\theta}_0)\}]^{\prime }V_M^{-1}[E\{\nabla_{\boldsymbol{\theta}}g_t(\boldsymbol{\theta}_0)\}]([E\{\nabla_{\boldsymbol{\theta}}g_t(\boldsymbol{\theta}_0)\}]^{\prime }V_M^{-1}V_nV_M^{-1}[E\{\nabla_{\boldsymbol{\theta}}g_t(\boldsymbol{\theta}_0)\}])^{-1} \\ &~~~~~~~~~~~~~~~~\times[E\{\nabla_{\boldsymbol{\theta}}g_t(\boldsymbol{\theta}_0)\}]^{\prime }V_M^{-1}[E\{\nabla_{\boldsymbol{\theta}}g_t(\boldsymbol{\theta}_0)\}]=\left( \begin{array}{cc} ({\mathbf{S}}_{11}-{\mathbf{S}}_{12}{\mathbf{S}}_{22}^{-1}{\mathbf{S}}_{21})^{-1} & \ast \\ \ast & \ast \\ & \end{array} \right). \end{split}$$ Let $$[E\{\nabla_{\boldsymbol{\theta}}g_t(\boldsymbol{\theta}_0)\}]^{\prime }V_M^{-1}[E\{\nabla_{\boldsymbol{\theta}}g_t(\boldsymbol{\theta}_0)\}]([E\{\nabla_{\boldsymbol{\theta}}g_t(\boldsymbol{\theta}_0)\}]^{\prime }V_M^{-1}V_nV_M^{-1}[E\{\nabla_{\boldsymbol{\theta}}g_t(\boldsymbol{\theta}_0)\}])^{-1/2}=\left( \begin{array}{cc} {\mathbf{U}} & {\mathbf{V}} \\ {\mathbf{V}}^{\prime } & \ast \\ & \end{array} \right),$$ where ${\mathbf{U}}$ is a $s\times s$ symmetric matrix, then ${\mathbf{U}}{\mathbf{U}}^{\prime }+{\mathbf{V}}{\mathbf{V}}^{\prime }=({\mathbf{S}}_{11}-{\mathbf{S}}_{12}{\mathbf{S}}_{22}^{-1}{\mathbf{S}}_{21})^{-1}$. For any $\boldsymbol{\alpha}_n\in\mathbb{R}^s$ such that $\|\boldsymbol{\alpha}_n\|_2=1$, define $$\widetilde{\boldsymbol{\alpha}}_n=\left( \begin{array}{c} {\mathbf{U}}^{\prime } \\ {\mathbf{V}}^{\prime } \\ \end{array} \right)({\mathbf{S}}_{11}-{\mathbf{S}}_{12}{\mathbf{S}}_{22}^{-1}{\mathbf{S}}_{21})^{1/2}\boldsymbol{\alpha}_n.$$ Then, $$\widetilde{\boldsymbol{\alpha}}_n^{\prime }\widetilde{\boldsymbol{\alpha}}_n=\boldsymbol{\alpha}_n^{\prime }({\mathbf{S}}_{11}-{\mathbf{S}}_{12}{\mathbf{S}}_{22}^{-1}{\mathbf{S}}_{21})^{1/2}({\mathbf{U}}{\mathbf{U}}^{\prime }+{\mathbf{V}}{\mathbf{V}}^{\prime })({\mathbf{S}}_{11}-{\mathbf{S}}_{12}{\mathbf{S}}_{22}^{-1}{\mathbf{S}}_{21})^{1/2}\boldsymbol{\alpha}_n=1.$$ Following the same argument for Proposition 2, we know it still holds for $\widehat{\boldsymbol{\theta}}_n^{(\mathrm{pe})}$. Note that $$\begin{split} &\widetilde{\boldsymbol{\alpha}}_n^{\prime }([E\{\nabla_{\boldsymbol{\theta}}g_t(\boldsymbol{\theta}_0)\}]^{\prime }V_M^{-1}V_nV_M^{-1}[E\{\nabla_{\boldsymbol{\theta}}g_t(\boldsymbol{\theta}_0)\}])^{-1/2}[E\{\nabla_{\boldsymbol{\theta}}g_t(\boldsymbol{\theta}_0)\}]^{\prime }V_M^{-1}[E\{\nabla_{\boldsymbol{\theta}}g_t(\boldsymbol{\theta}_0)\}](\widehat{\boldsymbol{\theta}}_n^{(\mathrm{pe})}-\boldsymbol{\theta}_0) \\ &~~~~~~~~~~~~~~~~=\boldsymbol{\alpha}_n^{\prime }({\mathbf{S}}_{11}-{\mathbf{S}}_{12}{\mathbf{S}}_{22}^{-1}{\mathbf{S}}_{21})^{-1/2}(\widehat{\boldsymbol{\theta}}_n^{(1)}-\boldsymbol{\theta}_0^{(1)}), \end{split}$$ then we establish the second result following Proposition 2. $\hfill\Box$ [Hjort, McKeague and Van Keilegom(2009)]{} Ai, C. and Chen, X. (2003). Efficient estimation of models with conditional moment restrictions containing unknown functions, *Econometrica*, **71**, 1795–1843. Anatolyev, S. (2005). 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Tel.: +86 10 62760736; fax: +86 10 62760736. E-mail address: csx@gsm.pku.edu.cn [^3]: Department of Economics, Yale University, New Haven, CT, 06520, U.S.A.

--- abstract: 'We give a characterization, in terms of the residue field, of those henselian valuation rings and those henselian valuation ideals that are diophantine. This characterization gives a common generalization of all the positive and negative results on diophantine henselian valuation rings and diophantine valuation ideals in the literature. We also treat questions of uniformity and we apply the results to show that a given field can carry at most one diophantine nontrivial equicharacteristic henselian valuation ring or valuation ideal.' address: - 'Jeremiah Horrocks Institute, University of Central Lancashire, Preston PR1 2HE, United Kingdom' - 'Fachbereich Mathematik und Statistik, University of Konstanz, 78457 Konstanz, Germany' author: - Sylvy Anscombe and Arno Fehm title: | Characterizing diophantine henselian\ valuation rings and valuation ideals --- [^1] Introduction ============ We study diophantine subsets of a field $K$, that is, sets $X\subseteq K$ that are the projection of the common zero set $Z\subseteq K^n$ of finitely many polynomials $$f_1,\dots,f_r\in\mathbb{Z}[x_1,\dots,x_n]$$ onto the first coordinate. Diophantine subsets of fields have been studied for a long time in various areas, like number theory, arithmetic geometry and mathematical logic, and a wide variety of methods has been developed. (From a model theoretic point of view, a subset of $K$ is diophantine if it is definable by an existential parameter-free formula in the language $\mathcal{L}_{\rm ring}$ of rings.) See for example [@Denef; @Shlapentokh; @Kollar; @Koenigsmann] and the references therein for problems and results on diophantine subsets of number fields, function fields, and certain infinite algebraic extensions of $\mathbb{Q}$. The subsets $X\subseteq K$ we are mainly concerned with are the valuation ring $X=\mathcal{O}_v$ and the valuation ideal $X=\mathfrak{m}_v$ of some henselian valuation $v$ on $K$. As an example, already Julia Robinson observed in [@Robinson] that the valuation ring $\mathbb{Z}_p$ inside the field of $p$-adic numbers $\mathbb{Q}_p$ is diophantine: For $p>2$, it is the projection of the zero set $$Z=\left\{ (x,y)\in\mathbb{Q}_p^2 : 1+px^2-y^2=0 \right\}$$ onto the first coordinate. The analogous result for finite extensions of $\mathbb{Q}_p$ requires more care and was worked out much later in [@Cluckersetal]. Similarly, an analogous definition for the local fields $\mathbb{F}_p((t))$ is more complicated and was obtained only recently in [@AnscombeKoenigsmann]. In [@Fehm] it was observed that this last result does not make use of the fact that the value group of the valuation is discrete but can be extended to arbitrary henselian valuations with residue field $\mathbb{F}_p$. Moreover, the methods were extended to certain henselian valued fields with residue field pseudo-algebraically closed (PAC) or pseudo-real closed (PRC). Our first result is that the observation in [@Fehm] reflects in fact a general principle: The question of whether a henselian valuation ring is diophantine never depends on its value group, but only on its residue field (see below for the precise statement). Moreover, we are able to isolate a condition that characterizes those residue fields for which the valuation ring, respectively the valuation ideal, of the henselian valuation is diophantine: \[thm:introthm\] Let $F$ be a field. Then the following are equivalent: 1. There is an $\exists$-$\mathcal{L}_{\mathrm{ring}}$-formula that defines $\mathcal{O}_v$ \[respectively, $\mathfrak{m}_v$\] in $K$ for [*some*]{} equicharacteristic henselian nontrivially valued field $(K,v)$ with residue field $F$. 2. There is an $\exists$-$\mathcal{L}_{\mathrm{ring}}$-formula that defines $\mathcal{O}_v$ \[respectively, $\mathfrak{m}_v$\] in $K$ for [*every*]{} henselian valued field $(K,v)$ with residue field elementarily equivalent to $F$. 3. There is no elementary extension $F\preceq F^*$ with a nontrivial valuation $v$ on $F^*$ for which the residue field $F^*v$ embeds into $F^*$ \[respectively, with a nontrivial henselian valuation $v$ on a subfield $E$ of $F^*$ with $Ev\cong F^*$\]. In fact, we get further interesting equivalent statements and also treat definitions with parameters from a subfield of $F$, as well as questions of uniform definitions for classes of fields $F$. See Section \[section:main.theorem\] for a summary of the main results, including a proof of \[thm:introthm\]. Using \[thm:introthm\], we can easily reprove all the definability results mentioned above (namely where the residue field is finite, PAC or PRC), get new definability results (for example in the case where the residue field is P$p$C or equals $\mathbb{Q}$), and also acquire several new negative results (\[cor:positive.applications\], \[cor:negative.applications\], \[cor:positive.large\], \[cor:negative.applications.large\]). Using the more general results, we for example gain new insight into the question for which sets of prime numbers $P$ there are uniform $\exists$-$\emptyset$-definitions for the valuation rings in the families $\{\mathbb{Q}_p:p\in P\}$ and $\{\mathbb{F}_p((t)):p\in P\}$ (\[cor:QpZp\]). We finally combine \[thm:introthm\] and the negative results to conclude that a given field admits at most one nontrivial equicharacteristic henselian valuation with diophantine valuation ring or diophantine valuation ideal (\[thm:classification\]). The paper is organized as follows: After some preliminaries in Section \[sec:prelim\], we start in Section \[sec:embedded\] with the proof of $(2)\Leftrightarrow(3)$, which is based on a simple non-constructive but powerful characterization result of Prestel. The proof of $(1)\Leftrightarrow(2)$, which we present in Section \[sec:uniform\], is more technical and builds on our work [@AnscombeFehm] on the existential theory of equicharacteristic henselian valued fields. After putting everything together in Section \[section:main.theorem\], we give the above mentioned applications and a few further corollaries in Section \[section:examples\]. Notation and preliminaries {#sec:prelim} ========================== Valued fields ------------- Let $K$ be a field and $v:K\rightarrow\Gamma\cup\{\infty\}$ a valuation on $K$. We denote by $vK=v(K^\times)$ the value group of $v$, by $\mathcal{O}_v=\{x\in K:v(x)\geq0\}$ the valuation ring of $v$, by $\mathfrak{m}_v=\{x\in K:v(x)>0\}$ the valuation ideal of $v$, and by $Kv=\mathcal{O}_v/\mathfrak{m}_v$ the residue field of $v$. For $\gamma\in vK$ and $a\in K$ we denote by $B_K(\gamma,a):=\{x\in K:v(x-a)\geq\gamma\}$ the ball of radius $\gamma$ around $a$. We denote by $K^{\mathrm{alg}}$ an algebraic closure of $K$ and by $\mathrm{Gal}(K)$ the absolute Galois group of $K$, i.e. the group of automorphisms of $K^{\mathrm{alg}}$ that fix $K$ pointwise. We will make use of the following well-known valuation theoretic facts: \[lem:complete.the.square\] Let $(K,v)$ be a valued field and let $F/Kv$ be any field extension. Then there is an extension of valued fields $(L,w)/(K,v)$ such that 1. $L/K$ is separable, 2. $Lw/Kv$ is isomorphic to the extension $F/Kv$, 3. $w$ is nontrivial, and 4. $w$ is henselian. Embed $vK$ into a nontrivial ordered abelian group $\Gamma$. By Theorem 2.14 from [@Kuhlmann04a], there is an extension $(L,w)/(K,v)$ such that $L/K$ is separable, $Lw/Kv$ is isomorphic to $F/Kv$, and $wL=\Gamma$ (thus $w$ is nontrivial). By passing to the henselisation, which is an immediate separable extension, we may also suppose that $(L,w)$ is henselian. \[rem:henselian.ext\] We will on several occasions have to construct a nontrivial henselian valued extension $(K,v)$ of a trivially valued field $F$ with $Kv=F$. One way to obtain such an extension is via , but we could as well take a concrete extension like $K=F((t))$ with $v=v_t$ the $t$-adic valuation. \[lem:extend.with.alg.res.field\] Let $(K,v)$ be a valued field and $L/K$ a field extension. Then there exists an extension of $v$ to a valuation $w$ on $L$ such that $Lw/Kv$ is algebraic. This follows for example from the version of Chevalley’s theorem in [@LangAlgebraicGeometry p. 8 Thm. 1]. \[lem:residue.of.sep.closed\] Let $F$ be separably closed and let $v$ be a nontrivial valuation on $F$. Then $Fv$ is algebraically closed. See [@EP Theorem 3.2.11]. The category of $C$-fields -------------------------- In this work, we always fix a ring $C$ and work in the category of [*$C$-fields*]{}, i.e. fields $F$ together with a fixed (structure) homomorphism $C\rightarrow F$. All fields are understood to be $C$-fields, all embeddings of fields are understood to be $C$-embeddings, i.e. to commute with the structure homomorphism, and all valuations on fields are understood to be $C$-valuations, i.e. nonnegative on (the image of) $C$. The residue field of such a valuation naturally admits the structure of a $C$-field, simply by composing the structure homomorphism with the residue map, and we will tacitly view them as such. Similarly, also any field extension of a $C$-field is naturally again a $C$-field. For a $C$-field $F$ we denote by $C_F\subseteq F$ the quotient field of the image of the structure homomorphism $C\rightarrow F$. An important example is the case $C=\mathbb{Z}$: In this case, any field $F$ can be turned into a $C$-field in a unique way, and $C_F$ is the prime field of $F$. Another important example is the case where $C$ is a field: In this case, the $C$-fields are exactly the field extensions of $C$. Model theory of valued fields ----------------------------- We now fix the languages in which we are going to work. Let $$\mathcal{L}_{\mathrm{ring}}=\{+,-,\cdot,0,1\}$$ be the language of rings and let $$\mathcal{L}_{\rm vf}=\{+^K,-^K,\cdot^K,0^K,1^K,+^k,-^k,\cdot^k,0^k,1^k,+^\Gamma,<^\Gamma,0^\Gamma,\infty^\Gamma,v,{\rm res}\}$$ be a three sorted language for valued fields with a sort $K$ for the field itself, a sort $\Gamma\cup\{\infty\}$ for the value group with infinity, and a sort $k$ for the residue field, as well as both the valuation map $v$ and the residue map ${\rm res}$, which we interpret as the constant $0^k$ map outside the valuation ring. For a ring $C$, we let $\mathcal{L}_{\mathrm{ring}}(C)$ and $\mathcal{L}_{\rm vf}(C)$ be the languages obtained by adding symbols for elements of $C$, where in the case of $\mathcal{L}_{\rm vf}(C)$, the constant symbols are added to the field sort $K$. A valued $C$-field $(K,v)$ gives rise in the usual way to an $\mathcal{L}_{\rm vf}(C)$-structure $$(K,vK\cup\{\infty\},Kv,v,\mathrm{res},c^K)_{c\in C},$$ where $vK$ is the value group, $Kv$ is the residue field, and $\mathrm{res}$ is the residue map. For notational simplicity, we will usually write $(K,v)$ to refer to the $\mathcal{L}_{\rm vf}(C)$-structure it induces. The class of $C$-fields forms an elementary class in $\mathcal{L}_{\rm ring}(C)$, and the class of valued $C$-fields forms an elementary class in $\mathcal{L}_{\rm vf}(C)$. We remind the reader that most model theoretic constructions, like ultraproducts, can be carried out in a three-sorted language just like for the usual one-sorted languages, see e.g. [@Chatzidakis Section 1.7]. In particular, for ultraproducts of valued fields one has: \[lem:residue.ultraproduct\] Let $(K_i,v_i)_{i\in I}$ be a family of valued fields and $\mathcal{F}$ an ultrafilter on $I$. Then the ultraproduct $(K,v)=\prod_{i\in I}(K_i,v_i)/\mathcal{F}$ is a valued field with residue field $Kv=\prod_{i\in I}K_iv_i/\mathcal{F}$. Definable sets -------------- Let $\mathcal{L}$ be a first order language. For an $\mathcal{L}$-formula $\phi(x)$ in one free variable $x$ and an $\mathcal{L}$-structure $K$ we denote by $\phi(K)=\{x\in K: K\models\phi(x)\}$ the set defined by $\phi$ in $K$. A subset $X\subseteq K$ is [*definable*]{} if there exists some $\phi$ with $X=\phi(K)$, in which case we also call $\phi$ a [*definition*]{} for $X$. As usual, we say that an $\mathcal{L}$-formula is an [*$\exists$-$\mathcal{L}$-formula*]{} (resp. [*$\forall$-$\mathcal{L}$-formula*]{}) if it is logically equivalent to a formula in prenex normal form with only existential (resp. universal) quantifiers. We stress that for the three-sorted language $\mathcal{L}_{\rm vf}(C)$ the existential (resp. universal) quantifiers in the above-mentioned prenex normal form may quantify over any of the sorts. We say that an $\mathcal{L}_{\rm vf}(C)$-sentence is an [*$\forall^{k}\exists$-$\mathcal{L}_{\rm vf}(C)$-sentence*]{} if it is logically equivalent to a sentence of the form $\forall\mathbf{x}\,\psi(\mathbf{x})$, where $\psi$ is an $\exists$-$\mathcal{L}_{\rm vf}(C)$-formula and the universal quantifiers range over the residue field sort. If we say that a subset $X$ of a $C$-field $K$ is [*$\exists$-$C$-definable*]{} (resp. [*$\forall$-$C$-definable*]{}), we always mean that it is definable by an $\exists$-$\mathcal{L}_{\mathrm{ring}}(C)$-formula (resp. $\forall$-$\mathcal{L}_{\mathrm{ring}}(C)$-formula); and if we say that $X$ is [*$\exists$-$\emptyset$-definable*]{} (resp. [*$\forall$-$\emptyset$-definable*]{}), we mean that it is definable by an $\exists$-$\mathcal{L}_{\mathrm{ring}}$-formula (resp. $\forall$-$\mathcal{L}_{\mathrm{ring}}$-formula). For a class $\mathcal{K}$ of valued $C$-fields, we say that the valuation ring resp. the valuation ideal is [*uniformly $\exists$-$C$-definable*]{} in $\mathcal{K}$ if there is an $\exists$-$\mathcal{L}_{\mathrm{ring}}(C)$-formula $\phi(x)$ such that $\mathcal{O}_{v}=\phi(K)$ (resp. $\mathfrak{m}_v=\phi(K)$) for each $(K,v)\in\mathcal{K}$. The term [*uniformly $\forall$-$C$-definable*]{} is used analogously. \[lem:EAOm\] Let $\mathcal{K}$ be a class of valued $C$-fields. Then the valuation ring is uniformly $\exists$-$C$-definable in $\mathcal{K}$ if and only if the valuation ideal is uniformly $\forall$-$C$-definable in $\mathcal{K}$, and the valuation ring is uniformly $\forall$-$C$-definable in $\mathcal{K}$ if and only if the valuation ideal is uniformly $\exists$-$C$-definable in $\mathcal{K}$. Let $(K,v)\in\mathcal{K}$. If $\phi(x)$ defines $\mathcal{O}_v$ (resp. $\mathfrak{m}_v$), then both $$x=0\vee\exists y\;(xy=1\wedge\neg\phi(y))$$ and $$x=0\vee\forall y\;(xy=1\rightarrow\neg\phi(y))$$ define $\mathfrak{m}_v$ (resp. $\mathcal{O}_v$). \[rem:EAOm\] Due to this observation we can either talk about $\exists$-$C$-definable valuation rings and valuation ideals (which we refer to as the [*$\mathcal{O}$-case*]{} and the [*$\mathfrak{m}$-case*]{}), or, [*equivalently*]{}, about $\exists$-$C$-definable and $\forall$-$C$-definable valuation rings (which we refer to as the [*$\exists$-case*]{} and the [*$\forall$-case*]{}). We will frequently switch between these two viewpoints. We will use the following criterion which is a special case of [@Prestel Characterization Theorem]. \[thm:Prestel\] Let $\mathcal{K}$ be an $\mathcal{L}_{\mathrm{vf}}(C)$-elementary class of valued $C$-fields. 1. The valuation ring is uniformly $\exists$-$C$-definable in $\mathcal{K}$ if and only if $$(K_{1}\subseteq K_{2}\implies\mathcal{O}_{v_{1}}\subseteq\mathcal{O}_{v_{2}}),$$ for all $(K_{1},v_{1}),(K_{2},v_{2})\in\mathcal{K}$. 2. The valuation ring is uniformly $\forall$-$C$-definable in $\mathcal{K}$ if and only if $$(K_{1}\subseteq K_{2}\implies\mathcal{O}_{v_{2}}\cap K_{1}\subseteq\mathcal{O}_{v_{1}}),$$ for all $(K_{1},v_{1}),(K_{2},v_{2})\in\mathcal{K}$. In subsequent arguments, rather than use the ‘only if’ direction of \[thm:Prestel\] we will instead use the basic fact that existential sentences ‘pass up’. We conclude this section with a probably well-known description of sets defined by quantifier-free formulas: \[Lemma:qfdefinable\] Let $(K,v)$ be a valued $C$-field and $\phi(x)$ a quantifier-free $\mathcal{L}_{\rm vf}(C)$-formula with free variable $x$ belonging to the field sort. Then the set defined by $\phi(x)$ in $K$ is of the form $\phi(K)=A\cup U$ with $A$ finite and algebraic over $C_K$, and $U$ open. Let $(K^{\mathrm{alg}},v)$ be an algebraic closure of $(K,v)$, viewed as a valued $C$-field. Since $\phi(x)$ is quantifier-free, $\phi(K)=\phi(K^{\mathrm{alg}})\cap K$. Let $\mathcal{L}_{\rm div}=\mathcal{L}_{\rm ring}\cup\{|\}$ be the expansion of $\mathcal{L}_{\mathrm{ring}}$ by the divisibility predicate $x|y\Leftrightarrow v(x)\leq v(y)$. There exists an $\mathcal{L}_{\rm div}(C)$-formula $\psi(x)$ such that $\psi(K^{\mathrm{alg}})=\phi(K^{\mathrm{alg}})$. By [@Holly95 Theorem 3.26], the set $\psi(K^{\mathrm{alg}})$ defined by $\psi(x)$ is a boolean combination of balls and singletons (in fact, it is a finite union of “Swiss cheeses” in Holly’s picturesque terminology). Moreover, the singletons are algebraic over $C_{K^{\mathrm{alg}}}=C_{K}$; this can be seen either by a straightforward analysis of $\mathcal{L}_{\mathrm{div}}(C)$-formulas in $(K^{\mathrm{alg}},v)$, or by the fact that ‘model theoretic algebraic closure’ in $(K^{\mathrm{alg}},v)$ is equal to ‘field theoretic algebraic closure’. Therefore $\phi(K^{\mathrm{alg}})=\psi(K^{\mathrm{alg}})$ is equal to $A'\cup U'$ for some open set $U'$ and a finite set $A'$ which is algebraic over $C_{K}$. Since quantifier-free formulas ‘pass down’, $\phi(K)$ is of the required form. Characterizing uniform definitions {#sec:embedded} ================================== We fix a ring $C$ and a class of $C$-fields $\mathcal{F}$. Let $$\mathcal{H}(\mathcal{F})=\left\{(K,v) \;|\; (K,v)\mbox{ henselian valued $C$-field}, Kv\in\mathcal{F} \right\}$$ denote the class of henselian valued $C$-fields with residue field in $\mathcal{F}$, and $$\mathcal{H}'(\mathcal{F})=\left\{(K,v)\in\mathcal{H}(\mathcal{F}) \;|\; v\mbox{ nontrivial}\right\}$$ the subclass of nontrivially valued fields. Moreover, let $$\mathcal{H}_e(\mathcal{F})=\{(K,v)\in\mathcal{H}(\mathcal{F})\;|\;\mathrm{char}(K)=\mathrm{char}(Kv)\}$$ be the subclass of equicharacteristic fields, and $$\mathcal{H}_0(\mathcal{F})=\{(K,v)\in\mathcal{H}(\mathcal{F})\;|\;\mathrm{char}(K)=0\}$$ the subclass of fields of characteristic zero, and define $\mathcal{H}_e'(\mathcal{F})$ and $\mathcal{H}_0'(\mathcal{F})$ accordingly. For a single $C$-field $F$, we let $\mathcal{H}(F)$ be $\mathcal{H}(\mathcal{F})$ with $\mathcal{F}$ the class of $C$-fields elementarily equivalent to $F$. Again we define $\mathcal{H}'(F)$, $\mathcal{H}_{e}(F)$, $\mathcal{H}_{0}(F)$, $\mathcal{H}_{e}'(F)$, and $\mathcal{H}_{0}'(F)$ accordingly. We have that $\mathcal{H}(\mathcal{F})=\mathcal{H}_e(\mathcal{F})\cup\mathcal{H}_0(\mathcal{F})$ and $\mathcal{H}'(\mathcal{F})=\mathcal{H}_e'(\mathcal{F})\cup\mathcal{H}_0'(\mathcal{F})$. Note that if $\mathcal{F}$ is an $\mathcal{L}_{\rm ring}(C)$-elementary class of $C$-fields, then the classes $\mathcal{H}(\mathcal{F})$, $\mathcal{H}_e(\mathcal{F})$, $\mathcal{H}_0(\mathcal{F})$, $\mathcal{H}'(\mathcal{F})$, $\mathcal{H}_e'(\mathcal{F})$ and $\mathcal{H}_0'(\mathcal{F})$ are $\mathcal{L}_{\rm vf}(C)$-elementary classes of valued $C$-fields. In [@Prestel], \[thm:Prestel\] is used to reprove [@Fehm Theorem 1.1]: there is a uniform $\exists$-$\emptyset$-definition of the valuation ring for henselian fields with residue field elementarily equivalent to a fixed finite or PAC field (not containing an algebraically closed subfield). We extend this idea to characterize those elementary classes of $C$-fields $\mathcal{F}$ for which the valuation ring and the valuation ideal are uniformly $\exists$-$C$-definable in $\mathcal{H}(\mathcal{F})$, in $\mathcal{H}_e(\mathcal{F})$, and (under certain conditions on $C$) in $\mathcal{H}_0(\mathcal{F})$. Most definitions and statements in this section can be phrased both in an “existential” and in a “universal” setting (see also \[rem:EAOm\]), and both for “arbitrary” valuations and for “equicharacteristic” valuations. The “existential” and “universal” definitions and statements show a huge formal similarity, but the proofs are sometimes different for both cases. On the other hand, the proofs for the “arbitrary” and “equicharacteristic” versions are so similar that we combine them by putting the necessary changes in the equicharacteristic case in brackets. Embedded residue and large classes of fields -------------------------------------------- \[def:embedded.residue.large\] 1. We say that $\mathcal{F}$ has *\[equicharacteristic\] embedded residue if there exist $F_{1},F_{2}\in\mathcal{F}$ and a nontrivial \[equicharacteristic\] valuation $w$ on $F_{1}$ with an embedding of $F_1w$ into $F_2$. For a single $C$-field $F$, we say $F$ has *embedded residue if the class of $C$-fields elementarily equivalent to $F$ has embedded residue.** 2. We say that $\mathcal{F}$ is *\[equicharacteristic\] large if there exist $F_{1},F_{2}\in\mathcal{F}$, a $C$-subfield $E\subseteq F_{2}$, and a nontrivial \[equicharacteristic\] henselian valuation $w$ on $E$ such that $Ew$ is isomorphic to $F_{1}$. For a single $C$-field $F$, we say $F$ is [*large[^2]*]{} if the class of $C$-fields elementarily equivalent to $F$ is large.* \[rem:unmixed\] Note that if $\mathcal{F}$ has equicharacteristic embedded residue, then $\mathcal{F}$ has embedded residue. Moreover, note that if $\mathcal{F}$ is [*unmixed*]{}, i.e.  $\mathcal{F}$ does not contain both fields of characteristic zero and of positive characteristic, then $\mathcal{F}$ has embedded residue if and only if it has equicharacteristic embedded residue. This applies in particular if $\mathcal{F}$ is a class of $C$-fields elementarily equivalent to a fixed $C$-field $F$; in other words, if $\mathcal{F}$ is an elementary class of $C$-fields and one $F\in\mathcal{F}$ has embedded residue, then $\mathcal{F}$ has equicharacteristic embedded residue. In the case that $C$ itself has positive characteristic or is a field, every class of $C$-fields is unmixed. The analogous statements hold for “large” instead of “embedded residue”. \[lemma:Schoutens\] Let $F$ be a $C$-field. 1. $F$ has embedded residue iff there is an elementary extension $F\preceq F^*$ and a nontrivial valuation $v$ on $F^*$ with an embedding of $F^*v$ into $F^*$. 2. $F$ is large iff there is an elementary extension $F\preceq F^*$, a subfield $E\subseteq F^*$ and a nontrivial henselian valuation $v$ on $E$ such that $Ev$ is isomorphic to $F^*$. \(1) Let $F_1\equiv F_2\equiv F$ and $v$ a nontrivial valuation on $F_1$ such that $F_1v$ embeds into $F_2$. By [@Shelah] there exists a cardinal $\lambda$ and an ultrafilter $D$ on $\lambda$ such that the ultrapowers $F_1^\lambda/D$ and $F_2^\lambda/D$ are isomorphic and $|F|^+$-saturated. Thus, if we let $(F^*,v^*)=(F_1,v)^\lambda/D$, then $F$ can be embedded elementarily into $F^*$, and $F^*v^*\cong (F_1v)^\lambda/D$ embeds into $F_2^\lambda/D\cong F^*$. \(2) Let $F_1\equiv F_2\equiv F$, $E$ a subfield of $F_1$ and $v$ a nontrivial henselian valuation on $E$ with $Ev\cong F_2$. Again there exist $\lambda$ and $D$ such that $F_1^\lambda/D\cong F_2^\lambda/D$ is $|F|^+$-saturated. If we equip $F_1$ with a predicate for $E$ and let $(F^*,E^*,v^*)=(F_1,E,v)^\lambda/D$, then $E^*$ is a subfield of $F^*$ and $v^*$ is a nontrivial henselian valuation on $E^*$ such that $E^*v^*=(Ev)^\lambda/D\cong F_2^\lambda/D\cong F^*$. \[lem:sep.cl\] Let $F$ be a $C$-field that contains an algebraically closed $C$-field $D$. Then 1. $F$ has embedded residue. 2. $F$ is large. \(1) By passing to an elementary extension, we may assume that $F$ is transcendental over $D$. By \[lem:extend.with.alg.res.field\] there exists an extension of the trivial valuation on $D$ to a valuation $v$ on $F$ with $Fv= D\subset F$. It is necessarily nontrivial. \(2) Let $(E,v)$ be a nontrivial henselian extension of the trivially valued field $F$ with $Ev\cong F$ (\[rem:henselian.ext\]). Since $D$ is algebraically closed, $D\preceq_\exists E$. Let $(F^*,D^*)$ be an $|E|^+$-saturated elementary extension of $F$ with a predicate for $D$. Then there exists a $D$-embedding $E\rightarrow D^*\subseteq F^*$. Existential definitions ----------------------- \[lem:ER.implies.not.uniform\] Suppose that $\mathcal{F}$ has \[equicharacteristic\] embedded residue. Then the valuation ring **is not *uniformly $\exists$-$C$-definable in $\mathcal{H}'(\mathcal{F})$ \[respectively, $\mathcal{H}_e'(\mathcal{F})$\].*** By assumption (in both cases), there exist $F_{1},F_{2}\in\mathcal{F}$ and a nontrivial valuation $w$ on $F_{1}$ such that $F_{1}w\subseteq F_{2}$. Let $(K_1,v_1)$ be a nontrivial henselian extension of the trivially valued field $F_1$ with $K_1v_1= F_{1}$ (\[rem:henselian.ext\]). Then $(K_1,v_1)\in\mathcal{H}_e'(\mathcal{F})$. $$\xymatrix{ K_2\ar@{-}[d]\ar@{.>}[rr]^{v_2} && F_2\ar@{-}[d] \\ K_1\ar@{.>}[r]^{v_1}\ar@/^1.6pc/@{.>}[rr]^{u} & F_1\ar@{.>}[r]^w & F_1w\\ &C\ar@{->}[ul]\ar@{->}[u]\ar@{->}[ur]& }$$ Let $u$ denote the composition $w\circ v_1$ on $K_1$. Observe that $u$ is a proper refinement of $v_1$ and that $K_1u=F_{1}w$ is a subfield of $F_{2}$. By \[lem:complete.the.square\] there exists a henselian extension $(K_2,v_2)$ of $(K_1,u)$ such that $K_2v_2$ is isomorphic to $F_{2}$. Then $(K_2,v_2)\in\mathcal{H}'(\mathcal{F})$, but $v_2$ restricted to $K_1$ is a proper refinement of $v_1$, hence $\mathcal{O}_{v_1}\not\subseteq\mathcal{O}_{v_2}$. Since existential formulas ‘pass up’, the valuation ring is not uniformly $\exists$-$C$-definable in $\mathcal{H}'(\mathcal{F})$. If in addition $w$ is equicharacteristic, then so is $u$, and therefore also $v_2$, hence $(K_2,v_2)\in\mathcal{H}_e'(\mathcal{F})$, showing that the valuation ring is not uniformly $\exists$-$C$-definable in $\mathcal{H}_e'(\mathcal{F})$. \[lem:no.ER.implies.uniform\] Suppose that $\mathcal{F}$ is an elementary class and that $\mathcal{F}$ does not have \[equicharacteristic\] embedded residue. Then the valuation ring **is *uniformly $\exists$-$C$-definable in $\mathcal{H}(\mathcal{F})$ \[respectively, $\mathcal{H}_e(\mathcal{F})$\].*** We test the hypothesis of \[thm:Prestel\] for the class $\mathcal{K}=\mathcal{H}(\mathcal{F})$ \[respectively, $\mathcal{K}=\mathcal{H}_e(\mathcal{F})$\]. Let $(K_1,v_1),(K_2,v_2)\in\mathcal{H}(\mathcal{F})$ and suppose that $K_1$ is an $\mathcal{L}_{\mathrm{ring}}(C)$-substructure of $K_2$. Let $L\subseteq K_2$ be the henselisation of $K_1$ with respect to the restriction $u$ of $v_2$ to $K_1$. Let $w_1$ be the unique extension of $v_1$ to $L$ and $w_2$ the restriction of $v_2$ to $L$. We now have a trichotomy by comparing $v_1$ and $u$: either $v_1$ and $u$ are incomparable, or $v_{1}$ is a proper coarsening of $u$, or $v_{1}$ is a refinement of $u$. $$\xymatrix{ &K_2\ar@{-}[d] & & v_2\ar@{-}[d] \\ &L\ar@{-}[d] & w_1\ar@{-}[d] & w_2\ar@{-}[d]\\ C\ar@{->}[r]&K_1 & v_1 & u %C\ar@{->}[u]&& }$$ If $v_1$ and $u$ are incomparable, then so are $w_1$ and $w_2$, hence the residue fields of $w_1$, $w_2$, and their finest common coarsening $w$ are all separably closed. Furthermore $w_2$ induces a nontrivial valuation $\bar{w}_2$ on the separably closed field $Lw$. By \[lem:residue.of.sep.closed\], $(Lw)\bar{w}_2=Lw_2$ is algebraically closed, hence, by \[lem:sep.cl\], $K_2v_2\supseteq Lw_2$ has embedded residue. Thus, $\mathcal{F}$ has equicharacteristic embedded residue, see \[rem:unmixed\]. This contradicts our assumption (in both cases). Therefore $v_1$ and $u$ are comparable. If $v_{1}$ is a proper coarsening of $u$ then $u$ induces a nontrivial valuation $\bar{u}$ on $K_1v_1$ and $(K_1v_1)\bar{u}=K_1u\subseteq K_2v_2$. Thus $\mathcal{F}$ has embedded residue. If in addition, $(K_2,v_2)\in\mathcal{H}_e(\mathcal{F})$, then $u$ is equicharacteristic, and then so is $\bar{u}$, hence $\mathcal{F}$ has equicharacteristic embedded residue. Thus, in both cases this is a contradiction. Therefore $v_{1}$ is a refinement of $u$, i.e. $\mathcal{O}_{v_{1}}\subseteq\mathcal{O}_{u}$, as required. Applying \[thm:Prestel\], we are done. \[thm:existential.characterisation\] Let $\mathcal{F}$ be an elementary class of $C$-fields. The following are equivalent: 1. The valuation ring is uniformly $\exists$-$C$-definable in $\mathcal{H}'(\mathcal{F})$. 2. The valuation ring is uniformly $\exists$-$C$-definable in $\mathcal{H}(\mathcal{F})$. 3. $\mathcal{F}$ does not have embedded residue. Moreover, also the following are equivalent: 1. The valuation ring is uniformly $\exists$-$C$-definable in $\mathcal{H}'_e(\mathcal{F})$. 2. The valuation ring is uniformly $\exists$-$C$-definable in $\mathcal{H}_e(\mathcal{F})$. 3. $\mathcal{F}$ does not have equicharacteristic embedded residue. $(1^{\exists})\implies(1^{'\exists})$ and $(1^{\exists}_{e})\implies(1^{'\exists}_{e})$: This follows from the inclusions $\mathcal{H}^{'}(\mathcal{F})\subseteq\mathcal{H}(\mathcal{F})$ and $\mathcal{H}_e^{'}(\mathcal{F})\subseteq\mathcal{H}_e(\mathcal{F})$. $(1^{'\exists})\implies(2^{\exists})$ and $(1^{'\exists}_{e})\implies(2^{\exists}_e)$: Apply \[lem:ER.implies.not.uniform\]. $(2^{\exists})\implies(1^{\exists})$ and $(2^{\exists}_{e})\implies(1^{\exists}_{e})$: Apply \[lem:no.ER.implies.uniform\]. \[cor:existential.characterisation.one.characteristic\] Let $\mathcal{F}$ be an elementary class of $C$-fields which is unmixed. Then all six conditions $(1^{'\exists})$, $(1^{\exists})$, $(2^\exists)$, $(1^{'\exists}_{e})$, $(1^{\exists}_{e})$, and $(2^{\exists}_{e})$ are equivalent. The equivalence between $(1^{'\exists})$ and $(1^{\exists})$ (resp. $(1^{'\exists}_{e})$ and $(1^{\exists}_{e})$) does not seem obvious. In general we may require a different definition: an $\exists$-$\mathcal{L}_{\mathrm{ring}}(C)$-formula that uniformly defines the valuation ring in $\mathcal{H}^{'}(\mathcal{F})$ may not also define the trivial valuation ring in the trivially valued $C$-fields with residue field in $\mathcal{F}$. For example, let $C=\mathbb{Z}$ and $\mathcal{F}=\{\mathbb{F}_{p}\}$, for a prime $p$. As discussed below in \[prop:positive.applications\](1), $\mathcal{F}$ does not have embedded residue. Therefore there is an $\exists$-$\mathcal{L}_{\mathrm{ring}}(C)$-formula $\phi(x)$ that uniformly defines the valuation ring in $\mathcal{H}(\mathcal{F})$. However, it is easy to see that the formula $$\phi(x)\wedge\exists y_{0},...,y_{p}\;\bigwedge_{i\neq j}y_{i}\neq y_{j}$$ also uniformly defines the valuation ring in $\mathcal{H}^{'}(\mathcal{F})$ but does not define the trivial valuation ring in $\mathbb{F}_{p}$. Also note that as explained in \[rem:unmixed\], the assumption of the corollary is satisfied in particular if $\mathcal{F}$ consists of $C$-fields elementarily equivalent to a fixed $C$-field $F$, and also if $C$ has positive characteristic or is a field. If we put a different condition on $C$, then we get a further equivalent characterization: \[cor:existential.characterisation.characteristic.zero\] Let $C$ be a Dedekind domain[^3] of characteristic zero and $\mathcal{F}$ an elementary class of $C$-fields. Then $(1^{'\exists})$, $(1^{\exists})$, and $(2^{\exists})$ are each equivalent to the following: 1. The valuation ring is uniformly $\exists$-$C$-definable in $\mathcal{H}_0(\mathcal{F})$. $(1^{'\exists})$ $(1^{\exists})$, and $(2^{\exists})$ are equivalent by \[thm:existential.characterisation\], and trivially $(1^{\exists})\implies (1^{\exists}_0)$. We repeat and adapt the proof of $(1^{\exists})\implies(2^{\exists})$ (i.e. the proof of \[lem:ER.implies.not.uniform\]) to obtain a proof of $(1^{\exists}_0)\implies(2^{\exists})$: Suppose there exist $F_{1},F_{2}\in\mathcal{F}$ and a nontrivial valuation $w$ on $F_{1}$ such that $F_{1}w\subseteq F_{2}$. Instead of taking an extension $K_1$ of $F_1$, we note that the localization of $C$ at the kernel $\mathfrak{p}$ of the structure homomorphism $C\rightarrow F_1$ is the valuation ring of a valuation $v_0$ on a $C$-field $K_0$ of characteristic zero with residue field $K_0v_0\cong {\rm Quot}(C/\mathfrak{p})\subseteq F_1$. By \[lem:complete.the.square\], there is a henselian extension $(K_1,v_1)$ of $(K_0,v_0)$ with $K_1v_1\cong F_1$, so $(K_1,v_1)\in\mathcal{H}_0(\mathcal{F})$. Let $u$ denote the composition $w\circ v_1$ on $K_1$. Observe that $u$ is a proper refinement of $v_1$ and that $K_1u=F_{1}w$ is a subfield of $F_{2}$. By \[lem:complete.the.square\] there exists a henselian extension $(K_2,v_2)$ of $(K_1,u)$ such that $K_2v_2$ is isomorphic to $F_{2}$. Then $(K_2,v_2)\in\mathcal{H}_0(\mathcal{F})$, but $v_2$ restricted to $K_1$ is a proper refinement of $v_1$, hence $\mathcal{O}_{v_1}\not\subseteq\mathcal{O}_{v_2}$. Since existential formulas ‘pass up’, the valuation ring is not uniformly $\exists$-$C$-definable in $\mathcal{H}_0(\mathcal{F})$. Note that \[cor:existential.characterisation.characteristic.zero\] applies for example to $C=\mathbb{Z}$, and $\exists$-$\mathbb{Z}$-definable is the same as $\exists$-$\emptyset$-definable. Moreover, if we apply \[cor:existential.characterisation.one.characteristic\] to $C=\mathbb{Z}/n\mathbb{Z}$ for $n\in\mathbb{N}$, then $\mathcal{F}$ can be any elementary class of fields $F$ with ${\rm char}(F)$ dividing $n$, and here again, $\exists$-$\mathbb{Z}/n\mathbb{Z}$-definable is the same as $\exists$-$\emptyset$-definable. Universal definitions --------------------- \[lem:large.implies.not.uniform\] Suppose that $\mathcal{F}$ is \[equicharacteristic\] large. Then the valuation ring **is not *uniformly $\forall$-$C$-definable in $\mathcal{H}'(\mathcal{F})$ \[respectively, $\mathcal{H}_e'(\mathcal{F})$\].*** By assumption (in both cases), there exist $F_{1},F_{2}\in\mathcal{F}$, a subfield $E\subseteq F_{2}$ and a nontrivial henselian valuation $w$ on $E$ such that $Ew$ is isomorphic to $F_{1}$. Let $(K_1,u)$ be a nontrivial henselian extension of the trivially valued field $E$ with $K_1u=E$ (\[rem:henselian.ext\]). Let $v_1$ denote the composition $w\circ u$, so that $K_1v_1=F_{1}$. Then $(K_1,v_1)\in\mathcal{H}'(\mathcal{F})$. $$\xymatrix{ K_2\ar@{-}[d]\ar@{.>}[r]^{v_2} & F_2\ar@{-}[d]&\\ K_1\ar@{.>}[r]^{u}\ar@/^1.6pc/@{.>}[rr]^{\qquad v_{1}} & E\ar@{.>}[r]^w & F_1\\ &C\ar@{->}[ul]\ar@{->}[u]\ar@{->}[ur] }$$ Using \[lem:complete.the.square\], we may construct a henselian extension $(K_2,v_2)$ of $(K_1,u)$ of valued fields so that $K_2v_2$ is isomorphic to $F_{2}$. Thus $(K_2,v_2)\in\mathcal{H}_e'(\mathcal{F})$ and $K_1\subseteq K_2$. Since $w$ is nontrivial, the restriction of $v_2$ to $K_1$ is a proper coarsening of $v_1$. Since universal formulas ‘go down’, the valuation ring is not uniformly $\forall$-$C$-definable in $\mathcal{H}'(\mathcal{F})$. If in addition $w$ is equicharacteristic, then so is $v_1$, hence $(K_1,v_1)\in\mathcal{H}_e'(\mathcal{F})$, showing that the valuation ring is not uniformly $\forall$-$C$-definable in $\mathcal{H}_e'(\mathcal{F})$. \[lem:not.large.implies.uniform\] Suppose that $\mathcal{F}$ is an elementary class and that $\mathcal{F}$ is not \[equicharacteristic\] large. Then the valuation ring **is *uniformly $\forall$-$C$-definable in $\mathcal{H}(\mathcal{F})$ \[respectively, in $\mathcal{H}_e(\mathcal{F})$\].*** We test the hypothesis of \[thm:Prestel\]. Let $(K_1,v_1),(K_2,v_2)\in\mathcal{H}(\mathcal{F})$ and suppose that $K_1$ is an $\mathcal{L}_{\mathrm{ring}}(C)$-substructure of $K_2$. Let $u$ denote the restriction of $v_2$ to $K_1$. As in the proof of \[lem:no.ER.implies.uniform\] we see that if $v_1$ and $u$ are incomparable, then $K_2v_2$ would contain an algebraically closed $C$-field. Since by \[lem:sep.cl\] this would also imply that $K_2v_2$ is large, and therefore $\mathcal{F}$ is equicharacteristic large (\[rem:unmixed\]), which is a contradiction (in both cases), we conclude that $v_1$ and $u$ are comparable. If $v_{1}$ is a proper refinement of $u$ then $v_1$ induces a nontrivial henselian valuation $\bar{v}_1$ on $K_1u\subseteq K_2v_2$ and $(K_1u)\bar{v}_1=K_1v_1$. Thus $\mathcal{F}$ is large. If in addition $(K_1,v_1)\in\mathcal{H}_e(\mathcal{F})$, then also $\bar{v}_1$ is equicharacteristic, so $\mathcal{F}$ is equicharacteristic large. Thus, in both cases this is a contradiction. Therefore $v_{1}$ is a coarsening of $u$, i.e. $\mathcal{O}_{v_{1}}\supseteq\mathcal{O}_{u}$, as required. Applying \[thm:Prestel\], we are done. \[thm:universal.characterisation\] Let $\mathcal{F}$ be an elementary class of $C$-fields. The following are equivalent: 1. The valuation ring is uniformly $\forall$-$C$-definable in $\mathcal{H}'(\mathcal{F})$. 2. The valuation ring is uniformly $\forall$-$C$-definable in $\mathcal{H}(\mathcal{F})$. 3. $\mathcal{F}$ is not large. Moreover, also the following are equivalent: 1. The valuation ring is uniformly $\forall$-$C$-definable in $\mathcal{H}'_e(\mathcal{F})$. 2. The valuation ring is uniformly $\forall$-$C$-definable in $\mathcal{H}_e(\mathcal{F})$. 3. $\mathcal{F}$ is not equicharacteristic large. $(1^{\forall})\implies(1^{'\forall})$ and $(1^{\forall}_{e})\implies(1^{'\forall}_{e})$: This follows from the inclusions $\mathcal{H}'(\mathcal{F})\subseteq\mathcal{H}(\mathcal{F})$ and $\mathcal{H}_e'(\mathcal{F})\subseteq\mathcal{H}_e(\mathcal{F})$. $(1^{'\forall})\implies(2^{\forall})$ and $(1^{'\forall}_{e})\implies(2^{\forall}_{e})$: Apply \[lem:large.implies.not.uniform\]. $(2^{\forall})\implies(1^{\forall})$ and $(2^{\forall}_e)\implies(1^{\forall}_e)$: Apply \[lem:not.large.implies.uniform\]. \[cor:universal.characterisation.one.characteristic\] Let $\mathcal{F}$ be an elementary class of $C$-fields which is unmixed. Then all six conditions $(1^{'\forall})$, $(1^{\forall})$, $(2^{\forall})$, $(1^{'\forall}_{e})$, $(1^{\forall}_{e})$, and $(2^{\forall}_{e})$ are equivalent. \[cor:universal.characterisation.characteristic.zero\] Let $C$ be a Dedekind domain of characteristic zero and $\mathcal{F}$ an elementary class of $C$-fields. Then $(1^{'\forall})$, $(1^{\forall})$, and $(2^{\forall})$ are each equivalent to the following: 1. The valuation ring is uniformly $\forall$-$C$-definable in $\mathcal{H}_0(\mathcal{F})$. $(1^{'\forall})$, $(1^{\forall})$, and $(2^{\forall})$ are equivalent by \[thm:universal.characterisation\], and trivially $(1^{\forall})\implies(1^{\forall}_0)$. We repeat and adapt the proof of $(1^{\forall})\implies(2^{\forall})$ (i.e. the proof of \[lem:large.implies.not.uniform\]) to obtain a proof of $(1^{\forall}_0)\implies(2^{\forall})$: Suppose there exist $F_{1},F_{2}\in\mathcal{F}$, a subfield $E\subseteq F_{2}$ and a nontrivial henselian valuation $w$ on $E$ such that $Ew$ is isomorphic to $F_{1}$. The localization of $C$ at the kernel $\mathfrak{p}$ of the structure homomorphism $C\rightarrow E$ is the valuation ring of a valuation $v_0$ on a $C$-field $K_0$ of characteristic zero with residue field $K_0v_0\cong {\rm Quot}(C/\mathfrak{p})\subseteq E$. By \[lem:complete.the.square\], there is a henselian extension $(K_1,u)$ of $(K_0,v_0)$ with $K_1u\cong E$. Let $v_1$ denote the composition $w\circ u$, so that $K_1v_1=F_{1}$. Then $(K_1,v_1)\in\mathcal{H}_0(\mathcal{F})$. Using \[lem:complete.the.square\], we may construct a henselian extension $(K_2,v_2)$ of $(K_1,u)$ of valued fields so that $K_2v_2$ is isomorphic to $F_{2}$. Thus $(K_2,v_2)\in\mathcal{H}_0(\mathcal{F})$ and $K_1\subseteq K_2$. Since $w$ is nontrivial, the restriction of $v_2$ to $K_1$ is a proper coarsening of $v_1$. Since universal formulas ‘go down’, the valuation ring is not uniformly $\forall$-$C$-definable in $\mathcal{H}_0(\mathcal{F})$. Making definitions uniform for equicharacteristic fields {#sec:uniform} ======================================================== The goal of this section is to show that existential definitions of the valuation ring or valuation ideal of an equicharacteristic henselian nontrivially valued field can be modified to work for all such valued fields with elementarily equivalent residue field. Depending on the parameters $C$, we may have to restrict the residue fields that we consider: We say that a $C$-field $F$ satisfies $(*)$ if $$\label{star} \begin{minipage}{11cm} {\it \begin{enumerate} \item[(a)] $C$ is integral over its prime ring, \underline{or} \item[(b)] $C$ is a perfect field and $F$ is perfect. \end{enumerate} } \end{minipage}$$ We say that a class of $C$-fields $\mathcal{F}$ satisfies $(*)$ if each $F\in\mathcal{F}$ satisfies $(*)$. Note that in the case $C=\mathbb{Z}$ (that is, when we consider $\exists$-$\emptyset$-definitions), all classes of $C$-fields satisfy $(*)$. \[prp:GDM.version.1\] Let $\mathcal{F}$ be an elementary class of $C$-fields that satisfies $(*)$. Let $\phi(x)$ be an $\exists$-$\mathcal{L}_{\mathrm{ring}}(C)$-formula. 1. If for all $F\in\mathcal{F}$ there exists $(K,v)\in\mathcal{H}_e'(F)$ with $\phi(K)=\mathcal{O}_v$, then there is an $\exists$-$\mathcal{L}_{\mathrm{ring}}(C)$-formula $\psi(x)$ with $\psi(K)=\mathcal{O}_v$ for all $(K,v)\in\mathcal{H}_e'(\mathcal{F})$. 2. If for all $F\in\mathcal{F}$ there exists $(K,v)\in\mathcal{H}_e'(F)$ with $\phi(K)=\mathfrak{m}_v$, then there is an $\exists$-$\mathcal{L}_{\mathrm{ring}}(C)$-formula $\chi(x)$ with $\chi(K)=\mathfrak{m}_v$ for all $(K,v)\in\mathcal{H}_e'(\mathcal{F})$. We note that in general we cannot take $\psi(x)$ (resp. $\chi(x)$) to be simply $\phi(x)$, as the following example in the $\mathfrak{m}$-case shows: If the $\exists$-$\mathcal{L}_{\rm ring}(C)$-formula $\chi(x)$ uniformly defines the valuation ideal in $\mathcal{H}_e'(F)$, then the $\exists$-$\mathcal{L}_{\rm ring}(C)$-formula $\phi(x)$ given by $$\exists y \exists z (x=yz\wedge\chi(y)\wedge\chi(z))$$ defines the valuation ideal in any $(K,v)\in\mathcal{H}_e'(F)$ with divisible value group, but for example not in $(F((t)),v_t)\in\mathcal{H}_e'(F)$. The proof of \[prp:GDM.version.1\] below however does give a quite explicit construction of $\psi$ and $\chi$ from $\phi$. Note that the conclusion of (1) (respectively, (2)) is $(1'^{\exists}_e)$ (resp., $(1'^{\forall}_e)$) from \[thm:existential.characterisation\] (resp. \[thm:universal.characterisation\]), see also \[rem:EAOm\]. In order to prove this proposition, we first break the statement $\phi(K)=\mathcal{O}_v$ (respectively $\phi(K)=\mathfrak{m}_v$) into several parts which we then treat separately: \[def:important.sentences\] For an $\mathcal{L}_{\rm vf}(C)$-formula $\phi(x)$ with free variable $x$ belonging to the field sort we define the following $\mathcal{L}_{\rm vf}(C)$-sentences: 1. $\forall x\;(\phi(x)\rightarrow v(x)\geq0)$ 2. $\forall x\;(v(x)\geq0 \rightarrow \phi(x))$ 3. $\forall x\;(\phi(x)\rightarrow v(x)>0)$ 4. $\forall x\;(v(x)>0\rightarrow\phi(x))$ 5. $\exists x\;(v(x)>0\wedge x\neq0\wedge\phi(x))$ 6. $\forall^{k}x\exists y\;(\mathrm{res}(y)=x\wedge\phi(y))$ In the following table we summarize what it means for a valued $C$-field $(K,v)$ to satisfy one of these sentences, as well as the quantifier complexity for the case that $\phi(x)$ is an $\exists$-$\mathcal{L}_{\rm vf}(C)$-formula: holds in $(K,v)$ iff quantifiers ------------------- ---------------------------------------------------------- -------------------- $({\rm SO}_\phi)$ $\phi(K)\subseteq\mathcal{O}_v$ $\forall$ $({\rm CO}_\phi)$ $\phi(K)\supseteq\mathcal{O}_v$ $\forall\exists$ $({\rm SM}_\phi)$ $\phi(K)\subseteq\mathfrak{m}_v$ $\forall$ $({\rm CM}_\phi)$ $\phi(K)\supseteq\mathfrak{m}_v$ $\forall\exists$ $({\rm IM}_\phi)$ $\phi(K)\cap(\mathfrak{m}_v\setminus\{0\})\neq\emptyset$ $\exists$ $({\rm R}_\phi)$ ${\rm res}(\phi(K))=Kv$ $\forall^k\exists$ With these interpretations, the following lemma is obvious: \[lem:relationships.between.important.sentences\] Let $\phi(x)$ be an $\mathcal{L}_{\rm vf}(C)$-formula and $(K,v)$ a nontrivially valued $C$-field. 1. $\phi(K)=\mathcal{O}_v$ $\Leftrightarrow$ $(K,v)\models({\rm SO}_\phi)\wedge({\rm CO}_\phi)$ $\Rightarrow$ $(K,v)\models({\rm SO}_\phi)\wedge({\rm R}_\phi)\wedge({\rm IM}_\phi)$ 2. $\phi(K)=\mathfrak{m}_v$ $\Leftrightarrow$ $(K,v)\models({\rm SM}_\phi)\wedge({\rm CM}_\phi)$ $\Rightarrow$ $(K,v)\models({\rm SM}_\phi)\wedge({\rm IM}_\phi)$ \[rem:strategy\] The strategy now is to use the results of [@AnscombeFehm], which allow us to ‘transfer’ the truth of $\forall^k\exists$-$\mathcal{L}_{\rm vf}(C)$-sentences between equicharacteristic henselian nontrivially valued fields with the same residue field (see \[cor:transfer\]). According to the table above, if $\phi$ is an $\exists$-$\mathcal{L}_{\rm vf}(C)$-formula, these results can be applied directly to $({\rm SO}_\phi)$ and $({\rm SM}_\phi)$, but unfortunately not to $({\rm CO}_\phi)$ and $({\rm CM}_\phi)$. Therefore, we instead have to work with the weaker statements $({\rm R}_\phi)$ and $({\rm IM}_\phi)$, to which the results can be applied, and modify the formula $\phi$ suitably. For $\mathcal{F}$ an elementary class of $C$-fields, we recall that $\mathcal{H}_e'(\mathcal{F})$ denotes the class of equicharacteristic henselian nontrivially valued $C$-fields with residue field in $\mathcal{F}$. So far we have formulated our results (in particular \[prp:GDM.version.1\]) in terms of the elementary class $\mathcal{H}_e'(\mathcal{F})$. However, for the rest of the section it will be convenient to make the following notational change: instead of $\mathcal{H}_e'(\mathcal{F})$, we formulate our results in terms of the theory $\mathbf{T}_{\mathcal{F}}$, which is defined as follows. \[def:T\_F\] We let $\mathbf{T}_{\mathcal{F}}$ be the $\mathcal{L}_{\rm vf}(C)$-theory of equicharacteristic henselian nontrivially valued $C$-fields with residue field a member of $\mathcal{F}$. For a single $C$-field $F$, we let $\mathbf{T}_{F}$ denote the theory of equicharacteristic henselian nontrivially valued $C$-fields with residue field elementarily equivalent to $F$. Thus $\mathbf{T}_{\mathcal{F}}$ is the theory of $\mathcal{H}_e'(\mathcal{F})$ and $\mathcal{H}_e'(\mathcal{F})$ is the class of models of $\mathbf{T}_{\mathcal{F}}$. The rest of this section is devoted to proving , which is simply rewritten using our new notation. Existential transfer principle ------------------------------ First we recall the ‘transfer principle’ from [@AnscombeFehm] which, as we have indicated, will be our main tool in this section. Let $F/C'$ be a separable field extension. In [@AnscombeFehm], $\mathbf{T}_{F/C'}$ denotes the theory of equicharacteristic henselian nontrivially valued fields $(K,v,d_c)_{c\in C'}$ in the language $\mathcal{L}_{\rm vf}(C')$ for which $c\mapsto d_c$ gives a homomorphism $C'\rightarrow K$, the valuation $v$ is trivial on $D:=\{d_c:c\in C'\}$, and $(Kv,d_cv)_{c\in C'}\equiv(F,c)_{c\in C'}$. That is, $\mathbf{T}_{F/C'}$ is the same as $\mathbf{T}_{F}$ where we view $F$ as a $C'$-field. \[fact:transfer\] Let $\psi(\mathbf{x})$ be an $\exists$-$\mathcal{L}_{\rm vf}(C')$-formula with free variables $\mathbf{x}$ belonging to the residue field sort. Suppose there exists $(K,v)\models\mathbf{T}_{F/C'}\cup\{\forall^{k}\mathbf{x}\;\psi(\mathbf{x})\}$. Then there exists $n\in\mathbb{N}$ such that, for all $(L,w)\models\mathbf{T}_{F/C'}$, we have ${}^{\mathbf{x}}Lw\subseteq\psi(L^{p^{-n}})$, where $p\geq1$ is the characteristic exponent of $F$. \[fact:transfer.perfect\] Suppose that $F$ is perfect. Then for any $\forall^{k}\exists$-$\mathcal{L}_{\rm vf}(C')$-sentence $\phi$, either $\mathbf{T}_{F/C'}\models\phi$ or $\mathbf{T}_{F/C'}\models\neg\phi$. For convenience, we collect these results together into one corollary which covers almost all of the applications and is expressed using the new notation. For a $C$-field $F$ let $C'=C_F$ and let $\alpha:C\rightarrow C'$ denote the structure homomorphism of $F$. If $(K,v)\models\mathbf{T}_{F/C'}$, then composing the structure homomorphism $C'\rightarrow K$ with $\alpha$ turns $K$ into a $C$-field $K^\circ$ with $(K^\circ,v)\models\mathbf{T}_F$. To a $\mathcal{L}_{\rm vf}(C)$-formula $\phi(\mathbf{x})$ we assign a $\mathcal{L}_{\rm vf}(C')$-formula $\phi'(\mathbf{x})$ by applying $\alpha$ to all the constants. \[rem:translate\] Note that for $(K,v)\models\mathbf{T}_{F/C'}$, an $\mathcal{L}_{\rm vf}(C)$-formula $\phi(\mathbf{x})$ and $\mathbf{a}\in K^n$ we trivially have $(K^\circ,v)\models\phi(\mathbf{a})$ if and only if $(K,v)\models\phi'(\mathbf{a})$. \[lem:star\] Let $F$ be a $C$-field that satisfies $(*)$. Then $C_F$ is perfect and if $(K,v)\models\mathbf{T}_F$, then $v$ is trivial on $C_K$ and the residue map induces an isomorphism $C_K\rightarrow C_F$. In case (a), $C_F$ and $C_K$ are algebraic extensions of the prime field; in case (b), $C\cong C_F\cong C_K$ is a perfect field. In particular, $C_F$ is perfect in both cases. In case (a), the assumption that $v$ is equicharacteristic implies that it is trivial on the prime field, hence also on $C_K$. In case (b), since $v$ is nonnegative on the image $C_K$ of $C$, which is a field, it is trivial on $C_K$. \[rem:star\] Let $F$ be a $C$-field that satisfies $(*)$. Then the map $(K,v)\mapsto (K^\circ,v)$ from the models of $\mathbf{T}_{F/C'}$ to the models of $\mathbf{T}_{F}$ has an inverse $(K,v)\mapsto (K',v)$, and for $(K,v)\models\mathbf{T}_F$, an $\mathcal{L}_{\rm vf}(C)$-formula $\phi(\mathbf{x})$ and $\mathbf{a}\in K^n$ we have $$(K,v)\models\phi(\mathbf{a}) \quad\Longleftrightarrow\quad (K',v)\models\phi'(\mathbf{a}).$$ If $(K,v)\models\mathbf{T}_F$, then gives an isomorphism $\beta:C_K\rightarrow C_F=C'$, so $\beta^{-1}:C'\rightarrow K$ turns $K$ into a $C'$-field $K'$ with $(K',v)\models\mathbf{T}_{F/C'}$. Obviously this is inverse to $(K,v)\mapsto (K^\circ,v)$. The second claim is then immediate from . \[cor:transfer\] Let $F$ be a $C$-field of characteristic exponent $p\geq 1$ that satisfies $(*)$. 1. Let $\psi(x)$ be an $\exists$-$\mathcal{L}_{\rm vf}(C)$-formula with free variable $x$ belonging to the residue field sort. If $\mathbf{T}_{F}\cup\{\forall^{k}x\;\psi(x)\}$ is consistent then there exists $n\in\mathbb{N}$ such that for all $(L,w)\models\mathbf{T}_{F}$ we have $(Lw)^{p^{n}}\subseteq\psi(L)$. 2. Let $\phi$ be an $\exists$-$\mathcal{L}_{\rm vf}(C)$-sentence. If $\mathbf{T}_{F}\cup\{\phi\}$ is consistent, then $\mathbf{T}_{F}\models\phi$. 3. Let $\phi$ be an $\forall$-$\mathcal{L}_{\rm vf}(C)$-sentence. If $\mathbf{T}_{F}\cup\{\phi\}$ is consistent, then $\mathbf{T}_{F}\models\phi$. We first show (1). Let again $C'=C_F$. If $\mathbf{T}_{F}\cup\{\forall^{k}x\;\psi(x)\}$ is consistent, then by \[rem:star\] also $\mathbf{T}_{F/C'}\cup\{\forall^{k}x\;\psi'(x)\}$ is consistent, and it suffices to show that there exists $n\in\mathbb{N}$ such that $(Lw)^{p^{n}}\subseteq\psi'(L)$ for all $(L,w)\models\mathbf{T}_{F/C'}$. If $F$ satisfies (b), then \[fact:transfer.perfect\] shows that the claim holds for $n=0$. Now suppose instead that $F$ satisfies (a), i.e. $C$ is integral over its prime ring. There exists a quantifier-free $\mathcal{L}_{\rm vf}$-formula $q(x,\mathbf{y},\mathbf{z})$ such that $\psi'(x)$ is equivalent to $\exists\mathbf{y}\;q(x,\mathbf{y},\mathbf{c})$ for some finite tuple $\mathbf{c}\subseteq C'$ of parameters. Let $(L,w)\models\mathbf{T}_{F/C'}$. By \[fact:transfer\], there exists $n_{1}\in\mathbb{N}$ such that $Lw\subseteq\psi'(L^{p^{-n_{1}}})$, i.e. $Lw$ is contained in the projection onto the $x$-coordinate of the set of realizations $(a,\mathbf{b})$ of $q(x,\mathbf{y},\mathbf{c})$ in $L^{p^{-n_{1}}}$, where we view $C'\subseteq L\subseteq L^{p^{-n_1}}$. There exists $m\in\mathbb{N}$ such that $\mathbf{c}$ is fixed pointwise by the $m$-th power of the Frobenius endomorphism, i.e. by the map $f:x\longmapsto x^{p^{m}}$. Note that the $n_{1}$-th iterated composition of the map $f$ with itself is the map $f^{n_{1}}:x\longmapsto x^{p^{mn_{1}}}$. Thus $\mathbf{c}=\mathbf{c}^{p^{mn_{1}}}$. Now let $a\in\mathcal{O}_w$. Then $aw\in\psi'(L^{p^{-n_{1}}})$, so there exists $\mathbf{b}\subseteq L^{p^{-n_{1}}}$ such that $$(L^{p^{-n_1}},w)\models q(aw,\mathbf{b},\mathbf{c}).$$ Applying the endomorphism $x\longmapsto x^{p^{mn_{1}}}$ of $L^{p^{-n_1}}$, we get $$(L^{p^{-n_1}},w)\models q(aw^{p^{mn_{1}}},\mathbf{b}^{p^{mn_{1}}},\mathbf{c}).$$ Since $\mathbf{b}^{p^{mn_{1}}}=(\mathbf{b}^{p^{n_{1}}})^{p^{(m-1)n_{1}}}\subseteq L$ and $q$ is quantifier free, we have $aw^{p^{mn_{1}}}\in\psi'(L)$. Setting $n:=mn_{1}$ completes the proof of $(1)$. Having shown $(1)$, for $(2)$ we simply view the $\exists$-sentence $\phi$ as an $\forall^{k}\exists$-formula. More precisely, let $x$ be a variable that does not appear in $\phi$. Then $\phi$ is logically equivalent to both $\forall^{k}x\;\psi(x)$ and $\exists^{k}x\;\psi(x)$, with $\psi(x)=\phi$. Our assumption can be restated as: $\mathbf{T}_{F}\cup\{\forall^{k}x\;\psi(x)\}$ is consistent. Now let $(L,w)\models\mathbf{T}_{F}$. Part $(1)$ implies that $(Lw)^{p^{n}}\subseteq\psi(L)$. In particular $\psi(L)$ is non-empty, i.e. $(L,w)\models\exists^{k}x\;\psi(x)$. Therefore $(L,w)\models\phi$ as required. Finally, $(3)$ follows immediately from $(2)$, since if $\mathbf{T}_F\not\models\phi$, then $\mathbf{T}_F\cup\{\neg\phi\}$ is consistent, so $\mathbf{T}_F\models\neg\phi$, hence $\mathbf{T}_F\cup\{\phi\}$ is inconsistent. The valuation ideal ------------------- As before, $\mathcal{F}$ denotes an elementary class of $C$-fields that satisfies $(*)$. In this subsection we focus on the property $({\rm CM}_{\phi})$. As remarked above, we cannot simply apply to conclude from $(K,v)\models({\rm CM}_{\phi})$ that $(L,w)\models({\rm CM}_{\phi})$ for all $(L,w)\models\mathbf{T}_F$. Instead we make use of the weaker property $({\rm IM}_{\phi})$ and proceed as follows. First we show in that $$(F(t)^{h},v_{t})\models({\rm CM}_{\phi})\implies\mathbf{T}_{F}\models({\rm CM}_{\phi}).$$ Then, for each $F\in\mathcal{F}$, we use to ‘transfer’ $({\rm IM}_{\phi})$, so that $(F(t)^{h},v_{t})\models({\rm IM}_{\phi})$. This allows us to identify a formula $\phi_{n}$ such that $(F(t)^{h},v_{t})\models({\rm CM}_{\phi_{n}})$. Finally, we use a compactness argument to find $\phi_{n}$ uniformly across $\mathcal{F}$. \[lem:embedding.t\] Let $F$ be a $C$-field satisfying $(*)$ and let $\phi(x)$ be an $\exists$-$\mathcal{L}_{\rm vf}(C)$-formula with free variable $x$ belonging to the field sort. Suppose that $(F(t)^{h},v_{t})\models\phi(t)\wedge\phi(0)$. Then $\mathbf{T}_{F}\models({\rm CM}_\phi)$. Let $(K,v)\models\mathbf{T}_{F}$. We aim to show that $(K,v)\models({\rm CM}_{\phi})$, i.e $\mathfrak{m}_{v}\subseteq\phi(K)$. By passing, if necessary, to an elementary extension, we may assume that $(K,v)$ is $|F|^{+}$-saturated. Since $F\equiv Kv$, there is an $\mathcal{L}_{\mathrm{ring}}(C)$-embedding $F\longrightarrow Kv$. By $(*)$, the residue map induces an isomorphism $C_K\rightarrow C_F$ (see \[lem:star\]). Let $f:C_F\rightarrow C_K$ denote its inverse. By [@AnscombeFehm Lemma 2.3], we can extend $f$ as a partial section of the residue map to any finitely generated subextension of $F/C_F$. By saturation, $f$ extends to an $\mathcal{L}_{\rm vf}(C)$-embedding $f:(F,v_{0})\longrightarrow(K,v)$, which is also a partial section of the residue map, where $v_{0}$ denotes the trivial valuation on $F$. Let $s\in\mathfrak{m}_{v}\setminus\{0\}$ and note that $s$ is transcendental over $f(F)$. By sending $t\longmapsto s$, we may extend $f$ to an $\mathcal{L}_{\rm vf}(C)$-embedding $(F(t),v_{t})\longrightarrow(K,v)$. Since $(K,v)$ is henselian, we may again extend $f$ to an $\mathcal{L}_{\rm vf}(C)$-embedding $(F(t)^{h},v_{t})\longrightarrow(K,v)$. Since existential sentences ‘go up’, we have that $s\in\phi(K)$, and also that $0\in\phi(K)$. This shows that $\mathfrak{m}_{v}\subseteq\phi(K)$. \[lem:henselian.approx\] Let $(K,v)$ be a henselian nontrivially valued field, let $E\subseteq K$ be a subfield, and let $b\in K$ be separably algebraic over $E$. Suppose that $(K,v)$ is $|E|^{+}$-saturated. Then there exists $b'\in K$ which is transcendental over $E$ such that $b\in E(b')^{h}$, the henselisation of $E(b')$ with respect to $v$. Let $v$ denote the unique extension of $v$ to $K^{\rm alg}$ and let $$\gamma:=\max\{v(\sigma b-b): \sigma\in{\rm Gal}(E),\sigma b\neq b\} \in v E^{\rm alg}.$$ Since $vE\subseteq vK$ is cofinal in $v E^{\rm alg}$, there exists $\gamma'\in vE$ such that $\gamma\leq\gamma'$. Then ${\rm B}_K(\gamma',b)=\{x\in K\;|\;v(x-b)\geq\gamma'\}$ is an infinite definable subset of $K$, so by saturation, there exists $b'\in {\rm B}_K(\gamma',b)$ which is transcendental over $E$. Then $b$ is also separably algebraic over $E(b')^h$, and $$\max\{v(\sigma b -b):\sigma\in{\rm Gal}(E(b')^h) ,\sigma b\neq b\} \leq \gamma\leq\gamma'\leq v(b'-b),$$ so Krasner’s Lemma [@EP Theorem 4.1.7] implies that $b\in E(b')^h(b')=E(b')^h$. \[lem:relative.inseparable.closure.of.singleton.over.perfect.subfield\] Let $C'\subseteq F\subseteq K$ be a tower of fields of characteristic exponent $p$ and let $a\in K$. Suppose that $C'$ is perfect and $F/C'$ is finitely generated. Then there exists $m\in\mathbb{N}$ such that $a^{p^{-m}}\in F(a)$ and $F(a)$ is separable over $C'(a^{p^{-m}})$. Let $E_{a}$ (respectively, $E_{s}$) denote the relative algebraic (resp., separable algebraic) closure of $C'(a)$ in $F(a)$. Since $F/C'$ is finitely generated, so is $E_a/C'$, hence [@FriedJarden 2.7.2 and 2.7.5] shows that $F(a)/E_{a}$ is regular, in particular separable. By [@Sti 3.10.2], $E_{s}=E_{a}^{p^{m}}$, for some $m\geq 0$. Since $E_{s}$ is separable over $C'(a)$, by the Frobenius isomorphism we have that $E_{a}$ is separable over $C'(a)^{p^{-m}}=C'(a^{p^{-m}})$. $$\xymatrix{ F\ar@{-}[rr]\ar@{-}[dd] & & F(a)\ar@{-}[d]^{\rm reg.} \\ & E_s\ar@{-}[r]^{\rm p.i.}\ar@{-}[d]^{\rm sep.} & E_a\ar@{-}[d]^{\rm sep.} \\ C'\ar@{-}[r] & C'(a)\ar@{-}[r]^{\rm p.i.} & C'(a^{p^{-m}}) \\ }$$ Therefore $F(a)$ is separable over $C'(a^{p^{-m}})$. For a field $K$, subsets $X,Y\subseteq K$ and $n\in\mathbb{N}$ we use the notations $$\begin{aligned} X^{(n)} &:=& \{ x^n : x\in X\},\\ X\pm Y &:=& \{x\pm y:x\in X,y\in Y\}.\end{aligned}$$ \[prp:monkey.lemma\] Let $F$ be a $C$-field that satisfies $(*)$ and let $(K,v)$ be an equicharacteristic henselian valued $C$-field with value group $vK=\mathbb{Z}$ and residue field $Kv=F$. For any $\exists$-$\mathcal{L}_{\rm vf}(C)$-formula $\phi(x)$ with free variable from the field sort and $(K,v)\models({\rm IM}_\phi)$ there exists $n\in\mathbb{N}$ such that $$\mathfrak{m}_{v}^{(n)}\subseteq\phi(K)-\phi(K).$$ By the assumptions, $\phi(x)$ is logically equivalent to $\exists\mathbf{y}\;q(x;\mathbf{y})$ with a quantifier-free $\mathcal{L}_{\rm vf}(C)$-formula $q(x;\mathbf{y})$, and there exist $0\neq a\in\mathfrak{m}_{v}$ and a $\mathbf{y}$-tuple $\mathbf{b}\subset K$ such that $(K,v)\models q(a;\mathbf{b})$. Let $C'=C_F$ and note that $a$ is transcendental over $C'$, by $(*)$. Since $C'$ is perfect, we may apply to the tower $C'\subseteq C'(\mathbf{b})\subseteq K$ of field extensions and the element $a\in K$ to find $m\in\mathbb{N}$ such that $a':=a^{p^{-m}}\in C'(a,\mathbf{b})$ and $C'(a',\mathbf{b})$ is separable over $C'(a')$, where $p$ denotes the characteristic exponent of $F$. By separability and reordering the tuple $\mathbf{b}$ if necessary, we may write $\mathbf{b}=(\mathbf{b}_{1},\mathbf{b}_{2})$ such that $\mathbf{b}_{1}$ is algebraically independent over $C'(a')$ and $C'(a',\mathbf{b})$ is separably algebraic over $E:=C'(a',\mathbf{b}_{1})$. Let $b\in C'(a',\mathbf{b})$ be a primitive element of this extension, so that $C'(a',\mathbf{b})=E(b)$. Applying to $E$, there is an elementary extension $(K,v)\preceq(K^{*},v^{*})$ and an element $b'\in K^{*}$ which is transcendental over $E$ such that $b\in E(b')^{h}$, where the henselisation is taken with respect to the restriction of $v^{*}$. Consequently $C'(a',\mathbf{b})\subseteq E(b')^{h}$. $$\xymatrix{ D\ar@{-}[r]\ar@{-}[d] & E(b')\ar@{-}[r]\ar@{-}[d]^{\rm p.tr.} & E(b')^h\ar@{-}[r]\ar@{-}[d] & K^*\ar@{-}[d] \\ C'(\mathbf{b}_1)\ar@{-}[r]\ar@{-}[d] & E\ar@{-}[r]^{\rm sep. alg.}\ar@{-}[d]^{\rm p.tr.} & E(b)\ar@{-}[r] & K \\ C'\ar@{-}[r]^{\rm p.tr.} & C'(a') & & }$$ Let $D:=C'(\mathbf{b}_{1},b')$ and note that $a'$ is transcendental over $D$. Let $\Psi(x)$ be the quantifier-free $\mathcal{L}_{\rm vf}$-type of $a'$ over $D$, i.e. the set of quantifier-free $\mathcal{L}_{\rm vf}(D)$-formulas $\psi(x)$ in one free variable $x$ belonging to the field sort that satisfy $(K^*,v^*)\models \psi(a')$. If some $\hat{a}\in K^{*}$ realises $\Psi(x)$ then there is an $\mathcal{L}_{\rm vf}(D)$-isomorphism $D(a')\longrightarrow D(\hat{a})$ with $a'\longmapsto\hat{a}$. By the universal property of the henselisation, it extends to an $\mathcal{L}_{\rm vf}(D)$-isomorphism $D(a')^{h}\longrightarrow D(\hat{a})^{h}$. Since $C'(a',\mathbf{b})\subseteq E(b')^{h}=D(a')^{h}$ and $D(\hat{a})^{h}\subseteq K^{*}$, this implies $(K^{*},v^{*})\models\phi(\hat{a}^{p^{m}})$. Thus, by compactness, there is a single quantifier-free $\mathcal{L}_{\rm vf}(D)$-formula $\psi(x)\in\Psi(x)$ such that $$(K^{*},v^{*})\models\;\forall x\;(\psi(x)\longrightarrow\phi(x^{p^{m}})).$$ By \[Lemma:qfdefinable\], $\psi(K^*)=A\cup U$ with $A$ algebraic over $D$ and $U$ open. As $a'\in \psi(K^*)$ is transcendental over $D$, we get that the set defined by $\phi(x^{p^m})$ in $K^*$ contains a ball ${\rm B}_{K^*}(\gamma,a')$ for some $\gamma\in v^*K^*$. As $(K,v)\prec(K^*,v^*)$, the set defined by $\phi(x^{p^m})$ in $K$ therefore contains a ball ${\rm B}_K(l,a')$ for some $l\in vK=\mathbb{Z}$. Without loss of generality, $l\in\mathbb{N}$. Finally, let $n:=lp^{m}$ and let $c\in\mathfrak{m}_{v}^{(n)}$. Then $$c':=c^{p^{-m}}\in\mathfrak{m}_{v}^{(l)}\subseteq {\rm B}_K(l,a')-a',$$ and so $(K,v)\models\phi((c'+a')^{p^m})$, i.e. $(K,v)\models\phi(c+a)$. Therefore $c\in\phi(K)-a\subseteq\phi(K)-\phi(K)$. \[rem:phi\_n\] In the setting of , if we denote by $\phi_n(x)$ the $\exists$-$\mathcal{L}_{\rm vf}(C)$-formula $$\exists y \exists z\; (x^n=y-z\wedge\phi(y)\wedge\phi(z)),$$ then $$(K,v)\models({\rm CM}_{\phi_{n}}).$$ Furthermore, $\phi_n(x)$ defines the set of $n$-th roots of the set of differences between elements in the set defined by $\phi(x)$. Thus if $\phi(x)$ defines a subset of $\mathcal{O}_{v}$ (respectively, $\mathfrak{m}_{v}$) then so does $\phi_{n}(x)$, i.e. for all $n\in\mathbb{N}$ the theory of valued fields entails $$({\rm SO}_{\phi})\longrightarrow({\rm SO}_{\phi_{n}})$$ and $$({\rm SM}_{\phi})\longrightarrow({\rm SM}_{\phi_{n}}).$$ This fact is used several times in the proof of . Making a definition uniform --------------------------- In the rest of this section we allow ourselves to write $({\rm CM}_{\phi(x)})$ (rather than simply $({\rm CM}_{\phi})$), when needed, to highlight the rôle of the variable $x$ and to allow us the flexibility to substitute other terms in its place. \[lem:bounded.roots.compactness\] Let $\mathcal{F}$ be an elementary class of $C$-fields. Let $\phi(x)$ be an $\exists$-$\mathcal{L}_{\rm vf}(C)$-formula with free variable $x$ belonging to the field sort. Suppose that for all $F\in\mathcal{F}$ there exists $n_{F}\in\mathbb{N}$ such that for all $(K,v)\models\mathbf{T}_{F}$ we have $$\mathfrak{m}_{v}^{(n_{F})}\subseteq\phi(K)\qquad\left(\text{resp. }\mathcal{O}_{v}^{(n_{F})}\subseteq\phi(K)\right).$$ Then there exists $n\in\mathbb{N}$ such that for all $(K,v)\models\mathbf{T}_{\mathcal{F}}$ we have $$\mathfrak{m}_{v}^{(n)}\subseteq\phi(K)\qquad\left(\text{resp. }\mathcal{O}_{v}^{(n)}\subseteq\phi(K)\right).$$ This is a straightforward compactness argument. We first note that, for a valued field $(K,v)$, we have $\mathfrak{m}_{v}^{(n)}\subseteq\phi(K)$ if and only if $(K,v)\models({\rm CM}_{\phi(x^{n})})$. Next we consider the $\mathcal{L}_{\rm vf}(C)$-theory $$\mathbf{T}_{\rm CM}:=\left\{\neg({\rm CM}_{\phi(x^{n})})\;\Big|\;n\in\mathbb{N}\right\}.$$ Now let $(K,v)\models\mathbf{T}_{\mathcal{F}}$. Then $F:=Kv\in\mathcal{F}$ and, by assumption, we have $\mathfrak{m}_{v}^{(n_{F})}\subseteq\phi(K)$, i.e. $(K,v)\models({\rm CM}_{\phi(x^{n_{F}})})$. In particular we have $(K,v)\not\models\mathbf{T}_{\rm CM}$. This shows that $\mathbf{T}_{\mathcal{F}}\cup\mathbf{T}_{\rm CM}$ is inconsistent. By compactness, $\mathbf{T}_{\rm CM}$ is finitely inconsistent; thus there exists $M\in\mathbb{N}$ such that $$\mathbf{T}_{\mathcal{F}}\models\bigvee_{m<M}({\rm CM}_{\phi(x^{m})}).$$ Finally, we let $n:=M!$. For $(K,v)\models\mathbf{T}_{\mathcal{F}}$ there exists $m<M$ such that $(K,v)\models({\rm CM}_{\phi(x^{m})})$, so since $m|n$, we have that $$\mathfrak{m}_{v}^{(n)}\subseteq\mathfrak{m}_{v}^{(m)}\subseteq\phi(K),$$ as required to complete the proof of the first statement. The second statement (i.e. the ‘respectively’ version) is entirely analogous and the proof is exactly the same except we exchange $\mathfrak{m}$ for $\mathcal{O}$, and $({\rm CM})$ for $({\rm CO})$, etc. Now we prove the main result of this section, , by repeatedly using to ‘transfer’ several key properties of $\phi$ which were introduced in . We will frequently use without further comment. As noted earlier, the following is simply a rephrasing of using different notation. \[prp:GDM.version.2\] Let $\mathcal{F}$ be an elementary class of $C$-fields that satisfies $(*)$. Let $\phi(x)$ be an $\exists$-$\mathcal{L}_{\mathrm{ring}}(C)$-formula. 1. [**($\mathcal{O}$-case)**]{} If for all $F\in\mathcal{F}$ the theory $\mathbf{T}_{F}\cup\{({\rm SO}_{\phi}),({\rm CO}_{\phi})\}$ is consistent, then there is an $\exists$-$\mathcal{L}_{\mathrm{ring}}(C)$-formula $\psi(x)$ such that $$\mathbf{T}_{\mathcal{F}}\models({\rm SO}_{\psi})\wedge({\rm CO}_{\psi}).$$ 2. [**($\mathfrak{m}$-case)**]{} If for all $F\in\mathcal{F}$ the theory $\mathbf{T}_{F}\cup\{({\rm SM}_{\phi}),({\rm CM}_{\phi})\}$ is consistent, then there is an $\exists$-$\mathcal{L}_{\mathrm{ring}}(C)$-formula $\chi(x)$ such that $$\mathbf{T}_{\mathcal{F}}\models({\rm SM}_{\chi})\wedge({\rm CM}_{\chi}).$$ **Step 1. For the moment we work in both cases. Our aim is to find a formula $\chi(x)$ (by a simple adaptation of the formula $\phi(x)$) such that $\mathbf{T}_{\mathcal{F}}\models({\rm CM}_{\chi})$.** Let $F\in\mathcal{F}$. Recall the $\exists$-$\mathcal{L}_{\rm vf}(C)$-sentence $({\rm IM}_{\phi})$. In each case our assumption entails that $\mathbf{T}_{F}\cup\{({\rm IM}_{\phi})\}$ is consistent, hence $\mathbf{T}_{F}\models({\rm IM}_{\phi})$ by (2). In particular we have $(F(t)^{h},v_{t})\models({\rm IM}_{\phi})$. By applying we find $n_{F}\in\mathbb{N}$ such that $$\mathfrak{m}_{v_{t}}^{(n_{F})}\subseteq\Phi(F(t)^{h}),$$ where $\Phi(x)$ denotes the $\exists$-$\mathcal{L}_{\rm ring}(C)$-formula $$\exists y\exists z\left(x=y-z\wedge\phi(y)\wedge\phi(z)\right).$$ In particular, we have $$(F(t)^{h},v_{t})\models\Phi(t^{n_{F}})\wedge\Phi(0^{n_{F}}).$$ Therefore, by , $$\mathbf{T}_{F}\models\left({\rm CM}_{\Phi(x^{n_{F}})}\right),$$ i.e. for all $(K,v)\models\mathbf{T}_{F}$ we have $$\mathfrak{m}_{v}^{(n_{F})}\subseteq\Phi(K).$$ Next, by applying , there exists $n\in\mathbb{N}$ such that for all $(K,v)\models\mathbf{T}_{\mathcal{F}}$ we have $$\mathfrak{m}_{v}^{(n)}\subseteq\Phi(K).$$ Finally, we let $\chi(x)$ be the $\exists$-$\mathcal{L}_{\rm ring}(C)$-formula $\Phi(x^{n})$ and rewrite the previous statement as $$\mathbf{T}_{\mathcal{F}}\models\left({\rm CM}_{\chi}\right),$$ which is the desired conclusion. **Step 2 ($\mathfrak{m}$-case). Let $F\in\mathcal{F}$. Consider the $\forall$-$\mathcal{L}_{\rm vf}(C)$-sentence $({\rm SM}_{\phi})$. Trivially, our assumption entails that $\mathbf{T}_{F}\cup\{({\rm SM}_{\phi})\}$ is consistent, hence $\mathbf{T}_{F}\models({\rm SM}_{\phi})$ by (3). Since every model of $\mathbf{T}_{\mathcal{F}}$ is a model of $\mathbf{T}_{F}$ for some $F\in\mathcal{F}$, we deduce that $\mathbf{T}_{\mathcal{F}}\models({\rm SM}_{\phi})$.** As noted in , the theory of valued fields entails $({\rm SM}_{\phi})\longrightarrow({\rm SM}_{\chi})$. Therefore $\mathbf{T}_{\mathcal{F}}\models({\rm SM}_{\chi})$. This completes the ($\mathfrak{m}$-case). **Step 2 ($\mathcal{O}$-case). To begin with, this step is similar to the ($\mathfrak{m}$-case). Let $F\in\mathcal{F}$. Consider the $\forall$-$\mathcal{L}_{\rm vf}(C)$-sentence $({\rm SO}_{\phi})$. Trivially, our assumption entails that $\mathbf{T}_{F}\cup\{({\rm SO}_{\phi})\}$ is consistent, hence $\mathbf{T}_{F}\models({\rm SO}_{\phi})$ by (3). Since every model of $\mathbf{T}_{\mathcal{F}}$ is a model of $\mathbf{T}_{F}$ for some $F\in\mathcal{F}$, we deduce that $\mathbf{T}_{\mathcal{F}}\models({\rm SO}_{\phi})$. As noted in , the theory of valued fields entails $({\rm SO}_{\phi})\longrightarrow({\rm SO}_{\chi})$, hence $\mathbf{T}_{\mathcal{F}}\models({\rm SO}_{\chi})$. Combining this with Step 1, we have that $$\mathbf{T}_{\mathcal{F}}\models({\rm SO}_{\chi})\wedge({\rm CM}_{\chi}).$$ Let $F\in\mathcal{F}$ and let $p\geq1$ be the characteristic exponent of $F$. Consider the $\forall^{k}\exists$-$\mathcal{L}_{\rm vf}(C)$-sentence $({\rm R}_{\phi})$. Our assumption entails that $\mathbf{T}_{F}\cup\{({\rm R}_{\phi})\}$ is consistent. By (1) there exists $m_{F}\in\mathbb{N}$ such that for all $(K,v)\models\mathbf{T}_{F}$ we have $$(Kv)^{(p^{m_{F}})}\subseteq\phi(K)v.$$** Let $\psi'(x)$ be the $\exists$-$\mathcal{L}_{\rm ring}(C)$-formula $$\exists y\exists z\;(x=y+z\wedge\phi(y)\wedge\chi(z)).$$ We will show that $\mathcal{O}_{v}^{(p^{m_{F}})}\subseteq\psi'(K)$ for all $(K,v)\models\mathbf{T}_{F}$. Let $a\in\mathcal{O}_{v}$. Since $(Kv)^{(p^{m_{F}})}\subseteq\phi(K)v$, there exists $b\in\phi(K)$ such that $(av)^{p^{m_{F}}}=bv$. Let $c:=a^{p^{m_{F}}}-b$. Then $c\in\mathfrak{m}_{v}\subseteq\chi(K)$. Thus $a^{p^{m_{F}}}=b+c\in\phi(K)+\chi(K)=\psi'(K)$, as required. By applying , there exists $m\in\mathbb{N}$ such that $$\mathcal{O}_{v}^{(m)}\subseteq\psi'(K),$$ for all $(K,v)\models\mathbf{T}_{\mathcal{F}}$. If we let $\psi(x)$ be the formula $\psi'(x^{m})$ then we may rewrite the previous statement as $\mathbf{T}_{\mathcal{F}}\models({\rm CO}_{\psi})$. Finally, since valuation rings are integrally closed and we have already seen that $\mathbf{T}_{\mathcal{F}}\models({\rm SO}_{\phi})\wedge({\rm SO}_{\chi})$, we deduce that $\mathbf{T}_{\mathcal{F}}\models({\rm SO}_{\psi})$, as required. For a class $\mathcal{C}$ of $C$-fields or valued $C$-fields we denote by $U\mathcal{C}$ the closure of $\mathcal{C}$ under ultraproducts and by $E\mathcal{C}$ the closure of $\mathcal{C}$ under elementary equivalence, and in the case of valued $C$-fields we let $\mathcal{C}v:=\{Kv:(K,v)\in\mathcal{C}\}$. \[Cor:KtoHeF\] Let $\mathcal{K}$ be a class of equicharacteristic henselian nontrivially valued $C$-fields and let $\mathcal{F}$ be the smallest elementary class of $C$-fields that contains $\mathcal{K}v$. Suppose that $\mathcal{F}$ satisfies $(*)$. Let $\phi(x)$ be an $\exists$-$\mathcal{L}_{\mathrm{ring}}(C)$-formula. 1. If $\phi(K)=\mathcal{O}_v$ for all $(K,v)\in\mathcal{K}$, then there exists an $\exists$-$\mathcal{L}_{\mathrm{ring}}(C)$-formula $\psi(x)$ with $\psi(K)=\mathcal{O}_v$ for all $(K,v)\in\mathcal{H}_e'(\mathcal{F})$. 2. If $\phi(K)=\mathfrak{m}_v$ for all $(K,v)\in\mathcal{K}$, then there exists an $\exists$-$\mathcal{L}_{\mathrm{ring}}(C)$-formula $\chi(x)$ with $\chi(K)=\mathfrak{m}_v$ for all $(K,v)\in\mathcal{H}_e'(\mathcal{F})$. We show how to deduce (1) from (1). The proof of (2) from (2) is completely analogous. Let $\mathcal{K}^{\dagger}$ be the class of equicharacteristic henselian nontrivially valued $C$-fields $(K,v)$ for which $$\label{eqn:1} (\dagger)\qquad\mathcal{O}_{v}=\phi(K)$$ holds, and let $\mathcal{F}^{\dagger}$ denote the smallest elementary class of $C$-fields containing $\mathcal{K}^{\dagger}v$. Then $\mathcal{K}\subseteq\mathcal{K}^{\dagger}$ and $\mathcal{F}\subseteq\mathcal{F}^{\dagger}=EU(\mathcal{K}^{\dagger}v)$, cf. [@CK Exercise 4.1.18]. Note that $\mathcal{K}^\dagger$ is closed under ultraproducts (i.e. $U\mathcal{K}^\dagger=\mathcal{K}^\dagger$) since $(\dagger)$ is an $\mathcal{L}_{\rm vf}(C)$-elementary property and $\mathcal{K}^{\dagger}$ is an $\mathcal{L}_{\rm vf}(C)$-elementary class. Thus, $$\mathcal{F}\subseteq\mathcal{F}^{\dagger}=EU(\mathcal{K}^{\dagger}v)\stackrel{\autoref{lem:residue.ultraproduct}}{=}E((U\mathcal{K}^{\dagger})v)=E(\mathcal{K}^{\dagger}v).$$ That is, for every $F\in\mathcal{F}$ there exists $(K,v)\in\mathcal{K}^{\dagger}$ such that $F\equiv Kv$. Then $(K,v)\models\mathbf{T}_{F}\cup\{{\rm(SO_{\phi}))},{\rm(CO_{\phi})}\}$, so the hypotheses of (1) are satisfied, the conclusion of which is $(1)$. Main Theorem {#section:main.theorem} ============ In this section we bring together the results developed so far into one main theorem and draw some first conclusions. We state both cases (i.e. $\exists$- and $\forall$-definitions of the valuation ring) simultaneously. Recall that $\forall$-definitions of the valuation ring correspond to $\exists$-definitions of the valuation ideal, cf. \[rem:EAOm\]. \[thm:main\] Let $\mathcal{K}$ be a class of equicharacteristic henselian nontrivially valued $C$-fields, let $\mathcal{F}$ be the smallest elementary class of $C$-fields that contains $\mathcal{K}v$, and suppose that $$\label{star} \begin{minipage}{11cm} {\it \begin{enumerate} \item[(a)] $C$ is integral over its prime ring, \underline{or} \item[(b)] $C$ is a perfect field and every $F\in\mathcal{F}$ is perfect. \end{enumerate} } \end{minipage}$$ Then for $Q\in\{\exists,\forall\}$ the following holds: 1. The following properties are equivalent: 1. The valuation ring is uniformly $Q$-$C$-definable in $\mathcal{K}$. 2. The valuation ring is uniformly $Q$-$C$-definable in $\mathcal{H}^{'}_{e}(\mathcal{F})$. 3. The valuation ring is uniformly $Q$-$C$-definable in $\mathcal{H}_{e}(\mathcal{F})$. 4. $\mathcal{F}$ does not have equicharacteristic embedded residue, if $Q=\exists$ (resp. is not equicharacteristic large, if $Q=\forall$). 2. Also the following properties are equivalent: 1. The valuation ring is uniformly $Q$-$C$-definable in $\mathcal{H}^{'}(\mathcal{F})$. 2. The valuation ring is uniformly $Q$-$C$-definable in $\mathcal{H}(\mathcal{F})$. 3. $\mathcal{F}$ does not have embedded residue, if $Q=\exists$ (resp. is not large, if $Q=\forall$). 3. Moreover, if $\mathcal{F}$ is unmixed, i.e. does not both contain fields of characteristic zero and fields of positive characteristic, then all seven conditions $(0^{Q}_{e})$, $(1^{'Q}_{e})$, $(1^{'Q})$, $(1^{Q}_{e})$, $(1^{Q})$, $(2^{Q}_{e})$, and $(2^{Q})$ are equivalent. The strategy of proof is summarized in the following diagram: $$\xymatrix{ (0^{Q}_{e}) \ar@/^0.5pc/@{=>}[rr]^{\ref{Cor:KtoHeF}} \ar@/_0.5pc/@{<=}[rr]_{\text{trivial}} & & (1^{'Q}_{e}) \ar@/^1.0pc/@{=>}[rrrr]^{\ref{lem:ER.implies.not.uniform}\;/\;\ref{lem:large.implies.not.uniform}} \ar@/_0.5pc/@{<=}[rr]_{\text{trivial}} \ar@/^0.0pc/@{<=}[dd]_{\text{trivial}} & & (1^{Q}_{e}) \ar@/_0.5pc/@{<=}[rr]_{\ref{lem:no.ER.implies.uniform}\;/\;\ref{lem:not.large.implies.uniform}} \ar@/^0.0pc/@{<=}[dd]_{\text{trivial}} & & (2^{Q}_{e}) \ar@/_0.5pc/@{<=}[dd]_{\text{trivial}} \ar@/^0.5pc/@{=>}[dd]^{\mathcal{F}\text{ unmixed: } \ref{cor:existential.characterisation.one.characteristic}\;/\;\ref{cor:universal.characterisation.one.characteristic}} \\ \\ & & (1^{'Q}) \ar@/_1.0pc/@{=>}[rrrr]_{\ref{lem:ER.implies.not.uniform}\;/\;\ref{lem:large.implies.not.uniform}} \ar@/^0.5pc/@{<=}[rr]^{\text{trivial}} & & (1^{Q}) \ar@/^0.5pc/@{<=}[rr]^{\ref{lem:no.ER.implies.uniform}\;/\;\ref{lem:not.large.implies.uniform}} & & (2^{Q}) }$$ Since $\mathcal{K}\subseteq\mathcal{H}^{'}_{e}(\mathcal{F})\subseteq\mathcal{H}_{e}(\mathcal{F})\subseteq\mathcal{H}(\mathcal{F})$ and $\mathcal{H}^{'}_{e}(\mathcal{F})\subseteq\mathcal{H}'(\mathcal{F})\subseteq\mathcal{H}(\mathcal{F})$, we have five of the trivial implications $(1^{'Q}_{e})\Longrightarrow(0^{Q}_{e})$, $(1^{Q}_{e})\Longrightarrow(1^{'Q}_{e})$, $(1^{Q})\Longrightarrow(1^{Q}_{e})$, $(1^{'Q})\Longrightarrow(1^{'Q}_{e})$, and $(1^{Q})\Longrightarrow(1^{'Q})$. Also $(2^{Q})\Longrightarrow(2^{Q}_{e})$ is immediate from the definition (cf. \[rem:unmixed\]). By applying \[Cor:KtoHeF\] we have the implication $(0^{Q}_{e})\Longrightarrow(1^{'Q}_{e})$. For $Q=\exists$ (resp. $Q=\forall$), both $(1^{'Q}_{e})\implies(2^{Q}_{e})$ and $(1^{'Q})\implies(2^{Q})$ follow from \[lem:ER.implies.not.uniform\] (resp. \[lem:large.implies.not.uniform\]); and both $(2^{Q}_{e})\implies(1^{Q}_{e})$ and $(2^{Q})\implies(1^{Q})$ follows from \[lem:no.ER.implies.uniform\] (resp. \[lem:not.large.implies.uniform\]). Finally, by \[cor:existential.characterisation.one.characteristic\] (resp. \[cor:universal.characterisation.one.characteristic\]), if $\mathcal{F}$ is unmixed then we have $(2^{Q}_{e})\Longrightarrow(2^{Q})$. Note that the theorem can also be applied if we start with an elementary class of $C$-fields $\mathcal{F}$, as we can always find a suitable class of valued $C$-fields $\mathcal{K}$ as in the theorem, e.g. $\mathcal{K}=\mathcal{H}^{'}_{e}(\mathcal{F})$. The case of a single $C$-field $F$ ---------------------------------- If we restrict our attention to henselian nontrivially valued fields with residue field elementarily equivalent to a given $F$, the situation becomes particularly nice. The following corollary is immediate from . \[cor:single.F\] Let $F$ be any $C$-field that satisfies $(*)$.\ The following are equivalent. 1. The valuation ring is $\exists$-$C$-definable in some $(K,v)\in\mathcal{H}_{e}'(F)$. 2. The valuation ring is uniformly $\exists$-$C$-definable in $\mathcal{H}(F)$. 3. $F$ does not have embedded residue. Also the following are equivalent. 1. The valuation ring is $\forall$-$C$-definable in some $(K,v)\in\mathcal{H}_{e}'(F)$. 2. The valuation ring is uniformly $\forall$-$C$-definable in $\mathcal{H}(F)$. 3. $F$ is not large. Let $\mathcal{F}$ be the class of $C$-fields elementarily equivalent to $F$. As explained in \[rem:unmixed\], $\mathcal{F}$ is unmixed. The result is immediate from \[thm:main\]. Note that Theorem from the introduction follows immediately from \[cor:single.F\] and \[lemma:Schoutens\] in the special case $C=\mathbb{Z}$. See also \[rem:EAOm\]. The necessity of the assumptions -------------------------------- The assumption ‘unmixed’ cannot simply be removed from part (iii) of , as the following example shows. \[eg:unmixed.necessary.E\] In this example we set $Q:=\exists$ and $C:=\mathbb{Z}$; that is, we are studying the definability of valuation rings by existential formulas without parameters. Let $p$ be a fixed prime. Trivially, $\mathbb{F}_p$ does not have embedded residue, see also (1) below. Later, in \[prop:positive.applications\](6) we will show that also $\mathbb{Q}$ does not have embedded residue, i.e. $\mathcal{F}_0:=\{F:F\equiv\mathbb{Q}\}$ does not have embedded residue. So, since $\mathbb{F}_p$ and $\mathbb{Q}$ have different characteristics, the elementary class $\mathcal{F}=\{\mathbb{F}_p\}\cup\mathcal{F}_0$ does not have equicharacteristic embedded residue. On the other hand, the $p$-adic valuation on $\mathbb{Q}$ has residue field $\mathbb{F}_p$, so $\mathcal{F}$ does have embedded residue. Therefore, $\mathcal{F}$ satisfies $(2_e^\exists)$ but not $(2^\exists)$. Also the assumption in that the valued fields in $\mathcal{K}$ are ‘equicharacteristic’ cannot be omitted, as the following example shows: Again we let $Q:=\exists$ and $C:=\mathbb{Z}$. Fix a prime $p>2$ and let $F:=\mathbb{F}_p^{\rm alg}$ be an algebraic closure of $\mathbb{F}_p$. By \[lem:sep.cl\], $F$ has embedded residue, so the valuation ring is not $\exists$-$\emptyset$-definable in any $(K,v)\in\mathcal{H}_e'(F)$, by \[cor:single.F\]. However, it is $\exists$-$\emptyset$-definable in some $(K,v)\in\mathcal{H}'(F)$: For example, take the maximal unramified extension $K:=\mathbb{Q}_p^{\rm ur}$ of $\mathbb{Q}_p$ with the extension $v_p$ of the $p$-adic valuation. Then $Kv_{p}=F$ and Julia Robinsons’s formula $\exists y\;(y^2=1+px^2)$, which defines the valuation ring in $(\mathbb{Q}_p,v_{p})$ (see the introduction), also defines the valuation ring in $(K,v_{p})$. That is, for $\mathcal{K}=\{(K,v_p)\}$, $(0_e^Q)$ holds but $(1_e^Q)$ does not hold. Mixed classes and families of local fields ------------------------------------------ We now focus on a special case in which the ‘unmixed’ assumption in part (iii) of \[thm:main\] can be removed, and we apply this to draw a conclusion on uniform existential definability of valuation rings in local fields, continuing the theme of results developed in [@Cluckersetal] and [@Fehm]. We work in the existential case, i.e. $Q=\exists$. Let $\mathcal{F}$ be an elementary class of $C$-fields. If $\mathcal{F}$ is assumed to be unmixed then claim (iii) of \[thm:main\] shows that all seven conditions (i.e. $(0^{\exists}_{e})$, $(1^{'\exists}_{e})$, $(1^{'\exists})$, $(1^{\exists}_{e})$, $(1^{\exists})$, $(2^{\exists}_{e})$, and $(2^{\exists})$) are equivalent. On the other hand, \[eg:unmixed.necessary.E\] shows that the seven conditions may be inequivalent if we do not assume that $\mathcal{F}$ is unmixed. However, if $\mathcal{F}$ is an elementary class of PAC fields then all seven conditions are necessarily equivalent, as the following observation shows: \[lem:PAC.nearly.unmixed\] Let $\mathcal{F}$ be an elementary class of finite or PAC $C$-fields. If $\mathcal{F}$ has embedded residue, then some $F\in\mathcal{F}$ has embedded residue. Suppose that $\mathcal{F}$ has embedded residue, i.e. there are $F_{1},F_{2}\in\mathcal{F}$ and a nontrivial valuation $v$ on $F_{1}$ such that $F_{1}v$ embeds into $F_{2}$. As $v$ is nontrivial, $F_1$ is not finite, hence $F_1$ is PAC, and so $F_1v$ is an algebraically closed $C$-field, cf. \[lem:residue.of.sep.closed\] and [@FriedJarden Corollary 11.5.5]. So since $F_1v$ embeds into $F_2$, shows that $F_{2}$ has embedded residue. \[cor:QpZp\] Let $P$ be a set of prime numbers. The following are equivalent. 1. There exists an $\exists$-$\mathcal{L}_{\mathrm{ring}}$-formula $\phi(x)$ such that $\phi(\mathbb{Q}_{p})=\mathbb{Z}_{p}$ for all $p\in P$. 2. There exists an $\exists$-$\mathcal{L}_{\mathrm{ring}}$-formula $\phi(x)$ such that $\phi(\mathbb{F}_{p}((t)))=\mathbb{F}_{p}[[t]]$ for all $p\in P$. 3. There exists an $\exists$-$\mathcal{L}_{\mathrm{ring}}$-formula $\phi(x)$ such that $\phi(\mathbb{Q}_{p})=\mathbb{Z}_{p}$ and $\phi(\mathbb{F}_{p}((t)))=\mathbb{F}_{p}[[t]]$ for all $p\in P$. Clearly $(3)\Longrightarrow(1),(2)$. $(1)\Longrightarrow (2)$: By the Ax-Kochen transfer principle [@AxKochenI], $\phi(\mathbb{F}_{p}((t)))=\mathbb{F}_{p}[[t]]$ for $p\in P_0$, with $P_0$ cofinite in $P$. By \[thm:main\](i), the smallest elementary class $\mathcal{F}_0$ containing all $\mathbb{F}_p$, $p\in P_0$, does not have equicharacteristic embedded residue. Therefore, also the elementary class $\mathcal{F}:=\mathcal{F}_0\cup\{\mathbb{F}_p:p\in P\setminus P_0\}$ does not have equicharacteristic embedded residue, so \[thm:main\](i) shows that (2) holds. $(2)\Longrightarrow (3)$: By \[thm:main\](i), the smallest elementary class $\mathcal{F}$ containing all $\mathbb{F}_p$, $p\in P$, does not have equicharacteristic embedded residue. Since it consists of fields that are finite or pseudofinite (hence PAC, by [@FriedJarden Corollary 11.3.4]), \[lem:PAC.nearly.unmixed\] gives that $\mathcal{F}$ does not have embedded residue. Therefore, \[thm:main\](ii) shows that (3) holds. Note that by the Ax-Kochen transfer principle, \[cor:QpZp\] is well-known if we replace ‘for all’ by ‘for almost all’. The implication $(1)\Longrightarrow(2)$ can be derived explicitly by combining this with the results from [@AnscombeKoenigsmann]. We do, however, not know of any other proof of the implication $(2)\Longrightarrow(1)$. We remark that one could use the explicit proof of $(1)\Longrightarrow(2)$ to construct the formula in $(3)$ to satisfy in addition that $\phi(\mathbb{Q}_{p})=\mathbb{Z}_{p}$ if and only if $\phi(\mathbb{F}_{p}((t)))=\mathbb{F}_{p}[[t]]$. We will explore this and further questions of uniform definability of valuation rings in local fields in a forthcoming paper. Examples and applications {#section:examples} ========================= Diophantine valuation rings: examples and counterexamples --------------------------------------------------------- We now explore $\mathbb{Z}$-fields with and without embedded residue. We begin by collecting all the known examples of fields without embedded residue, which gives a common generalization of all the results in the literature of fields with nontrivial diophantine henselian valuation rings. \[prop:positive.applications\] In each of the following cases, a $\mathbb{Z}$-field $F$ does not have embedded residue: 1. $F$ is finite, 2. $F$ is PAC and does not contain a separably closed subfield, 3. $F$ is PRC of characteristic zero and does not contain a real closed subfield, 4. $F$ is P$p$C of characteristic zero and does not contain a $p$-adically closed subfield, for some prime number $p$, 5. $F$ is pseudo-classically closed[^4] of characteristic zero and does not contain any real closed or $p$-adically closed subfield (for any $p$), 6. $F=\mathbb{Q}$. (1): The class $\{F\}$ is elementary and does not have embedded residue since finite fields do not admit nontrivial valuations. (2): The residue field $v$ of any nontrivial valuation $v$ on $F_1\equiv F$ is separably closed, see [@FriedJarden Corollary 11.5.5], hence cannot be embedded into any $F_2\equiv F$. (5): Let $w$ be a nontrivial valuation on some $F_1\equiv F$. By [@FehmLGP Corollary 1.5], also $F_1$ is PCC, i.e. it satisfies a local-global principle with respect to a family $\mathcal{F}$ of separably closed, real closed or $p$-adically closed algebraic extensions. Let $F_1^h$ be a henselization of $F_1$ with respect to $w$. Then [@Pop Theorem 2.9] implies that $F_1^h\in\mathcal{F}$, so $F_1w=F_1^hw$ is the residue field of a separably, real or $p$-adically closed field with respect to some henselian valuation and is therefore finite, separably closed, real closed or $p$-adically closed itself. The same then applies to the algebraic part of $F_1w$. Thus, since $F$ is by assumption of characteristic zero and does not contain a separably closed, real closed or $p$-adically closed subfield, there is no embedding of $F_1w$ into any $F_2\equiv F$. (3): If $F$ is PAC, the claim follows from (2). If $F$ is PRC but not PAC, then $F$ is contained in a real closed field. As real closed fields have no $p$-adically closed subfields, the claim then follows from (5). (4): If $F$ is PAC, the claim follows from (2). If $F$ is P$p$C but not PAC, then $F$ is contained in a $p$-adically closed field. As $p$-adically closed fields have no real closed subfields, the claim then follows from (5). (6): Let $v$ be a nontrivial valuation on $F_1\equiv\mathbb{Q}$. By Lagrange’s Four Squares Theorem [@Hardy-Wright79 Theorem 369], the sums of four squares in $F_1$ form the positive cone of an ordering $<_{F_1}$ (which is then necessarily the only ordering of $F_1$), so that we have $(\mathbb{Q},<)\equiv(F_1,<_{F_1}$). If $F_1v$ can be embedded into $F_2\equiv \mathbb{Q}$, then $F_1v$ carries an ordering (namely the restriction of $<_{F_2}$), from which we deduce by the Baer-Krull Theorem [@EP Theorem 2.2.5] that $\mathfrak{m}_v$ is convex with respect to $<_{F_1}$. Since $\mathbb{Q}$ is dense in $\mathbb{R}$ and $(F_1,<_{F_1})\equiv(\mathbb{Q},<)$, we have $$(F_1,<_{F_1})\models\;\forall\varepsilon\!>\!0\;\exists a (0\leq 2-a^{2}<\varepsilon).$$ So, for $\varepsilon\in\mathfrak{m}_{v}$ with $\varepsilon>_{F_1}0$ there exists $a\in F_1$ such that $0\leq_{F_1}2-a^{2}<_{F_1}\varepsilon$. Since $\mathfrak{m}_{v}$ is convex, $2-a^{2}\in\mathfrak{m}_{v}$, so applying the residue map gives $(av)^{2}=2$. Thus $F_1v\subseteq F_2\equiv\mathbb{Q}$ contains a square root of $2$, a contradiction. Combining this with we immediately get the following corollary: \[cor:positive.applications\] For each field $F$ in the list of , there is an $\exists$-$\mathcal{L}_{\rm ring}$-formula which uniformly defines the valuation ring in $\mathcal{H}(F)$. This corollary was known before in special cases $(K,v)\in\mathcal{H}(F)$ (without the uniformity statement): For fields $F$ in (1) by [@Fehm Theorem 2.6] (generalizing the earlier special cases $K=\mathbb{Q}_p$ by [@Robinson p. 303], $K$ a finite extension of $\mathbb{Q}_p$ by [@Cluckersetal Theorem 6], and $K=\mathbb{F}_q((t))$ by [@AnscombeKoenigsmann Theorem 1.1]), for fields $F$ in (2) by [@Fehm Theorem 3.5] and for fields $F$ in (3) by [@Fehm Corollary 3.6]. To the best of our knowledge, it is new for the fields $F$ in (4)-(6). We note that for certain special cases of (6), for example for $K=\mathbb{Q}((t))$, it follows from more general results in [@Prestel] that the valuation ring is both $\exists\forall$-$\emptyset$-definable and $\forall\exists$-$\emptyset$-definable. In the forthcoming work [@AF3] we extend the method used in the proof of (6) of to further study residue fields of valuations on nonstandard number fields. In particular, we show that no number field has embedded residue. \[prp:perfect.hull.embedded.residue\] Let $F$ be a $\mathbb{Z}$-field. If the perfect hull $F^{\mathrm{perf}}$ has embedded residue then $F$ has embedded residue. If $F$ does not have embedded residue, then by \[cor:single.F\] there is an $\exists$-$\mathcal{L}_{\rm ring}$-formula $\phi(x)$ that defines $\mathcal{O}_{v_t}$ in $K:=F((t))$. Since $K^{\rm perf}=\bigcup_{n\in\mathbb{N}}K^{p^{-n}}$ and $K^{p^{-n}}\cong K$, the same formula $\phi(x)$ defines the valuation ring of the unique extension $v_t^{\rm perf}$ of $v_t$ to $K^{\rm perf}$. Since $K^{\rm perf}v_t^{\rm perf}=F^{\rm perf}$, \[cor:single.F\] gives that $F^{\rm perf}$ does not have embedded residue. \[prop:negative.applications\] A $\mathbb{Z}$-field $F$ has embedded residue in each of the following cases: 1. $F$ contains the separable closure of the prime field. 2. $F$ admits a nontrivial henselian valuation. 3. $F$ is separably closed, real closed, or $p$-adically closed. 4. $F$ is a proper purely transcendental extension of some subfield $F_0$. \(1) is a restatement of in the special case $C=\mathbb{Z}$. (2): Let $(F,v)$ be a henselian nontrivially valued field. By , we may assume that $F$ is perfect. We can furthermore assume that $(F,v)$ is $\aleph_0$-saturated, so if $v$ is of mixed characteristic, then $v(\mathbb{Z}\setminus\{0\})$ is contained in a proper convex subgroup of $vF$, hence we can replace $v$ by an equicharacteristic henselian coarsening. If $\mathcal{X}$ is a transcendence base of $Fv$ over its prime field $\mathbb{F}$, then the embedding $\mathbb{F}\rightarrow F$ extends to the relative separable closure of $\mathbb{F}(\mathcal{X})$ in $Fv$, cf. [@AnscombeFehm Lemma 2.3]. Since $F$ is perfect, this extends further to an embedding of $Fv$ into $F$, hence $F$ has embedded residue. (3): If $F$ is separably closed, then the claim follows from (1). If $F$ is $p$-adically closed, then the claim follows from (2). If $F$ is real closed, then any sufficiently saturated elementary extension of $F$ admits a henselian valuation, so the claim follows from (2). (4): Without loss of generality, $F=F_0(t)$, in which case $Fv_t=F_0\subseteq F$. \[cor:negative.applications\] For each field $F$ in the list of , there is no $\exists$-$\mathcal{L}_{\rm ring}$-formula which defines the valuation ring in any $(K,v)\in\mathcal{H}_e'(F)$. This corollary was known at least for certain $(K,v)\in\mathcal{H}_e'(F)$ for fields $F$ in (1) by [@Fehm Remark 3.8], and for fields $F$ in (2) and (3) by [@AnscombeKoenigsmann Observation A.1] and [@FehmPrestel Proposition 4.6 and Example 5.4]. Diophantine valuation ideals: Examples and counterexamples, and Pop’s large fields {#sec:large} ---------------------------------------------------------------------------------- In this section we drop our general assumption that all fields are $C$-fields. Let $F$ be a field. By an $F$-variety we mean a separated scheme $X$ of finite type over ${\rm Spec}(F)$. For a field extension $E$ of $F$ we denote by $$X(E)={\rm Hom}_{{\rm Spec}(F)}({\rm Spec}(E),X)$$ the set of $E$-rational points of $X$. Each $x\in X(E)$ gives a scheme theoretic point $\tilde{x}=x(\mathfrak{p})\in X$, where $\mathfrak{p}$ is the unique prime ideal of $E$, and when we speak of closure or denseness of a subset $A\subseteq X(E)$ in $X$ we in fact mean the closure or denseness of the corresponding set $\tilde{A}=\{\tilde{x}:x\in A\}\subseteq X$. Recall that $F$ is [*large*]{} (in the sense of Pop) if it satisfies any of the following equivalent conditions, cf. [@Pop96 Proposition 1.1]: \[fact:Pop.large\] Let $F$ be a field. Then the following are equivalent: 1. Every integral $F$-curve with a smooth $F$-rational point has infinitely many such points. 2. For every smooth integral $F$-variety $X$, the set $X(F)$ is empty or dense in $X$. 3. $F$ is existentially closed in the henselization $F(t)^{h}$. 4. $F$ is existentially closed in the Laurent series field $F((t))$. For further information on large fields and their relevance in various areas see for example [@Jarden; @BF; @Popsurvey]. As we will see in , $F$ is a large field if and only if $F$ is a large $F$-field in the sense of (where we view $F$ as an $F$-field via the identity map). Recall that for a $C$-field $F$ we denote by $C_F$ the quotient field of the image of $C\rightarrow F$. We have to restrict to cases in which sufficiently general versions of resolution of singularities or its local form, local uniformization, have been proven, and here we do not strive for maximal generality: \[lem:localuniformization\] Let $F$ be a $C$-field and let $E\subseteq F(t)^h$ be a finitely generated extension of $C_F$ not contained in $F$. Then there exists a finite extension $E'$ of $E$ which is the function field of a smooth integral $C_F$-variety $X$ with $X(F)\neq\emptyset$ if one of the following conditions holds: ** 1. $C$ is integral over its prime ring 2. $C$ is a perfect field and $F$ is perfect 3. $F=C_F$ We start with case (c). In this case, $F(t)^h$ is regular and of transcendence degree $1$ over $F=C_F$, hence $E$ is the function field of a geometrically integral $C_F$-curve $X_0$, which always has a smooth projective model $X$. The restriction of the $t$-adic valuation to $E$ corresponds to an $F$-rational point on $X$. In cases (a) and (b), $C_F$ is perfect and the restriction of the $t$-adic valuation to $E$ gives a place with residue field $F_0$ contained in $F$. By Temkin’s inseparable local uniformization [@Temkin Theorem 1.3.2] there exists a finite purely inseparable extension $E'$ of $E$ such that $E'$ is the function field of a smooth integral $C_F$-variety $X$ with an $F_0'$-rational point, where $F_0'$ is a finite purely inseparable extension of $F_0$. If (a) holds and ${\rm char}(F)=p>0$, then $X$ is defined over some finite field $\mathbb{F}_q$, hence the $q$-Frobenius fixes $X$; but $(F_0')^{q^k}\subseteq F$ for sufficiently large $k$, hence $X(F)\neq\emptyset$. If ${\rm char}(F)=0$ or (b) holds, then $F_0'\subseteq F$, so $X(F)\neq\emptyset$. \[prp:large.1\] Let $F$ be a $C$-field and suppose that one of the conditions (a), (b) or (c) of \[lem:localuniformization\] holds. Then the following are equivalent: 1. $F$ is a large $C$-field (see ). 2. For every smooth integral $C_F$-variety $X$, the set $X(F)$ is empty or dense in $X$. 3. $F$ and $F(t)^{h}$ have the same $\exists$-$\mathcal{L}_{\rm ring}(C)$-theory. 4. $F$ and $F((t))$ have the same $\exists$-$\mathcal{L}_{\rm ring}(C)$-theory. 5. There is a $C$-embedding $F(t)^{h}\longrightarrow F^{*}$, for some elementary extension $F\preceq F^*$. 6. There is a $C$-embedding $F((t))\longrightarrow F^{*}$, for some elementary extension $F\preceq F^*$. $(3)\Longleftrightarrow (4)$: This follows from $F(t)^{h}\preceq_{\exists}F((t))$, cf. [@AnscombeFehm Lemma 4.5]. $(3)\Longleftrightarrow (5)$ and $(4)\Longleftrightarrow (6)$: These are simple compactness arguments. $(5)\Longrightarrow(1)$: This is immediate from the definition , as $E=F(t)^h$ admits the henselian valuation $v_t$ with residue field $F$. $(1)\Longrightarrow(5)$: Suppose that $F$ is a large $C$-field, i.e. there exist $C$-fields $F_{1},F_{2}\equiv_{C}F$, a $C$-subfield $E\subseteq F_{1}$, and a nontrivial henselian $C$-valuation $v$ on $E$ with residue field $F_{2}$. By passing to elementary extensions if necessary, we may assume that $E$, $F_{1}$ and $F_{2}$ are $|F|^{+}$-saturated. In particular we may assume that both $F_1$ and $F_{2}$ are elementary extensions of the $C$-field $F$, in particular $C_{F_1}=C_{F_2}=C_F$. Note that $v$ is trivial on $C_{F_1}$ (cf. ), so the residue map is the identity on $C_{F_1}=C_{F}$. Since $F/C_{F}$ is separable in each of the cases (a)-(c), and $E$ is $|F|^{+}$-saturated, the identity $C_F\rightarrow C_{F_1}$ extends to a partial section $f:F\longrightarrow E$ of the residue map, cf. [@AnscombeFehm Lemma 2.3]; this means in particular that $f$ is an $\mathcal{L}_{\rm vf}(C)$-homomorphism $(F,v_0)\rightarrow (E,v)$, where $v_0$ is the trivial valuation. Since $v$ is nontrivial on $E$, there exists $s\in\mathfrak{m}_{v}\setminus\{0\}$, and $f$ extends to a homomorphism $f:(F(t),v_t)\rightarrow(E,v)$ by $f(t):=s$. Since $v$ is henselian, this extends further to a homomorphism $F(t)^h\rightarrow E\subseteq F_1$. $(2)\Longrightarrow(3)$: Suppose that $(2)$ holds and let $\varphi$ be an $\exists$-$\mathcal{L}_{\rm ring}(C)$-sentence. Without loss of generality, $\varphi$ is of the form $(\exists\mathbf{x})\bigwedge_{i=1}^r f_i(\mathbf{x})=0$ with $f_1,\dots,f_r\in C[X_1,\dots,X_n]$, so there is some closed subset $X_0\subseteq\mathbb{A}^n_{C_F}$ so that $\varphi$ holds in an extension $E$ of $C_F$ if and only if $X_0(E)\neq\emptyset$. Trivially, if $X_0(F)\neq\emptyset$, then $X_0(F(t)^h)\neq\emptyset$. Conversely, let $x\in X_0(F(t)^h)$. Then $\tilde{x}\in X_0$ is the generic point of an integral $C_F$-variety $X_1\subseteq X_0$. If $x$ is $F$-rational, i.e. $x$ factors through ${\rm Spec}(F(t)^h)\rightarrow{\rm Spec}(F)$, then $X_0(F)\neq\emptyset$ and we are done. Otherwise, the residue field $E:=\kappa(\tilde{x})\hookrightarrow F(t)^h$ of $x$ (which is the function field of $X_1$) satisfies the assumptions of \[lem:localuniformization\], so there exists a smooth integral $C_F$-variety $X$ with $X(F)\neq\emptyset$ and a dominant rational map $X\dashrightarrow X_1$, i.e. a non-empty open subvariety $X'$ of $X$ and a morphism $\pi:X'\rightarrow X_1$. By $(2)$, $X(F)$ is dense in $X$, hence $X'(F)\neq\emptyset$, and thus $X_0(F)\supseteq X_1(F)\supseteq\pi(X'(F))\neq\emptyset$. $(4)\Longrightarrow(2)$: Suppose that $(4)$ holds, let $X$ be a smooth integral $C_F$-variety with $x\in X(F)$, and let $X_0\subseteq X$ be a non-empty open subvariety. Without loss of generality assume that $X_0$ is affine, so there exists an $\exists$-$\mathcal{L}_{\rm ring}(C)$-sentence $\varphi$ such that $\varphi$ holds in an extension $E$ of $C_F$ if and only if $X_0(E)\neq\emptyset$. The base change $X_F:=X\times_{{\rm Spec}(C_F)}{\rm Spec}(F)$ is a smooth $F$-variety, and $x$ induces an $F$-rational point $x_F$ on $X_F$, which lies on some irreducible component $Y$ of $X_F$. Note that $Y\cap X_0$ is a non-empty open subset of $Y$. The local ring $\mathcal{O}_{Y,\tilde{x}_F}$ is contained in a discrete valuation ring $\mathcal{O}$ of the function field $F(Y)$ with residue field $F$, cf. [@JardenRoquette Lemma A.1]. The completion of $\mathcal{O}$ is then $F$-isomorphic to $F[[t]]$. Since $Y(F(Y))$ is dense in $Y$, so is $Y(F((t)))$, and therefore $(Y\cap X_0)(F((t)))\neq\emptyset$, in particular $F((t))\models\varphi$. Thus, $(4)$ implies that also $F\models\varphi$, i.e. $X_0(F)\neq\emptyset$. \[cor:large\] Let $F$ be a field. Then $F$ is a large field if and only if $F$ is a large $F$-field. In particular, if $F$ is a $C$-field which is large as a field, then $F$ is a large $C$-field. The first claim follows by comparing (4) and (4) for $C=F$, in which case (c) is satisfied. The second claim is a direct consequence of this, since if $F$ is large as an $F$-field, then trivially also as a $C$-field. Let $F$ be a field. Then there exists a $\exists$-$\mathcal{L}_{\rm ring}(F)$-formula that defines $\mathfrak{m}_{v_t}$ in $F((t))$ if and only if $F$ is not a large field. If we assume that $F$ is perfect, then this follows immediately from and . Without this assumption, we can argue as follows: If $F$ is large, then $F$ is existentially closed in $F((t))$ by (4), so if $\varphi(x)$ is an $\exists$-$\mathcal{L}_{\rm ring}(F)$-formula that defines $\mathfrak{m}_{v_t}$ in $F((t))$, then $F((t))\models \exists x(x\neq 0\wedge\varphi(x))$ implies that there exists $0\neq x\in F\cap \mathfrak{m}_{v_t}$, a contradiction. If $F$ is not large, then $F$ is not a large $F$-field by , hence the valuation ideal is uniformly $\exists$-$F$-definable in $\mathcal{H}_e(F)$ by , so in particular in $(F((t)),v_t)\in\mathcal{H}_e(F)$. We note that in cases (a) and (b), $F(t)^h$ and $F((t))$ have the same $\exists$-$\mathcal{L}_{\rm ring}(C)$-theory as any $(K,v)\in\mathcal{H}_e'(F)$, by . Therefore, in this case, (3) and (4) of allow several further equivalent formulations. We do not know whether the statements in \[prp:large.1\] are also equivalent to the following statement analogous to (1) of \[fact:Pop.large\]: 1. Every $C_F$-curve with a smooth $F$-rational point has infinitely many such points. We do not intend to give a comprehensive study of large $C$-fields here, but we do want to give one sufficient condition for a $C$-field to be large, which also serves as an illustration of the differences between large fields and large $C$-fields. \[lem:large.is.E-C.closed\] Let $E\subseteq F$ be an extension of $C$-fields with the same existential $\mathcal{L}_{\rm ring}(C)$-theory. Then $E$ is a large $C$-field if and only if $F$ is a large $C$-field. Note that $C_E=C_F$ and let $X$ be a smooth integral $C_F$-variety. For any open subvariety $X_0$ of $X$ (including $X_0=X$), there is an existential $\mathcal{L}_{\rm ring}(C)$-sentence $\varphi_{X_0}$ such that for an extension $F'$ of $C_F$, $F'\models\varphi_{X_0}$ if and only if $X_0(F')\neq\emptyset$. &gt;From this the claim follows immediately. \[lem:PAC.large\] Let $F/E$ be a separable extension of $C$-fields and suppose that $E$ is PAC. Then $F$ is a large $C$-field. Let $\tilde{E}$ denote the relative algebraic closure of $E$ in $F$. Since $E$ is PAC, so is $\tilde{E}$, and since $F/E$ is separable, $F/\tilde{E}$ is regular. Thus, $\tilde{E}\preceq_{\exists}F$, see [@FriedJarden 11.3.5], so in particular $\tilde{E}$ and $F$ have the same existential $\mathcal{L}_{\rm ring}(C)$-theory. Since PAC fields are large, $\tilde{E}$ is a large $C$-field (). The conclusion now follows from . We continue by collecting some examples of $\mathbb{Z}$-fields that are or are not large. \[prop:positive.large\] In each of the following cases, a $\mathbb{Z}$-field $F$ is not large: 1. $F$ is finite. 2. $F=\mathbb{Q}$ 3. $F$ is finitely generated over its prime field. We prove (3), of which (1) and (2) are special cases. Let $\mathbb{F}$ denote the prime field of $F$, and $F_0$ the relative algebraic closure of $\mathbb{F}$ in $F$, and choose a tower of fields $F_0\subseteq\dots\subseteq F_n=F$ such that $F_i=F_{i-1}(X_i)$ with a smooth projective geometrically integral $F_{i-1}$-curve $X_i$, for $i=1,\dots,n$. Choose a smooth projective geometrically integral $\mathbb{F}$-curve $X$ with $X(\mathbb{F})\neq\emptyset$ of genus $g_X>1$ and $g_X>g_{X_i}$ for $i=1,\dots,n$. Then $X(F)\supseteq X(\mathbb{F})\neq\emptyset$, and $X(F)=X(F_n)=\dots=X(F_0)$ by the Riemann-Hurwitz theorem (cf. [@Jarden Lemma 6.1.3]). This latter set of points is finite and not dense, since either $F_0$ is finite, or $F_0$ is a number field, in which case this follows from Falting’s theorem. Since $\mathbb{Z}_F=\mathbb{F}$, (2) shows that the $\mathbb{Z}$-field $F$ is not large. \[cor:positive.large\] For each field $F$ in the list of , there is an $\exists$-$\mathcal{L}_{\rm ring}$-formula which uniformly defines the valuation ideal in $\mathcal{H}(F)$. Again this corollary was known before for certain fields in (1): For $K$ a finite extension of $\mathbb{Q}_p$ by [@Cluckersetal Theorem 6], and for $K=\mathbb{F}_q((t))$ by [@AnscombeKoenigsmann proof of Proposition 3.3]). \[prop:negative.applications.large\] A $\mathbb{Z}$-field $F$ is large in each of the following cases: 1. $F$ contains the separable closure of the prime field. 2. $F$ admits a nontrivial henselian valuation. 3. $F$ is separably closed, real closed, or $p$-adically closed. 4. $F$ is pseudo-classically closed. \(1) is a restatement of in the special case $C=\mathbb{Z}$. (2) and (4) follow from since both henselian fields and pseudo-classically closed fields are large fields, see e.g. [@Jarden Example 5.6.2, Example 5.6.4]. (3) is a special case of (4). \[cor:negative.applications.large\] For each field $F$ in the list of , there is no $\exists$-$\mathcal{L}_{\rm ring}$-formula which defines the valuation ideal in any $(K,v)\in \mathcal{H}_e'(F)$. Application: Diophantine henselian valuation rings and valuation ideals on a given field ---------------------------------------------------------------------------------------- In this final subsection we combine our main result with some of the examples that we just collected to show that any given field admits at most one nontrivial equicharacteristic henselian valuation for which the valuation ring or the valuation ideal is diophantine. For this we use the notion of the [*canonical henselian valuation*]{} as defined in [@EP §4.4]: If $H_1(K)$ resp. $H_2(K)$ denotes the set of henselian valuations on $K$ with residue field not separably closed resp. separably closed, then the canonical henselian valuation $v_K$ is the unique coarsest valuation in $H_2(K)$, if $H_2(K)\neq\emptyset$, and otherwise the unique finest valuation in $H_1(K)$. \[thm:classification\] Let $K$ be a field and $v$ a nontrivial equicharacteristic henselian valuation on $K$. If $\mathcal{O}_v$ or $\mathfrak{m}_v$ are $\exists$-$\emptyset$-definable, then 1. $v$ is the canonical henselian valuation on $K$, 2. $v\in H_{1}(K)$ and $H_{2}(K)=\emptyset$, and 3. no other nontrivial henselian valuation on $K$ has $\exists$-$\emptyset$-definable valuation ring or valuation ideal. We treat the $\exists$-case and the $\forall$-case simultaneously. By , the $\mathbb{Z}$-field $Kv$ does not have embedded residue resp. is not large, so (3) resp. (3) shows that $Kv$ cannot be separably closed. Thus, $v\in H_{1}(K)$. Furthermore, by (2) resp. (2), $Kv$ cannot admit a nontrivial henselian valuation, so $v$ does not admit any proper henselian refinements. Thus $v$ is the canonical henselian valuation, so $v_K=v\in H_1(K)$, which implies by definition that $H_{2}(K)=\emptyset$. Moreover, all other henselian valuations on $K$ are coarsenings of $v$, in particular also equicharacteristic, so the argument applies to them as well. We remark that all four possibilities of $\mathcal{O}_{v_K}$ and $\mathfrak{m}_{v_K}$ being diophantine or not diophantine can occur. The following examples of fields $K=F((t))$ with $v_K=v_t$ demonstrate this. $F$ $K$ $\mathcal{O}_{v_K}$ diophantine $\mathfrak{m}_{v_K}$ diophantine ----------------------- ---------------------------- --------------------------------- ---------------------------------- $\mathbb{Q}$ $\mathbb{Q}((t))$ yes ((6)) yes ((2)) $\mathbb{Q}(x)$ $\mathbb{Q}(x)((t))$ no ((4)) yes ((3)) $\mathbb{Q}^{\rm tr}$ $\mathbb{Q}^{\rm tr}((t))$ yes ((3)) no ((4)) $\mathbb{C}$ $\mathbb{C}((t))$ no ((3)) no ((3)) Here, $\mathbb{Q}^{\rm tr}$ is the field of totally real algebraic numbers, which is a PRC field. We end this work with a discussion of the equicharacteristic assumption in , for which we restrict our attention to the valuation ring: If a $\mathbb{Z}$-field $F$ does not have embedded residue, then the valuation ring is uniformly $\exists$-$\emptyset$-definable in $\mathcal{H}(F)$. In particular, the valuation ring is uniformly $\exists$-$\emptyset$-definable in the mixed-characteristic henselian valued fields $(K,v)$ with $Kv\equiv F$. However, in mixed characteristic $(0,p)$ the converse does not hold and there are other reasons why a valuation ring might be $\exists$-$\emptyset$-definable: For example, if $v(p)$ is minimal positive, then Julia Robinson’s formula mentioned in the introduction defines the valuation ring, and this can easily be extended to valuations with so-called finite initial ramification. The following example shows that a field can indeed admit more than one $\exists$-$\emptyset$-definable nontrivial mixed characteristic henselian valuation: Let $F=\mathbb{F}_p((t))^{\rm perf}$ and let $K$ be the field of fractions of the Witt vectors over $F$, see e.g. [@Serre Chapter II §5]. Then $K$ carries a discrete henselian valuation $u$ with uniformizer $p$ and residue field $F$, and $\mathcal{O}_u$ is $\exists$-$\emptyset$-definable using Julia Robinson’s formula. But $v_t\circ u$ is a henselian valuation on $K$ with residue field $\mathbb{F}_p$, hence $\mathcal{O}_{v_t\circ u}$ is $\exists$-$\emptyset$-definable by \[cor:positive.applications\](1). Thus, all [*three*]{} henselian valuation rings on $K$ (including the trivial one) are diophantine. Since all $\exists$-$\emptyset$-definitions of nontrivial henselian valuation rings in the literature exploit either properties of the residue field or finite initial ramification, it seems natural to suspect that non-embedded residue and finite initial ramification are the only two reasons why such a valuation ring can be diophantine. Since at most one of the henselian valuations on a field can have residue field without embedded residue, and at most one of them can have finite initial ramification, we would like to pose the following question: Do all fields admit [*at most three*]{} diophantine henselian valuation rings? Acknowledgements {#acknowledgements .unnumbered} ================ The authors would like to thank Hans Schoutens for suggesting Lemma \[lemma:Schoutens\], and Florian Pop for suggesting Proposition \[prop:positive.applications\](5). They would also like to thank Immanuel Halupczok, Franziska Jahnke, Jochen Koenigsmann, Dugald Macpherson, Sebastian Petersen, and Alexander Prestel for frequent helpful discussions and encouragement. [CDLM13]{} Sylvy Anscombe and Arno Fehm. The existential theory of equicharacteristic henselian valued fields. arXiv:1501.04522 \[math.LO\], 2015. Sylvy Anscombe and Arno Fehm. Residue fields of valuations on nonstandard number fields. Manuscript, 2016. 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Every two elementarily equivalent models have isomorphic ultrapowers. 10(2):224–233, 1971. Alexandra Shlapentokh. On diophantine definability and decidability in some infinite totally real extensions of $\mathbb{Q}$. 356(8):3189–3207, 2003. Henning Stichtenoth. . Springer, 2009. Michael Temkin. Inseparable local uniformization. 373:65–119, 2013. [^1]: \ During this research the first author was funded by EPSRC grant EP/K020692/1. [^2]: This notion of a large $C$-field is related to the notion of a large field in the sense of Pop [@Pop96]. We discuss this connection in Section \[sec:large\]. [^3]: It would suffice that for every $\mathfrak{p}\in{\rm Spec}(C)$ the local ring $(C_\mathfrak{p},\mathfrak{p}C_\mathfrak{p})$ is dominated by a valuation ring $(\mathcal{O},\mathfrak{m})$ with $\mathcal{O}/\mathfrak{m}=C_\mathfrak{p}/\mathfrak{p}C_\mathfrak{p}$. This condition is satisfied more generally for Prüfer domains (like the ring of algebraic integers $C=\mathbb{Z}^{\rm alg}$) and for regular Noetherian domains (like $C=\mathbb{Z}[t]$). [^4]: See [@Pop] for the definition of a PCC field. We note that the class of PCC fields contains in particular all PRC fields and all P$p$C fields.

--- abstract: | The main aim of the present set of notes is to give new, short and essentially self-contained proofs of some classical, as well as more recent, results about random walks on groups. For instance, we shall see that the drift characterization of Liouville groups, due to Kaimanovich-Vershik and Karlsson-Ledrappier (and to Varopoulos in some important special cases) admits a very short and quite elementary proof. Furthermore, we give a new, and rather short proof of (a weak version of) an observation of Kaimanovich (as well as a small strengthening thereof) that the Poisson boundary of any symmetric measured group $(G,\mu)$, is doubly ergodic, and the diagonal $G$-action on its product is ergodic with unitary coefficients. We also offer a characterization of weak mixing for ergodic $(G,\mu)$-spaces parallel to the measure-preserving case. We shed some new light on Nagaev’s classical technique to prove central limit theorems for random walks on groups. In the interesting special case when the measured group admits a product current, we define a Besov space structure on the space of bounded harmonic functions with respect to which the the associated convolution operator is quasicompact without any assumptions on finite exponential moments. For Gromov hyperbolic measured groups, this gives an alternative proof of the fact that every Hölder continuous function with zero integral with respect to the unique stationary probability measure on the Gromov boundary is a co-boundary. Finally, we give a new and almost self-contained proof of a special case of a recent combinatorial result about piecewise syndeticity of product sets in groups by the author and A. Fish. address: 'Department of Mathematics, Chalmers, Gothenburg, Sweden' author: - Michael Björklund title: Five remarks about random walks on groups --- [^1] Drifts of random walks and the Liouville property ================================================= The study of random walks on countable groups is to a large extent concerned with the asymptotic behavior of convolution powers of some fixed probability measures on the groups. One is particularly interested in the growth of the integrals of certain geometrically defined functions against these convolution powers. For instance, let $G$ be a countable group and let $\mu$ be a probability measure on $G$ with the property that the support of $\mu$ generates $G$ as a semigroup. We shall refer to $(G,\mu)$ as a *measured group*. Given a left $G$-invariant and $\mu$-integrable metric $d$ on $G$, we define the *drift* by $$\ell_d(\mu) = \lim_n \frac{1}{n} \int_G d(g,e) \, d\mu^{*n}(g),$$ where $\mu^{*n}$ denotes the $n$-th convolution power of $\mu$. An elementary sub-additivity argument (Fekete’s Lemma) guarantees that the limit exists and is finite. From a probabilistic point of view, the drift $\ell_d(\mu)$ measures the asymptotic linear speed (with respect to $d$) of a sequence of products of independent and $\mu$-distributed elements in $G$. Since metrics are symmetric functions on $G \times G$, we always have $\ell_d({\check{\mu}}) = \ell_d(\mu)$, where ${\check{\mu}}(g) = \mu(g^{-1})$ for all $g$ in $G$. We say that $\mu$ is a *symmetric* probability measure (and $(G,\mu)$ is a *symmetric measured group*) if ${\check{\mu}}= \mu$. We note that if $G$ is generated by a finite (symmetric) set $S$, then the word metric $d_S$ with respect to $S$ has the property that for every left $G$-invariant metric $d$, there exists a constant $C$ such that the inequality $d(g,e) \leq C \cdot d_S(g,e)$ holds for all $g \in G$, so in particular, if $\ell_{d_S}(\mu) = 0$ for some probability measure $\mu$, then $\ell_d(\mu) = 0$ as well. Our aim in this section is to give a characterization of those finitely generated and symmetric measured groups $(G,\mu)$ with $\ell_d(\mu) = 0$ for *some* (and hence any) word-metric on $G$ in terms of bounded left $\mu$-harmonic functions. Recall that a $\mu$-integrable complex-valued function $f$ on $G$ is *left $\mu$-harmonic* if it satisfies $$\label{defharmonic} ({\check{\mu}}* f)(g) = \int_G f(sg) \, d{\check{\mu}}(s), \quad \forall \, g \in G,$$ and it is *right $\mu$-harmonic* if $$\label{defharmonicright} (f * \mu)(g) = \int_G f(gs) \, d\mu(s), \quad \forall \, g \in G.$$ If $f$ is both right *and* left $\mu$-harmonic, then we say that $f$ is *bi-$\mu$-harmonic*. Of special interest to us are the *bounded* left $\mu$-harmonic functions on $G$. Clearly, constant functions on $G$ are left $\mu$-harmonic for every choice of $\mu$, and we say that $(G,\mu)$ is (left) *Liouville* if there are no non-constant left $\mu$-harmonic functions. Since the function $\check{f}(g) = f(g^{-1})$ is right $\mu$-harmonic if and only if $f$ is left-$\mu$-harmonic, we see that the notions of left and right Liouville coincide. The original proof of the following theorem combined a series of fundamental observations of Avez [@A72], Derriennic [@D80], Kaimanovich-Vershik [@KV83] and Karlsson-Ledrappier [@KL07] respectively. In the special case when $\mu$ is finitely supported, Varopoulos established this theorem in [@V85]. We shall give a short proof of the general theorem below. \[drift\] Let $(G,\mu)$ be a finitely generated and symmetric measured group and suppose $d$ is a word metric with respect to a finite symmetric generating set for $G$. Then $\ell_d(\mu) = 0$ if and only if $(G,\mu)$ is Liouville. Recall that a countable group $G$ is *amenable* if every action of $G$ by homeomorphims on a compact hausdorff space $X$ admits a $G$-invariant probability measure. It is not hard to see that finite groups and the group of integers are amenable, and that the class of amenable groups is closed under extensions and direct unions, which immediately shows that every finite extension of a solvable group is amenable. Furthermore, every finitely generated group of sub-exponential growth can be shown to be amenable, while free groups on at least two generators, and countable supergroups thereof are non-amenable. Suppose $(G,\mu)$ is a countable *non-amenable* measured group and let $X$ denote a compact Hausdorff space, equipped with an action of $G$ by homeomorphisms with no $G$-invariant probability measures. A simple application of Kaktuani’s fixed point argument shows that there is always a probability measure $\nu$ on $X$ which satisfies the equation $$\int_G \int_X \phi(g^{-1} x) \, d\nu(x) d\mu(g) = \int_X \phi(x) \, d\nu(x)$$ for all $\phi \in C(X)$. Since, by assumption, $\nu$ cannot be $G$-invariant, there exists at least one $\phi \in C(X)$ such that the function $$f(g) = \int_X \phi(g^{-1}x) \, d\nu(x), \quad g \in G,$$ is *not* constant. It is readily verified that $f$ is left $\mu$-harmonic and thus $(G,\mu)$ is *not* Liouville. In particular, in view of Theorem \[drift\], if $G$ is a finitely generated *non-amenable* group, then $\ell_d(\mu) > 0$ for *every* word-metric $d$ on $G$ and (symmetric) probability measure $\mu$. Liouville implies zero drift {#subsecLD} ---------------------------- The proof of Theorem \[drift\] splits naturally into two parts. The first one concerns the “if”-direction, for which the relevant result can be stated as follows. \[KL\] If $(G,\mu)$ is a measured Liouville group and $d$ is a left invariant and $\mu$-integrable metric $d$ on $G$, then there exists a real-valued and $\mu$-integrable homomorphism $u$ on $G$ such that $$\ell_d(\mu) = \int_G u(g) \, d{\check{\mu}}(g).$$ In particular, if $\mu$ is a symmetric probability measure on $G$, then $\ell_d(\mu) = 0$ for every left invariant and $\mu$-integrable metric $d$ on $G$. Let us now outline a simple proof of this theorem, which has some similarities with the arguments in the paper [@EK10], where are more quantitative version of Theorem \[KL\] is proved. We first note that by the triangle-inequality, the sequence $$n \mapsto \int_G \big( d(g,x) - d(x,e) \big) \, d\mu^{*k}(x)$$ is bounded for every fixed $g$ in $G$, so by a simple diagonal argument, there exists a sub-sequence $(n_j)$ such that the limit $$u(g) = \lim_{j {\rightarrow}\infty} \frac{1}{n_j} \sum_{k=0}^{n_j-1} \int_G \big( d(g,x) - d(x,e) \big) \, d\mu^{*k}(x)$$ exists for all $g \in G$ and thus, $$\int_G u(sg) \, d{\check{\mu}}(s) = u(g) + \int_G u(s) \, d{\check{\mu}}(s), \quad \forall \, g \in G.$$ We shall refer to functions $u$ with this property as *left quasi-$\mu$-harmonic*. By dominated convergence, we have $$\begin{aligned} \int_G u(s) \, d{\check{\mu}}(s) &=& \lim_{j {\rightarrow}\infty} \frac{1}{n_j} \sum_{k=0}^{n_j-1} \Big( \int_G d(x,e) \, d\mu^{*(k+1)}(x) - \int_G d(x,e) \, d\mu^{*k}(x) \Big) \\ &=& \lim_{j {\rightarrow}\infty} \frac{1}{n_j} \int_G d(x,e) \, d\mu^{*n_j}(x) = \ell_d(\mu).\end{aligned}$$ Furthermore, the triangle inequality guarantees that the function $u$ is left Lipschitz, i.e. for every element $g$ in $G$, we have $$\sup_{x \in G} |u(xg) - u(x)| < + \infty.$$ The theorem of Karlsson-Ledrappier is now an immediate consequence of the following proposition, which is interesting in its own right. If $(G,\mu)$ is a Liouville measured group, then every left Lipschitz and left $\mu$-quasi-harmonic function on $G$ which vanishes at the identity is a homomorphism. Note that if $u$ is a left Lipschitz and left $\mu$-quasi-harmonic function on $G$, then for every $g \in G$, the function $$v_g(x) = u(xg) - u(x), \quad x \in G,$$ is a *bounded* left $\mu$-harmonic function on $G$, and hence constant. Since $u(e) = 0$, we conclude that $$u(xg) - u(x) = u(g)$$ for all $g, x \in G$, that is to say, $u$ is a homomorphism. In particular, a Liouville group $(G,\mu)$ without homomorphisms into the additive group of the real numbers cannot admit any non-constant *left Lipschitz* and *left (quasi) $\mu$-harmonic* functions. However, we shall see in Appendix I, that *every* countable symmetric measured group $(G,\mu)$ always admits a non-constant *left Lipschitz* and *right $\mu$-harmonic* function. Zero drift implies Liouville ---------------------------- We now tend to the proof of the “only if”-direction in Theorem \[drift\], for which the relevant result can be stated as follows. \[KV\] Let $(G,\mu)$ be a finitely generated measured group and suppose $d$ is a left-invariant word metric on $G$ with respect to a finite symmetric generating set. If $\ell_d(\mu) = 0$, then $(G,\mu)$ is Liouville. The classical route to this theorem employs the entropy theory of measured groups. One first shows that if $\ell_d(\mu) < +\infty$ for some (and hence any) word metric $d$ on $G$, then the limit $$h(G,\mu) = \lim_{n {\rightarrow}\infty} -\frac{1}{n} \sum_{g \in G} \mu^{*n}(g) \cdot \log \, \mu^{*n}(g)$$ exists and is finite. We refer to $h(G,\mu)$ as the (Avez) *entropy* of the measured group $(G,\mu)$, and the main result of Kaimanovich-Vershik in [@KV83] asserts (under the assumption that $\mu$ is symmetric and the Avez entropy is finite) that $h(G,\mu) = 0$ if and only if $(G,\mu)$ is Liouville. One is thus left with the task of showing that $\ell_d(\mu) = 0$ implies $h(G,\mu) = 0$. This is taken care of what is sometimes referred to as the “fundamental” inequality (see e.g. Section 4 in [@KL07]), which we now formulate. Let $S$ be a finite and symmetric generating set for $G$ and let $d$ be the word metric associated to $S$. If one defines the *exponential volume growth* of $(G,S)$ by $$v(G,S) = \varlimsup_{n {\rightarrow}\infty} \frac{\log |S^n|}{n} \leq \log|S|$$ then $$\label{fund} h(G,\mu) \leq v(G,S) \cdot \ell_d(\mu) = 0,$$ which finishes the (classical) proof of Theorem \[KV\]. Note that the argument gives a bit more, namely that if $G$ has subexponential growth, i.e. $v(G,S) = 0$, then $h(G,\mu) = 0$ and thus $(G,\mu)$ is Liouville. By Theorem \[KL\] and the remark following its statement, we conclude that $G$ is amenable and $\ell_d(\mu) = 0$. We shall now give a new alternative (and self-contained) proof of Theorem \[KV\] which avoids the use of entropy theory. Let $(G,\mu)$ be a measured group and denote by ${\mathcal{H}}^\infty_l(G,\mu)$ the space of all bounded left $\mu$-harmonic functions on $G$. We say that a Borel probability space $(X,\nu)$ is a *$(G,\mu)$-space* if $X$ is equipped with an action of $G$ by bi-measurable maps, which all preserve the measure class of $\nu$, such that $$\int_G \int_X \phi(g^{-1}x) \, d\nu(x) \, d\mu(g) = \int_X \phi(x) \, d\nu(x)$$ for all $\phi \in L^\infty(X,\nu)$. Probability measures with this property are often referred to as *$\mu$-harmonic* (or *$\mu$-stationary*). Given a $\mu$-harmonic probability measure $\nu$ on $X$, one readily checks that the association $$P_\nu\phi(g) = \int_X \phi(g^{-1}x) \, d\nu(x), \quad g \in G.$$ defines an element in ${\mathcal{H}}^\infty_l(G,\mu)$ for every $\phi \in L^\infty(X,\nu)$. We note that if $\nu$ is $G$-invariant, then such elements are all constants. A remarkable fact, often attributed to Furstenberg, is that every measured group $(G,\mu)$ admits a $(G,\mu)$-space $(B,m)$, which we shall refer to to as the *Poisson boundary of $(G,\mu)$*, for which the linear map $P_m$ above is in fact *isometric and onto* ${\mathcal{H}}^\infty_l(G,\mu)$. There are many constructions of the Poisson boundary of a measured group in the literature. We refer the reader to [@Fu] for a detailed exposition of one of the more elementary constructions. \[furstenberg\] For every measured group $(G,\mu)$ there exists an ergodic $(G,\mu)$-space $(B,m)$, which we shall refer to as the Poisson boundary of $(G,\mu)$, such that the Poisson transform $P : L^\infty(B,m) {\rightarrow}{\mathcal{H}}_l^{\infty}(G,\mu)$ defined by $$P\phi(g) = \int_B \phi(g^{-1}b) \, dm(b), \quad g \in G,$$ is an isometric isomorphism. In particular, $(B,m)$ is trivial if and only if $m$ is $G$-invariant. As was pointed out by Jaworski in [@Ja94], the Poisson boundary $(G,m)$ is strongly approximately transitive (SAT), i.e. for every measurable subset $A \subset B$ of positive $m$-measure and for every ${\varepsilon}> 0$, there exists $g \in G$ such that $m(gA) > 1 - {\varepsilon}$. Indeed, since $P$ is isometric, if $A \subset B$ has positive $m$-measure, then $$\sup_{g \in G} P\chi_A(g) = \sup_{g \in G} \int_B \chi_A(g^{-1}b) \, dm(b) = \sup_{g \in G} m(gA) = 1,$$ from which the SAT-property follows. Since the $G$-action on $(B,m)$ preserves the measure class of $m$ (i.e. the set of all null-sets for $m$), the Radon-Nikodym derivative $$\sigma_m(g,b) = \frac{dg^{-1}m}{dm}(b)$$ is a well-defined non-negative element in $L^1(B,m)$ for every $g \in G$, and one readily checks that $$\label{multcocycle} \sigma_m(g_1g_2,b) = \sigma_m(g_1,b) \, \sigma_m(g_2,g_1b)$$ for all $g_1, g_2 \in G$ and for almost every $b$ with respect to $m$, that is to say $\sigma_m$ is a multiplicative cocycle for the $G$-action on $(B,m)$. A crucial feature with this cocycle is its $\mu$-harmonicity, namely $$\label{cocycleharm} \int_G \sigma_m(g,b) \, d\mu^{*n}(g) = 1, \quad \textrm{for a.e. $b$ and for all $n \geq 1$}.$$ Indeed, for every $\phi \in L^\infty(B,m)$, one has $$\int_B \Big( \int_G \sigma_m(g,b) \, d\mu^{*n}(g) \Big) \, \phi(b) \, dm(b) = \int_G \int_B \phi(g^{-1}b) \, dm(b) \, d\mu^{*n}(g) = \int_B \phi \, dm,$$ for all $n$, which immediately yields .\ After a simple use of Jensen’s inequality and , Theorem \[KV\] can now be reformulated as follows. Let $(G,\mu)$ be a finitely generated measured group with Poisson boundary $(B,m)$ and suppose $\ell_d(\mu) = 0$ for some (and hence any) word-metric with respect to a finite symmetric generating set. Then $m$ is $G$-invariant, or equivalently $$\int_G \int_B \log \frac{dg^{-1}m}{dm}(b) \, dm(b) \, d\mu(g) = 0.$$ We note that equation implies that $\sigma_m(g,\cdot)$ is not only in $L^1(B,m)$ for every $g$, but is in fact essentially bounded. This can be seen as follows. Since we assume that the support of $\mu$ generates $G$ as a semigroup, there exists for every $s$ in $G$, an integer $n$ such that the measure $\mu^{*n}(s)$ is positive, and thus $$\sigma_m(s,b) \mu^{*n}(s) \leq \int_G \sigma_m(g,b) \, d\mu^{*n}(g) = 1$$ for $m$-almost every $b$ in $B$. We conclude that $$\|\sigma_m(s,\cdot)\|_\infty \leq \frac{1}{\mu^{*n}(s)} < +\infty,$$ where $n$ is chosen as above, and $$\sigma_m(s,\cdot) \geq \frac{1}{\sigma_m(s^{-1},s \cdot )} \geq \frac{1}{\|\sigma_m(s^{-1},\cdot)\|_\infty}, \quad \forall \, s \in G.$$ In particular, if we define $$c_m(g,b) = -\log \sigma_m(g,b),$$ then one can think of $c_m$ as a map from $G$ into $L^\infty(B,m)$, which satisfies the equations $$c_m(g_1 g_2) = c_m(g_1) +g_1^{-1} c_m(g_2), \quad \forall \, g_1, g_2 \in G,$$ where $G$ acts on $L^\infty(B,m)$ via the left regular representation. We shall refer to such maps $c$ from $G$ into $L^\infty(B,m)$ as *cocycles*, and one readily checks that if $c$ is a cocycle, then $$\rho_c(g) = \|c(g)\|_\infty, \quad g \in G,$$ defines a semi-metric on $G$, i.e. $$\rho_c(g_1,g_2) \leq \rho_c(g_1) + \rho_c(g_2), \quad \forall \, g_1, g_2 \in G.$$ We reserve the notation $\rho_m$ for the semi-metric associated to the cocycle $c_m$ above and refer to $\rho_m$ as the *canonical semi-metric on $G$ associated with $(G,\mu)$*. In order to get a better feeling for the canonical semi-metric of a measured group, let us consider the case when $G = {\mathbb{F}}_2$, the free group on two free generators $a$ and $b$, equipped with the (symmetric) probability measure $$\mu = \frac{1}{4}\big( \delta_a + \delta_b + \delta_{a^{-1}} + \delta_{b^{-1}} \big).$$ Let $\partial {\mathbb{F}}_2$ denote the compact space of all infinite one-sided reduced words in $a$ and $b$ and their inverses and note that the action of ${\mathbb{F}}_2$ on itself extends to an action by homeomorphisms on $\partial {\mathbb{F}}_2$. One readily checks that the Borel probability measure $m$ on $\partial {\mathbb{F}}_2$ which assigns the same measure to all cylinder sets in $\partial {\mathbb{F}}_2$ corresponding to words of the same word length is $\mu$-harmonic, and one can prove that $(\partial {\mathbb{F}}_2,m)$ realizes the Poisson boundary for $({\mathbb{F}}_2,\mu)$. Furthermore, the Radon-Nikodym cocycle of $m$ is given by $$\sigma_m(g,\xi) = 3^{-(\|g\|-2(g,\xi))}, \quad (g,\xi) \in {\mathbb{F}}_2 \times \partial {\mathbb{F}}_2,$$ where $\|\cdot\|$ denotes the word metric on ${\mathbb{F}}_2$ (with respect to $a$ and $b$ and their inverses) and $(g,\xi)$ is the length of the longest common sub-word of $g$ and $\xi$ (this is often referred to as the *confluent* or *Gromov product* in the literature). A straightforward calculation now yields $$\rho_m(g) = \|g\| \cdot \ln 3, \quad \forall \, g \in G,$$ which in particular shows that in this special case, $\rho_m$ is in fact a metric. The following simple proposition relates the asymptotic behavior of a cocycle $c$ to the vanishing of the drift of $(G,\mu)$ with respect to word-metrics on finitely generated groups, and finishes the proof of our version of the theorem of Kaimanovich-Vershik. Recall that a *$\mu$-harmonic mean $\lambda$ on $L^\infty(B,m)$* is a functional on $L^\infty(B,m)$ which is positive, i.e. gives non-negative values to non-negative elements in $L^\infty(B,m)$, normalized, i.e. $\lambda(1) = 1$ and satisfies $$\int_G \lambda(g \cdot \phi) \, d\mu(g) = \lambda(\phi), \quad \forall \, \phi \in L^\infty(B,m)$$ where $G$ acts on $L^\infty(B,m)$ via the left regular representation. In particular, the measure $m$ is a $\mu$-harmonic mean on $L^\infty(B,m)$. Let $(G,\mu)$ be a finitely generated measured group such that $\ell_d(\mu) = 0$ for some left invariant word-metric $d$ with respect to a finite symmetric generating set of $G$. Let $(B,m)$ denote the Poisson boundary of $(G,\mu)$. If $c : G {\rightarrow}L^\infty(B,m)$ is a cocycle, then $$\int_G \lambda(c(g)) \, d\mu(g) = 0$$ for every $\mu$-harmonic mean $\lambda$ on $L^\infty(B,m)$. In particular, we have $$\int_B \int_G \log \frac{dg^{-1}m}{dm}(b) \, d\mu(g) dm(b) = 0,$$ so $m$ is $G$-invariant, and thus $(B,m)$ is trivial. Since $G$ is finitely generated, there exists for every cocycle $c : G {\rightarrow}L^\infty(B,m)$ a constant $C_c$ such that $$\rho_c(g) \leq C_c \cdot d(g,e), \quad \forall \, g \in G,$$ where $d$ is a word-metric on $G$ with respect to a finite symmetric generating set of $G$. We assume that $\ell_d(\mu) = 0$, and thus $$\lim_{n} \frac{1}{n} \int_G \rho_c(g) \, d\mu^{*n}(g) = 0$$ for every cocycle $c$. If $\lambda$ is a $\mu$-harmonic mean on $L^\infty(B,m)$, one readily checks that $$\int_G \lambda(c(g)) \, d\mu^{*n}(g) = n \cdot \int_G \lambda(c(g)) \, d\mu(g)$$ for all $n$, and thus $$\Big| \int_G \lambda(c(g)) \, d\mu(g) \Big| = \Big|\frac{1}{n} \int_G \lambda(c(g)) \, d\mu^{*n}(g) \Big| \leq \frac{1}{n} \int_G \| c(g)\|_\infty \, d\mu^{*n}(g) {\rightarrow}0,$$ which finishes the proof. Ergodicity with unitary coefficients ==================================== We now turn to some ergodic-theoretical aspects of random walks on groups. As we have seen, to every measured group $(G,\mu)$ one can associate an ergodic $(G,\mu)$-space $(B,m)$, called the Poisson boundary of $(G,\mu)$ with the remarkable property that the linear map $P : L^\infty(B,m) {\rightarrow}{\mathcal{H}}^\infty_l(G,\mu)$ defined by $$P\phi(g) = \int_B \phi(g^{-1}b) \, dm(b), \quad g \in G,$$ is an isometric isomorphism. The aim of this section is to give a short proof of (a weak version) of an observation of Kaimanovich in [@K03], that the mere fact that this is an isomorphism onto ${\mathcal{H}}_l^\infty(G,\mu)$ automatically forces significantly stronger ergodicity properties. \[Kde\] Let $(G,\mu)$ be a measured group and denote by $(B,m)$ and $(\check{B},\check{m})$ the Poisson boundaries of $(G,\mu)$ and $(G,{\check{\mu}})$ respectively. If $(Y,\eta)$ is any ergodic probability measure preserving $G$-space, then the diagonal action on the triple $(B \times \check{B} \times Y, m \otimes \check{m} \otimes \eta)$ is ergodic. It is not hard to show (see for instance the recent survey by Glasner-Weiss [@GW13]) that the theorem above can be equivalently formulated as follows: For every unitary $G$-representation $({\mathcal{H}},\pi)$ on a Hilbert space ${\mathcal{H}}$, any measurable $G$-equivariant map $F : B \times \check{B} {\rightarrow}{\mathcal{H}}$ must be essentially constant. Kaimanovich proves in [@K03] the a priori stronger statement that one can assert the same thing about $G$-equivariant and weak\*-measurable maps from $B \times \check{B}$ into *any* isometric $G$-representation on any (separable) Banach space. For certain applications in bounded cohomology, this seemingly stronger statement is needed. Since we wish to keep the discussions in this paper fairly short, we shall confine ourselves to the setting of Theorem \[Kde\], although many of the techniques we shall describe can be used to give a complete proof of the main result in [@K03]. To start the proof of Theorem \[Kde\], we first observe that if $(\check{B},\check{m})$ is the Poisson boundary of $(G,{\check{\mu}})$ and $(Y,\eta)$ is any probability measure preserving $G$-space, then the diagonal $G$-action on $(\check{B} \times Y,m \otimes \eta)$ is a $(G,{\check{\mu}})$-space. Hence Theorem \[Kde\] follows immediately from the following result, which we have not been able to directly locate in the literature. \[K\] Let $(G,\mu)$ be a measured group with Poisson boundary $(B,m)$ and suppose $(X,\nu)$ is an ergodic $(G,{\check{\mu}})$-space. Then the diagonal $G$-action on the product space $(B \times X,m \otimes \nu)$ is ergodic. In order to see how Theorem \[Kde\] follows this statement, we argue in two steps. First note that if $(Y,\eta)$ is an *ergodic* probability measure preserving $G$-space, then it is an ergodic $(G,{\check{\mu}})$-space as well, and Theorem \[K\] implies that $X = \check{B} \times Y$, with the probability measure $\nu = \check{m} \otimes \eta$, is an *ergodic* $(G,{\check{\mu}})$-space. If we now apply Theorem \[K\] to the diagonal $G$-action on the direct product $$(B \times X,m \otimes \nu) = (B \times \check{B} \times Y,m \otimes \check{m} \otimes \eta),$$ then we conclude that it is also ergodic, which is exactly the assertion of Theorem \[Kde\].\ In the case when $(X,\nu)$ is an ergodic probability measure preserving $G$-space, Theorem \[K\] is due to Aaronson and Lemańczyk in [@AL05]. Their proof however follows quite different lines than ours. We now begin our proof of Theorem \[K\]. Let $(X,\mu)$ be an ergodic $(G,{\check{\mu}})$-space, and suppose $f$ is a $G$-invariant essentially bounded function on $B \times X$, which we may without loss of generality assume to have zero integral. We wish to prove that $f$ vanishes identically. For this purpose, we shall show that $$\label{zeroint} \int_B f(b,x) \, \phi(b) \, dm(b) = 0, \quad \textrm{for $\nu$-a.e. $x$ in $X$}$$ for all $\phi \in L^1(B,m)$, and thus $$\int_X \int_B f(b,x) \phi(b) \psi(x) \, dm(b) \, d\nu(x) = 0$$ for all $\phi \in L^1(B,m)$ and $\psi \in L^1(X,\nu)$, which shows that $f$ must vanish identically, establishing ergodicity for the diagonal action on $(B \times X,m \otimes \nu)$. To prove , we argue as follows. Define $$s(x) = \int_B f(b,x) \, dm(b)$$ and note that $$\begin{aligned} \int_G s(gx) \, d\mu(g) &=& \int_G \int_B f(b,gx) \, dm(b) \, d\mu(g) \\ &=& \int_G \int_B f(g^{-1}b,x) \, dm(b) \, d\mu(g) \\ &=& \int_B f(b,x) \, dm(b) = s(x),\end{aligned}$$ since $m$ is $\mu$-harmonic. The following lemma now shows that $s$ must vanish almost everywhere with respect to the measure $\nu$ (recall that $(X,\nu)$ is an ergodic $(G,{\check{\mu}})$-space). \[harmonic constant\] Let $(X,\nu)$ be a $(G,\mu)$-space and suppose that $s \in L^\infty(X,\nu)$ satisfies the equation $$s = \int_G s(g^{-1} \cdot ) \, d\mu(g) \quad \textrm{in $L^\infty(X,\nu)$.}$$ Then $f$ is essentially $G$-invariant. In particular, if $(X,\nu)$ is ergodic, then $f$ equals its $\nu$-integral almost everywhere. We may assume that $s$ is real-valued. Since the support of $\mu$ is assumed to generate $G$, it suffices to show that $$\int_G \int_X \big| s(g^{-1}x) - s(x) \big|^2 \, d\mu^{*k}(g) \, d\nu(x) = 0$$ for all $k$. However, upon expanding the square, and using the harmonicity of $\nu$, we see that $$\int_G \int_X \big| s(g^{-1}x) - s(x) \big|^2 \, d\mu^{*k}(g) \, d\nu(x) = 2 \cdot \Big( \int_X |s(x)|^2 \, d\nu(x) - \int_X s(x) \, \Big( \int_G s(g^{-1}x) \, d\mu^{*k}(g) \Big) \, d\nu(x) \Big),$$ which clearly vanishes by our assumption on $s$. Going back to the proof of Theorem \[K\], we can now conclude that $$\int_B f(b,x) \, dm(b) = 0 \quad \textrm{for $\nu$-a.e. $x$ in $X$},$$ and thus $$\int_B f(b,gx) \, dm(b) = \int_B f(g^{-1}b,x) \, dm(b) = \int_B f(b,x) , \sigma_m(g,b) \, dm(b) = 0,$$ for almost every $x$ in $X$ and for all $g$ in $G$. In particular, we have $$\int_B f(b,x) \, \phi(b) \, dm(b) = 0, \quad \textrm{for $\nu$-a.e. $x$ in $X$},$$ and for all $\phi$ in the linear span of all $\sigma_m(g,\cdot)$ as $g$ ranges over $G$. Hence Theorem \[K\] will follow from the following simple lemma, which is essentially just a reformulation of Proposition \[furstenberg\]. Let $(G,\mu)$ be a measured group and denote by $(B,m)$ its Poisson boundary. Then the linear span $${\mathcal{R}}_m = \operatorname{span}\Big\{ \frac{dg^{-1}m}{dm} \, : \, g \in G \Big\} \subset L^1(B,m)$$ is dense in $L^1(B,m)$. If this span would not be dense, then by Hahn-Banach’s Theorem, there exists a non-zero functional $\phi \in L^1(B,m)^* = L^\infty(B,m)$ such that $$\int_X \phi(b) \, \frac{dg^{-1}m}{dm}(b) \, dm(b) = \int_X \phi(g^{-1}b) \, dm(b) = 0, \quad \forall \, g \in G,$$ or equivalently, $P\phi(g) = 0$, where $P$ is as in Proposition \[furstenberg\]. Since $P$ is an isomorphism, we conclude that $\phi$ vanishes identically, which is a contradiction. We finish this section with yet another consequence of Proposition \[furstenberg\] which seems to be rarely stressed in the literature. It was first observed by Kaimanovich in [@Ka92], but the analogous case (in fact, concerning positive bi-harmonic functions) for amenable *connected* measured group goes back to Raugi in [@Ra88]). \[CD\] For every probability measure $\mu$ on a countable group $G$, there are no non-constant *bounded* functions which are both left and right $\mu$-harmonic. In particular, measured *abelian* groups do not admit any non-constant bounded harmonic functions. The last assertion is immediate if $\mu$ is symmetric. If it is not and $\phi$ is a bounded left $\mu$-harmonic function with $\phi(e) = 0$ (and hence right ${\check{\mu}}$-harmonic), then the symmetrized function $$\psi(g) = \phi(g) + \phi(g^{-1}), \quad g \in G,$$ is a bounded left and right $\mu$-harmonic and thus identically zero by the corollary above. This forces the identities $\phi(g) = - \phi(g^{-1})$ for all $g \in G$, and thus $\phi$ is both left and right $\mu$-harmonic, and hence constant. Let $(B,m)$ denote the Poisson boundary of $(G,\mu)$ and suppose $f$ is a bounded left and right $\mu$-harmonic function on $G$. Let $\phi$ denote the unique element in $L^\infty(B,m)$ such that $f = P\phi$, where $P$ is the linear map defined in Proposition \[furstenberg\]. Note that the uniqueness of $\phi$ forces the identity $$\int_X \phi(g^{-1} \cdot ) \, d\mu(g) = \phi \quad \textrm{in $L^\infty(B,m)$}.$$ By the last lemma, we conclude that $\phi$ is $G$-invariant and thus constant, since $(B,m)$ is ergodic. Weak mixing for $(G,\mu)$-spaces ================================ The aim of this section is to characterize weakly mixing $(G,\mu)$-spaces as exactly those which do not admit any non-trivial probability measure preserving factors with discrete spectrum. The author has not been able to locate an explicit formulation of this equivalence in the literature, although it should be stressed that the characterization does follow from applying a series of classical and well-known techniques combined with the fact that WAP-actions are $\mu$-stiff in the sense of Furstenberg, which was established in [@FG10]. However, as the proof of this fact utilizes some serious machinery from the theory of Ellis semigroups and weakly almost periodic functions, the route to the characterization of weakly mixing $(G,\mu)$-spaces (following these lines) is not very direct. We shall try to outline below a more direct approach. First recall that a non-singular $G$-space $(X,\nu)$ is *weakly mixing* if for every ergodic probability measure preserving $G$-space $(Y,\eta)$, the diagonal $G$-action on $(X \times Y,\nu \otimes \eta)$ is ergodic. If $\nu$ is $G$-invariant, then this is equivalent to the absence of a non-trivial factor with discrete spectrum, that is to say, a probability measure preserving $G$-space $(Z,\xi)$ with the property that the corresponding unitary (Koopman) representation on $L^2(Z,\xi)$ decomposes into a *direct sum* of finite dimensional sub-representations. By a classical theorem of Mackey in [@Ma], an ergodic probability measure preserving $G$-space with discrete spectrum is very special. Indeed, it is always isomorphic to an isometric $G$-action on a compact homogeneous space, that is to say, there exists a compact group $K$ and a closed subgroup $K_o < K$ and a homomorphism $\tau : G {\rightarrow}K$ with dense image such that the $\tau(G)$-action on $K/K_o$ (with the Haar probability measure) is isomorphic (as a $G$-space) to $(Z,\xi)$. In particular, if $G$ is a minimally almost periodic group (which means that there are no non-trivial finite dimensional unitary $G$-representations whatsoever), then every ergodic probability measure preserving $G$-space is automatically weakly mixing. It is not true in general that an ergodic non-weakly mixing non-singular $G$-space admits a probability measure preserving factor with discrete spectrum. In fact, Aaronson and Nadkarni constructs in [@AN87] a probability measure on a compact group, which is non-singular and ergodic with respect to dense cyclic subgroup, so that the corresponding ergodic non-singular ${\mathbb{Z}}$-space (which is certainly not weakly mixing) does not admit any non-trivial probability measure preserving factors whatsoever. However, in the category of $(G,\mu)$-spaces the situation is much nicer and the aim of this section is to give a self-contained proof of the following theorem, which is not new and certainly known to experts. \[WM\] Let $(G,\mu)$ be a measured group and suppose $(X,\nu)$ is an ergodic $(G,\mu)$-space. Then $(X,\nu)$ is weakly mixing if and only if $(X,\nu)$ does not admit a non-trivial probability measure preserving factor with discrete spectrum. In particular, if $G$ is minimally almost periodic, then every ergodic $(G,\mu)$-space is weakly mixing. In particular this theorem applies to the Poisson boundary of $(G,\mu)$, which certainly does not have any probability measure preserving factors whatsoever, and thereby giving yet another proof of the weak mixing of Poisson boundaries, originally due to Aaronson and Lemańczyk in [@AL05]. Note however that the theorem does *not* directly apply to the setting of Theorem \[Kde\] since products of Poisson boundaries are not $(G,\mu)$-spaces in general (unless of course, they are trivial). Let us now begin the proof of Theorem \[WM\], which naturally falls into two steps, both of which are essentially classical, and only the first step needs to be complemented with a less classical argument concerning probability measure preserving factors. As we have already mentioned above, this argument could be replaced by a nice, but not very elementary observation of Furstenberg and Glasner in [@FG10] about $\mu$-harmonic measures on WAP-spaces. However, since no self-contained proof of Theorem \[WM\] seems to exist in the literature, it makes sense to outline a more direct route in this paper, and to collect here all the necessary arguments, although we do allow ourselves to be a bit sketchy in the more classical arguments. For the first step, we let $(G,\mu)$ be a measured group and $(X,\nu)$ is an ergodic $(G,\mu)$-space. Suppose there exists an ergodic probability measure preserving $G$-space $(Y,\eta)$ such that the diagonal $G$-action on $(X \times Y,\nu \otimes \eta)$ is *not* ergodic, that is to say, there exists a non-constant essentially bounded real-valued function $f$ on $X \times Y$. Without loss of generality, we can assume that $f$ is bounded by one, so that the map $$\pi_f : X {\rightarrow}B_1(L^2(Y,\eta))$$ given by $\pi_f(x) = f(x,\cdot) \in B_1(L^2(Y,\eta))$ is well-defined for almost every $x$ in $X$, where $B_1(L^2(Y,\eta))$ denotes the unit ball in the Hilbert space $L^2(Y,\eta)$. Since $f$ is assumed to be $G$-invariant, one can readily verify that $\pi_f$ is a (weakly measurable) factor map from $X$ into the $(G,\mu)$-space $(B_1(L^2(Y,\eta),\pi_*\nu)$, where $G$ acts on the unit ball $B_1(L^2(Y,\eta))$ via the (unitary) Koopman operator on $L^2(Y,\eta)$ (recall that $(Y,\eta)$ is measure-preserving). We note that since $f$ is non-constant, the corresponding factor is non-trivial. More generally, suppose $({\mathcal{H}},\pi)$ is a unitary $G$-representation on a Hilbert space ${\mathcal{H}}$. Then $\pi$ induces a (weakly continuous) action of $G$ on the unit ball $B_1({\mathcal{H}})$, and if $\nu$ is a $\mu$-harmonic probability measure (with respect to this $G$-action) on $B_1({\mathcal{H}})$, then we shall refer to $(B_1({\mathcal{H}}),\nu)$ as a *Hilbertian $(G,\mu)$-space*. Theorem \[WM\] will then follow from the following proposition. Every Hilbertian $(G,\mu)$-space is measure-preserving and has discrete spectrum. We begin by proving that Hilbertian $(G, \mu)$-spaces are measure-preserving. To do so, we first note that by Stone-Weierstrass Theorem, the linear span of the constants and all functions of the form $$\label{prods} \phi(x) = \langle y_1, x\rangle \cdots \langle y_k, x\rangle , \quad x \in B_1({\mathcal{H}}), \quad y_1, \ldots, y_k \in {\mathcal{H}}$$ is dense in $C(B_1({\mathcal{H}}))$, when $B_1({\mathcal{H}})$ is equipped with the weak topology, and we wish to prove that $$\label{harmeq} \int_X \phi(g^{-1}x) \, d\nu(x) = \int_X \phi(x) \, d\nu(x), \quad \forall \, g \in G,$$ for all $y_1,\ldots,y_k \in {\mathcal{H}}$. Since $$\begin{aligned} \int_G \int_{B_1({\mathcal{H}})} \phi(g^{-1}x) \, d\nu(x) \, d\mu(g) &=& \Big\langle y_1 \otimes \cdots \otimes y_k, \int_G \pi^{\otimes k}(g) \xi_\nu \, d\mu(g) \Big\rangle, \\ &=& \Big\langle y_1 \otimes \cdots \otimes y_k, \xi_\nu \Big\rangle,\end{aligned}$$ for all $y_1,\ldots,y_k \in {\mathcal{H}}$, where $\pi^{\otimes k}$ denotes the $k$-th tensor product representation of $({\mathcal{H}},\pi)$, and $$\xi_\nu = \int_X x \otimes \cdots \otimes x \, d\nu(x),$$ we can conclude that $$\pi^{\otimes k}(\mu) \xi_\nu = \int_G \pi^{\otimes k}(g) \xi_\nu \, d\mu(g) = \xi_\nu.$$ Hence will follow from the following simple lemma (applied to all finite tensor product representations of $({\mathcal{H}},\pi)$). Let $({\mathcal{H}},\pi)$ be a unitary $G$-representation and suppose $\xi \in {\mathcal{H}}$ satisfies $\pi(\mu)\xi = \xi$. Then $\xi$ is $\pi(G)$-invariant. We may without loss of generality assume that $\|\xi\| = 1$. Since $\pi$ is unitary, the equation $\pi(\mu)\xi = \xi$ simply means that a convex average of points of the form $\pi(g)\xi$, for $g$ in the support of $\mu$, equals $\xi$. However, by the strict convexity of the unit ball in ${\mathcal{H}}$, this can only happen if $\pi(g)\xi = \xi$ for all $g$ in the support of $\mu$. Since the support of $\mu$ is assumed to generate $G$, we conclude that $\xi$ is $\pi(G)$-invariant. It remains to show that the measure-preserving $G$-space $(B_1({\mathcal{H}}),\nu)$ has discrete spectrum. For this purpose, we define the closed linear subspace $${\mathcal{H}}_o = \overline{\Big\{ v \in {\mathcal{H}}\, : \, \textrm{the cyclic span of $v$ is finite-dimensional} \Big\}} \subset {\mathcal{H}}.$$ One readily checks that ${\mathcal{H}}_o$ is a sub-representation of ${\mathcal{H}}$, and thus its orthogonal complement ${\mathcal{H}}_1$ is a sub-representation with the property that it does not have any finite-dimensional sub-representations whatsoever. Furthermore, we have $$B_1({\mathcal{H}}) = \Big\{ (\xi,\eta) \in {\mathcal{H}}_o \oplus {\mathcal{H}}_1 \, : \, \|\xi\|_0^2 + \|\eta\|_1^2 \leq 1 \Big\} \subset B_1({\mathcal{H}}_o) \times B_1({\mathcal{H}}_1).$$ We have canonical continuous $G$-equivariant projections $\pi_o$ and $\pi_1$ from $B_1({\mathcal{H}})$ onto $B_1({\mathcal{H}}_o)$ and $B_1({\mathcal{H}}_1)$ respectively, and it is not hard to show that the $G$-space $(B_1({\mathcal{H}}_o),\pi_o)_*\nu)$ has discrete spectrum. Hence it suffices to show the following lemma. Suppose $({\mathcal{H}},\pi)$ is a unitary $G$-representation with no (non-trivial) finite-dimensional sub-representations. If $\nu$ is a $G$-invariant probability measure on $B_1({\mathcal{H}})$, then it is concentrated at zero. Since $C(B_1({\mathcal{H}}))$ is generated by limits of linear combinations of products of the form as in , it suffices to show that $$\label{suff} \int_{B_1({\mathcal{H}})} \big|\langle y,x\rangle\big|^2 \, d\nu(x) = 0, \quad \forall \, y \in {\mathcal{H}}.$$ In order to establish , we note that $$\int_{B_1({\mathcal{H}})} \big|\langle y,x\rangle\big|^2 \, d\nu(x) = \big\langle y \otimes y^*, \int_{B_1({\mathcal{H}})} x \otimes x^* \, d\nu(x) \big\rangle, \quad \forall \, y \in {\mathcal{H}},$$ where the $*$ refers to complex conjugation, and since $\nu$ is $G$-invariant, the vector $$\xi = \int_{B_1({\mathcal{H}})} (x \otimes x^*) \, d\nu(x) \in {\mathcal{H}}\otimes {\mathcal{H}}^*$$ is invariant under $\pi \otimes \pi^*(G)$. We wish to show that $\xi$ is zero. To do so, we note that $\xi$ induces a *compact and self-adjoint* linear map $K_\xi : {\mathcal{H}}{\rightarrow}{\mathcal{H}}$ which is uniquely determined by $$\langle y,K_\xi z\rangle = \langle y \otimes z^*, \xi \rangle, \quad \forall \, y, z \in {\mathcal{H}}.$$ One readily checks that $K_\xi$ intertwines the representation $\pi$. By the spectral theorem for compact and self-adjoint linear maps, ${\mathcal{H}}$ decomposes into a direct sum of the kernel of $K_\xi$ and *finite*-dimensional eigenspaces for $K_\xi$. Since $\pi(g)$ commutes with $K_\xi$ for every $g$, each of these finite-dimensional subspaces must be invariant under $\pi$. However, since ${\mathcal{H}}$ is assumed to completely lack finite-dimensional sub-representations, only the kernel of $K_\xi$ remains and we conclude that $K_\xi$ is trivial, i.e. $\xi$ is zero, which finishes the proof. Biharmonic functions, coboundaries and central limit theorems ============================================================= In this section we shall discuss a novel perspective on a powerful classical technique, often attributed to S.V. Nagaev [@Na1], which is designed to prove central limit theorems for certain classes of Markov chains. This technique is discussed at length in the book [@HeHe], but in this paper we shall approach it in a slightly different way in the setting of random walks on groups. We begin by describing a motivating example. Let $(G,\mu)$ be a measured group and suppose $d$ is a left invariant distance function on $G$ which satisfies the moment condition $$\int_G d(g,e)^{2+{\varepsilon}} \, d\mu(g) < \infty, \quad \textrm{for some ${\varepsilon}> 0.$}$$ Let $(\Omega,{\mathcal{P}}) = (G^{{\mathbb{Z}}},\mu^{{\mathbb{Z}}})$ and if $\omega$ is an element in $\Omega$, then we denote by $\omega_n$ the $n$’th coordinate of $\omega$. One readily checks that $(\omega_n)$ is a sequence of independent $\mu$-distributed random variables on $G$, and we define $(z_n)$ to be the corresponding random walk, i.e. $$z_n(\omega) = \omega_o \cdots \omega_{n-1}, \quad n \geq 1.$$ We note that the limit $$\ell_d(\mu) = \lim_n \frac{1}{n} \int_\Omega d(z_n(\omega),e) \, d{\mathbb{P}}(\omega) = \lim_n \frac{1}{n} \int_G d(g,e) \, d\mu^{*n}(g)$$ exists and coincides with the drift of $(G,\mu,d)$ defined in the first section of this paper. It follows from Theorem \[KL\] that $\ell_d(\mu)$ is positive whenever $(G,\mu)$ is not Liouville, so in particular the drift is positive if $G$ is non-amenable. In this case, the sequence $$\label{normseq} Y_n = \frac{d(z_n,e) - n\ell_d(\mu)}{\sqrt{n}}, \quad n \geq 1,$$ of random variables fluctuates around zero, and it makes sense to ask whether it has a non-trivial distributional limit. The aim of this section is to outline a technique which isolates a class of triples $(G,\mu,d)$ for which the sequence $(Y_n)$ defined in \[normseq\] converges weakly to a non-degenerate Gaussian distribution on the real line, that is to say, we wish to impose natural conditions on $G$, $\mu$ and $d$ such that for every continuous function $\varphi$ on ${\mathbb{R}}$ with compact support, we have $$\label{clt} \lim_n \int_\Omega \varphi\Big(\frac{d(z_n,e) - n\ell_d(\mu)}{\sqrt{n}}\Big) \, d{\mathbb{P}}= \frac{1}{\sigma \sqrt{2\pi}} \int_{-\infty}^\infty \varphi(t) e^{-t^2/2\sigma^2} \, dt,$$ for some constant $\sigma > 0$. In probability theory, this convergence is usually denoted by $Y_n \Rightarrow N(0,\sigma^2)$, and we shall adopt this notation in this paper. Biharmonicity and central limit theorems ---------------------------------------- Let us now briefly outline how the technique of S.V. Nagaev works in this setting. Its starting point is the fundamental observation (Proposition \[fund\] below) that the values of *biharmonic functions* along random walks always satisfy, under very weak assumptions, a central limit theorem. To make this observation precise, we first recall that a real-valued function $\phi$ on $G$ is *left Lipschitz* if the function $$\rho_\phi(g) = \sup_{s} \big|\phi(sg) - \phi(s)|,$$ is finite for every $g$ in $G$. One observes that if $\phi$ is left Lipschitz, then $\rho_\phi$ satisfies the triangle inequality $$\rho_\phi(g_1g_2) \leq \rho_{\phi}(g_1) + \rho_{\phi}(g_2), \quad \forall \, g_1, g_2 \in G,$$ and thus, if $G$ is finitely generated, it is bounded from above by any word metric on $G$. Furthermore, recall that a $\mu$-integrable function $\phi$ on $G$ is *left $\mu$-quasiharmonic* if there exists a constant $\ell(\phi)$ such that $$\int_G \phi(sg) \, d{\check{\mu}}(s) = \phi(g) + \ell(\phi), \quad \forall \, g \in G,$$ and *right $\mu$-quasiharmonic* if there exists a constant $r(\phi)$ such that $$\int_G \phi(gs) \, d\mu(s) = \phi(g) + r(\phi), \quad \forall \, g \in G.$$ Finally, we say that $\phi$ is *bi-$\mu$-quasiharmonic* if it is left and right $\mu$-quasiharmonic. By letting $g = e$ in the formulas above, we see that if $\phi$ is bi-$\mu$-quasiharmonic, then $r(\phi) = \ell(\phi)$.\ The fundamental observation upon which the technique of S.V. Nagaev hings can now be formulated as follows. \[fund\] Let $(G,\mu)$ be a symmetric measured group and suppose $\phi$ is a left Lipschitz, bi-$\mu$-quasiharmonic function on $G$ such that $$\int_G \rho_\phi(g)^{2+{\varepsilon}} \, d\mu(g) < + \infty, \quad \textrm{for some ${\varepsilon}> 0$.}$$ If $\phi$ is not identically equal to $\ell(\phi)$, then there exists $\sigma > 0$ such that $$\frac{\phi(z_n) - n\ell(\phi)}{\sqrt{n}} \Rightarrow N(0,\sigma^2).$$ Since $\phi$ is a right $\mu$-quasiharmonic function, the sequence $$M_n = \phi(z_n) - n \ell(\phi), \quad n \geq 1,$$ of measurable functions on $\Omega$ forms a martingale with respect to the filtration generated by the coordinates up to $n-1$. According to the martingale central limit theorem by McLeish in [@Mc74], in order to prove the distributional convergence asserted in the proposition, it suffices to show that the sequence $(\psi_n)$ defined by $$\psi_n(\omega) = \frac{1}{\sqrt{n}} \max \Big\{ \big| \phi(z_{j+1}(\omega)) - \phi(z_j(\omega)) - \ell(\phi)\big| \, : \, j=1,\ldots,n-1 \Big\}$$ is uniformly integrable and $\int_\Omega \psi_n \, d{\mathbb{P}}{\rightarrow}0$ as $n$ tends to infinity, and $$\label{convergence} \lim_{n} \frac{1}{n} \sum_{j=1}^{n-1} \big| \phi(z_{j+1}(\omega)) - \phi(z_j(\omega)) - \ell(\phi)\big|^2 = \sigma^2,$$ almost everywhere with respect to ${\mathbb{P}}$, where $\sigma$ is a positive constant.\ Recall that by de la Vallée-Poussin Theorem, $(\psi_n)$ is uniformly integrable if, but not only if, $$\label{goodbound} \sup_n \int_\Omega \big|\psi_n\big|^{2+{\varepsilon}} \, d{\mathbb{P}}< \infty,$$ and thus to prove the two first assertions it suffices to show that $$\int_\Omega |\psi_n|^{2+{\varepsilon}} \, d{\mathbb{P}}\leq \frac{1}{n^{{\varepsilon}/2}} \int_{G} \rho_\phi(g)^{2+{\varepsilon}} \, d\mu(g),$$ since the last integral is finite by assumption. First note that the shift map $\tau : \Omega {\rightarrow}\Omega$ given by $\tau(\omega)_n = \omega_{n+1}$ preserves the probability measure ${\mathbb{P}}$ on $\Omega$ and is ergodic. Secondly, we have $$\psi_n(\omega) \leq \frac{1}{\sqrt{n}} \cdot \max\Big\{ \rho_\phi(\omega_j) \, : \, 1 \leq j \leq n-1 \Big\}$$ for all $n$, so that if we define $v(\omega) = \rho_\phi(\omega_o)$, then $v \in L^{2+{\varepsilon}}(\Omega,{\mathbb{P}})$, and it is a straightforward exercise to show that $$\int_\Omega \max_{1 \leq j \leq n-1} |v(\tau^j \omega)|^{2+{\varepsilon}} \, d{\mathbb{P}}\leq \frac{1}{n^{{\varepsilon}/2}} \int_{\Omega} |v(\omega)|^{2+{\varepsilon}} \, d{\mathbb{P}}(\omega) = \frac{1}{n^{{\varepsilon}/2}} \int_{G} \rho_\phi(g)^{2+{\varepsilon}} \, d\mu(g) {\rightarrow}0,$$ Hence it remains to show the convergence in . For this purpose, we define the sequence $$u_j(\omega) = \big| \phi(\omega_{-j} \cdots \omega_o) - \phi(\omega_{-j}\cdots \omega_{-1}) - \ell(\phi)\big|^2,$$ so that we can write $$u_j(\tau^{j}\omega) = \big| \phi(z_{j+1}(\omega)) - \phi(z_j(\omega)) - \ell(\phi)\big|^2, \quad \forall \, j \geq 1.$$ We wish to prove that there exists a positive constant $\sigma > 0$ such that $$\sigma^2 = \lim_n \frac{1}{n} \sum_{j=1}^n u_j(\tau^j\omega)$$ almost everywhere. By Breiman’s Lemma (see e.g. Lemma 14.34 in [@Gla]), it suffices to show that $$\int_\Omega \sup_j u_j \, d{\mathbb{P}}< \infty$$ and that there exists a function $u \in L^1(\Omega,{\mathbb{P}})$ such that $u_j {\rightarrow}u$ almost surely and in the $L^1$-norm. Indeed, if this is the case, then $$\sigma^2 = \lim_n \frac{1}{n} \sum_{j=1}^n u_j(\tau^j\omega) = \int_\Omega u \, d{\mathbb{P}},$$ almost surely and $\sigma = 0$ if and only if $u$ vanishes almost everywhere. To prove the existence of a function $u$ as above, we define the sequence $$N_j = \phi(\omega_{-j} \cdots \omega_o) - \phi(\omega_{-j}\cdots \omega_{-1}) - \ell(\phi),$$ so that $u_j = |N_j|^2$, and since $\phi$ is *left* $\mu$-quasiharmonic, we see that $(N_j)$ is a martingale with respect to the filtration generated by the coordinates from $-j$ to $0$. Furthermore, since $\phi$ is left Lipschitz, we also have that $$C = \sup_j \int_\Omega |N_j|^{2} \, d{\mathbb{P}}\leq \int_G \rho_\phi(g)^{2} \, d\mu(g) + \ell(\phi)^2 + 2 \cdot \ell(\phi) \cdot \int_G \rho_\phi(g) \, d\mu(g),$$ and thus $(N_j)$ is a $L^2$-bounded martingale. In particular, by the classical Martingale Convergence Theorem, there exists a function $N_\infty$ in $L^2(\Omega,{\mathbb{P}})$ such that $N_j {\rightarrow}N_\infty$ almost everywhere and $$\lim_j \int_\Omega \big|N_j - N_\infty|^2 \, d{\mathbb{P}}= 0$$ and thus, with $u = |N_\infty|^2$, we have $u_j {\rightarrow}u$ almost everywhere and $$\int_\Omega \big|u_j - u\big| \, d{\mathbb{P}}\leq \int_\Omega \big|(N_j-N_\infty)(N_j + N_\infty)\big| \, d{\mathbb{P}}\leq 4 \cdot C \cdot \int_\Omega \big|N_j - N_\infty|^2 \, d{\mathbb{P}}{\rightarrow}0.$$ Finally, we need to show that $u$ does not vanish almost everywhere with respect to ${\mathbb{P}}$. Note that if $u$ vanishes almost everywhere, then so does $N_\infty$ and thus $$\lim_j \phi(\omega_{-j} \cdots \omega_o) - \phi(\omega_{-j}\cdots \omega_{-1}) = \ell(\phi)$$ almost everywhere. Hence, for any fixed $j_o$, by calculating the conditional expectation of the limit with respect to the $\sigma$-algebra generated by all coordinates strictly below $-j_o$, we conclude that $$\phi(\omega_{-j_o} \cdots \omega_{o}) - \phi(\omega_{-j_o} \cdots \omega_{-1}) = \ell(\phi),$$ almost everywhere. In particular, since $\mu$ is assumed to generate $G$ as a semigroup, we have $\phi = \ell(\phi)$ everywhere, which we have assumed is not the case. Constructing bi-quasiharmonic functions --------------------------------------- We now return to our motivating example. As we have seen in Subsection \[subsecLD\], given any triple $(G,\mu,d)$, there exists a sequence $(n_j)$ such that the limit $$\phi(g) = \lim_{j {\rightarrow}\infty} \frac{1}{n_j} \sum_{k=0}^{n_j-1} \int_G \big(d(g,x)-d(x,e)\big) \, d\mu^{*k}(x)$$ exists for all $g$ in $G$, and the function $\phi$ satisfies $$\int_G \phi(sg) \, d{\check{\mu}}(s) = \phi(g) + \ell_d(\mu)$$ and $$\phi(g) \leq d(g,e) {\quad \textrm{and} \quad}\rho_\phi(g) = \sup_{s} \big|\phi(sg) - \phi(s)\big| \leq d(g,e), \quad \forall \, g \in G.$$ In particular, $\phi$ is left Lipschitz and left $\mu$-quasiharmonic. Furthermore, if we write $$\frac{d(z_n,e) - n\ell_d(\mu)}{\sqrt{n}} = \frac{d(z_n,e) - \phi(z_n)}{\sqrt{n}} + \frac{\phi(z_n) - n\ell_d(\mu)}{\sqrt{n}},$$ then the first term is non-negative and converges to zero in the $L^1$-norm if and only if $$\label{neglect} \lim_n \frac{1}{\sqrt{n}} \Big( \int_G d(g,e) \, d\mu^{*n}(g) - n \ell_d(\mu) \Big) = 0.$$ Hence, under condition \[neglect\], the question whether \[clt\] holds is completely reduced to the question whether $$\label{cltqh} \frac{\phi(z_n) - n\ell_d(\mu)}{\sqrt{n}} \Rightarrow N(0,\sigma^2)$$ for some positive constant $\sigma$.\ Unfortunately, there is no reason in general to expect that $\phi$ is also right $\mu$-quasiharmonic so that Proposition \[fund\] can be directly applied. We approach this serious problem as follows. Let $(B,m)$ be the Poisson boundary of $(G,\mu)$ and note that for every $u \in L^\infty(B)$, the function $$\phi_u(g) = \phi(g) + \int_G u(g^{-1}b) \, dm(b), \quad g \in G,$$ is again left Lipschitz and left $\mu$-quasiharmonic. Furthermore, holds for $\phi_u$ if and only if it holds for $\phi$. Hence it makes sense to ask whether we can find $u \in L^\infty(B,m)$ such that $\phi_u$ is right $\mu$-quasiharmonic. It turns out that there is a simple criterion for this. Indeed, since $\phi$ is left Lipschitz, one can readily check that the function $$\widehat{\psi}(s) = \int_G \big( \phi(sg) - \phi(s) \big) \, d\mu(g),$$ is bounded and left $\mu$-harmonic, and thus it corresponds via the Poisson transform (discussed in the first section of this paper) to an element $\psi$ in $L^\infty(B)$ (which we shall refer to as the *right $\mu$-obstruction*), with the property that $$\int_B \psi(b) \, dm(b) = \ell_d(\mu).$$ We observe that $\phi_u$ is right $\mu$-quasiharmonic if and only if $u$ satisfies the “cohomological equation” $$\label{cohomo} u(b) - \int_B u(s^{-1}b) \, d\mu(s) = \psi(b) - \ell_d(\mu), \quad \textrm{a.e. $[m]$.}$$ For many triples $(G,\mu,d)$ of interest, such as Gromov hyperbolic groups equipped with symmetric probability measures with finite exponential moments, one can show that $\psi -\ell_d(\mu)$ must belong to a certain subspace ${\mathcal{B}}\subset L^\infty(B,m)$ consisting of “smooth” functions with zero $m$-integrals, which admits a seminorm $N_o$ with the property that $$N(u) = \|u\|_\infty + N_o(u)$$ is a norm on ${\mathcal{B}}$ and there exist $0 < \tau < 1$ and an integer $n_o$ such that the convolution operator $$Q_\mu u(b) = \int_G u(g^{-1} b) \, d\mu(g)$$ satisfies the contraction bound $$\label{boundN} N_o(Q_\mu^{n_o}u) \leq \tau \cdot N_o(u) \quad \forall \, u \in {\mathcal{B}}.$$ Note that once such a bound has been established, it is not hard to show that the von Neumann series $$u = \sum_{n \geq 0} Q_{\mu}^{*n}\big(\psi-\ell_d(\mu)\big)$$ is a well-defined element in ${\mathcal{B}}$ which solves the equation \[cohomo\]. The main aim of the rest of this section will be to single out a class of symmetric measured groups which comes equipped with a “natural” weakly dense semi-normed subspace of $L^\infty(B,m)$, which one should think of as “measurably Hölder continuous” functions, on which $Q_\mu$ satisfies the above contraction bound. Besov spaces defined by product currents ---------------------------------------- Let $(G,\mu)$ be a countable symmetric measured group and suppose $(X,\nu)$ is a compact $(G,\mu)$-space, that is to say, $X$ is a compact metrizable space equipped with an action of $G$ by homeomorphisms such that $\nu$ satisfies the equation $$\int_G \int_X \phi(s^{-1}x) \, d\nu(s) \, d\mu(s) = \int_X \phi(x) \, d\nu(x)$$ for all $\phi \in C(X)$. If $\nu$ is non-atomic, then we can think of the product measure $\nu \otimes \nu$ as a probability measure on the (in general) *non-compact* space $\partial^2 X = X \times X \setminus \Delta X$, where $\Delta X$ denotes the (closed) diagonal subspace in $X \times X$. A non-negative Borel measurable function $\rho$ on $\partial^2 X$ is called a *product current* if the (possibly infinite) Borel measure $\eta$ on $\partial^2 X$ defined by $$\int_{\partial^2 X} \phi(x,y) \, d\eta(x,y) = \int_{\partial^2 X} \phi(x,y) \, \rho(x,y) \, d\nu(x) \, d\nu(y)$$ is invariant with respect to the diagonal action of $G$ on $\partial^2 X \subset X \times X$. One can readily check that this condition simply translates to the validity of the equation $$\label{equivariance} \rho(gx,gy) \, \sigma_{\nu}(g,x) \, \sigma_{\nu}(g,y) = \rho(x,y)$$ for all $g$ in $G$ and for almost every $(x,y)$ with respect to the product measure $\nu \otimes \nu$. We stress that not every $(G,\mu)$-space admits a product current. However, certain classes of countable groups, such as Gromov hyperbolic groups and lattices in higher rank Lie groups, carry symmetric probability measures with the property that their Poisson boundaries (in some compact model) admit “natural” and “geometrically defined” product currents. We refer the reader to Section 5 of the paper [@BHM] for a detailed discussion about product currents for Gromov hyperbolic measured groups. In this case, $X$ is the Gromov boundary of the hyperbolic group $G$, equipped with a certain distance function $d_o$, and $\rho$ is roughly proportional to $d_o(x,y)^{-D}$, where $D$ is a constant related to the Hausdorff dimension of $X$. Given a product current $\rho$ for a $(G,\mu)$-space $(X,\nu)$ and given ${\varepsilon}> 0$, we define a semi-norm $N_{\rho,{\varepsilon}}$ on a subspace ${\mathcal{B}}_{\rho,{\varepsilon}} \subset L^\infty(X,\nu)$ by $$N_{\rho,{\varepsilon}}(u) = \int_{\partial^2 X} \big|u(x) - u(y)\big| \, \rho(x,y)^{\frac{1}{2} + {\varepsilon}} \, d\nu(x) \, d\nu(y),$$ where ${\mathcal{B}}_{\rho,{\varepsilon}}$ consists of those elements in $L^\infty(X,\nu)$ with finite $N_{\rho,{\varepsilon}}$-seminorms. We shall refer to linear space $({\mathcal{B}}_{\rho,{\varepsilon}},N_{\rho,{\varepsilon}})$ as the *Besov space associated to $\rho$ of order ${\varepsilon}$*, and since $\rho$ usually blows up close to the diagonal, we may think of ${\mathcal{B}}_{\rho,{\varepsilon}}$ as a “measurable” replacement of Hölder continous functions on $X$. As the following proposition will show, there is a simple criterion for the validity of the contraction bound \[boundN\] for $Q_\mu$ acting on the space ${\mathcal{B}}_{\rho,{\varepsilon}}$. Let $(G,\mu)$ be a countable measured group and suppose $(X,\nu)$ is a $(G,\mu)$-space which admits a product current $\rho$. Given ${\varepsilon}> 0$ and an integer $n$, we define $$\tau_{{\varepsilon},n} = \operatorname{ess \, sup}\int_G \sigma_\nu(g,\cdot)^{1-2{\varepsilon}} \, d\mu^{*n}(g).$$ Then $N_{\rho,{\varepsilon}}(Q_\mu^{n} u) \leq \tau_{{\varepsilon},n} \cdot N_{\rho,{\varepsilon}}(u)$ for all $u \in {\mathcal{B}}_{\rho,{\varepsilon}}$. First recall that $$\rho(sx,sy) \, \sigma_\nu(s,x) \, \sigma_\nu(s,y) = \rho(x,y)$$ for almost every $(x,y)$ with respect to $\nu \otimes \nu$. Hence, we have $$\begin{aligned} N_{\rho,{\varepsilon}}(Q_{\mu}^{*n}u) &\leq & \int_G \int_{\partial^2 X} \big|u(s^{-1}x) - u(s^{-1}y)\big| \, \rho(x,y)^{\frac{1}{2}+{\varepsilon}} \, d\nu(x) \, d\nu(y) \, d\mu^{*n}(s) \\ &= & \int_G \int_{\partial^2 X} \big|u(x) - u(y)\big| \, \rho(sx,sy)^{\frac{1}{2}+{\varepsilon}} \, \sigma_\nu(s,x) \, \sigma_\nu(s,y) \, d\nu(x) \, d\nu(y) \, d\mu^{*n}(s) \\ &= & \int_G \int_{\partial^2 X} \big|u(x) - u(y)\big| \, \rho(x,y)^{\frac{1}{2}+{\varepsilon}} \, \sigma_\nu(s,x)^{\frac{1}{2}-{\varepsilon}} \, \sigma_\nu(s,y)^{\frac{1}{2}-{\varepsilon}} \, d\nu(x) \, d\nu(y) \, d\mu^{*n}(s) \\ &\leq & \Big( \operatorname{ess \, sup}\int_G \sigma_\nu(s,\cdot)^{\frac{1}{2}-2{\varepsilon}} \, d\mu^{*n}(s) \Big) \cdot N_{\rho,{\varepsilon}}(u),\end{aligned}$$ where we in the last line used Hölder’s inequality twice. Minimality and non-invariance force contraction ----------------------------------------------- The aim of the final subsection of this section will be to isolate natural conditions on a compact $(G,\mu)$-space $(X,\nu)$ which will force the existence of an integer $n$, for every given ${\varepsilon}> 0$, such that $$\label{cocyclebnd} \tau_{n,{\varepsilon}} = \operatorname{ess \, sup}\int_G \sigma_{\nu}(s,\cdot)^{1-2{\varepsilon}} \, d\mu^{*n}(s) < 1.$$ We shall henceforth assume that the functions $x \mapsto \sigma_\nu(s,x)$ are continuous for every $s$ in $G$. Although this assumption is not absolutely necessary, it will simplify many of the arguments below. Furthermore, we may without loss of generality assume that the identity belongs to the support of $\mu$. Indeed, if not, then we can replace $\mu$ with the probability measure $$\mu_o = \frac{1}{2}\delta_e + \frac{1}{2} \mu,$$ with respect to which $\nu$ is still stationary, and holds for $\mu$ if and only if it holds for $\mu_o$. Note that the supports of $\mu_o^{*n}$ forms an *increasing* family of sets in $G$ which asymptotically exhausts $G$.\ First recall that by by (which holds for every $(G,\mu)$-space), we have $$\int_G \sigma_\nu(s,x) \, d\mu^{*n}(s) = 1$$ for all $n$ and for almost every $x$ in $X$, In particular, $\tau_{n,{\varepsilon}}$ is always bounded by one for all $n$ and ${\varepsilon}$, and fails if and only if for every $n$, there exists $x_n \in X$ such that $$\int_G \sigma_\nu(s,x_n)^{1-2{\varepsilon}} \, d\mu^{*n}(s) = 1.$$ In other words, for every $n$, we have equality in Hölder’s inequality (when integrating against $\mu^{*n}$), which clearly forces the identities $$\sigma_\nu(s,x_n) = 1 \quad \forall \, s \in \operatorname{supp}\mu^{*n}$$ for all $n$. Let $x_\infty$ be an accumulation point of the sequence $(x_n)$ in $X$. Since $\sigma_\nu(s,\cdot)$ is continuous for every $s$ and the supports of $\mu^{*n}$ is an increasing exhausting family of sets in $G$, we conclude that $$\sigma_\nu(s,x_\infty) = 1, \quad \forall \, s \in G.$$ Let us now further assume that the $G$-action on $X$ is *minimal*, i.e. every $G$-orbit is dense. Then, by the cocycle equation , which holds for every $(G,\mu)$-space, we have $\sigma_\nu(s,tx_\infty) = 1$ for all $s,t$ in $G$, and since $Gx_\infty$ is dense and $\sigma_\nu(s,\cdot)$ is continuous, we conclude that $\sigma_\nu(s,x) = 1$ for all $s$ in $G$ and $x$ in $X$, or equivalently, $\nu$ is $G$-invariant. We summarize the above discussion in the following proposition. \[cont\] Let $(G,\mu)$ be a countable measured group and suppose $(X,\nu)$ is a compact minimal $(G,\mu)$-space such that $\sigma_\nu(s,\cdot)$ is continuous for every $s$ in $G$. If $\nu$ is not $G$-invariant, then for every ${\varepsilon}> 0$, there exists an integer $n$ such that $$\sup \int_G \sigma_\nu(s,\cdot)^{1-2{\varepsilon}} \, d\mu^{*n}(s) < 1.$$ The assumptions in the last proposition are satisfied for every symmetric probability measure $\mu$ with finite exponential moments (with respect to the any word metric) on any non-elementary Gromov hyperbolic group, where $(X,\nu)$ denotes its Gromov boundary and $\nu$ is the unique $\mu$-stationary measure on $X$. Hence Proposition \[cont\] gives a new proof of the main technical estimate in the author’s paper [@Bj08]. Although Proposition \[cont\] assumes a lot about the topological and dynamical structure of $(X,\nu)$, there is no assumption about the moments of $\mu$. In particular, Proposition \[cont\], as well as the discussions about product currents proceeding it, also apply to the Furstenberg boundary action of a lattice $G$ in a simple Lie group $H$, at least when the $\mu$-stationary measure $\nu$ on the Furstenberg boundary $H/P$ (here $P$ is a minimal parabolic subgroup of $H$) belong to the Haar measure class. Such probability measures on the lattice always exist (see e.g. [@Fu]), but they tend to have very heavy tails. Since the Furstenberg boundary of a simple Lie group, equipped with the Haar measure, always admits a product current (upon identifying a conull subset of $H/P \times H/P$ with $H/A$, where $A$ is the (unimodular) split torus of $G$), Proposition \[cont\] in particular implies that every function in the associated Besov space ${\mathcal{B}}_{\rho,{\varepsilon}}$ is in fact of the form $\phi - \mu * \phi$ for some $\phi \in {\mathcal{B}}_{\rho,{\varepsilon}}$. Product sets in groups ====================== This final section is concerned with the structure of difference sets in free groups, and the aim here is to give a short and rather elementary proof of a weaker version of a recent theorem by the author and A. Fish (Theorem 1.1 in [@BF3]). We begin by providing some background and motivation.\ A significant part of additive combinatorics is concerned with special instances of the following phenomenology: If $G$ is a countable group and $A, B \subset G$ are “large” subsets, then the product set $AB$ should exhibit “substantial sub-structures”. The exact meanings of these notions varies a lot depending on the context, and in this section we shall only be concerned with (partially) extending the following result by Khintchine [@Kh35] and Følner [@Fo], which was one of the first observations of this phenomenology (at least in the setting of discrete groups). \[FO\] Suppose $A_1, \ldots, A_k \subset {\mathbb{Z}}$ are subsets which are “large” in the sense that $$\varlimsup_{n {\rightarrow}\infty} \frac{|A_i \cap [-n,n]|}{2n+1} > 0, \quad \forall \, i=1,\ldots,k.$$ Then their difference sets contain “substantial sub-structures”, in the sense that there exists a finite set $F \subset {\mathbb{Z}}$ such that $$F + \bigcap_{i=1}^k (A_i-A_i) = {\mathbb{Z}}.$$ The additive group of integers is of course nothing but the free group on one generator. A first naive attempt to extend Theorem \[FO\] to free group on two or more generators could be devised along the following lines. Let ${\mathbb{F}}_2$ denote the free group on two (free) generators $a$ and $b$ and let $B_n$ denote the ball of radius $n$ with respect to these generators, that is to say, $B_n$ consists of all the words in $a$ and $b$ and their inverses whose reduced form have length at most $n$. In analogy with Theorem \[FO\] (where the “balls” with respect to the one free generator $1$ are simply given by the interval $[-n,n]$), we say that a set $A \subset {\mathbb{F}}_2$ is *upper large* if $$\varlimsup_{n {\rightarrow}\infty} \frac{|A \cap B_n|}{|B_n|} > 0.$$ However, we warn the reader that upper large sets could be *very* sparse in ${\mathbb{F}}_2$; for instance, given any increasing sequence $(r_i)$ of positive integers, the set $$\label{sparse} A = \bigcup_{i=1}^\infty \big(B_{r_{i}+1} \setminus B_{r_i}\big) \subset {\mathbb{F}}_2$$ satisfies $$\varlimsup_{n {\rightarrow}\infty} \frac{|A \cap B_n|}{|B_n|} \geq \frac{2}{3}.$$ We do not expect to say anything intelligent about differences of sets like these, so we slightly modify our notion of largeness to exclude too sparse examples. Define the *sphere* $S_n$ of radius $n$ by $S_n = B_n \setminus B_{n-1}$ and say that a set $A \subset G$ is *large* if $$\varlimsup_{m {\rightarrow}\infty} \frac{1}{m} \sum_{n=1}^m \frac{|A \cap S_n|}{|S_n|} > 0.$$ We see that for a set as in to be large in this sense, serious growth constraints on the sequence $(r_i)$ have to be imposed, so the notion of largeness is *strictly* weaker than upper largeness. One can now ask whether something like Theorem \[FO\] could be true for large sets. However, already simple considerations show that great care has to be taken to even formulate the right statement. Indeed, it is not hard to construct (and we refer to [@BF3] for details) large subsets $A_1,A_2,A_3 \subset {\mathbb{F}}_2$ such that $$A_1 A_1^{-1} \cap A_2 A_2^{-1} \cap A_3 A_3^{-1} = \{0\}$$ and for which there is no finite subset $F \subset G$ such that $FA_iA_i^{-1} = {\mathbb{F}}_2$ for *some* $i = 1,2,3$. However, the situation is not completely hopeless if one is willing to slightly weaken the notion of “substantial sub-structure” as the following recent observation (see Corollary 1.2 in [@BF3]) by the author and A. Fish shows. \[BF\] Suppose $A \subset {\mathbb{F}}_2$ is “large” in the sense that $$\varlimsup_{m {\rightarrow}\infty} \frac{1}{m} \sum_{n=1}^m \frac{|A \cap S_n|}{|S_n|} > 0.$$ Then there exists a finite set $F \subset {\mathbb{F}}_2$ such that $FAA^{-1}$ contains a right translate of every finite subset of $G$. We stress that this is not the formulation of Corollary 1.2. in [@BF3], so we first take a moment to rewrite Theorem \[BF\] in a language which better align with the present paper (and with [@BF3]). Let $G = {\mathbb{F}}_2$ and define the probability measures $(\sigma_n)$ on $G$ (uniform sphere averages) by $$\sigma_o = \delta_e {\quad \textrm{and} \quad}\sigma_n = \frac{1}{|S_n|} \sum_{s \in S_n} \delta_s, \quad \textrm{for $n \geq 1$}.$$ It is not hard to verify the relations $$\label{rec} \sigma_n * \sigma_1 = \frac{3}{4} \cdot \sigma_{n+1} + \frac{1}{4} \cdot \sigma_{n-1}, \quad \forall \, n \geq 1,$$ which in particular shows that every $\sigma_n$ can be written as a convex combination of convolution powers of $\sigma_1$ and $\delta_e$. Let $M(G)$ denote the convex set of all means on $G$, i.e. the set of all linear functionals on $\ell^\infty(G)$ which are positive and unital (i.e. $\lambda(1) = 1)$. We note that every mean $\lambda$ gives rise to a *finitely additive* probability measure $\lambda'$ on $G$ via the formula $$\lambda'(C) = \lambda(\chi_C), \quad C \subset G,$$ and by the Banach-Alaoglo’s Theorem, the set $M(G)$ is compact with respect to the weak\*-topology. In particular, every sequence $(\lambda_i)$ of the form $$\lambda_i = \frac{1}{m_i} \sum_{n=1}^{m_i} \sigma_n, \quad i \geq 1,$$ for some increasing sequence $(m_i)$, must have at least one cluster point $\lambda$, which by the relations in is necessarily left $\sigma_1$-harmonic (note that $\sigma_1$ is symmetric), that is to say $$\int_G g \cdot \lambda(\varphi) \, d\sigma_1(g) = \lambda(\varphi), \quad \forall \, \varphi \in \ell^\infty(G),$$ where $G$ acts on $\ell^\infty(G)$ (and hence on its dual via the transpose map) by the left regular representation. In particular, if we choose a sequence $(m_i)$ such that $$\lim_{i {\rightarrow}\infty} \frac{1}{m_i} \sum_{n=1}^{m_i} \frac{|A \cap S_n|}{|S_n|} > 0,$$ and a cluster point $\lambda$ of the corresponding sequence of means as above, then $\lambda'(A) > 0$. The aim is now to show that this condition automatically forces the existence of a finite set $F \subset G$ such that $FAA^{-1}$ contains a right translate of of every finite subset of $G$. It will be convenient to adopt a slightly more general perspective on these matters. Let $(G,\mu)$ be a countable symmetric measured group. We say that an element $\lambda \in {\mathcal{M}}(G)$ is *left $\mu$-harmonic* if $$\int_G \lambda(\varphi(g^{-1} \cdot)) \, d\mu(g) = \lambda(\varphi), \quad \forall \, \varphi \in \ell^\infty(G).$$ We say that a set $T \subset G$ is *right thick* if it contains a right translate of every finite subset of $G$, that is to say, if for every finite subset $F \subset G$, there exists $g \in G$ such that $Fg \subset T$. It is not hard to see that a set $T \subset G$ is right thick if and only if for every finite set $F \subset$, the intersection of all left translates of the form $fT$, with $f \in F$, is non-empty. In particular, if $\lambda$ is a left $\mu$-harmonic mean on $G$ such that $\lambda'(T) = 1$, then $$\int_G \lambda'(gT) \, d\mu^{*k}(g) = \lambda'(T) = 1, \quad \forall \, k \geq 1,$$ which shows that $\lambda'(gT) = 1$ for all $g \in G$. Since $\lambda'$ is a finitely additive measure, we conclude that for every finite set $F \subset G$, the intersection of all left translates $fT$, with $f \in F$, still has full $\lambda'$-measure (so in particular it is non-empty), which shows that $T$ must be right thick.\ Theorem \[BF\] will now follow from the following proposition. \[mainBF\] Let $(G,\mu)$ be a measured group and suppose $A \subset G$ has positive measure with respect to some left $\mu$-harmonic mean on $G$. Then there exists a finite set $F \subset A$ such that $FAA^{-1}$ has measure one with respect to some left $\mu$-harmonic mean on $G$. To prove this proposition, we will need the following result, which is not hard, and follows from quite standard correspondence principles. However, the author is not aware of a (short) proof which avoids various technical manipulations with extreme points in the simplex of $\mu$-harmonic measures on compact $G$-spaces. We shall therefore omit the proof, and refer the interested reader to Proposition 1.2 in [@BF3], where a much stronger result is proven. Fix ${\varepsilon}> 0$ and suppose $A \subset G$ has positive measure with respect to some left $\mu$-harmonic mean. Then there exists a finite set $F \subset G$ and a (possibly different) left $\mu$-harmonic mean $\eta$ on $G$ such that $\eta(FA) \geq 1 - {\varepsilon}$. If one is willing to take this lemma for granted, then we argue as follows. Suppose $A \subset G$ has positive $\lambda'$-measure for some left $\mu$-harmonic mean $\lambda$ on $G$. Fix ${\varepsilon}> 0$ and find, by the previous lemma, a finite set $F \subset G$ and a left $\mu$-harmonic mean $\eta$ on $G$ such that $$\eta(FA) \geq 1 - {\varepsilon}\cdot \eta(A).$$ We note that $$FAA^{-1} \supset \big\{ g \in G \, : \, \eta(FA \cap gA) > 0 \big\} \supset \big\{ g \in G \, : \, \eta(gA) > {\varepsilon}\cdot \eta(A) \Big\},$$ and the function $$u(g) = \eta(gA) - {\varepsilon}\cdot \eta(A), \quad g \in G,$$ is a real-valued bounded left $\mu$-harmonic function on $G$. Furthermore, if $0 < {\varepsilon}< 1$, then $u(e) > 0$ and $u$ is *positively correlated* in the sense that $$\|u\|_\infty = \sup\big\{ u(g) \, : \, g \in G \big\},$$ so Proposition \[mainBF\] will follow from the “zero-one law” stated below. \[measone\] Let $(G,\mu)$ be a measured group and suppose $u$ is a bounded real-valued left $\mu$-harmonic function on $G$. Define the set $$S_u = \Big\{ g \in G \, : \, u(g) > 0 \Big\} \subset G.$$ If $u$ is positively correlated, then there exists a left $\mu$-harmonic mean which gives measure one to the set $S_u$. To prove this lemma, we first note that for any mean $\lambda$ on $G$, for any ${\varepsilon}> 0$ and for every bounded function $u$ on $G$, we have $$\begin{aligned} \lambda'\big(\big\{ g \in G \, : \, u(g) > 0 \big\} &\geq& \lambda'\big(\big\{ g \in G \, : \, u(g) \geq (1-{\varepsilon}) \cdot \|u\|_\infty \big\} \\ &=& 1 - \lambda'\big(\big\{ g \in G \, : \, u(g) < (1-{\varepsilon}) \cdot \|u\|_\infty \big\} \\ &=& 1 - \lambda'\big(\big\{ g \in G \, : \, \|u\|_\infty - u(g) > {\varepsilon}\cdot \|u\|_\infty \big\} \\ &\geq& 1 - \frac{1}{{\varepsilon}\cdot \|u\|_\infty} \cdot (\|u\|_\infty - \lambda(u)),\end{aligned}$$ by Chebyshev’s inequality (which works equally well for finitely additive probability measures). Hence it suffices to show that whenever $u$ is a positively correlated left $\mu$-harmonic function, there exists a left $\mu$-harmonic mean $\lambda$ such that $\lambda(u) = \|u_\infty\|$. To prove this, we fix a sequence $(g_n)$ such that $$\lim_n u(g_n) = \sup\big\{ u(g) \, : \, g \in G \big\},$$ and define the sequence $(\lambda_m)$ of means on $G$ by $$\lambda_m(\phi) = \frac{1}{m} \sum_{n=1}^m \int_G \phi(xg_m) \, d{\check{\mu}}^{*n}(x), \quad \phi \in \ell^\infty(G).$$ Since $u$ is left $\mu$-harmonic, we have $\lambda_m(u) = u(g_m)$ for all $m$, and one readily checks that any cluster point $\lambda$ of the sequence $(\lambda_m)$ in $M(G)$ is left $\mu$-harmonic and satisfies $\lambda(u) = \lim_m u(g_m)$. Appendix I: Harmonic functions and affine isometric actions on Hilbert spaces ============================================================================= As part of Theorem \[KL\], we proved that if $(G,\mu)$ is a measured Liouville group and $u$ is a left Lipschitz and left (quasi-)$\mu$-harmonic function on $G$, then $u$ must be a homomorphism. The aim of this appendix is to show that the combination “LEFT Lipschitz” and “LEFT quasi-$\mu$-harmonic” is crucial, and if one (but not both) is replaced by a “RIGHT”, then the situation is quite different. Indeed, we shall prove the following theorem, whose origin is hard to track down, but which is well-known to experts. Every infinite, finitely generated and symmetric measured group $(G,\mu)$, where $\mu$ is assumed to be finitely supported, admits a non-trivial left Lipschitz and *right* $\mu$-harmonic function. Since every *non-amenable* measured group $(G,\mu)$ admits a wealth of non-trivial *bounded* right $\mu$-harmonic functions, the theorem is perhaps most interesting for amenable groups. However, the construction which we will describe below works for a larger class of groups, namely those which admit affine isometric actions on (real) Hilbert spaces with *unbounded* orbits. It is well-known (see e.g. Theorem 13.10 in [@Gla]) that this is equivalent to assuming that the group $G$ does *not* have Kazhdan’s Property (T). In particular, our construction will work for every countable (infinite) *amenable* group. Recall that if ${\mathcal{H}}$ is a real Hilbert space, then a map $T : {\mathcal{H}}{\rightarrow}{\mathcal{H}}$ is an *affine isometry* $T$ if it can be written on the form $$Tx = Ux + b, \quad x \in {\mathcal{H}},$$ for some *linear* isometry $U$ of ${\mathcal{H}}$ and $b \in {\mathcal{H}}$. Clearly, the set of affine isometries of ${\mathcal{H}}$ forms a group $\operatorname{Aff}({\mathcal{H}})$ under composition, and a homomorphism $\alpha : G {\rightarrow}\operatorname{Aff}({\mathcal{H}})$ is called an *affine isometric action* of $G$ on ${\mathcal{H}}$. Explicitly, we have $$\alpha(g)x = \pi(g)x + b(g)$$ for some linear isometric representation $\pi$ of $G$ and a map $b : G {\rightarrow}{\mathcal{H}}$ which satisfies $$b(gh) = b(g) + \pi(g)b(h), \quad \forall \, g, h \in G.$$ We shall refer to such maps as *$\pi$-cocycles*, and we note that the action $\alpha$ has bounded orbits if and only if the corresponding $b$ is a norm-bounded function on $G$. Let $(G,\mu)$ be a finitely generated and symmetric measured group, where $\mu$ is assumed to be finitely supported, and suppose $\alpha$ is an affine isometric action of $G$ on a *real* Hilbert space ${\mathcal{H}}$ without unbounded orbits. Then there exists $x_o$ and $y$ in ${\mathcal{H}}$ such that $$f(g) = \langle y, \alpha(g) \cdot x_o \rangle_{\mathcal{H}}, \quad g \in G,$$ is an *unbounded*, left Lipschitz and *right* $\mu$-harmonic function on $G$. Recall that $\alpha$ can be written on the form $$\alpha(g)x = \pi(g)x + b(g),$$ for some linear isometric representation $\pi$ of $G$ and a $\pi$-cocycle $b : G {\rightarrow}{\mathcal{H}}$. The assumption that the $\alpha$ has unbounded orbits simply means that $$\sup_{g \in G} \|\alpha(g)x\| = \infty, \quad \forall \, x \in {\mathcal{H}},$$ and we shall prove that there exists $x_o \in {\mathcal{H}}$ such that the orbit map $$F(g) = \alpha(g)x_o, \quad g \in G,$$ satisfies $F * \mu = F$ in ${\mathcal{H}}$, or equivalently (after some easy manipulations) $$\label{eqharmonic} \int_G \big( x_o - \alpha(s) x_o\big) \, d\mu(s) = 0.$$ By the uniform boundedness principle, if $$\sup_{g \in G} \big| \langle y, \alpha(g) x_o \rangle\big| < +\infty$$ for all $y \in {\mathcal{H}}$, then $$\sup_{g \in G} \|\alpha(g) x_o\|< +\infty,$$ which is a contradiction, and we conclude that there must exist $y \in {\mathcal{H}}$ such that the function $$f(g) = \langle y, \alpha(g)x_o \rangle, \quad g \in G,$$ is an *unbounded* (and hence non-constant) right $\mu$-harmonic function on $G$. Also note that $$\big|f(sg) - f(s)\big| = \big| \langle y,\pi(s)\big(\pi(g)x_o - x_o + b(g)\big)\rangle\big| \leq \|y\| \cdot \big(2 \|x_o\| + \|b(g)\|\big),$$ for all $g$ and $s$, which shows that $f$ is left Lipschitz.\ To establish the existence of $x_o \in {\mathcal{H}}$ such that holds, we argue as follows. Consider the “energy functional” $$E(x) = \int_G \big\| \alpha(s) x - x\big\|^2 \, d\mu(s), \quad x \in {\mathcal{H}},$$ which is well-defined since $\mu$ is finitely supported (but clearly this assumption can be substantially weakened). One readily checks that $E$ admits a local minimum $x_o$, and thus $$\frac{d}{dt}E(x_o + tv) \Big|_{t = 0} = 0, \quad \forall \, v \in {\mathcal{H}}.$$ The left hand side can be easily calculated. Indeed, after a series of calculations, using the assumptions that ${\mathcal{H}}$ is a real Hilbert space and $\mu$ is a symmetric measure on $G$, we arrive at the identities, $$\frac{d}{dt}E(x_o + tv) \Big|_{t = 0} = 4 \cdot \langle v, \int_G \big( x_o - \alpha(s) x_o\big) \, d\mu(s) \rangle = 0,$$ for all $v \in {\mathcal{H}}$, from which follows. Appendix II: Open problems and remarks ====================================== We collect in this appendix some questions and remarks relating to the topics discussed in this paper. Drifts of random walks on homogeneous spaces -------------------------------------------- Let $(G,\mu)$ be a countable measured group and denote by $(B,m)$ its Poisson boundary. Clearly, if $H < G$ is a subgroup which acts ergodically on $(B,m)$, then there are no non-constant *bounded* left $\mu$-harmonic functions on the quotient space $G/H$. However, it certainly also makes sense to ask whether *unbounded* left $\mu$-harmonic functions can exist on the quotient space $G/H$, at least when $H$ has infinite index in the group $G$. For instance, in the extreme case when $\mu$ is symmetric and finitely supported such that $(G,\mu)$ is a Liouville group (that is to say, $(B,m)$ is just a singleton space) and $H$ is the trivial subgroup, then the construction in Appendix I, shows that there are always unbounded left $\mu$-harmonic functions. A less extreme case is suggested by Corollary \[CD\], which can be equivalently stated as the assertion that there are no non-constant bounded left $\mu_o \otimes {\check{\mu}}_o$-harmonic functions on the quotient $G_o \times G_o/\Delta_2 G_o$ for any measured group $(G_o,\mu_o)$. In this setting, the problem above can be equivalently formulated as follows. Does every measured group $(G,\mu)$ admit a non-trivial bi-$\mu$-harmonic (left and right $\mu$-harmonic) function? The problem for general quotient spaces seems intractable, and there could very well be obvious counter-examples. Construct a countable measured group $(G,\mu)$ and an infinite index subgroup $H < G$ such that the quotient space $G/H$ does not admit *any* non-constant left (quasi-)$\mu$-harmonic functions whatsoever. It is clear that the notion of drift can be generalized to invariant metrics on more general $G$-spaces (in particular coset spaces). An affirmative answer to the following question would generalize the Karlsson-Ledrappier Theorem (Theorem \[KL\]) to this setting. Let $(G,\mu)$ be a measured group and $H < G$ a subgroup which acts ergodically on the Poisson boundary of $(G,\mu)$. If $d$ is a left $G$-invariant metric on the quotient space $G/H$, is it then true that $$\lim_{n {\rightarrow}\infty} \frac{1}{n} \sum_{k=1}^n \int_G d(gH,H) \, d\mu^{*k}(g) = 0?$$ One could start by analyzing the following special case which corresponds to the case when $G = G_o \times G_o$ and $H = \Delta_2 G_o$ and $\mu = \mu_o \otimes \mu_o$, for some countable group $G_o$ and some symmetric measure $\mu_o$ on $G_o$. Let $(G,\mu)$ be a symmetric measured group and suppose there exists a *bi-invariant* (conjugation-invariant) and $\mu$-integrable (semi-)metric $d$ on $G$. Is $\ell_d(\mu) = 0$? For instance, as a first test case, one could focus on the commutator subgroup $G$ of the free group on two generators and the stable commutator length on $G$. Harmonic Kronecker factors -------------------------- Let $G$ be a countable group and $(X,\nu)$ a non-singular ergodic $G$-space. Let ${\mathcal{K}}$ denote the smallest $G$-invariant sub-$\sigma$-algebra of the Borel $\sigma$-algebra on $X$ with the property that ${\mathcal{K}}\times {\mathcal{K}}$ contains the $\sigma$-algebra of all $G$-invariant subsets in $X \times X$. When $\nu$ is $G$-invariant, this $G$-invariant $\sigma$-algebra (or its corresponding factor) is usually called the *Kronecker factor*, and it is a classical fact that the factor $G$-space is isomorphic to an action by rotations of $G$ on a compact homogeneous space. Except for some remarks in [@AN87], this factor does not seem to have attracted much attention, and it seems hard to say anything significant about it in this generality. However, it could be that the situation for $(G,\mu)$-spaces is more amenable for a closer analysis. Let $(G,\mu)$ be a symmetric measured group with Poisson boundary $(B,m)$ and suppose $(X,\nu)$ is an ergodic $(G,\mu)$-space which admits $(B,m)$ as a factor. Assume that the product of $(X,\nu)$ with itself is *not* ergodic. Does this mean that there exists a factor $(Y,\eta)$ of $(X,\nu)$ which is a non-trivial isometric extension of $(B,m)$? Put differently, is $(Y,\eta)$ isomorphic (as a $G$-space) to a skew product of the form $(B \times K/K_o,m \otimes \eta)$, where $K$ is a compact group and $K_o$ a closed subgroup and $\eta$ is the Haar probability measure on $K/K_o$, such that the $G$-action can be written as $$g(b,z) = (gb,c(g,b)z), \quad (b,z) \in B \times K/K_o,$$ where $c : G \times B {\rightarrow}K$ is a measurable cocycle? Acknowledgments =============== The author would like to thank Vadim Kaimanovich for encouraging him to write up the present collection of random observations. His warm thanks also go to Uri Bader, Alex Furman, Yvés Guivarc’h, Yair Hartman, Anders Karlsson, Gady Kozma and Amos Nevo for their never-ending willingness to discuss various problems relating to harmonic functions and $(G,\mu)$-spaces. 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--- abstract: 'We propose a novel multi-layered nonlinear model that is able to capture and predict the housing-demographic dynamics of the real-state market by simulating the transitions of owners among price-based house layers. This model allows us to determine which parameters are most effective to smoothen the severity of a potential market crisis. The International Monetary Fund (IMF) has issued severe warnings about the current real-state bubble in the United States, the United Kingdom, Ireland, the Netherlands, Australia and Spain in the last years. Madrid (Spain), in particular, is an extreme case of this bubble. It is, therefore, an excellent test case to analyze housing dynamics in the context of the empirical data provided by the Spanish [*National Institute of Statistics*]{} and other sources of data. The model is able to predict the mean house occupancy, and shows that i) the house market conditions in Madrid are unstable but not critical; and ii) the regulation of the construction rate is more effective than interest rate changes. Our results indicate that to accommodate the construction rate to the total population of first-time buyers is the most effective way to maintain the system near equilibrium conditions. In addition, we show that to raise interest rates will heavily affect the poorest housing bands of the population while the middle class layers remain nearly unaffected.' author: - | Ramón Huerta$^{a}$, Fernando Corbacho$^{b,c}$, Luis F. Lago-Fernández$^{b,c,}$[^1]\ \ $^{a}$ Institute for Nonlinear Science, University of California, San Diego,\ La Jolla, CA 92093-0402\ $^{b}$ Escuela Politécnica Superior, Universidad Autónoma de Madrid,\ 28049 Madrid (SPAIN)\ $^{c}$ Cognodata Consulting, Caracas 23, 28010 Madrid (SPAIN)\ title: 'A housing-demographic multi-layered nonlinear model to test regulation strategies' --- Introduction ============ A crisis on the housing market could heavily reverse the current positive economical indicators [@crashbook; @garber2000] and have a negative general impact in the world economy, as shown in the Asian crisis in 1997 [@asian]. Real-state prices have increased more than fifty percent in Australia, Ireland, United Kingdom and Spain from 1997 to 2004. These increases are hardly explainable in terms of economic fundamentals alone, even including record-low interest rates [@imf2]. Spain, in particular, holds high risk figures in terms of ratios of house prices to disposable income per worker (RPI) and house prices to rent (RPR). According to IMF calculations Spain has $3.6$, $3.6$, $2.5$, $2.2$, $1.8$ times the RPI of Germany, Japan, United States, France and United Kingdom, respectively [@imf2]. Despite the overwhelming cost to acquire a house in Spain compared to all developed countries, house unit prices currently remain unaffected. In addition, in terms of the RPR, which is a good estimation of the yield of the investment, Spain has $3.42$, $3.32$, $1.9$, $1.8$, and $1.28$ times the RPR of Germany, Japan, France, United States and United Kingdom, respectively [@imf2]. The RPI index has been proved to be a good reference of the distance to market equilibrium [@overprize]. Therefore, Spain is a valid scenario to analyze the conditions to understand and control a housing market by means of nonlinear dynamical models. In this paper we develop a mean field model of the house occupancy derived from a stochastic process as previously used in the study of epidemic dynamics [@epidemic]. This nonlinear model is employed as a tool to ascertain the possibility of sudden changes in the dynamics as a function of the control parameters. The model is based on dissecting the house population in several groups or layers. Each layer has a given number of houses, which can be either occupied or empty. Every family occupying a house in a given layer has a certain probability rate to migrate to any other layer. In addition, there exists a base pool of nonowners, which can jump to the housing layers at a certain probability rate. Thus the model captures the dynamics of the mean migrations between housing layers. As a particular case, we analyze the house market in the city of Madrid in the framework of the model. The parameters of the model have been obtained from different sources such as the Spanish National Institute of Statistics, the Spanish Ministry, and some valuation companies [@censo91; @epa; @ine; @tasacion]. Our main results may be summarized as follows: - The model is able to predict the occupancy levels in 2001 given the parameters obtained from 1991 data. This is a good indication that the mean description is able to model the real dynamics. - The model shows that the house market for the city of Madrid asymptotically evolves to an out-of-equilibrium condition. It is rather worrisome that the new housing units cannot be replenished by first-time buyers. - Critical phenomena do not exist in the context of this model. Only smooth changes can be expected. - A sudden increase in the interest rate will seriously affect the occupancy level of the lowest layer of the market, i.e. the poorest housing unit sector, while the middle layers remain nearly unaffected. - According to the model, the most effective way to control the out-of-equilibrium situation of the Madrid house market is to down regulate the construction rate. There is a large body of research in the housing market [@review; @review2], most of the analyses aiming at predicting house prices. We do not intend to estimate prices, our modeling efforts fit better with housing-demographic models [@population1; @population2]. The novelty of our approach lies on building a multi-layered nonlinear dynamical model that includes family migrations across layers of housing units. The whole population is separated in non-overlapping housing bands, which are estimated from census data. Our model neglects the random component of the stochastic process, because we use a mean field approach under the assumption of random mixing. The random mixing approximation basically states that any family can, in principle, uniformly access any house. This random mixing approximation allows to obtain a set of ordinary differential equations (ODEs) derived from a stochastic process. The formalism that we use here sets us apart from previous approaches. Overview of the model ===================== In this section we provide an overall description of the non-linear dynamical model for the house occupancy. The model equations will be explicitly derived in the next section. We divide the total housing population into different price bands or layers, and model the transitions of owners among these layers. Note that, in this paper, we only use the price as a tool to determine the different house bands. Let us assume that there are $N_{layers}$ layers. We define $N_{i}(t)$ as the [*total*]{} number of houses, or units, in layer $i$ at time $t$. Each of the layers has a certain number of [*occupied*]{} houses, $O_{i}(t)$. These are defined as houses occupied by owners. In addition to the house price bands, we consider a source of new buyers, which we call the [*base pool*]{}. It accounts for non-emancipated people, immigrants, and rented units. A group of people living in the same house will be called a [*family*]{}. The family is the basic people unit in our model, and so we are modeling family jumps among house layers. Note that the number of families in a given layer $i$ equals the number of occupied houses in that layer, $O_{i}(t)$. The concept of family applies to non-emancipated people and immigrants as well. However, in these cases information concerning the total number of single individuals is more frequently available in census databases, so we must apply a correction factor when calculating the number of families ($\Sigma(t)$) in the base pool. The migrations among house layers are modeled in terms of transition probabilities. We denote by $\mu_{ij}$ the probability rate for a family in house layer $j$ to move to house layer $i$. Equivalently, we denote the probability rate to move from the base pool to any of the house layers $i$ by $\eta_{i}$. Finally, there is a probability for any family to disappear, or die, leaving its house empty. We call this probability the [*death rate*]{}, $\lambda_{i}$, which we assume to be constant for each layer $i$. Figure \[fig2\]A shows an overall scheme of the model; figure \[fig2\]B displays a list of the variables and parameters involved. With all the above ingredients we pose a set of non-linear coupled equations for the mean occupancy of each level: $o_{i}(t) = O_{i}(t) / N_{i}(t)$. The model equations are derived from the mean field approximation to a stochastic process, as used in epidemic dynamics. Details are provided in the next section. Derivation of the model equations ================================= The model proposed here is similar in derivation to the ones used in epidemic modeling [@epidemic; @huerta] where the stochastic epidemic process is reduced to a set of ordinary differential equations (ODEs). These ODEs capture quite faithfully the behavior of the stochastic process behind epidemics. The main advantage of this approach is that the complexity of the process is simplified to a formalism that allows an easier understanding of the qualitative behavior. The parameters can be easily related to the end result of the stochastic process. In most cases the ODEs match well the stochastic process, although there are some others where the ODE description fails, for example, for finite-size effects. Overall, the ODE description is a very good framework to gain understanding that can complement very well stochastic modeling. Our main contribution is to bring these convenient tools to housing-demographic modeling. To model the dynamics across housing layers we use two possible states for any house: occupied and empty. The total number of houses in each layer $i$, $N_{i}(t)$, evolves in time according to a function estimated from the census data. Given the number of occupied units in layer $i$, $O_{i}(t)$, and the transition probability rate from layer $j$ to layer $i$, $\mu_{ij}$, the probability that a family jumps from layer $j$ to layer $i$ in the time interval $dt$ is $\mu_{ij}\left\{1-O_i(t)/N_i(t)\right\}\, dt$. This probability already assumes that a family can only occupy an empty house in layer $i$. This is equivalent to the random mixing approximation widely used in epidemic modeling [@epidemic]. The net flow into layer $i$ is the difference between the number of incoming and outgoing families in the time interval $dt$: $$F_{i}=\sum_{j=1}^{N_{layers}} \left[O_j(t) \mu_{ij}\left\{1-\frac{O_i(t)}{N_i(t)}\right\} -O_i(t) \mu_{ji}\left\{1-\frac{O_j(t)}{N_j(t)}\right\}\right] dt$$ This flow equals the variation in the number of occupied states in layer $i$ during the time interval $dt$, so we can write the following set of ordinary differential equations (ODEs) for the layer occupancy: $$\frac{d O_i (t)}{dt}=\sum_{j=1}^{N_{layers}} \left[\mu_{ij} O_j (t)\left\{1- \frac{O_i (t)}{N_i(t)}\right\} -\mu_{ji}O_i (t)\left\{1-\frac{O_j (t)}{N_j(t)}\right\}\right]$$ It is critical to include the dynamical contribution of the base pool. The probability for a family to jump from the base pool to the layer $i$ in the time interval $dt$ is given by $\eta_i \left\{1-(O_i(t)/N_i(t))\right\} \, dt$. Then, the net change in layer $i$ due to the flow from the base pool is simply $ \Sigma (t) \eta_i \left\{1-(O_i(t)/N_i(t))\right\} \, dt$. Finally, we will consider a family death rate for each layer, $\lambda_i$, which contributes to the variation in layer occupancy with the term $-\lambda_i O_i(t) dt$. The final model equations can be written as: $$\frac{d O_i}{dt}=\eta_i \left(1-\frac{O_i}{N_i(t)}\right)\Sigma(t)- \lambda_i O_i+ \sum_{j=1}^{N_{layers}} \left[\mu_{ij} O_j\left\{1-\frac{O_i}{N_i(t)}\right\} -\mu_{ji}O_i\left\{1-\frac{O_j}{N_j(t)}\right\}\right] \label{basic}$$ Since we plan to analyze the asymptotic behavior of the equations, we define a new variable, $o_i(t)\equiv O_i(t)/N_i(t)$, which is the normalized occupancy level, bounded between $0$ and $1$. We can rewrite the set of equations \[basic\] in terms of the normalized occupancy as: $$\frac{d o_i}{dt}=\eta_i (1-o_i)\frac{\Sigma(t) }{N_i(t)}-o_i\frac{d}{dt}\log N_i(t)-\lambda_i o_i+\sum_{j=1}^{N_{layers}} \left(\mu_{ij} o_j (1-o_i)\frac{N_j(t)}{N_i(t)} -\mu_{ji}o_i (1-o_j)\right) \label{normalized}$$ There are three terms in these ODEs with explicit dependence on time. The first one is the drive from the base pool to saturation levels. If $\Sigma (t)$ grows faster than single layers do, then all layers will saturate. The third term with explicit dependence on $t$ is multiplied by $N_j(t)/N_i(t)$. This term implies that, if the size of layer $j$ grows much faster than layer $i$, in the asymptotic limit the layer $i$ will be totally full, with a huge demand in that layer that will quickly change the band location in the whole distribution of layers. Test case: the city of Madrid ============================= Madrid is a particularly extreme case of the real-state bubble in Spain [@imf2]. This fact, together with the availability of data to estimate model parameters, makes Madrid an interesting test case to be analyzed in the context of our mean field model. The two main sources of data used to feed the model parameters are the Spanish National Institute of Statistics (INE) and the Spanish Ministry, but we have also used data provided by valuation companies and real-state internet sites. First we will provide an overall description of the model parameter estimation in section \[sec\_parameters\]. Then, in section \[sec\_full\_model\], we will use these parameters in the model equations (\[normalized\]) to understand the implications of current market conditions. Parameter estimation {#sec_parameters} -------------------- First of all we must determine the price layers. Figure \[fig1\]A represents the distribution of house prices in the city of Madrid in 1991 (data from [@censo91; @tasacion]). This distribution provides the total number of houses per layer, $N_{i}$, using $7$ layers that account for the $99\%$ of the total number of houses. The number of occupied houses in each layer, $O_{i}$, is also provided by [@censo91; @tasacion]. To choose the price size of each layer we find a compromise between two opposing criteria: i) the maximum number of layers in order to have a detailed distribution of the housing stock; and ii) the widest price size per band such that the transition probability rates between layers are not very small when normalized to the integration time scale. This second criterion intends to avoid finite size effect problems. The time evolution of the number of houses per layer, $N_i(t)$, is hardly available in public databases. Nevertheless, as shown in fig. \[fig1\]B, the total population of houses is available at five different years since 1970 [@ine]. In that figure we can see that the total population of houses is very well fit by an exponential function of time. We will assume that the shape of the layers distribution is time invariant, and will apply the same exponential growth to all layers. This assumption is supported by two facts: (i) the evidence of self-regulation of each of the layers as shown in [@regulation] for the city of Philadelphia, [*i.e.*]{}, undervalued houses compared to similar type of houses get more appreciated than the average; and (ii) the similarity of price distributions in the cities of Pitt County (North Carolina) [@pitt] and Madrid (figure \[fig1\]A). The number of families in the base pool ($\Sigma (t)$), [*i.e.*]{}, the subset of families that do not own a property, is estimated from the INE databases as the sum of: (i) the number of people that are not emancipated within the range of age where people usually emancipate (data from [@epa; @objovi]); (ii) the number of families that rent an apartment (data provided by [@ine]); and (iii) the number of incoming families due to the immigration (data from [@ine]). These statistics do not provide family units, but they report single people data instead. So a headship factor of $2$ has been applied when calculating $\Sigma (t)$. In fig. \[fig1\]C we can see the time evolution of the three subsets that conform the base pool. The two main contributions to the pool have been decreasing in the last few years. On the other hand the immigrant population has undergone a sharp increase. However, in contrast to the popular view, this subset of the population is not officially large, and it might just compensate for the negative tendency in the base pool size time evolution. In contrast to the total number of houses (fig. \[fig1\]B), there is no exponential growth of $\Sigma (t)$. In fact, we will assume it is a constant. This situation, if maintained, would lead the system to complete depletion as we will discuss bellow in the context of the model. Other important parameters to be estimated are the transition probability rates from the base pool to the different layers, $\eta_{i}$, and the transition probability rates among layers, $\mu_{ij}$. The average transition probability rate from the base pool to any of the layers, $\bar{\eta}$, can be estimated from the total number of new occupancies in 1981, 1991 and 2001, as follows. The total number of occupied houses in 1981 was $699,557$, in 1991 was $789,444$ and in 2001 was $908,790$ [@ine]. Therefore the occupancy rate from 1981 to 1991 was $8,989$ new occupancies per year, and from 1981 to 1991 it was $11,935$ new occupancies per year. The base pool population in 1981 is estimated in $568,300$ families (we have data for the non-emancipated population since 1986 and we apply a headship per future household of $2$), in 1991 it was $532,717$ and in 2001 it was $543,323$. Then an estimation of the probability rate of having one base pool family accessing any of the layers is $0.016$ year$^{-1}$ from 1981 to 1991, and $0.022$ year$^{-1}$ from 1991 to 2001. This rate has been increasing in the last few years, maybe due to the decrease in the interest rates. To fit the probabilities $\mu_{ij}$, we assume that the occupancy levels in $1991$ are stationary, and search for the values of $\mu_{ij}$ that provide these levels at equilibrium. The available data do not permit direct calculation of these probabilities, nevertheless it is possible to estimate their average by extrapolating the number of housing transactions in Spain (provided by analysis office by [@bbva] and [@objovi]) to the city of Madrid. If we take this total number of transactions and discount the probability of transition from the base pool and the estimation of housing transactions due to investments, we obtain $\bar{\mu}=0.052$ year$^{-1}$. This basically states that the time it takes a family to change to a new house is 20 years on average for the city of Madrid. This is imposed as a constraint in the following calculations. Therefore, we will assume that the occupancy levels are slightly off from the equilibrium state and that the growth of the population is compensated with the growth of the total number of housing units. The occupancy level of each of the $7$ layers at 1991 is given by $\hat{{\bf o}}= (0.77, 0.56, 0.68, 0.62, 0.85, 0.62, 0.39)$. The parameter search is performed by means of a genetic algorithm with a fitness function that measures the Euclidean distance between $\hat{{\bf o}}$ and the model equilibrium solution ${\bf o}$. To reduce the search space we apply two constraints: (i) the average value of $\mu_{ij}$ is, as stated above, $\bar{\mu}=0.052$ year$^{-1}$; and (ii) the derivative between close parameters is as small as possible. A total of $50$ different simulations were run, the outcomes of three of them are shown in figure \[fig\_probs\]. Although the specific values of the probabilities do not match for the three simulations, the shape and tendency maintain qualitative agreement. We use their average value when solving the model equations. The last important estimation in our model is the family death rate for each layer. We have data of the absolute number of deaths per age and the age distributions per housing layer [@ine]. With these data, assuming a family is composed of two people very close in age, we can calculate the instantaneous family death rate per layer (fig. \[fig1\]D). We can see that the lower layers have a higher death rate, because the average age of families in these layers is higher. This can be explained by the slow integration of the old houses into the lower price layers. On the other hand, newly constructed houses tend to fill higher layers. =0.9 Results {#sec_full_model} ------- For the problem under analysis, and taking into account the exponential growth of the total number of houses for the city of Madrid, the model equations (\[normalized\]) can be rewritten as: $$\frac{d o_i}{dt}=\eta_i (1-o_i)\frac{\Sigma }{\hat{N}_i \exp{k\, t}}-o_i \, (k +\lambda_i)+\sum_{j=1}^{N_{layers}} \left(\mu_{ij} o_j (1-o_i)\frac{\hat{N}_j}{\hat{N}_i} -\mu_{ji}o_i (1-o_j)\right) \label{eq_madrid}$$ where $k$ is the constant of the exponential growth fitted in figure \[fig1\]B, and $\hat{N}_i$ is the total number of houses in layer $i$ in 1991 (figure \[fig1\]A). Integrating the equations with the parameters of previous section, we can make predictions on the mean occupancy level. In particular, the decreasing trend of the occupancy levels from 1991 to 2001 is well predicted by the model using data from 1971 to 1991 (see figure \[fig5\]A). A sufficient condition to keep the system asymptotically away from $0$ (empty) or $1$ (full) is the following $$N(t)=\left(N(0)+\int_{0}^{t}\left(\sum_i {\eta_i}\right) \frac{1-\bar{o}}{\bar{o}}\,e^{\bar{\lambda}\, \tau} \Sigma (\tau)\, d\tau\right)e^{-\bar{\lambda}\,t},\label{eqsimple}$$ where $\bar{o}$ is the mean occupancy level, and $\bar{\lambda}$ is the mean instantaneous household death rate. If the time evolution of the base population is constant, as it appears according to figure \[fig1\]C, then $N(t)\propto exp(-\bar{\lambda} t)$. The optimal equilibrium occupancy levels in the housing market is out of the scope of this paper. Nevertheless it is certainly desirable to steer the system to a equilibrium condition whose asymptotic state is far from the $1$-$0$ extremes. Equation (\[eqsimple\]) intends to provide a simple framework for policy makers to regulate the system in a smooth way from a housing-demographic perspective, disregarding price value of the housing units. Nevertheless, although equation (\[eqsimple\]) is useful due to its simplicity, the intrinsic dynamics of the system is more complicated. Each layer follows a different trend due to the integration of the ODEs. In order to provide a vacancy rate for each of the layers, [*i.e.*]{}, the pace at which each layer becomes empty, we fit an exponential function of time to the predicted occupancy levels, such as $o_i(t)\propto exp(\kappa_i \, t)$, for 20 years since 1991. Note that $o_i(t)$ is not an exponential decay function, but since we are mostly interested in short periods of time ($10$ to $50$ years), the exponential fits are adequate. As it can be seen in Fig. \[fig5\]B, layers 2 and 3 undergo the heaviest vacancy rate at current market conditions, while the other layers are in a more stable situation. This indicates that the middle class layers are subject to the heaviest speculative process, while the other layers may have a lower demand. =0.9 Economical conditions and temperature control parameter ======================================================= The economical conditions that may affect the transition probabilities among layers are incorporated into the model by means of a single parameter $T$, that plays a role similar to the temperature in statistical physics. For example, if the interest rates are raised, the probability of transition from one layer to the rest or from the base pool to any layer is decreased, which is modeled by a temperature decrement. Another example, if the RPI grows excessively, then the temperature is also lowered, with the consequent decrease of the transition probabilities. Unfortunately, we do not have data to quantify the dependence of transition probabilities on economical conditions. To be able to estimate them we would need data that indicates the number of mortgage requests and the amount loaned as a function of time. This has to be provided by banks and we currently do not have such data. The general dependence of the transition probability rates are therefore unknown. Initially, we assume that the transition rates ($\mu_{ij}$ and $\eta_i$) have a linear dependence on the parameter $T$. Therefore, the parameters $\mu_{ij}$ and $\eta_i$ in equation \[eq\_madrid\] are replaced by $T \mu_{ij}$ and $T \eta_i$. This dependence basically implies that the economical conditions uniformly affect all the probability rates in the same way. The goal is to determine whether there is a value of the temperature parameter that leads to a sudden change on the occupancy levels. As long as the dependence of the probability rates on the temperature is an analytic function, the existence of critical behavior in the dynamical system should be preserved. In other words, if $\mu_{ij}(T)$ and $\eta_i(T)$ are discontinuous, then the criticality emerges from parameter dependence on $T$, not from the dynamical system under consideration. In this work, we can only concentrate on the dynamical system behavior. First, to determine the effects of freezing the system by temperature reduction we fit the decay rate of the occupancy level to a exponential function as shown in the previous section (see Fig. \[temperature\]A). The middle layers are more robust to temperature changes. The middle layers show more resilience to critical economical conditions while both extremes of the multilayer model are highly sensitive. This contrasts with the depletion resistance observed in Fig. \[fig5\]B for layer one. Thus, the lowest layer is the most sensitive to changes in the economical conditions. =0.9 As an example of the sensitivity of the lowest layer we introduce in 2005 a sudden decrease in the temperature that can be seen as a strong increase in the interest rates. In Fig. \[temperature\]B we can see that a sudden change in the temperature yields a sharp slope modification in layer 1. On the other hand, it is striking to see that layers 2 and 3 are not as heavily impaired. In summary, using the 1991 market parameters and the nonlinear dynamics formalism proposed in this article, the housing market of Madrid is not subject to critical (non-continuous) behavior on the temperature control parameter. It is clear that layers (2 and 3) are very resilient to temperature changes. Nevertheless, although criticality is absent, layer 1 undergoes the most rapid changes to variable economical conditions. The layers with most expensive houses can potentially undergo serious reversals under difficult economical conditions. By means of this multilayer model we can identify which sectors of the housing population would be more affected. An overall regulation of the market following equation (\[eqsimple\]) may not lead to uniform stabilization of the system. Multi-layer models can be used to detect the individual impact of global policies. Testing regulation strategies ============================= The temperature parameter is not a strong control parameter, since it does not lead the system into an asymptotic equilibrium. Therefore, the question of whether there is a parameter that can globally keep the system in a nearly asymptotic equilibrium remains unanswered. A possible candidate is the growth rate of the number of houses, $k$. As we have seen in section 4 and fig. \[fig1\]B the house growth rate is exponential for the city of Madrid, which is in contrast with the nearly flat increase of the base pool population (fig. \[fig1\]C). This situation is not sustainable, and the model can be useful to determine what to expect when regulating the house growth. We have integrated equations \[eq\_madrid\] with the parameter values obtained for 1991, suddenly changing the growth factor $k$ at 2005. Time to $10\%$ depletion versus $k$ is shown in figure \[Kfactor\]. As we can see, there is a power law dependence. Basically, the time required to a 10% depletion is proportional to $(1/k)$ (fits actually yield $k^{-1.01}$ to $k^{-1.04}$). This simple solution allows to easily estimate what the effects of house growth regulation will have in the market. This housing regulation policy contradicts the intuition about a voiced general opinion that the land in Madrid should be deregulated to build more housing units and, therefore, decrease the overall prices. This suggestion might reduce the prices of the housing units, yet it could worsen the current situation in Madrid in the long term. From the housing-demographic point of view, and according to this nonlinear model, more deregulation of land can push the system even farther out of equilibrium. It is interesting to note that regulation policies for the lowest and the higher layers are closer to each other than the middle ones. Policies to regulate the poorest layer will also contribute positively in regulating the upper ones. The middle layers follow a different dynamics on its own mostly due to the fact that they receive and send family units from both sides of the distribution. Conclusion ========== As pointed out by Brian Arthur [@arthur]:“...complexity economics, is not an adjunct to standard economic theory, but theory at a more general, out-of-equilibrium level.” In this paper, we develop a novel multi-layered nonlinear dynamical framework for modeling the housing market dynamics. Using realistic data with the highest available precision, we show that the housing market for the city of Madrid is currently driving away from equilibrium. This model can be used as a testing tool to determine the global effects of policy changes by governments. It is a tool to determine the effect of control parameters for all potential types of dynamics even for out-of-equilibrium conditions. Traditional econometric tools approximate the dynamics near set points, which are estimated (sometimes believed) to be the equilibrium points. When the system gets out of equilibrium there is not much guidance about what to expect, except waiting till it gets near the equilibrium point again. General tools to estimate effects of policy changes in the long run can be useful. Here we give an example that is able to provide a good prediction of the global level of occupancy in Madrid in 2001 (see Fig. \[fig5\]A). Our original goal was to find out whether the current house-market conditions in Madrid could lead to a critical condition such that, while slowly moving a parameter value as the temperature, the levels of occupancy suddenly drop to low levels. Fortunately, we did not find the existence of such criticality in the context of this model that uses to the maximum possible extent realistic data. When the interest rate is increased, which reduces the temperature parameter in our model, the occupancy levels of the medium range housing are not seriously affected. The first layer suffers dramatic consequences, and the wealthier layers undergo serious readjustments. Although this result appears to be good news (except for the poorest sector of the housing units), the fact is that the city of Madrid is in a serious out-of-equilibrium condition that asymptotically drives the levels of occupancy to $0$. According to our model, interest rate corrections will not modify this condition. The pragmatical way to control this situation is to individually regulate the amount of new construction for each of the housing layers. Smooth changes in the construction rate can have a positive effect slowing down the out-of-equilibrium condition. Acknowledgments =============== We want to thank Montserrat Martínez and Cesar Peñas for much of the data mining. This work has been funded by Ministerio de Industria (Spain) PROFIT FIT 340000-2004-103. [99]{} Arthur, W.B. (1999). Complexity and the economy. Science, 284, 107-109 Bailey, N.T.J. (1975). The mathematical theory of infectious diseases. (London: Charles Griffin & Co.) Banco Bilbao Vizcaya Argentaria, Servicio de Estudios (2003). Situación Inmobiliaria, Abril Bin, O. (2004). A prediction comparison of housing sale prices by parametric versus semi-parametric regressions. Journal of Housing Economics, 13, 68-84 Crone, T.M. & Mills, L.O. (1991). Forecasting trends in the housing stock using age-specific demographic projections. Journal of Housing Research, 2, 1-20 Garber P.M. (2000). Famous first bubbles: the fundamentals of early manias. (Cambridge, Massachusetts: The MIT Press) Green R.K. & Malpezzi, S. (2003). A primer on U.S. housing markets and policies. (Washington DC: The Urban Institute Press) Huerta, R. & Tsimring, L.S. (2002). Contact tracing and epidemics control in social networks. Physical Review E, 66, 056115-4. Instituto Nacional de Estadística (1991). Censo de Población y Viviendas Instituto Nacional de Estadística (1987-2002). Encuesta de Población Activa Instituto Nacional de Estadística. `http://www.ine.es/` Kindleberger, C.P. (2000). Manias, panics, and crashes: a history of financial crises. (New York: John Wiley & Sons) Linneman, P. (1986). An empirical test of the efficiency of the housing market. Journal of Urban Economics, 20, 140-154 Malpezzi, S. (1990). Urban housing and financial markets: some international comparisons. Urban Studies, 27, 971-1022 Malpezzi, S. (1999). A simple error correction model of house prices. Journal of Housing Economics, 8, 27-62 Mankiw, G.N. & Weil, D.N. (1989). The baby boom, the baby bust, and the housing market. Regional Science and Urban Economics, 19, 235-58 Observatorio Joven de la Vivienda en España\ `http://www.cje.org/C14/C6/OBJOVI/default.aspx?lang=es-ES` Quigley, J.M. (2001). Real state and the Asian crisis. Journal of Housing Economics, 10, 129-161 Sociedad de Tasación S.A.\ `http://web.st-tasacion.es/html/index.php` Terrones, M., Otrok, C. & Carcenac, N. (2004). The global house price boom. World Economic Outlook, chapter II, September 22, 2004, 71-89. Retrieved December 20, 2004, from `http://www.imf.org/external/pubs/ft/weo/2004/02/` [^1]: Corresponding author. E-mails: rhuerta@ucsd.edu (R. Huerta), fernando.corbacho@cognodata.com (F. Corbacho), luis.lago@uam.es (Luis F. Lago-Fernández).

--- abstract: 'We demonstrate a spatially resolved autocorrelation measurement with a Bose-Einstein condensate (BEC) and measure the evolution of the spatial profile of its quantum mechanical phase. Upon release of the BEC from the magnetic trap, its phase develops a form that we measure to be quadratic in the spatial coordinate. Our experiments also reveal the effects of the repulsive interaction between two overlapping BEC wavepackets and we measure the small momentum they impart to each other.' address: | $^{1}$National Institute of Standards and Technology, Gaithersburg, MD 20899\ $^{2}$Georgia Southern University, Statesboro, GA 30460-8031 author: - 'J. E. Simsarian,$^{1}$ J. Denschlag,$^{1}$ Mark Edwards,$^{1,2}$ Charles W. Clark,$^{1}$ L. Deng,$^{1}$ E. W. Hagley,$^{1}$ K. Helmerson,$^{1}$ S. L. Rolston,$^{1}$ and W.D. Phillips$^{1}$' title: ' Imaging the Phase of an Evolving Bose-Einstein Condensate Wavefunction' --- = 10000 A trapped Bose-Einstein condensate [@anderson0] has unique value as a source for atom lasers [@mewes] and matter-wave interferometry [@berman] because its atoms occupy the same quantum state, with uniform spatial phase. However, when released from the trapping potential, a BEC with repulsive atom-atom interactions expands, developing a non-uniform phase profile. Understanding this phase evolution will be important for applications of coherent matter waves. We have developed a new interferometric technique using spatially resolved autocorrelation to measure the functional form and time evolution of the phase of a BEC wavepacket expanding under the influence of its mean field repulsion. In 1997, the coherence of weakly interacting BECs was demonstrated by releasing two spatially separated condensates and observing their interference [@andrews]. Subsequent experiments have further investigated condensate coherence properties. One [@stenger] used velocity-resolved Bragg diffraction [@kozuma] to probe the momentum spectrum of trapped and released BECs. A complementary experiment [@hagley2] that used matter-wave interferometry can be interpreted as a measurement of the spatial correlation function, whose Fourier transform is the momentum spectrum. These experiments showed that a trapped condensate has a uniform phase, and a released condensate develops a non-uniform phase profile. (Recently the influence of non-zero temperature on coherence properties was also investigated [@bloch]). The experiments reported in this Letter combine spatial resolution and interferometry to measure the functional form of the time-dependent phase profile of a released condensate. We also make the first measurement of the velocity imparted to two equal BEC wavepackets from their mutual mean-field repulsion [@ketterle]. We perform our experiments with a condensate of $1.8(4) \times 10^6$ [@uncertainty] sodium atoms in the $3S_{1/2}$, $F=1$, $m_{F}=-1$ state. The sample has no discernable non-condensed (i.e. thermal) component. The condensate is prepared following the method of Ref. [@kozuma] and is held in a magnetic trap with trapping frequencies $\omega_x = \sqrt{2}\omega_y = 2 \omega_z = 2\pi \times $27 Hz. Using a scattering length of $a = 2.8$ nm, the calculated Thomas-Fermi diameters [@dalfovo] are 47 $\mu$m, 66 $\mu$m, and 94 $\mu$m, respectively. We release the BEC from the magnetic trap and it expands, driven mostly by the mean-field repulsion of the atoms. This expansion implies the development of a nonuniform spatial phase profile (recall that the velocity field is proportional to the gradient of the quantum phase). After an expansion time $T_{0}$, we probe the phase profile with matter-wave Bragg interferometry [@giltner; @torii; @denschlag]. Our interferometer splits the BEC into two wavepackets and recombines them with a chosen overlap, producing interference fringes, which we measure with absorption imaging [@imaging]. From the dependence of the fringe spacing on the overlap, we extract the phase profile of the wavepackets. =3.2in Our atom interferometer [@denschlag] consists of three optically-induced Bragg-diffraction pulses applied successively in time (Fig. 1). Each pulse consists of two counter-propagating laser beams whose frequencies differ by 100 kHz. They are detuned by about $-2$ GHz from atomic resonance ($\lambda = 2\pi/k = 589$ nm) so that spontaneous emission is negligible. The first pulse has a duration of 6 $\mu$s and intensity sufficient to provide a $\pi/2$ pulse, which coherently splits the BEC into two wavepackets, $\psi_{{ A}}$ and $\psi_{{ B}}$. The wavepackets have about the same number of atoms and only differ in their momenta: $p=0$ and $p=2\hbar k$. At a time $T_{1}=1$ ms after the first Bragg pulse, the two wavepackets are completely separated and a second Bragg pulse (a $\pi$ pulse) of 12 $\mu$s duration transfers $\psi_{{ B}}$ to a state with $p\approx 0$ and $\psi_{{ A}}$ to $p\approx 2\hbar k$ [@almost]. After a variable time $T_{2}$ the wavepackets partially overlap again and we apply a third pulse, of 6 $\mu$s duration (a $\pi/2$ pulse). This last pulse splits each wavepacket into the two momentum states. The interference of the overlapping wavepackets in each of the two momentum states allows the determination of the local phase difference between them. By changing the time $T_{2}$ we vary $\delta x = x_{ A} - x_{ B}$, the separation of $\psi_{{A}}$ and $\psi_{{B}}$ at the time of the final Bragg pulse. The set of data at different $\delta x$ constitutes a new type of spatial autocorrelation measurement that is similar to the “FROG” technique [@frog] used to measure the complete field of ultrafast laser pulses. From these measurements we obtain the phase profile of the wavepackets in the $x$ direction. =3.3in Figure \[fig2\]a-e shows one interferometer output port for different $\delta x$ (different $T_2$) after an expansion time $T_{0}$ = 4 ms. In general, we observe straight, evenly spaced fringes (although for small $T_{0}$ and $T_{2}$ the fringes may be somewhat curved). There is a value of $\delta x = x_{0} \neq 0$ where we observe no fringes (Fig. \[fig2\]c) and the fringe spacing decreases as $|\delta x - x_{0}|$ increases. Figure \[fig2\]f, a cut through Fig. \[fig2\]d, shows the high-contrast fringes [@contrast]. Our data analysis uses the average fringe period $d$, obtained from plots like Fig. \[fig2\]f. The fringes come from two different effects: the interference of two wavepackets with quadratic phase profile, and a relative velocity between the wavepackets’ centers. The data can be understood by calculating the fringe spacing along $x$ at output port 1 [@separable]. We assume that the phase $\phi$ of the wavefunction $f {\rm e}^{i\phi}$ can be written as $\phi = \frac{\alpha}{2}x^{2}+\beta x$. The equal spacing of the fringes implies, as predicted in the Thomas-Fermi limit [@castin], that $\phi$ has no significant higher-order terms [@polynomials]. The curvature coefficient $\alpha$ describes the mean-field expansion of the wavepackets and $\beta$ describes a relative repulsion velocity. The velocity arises because the wavepackets experience a repulsive push as they first separate and again as they recombine. The density at port 1 (see Fig. 1) just after the final interferometer pulse is the interference pattern $|{\psi_{A1} + \psi_{B1}}|^{2}$ of the wavepackets $\psi_{{A1}}$ and $\psi_{{B1}}$: $$|f(x-\delta x)e^{i(\frac{\alpha}{2}(x-\delta x)^{2} - \beta(x-\delta x))} + f (x)e^{i(\frac{\alpha}{2} x^{2} + \beta x)}|^{2}, \label{wavefunctions}$$ where we assume that the amplitudes and curvatures of the wavepackets are equal and their velocities have equal magnitude and opposite direction. The cross term of (\[wavefunctions\]) is $$2f(x-\delta x)f(x){\rm cos}\left[ \left(\alpha \, \delta x + \frac{{M}\, \delta v}{\hbar}\right)x + C\right], \label{fringes}$$ where $M$ is the sodium mass, ${M}\, \delta v/\hbar \equiv 2 \beta$, and $C$ is independent of $x$ [@constant]. $\delta v = v_{{B}} - v_{{A}}$ is the relative repulsion velocity between the wavepackets $\psi_{{A1}}$ and $\psi_{{B1}}$. Expression (\[fringes\]) predicts fringes with spatial frequency, $$\kappa = \alpha \, \delta x + \frac{{M}\, \delta v}{\hbar}, \label{line}$$ where $|\kappa| = 2 \pi/d$. When there are no fringes, $\kappa$ = 0 and the wavepacket separation $\delta x = x_{0} \equiv -{M}\, \delta v/ \alpha \hbar$. =3.3in Figure \[fig3\] plots the measured $\kappa $ vs. $\delta x$ [@separation] for $T_{0}$ = 1 and 4 ms. The data are well fit by a straight line as expected from Eq. (3) in the approximation that $\alpha$ and $\delta v$ are independent of $\delta x$. The slopes of the lines are the phase curvatures $\alpha$, and the $\kappa$ intercepts give the relative velocities $\delta v$. We checked the validity of the data analysis procedure by analyzing data simulated with a 1-D Gross-Pitaevskii (GP) treatment. Despite variations of $\delta v$ and $\alpha$ with $\delta x$ (due to their continued evolution during the variable time $T_{2}$), we find that $\kappa$ is still linear in $\delta x$. The slopes and intercepts in general are averages over the range of $\delta x$ used in the experiment. The interference fringes used to determine $\alpha$ and $\delta v$ are created at the time of the final interferometer pulse. Because the two outputs overlap at that moment, we wait a time $T_{3}$ for them to separate before imaging. During this time, the wavepackets continue to expand. The 1-D simulations show that the fringe spacings and the wavepackets expand in the same proportion. We correct $\kappa$ (by typically 15 $\%$) for this, using the calculated expansion from a 3-D solution of the GP equation described below. =3.3in The different slopes and intercepts of the two lines in Fig.  \[fig3\] show that the curvature $\alpha$ and relative velocity $\delta v$ of the wavepackets depend on the release time $T_{0}$ before the first interferometer pulse. Figure \[fig4\] plots the dependence of $\alpha$ and $\delta v$ on various release times $T_{0}$. The condensate initially has a uniform phase so that immediately after its release from the trap $\alpha = 0$. We nevertheless measure a nonzero $\alpha$ for $T_{0}$ = 0 ms because the BEC expands during $T_{1}$ and $T_{2}$. As a function of time, $\alpha$ behaves as $\dot D$/$D$ where $D$ is the wavepacket diameter and $\dot D$ is its rate of change [@castin]. At early times when the mean-field energy is being converted to kinetic energy, $\dot D$ increases rapidly, [*increasing*]{} $\alpha$. At late times, after the mean-field energy has been converted, $D$ increases while $\dot D$ is nearly constant, [*decreasing*]{} $\alpha$. We predict the time evolution of $\alpha$ using the Lagrangian Variational Method (LVM) [@perez]. The LVM uses trial wavefunctions with time dependent parameters to provide approximate solutions of the 3-D time-dependent GP equation. In the model, the effect of the interferometer pulses is to replace the original wavepacket with a superposition of wavepackets having different momenta; e.g., the action of our first interferometer pulse is $\psi_{0} \rightarrow \left(\psi_{0} + e^{i2kx} \psi_{0}\right)/\sqrt{2}$. We use Gaussian trial wavefunctions in the LVM and, for simplicity, neglect the interaction between the wavepackets, to calculate the phase curvature $\alpha$ at the time of the last interferometer pulse. This result, with $T_{1} = T_{2}$, is the solid line of Fig. 4a. We use energy conservation to calculate the relative repulsion velocity $\delta v$ between $\psi_{A1}$ and $\psi_{B1}$ because we neglect wavepacket interactions in the LVM. In the Thomas-Fermi approximation, we can calculate the amount of energy available for repulsion when $T_0$ = 0. A trapped condensate has $\frac{5}{7}\mu$ average total energy per particle, where $\mu$ is the chemical potential [@dalfovo]. After release from the trap, it has $\frac{2}{7} \mu$ average mean-field energy per particle. Applying a $\pi/2$ Bragg pulse to the BEC causes a density corrugation, which increases the mean-field energy to $\frac{3}{7}\mu$ per particle. In the approximation that the wavepackets do not deform as they separate and recombine, one can show that 1/3 of the total mean-field energy goes into expansion of the wavepackets, and 2/3 is available for kinetic energy of center-of-mass motion. Therefore $\frac{2}{7}\mu$ of mean-field energy per particle is available for repulsion. The corresponding repulsion velocity is only about $10^{-2}$ of a photon recoil velocity. The repulsion energy and $\delta v$ decrease for larger $T_0$ because both are inversely proportional to the condensate volume, which we calculate with the LVM. The two curves shown in Fig. \[fig4\]b are the calculated $\delta v$ when $\delta x = 0$ (solid curve) and $\delta v$ averaged over the different $\delta x$ used in the experiment (dashed curve). The 1-D GP simulations suggest that for small $T_{0}$, the results of the experiment should be closer to the solid curve; and for large $T_0$, closer to the dashed curve. The data is consistent with this trend. =3.4in In a related set of experiments we performed interferometry in the trap. This differs from the experiments on a released BEC because there is no expansion before the first interferometer pulse [@contraction] and the magnetic trap changes the relative velocity of the wavepackets between the interferometer pulses (Fig. 5a). To better reveal the velocity differences, we choose $T_{1}$ = $T_{2} = T$ to suppress fringes arising from the phase curvature. As with the released BEC measurements, we observe equally spaced fringes at the output of the interferometer, although the fringes are almost entirely due to a relative velocity $v$ between the wavepackets $\psi_{{A1}}$ and $\psi_{{B1}}$ at the time of the third interferometer pulse. We obtain $v$ from the fringe periodicity after a small correction for residual phase curvature [@velcorrection]. Two effects contribute to $v$: the mutual repulsion between the wavepackets $\psi_{{A}}$ and $\psi_{{B}}$ and the different action of the trapping potential on the two wavepackets in the interferometer. The latter effect occurs because after the first Bragg pulse, $\psi_{{A}}$ remains at the minimum of the magnetic potential while $\psi_{{B}}$ is displaced. Wavepacket $\psi_{{B}}$ therefore spends more time away from the center of the trap and experiences more acceleration than $\psi_{{A}}$. Following the last Bragg pulse, $\psi_{{A1}}$ and $\psi_{{B1}}$ have a velocity difference which for our parameters can be approximated by $v \approx -\frac{2\hbar k}{M}{\rm sin^{2}}(\omega_{x} {T}) + \delta v$ [@approx]. Figure \[fig5\]b plots $v$ versus $T$, and the curve is a fit to the above expression. We obtain the trap frequency $\omega_x /2\pi = 26.7(15)$ Hz, in excellent agreement with an independent measurement. We also obtain the relative velocity from the mean-field repulsion $\delta v = 0.49(12)$ mm/s, which we expect to be somewhat larger than for the released measurements because the wavepackets contract, producing a larger mean field. In conclusion, we demonstrate an autocorrelating matter-wave interferometer and use it to study the evolution of a BEC phase profile by analyzing spatial images of interference patterns. We study how the phase curvature of the condensate develops in time and measure the repulsion velocity between two BEC wavepackets. Our interferometric method should be useful for characterizing other interesting condensate phase profiles. For example, it can be applied to detect excitations of a BEC with characteristic phase patterns, such as vortices and solitons [@denschlag; @burger; @matthews; @madison; @jackson]. The method should be useful for further studies of the interaction of coherent wavepackets and to study the coherence of atom lasers. We thank T. Busch, D. Feder, and L. Collins for helpful discussions. This work was supported in part by the US Office of Naval Research and NASA. J.D. acknowledges support from the Alexander von Humboldt foundation. M.E. and C.W.C. acknowledge partial support from NSF grant numbers 9802547 and 9803377. M. H. Anderson [*et al.*]{}, Science [**269**]{}, 198 (1995); K. B. Davis [*et al.*]{}, Phys. Rev. Lett. [**75**]{}, 3969 (1995); C. C. Bradley, C. A. Sackett, and R. G. Hulet, Phys. Rev. Lett. [**78**]{}, 985 (1997); see also C. C. Bradley [*et al.*]{}, Phys. Rev. Lett. 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All uncertainties reported here are 1 standard deviation combined statistical and systematic uncertainties. F. Dalfovo [*et al.*]{}, Rev. Mod. Phys. [**71**]{}, 463 (1999). D. M. Giltner, R. W. Mc Gowan, and S. A. Lee, Phys. Rev. Lett. [**75**]{}, 2638 (1995). Y. Torii [*et al.*]{}, Phys. Rev. A [**61**]{}, 041602(R) (2000). J. Denschlag [*et al.*]{}, Science [**287**]{}, 97 (2000). The condensate was imaged by first optically pumping the atoms to the $F=2$ ground state and then imaging the absorption of a probe beam on the $F = 2 \rightarrow F' = 3$ transition. The pulse had a 5 $\mu$s duration, $\approx$ 170 mW/cm$^{2}$ intensity, and was detuned 15 MHz from resonance. The momenta are not exactly $p=0$ and $p= 2 \hbar k$ because of repulsion effects that will be discussed. R. Trebino [*et al.*]{}, Rev. Sci. Instrum. [**68**]{}, 3277 (1997). The observation is consistent with the full predicted fringe contrast when we include the finite imaging resolution. We can treat the $x$ direction independently of $y$ and $z$ when the wavefunction $\psi= f e^{i\phi(x,y,z)}$ is separable, i.e. $\phi(x,y,z) = \varphi(x) + \eta(y,z)$. Straight fringes imply this separability. Y. Castin and R. Dum, Phys. Rev. Lett. [**77**]{}, 5315 (1996). For $T_{0} = 6$ ms and an interference pattern with many fringes, we found that the coefficients of third and fourth order terms are smaller than 1$\times 10^{-5}$ $\mu$m$^{-3}$ and 1$\times 10^{-7}$ $\mu$m$^{-4}$ respectively. In practice, $C$ includes a random phase from mirror vibrations. We use $\delta x =\frac{2\hbar k}{M}{(T_{2} - T_{1}})-x_{\epsilon}$ where $\frac{2\hbar k}{M}$ = 5.9 cm/s is the two photon recoil velocity of sodium and $x_{\epsilon}$ is a small correction of the order $\delta v \, {T_{1}}$ due to the repulsion of the wavepackets. We include the correction in our data analysis in a self-consistent manner. The correction modifies $\alpha$ insignificantly, but increases the final values of $\delta v$ by $\approx$ 0.05 mm/s. V. M. Pérez–Garcia [*et al.*]{}, Phys. Rev. Lett. [**77**]{}, 5320 (1996). In fact, after the first pulse the wavepackets contract because the reduced mean-field energy can no longer support the wavepacket size. We also correct the fringe spacings for the contraction of the wavepackets between the final interferometer pulse and when the image is taken. We assume $\delta v<<\frac{\hbar k}{M}$, $T\omega_{x}<<1$, and harmonic motion in the trap. S. Burger [*et al.*]{}, Phys. Rev. Lett. [**83**]{}, 5198 (1999). M. R. Matthews [*et al.*]{}, Phys. Rev. Lett. [**83**]{}, 2498 (1999). K. W. Madison [*et al.*]{}, Phys. Rev. Lett. [**84**]{}, 806 (2000). A. D. Jackson, G.M. Kavoulakis, and C. J. Pethick, Phys. Rev. A [**58**]{}, 2417 (1998).

--- abstract: 'Inclusive associated production of a light Higgs boson ($m_{\rm H}\le m_t$) with one jet in $pp$ collisions is studied in next-to-leading order QCD. Transverse momentum ($p_{\rm T}\geq 30\gev$) and rapidity distributions of the Higgs boson are calculated for the LHC in the large top-quark mass limit. It is pointed out that, as much as in the case of inclusive Higgs production, the $K$-factor of this process is large ($\approx 1.6$) and depends weakly on the kinematics in a wide range of transverse momentum and rapidity intervals. Our result confirms previous suggestions that the production channel $p+p\to H+{\rm jet}$ $\to \gamma+\gamma\, +$ jet gives a measurable signal for Higgs production at the LHC in the mass range $100-140\gev$, crucial also for the ultimate test of the Minimal Supersymmetric Standard Model.' address: 'Institute of Theoretical Physics, ETH, CH-8093 Zürich, Switzerland' author: - 'D. de Florian, M. Grazzini and Z. Kunszt' title: | Higgs production with large transverse momentum\ in hadronic collisions at next-to-leading order [^1] --- epsf.tex (\#1 width \#2)[=\#2 ]{} Recent results from LEP and the SLC indicate that the Higgs boson of the Standard Model might be light. A fit to the precision data has given the values $m_{\rm H}=76^{\small+85}_{\small-47} \gev$, corresponding to $m_{\rm H}\le 262\gev$ at the $95\%$ confidence level, whereas a direct search at LEP200 gives the lower limit as $90\gev\le m_{\rm H}$ [@lepsld]. In addition, a crucial theoretical upper limit exists on the mass of the light neutral scalar Higgs boson of the Minimal Supersymmetric Standard Model $m_h\le 130\GeV$. It is, therefore, significant that one attempts to get the best possible signals in the light mass range of $100\gev\le m_{\rm H} \le 140\gev$ at the LHC. Simulation studies carried out by ATLAS and CMS have shown that assuming a low integrated luminosity of $3\cdot 10^4~{\rm pb}^{-1}$, even in the case of the “gold-plated” decay channel into two photons, the signal significance ($S/\sqrt{B}$) is only around 5 [@lhc]. This conclusion depends on the value of the $K$-factor (a conservative value $K$=1.5 was used) and some plausible assumptions on the size of the background. The calculation of the next-to-leading order (NLO) corrections to the background is not yet complete [@aurenche] and the contribution of the NNLO subprocess $gg\to\gamma\gamma$ is large [@back]. The complete NNLO analysis is extremely laborious, but appears to be feasible. Actually, for the full numerical control the background has to be calculated in NNNLO, which is completely beyond the scope of presently available techniques. Fortunately, the ambiguity in the value of the background to the signal is suppressed by the square root appearing in the definition of the signal significance. This situation, which is not completely satisfactory, can be improved by studying the $\gamma+\gamma{\rm \ + \ jet(s)}$ final states[^2]; this offers several advantages. The photons are more energetic than in the case of the inclusive channel and the reconstruction of the jet in the calorimeter allows a more precise determination of the interaction vertex, improving the efficiency and mass resolution. Furthermore the existence of a jet in the final state allows for a new type of event selection and a more efficient background suppression. In addition, the necessary control of the background contributions can probably be already achieved by the inclusion of the NLO corrections (the matrix elements are known [@sigbern]). In a recent phenomenological study [@adikss] it has been found that these advantages appear to be able to compensate the loss in production rates, provided one gets a large $K$-factor also for this process. The presentation of the NLO QCD corrections for this process is the main purpose of this letter. The production process $gg\to H$ is given by loop diagrams in the Born approximation, since the gluons interact with the Higgs boson via virtual quark loops [@lo]. The exact calculation of the NLO corrections is rather complex [@spira]. Fortunately, the effective field theory approach [@approx] obtained in the large top mass limit with effective gluon–gluon–Higgs coupling gives an accurate approximation (with or without QCD corrections) with an error less than 5%, provided $m_{\rm H}\le 2\, m_{\rm t}$ [@spira; @five]. It has been checked in LO, by an explicit calculation, that the approximation remains valid also for the production of Higgs bosons with large transverse momentum, provided both $m_{\rm H}$ and $p_{\rm T}$ are smaller than $m_{\rm t}$ [@glover]. It is therefore plausible to assume that the approximation remains valid also if we include NLO QCD corrections. Recently, in this approximation and using the helicity method, the transition amplitudes relevant to the NLO corrections have been analytically calculated for all the contributing subprocesses (loop corrections [@virt] and bremsstrahlung [@real]). The available NLO matrix elements contain soft and collinear singularities and therefore do not allow for a direct numerical evaluation of the physical cross section. In the past few years, exploiting the universal structure of the soft and collinear contributions, several efficient algorithms have been suggested to obtain finite cross section expressions from the singular NLO matrix elements. We have used the method of ref. [@fks] and implemented it into a numerical Monte Carlo style program which allows to calculate any infrared-safe physical quantity for the inclusive production of a Higgs boson with one jet in NLO accuracy. In this paper we report some of our results obtained for proton–proton collisions with $\sqrt{S}=14\tev$. For the strong coupling constant at NLO (LO) we use the standard two-loop (one-loop) form with $\Lambda_{QCD}$ set to the value used in the analysis of the parton distribution function under consideration. Our default choice for the factorization and renormalization scales is $Q_0^2= (m_{\rm H}^2+p_{\rm T}^2)$, where $p_{\rm T}$ is the transverse momentum of the Higgs boson. Here, unless stated, we consider the case of $m_{\rm H}=120$ GeV and $p_{\rm T}>30$ GeV in the kinematical region where the perturbative result can be applied without having to consider low-$p_{\rm T}$ resummation effects. Most of our curves have been obtained with MRST (ft08a) parton distribution functions, but we will also show some results using CTEQ(4M) and GRV98 [@pdf]. To compare the leading with the NLO results, for consistency, we use the corresponding LO parton distributions from each set. We shall discuss only results for the inclusive production cross section of a Higgs boson with large transverse momentum, although as we mentioned above the Monte Carlo program allows to study any infrared-safe quantity, including the implementation of different jet algorithms and experimental cuts. (pt.eps width 18 cm) In Fig. 1(a) we show the $p_{\rm T}$ distribution of the NLO and LO cross sections using MRST parton densities at three different scales $Q=\mu \, Q_0$, with $\mu=0.5,1,2$. In this figure one can see three important points. First, the radiative corrections are large; second, there is a reduction in the scale dependence when going from LO to NLO; third, the improvement in the scale dependence is still not completely satisfactory. The same features can be observed in more detail in Fig. 1(b), where the LO and NLO cross sections integrated for $p_{\rm T}$ larger than 30 and 70 GeV are shown as a function of the renormalization/factorization scale. Both the LO and NLO cross sections increase monotonically with decreasing $\mu$ scale, down to the limiting value where perturbative QCD can still be applied. In NLO the scale dependence has a maximum and its position characterizes the stability of the NLO perturbative results. In our case, as a result of the very large positive radiative corrections, the position of the maximum is shifted down to small $\mu$ values where the perturbative treatment is not valid, indicating that the stability of the NLO result is not completely satisfactory. In the usual range of variation of $\mu$ from 0.5 to 2, however, the LO scale uncertainty amounts up to $\pm 35\%$, whereas at NLO this is reduced to $\pm 20\%$, indicating the relevance of the QCD corrections. This feature of the scale dependence in NLO is very much the same as the one in the case of inclusive Higgs production [@spira]. (k.eps width 18 cm) In Fig. 2 we show the ratio $$\label{K} K=\frac{\Delta \sigma_{NLO}}{\Delta \sigma_{LO}}$$ of the next-to leading and the LO cross sections ($K$-factor) for the three different sets of parton distributions: MRST, CTEQ and GRV98, as a function of the transverse momentum and the rapidity of the Higgs boson. We can see that the $K$-factor is in the range 1.5–1.6 and it is almost constant (within 15% accuracy) for a large range of $p_{\rm T}$ and $y$. In the $p_{\rm T}$ distribution the variation never exceeds 10%, whereas it is a bit larger for the $y$ distribution at large $|y|$. The ratios of the NLO cross sections $$\label{R} R=\frac{\Delta \sigma _{\rm CTEQ,GRV}}{\Delta \sigma _{\rm MRST}}$$ computed by using CTEQ and GRV98 over the one obtained by using MRST parton densities are also shown. From there, it is possible to see that the differences in the $K$-factors basically come from variations in the LO cross sections, mostly because of the value of $\Lambda_{QCD}$ used in each set. We conclude that the properties of the $K$-factors found for large transverse momentum Higgs production are very similar to the ones obtained for the total inclusive Higgs production. They are about the same size, they show the same scale dependence, and the $K$-factor changes mildly with changing $m_{\rm H},\, y$ and $p_{\rm T}$. Its large value and its surprising independence from the kinematics might be interpreted as an evidence for some universal origin of the large radiative corrections [@resum]. This requires further theoretical understanding. (mass.eps width 18 cm) We have not included in our analysis the contributions from electroweak reactions, which can increase the cross section about 10% when suitable cuts are applied [@adikss]. Nevertheless it is worth noticing that since QCD corrections to electroweak boson fusion are substantially smaller than the ones corresponding to gluon fusion, the significance of the electroweak contributions is reduced at NLO. In Fig. 3 we show NLO cross section values for the physics signal $p+p\to H+{\rm jet}\to \gamma+\gamma+{\rm jet}$ as a function of the Higgs mass (with a reference value for the branching ratio given by Br$(H\to \gamma \gamma)= 2.18\cdot 10^{-3}$ for $m_{\rm H}=120$ GeV [@spiraprog]). For comparison, the cross section values of the physics signal $p+p\to H\to \gamma+\gamma$ are also shown. From there it is possible to see that the loss in production rate due to the transverse momentum cut of $p_{\rm T}>30$ GeV is less than a factor of 2 for the range of masses considered. In conclusion, we have pointed out that, much as in the case of inclusive Higgs production, the cross section values of the associated production of a Higgs boson with a jet are increased by a $K$-factor of 1.5–1.6 given by NLO QCD radiative corrections. Our result confirms previous suggestions that the production channel $p+p\to H+{\rm jet} $ $\to \gamma+\gamma\, +$ jet gives a measurable signal for Higgs production at the LHC in the mass range $100$–$140\gev$, crucial also for the ultimate test of the Minimal Supersymmetric Standard Model.\ We are grateful to M. Spira for discussions. One of us (DdeF) would like to thank\ S. Frixione for helpful comments. LEP Collaborations and Electroweak Working Group, SLD heavy flavour and electroweak working groups, CERN-EP/99-15. ATLAS Collaboration, Technical Proposal, CERN/LHCC 94-93 (1994);\ CMS Collaboration, Technical Design Report, CERN/LHCC 97-33 (1997) 27;\ C. Seez [*et al.*]{}, Proc. Large Hadron Collider Workshop, Aachen, CERN-90-10-B/ECFA 90-133, Vol.II, p.474 (1990);\ D. Froidevaux, F. Gianotti and E. Richter-Was, ATLAS Internal Note, Phys-No-64 (1995). P. Aurenche, A. Donini, R. Baier, M. Fontannaz and D. Schiff, Z. Phys. C [ **29**]{}, 459 (1985). D. Dicus and S. 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[^2]: The study of Higgs production in association with a jet was first suggested in the context of improving $\tau$ reconstruction in the $\tau^+\tau^-$ decay channel [@tau].

--- abstract: | We show that zero-energy flows appear in many particle systems as same as in single particle cases in 2-dimensions. Vortex patterns constructed from the zero-energy flows can be investigated in terms of the eigenstates in conjugate spaces of Gel’fand triplets. Stable patterns are written by the superposition of zero-energy eigenstates. On the other hand vortex creations and annihilations are described by the insertions of unstable eigenstates with complex-energy eigenvalues into the stable patterns. Some concrete examples are presented in the 2-dimensional parabolic potential barrier case. We point out three interesting properties of the zero-energy flows; (i) the absolute economy as for the energy consumption, (ii) the infinite variety of the vortex patterns, and (iii) the absolute stability of the vortex patterns . Keywords: Zero-energy solutions, vortex creations and annihilations, quantum mechanics, Gel’fand triplets, author: - | Tsunehiro Kobayashi[^1]\ [*Department of General Education for the Hearing Impaired,*]{} [*Tsukuba College of Technology*]{}\ [*Ibaraki 305-0005, Japan*]{} title: ' **Zero-Energy Flows and Vortex Patterns in Quantum Mechanics** ' --- [**1. Introduction**]{} Vortices play interesting roles in various aspects of present-day physics such as vortex matters (vortex lattices) in condensed matters [@blat; @crab], quantum Hall effects \[3-5\], various vortex patterns of non-neutral plasma \[6-9\] and Bose-Einstein gases \[10-14\]. Some fundamental properties and applications of vortices in quantum mechanics were examined by many authors \[15-23\]. Recently we have proposed a way to investigate vortex patterns in terms of zero-energy solutions of Schrödinger equations in 2-dimensions, which are infinitely degenerate and eigenfunctions in conjugate spaces of Gel’fand triplets (CSGT) [@k1; @ks8]. It should be noted that the eigenfunctions in CSGT represent scattering states, and thus they are generally not normalizable \[26\]. Therefore, the probability density ($|\psi|^2$) and the probability current (${{\boldsymbol{j}}}={\rm Re}[\psi^*(-i\hbar \nabla)\psi]/m$) for the eigenfunction ($\psi$), which are defined in usual Hilbert spaces, cannot be introduced to the eigenfunctions in CSGT. Instead of the probability current, however, the velocity which is defined by ${{\boldsymbol{v}}}={{\boldsymbol{j}}}/|\psi|^2$ can have a well-defined meaning, because the ambiguity due to the normalization of the eigenfunctions disappears in the definition of the velocity. Actually we have shown that many interesting objects used in hydrodynamics such as the complex velocity potential can be introduced in the 2-dimensions of CSGT \[27\]. We can expect that the hydrodynamical approach is a quite hopeful framework in the investigation of phenomena described in CSGT. One should pay attention to two important facts obtained in the early works \[24,25\]. One is the fact that the zero-energy solutions are common over the two-dimensional central potentials such that $V_a(\rho)=-a^2g_a\rho^{2(a-1)}$ with $\rho=\sqrt{x^2+y^2}$ except $a=0$ and then similar vortex patters described by the zero-energy solutions appear in all such potentials. Actually zero-energy solutions for a definite number of $a$ can be transformed to solutions for arbitrary number of $a$ by conformal transformations [@k1; @ks8]. The other is that the zero-energy solutions are infinitely degenerate, and thus all energy eigenvalues in CSGT with the potentials $V_a(\rho)$ are infinitely degenerate because the addition of the infinitely degenerate zero-energy solutions to arbitrary eigenfunctions does not change the energy eigenvalues at all. We, however, have to know that these results are obtained in equations for the single particle. We can, of course, apply the results in scattering processes where injected particles can be treated as individual particles. In the above-mentioned processes where vortices are observed \[1-14\], however, correlations among constituent particles cannot be ignored. We have to study whether such zero-energy flows appear in many particle systems. Furthermore we also have to study time evolutions of vortex patterns that have already been observed in experiments \[6-14\]. As for the time evolutions we have to take account of non-zero energy solutions in CSGT. It is known that the non-zero energy solutions generally have complex energy eigenvalues like ${\cal E}=E \mp i \Gamma$ with $E$, $\Gamma \in {\mathbb{R}}$, and then they have the time developments described by the factors $e^{-(i E/\hbar \pm \Gamma/\hbar)t}$ [@bohm]. Considering that stationary flows and time-dependent flows are, respectively, represented by the zero-energy solutions and the non-zero energy solutions in CSGT, we can expect that in CSGT general time-dependent flows are described by linear combinations of the stationary flows written by the zero-energy solutions and non-stationary flows (time-dependent flows) described by non-zero (complex) energy solutions. In hydrodynamics it is well-known that the vortex patterns are important objects to identify the situations of the flows. In the present model we can image two different types of the vortex patterns. One is the stationary vortex patterns, and the other is the vortex patterns varying in the time evolutions. As presented in our early works  [@k1; @ks8], the stationary vortex patterns can be described by the superposition of the zero-energy solutions, while the time-dependent ones will be done by putting the non-zero energy solutions into the superposition. In this paper we shall study two problems; one is the zero-energy flows in many particle systems in section 2, and the other is time-developments of vortex patterns in section 3. In order to obtain concrete examples of time-dependent vortex patterns, we shall use the eigenfunctions of the 2-dimensional parabolic potential barrier (2D-PPB) \[27\] in section 3, because the eigenfunctions with non-zero energy (complex energy) solutions in CSGT are known only in the case of the PPB. It is, however, noticed that the stable vortex patterns obtained in the 2D-PPB can easily be transformed those of the potentials $V_a(\rho)$ by the conformal transformations \[24,25\]. From these concrete analyses we shall point out three interesting properties of zero-energy flows in section 4. In section 5 some remarks will be done. Throughout these investigations we shall see that this approach can be one interesting possibility to analyze various time-dependent vortex patterns in a rigorous framework of quantum mechanics in CSGT. Let us briefly see the arguments for the zero-energy flows in the single particle motions. (For details, see refs. 24 and 25.) In 2-dimensions the eigenvalue problems with the energy eigenvalue ${\cal E}$ are explicitly written by $$[-{\hbar^2 \over 2m}{\boldsymbol{\vartriangle}}+V_a(\rho)]\ \psi(x,y) = {\cal E}\ \psi(x,y), \label{1}$$ where $$ =\^2/ x\^2+\^2 / y\^2, $ $ the central potentials are generally given by $ V_a(\rho)=-a^2 g_a\rho^{2(a-1)}, $ with $\rho=\sqrt{x^2+y^2}$, $a \in {\mathbb{R}}$ ($a\not=0$), and $m$ and $g_a$ are, respectively, the mass of the particle and the coupling constant. Note here that the eigenvalues ${\cal E}$ should generally be complex numbers in CSGT. Let us consider the conformal mappings $ \zeta_a=z^a,\ \ \ \ \ \ {\rm with}\ z=x+iy. $ We use the notations $u_a$ and $v_a$ defined by $$ \_a=u\_a+iv\_a $ $ that are written as $ u_a=\rho^a\cos a\varphi,\ \ v_a=\rho^a\sin a\varphi, $ where $\varphi=\arctan (y/x)$. In the $(u_a,v_a)$ plane the equations are written down as $$a^2\rho_a^{2(a-1) / a} [-{\hbar^2 \over 2m}{\boldsymbol{\vartriangle}}_a-g_a]\ \psi(u_a,v_a)= {\cal E}\ \psi(u_a,v_a), \label{2}$$ where $ $ \_a=\^2/ u\_a\^2+\^2/ v\_a\^2. $ $ We see that the equations become same for all values of $a$ (except $a=0$) for the energy eigenvalue ${\cal E}=0$. In fact, for ${\cal E}=0$ the equations have the same form as that for the free particle with the constant potentials $g_a$ as $$[-{\hbar^2 \over 2m}{\boldsymbol{\vartriangle}}_a-g_a]\ \ \psi(u_a,v_a)=0. \label{3}$$ It should be noticed that in the case of $a=1$ where the original potential is a constant $g_1$ the energy does not need to be zero but can take arbitrary real numbers, because the right-hand side of has no $\rho$ dependence. In the $a=1$ case, therefore, we should take $g_1+{\cal E}$ instead of $g_a$. This means that all plane wave solutions have the same infinite degeneracy discussed below. It is trivial that the equations for all $a$ have the particular solutions $ \psi_0^\pm(u_a)=N_a e^{\pm ik_au_a} $ and $\psi_0^\pm(v_a)=N_a e^{\pm ik_av_a} $ with $k_a=\sqrt{2mg_a}/\hbar$ for $g_a>0. $ We notice here the degeneracy of the solutions that have already been known in the 2D-PPB \[27\]. By putting the wave function $f^\pm (u_a;v_a)\psi_0^\pm(u_a)$ into  where $f^\pm (u_a;v_a)$ is a polynomial function of $u_a$ and $v_a$, we obtain the equation $$[{\boldsymbol{\vartriangle}}_a \pm2ik_a{\partial \over \partial u_a}]f^\pm (u_a;v_a)=0. \label{4}$$ A few examples of the functions $f$ are given by $$\begin{aligned} f_0^\pm (u_a;v_a)&=1, \nonumber \\ f_1^\pm (u_a;v_a)&=4k_a v_a, \nonumber \\ f_2^\pm (u_a;v_a)&=4(4k_a^2 v_a^2+1\pm 4 i k_a u_a). \label{5}\end{aligned}$$ We can obtain the general forms of the polynomials in the 2D PPB, which are generally written by the multiple of the polynomials of degree $n$, $H_n^\pm(\sqrt{2k_2} x)$, such that $ f_{n}^{\pm}(u_2;v_2)=H_{n}^\pm(\sqrt{2k_2} x) \cdot H_{n}^\mp(\sqrt{2k_2} y), $ where $x$ and $y$ in the right-hand side should be considered as the functions of $u_2$ and $v_2$ [@sk4]. Since the form of the equations  is common for all $a$, the solutions can be written by the same polynomial functions that are obtained in the PPB. The states expressed by these wave functions belong to the conjugate spaces of Gel’fand triplets of which nuclear space is given by Schwarz space. Actually we easily see that the wave functions cannot be normalized in terms of Dirac’s delta functions except the lowest polynomial solutions. Here it should be stressed that the existence of the infinitely degenerate zero-energy solutions brings the infinite degeneracy to all the eigenstates. This fact means that the energy and the other quantum numbers like angular momentums, which are related to the determination of the energy eigenvalues, are not enough to discriminate the eigenstates. What are good quantum numbers to characterize the infinite degeneracy? An interesting candidate to characterize the states is vortex patterns that tell us topological properties of the staes. Note that the voertex patterns have been observed in experiments \[8-14\]. Time developments of the patterns can also be good observables in those processes. Let us here briefly note how vortices are interpreted in quantum mechanics. The probability density $\rho(t,x,y)$ and the probability current ${\boldsymbol{j}}(t,x,y)$ of a wavefunction $\psi(t,x,y)$ in non-relativistic quantum mechanics are, respectively, defined by $ \rho(t,x,y)\equiv\left| \psi(t,x,y)\right|^2 $ and $ {\boldsymbol{j}}(t,x,y)\equiv{\rm Re}\left[\psi(t,x,y)^* \left(-i\hslash\nabla\right)\psi(t,x,y)\right]/m. $ They satisfy the equation of continuity $ \partial\rho/\partial t+\nabla\cdot{\boldsymbol{j}}=0.$ Following the analogue of the hydrodynamical approach, the fluid can be represented by the density $\rho$ and the fluid velocity ${\boldsymbol{v}}$. They satisfy Euler’s equation of continuity $ \partial\rho/\partial t+\nabla\cdot(\rho{\boldsymbol{v}})=0. $ Comparing this equation with the continuity equation, the following definition for the quantum velocity of the state $\psi(t,x,y)$ is led in the hydrodynamical approach; $${\boldsymbol{v}}\equiv\frac{{\boldsymbol{j}}(t,x,y)}{\left| \psi(t,x,y)\right|^2}. $$ Now it is obvious that vortices appear at the zero points of the density, that is, the nodal points of the wavefunction. At the vortices, of course, the current ${\boldsymbol{j}}$ must not vanish. When we write the wavefunction $\psi(t,x,y)=\sqrt{\rho (x,y)} e^{iS(x,y)/\hbar }$, the velocity is given by ${{\boldsymbol{v}}}=\nabla S/m$. We should here remember that the solutions degenerate infinitely. This fact indicates that we can construct wavefunctions having the nodal points at arbitrary positions in terms of linear combinations of the infinitely degenerate solutions [@k1; @ks8; @sk4]. The strength of vortex is characterized by the circulation ${\varGamma}$ that is represented by the integral round a closed contour $C$ encircling the vortex such that $${\varGamma}=\oint_C {\boldsymbol{v}}\cdot d{\boldsymbol{s}}$$ and it is quantized as $${\varGamma}=2\pi l\hbar/m, $$ where the circulation number $l$ is an integer [@joh2; @joh5; @wu-sp; @bb2]. Let us consider a simple system composed of $N$ number of the same particles with the mass $m$. The interactions between two constituent particles are supposed to be written by the same potential $V(\rho_{ij})$, where $\rho_{ij}=|\vec{\rho_i}-\vec{\rho_j}| $ stands for the relative distance between two particles. The Schr$\ddot o$dinger equation for the $N$ particle system is written as $$[-{\hbar^2 \over 2m}\sum_{i=1}^N{\boldsymbol{\vartriangle}}_i +\sum_{i>j}^N \sum_{j=1}^{N-1}V(\rho_{ij})]\ \Psi(t,\vec{\rho}_1,\cdots,\vec{\rho}_N) = {\cal E}\ \Psi(t,\vec{\rho}_1,\cdots,\vec{\rho}_N). $$ Introducing a centre of mass coordinate $\vec{\rho}_C$ and $N-1$ relative coordinates $\vec{\rho}_{r_i}$ with $i=1,\cdots,N-1$, the equation is rewritten as $$[-{\hbar^2 \over 2M}{\boldsymbol{\vartriangle}}_C-H_r(\rho_{r_1},\cdots,\rho_{r_{N-1}})] \Phi_C(t,\vec{\rho_C})\phi_r (t,\vec{\rho}_{r_1},\cdots,\vec{\rho}_{r_{N-1}}) = {\cal E}\ \Phi_C(t,\vec{\rho}_C)\phi_r (t,\vec{\rho}_{r_1},\cdots,\vec{\rho}_{r_{N-1}}), $$ where $M=Nm$ and $H_r(\rho_{r_1},\cdots,\rho_{r_{N-1}})$ stands for the Hamiltonian for the relative coordinates. It should be noted that the centre of mass coordinate can always be separable from the relative ones, and then the energy eigenvalue ${\cal E}$ are written by the sum of the eigenvalue of centre of mass system $E_C$ and that of the relative ones $E_r$ as ${\cal E}=E_C +E_r$. We can consider that the zero-energy solutions can appear in the relative motions, e.g., when the relative interactions are written by the same PPB, $H_r$ are separable for all the relative coordinates, for which interatcions are written by the PPB. Here we shall, however, discuss the case where the flows discussed in the section 2.1 appear in the centre of mass motions, that is, the total flows. Now let us consider the zero-energy solutions of the total system represented by ${\cal E}=0$. Considering the equation ${\cal E}=E_C +E_r$, the flows discussed in the section 2.1 appear for $E_r<0$, because $E_C>0$ is required. This means that the relative interactions have to be totally described by a kind of attractive potential. In this case the centre of mass motions $\Phi_C(t,\vec{\rho_C})$ are described by the plane waves with the infinite degeneracy, which are obtained from the solutions in the section 2.1 by putting $a=1$. If the Hamiltonian $H_r$ has levels with different negative-energy eigenvalues, we have different zero-energy flows characterized by different wave numbers $k=\sqrt{2M|E_r|}/\hbar $ corresponding to the negative-energy eigenvalues $E_r<0$. Provided that some external forces like electro-magnetic forces are put in the systems, the central motions possibly have potentials ($V_C(\rho_C)$). In this case the zero-energy solutions appear when $V_C(\rho_C)$ is written by one of the central potentials $V_a(\rho_C)$. Here we note that in cases where the potentials of all the constituents have a negative constant potential $V_0<0$ in a common area such as in the region surrounded by repulsive potentials like short distance attractive potentials as shown in fig. \[fig:1.1\], the zero-energy solutions can appear in the area. In these cases we have to solve the zero-energy problems under some special boundary conditions. We will be able to find out the solutions fulfilling the boundary conditions in terms of the infinitely degenerate zero-energy solutions. It is obvious that we can introduce the velocity for the centre of mass system as same as that for the single particle given in section 2.1 such that $${\boldsymbol{v}}_C={\boldsymbol{j}}_C(t,\vec{\rho_C})/|\Phi_C(t,\vec{\rho_C})|^2,$$ where the current for the centre of mass system is defined by $ {\boldsymbol{j}}_C(t,\vec{\rho}_C)= \break {\rm Re}\left[\Phi_C^* \left(-i\hslash\nabla_{\rho_C}\right) \Phi_C\right]/m. $ Since the derivative with respect to the centre of mass coordinate is just written by the sum of the derivatives of all the constituents, the velocity ${\boldsymbol{v}}_C$ is understood as the mean velocity of the constituents. We may consider that ${\boldsymbol{v}}_C$ represents the velocity that is used in hydrodynamics. Now it is trivial that in the case of the centre of mass motions we can follow the discussion on the vortices for the single particle motions presented in section 2.1. We note that the velocity for a relative coordinate $\rho_i$ can also be defined as same as that for the centre of mass coordinate such that $${\boldsymbol{v}}_i={\boldsymbol{j}}_{\rho_i}/|\phi_r|^2,$$ where $ {\boldsymbol{j}}_{\rho_i}={\rm Re}\left[\phi_r^* \left(-i\hslash\nabla_{\rho_i}\right)\phi_r \right]/m. $ We easily see that in independent particle models the above velocity coincides with that of the single particle given in section 2.1. If the central potentials of the type $V_a(\rho)$ appear for some relative coordinates, the zero-energy flows appear, and then the zero-energy vortices are produced in the relative motions. In order to see some concrete examples of stable and time-dependent vortex patterns we shall make some vortex patterns in terms of the eigenfunctions of the 2D-PPB \[27\]. As already noted, the stable patterns can be transformed to those of arbitrary potentials of the type $V_a(\rho)$ by the conformal transformations \[24,25\]. Let us start from the short review on the eigenfunctions of the 2D-PPB $V=-m\gamma^2(x^2+y^2)/2$ that will be used in the following analyses. The infinitely degenerate eigenfunctions with the zero-energy for the potentials $V_a(\rho)$ have explicitly been given in (5), whereas the eigenfunctions with non-zero energies are known only in the case of PPB  \[27-33\]. It is trivial that the eigenfunctions in the 2D-PPB are represented by the multiples of those of the 1D-PPB. The eigenfunctions of the 1D-PPB for $x$, which have pure imaginary energy eigenvalues $ \mp i( n_x+{1 \over 2})\hbar \gamma$, are given by $$u^{\pm}_{n_x}(x)=e^{\pm i\beta^2x^2/2}H^{\pm}_{n_x}(\beta x)\ \ \ \ (\beta \equiv \sqrt{m\gamma/\hbar}),$$ where $ H^{\pm}_{n_x}(\beta x)$ are the polynomials of degree $n_x$ written in terms of Hermite polynomials $H_{n}(\xi)$ with $\xi =\beta x$ as [@sk; @s2] $$H_{n}^\pm(\xi)=e^{\pm i n\pi/4} H_{n}(e^{\mp i\pi/4}\xi). $$ We have four different types of the eigenfunctions in the 2D-PPB [@sk4]. Two of them $$U^{\pm\pm}_{n_xn_y}(x,y)\equiv u^{\pm}_{n_x}(x)u^{\pm}_{n_y}(y)$$ with the energy eigenvalues ${\cal E}^{\pm\pm}_{n_xn_y}=\mp i(n_x+n_y+1)$, respectively, represent flows diverging from the origin and flows converging towards the origin. Some examples for the low degrees are obtained as follows; $$\begin{aligned} U^{\pm\pm}_{00}(x,y)&=e^{\pm i\beta^2(x^2+y^2)/2} , \nonumber \\ U^{\pm\pm}_{10}(x,y)&=2\beta x e^{\pm i\beta^2(x^2+y^2)/2} , \nonumber \\ U^{\pm\pm}_{20}(x,y)&=(4\beta^2x^2 \mp 2i) e^{\pm i\beta^2(x^2+y^2)/2}, \nonumber \\ U^{\pm\pm}_{11}(x,y)&=4\beta^2xy e^{\pm i\beta^2(x^2+y^2)/2} . $$ Note that $U^{\pm\pm}_{01}(x,y)$ and $U^{\pm\pm}_{02}(x,y)$ are ,respectively, obtained by exchanging $x$ and $y$ in $U^{\pm\pm}_{10}(x,y)$ and $U^{\pm\pm}_{20}(x,y)$. It is transparent that the eigenfunctions of angular momentums are constructed in terms of the linear combinations of these diverging and converging flows [@sk4]. The other two $$U^{\pm\mp}_{n_xn_y}(x,y)\equiv u^{\pm}_{n_x}(x)u^{\mp}_{n_y}(y)\ \ {\rm with}\ \ {\cal E}^{\pm\mp}_{n_xn_y}(x,y)=\mp i(n_x-n_y)$$ are corner flows round the center. (See figs. \[fig:1.2\] and \[fig:1.3\].) The zero-energy solutions that are common in the potentials $V_a(\rho)$ appear when $n_x=n_y$ is satisfied. A few examples of the zero-energy eigenfunctions are explicitly obtained as follows; $$\begin{aligned} U^{\pm\mp}_{00}(x,y)&=e^{\pm i\beta^2(x^2-y^2)/2} , \nonumber \\ U^{\pm\mp}_{11}(x,y)&=4\beta^2 xy e^{\pm i\beta^2(x^2-y^2)/2} , \nonumber \\ U^{\pm\mp}_{22}(x,y)&=4[4\beta^4x^2 y^2+1\pm 2i\beta^2(x^2-y^2)] e^{\pm i\beta^2(x^2-y^2)/2}.\end{aligned}$$ It is obvious that the eigenfunctions of (15) and (16) are not normalizable. The proof that they are the eigenfunctions of CSGT are presented in refs. \[32,33\]. [**3.2 Stable vortex-patterns**]{} Let us investigate vortex patterns in terms of the 2D-PPB eigenfunctions. Since the zero-energy solution given in the section 3.1 have no nodal point with non-vanishing currents, they have no vortex. However, it has been shown that some vortex patterns having infinite numbers of vortices, like vortex lines and vortex lattices, can be made in terms of simple linear combinations of those low lying stationary states in the early works [@k1; @ks8]. We shall here study linear combinations having a few or some vortices observed in experiments \[6-14\]. Let us start from compositions of stable vortex patterns. For the convenience in the following discussions we take the flow without any vortices described by $$\begin{aligned} \Phi_B&={1 \over 4} U_{22}^{+-}(x,y) -U_{00}^{+-}(x,y) \nonumber \\ &=(4\beta^4x^2y^2+2i\beta^2(x^2-y^2))e^{i\beta^2(x^2-y^2)/2}, \end{aligned}$$ which will be called the basic flow hereafter. Note that the nodal point of the basic flow at the origin does not produce vortices, because the current vanishes there. Here we shall show three stable patterns, which will be used in the discussions on vortex creations and annihilations. The linear combination of the basic flow and $U^{+-}_{11}$ such that $$\begin{aligned} \Phi^{+-}_{012}(x,y)&=\Phi_B-c^2 U^{+-}_{11}(x,y) \nonumber \\ &=[4\beta^2xy(\beta^2xy-c^2)+2i\beta^2(x^2-y^2)] e^{i\beta^2(x^2-y^2)/2},\end{aligned}$$ with $c \in {\mathbb{R}}$ has three vortices at the origin and the two points $( \pm c/\beta, \pm c/\beta)$ as shown in fig. \[fig:1.4\]. If we take $-c^2$ instead of $c^2$, three vortices appear at the origin and the points $( \pm c/\beta, \mp c/\beta)$. Let us consider the linear combinations of two flows that are represented by the above eigenfunction $\Phi^{+-}_{012}(x,y)$ and coming from two different directions such that $$\begin{aligned} \Phi^{+-}_{012}(x,y)+\Phi^{+-}_{012}(\xi,\eta),\end{aligned}$$ where $\xi={\rm cos}\alpha \cdot x+{\rm sin}\alpha \cdot y$ and $\eta=-{\rm sin}\alpha \cdot x+{\rm cos}\alpha \cdot y $ with $0<\alpha <2\pi $. It is apparent that the two functions have only one common zero-point at the origin for $0<\alpha <2\pi $. We see that the vortex at the origin has the circulation number $l=-2$. The linear combinations given by $$\begin{aligned} \Phi^{+-}_{02}(x,y)&={1 \over 4}\Phi_B-c^4U^{+-}_{00}(x,y) \nonumber \\ &=[(\beta^2xy-c^2)(\beta^2xy+c^2)+i\beta^2(x^2-y^2)/2] e^{i\beta^2(x^2-y^2)/2},\end{aligned}$$ has four vortices at the points $( \pm c/\beta, \pm c/\beta)$ and $( \pm c/\beta, \mp 1c/\beta)$ as shown in fig. \[fig:1.5\]. Let us go to the study of time-dependent vortex patterns, where vortices can move, and sometimes be created and annihilated. Such patterns are obtained by linear combinations of stable flows and time-dependent ones. In the following considerations the time dependent flows are put in the stable flows at $t=0$. Let us consider the following linear combination; $$\begin{aligned} \Phi^{+-}_{02,1}(x,y,t)&={1 \over 2}\Phi_B-\theta(t)c^3U^{+-}_{10}(x,y) e^{-\gamma t} \nonumber \\ &=[2\beta x(\beta^3xy^2-\theta(t)c^3e^{-\gamma t})+i\beta^2(x^2-y^2)] e^{i\beta^2(x^2-y^2)/2}, \end{aligned}$$ where the theta function is taken as $\theta (t)=0$ for $t<0$ and $=1$ for $t\geq 0$. It has two nodal points at $(c e^{-\gamma t/3}/\beta,\pm c e^{-\gamma t/3}/\beta)$ for $t\geq 0$, where two vortices with opposite circulation numbers exist. The nodal points go to the origin as the time $t$ goes to infinity as shown in fig. \[fig:1.6\]. Since the contribution of the unstable flow decreases as $t \rightarrow \infty $ because of the time factor $e^{-\gamma t}$, the wavefunction $\Phi^{+-}_{02,1}(x,y,t)$ goes to $\Phi_B/2$ as $t \rightarrow \infty$. Thus the flow has no nodal point in the limit. This means that the pair of vortices which are created at $t=0$ disappears at origin in the limit $t \rightarrow \infty $. We can say that this wavefunction describes the pair annihilation of two vortices. The time development of this process can be described as follows: (i) Before the time-dependent flow is put in the basic flow, i.e., $t<0$, there is no vortex. (ii) At $t=0$ when the time-dependent flow is put in the basic flow, a pair of vortices are suddenly created. (iii) The pair moves toward the origin, and then they annihilate at the origin, that is, the flow turns back to the basic flow $\Phi_B$ having no vortex. The linear combination given by $$\begin{aligned} \Phi^{+-}_{02,2}(x,y,t)&=\Phi_B-\theta(t)c^2[2iU^{+-}_{00}(x,y) -U^{+-}_{20}(x,y)e^{-2\gamma t}] \nonumber \\ &=[4\beta^2x^2(\beta^2y^2-\theta(t)c^2e^{-2\gamma t})- 2i(\theta(t)c^2(1-e^{-2\gamma t})-\beta^2(x^2-y^2))] e^{i\beta^2(x^2-y^2)/2}\end{aligned}$$ has four nodal points at $(\pm c/\beta, c e^{-\gamma t}/\beta)$ and $(\pm c/\beta, -c e^{-\gamma t}/\beta)$ for $t\geq 0$. In this case we easily see that the pair of vortices at $(\pm c/\beta, c e^{-\gamma t}/\beta)$ and that at $(\pm c/\beta, -c e^{-\gamma t}/\beta)$ annihilate as $t \rightarrow \infty $ as shown in fig. \[fig:1.7\]. The stable flow $\Phi_B-2ic^2U^{+-}_{00}(x,y)$ that appears in the limit has two nodal points at $(\pm c/\beta,0)$, but it has no vortex but two vortex dipoles, because the current also vanish at the points. The time development of this process is interpreted similarly as the case (M-1). In these two cases all vortices move on straight lines in the pair annihilation processes. The linear combinations of stationary flows and diverging or converging ones make different types of annihilation processes. For an example, let us consider the linear combination of $ \Phi_B$ and the lowest order diverging flow described by $$U_{00}^{++}(x,y,t)=e^{i\beta^2(x^2+y^2)/2}e^{-\gamma t}$$ having the energy eigenvalue $-i \gamma \hbar$. Let us consider the linear combination described by $$\begin{aligned} \Phi^{++}_{02,0}(x,y,t)&={1 \over 4}\Phi_B -\theta(t)c^2 U^{++}_{00}(x,y,t) \nonumber \\ &=[\beta^4x^2y^2-\theta(t)c^2 e^{-\gamma t}e^{i\beta^2y^2}+i\beta^2(x^2-y^2)/2] e^{i\beta^2(x^2-y^2)/2}.\end{aligned}$$ For $t\geq 0$ it has two nodal points at the points where the following relations are fulfilled; $$XY=c(t) {\rm cos}Y,\ \ \ {1 \over 2}(X-Y)-c(t){\rm sin}Y=0,$$ where $X=\beta^2 x^2$, $Y=\beta^2 y^2$ and $c(t)=c^2 e^{-\gamma t}$. From these relations we have an equation for $Y$ $$Y^2-c(t) {\rm cos}Y +2c(t)Y{\rm sin}Y=0.$$ The solutions are obtained from the cross points of two functions $f(Y)=Y^2$ and $g(Y)=c(t)({\rm cos}Y-2Y{\rm sin}Y)$. We easily see that a solution for $Y\geq 0$ exists in the region $0<Y<\pi /2$ for arbitrary positive numbers of $c(t)$. Four vortices appear at the four points expressed by the combinations of $x=\pm \sqrt{X}/\beta $ and $y=\pm \sqrt{Y}/\beta$, where $X$ is obtained by using the first relation of (25). Since $c(t)$ goes to $0$ as $t \rightarrow \infty $, we see that $X$ and $Y$ simultaneously go to $0$ in the limit such that $$X\simeq Y\rightarrow |c|e^{-\gamma t/2} \rightarrow 0, \ \ \ {\rm for}\ \ t \rightarrow \infty .$$ Since the flow turns back to the basic flow, the four vortices annihilate at the origin in the limit. From the second relation of (25), we have $$X=Y+2c(t) {\rm sin}Y.$$ This equation show us that the vortex points do not move along straight lines. Here we consider a somewhat complicated processe. Here we take $\Phi^{+-}_{02}(x,y)$ of (20) as the stable flow, which has four vortices. For the simplicity $c=1/2$ is taken in the following discussions. Here the lowest order diverging flow $U_{00}^{++}(x,y,t)$ are put into the stationary flow at $t=0$. The wavefunction are given by $$\begin{aligned} \Phi_{02,0}^{++}(x,y;t)=&16\Phi^{+-}_{02}(x,y)+\theta(t)b^2U_{00}^{++}(x,y,t) \nonumber \\ =&[16\beta^4x^2y^2-1+\theta(t)b^2 e^{-\gamma t}e^{i\beta^2y^2}+ 8i\beta^2(x^2-y^2)] e^{i\beta^2(x^2-y^2)/2} \nonumber \\ =&[16\beta^4x^2y^2-1+\theta(t)b(t) e^{-\gamma t}{\rm cos}(\beta^2y^2) +i(8\beta^2(x^2-y^2) \nonumber \\ &+ \theta(t)b(t) e^{-\gamma t}{\rm sin}(\beta^2y^2))] e^{i\beta^2(x^2-y^2)/2},\end{aligned}$$ where $b\in {\mathbb{R}}$ and $b(t) =b^2e^{-\gamma t}$. Using $X=\beta^2x^2$ and $Y=\beta^2y^2$, we have two relations for nodal points of the wavefunction for $t>0$ as follows; $$16X Y +b(t) {\rm cos}Y-1=0, \ \ \ \ 8(X-Y)+b(t){\rm sin}Y=0.$$ From these relations we obtain an equation for the nodal points $$1-b(t){\rm cos}Y -16Y^2+2b(t)Y{\rm sin}Y=0.$$ Examining the cross point of the two functions $F(Y)=16Y^2-1$ and $G(Y)=-b(t)({\rm cos}Y -2Y{\rm sin}Y)$, we obtain the following results: (1) In the case of $b(t)<1$ the two functions always have a cross-point in the region satisfying $Y\geq 0$ (note that $Y=\beta^2y^2$). The wavefunction, therefore, has four nodal points. This means that the flow always has four vortices that move toward the stationary points fulfilling $|x|=|y|=(2 \beta)^{-1}$ as $t$ increases. (2) In the case of $b(t)>1$, eq.(29) has an even number of solutions like $n=0,2,4,\cdots$. Since one solution brings four vortices on a circle with the centre at the origin, the vortex number is given by $4n$. Note that the number $n$ increases as $b(t)$ increases. This fact means that, since $b(t)$ decreases as $t$ increases, the vortex number decreases as $t$ increases, until $b(t)$ gets to 1. Considering that the change of $n$ is always 2, we see that the reduction of the vortex number caused by the change of $n$ is always 8. That is to say, we observe that four vortex-pairs simultaneously annihilate at four different points on a circle. As a simple example, let us consider the case of $n=2$ at $t=0$. We observe the following time development of the flow: (i) For $t<0$ the stationary flow has the four vortices as shown in fig.5. (ii) At $t=0$ the original four vortices are disappear and eight vortices are newly created. Then we observe the flow having eight vortices. (See fig. \[fig:1.8\].) (iii) In the time-evolution the eight vortices disappear simultaneously. We observe the process as the annihilations of four vortex-pairs. Thus the flow having no vortex appears. (iv) At the critical time $t_c= {\rm ln} b^2 /\gamma$ when $b(t_c)=1$ is fulfilled four vortices are created at the origin, and then they move toward the stationary points. The vortex state at $t=t_c$ can be understood as a vortex quadrupole [@k1]. If, instead of the diverging flow $U_{00}^{++}(x,y,t)$, the converging flow $U_{00}^{--}(x,y,t)$ is put in the stationary flow, we observe a flow continuously creating 8 vortices for any choices of $b$. Of course, the time dependent flow blows up the magnitude in the limit of $t\rightarrow \infty$, and thus the original stable flow can not be observed in the limit. Let us here consider interesting properties of the zero-energy flows. The first interesting property is due to the fact that the use of the zero-energy flows is very useful and economical from the viewpoint of energy consumption. For example it can be a very economical step for the transmission of information. Considering the huge variety arising from the infinite degeneracy, the transmission by the use of the zero-energy flows enable us to transmit an enormous amount of information without any energy loss. The flows are stable and then they can also be a very useful step for making mechanisms to preserve such information, e.g., for memories in living beings. The huge variety of vortex patterns can possibly discriminate the enormous amount of information. The addition of new memories and also the change of preserved memories can easily be carried out by pouring some zero-energy (stable) flows in the preserved ones. Furthermore, as shown in section 3.3, in all the time-dependent processes induced by pouring the unstable flows with complex energy eigenvalues the time-dependent flows always turn back to the stable flows in the long time scales, and then the initial flow patterns are recovered. That is to say, the initial patterns are kept in all such time-dependent processes. This stability of the flow patterns seems to be a very interesting property for the interpretation of the stability of memories not only in their preservations but in their applications as well. The applications, of course, mean thinking processes. These flows will possibly be workable in the steps for thinking in living beings. The use of the zero-energy and complex-energy solutions enables living beings to make up many functions in their bodies very economically on the basis of energy consumption. Anyway the zero-energy and complex-energy solutions are interesting objects to describe mechanisms working very economically as for the energy consumption. Especially, in the 2D-PPB case we can do it without any energy loss, because all the solutions of PPB have no real energy eigenvalue, i.e., the energy eigenvalues are zero or pure imaginary. Here we would like to summarize the property of flows in CSGT. As for the zero-energy flows we can stress the following three properties; they can be (i) the absolutely energy-saving mechanism, (ii) the mechanism including an enormous topological variety in terms of vortex patterns, and also (iii) the perfect mechanism to recover the initial flow patterns in any disturbance by pouring arbitrary decaying flows with complex-energy eigenvalues. The role of the flows with complex energies will be understood as short excitement mechanisms of the vortex patterns. We still have a lot of problems to overcome the present situation, but we may expect that the study of the zero- and complex-energy solutions in CSGT will open a new site in physics. We have shown concrete examples of different types of vortex patterns accompanied by creations and annihilations of vortices by using only some low degree solutions of the 2D-PPB. We can, of course, present more complicated patterns by introducing the higher degree solutions, but the examples presented in the sections 3 will be enough to show the fact that various vortex patterns can be reproducible in terms of the eigenfunctions of the 2D-PPB. As already noted that the zero-energy solutions in the 2D-PPB can be transformed to those in the potentials $V_a(\rho )$ by the conformal transformations \[24,25\], the stable patterns given in section 3.2 can be transformed into the stable patterns of arbitrary potentials. This fact means that, as far as the stable patterns are concerned, there is an exact one-to-one correspondence between the patterns of the PPB and those of the other potentials. As for the time-dependent patterns we cannot present any concrete examples except the case of the 2D-PPB at this moment, but we may expect that similar vortex patterns as those given in section 3.3 for the PPB will appear in other potentials, since all energy eigenstates with complex eigenvalues degenerate infinitely in all the potentials $V_a(\rho )$ as same as in the 2D-PPB. Anyhow we cannot exactly say about the problem before we find any solutions with complex eigenvalues in the other cases. Here we would like to note 2D-PPB. We do not know any physical phenomena that are described by 2D-PPB. We can, however, expect that most of weak repulsive forces in matters composed of many constituents will be approximated by PPBs as most of weak attractive forces are well approximated by harmonic oscillators. In general flows that go round a smooth hill of potential feel a weak repulsive force represented by a PPB \[34-36\]. Actually we see that when a charged particle is put in an infinitely long tube where same charged particles are uniformly distributed, the charged particle feels 2D-PPB. In non-neutral plasma electrons being near the center will possibly be in a similar situation. In the plasma electro-magnetic interactions must be introduced. It should be noted that in the case of a charged particle in a magnetic field the vortex quantization given by (6) can be read as $$\begin{aligned} m {\varGamma}&=\oint _c (\nabla S -q{\bf A})\cdot d{\bf s} \nonumber \\ &=\oint _c {\bf p}\cdot d{\bf s} - q\Phi , \end{aligned}$$ where $q$ is the charge of the particle and $\Phi$ is the magnetic flux passing through the enclosed surface. Analyzing vortex phenomena of non-neutral plasma in terms of the eigenfunctions of the 2D-PPB will be an interesting application. The infinite freedom arising from the infinite degeneracy of the zero-energy solutions should be noticed. Such a freedom has never appeared in the statistical mechanics describing thermal equilibrium. The freedom is different from that generating the usual entropy and then temperatures, because the freedom does not change real energy observed in experiments at all. A model of statistical mechanics for the new freedom has been proposed and some simple applications have been performed in the case of 1D-PBB \[37-39\]. The model is applicable to slowly changing phenomena in the time evolutions, because the PPB has only pure imaginary energy eigenvalues in the 1-dimension. In the present model of the 2-dimensions, however, we have the infinite degree of freedom arising from the zero-energy solutions that have no time evolution. The huge degeneracy of the zero-energy solutions can provide the huge variety in every energy eigenstate, which will be identified by the vortex patterns. In such a consideration the vortex patterns will be understood as the topological properties of flows. How this freedom should be counted in statistical mechanics is an important problem in future considerations. Finally we would like to comment on the meaning of the eigenstates in CSGT. As already noted, the eigenfunctions in CSGT are generally not normalizable, and then the probability and the probability current cannot have a definite meaning. This fact means that the probabilistic interpretation for the eigenfunctions cannot be introduced in CSGT. How should we interpret the eigenfunctions in CSGT? From the discussions presented in this paper we find out a possible idea that the quantization in terms of Gel’fand triplets describes the quantization of flows. Flows are, of course, composed of many particles, and then the probability used in the description of one particle motions cannot be introduced. The magnitudes of the eigenfunctions should be considered to be proportional to the densities of the flows like the intensity of beams in scattering processes. Thus the normalizations of wavefunctions expessed by the linear combinations of the eigenfunctions lose the meaning in CSGT. We can, however, fond out that the energies are quantized as discrete or continuous numbers including complex numbers, and also the flows expressed by the eigenfunctions interfere. Though quantities in CSGT are in general not directly observed except eigenvalues such as energies, we see that velocities have a special role that they are observables being definable only on CSGT. Vortex patterns that are determined only from nodal points of wavefunctions are also good observables to investigate solutions in CSGT. As shown in section 3, we can actually see the interferences among flows through the investigation of the vortex patterns. We may say that hydrodynamical approach will be an interesting trial to investigate physics in CSGT. [99]{} G. Blatter [*et al.*]{}, Rev. Mod. Phys. [**66**]{}, 1125 (1994). G. W. Crabtree and D. R. Nelson, Phys. 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(300,300) (0,100)[(1,0)[300]{}]{} (150,0)[(0,1)[250]{}]{} (140,88)[$0$]{} (305,98)[$x({\rm or}\ y)$]{} (148,255)[$V$]{} (230,200)(230,103)(300,103) (70,200)(70,103)(0,103) (70,50)[(0,1)[150]{}]{} (230,50)[(0,1)[150]{}]{} (70,50)[(1,0)[160]{}]{} (137,38)[$V_0$]{} (200,200) (0,100)[(1,0)[200]{}]{} (100,0)[(0,1)[200]{}]{} (90,88)[$0$]{} (205,98)[$x$]{} (98,205)[$y$]{} (105,200)(105,105)(200,105) (95,200)(95,105)(0,105) (95,0)(95,95)(0,95) (105,0)(105,95)(200,95) (200,105)[(1,0)[1]{}]{} (0,105)[(-1,0)[1]{}]{} (0,95)[(-1,0)[1]{}]{} (200,95)[(1,0)[1]{}]{} (200,200) (0,100)[(1,0)[200]{}]{} (100,0)[(0,1)[200]{}]{} (90,88)[$0$]{} (205,98)[$x$]{} (98,205)[$y$]{} (105,200)(105,105)(200,105) (95,200)(95,105)(0,105) (95,0)(95,95)(0,95) (105,0)(105,95)(200,95) (105.5,200)[(0,1)[1]{}]{} (95.5,200)[(0,1)[1]{}]{} (95.5,0)[(0,-1)[1]{}]{} (105.5,0)[(0,-1)[1]{}]{} (200,200) (0,100)[(1,0)[200]{}]{} (100,0)[(0,1)[200]{}]{} (90,88)[$0$]{} (205,98)[$x$]{} (98,205)[$y$]{} (148,98)[$\cdot $]{} (50,98)[$\cdot $]{} (140,105)[$|c|/\beta $]{} (35,105)[$-|c|/\beta $]{} (98,147)[$\cdot $]{} (98,50)[$\cdot $]{} (106,150)[$|c|/\beta $]{} (104,50)[$-|c|/\beta $]{} (147,147)[$\bullet$]{} (97.4,97.8)[$\bullet$ ]{} (50,50)[$\bullet $]{} (200,200) (0,100)[(1,0)[200]{}]{} (100,0)[(0,1)[200]{}]{} (90,88)[$0$]{} (205,98)[$x$]{} (98,205)[$y$]{} (150,98)[$\cdot $]{} (50,98)[$\cdot $]{} (141,105)[$|c|/\beta $]{} (35,105)[$-|c|/\beta $]{} (98,150)[$\cdot $]{} (98,50)[$\cdot $]{} (106,150)[$|c|/\beta $]{} (104,50)[$-|c|/\beta $]{} (148,149)[$\bullet $]{} (148,49)[$\bullet $]{} (49,149)[$\bullet $]{} (49,49)[$\bullet $]{} (200,200) (0,100)[(1,0)[200]{}]{} (100,0)[(0,1)[200]{}]{} (90,88)[$0$]{} (205,98)[$x$]{} (98,205)[$y$]{} (150,150)[$\bullet$]{} (150,46)[$\bullet $]{} (150,98)[$\cdot $]{} (146,105)[$c/\beta $]{} (98,150)[$\cdot $]{} (98,46)[$\cdot $]{} (106,150)[$c/\beta $]{} (103,46)[$-c/\beta $]{} (151,151)[(-1,-1)[47.65]{}]{} (150,50)[(-1,1)[47]{}]{} (200,200) (0,100)[(1,0)[200]{}]{} (100,0)[(0,1)[200]{}]{} (90,88)[$0$]{} (205,98)[$x$]{} (98,205)[$y$]{} (150,150)[$\bullet $]{} (150,50)[$\bullet $]{} (48,150)[$\bullet $]{} (48,50)[$\bullet $]{} (151,98)[$\cdot $]{} (48.5,98)[$\cdot $]{} (156,105)[$|c|/\beta $]{} (14,105)[$-|c|/\beta $]{} (98,150)[$\cdot $]{} (98,50)[$\cdot $]{} (106,150)[$|c|/\beta $]{} (103,50)[$-|c|/\beta $]{} (152.4,150)[(0,-1)[47]{}]{} (152.5,52)[(0,1)[46]{}]{} (50.4,52)[(0,1)[46]{}]{} (50.4,150)[(0,-1)[47]{}]{} (200,200) (0,100)[(1,0)[200]{}]{} (100,0)[(0,1)[200]{}]{} (90,88)[$0$]{} (205,98)[$x$]{} (98,205)[$y$]{} (183,159)[$\bullet$]{} (15,40)[$\bullet $]{} (185.9,161.5)[(-3,-2)[7]{}]{} (17.5,43.3)[(3,2)[7]{}]{} (153,139)[$\bullet$]{} (45,60)[$\bullet $]{} (156,141.8)[(3,2)[7]{}]{} (48.2,62.4)[(-3,-2)[7]{}]{} (15,159)[$\bullet$]{} (153,60)[$\bullet $]{} (17.9,161.7)[(3,-2)[7]{}]{} (155.5,62.3)[(3,-2)[7]{}]{} (45,139)[$\bullet$]{} (183,40)[$\bullet $]{} (48.1,142.1)[(-3,2)[7]{}]{} (185.9,43.0)[(-3,2)[7]{}]{} [^1]: E-mail: kobayash@a.tsukuba-tech.ac.jp

--- abstract: 'The lithium abundance in 62 halo dwarfs is determined from accurate equivalent widths reported in the literature and an improved infrared flux method (IRFM) temperature scale. The Li abundance of 41 plateau stars (those with [T$_{\rm eff}$ ]{}$>$ 6000 K) is found to be independent of temperature and metallicity, with a star-to-star scatter of only 0.06 dex over a broad range of temperatures (6000 K $<$ [T$_{\rm eff}$ ]{}$<$ 6800 K) and metallicities ($-3.4 <$ \[Fe/H\] $< -1$), thus imposing stringent constraints on depletion by mixing and production by Galactic chemical evolution. We find a mean Li plateau abundance of $A_{Li}$ = 2.37 dex ($^7$Li/H = 2.34 $\times 10^{-10}$), which, considering errors of the order of 0.1 dex in the [*absolute*]{} abundance scale, is just in borderline agreement with the constraints imposed by the theory of primordial nucleosynthesis and WMAP data (2.51 $< A_{Li}^{\rm WMAP}$ $<$ 2.66 dex).' author: - Jorge Meléndez and Iván Ramírez title: 'Reappraising the Spite Lithium Plateau: Extremely Thin and Marginally Consistent with WMAP ' --- = ==1=1=0pt =2=2=0pt =2=2=0pt Introduction ============ The Li plateau was discovered by Spite & Spite (1982), who showed that the $^7$Li abundance obtained from the Li doublet at 6708 Å in F and early G halo dwarfs is independent of temperature and metallicity, suggesting that the Li abundance determined in halo stars represents the primordial abundance from Big Bang nucleosynthesis (BBN). The standard theory of BBN predicts the abundance of light elements, in particular $^7$Li, as a function of the baryon-to-photon ratio $\eta$. Thus, if the plateau is primordial, an inverse analysis could be used to determine $\eta$ (Spite & Spite 1982). On the other hand, Ryan and collaborators (Ryan et al. 1996, 1999, 2001; hereafter jointly referred as R01) suggested that the Li abundance in halo stars is postprimordial, with a 0.3 dex increase from \[Fe/H\] = $-$3.5 to \[Fe/H\] = $-$1.[^1] Those results rely on the temperature scale of Magain (1987), which was calibrated with only 3 metal-poor (\[Fe/H\] $<$ -2) stars. Considering the metallicity dependence of the Li abundances, R01 extrapolated a primordial Li abundance of $A_{Li}$ $\approx$ 2.0 - 2.1 dex.[^2] Recently, precise measurements of the cosmic background anisotropies by the Wilkinson Microwave Anisotropy Probe (WMAP, Spergel et al. 2003) have been used to constrain $\eta$, and employing the standard theory of BBN, the primordial Li abundance was predicted to be $A_{Li}^{\rm WMAP}$ = 2.58$_{-0.07}^{+0.08}$ dex (Cyburt et al. 2003), a value that is much higher than the primordial abundance proposed by R01. The large difference of about 0.5 dex between the results of Ryan et al. and WMAP has stimulated theoretical work on non-standard BBN (e.g. Feng et al. 2003; Ichikawa & Kawasaki 2004). Bonifacio & Molaro (1997, hereafter BM97) found that when the [T$_{\rm eff}$ ]{}scale based on the infrared flux method (IRFM) is employed (that of Alonso et al. 1996), the Li abundance in metal-poor stars does not depend on metallicity, and $A_{Li}$ = 2.24 dex is obtained, higher than the primordial abundance proposed by R01, but still lower by a factor of 2 than the abundance suggested by the WMAP data. Furthermore, Bonifacio et al. (2002) analyzed 5 metal-poor field stars and 12 turn-off stars in the globular cluster NGC 6397, obtaining a Li abundance 0.05 and 0.1 dex higher than in BM97, respectively. Ford et al. (2002) have determined the Li abundance in a few metal-poor stars employing the extremely weak ($\approx$ 2-4 mÅ) Li lines at 6104 Å, resulting in $A_{Li}$ $\approx$ 2.5 dex, but note that the observational errors are very large (about 30%), not allowing them to draw any firm conclusions. In fact, when the Li lines at 6708 Å were used by Ford et al. (2002), $A_{Li}$ $\approx$ 2.2 dex was found. The temperature is the most important atmospheric parameter required to obtain the Li abundance, and since metal-poor stars are too distant to measure their angular diameters, indirect methods have to be used to estimate their temperatures. The closest (and less model-dependent) approach to the direct method is the IRFM, where the temperature is found by a comparison between the observed and theoretical ratio of the bolometric flux to the infrared flux. In a pioneer work, Magain (1987) calibrated the [T$_{\rm eff}$ ]{}scale of metal-poor dwarfs (mostly [T$_{\rm eff}$ ]{}$<$ 6000 K and \[Fe/H\] $> -2$) employing IRFM temperatures of 11 dwarfs. The titanic work by Alonso et al. (1996) improved the [T$_{\rm eff}$ ]{}scale including a larger number of metal-poor stars (down to \[Fe/H\] $\approx$ -3), employing spectroscopic metallicities (when available) determined before 1992. In the meantime, new spectroscopic and photometric surveys have appeared in the literature (a milestone was the completion of 2MASS), providing the necessary input data for a new revision of the [T$_{\rm eff}$ ]{}scale (Ramírez & Meléndez 2004b,c, hereafter RM04b,c). As an example, only from January 2001 to June 2003, more than 1500 new spectroscopic metallicity determinations for field FGK stars have appeared in the literature, increasing significantly the number of calibrating stars with reliable \[Fe/H\] values, and hence allowing to better define the metallicity dependence of the [T$_{\rm eff}$ ]{}scale. We have determined the IRFM [T$_{\rm eff}$ ]{}of more than 10$^3$ stars (RM04b) using updated input data, and metallicity-dependent [T$_{\rm eff}$ ]{}vs. color calibrations were derived in the metallicity range $-3.5 <$ \[Fe/H\] $< +0.4$ (RM04c). Employing our improved IRFM temperature scale, we show that the Spite plateau has essentially zero scatter and that the derived abundances are now consistent, although only marginally, with WMAP data. Data and Atmospheric Parameters =============================== The sample was selected from relatively unevolved halo stars (log $g \geq$ 3.5, \[Fe/H\] $<$ $-$1) with previous Li observations and having either an IRFM effective temperature determined by us (RM04b) and/or with a previous determination of metallicity as reported in the “2003 updated” version of the Cayrel de Strobel et al. (2001, hereafter C03) catalog (see $\S$2 in Ramírez & Meléndez 2004a). For most works previous to 1997, the equivalent widths ($W_\lambda$) of the Li doublet at 6708 Å  were taken from the compilation of Ryan et al. (1996), adopting a minimum error of 1.5 mÅ for the individual measurements. Other $W_\lambda$ considered are those by Molaro et al. (1995), Spite et al. (1996), Nissen & Schuster (1997), Ryan & Deliyannis (1998), Smith et al. (1998), Gutiérrez et al. (1999), Hobbs et al. (1999), King (1999), Ryan et al. (1999, 2001), Bonifacio et al. (2002), and Ford et al. (2002). The weighted averages of the equivalent widths and errors ($W_\lambda$ and $\delta W_\lambda$) were computed, as well as the standard errors ($\sigma_m$). The adopted error is $\sigma_W \equiv$ max($\delta W_\lambda$, $\sigma_m$, 1 mÅ). A star was rejected if $\sigma_W \geq 0.1 W_\lambda$. Stars with E(B-V) $>$ 0.05 dex were also discarded. Sixty two dwarfs satisfied our selection criteria. The key parameter for a reliable Li abundance determination in metal-poor stars is T$_{\rm eff}$; errors in the other atmospheric parameters (log $g$, \[Fe/H\], $v_t$) result in abundance errors of the order of 0.01 dex (e.g. Ryan et al. 1996). Surface gravities and metallicities were taken from the C03 catalog, and a microturbulence $v_t$ = 1 km s$^{-1}$ was adopted. For 3 stars not included in the C03 catalog, we adopted metallicities from a photometric calibration (RM04b) that is based on C03 \[Fe/H\] values. The effective temperatures are based on our IRFM [T$_{\rm eff}$ ]{}scale. Individual IRFM temperatures were taken from RM04b or recomputed employing new reddening corrections, as well as calculated employing the IRFM [T$_{\rm eff}$ ]{}calibrations of RM04c. We used up to 17 colors in the [*BVRI*]{}(Johnson-Cousins), Tycho-Hipparcos, Strömgren, DDO, Vilnius, Geneva and 2MASS systems. Most of the optical photometry was taken from the General Catalogue of Photometric Data (Mermilliod et al. 1997) and the Tycho catalog, while the infrared photometry was taken from 2MASS. E(B-V) values were computed employing several extinction maps, giving a higher weight to the maps by Schlegel et al. (1998) for high latitude stars (RM04b). The use of several colors improves the determination of the effective temperature, since it alleviates the impact of photometric errors. Following RM04c, the adopted [T$_{\rm eff}$ ]{} was the mean of the IRFM determination and the temperature from the IRFM calibrations (T$_{\rm eff}^{\rm cal}$): $${\rm T_{\rm eff}} = \case{1}{n+m} (n {\rm T_{eff}^{IRFM}} + m {\rm T_{eff}^{cal}})$$ where the weights we adopted were $n$ = 1 and $m$ equal to the square root of the number of colors employed (typically 10, thus $m \approx 3$). T$_{\rm eff}^{\rm cal}$ was calculated taking into account the errors of each IRFM color calibration. The mean internal error of the [T$_{\rm eff}$ ]{}derived in this work is $\delta$[T$_{\rm eff}$ ]{}$\approx$ 66 K. It is important to note that the use of eq. (1) results in a [T$_{\rm eff}$ ]{}that is only $\approx$ 22 K lower than the temperature directly determined from the IRFM. If only T$_{\rm eff}^{\rm IRFM}$ were used, A$_{Li}$ would be about 0.015 dex higher. Li Abundances ============= In order to determine the Li abundances, we computed a grid of theoretical equivalent widths of the Li doublet in the parameters space: [T$_{\rm eff}$ ]{}= \[5250 K, 5500 K, ..., 6750 K\], log $g$ = \[3.5, 4.0, 4.5\], \[Fe/H\] = \[-1.0, -1.5, ..., -3.5\], $A_{Li}$ = \[1.7, 1.9, ..., 2.5\]. We took into account the fine and hyperfine structure of the doublet, with wavelengths and $gf$-values as given in Andersen et al. (1984). Note that Smith et al. (1998) and Hobbs et al. (1999) reported new $gf$-values based on theoretical calculations, but the difference with our adopted (laboratory) values is only $\approx$ 1% (Smith et al. 1998). The collisional broadening constant was obtained from Barklem et al. (2000). A grid of 630 synthetic spectra was computed employing the 2002 version of the LTE spectral synthesis code MOOG (Sneden 1973) and Kurucz overshooting models. The theoretical $W_\lambda$ were measured by integration. Using the stellar parameters and observed $W_\lambda$ as input, the Li abundance was determined from a full 4-dimensional interpolation in the synthetic grid $A_{Li}$([T$_{\rm eff}$ ]{}, log $g$, \[Fe/H\], log $W_\lambda$). Note that in some previous studies (Thorburn 1994; Ryan et al. 1996, 1999) a more restricted approach was adopted, considering only two fixed log $g$ values (no interpolation) and metallicity interpolation only for models within $-$2 $<$ \[Fe/H\] $<$ $-$1 (Ryan et al. 1996) or $-$3 $<$ \[Fe/H\] $<$ $-$2 (Thorburn 1994). Our full interpolation results in lower internal abundance uncertainties, which is very important to determine the true scatter of the Li abundances. NLTE effects are small for the Li plateau stars. We applied the NLTE corrections ($\Delta x \equiv$ NLTE - LTE) by Carlsson et al. (1994), which are $\Delta x$ $\approx$ $-$0.04 and 0.0 dex for the hottest and coolest plateau dwarfs, respectively. Although these NLTE corrections were not specifically calculated for our set of model atmospheres, their relative dependence with [T$_{\rm eff}$ ]{}and \[Fe/H\] is still useful, and should minimize the scatter of the derived abundances. Since the [*absolute*]{} NLTE corrections may be different, we prefer to preserve the mean $A_{Li}$(LTE) of the plateau stars by adopting a NLTE correction of $\Delta x$ + 0.02 dex. The final Li abundances are reported in Table 1. In Fig. 1 are shown the Li abundances obtained in this work. In the lower panel $A_{Li}$ vs. [T$_{\rm eff}$ ]{}is plotted, where it is clearly seen that stars with [T$_{\rm eff}$ ]{}$>$ 6000 K have approximately the same Li abundance (plateau stars), as no trend with [T$_{\rm eff}$ ]{}(slope of $4 \pm 5 \times 10^{-3}$ per 100 K) is observed. In the upper panel are plotted 41 plateau stars ([T$_{\rm eff}$ ]{}$>$ 6000 K) as a function of \[Fe/H\], showing an impressive constancy in their Li abundance from \[Fe/H\] $\approx$ $-$3.4 to $-$1, with an amazingly small scatter and essentially zero slope (0.013 $\pm$ 0.018 per dex). This figure strongly supports the primordial origin of the Li plateau (Spite & Spite 1982), and imposes stringent constraints on stellar mixing and Galactic chemical evolution. The very small star-to-star [*observed*]{} scatter ($\sigma_{obs}$ = 0.06 dex) is fully explained by errors in [T$_{\rm eff}$ ]{} and $W_\lambda$. Adding in quadrature the typical errors of $\delta$T = 66 K ($\delta A_{Li} \approx$ 0.045 dex) and $\delta W_\lambda$ = 5.5% ($\delta A_{Li} \approx$ 0.025 dex) (as well as errors of 0.3 dex in log $g$ and \[Fe/H\]), a predicted error $\sigma_{pred}$ = 0.053 dex is obtained, in excellent agreement with the observed scatter ($\sigma_{obs}$ = 0.06 dex). Note that if the 2 plateau stars with the highest Li abundances are discarded, then $\sigma_{obs}$ = 0.05 dex. Discussion ========== The abundances obtained in this work for plateau stars show a very small star-to-star-scatter. This is a consequence of employing very accurate temperatures based on the IRFM and their photometric calibrations. RM04b,c have shown that the IRFM [T$_{\rm eff}$ ]{}scale is in almost perfect agreement (at the 10 K level) with interferometric measurements of dwarfs and giants with \[Fe/H\] $> -0.6$. Provided that the plateau is truly primordial, the constancy of the Li abundance in our plateau stars indicates that our IRFM [T$_{\rm eff}$ ]{}scale is also highly reliable when we approach low metallicities, and this would be consistent with the small dependence of the IRFM on the detailed assumptions of model atmospheres. Although the Li abundances derived in this work have very small internal errors, systematic errors dominate the [*absolute*]{} abundances. For example, the use of Kurucz models with no overshooting (NOVER, Castelli et al. 1997) results in smaller $A_{Li}$ by about 0.08 dex. Note that the Kurucz solar overshooting model reproduces more solar observables than the NOVER model (Castelli et al. 1997). Interestingly, the [*(R-I)*]{}$_{C}$ synthetic colors computed from overshooting models are in better agreement with the IRFM [T$_{\rm eff}$ ]{}scale than colors computed with NOVER models, at least for solar metallicity dwarfs (6000 K $<$ [T$_{\rm eff}$ ]{}$<$ 6800 K, Meléndez & Ramírez 2003) and giants (4000 K $<$ [T$_{\rm eff}$ ]{}$<$ 6300 K, Ramírez & Meléndez 2004a). The mean Li abundance obtained for the plateau stars, $A_{Li}$ = 2.37 dex, is considerably higher than the primordial Li abundance proposed by R01 ($A_{Li} \approx$ 2.0 - 2.1 dex). As shown in Fig. 2, the reason for this are the lower temperatures adopted by R01. The temperature difference (between our [T$_{\rm eff}$ ]{}and those adopted by R01) is metallicity dependent, and at very low metallicities (\[Fe/H\] $< -3)$ the [T$_{\rm eff}$ ]{}adopted by R01 is about 400 K lower than ours. When the temperatures adopted by R01 are used, we also find a trend of Li abundance with metallicity. If we use the same [T$_{\rm eff}$ ]{}(and $W_\lambda$) as those employed by R01, $<A_{Li}>$ = 2.13 dex is found for stars with [T$_{\rm eff}$ ]{}$>$ 6000 K and \[Fe/H\] $<$ -3, whereas R01 found $<A_{Li}>$ = 2.07 dex for the same stars (the small difference is due to the different model atmospheres adopted). Note that when Magain (1987) calibration is used in its validity range ([T$_{\rm eff}$ ]{}$<$ 6000 K, \[Fe/H\] $> -2$), the agreement with our [T$_{\rm eff}$ ]{}scale is satisfactory (open circles, Fig. 2); outside that range it may not be appropriate to use it. Most (0.07 dex) of the 0.13 dex difference between the present Li plateau abundance and that obtained by BM97 is due to different model atmospheres, and the remaining difference is because our [T$_{\rm eff}$ ]{}scale is hotter for metal-poor F dwarfs. We have compared our IRFM temperatures with those given by Alonso et al. (1996) (employing the same reddening adopted by Alonso et al.) The agreement is good for metal-poor stars with [T$_{\rm eff}$ ]{}$<$ 6500 K, and our temperatures are just $\approx$ 50 K higher in the range 6000 $<$ [T$_{\rm eff}$ ]{}$<$ 6500. For hotter stars the differences are larger. Since only 5 stars with [T$_{\rm eff}$ ]{}$>$ 6500 K are included in the present work, their exclusion does not alter our results (Fig. 1), but restricts our claim of the constancy of the plateau to $-3 <$ \[Fe/H\] $< -1$ and 6000 K $<$ [T$_{\rm eff}$ ]{}$<$ 6500 K. Ryan et al. (1999) have criticized Alonso’s calibrations for turn-off stars because of their strong metallicity dependence. Nevertheless, this effect is predicted by both Kurucz and MARCS model atmospheres. In fact, for metal-poor (\[Fe/H\] = -3) F dwarfs, our ($V-K$) IRFM calibration is in good agreement with $V-K$ colors predicted from MARCS models (Houdashelt et al. 2000). At \[Fe/H\] = -3, our calibration (RM04c) gives 6250 and 6500 K for $V-K_{2MASS}$ = 1.31 and 1.21, while Houdashelt calibration results in [T$_{\rm eff}$ ]{}[*higher*]{} by +76 and +7 K, respectively. The same exercise employing Kurucz models at $-2.5$ (M. S. Bessell 2004, private communication) gives a similar result. Asplund et al. (2003) have shown that 3D hydrodynamical models with NLTE computations give similar Li abundances than 1D + NLTE. However, that result was obtained neglecting collisions with H, and taking them into account, the Li abundance is lowered (Barklem et al. 2003). It is possible to compare 1D calculations (considering relative NLTE corrections) with 3D NLTE results by using the plateau star HD 84937, which has predicted equivalent widths in 3D NLTE (Barklem et al. 2003). Employing the same atmospheric parameters and Li abundance as given by Barklem et al. (2003), we obtain a 1D $W_\lambda$ = 11.4 mÅ, which is very close to the 3D NLTE results, both without collisions (11.1 mÅ) and when collisions are included (12.6 mÅ). Within 0.05 dex, both 3D NLTE calculations are in (probably fortuitous) agreement with the 1D calculation. Nevertheless, this result is by any means conclusive, as the 3D results are based on only one snapshot (Barklem et al. 2003). The observational star-to-star scatter found in this work ($\sigma_{obs}$ $\approx$ 0.05-0.06 dex) is considerably lower than those found in previous analyses of large samples: 0.13, 0.06-0.08, 0.089 and 0.094 dex according to Thorburn (1994), Spite et al. (1996), Ryan et al. (1996) and BM97, respectively. This extremely thin scatter of the plateau stars is fully explained by small errors in [T$_{\rm eff}$ ]{}and $W_\lambda$, leaving little room (if any) for mixing or Galactic evolution. Considering the constancy of the Li abundance of the plateau stars, depletion is constrained probably at a level of 0.05 dex, and if rotational mixing plays a role, then our mean $A_{Li}$ should be slightly increased. Within the [*absolute*]{} abundance uncertainties involved (of the order of 0.1 dex) in our simple 1D modeling, we conclude that our results are just in marginal agreement with the constraint imposed by WMAP, and if new physics or non-standard Big Bang nucleosynthesis are invoked (e.g. Feng et al. 2003; Ichikawa & Kawasaki 2004), the present work constraints their effects to a level of 0.1-0.15 dex. We thank the referee for his constructive comments, and JG Cohen for her suggestions. JM acknowledges support from NSF grant AST-0205951 to JGC. 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A. 1994, ApJ, 421, 318 [lccccrcrrrrl]{} \[tab1\] BD+002058 & 6092 & 4.19 & -1.16 & 41.9 & 2.49\ BD+023375 & 6124 & 3.97 & -2.29 & 32.5 & 2.34\ BD+030740 & 6443 & 3.76 & -2.68 & 19.5 & 2.31\ BD+090352 & 6075 & 4.38 & -2.03 & 34.0 & 2.34\ BD+092190 & 6487 & 4.11 & -2.05 & 17.3 & 2.28\ BD+174708 & 6154 & 3.93 & -1.64 & 25.8 & 2.28\ BD+203603 & 6248 & 4.15 & -2.12 & 26.2 & 2.32\ BD+241676 & 6510 & 3.91 & -2.32 & 25.3 & 2.46\ BD+262606 & 6157 & 4.20 & -2.37 & 30.5 & 2.33\ BD+282137 & 6229 & 3.50 & -2.20 & 27.2 & 2.33\ BD+342476 & 6433 & 3.92 & -2.10 & 22.6 & 2.37\ BD+422667 & 6086 & 4.13 & -1.51 & 30.6 & 2.32\ BD+592723 & 6121 & 4.00 & -1.83 & 30.5 & 2.33\ BD+710031 & 6331 & 4.05 & -1.96 & 31.0 & 2.44\ BD-043208 & 6404 & 3.77 & -2.36 & 25.4 & 2.40\ BD-100388 & 6240 & 3.61 & -2.37 & 27.0 & 2.33\ BD-133442 & 6484 & 3.98 & -2.87 & 20.5 & 2.35\ CD-2417504 & 6507 & 4.12 & -3.36 & 18.6 & 2.31\ CD-3018140 & 6336 & 4.08 & -1.95 & 30.3 & 2.44\ CD-3301173 & 6674 & 4.19 & -2.99 & 16.3 & 2.35\ CD-7101234 & 6375 & 4.43 & -2.35 & 26.0 & 2.38\ G059-024 & 6191 & 4.36 & -2.38 & 36.0 & 2.42\ G064-012 & 6682 & 4.10 & -3.31 & 23.4 & 2.53\ G064-037 & 6775 & 4.05 & -3.13 & 15.6 & 2.39\ G075-031 & 6036 & 4.07 & -1.04 & 48.0 & 2.54\ G192-043 & 6184 & 4.30 & -1.61 & 34.5 & 2.42\ G201-005 & 6370 & 3.79 & -2.51 & 25.0 & 2.37\ G206-034 & 6310 & 4.18 & -2.53 & 27.2 & 2.37\ HD016031 & 6216 & 4.07 & -1.82 & 27.0 & 2.33\ HD074000 & 6392 & 4.27 & -1.96 & 23.9 & 2.36\ HD084937 & 6345 & 3.96 & -2.13 & 24.8 & 2.35\ HD102200 & 6104 & 4.14 & -1.21 & 32.8 & 2.38\ HD108177 & 6099 & 4.30 & -1.64 & 31.4 & 2.33\ HD160617 & 6099 & 3.68 & -1.79 & 40.0 & 2.43\ HD166913 & 6096 & 4.08 & -1.58 & 37.0 & 2.40\ HD181743 & 6038 & 4.47 & -1.81 & 38.0 & 2.38\ HD218502 & 6222 & 3.94 & -1.79 & 27.1 & 2.34\ HD284248 & 6133 & 4.21 & -1.61 & 29.9 & 2.33\ HD338529 & 6504 & 3.90 & -2.27 & 23.9 & 2.43\ LP0056-0075 & 6197 & 4.36 & -2.67 & 25.5 & 2.27\ LP0831-0070 & 6521 & 4.23 & -3.07 & 23.8 & 2.42\ [^1]: \[A/B\] $\equiv$ log ($N_A/N_B$) - log($N_A/N_B$)$_\odot$ [^2]: $A_{X}$ $\equiv$ log ($N_X$/$N_H$) + 12

--- author: - 'Monika Dryl and Delfim F. M. Torres' title: 'Direct and Inverse Variational Problems on Time Scales: A Survey[^1]' --- Introduction ============ The theory of time scales is a relatively new area, which was introduced in 1988 by Stefan Hilger in his Ph.D. thesis [@phdHilger; @Hilger; @Hilger2]. It bridges, generalizes and extends the traditional discrete theory of dynamical systems (difference equations) and the theory for continuous dynamical systems (differential equations) [@BohnerDEOTS] and the various dialects of $q$-calculus [@Ernst; @MyID:266] into a single unified theory [@BohnerDEOTS; @MBbook2003; @Lakshmikantham]. The calculus of variations on time scales was introduced in 2004 by Martin Bohner [@BohnerCOVOTS] (see also [@MR1908827; @HilscgerZeidan]) and has been developing rapidly in the past ten years, mostly due to its great potential for applications, e.g., in biology [@BohnerDEOTS], economics [@Almeida:iso; @MR2218315; @Atici:HMMS; @MR2405376; @natalia:CV] and mathematics [@MR2662835; @MR2668257; @MR2879335; @Torres]. In order to deal with nontraditional applications in economics, where the system dynamics are described on a time scale partly continuous and partly discrete, or to accommodate nonuniform sampled systems, one needs to work with variational problems defined on a time scale [@MR2218315; @Atici:comparison; @PhD:thesis:Monia; @MyID:267]. This survey is organized as follows. In Section \[preliminaries\] we review the basic notions of the time-scale calculus: the concepts of delta derivative and delta integral (Section \[sec:prelim:delta\]); the analogous backward concepts of nabla differentiation and nabla integration (Section \[sec:prelim:nabla\]); and the relation between delta/forward and nabla/backward approaches (Section \[subsec:relation:d:n\]). Then, in Section \[CoV\], we review the central results of the recent and powerful calculus of variations on time scales. Both delta and nabla approaches are considered (Sections \[sec:calculus:of:variations\] and \[subsec:nabla:CoV\], respectively). Our results begin with Section \[chapter\_5\_P3+P4\], where we investigate inverse problems of the calculus of variations on time scales. To our best knowledge, and in contrast with the direct problem, which is already well studied in the framework of time scales [@Torres], the inverse problem has not been studied before. Its investigation is part of the Ph.D. thesis of the first author [@PhD:thesis:Monia]. Let here, for the moment and just for simplicity, the time scale $\mathbb{T}$ be the set $\mathbb{R}$ of real numbers. Given $L$, a Lagrangian function, in the ordinary/direct fundamental problem of the calculus of variations one wants to find extremal curves $y : [a, b] \rightarrow \mathbb{R}^n$ giving stationary values to some action integral (functional) $$\mathcal{I}(y)=\int\limits_{a}^{b} L(t,y(t),y'(t)) dt$$ with respect to variations of $y$ with fixed boundary conditions $y(a)=y_{a}$ and $y(b)=y_{b}$. Thus, if in the direct problem we start with a Lagrangian and we end up with extremal curves, then one might expect as inverse problem to start with extremal curves and search for a Lagrangian. Such inverse problem is considered, in the general context of time scales, in Section \[P3\]: we describe a general form of a variational functional having an extremum at a given function $y_{0}$ under Euler–Lagrange and strengthened Legendre conditions (Theorem \[P3:th:integrand:2\]). In Corollary \[P3:cor:isolated\] the form of the Lagrangian $L$ on the particular case of an isolated time scale is presented and we end Section \[P3\] with some concrete cases and examples. We proceed with a more common inverse problem of the calculus of variations in Section \[P4\]. Indeed, normally the starting point are not the extremal curves but, instead, the Euler–Lagrange equations that such curves must satisfy: $$\label{eq:EL} \frac{\partial L}{\partial y} - \frac{d}{dt} \frac{\partial L}{\partial y'} = 0 \Leftrightarrow \frac{\partial L}{\partial y} - \frac{\partial^2 L}{\partial t \partial y'} - \frac{\partial^2 L}{\partial y \partial y'} y' - \frac{\partial^2 L}{\partial y' \partial y'} y'' = 0$$ (we are still keeping, for illustrative purposes, $\mathbb{T} = \mathbb{R}$). This is what is usually known as the inverse problem of the calculus of variations: start with a second order ordinary differential equation and determine a Lagrangian $L$ (if it exists) whose Euler–Lagrange equations *are the same as* the given equation. The problem of variational formulation of differential equations (or the inverse problem of the calculus of variations) dates back to the 19th century. The problem seems to have been posed by Helmholtz in 1887, followed by several results from Darboux (1894), Mayer (1896), Hirsch (1897), Volterra (1913) and Davis (1928, 1929) [@MR2732180]. There are, however, two different types of inverse problems, depending on the meaning of the phrase “are the same as”. Do we require the equations to be the same or do we allow multiplication by functions to obtain new but equivalent equations? The first case is often called *Helmholtz’s inverse problem*: find conditions under which a given differential equation is an Euler–Lagrange equation. The latter case is often called the *multiplier problem*: given $f(t,y,y',y'') = 0$, does a function $r(t,y,y')$ exist such that the equation $r(t,y,y') f(t,y,y',y'') = 0$ is the Euler–Lagrange equation of a functional? In this work we are interested in Helmholtz’s problem. The answer to this problem in ${\mathbb{T}}= {\mathbb{R}}$ is classical and well known: the complete solution to Helmholtz’s problem is found in the celebrated 1941 paper of Douglas [@Douglas]. Let $O$ be a second order differential operator. Then, the differential equation $O(y) = 0$ is a second order Euler–Lagrange equation if and only if the Fréchet derivatives of $O$ are self-adjoint. A simple example illustrating the difference between both inverse problems is the following one. Consider the second order differential equation $$\label{eq:EL:ex:introd:fric} m y'' + h y' + k y = f.$$ This equation is not self-adjoint and, as a consequence, there is no variational problem with such Euler–Lagrange equation. However, if we multiply the equation by $p(t) = \exp(ht/m)$, then $$\label{eq:EL:ex:introd} m \frac{d}{dt} \left[\exp(ht/m) y'\right] + k \, \exp(ht/m) y = \exp(ht/m) f$$ and now a variational formulation is possible: the Euler–Lagrange equation associated with $$\mathcal{I}(y)=\int\limits_{t_{0}}^{t_{1}} \exp(ht/m) \left[ \frac{1}{2} m y'^2 - \frac{1}{2} k y^2 - f y \right] dt$$ is precisely . A recent theory of the calculus of variations that allows to obtain directly has been developed, but involves Lagrangians depending on fractional (noninteger) order derivatives [@livro:FC:Comput; @livro:Adv:CoV; @livro:FC:Int]. For a survey on the fractional calculus of variations, which is not our subject here, we refer the reader to [@MyID:288]. For the time scale ${\mathbb{T}}= {\mathbb{Z}}$, available results on the inverse problem of the calculus of variations are more recent and scarcer. In this case Helmholtz’s inverse problem can be formulated as follows: find conditions under which a second order difference equation is a second order discrete Euler–Lagrange equation. Available results in the literature go back to the works of Crăciun and Opriş (1996) and Albu and Opriş (1999) and, more recently, to the works of Hydon and Mansfeld (2004) and Bourdin and Cresson (2013) [@MR1770981; @Helmholtz; @MR1609049; @MR2049870]. The main difficulty to obtain analogous results to those of the classical continuous calculus of variations in the discrete (or, more generally, in the time-scale) setting is due to the lack of chain rule. This lack of chain rule is easily seen with a simple example. Let $f,g: \mathbb{Z} \rightarrow \mathbb{Z}$ be defined by $f\left( t\right) =t^{3}$, $g\left( t\right) =2t$. Then, $\Delta\left( f\circ g\right) = \Delta\left( 8t^{3}\right)=8\left( 3t^{2}+3t+1\right) =24t^{2}+24t+8$ and $\Delta f\left( g\left( t\right) \right) \cdot \Delta g\left( t\right) =\left( 12t^{2}+6t+1\right) 2=24t^{2}+12t+2$. Therefore, $\Delta \left( f\circ g\right) \neq \Delta f\left( g\left( t\right) \right) \Delta g\left( t\right)$. The difficulties caused by the lack of a chain rule in a general time scale ${\mathbb{T}}$, in the context of the inverse problem of the calculus of variations on time scales, are discussed in Section \[final remarks\]. To deal with the problem, our approach to the inverse problem of the calculus of variations uses an integral perspective instead of the classical differential point of view. As a result, we obtain a useful tool to identify integro-differential equations which are not Euler–Lagrange equations on an arbitrary time scale $\mathbb{T}$. More precisely, we define the notion of self-adjointness of a first order integro-differential equation (Definition \[P4:def:self:adj:int:diff\]) and its equation of variation (Definition \[P4:def:eq:var\]). Using such property, we prove a necessary condition for an integro-differential equation on an arbitrary time scale $\mathbb{T}$ to be an Euler–Lagrange equation (Theorem \[P4:th:necess:EL:int:diff\]). In order to illustrate our results we present Theorem \[P4:th:necess:EL:int:diff\] in the particular time scales $\mathbb{T}\in\left\{ \mathbb{R},h\mathbb{Z},\overline{q^{\mathbb{Z}}}\right\}$, $h > 0$, $q>1$ (Corollaries \[P4:cor:R\], \[P4:cor:hZ\], and \[P4:cor:qZ\]). Furthermore, we discuss equivalences between: (i) integro-differential equations and second order differential equations (Proposition \[P4:prop:1\]), and (ii) equations of variations of them on an arbitrary time scale $\mathbb{T}$ ( and , respectively). As a result, we show that it is impossible to prove the latter equivalence due to lack of a general chain rule on an arbitrary time scale [@BohGus1; @BohnerDEOTS]. In Section \[sec:mr\] we address the direct problem of the calculus of variations on time scales by considering a variational problem which may be found often in economics (see [@MalinowskaTorresCompositionDelta] and references therein). We extremize a functional of the calculus of variations that is the composition of a certain scalar function with the delta and nabla integrals of a vector valued field, possibly subject to boundary conditions and/or isoperimetric constraints. In Section \[subsec:EL\] we provide general Euler–Lagrange equations in integral form (Theorem \[P6:th:main\]), transversality conditions are given in Section \[sec:nbc\], while Section \[sub:sec:iso:p\] considers necessary optimality conditions for isoperimetric problems on an arbitrary time scale. Interesting corollaries and examples are presented in Section \[sec:examples\]. We end with Section \[sec:conc\] of conclusions and open problems. Preliminaries ============= A time scale $\mathbb{T}$ is an arbitrary nonempty closed subset of $\mathbb{R}$. The set of real numbers ${\mathbb{R}}$, the integers ${\mathbb{Z}}$, the natural numbers ${\mathbb{N}}$, the nonnegative integers ${\mathbb{N}}_{0}$, an union of closed intervals $[0,1]\cup [2,7]$ or the Cantor set are examples of time scales, while the set of rational numbers $\mathbb{Q}$, the irrational numbers ${\mathbb{R}}\setminus\mathbb{Q}$, the complex numbers $\mathbb{C}$ or an open interval like $(0,1)$ are not time scales. Throughout this survey we assume that for $a,b\in\mathbb{T}$, $a<b$, all intervals are time scales intervals, i.e., $[a,b]=[a,b]_{\mathbb{T}}:=[a,b]\cap\mathbb{T} =\left\{t\in\mathbb{T}: a\leq t\leq b\right\}$. \[def:jump:op\] Let ${\mathbb{T}}$ be a time scale and $t\in{\mathbb{T}}$. The forward jump operator $\sigma:\mathbb{T} \rightarrow \mathbb{T}$ is defined by $\sigma(t):=\inf\left\{ s\in\mathbb{T}: s>t\right\}$ for $t\neq \sup\mathbb{T}$ and $\sigma(\sup\mathbb{T}) := \sup\mathbb{T}$ if $\sup\mathbb{T}<+\infty$. Accordingly, we define the backward jump operator $\rho:\mathbb{T} \rightarrow \mathbb{T}$ by $\rho(t):=\sup\left\{ s\in\mathbb{T}: s<t\right\}$ for $t\neq \inf\mathbb{T}$ and $\rho(\inf\mathbb{T}) := \inf\mathbb{T}$ if $\inf\mathbb{T}>-\infty$. The forward graininess function $\mu:\mathbb{T} \rightarrow [0,\infty)$ is defined by $\mu(t):=\sigma(t)-t$ and the backward graininess function $\nu:\mathbb{T} \rightarrow [0,\infty)$ by $\nu(t):=t-\rho(t)$. The two classical time scales are $\mathbb{R}$ and $\mathbb{Z}$, representing the continuous and the purely discrete time, respectively. Other standard examples are the periodic numbers, $h\mathbb{Z}=\left\{ hk: h>0, k\in{\mathbb{Z}}\right\}$, and the $q$-scale $$\overline{q^{\mathbb{Z}}} :=q^{{\mathbb{Z}}}\cup\left\{ 0\right\}=\left\{ q^{k}:q>1, k\in\mathbb{Z}\right\}\cup\left\{ 0\right\}.$$ Sometimes one considers also the time scale $q^{\mathbb{N}_{0}}=\left\{ q^{k}:q>1, k\in\mathbb{N}_{0}\right\}$. The following time scale is common: $\mathbb{P}_{a,b}=\bigcup\limits_{k=0}^{\infty} [k(a+b),k(a+b)+a]$, $a,b>0$. Table \[tbl:1\] and Example \[ex:1\] present different forms of jump operators ${\sigma}$ and $\rho$, and graininiess functions $\mu$ and $\nu$, in specified time scales. ${\mathbb{T}}$ ${\mathbb{R}}$ $h{\mathbb{Z}}$ $\overline{q^{{\mathbb{Z}}}}$ ---------------- ---------------- ----------------- ------------------------------- $\sigma(t)$ $t$ $t+h$ $qt$ $\rho(t)$ $t$ $t-h$ $\frac{t}{q}$ $\mu(t)$ $0$ $h$ $t(q-1)$ $\nu(t)$ $0$ $h$ $\frac{t(q-1)}{q}$ : \[tbl:1\]Examples of jump operators and graininess functions on different time scales. \[ex:1\] Let $a,b>0$ and consider the time scale $$\mathbb{P}_{a,b}=\bigcup\limits_{k=0}^{\infty} [k(a+b),k(a+b)+a].$$ Then, $$\sigma(t)= \begin{cases} t &\text{if } t\in A_{1}, \\ t+b &\text{if } t\in A_{2}, \end{cases} \quad\quad\quad \rho(t)= \begin{cases} t-b &\text{if } t\in B_{1}, \\ t &\text{if } t\in B_{2} \end{cases}$$ (13,35)(0,-13) (-15,-15)[(0,2)[30]{}]{} (-15,-15)[(1,0)[6]{}]{} (-6,-15)[(1,0)[6]{}]{} (3,-15)[(1,0)[6]{}]{} (12,-15)[(1,0)[6]{}]{} (16,-15)[(1,0)[2]{}]{} (-15,-17)[(0,0)\[cc\][$0$]{}]{} (-9,-17)[(0,0)\[cc\][$a$]{}]{} (17,-17)[(0,0)\[cc\][$t$]{}]{} (-15.5,-9)[(1,0)[1]{}]{} (-15.5,-6)[(1,0)[1]{}]{} (-15.5,0)[(1,0)[1]{}]{} (-15.5,3)[(1,0)[1]{}]{} (-15.5,9)[(1,0)[1]{}]{} (-15.5,12)[(1,0)[1]{}]{} (-17,-9)[(0,0)\[rc\][$a$]{}]{} (-17,-6)[(0,0)\[rc\][$a+b$]{}]{} (-17,0)[(0,0)\[rc\][$2a+b$]{}]{} (-15,-15)[(1,1)[5.55]{}]{} (-6,-6)[(1,1)[5.55]{}]{} (3,3)[(1,1)[5.55]{}]{} (12,12)[(1,1)[4]{}]{} (-9,-9) (-9,-6) (-6,-6) (0,0) (0,3) (3,3) (9,9) (9,12) (12,12) (12,3)[(0,0)\[cc\][$\sigma(t)$]{}]{} (13,25)(0,-13) (-15,-15)[(0,2)[30]{}]{} (-15,-15)[(1,0)[6]{}]{} (-6,-15)[(1,0)[6]{}]{} (3,-15)[(1,0)[6]{}]{} (12,-15)[(1,0)[6]{}]{} (16,-15)[(1,0)[2]{}]{} (-15,-17)[(0,0)\[cc\][$0$]{}]{} (-9,-17)[(0,0)\[cc\][$a$]{}]{} (17,-17)[(0,0)\[cc\][$t$]{}]{} (-15.5,-9)[(1,0)[1]{}]{} (-15.5,-6)[(1,0)[1]{}]{} (-15.5,0)[(1,0)[1]{}]{} (-15.5,3)[(1,0)[1]{}]{} (-15.5,9)[(1,0)[1]{}]{} (-15.5,12)[(1,0)[1]{}]{} (-17,-9)[(0,0)\[rc\][$a$]{}]{} (-17,-6)[(0,0)\[rc\][$a+b$]{}]{} (-17,0)[(0,0)\[rc\][$2a+b$]{}]{} (-15,-15)[(1,1)[5.6]{}]{} (-5.55,-5.55)[(1,1)[5.6]{}]{} (3.45,3.45)[(1,1)[5.6]{}]{} (12.45,12.45)[(1,1)[4]{}]{} (-9,-9) (-6,-9) (-6,-6) (0,0) (3,0) (3,3) (9,9) (12,9) (12,12) (12,3)[(0,0)\[cc\][$\rho(t)$]{}]{} (see Figure \[Fig1\]) and $$\mu(t)= \begin{cases} 0 &\text{if } t\in A_{1}, \\ b &\text{if } t\in A_{2}, \end{cases} \quad\quad\quad \nu(t)= \begin{cases} b &\text{if } t\in B_{1}, \\ 0 &\text{if } t\in B_{2}, \end{cases}$$ where $$\bigcup\limits_{k=0}^{\infty} [k(a+b),k(a+b)+a]=A_{1}\cup A_{2}=B_{1}\cup B_{2}$$ with $$A_{1}=\bigcup\limits_{k=0}^{\infty} [k(a+b),k(a+b)+a), \quad B_{1}=\bigcup\limits_{k=0}^{\infty} \left\{ k(a+b)\right\},$$ $$A_{2}=\bigcup\limits_{k=0}^{\infty} \left\{ k(a+b)+a\right\}, \quad B_{2}=\bigcup\limits_{k=0}^{\infty} (k(a+b),k(a+b)+a].$$ In the time-scale theory the following classification of points is used: - A point $t\in\mathbb{T}$ is called *right-scattered* or *left-scattered* if $\sigma(t)>t$ or $\rho(t)<t$, respectively. - A point $t$ is *isolated* if $\rho(t)<t<\sigma(t)$. - If $t<\sup{\mathbb{T}}$ and $\sigma(t)=t$, then $t$ is called *right-dense*; if $t>\inf {\mathbb{T}}$ and $\rho(t)=t$, then $t$ is called *left-dense*. - We say that $t$ is *dense* if $\rho(t)=t=\sigma(t)$. A time scale $\mathbb{T}$ is said to be an isolated time scale provided given any $t \in \mathbb{T}$, there is a $\delta > 0$ such that $(t - \delta, t+\delta) \cap \mathbb{T} = \{t\}$. \[def:regular\] A time scale $\mathbb{T}$ is said to be regular if the following two conditions are satisfied simultaneously for all $t\in\mathbb{T}$: $\sigma(\rho(t))=t$ and $\rho(\sigma(t))=t$. The delta derivative and the delta integral {#sec:prelim:delta} ------------------------------------------- If $f:{\mathbb{T}}\rightarrow {\mathbb{R}}$, then we define $f^{\sigma}:{\mathbb{T}}\rightarrow{\mathbb{R}}$ by $f^{{\sigma}}(t):=f({\sigma}(t))$ for all $t\in{\mathbb{T}}$. The delta derivative (or *Hilger derivative*) of function $f:{\mathbb{T}}{\rightarrow}{\mathbb{R}}$ is defined for points in the set ${\mathbb{T}}^{\kappa}$, where $${\mathbb{T}}^{\kappa} := \begin{cases} {\mathbb{T}}\setminus\left\{\sup{\mathbb{T}}\right\} & \text{ if } \rho(\sup{\mathbb{T}})<\sup{\mathbb{T}}<\infty,\\ {\mathbb{T}}& \text{ otherwise. } \end{cases}$$ Let us define the sets ${\mathbb{T}}^{\kappa^n}$, $n\geq 2$, inductively: ${\mathbb{T}}^{\kappa^1} :={\mathbb{T}}^\kappa$ and ${\mathbb{T}}^{\kappa^n} := \left({\mathbb{T}}^{\kappa^{n-1}}\right)^\kappa$, $n\geq 2$. We define delta differentiability in the following way. \[def:differ:delta\] Let $f:\mathbb{T}\rightarrow\mathbb{R}$ and $t\in\mathbb{T}^{\kappa}$. We define $f^{\Delta}(t)$ to be the number (provided it exists) with the property that given any $\varepsilon >0$, there is a neighborhood $U$ ($U=(t-\delta, t+\delta)\cap{\mathbb{T}}$ for some $\delta>0$) of $t$ such that $$\left|f^{\sigma}(t)-f(s)-f^{\Delta}(t)\left(\sigma(t)-s\right)\right| \leq \varepsilon \left|\sigma(t)-s\right| \mbox{ for all } s\in U.$$ A function $f$ is delta differentiable on $\mathbb{T}^{\kappa}$ provided $f^{\Delta}(t)$ exists for all $t\in\mathbb{T}^{\kappa}$. Then, $f^{\Delta}:\mathbb{T}^{\kappa}\rightarrow\mathbb{R}$ is called the delta derivative of $f$ on $\mathbb{T}^{\kappa}$. \[th:differ:delta\] Let $f:\mathbb{T} \rightarrow \mathbb{R}$ and $t\in\mathbb{T}^{\kappa}$. The following hold: 1. If $f$ is delta differentiable at $t$, then $f$ is continuous at $t$. 2. If $f$ is continuous at $t$ and $t$ is right-scattered, then $f$ is delta differentiable at $t$ with $$f^{\Delta}(t)=\frac{f^\sigma(t)-f(t)}{\mu(t)}.$$ 3. If $t$ is right-dense, then $f$ is delta differentiable at $t$ if and only if the limit $$\lim\limits_{s\rightarrow t}\frac{f(t)-f(s)}{t-s}$$ exists as a finite number. In this case, $$f^{\Delta}(t)=\lim\limits_{s\rightarrow t}\frac{f(t)-f(s)}{t-s}.$$ 4. If $f$ is delta differentiable at $t$, then $$f^\sigma(t)=f(t)+\mu(t)f^{\Delta}(t).$$ The next example is a consequence of Theorem \[th:differ:delta\] and presents different forms of the delta derivative on specific time scales. Let ${\mathbb{T}}$ be a time scale. 1. If $\mathbb{T}=\mathbb{R}$, then $f:\mathbb{R} \rightarrow \mathbb{R}$ is delta differentiable at $t\in\mathbb{R}$ if and only if $$f^\Delta(t)=\lim\limits_{s\rightarrow t}\frac{f(t)-f(s)}{t-s}$$ exists, i.e., if and only if $f$ is differentiable (in the ordinary sense) at $t$ and in this case we have $f^{\Delta}(t)=f'(t)$. 2. If $\mathbb{T}=h\mathbb{Z}$, $h > 0$, then $f:h\mathbb{Z} \rightarrow \mathbb{R}$ is delta differentiable at $t\in h\mathbb{Z}$ with $$f^{\Delta}(t)=\frac{f(\sigma(t))-f(t)}{\mu(t)}=\frac{f(t+h)-f(t)}{h}=:\Delta_{h}f(t).$$ In the particular case $h=1$ we have $f^{\Delta}(t)=\Delta f(t)$, where $\Delta$ is the usual forward difference operator. 3. If $\mathbb{T}=\overline{q^{\mathbb{Z}}}$, $q>1$, then for a delta differentiable function $f:\overline{q^{\mathbb{Z}}}{\rightarrow}{\mathbb{R}}$ we have $$f^{\Delta}(t)=\frac{f(\sigma(t))-f(t)}{\mu(t)}=\frac{f(qt)-f(t)}{(q-1)t}=:\Delta_{q}f(t)$$ for all $t\in\overline{q^{\mathbb{Z}}}\setminus \left\{ 0\right\}$, i.e., we get the usual Jackson derivative of quantum calculus [@QC; @MyID:266]. Now we formulate some basic properties of the delta derivative on time scales. \[th:differ:prop:delta\] Let $f,g:\mathbb{T} \rightarrow \mathbb{R}$ be delta differentiable at $t\in\mathbb{T^{\kappa}}$. Then, 1. the sum $f+g:\mathbb{T} \rightarrow \mathbb{R}$ is delta differentiable at $t$ with $$(f+g)^{\Delta}(t)=f^{\Delta}(t)+g^{\Delta}(t);$$ 2. for any constant $\alpha$, $\alpha f:\mathbb{T}\rightarrow\mathbb{R}$ is delta differentiable at $t$ with $$(\alpha f)^{\Delta}(t)=\alpha f^{\Delta}(t);$$ 3. the product $fg:\mathbb{T}\rightarrow\mathbb{R}$ is delta differentiable at $t$ with $$(fg)^{\Delta}(t)=f^{\Delta}(t)g(t)+f^{\sigma}(t)g^{\Delta}(t) =f(t)g^{\Delta}(t)+f^{\Delta}(t)g^{\sigma}(t);$$ 4. if $g(t)g^{\sigma}(t)\neq 0$, then $f/g$ is delta differentiable at $t$ with $$\left(\frac{f}{g}\right)^{\Delta}(t) =\frac{f^{\Delta}(t)g(t)-f(t)g^{\Delta}(t)}{g(t)g^{\sigma}(t)}.$$ Now we introduce the theory of delta integration on time scales. We start by defining the associated class of functions. A function $f:\mathbb{T}\rightarrow \mathbb{R}$ is called rd-continuous provided it is continuous at right-dense points in $\mathbb{T}$ and its left-sided limits exist (finite) at all left-dense points in $\mathbb{T}$. The set of all rd-continuous functions $f:\mathbb{T} \rightarrow \mathbb{R}$ is denoted by $C_{rd}=C_{rd}(\mathbb{T})=C_{rd}(\mathbb{T},\mathbb{R})$. The set of functions $f:\mathbb{T}\rightarrow \mathbb{R}$ that are delta differentiable and whose derivative is rd-continuous is denoted by $C^{1}_{rd}=C_{rd}^{1}(\mathbb{T})=C^{1}_{rd}(\mathbb{T},\mathbb{R})$. A function $F:\mathbb{T} \rightarrow \mathbb{R}$ is called a delta antiderivative of $f:\mathbb{T} \rightarrow \mathbb{R}$ provided $F^{\Delta}(t)=f(t)$ for all $t\in\mathbb{T}^{\kappa}$. Let $\mathbb{T}$ be a time scale and $a,b\in\mathbb{T}$. If $f:\mathbb{T}^{\kappa} \rightarrow \mathbb{R}$ is a rd-continuous function and $F:\mathbb{T} \rightarrow \mathbb{R}$ is an antiderivative of $f$, then the Cauchy delta integral is defined by $$\int\limits_{a}^{b} f(t)\Delta t := F(b)-F(a).$$ Every rd-continuous function $f$ has an antiderivative $F$. In particular, if $t_{0}\in\mathbb{T}$, then $F$ defined by $$F(t):=\int\limits_{t_{0}}^{t} f(\tau)\Delta \tau, \quad t\in\mathbb{T},$$ is an antiderivative of $f$. \[th:int:lim:sigma\] If $f\in C_{rd}$, then $\displaystyle \int\limits_{t}^{\sigma(t)}f(\tau)\Delta \tau=\mu(t)f(t)$, $t\in \mathbb{T}^{\kappa}$. Let us see two special cases of the delta integral. \[ex:int:R:hZ:qN\] Let $a,b\in\mathbb{T}$ and $f:\mathbb{T} \rightarrow \mathbb{R}$ be rd-continuous. 1. If $\mathbb{T}=\mathbb{R}$, then $$\int\limits_{a}^{b}f(t)\Delta t=\int\limits_{a}^{b}f(t)dt,$$ where the integral on the right hand side is the usual Riemann integral. 2. If $[a,b]$ consists of only isolated points, then $$\int\limits_{a}^{b} f(t)\Delta t= \begin{cases} \sum\limits_{t\in[a,b)}\mu(t)f(t), & \hbox{ if } a<b, \\ 0, & \hbox{ if } a=b,\\ -\sum\limits_{t\in[b,a)}\mu(t)f(t), & \hbox{ if } a>b. \end{cases}$$ Now we present some useful properties of the delta integral. \[th:int:prop:delta\] If $a,b,c\in\mathbb{T}$, $a < c < b$, $\alpha\in\mathbb{R}$, and $f,g \in C_{rd}(\mathbb{T}, \mathbb{R})$, then: 1. $\int\limits_{a}^{b}(f(t)+g(t))\Delta t =\int\limits_{a}^{b} f(t)\Delta t+\int\limits_{a}^{b}g(t)\Delta t$, 2. $\int\limits_{a}^{b}\alpha f(t)\Delta t=\alpha \int\limits_{a}^{b} f(t)\Delta t$, 3. $\int\limits_{a}^{b}f(t)\Delta t =-\int\limits_{b}^{a}f(t)\Delta t$, 4. $\int\limits_{a}^{b} f(t)\Delta t =\int\limits_{a}^{c} f(t)\Delta t +\int\limits_{c}^{b} f(t)\Delta t$, 5. $\int\limits_{a}^{a} f(t)\Delta t=0$, 6. $\int\limits_{a}^{b}f(t)g^{\Delta}(t)\Delta t =\left.f(t)g(t)\right|^{t=b}_{t=a} -\int\limits_{a}^{b} f^{\Delta}(t)g^\sigma(t)\Delta t$, 7. $\int\limits_{a}^{b} f^\sigma(t) g^{\Delta}(t)\Delta t =\left.f(t)g(t)\right|^{t=b}_{t=a} -\int\limits_{a}^{b}f^{\Delta}(t)g(t)\Delta t$. The nabla derivative and the nabla integral {#sec:prelim:nabla} ------------------------------------------- The nabla calculus is similar to the delta one of Section \[sec:prelim:delta\]. The difference is that the backward jump operator $\rho$ takes the role of the forward jump operator ${\sigma}$. For a function $f:{\mathbb{T}}{\rightarrow}{\mathbb{R}}$ we define $f^{\rho}:{\mathbb{T}}{\rightarrow}{\mathbb{R}}$ by $f^{\rho}(t):=f(\rho(t))$. If $\mathbb{T}$ has a right-scattered minimum $m$, then we define $\mathbb{T}_{\kappa}:=\mathbb{T}-\left\{ m\right\}$; otherwise, we set $\mathbb{T}_{\kappa}:=\mathbb{T}$: $${\mathbb{T}}_{\kappa} := \begin{cases} {\mathbb{T}}\setminus\left\{\inf{\mathbb{T}}\right\} & \text{ if } -\infty<\inf{\mathbb{T}}<{\sigma}(\inf{\mathbb{T}}),\\ {\mathbb{T}}& \text{ otherwise}. \end{cases}$$ Let us define the sets ${\mathbb{T}}_{\kappa}$, $n\geq 2$, inductively: $\mathbb{T}_{\kappa^1} := \mathbb{T}_\kappa$ and $\mathbb{T}_{\kappa^n} := (\mathbb{T}_{\kappa^{n-1}})_\kappa$, $n\geq 2$. Finally, we define $\mathbb{T}_{\kappa}^{\kappa} := \mathbb{T}_{\kappa} \cap \mathbb{T}^{\kappa}$. The definition of nabla derivative of a function $f:{\mathbb{T}}{\rightarrow}{\mathbb{R}}$ at point $t\in{\mathbb{T}}_{\kappa}$ is similar to the delta case (cf. Definition \[def:differ:delta\]). We say that a function $f:\mathbb{T} \rightarrow \mathbb{R}$ is nabla differentiable at $t\in\mathbb{T}_{\kappa}$ if there is a number $f^{\nabla}(t)$ such that for all $\varepsilon >0$ there exists a neighborhood $U$ of $t$ (i.e., $U=(t-\delta, t+\delta)\cap\mathbb{T}$ for some $\delta>0$) such that $$|f^{\rho}(t)-f(s)-f^{\nabla}(t)(\rho(t)-s)| \leq \varepsilon |\rho(t)-s| \, \mbox{ for all } s\in U.$$ We say that $f^{\nabla}(t)$ is the nabla derivative of $f$ at $t$. Moreover, $f$ is said to be nabla differentiable on $\mathbb{T}$ provided $f^{\nabla}(t)$ exists for all $t\in\mathbb{T}_{\kappa}$. The main properties of the nabla derivative are similar to those given in Theorems \[th:differ:delta\] and \[th:differ:prop:delta\], and can be found, respectively, in Theorems 8.39 and 8.41 of [@BohnerDEOTS]. If ${\mathbb{T}}={\mathbb{R}}$, then $f^{{\nabla}}(t)=f'(t)$. If ${\mathbb{T}}=h{\mathbb{Z}}$, $h>0$, then $$f^{{\nabla}}(t)=\frac{f(t)-f(t-h)}{h} =: {\nabla}_{h} f(t).$$ For $h=1$ the operator ${\nabla}_{h}$ reduces to the standard backward difference operator $\nabla f(t) = f(t) - f(t-1)$. We now briefly recall the theory of nabla integration on time scales. Similarly as in the delta case, first we define a suitable class of functions. Let $\mathbb{T}$ be a time scale and $f:\mathbb{T}\rightarrow \mathbb{R}$. We say that $f$ is ld-continuous if it is continuous at left-dense points $t\in{\mathbb{T}}$ and its right-sided limits exist (finite) at all right-dense points. If ${\mathbb{T}}={\mathbb{R}}$, then $f$ is ld-continuous if and only if $f$ is continuous. If ${\mathbb{T}}={\mathbb{Z}}$, then any function is ld-continuous. The set of all ld-continuous functions $f:\mathbb{T}\rightarrow \mathbb{R}$ is denoted by $C_{ld}=C_{ld}(\mathbb{T})=C_{ld}(\mathbb{T},\mathbb{R})$; the set of all nabla differentiable functions with ld-continuous derivative by $C^{1}_{ld}=C^{1}_{ld}(\mathbb{T})=C^{1}_{ld}(\mathbb{T},\mathbb{R})$. Follows the definition of nabla integral on time scales. A function $F : \mathbb{T} \rightarrow \mathbb{R}$ is called a nabla antiderivative of $f : \mathbb{T} \rightarrow \mathbb{R}$ provided $F^\nabla(t) = f(t)$ for all $t \in \mathbb{T}_\kappa$. In this case we define the nabla integral of $f$ from $a$ to $b$ ($a, b \in \mathbb{T}$) by $$\int_a^b f(t) \nabla t := F(b) - F(a).$$ Every ld-continuous function $f$ has a nabla antiderivative $F$. In particular, if $a \in \mathbb{T}$, then $F$ defined by $$F(t) = \int_a^t f(\tau) \nabla \tau, \quad t \in \mathbb{T},$$ is a nabla antiderivative of $f$. \[th:int:lim:rho\] If $f:{\mathbb{T}}\rightarrow{\mathbb{R}}$ is ld-continuous and $t\in{\mathbb{T}}_{\kappa}$, then $$\int_{\rho(t)}^{t}f(\tau)\nabla \tau=\nu(t)f(t).$$ Properties of the nabla integral, analogous to the ones of the delta integral given in Theorem \[th:int:prop:delta\], can be found in Theorem 8.47 of [@BohnerDEOTS]. Here we give two special cases of the nabla integral. Assume $a,b\in{\mathbb{T}}$ and $f:{\mathbb{T}}{\rightarrow}{\mathbb{R}}$ is ld-continuous. 1. If ${\mathbb{T}}={\mathbb{R}}$, then $$\int\limits_{a}^{b}f(t){\nabla}t=\int\limits_{a}^{b} f(t)dt,$$ where the integral on the right hand side is the Riemann integral. 2. If ${\mathbb{T}}$ consists of only isolated points, then $$\int\limits_{a}^{b}f(t){\nabla}t = \begin{cases} \sum\limits_{t\in (a,b]}\nu(t)f(t), & \hbox{ if } a<b, \\ 0, & \hbox{ if } a=b,\\ -\sum\limits_{t\in (b,a]}\nu(t)f(t), & \hbox{ if } a>b. \end{cases}$$ Relation between delta and nabla operators {#subsec:relation:d:n} ------------------------------------------ It is possible to relate the approach of Section \[sec:prelim:delta\] with that of Section \[sec:prelim:nabla\]. \[th:differ:delta:nabla\] If $f:\mathbb{T}\rightarrow\mathbb{R}$ is delta differentiable on $\mathbb{T}^{\kappa}$ and if $f^{\Delta}$ is continuous on $\mathbb{T}^{\kappa}$, then $f$ is nabla differentiable on $\mathbb{T}_{\kappa}$ with $$f^{\nabla}(t)=\left(f^{\Delta}\right)^{\rho}(t) \, \textrm{ for all } t\in\mathbb{T}_{\kappa}.$$ If $f:\mathbb{T}\rightarrow\mathbb{R}$ is nabla differentiable on $\mathbb{T}_{\kappa}$ and if $f^{\nabla}$ is continuous on $\mathbb{T}_{\kappa}$, then $f$ is delta differentiable on $\mathbb{T}^{\kappa}$ with $$\label{eq:delta:nabla:sigma} f^{\Delta}(t)=\left(f^{\nabla}\right)^{\sigma}(t) \, \textrm{ for all } t\in\mathbb{T}^{\kappa}.$$ \[th:int:delta:nabla\] If function $f:\mathbb{T}\rightarrow\mathbb{R}$ is continuous, then for all $a,b\in\mathbb{T}$ with $a<b$ we have $$\begin{gathered} \int\limits_{a}^{b}f(t)\Delta t =\int\limits_{a}^{b}f^{\rho}(t)\nabla t,\\ \int\limits_{a}^{b}f(t)\nabla t =\int\limits_{a}^{b}f^{\sigma}(t)\Delta t.\end{gathered}$$ For a more general theory relating delta and nabla approaches, we refer the reader to the duality theory of Caputo [@cc:dual]. Direct problems of the calculus of variations on time scales {#CoV} ============================================================ There are two available approaches to the (direct) calculus of variations on time scales. The first one, the delta approach, is widely described in literature (see, e.g., [@BohnerCOVOTS; @mb:gg:ap; @BohnerDEOTS; @MBbook2003; @FerreiraTorres; @MR2405376; @HilscgerZeidan; @TorresDeltaNabla; @Generalizing_the_variational_theory; @Torres; @Wyrwas]). The latter one, the nabla approach, was introduced mainly due to its applications in economics (see, e.g., [@MR2218315; @AticiGreen's_functions; @Atici:comparison; @Atici:HMMS]). It has been shown that these two types of calculus of variations are dual [@cc:dual; @MR2957726; @MalinowskaTorresCompositionNabla]. The delta approach to the calculus of variations {#sec:calculus:of:variations} ------------------------------------------------ In this section we present the basic information about the delta calculus of variations on time scales. Let $\mathbb{T}$ be a given time scale with at least three points, and $a,b\in{\mathbb{T}}$, $a<b$, $a=\min{\mathbb{T}}$ and $b=\max{\mathbb{T}}$. Consider the following variational problem on the time scale $\mathbb{T}$: $$\label{eq:var:probl:delta:sigma} \mathcal{L}[y]=\int\limits_{a}^{b} L\left(t, y^{\sigma}(t),y^{\Delta}(t)\right)\Delta t \longrightarrow \min$$ subject to the boundary conditions $$\label{eq:bound:conds:1} y(a)=y_{a}, \quad y(b)=y_{b}, \quad y_{a},y_{b}\in{\mathbb{R}}^{n}, \quad n\in{\mathbb{N}}.$$ A function $y\in C_{rd}^{1}(\mathbb{T},{\mathbb{R}}^{n})$ is said to be an admissible path (function) to problem – if it satisfies the given boundary conditions $y(a)=y_{a}$, $y(b)=y_{b}$. In what follows the Lagrangian $L$ is understood as a function $L:{\mathbb{T}}\times{\mathbb{R}}^{2n}{\rightarrow}{\mathbb{R}}$, $(t,y,v) \rightarrow L(t,y,v)$, and by $L_y$ and $L_v$ we denote the partial derivatives of $L$ with respect to $y$ and $v$, respectively. Similar notation is used for second order partial derivatives. We assume that $L(t,\cdot,\cdot)$ is differentiable in $(y,v)$; $L(t,\cdot,\cdot)$, $L_{y}(t,\cdot,\cdot)$ and $L_{v}(t,\cdot,\cdot)$ are continuous at $\left(y^{{\sigma}}(t),y^{\Delta}(t)\right)$ uniformly at $t$ and rd-continuous at $t$ for any admissible path $y$. Let us consider the following norm in $C_{rd}^{1}$: $$\|y\|_{C^{1}_{rd}} = \sup_{t\in[a,b]}\|y(t)\| +\sup_{t\in[a,b]^{\kappa}}\|y^{\triangle}(t)\|,$$ where $\|\cdot\|$ is the Euclidean norm in $\mathbb{R}^n$. We say that an admissible function $\hat{y}\in C^{1}_{rd}({\mathbb{T}};{\mathbb{R}}^{n})$ is a local minimizer (respectively, a local maximizer) to problem – if there exists $\delta >0$ such that $\mathcal{L}[\hat{y}] \le\mathcal{L}[y]$ (respectively, $\mathcal{L}[\hat{y}]\geq\mathcal{L}[y]$) for all admissible functions $y\in C^{1}_{rd}({\mathbb{T}}; {\mathbb{R}}^{n})$ satisfying the inequality $||y-\hat{y}||_{C_{rd}^{1}}<\delta$. Local minimizers (or maximizers) to problem – fulfill the delta differential Euler–Lagrange equation. \[th:EL:1\] If $\hat{y}\in C^{2}_{rd}({\mathbb{T}}; {\mathbb{R}}^{n})$ is a local minimizer to –, then the Euler–Lagrange equation (in the delta differential form) $$L^{\Delta}_{v}\left(t,\hat{y}^{{\sigma}}(t),\hat{y}^{\Delta}(t)\right) =L_{y}\left(t,\hat{y}^{{\sigma}}(t),\hat{y}^{\Delta}(t)\right)$$ holds for $t\in [a,b]^{\kappa}$. The next theorem provides the delta integral Euler–Lagrange equation. \[th:EL:delta:sigma\] If $\hat{y}(t)\in C_{rd}^{1}({\mathbb{T}}; {\mathbb{R}}^{n})$ is a local minimizer of the variational problem –, then there exists a vector $c\in\mathbb{R}^{n}$ such that the Euler–Lagrange equation (in the delta integral form) $$\label{eq:EL:delta:sigma} L_{v}\left(t, \hat{y}^{\sigma}(t),\hat{y}^{\Delta}(t)\right) =\int\limits_{a}^{t}L_{y}(\tau, \hat{y}^{\sigma}(\tau),\hat{y}^{\Delta}(\tau))\Delta\tau +c^{T}$$ holds for $t\in[a,b]^{\kappa}$. In the proof of Theorem \[th:EL:1\] and Theorem \[th:EL:delta:sigma\] a time scale version of the Dubois–Reymond lemma is used. \[lem:Dubois:Reymond:delta\] Let $f\in C_{rd}$, $f:[a,b]{\rightarrow}{\mathbb{R}}^{n}$. Then $$\int\limits_{a}^{b}f^{T}(t)\eta^{\Delta}(t)\Delta t=0$$ holds for all $\eta \in C^{1}_{rd}([a,b],\mathbb{R}^{n})$ with $\eta(a)=\eta(b)=0$ if and only if $f(t) = c$ for all $t\in [a,b]^{\kappa}$, $c\in{\mathbb{R}}^{n}$. The next theorem gives a second order necessary optimality condition for problem –. \[th:Legendre\] If $\hat{y}\in C^{2}_{rd}({\mathbb{T}}; {\mathbb{R}}^{n})$ is a local minimizer of the variational problem –, then $$\label{eq:Legendre} A(t)+\mu(t)\left\lbrace C(t)+C^{T}(t)+\mu(t)B(t) +(\mu(\sigma(t)))^{\dag}A(\sigma(t))\right\rbrace\geq 0,$$ $t\in[a,b]^{\kappa^{2}}$, where $$\begin{split} & A(t)=L_{vv}\left(t,\hat{y}^{\sigma}(t),\hat{y}^{\Delta}(t)\right),\\ & B(t)=L_{yy}\left(t,\hat{y}^{\sigma}(t),\hat{y}^{\Delta}(t)\right),\\ & C(t)=L_{yv}\left(t,\hat{y}^{\sigma}(t),\hat{y}^{\Delta}(t)\right) \end{split}$$ and where $\alpha^{\dag}=\frac{1}{\alpha}$ if $\alpha\in\mathbb{R}\setminus\lbrace 0 \rbrace$ and $0^{\dag}=0$. If holds with the strict inequality “$>$”, then it is called *the strengthened Legendre condition*. The nabla approach to the calculus of variations {#subsec:nabla:CoV} ------------------------------------------------ In this section we consider a problem of the calculus of variations that involves a functional with a nabla derivative and a nabla integral. The motivation to study such variational problems is coming from applications, in particular from economics [@MR2218315; @Atici:HMMS]. Let ${\mathbb{T}}$ be a given time scale, which has sufficiently many points in order for all calculations to make sense, and let $a,b\in{\mathbb{T}}$, $a<b$. The problem consists of minimizing or maximizing $$\label{eq:var:probl:nabla} \mathcal{L}[y]=\int\limits_{a}^{b} L\left(t, y^{\rho}(t), y^{{\nabla}}(t)\right){\nabla}t$$ in the class of functions $y\in C^{1}_{ld}({\mathbb{T}};{\mathbb{R}}^{n})$ subject to the boundary conditions $$\label{eq:bound:conds:nabla} y(a)=y_{a}, \quad y(b)=y_{b}, \quad y_{a}, y_{b}\in{\mathbb{R}}^{n}, \quad n\in{\mathbb{N}}.$$ A function $y\in C_{ld}^{1}(\mathbb{T},{\mathbb{R}}^{n})$ is said to be an admissible path (function) to problem – if it satisfies the given boundary conditions $y(a)=y_{a}$, $y(b)=y_{b}$. As before, the Lagrangian $L$ is understood as a function $L:{\mathbb{T}}\times{\mathbb{R}}^{2n}{\rightarrow}{\mathbb{R}}$, $(t,y,v) \rightarrow L(t,y,v)$. We assume that $L(t,\cdot,\cdot)$ is differentiable in $(y,v)$; $L(t,\cdot,\cdot)$, $L_{y}(t,\cdot,\cdot)$ and $L_{v}(t,\cdot,\cdot)$ are continuous at $\left(y^{\rho}(t),y^{\nabla}(t)\right)$ uniformly at $t$ and ld-continuous at $t$ for any admissible path $y$. Let us consider the following norm in $C_{ld}^{1}$: $$\|y\|_{C^{1}_{ld}} = \sup_{t\in[a,b]}\|y(t)\| +\sup_{t\in[a,b]_{\kappa}}\|y^{\nabla}(t)\|$$ with $\|\cdot\|$ the Euclidean norm in $\mathbb{R}^n$. We say that an admissible function $y\in C^{1}_{ld}({\mathbb{T}};{\mathbb{R}}^{n})$ is a local minimizer (respectively, a local maximizer) for the variational problem – if there exists $\delta >0$ such that $\mathcal{L}[\hat{y}]\leq\mathcal{L} [y]$ (respectively, $\mathcal{L}[\hat{y}]\geq\mathcal{L} [y]$) for all $y\in C^{1}_{ld}({\mathbb{T}}; {\mathbb{R}}^{n})$ satisfying the inequality $||y-\hat{y}||_{C^{1}_{ld}} <\delta$. In case of the first order necessary optimality condition for nabla variational problem on time scales –, the Euler–Lagrange equation takes the following form. \[th:var:probl:nabla\] If a function $\hat{y}\in C_{ld}^{1}({\mathbb{T}};{\mathbb{R}}^{n})$ provides a local extremum to the variational problem –, then $\hat{y}$ satisfies the Euler–Lagrange equation (in the nabla differential form) $$L^{\nabla}_{v}\left(t, y^{\rho}(t), y^{{\nabla}}(t)\right) =L_{y}\left(t, y^{\rho}(t), y^{{\nabla}}(t)\right)$$ for all $t\in [a,b]_{\kappa}$. Now we present the fundamental lemma of the nabla calculus of variations on time scales. \[lem:Dubois:Reymond:nabla\] Let $f\in C_{ld}([a,b],\mathbb{R}^{n})$. If $$\int\limits_{a}^{b}f(t)\eta^{\nabla}(t)\nabla t=0$$ for all $\eta \in C^{1}_{ld}([a,b],\mathbb{R}^{n})$ with $\eta(a)=\eta(b)=0$, then $f(t)=c$ for all $t\in [a,b]_{\kappa}$, $c\in{\mathbb{R}}^{n}$. For a good survey on the direct calculus of variations on time scales, covering both delta and nabla approaches, we refer the reader to [@Torres]. Inverse problems of the calculus of variations on time scales {#chapter_5_P3+P4} ============================================================= This section is devoted to inverse problems of the calculus of variations on an arbitrary time scale. To our best knowledge, the inverse problem has not been studied before 2014 [@PhD:thesis:Monia; @Dryl:Torres:1; @MyID:291] in the framework of time scales, in contrast with the direct problem, that establishes dynamic equations of Euler–Lagrange type to time-scale variational problems, that has now been investigated for ten years, since 2004 [@BohnerCOVOTS]. To begin (Section \[P3\]) we consider an inverse extremal problem associated with the following fundamental problem of the calculus of variations: to minimize $$\label{P3:eq:funct:1} \mathcal{L}[y]=\int\limits_{a}^{b} L\left(t,y^{\sigma}(t),y^{\Delta}(t)\right)\Delta t$$ subject to boundary conditions $y(a)=y_{0}(a)$, $y(b)=y_{0}(b)$ on a given time scale $\mathbb{T}$. The Euler–Lagrange equation and the strengthened Legendre condition are used in order to describe a general form of a variational functional that attains an extremum at a given function $y_0$. In the latter Section \[P4\], we introduce a completely different approach to the inverse problem of the calculus of variations, using an integral perspective instead of the classical differential point of view [@Helmholtz; @Davis]. We present a sufficient condition of self-adjointness for an integro-differential equation (Lemma \[P4:lem:suff:self:adj\]). Using this property, we prove a necessary condition for an integro-differential equation on an arbitrary time scale $\mathbb{T}$ to be an Euler–Lagrange equation (Theorem \[P4:th:necess:EL:int:diff\]), related to a property of self-adjointness (Definition \[P4:def:self:adj:int:diff\]) of its equation of variation (Definition \[P4:def:eq:var\]). A general form of the Lagrangian {#P3} -------------------------------- The problem under our consideration is to find a general form of the variational functional $$\label{P3:eq:var:probl:delta:sigma} \mathcal{L}[y]=\int\limits_{a}^{b} L\left(t,y^{\sigma}(t),y^{\Delta}(t)\right)\Delta t$$ subject to boundary conditions $y(a)=y(b)=0$, possessing a local minimum at zero, under the Euler–Lagrange and the strengthened Legendre conditions. We assume that $L(t,\cdot,\cdot)$ is a $C^{2}$-function with respect to $(y,v)$ uniformly in $t$, and $L$, $L_{y}$, $L_{v}$, $L_{vv}\in C_{rd}$ for any admissible path $y(\cdot)$. Observe that under our assumptions, by Taylor’s theorem, we may write $L$, with the big $O$ notation, in the form $$\label{P3:eq:pre:integrand} L(t,y,v)=P(t, y) +Q(t, y) v +\frac{1}{2} R(t, y,0)v^{2} + O(v^3),$$ where $$\label{P3:eq:notation:P:Q:R} \begin{split} P(t, y) &= L(t, y,0),\\ Q(t, y) &= L_{v}(t, y,0),\\ R(t, y,0) &= L_{vv}(t, y,0). \end{split}$$ Let $R(t, y, v) = R(t, y,0) + O(v)$. Then, one can write as $$L(t, y,v)=P(t, y) +Q(t, y) v +\frac{1}{2} R(t, y, v) v^{2}.$$ Now the idea is to find general forms of $P(t, y^{\sigma}(t))$, $Q(t, y^{\sigma}(t))$ and $R(t, y^{\sigma}(t), y^{\Delta}(t))$ using the Euler–Lagrange and the strengthened Legendre conditions with notation . Then we use the Euler–Lagrange equation and choose an arbitrary function $P(t,y^{\sigma}(t))$ such that $P(t,\cdot)\in C^{2}$ with respect to the second variable, uniformly in $t$, $P$ and $P_y$ rd-continuous in $t$ for all admissible $y$. We can write the general form of $Q$ as $$Q(t,y^{\sigma}(t))=C+\int\limits_{a}^{t}P_{y}(\tau,0)\Delta \tau +q(t,y^{\sigma}(t))-q(t,0),$$ where $C\in\mathbb{R}$ and $q$ is an arbitrarily function such that $q(t,\cdot)\in C^{2}$ with respect to the second variable, uniformly in $t$, $q$ and $q_y$ are rd-continuous in $t$ for all admissible $y$. From the strengthened Legendre condition , with notation , we set $$\label{P3:eq:1} R(t,0,0)+\mu(t)\left\lbrace 2Q_{y}(t,0)+\mu(t)P_{yy}(t,0) +\left(\mu^{\sigma}(t)\right)^{\dag}R(\sigma(t),0,0)\right\rbrace = p(t)$$ with $p\in C_{rd}([a,b])$, $p(t)>0$ for all $t\in [a,b]^{\kappa^{2}}$, chosen arbitrary, where $\alpha^{\dag}=\frac{1}{\alpha}$ if $\alpha\in\mathbb{R}\setminus\lbrace 0 \rbrace$ and $0^{\dag}=0$. We obtain the following theorem, which presents a general form of the integrand $L$ for functional . \[P3:th:integrand:1\] Let $\mathbb{T}$ be an arbitrary time scale. If functional with boundary conditions $y(a)=y(b)=0$ attains a local minimum at $\hat{y}(t)\equiv 0$ under the strengthened Legendre condition, then its Lagrangian $L$ takes the form $$\begin{split} &L\left(t,y^{\sigma}(t),y^{\Delta}(t)\right) = P\left(t,y^{\sigma}(t)\right)\\ &+\left(C+\int\limits_{a}^{t}P_{y}(\tau,0)\Delta \tau +q(t,y^{\sigma}(t))-q(t,0)\right)y^{\Delta}(t)\\ &+\Biggl(p(t)-\mu(t)\left\lbrace 2Q_{y}(t,0)+\mu(t) P_{yy}(t,0) +\left(\mu^{\sigma}(t)\right)^{\dag}R(\sigma(t),0,0)\right\rbrace\\ &\qquad +w(t,y^{\sigma}(t),y^{\Delta}(t)) -w(t,0,0)\Biggr)\frac{(y^{\Delta}(t))^{2}}{2}, \end{split}$$ where $R(t,0,0)$ is a solution of equation , $C\in\mathbb{R}$, $\alpha^{\dag}=\frac{1}{\alpha}$ if $\alpha\in\mathbb{R}\setminus\lbrace 0 \rbrace$ and $0^{\dag}=0$. Functions $P$, $p$, $q$ and $w$ are arbitrary functions satisfying: - $P(t,\cdot),q(t,\cdot)\in C^{2}$ with respect to the second variable uniformly in $t$; $P$, $P_y$, $q$, $q_y$ are rd-continuous in $t$ for all admissible $y$; $P_{yy}(\cdot,0)$ is rd-continuous in $t$; $p\in C^{1}_{rd}$ with $p(t)>0$ for all $t\in [a,b]^{\kappa^{2}}$; - $w(t,\cdot,\cdot)\in C^2$ with respect to the second and the third variable, uniformly in $t$; $w, w_y, w_v, w_{vv}$ are rd-continuous in $t$ for all admissible $y$. See [@Dryl:Torres:1]. Now we consider the general situation when the variational problem consists in minimizing subject to arbitrary boundary conditions $y(a)=y_{0}(a)$ and $y(b)=y_{0}(b)$, for a certain given function $y_{0}\in C_{rd}^{2}([a,b])$. \[P3:th:integrand:2\] Let $\mathbb{T}$ be an arbitrary time scale. If the variational functional with boundary conditions $y(a)=y_{0}(a)$, $y(b)=y_{0}(b)$, attains a local minimum for a certain given function $y_{0}(\cdot)\in C^{2}_{rd}([a,b])$ under the strengthened Legendre condition, then its Lagrangian $L$ has the form $$\begin{split} &L\left(t,y^{\sigma}(t),y^{\Delta}(t)\right) = P\left(t,y^{\sigma}(t)-y^{\sigma}_{0}(t)\right) + \left(y^{\Delta}(t)-y_{0}^{\Delta}(t)\right)\\ &\times \left(C+\int\limits_{a}^{t}P_{y}\left(\tau,-y_{0}^{\sigma}(\tau)\right)\Delta \tau +q\left(t,y^{\sigma}(t)-y^{\sigma}_{0}(t)\right) -q\left(t,-y_{0}^{\sigma}(t)\right)\right) +\frac{1}{2}\Biggl(p(t)\\ & -\mu(t) \left\lbrace 2Q_{y}(t,-y_{0}^{\sigma}(t))+\mu(t) P_{yy}(t,-y_{0}^{\sigma}(t)) +\left(\mu^{\sigma}(t)\right)^{\dag}R(\sigma(t),-y_{0}^{\sigma}(t),-y_{0}^{\Delta}(t))\right\rbrace\\ &+w(t,y^{\sigma}(t)-y^{\sigma}_{0}(t),y^{\Delta}(t)-y_{0}^{\Delta}(t)) -w\left(t,-y_{0}^{\sigma}(t),-y_{0}^{\Delta}(t)\right)\Biggr) \left(y^{\Delta}(t)-y_{0}^{\Delta}(t)\right)^{2}, \end{split}$$ where $C\in\mathbb{R}$ and functions $P$, $p$, $q$, $w$ satisfy conditions (i) and (ii) of Theorem \[P3:th:integrand:1\]. See [@Dryl:Torres:1]. For the classical situation $\mathbb{T}=\mathbb{R}$, Theorem \[P3:th:integrand:2\] gives a recent result of [@orlov; @MR2907362]. \[P3:cor:R\] If the variational functional $$\mathcal{L}[y]=\int\limits_{a}^{b} L(t,y(t),y'(t))dt$$ attains a local minimum at $y_{0}(\cdot)\in C^{2}[a,b]$ when subject to boundary conditions $y(a)=y_{0}(a)$ and $y(b)=y_{0}(b)$ and the classical strengthened Legendre condition $$R(t,y_{0}(t),y'_{0}(t))>0, \quad t\in[a,b],$$ then its Lagrangian $L$ has the form $$\begin{split} &L(t,y(t),y'(t))=P(t,y(t)-y_{0}(t))\\ & +(y'(t)-y'_{0}(t))\left(C+\int\limits_{a}^{t} P_{y}(\tau,-y_{0}(\tau))d\tau+q(t,y(t)-y_{0}(t))-q(t,-y_{0}(t))\right)\\ & +\frac{1}{2}\left(p(t)+w(t,y(t)-y_{0}(t),y'(t)-y'_{0}(t)) -w(t,-y_{0}(t),-y'_{0}(t))\right)(y'(t)-y'_{0}(t))^{2}, \end{split}$$ where $C\in\mathbb{R}$. In the particular case of an isolated time scale, where $\mu(t) \neq 0$ for all $t\in\mathbb{T}$, we get the following corollary. \[P3:cor:isolated\] Let $\mathbb{T}$ be an isolated time scale. If functional subject to the boundary conditions $y(a)=y(b)=0$ attains a local minimum at $\hat{y}(t) \equiv 0$ under the strengthened Legendre condition, then the Lagrangian $L$ has the form $$\begin{split} &L\left(t,y^{\sigma}(t),y^{\Delta}(t)\right) = P\left(t,y^{\sigma}(t)\right)\\ &+\left(C+\int\limits_{a}^{t}P_{y}(\tau,0)\Delta \tau +q(t,y^{\sigma}(t))-q(t,0)\right)y^{\Delta}(t)\\ &+\left(e_{r}(t,a)R_{0} +\int\limits_{a}^{t}e_{r}(t,\sigma(\tau))s(\tau)\Delta\tau +w(t,y^{\sigma}(t),y^{\Delta}(t)) -w(t,0,0)\right)\frac{(y^{\Delta}(t))^{2}}{2}, \end{split}$$ where $C,R_{0}\in\mathbb{R}$ and $r(t)$ and $s(t)$ are given by $$\label{P3:eq:funct:r:s} r(t) := -\frac{1+\mu(t)(\mu^{\sigma}(t))^{\dag}}{\mu^{2}(t)(\mu^{\sigma}(t))^{\dag}}, \quad s(t) := \frac{p(t) -\mu(t)[2Q_{y}(t,0)+\mu(t)P_{yy}(t,0)]}{\mu^{2}(t)(\mu^{\sigma}(t))^{\dag}}$$ with $\alpha^{\dag} = \frac{1}{\alpha}$ if $\alpha \in\mathbb{R}\setminus\lbrace 0 \rbrace$ and $0^{\dag}=0$, and functions $P$, $p$, $q$, $w$ satisfy assumptions of Theorem \[P3:th:integrand:1\]. Based on Corollary \[P3:cor:isolated\], we present the form of Lagrangian $L$ in the periodic time scale $\mathbb{T}=h\mathbb{Z}$. \[P3:ex:hZ\] Let $\mathbb{T}=h\mathbb{Z}$, $h > 0$, and $a, b\in h\mathbb{Z}$ with $a<b$. Then $\mu(t) \equiv h$. Consider the variational functional $$\label{P3:eq:funct:hZ} \mathcal{L}[y]=h\sum_{k=\frac{a}{h}}^{\frac{b}{h}-1} L\left(kh,y(kh+h),\Delta_h y(kh)\right)$$ subject to the boundary conditions $y(a)=y(b)=0$, which attains a local minimum at $\hat{y}(kh)\equiv 0$ under the strengthened Legendre condition $$R(kh,0,0)+2hQ_{y}(kh,0)+h^{2}P_{yy}(kh,0)+R(kh+h,0,0)>0,$$ $kh\in [a,b-2h]\cap h\mathbb{Z}$. Functions $r(t)$ and $s(t)$ (see ) have the following form: $$r(t) \equiv -\frac{2}{h}, \quad s(t)=\frac{p(t)}{h} -\left(2Q_{y}(t,0) + h P_{yy}(t,0)\right).$$ Hence, $$\int\limits_{a}^{t}P_{y}(\tau,0)\Delta \tau =h\sum\limits_{i=\frac{a}{h}}^{\frac{t}{h}-1}P_{y}(ih,0),$$ $$\int\limits_{a}^{t}e_{r}(t,\sigma(\tau))s(\tau)\Delta \tau =\sum_{i=\frac{a}{h}}^{\frac{t}{h}-1}(-1)^{\frac{t}{h}-i-1} \left(p(ih)-2hQ_{y}(ih,0)-h^{2}P_{yy}(ih,0)\right).$$ Thus, the Lagrangian $L$ of the variational functional on $\mathbb{T}=h\mathbb{Z}$ has the form $$\begin{split} L&\left(kh,y(kh+h),\Delta_h y(kh)\right)=P\left(kh,y(kh+h)\right)\\ &+\left(C+\sum\limits_{i=\frac{a}{h}}^{k-1}hP_{y}(ih,0) +q(kh,y(kh+h))-q(kh,0)\right)\Delta_h y(kh)\\ &+\frac{1}{2}\Biggl((-1)^{k-\frac{a}{h}}R_{0} +\sum_{i=\frac{a}{h}}^{k-1}(-1)^{k-i-1} \left(p(i h)-2hQ_{y}(ih,0)-h^{2}P_{yy}(ih,0)\right)\\ &\qquad +w(kh,y(kh+h),\Delta_h y(kh))-w(kh,0,0)\Biggr) \left(\Delta_h y(kh)\right)^{2}, \end{split}$$ where functions $P$, $p$, $q$, $w$ are arbitrary but satisfy assumptions of Theorem \[P3:th:integrand:1\]. Necessary condition for an Euler–Lagrange equation {#P4} -------------------------------------------------- This section provides a necessary condition for an integro-differential equation on an arbitrary time scale to be an Euler–Lagrange equation (Theorem \[P4:th:necess:EL:int:diff\]). For that the notions of self-adjointness (Definition \[P4:def:self:adj:int:diff\]) and equation of variation (Definition \[P4:def:eq:var\]) are essential. \[P4:def:self:adj:int:diff\] A first order integro-differential dynamic equation is said to be *self-adjoint* if it has the form $$\label{P4:eq:self:adj:int:diff} Lu(t)=const, \hbox{ where } Lu(t)=p(t)u^{\Delta}(t)+\int\limits_{t_{0}}^{t}\left[r(s)u^{\sigma}(s)\right]\Delta s$$ with $p,r\in C_{rd}$, $p\neq 0$ for all $t\in\mathbb{T}$ and $t_{0}\in{\mathbb{T}}$. Let $\mathbb{D}$ be the set of all functions $y:\mathbb{T}\rightarrow \mathbb{R}$ such that $y^{\Delta}:\mathbb{T}^{\kappa}\rightarrow\mathbb{R}$ is continuous. A function $y\in\mathbb{D}$ is said to be a solution of provided $Ly(t)=const$ holds for all $t\in\mathbb{T^{\kappa}}$. For simplicity, we use the operators $[\cdot]$ and $\langle \cdot\rangle$ defined as $$\label{eq:2convenientoper:++} [y](t):=(t, y^{{\sigma}}(t), y^{\Delta}(t)), \quad\quad \langle y\rangle (t):=(t, y^{{\sigma}}(t), y^{\Delta}(t), y^{\Delta\Delta}(t)),$$ and partial derivatives of function $(t,y,v,z){\rightarrow}L(t,y,v,z)$ are denoted by $\partial_{2}L=L_{y}$, $\partial_{3}L=L_{v}$, $\partial_{4}L=L_{z}$. \[P4:def:eq:var\] Let $$\label{P4:eq:int:diff:1} H[y](t)+\int\limits_{t_{0}}^{t}G[y](s)\Delta s = const$$ be an integro-differential equation on time scales with $H_{v}\neq 0$, $t{\rightarrow}F_{y}[y](t)$, $t{\rightarrow}F_{v}[y](t) \in C_{rd}({\mathbb{T}},{\mathbb{R}})$ along every curve $y$, where $F\in{\lbrace}G,H{\rbrace}$. The *equation of variation* associated with is given by $$\label{P4:eq:eq:var} H_{y}[u](t)u^{\sigma}(t)+H_{v}[u](t) u^{\Delta}(t) +\int\limits_{t_{0}}^{t}G_{y}[u](s)u^{\sigma}(s)+G_{v}[u](s) u^{\Delta}(s)\Delta s=0.$$ \[P4:lem:suff:self:adj\] Let be a given integro-differential equation. If $$H_{y}[y](t)+G_{v}[y](t)=0,$$ then its equation of variation is self-adjoint. See [@MyID:291]. Now we provide an answer to the general inverse problem of the calculus of variations on time scales. \[P4:th:necess:EL:int:diff\] Let $\mathbb{T}$ be an arbitrary time scale and $$\label{P4:eq:int:diff:2} H(t,y^{\sigma}(t),y^{\Delta}(t))+\int\limits_{t_{0}}^{t}G(s,y^{\sigma}(s),y^{\Delta}(s))\Delta s=const$$ be a given integro-differential equation. If is to be an Euler–Lagrange equation, then its equation of variation is self-adjoint in the sense of Definition \[P4:def:self:adj:int:diff\]. See [@MyID:291]. \[P4:rem:not:EL\] In practical terms, Theorem \[P4:th:necess:EL:int:diff\] is useful to identify equations that are not Euler–Lagrange: if the equation of variation of a given dynamic equation is not self-adjoint, then we conclude that is not an Euler–Lagrange equation. Now we present an example of a second order differential equation on time scales which is not an Euler–Lagrange equation. Let us consider the following second-order linear oscillator dynamic equation on an arbitrary time scale $\mathbb{T}$: $$\label{P4:eq:3} y^{\Delta\Delta}(t)+y^{\Delta}(t)-t=0.$$ We may write equation in integro-differential form : $$\label{P4:eq:4} y^{\Delta}(t)+\int\limits_{t_{0}}^{t}\left(y^{\Delta}(s)-s\right)\Delta s=const,$$ where $H[y](t)=y^{\Delta}(t)$ and $G[y](t)=y^{\Delta}(t)-t$. Because $$H_{y}[y](t)=G_{y}[y](t)=0, \quad H_{v}[y](t)=G_{v}[y](t)=1,$$ the equation of variation associated with is given by $$\label{P4:eq:5} u^{\Delta}(t)+\int\limits_{t_{0}}^{t}u^{\Delta}(s)\Delta s=0 \iff u^{\Delta}(t)+u(t)=u(t_{0}).$$ We may notice that equation cannot be written in form , hence, it is not self-adjoint. Following Theorem \[P4:th:necess:EL:int:diff\] (see Remark \[P4:rem:not:EL\]) we conclude that equation is not an Euler–Lagrange equation. Now we consider the particular case of Theorem \[P4:th:necess:EL:int:diff\] when $\mathbb{T}=\mathbb{R}$ and $y\in C^{2}([t_{0},t_{1}];\mathbb{R})$. In this case operator $[\cdot]$ of has the form $$[y](t)=(t,y(t),y'(t))=:[y]_{{\mathbb{R}}}(t),$$ while condition can be written as $$\label{P4:eq:self:adj:R} p(t)u'(t)+\int\limits_{t_{0}}^{t}r(s)u(s)ds=const.$$ \[P4:cor:R\] If a given integro-differential equation $$H(t,y(t),y'(t))+\int\limits_{t_{0}}^{t}G(s,y(s),y'(s))ds=const$$ is to be the Euler–Lagrange equation of the variational problem $$\label{P4:eq:funct:R} \mathcal{I}[y]=\int\limits_{t_{0}}^{t_{1}} L(t,y(t),y'(t))dt$$ (cf., e.g., [@MR2004181]), then its equation of variation $$H_{y}[u]_{\mathbb{R}}(t)u(t)+H_{v}[u]_{\mathbb{R}}(t)u'(t) +\int\limits_{t_{0}}^{t}G_{y}[u]_{\mathbb{R}}(s)u(s)+G_{v}[u]_{\mathbb{R}}(s)u'(s)ds=0$$ must be self-adjoint, in the sense of Definition \[P4:def:self:adj:int:diff\] with given by . Follows from Theorem \[P4:th:necess:EL:int:diff\] with $\mathbb{T}=\mathbb{R}$. Now we consider the particular case of Theorem \[P4:th:necess:EL:int:diff\] when $\mathbb{T}=h\mathbb{Z}$, $h>0$. In this case operator $[\cdot]$ of has the form $$[y](t)=(t,y(t+h),\Delta_{h} y(t))=:[y]_{h}(t),$$ where $$\Delta_{h}y(t)=\frac{y(t+h)-y(t)}{h}.$$ For $\mathbb{T}=h\mathbb{Z}$, $h>0$, condition can be written as $$\label{P4:eq:self:adj:hZ} p(t)\Delta_{h}u(t)+\sum\limits_{k=\frac{t_{0}}{h}}^{\frac{t}{h}-1}hr(kh)u(kh+h)=const.$$ \[P4:cor:hZ\] If a given difference equation $$H(t,y(t+h),\Delta_{h} y(t)) +\sum\limits_{k=\frac{t_{0}}{h}}^{\frac{t}{h}-1}hG(kh,y(kh+h),\Delta_{h} y(kh))=const$$ is to be the Euler–Lagrange equation of the discrete variational problem $$\label{P4:eq:funct:hZ} \mathcal{I}[y]=\sum\limits_{k=\frac{t_{0}}{h}}^{\frac{t_{1}}{h}-1}hL(kh,y(kh+h),\Delta_{h} y(kh))$$ (cf., e.g., [@MyID:179]), then its equation of variation $$\begin{gathered} H_{y}[u]_{h}(t)u(t+h)+H_{v}[u]_{h}(t)\Delta_{h}u(t)\\ +h\sum\limits_{k=\frac{t_{0}}{h}}^{\frac{t}{h}-1}\left( G_{y}[u]_{h}(kh)u(kh+h)+G_{v}[u]_{h}(kh)\Delta_{h}u(kh)\right)=0\end{gathered}$$ is self-adjoint, in the sense of Definition \[P4:def:self:adj:int:diff\] with given by . Follows from Theorem \[P4:th:necess:EL:int:diff\] with $\mathbb{T}=h\mathbb{Z}$. Finally, let us consider the particular case of Theorem \[P4:th:necess:EL:int:diff\] when $\mathbb{T}=\overline{q^{\mathbb{Z}}}=q^{\mathbb{Z}}\cup\left\{0\right\}$, where $q^{\mathbb{Z}}=\left\{q^{k}: k\in\mathbb{Z}, q>1\right\}$. In this case operator $[\cdot]$ of has the form $$[y]_{\overline{q^{\mathbb{Z}}}}(t)=(t,y(qt),\Delta_{q} y(t))=:[y]_{q}(t),$$ where $$\Delta_{q}y(t)=\frac{y(qt)-y(t)}{(q-1)t}.$$ For $\mathbb{T}=\overline{q^{\mathbb{Z}}}$, $q>1$, condition can be written as (cf., e.g., [@Rahmat]): $$\label{P4:eq:self:adj:qZ} p(t)\Delta_{q}u(t)+ (q-1)\sum\limits_{s\in [t_{0},t) \cap\mathbb{T}}sr(s)u(qs)=const.$$ \[P4:cor:qZ\] If a given $q$-equation $$H(t,y(qt),\Delta_{q} y(t))+(q-1)\sum\limits_{s\in [t_{0},t) \cap\mathbb{T}}sG(s,y(qs),\Delta_{q}y(s))=const,$$ $q>1$, is to be the Euler–Lagrange equation of the variational problem $$\label{P4:eq:funct:qZ} \mathcal{I}[y]=(q-1)\sum\limits_{t\in [t_{0},t_{1}) \cap\mathbb{T}}tL(t,y(qt),\Delta_{q}y(t)),$$ $t_{0}, t_{1}\in \overline{q^{\mathbb{Z}}}$, then its equation of variation $$\begin{gathered} H_{y}[ u]_{q}(t)u(qt)+H_{v}[u]_{q}(t)\Delta_{q}u(t)\\ +(q-1)\sum\limits_{s\in [t_{0},t)\cap\mathbb{T}} s\left( G_{y}[u]_{q}(s)u(qs)+G_{v}[u]_{q}(s)\Delta_{q}u(s) \right)=0\end{gathered}$$ is self-adjoint, in the sense of Definition \[P4:def:self:adj:int:diff\] with given by . Choose $\mathbb{T}=\overline{q^{\mathbb{Z}}}$ in Theorem \[P4:th:necess:EL:int:diff\]. More information about Euler–Lagrange equations for $q$-variational problems may be found in [@FerreiraTorres; @MyID:266; @MR2966852] and references therein. Discussion {#final remarks} ---------- On an arbitrary time scale $\mathbb{T}$, we can easily show equivalence between the integro-differential equation and the second order differential equation below (Proposition \[P4:prop:1\]). However, when we consider equations of variations of them, we notice that it is not possible to prove an equivalence between them on an arbitrary time scale. The main reason of this impossibility, even in the discrete time scale $\mathbb{Z}$, is the absence of a general chain rule on an arbitrary time scale (see, e.g., Example 1.85 of [@BohnerDEOTS]). However, on $\mathbb{T}=\mathbb{R}$ we can present this equivalence (Proposition \[P4:prop:2\]). \[P4:prop:1\] The integro-differential equation is equivalent to a second order delta differential equation $$\label{P4:eq:6} W\left(t,y^{\sigma}(t), y^{\Delta}(t), y^{\Delta\Delta}(t)\right)=0.$$ Let $\mathbb{T}$ be a time scale such that $\mu$ is delta differentiable. The equation of variation of a second order differential equation is given by $$\label{P4:eq:eq:var:1} W_{z}\langle u\rangle (t)u^{\Delta\Delta}(t) +W_{v}\langle u\rangle (t) u^{\Delta}(t) +W_{y}\langle u\rangle (t) u^{\sigma}(t)=0.$$ On an arbitrary time scale it is impossible to prove the equivalence between the equation of variation and . Indeed, after differentiating both sides of equation and using the product rule given by Theorem \[th:differ:prop:delta\], one has $$\begin{gathered} \label{P4:eq:9} H_{y}[u](t)u^{\sigma\Delta}(t)+H_{y}^{\Delta}[u](t)u^{\sigma\sigma}(t) +H_{v}[u](t)u^{\Delta\Delta}(t)+H_{v}^{\Delta}[u](t)u^{\Delta\sigma}(t)\\ +G_{y}[u](t)u^{\sigma}(t)+G_{v}[u](t)u^{\Delta}(t)=0.\end{gathered}$$ The direct calculations - $H_{y}[u](t)u^{\sigma\Delta}(t)=H_{y}[u](t)(u^{\Delta}(t) +\mu^{\Delta}(t)u^{\Delta}(t)+\mu^{\sigma}(t)u^{\Delta\Delta}(t))$, - $H_{y}^{\Delta}[u](t)u^{\sigma\sigma}(t) =H_{y}^{\Delta}[u](t)(u^{\sigma}(t)+\mu^{\sigma}(t)u^{\Delta}(t) +\mu(t)\mu^{\sigma}(t)u^{\Delta\Delta}(t))$, - $H_{v}^{\Delta}[u](t)u^{\Delta\sigma}(t) =H_{v}^{\Delta}[u](t)(u^{\Delta}(t)+\mu u^{\Delta\Delta}(t))$, and the fourth item of Theorem \[th:differ:delta\], allow us to write equation in form $$\begin{gathered} \label{P4:eq:11} u^{\Delta\Delta}(t)\left[\mu(t)H_{y}[u](t)+H_{v}[u](t)\right]^{\sigma}\\ +u^{\Delta}(t)\left[H_{y}[u](t)+(\mu(t)H_{y}[u](t))^{\Delta} +H_{v}^{\Delta}[u](t)+G_{v}[u](t)\right]\\ +u^{\sigma}(t)\left[H_{y}^{\Delta}[u](t)+G_{y}[u](t)\right]=0.\end{gathered}$$ We are not able to prove that the coefficients of are the same as in , respectively. This is due to the fact that we cannot find the partial derivatives of , that is, $W_{z}\langle u\rangle(t)$, $W_{v}\langle u\rangle(t)$ and $W_{y}\langle u\rangle(t)$, from equation because of lack of a general chain rule in an arbitrary time scale [@BohGus1]. The equivalence, however, is true for $\mathbb{T}=\mathbb{R}$. Operator $\left\langle \cdot\right\rangle$ has in this case the form $\left\langle y\right\rangle (t)=(t, y(t), y'(t), y''(t)) =: \left\langle y\right\rangle_{{\mathbb{R}}} (t)$. \[P4:prop:2\] The equation of variation $$H_{y}[u]_{\mathbb{R}}(t)u(t)+H_{v}[u]_{\mathbb{R}}(t)u'(t) +\int\limits_{t_{0}}^{t}G_{y}[u]_{\mathbb{R}}(s)u(s) +G_{v}[u]_{\mathbb{R}}(s)u'(s)ds=0$$ is equivalent to the second order differential equation $$W_{z}\langle u\rangle_{\mathbb{R}}(t)u''(t) +W_{v}\langle u\rangle_{\mathbb{R}}(t) u'(t) +W_{y}\langle u\rangle_{\mathbb{R}}(t)u(t)=0.$$ Proposition \[P4:prop:2\] allows us to obtain the classical result of [@Davis Theorem II] as a corollary of our Theorem \[P4:th:necess:EL:int:diff\]. The absence of a chain rule on an arbitrary time scale (even for $\mathbb{T}=\mathbb{Z}$) implies that the classical approach [@Davis] fails on time scales. This is the reason why we use here a completely different approach to the subject based on the integro-differential form. The case $\mathbb{T}=\mathbb{Z}$ was recently investigated in [@Helmholtz]. However, similarly to [@Davis], the approach of [@Helmholtz] is based on the differential form and cannot be extended to general time scales. The delta-nabla calculus of variations for composition functionals {#sec:mr} ================================================================== The delta-nabla calculus of variations has been introduced in [@TorresDeltaNabla]. Here we investigate more general problems of the time-scale calculus of variations for a functional that is the composition of a certain scalar function with the delta and nabla integrals of a vector valued field. We begin by proving general Euler–Lagrange equations in integral form (Theorem \[P6:th:main\]). Then we consider cases when initial or terminal boundary conditions are not specified, obtaining corresponding transversality conditions (Theorems \[P6:th:trans:initial\] and \[P6:th:trans:terminal\]). Furthermore, we prove necessary optimality conditions for general isoperimetric problems given by the composition of delta-nabla integrals (Theorem \[P6:th:conds:iso\]). Finally, some illustrating examples are presented (Section \[sec:examples\]). The Euler–Lagrange equations {#subsec:EL} ---------------------------- Let us begin by defining the class of functions $C_{k,n}^{1}([a,b];\mathbb{R})$, which contains delta and nabla differentiable functions. \[P6:class\] By $C_{k,n}^{1}([a,b];\mathbb{R})$, $k,n\in{\mathbb{N}}$, we denote the class of functions $y:[a,b]\rightarrow\mathbb{R}$ such that: if $k\neq 0$ and $n\neq 0$, then $y^{\Delta}$ is continuous on $[a,b]^{\kappa}_{\kappa}$ and $y^{\nabla}$ is continuous on $[a,b]_{\kappa}^{\kappa}$, where $[a,b]^{\kappa}_{\kappa}:=[a,b]^{\kappa}\cap [a,b]_{\kappa}$; if $n=0$, then $y^{\Delta}$ is continuous on $[a,b]^{\kappa}$; if $k=0$, then $y^{\nabla}$ is continuous on $[a,b]_{\kappa}$. Our aim is to find a function $y$ which minimizes or maximizes the following variational problem: $$\begin{gathered} \label{P6:eq:problem} \mathcal L[y]=H\left(\int\limits_{a}^{b}f_{1}(t,y^{\sigma}(t),y^{\Delta}(t))\Delta t, \ldots,\int\limits_{a}^{b}f_{k}(t,y^{\sigma}(t),y^{\Delta}(t))\Delta t,\right.\\ \left.\int\limits_{a}^{b}f_{k+1}(t,y^{\rho}(t),y^{\nabla}(t))\nabla t,\ldots, \int\limits_{a}^{b}f_{k+n}(t,y^{\rho}(t),y^{\nabla}(t))\nabla t\right),\end{gathered}$$ $$\label{P6:eq:bound:conds} (y(a)=y_{a}), \quad (y(b)=y_{b}).$$ The parentheses in , around the end-point conditions, means that those conditions may or may not occur (it is possible that one or both $y(a)$ and $y(b)$ are free). A function $y\in C_{k,n}^{1}$ is said to be admissible provided it satisfies the boundary conditions (if any is given). For $k = 0$ problem – becomes a nabla problem (neither delta integral nor delta derivative is present); for $n = 0$ problem – reduces to a delta problem (neither nabla integral nor nabla derivative is present). For simplicity, we use the operators $[\cdot]$ and ${\lbrace}\cdot{\rbrace}$ defined by $$[y](t):=(t,y^{{\sigma}}(t),y^{\Delta}(t)),\quad {\lbrace}y{\rbrace}(t):=(t,y^{\rho}(t),y^{{\nabla}}(t)).$$ We assume that: 1. the function $H:\mathbb{R}^{n+k}\rightarrow\mathbb{R}$ has continuous partial derivatives with respect to its arguments, which we denote by $H_{i}^{'}$, $i=1, \ldots, n+k$; 2. functions $(t,y,v)\rightarrow f_{i}(t,y,v)$ from $[a,b]\times\mathbb{R}^{2}$ to $\mathbb{R}$, $i=1,\ldots, n+k$, have partial continuous derivatives with respect to $y$ and $v$ uniformly in $t \in [a,b]$, which we denote by $f_{iy}$ and $f_{iv}$; 3. $f_{i}$, $f_{iy}$, $f_{iv}$ are rd-continuous on $[a,b]^{\kappa}$, $i=1,\ldots,k$, and ld-continuous on $[a,b]_{\kappa}$, $i=k+1,\ldots,k+n$, for all $y\in C_{k,n}^{1}$. \[def:loc:extr\] We say that an admissible function $\hat{y}\in C_{k,n}^{1}([a,b];\mathbb{R})$ is a local minimizer (respectively, local maximizer) to problem –, if there exists $\delta >0$ such that $\mathcal{L}[\hat{y}]\leq \mathcal{L} [y]$ (respectively, $\mathcal{L}[\hat{y}]\geq \mathcal{L}[y]$) for all admissible functions $y\in C_{k,n}^{1}([a,b];\mathbb{R})$ satisfying the inequality $|| y-\hat{y}||_{1,\infty}<\delta$, where $$||y||_{1,\infty}:=||y^{\sigma}||_{\infty}+||y^{\Delta}||_{\infty} +||y^{\rho}||_{\infty}+||y^{\nabla}||_{\infty}$$ with $||y||_{\infty}:= \sup_{t\in[a,b]_{\kappa}^{\kappa}} |y(t)|$. For brevity, in what follows we omit the argument of $H_{i}^{'}$. Precisely, $$H_{i}^{'}:=\frac{\partial H}{\partial \mathcal{F}_{i}}(\mathcal{F}_{1}(y),\ldots,\mathcal{F}_{k+n}(y)), \quad i=1,\ldots,n+k,$$ where $$\begin{split} \mathcal{F}_{i}(y) &=\int\limits_{a}^{b} f_{i}[y](t)\Delta t, \hbox{ for } i=1,\ldots,k,\\ \mathcal{F}_{i}(y) &=\int\limits_{a}^{b}f_{i}{\lbrace}y{\rbrace}(t)\nabla t, \hbox{ for } i=k+1,\ldots,k+n. \end{split}$$ Depending on the given boundary conditions, we can distinguish four different problems. The first one is the problem $(P_{ab})$, where the two boundary conditions are specified. To solve this problem we need an Euler–Lagrange necessary optimality condition, which is given by Theorem \[P6:th:main\] below. Next two problems — denoted by $(P_{a})$ and $(P_{b})$ — occur when $y(a)$ is given and $y(b)$ is free (problem $(P_{a})$) and when $y(a)$ is free and $y(b)$ is specified (problem $(P_{b})$). To solve both of them we need an Euler–Lagrange equation and one proper transversality condition. The last problem — denoted by $(P)$ — occurs when both boundary conditions are not present. To find a solution for such a problem we need to use an Euler–Lagrange equation and two transversality conditions (one at each time $a$ and $b$). \[P6:th:main\] If $\hat{y}$ is a local solution to problem –, then the Euler–Lagrange equations (in integral form) $$\begin{gathered} \label{P6:eq:EL:nabla} \sum\limits_{i=1}^{k}H_{i}^{'}\cdot \left(f_{iv}[\hat{y}](\rho(t))-\int\limits_{a}^{\rho(t)} f_{iy}[\hat{y}](\tau)\Delta \tau \right)\\ +\sum\limits_{i=k+1}^{k+n}H_{i}^{'}\cdot \left(f_{iv}{\lbrace}\hat{y}{\rbrace}(t)-\int\limits_{a}^{t} f_{iy}{\lbrace}\hat{y}{\rbrace}(\tau)\nabla \tau \right) = c, \quad t\in\mathbb{T}_{\kappa},\end{gathered}$$ and $$\begin{gathered} \label{P6:eq:EL:delta} \sum\limits_{i=1}^{k}H_{i}^{'}\cdot \left(f_{iv}[\hat{y}](t)-\int\limits_{a}^{t} f_{iy}[\hat{y}](\tau)\Delta \tau\right)\\ +\sum\limits_{i=k+1}^{k+n}H_{i}^{'}\cdot \left(f_{iv}{\lbrace}\hat{y}{\rbrace}(\sigma(t))-\int\limits_{a}^{\sigma(t)} f_{iy}{\lbrace}\hat{y}{\rbrace}(\tau)\nabla \tau\right) = c, \quad t\in\mathbb{T}^{\kappa},\end{gathered}$$ hold. See [@MR3040923]. For regular time scales (Definition \[def:regular\]), the Euler–Lagrange equations and coincide; on a general time scale, they are different. Such a difference is illustrated in Example \[P6:ex:5\]. For such purpose let us define $\xi$ and $\chi$ by $$\label{P6:eq:xi:chi} \begin{gathered} \xi(t):=\sum\limits_{i=1}^{k}H_{i}^{'}\cdot \left(f_{iv}[\hat{y}](t)-\int\limits_{a}^{t} f_{iy}[\hat{y}](\tau)\Delta \tau \right),\\ \chi(t):=\sum\limits_{i=k+1}^{k+n}H_{i}^{'}\cdot \left(f_{iv}{\lbrace}\hat{y}{\rbrace}(t)-\int\limits_{a}^{t} f_{iy}{\lbrace}\hat{y}{\rbrace}(\tau)\nabla \tau \right). \end{gathered}$$ \[P6:ex:5\] Let us consider the irregular time scale $\mathbb{T}=\mathbb{P}_{1,1}=\bigcup\limits_{k=0}^{\infty}\left[2k,2k+1\right]$. We show that for this time scale there is a difference between the Euler–Lagrange equations and . The forward and backward jump operators are given by $$\sigma(t)= \begin{cases} t,\quad t\in\bigcup\limits_{k=0}^{\infty}[2k,2k+1),\\ t+1, \quad t\in \bigcup\limits_{k=0}^{\infty}\left\{2k+1\right\}, \end{cases} \quad \rho(t)= \begin{cases} t,\quad t\in\bigcup\limits_{k=0}^{\infty}(2k,2k+1],\\ t-1, \quad t\in \bigcup\limits_{k=1}^{\infty}\left\{2k\right\},\\ 0, \quad t = 0. \end{cases}$$ For $t = 0$ and $t\in \bigcup\limits_{k=0}^{\infty}\left(2k,2k+1\right)$, equations and coincide. We can distinguish between them for $t\in \bigcup\limits_{k=0}^{\infty}\left\{2k+1\right\}$ and $t\in \bigcup\limits_{k=1}^{\infty}\left\{2k\right\}$. In what follows we use the notations . If $t\in \bigcup\limits_{k=0}^{\infty}\left\{2k+1\right\}$, then we obtain from and the Euler–Lagrange equations $\xi(t) + \chi(t) = c$ and $\xi(t) + \chi(t+1) = c$, respectively. If $t\in \bigcup\limits_{k=1}^{\infty}\left\{2k\right\}$, then the Euler–Lagrange equation has the form $\xi(t-1) + \chi(t) = c$ while takes the form $\xi(t) + \chi(t) = c$. Natural boundary conditions {#sec:nbc} --------------------------- In this section we minimize or maximize the variational functional , but initial and/or terminal boundary condition $y(a)$ and/or $y(b)$ are not specified. In what follows we obtain corresponding transversality conditions. \[P6:th:trans:initial\] Let $\mathbb{T}$ be a time scale for which $\rho(\sigma(a))=a$. If $\hat{y}$ is a local extremizer to with $y(a)$ not specified, then $$\sum\limits_{i=1}^{k}H_{i}^{'} \cdot f_{iv}[\hat{y}](a) +\sum\limits_{i=k+1}^{k+n}H_{i}^{'}\cdot \left( f_{iv}{\lbrace}\hat{y}{\rbrace}(\sigma(a)) - \int\limits^{\sigma(a)}_{a}f_{iy}{\lbrace}\hat{y}{\rbrace}(t)\nabla t \right) = 0$$ holds together with the Euler–Lagrange equations and . See [@MR3040923]. \[P6:th:trans:terminal\] Let $\mathbb{T}$ be a time scale for which $\sigma(\rho(b))=b$. If $\hat{y}$ is a local extremizer to with $y(b)$ not specified, then $$\sum\limits_{i=1}^{k}H_{i}^{'}\cdot \left( f_{iv}[\hat{y}](\rho(b)) + \int\limits_{\rho(b)}^{b}f_{iy}[\hat{y}](t)\Delta t \right) +\sum\limits_{i=k+1}^{k+n}H_{i}^{'} \cdot f_{iv}{\lbrace}\hat{y}{\rbrace}(b) = 0$$ holds together with the Euler–Lagrange equations and . See [@MR3040923]. Several interesting results can be immediately obtained from Theorems \[P6:th:main\], \[P6:th:trans:initial\] and \[P6:th:trans:terminal\]. An example of such results is given by Corollary \[P6:cor:quotient\]. \[P6:cor:quotient\] If $\hat{y}$ is a solution to the problem $$\begin{gathered} \mathcal{L}[y] =\frac{\int\limits_{a}^{b}f_{1}(t,y^{\sigma}(t),y^{\Delta}(t)) \Delta t}{\int\limits_{a}^{b}f_{2}(t,y^{\rho}(t),y^{\nabla}(t))\nabla t} \longrightarrow \mathrm{extr},\\ (y(a)=y_{a}), \quad (y(b)=y_{b}),\end{gathered}$$ then the Euler–Lagrange equations $$\frac{1}{\mathcal{F}_{2}} \left( f_{1v}[\hat{y}](\rho(t))-\int\limits_{a}^{\rho(t)} f_{1y}[\hat{y}](\tau)\Delta \tau\right) - \frac {\mathcal{F}_{1}} {\mathcal{F}_{2}^{2}} \left(f_{2v}{\lbrace}\hat{y}{\rbrace}(t)-\int\limits_{a}^{t} f_{2y}{\lbrace}\hat{y}{\rbrace}(\tau)\nabla \tau\right) = c,$$ $t\in{\mathbb{T}}_{\kappa}$, and $$\frac{1}{\mathcal{F}_{2}} \left( f_{1v}[\hat{y}](t)-\int\limits_{a}^{t} f_{1y}[\hat{y}](\tau)\Delta \tau\right) -\frac{\mathcal{F}_{1}}{\mathcal{F}_{2}^{2}} \left(f_{2v}{\lbrace}\hat{y}{\rbrace}(\sigma(t)) -\int\limits^{\sigma (t)}_{a} f_{2y}{\lbrace}\hat{y}{\rbrace}(\tau)\nabla \tau\right)=c,$$ $t\in{\mathbb{T}}^{\kappa}$, hold, where $$\mathcal{F}_{1}:={\int\limits_{a}^{b} f_{1}(t,\hat{y}^{\sigma}(t),\hat{y}^{\Delta}(t))\Delta t} \quad \text{ and } \quad \mathcal{F}_{2}:={\int\limits_{a}^{b} f_{2}(t,\hat{y}^{\rho}(t),\hat{y}^{\nabla}(t))\nabla t}.$$ Moreover, if $y(a)$ is free and $\rho(\sigma(a))=a$, then $$\frac{1}{\mathcal{F}_{2}} f_{1v}[\hat{y}](a) -\frac{\mathcal{F}_{1}}{\mathcal{F}_{2}^{2}} \left(f_{2v}{\lbrace}\hat{y}{\rbrace}(\sigma(a))-\int\limits_{a}^{\sigma(a)} f_{2y}{\lbrace}\hat{y}{\rbrace}(t)\nabla t\right) =0;$$ if $y(b)$ is free and $\sigma(\rho(b))=b$, then $$\frac{1}{\mathcal{F}_{2}} \left(f_{1v}[\hat{y}](\rho(b))+\int\limits^{b}_{\rho(b)} f_{1y}[\hat{y}](t)\Delta t\right) -\frac{\mathcal{F}_{1}}{\mathcal{F}_{2}^{2}} f_{2v}{\lbrace}\hat{y}{\rbrace}(b)=0.$$ Isoperimetric problems {#sub:sec:iso:p} ---------------------- Let us now consider the general delta–nabla composition isoperimetric problem on time scales subject to boundary conditions. The problem consists of extremizing $$\begin{gathered} \label{P6:eq:iso} \mathcal L[y]=H\left(\int\limits_{a}^{b}f_{1}(t,y^{\sigma}(t),y^{\Delta}(t))\Delta t, \ldots,\int\limits_{a}^{b}f_{k}(t,y^{\sigma}(t),y^{\Delta}(t))\Delta t,\right.\\ \left.\int\limits_{a}^{b}f_{k+1}(t,y^{\rho}(t),y^{\nabla}(t))\nabla t,\ldots, \int\limits_{a}^{b}f_{k+n}(t,y^{\rho}(t),y^{\nabla}(t)) \nabla t \right)\end{gathered}$$ in the class of functions $y\in C^1_{k+m,n+p}$ satisfying given boundary conditions $$\label{P6:eq:bound:conds:iso} y(a)=y_{a},\quad y(b)=y_{b},$$ and a generalized isoperimetric constraint $$\begin{gathered} \label{P6:eq:iso:constraint} \mathcal{K}[y] =P\left(\int\limits_{a}^{b}g_{1}(t,y^{\sigma}(t),y^{\Delta}(t))\Delta t, \ldots,\int\limits_{a}^{b}g_{m}(t,y^{\sigma}(t),y^{\Delta}(t))\Delta t,\right.\\ \left.\int\limits_{a}^{b}g_{m+1}(t,y^{\rho}(t),y^{\nabla}(t))\nabla t,\ldots, \int\limits_{a}^{b}g_{m+p}(t,y^{\rho}(t),y^{\nabla}(t)) \nabla t \right)=d,\end{gathered}$$ where $y_{a},y_{b},d\in\mathbb{R}$. We assume that: 1. the functions $H:\mathbb{R}^{n+k}\rightarrow\mathbb{R}$ and $P:\mathbb{R}^{m+p}\rightarrow\mathbb{R}$ have continuous partial derivatives with respect to all their arguments, which we denote by $H_{i}^{'}$, $i=1,\ldots,n+k$, and $P_{i}^{'}$, $i=1,\ldots,m+p$; 2. functions $(t,y,v)\rightarrow f_{i}(t,y,v)$, $i=1,\ldots, n+k$, and $(t,y,v)\rightarrow g_{j}(t,y,v)$, $j=1,\ldots,m+p$, from $[a,b]\times\mathbb{R}^{2}$ to $\mathbb{R}$, have partial continuous derivatives with respect to $y$ and $v$ uniformly in $t\in [a,b]$, which we denote by $f_{iy}$, $f_{iv}$, and $g_{jy}, g_{jv}$; 3. for all $y\in C_{k+m,n+p}^{1}$, $f_{i}$, $f_{iy}$, $f_{iv}$ and $g_{j},g_{jy}$, $g_{jv}$ are rd-continuous in $t\in [a,b]^{\kappa}$, $i=1,\ldots,k$, $j=1,\ldots,m$, and ld-continuous in $t\in [a,b]_{\kappa}$, $i=k+1,\ldots,k+n$, $j=m+1,\ldots,m+p$. A function $y\in C^{1}_{k+m, n+p}$ is said to be admissible provided it satisfies the boundary conditions and the isoperimetric constraint . For brevity, we omit the argument of $P_{i}^{'}$: $P_{i}^{'}:=\frac{\partial P}{\partial \mathcal{G}_{i}}(\mathcal{G}_{1}(\hat{y}), \ldots,\mathcal{G}_{m+p}(\hat{y}))$ for $i=1,\ldots,m+p$, with $$\mathcal{G}_{i}(\hat{y})=\int\limits_{a}^{b} g_{i}(t,\hat{y}^{\sigma}(t),\hat{y}^{\Delta}(t))\Delta t, \quad i=1,\ldots,m,$$ and $$\mathcal{G}_{i}(\hat{y})=\int\limits_{a}^{b} g_{i}(t,\hat{y}^{\rho}(t),\hat{y}^{\nabla}(t))\nabla t, \quad i=m+1,\ldots,m+p.$$ We say that an admissible function $\hat{y}$ is a local minimizer (respectively, a local maximizer) to the isoperimetric problem –, if there exists a $\delta >0$ such that $\mathcal{L}[\hat{y}]\leqslant \mathcal{L}[y]$ (respectively, $\mathcal{L}[\hat{y}]\geqslant \mathcal{L}[y]$) for all admissible functions $y\in C_{k+m,n+p}^{1}$ satisfying the inequality $||y-\hat{y}||_{1,\infty}<\delta$. Let us define $u$ and $w$ by $$\label{P6:eq:u:w} \begin{gathered} u(t):= \sum\limits_{i=1}^{m}P_{i}^{'}\cdot \left(g_{iv}[\hat{y}](t)-\int\limits_{a}^{t} g_{iy}[\hat{y}](\tau)\Delta \tau \right),\\ w(t):= \sum\limits_{i=m+1}^{m+p}P_{i}^{'}\cdot \left(g_{iv}{\lbrace}\hat{y}{\rbrace}(t)-\int\limits_{a}^{t} g_{iy}{\lbrace}\hat{y}{\rbrace}(\tau)\nabla \tau \right). \end{gathered}$$ An admissible function $\hat{y}$ is said to be an extremal for $\mathcal{K}$ if $u(t) + w(\sigma(t)) = const$ and $u(\rho(t)) + w(t) = const$ for all $t\in[a,b]_\kappa^\kappa$. An extremizer (i.e., a local minimizer or a local maximizer) to problem – that is not an extremal for $\mathcal{K}$ is said to be a normal extremizer; otherwise (i.e., if it is an extremal for $\mathcal{K}$), the extremizer is said to be abnormal. \[P6:th:conds:iso\] Let $\chi$ and $\xi$ be given as in , and $u$ and $w$ be given as in . If $\hat{y}$ is a normal extremizer to the isoperimetric problem –, then there exists a real number $\lambda$ such that 1. $\xi^{\rho}(t)+\chi(t)-\lambda\left(u^{\rho}(t)+w(t)\right)= const$; 2. $\xi(t)+\chi^{\sigma}(t)-\lambda\left(u^{\rho}(t)+w(t)\right)= const$; 3. $\xi^{\rho}(t)+\chi(t)-\lambda\left(u(t)+w^{\sigma}(t)\right)= const$; 4. $\xi(t)+\chi^{\sigma}(t)-\lambda\left(u(t)+w^{\sigma}(t)\right)= const$; for all $t\in [a,b]^{\kappa}_{\kappa}$. See proof of Theorem 3.9 in [@MR3040923]. Illustrative examples {#sec:examples} --------------------- In this section we consider three examples which illustrate the results of Theorem \[P6:th:main\] and Theorem \[P6:th:conds:iso\]. We begin with a nonautonomous problem. \[P6:ex:1\] Consider the problem $$\label{P6:eq:9} \begin{gathered} \mathcal{L}[y]= \frac{\int\limits_{0}^{1} t y^{\Delta}(t) \Delta t} {\int\limits_{0}^{1}(y^{\nabla}(t))^{2}\nabla t} \longrightarrow \min, \\ y(0)=0, \quad y(1)=1. \end{gathered}$$ If $y$ is a local minimizer to problem , then the Euler–Lagrange equations of Corollary \[P6:cor:quotient\] must hold, i.e., $$\frac{1}{\mathcal{F}_{2}}\rho(t)-2\frac{\mathcal{F}_{1}}{\mathcal{F}_{2}^{2}} y^{\nabla}(t)=c, \quad t \in \mathbb{T}_{\kappa},$$ and $$\frac{1}{\mathcal{F}_{2}}t-2\frac{\mathcal{F}_{1}}{\mathcal{F}_{2}^{2}} y^{\nabla}(\sigma(t))=c, \quad t \in \mathbb{T}^{\kappa},$$ where $\mathcal{F}_{1}:=\mathcal{F}_{1}(y)=\int\limits_{0}^{1}t y^{\Delta}(t)\Delta t$ and $\mathcal{F}_{2}:=\mathcal{F}_{2}(y)=\int\limits_{0}^{1}(y^{\nabla}(t))^{2}\nabla t$. Let us consider the second equation. Using of Theorem \[th:differ:delta:nabla\], it can be written as $$\label{P6:eq:10} \frac{1}{\mathcal{F}_{2}}t-2\frac{\mathcal{F}_{1}}{\mathcal{F}_{2}^{2}}y^{\Delta}(t)=c, \quad t \in \mathbb{T}^{\kappa}.$$ Solving subject to the boundary conditions $y(0)=0$ and $y(1)=1$ gives $$\label{P6:eq:ex:1:sol} y(t)= \frac{1}{2Q}\int\limits_{0}^{t}\tau\Delta\tau -t\left(\frac{1}{2Q}\int\limits_{0}^{1}\tau\Delta\tau -1\right), \quad t \in \mathbb{T}^{\kappa},$$ where $Q:=\frac{\mathcal{F}_{1}}{\mathcal{F}_{2}}$. Therefore, the solution depends on the time scale. Let us consider two examples: $\mathbb{T}=\mathbb{R}$ and $\mathbb{T}=\left\{0,\frac{1}{2},1\right\}$. On $\mathbb{T}=\mathbb{R}$, from we obtain $$\label{P6:eq:11} y(t)=\frac{1}{4Q}t^{2}+\frac{4Q-1}{4Q}t, \quad\quad y^{\Delta}(t) = y^{\nabla}(t) = y'(t)=\frac{1}{2Q}t+\frac{4Q-1}{4Q}$$ as solution of . Substituting into $\mathcal{F}_{1}$ and $\mathcal{F}_{2}$ gives $\mathcal{F}_{1}=\frac{12Q+1}{24Q}$ and $\mathcal{F}_{2}=\frac{48Q^{2}+1}{48Q^{2}}$, that is, $$\label{P6:eq:Q:1} Q=\frac{2Q(12Q+1)}{48Q^{2}+1}.$$ Solving equation we get $Q\in\left\{\frac{3-2\sqrt{3}}{12},\frac{3+2\sqrt{3}}{12}\right\}$. Because is a minimizing problem, we select $Q=\frac{3-2\sqrt{3}}{12}$ and we get the extremal $$\label{P6:eq:12} y(t)=-(3+2\sqrt{3}) t^{2} + (4 + 2 \sqrt{3}) t.$$ If $\mathbb{T}=\left\{0,\frac{1}{2},1\right\}$, then from we obtain $y(t)=\frac{1}{8Q}\sum\limits_{k=0}^{2t-1}k+\frac{8Q-1}{8Q}t$, that is, $$y(t)= \begin{cases} 0, & \text{ if } t=0,\\ \frac{8Q-1}{16Q}, & \text{ if } t=\frac{1}{2},\\ 1, & \text{ if } t=1. \end{cases}$$ Direct calculations show that $$\label{P6:eq:13} \begin{gathered} y^{\Delta}(0)=\frac{y(\frac{1}{2})-y(0)}{\frac{1}{2}}=\frac{8Q-1}{8Q}, \quad y^{\Delta}\left(\frac{1}{2}\right) =\frac{y(1)-y(\frac{1}{2})}{\frac{1}{2}}=\frac{8Q+1}{8Q},\\ y^{\nabla}\left(\frac{1}{2}\right) =\frac{y(\frac{1}{2})-y(0)}{\frac{1}{2}}=\frac{8Q-1}{8Q}, \quad y^{\nabla}(1)=\frac{y(1)-y(\frac{1}{2})}{\frac{1}{2}} =\frac{8Q+1}{8Q}. \end{gathered}$$ Substituting into the integrals $\mathcal{F}_{1}$ and $\mathcal{F}_{2}$ gives $$\mathcal{F}_{1}=\frac{8Q+1}{32Q}, \quad \mathcal{F}_{2}=\frac{64Q^{2}+1}{64Q^{2}}, \quad Q=\frac{\mathcal{F}_{1}}{\mathcal{F}_{2}}=\frac{2Q(8Q+1)}{64Q^{2}+1}.$$ Thus, we obtain the equation $64Q^{2}-16Q-1=0$. The solutions to this equation are: $Q\in \left\{\frac{1-\sqrt{2}}{8}, \frac{1+\sqrt{2}}{8}\right\}$. We are interested in the minimum value $Q$, so we select $Q = \frac{1+\sqrt{2}}{8}$ to get the extremal $$\label{P6:eq:14} y(t) =\begin{cases} 0, & \hbox{ if } t=0,\\ 1-\frac{\sqrt{2}}{2}, & \hbox{ if } t=\frac{1}{2},\\ 1, & \hbox{ if }t=1. \end{cases}$$ Note that the extremals and are different: for one has $x(1/2) = \frac{5}{4} + \frac{\sqrt{3}}{2}$. In the previous example, the variational functional is given by the ratio of a delta and a nabla integral. Now we discuss a variational problem where the composition is expressed by the product of three time-scale integrals. \[P6:ex:3\] Consider the problem $$\label{P6:eq:18} \begin{gathered} \mathcal{L}[y]= \left(\int\limits_{0}^{3} t y^{\Delta}(t) \Delta t\right) \left(\int\limits_{0}^{3} y^{\Delta}(t)\left(1+t\right)\Delta t\right) \left(\int\limits_{0}^{3}\left[\left(y^{\nabla}(t)\right)^{2}+t\right]\nabla t\right) \longrightarrow \min,\\ y(0)=0,\quad y(3)=3. \end{gathered}$$ If $y$ is a local minimizer to problem , then the Euler–Lagrange equations must hold, and we can write that $$\label{P6:eq:19} \left(\mathcal{F}_{1}\mathcal{F}_{3}+\mathcal{F}_{2}\mathcal{F}_{3}\right)t +\mathcal{F}_{1}\mathcal{F}_{3}+2\mathcal{F}_{1}\mathcal{F}_{2} y^{\nabla}(\sigma(t))=c, \quad t \in \mathbb{T}^{\kappa},$$ where $c$ is a constant, $\mathcal{F}_{1}:=\mathcal{F}_{1}(y) =\int\limits_{0}^{3} t y^{\Delta}(t) \Delta t$, $\mathcal{F}_{2}:=\mathcal{F}_{2}(y) =\int\limits_{0}^{3} y^{\Delta}(t)\left(1+t\right)\Delta t$, and $\mathcal{F}_{3}:=\mathcal{F}_{3}(y) =\int\limits_{0}^{3}\left[\left(y^{\nabla}(t)\right)^{2}+t\right]\nabla t$. Using relation , we can write as $$\label{P6:eq:20} \left(\mathcal{F}_{1}\mathcal{F}_{3}+\mathcal{F}_{2}\mathcal{F}_{3}\right)t +\mathcal{F}_{1}\mathcal{F}_{3}+2\mathcal{F}_{1}\mathcal{F}_{2}y^{\Delta}(t)=c, \quad t \in \mathbb{T}^{\kappa}.$$ Using the boundary conditions $y(0)=0$ and $y(3)=3$, from we get that $$\label{P6:eq:ex:3:sol} y(t)=\left(1+\frac{Q}{3}\int\limits_{0}^{3}\tau\Delta \tau\right) t -Q \int\limits_{0}^{t}\tau\Delta \tau, \quad t \in \mathbb{T}^{\kappa},$$ where $Q=\frac{\mathcal{F}_{1}\mathcal{F}_{3} +\mathcal{F}_{2}\mathcal{F}_{3}}{2\mathcal{F}_{1}\mathcal{F}_{2}}$. Therefore, the solution depends on the time scale. Let us consider $\mathbb{T}=\mathbb{R}$ and $\mathbb{T} =\left\{0,\frac{1}{2},1,\frac{3}{2},2,\frac{5}{2},3\right\}$. On $\mathbb{T}=\mathbb{R}$, expression gives $$\label{P6:eq:21} y(t)=\left(\frac{2+3Q}{2}\right) t - \frac{Q}{2}t^{2}, \quad y^{\Delta}(t)= y^{\nabla}(t) = y'(t) = \frac{2+3Q}{2}-Qt$$ as solution of . Substituting into $\mathcal{F}_{1}$, $\mathcal{F}_{2}$ and $\mathcal{F}_{3}$ gives: $$\mathcal{F}_{1}=\frac{18-9Q}{4}, \quad \mathcal{F}_{2}=\frac{30-9Q}{4}, \quad \mathcal{F}_{3}=\frac{9Q^{2}+30}{4}.$$ Solving equation $9Q^{3} - 36 Q^{2} + 45 Q - 40=0$, one finds the solution $$Q = \frac{1}{27}\left[ 36+ \sqrt[3]{24786-729\sqrt{1155}} +9\sqrt[3]{34+\sqrt{1155}}\right]\approx 2,7755$$ and the extremal $y(t)=5,16325t-1,38775t^{2}$. Let us consider now the time scale $\mathbb{T} =\left\{0,\frac{1}{2},1,\frac{3}{2},2,\frac{5}{2},3\right\}$. From , we obtain $$\label{P6:eq:22} y(t)=\left(\frac{4+5Q}{4}\right)t-\frac{Q}{4} \sum\limits_{k=0}^{2t-1}k = \begin{cases} 0, & \hbox{ if } t=0,\\ \frac{4+5Q}{8}, & \hbox{ if } t=\frac{1}{2},\\ 1+Q, & \hbox{ if } t=1,\\ \frac{12+9Q}{8}, & \hbox{ if } t=\frac{3}{2},\\ 2+Q, & \hbox{ if } t=2,\\ \frac{20+5Q}{8}, & \hbox{ if } t=\frac{5}{2},\\ 3, & \hbox{ if } t=3 \end{cases}$$ as solution of . Substituting into $\mathcal{F}_{1}$, $\mathcal{F}_{2}$ and $\mathcal{F}_{3}$, yields $$\mathcal{F}_{1}=\frac{60-35Q}{16}, \quad \mathcal{F}_{2}=\frac{108-35Q}{16}, \quad \mathcal{F}_{3}=\frac{35Q^{2}+132}{16}.$$ Solving equation $245Q^{3}-882Q^{2}+1110-\frac{5544}{5}=0$, we get $Q\approx 2,5139$ and the extremal $$\label{eq:extremal:50} y(t)= \begin{cases} 0, & \hbox{ if } t=0,\\ 2,0711875, & \hbox{ if } t=\frac{1}{2},\\ 3,5139, & \hbox{ if } t=1,\\ 4,3281375, & \hbox{ if } t=\frac{3}{2},\\ 4,5139, & \hbox{ if } t=2,\\ 4,0711875, & \hbox{ if } t=\frac{5}{2},\\ 3, & \hbox{ if } t=3 \end{cases}$$ for problem on $\mathbb{T}=\left\{0,\frac{1}{2},1,\frac{3}{2},2,\frac{5}{2},3\right\}$. In order to illustrate the difference between composition of mixed delta-nabla integrals and pure delta or nabla situations, we consider now two variants of problem : (i) the first consisting of delta operators only: $$\label{survey:1} \begin{gathered} \mathcal{L}[y]= \left(\int\limits_{0}^{3} t y^{\Delta}(t) \Delta t\right) \left(\int\limits_{0}^{3} y^{\Delta}(t)\left(1+t\right)\Delta t\right) \left(\int\limits_{0}^{3}\left[\left(y^{\Delta}(t)\right)^{2}+t\right]\Delta t\right) \longrightarrow \min; \end{gathered}$$ (ii) the second of nabla operators only: $$\label{survey:2} \begin{gathered} \mathcal{L}[y]= \left(\int\limits_{0}^{3} t y^{\nabla}(t) \nabla t\right) \left(\int\limits_{0}^{3} y^{\nabla}(t)\left(1+t\right)\nabla t\right) \left(\int\limits_{0}^{3}\left[\left(y^{\nabla}(t)\right)^{2}+t\right]\nabla t\right) \longrightarrow \min. \end{gathered}$$ Both problems (i) and (ii) are subject to the same boundary conditions as in : $$\label{eq:boundaryConditions} y(0)=0,\quad y(3)=3.$$ All three problems , and , and –, coincide in $\mathbb{R}$. Consider, as before, the time scale $\mathbb{T}=\left\{0,\frac{1}{2},1,\frac{3}{2},2,\frac{5}{2},3\right\}$. Recall that problem has extremal . (i) Now, let us consider the delta problem and . We obtain $$\mathcal{F}_{1}=\frac{60-35Q}{16}, \quad \mathcal{F}_{2}=\frac{108-35Q}{16}, \quad \mathcal{F}_{3}=\frac{35Q^{2}+108}{16}$$ and the equation $245Q^{3}-882Q^{2}+1026-\frac{5436}{5}=0$. Its numerical solution $Q\approx 2,5216$ entails the extremal $$y(t)= \begin{cases} 0, & \hbox{ if } t=0,\\ 2,076, & \hbox{ if } t=\frac{1}{2},\\ 3,5216, & \hbox{ if } t=1,\\ 4,3368, & \hbox{ if } t=\frac{3}{2},\\ 4,5216, & \hbox{ if } t=2,\\ 4,076, & \hbox{ if } t=\frac{5}{2},\\ 3, & \hbox{ if } t=3. \end{cases}$$ (ii) In the latter nabla problem – we have $$\mathcal{F}_{1}=\frac{84-35Q}{16}, \quad \mathcal{F}_{2}=\frac{132-35Q}{16}, \quad \mathcal{F}_{3}=\frac{35Q^{2}+132}{16}$$ and the equation $175Q^{3}-810Q^{2}+1122-\frac{7128}{7}=0$. Using its numerical solution $Q\approx 3,1097$ we get the extremal $$y(t)= \begin{cases} 0, & \hbox{ if } t=0,\\ 2,4942, & \hbox{ if } t=\frac{1}{2},\\ 4,1907, & \hbox{ if } t=1,\\ 5,0895, & \hbox{ if } t=\frac{3}{2},\\ 5,1907, & \hbox{ if } t=2,\\ 4,4942, & \hbox{ if } t=\frac{5}{2},\\ 3, & \hbox{ if } t=3. \end{cases}$$ Finally, we apply the results of Section \[sub:sec:iso:p\] to an isoperimetric problem. \[P6:ex:4\] Let us consider the problem of extremizing $$\mathcal{L}[y]= \frac{ \int\limits_{0}^{1}(y^{\Delta}(t))^{2}\Delta t} {\int\limits_{0}^{1} ty^{\nabla}(t)\nabla t}$$ subject to the boundary conditions $y(0)=0$ and $y(1)=1$ and the isoperimetric constraint $$\mathcal{K}[y]=\int\limits_{0}^{1} ty^{\nabla}(t)\nabla t=1.$$ Applying Theorem \[P6:th:conds:iso\], we get the nabla differential equation $$\label{P6:eq:23} \frac{2}{\mathcal{F}_{2}}y^{\nabla}(t) - \left(\lambda + \frac{\mathcal{F}_{1}}{(\mathcal{F}_{2})^{2}}\right) t = c, \quad t \in \mathbb{T}^{\kappa}_{\kappa}.$$ Solving this equation, we obtain $$\label{P6:eq:ex:4:sol} y(t)=\left(1-Q\int\limits_{0}^{1}\tau\nabla\tau\right)t +Q\int\limits_{0}^{t}\tau\nabla \tau,$$ where $Q=\frac{\mathcal{F}_{2}}{2}\left(\frac{\mathcal{F}_{1}}{(\mathcal{F}_{2})^{2}}+\lambda\right)$. Therefore, the solution of equation depends on the time scale. Let us consider $\mathbb{T}=\mathbb{R}$ and $\mathbb{T}=\left\{0,\frac{1}{2},1\right\}$. On $\mathbb{T}=\mathbb{R}$, from we obtain that $y(t)=\frac{2-Q}{2}t+\frac{Q}{2}t^{2}$. Substituting this expression for $y$ into the integrals $\mathcal{F}_{1}$ and $\mathcal{F}_{2}$, gives $\mathcal{F}_{1}=\frac{Q^{2}+12}{12}$ and $\mathcal{F}_{2}=\frac{Q+6}{12}$. Using the given isoperimetric constraint, we obtain $Q=6$, $\lambda =8$, and $y(t)=3t^{2}-2t$. Let us consider now the time scale $\mathbb{T}=\left\{0,\frac{1}{2},1\right\}$. From , we have $$y(t)=\frac{4-3Q}{4}t+Q\sum\limits_{k=1}^{2t}\frac{k}{4} = \begin{cases} 0, & \hbox{ if } t=0,\\ \frac{4-Q}{8}, & \hbox{ if } t=\frac{1}{2},\\ 1, & \hbox{ if } t=1. \end{cases}$$ Simple calculations show that $$\begin{split} & \mathcal{F}_{1}=\sum\limits_{k=0}^{1}\frac{1}{2} \left(y^{\Delta}\left(\frac{k}{2}\right)\right)^{2} =\frac{1}{2}\left(y^{\Delta}(0)\right)^{2} +\frac{1}{2}\left(y^{\Delta}\left(\frac{1}{2}\right)\right)^{2}=\frac{Q^{2}+16}{16}, \\ & \mathcal{F}_{2}=\sum\limits_{k=1}^{2}\frac{1}{4}k y^{\nabla}\left(\frac{k}{2}\right) =\frac{1}{4} y^{\nabla}\left(\frac{1}{2}\right)+\frac{1}{2}y^{\nabla}(1)=\frac{Q+12}{16} \end{split}$$ and $\mathcal{K}(y)=\frac{Q+12}{16}=1$. Therefore, $Q=4$, $\lambda=6$, and we have the extremal $$y(t)= \begin{cases} 0, & \hbox{ if } t \in \left\{0, \frac{1}{2}\right\},\\ 1, & \hbox{ if } t=1. \end{cases}$$ Conclusions {#sec:conc} =========== In this survey we collected some of our recent research on direct and inverse problems of the calculus of variations on arbitrary time scales. For infinity horizon variational problems on time scales we refer the reader to [@MyID:254; @MR2994055]. We started by studying inverse problems of the calculus of variations, which have not been studied before in the time-scale framework. First we derived a general form of a variational functional which attains a local minimum at a given function $y_{0}$ under Euler–Lagrange and strengthened Legendre conditions (Theorem \[P3:th:integrand:2\]). Next we considered a new approach to the inverse problem of the calculus of variations by using an integral perspective instead of the classical differential point of view. In order to solve the problem, we introduced new definitions: (i) self-adjointness of an integro-differential equation, and (ii) equation of variation. We obtained a necessary condition for an integro-differential equation to be an Euler–Lagrange equation on an arbitrary time scale $\mathbb{T}$ (Theorem \[P4:th:necess:EL:int:diff\]). It remains open the question of sufficiency. Finally, we developed the direct calculus of variations by considering functionals that are a composition of a certain scalar function with delta and nabla integrals of a vector valued field. For such problems we obtained delta-nabla Euler–Lagrange equations in integral form (Theorem \[P6:th:main\]), transversality conditions (Theorems \[P6:th:trans:initial\] and \[P6:th:trans:terminal\]) and necessary optimality conditions for isoperimetric problems (Theorem \[P6:th:conds:iso\]). To consider such general mixed delta-nabla variational problems on unbounded time scales (infinite horizon problems) remains also an open direction of research. Another interesting open research direction consists to study delta-nabla inverse problems of calculus of variations for composition functionals and their conservation laws [@MR2098297]. Acknowledgments {#acknowledgments .unnumbered} =============== This work was partially supported by Portuguese funds through the *Center for Research and Development in Mathematics and Applications* (CIDMA), and *The Portuguese Foundation for Science and Technology* (FCT), within project UID/MAT/ 04106/2013. The authors are grateful to two anonymous referees for their valuable comments and suggestions. [99]{} Ahlbrandt, C. D., Morian, C.: Partial differential equations on time scales. J. Comput. Appl. Math. **141**, no. 1-2, 35–55 (2002) Albu, I. D., Opriş, D.: Helmholtz type condition for mechanical integrators. Novi Sad J. 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--- abstract: 'Methods based on propagation of the one-body reduced density-matrix hold much promise for the simulation of correlated many-electron dynamics far from equilibrium, but difficulties with finding good approximations for the interaction term in its equation of motion have so far impeded their application. These difficulties include the violation of fundamental physical principles such as energy conservation, positivity conditions on the density, or unchanging natural orbital occupation numbers. We review some of the recent efforts to confront these problems, and explore a semiclassical approximation for electron correlation coupled to time-dependent Hartree-Fock propagation. We find that this approach captures changing occupation numbers, and excitations to doubly-excited states, improving over TDHF and adiabatic approximations in density-matrix propagation. However, it does not guarantee $N$-representability of the density-matrix, consequently resulting sometimes in violation of positivity conditions, even though a purely semiclassical treatment preserves these conditions.' author: - Peter Elliott - 'Neepa T. Maitra' title: 'Density-Matrix Propagation Driven by Semiclassical Correlation' --- INTRODUCTION {#sec:intro .unnumbered} ============ Time-resolved dynamics of electrons in atoms, molecules, and solids are increasingly relevant for a large class of problems today. The electrons and ions are excited far from their ground states in photo-induced processes such as in photovoltaics or laser-driven dynamics, and control of the dynamics on the attosecond time-scale is now experimentally possible. To theoretically model these processes an adequate accounting of electron correlation is required. Clearly solving the full time-dependent Schrödinger equation (TDSE) is impossible for more than a few electrons, and, moreover, the many-electron wavefunction contains much more information than is needed. Most observables of interest involve one- or two-body operators, suggesting that a description in terms of reduced variables would be opportune: in particular, obtaining directly just the one- and two-body time-dependent reduced density matrices (TD RDMs) [@RDMbook] would enable us to obtain any one- or two-body observable (e.g. electron densities, momentum profiles, double-ionization probabilities, etc). Even simpler, the theorems underlying time-dependent density functional theory (TDDFT), prove that [*any*]{} observable can be obtained from knowledge of simply the one-body density. However, hiding in any of these reduced descriptions is the complexity of the full many-body interacting electron problem, in the form of reconstruction functionals for the RDM case and exchange-correlation potentials as well as observable-functionals in the TDDFT case. In practice, these terms must be approximated, and intense research has been underway in recent years to determine approximations that are accurate but practically efficient. The temptation to simply use approximations that were developed for the ground-state cases, whose properties for the ground-state have been well-studied and understood, has proved profitable in some cases giving, for example, usefully accurate predictions of excitation spectra [@EFB07; @TDDFTbook2; @PG15]. But when used for non-perturbative dynamics, these same approximations can become rapidly unreliable. These problems will be reviewed in the next section. Solving the full TDSE scales exponentially with the number of electrons in the system, while the computational cost of propagating RDMs is in principle independent of the system-size. Classical dynamics of many-body systems on the other hand scales linearly with the number of particles, which raises the question of using semiclassical approaches to many-electron systems. Usually used for nuclear dynamics, a semiclassical wavefunction is built using classical dynamical information alone, in particular the phase arises from the classical action along the trajectory [@V28; @S81; @H81; @M98; @TW04; @K05]; in this way semiclassical methods can capture essential quantum phenomena such as interference, zero-point energy effects, and to some extent tunneling. We present some results from an approach that uses semiclassical electron dynamics to evaluate the correlation term in the propagation of the reduced density-matrix, with all the other terms in the equation of motion treated exactly, as introduced in Refs. ; thus it is a semiclassical-correlation driven time-dependent Hartree-Fock (TDHF). We study dynamics in perturbative and non-perturbative fields in two one-dimensional ($1d$) model systems of two electrons: one is a Hooke’s atom, and the other a soft-Coulomb Helium atom. The method improves over both TDHF and the pure semiclassical method for the dynamics and excitation spectra in the Hooke’s atom case, but gives unphysical negative density regions in the soft-Coulomb case. This is due to violation of $N$-representability conditions, even though the pure semiclassical dynamics and the TDHF on their own preserve these conditions. Propagating Reduced Density Matrices: A Brief Review {#sec:review .unnumbered} ==================================================== We start with the time-dependent Schrödinger equation for the electron dynamics of a given system (defined by the external potential, $v\ext(\br)$): i(x\_1..x\_N,t) = ( \_j -\_j\^2/2 + \_j v(\_j) + \_[i&lt;j]{}v(\_i,\_j) )(x\_1..x\_N,t) \[tdse\] where $x=(\br,\sigma)$ is a combined spatial and spin index, $\Psi$ is the wavefunction, and $v\inter$ is the $2$-body Coulomb interaction between the electrons, $v\inter(\br,\br') = 1/\vert\br-\br'\vert$. Atomic units are used throughout this paper, ($m_e = \hbar = e^2 = 1$). Additionally the initial wavefunction $\Psi_0$ must be specified in order to begin the propagation. However solving Eq. (\[tdse\]) is computationally an extremely costly exercise and becomes intractable as the number of electrons in the system grows. Thus we must seek an alternative approach that aims to reproduce the result of Eq. (\[tdse\]) but at a much more reasonable computational cost. Further, as mentioned in the introduction, the many-electron wavefunction contains far more information than one usually needs. Most observables that are experimentally measurable or of interest involve one- and two-body operators, such as the density e.g. in dipole/quadrupole moments, the momentum-density e.g. in Compton profiles, and pair-correlation functions in e.g. double-ionization. So a formulation directly in terms of one- and two-body density-matrices, bypassing the need to compute the many-electron wavefunction, would be more useful. This leads to the concept of reduced density-matrices, where the $p$-RDM involves tracing the full $N$-electron wavefunction over $N-p$ degrees of freedom: \_[p]{}(x’\_1..x’\_p, x\_1...x\_p, t) = dx\_[p+1]{}...dx\_N \^\*(x’\_1..x’\_p,x\_[p+1]{}...x\_N,t)(x\_1..x\_p,x\_[p+1]{}...x\_N,t) \[pRDM\] The diagonal of the $p$-RDM gives the $p$-body density, $\Gamma(x_1..x_p,t)$, the probability of finding any $p$ electrons at points $\br_1..\br_p$ with spin $\sigma_1..\sigma_p$ at time $t$. One can also define spin-summed RDMs: e.g. $\rho_1(\br',\br,t) = \sum_{\sigma, \sigma_2...\sigma_N}\rho_1(x',x,t)$. The Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy of equations of motion for the RDMs were written down sixty years ago [@BBGKY; @Bonitz]: The first in the hierarchy is that for $\rho_1$, which spin-summed is: i([**[r’]{}**]{},[**[r]{}**]{},t) = (-\^2/2 + v([**[r]{}**]{},t)+ ’\^2/2 - v([**[r’]{}**]{},t))\_1([**[r’]{}**]{},[**[r]{}**]{},t) + d\^3r\_2 f(,’,\_2) \_2(’,\_2,,\_2,t) \[rho1dot\] where f(,’,\_2) = v(,\_2)-v(’,\_2) The electron-interaction term in the equation for the 1RDM involves the 2RDM, whose equation of motion, the second in the hierarchy, i(’\_1,’\_2,\_1,\_2,t) = (\_[i=1,2]{}((-\_i\^2/2 + v([**[r\_i]{}**]{},t)) - (-\_i’\^2/2 + v([**[r\_i’]{}**]{},t)))+v(\_1,\_2)\ -v(\_1’,\_2’))\_2(’\_1,’\_2,\_1,\_2,t) + dr\_3(f(\_1,\_1’,\_3)+f(\_2,\_2’,\_3))\_3(\_1,\_2,\_3,\_1’,\_2’,\_3,t) involves the 3RDM, and so on. Solving the full hierarchy is equivalent to solving Eq. (\[tdse\]) and no less impractical for many-electron systems. The hierarchy is usually therefore truncated, typically using a “cluster expansion” where one reconstructs higher-order RDMs as antisymmetrized products of lower order ones plus a correlation term, sometimes referred to as a cumulant. Putting the correlation term to zero becomes exact for the case when the underlying wavefunction is a single Slater determinant (SSD). For example, in the case of truncation at the first equation, the equations reduce to TDHF, and, for a spin singlet of a closed shell system, $\rho_2$ in Eq. (\[rho1dot\]) is replaced by, \_2\^[SSD]{}(’,\_2,,\_2,t)=(\_2,t)\_1(’,,t)-\_1(’,\_2,t)\_1(\_2,,t) \[rho2SSD\] If instead, the truncation is done at the second equation in the BBGKY heirarchy, putting the correlation term to zero in $\rho_3$, one obtains the Wang and Cassing approximation [@WC85; @CVV93]. One has then the choice of propagating the equation for $\rho_2$ alone or propagating it simultaneously alongside the equation for $\rho_1$ [@AHNRL12]. However, recently it was found that, in the former case, propagating while neglecting the three-particle correlation term leads to the eventual violation of energy conservation [@LBSI15; @AHNRL12]. Now the $p$-RDM can be obtained from higher-order RDMs via contraction (i.e. partial trace), \_p(x\_1’..x\_p’, x\_1..x\_p,t) = dx\_[p+1]{} \_[p+1]{}(x\_1’..x\_p’,x\_[p+1]{}, x\_1..x\_p,x\_[p+1]{}, t), \[contcons\] as follows from the definition Eq. (\[pRDM\]). So an important condition to consider when formulating reconstructions is whether they are contraction-consistent, i.e. whether they satisfy Eq. (\[contcons\]). In fact, in Ref. , it was shown that the reconstruction approximation of Refs.  violated this condition; the underlying reason was that the spin-decomposed three-particle cumulant[@Mazz98], neglected in this approximation, has non-zero contraction. This realization enabled the authors of Ref.  to derive a “contraction-consistent” reconstruction for $\rho_3$ by including the part of the three-particle cumulant that has non-zero contraction, which fortunately is exactly known as a functional of the 2RDM. This was able to conserve energy in the dynamical simulations. However one cannot breathe easy just yet: propagation with this contraction-consistent reconstruction violated $N$-representability, another fundamental set of conditions that wreak havoc if not satisfied. Ref.  showed that contraction-consistency, and energy conservation, can be enforced if both 1RDM and 2RDM are propagated simultaneously, even while neglecting the three-particle correlation term. However again, $N$-representability was violated in this approach, leading to unphysical dynamics, instabilities and regions of negative densities. $N$-representability means that there exists an underlying many-electron wavefunction whose contraction via Eq. (\[pRDM\]) yields the matrix in question [@C63; @Mazz12]. For 1RDMs, the $N$-representability conditions are simple, and usually expressed in terms of its eigenvalues $\eta_j$, called natural orbital (NO) occupation numbers, as defined via: \_1(’,,t) = \_[j]{} \_j\^\*\_[j]{}(’,t)\_[j]{}(,t) \[NOexp\] for the spin-summed singlet case, where $\xi_{j}(\br,t)$ are natural orbitals. The $N$-representability conditions are that $0\le \eta_j \le 2$, and $\sum_j \eta_j = N$. (For the spin-resolved case, the first condition becomes $0\le \eta_j \le 1$). The 1RDM should be positive semi-definite, with trace equal N, and each eigenvalue bounded above by 2. If this is violated, densities can become negative in places, even when the norm is conserved. (Note that it can be shown that particle number is always conserved by any approximation [@AHNRL12]). For 2RDMs, only very recently has a complete set of conditions for ensemble $N$-representability been discovered [@Mazz12]; for pure states, not all the conditions are known, although some are. One important condition regards positive semi-definiteness of the 2RDM, which is challenging to maintain in dynamics when using approximate reconstructions. Yet without positive semi-definiteness, the propagation becomes unstable. The condition is in fact violated even by the contraction-consistent reconstruction introduced in Ref  and by the joint 1RDM and 2RDM propagation in Ref. . Even for ground-state problems (where the analog of the BBGKY equations is referred to as the contracted Schrödinger equation), the reconstruction functionals can violate such conditions, and iterative “purification” schemes have been introduced to yield self-consistent $N$-representable ground-state solutions [@Mazz02; @ACTPV05]. Even when the initial RDM satisfies $N$-representability conditions, one can find violations building up at later times in the propagation when approximate reconstruction functionals are used [@AHNRL12; @LBSI15]. Ref.  presented promising results where a dynamical purification scheme was applied at each time-step in the dynamics of molecules in strong-fields, leading to stable and accurate propagation. A similar method [@JD14] uses an energy-optimization procedure to obtain the 2RDM at each time-step while also enforcing various $N$-representability conditions, but the resulting dynamics is unable to change occupation numbers. From a different angle, it has been recently shown that the BBGKY equations can be recast into a Hamiltonian formulation [@R12], that opens the possibility of using advanced approximate methods of classical mechanics to analyze the equations and derive different reconstructions, in terms of equivalent classical variables. On the other hand, “time-dependent density-matrix functional theory”(TDDMFT) [@PGB07], which deals only with Eq. (\[rho1dot\]), proceeds from a somewhat different philosophy: the idea is that the 2RDM and all observables can in principle be obtained [*exactly*]{} from the time-dependent 1RDM due to the Runge-Gross theorem of TDDFT. The latter theorem [@RG84; @TDDFTbook2] proves that given an initial state, there is a one-to-one mapping between the time-evolving one-body density ($\rho(\br,t)$, diagonal of the 1RDM), and the externally applied potential. This means that, in principle, knowledge of $\rho(\br,t)$ is enough to determine the many-electron wavefunction, up to a purely time-dependent phase, and hence all pRDMs also. Since $\rho_1(\br,\br,t) = \rho(\br,t)$, this means in turn that $\rho_1(\br',\br,t)$ determines all properties of the system. The only assumption is that time-evolution of $\rho_1$ occurs in a local potential, meaning a multiplicative operator in space, which raises questions about $v$-representability [@G15]. There has been significant effort to approximate the 2RDM of Eq. (\[rho1dot\]) as a functional of $\rho_1$, or of its NOs and occupation numbers (Eq. (\[NOexp\])). A natural starting point is to insert the time-evolving 1RDM into an approximation developed for ground-state density-matrix functional theory [@Mueller84; @GU98; @BB02; @GPB05; @LeivaPiris05; @SDLG08], thus making an “adiabatic” approximation. These functionals can give very good approximations for ground-state properties, especially important for strongly-correlated systems where common approximations in alternative scalable methods like density-functional theory fail. However, when used in time-propagation, these same functionals keep the occupation numbers fixed [@GBG08; @RP10; @RP11; @AG10; @GGB10], which leads to erroneous dynamics. The first real-time non-perturbative application of TDDMFT [@RP10] resorted to an extra energy-minimizing procedure to determine occupation numbers at each time-step that resulted in time-evolving occupation numbers. By considering perturbations around the ground-state, a frequency-dependent response theory can be formalized [@GBG08] from which excitation energies can be computed, and it was shown that adiabatic functionals cannot capture double-excitations. Phase-including NO (PINO) functional theory [@GGB10; @GGB12; @MGGB13] has been introduced to overcome this problem. Here the functional depends on the phase of the NO, which extends out of the realm of TDDMFT since any phase-dependence of the NOs cancels out when $\rho_1$ is formed. Computationally, it has been argued that there is an advantage to propagating the NOs and occupation numbers directly instead of working with Eq. (\[rho1dot\]) [@AG10; @BB13]. By renormalizing the NOs via their occupation numbers, $\tilde\xi(t)\rangle = \sqrt{\eta_i(t)}\vert \xi(t)\rangle$, Refs.  showed that the equations of motion for the orbitals and those for the occupation numbers can be instead combined into one for each renormalized-orbital, which is numerically far more stable than the coupled equations for $\eta_i(t)$ and $\vert\xi_i(t)\rangle$. By studying model two-electron systems, for which the exact 2RDM is known in terms of the NOs and occupation numbers [@LS56], Refs.  could propagate the renormalized NOs in strong fields, with the only approximation being propagating a finite number of orbitals. Even relatively few orbitals gave very good results for challenging phenomena in correlated strong-field dynamics: autoionization, Rabi oscillations, and non-sequential double-ionization. For more than two electrons, one will however run again into the challenge of finding an accurate approximation for the 2RDM in terms of the renormalized NOs. The progress and applications in time-propagating RDMs as described above has been relatively recent (the use of RDMs in static electronic structure theory is much older and more established), although the BBGKY equations were written down sixty years ago. This is partly because of the instabilities stemming from violating $N$-representability when one truncates the hierarchy, or the inability of the adiabatic approximations for the functionals $\rho_2[\rho_1]$ to change occupation numbers, as reviewed above. TDDFT, on the other hand, formulated about thirty years ago [@RG84; @TDDFTbook2], has seen significant applications, especially for the calculations of excitations and response, while the past decade has witnessed exciting explorations into strong-field regime. As discussed above, the Runge-Gross states that all observables can be obtained from the one-body density, but, instead of working directly with the density, TDDFT operates by propagating a system of non-interacting electrons, the Kohn-Sham system, that reproduces the exact interacting density. The potential in the equation for the Kohn-Sham orbitals is defined such that $N$ non-interacting electrons evolving in it have the same time-dependent one-body density as the true interacting problem. One component of this potential is the exchange-correlation potential, a functional of the density $\rho$, the initial interacting state $\Psi$, and the initial choice of Kohn-Sham orbitals $\Phi$ in which to begin the propagation, $v\xc[\rho; \Psi_0, \Phi_0](\br, t)$. In almost all of the real-time non-perturbative calculations, an adiabatic approximation is used, in which the time-evolving density is inserted in a ground-state functional approximation, neglecting the dependence on the initial-states and the history of the density. This has produced usefully accurate results in a range of situations, e.g. modeling charge-transfer dynamics in photovoltaic candidates [@Carlo], ultrafast demagnetization in solids [@peter], dynamics of molecules in strong fields [@Bocharova]. Yet, there are errors, sometimes quite large [@RB09; @RN11; @RN12; @RN12b; @HTPI14], and investigation of the behavior of the [ *exact*]{} exchange-correlation potential reveals non-adiabatic features that are missing in the approximations in use today [@EFRM12; @FERM13; @RG12]. Further, when one is interested in observables that are not directly related to the density, additional “observable-functionals” are need to extract the information from the Kohn-Sham system: simply evaluating the usual operators on the Kohn-Sham wavefunction is not correct, even when the exact exchange-correlation potential functional is used [@WB06; @WB07; @RHCM09; @Henkel09]. Although it is in principle possible to extract all observables from the Kohn-Sham system, it is not known how. A final challenge is that Kohn-Sham evolution maintains constant occupation numbers, even with the exact functional, which results in strong exchange-correlation effects. The one-body nature of the Kohn-Sham potential means that the Kohn-Sham state remains a SSD throughout the evolution, even though the interacting system that it is modeling can dramatically change occupation numbers [@AG10; @LFSEM14], evolving far from an SSD (e.g. if a singlet single excitation gets appreciably populated during the dynamics). This leads to large features in the exact exchange-correlation potential that are difficult to model accurately. So, although computationally attractive, one could argue that operating via a non-interacting reference leads to a more difficult task for functionals. This has motivated the revisiting of the 1RDM dynamics in recent years as discussed above: any one-body observable can be directly obtained from the 1RDM using the usual operators, and one does not need the effective potential to “translate” from a non-interacting system to an interacting system. This suggests the terms in the equation that contain the many-body physics could be easier to model. As discussed in the previous paragraphs, the difficulty then is to come up with approximations for the 2RDM that can change occupation numbers and maintain $N$-representability of the 1RDM. In this work, we implement the idea first introduced in Ref. , using a semiclassical approximation for the correlation term in Eq. (\[rho1dot\]). Semiclassical-Correlation Driven TDHF {#sec:formalism .unnumbered} ===================================== From now on we deal only with singlet states and consider only the spin-summed RDMs. We begin by extracting the correlation component of Eq. (\[rho1dot\]), by decomposing $\rho_2$ via an SSD-contribution from Eq. (\[rho2SSD\]), plus a correlation correction: \_2(’,\_2,,\_2,t) = \_2\^[SSD]{}(’,\_2,,\_2,t) + c(’,\_2,,\_2,t) \[rho2\] Then the last term of Eq. (\[rho1dot\]) can be written d\^3r\_2 f(,’,\_2) \_2(’,\_2,,\_2,t) = (v(,t)-v(’,t))\_1(’,,t) + F(’,,t) + vc(’,,t) \[eepart\] where v(,t) = d\^3r’   is the familiar Hartree potential of DFT and \[fock\] F(’,,t) = -d\^3r\_2 f(,’,\_2) \_1(’,\_2,t)\_1(\_2,,t) is the Fock exchange matrix. The final term of Eq. (\[eepart\]) we refer to as the correlation potential: vc(’,,t) = d\^3r\_2 f(,’,\_2)c(’,\_2,,\_2,t) \[v2c\] Without $v\2c$, the propagation of Eq. \[rho1dot\] using the first two terms of Eq. \[eepart\], reduces to TDHF. In the present work we will evaluate $v\2c$ via semiclassical Frozen Gaussian dynamics, so turn now to a short review of this. Frozen Gaussian Dynamics {#frozen-gaussian-dynamics .unnumbered} ------------------------ Semiclassical methods aim to approximate the solution of Eq. (\[tdse\]) via an expansion in $\hbar$; the zeroth order recovers the classical limit while the first-order $O(\hbar)$ terms are referred to as the semiclassical limit. For propagation, a popular example is the Heller-Herman-Kluk-Kay (HHKK) propagator [@H81; @HK84; @KHD86; @TW04; @K05; @M98; @GX98; @S81; @V28] where the $N$-particle wavefunction at time $t$ as a function of the $3N$ coordinates, denoted ${\underline{\underline{\br}}}=\{ \br_1,...,\br_N\}$, is: \[frozg\] \^[FG]{}(,t) = \_t\_tC\_[,,t]{} e\^[iS\_t/]{}\_0\_0 \_0where $\{\bq_t,\bp_t\}$ are classical phase-space trajectories at time $t$ in $6N$-dimensional phase-space, starting from initial points $\{\bq_0,\bp_0\}$. In Eq. (\[frozg\]), $\langle{\underline{\underline{\br}}}\vert\bq\bp\rangle$ denotes the coherent state: = \_[j=1]{}\^[3N]{}()\^[1/4]{}e\^[-(r\_j-q\_j)\^2 + ip\_j(r\_j-q\_j)/]{} where $\gamma_j$ is a chosen width parameter. $S_t$ is the classical action along the trajectory $\{\bq_t,\bp_t\}$. Finally, each trajectory in the integrand is weighted by a complex pre-factor based on the monodromy (stability) matrix, $C_{\bq,\bp,t}$ which guarantees the solution is exact to first order in $\hbar$. Computing this pre-factor is the most time-consuming element in the integral, scaling cubically with the number of degrees of freedom. When the prefactor in Eq. (\[frozg\]) is set to unity, HHKK reduces to the simpler Frozen gaussian (FG) propagation [@H81], which is more efficient. As a consequence, it is no longer exact to order $\hbar$ and the results are no longer independent of the choice of width parameter $\gamma_j$, unlike in HHKK. For our calculations we take $\gamma_j=1$. Neither the HHKK propagation nor FG are unitary; typically we find the norm of the FG wavefunction decreases with time, and so we must renormalize at every time-step. In previous work[@EM11], the FG dynamics of electrons was investigated and found to give reasonable results for a number of different quantities and systems. Some of these will be referred to in the Results presented here. TDDMFG {#tddmfg .unnumbered} ------ In this work we will implement the idea of Ref. whereby a FG propagation, running parallel to a propagation of the 1RDM, is used to construct $v\2c$, which is then used in Eq. (\[eepart\]) and Eq. (\[rho1dot\]) to propagate the 1RDM. From the FG wavefunction given by Eq. (\[frozg\]), the 1RDM and 2RDM are computed and then used to construct: \[v2cfg\] vc(’,, t) = d\^3r\_2 f(,’,\_2)c(’,\_2,,\_2,t) where $\rho\FG\2c$ is found by inverting Eq. (\[rho2\]): c(’,\_2,,\_2,t)=\_2(’,\_2,,\_2,t)-(\_2,t)\_1(’,,t)+\_1(’,\_2,t)\_1(\_2,,t) We then insert this into Eq. \[v2cfg\], and propagate Eq. \[rho1dot\] with the last term evaluated using Eq. \[v2cfg\]. We refer to this coupled dynamics as TDDMFG, meaning time-dependent density-matrix propagation with frozen-gaussian correlation. The scheme of Ref.  takes advantage of the “forward-backward” nature of the propagation of the 2RDM (i.e there is both a $\Psi(t)$ and a $\Psi^*(t)$), which leads to some cancellation of the oscillatory phase for more than two electrons. We also observe that the spatial permutation symmetry of the initial-state is preserved during the evolution (since the Hamiltonian is for identical particles, exchanging coordinate-momentum pairs of two electrons does not change the action). We will study here two-electron singlet states where the wavefunction is spatially-symmetric under exchange of particles. In the FG propagation, although the energy of each classical trajectory is conserved in the absence of external fields, the energy of the FG wavefunction constructed from these trajectories is not guaranteed to be. As noted earlier, the norm is not conserved either and the wavefunction must be renormalized at each time. Thus, in general it is unlikely that energy will be conserved in the TDDMFG scheme. Computational Details {#computational-details .unnumbered} --------------------- The phase-space integral in Eq. (\[frozg\]) is performed using Monte Carlo integration, with the distribution of $M$ initial phase-space points weighted according to a simple gaussian initial distribution. In principle this method scales as $\sqrt{M}$, however the oscillatory phase from the action $S_t$ can make the FG propagation difficult to converge and thus a large number of trajectories are often needed. This, in turn, means that parallelization of the numerical computation of Eq. (\[frozg\]) is needed. Fortunately the main task is “embarrassingly parallel” as each classical trajectory can be calculated separately, however construction in real space of the FG wavefunction, 2RDM, 1RDM, and Eq. (\[v2cfg\]) is time-consuming and also required parallelization (in a manner similar to the Fock exchange matrix calculations discussed below). To avoid performing these costly procedures at every time step, we used a linear-interpolation of Eq. (\[v2cfg\]) which only required its construction every $D_V$ timesteps. We tested that the results were converged with respect to $D_V$, finding accurate results for values as high as $D_V=200$ for a timestep of $dt=0.001$ au for the cases we studied. The 1RDM propagation was performed via the predictor-corrector method combined with an Euler forward-stepping algorithm. All quantities were calculated on a real-space grid and the derivatives in Eq. (\[rho1dot\]) were done using a $3$-point finite difference rule. Calculation of the Fock exchange matrix, Eq. (\[fock\]), also required parallelization, as it has the worst scaling (cubic) with respect to the number of grid points of the remaining quantities. Parallelization often contains additional subtleties which can make the problem non-trivial, thus in order to parallelize efficiently, the problem was first transformed to resemble a more typical problem in parallel computing. To detail this procedure, it is convenient to switch to a matrix representation: \_[nm]{}=\_1(\_n,\_m) where $\br_n$ is the $n$^th^ point of the real-space grid. We then define a new matrix \_[nm]{} = A\_[nm]{}\_[nm]{} where A\_[nm]{} = v(\_n,\_m) which is then used to construct = C\_[nm]{} = \_k \_[nk]{}\_[km]{} The Fock exchange matrix can then be written as = (\^- ) in the case when the Fock integral is evaluated via quadrature and $\Delta x$ is the grid spacing. Thus, we have reduced the calculation of the Fock exchange integral to the calculation of $\mathbf{C}$, which only involves a matrix-matrix multiplication. In parallel computing, matrix-matrix multiplication is a well-studied problem for which standard solutions exist and thus could be easily implemented in our code without additional difficulty. RESULTS {#sec:results .unnumbered} ======= In this section we present the results of the TDDMFG formulation for various time-dependent problems and compare the results to the TDHF (i.e. $v\2c=0$), the pure FG, and the exact cases. In order to compare to exact results we work in $1d$ and focus on two-electron systems, as it allows us to solve the full TDSE in a reasonable time with reasonable computational resources. We first tested our TDDMFG propagation algorithm by coupling to the exact dynamics, i.e. we used the exact wavefunction to calculate the exact $v\2c$ at each time which is then used within the 1RDM propagation to verify it recovers the exact dynamics. To remove any error due to the initial ground state we start the FG dynamics in the exact GS wavefunction and the 1RDM propagation in the exact GS density matrix. Each of TDHF and FG calculations on their own can yield reasonably good results for particular cases, thus the goal of the coupled TDDMFG propagation should be to either improve upon both, or at least, improve the results in scenarios where one or the other performs poorly. Hooke’s atom {#hookes-atom .unnumbered} ------------ (12,6.2) (-4.9,-3.6) (12,6.2) We begin by studying Hooke’s atom in $1d$, which consists of a harmonic external potential: v(x) = k\_0x\^2 and a softened Coulomb interaction between the electrons: v(x’,x) = In our first application we drive the system by applying an oscillating quadrupole field v(x,t) = k(t)x\^2 where $k(t)=-0.025 \sin(2t)$ and $k_0=1$. The frequency of this perturbation is chosen to be resonant with an allowed excitation of the system. We then compare in Fig. \[f:hquad\] the change in the quadrupole moment relative to the ground state quadrupole ($Q_0$), Q(t) = dx  (x)x\^2 - Q\_0 as computed via exact propagation, TDHF, FG semiclassical dynamics, and TDDMFG propagation. In this case we see an improvement to both the TDHF and the FG calculations. While the exact quadrupole is seen to continue increasing in amplitude over time, the TDHF does not, and in fact oscillates in a beating pattern. This is partly due to the fact that TDHF spectra cannot describe this particular excitation (which will be discussed in more detail later) and leads to an off-resonance Rabi oscillation. However even running TDHF at the resonance frequency of TDHF does not show Rabi oscillations either; although the quadrupole begins to grow, it ultimately fails because of the spurious detuning effect explained in Ref. . The FG, in contrast, overestimates the amplitude of the oscillations, while the TDDMFG coupled dynamics lies in between these two extremes and is much closer to the exact result. In previous work we found that the quadrupole is more sensitive than other quantities to the number of trajectories used in our FG calculation, in this case $100224$. Thus we could expect better agreement if we further increase the number of trajectories. It is interesting to note that all three approximate calculations work reasonably well for the first $10$ au. (12,6.2) (-4.9,-3.6) (12,6.2) Buoyed on by this success, we next examine the NO occupation numbers to probe the 1RDM in more detail than the quadrupole, an averaged expectation value, can provide. As noted in the earlier review, a major shortcoming of adiabatic functionals in TDDMFT is their inability to change these occupations. As can be seen in Fig. \[f:hNOs\] this behavior manifests itself as straight lines, labelled ATDDM, constant at the initial state NO occupations. This is also true for the TDHF case shown previously. Since we spin-sum, the NOs go between 0 and 2, and we plot the highest occupied NO occupation, which this case begins very close to $2$, indicating the initial state is strongly of an SSD character. The time-dependence of this occupation is shown in Fig. \[f:hNOs\]. The exact value decreases towards a value of $1$ as the system becomes excited and the wavefunction moves away from SSD-like state. In Ref. , it was shown that FG propagation can quite accurately capture the evolution of the NO occupations, so the question is whether the coupled dynamics of TDDMFG is also able to do so. Examining the TDDMFG values, we see that TDHF coupled to the FG correlation is able to evolve the occupations accurately. In fact TDDMFG is slightly better than the pure FG values where FG has sometimes spuriously large oscillations. Although the amplitude of the oscillations of the TDDMFG quadrupole are closer to those of the exact than the FG oscillations, the phase of the oscillations of the latter is closer to the exact case than the TDDMFG case. This becomes more evident carrying the propagation out to longer times. This can be understood from considering the resonant frequencies of the system: the frequency of the perturbation $k(t)$ is on-resonance with an excitation of the exact system which FG in fact correctly describes. The TDDMFG excitation frequency is however shifted slightly leading to the difference in phase observed here. We will now examine the respective excitation frequencies of each method in more detail. EXACT TDHF FG TDDMFG ---------- ------ ----- -------- 1.000000 0.99 1.0 0.99 1.734522 1.86 1.6 1.58 2.000000 - 2.0 1.86 2.734522 2.79 2.6 2.58 3.000000 - 3.0 2.77 : \[t:hooke\]The singlet excitation frequencies $\omega_n=E_n-E_0$, where the ground-state energy is $E_0 = 1.774040$ a.u., solved exactly for Hooke’s atom, and the corresponding TDHF,FG, and TDDMFG values as calculated by real-time linear response. Turning to how well the TDDMFG captures excitation spectra, we focus in particular on so-called double excitations (defined loosely as excitations which have a large fraction of a doubly excited state character, with respect to the SSD ground state of a non-interacting reference system [@EGCM11]). It is known that these excitations are missing from all TDDFT spectra calculations within the adiabatic approximation. Since TDHF is equivalent to adiabatic exact exchange in TDDFT for the $2$-electron case studied here, we do not see double excitations in the uncoupled 1RDM TDHF spectra. This point is illustrated in the upper panel of Fig. \[f:hqspec\] which shows the TDHF power spectra. The spectra are calculated via the linear response method, utilizing a ’kicked’ initial state defined as: \_0(x,y) = e\^[ik(x\^n+y\^n)]{}(x,y) where $\Psi\GS$ is the ground-state wavefunction, $k$ is a small constant, and $n$ is an integer (we define a quadratic kick as $n=2$ and a cubic kick as $n=3$). An expression for the initial 1RDM can be easily derived from this. A dipole kick ($n=1$) is commonly used when calculating optical spectra as it corresponds to an electric field consisting of a $\delta-$function at time $t=0$ which excites all dipole allowed excitations [@YNIB06]. However due to symmetries of Hooke’s atom, to access the double excitations, higher moment kicks were necessary [@EM11]. To obtain the spectra, for each run we calculate the appropriate moment (e.g. quadrupole moment for quadratic kicks) and Fourier transform to frequency space to reveal the excitation peaks [@EM11]. In the TDHF case, we do not see the pair of excitations peaks at frequencies (1.73,2.0), nor the pair (2.73,3.0) but instead see a single peak in between. This behavior is commonly seen for TDDFT calculations with an adiabatic approximation, where a frequency-dependent XC kernel is required to split the peak into two separate excitations [@MZCB04; @EGCM11]; any adiabatic approximation in TDDMFT will also only display one peak [@GBG08]. Moving to the lower panel of Fig. \[f:hqspec\], we plot the TDDMFG spectra calculated in the same manner. It can be seen that including $v\FG\2c$ into the TDHF propagation correctly splits the single peak into two peaks, for both the quadratic and cubic kick cases. Thus we have demonstrated that our coupled dynamics does indeed capture double excitations. Identifying the position of the peaks, we compare in Table \[t:hooke\] the values given by each method for the lowest 5 excitations of Hooke’s atom. It was found in Ref. that FG on its own also describes double excitations quite well, and in fact is exact for certain excitations in Hooke’s atom. This is due to the fact that the Hamiltonian becomes separable in center-of-mass and relative coordinates, and that in the center-of-mass coordinate is a simple harmonic oscillator [@EM11]. It is well-known that harmonic potentials are a special case for semiclassical methods as they often perform exactly. With this in mind, although the value of the TDDMFG frequencies are worse than the pure FG values, they are competing with a special case, but in fact the splitting between the peaks is better described by the TDDMFG. In particular, in the second multiplet, the exact splitting is 0.27, while that of the TDDMFG is 0.28, improved over the bare FG result of 0.4 (and obviously over TDHF where there is no peak). We emphasize again that these excitations are completely missing in the TDHF case, or in any adiabatic TDDFT or TDDMFT functional. It is better to have the excitations shifted slightly from the exact result than to not describe them at all. (12,15.1) (-4.9,-3.4) (12,6.2) (-4.9,4.1) (12,6.2) Soft-Coulomb Helium {#soft-coulomb-helium .unnumbered} ------------------- We now move to the more realistic case of soft-Coulomb Helium where the external potential is: v(x) = - which mimics the $3d$ case as for large $x$ it decays as $-1/|x|$. In the previous case of Hooke’s atom, we saw that the problem was well described by the FG dynamics whereas the TDHF performed poorly (i.e. not changing the NO occupations or capturing double excitations). Driving the TDHF with FG correlation in TDDMFG interestingly improved over FG for the NO occupations and quadrupole moment, with slightly worse performance for the double-excitations. For dynamics in the soft-Coulomb Helium case we will see, in contrast, that the FG is worse than the TDHF for some quantities. Will the coupled dynamics of TDDMFG improve the situation? As detailed in Ref. , this case is much more difficult for the FG method due to classically auto-ionizing trajectories (where one electron gains energy from the other and ionizes while the other slips below the zero-point-energy), thus a far greater number of trajectories are required: . Ref.  discussed how in a truly converged Frozen Gaussian calculation, the contributions from these unphysical trajectories cancel each other out, but a very large number of trajectories are required; otherwise methods to cut out their contribution to the semiclassical integral can be used. In the presented calculations, $2000448$ trajectories were used and all were kept. We apply a strong laser pulse with a linearly-switched-on electric field: (t) = (0.5t) { [c l]{} & t20\ 1 & t&gt;20 . \[eq:trapeps\] which is included in our Hamiltonian via the dipole approximation, i.e. $\delta v\ext(x) = \epsilon(t)x $. We begin by examining the 1RDM itself at time $T=10$ au: both the real and imaginary parts are shown in Fig. \[f:dmR10\]. At this time, while the structure of the FG 1RDM is broadly correct, it can be seen that the TDHF 1RDM is much closer to the exact. The TDDMFG 1RDM also captures more of the correct structure compared to the pure FG case, although it generally overestimates the peaks and valleys. Thus, while the $v\FG\2c$ is constructed from the poorer FG calculation, the TDDMFG follows more closely the more accurate TDHF description. (16.15,8) (-4.0,-4.6) (7,7) (4.2,-4.6) (7,7) We next turn to the dipole moment which is plotted in Fig. \[f:sdips\]. The TDHF (not shown) essentially matches the exact case, whereas the FG performs quite poorly, particularly during the second optical cycle. The TDDMFG, in contrast, is performing particularly well and follows very closely the exact result, even at times greater than $T=10$ au, likely due to the guidance of the TDHF component in the evolution. At this point one might conclude that the TDDMFG is behaving correctly, however the good results for the 1RDM and dipole moment are masking the fact that the underlying description suffers from a major error, described below. (12,6.2) (-4.9,-3.6) (12,6.2) As was the case for Hooke’s atom, a more thorough examination of the method is given by studying the NO occupations. The highest two NO occupations are plotted in Fig. \[f:sNOs\] where the strong field causes a large change in their values. In fact we see that the FG description of the NO occupations is working better than we previously anticipated, albeit overestimating their change. Again the TDHF occupations (not shown) are constant, fixed at their initial values. At $T=10$ au, the exact 1RDM is still dominated by the highest occupied natural orbital, explaining why the TDHF appeared so good at this time. Unfortunately in the TDDMFG case, we see that the highest occupation rises above $2$ thus violating the exact condition for $N$-representability of the 1RDM (positive semidefiniteness). The effect of this is quite drastic as the density develops negative regions, due to having negative occupations (the sum of the occupations remains $2$, thus an increase above $2$ is accompanied by negative values). At $T=10$ au the value of the highest NO is only slightly above $2$ and so this bad behavior has yet to truly manifest itself. The FG NO occupation at $T=10$ au is underestimated, consistent with the FG 1RDM being not so accurate. At later times, none of the methods provide a particularly good description of the 1RDM structure, with the TDHF remaining the closest, despite its constant occupation number, but the TDHF momentum densities are not good. The TDDMFG 1RDM resembles the TDHF but with exaggerated highs and lows, and unphysical negative regions, manifest also in the momentum density. (12,6.2) (-4.9,-3.6) (12,6.2) It is a frustrating situation where neither the FG and TDHF on their own violate the N-representability condition, and FG does evolve the occupation numbers unlike any adiabatic approximation, however coupling FG correlation to TDHF in the TDDMFG leads violation of $N$-representability. We speculate this is due to a mismatch in correlation potential and the Hartree-exchange terms as there is no mechanism to provide feedback between the two calculations. Implementing a dynamical purification scheme along the lines of that in Ref.  that iteratively decreases the magnitude of the negative occupation numbers should be investigated. CONCLUSIONS {#sec:Conclusions .unnumbered} =========== The TDDMFG dynamics which uses a Frozen Gaussian calculation to construct an approximation to the correlation potential in 1RDM propagation gives mixed results. On one hand, it was shown for Hooke’s atom to be a significant improvement over all adiabatic 1RDM functionals as it can vary the natural orbital occupations reasonably accurately. Furthermore it was shown that double excitations, which are difficult to capture with the commonly-used TDDFT method and adiabatic TDDMFT, can be accurately described with the TDDMFG formalism. On the other hand, for soft-Coulomb Helium the method was seen to fail drastically giving the unphysical result of negative density due to violation of an exact constraint. Further work is required to understand why this violation occurs and then hopefully to then use this information to prevent it from occurring. This same problem, violation of the positive semi-definiteness, was also found in other recent approaches propagating RDMs with approximate correlation terms [@AHNRL12; @LBSI15]. Dynamical purification schemes along the line of that successfully used in Ref.  could be very helpful here. Indeed understanding how the FG correlation potential changes NO occupation or includes double excitations could be used to construct better approximations for TDDMFT. Finally, in this work we studied systems of only $2$ electrons, whereas we might expect that (semi)classical methods work best for large numbers of particles, and it remains to be seen whether the problems we encountered become less significant for larger systems. An advantage of using a semiclassical scheme to evaluate the correlation term, is that initial state dependence is automatically taken care of: in general reconstructions, whether one begins in an excited state or ground-state, the same approximation for the 2RDM in terms of the 1RDM is assumed. This is known to be incorrect; several different initial wavefunctions and different initial 2RDM’s can give rise to the same initial 1RDM [@EM12; @MB01]. The resulting correlation effect is clearly different depending on the wavefunction, but this effect is ignored in all reconstructions in use today. For this reason, it seems worthwhile to pursue further investigations of a semiclassical-correlation driven TDHF, once the $N$-representability problem is taken care of, especially since in many simulations of non-equilibrium dynamics of interest today, the problem starts in a photo-excited state. 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--- abstract: 'There is renewed interest in formulating integration as a statistical inference problem, motivated by obtaining a full distribution over numerical error that can be propagated through subsequent computation. Current methods, such as Bayesian Quadrature, demonstrate impressive empirical performance but lack theoretical analysis. An important challenge is therefore to reconcile these probabilistic integrators with rigorous convergence guarantees. In this paper, we present the first probabilistic integrator that admits such theoretical treatment, called Frank-Wolfe Bayesian Quadrature (FWBQ). Under FWBQ, convergence to the true value of the integral is shown to be up to exponential and posterior contraction rates are proven to be up to super-exponential. In simulations, FWBQ is competitive with state-of-the-art methods and out-performs alternatives based on Frank-Wolfe optimisation. Our approach is applied to successfully quantify numerical error in the solution to a challenging Bayesian model choice problem in cellular biology.' author: - | François-Xavier Briol\ Department of Statistics\ University of Warwick\ `f-x.briol@warwick.ac.uk`\ Chris J. Oates\ School of Mathematical and Physical Sciences\ University of Technology, Sydney\ `christopher.oates@uts.edu.au`\ Mark Girolami\ Department of Statistics\ University of Warwick\ & The Alan Turing Institute for Data Science\ `m.girolami@warwick.ac.uk`\ Michael A. Osborne\ Department of Engineering Science\ University of Oxford\ `mosb@robots.ox.ac.uk`\ bibliography: - 'NIPS\_bib.bib' title: 'Frank-Wolfe Bayesian Quadrature: Probabilistic Integration with Theoretical Guarantees' --- Introduction ============ Computing integrals is a core challenge in machine learning and numerical methods play a central role in this area. This can be problematic when a numerical integration routine is repeatedly called, maybe millions of times, within a larger computational pipeline. In such situations, the cumulative impact of numerical errors can be unclear, especially in cases where the error has a non-trivial structural component. One solution is to model the numerical error statistically and to propagate this source of uncertainty through subsequent computations. Conversely, an understanding of how errors arise and propagate can enable the efficient focusing of computational resources upon the most challenging numerical integrals in a pipeline. Classical numerical integration schemes do not account for prior information on the integrand and, as a consequence, can require an excessive number of function evaluations to obtain a prescribed level of accuracy [@OHagan1984]. Alternatives such as Quasi-Monte Carlo (QMC) can exploit knowledge on the smoothness of the integrand to obtain optimal convergence rates [@Dick2010]. However these optimal rates can only hold on sub-sequences of sample sizes $n$, a consequence of the fact that all function evaluations are weighted equally in the estimator [@Owen2014]. A modern approach that avoids this problem is to consider arbitrarily weighted combinations of function values; the so-called *quadrature rules* (also called cubature rules). Whilst quadrature rules with non-equal weights have received comparatively little theoretical attention, it is known that the extra flexibility given by arbitrary weights can lead to extremely accurate approximations in many settings (see applications to image de-noising [@Chen2015] and mental simulation in psychology [@Hamrick2013mental]). Probabilistic numerics, introduced in the seminal paper of [@Diaconis1988], aims at re-interpreting numerical tasks as inference tasks that are amenable to statistical analysis.[^1] Recent developments include probabilistic solvers for linear systems [@Hennig2015solvers] and differential equations [@Conrad2015; @Schober2014]. For the task of computing integrals, Bayesian Quadrature (BQ) [@OHagan1991] and more recent work by [@Oates2015] provide probabilistic numerics methods that produce a full posterior distribution on the output of numerical schemes. One advantage of this approach is that we can propagate uncertainty through all subsequent computations to explicitly model the impact of numerical error [@Hennig2015ProbNum]. Contrast this with chaining together classical error bounds; the result in such cases will typically be a weak bound that provides no insight into the error structure. At present, a significant shortcoming of these methods is the absence of theoretical results relating to rates of posterior contraction. This is unsatisfying and has likely hindered the adoption of probabilistic approaches to integration, since it is not clear that the induced posteriors represent a sensible quantification of the numerical error (by classical, frequentist standards). This paper establishes convergence rates for a new probabilistic approach to integration. Our results thus overcome a key perceived weakness associated with probabilistic numerics in the quadrature setting. Our starting point is recent work by [@Bach2012], who cast the design of quadrature rules as a problem in convex optimisation that can be solved using the Frank-Wolfe (FW) algorithm. We propose a hybrid approach of [@Bach2012] with BQ, taking the form of a quadrature rule, that (i) carries a full probabilistic interpretation, (ii) is amenable to rigorous theoretical analysis, and (iii) converges orders-of-magnitude faster, empirically, compared with the original approaches in [@Bach2012]. In particular, we prove that super-exponential rates hold for posterior contraction (concentration of the posterior probability mass on the true value of the integral), showing that the posterior distribution provides a sensible and effective quantification of the uncertainty arising from numerical error. The methodology is explored in simulations and also applied to a challenging model selection problem from cellular biology, where numerical error could lead to mis-allocation of expensive resources. Background {#section:sigmapoint} ========== Quadrature and Cubature Methods ------------------------------- Let $\mathcal{X} \subseteq \mathbb{R}^d$ be a measurable space such that $d \in \mathbb{N}_{+}$ and consider a probability density $p(x)$ defined with respect to the Lebesgue measure on $\mathcal{X}$. This paper focuses on computing integrals of the form $\int f(x) p(x) \mathrm{d}x$ for a test function $f:\mathcal{X} \rightarrow \mathbb{R}$ where, for simplicity, we assume $f$ is square-integrable with respect to $p(x)$. A *quadrature rule* approximates such integrals as a weighted sum of function values at some design points $\{x_i\}_{i=1}^n \subset \mathcal{X}$: $$\int_\mathcal{X} f(x) p(x) \mathrm{d}x \approx \sum_{i=1}^n w_i f(x_i).$$ Viewing integrals as projections, we write $p[f]$ for the left-hand side and $\hat{p}[f]$ for the right-hand side, where $\hat{p} = \sum_{i=1}^n w_i \delta(x_i)$ and $\delta(x_i)$ is a Dirac measure at $x_i$. Note that $\hat{p}$ may not be a probability distribution; in fact, weights $\{w_i\}_{i=1}^n$ do not have to sum to one or be non-negative. Quadrature rules can be extended to multivariate functions $f:\mathcal{X} \rightarrow \mathbb{R}^d$ by taking each component in turn. There are many ways of choosing combinations $\{x_i,w_i\}_{i=1}^n$ in the literature. For example, taking weights to be $w_i = 1/n$ with points $\{x_i\}_{i=1}^n$ drawn independently from the probability distribution $p(x)$ recovers basic Monte Carlo integration. The case with weights $w_i= 1/n$, but with points chosen with respect to some specific (possibly deterministic) schemes includes kernel herding [@Chen2010] and Quasi-Monte Carlo (QMC) [@Dick2010]. In Bayesian Quadrature, the points $\{x_i\}_{i=1}^n$ are chosen to minimise a posterior variance, with weights $\{w_i\}_{i=1}^n$ arising from a posterior probability distribution. Classical error analysis for quadrature rules is naturally couched in terms of minimising the worst-case estimation error. Let $\mathcal{H}$ be a Hilbert space of functions $f: \mathcal{X}\rightarrow \mathbb{R}$, equipped with the inner product $\langle \cdot,\cdot \rangle_{\mathcal{H}}$ and associated norm $\|\cdot\|_{\mathcal{H}}$. We define the *maximum mean discrepancy* (MMD) as: $$\text{MMD}\big(\{x_i,w_i\}_{i=1}^n \big) {\coloneqq}\sup_{\substack{f \in \mathcal{H}: \|f\|_{\mathcal{H}}=1}} \big|p[f] - \hat{p}[f] \big|.$$ The reader can refer to [@Sriperumbudur2009] for conditions on $\mathcal{H}$ that are needed for the existence of the MMD. The rate at which the MMD decreases with the number of samples $n$ is referred to as the ‘convergence rate’ of the quadrature rule. For Monte Carlo, the MMD decreases with the slow rate of $\mathcal{O}_P(n^{-1/2})$ (where the subscript $P$ specifies that the convergence is in probability). Let $\mathcal{H}$ be a RKHS with reproducing kernel $k: \mathcal{X}\times \mathcal{X} \rightarrow \mathbb{R}$ and denote the corresponding canonical feature map by $\Phi(x) = k(\cdot,x)$, so that the mean element is given by $\mu_p(x) = p[\Phi(x)] \in \mathcal{H}$. Then, following [@Sriperumbudur2009] $$\text{MMD}\big(\{x_i,w_i\}_{i=1}^n \big) = \| \mu_p - \mu_{\hat{p}} \|_{\mathcal{H}}.$$ This shows that to obtain low integration error in the RKHS $\mathcal{H}$, one only needs to obtain a good approximation of its mean element $\mu_p$ (as $\forall f \in \mathcal{H}$: $p[f] = \langle f , \mu_p \rangle_{\mathcal{H}}$). Establishing theoretical results for such quadrature rules is an active area of research [@Bach2015]. Bayesian Quadrature {#subsec:BQ} ------------------- Bayesian Quadrature (BQ) was originally introduced in [@OHagan1991] and later revisited by [@Rasmussen2003; @Gunter2014] and [@Osborne2012]. The main idea is to place a functional prior on the integrand $f$, then update this prior through Bayes’ theorem by conditioning on both samples $\{x_i\}_{i=1}^n$ and function evaluations at those sample points $\{\mathrm{f}_i\}_{i=1}^n$ where $\mathrm{f}_i = f(x_i)$. This induces a full posterior distribution over functions $f$ and hence over the value of the integral $p[f]$. The most common implementation assumes a Gaussian Process (GP) prior $f \sim \mathcal{GP}(0,k)$. A useful property motivating the use of GPs is that linear projection preserves normality, so that the posterior distribution for the integral $p[f]$ is also a Gaussian, characterised by its mean and covariance. A natural estimate of the integral $p[f]$ is given by the mean of this posterior distribution, which can be compactly written as $$\hat{p}_{\text{BQ}}[f] = \mathrm{z}^T K^{-1} \mathrm{f}. \label{BQmeaneq}$$ where $\mathrm{z}_i = \mu_p(x_i)$ and $K_{ij} = k(x_i,x_j)$. Notice that this estimator takes the form of a quadrature rule with weights $\mathrm{w}^{\text{BQ}} =\mathrm{z}^T K^{-1}$. Recently, [@Sarkka2015] showed how specific choices of kernel and design points for BQ can recover classical quadrature rules. This begs the question of how to select design points $\{x_i\}_{i=1}^n$. A particularly natural approach aims to minimise the posterior uncertainty over the integral $p[f]$, which was shown in [@Huszar2012 Prop. 1] to equal: $$v_{\text{BQ}}\big(\{x_i\}_{i=1}^n \big) \; = \; p [\mu_p] - \mathrm{z}^T K^{-1} \mathrm{z} \; = \; \text{MMD}^2\big(\{x_i,w_i^{\text{BQ}}\}_{i=1}^n \big). \label{eq:variance}$$ Thus, in the RKHS setting, minimising the posterior variance corresponds to minimising the worst case error of the quadrature rule. Below we refer to Optimal BQ (OBQ) as BQ coupled with design points $\{x_i^\text{OBQ}\}_{i=1}^n$ chosen to globally minimise . We also call Sequential BQ (SBQ) the algorithm that greedily selects design points to give the greatest decrease in posterior variance at each iteration. OBQ will give improved results over SBQ, but cannot be implemented in general, whereas SBQ is comparatively straight-forward to implement. There are currently no theoretical results establishing the convergence of either BQ, OBQ or SBQ. [*Remark:*]{} is independent of observed function values $\mathrm{f}$. As such, no active learning is possible in SBQ (i.e. surprising function values never cause a revision of a planned sampling schedule). This is not always the case: For example [@Gunter2014] approximately encodes non-negativity of $f$ into BQ which leads to a dependence on $\mathrm{f}$ in the posterior variance. In this case sequential selection becomes an [*active*]{} strategy that outperforms batch selection in general. Deriving Quadrature Rules via the Frank-Wolfe Algorithm {#section:introFW} ------------------------------------------------------- Despite the elegance of BQ, its convergence rates have not yet been rigorously established. In brief, this is because $\hat{p}_{\text{BQ}}[f]$ is an orthogonal projection of $f$ onto the [*affine*]{} hull of $\{\Phi(x_i)\}_{i=1}^n$, rather than e.g. the [*convex*]{} hull. Standard results from the optimisation literature apply to bounded domains, but the affine hull is not bounded (i.e. the BQ weights can be arbitrarily large and possibly negative). Below we describe a solution to the problem of computing integrals recently proposed by [@Bach2012], based on the FW algorithm, that restricts attention to the (bounded) convex hull of $\{\Phi(x_i)\}_{i=1}^n$. The Frank-Wolfe (FW) algorithm (Alg. \[alg:FWalgorithm\]), also called the conditional gradient algorithm, is a convex optimization method introduced in [@Frank1956]. It considers problems of the form $\min_{g \in \mathcal{G}} J(g)$ where the function $J:\mathcal{G}\rightarrow\mathbb{R}$ is convex and continuously differentiable. A particular case of interest in this paper will be when the domain $\mathcal{G}$ is a compact and convex space of functions, as recently investigated in [@Jaggi2013]. These assumptions imply the existence of a solution to the optimization problem. At each iteration $i$, the FW algorithm computes a linearisation of the objective function $J$ at the previous state $g_{i-1} \in \mathcal{G}$ along its gradient $(DJ)(g_{i-1})$ and selects an ‘atom’ $\bar{g}_i \in \mathcal{G}$ that minimises the inner product a state $g$ and $(DJ)(g_{i-1})$. The new state $g_{i} \in \mathcal{G}$ is then a convex combination of the previous state $g_{i-1}$ and of the atom $\bar{g}_{i}$. This convex combination depends on a step-size $\rho_i$ which is pre-determined and different versions of the algorithm may have different step-size sequences. function $J$, initial state $g_1=\bar{g}_1 \in \mathcal{G}$ (and, for FW only: step-size sequence $\{\rho_i\}_{i=1}^n$). Compute $\bar{g}_i = \text{argmin}_{g \in \mathcal{G}} \big\langle g , (DJ)(g_{i-1}) \big\rangle_{\times} $ Update $g_{i} = (1 - \rho_i) g_{i-1} + \rho_i \bar{g}_i$ Our goal in quadrature is to approximate the mean element $\mu_p$. Recently [@Bach2012] proposed to frame integration as a FW optimisation problem. Here, the domain $\mathcal{G} \subseteq \mathcal{H}$ is a space of functions and taking the objective function to be: $$J(g) = \frac{1}{2}\big\| g - \mu_p \big\|^2_{\mathcal{H}}.$$ This gives an approximation of the mean element and $J$ takes the form of half the posterior variance (or the MMD$^2$). In this functional approximation setting, minimisation of $J$ is carried out over $\mathcal{G} = \mathcal{M}$, the marginal polytope of the RKHS $\mathcal{H}$. The marginal polytope $\mathcal{M}$ is defined as the closure of the convex hull of $\Phi(\mathcal{X})$, so that in particular $\mu_p \in \mathcal{M}$. Assuming as in [@Lacoste-Julien2015] that $\Phi(x)$ is uniformly bounded in feature space (i.e. $\exists R>0: \forall x \in \mathcal{X}$, $\|\Phi(x)\|_{\mathcal{H}} \leq R$), then $\mathcal{M}$ is a closed and bounded set and can be optimised over. In order to define the algorithm rigorously in this case, we introduce the Fréchet derivative of $J$, denoted $DJ$, such that for $\mathcal{H}^*$ being the dual space of $\mathcal{H}$, we have the unique map $DJ:\mathcal{H} \rightarrow \mathcal{H}^*$ such that for each $g \in \mathcal{H}$, $(DJ)(g)$ is the function mapping $h \in \mathcal{H}$ to $(DJ)(g)(h) = \big\langle g - \mu, h \big\rangle_\mathcal{H}$. We also introduce the bilinear map $\langle \cdot, \cdot \rangle_{\times}: \mathcal{H} \times \mathcal{H}^* \rightarrow \mathbb{R}$ which, for $F \in \mathcal{H}^*$ given by $F(g) = \langle g, f \rangle_\mathcal{H}$, is the rule giving $\langle h, F \rangle_{\times} = \langle h, f \rangle_{\mathcal{H}}$. A particular advantage of this method is that it leads to ‘sparse’ solutions which are linear combinations of the atoms $\{\bar{g}_i\}_{i=1}^n$ [@Bach2012]. In particular this provides a weighted estimate for the mean element: $$\label{eq:FWsparse} \hat{\mu}_{\text{FW}} {\coloneqq}g_n = \sum_{i=1}^n \Big( \prod_{j=i+1}^{n} \big( 1 - \rho_{j-1} \big) \rho_{i-1} \Big) \bar{g}_i {\coloneqq}\sum_{i=1}^n w_i^{\text{FW}} \bar{g}_i ,$$ where by default $\rho_0 = 1$ which leads to all $w_i^{\text{FW}} \in [0,1]$ when $\rho_i = 1/(i+1)$. A typical sequence of approximations to the mean element is shown in Fig. \[fig:designpoints\] (left), demonstrating that the approximation quickly converges to the ground truth (in black). Since minimisation of a linear function can be restricted to extreme points of the domain, the atoms will be of the form $\bar{g}_i = \Phi(x_i^{\text{FW}}) = k(\cdot ,x_i^{\text{FW}})$ for some $x_i^{\text{FW}} \in \mathcal{X}$. The minimisation in $g$ over $\mathcal{G}$ from step 2 in Algorithm \[alg:FWalgorithm\] therefore becomes a minimisation in $x$ over $\mathcal{X}$ and this algorithm therefore provides us design points. In practice, at each iteration $i$, the FW algorithm hence selects a design point $x_i^{\text{FW}} \in \mathcal{X}$ which induces an atom $\bar{g}_i$ and gives us an approximation of the mean element $\mu_p$. We denote by $\hat{\mu}_{\text{FW}}$ this approximation after $n$ iterations. Using the reproducing property, we can show that the FW estimate is a quadrature rule: $$\hat{p}_{\text{FW}}[f] {\coloneqq}\big\langle f,\hat{\mu}_{\text{FW}} \big\rangle_{\mathcal{H}} = \Big\langle f , \sum_{i=1}^n w_i^{\text{FW}} \bar{g}_i \Big\rangle_{\mathcal{H}} = \sum_{i=1}^n w_i^{\text{FW}} \big\langle f , k(\cdot,x_i^{\text{FW}}) \big\rangle_{\mathcal{H}} = \sum_{i=1}^n w_i^{\text{FW}} f(x_i^{\text{FW}}).$$ The total computational cost for FW is $\mathcal{O}(n^2)$. An extension known as FW with Line Search (FWLS) uses a line-search method to find the optimal step size $\rho_i$ at each iteration (see Alg. \[alg:FWalgorithm\]). Once again, the approximation obtained by FWLS has a sparse expression as a convex combination of all the previously visited states and we obtain an associated quadrature rule. FWLS has theoretical convergence rates that can be stronger than standard versions of FW but has computational cost in $\mathcal{O}(n^3)$. The authors in [@Garber2015] provide a survey of FW-based algorithms and their convergence rates under different regularity conditions on the objective function and domain of optimisation. [*Remark:*]{} The FW design points $\{x_i^{\text{FW}}\}_{i=1}^n$ are generally not available in closed-form. We follow mainstream literature by selecting, at each iteration, the point that minimises the MMD over a finite collection of $M$ points, drawn i.i.d from $p(x)$. The authors in [@Lacoste-Julien2015] proved that this approximation adds a $\mathcal{O}(M^{-1/4})$ term to the MMD, so that theoretical results on FW convergence continue to apply provided that $M(n) \rightarrow \infty$ sufficiently quickly. Appendix A provides full details. In practice, one may also make use of a numerical optimisation scheme in order to select the points. A Hybrid Approach: Frank-Wolfe Bayesian Quadrature {#section:BayesianFW} ================================================== To combine the advantages of a probabilistic integrator with a formal convergence theory, we propose Frank-Wolfe Bayesian Quadrature (FWBQ). In FWBQ, we first select design points $\{x_i^{\text{FW}}\}_{i=1}^n$ using the FW algorithm. However, when computing the quadrature approximation, instead of using the usual FW weights $\{w_i^{\text{FW}}\}_{i=1}^n$ we use instead the weights $\{w_i^{\text{BQ}}\}_{i=1}^n$ provided by BQ. We denote this quadrature rule by $\hat{p}_{\text{FWBQ}}$ and also consider $\hat{p}_{\text{FWLSBQ}}$, which uses FWLS in place of FW. As we show below, these hybrid estimators (i) carry the Bayesian interpretation of Sec. \[subsec:BQ\], (ii) permit a rigorous theoretical analysis, and (iii) out-perform existing FW quadrature rules by orders of magnitude in simulations. FWBQ is hence ideally suited to probabilistic numerics applications. For these theoretical results we assume that $f$ belongs to a finite-dimensional RKHS $\mathcal{H}$, in line with recent literature [@Bach2012; @Garber2015; @Jaggi2013; @Lacoste-Julien2015]. We further assume that $\mathcal{X}$ is a compact subset of $\mathbb{R}^d$, that $p(x) > 0$ $\forall x \in \mathcal{X}$ and that $k$ is continuous on $\mathcal{X} \times \mathcal{X}$. Under these hypotheses, Theorem \[theorem:posteriormean\] establishes consistency of the posterior mean, while Theorem \[theo1\] establishes contraction for the posterior distribution. \[theorem:posteriormean\] The posterior mean $\hat{p}_{\emph{FWBQ}}[f]$ converges to the true integral $p[f]$ at the following rates: $$\Big|p[f] - \hat{p}_{\emph{FWBQ}}[f]\Big| \leq \text{MMD}\big(\{x_i,w_i\}_{i=1}^n\big) \leq \left\{ \begin{array}{cl} \frac{2 D^2}{R} n^{-1} & \text{for FWBQ}\\ \sqrt{2}D \exp(-\frac{R^2}{2 D^2} n) & \text{for FWLSBQ} \end{array} \right.$$ where the FWBQ uses step-size $\rho_i = 1/(i+1)$, $D \in (0,\infty)$ is the diameter of the marginal polytope $\mathcal{M}$ and $R \in (0,\infty)$ gives the radius of the smallest ball of center $\mu_p$ included in $\mathcal{M}$. Note that all the proofs of this paper can be found in Appendix B. An immediate corollary of Theorem \[theorem:posteriormean\] is that FWLSBQ has an asymptotic error which is exponential in $n$ and is therefore superior to that of any QMC estimator [@Dick2010]. This is not a contradiction - recall that QMC restricts attention to uniform weights, while FWLSBQ is able to propose arbitrary weightings. In addition we highlight a robustness property: Even when the assumptions of this section do not hold, one still obtains atleast a rate $\mathcal{O}_P (n^{-1/2})$ for the posterior mean using either FWBQ or FWLSBQ [@Dunn1980]. [*Remark*]{}: The choice of kernel affects the convergence of the FWBQ method [@Hennig2015ProbNum]. Clearly, we expect faster convergence if the function we are integrating is ‘close’ to the space of functions induced by our kernel. Indeed, the kernel specifies the geometry of the marginal polytope $\mathcal{M}$, that in turn directly influences the rate constant $R$ and $D$ associated with FW convex optimisation. Consistency is only a stepping stone towards our main contribution which establishes posterior contraction rates for FWBQ. Posterior contraction is important as these results justify, for the first time, the probabilistic numerics approach to integration; that is, we show that the [*full*]{} posterior distribution is a sensible quantification (at least asymptotically) of numerical error in the integration routine: \[theo1\] Let $S \subseteq \mathbb{R}$ be an open neighbourhood of the true integral $p[f]$ and let $\gamma = \inf_{r \in S^C} | r - p[f]| >0$. Then the posterior probability mass on $S^c = \mathbb{R} \setminus S$ vanishes at a rate: $$\emph{prob}(S^c) \leq \left\{ \begin{array}{cl} \frac{2\sqrt{2}D^2}{\sqrt{\pi}R \gamma} n^{-1} \exp \Big(- \frac{\gamma^2 R^2}{8 D^4} n^2 \Big) & \text{for FWBQ} \\ \frac{2 D}{\sqrt{\pi} \gamma} \exp\Big( - \frac{R^2}{2 D^2} n - \frac{\gamma^2}{2\sqrt{2}D} \exp\big( \frac{R^2}{2D^2} n\big)\Big) & \text{for FWLSBQ} \end{array} \right.$$ where the FWBQ uses step-size $\rho_i = 1/(i+1)$, $D \in (0,\infty)$ is the diameter of the marginal polytope $\mathcal{M}$ and $R \in (0,\infty)$ gives the radius of the smallest ball of center $\mu_p$ included in $\mathcal{M}$. The contraction rates are exponential for FWBQ and super-exponential for FWLBQ, and thus the two algorithms enjoy both a probabilistic interpretation and rigorous theoretical guarantees. A notable corollary is that OBQ enjoys the same rates as FWLSBQ, resolving a conjecture by Tony O’Hagan that OBQ converges exponentially \[personal communication\]: \[theorem:OBQ\_rates\] The consistency and contraction rates obtained for FWLSBQ apply also to OBQ. Experimental Results {#section:Experimental_Results} ==================== Simulation Study ---------------- To facilitate the experiments in this paper we followed [@Bach2015; @Bach2012; @Rasmussen2003; @Lacoste-Julien2015] and employed an exponentiated-quadratic (EQ) kernel $k(x, x') {\coloneqq}\lambda^{2} \exp ( -\nicefrac{1}{2 \sigma^2} \|x-x'\|^2_2)$. This corresponds to an infinite-dimensional RKHS, not covered by our theory; nevertheless, we note that all simulations are practically finite-dimensional due to rounding at machine precision. See Appendix E for a finite-dimensional approximation using random Fourier features. EQ kernels are popular in the BQ literature as, when $p$ is a mixture of Gaussians, the mean element $\mu_p$ is analytically tractable (see Appendix C). Some other $(p,k)$ pairs that produce analytic mean elements are discussed in [@Bach2015]. For this simulation study, we took $p(x)$ to be a 20-component mixture of 2D-Gaussian distributions. Monte Carlo (MC) is often used for such distributions but has a slow convergence rate in $\mathcal{O}_P(n^{-1/2})$. FW and FWLS are known to converge more quickly and are in this sense preferable to MC [@Bach2012]. In our simulations (Fig. \[fig:sim study\], left), both our novel methods FWBQ and FWLSBQ decreased the MMD much faster than the FW/FWLS methods of [@Bach2012]. Here, the same kernel hyper-parameters $(\lambda,\sigma) = (1,0.8)$ were employed for all methods to have a fair comparison. This suggests that the best quadrature rules correspond to elements [*outside*]{} the convex hull of $\{\Phi(x_i)\}_{i=1}^n$. Examples of those, including BQ, often assign negative weights to features (Fig. S1 right, Appendix D). The principle advantage of our proposed methods is that they reconcile theoretical tractability with a fully probabilistic interpretation. For illustration, Fig. \[fig:sim study\] (right) plots the posterior uncertainty due to numerical error for a typical integration problem based on this $p(x)$. In-depth empirical studies of such posteriors exist already in the literature and the reader is referred to [@Chen2015; @Hamrick2013mental; @OHagan1991] for details. Beyond these theoretically tractable integrators, SBQ seems to give even better performance as $n$ increases. An intuitive explanation is that SBQ picks $\{x_i\}_{i=1}^n$ to minimise the MMD whereas FWBQ and FWLSBQ only minimise an approximation of the MMD (its linearisation along $DJ$). In addition, the SBQ weights are optimal at each iteration, which is not true for FWBQ and FWLSBQ. We conjecture that Theorem \[theorem:posteriormean\] and \[theo1\] provide upper bounds on the rates of SBQ. This conjecture is partly supported by Fig. \[fig:designpoints\] (right), which shows that SBQ selects similar design points to FW/FWLS (but weights them optimally). Note also that both FWBQ and FWLSBQ give very similar result. This is not surprising as FWLS has no guarantees over FW in infinite-dimensional RKHS [@Jaggi2013]. Quantifying Numerical Error in a Proteomic Model Selection Problem ------------------------------------------------------------------ A topical bioinformatics application that extends recent work by [@Oates2014] is presented. The objective is to select among a set of candidate models $\{M_i\}_{i=1}^m$ for protein regulation. This choice is based on a dataset $\mathcal{D}$ of protein expression levels, in order to determine a ‘most plausible’ biological hypothesis for further experimental investigation. Each $M_i$ is specified by a vector of kinetic parameters $\theta_i$ (full details in Appendix D). Bayesian model selection requires that these parameters are integrated out against a prior $p(\theta_i)$ to obtain marginal likelihood terms $L(M_i) = \int p(\mathcal{D}|\theta_i) p(\theta_i) \mathrm{d}\theta_i$. Our focus here is on obtaining the [*maximum a posteriori*]{} (MAP) model $M_j$, defined as the maximiser of the posterior model probability $L(M_j) / \sum_{i=1}^m L(M_i)$ (where we have assumed a uniform prior over model space). Numerical error in the computation of each term $L(M_i)$, if unaccounted for, could cause us to return a model $M_k$ that is different from the true MAP estimate $M_j$ and lead to the mis-allocation of valuable experimental resources. The problem is quickly exaggerated when the number $m$ of models increases, as there are more opportunities for one of the $L(M_i)$ terms to be ‘too large’ due to numerical error. In [@Oates2014], the number $m$ of models was combinatorial in the number of protein kinases measured in a high-throughput assay (currently $\sim 10^2$ but in principle up to $\sim 10^4$). This led [@Oates2014] to deploy substantial computing resources to ensure that numerical error in each estimate of $L(M_i)$ was individually controlled. Probabilistic numerics provides a more elegant and efficient solution: At any given stage, we have a fully probabilistic quantification of our uncertainty in each of the integrals $L(M_i)$, shown to be sensible both theoretically and empirically. This induces a full posterior distribution over numerical uncertainty in the location of the MAP estimate (i.e. ‘Bayes all the way down’). As such we can determine, on-line, the precise point in the computational pipeline when numerical uncertainty near the MAP estimate becomes acceptably small, and cease further computation. The FWBQ methodology was applied to one of the model selection tasks in [@Oates2014]. In Fig. \[fig:model posteriors\] (left) we display posterior model probabilities for each of $m = 352$ candidates models, where a low number ($n = 10$) of samples were used for each integral. (For display clarity only the first 50 models are shown.) In this low-$n$ regime, numerical error introduces a second level of uncertainty that we quantify by combining the FWBQ error models for all integrals in the computational pipeline; this is summarised by a box plot (rather than a single point) for each of the models (obtained by sampling - details in Appendix D). These box plots reveal that our estimated posterior model probabilities are completely dominated by numerical error. In contrast, when $n$ is increased through 50, 100 and 200 (Fig. \[fig:model posteriors\], right and Fig. S2), the uncertainty due to numerical error becomes negligible. At $n = 200$ we can conclude that model $26$ is the true MAP estimate and further computations can be halted. Correctness of this result was confirmed using the more computationally intensive methods in [@Oates2014]. In Appendix D we compared the relative performance of FWBQ, FWLSBQ and SBQ on this problem. Fig. S1 shows that the BQ weights reduced the MMD by orders of magnitude relative to FW and FWLS and that SBQ converged more quickly than both FWBQ and FWLSBQ. Conclusions =========== This paper provides the first theoretical results for probabilistic integration, in the form of posterior contraction rates for FWBQ and FWLSBQ. This is an important step in the probabilistic numerics research programme [@Hennig2015ProbNum] as it establishes a theoretical justification for using the posterior distribution as a model for the numerical integration error (which was previously assumed [@Rasmussen2003; @Gunter2014; @Oates2015; @Osborne2012; @Sarkka2015 e.g.]). The practical advantages conferred by a fully probabilistic error model were demonstrated on a model selection problem from proteomics, where sensitivity of an evaluation of the MAP estimate was modelled in terms of the error arising from repeated numerical integration. The strengths and weaknesses of BQ (notably, including scalability in the dimension $d$ of $\mathcal{X}$) are well-known and are inherited by our FWBQ methodology. We do not review these here but refer the reader to [@OHagan1991] for an extended discussion. Convergence, in the classical sense, was proven here to occur exponentially quickly for FWLSBQ, which partially explains the excellent performance of BQ and related methods seen in applications [@Gunter2014; @Osborne2012], as well as resolving an open conjecture. As a bonus, the hybrid quadrature rules that we developed turned out to converge much faster in simulations than those in [@Bach2012], which originally motivated our work. A key open problem for kernel methods in probabilistic numerics is to establish protocols for the practical elicitation of kernel hyper-parameters. This is important as hyper-parameters directly affect the scale of the posterior over numerical error that we ultimately aim to interpret. Note that this problem applies equally to BQ, as well as related quadrature methods [@Bach2012; @Rasmussen2003; @Gunter2014; @Oates2015] and more generally in probabilistic numerics [@Schober2014]. Previous work, such as [@Hamrick2013mental], optimised hyper-parameters on a per-application basis. Our ongoing research seeks automatic and general methods for hyper-parameter elicitation that provide good frequentist coverage properties for posterior credible intervals, but we reserve the details for a future publication. ### Acknowledgments {#acknowledgments .unnumbered} The authors are grateful for discussions with Simon Lacoste-Julien, Simo S[ä]{}rkk[ä]{}, Arno Solin, Dino Sejdinovic, Tom Gunter and Mathias Cronj[ä]{}ger. FXB was supported by EPSRC \[EP/L016710/1\]. CJO was supported by EPSRC \[EP/D002060/1\]. MG was supported by EPSRC \[EP/J016934/1\], an EPSRC Established Career Fellowship, the EU grant \[EU/259348\] and a Royal Society Wolfson Research Merit Award. Supplementary Material {#supplementary-material .unnumbered} ====================== Appendix A: Details for the FWBQ and FWLSBQ Algorithms {#appendix-a-details-for-the-fwbq-and-fwlsbq-algorithms .unnumbered} ------------------------------------------------------ A high-level pseudo-code description for the Frank-Wolfe Bayesian Quadrature (FWBQ) algorithm is provided below. function $f$, reproducing kernel $k$, initial point $x_0 \in \mathcal{X}$. Compute design points $\big\{x_i^{\text{FW}} \big\}_{i=1}^n$ using the FW algorithm (Alg. 1). Compute associated weights $\big\{w_i^{\text{BQ}}\big\}_{i=1}^n$ using BQ (Eqn. 4). Compute the posterior mean $\hat{p}_{\text{FWBQ}}[f]$, i.e. the quadrature rule with $\big\{ x_i^{\text{FW}}, w_i^{\text{BQ}} \big\}_{i=1}^n$. Compute the posterior variance $v_{\text{BQ}}\big(\{x_i^{\text{FW}}\}_{i=1}^n \big)$ using BQ (Eqn. 5). Return the full posterior $\mathcal{N} \big(\hat{p}_{\text{FWBQ}},v_{\text{BQ}}(\{x_i^{\text{FW}}\}_{i=1}^n)\big)$ for the integral $p[f]$. Frank-Wolfe Line-Search Bayesian Quadrature (FWLSBQ) is simply obtained by substituting the Frank-Wolfe algorithm with the Frank-Wolfe Line-Search algorithm. In this appendix, we derive all of the expressions necessary to implement both the FW and FWLS algorithms (for quadrature) in practice. All of the other steps can be derived from the relevant equations as highlighted in Algorithm \[alg:FWBQalgorithm\] above. The FW/FWLS are both initialised by the user choosing a design point $x_1^{\text{FW}}$. This can be done either at random or by choosing a location which is known to have high probability mass under $p(x)$. The first approximation to $\mu_p$ is therefore given by $g_1 = k(\cdot,x_1^{\text{FW}})$. The algorithm then loops over the next three steps to obtain new design points $\{x_i^{\text{FW}}\}_{i=2}^n$: [*Step 1) Obtaining the new Frank-Wolfe design points $x_{i+1}^{\text{FW}}$.*]{} At iteration $i$, the step consists of choosing the point $\bar{x}_i^{\text{FW}}$. Let $\{w_l^{(i)}\}_{l=1}^{i-1}$ denote the Frank-Wolfe weights assigned to each of the previous design points $\{x_l^{\text{FW}}\}_{l=1}^{i-1}$ at this new iteration, given that we choose $x$ as our new design point. The choice of new design point is done by computing the derivative of the objective function $J(g_{i-1})$ and finding the point $x^*$ which minimises the inner product: $${\arg\min}_{g \in \mathcal{G}} \big\langle g , (DJ)(g_{i-1}) \big\rangle_{\times}$$ To do so, we need to obtain an equivalent expression of the minimisation of the linearisation of $J$ (denoted $DJ$) in terms of kernel values and evaluations of the mean element $\mu_p$. Since minimisation of a linear function can be restricted to extreme points of the domain, we have that $${\arg\min}_{g \in \mathcal{G}} \big\langle g , (DJ)(g_{i-1}) \big\rangle_{\times} \; = \; {\arg\min}_{x \in \mathcal{X}} \big\langle \Phi(x) , (DJ)(g_{i-1}) \big\rangle_{\times}.$$ Then using the definition of $J$ we have: $${\arg\min}_{x \in \mathcal{X}} \big\langle \Phi(x) , (DJ)(g_{i-1}) \big\rangle_{\times} \; = \; {\arg\min}_{x \in \mathcal{X}} \big\langle \Phi(x) , g_{i-1} - \mu_p \big\rangle_{\mathcal{H}} ,$$ where $$\begin{split} \big\langle \Phi(x) , g_{i-1} - \mu_p \big\rangle_{\mathcal{H}} \quad & = \quad \Big\langle \Phi(x), \sum_{l=1}^{i-1} w_l^{(i-1)} \Phi(x_l) - \mu_p \Big\rangle_{\mathcal{H}} \\ & = \quad \sum_{i=1}^{i-1} w_l^{(i-1)} \big\langle \Phi(x) , \Phi(x_l) \big\rangle_{\mathcal{H}} - \big\langle \Phi(x) , \mu_p \big\rangle_{\mathcal{H}} \\ & = \quad \sum_{l=1}^{i-1} w_l^{(i-1)} k( x,x_l) - \mu_p(x). \end{split}$$ Our new design point $x_i^{\text{FW}}$ is therefore the point $x^*$ which minimises this expression. Note that this equation may not be convex and may require us to make use of approximate methods to find the minimum $x^*$. To do so, we sample $M$ points (where $M$ is large) independently from the distribution $p$ and pick the sample which minimises the expression above. From [@Lacoste-Julien2015] this introduces an additive error term of size $\mathcal{O}(M^{-1/4})$, which does not impact our convergence analysis provided that M(n) vanishes sufficiently quickly. In all experiments we took $M$ between $10,000$ and $50,000$ so that this error will be negligible. It is important to note that sampling from $p(x)$ is likely to not be the best solution to optimising this expression. One may, for example, be better off using any other optimisation method which does not require convexity (for example, Bayesian Optimization). However, we have used sampling as the result from [@Lacoste-Julien2015] discussed above allows us to have a theoretical upper bound on the error introduced. [*Step 2) Computing the Step-Sizes and Weights for the Frank-Wolfe and Frank-Wolfe Line-Search Algorithms.*]{} Computing the weights $\{w_l^{(i)}\}_{l=1}^n$ assigned by the FW/FWLS algorithms to each of the design points is obtained using the equation: $$w_l^{(i)}= \prod_{j=l+1}^{i} \big( 1 - \rho_{j-1} \big) \rho_{l-1}$$ Clearly, this expression depends on the choice of step-sizes $\{\rho_l\}_{l=1}^i$. In the case of the standard Frank-Wolfe algorithm, this step-size sequence is a an input from the algorithm and so computing the weights is straightforward. However, in the case of the Frank-Wolfe Line-Search algorithm, the choice of step-size is optimized at each iteration so that $g_i$ minimises $J$ the most. In the case of computing integrals, this optimization step can actually be obtained analytically. This analytic expression will be given in terms of values of the kernel values and evaluations of the mean element. First, from the definition of $J$ $$\begin{split} J \big( (1-\rho) g_{i-1} + \rho \Phi(x_{i}) \big) \quad & = \quad \frac{1}{2} \big\langle (1-\rho) g_{i-1} + \rho \Phi(x_{i}) - \mu_p , (1-\rho) g_{i-1} + \rho \Phi(x_{i}) - \mu_p \big\rangle_{\mathcal{H}} \\ & = \quad \frac{1}{2} \Big[ (1- \rho)^2 \big\langle g_{i-1},g_{i-1} \big\rangle_{\mathcal{H}} + 2 (1 - \rho) \rho \big\langle g_{i-1}, \Phi(x_{i}) \big\rangle_{\mathcal{H}} \\ & \qquad + 2 \rho^2 \big\langle \Phi(x_{i}), \Phi(x_{i}) \big\rangle_{\mathcal{H}} - 2 (1 - \rho) \big\langle g_{i-1}, \mu_p \big\rangle_{\mathcal{H}} \\ & \qquad - 2\rho \big\langle \Phi(x_{i}), \mu_p \big\rangle_{\mathcal{H}} + \big\langle \mu_p,\mu_p \big\rangle_{\mathcal{H}} \Big]. \end{split}$$ Taking the derivative of this expression with respect to $\rho$, we get: $$\begin{split} \frac{\partial J \big( (1-\rho) g_{i-1} + \rho \Phi(x_{i}) \big) }{\partial \rho} \quad & = \quad \frac{1}{2} \Big[ - 2(1- \rho) \big\langle g_{i-1} , g_{i-1} \big\rangle_{\mathcal{H}} + 2 (1 - 2 \rho) \big\langle g_{i-1}, \Phi(x_{i})\big\rangle_{\mathcal{H}} \\ & \qquad + 2 \rho \big\langle \Phi(x_{i}), \Phi(x_{i}) \big\rangle_{\mathcal{H}} + 2 \big\langle g_{i-1} , \mu_p \big\rangle_{\mathcal{H}} - 2 \big\langle \Phi(x_{i}), \mu_p \big\rangle_{\mathcal{H}} \Big] \\ & = \quad \rho \Big[ \big\langle g_{i-1}, g_{i-1} \big\rangle_{\mathcal{H}} - 2 \big\langle g_{i-1} , \Phi(x_{i}) \big\rangle_{\mathcal{H}} + \big\langle \Phi(x_{i}) , \Phi(x_{i}) \big\rangle_{\mathcal{H}} \\ & = \quad \rho \big\| g_{i-1} - \Phi(x_{i}) \big\|^2_{\mathcal{H}} - \big\langle g_{i-1} - \Phi(x_{i}) , g_{i-1} - \mu_p \big\rangle_{\mathcal{H}} .\\ \end{split}$$ Setting this derivative to zero gives us the following optimum: $$\rho^* \quad = \quad \frac{\Big\langle g_{i-1} - \mu_p , g_{i-1} - \Phi(x_{i}) \Big\rangle_{\mathcal{H}}}{\Big\| g_{i-1} - \Phi(x_{i}) \Big\|^2_{\mathcal{H}} }.$$ Clearly, differentiating a second time with respect to $\rho$ gives $\| g_{i-1} - \Phi(x_{i}) \|_{\mathcal{H}}^2$, which is non-negative and so $\rho^*$ is a minimum. One can show using geometrical arguments about the marginal polytope $\mathcal{M}$ that $\rho^*$ will be in $[0,1]$ [@Jaggi2013]. The numerator of this line-search expression is $$\begin{split} \Big\langle g_{i-1} - \mu_p, g_{i-1} - \Phi(x_{i})\Big\rangle_{\mathcal{H}} \quad & = \quad \big\langle g_{i-1}, g_{i-1} \big\rangle_{\mathcal{H}} - \big\langle \mu_p, g_{i-1} \big\rangle_{\mathcal{H}} \\ & \qquad - \sum_{l=1}^{i-1} w_l^{(i-1)} k(x_{l},x_{i}) + \mu_p(x_{i}) \\ & = \quad \sum_{l=1}^{i-1} \sum_{m=1}^{i-1} w_l^{(i-1)} w_m^{(i-1)} k(x_l,x_m) \\ & \qquad - \sum_{l=1}^{i-1} w_l^{(i-1)} \Big[ k(x_l,x_{i}) + \mu_p(x_l) \Big] + \mu_p(x_{i}). \\ \end{split}$$ Similarly the denominator is $$\begin{split} \big\| g_{i-1} - \Phi(x_{i})\big\|^2_{\mathcal{H}} \quad & = \quad \big\langle g_{i-1} - \Phi(x_{i}), g_{i-1} - \Phi(x_{i}) \big\rangle_{\mathcal{H}} \\ & = \quad \big\langle g_{i-1}, g_{i-1} \big\rangle_{\mathcal{H}} - 2 \big\langle g_{i-1} , \Phi(x_{i}) \big\rangle_{\mathcal{H}} + \big\langle \Phi(x_{i}), \Phi(x_{i}) \big\rangle_{\mathcal{H}} \\ & = \quad \sum_{l=1}^{i-1} \sum_{m=1}^{i-1} w_l^{(i-1)} w_m^{(i-1)} k(x_l,x_m) - 2 \sum_{l=1}^{i-1} w_l^{(i-1)} k(x_l,x_{i}) + k(x_{i},x_{i}).\\ \end{split}$$ Clearly all expressions provided here can be vectorised for efficient computational implementation. [*Step 3) Computing a new approximation of the mean element.*]{} The final step consists of updating the approximation of the mean element, which can be done directly by setting: $$g_{i} = (1 - \rho_i) g_{i-1} + \rho_i \bar{g}_i$$ Appendix B: Proofs of Theorems and Corollaries {#appendix-b-proofs-of-theorems-and-corollaries .unnumbered} ---------------------------------------------- The posterior mean $\hat{p}_{\emph{FWBQ}}[f]$ converges to the true integral $p[f]$ at the following rates: $$\Big|p[f] - \hat{p}_{\emph{FWBQ}}[f]\Big| \leq \text{MMD}\big(\{x_i,w_i\}_{i=1}^n\big) \leq \left\{ \begin{array}{cl} \frac{2 D^2}{R} n^{-1} & \text{for FWBQ}\\ \sqrt{2}D \exp(-\frac{R^2}{2 D^2} n) & \text{for FWLSBQ} \end{array} \right.$$ where the FWBQ uses step-size $\rho_i = 1/(i+1)$, $D \in (0,\infty)$ is the diameter of the marginal polytope $\mathcal{M}$ and $R \in (0,\infty)$ gives the radius of the smallest ball of center $\mu_p$ included in $\mathcal{M}$. The posterior mean in BQ is a Bayes estimator and so the MMD takes a minimax form [@Huszar2012]. In particular, the BQ weights perform no worse than the FW weights: $$\text{MMD} \Big( \big\{x_i^{\text{FW}} ,w_i^{\text{BQ}} \big\}_{i=1}^n \Big) \; = \; \inf_{\textrm{w} \in \mathbb{R}^n} \text{MMD} \Big( \big\{x_i^{\text{FW}},w_i \big\}_{i=1}^n \Big) \; \leq \; \text{MMD} \Big( \big\{x_i^{\text{FW}},w_i^{\text{FW}} \big\}_{i=1}^n \Big). \label{eq:minimax}$$ Now, the values attained by the objective function $J$ along the path $\{g_i\}_{i=1}^n$ determined by the FW(/FWLS) algorithm can be expressed in terms of the MMD as follows: $$J(g_n) = \frac{1}{2} \big\|\hat{\mu}_{\text{FW}} - \mu_p \big\|^2_{\mathcal{H}} = \frac{1}{2} \text{MMD}^2\Big( \big\{x_i^{\text{FW}},w_i^{\text{FW}} \big\}_{i=1}^n \Big). \label{FWobjvals}$$ Combining and gives $$\Big|p[f] - \hat{p}_{\text{FWBQ}}[f] \Big| \; \leq \; \text{MMD}\Big( \big\{x_i^{\text{FW}},w_i^{\text{BQ}} \big\}_{i=1}^n \Big) \big\|f \big\|_{\mathcal{H}} \; \leq \; 2^{1/2} J^{1/2}(g_n),$$ since $\|f\|_{\mathcal{H}} \leq 1$. To complete the proof we leverage recent analysis of the FW algorithm with steps $\rho_i = 1/(n+1)$ and the FWLS algorithm. Specifically, from [@Bach2012 Prop. 1] we have that: $$J(g_n) \leq \left\{ \begin{array}{cl} \frac{2D^4}{R^2} n^{-2} & \text{for FW with step size } \rho_i = 1/(i+1) \\ D^2 \exp(-R^2 n/D^2 ) & \text{for FWLS} \end{array} \right.$$ where $D$ is the diameter of the marginal polytope $\mathcal{M}$ and $R$ is the radius of the smallest ball centered at $\mu_p$ included in $\mathcal{M}$. Let $S \subseteq \mathbb{R}$ be an open neighbourhood of the true integral $p[f]$ and let $\gamma = \inf_{r \in S^C} | r - p[f]| >0$. Then the posterior probability mass on $S^c = \mathbb{R} \setminus S$ vanishes at a rate: $$\emph{prob}(S^c) \leq \left\{ \begin{array}{cl} \frac{2\sqrt{2}D^2}{\sqrt{\pi}R \gamma} n^{-1} \exp \Big(- \frac{\gamma^2 R^2}{8 D^4} n^2 \Big) & \text{for FWBQ, } \rho_i = 1/(i+1) \\ \frac{2 D}{\sqrt{\pi} \gamma} \exp\Big( - \frac{R^2}{2 D^2} n - \frac{\gamma^2}{2\sqrt{2}D} \exp\big( \frac{R^2}{2D^2} n\big)\Big) & \text{for FWLSBQ} \end{array} \right.$$ where $D \in (0,\infty)$ is the diameter of the marginal polytope $\mathcal{M}$ and $R \in (0,\infty)$ gives the radius of the smallest ball of center $\mu_p$ included in $\mathcal{M}$. We will obtain the posterior contraction rates of interest using the bounds on the MMD provided in the proof of Theorem 1. Given an open neighbourhood $S \subseteq \mathbb{R}$ of $p[f]$, we have that the complement $S^c = \mathbb{R} \setminus S$ is closed in $\mathbb{R}$. We assume without loss of generality that $S^c \neq \emptyset$, since the posterior mass on $S^c$ is trivially zero when $S^c = \emptyset$. Since $S^c$ is closed, the distance $\gamma = \inf_{r \in S^c} \bigl|r - p[f]\bigr| > 0$ is strictly positive. Denote the posterior distribution by $\mathcal{N}(m_n,\sigma_n^2)$ where we have that $m_n {\coloneqq}\hat{p}_{\text{FWBQ}}[f]$ where $\hat{p}_{\text{FWBQ}} = \sum_{i=1}^n w_i^{\text{BQ}} \delta(x_i^{\text{FW}})$ and $\sigma_n {\coloneqq}\text{MMD}(\{x_i^{\text{FW}},w_i^{\text{BQ}}\}_{i=1}^n)$. Directly from the supremum definition of the MMD we have: $$\Big|p\big[f\big] - m_n \Big| \leq \sigma_n \big\|f\big\|_{\mathcal{H}}. \label{eq:CS}$$ Now the posterior probability mass on $S^c$ is given by $$M_n = \int_{S^c} \phi(r|m_n,\sigma_n) \mathrm{d}r,$$ where $\phi(r|m_n,\sigma_n)$ is the p.d.f. of the posterior normal distribution. By the definition of $\gamma$ we get the upper bound: $$\begin{aligned} M_n & \leq & \int_{-\infty}^{p[f] - \gamma} \phi(r|m_n,\sigma_n) \mathrm{d}r + \int_{p[f] + \gamma}^\infty \phi(r|m_n,\sigma_n) \mathrm{d}r \\ & = & 1 + \Phi\Big(\underbrace{\frac{p[f] - m_n}{\sigma_n}}_{(*)} - \frac{\gamma}{\sigma_n}\Big) - \Phi\Big(\underbrace{\frac{p[f] - m_n}{\sigma_n}}_{(*)} + \frac{\gamma}{\sigma_n}\Big).\end{aligned}$$ From we have that the terms $(*)$ are bounded by $\|f\|_{\mathcal{H}} \leq 1 <\infty$ as $\sigma_n \rightarrow 0$, so that asymptotically we have: $$\begin{aligned} M_n & \lesssim & 1 + \Phi\big(- \gamma / \sigma_n\big) - \Phi\big(\gamma / \sigma_n \big) \\ & = & \text{erfc}\big(\gamma/\sqrt{2}\sigma_n\big) \sim \big(\sqrt{2}\sigma_n / \sqrt{\pi} \gamma \big) \exp\big(- \gamma^2 / 2 \sigma_n^2 \big). \label{eq:final}\end{aligned}$$ Finally we may substitute the asymptotic results derived in the proof of Theorem 1 for the MMD $\sigma_n$ into to complete the proof. The consistency and contraction rates obtained for FWLSBQ apply also to OBQ. By definition, OBQ chooses samples that globally minimise the MMD and we can hence bound this quantity from above by the MMD of FWLSBQ: $$\text{MMD}\Big(\big\{x_i^{\text{OBQ}},w_i^{\text{BQ}}\big\}_{i=1}^n \Big) = \inf_{\{x_i\}_{i=1}^n \in \mathcal{X}} \text{MMD}\Big(\big\{x_i,w_i^{\text{BQ}}\big\}_{i=1}^n \Big) \leq \text{MMD}\Big( \big\{x_i^{\text{FW}},w_i^{\text{BQ}} \big\}_{i=1}^n \Big).$$ Consistency and contraction follow from inserting this inequality into the above proofs. Appendix C: Computing the Mean Element for the Simulation Study {#appendix-c-computing-the-mean-element-for-the-simulation-study .unnumbered} --------------------------------------------------------------- We compute an expression for $\mu_p(x) = \int_{- \infty}^{\infty} k(x,x') p(x') \mathrm{d}x'$ in the case where $k$ is an exponentiated-quadratic kernel with length scale hyper-parameter $\sigma$: $$k \big(x,x' \big) \; := \; \lambda^2 \exp \Big( \frac{- \sum_{i=1}^d (x_{i} - x_{i}')^2 }{2 \sigma^2 } \Big) \; = \; \lambda^2 (\sqrt{2 \pi} \sigma)^d \phi \big(x \big| x', \Sigma_{\sigma} \big),$$ where $\Sigma_\sigma$ is a d-dimensional diagonal matrix with entries $\sigma^2$, and where $p(x)$ is a mixture of d-dimensional Gaussian distributions: $$p(x) \quad = \quad \sum_{l=1}^L \rho_l \hspace{1mm} \phi \big(x\big|\mu_l,\Sigma_l \big).$$ (Note that, in this section only, $x_i$ denotes the $i$th component of the vector $x$.) Using properties of Gaussian distributions (see Appendix A.2 of [@Rasmussen2006]) we obtain $$\begin{split} \mu_p(x) \quad & = \quad \int_{- \infty}^{\infty} k(x,x') p(x') \mathrm{d}x' \\ & = \quad \int_{- \infty}^{\infty} \lambda^2 (\sqrt{2 \pi} \sigma)^d \phi\big(x' \big| x, \Sigma_{\sigma} \big) \times \Big( \sum_{l=1}^L \rho_l \hspace{1mm} \phi\big(x'\big|\mu_l,\Sigma_l \big)\Big) \mathrm{d}x' \\ & = \quad \lambda^2 (\sqrt{2 \pi} \sigma)^d \sum_{l=1}^L \rho_l \int_{- \infty}^{\infty} \phi\big(x' \big| x, \Sigma_{\sigma} \big) \times \phi\big(x'\big|\mu_l,\Sigma_l \big) \mathrm{d}x' \\ & = \quad \lambda^2 (\sqrt{2 \pi} \sigma)^d \sum_{l=1}^L \rho_l \int_{- \infty}^{\infty} a_l^{-1} \phi\big(x'\big|c_l,C_l \big) \mathrm{d}x' \\ & = \quad \lambda^2 (\sqrt{2 \pi} \sigma)^d \sum_{l=1}^L \rho_l a_l^{-1} .\\ \end{split}$$ where we have: $$a_l^{-1} \; = \; (2 \pi)^{-\frac{d}{2}} \big| \Sigma_{\sigma} + \Sigma_{l} \big|^{-\frac{1}{2}} \exp \big( -\frac{1}{2} \big(x - \mu_l \big)^T \big( \Sigma_{\sigma} + \Sigma_{l} \big)^{-1} \big(x - \mu_l \big) \big).$$ This last expression is in fact itself a Gaussian distribution with probability density function $\phi(x|\mu_l, \Sigma_l + \Sigma_\sigma)$ and we hence obtain: $$\mu_p(x) \quad := \quad \lambda^2 \big(\sqrt{2 \pi} \sigma \big)^d \sum_{l=1}^L \rho_l \text{ } \phi\big(x|\mu_l, \Sigma_l + \Sigma_\sigma\big).$$ Finally, we once again use properties of Gaussians to obtain $$\begin{split} \int_{- \infty}^{\infty} \mu_p(x) p(x) \mathrm{d}x \quad & = \quad \int_{- \infty}^{\infty} \Big[ \lambda^2 \big(\sqrt{2 \pi} \sigma \big)^d \sum_{l=1}^L \rho_l \text{ } \phi\big(x|\mu_l, \Sigma_l + \Sigma_\sigma\big) \Big] \\ & \quad \times \Big[ \sum_{m=1}^L \rho_m \hspace{1mm} \phi\big(x\big|\mu_m,\Sigma_m \big) \Big] \mathrm{d}x \\ & = \quad \lambda^2 \big(\sqrt{2 \pi} \sigma \big)^d \sum_{l=1}^L \sum_{m=1}^L \rho_l \rho_m \int_{- \infty}^{\infty} \phi\big(x|\mu_l, \Sigma_l + \Sigma_\sigma\big) \phi\big(x\big|\mu_m,\Sigma_m \big) \mathrm{d}x \\ & = \quad \lambda^2 \big(\sqrt{2 \pi} \sigma \big)^d \sum_{l=1}^L \sum_{m=1}^L \rho_l \rho_m a_{lm}^{-1} \\ & = \quad \lambda^2 \big(\sqrt{2 \pi} \sigma \big)^d \sum_{l=1}^L \sum_{m=1}^L \rho_l \rho_m \phi\big(\mu_l|\mu_m,\Sigma_l+\Sigma_m+\Sigma_{\sigma} \big). \end{split}$$ Other combinations of kernel $k$ and density $p$ that give rise to an analytic mean element can be found in the references of [@Bach2015]. Appendix D: Details of the Application to Proteomics Data {#appendix-d-details-of-the-application-to-proteomics-data .unnumbered} --------------------------------------------------------- [*Description of the Model Choice Problem*]{} The ‘CheMA’ methodology described in [@Oates2014] contains several elements that we do not attempt to reproduce in full here; in particular we do not attempt to provide a detailed motivation for the mathematical forms presented below, as this requires elements from molecular chemistry. For our present purposes it will be sufficient to define the statistical models $\{M_i\}_{i=1}^m$ and to clearly specify the integration problems that are to be solved. We refer the reader to [@Oates2014] and the accompanying supplementary materials for a full biological background. Denote by $\mathcal{D}$ the dataset containing normalised measured expression levels $y_S(t_j)$ and $y_S^*(t_j)$ for, respectively, the unphosphorylated and phosphorylated forms of a protein of interest (‘substrate’) in a longitudinal experiment at time $t_j$. In addition $\mathcal{D}$ contains normalised measured expression levels $y_{E_i}^*(t_j)$ for a set of possible regulator kinases (‘enzymes’, here phosphorylated proteins) that we denote by $\{E_i\}$. An important scientific goal is to identify the roles of enzymes (or ‘kinases’) in protein signaling; in this case the problem takes the form of variable selection and we are interested to discover which enzymes must be included in a model for regulation of the substrate $S$. Specifically, a candidate model $M_i$ specifies which enzymes in the set $\{E_i\}$ are regulators of the substrate $S$, for example $M_3 = \{E_2,E_4\}$. Following [@Oates2014] we consider models containing at most two enzymes, as well as the model containing no enzymes. Given a dataset $\mathcal{D}$ and model $M_i$, we can write down a likelihood function as follows: $$\begin{aligned} L(\theta_i,M_i) = \prod_{n=1}^N \phi\left( \frac{y_S^*(t_{n+1}) - y_S^*(t_n)}{t_{n+1} - t_n} \left| \frac{-V_0 y_S^*(t_n)}{y_S^*(t_n) + K_0} + \sum_{E_j \in M_i} \frac{V_j y_{E_j}^*(t_n) y_S(t_n)}{y_S(t_n) + K_j} , \sigma_{\text{err}}^2 \right. \right). \label{chemalike}\end{aligned}$$ Here the model parameters are $\theta_i = \{\mathrm{K} , \mathrm{V}, \sigma_{\text{err}} \}$, where $(\mathrm{K})_j = K_j$, $(\mathrm{V})_j = V_j$, $\phi$ is the normal p.d.f. and the mathematical forms arise from the Michaelis-Menten theory of enzyme kinetics. The $V_j$ are known as ‘maximum reaction rates’ and the $K_j$ are known as ‘Michaelis-Menten parameters’. This is classical chemical notation, not to be confused with the kernel matrix from the main text. The final parameter $\sigma_{\text{err}}$ defines the error magnitude for this ‘approximate gradient-matching’ statistical model. The prior specification proposed in [@Oates2014] and followed here is $$\begin{aligned} \mathrm{K} & \sim & \phi_T \big(K \big| 1, 2^{-1} \mathrm{I} \big), \\ \sigma_{\text{err}} | \mathrm{K} & \sim & p(\sigma_{\text{err}}) \propto 1/\sigma_{\text{err}}, \\ \mathrm{V} | \mathrm{K},\sigma & \sim & \phi_T \big(V \big| 1, N \sigma_{\text{err}}^2 \big(\mathrm{X}(\mathrm{K})^T\mathrm{X}(\mathrm{K})\big)^{-1} \big),\end{aligned}$$ where $\phi_T$ denotes a Gaussian distribution, truncated so that its support is $[0,\infty)$ (since kinetic parameters cannot be non-negative). Here $\mathrm{X}(\mathrm{K})$ is the design matrix associated with the linear regression that is obtained by treating the $\mathrm{K}$ as known constants; we refer to [@Oates2014] for further details. Due to its careful design, the likelihood in Eqn. \[chemalike\] is partially conjugate, so the following integral can be evaluated in closed form: $$L(\mathrm{K},M_i) = \int_{0}^{\infty} \int_{0}^{\infty} L(\theta_i,M_i) p(\mathrm{V},\sigma_{\text{err}} | \mathrm{K}) \mathrm{d}\mathrm{V} \mathrm{d}\sigma_{\text{err}}.$$ The numerical challenge is then to compute the integral $$L(M_i) = \int_{0}^{\infty} L(\mathrm{K},M_i) p(\mathrm{K}) \mathrm{d}\mathrm{K},$$ for each candidate model $M_i$. Depending on the number of enzymes in model $M_i$, this will either be a 1-, 2- or 3-dimensional numerical integral. Whilst such integrals are not challenging to compute on a per-individual basis, the nature of the application means that the values $L(M_i)$ will be similar for many candidate models and, when the number of models is large, this demands either a very precise calculation per model or a careful quantification of the impact of numerical error on the subsequent inferences (i.e. determining the MAP estimate). It is this particular issue that motivates the use of probabilistic numerical methods. [*Description of the Computational Problem*]{} We need to compute integrals of functions with domain $\mathcal{X} = [0,\infty)^d$ where $d \in \{1,2,3\}$ and the sampling distribution $p(x)$ takes the form $\phi_T(x | 1,2^{-1} \mathrm{I})$. The test function $f(x)$ corresponds to $L(\mathrm{K},M_i)$ with $x = \mathrm{K}$. This is given explicitly by the $g$-prior formulae as: $$\begin{aligned} L(\mathrm{K},M_i) & = & \frac{1}{(2 \pi)^{N/2}} \frac{1}{(N+1)^{d/2}} \Gamma\left(\frac{N}{2}\right) b_N^{-\frac{N}{2}}, \\ b_N & = & \frac{1}{2} \left( \mathrm{Y}^T\mathrm{Y} + \frac{1}{N} 1^T \mathrm{X}^T\mathrm{X} 1 - \mathrm{V}_N^T \mathrm{\Omega}_N \mathrm{V}_N \right), \\ \mathrm{V}_N & = & \mathrm{\Omega}_N^{-1} \left( \frac{1}{N} \mathrm{X}^T\mathrm{X} 1 + \mathrm{X}^T\mathrm{Y} \right), \\ \mathrm{\Omega}_N & = & \left(1 + \frac{1}{N} \right) \mathrm{X}^T \mathrm{X}, \\ (\mathrm{Y})_n & = & \frac{y_S^*(t_{n+1}) - y_S^*(t_n)}{t_{n+1} - t_n}, \\\end{aligned}$$ where for clarity we have suppressed the dependence of $\mathrm{X}$ on $\mathrm{K}$. For the Frank-Wolfe Bayesian Quadrature algorithm, we require that the mean element $\mu_p$ is analytically tractable and for this reason we employed the exponentiated-quadratic kernel with length scale $\lambda$ and width scale $\sigma$ parameters: $$k(x,x') = \lambda^2 \exp\left(- \frac{\sum_{i=1}^d (x_i - x_i')^2}{2 \sigma^2}\right).$$ For simplicity we focussed on the single hyper-parameter pair $\lambda = \sigma = 1$, which produces: $$\begin{aligned} \mu_p(x) & = & \int_{0}^{\infty} k(x,x') p(x') \mathrm{d}x' \\ & = & \int_{0}^{\infty} \exp\left(-\sum_{i=1}^d (x_i - x_i')^2\right) \phi_T \big(x' \big|1,2^{-1}\mathrm{I} \big) \mathrm{d}x' \\ & = & 2^{-d/2} \big(1 + \text{erf}(1) \big)^{-d} \prod_{i=1}^d \exp\left(-\frac{(x_i-1)^2}{2}\right) \left(1 + \text{erf}\left(\frac{x_i + 1}{\sqrt{2}}\right)\right),\end{aligned}$$ where $\phi_T$ is the p.d.f. of the truncated Gaussian distribution introduced above and $\text{erf}$ is the error function. To compute the posterior variance of the numerical error we also require the quantity: $$\int_{0}^{\infty} \int_{0}^{\infty} k(x,x') p(x) p(x') \mathrm{d}x \mathrm{d}x' = \int_{0}^{\infty} \mu_p(x) p(x) \mathrm{d}x = \left\{ \begin{array}{cl} 0.629907... & \text{for } d = 1 \\ 0.396783... & \text{for } d = 2 \\ 0.249937... & \text{for } d = 3 \end{array} \right. ,$$ which we have simply evaluated numerically. We emphasise that principled approaches to hyper-parameter elicitation are an important open research problem that we aim to address in a future publication (see discussion in the main text). The values used here are scientifically reasonable and serve to illustrate key aspects of our methodology. FWBQ provides posterior distributions over the numerical uncertainty in each of our estimates for the marginal likelihoods $L(M_i)$. In order to propagate this uncertainty forward into a posterior distribution over posterior model probabilities (see Figs. 3 in the main text and \[fig:model posteriors2\] below), we simply sampled values $\hat{L}(M_i)$ from each of the posterior distributions for $L(M_i)$ and used these samples values to construct posterior model probabilities $\hat{L}(M_i) / \sum_j \hat{L}(M_j)$. Repeating this procedure many times enables us to sample from the posterior distribution over the posterior model probabilities (i.e. two levels of Bayes’ theorem). This provides a principled quantification of the uncertainty due to numerical error in the output of our primary Bayesian analysis. [*Description of the Data*]{} The proteomic dataset $\mathcal{D}$ that we considered here was a subset of the larger dataset provided in [@Oates2014]. Specifically, the substrate $S$ was the well-studied 4E-binding protein 1 (4EBP1) and the enzymes $E_j$ consisted of a collection of key proteins that are thought to be connected with 4EBP1 regulation, or at least involved in similar regulatory processes within cellular signalling. Full details, including experimental protocols, data normalisation and the specific choice of measurement time points are provided in the supplementary materials associated with [@Oates2014]. For this particular problem, biological interest arises because the data-generating system was provided by breast cancer cell lines. As such, the textbook description of 4EBP1 regulation may not be valid and indeed it is thought that 4EBP1 dis-regulation is a major contributing factor to these complex diseases (see [@Weinberg2006]). We do not elaborate further on the scientific rationale for model-based proteomics in this work. ![ Comparison of quadrature methods on the proteomics dataset. *Left:* Value of the MMD$^2$ for FW (black), FWLS (red), FWBQ (green), FWLSBQ (orange) and SBQ (blue). Once again, we see the clear improvement of using Bayesian Quadrature weights and we see that Sequential Bayesian Quadrature improves on Frank-Wolfe Bayesian Quadrature and Frank-Wolfe Line-Search Bayesian Quadrature. *Right:* Empirical distribution of weights. The dotted line represent the weights of the Frank-Wolfe algorithm with line search, which has all weights $w_i=1/n$. Note that the distribution of Bayesian Quadrature weights ranges from $-17.39$ to $13.75$ whereas all versions of Frank-Wolfe have weights limited to $[0,1]$ and have to sum to $1$.[]{data-label="fig:proteinsignalling"}](MMD2protein4.pdf){width="49.00000%"} Appendix E: FWBQ algorithms with Random Fourier Features {#appendix-e-fwbq-algorithms-with-random-fourier-features .unnumbered} -------------------------------------------------------- In this section, we will investigate the use of random Fourier features (introduced in [@Rahimi2007]) for the FWLS and FWLSBQ algorithms. An advantage of using this type of approximation is that the cost of manipulating the Gram matrix, and in particular of inverting it, goes down from $\mathcal{O}(n^3)$ to $\mathcal{O}(nD^2)$ for some user-defined constant $D$ which controls the quality of approximation. This could make Bayesian Quadrature more competitive against other integration methods such as MCMC or QMC. Furthermore, the kernels obtained using this method lead to finite-dimensional RKHS, which therefore satisfy the assumptions required for the theory in this paper to hold. This will be the aspect that we will focus on. In particular, we will show empirically that exponential convergence may be possible even when the RKHS is infinite-dimensional. We will re-use the $20$-component mixture of Gaussians example with $d=2$ from our simulation studies, but using instead a random Fourier approximation of the exponentiated-quadratic (EQ) kernel $k(x,x'):= \lambda^2\exp(-1/2\sigma^2\|x-x'\|_2^2)$ with $(\lambda,\sigma)=(1,0.8)$ and $M=10000$. Following Bochner’s theorem, we can always express translation invariant kernels in Fourier space: $$k(x,x') = \int_\mathcal{W} g(w) \exp\big(jw(x-x')\big)\mathrm{d}w = \mathbb{E}\Big[\exp\big(jw^Tx \big)\exp\big(jw^Tx' \big)\Big]$$ where $w \sim g(w) $ for $g(w)$ being the Fourier transform of the kernel. One can then use a Monte Carlo approximation of the kernel’s Fourier expression with $D$ samples whenever $g$ is a p.d.f.. Our approximated kernel will then lead to a $D$-dimensional RKHS and will be given by: $$k(x,x') \approx \frac{1}{D} \sum_{j=1}^D z_{w_j,b_j}(x)z_{w_j,b_j}(x') = \hat{k}_D(x,x')$$ where $z_{w_j,b_j}(x)=\sqrt{2}\cos(w_j^Tx+b_j)$ and $b_j \sim [0,2\pi]$ uniformly. Random Fourier features approximations are unbiased and, in the specific case of a $d$-dimensional EQ kernel with $\lambda=1$, we have to samples from the following Fourier transform: $$g(w) = \Big(\frac{2\pi}{\sigma^2}\Big)^{-\frac{d}{2}} \exp \Big(-\frac{\sigma^2\|w\|_2^2}{2} \Big)$$ which is a $d$-dimensional Gaussian distribution with zero mean and covariance matrix with all diagonal elements equal to $(1/\sigma^2)$. The impact on the MMD from the use of random Fourier features to approximate the kernel for both the FWLS and FWLSBQ algorithms is demonstrated in Figure \[fig:RFF\_MMD\]. In this example, the quadrature rule uses the kernel with random features but the MMD is calculated using the original $\mathcal{H}$-norm. The reason for using this $\mathcal{H}$-norm is to have a unique measure of distance between points which can be compared. Clearly, we once again have that the rate of convergence of the FWLSBQ is much faster than FWLS when using the exact kernel. The same phenomena is observed for the method with high number of random features ($D=5000$). This suggests that both the choice of design points and the calculation of the BQ weights is not strongly influenced by the approximation. It is also interesting to notice that the rates of convergence is very close for the exact and $D=5000$ methods (atleast when $n$ is small), potentially suggesting that exponential convergence is possible for the exact method. This is not so surprising in itself since using a Gaussian kernel represents a prior belief that the integrand of interest is very smooth, and we can therefore expect fast convergence of the method. However, in the case with a smaller number of random features is used ($D=1000$), we actually observe a very poor performance of the method, which is mainly due to the fact that the weights are not well approximated anymore. In summary, the experiments in this section suggest that the use of random features is a potential alternative for scaling up Bayesian Quadrature, but that one needs to be careful to use a high enough number of features. The experiments also give hope of having very similar convergence for infinite-dimensional and finite-dimensional spaces. [^1]: A detailed discussion on probabilistic numerics and an extensive up-to-date bibliography can be found at <http://www.probabilistic-numerics.org>.

--- abstract: 'When rapidity gaps in high-$p_T$ dijet events are identified by energy flow in the central region, they may be calculated from factorized cross sections in perturbative QCD, up to corrections that behave as inverse powers of the central region energy. Although power-suppressed corrections may be important, a perturbative calculation of dijet rapidity gaps in ${\rm p}\bar{\rm p}$ scattering successfully reproduces the overall features observed at the Tevatron. In this formulation, the average color content of the hard scattering is well-defined. We find that hard dijet rapidity gaps in quark-antiquark scattering are not due to singlet exchange alone.' --- 0.0in 0.0in 6.in 9.in -1.9 cm 24 mm ITP-SB-98-46 [**Energy and Color Flow in Dijet Rapidity Gaps**]{} 1.2cm Gianluca Oderda and George Sterman\ 0.5cm [*Institute for Theoretical Physics\ SUNY at Stony Brook, Stony Brook, NY 11794-3840, USA*]{}\ PACS Nos.: 12.38.Aw, 12.38.Cy, 13.85.-t, 13.87.-a Among the most intriguing recent experimental results in quantum chromodynamics is the observation of dijet rapidity-gap events, with anomalously low radiation in a wide interjet rapidity region [@D0; @CDF; @ZEUS]. These events are typically identified by low or zero hadron multiplicity in the central region, despite the high momentum transfer necessary to produce the jets. The existence of such events was originally suggested on the basis of color flow considerations in QCD [@DKT; @Bjork]. If forward jets are produced by exchanging a pair of gluons in a color singlet state, color can be recombined independently in each forward region. Then much less radiation is expected between the jets than when the exchange is a color octet gluon, which requires recombining color between particles moving in nearly opposite directions. Rapidity gap events have special interest as clear illustrations of color coherence and its interplay with hadronization. In addition, because their observation requires large rapidity intervals, they offer a new window into a perturbative, yet Regge-like limit of QCD. Nonetheless, despite their intuitive appeal, the theoretical understanding of rapidity gaps has been somewhat hampered by two problems. One of these is the issue of “survival" [@Bjork; @GLM]. In any high-energy scattering, multiple soft interactions between spectators of the hard interaction may fill the gap by processes unrelated to the color content of the hard interaction. The second is that, since even the softest gluon carries color in the octet representation, it is not immediately obvious how the color of the hard scattering is to be defined. In this paper, we observe that it is possible to overcome these problems, at least in part, by identifying rapidity gaps in terms of energy flow, rather than multiplicity. The energy flow $Q_c$ into the central rapidity interval between a pair of jets is an infrared safe observable. That is, $d\sigma/dQ_c$ can be written as a convolution of parton distributions with a perturbative hard-scattering function, which depends on $Q_c$. The issue of color flow may then be formulated self-consistently in the hard-scattering function. Corrections to the factorized cross section are proportional to powers of $\Lambda/Q_c$, with $\Lambda$ the scale of the QCD coupling, and may become large for small $Q_c$. As we shall see, however, the purely perturbative cross section remains well-defined, and energy flow gaps appear in this limit, once soft radiation is resummed including color effects [@BottsSt; @KOS1; @KOS2]. In the conventional formulation for rapidity gaps, one writes $f_{\rm gap}=f_{\rm singlet}P_S$, with $f_{\rm gap}$ the fraction of gap events, $f_{\rm singlet}$ the fraction of “hard singlet" exchanges, and $P_S$ the survival probability. Compared to this, we generalize $f_{\rm singlet}$, which then necessarily incorporates a perturbative survival probability. Nonperturbative survival considerations may reappear as we approach zero energy in the gap, but their importance should be reduced in a calorimetric measurement. Our results below support this possibility. To be specific, we will study the process $p(p_A)+\bar{p}(p_B) \rightarrow J_1(p_1)+J_2(p_2)+X_{{\rm gap}}$, where we sum inclusively over final states, while measuring the energy that flows into the intermediate region between two forward jets. For simplicity, we restrict ourselves to valence quarks and antiquarks, $q(k_A)+\bar{q}(k_B) \rightarrow q(k_1)+\bar{q}(k_2)+X$. We will begin by deriving a cross section for this process, specific to the geometry described by the D0 and CDF collaborations [@D0; @CDF; @D0fig]. We go on to evaluate the cross section as a function of $Q_c$, and to view the results in the light of what we have learned from experiment. We will close with a few comments on the relation of our approach to previous work, and on prospects for further development in this problem. Following CDF and D0, we require the two jets, and therefore the outgoing partons coming from the hard scattering, $q(k_1)$, $\bar{q}(k_2)$, to be directed into fixed forward and backward (collectively denoted “forward") regions of the calorimeter, defined by $|y|>y_0$, where $y$ is the (pseudo)rapidity $y=(1/2)\ln\cot(\theta/2)$, with $\theta$ the polar angle. In addition we require the jets to have transverse energies above an experimental threshold, $E_T$. We will discuss cross sections for measured energy in a symmetric central region, spanning rapidity $\Delta y=2y_0$. This geometry is presented schematically in Fig.  \[geometry\]. The inclusive dijet cross section for all events with energy in the central region equal to $Q_c$ is a typical factorizable jet cross section, which may be written as &&(S,E\_[T]{},y) = \_[f\_A,f\_B=u,d]{} d\ && \_0\^1 dx\_A \_0\^1 dx\_B \_[f\_A/p]{}(x\_A,-) \_[|[f]{}\_B/|[p]{}]{}(x\_B,-)\ && \_[f\_1,f\_2=u,d]{} (,,y\_[JJ]{},y,\_s()) ,\ \[crosssec\] where $\phi_{f_A/p}$, $\phi_{\bar{f}_B/\bar{p}}$ are valence parton distributions, evaluated at scale $-\hat t$, the dijet momentum transfer. ${d\hat{\sigma}^{(\a)}}/{dQ_c \, d\cos\hat{\theta}}$ is a hard scattering function, starting with the Born cross section at lowest order. The index $\a$ denotes $f_A+\bar{f}_B \rightarrow f_1 +\bar{f}_2$. The detector geometry determines the phase space for the dijet total rapidity, $y_{JJ}$, the partonic center-of-mass (c.m.) energy squared, $\hat{s}$, and the partonic c.m. scattering angle $\hat{\theta}$, with $-\frac{\hat{s}}{2}\left(1-\cos \hat{\theta} \right)=\hat{t}$. For simplicity of presentation, we take $Q_c$ to be the energy in the dijet c.m. . In the spirit of Refs.[@BottsSt; @KOS1; @KOS2], we now observe that we may perform a further factorization on the partonic hard-scattering function ${d\hat{\sigma}^{(\a)}}/{dQ_c \, d\cos\hat{\theta}}$. The underlying observation is that for $Q_c\ll \sqrt{-\hat{t}}$, the soft gluon radiation that appears in the central region decouples from the dynamics of the hard interaction that produces the dijet event. In technical terms, soft gluon emission may be approximated by an effective cross section, in which the hard scattering is replaced by a product of recoilless color sources (specifically, Wilson lines [@KOS1; @KOS2]) in the directions of the incoming partons and the outgoing jets. The refactorized hard-scattering function then takes the form [@StOd] Q\_c(,,y\_[JJ]{},y,\_s(-t)) &=& H\_[IL]{}( , ,,\_s(\^2) )\ && S\_[LI]{} ( ,y\_[JJ]{},y). \[factor\] The functions $S_{LI}$ and $H_{IL}$ contain the dynamics of soft radiation from Wilson lines (at measured $Q_c)$, and the hard interaction, respectively, with $\mu$ a new factorization scale. The product itself must be independent of how we choose the scale, so long as $\mu>Q_c$. The indices $I$ and $L$ label the possible color structures of the hard interaction, which correspond in $S_{LI}$ to the color matrices that couple the four Wilson lines representing the $2\rightarrow 2$ hard subprocess. One index refers to the hard scattering in the amplitude, the other to the hard scattering in the complex conjugate. For quark-antiquark scattering, the Wilson lines are in the 3 (quark) and 3$^*$ (antiquark) representations of $SU(3)$, respectively, and their product may be characterized by either singlet or octet color exchange in the $t$- or $s$-channel. For the physical reasons outlined above, we will choose a $t$-channel color basis. Corrections to Eq. (\[factor\]) are expected from [*three-jet*]{} final states, for which the analysis below must, in principle, be repeated. We expect these corrections to be relatively small. Because the left-hand side of Eq. (\[factor\]) is independent of the precise choice of factorization scale $\mu$, the matrices $H_{IL}$ and $S_{LI}$ must satisfy evolution equations, in which their variations with $\mu$ cancel each other. The only variable that $H$ and $S$ hold in common is $\alpha_s(\mu^2)$, and, as a result, the evolution equation for $S$ is (+(g) )S\_[LI]{}= -(\_S\^)\_[LB]{}S\_[BI]{}-S\_[LA]{}(\_S)\_[AI]{} , \[eq:resoft\] and similarly for $H$, with $\Gamma_S(\alpha_s)$ an anomalous dimension matrix. Consider a $t$-channel singlet-octet basis, with color vertices schematically given by c\_1=II , c\_2=\_a T\^aT\^a , \[eq:basqqbar\] with $I$ the identity and $T^a$ the generators of $SU(3)$ in the quark representation. A one-loop calculation in this basis gives [@StOd] \_S(y\_[JJ]{},y,) = ( [cc]{} +& -4i\ -8i& - ), \[rapanodim\] where $N_c$ is the number of colors. The functions $\g$ and $\w$ are (y)&=&-2N\_c y +2i, \[defntg\]\ &&(y\_[JJ]{}, y, )=\ &&\ & & +y-2i . \[defntw\] While $\rho$ depends on the jet rapidities and on the partonic scattering angle, $\xi$ depends on the geometry only, through $\Delta y$. We note that the off-diagonal components of $\Gamma_S$ are purely imaginary. This interesting feature is due to strong coherence effects in the one-loop calculation, related to angular ordering [@cohere]. To study the $Q_c$-dependence of $S$, it is convenient to diagonalize $\Gamma_S$. In the basis in which $\Gamma_S$ is diagonal, Eq. (\[eq:resoft\]) implies that the components of $S$ evolve independently in $\mu$. In this basis we may calculate unambiguously the dependence on the central energy flow $Q_c$. This is the technique that we summarize in the following. The eigenvectors of $\Gamma_S$ in Eq. (\[rapanodim\]) may be chosen as e\_1&=&( [c]{} 1\ (-)\^[-1]{}\ )\ e\_2&=&( [c]{} (+) ,\ 1\ ), \[eigenvectors\] where we define (y) . \[ydef\] A very useful feature is that these eigenvectors are independent of the jet rapidities, and depend only on $\Delta y$. The corresponding eigenvalues of $\Gamma_S$ are in general complex, \_1&=&\ \_2&=&. \[eigenvalues\] In the limit of a large central region, $\Delta y \gg 1$, the function $\g$ has a large negative real part, while the real part of $\eta$ is positive, and grows with $\Delta y$. From Eq. (\[eigenvectors\]), we see that, as $\Delta y\rightarrow \infty$, $e_1$ reduces to a color “quasi-singlet”, and $e_2$ to a color “quasi-octet”. As a realistic example, we take the value $\Delta y=4$, and find e\_1&=&( [c]{} 1\ 0.455 e\^[2.161 i]{}\ )\ e\_2&=&( [c]{} 0.101 e\^[-0.981 i]{}\ 1\ ). \[eigenvnumb\] For this configuration, the second eigenvector is close to a color octet, but the first is still a mixture of octet and singlet, with the latter only slightly predominant. In the following, however, we find it suggestive to retain the names “quasi-singlet” and “quasi-octet” for the elements of the diagonal basis. In the limit of large $\Delta y$, the eigenvalue for the quasi-octet grows with $\Delta y$, while the eigenvalue of the quasi-singlet does not. This will produce the expected enhancement of the latter relative to the former in the resummed cross section at small $Q_c$. We shall use Greek indices to identify the basis in which $\Gamma_S$ is diagonal. We can now write down a resummed cross section, working to lowest order in $\alpha_s(-\hat{t})$, but resumming all leading logarithms in $Q_c$. We transform Eq. (\[factor\]) to the diagonal basis, and solve the evolution equation for $S$, to get (,,y\_[JJ]{},y,\_s(-t)) &=&\ && H\^[(1)]{}\_( y,,, \_s(-) ) S\^[[(0)]{}]{}\_ ( y )\ && \^[E\_-1]{} \^[-E\_]{} .\ \[factor2\] The coefficients $E_{\gamma \beta}$ are given by &&E\_(y\_[JJ]{},,y ) = , \[expon\] where $\beta_1$ is the first coefficient in the expansion of the QCD $\beta$-function, $\beta_1=\frac{11}{3}N_c-\frac{2}{3}n_f$, and where we define $\hat{ \lambda}_{\beta}$ by $\lambda_{\beta}=\alpha_s \hat \lambda_\beta+\cdots$. In accordance with our approximation, the matrix $S^{{(0)}}_{\gamma \beta}$ is obtained by transforming the zeroth order $S^{{(0)}}_{LI}$ of Eq. (\[factor\]) to the new basis. The matrix $S^{{(0)}}_{LI}$ is just a set of color traces, S\^[[(0)]{}]{}\_[LI]{}=( [cc]{} N\^2\_c & 0\ 0 & ( N\^2\_c-1 ) ), \[softcol\] and is transformed to the diagonal basis by the matrix ${\left( R^{-1} \right)}_{K \beta} \equiv \left( {e_{\beta}} \right)_K$ [@KOS2], S\^[(0)]{}\_ \_[M]{} S\^[(0)]{}\_[MN]{} [( R\^[-1]{} )]{}\_[N ]{} . \[eq:newbasS\] Analogously, we take for $H^{(1)}_{IL}$ the square of the single-gluon exchange amplitude, represented in the color basis. Considering the dominant $t$-channel Born-level amplitude alone, which is purely octet, we have $H^{(1)}_{IL}=\delta_{I2}\, \delta_{L2} \, \hat{\sigma}_t$, where $\hat{\sigma}_t$ is the $t$-channel partonic cross section, including the coupling $\alpha_s(-\hat{t})$. The contribution of $s$-channel diagrams has a relatively small effect, and will be described elsewhere [@StOd]. In the diagonal basis the hard matrix $H^{(1)}_{IL}$ becomes $H^{(1)}_{\beta\gamma}$, defined as H\^[(1)]{}\_=[( R )]{}\_[ L]{} H\^[(1)]{}\_[LK]{} [( R\^ )]{}\_[K ]{}. \[newbasH\] Observe that $S^{(0)}$ and $H^{(1)}$ both acquire a $\Delta y$-dependence through the change of basis. From Eqs. (\[factor2\])-(\[expon\]), using the results described above, it is possible to evaluate Eq. (\[crosssec\]). For the valence partons we have taken the leading order CTEQ4L distributions [@cteq4]. In Fig. \[results\] we plot the shapes of the cross sections obtained in this way, as a function of the radiation into the central region, $Q_c$, for two different sets of conditions, $\sqrt{S}=630$ GeV, $\Delta y=3.2$, and $\sqrt{S}=1800$ GeV, $\Delta y=4.0$. We also show the contributions of quasi-octet and quasi-singlet terms. There is in addition a negative interference term, not exhibited separately. As anticipated above, we find a strong suppression of the quasi-octet component for very small values of $Q_c$, contrasted to a peak for the quasi-singlet in this limit. The reason for this difference is easily found. For most kinematic configurations, the coefficient $E_{11}$ from Eq. (\[expon\]) is less than one, so that the quasi-singlet cross section in Eq. (\[factor2\]) decreases monotonically with increasing $Q_c$. For the quasi-octet, on the other hand, $E_{22}$ is always greater than unity, so that its contribution grows with $Q_c$, until the power of the logarithm is overcome by the dimensional factor of $1/Q_c$. These results can be compared with the experimental data in Fig. 1 of Ref. [@D0fig], showing the measured number of events as a function of the number of towers counted in the central region of the calorimeter, clearly related to $Q_c$. We can understand the similarity of shapes in terms of the $Q_c$ dependence in Eq. (\[factor2\]), discussed above. This similarity is suggestive; indeed, from our simulation we have evaluated the minimum-maximum ratio of the cross section, finding about $30\%$ at $\sqrt{S}=630 \, {\rm GeV}$ and about $15 \%$ at $\sqrt{S}=1800 \, {\rm GeV}$, close to the analogous ratios in Fig. 1 of Ref. [@D0fig]. We have also determined an analog of a “hard singlet fraction" [@D0; @CDF], as the ratio of the area under the quasi-singlet curve to the area under the overall curve. It is about $5\%$ at $\sqrt{S}=630 \, {\rm GeV}$ and about $3 \%$ at $\sqrt{S}=1800\, {\rm GeV}$. The order of magnitude of the result is reasonable, although higher than the roughly $1\%$ found at the Tevatron using track or tower multiplicities. How much of this difference is due to our new definition of the gap and how much to the lack of a nonperturbative survival probability remains to be explored. The sharp upturn that we observe below $1 \, {\rm{GeV}}$ is due to the divergence of the perturbative running coupling at $Q_c=\Lambda$; nonperturbative effects will attenuate this rise. Previous analysis of rapidity gaps in dijet events has tended to emphasize either the short-distance [@Bjork; @DDucaT] or long-distance [@BuchHeb; @eboli; @zepp] aspects of the problem. (The role of Sudakov logarithms in double-rapidity gap events has been discussed in [@MRK].) Here, we have argued that by factorizing short- and long-distance effects, we may treat both dependences systematically. In our formalism, the mixing of color states begins at short distances precisely with two-gluon exchange [@Bjork; @DDucaT], summarized through the anomalous dimension $\Gamma_S$, while long-distance color (“bleaching") effects [@BuchHeb; @eboli; @zepp] follow the evolution of the different color components between the short-distance scale $\sqrt{-\hat{t}}$ and the long-distance scale $Q_c$. As we have observed above, our formalism does not include a nonperturbative survival probability [@Bjork; @GLM] associated with the interaction of spectator partons. Clearly, a full phenomenological analysis will also require the inclusion of processes involving gluons (including $q\bar q\rightarrow gg$) and sea quarks. The treatment of gluon-gluon scattering [@eboli; @zepp] should be particularly interesting [@KOS2]. Nevertheless, we believe that the basic features shown in the valence-quark analysis outlined above will appear as well in a more complete discussion. A calorimetric analysis of dijet rapidity gap events, if possible experimentally, could shed valuable light on the dynamics of QCD. Acknowledgments {#acknowledgments .unnumbered} --------------- We are indebted to Jack Smith for his valuable advice in the implementation of the numerical simulation. This work was supported in part by the National Science Foundation, grant PHY9722101. 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--- abstract: 'Experimental studies of systems containing active proteins that undergo conformational changes driven by catalytic chemical reactions have shown that the diffusion coefficients of passive tracer particles and active molecules are larger than the corresponding values when chemical activity is absent. Various mechanisms have been proposed for such behavior, including, among others, force dipole interactions of molecular motors moving on filaments and collective hydrodynamic effects arising from active proteins. Simulations of a multi-component system containing active dumbbell molecules that cycle between open and closed states, a passive tracer particle and solvent molecules are carried out. Consistent with experiments, it is shown that the diffusion coefficients of both passive particles and the dumbbells themselves are enhanced when the dumbbells are active. The dependence of the diffusion enhancement on the volume fraction of dumbbells is determined, and the effects of crowding by active dumbbell molecules are shown to differ from those due to inactive molecules.' author: - 'Matthew Dennison$^{1}$' - 'Raymond Kapral$^{1,2}$' - 'Holger Stark$^{1}$' bibliography: - 'main.bib' title: 'Diffusion in systems crowded by active force-dipole molecules' --- Introduction {#sec:intro} ============ A body of evidence points to the existence and importance of nonthermal fluctuations in the cell that are driven by chemical activity that maintains the cell in a nonequilibrium state. [@ref:Caspi; @ref:Lau; @ref:Fred_motor; @ref:Wilhelm08; @ref:Gallet; @ref:Weber; @ref:Bruinsma; @ref:Guo] Such fluctuations have been measured and characterized using various experimental probes and are often attributed to the forces generated by molecular motors when they interact with filaments comprising the cellular cytoskeletal network. The results of [*in vitro*]{} experiments on systems containing actin networks and active myosin motors also suggest that nonthermal fluctuations play a significant role in the systems’ dynamical response to deformation. [@ref:Mizuno; @ref:Toyota] Support for such effects is provided by the observation that the mean square displacements of passive molecules and active species are smaller when the production of ATP is inhibited. For example, the diffusive dynamics of chromosomal loci in prokaryotic cells is sensitive to metabolic activity; when ATP synthesis is inhibited the apparent diffusion coefficient decreases. [@ref:Weber] Force-spectrum-microscopy studies have shown that force fluctuations in eukaryotic cells enhance the movement of large and small molecules; when the activity of myosin II motors is selectively inhibited diffusive motion decreases but not to the degree when all ATP synthesis is suppressed. [@ref:Guo] These studies have concluded that it is the aggregate of all metabolic activity, and not just that of motor proteins, that contributes to enhanced diffusive motion. Enhanced diffusion of enzymes and passive particles has also been observed in [*in vitro*]{} studies of active enzymes in solution where motor proteins are not present [@ref:Sen-enz10; @ref:Sengupta13; @ref:Sen-grubbs], and the possible origins of these effects have been discussed [@ref:Golestanian]. Proteins executing conformational changes as a result of catalytic chemical activity can give rise to collective hydrodynamic effects that enhance the diffusion of both passive particles and enzymes. [@ref:Mikhailov15; @Kapral16; @ref:Mikhailov16] On more macroscopic scales, and in a somewhat different context, experimental and theoretical investigations of the diffusion coefficients of passive particles in suspensions of active microorganisms have shown diffusion enhancement due to the hydrodynamic flow fields generated by their swimming motions. [@Kim_Breuer_04; @Leptos_09; @Mino_13; @saintillan2012; @Kasyap_14] In this paper we investigate diffusive dynamics in a system containing active dumbbell molecules, a passive particle and solvent. The active dumbbell molecules cycle between open and closed conformations and act as nonequilibrium fluctuating force dipoles. The microscopic dynamics accounts for direct hydrodynamic interactions as well as direct interactions among the dumbbell molecules. Molecular crowding is known to influence the diffusive properties of tracer particles in solutions where the concentration of crowding species is high: subdiffusive dynamics is observed on intermediate time scales and long-time diffusion coefficients decrease as the concentration of crowding elements increases. [@Metzler00; @Schnell04; @HF13; @Nakano14; @Kuznetsova14] Our investigations show how the diffusive dynamics of passive particles, and the dumbbells themselves, vary with the volume fraction of active dumbbell molecules. Comparisons with the results for the diffusive dynamics in systems containing inactive dumbbells allow us to analyze and describe the effects of dumbbell activity. The outline of the paper is as follows. Section \[sec:method\] describes the system under study, including the active dumbbell-shaped “molecules", the interaction potentials among species and the dynamical method used to evolve the system. The properties of single active and inactive dumbbell molecules are presented in Sec. \[sec:single\]. Systems containing many active dumbbell molecules are considered in Sec. \[sec:diffusion\], where simulation results for the self-diffusion coefficients of the passive particle and dumbbell molecules are given as a function of the dumbbell force constants and volume fractions. The conclusions of the study are given in Sec. \[sec:Conclusions\]. System and dynamical model {#sec:method} ========================== The entire system is comprised of dumbbell-shaped active molecules, a passive particle and solvent molecules. The dumbbell-shaped molecules consist of two beads linked by a harmonic bond, with interactions between beads of different dumbbells described by the steep repulsive potential function [@ref:Padding], $$\label{eq:Ucc} U_{cc} = 4\epsilon\left[\left(\frac{\sigma_{c}}{r}\right)^{48}-\left(\frac{\sigma_{c}}{r}\right)^{24}+\frac{1}{4} \right] \;\; r \le 2^{1/24}\sigma_{c},$$ and zero otherwise, where $\epsilon$ sets the energy scale, which we take to be $\epsilon=2.5 \; k_{B}T$ throughout, with $k_{B}$ the Boltzmann constant and $T$ the temperature. The passive particle is a structureless bead of diameter $\sigma_{c}$ and mass $m_{c}$, and interacts with the dumbbell beads through the potential in Eq. (\[eq:Ucc\]). The passive particle interacts with the solvent particles through the repulsive Lennard-Jones interaction potential $U_{cf}$, given by $$\label{eq:Ucf} U_{cf} = 4\epsilon\left[\left(\frac{\sigma_{cf}}{r}\right)^{12}-\left(\frac{\sigma_{cf}}{r}\right)^{6}+\frac{1}{4} \right] \;\; r \le 2^{1/6}\sigma_{cf},$$ and zero otherwise, where $\sigma_{cf}$ is the passive particle-solvent interaction distance. Other interactions involving the solvent are taken into account through multiparticle collision (MPC) dynamics, comprising streaming and collision steps. [@ref:SRD; @*ref:SRD2; @Kapral_08; @ref:Gompper] The solvent molecules are represented by point particles of mass $m_f$ which are evolved in the streaming step, either ballistically or, when potential interactions are present, by Newton’s equations of motion. In the collision steps, which occur at time intervals $\tau_{{\text MPC}}$, the solvent particles are sorted into cubic collision cells with length $a$, in which they interact with each other according to multiparticle collisions. The coupling of the dumbbell to the solvent also can be accounted for in this way, where the constituent spheres of the dumbbell are included with the solvent particles in the collision step, in the same manner as for single polymers [@ref:polymer2; @ref:polymer3]. Further simulation details on the implementation of the MPC algorithm, along with parameter values, are given in Appendix A. Dumbbell molecule {#dumbbell-molecule .unnumbered} ----------------- While the dumbbells are fictitious “molecules” their dynamics is constructed to mimic the conformational changes that occur in active enzymes. [@kogler12; @ref:Mikhailov15] Many catalytically active proteins cycle between open and closed conformations: substrate binding triggers passage from the open to closed state, while substrate unbinding or product release causes the protein to return to its open conformation. Such systems are maintained in a nonequilibrium state by input of substrate and removal of product. The dumbbell beads, each with mass $m_b$, are linked by a harmonic bond that specifies open (large bond rest length) and closed (small bond rest length) conformations. The bond potential energy function has the form, $$\label{eq:En_spring} U=\displaystyle\frac{1}{2}k_0(t)\left(\ell-\ell_{0}(t)\right)^{2},$$ where the bond rest length $\ell_0(t)$ and force constant $k_0(t)$ are dichotomous random variables that take the two values $\{\ell_o,\ell_c\}$ and $\{k_o,k_c\}$. These correspond to the values for the open and closed configurations, respectively. A stochastic process that switches the dumbbell between the open and closed states is as follows: suppose the current rest bond length is $\ell_c$. If during the evolution the bond length $\ell$ crosses a threshold and satisfies the condition $\ell<\ell_{c}+\delta\ell_{c}$, a random time $t_{h}$ is drawn from a log-normal distribution with average $t_{c}$. The rest length and force constant will remain as $\ell_{0}(t)=\ell_{c}$ and $k_{0}(t)=k_{c}$ for this time, after which $\ell_0(t)$ is set to $\ell_{o}$ and the force constant to $k_{0}(t)=k_{o}$. Similarly, if $\ell>\ell_{o}-\delta\ell_{o}$, $\ell_0(t)$ is set to $\ell_{0}(t)=\ell_{c}$ and $k_0(t)$ to $k_c$ after a randomly chosen time $t_{h}$ with average $t_{o}$. This model captures the gross features of active proteins that adopt open and closed metastable conformations and operate though Michaelis-Menten kinetics, $E+S\mathrel{ \mathop{\kern0pt {\rightleftharpoons}}\limits^{{\mathrm{k}_1}}_{\mathrm{k}_{-1}}} C \stackrel{\mathrm{k}_{cat}}{\rightarrow } E+P$, where $E,\;S,\;C$ and $P$ represent the enzyme, substrate, enzyme-substrate complex and product, respectively, with excess substrate supplied and product removed. We shall call dumbbells that undergo such nonequilibrium cyclic conformational changes [*active*]{} dumbbells. If instead the stochastic mechanism responsible for these changes is absent and only thermal fluctuations are present, the dumbbells will be termed [ *inactive*]{} dumbbells. In our model, the inactive dumbbells will simply fluctuate around the open conformation. This corresponds to a system where enzymes are not supplied with substrate and remain in open conformations. Units and parameters {#units-and-parameters .unnumbered} -------------------- Results are reported in dimensionless units: lengths are scaled by the MPC cell size $a$, masses by the solvent particle mass $m_f$, energy by $k_BT$ and time by $(a^2 m_f/k_BT)^{1/2}$. The spring constant $k$ is in units of $k_BT/a^2$. In the simulations presented below we set $t_{o}=0$ and vary $t_c$, the average time spent in the closed conformation. Furthermore, we let $k_o=k$ and choose $k_c= 2k$ and $\ell_c = \ell_o/2$. Our choice of $t_{o}=0$ corresponds to a system with excess substrate and reaction rates $\mathrm{k}_1, \; \mathrm{k}_{cat} \gg \mathrm{k}_{-1}$. The closed and open dumbbell bond lengths used in all of the simulation results are $\ell_{c}=2$ and $\ell_{o}=4$, respectively, and $\delta\ell_{c}=\delta\ell_{o}=0.05( \ell_{o}-\ell_{c})=0.1$. Solvent conditions will be indicated by the value of $\tau_{{\text MPC}}$. Unless stated otherwise, simulations use $\tau_{{\text MPC}}=0.01$, but some results will be presented for $\tau_{{\text MPC}}=0.05$ to explore the effects of different solvent conditions (see Appendix A). Properties of single active and inactive dumbbells {#sec:single} ================================================== ![(a) Plot of the instantaneous bond length $\ell(t)$ against time $t$ for an active dumbbell with force constant $k=20$. Results are presented for system with average hold times $t_{c}=0$ (top) and $t_{c}=100$ (bottom). (b) Probability density of lengths $P(\ell)$ against length $\ell$ for the above systems with various average hold times $t_{c}$ indicated in the plot; $P(\ell)$ for inactive dumbbells is also shown. The solid black line on the left peak shows a Gaussian distribution centred around the closed configuration, with mean $\ell_c$ and variance $\sigma^2=k_BT/k_c$, and the solid black line on the right peak shows a Gaussian distribution centred around the open configuration, with mean $\mu=\ell_o$ and variance $\sigma^2=k_BT/k_o$.[]{data-label="fig:l_t"}](fig1.eps){height="1.0\columnwidth"} Figure \[fig:l\_t\](a) shows how the bond length $\ell(t)$ of a single active dumbbell varies with time as the dumbbell cycles between open and closed conformations. The open and closed rest lengths are indicated by the solid horizontal lines. Data for two values of the average time spent in the closed conformation, $t_c=0$ and $t_c=100$, are presented. When the dumbbell is in the metastable open or closed states its dynamics will be controlled by thermal fluctuations about the rest values $\ell_o$ and $\ell_c$ of these states. The average time for a complete open-close cycle, $t_{\rm{cy}}$, is dominated by $t_{c}$ when $t_{c}>t_{t}$, where $t_{t}$ is the time taken to pass from one metastable state to the other. We note that $t_{t}$ will depend on both the force constants $\{k_o,k_c\}$ and the rest bond lengths $\{\ell_o,\ell_c\}$. In Fig. \[fig:l\_t\](b) we show the probability density $P(\ell)$ of bond lengths for a range of average hold times $t_{c}$. For $t_c=0$ we see two peaks of roughly equal size centered near to but smaller than the open and closed configurations, as the rest length switches between the two values $\{\ell_o =4, \ell_c = 2\}$. As we increase $t_c$ the peak close to $\ell_o$ decreases while the one about $\ell_c$ becomes more pronounced, approaching that of a Gaussian distribution with mean $\mu=\ell_c$ and variance $\sigma^2=k_BT/k_c$, resulting from thermal motion about $\ell_c$. We also present data for an inactive dumbbell, corresponding to a protein in the absence of substrate that remains in the open configuration, which also exhibits a Gaussian distribution with mean $\mu=\ell_o$ and variance $\sigma^2=k_BT/k_o$. The orientational dynamics of the dumbbell molecules can be characterized by the time $t_r$ it takes the orientational correlation function, $C_{S}(t) = \langle \hat{{\bf e}}(t) \cdot \hat{{\bf e}}(0) \rangle=\langle \cos\theta(t)\rangle$, to decay to 1/e of its initial value. Here $\theta$ is the angle between the dumbbell’s initial orientation $\hat{{\bf e}}(0)$ and its orientation $\hat{{\bf e}}(t)$ at time $t$. This function is plotted in Fig. \[fig:S\_t\] for both inactive and active dumbbells. For inactive dumbbells $t_r \approx 300$. When the dumbbells are active, $t_r$ is shorter, particularly for small $t_{c}$ and large $k$, with $t_r \approx 10$ for $t_{c}=20$ and $k=90$, and $t_r \approx 40$ for $t_{c}=200$ and $k=20$. ![Plot of the orientational correlation function $C_{S}(t) = \langle \cos\theta(t)\rangle$, where $\theta$ is the angle between the dumbbell’s initial orientation and its orientation at time $t$, against $t$ for a single dumbbell in solution. Parameters are indicated in the legend.[]{data-label="fig:S_t"}](fig2.eps){height="1.0\columnwidth"} The force dipole for a dumbbell molecule is $m(t) = -k_0\ell(t)(\ell(t)-\ell_{0})$, where $k_0$ and $\ell_{0}$ stand for, respectively, the spring constant and bond rest length at the time when $m(t)$ is measured. We define the normalized temporal force-dipole autocorrelation function by $C_m(t)=\langle \Delta m(t)\Delta m(0) \rangle / \langle \Delta m^2 \rangle$, with $\Delta m =m -\langle m \rangle$. It has an initial value of unity and decays to zero at long times, since the asymptotic value of $\langle m(t)m(0) \rangle$ is $\langle m \rangle^2$. This correlation function is plotted in Fig. \[fig:mcf\_t\](a) for several values of the force constant $k$. The force dipole correlations decay with a strongly damped oscillatory tail at longer times that is due to the changes in sign when the forces that trigger closing or opening change their sign. The force dipole correlation time $t_m$, defined as the time for $C_m(t)$ to decay to 1/e of its initial value, ranges from $t_m \approx 2-7$ for the data in the figure. These times are less than an order of magnitude shorter than the orientational correlation times. For active dumbbells $\langle \Delta m^{2}\rangle$ depends on $k$, $\ell_{o,c}$ and $t_{c}$, and its magnitude decreases with increasing $t_{c}$. We find that it scales with the force constant as $\langle \Delta m^{2}\rangle \sim k^{\alpha}$, where $\alpha$ also changes with $t_{c}$. For the results shown in the inset to Fig. \[fig:mcf\_t\](a), we find $\alpha \sim 1.6$ for $t_{c}=20$ and $\alpha\sim1.2$ for $t_{c}=100$. In the limit $t_{c} \to \infty$, $\alpha=1$. ![(a) Plot of $C_m(t)= \langle \Delta m(t)\Delta m(0)\rangle / \langle \Delta m^2 \rangle$, with $m = -k_0\ell(\ell-\ell_{0})$ the force dipole for a dumbbell, against $t$ for an active dumbbell with $t_{c}=100$. The value of $k$ is indicated in the legend. Also shown is data for $t_{c}=20$ at $k=70$. The value of $\langle \Delta m^{2}\rangle$ is plotted in the inset as a function of $k$ for $t_{c}=100$ (red circles) and $t_{c}=20$ (green triangles). The solid lines show the dependence of this quantity on $k^{\alpha}$ with $\alpha$ indicated in the plot. (b) Plot of $C_m(t)$ against $t$ for an inactive dumbbell. Solid black lines show the theoretical prediction given by Eq. (\[eq:langevin2\]). The value of $\langle \Delta m^{2}\rangle$ is shown in the inset as a function of $k$, where the solid line shows a $k^{\alpha}$ dependence with $\alpha=1$. []{data-label="fig:mcf_t"}](fig3.eps){height="1.0\columnwidth"} The results for an active dumbbell may be contrasted with those for an inactive dumbbell that simply experiences thermal fluctuations about its open conformation. The correlation function $C_m(t)$ for this situation is plotted in Fig. \[fig:mcf\_t\](b). It decays monotonically to its long-time value, signalling the absence of anti-correlation effects that arise from the active dumbbell conformational changes. In addition, $\langle \Delta m^{2}\rangle$ now scales as $\langle \Delta m^{2}\rangle \sim k$. A simple Langevin model, $$\label{eq:langevin} \mu \frac{d^2 \ell(t)}{dt^2}= -\zeta \frac{d \ell(t)}{dt} -\mu \omega_o^2 (\ell(t)-\ell_o) +f(t),$$ can be used to compute $C_m(t)$ for an inactive dumbbell. In this equation $\zeta$ is the friction coefficient, $\mu=m_b/2$ is the relative dumbbell mass, $\omega_o^2=k_o/\mu$ and $f(t)$ is a Gaussian white-noise random force with correlation function $\langle f(t)f\rangle= 2 k_B T\zeta \delta(t)$. The force dipole here takes the form $m(t)=-k_o \ell(t)(\ell(t)-\ell_o)$ with $k_o=k$. Using the solution of Eq. (\[eq:langevin\]), the unnormalized force dipole correlation function is given by $$\begin{aligned} \label{eq:langevin2} &&\langle m(t) m(0)\rangle =(k_BT)^2 \Big\{1 + \nonumber \\ && 2e^{-\gamma t}\big( 1+\big(2 + \big(\frac{\omega_o}{\omega} \big)^2\big) \sinh^2 \omega t + \frac{\gamma}{2\omega} \sinh 2\omega t\big) + \nonumber \\ && \frac{k \ell_o^2}{k_B T} e^{-\gamma t/2 }\big( \frac{\gamma}{2 \omega} \sinh \omega t +\cosh \omega t \big)\Big\},\end{aligned}$$ where $\gamma=\zeta/\mu$, $\omega=\sqrt{\gamma^2/4- \omega_o^2}$ with $\gamma > 2 \omega_o$ for overdamped dynamics. Its limiting form is $\lim_{t \to \infty}\langle m(t) m(0)\rangle= \langle m\rangle^2=(k_BT)^2$, which we may use to calculate $C_m(t)$. The only unknown parameter in the expression for $\langle m(t) m(0)\rangle$ is the friction coefficient $\zeta$ that appears in the ratio $\gamma=\zeta/\mu$. By fitting to the data for a single force constant, we obtain a value of $\gamma\sim3.19$, from which one obtains good agreement with the simulation results for all values of the force constant shown in Fig. \[fig:mcf\_t\](b). In the strongly overdamped limit Eq. (\[eq:langevin2\]) takes the simpler form, $$\langle m(t) m(0)\rangle =(k_BT)^2 \Big\{1 + 2 e^{-2kt/\zeta} + \frac{k \ell_o^2}{k_B T} e^{-kt/\zeta}\Big\}.$$ For inactive dumbbells $ \langle \Delta m^2\rangle=(k_BT)^2\left(2+k\ell_o^2/k_BT \right)$, which shows the linear scaling of $ \langle \Delta m^2\rangle$ with $k$ seen in the inset to Fig. \[fig:mcf\_t\](b). Since $k \ell_o^2/2k_BT \gg 1$, to a good approximation we may write $ \langle m^2\rangle/k \ell_o^2 \approx k_BT$ and this ratio is approximately independent of the force constant magnitude. The decay from this initial value of $C_m(t)$ depends on the value of the $k$, with a larger value resulting in a faster decay, as can be seen in Fig. \[fig:mcf\_t\](b). The decay time $t_m$ varies between $t_m \approx 2-8$, comparable to that for active dumbbells, albeit from a smaller initial value. Diffusion in a field of active dumbbells {#sec:diffusion} ======================================== The diffusion of a passive particle as well as the self-diffusion of an active dumbbell, in a field of active dumbbell molecules, are discussed in this section. A visual representation of the system under study is given in Fig. \[fig:diagrams\], which shows an instantaneous configuration of the dumbbells and passive particle drawn from the dynamics. Solvent particles are not displayed due to their large number. ![Instantaneous configuration of the system showing active dumbbells (green) and the single passive particle (red) for a system with dumbbell volume fraction $\phi=0.133$. []{data-label="fig:diagrams"}](fig4.eps){height="0.8\columnwidth"} All our simulations start from an isotropic configuration of dumbbell particles, and for all the system parameters studied here we find no evidence of positional or orientational ordering of the dumbbells. Passive particle diffusion {#passive-particle-diffusion .unnumbered} -------------------------- As briefly described in the Introduction, the diffusion coefficients of passive particles are enhanced when the medium in which they move contains active enzymes or swimmers. Recent experimental studies have shown that even on microscopic scales the diffusion of passive molecular tracers are enhanced in the presence of active catalyst molecules. [@ref:Sen-tracer] In this subsection, we determine how the diffusion coefficient of a passive tracer particle varies as a function of the volume fraction of active dumbbells. Although our study is motivated by the diffusive dynamics of active enzyme systems, no specific enzymatic system is considered. Rather, we explore the dependence of the diffusion coefficients on a wide range of dumbbell and other system parameters. Consider a single passive particle immersed in a system of volume $V$ containing $n_{\mathrm{db}}$ dumbbells with volume fraction $\phi=v_{\mathrm{db}}n_{\mathrm{db}}/V=v_{\mathrm{db}}c$, where $v_{\mathrm{db}}$ is the volume of a dumbbell and $c=n_{\mathrm{db}}/V$ is the dumbbell concentration. We define this volume to be that of overlapping monomer spheres with radius $\sigma_c/2$ in the open configuration, $v_{\mathrm{db}}=2\pi[\sigma^{3}_{c}/2-(\sigma_{c}-l_{o})^2(2\sigma_{c}+l_{o})/8]/3$ with $v_{\mathrm{db}}=82.96$ for the parameters used here. The effective volume will also vary as the dumbbell undergoes conformational changes, so the volume fraction obtained using this value of $v_{\mathrm{db}}$ simply provides a convenient way to specify the dumbbell concentration, $c$. The diffusion coefficient $D$ may be determined from the long-time limit of the mean square displacement [^1] $\Delta R^2(t)= \langle |{\bf R}(t)-{\bf R}(0)|^2 \rangle \sim 6D t$, where ${\bf R}$ is the position of the passive particle and the angular brackets denote a time and ensemble average. In general the diffusion coefficient will depend on the dumbbell volume fraction, force constants and average hold time, $D=D(\phi,k,t_c)$. In the absence of dumbbells ($\phi=0$) it will be denoted by $D_0$ and for our system parameters this has the value $D_0=1.14 \times 10^{-3}$. The thermal diffusion coefficient of the passive particle in a solution containing inactive dumbbells, denoted by $D_T(\phi)$, will also be a function of the dumbbell volume fraction, since crowding by dumbbells will alter its value. ![Diffusion coefficient $D$ of the passive particle versus dumbbell volume fraction $\phi$, in a system of active dumbbells with $t_{c}=20$ and two values of the force constant $k$ indicated in the figure. Also shown is $D_T$, the diffusion coefficient of a passive particle in a system of inactive dumbbells. Data is normalized by $D_0$, the diffusion coefficient of a single passive particle in the absence of dumbbells.[]{data-label="fig:D_c"}](fig5.eps){height="\columnwidth"} Figure \[fig:D\_c\] compares the dependence of $D$, for two values of $k$ and fixed $t_c=20$, and $D_T$ on $\phi$. (Additional data is given in the Appendix B.) For inactive dumbbells $D_T(\phi)$ decreases with increasing $\phi$, consistent with the fact that the inactive dumbbells act as crowding agents and inhibit the diffusive dynamics of the passive particle. The solid line in the figure shows that the diffusion coefficient varies linearly with $\phi$ over the range of volume fractions presented. Expressed as a function of $\phi$, we have $D_T(\phi)=D_0(1+\kappa_o \phi)$, where the constant $\kappa_o=-0.321$, which is independent of $k$, is obtained from a fit to the data. The dependence on the dumbbell volume fraction is different when the dumbbells are active: now $D$ increases with increasing $\phi$ over the same range of $\phi$ values, rather than decreasing as is the case for crowding by inactive dumbbells. We may write $D(\phi,k,t_c)=D_0(1+\kappa(k,t_c) \phi)$, where $\kappa$ depends on $k$ and $t_c$, with $\kappa=0.701$ and 0.067 for $k=90$ and $35$, respectively, for the data in Fig. \[fig:D\_c\]. ![Active contribution to the diffusion coefficient, $D_{A}/D_0$, as a function of $k$ for several values of $t_{c}$ and $\phi=0.133$. Points show simulation data, lines show fits of the form $D_{A}/D_0 = \lambda k^{\delta}\phi$ to the data. []{data-label="fig:D_k_tc"}](fig6.eps){height="1.0\columnwidth"} The active contribution to the diffusion coefficient, $D_A$, is defined by the equation $D= D_T+D_A$. Extensive computations of $D_A(\phi,k,t_c)$ have been carried out to determine its dependence on the dumbbell volume fraction, force constant and average hold time. Figure \[fig:D\_k\_tc\] shows that for fixed $\phi$, $D_A$ has a power-law dependence on $k$ of the form $D_A \sim k^{\delta}$, where the exponent $\delta$ depends on $t_c$. For a fixed value of $k$, as $t_{c}$ increases $D_{A}$ decreases. The larger $t_c$ is, the slower $D_A$ increases with $k$. The results of these simulations may be summarized in the following form for the diffusion coefficient: $$\label{eq:D_l_phi} D(\phi,k,t_c)= D_T(\phi) +D_0\kappa_A(k,t_c)\phi,$$ where $\kappa_A(k,t_c)=\kappa(k,t_c)-\kappa_o=\lambda(t_c) k^{\delta(t_c)}$. Additional information on the dependence of $\lambda$ and $\delta$ on $t_c$ is given in Appendix B. These results are applicable provided $k$ is not too small since as $k \to 0$ the dumbbell bond will soften and the dumbbell will dissociate. Note that as $t_c \to \infty$ we have $\kappa(k,t_c) \to \kappa_c$, its value for a dumbbell that fluctuates about its closed conformation. Depending on the system parameters, it is possible that the decrease in diffusion due to crowding-induced hindered motion may be larger than the increase due to dumbbell activity. In such a circumstance $D(\phi)/D_0$ may be less than unity ($\kappa (k,t_c)<0$), although the ratio will be larger than $D_T(\phi)/D_0$ for systems with inactive dumbbells. In our simulations we found that $\kappa(k,t_c) >0$ for most values of $k$ and $t_c$, although for a given $t_c$ there is a $k$ value at which $\kappa(k,t_c)$ will change sign. This will occur when $\kappa(k,t_c)=0$, which corresponds to $k=(-\kappa_o/\lambda(t_c))^{1/\delta(t_c)}$, given the scaling forms below Eq. (\[eq:D\_l\_phi\]). The magnitude of the enhancement of the diffusion coefficient as measured by $D_A/D_T$ depends strongly on the system parameters, $\phi$, $k$ and $t_c$, as well as $\tau_{{\text MPC}}$, which determines the solvent properties. The largest enhancements for $k=90$, $t_c=20$, $\tau_{{\text MPC}}=0.01$ in Fig. \[fig:D\_c\] are $D_A/D_T \approx 0.15$ and $D_A/D_T \approx 0.3$ for $\phi=0.133$ and $0.266$, respectively. For another set of system parameters with $\tau_{{\text MPC}}=0.05$, corresponding to a smaller solvent viscosity, $t_c=20$ and $k=9$, we find $D=4.68 \times 10^{-3}$ and $D_0=4.44 \times 10^{-3}$ for $\phi=0.133$ giving $D_A/D_T \approx 0.05$. In addition to the quantitative estimates of the diffusion enhancement, several qualitative features of $D_A$ are worth summarizing. The coefficient $\kappa_A$ differs from $\kappa_o$ since it depends strongly on $k$ and $t_c$. For fixed $t_c$, $\kappa_A \sim k^\delta$ where the exponent $\delta$ decreases with increasing $t_c$. These qualitative features of $D_A$ differ markedly from those that characterize the behavior of $D_T$. Active contributions to passive particle diffusion from the collective hydrodynamic interactions of many active proteins were discussed earlier using a Langevin model. [@ref:Mikhailov15] The result was the following estimate for $D_A$: $$\label{eq:D_A_th} D_{A}^{\text{th}}= \frac{S_{A}}{60\pi\ell_{{\text cut}}\eta^{2}v_{\mathrm{ex}}}\phi\equiv D_0\kappa_A^{\text{th}}(k,t_c) \phi,$$ for a uniform distribution of proteins with concentration $c=\phi/v_{\mathrm{ex}}$. This order-of-magnitude estimate was derived assuming a random static distribution of protein orientations, slow protein translational dynamics, and Oseen interactions with a short-distance cut-off, $\ell_{{\text cut}}$, taken to be the sum of the effective radii of the passive particle and protein. In this equation $S_A$ characterizes the strength of the force dipole correlations, $\langle\Delta m(t) \Delta m(0)\rangle $. The last equality defines the theoretical estimate of $\kappa_A$. Evaluation of Eq. (\[eq:D\_A\_th\]) for a range of $k$ and $t_c$ values shows that $\kappa_A^{\text{th}}(k,t_c) \sim k^{\sigma}$ for fixed $t_c$, where $\sigma < \delta$, smaller than the exponent $\delta$ found from simulation. The predicted values of the diffusion enhancement are consistent with Eq. (\[eq:D\_A\_th\]) since they differ by less than an order-of-magnitude from those in our microscopic simulations; for example, for $\phi=0.133$, $k=35$ and $t_c=20$ we have $D_{A}^{\text{th}}/D_T=0.01$ while from simulation $D_{A}/D_T=0.051$. Active dumbbell self-diffusion {#active-dumbbell-self-diffusion .unnumbered} ------------------------------ The self-diffusion coefficients of the dumbbells themselves, $D_{\mathrm{db}}$, are also modified due to crowding by other dumbbells, and the effects of crowding differ depending on whether the dumbbells are active or inactive. Figure \[fig:D\_c\_db\](a) shows $D_{\mathrm{db}}^T$ and $D_{\mathrm{db}}$, the diffusion coefficients for inactive and active dumbbells, respectively, as a function of $\phi$, normalized by the diffusion coefficient of a single inactive dumbbell, $D_{\mathrm{db}}^0 = 1.625 \times 10^{-3}$. For inactive dumbbells $D_{\mathrm{db}}^T$ decreases with increasing $\phi$, as expected for a crowded environment. A similar trend is seen for active dumbbells, with $D_{\mathrm{db}}$ also decreasing with an increase in $\phi$, although the decrease is much smaller than that for inactive dumbbells. As discussed above, recall that even for the passive particle in a system of active dumbbells, depending on the system parameters, the diffusion coefficient $D$ may decrease as $\phi$ is increased, but this decrease will be less strong than that for a system of inactive dumbbells. In these cases the effects of activity are not sufficient to completely overcome the tendency for crowding to decrease the diffusion coefficient. Nevertheless, $D_{\mathrm{db}}/D_{\mathrm{db}}^T> 1$ and the magnitudes of the diffusion coefficient changes are comparable to those for the passive particle. For example, for a volume fraction of $\phi=0.133$ and $k=90$, $t_c=20$, $\tau_{{\text MPC}}=0.01$ we have $D_{\mathrm{db}}/D_{\mathrm{db}}^T \approx 1.19$, while for $k=9$, $t_c=20$, $\tau_{{\text MPC}}=0.05$ we have $D_{\mathrm{db}}/D_{{\text db}}^T \approx 1.05$. ![(a) Dumbbell self-diffusion coefficient $D_{\mathrm{db}}$ versus $\phi$, for a system of active dumbbells with $t_{c}=20$ and $k=90$ (green triangles), and $D_{\mathrm{db}}^T$ for a system of inactive dumbbells (red circles). Data is normalized by $D_{\mathrm{db}}^0$, the diffusion coefficient of a single inactive dumbbell. (b) Active contribution to the dumbbell self-diffusion coefficient $D^A_{\mathrm{db}}$ versus $\phi$.[]{data-label="fig:D_c_db"}](fig7.eps){height="0.9\columnwidth"} Although the tendency for $D_{\mathrm{db}}$ to increases or decrease with increasing $\phi$ depends on the system parameters, one can see that the values of the dumbbell self-diffusion coefficients differ markedly depending on whether they are active or inactive. In contrast to $D$ for passive particle diffusion, which tends to a common value of $D_0$ as $\phi \to 0$ regardless of whether the dumbbells are active or inactive, even for a single dumbbell in solution $D_{\mathrm{db}}$ will be different if it is active or inactive. For example, for the data in Fig. \[fig:D\_c\_db\](a), $D_{\mathrm{db}}(\phi=0)=2.0 \times 10^{-3}$ and $D_{{\text db}}/D_{\mathrm{db}}^0 \approx 1.15$ for a single dumbbell in solution. Since $D_{\mathrm{db}}$ depends on the dumbbell conformation and is larger when the dumbbell is in the compact closed form, one expects, and finds, $D_{\mathrm{db}}> D_{{\text db}}^T$.  Furthermore, when the average hold time $t_{c}$ is smaller, $D_{\mathrm{db}}(\phi=0)$ becomes smaller, as the dumbbell spends less time in its closed conformation. This results in a measured $D_{\mathrm{db}}$ that is smaller at low $k$ values for dumbbells with shorter average hold times than for ones with larger $t_{c}$ values, as can be seen in Table \[tab:3\] in Appendix B. Accounting for the fact that $D_{\mathrm{db}}$ is different for single active and inactive dumbbells in solution, we define the active dumbbell contribution to the self-diffusion coefficient by the equation, $D_{\mathrm{db}}=D^T_{\mathrm{db}}+D^A_{{\text db}}+D^{A,0}_{\mathrm{db}}$, where $D^{A,0}_{\mathrm{db}}=D_{\text{ db}}(\phi=0)-D^0_{\mathrm{db}}$. Figure \[fig:D\_c\_db\](b) plots $D^{A}_{\mathrm{db}}$ versus $\phi$ and shows that the active contribution increases with increasing $\phi$. The results presented above show that hydrodynamic interactions resulting from nonequilibrium force dipole fluctuations of the dumbbell molecules give rise to enhanced diffusion of both the passive particle and the dumbbells themselves when compared to their values for systems containing only inactive dumbbells. Direct intermolecular interactions also play a role but estimates based on dumbbell sizes and ranges of intermolecular forces suggest that these interactions are important only at the highest volume fractions considered in this study. Contributions from direct intermolecular interactions will increase in importance as the average separation between dumbbells $l_{\mathrm{sep}}=(v_{\mathrm{ex}}/\phi)^{1/3}$ approaches the maximum length of a dumbbell $l_{\mathrm{db}}=\ell_{o} + 2 \sigma_{c} =8.3$. This is the case only for systems with the largest two volume fractions studied, $\phi=0.133$ and $\phi=0.266$, where $l_{\mathrm{sep}}=8.54$ and $6.78$, respectively. Conclusions {#sec:Conclusions} =========== The microscopic simulation systems containing a passive particle and either active or passive dumbbell molecules has allowed us to explore how diffusive dynamics varies with dumbbell activity and volume fraction. The results showed that the diffusive dynamics of passive particles in systems crowded by active molecules that change their conformations differs markedly from that when the crowding molecules are inactive. The self-diffusion coefficients of the crowding molecules themselves also display properties that depend on their activity. While crowding by molecules that thermally fluctuate about their open (or closed) metastable states leads to well-known subdiffusive dynamics and diffusion coefficients that decrease with increasing volume fraction, diffusion coefficients are enhanced, or decrease more slowly, when the crowding agents are active. Hydrodynamic interactions induced by active force dipole fluctuations are responsible for the observed diffusion coefficient increases, and direct intermolecular interactions contribute at the highest volume fractions. The particle-based dynamical model used in this investigation accounts for both of these effects and permits a detailed analysis of the phenomena. The magnitudes of the changes to the diffusion coefficient were shown to depend not only on the volume fraction of dumbbells, but also on the force dipole strength and the mean times spent in the open or closed conformations. The diffusion enhancement in experiments and in our simulations is not large but its existence signals that conformational changes arising from catalytic activity play a role in transport in active systems. The dumbbell conformational changes in this study were specified by a stochastic model that was chosen to mimic some features of the cyclic dynamics of enzymes undergoing conformational changes during their catalytic operation. Our dynamical model can be generalized to include a more detailed description of the substrate binding, unbinding and reaction processes with the enzyme so that the dependence of the effects on substrate concentration can be investigated. More realistic enzyme models may also be employed. [@carlos2011; @inder2014] The results in this paper provide the basis for the development and further study of more realistic models to probe transport properties in systems crowded by chemically active and inactive molecules and their relevance to biochemical processes in the cell. Acknowledgements: We would like to thank Alexander Mikhailov for useful discussions on this topic. The research of RK was supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada. MD thanks the Humboldt foundation for financial support. RK and MD were partially supported through the research training group GRK 1558 funded by Deutsche Forschungsgemeinschaft. Appendix A {#sec:appendixA .unnumbered} ========== Multiparticle collisions were implemented using the MPC-AT+a rule that employs the Anderson thermostat and conserves linear and angular momentum. [@ref:cell1; @*ref:noguchi2007; @*ref:noguchi2007b] Allowing the solvent particles to interact and exchange momentum only in the collision step, the frequency of which can be chosen according to the desired properties of the solvent, makes the method computationally efficient. Since the collision rule conserves mass and momentum at the cell level, the hydrodynamic flow fields will be described correctly, a feature that is essential for hydrodynamic interactions. The solvent viscosity $\eta$ can be controlled by varying the number of solvent particles $n_{f}$ per collision cell and/or the MPC time, $\tau_{{\text MPC}}$. [@Kapral_08; @ref:Gompper] In our simulations, we use a cubic simulation box of linear size $n_{x}=50$ MPCD cells in each direction, with $n_{f}=10$ fluid particles per cell, and an MD time step of $\Delta t=0.001$. The choice of parameters was based on several criteria. We wish to study a large range of force constants in order to identify the underlying trends in the system behaviour. Since inertia does not play a significant role in protein dynamics in solution, the dumbbell dynamics should be overdamped. This condition sets a limit on the maximum value of the spring constant. Furthermore, for spring energies which are comparable to the thermal energy ($k\sim k_{B}T/a^2$), thermal fluctuations dominate the dumbbell motion. This lower bound depends only on the value of $k_{B}T$, and not on any other fluid parameters. We consider two values of the MPC time, $\tau_{{\text MPC}}=0.01$ which gives a large fluid viscosity of $\eta\sim 36$, and $\tau_{{\text MPC}}=0.05$ which gives a lower fluid viscosity of $\eta\sim 8.4$. In simulations with the higher viscosity a large range of force constant values may be used while still remaining in the regime of overdamped dynamics. The dumbbell mass is chosen so that it is neutrally buoyant. Since the dumbbell interacts with the solvent only in the MPC collision step, we define the volume of interaction of a dumbbell with the fluid by $v_{\mathrm{f,db}}=4\pi r_{\mathrm{f,db}}^{3}/3$, where $r_{\mathrm{f,db}}=(\ell_{o}+\ell_{c})/2$, which corresponds to a sphere with a diameter equal to the average dumbbell spring rest length. Note that this is distinct from the dumbbell volume $v_{\mathrm{db}}$ defined earlier. This then gives a total dumbbell mass of $m_{\mathrm{db}}=v_{\mathrm{f,db}}n_{f}$. Throughout we set $\ell_{c}=2$ and $\ell_{o}=4$, such that we have $m_{\mathrm{db}}=141.37$. From our fluid and dumbbell parameters we then obtain a cross-over spring constant between overdamped and underdamped motion of $k_{o}> 90$ for $\tau_{{\text MPC}}=0.01$ and $k_{o}>11$ for $\tau_{{\text MPC}}=0.05$. To control the switching between the two spring rest lengths we must set the cut-off parameter that the length must cross before the hold time is chosen. We set $\delta\ell_{o}=\delta\ell_{c}= 0.05(\ell_{o}-\ell_{c})=0.1$. The hold times are chosen from log normal distributions with averages for the open and closed configurations of $t_{o}$ and $t_{c}$, and with scale parameter $\sigma=0.5$. Throughout the time spent in the open configuration is $t_{o}=0$. For the passive particle we set the passive particle-fluid interaction radius to be $\sigma_{cf}=2$ and the passive particle-dumbbell bead interaction diameter to be $\sigma_{c}=4.30$. The passive particle is also chosen to be neutrally buoyant, such that its mass is given by $m_{c}=4\pi\sigma^{3}_{cf}n_{f}/3$ which for the parameters given here is $m_{c}=335.103$. Dimensionless units can be mapped approximately onto physical units by matching dimensions and time scales. [@ref:Padding] Consider $\tau_{{\text MPC}}=0.05$ for which $D_0=4.44 \times 10^{-3}\; a^2/t_0$. Taking a radius of 5 nm for the passive particle and assuming $D_0$ is given by its Stokes-Einstein value one finds $t_0 \approx 0.1$ ns. For $k=9\; k_BT/a^2$ we then have for the forces corresponding to active opening and closing $F \approx 20$ pN. Appendix B {#sec:appendixB .unnumbered} ========== This Appendix provides some of the numerical values of the data in the plots, along with parameters that enter in the phenomenological forms for the diffusion coefficients. The data used to construct Fig. \[fig:D\_c\] is in Table \[tab:0\]. [8cm]{}[|C|C|C|C|]{} $\phi/10^{-2}$ & $k$ & &\ $0.66$ & $90$ & $1.145$ & $1.137$\ $1.66$ & $90$ & $1.155$ & $1.134$\ $3.32$ & $90$ & $1.163$ & $1.128$\ $6.64$ & $90$ & $1.182$ & $1.115$\ $13.3$ & $90$ & $1.261$ & $1.092$\ $26.6$ & $90$ & $1.350$ & $1.043$\ $0.66$ & $35$ & $1.140$ & $1.137$\ $1.66$ & $35$ & $1.141$ & $1.134$\ $3.32$ & $35$ & $1.142$ & $1.128$\ $6.64$ & $35$ & $1.145$ & $1.115$\ $13.3$ & $35$ & $1.148$ & $1.092$\ $26.6$ & $35$ & $1.160$ & $1.043$\ Table \[tab:2a\] gives representative values of the parameters that enter in Eq. (\[eq:D\_l\_phi\]) for the passive particle diffusion coefficient. [8.5cm]{}[|C|C|C|C|C|C|]{} $\phi/10^{-2}$ & $t_c$ & $\lambda/10^{-3}$ & $\delta$ & &\ $0.66$ & $20$ & $5.18$ & $1.13$ & $1.137$ & $1.137$\ $1.66$ & $20$ & $4.74$ & $1.12$ & $1.133$ & $1.134$\ $3.32$ & $20$ & $3.60$ & $1.16$ & $1.128$ & $1.128$\ $6.64$ & $20$ & $4.30$ & $1.12$ & $1.112$ & $1.115$\ $13.3$ & $20$ & $4.74$ & $1.16$ & $1.086$ & $1.092$\ $26.6$ & $20$ & $4.82$ & $1.18$ & $1.052$ & $1.043$\ $13.3$ & $30$ & $11.0$ & $0.98$ & $1.087$ & $1.092$\ $13.3$ & $50$ & $19.5$ & $0.73$ & $1.083$ & $1.092$\ $13.3$ & $100$ & $37.1$ & $0.47$ & $1.084$ & $1.092$\ $13.3$ & $200$ & $78.6$ & $0.20$ & $1.097$ & $1.092$\ Table \[tab:3\] presents data for the dumbbell self-diffusion coefficient for several values of the hold time $t_c$ and force constant $k$. [8cm]{}[|C|C|C|]{} $t_c$ & $k$ & $D_{\mathrm{db}}/10^{-3}$\ $30$ & $5$ & $1.707$\ $30$ & $10$ & $1.727$\ $30$ & $20$ & $1.758$\ $30$ & $35$ & $1.783$\ $30$ & $90$ & $1.908$\ $50$ & $5$ & $1.717$\ $50$ & $10$ & $1.729$\ $50$ & $20$ & $1.743$\ $50$ & $35$ & $1.774$\ $50$ & $90$ & $1.847$\ $100$ & $5$ & $1.724$\ $100$ & $10$ & $1.741$\ $100$ & $20$ & $1.753$\ $100$ & $35$ & $1.770$\ $100$ & $90$ & $1.817$\ ${\text{ inactive}}$ & $90$ & $1.625$\ [^1]: As the volume fraction increases the mean square displacement will generally exhibit subdiffusive dynamics on intermediate time scales. Here we focus on the diffusion coefficient determined in the long-time regime where normal diffusion again observed.

--- author: - | [Jiachen Sun]{}\ University of Michigan\ <jiachens@umich.edu> - | [Yulong Cao]{}\ University of Michigan\ <yulongc@umich.edu> - | [Qi Alfred Chen]{}\ UC Irvine\ <alfchen@uci.edu> - | [Z. Morley Mao]{}\ University of Michigan\ <zmao@umich.edu> bibliography: - 'refs.bib' title: '**Towards Robust LiDAR-based Perception in Autonomous Driving: General Black-box Adversarial Sensor Attack and Countermeasures**' --- =1

--- author: - | [^1]\ H. Niewodniczański Institute of Nuclear Physics, Polish Academy of Sciences,\ ul. Radzikowskiego 152, 31-342 Krakow, Poland\ E-mail: - | A. Kusina\ Southern Methodist University, Dallas, TX 75275, USA\ E-mail: - | M. Skrzypek\ H. Niewodniczański Institute of Nuclear Physics, Polish Academy of Sciences,\ ul. Radzikowskiego 152, 31-342 Krakow, Poland\ E-mail: - | M. Slawinska\ H. Niewodniczański Institute of Nuclear Physics, Polish Academy of Sciences,\ ul. Radzikowskiego 152, 31-342 Krakow, Poland\ E-mail: title: | NLO parton shower for LHC physics -\ hard processes and beyond --- Introduction ============ The Large Hadron Collider (LHC) at CERN provides rich harvest of experimental data. The proper understanding and interpretation of these data, possibly leading to discovery of new phenomena, requires perfect mastering of the “trivial” effects due to the multiple emissions of soft and collinear gluons and quarks. Perturbative Quantum Chromodynamics (pQCD) [@GWP; @Gross:1974cs; @Georgi:1951sr], supplemented with clever modelling of the low energy nonperturbative effects, is an indispensable tool for disentangling the Standard Model physics component in the data. This work presents part of the global effort of improving quality of the pQCD calculations for LHC experiments. Most of the results presented here are described in refs. [@Jadach:2011cr] and [@Jadach:2012vs]. Although this work elaborates on the improved method of the pQCD calculation combining NLO-corrected hard process and LO parton shower Monte Carlo (MC), it should be regarded as the first step towards NNLO-corrected hard process combined with the NLO parton shower MC [@IFJPAN-IV-2012-7]. Basic LO parton shower MC ========================= The multigluon distribution of the single initial state ladder, which is a building block of our parton shower MC, is represented by the integrand of the “exclusive/unintegrated PDF”, which in the LO approximation is the following: $$\label{eq:LOMC} \begin{split} & D(t,x) =\int dx_0\; dZ\; \delta_{x=x_0 Z}\; d_0(\hat{t}_0,x_0)\; G(t, \hat{t}_0- \ln x_0 | Z), \\& G(t, t_0 | Z)= e^{-S_F} \sum_{n=0}^\infty \bigg( \prod_{i=1}^n \int d^3{{\cal E}}(\bar{k}_i)\; \theta_{\xi_i>\xi_{i-1}} \frac{2C_F\alpha_s}{\pi^2} \bar{P}(z_i) \bigg) \\& \qquad\qquad\quad\times \theta_{t>\xi_n} \delta_{Z=\prod_{j=1}^n z_j}, \end{split}$$ where evolution kernel is $\bar{P}(z)=\frac{1}{2}(1+z^2)$, evolution time is $\hat{t}_0=\ln(q_0/\Lambda)$ and the “eikonal” phase space integration element is $ d^3{{\cal E}}(k)=\frac{d^3 k}{2k^0}\;\frac{1}{{{\bf{k}}}^2} =\pi \frac{d\phi}{2\pi} \frac{d k^+}{k^+} d \xi $ and $k^\pm = k^0\pm k^3$. We use rapidities $\xi_i =\frac{1}{2}\ln\frac{k^-_i}{k^+_i}\big|_{\rm Rh}$ in the hadron beam rest frame (Rh), and $\eta_i=\frac{1}{2}\ln\frac{k^+_i}{k^-_i}\big|_{\rm RFHP}$ defined in hard process rest frame (RFHP). They are related by $\xi_i=\ln\frac{\sqrt{s}}{m_h}-\eta_i$. Rapidity ordering is now $t=\xi_{\max}>\xi_n>\dots>\xi_i>\xi_{i-1}>\dots>\xi_0=t_0$, where $t_0=\xi_0=\ln(q_0/m_h)-\ln x_0$. The direction of the $z$ axis in the RFHP is pointing out towards the hadron momentum. A lightcone variable of the emitted gluon is defined as $\alpha_i= \frac{2k_i^+}{\sqrt{s}}$ and of the emitter parton (quark) as $x_i=x_0-\sum_{j=0}^{i}\; \alpha_j$ (after $i$ emissions). We also use fractions $z_i=x_i/x_{i-1}$. The Sudakov formfactor $S_F$ comes from the “unitarity” condition[^2] $ \int_0^1 dZ\; G(t, t_0 | Z)=1, $ which is also instrumental in the Markovian MC implementation used to obtain $D(t,x)$ at any value of $t>t_0$. The initial distribution $d_0(q_0,x_0)$ related to experiment, to previous steps in the MC ladder, or to PDF in the standard $\overline{MS}$ system is not essential for the following discussion, we only note that the unitarity condition provides baryon number conservation sum rule $\int_0^1 dx\; D(t,x) = \int_0^1 dx_0\; d_0(t_0,x_0)$. For testing our new method of correcting hard process to the NLO level we use the following simplified MC parton shower, implementing the DY process with two ladders and the hard process: [^3] $$\label{eq:LOMCFBmaster} \begin{split} &\sigma_0= \int d x_{0F} d x_{0B}\;\; d_0(\hat{t}_0,x_{0F}) d_0(\hat{t}_0,x_{0B}) \sum_{n_1=0}^\infty\; \sum_{n_2=0}^\infty \int dx _F\; dx_B\; \\&~~~~~~~~~\times e^{-S_{_F}} \int_{\Xi<\eta_{n_1}} \bigg( \prod_{i=1}^{n_1} d^3{{\cal E}}(\bar{k}_i) \theta_{\eta_i<\eta_{i-1}} \frac{2C_F\alpha_s}{\pi^2} \bar{P}(z_{Fi}) \bigg) \delta_{x_F = x_{0F}\prod_{i=1}^{n_1} z_{Fi}} \\&~~~~~~~~~\times e^{-S_{_B}} \int_{\Xi>\eta_{n_2}} \bigg( \prod_{j=1}^{n_2} d^3{{\cal E}}(\bar{k}_j) \theta_{\eta_j>\eta_{j-1}} \frac{2C_F\alpha_s}{\pi^2} \bar{P}(z_{Bj}) \bigg) \delta_{x_B = x_{0B}\prod_{j=1}^{n_2} z_{Bj}} \\&~~~~~~~~~\times d\tau_2(P-\sum_{j=1}^{n_1+n_2} k_j ;q_1,q_2)\; \frac{d\sigma_B}{d\Omega}(sx_Fx_B,\hat\theta)\; W^{NLO}_{MC}. \end{split}$$ In the LO approximation $W^{NLO}_{MC}=1$. Rapidity $\xi$ is translated into $\eta$ – the center of mass system rapidity, in the forward part (F) of the phase space as $\xi_i=\ln\frac{\sqrt{s}}{m_h}-\eta_i$, $\eta_{0F}>\eta_i>\Xi$, and in the backward (B) part as $\xi_i=-\ln\frac{\sqrt{s}}{m_h}+\eta_i$, $\Xi>\eta_i>\eta_{0B}$. The rapidity boundary between the two hemispheres $\Xi=0$ is used, until a more sophisticated version related to rapidity of the produced $W/Z$ is introduced. Analytical integration of eq. (\[eq:LOMCFBmaster\]) results in the standard factorization formula ($W^{NLO}_{MC}=1$) $$\label{eq:LOfactDY2LO} \sigma_0 = \int_0^1 dx_F\;dx_B\; D_F(t, x_F)\; D_B(t, x_B)\; \sigma_B(sx_Fx_B).$$ The distributions $D_F(t, x_F)=(d_0\otimes G_F) (t, x_F)$ and $D_B(t, x_B)=(d_0\otimes G_B) (t, x_B)$ are obtained from separate Markovian LO Monte Carlo runs. The above LO formula is exact, and can be tested with an arbitrary numerical precision. Figure \[fig:etaW\_LO\_7TeV\] represents a “calibration benchmark” for the overall normalization at the LO level. We show there the properly normalized distribution of the variable $\eta_W^*=\frac{1}{2}\ln(x_F/x_B)$, which in the collinear limit approximates the rapidity of $W$ boson. The distribution in the upper plot of Fig. \[fig:etaW\_LO\_7TeV\], representing eq. (\[eq:LOfactDY2LO\]), is obtained using the general purpose MC program FOAM [@foam:2002]. The collinear PDF $D(t,x)$ there has been obtained from a separate high statistics run ($10^{10}$ events) of a Markovian MC (MMC), creating $D(t,x)$ in a form of the 2-dimensional look-up table[^4]. The other distribution in the upper plot of Fig. \[fig:etaW\_LO\_7TeV\] represents eq. (\[eq:LOMCFBmaster\]) in LO approximation. It comes from the full scale MC generation (with four-momenta conservation). The MC run with $10^8$ events was used. The constrained MC (CMC) technique of ref. [@Jadach:2007qa] is used here because of the narrow Breit-Wigner peak due to a heavy boson propagator[^5]. Two CMC modules and FOAM are combined into one MC generating gluon emissions and the $W$ boson production. FOAM is taking care of the generation of the variables $x_F,x_B,x_{F0},x_{B0}$ and the sharp Breit-Wigner peak in $\hat{s}=s x_F x_B$, then two CMC modules are initialized and generate the gluon four-momenta $\bar{k}^\mu_j$. They are mapped into $k^\mu_j$, following the prescription defined in ref. [@Jadach:2011cr], such that the overall energy-momentum conservation is achieved. Figure \[fig:etaW\_LO\_7TeV\] demonstrates a very good numerical agreement between $d\sigma/ d \eta_W^*$ from our full scale LO parton shower MC of eq. (\[eq:LOMCFBmaster\]) and the simple formula of eq. (\[eq:LOfactDY2LO\]), to within 0.5%, as seen from the ratio of the two results in the lower part of the figure. Introducing NLO corrections to hard process =========================================== The NLO corrections to hard process are imposed on top of the LO distributions of eq. (\[eq:LOMCFBmaster\]) using a single “monolithic” weight $W^{NLO}_{MC}$ defined exactly as in ref. [@Jadach:2011cr]: $$\label{eq:NLODYMCwt} \begin{split} &W^{NLO}_{MC}= 1+\Delta_{S+V} +\sum_{j\in F} \frac{{{\tilde{\beta}}}_1(q_1,q_2,\bar{k}_j)} {\bar{P}(z_{Fj})\;d\sigma_B(\hat{s},\hat\theta)/d\Omega} +\sum_{j\in B} \frac{{{\tilde{\beta}}}_1(q_1,q_2,\bar{k}_j)} {\bar{P}(z_{Bj})\;d\sigma_B(\hat{s},\hat\theta)/d\Omega}, \end{split}$$ the NLO soft+virtual correction is $ \Delta_{V+S} =\frac{C_F \alpha_s}{\pi}\; \left( \frac{2}{3}\pi^2 -\frac{5}{4} \right) $, and the real correction reads: $$\label{eq:DYbeta1FB} \begin{split} &{{\tilde{\beta}}}_1(q_1,q_2,k)= \Big[ \frac{(1-\beta)^2}{2} \frac{d\sigma_{B}}{d\Omega_q}(\hat{s},\theta_{F}) +\frac{(1-\alpha)^2}{2} \frac{d\sigma_{B}}{d\Omega_q}(\hat{s},\theta_{B}) \Big] \\&~~~~~~~~~~~~~~~~~~~~ -\theta_{\alpha>\beta} \frac{1+(1-\alpha-\beta)^2}{2} \frac{d\sigma_{B}}{d\Omega_q}(\hat{s},\hat\theta) -\theta_{\alpha<\beta} \frac{1+(1-\alpha-\beta)^2}{2} \frac{d\sigma_{B}}{d\Omega_q}(\hat{s},\hat\theta). \end{split}$$ The above is the exact ME of the quark-antiquark annihilation into a heavy vector boson with additional single real gluon emission[^6]. The LO component, which is already included in the LO MC, is subtracted here. The variable $\hat{s}=s x_F x_B = (q_1+q_2)^2$ is the effective mass squared of the heavy vector boson. The definition of angle $\hat\theta$ in the LO component is rather arbitrary. We define it in the rest frame of the heavy boson, where $\vec{q}_1+\vec{q}_2=0$, as an angle between the decay lepton momentum $\vec{q}_1$ and the difference of momenta of the incoming quark and antiquark $\hat\theta=\angle(\vec{q}_1,\vec{p}_{0F}-\vec{p}_{0B})$. On the other hand the two angles in the NLO ME are defined quite unambiguously as $\hat\theta_F=\angle(\vec{q}_1,-\vec{p}_{0B})$ and $\hat\theta_B=\angle(\vec{q}_1, \vec{p}_{0F})$. In the above we only need directions of the $\vec{p}_{0F}$ and $\vec{p}_{0B}$ vectors, which are the same as the directions of the hadron beams. The lightcone variables $\alpha_j$ and $\beta_j$ of the emitted gluon are defined in the F and B parts of the phase space as follows[^7]: $$\begin{split} &\alpha_j=1-z_{Fj},\quad \beta_j= \alpha_j\; e^{2(\eta_j-\Xi)},\quad ~~ {\rm for}~~~ j\in F, \\& \beta_j=1-z_{Bj},\quad \alpha_j=\beta_j\; e^{-2(\eta_j-\Xi)},\quad {\rm for}~~~ j\in B. \end{split}$$ Again, the exact phase space integration of eq. (\[eq:LOMCFBmaster\]) including $W^{NLO}_{MC}$ of eq. (\[eq:NLODYMCwt\]) is feasible, and the resulting compact expression for the total cross section is obtained [@Jadach:2011cr]: $$\label{eq:DYanxch} \begin{split} \sigma_1 &= \int_0^1 dx_F\;dx_B\; dz\; D_F(t, x_F)\;D_B(t, x_B)\; \sigma_B(szx_Fx_B) \big\{ \delta_{z=1}(1+\Delta_{S+V}) +C_{2r}(z) \big\}, \end{split}$$ where $ C_{2r}(z) =\frac{2C_F \alpha_s}{\pi}\; \left[ -\frac{1}{2}(1-z) \right]. $ Numerical test of NLO correction -------------------------------- Figure \[fig:etaW\_NLOglu\_7TeV\] represents a principal [*proof of concept*]{} of our new methodology for implementing the NLO corrections to the hard process in the parton shower MC. The plotted NLO correction to the $\eta_W^*$ distribution[^8] comes from the parton shower MC with the NLO-corrected hard process according to eqs. (\[eq:LOMCFBmaster\]) and (\[eq:NLODYMCwt\]). Additionally we also plot there result of a simple collinear formula of eq. (\[eq:DYanxch\]), where two collinear PDFs are convoluted with the analytical coefficient function $C_{2r}(z)$ for the hard process. Both results coincide within the statistical error, see their ratio in the lower part of Fig. \[fig:etaW\_NLOglu\_7TeV\]. Technically, the inclusion of the NLO correction in our parton shower MC is rather straightforward, and is obtained by including $W^{NLO}_{MC}$ weight of eq. (\[eq:NLODYMCwt\]). MC is providing both LO and NLO-corrected results in a single run with weighted events. The NLO weight is strongly peaked near $W^{NLO}_{MC}=1$, positive, and without long-range tails. Its distribution is shown in Fig. \[fig:Canv1w\_7TeV\]. In all numerical results we have set $\Delta_{V+S}=0$, as it is completely unimportant for the presented analysis. The initial distributions $d_0(q_0,x_0)$ are defined in ref. [@Jadach:2012vs]. Simplification of the method and comparison with other methodologies ==================================================================== {width="75mm"} {width="75mm"} \[fig:mcRho1glu\] Our new method for introducing NLO corrections in the hard process, proposed in ref. [@Jadach:2011cr] and tested in ref. [@Jadach:2012vs], is an alternative to the two well established MC@NLO [@Frixione:2002ik] and POWHEG [@Nason:2004rx; @Frixione:2007vw] methodologies. With MC numerical implementation at hand, let us elaborate on the differences with the above two techniques in particular with the POWHEG technique. We shall also see that it is possible to make our method more efficient in terms of CPU time consumption. This improvement is not so critical in the present case of NLO corrected hard process, but may be quite useful in the case of correcting evolution kernels to the NLO in the ladder parts of the MC [@IFJPAN-IV-2012-7]. The most important differences with the POWHEG and MC@NLO techniques are: - The summation over all emitted gluons, without deciding which gluon is the one involved in the NLO correction and which ones are merely “LO spectators” in the parton shower. - The absence of $(1/(1-z))_+$ distributions in the real part of the NLO corrections (virtual+soft correction is kinematically independent). To explain more clearly how $W^{NLO}_{MC}$ of eq. (\[eq:NLODYMCwt\]) is distributed over the multigluon phase space, we restrict now to single ladder (hemisphere) with a simplified weight: $$\label{eq:NLODYMCwt_sim} \begin{split} &W^{NLO}_{MC}= 1+\sum_{j\in F} W^{NLO}_j,\qquad W^{NLO}_j= \frac{{{\tilde{\beta}}}_1(q_1,q_2,\bar{k}_j)} {\bar{P}(z_{Fj})\;d\sigma_B(\hat{s},\hat\theta)/d\Omega}. \end{split}$$ In order to find out the phase space regions specific for NLO corrections we consider inclusive distributions of gluons on the Sudakov logarithmic plane of rapidity $\xi$ and variable $v=\ln(1-z)$. In the left hand side (LHS) of Fig. \[fig:mcRho1glu\] we show gluons inclusive distribution in the LO approximation. The flat plateau there represents IR/collinear singularity[^9] $2C_F\frac{\alpha_S}{\pi} d\xi\frac{dz}{1-z}$ with the drop by factor 1/2 towards $z=0$, due to $\frac{1+z^2}{2}$ factor in the LO kernel. In the right hand side (RHS) of Fig. \[fig:mcRho1glu\] we show contributions from all gluons weighted with the component weight[^10] $-W^{NLO}_j$ of eq. (\[eq:NLODYMCwt\_sim\]). The NLO contribution is concentrated in the area near the hard process rapidity $t=\xi_{\max}$, which has to be true for the genuine NLO contribution [^11]. The completeness of the phase space near this important region ($z=0$, $\xi_{\max}$) is critical for the completeness of the NLO corrections. Both POWHEG and MC@NLO use standard LO MCs which feature an empty “dead zone” in this phase space corner. Figure \[fig:mcRho1glu\] suggests that the dominant contribution to $\sum_j W^{NLO}_j$ could be from the gluon with the maximum $\ln k_j^T\sim \xi_j+\ln(1-z_j)$, which is closest to the hard process phase space corner. In the MC we may easily relabel generated gluons using new index $K$ such that they are ordered in the variable $\kappa_K=\xi_K+\ln(1-z_K),\; \kappa_{K+1}<\kappa_K$ with $K=1$ being the hardest one. Figure \[fig:mcCanv1k\] demonstrates a split of the LO inclusive distribution of Fig. \[fig:mcRho1glu\] into the $K=1$ component and the rest $K>1$. The important point is that the $K=1$ component reproduces the original complete distribution over the whole region where the NLO correction is non-negligible! This is exactly the observation on which POWHEG technique is built. According to the POWHEG authors, taking the $K=1$ component is sufficient to reproduce the complete NLO correction (modulo NNLO). The above statement is checked numerically in Fig. \[fig:mcCanV8k\], where we compare the NLO correction to the $x=\prod_j z_j$ distribution from the complete sum $\sum_j W^{NLO}_j$ and from $ W^{NLO}_{K=1}$. As we see the $K=1$ component saturates the complete sum very well, with the $K=2$ component being negligible in the first approximation. We can therefore speed up the calculation by means of taking only the $K=1$ contribution. The price will be that the formula of eq. (\[eq:DYanxch\]) will not be exact any more. Our method differs, however, from the POWHEG scheme, where the $K=1$ gluon is generated separately in the first step, and other gluons are generated (by the LO parton shower MC) in the next step. That is easy for LO MC with $k^T$-ordering, while in case of the LO MC with angular-ordering POWHEG requires additional effort of generating the so called vetoed and truncated showers. In our method, there is no need for such vetoed/truncated showers in case of angular ordering. The reason why POWHEG technique is complicated in case of the angular ordering is illustrated in Fig. \[fig:mcRhoOrdLO\]. We show there the distribution of gluons ordered in rapidity, starting from the gluon with the maximum rapidity, the closest to hard process. The gluon distribution with the highest rapidity $\xi\sim \xi_{\max}$ ($J=1$) has a ridge extending towards the soft region. Notice that, when the IR cut-off $\epsilon\to0$ in $(1-z)<\epsilon$, the width of this ridge also goes to zero. Consequently, the gluon with the highest $\xi$ is unable to reproduce the gluon distribution in the NLO corner, close to hard process. This is why in this case POWHEG requires truncated and vetoed showers, which are not needed in our method. Summary and outlook =================== A new method of adding the QCD NLO corrections to the hard process in the initial state Monte Carlo parton shower is tested numerically showing that the basic concept of the new methodology works correctly in the numerical environment of a Monte Carlo parton shower. The differences with the well established methods of MC@NLO and POWHEG are briefly discussed. Also, variants of the new method with better efficiency in terms of CPU time are proposed. Acknowledgement {#acknowledgement .unnumbered} =============== This work is partly supported by the Polish National Science Centre grant UMO-2012/04/M/ST2/00240, Foundation for Polish Science grant Homing Plus/2010-2/6, the Research Executive Agency (REA) of the European Union Grant PITN-GA-2010-264564 (LHCPhenoNet), the U.S. Department of Energy under grant DE-FG02-04ER41299 and the Lightner-Sams Foundation. [99]{} D. J. Gross and F. Wilczek, [*Phys. Rev. Lett.*]{} [**30**]{} (1973) 1343;\ H. D. Politzer, [*Phys. Rev. Lett.*]{} [**30**]{} (1973) 1346;\ D. J. Gross and F. Wilczek, [*Phys. Rev.*]{} [**D8**]{} (1973) 3633;\ H. D. Politzer, [ *Phys. Rep.*]{} [**14**]{} (1974) 129. D. J. Gross and F. Wilczek, [*[Asymptotically Free Gauge Theories. 2]{}*]{}, [*Phys. Rev.*]{} [**D9**]{} (1974) 980–993. 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[^1]: The partial support of the TH Unit of the CERN PH Division for this author is acknowledged. [^2]: The usual cutoff $1-z<\epsilon$ regularizing the IR singularity is implicit. [^3]: Following ref. [@Jadach:2011cr], we adopt $d\tau_2(P;q_1,q_2) = \delta^{(4)}(P-q_1-q_2)\frac{d^3q_1}{2q_1^0}\frac{d^3q_2}{2q_2^0}$. [^4]: This MMC run solves the LO DGLAP equation using the MC method, as in refs. [@Jadach:2008nu; @GolecBiernat:2006xw]. [^5]: A backward evolution algorithm of ref. [@Sjostrand:1985xi] could be also used here. [^6]: We employ here the compact representation of ref. [@Berends:1980jk], which has also been used in POWHEG [@Alioli:2011nr]. [^7]: See ref. [@Jadach:2011cr] for more explanations. [^8]: Extra minus sign introduced to facilitate visualization. [^9]: We use constant $\alpha_S$. [^10]: We again insert a minus sign in order to facilitate visualization. [^11]: It also vanishes towards the soft limit $z\to 1$.

--- abstract: 'This paper deals with the routing protocols for distributed wireless sensor networks. The conventional protocols for WSNs like Low Energy adaptive Clustering Hierarchy (LEACH), Stable Election Protocol (SEP), Threshold Sensitive Energy Efficient Network (TEEN), Distributed Energy Efficient Clustering Protocol (DEEC) may not be optimal. We propose a scheme called Away Cluster Head (ACH) which effectively increases the efficiency of conventional clustering based protocols in terms of stability period and number of packets sent to base station (BS). We have implemented ACH scheme on LEACH, SEP, TEEN and DEEC. Simulation results show that LEACH-ACH, SEP-ACH, TEEN-ACH and DEEC-ACH performs better than LEACH, SEP, TEEN and DEEC respectively in terms of stability period and number of packets sent to BS. The stability period of the existing protocols prolongs by implementing ACH on them.' author: - | N. Javaid$^{\ddag}$, M. Waseem$^{\ddag}$, Z. A. Khan$^{\$}$, U. Qasim$^{\pounds}$, K. Latif$^{\ddag}$, A. Javaid$^{\natural}$\ COMSATS Institute of IT, $^{\ddag}$Islamabad, $^{\natural}$Wah Cantt, Pakistan.\ $^{\$}$Faculty of Engineering, Dalhousie University, Halifax, Canada.\ $^{\pounds}$University of Alberta, Alberta, Canada. title: 'ACH: Away Cluster Heads Scheme for Energy Efficient Clustering Protocols in WSNs' --- Wireless sensor networks, Distributed networks, Clustering Protocol. Background ========== In Direct Transmission \[1\], each node in the sensor network communicates directly to BS. In the aforementioned protocol, farthest nodes die faster than the nearest nodes. In Minimum transmission energy \[2\] routing protocol each node transmits to its nearest node so the nearest nodes die at a faster rate because they receive data from the farther nodes. In the current body of research going in the field of WSNs clustering based protocols have attain significant attraction. In clustering based routing protocols the sensor nodes form clusters. In these clusters, one node is selected as CH. The nodes sense data and send to their respective CHs which aggregate and fuse the data, thus saving the energy as global communication is reduced due to local compression. Once the CH receives data from its nodes it aggregates and fuses the data into a small set and sends to BS. Unbalanced energy consumption among the sensor nodes may cause network partition and node failures where transmission from some sensors to the sink node becomes blocked. Therefore, construction of a stable backbone is one of the challenges in sensor network applications. LEACH \[3\] proposes a clustering based routing protocol for homogenous networks in which a node becomes CH by a probabilistic equation and forms a cluster of those nodes which receive strong signal to noise ratio from it. The nodes sense the environment and send data to CH where it is aggregated and finally send to BS. In LEACH there is a localized coordination amongst the nodes for cluster set up and locally compress the data to reduce global communication. CHs in LEACH are rotated randomly. Heterogeneous networks are more stable and beneficiary than homogenous networks. A number of protocols like SEP, DEEC and Threshold Distributed Energy-Efficient Clustering protocol (T-DEEC) have been proposed for WSNs. SEP \[4\] has two level of heterogeneity. In DEEC \[5\], CH selection is based on the ratio of residual energy and average energy of the network. The high energy nodes have more chances to become CH. In this way the energy is evenly distributed in the network. These routing protocols have some limitation due to their design and performance. The ACH Scheme ============== Optimal Number Of CHs --------------------- The optimal probability of a node to take part in election for selection of CHs is a function of the spatial density when the nodes are uniformly distributed over the sensors’ field. When the total energy consumption is minimum and energy consumption is well distributed over all sensors, the clustering is then called optimal clustering. The energy model we use for our simulation effect the optimal number of CHs. We use similar energy model as proposed in LEACH, SEP and DEEC. We have been giving particular attention to distribution of CHs in network so as energy in the network. Once nodes are deployed in region of interest the nodes locally coordinate for cluster set up and operation. Each node decides whether to become a CH or not. The node generates a random number and compares it with the threshold value. If the number generated is less than or equal to the threshold value and the node has not been CH for the last $\frac{1}{p}$ round the node is marked as to be one of the CH. p is the probability of a node to become CH. In ACH scheme, once the CHs have been formed the CHs send a confirmation message to one another using CSMA-MAC protocol. The CH which receives a strong SNR from its adjacent CH will be marked as a normal node. For simplicity in our simulations we replace SNR by distance. We assume the CH will be made unmark and will become a normal node if its distance from the nearest CH is less than $12m$. The distance between CHs less than $12m$ is shown by “a” otherwise “b”. After confirmation of CHs, nodes receive an association message from CHs and respond according to the strength of SNR. The clusters are thus formed and the CHs are well distributed in the network. This makes clusters even in terms of number of nodes in each cluster. In this way energy of CH dissipated in each round is comparably equal. The distant CHs’ network is shown in fig. 2. We implement ACH scheme on LEACH, SEP, TEEN and DEEC. Simulation results show that ACH scheme performs better with all of 4 selected protocols. We initialize parameters for simulation, randomly deploy our nodes and start network’s operation. In network’s operation each node is checked whether eligible to become CH. We call this operation for epoch. If a node successfully pass through this test energy of the node is checked. Next comes the turn of CH formation. A node becomes CH if it has energy and is eligible to be CH which is shown in CH formation block. The CH then goes through neighbors’ association phase and data transmission phase as shown in fig. 1. {height="7cm" width="9cm"} {height="7cm" width="8cm"} Simulations =========== We have implemented our protocol on Matlab \[6\] to evaluate its performance with LEACH, SEP, TEEN and DEEC. We have proposed LEACH-ACH, SEP-ACH, TEEN-ACH and DEEC-ACH. Our goals in conducting the simulation are as follows: - Compare the performance of LEACH, SEP, TEEN, DEEC and their ACH versions on the basis of longevity of the network. - Compare throughput of LEACH, SEP, TEEN, DEEC and the respective ACH schemes. We have performed our simulation on $100$ nodes and a fixed BS located in the center of the field. We randomly distributed $100$ nodes in a $100m×100m$ field. The most distant node from BS is at $70.7m$. The nodes have their horizontal and vertical coordinates located between $0$ and maximum value of the dimension which is $100$. All the nodes have different energies as the environment is heterogeneous. We simulate our protocol on the basis of initial energy as follows: - The maximum energy of a node in the field is not more than 0.5J/bit. The parameters used in our simulation are summarized in Table. 1. Parameter Value ----------------------------------------- ------------ Simulation Area 100m 100m Location of BS (50m, 50m) Number Of Nodes 100 Initial Energy of Nodes (Maximum Value) 0.5 J/node Packet Size 4000 bits : Parameters used in our simulations To analyze and compare the performance of our protocol with LEACH, SEP, TEEN and DEEC we have used two metrics. They are: - Total number of dead nodes: This metric show the overall lifetime of the network. It gives us an idea about the stability period and instability period. This metric is an indication of the number of dead nodes with time. - Through put: This metric is an indication of the rate of packets sent to BS. Implementing ACH Scheme on LEACH, TEEN, DEEC, and SEP ----------------------------------------------------- ### LEACH-ACH LEACH is a clustering based routing protocol for homogenous networks. In LEACH probability is same for all nodes to become CHs. The nodes compare the random value generated at each round with the threshold equation and become CH if the threshold value is less than the random number. The threshold formula is given by: $ T(\emph{n})=\begin{array}{cc} \{ & \begin{array}{cc} \frac{P}{1-P(rmod\frac{1}{P})} & n\in G \\ 0 & {else} \end{array} \end{array} $ We implement ACH scheme in LEACH and found its results better than LEACH as shown in fig. 3. {height="6cm" width="8cm"} ### TEEN-ACH TEEN is a routing protocol for reactive networks. In TEEN two thresholds: hard and soft have been introduced to reduce number of communications. After deployment of nodes CH set up phase starts in which CHs are formed. Once the CHs are confirmed the nodes sense environment and on their transmitter. When the the sense value reach hard threshold the transmitter is on and data is send to CH. This value is stored in an internal variable. Next time when the sense value reach the hard threshold, difference of stored value and sense value is obtained, if this value is greater or equal to soft threshold transmission is done otherwise transmitter are kept off. In TEEN-ACH we make CHs distant which makes energy dissipation even among CHs and thus very less energy is consumed in each round. Fig. 4 shows the behavior of TEEN against TEEN-ACH. {height="6cm" width="8cm"} ### DEEC-ACH DEEC-ACH is an extension of DEEC protocol which enhances stability period of DEEC. The CH selection criterion is based on DEEC protocol however we introduce ACH scheme which enhances the performance of DEEC. DEEC is a heterogeneous routing protocol in which nodes have different initial energy as the network starts. DEEC uses initial and residual energy level of nodes to form CHs. Once sensor nodes are deployed in the region they locally coordinate for cluster set up and operation. Let $\emph{ni}$ denotes the number of rounds for which the node $\emph{Si}$ is CH often referred as the rotating epoch. $\emph{Popt}$ is our desired percentage of CHs and $\emph{ni}=1/\emph{popt}$ is the rotating epoch. By epoch we means that a node once becomes CH will not take part in CH formation for the next $1/\emph{popt}$ rounds. As in DEEC nodes have different energy levels the CH selection probability is different for each node and we call it average probability $\emph{pi}$. $\emph{pi}$ of nodes with more energy is greater. The average energy of network denoted by $\overline{E(r)}$ is given by eq.(1): $$\overline{E}(r)=\frac{1}{N}\sum_{i=1}^{N}E_i(r)$$ The average probability of CHs per round per epoch is represented in eq.(2): $$\emph{pi}=\emph{popt}\,\,[1-\frac{\overline{E}(r)-Ei(r)}{\overline{E}(r)}]$$ $$\sum_{i=1}^{N}pi=\sum_{i=1}^{N}popt\frac{Ei(r)}{{\overline{E}(r)}}=popt\sum_{i=1}^{N}\frac{Ei(r)}{{\overline{E}(r)}}=Npopt$$ Eq.(3) shows the optimal number of CHs we want to achieve. The probability of nodes in the network to become CHs is based on the ratio of their residual energy and average residual energy of the network. The probability equation for nodes to become CH is given by eq.(4): $$pi=\frac{potp N (1+a) E_i(r)}{(N+\sum_{i=1}^{N}a_i)\overline{E(r)}}$$ Where $\emph{popt}$ is the desired percentage of CHs, $N$ is number of nodes, $\emph{Ei(r)}$ is residual energy of a node and $\overline{E}(r)$ is network’s average energy. $Pi$ is average probability of a node to become CH. Each node creates a random number for itself and compares it with threshold equation, if the number generated is less than or equal to the threshold value the node is selected as CH for that round. The threshold equation is given by: $ T(\emph{Si})=\begin{array}{cc} \{ & \begin{array}{cc} \frac{pi}{1-pi(rmod\frac{1}{pi})} & Si\in G \\ 0 & {else} \end{array} \end{array} $ Where $G$ represents the set of nodes eligible to take part in CH selection at round $r$. $\emph{Si}\,\,\,$$\epsilon$ $G$ consists of all those nodes which have not been CHs for the most recent $ni$ rounds. Once a node becomes CH it sends a confirmation message to all CHs. As soon as the CH is confirmed it sends an association message to all the nodes using CSMA-MAC protocol. The nodes respond according to strength of SNR received. The nodes associate themselves to that CH whose SNR is stronger. The CH then allocates TDMA slot to each node in the cluster. The nodes sense the environment and send data to their respective CHs in the TDMA slots allocated to them. CH formation depends on random number generated, threshold value and energy of nodes. At some stages two or more very close (or intersecting) nodes become CHs and energy dissipation is even more and unbalance. We make CHs away in our protocol. Once a node becomes a CH it virtually take part in election of next CH. The CH decides area of next node taking part in election for CH. We assume that no node can become CH in (15m, 15m). A node which has become CH at (5m, 5m) will force the nodes to be normal nodes in the area ((5+10)m, (5+10)m) . In this way the CHs are made distant and we are able to achieve $Npopt$ CHs each round. Fig. 5 shows the comparison of DEEC with DEEC-ACH. {height="6cm" width="8cm"} ### SEP-ACH In this section we implement ACH scheme on SEP. In SEP we have two level heterogeneity. The normal nodes in SEP have $a$ times less energy than advance nodes. The probability of normal nodes in SEP differs from advance nodes as follows: $$pnrm=\frac{p_{opt}}{1+a.m}$$ $$padv=\frac{p_{opt}(1+a)}{1+a.m}$$ where $pnrm$ in eq.(5) and $padv$ in eq.(6) is the probability equation for normal and advance nodes respectively. m is the fraction of advance nodes. Eq.(6) shows greater probability of advance nodes to become CHs. Each node in the network generates a random number for itself, compares itself with the threshold value and become CH if the number is less than or equal to threshold value. After the CH formation CH confirmation phase starts in which the CHs are made distant. We introduce ACH scheme in SEP which makes CHs formed in SEP distant. The energy in the nodes are thus conserved and the stability period of SEP is enhanced. Fig. 6 shows that SEP-ACH performs better than SEP. The $1st$ node in SEP dies at round $1130$ whereas in SEP-ACH the $1st$ node dies at round $2004$. {height="6cm" width="8cm"} Conclusion ========== This paper deals with ACH scheme, a clustering technique for WSNs that enhances life time of LEACH, TEEN, DEEC, SEP and minimizing global energy consumption by distributing the load to all the nodes at different points in time. In DEEC-ACH high energy nodes are made CHs frequently than low energy nodes, making energy distribution evenly in the network. Also a CH take part in the selection of next CH thus the number of CHs are reduced and the CHs are made distant. We have forced those nodes which have not become CHs and are close to each other or intersecting. The energy of nodes are conserved in this way and stability period of the network is prolonged. DEEC-ACH outperforms LEACH as LEACH is not suited with heterogeneous environment. DEEC-ACH distribute the energy evenly in the network by giving high priority to high energy nodes in election for CHs and making CHs away from one another. DEEC-ACH also perform well than DEEC as the adjacent and very close nodes are made CHs in DEEC. This consume much energy of the nodes in the process of aggregation and fusion. The global communication is increased and the nodes die at a faster rate. The stability period and throughput of the network is decreased. [1]{} I. Akyildiz, W. Su, Y. Sankarasubramaniam, and E. Cayirci. A survey on sensor networks. IEEE Communications Magazine, 40(8):102–114, August 2002. T. J. Shepard. A channel access scheme for large dense packet radio networks. In Proccedings of ACM SIGCOMM, pages 219–230, September 1996. W. Heinzelman, A. Chandrakasan, and H. Balakrishnan. “Energy-Ef?cient Communication Protocols for Wireless Microsensor Networks”. In Proceedings of Hawaiian In- ternational Conference on Systems Science, January 2000. G. Smaragdakis, I. Matta, A. Bestavros, “SEP: A Stable Election Protocol for clustered heterogeneous wireless sensor networks”, in: Second International Workshop on Sensor and Actor Network Protocols and Applications (SANPA 2004), 2004. L. Qing, Q. Zhu, M. Wang, ”Design of a distributed energy-efficient clustering algorithm for heterogeneous wireless sensor networks”. In ELSEVIER, Computer Communications 29 (2006) 22302237. MATLAB 7.4.0(R2007a) www.mathworks.com

--- abstract: 'Malicious software are categorized into families based on their static and dynamic characteristics, infection methods, and nature of threat. Visual exploration of malware instances and families in a low dimensional space helps in giving a first overview about dependencies and relationships among these instances, detecting their groups and isolating outliers. Furthermore, visual exploration of different sets of features is useful in assessing the quality of these sets to carry a valid abstract representation, which can be later used in classification and clustering algorithms to achieve a high accuracy. In this paper, we investigate one of the best dimensionality reduction techniques known as t-SNE to reduce the malware representation from a high dimensional space consisting of thousands of features to a low dimensional space. We experiment with different feature sets and depict malware clusters in 2-D. Surprisingly, t-SNE does not only provide nice 2-D drawings, but also dramatically increases the generalization power of SVM classifiers. Moreover, obtained results showed that cross-validation accuracy is much better using the 2-D embedded representation of samples than using the original high-dimensional representation.' author: - Mohamed Nassar - Haidar Safa bibliography: - 'malwareViz.bib' title: Throttling Malware Families in 2D --- Introduction ============ Security breaches are executed through malwares and are a major threat to the Internet today. There are several forms of malware ranging from viruses and spam bots to trojan horses and rootkits [@stallings2016cryptography]. Recently, the Petya ransomware [@akkas2017malware] crashed shipping companies, ports, and law agencies. This malware targets the master boot record of a machine and prohibits the operating system from normal execution. It then spreads and encrypts all the system files. A message appears on the screen stating the amount of ransom to decrypt the files. The payment is through crypto-currencies. The prominent expansion of malwares is due to their metamorphic and polymorphic techniques that give the ability to change their code as they propagate. In addition, malwares adopt new ways to detect the environments where they are running, hence hindering their detection and making dynamic analysis difficult if not impossible. Visual analytics provides approaches to obtain an understanding from complex data. It aims at developing methods that allow analysts to examine the processes underlying the data [@ellis2010mastering]. Visual exploration of malware families is a pre-processing step of a more in-depth malware family analysis, as it allows for the development of intuitions and hypotheses about the discriminative power of a set of contextual or behavioral features. However, visualizing malware families in low dimensional space (2-D, 3-D) is a topic that received little attention in the literature. Malware data are fundamentally different than text and images[^1] which motivates investigating ways to adapt existing approaches or inventing new ones. For instance a byte in malware has different meanings in different contexts, in contrast to a byte representing pixel intensity in an image. In this paper, we experiment with the best low-dimensional embedding technique known as t-SNE (Student-t distribution – Stochastic Neighborhood Embedding) for depicting malware clusters and features. We propose a pipeline for feature extraction and selection, followed by visualization. We compare the raw classification accuracy at the high-dimensional and low-dimensional spaces for n-grams features. Finally We propose a new first-insight classifier based on t-SNE and SVM. Note that our goal is not to propose a very high accuracy classifier or to compete with extensive feature selection approaches. Instead, we aim at exploring the visualization space of malware families and to which extent such a pre-processing procedure might be useful for analyzing a typical malware dataset. The remaining of the paper is organized as follows. Section 2 surveys relevant related work. Section 3 presents our proposed methodology. In Section 4, we discuss implementation and results. We finally conclude in Section 5 and present future research directions. Background and Related Work =========================== Classifying and Clustering malware families were addressed in many recent work in the literature. In [@li2010challenges], the current automated approaches for malware clustering were summarized. The paper considered the high accuracy obtained by six commercial anti-viruses as biased since unbalanced datasets were used where most malware instances are easy to classify. A plagiarism detector algorithm was applied on the same dataset and yielded the same accuracy results compared to those of the anti-viruses, though the plagiarism detector does not have any expert knowledge about malwares. In [@bayer2009scalable] a scalable, behavior-based malware clustering approach was proposed. This approach aims to isolate outliers that exhibit a novel behavior to be further analyzed. It used a recent technique called taint tracking to build behavioral profiles and locality sensitive hashing for clustering these profiles. Malware instance visualization was proposed in [@nataraj2011malware]. This approach suggested transforming the binary into a vector of 8-bit integers, which can be reshaped into a matrix and therefore viewed as a gray-scale image. This technique proves useful in increasing the accuracy of malware classifiers. The work presented in this paper is different such that we focus on visualizing families of malwares as scatter plots using t-SNE [@maaten2008visualizing], an embedding technique that allows visualizing high-dimensional data by giving each datapoint a location in a two or three-dimensional map. The technique is a variation of Stochastic Neighbor Embedding (SNE). SNE starts by converting the high-dimensional Euclidean distances between datapoints into conditional probabilities that represent similarities. The similarity of datapoint $x_j$ to datapoint $x_i$ is the conditional probability, $p_{j|i}$ , that $x_i$ would pick $x_j$ as its neighbor if neighbors were picked in proportion to their probability density under a Gaussian centered at $x_i$. For the low-dimensional counterparts $y_i$ and $y_j$ of $x_i$ and $x_j$, it is possible to compute a similar conditional probability $q_{j|i}$. SNE aims to find a low-dimensional data representation that minimizes the mismatch between $p_{j|i}$ and $q_{j|i}$. To do so, SNE minimizes the sum of Kullback-Leibler divergences over all datapoints using a gradient descent method. t-SNE differs from the old SNE in two ways: (1) it uses a symmetrized version of the SNE cost function with simpler gradients and (2) it uses a heavy-tailed Student-t distribution rather than a Gaussian distribution to compute the similarity between two points in the low-dimensional space. This allows t-SNE to alleviate both the so-called crowding problem and the optimization problems of SNE [@maaten2008visualizing]. The dataset we use is the training set from Microsoft Malware Classification Challenge [@ronen2018microsoft] released in 2015, which since has been studied by researchers in more than 50 publications targeting feature engineering, deep learning, clustering and classification approaches. For instance, [@gibert2016convolutional] proposes using convolutional neural networks for feature extraction and classification based on the binary and reconstructed assembly files. Our work takes another direction and focuses on the visualization question. Proposed Methodology ==================== In this section we describe our proposed methodology which copes with relatively large data sets. We have used the training dataset from Microsoft malware classification challenge (Big 2015) having 10868 labeled instances [@ronen2018microsoft]. The standard version of t-SNE having quadratic complexity in terms of the number of instances $O(N^2)$ might be applied. For larger data-sets, other versions of t-SNE are proposed such as random-walk based sampling of landmark points or using specialized data structures leading to $O(N \log N)$ complexity [@van2014accelerating]. We particularly used the version of scikit-learn with the Barnes-Hut approximation running in $O(N \log N)$ time. Fig. \[fig1\] shows the pipeline of our proposed methodology. Starting from a labeled corpus of files containing the malwares payload in hexadecimal format, we extract n-byte grams (3, 4 and 5). Note that the number of features grow exponentially such that for 3-byte grams we have $2^{8*3}$ possible words, for 4-byte grams it is $2^{8*4}$ and so on. Most current machine learning libraries cannot handle this number of features even in sparse format. For example, Scikit-learn accepts feature indices less than a positive 4-bytes signed integer ($2^{31}-1$). Therefore, in the proposed methodology, we hash the feature indices to 22-bits integers. We also proceed by early removal of rare words that appear less than k times (we assumed k = 3). These words probably represent addresses in memory or literals and have little differentiation power. This technique is very efficient in reducing the storage amount of the features set. We store the resulting features along with the instance label in sparse format (LibSVM/SVMLight format) as one line per instance in the output text file. Each feature is represented by an index (the hash of the n-bytes gram) and a value which is the number of occurrences of this n-bytes gram in the malware instance. This stage is implemented by using the Sally tool [@rieck2012sally] which is an efficient feature extraction tool that generates n-grams besides other features such as TF-IDF [@chowdhury2010introduction]. We proceed with a first feature selection stage using the $\chi^2$ statistical test-based selector where our target is to reduce the number of features to 1,000. This limit reduces the complexity of computing the pair-wise distances in t-SNE. However, we are starting with a much larger number ($2^{22} = 4,194,304$ feature-space). The time complexity of the $\chi^2$ selector is $O(n_\text{classes} * n_\text{features}$). The features that are the most likely to be independent of class and therefore irrelevant for classification are removed. We also can reduce the space complexity of this stage (mainly because of memory limitations) by sampling the instances in equal proportions to their family sizes. If sampling is used than we must use the generated $\chi^2$ selector model to transform the complete dataset and keep the top $1,000$ features of each instance. For the dataset in question we did not use sampling. ![Pipeline for malware family visualization.[]{data-label="fig1"}](pipeline2){width="\textwidth"} Optionally, we apply a PCA (Principal Component Analysis) transformation to reduce further the number of features to the range of 30-50 features. This speeds up the computation of pairwise distances between the data-points in the next stage and suppresses some noise without severely distorting the inter-point distances [@maaten2008visualizing]. t-SNE then embeds these features in 2 dimensions. The malware instances are depicted as scatter plots. t-SNE outperforms other data embedding techniques such as PCA, Sammon mapping, Isomap and LLE. Implementation, Preliminary Results, and Interpretation ======================================================== In this section, we describe the implementation environment and setup, the dataset used, the hyper parameters then we discuss the obtained results. Setup, implementation, and tools -------------------------------- All our experiments were performed using a commercial off-the-shelf laptop with a 64-bit Ubuntu 16.04 LTS operating system, an Intel core i5-5200U CPU (4 cores, 2.20GHz) 8 GB RAM and 1 TB Hard disk. Our implementation was based on Python v3.6, numpy v1.13, scipy v0.19 and the Scikit-learn v0.19 library. The plots are generated using the BokehJS v0.12 library. We have used Sally [@rieck2012sally] for feature extraction. Our code is available at <https://github.com/mnassar/malware-viz> under form of Python Jupyter notebooks for further exploration and result reproducibility. Dataset ------- The dataset is the training set from Microsoft Malware Classification Challenge [@ronen2018microsoft], which includes 10868 labeled samples. For each sample, the raw data and meta data are provided. The raw data contains the hexadecimal representation of the file’s binary content, without the Portable Executable (PE) header to ensure sterility. The metadata manifest is a log containing various metadata information extracted from the binary, such as function calls, strings, etc. This was generated using the IDA disassembler tool. In our implementation, we focused solely on the raw data. However, we consider augmenting our visualization with the meta data for future work. The raw training data volume is about 41 GBytes. However, our feature selection method reduces this to about 6.4 GBytes for the sum of three feature sets (3, 4 and 5-grams). The dataset contains malwares belonging to the following 9 families: Ramnit, Lollipop, Kelihos Ver. 3, Vundo, Simda, Tracur, Kelihos Ver. 1, Obfuscator.ACY and Gatak. One challenge of this data set is the unbalanced sizes of different families. The distribution of instances for the training dataset is shown in Table \[tab1\]. It is shown that classes 4, 5, 6 and 7 are underrepresented as compared to the other families. Class Family Type Nb. Of Instances Percentage (%) ------- ---------------- -------------------- ------------------ ---------------- 1 Ramnit Worm 1541 14.20 2 Lollipop Adware 2478 22.80 3 Kelihos\_ver 3 Backdoor 2942 27.07 4 Vundo Trojan 475 4.37 5 Simda Backdoor 42 0.39 6 Tracur Trojan Downloader 751 6.91 7 Kelihos\_ver 1 Backdoor 398 3.66 8 Obfuscator.ACY Obfuscated malware 1228 11.30 9 Gatak Backdoor 1013 9.32 : Distribution of samples among the families.[]{data-label="tab1"} t-SNE parameters ---------------- t-SNE implementation in Sklearn has many tunable parameters. We show the most important ones with a short description in Table \[tab2\]. Note that the t-SNE method is known to be little sensitive to these parameters. Nevertheless, a good tuning enhances the quality of the obtained clusters sometimes. **Parameter** **Description** **Typical/default value** ---------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------- *Random\_state* This is the seed of random initialization in the embedded space. It is an important parameter to accomplish comparisons under same initial conditions (used with init = ’random’) No typical value. We have chosen 42 as our default. *N\_iter* Maximum number of iterations for the optimization. Default: 1000 *Perplexity* The perplexity can be interpreted as a smooth measure of the effective number of neighbors. 5 – 50 Our default is 40 *Early exaggeration* Larger values of this parameter tend to start with distant clusters in the embedded space Default: 12 *metric* The distance between instances where each instance is a feature array. Euclidean *Learning rate (lr)* It must not be too low or too high. If the cost function gets stuck in a bad local minimum increasing the learning rate may help. 10 – 1000 We have chosen 200 as our default. : Tunable Parameters in t-SNE.[]{data-label="tab2"} Sample of results ----------------- Due to lack of space we do not show all obtained visual plots. However, we depict and interpret the most important ones. Fig. \[fig2\] shows the scatter plot of the 9 classes for 4-byte grams (no PCA). In this figure, it is directly noticed that each family is composed of one or multiple clusters. No one clear cluster per family exists as in the case of the MNIST dataset [@lecun1998gradient]. This is expected due to the polymorphic nature of malware data which has much more noise than what can be carried by handwritten digits or letters. Common evasion and obfuscation techniques may also explain why some families have intersection regions in between their clusters. Still we can identify big clusters for each of the big families (1, 2, 3, 8, 9) and even some smaller clusters for the families 4 (Vundo) and 7 (Kelihos\_ver 1). It is also important to notice the existence of some outliers. To give a chance to the underrepresented families, we started over with only their instances in the pipeline. Obtained results are shown in Fig. \[fig3\]. These results are much better, and we can observe clear groups for each of these small families. Next, we take only the largest two classes (2 and 3) to compare plots between different sets of features (3-bytes grams, 4-bytes grams and 5-bytes grams). Obtained results are illustrated in Fig. \[fig4\]. They show that to some extent, 5-grams (Fig. \[fig4\](c)) have less overlap between clusters than 3-grams (Fig. \[fig4\](a)) or 4-grams (Fig. \[fig4\](b)). We wanted to correlate this with the classification accuracy on the original dataset (the one after feature selection which is 5420 instances \* 1000 features) and the transformed dataset (5420 instances \* 2 features). Results in Table \[tab3\] shows that training accuracy is almost perfect for all three feature sets. However, training accuracy is often not a good metric since the classifier might overfit the training set. ![4-grams, no-PCA, 9 classes, perplexity=40, lr=200.[]{data-label="fig2"}](4grams-no-pca-9classes){width="70.00000%"} ![4-grams, no-PCA, classes 4,5,6,7, perplexity=40, lr=200.[]{data-label="fig3"}](4grams-no-pca-4567classes){width="70.00000%"} ![(a) Classes 2 and 3 (3-grams, perplexity=40, lr=200); (b) Classes 2 and 3 (4-grams, perplexity=40, lr=200); (c) Classes 2 and 3 (5-grams, perplexity=40, lr=200) []{data-label="fig4"}](3grams "fig:"){width="48.00000%"} ![(a) Classes 2 and 3 (3-grams, perplexity=40, lr=200); (b) Classes 2 and 3 (4-grams, perplexity=40, lr=200); (c) Classes 2 and 3 (5-grams, perplexity=40, lr=200) []{data-label="fig4"}](4grams "fig:"){width="48.00000%"}\ (a) (b)\ ![(a) Classes 2 and 3 (3-grams, perplexity=40, lr=200); (b) Classes 2 and 3 (4-grams, perplexity=40, lr=200); (c) Classes 2 and 3 (5-grams, perplexity=40, lr=200) []{data-label="fig4"}](5grams "fig:"){width="48.00000%"}\ (c) We compute the two-fold cross validation accuracy which is much more indicative of the generalization power of the classifier. Astonishingly, the two-fold cross validation accuracy is very poor on the original dataset and dramatically much better on the embedded dataset. This can be explained by the fact that t-SNE groups the datapoints into separate clusters in a low dimensional space, which is at the origin of the design of support vector classifiers with the Radial Basis Function (RBF) kernel. These classifiers shine under this kind of settings. The SVC accuracy (%) for the three feature sets with default Sklearn parameters (RBF kernel, $C=1.0$, $\gamma=1/n_\text{features}$) are shown in Table \[tab3\]. Better SVC accuracy results are usually obtained after a grid search for the best hyperparameters $C$ and $\gamma$. -------------- ---------- ---------- --------------------- --------------------- Feature Set Training Training Two-fold cross- Two-fold cross- Accuracy Accuracy Validation Accuracy Validation Accuracy (1000-d) (2-d) (1000-d) (2-d) 3-byte grams 99.98 99.98 67.91 99.57 4-byte grams 100.00 99.96 64.06 99.88 5-byte grams 99.98 99.98 68.13 99.92 -------------- ---------- ---------- --------------------- --------------------- : SVC Accuracy on different feature-sets (classes 2 and 3).[]{data-label="tab3"} Next, we examine the performance of t-SNE on unbalanced families. We take the extreme case by choosing the largest family (Kelihos\_ver 3 – class 3) and the smallest family (Simda – class 5). Results that are presented in Fig. \[fig5\] show that t-SNE can isolate the class 5 in a small cluster. They also show that 4-grams and 5-grams perform better than 3-grams in isolating clusters of the two classes. Fig. \[fig5\](d) shows that choosing a bad perplexity value might degrade the clustering quality. ![(a) Classes 3 and 5 (3-grams, perplexity=40, lr=20); (b) Classes 3 and 5 (4-grams, perplexity=40, lr=200); (c) Classes 3 and 5 (5-grams, perplexity=40, lr=200); (d) Classes 3 and 5 (5-grams, perplexity=5, lr=200) []{data-label="fig5"}](largest_vs_smallest_3grams "fig:"){width="45.00000%"} ![(a) Classes 3 and 5 (3-grams, perplexity=40, lr=20); (b) Classes 3 and 5 (4-grams, perplexity=40, lr=200); (c) Classes 3 and 5 (5-grams, perplexity=40, lr=200); (d) Classes 3 and 5 (5-grams, perplexity=5, lr=200) []{data-label="fig5"}](largest_vs_smallest_4grams "fig:"){width="45.00000%"}\ (a) (b)\ ![(a) Classes 3 and 5 (3-grams, perplexity=40, lr=20); (b) Classes 3 and 5 (4-grams, perplexity=40, lr=200); (c) Classes 3 and 5 (5-grams, perplexity=40, lr=200); (d) Classes 3 and 5 (5-grams, perplexity=5, lr=200) []{data-label="fig5"}](largest_vs_smallest_5grams "fig:"){width="45.00000%"} ![(a) Classes 3 and 5 (3-grams, perplexity=40, lr=20); (b) Classes 3 and 5 (4-grams, perplexity=40, lr=200); (c) Classes 3 and 5 (5-grams, perplexity=40, lr=200); (d) Classes 3 and 5 (5-grams, perplexity=5, lr=200) []{data-label="fig5"}](largest_vs_smallest_perp_5 "fig:"){width="45.00000%"}\ (c) (d)\ t-SNE shows similar performance in enhancing the classification accuracy as shown in Table \[tab4\]. Note that since the classes are severely unbalanced, an accuracy of 98.60 would be simply obtained if the classifier considers all the data points a belonging to the majority class 5. The two-fold cross validation accuracies on the original dataset are bad in this sense. However, we notice that the two-fold cross validation accuracy is much better in the embedded space. 3-grams features perform the worst as it can be expected by examining the corresponding scatter plot. -------------- ---------- ---------- --------------------- --------------------- Feature Set Training Training Two-fold cross- Two-fold cross- Accuracy Accuracy Validation Accuracy Validation Accuracy (1000-d) (2-d) (1000-d) (2-d) 3-byte grams 100.00 99.93 98.52 (bad) 99.83 4-byte grams 100.00 99.93 98.62 (bad) 99.90 5-byte grams 100.00 99.97 98.62 (bad) 99.90 -------------- ---------- ---------- --------------------- --------------------- : SVC Accuracy on different feature sets (classes 3 and 5)[]{data-label="tab4"} Finally, we want to validate this hypothesis on the complete dataset (9-classes). Results are shown in Table \[tab5\]. The training accuracy in 2-d is a bit smaller than in 1000-d but allows much better generalization of the classification model as clearly inferred from the cross-validation accuracy results. -------------- ---------- ---------- --------------------- --------------------- Feature Set Training Training Two-fold cross- Two-fold cross- Accuracy Accuracy Validation Accuracy Validation Accuracy (1000-d) (2-d) (1000-d) (2-d) 3-byte grams 99.66 96.84 56.76 94.26 4-byte grams 99.58 96.22 55.26 93.13 5-byte grams 98.25 95.69 60.24 92.58 -------------- ---------- ---------- --------------------- --------------------- : SVC Accuracy on different feature sets (all classes)[]{data-label="tab5"} Testing Accuracy ---------------- In this section we further assess the idea of squeezing the dimensions into a small hyperspace using t-SNE than expanding it back to infinite dimensional space using the RBF kernel with SVM. We wanted to estimate the testing accuracy using the raw unlabeled test dataset (10873 instances). Note that t-SNE is a non-parametric mapping, therefore we cannot use the learnt model to map the test datapoints to the embedded space which is formed by the train datapoints. As an alternative we run t-SNE on the full dataset composed of both train and test datapoints (Another approach is to train a multivariate regressor to predict the map location from the input data [@van2009learning]). We then fit an SVC model solely based on the train embedded datapoints. This SVC model is used to estimate probabilistic predictions of the membership of each embedded test point to each of the 9 possible classes. The pipeline of this approach is depicted in Fig. \[fig6\]. ![Testing Accuracy Pipeline[]{data-label="fig6"}](testing_){width="100.00000%"} Note that the labels of these instances are not available, and the only way of evaluation is to obtain the multi-class logarithmic loss by submitting our predictions in probabilistic form to the dataset hosting platform (Kaggle) online. The equation for the logarithmic loss is: $$\text{logloss}= -\frac{1}{10873}\sum_{i=1}^{10873}\sum_{j=1}^{9}y_{ij}\log p_{ij}$$ where $y_{ij}=1$ if $i$ belongs to class $j$ and $0$ otherwise, $\log$ is the natural logarithm and $p_{ij}$ is the probability that $i$ belongs to class $j$ as given by the classifier. Our classifier achieves a testing logloss of 0.1719. This is fairly acceptable given the simplicity of the employed feature set (1000 best features among 1,2,3,4 and 5 n-bytes grams) and without recurring to any involved feature engineering. A clueless classifier scores 2.1972. For this experiment we have used an m5.4xlarge AWS EC2 instance (64 GB RAM) and a 500 GB volume. Conclusion ========== In this paper, we have successfully applied feature extraction, selection, embedding and visualization over a recent malware dataset. We have proposed a pipeline that can cope with dataset of similar or larger size. We use the t-SNE algorithm to embed the malware datapoints in 2D and visualize them as scatter plots. A very interesting result is that compressing the data using t-SNE dramatically enhances the cross-validation accuracy of Support Vector Machines classifiers. t-SNE shapes the clusterability of datapoints in the embedded space, which is very appealing to SVM classifiers with the RBF kernel. In future work, we aim to experiment with other feature sets, for instance by analyzing the assembly data files. We also want to assess the viability of the SVM–t-SNE classifier over other data sets. Another direction is to work on implementing t-SNE in a 3D WebGL framework and integrate it in Jupyter notebooks. Available GPUs can also be used to bear some of the tedious computations. Acknowledgement {#acknowledgement .unnumbered} =============== This work was supported in part by a grant from the University Research Board of the American University of Beirut, Lebanon. [^1]: <http://www.jsylvest.com/blog/2017/12/malconv/>

--- abstract: | [Let $Y$ be a regular covering of a complex projective manifold $M\hookrightarrow\Co\P^{N}$ of dimension $n\geq 2$. Let $C$ be intersection with $M$ of at most $n-1$ generic hypersurfaces of degree $d$ in $\Co\P^{N}$. The preimage $X$ of $C$ in $Y$ is a connected submanifold. Let $H^{\infty}(Y)$ and $H^{\infty}(X)$ be the Banach spaces of bounded holomorphic functions on $Y$ and $X$ in the corresponding supremum norms. We prove that the restriction $H^{\infty}(Y)\longrightarrow H^{\infty}(X)$ is an isometry for $d$ large enough. This answers the question posed in \[L\].\ ]{} author: - | Alexander Brudnyi[^1]\ Department of Mathematics and Statistics\ University of Calgary\ Calgary, Canada title: 'A Uniqueness Property for $H^{\infty}$ on Coverings of Projective Manifolds' --- -10mm =msbm10 scaled 1 =msbm7 scaled 1 =msbm5 scaled 1 === ¶[[P]{}]{} \[section\] \[Th\][Lemma]{} \[Th\][Corollary]{} \[Th\][Definition]{} \[Th\][Proposition]{} \[Th\][Remark]{} \[Th\][Example]{} . Introduction and Formulation of the Result. ============================================= [**1.1. An Extension Theorem.**]{} Let $M$ be a complex projective manifold of -10000 dimension $n\geq 2$ with a Kähler form $\omega$ and let $L$ be a positive line bundle on $M$ with canonical connection $\nabla$ and curvature $\Theta$ in a hermitian metric $h$. Let $C$ be the common zero locus of holomorphic sections $s_{1},...,s_{k}$, $k<n$, of $L$ over $M$ which, in a trivialization, can be completed to a set of local coordinates at each point $C$. Then $C$ is a (possibly disconnected) $k$-dimensional submanifold of $M$ which will be referred to as an [*$L$-submanifold of $M$*]{}. Let $\pi:Y_{G}\longrightarrow M$ be a regular covering of $M$ with a transformation group $G$ and $X_{G}=\pi^{-1}(C)$. We denote the pullbacks to $Y_{G}$ of $\omega$ and $\Theta$ by the same letters. \[e0\] [If $L$ is very ample, then it is pullback of the hyperplane bundle by an embedding of $M$ into some projective space $\Co\P^{N}$. Further, zero loci of holomorphic sections of $L$ are hyperplane sections of $M$. By Bertini’s theorem, the generic linear subspace of codimension $n-k$, $k<n$, intersects $M$ transversely in a smooth manifold $C$ of dimension $k$, and by the Lefschetz hyperplane theorem, $C$ is connected and the induced homomorphism $\pi_{1}(C)\longrightarrow \pi_{1}(M)$ of fundamental groups is surjective. Thus in this case $X_{G}\subset Y_{G}$ is a connected submanifold.]{} Further, let $dist(\cdot,\cdot)$ be the distance on $Y_{G}$ induced by $\omega$. Consider a function $\phi: Y_{G}\longrightarrow\Re$ such that $d\phi$ is bounded, i.e. $$|\phi(x)-\phi(y)|\leq a\cdot dist(x,y)\ \ \ \ {\rm for\ some}\ \ \ a>0.$$ By ${\cal O}_{\phi}(X_{G})$ we denote the vector space of holomorphic functions on $X_{G}$ such that $|f|^{2}e^{-\phi}$ is integrable on $X_{G}$ with respect to the volume form of the induced Kähler metric on $X_{G}$. This is a Hilbert space with respect to the inner product $$(f,g)\mapsto\int_{X_{G}}f\overline{g}e^{-\phi}\omega^{k}\ .$$ We define ${\cal O}_{\phi}(Y_{G})$ similarly. By $|\cdot|_{\phi,X_{G}}$ and $|\cdot|_{\phi,Y_{G}}$ we denote the corresponding norms. It was shown in \[L\] that the restriction determines a continuous linear map $$\rho: {\cal O}_{\phi}(Y_{G})\longrightarrow {\cal O}_{\phi}(X_{G}),\ \ \ f\mapsto f|_{X_{G}}\ .$$ The following remarkable result was proved by Lárusson \[L, Th.1.2\]. \[la\] Suppose $$\Theta\geq i\partial\overline{\partial}\phi+\epsilon\omega$$ for some $\epsilon>0$ in the sense of Nakano. Then $\rho$ is an isomorphism. [**1.2. Formulation of the Main Result.**]{} An important example of a function $\phi$ as above is obtained by smoothing the distance $\delta$ from a fixed point $o$ in $Y_{G}$. By a result of Napier \[N\], there is a smooth function $\tau$ on $Y_{G}$ such that [(A)]{} : $c_{1}\delta\leq\tau\leq c_{2}\delta+c_{3}$ for some $c_{1},c_{2},c_{3}>0$, [(B)]{} : $d\tau$ is bounded, and [(C)]{} : $i\partial\overline\partial\tau$ is bounded. Furthermore, by (A) and since the curvature of $Y_{G}$ is bounded below, there is $c>0$ such that $e^{-c\tau}$ is integrable on $Y_{G}$. Then $e^{-c\tau}$ is also integrable on $X_{G}$. We set $$\label{A} A:=\frac{cc_{2}}{c_{1}}\ .$$ Let $\tilde L$ be any positive line bundle on $M$ with curvature $\tilde\Theta$. By (C) there is a non-negative integer $m_{0}$ such that for any integer $m>m_{0}$ $$\label{e1} m\tilde\Theta> i\partial\overline\partial(A\tau)\ .$$ We set $L:=\tilde L^{\otimes m}$. Then Lárusson’s theorem holds for coverings $X_{G}:=\pi^{-1}(C)\subset Y_{G}$ of $L$-submanifolds $C\subset M$ with $\phi:=A\tau$ and with $\phi:=c\tau$ (because $A\geq c$). Let $H^{\infty}(Y_{G}),\ H^{\infty}(X_{G})$ be the Banach spaces of bounded holomorphic functions on $Y_{G}$ and $X_{G}$ in the corresponding supremum norms. \[te1\] The map $\rho: H^{\infty}(Y_{G})\longrightarrow H^{\infty}(X_{G})$, $f\mapsto f|_{X_{G}}$, is an isometry. [**1.3. Corollaries and Examples.**]{} Let $X$ be a complex manifold and $H^{\infty}(X)$ be the Banach algebra (in the supremum norm) of bounded holomorphic functions on $X$. The maximal ideal space ${\cal M}={\cal M}(H^{\infty}(X))$ is the set of all nontrivial linear multiplicative functionals on $H^{\infty}(X)$. The norm of any $\phi\in {\cal M}$ is $\leq 1$ and so ${\cal M}$ is embedded into the unit ball of the dual space $(H^{\infty}(X))^{*}$. Then ${\cal M}$ is a compact Hausdorff space in the weak $*$ topology induced by $(H^{\infty}(X))^{*}$ (i.e. the [*Gelfand topology*]{}). Further, there is a continuous map $i:X\longrightarrow {\cal M}$ taking $x\in X$ to the evaluation homomorphism $f\mapsto f(x)$. This map is an emebedding if $H^{\infty}(X)$ separates points of $X$. Recall also that the complement to the closure of $i(X)$ in ${\cal M}$ is called [*the corona*]{}. The [*corona problem*]{} is to determine those $X$ for which the corona is empty. For example, according to Carleson’s celebrated Corona Theorem \[C\] this is true for $X$ being the open unit disk $\Di\subset\Co$. Also there are non-planar Riemann surfaces for which the corona is non-trivial (see e.g. \[JM\], \[G\], \[BD\] and references therein). The general problem for planar domains is still open, as is the problem in several variables for the ball and polydisk. In \[L, Th. 2.1\] Lárusson discovered simplest examples of Riemann surfaces with big corona. Namely, using his Theorem \[la\] he proved that if $Y_{G}\subset\Co^{n}$ is a bounded domain and $X_{G}\subset Y_{G}$ is a Riemann surface satisfying assumptions of Theorem \[te1\] then the natural map $i:X_{G}\hookrightarrow {\cal M}(H^{\infty}(X_{G}))$ extends to an embedding $Y_{G}\hookrightarrow {\cal M}(H^{\infty}(X_{G}))$. The next statement is an extension of his result. \[cor1\] Under the assumptions of Theorem \[te1\], the transpose map -10000 $\rho^{*}:{\cal M}(H^{\infty}(X_{G}))\longrightarrow {\cal M}(H^{\infty}(Y_{G}))$, $\phi\mapsto\phi\circ\rho$, is a homeomorphism. This follows from the fact that $\rho: H^{\infty}(Y_{G})\longrightarrow H^{\infty}(X_{G})$ is an isometry of Banach algebras.     $\Box$ \[e1\] [(1) (The references for this example are in \[L, Sect. 4\].) Let $M$ be a projective manifold covered by the unit ball $\B\subset\Co^{n}$ with a positive line bundle $L$ with curvature $\Theta$, and $X\subset\B$ be the preimage of an $L$-submanifold $C\subset M$. Let $\delta$ be the distance from the origin in the Bergman metric of $\B$. By $\omega$ we denote the Kähler form of the Bergman metric. It was shown in \[L, Sect. 4\] that there is a nonnegative function $\tau$ on $\B$ such that $i\partial\overline{\partial}\tau=\omega$, $d\tau$ is bounded, and $$\sqrt{n+1}\delta\leq\tau\leq\sqrt{n+1}\delta+(n+1)\log\ 2 .$$ Moreover, $$\int_{\B}e^{-c\tau}\omega^{n}<\infty\ \ \ \ {\rm if\ and\ only\ if}\ \ \ c>\frac{2n}{n+1}\ .$$ Applying Theorem \[te1\] (with $c_{2}=c_{1}=\sqrt{n+1}$) we obtain that [*$\rho:H^{\infty}(\B)\rightarrow H^{\infty}(X)$ is an isometry if $\Theta>\frac{2n}{n+1}\omega$.*]{} This holds for instance if $L=K^{\otimes m}$ with $m\geq 2$ where $K$ is the canonical bundle of $M$.\ (2) Let $S$ be a compact complex curve of genus $g\geq 2$ and $\Co\To$ be a one-dimensional complex torus. Consider an $L$-curve $C$ in $M:=S\times\Co\To$ with a very ample $L$ satisfying the assumptions of Theorem \[te1\]. Let $\pi:\Di\times\Co\longrightarrow M$ be the universal covering. Then Theorem \[te1\] is valid for the connected curve $X:=\pi^{-1}(C)\subset\Di\times\Co$. This implies that any $f\in H^{\infty}(X)$ is constant on each $S_{y}:=(\{y\}\times\Co)\cap X$, $y\in\Di$. Note that $S_{y}$ is union of the orbits of some $z_{iy}\in X$, $i=1,...,k$, under the natural action of the group $\pi_{1}(\Co\To)\ (\cong\Z\oplus\Z)$ on $\Di\times\Co$.]{} . Proof of Theorem 1.3. ======================= Let us fix a fundamental compact $K$ of the action of $G$ on $Y_{G}$, i.e., $Y_{G}=\cup_{g\in G}\ g(K)$. Consider finite covers ${\cal U}=(U_{i})$ and ${\cal V}=(V_{j})$ of $K$ by compact coordinate polydisks such that each $V_{j}$ belongs to the interior of some $U_{i_{j}}$. \[le1\] Let $f$ be a holomorphic function defined in an open neighbourhood $O$ of $\cup_{i}\ U_{i}$. Assume that $$\int_{O}|f|^{2}\omega^{n}=B<\infty\ .$$ Then there is a constant $b>0$ (depending on ${\cal U}$ and ${\cal V}$ only) such that $$\label{e2} \max_{K}|f|\leq b\sqrt{B}\ .$$ The proof of the lemma is the consequence of the following facts:\ (a) after the identification of $U_{i_{j}}$ with the closed unit polydisk $D$ and of $V_{j}$ with a compact subset $D_{j}\subset D$, the volume form $\omega^{n}$ restricted to each $U_{i_{j}}$ is equivalent to the Euclidean volume form $do:=dz_{1}\wedge d\overline{z}_{1}\wedge\dots\wedge dz_{n}\wedge d\overline{z}_{n}$ ;\ (b) the Bergman inequality (see \[GR, Ch.6, Th.1.3\]) $$\max_{D_{j}}|f|\leq\frac{(\sqrt{n})^{n}}{(\sqrt{\pi}d)^{n}}\cdot \left(\int_{D}|f|^{2}do\right)^{1/2},$$ where $d$ is the Euclidean distance from $D_{j}$ to the boundary of $D$.\ We leave the details to the reader.      $\Box$ Let $dist(\cdot,\cdot)$ be the distance on $Y_{G}$ in the metric induced by $\omega$. In particular, $\delta(x):=dist(x,o)$. Since $\omega$ is invariant with respect to the action of $G$ we also have $dist(g(x),g(y))=dist(x,y)$ for any $g\in G$. From inequalities (A) for $\tau$ and the triangle inequality for the distance we obtain $$\label{e3} \begin{array}{c} \tau(g(x))\geq c_{1}dist(g(x),o)\geq c_{1}[dist(g(x),g(o))-dist(g(o),o)]=\\ \\ c_{1}[dist(x,o)-dist(g(o),o)]\geq (c_{1}/c_{2})\tau(x)-(c_{1}c_{3}/c_{2})- c_{1}\delta(g(o))\ . \end{array}$$ Further, if $x\in K$ then $$\label{e4} a_{1}\leq\tau(x)\leq a_{2}\ \ {\rm for\ some}\ \ a_{1},a_{2}>0\ .$$ Below by $|\cdot|_{\infty, X_{G}}$ and $|\cdot|_{\infty, Y_{G}}$ we denote the corresponding $H^{\infty}$-norms. Let $f\in H^{\infty}(X_{G})$. Then $f\in {\cal O}_{A\tau}(X_{G})\cap {\cal O}_{c\tau}(X_{G})$ and there is $a_{3}>0$ such that $$\max\{|f|_{A\tau, X_{G}}, |f|_{c\tau, X_{G}}\}\leq a_{3}\sup_{X_{G}}|f|:= a_{3}|f|_{\infty, X_{G}}\ .$$ By Theorem \[la\], there is a unique $\tilde f\in {\cal O}_{A\tau}(Y_{G})\cap {\cal O}_{c\tau}(Y_{G})$ such that $\tilde f|_{X_{G}}=f$ and $$\max\{|\tilde f|_{A\tau, Y_{G}}, |\tilde f|_{c\tau, Y_{G}}\}\leq a_{4} \max\{|f|_{A\tau, X_{G}}, |f|_{c\tau, X_{G}}\}\ \ {\rm for\ some}\ \ a_{4}>0 .$$ Combining these inequalities with (\[e4\]) and (\[e2\]) yields $$\max_{K}|\tilde f|\leq a_{5}|f|_{\infty, X_{G}},$$ with some $a_{5}>0$ depending on $X_{G},Y_{G}$ only. For a fixed $g\in G$ consider $(g^{*}f)(z):=f(g(z))$. As above, there is a unique function $\tilde f_{g}\in {\cal O}_{A\tau}(Y_{G})\cap {\cal O}_{c\tau}(Y_{G})$, $\tilde f_{g}|_{X_{G}}=g^{*}f$, such that $$\max_{K}|\tilde f_{g}|\leq a_{5}|f|_{\infty, X_{G}},$$ But according to (\[e3\]) and (\[A\]), function $(g^{*}\tilde f)(z):=\tilde f(g(z))$ belongs to ${\cal O}_{A\tau}(Y_{G})$ and $(g^{*}\tilde f-\tilde f_{g})|_{X_{G}}\equiv 0$. Thus by Theorem \[la\] we have $\tilde f_{g}=g^{*}\tilde f$. Since $K$ is the fundamental compact, the above inequality for each $\tilde f_{g}$ implies that $$\label{e5} |\tilde f|_{\infty, Y_{G}}\leq a_{5}|f|_{\infty, X_{G}}\ .$$ We will prove now that $a_{5}=1$ which gives the required statement. Indeed, the same arguments as above show that for any integer $n\geq1$ the function $(\tilde f)^{n}$ is the unique extension of $f^{n}$ satisfying (\[e5\]): $$|(\tilde f)^{n}|_{\infty, Y_{G}}\leq a_{5}|f^{n}|_{\infty, X_{G}}\ .$$ Thus $$|\tilde f|_{\infty, Y_{G}}\leq \lim_{n\to\infty}(a_{5})^{1/n}|f|_{\infty, X_{G}}=|f|_{\infty, X_{G}}$$ The proof of the theorem is complete.      $\Box$ [ ]{} D. E. Barett and J. Diller, A new construction of Riemann surfaces with corona. J. Geom. Anal. [**8**]{} (1998), 341-347. L. Carleson, Interpolation of bounded analytic functions and the corona problem. Ann. of Math. [**76**]{} (1962), 547-559. T. W. Gamelin, Uniform algebras and Jensen measures. London Math. Soc. Lecture Notes Series [**32**]{}, Cambridge University Press, 1978. H. Grauert and R. Remmert, Theorie der Steinschen Räume. Springer-Verlag, Berlin, 1977. P. Jones and D. Marshall, Critical points of Green’s functions, harmonic measure and the corona theorem. Ark. Mat. [**23**]{} no.2 (1985), 281-314. F. Lárusson, Holomorphic functions of slow growth on nested covering spaces of compact manifolds. Canad. J. Math. [**52**]{} (5) (2000), 982-998. T. Napier, Convexity properties of coverings of smooth projective varieties. Math. Ann. [**286**]{} (1990), 433-479. [^1]: Research supported in part by NSERC. 2000 [*Mathematics Subject Classification*]{}. Primary 32A10. Secondary 14E20. . Bounded holomorphic function, corona problem, transformation group.

--- abstract: 'The effects of the initial state interactions on the $K^--p$ radiative capture branching ratios are examined and found to be quite sizable. A general coupled-channel formalism for both strong and electromagnetic channels using a particle basis is presented, and applied to all the low energy $K^--p$ data with the exception of the [*1s*]{} atomic level shift. Satisfactory fits are obtained using vertex coupling constants for the electromagnetic channels that are close to their expected SU(3) values.' address: - 'California State Polytechnic University, Pomona CA 91768' - | Service de Physique Nucléaire, CEA/DSM/DAPNIA,\ Centre d’Études de Saclay, F-91191 Gif–sur–Yvette, France author: - 'Peter B. Siegel' - Bijan Saghai title: ' Initial State Interactions for $K^-$-Proton Radiative Capture' --- =1.7 in PACS numbers: 13.75.Jz, 24.10.Eq, 25.40.Lw, 25.80.Nv **Introduction** ================ The $K^-$-proton interaction is a strong multichannel process \[1\], with the $\Lambda (1405)$ resonance just below the $K^--p$ threshold at 1432 MeV. At low energies, the $K^-$ can elastically scatter off the proton, charge exchange to $\overline{K^0}-n$, or scatter to ($\Sigma^+ \pi^-$), ($\Sigma^0 \pi^0$), ($\Sigma^- \pi^+$) or ($\Lambda \pi^0$) final states. In addition, the electromagnetic radiative capture processes $K^- p \rightarrow \Lambda \gamma (\Sigma^0 \gamma)$ are also possible \[2\]. Besides, the amplitudes of these latter reactions can be related to those of the associated strangeness photoproduction, i.e. $\gamma p \rightarrow K^+ \Lambda (K^+ \Sigma^0)$, by the crossing symmetry. Much effort has been done experimentally and theoretically to understand this system. In particular, experiments to measure the branching ratios of the radiative capture reactions $K^-p \rightarrow \Lambda \gamma$ and $K^-p \rightarrow \Sigma^0 \gamma$ were recently performed \[3\] to help clarify the details of the reaction mechanism, with a special interest in the nature of the $\Lambda (1405)$ resonance \[4\]. However, on one hand the result for the $\Lambda \gamma$ channel is unexpectably smaller than both the previous measured value \[5\] and those obtained through phenomenological models \[2\], and on the other hand the measured branching ratio for the $\Sigma^0 \gamma$ final states comes out significantly higher than the one for the other channel, producing yet another mystery to this already complicated problem. The most recent measurements of the threshold branching ratios with stopped kaons, done at Brookhaven \[3\] are: $$R_{\Lambda \gamma} = {{\Gamma(K^-p \rightarrow \Lambda \gamma)} \over {\Gamma(K^-p \rightarrow all)}} = .86 \pm .07 \pm .09 \times 10^{-3}~,$$ and $$R_{\Sigma \gamma} = {{\Gamma(K^-p \rightarrow \Sigma^0 \gamma)} \over {\Gamma(K^-p \rightarrow all)}} = 1.44 \pm .20 \pm .11 \times 10^{-3}.$$ Existing calculations \[2\] overestimate $R_{\Lambda \gamma}$ by a factor of 3 or 4 (except a few recent phenomenological analysis \[6,7\] of the kaon photoproduction processes). The pioneer calculations considered the $\Lambda (1405)$ in two different ways: as an [*s*]{}-channel resonance \[8\] or as a quasi-bound ($K-N,\Sigma \pi$) state \[9-11\]. In the quark-model approaches, this hyperon is considered as a pure $q^3$ state \[12,13\], a quasi-bound $\overline{K}$N state \[14,15\] or still as a hybrid ($q^3$+$q^4$$\overline{q}$...) state \[16-18\]. A number of potential model fits to the scattering data incorporate the $\Lambda (1405)$ as a quasi-bound ($K-N,\Sigma \pi$) resonance \[19-21\]. It would be interesting to see if the radiative capture channels can also be understood within a single model. Specially, since the radiative capture data were taken to help distinguish between these two possibilities. The diversified nature of the low energy data challenges theoretical models. Even if the analysis is restricted to the hadronic sector, difficulties arise when trying to understand the $K^-p$ [*1s*]{} atomic level shift. The sign of the $K^-p$ scattering length, extracted from this experiment, is opposite to that determined from K-matrix and potential model fits to the other hadronic data. This conflict poses interesting questions which are discussed in Ref.\[19,20,22\]. Only one potential, Ref. \[20\], has been published which is compatible with all the low energy hadronic data. We will examine the initial state interactions for the radiative capture branching ratio for this potential. Calculations which focus on the radiative capture branching ratios usually do not include the initial state interactions. Only one group \[15\], which uses the cloudy bag quark model, has included these in the electromagnetic branching ratio calculation. Since the interactions are strongly coupled among the various channels, any meaningful comparison with the data needs to include channel couplings. We find here that the effect of the initial state interactions is far from being negligible. One limitation with this latter calculation is that no comparison is made with the strong branching ratio data. The other threshold branching ratios are \[23,24\]: $$\gamma = {{\Gamma(K^-p \rightarrow \pi^+ \Sigma^-)} \over {\Gamma(K^-p \rightarrow \pi^- \Sigma^+)}} = 2.36 \pm .04 ,$$ $$R_c = {{\Gamma(K^-p \rightarrow charged ~particles)} \over {\Gamma(K^-p \rightarrow all)}} = .664 \pm .011 ,$$ and $$R_n = {{\Gamma(K^-p \rightarrow \pi^0 \Lambda)} \over {\Gamma(K^-p \rightarrow ~all~ neutral~ states)}} = .189 \pm .015 .$$ They put tight constraints on the threshold amplitudes and potential coupling strengths \[17,20,21\]. In fact, the five branching ratios are amongst the most precise data in the strangeness sector. However, at present there is no comprehensive analysis which includes both the hadronic and electromagnetic branching ratios of the $K^-p$ system. The aim of this paper is to examine all the low energy data within a single model and determine if it can be understood using known coupling strengths and minimal SU(3) symmetry breaking for relevant vertices in the electromagnetic channels. In doing so, we focus on how the two radiative capture branching ratios are affected by the initial state interactions among the different channels. In order to unravel the essential physics from the many channel system, in section 2 we will first set up a general procedure to separate the strong (or initial state) interactions from the electromagnetic ones. As shown in section 2, the initial state hadronic interactions can be described with 6 complex numbers. Thus we need a model of the interaction between the strong channels to produce these 6 numbers. In section 3 we examine two phenomenological potential models, each of which fit the low-energy scattering data, the resonance at 1405 MeV, and the hadronic branching ratios at threshold. In one potential, present work, the relative potential strengths between the various channels are guided by SU(3) symmetry. For this potential, the parameters are adjusted to fit all the low energy $K^-p$ data with the exception of the [*1s*]{} atomic level shift value of the $K^-p$ scattering length. The other potential is from Ref.\[20\], in which the scattering length is compatible with the atomic level shift data. To clarify our discussion, we wish to underline here the nature of the fitting parameters in the potential guided by SU(3) symmetry. For the hadronic channels, the relative potential strengths are given by a value determined from SU(3) symmetry times a “breaking factor”, which is equal to 1, if the the relative channel couplings are SU(3) symmetric. This SU(3) structure is motivated by chiral symmetry \[14\]. For a good fit to the low energy data we need to vary the relative strengths somewhat, allowing the breaking factor to deviate from 1. The final values of this factor have no obvious physical significance. The potential enables one to estimate the effects of the initial state interactions from a potential which gives a good fit to the low-energy data. In the electromagnetic channels, the radiative capture amplitudes are derived from first order “Born” photoproduction processes, which involve the meson-baryon-baryon coupling constants, $g_{Kp \Lambda}$, $g_{Kp \Sigma}$, $g_{\pi \Sigma \Sigma}$, and $g_{\pi \Sigma \lambda}$. These coupling constants are related by SU(3) symmetry to the well known $\pi NN$ coupling constant. For the fit, these coupling constants are allowed to vary up to $\pm 50 \%$ from their SU(3) values. Here the final values of these 4 parameters will have physical significance and can be compared to values derived from other analyses. Thus we will examine if the radiative capture branching ratio data can be understood using vertex coupling constants for the electromagnetic channels that are close to their expected SU(3) values when initial state interactions are included from a hadronic interaction which fits the low energy hadronic data. **General Formalism** ===================== Consider a coupled-channel system consisting of [*n*]{} hadronic channels and one electromagnetic channel. We assume that the interaction between channels can be represented for each partial wave $l$ by real potentials of the form $V_{ij}^l(\sqrt{s},k_i,k_j)$ where $\sqrt{s}$ is the total energy and $k_i$ is the momentum of channel [*i*]{} in the center-of-mass frame. We will use the notation where the indices [*i,j*]{} are integers for the strong channels and is $\gamma $ for the electromagnetic channels. We will also assume that the transition matrix element for each partial wave from channel [*i*]{} to channel [*j*]{} can be derived from a coupled-channel Lippmann-Schwinger equation: $$T_{ij}(\sqrt{s},k_i,k_j) = V_{ij}(\sqrt{s},k_i,k_j) + \sum\limits_{m} \int V_{im}(\sqrt{s},k_i,q) G_m(\sqrt{s},q) T_{mj}(\sqrt{s},q,k_j) q^2 dq, \eqno(1)$$ where $G_i(\sqrt{s},q)$ is the propagator for channel [*i*]{}. We suppress the index $l$, since for our problem only the $l=0$ partial wave is of interest. The electromagnetic coupling is weak, and to a very good approximation we can neglect the back coupling of the photon channels. Thus the T-matrix for radiative capture can be written as: $$T_{i \gamma}(\sqrt{s},k_i,k_{\gamma}) = V_{i \gamma}(\sqrt{s},k_i,k_{\gamma}) + \sum\limits_{m\ne \gamma} \int V_{im}(\sqrt{s},k_i,q) G_m(\sqrt{s},q) T_{m \gamma}(\sqrt{s},q,k_{\gamma}) q^2 dq . \eqno(2)$$ Note that in the above equation there is no integration over the photon’s momentum. There is only an integration over the hadronic momentum $k_i$ in the $V_{i \gamma}$ potential. This means that only half off-shell information is needed for the hadron-photon potential. Since $\sqrt{s}$ and $k_\gamma$ are fixed in the integral, we can write $V_{i \gamma}(\sqrt{s},q,k_{\gamma})$ as: $$V_{i \gamma}(\sqrt{s},q,k_\gamma) = {{V_{i \gamma}(\sqrt{s},q,k_\gamma)} \over {V_{i \gamma}(\sqrt{s},k_i,k_\gamma)}} V_{i \gamma}(\sqrt{s},k_i,k_\gamma),$$ or $$V_{i \gamma}(\sqrt{s},q,k_\gamma) = v_{i \gamma}(q) V_{i \gamma}(\sqrt{s},k_i,k_\gamma). \eqno(3)$$ Substituting this form for $V_{i \gamma}$ into equation (2) we obtain for the T-matrix: $$T_{i \gamma}(\sqrt(s),k_i,k_\gamma) = \sum \limits_{m \ne \gamma} M_{im}(\sqrt{s}) V_{m \gamma}(\sqrt{s},k_m,k_\gamma),$$ with the matrix $M_{im}$ defined as $$\begin{aligned} M_{im} &\equiv& \delta_{im} + \int V_{i,m}(\sqrt{s},k_i,q) G_m(\sqrt{s},q) v_{m \gamma}(q) q^2 dq \\ & & + \sum_{n \ne \gamma} \int \int V_{in}(\sqrt{s},k_i,q') G_n(\sqrt{s},q') V_{nm}(\sqrt{s},q',q) G_m(\sqrt{s},q) v_{m \gamma}(q) q'^2 dq' q^2 dq + \cdots .\end{aligned}$$ The state “m” is the last hadronic state before the photon is produced. Since all the on-shell momenta are determined from $\sqrt{s}$ we have $$T_{i \gamma}(\sqrt{s}) = \sum \limits_{m \ne \gamma} M_{im}(\sqrt{s}) V_{m \gamma}(\sqrt{s}). \eqno(4)$$ This form for the transition matrix to the photon channels is very convenient, since it separates out the strong part from the electromagnetic part of the interaction. The matrix [*M*]{} is determined entirely from the hadronic interactions and vertices. In the absence of channel-coupling M is the unit matrix. Any deviation from unity is related to the initial state interactions. Note that [*no assumptions were made on the form of the propagator or the potentials connecting the hadronic channels*]{}. They need not be separable. Labeling the $K^--p$ channel as number 5, and defining $A_m(\sqrt{s})$ as $M_{5m}(\sqrt{s})$ we can write the scattering amplitude to the photon channels as: $$F_{K^-p \rightarrow \Lambda \gamma (\Sigma^0 \gamma)} = \sum A_m(\sqrt{s}) f_{m \rightarrow \Lambda \gamma (\Sigma^0 \gamma)}. \eqno(5)$$ where the $f_m$’s are the amplitudes to go from the hadronic channel [*m*]{} to the appropriate photon channel. These amplitudes are derivable from diagrams representing the photoproduction process. The quantities $A_m$ are unitless complex numbers, and contain all the information about the initial state interactions for radiative capture. Generally the sum over m is restricted to states which have charged hadrons. For the $K^--p$ process the problem is greatly simplified, since there are only 3 channels which have charged hadrons: $\pi^+ \Sigma^-$, $\pi^- \Sigma^+$ and $K^-p$. To a very good approximation (see section 3), the $A_m$’s are the same for both the $\Lambda \gamma$ and the $\Sigma^0 \gamma$ channels. Thus, three complex numbers, determined from the hadronic interactions, describe all the initial state interactions for decays to both $\Lambda \gamma$ and $\Sigma^0 \gamma$ final states. The result of Eq. (5) is essentially Watson’s Theorem \[25\] using a particle basis. Watson’s Theorem, which also relates information about the strong interaction to that of the electromagnetic process, uses an isospin basis. The photoproduction amplitude is shown to have a phase equal to the hadronic phase shift for a given isospin. Eq. (5) reduces to this result if there is only one strong channel. In this case, $A$ is proportional to $e^{i \delta}$ where $\delta$ is the phase-shift of the strong channel. For pion-nucleon photoproduction it is useful to use an isospin basis since the T-matrix is diagonal and both the photoproduction amplitude and the hadronic phase shift can be determined from experiment. It is especially useful if one isospin dominates (i.e. the $P_{33}$). However, the T-matrix (or potential) for $K-N$, $\Sigma \pi$, $\Lambda \pi$ system is not diagonal in an isospin basis. Watson’s Theorem would apply to the eigenphases of the coupled $K-N$, $\Sigma \pi$ system for I=0, and the coupled $K-N$, $\Sigma \pi$, $\Lambda \pi$ system for I=1. Since these phases are not easily determined from experiment the results of Watson’s Theorem are not as useful in this case. Also, in the next section we point out that isospin breaking effects are very important at low $K^--p$ energies. Thus, in analyzing threshold branching ratios, a particle basis is necessary. Another advantage of using Eq. (5) is that the interference of the “Born Amplitudes” $f$ due to the initial state interactions of the hadrons is made transparent. **Results and Discussion** ========================== The potentials for the strong channels -------------------------------------- The two parts in determining the photoproduction rates in Eq. (4) are the $A_m$, which are determined from the strong part of the interaction, and the channel amplitudes $f_{m \rightarrow \Lambda \gamma (\Sigma \gamma)}$. For notation, we will label the channels 1-8 as $\pi^+ \Sigma^-$, $\pi^0 \Sigma^0$, $\pi^- \Sigma^+$, $\pi^0 \Lambda$, $K^- p$, $\overline{K^0} n$, $\Lambda \gamma$, and $\Sigma^0 \gamma$ respectively. We begin by discussing the determination of the $A_m$. These were obtained by using a separable potential and fitting to the available low energy data on the strong channels. We took $v_{i \gamma}$ in Eq. 3 to be equal to $v_i$ in Eq. 6 below. Two different separable potentials were used: one guided by $SU(3)$ symmetry for the relative channel couplings which fits all the low energy data except the [*1s*]{} atomic-level shift, and one from Ref. \[20\] which fits all the low energy data including the sign of the scattering length from the [*1s*]{} atomic-level shift. Values for the $A_m$ at the $K^-p$ threshold for each fit are listed in Table I. [[**TABLE I.**]{} The $A_i$ values from Eq. (5) for two different strong potentials. The potential with approximate SU(3) symmetry fits all low energy hadronic data except the [*1s*]{} $K^-p$ atomic level shift. The potential of Ref. \[20\] fits the atomic level shift as well. ]{} ------- ---------------------------- -------------------------- $A_i$ Potential with Approximate Potential of $SU(3)$ Symmetry Tanaka and Suzuki \[22\] $A_1$ (1.20, 0.52) (1.49, -0.28) $A_2$ (-1.02, -.14) (-1.10, 0.52) $A_3$ (0.83, -.23) (0.71, -0.75) $A_4$ (-0.16, -0.34) (-0.30, -0.39) $A_5$ (-0.15, 1.06) (2.01, 2.55) $A_6$ (1.18, -0.41) (2.08, -1.12) ------- ---------------------------- -------------------------- Following Ref. \[21\] the separable potentials for the strong channels are taken to be of the form: $$V^I_{ij}(k,k') = {{g^2} \over {4 \pi}} C^I_{ij} b^I_{ij} v_i(k) v_j(k'), \eqno(6)$$ where the $C^I_{ij}$ are determined from $SU(3)$ symmetry. The $b^I_{ij}$ are “breaking parameters” which are allowed to vary slightly from unity. The $v_i(k)$ are form factors, taken for this analysis to be equal to $\alpha_i^2 / (\alpha_i^2 + k^2)$, and [*g*]{} is an overall strength constant. These potentials are used in a coupled-channel Lippmann-Schwinger equation with a non-relativistic propagator to solve for the cross-sections to the various channels. The data used in the fit are from Refs. \[26-30\]. The resonance at an energy of 1405 MeV was also fitted. As discussed in Ref. \[21\] it was not possible to fit all the low-energy data using potentials that had $b^I_{ij}=1$ for all i and j. To get an acceptable fit without including the radiative capture data, it is necessary to vary the $b^I_{ij}$ by at least $\pm 15 \%$ from unity. To get a very good fit to all the data and determine the range of the $A_m$, we let the $b^I_{ij}$ vary from $0.5$ to $1.5$. In Table II we list the values we used for the $I=0$ and $I=1$ potentials for our “best fit”. This “best fit” also included the  radiative capture data, and is discussed in [[**TABLE II.**]{} The “best fit” values of $C^I_{ij}(b^I_{ij})$ for the potential of Eq. (6). ]{} ---------------------------------------------------------------------------------------------------------------------- $C^{I=0}_{ij}$ $\Sigma \pi$ $KN$ ---------------------------- -------------------------------- -------------------------------- ----------------------- $\Sigma \pi$ $-2$ (0.50) $-{{\sqrt{6}} \over 4}$ (1.29) $KN$ $-{{\sqrt{6}} \over 4}$ (1.29) $-{3 \over 2}$ (1.43) $C^{I=1}_{ij}$ $\Sigma \pi$ $\Lambda \pi$ $KN$ $\Sigma \pi$ $-1$ (0.50) 0 $-{1 \over 2}$ (1.37) $\Lambda \pi$ 0 0 $ {\sqrt{6} \over 4}$ $KN$ $-{1 \over 2}$ (1.37) $ {\sqrt{6} \over 4}$ $-{1 \over 2}$ (0.50) $\alpha_{\Sigma \i} = 974$ $\alpha_{\Lambda \pi} = 886$ $\alpha_{KN} = $g^2 = 1.19 fm^2$ 445 $ ---------------------------------------------------------------------------------------------------------------------- the next section. The elements are listed as a product of the $C^{I}_{ij}$ values from SU(3) times $b^I_{i}$ which was allowed to vary from $0.5$ to $1.5$. Also listed are the values for $\alpha_i$ in MeV/c and the overall strength $g^2$ from Eq. (6). We note that for this fit the $\Lambda (1405)$ is produced as a $K-N (\Sigma \pi)$ bound state resonance \[21\] (see Fig. 4). The $A_i$ are a measure of how much the initial state interactions enhance the single scattering amplitude. Not all the $A_i$ are needed in the $K^-p$ radiative decay calculation, since only channels which have charged particles contribute. Thus only $A_1$, $A_3$, and $A_5$ enter the calculation. Also due to isospin symmetry in the $\Sigma \pi$ sector $A_1$, $A_2$, and $A_3$ must satisfy the relation: $A_1 + A_3 = -2A_2$. In the absence of initial state interactions, $A_1=A_3=0$ and $A_5=1$. As can be seen in Table I, the magnitude of $A_1$, $A_3$, and $A_5$ are between 0.8 and 1.3. Since $F_{K^-p \rightarrow \Lambda \gamma} = A_1 f_{\Sigma^- \pi^+ \rightarrow \Lambda \gamma} + A_3 f_{\Sigma^+ \pi^- \rightarrow \Lambda \gamma} + A_5 f_{K^-p \rightarrow \Lambda \gamma}$, cancellations amongst the various amplitudes can make the radiative capture probability very sensitive to the initial state interactions. Unfortunately, the $A_i$ cannot be directly determined experimentally, and will have some model dependencies. We tried to estimate the model dependency for the potential of Eq. 6 by allowing the $b^I_{ij}$ to vary different amounts between 0.5 and 1.5 and see how much the $A_i$ changed. For acceptable fits to the data, excluding the atomic [*1s*]{} level shift, the $A_i$ varied only $\pm 20 \%$ in magnitude. An important aspect of the problem is to include the appropriate isospin breaking effects due to the mass differences of the particles. This was done as described in Ref. \[21\] by using the correct relativistic momenta and reduced energies in the propagator. The effects are very important in calculating the threshold branching ratios, since the masses of $\overline{K^0}-n$ are 7 MeV greater than the masses of $K^-p$. As shown in Ref.\[31\], the Coulomb potential can be neglected when calculating the branching ratios. In Fig. 1 we plot the branching ratios as a function kaon laboratory momentum $P_{Lab}$. The three different curves for each ratio correspond to different types of SU(3) breaking to be discussed later. Note that the energy dependence is particularly strong for branching ratios $\gamma$, $R_n$, $R_{\Lambda \gamma}$ and $R_{\Sigma \gamma}$. For $\gamma$, which is the ratio of $\Sigma^- \pi^+$ production to $\Sigma^+ \pi^-$ the energy dependence is easy to understand. For a model as the one presented here which does not include the $\Lambda (1405)$ as an [*s*]{}-channel resonance, the reaction $K^-p \rightarrow \pi^+ \Sigma^-$ cannot occur in a single step. This is a double charge exchange reaction, and needs to undergo two single charge exchange steps with the middle one being neutral. From the total cross-section data (See Fig. 2), the most important neutral channel in low-energy $K^-p$ scattering comes out to be $\overline{K^0} n$. This causes the ratio $\gamma$ to have a strong energy dependence near the $\overline{K^0} n$ threshold. Since the $\overline{K^0} n$ channel is also important in $\Lambda \pi$ production, the ratio $R_n$ also varies rapidly with energy near threshold. Thus for an accurate comparison with the data, one needs to use a particle basis in calculating the photoproduction branching ratios at threshold. We note that if an [*s*]{}-channel resonance was the dominating process in the $\Sigma \pi$ reaction, then the ratio $\gamma$ would not have as rapid an energy dependence near the $K^-n$ threshold. It is also interesting that $\Gamma (K^-p \rightarrow \Lambda \gamma)$ is substantially less than $\Gamma (K^-p \rightarrow \Sigma^0 \gamma)$ at energies below the $\overline{K^0} n$ threshold and greater at energies above. Experimental data of these branching ratios near the $K^-p$ threshold would help clarify the nature of the $\Lambda (1405)$. The electromagnetic channels ---------------------------- We now turn our attention to the most important part of the calculation, the amplitudes for the $\Lambda \gamma$ and $\Sigma^0 \gamma$ channels. As discussed previously, only the amplitudes for the three charged channels will contribute to the radiative capture amplitude in Eq. (5): $K^-p \rightarrow \Lambda \gamma (\Sigma^0 \gamma)$, $\Sigma^{\pm} \pi^{\mp} \rightarrow \Lambda \gamma (\Sigma^0 \gamma)$. Here we will use the amplitudes obtained from the diagrams shown in Fig. 3. These diagrams are the leading order contributions to photoproduction \[32\]. We include the most important amplitudes which are the “extended Born terms”, including the $\Lambda$ and the $\Sigma^0$ exchange terms, and the vector meson exchange terms ($K^*$, $\rho$). The expressions for these terms and their relative importance are given in Appendix I. There are four coupling constants which enter in the photoproduction amplitudes: $g_{KN \Lambda}$, $g_{KN \Sigma}$, $g_{\pi \Sigma \Sigma}$, and $g_{\pi \Sigma \Lambda}$. Since there are only two branching ratios to fit, we need to limit the method of our search. We investigated three cases: [*a*]{}) assume exact SU(3) symmetry for the coupling constants with $g_{\pi NN} = 13.4$ and vary the F-D mixing ratio $\alpha$ to best fit the data, [*b)*]{} assume $\alpha = 0.644$, $g_{\pi NN} = 13.4$ and vary the coupling constants slightly from their SU(3) values for a best fit, and [*c*]{}) use the $A_m$ from the potential of Ref. \[20\] and SU(3) symmetry for the coupling constants to see if a fit of the radiative decay branching ratios is possible. For the search, we weighted each data point equally, and thus the two radiative capture branching ratios did not have a great affect on the hadronic parameters. In the first case, we assumed exact SU(3) symmetry with $g_{\pi NN} = 13.4$ and varied the F-D mixing ratio $\alpha$ for a best fit to the data. We found an acceptable fit with a $\chi^2$ per data point of 2.47 for $\alpha = 1.0$. The branching ratios for this fit are $ \gamma$ = 2.25, $ R_c$ = 0.66 and $ R_n$ = 0.17 for the strong channels, and $ R_{\Lambda \gamma}$ =$1.22 \times 10^{-3}$ and $ R_{\Sigma \gamma}$ =$1.47 \times 10^{-3}$ for the electromagnetic channels. Although this is not the accepted value for $\alpha$, it is remarkable to get a fit with only one adjustable variable. In the next case, we fix $\alpha$ to be 0.644. The search is done using MINUIT code \[33\] on 13 parameters: the three ranges for the strong channels, the six breaking factors for the strong channels $b^I_{ij}$, $g_{Kp \Lambda}$, $g_{Kp \Sigma}$, $g_{\pi \Sigma \Sigma}$, and $g_{\pi \Sigma \lambda}$. We allowed the $b^I_{ij}$ to vary $\pm 50 \%$, $\pm 40 \%$, and $\pm 30 \%$ from unity while the 4 coupling constants $g_{Kp \Lambda}$, $g_{Kp \Sigma}$, $g_{\pi \Sigma \Sigma}$, and $g_{\pi \Sigma \lambda}$ varied by $\pm 50 \%$, $\pm 40 \%$, and $\pm 30 \%$ from their SU(3) values respectively. The range parameters $\alpha$ were allowed to vary from 200 to 1000 MeV/c. The results for the branching ratios and the reduced $\chi^2$ are listed in Table III. The first column lists the percentage that the parameters, except $g_{Kp \Lambda}$, were allowed to vary from their SU(3) values (or in the case of the $b^I_{ij}$’s from unity). The second column lists the percentage that $g_{Kp \Lambda}$ was allowed to vary from its SU(3) value of -13.2. We also tried to find a satisfactory fit in which $g_{Kp \Lambda}$ was as close to -13.2 as possible. A fit was found in which $g_{Kp \Lambda}$ was varied only $\pm 20 \%$, while the other parameters were allowed to vary $\pm 50 \%$. The first row of Table III shows these results. We call this our “best fit” since the most well determined coupling constants, $g_{KN \Lambda}$ and $g_{KN \Sigma}$ are close to their SU(3) values, with $g_{KN \Sigma}$ only $50 \%$ high. Our best fit values for the coupling constants are $g_{KN \Lambda}=-10.6$, $g_{KN \Sigma}=5.8$, $g_{\pi \Sigma \Sigma}=-7.2$, and $g_{\pi \Sigma \Lambda}=-5.0$. Notice that our values for the two first coupling constants are in agreement with those obtained from strangeness photoproduction \[7,34\] and hadronic sector \[35,36\] analyses. The electromagnetic branching ratios change drastically if the initial state interactions are excluded from the calculation. We obtain $R_{\Lambda \gamma}=.56 \times 10^{-3}$ and $R_{\Sigma \gamma} = .12 \times 10^{-3}$ without the initial state interactions. The two branching ratios are hence decreased by roughly a factor of 2 and more than one order of magnitude, respectively, by switching off the initial state interactions. Graphs of the different fits for the five branching ratios and total cross sections as a function of kaon laboratory momentum are shown in Fig. 1 and 2, respectively. In Fig. 4 we plot the $\Sigma \pi$ spectrum normalized to the data of Hemmingway \[37\]. As in Ref.\[14\], we plot $k^\pi_{c.m.} |T_{\Sigma \pi \rightarrow \Sigma \pi}|^2$, where $T_{\Sigma \pi \rightarrow \Sigma \pi}$ is the T-matrix in the I=0 sector for $\Sigma \pi \rightarrow \Sigma \pi$ scattering. In each figure, the solid line corresponds to the “best fit” parameters, the dotted line to $\pm 40 \%$ SU(3) breaking for all the parameters, and the dashed line to $\pm 30 \%$ breaking for all the parameters. In each case the $\Lambda (1405)$ is produced as a bound state resonance as in Ref \[21\]. The $K^-p$ scattering length obtained from our best fit is (-.63 + .76 i) fm. This compares closely with the value from Ref. \[32\] of (-.66 + .64 i) fm. These values, however, have the opposite sign for the real part from that extracted from the [*1s*]{} $K^-p$ atomic level shift data \[38\]. Since the atomic level shift data is still puzzling \[39\], we did not try to fit it in our search. This discrepancy has been discussed in detail in Ref. \[20\] with some interesting results. Hence, we used the $A_m$ obtained from the potential of Ref. \[20\] which fit all the hadronic low energy data and has the same sign for the scattering length as the atomic level shift data. We were able to reproduce their results using their non-relativistic potentials. From Table I we see that $A_1$ and $A_3$ do not differ too much from those obtained with the “SU(3) guided” potential. However, $A_5$ is much different in magnitude and its real part has the opposite sign. Perhaps this is because the atomic [*1s*]{} shift and hence the scattering length has the opposite sign. For the potential of \[20\] the resulting radiative capture branching ratios using coupling constants from SU(3) symmetry are $R_{\Lambda \gamma} = 17.5 \times 10^{-3}$ and $R_{\Sigma \gamma} = 3.29 \times 10^{-3}$, which are far from the experimental values. For satisfactory agreement with the radiative capture branching ratios, the coupling constants would have to deviate from their SU(3) values by an unreasonable amount. The reason for the bad agreement is that $A_5$ is very large and its real part is positive. In order to obtain a small value for $\Lambda \gamma$ production, the amplitudes have to cancel in Eq. (5). Since the relative signs of the $f_{m \rightarrow \Lambda \gamma (\Sigma^0 \gamma)}$ are fixed by SU(3) symmetry, the $A_m(\sqrt{s})$ have to have appropriate relative phases to cause this cancellation. The $A_m$ from the potential guided by SU(3) symmetry have this feature. [[**TABLE III.**]{} Branching ratios and $\chi^2$ per data point for different amounts of SU(3) breaking. Column 2 lists the variation in the coupling constant $g_{Kp \Lambda}$. Column 1 lists the variation in the other parameters. ]{} All except $g_{Kp \Lambda}$ $g_{Kp \Lambda}$ $\chi^2/N$ $\gamma$ $R_c$ $R_n$ $R_{\Lambda \gamma} \times 10^{3}$ $R_{\Sigma \gamma} \times 10^{3}$ ----------------------------- ------------------ ------------ ---------------- ----------------- ----------------- ------------------------------------ ----------------------------------- $\pm 50 \%$ $\pm 20 \%$ 1.76 2.31 .661 .164 1.09 1.55 $\pm 50 \%$ $\pm 50 \%$ 1.21 2.35 .659 .194 0.89 1.46 $\pm 40 \%$ $\pm 40 \%$ 1.54 2.32 .659 .179 1.04 1.53 $\pm 30 \%$ $\pm 30 \%$ 2.94 2.20 .652 .174 1.31 1.65 Experiment 2.36 $\pm$ .04 .664 $\pm$ .011 .189 $\pm$ .015 .86 $\pm$.07 1.44 $\pm$ .20 **Conclusions** =============== We have done a comprehensive analysis of all the low energy data, except the [*1s*]{} atomic level shift, on the $K^-p$ system. To facilitate the analysis, we derived an expression for the radiative capture cross section which separates out the strong interaction from the electromagnetic ones. The initial state interactions can be described by six complex amplitudes, $A_m$, with only three of them relevant to the radiative capture process. For the strong part of the interaction we choose a separable potential whose relative potential strengths were guided by SU(3) symmetry. This potential is phenomenological and serves to produce appropriate $A_m$ from the low energy scattering and resonance data. The radiative capture amplitudes are derived from first order “Born” photoproduction processes, and are determined from meson-baryon-baryon coupling constants, whose values are related by SU(3) symmetry to the well known $\pi NN$ coupling constant. We found a number of good fits in which the coupling constants were close to their expected SU(3) values. For these fits, the relative coupling strengths in the strong channels were guided by $SU(3)$ symmetry. In all of the fits, the $\Lambda (1405)$ is produced as a bound $K-N (\Sigma \pi)$ resonance, and the initial state interactions were very important for the radiative capture branching ratios. The ratio $R_{\Lambda \gamma}$ varies roughly by a factor of 2, and the ratio $R_{\Sigma \gamma}$ by more than a factor of 10 due to the initial state interactions. Results presented in this paper, reproduce well enough the existing strong and electromagnetic data from threshold up to $P_{K}^{\rm lab} \approx$ 200 MeV/c. Our predictions, specially for the branching ratios, show clearly the need for more experimental investigations ; one of the main motivations being to clarify the nature of the $\Lambda (1405)$ resonance. Such measurements are planned at DA$\Phi$NE \[40\] using the tagged low energy kaon beam and may also be achieved at Brookhaven and KEK. **Acknowledgements** ==================== We would like to thank J.C. David, C. Fayard, G.H. Lamot, Andreas Steiner and Wolfram Weise for many helpful discussions and suggestions regarding this work. We are grateful to the Institute for Nuclear Theory (Seattle), for an stimulating and pleasant stay, where the idea of this collaboration emerged. One of us (PS) would like to thank the Centre d’Etudes de Saclay for the hospitality extended to him. \ In this Appendix we will summarize the contributions to the photoproduction amplitudes shown in Fig. 3. Here we will write the expressions for the $K^- p \rightarrow \Lambda \gamma$ amplitude. The amplitudes for the $K^- p \rightarrow \Sigma^0 \gamma$, $\Sigma^{\pm} \pi^{\mp} \rightarrow \Lambda \gamma$, and $\Sigma^{\pm} \pi^{\mp} \rightarrow \Sigma^0 \gamma$ processes will be the same with appropriate masses and coupling constants. To lowest order the amplitude for the $K^-p \rightarrow \Lambda \gamma$ reaction f is the sum of three amplitudes: $$f_{K^-p \rightarrow \Lambda \gamma} = F_{Born} + F_\Sigma + F_{K^*},$$ which correspond to the the Born, the $\Sigma^0$, and the $K^*$ diagrams shown in Fig 3. The Born amplitude is derived in Ref. \[32\] and is given by: $$F_{Born} = - \sqrt{{E_{\Lambda} + m_{\Lambda}} \over {2 m_{\Lambda}}} {{g_{Kp \Lambda} e} \over {2 m_p}} (1 + {{k_{\gamma}} \over {E_{\Lambda}+m_{\Lambda}}} (1 + \kappa_p + \kappa_{\Lambda})),$$ at the $K^-p$ threshold. The “$\Sigma$” term is also derived in Ref. \[32\] and is given by: $$F_{\Sigma} = - \sqrt{{E_{\Lambda} + m_{\Lambda}} \over {2 m_{\Lambda}}} {{g_{Kp \Sigma} e} \over {2 m_p}} \kappa_{\Sigma \Lambda} {{\sqrt{s}-m_\Lambda} \over {\sqrt{s} + m_\Sigma}}.$$ In a similar manner, the $K^*$ exchange term can be evaluated. In this case, there is a vector and a tensor piece. We obtain for the amplitude: $$F_{K^*} = - \sqrt{{E_{\Lambda} + m_{\Lambda}} \over {2 m_{\Lambda}}} [ {{g^V_{K^*p \Lambda} e} \over {2 m_p}} {{\kappa_{K^*K} k^2_\gamma (\sqrt{s}-m_p)} \over {(t - m^2_{K^*})(E_\Lambda + m_\Lambda)}} + {{g^T_{K^*p \Lambda} e} \over {2 m_p}} {{\kappa_{K^*K} k^2_\gamma (\sqrt{s}-m_p)} \over {(t - m^2_{K^*})2 m_p}} ({{m_\Lambda + m_p} \over {m_\Lambda + E_\Lambda}})].$$ In the absence of initial state interactions, the differential cross section is given by: $${{d \sigma} \over {d \Omega}} = {{(E_\Lambda + m_\Lambda) (E_p + m_p)} \over {64 \pi^2 s}} {{P_\gamma} \over {P_K}} |f_{K^-p \rightarrow \Lambda \gamma}|^2.$$ The first part of $F_{Born}$ has the largest magnitude. The other pieces are reduced by kinematical factors, with $F_{K^*}$ giving the smallest contribution. The $K^*$ exchange makes up about $2 \%$ of the amplitude. Thus, the uncertainty in the vector and tensor coupling constants are not so important, and we fixed them to be the SU(3) values. Also, since the calculation is not particularly sensitive to the values of the electromagnetic couplings, we fixed them at their excepted values. Values for the constants which were held fixed during the search are listed in Table IV. [[**TABLE IV.**]{} Values of the coupling constants which were held constant. ]{} --------------------------------- ------------------------------------ $\kappa_p = 1.793$ $g^V_{K^*p \Lambda} = -4.5 $ $\kappa_\Lambda = -.613$ $g^T_{k^*p \Lambda} = -16.6 $ $\kappa_{\Sigma \Lambda} = 1.6$ $g^V_{K^*p \Sigma} = -2.6 $ $\kappa_{K^*K} = 1.58$ $g^T_{K^*p \Sigma} = 3.2 $ $\kappa_{\rho \pi} = 1.41$ $g^V_{\rho \Sigma \Lambda} = 0$ $\kappa_{\Sigma^-} = -2.157$ $g^T_{\rho \Sigma \Lambda} = 11.1$ $\kappa_{\Sigma^+} = 1.42$ $g^V_{\rho \Sigma \Sigma} = -5.2$ $\kappa_{\Sigma^0} = .619$ $g^T_{\rho \Sigma \Sigma} = -12.8$ --------------------------------- ------------------------------------ The search was done on the more important coupling constants, $g_{K p \Lambda}$, $g_{K p \Sigma}$, $g_{\pi \Sigma \Sigma}$, and $g_{\pi \Sigma \Lambda}$. At the $K^-p$ threshold, using the coupling constants of Table III, the radiative capture amplitudes are: $$f_{K^-p \rightarrow \Lambda \gamma} = {e \over {2 m_p}} (-g_{Kp \Lambda}(1.28) - g_{Kp \Sigma}(.2) -.75),$$ $$f_{K^-p \rightarrow \Sigma^0 \gamma} = {e \over {2 m_p}} (-g_{Kp \Sigma}(1.32) - g_{Kp \Lambda}(.15) + .02),$$ $$f_{\pi^+ \Sigma^- \rightarrow \Lambda \gamma} = {e \over {2 m_\Sigma}} (-g_{\pi \Sigma \Lambda}(.835) + g_{\pi \Sigma \Sigma}(.19) + .21),$$ $$f_{\pi^+ \Sigma^- \rightarrow \Sigma^0 \gamma} = {e \over {2 m_\Sigma}} (-g_{\pi \Sigma \Sigma}(.95) + g_{\pi \Sigma \Lambda}(.153) + .19),$$ $$f_{\pi^- \Sigma^+ \rightarrow \Lambda \gamma} = {e \over {2 m_\Sigma}} (g_{\pi \Sigma \Lambda}(1.16) - g_{\pi \Sigma \Sigma}(.19) -.21),$$ $$f_{\pi^- \Sigma^+ \rightarrow \Sigma^0 \gamma} = {e \over {2 m_\Sigma}} (-g_{\pi \Sigma \Sigma}(1.28) + g_{\pi \Sigma \Lambda}(.153) + .19),$$ where the three terms in parenthesis correspond to the three amplitudes described above. The above equations show the relative importance of the different contributions to radiative capture at the $K^-p$ threshold. The best fit values for these coupling constants are summarized in Table V. [[**TABLE V.**]{} “Best fit” values of the coupling constants for the electromagnetic amplitudes. ]{} -------------------------------- --------------------------------- $g_{Kp \Lambda} = -10.6$ $g_{Kp \Sigma} = 5.8$ $g_{\pi \Sigma \Sigma} = -7.2$ $g_{\pi \Sigma \Lambda} = -5.0$ -------------------------------- --------------------------------- \  \[1\] See, e.g., A.J.G. Hey and R.L. 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--- abstract: 'We examine dark energy models in which a phantom field $\phi$ is rolling near a local minimum of its potential $V{\left(}\phi{\right)}$. We require that $(1/V)(dV/d\phi) \ll 1$, but $(1/V)(d^2 V/d\phi^2)$ can be large. Using techniques developed in the context of hilltop quintessence, we derive a general expression for $w$ as a function of the scale factor, and as in the hilltop case, we find that the dynamics of the field depend on the value of ${\left(}1/V{\right)}{\left(}d^2 V/d\phi^2{\right)}$ near the mimimum. Our general result gives a value for $w$ that is within 1% of the true (numerically-derived) value for all of the particular cases examined. Our expression for $w(a)$ reduces to the previously-derived phantom slow-roll result of Sen and Scherrer in the limit where the potential is flat, $(1/V)(dV/d\phi) \ll 1$.' author: - 'Sourish Dutta and Robert J. Scherrer' title: Dark Energy from a Phantom Field Near a Local Potential Minimum --- Introduction ============ Considerable evidence [@Knop; @Riess1] has accumulated suggesting that at least 70% of the energy density in the universe is in the form of an exotic, negative-pressure component, called dark energy. (See Ref. [@Copeland] for a recent review). A parameter of considerable importance is the equation of state (EoS) of the dark energy component, defined as the ratio of its pressure to its density: w=p\_[DE]{}/\_[DE]{}. Observations constrain $w$ to be very close to $-1$. If $w$ is assumed to be constant, then $-1.1 {\ {\raise-.5ex\hbox{$\buildrel<\over\sim$}}\ }w {\ {\raise-.5ex\hbox{$\buildrel<\over\sim$}}\ }-0.9$ [@Wood-Vasey; @Davis]. On the other hand, a variety of models have been proposed in which $w$ is time varying. A common approach is to use a scalar field as the dark energy component. The class of models in which the scalar field is canonical is dubbed quintessence [@RatraPeebles; @TurnerWhite; @CaldwellDaveSteinhardt; @LiddleScherrer; @SteinhardtWangZlatev] and has been extensively studied. A related, yet somewhat different approach is phantom dark energy, i.e., a component for which $w<-1$. The simplest way to realize such a component is to use a scalar field with a negative kinetic term in its Lagrangian, as first proposed by Caldwell [@Caldwell]. Such models have well-known problems [@CarrollHoffmanTrodden; @ClineJeonMoore; @BuniyHsu; @BuniyHsuMurray], but nevertheless have been widely studied as potential dark energy candidates [@Caldwell; @Guo; @ENO; @NO; @Hao; @Aref; @Peri; @Sami; @Faraoni; @Chiba; @KSS]. Given the considerable freedom that exists in choosing the potential function of the scalar field, it is useful to develop general expressions for the evolution of $w$ which cover a wide range of models. One such approach is to begin with the observational result that $w$ for dark energy is very close to $-1$ today, and to assume that $w$ was always close to $-1$ in the redshift regime of interest. With this assumption, one can assume an expanding background that is very close to $\Lambda$CDM and solve for the evolution of the scalar field for this case. This assumption alone is not sufficient to derive a general solution for the evolution of the scalar field or its equation of state. However, certain fairly general classes of such models can be solved exactly. The simplest such models, explored in Ref. [@ScherrerSen1], assume a scalar field initially at rest in a potential satisfying the “slow-roll" conditions: \[SR1\]$$\frac{1}{V}\frac{dV}{d\phi}$$\^21,\ \[SR2\]||1. The first condition insures that $w$ is close to $-1$, while the two conditions taken together indicate that $(1/V)(dV/d\phi)$ is nearly constant. In the terminology of Ref. [@CL], these are “thawing" models. For all potentials satisfying these conditions, it was shown in [@ScherrerSen1] that the behavior of $w$ can be accurately described by a unique expression depending only on the present-day values of $\Omega_\phi$ and the initial value of $w$. In [@ScherrerSen2] this result was extended to phantom models satisfying Eqs. (\[SR1\]-\[SR2\]), and the $w$ dependence of these phantom models was shown to be described by the same expression as in the quintessence case. The slow roll conditions, Eqs. (\[SR1\]-\[SR2\]), while sufficient to ensure $w\simeq-1$ today, are not necessary. In [@DuttaScherrer], a second possibility was considered, in which equation (\[SR1\]) holds, but equation (\[SR2\]) is relaxed. This corresponds to a quintessence field rolling near a local maximum of its potential. As in the case of slow-roll quintessence, this case can be solved analytically. In this case, there is an extra degree of freedom, the value of $(1/V)(d^2 V/d\phi^2)$, so that instead of a single solution for the evolution of $w$, one obtains a family of solutions that depend on the present-day values of $\Omega_\phi$ and $w$ and the value of $(1/V)(d^2 V/d\phi^2)$ at the maximum of the potential. This family of solutions includes the slow-roll solution as a special case in the limit where $(1/V)(d^2 V/d\phi^2) \rightarrow 0$. Here we complete this series of studies by extending the above result to phantom dark energy models. Since phantom fields roll up their potentials, the analogous situation is a phantom rolling close to a local minimum in its potential. We find that a unique family of solutions, very similar to the one derived in Ref. [@DuttaScherrer], can be used to approximate the behavior of $w$ in these models. Phantom evolution near a minimum {#wexp} ================================ In this section we derive a general expression for the evolution of $w$ for a phantom field near its minimum. Our treatment is similar to [@DuttaScherrer]. First consider a minimally coupled phantom field $\phi$ in a potential $V{\left(}\phi{\right)}$. The phantom field has a negative kinetic term in its Lagrangian, leading to an equation of motion \[KG\] +3H-=0, where $a$ is the scale factor and $H\equiv\dot{a}/a$ is the expansion rate. Dots denote derivatives with respect to time and primes denote derivatives with respect to the field $\phi$. In a flat universe, the expansion rate is linked to the total density $\rho_{\rm T}$ via the Friedman equation (in units where $8\pi G=1$) as H\^2=\_[T]{}/3. The evolution of the scale factor is given by: =-$\rho_{\rm T}+p_{\rm T}$, where $p_{\rm T}$ is the total pressure Using the transformation (t)=u(t)/a(t)\^[3/2]{}, Eq. (\[KG\]) becomes \[KGnew\] +p\_[T]{}u-a\^[3/2]{}V’$u/a^{3/2}$=0. Now consider a universe consisting of pressureless matter and a phantom field, where the phantom plays the role of the dark energy. In order to realistically mimic the observed dark energy, the phantom must have $w$ close to $-1$ and its energy density must be roughly constant. The total pressure $p_{\rm T}$ is then simply given by $p_{\rm T}\approx -\rho_{\phi0}$, where $\rho_{\phi0}$ is the present day density of the dark energy. (In what follows, a subscript $0$ always indicates a present day value). Under this approximation, [Eq. (\[KGnew\])]{} becomes: \[KGu\] -\_[0]{}u-a\^[3/2]{}V’$u/a^{3/2}$=0. We now apply [Eq. (\[KGu\])]{} to a phantom rolling near a local minimum $\phi_{*}$ in its potential. For any $\phi$ close to the minimum, the potential can be expanded in a Taylor series: V$\phi$=V$\phi_{*}$+$1/2$V”$\phi_{*}$$\phi-\phi_*$\^2+O${\left(}\phi-\phi_*$\^3[)]{}. Substituting the above expansion into [Eq. (\[KGu\])]{}, and taking $V{\left(}\phi_{*}{\right)}=\rho_{\phi 0}$ we obtain \[unew\] -$$V''{\left(}\phi_{*}{\right)}+{\left(}3/4{\right)}V{\left(}\phi_{*}{\right)}$$u=0. With the definition \[k\] k, we obtain the general solution to [Eq. (\[unew\])]{} to be u=A$kt$+B$kt$. The requirement of $w\approx -1$ implies that the potential term dominates the kinetic term in the phantom’s energy density. In such a scenario the evolution of the scale factor can be well-approximated by $\Lambda$CDM (see e.g. [@gron]): \[LCDM a\] a$t$=$$\frac{1-\Omega_{\phi 0}}{\Omega_{\phi 0}}$$\^[1/3]{} \^[2/3]{}$t/t_\Lambda$, where $\Omega_{\phi 0}$ is the present-day value of the phantom density parameter $\Omega_{\phi}$ and $a=1$ at present. The time $t_{\Lambda}$ is defined as t\_2/=2/. These results give the general solution to equation [Eq. (\[KG\])]{} (under the approximations described above) as: $t$=$$\frac{1-\Omega_{\phi 0}}{\Omega_{\phi 0}}$$\^[1/2]{}, where $A$ and $B$ are arbitrary constants. If we require $\phi{\left(}t=0{\right)}=\phi_{i}$, then $B=0$ and \[phishort\] =. This is identical to the evolution of $\phi(t)$ for quintessence near a local maximum in the potential [@DuttaScherrer]. The EoS parameter $w$ for a phantom is given by \[wshort\] 1+w=-. Our requirement that $w\approx-1$ implies that $\rho_\phi\approx\rho_{\phi 0}\approx V{\left(}\phi_{*}{\right)}$. This, together with [Eq. (\[phishort\])]{} and [Eq. (\[wshort\])]{} gives an expression for $w(a)$ (normalized to $w_0$, the present-day value of $w$): \[final EOS\] 1+w(a)=(1+w\_0)a\^[-3]{} , where $t(a)$ and $t_0$ can be derived from Eq. (\[LCDM a\]): $$\label{ta} t(a) = t_\Lambda \sinh^{-1}\sqrt{\left(\frac{\Omega_{\phi 0} a^3} {1-\Omega_{\phi 0}}\right)}$$ and $$\label{t0} t_0 = t_\Lambda \tanh^{-1} \left(\sqrt{\Omega_{\phi 0}}\right).$$ For convenience, we now switch to the constant $K\equiv kt_{\Lambda}$. In terms of the phantom potential, $K$ can be written as $$\label{Kdef} K = \sqrt{1 + (4/3)V^{\prime \prime}(\phi_*)/V(\phi_*)}.$$ indicating that $K$ (which is always $>1$ since $V{\left(}\phi_*{\right)}>0$ and $V''{\left(}\phi_*{\right)}>0$) depends only on the value of the potential and its second derivative at its minimum. In terms of $K$ we can express the evolution of $w$ in the following form: $$\label{finalfinal} 1 + w(a) = (1+w_0)a^{3(K-1)}\frac{[(F(a)+1)^K(K-F(a)) +(F(a)-1)^K(K+F(a))]^2} {[(\Omega_{\phi0}^{-1/2}+1)^K(K-\Omega_{\phi0}^{-1/2}) +(\Omega_{\phi0}^{-1/2}-1)^K (K+\Omega_{\phi0}^{-1/2})]^{2}},$$ where $F(a)$ is given by F(a) = . (Note that $F(a) = 1/\sqrt{\Omega_\phi(a)}$, where $\Omega_\phi(a)$ is the value of $\Omega_\phi$ as a function of redshift, so that $F(a=1) = \Omega_{\phi0}^{-1/2}$.) [Eq. (\[finalfinal\])]{} is our central result. We therefore find that, in the limit where $w$ is slightly less than $-1$ (i.e., where the phantom potential energy dominates its kinetic energy), all phantom models with a given value of $w_0$ and $\Omega_{\phi 0}$ converge to a unique evolution of $w$ characterised by the value of $K$ (which depends on $V''{\left(}\phi_*{\right)}/V{\left(}\phi_*{\right)}$, i.e., loosely speaking, on the curvature of the potential at its minimum). For phantom potentials which are very flat, i.e., which satisfy both slow-roll conditions, a similar analysis was shown to yield only a single form of $w(a)$ (Eqn. (18) of [@ScherrerSen2]). [Eq. (\[finalfinal\])]{} captures the more complex behavior introduced by the dependence of $w(a)$ on $K$. It is straightforward to show analytically that [Eq. (\[finalfinal\])]{} goes over to Eqn. (18) of [@ScherrerSen2] in the slow-roll limit, $K\rightarrow1$. Note that [Eq. (\[finalfinal\])]{} has an identical functional form to the corresponding expression for $w(a)$ for quintessence models derived in Ref. [@DuttaScherrer]. The only difference for the phantom models is that $1+w_0 < 0$, producing a corresponding $w(a) < -1$. Of course, this is a significant difference in deriving observational constraints on the models. Further, there is a sign difference between the definition of $K$ for phantom models (Eq. \[Kdef\]) and for quintessence models [@DuttaScherrer], corresponding to the fact that here we are dealing with a minimum in the potenial, rather than a maximum. Interestingly, [Eq. (\[finalfinal\])]{} reduces to simple polynomial forms if $K$ is a small integer. Some of these forms are given in [@DuttaScherrer]. While it would require fine tuning of the potential for $K$ to be equal to a small integer, these forms are still useful in obtaining qualitative insight into the behavior of $w{\left(}a{\right)}$ as a function of $K$. Comparison to exact solutions ============================= We now turn to comparing our analytic expression for the evolution of $w$ to the numerically computed exact evolution for a few different models. In each case we have a perfect fluid dark matter and a phantom field $\phi$ dynamical dark energy. We now consider three different phantom potentials which have local minima. The phantom analog of the PNGB model [@Frieman], has a potential given by $$\label{PNGB} V(\phi) = \rho_{\phi 0}+M^4 {\left[}1-\cos{\left(}\phi/f{\right)}{\right]},$$ where $M$ and $f$ are constants. Other models with a local minimum in the potential include the Gaussian potential, $$\label{Gaussian} V(\phi) = \rho_{\phi 0}+M^4 {\left[}1-e^{-\phi^2/\sigma^2}{\right]},$$ and the quadratic potential $$\label{quadratic} V(\phi) = \rho_{\phi 0}+ V_2\phi^2.$$ where $\sigma$ and $V_2$ are constants. We set initial conditions deep within the matter-dominated regime. We choose $w_0 = -1.1$ so that our results will give an upper bound on the error in our approximation for $-1 > w_0 > -1.1$. The value of the potential at the minimum, $V{\left(}\phi_{*}{\right)}$, is chosen to be equal to the energy of the cosmological constant. The initial value of the field $\phi_i$ is taken to be slightly displaced from its minimum $\phi_*$, and is fixed to give $w_0=-1.1$. The initial velocity of the field is taken to be zero. As discussed above, our formalism applies to phantom models for which [Eq. (\[SR1\])]{} is satisfied, but [Eq. (\[SR2\])]{} is not. The latter clearly holds when $K\simeq1$ but is violated as $K$ departs from $1$. For our specific examples, we focus on the cases $K=3$ and $K=4$. For $K=3$, for all the potentials, $$\begin{aligned} {\left[}\frac{1}{V}\frac{dV}{d\phi}{\right]}_{a\rightarrow0}^2&\simeq& O{\left[}10^{-1}{\right]}\\ \left|\frac{1}{V}\frac{d^2 V}{d\phi^2}\right|_{a\rightarrow0}&\simeq&O{\left[}1{\right]}\end{aligned}$$ For $K=4$, for all the potentials, $$\begin{aligned} {\left[}\frac{1}{V}\frac{dV}{d\phi}{\right]}_{a\rightarrow0}^2&\simeq& O{\left[}10^{-2}{\right]}\\ \left|\frac{1}{V}\frac{d^2 V}{d\phi^2}\right|_{a\rightarrow0}&\simeq&O{\left[}10{\right]}\end{aligned}$$ In Figs. (\[quad-fig\]-\[gaussian-fig\]), the evolution of $w$ from [Eq. (\[finalfinal\])]{} is shown in comparison to the exact evolution for the three different models. The agreement, in all three cases, between Eq. (\[finalfinal\]) and the exact numerical evolution is excellent, with errors $\delta w \alt 0.2\%$ for the quadratic potential, $\delta w \alt 0.6\%$ for the PNGB potential, and $\delta w \alt 0.8\%$ for the Gaussian potential. We note that the accuracy is larger for the quadratic potential than it is for the other two potentials. This is expected, since the derivation of [Eq. (\[finalfinal\])]{} was based on the quadratic potential. This is however not the case for hilltop quintessence [@DuttaScherrer], where in spite of a similar derivation, the highest accuracy was for the PNGB potential. This is most likely due to an accidental cancellation of errors. Finally, in Figs. \[K3-fig\] and \[K4-fig\], we use our approximation (Eq. ) to construct a $\chi^2$ likelihood plot for $w_0$ and $\Omega_{\phi0}$ with $K = 3, 4$, using the recent Type Ia Supernovae standard candle data (ESSENCE+SNLS+HST from [@Davis]). Clearly, these models are not ruled out by current supernova data. As in the case of hilltop quintessence [@DuttaScherrer], we find that a larger $K$ increases the size of the allowed region. [Eq. (\[finalfinal\])]{} therefore allows one to map the $w(a)$ behavior of a wide range of phantom dark energy models on to a common model-independent evolution. Clearly, this result is not well described by popular linear (CPL) parameterization of $w$ [@Chevallier:2000qy; @Linder:2002et], i.e., $w(a)=w_0+w_1(1-a)$, except in the special case of $K\rightarrow 1$. However, since our treatment demonstrates that the evolution of $w$ is completely generic to any model satisfying [Eq. (\[SR1\])]{}, and depends only on $\Omega_{\phi 0}$, $w_0$ and the curvature of the potential at its minimimum ($K$), [Eq. (\[finalfinal\])]{} allows for an excellent parameterization of these models in terms of $\Omega_{\phi 0}$, $w_0$ and $K$. Such a parameterization is particularly interesting in the light of recent work implying that the CPL parameterization does not satisfactorily fit the SN+BAO+CMB data simultaneously at both low and high redshifts [@Shafieloo:2009ti]. Subsequent investigations have shown that [Eq. (\[finalfinal\])]{} has a much wider applicability than the models described in this work and [@DuttaScherrer]. It has been found to describe quintessence (phantom) models in which the field rolls into a minumum (maximum) of the potential, leading to a richer set of behaviors including oscillatory solutions [@Dutta:2009yb]. In [@Chiba] it was shown that [Eq. (\[finalfinal\])]{} is not limited to models evolving near extrema of their potentials and can be applied to a broader class of slow-roll quintessence models. [@Chiba:2009nh] showed that [Eq. (\[finalfinal\])]{} also applies to some slow-roll k-essence models. Conclusion ========== Using techniques previously applied to quintessence, we have derived a general expression for the evolution of $w$, which is valid for a wide class of phantom dark energy models in which is the field is rolling close to a local minimum. Such models provide a mechanism to produce a value of $w$ that is slightly less than $-1$. We have tested our expression against the (numerically determined) exact evolution for three different models and in each case it replicated the exact evolution studied with an accuracy greater than $1\%$. A comparison between our generic approximation and the observational data indicates that these models are allowed by SNIa data, and that the size of the allowed region increases with the curvature ($V''/V$) of the potential at the minimum. We thank Emmanuel Saridakis for pointing out a typo in one of our equations. R.J.S. was supported in part by the Department of Energy (DE-FG05-85ER40226). 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--- abstract: 'In this paper we present a new error bound on sampling algorithms for frequent itemsets mining. We show that the new bound is asymptotically tighter than the state-of-art bounds, i.e., given the chosen samples, for small enough error probability, the new error bound is roughly half of the existing bounds. Based on the new bound, we give a new approximation algorithm, which is much simpler compared to the existing approximation algorithms, but can also guarantee the worst approximation error with precomputed sample size. We also give an algorithm which can approximate the top-$k$ frequent itemsets with high accuracy and efficiency.' author: - | Shiyu Ji, Kun Wan\ {shiyu,kun}@cs.ucsb.edu bibliography: - './cs273.bib' title: '**An Asymptotically Tighter Bound on Sampling for Frequent Itemsets Mining**' --- Introduction ============ Frequent Itemsets (FI) mining has been popular in research recently [@AIS93; @HCX07; @RU15]. The goal of FI mining is to find out the items that most frequently appear in the observed transactions, e.g., the researchers who are the most prolific in writing papers with others, the patterns that appear frequently in long pieces of genetic code, etc. In the era of big data, to compute the exact frequencies can be very time consuming. Thus in many cases approximate values are also acceptable [@AIS93; @PCY95; @FSG99; @HCX07; @LRU14; @RU15]. For FI mining in large scale transactional datasets, we often take samplings on the transactions, and compute the frequencies of the itemsets among the sampled transactions as approximate results of their true frequencies among all the transactions. Usually the sampling size is much less than the scale of all the transactions, and the approximations can achieve acceptable precision. Also in reality we often only want to know the most frequent itemsets without the need of their actual frequencies, and there are already many works [@AIS93; @PCY95; @FSG99; @HCX07; @LRU14] on this area. Thus FI approximation can be useful in practice. The state-of-art progressive sampling based FI approximation algorithms [@RU15] need an upper bound of the approximation error for the worst case, i.e., the maximum error the algorithm can generate among all the items. The algorithms keep taking new samples until the upper bound is less than the acceptable threshold. Hence how to bound the maximum error as tightly as possible is an interesting problem. The current bounds use some results of Rademacher average in statistical learning theory [@Vap98; @Vap13; @BBL04; @BBL05]. However, we find that based on the ideas given by [@BBL04; @Toi96], we can develop a new upper bound *without* Rademacher average. We also find that this new bound is asymptotically tighter than the existing bounds, i.e., given the chosen samples, as the allowed error probability approaches zero, the new bound is roughly only half of the existing ones. This implies that by using the new bound, a progressive sampling based FI approximation algorithm can reach the guaranteed accuracy with much fewer samples. We also notice that there is no parameter in the new bound that needs to be progressively computed. Hence the sample size that will guarantee the worst error can be precomputed. Based on the similar idea, we also consider the top-$k$ FI mining problem, which seeks for the $k$ most frequent itemsets in the observed ones. We need to decide when the sampling should stop. The number of the sampled transactions is enough if the worst-case error upper bound is less than the frequency gap between the $k$-th and the $(k+1)$-th most frequent itemsets. Hence we propose a progressive approximation algorithm to address the top-$k$ FI mining problem. [**Our Contributions**]{}. We give a worst-case error upper bound that is asymptotically tighter than the state-of-art bounds, and propose an approximation algorithm which can guarantee the worst-case error upper bound with precomputed sample size. We also give a progressive sampling algorithm to find the top-$k$ most frequent itemsets. Combining with existing methods, our algorithms can approximate the frequent itemsets accurately and efficiently. The rest of this paper is organized as follows. Section \[sec:rw\] reviews the related research works. Section \[sec:prlm\] introduces the notations and preliminaries throughout this paper. Section \[sec:refine\] gives the worst-case error upper bound without Rademarcher average and compares it with the existing ones. Section \[sec:algs\] proposes our approximation algorithms based on our upper bounds. Section \[sec:eval\] gives our evaluation results, which compare our algorithms with the state-of-art. Related Works {#sec:rw} ============= Frequent Itemset Mining has been very popular in the communities of information retrieval and data mining [@LRU14]. Unsurprisingly, many algorithms that can compute the exact frequencies have been proposed, e.g., A-Priori algorithm [@AIS93], Park-Chen-Yu’s algorithm [@PCY95], Multistage algorithms [@FSG99]. However it is very challenging to deal with large scaled data sets with limited main memory. Thus the classical exact algorithms may not fit well in practice. As a result, how to approximate the frequent itemsets by sampling has become interesting, since usually the sample size is much less than the entire data scale. Toivonen [@Toi96] was among the first to study sampling on FI approximation, and suggested the first worst-case error bound on frequencies. However his algorithm did not directly use the bound and still needed to parse all the dataset. Thus for scenarios like streams where the size of dataset is unbounded, we cannot use Toivonen’s algorithm directly. Sampling-based frequent itemset approximation has been studied extensively by the researchers. The first works on this problem used heuristic methods to progressively approximate the frequencies [@CHS02; @CCY05; @Parth02]. There were no guarantee on the worst-case error upper bound. To fix this, Riondato and Upfal were the first to propose FI approximation algorithms that could guarantee the worst error bounds by using results of Vapnik-Chervonenkis (VC) dimension [@RU12; @RU14] and Rademacher average [@RU15]. Note that in statistical learning theory VC dimension and Rademacher average are usually used to address the worst-case error upper bound for *infinite case*, i.e., the number of possible functions in the learning model is infinite [@BBL04]. However in the case of FI mining, since there are only *finite* itemsets, it is possible to develop bounds without VC dimension or Rademacher average [@BBL04]. In this paper we apply this idea on FI mining problem. Riondato et al. also considered using parallelism in FI mining [@RDF12], which is an orthogonal topic to sampling-based FI approximation. In practice we are often only interested in the most frequent itemsets. Thus top-$k$ FI mining is a popular research topic with many research works [@PRU10; @SW02; @RU15; @RV14]. Another interesting question is to find all itemsets with frequencies larger than a threshold. Savasere, Omiecinski, and Navathe [@SON95] give an two-pass algorithm (called SON algorithm) that can find the exact solutions. We will use SON algorithm to significantly reduce the number of itemsets to be observed, and then apply our algorithms to approximate the frequencies and select the top $k$ ones. Also Toivonen’s Algorithm [@Toi96] is an alternative way to find the most frequent itemsets given a threshold. Preliminaries {#sec:prlm} ============= Frequency of Itemset -------------------- In this paper we use the notations and definitions from Riondato and Upfal’s pioneering work [@RU15]. Let ${\mathcal{I}}$ be the set of items. A transaction $\tau$ is a subset of ${\mathcal{I}}$ (i.e., $\tau \subseteq {\mathcal{I}}$). An itemset $A$ is a set of items that appear together in a transaction $\tau$, i.e., $A \subseteq \tau$. Clearly any itemset is also a subset of ${\mathcal{I}}$. Let transactional dataset ${\mathcal{D}}$ be the set of all the transactions. In this paper we always assume ${\mathcal{D}}$ is a finite set. Denote by $T_{\mathcal{D}}(A)$ the set of all the transactions in ${\mathcal{D}}$ that contain the itemset $A$. $T_{\mathcal{D}}(A)$ is also known as the support set of $A$ in ${\mathcal{D}}$. If ${\mathcal{D}}$ is a finite set, we can define the frequency of itemset $A$ in ${\mathcal{D}}$ as the fraction of transactions in ${\mathcal{D}}$ that contain $A$. $$f_{\mathcal{D}}(A) = |T_{\mathcal{D}}(A)|/|{\mathcal{D}}|.$$ Clearly $0 \leq f_{\mathcal{D}}(A) \leq 1$ for any $A \subseteq {\mathcal{I}}$. The goal of our sampling algorithm is to approximate $f_{\mathcal{D}}(A)$ given an itemset $A$ as accurately as possible. Approximation Algorithms ------------------------ An $(\epsilon,\delta)$-approximation algorithm of the frequencies $f_{\mathcal{D}}(\cdot)$ takes as input all the items ${\mathcal{I}}$ and outputs a sampled average $f_{\mathcal{S}}(A)$ for each $A\subseteq{\mathcal{I}}$ such that with probability at least $1-\delta$, $$\max_{A\subseteq{\mathcal{I}}}|f_{\mathcal{D}}(A) - f_{\mathcal{S}}(A)| \leq \epsilon.$$ We often use progressive sampling [@RU15; @RU16], i.e., to keep taking more samples until a stopping condition is reached. A stopping condition usually takes the form $\Delta(n, \delta) \leq \epsilon$, where $n$ is the number of samples that have been taken, and $\Delta$ is an upper bound of the worst approximation error given by statistical learning theory. Note that $\Delta$ is usually a function of $n$ and $\delta$. There is a variant called top-$k$ approximation, which returns the $k$ most frequent itemsets among the observed ones based on the approximated frequencies. This is quite popular in practice since we are often only interested in the most common itemsets. Risk Bounds {#sec:rb} ----------- We briefly review some risk bounds in statistical learning theory [@BBL05] with the background of frequent itemsets mining. For each itemset $A\subseteq{\mathcal{I}}$, define the indicator function $\phi_A : 2^{\mathcal{I}}\to \{0, 1\}$ as follows. $$\phi_A(\tau) = \begin{cases} 1 & \textrm{if $A\subseteq \tau$}\\ 0 & \textrm{otherwise}\\ \end{cases},\quad \tau\subseteq{\mathcal{I}}.$$ Clearly, the frequency $f_{\mathcal{D}}(A)$ equals to the *true* average of $\phi_A(\tau)$ where $\tau$ goes over all the transactions in ${\mathcal{D}}$. $$f_{\mathcal{D}}(A) = \frac{1}{|{\mathcal{D}}|} \sum_{\tau\in{\mathcal{D}}} \phi_A(\tau).$$ Similarly let ${\mathcal{S}}$ be the set of the sampled transactions. Then the *sampled* average of $\phi_A(\tau)$ can be defined as $$f_{\mathcal{S}}(A) = \frac{1}{|{\mathcal{S}}|} \sum_{\tau\in{\mathcal{S}}} \phi_A(\tau).$$ Clearly $f_{\mathcal{S}}(A)$ is the frequency of $A$ appearing in the sampled transactions ${\mathcal{S}}$. Assume $|{\mathcal{S}}| = n$. For each transaction $\tau_i \in {\mathcal{S}}$, let $\sigma_i$ be a Rademacher random variable taking value from $\{-1, 1\}$ with uniform probability distribution. The $\sigma_i$’s are independent. Assuming ${\mathcal{I}}$ is finite, we define the sample conditional Rademacher average as follows. $${\mathcal{R}}_{\mathcal{S}}= \mathbb{E}_\sigma \left[\max_{A\subseteq{\mathcal{I}}}\frac{1}{n}\sum_{i=1}^n \sigma_i\phi_A(\tau_i)\right],$$ where $\mathbb{E}_\sigma$ denotes the expectation taken over all the random variables $\sigma_i$’s, conditionally on the sample ${\mathcal{S}}$. The following theorem tells us that Rademacher average can be used to upper bound the approximation error, even for the worst case. \[thm:old\] (Theorem 3.2, [@BBL05]) For any $\delta>0$, with probability at least $1-\delta$, $$\max_{A\subseteq{\mathcal{I}}} |f_{\mathcal{D}}(A) - f_{\mathcal{S}}(A)|\leq 2{\mathcal{R}}_{\mathcal{S}}+ \sqrt{\frac{2\log(2/\delta)}{n}}.$$ If we want to use the upper bound given in Theorem \[thm:old\] in an approximation algorithm, we still need to upper bound the ${\mathcal{R}}_{\mathcal{S}}$. A classical result is given by Massart [@Mas00]. \[thm:massart\] (Lemma 5.2, [@Mas00]) Let $\ell = \max_{A\subseteq{\mathcal{I}}} [\sum_{i=1}^n\phi_A(\tau_i)^2]^{1/2}$ where each $\tau_i\in{\mathcal{S}}$. Then $${\mathcal{R}}_{\mathcal{S}}\leq \frac{\ell}{n}\sqrt{2\log N},$$ where $N = 2^{|{\mathcal{I}}|}$ and $n = |{\mathcal{S}}|$. Hence we have the following stopping condition for an $(\epsilon,\delta)$-approximation sampling algorithm. $$\Delta_1 := \frac{2\ell}{n}\sqrt{2\log N} + \sqrt{\frac{2\log(2/\delta)}{n}} \leq \epsilon.$$ However for many applications the above bound is not tight enough [@RU15; @RU16]. In the next section we will first review the state-of-art bound on the worst approximation error, and then give an asymptotically tighter bound. Refining the Upper Bound {#sec:refine} ======================== The reason why the bound given in the previous section is often not tight enough in practice is that the $\ell$ defined in Theorem \[thm:massart\] can be quite large. Suppose there is an itemset $A$ that almost always appears in every transaction in ${\mathcal{D}}$. Then no matter which sample the algorithm chooses, $\ell$ is roughly $\sqrt{n}$. For $\delta=0.01$, $N = 2^{1000}$, even 100,000 samples are taken, the upper bound is still larger than 0.15. For many applications such an upper bound cannot be acceptable and thus we need to take more samples. Clearly if the upper bound is tighter, a lot of samples can be saved. A Brief Review on the Existing Results -------------------------------------- Riondato and Upfal [@RU15] attempted to give a tighter bound of the Rademacher average ${\mathcal{R}}_{\mathcal{S}}$. \[thm:ru\] (Theorem 3, [@RU15], revised) Let $w : \mathbb{R}^+ \to \mathbb{R}^+$ be the function defined as $$w(s) = \frac{1}{s}\log \sum_{A\subseteq{\mathcal{I}}}\exp\left(\frac{s^2 \sum_{i=1}^n \phi_A(\tau_i)^2}{2n^2}\right).$$ Then ${\mathcal{R}}_{\mathcal{S}}\leq \min_{s>0} w(s)$. [**Remark**]{}. Note that in Theorem \[thm:ru\], the summation in $w(s)$ takes *exactly* $2^{|{\mathcal{I}}|}$ terms. However in the original version in [@RU15], the authors claimed that the summation could take much less than $2^{|{\mathcal{I}}|}$ terms. We argue that there is a gap between these two versions. Based on the proof given in [@RU15], one can reach the inequality as follows. $$\label{eqn:ru}\exp(s{\mathcal{R}}_{\mathcal{S}}) \leq \sum_{A\subseteq{\mathcal{I}}}\exp\left(\frac{s^2\sum_{i=1}^n \phi_A(\tau_i)^2}{2n^2}\right).$$ Note that on the right hand side, each term in the summation is no less than 1. Hence when taking the logarithm on both sides and dividing by $s$, each of the $2^{|{\mathcal{I}}|}$ terms cannot be eliminated. Thus the range of the summation cannot be compressed. Formally, suppose there is a set $\mathcal{V}\subseteq 2^{\mathcal{I}}$, where $2^{\mathcal{I}}$ denotes the power set of ${\mathcal{I}}$, such that $$\alpha(s) := \sum_{A\in 2^{\mathcal{I}}}\exp\left(\frac{s^2\sum_{i=1}^n \phi_A(\tau_i)^2}{2n^2}\right) \leq \sum_{A\in \mathcal{V}}\exp\left(\frac{s^2\sum_{i=1}^n \phi_A(\tau_i)^2}{2n^2}\right) :=\beta(s).$$ We take the limits as $s$ approaches 0. $$2^{|{\mathcal{I}}|}=\lim_{s\to 0}\alpha(s) \leq \lim_{s\to 0}\beta(s) = |\mathcal{V}|.$$ Hence $\mathcal{V} = 2^{\mathcal{I}}$. This implies any summation over only a part of $2^{\mathcal{I}}$ must be less than the summation over all of $2^{\mathcal{I}}$. Thus one cannot use Inequality (\[eqn:ru\]) to reach Theorem 3 in [@RU15]. Tighter Bound Without Rademacher Average ---------------------------------------- In statistical learning theory, the upper bound given by Theorem \[thm:old\] is for the general case, i.e., the set of itemsets can be infinite or finite. However, for frequent itemsets mining, the number of itemsets is always finite (at most $2^{|{\mathcal{I}}|}$). Given this assumption, can we establish any upper bound without using the Rademacher average? Following the similar lines given by Boucheron, Bousquet and Lugosi [@BBL04] and Toivonen [@Toi96], we can give a positive answer. For any $\epsilon > 0$, $$\begin{aligned} & \Pr[\max_{A\subseteq{\mathcal{I}}}|f_{\mathcal{D}}(A)-f_{\mathcal{S}}(A)|>\epsilon] \\ = & \Pr[\exists A\subseteq{\mathcal{I}}, f_{\mathcal{D}}(A)-f_{\mathcal{S}}(A)>\epsilon \vee f_{\mathcal{D}}(A)-f_{\mathcal{S}}(A)<-\epsilon] \\ \leq & \Pr[\exists A\subseteq{\mathcal{I}}, f_{\mathcal{D}}(A)-f_{\mathcal{S}}(A)>\epsilon] +\Pr[\exists A\subseteq{\mathcal{I}}, f_{\mathcal{D}}(A)-f_{\mathcal{S}}(A)<-\epsilon] && \textrm{(union bound)}\\ \leq & \sum_{A\subseteq{\mathcal{I}}} \Pr[f_{\mathcal{D}}(A)-f_{\mathcal{S}}(A)>\epsilon] + \sum_{A\subseteq{\mathcal{I}}} \Pr[f_{\mathcal{S}}(A)-f_{\mathcal{D}}(A)>\epsilon] && \textrm{(union bound)}. \end{aligned}$$ Recall Hoeffding’s inequalities [@H63]. Let $X_1, \cdots, X_n$ be independent random variables bounded by the intervals $[a_i, b_i]$. Define the sampled average of them as $$\overline{X} = \frac{1}{n}\sum_{i=1}^n X_i.$$ Then for any $t>0$, $$\Pr[\overline{X} - \mathbb{E}[\overline{X}] > t] \leq \exp\left(-\frac{2n^2t^2}{\sum_{i=1}^n (b_i-a_i)^2}\right),$$ and $$\Pr[\mathbb{E}[\overline{X}] - \overline{X} > t] \leq \exp\left(-\frac{2n^2t^2}{\sum_{i=1}^n (b_i-a_i)^2}\right).$$ Note that if we set $X_i = \phi_A(\tau_i)$, then $X_i$’s are independent since $\tau_i$’s are independent, and thus $f_{\mathcal{D}}(A) = \mathbb{E}[\overline{X}]$ and $f_{\mathcal{S}}(A) = \overline{X}$. Based on Hoeffding’s inequalities, $$\Pr[f_{\mathcal{D}}(A) - f_{\mathcal{S}}(A) > \epsilon] \leq \exp\left(-2n\epsilon^2\right),$$ and $$\Pr[f_{\mathcal{S}}(A) - f_{\mathcal{D}}(A) > \epsilon] \leq \exp\left(-2n\epsilon^2\right).$$ Putting the above results together, we have $$\begin{aligned} & \Pr[\max_{A\subseteq{\mathcal{I}}}|f_{\mathcal{D}}(A)-f_{\mathcal{S}}(A)|>\epsilon] \\ \leq & 2\sum_{A\subseteq{\mathcal{I}}} \exp\left(-2n\epsilon^2\right) \\ = & 2N \exp\left(-2n\epsilon^2\right), \end{aligned}$$ where $N = 2^{|{\mathcal{I}}|}$. Equivalently for any $\delta>0$, with probability at least $1-\delta$, $$\max_{A\subseteq{\mathcal{I}}}|f_{\mathcal{D}}(A)-f_{\mathcal{S}}(A)| \leq \sqrt{\frac{\log(2N) + \log(1/\delta)}{2n}} =: \Delta_2.$$ Note that the above bound $\Delta_2$ is very similar to the result in Section 3.4, [@BBL04]. Now the bound $\Delta_2$ can generate a new stopping condition for an approximation algorithm. Recall the classical upper bound given in Section \[sec:rb\]. $$\Delta_1 := \frac{2\ell}{n}\sqrt{2\log N} + \sqrt{\frac{2\log(2/\delta)}{n}}.$$ Clearly $\lim_{\delta\to 0}\Delta_1/\Delta_2 = 2$, i.e., when $\delta$ is very small, the bound $\Delta_1$ is roughly twice of $\Delta_2$ given the sample size $n$. This assures us that the bound $\Delta_2$ is highly competitive. Theorem \[thm:ru\] can give another upper bound on the worst approximation error. However, since the number of terms in the summation grows exponentially on $|{\mathcal{I}}|$, to find the minimum is computationally infeasible. Furthermore, even if the minimum $w(s^*)$ is found, let $\Delta_1'$ be the upper bound of this variant defined as $$\Delta_1' := w(s^*) + \sqrt{\frac{2\log(2/\delta)}{n}}.$$ By fixing the sample ${\mathcal{S}}$, we still have $\lim_{\delta\to 0}\Delta_1'/\Delta_2 = 2$. For small $\delta$, the bound without Rademacher average still outperforms the existing ones. Our Frequent Itemset Approximation Algorithm {#sec:algs} ============================================ Approximating with Precomputed Sample Size {#sec:nonprg} ------------------------------------------ We observe the bound given in the previous section: $$\Delta_2 := \sqrt{\frac{\log(2N) + \log(1/\delta)}{2n}},$$ where $N = 2^{|{\mathcal{I}}|}$. The upper bound $\Delta_2$ can be treated as a function of allowed error probability $\delta$, sample size $n$ and $N = 2^{|{\mathcal{I}}|}$, all of which are already given. A good news is that there is no parameter that needs to be progressively computed (e.g., $\ell$ in $\Delta_1$). Thus to guarantee an worst approximation error at most $\epsilon$, we only need to make sure $\Delta_2 \leq \epsilon$. By solving it we have $$n \geq \frac{1}{2\epsilon^2}(\log (2N) + \log (1/\delta)).$$ Note that this sampling bound agrees with Toivonen’s result (Corollary 2 in [@Toi96]). Hence an $(\epsilon,\delta)$-approximation algorithm takes a very simple form. We first consider a brute-force algorithm to approximate frequencies for *all* the itemsets. Note that since the number of subsets (itemsets) in ${\mathcal{I}}$ is exponential (i.e., $2^{|{\mathcal{I}}|}$), the brute-force algorithm is not efficient. Since the brute-force algorithm above is computationally infeasible when $|{\mathcal{I}}|$ is large, in practice, we often only consider the frequencies of a few itemsets, e.g., most popular pairs of complementary goods, influential coauthoring in a community, etc. For this case, we do not have to consider the itemsets, which do not appear frequently enough. Denote by ${\mathbf{Ob}}$ the set of the itemsets to be observed. Then the worst approximation error is defined as the maximum error on every itemset in ${\mathbf{Ob}}$. By the same reasoning in the derivation of $\Delta_2$, we have the adjusted new bound: $$\Delta_2' := \sqrt{\frac{\log(2|{\mathbf{Ob}}|) + \log(1/\delta)}{2n}}.$$ Since ${\mathbf{Ob}}$ is a subset of $2^{{\mathcal{I}}}$, this bound $\Delta_2'$ is tighter than $\Delta_2$. Note that Toivonen (Corollary 2, [@Toi96]) also found a similar result as our bound here. The approximation algorithm will also be revised as follows. Note that we do not have to estimate for any itemset which is out of the observed ones ${\mathbf{Ob}}$. Also we need the size of ${\mathbf{Ob}}$ to be as small as possible. Depending on the practical requirements, the choice of ${\mathbf{Ob}}$ can vary a lot. We will give a SON-based idea in the next section. However many other methods can be tried, e.g., most potentially frequent itemsets can be suggested by the users’ experience or historic records. Approximating Top-$k$ Frequent Itemsets {#sec:prg} --------------------------------------- In practice we often need to find out the top-$k$ frequent itemsets among the given candidates ${\mathbf{Ob}}$. We can slightly revise the algorithm given in the previous section to approximate the $k$ most frequent itemsets. A new problem here is how to give the stopping condition. Note that if we only need the top-$k$ frequent itemsets, then our approximation can stop when the members of top-$k$ FIs are fixed with high probability (i.e., at least $1-\delta$). In particular, if with probability at least $1-\delta$, the true frequency of any itemset will not surpass the middle point of the $k$-th and $(k+1)$-th largest approximated frequencies, the $k$ itemsets with largest approximated frequencies should probably be the correct top $k$ ones. By Hoeffding’s inequality and union bounds, given the approximate frequencies with $n$ samples, the probability $p$ that there exists an itemset, whose approximated frequency and true frequency are on the different sides of the middle point of the $k$-th and $(k+1)$-th largest approximated frequencies, can be upper bounded as follows: $$\begin{aligned} p&=\Pr[\bigvee_{A\in{\mathbf{Ob}}} (f_{\mathcal{D}}(A) < m < \hat{f}_{\mathcal{D}}(A)) \vee (f_{\mathcal{D}}(A) > m > \hat{f}_{\mathcal{D}}(A)) ] \\ &\leq \sum_{A\in{\mathbf{Ob}}} \exp\left(-2n(\hat{f}_{\mathcal{D}}(A) - m)^2\right), \end{aligned}$$ where $m$ is the frequency middle point as described above. Hence we can let the sampling stop when the upper bound of $p$ is less than $\delta$. Combining these ideas, a progressive sampling approximation algorithm can be given as follows: Note that in out top-$k$ approximation algorithm, the stopping condition depends on the $k$-th and $(k+1)$-th largest frequencies. If these two frequencies tie, it is likely that many samples will be needed since we cannot distinguish them based on approximated frequencies. Hence we require the number of samples should not exceed $N$, the number of samples that can guarantee the $(\epsilon, \delta)$-approximation. If more than $N$ samples are needed, we can assume that the $k$-th and $(k+1)$-th largest frequencies tie or are very close. Then we directly output the approximated top FIs, since further computations to distinguish the very close FIs on the boundary are often unnecessary in practice. To be efficient, we must ensure the size of ${\mathbf{Ob}}$ is small enough. One possible way, which is similar to A-Priori algorithm [@AIS93], is given as follows: 1. We first only consider the itemsets with single item. The item size is usually small enough (i.e., $|{\mathcal{I}}|$) that can be put in main memory. We approximate their frequencies, and take the threshold $T$ as the $k$-th largest frequency among the single items. Usually people are only interested in small $k$, e.g., 10 to 100, which is much less than $|{\mathcal{I}}|$. 2. Then we use SON algorithm $\cite{SON95}$ to exactly find the itemsets with frequencies at least $T$. For efficiency, in SON we only consider the itemsets with 2 items, like what [@LRU14] did. The reason is that the itemsets of sizes larger than 2 usually have much lower frequencies than pairs. Clearly SON algorithm can find at most $k^2$ candidate itemsets. 3. We use our (top-$k$) approximation algorithms to estimate the frequencies of the candidate itemsets (or select the top-$k$ ones). Since we only consider frequent pairs, we can also build our scheme on PCY or Multistage algorithms. The major difference between PCY, Multistage and A-Priori is how to fully use the main memory for the passes, in which we select the most frequent items or itemsets. Hence the difference does not affect our sampling. Also we can use Toivonen’s algorithm instead of SON to mine frequent itemsets with more than 2 items. Evaluation {#sec:eval} ========== In this section we present our evaluation results. We try to find the top-$K$ frequent itemsets (in pairs) by two algorithms proposed in this paper: - [**A-Priori + Precomputed Sample Size**]{}. We first use A-Priori algorithm to find the top-$K$ most frequent items, and then use the algorithm (discussed in Section \[sec:nonprg\]) to approximate the frequencies of all the pairs formed by the $K$ items. At last we sort the frequencies and find the top-$K$ pairs with the highest approximate frequencies. Note that the sample size can be precomputed. - [**A-Priori + Progressive Sampling**]{}. This is given in Section \[sec:prg\]. Note that the sampling is progressive, i.e., there is no precomputed sample size. We compare our algorithms with the state-of-art [@RU15], which is a progressive sampling approximation. For each comparison, our code for each algorithm is similarly organized except that the bounds are different. For the performance, we consider time complexity (running time), sample size, and precision/recall. Setup ----- We implemented our algorithms by Python 3.4.3 and ran the programs on knot cluster (one DL580 nodes with 4 Intel X7550 eight core processors and 512GB RAM) at UCSB Center for Scientific Computing. To reduce the entire running time in the experiments (it is very time consuming to compute the exact solutions), for each dataset, we selected the first 70 items and then approximate the top 10 frequencies of their pairs. Each approximation was repeated by 10 times and we took the averaged results. By default we chose that the sample size increases by 100 for each round, and $\epsilon = 0.05$, $\delta = 0.01\%$. Datasets -------- [l | r r]{} Name & No. of Transactions & No. of items\ accidents & 340183 & 468\ chess & 3196 & 75\ connect & 67557 & 129\ kosarak & 990002 & 41270\ mushroom & 8124 & 119\ pumsb & 49046 & 7116\ pumsb star & 49046 & 7116\ retail & 88162 & 16470\ For consistency, we choose FIMI’03 data repository [@GZ04], the real-world data set from [@RU15] (the data repository is available at `http://fimi.ua.ac.be/data/`). The item and transaction data sizes of the FIMI datasets are given in Table \[tab:data\]. We will use these data sets to evaluate the samples sizes and worst case errors of our algorithms, and compare our results with the state-of-art algorithms. Worst-case Error Upper Bound Comparison --------------------------------------- We compare our new worst-case error bound with the state-of-art [@RU15]. Figure \[fig:b1\] and Figure \[fig:b2\] give our bounds on worst-case errors and [@RU15]’s for different combinations of $\epsilon$, $\delta$ and sample size. Clearly our new bound outperforms the state-of-art, implying that to achieve a certain degree of accuracy, compared to [@RU15]’s estimation, actually much fewer samples are needed. This key result gives the basic motivation of our algorithms. ![$\epsilon = 0.001$, $\delta = 0.00001$.[]{data-label="fig:b2"}](e01d0001.eps){width="\textwidth"} ![$\epsilon = 0.001$, $\delta = 0.00001$.[]{data-label="fig:b2"}](e001d00001.eps){width="\textwidth"} Comparison Results ------------------ We compare the performance between our methods and the state-of-art [@RU15]. Table \[tab:res1\] gives the evaluation results for our algorithms discussed in Section \[sec:nonprg\]. The algorithms try to find the top-100 most frequent itemsets with two items. Our algorithm will first find the 100 most frequent items, and then approximate the frequencies of the pairs between the 100 items. Thus the number of the observed itemsets is $|{\mathbf{Ob}}| = 100*99/2 = 4950$ for the second pass. Given the default $\epsilon$ and $\delta$, the precomputed sample size $n$ for the second pass is fixed, except the dataset chess, whose transactional size is less than the precomputed sample size. Note that we have significantly confined the observed itemsets, and thus it is efficient to compute the upper bound in [@RU15], i.e., we do not need to consider each subset of ${\mathcal{I}}$. In the table we have included all the samples taken in both the first and second passes. By using our new error upper bound, the running time and sample size are significantly reduced compared to [@RU15], while the accuracy is still quite competitive, i.e., all are larger than 90%. This is natural since [@RU15] takes more samples and thus the estimation should be more accurate. In the dataset chess, every transaction is sampled since its data volume is very small. In the dataset connect, our sample size is only 1/20 of [@RU15]’s, but the accuracy is almost the same. [l | r r r | r r r]{} Item & ----------------- Used time (sec) (ours) ----------------- : Results of our approximation algorithms with precomputed sample size and [@RU15].[]{data-label="tab:res1"} & ------------- Sample size (ours) ------------- : Results of our approximation algorithms with precomputed sample size and [@RU15].[]{data-label="tab:res1"} & ----------- Precision (ours) ----------- : Results of our approximation algorithms with precomputed sample size and [@RU15].[]{data-label="tab:res1"} & ----------------- Used time (sec) [@RU15] ----------------- : Results of our approximation algorithms with precomputed sample size and [@RU15].[]{data-label="tab:res1"} & ------------- Sample size [@RU15] ------------- : Results of our approximation algorithms with precomputed sample size and [@RU15].[]{data-label="tab:res1"} & ----------- Precision [@RU15] ----------- : Results of our approximation algorithms with precomputed sample size and [@RU15].[]{data-label="tab:res1"} \ accidents & 48.78 & 6894 & 97% & 2683.74 & 387782 & 99%\ chess & 21.44 & 3196 & 100% & 21.86 & 3196 & 100%\ connect & 50.34 & 6636 & 98% & 1097.70 & 132751 & 99%\ kosarak & 104.31 & 7657 & 90% & 11735.33 & 949155 & 98%\ mushroom & 43.32 & 6620 & 98% & 125.56 & 18646 & 99%\ pumsb & 193.39 & 7438 & 98% & 2998.37 & 115939 & 98%\ pumsb star & 155.15 & 7438 & 95% & 2595.17 & 133668 & 99%\ retail & 102.93 & 7606 & 91% & 1909.02 & 128544 & 96%\ Table \[tab:res2\] gives the evaluation results for our algorithms discussed in Section \[sec:prg\]. Similarly to the above non-progressive version, our progressive method is still quite efficient and accurate. The comparison between ours and [@RU15] show that our method is competitive. [l | r r r | r r r]{} Item & ----------------- Used time (sec) (ours) ----------------- : Comparisons on our progressive approximation algorithms and [@RU15].[]{data-label="tab:res2"} & ------------- Sample size (ours) ------------- : Comparisons on our progressive approximation algorithms and [@RU15].[]{data-label="tab:res2"} & ----------- Precision (ours) ----------- : Comparisons on our progressive approximation algorithms and [@RU15].[]{data-label="tab:res2"} & ----------------- Used time (sec) [@RU15] ----------------- : Comparisons on our progressive approximation algorithms and [@RU15].[]{data-label="tab:res2"} & ------------- Sample size [@RU15] ------------- : Comparisons on our progressive approximation algorithms and [@RU15].[]{data-label="tab:res2"} & ----------- Precision [@RU15] ----------- : Comparisons on our progressive approximation algorithms and [@RU15].[]{data-label="tab:res2"} \ accidents & 18.69 & 3600 & 98% & 357.37 & 60600 & 99%\ chess & 21.96 & 3200 & 98% & 22.33 & 3200 & 99%\ connect & 43.04 & 6700 & 99% & 748.37 & 110200 & 100%\ kosarak & 110.84 & 7700 & 88% & 1367.20 & 86000 & 97%\ mushroom & 46.13 & 6700 & 98% & 100.68 & 16400 & 98%\ pumsb & 170.12 & 7500 & 98% & 2225.19 & 98200 & 98%\ pumsb star & 130.49 & 7500 & 98% & 1708.80 & 98200 & 98%\ retail & 98.24 & 7700 & 94% & 1152.59 & 83700 & 97%\ Conclusion ========== We have proposed a new upper bound for the worst-case errors on sampling-based approximate frequent itemsets mining. Our new bound is tighter than the state-of-art result. Based on our new bound, two approximation algorithms have been proposed. We have used real-world datasets to evaluate our results and the performance of our algorithms. The evaluation results have shown that our algorithms are not only competitive but also efficient.

--- abstract: 'We predict and demonstrate that a disorder induced carrier density inhomogeneity causes magnetoresistance (MR) in a two-dimensional electron system. Our experiments on graphene show a quadratic MR persisting far from the charge neutrality point. Effective medium calculations show that for charged impurity disorder, the low-field MR is a universal function of the ratio of carrier density to fluctuations in carrier density, a power-law when this ratio is large, in excellent agreement with experiment. The MR is generic and should occur in other materials with large carrier density inhomogeneity.' author: - Jinglei Ping - Indra Yudhistira - Navneeth Ramakrishnan - Sungjae Cho - Shaffique Adam - 'Michael S. Fuhrer' bibliography: - 'shaffiquebib.bib' title: Disorder induced magnetoresistance in a two dimensional electron system --- [*Introduction –* ]{} The classical magnetoresistance of a material arises when the Lorentz force caused by an applied magnetic field has a component acting against the direction of electron motion thereby decreasing the conductivity of an electronic material. This property has long been of interest both as a tool to probe the fundamental properties of an electronic material (such as the topology of the electron bands) [@kn:ashcroft1976] and also for technological applications such as its use in magnetic memory read-heads [@kn:nickel1995]. A well known result is that single electronic bands (in systems with a spatially homogenous carrier density) will have no magnetoresistance, while the presence of two or more electronic bands with different carrier mobilities readily gives rise to a classical magnetoresistance. This classical effect is different from weak localization [@kn:altshuler1980] (a quantum interference effect present at low temperatures) or Abrikosov’s quantum magnetoresistance [@kn:abrikosov1998] (that occurs in gapless semiconductors in the high field Landau quantized regime). Graphene is an example of a material with more than one electronic band. This single-atom-thick sheet of carbon comprises an electron band and a hole band each with a linear dispersion that touch at a topologically protected Dirac point [@kn:neto2009; @kn:dassarma2011a]. It was shown by Ref. [@kn:hwang2006e] that if both bands are occupied, one then expects a classical magnetoresistance even if the electron and hole bands have the same electronic mobility (this is by virtue of the Lorentz force being of opposite sign for the electron and hole carriers). While this two-channel model has been reasonably successful at modeling the density dependence of graphene magnetotransport at fixed magnetic field, it is unable to quantitatively explain the magnetic field dependence at fixed carrier density. In this context, Ref. [@kn:guttal2005] and Ref. [@kn:tiwari2009] developed an effective medium approximation where the planar landscape is broken up into electron regions and hole regions with different area fractions. Assuming that all electron regions had a uniform conductivity $\sigma_e$ and all hole regions had a uniform conductivity $\sigma_h$, they could calculate the magnetoresistance of this effective medium. While this description works well at the charge neutrality point (where the electrons and holes regions have equal area fractions), it fails to adequately describe the experiments away from this symmetry point. We refer the reader to Ref. [@kn:cho2008] for a detailed discussion of these earlier theoretical and experimental results. However, we note that both the two-channel model of Hwang [*et al.*]{} and the effective medium calculation of Stroud and collaborators both predict that the magnetoresistance should vanish away from the Dirac point when only a single band is occupied. By contrast, in this work we discuss a carrier density inhomogeneity contribution to the magnetoresistance that persists away from the Dirac point and exists even if only one electronic band is occupied. For concreteness we focus our discussion on graphene which has a linear dispersion, but the mechanism itself does not rely on the linear dispersion and should therefore be observable in other materials with large spatial inhomogeneity in the carrier density distribution. The idea of a disorder induced magnetoresistance is not new. While working on silver chalcogenides, Parish and Littlewood [@kn:parish2003; @kn:parish2005] predicted such an effect by mapping the problem onto a random resistor network and solving it numerically. They focussed on the high magnetic field regime and found a linear magnetoresistance, but were otherwise unable to make quantitative comparisons with experiments. By contrast, in this work, we use an effective medium theory to study the low field regime and make quantitative predictions that are then compared to experimental results. The basic mechanism of inhomogeneity induced magnetoresistance can be seen in Fig. \[Fig:fig1\]. We schematically show two regions where the local carrier densities (and local conductivities) are smaller and larger than the bulk average respectively. In both regions the Hall field is the same and perpendicular to both the applied electric field and the applied magnetic field. The magnetic field results in a Lorentz force acting on the charge carriers as well as an adjustment of the local electric fields such that current remains conserved in the direction of the applied electric field. Requiring that the net adjustment of the electric field over the sample be zero results in a reduction of the drift velocity in both the high and low conductivity regions in the presence of magnetic field. Below, we develop a full effective medium theory that quantitatively captures this effect. [*Experimental procedure –* ]{} We measure the magnetotransport in single-crystal graphene synthesized by chemical vapor deposition on Pt foil [@kn:gao2012; @kn:ping2013]. The three CVD-grown samples 1, 2, and 3 were prepared at temperatures of 1000 $^\circ$[C]{}, 950 $^\circ$[C]{} and 900 $^\circ$[C]{}, and hydrogen mass flow rates of 700 sccm, 500 sccm and 380 sccm, respectively. The synthesized graphene is then coated with PMMA with a spinning speed of 2000 rpm and then transferred to a 300 nm SiO$_2$ on Si substrate by electrolysis method [@kn:gao2012]. Electron-beam lithography using poly(methyl methacrylate) resist is used to establish Cr/Au contacts via liftoff and again to define the graphene in a Hall-bar geometry via oxygen plasma etching. All three devices have the same geometry shown in the inset of Fig. \[Fig:fig2\]b. As discussed in detail elsewhere [@kn:ping2013], the differences between the Raman spectra of the three samples suggest the presence of nanocrystalline carbon impurities on the continuous crystalline graphene layer; sample 1 has the greatest concentration of impurities and sample 3 has the least. These impurities do not correlate with mobility, and so for the purposes of this work, the samples simply have varying amounts of disorder. {width="6.4in"} Fig. \[Fig:fig2\]a shows the zero-field conductivity as a function of back gate induced carrier density $n_0$ for samples 1, 2 and 3. We observe the typical approximately linear dependence of conductivity as a function of carrier density [@kn:dassarma2011a], and from the data we can extract the charge-impurity limited mobility $\mu$ of the three samples as 8,300 [cm]{}$^2$/[Vs]{}, 8,100 [cm]{}$^2$/[Vs]{}, and 10,700 [cm]{}$^2$/[Vs]{} for samples 1, 2 and 3 respectively (see supplementary material for more details). These values of mobility are among the highest for CVD-grown graphene transferred to method. Also shown in Fig. \[Fig:fig2\]a are data from an exfoliated graphene sample [@kn:cho2008], which has charge-impurity limited mobility $\mu$ of 18,200 [cm]{}$^{2}$/[Vs]{}. We also experimentally measure the minimum conductivity $\sigma_{\rm min}$, and use this to extract the disorder induced carrier density fluctuations $n_{\rm rms}$ using $\sigma_{\rm min} = n_{\rm rms} e \mu/\sqrt{3}$. The values for $n_{\rm rms}$ extracted this way are between $20\%$ and $40\%$ lower than what one would expect from the self-consistent theory for graphene transport [@kn:adam2007a] that is normally [@kn:dassarma2011a] used to understand the graphene minimum conductivity. We attribute this discrepancy to the non-perfect transmission across p-n junctions separating the electron and hole regions or due to additional scattering by the nanocystaline grain boundaries. As will become clearer later, we parameterize our data as a function of the ratio $n_0/n_{\rm rms}$, where both the average carrier density $n_0$ and the density fluctuations $n_{\rm rms}$ are measured independently. Fig. \[Fig:fig2\]b shows the longitudinal resistance $R_{xx}$ and Hall conductivity $\sigma_{xy}$ as a function of back gate voltage for all three samples at $T=4.2~{\rm K}$ and $B = 8~{\rm T}$. The data clearly shows Subnikov-de Haas oscillations and quantum Hall plateaus with $\sigma_{xy} = 4(n +1/2)e^2/h$ (where the factor 1/2 is the fingerprint of the $\pi$ Berry’s phase in monolayer graphene) [@kn:novoselov2005; @kn:zhang2005; @kn:wu2007]. Fig. \[Fig:fig2\] shows the magnetoresistance. At high magnetic field and low carrier densities, we sometimes observe a linear magnetoresistance. However, for sufficiently low magnetic fields, we always observe a quadratic magnetoresistance, where for different values of carrier density $n_0$ and density fluctuation $n_{\rm rms}$, we can fit our data to $$\label{Aeqn} \rho_{xx}(B) = \rho_{xx}(B=0) \left[ 1 + A (\mu B)^2 \right]$$ and extract the dimensionless coefficient $A[n_0, n_{\rm rms}]$ from our data. We can also fit the data over a larger range of B using the phenomenological formula of Ref. [@kn:cho2008], $\rho_{xx}(B) = \rho_{xx}(0)\left[1-\alpha+\frac{\alpha}{\sqrt{1+\frac{2A(\mu B)^{2}}{\alpha}}}\right]^{-1}$, where $\alpha$ is a fitting parameter. Notice that for $\mu B<<1$, this phenomenological expression gives the same value for the quadratic magnetoresistance as Eq. \[Aeqn\]. These fits are shown in Fig. \[Fig:fig2\]. We find experimentally that $A[n_0, n_{\rm rms}]$ scales as a function of the ratio $n_0/n_{\rm rms}$ and that for large $n_0/n_{\rm rms}$ it follows a power law $A \sim (n_0/n_{\rm rms})^{-2}$. Our experimental results shown in Fig. \[Fig:fig3\] suggest that the magnetoresistance persists far away from the Dirac point (i.e. $n_0 > n_{\rm rms}$) and is caused by carrier density inhomogeneity and not due to the presence of both electrons and holes close to the Dirac point. [*Theoretical analysis –* ]{} The starting point for this analysis is to assume that the carrier density $n$ is Gaussian distributed centered at an average carrier density $n_0$ with an [rms]{} fluctuation given by $n_{\rm rms}$ (we denote this distribution as $P[n, n_0, n_{\rm rms}]$. For the specific case of graphene, Ref. [@kn:adam2009] justified theoretically the use of a Gaussian distribution, and at least close to charge neutrality, this has been seen in several experiments starting with Ref. [@kn:martin2008]. In this context ${\rm sign}(n_0) = \pm 1$ represents the electron and hole bands respectively. For the case where charged impurities dominate the transport properties, knowing the impurity concentration $n_{\rm imp}$ and the distance $d$ away from the graphene sheet, one can calculate $n_{\rm rms}$ both analytically [@kn:adam2007a] (using the self-consistent approximation) and numerically [@kn:rossi2008] (using the mesoscopic density functional approach). The carrier mobility $\mu$ is calculated using the semi-classical Boltzmann transport theory [@kn:dassarma2011a]. In what follows, for simplicity, we assume that $\mu$ is density independent, and we show in the supplementary material that the weak density dependence of the carrier mobility hardly changes any of our results. Since $n_0$ is controllably tuned by a back gate, and the parameters $n_{\rm imp}$ and $d$ can be obtained from the conductivity at zero magnetic field, all the parameters used in our theory for magnetoresistance can be fixed by measurements done before applying a magnetic field. Before discussing our inhomogeniety induced magnetoresistance, we first briefly comment on previous theories for graphene magnetoresistance in the context of our framework. For a single channel model, the transport in the presence of a magnetic field is given by [@kn:ashcroft1976] $$\label{Eq:onechannel} \sigma(B) = \frac{|n_0| e \mu}{1 + (\mu B)^2} \left( \begin{array}{cc} 1 & +{\rm sign}(n_0) \mu B \\ -{\rm sign}(n_0) \mu B & 1 \end{array} \right),$$ where it is easy to verify that there is no magnetoresistance i.e. $\rho_{xx}(B) = \sigma_{xx} /(\sigma_{xx}^2 + \sigma_{xy}^2) = \rho_{xx}(0)$. The two channel model [@kn:hwang2006e] assumes that the total conductivity is the sum of the electron and hole channels, $\sigma_{xx} = \sigma_{xx}^e + \sigma_{xx}^h$, and similarly for the transverse conductivity, $\sigma_{xy} = \sigma_{xy}^e + \sigma_{xy}^h$. Defining $ = n_0/n_{\rm rms}$, a straightforward calculation gives the quadratic coefficient of the magnetoresistance (see definition above) as $$A \left[ \eta = \frac{n_0}{n_{\rm rms}} \right] = 1 - \left( \frac{\eta \sqrt{2 \pi}}{2e^{-\eta^2/2} + \eta \sqrt{2 \pi} {\rm Erf}(\eta/\sqrt{2})} \right)^2 \label{Eq:twochannel}$$  Theoretical and experimental results for the dependence of the coefficient of quadratic magnetoresistance (a) plotted as a function of the carrier density $n_0$ and (b) as a function of the ratio between carrier density $n_0$ and carrier density fluctuations $n_{\rm rms}$. The magnetoresistance in the earlier theoretical models including the two channel model by and the area fraction effective medium theory by . In both earlier models, the magnetoresistance vanishes quickly once $n_0 > n_{\rm rms}$. By contrast, the magnetoresistance discussed in this work persists away from the Dirac point. For $n >> n_{\rm. rms}$, we find both theoretically and experimentally a power law dependance: $A = (1/2)(n_0/n_{\rm rms})^{-2}$.](Fig3.pdf){width="2.8in"} Here ${\rm Erf}(x)$ is the error function [@kn:gradshteyn1994]. Notice that $A[\eta]$ is independent of the carrier mobility $\mu$ and depends only on the ratio of the carrier density and density fluctuation. This remains true so long as $\mu$ is independent of carrier density, and in this sense $A[\eta]$ becomes a universal function (where the different theoretical models each give a different functional form for $A[\eta]$). The two-channel result Eq. \[Eq:twochannel\] is shown in Fig. \[Fig:fig3\]; it has the value $A=1$ at the Dirac point, stays roughly constant for $n_0 < n_{\rm rms}$, and then rapidly decreases for $n>n_{\rm rms}$ as the second channel becomes depopulated. The inadequacy of this model to explain experimental data led Ref. [@kn:tiwari2009] to develop an area-fraction effective medium theory. This model assumes that there are electron regions with area-fraction $f_e$ and conductivity $\sigma_e = n_e e \mu$, and hole regions with $f_h$ and $\sigma_h = n_h e \mu$. The effective medium conductivity tensor $\sigma_{\rm EMT}$ is obtained by solving $\sum_{i = e,h} f_i \delta \sigma_i (\openone_2 - \Gamma \delta \sigma_i)^{-1} = 0$, where the shorthand notation $\delta \sigma_i = \sigma_i - \sigma_{\rm EMT}$ is used. In the case where the electron and hole puddles can be assumed to be nearly circular, the depolarization tensor $\Gamma = -\openone_2 /(2 \sigma_{\rm EMT}^{xx})$ takes a simple scalar form (see Ref. [@kn:stroud1975] for details). In this case, a remarkable result [@kn:guttal2005] is that when $n_0 = 0$ (and hence $f_e = f_h$), the magnetoresistance is given by $\rho_{xx}(B) = \rho_{xx} (0) \sqrt{1 + (\mu B)^2}$. Since the self-consistent theory [@kn:adam2007a] gives $\rho_{xx}(0) = \sqrt{3}/(n_{\rm rms} e \mu)$ and $\mu [{\rm m}^2/{\rm Vs}] \approx 50/(n_{\rm imp}[10^{10} {\rm cm}^{-2}])$, the full magnetoresistance at the Dirac point is completely specified. In particular, we have $A[0] = 1/2$. This model can be solved numerically away from the Dirac point, and the results are shown in Fig. \[Fig:fig3\]. Notice again that for $n_0>n_{\rm rms}$, the area fraction of the hole channel vanishes and the magnetoresistance drops rapidly. The inadequacy of the two-channel model is that it does not account for the spatial inhomogeneity of the carrier density, and the inadequacy of the area-fraction EMT is that although it allows for two dimensional space to broken up into regions of electron and hole puddles, all electron and hole regions are assumed to be uniform. What is required is an effective medium approach with a continuous distribution of carrier density (similar to what has been developed in Refs. [@kn:rossi2008b; @kn:fogler2008b] for transport in zero-magnetic field). Using the form of the depolarization tensor derived in Ref. [@kn:stroud1975], for our system, we can derive a set of coupled equations \[Eq:main\] $$\int dn P[n,n_0, n_{\rm rms}] \frac{\sigma_{xx}^2[n] - (\sigma^{\rm EMT}_{xx})^2 + (\sigma^{\rm EMT}_{xy} - \sigma_{xy}[n] )^2}{(\sigma^{\rm EMT}_{xx} + \sigma_{xx}[n] )^2 + (\sigma^{\rm EMT}_{xy} - \sigma_{xy}[n] )^2} = 0$$ $$\int dn P[n,n_0, n_{\rm rms}] \frac{\sigma_{xy}[n] - \sigma^{\rm EMT}_{xy}}{(\sigma^{\rm EMT}_{xx} + \sigma_{xx}[n] )^2 + (\sigma^{\rm EMT}_{xy} - \sigma_{xy}[n] )^2}=0.$$ It is understood from Eq. \[Eq:main\] that $\sigma_{xx}[n]$ and $\sigma_{xy}[n]$ are obtained from some homogenous density model (such as Eq. \[Eq:onechannel\]) and then these coupled integral equations give the correct averaging over the density inhomogeneity. One can verify that for $B=0$, we get $\sigma^{\rm EMT}_{xy} =0$, and the equation for $\sigma^{\rm EMT}_{xx}$ reproduces the zero-magnetic field effective medium theory results of  [@kn:adam2009] and  [@kn:rossi2008b]. Moreover $\sigma^{\rm EMT}_{xy} =0$ also for $n_0 = 0$, and a numerical solution of the $\sigma^{\rm EMT}_{xx}$ gives results very close to $\rho_{xx}(B) = \rho_{xx} (0) \sqrt{1 + (\mu B)^2}$ (although, technically, it need not have given the same result since our model allows for carrier density inhomogeneity inside each puddle). We emphasize that Eq. \[Eq:main\] can be solved with any model for the density profile $P[n,n_0,n_{rms}]$ and scaterring potential as input for $\mu$. In the simplified case where $\mu$ is independent of density, e.g. for charged impurity scattering, Eq. \[Eq:main\] simplifies considerably and the normalized magnetoresitance $\rho_{xx}(B)/\rho_{xx}(0)$ depends only on the ratios $n_{0}/n_{\rm rms}$ and $\mu B$. In this case, both $\sigma_{xx}^{\rm EMT}$ and $\sigma_{xy}^{\rm EMT}$ can be written in term of dimensionless coefficient $y_1$ and $y_2$ as $$\sigma_{xx}^{\rm EMT} = y_1~n_{\rm rms} e \mu/(1+ (\mu B)^2)$$ $$\sigma_{xy}^{\rm EMT} = y_2~n_{\rm rms} e \mu^2 B/(1+ (\mu B)^2),$$ where $y_{1}=y_{1}\left[\frac{n_{0}}{n_{{\rm rms}}},\mu B\right]$ and $y_{2}=y_{2}\left[\frac{n_{0}}{n_{{\rm rms}}},\mu B\right]$ are computed in the supplementary material. The coefficient of quadratic magnetoresistance obtained by solving these equations numerically has been plotted in Fig. \[Fig:fig3\]. It is important to notice that our inhomogenous carrier density driven magnetoresistance persists far away from the Dirac point, and is not specific to the linear dispersion of graphene. Generally, there is remarkable agreement between the theoretical and experimental results presented here. As we explain in the supplementary material, the discrepancy close to the Dirac point (for small values of $\eta$) can be directly traced to the overestimating of $\sigma_{\rm min}$ in the self-consistent theory and therefore a difference between the theoretical and experimental values used for $n_{\rm rms}$ close to the Dirac point. In summary we have shown both theoretical and experimental results for an inhomogeneity induced quadratic magnetoresistance that scales as a power law of the ratio $n_0/n_{\rm rms}$. While we focused on the case of charged impurities in graphene, the mechanism itself requires only spatial fluctuations in the carrier density and should therefore be observable in other systems. [*Acknowledgement:*]{} The experimental work was supported by the University of Maryland NSF-MRSEC under Grant No. DMR 05-20471 and the US ONR MURI program. MSF acknowledges support from an ARC Laureate fellowship. The theoretical work in Singapore was supported by the National Research Foundation Singapore under its Fellowship program (NRF-NRFF2012-01). [**SUPPLEMENTARY MATERIAL**]{}\ Conductivity fitting at zero magnetic field ------------------------------------------- ![\[Fig:S1\] (Color online) Fitting of effective medium theory conductivity to experimental results at zero magnetic field.](S1.pdf){width="3.2in"} Following a standard procedure [@ref1], we fit conductivity of experiment to conductivity of effective medium theory at zero magnetic field as shown in Fig. \[Fig:S1\]. There are three parameters used in the fitting, which are short-range scatterers $\sigma_{\mathrm{s}}$, charged impurity density $n_{\mathrm{imp}}$ and the distance between graphene and the substrate $d$. These parameters enter the EMT equations through the RPA-Boltzmann conductivity $\sigma_{\mathrm{B}}[\sigma_\mathrm{{s}},n_{\mathrm{imp}},d,r_{\mathrm{s}}]$, where $r_{\mathrm{s}}=0.8$ is used for graphene on . The RPA-Boltzmann conductivity is used in the zero magnetic field EMT equations $$\int_{-\infty}^{\infty}dn\, P[n,n_{0},n_{\rm rms}]\frac{\sigma_{\mathrm{B}}-\sigma_{\mathrm{EMT}}}{\sigma_{\mathrm{B}}+\sigma_{\mathrm{EMT}}}=0$$ where $P[n,n_{0},n{\mathrm{rms}}]$ is a Gaussian distribution centered at an average carrier density $n_0$ with an [rms]{} fluctuation given by $n_{\rm rms}$. The fitting parameters for the three samples are shown in Table \[tab:fitparam\]. [cccc]{} & $\sigma_{\mathrm{s}}$ (e$^{2}$/h) & $n_{\mathrm{imp}}$ ($10^{10}$ cm$^{-2}$) & $d$ (nm)CVD Sample 1 & 137.4 & 58.4 & 0.52 CVD Sample 2 & 79.1 & 59.6 & 0.80 CVD Sample 3 & 121.9 & 45.3 & 0.72Exfoliated Sample & 729.0 & 26.5 & 0.14 The charge-impurity limited mobility of the three CVD-grown graphene samples are then calculated to be 8,300 [cm]{}$^2$/[Vs]{}, 8,100 [cm]{}$^2$/[Vs]{}, and 10,700 [cm]{}$^2$/[Vs]{} for samples 1, 2, and 3 respectively and 18,200 [cm]{}$^{2}$/[Vs]{} for exfoliated graphene. The charged impurity limited conductivity seems uncorrelated with the island like impurities that are observed in an optical image. See Ref.  [@ref2; @ref3] for details about the island like impurities. Effective Medium Theory for constant mobility --------------------------------------------- For the case where mobility $\mu$ is independent of carrier density, we introduce a dimensionless magnetic field ${\tilde b} = \mu B$ and dimensionless functions $y_1$ and $y_2$ such that $\sigma_{xx}^{{\rm EMT}}=y_{1}~n_{{\rm rms}}e\mu/\left[1+(\mu B)^{2}\right]$ and $\sigma_{xy}^{{\rm EMT}}=y_{2}~n_{{\rm rms}}e\mu^{2}B/\left[1+(\mu B)^{2}\right]$. Equation 3 of the main text then simplifies to $$\int_{-\infty}^{\infty} dy \ e^{-(y - \eta)^2/2} \frac{y^2 - y_1^2 + {\tilde b}^2 (y_2 - y)^2 }{(|y| + y_1)^2 + {\tilde b}^2 (y_2 - y)^2} = 0$$ $$\int_{-\infty}^{\infty} dy \ e^{-(y - \eta)^2/2} \frac{y_2 -y }{(|y| + y_1)^2 + {\tilde b}^2 (y_2 - y)^2} = 0.$$ These equations can be easily solved numerically both for constant $\tilde{b}$ and also for constant $n_0/n_{\mathrm rms}$. Shown in Fig. \[Fig:S2\] are the dependence of dimensionless constants $y_1$ and $y_2$ on $\tilde{b}$ and $n_0/n_{\mathrm rms}$. ![\[Fig:S2\]Dependence of dimensionless functions $y_1$ and $y_2$ on dimensionless magnetic field $\tilde{b}$ and $n_0/n_{\mathrm rms}$. The upper panel shows $y_1$ and $y_2$ as a function of $n_0/n_{\mathrm rms}$ at $\tilde{b}=0.1$ and the lower panel shows $y_1$ and $y_2$ as a function of $\tilde{b}$ at $n_0/n_{\mathrm rms}=1$.](S2.pdf){width="3.43in"} From the data in Fig. \[Fig:S2\], we can obtain the coefficient of quadratic magnetoresistance through $$A=1-\left[\left(\frac{y_{2}(0)}{y_{1}(0)}\right)^{2}+\frac{1}{2\mu^{2}}\frac{\partial_{B}^{2}y_{1}(0)}{y_{1}(0)}\right]$$ Non-constant mobility --------------------- In reality, graphene has a mobility that is weakly dependent on carrier density due to the present of short-range scatterers $\sigma_{\mathrm{s}}$ and finite substrate distance $d$. In this case, we need to employ the full EMT equations of Eq. 3 of the main text. $\sigma_{B}[n_{0}]$ and $\mu_{B}[n_{0}]$ enter the EMT equation through longitudinal and transverse conductivity of the single channel model, as shown below. $$\sigma[n_{0},B]=\frac{\sigma_{B}[n_{0}]}{1+(\mu_{B}[n_{0}]B)^{2}}\begin{pmatrix}1 & \pm\mu_{B}[n_{0}]B\\ \mp\mu_{B}[n_{0}]B & 1 \end{pmatrix}$$ This is essentially Eq. 1 of the main text, except that we have the RPA–Boltzmann conductivity. $\sigma_{B}[n_{0}]=|n_{0}|e\mu_{B}[n_{0}]$ instead of $n_{0}e\mu$.  Theoretical results for the dependence of the coefficient of quadratic magnetoresistance $A$, on the ratio of carrier density $n_{0}$ and carrier density fluctuations $n_{\mathrm{rms}}$ for $\mu$ independent on carrier density and $\mu$ of the graphene samples, where it is weakly dependent on carrier density. $A$ determines the magnetoresistance through $\rho_{xx}[n_{0,}B]=\rho_{xx}(0)\left[1+A(\mu[n_{0}]B)^{2}\right]$. We find that the non-constant $\mu$ hardly affects $A$, and thus the power law dependence $A=(1/2)(n_{0}/n_{\mathrm{rms}})^{-2}$ still holds for $n\gg n_{\mathrm{rms}}$ for all samples.](S3.pdf){width="3.2in"} We calculate the theoretical values of coefficient of the quadratic magnetoresistance $A$ of these samples. We find that the density dependent carrier mobility hardly affects $A$, as shown in Fig. \[Fig:S3\]. Since the density dependence of $\mu$ does not change the result, we use the simple density independent calculation of $A$ in Fig. 3 of the main text. Solving Eq. 3 of the main text numerically gives $\sigma_{xx}^{\mathrm{EMT}}$ and $\sigma_{xy}^{\mathrm{EMT}}$. The magnetoresistance and quadratic coefficient of magnetoresistance can then be obtained through $$\rho_{xx}^{\mathrm{EMT}}=\frac{\sigma_{xx}^{\mathrm{EMT}}}{\left(\sigma_{xx}^{\mathrm{EMT}}\right)^{2}+\left(\sigma_{xy}^{\mathrm{EMT}}\right)^{2}}$$ and $$\label{Eq:rhoxxEMT} \rho_{xx}^{\mathrm{EMT}}[n_{0,}B]=\rho_{xx}^{\mathrm{EMT}}(B=0)\left[1+A(\mu[n_{0}]B)^{2}\right]$$ Discrepancy of coefficient of quadratic magnetoresistance close to Dirac point ------------------------------------------------------------------------------ Close to the Dirac point, we see that there are discrepancies between the theoretical and experimental values of the coefficient of quadratic magnetoresistance $A$. This discrepancy can be traced back to an overestimation of the value of self-consistent RPA–Boltzmann conductivity $\sigma_{B}$ (for small $\eta$) used in our model. To see this, we compare the theoretical and experimental values of $\sigma_{\mathrm{min}}$ at zero magnetic field and find that the theoretical $\sigma_{\mathrm{min}}$ in our model are larger than the experimental values by 20 % to 45 % for CVD-grown sample 1, sample 2, and sample 3. This overestimation of $\sigma_{B}$ is equivalent to an overestimation of mobility $\mu_{B}$ close to the Dirac point. The lower experimental mobility close to the Dirac point is not captured by the fit value for $\mu[n_{0}]$ used in Eq. \[Eq:rhoxxEMT\]. This zero magnetic field effect can be attributed either to effect of island impurities found in the CVD-grown samples or to the finite resistance caused by p-n junctions between electron and hole puddles. We have checked that if we empirically modify $\mu[n_{0}]$ to account for scattering of p-n junctions and line defects, this discrepancy between theory and experiment at the Dirac point vanishes. However, this is just a phenomenological modification of $\mu[n_{0}]$ and a full theory of scattering by p-n junctions is beyond the scope of this work. In principle, we expect this discrepancy to disappear if one had a better way to measure $n_{\rm rms}$ directly in the experiment e.g. through scanning tunneling microscopy. Derivation of -------------- We can derive this result using both the schematic discussed in Fig. 1 of the main text and using a simplified effective medium theory. ### Simple Physical Model Using the model established in Fig. 1 of the main text, we assume a magnetic field along the $z$ direction and an applied electric field along the $x$ direction. Examining two regions of electrons with carrier concentrations $n_{1}$ and $n_{2}$, we can relate the local currents to the local electric fields. For $i=1,2$ we have $$\label{eq:conductivity} \begin{pmatrix}J_{x}^{i}\\ J_{y}^{i} \end{pmatrix}=\frac{n_{i}e\mu}{1+\mu^{2}B^{2}}\begin{pmatrix}1 & -\mu B\\ \mu B & 1 \end{pmatrix}\begin{pmatrix}E_{x}^{i}\\ E_{y}^{i} \end{pmatrix}$$ Noting that the steady state Hall field ensures no net current in the $y$ direction and assuming that the Hall field is the same in the two regions, we have $$\begin{aligned} E_{y} & =-\frac{\mu BJ_{x}}{n_{\rm avg}e\mu}\label{eq:Ey}\end{aligned}$$ Enforcing current conservation in the $x$ direction, we have $$\begin{aligned} E_{x}^{i}=\frac{J_{x}}{n_{i}e\mu}+\frac{J_{x}\mu^{2}B^{2}}{e\mu}\left(\frac{1}{n_{i}}-\frac{1}{n_{\rm avg}}\right)\label{eq:Exfinal}\end{aligned}$$ Assuming that the electric field across the sample may be redistributed over the regions to get different local field values but the net change in electric field due to the applied B field must be zero, we get $$\begin{aligned} \frac{J_{0}}{n_{1}e\mu}+\frac{J_{0}}{n_{2}e\mu} & =\frac{J_{x}}{n_{1}e\mu}+\frac{J_{x}\mu^{2}B^{2}}{e\mu}\left(\frac{1}{n_{1}}-\frac{1}{n_{\rm avg}}\right)\nonumber \\ & +\frac{J_{x}}{n_{2}e\mu}+\frac{J_{x}\mu^{2}B^{2}}{e\mu}\left(\frac{1}{n_{2}}-\frac{1}{n_{\rm avg}}\right)\end{aligned}$$ For $\mu B\ll 1$, $$J_{x}=J_{0}(1-\alpha^{2}\mu^{2}B^{2})$$ We have defined $\alpha=\frac{(n_{1}-n_{2})}{(n_{1}+n_{2})}$. More generally, we see that $A\sim\alpha^{2}\sim\left(\frac{\Delta n}{n_{\rm avg}}\right)^{2}\sim(\frac{n_{0}}{n_{{\rm rms}}})^{-2}$ as calculated in the full EMT model. ### Simplified effective medium theory The coefficient of quadratic magnetoresistance $A$ for $n_{0}\gg n_{\mathrm{rms}}$ can also be obtained from a one–band model for which the zero field conductivity is given by $\sigma_{0}=n_{0}e\mu$ with constant $\mu$. Carrier density inhomogeneity is represented by $P[n_{0}]=(1/2)\delta(n_{0}-n_{\mathrm{rms}})+(1/2)\delta(n_{0}+n_{\mathrm{rms}})$. This model can be solved analytically for $|\eta|\gg1$ and $\mu B\ll1$. The longitudinal and transverse conductivities are given by $$\begin{aligned} \sigma_{xx}^{\rm 2ch,EMT} & =\frac{|n_{0}|e\mu}{1+(\mu B)^{2}}\sqrt{\left(1-\eta^{-2}\right)\left[1+\eta^{-2}(\mu B)^{2}\right]}\label{eq:sigmaxx2chemt}\\ \sigma_{xy}^{\rm 2ch,EMT} & =-\frac{n_{0}\left(1-\eta^{-2}\right)e\mu}{1+(\mu B)^{2}}(\mu B)\label{eq:sigmaxy2chemt}\end{aligned}$$ Thus, we have magnetoresistance $$\rho_{xx}^{\rm 2ch,EMT} =\frac{\sqrt{1+\eta^{-2}(\mu B)^{2}}}{n_{0}\sqrt{1-\eta^{-2}}e\mu}$$ which in turn gives $$\ A=\frac{1}{2}\eta^{-2}$$ as what is obtained in the full effective medium theory. [99]{} S. Das Sarma, S. Adam, E. H. Hwang and E. Rossi, Rev. Mod. Phys. **83**, 407 (2011) J. Ping and M. S. Fuhrer, arXiv preprint arXiv:1304.5123 (2013) J. Ping and M. S. Fuhrer, Nano letters **12**, 4635 (2012)

--- abstract: 'We revise the cosmological standard model presuming that matter, i.e. baryons and cold dark matter, exhibits a non-vanishing pressure mimicking the cosmological constant effects. In particular, we propose a scalar field Lagrangian $\mathcal L_1$ for matter with the introduction of a Lagrange multiplier as constraint. We also add a symmetry breaking effective potential accounting for the classical cosmological constant problem, by adding a second Lagrangian $\mathcal{L}_2$. Investigating the Noether current due to the shift symmetry on the scalar field, $\varphi\rightarrow\varphi+c^0$, we show that $\mathcal{L}_1$ turns out to be independent from the scalar field $\varphi$. Further we find that a positive Helmotz free-energy naturally leads to a negative pressure without introducing by hand any dark energy term. To face out the fine-tuning problem, we investigate two phases: before and after transition due to the symmetry breaking. We propose that during transition dark matter cancels out the quantum field vacuum energy effects. This process leads to a negative and constant pressure whose magnitude is determined by baryons only. The numerical bounds over the pressure and matter densities are in agreement with current observations, alleviating the coincidence problem. Finally assuming a thermal equilibrium between the bath and our effective fluid, we estimate the mass of the dark matter candidate. Our numerical outcomes seem to be compatible with recent predictions on WIMP masses, for fixed spin and temperature. In particular, we predict possible candidates whose masses span in the range $0.5-1.7$ TeV.' author: - Orlando Luongo - Marco Muccino title: Speeding up the universe using dust with pressure --- Introduction {#sec1} ============ The $\Lambda$CDM concordance paradigm is described by the fewest number of assumptions possible. In particular, the universe is approximated at late times by two fluids: pressureless matter and a cosmological constant $\Lambda$. Both baryonic matter (BM) and cold dark matter (DM) are unable to push the universe to accelerate [@2006IJMPD..15.1753C]. Thus, besides dust-like fluids, one needs to include $\Lambda$ to account for the observed speed up. The simplicity of the concordance paradigm turns out to be the strong suit to admit its validity. However, the magnitude of $\Lambda$ predicted by quantum fluctuations of flat space-times leads to a severe fine-tuning problem with the observed value of $\Lambda$. Even considering a curved space-time one cannot remove the problem [@1989RvMP...61....1W]. Further, both matter and $\Lambda$ magnitudes are extremely close today, leading to the well-known *coincidence problem* . Under these aspects the $\Lambda$CDM model seems to be incomplete, whereas from a genuine observational point of view it well adapts to data. In this work, we revise the cosmological standard model assuming an effective a scalar field $\varphi$ Lagrangian for baryons and cold DM. We require that matter provides a *non-vanishing pressure term* and we wonder whether it can accelerate the universe alone, i.e. without the need of $\Lambda$. To do so, we propose the most general Lagrangian, depending upon a *kinetic term and Lagrange multiplier*, with the inclusion of a potential term due to the vacuum energy cosmological constant, inducing a *phase transition*. During such an early-time phase transition the DM pressure counterbalances the $\Lambda$ pressure, leaving as unique contribute the pressure of baryons[^1]. In particular, the baryonic pressure turns out to be negative to guarantee a positive Helmotz free-energy for the whole system. In this picture, we find a Noether current, coinciding with the entropy density current and providing the Lagrangian to be independent from $\varphi$. We thus write up the thermodynamics associated to the model and we investigate small perturbations, finding the adiabatic and non-adiabatic sound speeds naturally vanish in analogy to the $\Lambda$CDM approach. Our paradigm candidates as an alternative to the concordance model and predicts the existence of a single fluid, composed of baryons and cold DM with pressure. The fluid cancels out the quantum contribution due to $\Lambda$, driving the universe today with a constant and negative pressure. This process does not set $\Lambda$ to zero, but removes it naturally. This is possible if DM constituents lie on the mass interval $\sim0.5$–$1.7$ TeV. To show this, we relate the predictions of our model to the thermal history of the primordial universe and to the expected DM relic abundance. The paper is structured as follows. In Sec. \[sec:action\], we propose the effective representation for matter with pressure. We thus write the equations of motion and we discuss the introduction of the potential term due to the vacuum energy cosmological constant. In Sec. \[sec:thermodynamics\], we describe the thermodynamics of our matter fluid, which *naturally* suggests an emergent negative pressure, and demonstrate that our Lagrangian does not depend upon $\varphi$. In Sec. \[pert\] we investigate the small perturbations and we find that both the adiabatic and non-adiabatic sound speeds naturally vanish, leading to a constant pressure, in analogy to the $\Lambda$CDM approach. In Sec. \[vacuumenergy\] we closely analyze the role of the effective potential $V^{\rm eff}$. This term induces a first order transition phase during which the quantum field vacuum energy density mutually cancels with the DM pressure. Soon after the transition the emergent $\Lambda$ is given by the (negative) pressure of baryons. We show how our mechanism overcomes the fine-tuning and the coincidence issues affecting the $\Lambda$CDM model. In Sec. \[DMparticle\] we relate the predictions of our paradigm to observable quantities. We thus obtain that, almost independently from the spin, the DM mass ranges within the interval: $0.5\lesssim M{\rm c^2/ TeV} \lesssim1.7$. In Sec. \[predictions\] we summarize the main predictions of our model and then in Sec. \[conclusions\] we propose conclusions and perspectives of our work. The effective representation of matter with pressure {#sec:action} ==================================================== We discussed in Sec. \[sec1\] that our model pushes up the universe to accelerate by means of matter with pressure. In particular, we want to demonstrate that the DM pressure may counterbalance the effects of $\Lambda$, if a transition phase is involved in our picture. To show that, let us start from a few number of hypotheses that we will use later on, summarized below: - there exists only *one fluid*, composed of BM and DM; - matter is coupled to $\Lambda$ and the coupling effect cancels the cosmological constant density which does not enter the dynamical equations; - the process which cancels the effects of $\Lambda$ is due to a *first order phase transition*; - the whole kinetic energy of matter is constrained through a Lagrange multiplier; - the thermodynamics of matter *naturally* suggests an emergent negative pressure; - the model mimes the $\Lambda$CDM effects, without departing from observations at both late and early stages of universe’s evolution. This suggests an effective representation of *dust with pressure* in a curved space-time given by Lagrangian density $\mathcal{L}=\mathcal{L}_1+\mathcal{L}_2$, where $$\begin{aligned} \label{eq:1} \mathcal{L}_1 &= K\left(X,\varphi\right) +\lambda Y\left[X,\nu\left(\varphi\right)\right]\,,\\ \label{eq:1bi} \mathcal{L}_2 &= -V^{\rm eff}\left(X,\varphi\right)\,,\end{aligned}$$ depend upon the scalar field $\varphi$ and its first covariant derivatives[^2] in the form of the standard kinetic term $$\label{eq:2} X = \frac{1}{2}g^{\alpha\beta}\nabla_\alpha \varphi \nabla_\beta\varphi\,,$$ where $g^{\alpha\beta}$ is the metric tensor and $\nu(\varphi)$ plays the role of the specific inertial mass [@1970PhRvD...2.2762S]. The Lagrangian $\mathcal{L}_1$ represents a dust component with pressure. It is written in the most generic form without indicating a priori the functional forms of the functions $Y$ and $K$, while the Lagrange multiplier $\lambda$ constraints the kinetic energy with the potential term in $\nu$. The physical motivation behind $\mathcal{L}_1$ supports the idea of BM and DM with pressure [@2000ApJ...535L..21M; @2006MNRAS.372..136F; @2017arXiv170701059S; @bet]. It is important to stress that our fluid consists of BM and DM, so that in principle the Lagrangian $\mathcal{L}_1$ can be written as $$\label{BM+DM} \mathcal{L}_1=K_{\rm BM}+ K_{\rm DM} + \lambda \left(Y_{\rm BM} + Y_{\rm DM}\right)\,,$$ where $K\equiv K_{\rm BM}+K_{\rm DM}$ and $Y\equiv Y_{\rm DM}+Y_{\rm DM}$. The Lagrangian $\mathcal{L}_2$ models the coupling with the standard cosmological constant through an interacting potential $V^{\rm eff}$ used to investigate the phase transition. We write down the simplest form of $V^{\rm eff}$ by $$\label{eq:vint} V(\varphi,\psi)=V_0+\frac{\chi}{4}\left(\varphi^2-\varphi^2_0\right)^2+\frac{\bar{g}}{2}\varphi^2\psi^2,$$ in which the first two terms describe the self-interacting potential, with a dimensionless coupling constant $\chi$, of the scalar field $\varphi$, and the last one the interacting potential, with a dimensionless coupling constant $\bar{g}$, between $\varphi$ and another scalar field $\psi$. The quantity $V_0$ denotes the classical off-set, while $\varphi^2_0$ is the value of $\varphi$ at the minimum of its potential without interactions with $\psi$. We can thus assume that $\psi$ is in thermal equilibrium. In such a case, $\psi^2$ can be replaced through its average in a thermal state. In a thermal state there exists a correspondence between the thermal average state and the temperature. Following [@2006ftft.book.....K], we redefine the coupling constant $\bar{g}$ to account for the proportionality between the thermal average and the temperature, i.e. we have $$\langle\psi^2\rangle_{T}\propto T^2\,.$$ After some manipulations, we simple have $$\label{eq:pottemperature} V^{\rm eff}(X,\varphi)=V_0+\frac{\chi}{4}\left(\varphi^2-\varphi_0^2\right)^2+\frac{\chi}{2}\varphi_0^2\varphi^2 \frac{T^2(X)}{T_{\rm c}^2}\,,$$ where $T_{\rm c}=\varphi_0\sqrt{\chi/\bar{g}}$ is the critical temperature, discriminating as transition starts. Before the transition when $T>T_{\rm c}$, the minimum of $V^{\rm eff}$ is located at $\varphi=0$ and the corresponding value is $V_0+\chi \varphi_0^4/4$. After the transition, when $T<T_{\rm c}$ the minimum is at $\varphi=\varphi_0$ with a value $V_0$. From Eqs. – we define the action $ S = \int \mathcal{L}\,\sqrt{-g}{\rm d}^4x$, where $g$ is the determinant of $g^{\alpha\beta}$. Assuming a standard minimal coupling with gravity, from the variation of the action with respect to $\lambda$, $\varphi$ and the metric tensor we obtain a *constraint* and a *dynamical* equation and the energy-momentum tensor respectively (details of calculations are reported in Appendix \[appA\]) $$\begin{aligned} \label{eq:no5a} &\,Y = 0\,, \\ \label{eq:no5b} &\,\mathcal{L}_\varphi - \nabla_\alpha \left( \mathcal{L}_X \nabla^\alpha\varphi \right) = 0\,,\\ \label{eq:no6} &\, T_{\alpha\beta} = \mathcal{L}_X \nabla_\alpha\varphi \nabla_\beta \varphi - \left(K - V^{\rm eff} \right) g_{\alpha\beta}\,,\end{aligned}$$ where the subscripts label the partial derivatives, so that $\mathcal{L}_X=K_X-V^{\rm eff}_X+\lambda Y_X$ and $\mathcal{L}_\varphi = K_\varphi-V^{\rm eff}_\varphi+\lambda Y_\nu \nu_\varphi$. For time-like derivatives it holds $X>0$ and, from Eq. (\[eq:2\]), we can introduce an effective $4$-velocity $$\label{eq:no7} v_\alpha = \frac{\nabla_\alpha\varphi}{\sqrt{2X}}\ ,$$ while the $4$-acceleration identically vanishes $$\label{eq:no8} a_\beta = \dot{v}_\beta = v_\gamma \nabla^\gamma v_\beta = 0\,,$$ where $\dot{y}=v^\alpha\nabla_\alpha y$ is the Lie derivative of $y$ along $v^\alpha$, which is tangent to time-like geodesics. Using Eq. (\[eq:no7\]), the energy-momentum tensor can be written as $$\label{eq:no10} T_{\alpha\beta} = 2X \mathcal{L}_X v_\alpha v_\beta - \left(K - V^{\rm eff} \right) g_{\alpha\beta}\ ,$$ which is of the perfect fluid form for an energy density and a pressure, respectively $$\begin{aligned} \label{eq:no11} \rho\left(\lambda,X,\varphi\right) =\,& 2X \mathcal{L}_X - \left(K - V^{\rm eff} \right)\,,\\ \label{eq:no12} P\left(X,\varphi\right) =\,& K - V^{\rm eff}\,.\end{aligned}$$ Thus, from the above definitions one wonders whether it is possible to fulfill the weak energy conditions $T_{\alpha\beta}k^\alpha k^\beta \geq0, \rho \geq 0, \rho + P \geq 0$, where $k^\alpha$ is a time-like vector field. From the above conditions and the fact that $X>0$, it follows that $$\label{weaks} 2X\mathcal{L}_X \geq K - V^{\rm eff}\,,\qquad \mathcal{L}_X \geq 0\,.$$ The energy-momentum tensor conservation gives $$\label{eq:no13} \nabla_\alpha T^{\alpha\beta} = \left[\dot{\rho} + \theta \left(\rho + P\right) \right] v^\beta = 0\ ,$$ leading to the energy conservation $ \dot{\rho} + \theta\left(P + \rho \right) = 0$ where we defined the *expansion* $$\label{eq:no15} \theta = \nabla_\alpha v^\alpha = \frac{\nabla_\alpha\nabla^\alpha \varphi-X_\varphi}{\sqrt{2X}}\,.$$ The energy flux $T^{\alpha\beta}v_\beta=\rho v^\alpha$ always follows time-like geodesics, as for the perfect fluid with no pressure. By means of this position $$\label{eq:no16} \eta_\varphi = 2X \left(\mathcal{L}_{XX} X_\varphi + \mathcal{L}_{X\varphi} \right) + \mathcal{L}_X X_\varphi -\mathcal{L}_\varphi\,,$$ the constraint in Eq. (\[eq:no5b\]) can be written as follows $$\label{eq:no9b} \dot{\lambda} = -\frac{1}{2X Y_X} \left[\sqrt{2X}\eta_\varphi + \theta\left(P + \rho \right) \right] \,.$$ Eqs. (\[eq:no5a\]) and (\[eq:no9b\]) represent the equations of motion for a perfect fluid ruled by two first-order ordinary differential equations for the scalar fields $\varphi$ and $\lambda$ [@LSV10]. Thermodynamics of matter with pressure {#sec:thermodynamics} ====================================== Now we address the thermodynamics of the perfect fluid described by the effective Lagrangian in Sec. \[sec1\]. Non-dissipative fluids are described by virtue of the *pullback* formalism [@1993CQGra..10.2317C; @1994CQGra..11..709C; @2007LRR....10....1A] through Carter’s covariant formulation [@1989LNM..1385....1C] in a relativistic effective field theory. In this formulation, an observer is attached to a particular fluid element by introducing a matter space such that its worldline is identified with a unique point in this space. The coordinates of each matter space serve as labels that distinguish fluid element worldlines and remain unchanged throughout the evolution. The matter space coordinates can be considered as scalar fields on spacetime, with a unique map relating them to the spacetime coordinates. Generally fluids are framed with four scalar fields, namely $\phi^a$. In this puzzle three scalar fields, corresponding to $a=1,\,2,\,3$, become fluid comoving coordinates as they propagate in space, whereas $\phi^0$ is interpreted as an internal time coordinate [@2016PhRvD..94l4023B; @2016PhRvD..94b5034B; @2017arXiv171101961C]. These scalars can be viewed as Stückelberg fields[^3] that allow to restore broken diffeomorphisms in four-dimensional spacetimes [@2016PhRvD..94l4023B; @2003AnPhy.305...96A; @2004JHEP...10..076D; @2008PhyU...51..759R]. So that, the fluid physical properties are encoded within a set of symmetries of the scalar field action. We here are interested in cosmological perfect fluids describing the matter sector only. Such a framework may be seen as in Sec. \[sec:action\]. Moreover we only deal with the temporal Stückelberg field [@Matarrese:1984zw; @2016PhRvD..94l4023B; @2016PhRvD..94b5034B], hereafter renamed $\varphi$. In other words, providing the cosmological principle, it is licit to take into account that our model is well motivated if one fluid is accounted, say $\phi^0\equiv\varphi$. Without considering the spatial fields implies that the corresponding Lagrangian respects the global shift symmetry $$\varphi\rightarrow \varphi+c^0\,,$$ with $c^0$ an arbitrary constant [@2016PhRvD..94l4023B; @2016PhRvD..94b5034B]. The scenario defined in Sec. \[sec:action\] turns out to describe a barotropic matter fluid, i.e., its pressure is completely defined by knowing its energy density and viceversa, see Eqs. (\[eq:no11\]) and (\[eq:no12\]). To provide its thermodynamic interpretation, we choose the particle number density $n$ and the temperature $T$ of the fluid as thermodynamic variables to find correspondence with the field $X$ [@2016PhRvD..94l4023B; @2016PhRvD..94b5034B]. To account for perfect fluid thermodynamics we use the first principle, the Gibbs-Duhem relation and the Helmotz free-energy density $f=\rho-Ts$, respectively, $$\begin{aligned} \label{eq:1pric} d\rho =\,& T\,ds+\mu\,dn\,, \\ \label{eq:gibbsduhem} dP =\,& s\,dT +n\,d\mu\,,\\ \label{eq:dhelmotz} df =\,& \mu\,dn - s\,dT\,.\end{aligned}$$ where $s$ is the entropy density and $\mu$ the chemical potential. Combining Eqs. (\[eq:1pric\])–(\[eq:gibbsduhem\]) and (\[eq:no11\])–(\[eq:no12\]) with the definitions of $f$ we get $$\begin{aligned} \label{eq:solutions11} d\left(\mu n\right) =&\, d\left(2X\mathcal{L}_X\right) - d\left(Ts\right)\,,\\ \label{eq:solutions22} df =&\, d\left(2X\mathcal{L}_X\right) - d\left(K-V^{\rm eff}\right) - d\left(Ts\right)\,.\end{aligned}$$ Keeping in mind that in view of Eq. (\[eq:no5a\]) $d\left(K-V^{\rm eff}\right)\equiv d\mathcal{L}$, the above two relations admit as solutions: $$\begin{aligned} \label{eq:solutions1} f =\,& -\mathcal{L}\,,\\ \label{eq:solutions2} s=\,&\sqrt{2X}\mathcal{L}_X\,,\\ \label{eq:solutions3} T=\,&\sqrt{2X}\,,\\ \label{eq:solutions4} \mu=\,&0\,.\end{aligned}$$ Eq. (\[eq:solutions1\]) fulfills the thermodynamic relations: $$\begin{aligned} \label{dFdTV} \frac{\partial\left(fV\right)}{\partial T}\bigg\rvert_V =\,& -\sqrt{2X} \mathcal{L}_X V = -sV\,,\\ \label{dFdVT} \frac{\partial\left(fV\right)}{\partial V}\bigg\rvert_T =\,& -\mathcal{L}= -\left(K-V^{\rm eff}\right) = -P\,.\end{aligned}$$ From Eq.  it follows that $df=-sdT$. The condition $X>0$ and Eq.  automatically define the sign of the entropy density in Eq. , i.e. $s\geq0$. Since for expanding systems it has to be $dV>0$ and $dT<0$, necessarily we have that $df>0$. Thence, from Eq.  we deduce that $P$ turns out to be *naturally* negative: $$\label{dfvsP} df>0\Leftrightarrow P<0\,.$$ Conversely, this is even in agreement with the naive fact that for an the expanding fluid the work is positive. Under our convention of the first principle signs, one thus has: $-PdV>0$ which implies $P<0$. Eq.  implies that *dust-like matter having pressure naturally fixes the sign of $P$ to be negative*. This ensures no need of putting by hand the sign of $P$ inside Einstein’s equations to guarantee the universe speed up. Now we define the densities of internal energy $u$, enthalpy $h$, and Gibbs free-energy $g$ respectively by: $$\begin{aligned} \label{inendenty} u = & \rho =\, 2X \mathcal{L}_X - \left(K-V^{\rm eff}\right)\,,\\ \label{enthalpydens} h = & u + P =\, 2X \mathcal{L}_X\,,\\ \label{gibbsdens} g = & f + P =\, 0\,.\end{aligned}$$ Invoking the Noether’s theorem we notice the global shift symmetry changes the matter Lagrangian density $\mathcal{L}_1$ mostly by a total divergence. We explicitly get: $$\begin{aligned} \nonumber &\mathcal{L}_1\left(X^\prime,\varphi^\prime\right) =\mathcal{L}_1\left(\frac{1}{2}\nabla_\alpha\varphi \nabla^\alpha\varphi,\varphi+c^0\right)=\\ \nonumber &\mathcal{L}_1\left(X,\varphi\right) + c^0\left[\frac{\partial\mathcal{L}_1}{\partial\varphi} - \nabla_\alpha\frac{\partial\mathcal{L}_1}{\partial\left(\nabla_\alpha\varphi\right)}\right] + c^0 \nabla_\alpha\frac{\partial\mathcal{L}_1}{\partial\left(\nabla_\alpha\varphi\right)}=\\ \label{eq:ap5} &\mathcal{L}_1\left(X,\varphi\right) + c^0 \nabla_\alpha\left(\mathcal{L}_{1,X} \nabla_\alpha\varphi\right)\,,\end{aligned}$$ where, in the second line of Eq. (\[eq:ap5\]), the quantity in the brackets identically vanishes in view of the Euler–Lagrange equation. The conserved current $\mathcal{J}_1^\alpha$ corresponds to the total divergence of Eq. (\[eq:ap5\]), i.e., $$\label{eq:conscurr1} \mathcal{J}_1^\alpha = \sqrt{2X} \left(K_X+\lambda Y_X\right) v^\alpha\,.$$ At this point it behooves us to discuss how to deal with $V^{\rm eff}$. Its behavior during the phase transition is not trivial, since it depends on both $\varphi$ and its covariant derivatives via the kinetic term $X$. This implies that in general, the potential $V^{\rm eff}$ is not invariant under global shift symmetry. On the contrary, $V^{\rm eff}$ is well defined in its minima. *Before transition* (**BT**, with $V^{\rm eff}=V_0+\chi \varphi_0^4/4$ at $\varphi=0$) the effective potential is a constant. *After transition* (**AT**, with $V^{\rm eff}=V_0$ at $\varphi=\varphi_0$), the effective potential is a function of $X$ only, i.e., $V^{\rm eff}\equiv V^{\rm eff}(X)$, which is invariant under global shift symmetry. For the above reasons, in the following we limit our investigation to BT and AT, where $V^{\rm eff}$ is invariant under shift symmetry. Thus, we do not need to assess the intermediate cases, i.e. during transition. Therefore, during the BT and the AT phases, the Noether’s theorem implies that $$\label{eq:ap5bis} \mathcal{L}_2\left(X^\prime\right) = - V^{\rm eff}(X)- c^0 \nabla_\alpha\left(V^{\rm eff}_X \nabla_\alpha\varphi\right)\,,$$ where we can define another conserved current from the total divergence of Eq. (\[eq:ap5bis\]), i.e., $$\label{eq:conscurr1bis} \mathcal{J}_2^\alpha = -\sqrt{2X} V^{\rm eff}_X v^\alpha\,.$$ Hence the total conserved current $\mathcal{J^\alpha}$ is given by combining Eqs.  and , i.e., $$\label{eq:conscurr} \mathcal{J}^\alpha \equiv \mathcal{J}_1^\alpha + \mathcal{J}_2^\alpha = \sqrt{2X} \mathcal{L}_X v^\alpha = s^\alpha\,,$$ and coincides with the entropy density current $s_\alpha=sv_\alpha$. Eq.  simplifies Eq. (\[eq:no5b\]) into $\mathcal{L}_\varphi = 0$, *implying that the Lagrangian does not depend upon $\varphi$*. We thus have: $$\label{eq:1new} \mathcal{L}\left(\lambda,X,\nu\right) = K(X) - V^{\rm eff}(X) + \lambda Y \left(X,\nu\right)\,.$$ By combining Eqs. (\[eq:1pric\])–(\[eq:gibbsduhem\]) we get the Euler relation $$\label{eq:euler} P+\rho = Ts + \mu n\,,$$ and recast the energy-momentum tensor as $$\label{eq:tnesenimp2} T_{\alpha\beta}=\left(Ts_\alpha +\mu n_\alpha\right)v_\beta + P g_{\alpha\beta}\,,$$ where $n_\alpha=nv_\alpha$ is the particle number density current. The projection of the energy-momentum tensor conservation along $v^\alpha$, i.e., $v^\alpha\nabla^\beta T_{\alpha\beta}=0$, leads to $$\label{eq:tnesenimp3} T\nabla^\alpha s_\alpha + \mu \nabla^\alpha n_\alpha = 0\,,$$ and by virtue of the existence of $\mathcal{J}^\alpha$, it reduces to $$\label{eq:tnesenimp4} \mu \nabla^\alpha n_\alpha = 0\,,$$ which represents and identity, since $\mu=0$. However, one can also safely assume that the particle number density current is also conserved, i.e., $\nabla^\alpha n_\alpha = 0$. The conservation of the energy-momentum tensor can be recast as the Carter-Lichnerowicz equations [@2016PhRvD..94b5034B] $$\label{eq:cl} n\mathcal{W}_{\alpha\nu}v^\nu=nT\nabla_\alpha\sigma-\varsigma_\alpha\nabla^\nu n_\nu\,,$$ where $\mathcal{W}_{\alpha\nu}=\nabla_\nu \varsigma_\alpha -\nabla_\alpha \varsigma_\nu$ is the vorticity tensor [@2013rehy.book.....R], $\varsigma^\alpha=h/nv^\alpha$ the current of the enthalpy per particle, and $\sigma=s/n$ the entropy per particle. Since the $4$-velocity is the derivative of the scalar field $\varphi$ and $\nabla^\alpha n_\alpha = 0$, we infer from Eq. (\[eq:cl\]) that: $$\begin{aligned} \mathcal{W}_{\alpha\nu}=0\quad&\Rightarrow\quad{\rm the\ fluid\ is\ irrotational}\,,\\ \nabla_\alpha\sigma=0\quad&\Rightarrow\quad{\rm the\ fluid\ is\ isentropic}\,,\end{aligned}$$ respectively from the first and second conditions [@2016PhRvD..94b5034B]. Cosmological perturbations and the role of sound speed {#pert} ====================================================== In the previous sections we demonstrated that our matter fluid is irrotational and insentropic. We now discuss the cosmological perturbations taking into account our Lagrangian, as in Eq. . We thus unveil additional features characterizing our matter fluid concerning the magnitudes of the pressure $P$ and the sound speed. In the conformal Newtonian gauge, in absence of any anisotropic stress, we consider [@1999PhLB..458..219G; @Gao] $$\label{pert1} ds^2=a(\tau)^2\left[\left(1+2\Phi\right)d\tau^2-\left(1-2\Phi\right)dx^2\right]\,,$$ where $\Phi$ is the Newtonian potential, $\tau=a(t)t$ the conformal time and $a(t)$ the scale factor. The first order $\left(0,0\right)$, $\left(0,i\right)$ and $\left(i,j\right)$ components of Einstein’s equations in the Friedmann-Robertson-Walker model are respectively, $$\begin{aligned} \label{pert2} &\nabla^2\Phi-3\mathcal{H}\left(\Phi^\prime+\mathcal{H}\Phi\right)=4\pi a^2G\delta\rho\,,\\ \label{pert3} &\nabla_i\left(\Phi^\prime+\mathcal{H}\Phi\right)=4\pi a^2G\left(P+\rho\right) \delta v_i\,,\\ \label{pert4} &\Phi^{\prime\prime}+3\mathcal{H}\Phi^\prime+\left(2\mathcal{H}^\prime+\mathcal{H}^2\right)\Phi=4\pi a^2G\delta P\,,\end{aligned}$$ where the prime denotes the derivatives with respect to the conformal time and $\mathcal{H}=a^\prime/a$. The perturbation of the 3-velocity $\delta v_i=\nabla_i\delta\varphi/(a\varphi^\prime)$ depends upon the perturbation of the scalar field $\delta\varphi$ which depends also on the spatial coordinates [@2014CQGra..31e5006P]. The density perturbations depend on the kinetic term and on the Lagrange multiplier perturbations, respectively $\delta X$ and $\delta\lambda$, whereas the pressure perturbations depend only upon $\delta X$, so that: $$\begin{aligned} \label{pert5} \delta\rho=&\,A(X)\delta X+2X\lambda Y_{X\nu}\delta\nu+2XY_X\delta\lambda\,,\\ \label{pert6} \delta P=&\,B(X)\delta X\,.\end{aligned}$$ To infer the explicit expressions of $A(X)$ and $B(X)$, we discriminate between two regimes [@2012CRPhy..13..566M]: - BT, when the minimum is at $\varphi=0$ and the potential is $V^{\rm eff}=V_0+\chi \varphi_0^4/4$; - AT, when the minimum is at $\varphi=\varphi_0$ and the potential is $V^{\rm eff}=V_0$. Hence, the pressure and density become $$\begin{aligned} \label{s8} P(X)&= \left\{ \begin{array}{ll} K-V_0-\chi \varphi_0^4/4\quad &\quad{\rm (BT)}\\ K-V_0\quad&\quad{\rm (AT)} \end{array} \right.,\\ \label{s8bisaaa} \rho(X)&= \left\{ \begin{array}{ll} 2X\mathcal{L}_X-K+V_0+\chi \varphi_0^4/4\quad &\quad{\rm (BT)}\\ 2X\mathcal{L}_X-K+V_0\quad&\quad{\rm (AT)} \end{array} \right.,\end{aligned}$$ From the above definitions it follows that $$\begin{aligned} \label{pert5a} A(X)&= \left(2X\mathcal{L}_X\right)_X-K_X\,,\\ \label{pert5b} B(X)&= K_X\,.\end{aligned}$$ Combining Eqs.  and we get $$\begin{aligned} \nonumber \Phi^{\prime\prime}&+3\mathcal{H}\left(1+c_{\rm X}^2\right)\Phi^\prime+\left[2\mathcal{H}^\prime+\left(1+3c_{\rm X}^2\right)\mathcal{H}^2\right]\Phi+\\ \label{pert7} &-c_{\rm X}^2\nabla^2\Phi=4\pi a^2G\left[D(X)\delta\nu+E(X)\delta\lambda\right]\,,\end{aligned}$$ in which $$\begin{aligned} c_{\rm X}^2&\equiv& B(X)/A(X)\,,\\ D(X)&\equiv&-2X\lambda Y_{X\nu}c_{\rm X}^2\,,\\ E(X)&\equiv& -2XY_Xc_{\rm X}^2\,.\end{aligned}$$ The evolution of $\Phi$ in terms of $\rho$ and $\sigma$ perturbations can be written as [@1999PhLB..458..219G] $$\begin{aligned} \nonumber \Phi^{\prime\prime}&+3\mathcal{H}\left(1+c_{\rm s}^2\right)\Phi^\prime+\left[2\mathcal{H}^\prime+\left(1+3c_{\rm s}^2\right)\mathcal{H}^2\right]\Phi+\\ \label{pert8} &-c_{\rm s}^2\nabla^2\Phi=4\pi a^2G\zeta\delta\sigma\,,\end{aligned}$$ where $$\begin{aligned} c_{\rm s}^2\equiv\partial P/\partial\rho|_\sigma\end{aligned}$$ is the square of the adiabatic speed of sound and $\zeta\equiv\partial P/\partial\sigma|_\rho$. From the above considerations, one can define the entropy perturbation shift, $\Delta$, which quantifies how much $\delta P/\delta \rho$ departs from $c_{\rm s}^2$ [@2014CQGra..31e5006P]. It can be written as: $$\label{pert9} \Delta=\left(\frac{\delta P}{\delta \rho}-c_{\rm s}^2\right)\frac{\delta\rho}{P}=-\frac{D(X)\delta\nu+E(X)\delta\lambda}{P}\,.$$ For isentropic fluids (see Sec. \[sec:thermodynamics\]), it immediately follows that $\zeta\equiv0$. This is in agreement with our previous outcomes, since from Eq.  we require $Y_X\neq0$,[^4] and so $c_{\rm s}^2\equiv0$ as one assumes $P={\rm const}$ and vice-versa . Taking into account that $P={\rm const}$, we may draw relevant consequences on our fluid temperature. Indeed, according to Eq. (\[gibbsdens\]) we find our fluid to lie on the minimum of the Gibbs energy, i.e. at an equilibrium state. By combining Eqs. (\[eq:gibbsduhem\]), (\[eq:dhelmotz\]), and (\[eq:solutions4\]) we get $$\label{equilibrium} dg = dP - s\,dT = 0\,.$$ Since $P={\rm const}$, one necessarily has $T={\rm const}$ *in the proximity of each minimum of the effective potential*. Last but not least, it is worth noticing that an isentropic fluid can be even attained from Eq.  by setting $\lambda\rightarrow0$. However, this would represent a particular case for which the Lagrangian term $Y$ has no longer relevance. For the sake of generality, this case is thus excluded into our picture. Considerations on quantum vacuum energy {#vacuumenergy} ======================================= We now analyze in more details the role played by the effective potential $V^{\rm eff}$. In particular, we wonder whether the two possible choices of the off-set $V_0$ provide different physical considerations. Hence, to alleviate the degeneracy between the two approaches, we need to fix the magnitude associated to $K$. Our target is to bound $K$ in order to heal the fine-tuning issue associated to the cosmological constant $\Lambda$. We thus explore two possibilities: - [$V_0= -\chi\varphi_0^4/4$, so BT we have $V^{\rm eff}=0$ and hence $$\begin{aligned} \label{s8bbb} P_1&= \left\{ \begin{array}{ll} K\quad &\quad{\rm (BT)}\\ K+\chi \varphi_0^4/4\quad&\quad{\rm (AT)} \end{array} \right.,\\ \label{s8bisbbb} \rho_1&= \left\{ \begin{array}{ll} 2X\lambda Y_X-K\quad &\quad{\rm (BT)}\\ 2X\lambda Y_X-K-\chi \varphi_0^4/4\quad&\quad{\rm (AT)} \end{array} \right.,\end{aligned}$$ and by virtue of Eq. , then $K<-\chi \varphi_0^4/4$.]{} - [$V_0=0$, so AT we have $V^{\rm eff}=0$ and hence $$\begin{aligned} \label{s8ccc} P_2&= \left\{ \begin{array}{ll} K-\chi \varphi_0^4/4\quad &\quad{\rm (BT)}\\ K\quad&\quad{\rm (AT)} \end{array} \right.,\\ \label{s8bisccc} \rho_2&= \left\{ \begin{array}{ll} 2X\lambda Y_X-K+\chi \varphi_0^4/4\quad &\quad{\rm (BT)}\\ 2X\lambda Y_X-K\quad&\quad{\rm (AT)} \end{array} \right.,\end{aligned}$$ and again, by virtue of Eq. , then, $K<0$.]{} In both cases $K<0$, but with different magnitudes. ### The case $V_0= -\chi\varphi_0^4/4$ Since $X_\varphi=0$ and $P={\rm const}$, from Eq. (\[eq:no16\]) we get that $\eta_\varphi=0$, and, therefore, Eq. (\[eq:no9b\]) reduces to $$\label{eq:no9bb} \dot{\lambda} = - \theta\lambda\,.$$ In the Friedmann-Robertson-Walker spacetime, $\varphi$ is a function of the time only, thus it is easy to demonstrate that $X\equiv\dot{\varphi}^2/2$ and $\theta\equiv 3\dot{a}/a$. Finally, the solution of Eq. (\[eq:no9bb\]) becomes $$\label{lambdaphia} \lambda = \lambda_0 a^{-3}\,,$$ where $\lambda_0$ is a constant. Further, in the Friedmann-Robertson-Walker scenario the simplest choice for the (adiabatic) volume may be $\mathcal V=\mathcal V_0a^3$, where $\mathcal V_0$ the initial volume. Recalling that our fluid is isentropic, with constant $P$ and $T$, by using Eqs. (\[eq:solutions2\]) and (\[lambdaphia\]), we get $$\label{isentropic} s\mathcal{V}=\sqrt{2X}\lambda_0\mathcal V_0 Y_X={\rm const}\,,$$ from which it follows that $Y_X=Y_{{\rm BM},X}+Y_{{\rm DM},X}={\rm const}$. We propose the following assumptions: $$\begin{aligned} K_{\rm DM}&\approx&-\chi\varphi_0^4/4\,,\\ K_{\rm BM}&\ll& K_{\rm DM}\,.\end{aligned}$$ These positions and the fact that $a=(1+z)^{-1}$ (where $z$ is the redshift) allow us to rewrite Eqs. – as $$\begin{aligned} \label{s8bbbb} P_1&\approx \left\{ \begin{array}{ll} K_{\rm DM}\quad&\quad{\rm (BT)}\\ K_{\rm BM}\quad&\quad{\rm (AT)} \end{array} \right.,\\ \label{s8bisbbbb} \rho_1&\approx \left\{ \begin{array}{ll} \left(\rho_{\rm DM}+\rho_{\rm BM}\right)\left(1+z\right)^3-K_{\rm DM}\quad &\quad{\rm (BT)}\\ \left(\rho_{\rm DM}+\rho_{\rm BM}\right)\left(1+z\right)^3-K_{\rm BM}\quad&\quad{\rm (AT)} \end{array} \right.,\end{aligned}$$ where $\rho_{\rm BM}=2X\lambda_0Y_{{\rm BM},X}$ and $\rho_{\rm DM}=2X\lambda_0Y_{{\rm DM},X}$ are constants. This mechanism elides the vacuum energy cosmological constant contribution through the use of DM. As $\chi>0$, the sign of $K_{\rm DM}$ is opposite to the vacuum energy term. Hence, from the one hand the DM fluid pushes the universe up to accelerate, while on the other hand vacuum energy provides the opposite contribution in the net pressure. Then, AT the universe accelerates *because of the presence of a negative baryonic pressure*. This plays the role of *emergent cosmological constant*, which is is negligible with respect to the vacuum energy BT, whereas becomes dominant AT. Since its magnitude is due to the baryon pressure, this alleviates the coincidence problem. In addition the fine-tuning problem is clearly removed because the high value of the predicted vacuum energy density is suppressed and does not enter our framework AT. In Fig. \[fig:1\] we compare the observational Hubble parameter data (OHD) $H(z)$ (see the black datapoints) with the predictions of our model in Eq. . The OHD are model-independent measurements of the evolution of the Hubble parameter with redshift from differential age of two galaxies at the same redshift. The most updated OHD values have been taken from [@2018MNRAS.476.3924C]. From Eq. , we have AT the Hubble parameter can be written as $$\label{parOL} H(z)\equiv H_0\sqrt{\frac{2X\lambda_0Y_X}{\rho_{\rm c,0}}\left(1+z\right)^3+\frac{K_{\rm BM}}{\rho_{\rm c,0}}}\,,$$ where we can identify the BM+DM density parameter with $\Omega_{\rm m}\equiv 2X\lambda_0Y_X/\rho_{\rm c,0}$ and the dark energy density parameter with $\Omega_\Lambda\equiv -K_{\rm BM}/\rho_{\rm c,0}$. Our predictions can be constrained with the most recent results on the Hubble constant $H_0=(67.74\pm0.46)$ km s$^{-1}$ Mpc$^{-1}$, the density parameters $\Omega_{\rm m}=0.3089\pm0.0062$ and $\Omega_\Lambda=0.6911\pm0.0062$, and current value of the universe critical density $\rho_{\rm c,0}=(8.62\pm0.12)\times10^{-30}$ g/cm$^3$ obtained by *Planck* . These constraints result in the solid blue curve and the $1$–$\sigma$ error limits (the dashed blue curves) shown in Fig. \[fig:1\]. ![$H(z)$ dataset from [@2018MNRAS.476.3924C] (black data) compared with the results of our model described by Eq.  (solid blue curve). The $1$–$\sigma$ error limits (dashed blue curves) have been obtained by using the best-fit parameters from .[]{data-label="fig:1"}](Hz){width="\hsize"} ### The case $V_0=0$ Eqs. (\[eq:no9bb\])–(\[isentropic\]) still hold and retain the same form. However, in this case the only needed assumption to get the measured cosmological constant is that $K\ll \chi\varphi_0^4/4$. Therefore we obtain $$\begin{aligned} \label{s8cccc} P_2&\approx \left\{ \begin{array}{ll} -\chi \varphi_0^4/4\quad &\quad{\rm (BT)}\\ K\quad&\quad{\rm (AT)} \end{array} \right.,\\ \label{s8biscccc} \rho_2&= \left\{ \begin{array}{ll} \left(\rho_{\rm DM}+\rho_{\rm BM}\right)\left(1+z\right)^3+\chi \varphi_0^4/4\ &\ {\rm (BT)}\\ \left(\rho_{\rm DM}+\rho_{\rm BM}\right)\left(1+z\right)^3-K\ &\ {\rm (AT)} \end{array} \right.,\end{aligned}$$ where the BM can be considered even pressureless. In this case the vacuum energy density cancels without the effect of any matter component. This occurrence is due to the discontinuity of the effective potential introduced by the phase transition only. The emergent cosmological constant appears soon after the transition as related to the DM sector of the universe and holds the *ad hoc* value to justify the observed acceleration of the universe. Therefore, this case still suffers from the coincidence problem, which affects the $\Lambda$CDM model. Moreover, differently from the previous case and in analogy with the concordance model, the baryons do not play a significant role in speeding up the universe. Indeed, they can be viewed as pressureless particles. Temperature and mass of the DM candidate {#DMparticle} ======================================== As discussed above, the $V_0= -\chi\varphi_0^4/4$ case is preferred over $V_0=0$, to avoid discontinuities in the pressure contribution. In so doing, one may break the degeneracy between the two approaches, choosing the case $V_0= -\chi\varphi_0^4/4$ which corresponds to a dark fluid defined by matter with pressure. Thus, limiting on $V_0= -\chi\varphi_0^4/4$ we draw in the thermal universe the bounds over the DM constituent as particle candidate for DM enabling the process for that the DM pressure elides the vacuum energy contribution.[^5] The energy and number densities, together with the pressure of each particles having mass $m$, momentum $p$ and equilibrium temperature $T$, can be computed as [@2008cosm.book.....W] $$\begin{aligned} \label{fd_endens} \epsilon=&\,g\frac{(k_{\rm B}T)^4}{2\pi^2\hbar^3c^3}\int^{\infty}_{0}\frac{\xi^2\sqrt{\xi^2+A^2}}{e^{\sqrt{\xi^2+A^2}}\pm1}d\xi\,,\\ \label{fd_partdens} n=&\,g\frac{(k_{\rm B}T)^3}{2\pi^2\hbar^3c^3}\int^{\infty}_{0}\frac{\xi^2}{e^{\sqrt{\xi^2+A^2}}\pm1}d\xi\,,\\ \label{fd_pressdens} P=&\,g\frac{(k_{\rm B}T)^4}{2\pi^2\hbar^3c^3}\int^{\infty}_{0}\frac{\xi^4}{3\sqrt{\xi^2+A^2}}\frac{d\xi}{e^{\sqrt{\xi^2+A^2}}\pm1}\,,\end{aligned}$$ where $\xi=pc/(k_{\rm B}T)$, $A=mc^2/(k_{\rm B}T)$, $g=2s+1$ is the spin $s$ degeneracy parameter, $c$ the speed of light, $\hbar$ the reduced Planck constant and $k_{\rm B}$ the Boltzmann constant. Here the choice “$\pm$” distinguishes fermions and bosons, respectively. The entropy density is simply given by $$\label{fd_entrdens} s=\frac{g k_{\rm B}^4T^3}{6\pi^2\hbar^3c^3}\int^{\infty}_{0}\frac{4\xi^2+3A^2}{\sqrt{\xi^2+A^2}}\frac{\xi^2d\xi}{e^{\sqrt{\xi^2+A^2}}\pm1}\,.$$ We focus on bosons since $\mathcal L_1$ has been written for bosons only. The DM constituents are in our picture bosons that at early times behave as relativistic particles ($mc^2\ll k_{\rm B}T$), in thermal equilibrium. The energy density of all bosons (b) and fermions (f) species comes by summing up Eq.  for each of them, i.e., $$\label{endenstot2} \epsilon_{\rm BT}=g_{*}\frac{\pi^2(k_{\rm B}T_{\rm p})^4}{30(\hbar c)^3}\,,$$ where $g_{*}$ is the sum of the standard term $g_{*}^{\rm ST}=\sum_b g_b+\frac{7}{8}\sum_f g_f\approx 106.75$ [@2008cosm.book.....W] and our DM particle term $g_{\rm DM}=2s_{\rm DM}+1$ with spin $s_{\rm DM}$. Independently from the offset on $V_0$, the BT total energy density in Eqs. – is given by $$\label{endenstot} \epsilon_{\rm BT}=\left[\Omega_{\rm r}\left(\frac{T_{\rm p}}{T_0}\right)^4 + \Omega_{\rm m}\left(\frac{T_{\rm p}}{T_0}\right)^3 + \Omega_\Lambda \right]\epsilon_{\rm c} + \epsilon_{\rm v}\,,$$ where $\Omega_{\rm r}=(9.16\pm0.19)\times10^{-5}$ is the radiation density parameter , and $\epsilon_{\rm c}=\rho_{\rm c}c^2$, where $\rho_{\rm c}=3H^2/(8\pi G)$ is the universe critical density, in which $H$ is the Hubble parameter and $G$ the gravitational constant. As already stated in Sec. \[vacuumenergy\], the current value of the universe critical density is $\rho_{\rm c,0}=(8.62\pm0.12)\times10^{-30}$ g/cm$^3$ . With respect to Eqs. –, we include the radiation, which is not negligible at early times, and use the relation $T_{\rm p}/T_0=(1+z)$, in which $T_{\rm p}$ is the cosmic plasma temperature and $T_0=2.725$ K the current *Cosmic Microwave Background* temperature.[^6] Finally, $\epsilon_{\rm v}=7.74\times10^{46}$ erg/cm$^3$ is the vacuum energy density. By equating Eq.  and Eq.  and solving numerically, we get the plasma temperature $$T_{\rm p}=(6.6559\pm0.0019)\times10^{14}h(s_{\rm DM})~K\,,$$ where $h(0)=1$, $h(1)=0.995$, and $h(2)=0.991$. The primordial DM interactions can be viewed as the annihilation of a heavier DM particle $Q$ and its antiparticle $\bar{Q}$, both with masses $M$, to produce two lighter particles $q$ and $\bar{q}$. Assuming no initial asymmetry between the particles $Q$ and $\bar{Q}$, their comoving density must be the same, i.e., $n_{Q}\equiv n_{\bar{Q}}\equiv n$; on the other hand $q$ and $\bar{q}$ are tightly coupled to the cosmic plasma. Therefore the Boltzmann equation for the evolution of $n$ writes as $$\label{NQ} \frac{1}{a^3}\frac{d \left(a^3n\right)}{dt}=-\langle \varkappa v\rangle \left(n^2-n_{\rm eq}^2\right)\,,$$ where $n_{\rm eq}$ is the equilibrium number density and $\langle \varkappa v\rangle$ the thermally averaged cross-section. From the entropy conservation we write the number density as an adimensional quantity $N=n k_{\rm B}/s$. Then, we note that the comoving time $t$ is related to $A$ by: $dA=HA\,dt$. Before neutrino decoupling, the entropy density degeneracy parameters is $g_{\rm s}^{*}\equiv g_{*}$ [@2008cosm.book.....W], therefore the total entropy density is given by $s=2\pi^2 k_{\rm B}^4 g_{*} T^3/[45(\hbar c)^3]$. From the identity $\epsilon_{\rm BT}\equiv\rho_{\rm c}c^2$, we obtain $$\label{NQ4} H\equiv\left(\frac{\dot a}{a}\right)=\sqrt{\frac{4\pi^3c^3g_{*}G}{45\hbar^3}}\left(\frac{M}{A}\right)^2\,.$$ From the above definitions, we can recast the Boltzmann equation to obtain a Riccati-like equation $$\label{NQ3} \frac{d N}{dA}=-\frac{\Gamma}{A^2}\left(N^2-N_{\rm eq}^2\right)\,,$$ where we defined the interaction rate $$\label{Gamma} \Gamma\equiv \sqrt{\frac{g_{*}\pi c^3}{45G\hbar^3}} \langle\varkappa v\rangle M\,.$$ Fig. \[fig:2\], shows $N(A)$ for $\Gamma=10^5$, $10^8$, $10^{11}$, and $10^{14}$. The value of the DM relic abundance is given by $$N_\infty\approx A_{\rm f}/\Gamma\,,$$ where $A_{\rm f}$ marks the transition from the relativistic regime to the non-relativistic one. For the above wide range of $\Gamma$, we can safely assume that the non-relativistic regime is attained for $A_{\rm f}=10$–$30$. We assume that at this stage the temperature is approximately the above equilibrium temperature $T_{\rm p}$. For this choice the DM particle mass stays approximately in the range of values $0.5\lesssim M{\rm c^2/ TeV} \lesssim 1.7$, in agreement with the most recent predictions over the WIMPs [@2002JHEP...12..034K; @PhysRevLett.84.5699; @2002PhLB..545...43B]. The precise values depend upon on the value of $h$, which is quite insensitive to $s_{\rm DM}$, as summarized in Tab. \[tab:table1\]. ![Plot of $N(A)$ for $\Gamma=10^5$,$10^8$,$10^{11}$,$10^{14}$. The freeze-out occurs at $A_{\rm f}=11$,$17$,$25$,$32$, respectively.[]{data-label="fig:2"}](DM.eps){width="\hsize"} $s_{\rm DM}$ h($s_{\rm DM}$) M (TeV) -------------- ----------------- ------------- 0 1.000 0.574–1.723 1 0.995 0.572–1.715 2 0.991 0.569–1.708 : \[tab:table1\]The mass range of the DM boson particle candidate depending on the spin particle. Columns list DM spin $s_{\rm DM}$, the function $h(s_{\rm DM})$, and the mass range of the DM particle. Using the above definitions, we now relate the freeze-out abundance of DM relics to its density today, i.e., $\rho_{\rm Q,0}=N_\infty s_0 M$. The DM density parameter is $$\label{relic} \Omega_{\rm Q}=\frac{\rho_{\rm Q,0}}{\rho_{\rm c,0}}=\frac{16g^{*}_{\rm s,0}}{3H_0^2}\sqrt{\frac{G^3\pi^5\hbar^3}{45g_{*}c^3}}\left(\frac{k_{\rm B}T_0}{\hbar c}\right)^3\frac{A_{\rm f}}{\langle \varkappa v\rangle}\,,$$ where $g^{*}_{\rm s,0}=3.91$, and $A_{\rm f}=10$–$30$. Within the proposed case $V_0= -\chi\varphi_0^4/4$, by looking at Eq.  we can impose $\Omega_{\rm Q}\equiv\Omega_{\rm dm}=0.2589\pm0.0057$ in Eq. . This position provides a range of values for the thermally averaged cross-section $0.81\leq\langle \varkappa v\rangle/(10^{-26}{\rm cm}^3{\rm s}^{-1})\leq2.42$.[^7] Predictions of our paradigm {#predictions} =========================== We here sum up the main results of our paradigm. We revise the concordance model, assuming the most general Lagrangian for matter with pressure. To do so, we consider a transition phase induced by the effective potential of a vacuum energy cosmological constant, with a mechanism in which the DM pressure elides the vacuum energy pressure itself. So that we obtain: $$\begin{aligned} &P={\rm const}\ {\rm (always)}\,\Rightarrow\,c_{\rm s}^{\rm DM}\equiv c_{\rm s}^{\rm BM}\equiv0\,,\\ &P<0\ {\rm (from\ thermodynamics)}\,,\\ &T={\rm const}\ {\rm (during\ the\ transition)}\,,\\ &P_{\rm DM}\gg P_{\rm BM}\ ,\ P_{\rm DM}\approx\epsilon_{\rm v}\,,\\ &\rho_\Lambda\equiv P_{\rm DM}\ {\rm (BT)}\,,\\ &\rho_\Lambda\equiv P_{\rm BM}\ {\rm (AT)}\,,\\ % \rho_{\Lambda_M}(z=z_{tr})&\quad perfect\,\, fluid\\ % \rho_{\Lambda_M}(z=0)&\quad perfect\,\, gas\\ &0.5\lesssim M{\rm c^2/ TeV} \lesssim1.7 \ {\rm (Cold\ Dark\ Matter)}\,,\\ &0.81\leq\langle \varkappa v\rangle/(10^{-26}{\rm cm}^3{\rm s}^{-1})\leq2.42\,.\end{aligned}$$ Hence, in our scheme there exists only *one perfect, irrotational, and isentropic fluid*, composed of BM and DM. $\Lambda$ is coupled with the matter. The thermodynamics of such a fluid *naturally* suggests an emergent negative pressure. The effective potential $V^{\rm eff}$ induces a transition phase during which the quantum vacuum energy density mutually cancels with the DM pressure. Soon after the transition the emergent cosmological constant is given by the (negative) pressure of baryons. This overcomes the fine-tuning problem between the predicted and observed values of $\Lambda$ and the coincidence problem, due to the fact that it is the matter which induces the effective cosmological constant at late times and, therefore, it is natural that their magnitudes are extremely close today. The model mimes the $\Lambda$CDM effects, without departing from observations made at both late and early stages of universe’s evolution . As principal responsible for DM in the universe, our predictions on the mass constituents, i.e. $0.5\lesssim M\lesssim1.7$ TeV, leave open the possibility to detect in laboratory additional heavier bosons, e.g. for example additional $Z^\prime$ or $W^\prime$ bosons or Leptoquarks as potentially predicted by extensions of the particle standard model. Final outlooks and perspectives {#conclusions} =============================== In this work, we proposed an alternative model to the standard $\Lambda$CDM paradigm. We assumed the existence of a single fluid composed by matter only, i.e. baryons and cold DM. The fluid pushes the universe up, canceling the quantum contribution due to the cosmological constant through the assumption that matter shows a non-vanishing pressure. In particular, we proposed that both DM and BM are collisional, through a generalized scalar field $\varphi$ representation of the matter fluid Lagrangian $\mathcal{L}_1$ depending upon a kinetic term, $X$, and a Lagrange multiplier, $\lambda$. We even included a potential, $V^{\rm eff}$, which models the coupling with the standard cosmological constant and induces a phase transition. We described the thermodynamics of our matter fluid, showing that it is *perfect, irrotational, and isentropic*. Moreover, we demonstrated that the positiveness of the Helmotz energy *naturally* suggests a negative pressure. We showed the existence of a Noether current due to the shift symmetry, which coincided with the entropy density current $s^\alpha$, making the Lagrangian independent from $\varphi$. Thus, we assumed a homogeneous and isotropic space-time to investigate small perturbations and we found that the adiabatic sound speed naturally vanishes, leading to a constant pressure, but with an evolving energy density, differently from the standard $\Lambda$CDM model. To this end, we mostly analyzed the role of the effective potential, $V^{\rm eff}$. To do so, we managed the off-set $V_0$ by analyzing two possibilities: $V_0= -\chi\varphi_0^4/4$ and $V_0=0$. In the first case ($V_0= -\chi\varphi_0^4/4$), the effective potential induced a phase transition during which the quantum vacuum energy density mutually cancels with the DM pressure. This mechanism has consequences even as the transition stops. Indeed, soon AT the emergent cosmological constant, able to accelerate the universe today, is given by the (negative) pressure of BM. This achievement overcomes both the fine-tuning and the coincidence problem. The fine-tuning problem is overcome since the contributions due to the vacuum energy is canceled out through DM. The coincidence problem is healed since it is the matter which induces the effective cosmological constant at late times. Thus, it is natural to presume that their magnitudes are extremely close today. In the second case ($V_0=0$), the DM does not play an active role in erasing the quantum field vacuum energy density and BM can be viewed as pressureless. Hence, this landscape does not offer solutions for the fine-tuning and the coincidence caveats. As a consequence it degenerates with the previous case and can be identified with the standard $\Lambda$CDM paradigm. The so-obtained dark fluid is thus mimed by a matter fluid with pressure in which the minimum favors the first case. Further, both cases manifested constant pressure and constant Gibbs free-energy during the universe evolution, with $T={\rm const}$ during transition as naturally expected for first order phase transition. In addition, we related the predictions of our model to observations by directly comparing the energy density of the cosmic plasma with the one described by our matter fluid. The temperature at which the transition occurred is in quite good agreement with early-time temperatures of hot plasma. Afterwards, from the study of the DM relic abundance for the preferred case with $V_0= -\chi\varphi_0^4/4$, we posed stringent limits on the mass, $0.5\lesssim M{\rm c^2/ TeV} \lesssim1.7$, and the thermally averaged cross-section, $0.81\leq\langle \varkappa v\rangle/(10^{-26}{\rm cm}^3{\rm s}^{-1})\leq2.42$, of the DM particle candidate. These estimates are quite independent from the spin of DM particles. In future works, we will study inflationary scenarios, naturally arising from Eqs. –, through our hypothesis of matter with non-vanishing pressure. We will better analyze also additional symmetries of our Lagrangians and we will put more stringent constraints on the DM particle, bounding the cross-section from current DM experiments. O. L. thanks Danilo Babusci, Stefano Bellucci, Salvatore Capozziello, Raffaele Marotta, Hernando Quevedo, Luigi Rosa and Patrizia Vitale for their suggestions. He is also grateful to Peter K. S. Dunsby for the discussions on the topic of adiabatic fluids in cosmology, made at the University of Cape Town. M. M thanks Enrico Nardi for enlightening discussions about the topic of dark matter relic abundance. Derivation of the equations in Sec. \[sec:action\] {#appA} ================================================== From the variation of the action in Sec. \[sec:action\] we get $$\begin{aligned} \nonumber \delta S = &\int \left[ \left(K_\varphi - V^{\rm eff}_\varphi + \lambda Y_\nu\nu_\varphi \right) \delta\varphi + \mathcal{L}_X \nabla_\alpha\varphi \nabla^\alpha \delta \varphi + Y \delta\lambda + \frac{1}{2} g_{\alpha\beta} \left(\mathcal{L}_X \nabla_\alpha\varphi \nabla^\alpha \varphi - K + V^{\rm eff} - \lambda Y \right) \delta g^{\alpha\beta}\right] \sqrt{-g}{\rm d}^4x =\\ \label{eq:ap1} = & \int \left\{ \left[ \mathcal{L}_\varphi - \nabla_\alpha \left(\mathcal{L}_X \nabla^\alpha \varphi \right) \right] \delta\varphi + Y \delta\lambda + \frac{1}{2} g_{\alpha\beta} \left( \mathcal{L}_X \nabla_\alpha\varphi \nabla^\alpha \varphi - K + V^{\rm eff} - \lambda Y \right) \delta g^{\alpha\beta} \right\} \sqrt{-g}{\rm d}^4x\,.\end{aligned}$$ The variations with respect to $\lambda$, $\varphi$ and $g_{\alpha\beta}$, respectively, lead to Eqs. (\[eq:no5a\]), (\[eq:no5b\]), and (\[eq:no6\]). Eq. (\[eq:no13\]), instead, is obtained trought simple calculations including the vanishing acceleration in Eq. (\[eq:no8\]) $$\begin{aligned} \nonumber \nabla_\alpha T^{\alpha\beta} =\,& \dot{\rho} v^\beta + \nabla_\alpha P v^\alpha v^\beta + \theta \left( \rho + P \right) v^\beta - \nabla^\beta P =\\ \label{eq:ap2} =\,& \left[ \dot{\rho} + \theta \left( \rho + P \right) \right] v^\beta\,,\end{aligned}$$ and the expansion $\theta$ in Eq. (\[eq:no15\]) is obtained as $$\label{eq:ap3} \theta = \frac{\nabla_\alpha \nabla^\alpha \varphi}{\sqrt{2X}} - \frac{X_\varphi}{\sqrt{2X}}\frac{\nabla_\alpha \varphi \nabla^\alpha \varphi}{2X} = \frac{\nabla_\alpha \nabla^\alpha \varphi - X_\varphi}{\sqrt{2X}}\,.$$ By using Eqs. (\[eq:no5a\]) and (\[eq:no16\]), we can recast Eq.  to obtain Eq. (\[eq:no9b\]) $$\begin{aligned} \nonumber &K_\varphi - V^{\rm eff}_\varphi + \lambda Y_\nu \nu_\varphi - \nabla_\alpha \left( K_X - V^{\rm eff}_X + \lambda Y_X \right)\nabla^\alpha \varphi - \left( K_X - V^{\rm eff}_X + \lambda Y_X \right)\nabla_\alpha\nabla^\alpha \varphi =\\ \nonumber =\,&\mathcal{L}_\varphi- \left[ K_{XX}X_\varphi + K_{X\varphi} - V^{\rm eff}_{XX}X_\varphi - V^{\rm eff}_{X\varphi} + \lambda\left(Y_{XX}X_\varphi+Y_{X\nu}\nu_\varphi\right)\right]\nabla_\alpha\varphi\nabla^\alpha\varphi - \mathcal{L}_X \nabla_\alpha\nabla^\alpha \varphi - \nabla_\alpha\lambda Y_X \nabla^\alpha \varphi =\\ \label{eq:ap4} =\,&\mathcal{L}_\varphi - 2X \left(\mathcal{L}_{XX} X_\varphi + \mathcal{L}_{X\varphi} \right) - \left(\sqrt{2X}\theta+X_\varphi\right) \mathcal{L}_X - \sqrt{2X}\dot{\lambda} Y_X = -\eta_\varphi - \frac{\theta}{\sqrt{2X}}\left(\rho+P\right) - \sqrt{2X}\dot{\lambda} Y_X =0\,.\end{aligned}$$ [37]{} E. J. 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[^2]: Higher order derivatives are excluded because of the *Ostrogradski’s theorem*: systems characterized by a non-degenerate Lagrangian dependent on time derivatives of higher than the first leads to a linearly unstable Hamiltonian function. [^3]: This name commonly designates a field that makes explicit a (spontaneously broken) gauge symmetry. [^4]: That is requested to guarantee the validity of Eq. . [^5]: For the sake of clearness, we hereafter restore the usual physical constants, previously set to $1$. [^6]: AT till today the universe is in equilibrium. Therefore, as in Eq. , the entropy conservation implies that $s\propto a^{-3}$ and $T\propto a^{-1}$, as follows from Eq. , whereas during the phase transition the temperature has a constant value $T_{\rm p}$. [^7]: For completeness, one may also deal with the case $V_0=0$, which corresponds to the $\Lambda$CDM case. By looking at Eq. , this time we are forced to impose $\Omega_{\rm Q}\equiv\Omega_{\rm dm}+\Omega_\Lambda$ in Eq. . This position gives as range $2.20\leq\langle \varkappa v\rangle/(10^{-27}{\rm cm}^3{\rm s}^{-1})\leq6.61$. This case however, albeit degenerating with the previous one, is not favored for the requests we made in the previous section.

--- abstract: 'This survey paper is written with the intention of giving a mathematical introduction to filtering techniques for intermittent data assimilation, and to survey some recent advances in the field. The paper is divided into three parts. The first part introduces Bayesian statistics and its application to statistical inference and estimation. Basic aspects of Markov processes, as they typically arise from scientific models in the form of stochastic differential and/or difference equations, are covered in the second part. The third and final part describes the filtering approach to estimation of model states by assimilation of observational data into scientific models. While most of the material is of survey type, very recent advances in the field of nonlinear data assimilation covered in this paper include a discussion of Bayesian inference in the context of optimal transportation and coupling of random variables, as well as a discussion of recent advances in ensemble transform filters. References and sources for further reading material will be listed at the end of each section.' author: - | Sebastian Reich\ Department of Mathematics, University of Potsdam\ \ and\ \ Colin Cotter\ Department of Aeronautics, Imperial College London bibliography: - 'survey.bib' title: 'Ensemble filter techniques for intermittent data assimilation - a survey' ---

--- author: - Guillaume Rivier - Aurélien Crida - Alessandro Morbidelli - Yann Brouet bibliography: - 'ref.bib' date: 'Accepted 29/10/2012' title: | Circum-planetary discs as bottlenecks\ for gas accretion onto giant planets --- Introduction {#sec:Introduction} ============ The detection of exoplanets is probably one of the most striking discoveries of the past 50 years in astrophysics. Up to now, more than 700 exoplanets have been found. This constantly increasing number allows a statistical approach to the analysis of exoplanets. The resulting statistical distributions are then crucial benchmarks for planetary formation models. For an exoplanet, the two easiest parameters to determine are the mass and the semi-major axis of its orbit. Low mass planets are still hard to detect, but it seems that the mass distribution of planets more than 10 Earth masses is double-peaked, with a peak close to the mass of Neptune, and another peak close to the mass of Jupiter [^1]. The first peak is probably an effect of observational biases as low mass planets are still hard to detect, but the second peak seems to be real [@Mayor_2011]. The most popular model for giant planet formation is the core accretion model of @Bodenheimer_1986. In this model, a solid core grows, embedded in a gaseous proto-planetary disc. The gas within the Bondi radius [@Bodenheimer_2000] is gravitationally bound to the core, and forms an envelope. When the planet reaches a critical core mass of about $12-16\ M_{\oplus}$ [@Pollack_1996; @Tajima_1997], the envelope starts to contract quasi-statically and gas accretion rate increases, for an evolution timescale of a few million years. This long phase is followed by a runaway gas accretion stage occurring as soon as both core and envelope masses are approximately equal, during which quasi-static contraction and thus gas accretion rate dramatically increase. This leads to an exponential growth of the planet, in a few $10^5$ yr, which proceeds as long as there is gas available. This raises the question: what sets the terminal mass of a giant planet? When a planet becomes massive enough (typically a Saturn mass), it starts to open a gap in the disc around its orbit [@Papaloizou-Lin-1984; @Lin_1986; @Crida_2006]. However, gap opening does not necessarily imply that the gasflow onto the planet has ceased [@Artymowicz_1996] because the gap is not totally empty. Hydro-dynamical simulations [@Bryden_1999; @Kley_1999; @Lubow_1999; @Lubow_2006] show that a Jupiter-mass planet keeps accreting gas almost as fast as if gap-opening had not occurred. Accretion stalls by gap-opening only once a mass in the range of $5-10$ Jupiter masses is achieved. This is at odds with the masses of the giant planets in the solar system and of many of the extra-solar giant planets. In particular, the existence of Saturn-mass planets is a mystery in this scenario. An often proposed solution [e.g.: @Thommes_2008] is that the giant planets form in “dead zones” – regions in the disc with very low viscosity – and, consequently, open gaps that are much wider and cleaner than previously thought. These gaps could inhibit accretion for planet masses of order of a Jupiter mass or less. However, this idea fails because, as demonstrated by @Crida_2006, the depth and width of a gap does not depend just on viscosity but also on the temperature (i.e. scale height) of the disc. Thus, even in a dead zone the gap opened by a Jupiter-mass planet should not inhibit significantly the accretion of gas into the planet’s Hill sphere. Thus, we believe that the question on the terminal mass of giant planets is still open. We notice, however, that the @Pollack_1996 model considered an omnidirectional gas accretion towards the planet; yet, it is physically impossible to reach this state due to angular momentum conservation of the inflowing gas. In fact, gas must form a consistent circum-planetary disc (CPD) once the mass of the planet is higher than about a hundred Earth masses and the planetary atmosphere has shrunk well inside the planet’s Hill radius [@Ward_2010]. This has been confirmed by numerical simulations [e.g. @Ayliffe_2009a; @Ayliffe_2009b]. Once a CPD is formed, the accretion rate of the planet depends on the ability of gas to lose angular momentum, either by its re-distribution within the disc (through viscosity) or exchange with external perturbers (e.g. the star). Most of the previous simulations considered a significant viscosity in the CPD, comparable to that of the active zones of the circum-stellar disc. In this case, angular momentum is redistributed very quickly and the accretion of gas from the CPD onto the planet is extremely fast, so that the role of the disc can generally be neglected as shown by @Papaloizou_2005 who assumed a viscosity about $\alpha = 10^{-3}$. However, it is possible that the CPD is in an MRI dead state as suggested by @Turner_2010, leading to vanishing turbulence and a very low viscosity. Indeed, if the circum-planetary disc has a very low viscosity, then the transport of angular momentum through this disc can be very inefficient and gas can only accrete onto the planet at the rate allowed by the removal of angular momentum by external perturbations, such as the stellar tide. Consequently, the circum-planetary disc may act as a bottleneck for gas accretion and dramatically slow down the planet’s growth. If, as a result, the accretion of a planet does not enter the runaway phase but proceeds at a more regular pace on a timescale comparable to the circum-stellar disc lifetime (a few million years) [@Haisch_2001; @Hillenbrand_2008] then the observed mass spectrum of the giant planets may be the consequence of the competition between gas accretion and gas dissipation. To test whether the idea of a slow accretion rate through a low-viscosity disc is realistic, we study in this paper the effect of the stellar tides. We are aware that Reynolds stresses from waves driven from the circum-solar disc may allow angular momentum losses in the CPD as well. However, the accretion rate due solely to the solar tide provides a lower-bound to the real accretion rate for a CPD of vanishing viscosity. If the result is encouraging —i.e. the mass-doubling time for a Jupiter-mass planet from solar tides is longer than the proto-planetary disc lifetime— then the idea of a low-viscosity CPD as a regulator of the gas accretion rate onto a giant planet is promising. In this case, future work will have to evaluate in detail the gas accretion rate with realistic 3D hydro-dynamical simulations. Thus, our present paper should be considered as a first step in a long research plan. The structure of the paper is the following. In Sect. \[sec:model\] we elaborate an idealised semi-analytical model to evaluate the steady state accretion rate of a planet that is surrounded by an inviscid disc which undergoes vertical gas inflow and is submitted to a tidal torque from a distant star. Then we evaluate the two key parameters that enter in this model. The vertical gas influx rate is estimated in Sect. \[sec:gas-inflow\] from 3D simulations available in the literature. To compute the stellar torque, in Sect. \[sec:numerical-simulations\] we perform 2D hydrodynamical simulations of a disc centred on a Jupiter-mass planet. The disc feels the gravitational perturbation from the star, assumed to be on a distant, circular orbit. We measure the torque due to the stellar tides, after a steady-state is reached. With these two values in hand, in Sect. \[sec:results\] we estimate the planet accretion rate, expressed as a timescale for the doubling of the mass of a Jupiter-planet. Finally, we interpret these results and discuss their possible implications for giant planet formation theories in Sect. \[sec:conclusion\]. Model {#sec:model} ===== The tidal torque and the accretion rate are not related in a trivial way in an inviscid disc. In this section, we present a simple model to derive an accretion rate on the planet from the tidal torque that we will measure in Sect. \[sec:results\]. Depletion of the CPD {#sec:depletion-cpd} -------------------- The torque exerted by the star on a ring of the disc is not entirely deposited locally in the disc. Only a fraction of the torque is deposited. This fraction depends on the disc viscosity and tends to zero for vanishing viscosity [@Martin_2011]. The rest (i.e. the totality of the torque in the case of an inviscid disc) is passed to the adjacent rings through pressure effects [@Crida_2006 Appendix C]. If no torque is deposited on a ring, then the gas in the ring does not lose angular momentum and does not move towards the central planet. It would be incorrect, though, to expect that the wave raised by the star does not promote gas accretion from an inviscid disc. The torque is transmitted from each annulus of the disc to its internal neighbour by pressure all the way to the inner edge of the disc (at the boundary with the atmosphere). This inner annulus, not being supported by anything interior to it on a Keplerian motion, then loses angular momentum and falls onto the planet. Once the innermost annulus is emptied, the same fate occurs to the second innermost annulus and so forth. Thus, an inviscid disc is depleted from the inside out as gas is accreted onto the planet’s atmosphere. More precisely, let us note $X$ the radius of the region of the CPD that is emptied by the torque over a time $t$. The total orbital angular momentum of the gas present within $r<X$ is given by: $$\begin{aligned} L & = & \int_0^X \Sigma 2\pi r \times r(r\Omega) \,dr\\ & = & \frac{4\pi}{5} \Sigma \sqrt{G M_p} X^{5/2}\end{aligned}$$ (assuming the surface density of the gas, $\Sigma$ is independent of $r$). Here, $ \Omega = \sqrt{\frac{GM_{p}}{r^3}} $ is the angular speed of the gas in Keplerian motion at a radius $r$, with $M_p$ the mass of the planet. By definition of $X$, $L$ should be exactly opposite to the total angular momentum taken by the star, which is the torque $T$, accumulated over time: $T\times t$. This gives: $$X(t) = \left(\frac{5}{4\pi}\frac{|T|}{\Sigma \sqrt{G M_p} }\right)^{2/5}t^{2/5}\ . \label{eq:Xt}$$ We see that the radius $X$ increases with time, as expected. Infilling of the CPD {#sec:filling-cpd} -------------------- Here, we assume that the CPD is non-viscous. Thus, the CPD doesn’t spread radially, and any gas input at its outer edge just stays there. However, @Machida_2008 and @Tanigawa_2011 have shown that vertical inflow onto the CPD is the main source mass of the CPD. Let us call $C$ this vertical mass flux per surface unit of the CPD. Then, the surface density in an initially void region is $\Sigma = Ct$. Input into Eq. (\[eq:Xt\]), this gives: $$X_C = \left(\frac{5}{4\pi}\frac{|T|}{C \sqrt{G M_p} }\right)^{2/5}\ . \label{eq:XC}$$ This implies that there is a radius $X_C$ (independent of time) such that inside $r<X_C$, depletion due to the stellar torque $T$ and infilling due to the vertical inflow $C$ exactly balance out. In an arbitrary time span, all the gas input within a circle of radius $X_C$ around the planet has exactly the orbital angular momentum taken by the stellar torque. As a consequence, the whole gas of this region falls onto the planet while the vertical inflow completely refills it in the same time. Thus, the region within $X_C$ reaches a steady-state density. The disc outside $X_C$, however, does not accrete and gas piles up. This issue will be discussed later in Sect. \[sec:gas-piling\]. Accretion rate estimate {#sec:accretion-rate-estim} ----------------------- We can now write the planet accretion rate as a function of the torque. The mass flow on the planet is simply the mass flow inside $r<X_C$: $$\begin{aligned} \dot M_p & = & C \pi {X_C}^2\end{aligned}$$ $$\begin{aligned} \dot M_p & = & \pi \left(\frac{5}{4\pi \sqrt{G M_p}}\right)^{4/5} T^{4/5} C^{1/5} \label{accr-rate}\end{aligned}$$ Note that the accretion rate depends weakly on the gas injection flux $C$, whereas it depends almost linearly on the torque exerted by the star $T$. This equation is central in this paper, as it links the stellar torque to the planetary accretion rate. The issue of gas-piling beyond $X_C$ {#sec:gas-piling} ------------------------------------ The model introduced in Sect. \[sec:filling-cpd\] raises the issue of gas-piling in the outer part of the disc. Indeed, if the density in the outer disc continued to increase, the total torque $T$ exerted through the wave would increase as well, as it is proportional to $\Sigma$. This would imply an increase of the accretion radius $X$, and of the accretion rate $\dot M_p$. Moreover, the density in the outer disc could reach the gravitational instability limit, changing completely the structure of the CPD and leading to outbursts of accretion, as suggested by @Martin_2011. In reality, though, gas cannot pile-up indefinitely. If the density in the CPD becomes too large, gas is repelled by pressure and the inflow from the circum-stellar disc has to stop. Thus, in this section we estimate the density in the outer CPD, by imposing the pressure equilibrium between the CPD and the circum-stellar disc (CSD). Our approach is detailed below. We assume the CSD is much thicker than the CPD. This, and the fact that the inflow from the CSD to the CPD is along the vertical direction [@Tanigawa_2011] suggests that the CSD “weights” on top of the surface of the CPD. Then, at equilibrium, the pressure in the CPD at a height $H_{\rm CPD}$ has to be equal to the pressure of the CSD at the same height. Let us now assume a standard Gaussian pressure profile in both discs: $P(z) = P_0 \exp \left(-\frac{1}{2} (\frac{z}{H})^2 \right)$ where $P_0$ is the pressure at the mid-plane (denoted hereafter $P_0^{\rm CSD}$ and $P_0^{\rm CPD}$ for the CSD and CPD respectively). At height $H_{\rm CPD}$, the pressure in the CSD is $P_0^{\rm CSD}$, given our assumption that $H_{\rm CSD} \gg H_{\rm CPD}$. In the CPD, $P(H_{\rm CPD})=P_0^{\rm CPD}/\sqrt{e}$. $P_0$ and $\Sigma$ are linked by the equation of state $P_0 = c_s^2 \Sigma$ (where $c_s=H\Omega$ is the sound speed). The equation $P_{\rm CPD}(H_{\rm CPD})=P_{\rm CSD}(0)$ can then be solved for the sole unknown of the problem: $$\Sigma_{\rm CPD}(r) = \sqrt{e} \left( \frac{(H/r_\odot)_{\rm CSD}}{(H/r)_{\rm CPD}} \right)^2 \frac{M_\odot}{M_p} \frac{r}{r_\odot} \Sigma_{\rm CSD}$$ where $M_\odot$ is the mass of the star and $r_\odot$ the semi-major axis of the planet. For $(H/r_\odot)_{\rm CSD}=0.05$ [@Pietu_2007], $(H/r)_{\rm CPD}=0.3$ [@Ayliffe_2009b], $M_\odot=10^3M_p$, $r_\odot = 14.4\ R_H$ (see simulation set-up in \[sec:code\]) and $r = 0.3\ R_H$ which corresponds to the truncation radius of the CPD, as explained in \[sec:numerical-parameters\], one gets $H_{\rm CSD}=8\times H_{\rm CPD}$, justifying our assumption. This gives: $$\Sigma_{\rm CPD}(r) = \left(\frac{r}{0.31\,R_H}\right) \Sigma_{\rm CSD} \label{eq:Sigma_CPD}$$ with $R_H$ the Hill radius of the planet: $R_H = r_\odot\left(\frac{M_{p}}{3M_{\odot}}\right)^{1/3}$. So, we estimate that the surface density of the CPD at $0.3\,R_H$ is about equal to the local surface density of the CSD, and can not exceed this value. This value of the density will be used in our simulations for tidal torque calculation in Sect.\[sec:gas-parameters\]. Gas inflow estimate {#sec:gas-inflow} =================== To apply the method outlined in Sect. \[sec:model\], we need to estimate $C$: the flux per unit area of the gas injected onto the CPD. To do that, we use the results from @Ayliffe_2009b. Those simulations give the flux of gas through the Hill sphere ($M_H$, identified in that paper as the “planet accretion rate”). For a planet of mass $M_p = 333\,M_{\oplus}$ (which is consistent with our choice of parameters) and a locally isothermal disc, @Ayliffe_2009b find $\dot M_H = 8 \times 10^{-5}\ \mathrm{M_{\mathrm{Jup}}/yr}$. Given that almost all of this flux is vertical [@Tanigawa_2011], we can set $\dot M_H=C \pi R_{H}^2$, which gives $$C = 2.54 \times 10^{-5}\ \mathrm{M_{Jup}/yr/R_H^{\ 2}}\ . \label{eq:C_num}$$ Numerical simulations and Torque computation {#sec:numerical-simulations} ============================================ The model introduced in Sect. \[sec:model\] allows us to derive analytically an accretion rate onto the planet, given a gas inflow on the whole disc and an accurate knowledge of tidal effects from the star, i.e. an estimate of the stellar torque on the disc. Having evaluated the former in the previous section, here we describe how we compute the torque numerically. We start by reviewing the set-up of our simulations. Simulation set-up {#sec:simulation-set-up} ----------------- ### Code {#sec:code} To run simulations, we have used the code FARGO[^2] from @Masset_2000a [@Masset_2000b], which is a 2D Eulerian, non self-gravitating, isothermal code, with a polar grid. FARGO is very suitable for simulations of the interactions between planets and the proto-planetary disc. Here, we focus on the environment of the planet. Thus, we have slightly modified FARGO to adapt it to our case of study. From now on, the central body will be the planet, while the star will be the orbiting body around the planet and its disc. Simulations take place in a planetocentric frame, corotating with the star. The orbit of the star is circular and fixed. The mass unit is the mass of the central body: the planet, $M_p$. The distance unit is $R_H$. The time unit is $\frac{1}{\Omega_H}$, where $\Omega_H = \sqrt{GM_{p}/R_H^{\,3}}$ is the angular speed at $r=R_H$. Note that the star’s angular speed is thus $\Omega_{\odot} \sim \sqrt{\frac{GM_{\odot}}{r_{\odot}^3}} = \frac{ \Omega_H}{\sqrt{3}}$. The mass of the star is taken to be $ M_\odot = 1000\ M_p$, which makes our system scaled like the Jupiter–Sun system. This gives $r_\odot = 14.4\ R_H$. The star being out of the grid, we do not apply any smoothing in the computation of the force exerted by the star onto the disc. This is actually necessary for an accurate computation of the differential, tidal forces. ### Gas parameters {#sec:gas-parameters} In our simulations, we take an initial density profile $\Sigma(r)=\Sigma_0 \left(\frac{r}{R_H}\right)^{-\alpha}$, with $\Sigma_0 = 5\times 10^{-4}\ M_{p}/R_H^2$. This is in agreement with the global simulations from @Ayliffe_2009a. Moreover, the surface density evaluated at $0.3 R_H$ is $\sim 9\times 10^{-4}\ M_{p}/R_H^2$ as $\alpha=1/2$ (see below). This should be compared to the surface density at the location of Jupiter: $\Sigma_{\rm CSD} \approx 10^{-3}\ M_{\mathrm{Jup}}/R_H^2$ in the Minimum Mass Solar Nebula [@Weidenschilling_1977; @Hayashi_1981]. Thus, our disc fulfils the condition of maximum pile-up, that we evaluated in Sect. \[sec:gas-piling\]. Due to numerical reasons, FARGO does not allow the modelling of a perfectly inviscid disc. Thus, we take a low but non-zero kinematic viscosity $\nu = 10^{-5}\ R_H^2\Omega_H$, unless specified otherwise and adopt a surface density profile $\alpha=1/2$ so that the viscous torque between adjacent annuli is zero. Notice, moreover, that our method to compute the gas accretion onto the planet does not use the actual flow of the gas, but only the shape of the spiral wave generated by the star (whose structure is detailed in Sect. \[sec:disc-structure\]). Thus, our results should not be significantly affected by our choice of $\nu$, as we will check in Sect. \[sec:torque-results\]. The Equation Of State in FARGO is locally isothermal, with $P={c_s}^2\,\Sigma$, where $c_s$ is the sound speed and $c_s=H\Omega$. The aspect ratio, $H/r$, is constant in all of our simulations and independent of $r$. The $H/r$ of the CPD has been measured to be $0.3-0.4$ in @Ayliffe_2009a. Unfortunately, the code FARGO is not stable for values of the scale height that are so large. Thus, we have run simulations with $H/r$ values ranging from $0.05$ to $0.2$, in order to extrapolate the results to $H/r=0.4$. Fortunately, as we will see below, the cumulated torque at the inner edge of the disc turns out to be independent of the $H/r$ value, which makes the extrapolation trivial. ### Numerical parameters {#sec:numerical-parameters} In several previous studies, simulations have shown that the disc reaches an equilibrium distribution with a sharp truncation at its outer edge [@Ayliffe_2009a; @Ayliffe_2009b; @Martin_2011]. This truncation radius is about $0.3 - 0.4 \ R_H$ and is likely set by tidal truncation effects as outlined by @Martin_2011. Indeed, we observed ourselves such a truncation radius in simulations with a grid extended to the Hill radius. Outside the truncation radius, the motion of the gas relative to the planet is strongly non-Keplerian, as noticed by @Ayliffe_2009a. The interaction between this region and the CPD is not clearly understood. Therefore, as we just want to measure the effect of stellar tides on the CPD, we consider in this paper a limit case where the CPD is completely isolated and we truncate the grid at $R_{\rm out} = 0.3\ \mathrm{R_H}$. The inner border radius is $R_{\rm in}=0.01\ \mathrm{R_H}$; for a Jupiter mass planet at $5.2$ A.U., it corresponds to about $540\,000$ kilometres, which is smaller than the semi-major axis of Europa. We choose a logarithmic radial spacing for the grid (the width of a ring $\delta r$ is proportional to its radius $r$), which is suitable to have an accurate resolution very close to the planet. The resolution is taken such that $\delta \theta=\delta r/r < H/5r$ (see Table \[tab:results\]). This resolution is appropriate to resolve the pressure density wave exerted by the star, as $H$ is the typical pressure scale length. ### Boundary conditions {#sec:boundary-conditions} We use a non-reflecting boundary condition for both the inner and the outer borders of the disc. This prevents density waves from reflecting at the border of the grid. It is important for an accurate evaluation of the torque onto the disc through the wave. It also does not allow gas inflow or outflow and keeps the mean density on the first and last rings constant with time, which is consistent with our model of an isolated disc. Torque computation {#sec:torque-computation} ------------------ The gravitational torque exerted by the star on each ring of the CPD is computed as follows [see @Martin_2011]. First, it is computed on each grid-cell. To the direct torque $T_g$, we have to subtract an indirect torque $T_i$ due to the acceleration of the frame (i.e. of the planet). Denoting by $\vec r = (x_c, y_c)$ the coordinates of a cell of mass $m_{\rm cell}$, by $\vec r_\odot = (x_\odot, y_\odot)$ the coordinates of the star and by $\vec d = \vec r - \vec r_\odot = (x_d, y_d)$ the mutual distance vector, the direct and indirect forces exerted by the star on the cell write: $$\vec f_g = -\frac{G M_{\odot} m_{\rm cell}}{d^2} \, \frac{\vec d}{\Vert \vec d \Vert}$$ $$\vec f_i = -\frac{G M_{\odot} m_{\rm cell}}{r_\odot^2} \, \frac{\vec r_\odot}{\Vert \vec r_\odot \Vert}$$ Thus, the gravitational torque exerted from the star on one cell is : $$\begin{aligned} \Vert \vec T \Vert & = &\Vert \vec T_g - \vec T_i \Vert \\ & = & \Vert \vec r \wedge (\vec f_g - \vec f_i) \Vert \\ & = & - [x_c (f_g.y_d - f_i.y_\odot) - y_c (f_g.x_d - f_i.x_\odot)]\end{aligned}$$ Then, we compute the azimuthal sum over all the cells of an annulus, to obtain the gravitational torque felt by one ring. Finally, the total torque on the disc is the sum of the torques felt by the individual rings. Thus, it is useful to introduce the notion of [ *cumulative torque*]{}, which is the sum of the torques felt from the first ring to the ring in consideration. The total torque is therefore the cumulative torque at the last ring. For our purposes in this paper we consider the outermost ring to be the first and the innermost one the last (unlike @Martin_2011, who adopted the opposite convention). Results {#sec:results} ======= In this section, we describe the results of numerical simulations on the structure of the CPD and present our estimate of the stellar torque. Then, we derive results for the accretion rate according to our analytical model. Disc structure {#sec:disc-structure} -------------- As expected, we see a two-armed spiral density wave, created by the tides of the star. The wave propagates radially inwards with the speed of sound. As $c_s=H\Omega$, its radial velocity is $H/r$ times its azimuthal velocity. Thus, the wave has the shape of a logarithmic spiral, with its pitch angle given by $H/r$. This is clearly seen in Fig. \[fig:disc-structure\]. ![Density map for an aspect ratio of $0.15$; the color corresponds to the logarithm of $\Sigma/(M_{\mathrm{Jup}}R_H^{-2})$. The disc is extended from $r = 0.01$ to $0.3\ \mathrm{R_H}$. []{data-label="fig:disc-structure"}](Rivier-etal-fig1.eps){width="50.00000%"} Torque {#sec:torque-results} ------ We measure the torques after the simulations have reached a steady-state (after $150$ orbits at $0.3\,R_H$). @Martin_2011 showed that the total torque is negative and that most of it is exerted on the outer region of the disc (which represents less than 10 % of the disc mass). Therefore, the variations of the cumulative torque in the inner part of the disc ($r \leq 0.1\ R_H$) are negligible , as can be seen on Fig. \[fig:torques\]. This figure shows the cumulative torques for various $H/r$, after 250 orbits at $r=0.3\ R_H$. They oscillate from $r = 0.3$ to $0.1\ R_H$. As explained by @Martin_2011, the disc can be divided in four quadrants, starting from the line connecting the planet to the star, rotating in the anti-clockwise direction. When the two-arm wave is in the first and third radiant, the torque is negative; when it is in the second and fourth one, the torque is positive. Thus, the oscillations of the torque correspond to the passage of the wave from even to odd quadrants and vice versa, due to its spiral shape. The wave-length of these pseudo-periodic oscillations is the radial propagation of the wave during half a revolution around the planet, that is $\pi\,r\,(H/r)$. ![Cumulated torques (in $M_{\rm Jup}R_H^2\Omega_H^2$) as a function of radius (in $R_H$) for different aspect ratios.[]{data-label="fig:torques"}](Rivier-etal-fig2.eps){width="50.00000%"} Notice that, although the radial profile of the cumulative torque is different from simulation to simulation given that the wavelength of its oscillation is a function of $H/r$, the limit value at the inner edge of the disc is remarkably insensitive on $H/r$. Thus we can safely assume that the total cumulative torque for a $H/r=0.3-0.4$ disc (the realistic value of the scale height according to @Ayliffe_2009a is also $\sim 4\times 10^{-8}$. We can now check whether our choice of the disc viscosity $\nu$ impacts significantly the total torque that we measure. For this purpose, Table \[tab:results\] shows the torques measured in simulations with various viscosities. Some differences are visible, which is a sign that our simulations are not dominated by numerical viscosity. However, the total torques are very similar to each other (within $10\%$ for an order of magnitude change in $\nu$). This gives confidence that the total torque that we estimate is valid also in the limit of an inviscid disc. Accretion rate {#sec:accr-rate} -------------- In Sect. \[sec:gas-inflow\], we obtained a value for the influx of gas onto the CPD $C$. Then, thanks to numerical simulations, we measured the stellar torque for different aspect ratios and low viscosities and obtained its order of magnitude in Sect. \[sec:torque-results\]. Given these values, we can derive analytically the planet accretion rate $\dot M$ from Eq. . The resulting values are listed in Table \[tab:results\], along with results on the torques and the corresponding steady-state radius $X_C$. We find that $X_C$ is of the order of $\sim 0.05\ R_H$, approximately 50 times the current physical radius of Jupiter. As can be seen in Table \[tab:results\], the accretion rate hardly depends on the aspect ratio, and only increases by about $5\%$ when $H/r$ doubles. This is because, as we said above, the total cumulative torque is insensitive to the $H/r$ value of the disc. The viscosity of the gas in the simulation also has very little influence on the measured torque. In all the cases, we find that the accretion rate is of the order of: $$\dot M_p \approx 2 \times 10^{-7}\ M_{\rm Jup}/\mathrm{yr}\ .$$ This accretion rate is low, making Jupiter double its mass in 5 million years. The torque that we measure is of course proportional to the surface density of the CPD in the simulations, but we have shown that the adopted value is realistic, and could hardly be larger. The parameter $C$ may be poorly constrained, but the dependence of $\dot M_p$ on $C$ is such that two orders of magnitude difference in $C$ would only change $\dot M_p$ by a factor $3$. Therefore, this order-of-magnitude estimate of the accretion rate in an inviscid CPD solely perturbed by the star, is robust. As a consequence, the accretion of giant planets could be much slower than expected, thus preventing the runaway accretion phase of the planet. ------- ------- ------- ------------------- -------------------------------- ---------------------- ---------------------- $H/r$ $N_r$ $N_s$ $\nu$ $T$ $X_C$ $\dot M_p$ ($R_H^2\Omega_H$) ($M_{\rm Jup}R_H^2\Omega_H^2$) ($R_H$) ($M_{\rm Jup}$/yr) 0.05 342 632 $10^{-5}$ $-3.39\times 10^{-8}$ $5.11\times 10^{-2}$ $1.87\times 10^{-7}$ 0.1 170 314 $3\times 10^{-5}$ $-3.76\times 10^{-8}$ $5.33\times 10^{-2}$ $2.03\times 10^{-7}$ 0.1 170 314 $10^{-5}$ $-3.64\times 10^{-8}$ $5.26\times 10^{-2}$ $1.98\times 10^{-7}$ 0.1 170 314 $3\times 10^{-6}$ $-3.49\times 10^{-8}$ $5.18\times 10^{-2}$ $1.92\times 10^{-7}$ 0.1 170 314 $10^{-6}$ $-3.32\times 10^{-8}$ $5.08\times 10^{-2}$ $1.84\times 10^{-7}$ 0.15 114 212 $10^{-5}$ $-3.70\times 10^{-8}$ $5.30\times 10^{-2}$ $2.01\times 10^{-7}$ 0.2 86 160 $3\times 10^{-5}$ $-3.88\times 10^{-8}$ $5.40\times 10^{-2}$ $2.09\times 10^{-7}$ 0.2 86 160 $ 10^{-5}$ $-4.02\times 10^{-8}$ $5.47\times 10^{-2}$ $2.15\times 10^{-7}$ ------- ------- ------- ------------------- -------------------------------- ---------------------- ---------------------- Conclusion {#sec:conclusion} ========== The classic model for giant planet formation [@Pollack_1996] predicts that the final phase of gas accretion occurs in a very fast runaway mode. Consequently, planets should keep accreting gas until they are so massive they open very wide gaps in the disc, which occurs when they reach a mass equal to multiple times the mass of Jupiter [@Bryden_1999; @Kley_1999; @Lubow_1999; @Lubow_2006]. In this paper we considered that, during the alleged runaway growth phase, a planet should be surrounded by a circum-planetary disc (CPD) due to angular momentum conservation of the gas global flow. Thus, most of the gas that it accretes should have passed through this disc. The viscosity in the CPD may be very low, if the planet is located in a dead zone. Therefore, we have investigated whether a non-viscous CPD could act as a bottleneck for gas accretion. If, consequently, the gas accretion timescale can become comparable to the timescale of gas removal from the proto-planetary nebula, the observed large spread in giant planet masses could stem from the competitions between these two timescales. Considering that there is no radial drift in a non-viscous disc, we have developed a model for a steady-state non-viscous CPD. The disc is fed by a vertical gas inflow from the surrounding environment [as previously observed in 3D numerical simulations, see @Machida_2008; @Tanigawa_2011; @Ayliffe_2012], until a pressure equilibrium is reached. The surface density of the CPD is thus analytically determined and found in agreement with previous numerical simulations [@Ayliffe_2009a]. However, the CPD is perturbed by the star. This results in the formation of a two-armed logarithmic spiral density wave, that propagates all the way inwards down to the inner edge of our disc. As a consequence, a negative torque from the star is deposited in the very inner regions of the disc, where gas consequently falls onto the planet. We find that the planetary accretion rate depends almost linearly on the cumulated stellar torque, and weakly on the vertical gas inflow (Eq. (\[accr-rate\]) ). Running 2D simulations of an isolated disc, in a planetocentric frame, extended to $0.3$ Hill radius, we have studied the effect of the star on the disc, and measured the torque. We find that the torque is negative and basically independent on the disc’s aspect ratio. This allowed us to derive the accretion rate of a giant planet surrounded by a non-viscous CPD perturbed by the star. We find that the mass doubling time for a Jupiter-mass planet is about 5 Myr. This timescale is much longer than that in the runaway accretion mode of @Pollack_1996. However, such a low accretion rate is valid in the limit condition of an inviscid disc. In reality, even in a dead zone the viscosity is not null, though it is still extremely hard to estimate quantitatively its value. Furthermore, viscosity, if related to ionisation, can increase with time as the disc becomes less optically thick [@Turner_2010]. As a consequence, the broad mass range of observed exoplanets may come from a range of viscosities (or viscosity evolutions) in their original CPDs. Moreover, we have only studied the effect of the star on gas accretion inside the CPD. Several other effects may intervene to make the CPD lose angular momentum, which need to be investigated in the near future. In particular, interaction with the gas beyond $0.3\,R_H$ from the planet and outside the Hill sphere may perturb the flow in the CPD. The study of such interactions would require further investigations in global simulations and is beyond the scope of this paper. In conclusion, it emerges from this paper that the accretion history for planets more massive than Saturn [the mass beyond which a CPD forms @Ayliffe_2009b] may be dominated by the viscous evolution of the CPD. We speculate that planets with masses between Saturn’s and a few times Jupiter’s may have formed in “dead zones” with different levels of low viscosity in their CPDs. Instead, very massive planets (5-10 Jupiter masses), which reached the mass limit of the runaway growth process imposed by gap opening in the disc, should have formed in active zones, where CPDs did not act as bottlenecks to accretion. G.R. and Y.B.’s internships at O.C.A. were funded by the PPF OPERA.\ We thank the CRIMSON team, who manages the mesocentre SIGAMM of the O.C.A., on which most simulations were performed. [^1]: http://exoplanet.eu/ [^2]: http://fargo.in2p3.fr/

--- abstract: 'In order to study short timescale optical variability of $\gamma$-ray blazar S5 0716+714, quasi-simultaneous spectroscopic and multi-band photometric observations were performed from 2018 November to 2019 March with the 2.4 m optical telescope located at Lijiang Observatory of Yunnan Observatories. The observed spectra are well fitted with a power-law $F_{\lambda}=A\lambda ^{-\alpha}$ (spectral index $\alpha >0$). Correlations found between $\dot{\alpha}$, $\dot{A}$, $\dot{A}/A$, $\dot{F_{\rm{\lambda}}}$, and $\dot{F_{\rm{\lambda}}}/F_{\rm{\lambda}}$ are consistent with the trend of bluer-when-brighter (BWB). **The same case is for colors, magnitudes, color variation rates, and magnitude variation rates of photometric observations.** The variations of $\alpha$ lead those of $F_{\rm{\lambda}}$. Also, the color variations lead the magnitude variations. The observational data are mostly distributed in the I(+,+) and III(-,-) quadrants of coordinate system. Both of spectroscopic and photometric observations show BWB behaviors in S5 0716+714. The observed BWB may be explained by the shock-jet model, and its appearance may depend on the relative position of the observational frequency ranges with respect to the synchrotron peak frequencies, e.g., at the left of the peak frequencies. **Fractional variability amplitudes are $F_{\rm{var}}\sim 40\%$ for both of spectroscopic and photometric observations. Variations of $\alpha$ indicate variations of relativistic electron distribution producing the optical spectra.**' author: - 'Hai-Cheng Feng, H. T. Liu, J. M. Bai, L. F. Xing, Y. B. Li, M. Xiao, Y. X. Xin' title: 'Quasi-simultaneous Spectroscopic and Multi-band Photometric Observations of Blazar S5 0716+714 during 2018-2019' --- Introduction ============ **Blazars are a subclass of active galactic nuclei (AGNs)** and usually exhibit extreme variability in the whole electromagnetic spectrum [e.g., @Ul97]. **Depending on the rest-frame equivalent widths (EWs), blazars can be divided into BL Lacertae objects (BL Lacs) and flat-spectrum radio quasars (FSRQs). The EWs of BL Lacs and FSRQs are $<$ 5 $\mathring{\rm{A}}$ and $>$ 5 $\mathring{\rm{A}}$, respectively [e.g., @Gh11; @GT15].** Generally, the continuum radiation of BL Lacs is believed to be relativistically boosted along the line of sight by relativistic jets with small viewing angles [e.g., @UP95; @Ul97] and shows observational characteristics, such as featureless optical spectra, strong non-thermal emission, and high polarization, etc. There are two peaks in broadband spectral energy distributions (SEDs) of **blazars** [e.g., @Gh98; @Ul97]. Their low and high energy peaks are located around **from infrared-**optical-ultraviolet (UV) **to** X-ray bands and **around MeV-GeV-TeV** $\gamma$-ray bands, respectively. The low energy peak is the synchrotron radiation from relativistic electrons in the relativistic jets and the high energy peak, the $\gamma$-ray emission, is generally interpreted as the inverse-Compton (IC) scattering of the synchrotron soft photons **for blazars** and the external soft photons **for FSRQs** by the same electron distribution that radiates the synchrotron photons [e.g., @Ul97; @Gh98; @CG08; @Ta10; @Ne12; @Zh12; @MS16; @Zh17]. Various variability timescales, e.g., from minutes to decades, have been found in most BL Lacs and these timescales can help us to investigate the properties of radiation region [e.g., @Xi99; @Xi02; @Xi05; @Co15; @Li15; @Wi15; @Fe17; @Li19]. The variability timescales are usually divided into three classes: the timescales less than one night are regarded as intra-day variability (IDV) or micro-variability [e.g., @WW95; @Fa14]; the timescales from days to a few months are short-term variability (STV) [e.g., @Li17]; and the timescales larger than several months are known as long-term variability (LTV) [e.g., @Da15]. Different variability timescales may be originated from different emission regions. Thus, we can study different radiation mechanisms via variability with different timescales. Furthermore, the flux variability often follows different spectral behavior and the correlation between the variability of flux and spectral index (or magnitude and color) will shed light on the physical processes of radiation for BL Lacs. A common phenomenon has been found in most BL Lacs. The bluer spectral index usually arises at the brighter phase in most BL Lacs [e.g., @Vi04; @Bo12], i.e., bluer-when-brighter (BWB). The BWB trend is often regarded as evidence of shock-in-jet model [e.g., @MG85; @Gu08; @Bo12]. However, many observations do not show any correlation between colors and magnitudes [e.g., @Ag16; @Ho17], or show only weak correlations [e.g., @Wi15]. The discrepancy of the color-magnitude correlations is a crucial issue that can help us to understand more detailed radiation properties in jets. S5 0716+714 is a typical BL Lac object at a redshift of $0.31\pm0.08$ [@Ni08]. It was first discovered by @Ku81 and was widely studied on the whole electromagnetic spectrum [e.g., @Os06; @Ab10; @Hu14; @Da15; @Ga15; @Fe17; @Ho17; @Sa17; @Li19]. It is one of the most active and bright BL Lacs in the optical band and shows a completely featureless spectrum [e.g., @Bi81; @Da13]. A number of groups have focused on the broadband photometric study of S5 0716+714 in the optical regime. Almost, all of them have found the variability with timescales from mins to years [e.g., @Ne02; @Hu14; @Da15; @Ag16; @Ho17; @Li17]. Many studies reported high IDV duty cycles [DCs; @WW95], i.e., DCs $\geq 70\%$ for S5 0716+714. Variation amplitudes are larger than 0.05 mag for 80% of 52 nights [@Ne02]. @Hu14 gave a DC of 83.9% on 42 nights. @Ag16 obtained a DC of $\sim$90% by 23 night observations. The probability of variability in S5 0716+714 is nearly daily. The various (strong, weak, or non) BWB trends have been also reported in many observations. @Dai13 found that the source exhibited strong BWB chromatism in LTV, STV, and IDV. @Hu14 showed strong and mild BWB trends on IDV and STV, respectively. @Ag16 did not found any correlations between colors and magnitudes. Recently, @Ho17 reported an outburst state during 2012 and they found both BWB chromatism and weak BWB trend in most nights. However, in few nights, the data did not show any correlations between colors and magnitudes. The observational characteristics mentioned above indicate that S5 0716+714 is a natural laboratory for studying the radiation properties of BL Lacs. **Almost all of the previous studies only used a few of broadband photometric observations. Bandwidths of broadband filters are usually larger than 1000 $\mathring{\rm{A}}$ and different filters have different bandwidths. Therefore, the relationship between brightness and spectral behavior is only roughly studied. The broad bandwidths might also influence the relationship during some phases (e.g., might decrease the correlation coefficient during weak phases). Moreover, the adjacent bands will partly overlap each other, which will further influence the correlation between the brightness and spectral behavior.** In order to investigate the relationships of index–flux, index variability–flux variability, color–magnitude, and color variability–magnitude variability, and shed some light on the radiation processes of BL Lacs, we simultaneously monitored S5 0716+714 with spectroscopic observations and broadband photometry. **The spectral data can provide the light curves (LCs) at narrow enough wavelength coverage which allow us to study the above relationships in details. Besides, comparing photometric LCs to spectral integral LCs will help us probe the effect of bandwidth.** The correlations of variability among different bands and different wavelength ranges could also help us to limit the relative location of radiation. In Section 2, we describe the detailed information of observations and data reductions. The results and our analyses are presented in Section 3. Finally, discussion and conclusion are presented in Section 4. Observations and Data Reduction =============================== All the spectroscopic and photometric observations of S5 0716+714 were carried out with the 2.4 m alt-azimuth telescope, which is located at Lijiang Observatory of Yunnan Observatoris, Chinese Academy of Science. The longitude, latitude, and altitude of the observatory are 100$^{\circ}$01$^{\prime}$48$^{\prime\prime}$, 26$^{\circ}$42$^{\prime}$42 $^{\prime\prime}$, and 3193 m, respectively. From mid-September to May, the observatory is dry and most nights are clear. The average seeing of the telescope obtained by the full width at half maximum (FWHM) of stars is $\sim$ 1$^{\prime\prime}_{\cdot}$5 [e.g., @Du14]. For the 2.4 m telescope, the pointing accuracy is about 2$^{\prime\prime}$, and the closed-loop tracking accuracy is better than 0$^{\prime\prime}_{\cdot}$5 hr$^{-1}$. In 2010, the telescope was mounted with an Yunnan Faint Object Spectrograph and Camera (YFOSC) at Cassegrain focus. This is an all-purpose CCD for low/medium dispersion spectroscopy and photometry. The CCD can keep low readout noise under high readout speed, which benefits from all-digital hyper-sampling technology. During our observations, the readout noise and gain are 9.4 electrons and 0.35 electrons/ADU, respectively. The CCD chip covers a field of view (FOV) of 9$^{\prime}_{\cdot}$6$\times$ 9$^{\prime}_{\cdot}$6 with 2048 $\times$ 4096 pixels, and the pixel scale is 0$^{\prime\prime}_{\cdot}$283 pixel$^{-1}$. YFOSC can quickly switch from photometry to spectroscopy ($\leq$ 1 s), and we can also choose the binning mode to reduce the photometric readout time. The detailed parameters of the telescope and YFOSC were described in @Wa19. The monitoring campaign started in 2018 November and spanned $\sim$ 106 days. For most clear dark or grey nights, we basically performed photometric and spectroscopic observations of S5 0716+714 within 10 minutes. Thus, the photometry and spectroscopy can be considered to be quasi-simultaneous. During our observations, we successfully obtained the photometric data in 42 nights and spectral data in 47 nights. The cadence of spectroscopy is $\sim$ 2.08 days. The complete observation information is listed in Table 1. Photometry ---------- The photometric observations were performed using Johnson $BV$ and Cousins $RI$ filters. In order to obtain the accurate magnitude calibration of the target, we always set several comparison stars in the observed FOV. The comparison stars were presented in @Vi98, who have calibrated the magnitudes in the $BVR$ bands. We found that star2, star3, star5, and star6 are closest to the target [see Figure 3 in @Vi98]. Besides, the four comparison stars were also used in @Gh97, who gave the data of the $I$ band. Thus, these stars are selected as comparison stars in our observations. The magnitude of S5 0716+714 is calibrated as follows: $$Mag = -2.5\; \log\frac{1}{N}\sum_{i}^{N}10^{-0.4(M^{i}_{\rm{std}} + M_{\rm{o}} - M_{i})}, \label{equation 1}$$ where $N$ is the number of comparison stars, $M^{i}_{\rm{std}}$ is the standard magnitude of the $i$th comparison star, and $M_{\rm{o}}$ and $M_{i}$ are the instrumental magnitudes of the target and the $i$th comparison star, respectively. **Figure 1** shows the calibrated LCs of S5 0716+714. The calibration errors include two components. The first is the Poisson errors of the target and comparison stars, and it can propagate through Equation (1). The second is from the systematic uncertainties which might be caused by the phase of the moon, weather condition, etc. We calibrated one of the comparison stars (star3) using Equation (1) and the variability of the star can be regarded as the systematic error. The different band calibrated magnitudes of S5 0716+714 and star3 are listed in Tables 3–6. The systematic error is calculated by $$\sigma_{\rm{sys}} = Mag_{3} - \overline{Mag_{3}}, \label{equation 2}$$ where $Mag_{3}$ is the calibrated magnitude of star3. Finally, the errors are $\leq 1\%$ in most nights. The errors are also listed in Tables 3–6. All the photometric data were reduced using standard Image Reduction and Analysis Facility (IRAF) software. After the bias and flat-field corrections, we extracted the instrumental magnitudes of the target and comparison stars with different apertures. To avoid the contamination of the host galaxy mentioned in @Fe17, we tested two different apertures: dynamic apertures (several times FWHM) and fixed apertures. For each type of aperture, we chose 10 different apertures. The aperture radii of fixed apertures and dynamic apertures are 1$^{\prime\prime}_{\cdot}$5–8$^{\prime\prime}_{\cdot}$0 and 1.3–3.5 $\times$ FWHM, respectively. The results are almost the same in different apertures. However, the best signal-to-noise ratio (S/N) could be obtained with the aperture radius of 6$^{\prime\prime}_{\cdot}$0, and we adopted the photometry under this aperture as the final result. Spectroscopy ------------ Considering the featureless spectra of BL Lacs, the spectroscopic observations were carried out with Grism 3, which provides a relatively low dispersion (2.93 $\mathring{\rm{A}}$ pixel$^{-1}$) and wide wavelength coverage (3400–9100 $\mathring{\rm{A}}$). We found that the spectrum of Grism 3 might be slightly contaminated by the 2nd order spectrum as wavelength is longer than $\sim$ 7000 $\mathring{\rm{A}}$, and the 2nd order spectrum is $\sim 5\%$ times intensity of the 1st order spectrum. To avoid the effect of the 2nd order spectrum, we use a UV-blocking filter which cuts off at $\sim$ 4150 $\mathring{\rm{A}}$. Thus, the secondary spectrum will be rejected shorter than $\sim$ 8300 $\mathring{\rm{A}}$. The final spectra cover the observed-frame of 4250–8050 $\mathring{\rm{A}}$. To improve the flux calibration, we simultaneously put the target and star3 in the long slit. This method was used widely , and can obtain the relatively high quality spectra even in poor weather. To minimize the effects of seeing, we use a wide slit with a projected width of 5$^{\prime\prime}_{\cdot}$05. For each night, we also observe a spectrophotometric standard star, which can calibrate the absolute fluxes of the target and comparison star. The raw spectral data are also reduced with IRAF. After correcting the bias and flat-field, we calibrate the wavelength of two-dimensional spectral image using standard helium and neon lamps. We extract the spectra of the target and star3 after removing the cosmic-rays. The extraction aperture radius is 21 pixels ($\sim$ 5$^{\prime\prime}_{\cdot}$943), nearly the same with photometry. We calibrate the absolute fluxes of the target and star3 using the spectrophotometric standard star. Note that miscentering of the object in slit will cause the shift of wavelength and then will influence the calibration of flux. We correct the shift by the absorption lines from 6400–7100 $\mathring{\rm{A}}$. In the end, we re-calibrate the spectra using the template spectrum of comparison star. The template spectrum is obtained by averaging the spectra of star3, which are observed in the nights with good weather conditions. The absorption lines of atmosphere are also corrected by the comparison star. Figure 2 is the mean spectrum and an individual spectrum. We bin individual spectra to obtain the spectroscopic LCs, and the bin width is 50 $\mathring{\rm{A}}$. The flux and error of each bin are obtained by the mean and standard deviation of the fluxes in the corresponding bin, respectively. We find that the LC of each bin is nearly the same with each other, and then only 6 bins with the centers of 4425, 5125, 5825, 6525, 7225, and 7925 $\mathring{\rm{A}}$ are used for analysis. The 6 bins are denoted in the top panel of Figure 2, and the relevant LCs are shown in Figure 3. Fractional Variability Amplitude and Spectral Index --------------------------------------------------- The variability amplitude of each light curve is calculated by the root-mean-square (RMS) fractional variability amplitude $F_{\rm{var}}$ [e.g., @Ro97; @Ed02; @Va03]. The fractional variability amplitude $F_{\rm{var}}$ is defined as $$F_{\rm{var}}=\frac{\sqrt{S^2-<\sigma^2_{\rm{err}}>}}{<F>}, \label{equation 3}$$ where $S^2$ denotes the total variance for the $N$ data points in a light curve, $<F>$ is the mean flux of the light curve, and $<\sigma^2_{\rm{err}}>$ denotes the measured mean square error of the $N$ data points: $$\begin{aligned} S^2 = \frac{1}{N-1}\sum^{N}_{\rm{i=1}}(F_{\rm{i}}-<F>)^2,\\ <F> = \frac{1}{N}\sum_{\rm{i}}^{N}F_{\rm{i}},\\ <\sigma^2_{\rm{err}}> = \frac{1}{N}\sum^{N}_{\rm{i=1}}\sigma^2_{\rm{err,i}}. \end{aligned}$$ \[equation 4\] @Ed02 gave the error $\sigma_{F_{\rm{var}}}$ on $F_{\rm{var}}$: $$\sigma_{F_{\rm{var}}}=\frac{1}{F_{\rm{var}}}\sqrt{\frac{1}{2N}}\frac{S^2}{<F>^2}. \label{equation 5}$$ First, we convert all the photometric data to flux. Then, we measure both spectral and photometric variability amplitude. The variability amplitudes of different LCs are listed in Table 2. The spectral indices and amplitudes of S5 0716+714 are obtained by fitting the spectra via a power-law ($f_{\lambda} = A \lambda^{-\alpha}$). Figure 2 shows the best fit to the mean spectrum and individual spectrum. The variability of spectral index is shown in the left top panel of Figure 3. Results and Analysis ==================== During our observations, the amplitudes of variability are $\sim$40%, calculated from Equation (3). The photometric and spectroscopic results of $F_{\rm{var}}$ are consistent with each other and show that the variability amplitudes of S5 0716+714 in the blue side are consistent with those in the red side as considering the relevant uncertainties (see Table 2). The band widths of the filters are hundreds to thousands angstroms, and the variability amplitudes of photometry are the average results of broad bands. The width of the spectral bin is much narrow than the filter band width. Though there are the differences between the photometric and spectroscopic bandwidths and bins, the very close wavelength coverage should result in their consistent $F_{\rm{var}}$ for the photometric and spectroscopic observations. To compare the variability of different bands, we shift each photometric LC to the same level depending on the magnitude at JD$\sim$ 2458545.12 (the median magnitude of each LC). Figure 4 shows the shifted results. In addition to the differences of the variability amplitudes of valleys, the LCs of different filters are nearly the same as each other. We measure the time delay among different photometric LCs. However, we do not find any reliable time lags. The result of interpolated cross-correlation function [ICCF, @WP94; @Wa16] between $I$ and $B$ is shown in the right bottom panel of Figure 5. We also test the time delays between the photometric and spectroscopic LCs (see Figure 3), and the LCs are consistent with each other. Therefore, the variability in different wavelength ranges should originate from the same region, and the variations of brightness might cause the changes of color and spectral index. We find that the variability of different colors is similar to that of the photometric LCs (see Figure 5). The spectral index variability is also similar to that of the LC of each bin (see Figure 3). We test correlations between different colors and different magnitudes. Figures 5 and 6 show the test results. The results indicate that the bluer spectra usually occur at brighter phases, i.e., BWB. The Spearman rank correlation between $B-I$ and $B$ is significant, and other colors are also correlated with $B$. The BWB trend was often found in S5 0716+714 (see Section 1) and can be explained with a shock-in-jet model. The larger variability amplitude is inclined to occur at the shorter wavelength. Thus, the BWB tend will be more significant when the interval of effective wavelengths between two bands is larger. As mentioned in Section 1, there are some groups which do not find any correlations between the colors and magnitudes. The discrepancy might be caused by the following reasons: 1\. For some extended sources, the contamination of the host galaxies might lead to some fake variability because of the change of seeing [e.g., @Fe17; @Fe18]. As a result, the observed correlation of the color-magnitude may not be related to the radiation processes. For point sources, the strong host galaxies may dilute the variability amplitudes of AGNs, and then influence the correlation between flux and spectral index, especially during the weak states. S5 0716+714 is a point source and its host galaxy is more than four times darker than the target itself [@Ni08]. Thus, the discrepancy should not be caused by the effect of the host galaxy. 2\. The accuracy of photometry may also influence the variability of colors. Most photometric studies are based on the small telescopes ($\leq$ 1 m). For most BL Lacs, the typical variability amplitudes of colors are $\sim$0.05 mag [e.g., @St06; @Hu14; @AG15]. Furthermore, the variability amplitudes might be less than 0.02 mag for some adjacent bands. When the photometric accuracy is larger than 0.01 mag, the color-magnitude correlations will be seriously affected. The accuracy of our photometric measurement is less than 1% in most nights. So, the adjacent bands can show the mild BWB trends (see Figure 6). During our observations, the entire data show that the BWB trend exists in S5 0716+714. The S/N and sampling frequency of the data are high enough. Therefore, the BWB trend may be an intrinsic phenomenon of the source. The color-magnitude data roughly obey the BWB trend, but the data scatter is visible as well (see Figure 6). The variability of flux density and spectral index are similar to each other (see Figures 3 and 5). Thus, the variability rate of flux might influence the variability of spectral index. Another possibility is that the variability of flux density and spectral index may result from changes of relativistic electron distribution emitting the observed photons and may have a correlation between the relevant variability rates. Thus, we test whether a correlation exists between the variability rates of flux density and spectral index. The sampling of observational data is nearly homogeneous and the variability rates of flux density $F_{\lambda}$, spectral index $\alpha$, and spectral amplitude $A$ are defined as $$\begin{aligned} \dot{F_{\rm{\lambda}}} = \frac{F^{\rm{\lambda}}_{\rm{i+1}} - F^{\rm{\lambda}}_{\rm{i}}}{T_{\rm{i+1}} - T_{\rm{i}}},\\ \dot{\alpha} = \frac{\alpha_{\rm{i+1}} - \alpha_{\rm{i}}}{T_{\rm{i+1}} - T_{\rm{i}}},\\ \dot{A}= \frac{A_{\rm{i+1}}-A_{\rm{i}}}{T_{\rm{i+1}} - T_{\rm{i}}}, \end{aligned}$$ where$F^{\rm{\lambda}}_{\rm{i}}$, $\alpha_{\rm{i}}$, and $A_{\rm{i}}$ are the flux density, spectral index, and spectral amplitude observed at the time series $T_{\rm{i}}$, respectively. Figure 7 shows a positive correlation between $\dot{\alpha}$ and $\dot{F_{\rm{\lambda}}}$ for Bin1. The most of data of BWB behavior is distributed in I and III quadrants of coordinate system (see Figure 7). At the same time, there are strong positive correlations of $\dot{\alpha}$–$\dot{A}$ and $\dot{F_{\rm{\lambda}}}$–$\dot{A}$ for Bin1 (see Table 8). Also, there is a correlation between the variability rates of $B$ and $B - I$ and nearly the data of BWB behavior are distributed in I and III quadrants (see Figure 8). Hereafter, spectral index-flux density and color-magnitude relations are called as “color-brightness” relation. These correlations indicate that the variability rates of color and brightness are likely dominated by the cooling and accelerating processes of the relativistic electrons that generate the observed photons and the relevant variability. In Equations (6a)–(6c), the variability rates are calculated from the differences of adjacent data points. The adjacent data points may be considered to originate from the same flare. In order to compare the color-magnitude variability rate correlations with the spectral index-flux density variability rate correlation, a relative variability rate of flux density is defined as $$\frac{\dot{F_{\rm{\lambda}}}}{F_{\rm{\lambda}}} =\left(\frac{F^{\rm{\lambda}}_{\rm{i+1}}}{F^{\rm{\lambda}}_{\rm{i}}}-1\right) \frac{1}{T_{\rm{i+1}} - T_{\rm{i}}}.$$ If the flux variability is mainly caused by the variability of spectrum $F_{\rm{\lambda}}=A\lambda ^{-\alpha}$, $\dot{F_{\rm{\lambda}}}/F_{\rm{\lambda}}$ will be a function of $\dot{A}/A$ and $\dot{\alpha}$, where $$\frac{\dot{A}}{A} =\left(\frac{A_{\rm{i+1}}}{A_{\rm{i}}}-1\right) \frac{1}{T_{\rm{i+1}} - T_{\rm{i}}}.$$ The observational data of $\dot{F_{\rm{\lambda}}}/F_{\rm{\lambda}}$ and $\dot{\alpha}$ can be linearly fitted with $\dot{F_{\rm{\lambda}}}/F_{\rm{\lambda}}=B+C\dot{\alpha}$. The Spearman’s rank correlation test shows a strong positive correlation between $\dot{F_{\rm{\lambda}}}/F_{\rm{\lambda}}$ and $\dot{\alpha}$ (see Table 8), and the BWB data of S5 0716+714 are mostly distributed in I and III quadrants (see Figure 9). $B$ is almost close to zero, and $C=3.29\pm0.23$. The Spearman’s rank correlation analyses show strong positive correlations of $\dot{\alpha}$–$\dot{A}/A$ and $\dot{A}/A$–$\dot{F_{\rm{\lambda}}}/F_{\rm{\lambda}}$ (see Table 8). Since three correlations exist among $\dot{\alpha}$, $\dot{F_{\rm{\lambda}}}/F_{\rm{\lambda}}$, and $\dot{A}/A$, there should be a correlation like as $\dot{F_{\rm{\lambda}}}/F_{\rm{\lambda}}(\dot{A}/A,\dot{\alpha})$ (see Figure 10). In fact, there is a correlation among $\dot{\alpha}$, $\dot{A}/A$, and $\dot{F_{\rm{\lambda}}}/F_{\rm{\lambda}}$ at the confidence level of $>99.99\%$, $\dot{F_{\rm{\lambda}}}/F_{\rm{\lambda}}=0.001 + 0.012\dot{A}/A+ 1.839\dot{\alpha}$. Since I and III quadrants in Figures 7–9 correspond to the BWB, II and IV quadrants in Figures 7–9 should correspond to redder-when-brighter (RWB), which likely have $F_{\rm{\lambda}}=D\lambda^{\alpha}$ in the optical band. Spectroscopic and photometric observations show consistent BWB trends in the color-brightness diagrams (Figures 7–9). **In order to confirm the Spearman’s rank test results listed in Table 8, a Monte Carlo (MC) simulation is used to reproduce these parameters presented in Table 8. For each pair of these parameters, each data array generated by the MC simulation is fitted with the SPEAR [@Pr92] and the fitting gives the relevant $r_{\rm{s}}$ and $P_{\rm{s}}$, the Spearman’s rank correlation coefficient and the p-value of hypothesis test. Considering the errors of X and Y and assuming Gaussian distributions of X and Y, $r_{\rm{s}}$ and $P_{\rm{s}}$ distributions are generated by the SPEAR fitting to the data of X and Y from $10^4$ realizations of the MC simulation. Averages, $r_{\rm{s}}$(MC) and $P_{\rm{s}}$(MC), are calculated by the $r_{\rm{s}}$ and $P_{\rm{s}}$ distributions, respectively. Standard deviations of these two distributions are taken as the relevant uncertainties of $r_{\rm{s}}$(MC) and $P_{\rm{s}}$(MC) (see Table 8). These results given by the MC simulation confirm the ordinary Spearman’s rank test results listed in Table 8. Thus, these correlations will be reliable.** Discussion and Conclusion ========================= We also test the BWB trend using the bin flux and spectral index (see Figure 11). This BWB trend is slightly different from that of color-magnitude. The data are fitted with a fifth-order polynomial and a monotonically increasing trend appears in Figure 11. Figure 11 shows that the BWB trend might depend on the brightness. Thus, the relevant radiation of the BWB at least includes two components: one component is caused by the propagation of shocks in jet; another component is the underlying radiation which is not related with the shock process. If the particles in jet is homogeneous, the variability of BL Lacs should be caused by the disturbance of magnetic field [e.g., @Ch15], the precession of jet [e.g., @CK92], the inhomogeneous region of jet, etc. The variations of the underlying radiation of jet may not cause the change of spectral index. But, during a weaker phase the BWB trend caused by the shock will be more significant and during a brighter phase the underlying radiation might dilute the BWB trend. This possibility needs more observation evidences to test. **There is a possible discrepant point, the one at the lower left quarter in Figure 11, that might affect the fitting result. We exclude this point and re-fit the rest data. The result is very similar to the previous one. The reason that one flux may correspond to several $\alpha$ values is that the spectrum fitting includes two parameters $A$ and $\alpha$. Different $A$ and $\alpha$ combinations may give the same flux. This will result in the data point scatter of BWB for both of spectroscopic and photometric observations. Though the dispersion of $\alpha$ exists, the BWB trend roughly holds (see the best fittings in Figure 11).** The BWB behavior is observed in our monitoring epoch with the 2.4 m optical telescope located at Lijiang Observatory of Yunnan Observatories. The BWB behavior can be explained by the shock-jet model. A relativistic shock propagating down a jet will accelerate electrons to higher energies, where the shock interacts with a nonuniform region of high magnetic field and/or electron density, likely observed to be knots in jets. The shock acceleration will cause radiations at different frequencies being produced at different distances. The synchrotron peak frequency depends on the relativistic electron distribution and the magnetic field, i.e., the distances behind the shock front, and the radiation cooling will make the synchrotron radiation peak decrease at the intensity and the frequency. Thus, frequency dependence of the duration of a flare corresponds to an energy-dependent cooling length behind the shock front, which will cause colour variations in blazars. @Pa07 proposed that the observations during early rising phase of the flux will give a bluer colour while those taken during later phases of the same flare will show more enhanced redder fluxes. The synchrotron peak of SED of S5 0716+714 is located very close to the optical wavelengths, and the corresponding broadband SED can be well explained by the synchrotron self-Compton (SSC) and the external radiation Compton (ERC) models, where the SSC soft photons are the synchrotron photons and the ERC soft photons in the IC scattering are emission from a broad-line region (BLR) and/or infrared (IR) emission from a dust torus [e.g., @Li14]. No emission lines were detected in the IR, optical, and UV spectra of S5 0716+714 [@Ch11; @Sh09; @Da13], and this may from the fact that thermal emission from accretion disk is not found in multiwavelength SED of S5 0716+714 [e.g., @Li14]. The ionizing radiation from accretion disk is so weak that broad emission lines are not observable, even though a BLR exists in S5 0716+714. Also, the dust emission is not observable because of very weak emission of accretion disk, even though a dust torus exists in S5 0716+714. The observational frequency band is at the left of the synchrotron radiation peak because $F_{\rm{\lambda}}=A\lambda ^{-\alpha}$ ($\alpha >0$). This corresponds to the BWB behavior data in the I(+,+) and III(-,-) quadrants of coordinate system. If the observational frequency band is at the right of the synchrotron radiation peak, we may have $F_{\rm{\lambda}} =D\lambda ^{\alpha}$ ($\alpha >0$). This may correspond to the RWB behavior in the II(-,+) and IV(+,-) quadrants of coordinate system. The first case is observed in our observations and the second one is not observed in our observations. The BWB trends usually arise in most BL Lacs [e.g., @Vi04; @Bo12], and this is probably because that the synchrotron peaks are at optical-UV-X-ray bands for most BL Lacs and that the optical observations are usually at the left of the synchrotron radiation peak. No or only weak BWB trends are observed in many observations [e.g., @Wi15; @Ag16; @Ho17], and this may result from that the observational frequency ranges span the synchrotron peak frequencies. Also, the optical variability may be produced by a superposition of optical variability from different regions in jets for BL Lacs without the color-brightness correlations. The BL Lacs with the BWB trends may have a single emitting region of optical variability. The relativistic electrons in a single emitting region can produce the broadband SED containing the synchrotron and IC components [e.g., @Li14]. This single emitting region of optical variability will avoid superposing of optical variability from different regions and weakening of the color-brightness correlations. Thus, the variability of brightness, color, and spectral index is likely caused by the change of the underlying relativistic electron distribution that generates the relevant radiation behavior observed in S5 0716+714 as a shock passes through a high density region in jet. This passing of shock will produce SED’s variability, such as SED’s shape, peak frequency, and peak intensity. In order to research short timescale optical variability of $\gamma$-ray blazar S5 0716+714, quasi-simultaneous spectroscopic and multi-band photometric observations were performed from 2018 November to 2019 March with the 2.4 m optical telescope located at Lijiang Observatory of Yunnan Observatories. As the BWB trends are detected in the photometric observations, what will the optical spectra show and how will vary? First, the observed spectra can be well fitted with a power-law $F_{\lambda}=A\lambda ^{-\alpha}$. Then we study $\dot{\alpha}$, $\dot{A}$, $\dot{A}/A$, $\dot{F_{\rm{\lambda}}}$, and $\dot{F_{\rm{\lambda}}}/F_{\rm{\lambda}}$ for spectroscopic observations. We find correlations between these quantities, **which** are consistent with the BWB trends. Interestingly, $\alpha$ is correlated to $F_{\rm{\lambda}}$ and the variations of $\alpha$ lead those of $F_{\rm{\lambda}}$. **The variations of $\alpha$ indicate variations of relativistic electron distribution producing these optical spectra.** A correlation among $\dot{\alpha}$, $\dot{A}/A$, and $\dot{F_{\rm{\lambda}}}/F_{\rm{\lambda}}$ is found as well. Colors, magnitudes, color variation rates, and magnitude variation rates are studied for photometric observations. We also find correlations between these quantities, which are consistent with the BWB trends. Moreover, the color variations lead the magnitude variations. 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C. 2017, ApJS, 228, 1 [lcccccclccccc]{} 2018-11-29 & & 30 & 20 & 15 & 15 & & 2019-01-21 & 200 & & & &\ 2018-12-07 & 120 & & & & & & 2019-01-24 & 120 & 30 & 20 & 15 & 10\ 2018-12-10 & 200 & 30 & 20 & 15 & 15 & & 2019-01-25 & 120 & 30 & 20 & 15 & 10\ 2018-12-12 & 120 & 30 & 20 & 15 & 10 & & 2019-01-27 & 120 & 30 & 20 & 15 & 10\ 2018-12-13 & 120 & 30 & 20 & 15 & 10 & & 2019-01-31 & 120 & 30 & 20 & 15 & 10\ 2018-12-15 & 120 & 30 & 20 & 15 & 10 & & 2019-02-02 & 120 & 30 & 20 & 15 & 10\ 2018-12-16 & & 30 & 20 & 15 & & & 2019-02-05 & 120 & 30 & 20 & 15 & 10\ 2018-12-19 & 120 & 30 & 20 & 15 & 10 & & 2019-02-08 & 120 & 30 & 20 & 15 & 10\ 2018-12-20 & 120 & 30 & 20 & 15 & 10 & & 2019-02-10 & 120 & 30 & 20 & 15 & 10\ 2018-12-21 & 120 & 30 & 20 & 15 & 10 & & 2019-02-12 & 120 & 30 & 20 & 15 & 10\ 2018-12-22 & & 30 & 20 & 15 & 10 & & 2019-02-15 & 120 & 30 & 20 & 15 & 10\ 2018-12-23 & & 30 & 20 & 15 & 10 & & 2019-02-19 & 200 & & & &\ 2018-12-24 & 120 & 30 & 20 & 15 & 10 & & 2019-02-20 & 200 & 30 & 20 & 15 & 10\ 2018-12-25 & 120 & 30 & 20 & 15 & 10 & & 2019-02-22 & 200 & 30 & 20 & 15 & 10\ 2018-12-26 & 120 & 30 & 20 & 15 & 10 & & 2019-02-24 & 300 & 30 & 20 & 15 & 10\ 2018-12-30 & 120 & 30 & 20 & 15 & 10 & & 2019-02-26 & 300 & 30 & 20 & 15 & 10\ 2018-12-31 & 120 & 30 & 20 & 15 & 10 & & 2019-03-02 & 300 & 30 & 20 & 15 & 10\ 2019-01-02 & 120 & 30 & 20 & 15 & 10 & & 2019-03-03 & 180 & 30 & 20 & 15 & 10\ 2019-01-06 & 120 & 30 & 20 & 15 & 10 & & 2019-03-06 & 180 & 30 & & 15 & 30\ 2019-01-10 & 120 & 30 & 20 & 15 & 10 & & 2019-03-09 & 120 & 30 & 20 & 15 & 10\ 2019-01-11 & 120 & 30 & 20 & 15 & 10 & & 2019-03-11 & 180 & 30 & 20 & 15 & 10\ 2019-01-13 & 120 & 30 & 20 & 15 & 10 & & 2019-03-13 & 120 & 30 & 20 & 15 & 10\ 2019-01-16 & 120 & 30 & 20 & 15 & 10 & & 2019-03-14 & 900 & 30 & 20 & 15 & 10\ 2019-01-18 & 200 & & & & & & 2019-03-15 & 180 & 30 & 20 & 15 & 10\ 2019-01-19 & 200 & & & & & & & & & & &\ \[Table1\] [ccccc]{} B & 41.8$\pm$4.6 & & Bin1 & 41.2$\pm$4.3\ V & 41.4$\pm$4.6 & & Bin2 & 40.4$\pm$4.2\ R & 40.1$\pm$4.4 & & Bin3 & 39.8$\pm$4.1\ I & 39.5$\pm$4.4 & & Bin4 & 39.3$\pm$4.1\ & & & Bin5 & 39.0$\pm$4.1\ & & & Bin6 & 37.9$\pm$3.8\ \[Table2\] [cccc]{} 2458452.42872 & 14.563 $\pm$ 0.009 & & 13.286\ 2458463.25048 & 14.339 $\pm$ 0.002 & & 13.295\ 2458465.23931 & 14.189 $\pm$ 0.002 & & 13.295\ 2458466.24818 & 14.047 $\pm$ 0.001 & & 13.294\ 2458468.34552 & 14.219 $\pm$ 0.008 & & 13.302\ .... & ... & & ...\ \[Table3\] [cccc]{} 2458452.42915 & 13.796 $\pm$ 0.014 & & 12.424\ 2458463.25093 & 13.529 $\pm$ 0.002 & & 12.438\ 2458465.23978 & 13.393 $\pm$ 0.001 & & 12.437\ 2458466.24862 & 13.258 $\pm$ 0.001 & & 12.437\ 2458468.34611 & 13.402 $\pm$ 0.005 & & 12.442\ ... & ... & & ...\ \[Table4\] [cccc]{} 2458452.42950 & 13.396 $\pm$ 0.011 & & 12.056\ 2458463.25130 & 13.089 $\pm$ 0.002 & & 12.064\ 2458465.24019 & 12.967 $\pm$ 0.003 & & 12.064\ 2458466.24899 & 12.849 $\pm$ 0.005 & & 12.062\ 2458468.34655 & 12.965 $\pm$ 0.004 & & 12.063\ ... & ... & & ...\ \[Table5\] [cccc]{} 2458452.42980 & 12.911 $\pm$ 0.008 & & 11.779\ 2458463.25164 & 12.536 $\pm$ 0.025 & & 11.762\ 2458465.24049 & 12.459 $\pm$ 0.001 & & 11.787\ 2458466.24932 & 12.352 $\pm$ 0.002 & & 11.788\ 2458468.34704 & 12.437 $\pm$ 0.004 & & 11.786\ ... & ... & & ...\ \[Table6\] [ccccccc]{} 2458460.226840 & 1.361 $\pm$ 0.036 & 1.224 $\pm$ 0.036 & 1.136 $\pm$ 0.020 & 1.030 $\pm$ 0.027 & 0.988 $\pm$ 0.023 & 0.892 $\pm$ 0.024\ 2458463.255093 & 1.682 $\pm$ 0.019 & 1.525 $\pm$ 0.022 & 1.395 $\pm$ 0.024 & 1.280 $\pm$ 0.018 & 1.228 $\pm$ 0.029 & 1.117 $\pm$ 0.023\ 2458465.244306 & 1.928 $\pm$ 0.041 & 1.677 $\pm$ 0.023 & 1.550 $\pm$ 0.024 & 1.408 $\pm$ 0.027 & 1.330 $\pm$ 0.029 & 1.200 $\pm$ 0.035\ 2458466.257454 & 2.277 $\pm$ 0.058 & 1.953 $\pm$ 0.029 & 1.798 $\pm$ 0.030 & 1.610 $\pm$ 0.027 & 1.521 $\pm$ 0.037 & 1.397 $\pm$ 0.037\ 2458468.356979 & 1.922 $\pm$ 0.043 & 1.705 $\pm$ 0.031 & 1.570 $\pm$ 0.025 & 1.453 $\pm$ 0.033 & 1.347 $\pm$ 0.025 & 1.233 $\pm$ 0.028\ ... & ... & ... & ... & ... & ... & ...\ \[Table7\] [cccccc]{} $\dot{\alpha}$ & $\dot{F_{\lambda}}$ & 0.800 & $< 10^{-4}$ & 0.70$\pm$0.05 & 7.3$\pm$1.5\ $\dot{A}$ &$\dot{F_{\lambda}}$ & 0.864 & $< 10^{-4}$ & 0.79$\pm$0.05 & 10.5$\pm$1.9\ $\dot{\alpha}$ & $\dot{A}$ & 0.856 & $< 10^{-4}$ & 0.76$\pm$0.05 & 9.4$\pm$1.8\ $\dot{\alpha}$ &$\dot{F_{\lambda}}/F_{\lambda}$ & 0.787 & $< 10^{-4}$ & 0.69$\pm$0.06 & 7.2$\pm$1.5\ $\dot{A}/A$ & $\dot{F_{\lambda}}/F_{\lambda}$ & 0.875 & $< 10^{-4}$ & 0.77$\pm$0.05 & 9.8$\pm$2.0\ $\dot{\alpha}$ & $\dot{A}/A$ & 0.971 & $< 10^{-4}$ & 0.85$\pm$0.05 & 13.4$\pm$2.8\ $\Delta B/ \Delta T$ & $ \Delta(B-I)/ \Delta T$ & 0.781 & $< 10^{-4}$ & 0.78$\pm$0.02 & 8.6$\pm$0.8\ $B$ & $B - I$ & 0.553 & 2 $\times 10^{-4}$ & 0.54$\pm$0.02 & 3.6$\pm$0.2\ $B$ & $B - V$ & 0.395 & 1 $\times 10^{-2}$ & 0.41$\pm$0.06 & 2.2$\pm$0.6\ $B$ & $V - R$ & 0.470 & 2 $\times 10^{-3}$ & 0.44$\pm$0.06 & 2.5$\pm$0.6\ $B$ & $R - I$ & 0.492 & 1 $\times 10^{-3}$ & 0.49$\pm$0.04 & 3.0$\pm$0.5\ \[Table8\]  

--- abstract: | We develop a gauge theory for diffusive and precessional spin dynamics in two-dimensional electron gas with disorder. Our approach reveals a direct connections between the absence of the equilibrium spin current and strong anisotropy in the spin relaxation: both effects arise if the spin-orbit coupling is reduced to a pure gauge $SU(2)$ field. In this case, by a gauge transformation in the form of a local $% SU(2)$ rotation in the spin subspace the spin-orbit coupling can be removed. The resulting spin dynamics is exactly described in terms of two kinetic coefficients: the spin diffusion and electron mobility. After the inverse transformation, full diffusive and precessional spin density dynamics, including the anisotropic spin relaxation, formation of stable spin structures, and spin precession induced by a macroscopic current, is restored. Explicit solutions of the spin evolution equations are found for the initially uniform spin density and for stable nonuniform structures. Our analysis demonstrates a universal relation between the spin relaxation rate and spin diffusion coefficient. author: - 'I.V. Tokatly$^{1,2,4}$ and E. Ya. Sherman$^{3,4}$' title: 'Diffusive and precessional spin dynamics in a two-dimensional electron gas with disorder: a gauge theory view' --- Introduction. ============= Description of the spin dynamics of a two-dimensional electron gas is one of the most important problems for fundamental and applied modern spintronics.[@Zutic04; @Fabian07; @Dyakonov08] Two mutually related problems in this field attract a great deal of attention: spin current and spin relaxation. Spin currents describe how the spin density pattern changes with time mainly due to the spin transfer between different parts of the electron gas. Since the spin dynamics of interest occurs usually in systems out of equilibrium, spin relaxation becomes important and contributes strongly into the evolution of spin density pattern. The key for understanding these properties is the spin-orbit coupling, making the orbital and spin degrees of freedom mutually dependent. Spin-orbit coupling has many crucial influences on the properties of the systems where it occurs: the typical examples are nuclei, elementary particles, atoms, and electrons in solids. Spin-orbit coupling, in general, makes the spin a non-conserved quantity, thus leading to a spin relaxation. It causes mutually dependent spin, charge, and mass flows in solids and quantum liquids.[@Leurs08; @Kleinert; @Galitski] In addition, spin-orbit coupling leads to a spin response to an external electric field, providing an ability of spin manipulation by the electric field driving the dynamics in the orbital degrees of freedom.[@Rashba03a] The general techniques for calculation of spin relaxation and spin current out of equilibrium are the classical or quantum Boltzmann-like equation for the spin-density matrix [@Mishchenko04; @Culcer07; @Glazov07; @Wu09; @Bronold04] and nonequilibrium Green functions.[@Raimondi; @Bleibaum] In this approach, the description of the electron dynamics takes into account possible relaxation processes due to the presence of spin-orbit coupling, disorder, phonons, and electron-electron collisions. The experimentally observable spin dynamics is due to the spin-orbit (spin-momentum) coupling. At the equilibrium, the expectation value of the spin current can be calculated directly. Surprisingly, such a direct calculation demonstrated that the spin current can exist even in the equilibrium state of a two-dimensional electron gas with spin-orbit coupling.[@Rashba03] This observation brought a puzzle for the understanding of the basic phenomena in spin transport since the equilibrium spin current is not related to any spatial spin accumulation that can be seen experimentally. On the other hand, the spin dynamics due to the spin-orbit coupling can be understood in terms of a theory where the coupling is treated as a non-Abelian gauge field, and the corresponding formalism can be applied [@Mineev92; @Frolich93]. On a single-electron scale, for example, for electrons in quantum dots, the gauge transformation of the spin-orbit field was employed in Refs.[@Aleiner01; @Levitov03] and used for the analysis of experimental results on spin manipulation by electric field in quantum dots in Ref.[@Nowack07]. Other interesting theoretical examples of applications of the non-Abelian gauge field approach for single-electron spin transport and electrons in quantum dots, were found and studied (for example, Refs.[@Yang08]-[@Yang06]). When applied to the two-dimensional electron gas, the approach based on a formal $SU(2)$ gauge invariance of the spin-orbit Hamiltonian (i. e. the symmetry with respect to local rotations in the spin subspace) proved that the equilibrium spin current is the diamagnetic response to the effective non-Abelian spin-orbit magnetic field.[@Tokatly08] If the spin-orbit field is a pure gauge and, thus, can be removed by a gauge transformation, the effective $SU(2)$ magnetic field is zero, and the equilibrium spin current vanishes. Here we present a theory based on the gauge transformation, for spin dynamics in a two-dimensional electron gas in the case when the spin-orbit field can be completely removed by such a gauge transformation. We show that the absence of the equilibrium spin current is directly related to the giant anisotropy in the spin relaxation rate, when the relaxation does not occur for certain spin directions.[@Averkiev99; @Schliemann03] After gauging away the spin-orbit coupling, the entire nonequilibrium dynamics of a [*transformed*]{} spin becomes almost trivial and can be described phenomenologically exactly by only two transport coefficients which can be determined experimentally, or calculated theretically to any desired level of accuracy. The first is the spin diffusion coefficient and the second is the electron mobility required only when a constant electric field is applied. With the inverse transformation to the initial dynamical variables, we recover the full nontrivial dynamics of the [*physical*]{} spin, including the absence of the spin relaxation for certain spin directions, that is a strong anisotropy in the spin relaxation, stable spin configurations forming persistent spin helices, and spin precession due to a charge current in a constant external electric field. In addition, this approach allows making predictions for more general cases of spin-orbit coupling, including nonuniform spin-orbit fields. Spin current and spin relaxation: the conventional approach. ============================================================ We begin with the conventional Hamiltonian of spin-orbit coupling in two-dimensional electron gas: $$H_{\rm so}=\frac{1}{2}\sum_{j}\left( \alpha _{ja}({\bm \rho })k_{j}+k_{j}\alpha _{ja}({\bm \rho })\right) \sigma ^{a}, \label{Hso}$$ where $\alpha _{ja}$ is the coordinate-dependent spin-orbit coupling field, $% k_{j}=-i\partial /\partial x_{j}$ is the momentum operator, Cartesian subscript indices $j=x,y$ correspond to the electron coordinate ${\bm \rho }=\left( x,y\right) ,$ and $\sigma ^{a}$ are the Pauli matrices with the upper Cartesian indices corresponding to three directions $x,y,z$ in the spin subspace. We use the system of units with $\hbar \equiv 1$ and sum up over repeating indices. Interaction $H_{\rm so}$ arises in a two-dimensional electron gas from various sources. Two origins are considered as the most important. The first one, arising due to the inversion asymmetry of the crystal unit cell, is described by the Dresselhaus model. The other one is the Rashba field,[@Rashba84] where the coupling originates from the macroscopic asymmetry of the structure hosting the two-dimensional electron gas.[@Winkler03] Due to various physical origins, including material, structure, doping, and possible mechanical strain, numerical values of parameters $\alpha _{ja}$ vary strongly from system to system ranging from $10^{-12}$ eV$\cdot $cm for Si- to $10^{-9}$ eV$\cdot $cm for GaAs-based structures and will not be discussed here. For the coordinate-independent spin-orbit field $k_{j}\alpha _{ja}({\bm \rho })=0$, and the Hamiltonian (\[Hso\]) can be presented as: $$H_{\rm so}=\sum_{j}\alpha _{j}\left( \mathbf{h}_{[j]}\cdot{\bm \sigma} \right) k_{j}, \label{Hso1}$$ where $\mathbf{h}_{[j]}$ is a unit-length vector, and $\alpha _{j}$ is the corresponding spin-orbit coupling constant for given component of momentum; its contribution to the Hamiltonian is, therefore, proportional to the spin projection onto the $% {\mathbf h}_{[j]}$ axis. The coupling leads to a momentum-dependent spin splitting of the electron states. As a results, the Fermi line of the electron gas becomes spin-dependent and two Fermi lines in the electron gas appear. The $H_{\rm so}$ Hamiltonian makes the velocity spin-dependent: $$v_{j}=\frac{k_{j}}{m}+i\left[ H_{\rm so},\rho _{j}\right] =\frac{k_{j}}{m}+\alpha _{j}\left( \mathbf{h}_{[j]}\cdot{\bm \sigma} \right), \label{vj}$$ with $m$ being the electron effective mass. With the spin-dependent velocity in Eq.(\[vj\]) we define the operator of the spin current in the form: $$J_{j}^{a}=\frac{1}{2}\sum_{\mathbf{k}}C_{\mathbf{k}}^{\dagger } \left(v_{j}\tau ^{a}+\tau ^{a}v_{j}\right) C_{\mathbf{k}},$$ where the $SU(2)$ group generators $\tau ^{a}=\sigma ^{a}/2,$ and $C_{% \mathbf{k}}^{\dagger },C_{\mathbf{k}}$ are the corresponding spinors. For example, let $\left|\Phi\right>$ be the ground state wave function. The resulting expectation value of the total spin current, summed up over all electrons in the gas is: $$\left\langle J_{j}^{a}\right\rangle =\left\langle \Phi \right|J_{j}^{a}\left| \Phi \right\rangle.$$ In the absence of special symmetry relations between the components of the Hamiltonian $H_{\rm so},$ the expectation values of spin current $\left\langle J_{j}^{a}\right\rangle $ are not zero, leading to the puzzling equilibrium spin current without any measurable spin transport. Therefore, spin current can be a characteristic of the equilibrium states of two-dimensional electron gas. In the conventional calculation of $% \left\langle J_{j}^{a}\right\rangle $ due to the spin-doubling of the Fermi line, contributions to $\left\langle J_{j}^{a}\right\rangle $ come from two subsystems: single- and double occupied states at a given electron momentum. These two contributions have opposite signs and almost compensate each other, yielding the results in the third order of the coupling constants $\alpha _{j}^{3}$. This third-power dependence is expected from perturbation theory: $\left\langle J_{j}^{a}\right\rangle $ should be an odd function of the spin-orbit coupling and vanish in the first order since in the ground state without spin-orbit coupling the Fermi-line is not spin-split and all states with given $\mathbf{k}$ are doubly occupied. Another important feature of the electron gas with spin-orbit coupling is the spin relaxation. Assume that one has initially produced a nonequilibrium state $\Phi _{S}$ of a uniform spin density with the components: $$S^{a}=\left\langle\Phi_{S}\right|\sum_{\mathbf{k}}C_{\mathbf{k}}^{\dagger}\tau^{a}C_{\mathbf{k}}\left|\Phi_{S}\right\rangle .$$ Then, the state $\left|\Phi_{S}\right\rangle$ will relax to the equilibrium through all possible interactions and spin-orbit coupling. The first stage of the process, the momentum relaxation, is fast. If the spin-orbit coupling is weak compared to the random interactions causing the momentum relaxation, as it is assumed for the rest of this paper, the following spin relaxation is relatively slow. As a result, at the second stage the spin components decrease with relaxation rates described by a symmetric tensor $\Gamma^{ab}$: $$\frac{dS^{a}}{dt}=-\Gamma^{ab}S^{b}.$$ The components of $\Gamma ^{ab}$ depend on the spin-orbit coupling and all possible interactions of electrons with disorder, phonons, and other electrons in the system.[@Ivchenko; @Wu] If the spin-orbit coupling vanishes, $\Gamma^{ab}=0$. Spin-orbit coupling as a gauge field: pure gauge. ================================================= Now we write the Hamiltonian of two-dimensional electron gas in the presence of spin-orbit coupling as: $$\label{H} H=\frac{1}{2m}\int dxdy\Psi ^{+}\left(i\partial_{i}+\mathcal{A}_{i}\right)^{2}\Psi + W\left(\Psi ^{+},\Psi\right)$$ where $W\left(\Psi^{+},\Psi\right) $ contains all explicitly spin-independent terms, including electron-electron interactions and possibly, the effect of the external potential. The general non-Abelian two-component potential is given by $2\times 2$ matrices: $$\mathcal{A}_{j}\equiv A_{j}^{a}\tau^{a}= 2m\alpha _{j}({\bm\rho})h_{[j]}^{a}({\bm\rho})\tau^{a}. \label{Aj}$$ The expression (\[Aj\]) is valid for any arbitrary nonuniform spin-orbit coupling. Let us now perform at a given ${\bm\rho-}$point a local $SU(2)-$rotation [@Tokatly08] in the spin subspace by $$\label{U} \mathbf{U}=\exp[i\theta^{a}({\bm\rho})\tau^{a}],$$ with the transformation of the field operators: $$\label{tildePsi} \widetilde{\Psi }^{+}\mathbf{U}^{-1}=\Psi ^{+},\qquad \widetilde{\Psi }=\mathbf{U}\Psi .$$ This transformation renders the spin-independent quantities such as the charge density and the charge current density, invariant. In contrast, the spin density operators, $$\mathcal{S}=S^{a}\tau^{a},$$ transforming as $$\mathcal{S}=\mathbf{U}\widetilde{\mathcal{S}}\mathbf{U}^{-1}, \label{transform1}$$ exemplify covariant observable quantities. This difference between the physical quantities which transform invariantly and covariantly under a local $SU(2)$ rotation is crucial for the understanding of the spin dynamics. For the matrix $\mathbf{U}=\exp\left[i\theta\left(\mathbf{h}\cdot{\bm\tau}\right)\right],$ where $\mathbf{% h}$ is a unit length vector, the $\tau ^{b}-$matrices, transformed according to Eq.(\[transform1\]), acquire the form: $$\widetilde{\tau }^{b}= h^{b}\left(\mathbf{h}\cdot{\bm\tau}\right)+ \cos \theta [\tau ^{b}-h^{b}\left(\mathbf{h}\cdot{\bm\tau}\right)]+ \sin \theta \varepsilon^{abc}h^{a}\tau^{c}, \label{matr_transform}$$ where $\varepsilon ^{abc}$ is the Levi-Civita tensor. This equation shows that the product $\mathbf{h}\cdot{\bm\tau }$ is unaffected by the transformation (\[U\]). Therefore, if we present $\mathcal{S}$ as the sum of longitudinal and transverse components $\mathcal{S=S}_{\parallel }\mathcal{+S}_{\perp }$ with $\mathcal{S}_{\parallel}=\mathbf{h}\left(\mathcal{S}\cdot\mathbf{h}\right) ,$ the longitudinal component (spin projection onto the $% \mathbf{h-}$axis) remains constant, while the $\mathcal{S}_{\perp }$ does not; it rotates by the angle $\theta$ around the $\mathbf{h-}$axis). This simple observation will be important for the further analysis in this paper. The Hamiltonian preserves its form under a local $SU(2)$ rotation of the fermionic fields if the vector-potential is transformed as follows $$\label{tildeA} \widetilde{\mathcal{A}}_{i}= \mathbf{U}^{-1}\left(i\partial_{i}\mathbf{U}\right)+\mathbf{U}^{-1}\mathcal{A}_{i}\mathbf{U}.$$ Indeed, after the transformation the Hamiltonian acquires the form: $$H=\frac{1}{2m}\int dxdy\widetilde{\Psi }^{+} \left(i\partial_{i}+\widetilde{\mathcal{A}}_{i}\right)^{2}\widetilde{\Psi } +W\left( \widetilde{\Psi }^{+},\widetilde{\Psi}\right) ,$$ which is identical to that of Eq. (\[H\]), but with $\Psi$ and $\mathcal{A}_{i}$ being replaced by the transformed quantities, $\widetilde{\Psi}$ and $\widetilde{\mathcal{A}}_{i}$, respectively. Assume now that $\mathcal{A}_{i}$ in the original Hamiltonian (\[H\]) corresponds to a pure gauge vector-potential, that is both $\mathcal{A}% _{x}$ and $\mathcal{A}_{y}$ can be removed by the above transformation such that $\widetilde{\mathcal{A}}_{x}=\widetilde{\mathcal{A}}_{y}=0$. In this case there exists a local rotation determined by three coordinate-dependent functions $\theta _{\mathcal{A}}^{a}(x,y)$: $$\mathbf{U}_{\mathcal{A}}=\exp[i\theta _{\mathcal{A}}^{a}({\bm \rho })\tau ^{a}],$$ such that the initial components $\mathcal{A}_{i}$ can be presented in the form $$\mathcal{A}_{i}=\mathbf{U}_{\mathcal{A}} \left(i\partial_{i}\mathbf{U}_{\mathcal{A}}^{-1}\right) .$$ A vector-potential of this form is gauged away by the transformation (\[U\]) with $\mathbf{U}=\mathbf{U}_{\mathcal{A}}$: $$\widetilde{\mathcal{A}}_{i}=\mathbf{U}_{\mathcal{A}}^{-1} \left( i\partial_{i}+\mathbf{U}_{\mathcal{A}}\left(i\partial_{i}\mathbf{U}_{\mathcal{A}}^{-1}\right) \right) \mathbf{U}_{\mathcal{A}}=0.$$ If the spin-orbit field can be removed by a gauge transformation, the subsequent spin dynamics is simplified considerably and in certain regimes, like the drift-diffusion processes considered below, the problem becomes elementary. The inverse $SU(2)-$rotation transforms the spin components to the actual values and we recover the full dynamics of the physical spin. We will follow this procedure in the present paper. We mention a textbook example of a similar approach. When the motion of a relativistic electron in static perpendicular electric field $\mathbf{E}$ and magnetic field $\mathbf{H}$ is considered, there exists a reference frame, where, after the Lorentz transformation, the smaller of these fields vanishes. In this frame the equations of the electron motion are very simple, and in the case $H<E,$ they are essentially, one-dimensional. The inverse Lorentz transformation provides the full description of the electron motion in the presence of both fields.[@Jackson] In the pure gauge field after the local $SU(2)$ transformation $\mathbf{U}_{\mathcal{A}}=\exp[i\theta_{\mathcal{A}}^{a}({\bm \rho })\tau ^{a}]$ the Hamiltonian takes the form: $$H=-\frac{1}{2m}\int dxdy\widetilde{\Psi }^{+}\Delta \widetilde{\Psi }% +W\left( \widetilde{\Psi }^{+},\widetilde{\Psi }\right),$$ with no spin-orbit coupling present. As mentioned above, the spin dynamics with this Hamiltonian can be formulated in general terms phenomenologically and then by inverse transformation, returned to the form where the coupling and full spin dynamics are restored. Vector-potential is a pure gauge, allowing removal six terms in $\mathcal{A% }_{x},\mathcal{A}_{y},$ with the transformation $\mathbf{U}_{\mathcal{A}}$ based on the three functions $\theta _{\mathcal{A}}^{a}({\bm \rho })$ given certain relations between the $\mathcal{A}_{x}$ and $\mathcal{A}_{y}$ components. The corresponding conditions are naturally formulated in terms of a non-Abelian field strength tensor $\mathcal{F}_{ij}$ : the vector potential is locally a pure gauge if the field strength is zero, $$\mathcal{F}_{ij}=\partial _{i}\mathcal{A}_{j}-\partial _{j}\mathcal{A}_{i}- i\left[ \mathcal{A}_{i},\mathcal{A}_{j}\right] =0.$$ =6.5cm For the spatially uniform case, using Eq.(\[Aj\]) this condition is reduced to $\left[ \mathcal{A}_{i},\mathcal{A}_{j}\right] =0,$ that is: \(i) either $\alpha _{i}\alpha _{j}=0,$ or \(ii) $\left[ \mathbf{h}_{[i]}\cdot{\bm \tau },\mathbf{h}_{[j]}\cdot{\bm \tau }\right] =0$ if $\alpha _{i}\alpha _{j}\neq 0.$ The commutation relation $$\left[ \mathbf{h}_{[i]}{\bm \tau },\mathbf{h}_{[j]}{\bm \tau }\right] = i{\bm \tau }\cdot \left(\mathbf{h}_{[i]}\times \mathbf{h}_{[j]}\right), \label{commutator}$$ demonstrates that the spin projections commute only for the same axis, that is $% \mathbf{h}_{[i]}=\pm \mathbf{h}_{[j]}$. Therefore, the solution to Eq. (\[commutator\]) has the form (we assume below $\mathbf{h}_{[i]}=\mathbf{h}_{[j]}$ in the case (ii) for definiteness): $$\mathcal{A}_{j}=2m\alpha \nu _{j}\left( \mathbf{h}\cdot{\bm \tau }\right)$$ where $\mathbf{h}=$ $\mathbf{h}_{[i]}=\mathbf{h}_{[j]}$ if $\alpha _{i}\alpha _{j}\neq 0$ or $\mathbf{h}=$ $\mathbf{h}_{[f]}$ for nonzero $% \alpha _{f}$, where $f=x$ or $f=y$, as illustrated in Fig.(1). Here ** **$\alpha =\left( \alpha _{x}^{2}+\alpha _{y}^{2}\right) ^{1/2},$ and ${\bm \nu }$ is a unit vector. The corresponding gauge transformation is: $$\mathbf{U}_{\mathcal{A}}={\exp }\left[ 2im\alpha \rho _{j}\nu _{j}\left( \mathbf{h}\cdot{\bm \tau }\right) \right] {\exp }\left[ 2im\alpha \rho _{i}\nu_{i}\left( \mathbf{h}\cdot{\bm \tau }\right) \right] =\exp \left[ 2im\alpha\left( \mathbf{h}\cdot{\bm \tau}\right) \left({\bm \rho}\cdot{\bm\nu}\right) \right]. \label{Uuniform}$$ From this condition we immediately conclude that the projection of the total spin at the $\mathbf{h}_{[i]}=\mathbf{h}_{[j]}$ axis commutes with $% H_{\rm so}$, and, therefore, remains constant with time for arbitrary dynamics. Experimentally, this fact corresponds to the vanishing relaxation for this spin direction; this conclusion crucial for the design and application of spin-based devices. If $\alpha_{i}\alpha _{j}=0,$ the problem immediately becomes one-dimensional, trivial from the diamagnetic response interpretation of the equilibrium spin current, [@Tokatly08] since one-dimensional systems do not demonstrate this kind of response. The same situation occurs in quantum wires, where the motion of electrons is strictly one-dimensional, no equilibrium spin current exists, and the spin projection along the $\mathbf{h}_{[f]}$ axis is conserved. In the Appendix, for illustration, we perform a conventional calculation of the equilibrium spin current in a two-dimensional electron gas with the pure gauge spin-orbit coupling and in a quantum wire, and demonstrate that the spin current vanishes in both systems. =6.5cm There are two widely studied realizations of the above discussed pure gauge field. The $\alpha_{i}\alpha_{j}=0$ case is realized for the Dresselhaus model for the electron gas confined in the quantum wells of GaAs grown along the $[110]$ crystal axis. The coupling constant $\alpha$ in this system [@Dyakonov86] is approximately inversely proportional to the square of the quantum well width $w$. In this case the vector-potential and the corresponding transformations are: $$\left( \mathcal{A}_{x},\mathcal{A}_{y}\right) = \left( 2m\alpha \tau^{z},0\right) ,\qquad \mathbf{U}_{\mathcal{A}}=\exp \left[2im\alpha x\tau^{z}\right],$$ where the $z-$ axis is oriented along the growth direction and the $x$-axis is that of the unit cell. Here we use transformation (\[Uuniform\]) with $\mathbf{h=}(0,0,1),$ ${\bm \nu}=(1,0),$ and $\theta (x,y)=2m\alpha x$ to obtain: $$\widetilde{\tau }^{z}=\tau ^{z}, \qquad \widetilde{\tau }^{x}=\cos \theta\tau ^{x}+\sin \theta \tau ^{y}, \qquad \widetilde{\tau }^{y}=\cos \theta\tau ^{y}-\sin \theta \tau ^{x}. \label{taus:011}$$ We illustrate the resulting relations between $\mathbf{S}$ and $\widetilde{% \mathbf{S}}$ for this simple situation in Fig.(2): when $\widetilde{\mathbf{S% }}$ remains constant in space, $\mathbf{S}$ turns by the angle $\theta (x,y) $ around the $z-$axis. The $\alpha _{i}\alpha _{j}\neq 0$ case is realized in the compensated Dresselhaus-Rashba model for the GaAs structure grown along the $[001]$ crystal axis. Here $$\begin{aligned} \left(\mathcal{A}_{x},\mathcal{A}_{y}\right) &=& \left( 2m\alpha \left(\tau ^{x}-\tau ^{y}\right) ,2m\alpha \left( \tau ^{x}-\tau ^{y}\right)\right),\\ \mathbf{U}_{\mathcal{A}} &=& \exp \left[ 2im\alpha \left( x+y\right)\left(\tau ^{x}-\tau ^{y}\right) \right] . \label{trans011}\end{aligned}$$ Here we obtain with $\mathbf{h=}(1,-1,0)/\sqrt{2},$ ${\bm \nu}=(1,1)/\sqrt{2},$ and $\theta =2\sqrt{2}m\alpha\left( x+y\right)$: $$\begin{aligned} \widetilde{\tau }^{z} &=&\cos \theta \tau ^{z}-\frac{1}{\sqrt{2}}\sin \theta \left( \tau ^{x}+\tau ^{y}\right), \label{taus:001} \\ \widetilde{\tau }^{x} &=&\cos^2\frac{\theta}{2}\tau^{x} - \sin ^{2}\frac{\theta }{2}\tau ^{y} +\frac{1}{\sqrt{2}}\sin \theta \tau ^{z}, \qquad \widetilde{\tau }^{y}=\cos^2\frac{\theta}{2}\tau^{y} - \sin ^{2}\frac{\theta }{2}\tau^{x} +\frac{1}{\sqrt{2}}\sin \theta \tau ^{z} . \nonumber\end{aligned}$$ Equations (\[taus:011\]), (\[taus:001\]) illustrate a general feature of the relations between the original $\mathbf{S}=(S^{x},S^{y},S^{z})$ and gauge-transformed $\widetilde{\mathbf{S}}$ spin densities and *vice versa*. For the spin-orbit field characterized by the direction $\mathbf{h,}$ for a uniform coordinate-independent $\mathbf{S,}$ the components $\mathbf{S}_{\parallel }$ and $\widetilde{\mathbf{S}}_{\parallel }$ coincide. The $\widetilde{\mathbf{S}}_{\perp }$-component forms a periodic structure on the spatial scale of the order of $L_{\rm so}=1/m\alpha$, or $\hbar ^{2}/m\alpha $ when the units are restored. The mean value $\left\langle \widetilde{\mathbf{S}}_{\perp}(x,y)\right\rangle =0$ for the infinitely large systems considered here, where the boundary conditions do not change the spin dynamics. The meaning of the length $L_{\rm so}$ can be understood as follows. Hamiltonian (\[Hso1\]) shows that the spin-orbit coupling $H_{\rm so}$ causes for an electron moving with the velocity ${\bf v}$, spin precession around $\mathbf{h}$ with the rate of the order of $\alpha mv.$ The corresponding precession angle is of the order of $\alpha mL,$ where $L=vt$ is the electron displacement. Thefore, $L_{\rm so}$ can be viewed as the travel distance at which the electron spin can undergo a full rotation. Another circumstance is, however, more important: the spin rotation angle depends only on the electron displacement and not on the details of its motion between initial and final points, leading to the appearance of stable spin structures, discussed below. Here a numerical value of typical $L_{\rm so}$ can be of interest. For GaAs with $% m=0.067m_{0},$ where $m_{0}$ is the free electron mass, and $\alpha $ of the order of $10^{-7}$ meV$\cdot$cm, $L_{\rm so}$ is of the order of several microns. In both these systems, the observed spin relaxation rate is strongly anisotropic with one spin component having lifetime orders of magnitude longer than the others. The weak relaxation rate for these components is determined by the mechanisms different from the homogeneous spin-orbit coupling, most probably, related to the disorder in the spin-orbit coupling. [@Muller08; @Glazov05; @Dugaev09] Spin dynamics: diffusion, precessional behavior, and drift contributions. ========================================================================= =6.5cm After the gauge transformation, the spin-orbit interactions is switched off. Therefore, on a time scale much longer than the momentum relaxation time, the spin dynamics becomes combination of pure spin diffusion and spin drift: $$\label{tilde-diff} \partial _{t}\widetilde{\mathcal{S}}= D\Delta \widetilde{\mathcal{S}}+ \mu E_{j}\partial_{j}\widetilde{\mathcal{S}},$$ where $D$ is the spin-diffusion coefficient, $\mu $ is the electron mobility, and $\mathbf{E}$ is the two-dimensional applied electric field [@Amico01] as illustrated in Fig.(3). In Eq.(\[tilde-diff\]) we have taken into account that the uniform velocity of electrons is $-\mu\mathbf{E}$. These two parameters fully describe the drift-diffusive spin dynamics in the absence of spin-orbit coupling. Macroscopic motion of electrons (electric currents) can drag nonuniform spin density between different parts of the electron gas. This effect leads to the $\mu E_{j}\partial_{j}\widetilde{\mathcal{S}}$ term in $\partial _{t}\widetilde{S}$. The initial spin density eventually vanishes due to diffusion, however, the total integrated spin polarization will remain constant. The diffusive evolution of the transformed spin density $D\Delta \widetilde{\mathcal{S}}$ occurs if the electron free path of the order of $\ell =v\tau_{p}$ is much less than the spatial scale of the inhomogeneity: $\ell\ll L_{\rm so}.$ This condition can be formulated as $\Omega_{\rm so}\tau_{p}\ll 1,$ meaning that the spin-orbit coupling is relatively weak. The spatial inhomogeneity of the order of $L_{\rm so}$ and $D$ of the order of $v^{2}\tau _{p}$ set the time scale of the diffusion smearing of the $\widetilde{\mathcal{S}}$ as $\widetilde{t}_{D}\sim L_{\rm so}^{2}/D$ on the order of $\Omega _{\rm so}^{-2}\tau_{p}^{-1},$ and, therefore, the same spin relaxation time for real spin $\mathcal{S}.$ The evolution of the physical measurable spin density: $$\label{inverse-transf} \mathcal{S}=\mathbf{U}_{\mathcal{A}}\widetilde{\mathcal{S}}\mathbf{U}_{\mathcal{A}}^{-1},$$ is due to the diffusion and spin precession since the transition of electron from point ${\bm \rho}_{1}$ to point ${\bm \rho}_{2}$ is accompanied by the rotation of its spin, dependent only on the displacement ${\bm \rho}_{2}-{\bm \rho}_{1}$. Irregular motion in the diffusion process is seen in the spin relaxation, and regular drift causes spin precession, with these two processes being mutually related. Motion of $\mathcal{S}$ is described, therefore, by the following equations for the time evolutions of the spin density, which are obtained by applying the inverse transformation (\[inverse-transf\]) to the drift-diffusion equation (\[tilde-diff\]), $$\partial _{t}\mathcal{S}= D\mathbf{U}_{\mathcal{A}}\left[ \Delta \left( \mathbf{U}_{\mathcal{A}}^{-1}\mathcal{S}\mathbf{U}_{\mathcal{A}}\right) \right]\mathbf{U}_{\mathcal{A}}^{-1} +\mu E_{j}\mathbf{U}_{\mathcal{A}} \left[ \partial _{j}\left( \mathbf{U}_{\mathcal{A}}^{-1}\mathcal{S} \mathbf{U}_{\mathcal{A}}\right)\right]\mathbf{U}_{\mathcal{A}}^{-1}.$$ The resulting most general equation of motion valid for any pure gauge spin-orbit field takes the form $$\begin{aligned} &&\partial _{t}\mathcal{S}-D\Delta \mathcal{S} -\mu E_{j}\partial _{j}\mathcal{S} \nonumber \\ &=&D\left\{ 2\left[\mathbf{U}_{\mathcal{A}}\Bbb{\nabla }\mathbf{U}_{% \mathcal{A}}^{-1},\Bbb{\nabla }\mathcal{S}\right] - 2\left( \mathbf{U}_{\mathcal{A}}\Bbb{\nabla }\mathbf{U}_{\mathcal{A}}^{-1}\right) \mathcal{S}% \left( \mathbf{U}_{\mathcal{A}}\Bbb{\nabla }\mathbf{U}_{\mathcal{A}}^{-1}\right) +\left( \mathbf{U}_{\mathcal{A}}\Delta \mathbf{U}_{\mathcal{A}}^{-1}\right) \mathcal{S} +\mathcal{S}\left( \Delta \mathbf{U}_{\mathcal{A}}\right)\mathbf{U}_{\mathcal{A}}^{-1}\right\} \nonumber \\ &&+\mu E_{j}\left[ \left( \mathbf{U}_{\mathcal{A}}\partial_{j}\mathbf{U}_{\mathcal{A}}^{-1}\right) ,\mathcal{S}\right] .\end{aligned}$$ The total expression for local evolution of spin density components can be obtained from this equation by multiplying both sides by $\tau ^{a}$ and taking the trace using the identity: $\mathrm{tr}\left( \tau ^{a}\tau ^{b}\right) =\delta ^{ab}/2.$ The result can be presented as the sum: $$\partial _{t}S^{a}=D\Delta S^{a}+ \mu E_{j}\partial_{j}S^{a}+B_{j}^{ab}\partial _{j}S^{b}-H^{ab}S^{b}-\Gamma ^{ab}S^{b}.$$ The general expressions for non-diagonal tensors of kinetic coefficients entering this equation are: $$\begin{aligned} B_{j}^{ab} &=&-B_{j}^{ba}=4D\mathrm{tr} \left\{ \tau^{a}\left[\mathbf{U}_{\mathcal{A}}\partial _{j}\mathbf{U}_{\mathcal{A}}^{-1},\tau ^{b}\right] \right\}, \\ H^{ab} &=&-H^{ba}= 2\mu E_{j}\mathrm{tr} \left\{ \tau ^{a}\left[ \mathbf{U}_{\mathcal{A}}\partial _{j}\mathbf{U}_{\mathcal{A}}^{-1},\tau ^{b}\right] \right\}, \\ \Gamma ^{ab} &=&4D\left( \mathrm{tr}\left\{ \tau ^{a}\left( \mathbf{U}_{% \mathcal{A}}\partial _{j}\mathbf{U}_{\mathcal{A}}^{-1}\right) \tau ^{b}\left( \mathbf{U}_{\mathcal{A}}\partial _{j} \mathbf{U}_{\mathcal{A}}^{-1}\right) \right\} - \frac{1}{2}\mathrm{tr}\left\{\tau^{a}\left( \mathbf{U}_{\mathcal{A}}\Delta \mathbf{U}_{\mathcal{A}}^{-1}\right) \tau ^{b}+\tau ^{a}\left( \Delta \mathbf{U}_{\mathcal{A}} \right) \mathbf{U}_{\mathcal{A}}^{-1}\tau ^{b}\right\} \right) \nonumber \\ &&\end{aligned}$$ Now we can study the physical meaning of the obtained non-diagonal tensors and simplify the expressions for the time derivatives for uniform spin-orbit coupling with: $$\mathbf{U}_{\mathcal{A}}=\exp \left[ 2im\alpha \left( \mathbf{h}\cdot{\bm \tau }\right) ({\bm \nu}\cdot{\bm \rho}) \right] , \qquad \mathbf{U}_{\mathcal{A}}^{-1}=\mathbf{U}_{\mathcal{A}}^{+} =\exp \left[ -2im\alpha \left( \mathbf{h}{\bm \tau }\right) ({\bm \nu}\cdot{\bm \rho}) \right], \label{UA}$$ With the given form in Eq.(\[UA\]) of $\mathbf{U}_{\mathcal{A}}$ and $\mathbf{U}_{\mathcal{A% }}^{-1}$ we obtain: $$\mathbf{U}_{\mathcal{A}}\Bbb{\nabla }\mathbf{U}_{\mathcal{A}}^{-1}= -2im\alpha \left( \mathbf{h}\cdot{\bm \tau }\right) {\bm \nu ,} \qquad \Delta \mathbf{U}_{\mathcal{A}}=-4m^{2}\alpha ^{2}\mathbf{U_{\mathcal{A}}}, \qquad \Delta \mathbf{U}^{-1}=-4m^{2}\alpha ^{2}\mathbf{U}_{\mathcal{A}}^{-1}. \label{UAdiff}$$ With formulas (\[UA\]),(\[UAdiff\]) we obtain for the diffusion-related coefficients: $$\begin{aligned} B_{j}^{ab} &=&-2m\alpha _{j}D\varepsilon ^{abc}h^{c}, \\ \Gamma ^{ab} &=&4m^{2}\alpha ^{2}D\left( \delta ^{ab}-h^{a}h^{b}\right) .\end{aligned}$$ The corresponding drift-dependent contribution: $$H^{ab}=-m\alpha \mu \left( {\bm \nu }\cdot \mathbf{E}\right) \varepsilon ^{abc}h^{c},$$ describes the spin precession. The answer for the spin density $\mathbf{S}$ with components $\left( S^{x},S^{y},S^{z}\right)$ has the form of three terms of different order in $\alpha $: $$\label{diff} \partial _{t}\mathbf{S}=\left. \partial _{t}\mathbf{S}\right| _{0}+ \left.\partial _{t}\mathbf{S}\right| _{1}+ \left.\partial_{t}\mathbf{S}\right|_{2}.$$ These terms have different meaning and can be expressed as: $$\begin{aligned} \left. \partial _{t}\mathbf{S}\right| _{0} &=& D\Delta \mathbf{S}+\mu E_{j}\partial _{j}\mathbf{S}, \label{dsdt1} \\ \left. \partial _{t}\mathbf{S}\right| _{1} &=& 4mD\alpha \left( {\bm\nu }\cdot \nabla \right) \left( \mathbf{h}\times \mathbf{S}\right) +2m\alpha \mu \left( {\bm \nu }\cdot \mathbf{E}\right) \left( \mathbf{h}\times \mathbf{S}\right), \label{dsdt2} \\ \left. \partial _{t}\mathbf{S}\right| _{2} &=& -4m^{2}\alpha ^{2}D\left(\mathbf{S}-\mathbf{h}(\mathbf{h}\cdot\mathbf{S})\right). \label{dsdt3}\end{aligned}$$ The $\left. \partial _{t}\mathbf{S}\right| _{0}$ term describes the standard drift-diffusion spin dynamics for zero spin-orbit coupling. The $\left. \partial _{t}\mathbf{S}\right| _{1}$ term corresponds to the spin precession due to the spin-orbit coupling. The mobility-determined contribution in $\left. \partial _{t}\mathbf{S}\right| _{1}$ is the precession in the macroscopic spin-orbit field arising due to the uniform velocity of electrons. When the electric current is induced, the momentum distribution function is shifted such that the momentum has a nonzero value. As a result, the Hamiltonian $H_{\rm so}$ forms a macroscopic spin-orbit Zeeman field [@Wilamowski08] and, as a result, a regular spin precession $\partial_{t}\mathbf{S=}2m\alpha \mu \left( {\bm \nu }\cdot \mathbf{E}\right) \left( \mathbf{h}\times \mathbf{S}% \right).$ If $\left( {\bm \nu }\cdot \mathbf{E}\right)=0$, contributions of the momentum changes along the $x$ and $y$-axes in the macroscopic spin-orbit “magnetic” field compensate each other, and no regular precession occurs. Thus, Eq.(\[dsdt2\]) reproduces the diffusive and non-diffusive spin precession. The $\left.\partial_{t}\mathbf{S}\right|_{2}$ term is the Dyakonov-Perel’ mechanism of spin relaxation,[@Dyakonov73] which can be seen from the fact that $D$ is determined by $\left\langle v^{2}\right\rangle \tau _{p}$, where $v$ is the electron velocity (see, also in Ref. [@Raimondi]). Taking into account that electron momentum is $mv,$ one can see that $% \left. \partial _{t}\mathbf{S}\right| _{2}$ corresponds to the Dyakonov-Perel relaxation with the relaxation rate on the order of $\alpha ^{2}k^{2}\tau_p$. The obtained relation between the spin relaxation rate and diffusion coefficient is universal. For two different systems with the same sample-dependent $m\alpha $ parameter, the ratio of $\Gamma ^{ab}/D$ remains constant. Since the parameters $\Gamma ^{ab}$ and $D$ can be measured independently, this universality can be verified experimentally. For example, in the measurements performed at the same sample at different temperatures, the ratio $\Gamma ^{ab}/D$ is expected to remain constant for degenerated and non-degenerated electron gas. Eqs.(\[dsdt2\]) and (\[dsdt3\]) show that $\mathbf{S}_{\parallel}=\mathbf{h}\cdot\mathbf{S}$ does not change with time, as expected, and the entire dynamics is solely due to the $\mathbf{S}% _{\perp }-$component. As a simple illustration we consider the evolution of an initially homogeneous spin density. By solving equations (\[diff\])-(\[dsdt3\]) with the initial condition ${\bf S}(\bm\rho,t=0)={\bf S}_0$ we find the spin dynamics $$\label{decay} {\bf S}(t) = {\bf h}({\bf h}\cdot{\bf S}_0) + \left\{\cos(\Omega_Et)[{\bf S}_0-{\bf h}({\bf h}\cdot{\bf S}_0)] + \sin(\Omega_Et) ({\bf h}\times{\bf S}_0) \right\}e^{-\Gamma t},$$ where $\Omega_E = 2\alpha m\mu({\bm\nu}\cdot{\bf E})$ is the precession frequency in a drift-induced spin-orbit Zeeman field, and $\Gamma=4\alpha^2m^2D$ is the diffusion related relaxation rate. From Eq. (\[decay\]) we see that the spin precesses with the frequency $\Omega_E$ about the ${\bf h}$-axis and its transverse component decays at the rate $\Gamma$ in such a way that the projection of the spin at ${\bf h}$ remains stationary. By comparing the characteristic time scales of the drift-induced precession and the diffusion-induced relaxation, we can estimate the external electric field at which the role of the drift-dependent terms becomes important; in particular, the precession becomes visible at the scale of the relaxation time. From the condition of still visible precession $\Omega_E\sim\Gamma$ we find the corresponding electric field $E\sim \alpha mD/\mu$. In this field, the precession rate $\Omega_E$ is of the order of $\Omega_{\rm so}^{2}\tau_p$, making the contributions of regular and random motion in the precession angle of the same order. If the spin diffusion is dominated by the impurity scattering, then $D$ and $\mu$ are proportional to the momentum relaxation time $\tau_{p}$, and this electric field is disorder-independent. However, it can change with the temperature since in the non-degenerated gas $D$ approaches the electron diffusion coefficient [@Amico01] and, therefore, by the Einstein relation $D=\mu T$. Another interesting effect of a spin-orbit coupling, which follows straightforwardly from our formulation – the existence of stable spatially inhomogeneous spin configuration. It is easy to verify that a general stationary ($\partial _{t}\mathbf{S}=\mathbf{0}$) solution to the equations (\[diff\])-(\[dsdt3\]) is of the form $$\label{helix} {\bf S}(\bm\rho) = {\bf h}({\bf h}\cdot{\bf S}_0) + \cos (2m\alpha(\bm{\nu}\cdot{\bm\rho}))[{\bf S}_0-{\bf h}({\bf h}\cdot{\bf S}_0)]- \sin(2m\alpha(\bm{\nu}\cdot{\bm\rho}))({\bf h}\times{\bf S}_0),$$ where ${\bf S}_0$ is an arbitrary constant vector. This spatially inhomogeneous stationary solution to the drift-diffusion equation arises due to the symmetry of the system. As it was demonstrated for the particular case of the model with balanced Rashba and Dresselhaus couplings, the symmetry can protect electron spins from relaxation [@Schliemann03] and allows for the persistent spin helix structures [@Bernevig06; @Koralek09] of the form of Eq.(\[helix\]). The fact that the shape of this configuration does not depend on the diffusion coefficient $D$ shows that the persistent spin structure is insensitive to the spin-independent disorder, in agreement with Ref.[@Bernevig06] The analysis of the spin helix stability in the presence of disordered spin-orbit coupling can be found in Ref.[@Liu06] It is interesting to note that the helix structure is also insensitive to the presence of the electric field and, therefore, to the mobility and presence of a transport charge current, at least in the linear Ohm’s law regime. This seemingly counterintuitive result follows from the fact that the drift of the helix governed by the second term in (\[dsdt1\]) is exactly compensated by the spin precession in the current-induced effective Zeeman field, the second term in (\[dsdt2\]). A similar cancellation occurs in the diffusion channel. A diffusive spreadout of the helix, the first term in (\[dsdt1\]), and the relaxation of the transverse component of the spin, (\[dsdt3\]), are balanced by the “gradient-precession“ contribution, the first term in (\[dsdt2\]). The persistent spin helix configuration (\[helix\]) has an extremely simple interpretation in terms of the transformed spin density $\widetilde{\bf S}$. The general stationary solution of the standard drift-diffusion equation (\[tilde-diff\]) is simply a constant $\widetilde{\bf S}=\widetilde{\bf S}_0$. The relation between the physical and transformed spin density components yield the conservation $\widetilde{\bf S}_0\cdot\mathbf{h}={\bf S}\cdot\mathbf{h}$. The perpendicular $\widetilde{\bf S}_{0,\perp}$ is transformed according to Eq. (\[inverse-transf\]) as $\mathcal{S}=\mathbf{U}_{\mathcal{A}}\widetilde{\mathcal{S}}\mathbf{U}_{\mathcal{A}}^{-1}$ with $\mathbf{U}_{\mathcal{A}}=\exp\left[2im\alpha\left(\mathbf{h}\cdot{\bm \tau}\right)\left({\bm \rho}\cdot{\bm\nu}\right)\right]$ from Eq. (\[Uuniform\]) according to Eq. (\[matr\_transform\]). The sum of the transformed terms is precisely the persistent spin helix of Eq. (\[helix\]). It is also instructive to look at the precession and relaxation of a spatially homogeneous spin ${\bf S}(t)$, Eq. (\[decay\]), from the point of view of the dynamics of the transformed spin density $\widetilde{\bf S}$. The initial condition ${\bf S}(\bm\rho,t=0)={\bf S}_0$ for the physical spin is mapped to the initial configuration for $\widetilde{\bf S}$ in a form of a spin helix that is similar to Eq. (\[helix\]). The subsequent evolution of $\widetilde{\bf S}$ is governed by the standard drift-diffusion equation (\[tilde-diff\]). Therefore the dynamical behavior is obvious -the initial helix for the transformed spin moves with the drift velocity $v_{\rm drift}=m\alpha\mu(\bm{\nu}\cdot{\bf E})$, and washes out diffusively. When transformed to the physical spin, the drift of the helix translates to the precession, while its diffusive decay is mapped to the relaxation of the physical spin. This interpretation clearly explains why the relaxation time of the transverse components of the spin is universally determined by the diffusion coefficient. The relaxation of the physical spin components is gauge-equivalent to a purely diffusive process of washing out the initial helix configuration of the transformed spin. Conclusions. ============ We developed a gauge theory of macroscopic spin dynamics in a two-dimensional electron gas when the spin-orbit coupling can be described as a pure gauge, and, therefore, removed by a local $SU(2)$ rotation in the spin subspace. We have shown that for a pure spin gauge, equilibrium spin current vanishes and a selected axis of conserved spin projection appears simultaneously, demonstrating gauge-related symmetry relation of these effects. After removing the spin-orbit coupling, we considered macroscopic phenomenological equations of spin dynamics, including spin diffusion and spin drift in an external electric field. By the inverse $SU(2)$ rotation we obtained the full system of partial differential equations for the time- and spatial measured spin density evolution. This system reproduces the physics of spin precession, stable spin configurations such as the persistent spin helix, and the resulting strongly anisotropic spin relaxation. Since we described the system without spin-orbit coupling phenomenologically, our approach is valid at any temperature and electron concentration. It predicts that the ratio of the spin relaxation rate to the spin diffusion coefficient remains temperature- and electron concentration-independent if the coupling constants do not depend on these two system parameters. We presented explicit equations for the spatially uniform spin-orbit coupling and their solutions describing stable nonuniform structures, the precession and the relaxation of uniform spin polarization. These equations can be explicitly generalized for nonuniform two-dimensional electron gas in macroscopic systems. We mention two of them. The first one is the GaAs quantum well grown along the $\left[110\right]$ direction with a modulated width $w(x)$, where the spin-orbit $\alpha(x)$ originated from the Dresselhaus coupling, varies as $1/w^{2}(x)$. The corresponding spin-orbit field $\mathcal{A}_{x}=2m\alpha(x)\tau^{z}$, $\mathcal{A}_{y}=0$, with $\partial\mathcal{A}_{x}/\partial y=0$ remains a pure gauge. The other example is the balanced Rashba-Dresselhaus model with the coordinate-dependent Rashba and Dresselhaus parameters remaining exactly equal or exactly opposite everywhere. As in the $\left[110\right]$ structure, variation in the Dresselhaus term is due to the controlled variation in the structure width, while the control of the Rashba coupling is achieved by a coordinate-dependent bias across the well. A different kind of inhomogeneity occurs in mesoscopic systems where the effect of the boundary conditions for the coupled spin-charge dynamics becomes important.[@Tserkovnyak07; @Bleibaum06; @Duckheim09] Generalization of the gauge theory approach for the dynamics at the sample boundaries can be an interesting extension of our analysis for the infinite systems. Spin dynamics in these systems is of interest for the fundamental understanding of spin transport and for applications in spintronics devices. Acknowledgment. =============== IVT acknowledges funding by the Spanish MEC (FIS2007-65702-C02-01), “Grupos Consolidados UPV/EHU del Gobierno Vasco” (IT-319-07), and the European Community through e-I3 ETSF project (Grant Agreement: 211956). EYS is grateful to the University of Basque Country UPV/EHU for support by the Grant GIU07/40. Appendix. ========= Here we show by a conventional calculation of the spin current that it vanishes at the equilibrium in the considered above pure gauge spin-orbit coupling in a two-dimensional electron gas and, similarly, in one-dimensional quantum wires. We begin with the pure-gauge two-dimensional Hamiltonian, $$H=\frac{k_{x}^{2}}{2m}+\alpha _{x}\left( h^{a}\sigma ^{a}\right) k_{x}+ \frac{k_{y}^{2}}{2m}+\alpha _{y}\left( h^{a}\sigma ^{a}\right) k_{y}, \label{H2}$$ where $\mathbf{h}$ is unit length vector and $\alpha _{x},\alpha _{y}$ are the corresponding spin-orbit coupling constants. The spectrum of electrons described by Eq.(\[H2\]) is the sum of $k_x$ and $k_y$-dependent terms for the two spin branches ”+“ and ”-“: $$\varepsilon_{\pm }\left( k_{x},k_{y}\right) = \frac{k_{x}^{2}+k_{y}^{2}}{2m}\pm \left( \alpha _{x}k_{x}+\alpha _{y}k_{y}\right). \label{S2}$$ Eq. (\[H2\]) demonstrates that the system with pure gauge spin-orbit coupling remains in a certain sense, one dimensional: spin-orbit coupling does not couple different components of momentum in the spectrum. For illustration we consider only the $x-$component of momentum and velocity: $$\frac{\partial \varepsilon _{\pm }\left(\mathbf{k}\right) }{\partial k_{x}}% =\frac{k_{x}}{m}\,\pm \alpha _{x},\qquad v_{x}= i\left[ H,x\right] =\frac{k_{x}}{m}+\alpha _{x}\left( h^{a}\sigma ^{a}\right) ,$$ yielding the spin current component: $$J_{x}^{b}=\frac{1}{4} \sum_{\mathbf{k}}C_{\mathbf{k}}^{\dagger }\left\{v_{x},\sigma ^{b}\right\} C_{\mathbf{k}}.$$ Taking into account that $\left\{\sigma ^{a},\sigma ^{b}\right\}=2\delta^{ab}$, we obtain the anticommutator: $$\frac{1}{2}\left\{ v_{x},\sigma ^{b}\right\} = \frac{1}{2}\left\{\frac{k_{x}}{m}+\alpha _{x}\left( h^{a}\sigma ^{a}\right),\sigma ^{b}\right\} =\frac{k_{x}}{m}\sigma ^{b}+\alpha _{x}h^{b}.$$ The total spin current is the sum of contributions of two subsystems $% \left\langle J_{x}^{b}\right\rangle =\left\langle J_{x}^{b}\right\rangle _{+}+\left\langle J_{x}^{b}\right\rangle _{-}=2\left\langle J_{x}^{b}\right\rangle _{+}.$ Taking into account that for given branch $% \left\langle \sigma ^{b}\right\rangle _{\pm }=\pm h^{b}/h,$ the $% \left\langle J_{x}^{b}\right\rangle _{+}$ spin current component becomes: $$\left\langle J_{x}^{b}\right\rangle _{+}=\frac{1}{2}h^{b}\int dk_{y}\int \frac{\partial \varepsilon \left(\mathbf{k}\right) }{\partial k_{x}}dk_{x},$$ where the integration is perfromed over the area in momentum space occupied by electrons from the branch. The value of this integral is zero since this area is restricted by the line of the constant Fermi energy $E_{F}.$ For one-dimensional case the situation is the same. We take the Hamiltonian: $$H=\frac{k^{2}}{2m}+\alpha \left( h^{a}\sigma ^{a}\right) k.$$ The eigenstates of this Hamiltonian form two branches: $$\varepsilon_{\pm }=\frac{k^{2}}{2m}\pm \alpha k,$$ corresponding to two parabolas with the minima at $-k_{0}$ and $k_{0}=\alpha m,$ respectively, as shown in Fig.(4). =6.5cm To calculate the spin current directly, we perform integration over momenta and summation over spin branches. The ground state expectation value is: $$\left\langle J^{b}\right\rangle = \frac{1}{2}\left[ \int_{-k_{0}-k_{F}}^{-k_{0}+k_{F}}\left( \frac{k}{m}\left\langle \sigma^{b}\right\rangle _{+}+h^{b}\right)dk+ \int_{k_{0}-k_{F}}^{k_{0}+k_{F}}\left( \frac{k}{m}\left\langle \sigma^{b}\right\rangle _{-}+h^{b}\right) dk \right] ,$$ where $k_{F}$ is the Fermi momentum determined by the total concentration of electrons $n$ as $k_{F}=\pi n/2$. We obtain $$\begin{aligned} \left\langle J^{b}\right\rangle &=&2k_{F}h^{b}+\frac{1}{2m} \left[ \int_{-k_{0}-k_{F}}^{k_{0}-k_{F}}k\frac{h^{b}}{h}dk-% \int_{-k_{0}+k_{F}}^{k_{0}+k_{F}}k\frac{h^{b}}{h}dk \right] \nonumber \\ &=&2k_{F}h^{b}-\frac{2}{m}\frac{h^{b}}{h}k_{F}k_{0}.\end{aligned}$$ The minimum position $k_{0}=\alpha m,$ yields $\left\langle J^{b}\right\rangle =0,$ as expected. 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--- abstract: 'We construct KMS-states from $\mathrm{Li}_1$-summable semifinite spectral triples and show that in several important examples the construction coincides with well-known direct constructions of KMS-states for naturally defined flows. Under further summability assumptions the constructed KMS-state can be computed in terms of Dixmier traces. For closed manifolds, we recover the ordinary Lebesgue integral. For Cuntz-Pimsner algebras with their gauge flow, the construction produces KMS-states from traces on the coefficient algebra and recovers the Laca-Neshveyev correspondence. For a discrete group acting on its Stone-Čech boundary, we recover the Patterson-Sullivan measures on the Stone-Čech boundary for a flow defined from the Radon-Nikodym cocycle.' author: - | Magnus Goffeng${}^*$, Adam Rennie, Alexandr Usachev[^1]\ ${}^*$ Department of Mathematical Sciences,\ Chalmers University of Technology and University of Gothenburg,\ Gothenburg, Sweden\ School of Mathematics and Applied Statistics,\ University of Wollongong, Northfields Ave\ Wollongong, Australia\ title: ' Constructing KMS states from infinite-dimensional spectral triples' --- Introduction {#sec:intro} ============ The construction of the JLO cocycle [@GS; @JLO1; @JLO2] from $\theta$-summable spectral triples [@Con-trace] has from the start been closely linked with the idea of KMS states. A $\theta$-summable spectral triple $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}})$ on a $C^*$-algebra $A$ gives rise to a state $\phi(a):= \mathrm{Tr}(a{\mathrm{e}}^{-{\mathcal{D}}^2})$ on $A$ and under suitable conditions this is a KMS-state on the saturation of $A$ by the ${\mathbb{R}}$-action defined from the wave operators ${\mathrm{e}}^{it{\mathcal{D}}^2}$. By [@JLO2] the JLO-cocycle can be defined starting from this KMS-state. On the other hand, [@Con-trace] shows that a finitely summable spectral triple $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}})$ on a $C^*$-algebra $A$ defines a tracial state on $A$. Similar constructions were studied in [@Voics]. The idea since then has been to understand the measure theory associated to $\theta$-summable spectral triples in terms of ‘twisted traces’, and more specifically KMS states. Indeed this idea was present early in the development, [@JLO2]. Two viewpoints make it interesting to study states associated with spectral triples having specified summability degrees: the associated states obstructs summability degrees, and the states provide a notion of measure theory. In this paper we present a construction of KMS states from $\mathrm{Li}_1$-summable spectral triples. By definition, a spectral triple $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}})$ is $\mathrm{Li}_1$-summable if and only if ${\mathrm{e}}^{-t|{\mathcal{D}}|}$ is trace class for $t$ large enough – a slight strengthening of being $\theta$-summable. [*It is an important observation that large classes of examples of $\theta$-summable spectral triples are also $\mathrm{Li}_1$-summable.*]{} For the spectral triple defined from a Dirac operator on a closed manifold, our construction recovers the Lebesgue integral. For Cuntz-Pimsner algebras we also relate our construction to previous work of Laca and Neshveyev [@LN], and the authors [@GMR; @RRS]. We also examine spectral triples arising from certain Hilbert space valued cocycles on discrete groups. In the examples we consider, the KMS-states are associated to flows that are well-suited to the geometries. This is usually not the case for the KMS-state $\phi(a)= \mathrm{Tr}(a{\mathrm{e}}^{-{\mathcal{D}}^2})$ associated with a $\theta$-summable spectral triple. It is our hope that our construction provides a more natural approach to the KMS-states appearing in the JLO-cocycle and that in the future it will have a bearing on the index theory of $\mathrm{Li}_1$-summable spectral triples. Main results ------------ We now state our main results. All our results make sense for general semifinite spectral triples, and so we fix a semifinite trace ${\mathcal{T}}$ for this discussion. First, we state the main technical construction of KMS-states from $\mathrm{Li}_1$-summable spectral triples. After that, we state the implications of this construction to more specific examples. We use the notation $P_{\mathcal{D}}$ for the non-negative spectral projection of ${\mathcal{D}}$, i.e. $P_{\mathcal{D}}:=\chi_{[0,\infty)}({\mathcal{D}})$. If for some $\beta_{\mathcal{D}}\geq 0$, [**${\mathcal{T}}(P_{\mathcal{D}}{\mathrm{e}}^{-t{\mathcal{D}}})$ is finite for $t>\beta_{\mathcal{D}}$ and diverges as $t\searrow \beta_{\mathcal{D}}$**]{}, we say that ${\mathcal{D}}$ has positive ${\mathcal{T}}$-essential spectrum. We define the $C^*$-algebra $A_{\mathcal{D}}$ as the saturation of $A$ under the action of the wave group ${\mathrm{e}}^{it{\mathcal{D}}}$, that is $$A_{\mathcal{D}}:=C^*\left(\cup_{t\in {\mathbb{R}}} \sigma_t(A)\right),\quad\mbox{where}\quad \sigma_t(a):={\mathrm{e}}^{it{\mathcal{D}}}a{\mathrm{e}}^{-it{\mathcal{D}}}.$$ At this stage, we formulate our results in terms of $A_{\mathcal{D}}$. In Subsection \[subsec:toplitz\] we refine the construction to a smaller $C^*$-algebra. In examples, the construction often applies to $A$ directly. Recall from [@BRII Definition 5.3.1] that a state $\phi$ on an ${\mathbb{R}}$-$C^*$-algebra $\sigma:{\mathbb{R}}\curvearrowright A$ is said to be KMS at inverse temperature $\beta$ if $\phi(ab)=\phi(\sigma_{-i\beta}(b)a)$ for $a,b$ from an ${\mathbb{R}}$-invariant norm dense $*$-subalgebra of $A$. If $\phi$ is a state on an ${\mathbb{R}}$-von Neumann algebra $\sigma:{\mathbb{R}}\curvearrowright A$ we say that it is KMS if the same condition holds on an ${\mathbb{R}}$-invariant $\sigma$-weakly dense $*$-subalgebra of $A$. The following theorem is the main result of the paper. \[mainthmconstr\] Let $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}},{\mathcal{N}},{\mathcal{T}})$ be a unital $\mathrm{Li}_1$-summable semifinite spectral triple such that ${\mathcal{D}}$ has positive ${\mathcal{T}}$-essential spectrum (see Definition \[ass:minus-one\] on page ) and is $\beta$-analytic (see Definition \[ass:zero\] on page ). Define $$\beta_{\mathcal{D}}:=\inf\{t>0: {\mathcal{T}}(P_{\mathcal{D}}{\mathrm{e}}^{-t{\mathcal{D}}})<\infty\}.$$ For any extended limit $\omega\in L^\infty(\beta_{\mathcal{D}},\infty)^*$ as $t\to\beta_{\mathcal{D}}$ (see Definition \[extendedlimitdef\] on page ), we define the state $\phi_\omega$ on $A_{\mathcal{D}}$ as $$\phi_\omega(a):=\lim_{t\to \omega} \frac{{\mathcal{T}}(P_{\mathcal{D}}a{\mathrm{e}}^{-t{\mathcal{D}}})}{{\mathcal{T}}(P_{\mathcal{D}}{\mathrm{e}}^{-t{\mathcal{D}}})}.$$ Then $\phi_\omega$ is a KMS-state at inverse temperature $\beta_{\mathcal{D}}$ for the ${\mathbb{R}}$-action defined from $\sigma_t$. In particular, if $\beta_{\mathcal{D}}=0$ then $\phi_\omega$ is a tracial state on $A$. If $\beta_{\mathcal{D}}=0$, and there is a decreasing function $\psi:[0,\infty)\to (0,\infty)$ with regular variation of index $-1$, satisfying the conditions  and , and for some $d>0$ we have that $\mu_{\mathcal{T}}(t,P_{\mathcal{D}}{\mathcal{D}})\sim \psi(t)^{-1/d}$ as $t\to \infty$, then for any exponentiation invariant extended limit $\omega$ as $t\to\infty$, $$\phi_{\tilde{\omega}}(a)={\mathcal{T}}_{\omega,\psi}(P_{\mathcal{D}}a(1+{\mathcal{D}}^2)^{-d/2}),$$ where $\tilde{\omega}$ is an extended limit as $t\to 0$ defined in Theorem \[Frohlich\] (see page ), and ${\mathcal{T}}_{\omega,\psi}$ is the Dixmier trace defined from ${\mathcal{T}}$ and $\omega$ on the weak ideal $\mathcal{L}_\psi({\mathcal{N}}):=\{T\in {\mathbb{K}}_{\mathcal{N}}: \mu_{\mathcal{T}}(t,T)=O(\psi(t))\}$. The first part of this result can be found as Corollary \[cor:phi-omega\] (see page ) in the body of the text and the second part as Corollary \[dixmiercorforphiom\] (see page ). \[rmarkonbetazero\] If $\beta_{\mathcal{D}}=0$, any unital $\mathrm{Li}_1$-summable semifinite spectral triple with ${\mathcal{T}}(P_{\mathcal{D}})=\infty$ has positive ${\mathcal{T}}$-essential spectrum and is $\beta$-analytic. Therefore, Theorem \[mainthmconstr\] shows that any unital $\mathrm{Li}_1$-summable semifinite spectral triple $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}},{\mathcal{N}},{\mathcal{T}})$ with ${\mathcal{T}}(P_{\mathcal{D}}{\mathrm{e}}^{-t{\mathcal{D}}})<\infty$ for $t>0$ and ${\mathcal{T}}(P_{\mathcal{D}})=\infty$ gives rise to a tracial state on $A$. This extends a result of Voiculescu [@Voics Proposition 4.6]. For details on this case, see Theorem \[thm:voics\] (see page ). The following three results compute the KMS-state in specific examples. Let $M$ be a closed Riemannian manifold, ${\mathcal{A}}:=C^\infty(M)$, ${\mathcal{D}}$ be a Dirac operator on a Clifford bundle $S\to M$ and ${\mathcal{H}}:=L^2(M,S)$. Then the KMS-state $\phi_\omega$ constructed in Theorem \[mainthmconstr\] is independent of $\omega$ and is a tracial state on $C(M)$ that takes the form $$\phi_\omega(a)=\displaystyle\stackinset{c}{}{c}{}{-\mkern4mu}{\displaystyle\int_M} a\,\mathrm{d}V,$$ where $\mathrm{d}V$ denotes the volume measure defined from the Riemannian metric on $M$ and $\displaystyle\stackinset{c}{}{c}{}{-\mkern4mu}{\displaystyle\int}$ the normalized integral. This result appears as Theorem \[simpleacse\] (see page ) in the body of the text. Let $A$ be a unital $C^*$-algebra, $E$ be a strictly W-regular fgp bi-Hilbertian bimodule (see Definitions \[cond:one\] and \[ass:two\] on pages and , respectively) and $(\mathcal{O}_E,\Xi_A,{\mathcal{D}})$ the associated unbounded $(O_E,A)$-cycle as in [@GMR]. If $\tau$ is a positive trace on $A$, then the semifinite spectral triple $(\mathcal{O}_E,\Xi_A\otimes_A L^2(A,\tau),{\mathcal{D}}\otimes 1_A, (\operatorname{End}^*_A(\Xi_A)\otimes 1)'',\mathrm{Tr}_\tau)$ is $\mathrm{Li}_1$-summable. Moreover, if $\tau$ is critical for $E$ (see Definition \[defn:critical\] on page ), the assumptions in Theorem \[mainthmconstr\] are satisfied and the state $\phi_\omega$ is KMS for the gauge action on $O_E$. If $\tau$ satisfies the Laca-Neshveyev condition for $\alpha\geq 0$ (see Definition \[ass:3.5\] on page ), then $\phi_\omega$ is independent of $\omega$ and takes the form $\phi_\omega=\phi_{LN,\tau}$ where $\phi_{LN,\tau}$ is the KMS-state defined from $\tau$ via the Laca-Neshveyev correspondence. This result is found in Section \[diraccpkms\] (starting on page ). We also discuss extensions of these results to more general $A-A$-correspondences in Subsection \[sub:no-left\] (starting on page ) dispensing the assumption of strict W-regularity. \[mainthmongamma\] Let $\Gamma$ be a discrete group and $c:\Gamma\to {\mathcal{H}}_0$ a Hilbert space valued proper $1$-cocycle defining a length function of at most exponential growth. The semifinite spectral triple $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}},{\mathcal{N}},{\mathcal{T}})$ constructed from $c$ in Subsection \[groupcstarexam\] is an $\mathrm{Li}_1$-summable semifinite spectral triple on $C_b(\Gamma)\rtimes\Gamma$. Moreover, if $c$ is critical (see Definition \[criticaldefn\] on page ) the assumptions of Theorem \[mainthmconstr\] are satisfied and the associated KMS-state $\phi_\omega$ on $C(\partial_{SC} \Gamma)\rtimes\Gamma$ is given by $$\phi_\omega\left(\sum_{g\in \Gamma} a_g\lambda_g\right) =\int_{\partial \Gamma} a_e\,\mathrm{d}\mu_\omega,$$ where $\mu_\omega$ is a quasi-invariant Patterson-Sullivan measure on the Stone-Cech boundary $\partial_{SC}\Gamma$. The state $\phi_\omega$ extends to a KMS-state on the von Neumann algebra $L^\infty(\partial_{SC}\Gamma,\mu_\omega)\overline{\rtimes}\Gamma$ where it is KMS with inverse temperature $1$ for the ${\mathbb{R}}$-action defined from the Radon-Nikodym cocycle $$\sigma_t\left( \sum_{g\in \Gamma} a_g\lambda_g\right) := \sum_{g\in \Gamma} \left(\frac{\mathrm{d}g_*\mu}{\mathrm{d}\mu}\right)^{it}a_g\lambda_g.$$ This result appears as Theorem \[themforgamma\] (see page ) below. Our method extends to proper quasi-cocycles, and as such would allow for the construction of KMS-states from semifinite spectral triples with possible $K$-homological content on a-TT-menable groups. We will prove that the spectral triple of a length function (which is $K$-homologically trivial) gives rise to the same KMS state as that appearing in Theorem \[mainthmongamma\]. Connection to some earlier work {#sub:Connes-lift} ------------------------------- Here we show how our approach relates to some results obtained by Connes in [@BRB Section IV.8.$\alpha$, Theorem 4]. Connes proves that $\theta$-summable Fredholm modules can be lifted to $\theta$-summable spectral triples. We show that Connes’ result can be extended to $\mathrm{Li}_s$-summability for $0<s\leq 1$, and discuss obstructions to summability properties of $K$-homology classes. For terminology and notations concerning summability and operator ideals, the reader is referred forward to Subsection \[subsecsemiffe\]. Recall [@CC; @CGRS2] that a semifinite Fredholm module is a collection $({\mathcal{A}},{\mathcal{H}},F,{\mathcal{N}},{\mathcal{T}})$ where ${\mathcal{A}}$ acts on the Hilbert space ${\mathcal{H}}$ by operators from ${\mathcal{N}}$ and $F\in {\mathcal{N}}$ is an operator with $a(F-F^*), a(F^2-1),[F,a]\in {\mathbb{K}}_{\mathcal{T}}$ for all $a\in {\mathcal{A}}$. We say that $({\mathcal{A}},{\mathcal{H}},F,{\mathcal{N}},{\mathcal{T}})$ is unital if ${\mathcal{A}}$ acts unitally. A unital semifinite Fredholm is said to be $\mathrm{Li}_s$-summable if $[F,a]\in \mathrm{Li}_s({\mathcal{T}})$ for all $a\in {\mathcal{A}}$ and $F^2-1,F-F^*\in \mathrm{Li}_{2s}({\mathcal{T}})$. If the same conditions holds with $ \mathrm{Li}_s({\mathcal{T}})$ replaced by $\mathcal{L}^p({\mathcal{T}})$, and $\mathrm{Li}_{2s}({\mathcal{T}})$ by $\mathcal{L}^{p/2}({\mathcal{T}})$, we say that $({\mathcal{A}},{\mathcal{H}},F,{\mathcal{N}},{\mathcal{T}})$ is $p$-summable. If $({\mathcal{A}},{\mathcal{H}},F,{\mathcal{N}},{\mathcal{T}})$ is a semifinite Fredholm module we say that a semifinite spectral triple $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}},{\mathcal{N}},{\mathcal{T}})$ is a lift if $F-\mathrm{sign}({\mathcal{D}})\in {\mathbb{K}}_{\mathcal{T}}$. \[liftinglis\] Let $s\in (0,1]$ and $({\mathcal{A}},{\mathcal{H}},F,{\mathcal{N}},{\mathcal{T}})$ be a unital semifinite $\mathrm{Li}_s$-summable Fredholm module with $F^2=1$ and $F=F^*$. Assume that ${\mathcal{A}}$ is countably generated. Then there is a self-adjoint operator ${\mathcal{D}}$ affiliated with ${\mathcal{N}}$ making $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}},{\mathcal{N}},{\mathcal{T}})$ into a unital semifinite $\mathrm{Li}_s$-summable spectral triple with $$F=F_{\mathcal{D}}:={\mathcal{D}}|{\mathcal{D}}|^{-1}.$$ Moreover, $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}},{\mathcal{N}},{\mathcal{T}})$ satisfies that $a\operatorname{Dom}(|{\mathcal{D}}|^{1/s})\subseteq \operatorname{Dom}(|{\mathcal{D}}|^{1/s})$ and $[|{\mathcal{D}}|^{1/s},a]$ has a bounded extension for all $a\in {\mathcal{A}}$. This theorem is found in [@BRB Section IV.8.$\alpha$, Theorem 4] in the special case $s=1/2$ and ${\mathcal{N}}=\mathbb{B}({\mathcal{H}})$. We will not give the full details of the proof in the general case, but merely indicate how Connes’ proof extends. The starting point of Connes’ proof is a reduction to the case that $\mathcal{A}$ contains $F$ and is generated by a countable group of unitaries $\Gamma$ generated by a countable set of unitaries $(u^\mu)_{\mu\in {\mathbb{N}}}$. This argument extends to a general von Neumann algebra ${\mathcal{N}}$. Connes introduces the operator $$G:=\sum_{\mu\in {\mathbb{N}}} \frac{[F,u^\mu]^*[F,u^\mu]}{2^\mu \|[F,u^\mu]^*[F,u^\mu]\|_{\mathrm{Li}_1}}.$$ Since $[F,u^\mu]\in \mathrm{Li}_{1/2}$ for all $\mu$, the series converges in $\mathrm{Li}_1$. The proof proceeds by using an average procedure $\Theta$ over the group $\Gamma$ applied to $G$ and Connes proves that ${\mathcal{D}}:=F\Theta(G)^{-1/2}$ fulfils the statement of the theorem. For general $s\in (0,1]$, the proof goes mutatis mutandis using the operator $$G_s:=\sum_{\mu\in {\mathbb{N}}} \frac{([F,u^\mu]^*[F,u^\mu])^{2s}}{2^\mu \|([F,u^\mu]^*[F,u^\mu])^{2s}\|_{\mathrm{Li}_1}}\in \mathrm{Li}_1({\mathcal{T}}),$$ and setting ${\mathcal{D}}:=F\Theta(G)^{-s}$. In the special case $s=1$, we obtain that $({\mathcal{A}},{\mathcal{H}},F,{\mathcal{N}},{\mathcal{T}})$ lifts to a unital semifinite $\mathrm{Li}_s$-summable spectral triple $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}},{\mathcal{N}},{\mathcal{T}})$ which is Lipschitz regular, i.e. for all $a\in{\mathcal{A}}$ the commutator $[|{\mathcal{D}}|,a]$ is bounded. The lifting theorem for $\mathrm{Li}_s$-summable spectral triples (Theorem \[liftinglis\]) stands in sharp contrast to the finitely summable setup, or even the $\mathrm{Li}_{(0),s}$-summable setup. The two upcoming theorems show that a statement as in Theorem \[liftinglis\] could not extend to the ideal $\mathrm{Li}_{(0),1}$. \[tracialobs\] Let $A$ be a unital $C^*$-algebra with no tracial states and $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}},{\mathcal{N}},{\mathcal{T}})$ be a unital semifinite spectral triple on $A$ defining a non-trivial class in $KK_1(A,{\mathbb{K}}_{\mathcal{T}})$. Then $P_{\mathcal{D}}(i\pm {\mathcal{D}})^{-1}\notin \mathrm{Li}_{(0),1}({\mathcal{T}})$. Consider a unital $C^*$-algebra $A$ and an $\mathrm{Li}_{(0),1}$-summable unital semifinite spectral triple $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}},{\mathcal{N}},{\mathcal{T}})$ on $A$ defining a non-trivial class in $KK_1(A,{\mathbb{K}}_{\mathcal{T}})$. In particular, ${\mathcal{T}}(P_{\mathcal{D}})=\infty$; otherwise $P_{\mathcal{D}}\in {\mathbb{K}}_{\mathcal{T}}$ which contradicts the non-triviality of the $KK_1(A,{\mathbb{K}}_{\mathcal{T}})$-class defined by $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}},{\mathcal{N}},{\mathcal{T}})$. By Remark \[rmarkonbetazero\] (see page ) all assumptions of Theorem \[mainthmconstr\] reduces to ${\mathcal{T}}(P_{\mathcal{D}})=\infty$ in the $\mathrm{Li}_{(0),1}$-summable case. Therefore, the existence of $\mathrm{Li}_{(0),1}$-summable unital semifinite spectral triples on $A$ being non-trivial in $KK$ implies that $A$ admits a tracial state. This argument shows that if $A$ admits no tracial states, it admits no $\mathrm{Li}_{(0),1}$-summable unital semifinite spectral triple. In fact, a careful inspection of the results used show that as soon as there is a unital semifinite spectral triple on $A$ with $P_{\mathcal{D}}(i\pm {\mathcal{D}})^{-1}\in \mathrm{Li}_{(0),1}({\mathcal{T}})$, there is an associated tracial state on $A$. The theorem follows. There are several $C^*$-algebras carrying no traces, for instance any purely infinite $C^*$-algebra. Using Theorems \[mainthmconstr\] and \[tracialobs\], we will give an example of a finitely summable Fredholm module that can not lift to an $\mathrm{Li}_{(0),s}$-summable spectral triple. In particular, lifting of finite summability and $\mathrm{Li}_{(0),s}$-summability fails in general. \[liftingfails\] There is a $C^*$-algebra $A$ with $K^1(A)\neq 0$, such that for a dense $*$-subalgebra ${\mathcal{A}}\subseteq A$ we can represent any $x\in K^1(A)$ by a Fredholm module $({\mathcal{A}},{\mathcal{H}}_x,F_x)$ with $F_x^2=1$, $F_x^*=F_x$ and for any $a\in \mathcal{A}$, $[F_x,a]$ is of finite rank. Moreover, any lift $({\mathcal{A}},{\mathcal{H}}_x,{\mathcal{D}}_x)$ of $({\mathcal{A}},{\mathcal{H}}_x,F_x)$ will satisfy that $(1+{\mathcal{D}}_x^2)^{-1/2}\notin \mathrm{Li}_{(0),1}({\mathcal{H}})$. Consider the Cuntz algebra $A=O_N$ and ${\mathcal{A}}$ the $*$-algebra generated by isometries $S_1,S_2,\ldots, S_N\in O_N$ with orthogonal ranges. It is a well known fact that $K^1(O_N)\cong {\mathbb{Z}}/(N-1){\mathbb{Z}}\neq 0$. By [@GM], we can represent the generator of $K^1(O_N)\cong {\mathbb{Z}}/(N-1){\mathbb{Z}}$ by the Fredholm module $({\mathcal{A}},L^2(O_N,\phi),2P-1)$ where $\phi$ is the KMS-state on $O_N$ and $P$ is the orthogonal projection onto the closed linear span of $S_\mu$, where $\mu$ ranges over all finite words on the alpabet $\{1,\ldots, N\}$. By the results of [@GM Section 2.2], $[2P-1,a]=2[P,a]$ is finite rank for all $a\in {\mathcal{A}}$. The first statement of the theorem follows. There are no tracial states on $O_N$ since $$1_{O_N}=\frac{1}{N-1}(N-1)1_{O_N}=\frac{1}{N-1}\left(\sum_{j=1}^N S_j^*S_j-\sum_{j=1}^N S_jS_j^*\right)=\frac{1}{N-1}\sum_{j=1}^N [S_j^*,S_j].$$ We can now deduce the second statement of the theorem from Theorem \[tracialobs\]. It is not of importance that $K^1(O_N)$ is torsion for the argument in Theorem \[liftingfails\] to work. In [@GM], non-torsion examples satisfying the conclusions of Theorem \[liftingfails\] can be found. The reader should also note that the proof of Theorem \[liftingfails\] obstructs all lifts $({\mathcal{A}},{\mathcal{H}}_x,{\mathcal{D}}_x)$ of $({\mathcal{A}},{\mathcal{H}}_x,F_x)$ with $P_{{\mathcal{D}}_x}(1+{\mathcal{D}}_x^2)^{-1/2}\in \mathrm{Li}_{(0),1}({\mathcal{H}})$. In the nonunital case, the techniques of [@CGRS1] will likely be required. The substantial technical considerations in the nonunital case goes beyond this paper, and is left to future work. Structure of the paper ---------------------- Section \[subsec:defns\] recalls the basics of (unital) semifinite spectral triples and their summability. We also recall our main examples from the literature in this section for later use. Section \[sec:KMS\] presents our construction of KMS states from $\mathrm{Li}_1$-summable spectral triples. We close the section by discussing connections to modular spectral triples. We consider the case $\beta_{\mathcal{D}}=0$ in Section \[kmsanddix\] and compute the tracial states constructed in Section \[sec:KMS\] by means of Dixmier traces. In Section \[kmsinexamplesec\] we apply the techniques of Section \[sec:KMS\] to the examples. The final Section \[diraccpkms\] examines the construction of KMS states for Cuntz-Pimsner algebras. In this case we apply our ideas to derive obstructions to the existence of fgp bi-Hilbertian bimodule structures compatible with the underlying correspondence of the Cuntz-Pimsner algebra. Notations --------- -------------------------------------------------------------------------- --------------------------------------------------------------------------------------- ${\mathcal{N}}$ semifinite von Neumann algebra ${\mathcal{T}}$ positive, faithful, normal, semifinite trace on ${\mathcal{N}}$ ${\mathbb{K}}_{\mathcal{N}}$ ideal of ${\mathcal{T}}$-compact operators $\operatorname{End}^*_A(X)$ $C^*$-algebra of adjointable endomorphisms of an $A$-Hilbert $C^*$-module X ${\mathbb{K}}_A(X)$ $C^*$-algebra of compact endomorphisms of an $A$-Hilbert $C^*$-module X ${\mathcal{A}}$ $*$-algebra ${\mathcal{A}}'$ the commutant of an algebra ${\mathcal{A}}$ $A$ $C^*$-closure of an algebra ${\mathcal{A}}$ $A_{\mathcal{D}}$ saturation of $A$ under the action of the wave group ${\mathrm{e}}^{it{\mathcal{D}}}$ $P_{\mathcal{D}}:=\chi_{[0,\infty)}({\mathcal{D}})$ non-negative spectral projection of an operator ${\mathcal{D}}$ $F_{\mathcal{D}}:=2 P_{\mathcal{D}}-1$ $\mu_{\mathcal{T}}(\cdot, T)$ singular values function of an operator $T$ affiliated with ${\mathcal{N}}$ $n_{\mathcal{T}}(\cdot,T)$ distribution function of an operator $T$ affiliated with ${\mathcal{N}}$ $\mathrm{Li}_s({\mathcal{T}}), \mathrm{Li}_{(0),s}({\mathcal{T}})$ ideals of compact operators in Definition \[derjugendvonheutebrauchidealen\] on page  $\mathcal{L}_\psi({\mathcal{T}}), \mathcal{L}_{(0),\psi}({\mathcal{T}})$ ideals of compact operators in Definition \[derjugendvonheutebrauchidealen\] on page  ${\mathcal{T}}_{\omega, \psi}(T)$ Dixmier trace on $\mathcal L_\psi({\mathcal{T}})$ $\mathfrak{T}(A)$ set of positive traces on a unital $C^*$-algebra $A$ $L^\infty(a,\infty), \ a\ge0$ space of essentially bounded functions on $(a,\infty)$ equipped with the essential supremum norm $C_0(a,\infty), \ a\ge0$ subspace of $L^\infty(a,\infty)$ of all continuous functions vanishing at infinity $\lim_{t\to \omega} f(t)$ value of an extended limit $\omega$ on a function $f$ $\ell^\infty({\mathbb{N}})$ space of bounded sequences equipped with the supremum norm $\lim_{k\to \omega} x_k$ value of an extended limit $\omega$ on a sequence $x$ $\mathfrak{L}_g$ transfer operator defined by formula  on page  $f \sim g$ for two functions or sequences $f$ and $g$ if $f=g+o(f)$ and $g=f+o(g)$ -------------------------------------------------------------------------- --------------------------------------------------------------------------------------- Acknowledgements ---------------- A. R. thanks the Gothenburg Centre for Advanced Studies in Science and Technology for funding and the University of Gothenburg and Chalmers University of Technology for their hospitality in 2017 when this work was begun. M. G. and A. U. were supported by the Swedish Research Council Grant 2015-00137 and Marie Sklodowska Curie Actions, Cofund, Project INCA 600398. The authors acknowledge the support of the Erwin Schrödinger Institute where part of this work was conducted. The authors are grateful to Alan Carey, Heath Emerson and Bram Mesland for inspiring discussions. We also thank Branimir Ćaćić for sharing his construction of semifinite spectral triples from proper group cocycles, and Edward McDonald for references on previous work in that direction. Preliminaries {#subsec:defns} ============= Before entering into the body of the paper, we recall some basic definitions that we will require and provide some examples that motivated this work. These examples will be studied further in the later sections of the paper. The results will be formulated for semifinite spectral triples. We do however remark that there are several examples of ‘vanilla’ spectral triples that will be used throughout the paper. Semifinite spectral triples and summability {#subsecsemiffe} ------------------------------------------- To set the stage for the paper, we summarize the basic definitions and properties of semifinite spectral triples. The reader familiar with semifinite spectral triples and symmetrically normed operator ideals can skip this subsection. We let ${\mathcal{N}}$ denote a semifinite von Neumann algebra and we fix a positive, faithful, normal, semifinite trace ${\mathcal{T}}$ on ${\mathcal{N}}$. The ${\mathcal{T}}$-compact operators are denoted by ${\mathbb{K}}_{\mathcal{N}}$. The $C^*$-algebra ${\mathbb{K}}_{\mathcal{N}}$ can be defined as the norm closed ideal generated by the projections $E\in\mathcal N$ with ${\mathcal{T}}(E)<\infty$. Equivalently, one can define ${\mathbb{K}}_{\mathcal{N}}:=\{T\in {\mathcal{N}}: \mu_{\mathcal{T}}(t,T)=o(1)$ as $t\to \infty\}$ where the singular value function $\mu_{\mathcal{T}}(t,T)$ is defined as $$\label{mu} \mu_{\mathcal{T}}(t,T):=\inf\big\{\|T(1-E)\|_{\mathcal{N}}: \mbox{ where $E\in {\mathcal{N}}$ is a projection with ${\mathcal{T}}(E)\leq t$}\big\}.$$ \[thedefn\] A semifinite spectral triple $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}},{\mathcal{N}},{\mathcal{T}})$ consists of - A $*$-algebra ${\mathcal{A}}$ represented on a Hilbert space ${\mathcal{H}}$ as operators in ${\mathcal{N}}\subseteq \mathbb{B}({\mathcal{H}})$, that is, we have a specified $*$-homomorphism $\pi:{\mathcal{A}}\to{\mathcal{N}}$. - A densely defined self-adjoint operator $ {\mathcal{D}}:{\rm dom}\ {\mathcal{D}}\subset{\mathcal{H}}\to{\mathcal{H}}$ which is affiliated with ${\mathcal{N}}$ such that for all $a\in {\mathcal{A}}$ we have $a\cdot\mathrm{dom}{\mathcal{D}}\subset\mathrm{dom}{\mathcal{D}}$ and 1. $[{\mathcal{D}},\pi(a)]:={\mathcal{D}}\pi(a)-\pi(a){\mathcal{D}}$ initially defined on $\mathrm{dom}({\mathcal{D}})$ is bounded in operator norm. 2. $\pi(a)(1+{\mathcal{D}}^2)^{-1/2}\in {\mathbb{K}}_{\mathcal{N}}$. Sometimes we write a spectral triple as a collection of three objects $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}})$. In this case, it is implicitly assumed that ${\mathcal{N}}=\mathbb{B}(L^2(M,S))$ and ${\mathcal{T}}$ is the standard trace. If in addition to the data $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}},{\mathcal{N}},{\mathcal{T}})$ we have specified an operator $\gamma\in{\mathcal{B}}({\mathcal{H}})$ with $\gamma=\gamma^*$, $\gamma^2=1$, ${\mathcal{D}}\gamma+\gamma{\mathcal{D}}=0$ on $\operatorname{Dom}({\mathcal{D}})$, and for all $a\in{\mathcal{A}}$ we have $\gamma\pi(a)=\pi(a)\gamma$, we call the semifinite spectral triple even, or sometimes graded. If $\gamma$ has not been specified, we say that the semifinite spectral triple is odd, or ungraded. This distinction plays an important role in the topological properties of the spectral triple, but since this paper deals with measure theory it will not play a role in this paper. We will nearly always dispense with the representation $\pi$, treating ${\mathcal{A}}$ as a subalgebra of ${\mathcal{N}}\subseteq {\mathcal{B}}({\mathcal{H}})$. In the sequel we assume that the algebra ${\mathcal{A}}$ is unital and that $1\in{\mathcal{A}}$ acts as the identity of the Hilbert space. In particular, the operator $(1+{\mathcal{D}}^2)^{-1/2}$ is a ${\mathcal{T}}$-compact operator. To emphasize this assumption, we refer to the data $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}},{\mathcal{N}},{\mathcal{T}})$ as a *unital* semifinite spectral triple. Examples of semifinite spectral triples often satisfy a finer summability structure, i.e. a refinement of the condition $(1+{\mathcal{D}}^2)^{-1/2}\in {\mathbb{K}}_{\mathcal{N}}$. We formulate such conditions in terms of symmetrically quasi-normed operator ideals. We will use the Schatten ideals, the $\mathrm{Li}$-ideals and more generally weak ideals. \[derjugendvonheutebrauchidealen\] Let ${\mathcal{N}}$ denote a semifinite von Neumann algebra and ${\mathcal{T}}$ a positive, faithful, normal, semifinite trace on ${\mathcal{N}}$. For parameters $p,d\in [1,\infty)$ and $s>0$ we define the following operator ideals. - $\mathcal{L}^p({\mathcal{T}}):=\{T\in {\mathbb{K}}_{\mathcal{N}}: \mu_{\mathcal{T}}(\cdot,T)\in L^p(0,\infty)\}$. - $\mathcal{L}^{(d,\infty)}({\mathcal{T}}):=\{T\in {\mathbb{K}}_{\mathcal{N}}: \mu_{\mathcal{T}}(\cdot,T)=O(t^{-1/d})\mbox{ as $t\to \infty$}\}$. - $\mathrm{Li}_s({\mathcal{T}}):=\{T\in {\mathbb{K}}_{\mathcal{N}}:\mu_{\mathcal{T}}(t,T)=O((\log(t))^{-s})\mbox{ as $t\to \infty$}\}$. - $\mathrm{Li}_{(0),s}({\mathcal{T}}):=\{T\in {\mathbb{K}}_{\mathcal{N}}:\mu_{\mathcal{T}}(t,T)=o((\log(t))^{-s})\mbox{ as $t\to \infty$}\}$. - If $\psi:[0,\infty)\to (0,\infty)$ is a decreasing function satisfying that $\sup_{t>0}\frac{\psi(t)}{\psi(2t)}<\infty$, we define the associated weak ideal $$\mathcal{L}_\psi({\mathcal{T}}):=\{T\in {\mathbb{K}}_{\mathcal{N}}: \mu_{\mathcal{T}}(t,T)=O(\psi(t))\},$$ and its separable subspace $$\mathcal{L}_{(0),\psi}({\mathcal{T}}):=\{T\in {\mathbb{K}}_{\mathcal{N}}: \mu_{\mathcal{T}}(t,T)=o(\psi(t))\},$$ The condition $\sup_{t>0}\frac{\psi(t)}{\psi(2t)}<\infty$ guarantees that $\mathcal{L}_\psi({\mathcal{T}})$ is a vector space, and in fact even a quasi-Banach space in the quasi-norm $$\|T\|_{\mathcal{L}_\psi}:=\sup_{t>0}\frac{\mu_{\mathcal{T}}(t,T)}{\psi(t)}.$$ Note that $\mathrm{Li}_s({\mathcal{T}})=\mathcal{L}_\psi({\mathcal{T}})$ and $\mathrm{Li}_{(0),s}({\mathcal{T}})=\mathcal{L}_{(0),\psi}({\mathcal{T}})$ for $\psi(t):=(\log(2+t))^{-s}$. It is immediate from the definition that $\mathcal{L}^p({\mathcal{T}})\subseteq \mathrm{Li}_{(0),s}({\mathcal{T}})$ for any $p$ and $s$. More generally, if $\psi_1,\psi_2:[0,\infty)\to (0,\infty)$ are two decreasing functions satisfying that $\sup_{t>0}\frac{\psi_j(t)}{\psi_j(2t)}<\infty$, then $\mathcal{L}_{\psi_1}({\mathcal{T}})\subseteq \mathcal{L}_{\psi_2}({\mathcal{T}})$ as soon as $\psi_1=O(\psi_2)$. Our definition of symmetrically normed operator ideals in the semifinite setting differs slightly from the standard definition unless ${\mathcal{N}}$ is atomic. In the usual definition, the symmetrically normed operator ideals are defined from operators affiliated with ${\mathcal{N}}$ that potentially are unbounded. Since we only use bounded operators from these ideals, we have incorporated this fact in our definition. \[summdefn\] Let $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}},{\mathcal{N}},{\mathcal{T}})$ be a unital semifinite spectral triple. - $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}},{\mathcal{N}},{\mathcal{T}})$ is said to be $p$-summable if $(1+{\mathcal{D}}^2)^{-1/2}\in \mathcal{L}^p({\mathcal{T}})$. - $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}},{\mathcal{N}},{\mathcal{T}})$ is said to be $(d,\infty)$-summable if $(1+{\mathcal{D}}^2)^{-1/2}\in \mathcal{L}^{(d,\infty)}({\mathcal{T}})$. - $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}},{\mathcal{N}},{\mathcal{T}})$ is said to be $\mathrm{Li}_s$-summable if $(1+{\mathcal{D}}^2)^{-1/2}\in \mathrm{Li}_s({\mathcal{T}})$. - $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}},{\mathcal{N}},{\mathcal{T}})$ is said to be $\mathrm{Li}_{(0),s}$-summable if $(1+{\mathcal{D}}^2)^{-1/2}\in \mathrm{Li}_{(0),s}({\mathcal{T}})$. - $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}},{\mathcal{N}},{\mathcal{T}})$ is said to be $\psi$-summable if $(1+{\mathcal{D}}^2)^{-1/2}\in \mathcal{L}_\psi({\mathcal{T}})$. The standard terminology in the literature for the special case $s=1/2$ is to refer to $\mathrm{Li}_{1/2}$-summability as weak $\theta$-summability and to $\mathrm{Li}_{(0),1/2}$-summability as $\theta$-summability. Since $\mathcal{L}^{(d,\infty)}({\mathcal{T}})\subseteq \mathcal{L}^{p}({\mathcal{T}})$ for all $p>d$, $(d,\infty)$-summability refines $p$-summability. The notion of $(d,\infty)$-summability is a noncommutative generalization of being $d$-dimensional as the spectral triple defined from a Dirac operator on a closed $d$-dimensional manifold (as in Subsection \[diracmfdfirst\]) is $(d,\infty)$-summable. We shall see an abundance of $\mathrm{Li}_1$-summable, truly noncommutative, examples where $p$-summability and $(d,\infty)$-summability fails for all $p$ and $d$. The notion of $\psi$-summability generalizes both $(d,\infty)$-summability and $\mathrm{Li}_1$-summability, and appears naturally in examples of (semi-) group actions on manifolds (see Subsection \[diracmfdkms\] and [@DGMW; @GU]). We will make use of this notion in Section \[kmsanddix\] where certain conditions on $\psi$ allows one to compute the tracial state defined from a $\psi$-summable unital semifinite spectral triples in terms of Dixmier traces on $\mathcal{L}_\psi({\mathcal{T}})$. It is readily verified that $\mathrm{Li}_s$-summability is equivalent to $${\mathcal{T}}({\mathrm{e}}^{-t|{\mathcal{D}}|^{1/s}})<\infty, \quad\mbox{for $t>t_0$ for some critical value $t_0$},$$ and that $\mathrm{Li}_{(0),s}$-summability is equivalent to $${\mathcal{T}}({\mathrm{e}}^{-t|{\mathcal{D}}|^{1/s}})<\infty, \quad\mbox{for $t>0$}.$$ In particular, $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}},{\mathcal{N}},{\mathcal{T}})$ is $\theta$-summable if and only if $${\mathcal{T}}({\mathrm{e}}^{-t{\mathcal{D}}^2})<\infty, \quad\mbox{for all $t>0$}.$$ Historically, $\theta$-summability has been studied more in depth than $\mathrm{Li}_1$-summability. This can in part be explained from the two facts that the JLO-cocycle only requires $\theta$-summability and classically, the heat operator $\mathrm{e}^{-t{\mathcal{D}}^2}$ is geometrically more interesting than $\mathrm{e}^{-t|{\mathcal{D}}|}$ to study on a manifold. The two operators $\mathrm{e}^{-t{\mathcal{D}}^2}$ and $\mathrm{e}^{-t|{\mathcal{D}}|}$ can be compared by explicit integral formulas, see [@ggneumann Chapter 4]. We will exploit the observation that large classes of examples of $\theta$-summable spectral triples are also $\mathrm{Li}_1$-summable. In the bulk of the paper, we are interested in computing asymptotics of heat traces of the form ${\mathcal{T}}(B{\mathrm{e}}^{-t|{\mathcal{D}}|^{1/s}})$ for $B\in {\mathcal{N}}$ as $t$ approaches a critical value. When $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}},{\mathcal{N}},{\mathcal{T}})$ is $\mathrm{Li}_{(0),s}$-summable, the critical value of $t$ is $0$, and in several classical examples (e.g. on closed manifolds) the heat trace ${\mathcal{T}}(B{\mathrm{e}}^{-t|{\mathcal{D}}|^{1/s}})$ admits an asymptotic expansion. The following result is useful for relating heat trace asymptotics to zeta function asymptotics in the case of the nice behaviour appearing when $t_0=0$. \[heatvszeta\] Let $s\in(0,1]$. Let $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}},{\mathcal{N}},{\mathcal{T}})$ be an $\mathrm{Li}_{(0),s}$-summable semifinite spectral triple and $B\in {\mathcal{N}}$. The following are equivalent. 1. There are constants $p^{\rm heat}>0$, $\epsilon>0$ and $c_B^{\rm heat}\in {\mathbb{C}}$ such that $${\mathcal{T}}(B{\mathrm{e}}^{-t|{\mathcal{D}}|^{1/s}})=c_B^{\rm heat}t^{-sp^{\rm heat}}+O(t^{-sp^{\rm heat}+\epsilon}), \quad\mbox{as $t\to 0$}.$$ 2. The $\zeta$-function $\zeta(z;B,|{\mathcal{D}}|^{1/s}):={\mathcal{T}}(B|{\mathcal{D}}|^{-z/s})$ is well-defined for large $\mathrm{Re}(z)$ and there are constants $p^{\zeta}>0$, $\epsilon'>0$, $c_B^{\zeta}\in {\mathbb{C}}$ and a function $f=f(z)$ holomorphic in the region $\mathrm{Re}(z)>sp^{\zeta}-\epsilon'$ such that $$\zeta(z;B,|{\mathcal{D}}|^{1/s})=\frac{1}{\Gamma(sp^\zeta)}\frac{c_B^{\zeta}}{z-sp^\zeta}+f(z).$$ In this case, $p^{\rm heat}=p^{\zeta}$ and $c_B^{\rm heat}=c_B^\zeta$. Moreover, if the conditions above hold for one $s\in (0,1]$, it holds for all $s\in (0,1]$. If $B=1$ and either of the conditions above hold, then $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}},{\mathcal{N}},{\mathcal{T}})$ is $p^\zeta$-summable. The proof of Theorem \[heatvszeta\] follows by noting that $\Gamma(z)\zeta(z;B,|{\mathcal{D}}|^{1/s})$ is the Mellin transform of ${\mathcal{T}}(B{\mathrm{e}}^{-t|{\mathcal{D}}|^{1/s}})$ and using [@grubbsee Proposition 5.1]. Recall that we use the notation $P_{\mathcal{D}}:=\chi_{[0,\infty)}({\mathcal{D}})$ for the non-negative spectral projection of ${\mathcal{D}}$. Let us state a fundamental lemma on the commutators of $A$ with the function of ${\mathcal{D}}$ defined by $$F_{\mathcal{D}}:= 2P_{\mathcal{D}}-1.$$ Note that $F_{\mathcal{D}}$ differs from the phase ${\mathcal{D}}|{\mathcal{D}}|^{-1}$ by the ${\mathcal{T}}$-finite kernel projection of ${\mathcal{D}}$. \[bddcommsum\] Let $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}},{\mathcal{N}},{\mathcal{T}})$ be a semifinite spectral triple. Then for any $a\in A$ $$[F_{\mathcal{D}},a]\in {\mathbb{K}}_{\mathcal{N}}.$$ Moreover if $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}},{\mathcal{N}},{\mathcal{T}})$ is unital, then if $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}},{\mathcal{N}},{\mathcal{T}})$ is $p$-summable, then $[F_{\mathcal{D}},a]\in \mathcal{L}^p({\mathcal{T}})$ for all $a\in {\mathcal{A}}$, and if $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}},{\mathcal{N}},{\mathcal{T}})$ is $\mathrm{Li}_s$-summable, then $[F_{\mathcal{D}},a]\in \mathrm{Li}_s({\mathcal{T}})$ for all $a\in {\mathcal{A}}$. More generally, if $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}},{\mathcal{N}},{\mathcal{T}})$ is $\psi$-summable then $[F_{\mathcal{D}},a]\in \mathcal{L}_\psi({\mathcal{T}})$ for all $a\in {\mathcal{A}}$. The proof of the operator inequality $-\|[{\mathcal{D}},a]\||{\mathcal{D}}|^{-1} \leq [F_{\mathcal{D}},a]\leq\|[{\mathcal{D}},a]\||{\mathcal{D}}|^{-1}$ for invertible ${\mathcal{D}}$ and $a=-a^*$ is found in the proof of [@sww Proposition 1]. The assertion follows from the definition of $p$-, $\rm{Li}_s$- and $\psi$-summability, resp. In the non-invertible case we replace $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}},{\mathcal{N}},{\mathcal{T}})$ by $$\label{double-trick} \left(\begin{pmatrix}{\mathcal{A}}&0\\ 0 & 0\end{pmatrix},{\mathcal{H}}\oplus {\mathcal{H}},{\mathcal{D}}_\mu=\begin{pmatrix} {\mathcal{D}}&\mu\\ \mu&-{\mathcal{D}}\end{pmatrix},M_2({\mathcal{N}}),{\mathcal{T}}\otimes \mathrm{Tr}_{M_2}\right),$$ for $\mu\in[0,1]$. When $\mu>0$ we are back in the invertible case. In [@CGRS2 Proposition 2.25] it is shown that for any $a\in{\mathcal{A}}$ we have the norm limit $$\begin{pmatrix} [F_{\mathcal{D}},a] & 0\\ 0 & 0\end{pmatrix}=\lim_{\mu\to 0}[F_{{\mathcal{D}}_\mu},a]$$ and so $[F_{\mathcal{D}},a]$ is compact. Indeed, the proof of [@CGRS2 Proposition 2.25] shows that if $(1+{\mathcal{D}}^2)^{-1/2}$ is in a symmetrically quasi-normed ideal $J$ of ${\mathcal{T}}$-compact operators then $[F_{\mathcal{D}},a]\in J$ for all $a\in{\mathcal{A}}$. Again the assertion follows from the definitions of summability. ### Semifinite spectral triples from unbounded Kasparov modules For several kinds of $C^*$-algebras one can capture the noncommutative geometry through an unbounded Kasparov module. This is a bivariant generalization of spectral triples. Localizing an unbounded Kasparov module in a positive trace gives rise to a semifinite spectral triple as in Theorem \[localintra\] below. Several of the examples in this paper arises in this way. We briefly recall this construction, which has been informally used for some years. Let $A$ and $B$ be unital $C^*$-algebras. A unital unbounded $(B,A)$-Kasparov module is a collection $(\mathcal{B},X,{\mathcal{D}})$ where - $\mathcal{B}\subseteq B$ is a dense $*$-subalgebra, - $X$ is an $A$-Hilbert $C^*$-module carrying a left action of $B$ as adjointable operators, - ${\mathcal{D}}$ is an $A$-linear, densely defined, self-adjoint, regular operator on $X$ with $A$-compact resolvent $(i\pm {\mathcal{D}})^{-1}\in {\mathbb{K}}_A(X)$ and - for $a\in \mathcal{B}$ the operator $[{\mathcal{D}},a]$ is defined on $\mathrm{dom}({\mathcal{D}})$ and is bounded in the norm on $X$. If $\tau$ is a positive trace on $A$, we write $L^2(X,\tau):=X\otimes_A L^2(A,\tau)$ where $L^2(A,\tau)$ is the GNS-representation associated with $\tau$. For $\xi,\eta\in X$, we write $\Theta_{\xi,\eta}$ for the rank one operator $\Theta_{\xi,\eta}(\nu)=\xi(\eta|\nu)_A$. The von Neumann algebra ${\mathcal{N}}_\tau(X):=(\operatorname{End}^*_A(X)\otimes 1_A)''\subseteq \mathbb{B}(L^2(X,\tau))$ coincides with the weak closure of the set of operator spanned by $\{\Theta_{\xi,\eta}\otimes 1_A: \xi,\eta\in X\}$ and carries a positive, normal, semifinite, faithful trace $\operatorname{Tr}_\tau$ characterized by $\operatorname{Tr}_\tau(\Theta_{\xi,\eta}\otimes 1_A):=\tau((\eta|\xi)_A)$, see [@LN Section 3]. The following theorem also appears in [@MacR]. \[localintra\] Let $({\mathcal{B}},X_A,{\mathcal{D}})$ be a unital unbounded $(B,A)$-Kasparov module and $\tau:\,A\to{\mathbb{C}}$ a faithful norm densely defined norm lower semicontinuous tracial weight. Then the data $({\mathcal{B}},L^2(X,\tau),{\mathcal{D}}{\otimes}1,{\mathcal{N}}_\tau(X),\operatorname{Tr}_\tau)$ defines a semifinite spectral triple. The von Neumann algebra is ${\mathcal{N}}_\tau(X)=(\operatorname{End}_A^*(X){\otimes}1)''$ and $\operatorname{Tr}_\tau:\,{\mathcal{N}}\to{\mathbb{C}}$ is the (positive faithful semifinite normal) trace dual to the normal extension of $\tau$ to $A''\subseteq \mathbb{B}(L^2(A,\tau))$. The operator ${\mathcal{D}}{\otimes}1$ is self-adjoint by [@lancesbook Proposition 9.10]. The commutant of ${\mathcal{N}}_\tau(X)$ in $X{\otimes}_AL^2(A,\tau)$ is the algebra $A''$ (acting by right multiplication). Every unitary in $A''$ thus preserves the domain of ${\mathcal{D}}{\otimes}1$ and so ${\mathcal{D}}{\otimes}1$ is affiliated to ${\mathcal{N}}_\tau(X)$. Plainly commutators of ${\mathcal{D}}{\otimes}1$ with ${\mathcal{B}}$ remain bounded. Since we start with an unbounded Kasparov module, the operator $(1+{\mathcal{D}}^2)^{-1/2}\in {\mathbb{K}}_A(X)$. So we can approximate $(1+{\mathcal{D}}^2)^{-1/2}$ in norm by finite rank operators $\sum_j\Theta_{x_j,y_j}$ and we can take the $x_j,\,y_j\in X$. Hence $(1+({\mathcal{D}}{\otimes}1)^2)^{-1/2}=(1+{\mathcal{D}}^2)^{-1/2}{\otimes}1$ is in the norm closure of the finite trace operators in ${\mathcal{N}}_\tau(X)$, and so ${\mathcal{T}}$-compact. A conceptual viewpoint is that $(\mathcal{B},L^2(X,\tau),{\mathcal{D}}\otimes 1_A, (\operatorname{End}^*_A(X)\otimes 1_A)'', \operatorname{Tr}_\tau)$ is a semi-finite refinement of the unbounded Kasparov product of $(\mathcal{B},X,{\mathcal{D}})$ with the Morita morphism $A\to {\mathbb{K}}_A(X)$ and the $*$-homomorphism ${\mathbb{K}}_A(X) \to \mathbb{K}_{(\operatorname{End}^*_A(X)\otimes 1_A)''}$. There is a close relationship between the semifinite index and the Kasparov product, described in [@CGRS2; @KNR]. We remark at this stage that there is to date no general theory of symmetrically quasi-normed operator ideals in Hilbert $C^*$-modules, and, as such, no satisfactory way of describing summability. In concrete applications, it is possible to circumvent this problem by choosing a frame on $X$, implicitly using the machinery of [@nistann]. In the examples of most relevance to this paper the spectrum of ${\mathcal{D}}$ is discrete, and we can use the following proposition to study summability. Let $(\mathcal{B},X,{\mathcal{D}})$ be a unital unbounded $(B,A)$-Kasparov module, where ${\mathcal{D}}$ has discrete spectrum $\sigma({\mathcal{D}})\subseteq {\mathbb{R}}$, and $\tau$ be a positive trace on $A$. Set $P_\lambda:=\chi_{\{\lambda\}}({\mathcal{D}})\in {\mathbb{K}}_A(X)$ for $\lambda\in \sigma({\mathcal{D}})$. Then it holds that 1. $(\mathcal{B},L^2(X,\tau),{\mathcal{D}}\otimes 1_A, {\mathcal{N}}_\tau(X), \operatorname{Tr}_\tau)$ is $\mathrm{Li}_s$-summable if and only if $$\sum_{\lambda\in \sigma({\mathcal{D}})} {\mathrm{e}}^{-t|\lambda|^{1/s}}\operatorname{Tr}_\tau(P_\lambda)<\infty,$$ for $t$ large enough. 2. $(\mathcal{B},L^2(X,\tau),{\mathcal{D}}\otimes 1_A, {\mathcal{N}}_\tau(X), \operatorname{Tr}_\tau)$ is $p$-summable if and only if $$\sum_{\lambda\in \sigma({\mathcal{D}})} (1+\lambda^2)^{-p/2}\operatorname{Tr}_\tau(P_\lambda)<\infty.$$ The proof follows from the following formula: $$\operatorname{Tr}_\tau(f({\mathcal{D}}))=\sum_{\lambda\in \sigma({\mathcal{D}})} f(\lambda)\operatorname{Tr}_\tau(P_\lambda),$$ which holds for every positive Borel function $f$. Examples {#subsec:examples} -------- To give some further context before entering into the main construction of this paper, let us recall some well known examples that we will further explore later on in the paper. The focus in our presentation is on $\mathrm{Li}_1$-summability and heat traces. We remark that the constructions in this subsection are rather lengthy, and the reader familiar with the literature can at a first read restrict themself to glancing through this subsection. ### Dirac operators on closed manifolds {#diracmfdfirst} The prototypical example of a spectral triple arises from Dirac operators on a closed Riemannian manifold $M$. We can work with a rather general type of Dirac operators: if $S\to M$ is a Clifford module on $M$ and $\slashed{D}$ is a first order elliptic operator acting on $C^\infty(M,S)$ being symmetric in the $L^2$-inner product and $\slashed{D}^2$ is a Laplacian type operator[^2], we say that $\slashed{D}$ is a Dirac operator. In this case, the closure of $\slashed{D}$ in its graph norm defines a self-adjoint operator on $L^2(M,S)$ that we by an abuse of notation also denote by $\slashed{D}$. It is well-known that $(C^\infty(M),L^2(M,S),\slashed{D})$ is a spectral triple on $C^\infty(M)$. We summarize the main properties of its heat traces in the following proposition. \[heatasumfd\] Let $M$ be an $n$-dimensional Riemannian closed manifold, $\slashed{D}$ a Dirac operator on $M$, and $(C^\infty(M),L^2(M,S),\slashed{D})$ the associated spectral triple. This spectral triple is $(n,\infty)$-summable and for any classical zero-th order pseudo-differential operator $A$ on $S$ with principal symbol $a\in C^\infty(S^*M, \mathrm{End}(S))$ and every $s\in(0,1]$ we have $$\operatorname{Tr}_{L^2(M,S)}(A{\mathrm{e}}^{-t|\slashed{D}|^{1/s}}) =\Gamma(sn+1)c_n^st^{-sn}\int_{S^*M} \mathrm{Tr}_S(a)\,\mathrm{d}V +O(t^{-sn+\epsilon}), \quad\mbox{as $t\to 0$},$$ for some dimensional constant $c_n>0$ and $\epsilon>0$. Here $\mathrm{Tr}_S(a)\in C^\infty(S^*M)$ denotes the fibrewise trace of $a$. The dimensional constant $c_n$ is determined by the Weyl law for $|\slashed{D}|$ describing the ordered sequence $(\lambda_k(|\slashed{D}|))_{k\in {\mathbb{N}}}$ of eigenvalues as $$\lambda_k(|\slashed{D}|)=\frac{1}{c_n\mathrm{vol}(M)^{1/n}\mathrm{rank}(S)^{1/n}}k^{1/n}+O(k^{1/n-\epsilon_0}),$$ for some $\epsilon_0>0$. The Weyl law is proven in many places, for instance [@Gilkey]. The general heat trace asymptotics follows from Theorem \[heatvszeta\] and [@Gilkey]. We return to this example below in Example \[diracmfdhea\] (see page ) and Subsection \[diracmfdkms\] (see page ). Later on in the paper, we will make use of a modification of the spectral triple coming from a Dirac operator that is also compatible with non-isometric semigroup actions. Similar constructions were previously considered in [@DGMW; @GU]. \[regcariaden\] Let $\psi:[0,\infty)\to (0,\infty)$ be a positive measurable function. - We say that $\psi$ is regularly varying of index $\rho$ if for all $\lambda>0$ $$\label{varfrho} \lim_{t\to \infty} \frac{\psi(\lambda t)}{\psi(t)}=\lambda^\rho.$$ - We say that $\psi$ is smoothly regularly varying of index $\rho$ if $\psi\in C^\infty$ and for any $k\in {\mathbb{N}}$, $$\label{smoothvarfrho} \lim_{t\to \infty} \frac{t^k \psi^{(k)}(t)}{\psi(t)}=\rho(\rho-1)\cdots(\rho-k+1).$$ A regularly varying function satisfies $\sup_{t>0}\frac{\psi(t)}{\psi(2t)}<\infty$ and there is an associated weak ideal $\mathcal{L}_\psi$ as in Definition \[derjugendvonheutebrauchidealen\]. By [@GU Lemma 7.1], any smoothly regularly varying function $\psi$ satisfies that $\partial_t^k\psi(t)=O(\psi(t)(1+t^2)^{-k/2})$ so if $\psi$ additionally is bounded, $\psi$ belongs to the Hörmander class $S^0$. Smooth regular variation is a strengthening of having regular variation – a condition used below in Section \[kmsanddix\] in the context of defining and computing Dixmier traces. See more also in [@GU]. By [@RegVar Theorem 1.8.2], any regularly varying function asymptotically behaves like a smoothly regularly varying function. Smooth regular variation allows for defining associated classes of pseudo-differential operators and computing Dixmier traces of geometric operators by means of a Connes trace theorem, see [@GU Section 7 and 9]. For a decreasing smoothly varying function $\psi:[0,\infty)\to (0,\infty)$ with $\lim_{t\to 0}\psi(t)=0$, we define the self-adjoint operator $$\slashed{D}_\psi:=F_{\slashed{D}}\psi(|\slashed{D}|^n)^{-1}.$$ It follows from [@GU Proposition 10.1] that $\slashed{D}_\psi\in L^0_{\psi^{-1}}(M,S)$ (see [@GU] for the meaning of this symbol) and its $\psi$-principal symbol is $c_S(\xi)|\xi|^{-1}\psi(|\xi|)^{-1}$, where $c_S:T^*M\to \operatorname{End}(S)$ denotes Clifford multiplication. The next result follows from [@GU Proposition 10.3]. \[heatasumfdwithapsi\] Let $M$ be an $n$-dimensional Riemannian closed manifold, $\slashed{D}$ a Dirac operator on $M$, and $\psi$ as above with $$\psi(t)^{-1}=O(t^{1/n}),\quad\mbox{as $t\to \infty$}.$$ Then $(C^\infty(M),L^2(M,S),\slashed{D}_\psi)$ is a $\psi$-summable spectral triple whose associated $K$-homology class coincides with that of $(C^\infty(M),L^2(M,S),\slashed{D})$. If $\psi(t)^{-1}=o(t^{1/n})$, then for any $a\in C^\infty(M)$, the operator $[\slashed{D}_\psi,a]$ is compact with $$\mu(t,[\slashed{D}_\psi,a])=O(t^{-1/n}\psi(t)^{-1}),\quad\mbox{as $t\to \infty$}.$$ We return to the problem of computing heat traces involving $\slashed{D}_\psi$ below in the Section \[kmsanddix\] (see page ) and Subsection \[diracmfdkms\] (see page ). The spectral triple $(C^\infty(M),L^2(M,S),\slashed{D}_\psi)$ does in some cases extend to a crossed product by a semigroup action. For simplicity, we consider an action of ${\mathbb{N}}$ by a local diffeomorphism $g:M\to M$. Following [@GU], we say that $g$ acts conformally if there is a function $c_g\in C^\infty(M,{\mathbb{R}}_{>0})$ such that $g^*g_M=c_gg_M$ where $g_M$ denotes the Riemannian metric on $M$. We say that $g$ lifts to the Clifford bundle $S\to M$ if there is a unitary Clifford linear morphism $u_g:g^*S\to S$. For simplicity, we assume $M$ to be connected and define $N:=\#g^{-1}(\{x\})$ for some $x\in M$. Since $g$ is a local diffeomorphism and $M$ is connected, $N$ is independent of the choice of $x$. If $g:M\to M$ is a surjective local diffeomorphism, acting conformally and lifting to $S$, we can define the isometry $$\label{vgdef} V_g:L^2(M,S)\to L^2(M,S), \quad V_g\xi:=c_g^{n/4}N^{-1/2} u_g(\xi\circ g).$$ The isometry $V_g$ satisfies the following for $a\in C(M)$: $$V_gaV_g^*=(a\circ g)V_gV_g^*,\quad\mbox{and}\quad V_g^*aV_g=\mathfrak{L}_g(a),$$ where $\mathfrak{L}_g(a)(x):=\sum_{g(y)=x}a(y)$. See more in [@DGMW Proposition 8.3]. Define $\mathcal{A}$ as the $*$-algebra generated by $C^\infty(M)$ and $V_g$. By [@DGMW Proposition 8.6], the $C^*$-closure of ${\mathcal{A}}$ is the image in a representation on $L^2(M,S)$ of the Cuntz-Pimsner algebra $O_{E_g}$ defined from $E_g:=C(M)V_g$ (see more in Example \[localhomeofirst\] (see page ) and Example \[localhomeoex\] (see page )). The space $E_g=C(M)V_g\subseteq \mathbb{B}(L^2(M,S))$ is considered as a bimodule over $C(M)$ and is an fgp bi-Hilbertian $C(M)$-bimodule because $E_g$ can be identified with $C(M)$ with the bimodule structure $(afb)(x)=a(x)f(x)b(g(x))$ for $a,b,f\in C(M)$, for details, see Example \[localhomeofirst\] below on page or [@DGMW Section 8]. Let $\psi:[0,\infty)\to (0,\infty)$ be a decreasing function and $g:M\to M$ a local diffeomorphism of a Riemannian manifold. - We say that $\psi$ and $g$ are compatible if there is a constant $C\geq 0$ such that for all $x\in M$ and $\xi\in T^*_xM$, the differential $Dg$ satisfies $$\left|\psi(|(Dg)^T_x\xi|)-\psi(|\xi|)\right|\leq C\psi(|\xi|)^2.$$ - We say that $g$ acts isometrically if $Dg$ acts isometrically on each fibre. The following results poses restrictions on a function $\psi$ compatible with a local diffeomorphism which acts non-isometrically. \[deathorgladiolis\] Let $\psi:[0,\infty)\to (0,\infty)$ be a decreasing function with $\lim_{t\to \infty}\psi(t)=0$ and $g:M\to M$ a compatible local diffeomorphism of a Riemannian manifold. If $\psi$ has regular variation, then either $g$ acts isometrically or $\psi$ has regular variation of index $0$. Assume that $g$ acts non-isometrically, we shall prove that in this case $\psi$ has regular variation of index $0$. Since $\psi$ is decreasing, it suffices to show that $\lim_{t\to \infty} \frac{\psi(r t)}{\psi(t)}=1$ for some $r\neq 1$ by [@GU Proposition 2.15]. If $g$ acts non-isometrically, there is a point $x\in M$ and a unit vector $\xi_0\in T^*_xM$ such that $|(Dg)_x^T\xi_0|_{g(x)}\neq 1$. Set $$r:=|(Dg)_x^T\xi_0|_{g(x)}\neq 1.$$ Since $\psi$ is compatible with $g$, there is a constant $C\geq 0$ such that $$\left|\psi(|(Dg)^T_x\xi|)-\psi(|\xi|)\right|\leq C\psi(|\xi|)^2$$ and by setting $\xi=t\xi_0$, we arrive at the inequality $$\left|\psi(rt)-\psi(t)\right|\leq C\psi(t)^2.$$ After dividing by $\psi(t)$, taking the limit $t\to \infty$ and using that $\lim_{t\to \infty}\psi(t)=0$ we arrive at the desired equality $\lim_{t\to \infty} \frac{\psi(r t)}{\psi(t)}=1$. A prototypical example of a function with regular variation of index $0$ is $\psi(t):=\log(1+t)$. This function is compatible with any conformal local diffeomorphism and has been considered in the context of constructing spectral triples in [@DGMW; @GM; @GM2; @GU; @MHaluk]. \[psidiracprop\] Let $M$ be an $n$-dimensional Riemannian closed manifold, $\slashed{D}$ a Dirac operator on $S\to M$, $g:M\to M$ a surjective local diffeomorphism acting conformally and lifting to $S$ and $\psi:[0,\infty)\to (0,\infty)$ a decreasing function compatible with $g$, having smooth regular variation and satisfying $\lim_{t\to \infty}\psi(t)=0$ and $\psi(t)^{-1}=O(t^{1/n})$ as $t\to \infty$. Let ${\mathcal{A}}$ denote the $*$-algebra generated by $C^\infty(M)$ and the isometry $V_g$ from Equation (see page ). Then $({\mathcal{A}},L^2(M,S), \slashed{D}_\psi)$ is a unital $\psi$-summable spectral triple. This result follows in the same manner as in [@DGMW Section 8] and [@GU Theorem 10.6]. Clearly, if $\psi(t)^{-1}=O(\log(t))$ as $t\to \infty$, then $({\mathcal{A}},L^2(M,S), \slashed{D}_\psi)$ is $\mathrm{Li}_1$-summable. The reader should note that the $K$-homology class $[({\mathcal{A}},L^2(M,S), \slashed{D}_\psi)]\in K^*(O_{E_g})$, where $O_{E_g}$ is the Cuntz-Pimsner algebra of the module $E_g=C(M)V_g$, discussed later. The class $[({\mathcal{A}},L^2(M,S), \slashed{D}_\psi)]$ restricts to $[(C^\infty(M),L^2(M,S), \slashed{D})]\in K^*(C(M))$. Thus the class $[(C^\infty(M),L^2(M,S), \slashed{D})]\in K^*(C(M))$ obstructs $[({\mathcal{A}},L^2(M,S), \slashed{D}_\psi)]\in K^*(O_{E_g})$ being a Kasparov product with the Cuntz-Pimsner boundary extension in $KK^1(O_{E_g},C(M))$ – we will return to study this boundary extension and its associated semifinite spectral triples below in Subsection \[cpalgexam\]. ### Graph $C^*$-algebras {#graphcstarsubsec} A class of examples carrying interesting noncommutative geometries with a well-studied set of KMS-states is that of graph $C^*$-algebras. With a finite directed graph $G=(V,E)$, with edge set $E$ and vertex set $V$, one associates a $C^*$-algebra $C^*(G)$ [@Raeburn]. For simplicity we suppose that we have no sources nor sinks. The $C^*$-algebra $C^*(G)$ is generated by partial isometries $(S_e)_{e\in E}$ and projections $(p_v)_{v\in V}$ satisfying the relations $$S^*_eS_e=p_{r(e)}, \quad \mbox{and}\quad p_v=\sum_{s(e)=v} S_eS_e^*,$$ where $r(e)$ denotes the range of the vertex $e$ and $s(e)$ its source. The $C^*$-algebra $C^*(G)$ can be described as a Cuntz-Pimsner algebra in several ways, a class of $C^*$-algebras carrying noncommutative geometries that we will study in the next subsection. In this subsection, we focus on a construction of noncommutative geometries along an orbit in the infinite path space of $G$ – a construction based in the model of $C^*(G)$ as a groupoid $C^*$-algebra. The associated noncommutative geometries come from [@GM]. As explained in [@GMR] the groupoid model can be seen as a Cuntz-Pimsner model of $C^*(G)$ using the one-sided infinite path space $$\Omega_G :=\{x=e_1e_2\cdots \in E^{\mathbb{N}}: \,s(e_j)=r(e_{j+1})\ \forall j\}.$$ The path space $\Omega_G$ is a compact Hausdorff space in the subspace topology $\Omega_G\subseteq E^{\mathbb{N}}$. It carries a shift mapping $\sigma_G:\Omega_G\to \Omega_G$, $\sigma_G(e_1e_2e_3\cdots):=e_2e_3\cdots$. The shift mapping is a surjective local homeomorphism. We can define an étale groupoid $\mathcal{G}_G$ over $\Omega_G$ by $$\mathcal{G}_G:=\big\{(x,n,y)\in \Omega_G\times {\mathbb{Z}}\times \Omega_G: \,\exists k\geq \max(0,-n) \ \mbox{such that}\ \sigma_G^{n+k}(x)=\sigma_G^k(y)\big\}.$$ The range mapping is defined by $r(x,n,y)=x$, the source mapping as $s(x,n,y):=y$ and the product by $$(x,n,y)(y,m,z):=(x,n+m,z).$$ The topology of $\mathcal{G}_G$ is uniquely determined by declaring the groupoid to be étale and the mappings $(x,n,y)\mapsto n$ and $$\kappa_G(x,n,y):=\min\{k\geq \max(0,-n): \sigma_G^{n+k}(x)=\sigma_G^k(y)\},$$ to be continuous. There is an isomorphism $\pi_G:C^*(G)\to C^*(\mathcal{G}_G)$ determined by defining $\pi_G(S_e)$ to be the characteristic function of the set $\{(x,1,\sigma_G(x)): x\in C_e\}$ where $C_e:=\{e_1e_2\cdots \in \Omega_G: e_1=e\}$. See [@deaacaneudoaod]. Define the function $\psi_0:\{(n,k)\in {\mathbb{Z}}\times {\mathbb{N}}: n+k\geq 0\}\to {\mathbb{Z}}$ by $$\psi_0(n,k):= \begin{cases} n, \; &k=0\\ -|n|-k,\; &k>0. \end{cases}$$ For a point $y\in \Omega_G$, we define the discrete set $$\mathcal{V}_y:=d^{-1}(\{y\})=\{(x,n): (x,n,y)\in \mathcal{G}_G\}.$$ The set $\mathcal{V}_y$ is the union of all forward orbits of all backward orbits of $y$ under $\sigma_G$ where we keep track of the lag $n$. Since $\mathcal{V}_y$ is a fibre of the domain mapping, point evaluation in $y$ induces a representation $\pi_y:C^*(G)\to \mathbb{B}(\ell^2(\mathcal{V}_y))$. If $G$ is primitive $C^*(G)$ is simple and $\pi_y$ is faithful. At the level of the generators, $$\pi_y(S_e)\delta_{(x,n)}= \begin{cases} \delta_{(ex,n+1)},\; &r(x)=s(e),\\ 0,\; &r(x)\neq s(e).\end{cases}$$ \[localizeckdirinpoint\] Define the operator ${\mathcal{D}}_y$ densely on $\ell^2(\mathcal{V}_y)$ as the self-adjoint operator with $${\mathcal{D}}_y\delta_{(x,n)}:=\psi_0(n,\kappa(x,n,y))\delta_{(x,n)}.$$ The triple $(C_c(\mathcal{G}_G,\ell^2(\mathcal{V}_y),{\mathcal{D}}_y)$ is an $\mathrm{Li}_1$-summable spectral triple. Moreover, under the isomorphism $K^1(C^*(G))\cong {\mathbb{Z}}^E/(1-A_{\rm edge}){\mathbb{Z}}^E$, of odd $K$-homology with the cokernel of the edge adjacency matrix, the class $[(C_c(\mathcal{G}_G,\ell^2(\mathcal{V}_y),{\mathcal{D}}_y)]$ is mapped to the element $\delta_e\mod (1-A_{\rm edge}){\mathbb{Z}}^E$ where $\delta_e\in {\mathbb{Z}}^E$ denotes the basis element corresponding to $e\in E$. It is proven in [@GM Theorem 5.2.3] that $(C_c(\mathcal{G}_G,\ell^2(\mathcal{V}_y),{\mathcal{D}}_y)$ is a spectral triple whose $K$-homology class corresponds to the element $\delta_e\mod (1-A_{\rm edge}){\mathbb{Z}}^E$ under $K^1(C^*(G))\cong {\mathbb{Z}}^E/(1-A_{\rm edge}){\mathbb{Z}}^E$. It remains to prove that $(C_c(\mathcal{G}_G,\ell^2(\mathcal{V}_y),{\mathcal{D}}_y)$ is $\mathrm{Li}_1$-summable. We compute that $$\operatorname{Tr}({\mathrm{e}}^{-t|{\mathcal{D}}_y|})=\sum_{(x,n)\in \mathcal{V}_y} {\mathrm{e}}^{-t(|n|+\kappa_G(x,n,y))} =\sum_{n\in {\mathbb{Z}}}\sum_{k=\max(0,-n)}^\infty \#\{(x,n): \kappa_G(x,n,y)=k\} {\mathrm{e}}^{-t(|n|+k)}.$$ However, if $(x,n)$ is such that $\kappa_G(x,n,y)=k$ the path $x$ is determined by $y$ except for its first $n+k$ steps so $\#\{(x,n): \kappa_G(x,n,y)=k\}\leq |E|^{n+k}\leq {\mathrm{e}}^{\log(|E|)(|n|+k)}$. We conclude that $$\operatorname{Tr}({\mathrm{e}}^{-t|{\mathcal{D}}_y|})<\infty \quad\mbox{if}\quad t>\log(|E|). \qedhere$$ We return to this example below in Example \[diracgraphhea\] (see page ) and Subsection \[diracgraphkms\] (see page ). ### Cuntz-Pimsner algebras {#cpalgexam} In this subsection we consider the construction of semi-finite spectral triples on Cuntz-Pimsner algebras – a broad class of examples which include both Cuntz-Krieger algebras and crossed products by ${\mathbb{Z}}$. Quite general techniques for constructing spectral triples for these algebras were developed in a series of papers (in rough chronological order) [@GM; @RRS; @GMR; @GM2; @RRSmore]. We consider the set up of [@RRS] and [@GMR], which provide a means of lifting data from the (unital) coefficient algebra of a bi-Hilbertian bimodule to its Cuntz-Pimsner algebra. We start with a unital, separable $C^*$-algebra $A$, and a finitely generated projective (fgp) bi-Hilbertian bimodule $E$ over $A$, i.e. a module fulfilling the conditions of the following definition. \[cond:one\] An fgp bi-Hilbertian bimodule $E$ over $A$ is an $A$-bimodule equipped with the following structures: - $E$ has both left and right $A$-valued inner products which induce equivalent norms on $E$. - The left and right actions are both injective and adjointable. - $E$ is finitely generated and projective as both a left and right module. To separate the left and the right structures, we write $E_A$ when we want to emphasize the right module structure and $(\cdot|\cdot)_A$ for the right inner product. Similarly, ${}_AE$ denotes the left module defined from $E$ and ${}_A(\cdot|\cdot)$ the left inner product. The algebraic Fock space ${\mathcal{F}_E}^{\rm alg}$ is the algebraic direct sum of the $A$-modules $E^{\otimes_A k}$. The Fock space ${\mathcal{F}_E}$ is defined as the right $A$-Hilbert $C^*$-module completion of ${\mathcal{F}_E}^{\rm alg}$. The Cuntz-Toeplitz algebra $\mathcal{T}_E\subseteq \operatorname{End}^*_A({\mathcal{F}_E})$ is the $C^*$-algebra generated by the creation operators $T_\mu \xi:=\mu\otimes \xi$ for $\mu\in {\mathcal{F}_E}^{\rm alg}$. The Cuntz-Pimsner algebra $O_E$ is defined from the short exact sequence $$0\to \mathbb{K}_A({\mathcal{F}_E})\to \mathcal{T}_E\to O_E\to 0.$$ We call this short exact sequence the defining extension of $O_E$. A set $(e_j)_{j=1}^N\subset E$ of vectors is a frame for the right module $E_A$ if for all $e\in E$ we have $e=\sum_je_j(e_j|e)_A$, and similarly for a left frame $(f_k)_{k=1}^N$. Since $E$ is finitely generated and projective, there exists left and right frames and we can for simplicity assume that they have the same cardinality. For $e$ and $f$ in the right Hilbert module $E_A$, we denote the associated rank-one operator by $\Theta_{e,f}:=e( f|\cdot)$. Then the frame condition can be expressed as $$\sum_{j=1}^N\Theta_{e_j,e_j}={\rm Id}_E$$ and similarly for $f_\sigma$. The frame $(e_j)_{j=1}^N$ induces a frame for $E_A^{{\otimes}k}$, namely $(e_\rho)_{|\rho|=k}$ where $\rho$ is a multi-index and $e_\rho=e_{\rho_1}{\otimes}\cdots{\otimes}e_{\rho_k}$. We define the right Watatani index of $E^{\otimes k}$ as the element of $A$ given by $$\label{dfn:Watatani} {\mathrm{e}}^{\beta_k}=\sum_{|\rho|=k}{}_A(e_\rho|e_\rho) =\sum_{|\rho'|=k-1}{}_A(e_{\rho'}{\mathrm{e}}^\beta|e_{\rho'}).$$ The right Watatani index is positive, central and since the left action is injective, also invertible. Therefore $\beta_k$ is a well defined self-adjoint central element in $A$. The key assumptions we make concern the asymptotic behaviour of the right Watatani indices. In [@RRS Section 3.2] we define an $A$-bilinear functional $\Phi_\infty:{\mathcal{O}}_E\to A$. This functional gives us an $A$-valued inner product on ${\mathcal{O}}_E$. The construction of $\Phi_\infty$ begins by defining $$\label{Phi_k} \Phi_k:\operatorname{End}_A^*(E^{{\otimes}k})\to A, \qquad \Phi_k(T)=\sum_{|\rho|=k}{}_A(Te_\rho|e_\rho).$$ Here we use the notation $\operatorname{End}_A^*(E^{{\otimes}k})$ for the $C^*$-algebra of $A$-linear adjointable operators on $E^{{\otimes}k}$. It follows from [@KajPinWat Lemma 2.16] that $\Phi_k$ does not depend on the choice of frame. We note that $\mathrm{e}^{\beta_k}=\Phi_k({\rm Id}_{E^{{\otimes}k}})$. Since $\Phi_k$ is independent of the choice of frame, so is $\mathrm{e}^{\beta_k}$. We extend the functional $\Phi_k$ to a mapping $\operatorname{End}^*_A({\mathcal{F}_E})\to A$ by compressing along the orthogonally complemented submodule $E^{{\otimes}k}\subseteq {\mathcal{F}_E}$. To obtain a good “limiting functional" $\Phi_\infty(T):=\lim_{k\to\infty}\Phi_k(T)\mathrm{e}^{-\beta_k}$ on the Cuntz-Toeplitz algebra, we impose the following condition on the Watatani indices. \[ass:one\] Let $E$ be an fgp bi-Hilbertian bimodule over the unital $C^*$-algebra $A$. We say that $E$ is W-regular if for every $k\in {\mathbb{N}}$ and $\nu\in E^{\otimes k}$ there exists a $\tilde{\nu}\in E^{\otimes k}$ satisfying $$\Vert \mathrm{e}^{-\beta_n}\nu \mathrm{e}^{\beta_{n-k}}-\tilde\nu\Vert_{E^{\otimes k}} \to 0 \quad\mbox{as $n\to \infty$}.$$ In [@RRS] the reader can find several examples of Cuntz-Pimsner algebras for which a stronger version of W-regularity as defined in Definition \[ass:one\] holds. There are no known examples of modules that are not W-regular. When $E$ is W-regular, [@RRS Proposition 3.5] guarantees that $\lim_{k\to\infty}\Phi_k(T)\mathrm{e}^{-\beta_k}$ is well defined for $T$ from the $*$-algebra generated by the set of creation operators $\{T_\nu: \;\nu\in {\mathcal{F}_E^{\textnormal{alg}}}\}$ and is continuous in the $C^*$-norm. In Section \[sub:no-left\] we shall see that there are ways around W-regularity, and even the existence of an $A$-valued left inner product, when constructing semifinite spectral triples giving rise to KMS-states. We thus obtain a unital positive $A$-bilinear functional $\Phi_\infty: {\mathcal{T}}_E\to A$. The functional $\Phi_\infty$ annihilates the compact endomorphisms, and descends to a well-defined functional on the Cuntz-Pimsner algebra ${\mathcal{O}}_E$. By an abuse of notation, we also denote this functional by $\Phi_\infty:{\mathcal{O}}_E\to A$. Since $\Phi_k$ and $\mathrm{e}^{\beta_k}$ do not depend on the choice of frame, neither does $\Phi_\infty$. We define the inner product $$(S_1|S_2)_A:=\Phi_\infty(S_1^*S_2),\qquad S_1,\,S_2\in {\mathcal{O}}_E.$$ When computing these inner products, the following fact is useful. \[computeinnerprod\] Let $E$ be a W-regular fgp bi-Hilbertian bimodule. For homogeneous elements $\mu,\,\nu\in {\mathcal{F}_E^{\textnormal{alg}}}$ we have $$\label{eq:mu-nu-beta}\Phi_\infty(S_\mu S_\nu^*) = \lim_{k\to\infty}{}_A(\mu|\mathrm{e}^{-\beta_k}\nu \mathrm{e}^{\beta_{k-|\nu|}})={}_A(\mu|\tilde{\nu}).$$ In particular, if $S\in {\mathcal{O}}_{E}$ is homogeneous of degree $n\neq 0$, then $\Phi_{\infty}(S)=0$. Completing ${\mathcal{O}}_E$ modulo the vectors of zero length (with respect to $\Phi_\infty$) yields a right $A$-Hilbert $C^*$-module that we denote by ${\Xi_A}$. The module ${\Xi_A}$ carries a left action of ${\mathcal{O}}_E$ given by extending the multiplication action of ${\mathcal{O}}_E$ on itself. By considering the linear span of the image of the generators $S_\nu$, $\nu\in {\mathcal{F}_E^{\textnormal{alg}}}$, inside the module ${\Xi_A}$, we obtain an isometrically embedded and complemented copy of the Fock space. We let $Q$ be the projection on this copy of the Fock space. Let $E$ be a W-regular fgp bi-Hilbertian bimodule over a unital $C^*$-algebra $A$. The tuple $({\mathcal{O}}_E,{\Xi_A},2Q-1)$ is an odd Kasparov module representing the class of the defining extension $$\begin{aligned} \label{defext} 0\to {\mathbb{K}}_A(F_E)\to \mathcal{T}_E\to {\mathcal{O}}_E\to 0.\end{aligned}$$ To construct an unbounded representative of $({\mathcal{O}}_E,{\Xi_A},2Q-1)$, we will add an additional assumption regarding the fine structure of the operation $\nu\mapsto \tilde{\nu}$ in the definition of W-regularity (see Definition \[ass:one\]). Assuming W-regularity, we can define the operator ${\mathfrak{q}}_k:E^{\otimes k}\to E^{\otimes k}$ by $${\mathfrak{q}}_k\nu:=\tilde{\nu}=\lim_{n\to\infty}\mathrm{e}^{-\beta_n}\nu \mathrm{e}^{\beta_{n-k}}.$$ \[ass:two\] Let $E$ be an fpg bi-Hilbertian bimodule over the unital $C^*$-algebra $A$. We say that $E$ is strictly W-regular if it is W-regular and for any $k$, we can write ${\mathfrak{q}}_k=c_kP_k=P_{k}c_{k}$ where $P_{k}\in \operatorname{End}^{*}_{A}(E^{\otimes k})$ is a (necessarily $A$-bilinear) projection and $c_k$ is given by left-multiplication by an element in $A$. \[justforpage\] As with W-regularity, the reader can in [@RRS] find several examples of Cuntz-Pimsner algebras for which strict W-regularity holds. There are no known examples of modules that are not strictly W-regular. \[centralityandasstwo\] If there is a decomposition ${\mathfrak{q}}_{k}=c_{k}P_{k}$ as in the definition of strict W-regularity, [@GMR Lemma 3.8] shows that it is unique and of a very specific form. Indeed, each $c_k$ is central, invertible and $c_k=\Phi_k(P_k)^{-1}$. Strict W-regularity is readily verified in practice using [@GMR Lemma 3.8]. For instance, if $\beta_1$ is central for the module action on $E$, $c_k=\mathrm{e}^{-\beta_k}=\mathrm{e}^{-k\beta_1}$ is central for the module action on $E$ and $P_k=1_{E^{{\otimes}k}}$. When $E$ is strictly W-regular, an unbounded self-adjoint regular operator ${\mathcal{D}}_\psi$ on ${\Xi_A}$ is constructed in [@GMR] making $({\mathcal{O}}_E,{\Xi_A},{\mathcal{D}}_\psi)$ into an unbounded Kasparov module representing the $KK$-class of the defining extension . The operator ${\mathcal{D}}_\psi$ is of the form $${\mathcal{D}}_\psi=\sum_{n\in{\mathbb{Z}}}\sum_{r\geq\max\{0,n\}}\psi(n,r)P_{n,r}$$ where $\psi:\,{\mathbb{Z}}\times{\mathbb{N}}\to[0,\infty)$ is a function with certain Lipschitz properties (see [@GMR Remark 3.20]), and the $P_{n,r}$ are projections on finitely generated projective subspaces, [@GMR]. While the particular choice of function $\psi$ does not matter much, we will take the function $$\psi(n,r)=\left\{\begin{array}{ll} n & n=r\\ -(2r-n) & \mbox{otherwise}\end{array}\right..$$ The projections $P_{n,r}$ form a sequence of mutually orthogonal projections satisfying that the direct sum $\oplus_{r=\max(0,n)}^{r_0} P_{n,r}$ is the projection onto the $A$-linear span of $\{S_\mu S_\nu^*: |\mu|-|\nu|=n, \max(0,n)\leq r\leq r_0\}$. In particular, $P_{n,n}$ is the projection onto $E^{\otimes n}$ for $n\in {\mathbb{N}}$. More precisely, the projections are defined by $$\label{onrqnr} P_{n,r}:= \begin{cases} Q_{n,r}-Q_{n,r-1},\; & r>\max\{0,n\}\\ Q_{n,r},\; & r=\max\{0,n\} \end{cases}$$ where the projections $Q_{n,r}$ are defined in terms of the right frame $(e_j)_{j=1}^N$ and the left frame $(f_j)_{j=1}^N$ as $$\label{qnrqnr} Q_{n,r}:=\sum_{|\rho|-|\sigma|=n,\,|\rho|=r} \Theta_{W_{e_\rho,c^{-1/2}_{|\sigma|}P_Ff_\sigma},W_{e_\rho,c^{-1/2}_{|\sigma|}P_Ff_\sigma}}$$ where $P_F=\oplus P_k$ is the projection on the Fock module coming from Definition \[ass:two\]. Here we have written $W_{\xi,\eta}\in {\Xi_A}$ for the element defined from $S_\xi S_\eta^*\in O_E$ where $\xi\in E^{\otimes r}$ and $\eta\in E^{\otimes k}$. For details, see [@GMR Lemma 3.10 and Proposition 3.11]. To obtain a semifinite spectral triple, we localize $({\mathcal{O}}_E,{\Xi_A},{\mathcal{D}}_\psi)$ in a positive trace on $A$. Following Proposition \[localintra\], we consider the semifinite spectral triple $$({\mathcal{O}}_E,L^2({\Xi_A},\tau),{\mathcal{D}}_\psi,{\mathcal{N}}_\tau({\Xi_A}),\operatorname{Tr}_\tau).$$ Here $L^2({\Xi_A},\tau):=\Xi_A\otimes_A L^2(A,\tau)$ and $\operatorname{Tr}_\tau$ is the dual trace on ${\mathcal{N}}_\tau({\Xi_A}):=(\operatorname{End}^*_A({\Xi_A})\otimes 1)''$ which satisfies $\operatorname{Tr}_\tau(\Theta_{e,f})=\tau((f|e)_A)$, [@LN; @Tak]. \[ximodsemi\] Assume that the fgp bi-Hilbertian bimodule $E$ is strictly W-regular and that $\tau$ is a positive trace on $A$. Then the semifinite spectral triple $$({\mathcal{O}}_E,L^2({\Xi_A},\tau),{\mathcal{D}}_\psi,{\mathcal{N}}_\tau({\Xi_A}),\operatorname{Tr}_\tau)$$ is $\mathrm{Li}_1$-summable. The assumptions on the existence of limiting behaviour for the Watatani indices are really just for convenience here. These assumptions relate to existence and behaviour of norm limits, but we have passed to the ‘measurable setting’ and so really only need weak limits. We will explore this point of view in Subsection \[sub:no-left\]. We need to prove that the following expression is finite for $t$ large enough: $$\operatorname{Tr}_\tau({\mathrm{e}}^{-t|{\mathcal{D}}_\psi|}) =\sum_{n\in{\mathbb{Z}}}\sum_{r\geq\max\{0,n\}}{\mathrm{e}}^{-t|\psi(n,r)|}\operatorname{Tr}_\tau(P_{n,r}).$$ By definition (see ), $P_{n,r}=Q_{n,r}-Q_{n,r-1}$ when $r>\max(0,n)$ and $P_{n,r}=Q_{n,r}$ when $r=\max(0,n)$ and $Q_{n,r}$ is defined as in . Using the computations of [@GMR Lemma 2.8], we see that $$\begin{aligned} \label{tracompqnr} \operatorname{Tr}_\tau(Q_{n,r})&=\sum_{|\rho|-|\sigma|=n,\,|\rho|=r}\tau\left(( W_{e_\rho,c^{-1/2}_{|\sigma|}P_Ff_\sigma}|W_{e_\rho,c^{-1/2}_{|\sigma|}P_Ff_\sigma})_{A}\right)\\ \nonumber &=\sum_{|\rho|-|\sigma|=n,\,|\rho|=r}\tau\circ \Phi_\infty\left(S_{c^{-1/2}_{|\sigma|}P_Ff_\sigma} S_{e_\rho}^*S_{e_\rho}S_{c^{-1/2}_{|\sigma|}P_Ff_\sigma} ^*\right)\\ \nonumber &=\sum_{|\rho|-|\sigma|=n,\,|\rho|=r}\tau\left({}_A(P_Ff_\sigma|P_Ff_\sigma (e_\rho|e_\rho)_A)\right).\end{aligned}$$ Using the fact that the elements of the frame have norm bounded by $1$, we see that $$\begin{aligned} |\operatorname{Tr}_\tau(Q_{n,r})|&\leq \sum_{|\rho|-|\sigma|=n,\,|\rho|=r}|\tau\left({}_A(P_Ff_\sigma|P_Ff_\sigma (e_\rho|e_\rho)_A)\right)|\\ &\leq \sum_{|\rho|-|\sigma|=n,\,|\rho|=r}\|({}_A(P_Ff_\sigma|P_Ff_\sigma (e_\rho|e_\rho)_A)\|_A\leq N^{2r-n}\leq {\mathrm{e}}^{\log(N)|\psi(n,r)|},\end{aligned}$$ where $N$ is the number of elements in the left frame and the right frame. We can now estimate $$\begin{aligned} \left|\operatorname{Tr}_\tau({\mathrm{e}}^{-t|{\mathcal{D}}_\psi|})\right| =&\left|\sum_{n\in{\mathbb{Z}}}\sum_{r>\max\{0,n\}}{\mathrm{e}}^{-t|\psi(n,r)|}\operatorname{Tr}_\tau(Q_{n,r}-Q_{n,r-1})\right|\\ &+\left|\sum_{n\in{\mathbb{Z}}}\sum_{r=\max\{0,n\}}{\mathrm{e}}^{-t|\psi(n,r)|}\operatorname{Tr}_\tau(Q_{n,r})\right| \\ \leq& 2\sum_{n\in{\mathbb{Z}}}\sum_{r\geq\max\{0,n\}}{\mathrm{e}}^{-t|\psi(n,r)|}|\operatorname{Tr}_\tau(Q_{n,r})| \\ \leq& 2\sum_{n\in{\mathbb{Z}}}\sum_{r\geq\max\{0,n\}}{\mathrm{e}}^{-(t-\log(N))|\psi(n,r)|}<\infty,\end{aligned}$$ if $t>\log(N)$. We return to Cuntz-Pimsner algebras below in Example \[diraccphea\] (see page ) and Section \[diraccpkms\] (see page ). Let us discuss a special case of Cuntz-Pimsner algebras arising on a commutative coefficient algebra. \[localhomeofirst\] Let $Y$ be a compact Hausdorff space and $g:Y\to Y$ a surjective local homeomorphism. We consider the module $E_g=C(Y)$ with the bimodule action $$(afb)(x)=a(x)f(x)b(g(x)), \quad a,b\in C(Y),\ f\in E_g.$$ This is an fgp bi-Hilbertian bimodule in the inner products $${}_{C(Y)}(f_1,f_2)=f_1\overline{f_2}\quad\mbox{and}\quad (f_1|f_2)_{C(Y)}:=\mathfrak{L}_g(\overline{f_1}f_2),$$ where $\mathfrak{L}_g:C(Y)\to C(Y)$ is the transfer operator $$\label{transferoperator} \mathfrak{L}_g(f)(x):=\sum_{y\in g^{-1}(x)}f(y).$$ The Cuntz-Pimsner algebra $O_{E_g}$ can be realized as a groupoid $C^*$-algebra as in [@deaacaneudoaod] (see also [@DGMW Theorem 3.2]) over the solenoid $$X=\{x=y_1y_2\cdots \in Y^{\mathbb{N}}: \,g(y_{k+1})=y_k\ \forall k\}.$$ The case that $X$ equipped with the shift mapping is a Smale space was studied in [@DGMW]. The module $E_g$ has a right frame $(e_j)_{j=1}^N$ where $e_j=\sqrt{\chi_j}$ for a partition of unity $(\chi_j)_{j=1}^N$ subordinate to an open covering $(U_j)_{j=1}^N$ of $Y$ such that $g|_{U_j}$ is injective for all $j$. Using this partition of unity, one sees that $\beta_k=0$ for all $k$. It follows that $E_g$ is a strictly W-regular module. The case that $g:M\to M$ was a surjective local diffeomorphism acting conformally was considered in Subsection \[diracmfdfirst\] (see page ). However, the spectral triple considered Proposition \[psidiracprop\] on $O_{E_g}$ differs greatly from the semifinite spectral triples considered in Lemma \[ximodsemi\] – the latter are in the image of the boundary mapping in $KK_1(O_{E_g},C(M))$ defined from Equation while the former is not if $[\slashed{D}]\neq 0\in K^*(C(M))$. Another class of examples already considered arises from a finite graph $G$ as in Subsection \[graphcstarsubsec\] (see page ) where the shift mapping $\sigma_G:\Omega_G\to \Omega_G$ is a surjective local homeomorphism and $C^*(G)\cong O_{E_{\sigma_G}}$. The spectral triples in Proposition \[localizeckdirinpoint\] (see page ) arises from the construction of Lemma \[ximodsemi\] by taking the trace $\tau:C(\Omega_G)\to {\mathbb{C}}$ to be defined from point evaluation in $y$. ### Group $C^*$-algebras {#groupcstarexam} We now turn our attention to examples coming from the reduced group $C^*$-algebra of a discrete group. A well known construction associates a spectral triple with a length function on the group, we consider this example and a semifinite modification thereof which is possible for a-T-menable groups, i.e. groups with the Haagerup property. The methods extend to a-TT-menable groups, a class of groups containing all hyperbolic groups, see more in [@mimurathesis Chapter 7.2]. Let $\Gamma$ denote a countable discrete group. Recall that a length function $\ell:\Gamma\to {\mathbb{R}}_{\geq 0}$ is a function satisfying $\ell(e)=0$ for $e\in \Gamma$ the identity, and $\ell(\gamma \gamma')\leq \ell(\gamma)+\ell(\gamma')$ for all group elements $\gamma,\gamma'\in\Gamma$. We say that $\ell$ is a proper length function if $\ell$ is a proper function, i.e. the set $\{\gamma\in \Gamma: \ell(\gamma)\leq R\}$ is finite for any $R\geq 0$. If there exists a constant $\beta\geq 0$ such that $\#\{\gamma\in \Gamma: \ell(\gamma)\leq R\}=O({\mathrm{e}}^{\beta R})$ as $R\to \infty$ we say that $(\Gamma,\ell)$ has at most exponential growth. Define the operator ${\mathcal{D}}_\ell$ densely on $\ell^2(\Gamma)$ as the self-adjoint operator with $${\mathcal{D}}_\ell\delta_{\gamma}:=\ell(\gamma)\delta_{\gamma}.$$ The space of compactly supported functions $c_c(\Gamma)$ is a core for ${\mathcal{D}}_\ell$. We define the $*$-algebra $c_b(\Gamma)\rtimes ^{\rm alg}\Gamma$ as the $*$-algebra generated by multiplication operators (by bounded functions on $\Gamma$) $c_b(\Gamma)\subseteq \mathbb{B}(\ell^2(\Gamma))$ and all left translation operators. \[spectripfromlength\] Let $\ell$ be a proper length function on $\Gamma$. The triple $(c_b(\Gamma)\rtimes ^{\rm alg}\Gamma,\ell^2(\Gamma),{\mathcal{D}}_\ell)$ is a spectral triple defining the trivial class in the $K$-homology of the $C^*$-algebra $c_b(\Gamma)\rtimes_r \Gamma$. Moreover, if $(\Gamma,\ell)$ has at most exponential growth the spectral triple $(c_b(\Gamma)\rtimes ^{\rm alg}\Gamma,\ell^2(\Gamma),{\mathcal{D}}_\ell)$ is $\mathrm{Li}_1$-summable. Since $\ell$ is proper, it is clear that ${\mathcal{D}}_\ell$ has compact resolvent and if $(\Gamma,\ell)$ has at most exponential growth, then there is $C>0$ such that $$\operatorname{Tr}({\mathrm{e}}^{-t|{\mathcal{D}}_\ell|})=\sum_{\gamma\in \Gamma}{\mathrm{e}}^{-t\ell(\gamma)}=\sum_{n=0}^\infty \#\{\gamma\in \Gamma: \ell(\gamma)=n\}{\mathrm{e}}^{-tn}\leq C\sum_{n=0}^\infty {\mathrm{e}}^{-(t-\beta)n}=\frac{C}{1-{\mathrm{e}}^{\beta-t}}<\infty.$$ To show that $(c_b(\Gamma)\rtimes ^{\rm alg}\Gamma,\ell^2(\Gamma),{\mathcal{D}}_\ell)$ is a spectral triple, it remains to show that $c_b(\Gamma)\rtimes ^{\rm alg}\Gamma$ preserves the domain of ${\mathcal{D}}_\ell$ and has bounded commutators with ${\mathcal{D}}_\ell$. Domain preservation is clear. For an element $a\lambda_\gamma\in c_b(\Gamma)\rtimes ^{\rm alg}\Gamma$ and a function $f\in c_c(\Gamma)$ we compute that $$[{\mathcal{D}}_\ell,a\lambda_\gamma]f(g)=a(g)(\ell(g)-\ell(\gamma^{-1}g))f(\gamma^{-1}g).$$ It follows that $$\|[{\mathcal{D}}_\ell,a\lambda_\gamma]\|_{\mathbb{B}(\ell^2(\Gamma)}\leq \|a\|_{c_b(\Gamma)}\sup_{g\in \Gamma}|\ell(g)-\ell(\gamma^{-1}g)|\leq \|a\|_{c_b(\Gamma)}\ell(\gamma).\qedhere$$ The $K$-homology class of $(c_b(\Gamma)\rtimes ^{\rm alg}\Gamma,\ell^2(\Gamma),{\mathcal{D}}_\ell)$ is trivial. We shall now consider a topologically more interesting semifinite spectral triple that can be constructed on groups with the Haagerup property. We are grateful to Branimir Ćaćić for sharing this construction with us. Similar ideas appeared in [@juuunge Appendix B]. Let $\Gamma$ be a discrete group with the Haagerup property. Then there is a proper isometric action of $\Gamma$ on a real Hilbert space ${\mathcal{H}}_\Gamma$. By the Mazur-Ulam theorem there exists an orthogonal representation $$\pi_\Gamma:\,\Gamma\to O({\mathcal{H}}_\Gamma)$$ on the Hilbert space ${\mathcal{H}}_\Gamma$ and a proper cocycle $c_\Gamma$ for $\pi_\Gamma$, meaning that $c_\Gamma:\Gamma\to {\mathcal{H}}$ is a proper function satisfying the cocycle identity $$c_\Gamma(\gamma_1\gamma_2)=c_\Gamma(\gamma_1)-\pi_\Gamma(\gamma_1)c_\Gamma(\gamma_2). \label{eq:cocycle}$$ The cocycle identity allows us to define a length function on $\Gamma$ by $$\ell(\gamma):=\Vert c_\Gamma(\gamma)\Vert_{{\mathcal{H}}_\Gamma}.$$ Since $c_\Gamma$ is proper, so is $\ell$. The existence of a proper isometric action of a group $\Gamma$ on a Hilbert space is equivalent to $\Gamma$ having the Haagerup property, also known as a-T-menability. Our construction extends to the case when there exists an orthogonal representation $\pi_\Gamma:\,\Gamma\to O({\mathcal{H}}_\Gamma)$ and a proper quasi-cocycle $c_\Gamma:\Gamma\to {\mathcal{H}}_\Gamma$. That is, when $$Q(c_\Gamma):=\sup_{\gamma_1,\gamma_2}\|c_\Gamma(\gamma_1\gamma_2)-c_\Gamma(\gamma_1)+\pi_\Gamma(\gamma_1)c_\Gamma(\gamma_2)\|_{{\mathcal{H}}_\Gamma}<\infty.$$ In this case, $\ell(\gamma):=\Vert c_\Gamma(\gamma)\Vert_{{\mathcal{H}}_\Gamma}$ could fail to be a length function but still satisfies $\ell(\gamma \gamma')\leq \ell(\gamma)+\ell(\gamma')+Q(c_\Gamma)$ which suffices for our purposes. The existence of a proper quasi-cocycle on a Hilbert space is equivalent to $\Gamma$ being a-TT-menable. Hyperbolic groups are a-TT-menable. For notational simplicity, we restrict our attention to cocycles. Let $\operatorname{{{\mathbb{C}}\ell}}({\mathcal{H}}_\Gamma)$ denote the the complex Clifford algebra of ${\mathcal{H}}_\Gamma$ and assume that $\mathfrak{c}_S:\operatorname{{{\mathbb{C}}\ell}}({\mathcal{H}})\to \mathbb{B}(S_{\mathcal{H}})$ is a representation of $\operatorname{{{\mathbb{C}}\ell}}({\mathcal{H}})$. We say that a unitary representation $\pi_S:\,\Gamma\to U(S_{\mathcal{H}})$ is a lift of $\pi_\Gamma$ to $S_{\mathcal{H}}$ if for all $v\in{\mathcal{H}}$ and $g\in\Gamma$ we have $$\pi_S(g)\mathfrak{c}_S(v)\pi_S(g^{-1})=\mathfrak{c}_S(\pi_\Gamma(g)v). \label{eq:S-cl}$$ When ${\mathcal{H}}$ is finite dimensional this is just the well-known Clifford algebra, but when ${\mathcal{H}}$ is infinite dimensional we refer to [@CO; @W] for a description of this algebra. For a representation $S_{\mathcal{H}}$ of $\operatorname{{{\mathbb{C}}\ell}}({\mathcal{H}})$ we consider the new Hilbert space $\ell^2(\Gamma,S_{\mathcal{H}})$. Assuming that $\pi_S$ lifts $\pi_\Gamma$ to $S_{\mathcal{H}}$ the Hilbert space $\ell^2(\Gamma,S_{\mathcal{H}})$ carries a representation of $\Gamma$ defined by $$\label{tildepidef} \tilde{\pi}:\,\Gamma\to U(\ell^2(\Gamma,S_{\mathcal{H}})),\qquad \big(\tilde{\pi}(g)f\big)(\gamma)=\pi_S(g)f(g^{-1}\gamma).$$ On the Hilbert space $\ell^2(\Gamma, S_{\mathcal{H}})$ define a self-adjoint operator ${\mathcal{D}}_c$ by declaring $$({\mathcal{D}}_cf)(\gamma):=\mathfrak{c}_S(c_\Gamma\gamma))f(\gamma),\quad f\in c_c(\Gamma, S_{\mathcal{H}}). \label{eq:cliff}$$ Since $\mathfrak{c}_S(v)^2=\|v\|_{\mathcal{H}}^2$ for all $v\in {\mathcal{H}}$, the domain for ${\mathcal{D}}_c$ can be deduced from $$({\mathcal{D}}_c^2f)(\gamma)=\ell(\gamma)^2f(\gamma).$$ The compatibility requirement Equation and cocycle property Equation imply that for $g,\gamma\in\Gamma$ we have $$\pi_S(g)\mathfrak{c}_S(c_\Gamma(\gamma))=\mathfrak{c}_S(\pi(g)c_\Gamma(\gamma))\pi_S(g)=\mathfrak{c}_S(c_\Gamma(g))\pi_S(g)+\mathfrak{c}_S(c_\Gamma(g\gamma))\pi_S(g).$$ Then the commutator of ${\mathcal{D}}_c$ and a group element is $$\begin{aligned} ([{\mathcal{D}}_c,\tilde{\pi}(g)]f)(\gamma)&= \mathfrak{c}_S(c_\Gamma(\gamma))\pi_S(g)f(g^{-1}\gamma)-\pi_S(g)\mathfrak{c}_S(c_\Gamma(g^{-1}\gamma))f(g^{-1}\gamma)\\ &=\mathfrak{c}_S(c_\Gamma(g))\pi_S(g)f(g^{-1}\gamma)=(\mathfrak{c}_S(c_\Gamma(g))\tilde{\pi}(g))(f)(\gamma).\end{aligned}$$ Hence the commutators between ${\mathcal{D}}_c$ and group elements are bounded. It is moreover clear that these commutators lie in ${\mathcal{N}}_0\rtimes\Gamma$ where ${\mathcal{N}}_0=\mathfrak{c}_S(\operatorname{{{\mathbb{C}}\ell}}({\mathcal{H}}))''$. We define ${\mathcal{N}}$ as the von Neumann algebraic tensor product $\mathbb{B}(\ell^2(\Gamma))\bar{\otimes}{\mathcal{N}}_0$. Finally, $$(1+{\mathcal{D}}_c^2)^{-1}\in \mathbb{K}(\ell^2(\Gamma)){\otimes}1\subset \mathbb{K}(\ell^2(\Gamma)){\otimes}\mathbb{K}_\tau$$ where $\mathbb{K}_\tau$ is the compacts in ${\mathcal{N}}_0$ for a choice of normalized positive trace $\tau$. Let $\operatorname{Tr}_\tau$ be the trace on ${\mathcal{N}}$ defined from the trace $\tau$ on ${\mathcal{N}}_0$. We conclude that $(1+{\mathcal{D}}_c^2)^{-1}\in \mathbb{K}_{\operatorname{Tr}_\tau}$. Finally, if $\ell$ has at most exponential growth then $\operatorname{Tr}_\tau({\mathrm{e}}^{-t|{\mathcal{D}}_c|})=\sum_{\gamma\in \Gamma}{\mathrm{e}}^{-t\ell(\gamma)}<\infty$ for $t$ large enough. As such, $(i\pm {\mathcal{D}}_c)^{-1}\in \mathrm{Li}_1$ if $\ell$ has at most exponential growth. We conclude the following result. \[cssfst\] Assume that $\mathfrak{c}_S:\operatorname{{{\mathbb{C}}\ell}}({\mathcal{H}})\to \mathbb{B}(S_{\mathcal{H}})$ is a representation of $\operatorname{{{\mathbb{C}}\ell}}({\mathcal{H}})$ and that the unitary representation $\pi_S:\,\Gamma\to U(S_{\mathcal{H}})$ lifts $\pi_\Gamma$ to $S_{\mathcal{H}}$. Let $c_b(\Gamma)$ be the (continuous) bounded functions on $\Gamma$, and define a representation of $c_b(\Gamma)\rtimes \Gamma$ on $\ell^2(\Gamma, S_{\mathcal{H}})$ by $$\hat{\pi}_S(a\lambda_g)f(\gamma)=a(\gamma) [\tilde{\pi}(g)f](\gamma),$$ where $\tilde{\pi}$ is as in . Then the triple $(c_b(\Gamma)\rtimes^{\rm alg} \Gamma, \ell^2(\Gamma, S_{\mathcal{H}}), {\mathcal{D}}_c, {\mathcal{N}}, \operatorname{Tr}_\tau)$ is a semifinite spectral triple which is $\mathrm{Li}_1$-summable if $\ell$ has at most exponential growth. We return to the example of this subsubsection in Example \[diracgrouphea\] (see page ) and Subsection \[diracgroupkms\] (see page ). The following example of a proper group cocycle shows the construction’s geometric advantage compared to only using a length function. Consider the trivial action of the discrete group $\Gamma={\mathbb{Z}}^n$ on ${\mathcal{H}}_\Gamma={\mathbb{R}}^n$. The inclusion ${\mathbb{Z}}^n\hookrightarrow{\mathbb{R}}^n$ is additive and proper, and therefore a proper group cocycle for the trivial action. The semifinite spectral triple associated with a finite dimensional Clifford representation $\mathfrak{c}_S:{\mathbb{R}}^n\to \operatorname{End}_{\mathbb{C}}(S)$ can when restricted to $C^*({\mathbb{Z}}^n)\cong C(\mathbb{T}^n)$ be identified with the semifinite spectral triple $(C^\infty(\mathbb{T}^n),L^2(\mathbb{T}^n,S),\slashed{D}_{\mathbb{T}^n}\otimes 1_S,\mathbb{B}(L^2(\mathbb{T}^n))\otimes {\mathbb{C}}\ell_n,\operatorname{Tr}_\tau)$ using Fourier theory on the dual torus $\mathbb{T}^n=\widehat{{\mathbb{Z}}^n}$. We observe that to extract $K$-homological content from this construction we need to specify a grading if $n$ is even. In particular, it is unclear how to interpret the construction above in $K$-homology when ${\mathcal{H}}_\Gamma$ is infinite-dimensional. KMS states constructed from $\mathrm{Li}_1$-summable spectral triples {#sec:KMS} ===================================================================== This section contains the fundamental technical construction of the paper. Starting from a semifinite $\mathrm{Li}_1$-summable spectral triple, we use the associated algebra of Toeplitz operators to construct an action from the operator ${\mathcal{D}}$ and a KMS-state from the operator $|{\mathcal{D}}|$. The positive part of the spectrum and heat traces ------------------------------------------------- Throughout this section we suppose that $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}},{\mathcal{N}},{\mathcal{T}})$ is a unital semifinite spectral triple. The spectral triple can be semifinite, in which case we let ${\mathcal{T}}$ denote the given positive faithful normal semifinite trace. In general ${\mathcal{T}}$ is not unique, and coincides with a non-zero multiple of the operator trace in the “usual" non-semifinite case ${\mathcal{N}}={\mathcal{B}}({\mathcal{H}})$. We write $\mathbb{K}_{\mathcal{N}}$ for the compacts for ${\mathcal{T}}$. Again, ${\mathbb{K}}_{\mathcal{N}}$ coincides with the usual compacts in the case of the type I factor ${\mathcal{N}}={\mathcal{B}}({\mathcal{H}})$. We write ${\mathcal{N}}^+$ for the von Neumann algebra $P_{\mathcal{D}}{\mathcal{N}}P_{\mathcal{D}}$. By an abuse of notation, we write ${\mathcal{T}}$ also for the induced faithful normal semifinite trace on ${\mathcal{N}}^+$. The ${\mathcal{T}}$-compacts on ${\mathcal{N}}^+$ will be denoted by ${\mathbb{K}}_{\mathcal{N}}^+$. \[ass:minus-one\] The operator ${\mathcal{D}}$ is said to have positive ${\mathcal{T}}$-essential spectrum if for some $\beta\in [0,\infty)$, we have ${\mathcal{T}}(P_{\mathcal{D}}{\mathrm{e}}^{-t{\mathcal{D}}})<\infty$ for $t>\beta$ and ${\mathcal{T}}(P_{\mathcal{D}}{\mathrm{e}}^{-t{\mathcal{D}}})\nearrow \infty$ as $t\searrow\beta$. \[converatbeta\] Let ${\mathcal{D}}$ be a self-adjoint densely defined operator affiliated with ${\mathcal{N}}$ satisfying that $(1+{\mathcal{D}}^2)^{-1/2}\in \mathrm{Li}_1({\mathcal{T}})$. We define the number $$\label{invtempish} \beta_{\mathcal{D}}:=\inf\{t>0: {\mathcal{T}}(P_{\mathcal{D}}{\mathrm{e}}^{-t{\mathcal{D}}})<\infty\}.$$ Then $\beta_{\mathcal{D}}\in [0,\infty)$ and ${\mathcal{D}}$ has positive ${\mathcal{T}}$-essential spectrum if and only if $$\lim_{t\searrow\beta_{\mathcal{D}}}{\mathcal{T}}(P_{\mathcal{D}}{\mathrm{e}}^{-t{\mathcal{D}}})=\infty.$$ In particular, if $\beta_{\mathcal{D}}=0$ then ${\mathcal{D}}$ has positive ${\mathcal{T}}$-essential spectrum if and only if ${\mathcal{T}}(P_{\mathcal{D}})=\infty$. By definition, if $(1+{\mathcal{D}}^2)^{-1/2}\in \mathrm{Li}_1$ then ${\mathcal{T}}({\mathrm{e}}^{-t|{\mathcal{D}}|})<\infty$ for $t$ large enough. Therefore, ${\mathcal{T}}(P_{\mathcal{D}}{\mathrm{e}}^{-t{\mathcal{D}}})={\mathcal{T}}(P_{\mathcal{D}}{\mathrm{e}}^{-t|{\mathcal{D}}|})<\infty$ for $t$ large enough and $\beta_{\mathcal{D}}:=\inf\{t>0: {\mathcal{T}}(P_{\mathcal{D}}{\mathrm{e}}^{-t{\mathcal{D}}})<\infty\}$ will be a number in $[0,\infty)$. By definition, ${\mathcal{T}}(P_{\mathcal{D}}{\mathrm{e}}^{-t{\mathcal{D}}})<\infty$ for $t>\beta_{\mathcal{D}}$ and if $ \lim_{t\searrow\beta_{\mathcal{D}}}{\mathcal{T}}(P_{\mathcal{D}}{\mathrm{e}}^{-t{\mathcal{D}}})=\infty$ then ${\mathcal{D}}$ has positive ${\mathcal{T}}$-essential spectrum with $\beta=\beta_{\mathcal{D}}$. Conversely, if ${\mathcal{D}}$ has positive ${\mathcal{T}}$-essential spectrum there is a $\beta\in [0,\infty)$ with ${\mathcal{T}}(P_{\mathcal{D}}{\mathrm{e}}^{-t{\mathcal{D}}})<\infty$ for $t>\beta$ and ${\mathcal{T}}(P_{\mathcal{D}}{\mathrm{e}}^{-t{\mathcal{D}}})\nearrow \infty$ as $t\searrow\beta$, and in this case it is clear that $\beta=\beta_{\mathcal{D}}$. We will often impose the assumption of positive ${\mathcal{T}}$-essential spectrum. If for some $\beta\in [0,\infty)$, ${\mathcal{T}}((1-P_{\mathcal{D}}){\mathrm{e}}^{t{\mathcal{D}}})<\infty$ for $t>\beta$ and ${\mathcal{T}}((1-P_{\mathcal{D}}){\mathrm{e}}^{t{\mathcal{D}}})\nearrow \infty$ as $t\searrow\beta$, we can equally well use $-{\mathcal{D}}$ in our construction. Let ${\mathcal{D}}$ be a self-adjoint densely defined operator affiliated with ${\mathcal{N}}$ satisfying that $(1+{\mathcal{D}}^2)^{-1/2}\in \mathrm{Li}_1({\mathcal{T}})$. Then ${\mathcal{D}}$ has positive ${\mathcal{T}}$-essential spectrum if and only if both of the following conditions fail: 1. $P_{\mathcal{D}}$ has finite ${\mathcal{T}}$-trace. 2. There exists a $p>0$ such that $P_{\mathcal{D}}{\mathrm{e}}^{-|{\mathcal{D}}|}\in \mathcal{L}^p({\mathcal{T}})\setminus \cap_{q>p} \mathcal{L}^q({\mathcal{T}})$. The reader should note that conditions 1. and 2. are mutually exclusive. If ${\mathcal{D}}$ has positive ${\mathcal{T}}$-essential spectrum, then clearly 1. fails. Also 2. fails if ${\mathcal{D}}$ has positive ${\mathcal{T}}$-essential spectrum because condition 2. is equivalent to $\beta_{\mathcal{D}}=p$ and $\lim_{t\searrow p}{\mathcal{T}}(P_{\mathcal{D}}{\mathrm{e}}^{-t{\mathcal{D}}})$ being finite. Conversely, if Condition 1. and 2. fails, then either $\beta_{\mathcal{D}}=0$ and $\lim_{t\searrow 0}{\mathcal{T}}(P_{\mathcal{D}}{\mathrm{e}}^{-t{\mathcal{D}}})$ must be infinite not to violate ${\mathcal{T}}(P_{\mathcal{D}})$ being infinite or $\beta_{\mathcal{D}}>0$ and the set $\{p>0: P_{\mathcal{D}}{\mathrm{e}}^{-|{\mathcal{D}}|}\in \mathcal{L}^p({\mathcal{T}})\}$ is open (due to condition 2. failing) showing that ${\mathcal{T}}(P_{\mathcal{D}}{\mathrm{e}}^{-t{\mathcal{D}}})\nearrow \infty$ as $t\searrow\beta_{\mathcal{D}}$. \[diracmfdhea\] Dirac operators on closed manifolds, as considered in Subsection \[diracmfdfirst\] (see page ), have positive essential spectrum. In this case, we can compute $\beta_{\mathcal{D}}=0$ and the leading term in the heat trace asymptotics using Proposition \[heatasumfd\]. If $(C^\infty(M),L^2(M,S),\slashed{D})$ is the spectral triple associated with a Dirac operator, Proposition \[heatasumfd\] implies that $$\operatorname{Tr}_{L^2(M,S)}(P_{\mathcal{D}}{\mathrm{e}}^{-t|\slashed{D}|}) =n!c_nt^{-n}\int_{S^*M} \mathrm{Tr}_S(p_{\slashed{D}})\mathrm{d}V +O(t^{-n+\epsilon}), \quad\mbox{as $t\to 0$},$$ where $p_{\slashed{D}}$ is the principal symbol of the zeroth order pseudo-differential operator $P_\slashed{D}$. A direct computation shows that $p_{\slashed{D}}(x,\xi)=\frac{1}{2}\left(c_S(\xi)+1\right)$ for $x\in M$ and $\xi\in S^*_xM$. Here $c_S:T^*M\to \operatorname{End}(S)$ denotes Clifford multiplication. More generally, Proposition \[heatasumfd\] allows us to conclude that for $a\in C^\infty(M)$, $$\begin{aligned} \operatorname{Tr}_{L^2(M,S)}(P_{\mathcal{D}}a{\mathrm{e}}^{-t|\slashed{D}|}) &=n!c_nt^{-n}\int_{S^*M} \mathrm{Tr}_S(p_{\slashed{D}})a\,\mathrm{d}V +O(t^{-n+\epsilon})=\\ &=n!c_nt^{-n}\int_{M}\int_{S^*_xM} a(x)\mathrm{Tr}_S(p_{\slashed{D}}(x,\xi))\,\mathrm{d}V_{S^*_xM}(\xi)\mathrm{d}V(x) +O(t^{-n+\epsilon})=\\ &=\tilde{c}_nt^{-n}\int_{M}a\,\mathrm{d}V +O(t^{-n+\epsilon}), \quad\mbox{as $t\to 0$},\end{aligned}$$ for a new constant $\tilde{c}_n>0$, depending only on the dimension of $M$ and the rank of $S$. In the last equality we used that $p_{\slashed{D}}-1/2$ is an antisymmetric function under the involution $(x,\xi)\mapsto (x,-\xi)$ of $S^*M$ and therefore $$\begin{aligned} \int_{S^*_xM}\mathrm{Tr}_S(p_{\slashed{D}}(x,\xi))\,\mathrm{d}V_{S^*_xM}&=\int_{S^*_xM}\mathrm{Tr}_S(p_{\slashed{D}}(x,-\xi))\,\mathrm{d}V_{S^*_xM}=\\ &=\frac{\mathrm{rank}(S)}{2}\int_{S^*_xM} \,\mathrm{d}V_{S^*_xM}=\frac{\mathrm{rank}(S)\pi^{\dim(M)/2}}{\Gamma(\dim(M)/2)}.\end{aligned}$$ \[diracgraphhea\] The spectral triples for graph $C^*$-algebras from Proposition \[localizeckdirinpoint\] (see page ) also have positive essential spectrum. We compute $\beta_{\mathcal{D}}$ and the heat trace asymptotics assuming that $G$ is primitive. In this example, $P_{{\mathcal{D}}_y}$ is the projection onto the subspace $\ell^2(\mathcal{V}_y\cap \kappa_G^{-1}(0))$. We use the notation $$\label{vplusdef} \mathcal{V}_y^+:=\mathcal{V}_y\cap \kappa_G^{-1}(0) =\{(x,n)\in \mathcal{V}_y: n\geq 0, \; \sigma_G^n(x)=y\}.$$ The space $P_{{\mathcal{D}}_y}\ell^2(\mathcal{V}_y)$ is therefore spanned by the orthonormal basis $(\delta_{(x,n)})_{(x,n)\in \mathcal{V}_y^+}$. Note that if $\sigma_G^n(x)=y$ then $x$ is uniquely determined by $y$ and a finite path $\sigma=\sigma_1\sigma_2\cdots \sigma_n$ with $x=\sigma y$. Note that paths of the form $x=\sigma y$, with $s(\sigma_n)=r(y)$, exhaust all possible $x\in \sigma_G^{-n}(\{y\})$. Using that $P_{{\mathcal{D}}_y} {\mathcal{D}}_y\delta_{(x,n)}=n \delta_{(x,n)}$ for $(x,n)\in \mathcal{V}_y^+$, we compute that $$\operatorname{Tr}(P_{{\mathcal{D}}_y}{\mathrm{e}}^{-t|{\mathcal{D}}_y|}) =\sum_{n=0}^\infty \sum_{x\in \sigma_G^{-n}(\{y\})} {\mathrm{e}}^{-tn} =\sum_{n=0}^\infty \#\{\sigma\in E_n: s(\sigma_n)=r(y)\} {\mathrm{e}}^{-tn}.$$ Let $A$ denote the edge adjacency matrix of $G$ and $r_\sigma(A)$ its spectral radius. If $G$ is primitive, (i.e. all entries of $A^k$ are positive for some integer $k>0$) we let $w\in {\mathbb{C}}^E$ its $\ell^2$-normalized Perron-Frobenius vector. It follows from [@RRS Lemma 3.7] that there is an $\alpha_0\in [0,1)$ such that $$\#\{\sigma\in E_n: s(\sigma_n)=r(y)\} =\|w\|_{\ell^1}w_{r(y)} r_\sigma(A)^{n}+O((\alpha_0 r_\sigma(A))^n),$$ as $n\to \infty$. We can conclude that there is a function $f$ holomorphic in $\mathrm{Re}(t)>\log r_\sigma(A)+\log\alpha_0$ such that $$\operatorname{Tr}(P_{{\mathcal{D}}_y}{\mathrm{e}}^{-t|{\mathcal{D}}_y|})=\frac{\|w\|_{\ell^1}w_{r(y)}}{1-r_\sigma (A) {\mathrm{e}}^{-t}}+f(t).$$ Therefore, $\operatorname{Tr}(P_{{\mathcal{D}}_y}{\mathrm{e}}^{-t|{\mathcal{D}}_y|})-\frac{\|w\|_{\ell^1}w_{r(y)}}{t-\log(r_\sigma (A))}$ has a holomorphic extension to $\mathrm{Re}(t)>\log r_\sigma(A)+\log\alpha_0$ whenever $G$ is primitive. More generally, if $G$ is primitive, the method above shows that for two finite paths $\mu$ and $\nu$ we can compute that $$\begin{aligned} \operatorname{Tr}(P_{{\mathcal{D}}_y}S_\mu S_\nu^*{\mathrm{e}}^{-t|{\mathcal{D}}_y|}) =&\delta_{\mu,\nu}\sum_{n=0}^\infty \sum_{x\in \sigma_G^{-n}(\{y\})} \|S_\mu^*\delta_{(x,n)}\|_{\ell^2}^2{\mathrm{e}}^{-tn}\\ =&\sum_{n=|\mu|}^\infty \#\big\{\sigma\in E_{n-|\mu|}: r(\sigma_1)=s(\mu), \, s(\sigma_{n-|\mu|})=r(y)\big\} {\mathrm{e}}^{-tn}\\ &+\delta_{\mu,\nu}\sum_{n=0}^{|\mu|-1} \sum_{x\in \sigma_G^{-n}(\{y\})} \|S_\mu^*\delta_{(x,n)}\|_{\ell^2}^2{\mathrm{e}}^{-tn}\\ =&\delta_{\mu,\nu}w_{s(\mu)}w_{r(y)} \frac{{\mathrm{e}}^{-t|\mu|}}{1-r_\sigma(A){\mathrm{e}}^{-t}}+\delta_{\mu,\nu}f_{\mu,y}(t),\end{aligned}$$ for a function $f_{\mu,y}$ holomorphic in $\mathrm{Re}(t)>\log r_\sigma(A)+\log\alpha_0$. We conclude that $$\operatorname{Tr}(P_{{\mathcal{D}}_y}S_\mu S_\nu^*{\mathrm{e}}^{-t|{\mathcal{D}}_y|})-\delta_{\mu,\nu}w_{d(\mu)}w_{r(y)}\frac{r_\sigma(A)^{-|\mu|}}{t-\log(r_\sigma (A))},$$ has a holomorphic extension to $\mathrm{Re}(t)>\log r_\sigma(A)+\log\alpha_0$ whenever $G$ is primitive. As such, $\beta_{{\mathcal{D}}_y}=\log(r_\sigma(A))$ and ${\mathcal{D}}_y$ has positive essential spectrum. \[diraccphea\] Let $O_E$ be a Cuntz-Pimsner algebra defined from a strictly W-regular (recall Definition \[ass:two\]) finitely generated and projective bi-Hilbertian bimodule $E_A$ and a positive trace $\tau$ on the coefficient algebra $A$. The semifinite spectral triple considered in Lemma \[ximodsemi\] (see page ) also has positive $\operatorname{Tr}_\tau$-essential spectrum assuming a criticality condition on $\tau$ that we formulate below (see Definition \[defn:critical\] on page ). The heat trace asymptotics are slightly more involved, and we compute these explicitly in Subsection \[subsec:T-vs-LN\] under a condition on $\tau$ previously studied by Laca-Neshveyev [@LN] in the context of KMS-states. However, for a general $\tau$ we can proceed as in the proof of Lemma \[ximodsemi\] to deduce the following. \[posheatcomp\] For any strictly $W$-regular fgp bi-Hilbertian bimodule $E$ over the unital $C^*$-algebra and a positive trace $\tau$ on $A$, $$\operatorname{Tr}_\tau(P_{{\mathcal{D}}_\psi}{\mathrm{e}}^{-t{\mathcal{D}}_\psi})=\sum_{n=0}^\infty {\mathrm{e}}^{-tn} \tau_*(E^{\otimes_A n}),$$ where $\tau_*:K_0(A)\to {\mathbb{R}}$ denotes the map induced by $\tau$ on $K$-theory. In particular, $\operatorname{Tr}_\tau(P_{{\mathcal{D}}_\psi}{\mathrm{e}}^{-t{\mathcal{D}}_\psi})$ does not depend on the choice of inner products on $E$ but only on $\tau$ and the bimodule structure on $E$. We compute that $$\operatorname{Tr}_\tau(P_{\mathcal{D}}{\mathrm{e}}^{-t|{\mathcal{D}}_\psi|})=\sum_{n=0}^\infty {\mathrm{e}}^{-t|\psi(n,n)|}\operatorname{Tr}_\tau(P_{n,n})=\sum_{n=0}^\infty {\mathrm{e}}^{-tn}\operatorname{Tr}_\tau(Q_{n,n})=\sum_{n=0}^\infty \sum_{|\rho|=n}{\mathrm{e}}^{-tn}\tau((e_\rho|e_\rho)_A).$$ On the other hand, $\tau_*(E^{\otimes_A n})=(\tau\otimes \operatorname{Tr}_{M_{N(n)}})(p_{E^{\otimes_A n}})$ where $p_{E^{\otimes_A n}}\in M_{N(n)}(A)$ is a projection representing $E^{\otimes_A n}$. Using the choice of frame $(e_j)_{j=1}^N$, we can take $N(n):=N^n$ and $p_{E^{\otimes_A n}}:=((e_\mu|e_\nu)_A)_{|\mu|=|\nu|=n}$. In this choice of representing projection, $$\tau_*(E^{\otimes_A n})=(\tau\otimes \operatorname{Tr}_{M_{N(n)}})(p_{E^{\otimes_A n}})=\operatorname{Tr}_{M_{N(n)}}((\tau((e_\mu|e_\nu)_A)_{|\mu|=|\nu|=n}))=\sum_{|\rho|=n}\tau((e_\rho|e_\rho)_A).\qedhere$$ This computation shows that it is in general difficult to compute $\beta_{\mathcal{D}}$. In this case $\beta_{\mathcal{D}}$ depends on the asymptotic properties of the sequence $(\tau_*(E^{\otimes_A n}))_{n\in {\mathbb{N}}}$ as $n\to \infty$. For a simple tensor $\sigma\in E^{{\otimes}m}$ write $\sigma=\underline{\sigma}\overline{\sigma}$, where the initial segment $\underline{\sigma}$ will be of a length understood from context ($|\underline{\sigma}|=|\mu|$ in the next computation). With this notation, we can compute our functional on a typical $S_\mu S_\nu^*\in O_E$, where $\mu\in E^{\otimes k}$, $\nu\in E^{\otimes l}$ are simple tensors. We find $$\begin{aligned} \label{tracpcocomcme} \operatorname{Tr}_\tau(P_{\mathcal{D}}S_\mu S_\nu^* {\mathrm{e}}^{-t|{\mathcal{D}}_\psi|})&=\delta_{|\mu|,|\nu|}\sum_{n=0}^\infty {\mathrm{e}}^{-tn}\operatorname{Tr}_\tau(S_\mu S_\nu^* Q_{n,n})\\ \nonumber &=\delta_{|\mu|,|\nu|}\sum_{n=|\mu|}^\infty\sum_{|\sigma|=n} {\mathrm{e}}^{-tn}\tau ((S_\mu^*e_{\sigma}|S_\nu^*e_{\sigma})_{E^{\otimes (n-|\mu|)}})\\ \nonumber &=\delta_{|\mu|,|\nu|}\sum_{n=|\mu|}^\infty\sum_{|\sigma|=n} {\mathrm{e}}^{-tn}\tau (\,(\,(\mu|e_{\underline{\sigma}})_{E^{|\mu|}}e_{\overline{\sigma}}\,|\,(\nu|e_{\underline{\sigma}})_{E^{|\mu|}}e_{\overline{\sigma}})_{E^{\otimes (n-|\mu|)}}).\end{aligned}$$ \[diracgrouphea\] Consider a length function $\ell$ on a countable group $\Gamma$ as in Subsection \[groupcstarexam\]. \[criticaldefn\] We define the critical value of $(\Gamma,\ell)$ as $$\beta(\Gamma, \ell):=\inf\{t\geq 0:\sum_{\gamma\in \Gamma} {\mathrm{e}}^{-t\ell(\gamma)}<\infty \}.$$ If $\sum_{\gamma\in \Gamma} {\mathrm{e}}^{-t\ell(\gamma)}\nearrow \infty$ as $t\searrow \beta(\Gamma,\ell)$, we say that $\ell$ is critical. It follows directly from Definition \[ass:minus-one\] that the operator ${\mathcal{D}}_\ell$ appearing in Proposition \[spectripfromlength\] (see page ) has positive essential spectrum as long as $\ell$ is critical. The heat trace of an element $a\lambda_g\in c_b(\Gamma)\rtimes^{\rm alg}\Gamma$ is given by $$\operatorname{Tr}(P_{{\mathcal{D}}_\ell} a\lambda_g {\mathrm{e}}^{-t|{\mathcal{D}}_\ell|})=\operatorname{Tr}(a\lambda_g {\mathrm{e}}^{-t|{\mathcal{D}}_\ell|})=\delta_{e,g}\sum_{\gamma\in \Gamma} a(\gamma){\mathrm{e}}^{-t\ell(\gamma)}.$$ Similar computations can be carried out for the semifinite spectral triple constructed in Proposition \[cssfst\] using a Hilbert space valued cocycle $c_\Gamma$ (see page ). Note that $$P_{{\mathcal{D}}_c}f(g)=\frac{1}{2}\left(\frac{\mathfrak{c}_S(c_\Gamma(g))}{\|c_\Gamma(g)\|_{{\mathcal{H}}_\Gamma}}+1\right).$$ Therefore, the heat trace of an element $a\lambda_g\in c_b(\Gamma)\rtimes^{\rm alg}\Gamma$ is given by $$\begin{aligned} \operatorname{Tr}_\tau(P_{{\mathcal{D}}_c} a\lambda_g {\mathrm{e}}^{-t|{\mathcal{D}}_c|}) &=\frac{1}{2}\sum_{\gamma\in \Gamma}\left\langle \delta_\gamma, \tau\left(\frac{\mathfrak{c}_S(c_\Gamma(g))}{\|c_\Gamma(g)\|_{{\mathcal{H}}_\Gamma}}+1\right)a(g^{-1}\gamma)\delta_{g\gamma}\right\rangle {\mathrm{e}}^{-t\ell(\gamma)}=\\ &=\frac{1}{2}\delta_{e,g}\sum_{\gamma\in \Gamma} a(\gamma){\mathrm{e}}^{-t\ell(\gamma)}=\frac{1}{2}\operatorname{Tr}(P_{{\mathcal{D}}_\ell} a\lambda_g {\mathrm{e}}^{-t|{\mathcal{D}}_\ell|}).\end{aligned}$$ Here we use that $\tau(\mathfrak{c}_S(v))=0$ for any $v\in {\mathcal{H}}_\Gamma$ which holds due to the fact that we can pick a $w\in {\mathcal{H}}_\Gamma$ orthogonal to $v$ and compute that $$\tau(\mathfrak{c}_S(v))=\tau(\mathfrak{c}_S(w)\mathfrak{c}_S(v)\mathfrak{c}_S(w))=-\tau(\mathfrak{c}_S(w)^2\mathfrak{c}_S(v))=-\tau(\mathfrak{c}_S(v)).$$ If the length function $\ell(\gamma):=\Vert c_\Gamma(\gamma)\Vert_{{\mathcal{H}}_\Gamma}$ associated with $c_\Gamma$ is critical, we say that $c_\Gamma$ is critical. We conclude that the semifinite spectral triple from Proposition \[cssfst\] has positive essential spectrum if $c_\Gamma$ is critical. To-plitz or not To-plitz {#subsec:toplitz} ------------------------ We proceed under the same working conditions as in the previous section to construct states from spectral triples. Recall that ${\mathcal{N}}^+=P_{\mathcal{D}}{\mathcal{N}}P_{\mathcal{D}}$ and that ${\mathbb{K}}_{\mathcal{N}}^+=P_{\mathcal{D}}{\mathbb{K}}_{\mathcal{N}}P_{\mathcal{D}}$. \[defn:to-plitz\] Let $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}},{\mathcal{N}},{\mathcal{T}})$ be a semifinite spectral triple. We define the Toeplitz algebra of $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}},{\mathcal{N}},{\mathcal{T}})$ as $$T_A:=P_{\mathcal{D}}A P_{\mathcal{D}}+{\mathbb{K}}_{\mathcal{N}}^+\subseteq {\mathcal{N}}^+.$$ The saturated Toeplitz algebra of $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}},{\mathcal{N}},{\mathcal{T}})$ is defined by $$T_{A,{\mathcal{D}}}:=C^*\left(\bigcup_{s\in{\mathbb{R}}}{\mathrm{e}}^{is{\mathcal{D}}}T_A{\mathrm{e}}^{-is{\mathcal{D}}}\right)=C^*\left(\bigcup_{s\in{\mathbb{R}}}{\mathrm{e}}^{is{\mathcal{D}}}P_{\mathcal{D}}AP_{\mathcal{D}}{\mathrm{e}}^{-is{\mathcal{D}}}+{\mathbb{K}}_{\mathcal{N}}^+\right)\subseteq {\mathcal{N}}^+.$$ \[toepiscstar\] The Toeplitz algebra $T_A$ of a unital semifinite spectral triple $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}},{\mathcal{N}},{\mathcal{T}})$ is a $C^*$-algebra. It follows from Lemma \[bddcommsum\] that $[P_{\mathcal{D}},a]\in {\mathbb{K}}_{\mathcal{N}}$ for all $a\in A$. Therefore, the mapping $$\beta_{\mathcal{D}}:A\to {\mathcal{N}}^+/{\mathbb{K}}_{\mathcal{N}}^+, \quad a\mapsto P_{\mathcal{D}}aP_{\mathcal{D}}\mod{\mathbb{K}}_{\mathcal{N}}^+,$$ is a $*$-homomorphism and $\beta_{\mathcal{D}}(A)\subseteq {\mathcal{N}}^+/{\mathbb{K}}_{\mathcal{N}}^+$ is a closed $C^*$-subalgebra. By definition, $T_A$ is the preimage of $\beta_{\mathcal{D}}(A)$ under the quotient mapping ${\mathcal{N}}^+\to {\mathcal{N}}^+/{\mathbb{K}}_{\mathcal{N}}^+$ and is therefore a $C^*$-algebra. The reader should note that Proposition \[toepiscstar\] also holds in the non-unital setting because it only relies on Lemma \[bddcommsum\] which holds non-unitally. The saturated Toeplitz algebra of $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}},{\mathcal{N}},{\mathcal{T}})$ carries an ${\mathbb{R}}$-action $\sigma^+:{\mathbb{R}}\to \mathrm{Aut}(T_{A,{\mathcal{D}}})$ defined by $$\sigma_s^+(T):=P_{\mathcal{D}}{\mathrm{e}}^{is{\mathcal{D}}}T{\mathrm{e}}^{-is{\mathcal{D}}}P_{\mathcal{D}}=P_{\mathcal{D}}{\mathrm{e}}^{is|{\mathcal{D}}|}T{\mathrm{e}}^{-is|{\mathcal{D}}|}P_{\mathcal{D}},\qquad s\in {\mathbb{R}}, \; T\in T_{A,{\mathcal{D}}}.$$ This proposition is a consequence of that $T_{A,{\mathcal{D}}}$ is constructed as the saturation of $T_A$ under the action $\sigma^+$ extended to ${\mathcal{N}}^+$. We note that if we identify $T_{A,{\mathcal{D}}}$ with a subalgebra of ${\mathcal{N}}$, we can also write $\sigma_s^+(T):= {\mathrm{e}}^{is{\mathcal{D}}}T{\mathrm{e}}^{-is{\mathcal{D}}}= {\mathrm{e}}^{is|{\mathcal{D}}|}T{\mathrm{e}}^{-is|{\mathcal{D}}|}$. Define a one-parameter family of states $(\phi_{t,0})_{t>\beta}$ on $T_{A,{\mathcal{D}}}$ by $$\phi_{t,0}(T):=\frac{{\mathcal{T}}(P_{\mathcal{D}}T{\mathrm{e}}^{-t{\mathcal{D}}})}{{\mathcal{T}}(P_{\mathcal{D}}{\mathrm{e}}^{-t{\mathcal{D}}})}.$$ We shall compose the family $(\phi_{t,0})_{t>\beta}$ with an “extended limit" as $t\to \beta$: \[extendedlimitdef\] An extended limit as $t\to \beta$ is a state $\omega\in L^\infty(\beta,\infty)^*$ such that $\omega(f)=0$ whenever $\lim_{t\to \beta}f(t)=0$. For an extended limit $\omega$ and $f\in L^\infty(\beta,\infty)$, we write $$\lim_{t\to \omega} f:=\omega(f).$$ Let $\omega$ be any extended limit and define $$\phi_{\omega,0}:\,T_{A,{\mathcal{D}}}\to{\mathbb{C}},\qquad \phi_{\omega,0}(T):=\omega\circ \phi_{t,0}(T)=\lim_{t\to \omega} \phi_{t,0}(T).$$ \[vanishoncomp\] Let $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}},{\mathcal{N}},{\mathcal{T}})$ be a semifinite $\mathrm{Li}_1$-summable spectral triple with positive ${\mathcal{T}}$-essential spectrum (see Definition \[ass:minus-one\] on page ). For any extended limit $\omega$, the functional $\phi_{\omega,0}$ is a state on $T_{A,{\mathcal{D}}}$. Moreover $\phi_{\omega,0}(T)=0$ for all $T\in \mathbb{K}_{\mathcal{N}}^+$. It is immediate that $\phi_{\omega,0}$ is a state. For the statement that $\phi_{\omega,0}(T)=0$ for all $T\in \mathbb{K}_{\mathcal{N}}^+$, we observe that since $\phi_{\omega,0}$ is a state, it is also norm-continuous. It therefore suffices to prove that $\phi_{\omega,0}(T)=0$ for all projections $T\in{\mathcal{N}}^+$ with ${\mathcal{T}}(T)<\infty$. For such $T$, we can estimate that $$\phi_{t,0}(T)=\frac{{\mathcal{T}}(P_{\mathcal{D}}T{\mathrm{e}}^{-t{\mathcal{D}}})}{{\mathcal{T}}(P_{\mathcal{D}}{\mathrm{e}}^{-t{\mathcal{D}}})}\leq \frac{{\mathcal{T}}(T)}{{\mathcal{T}}(P_{\mathcal{D}}{\mathrm{e}}^{-t{\mathcal{D}}})}.$$ Since, ${\mathcal{T}}(P_{\mathcal{D}}{\mathrm{e}}^{-t{\mathcal{D}}})\nearrow \infty$ as $t\searrow \beta$, it follows that $\lim_{t\to \beta} \phi_{t,0}(T)=0$. We conclude that for all projections $T\in{\mathcal{N}}^+$, with ${\mathcal{T}}(T)<\infty$, and any extended limit $\omega$ as $t\to \beta$, $\omega\circ \phi_{t,0}(T)=0$. Due to Lemma \[vanishoncomp\], we can make the following definition. Define the $C^*$-algebra $A_{\mathcal{D}}:=T_{A,{\mathcal{D}}}/{\mathbb{K}}^+_{\mathcal{N}}$ and the state $\phi_\omega$ on $A_{\mathcal{D}}$ as $$\phi_\omega(T\!\!\!\mod {\mathbb{K}}^+_{\mathcal{N}}):=\phi_{\omega,0}(T).$$ The state $\phi_{\omega,0}$ also restricts to a state on $T_A$, and Lemma \[vanishoncomp\] implies that $\phi_{\omega,0}|_{T_A}$ descends to a state on $A$ via the $*$-epimorphism $\beta_{\mathcal{D}}:A\to T_A/{\mathbb{K}}^+_{\mathcal{N}}$ (see the proof of Proposition \[toepiscstar\] on page ). To analyse the situation of the state on $A_{\mathcal{D}}$ versus that on $A$, we consider the following ideal $$I:=\{a\in A:\,P_{\mathcal{D}}aP_{\mathcal{D}}\in \mathbb{K}_{\mathcal{N}}^+\}$$ so that $P_{\mathcal{D}}IP_{\mathcal{D}}=P_{\mathcal{D}}AP_{\mathcal{D}}\cap\mathbb{K}_{\mathcal{N}}^+$. Since $T_A\subseteq T_{A,{\mathcal{D}}}$, we obtain a commuting diagram $$\xymatrix{0\ar[r]& \mathbb{K}_{\mathcal{N}}^+\ar[r]\ar[d]^{=}&T_A\ar[r]\ar@{^{(}->}[d]&A/I\ar@{^{(}->}[d]\ar[r]&0\\ 0\ar[r]&\mathbb{K}_{\mathcal{N}}^+\ar[r]&T_{A,{\mathcal{D}}}\ar[r]&A_{\mathcal{D}}\ar[r]&0, }$$ with exact rows. The mapping $A/I\to A_{\mathcal{D}}$ is indeed injective by the four lemma. We identify $A/I$ with a subalgebra of $A_{\mathcal{D}}$. The induced mapping $\gamma: A\to A_{\mathcal{D}}$ is compatible with the states $\phi_\omega$ and $\phi_{\omega,0}$ in the sense that $\phi_{\omega,0}(P_{\mathcal{D}}aP_{\mathcal{D}})=\phi_\omega(\gamma(a))$ for $a\in A$. The ${\mathbb{R}}$-action $\sigma_s^+(T):={\mathrm{e}}^{is{\mathcal{D}}}T{\mathrm{e}}^{-is{\mathcal{D}}}$ on $T_{A,{\mathcal{D}}}$ induces an ${\mathbb{R}}$-action on $A/I$. The following proposition follows from the construction of $T_{A,{\mathcal{D}}}$ as the saturation of $T_A$ under $\sigma^+$. \[prop:3.13\] Let $\beta\in {\mathbb{R}}$. The algebra $A_{\mathcal{D}}$ carries an ${\mathbb{R}}$-action $\sigma:{\mathbb{R}}\to \mathrm{Aut}(A_{\mathcal{D}})$ defined by declaring the quotient mapping $T_{A,{\mathcal{D}}}\to A_{\mathcal{D}}$ to be equivariant. The $C^*$-algebra $A_{\mathcal{D}}$ is the saturation of $A/I$ under the action $\sigma$, i.e. $A_{\mathcal{D}}$ is generated in ${\mathcal{N}}^+/{\mathbb{K}}_{\mathcal{N}}^+$ by $\cup_{s\in {\mathbb{R}}}\sigma_s(A/I)$. The aim of our construction is to obtain a KMS state on $A$, or, failing that, on $A/I$. As a first step we introduce conditions ensuring that we at least get a KMS state on $T_{A,{\mathcal{D}}}$, and so on $A_{\mathcal{D}}$. To this end we make the following assumption \[ass:zero\] Let $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}},{\mathcal{N}},{\mathcal{T}})$ be a semifinite spectral triple. We say that a subset $S\subset{\mathcal{A}}$ is analytically generating at $\beta$ if the set $P_{\mathcal{D}}SP_{\mathcal{D}}+\mathbb{K}^+$ generates the Toeplitz algebra $T_A$ as a $C^*$-algebra and satisfies that ${\mathrm{e}}^{\beta{\mathcal{D}}}P_{\mathcal{D}}SP_{\mathcal{D}}{\mathrm{e}}^{-\beta{\mathcal{D}}}\subset {\mathcal{N}}^+$. We say that the semifinite spectral triple $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}},{\mathcal{N}},{\mathcal{T}})$ is $\beta$-analytic if it admits an analytically generating set at $\beta$. We note that this condition is empty if $\beta=0$. The condition of being $\beta$-analytic is just requiring that we have enough analytic elements in $T_{A,{\mathcal{D}}}$ to verify the KMS condition. Indeed we have the following. \[equivalencesbeta\] Let $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}},{\mathcal{N}},{\mathcal{T}})$ be a semifinite spectral triple and $\beta\in {\mathbb{R}}$. Consider the following statements: 1. The semifinite spectral triple $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}},{\mathcal{N}},{\mathcal{T}})$ is $\beta$-analytic. 2. There is a dense $\sigma^+$-invariant subspace $T^0_{A,{\mathcal{D}}}\subseteq T_{A,{\mathcal{D}}}$ of elements satisfying that for any $T\in T^0_{A,{\mathcal{D}}}$ the function $f_T:{\mathbb{R}}\to {\mathcal{N}}$, $f_T(t):=\sigma_t^+(T)$ has a bounded holomorphic extension to the strip $\{z\in {\mathbb{C}}: \; \mathrm{Im}(z)\in (-\beta,0)\}$. 3. There is a dense $\sigma$-invariant subspace $A^0_{{\mathcal{D}}}\subseteq A_{\mathcal{D}}$ of elements satisfying that for any $a\in A_{\mathcal{D}}$ the function $f_a:{\mathbb{R}}\to {\mathcal{N}}$, $f_a(t):=\sigma_t(a)$ has a bounded holomorphic extension to the strip $\{z\in {\mathbb{C}}: \; \mathrm{Im}(z)\in (-\beta,0)\}$. It holds that i) implies ii) which implies iii). If $I:=\{a\in A:\,P_{\mathcal{D}}aP_{\mathcal{D}}\in \mathbb{K}_{\mathcal{N}}^+\}=0$, iii) implies i). It is clear that i) implies ii) since for an analytically generating set $S$ at $\beta$ we can take $T^0_{A,{\mathcal{D}}}$ to be the $\sigma^+$-invariant $*$-algebra generated by $P_{\mathcal{D}}SP_{\mathcal{D}}\cup\mathcal{F}_{\mathcal{D}}$, where $\mathcal{F}_{\mathcal{D}}\subseteq \mathbb{K}^+_{\mathcal{N}}$ is the dense two sided ideal in ${\mathcal{N}}$ generated by the spectral projections of ${\mathcal{D}}$ over compact intervals in ${\mathbb{R}}$. The implication ii)$\Rightarrow$iii) is seen from taking $A^0_{\mathcal{D}}:=T^0_{A,{\mathcal{D}}}/ (T^0_{A,{\mathcal{D}}}\cap {\mathbb{K}}_{\mathcal{N}}^+)$. If $I=0$, the set $S=A^0_{\mathcal{D}}\cap A$ is dense in $A$, and for every $s\in S$ we find that ${\mathrm{e}}^{\beta |{\mathcal{D}}|}s{\mathrm{e}}^{-\beta|{\mathcal{D}}|}$ is a bounded operator in ${\mathcal{N}}$. Thus ${\mathrm{e}}^{\beta |{\mathcal{D}}|}P_{\mathcal{D}}SP_{\mathcal{D}}{\mathrm{e}}^{-\beta|{\mathcal{D}}|}\subset{\mathcal{N}}^+$ and $P_{\mathcal{D}}S P_{\mathcal{D}}+{\mathbb{K}}_{\mathcal{N}}^+$ plainly generates $T_A$. We conclude that $S$ is analytically generating at $\beta$ and that iii) implies i) if $I=0$. \[prop:no-work\] Let $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}},{\mathcal{N}},{\mathcal{T}})$ be a $\beta_{\mathcal{D}}$-analytic $\mathrm{Li}_1$-summable spectral triple with positive ${\mathcal{T}}$-essential spectrum. For any extended limit $\omega$ as $t\to \beta_{\mathcal{D}}$, the state $\phi_{\omega,0}$ is a $\beta_{\mathcal{D}}$-KMS state on $T_{A,{\mathcal{D}}}$ for the one-parameter group $\sigma^+$. It follows from Proposition \[equivalencesbeta\] that the dense subalgebra $T^0_{A,{\mathcal{D}}}\subseteq T_{A,{\mathcal{D}}}$ consists of $\beta_{\mathcal{D}}$-analytic elements of $T_{A,{\mathcal{D}}}$. The twisted trace property relative to the one-parameter group $\sigma^+$ holds on $T^0_{A,{\mathcal{D}}}$ by direct computation: for all $T_1,\,T_2\in T^0_{A,{\mathcal{D}}}$ $$\phi_\omega(T_1T_2)=\lim_{t\to \omega}\frac{{\mathcal{T}}(T_1T_2{\mathrm{e}}^{-t|{\mathcal{D}}|})}{{\mathcal{T}}(P_{\mathcal{D}}{\mathrm{e}}^{-t|{\mathcal{D}}|})}=\lim_{t\to \omega}\frac{{\mathcal{T}}({\mathrm{e}}^{t|{\mathcal{D}}|}T_2{\mathrm{e}}^{-t|{\mathcal{D}}|}T_1{\mathrm{e}}^{-t|{\mathcal{D}}|})}{{\mathcal{T}}(P_{\mathcal{D}}{\mathrm{e}}^{-t|{\mathcal{D}}|})}=\phi_\omega(\sigma_{-i\beta}(T_2)T_1).$$ By definition, $\phi_{\omega,0}$ is a $\beta_{\mathcal{D}}$-KMS-state for $\sigma^+$. \[cor:phi-omega\] Let $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}},{\mathcal{N}},{\mathcal{T}})$ be a $\beta_{\mathcal{D}}$-analytic $\mathrm{Li}_1$-summable spectral triple with positive ${\mathcal{T}}$-essential spectrum. For any extended limit $\omega$ as $t\to \beta_{\mathcal{D}}$, the state $\phi_{\omega}$ is a $\beta_{\mathcal{D}}$-KMS state on $A_{\mathcal{D}}$ for the one-parameter group $\sigma$, defined in Proposition \[prop:3.13\]. This follows from Proposition \[equivalencesbeta\] and Proposition \[prop:no-work\] because $\phi_{\omega,0}$ vanishes on the compacts and the fact that $\phi_\omega$ is induced from $\phi_{\omega,0}$. In practice, for a $\beta_{\mathcal{D}}$-analytic $\mathrm{Li}_1$-summable spectral triple with positive ${\mathcal{T}}$-essential spectrum, we will want to check that in fact $T_{A,{\mathcal{D}}}=T_A$. In this case $A_{\mathcal{D}}=A/I$ and $\phi_\omega$ induces a KMS-state on $A/I$. In the often occuring special case $P_{\mathcal{D}}AP_{\mathcal{D}}\cap\mathbb{K}_{\mathcal{N}}^+=0$, we obtain a KMS state on the algebra $A$. In practice, these things are all checkable and we will do so in several examples in the subsequent sections. The special case $\beta=0$ has been addressed by Voiculescu, [@Voics Proposition 4.6] under the assumption that ${\mathcal{D}}$ is positive. Exponential $\beta$-compatibility is superfluous when $\beta_{\mathcal{D}}=0$. By Proposition \[converatbeta\], when $\beta_{\mathcal{D}}=0$, positive ${\mathcal{T}}$-essential spectrum is equivalent to ${\mathcal{T}}(P_{\mathcal{D}})=\infty$. \[thm:voics\] Let $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}},{\mathcal{N}},{\mathcal{T}})$ be a unital $\mathrm{Li}_{1}$-summable semifinite spectral triple with $\beta_{\mathcal{D}}=0$ and ${\mathcal{T}}(P_{\mathcal{D}})=\infty$. Then $A$ has a tracial state. Indeed, for any extended limit $\omega$ as $t\to 0$, $$\phi_\omega(a) :=\lim_{t\to \omega}\frac{{\mathcal{T}}(P_{\mathcal{D}}a{\mathrm{e}}^{-t{\mathcal{D}}})}{{\mathcal{T}}(P_{\mathcal{D}}{\mathrm{e}}^{-t{\mathcal{D}}})}$$ is a tracial state. Theorem \[thm:voics\] applies to unital $\mathrm{Li}_{(0),1}$-summable semifinite spectral triples with ${\mathcal{T}}(P_{\mathcal{D}})=\infty$ since $\mathrm{Li}_{(0),1}$-summability implies $\beta_{\mathcal{D}}=0$. If the heat trace has an asymptotic expansion as in Theorem \[heatvszeta\], then Theorem \[thm:voics\] can be further simplified. Assume that there is a $p>0$ and that for any $a\in \mathcal{A}$, there is a $\phi_0(a)\in {\mathbb{C}}$ such that $\phi_0(1)\neq 0$ and $${\mathcal{T}}(P_{\mathcal{D}}a{\mathrm{e}}^{-t|{\mathcal{D}}|})=\phi_0(a)t^{-p}+O(t^{-p+\epsilon}), \quad\mbox{as $t\to 0$},$$ for some $\epsilon>0$ (which can depend on $a$). Since $\phi_0(1)\neq 0$, it follows that $\phi_0$ is continuous in the $C^*$-norm on ${\mathcal{A}}$ and $\phi_\omega(a)=\frac{\phi_0(a)}{\phi_0(1)}$ for all $a\in \mathcal{A}$. In fact, $\phi_0(a)=\Gamma(p)\mathrm{Res}_{z=p} \zeta(z; P_{\mathcal{D}}a, |{\mathcal{D}}|)$. The construction of the state $\phi_\omega$ in Corollary \[cor:phi-omega\] involves the operator $P_{\mathcal{D}}$. We shall now provide a result allowing us to remove $P_{\mathcal{D}}$ from the definition of $\phi_\omega$ in the presence of certain symmetries on the spectral triple. The result provides a checkable set of conditions to compute $\phi_\omega$ by means of asymptotics of ${\mathrm{e}}^{-t|{\mathcal{D}}|}$. \[gammajlem\] Let $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}},{\mathcal{N}},{\mathcal{T}})$ be a unital $\mathrm{Li}_{1}$-summable semifinite spectral triple and $\beta\geq 0$ a number such that ${\mathcal{T}}({\mathrm{e}}^{-t|D|})<\infty$ for $t>\beta$ and ${\mathcal{T}}({\mathrm{e}}^{-t|{\mathcal{D}}|})\nearrow \infty$ as $t\searrow\beta$. Assume that there exists self-adjoint operators $\gamma_1,\ldots, \gamma_N\in {\mathcal{N}}\cap {\mathcal{A}}'$ such that 1. $\sum_{j=1}^N \gamma_j^2$ is invertible; 2. $\gamma_j\operatorname{Dom}({\mathcal{D}})\subseteq \operatorname{Dom}({\mathcal{D}})$ and $[{\mathcal{D}},\gamma_j]_+:={\mathcal{D}}\gamma_j+\gamma_j {\mathcal{D}}$ has a bounded extension to ${\mathcal{H}}$; 3. For $j=1,\ldots, N$ and some $\epsilon>0$, the function $t\mapsto {\mathrm{e}}^{-t|{\mathcal{D}}|}\gamma_j{\mathrm{e}}^{t|{\mathcal{D}}|}$ extends to a norm continuous function from the interval $[\beta, \beta+\epsilon)$ to ${\mathcal{N}}$ with $$\lim_{t\to \beta}{\mathrm{e}}^{-t|{\mathcal{D}}|}\gamma_j{\mathrm{e}}^{t|{\mathcal{D}}|}=\gamma_j.$$ Then $\beta_{\mathcal{D}}=\beta$ and ${\mathcal{D}}$ has positive ${\mathcal{T}}$-essential spectrum. Moreover, for any extended limit $\omega$ as $t\to\beta_{\mathcal{D}}$, $$\phi_\omega(a)= \lim_{t\to \omega} \frac{{\mathcal{T}}(a{\mathrm{e}}^{-t|{\mathcal{D}}|})}{{\mathcal{T}}({\mathrm{e}}^{-t|{\mathcal{D}}|})}.$$ Before proceeding with the proof of the lemma, we give some examples of how the operators $\gamma_1,\ldots, \gamma_N\in {\mathcal{N}}\cap {\mathcal{A}}'$ can arise. The most trivial instance is when $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}},{\mathcal{N}},{\mathcal{T}})$ is even, in which case the grading $\gamma$ will satisfy the conditions of Lemma \[gammajlem\]. Here is a more geometric example. Let $(C^\infty(M),L^2(M,S),\slashed{D})$ denote the spectral triple defined from a Dirac operator on a closed Riemannian manifold $M$ as in Proposition \[heatasumfd\] (see page ). If we take a collection $X_1,\ldots, X_N\in C^\infty(M,TM)$ of vector fields spanning the tangent bundle $TM$ in all points, the collection of Clifford multiplication operators $\gamma_j:=c_S(X_j)$, $j=1,\ldots,N$ is readily verified to satisfy the conditions of Lemma \[gammajlem\]. This construction extends to semi-finite spectral triples defined from the fibrewise Dirac operator of a Riemannian spin$^c$ submersion $\pi:M\to B$ (see more in [@kaaaaadsuij]) and a measure on $B$ by taking $X_1,\ldots, X_N$ to be vertical vector fields spanning the vertical tangent bundle $\ker\mathrm{d}\pi$ in all points of $M$. Define $C:=\|(\sum_{j=1}^N \gamma_j^2)^{-1}\|$ and recall that $F_{\mathcal{D}}:={\mathcal{D}}|{\mathcal{D}}|^{-1}$ modulo a finite trace projection. For any self-adjoint $a\in {\mathcal{A}}$, we estimate $$\begin{aligned} {\mathcal{T}}(F_{\mathcal{D}}a{\mathrm{e}}^{-t|{\mathcal{D}}|})=&\frac{1}{2}{\mathcal{T}}\left((F_{\mathcal{D}}a+aF_{\mathcal{D}}){\mathrm{e}}^{-t|{\mathcal{D}}|}\right)\leq \frac{C}{2}{\mathcal{T}}\left(\sum_{j=1}^N\gamma_j(F_{\mathcal{D}}a+aF_{\mathcal{D}}){\mathrm{e}}^{-t|{\mathcal{D}}|}\gamma_j\right)=\\ =&\frac{C}{2}{\mathcal{T}}\left(\sum_{j=1}^N\gamma_j(F_{\mathcal{D}}a+aF_{\mathcal{D}})\gamma_j{\mathrm{e}}^{-t|{\mathcal{D}}|}\right)+\\ &+\frac{C}{2}{\mathcal{T}}\left(\sum_{j=1}^N\gamma_j(F_{\mathcal{D}}a+aF_{\mathcal{D}})({\mathrm{e}}^{-t|{\mathcal{D}}|}\gamma_j{\mathrm{e}}^{t|{\mathcal{D}}|}-\gamma_j){\mathrm{e}}^{-t|{\mathcal{D}}|}\right)=\\ =&-\frac{C}{2}{\mathcal{T}}\left(\sum_{j=1}^N\gamma_j^2(F_{\mathcal{D}}a+aF_{\mathcal{D}}){\mathrm{e}}^{-t|{\mathcal{D}}|}\right)+\\ &-\frac{C}{2}{\mathcal{T}}\left(\sum_{j=1}^N\gamma_j([F_{\mathcal{D}},\gamma_j]_+ a+a[F_{\mathcal{D}},\gamma_j]_+)\gamma_j{\mathrm{e}}^{-t|{\mathcal{D}}|}\right)\\ &+\frac{C}{2}{\mathcal{T}}\left(\sum_{j=1}^N\gamma_j(F_{\mathcal{D}}a+aF_{\mathcal{D}})({\mathrm{e}}^{-t|{\mathcal{D}}|}\gamma_j{\mathrm{e}}^{t|{\mathcal{D}}|}-\gamma_j){\mathrm{e}}^{-t|{\mathcal{D}}|}\right)\leq \\ \leq &-{\mathcal{T}}\left(F_{\mathcal{D}}a{\mathrm{e}}^{-t|{\mathcal{D}}|}\right)-\frac{C}{2}{\mathcal{T}}\left(\sum_{j=1}^N\gamma_j([F_{\mathcal{D}},\gamma_j]_+ a+a[F_{\mathcal{D}},\gamma_j]_+)\gamma_j{\mathrm{e}}^{-t|{\mathcal{D}}|}\right)\\ &+\frac{C}{2}{\mathcal{T}}\left(\sum_{j=1}^N\gamma_j(F_{\mathcal{D}}a+aF_{\mathcal{D}})({\mathrm{e}}^{-t|{\mathcal{D}}|}\gamma_j{\mathrm{e}}^{t|{\mathcal{D}}|}-\gamma_j){\mathrm{e}}^{-t|{\mathcal{D}}|}\right)\end{aligned}$$ Since $[{\mathcal{D}},\gamma_j]_+$ is bounded, $[F_{\mathcal{D}},\gamma_j]_+$ is compact and an approximation argument by finite ${\mathcal{T}}$-rank operators shows that $$\tau\left(\sum_{j=1}^N\gamma_j([F_{\mathcal{D}},\gamma_j]_+ a+a[F_{\mathcal{D}},\gamma_j]_+)\gamma_j{\mathrm{e}}^{-t|{\mathcal{D}}|}\right)=o({\mathcal{T}}({\mathrm{e}}^{-t|{\mathcal{D}}|})),$$ as $t\searrow \beta$. By norm continuity of $t\mapsto {\mathrm{e}}^{-t|{\mathcal{D}}|}\gamma_j{\mathrm{e}}^{t|{\mathcal{D}}|}$ we can also deduce that $$\tau\left(\sum_{j=1}^N\gamma_j(F_{\mathcal{D}}a+aF_{\mathcal{D}})({\mathrm{e}}^{-t|{\mathcal{D}}|}\gamma_j{\mathrm{e}}^{t|{\mathcal{D}}|}-\gamma_j){\mathrm{e}}^{-t|{\mathcal{D}}|}\right)=o({\mathcal{T}}({\mathrm{e}}^{-t|{\mathcal{D}}|})),$$ as $t\searrow \beta$. In conclusion, for a self-adjoint $a$, $${\mathcal{T}}(F_{\mathcal{D}}a{\mathrm{e}}^{-t|{\mathcal{D}}|})=-{\mathcal{T}}(F_{\mathcal{D}}a{\mathrm{e}}^{-t|{\mathcal{D}}|})+o({\mathcal{T}}({\mathrm{e}}^{-t|{\mathcal{D}}|})).$$ We can conclude that $\tau(F_{\mathcal{D}}a{\mathrm{e}}^{-t|{\mathcal{D}}|})=o({\mathcal{T}}({\mathrm{e}}^{-t|{\mathcal{D}}|}))$ as $t\searrow \beta$. Since $2P_{\mathcal{D}}=F_{\mathcal{D}}+1$, we have for any $a\in {\mathcal{A}}$ that $${\mathcal{T}}(a{\mathrm{e}}^{-t|{\mathcal{D}}|})=2{\mathcal{T}}(P_{\mathcal{D}}a{\mathrm{e}}^{-t|{\mathcal{D}}|})+o({\mathcal{T}}({\mathrm{e}}^{-t|{\mathcal{D}}|})), \quad\mbox{as $t\searrow \beta$.}$$ In particular $\beta=\beta_{\mathcal{D}}$ and ${\mathcal{T}}(P_{\mathcal{D}}{\mathrm{e}}^{-t|{\mathcal{D}}|})\nearrow \infty$ as $t\searrow \beta$. We compute that $$\frac{{\mathcal{T}}(a{\mathrm{e}}^{-t|{\mathcal{D}}|})}{{\mathcal{T}}({\mathrm{e}}^{-t|{\mathcal{D}}|})}= \frac{{\mathcal{T}}(P_{\mathcal{D}}a{\mathrm{e}}^{-t|{\mathcal{D}}|})}{{\mathcal{T}}(P_{\mathcal{D}}{\mathrm{e}}^{-t|{\mathcal{D}}|})}+o(1), \quad\mbox{as $t\searrow \beta$.}$$ In particular, for any extended limit $\omega$ as $t\to\beta$, $$\lim_{t\to \omega}\frac{{\mathcal{T}}(a{\mathrm{e}}^{-t|{\mathcal{D}}|})}{{\mathcal{T}}({\mathrm{e}}^{-t|{\mathcal{D}}|})}= \lim_{t\to \omega}\frac{{\mathcal{T}}(P_{\mathcal{D}}a{\mathrm{e}}^{-t|{\mathcal{D}}|})}{{\mathcal{T}}(P_{\mathcal{D}}{\mathrm{e}}^{-t|{\mathcal{D}}|})}.$$ This concludes the proof of the lemma. Modular spectral triples and modular index theory {#modularsubsec} ------------------------------------------------- Modular spectral triples and their (equivariant) index theory were considered in [@CPR2; @CRT; @CNNR], with the definition laid out most clearly in [@RenSen Definition 2.1]. These were defined in order to study the (equivariant) index theory of KMS weights associated to periodic flows, so one might wonder how modular spectral triples fit into our scheme. Given a KMS state $\psi:B\to{\mathbb{C}}$ with inverse temperature $\beta$ for a one-parameter group $\sigma:{\mathbb{R}}\to {\rm Aut}(B)$ on a unital $C^*$-algebra and a faithful expectation onto the fixed point algebra $\Phi:\,B\to B^\sigma$, we can emulate the constructions that inspired the definition of modular spectral triples. First we construct the right $C^*$-module $X$ over $B^\sigma$ by completing $B$ in the norm coming from the inner product $$(x|y)_{B^\sigma}=\Phi(x^*y).$$ Then we can use [@LN] to construct $\operatorname{Tr}_\psi:{\mathbb{K}}_{B^\sigma}(X)\to{\mathbb{C}}$ the trace dual to $\psi|_{B^\sigma}$. The action $\sigma$ induces a one parameter unitary group on $L^2(X,\psi)$, and we let ${\mathcal{D}}$ be the generator of this one parameter group. By [@CNNR], when the action $\sigma$ is periodic, the data $$(B,L^2(X,\psi),{\mathcal{D}},{\mathbb{K}}_{B^\sigma}(X)'',\phi_{\mathcal{D}}),$$ where $\phi_{\mathcal{D}}(a) :=\operatorname{Tr}_\psi(e^{-\beta {\mathcal{D}}/2}a e^{-\beta {\mathcal{D}}/2})$, gives us a modular spectral triple. The operator ${\mathcal{D}}$ is affiliated to the semifinite algebra $ ({\mathbb{K}}(X)^{\sigma^{\phi_{\mathcal{D}}}})''$ \[prop:Li-mod\] Let $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}},{\mathcal{N}},{\mathcal{T}})$ be a semifinite spectral triple such that that for all $t>\beta$ $${\mathcal{T}}(P_{\mathcal{D}}{\mathrm{e}}^{-t|{\mathcal{D}}|})<\infty$$ and $\lim_{t\searrow \beta}{\mathcal{T}}(P_{\mathcal{D}}{\mathrm{e}}^{-t{\mathcal{D}}})=\infty$. Assume that $P_{\mathcal{D}}AP_{\mathcal{D}}\cap{\mathbb{K}}_{\mathcal{N}}=\{0\}$ and the spectral triple is $\beta$-analytic. Then we obtain the KMS state $\phi_\omega:\,A_{\mathcal{D}}\to{\mathbb{C}}$, and provided that ${\mathcal{D}}$ has discrete spectrum we obtain an expectation $\Phi:\,A_{\mathcal{D}}\to A_{\mathcal{D}}^\sigma$ onto the fixed point subalgebra. Provided that the group $\sigma$ is periodic we then obtain a finitely summable modular spectral triple $$({\mathcal{A}}_{\mathcal{D}},L^2(A_{\mathcal{D}},\phi_\omega),{\mathcal{D}},({\mathbb{K}}_{A_{\mathcal{D}}^\sigma}(A_{\mathcal{D}}))'',\phi_{\mathcal{D}}),$$ where the operator ${\mathcal{D}}$ generates the one-parameter group induced by $\sigma$ on $L^2(A_{\mathcal{D}},\phi_\omega)$ and $\phi_{\mathcal{D}}:=\operatorname{Tr}_{\phi_\omega}(e^{-\beta {\mathcal{D}}/2}\cdot e^{-\beta {\mathcal{D}}/2})$. The spectral dimension is 1. The existence of the KMS state $\phi_\omega$ comes from Corollary \[cor:phi-omega\]. In general the action $\sigma$ is real, but assuming that the operator ${\mathcal{D}}$ has discrete spectrum, the action will factor through a (compact) torus. To see this, one takes a rational basis of the eigenvalues (possibly an infinite basis), and takes a product over the circles corresponding to these individual actions. Consequently, by averaging over this torus, there is an expectation $\Phi:\,{\mathcal{N}}\to {\mathcal{N}}^{\sigma}$ onto the fixed point algebra for $t\mapsto (T\mapsto {\mathrm{e}}^{it{\mathcal{D}}}T{\mathrm{e}}^{-it{\mathcal{D}}})$. Of course ${\mathcal{D}}$ is affiliated to the fixed-point algebra. Finally if the action $\sigma$ is periodic then [@CNNR] proves that we have a modular spectral triple, and that $\phi_{\mathcal{D}}((1+{\mathcal{D}}^2)^{-s/2})<\infty$ for $s>1$. For circle actions there is a local index formula in twisted cyclic theory, but for real actions factoring through a torus there is not. One serious issue that comes up in this more general setting is the compactness of the resolvent of ${\mathcal{D}}$, and determining summability. We leave this issue to another place. The KMS-state $\phi_\omega$ and Dixmier traces {#kmsanddix} ============================================== In the present section we discuss a relation between the trace $\phi_\omega$ from Theorem \[thm:voics\] and Dixmier traces. For a decreasing function $\psi: [0,\infty) \to (0,\infty)$ we denote $\Psi(t):=\int_0^t \psi(s) ds.$ Let $\mathcal{L}_\psi({\mathcal{T}})$ be the principal ideal defined as in Definition \[derjugendvonheutebrauchidealen\] (see page ). Let $E_T$ be the spectral projection of an operator $T$ affiliated with ${\mathcal{N}}$ and let $n_{\mathcal{T}}(s,T):={\mathcal{T}}(E_{|T|}(s,\infty))$ be its distribution function. The following result extends [@LSZ Lemma 12.6.3]. \[HTas\] Let $\psi$ be a regularly varying function of index $-1$. Let $T\in \mathcal{L}_\psi({\mathcal{T}})$ be strictly positive and $\mu_{\mathcal{T}}(t,T) \sim \psi(t)$, $t\to \infty$. For every $q>0$ we have $${\mathcal{T}}({\mathrm{e}}^{-{T^{-q}}/t}) \sim \Gamma(1+\frac{1}{q}) \psi^{-1}(t^{-\frac{1}{q}}), \ t\to \infty.$$ The assumption $\mu_{\mathcal{T}}(t,T) \sim \psi(t)$ implies $\mu_{\mathcal{T}}(t,T^{q}) \sim [\psi(t)]^{q}$, $t\to \infty$. Since the distribution function is an inverse of the singular values function, it follows that $n_{\mathcal{T}}(s, {T^{q}})\sim \psi^{-1}(s^{\frac{1}{q}}),$ $s\to0+$. Next we have ${\mathcal{T}}(E_{T^{-q}}(t)) \sim n_{\mathcal{T}}(1/t, {T^{q}})$, $t\to \infty$. Thus, $${\mathcal{T}}(E_{T^{-q}}(t)) \sim \psi^{-1}(t^{-\frac{1}{q}}), \ t\to \infty.$$ Since $\psi$ is varying regularly with index $-1$, [@RegVar Theorem 1.5.12] implies that $\psi^{-1}$ varies regularly with index $-1$, too. Thus, ${\mathcal{T}}(E_{T^{-q}})$ varies regularly with index $\frac{1}{q}$. Writing the heat trace as a Laplace transform $${\mathcal{T}}({\mathrm{e}}^{-{T^{-q}}/t}) = \int_0^\infty {\mathrm{e}}^{-{z}/{t}}\,\mathrm{d} {\mathcal{T}}(E_{T^{-q}}(z))$$ and using the Karamata theorem [@Korevaar Chapter IV, Theorem 8.1] we obtain $${\mathcal{T}}({\mathrm{e}}^{-{T^{-q}}/t}) \sim \Gamma(1+\frac{1}{q})\psi^{-1}(t^{-\frac{1}{q}}), \ t\to \infty.\qedhere$$ Let $P_a : L^\infty(0,\infty) \to L^\infty(0,\infty)$ be the exponentiation operator defined by $(P_a f)(t) := f(t^a), \ t>0.$ \[dfn:invariance\][@CPS2] An extended limit $\omega$ as $t\to\infty$ on $L^\infty(0,\infty)$ is said to be exponentiation invariant if $$\lim_{t\to\omega} (P_a f)(t) = \lim_{t\to\omega} f(t)$$ for every $f\in L^\infty(0,\infty)$ and every $a>0$. \[Dixmier traces\] Let $\psi: [0,\infty) \to (0,\infty)$ be a regularly varying function of index $-1$. For any extended limit $\omega$ as $t\to\infty$ on $L^\infty(0,\infty)$ a linear extension of the weight $${\mathcal{T}}_{\omega, \psi}(T):= \lim_{t\to\omega} \frac{1}{\Psi(t)}\int_0^t \mu_{\mathcal{T}}(s,T)\,\mathrm{d}s, \quad 0\le T\in \mathcal L_\psi({\mathcal{T}})$$ is said to be a Dixmier trace on $\mathcal L_\psi({\mathcal{T}})$. Usually Dixmier traces are defined on Lorentz ideals corresponding to the function $\Psi$ (which are strictly larger than $\mathcal L_\psi({\mathcal{T}})$) by exactly the same formula as in Defninition \[Dixmier traces\] (see e.g. [@Dixmier; @BRB; @LSZ]). Then, Dixmier traces on $\mathcal L_\psi({\mathcal{T}})$ are restrictions of those on Lorentz ideal to $\mathcal L_\psi({\mathcal{T}})$. Since we do not deal with Lorentz ideals here, it is convenient to define Dixmier traces directly on $\mathcal L_\psi({\mathcal{T}})$. Also, it should be pointed out that on Lorentz ideals to define Dixmier traces one needs an additional assumption on $\omega$: either dilation invariance [@Dixmier; @LSZ] or exponentiation invariance [@GaSu; @SUZ3]. As it was shown in [@Sed_Suk Theorem 17] these requirements are redundant on $\mathcal L_\psi({\mathcal{T}})$. The proof of the following theorem is the same as that of [@LSZ Theorem 8.5.1] and thus omitted. Note however, that in [@LSZ] the result was proved for Lorentz ideals and required dilation invariance of the extended limit $\omega$. For the case of $\mathcal L_\psi({\mathcal{T}})$ one can refer to [@Sed_Suk Lemma 15] to remove this assumption. \[zeta\] Let $f\in C^2[0,\infty)$ be a bounded function such that $f(0)=f'(0)=0$. Let $T\in \mathcal{L}_\psi({\mathcal{T}})$ be positive and let $B\in \mathcal N$. For every extended limit $\omega$ as $t\to\infty$ on $L^\infty(0,\infty)$ we have $$\lim_{t\to\omega}\left(\frac1{\Psi(t)} \int_1^t {\mathcal{T}}(f(sT)B) \frac{\mathrm{d}s}{s^2}\right) = \int_0^\infty f(s) \frac{\mathrm{d}s}{s^2} \cdot \lim_{t\to\omega}\left(\frac1{\Psi(t)} \int_1^t {\mathcal{T}}({\mathrm{e}}^{-(sTB)^{-1}}) \frac{\mathrm{d}s}{s^2}\right).$$ Below we will need the relation between generalised heat kernels and Dixmier traces on $\mathcal{L}_\psi({\mathcal{T}})$, which was proved in [@GaSu] under the additional assumption that $$\label{exp1} A_\Psi(\alpha):=\lim_{t\to\infty} \frac{\Psi(t^\alpha)}{\Psi(t)}\quad\mbox{exists for every $\alpha> 0$.}$$ Recall the notation $\Psi(t)=\int_0^t\psi(s) ds$. The corresponding (natural) assumption on $\psi$ is that $$\label{exp2} \alpha \cdot \lim_{t\to\infty} \frac{\psi(t^\alpha) t^{\alpha-1}}{\psi(t)} \quad\mbox{exists for every $\alpha> 0$.}$$ The number appearing in Equation equals $A_\Psi(\alpha)$. Note that condition  implies the condition  and regular variation of $\psi$ with index $-1$ [@GaSu Proposition 1.7]. Let $H: L^\infty(0,\infty) \to L^\infty(0,\infty)$ be the Cesaro mean defined as follows: $$(Hf)(t) := \frac1t \int_0^t f(s) \,\mathrm{d}s, \ t>0.$$ Let $M_\Psi: L^\infty(0,\infty) \to L^\infty(0,\infty)$ be the Cesaro mean twisted by $\Psi$, that is $$\begin{aligned} (M_\Psi f)(t):&= [(H (f\circ \Psi^{-1}))\circ \Psi](t) =\frac1{\Psi(t)} \int_0^t f(s) \psi(s) \,\mathrm{d}s, t>0.\end{aligned}$$ \[MP\] If $\psi$ satisfies condition , then $$M_\Psi \circ P_a - P_a \circ M_\Psi : L^\infty(0,\infty) \to C_0(0,\infty)$$ for every $a> 0.$ For $f\in L^\infty(0,\infty)$ we have $$(M_\Psi \circ P_a f)(t)= \frac1{\Psi(t)} \int_0^t f(s^a) \psi(s) \,\mathrm{d}s= \frac1{\Psi(t)} \int_0^{t^a} f(s) \psi(s^{1/a}) \frac{s^{1/a-1}}a \,\mathrm{d}s.$$ Using conditions  and  we obtain $$(M_\Psi \circ P_a f)(t)\in A_\Psi(a)A_\Psi(\frac1a) \frac1{\Psi(t^a)} \int_0^{t^a} f(s) \psi(s) \,\mathrm{d}s + C_0.$$ Direct calculations show that $A_\Psi(a)A_\Psi(\frac1a)=1$. Thus, $$(M_\Psi \circ P_a - P_a \circ M_\Psi )f \in C_0.\qedhere$$ \[ologdlemma\] If $\psi:[0,\infty)\to (0,\infty)$ is a decreasing function with regular variation of index $-1$. Then for any $d>0$, as $t\to \infty$, $$\psi(t)=o((\log(t))^{-d}).$$ Since $\psi$ is decreasing and $\lim_{t\to\infty} \frac{\psi(2t)}{\psi(t)}=2^{-1}$, we can for any $\epsilon\in (0,1)$ find a constant $C_\epsilon>0$ such that for $t\in (2^k,2^{k+1}]$, $$\psi(t)\leq C_\epsilon (2-\epsilon)^{-k}.$$ For details, see [@GU Proposition 2.21]. Therefore, $\psi(t)=o(t^{-1/2})$ and the lemma follows. For the next result we need to assume the (stronger) condition, that the function $\psi$ satisfies $$\label{invas} \lim_{t\to\infty} \frac{t^2 \psi(t)}{\psi^{-1}(1/t)}=c,$$ for some constant $c>0$. For every $k\in{\mathbb{Z}}$, the functions $\psi(t)= \frac{\log^k t}{t}$ and $\psi(t)= \frac{\log^k (\log t)}{t \cdot \log t}$ satisfy condition . The functions $\psi=\Psi'$ with $\Psi(t)= {\mathrm{e}}^{\log^\beta t}$ do not satisfy  for any $\beta>0$. In the following result we use a singular values function of an unbounded operator affiliated with ${\mathcal{N}}$. For such operators the formula  cannot be used. The singular values function of an operator $T$ affiliated with ${\mathcal{N}}$ is defined [@FK] as follows: $$\mu_{\mathcal{T}}(t,T):=\inf\{s\ge0 : n_{\mathcal{T}}(s,T)\le t\}.$$ \[Frohlich\] Let $d>0$ and $\psi:[0,\infty)\to (0,\infty)$ be a decreasing function satisfying conditions  and . Assume that $({\mathcal{A}}, {\mathcal{H}},{\mathcal{D}},{\mathcal{N}},{\mathcal{T}})$ is a unital semifinite spectral triple such that 1. ${\mathcal{T}}$ is infinite; 2. ${\mathcal{D}}$ is positive; 3. $\mu_{\mathcal{T}}(t,{\mathcal{D}})\sim \psi(t)^{-1/d}$. Then $({\mathcal{A}}, {\mathcal{H}},{\mathcal{D}},{\mathcal{N}},{\mathcal{T}})$ is $\mathrm{Li}_{(0),1}$-summable with positive ${\mathcal{T}}$-essential spectrum. Moreover, for every $a\in A_{\mathcal{D}}$ and every exponentiation invariant extended limit $\omega$ as $t\to\infty$ we have $$\phi_{\tilde{\omega}}(a) = {\mathcal{T}}_{\omega, \psi}(a (1+{\mathcal{D}}^2)^{-d/2}),$$ where $\tilde{\omega}:=\omega\circ (JM_\Psi J)$ and $J:L^\infty(0,\infty)\to L^\infty(0,\infty)$ is defined as the pullback along $t\mapsto t^{-1}$. Condition on $\psi$ implies that $\psi$ has regular variation of index $-1$ so $\psi(t)=o((\log(t))^{-d})$ by Lemma \[ologdlemma\]. Therefore, $\mu_{\mathcal{T}}(t,{\mathcal{D}})=o(\log(t))$ and the semifinite spectral triple $({\mathcal{A}}, {\mathcal{H}},{\mathcal{D}},{\mathcal{N}},{\mathcal{T}})$ is $\mathrm{Li}_{(0),1}$-summable. Since ${\mathcal{T}}$ is infinite, ${\mathcal{T}}({\mathrm{e}}^{-t{\mathcal{D}}})\nearrow \infty$ as $t\searrow 0$. The operator ${\mathcal{D}}$ therefore has positive ${\mathcal{T}}$-essential spectrum. Thus, $$\phi_{\tilde{\omega}}(a)= \lim_{t\to\tilde{\omega}}\frac{{\mathcal{T}}(a {\mathrm{e}}^{-t{\mathcal{D}}})}{{\mathcal{T}}({\mathrm{e}}^{-t{\mathcal{D}}})}$$ is a trace on ${\mathcal{A}}$ by Theorem \[thm:voics\]. The assumption on ${\mathcal{D}}$ implies that $\mu_{\mathcal{T}}(t, (1+{\mathcal{D}}^2)^{-d/2}) \sim \psi(t)$, $t\to\infty$. Applying Lemma \[HTas\] with $T=(1+{\mathcal{D}}^2)^{-d/2}$ and $q=1/d$ yields $${\mathcal{T}}({\mathrm{e}}^{-{\mathcal{D}}/t}) \sim \Gamma(1+d) \psi^{-1}(t^{-d}), \ t\to \infty.$$ Using the properties of extended limits, the definitions of $P_a$ and $J$ and Lemma \[MP\], we obtain $$\begin{aligned} \phi_{\tilde{\omega}}(a) &= \frac1{\Gamma(1+d)}\lim_{t\to\omega}(J\circ M_\Psi)\left(\frac{{\mathcal{T}}(a {\mathrm{e}}^{-{\mathcal{D}}/t})}{\psi^{-1}(t^{-d})} \right)\nonumber\\ &= \frac1{\Gamma(1+d)}\lim_{t\to\omega}(J\circ M_\Psi \circ P_{d})\left(\frac{{\mathcal{T}}(a {\mathrm{e}}^{-(t {\mathcal{D}}^{-d})^{-1/d}})}{\psi^{-1}(1/t)} \right)\nonumber\\ &= \frac1{\Gamma(1+d)} \lim_{t\to\omega}(J\circ P_{d} \circ M_\Psi )\left(\frac{{\mathcal{T}}(a {\mathrm{e}}^{-(t {\mathcal{D}}^{-d})^{-1/d}})}{\psi^{-1}(1/t)} \right). \label{eq10}\end{aligned}$$ Using the definition of $M_\Psi$ and assumption  we obtain $$\begin{aligned} M_\Psi \left(\frac{{\mathcal{T}}(a {\mathrm{e}}^{-(t {\mathcal{D}}^{-d})^{-1/d}})}{\psi^{-1}(1/t)} \right)&= \frac1{\Psi(t)} \int_0^t \frac{{\mathcal{T}}(a {\mathrm{e}}^{-(s {\mathcal{D}}^{-d})^{-1/d}})}{\psi^{-1}(1/s)} \psi(s) \,\mathrm{d}s\nonumber\\ &\in \frac1{\Psi(t)} \int_0^t {\mathcal{T}}(a {\mathrm{e}}^{-(s {\mathcal{D}}^{-d})^{-1/d}}) \frac{\mathrm{d}s}{s^2} + C_0(0,\infty). \label{eq11}\end{aligned}$$ Since $\omega$ is exponentiation invariant extended limit, combining  and  we obtain $$\phi_{\tilde{\omega}}(a) =\frac1{\Gamma(1+d)} \cdot \lim_{1/t\to\omega} \left(\frac1{\Psi(t)} \int_1^t {\mathcal{T}}(a {\mathrm{e}}^{-(s {\mathcal{D}}^{-d})^{-1/d}}) \frac{\mathrm{d}s}{s^2} \right).$$ Now we apply Lemma \[zeta\] twice with $T={\mathcal{D}}^{-d}$, $f(x)={\mathrm{e}}^{-x^{-1/d}}$ and then with $f(x)={\mathrm{e}}^{-x^{-1}}$. Since $$\int_0^\infty {\mathrm{e}}^{-x^{-q}}\frac{\mathrm{d}x}{x}=\Gamma(1+\frac{1}{q}),$$ we obtain $$\begin{aligned} \phi_{\tilde{\omega}}(a)&= \lim_{1/t\to\omega} J\left(\frac1{\Psi(t)} \int_1^t {\mathcal{T}}(a{\mathrm{e}}^{-(s{\mathcal{D}}^{-d})^{-1}}) \frac{\mathrm{d}s}{s^2}\right)={\mathcal{T}}_{\omega, \psi}\left(a (1+{\mathcal{D}}^2)^{-d/2}\right),\end{aligned}$$ where the last equality follows from [@GaSu Theorem 4.7]. We can now provide sufficient conditions on a general $\mathrm{Li}_{(0),1}$-summable unital semifinite spectral triples ensuring a relation between the trace $\phi_\omega$ of Theorem \[thm:voics\] and Dixmier traces. \[dixmiercorforphiom\] Let $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}},{\mathcal{N}},{\mathcal{T}})$ be a unital semifinite spectral triple with ${\mathcal{T}}(P_{\mathcal{D}})=\infty$. Assume that there is a number $d>0$ and a decreasing function $\psi:[0,\infty)\to (0,\infty)$ with regular variation of index $-1$ satisfying conditions  and  and $$\mu(t,P_{\mathcal{D}}{\mathcal{D}})\sim \psi(t)^{-1/d}.$$ Then, $\beta_{\mathcal{D}}=0$ and for any exponentiation invariant extended limit $\omega$ as $t\to\infty$ and $a\in A_{\mathcal{D}}$, $$\phi_{\tilde{\omega}}(a) = {\mathcal{T}}_{\omega, \psi}(P_{\mathcal{D}}a (1+{\mathcal{D}}^2)^{-d/2}),$$ where $\tilde{\omega}$ is as in Theorem \[Frohlich\]. The corollary follows immediately from Theorem \[Frohlich\] applied to the unital semifinite spectral triple $(\mathfrak{T},P_{\mathcal{D}}{\mathcal{H}},P_{\mathcal{D}}{\mathcal{D}},{\mathcal{N}}^+,{\mathcal{T}})$ where $\mathfrak{T}$ is the $*$-algebra generated by $P_{\mathcal{D}}{\mathcal{A}}P_{\mathcal{D}}$. Let us revisit the spectral triple constructed in Proposition \[heatasumfdwithapsi\] (see page ). If $\psi:[0,\infty)\to (0,\infty)$ is a smoothly varying function with $\lim_{t\to 0}\psi(t)=0$ and $\psi(t)^{-1}=O(t^{1/n})$ as $t\to \infty$, and $\slashed{D}$ a Dirac operator on a Riemannian closed $n$-dimensional manifold $M$, a $\psi$-summable spectral triple $(C^\infty(M),L^2(M,S),\slashed{D}_\psi)$ was constructed in Proposition \[heatasumfdwithapsi\], where $\slashed{D}_\psi:=F_{\slashed{D}} \psi(|\slashed{D}|^n)^{-1}$. The Weyl law for $|\slashed{D}|$ and Theorem \[heatvszeta\] applied to $B=P_{\slashed{D}}$ shows that the order of the spectral asymptotics of $|\slashed{D}|$ coincides with the order of the spectral asymptotics of $P_\slashed{D}\slashed{D}$ so $\mu(t,P_{\slashed{D}}\slashed{D}_\psi)\sim c\psi(t)^{-1}$ for some constant $c>0$. If $\psi$ has regular variation of index $-d$ for a $d>0$, Corollary \[dixmiercorforphiom\] shows that the tracial state on $C(M)$ defined from the spectral triple $(C^\infty(M),L^2(M,S),\slashed{D}_\psi)$ takes the form $$\phi_\omega(a)= c'\operatorname{Tr}_{\omega, \psi}(P_\slashed{D} a \psi(|\slashed{D}|^n)^{1/d}),$$ for some proportionality constant $c'$. Applying Connes’ trace theorem as in [@GU Theorem 9.1], it follows that $$\phi_\omega(a)=\displaystyle\stackinset{c}{}{c}{}{-\mkern4mu}{\displaystyle\int_M} a\mathrm{d}V,$$ where $\mathrm{d}V$ denotes the Riemannian volume measure on $M$ and $\displaystyle\stackinset{c}{}{c}{}{-\mkern4mu}{\displaystyle\int_M}$ the normalized integral. The computation above requires $\psi$ to have strictly negative index of regular variation. We note that by Proposition \[deathorgladiolis\] (see page ), the computation above can only extend to the spectral triple from Proposition \[psidiracprop\] (see page ) on a crossed product by a local diffeomorphism if the local diffeomorphism acts isometrically. The KMS-state $\phi_\omega$ in examples {#kmsinexamplesec} ======================================= We are now in a state where we can compute the KMS-states associated with the spectral triples considered in Subsection \[subsec:examples\] (see page ). The computations for the KMS-states associated with the unbounded Kasparov cycle on Cuntz-Pimsner algebras considered in Subsection \[cpalgexam\] (see page ) are more involved and dedicated a separate section, Section \[diraccpkms\] (see page ). Dirac operators on closed manifolds {#diracmfdkms} ----------------------------------- For a closed manifold $M$ with a Dirac operator $\slashed{D}$ acting on a Clifford bundle $S\to M$, we consider the spectral triple $(C^\infty(M),L^2(M,S),\slashed{D})$ as in Proposition \[heatasumfd\] (see page ). The following theorem can be deduced immediately from Example \[diracmfdhea\] (see page ) or from Corollary \[dixmiercorforphiom\] and Connes’ trace theorem for pseudo-differential operators. \[simpleacse\] Let $(C^\infty(M),L^2(M,S),\slashed{D})$ be the spectral triple associated with a Dirac operator on a closed manifold, $\omega$ an extended limit as $t\to0$ and $\phi_\omega$ the associated tracial state from Theorem \[thm:voics\] (see page ). The trace $\phi_\omega$ is the normalized volume integral on $M$, i.e. for $a\in C(M)$, $$\phi_\omega(a)=\displaystyle\stackinset{c}{}{c}{}{-\mkern4mu}{\displaystyle\int_M} a\,\mathrm{d} V.$$ For completeness, let us describe the Toeplitz algebras $T_{C(M)}$, $T_{C(M),\slashed{D}}$ and the flow $\sigma$ on $C(M)_{\slashed{D}}$ in this example. We remark that since $\phi_\omega$ is a trace in this case, the flow induced from our construction is irrelevant for the study of $\phi_\omega$ but it could nevertheless serve to clarify the constructions of Subsection \[subsec:toplitz\]. The relevant algebras are all contained in the $C^*$-algebra $\Psi^0_{C^*}(M,S)$ defined as the $C^*$-closure of the $*$-algebra $\Psi^0_{\rm cl}(M,S)$ of zeroth order classical pseudo-differential operators acting on $L^2(M,S)$. It is well known that $\Psi^0_{C^*}(M,S)$ fits into a short exact sequence $$0\to {\mathbb{K}}(L^2(M,S))\to \Psi^0_{C^*}(M,S)\xrightarrow{{\rm symb}}C(S^*M,\operatorname{End}(S))\to 0,$$ where ${\rm symb}$ denotes the continuous extension of the principal symbol mapping $\Psi^0_{\rm cl}(M,S)\to C^\infty(S^*M,\operatorname{End}(S))$. Since $P_{\slashed{D}}$ is a projection in $\Psi^0_{\rm cl}(M,S)$, we can consider the $C^*$-algebra $\Psi^0_{C^*,+}(M,S):=P_{\slashed{D}}\Psi^0_{C^*}(M,S)P_{\slashed{D}}$ and we obtain a short exact sequence $$0\to {\mathbb{K}}(P_{\slashed{D}}L^2(M,S))\to \Psi^0_{C^*,+}(M,S)\xrightarrow{{\rm symb}}C(S^*M,\operatorname{End}(p_{\slashed{D}}S))\to 0,$$ where $p_{\slashed{D}}\in C^\infty(S^*M,\operatorname{End}(S))$ denotes the principal symbol of $P_{\slashed{D}}$. The algebras $T_{C(M)}$ and $T_{C(M),\slashed{D}}$ are characterized by the following commuting diagram with exact rows $$\xymatrix{ 0\ar[r]& {\mathbb{K}}(P_{\slashed{D}}L^2(M,S))\ar[r]\ar[d]^{=}&T_{C(M)}\ar[rr]^{{\rm symb}}\ar@{^{(}->}[d]&&C(M)\ar@{^{(}->}[d]\ar[r]&0\\ 0\ar[r]& {\mathbb{K}}(P_{\slashed{D}}L^2(M,S))\ar[r]\ar[d]^{=}&T_{C(M),\slashed{D}}\ar[rr]^{{\rm symb}}\ar@{^{(}->}[d]&&C(M)_{\slashed{D}}\ar@{^{(}->}[d]\ar[r]&0\\ 0\ar[r]&{\mathbb{K}}(P_{\slashed{D}}L^2(M,S))\ar[r]&\Psi^0_{C^*,+}(M,S)\ar[rr]^{{\rm symb}}&&C(S^*M,\operatorname{End}(p_{\slashed{D}}S))\ar[r]&0, }$$ The composition of the mappings in the right most column coincides with the pull back homomorphism $C(M)\to C(S^*M)$ composed with the inclusion $C(S^*M)\subseteq C(S^*M,\operatorname{End}(p_{\slashed{D}}S))$. To describe the flow $\sigma$ on $C(M)_{\slashed{D}}$, we describe it on $C^\infty(S^*M,\operatorname{End}(p_{\slashed{D}}S))$. Surjectivity of the principal symbol mapping implies that any $a\in C^\infty(S^*M,\operatorname{End}(p_{\slashed{D}}S))$ is the symbol of an operator $A\in P_{\slashed{D}}\Psi^0_{\rm cl}(M,S)P_{\slashed{D}}$. By Egorov’s theorem [@egorovref] (see also [@horacta Section IV] and [@duissing]), ${\mathrm{e}}^{is\slashed{D}}A{\mathrm{e}}^{-is\slashed{D}}$ is again an element of $P_{\slashed{D}}\Psi^0_{\rm cl}(M,S)P_{\slashed{D}}$ and the expression $\sigma_s(a):={\rm symb}({\mathrm{e}}^{is\slashed{D}}A{\mathrm{e}}^{-is\slashed{D}})$ gives a well defined flow on $C^\infty(S^*M,\operatorname{End}(p_{\slashed{D}}S))$. Again by Egorov’s theorem, using that $\sigma_s(a)={\rm symb}({\mathrm{e}}^{is|\slashed{D}|}A{\mathrm{e}}^{-is|\slashed{D}|})$ for $A\in P_{\slashed{D}}\Psi^0_{\rm cl}(M,S)P_{\slashed{D}}$, we have that $\sigma_s(a)=g_s^*(a)$ where $g_s:S^*M\to S^*M$ is the Hamiltonian flow associated with the symbol $|\xi|$ of $|\slashed{D}|$. On the cosphere bundle, this Hamiltonian flow coincides with the geodesic flow. We conclude that the flow $\sigma$ is induced by geodesic flow and $C(M)_{\slashed{D}}\subseteq C(S^*M)$ is a closed subalgebra invariant under geodesic flow. This discussion should be compared with that in [@CSG]. As in Subsection \[diracmfdfirst\], we consider a local diffeomorphism $g:M\to M$. Assuming that $g$ acts conformally and lifts to $S\to M$, it is readily verified that $g$ is compatible with the decreasing function $\psi(t):=\frac{1}{\log(1+t^{2/n})}$. We use the notation $\slashed{D}_{\rm log}:=\slashed{D}_\psi$ for this particular choice of $\psi$. Note that $$\slashed{D}_{\rm log}=F_\slashed{D}\log(1+\slashed{D}^2) \quad\mbox{and}\quad {\mathrm{e}}^{-t|\slashed{D}_{\rm log}|}=(1+\slashed{D}^2)^{-t}.$$ By Proposition \[psidiracprop\], we arrive at a spectral triple $({\mathcal{A}},L^2(M,S),\slashed{D}_{\rm log})$ where ${\mathcal{A}}$ is the $*$-algebra generated by $C^\infty(M)$ and an isometry $V_g$. Let us compute KMS-state constructed from $({\mathcal{A}},L^2(M,S),\slashed{D}_{\rm log})$. Before diverting into this computation, we recall that the $C^*$-closure of ${\mathcal{A}}$ coincides with the image of a representation of $O_{E_g}$. As such, we can write elements of ${\mathcal{A}}$ as linear span of elements of the form $S_\mu S_\nu^*$ where $\mu=a_1V_g\cdots a_k V_g$ and $\nu=b_1V_g\cdots b_l V_g$, where $a_1,\ldots, a_k,b_1,\ldots, b_l\in C^\infty(M)$. Set $S:=C^\infty(M)\cup C^\infty(M)V_g\subseteq {\mathcal{A}}$. For any $\beta\in {\mathbb{R}}$, the set $S$ is an analytically generating set at $\beta$ for $({\mathcal{A}},L^2(M,S),\slashed{D}_{\rm log})$. For notational convenience, write ${\mathcal{D}}=\slashed{D}_{\rm log}$. The set $S$ generates ${\mathcal{A}}$, so $P_{\mathcal{D}}SP_{\mathcal{D}}+{\mathbb{K}}$ generates $T_A$. For any $\beta\in {\mathbb{R}}$, and $a\in S$ $${\mathrm{e}}^{-\beta{\mathcal{D}}}P_{\mathcal{D}}aP_{\mathcal{D}}{\mathrm{e}}^{\beta{\mathcal{D}}} =(1+\slashed{D}^2)^{-\beta}P_\slashed{D} aP_\slashed{D}(1+\slashed{D}^2)^{\beta}.$$ The proposition follows from that $\operatorname{Dom}((1+\slashed{D}^2)^\beta)=H^{2\beta}(M,S)$ as Banach spaces and any $a\in S$ extends to a continuous operator on the Sobolev spaces $H^{2\beta}(M,S)$ for all $\beta$. \[kmsforconfdiff\] Let $M$ be a connected $n$-dimensional Riemannian manifold, $\slashed{D}$ a Dirac operator on $S\to M$, and $g:M\to M$ a local diffeomorphism acting conformally and lifting to $S$. Then the spectral triple $({\mathcal{A}},L^2(M,S),\slashed{D}_{\rm log})$ is $\mathrm{Li}_1$-summable, has positive essential spectrum with $\beta_{\mathcal{D}}=n/2$ and is $n/2$-analytic. Assume for all $m\in {\mathbb{N}}_+$, that the set of fixed points $$\{x\in M: g^m(x)=x\},$$ has measure zero with respect to the volume measure. Then the state $\phi_\omega$ on $A$ constructed from Corollary \[cor:phi-omega\] (see page ) is independent of $\omega$ and takes the form $$\label{formforphiomforbg} \phi_\omega(S_\mu S_\nu^*) =\delta_{|\mu|,|\nu|}\displaystyle\stackinset{c}{}{c}{}{-\mkern4mu}{\displaystyle\int_M}\mathfrak{L}_g(c_g^{n/2}b_k^*\mathfrak{L}_g(c_g^{n/2}b_{k-1}^*\mathfrak{L}_g(\cdots \mathfrak{L}_g(c_g^{n/2}b_1^*a_1)a_2)\cdots a_{k-1})a_k) \,\mathrm{d}V,$$ For $\mu=a_1V_g\cdots a_k V_g$ and $\nu=b_1V_g\cdots b_l V_g$. Here $\mathrm{d}V$ denotes the Riemannian volume form. The state $\phi_\omega$ viewed as a state on $O_{E_g}$ via its representation on $L^2(M,S)$ is KMS for the action defined by $\gamma_t(aV_g):=c_g^{itn/2}aV_g$ with inverse temperature $1$. We note that the state Theorem \[kmsforconfdiff\] is a KMS-state on a Cuntz-Pimsner algebra, but not for its gauge action. If $c_g(x)<1$ for some $x\in M$, the generator of $\gamma$ is not positive on $E_g$ and the Laca-Neshveyev correspondence [@LN] does not apply in the context of $O_{E_g}$ with the action $\gamma$. In the case $k=l=0$, the formula should be interpreted as $\phi_\omega(a)=\displaystyle\stackinset{c}{}{c}{}{-\mkern4mu}{\displaystyle\int_M} a\,\mathrm{d}V$ for $a\in C^\infty(M)$. This special case follows from Connes’ trace theorem. For $t>n/2$, standard techniques of pseudo-differential operators show that the integral kernel $K_t$ of the trace class operator $P_\slashed{D} (1+\slashed{D}^2)^{-t}$ belongs to $C(M\times M,\mathrm{END}(S))\cap C^\infty(M\times M\setminus \Delta_M,\mathrm{END}(S))$. Here $\Delta_M\subseteq M\times M$ denotes the diagonal and $\mathrm{END}(S)$ denotes the big endomorphism bundle defined by $\mathrm{END}(S)_{(x,y)}:=\mathrm{Hom}(S_x,S_y)$ for $(x,y)\in M\times M$. By [@DGMW Proposition 8.3], $V_g^*$ takes the form $$V_g^*=N^{1/2}\mathfrak{L}_{S,g}c_g^{-n/4},$$ where $\mathfrak{L}_{S,g}$ is the $L^2$-extension of the operator $$\mathfrak{L}_{S,g}:C(M,S)\to C(M,S), \quad \mathfrak{L}_{S,g}\xi(x):=\sum_{g(y)=x} u_{g}(y)^{-1}\xi(y).$$ Take $a_1,\ldots, a_k,b_1,\ldots, b_l\in C^\infty(M)$ and write $\mu=a_1V_g\cdots a_k V_g$ and $\nu=b_1V_g\cdots b_l V_g$. We introduce the notation $\tilde{a}_j:=a_jc_g^{n/4}$ and $\tilde{b}_j=b_j c_g^{-n/4}$. We can compute for $t>n/2$ that $$\begin{aligned} \operatorname{Tr}&_{L^2(M,S)}(P_\slashed{D}S_\mu S_\nu^*{\mathrm{e}}^{-t|\slashed{D}_{\rm log}|})=\operatorname{Tr}_{L^2(M,S)}(a_1V_g\cdots a_k V_gV_g^*b_l^*V_g^*\cdots V_g^*b_1^*P_\slashed{D} (1+\slashed{D}^2)^{-t})\\ &{}\\ &=N^{-(k-l)/2}\operatorname{Tr}_{L^2(M,S)}(a_1c_g^{n/4}u_g g^*\cdots a_kc_g^{n/4}u_g g^*\mathfrak{L}_{S,g}c_g^{-n/4}b_l^*\cdots c_g^{-n/4}\mathfrak{L}_{S,g} b_1^*P_\slashed{D} (1+\slashed{D}^2)^{-t})\\ &{}\\ &=N^{-(k-l)/2}\int_{M}\sum_{(x_1,\ldots,x_{k+l})\in M(x,k,l)}\left(\prod_{j=1}^{k} \tilde{a}_j(x_j)\right)\left(\prod_{j=k+1}^{k+l} \tilde{b}_{j-k}(x_j)^*\right)K_t(x_{k+1},x_1) \mathrm{d}V(x),\end{aligned}$$ where $M(x,k,l)\subseteq M^{k+l}$ is defined as the $k+l$-tuples $(x_1,\ldots,x_{k+l})$ such that for $j=1,\ldots, k$, $x_j=g^{j-1}(x)$, $x_k=g(x_{k+l})$ and for $j=k+1,\ldots k+l-1$, $g(x_j)=x_{j+1}$. Note that $M(x,k,l)$ is finite for all $x$, because $g$ is a local homeomorphism, and that $x_1=x$ for $(x_1,\ldots, x_{k+l})\in M(x,k,l)$. Define $M^0(x,k,l)\subseteq M(x,k,l)$ as the $k+l$-tuples $(x_1,\ldots,x_{k+l})$ where $x_{k+1}=x_1$. The set $M^0(x,k,l)$ can be characterized as the $k+l$-tuples $(x_1,\ldots,x_{k+l})$ with $x=x_1=x_{k+1}$ and $x_k=g^l(x_{k+l})$ such that for $j=1,\ldots, k$, $x_j=g^{j-1}(x)$, and for $j=k+1,\ldots k+l-1$, $x_{j+1}=g(x_j)$. In particular, if $M^0(x,k,l)$ is non-empty then $g^k(x)=g^l(x)$. In other words, $(x_1,\ldots,x_{k+l})\in M^0(x,k,l)$ if and only if $g^k(x)=g^l(x)$, and $x_j=g^{j-1}(x)$ for $j=1,\ldots, k$ and $x_{k+j}=g^{j-1}(x)$ for $j=1,\ldots, l$. Therefore, $M^0(x,k,l)$ contains at most one element. In particular, since $M^0(x,k,l)$ is non-empty then $g^k(x)=g^l(x)$ and the set of fixed points $\{x\in M: g^m(x)=x\}$ has measure zero for all $m\in {\mathbb{N}}_+$ by assumptions, then $$M^0(x,k,l)=\emptyset\quad\mbox{ if $k\neq l$ a.e. in $x$.}$$ As $t$ approaches $n/2$, the integral kernel $K_t$ localizes (up to lower order term) to the diagonal and the leading order terms come from the sum over $M^0(x,k,l)$. The Weyl law for $\slashed{D}^2$ and an explicit pseudo-differential computation of the principal symbol of $P_\slashed{D}(1+\slashed{D}^2)^{-t}$ implies that there is a constant $c$ and an $\epsilon>0$ only depending on $\slashed{D}$ such that $$\begin{aligned} \operatorname{Tr}_{L^2(M,S)}&(P_\slashed{D}S_\mu S_\nu^*{\mathrm{e}}^{-t|\slashed{D}_{\rm log}|})=\\ =&c\delta_{k,l}(t-n/2)^{-1}\int_{M}\sum_{(x_1,\ldots,x_{k+l})\in M^0(x,k,l)}\left(\prod_{j=1}^{k} \tilde{a}_j(x_j)\right)\left(\prod_{j=k+1}^{k+l} \tilde{b}_{j-k}(x_j)^*\right) \mathrm{d}V(x)+f_{\mu,\nu}(t),\end{aligned}$$ where $f_{\mu,\nu}$ is holomorphic on a neighborhood of the intervall $[n/2-\epsilon,n/2]$. Recall the notation $\tilde{a}_j:=a_jc_g^{n/4}$ and $\tilde{b}_j:=b_jc_g^{-n/4}$. For $k=l$ and $(x_1,\ldots,x_{k+l})\in M^0(x,k,k)$ we write $$\begin{aligned} \Big(\prod_{j=1}^{k} &\tilde{a}_j(x_j)\Big)\left(\prod_{j=k+1}^{2k} \tilde{b}_{j-k}(x_j)^*\right)=\prod_{j=1}^{k} \tilde{a}_j(g^{j-1}(x))\tilde{b}_{j}^*(g^{j-1}(x))=\\ &=\prod_{j=1}^{k} a_j(g^{j-1}(x))b_{j}^*(g^{j-1}(x))=\left([a_1b_1^*][g^*(a_2b_2^*)][(g^2)^*(a_2b_2^*)]\cdots [(g^k)^*(a_kb_k^*)]\right)(x).\end{aligned}$$ By the same argument that $V_g^*=N^{-1/2}\mathfrak{L}_{S,g}c_g^{-n/4}$, we can partially integrate $\int_M ag^*(b)\mathrm{d}V=\int_M \mathfrak{L}_g(c_g^{n/2}a)b\mathrm{d}V$ for $a,b\in C(M)$. By partially integrating $k-1$ times we deduce that for some function $f_{\mu,\nu}$ holomorphic on a neighborhood of the intervall $[n/2-\epsilon,n/2]$. $$\begin{aligned} \operatorname{Tr}_{L^2(M,S)}&(P_\slashed{D}S_\mu S_\nu^*{\mathrm{e}}^{-t|\slashed{D}_{\rm log}|})=\\ =&c\delta_{k,l}(t-n/2)^{-1}\int_{M}[a_1b_1^*][g^*(a_2b_2^*)][(g^2)^*(a_2b_2^*)]\cdots [(g^k)^*(a_kb_k^*)] \mathrm{d}V+f_{\mu,\nu}(t)=\\ &=c\delta_{k,l}(t-n/2)^{-1}\int_{M}\mathfrak{L}_g(c_g^{n/2}b_k^*\mathfrak{L}_g(c_g^{n/2}b_{k-1}^*\mathfrak{L}_g(\cdots \mathfrak{L}_g(c_g^{n/2}b_1^*a_1)a_2)\cdots a_{k-1})a_k) \mathrm{d}V+f_{\mu,\nu}(t) \end{aligned}$$ We conclude that formula holds. It remains to show that $\phi_\omega$ defines a KMS-state on $O_{E_g}$ for the action defined by $\gamma_t(aV_g):=c_g^{itn/2}aV_g$. Let $\tau$ denote the tracial state on $C(M)$ defined from integrating against the volume form and $L\in \operatorname{End}^*_{C(M)}(E_g)$ the generator of $\gamma_t$, i.e. $L=\frac{n}{2}\log(c_g)$. Some yoga with inner products shows that for $\mu=a_1V_g\otimes \cdots \otimes a_k V_g, \nu=b_1V_g\otimes \cdots \otimes b_k V_g\in E_g^{\otimes_{C(M)} k}$, we have the computation $$\begin{aligned} \phi_\omega(S_\mu S_\nu^*)=\int_{M}&\mathfrak{L}_g(c_g^{n/2}b_k^*\mathfrak{L}_g(c_g^{n/2}b_{k-1}^*\mathfrak{L}_g(\cdots \mathfrak{L}_g(c_g^{n/2}b_1^*a_1)a_2)\cdots a_{k-1})a_k) \mathrm{d}V=\\ &=\tau\left(b_1V_g\otimes \cdots \otimes b_k V_g|{\mathrm{e}}^{-L}(a_1V_g)\otimes \cdots \otimes {\mathrm{e}}^{-L}(a_k V_g)\right)_{E_g^{\otimes_{C(M)} k}}=\phi_\omega(S_\nu^*\gamma_i(S_\mu)).\end{aligned}$$ We conclude that $\phi_\omega(ab)=\phi_\omega(b\gamma_i(a))$ for $a,b\in {\mathcal{A}}$ and $\phi_\omega$ is a KMS-state on $O_{E_g}$ in the action $\gamma$. Graph $C^*$-algebras {#diracgraphkms} -------------------- For a finite graph $G$ we consider the spectral triple on $C^*(G)$ constructed in Proposition \[localizeckdirinpoint\] from the choice of a point $y\in \Omega_G$ in the infinite path space. We will assume that $G$ is primitive, in which case $C^*(G)$ is simple. For an element $(x,n)\in \mathcal{V}_y$ and a finite path $\mu$, we compute that $${\mathrm{e}}^{is{\mathcal{D}}_y}P_{{\mathcal{D}}_y} S_\mu P_{{\mathcal{D}}_y}{\mathrm{e}}^{-is{\mathcal{D}}_y}\delta_{(x,n)}= {\mathrm{e}}^{is|\mu|}P_{{\mathcal{D}}_y} S_\mu P_{{\mathcal{D}}_y}\delta_{(x,n)}.$$ It follows that $\sigma_t(S_\mu S_\nu^*)={\mathrm{e}}^{is(|\mu|-|\nu|)}S_\mu S_\nu^*$ and $\sigma$ coincides with the gauge action on the graph $C^*$-algebra $C^*(G)$. We can conclude that $C^*(G)=C^*(G)_{{\mathcal{D}}_y}$ is closed under the flow $\sigma$. The reader can recall from Example \[diracgraphhea\] (see page ) that $P_{{\mathcal{D}}_y}\ell^2(\mathcal{V}_y)=\ell^2(\mathcal{V}_y^+)$ where $\mathcal{V}_y^+$ is defined in Equation . Moreover, $T_{C^*(G)}\subseteq \mathbb{B}(\ell^2(\mathcal{V}_y^+))$ is the $C^*$-algebra generated by the creation operators $$T_e \delta_{(x,n)}:=\begin{cases} \delta_{(ex,n)},\; &r(e)=s(x),\\ 0,\; &r(e)\neq s(x).\end{cases}$$ Let $\beta\in {\mathbb{R}}$. The set $S=\{S_e: e\in E\}\subseteq C_c(\mathcal{G}_G)$ is an analytically generating set at $\beta$ for $(C_c(\mathcal{G}_G),\ell^2(\mathcal{V}_y),{\mathcal{D}}_y)$. Since $P_{{\mathcal{D}}_y}S_eP_{{\mathcal{D}}_y}=T_e$, it is clear that $S$ satisfies that $P_{{\mathcal{D}}_y} SP_{{\mathcal{D}}_y}+\mathbb{K}(\ell^2(\mathcal{V}_y^+)$ generates the Toeplitz algebra $T_{C^*(G)}$ as a $C^*$-algebra. Moreover, ${\mathrm{e}}^{\beta{\mathcal{D}}_y}P_{\mathcal{D}}S_eP_{\mathcal{D}}{\mathrm{e}}^{-\beta{\mathcal{D}}_y}={\mathrm{e}}^{\beta}P_{\mathcal{D}}S_eP_{\mathcal{D}}$ is bounded and the proposition follows. We conclude the following theorem from Example \[diracgraphhea\]. \[kmsforgrpah\] Let $G$ be a finite primitive graph with edge adjacency matrix $A$ and $y\in \Omega_G$. For any extended limit $\omega$ as $t\to\log r_\sigma(A)$, the KMS-state $\phi_\omega$ on $C^*(G)$ associated with the spectral triple $(C_c(\mathcal{G}_G),\ell^2(\mathcal{V}_y),{\mathcal{D}}_y)$ (see Proposition \[localizeckdirinpoint\] on page ) as in Corollary \[cor:phi-omega\] (see page ) is given by $$\phi_\omega(S_\mu S_\nu^*)=\delta_{\mu,\nu}\frac{w_{s(\mu)}}{\|w\|_{\ell^1}} r_\sigma(A)^{-|\mu|},$$ where $\nu$ and $\mu$ are finite paths and $w\in {\mathbb{C}}^E$ is the $\ell^2$-normalized Perron-Frobenius vector. The state $\phi_\omega$ is KMS for the gauge action and its inverse temperature is $\log(r_\sigma(A))$. The KMS-state $\phi_\omega$ on $C^*(G)$ in Theorem \[kmsforgrpah\] is the unique KMS-state by [@EFW]. Numerous authors present constructions of this state and for more general graphs, eg [@aHLRS; @aHR; @CT; @KPgraph]. Group $C^*$-algebras {#diracgroupkms} -------------------- For the reduced group $C^*$-algebra of a countable discrete group we considered two types of (semifinite) spectral triples in Subsection \[groupcstarexam\] (see page ). We now compute the associated KMS-states. We fix a length function $\ell$ on the discrete countable group $\Gamma$. For technical simplicity, we assume that $\Gamma$ is an exact group ensuring that $\Gamma$ acts amenably on its Stone-Cech boundary $\partial_{SC}\Gamma$ (see [@ozawexact]). We assume that $(\Gamma,\ell)$ is of at most exponential growth and that $\ell$ is critical (see Definition \[criticaldefn\] on page ). Let $(c_b(\Gamma)\rtimes^{\rm alg} \Gamma, \ell^2(\Gamma), {\mathcal{D}}_\ell)$ denote the associated $\mathrm{Li}_1$-summable spectral triple as in Proposition \[spectripfromlength\]. Since ${\mathcal{D}}_\ell\geq 0$, we have that $$T_{c_b(\Gamma)\rtimes_r \Gamma}=c_b(\Gamma)\rtimes_r \Gamma+{\mathbb{K}}(\ell^2(\Gamma))=c_b(\Gamma)\rtimes_r \Gamma.$$ The last equality follows from that $c_0(\Gamma)\rtimes_r\Gamma={\mathbb{K}}(\ell^2(\Gamma))$. We conclude that we have a short exact sequence $$0\to {\mathbb{K}}(\ell^2(\Gamma))\to T_{c_b(\Gamma)\rtimes_r \Gamma}\to C(\partial_{SC}\Gamma)\rtimes_r \Gamma\to 0.$$ The flow $\sigma^+$ on $T_{c_b(\Gamma)\rtimes_r \Gamma}=c_b(\Gamma)\rtimes_r \Gamma$ is given on an element $a\lambda_g\in c_b(\Gamma)\rtimes \Gamma$ by $$\label{sigmapluonsc} \sigma^+_s(a\lambda_g)={\mathrm{e}}^{is(\ell(\cdot)-\ell(g^{-1}\cdot)}a\lambda_g,$$ and we conclude that $T_{c_b(\Gamma)\rtimes_r \Gamma}$ is invariant under $\sigma^+$. Therefore $T_{c_b(\Gamma)\rtimes_r \Gamma}=T_{c_b(\Gamma)\rtimes_r \Gamma,{\mathcal{D}}_\ell}$. Let $\beta\in {\mathbb{R}}$. The $*$-algebra $c_b(\Gamma)\rtimes^{\rm alg} \Gamma$ is an analytically generating set at $\beta$ for $(c_b(\Gamma)\rtimes^{\rm alg} \Gamma, \ell^2(\Gamma), {\mathcal{D}}_\ell)$. Note that $P_{{\mathcal{D}}_\ell}=1$ because ${\mathcal{D}}_\ell$ is positive. For $a\lambda_g\in c_b(\Gamma)\rtimes^{\rm alg} \Gamma$, we compute that $${\mathrm{e}}^{\beta{\mathcal{D}}_\ell}a\lambda_g{\mathrm{e}}^{-\beta{\mathcal{D}}_\ell}={\mathrm{e}}^{-\beta(\ell(\cdot)-\ell(g^{-1}\cdot))}a\lambda_g.$$ Since $\|{\mathrm{e}}^{-\beta(\ell(\cdot)-\ell(g^{-1}\cdot)}a\|_{c_b(\Gamma)}\leq {\mathrm{e}}^{|\beta|\ell(g^{-1})}\|a\|_{c_b(\Gamma)}$, it holds that ${\mathrm{e}}^{\beta{\mathcal{D}}_\ell}a\lambda_g{\mathrm{e}}^{-\beta{\mathcal{D}}_\ell}\in c_b(\Gamma)\rtimes^{\rm alg} \Gamma$ and the proposition follows. Let $\beta\in {\mathbb{R}}$. The $*$-algebra $c_b(\Gamma)\rtimes^{\rm alg} \Gamma$ is an analytically generating set at $\beta$ for $(c_b(\Gamma)\rtimes^{\rm alg} \Gamma, \ell^2(\Gamma, S_{\mathcal{H}}), {\mathcal{D}}_c, {\mathcal{N}}, \operatorname{Tr}_\tau)$. For $a\lambda_g\in c_b(\Gamma)\rtimes^{\rm alg} \Gamma$, we compute for $f\in \ell^2(\Gamma,S_{\mathcal{H}})$ that $${\mathrm{e}}^{\beta|{\mathcal{D}}_c|}\hat{\pi}_S(a\lambda_g){\mathrm{e}}^{-\beta|{\mathcal{D}}_c|}f(\gamma)={\mathrm{e}}^{-\beta(\ell(\gamma)-\ell(g^{-1}\gamma))}a(\gamma)[\tilde{\pi}(g)f](\gamma).$$ The estimate $\|{\mathrm{e}}^{-\beta(\ell(\cdot)-\ell(g^{-1}\cdot)}a\|_{c_b(\Gamma)}\leq {\mathrm{e}}^{|\beta|\ell(g^{-1})}\|a\|_{c_b(\Gamma)}$ shows that ${\mathrm{e}}^{\beta|{\mathcal{D}}_c|}\hat{\pi}_S(a\lambda_g){\mathrm{e}}^{-\beta|{\mathcal{D}}_c|}\in \hat{\pi}\left(c_b(\Gamma)\rtimes^{\rm alg} \Gamma\right)$. Therefore, $$\begin{aligned} &{\mathrm{e}}^{\beta{\mathcal{D}}_c}P_{{\mathcal{D}}_c} (c_b(\Gamma)\rtimes^{\rm alg} \Gamma) P_{{\mathcal{D}}_c}{\mathrm{e}}^{-\beta{\mathcal{D}}_c}\\&=P_{{\mathcal{D}}_c}{\mathrm{e}}^{\beta|{\mathcal{D}}_c|} (c_b(\Gamma)\rtimes^{\rm alg} \Gamma){\mathrm{e}}^{-\beta|{\mathcal{D}}_c|}P_{{\mathcal{D}}_c}\subseteq P_{{\mathcal{D}}_c} (c_b(\Gamma)\rtimes^{\rm alg} \Gamma) P_{{\mathcal{D}}_c}\subseteq {\mathcal{N}}^+,\end{aligned}$$ and the proposition follows. If $\omega$ is an extended limit as $t\to\beta(\Gamma,\ell)$, and $\ell$ is critical, we define the Patterson-Sullivan measure $\mu_\omega$ on the Stone-Čech boundary $\partial_{SC}\Gamma$ as $$\int_{\partial_{SC}\Gamma} a\,\mathrm{d}\mu_\omega:=\lim_{t\to \omega}\frac{\sum_{\gamma\in \Gamma} \tilde{a}(\gamma){\mathrm{e}}^{-t\ell(\gamma)}}{\sum_{\gamma\in \Gamma} {\mathrm{e}}^{-t\ell(\gamma)}},$$ for a function $a\in C(\partial_{SC}\Gamma)$ and where $\tilde{a}\in c_b(\Gamma)$ is any function with $a=\tilde{a}\mod c_0(\Gamma)$. It is possible to define the Patterson-Sullivan measure $\mu_\omega$ as an extended weak\*-limit of the family of probability measures on $\Gamma$ $$\mu_t=\frac{\sum_{\gamma\in \Gamma} \delta_\gamma{\mathrm{e}}^{-t\ell(\gamma)}}{\sum_{\gamma\in \Gamma} {\mathrm{e}}^{-t\ell(\gamma)}}.$$ In the literature, Patterson-Sullivan measures are usually defined as weak\* accumulation points of $(\mu_t)_{t>\beta(\Gamma, \ell)}$ but we allow for a slightly more general construction with extended limits. A priori, $\mu_\omega$ is a probability measure on the Stone-Čech compactification of $\Gamma$. Since the support of $\mu_\omega$ is contained in the closed subspace $\partial_{SC}\Gamma$ we consider $\mu_\omega$ as a measure on $\partial_{SC}\Gamma$. \[themforgamma\] Let $\Gamma$ be a discrete group and $\phi_\omega$ the KMS-state on $C(\partial_{SC}\Gamma)\rtimes_r \Gamma$ constructed as in Corollary \[cor:phi-omega\] (see page ) using an extended limit $\omega$ as $t\to\beta(\Gamma,\ell)$ and either of the following two semifinite spectral triples: - The spectral triple $$(c_b(\Gamma)\rtimes^{\rm alg} \Gamma, \ell^2(\Gamma), {\mathcal{D}}_\ell)$$ associated with a critical length function of at most exponential growth as in Proposition \[spectripfromlength\] (see page ). - The semifinite spectral triple $(c_b(\Gamma)\rtimes^{\rm alg} \Gamma, \ell^2(\Gamma, S_{\mathcal{H}}), {\mathcal{D}}_c, {\mathcal{N}}, \operatorname{Tr}_\tau)$ associated with a critical proper Hilbert space valued cocycle of at most exponential growth as in Proposition \[cssfst\] (see page ). Then $\phi_\omega$ is given in terms of the Patterson-Sullivan measure $\mu_\omega$ by $$\phi_\omega(a\lambda_g)=\delta_{e,g}\int_{\partial_{SC}\Gamma} a\,\mathrm{d}\mu_\omega.$$ The state $\phi_\omega$ is KMS at inverse temperature $\beta(\Gamma,\ell)$ for the flow on $C(\partial_{SC}\Gamma)\rtimes_r \Gamma$ induced by the action $\sigma^+$ on $c_b(\Gamma)\rtimes_r\Gamma$ given in Equation . Moreover, $\phi_\omega$ extends to a KMS-state at inverse temperature $1$ on the von Neumann algebra $L^\infty(\partial_{SC}\Gamma,\mu_\omega)\overline{\rtimes}\Gamma$ with its Radon-Nikodym flow $$\sigma^{RN}_s(a\lambda_g)=\left(\frac{\mathrm{d}g_*\mu_{\omega}}{\mathrm{d}\mu_{\omega}}\right)^{is} a\lambda_g.$$ By the computations of Example \[diracgrouphea\] (see page ), the spectral triple associated with a length function as in Proposition \[spectripfromlength\] have the same heat traces as the semifinite spectral triple associated with a proper Hilbert space valued cocycle as in Proposition \[cssfst\]. In both cases, Example \[diracgrouphea\] shows that for $\tilde{a}\lambda_g\in c_b(\Gamma)\rtimes^{\rm alg}\Gamma$ we have $$\phi_{t,0}(\tilde{a}\lambda_g)=\delta_{e,g}\frac{\sum_{\gamma\in \Gamma} \tilde{a}(\gamma){\mathrm{e}}^{-t\ell(\gamma)}}{\sum_{\gamma\in \Gamma} {\mathrm{e}}^{-t\ell(\gamma)}}.$$ It follows that $\phi_\omega(a\lambda_g)=\delta_{e,g}\int_{\partial_{SC}\Gamma} a\,\mathrm{d}\mu_\omega$ in both cases. To relate $\phi_\omega$ to the Radon-Nikodym flow, we first show that $\mu_\omega$ is strictly positive, i.e. that $\mu_\omega(U)>0$ for any open set $U\subset \partial_{SC}\Gamma$. For any open set $U\subset \partial_{SC}\Gamma$, the translates $(\gamma U)_{\gamma\in \Gamma}$ cover $\partial_{SC}\Gamma$. If $\mu_\omega(U)=0$, then by quasi-invariance $\mu_\omega(\gamma U)=0$ which contradicts $\mu_\omega$ being a probability measure. The fact that $\mu_{\omega}$ is strictly positive ensures that the Radon-Nikodym derivatives $\frac{\mathrm{d}g_*\mu_{\omega}}{\mathrm{d}\mu_{\omega}}$ are well defined and strictly positive. The mapping $g\mapsto \frac{\mathrm{d}g_*\mu_{\omega}}{\mathrm{d}\mu_{\omega}}$ is a cocycle, i.e. for $h,g\in \Gamma$, $$\frac{\mathrm{d}(gh)_*\mu_{\omega}}{\mathrm{d}\mu_{\omega}}=\frac{\mathrm{d}g_*\mu_{\omega}}{\mathrm{d}\mu_{\omega}}(g^{-1})^*\left[\frac{\mathrm{d}h_*\mu_{\omega}}{\mathrm{d}\mu_{\omega}}\right].$$ The proof is completed by computing that for $a\lambda_g, b\lambda_h\in C(\partial_{SC}\Gamma)\rtimes^{\rm alg}\Gamma$, we have the identity $$\begin{aligned} \phi_\omega(a\lambda_gb\lambda_h)&=\delta_{h,g^{-1}} \phi_\omega(ah^*(b))=\delta_{h,g^{-1}}\int_{\partial_{SC}\Gamma} ah^*(b)\mathrm{d}\mu_{\omega}=\\ &=\delta_{h,g^{-1}}\int_{\partial_{SC}\Gamma} h^*\left(g^*(a)b\right)\mathrm{d}\mu_{\omega}=\delta_{h,g^{-1}}\int_{\partial_{SC}\Gamma} g^*(a)b\mathrm{d}(h_*\mu_{\omega})=\\ &=\delta_{h,g^{-1}}\int_{\partial_{SC}\Gamma} bg^*(a)\frac{\mathrm{d}h_*\mu_{\omega}}{\mathrm{d}\mu_{\omega}}\mathrm{d}\mu_{\omega}=\\ &=\delta_{h,g^{-1}}\int_{\partial_{SC}\Gamma} b(h^{-1})^*\left(a\left(\frac{\mathrm{d}g_*\mu_{\omega}}{\mathrm{d}\mu_{\omega}}\right)^{-1}\right)\mathrm{d}\mu_{\omega}=\\ &=\phi_\omega\left(b\lambda_h\left(\frac{\mathrm{d}g_*\mu_{\omega}}{\mathrm{d}\mu_{\omega}}\right)^{-1}a\lambda_g\right)=\phi_\omega\left(b\lambda_h\sigma^{RN}_{s=i}(a\lambda_g)\right)\end{aligned}$$ In the third last identity we used the cocycle identity implying that if $hg=e$, then $$(h^{-1})^*\left(\frac{\mathrm{d}g_*\mu_\omega}{\mathrm{d}\mu_\omega}\right)\frac{\mathrm{d}h_*\mu_\omega}{\mathrm{d}\mu_\omega}=1.\qedhere$$ The reader should note that $\phi_\omega|_{C^*_r(\Gamma)}$ coincides with the $\ell^2$-trace. KMS-states on Cuntz-Pimsner algebras with their gauge action {#diraccpkms} ============================================================ In this section we consider the constructions of Corollary \[cor:phi-omega\] in a broad class of examples which include both Cuntz-Krieger algebras and crossed products by ${\mathbb{Z}}$. Here we use the techniques from Section \[sec:KMS\] in conjunction with those from Subsection \[cpalgexam\] to analyse the KMS states on Cuntz-Pimsner algebras, and compare them to the Laca-Neshveyev correspondence establishing a bijection between KMS-states on Cuntz-Pimsner algebras and tracial states on its coefficient algebra. KMS-states on Cuntz-Pimsner algebras from traces on the coefficient algebra --------------------------------------------------------------------------- Firstly, we shall show that a critical positive trace on $A$ (see Definition \[defn:critical\] below) gives rise to a KMS-state on the Cuntz-Pimsner algebra $O_E$ assuming that $E$ is strictly W-regular. Recall Lemma \[ximodsemi\] (see page ) giving the construction of the $\mathrm{Li}_1$-summable semifinite spectral triple $({\mathcal{O}}_E,L^2({\Xi_A},\tau),{\mathcal{D}}_\psi,{\mathcal{N}}_\tau(\Xi_A),\operatorname{Tr}_\tau)$, where ${\mathcal{N}}_\tau(\Xi_A):=(\operatorname{End}^*_A({\Xi_A})\otimes_A1)''$. \[betacrit\] Let $E$ be a strictly W-regular fgp bi-Hilbertian bimodule over $A$, $\beta\in {\mathbb{R}}$ and $\tau$ a positive trace on $A$. The set $S=\{S_{e}:\,e\in E\}\subset{\mathcal{O}}_E$ is an analytically generating set at $\beta$ for $({\mathcal{O}}_E,L^2({\Xi_A},\tau),{\mathcal{D}}_\psi,{\mathcal{N}}_\tau(\Xi_A),\operatorname{Tr}_\tau)$. Since $T_{{\mathcal{O}}_E,{\mathcal{D}}}$ is precisely the Toeplitz algebra $T_E$, it is immediate that the set of operators $\{ PS_eP : e\in E\}$ generates $T_{{\mathcal{O}}_E,{\mathcal{D}}}$. The analyticity condition on the Fock space is likewise obvious from the computation $${\mathrm{e}}^{\beta{\mathcal{D}}_\psi}P_{{\mathcal{D}}_\psi}S_eP_{{\mathcal{D}}_\psi}{\mathrm{e}}^{-\beta{\mathcal{D}}_\psi}= {\mathrm{e}}^\beta P_{{\mathcal{D}}_\psi}S_eP_{{\mathcal{D}}_\psi}.\qedhere$$ \[defn:critical\] Let $E$ be an $A$-bimodule which is fgp from the right and $\tau$ a positive trace on $A$. We define the critical value of $(E,\tau)$ as $$\beta(E,\tau):=\inf\{t\geq 0: \sum_{n=0}^\infty \tau_*(E^{\otimes_An}){\mathrm{e}}^{-tn}<\infty\}.$$ We say that $\tau$ is critical for $E$ if $$\lim_{t\searrow \beta(E,\tau)} \sum_{n=0}^\infty \tau_*(E^{\otimes_An}){\mathrm{e}}^{-tn}=\infty.$$ Note that the critical value of a trace and the notion of it being critical only depends on $[E]\in KK_0(A,A)$, in fact only on the sequence $(\mathrm{ch}_0(E^{\otimes_An}))_{n\in {\mathbb{N}}}\subseteq HC_0(A)$ in cyclic homology. Just as in the proof of Lemma \[ximodsemi\] we obtain the estimate $0\leq \beta(E,\tau)\leq \log(N)$, where $N$ is the number of elements in the left frame and the right frame. It follows from Definition \[ass:minus-one\] and Proposition \[posheatcomp\] that the $\mathrm{Li}_1$-summable semifinite spectral triple $({\mathcal{O}}_E,L^2({\Xi_A},\tau),{\mathcal{D}}_\psi,{\mathcal{N}}_\tau(\Xi_A),\operatorname{Tr}_\tau)$ of Lemma \[ximodsemi\] has positive $\operatorname{Tr}_\tau$-essential spectrum if and only if $\tau$ is critical. By construction, the projection $P_{{\mathcal{D}}_\psi}$ is the projection onto the Fock module ${\mathcal{F}_E}$ and therefore ${\mathcal{N}}_\tau(\Xi_A)^+= (\operatorname{End}^*_A({\mathcal{F}_E})\otimes_A1)''$ is a subalgebra of $\mathbb{B}(L^2({\mathcal{F}_E},\tau))$. We let $N$ denote the number operator on $L^2({\mathcal{F}_E},\tau)$ – the self-adjoint operator defined from $N|_{E^{\otimes n}\otimes_AL^2(A,\tau)}=n\mathrm{Id}_{E^{\otimes n}\otimes_AL^2(A,\tau)}$. The next proposition follows from the definition of the Toeplitz algebra of a semifinite spectral triple. \[toepliz\] The Toeplitz algebra $T_{O_E}$ of the semifinite spectral triple $$({\mathcal{O}}_E,L^2({\Xi_A},\tau),{\mathcal{D}}_\psi,{\mathcal{N}}_\tau(\Xi_A),\operatorname{Tr}_\tau)$$ is given by $$T_{O_E}=\mathcal{T}_E\otimes_A 1_A+{\mathbb{K}}_{(\operatorname{End}^*_A({\mathcal{F}_E}){\otimes}_A1)''},$$ where $\mathcal{T}_E\subseteq \operatorname{End}^*_A({\mathcal{F}_E})$ is the Cuntz-Toeplitz algebra of $E$. Moreover, the action $\sigma^+$ preserves $T_{O_E}$ and is generated by the number operator in the sense that for $\mu\in E^k$, $\nu\in E^{\otimes l}$ and $K\in {\mathbb{K}}_{(\operatorname{End}^*_A({\mathcal{F}_E}){\otimes}_A1)''}$, we have $$\sigma^+_s(T_\mu T_\nu^*+K)={\mathrm{e}}^{is(|\mu|-|\nu|)}T_\mu T_\nu^*+{\mathrm{e}}^{isN}K{\mathrm{e}}^{-isN}.$$ An immediate consequence of Proposition \[toepliz\] is that $O_E=T_{O_E}/ {\mathbb{K}}_{(\operatorname{End}^*_A({\mathcal{F}_E}))''}$ and that the action $\sigma$ on ${\mathcal{O}}_E$ coincides with the gauge action $\sigma_s(S_\mu S_\nu^*)={\mathrm{e}}^{is(|\mu|-|\nu|)}S_\mu S_\nu^*$. The following theorem is readily deduced from the computations of Example \[diraccphea\] (see page ). \[kmscp\] Let $E$ be a strictly W-regular fgp bi-Hilbertian bimodule over $A$. For any positive trace $\tau$ on $A$ which is critical for $E$ and any extended limit $\omega$ as $t\to\beta(E,\tau)$, the KMS-state $\phi_\omega$ on $O_E$ associated with the semifinite spectral triple $({\mathcal{O}}_E,L^2({\Xi_A},\tau),{\mathcal{D}}_\psi,{\mathcal{N}}_\tau(\Xi_A),\operatorname{Tr}_\tau)$ as in Theorem \[cor:phi-omega\] (see page ) is given by $$\phi_{\tau,\omega}(S_\mu S_\nu^*)=\delta_{|\mu|,|\nu|}{\mathrm{e}}^{-\beta(E,\tau)|\mu|}\lim_{t\to \omega}\frac{\sum_{n=0}^\infty\operatorname{Tr}^{E^{{\otimes}n}}_\tau\left((\nu|\mu)_{E^{|\mu|}}\right){\mathrm{e}}^{-tn}}{\sum_{n=0}^\infty \operatorname{Tr}^{E^{{\otimes}n}}_\tau(1){\mathrm{e}}^{-tn}} ,$$ where $\nu\in E^{\otimes k}$ and $\mu\in E^{\otimes l}$. The state $\phi_{\tau, \omega}$ is KMS for the gauge action on $O_E$ and its inverse temperature is $\beta(E,\tau)$. By Lemma \[betacrit\] the semifinite spectral triple $({\mathcal{O}}_E,L^2({\Xi_A},\tau),{\mathcal{D}}_\psi,{\mathcal{N}}_\tau(\Xi_A),\operatorname{Tr}_\tau)$ is $\beta$-analytic for any $\beta\in {\mathbb{R}}$. By Definition \[ass:minus-one\] and Proposition \[posheatcomp\] it has positive $\operatorname{Tr}_\tau$-essential spectrum. Thus the state $\phi_{\tau, \omega}$ as in Theorem \[cor:phi-omega\] is KMS for the gauge action on $O_E$ with inverse temperature $\beta(E,\tau)$. Using the computation in Equation , we see that $$\phi_{\tau,\omega}(S_\mu S_\nu^*)=\delta_{|\mu|,|\nu|}\lim_{t\to \omega}\frac{\sum_{n=|\mu|}^\infty\sum_{|\sigma|=n} {\mathrm{e}}^{-tn}\tau \big((\,(\mu|e_{\underline{\sigma}})_{E^{|\mu|}}\,e_{\overline{\sigma}}\,|\,(\nu|e_{\underline{\sigma}})_{E^{|\mu|}}\,e_{\overline{\sigma}})_{E^{\otimes (n-|\mu|)}}\big)}{\sum_{n=0}^\infty \sum_{|\rho|=n} \tau((e_\rho|e_\rho)_A){\mathrm{e}}^{-tn}} ,$$ for $\nu\in E^{\otimes k}$ and $\mu\in E^{\otimes l}$. However, this expression can be vastly simplified using that $\phi_{\tau,\omega}$ is KMS. Using $$\phi_{\tau,\omega}(S_\mu S_\nu^*)=\delta_{|\mu|,|\nu|}{\mathrm{e}}^{-\beta(E,\tau)|\mu|}\phi_{\tau,\omega}(S_\nu^*S_\mu)=\delta_{|\mu|,|\nu|}{\mathrm{e}}^{-\beta(E,\tau)|\mu|}\phi_{\tau,\omega}((\nu|\mu)_{E^{\otimes |\mu|}}),$$ we obtain $$\begin{aligned} \phi_{\tau,\omega}(S_\mu S_\nu^*)&=\delta_{|\mu|,|\nu|}{\mathrm{e}}^{-\beta(E,\tau)|\mu|}\phi_\omega((\nu|\mu)_{E^{\otimes |\mu|}})\\ &=\delta_{|\mu|,|\nu|}{\mathrm{e}}^{-\beta(E,\tau)|\mu|}\lim_{t\to \omega}\frac{\sum_{n=0}^\infty\sum_{|\sigma|=n} {\mathrm{e}}^{-tn}\tau \big((\,e_{\sigma}\,|\,(\nu|\mu)_{E^{|\mu|}}\,e_{\sigma})_{E^{\otimes n}}\big)}{\sum_{n=0}^\infty \sum_{|\rho|=n} \tau((e_\rho|e_\rho)_A){\mathrm{e}}^{-tn}}\\ &=\delta_{|\mu|,|\nu|}{\mathrm{e}}^{-\beta(E,\tau)|\mu|}\lim_{t\to \omega}\frac{\sum_{n=0}^\infty\operatorname{Tr}^{E^{{\otimes}n}}_\tau\left((\nu|\mu)_{E^{|\mu|}}\right){\mathrm{e}}^{-tn}}{\sum_{n=0}^\infty \operatorname{Tr}^{E^{{\otimes}n}}_\tau(1){\mathrm{e}}^{-tn}}.\end{aligned}$$ The reader should note that in the formula computing $\phi_{\tau,\omega}$, it is only the right inner product on $E$ that appears. \[localhomeoex\] Let us consider the construction from Theorem \[kmscp\] in a specific example. As in Example \[localhomeofirst\] (see page ), we consider a compact Hausdorff space $Y$, a surjective local homeomorphism $g:Y\to Y$ and the associated bimodule $E_g$. Let us compute $\phi_{\tau,\omega}$ starting from a positive trace $\tau$ on $C(Y)$, i.e. a positive measure $\lambda$ on $Y$. By the argument of Theorem \[kmscp\], the KMS-condition on $\phi_{\tau,\omega}$ guarantees that it suffices to describe $\phi_{\tau,\omega}(a)$ for $a\in C(Y)$. By the Riesz representation theorem, $$\phi_{\tau,\omega}(a)=\int_Y a\,\mathrm{d}\lambda_\omega,$$ for a probability measure $\lambda_\omega$. We compute that for $a\in C(Y)$, $$\operatorname{Tr}^{E^{{\otimes}n}}_\tau(a)=\sum_{|\sigma|=n}\tau( (e_\sigma|ae_\sigma)_{C(Y)}) =\int_Y \mathfrak{L}_g^n(a)\,\mathrm{d}\lambda =\int_Y a\,\mathrm{d}[(\mathfrak{L}_g^n)_*\lambda].$$ From Theorem \[kmscp\], we conclude that $\lambda_{\omega}$ is given by an extended weak\* limit of measures $$\lambda_\omega=\lim_{t\to \omega}\frac{\sum_{n=0}^\infty {\mathrm{e}}^{-tn}(\mathfrak{L}_g^n)_*\lambda}{\sum_{n=0}^\infty {\mathrm{e}}^{-tn}[(\mathfrak{L}_g^n)_*\lambda](Y)}.$$ The KMS-condition on $\phi_{\tau,\omega}$ translates into $(\mathfrak{L}_g)_*\lambda_\omega={\mathrm{e}}^{\beta(E,\tau)}\lambda_\omega$ which is readily verified for the measure $\lambda_\omega$. We remark that the construction above is reminiscent of the method in [@waltersruelle] to construct equilibrium measures. In the case that $g$ is mixing, i.e. for all open subsets $U,V\subseteq Y$ there is an $N\geq 0$ such that $g^n(U)\cap V\neq \emptyset$ for all $n\geq N$, then there exists a unique KMS-state on $O_{E_g}$ (see [@DGMW Theorem 6.1]). In particular, for mixing $g$, the KMS-state $\phi_{\tau,\omega}$ on $O_{E_g}$ does not depend on the choice of trace $\tau$. The Toeplitz construction vs the Laca-Neshveyev correspondence {#subsec:T-vs-LN} -------------------------------------------------------------- In the previous subsection we saw that there is a mapping from the set of positive critical traces on a $C^*$-algebra $A$ to the set of KMS-states on the Cuntz-Pimsner algebra $O_E$ when $E$ is strictly W-regular. As Example \[localhomeoex\] shows, this mapping is not injective in general, but it is surjective in some cases (e.g. when $g$ is mixing). We now compare our construction to the bijective correspondence between a certain set of tracial states on $A$ with KMS-states on the Cuntz-Pimsner algebra $O_E$ first discovered by Laca-Neshveyev [@LN]. \[ass:3.5\] The positive trace $\tau:A\to{\mathbb{C}}$ satisfies the Laca-Neshveyev condition for $\alpha\geq 0$ if $$\operatorname{Tr}^E_\tau(L_a)={\mathrm{e}}^\alpha\tau(a),$$ where $L_a$ denotes the left action of $a$ on $E$. For notational simplifity we often write $\operatorname{Tr}^E_\tau(a)$ instead of $\operatorname{Tr}^E_\tau(L_a)$. Given a positive trace $\tau:A\to{\mathbb{C}}$ satisfying the Laca-Neshveyev condition, it was proven by Laca-Neshveyev [@LN Theorem 2.1 and 2.5] that the expression $$\phi_{LN,\tau}(S_\mu S_\nu^*):= \delta_{|\mu|,|\nu|}\mathrm{e}^{-\alpha|\mu|}\tau((\nu|\mu)_A),$$ defines an $\alpha$-KMS state on $O_E$. Moreover, Laca-Neshveyev proved that the construction $\tau\mapsto \phi_{LN,\tau}$ is a bijection between tracial states on $A$ satisfying the Laca-Neshveyev condition for $\alpha\geq 0$ and $\alpha$-KMS states on $O_E$. The work of Laca-Neshveyev [@LN] gives more context to the construction in Theorem \[kmscp\] (see page ). For a unital $C^*$-algebra $A$, we let $\mathfrak{T}(A)$ denote the set of positive traces on $A$. If $E$ is an $A-A$-correspondence which is finitely generated and projective as a right module, we can also define $\mathfrak{CT}_{E,\alpha}(A)$ as the set of positive critical traces $\tau$ with $\beta(E,\tau)=\alpha$. We also define $\mathfrak{LN}_{E,\alpha}(A)$ as the set of positive traces satisfying the Laca-Neshveyev condition. Following [@LN Discussion proceeding Definition 2.3], we define $$\label{eq:F} F_{E,\alpha}:\mathfrak{T}(A)\to \mathfrak{T}(A), \quad F_{E,\alpha}\tau(a)=\operatorname{Tr}^E_\tau(a){\mathrm{e}}^{-\alpha}.$$ \[tracecompforf\] For any positive trace $\tau$ on a unital $C^*$-algebra and an $A$-$A$-correspondence $E$ which is fgp from the right, it holds that $$F^n_{E,\alpha}\tau(a)={\mathrm{e}}^{-\alpha n}\operatorname{Tr}^{E^{{\otimes}n}}_\tau(a), \quad n\in {\mathbb{N}}_+.$$ By definition, $F_{E,\alpha}\tau(a)={\mathrm{e}}^{-\alpha}\sum_{j=1}^N \tau ( e_j|ae_j)_A$ for a right frame $(e_j)_{j=1}^N$ of $E$. A direct computation shows that $$F^n_{E,\alpha}\tau(a)={\mathrm{e}}^{-\alpha n}\sum_{|\sigma|=n}\tau (e_\sigma|ae_\sigma)_A={\mathrm{e}}^{-\alpha n}\operatorname{Tr}^{E^{{\otimes}n}}_\tau(a). \qedhere$$ \[lnequitofixed\] Let $E$ be an $A$-$A$-correspondence which is fgp from the right. Then $\tau\in \mathfrak{T}(A)$ is a fixed point of $F_{E,\alpha}$ if and only if $\tau$ satisfies the Laca-Neshveyev condition from Definition \[ass:3.5\] (see page ). Proposition \[lnequitofixed\] is a direct consequence of Definition \[ass:3.5\] and the formula . \[quasicompu\]Let $E$ be an $A$-$A$-correspondence which is fgp from the right. If $\tau\in \mathfrak{T}(A)$ is a tracial state satisfying the Laca-Neshveyev condition for $\alpha\geq 0$, then $$\tau_*(E^{\otimes_A n})={\mathrm{e}}^{\alpha n} F_{E,\alpha}^n\tau(1)={\mathrm{e}}^{\alpha n}.$$ The proposition follows from the computation $$\tau_*(E^{\otimes_A n})=\mathrm{Tr}_\tau^{E^{\otimes n}}(1)={\mathrm{e}}^{\alpha n}F_{E,\alpha}^n(\tau)(1),$$ and Proposition \[lnequitofixed\]. \[someconforf\] Let $E$ be an $A$-$A$-correspondence which is fgp from the right and $\alpha\geq 0$. Then the following holds: 1. A positive trace $\tau\in \mathfrak{T}(A)$ is critical for $\alpha$ if and only if the positive trace $$S_{E,\alpha}^t\tau:=\sum_{n=0}^\infty {\mathrm{e}}^{-tn}F_{E,\alpha}^n\tau,$$ is finite for $t>0$ and satisfies $S_{E,\alpha}^t\tau(1)\to \infty$ as $t\to 0$. In particular, we have an inclusion of sets $$\mathfrak{LN}_{E,\alpha}(A)\subseteq \mathfrak{CT}_{E,\alpha}(A).$$ 2. For any extended limit $\omega$ as $t\to 0$, the mapping $$S_{E,\alpha}^\omega:\mathfrak{CT}_{E,\alpha}(A)\to \mathfrak{LN}_{E,\alpha}(A), \quad S_{E,\alpha}^\omega\tau:=\lim_{t\to \omega}\frac{S_{E,\alpha}^t\tau}{S_{E,\alpha}^t\tau(1)},$$ is well defined. Moreover, $(S_{E,\alpha}^\omega)^2\tau=\frac{S_{E,\alpha}^\omega\tau}{S_{E,\alpha}^\omega\tau(1)}$ and $S^\omega_{E,\alpha}$ surjects onto the set of tracial states in $\mathfrak{LN}_{E,\alpha}(A)$. Statement 1 is an immediate consequence of Proposition \[quasicompu\]. The first part of statement 2 follows from the computation $$S_{E,\alpha}^t\tau(1)-F_{E,\alpha}S_{E,\alpha}^t\tau(1)=\tau(1)=o(S_{E,\alpha}^t\tau(1)), \quad \mbox{as $t\to 0$ for $\tau \in \mathfrak{CT}_{E,\alpha}(A)$}.$$ The second part of statement 2 follows from the fact that Proposition \[lnequitofixed\] implies that for $\tau\in \mathfrak{LN}_{E,\alpha}(A)$, $$S_{E,\alpha}^t\tau:=(1-{\mathrm{e}}^{-t})^{-1}\tau. \qedhere$$ For Example \[localhomeoex\], the mapping $F_{E,\alpha}$ takes the form $F_{E,\alpha}\tau={\mathrm{e}}^{-\alpha}\mathfrak{L}_g^*\tau$. In particular, the computations of Example \[localhomeoex\] is a special case of the constructions in Proposition \[someconforf\]. We shall see that this holds in general below in Proposition \[restricprop\]. Using our previous results, Proposition \[posheatcomp\] (see page ) and Proposition \[quasicompu\] (see page ), we can deduce a computation of heat traces. \[specasuln\] Let $E$ be a strictly W-regular fpg bi-Hilbertian bimodule over a unital $C^*$-algebra $A$. If $\tau$ is a tracial state on $A$ satisfying the Laca-Neshveyev condition for $\alpha\geq 0$, then $$\operatorname{Tr}_\tau(P_{\mathcal{D}}{\mathrm{e}}^{-t|{\mathcal{D}}_\psi|})=\frac{1}{1-{\mathrm{e}}^{\alpha-t}}.$$ In particular, for any tracial state $\tau$ satisfying the Laca-Neshveyev condition for $\alpha\geq 0$ the semifinite spectral triple $({\mathcal{O}}_E,L^2({\Xi_A},\tau),{\mathcal{D}}_\psi,{\mathcal{N}}_\tau(\Xi_A),\operatorname{Tr}_\tau)$ has positive $\operatorname{Tr}_\tau$-essential spectrum with $\beta_{\mathcal{D}}=\beta(E,\tau)=\alpha$. We compute that $$\begin{aligned} \operatorname{Tr}_\tau(P_{\mathcal{D}}{\mathrm{e}}^{-t|{\mathcal{D}}_\psi|})&=\sum_{n=0}^\infty \tau_*(E^{\otimes_A n}){\mathrm{e}}^{-tn}=\sum_{n=0}^\infty {\mathrm{e}}^{-(t-\alpha)n}=\frac{1}{1-{\mathrm{e}}^{\alpha-t}}.\end{aligned}$$ In the first step we used Proposition \[posheatcomp\] and in the second step we used Proposition \[quasicompu\]. We can now reformulate Theorem \[kmscp\] in terms of the map $F_{E,\alpha}$ and the constructions of Proposition \[someconforf\]. \[restricprop\] Assume that $E$ is a strictly W-regular fgp bi-Hilbertian bimodule over $A$. Let $\alpha\geq 0$, $\omega$ be an extended limit as $t\to \alpha$ and $\tau\in \mathfrak{CT}_{E,\alpha}(A)$ a critical trace. The KMS-state $\phi_{\tau,\omega}$ defined from Theorem \[kmscp\] takes the following form: $$\phi_{\tau,\omega}(S_\mu S_\nu^*)=\delta_{|\mu|,|\nu|} {\mathrm{e}}^{-\alpha|\mu|}S_{E,\alpha}^{\omega_\alpha}\tau((\nu|\mu)_A),\quad \mu,\nu \in {\mathcal{F}_E}^{\rm alg},$$ where $\omega_\alpha$ is the extended limit at $0$ obtained from translating $\omega$ by $\alpha$. The KMS-condition on $O_E$ reduces the proof to showing that $\phi_{\tau,\omega}(a)=S_{E,\alpha}^{\omega_\alpha}\tau(a)$ for $a\in A$, just as in the proof of Theorem \[kmscp\]. Using Proposition \[tracecompforf\] we can compute for $a\in A$ that $$\begin{aligned} \phi_{\tau,\omega}(a) &=\lim_{t\to \omega_0} \frac{\sum_{n=0}^\infty \operatorname{Tr}_\tau^{E^{{\otimes}n}}(a){\mathrm{e}}^{-tn}}{\sum_{n=0}^\infty\operatorname{Tr}_\tau^{E^{{\otimes}n}}(1){\mathrm{e}}^{-tn}}=\lim_{t\to \omega} \frac{\sum_{n=0}^\infty F_{E,\alpha}^n\tau(a){\mathrm{e}}^{-(t-\alpha)n}}{\sum_{n=0}^\infty F_{E,\alpha}^n\tau(1){\mathrm{e}}^{-(t-\alpha)n}}=S_{E,\alpha}^{\omega_\alpha}\tau(a). \qedhere\end{aligned}$$ Let $KMS_\alpha(O_E)$ denote the set of $\alpha$-KMS states on $O_E$ for the gauge action and $\mathfrak{LNS}_{E,\alpha}(A)$ for the set of tracial states satisfying the Laca-Neshveyev condition. \[prop:tau-phi=LN\] Let $\alpha\geq 0$ and let $\omega$ be an extended limit as $t\to \alpha$. Assume that $E$ is a strictly W-regular fgp bi-Hilbertian bimodule over $A$. The mapping $$\mathfrak{LNS}_{E,\alpha}(A)\to KMS_\alpha(O_E), \quad \tau\mapsto \phi_{\tau,\omega},$$ defined from Theorem \[kmscp\], and revisited in Proposition \[restricprop\], is a well defined bijection of sets. More precisely, $\phi_{\tau,\omega}:O_E\to{\mathbb{C}}$ is a $KMS_\alpha$ state for the gauge action, which is independent of $\omega$ and coincides with $\phi_{LN,\tau}$. By Theorem \[kmscp\] and Proposition \[specasuln\], the mapping $\tau\mapsto \phi_{\tau,\omega}$ is a well defined mapping from the set of positive traces on $A$ satisfying the Laca-Neshveyev condition for $\alpha$ and $\alpha$-KMS-states on $O_E$ for the gauge action. By the Laca-Neshveyev correspondence, the KMS-state $\phi_{\tau,\omega}$ is uniquely determined by the trace $\phi_{\tau,\omega}|_A$. We can therefore deduce that $\phi_{\tau,\omega}=\phi_{LN,\tau}$ and the Theorem upon proving the identity $\phi_{\tau,\omega}|_A=\tau$. This statement follows immediately from Proposition \[restricprop\] and the second part of Proposition \[someconforf\]. There are some quasi-invariance assumptions on traces that allows us to compare the KMS-states constructed in Theorem \[kmscp\] and the Laca-Neshveyev correspondence to a simpler construction involving $\Phi_\infty$. While the construction of $\Phi_\infty$ depends on the left inner product on $E$, the quasi-invariance condition we impose also depends on the left inner product. The following quasi-invariance condition is a refinement of the notion of $E$-invariant functionals from [@RRS]. \[ass:three\] Let $\alpha\geq 0$ and $E$ a finitely generated projective bi-Hilbertian bimodule. We say that a positive trace $\tau$ on $A$ is $\alpha$-quasi-invariant with respect to $E$ and the extended limit $\omega_0\in \ell^\infty({\mathbb{N}})^*$ if for all $n\in {\mathbb{N}}$ and $\mu,\,\nu\in E^{\otimes n}$ we have $${\mathrm{e}}^{-\alpha |\mu|}\tau((\nu|\mu)_A) =\lim_{k\to\omega_0}\tau(\Phi_k(T_\mu T_\nu^*){\mathrm{e}}^{-\beta_k}) =\lim_{k\to\omega_0}\tau({}_A(\mu|\nu {\mathrm{e}}^{\beta_{k-|\nu|}}){\mathrm{e}}^{-\beta_k}). \label{eq:q-i}$$ If $\tau$ is $\alpha$-quasi-invariant with respect to $E$ and some extended limit, we simply say that $\tau$ is $\alpha$-quasi-invariant with respect to $E$. Note, that $\Phi_k$ are defined on page  (formula  and the paragraph after). Observe that if $E$ is W-regular then the limit in the definition of quasi-invariance exists, and so is independent of the extended limit $\omega_0$. Just like in [@RRS Lemma 4.2], if $E$ is full as a right module, then any positive functional $\tau:A\to{\mathbb{C}}$ which is quasi-invariant in the sense of Definition \[ass:three\] is a positive trace. To see this, observe that for all $\mu,\,\nu\in {\mathcal{F}_E}^{\rm alg}$ and $a\in A$, the centrality of the Watatani indices ${\mathrm{e}}^{\beta_k}\in A$ (see formula  for the definition) gives $$\begin{aligned} {\mathrm{e}}^{-\alpha|\mu|}\tau((\nu|\mu)_Aa) &={\mathrm{e}}^{-\alpha|\mu|}\tau((\nu|\mu a)_A) =\lim_{k\to \omega_0}\tau({}_A(\mu a|\nu {\mathrm{e}}^{\beta_{k-|\nu|}}){\mathrm{e}}^{-\beta_k})\\ &=\lim_{k\to \omega_0}\tau({}_A(\mu|\nu a^*{\mathrm{e}}^{\beta_{k-|\nu|}}){\mathrm{e}}^{-\beta_k}) ={\mathrm{e}}^{-\alpha|\mu|}\tau((\nu a^*|\mu )_A)\\ &={\mathrm{e}}^{-\alpha|\mu|}\tau(a(\nu|\mu )_A).\end{aligned}$$ We consider the module $E_g$ over $C(Y)$ defined from a surjective local homeomorphism $g:Y\to Y$ as in Example \[localhomeoex\] (see page ). In this case, $\beta_n=0$ for all $n$ and quasi-invariance of a positive trace $\tau$ given by a positive measure $\lambda$ on $Y$ is equivalent to the condition $$(\mathfrak{L}_g)_*\lambda={\mathrm{e}}^{\alpha} \lambda.$$ Another computation shows that this condition is equivalent to the Laca-Neshveyev condition. In particular, for the module $E_g$, quasi-invariance is equivalent to satisfying the Laca-Neshveyev condition. Let us consider the Cuntz algebra $O_N$ defined as the Cuntz-Pimsner algebra of the ${\mathbb{C}}$-bimodule ${\mathbb{C}}^N$. In this case, $(\nu|\mu)_A=\overline{{}_A(\mu|\nu)}$ for all $\mu,\nu \in ({\mathbb{C}}^N)^{\otimes n}$ and $\beta_n=n\log(N)$. Therefore, quasi-invariance of a trace $\tau$ on ${\mathbb{C}}$ is equivalent to $${\mathrm{e}}^\alpha\tau=N\tau.$$ That is, any non-zero trace on ${\mathbb{C}}$ is $\log(N)$-quasi invariant. Our immediate aim is to connect the quasi-invariance of Definition \[ass:three\] with the condition imposed by Laca-Neshveyev, [@LN]. \[lem:qi-LN\] Let $E$ be an fgp bi-Hilbertian bimodule. Suppose that $\tau$ satisfies the $\alpha$-quasi-invariance condition of Definition \[ass:three\] with respect to $E$. Then $\tau$ satisfies the Laca-Neshveyev condition for $\alpha$. This is a computation using Definition \[ass:three\] of quasi-invariance and a frame $(e_j)$ for $E_A$. So $$\begin{aligned} \operatorname{Tr}^E_\tau(L_a)&=\operatorname{Tr}_\tau\bigg(\sum_{j}a\Theta_{e_j,e_j}\bigg) =\sum_j\tau((e_j|ae_j)_A) ={\mathrm{e}}^\alpha\lim_{k\to \omega_0}\tau(\Phi_k(\sum_{j}T_{ae_j}T_{e_j}^*){\mathrm{e}}^{-\beta_k})\\ &={\mathrm{e}}^\alpha\lim_{k\to \omega_0}\tau(\sum_{j}a{}_A(e_j|e_j{\mathrm{e}}^{\beta_{k-1}}){\mathrm{e}}^{-\beta_k}) ={\mathrm{e}}^\alpha\lim_{k\to \omega_0}\tau(a{\mathrm{e}}^{\beta_{k}}{\mathrm{e}}^{-\beta_k}) ={\mathrm{e}}^\alpha\tau(a).\qedhere\end{aligned}$$ If $\tau:A\to {\mathbb{C}}$ satisfies the $\alpha$-quasi-invariance condition of Definition \[ass:three\], then we can rewrite the Laca-Neshveyev KMS state $\phi_{LN,\tau}:O_E\to{\mathbb{C}}$ as $$\begin{aligned} \phi_{LN,\tau}(T_\mu T_\nu^*) &=\delta_{|\mu|,|\nu|}{\mathrm{e}}^{-\alpha |\mu|}\tau((\mu|\nu)_A)=\lim_{k\to \omega_0}\tau(\Phi_k(T_\mu T_\nu^*){\mathrm{e}}^{-\beta_k}).\end{aligned}$$ This computation proves the following. Note that the next result does not require any W-regularity from the module $E$. \[prop:LN-tau-Phi\] Let $E$ be an fgp bi-Hilbertian bimodule and consider an $\alpha$-quasi-invariant positive trace $\tau:A\to{\mathbb{C}}$ with respect to $E$. The state on $O_E$ defined by $$\label{prestadefin} S_\mu S_\nu^*\mapsto \lim_{k\to \omega_0}\tau(\Phi_k(T_\mu T_\nu^*){\mathrm{e}}^{-\beta_k}),$$ is $\alpha$-KMS for the gauge action on $O_E$ and coincides with $\phi_{LN,\tau}$. If $E$ is W-regular, the definition $\Phi_\infty(S_\mu S_\nu^*):=\lim_{k\to \infty}\Phi_k(T_\mu T_\nu^*){\mathrm{e}}^{-\beta_k}$ shows that the state in Equation coincides with $\tau\circ \Phi_\infty$. We conclude the following. \[koroalala\] Let $\tau:A\to{\mathbb{C}}$ be an $\alpha$-quasi-invariant positive trace with respect to an fgp bi-Hilbertian bimodule $E$. Assume that $E$ is a W-regular. Then, the state $\tau\circ\Phi_\infty$ on $O_E$ is $\alpha$-KMS for the gauge action on $O_E$ and coincides with $\phi_{LN,\tau}$. By Theorem \[prop:tau-phi=LN\] and Corollary \[koroalala\], we have that $\phi_{\omega,\tau}=\tau\circ \Phi_\infty$ for any $\alpha$-quasi-invariant positive trace $\tau$ and extended limit $\omega$ at $\alpha$ assuming that $E$ is strictly regular. Obstructions to bi-Hilbertian bimodule structures {#sub:no-left} ------------------------------------------------- In the two previous subsections, we assumed our $A$-bimodule $E$ to be an fgp bi-Hilbertian bimodule, and imposed the additional assumption of strict W-regularity (see Definitions \[ass:one\] and \[ass:two\]). The assumptions allowed us to construct a semifinite spectral triple from a Kasparov module relying on both the left and the right inner product on $E$. Instead, we can just use [@LN] to proceed from a KMS-state directly to a semifinite spectral triple whose associated KMS-state as in Corollary \[cor:phi-omega\] coincides with the original KMS-state. In order to compare the indirect approach for strictly W-regular modules to the direct approach from the KMS-state we will need to extend our module to von Neumann algebra coefficients, and along the way we derive obstructions to having the structure of a strictly W-regular bi-Hilbertian bimodule structure on an $A-A$-correspondence. We suppose that we have a finitely generated projective right $A$-module $E_A$, with $A$ unital, and carrying a unital left action of $A$. Let $\tau:A\to{\mathbb{C}}$ be a faithful positive trace satisfying the Laca-Neshveyev condition for $\alpha\geq 0$ (see Definition \[ass:3.5\] on page ), and define the associated KMS-state on $O_E$ by $$\phi_{LN,\tau}(S_\mu S_\nu^*):=\delta_{|\mu|,|\nu|}\tau((\nu|\mu)_A){\mathrm{e}}^{-\alpha|\mu|},\quad \mu,\,\nu\in {\mathcal{F}_E}^{\rm alg}.$$ By construction, $\phi_{LN,\tau}|_A=\tau$. The Cuntz-Pimsner algebra $O_E$ acts on the GNS-space $L^2(O_E,\phi_{LN,\tau})$ by left multiplication and, since $\phi_{LN,\tau}$ restricts to a trace on $A$, $A$ acts by both left and right multiplication on $L^2(O_E,\phi_{LN,\tau})$. For notational simplicity, we identify $\tau$ with its normal extension to $A''$. Note that $A''$ is independent of whether we take the bicommutant in $L^2(O_E,\phi_{LN,\tau})$ or $L^2(A,\tau)$ and by faithfulness of $\tau$ we can identify $A$ with its image under the GNS-representation and obtain an inclusion $A\subseteq A''$. \[adoubprim\] Let $E$ be an fgp right $A$-Hilbert $C^*$-module with a left unital action of $A$, $\tau$ a faithful positive trace on $A$ satisfying the Laca-Neshveyev condition and let $P_0:L^2(O_E,\phi_{LN,\tau})\to L^2(A,\tau)\subset L^2(O_E,\phi_{LN,\tau})$ denote the orthogonal projection. It then holds that $$A''=P_0O_E''P_0.$$ It is clear that $A''\subseteq P_0O_E''P_0$. To prove the converse inclusion, take $T\in P_0O_E''P_0$ and write $T=P_0T_0P_0$ where $T_0$ is the WOT-limit of a net $T_\lambda=\sum_j S_{\mu_{\lambda,j}}S_{\nu_{\lambda,j}}^*\in O_E$. We have that $$P_0T_\lambda P_0=\sum_{j: |\mu_{\lambda,j}|=|\nu_{\lambda,j}|=0}S_{\mu_{\lambda,j}}S_{\nu_{\lambda,j}}^*,$$ so $P_0T_\lambda P_0\in A$ and $T\in A''$. We can define a conditional expectation $$\tilde{\Phi}_\infty:O_E''\to A'', \quad\tilde{\Phi}_\infty(S):=P_0SP_0,$$ which is well defined by Proposition \[adoubprim\]. Using the expectation $\tilde{\Phi}_\infty$ we can define a right module $\Xi_{A''}$ by completing $O_E$ in the norm defined by the inner product $$(S_1|S_2)_{A''}:=\tilde{\Phi}_\infty(S_1^*S_2),\quad S_1,\,S_2\in O_E.$$ It is clear that $L^2(O_E,\phi_{LN,\tau})=L^2(\Xi_{A''},\tau)$. The construction of $\Xi_{A''}$ does not require $E_A$ to be biHilbertian, just an $A$-$A$-correspondence. The following result follows from the relations defining the Cuntz-Pimnser algebra and the fact that $\tilde{\Phi}_\infty$ is a conditional expectation. \[phitildeinfocm\] For $\mu\in E^{{\otimes}k}$ and $\nu\in E^{{\otimes}l}$, $$\tilde{\Phi}_\infty(S_\mu^* S_\nu)=\delta_{|\mu|,|\nu|}( \mu|\nu)_A.$$ In particular, the map $\mu\mapsto S_\mu$ extends to an $A''$-linear isometric embedding ${\mathcal{F}_E}\otimes_A A''\to \Xi_{A''}$ of the Fock module. We now turn to describing the WOT-closure of $E$ inside $O_E''$. We will identify $E$ with an $A$-sub-bimodule of $O_E$ via $\mu\mapsto S_\mu$. \[lem:weak-biHilb\] Let $A$ be a unital $C^*$-algebra. Let $E_A$ be a finitely generated projective right $A$-module with a unital left action, and suppose that $\tau:A\to{\mathbb{C}}$ satisfies the Laca-Neshveyev condition on $E$ for $\alpha\geq 0$. Then $$E'':=\overline{E}^{\rm WOT}\subset O_E''$$ is an $A''$-bimodule. Moreover, the following hold: 1. As right $A''$-modules, $E''\cong E\otimes_A A''$ and the isomorphism is an isomorphism of $A''$-Hilbert $C^*$-modules when equipping $E''$ with the right inner product $$( \mu|\nu)_{A''}:=\tilde{\Phi}_\infty(S_\mu^* S_\nu), \quad\mu,\nu\in E''.$$ 2. The right $A''$-Hilbert $C^*$-module $E''$ is finitely generated and projective. 3. If $E$ is finitely generated and projective as a left $A$-module, then $E''$ is finitely generated and projective as a left $A''$-module. 4. If the implication $$\label{lefass} P_0ee^*P_0=0\Rightarrow e=0 \quad \forall e\in E'',$$ holds, the expression $${}_A(e|f)^{~}:=P_0S_eS_f^*P_0$$ gives a left inner product on $E''$ making it into a bi-Hilbertian bimodule. The right Watatani index of $E''$ is $1$ and ${\mathfrak{q}}_k={\rm Id}_{(E'')^{{\otimes}k}}$ for all $k$ (and so is invertible). 5. If $E''$ is a finitely generated projective module from the left and the implication holds, then $E''$ is a strictly $W$-regular fgp bi-Hilbertian bimodule over $A''$. The initial statement of the lemma is clear since $E$ is an $A$-sub-bimodule of $O_E$, so its WOT closure is a bimodule over $A''$. To prove statement 1. we note that $E\otimes_A A''\cong EA''\subseteq O_E''$. Moreover, using that $E$ is finitely generated and projective, it follows that $E''=EA''$ and therefore $E''\cong E\otimes_A A''$ follows. Statement 1. now follows from Proposition \[phitildeinfocm\]. Statement 2. follows from statement 1. because $E$ is finitely generated and projective over $A$. Statement 3. is proven in a similar way as Statement 1., indeed if $E$ is finitely generated and projective as a left $A$-module then $E''\cong A''\otimes_A E$ as a left $A$-module. Statement 4. is less trivial. Assuming that the implication holds, it is straight-forward to verify that the left and right actions are compatible, i.e. that the $A$-action from the left/right is adjointable for the right/left inner product. For $E''$ to be a bi-Hilbertian bimodule it remains to show that the norm arising from the left inner product is equivalent to the norm arising from the right inner product. For any $e\in E$, we compute that $$\begin{aligned} \nonumber \Vert{}_A(e|e)\Vert_A=\Vert P_0S_{e}S_{e}^*P_0\Vert_{L^2(A,\tau)} &=\Vert S_{e}^*P_0S_{e}\Vert_{L^2(E^*,\phi_\tau)} =\Vert S_{e}^*S_{e}\Vert_{L^2(E^*,\phi_\tau)}=\\ \label{relatingthetwoinnprod} &=\Vert(e|e)_A\Vert_{L^2(E^*,\phi_\tau)}=\Vert(e|e)_A\Vert_A\end{aligned}$$ where the norm of $S_{e}^*S_{e}$ is attained by $S_{e}^*P_0S_{e}$ on $L^2(E^*,\phi_\tau)$ by the assumption that ${}_A(\cdot|\cdot)$ is positive definite. Equation shows that the two norms $\sqrt{\Vert(\cdot |\cdot)_A\Vert_A}$ and $\sqrt{\Vert{}_A(\cdot |\cdot)\Vert_A}$ on $E$ are equivalent. To finalize the proof of statement 4., we compute the right Watatani index of $E$, which exists because $E''$ is finitely generated projective by statement 2. We compute on $L^2(A,\tau)\subset L^2(O_E,\phi_\tau)$ that $$\langle a, \sum_j{}_A(e_j|e_j)a\rangle =\langle a,\sum_jP_0\pi^{(1)}(\Theta_{e_j,e_j})P_0a\rangle= \phi_\tau(a^*P_0\pi^{(1)}({\rm Id_E})P_0a)=\phi_\tau(a^*1_Aa)=\tau(a^*a).$$ By the faithfulness of $\tau$ and Cuntz-Pimsner covariance we can now deduce that the right Watatani index is equal to $1_A$. This immediately implies that ${\mathfrak{q}}_k={\rm Id}_{(E'')^{{\otimes}k}}$ for all $k$. Finally, statement 5. follows from that under the stated assumptions, $E''$ is an fgp bi-Hilbertian bimodule (using statements 1., 2. and 4.) and by statement 4. ${\mathfrak{q}}_k$ satisfies the condition in Definition \[ass:two\], so $E''$ is strictly W-regular. \[leftdoubprime\] There are examples of $A$-$A$-correspondences $E$ that are fgp from the right but not fgp from the left such that $E''$ is fgp from the left. These examples come from (certain) self-similar dynamical systems, see [@KajWat]. Here is a simple example. Let $A=C([0,1])$, $$\gamma_1:[0,1]\to[0,1]\quad\gamma_1(x)=x/2,\qquad \gamma_2:[0,1]\to[0,1]\quad\gamma_2(x)=1/2+x/2,$$ and $E=C(\{(\gamma_1(x),x):x\in[0,1]\}\cup \{(\gamma_2(x),x):x\in[0,1]\})$. The correspondence structure is defined for $a,\,b\in A$ and $e\in E$ by $$(a\cdot e\cdot b)(\gamma_j(x),x)=a(\gamma_j(x))e(\gamma_j(x),x)b(x) \quad\mbox{and}\quad (e_1|e_2)_A(x)=\sum_{j=1,2}\overline{e_1(\gamma_j(x),x)}e(\gamma_j(x),x).$$ The graphs of $\gamma_1$ and $\gamma_2$ in $[0,1]\times [0,1]$ are disjoint and their respective characteristic function $\chi_1$ and $\chi_2$ are elements of $E$. One checks directly that $\{\chi_1,\chi_2\}$ is a right frame for $E_A$, and since $(\chi_1|\chi_2)=0$, there is an isomorphism of right Hilbert $C^*$-modules $E_A\cong C[0,1]\oplus C[0,1]$. We conclude that $E_A$ is fgp from the right. Using the frame $\{\chi_1,\chi_2\}$, we can identify ${}_AE\cong C[0,1/2]\oplus C[1/2,1]$ as a left $C[0,1]$-module. As a left module, ${}_AE$ is therefore finitely generated but clearly not projective as the rank of $E_x:=E/C_0([0,1]\setminus \{x\})E$ is discontinuous at $x=1/2$. We shall now see that it is even impossible for a left inner product compatible with the right inner product to exist. Let $0\leq \phi\in A$ be $1$ on $[0,1/2]$. Then if we have a compatible left $A$-valued inner product $${}_A(\chi_1|\chi_2)(x)={}_A(\phi\cdot\chi_1|\chi_2)(x)=\phi(x){}_A(\chi_1|\chi_2)(x) ={}_A(\chi_1|\chi_2)(x)\phi(x)={}_A(\chi_1|\phi\cdot\chi_2)(x).$$ Taking the infimum over such $\phi$ we see that the support of ${}_A(\chi_1|\chi_2)$ is contained in $\{1/2\}$. Then for arbitrary $a,\,b\in A$ $$\begin{aligned} &{}_A(a\chi_1+b\chi_2|a\chi_1+b\chi_2)\\ &=a{}_A(\chi_1|\chi_1)a^* +b{}_A(\chi_2|\chi_2)b^*+a(1/2)b^*(1/2){}_A(\chi_1|\chi_2)+b(1/2)a^*(1/2){}_A(\chi_2|\chi_1).\end{aligned}$$ From here one can show that any inner product taking values in the continuous functions takes values in the functions vanishing at $1/2$. Then one shows that the associated norm can not be equivalent to the right inner product. The situation is better for $E''$. We consider the trace $\tau(a):=\int_0^1 a(x)\mathrm{d}x$ on $C[0,1]$. A short computation shows that $\operatorname{Tr}_\tau^E=2\tau$ so $\tau$ satisfies the Laca-Neshveyev condition for $\alpha=\log(2)$ and extends to a KMS-state on $O_E$ at inverse temperature $\log(2)$. It is readily verified that $C[0,1]''=L^\infty[0,1]$, $E''\cong L^\infty[0,1]\oplus L^\infty[0,1]$ as a right module and $E''=L^\infty[0,1/2]\oplus L^\infty[1/2,1]$ as a left module. In particular, $E''$ is an fgp bi-Hilbertian bimodule over $L^\infty[0,1]$. The same discussion applies to any ‘graph separated’ iterated function system satisfying the open set condition. See [@KajWat] for more details. \[prop:left-noleft\] Let $E_A$ be a right $A$-Hilbert $C^*$-module with a unital left action and assume that $E$ is finitely generated both as a left and a right $A$-module and $E''$ is finitely generated and projective both as a left and a right $A''$-module. Then $E_A$ has a left inner product such that $E$ is an fgp bi-Hilbertian bimodule, has finite right Watatani index and is $W$-regular with ${\mathfrak{q}}_k$ invertible for all $k$ if and only if $\tilde{\Phi}_\infty$ is faithful and $\tilde{\Phi}_\infty(O_E)\subseteq A$. We remark that if $E$ is W-regular and ${\mathfrak{q}}_k$ is invertible for all $k$, then $E$ is strictly W-regular by [@GMR Lemma 3.8]. First suppose that $\tilde{\Phi}_\infty$ is faithful and $\tilde{\Phi}_\infty(O_E)\subseteq A$. Lemma \[lem:weak-biHilb\] shows that $E''$ has a left inner product making $E$ (not just $E''$) bi-Hilbertian with right Watatani index $1_{A}$ and (invertible) ${\mathfrak{q}}_k={\rm Id}_{(E)^{{\otimes}k}}$. Since $E$, both as a left and a right module, is finitely generated and admits a Hilbert $C^*$-module structure it is also projective. Conversely, if $E$ is an fgp bi-Hilbertian bimodule with finite right Watatani index and is W-regular with invertible ${\mathfrak{q}}_k$, then [@RRS] proves that the map $\Phi_\infty(S_\mu S_\nu^*) ={}_A(\mu|{\mathfrak{q}}_{|\nu|}(\nu))$ is a (faithful) conditional expectation. That it agrees with $\tilde{\Phi}_\infty$ is a computation. The issue with ${\mathfrak{q}}_k$ being non-invertible is as follows. Since ${\mathfrak{q}}_k={\mathfrak{q}}_1{\otimes}{\mathfrak{q}}_1{\otimes}\cdots{\otimes}{\mathfrak{q}}_1$, the non-invertibility occurs with ${\mathfrak{q}}_1$. Supposing $O_E$ to be strictly W-regular, we can define the right module $\Xi_A$ as the completion of $O_E$ for the norm coming from $\Phi_\infty$. Then $\tilde{\Phi}_\infty(S_eS_f^*)={}_A(e|{\mathfrak{q}}_1(f))$, and so if ${\mathfrak{q}}_1$ is not invertible, we do not get a left inner product in this way. See [@RRS Example 3.10] for an example where we have strict W-regularity with ${\mathfrak{q}}_1$ not being invertible. Heuristically, one should view the passage to $\Xi_A$ as erasing the information about the left inner product on $E$, corresponding to the kernel of ${\mathfrak{q}}_1$. On the other hand, if ${\mathfrak{q}}_k$ is invertible for all $k$ then we can replace our left inner products ${}_A(\cdot|\cdot)^{E^{{\otimes}k}}$ by ${}_A(\cdot|{\mathfrak{q}}_k(\cdot))^{E^{{\otimes}k}}$ and obtain an equivalent inner product structure with right Watatani index 1. We can now relate our constructions above back to KMS-states. Let $A$ be a unital $C^*$-algebra, $E$ a finitely generated projective right $A$-Hilbert $C^*$-module with a unital adjointable left $A$-action, $\alpha\geq 0$ and $\tau$ a positive trace on $A$. We assume that this data satisfies the following conditions: - $\tau$ satisfies the Laca-Neshveyev condition. - The $A''$-bimodule $E''$ is finitely generated and projective from the left. - The implication (see page ) holds. We define a semifinite spectral triple $(O_E, L^2(O_E,\phi_{LN,\tau}),{\mathcal{D}}_\psi, {\mathcal{N}}, {\mathcal{T}})$ using that $L^2(O_E,\phi_{LN,\tau})=L^2(O_{E''},\phi_{LN,\tau})=L^2(\Xi_{A''},\tau)$ and pulling back the semi-finite spectral triple defined from the fgp bi-Hilbertian $A''$-bimodule $E''$ as in Lemma \[ximodsemi\] along the inclusion $O_E\to O_{E''}$. 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--- abstract: 'An analytic solution for an uniaxial spherical resonator is presented using the method of Debye potentials. This serves as a starting point for the calculation of whispering gallery modes (WGM) in such a resonator. Suitable approximations for the radial functions are discussed in order to best characterize WGMs. The characteristic equation and its asymptotic expansion for the anisotropic case is also discussed, and an analytic formula with a precision of the order $O[\nu^{-1}]$ is also given. Our careful treatment of both boundary conditions and asymptotic expansions makes the present work a particularly suitable platform for a quantum theory of whispering gallery resonators.' author: - Marco Ornigotti - Andrea Aiello title: Theory of Anisotropic Whispering Gallery Resonators --- I. Introduction =============== London’s St. Paul’s Cathedral is famous for its rich history and architecture; one of the most unique aspect of this building is the whispering gallery that runs along the interior wall of its dome [@ref0]. When sounds are uttered in low voice against the wall, sound waves generated circulate around the wall many times before fading away. As these waves propagate, they bring with them sounds that are audible on the opposite side of the dome. On the contrary, if the same sounds are uttered at higher volume, the frequencies of these sounds waves will not match and a lot of noise is created, making the message difficult to be heard at any part of the wall. The physical explanation of this effect was firstly given more than a century ago in terms of reflection of acoustic *rays* from a surface near the dome apex. It was initially assumed that the rays that propagate along different large arcs of the dome in a form of a hemisphere should concentrate only at the point diametrically opposite to the source of the sound. Afterwards lord Rayleigh, in his *Theory of sound* [@ref1], provided a different explanation of the effect that he named *Whispering Gallery Waves*: sound clutches to the wall surface and creeps along it without diverging as fast as during the free space propagation: these sound waves then propagate within a narrow layer adjacent to the wall surface. It was then discovered, at the beginning of the last century, that optical whispering gallery waves can exist even in dielectric spheres [@ref2; @ref3]. An optical resonator that shows this particular wave structure was then called *Whispering Gallery Resonator* (WGR). In recent times whispering gallery waves have found new fame with the development of nano-optics, in particular with the ability to manufacture spherical and toroidal WGR with very high quality factors that ranges from $10^7$ to $10^{10}$ [@ref4bis; @ref5bis; @ref6bis]. This motivated a large theoretical and experimental work around these devices (see for example Ref. [@ref4; @ref5; @ref6; @ref7] and references therein). The ability to store light in microscopic spatial volumes for long periods of time (due to the high $Q$-factor) resulted in a significant enhancement of nonlinear interactions of various kinds like four-wave mixing [@ref8; @ref9], Raman [@ref10], parametric and Brillouin scattering [@ref7; @ref11], microwave up-conversion [@ref12], second and third order harmonic generation [@ref13; @ref14; @ref15]. Besides the field of nonlinear optics, WGRs were recently used even for cavity QED experiments [@ref15bis; @ref15ter]. For an exhaustive review on the applications of WGRs see Ref. [@ref15quater]. Since many of the applications of these resonators involve nonlinear optics, WGR are commonly fabricated using nonlinear materials or anisotropic crystals [@ref7]. Despite the wide scientific production in the theory of anisotropic spherical resonators, that ranges from generalization of scattering methods [@ref17; @ref18; @ref18bis; @ref18ter], potential method [@ref17bis], dyadic Green function approach [@ref17ter] and Fourier-based analysis [@ref19], and even though an extensive study of isotropic WGRs was done in the past [@ref26], detailed studies on anisotropic WGRs are still very few. To the knowledge of the authors anisotropic WGRs are mainly reported in literature as studied with FDTD models [@ref20; @ref21], cavity loading [@ref22] and direct solution of Maxwell’s equation with a surface nonlinear polarization as a forcing term [@ref6]. In this work, we intend to develop a suitable analytic theory for WGRs, starting from a review of the solutions of Maxwell’s equations in an uniaxial spherical resonator, then presenting and discussing its mode structure in the limit of small anisotropy, and finally obtaining the spectrum of whispering gallery modes sustained by the resonator and their structure, discussing how anisotropy influence those modes. A detailed discussion on the application of boundary conditions to this resonator is also presented, pointing out how to apply correctly these conditions and discussing some of their basic features that, to the knowledge of the authors, is not present in earlier works. It is opinion of the authors that this discussion is important in order to better understand the physics behind this problem. This is the first main result of this work. Finally, we introduce a more accurate approximation for the field outside the resonator when the index of the Hankel function tends to infinity, as we noticed that the commonly used power expansion (as, for example, the one presented in Ref. [@ref26]) does not match the exact function completely, i.e. it has an additional phase factor respect to the real function. Such phase factor becomes relevant when field amplitude, as opposed to field intensity, turn to be fundamental. This happens, for example, when one wants to quantize the electromagnetic field inside the resonator, as required for a proper treatment of spontaneous emissions processes. Thus, the present work may serve as basis for a quantum theory of WGRs. This is the second main result of this work. This paper is organized as follows: in section II the Debye method of potentials for solving Maxwell’s equations is briefly presented and then used in section III to develop the theory of an anisotropic spherical resonator for a dielectric uniaxial sphere. In section IV, *Whispering Gallery Modes* (WGMs) are obtained as limiting case of the normal modes of the dielectric sphere with high quantum numbers and their spectrum is discussed. II. Debye method of potentials ============================== A. Isotropic Solution --------------------- Before considering the problem of an uniaxial spherical resonator, it is pedagogical to briefly review the method of Debye potentials [@ref26], largely used to solve Maxwell’s equations in integrable systems. Let us consider a monochromatic field with an harmonic time dependence (i.e. , $\vec{E}(\vec{x},t)=\vec{E}(\vec{x})e^{-i\omega t}$) in an isotropic sourceless domain $\Omega$; Maxwell’s equations inside $\Omega$ can be written in the following symmetric form:\ \[maxeqGeneral\] $$\begin{aligned} \nabla \times \vec{E} & = & -i k \vec{H},\\ \nabla \times \vec{H} & = & i k \vec{E} ,\end{aligned}$$ where $k=\omega\sqrt{\varepsilon}/c$ is the wavevector in vacuum and and $\varepsilon$ is the dielectric constant inside $\Omega$. In order to fully determine the fields, it is necessary to specify their values on the domain boundary $\partial\Omega$. Electromagnetic boundaries are usually of two types: perfectly conducting walls (the field is zero on $\partial\Omega$), or open systems, where the field components inside $\Omega$ and the ones outside $\Omega$ must match on $\partial\Omega$. The boundary, together with symmetry considerations, gives a hint on which is the more suitable coordinate system to be used to solve the problem (e.g. spherical coordinates for spheres, cilindrical coordinates for wires etc.) [@ref23]. Let us specify our problem by considering an open system constituted by a sphere of dielectric constant $\varepsilon$ and radius $R$ surrounded by an isotropic medium (i.e. air) [@ref24]. The set of Maxwell’s equations in the spherical reference frame can be written in the following compact form:\ \[Maxwell\] $$\begin{aligned} \frac{\partial}{\partial\zeta_n}(L_mE_m)-\frac{\partial}{\partial\zeta_m}(L_nE_n) & = & - i k \epsilon_{lnm}L_n L_m H_l,\label{maxwellA}\\ \frac{\partial}{\partial\zeta_n}(L_mH_m)-\frac{\partial}{\partial\zeta_m}(L_nH_n) & = & i k \epsilon_{lnm} L_n L_m E_l,\label{maxwellB}\end{aligned}$$ where $\{l,n,m\} \in \{1, 2, 3\}$, $\zeta_m$ are the spherical coordinates ($\zeta_1=\varphi$, $\zeta_2=\theta$ and $\zeta_3=r$), and $L_m$ are the metric coefficients of the spherical reference frame ($L_1=r\sin\theta$, $L_2=r$, $L_3=1$). The Levi-Civita symbol $\epsilon_{lnm}$ on the right-hand side of Eqs. is equal to 1 if $\{l,n,m\}$ is equal to $\{1,2,3\}$ or any of its even permutation, is equal to $-1$ for any odd permutation of $\{1,2,3\}$ and equal to zero elsewhere. In this reference frame, the fields can be decomposed in the so-called transverse electric (TE) and transverse magnetic (TM) waves: TE waves are characterized by having $E_r=0$, i.e. the electric field is transveral with respect to the radial direction $r$. TM waves are instead characterized by having the magnetic field transveral with respect to the radial direction (i.e. $H_r=0$) [@ref23]. For the sake of simplicity, let us fix our attention on TM waves; the calculations for TE waves can be straightforward obtained by analogy. From Eqs. for $l=3$, by substituting $H_r=0$ it is possible to introduce the $W$ function such that\ \[potW\] $$\begin{aligned} r\sin\theta E_{\varphi} & = & \frac{\partial W}{\partial\varphi},\label{potWa}\\ rE_{\theta} & = & \frac{\partial W}{\partial\theta}.\label{potWb}\end{aligned}$$ By substituting relations into Eqs. for $l=1,2$ and writing $W=\partial U/\partial r$, where $U$ is the TM Debye potential, from we obtain\ $$\begin{aligned} H_{\varphi} & = & i k \frac{1}{r}\frac{\partial U}{\partial\theta},\\ H_{\theta} & = & - i k \frac{1}{r\sin\theta}\frac{\partial U}{\partial\varphi},\end{aligned}$$ and according to Eq , the $r$-component ($l=3$) of the electric field is given by\ $$\label{EradialTM} E_r = - \frac{1}{r^2\sin\theta}\Big[ \frac{\partial}{\partial\varphi}\Big( \frac{1}{\sin\theta}\frac{\partial}{\partial\varphi} \Big) + \frac{\partial}{\partial\theta} \Big( \sin\theta\frac{\partial}{\partial\theta} \Big) \Big]U.$$ Note that the differential operator that acts on the potential $U$ in this equation is the angular momentum operator $\hat{\vec{L}}= - (\vec{r}\times\nabla)$, that is the same operator that originates the centrifugal potential in the Hydrogen atom [@ref41]. Therefore, all the components of the electric field are expressed in terms of the $U$ potential solely. In order to explicit them, it is necessary to find the equation which the $U$ potential satisfy. To do this, we can use one of the last two equations left available from Eq. , i.e. the ones with $l=1,2$. By using one of them it is possible to obtain the following wave equation that $U$ must satisfy\ $$\label{waveU} \frac{\partial^2 U}{\partial r^2}+ \nabla^2_{\perp}U + k^2U=0,$$ where $\nabla^2_{\perp}$ is the angular part of the Laplace operator in spherical coordinates, i.e. the angular momentum operator $\hat{\vec{L}}$.\ The solution can be easily found with the method of separations of variables; writing the potential as $U(r,\theta,\varphi)=\Psi(r)\Theta(\theta)\Phi(\varphi)$ and substituting this into Eq. , we obtain the following equations for the functions $\Psi(r)$, $\Theta(\theta)$ and $\Phi(\varphi)$ [@ref23]:\ $$\begin{aligned} \frac{d^2\Psi}{dr^2} & + & \Big( k^2-\frac{c_3}{r^2} \Big)\Psi = 0,\label{parteRadiale}\\ \frac{1}{\sin\theta}\frac{d}{d\theta} \Big( \sin\theta\frac{d\Theta}{d\theta} \Big) & + & \Big( c_1 - \frac{c_2}{\sin^2\theta} \Big)\Theta = 0,\label{parteTheta}\\ \frac{d^2\Phi}{d\varphi^2} & + & c_2\Phi = 0,\label{partePhi}\end{aligned}$$ where $c_1$, $c_2$ and $c_3$ are the separation constants appearing in the equations by separating the variables, whose values must be $c_1=n(n+1)$ and $c_2=m^2$ in order to the solution to these equations to be unique, i.e. physically meaningful; $n$ and $m$ are integers, including zero. With these values the solution is straightforward. Equations and give rise to the so-called spherical harmonics $Y_{nm}(\theta,\varphi)=\mathscr{N}P_n^m(\cos\theta)e^{im\varphi}$, i.e. the eigensolutions of the angular momentum operator [@ref23], where $P_n^m(\cos\theta)$ are the associated Legendre functions of the first kind that are solution of Eq. , while the complex exponential is a solution of Eq. . $\mathscr{N}$ is a normalization constant that guarantees that the integral over the solid angle is unitary. The radial equation can be transformed into a Bessel equation by the substitution $\Psi(r)=\sqrt{kr}Z(kr)$ that brings to:\ $$\label{besselRadiale} \frac{d^2Z}{dx^2}+\frac{1}{x}\frac{dZ}{dx}+ \Big( 1- \frac{\nu^2}{x^2} \Big)Z=0,$$ where $x=kr$ and $\nu=n+1/2$; the solutions to this equation are the four Bessel functions $J_{\nu}(x)$, $N_{\nu}(x)$, $H^{(1)}_{\nu}(x)=J_{\nu}(x)+iN_{\nu}(x)$ and $H^{(2)}_{\nu}(x)=J_{\nu}(x)-iN_{\nu}(x)$. Physically, the solution inside the sphere must be finite at the origin, and the only plausible solution is $J_{\nu}(x)$ because $N_{\nu}(x)$ has a divergence at the origin. Outside the sphere, instead, the solution should have the form of a runaway wave with the Sommerfeld condition at the infinity (i.e. , the solution must drop at infinity as the inverse square of the distance). For this reason the correct solution in this domain is the Hankel function of the first kind $H^{(1)}_{\nu}(x)$ because its asymptotic form decreases to zero as the inverse square of the distance for $x\rightarrow\infty$. Putting everything together, the TM Debye potential reads as follows:\ $$\label{solution} U_{nm}^{int/ext}(r,\theta,\varphi) = C_{int/ext} \sqrt{k r}Z_{\nu}(k r)Y_{nm}(\theta,\varphi),$$ where $n,m$ are the angular quantum numbers that address the single mode of the resonator, $Z_{\nu}(k r)$ is the radial Bessel-type function that is equal to the Bessel function $J_{\nu}(k r)$ inside the dielectric sphere, and is equal to the Hankel function of the first kind $H_{\nu}^{(1)}(k_0 r)$ outside the dielectric sphere. The constants $C_{int/ext}$ are to be determined by applying suitable boundary conditions. Note that the argument of the Bessel function inside the sphere contains the sphere dielectric constant $\varepsilon$ via the wavevector $k=\omega\sqrt{\varepsilon}/c$ while the argument of the Hankel function outside the sphere contains only the vacuum wavevector $k_0=\omega/c$ because $\varepsilon_{air}=1$.\ The components of the electric and magnetic fields for TM waves can then be written as a function of $U$ as follows:\ \[TMcomponent\] $$\begin{aligned} E_r & = & \Big( \frac{\partial^2}{\partial r^2}+k^2 \Big)U,\\ E_{\theta} & = & \frac{1}{r}\frac{\partial^2 U}{\partial r\partial\theta},\\ E_{\varphi} & = & \frac{1}{r\sin\theta}\frac{\partial^2 U}{\partial r \partial\varphi},\\ H_r & = & 0,\\ H_{\theta} & = & -ik\frac{1}{r}\frac{\partial U}{\partial\varphi},\\ H_{\varphi} & = & ik\frac{1}{r}\frac{\partial U}{\partial\theta}.\end{aligned}$$ Note that in obtaining the expression of $E_r$ we have combined Eqs. and .\ If we proceed in a similar manner for TE waves, we obtain:\ \[TEcomponent\] $$\begin{aligned} E_r & = & 0,\\ E_{\theta} & = & -ik\frac{1}{r}\frac{\partial V}{\partial\varphi},\\ E_{\varphi} & = & ik\frac{1}{r}\frac{\partial V}{\partial\theta},\\ H_r & = & \Big( \frac{\partial^2}{\partial r^2}+k^2 \Big)V,\\ H_{\theta} & = & \frac{1}{r}\frac{\partial^2 V}{\partial r\partial\theta},\\ H_{\varphi} & = & \frac{1}{r\sin\theta}\frac{\partial^2 V}{\partial r \partial\varphi},\end{aligned}$$ where $V$ is the TE field potential obtained by Eqs. by substituting the TE ansatz $E_r=0$. B. Boundary Conditions ---------------------- Prior to investigate the structure of the modes for the anisotropic resonator, it is important to discuss the boundary conditions that have to be applied to this problem. At the resonator surface $r=R$, the wave vector $k$ inside the dielectric sphere has to match the wave vector $k_0=\omega/c$ outside the sphere and the constants $C_{int}$ and $C_{ext}$ should be chosen properly.\ There is not a unique way to fulfill boundary conditions: in fact, one could apply “pure" or “mixed" conditions: the former consists in applying the boundary conditions to all the components of only electric *or* magnetic field, while the latter applies the boundary to certain components of one field and certain other components of the other field. Obviously, these two different paths bring to the same physical solutions [@ref23]. Among these possibilities, in this work we chose to apply “pure" boundary condition, i.e. we impose that the tangential electric (magnetic) field components for TM (TE) waves has to be continuous at the resonator surface $r=R$, while the radial component of the displacement vector $\vec{D}=\varepsilon\vec{E}$ is continuous across the resonator surface. For TE fields, the radial condition is automatically fulfilled, since the resonator is non-magnetic (i.e. , $\mu=1$). The condition for the radial component of the displacement vector ($\varepsilon_{int}E_r^{int}=\varepsilon_{ext}E_r^{ext}$) across the resonator surface gives the ratio between the inner and outer coefficients\ $$\frac{C_{ext}}{C_{int}}=\varepsilon^{1/4}\frac{j_{\nu}(k_0\sqrt{\varepsilon}R)}{h_{\nu}^{(1)}(k_0 R)},$$ while the continuity of the tangential component $E_{\theta,\varphi}^{int}=E_{\theta,\varphi}^{ext}$ of the field gives rise to the so called characteristic equation, that allows to determine the allowed values for the wave vector $k$ (i.e. , to find the spectrum of the allowed modes) inside the resonator, and it turns out to be\ $$\label{eigenTM} \frac{[j_{\nu}(kR)]'}{j_{\nu}(kR)}=\sqrt{\varepsilon} \frac{[h_{\nu}^{(1)}(k_0R)]'}{h_{\nu}^{(1)}(k_0R)},$$ for TM waves and $$\label{eigenTE} \frac{[j_{\nu}(kR)]'}{j_{\nu}(kR)}=\frac{1}{\sqrt{\varepsilon}} \frac{[h_{\nu}^{(1)}(k_0R)]'}{h_{\nu}^{(1)}(k_0R)},$$ for TE waves. In these equations $j_{\nu}(x)=\sqrt{x}J_{\nu}(x)$ and $h_{\nu}^{(1)}(x)=\sqrt{x}H_{\nu}^{(1)}(x)$ are the Riccati-Bessel functions, and the prime indicates the total derivative with respect to the argument on which the functions depend, i.e. over $kR$ or $k_0R$. Although formally corrected, as they are these boundary conditions do not provide a unique solution to the determination of the mode patterns in the resonator. In order to better understand this not-uniqueness of the solution, let us consider the general structure of eqs. and . Let $T(x)$ be a piecewise function defined across an interface, placed at $x=1$, between two region of space, such that $T(x)=C^{(i)}f(x)$ for $x<1$ and $T(x)=C^{(e)}g(x)$ for $x>1$, with $f(x)$ and $g(x)$ two arbitrary real valued and regular functions. The constants $C^{(i,e)}$ are to be determined by the boundary conditions and they must be chosen in such a way that the following characteristic equation is satisfied:\ $$\label{generalB} \frac{f'(x)}{f(x)}=\alpha\frac{g'(x)}{g(x)},$$ where the apex stands for the derivative of the two functions with respect to their arguments. Since this is a generalization of the characteristic equations and , this equation must hold at the interface between the two region of space considered, i.e. its validity is limited to $x=1$. From eq. it is clear that if we admit that the derivatives $f'(x)$ and $g'(x)$ of the functions are equal at the separation interface $x=1$, then the functions themselves will be discontinuous with a jump that has the value of $1/\alpha$ . On the other hand if we now admit that the the functions $f(x)$ and $g(x)$ are equal at the separation interface $x=1$, then their derivatives must be discontinuous, and the magnitude of the discontinuity is precisely $\alpha$. The first situation corresponds to require that the derivative of the function $T(x)$ is continuous at the separation interface (i.e. $T'(1)^+ = T'(1)^-$, where the plus or minus superscript stands for the expression of $T(x)$ for $x>1$ and $x<1$ respectively). This implies that $C^{(e)}/C^{(i)} = f'(1)/g'(1)$ and the function $T(x)$ can be written as:\ $$T(x) = \left\{ \begin{array}{lc} \displaystyle{f(x)} & \displaystyle{x<1}, \\ \\ \\ \displaystyle{\Bigg[ \frac{f'(1)}{g'(1)} \Bigg] g(x)} & \displaystyle{x\geq 1}. \end{array} \right.$$\ It is then clear that taking $T'(x)$ to be continuous at the interface results in a discontinuity in the behavior of $T(x)$ while passing through $x=1$, whose magnitude is $f'(1)/g'(1)$, as is depicted in Fig.\[figuraBoundary1\]. ![(color online) The figure shows the behavior of the function $T(x)$ and its derivative $T'(x)$ when the condition of continuous derivative at the separation interface $x=1$ is considered. As can be noted, in this case the function shows a discontinuity while its derivative is (obviously) continuous. For this example we have used $f(x)=\mathrm{Ai}(x)$ and $g(x)=e^{-x}$, and the magnitude of the discontinuity in the function $T(x)$ at the separation interface is $f'(1)/g'(1) \simeq 0.5457$.[]{data-label="figuraBoundary1"}](fig1.png){width="50.00000%"} Conversely, the second condition on the functions $f(x)$ and $g(x)$ implies that the function $T(x)$ to be continuous at the separation interface (i.e. , $T(1)^+ = T(1)^-$), we have $C^{(e)}/C^{(i)} = f(1)/g(1)$ and the function $T(x)$ has the following form:\ $$T(x) = \left\{ \begin{array}{lc} \displaystyle{f(x)} & \displaystyle{x<1}, \\ \\ \\ \displaystyle{\Bigg[ \frac{f(1)}{g(1)} \Bigg]g(x)} & \displaystyle{x\geq 1}. \end{array} \right.$$\ In this case, taking $T(x)$ to be continuous at the interface results in a discontinuity in its derivative, whose magnitude is $f(1)/g(1)$, as Fig.\[figuraBoundary2\] underlines. ![(color online) The figure shows the behavior of the function $T(x)$ and its derivative $T'(x)$ when the condition of continuous function at the separation interface $x=1$ is considered. As can be noted, in this case the function itself is (obviously) continuous while its derivative shows a jump discontinuity. For this example we have used $f(x)=\mathrm{Ai}(x)$ and $g(x)=e^{-x}$ and the magnitude of the discontinuity in the derivative $T'(x)$ at the separation interface is $f(1)/g(1) \simeq 0.5457$.[]{data-label="figuraBoundary2"}](fig2.png){width="50.00000%"} In both cases, however, it is not possible to make the functions *and* the derivatives both continuous at the same time. This fact makes only possible to obtain the ratio between the two constants $C_{int}, C_{ext}$ and not their explicit value: in order to do that, another condition must be applied to the problem. This condition depends on the particular problem we are dealing on; in scattering problems, for example, the incoming field is known, and determines the field pattern on the resonator surface. In this case $C_{ext}$ is known and the ambiguity is removed. Another situation in which the ambiguity is overcome is by embedding the whole system (resonator plus surrounding medium) in an ideal perfectly reflective sphere of big, but finite radius $R_0$, in such a way that the boundary conditions at the metallic surface will completely determine the fields: this second approach is very useful if we are dealing with the quantization of the field in such a system. We want to end this discussion by pointing out that the first situation (derivative continuous at the interface) corresponds to the boundary condition for the electric field across a dielectric surface: the normal component with respect to the separation surface is discontinuous by a factor equal to the ratio of the two dielectric constants of the two regions, while the tangential components (i.e. , the derivative of the radial field in our spherical case) is continuous at the interface. The second situation, instead, corresponds to put continuous the normal component of the displacement vector across the separation surface, resulting in a discontinuity of the tangential component of the displacement vector at the interface. While the former correspond to the usual way of imposing boundary conditions in an electromagnetic problem, the latter is never used, but still valid. In this work, however, we are neither interested on scattering problems nor on field quantization, and so in the rest of the paper this ambiguity will not be removed. This does not create too much problems because we are only interested on the mode structure of the resonator. We leave this problem of not-uniqueness to future works.\ III. Normal modes of an uniaxial spherical resonator ==================================================== Let us consider the same dielectric spherical resonator of radius $R$ of the previous section, but with an uniaxial anisotropy along the $z$-axis described by the following dielectric tensor:\ $$\begin{aligned} \label{epsilon} \hat{\varepsilon}& =& \left( \begin{array}{ccc} \varepsilon_{xx} & 0 & 0\\ 0 & \varepsilon_{xx} & 0\\ 0 & 0 & \varepsilon_{zz}\\ \end{array} \right)= \nonumber \\ \vspace{1mm} & = &\varepsilon_{xx} (\mathbf{ \hat{x}\hat{x}}+\mathbf{\hat{y}\hat{y}})+\varepsilon_{zz}\mathbf{\hat{z}\hat{z}}.\end{aligned}$$ In order to use this dielectric tensor in Eqs. , it should be converted in spherical coordinates; this operation is simply done by converting the cartesian dyadics $\mathbf{\hat{x}\hat{x}}, \mathbf{\hat{y}\hat{y}}$ and $ \mathbf{\hat{z}\hat{z}}$ into the spherical dyadics $\mathbf{\hat{r}\hat{r}}, \boldsymbol{\hat{\theta}\hat{\theta}}$ and $\boldsymbol{\hat{\varphi}\hat{\varphi}}$ using the standard cartesian-to-spherical transformation relations [@ref23]. By performing this transformation, the dielectric tensor in spherical coordinates reads\ $$\hat{\varepsilon}=\left( \begin{array}{ccc} \varepsilon_{rr} & -\varepsilon_{r\theta} & 0\\ -\varepsilon_{r\theta} & \varepsilon_{\theta\theta} & 0\\ 0 & 0 & \varepsilon_{\perp}\\ \end{array} \right),$$ with $\varepsilon_{\pm} =(\varepsilon_{zz}\pm\varepsilon_{xx})/2$ and $\varepsilon_{\perp}=\varepsilon_{xx}$. We have then defined $\varepsilon_{rr}=\varepsilon_+ +\varepsilon_- \cos(2\theta)$, $\varepsilon_{\theta\theta}= \varepsilon_+ - \varepsilon_- \cos(2\theta)$ and $\varepsilon_{r\theta}=\varepsilon_- \sin(2\theta)$. The fact that the tensor components depend on the polar coordinate $\theta$ makes the problem to find the eigenmodes of the spherical resonator much more difficult. Moreover, in an anisotropic system it is in general no longer possible to divide the electric and magnetic fields in their TM and TE components. In order to overcome the latter problem, we will focus our attention on the case of small anisotropy, i.e. $\lambda=\varepsilon_-/\varepsilon_+\ll 1$ (this approximation is very good if we consider, for example, a dielectric sphere made of Lithium Niobate ($\mathrm{LiNbO_3}$) for which we have $\varepsilon_{xx}=5.3$, $\varepsilon_{zz}= 6.47$ and therefore $\lambda\simeq 0.01$). In such a way the fields can be decomposed in quasi-TE and quasi-TM oscillations, allowing us to solve the problem using the method of Debye potentials illustrated above. To this aim, and for the sake of clearness, let us rewrite the set of Eqs. for the anisotropic case as follows:\ \[AniMaxwell\] $$\begin{aligned} \frac{\partial}{\partial r}(r\sin\theta E_{\varphi})-\frac{\partial}{\partial\varphi}(E_r) & = & ik_0 \sin\theta (rH_{\theta}),\label{prima}\\ \frac{\partial}{\partial\theta}(E_r)-\frac{\partial}{\partial r}(rE_{\theta}) & = & ik_0 (rH_{\varphi}),\label{seconda}\\ \frac{\partial}{\partial\varphi}(r E_{\theta})-\frac{\partial}{\partial\theta}(r\sin\theta E_{\varphi}) & = & ik_0 r\sin\theta (rH_r),\label{terza}\end{aligned}$$ and\ \[AniMaxwell2\] $$\begin{aligned} \label{quarta} \frac{\partial}{\partial r}&(&r\sin\theta H_{\varphi})-\frac{\partial}{\partial\varphi}(H_r) = \nonumber \\ & = & ik_0 \sin\theta[ \varepsilon_{r\theta}(rE_r) - \varepsilon_{\theta\theta} (rE_{\theta})],\end{aligned}$$ $$\label{quinta} \frac{\partial}{\partial\theta}(H_r)-\frac{\partial}{\partial r}(rH_{\theta}) = -ik_0 \varepsilon_{\perp} (rE_{\varphi}),$$ $$\begin{aligned} \label{sesta} \frac{\partial}{\partial\varphi}(r H_{\theta})& -& \frac{\partial}{\partial\theta}(\sin\theta rH_{\varphi})= \nonumber \\ & = &- ik_0 r\sin\theta [\varepsilon_{rr}( rE_r) +\varepsilon_{r\theta} (rE_{\theta})].\end{aligned}$$ As in the previous section, we solve the problem for the quasi-TM component of the field (i.e. $H_r=0$); the quasi-TE solution is again obtained using similar arguments. We follow the solving procedure described in Ref.[@ref29]. Equation defines the $W$ function as in . Let us combine and the derivative with respect to $r$ of :\ $$\left\{ \begin{array}{l} \displaystyle{\frac{\partial}{\partial r}(rH_{\theta}) = \frac{1}{\sin\theta}ik_0\varepsilon_{\perp}\frac{\partial W}{\partial\varphi},} \\ \\ \\ \displaystyle{\frac{\partial^3}{\partial r^2\partial\varphi} (W) - \frac{\partial^2}{\partial r\partial\varphi}(E_r) = ik_0\sin\theta\frac{\partial}{\partial r}(rH_{\theta}).} \end{array} \right.$$\ If we substitute the expression of $\frac{\partial}{\partial r}(rH_{\theta})$ obtained from the first equation into the second one and if we define the differential operator $\hat{l}_x=\partial^2/\partial r^2 + k_0^2\varepsilon_{\perp}$ we obtain:\ $$\label{radial} \frac{\partial E_r}{\partial r}=\hat{l}_x W.$$ Combining now , and the derivative with respect to $r$ of gives:\ $$\left\{ \begin{array}{l} \displaystyle{\frac{\partial}{\partial r}(r\sin\theta H_{\varphi}) = ik_0r\sin\theta \Big( \varepsilon_{r\theta}E_r - \frac{1}{r}\varepsilon_{\theta\theta}\frac{\partial W}{\partial\theta} \Big),} \\ \\ \\ \displaystyle{\frac{\partial^2}{\partial r\partial\theta}(E_r) - \frac{\partial^2}{\partial r^2} (rE_{\theta}) = ik_0\frac{\partial}{\partial r} \Big( rH_{\varphi} \Big).}\\ \end{array} \right.$$ Again, by substituting the expression for $\frac{\partial}{\partial r}(rH_{\varphi})$ obtained from the first equation into the second one, and by defining the differential operator $\hat{l}_{\theta}=\partial^2/\partial r^2 + k_0^2\varepsilon_{\theta\theta}$ we obtain\ $$\label{intermedia} \Big[ k_0^2\varepsilon_{r\theta}r+\frac{\partial^2}{\partial r\partial\theta} \Big]E_r=\hat{l}_{\theta}\frac{\partial W}{\partial\theta}.$$ As in the isotropic case, we want to define $W$ as a function of the quasi-TM potential $U$, in order to fully determine the components of the fields as a function of the quasi-TM potential $U$ solely. In order to do this, let us compare Eqs. and . By noting that the differential operator $\hat{l}_{\theta}$ commutes with the operator $(k_0^2 \varepsilon_{r\theta}r + \partial^2/\partial r\partial\theta)$ that appears in , is it possible to define, after some simple algebra, the function $W$ as a function of the quasi-TM potential $U$ as follows:\ $$W=\frac{\partial}{\partial r} \Big( \hat{l}_{\theta} U \Big).$$ This allow us to write the components of the TM electric and magnetic field in terms of the quasi-TM potential $U$ as follows [@ref28]:\ \[TMfield\] $$\begin{aligned} E_r ^{TM} & = & \hat{ l}_x\hat{ l}_{\theta} U,\\ rE_{\varphi}^{TM} & = & \frac{1}{\sin\theta}\frac{\partial^2}{\partial r\partial\varphi}(\hat{l}_{\theta}U),\\ rE_{\theta}^{TM} & = & \Big( \varepsilon_{r\theta}k_o^2 r + \frac{\partial^2}{\partial r\partial\theta} \Big) \hat{l}_x U,\\ H_r^{TM} & = & 0,\\ rH_{\theta}^{TM} & = & \frac{i k_0 \varepsilon_{\perp}}{\sin\theta}\frac{\partial}{\partial\varphi}(\hat{l}_{\theta}U),\\ rH_{\varphi}^{TM} & = & i k_0 \Big[ \varepsilon_{\theta\theta}\frac{\partial}{\partial\theta} + \varepsilon_{r\theta} \Big( 1-r\frac{\partial}{\partial r} \Big) \Big] \hat{l}_x U.\end{aligned}$$ Again, if we proceed in a similar manner for the quasi-TE waves we obtain:\ \[TEfield\] $$\begin{aligned} H_r^{TE} & = & \hat{l}_x\hat{l}_{\theta} V,\\ rH_{\theta}^{TE} & = & \frac{\partial^2}{\partial r\partial\theta}(\hat{l}_{\theta}V),\\ rH_{\varphi}^{TE}& = & \frac{1}{\sin\theta}\frac{\partial^2}{\partial r\partial\varphi}(\hat{l}_x V),\\ E_r ^{TE} & = & 0,\\ rE_{\theta}^{TE} & = & \frac{ik_0}{\sin\theta}\frac{\partial}{\partial\varphi}(\hat{l}_x V),\\ rE_{\varphi}^{TE} & = & -ik_0\frac{\partial}{\partial\theta}(\hat{l}_x V),\end{aligned}$$ where $V$ represents the quasi-TE potential.\ Equations and represent the quasi-TM and quasi-TE components of electric and magnetic field inside an anisotropic spherical resonator in terms of the TM and TE quasi-potentials $U$ and $V$.\ The next step consists in constructing an equation that the quasi-potentials $U$ and $V$ satisfy, whose solutions give the mode fields of the resonator. To do that, let us consider a general electric and magnetic field, whose components are written as the superposition of the quasi-TE and quasi-TM oscillations, i.e. $E_i=E_i^{TM}+E_i^{TE}$ and $H_i=H_i^{TM}+H_i^{TE}$. Substituting this ansatz in Eqs. and , after some algebra we arrive at a set of two coupled equations for the quasi-potentials $U$ and $V$ that reads [@ref29]\ \[coupled\] $$\label{coupledU} \hat{L}_H V = 2 i \varepsilon_- k_0 \hat{\Upsilon}\frac{\partial U}{\partial\varphi},$$ $$\label{coupledV} \hat{L}_E U = 2 i \varepsilon_- k_0\hat{\Upsilon}\frac{\partial V}{\partial\varphi},$$ where we have defined:\ $$\begin{aligned} \hat{L}_H & = & (\nabla^2_{\perp}+r^2 \hat{l}_x)\hat{l}_{\theta}-2\varepsilon_- k_0^2\frac{\partial^2}{\partial\varphi^2},\nonumber\\ \hat{L}_E & = & \hat{\Xi}_0 + \lambda\hat{\Xi}\hat{l}_x,\nonumber\\ \nabla^2_{\perp} & = & \frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\sin\theta\frac{\partial}{\partial\theta}+\frac{1}{\sin^2\theta}\frac{\partial^2}{\partial\varphi^2},\nonumber\\ \hat{\Upsilon} & = & \Big( r \hat{l}_x -2\frac{\partial}{\partial r} \Big)\cos\theta -\sin\theta\frac{\partial^2}{\partial\theta\partial r},\nonumber\\ \hat{\Xi}_0 & = & \Big[ \nabla^2_{\perp} + r^2\frac{\partial^2}{\partial r^2}+ \varepsilon_+ \Big(1-\lambda^2 \Big)k_0^2r^2 \Big]\hat{l}_x +\nonumber \\ & + & 2\varepsilon_- (1-\lambda)k_0^2\frac{\partial^2}{\partial\varphi^2},\nonumber\\ \hat{\Xi} & = & \cos(2\theta) \Big[ r^2\frac{\partial^2}{\partial r^2} - \nabla^2_{\perp} + 3 \Big(1-r\frac{\partial}{\partial r} \Big) \Big] +\nonumber\\ & + & \Big( 3-2r\frac{\partial}{\partial r} \Big) \sin(2\theta)\frac{\partial}{\partial\theta} +\nonumber \\ & + & \Big( 1-r\frac{\partial}{\partial r} \Big)-2\frac{\partial^2}{\partial\varphi^2},\nonumber\end{aligned}$$ and $\lambda=\varepsilon_-/\varepsilon_+$ is the anisotropy parameter. From Eqs. it is evident that the anisotropy gives rise to a coupling between the two quasi-potentials $U$ and $V$; this coupling is absent in the isotropic case in which the two potentials are independent one each other. These equations, in fact, contain the isotropic solution in the limit of $\lambda=0$. Making this substitution in equation and using the definition of the operator $\hat{L}_H$, we obtain:\ $$(\nabla^2_{\perp}+r^2\hat{l}_x)\hat{l}_{\theta}U=0.$$ Because we set $\lambda=0$, the two differential operators $\hat{l}_x$ and $\hat{l}_{\theta}$ are equal, since $\varepsilon_{\theta\theta}=\varepsilon_+ - \varepsilon_-\cos2\theta = \varepsilon_+ =\varepsilon$ and $\varepsilon_{\perp}=\varepsilon$, i.e. no anisotropy is present anymore. We now define $\hat{l}_{\theta}U=A$ as the isotropic potential, and we assume that this potential can be written in a separable way, i.e. $A(r,\theta,\varphi)=\Psi(r)Y_{nm}(\theta,\varphi)$, where $Y_{nm}(\theta,\varphi)$ are the eigensolutions of the angular momentum operator, whose eigenvalues are $-n(n+1)$ (i.e. , $\nabla^2_{\perp}Y_{nm}=-n(n+1)Y_{nm}$). Substitution of this ansatz into the previous equation and consequent simplification of the angular part then brings to the following radial equation:\ $$\label{reduceTo} [n(n+1)-r^2\hat{l}_x]\Psi(r)=0,$$ that is precisely the radial equation for the isotropic potential, whose solutions are the Bessel functions given in the previous section. The same procedure applied to the quasi-potential $V$ brings to its isotropic counterpart.\ Since we stated that the anisotropy is small (i.e. , $\lambda\ll 1$), then we can use the method of separations of variable to solve the coupled equations . We then write the quasi-potentials as follows:\ \[ansatz\] $$U(r,\theta,\varphi)=\sum_p U_p(r,\theta,\varphi)=\sum_p u_p(r)Y_{nm}(\theta,\varphi),$$ $$V(r,\theta,\varphi)=\sum_p V_p(r,\theta,\varphi)=\sum_p v_p(r)Y_{nm}(\theta,\varphi),$$ where the subscript $p$ stands for the three indexes $n$,$m$ and $q$ on which the quasi-potential depends; the polar index $n$ determines the number of field nodes along the polar coordinate $\theta$, the azimuthal number $m$ characterizes the nodes in the $\varphi$ direction and, finally, the radial index $q$ gives the number of field oscillations along the radial direction $r$ that is related with the solution of the characteristic equation. Substituting into , using relationships (4) and (5) of Ref.[@ref29] and equating the terms with equal angular part $Y_{nm}(\theta,\varphi)$, we obtain the following set of differential equations for the radial components $u_p(r)$ and $v_p(r)$ of the quasi-potentials[@ref30]:\ \[radialpart\] $$\begin{aligned} a_{1,n}^{TM} u_n & - & a_{2,n-2}^{TM} u_{n-2} - a_{3,n+2}^{TM} u_{n+2} =\nonumber \\ & - & 2\varepsilon_- mk_0[ b_{1,n-1} v_{n-1} - b_{2,n+1} v_{n+1} ],\end{aligned}$$ $$\begin{aligned} a_{1,n}^{TE} v_n & - & \lambda(a_{2,n-2}^{TE} v_{n-2} + a_{3,n+2}^{TE} v_{n+2}) =\nonumber \\ & - & 2\lambda mk_0[b_{1,n-1} u_{n-1}- b_{2,n+1} u_{n+1}].\end{aligned}$$ For the sake of clarity, the expresison of the coefficients $a_i^{TM/TE}$ and $b_i$ are reported in Appendix A. These equations require, in general, a numerical approach to be solved. However, in the limit of small anisotropy, i.e. $\lambda \ll 1$, a solution to Eqs. can be searched in terms of power series in the factor $\lambda$. The zeroth order solution brings (as shown before) to the solution of the isotropic spherical resonator in terms of the Riccati-Bessel functions \[see Eq. \]. The first order solution, i.e. the anisotropic correction we are searching for, is obtained by neglecting the terms that are proportional to $\lambda^2$ in Eqs. : however, the resulting equations contain in the right-hand side a term that is not zero (like in the zeroth order solution) but depends on the quasi-potentials $u_{n\pm1}$ and $v_{n\pm1}$. This coupling among neighbor radial modes is a signature of the anisotropy, that on one hand breaks the azimuthal degeneracy \[the azimuthal quantum number appears in the definition of the coefficients of Eqs. \] and on the other hand results in a coupling between radial modes. Although this coupling results in an impossibility of an analytic solution, it can be demonstrated [@ref29] that these terms are of the order $\lambda^2$ and at the first order they can be neglected. With this argument, Eqs. at the leading order $\lambda$ read\ \[firstorder\] $$\Big[ r^2 \Big( \frac{d^2}{dr^2}+\frac{\gamma_1^2}{r^2} \Big) -n(n+1) \Big] l_+ u_n (r)=0$$ $$\Big[ r^2 \Big( \frac{d^2}{dr^2}+\frac{\gamma_2^2}{r^2} \Big) -n(n+1) \Big] l_+ v_n (r)=0$$ where $\gamma_1$ and $\gamma_2$ are the TM and TE (respectively) anisotropic factor given by\ $$\begin{aligned} \gamma_1^2 &= & k_0^2\varepsilon_+\Big\{ 1-\lambda \Big[ 1-\frac{2m^2}{n(n+1)} \Big] \Big\} \nonumber \\ \gamma_2^2 & = & k_0^2\varepsilon_+\Big\{ 1-\lambda \Big[ \frac{1-4m^2}{4n(n+1)-3}+\frac{2m^2}{n(n+1)} \Big] \Big\}\nonumber\end{aligned}$$ Eqs. have the same structure of the radial equation for the isotropic case [@ref23]. The only difference is the presence of the $\gamma_i$ terms that modify the arguments of the Riccati-Bessel functions and the quasi-potentials can be written as\ $$U_{nm}(r,\theta,\varphi)=C_{int/ext}z_{\nu}(x)Y_{nm}(\theta,\varphi)$$ $$V_{nm}(r,\theta,\varphi)=C_{int/ext}z_{\nu}(x)Y_{nm}(\theta,\varphi)$$ where $z_{\nu}(x)$ corresponds to $j_{\nu}(x)$ inside the sphere and to $h_{\nu}^{(1)}(x)$ outside the sphere. Note also that inside the sphere, where the anisotropy exists, $x=\gamma_i r$, while outside the sphere $x=k_0 r$ (the surrounding medium is still isotropic). Substituting these expressions in Eqs. and we obtain all the components of the electric and magnetic fields in an uniaxial anisotropic spherical resonator\ \[quasiTMComponents\] $$\begin{aligned} E_r &= & \frac{n(n+1)}{r^2} \Big[ \sqrt{\frac{\pi}{2}}z_{\nu}(\gamma_1 x)Y_{n,m}(\theta,\varphi) \Big],\\ rE_{\theta} &= & -\frac{\partial^2}{\partial r\partial\theta} \Big[ \sqrt{\frac{\pi}{2}}z_{\nu}(\gamma_1 x)Y_{n,m}(\theta,\varphi) \Big]\\ rE_{\varphi} &= & \frac{1}{\sin\theta}\frac{\partial^2}{\partial r\partial\varphi} \Big[ \sqrt{\frac{\pi}{2}}z_{\nu}(\gamma_1 x)Y_{n,m}(\theta,\varphi) \Big]\\ H_r &= & 0\\ rH_{\theta} &= & -\frac{ik_0\varepsilon_{\perp}}{\sin\theta}\frac{\partial}{\partial\varphi} \Big[ \sqrt{\frac{\pi}{2}}z_{\nu}(\gamma_1 x)Y_{n,m}(\theta,\varphi) \Big]\\ rH_{\varphi} &= & ik_0\varepsilon_{\perp}\frac{\partial}{\partial\theta} \Big[ \sqrt{\frac{\pi}{2}}z_{\nu}(\gamma_1 x)Y_{n,m}(\theta,\varphi) \Big]\\\end{aligned}$$ for quasi-TM fields. Similar expressions can be written for the quasi-TE fields by replacing $\gamma_1$ with $\gamma_2$, exchanging the role of the electric and magnetic field and setting $\varepsilon_{\perp}=1$. The characteristic equation can be found by applying the boundary conditions and it turns out to be ($\tilde{\gamma_i}=\gamma_i/k_0\sqrt{\varepsilon_+}$)\ $$\label{CEquasiTM} \tilde{\gamma_1}\frac{[j_{\nu}(\gamma_1 kR)]'}{j_{\nu}(\gamma_1 kR)}=\frac{\varepsilon_{\perp}}{\sqrt{\varepsilon_+}}\frac{[h_{\nu}^{(1)}(k_0 R)]'}{h_{\nu}^{(1)}(k_0 R)}$$ for the quasi-TM waves, and $$\label{CEquasiTE} \tilde{\gamma_2}\frac{[j_{\nu}(\gamma_2 kR)]'}{j_{\nu}(\gamma_2 kR)}=\frac{1}{\sqrt{\varepsilon_+}}\frac{[h_{\nu}^{(1)}(k_0 R)]'}{h_{\nu}^{(1)}(k_0 R)}$$ for the quasi-TE waves.\ As can be seen from the previous equations, in the small anisotropy regime, the only effect of the anisotropy is a rescaling of the radial coordinate; this is in accordance with the fact that an uniaxial crystal shows two different refractive indexes: one in-plane ($\sqrt{\varepsilon_{xx}}$) and the other out-of-plane ($\sqrt{\varepsilon_{zz}}$). Different refractive indexes correspond to different optical paths, and this is exactly reflected in the rescaling effect of the anisotropy onto the radial part of the modes of the resonator. Note also that at this level of analysis, the anisotropy doesn’t affect the angular structure ($\theta$ and $\varphi$) of the modes. Another difference respect to the isotropic case is the value of the coefficients on the right-hand side of the characteristic equations: while the coefficient for the quasi-TE wave is analogous to its isotropic counterpart (if we substitute the isotropic dielectric constant $\varepsilon$ with the anisotropy-averaged dielectric constant $\varepsilon_+$), the coefficient for the quasi-TM wave reveals the presence of the anisotropy, since it is a ratio between the in-plane dielectric constant and the anisotropy-averaged one. This is not so surprising because for a dielectric uniaxial crystal, only the TM component suffer directly anisotropy, while the TE component does not, because the crystal is magnetical isotropic. IV. Whispering gallery Modes ============================ A. Radial functions with large indices -------------------------------------- With the term whispering gallery mode (WGM) are commonly addressed the set of modes with a large index $n$; strictly speaking, the real WGMs are only those for which it results that $n=m$ and the radial wavefunction shows no roots inside the resonator. However, modes with indices $n\neq m$ and with $q > 1$, but close to unity, have properties that are close to those of WGMs: this means that there is no great difference between a “pure" WGM and other modes with nearest indices. To study such modes, the first thing we have to do is to find a suitable approximation of Riccati-Bessel functions for large index. This approximation is useful either from the numerical (where computing Bessel functions of large index is highly time-consuming) or analytical (where the approximation gives the possibility to work with easier functions that suit better onto the problem) point of view. The appropriate approximation, however, should be searched bearing in mind that the argument of the Bessel function for a WGM near the sphere surface is of the order of its index, i.e. $\nu/x \simeq 1$. By introducing the following change of variables\ $$\zeta=\Big( \frac{2}{\nu} \Big) ^{1/3}(\nu-x)\nonumber,$$ the Bessel function inside the dielectric resonator can be very well approximated by the Airy function of the first kind Ai as follows [@ref31]:\ \[besselApprox\] $$\begin{aligned} j_{\nu}(x) & \simeq & \sqrt{2} \Big( \frac{\nu}{2} \Big) ^{1/6} \textrm{Ai}(\zeta) ,\label{Approx_a}\\ \frac{d}{dx}[j_{\nu}(x)] & \simeq & - \sqrt{2} \Big( \frac{2}{\nu} \Big) ^{1/6} \frac{d}{d\zeta} [\textrm{Ai}(\zeta)].\end{aligned}$$ The accuracy of this approximation is of the order $\nu^{-1}$; if $\nu$ exceeds 1000, this accuracy is very satisfactory for many calculations. this can be seen in Fig.\[figuraApprox1\], where Bessel functions of high order are compared with their Airy approximation and in Fig.\[compareApprox\], where is shown that as $\nu$ grows, the accuracy of the approximation became satisfactory.\ ![(color online) Comparison between $j_{\nu}(x)$ (solid black line) and its Airy approximation from Eq (dashed red line) for $\nu=1000.5$. The approximation holds very well up to $x\simeq\nu$, while for $x$ larger than $\nu$ (say for $x>1020$) it starts to fail.[]{data-label="figuraApprox1"}](fig3.png){width="50.00000%"} ![(color online) The figure shows the accuracy $\sigma$ as a function of the Bessel index $\nu$; the accuracy is defined as the ration between the difference of the true function $j_{\nu}(\nu)$ and its Airy approximation $\mathrm{Ai}(\zeta)$ and their sum, i.e. $\sigma=[j_{\nu}(\nu)-\mathrm{Ai}(\zeta^*)]/[j_{\nu}(\nu)+\mathrm{Ai}(\zeta^*)]$, where $\zeta^*$ is $\zeta$ evaluated for $x=\nu$. As can be seen, as $\nu$ grows, the approximation becomes more precise.[]{data-label="compareApprox"}](fig4.png){width="50.00000%"} For the solution outside the resonator (the Hankel function of the first kind) various approximations are available. Here we use the following [@ref32]:\ $$\begin{aligned} \label{Happrox} h_{\nu}^{(1)}(x) & = & f(\eta)= \simeq \frac{e^{i[\nu(\tan\eta-\eta)-\frac{\pi}{4}]}}{\sqrt{\frac{\pi}{2}\tan\eta}} \Big\{ 1+\nonumber \\ & - &\frac{i}{\nu} \Big( \frac{1}{8\tan\eta}+\frac{5}{24}\frac{1}{\tan^3\eta} \Big)+ O[\nu^{-2}] \Big\}\end{aligned}$$ where $\cos\eta=\nu/x$; if $\nu$ is large enough (heuristically $\nu>1000$) the imaginary term inside the curly brackets can be neglected. The choice of this approximation rather than the one presented in Ref.[@ref26] reside in the fact that while the former is very good when the argument of the Hankel function is greater than the index (that is precisely the case of the outer functions), the latter is not suitable in this region, either for being out of phase with respect to Hankel function (as shown in Figs. \[approxH\_real\] and \[approxH\_imag\]) or to not approximate in the correct way the original function (Fig. \[approxH\_abs\]). Moreover, Fig. \[approxH\_abs\] shows that the field outside the resonator has all the characteristics of an evanescent wave, i.e. it decays exponentially as the distance from the resonator surface grows. In order to justify this evanescent behavior outside the resonator, one can directly solve Eq. in the limit $r>R$ (but still close to the resonator surface), where the terms $x=k_0 r$ outside the derivation symbol can be substituted with $k_0 R$, leading to the following equation:\ $$\label{evanescent} \frac{d^2Z}{dx^2}+\frac{1}{k_0 R}\frac{dZ}{dx}+\Big[1-\frac{\nu^2}{(k_0 R)^2}\Big]Z=0,$$ whose solution is:\ $$\label{evanescentSolution} Z_{\nu}(x)=C_0e^{-\delta x},$$ where\ $$\delta=\Bigg[\sqrt{\Big(\frac{\nu}{k_0R}\Big)^2-1+\frac{1}{4(k_0R)^2}}-\frac{1}{2k_0 R}\Bigg].\nonumber$$ This is, how we are expecting, the expression of an exponentially decreasing field, that is in perfect agreement with the hypothesis that the field outside the resonator is evanescent due to total internal reflection. It can be moreover noted that the oscillatory behavior of the field components outside the resonator (as depiscted in Figs. \[approxH\_real\] and \[approxH\_imag\] for the radial component of the electric field) is not in contrast with this hypothesis, since it only represent the behavior of the Hankel function as $r\rightarrow\infty$, i.e. it behaves like a runaway wave whose intensity is decreasing as $1/r^2$. In the case of WGM, however, no radiation will run away towards infinity since the external field is evanescent, i.e. the radiation is trapped inside the WGM and rapidly decreases toward zero when the field goes outside the resonator. ![(color online) Comparison between the real part of $h_{\nu}^{(1)}(x)$ (black solid line), Eq (dashed red line) and Eq. (31) of Ref.[@ref26] (dashed blue line) for $\nu=1000.5$. Our approximation works very well in the region in which the argument is grater than the index (i.e. $x>1010$), while the approximation presented in Ref.[@ref26] is out of phase respect to the Hankel function.[]{data-label="approxH_real"}](fig5.png){width="50.00000%"} ![(color online) Same as Fig. \[approxH\_real\] but the comparison is made for the imaginary part.[]{data-label="approxH_imag"}](fig6.png){width="50.00000%"} ![(color online) Same as Fig. \[approxH\_real\] but the comparison is made for the absolute value; note in this case how the approximation presented in Ref.[@ref26] completely fails to approximate the Hankel function.[]{data-label="approxH_abs"}](fig7.png){width="50.00000%"} B. Angular functions with large indices --------------------------------------- For large indices $n$, the WGM field is concentrated in a narrow interval of angles $\theta$ near $\theta_0=\pi/2$; this makes possible to approximate the associated Legendre functions (i.e. the $\theta$-part of the scalar spherical harmonics) with large indices , with Hermite polynomials with small indices as follows:\ $$\label{Yapprox} Y_{nm}(\theta,\varphi) \simeq \frac{\sqrt{m}}{2^{w}\sqrt{\pi}w !}H_{w}(\sqrt{m}\alpha)e^{-\frac{m}{2}\alpha^2}e^{im\varphi}.$$ Detailed calculations for obtaining this result are shown in Appendix B.\ C. Roots of characteristic equations ------------------------------------ The approximations exploited in the previous section are very useful in finding an analytical solution to the characteristic equation for the eigenfrequencies of the resonator; however, due to the anisotropy, some changes in the definition of the variables used above must be done. First of all, the $x$ appearing in Eqs. and has to be different for the inner and outer functions, due to the fact that the anisotropy is confined only inside the resonator; we can then define $x=k_0R$ as the outer variable and by consequence the inner variable results to be $y=\tilde{\gamma_i}\sqrt{\varepsilon_+}x$. Then, the definition of $\zeta$ must be changed into $\zeta=(2/\nu)^{1/3}(\nu-y)$. After that, by substituting Eqs. and into Eq , the characteristic equation for quasi-TM field gives [@ref33]\ $$\begin{aligned} \label{eigenWGM} \tilde{\gamma_1} \frac{1}{\textrm{Ai}(\zeta)}\frac{d\textrm{Ai}(\zeta)}{d\zeta}= \frac{\varepsilon_{\perp}}{\sqrt{\varepsilon_+}} \Big( \frac{\nu}{2} \Big) ^{1/3}\times \nonumber \\ \times \Bigg[ \frac{1}{4} \Big( \frac{2x}{x^2-\nu^2} \Big) - i\sqrt{1-\frac{\nu^2}{x^2}} \Bigg],\end{aligned}$$ the equation for the quasi-TE field can be deduced by this one upon changing $\tilde{\gamma_1}$ with $\tilde{\gamma_2}$ and putting $\varepsilon_{\perp}=1$.\ In order to find an approximate formula for the solutions of this equation, let us firstly analyze the limiting case in which $\nu\rightarrow\infty$; in this case the right-hand side of the equation goes to infinity and the only possible solution is that $\textrm{Ai}(\zeta)=0$, whose solutions are the zeros of the Airy function $\zeta_q$. Let us denote with $\Delta\zeta_q$ the first order correction to these roots; expanding both left-hand and right-hand size of Eq in power series with a first order accuracy to terms $\Delta\zeta_q$ we can obtain the first order correction to the roots $\zeta_q$, whose expression is\ $$\label{correction} \Delta\zeta_q=\frac{\tilde{\gamma_1}\sqrt{\varepsilon_+}}{\varepsilon_{\perp}\alpha} \Big( \frac{2}{\nu} \Big) ^{1/3},$$ where\ $$\alpha=\frac{x_q}{2(x_q^2-\nu^2)}-i\sqrt{1-\frac{\nu^2}{x_q^2}},$$ and $x_q$ is obtained by substituting the value of the first zero of the Airy function ($\zeta_q=-2.33811$) into the definition of $\zeta$ and inverting that relation with respect to $x$.\ Taking into account the definition of $\zeta$, the eigenvalues of the wave numbers for the anisotropic resonator can be represented in the following explicit form\ $$k_{0q}=\frac{\nu- \Big( \frac{2}{\nu} \Big) ^{1/3} (\zeta_q+\Delta\zeta_q)}{\tilde{\gamma_1}\sqrt{\varepsilon_+}R}.$$ Note that because the quantity $\Delta\zeta_q$ is complex, the eigenvalues of the wave number are also complex. The real part of the wave number then determines the eigenfrequencies of the mode. Complex eigenfrequencies are fully compatible with the open cavity. As can be seen from Eq. , this approximation has an accuracy of $\nu^{-1/3}$. more accurate asymptotic expressions that allow the calculation of the positions of resonances of the modes in an isotropic dielectric spherical resonator have been largely studied in literature (see for example Ref.[@ref34; @ref35; @ref36; @ref37; @ref38; @ref39] and references therein) and they were given with various accuracy with respect to the index $\nu$; in Ref.[@ref39] analytic calculations are carried out to the order $\nu^{-1/3}$, in Ref.[@ref35] the eigenfrequencies are calculated with an accuracy of $O[\nu^{-2/3}]$, while in Ref.[@ref38] the authors give an expression up to the order $O[\nu^{-8/3}]$. Here we report the anisotropic correction of the formula found in Ref.[@ref34] that gives the eigenfrequencies with a precision of the order of $O[\nu^{-1}]$\ $$\begin{aligned} \tilde{\gamma_i} x_{\nu}^{(q)} & = & \Big\{ \nu - \Big( \frac{\nu}{2} \Big) ^{1/3}\zeta_q - \sqrt{\frac{\varepsilon_+}{\varepsilon_+ -1}}P +\nonumber \\ & + &\frac{3}{10} \Big( \frac{1}{4\nu} \Big) ^{1/3} \zeta_q^2 - \Big( \frac{1}{2\nu^2} \Big) ^{1/3} \Big( \frac{\varepsilon_+}{\varepsilon_+ -1} \Big) ^{3/2}\times\\ & \times & P \Big(\frac{2}{3}P^2-1 \Big) \zeta_q\nonumber +O[\nu^{-1}] \Big\},\end{aligned}$$ where $P=1/(\tilde{\gamma_1}\varepsilon_+)$ for quasi-TM modes and $P=\varepsilon_{\perp}/ (\tilde{\gamma_2}\varepsilon_+)$ for quasi-TE modes.\ D. Whispering Gallery Modes --------------------------- We now have all the elements for writing the explicit expressions for the radial, polar and azimuthal components of the quasi-TM and quasi-TE WGMs. Taking the approximations , and , the equations for the components of the quasi-TM fields defined in Eqs. become\ \[quasi\_TM\_WGM\_in\] $$\begin{aligned} E_r & = &\frac{n(n+1)\sqrt{m}}{r^2 2^{w}\sqrt{\pi}w ! }\Big( \frac{\nu}{2} \Big) ^{1/6} \textrm{Ai}(\zeta)\times\nonumber \\ & \times & \mathrm{H}_{w}(\sqrt{m}\alpha)e^{-\frac{m}{2}\alpha^2}e^{im\varphi},\\ rE_{\theta} & = & \tilde{\gamma_1} k_0\frac{\sqrt{2\varepsilon_+ m}}{2^{w}w !}\Big( \frac{2}{\nu} \Big) ^{1/6}\frac{d\textrm{Ai}(\zeta)}{d\zeta}\times \nonumber \\ & \times & \frac{\partial}{\partial\theta} \Big[\mathrm{H}_{w}(\sqrt{m}\alpha)e^{-\frac{m}{2}\alpha^2} \Big] e^{im\varphi},\\ rE_{\varphi} & = & \frac{im^{3/2}\tilde{\gamma_1} k_0}{\sin\theta2^{w-1/2}w !}\Big( \frac{2}{\nu} \Big) ^{1/6} \frac{d\textrm{Ai}(\zeta)}{d\zeta}\times \nonumber \\ & \times & \mathrm{H}_{w}(\sqrt{m}\alpha)e^{-\frac{m}{2}\alpha^2}e^{im\varphi},\end{aligned}$$ $$\begin{aligned} H_r & = & 0,\\ rH_{\theta} & = & \frac{k_0\varepsilon_{\perp}m^{3/2}}{\sin\theta 2^{w}w !} \Big( \frac{\nu}{2} \Big) ^{1/6}\textrm{Ai}(\zeta)\times \nonumber \\ & \times & \mathrm{H}_{w}(\sqrt{m}\alpha)e^{-\frac{m}{2}\alpha^2}e^{im\varphi},\\ rH_{\varphi} & = & \frac{ik_0\varepsilon_{\perp}\sqrt{m}}{2^{w}w !} \Big( \frac{\nu}{2} \Big) ^{1/6}\textrm{Ai}(\zeta)\times \nonumber \\ & \times & \frac{\partial}{\partial\theta} \Big[ \mathrm{H}_{w}(\sqrt{m}\alpha)e^{-\frac{m}{2}\alpha^2} \Big] e^{im\varphi}.\end{aligned}$$ for the field inside the resonator, while for the field outside the resonator the expressions are the following:\ \[quasi\_TM\_WGM\_out\] $$\begin{aligned} E_r & = &\frac{n(n+1)\sqrt{m}}{r^2 2^{w}\sqrt{\pi}w ! }f(\eta)\times\nonumber \\ & \times & \mathrm{H}_{w}(\sqrt{m}\alpha)e^{-\frac{m}{2}\alpha^2}e^{im\varphi},\\ rE_{\theta} & = & \tilde{\gamma_1} k_0\frac{\sqrt{2\varepsilon_+ m}}{2^{w}w !}\Big(\frac{1}{\nu\sin\eta}\Big)\frac{df(\eta)}{d\eta}\times \nonumber \\ & \times & \frac{\partial}{\partial\theta} \Big[\mathrm{H}_{w}(\sqrt{m}\alpha)e^{-\frac{m}{2}\alpha^2} \Big] e^{im\varphi},\\ rE_{\varphi} & = & \frac{im^{3/2}\tilde{\gamma_1} k_0}{\sin\theta2^{w-1/2}w !}\Big(\frac{1}{\nu\sin\eta}\Big)\frac{df(\eta)}{d\eta}\times \nonumber \\ & \times & \mathrm{H}_{w}(\sqrt{m}\alpha)e^{-\frac{m}{2}\alpha^2}e^{im\varphi},\end{aligned}$$ $$\begin{aligned} H_r & = & 0,\\ rH_{\theta} & = & \frac{k_0\varepsilon_{\perp}m^{3/2}}{\sin\theta 2^{w}w !}f(\eta)\times \nonumber \\ & \times & \mathrm{H}_{w}(\sqrt{m}\alpha)e^{-\frac{m}{2}\alpha^2}e^{im\varphi},\\ rH_{\varphi} & = & \frac{ik_0\varepsilon_{\perp}\sqrt{m}}{2^{w}w !}f(\eta)\times \nonumber \\ & \times & \frac{\partial}{\partial\theta} \Big[ \mathrm{H}_{w}(\sqrt{m}\alpha)e^{-\frac{m}{2}\alpha^2} \Big] e^{im\varphi}.\end{aligned}$$ Similar expressions can be found for quasi-TE WGMs by interchanging the roles of the electric and magnetic field in the previous expressions, changing $\tilde{\gamma_1}$ with $\tilde{\gamma_2}$, putting $\varepsilon_{\perp}=1$ and changing $i$ with $-i$.\ Figures \[WGM\_fundamental\_radial\] and \[WGM\_fundamental\_polar\] show the behavior of the fundamental quasi-TM radial (no nodes in radial direction, i.e. $q=1$) and polar ( i.e. $n=m$) WGM component $r^2E_r$ and the “first excited" radial ($q=2$) and polar ($m=n+1$) mode for the same component of the quasi-TM field; the physical parameters have been set to be $\varepsilon_{xx}=5.30$, $\varepsilon_{zz}=6.47$ ($\mathrm{LiNbO_3}$) and $\lambda=1064$ nm. Note that the radial component has its maximum very close to the sphere surface (dashed vertical line in Fig. \[WGM\_fundamental\_radial\]), and its position shifts on the left, i.e. on the inner part of the resonator as the radial number $q$ increases. The polar part, instead, is localized around $\theta=\pi/2$ in its fundamental state and, as $m$ becomes smaller than $n$, the maxima of the polar component tent to repel each other from $\theta=\pi/2$. ![(color online) Radial part of the fundamental (black line) and first excited (red line) WGM for the anisotropic resonator; the vertical dashed line indicates the position of the resonator surface. The fundamental mode has indices $n=m=1000$ and $q=1$, while the first excited mode has the same $n$ and $m$ indices but $q=2$.[]{data-label="WGM_fundamental_radial"}](fig8.png){width="50.00000%"} ![(color online) Polar part of the fundamental (black line) and first polar-excited (red line) WGM for the anisotropic resonator. The fundamental mode has indices $n=m=1000$ and $q=1$, while the first polar-excited mode has $n=1000$, $m=n-1$ and $q=1$.[]{data-label="WGM_fundamental_polar"}](fig9.png){width="50.00000%"} In figures \[TF1\] to \[TF4\] the intensity distribution of the *total* electric field of a quasi-TM (i.e. $\mathbf{E}_{TM}=E_r\hat{r}+E_{\theta}\hat{\theta}+E_{\varphi}\hat{\varphi}$) is shown, where the components $E_k$ ($k=r, \theta, \varphi$) are given by Eqs. for the field inside the resonator and Eqs. for the field outside the resonator; $\hat{r}$,$\hat{\theta}$ and $\hat{\varphi}$ represent the unit vectors of the spherical basis ($r$,$\theta$,$\varphi$). In order to obtain the intensity distribution of such a field, one has to sum the square modulus of each component of the electric field; however, in this particular case, the contribution of $E_{\theta}$ and $E_{\varphi}$ is very small and localized at the resonator surface, and the total field is, with a good level of approximation, fully determined by its radial component. The intensity distribution for the magnetic field components of a quasi-TM mode can be straightforwardly obtained from Eqs. and or by nothing that the $H_{\theta}$ component of the magnetic field has the same intensity distribution as the radial electric field component $E_r$ and the $H_{\varphi}$ component, because of the presence of the derivative with respect to $\theta$, has the same intensity distribution as the one depicted in Fig. \[TF3\]. As the reader can see from these figures, the field is nonzero even after the resonator surface ($x=1$); this is not surprising because in this region the total field is evanescent due to the fact that it has been total internal reflected by the resonator, i.e. the field is confined in the resonator WGM. ![(color online) Intensity distribution of the electric field of the fundamental quasi-TM WGM. The WGM quantum numbers are $n=m=1000$,$q=1$.[]{data-label="TF1"}](fig10.png){width="50.00000%"} ![(color online) Same as Fig. \[TF1\] but for $q=2$; this mode represents the first radially excited WGM.[]{data-label="TF2"}](fig11.png){width="50.00000%"} Conclusions =========== In this work, we have developed a classical-optics theory for an uniaxial spherical whispering gallery resonator. We have presented and discussed the mode structure in the limit of small anisotropy for such resonator, and obtained its spectrum. Moreover, we have furnished a thorough discussion on the boundary conditions and asymptotic expressions for the electromagnetic field in WGRs. Our results may be easily generalized to achieve a quantum theory of WGRs. Acknowledgements ================ The authors want to thank Josef Fürst, Christoph Marquardt and Dmitry Strekalov for fruitful discussions. Appendix A: coefficients of Eqs. (22) ===================================== Here are reported the explicit expressions of the coefficients that appear on Eqs. . In order to express them in a compact form, let us introduce the following quantities:\ $$\begin{aligned} g_{\pm} & = & k_o^2\varepsilon_{\pm},\nonumber \\ f_n & = & \frac{1}{2n+2},\nonumber \\ T_n & = & r^2 \hat{l}_x -n(n+1),\nonumber \\ l_+ & = & \frac{\partial^2}{\partial r^2}+k_0^2\varepsilon_+.\nonumber\end{aligned}$$ With these parameters defined, the $a$ and $b$s coefficients of Eq. become:\ $$\begin{aligned} a_{1,n}^{TM} & = & T_n \Big[l_+ - \frac{1-4m^2}{4n(n+1)-3}g_- \Big]+2g_- m^2,\nonumber\\ a_{2,n}^{TM} & = & 2g_- (n-m+1)(n-m+2)f_n f_{n+1}T_n,\nonumber\\ a_{3,n}^{TM} & = & 2g_- (n+m)(n+m-1)f_n f_{n-1} T_n,\nonumber\\ b_{1,n} & = & f_n (n-m+1) \Big[ r \hat{l}_x -(n+2)\frac{d}{dr} \Big], \nonumber\\ b_{2,n} & = & f_n (n+m) \Big[ r \hat{l}_x + (n-1)\frac{d}{dr} \Big], \nonumber\end{aligned}$$ $$\begin{aligned} a_{1,n}^{TE} & = & \Big\{ \Big[ r^2\frac{d^2}{dr^2}-n(n+1) \Big] \Big[ 1+\lambda \Big(\frac{1-4m^2}{4n(n+1)-3} \Big) \Big] +\nonumber \\ & + & (1-\lambda)g_+ r^2 \Big\} \hat{l}_x - 2m^2g_- (1-\lambda),\nonumber\\ a_{n,2}^{TE} & = & 2f_n f_{n+1} (n-m+1)(n-m+2)\times\nonumber \\ & \times & \Big[ r^2\frac{d^2}{dr^2}-(2n+3)r\frac{d}{dr}+n(n+1) \Big],\nonumber\\ a_{n,3}^{TE} & = & 2f_n f_{n-1} (n+m)(n+m-1)\times\nonumber \\ & \times & \Big[ r^2\frac{d^2}{dr^2}-(2n-1)r\frac{d}{dr}+(n+1)(n-3) \Big].\nonumber\end{aligned}$$ ![(color online) Same as Fig. \[TF1\] but for $n-m=1$; this mode represents the first polar excited WGM.[]{data-label="TF3"}](fig12.png){width="50.00000%"} Appendix B: approximation of scalar spherical harmonics for large indices ========================================================================= The equation for the $\theta$-part of spherical harmonics is the following\ $$\frac{1}{\sin\theta}\frac{d}{d\theta} \Big( \sin\theta\frac{df}{d\theta} \Big) + \Big[ n(n+1) - \frac{m^2}{\sin^2\theta} \Big]f=0,\nonumber$$ whose solutions are the associated Legendre functions $f(\theta)=P_n^m(\cos\theta)$. Since WGMs are located near the equator of the resonator, the correspondent functions $f(\theta)$ will be peaked near the angle $\theta_0=\pi/2$; in order to find an approximate expression for the polar part of the spherical harmonics, let us introduce the new variable $\alpha=\pi/2-\theta$: substituting into equation above gives\ $$\frac{d^2}{d\alpha^2} - \tan\alpha\frac{df}{d\alpha} + \Big[ n(n+1) - \frac{m^2}{\cos^2\alpha} \Big] f=0.\nonumber$$ ![(color online) Same as Fig. \[TF1\] but for $n-m=1$ and $q=2$; this mode represents the first radially and polar excited WGM.[]{data-label="TF4"}](fig13.png){width="50.00000%"} We note that, since the modes are localized near the equator, $\alpha\ll 1$ and this allow us to expand in power series the trigonometric functions that appear in the previous equation, i.e. $\tan\alpha\simeq\alpha$ and $1/\cos^2\alpha\simeq 1+\alpha^2$. Substituting in the previous equation, writing $f(\alpha)=G(\alpha)e^{\alpha^2/4}$ and performing the change of variables $\alpha=x/[(m^2+1/4)^{1/4}]=\xi x$ we obtain\ $$\frac{d^2G}{dx^2} + \Big\{ \xi^2 \Big[ n(n+1) - m^2 \Big] -x^2 \Big\} G=0,\nonumber$$ Introducing the quantity $w=n-m$ and remembering that WGMs are characterized by high values of the indices, i.e. $n,m\gg1$, the first term that appears inside the curly brackets can be simplified as $2w+1$. With this substitution the last equation is precisely the Hermite-Gauss equation, whose solutions have the form $G(x)\simeq H_{w}(x)e^{-x^2/2}$. Function $f(\alpha)$ then becomes\ $$f(\alpha)= P_n^m(\cos\theta)\simeq NH_{w}(\sqrt{m}\alpha)e^{-\frac{m}{2}\alpha^2},\nonumber$$ where $N$ is a normalization factor whose expression could be found by requiring that the norm of $f(\alpha)$ integrated over the real axis is one. This equation gives the approximated form of the associated Legendre functions for WGMs; substituting it into the definition of the scalar spherical harmonics gives exactly Eq . [99]{} $http://en.wikipedia.org/wiki/St\_Paul's\_Cathedral$ Baron John William Strutt Rayleigh, *The Theory of Sound: Volume II*, Dover Publication (1945) G. Mie, Ann. Physik, **25**, 377 (1908) P. Debye, Ann. Physik, **30**, 57 (1909) V.S. Ilchenko *et.al.*, Phys. Rev. Lett. **92**, 049303(4)(2004) M.L. Gorodetsky *et.al.*, Opt. Lett **21**, 453(1996) I.S. Grudinin *et.al.*, Phys. Rev. A **74**, 063806(2006) A.A. Savchenkov *et.al.*, Phys. Rev. A **70**, 051804(R) (2004) V.S. Ilchenko *et.al.*, J. Opt. Soc. Am. B **20**, 1304(2003) G. Kozyreff *et.al.*, Phys. Rev. A **77**, 043817(2008) A.A. Savchenkov *et.al.*, Opt. Lett. **32**, 157(2007) A.A. Savchenkov *et.al.*, Phys. Rev. Lett. **93**, 243905(2004) P. Del’Haye *et.al.*, Nature (London) **450**, 1214(2007) A.A. Savchenkov *et.al.*, Phys. Rev. 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Strekalov *et. al.*, Phys. Rev. A **80**, 033810(2009) J.M. le Floch *et. al.*, Phys. Lett. A **359**, 1(2007) A.N. Oraevsky, Quantum Electronics **32**, 377(2002) J.D. Jackson, *Classical Electrodynamics*, Wiley (Third Edition) In this paper we consider a nonmagnetic resonator, i.e. $\mu=1$ surrounded by air ($\varepsilon_{ext}=1$). In obtaining the azimuthal component of the quasi-TM magnetic field following relation was used: $\Big[ \varepsilon_{\theta\theta}\frac{\partial}{\partial\theta} - \varepsilon_{r\theta}\frac{\partial}{\partial r} \Big( r \Big) \Big]\hat{l}_{\theta} = \hat{l}_{\theta} \Big[ \varepsilon_{\theta\theta}\frac{\partial}{\partial\theta} + \varepsilon_{r\theta} \Big(1-r\frac{\partial}{\partial r} \Big) \Big]$ Y.V. Proponenko *et. al.*, Technical Physics **49**, 459(2004) Eq. (5) of Ref.[@ref29] contains an error: the expresison for $\cos(2\theta)Y_{n,m}(\theta,\varphi)$ is incorrect. Here we report the correct relation, in which the argument of the spherical harmonics is omitted for the sake of simplicity: $\cos(2\theta)Y_{n,m}=A_{n,m}Y_{n+2,m}+B_{n,m}Y_{n-2,m}+C_{n,m}Y_{n,m}$ with $A_{n,m}=2(n-m+1)(n-m+2)f_nf_{n+1}$, $B_{n,m}=2(n+m)(n+m+1)f_n f_{f-1}$ and $C_{n,m}=(1-4m^2)/[4n(n+1)-3]$ M.A. Abramowitz and I. Stegun (Editors), *Handbook of Mathematical Functions: with Formulas, Graphs and Mathematical Tables*, Dover (1965) I.S. Gradshteyn and I.M. Ryzhik, *Table of Integrals, Series and Products*, Academy Press(2007) To write this equation we have used the expression in Eq stopped at the order $O[\nu^{-1}]$ S. Schiller and R.L. Byer, Opt. Lett **16**, 1138(1991) V.S. Ilchenko *et. al.*, J. Opt. Soc. Am. A **20**, 157(2003) M. Gadtine *et.al.*, IEEE Trans. Microwave Theory and Techniques **MTT-15**, 694(1997) B.R. Johnson, J. Opt. Soc. Am. A **10**, 343(1993) S. Schiller, Appl. Opt **32**, 2181(1993) C.C. Lam *et. al.*, J. Opt. Soc. Am. B **9**, 1585(1992) As can be seen from Eqs. , the radial component of the TM field has a dependence on the radial wavefunction itself, via the second derivative of the potential. The tangential component, instead, depends on the derivative (with respect to the radial variable) of the wavefunction. A. Messiah, *Quantum Mechanics*, Dover (1999)

[\ ]{} [**Canonical Structure of the ${{\rm{E}}_{10}}$ Model\ and Supersymmetry**]{}\ ${}^1$[*Max-Planck-Institut für Gravitationsphysik (Albert-Einstein-Institut)\ Am Mühlenberg 1, DE-14476 Potsdam, Germany*]{} 1 em ${}^2$[*International Solvay Institutes\ ULB-Campus Plaine CP231, BE-1050 Brussels, Belgium*]{} 1 em ${}^3$[*Indian Institute of Technology Madras, Department of Physics\ Chennai - 600036, India*]{} ------------------------------------------------------------------------ --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- [ A coset model based on the hyperbolic Kac–Moody algebra ${{\rm{E}}_{10}}$ has been conjectured to underly eleven-dimensional supergravity and M theory. In this note we study the canonical structure of the bosonic model for finite- and infinite-dimensional groups. In the case of finite-dimensional groups like $GL(n)$ we exhibit a convenient set of variables with Borel-type canonical brackets. The generalisation to the Kac–Moody case requires a proper treatment of the imaginary roots that remains elusive. As a second result, we show that the supersymmetry constraint of $D=11$ supergravity can be rewritten in a suggestive way using ${{\rm{E}}_{10}}$ algebra data. Combined with the canonical structure, this rewriting explains the previously observed association of the canonical constraints with null roots of ${{\rm{E}}_{10}}$. We also exhibit a basic incompatibility between local supersymmetry and the $K({{\rm{E}}_{10}})$ ‘R symmetry’, that can be traced back to the presence of imaginary roots and to the unfaithfulness of the spinor representations occurring in the present formulation of the ${{\rm{E}}_{10}}$ worldline model, and that may require a novel type of bosonisation/fermionisation for its resolution. This appears to be a key challenge for future progress with ${{\rm{E}}_{10}}$. ]{} --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------ Introduction ============ The conjectured ${{\rm{E}}_{10}}$ symmetry of the M Theory completion of $D=11$ supergravity [@Damour:2002cu] applies to both the bosonic and the fermionic sector. The one-dimensional spinning ${{\rm{E}}_{10}}$ model constructed and analysed in [@Damour:2005zs; @deBuyl:2005mt; @Damour:2006xu; @deBuyl:2005zy] has manifest symmetry under the hyperbolic Kac–Moody group ${{\rm{E}}_{10}}$ and its dynamics have been shown to match exactly the $D=11$ dynamics at the non-linear level, when both are suitably truncated. However, it has so far proved impossible to remove the truncation of this correspondence, one central obstacle being a dichotomy between the bosonic and fermionic variables on the ${{\rm{E}}_{10}}$ side. Whereas the bosonic variables are described in terms of [*infinitely*]{} many coordinates of the infinite-dimensional coset space ${{\rm{E}}_{10}}/K({{\rm{E}}_{10}})$, the fermionic variables are described by [*finitely*]{} many components of a finite-dimensional (unfaithful) spinor representation of $K({{\rm{E}}_{10}})$ [@Damour:2006xu]. This dichotomy is also reflected in the fact that the one-dimensional ${{\rm{E}}_{10}}$ model cannot be fully supersymmetric on its worldline, since in its presently known form it pairs an infinite number of bosonic with a finite number of fermionic degrees of freedom. In view of the fact that a detailed understanding of supersymmetry has often been central in advances regarding the structure of hidden symmetries, we initiate in this note a more detailed study of the worldline supersymmetry in the ${{\rm{E}}_{10}}$ context. Though we will not be able to present a new supersymmetric ${{\rm{E}}_{10}}$ model, our results bring the obstacles in the current formulation to the front and we hope they can serve as a first step to resolving the issues both in the physics and the mathematics associated with constructing a model that fully accommodates both supersymmetry and $K({{\rm{E}}_{10}})$ symmetry. In fact, progress towards solving the outstanding problems may well require some novel kind of bosonization/fermionization, and thus also involve quantisation in a crucial way. This is not only because the distinction between bosons and fermions becomes fluid in low dimensions and thus also in the (one-dimensional) worldline model, but also because the very meaning of what is a space-time boson and what is a space-time fermion, and hence also the ultimate relevance of [*space-time*]{} supersymmetry (as opposed to worldline supersymmetry), must be questioned in the context of emergent space-time scenarios. The present results can be viewed as a first step in this direction; in particular we identify the proper canonical variables on the bosonic side that couple naturally to the fermions, and hence will be an essential ingredient in approaching quantisation of the worldline model. We note that in the context of string theory the emergence of space-time fermions from bosonic fields was already suggested long ago in [@Casher:1985ra], and the relation of this construction to Kac–Moody algebras was discussed more recently in [@Englert:2008ft]. In the context of maximal supergravity in two dimensions (where $K({{\rm{E}}_{10}})$ is replaced by $K({\rm E}_9)$), it was already pointed out in [@Nicolai] that the associated linear system effectively constitutes a bosonisation of the supergravity fermions, especially in view of previous work in [@Witten; @GNO]. Our main tool is the detailed analysis of the canonical structure of one-dimensional coset models, starting with purely bosonic systems based on a coset $G/K$. We will exhibit explicitly a set of variables that makes the algebraic structure completely manifest and we propose that these variables are therefore also an appropriate starting point for quantum considerations extending the reduced quantum cosmological billiards of [@Kleinschmidt:2009cv] that should eventually lead to an implementation of the Wheeler-DeWitt equation for the full theory. For the case ${{\rm{E}}_{10}}/K({{\rm{E}}_{10}})$ our arguments remain somewhat formal since an explicit parametrisation of the group ${{\rm{E}}_{10}}$ similar to the one used in the proof for finite-dimensional $G$ is not available. Denoting the velocity type variables as ${P_{\alpha}}^r$, where $\alpha >0$ is a positive root of the ${{\rm{E}}_{10}}$ Borel algebra and $r$ labels an orthonormal basis of elements in the root space associated to the root $\alpha$ (this extra label is only required for imaginary roots), we will in particular argue, and [*prove*]{} for finite-dimensional $G$, that the canonical commutation relations of the ${P_{\alpha}}^r$ are exactly those of the ${{\rm{E}}_{10}}$ algebra itself. The bosonic expressions have to be completed by fermionic ones and in section \[sec:fermions\] we then look at $D=11$ supergravity [@Cremmer:1978km]. A rewriting of the supersymmetry constraint, inspired by recent studies in quantum supersymmetric cosmology in relation to Kac–Moody symmetries [@Damour:2013eua; @Damour:2014cba], suggests a very simple underlying algebraic formulation. We will here restrict attention to terms [*linear in the fermions*]{}, as the consideration of higher order fermionic terms does not affect our main conclusions.[^1] With every root $\alpha$ of ${{\rm{E}}_{10}}$ one can associate an element ${\tilde\Gamma}(\alpha)$ of the $SO(10)$ Clifford algebra and a polarisation of the fermionic field $\phi(\alpha)$. In [@Damour:2005zs] the supersymmetry constraint was analysed [*to linear order*]{} in fermions and shown to take the schematic form $$\begin{aligned} \label{PodotPsi} \mathcal{S} = P \odot \Psi \end{aligned}$$ where $P$ stands for the infinite component coset velocity of the ${{\rm{E}}_{10}}$ coset space model, and $\Psi$ for the [*finite*]{}-dimensional unfaithful spinor representation. The symbol $\odot$ is shorthand for the particular combination of the fermions and the bosonic coset velocities identified from the canonical supersymmetry constraint in [@Damour:2005zs]. In this paper, we will show how the above expression can (again schematically) be transformed into a sum $$\begin{aligned} \label{Sintro} \mathcal{S} = \ldots \, + \, \sum_\alpha {P_{\alpha}} \, {\tilde\Gamma}(\alpha) \phi(\alpha) +\ldots .\end{aligned}$$ One main goal of this paper will be to explore the validity, and more specifically the limit of validity, of this expression, and thereby attach a more concrete representation theoretic meaning to the symbol $\odot$. Indeed, already from the form of (\[Sintro\]) one may anticipate problems when trying to combine supersymmetry with the ‘R symmetry’ $K({{\rm{E}}_{10}})$: supersymmetry requires an [*equal*]{} number of bosons and fermions, whereas in (\[Sintro\]) an infinite number of bosonic degrees of freedom is to be paired with a finite number of fermionic degrees of freedom. To be sure, in the actual expression obtained from supergravity the above sum contains only finitely many bosonic contributions, as a result of ‘cutting off’ the sum over roots $\alpha$ at level $\ell =3$. Therefore the supersymmetry constraint ${\mathcal{S}}$ cannot, in its presently known form, be assigned to any known representation of $K({{\rm{E}}_{10}})$, even though separately, both $P$ and $\phi$ do transform properly (although it is not known whether $P$ transforms in an [*irreducible*]{} representation of $K({{\rm{E}}_{10}})$). The novel techniques introduced in this paper will allow us to analyse in considerable detail the terms by which the supersymmetry constraint fails to transform properly, and to highlight the differences between the finite-dimensional and infinite-dimensional cases. Our analysis thus identifies the terms that have to be dealt with differently in the construction of a supersymmetric ${{\rm{E}}_{10}}$ model, and we offer more comments in the concluding section. There we also explain that the failure to transform covariantly under $K({{\rm{E}}_{10}})$ cannot be cured by higher order fermionic terms. While the exact $D=11$ supersymmetry constraint can be transformed into a [*truncated*]{} expression of the type above, we thus encounter obstacles when trying to remove the truncation and to explore what the dots in the above formula could stand for. The expression above does provide a sensible object for $GL(n,{\mathbb{R}})$ and other finite-dimensional groups in the sense that it transforms covariantly as spinor as the supersymmetry should, but a similar result is no longer true for ${{\rm{E}}_{10}}$. From a more physical perspective, the mismatch between bosons and fermions in the latter case is also reflected in the fact that no fermionic analog of the gradient representations has been found so far, thus (so far, at least) precluding an expansion for the fermions à la Belinksi–Khalatnikov–Lifshitz (BKL). Canonical structure of bosonic worldline coset models {#sec:Can} ===================================================== In this section, we study the canonical structure of a coset model describing the motion of a point particle on a symmetric space $G/K$, with $G$ a split real simple Lie group and $K\equiv K(G)$ its maximal compact subgroup. To set the basic notations and conventions, we first discuss the case of finite dimensional $G$ where everything is well defined, and subsequently write down the corresponding expressions for Kac–Moody algebras and groups. In the latter case, of course, many expressions will remain formal. For previous work on the canonical structure of non-linear $\sigma$-models, see for example [@Matschull:1994vi]. Set-up in the finite-dimensional case ------------------------------------- To begin with, we restrict attention to finite-dimensional and simply-laced Lie group $G$. Then the Lie algebra $\mathfrak{g}=\mathrm{Lie}(G)$ is finite-dimensional and has a system of roots $\alpha\in\Delta=\Delta_-\cup\Delta_+$. The positive roots will also be written as $\alpha>0$, and we designate the Cartan subalgebra by $\mathfrak{h}$. We assume a Cartan–Weyl basis with basis vectors $H_{{\tt{a}}}$ and ${E_{\alpha}}$, where ${{\tt{a}}}=1,\ldots,\dim \mathfrak{h}$. The commutation relations are [@GoddardOlive] \[CWBasisFD\] $$\begin{aligned} \label{CWBasis1} {\left[}{E_{\alpha}}, {E_{\beta}} {\right]}&= \left\{ \begin{array}{cl} c_{\alpha,\beta} {E_{\alpha+\beta}} &\textrm{if $\alpha+\beta\in \Delta$,}\\[2mm] \alpha^{{\tt{a}}}H_{{\tt{a}}}& \textrm{if $\alpha=-\beta$,}\\[2mm] 0 & \textrm{otherwise,} \end{array}\right.\\[2mm] {\left[}H_{{\tt{a}}}, {E_{\alpha}} {\right]}&= \alpha_{{\tt{a}}}{E_{\alpha}}.\end{aligned}$$ with $c_{\alpha,\beta} = \pm 1$ or $= 0$ for simply laced finite-dimensional Lie algebras. There is a non-degenerate invariant bilinear form on $\mathfrak{g}$ that satisfies[^2] $$\begin{aligned} \label{Norm1} \langle {E_{\alpha}} | {E_{\beta}} \rangle&=\left\{\begin{array}{cl} 1&\textrm{if $\alpha=-\beta$},\\[1mm] 0 &\textrm{otherwise}, \end{array}\right.\\[2mm] \langle H_{{\tt{a}}}| H_{{\tt{b}}}\rangle &= G_{{{\tt{a}}}{{\tt{b}}}}.\end{aligned}$$ The metric $G_{{{\tt{a}}}{{\tt{b}}}}$ is positive definite for any simple finite dimensional Lie algebra $\mathfrak{g}$ but need not be positive definite for [*non-simple*]{} $\mathfrak{g}$. The inverse $G^{{{\tt{a}}}{{\tt{b}}}}$ of $G_{{{\tt{a}}}{{\tt{b}}}}$ has been used to raise the index in  according to $\alpha^{{\tt{a}}}=G^{{{\tt{a}}}{{\tt{b}}}} \alpha_{{\tt{b}}}$ and $\alpha_{{\tt{a}}}=\alpha(H_{{\tt{a}}})$. The compact subalgebra $K(\mathfrak{g})\equiv\mathfrak{k}\subset \mathfrak{g}$ is generated by ${{\rm k}_{\alpha}}\equiv{E_{\alpha}}-{E_{-\alpha}}$ with $\alpha>0$ and will be discussed in more detail in section \[sec:KE10\]. The structure constants $c_{\alpha,\beta}$ are antisymmetric and satisfy standard identities [@GoddardOlive], in particular $$\label{cab} c_{\alpha,\beta} = - c_{\beta,\alpha} = - c_{-\alpha, -\beta} \; \;, \quad c_{\alpha + \beta, -\beta} = c_{\alpha,\beta}.$$ The coset $G/K(G)\equiv G/K$ can be parametrised in a Borel gauge fixed form according to the Iwasawa decomposition $G=KAN$. For finite-dimensional $G$ any element of the coset $G/K$ can thus be written in the form[^3] $$\begin{aligned} \label{IwaPar} V (q^{{\tt{a}}}, A_\alpha) = \exp\left(q^{{\tt{a}}}H_{{\tt{a}}}\right) \exp\left(\sum_{\alpha> 0} {A_{\alpha}} {E_{\alpha}}\right).\end{aligned}$$ The worldline model describing the motion of a point particle on the coset manifold $G/K$ is then parametrised by a map $V: {\mathbb{R}}\rightarrow G/K$, where $t\in{\mathbb{R}}$ is the time coordinate. The Cartan derivative is (with $\partial \equiv d/dt$) $$\begin{aligned} \label{CF} \partial V V^{-1} = P+ Q = \partial q^{{\tt{a}}}H_{{\tt{a}}}+ \sum_{\alpha>0} e^{q^{{\tt{a}}}\alpha_{{\tt{a}}}}D{A_{\alpha}} {E_{\alpha}}\end{aligned}$$ where $Q\in\mathfrak{k}\, , P\in\mathfrak{k}^\perp$, and, schematically and due to $\partial e^X e^{-X}= \partial X +\frac12 {\left[}X,\partial X{\right]}+\ldots$, $$\begin{aligned} \label{DA} DA_\alpha = \partial A_\alpha + \frac12 \sum_{\beta>0\atop \alpha-\beta>0} c_{\alpha-\beta,\beta}{A_{\alpha-\beta}} \partial {A_{\beta}} + \ldots.\end{aligned}$$ Importantly, the Borel gauge implies a triangular expansion of $D {A_{\alpha}}$ where the factors contributing to the terms quadratic in ${A_{\gamma}}$ on the r.h.s. are of [*lower*]{} height, whence the sum on the r.h.s. of (\[DA\]) has only finitely many terms even for infinite-dimensional Kac–Moody algebras (a crucial fact for the calculability of the model). The invariant Lagrangian is given by $$\begin{aligned} \label{Lag} L = \frac12 \langle P | P \rangle = \frac12 \partial q^{{\tt{a}}}G_{{{\tt{a}}}{{\tt{b}}}}\partial q^{{\tt{b}}}+ \sum_{\alpha>0} {P_{\alpha}} {P_{\alpha}},\end{aligned}$$ where we have defined $$\begin{aligned} P = \partial q^{{\tt{a}}}H_{{\tt{a}}}+\sum_{\alpha} {P_{\alpha}} ({E_{\alpha}} + {E_{-\alpha}}),\quad\quad {P_{\alpha}} = \frac12 e^{q^{{\tt{a}}}\alpha_{{\tt{a}}}} D{A_{\alpha}}.\end{aligned}$$ The compact part is then given by $$\begin{aligned} Q = \sum_{\alpha>0} {Q_{\alpha}} ({E_{\alpha}}-{E_{-\alpha}}) = \sum_{\alpha>0} {P_{\alpha}} ({E_{\alpha}}-{E_{-\alpha}})\end{aligned}$$ where the equality ${Q_{\alpha}} = {P_{\alpha}}$ is a consequence of the triangular gauge choice. The model has global $G$ symmetry and local $K$ symmetry that we use to fix the triangular gauge  everywhere. The symmetries act by $$\begin{aligned} \label{symactions} V(t) \to k(t) V(t) g^{-1}\quad \Longrightarrow\quad P \to k P k^{-1},\quad Q\to k Q k^{-1} + \partial k k^{-1}.\end{aligned}$$ When the triangular gauge  is fixed, a local compensating $K$-transformation is required to restore the gauge for every $G$ transformation that throws $V$ out of the triangular gauge. The equations of motion of the coset model are $$\begin{aligned} \label{eomall} D P \equiv \partial P - [Q,P] = 0.\end{aligned}$$ (We note that this of course implies that the equations of the original coordinates $q^{{\tt{a}}}$ and ${A_{\alpha}}$ are second order differential equations.) For a given root component ${P_{\alpha}}$ this means $$\begin{aligned} \label{eomrt} \partial {P_{\alpha}} =-\partial q^{{\tt{a}}}\alpha_{{\tt{a}}}{P_{\alpha}}+2\sum_{\beta>0} c_{\alpha+\beta,-\beta} {P_{\beta}}{P_{\alpha+\beta}} \equiv -\partial q^{{\tt{a}}}\alpha_{{\tt{a}}}{P_{\alpha}}+2\sum_{\beta>0} c_{\alpha,\beta} {P_{\beta}}{P_{\alpha+\beta}}\end{aligned}$$ Note that (in contrast to (\[DA\])) the sum on the r.h.s. contains terms of [*ascending*]{} height. Changes in the Kac–Moody case ----------------------------- When the Lie algebra $\mathfrak{g}$ is an infinite-dimensional Kac–Moody algebra [@Ka90], the definition of the corresponding group $G$ requires more care, see for example [@KP; @Kumar]. Again we restrict to simply-laced algebras, and more specifically to symmetric generalised Cartan matrices with at most one line linking any two nodes. Of course, our primary interest here will be with ${{\rm{E}}_{10}}$ and its maximal compact subgroup $K({{\rm{E}}_{10}})$. There are now two types of roots of the algebra, called real and imaginary and they distinguished by their Cartan–Killing norm: Real roots $\alpha$ satisfy $\alpha^2=2$ and imaginary roots $\alpha^2\leq 0$. The generators corresponding to real roots are unique up to normalisation and can be denoted by ${E_{\alpha}}$ as above but the generators corresponding to imaginary roots can have non-trivial multiplicities and are more appropriately denoted by ${E_{\alpha}}^r$, where $r=1,\ldots,\mathrm{mult}(\alpha)$ labels an orthonormal basis (w.r.t. the Cartan–Killing metric) in the root space. We will write all generators in this way, keeping in mind that for real roots $r$ can only take one value. The commutation relations in the Cartan–Weyl basis (cf. ) then have to account also for the multiplicities and become \[CWBasisKM\] $$\begin{aligned} {\left[}{E_{\alpha}}^r, {E_{\beta}}^s {\right]}&= \left\{ \begin{array}{cl} \sum\limits_{t=1}^{\mathrm{mult}(\alpha+\beta)}c_{\alpha,\beta}^{rst} {E_{\alpha+\beta}}^t &\textrm{if $\alpha+\beta\in \Delta$,} \\[4mm] \delta^{rs} \alpha^{{\tt{a}}}H_{{\tt{a}}}& \textrm{if $\alpha=-\beta$,}\\[2mm] 0 & \textrm{otherwise,} \end{array}\right.\\[2mm] {\left[}H_{{\tt{a}}}, {E_{\alpha}}^r {\right]}&= \alpha_{{\tt{a}}}{E_{\alpha}}^r.\end{aligned}$$ We note that we still have $c_{\alpha,\beta} = \pm 1$ if $\alpha, \beta$ and $\alpha + \beta$ are all real, but this need no longer be true when any of these roots is imaginary. The bilinear form (\[Norm1\]) generalizes to \[Norm2\] $$\begin{aligned} \langle {E_{\alpha}}^r | {E_{\beta}}^s \rangle &=\left\{\begin{array}{cl} \delta^{rs} &\textrm{if $\alpha=-\beta$},\\[1mm] 0 &\textrm{otherwise}, \end{array}\right.\\[2mm] \langle H_{{\tt{a}}}| H_{{\tt{b}}}\rangle &= G_{{{\tt{a}}}{{\tt{b}}}}\end{aligned}$$ where the metric $G_{{{\tt{a}}}{{\tt{b}}}}$ is now indefinite (and Lorentzian for hyperbolic Kac–Moody algebras). The other important modification concerns the parametrisation of the elements of the formal coset space $G/K$ that, using the Iwasawa decomposition, could be given in the finite-dimensional case as in . Even at a purely formal level, and even if the sum in the exponent is truncated to a finite number of terms, it is not directly meaningful to parametrize a given element of the Kac–Moody group in the form $$\begin{aligned} V(q^{{\tt{a}}}, A_\alpha^r) \;\; ``\!=\!" \; \;\exp (q^{{\tt{a}}}H_{{\tt{a}}}) \, \exp\left( \sum_{\alpha>0}\sum_{r=1}^{\mathrm{mult}(\alpha)} {A_{\alpha}}^r {E_{\alpha}}^r \right)\end{aligned}$$ where $\big\{ q^{{\tt{a}}}, {A_{\alpha}}^r\big\} $ are local coordinates on the (infinite-dimensional) coset manifold, one coordinate for each Lie algebra element ${E_{\alpha}}^r$. The reason is that the step operators ${E_{\alpha}}^r$ associated with imaginary roots are not (locally) nilpotent in any standard representation, and therefore the exponential is [*a priori*]{} ill-defined.[^4] For this reason, standard approaches to Kac–Moody groups involve writing down only exponentials of [*real*]{} root generators (that are nilpotent) and then defining the Kac–Moody group as the group generated by the products of these real root exponentials [@KP]. Although such a treatment is mathematically well-defined, it does not solve by any means the problem of finding a manageable realisation of the Kac–Moody group, because different orderings of exponentials of a given set of real root generators will yield new group elements. Organizing these differently ordered exponentials is thus directly associated to the (unsolved) problem of classifying the independent elements of the associated root space (where the problem is to count and classify the inequivalent ways in which a given set of Chevalley generators can be ‘distributed’ over a multi-commutator). In particular, a parametrization in terms of fields associated only to real roots of ${{\rm{E}}_{10}}$, besides being incomplete, would also obscure the relation to the fields $\big\{ {A_{\alpha}}^r\big\}$, and therefore does not appear to lead to a convenient parametrisation of the coordinates on the coset space $G/K$.[^5] Irrespective of an explicit description of the coordinates on $G/K$ we can still generalise the worldline $\sigma$-model to infinite-dimensonal cosets, and consider the Cartan form $$\begin{aligned} \label{VdV} \partial V V^{-1} = P+Q = \partial q^{{\tt{a}}}H_{{\tt{a}}}\,+\, \sum_{\alpha>0} \sum_{r=1}^{\mathrm{mult}(\alpha)} {P_{\alpha}}^r {E_{\alpha}}^r\end{aligned}$$ without spelling out the explicit parametrisation of $V$ and ${P_{\alpha}}^r$ in terms of coordinates and their time derivatives. The triangular structure on $N$ implies, however, that the ${P_{\alpha}}^r$ are all *finite* combinations of coordinates and their derivatives, as we explained already after (\[VdV\]). The coset equations  take the same form if the Lagrangian is the formal extension of (\[Lag\]) to the infinite-dimensional Kac–Moody algebra, using the invariant bilinear form (\[Norm2\]) on the Kac–Moody algebra. Therefore  becomes $$\begin{aligned} \label{eomKMrt} \partial {P_{\alpha}}^t =-\partial{q}^{{\tt{a}}}\alpha_{{\tt{a}}}{P_{\alpha}}^t \,+\, 2\sum_{\beta>0} \sum_{r,s} c_{\alpha+\beta,-\beta}^{r\,s\,t} {P_{\beta}}^r{P_{\alpha+\beta}}^s.\end{aligned}$$ As we noted after (\[eomrt\]) the r.h.s. is a sum over terms of [*ascending*]{} height, and hence an infinite sum for infinite-dimensional $\mathfrak{g}$. This sum can be rendered finite and calculable only by consistently truncating the ${P_{\alpha}}^r$ to vanish beyond a given height, as is necessary for the comparison between supergravity and the ${{\rm{E}}_{10}}$ $\sigma$-model. More concretely, this can be done for example by choosing a grading on the root lattice and cutting off ${P_{\alpha}}^r$ after a certain degree [@Damour:2004zy]. Canonical treatment ------------------- We now analyse the canonical structure, by again considering the *finite-dimensional* coset model  first. The canonical momenta from  are $$\begin{aligned} \pi_{{\tt{a}}}&= \frac{\partial L}{\partial \partial q^{{\tt{a}}}} = G_{{{\tt{a}}}{{\tt{b}}}} \partial q^{{\tt{b}}},&{\nonumber}\\ {\Pi_{\alpha}} &= \frac{\partial L}{\partial \partial {A_{\alpha}}} = \frac12 \left(e^{2 q^{{\tt{a}}}\alpha_{{\tt{a}}}}D{A_{\alpha}} +\frac12 \sum_{\beta>0} c_{\beta,\alpha} e^{2 q^{{\tt{a}}}(\alpha+\beta)_{{\tt{a}}}}{A_{\beta}}D{A_{\alpha+\beta}} +\ldots\right),\end{aligned}$$ displaying again a triangular structure. This can be inverted to write the ${P_{\alpha}}$ in triangular form in terms of canonical coordinates and momenta: $$\begin{aligned} \label{pi2p} {P_{\alpha}} = e^{- q^{{\tt{a}}}\alpha_{{\tt{a}}}}\left( {\Pi_{\alpha}} -\frac12 \sum_{\beta>0} c_{\beta,\alpha}{A_{\beta}} {\Pi_{\alpha+\beta}}+\ldots\right).\end{aligned}$$ From this and the standard relations $$\big\{ q^{{\tt{a}}}, \pi_{{\tt{b}}}\big\} = \delta^{{\tt{a}}}_{{\tt{b}}}\;, \quad \big\{ {A_{\alpha}} , {\Pi_{\beta}} \big\} = \delta_{\alpha,\beta}$$ one can derive the canonical brackets among the $\pi^{{\tt{a}}}$ and ${P_{\alpha}}$: \[Pcan\] $$\begin{aligned} \label{pipi} \big\{ \pi_{{\tt{a}}}, \pi_{{\tt{b}}}\big\} &=0,&\\[2mm] \label{piP} \big\{ \pi_{{\tt{a}}}, {P_{\alpha}} \big\} &= \alpha_{{\tt{a}}}{P_{\alpha}},&\\[2mm] \label{PP} \big\{ {P_{\alpha}},{P_{\beta}} \big\} &= c_{\alpha,\beta} {P_{\alpha+\beta}}.&\end{aligned}$$ Only the first two of these relations are evident, while the third one is not and will be proven below. Of course, to the order given one can check the last relation easily from the expressions above, but the important point is that all the higher non-linear terms combine in the right way to produce such a simple result. Our main point here is that the ‘composite’ variables ${P_{\alpha}}$ are ‘good’ canonical variables because the canonical brackets between them assume a very simple form, and furthermore display a graded structure which is nothing but the Borel subalgebra. Equally important, the ${P_{\alpha}}$, being objects associated with the maximal compact subgroup $K$, couple naturally to the fermions. Let us note the relations $${\left\{} \newcommand{\rd}{\right\}}\pi_{{\tt{a}}}, V \rd = - H_{{\tt{a}}}V\; , \quad {\left\{} \newcommand{\rd}{\right\}}{P_{\alpha}} , V \rd = - {E_{\alpha}} V \,,\quad {\left\{} \newcommand{\rd}{\right\}}{P_{\alpha}}, q^{{\tt{a}}}\rd =0\,,\quad {\left\{} \newcommand{\rd}{\right\}}V, V \rd =0.$$ For any coset space $\sigma$ model the canonical conserved Noether current (or more properly, conserved [*charge*]{}) is given by general formula $$\label{Jcur} J = V^{-1} P V \equiv J^{{\tt{a}}}H_{{\tt{a}}}\, + \, \sum_{\alpha > 0} \big( J_{-\alpha} {E_{\alpha}} + J_{\alpha} {E_{-\alpha}} \big)$$ such that $\partial J = 0$ by the equations of motion (\[eomrt\]). Although the canonical commutation relations for this current reproduce the $GL(n)$ algebra (see below), we will see that the structure of its components is considerably more complicated, not least because $J$ has both upper [*and*]{} lower triangular pieces. For finite-dimensional $\mathfrak{g}$ the lower triangular half of the matrix $J$ takes a relatively simple form when one expresses the associated conserved components in terms of the momenta ${\Pi_{\alpha}}$. By contrast the components of the upper triangular half involve all canonical variables and become increasingly more complicated with growing $n$, see also [@Damour:2002et]. We will illustrate this explicitly with the example of the $GL(3)/SO(3)$ model in Appendix \[app:GL3\]. The conserved current $J$ of  generates the global $G$ transformations in  and, since we are working in fixed triangular gauge, the ‘lower triangular’ $G$ transformations induce a compensating $K$ transformation. That is, we expect the infinitesimal transformation of $V$ to be $$\begin{aligned} {\left\{} \newcommand{\rd}{\right\}}J, V \rd = \delta{V} = - V \delta{g} + \delta{k} V,\end{aligned}$$ where $\delta{g}$ and $\delta{k}$ are the infinitesimal versions of the group transformations in  and $\delta{k}$ is determined by $\delta{g}$ and $V$ such that the resulting $\delta{V}$ is in triangular gauge. This can be worked out in terms of the basis components in  and the canonical brackets (\[Pcan\]), with the result (for $\alpha >0$) \[JV\] $$\begin{aligned} \big\{ {J_{\alpha}} \,,\, V \big\} &= - V E_\alpha \\[2mm] \big\{ J_{{\tt{a}}}\,,\, V \big\} &= - V H_{{\tt{a}}}\\[2mm] \big\{ {J_{-\alpha}} \,,\, V \big\} &= - VE_{-\alpha} - \sum_{\beta>0} \big\langle VE_{-\alpha}V^{-1} \big| E_\beta \big\rangle \, (E_\beta - E_{-\beta}) V,\end{aligned}$$ where the extra term on the r.h.s. in the last line corresponds to the compensating transformation in $K(G)$ required to bring $V$ back into triangular gauge, so that $$\big\langle V^{-1} \{{J_{-\alpha}} , V \} \big| E_\beta \big\rangle = 0$$ and the explicit compensating element in $\mathfrak{k}$ is $$\begin{aligned} \delta{k}_\alpha = -\sum_{\beta>0} \big\langle VE_{-\alpha}V^{-1} \big| E_\beta \big\rangle \, (E_\beta - E_{-\beta}).\end{aligned}$$ In deriving the above brackets we made repeated use of the invariance of the invariant bilinear form (trace) and the orthonormality relations (\[Norm1\]).[^6] Relation  then implies directly the transformation of the velocity components $P$ under $J$. For $\alpha>0$ and $\beta>0$ one has $$\begin{aligned} {\left\{} \newcommand{\rd}{\right\}}J_{{\tt{a}}}, {P_{\beta}} \rd &=0,\\ {\left\{} \newcommand{\rd}{\right\}}{J_{\alpha}}, {P_{\beta}} \rd &=0 \\ {\left\{} \newcommand{\rd}{\right\}}{J_{-\alpha}}, {P_{\beta}} \rd &=\pi_{{\tt{a}}}\beta^{{\tt{a}}}\langle V {E_{-\alpha}} V^{-1} |{E_{\beta}}\rangle -\sum_{\gamma>0} c_{\beta-\gamma,\gamma} {P_{\gamma}} \langle V {E_{-\alpha}} V^{-1} |{E_{\beta-\gamma}}\rangle{\nonumber}\\ &\quad\quad -\sum_{\gamma>0} c_{\beta+\gamma,-\gamma} {P_{\gamma}} \langle V {E_{-\alpha}} V^{-1} |{E_{\beta+\gamma}}\rangle \end{aligned}$$ Because (\[JV\]) equivalently expresses the standard non-linear realisation of global symmetries in non-linear $\sigma$-models we can immediately infer the closure relations \[Jalgebra1\] $$\begin{aligned} \big\{ {J_{\alpha}}, {J_{\beta}} \big\} &= \left\{ \begin{array}{cl} c_{\alpha,\beta} {J_{\alpha+\beta}} &\textrm{if $\alpha+\beta\in \Delta$,} \\[2mm] \alpha^{{\tt{a}}}J_{{\tt{a}}}& \textrm{if $\alpha=-\beta$,}\\[2mm] 0 & \textrm{otherwise,} \end{array}\right.\\[2mm] \big\{ J_{{\tt{a}}}, {J_{\alpha}} \big\} &= \alpha_{{\tt{a}}}{J_{\alpha}}.\end{aligned}$$ in the non-linear realization of the $G$ symmetry acting on $V$. An explicit verification of the above relations for the $GL(3)/SO(3)$ model can be found in appendix \[app:GL3\]. An important aspect of the variables ${P_{\alpha}}$ concerns quantisation. When quantising a non-linear model there is always the question for which canonical variables one should perform the replacement of Poisson or Dirac brackets by quantum commutators (which in quantum field theory may yield [*inequivalent*]{} quantisations). Obviously, the variables ${P_{\alpha}}$ are ideally suited for this purpose; in particular, such a quantisation prescription eliminates all operator ordering ambiguities. Furthermore, we emphasise once again that the ${P_{\alpha}}$ are the natural variables coupling to fermions, as will be seen in more detail below. Canonical structure for $GL(n,\mathbb{R})/SO(n)$ {#sec:gln} ------------------------------------------------ We now prove , and in particular the crucial third relation, for $G= GL(n,{\mathbb{R}})$. This is a slight generalisation of the set-up of the preceding sections since $GL(n,{\mathbb{R}})$ is not simple, but it is the case of direct interest for cosmological billiards [@Damour:2002et]. Let us fix some notation. We denote the generators of $GL(n,{\mathbb{R}})$ by $K^a{}_b$ with $a,b=1,\ldots,n$ and commutation relations $$\begin{aligned} {\left[}K^a{}_b, K^c{}_d{\right]}= \delta^c_b K^a{}_d - \delta^a_d K^c{}_b.\end{aligned}$$ The symmetric and antisymmetric combinations are defined as $S^{ab} = K^a{}_b+K^b{}_a$ and $J^{ab} = K^a{}_b- K^b{}_a$. The positive roots of $GL(n,{\mathbb{R}})$ are denoted by $\alpha_{ab}$ with $a<b$ and will be written as tuples $\alpha_{ab} = (0\cdots 010\cdots 0 -\!\!1 0...)$ with $(+1)$ in the $a$-th and $(-1)$ in the $b$-th place. The generator corresponding to such a positive $\alpha_{ab}$ is then ${E_{\alpha_{ab}}}= K^a{}_b$ and the above commutation relations translate into (recall that $a<b$ and $c<d$) $${\left[}{E_{\alpha_{ab}}} , {E_{\alpha_{cd}}} {\right]}\,= \, c_{\alpha_{ab},\alpha_{cd}} {E_{\alpha_{ab}+\alpha_{cd}}} \, = \, \left\{ \begin{array}{cl} {E_{\alpha_{ad}}} &\textrm{if $b=c$,}\\[1mm] - {E_{\alpha_{cb}}} & \textrm{if $a=d$,}\\[1mm] 0 & \textrm{otherwise.} \end{array}\right.$$ So we read off the general formula $c_{\alpha_{ab},\alpha_{cd}}= \delta_{bc}-\delta_{ad}$. We write the coset element of $GL(n,{\mathbb{R}})/SO(n)$ in Borel gauge by an upper triangular $(n\times n$)-matrix through (as for notation, cf. footnote 1) $$\begin{aligned} V=A N \quad\textrm{with} \quad A=\mathrm{diag}(e^{q^1},\ldots,e^{q^n})\;, \quad N=N^a{}_i.\end{aligned}$$ Here, $a$ is the local (row) index, $i$ a global (column) index. The matrix $N^{a}{}_i$ is equal to $1$ on the diagonal and has vanishing entries for $a>i$. The inverse matrix $N^{-1}$ is also upper triangular, and has components $N^i{}_a$ which vanish for $i>a$. The Borel gauge implies that some of the summations below are restricted in the way indicated. For the coset velocities one finds $$\begin{aligned} \left(\partial V V^{-1}\right)^a{}_b = \partial q^a \delta^a_b + \sum_i e^{q^a- q^b}\partial N^a{}_i N^i{}_b.\end{aligned}$$ For a positive root $\alpha_{ab}$, i.e. $a<b$, we define the quantity $$\begin{aligned} {P_{\alpha_{ab}}} = \frac12 e^{q^a- q^b} \sum_{a<i\leq b} \partial N^a{}_i N^i{}_b\end{aligned}$$ This is just the variable ${P_{\alpha}}$ introduced in the previous section, except that we are now labelling the roots by indices $a,b$. In a convenient normalisation the Lagrangian can be written as $$\begin{aligned} \label{LGL} L \, = & \, \frac12 \mathrm{Tr}(P^2) - \frac12(\mathrm{Tr} P)^2 {\nonumber}\\[2mm] = & \, \frac12 \partial q^{{\tt{a}}}\partial q^{{\tt{b}}}G_{{{\tt{a}}}{{\tt{b}}}} + \sum_{a<b} {P_{\alpha_{ab}}} {P_{\alpha_{ab}}},\end{aligned}$$ where $G_{{{\tt{a}}}{{\tt{b}}}}$ is the DeWitt metric $$\begin{aligned} \sum_{{{\tt{a}}},{{\tt{b}}}}\partial q^{{\tt{a}}}\partial q^{{\tt{b}}}G_{{{\tt{a}}}{{\tt{b}}}} = \sum_{{\tt{a}}}(\partial q^{{\tt{a}}})^2 - \left(\sum_{{\tt{a}}}\partial q^{{\tt{a}}}\right)^2.\end{aligned}$$ The canonical momenta conjugate to $N^a{}_i$ are $$\begin{aligned} \Pi^i{}_a = \frac{\partial L}{\partial \partial N^a{}_i} = \sum_b e^{q^a- q^b} {P_{\alpha_{ab}}} N^i{}_b\end{aligned}$$ and vanish for $a\geq i$. In other words, for $a<b$, $$\begin{aligned} \label{Pab} {P_{\alpha_{ab}}}= \sum_i e^{q^b-q^a} N^a{}_i \Pi^i{}_b.\end{aligned}$$ The advantage of using the variables ${P_{\alpha}}$ is that they obey very simple commutation relations, [*to wit*]{}, $$\begin{aligned} \big\{ {P_{\alpha_{ab}}}, {P_{\alpha_{cd}}}\big\} = c_{\alpha_{ab},\alpha_{cd}} {P_{\alpha_{ab}+\alpha_{cd}}}\end{aligned}$$ whenever $\alpha_{ab}+\alpha_{cd}$ is a root, and we recall $c_{\alpha_{ab},\alpha_{cd}} =\delta_{bc}-\delta_{ad}$. This can be verified by straightforward computation using $\{ N^a{}_i , \Pi^j{}_b \} = \delta^a_b \delta^j_i$. It is equally easy to see that if $\alpha_{ab}+\alpha_{cd}$ is not a root, the canonical bracket vanishes, as does the structure constant. We thus have demonstrated the relation  for $GL(n,{\mathbb{R}})$. The other relations  and  are also evident for $GL(n,{\mathbb{R}})$, given the explicit form of the Lagrangian  and . The proof of (\[PP\]) can be extended to all other simple finite dimensional Lie algebras, either by direct computation, or more simply by looking at differently embedded $GL(n)$ subalgebras, and by observing that these commutation relations must be compatible with the action of the Weyl group, because all roots can be reached by Weyl transformations from the simple roots. As a simple example we discuss the case of $GL(3)/SO(3)$ in appendix \[app:GL3\]. The extension of the above results to [*infinite-dimensional*]{} Kac–Moody algebras, and more specifically to the ${{\rm{E}}_{10}}$ algebra, is more subtle, and here we do not have a complete picture. In particular, we do not have a proof that (\[piP\]) and (\[PP\]) remain valid for all roots. For instance, in the presence of imaginary roots (\[PP\]) would have to generalise to $$\label{PaPb1} \big\{ {P_{\alpha}}^r , {P_{\beta}}^s \big\} = \sum_{t=1}^{{\rm mult}(\alpha)} c^{rst}_{\alpha,\beta} {P_{\alpha + \beta}}^t.$$ where $\alpha$ and $\beta$ are any roots, and where the sum on $t$ ranges over the multiplicity of the root $(\alpha + \beta)$ if this is an imaginary root. While a general derivation by the above methods seems beyond reach, we can extend the argument at least to those roots $\alpha$ and $\beta$ for which $\alpha + \beta$ is also a real root, because the above commutation relation should respect the Weyl group, and because all real roots can be reached by ${{\rm{E}}_{10}}$ Weyl transformations. Hence at least for this special case, the above relation should also hold for ${{\rm{E}}_{10}}$. Hamiltonian analysis -------------------- The canonical Hamiltonian is $$\begin{aligned} H &= \pi_{{\tt{a}}}\partial q^{{\tt{a}}}+\sum_{\alpha>0} {\Pi_{\alpha}} \partial {A_{\alpha}} - L&{\nonumber}\\ &= \frac12 \pi_{{\tt{a}}}G^{{{\tt{a}}}{{\tt{b}}}} \pi_{{\tt{b}}}+ \sum_{\alpha>0} e^{-2q^{{\tt{a}}}\alpha_{{\tt{a}}}} {\Pi_{\alpha}}^2 + \ldots\end{aligned}$$ where the dots denote important non-linear terms. In terms of the coset velocities they can be summarised as (see also [@Matschull:1994vi] for a derivation in the finite-dimensional case) $$\begin{aligned} \label{HFD} H =\frac12 \pi_{{\tt{a}}}G^{{{\tt{a}}}{{\tt{b}}}} \pi_{{\tt{b}}}+ \sum_{\alpha>0} {P_{\alpha}}^2.\end{aligned}$$ Again, it is important that the non-linear terms combine in the right way to yield such a simple expression. We note that  implies that we can rewrite the Hamiltonian alternatively as $$\begin{aligned} \label{H=JJ} H= \frac12 \langle J | J\rangle.\end{aligned}$$ which we recognise as the standard bilinear form on the corresponding Lie algebra. Let us verify the consistency expression  with the coset equations of motion $$\begin{aligned} \partial {P_{\alpha}} = \left\{ {P_{\alpha}}, H \right\} = -\alpha_{{\tt{a}}}G^{{{\tt{a}}}{{\tt{b}}}}\pi_{{\tt{b}}}{P_{\alpha}} +2\sum_{\beta>0} c_{\alpha,\beta} {P_{\beta}} {P_{\alpha+\beta}}.\end{aligned}$$ Comparing the above relation with the general result  shows agreement. This shows that the Borel structure is correct, at least for all finite-dimensional algebras: any other algebra would not correctly reproduce the equations of motion. Staying at the formal level, an analogous argument also works for the infinite-dimensional case. Namely, we can similarly deduce a statement of the canonical brackets of the ${P_{\alpha}}^r$ in the Kac–Moody case. Starting from the same Lagrangian $$\begin{aligned} L = \frac12 \langle P | P \rangle\end{aligned}$$ as in the finite-dimensional case, but where $\langle\cdot|\cdot\rangle$ is now the standard invariant bilinear form on the Kac–Moody algebra, the Hamiltonian is given by the straight-forward formal extension of , using the arguments of [@Matschull:1994vi]: $$\begin{aligned} H=\frac12 \pi_{{\tt{a}}}G^{{{\tt{a}}}{{\tt{b}}}} \pi_{{\tt{b}}}+ \sum_{\alpha>0} \sum_r {P_{\alpha}}^r {P_{\alpha}}^r\end{aligned}$$ Because, formally, the conserved Noether current is still given by $$\label{JcurKM} J = V^{-1} P V \equiv J^{{\tt{a}}}H_{{\tt{a}}}\, + \, \sum_{\alpha > 0} \sum_{r=1}^{{\rm mult}(\alpha)} \big( J_{-\alpha}^r {E_{\alpha}}^r + J_{\alpha}^r {E_{-\alpha}}^r \big)$$ the Hamiltonian can again be cast into the form (\[H=JJ\]) with the bilinear form on the Kac–Moody algebra. When considered as a function of the phase space variables $\big\{ J^{{\tt{a}}}, J_{\pm \alpha}^r \big\}$ this is just the (unique) ${{\rm{E}}_{10}}$ invariant bilinear form. Let us mention, however, that in contrast to the finite-dimensional case a simple form of the lower triangular half can only be achieved by truncating the current components to $J_{-\alpha}^r =0$ for $\alpha$’s exceeding a given height. A related discussion can be found in [@Damour:2002et]. Compatibility of the canonical structure with the equations of motion  is then ensured by the canonical brackets $$\begin{aligned} \label{PcanE10} {\left\{} \newcommand{\rd}{\right\}}{P_{\alpha}}^r , {P_{\beta}}^s \rd =\sum_t c^{r\,s\,t}_{\alpha,\beta} {P_{\alpha + \beta}}^t.\end{aligned}$$ We thus see that [*if*]{} the Hamiltonian is given by the restriction of the ${{\rm{E}}_{10}}$ Casimir operator to the coset ${{\rm{E}}_{10}}/K({{\rm{E}}_{10}})$, the compatibility of the canonical structure with the equations of motion [*implies*]{} the extension of the Borel-like structure found in (\[Pcan\]) to the full Borel subalgebra of ${{\rm{E}}_{10}}$! However, it is known that beyond level $\ell = 3$ the canonical supergravity Hamiltonian starts to deviate from the Casimir operator, and therefore we will also have to eventually allow for modifications in the canonical algebra (\[PcanE10\]). Fermions and supersymmetry {#sec:fermions} ========================== The extension of the ${{\rm{E}}_{10}}$ coset model to include fermions was discussed in [@Damour:2005zs; @deBuyl:2005mt; @Damour:2006xu]. We briefly review the salient features of the resulting model and its relation to maximal $D=11$ supergravity in order to provide a self-contained presentation. ${{\rm{E}}_{10}}$, its level decomposition and the bosonic sector ----------------------------------------------------------------- The description of ${{\rm{E}}_{10}}$ that is most commonly used in connection with $D=11$ supergravity is that where the Lie algebra is presented in $GL(10)$ level decomposition [@Damour:2002cu]. In this presentation, the infinitely many generators of ${{\rm{E}}_{10}}$ are organised into $\mathfrak{gl}(10,{\mathbb{R}})$ tensor representations and graded by a level $\ell$ such that each level only contains finitely many $\mathfrak{gl}(10,{\mathbb{R}})$ representations. The Lie bracket is compatible with the level. At low non-negative levels one finds the following $\mathfrak{gl}(10,{\mathbb{R}})$ representations corresponding to the (spatial) components of the $D=11$ fields and their magnetic duals: Level $\ell$ Generator Representation of $\mathfrak{gl}(10,{\mathbb{R}})$ -------------- ----------------------------------------------------------------------------------- ---------------------------------------------------- $0$ $K^a{}_b$ $\mathbf{100}$ (adjoint; graviton) \[2mm\] $1$ $E^{abc}=E^{[abc]}$ $\mathbf{120}$ (three-form) \[2mm\] $2$ $E^{a_1\ldots a_6}= E^{[a_1\ldots a_6]}$ $\mathbf{210}$ (six-form) \[2mm\] $3$ $E^{a_0|a_1\ldots a_8}= E^{a_0|[a_1\ldots a_8]}$ with $E^{[a_0|a_1\ldots a_8]}=0$ $\mathbf{440}$ ((8,1)-hook; dual graviton) The ‘coset velocity’ $P$ of  can be similarly decomposed by level $$\begin{aligned} P &=& \sum_{\alpha>0} \sum_{r=1}^{{\rm mult} (\alpha)}{P_{\alpha}}^r ({E_{\alpha}}^r + {E_{-\alpha}}^r ) \equiv \sum_{\ell\geq 0} P^{(\ell)} * E^{(\ell)} {\nonumber}\\[2mm] &\equiv& \frac12 P^{(0)}_{ab} S^{ab} + \frac1{3!} P^{(1)}_{abc} S^{abc} + \frac1{6!} P^{(2)}_{abcdef} S^{abcdef} + \frac1{9!} P^{(3)}_{a_0| a_1\cdots a_8} S^{a_0|a_1\cdots a_8} + \cdots\end{aligned}$$ Here, the $P^{(\ell)}$ transform in the representation from the table branched to $SO(10)$ level (since $P$ transforms covariantly under the ‘compact’ subgroup $K({{\rm{E}}_{10}})$). The generators are defined by $$\begin{aligned} S^{ab} &= K^a{}_b + K^b{}_a,& S^{abc} &= E^{abc} + F_{abc},{\nonumber}\\[2mm] S^{a_1\ldots a_6} &= E^{a_1\ldots a_6} + F_{a_1\ldots a_6},& S^{a_0|a_1\ldots a_8} &= E^{a_0|a_1\ldots a_8} + F_{a_0|a_1\ldots a_8},\end{aligned}$$ where $F_{abc}$ etc. are the Chevalley transposed generators on the negative levels and correspond to the ${E_{-\alpha}}^r$ part in the general expression. As was shown in [@Damour:2002cu; @Damour:2006xu], the bosonic coset model with Lagrangian $L=\frac12 \langle P|P \rangle$, when restricted to levels $\ell\leq 3$, is equivalent to $D=11$ supergravity expanded about a fixed spatial point, ${\mathbf{x}_0}$ with the bosonic dictionary[^7] $$\begin{aligned} \label{dictbos} P^{(0)}_{ab} (t) &= -N \omega_{ab 0} (t,{\mathbf{x}_0}),& P^{(1)}_{abc}(t) &= N F_{0abc}(t,{\mathbf{x}_0}),\\[2mm] P^{(2)}_{a_1\ldots a_6}(t) &= \frac1{4!} N \epsilon_{a_1\ldots a_6 b_1\ldots b_4} F^{b_1\ldots b_4}(t,{\mathbf{x}_0}),& P^{(3)}_{a_0|a_1\ldots a_8}(t) &= \frac12 N \epsilon_{a_1\ldots a_8 b_1 b_2} \omega_{b_1 b_2 a_0}(t,{\mathbf{x}_0})\end{aligned}$$ when all higher order spatial gradients are neglected and the $SO(10)$ connection is traceless $\omega_{bba}=0$ (corresponding to the irreducibility condition of the $\ell=3$ generator in the table) and $t$ is the coordinate along the worldline that is identified with the physical time coordinate. The index $0$ is a flat index in the time-direction and $N$ is the lapse function in ADM gauge with zero shift. With the said truncations it can then be shown that the bosonic equations of motion of $D=11$ supergravity coincide with those of the worldline ${{\rm{E}}_{10}}$ sigma model. In order to re-express these $SO(10)$ objects in terms of ${{\rm{E}}_{10}}$ root data and ${P_{\alpha}}^r$ we need to explain how the roots at the various levels are related to the components. As in section \[sec:Can\], we work in the so-called ‘wall basis’ [@Damour:2009zc; @Damour:2013eua]. This means that we write a root $\alpha$ as $\alpha=\sum_{{\tt{a}}}\alpha_{{\tt{a}}}e^{{\tt{a}}}$ where $e^{{\tt{a}}}$ are the basis of the $\mathfrak{h}^*$ dual to the Cartan generators $H_{{\tt{a}}}$ such that $e^{{\tt{a}}}(H_{{\tt{b}}}) = \delta^{{\tt{a}}}_{{\tt{b}}}$ and hence $\alpha(H_{{\tt{a}}}) = \alpha_{{\tt{a}}}$. In the wall basis, the inner product is given by $$\label{eaeb} \langle e^{{\tt{a}}}| e^{{\tt{b}}}\rangle = G^{{{\tt{a}}}{{\tt{b}}}} = \delta^{{{\tt{a}}}{{\tt{b}}}}-\frac19$$ and agrees with the (inverse) DeWitt metric for diagonal metrics. In order to avoid confusion with the labelling of the simple roots it will sometimes be convenient to also use the notation $ p_{{\tt{a}}}\equiv \alpha_{{\tt{a}}}$ interchangeably for the component of $\alpha$ in the wall basis, and also write $\alpha=(p_1,\ldots, p_{10})$ as a row vector. The ten simple roots of ${{\rm{E}}_{10}}$ are explicitly given by $$\begin{aligned} \alpha_1 &= (1,-1,0,0,\ldots,0),{\nonumber}\\ \alpha_2 &= (0,1,-1,0,\ldots,0),{\nonumber}\\ \vdots&{\nonumber}\\ \alpha_9 &= (0,\ldots,0,0, 1,-1),{\nonumber}\\ \alpha_{10} &= (0,0,\ldots,0,1,1,1).\end{aligned}$$ The $\mathfrak{gl}(10)$ level of an arbitrary root $\alpha$ expanded on the simple roots as $\alpha= \sum_{j=1}^{10} m^j \alpha_j$ is $\ell\equiv\ell(\alpha) = m_{10}$. Roots on level $\ell=0$ are roots of $\mathfrak{gl}(10)$ and can be written as $\alpha_{ab}$ as above in section \[sec:gln\]. The components ${P_{\alpha}}$ for these roots are identified with $P_{ab}^{(0)}$ and we let $a<b$ for positive roots as before. The components of the Cartan subalgebra are identified via $\pi^{{\tt{a}}}= P^{(0)}_{aa}$ (no sum). Roots on level $\ell=1$ have three entries $1$ in the wall basis and the other entries $p_{{\tt{a}}}$ are zero, as for example in $\alpha_{10}$ above. Calling the three non-vanishing components $a$, $b$ and $c$ with $a<b<c$, we identify ${P_{\alpha}}$ with $P_{abc}^{(1)}$. Roots $\alpha_{a_1\ldots a_6}$ on level $\ell=2$ have six entries $p_{{\tt{a}}}=1$ and four vanishing entries. We assume $a_1<\ldots<a_6$ and then identify ${P_{\alpha_{a_1\ldots a_6}}}$ with the corresponding level $\ell=2$ coset velocity. Roots on level $\ell=3$ come in two varieties: They either have one entry $p_{{\tt{a}}}=2$, seven entries $p_{{\tt{a}}}=1$ and two $p_{{\tt{a}}}=0$ or they have nine $p_{{\tt{a}}}=1$ and one $p_{{\tt{a}}}=0$. In the first case, we let $a_0$ be the component with entry $p_{a_0}=0$ and assume again that the $p_{{\tt{a}}}=1$ components are ordered as $a_1<\ldots a_7$. Then we identify ${P_{\alpha}}$ with $P_{a_0|a_0a_1\ldots a_7}^{(3)}$. The second case corresponds to null roots (of multiplicity $8$) and schematically we distribute the ordered nine $p_{{\tt{a}}}=1$ components as ${P_{a_0|a_1\ldots a_8}}$. The multiplicity requires extra care and will be discussed in detail in section \[sec:gauge\]. In summary, we find that we can associate $$\begin{aligned} \label{newbosons} {P_{\alpha_{ab}}} = P^{(0)}_{ab},\quad {P_{\alpha_{abc}}} = P^{(1)}_{abc},\quad {P_{\alpha_{a_1\ldots a_6}}} = P^{(2)}_{a_1\ldots a_6},\quad {P_{\alpha_{a_0|a_1\ldots a_8}}} = P^{(3)}_{a_0|a_1\ldots a_8}.\end{aligned}$$ with root labels on the l.h.s., and with the associated $SO(10)$ tensors on the r.h.s. Up to $\ell\leq 3$, this correspondence rule allows us to rewrite any expression involving $P^{(\ell)}$ in terms of ${P_{\alpha}}$. Unfaithful spinor representations of $K({{\rm{E}}_{10}})$ {#sec:KE10} --------------------------------------------------------- Fermions are associated with the compact subalgebra $K({{\rm{E}}_{10}})$ of ${{\rm{E}}_{10}}$. This algebra is generated by the compact combinations ($\alpha>0$) $$\begin{aligned} \label{Jalpha} {{\rm k}_{\alpha}}^r = {E_{\alpha}}^r - {E_{-\alpha}}^r.\end{aligned}$$ We have chosen the letter ${{\rm k}_{\alpha}}^r$ for the $K({{\rm{E}}_{10}})$ generators, rather than using $J(\alpha)^r$ as in [@Kleinschmidt:2013eka] in order to avoid confusion with the components of the conserved current $J$ in . From  the $K({{\rm{E}}_{10}})$ elements satisfy $$\begin{aligned} \label{kcommutator} \big[ {{\rm k}_{\alpha}}^r\,,\, {{\rm k}_{\beta}}^s \big] = \sum_{t=1}^{{\rm mult}(\alpha + \beta)} c_{\alpha,\beta}^{rst} \,{{\rm k}_{\alpha +\beta}}^t \;-\; \sum_{t=1}^{{\rm mult}(\alpha - \beta)} c_{\alpha,-\beta}^{rst} {{\rm k}_{\alpha-\beta}}^t.\end{aligned}$$ In order to make sense of the above relation in general, and because $\alpha-\beta$ can be $<0$ for $\alpha,\beta>0$, one also requires a definition of ${{\rm k}_{\alpha}}^r$ for $\alpha<0$; from (\[Jalpha\]) we directly get $$\label{Jminus} {{\rm k}_{\alpha}}^r := - {{\rm k}_{-\alpha}}^r \qquad \mbox{ for $\alpha < 0$}.$$ which is also consistent with (\[cab\]). $K({{\rm{E}}_{10}})$ admits unfaithful finite-dimensional spinor representations [@Damour:2005zs; @deBuyl:2005mt; @Damour:2006xu; @deBuyl:2005zy; @Koehl1], but unfortunately no faithful spinor representations are known up to now. The unfaithful representations relevant to supergravity involve the vector-spinor (gravitino) and Dirac-spinor (supersymmetry parameter). The representations can be represented conveniently using the wall basis [@Damour:2009zc; @Damour:2013eua] and we use the same formalism as in [@Kleinschmidt:2013eka]. For the Dirac representation it is enough to restrict attention to real roots $\alpha,\beta,\dots$, and we will thus drop the multiplicity labels in the remainder of this section. Then to every element $v$ of the ${{\rm{E}}_{10}}$ root lattice $v = \sum n^j \alpha_j \equiv \sum_{{\tt{a}}}v_{{\tt{a}}}e^{{\tt{a}}}$ (which need not be a root for arbitrary $n^j \in {\mathbb{Z}}$) we associate an element of the $SO(10)$ Clifford algebra through $$\begin{aligned} \label{Gamma} \Gamma(v)= (\Gamma_1)^{v_1} \cdots (\Gamma_{10})^{v_{10}},\end{aligned}$$ where, of course, $\{ {\Gamma}_a , {\Gamma}_b\} = 2\delta_{ab}$ are the usual $SO(10)$ ${\Gamma}$-matrices. The product of two such matrices is given by $$\label{Guv} {\Gamma}(u) \, {\Gamma}(v) = {\varepsilon}_{u,v} {\Gamma}(u \pm v),$$ where we have defined the cocyle $$\begin{aligned} {\varepsilon}_{u,v} = (-1)^{\sum_{{{\tt{a}}}<{{\tt{b}}}} v_{{\tt{a}}}u_{{\tt{b}}}}.\end{aligned}$$ which obeys $$\label{cocycle} {\varepsilon}_{u,v} {\varepsilon}_{v,u} = (-1)^{u\cdot v} \quad , \qquad {\varepsilon}_{u,v} {\varepsilon}_{u+v,w} = {\varepsilon}_{u,v+w} {\varepsilon}_{v,w}$$ where $v\cdot w \equiv G^{{{\tt{a}}}{{\tt{b}}}} v_{{\tt{a}}}w_{{\tt{b}}}$. The cocycle ${\varepsilon}_{u,v}$ is defined only up to a co-boundary, that is, we can modify the above definition (\[Gamma\]) by $$\label{Gamma1} {\Gamma}(v) \quad \rightarrow \quad {\tilde\Gamma}(v) = {\sigma}_v {\Gamma}(v)$$ with ${\sigma}_v = \pm 1$ an (in principle) arbitrary sign factor; then $${\tilde\varepsilon}_{u,v} = {\sigma}_u {\sigma}_v {\sigma}_{u+v} {\varepsilon}_{u,v}$$ also obeys the cocycle relations (\[cocycle\]). Next we specialize to elements $v= \alpha\,,\, \beta \in \Delta$ which [*are*]{} roots, and choose the co-boundary such that $$\label{Gamma2} {\tilde\Gamma}(\alpha) := \sigma_\alpha \Gamma(\alpha) := \left\{\begin{array}{cl} {\Gamma}(\alpha) &\textrm{if $\alpha > 0$},\\[2mm] - {\Gamma}(\alpha)\equiv -\Gamma(-\alpha) &\textrm{if $\alpha < 0$}, \end{array}\right.$$ that is, ${\sigma}_\alpha = \pm 1$ according to whether $\alpha$ is positive or negative, whence ${\sigma}_\alpha {\sigma}_{-\alpha} = -1$. The sign switch between positive and negative roots in (\[Gamma2\]) is necessary to remain consistent with (\[Jminus\]). This definition can be extended to the whole root lattice by choosing ${\sigma}_v = \pm 1$ arbitrarily for non-roots $v$, but subject to the condition ${\sigma}_v {\sigma}_{-v} = -1$ (for $v\neq 0$). Indeed, in the relevant expressions in the supersymmetry constraint the matrix ${\tilde\Gamma}(\alpha)$ always comes with a factor ${P_{\alpha}}^r$ which vanishes when $\alpha$ is not a root. The extra sign in (\[Gamma2\]) leads to an important modification in the multiplication rule (\[Guv\]), [*viz.*]{} $$\label{Gamma3} {\tilde\Gamma}(\alpha) \, {\tilde\Gamma}(\beta) \,=\, {\tilde\varepsilon}_{\alpha,\beta} {\tilde\Gamma}(\alpha + \beta) \,=\, - {\tilde\varepsilon}_{\alpha,-\beta} {\tilde\Gamma}(\alpha - \beta).$$ With these definitions one can check that the map $$\label{Diracrep} {{\rm k}_{\alpha}}\; \mapsto \; \frac12\, {\tilde\Gamma}(\alpha)$$ for all real $\alpha$ provides a representation of $K({{\rm{E}}_{10}})$, when extended consistently by commutators. For consistency of this representation with (\[Jminus\]) the sign in  is crucial. This representation has a large kernel; for instance, for null roots $\delta$ one has ${{\rm k}_{\delta}}^r\mapsto 0$, and for time-like imaginary roots all elements of the corresponding root space either vanish or are represented by the [*same*]{} element of the Clifford algebra. The quotient algebra of $K({{\rm{E}}_{10}})$ by the kernel is isomorphic to $\mathfrak{so}(32)$ [@Damour:2006xu]. We will refer to this representation of $K({{\rm{E}}_{10}})$ as the Dirac-spinor representation, or just ‘Dirac representation’. (This type of representation can be straight-forwardly generalised to other simply-laced Kac–Moody algebras and also to arbitrary Kac–Moody algebras [@Koehl1].) Because (\[Diracrep\]) works for all real roots the comparison of (\[kcommutator\]) with (\[Gamma3\]) shows that, for real roots $\alpha$ and $\beta$, $$c_{\alpha,\beta} = -{\tilde\varepsilon}_{\alpha,\beta}$$ whenever $\alpha+ \beta$ or $\alpha-\beta$ is also a real root. This is furthermore consistent with the fact that for real $\alpha$ and $\beta$ only one of the terms on the r.h.s. of (\[kcommutator\]) can be non-zero. The minus sign in the above relation arises because below we will act on the components of the spinor rather than on the basis vectors. While the Dirac representation corresponds to the supersymmetry transformation parameter, the vector spinor representation derives from the $D=11$ gravitino, and is ‘less unfaithful’ than the Dirac representation. It was first obtained in [@Damour:2006xu] in terms of an $SO(10)$ covariant vector-spinor $\Psi^a_A$ with an $SO(10)$ vector index $a=1,\ldots,10$ and spinor indices $A,B,\ldots= 1,\ldots,32$. This vector-spinor is directly related via a fermionic dictionary to the spatial components of the $D=11$ gravitino $\psi_a$ through [@Damour:2006xu Eq. (5.1)]: $$\label{dictferm} \Psi_a(t) = g^{1/4} \psi_a(t,{\mathbf{x}_0}) ,$$ where $g=\det (g_{mn})$ is the determinant of the spatial part of the metric. For the time component of the gravitino (the Lagrange multiplier for the supersymmetry constraint), we adopt the gauge $\psi_0 = {\Gamma}_0 {\Gamma}^a \psi_a$, as in [@Damour:2006xu]. For the present purposes it is, however, advantageous to switch to a different description of the vector-spinor in terms of fermions $\phi^{{\tt{a}}}$ which are related to the $SO(10)$ covariant vector-spinor $\Psi^a$ of [@Damour:2006xu] above by the following crucial re-definition [@Damour:2009zc] $$\begin{aligned} \label{newfermions} \phi^{{\tt{a}}}= \Gamma^a \Psi^a \quad\quad \textrm{(no sum on $a$)}.\end{aligned}$$ This relation clearly breaks $SO(10)$ covariance, but has an important advantage: in this way the Lorentz group $SO(10)$ gets replaced by the $SO(1,9)$ symmetry acting on the space of diagonal scale factors $\{ q^{{\tt{a}}}\}$, which is also the invariance group of the DeWitt metric $G_{{{\tt{a}}}{{\tt{b}}}}$! It is for this reason that we adopt a different font (${{\tt{a}}}, {{\tt{b}}},...$), as we already did in [@Kleinschmidt:2013eka]; the latter indices are then covariant under the (Lorentzian) invariance group of the DeWitt metric $G_{{{\tt{a}}}{{\tt{b}}}}$. We will also use the notation $$\begin{aligned} \phi(\alpha) \equiv \alpha_{{\tt{a}}}\phi^{{\tt{a}}}.\end{aligned}$$ Like the Dirac representation the vector-spinor representation, now modeled by spinors $\phi^{{\tt{a}}}_A$, is obviously finite-dimensional (we will often suppress explicitly writing out the spinor indices). The vector-spinor $\phi^{{\tt{a}}}_A$ satisfies the canonical (Dirac) brackets [@Damour:2006xu; @Damour:2009zc]: $$\begin{aligned} \label{phispinors} {\left\{} \newcommand{\rd}{\right\}}\Psi^a_A , \Psi^b_B \rd = \delta^{ab}\delta_{AB} - \frac19 ({\Gamma}^a {\Gamma}^b)_{AB} \;\; \Rightarrow \quad {\left\{} \newcommand{\rd}{\right\}}\phi^{{\tt{a}}}_A, \phi^{{\tt{b}}}_B \rd = G^{{{\tt{a}}}{{\tt{b}}}} \delta_{AB}.\end{aligned}$$ (recall the definition of $G^{{{\tt{a}}}{{\tt{b}}}}$ in (\[eaeb\])). A canonical representation of $K({{\rm{E}}_{10}})$ is then obtained by defining for any real root $\alpha$ $$\begin{aligned} \label{Jreal} {{\rm k}_{\alpha}} = X_{{{\tt{a}}}{{\tt{b}}}}(\alpha)\phi^{{\tt{a}}}{\tilde\Gamma}(\alpha) \phi^{{\tt{b}}}, \quad X_{{{\tt{a}}}{{\tt{b}}}}\equiv X_{{{\tt{a}}}{{\tt{b}}}}(\alpha) = - \frac12 \alpha_{{\tt{a}}}\alpha_{{\tt{b}}}+ \frac14 G_{{{\tt{a}}}{{\tt{b}}}}.\end{aligned}$$ and this construction yields an unfaithful representation of $K({{\rm{E}}_{10}})$ [@Kleinschmidt:2013eka]. Note that we again have to employ the ${\tilde\Gamma}$-matrices from (\[Gamma2\]) in order to extend this definition to both positive and negative real roots. We also note that the unfaithful spinor representation can be used to deduce partial information about the unknown structure constants of $K({{\rm{E}}_{10}})$, and thus ${{\rm{E}}_{10}}$. With the bosonic dictionary  and the fermionic dictionary  one can now convert any supergravity expression into the ${{\rm{E}}_{10}}$-variables $P^{(\ell)}$ and $\Psi_a$. With the relations  and  we can then rewrite in the next step everything into ${P_{\alpha}}$ and $\phi^{{\tt{a}}}$ variables. This is the procedure we now apply to the supersymmetry constraint of $D=11$ supergravity. Supersymmetry constraint ------------------------ In terms of the original canonical variables of $D=11$ supergravity [@Cremmer:1978km], the canonical supersymmetry constraint is given by [@Damour:2006xu Eq. (3.12)] $$\begin{aligned} \label{cSt} \tilde{{\mathcal{S}}} &=& \Gamma^{ab} \Big[ \partial_a \psi_b + \frac14 \omega_{acd} \Gamma^{cd} \psi_b + \omega_{abc} \psi_c + \frac12 \omega_{ac0} \Gamma^c \Gamma^0 \psi_b \Big] {\nonumber}\\[1mm] && + \, \frac14 \, F_{0abc} \Gamma^0 \Gamma^{ab} \psi^c + \frac1{48} F_{abcd} \Gamma^{abcde} \psi_e,\end{aligned}$$ where $\omega_{ABC}$ are the components of the $D=11$ spin connection and $F_{ABCD}$ the components of the four-form (with flat indices $A,B,...=0,1,\dots,10$). Using the dictionaries  and  one can rewrite this expression in terms of ${{\rm{E}}_{10}}$ coset variables. The translation between the coset model and $D=11$ supergravity furthermore involves neglecting spatial gradients $\partial_a \psi_b$ on the fermions, terms of the form $\partial_a g\propto \omega_{bba}$, and all spatial gradients of second or higher order on the bosonic fields. It was then shown in [@Damour:2006xu Eq. (5.14)] that the supersymmetry constraint can be re-expressed in terms of the coset quantities $P^{(\ell)}$ and in an $SO(10)$ covariant manner as $$\begin{aligned} \label{SUSYconstr} {\mathcal{S}}= & \left( P^{(0)}_{ab}{\Gamma}^a - P^{(0)}_{cc} {\Gamma}_b\right)\Psi^b + \, \frac12 P^{(1)}_{abc} {\Gamma}^{ab} \Psi^c + \, \frac1{ 5!} P^{(2)}_{abcdef} {\Gamma}^{abcde} \Psi^f {\nonumber}\\[2mm] & + \, \frac1{ 6!} \left(P^{(3)}_{a|ac_1\cdots c_7} {\Gamma}^{c_1\cdots c_6} \Psi^{c_7} - \frac1{28} P^{(3)}_{a|c_1\cdots c_8} {\Gamma}^{c_1\cdots c_8} \Psi^a \right) . \end{aligned}$$ Compared to [@Damour:2006xu], we have rescaled the supersymmetry constraint by an overall factor of $2$ and we also recall the normalisation changes that we explained in footnote \[convchange\]. The notation ${\mathcal{S}}$ in place of $\tilde{{\mathcal{S}}}$ of  indicates that we have rescaled $\tilde{{\mathcal{S}}}$ and multiplied it by $\Gamma_0$. In this $SO(10)$ covariant form, repeated indices are summed over and indices are raised and lowered with the Euclidean metric $\delta_{ab}$. We will now rewrite this expression once more, in order to bring it into a form that conforms more closely with the new variables introduced in the foregoing section. A key fact here is that by so doing we will give up manifest spatial Lorentz covariance, and trade it for the Lorentzian $SO(1,9)$ symmetry on the space of scale factors exhibited above. In other words, the simplest form of the constraint is attained by trading a space-time symmetry for a symmetry in (a truncated version of) DeWitt superspace! To convert the expression  to the ${{\rm{E}}_{10}}$ covariant notation above, we change fermionic variables according to  and analyse the various terms. For the contributions from $\ell=0,1,2$, and now writing out the sums, we find $$\begin{aligned} \sum_a P^{(0)}_{aa} {\Gamma}^a\Psi^a - \sum_c P^{(0)}_{cc} \sum_a {\Gamma}_a \Psi^a &=& G_{{{\tt{a}}}{{\tt{b}}}} \pi^{{\tt{a}}}\phi^{{\tt{b}}}{\nonumber}\\[1mm] \sum_{a<b} P^{(0)}_{ab}{\Gamma}^a \Psi^b \, + \, \sum_{a>b} P^{(0)}_{ab}{\Gamma}^a \Psi^b &=& \sum_{a<b} P^{(0)}_{ab} {\Gamma}^{ab} (\phi^b - \phi^a) {\nonumber}\\[1mm] \sum_{a,b,c} P^{(1)}_{abc} {\Gamma}^{ab} \Psi^c &=& 2 \sum_{a<b<c} P_{abc}^{(1)} \Gamma^{abc}(\phi^a + \phi^b + \phi^c) {\nonumber}\\[1mm] \sum_{a,b,c,d,e,f} P^{(2)}_{abcdef} {\Gamma}^{abcde} \Psi^f &=& 5! \sum_{a<b<c<d<e<f} P^{(2)}_{abcdef} {\Gamma}^{abcdef} (\phi^a + \cdots + \phi^f) \end{aligned}$$ where we identified $\pi^{{\tt{a}}}= P^{(0)}_{aa}$. We now see that the expressions on the r.h.s. are already in the desired form; for instance, $$\begin{aligned} \label{sumneg} \sum_{a<b<c} P_{abc}^{(1)} \Gamma^{abc}(\phi^a + \phi^b + \phi^c) \, &\equiv \, \sum_{\alpha_{abc}} P_{\alpha_{abc}} {\Gamma}(\alpha_{abc}) \phi(\alpha_{abc}) \nonumber\\[2mm] &= \, \frac12 \sum_{\ell(\alpha) = \pm 1} {P_{\alpha}} \, {\tilde\Gamma}(\alpha) \phi(\alpha) \end{aligned}$$ where the middle sum on the r.h.s. runs over all level $\ell=1$ roots $\alpha_{abc}$ (which are positive), while the last sum includes positive and negative roots. The level $\ell =0 , 2$ contributions work in an analogous manner. At level $\ell=3$ we encounter not only real roots, but for the first time also null roots. To see this distinction one has to separately analyse those terms in $P^{(3)}_{a|c_1\cdots c_8}$ for which the index $a$ coincides with one of the $c_i$ (yielding real roots), and those terms for which all indices are different, i.e. $a\notin \{ c_1, \dots , c_8 \}$ (yielding null roots). In order to analyse these terms we thus have to split up the various sums. We start with $$\begin{aligned} \sum_{a,c_1,\ldots,c_7} P^{(3)}_{a|ac_1\cdots c_7} {\Gamma}^{c_1\cdots c_6} \Psi^{c_7} &\equiv \sum_{c_1,\ldots, c_7} \sum_{a\neq c_i} P^{(3)}_{a|ac_1\cdots c_7} {\Gamma}^{[c_1\cdots c_6} \Psi^{c_7]}{\nonumber}\\[2mm] &= 6! \sum_{c_1<\cdots<c_7}\sum_{a\neq c_i} P^{(3)}_{a|ac_1\cdots c_7} {\Gamma}^{c_1\cdots c_7} (\phi^{c_1}+\ldots +\phi^{c_7}),\end{aligned}$$ where the $c_i$ have been ordered in the second expression. The other contribution to the supersymmetry constraint  becomes $$\begin{aligned} &-\frac1{28}\sum_{a} \sum_{c_1,\ldots,c_8} P^{(3)}_{a|c_1\cdots c_8} {\Gamma}^{c_1\cdots c_8} \Psi^a =-2\cdot 6! \sum_{a}\sum_{c_1<\cdots<c_8} P^{(3)}_{a|c_1\cdots c_8} {\Gamma}^{c_1\cdots c_8} \Psi^a {\nonumber}\\[2mm] &\quad\quad\quad= 2\cdot 6!\sum_{c_1<\cdots<c_7} \sum_{a\neq c_i} P^{(3)}_{a|ac_1\cdots c_7} {\Gamma}^{c_1\cdots c_7} \phi^a -2\cdot 6! \sum_{c_1<\cdots<c_8}\sum_{a\neq c_i} P^{(3)}_{a|c_1\cdots c_8} {\Gamma}^{ac_1\cdots c_8} \phi^a. \end{aligned}$$ Combining the two parts one finds $$\begin{aligned} & \hspace{30mm}\frac1{6!} \left(P^{(3)}_{a|ac_1\cdots c_7} {\Gamma}^{c_1\cdots c_6} \Psi^{c_7} - \frac1{28} P^{(3)}_{a|c_1\cdots c_8} {\Gamma}^{c_1\cdots c_8} \Psi^a \right) {\nonumber}\\[2mm] &= \sum_{c_1<\cdots<c_7} \sum_{a\neq c_i} P^{(3)}_{a|ac_1\cdots c_7} {\Gamma}^{c_1\cdots c_7} (2\phi^a + \phi^{c_1}+\ldots +\phi^{c_7}) - 2 \, \sum_{c_1<\cdots<c_8}\sum_{a\neq c_i} P^{(3)}_{a|c_1\cdots c_8} {\Gamma}^{ac_1\cdots c_8} \phi^a. \end{aligned}$$ The first term is exactly the contribution from the $360$ (gravitational) real roots on level $\ell=3$, [*viz.*]{} $$\alpha = (2111111100) \quad \mbox{and permutations}$$ where the root shown is associated with the component $P^{(3)}_{1|12345678}$. The normalization is different from the one used previously since the level $\ell=3$ generators were normalised to $9$ rather than $1$ in [@Damour:2006xu], cf. also footnote \[convchange\]. The second term is a sum over the (gravitational) null roots $$\label{nullrts} \delta = (1111111110) \quad \mbox{and permutations}$$ where the first root is now associated with the component $P^{(3)}_{1|23456789}$. Note that the constraint as written above is overcounting them since there are $\begin{pmatrix}10\\8\end{pmatrix}\times 2 = 90$ instead of the required $80$. The reason is a new type of gauge invariance related to the irreducibility of the $\ell=3$ representation and that will be discussed in more detail in section \[sec:gauge\]. Let us summarise: altogether, the rewriting of the supersymmetry constraint  so far has led to the following expression up to and including all roots of $\ell\leq 3$: $$\begin{aligned} \label{SUSY3} \mathcal{S} &= \pi\cdot\phi + \sum_{\alpha^2=2\atop{\ell=0}, \alpha>0} {P_{\alpha}} {\tilde\Gamma}(\alpha) \phi(\alpha) + \sum_{\alpha^2=2\atop{\ell=1}} {P_{\alpha}}{\tilde\Gamma}(\alpha) \phi(\alpha) + \sum_{\alpha^2=2\atop{\ell=2}} {P_{\alpha}} {\tilde\Gamma}(\alpha) \phi(\alpha){\nonumber}\\ &\quad+ \sum_{\alpha^2=2\atop{\ell=3}} {P_{\alpha}} {\tilde\Gamma}(\alpha) \phi(\alpha) + \sum_{\delta^2=0\atop{\ell=3}} \sum_{r=1}^8 {P_{\delta}}^r {\tilde\Gamma}(\delta) \phi(\epsilon^r)\end{aligned}$$ where we have replaced ${\Gamma}$ by ${\tilde\Gamma}$ to underline that the sum can also be extended to run over negative roots as in . The ‘polarisation vectors’ $\epsilon^r$ appearing for the null roots will be discussed in detail in the next section. Null roots and gauge equivalences {#sec:gauge} --------------------------------- We now return to the counting issue mentioned after . The association of a particular index set $(a_1 c_1\ldots c_8)$ with all indices different to a null root component $P_{a_1|c_1\ldots c_8}^{(3)}$ is subject to the irreducibility constraint (Young symmetry) $$\begin{aligned} \ \label{irred} P_{[a_1|c_1\ldots c_8]}^{(3)} =0.\end{aligned}$$ This provides one linear relation between a priori nine different ways of distributing the nine indices on the hook tableau, bringing down the number of independent components to eight, in agreement with the multiplicity of null roots in ${{\rm{E}}_{10}}$. Let us discuss in more detail how this is implemented in the supersymmetry constraint. [**Gauge fixed form:**]{} To see this in more detail let us pick the particular null root corresponding to the indices $\{a,c_1,\ldots,c_8\}=\{1,\ldots, 9\}$. This root has contributions proportional to $\Gamma(\delta)$ through (reordering some of the indices) $$\begin{aligned} -2\left(P_{1|2\ldots 9} \phi^1 + P_{2|3\ldots 9 1} \phi^2 + \ldots +P_{9|1\ldots 8} \phi^9\right)\end{aligned}$$ Here, one could now trade the first term for a combination of the other terms by virtue of . This leads to $$\begin{aligned} -2\left(P_{2|3\ldots 9 1} (\phi^2-\phi^1) + \ldots +P_{9|1\ldots 8} (\phi^9-\phi^1)\right),\end{aligned}$$ that is, it can be written in the form $$\begin{aligned} \sum_{r=1}^8 {P_{\delta}}^r \Gamma(\delta) \phi(\epsilon^r)\end{aligned}$$ with $${P_{\delta}}^1 = P_{2|3\ldots 9 1}\;, \,\ldots,\; {P_{\delta}}^8=P_{9|1\ldots8 }$$ and polarization vectors $$\epsilon^1 = (2\,-\!2\, 0\, 0\, 0 \; 0\, 0\, 0\, 0\, 0),\,\ldots,\, \epsilon^8 = (2\, 0\, 0\, 0\, 0 \; 0\, 0\, 0\, -\!2\, 0)$$ These polarisation vectors are orthogonal to $\delta$ (as required) and correspond to positive $\ell=0$ roots associated with generators $K^1{}_{r+1}$ (or their negatives). [**Gauge unfixed form**]{}: We can avoid choosing a particular set of polarisation vectors by instead letting the ‘multiplicity sum’ run over an enlarged set $$\sum_{r=1}^9{P_{\delta}}^r\Gamma(\delta) \phi(\epsilon^r).$$ Here, ${P_{\delta}}^r$ denote the nine index arrangements and $\epsilon^r$ are nine independent polarisation vectors that are orthogonal to $\delta$. Shifting $\epsilon^r\to \epsilon^r+\delta$ leads to $$\begin{aligned} \sum_{r=1}^9 {P_{\delta}}^r\Gamma(\delta) \phi(\epsilon^r+\delta)= \sum_{r=1}^9 {P_{\delta}}^r\Gamma(\delta) \phi(\epsilon^r)+ \sum_{r=1}^9 {P_{\delta}}^r\Gamma(\delta) \phi(\delta) =\sum_{r=1}^9 {P_{\delta}}^r \Gamma(\delta) \phi(\epsilon^r)\end{aligned}$$ since $\sum_{r=1}^9 {P_{\delta}}^r=0$ by virtue of . Therefore, we have a gauge invariance in the expression that we could use to fix the gauge in the way we have done above. This gauge invariance no longer ‘lives’ in ordinary space-time, but rather in the DeWitt superspace of (diagonal) metrics. Properties of supersymmetry constraint ====================================== Having rewritten the supersymmetry constraint in terms of $K({{\rm{E}}_{10}})$ variables we will now re-investigate the canonical algebra of supersymmetry constraints and its $K({{\rm{E}}_{10}})$ covariance. As for the algebra we will recover the previously derived results according to which the canonical constraints of $D=11$ supergravity in the appropriate truncation are all associated with null roots of ${{\rm{E}}_{10}}$. As for the transformation properties of the superconstraint, we will exhibit its non-covariance under the full $K({{\rm{E}}_{10}})$ – a clear indication that the present construction is incomplete. Supersymmetry constraint algebra -------------------------------- The above calculations led to the following expression for the supersymmetry constraint $$\begin{aligned} \label{S} {\mathcal{S}}_A &= \pi_{{\tt{a}}}\phi^{{\tt{a}}}_A + \sum_{\alpha^2=2\atop{\ell\leq 3},\alpha>0} {P_{\alpha}} \big(\Gamma(\alpha) \phi(\alpha)\big)_A + \sum_{\delta^2=0\atop{\ell =3}} {P_{\delta}}^r \big(\Gamma(\delta)\phi(\epsilon^r)\big)_A $$ (recall that $\phi(\alpha)_A \equiv \alpha_{{\tt{a}}}\phi^{{\tt{a}}}_A$). As shown above, the terms written out in the above formula agree precisely with the supersymmetry constraint derived from supergravity by dropping terms containing spatial gradients as well as cubic terms in the fermions. In other words, apart from these omitted contributions, the full content of the supersymmetry constraint is captured by the $\ell\leq 3$ sector of the ${{\rm{E}}_{10}}$ model with fermions. However, from this restriction it is already clear that this expression cannot be the whole story, and we will make this point more explicit in the following section by showing that, contrary to first expectations, ${\mathcal{S}}$ does not transform in the Dirac representation, nor in any other known representation of $K({{\rm{E}}_{10}})$. Nevertheless, under the canonical brackets, the supersymmetry constraint in the above form should yield the Hamiltonian and all other constraints supergravity constraints in the gradient truncation (and ignoring higher order fermionic terms). Schematically, we see that $$\begin{aligned} \label{SS} \big\{ {\mathcal{S}}_A, {\mathcal{S}}_B\big\} = 2{\mathcal{H}}\delta_{AB} + \sum_{\delta^2=0} {\mathcal{C}}(\delta) \Gamma_{AB}(\delta) + \ldots\end{aligned}$$ Here, we have introduced the calligraphic letter ${\mathcal{H}}$ for the ‘Hamiltonian’ arising from the commutator of two supersymmetry constraints, to distinguish it notationally from the coset Hamiltonian $H$ discussed in the previous sections, since it is not clear a priori whether the two agree. Indeed, we will explain below that they do differ. The anti-commutator  contains many terms, but let us first concentrate on the ones containing no fermions (the ones bilinear in the fermions would also receive contributions from cubic fermionic terms, which are not included in the above formula for ${\mathcal{S}}$). Here we use (for roots $\alpha$ and $\beta$ that are real and hence have anti-symmetric $\Gamma(\alpha)$ and $\Gamma(\beta)$) $$\begin{aligned} \big\{ \pi\cdot\phi_A\, ,\,\pi\cdot \phi_B \big\} &= G_{{{\tt{a}}}{{\tt{b}}}} \pi^{{\tt{a}}}\pi^{{\tt{b}}}\delta_{AB} \\[2mm] \big\{ \pi\cdot \phi_A\,,\, {P_{\alpha}} \big(\Gamma(\alpha)\phi(\alpha)\big)_B \big\} &= (\alpha \cdot \pi) {P_{\alpha}} \Gamma(\alpha)_{BA} \, + \, \cdots \\[2mm] \label{SSpart3} \big\{ \pi\cdot \phi_A\,,\, {P_{\delta}}^r \big(\Gamma(\delta)\phi({\varepsilon}^r)\big)_B \big\} &= ({\varepsilon}^r \cdot \pi) {P_{\delta}}^r \Gamma(\delta)_{BA} \, + \, \cdots \\[2mm] \big\{ {P_{\alpha}} \big(\Gamma(\alpha)\phi(\alpha)\big)_A \, ,\, {P_{\beta}} \big(\Gamma(\beta)\phi(\beta)\big)_B \big\} &= - (\alpha \cdot \beta) {\varepsilon}_{\alpha,\beta} \, {P_{\alpha}} {P_{\beta}} \Gamma(\alpha + \beta)_{AB} \, + \cdots\end{aligned}$$ where dots stand for terms quadratic in the fermions. Now the anticommutator   is [*symmetric*]{} in $A,B$, hence the terms in the second line do not contribute because $\Gamma(\alpha)$ is antisymmetric for real roots $\alpha$.[^8] Consequently, the result will then contain only terms proportional to $\delta_{AB}$ (the Hamiltonian), and terms where $\alpha + \beta$ is light-like (the constraints), and more generally, for which $(\alpha + \beta)^2$ is a multiple of four. This is indeed the structure displayed in (\[SS\]). Let us first look at the Hamiltonian. The first kind of contribution will come from those terms with $\beta = \alpha$; in this case we use ${\varepsilon}_{\alpha,\alpha} = -1$ to get $$\label{Ham1} - (\alpha \cdot\alpha) {\varepsilon}_{\alpha,\alpha} \, {P_{\alpha}} {P_{\alpha}} \, \Gamma(2\alpha)_{AB} \, = \, + \, 2 \, {P_{\alpha}} {P_{\alpha}} \delta_{AB}$$ which is positive, and agrees with what we get from the ${{\rm{E}}_{10}}$ Casimir (see below). For the second kind we have $\alpha\neq \beta$, but such that $(\alpha + \beta)$ has only even components, such that again $\Gamma(\alpha + \beta) = {\bf 1}$; for example $\alpha+\beta=2\delta=(22222\,22220)$ with $$\alpha = (21111\,11100) \quad \mbox{and} \quad \beta = (01111\,11120)$$ In this case we still have ${\varepsilon}_{\alpha,\beta} = -1$ but $\alpha\cdot\beta = -2$, hence $$\label{Ham2} - (\alpha \cdot \beta) {\varepsilon}_{\alpha,\beta} \, {P_{\alpha}} {P_{\beta}} \, \Gamma(\alpha + \beta)_{AB} \, = \, - \, 2\, {P_{\alpha}} {P_{\beta}} \delta_{AB}$$ As one can easily check these are indeed associated with the [*negative definite*]{} terms in the bosonic part of the supergravity Hamiltonian. To see this more explicitly, we recall from [@Damour:2006xu Eq. (6.6)] the $SO(10)$ covariant expressions for ${\mathcal{H}}$ arising from the supersymmetry commutator: $$\begin{aligned} \label{HSUGRA} {\mathcal{H}}&= \frac12 P^{(0)}_{ab}P^{(0)}_{ab} - \frac12 P^{(0)}_{aa}P^{(0)}_{bb} +\frac1{ 3!}P^{(1)}_{abc}P^{(1)}_{abc} +\frac1{ 6!}P^{(2)}_{a_1\ldots a_6}P^{(2)}_{a_1\ldots a_6}{\nonumber}\\[2mm] &\quad+\frac2{8!} \Big( P^{(3)}_{a_0|a_1\ldots a_8}P^{(3)}_{a_0|a_1\ldots a_8} - 4 \, P^{(3)}_{b|ba_1\ldots a_7}P^{(3)}_{c|ca_1\ldots a_7} \Big){\nonumber}\\[2mm] &= \frac12 \pi_{{\tt{a}}}G^{{{\tt{a}}}{{\tt{b}}}} \pi_{{\tt{b}}}\;+ \sum_{\alpha>0\atop \alpha^2=2,\ell\leq 3} {P_{\alpha}} {P_{\alpha}} \; - \sum_{\alpha,\beta>0, \alpha+\beta=2\delta\atop \alpha^2=\beta^2=2,\ell= 3} {P_{\alpha}}{P_{\beta}}$$ (see footnote \[convchange\] for the normalisations of the level-2 and level-3 terms). Writing out the sums in last two terms we get exactly the two contributions (\[Ham1\]) and (\[Ham2\]) (plus the contribution from the null root). This result is to be contrasted with the coset Hamiltonian $H$ $$\begin{aligned} \label{HCOS} H= \frac12 \langle P | P\rangle &= \frac12 P^{(0)}_{ab}P^{(0)}_{ab} - \frac12 P^{(0)}_{aa}P^{(0)}_{bb} +\frac1{ 3!}P^{(1)}_{abc}P^{(1)}_{abc} +\frac1{ 6!}P^{(2)}_{a_1\ldots a_6}P^{(2)}_{a_1\ldots a_6}{\nonumber}\\[2mm] & \quad +\, \frac1{8!}P^{(3)}_{a_0|a_1\ldots a_8}P^{(3)}_{a_0|a_1\ldots a_8} \; + \, \ldots{\nonumber}\\[2mm] &= \frac12 \pi_{{\tt{a}}}G^{{{\tt{a}}}{{\tt{b}}}} \pi_{{\tt{b}}}\;+ \sum_{\alpha > 0\atop \alpha^2 =2,\ell\leq 3} {P_{\alpha}} {P_{\alpha}} \; + \; \ldots\end{aligned}$$ where the dots stand for higher level real roots, as well as imaginary roots. In $SO(10)$ form, this latter expression differs from the previous one not only by the appearance of the negative $\ell =3$ term in (\[HSUGRA\]) but also by the factor in front of the $\ell=3$ term with the correct sign. However, when writing out the Hamiltonian (\[HSUGRA\]) in terms of the $K({{\rm{E}}_{10}})$ variables in a manner completely analogous to the derivation in the foregoing section, we see that the terms at level $\ell=3$ are in one-to-one correspondence with the two types of terms exhibited in (\[Ham1\]) and (\[Ham2\]).[^9] We have thus isolated the source of the disagreement between the canonical Hamiltonian and the ${{\rm{E}}_{10}}$ Casimir that appears from level $\ell =3$ onwards, in terms of the $K({{\rm{E}}_{10}})$ covariant looking supersymmetry constraint (\[S\]). This disagreement is seen not only in the negativity, but also in the fact that the ${{\rm{E}}_{10}}$ Casimir does not pair ${P_{\alpha}}$ with ${P_{\beta}}$ for $\beta\neq \pm \alpha$. Clearly the source of the trouble resides in the unfaithfulness of the $K({{\rm{E}}_{10}})$ representation in terms of the $\Gamma(\alpha)$ matrices that we are dealing with here, and would seem to require a generalisation of the usual Clifford algebra. We also see that these troubles multiply when we extend the sum from roots with $\ell\leq 3$ to all real roots, as we will then have many more contributions proportional to $\delta_{AB}$, which would ruin the agreement with the supergravity Hamiltonian found above. The bosonic constraints identified in [@Damour:2006xu] and associated there with lightlike roots are also recovered from those combinations where $\alpha + \beta$ is a null root; as both $\alpha$ and $\beta$ can go up to level $\ell =3$, the resulting null roots go up to level $\ell =6$, in agreement with [@Damour:2006xu]. So the constraints are generically of the form $${\mathcal{C}}(\delta) = \sum_r {\varepsilon}^r \cdot {P_{\delta}}^r \; + \, \sum_{\alpha + \beta = \delta} {P_{\alpha}} {P_{\beta}} \, + \cdots$$ which agrees exactly with what was found before in [@Damour:2006xu]. Note, however, that starting from the supersymmetry constraint (\[S\]), the first term on the r.h.s. only appears for the null root at level $\ell =3$, whereas this term is missing for the higher level null roots, because the supersymmetry constraint only goes up to $\ell =3$. By contrast, the null roots appearing in the combinations $\alpha + \beta$ can go up to $\ell =6$. This is a clear signal of the incompleteness of the supersymmetry constraint (\[S\]) as derived from supergravity. We note also that there is only [*one*]{} constraint per null root $\delta$ whereas there are eight root generators ${E_{\delta}}^r$. We note that the fermion $\phi^{{\tt{a}}}$ appears as a [*matter fermion*]{} in the one-dimensional model even though it transforms in a vector-spinor representation and descends from the $D=11$ gravitino. This can for instance be seen by considering the transformation of $\phi^{{\tt{a}}}$ under ${\mathcal{S}}$ of  which does not contain any derivatives of the transformation parameter (these would come from $D_a \phi^{{\tt{a}}}$ terms that were truncated away in the derivation from supergravity). The one-dimensional gravitino that is the supersymmetry partner of the one-dimensional lapse function was set to zero. (In)compatibility of supersymmetry and $K({{\rm{E}}_{10}})$ {#sec:susytrm} ----------------------------------------------------------- We can now also investigate the transformation properties of the constraint ${\mathcal{S}}$ under $K({{\rm{E}}_{10}})$. Because ${\mathcal{S}}$ is ‘built’ out of objects that do transform properly under $K({{\rm{E}}_{10}})$, namely the coset quantities ${P_{\alpha}}$ on the one hand, and the unfaithful vector spinor representation $\phi^{{\tt{a}}}$ on the other, one would naively expect this constraint to transform in the Dirac representation, that is, $\delta_\alpha {\mathcal{S}}= \frac12 \Gamma(\alpha) {\mathcal{S}}$. However, there appears a basic clash: as we will now show very explicitly, ${\mathcal{S}}$ [*fails to transform properly under*]{} $K({{\rm{E}}_{10}})$. There are two reasons for this, namely first the presence of imaginary roots in ${{\rm{E}}_{10}}$ and $K({{\rm{E}}_{10}})$ (and thus the fact that both algebras are infinite-dimensional), and secondly the unfaithfulness of the vector-spinor representation. For the variation of ${\mathcal{S}}$ under a $K({{\rm{E}}_{10}})$ transformation generated by ${{\rm k}_{\alpha}}$ we use the formulas $$\begin{aligned} \delta_\alpha \pi^{{\tt{a}}}&=& -2\alpha^{{\tt{a}}}\, {P_{\alpha}} {\nonumber}\\[2mm] \delta_\alpha {P_{\beta}} &=& \delta_{\alpha,\beta} \alpha_{{\tt{a}}}\pi^{{\tt{a}}}\, +\, c_{\beta-\alpha,\alpha} {P_{\beta - \alpha}} - c_{\alpha + \beta, -\alpha} {P_{\alpha + \beta}} {\nonumber}\\[2mm] \delta_\alpha \phi^{{\tt{a}}}&=& \frac12 {\tilde\Gamma}(\alpha) \phi^{{\tt{a}}}- \alpha^{{\tt{a}}}{\tilde\Gamma}(\alpha) \phi(\alpha)\end{aligned}$$ restricting to positive real $\alpha, \beta$ for simplicity. For the first two lines we have evaluated ${\left[}P, {{\rm k}_{\alpha}}{\right]}$ and projected onto the $H_{{\tt{a}}}$ and ${E_{\beta}}+{E_{-\beta}}$ components.[^10] We emphasize that it is not known whether the ${P_{\alpha}}$, when supplemented by the higher root partners ${P_{\alpha}}^r$, transform in an irreducible representation of $K({{\rm{E}}_{10}})$, or whether this representation is reducible under $K({{\rm{E}}_{10}})$. Substituting these formulas into the variation of ${\mathcal{S}}$ some further calculation leads to $$\begin{aligned} \delta_\alpha {\mathcal{S}}&=& \frac12 {\tilde\Gamma}(\alpha) {\mathcal{S}}\; + \; \frac12 \sum_{\beta>0} {P_{\beta}} \big[ {\tilde\Gamma}(\beta) , {\tilde\Gamma}(\alpha)\big] \phi(\beta) {\nonumber}\\[2mm] && + \sum_{\beta>0\,,\, \beta\neq \alpha} \Big[ - (\alpha\cdot\beta) {P_{\beta}} {\tilde\Gamma}(\beta){\tilde\Gamma}(\alpha) \phi(\alpha) \, + \, c_{\beta-\alpha,\alpha} {P_{\beta - \alpha}} {\tilde\Gamma}(\beta) \phi(\beta) {\nonumber}\\[2mm] && \qquad \qquad \qquad\qquad\qquad - \, c_{\alpha + \beta, -\alpha} {P_{\alpha + \beta}} {\tilde\Gamma}(\beta) \phi(\beta) \Big]\end{aligned}$$ The result would thus have the desired structure if we could show that all terms on the r.h.s. cancel except for the first. However, as we will now demonstrate by explicit computation this is the case only for finite dimensional $K$, but no longer for $K({{\rm{E}}_{10}})$. To do so we first rewrite the last term in brackets as $$- \sum_{\beta>\alpha} c_{\beta,- \alpha} {P_{\beta}}{\tilde\Gamma}(\beta - \alpha) \phi(\beta - \alpha)$$ and shift the sums in the second term such a way that only ${P_{\beta}}$ with $\beta >0$ appear. So in the second term in brackets above we consider the partial sum[^11] $$\sum_{0<\beta <\alpha} c_{\beta-\alpha,\alpha} {P_{\beta- \alpha}} {\tilde\Gamma}(\beta) \phi(\beta) = \sum_{0 < \beta < \alpha} c_{-\beta,\alpha} {P_{-\beta}} {\tilde\Gamma}(- \beta+ \alpha) \phi(- \beta + \alpha)$$ Next, using ${P_{-\beta}} = {P_{\beta}}$ and $\phi( - \beta + \alpha) = - \phi(\beta - \alpha)$ as well as (\[cab\]) and not forgetting the extra minus sign from the definition of ${\tilde\Gamma}$ in (\[Gamma2\]) for negative roots this term becomes equal to $$- \sum_{0<\beta <\alpha} c_{\beta,- \alpha} {\tilde\Gamma}(\beta-\alpha) \phi(\beta-\alpha)$$ and therefore combines with the above term to give a full sum over $\beta >0$ (because $\phi(0) = 0$, there is no contribution for $\beta=\alpha$). Finally we obtain $$\begin{aligned} \delta_\alpha {\mathcal{S}}&=& \frac12 {\tilde\Gamma}(\alpha) {\mathcal{S}}\; + \; \frac12 \sum_{\beta>0} {P_{\beta}} \big[ {\tilde\Gamma}(\beta) , {\tilde\Gamma}(\alpha)\big] \phi(\beta) {\nonumber}\\[2mm] && + \sum_{\beta>0\,,\, \beta\neq \alpha} \Big[ - (\alpha\cdot\beta) {P_{\beta}} {\tilde\Gamma}(\beta){\tilde\Gamma}(\alpha) \phi(\alpha) \, + \, c_{\beta,\alpha} {P_{\beta}} {\tilde\Gamma}(\alpha + \beta) \phi(\alpha + \beta) {\nonumber}\\[2mm] && \qquad \qquad \qquad\qquad\qquad - \, c_{\beta, -\alpha} {P_{\beta}} {\tilde\Gamma}(\beta - \alpha) \phi(\beta- \alpha) \Big]\end{aligned}$$ where the sign in (\[Gamma2\]) is again essential. Let us now inspect the different terms here: using $\phi(\alpha + \beta) = \phi(\alpha) + \phi(\beta)$ the terms containing $\phi(\alpha)$ become $$-\sum_{\beta>0\, , \, \beta\neq \alpha} {P_{\beta}}\Big\{ (\alpha\cdot\beta) {\tilde\Gamma}(\beta){\tilde\Gamma}(\alpha)- c_{\beta,\alpha} {\tilde\Gamma}(\alpha+\beta) - c_{\beta,-\alpha}{\tilde\Gamma}(\beta-\alpha)\Big\} \phi(\alpha)$$ The expression inside brackets does indeed cancel if $\alpha$ and $\beta$ are real roots such that $(\alpha \pm \beta)$ are also real roots (in which case $\alpha\cdot\beta = \mp 1$); for $\alpha\cdot\beta =0$ all terms vanish. This covers all possible cases for $GL(n)$, but for indefinite $G$ there are infinitely many more possibilities because $\alpha\cdot\beta$ can assume any value, and then the extra terms no longer obviously cancel. We note that there is some room for modifications of the argument coming from the values of $c_{\alpha,\beta}$ when $\alpha+\beta$ is imaginary and also from terms associated with imaginary roots in the ansatz . The calculation in [@Damour:2006xu] shows that the (truncated) expression , involving some terms from null roots, does not transform covariantly. For the terms containing $\phi(\beta)$ the argument is similar; in this case we end up with $$\frac12 \sum_{\beta >0 \, , \, \beta\ne\alpha} {P_{\beta}}\Big\{ \big[ {\tilde\Gamma}(\beta) , {\tilde\Gamma}(\alpha)\big] + 2 c_{\beta,\alpha} {\tilde\Gamma}(\alpha + \beta) - 2 c_{\beta,-\alpha} {\tilde\Gamma}(\beta- \alpha) \Big\} \phi(\beta)$$ and a case by case analysis analogous to the one above shows again that these terms cancel under the same conditions as before. To sum up, the extra terms do cancel for finite dimensional $G$, when we need only consider the cases $\alpha\cdot\beta = \pm1$ or $=0$; in this case the supersymmetry constraint indeed transforms properly under $K$. This need no longer be true for infinite-dimensional $G$, where we have only insufficient knowledge of the structure constants $c_{\alpha,\beta}$. Let us also emphasize that this problem arises already at linear order in the fermions, so the addition of cubic or even higher order fermion terms cannot remedy this problem. Outlook ======= In this section, we discuss various possible extensions of our results. One pressing challenge is the correct treatment of the full ${{\rm{E}}_{10}}$ algebra beyond level $\ell=3$ when trying to construct a $K({{\rm{E}}_{10}})$ covariant supersymmetry constraint. This will be discussed in sections \[sec:beyond\]. Irrespective of the construction of a supersymmetric model one can consider the spinning particle of [@Damour:2006xu] and how the fermionic degrees of freedom influence the canonical structures discussed in section \[sec:Can\]. We offer some comments on this in section \[sec:ferm\] below. Tentative generalisation beyond $\ell=3$ {#sec:beyond} ---------------------------------------- The expression  is very suggestive of a generalisation beyond level $\ell=3$, so we are tempted to propose $$\begin{aligned} \label{Smod} {\mathcal{S}}_A &= \pi_{{\tt{a}}}\phi^{{\tt{a}}}_A + \sum_{\alpha^2=2\atop{\ell\leq 3}} {P_{\alpha}} \big(\Gamma(\alpha) \phi(\alpha)\big)_A + \sum_{\delta^2=0\atop{\ell =3}} {P_{\delta}}^r \big(\Gamma(\delta)\phi(\epsilon^r)\big)_A \, + \, \cdots $$ where the dots could stand for (at least) three kinds of additional terms, namely 1. additional terms linear in fermions associated with higher ($\ell>3$) level roots, coming either from real or imaginary roots; 2. additional terms cubic in fermions; 3. terms involving new ‘higher spin’ or other unfaithful realizations of $K({{\rm{E}}_{10}})$. In the following, we will concentrate only on the first extension. This already represents an extension beyond the truncated supergravity constraints. We note that the arguments of section \[sec:susytrm\] show that such a generalisation will not be $K({{\rm{E}}_{10}})$ covariant. Nevertheless we can find some constraints on the possible form by demanding at least Weyl invariance of the known terms. Since the Weyl group $W({{\rm{E}}_{10}})$ of ${{\rm{E}}_{10}}$ can be embedded in $K({{\rm{E}}_{10}})$ it would seem like a minimal requirement to extend the expression  by complete Weyl orbits of roots. As the real roots of ${{\rm{E}}_{10}}$ form a single Weyl orbit (${{\rm{E}}_{10}}$ is simply-laced) this would lead to the following expression for ${\mathcal{S}}_A$: $$\begin{aligned} \label{S1} {\mathcal{S}}_A &= \pi\!\cdot \!\phi_A\, + \frac12 \sum_{\alpha^2 =2} {P_{\alpha}} \big({\tilde\Gamma}(\alpha) \phi(\alpha)\big)_A + \cdots , {\nonumber}\\[2mm] &\equiv \pi\!\cdot \!\phi_A \; + \frac12 \sum_{w\in W({{\rm{E}}_{10}})} {P_{w(\alpha_0)}} \big({\tilde\Gamma}(w(\alpha_0)) \phi(w(\alpha_0))\big)_A + \cdots\end{aligned}$$ where the dots now indicate terms associated with imaginary roots. In the second line $\alpha_0$ represents an arbitrary real root. We see again that the minus sign in (\[Gamma1\]) is essential, otherwise the contributions from positive and negative roots would cancel in the sum. As we showed in section \[sec:susytrm\], this expression containing only the real roots is incompatible with $K({{\rm{E}}_{10}})$. The expression is, however, compatible with the ${{\rm{E}}_{10}}$ Weyl group. But the anticommutator would now give rise to an infinity of new terms that do not seem to make sense. We know from supergravity that we require also contributions from null imaginary roots ($\alpha^2=0$) and these would need to be covariantised under the Weyl group as well. We will not investigate the effect of this covariantisation here since already the real roots are problematic. A uniform treatment of all ${{\rm{E}}_{10}}$ roots requires also the inclusion of time-like imaginary roots ($\alpha^2<0$). These come with higher multiplicity and their addition to ${\mathcal{S}}_A$ might necessitate higher spin realizations of the type constructed in [@Kleinschmidt:2013eka], so as to be able to contract the relevant polarisation tensors with the fermions. Adding fermions {#sec:ferm} --------------- We now consider some aspects of the inclusion of fermionic degrees of freedom (at lowest order). Let $\Psi$ be a spinorial representation $\Psi$ of the compact subgroup and consider the Lagrangian $$\begin{aligned} \label{Lfermion} L = L_B + L_F = \frac12 \langle P| P\rangle - \frac{i}2 \langle \Psi| D\Psi\rangle,\end{aligned}$$ where $D\Psi$ is the $K$-covariant derivative with the composite connection $Q$ constructed out of $V$. In triangular gauge one has ${Q_{\alpha}}={P_{\alpha}}$ for all positive root components. We can write out the covariant derivative in the vector-spinor representation for $\alpha>0$ (real or imaginary) as follows $$\begin{aligned} L_F=-\frac{i}{2}\langle \Psi | D\Psi\rangle = -\frac{i}{2}G_{{{\tt{a}}}{{\tt{b}}}} \phi^{{\tt{a}}}\partial \phi^{{\tt{b}}}+\frac{i}2 \sum_{\alpha>0}\sum_{r=1}^{\mathrm{mult}(\alpha)} {P_{\alpha}}^r\, {{\rm j}_{\alpha}}^r,\end{aligned}$$ where ${{\rm j}_{\alpha}}^r$ denotes the fermion bilinear constructed out of the action of the ${{\rm k}_{\alpha}}^r$ in the vector-spinor representation and then contracted in the invariant bilinear form: ${{\rm j}_{\alpha}}^r = G_{{{\tt{a}}}{{\tt{b}}}} \phi^{{\tt{a}}}\delta_\alpha^r \phi^{{\tt{b}}}=G_{{{\tt{a}}}{{\tt{b}}}}\phi^{{\tt{a}}}({{\rm k}_{\alpha}}^r\cdot\phi^{{\tt{b}}})=-2{J_{\alpha}}^r$. We will suppress the multiplicity index $r$ in our schematic discussion below in order to avoid cluttering the expressions. The canonical fermionic momentum from  is $$\begin{aligned} \varpi_{{\tt{a}}}= \frac{\partial^L L}{\partial \partial\phi^{{\tt{a}}}} = \frac{i}2G_{{{\tt{a}}}{{\tt{b}}}} \phi^{{\tt{b}}},\end{aligned}$$ where we are using left Grassmann derivatives. The momentum satisfies the Poisson bracket $$\begin{aligned} {\left\{} \newcommand{\rd}{\right\}}\phi^{{\tt{a}}}, \varpi_{{\tt{b}}}\rd =-1.\end{aligned}$$ The corresponding (classical) Dirac bracket is therefore $$\begin{aligned} {\left\{} \newcommand{\rd}{\right\}}\phi^{{\tt{a}}}, \phi^{{\tt{b}}}\rd = i G^{{{\tt{a}}}{{\tt{b}}}}.\end{aligned}$$ Above we were using this bracket without the factor of $i$ by thinking of the $\phi^{{\tt{a}}}$ as quantum operators. The additional $i$ here implies that at the classical level $$\begin{aligned} \label{jj} {\left\{} \newcommand{\rd}{\right\}}{{\rm j}_{\alpha}},{{\rm j}_{\beta}} \rd =-2 i (c_{\alpha,\beta}\, {{\rm j}_{\alpha+\beta}}-c_{\alpha,-\beta}\, {{\rm j}_{\alpha-\beta}}).\end{aligned}$$ Let us denote the bosonic conjugate momenta in the theory with fermions by $\hat\Pi$. Then we get $$\begin{aligned} {\hat{\pi}}_{{\tt{a}}}&= \frac{\partial L}{\partial \partial q^{{\tt{a}}}}=G_{{{\tt{a}}}{{\tt{b}}}}\partial q^{{\tt{b}}}= \pi_{{\tt{a}}},\\ {\hat{\Pi}_{\alpha}} &= \frac{\partial L}{\partial \partial {A_{\alpha}}}=\sum_{\beta>0} \left(2{P_{\beta}}+\frac{i}2 {{\rm j}_{\beta}}\right)\frac{\partial {P_{\beta}}}{\partial\partial {A_{\alpha}}}.\end{aligned}$$ We note that the momenta conjugate to the Cartan subalgebra variables $q^{{\tt{a}}}$ do not change (since these do not couple to the fermions) and that the matrix $\frac{\partial {P_{\beta}}}{\partial\partial {A_{\alpha}}}$ relating the conjugate momenta to the ${P_{\beta}}$ is identical to the purely bosonic theory. This means that the inversion proceeds in exactly the same way as in , leading to $$\begin{aligned} {P_{\alpha}}+\frac{i}4 {{\rm j}_{\alpha}} = e^{-q^{{\tt{a}}}\alpha_{{\tt{a}}}} \left({\hat{\Pi}_{\alpha}}-\frac12\sum_\beta c_{\beta,\alpha} {A_{\beta}}{\hat{\Pi}_{\alpha+\beta}} + \ldots\right).\end{aligned}$$ We now introduce the notation $$\begin{aligned} {\hat{P}_{\alpha}} \equiv {P_{\alpha}} +\frac{i}4 {{\rm j}_{\alpha}}.\end{aligned}$$ Since ${\hat{\Pi}_{\alpha}}$ and ${A_{\alpha}}$ are conjugate variables as before, we deduce that we have the following canonical commutation relations $$\begin{aligned} \label{PhPh} {\left\{} \newcommand{\rd}{\right\}}{\hat{P}_{\alpha}} , {\hat{P}_{\beta}}\rd &= c_{\alpha,\beta} {\hat{P}_{\alpha+\beta}},{\nonumber}\\ {\left\{} \newcommand{\rd}{\right\}}{\hat{\pi}}_{{\tt{a}}},{\hat{P}_{\alpha}} \rd &= \alpha_{{\tt{a}}}{\hat{P}_{\alpha}}.\end{aligned}$$ The new ‘supercovariant’ velocity components ${\hat{P}_{\alpha}}$ therefore satisfy the same Borel algebra as the ${P_{\alpha}}$ in the purely bosonic theory. In terms of the original velocities, and in view of , one therefore has $$\begin{aligned} {\left\{} \newcommand{\rd}{\right\}}{P_{\alpha}}, {P_{\beta}} \rd &= c_{\alpha,\beta}{\hat{P}_{\alpha+\beta}} - \frac{i}{8} c_{\alpha,\beta} {{\rm j}_{\alpha+\beta}}+\frac{i}{8} c_{\alpha,-\beta} {{\rm j}_{\alpha-\beta}},{\nonumber}\\ {\left\{} \newcommand{\rd}{\right\}}\pi_{{\tt{a}}}, {P_{\alpha}}\rd &= \alpha_{{\tt{a}}}{\hat{P}_{\alpha}}.\end{aligned}$$ The appearance of ${\hat{P}_{\alpha}}$ on the r.h.s. in these equations is important. For deriving this, we used that ${P_{\alpha}}$ and ${{\rm j}_{\alpha}}$ commute while ${\hat{P}_{\alpha}}$ and ${{\rm j}_{\alpha}}$ satisfy $$\begin{aligned} \label{Phj} {\left\{} \newcommand{\rd}{\right\}}{\hat{P}_{\alpha}} , {{\rm j}_{\beta}} \rd = {\left\{} \newcommand{\rd}{\right\}}\frac{i}4 {{\rm j}_{\alpha}},{{\rm j}_{\beta}} \rd = \frac{1}{2} c_{\alpha,\beta} {{\rm j}_{\alpha+\beta}} -\frac{1}{2} c_{\alpha,-\beta} {{\rm j}_{\alpha-\beta}}.\end{aligned}$$ Let us verify the consistency of the relation  and  in the equations of motion. In the model  one has on the one hand the Euler–Lagrange equations $$\begin{aligned} \partial {P_{\alpha}} &= -\pi^{{\tt{a}}}\alpha_{{\tt{a}}}{P_{\alpha}}+2\sum_{\beta>0} c_{\alpha,\beta} {P_{\beta}}{P_{\alpha+\beta}} -\frac{i}4 (\alpha\cdot \pi) {{\rm j}_{\alpha}} +\frac{i}4 \sum_{\beta>0}{P_{\beta}} \left( c_{\alpha,\beta}{{\rm j}_{\alpha+\beta}} + c_{\alpha,-\beta} {{\rm j}_{\alpha-\beta}}\right){\nonumber}\\ &= -(\alpha\cdot \pi) {\hat{P}_{\alpha}} + 2\sum_{\beta>0} c_{\alpha,\beta}{P_{\beta}} {\hat{P}_{\alpha+\beta}} -\frac{i}{4} \sum_{\beta>0} {P_{\beta}}\left(c_{\alpha,\beta} {{\rm j}_{\alpha+\beta}}-c_{\alpha,-\beta} {{\rm j}_{\alpha-\beta}}\right)\end{aligned}$$ The Hamiltonian on the other hand is (as before) $$\begin{aligned} H = \frac12\langle P | P\rangle = \frac12 {\hat{\pi}}_{{\tt{a}}}G^{{{\tt{a}}}{{\tt{b}}}} {\hat{\pi}}_{{\tt{b}}}+ \sum_{\alpha>0} {P_{\alpha}} {P_{\alpha}},\end{aligned}$$ in terms of the ‘old’ purely bosonic $P$.[^12] The Hamiltonian equations of motion for ${P_{\alpha}}$ are then $$\begin{aligned} \partial {P_{\alpha}} &= {\left\{} \newcommand{\rd}{\right\}}{P_{\alpha}}, H \rd = -\pi_{{\tt{a}}}{\left\{} \newcommand{\rd}{\right\}}\pi^{{\tt{a}}}, {P_{\alpha}} \rd + 2 \sum_{\beta>0}{P_{\beta}} {\left\{} \newcommand{\rd}{\right\}}{P_{\alpha}},{P_{\beta}} \rd{\nonumber}\\ &= -(\alpha\cdot\pi) {\hat{P}_{\alpha}} +2\sum_{\beta>0} c_{\alpha,\beta} {P_{\beta}}{\hat{P}_{\alpha+\beta}} -\frac{i}{4} \sum_{\beta>0} {P_{\beta}}\left(c_{\alpha,\beta} {{\rm j}_{\alpha+\beta}}-c_{\alpha,-\beta} {{\rm j}_{\alpha-\beta}}\right)\end{aligned}$$ in complete agreement with the Lagrangian equations. Final comments -------------- The underlying problems of the non $K({{\rm{E}}_{10}})$ covariance of the supersymmetry constraint ${\mathcal{S}}$ appears to be the unfaithfulness of the spinor representation that was used to construct ${\mathcal{S}}$. A full understanding of this issue requires a more detailed understanding of the representation theory of $K({{\rm{E}}_{10}})$. This does not only involve the construction of faithful fermionic representations but also a study of the properties of the ‘coset representation’ $P$ and the decomposition of its tensor products with fermionic representations. Finding a supersymmetric ${{\rm{E}}_{10}}$ model might exhibit a feature similar to one of the hallmarks of superstring theory. In superstring theory, supersymmetry is implemented only on the two-dimensional worldsheet but the consistency conditions of the theory imply that there is also supersymmetry in the target space-time, leading to supergravity at low energies. It is not inconceivable that a supersymmetric ${{\rm{E}}_{10}}$ model on a worldline would similarly induce supersymmetry in the algebraically generated space-time. The close connection between the fermionic ${{\rm{E}}_{10}}$ model on the worldline and the space-time supergravity equations found in [@Damour:2005zs; @deBuyl:2005mt; @Damour:2006xu] could be viewed as evidence for this idea. The problem of finding a $K({{\rm{E}}_{10}})$ covariant supersymmetry constraint  can be phrased representation theoretically as follows. Both the coset velocity $P$ and the vector-spinor $\Psi$ are honest $K({{\rm{E}}_{10}})$ representations. Their tensor product $P \otimes \Psi$ is also a $K({{\rm{E}}_{10}})$ representation and the question is what the invariant subspaces of this tensor product are; in particular, if there is a Dirac-spinor representation $\mathcal{S}$ contained in it. To the best of our knowledge very little is known about these kinds of questions since $K({{\rm{E}}_{10}})$ is not a Kac–Moody algebra. Already the decomposability (or not) of the coset velocity $P$ itself is an open question. If $P$ [*was*]{} decomposable this could have important consequences for the construction of invariant Lagrangians. Finally, we note that similar issues already arise for the affine case [@NS1; @NS2], where $K({{\rm{E}}_{10}})$ is replaced by the simpler (but still infinite-dimensional) involutory subgroup $K({\rm E}_9)\subset {\rm E}_9$. In that case one is dealing with a [*field theory*]{} in two dimensions, rather than a worldline model, and the faithfulness of the $K({\rm E}_9)$ representations is ensured [*on-shell*]{} by the additional dependence on the space coordinate and the differential relations obeyed by the transformation coefficients. For the [*off-shell*]{} theory, however, the existence and construction of faithful representations remains an open problem there as well. **Acknowledgements:** We thank Thibault Damour, François Englert and Marc Henneaux for discussions related to this work, and Ralf Köhl for correspondence on parametrisations of the unipotent group $N$ in the Kac–Moody case. The second quantised vector-spinor for imaginary roots {#app:vsrep} ====================================================== The Dirac-spinor representation is insensitive to the polarisation (i.e., multiplicity) of the imaginary roots but the more faithful vector-spinor representation can be used to derive partial information on the structure constants of $K({{\rm{E}}_{10}})$. The more faithful higher-spin realisations of [@Kleinschmidt:2013eka] in principle capture even more information on the imaginary roots. In this appendix, we study in more detail the representation of the vector-spinor that was completely determined by its values on the real roots by . For any real root $\alpha$ of ${{\rm{E}}_{10}}$ we recall that the canonical $K({{\rm{E}}_{10}})$ generators are given by $$\begin{aligned} \label{Jreal2} {{\rm k}_{\alpha}} = X_{{{\tt{a}}}{{\tt{b}}}}(\alpha) \phi^{{\tt{a}}}{\tilde\Gamma}(\alpha) \phi^{{\tt{b}}},\quad \textrm{with $X_{{{\tt{a}}}{{\tt{b}}}}(\alpha) = -\frac12 \alpha_{{\tt{a}}}\alpha_{{\tt{b}}}+\frac14 G_{{{\tt{a}}}{{\tt{b}}}}$.}\end{aligned}$$ We seek to obtain similar general expression for null roots $\delta$, satisfying $\delta^2=0$ and timelike root $\Lambda$ with $\Lambda^2=-2$. Null roots $\delta$ {#null-roots-delta .unnumbered} ------------------- All null roots $\delta$ of ${{\rm{E}}_{10}}$ have multiplicity $\mathrm{mult}(\delta)=8$ and we therefore require the representation of eight generators ${{\rm k}_{\alpha}}^r$. To arrive at the expression, we decompose $\delta=\alpha+(\delta-\alpha)$ for a [*real*]{} root $\alpha$. Then $(\delta-\alpha)$ is also real and $\delta\cdot\alpha=0$. Employing then the commutator $$\begin{aligned} {\left[}{{\rm k}_{\alpha}}, {{\rm k}_{\delta-\alpha}} {\right]}={\tilde\varepsilon}_{\alpha,\delta-\alpha} {{\rm k}_{\delta}}^{(\alpha)},\end{aligned}$$ where we have indicated that there are different possibilities for ${{\rm k}_{\delta}}^{(\alpha)}$. One knows a priori that there are at most eight independent generators. Substituting in the explicit expression for the real root generators  one finds that in the (second quantised) vector-spinor representation $$\begin{aligned} {{\rm k}_{\delta}}^{(\alpha)} = -2\alpha_{[{{\tt{a}}}} \delta_{{{\tt{b}}}]} \phi^{{\tt{a}}}{\tilde\Gamma}(\delta) \phi^{{\tt{b}}},\end{aligned}$$ where $\alpha\cdot \delta=0$. To bring this into a form that brings out the multiplicity $\mathrm{mult}(\delta)=8$, we note that shifting $\alpha\to\alpha+\delta$ does not change the expression, so that we can also summarise it by $$\begin{aligned} \label{Jnull} {{\rm k}_{\delta}}^r = \epsilon^r_{[{{\tt{a}}}} \delta_{{{\tt{b}}}]} \phi^{{\tt{a}}}\Gamma(\delta) \phi^{{\tt{b}}},\end{aligned}$$ where the ‘polarisation vector’ $\epsilon^r$ is orthogonal (transverse) to $\delta$ in the DeWitt metric and there is also a gauge invariance $\epsilon^r\to \epsilon^r+\delta$. This leaves eight independent choices which agrees with the multiplicity of the null root of ${{\rm{E}}_{10}}$. Note that the transversality of the polarisation vector is *not* manifest in ; it is rather a consequence of the way the generator is constructed from commutators of real roots. Imaginary roots $\Lambda^2=-2$ {#imaginary-roots-lambda2-2 .unnumbered} ------------------------------ It is also possible to derive the general form of the $\Lambda^2=-2$ generators from commuting two real root generators in a way similar to above. Let $\Lambda=\alpha+(\Lambda-\alpha)$ with $\alpha^2=(\Lambda-\alpha)^2=2$, then $\alpha\cdot\Lambda =-1$. We know that $$\begin{aligned} \left[ {{\rm k}_{\alpha}}, {{\rm k}_{\beta}} \right] = {\tilde\varepsilon}_{\alpha,\beta} {{\rm k}_{\Lambda}}^{(\alpha)}.\end{aligned}$$ By substituting in the explicit form for the real root generators one finds $$\begin{aligned} {{\rm k}_{\Lambda}}^{(\alpha)} = 2 Y_{{{\tt{a}}}{{\tt{b}}}}(\alpha) \phi^{{\tt{a}}}\Gamma(\Lambda) \phi^{{\tt{b}}}\end{aligned}$$ with $$\begin{aligned} \label{Ytens} Y_{{{\tt{a}}}{{\tt{b}}}}(\alpha)= Y_{{{\tt{b}}}{{\tt{a}}}}(\alpha)&= -\alpha_{({{\tt{a}}}} \Lambda_{{{\tt{b}}})} + \alpha_{{\tt{a}}}\alpha_{{\tt{b}}}- \frac14 \Lambda_{{\tt{a}}}\Lambda_{{\tt{b}}}+\frac18 G_{{{\tt{a}}}{{\tt{b}}}} {\nonumber}\\ &= v_{({{\tt{a}}}} \Lambda_{{{\tt{b}}})} +a_{{{\tt{a}}}{{\tt{b}}}}\end{aligned}$$ for $$\begin{aligned} v_{{\tt{a}}}&= -\alpha_{{\tt{a}}}-\frac58 \Lambda_{{\tt{a}}},\\ a_{{{\tt{a}}}{{\tt{b}}}} &= \alpha_{{\tt{a}}}\alpha_{{\tt{b}}}+\frac38 \Lambda_{{\tt{a}}}\Lambda_{{\tt{b}}}+\frac18 G_{{{\tt{a}}}{{\tt{b}}}}.\end{aligned}$$ The separation of the $\Lambda_{(a} \Lambda_{b)}$ here was chosen in such a way that $$\begin{aligned} \Lambda^{{\tt{b}}}a_{{{\tt{b}}}{{\tt{a}}}} = v_{{\tt{a}}}\end{aligned}$$ and is motivated by vertex operator algebra (VOA) constructions. The gauge symmetries of the parametrisation  are $$\begin{aligned} a_{{{\tt{a}}}{{\tt{b}}}} &\to a_{{{\tt{a}}}{{\tt{b}}}} + 2\epsilon_{({{\tt{a}}}} \Lambda_{{{\tt{b}}})}\\ v_{{\tt{a}}}&\to v_{{\tt{a}}}- 2\epsilon_{{\tt{a}}}\end{aligned}$$ and also leave the above condition $\Lambda^{{\tt{b}}}a_{{{\tt{b}}}{{\tt{a}}}} = v_{{\tt{a}}}$ invariant. The parameter $\epsilon$ here is chosen orthogonal to $\Lambda$. We also note that $\Lambda^{{\tt{a}}}\Lambda^{{\tt{b}}}Y_{{{\tt{a}}}{{\tt{b}}}}=19/4$ and $G^{{{\tt{a}}}{{\tt{b}}}} Y_{{{\tt{a}}}{{\tt{b}}}}=-9/4$ are gauge invariant and constrain the tensor $Y_{{{\tt{a}}}{{\tt{b}}}}$. The count is ---- ------------------------------------------------------------- 55 components of $a_{{{\tt{a}}}{{\tt{b}}}}$ -9 components of $\epsilon$ such that $\epsilon\cdot\Lambda=0$ -2 norm conditions on $Y_{{{\tt{a}}}{{\tt{b}}}}$ 44 multiplicity of root space ---- ------------------------------------------------------------- This count does not completely parallel the VOA construction and it would be desirable to have an interpretation in terms of Young symmetries similar to the null case above. Conjectural form for any generator {#conjectural-form-for-any-generator .unnumbered} ---------------------------------- Similar to the formula for roots satisfying $\Lambda^2=-2$ as above, we can give a tentative form of the action of any generator ${{\rm k}_{\Lambda}}^r$ on the vector-spinor $\phi^{{\tt{a}}}$ for an arbitrary imaginary root $\Lambda$. This form rests on the assumption that any generator in the root space of $\Lambda$ can be written as the commutator of two [*real*]{} root generators, that is $$\begin{aligned} c_{\alpha,\beta} {{\rm k}_{\Lambda}}^{(\alpha)} = {\left[}{{\rm k}_{\alpha}},{{\rm k}_{\beta}}{\right]}\end{aligned}$$ for $\alpha,\beta>0$ and real with $\alpha+\beta=\Lambda$. (We note that there the second term in the commutation relation  vanishes automatically for the configuration chosen here since $\alpha-\beta$ is not a root.) The conjecture is that as $\alpha$ and $\beta$ traverse all possible decompositions of $\Lambda$, their commutators contain a basis of the root space of $\Lambda$. The number of decompositions of $\Lambda$ is larger than $\mathrm{mult}(\Lambda)$ and many of the commutators will be linearly dependent. What we require is that the space generated by all possible commutators is equal to the full root space of $\Lambda$: $$\begin{aligned} \left.\bigg\langle {{\rm k}_{\Lambda}}^{(\alpha)}\,\middle|\, \alpha>0,\, \alpha^2=2,\, (\Lambda-\alpha)^2=2 \right.\bigg\rangle =\left.\bigg\langle {{\rm k}_{\Lambda}}^r\,\middle|\, r=1,\ldots,\mathrm{mult}(\Lambda) \bigg\rangle\right.\end{aligned}$$ This is a stronger version of a conjecture already contained [@Brown:2004jb] which only addressed the decomposition of the imaginary root vector $\Lambda=\alpha+\beta$ into two real roots. In all cases that we checked the assumption we are making is true but we are not aware of a general proof. Under this assumption, we can find the following formula for ${{\rm k}_{\Lambda}}^{(\alpha)}$ [*in the vector-spinor representation*]{}. We have to distinguish the cases $\Lambda^2=-4k$ and $\Lambda^2=2-4k$ for integer $k\geq 0$ because of the (anti-)symmetry of ${\tilde\Gamma}(\Lambda)$. The condition that $\alpha$ and $\Lambda-\alpha$ be real implies that $\alpha\cdot(\Lambda-\alpha)=\frac12(\Lambda^2-4)$. Then the calculations are completely analogous to the two cases described above, leading to $$\begin{aligned} \Lambda^2&=-4k &:&& {{\rm k}_{\Lambda}}^{(\alpha)} &= 2(k-1)\alpha_{[{{\tt{a}}}} \Lambda_{{{\tt{b}}}]} \phi^{{\tt{a}}}{\tilde\Gamma}(\Lambda)\phi^{{\tt{b}}},\\ \Lambda^2&=2-4k &:&& {{\rm k}_{\Lambda}}^{(\alpha)} &= \left(-2k \alpha_{({{\tt{a}}}} \Lambda_{{{\tt{b}}})}+2k\alpha_{{\tt{a}}}\alpha_{{\tt{b}}}-\frac12\Lambda_{{\tt{a}}}\Lambda_{{\tt{b}}}+\frac14 G_{{{\tt{a}}}{{\tt{b}}}} \right)\phi^{{\tt{a}}}{\tilde\Gamma}(\Lambda)\phi^{{\tt{b}}}.\end{aligned}$$ These expressions were derived under the assumption that $c_{\alpha,\Lambda-\alpha}=-{\tilde\varepsilon}_{\alpha,\Lambda-\alpha}$. Some more explicit results for $GL(3,{\mathbb{R}})/SO(3)$ {#app:GL3} ========================================================= For $GL(3,{\mathbb{R}})$ the coset element is $$\begin{aligned} V=AN= \begin{pmatrix} e^{q^1}&&\\ &e^{q^2}&\\ &&e^{q^3} \end{pmatrix} \begin{pmatrix} 1 & N^1{}_{{\tilde{2}}}& N^1{}_{{\tilde{3}}}\\ &1 & N^2{}_{{\tilde{3}}}\\ &&1 \end{pmatrix}.\end{aligned}$$ The notation here is such that an index value with a tilde refers to a curved (world) index and an index value without a tilde to a flat (tangent space) direction. The inverse of $N$ is given by $$\begin{aligned} N^{-1} = \begin{pmatrix} 1 & N^{{\tilde{1}}}{}_2 & N^{{\tilde{1}}}{}_3\\ &1 & N^{{\tilde{2}}}{}_3\\&&1 \end{pmatrix} =\begin{pmatrix} 1 & -N^1{}_{{\tilde{2}}}& -N^1{}_{{\tilde{3}}}+ N^1{}_{{\tilde{2}}}N^2{}_{{\tilde{3}}}\\ &1&-N^2{}_{{\tilde{3}}}\\ &&1 \end{pmatrix}.\end{aligned}$$ With this parametrisation it is straightforward to compute the coset velocity from $\partial V V^{-1}$, $$\begin{aligned} P &= \begin{pmatrix} P_{11} & P_{12} & P_{13}\\ P_{21} & P_{22} & P_{23}\\ P_{31} & P_{32} & P_{33} \end{pmatrix}{\nonumber}\\ &=\begin{pmatrix} \partial{q}^1 & \frac12 e^{q^1-q^2} \partial{N}^1{}_{{\tilde{2}}}& \frac12 e^{q^1-q^3} \left(\partial{N}^1{}_{{\tilde{3}}}+\partial{N}^1{}_{{\tilde{2}}}N^{{\tilde{2}}}{}_3\right)\\ \frac12e^{q^1-q^2}\partial{N}^1{}_{{\tilde{2}}}& \partial{q}^2 & \frac12 e^{q^2-q^3} \partial{N}^2{}_{{\tilde{3}}}\\ \frac12 e^{q^1-q^3} \left(\partial{N}^1{}_{{\tilde{3}}}+\partial{N}^1{}_{{\tilde{2}}}N^{{\tilde{2}}}{}_3\right)& \frac12 e^{q^2-q^3} \partial{N}^2{}_{{\tilde{3}}}& \partial{q}^3 \end{pmatrix}\end{aligned}$$ where, of course, $P_{ab} = P_{ba}$. The bosonic Lagrangian is $$\begin{aligned} L &= \frac12 \big[{\textrm{Tr}\,}(P^2) - ({\textrm{Tr}\,}P)^2\big]{\nonumber}\\[2mm] &= \frac12 G_{{{\tt{a}}}{{\tt{b}}}} \partial{q}^{{\tt{a}}}\partial{q}^{{\tt{b}}}+ P_{12}^2 + P_{13}^2 + P_{23}^2{\nonumber}\\[2mm] & = \frac12 G_{{{\tt{a}}}{{\tt{b}}}} \partial{q}^{{\tt{a}}}\partial{q}^{{\tt{b}}}+ \frac14 e^{2q^1-2q^2} \left(\partial{N}^1{}_{{\tilde{2}}}\right)^2+ \frac14 e^{2q^2-2q^3} \left(\partial{N}^2{}_{{\tilde{3}}}\right)^2+ \frac14 e^{2q^1-2q^3} \left(\partial{N}^1{}_{{\tilde{3}}}+\partial{N}^1{}_{{\tilde{2}}}N^{{\tilde{2}}}{}_3\right)^2,\end{aligned}$$ where $\frac12 G_{{{\tt{a}}}{{\tt{b}}}} \partial{q}^{{\tt{a}}}\partial{q}^{{\tt{b}}}= -\partial{q}^1\partial{q}^2 -\partial{q}^1\partial{q}^3 -\partial{q}^2\partial{q}^3$. The conjugate momenta are $$\begin{aligned} \label{conm} \pi_{{\tt{a}}}&= G_{{{\tt{a}}}{{\tt{b}}}} \partial{q}^{{\tt{b}}},{\nonumber}\\[2mm] \Pi^{{\tilde{2}}}{}_1 &= \frac{\partial L}{\partial \partial{N}^1{}_{{\tilde{2}}}} = \frac12 e^{2q^1-2q^2} \partial{N}^1{}_{{\tilde{2}}}-\frac12e^{2q^1-2q^3} N^2{}_{{\tilde{3}}}\left(\partial{N}^1{}_{{\tilde{3}}}+\partial{N}^1{}_{{\tilde{2}}}N^{{\tilde{2}}}{}_3\right){\nonumber}\\[2mm] &= e^{q^1-q^2} P_{21} + e^{q^1-q^3} N^{{\tilde{2}}}{}_3 P_{31},\\[2mm] \Pi^{{\tilde{3}}}{}_1 &= \frac{\partial L}{\partial \partial{N}^1{}_{{\tilde{3}}}} = \frac12 e^{2q^1-2q^3} \left(\partial{N}^1{}_{{\tilde{3}}}+\partial{N}^1{}_{{\tilde{2}}}N^{{\tilde{2}}}{}_3\right) = e^{q^1-q^3} P_{31}\\ \Pi^{{\tilde{3}}}{}_2 &= \frac{\partial L}{\partial \partial{N}^2{}_{{\tilde{3}}}} = \frac12e^{2q^2-2q^3} \partial{N}^2{}_{{\tilde{3}}}= e^{q^2-q^3} P_{32}.\end{aligned}$$ These relations can be inverted to give $\partial{q}^{{\tt{a}}}= G^{{{\tt{a}}}{{\tt{b}}}} \pi_{{\tt{b}}}$ and $$\begin{aligned} \partial{N}^1{}_{{\tilde{2}}}&= 2 e^{-2q^1+2q^2} \big(\Pi^{{\tilde{2}}}{}_1+ N^2{}_{{\tilde{3}}}\Pi^{{\tilde{3}}}{}_1\big) {\nonumber}\\[2mm] \partial{N}^1{}_{{\tilde{3}}}&= 2e^{-2q^1+2q^3} \Pi^{{\tilde{3}}}{}_1 + 2e^{-2q^1+2q^2} N^2{}_{{\tilde{3}}}\big(\Pi^{{\tilde{2}}}{}_1+ N^2{}_{{\tilde{3}}}\Pi^{{\tilde{3}}}{}_1\big){\nonumber}\\[2mm] \partial{N}^2{}_{{\tilde{3}}}&= 2 e^{-2q^2+2q^3} \Pi^{{\tilde{3}}}{}_2.\end{aligned}$$ Equivalently, $$\begin{aligned} \label{invrels} P_{12} &= e^{-q^1+q^2} \big(\Pi^{{\tilde{2}}}{}_1 + N^2{}_{{\tilde{3}}}\Pi^{{\tilde{3}}}{}_1\big),{\nonumber}\\ P_{23} &= e^{-q^2+q^3} \Pi^{{\tilde{3}}}{}_2,{\nonumber}\\ P_{13} &= e^{-q^1+q^3} \Pi^{{\tilde{3}}}{}_1,\end{aligned}$$ in agreement with the general formula , up to an overall factor. Let us denote $$\begin{aligned} P(\alpha_{12}) \equiv P_{12}\, ,\quad P(\alpha_{23}) \equiv P_{23}\, , \quad P(\alpha_{13}) \equiv P_{13},\quad\end{aligned}$$ and $$\begin{aligned} \alpha_{12} = (1,-1,0),\quad \alpha_{13} = (1,0,-1),\quad \alpha_{23} = (0,1,-1).\end{aligned}$$ Then the Hamiltonian is $$\begin{aligned} H&= \frac12 \pi_{{\tt{a}}}G^{{{\tt{a}}}{{\tt{b}}}} \pi_{{\tt{b}}}+ e^{-2q^1+2q^2} \big(\Pi^{{\tilde{2}}}{}_1 + N^2{}_{{\tilde{3}}}\Pi^{{\tilde{3}}}{}_1\big)^2 + e^{-2q^1+2q^3} \big(\Pi^{{\tilde{3}}}{}_1\big)^2 + e^{-2q^2+2q^3} \big(\Pi^{{\tilde{3}}}{}_2\big)^2{\nonumber}\\[2mm] &=\frac12 \pi_{{\tt{a}}}G^{{{\tt{a}}}{{\tt{b}}}} \pi_{{\tt{b}}}+ \sum_{a<b} P(\alpha_{ab})^2.\end{aligned}$$ Using the canonical brackets $\left\{q,p\right\}=1$ between the conjugate variables we recover the relations already previously derived (for $a<b$ and any $\alpha>0$) $$\begin{aligned} \left\{ \pi_{{\tt{a}}},\pi_{{\tt{b}}}\right\} &= 0,{\nonumber}\\[2mm] \left\{ \pi_{{\tt{a}}},P(\alpha)\right\} &= \alpha_{{\tt{a}}}P(\alpha),{\nonumber}\\[2mm] \left\{ P(\alpha_{ab}),P(\alpha_{cd})\right\} &= \left\{\begin{array}{cl} \epsilon_{\alpha_{ab},\alpha_{cd}} P(\alpha_{ab}+\alpha_{cd}) & \textrm{if $\alpha_{ab}+\alpha_{cd}$ is a root,}\\ 0 & \textrm{otherwise.} \end{array}\right.\end{aligned}$$ The conserved current is $$\begin{aligned} J = V^{-1} P V = \begin{pmatrix} J^{{\tilde{1}}}{}_{{\tilde{1}}}& J^{{\tilde{1}}}{}_{{\tilde{2}}}& J^{{\tilde{1}}}{}_{{\tilde{3}}}\\ J^{{\tilde{2}}}{}_{{\tilde{1}}}& J^{{\tilde{2}}}{}_{{\tilde{2}}}& J^{{\tilde{2}}}{}_{{\tilde{3}}}\\ J^{{\tilde{3}}}{}_{{\tilde{1}}}& J^{{\tilde{3}}}{}_{{\tilde{2}}}& J^{{\tilde{3}}}{}_{{\tilde{3}}}\end{pmatrix}\end{aligned}$$ with $$\begin{aligned} J^{{\tilde{2}}}{}_{{\tilde{1}}}&= \Pi^{{\tilde{2}}}{}_1,{\nonumber}\\ J^{{\tilde{3}}}{}_{{\tilde{1}}}&= \Pi^{{\tilde{3}}}{}_1,{\nonumber}\\ J^{{\tilde{3}}}{}_{{\tilde{2}}}&= \Pi^{{\tilde{3}}}{}_2+ N^1{}_{{\tilde{2}}}\Pi^{{\tilde{3}}}{}_1 ,\end{aligned}$$ below the diagonal and the following diagonal and upper triangular components $$\begin{aligned} J^{{\tilde{1}}}{}_{{\tilde{1}}}&= G^{1{{\tt{a}}}}\pi_{{\tt{a}}}- N^1{}_{{\tilde{2}}}\Pi^{{\tilde{2}}}{}_1 - N^1{}_{{\tilde{3}}}\Pi^{{\tilde{3}}}{}_1{\nonumber}\\[1mm] J^{{\tilde{2}}}{}_{{\tilde{2}}}&= G^{2{{\tt{a}}}}\pi_{{\tt{a}}}+ N^1{}_{{\tilde{2}}}\Pi^{{\tilde{2}}}{}_1 - N^2{}_{{\tilde{3}}}\Pi^{{\tilde{3}}}{}_2{\nonumber}\\[1mm] J^{{\tilde{3}}}{}_{{\tilde{3}}}&= G^{3{{\tt{a}}}}\pi_{{\tt{a}}}+ N^1{}_{{\tilde{3}}}\Pi^{{\tilde{3}}}{}_1 + N^2{}_{{\tilde{3}}}\Pi^{{\tilde{3}}}{}_2\\[1mm] J^{{\tilde{1}}}{}_{{\tilde{2}}}&= e^{-2q^1+2q^2} (\Pi^{{\tilde{2}}}{}_1 + N^2{}_{{\tilde{3}}}\Pi^{{\tilde{3}}}{}_1) + N^1{}_{{\tilde{2}}}\left(\pi_{{\tilde{1}}}-\pi_{{\tilde{2}}}- N^1{}_{{\tilde{3}}}\Pi^{{\tilde{3}}}{}_1+ N^2{}_{{\tilde{3}}}\Pi^{{\tilde{3}}}{}_2\right) - N^1{}_{{\tilde{3}}}\Pi^{{\tilde{3}}}{}_2 - N^1{}_{{\tilde{2}}}N^1{}_{{\tilde{2}}}\Pi^{{\tilde{2}}}{}_1{\nonumber}\\[1mm] J^{{\tilde{1}}}{}_{{\tilde{3}}}&= e^{-2q^1+2q^3} \Pi^{{\tilde{3}}}{}_1 + e^{-2q^1+2q^2}N^2{}_{{\tilde{3}}}(\Pi^{{\tilde{2}}}{}_1 + N^2{}_{{\tilde{3}}}\Pi^{{\tilde{3}}}{}_1) -e^{-2q^2+2q^3} N^1{}_{{\tilde{2}}}\Pi^{{\tilde{3}}}{}_2{\nonumber}\\[1mm] &\quad + N^1{}_{{\tilde{3}}}\left( \pi_{{\tilde{1}}}-\pi_{{\tilde{3}}}- N^1{}_{{\tilde{2}}}\Pi^{{\tilde{2}}}{}_2- N^2{}_{{\tilde{3}}}\Pi^{{\tilde{3}}}{}_2\right) - N^1{}_{{\tilde{2}}}N^2{}_{{\tilde{3}}}\left(\pi_{{\tilde{2}}}- \pi_{{\tilde{3}}}-N^2{}_{{\tilde{3}}}\Pi^{{\tilde{3}}}{}_2\right) - N^1{}_{{\tilde{3}}}N^1{}_{{\tilde{3}}}\Pi^{{\tilde{3}}}{}_1{\nonumber}\\[1mm] J^{{\tilde{2}}}{}_{{\tilde{3}}}&= e^{-2q^2+2q^3} \Pi^{{\tilde{3}}}{}_2 + N^2{}_{{\tilde{3}}}(\pi_{{\tilde{2}}}-\pi_{{\tilde{3}}}) + N^1{}_{{\tilde{3}}}\Pi^{{\tilde{2}}}{}_1 - N^2{}_{{\tilde{3}}}N^2{}_{{\tilde{3}}}\Pi^{{\tilde{3}}}{}_2.{\nonumber}\end{aligned}$$ The relation of the components of the conserved charge to the canonical momenta was already discussed in [@Damour:2002et]. 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[**B533**]{} (1998) 210, \[hep-th/9804152\] H. Nicolai and H. Samtleben, “On $K(E_9)$", Q.J.Pure Appl. Math.[**1**]{} (2005) 180, \[hep-th/0407055\] J. Brown, O. J. Ganor and C. Helfgott, “M theory and E(10): Billiards, branes, and imaginary roots,” JHEP [**0408**]{} (2004) 063 \[hep-th/0401053\]. [^1]: By contrast, the supersymmetric Bianchi-type model recently (and impressively) analysed in [@Damour:2013eua; @Damour:2014cba] does retain all higher order fermionic terms, and is thus fully supersymmetric also at the quantum level, but with only partial manifestations of $K({\rm AE}_3)$ symmetry. [^2]: For finite-dimensional simple $\mathfrak{g}$ this is just the appropriately normalised matrix trace. [^3]: Here, we deviate from the standard notation in the cosmological billiards literature (see e.g. [@Damour:2002et]) where the diagonal degrees of freedom are denoted by $-\beta^{{\tt{a}}}\equiv + q^{{\tt{a}}}$. This is done in order to avoid confusion with the labeling of the components of the root $\beta$ below. [^4]: The notion of local nilpotency is defined as follows: an operator ${E_{\alpha}}^r$ is locally nilpotent in a representation $V$ of $\mathfrak{g}$ if for all $x\in V$ there exists an $n_0 = n_0(x)$ such that $$\big({E_{\alpha}}^r\big)^n (x) = 0 \;\; \mbox{for all $n> n_0$.}$$ (In the adjoint representation, the action is simply by commutators: $ {E_{\alpha}}^r (x)=\mathrm{ad}\, {E_{\alpha}}^r (x) \equiv \left[{E_{\alpha}}^r,x\right]$.) Clearly this holds for any real root $\alpha$ in the adjoint or any standard representation, but is no longer true for imaginary roots. The intuitive picture for this statement in the adjoint representation is that all roots lie in a solid hyperboloid $\{\alpha^2\leq 2\}$ in a Lorentzian space. Imaginary roots point [*into*]{} the light-cone where infinitely many roots of $\mathfrak{g}$ lie whereas real roots point outside the light-cone and eventually will leave the solid hyperboloid. [^5]: But let us note that the highest weights associated to the [*gradient representations*]{} of [@Damour:2002cu] are, in fact, real roots. [^6]: The invariance relation $\langle V^{-1}AV|B\rangle = \langle A|VBV^{-1}\rangle$ for $A,B\in {\mathfrak{g}}$ and $V\in G$ are also valid for infinite-dimensional $\mathfrak{g}$. In the finite-dimensional case, the relation can be viewed as the standard cyclic property of the matrix trace. [^7]: \[convchange\]We have here adjusted some normalisations relative to [@Damour:2006xu] in order to make subsequent expressions more uniform. The changes concern the dual fields on levels $\ell=2$ and $\ell=3$: The sign on level two here is opposite to that of [@Damour:2006xu], and $P^{(3)}_{\textrm{here}} = \frac13 P^{(3)}_{\textrm{DKN}}$. The reason for the rescaling is that we here are normalising the real root generators identically on all levels in conformity with (\[Norm2\]). [^8]: This is in agreement with structure of the diagonal components $\pi_{{\tt{a}}}=P_{aa}^{(0)}$ from the constraint $C_{[a_1\ldots a_9]}^{(3)}= P_{ca_1}^{(0)} P_{c|a_2\ldots a_9}^{(3)}+\ldots$ discussed in [@Damour:2007dt; @Damour:2009ww] that only couple to the null root components ${P_{\delta}}^r$ as determined by . [^9]: We stress that the coefficients of the ${P_{\alpha}}{P_{\alpha}}$ terms for real roots do agree in the two expressions. For $\ell=3$, this might seem surprising in view of the [*different*]{} coefficients in the $SO(10)$ covariant expressions. A simple way of seeing that they agree after the rewriting in $K({{\rm{E}}_{10}})$ variables is to look at a fixed particular real root, say ${P_{\alpha_{1|12345678}}}\equiv P^{(3)}_{1|12345678}$. In  this term has contributions from both expressions via $\frac{2}{8!}(8!-4\cdot 7!)=1$ and in  one similarly has $\frac1{8!}\cdot 8!=1$. This example illustrates well how the $K({{\rm{E}}_{10}})$ properties can be obscured by insisting on $SO(10)$ invariant expressions. [^10]: For other roots the first two lines would generalize to $$\begin{aligned} \delta^r_\alpha \pi^{{\tt{a}}}&=& - 2\alpha^{{\tt{a}}}\, {P_{\alpha}}^r {\nonumber}\\[2mm] \delta^r_\alpha {P_{\beta}}^s &=& \delta^{rs} \delta_{\alpha,\beta} \alpha_{{\tt{a}}}\pi^{{\tt{a}}}\, +\, \sum_t c_{\beta-\alpha,\alpha}^{rst} {P_{\beta - \alpha}}^t - \sum_t c_{\alpha + \beta, -\alpha}^{rst} {P_{\alpha + \beta}}^t \nonumber\end{aligned}$$ but no general formula is available for $\delta^r_\alpha \phi^{{\tt{a}}}$. A conjectural formula is given in appendix \[app:vsrep\]. [^11]: The inequalities in the sums are taken to imply that the corresponding elements are roots of the algebra (as an ordering cannot be generally defined for arbitrary pairs of roots $\alpha$ and $\beta$). [^12]: That this is true can be seen in the following simple example involving a derivative coupling. Let $L=\frac12 \dot{q}^2 + \dot{q} j$. The conjugate momentum is $\hat{p}=\dot{q}+j\equiv p+j$ and the Hamiltonian is $H=\hat{p}\dot{q} -L = \frac12\dot{q}^2=\frac12p^2$.

--- abstract: | Motivated by Tverberg-type problems in topological combinatorics and by classical results about embeddings (maps without double points), we study the question whether a finite simplicial complex $K$ can be mapped into ${\ensuremath{\mathbb{R}}}^d$ without higher-multiplicity intersections. We focus on conditions for the existence of *almost $r$-embeddings*, i.e., maps $f\colon K \to {\ensuremath{\mathbb{R}}}^d$ such that $f(\sigma_1)\cap \dots \cap f(\sigma_r) =\emptyset$ whenever $\sigma_1,\dots,\sigma_r$ are pairwise disjoint simplices of $K$. Generalizing the classical Haefliger-Weber embeddability criterion, we show that a well-known necessary *deleted product condition* for the existence of almost $r$-embeddings is sufficient in a suitable *$r$-metastable range* of dimensions: If $r d \geq (r+1) \dim K +3,$ then there exists an almost $r$-embedding $K\to {\ensuremath{\mathbb{R}}}^d$ if and only if there exists an equivariant map ${\ensuremath{{K}^{r}_{\Delta}}} \to_{{\ensuremath{\mathfrak{S}}}_r} S^{d(r-1)-1}$, where ${\ensuremath{{K}^{r}_{\Delta}}}$ is the *deleted $r$-fold product* of $K$ (the subcomplex of the $r$-fold cartesian product whose cells are products of pairwise disjoint simplices of $K$) and ${\ensuremath{\mathfrak{S}}}_r$ denotes the symmetric group. This significantly extends one of the main results of our previous paper (which treated the special case where $d=rk$ and $\dim K=(r-1)k$ for some $k\geq 3$), and settles one of the main open questions raised there. As a corollary, our result together with recent work of Filakovský and Vokřínek on the homotopy classification of equivariant maps under non-free actions imply that almost $r$-embeddability of simplicial complexes is algorithmically decidable in the $r$-metastable range (in polynomial time if $r$ and $d$ are fixed). bibliography: - 'EliminatingMultiplePoints.bib' --- [Eliminating Higher-Multiplicity Intersections, II.]{} 0.1in [ The Deleted Product Criterion in the $r$-Metastable Range]{} 0.15in \ `imabillard@ist.ac.at` 0.1in and 0.1in \ `uli@ist.ac.at` 0.15in [*IST Austria, Am Campus 1, 3400 Klosterneuburg, Austria 0.4in*]{} 0.1in

--- abstract: 'We present a quantum mechanical study of the bound states of a neutral scalar particle on the curved background described by the Reissner-Nordström (RN) spacetime that corresponds to a naked singularity. We show that there occurs both the metastable, *i.e.* decaying bound states and the stable ones. The corresponding energy spectra are calculated in the leading WKB approximation and the comparison with the RN black hole quasinormal mode spectrum is made. The metastable bound states on the naked singularity background turn out to be more long-living than the quasinormal modes for the black hole with the same mass.' author: - | V. D. Gladush[^1], D. A. Kulikov[^2]\ \ [*Theoretical Physics Department, Dniepropetrovsk National University* ]{}\ [*72 Gagarin av., Dniepropetrovsk 49010, Ukraine*]{} title: '**Stable and decaying bound states on the naked Reissner-Nordström background**' --- Introduction {#part1} ============ In general relativity, the probe with test particles and waves is an efficient tool to study the properties of space–time near gravitating mass [@Chandrasekhar; @Cohen; @Wald; @Horowitz; @Ishibashi99; @Pitelli]. When using quantum test particles, one observes some features that are unexpected from the viewpoint of classical theory. In particular, it is well established that quantum test particles cannot form stable stationary bound states in the field of the Schwarzschild and Reissner-Nordström (RN) black holes [@Deruelle]. Instead, there exist decaying quasistationary states known as the quasinormal modes (see Refs. [@Kokkotas; @Berti] for review). However, if in the RN setup the gravitating mass $M$ and the electric charge $Q$ obey the superextremality condition, $Q>M$, the space-time geometry corresponds not to a black hole but to a naked singularity. At the classical level, this results in a repulsive potential barrier near the origin that prevents the absorbtion of neutral test particles [@Qadir; @Gladush; @Pugliese]. Therefore one may anticipate that the bound states are stable at the quantum level as well. The main goal of this article is to verify the above proposition and to derive the energy spectrum of the bound states of a quantum test particle on the naked RN background. Probing the naked RN singularity, one faces two key problems. The first one concerns physical relevance. According to the cosmic censorship conjecture by Penrose [@Penrose], any singularity originated from gravitational collapse must be hidden inside an event horizon. Besides, since the RN singularity is electrically charged, it may be neutralized by spontaneous pair creation provided that charges are sufficiently high [@Damour]. Nevertheless, one cannot exclude scenarios in which the RN geometry with $Q>M$ is an exterior solution connected with a regular interior solution with matter, for instance, the Friedmann solution in the model of friedmon [@Markov]. Such models can be traced back to Dirac’s suggestion to describe leptons as charged shells [@Dirac]. Recall that it is $Q/M=2.0\cdot 10^{21}$ for the electron, so that the exterior geometry must correspond to the naked singularity. The second problem is the non-uniqueness of the time evolution for test particles and waves on the naked RN background [@Ishibashi99; @Ishibashi03; @Stalker]. This means one has to specify an additional boundary condition at the singularity to obtain a fully unique dynamics. In the present work, we overcome this shortcoming by employing the WKB approach, which turns out to be insensible to behavior in the vicinity of the singularity. The work uses a simple quantum mechanical model. We consider a neutral scalar particle on the background geometry; the word “particle” may refer equally to the test scalar field perturbation. Both massless and massive cases are studied. This investigation expands our previous work [@Kulikov], which was restricted to the stable bound states of a massive particle. The plan of the article is as follows. In Section \[part2\] we examine how particle wave functions behave in the vicinity of the singularity and advocate the WKB approximation. In Section \[part3\] the effective potential is analyzed to distinguish between stable and decaying bound states. Section \[part4\] is devoted to the calculation of the energy spectra through the WKB method. The energies obtained are then discussed and compared with the known quasinormal mode spectra for the RN black holes in Section \[part5\]. Finally, Section \[part6\] presents our conclusion. Behavior in the vicinity of a RN singularity {#part2} ============================================ To start with, we establish the behavior of a particle wave function in the vicinity of a RN singularity and study how it is accommodated in the WKB approximation. The external RN geometry is described by the metric (in units with $G=\hbar=c=1$) $$\label{RNmetric} ds^{2} =Fdt^{2}-F^{-1}dr^{2}-r^{2}(d\theta ^{2}+\sin ^{2}{\theta }{d\phi }^{2})\,, \quad F =1-\frac{2 M}{r}+\frac{Q^{2}}{r^{2}}.$$ For concreteness, we assume that $Q>0$. The Klein-Gordon equation for a neutral scalar particle of mass $m$ on this background $$\label{KG} \frac{1}{\sqrt{-g}}\frac{\partial }{\partial x^{\mu }}(\sqrt{-g}g^{\mu \nu } \frac{\partial \Psi }{\partial x^{\nu }})+m^{2} \Psi =0$$ has stationary state solutions, which we decompose into the spherical harmonics $$\label{Phi} \Psi (t,r,\theta ,{\phi })=\exp (-\frac{i\omega t}{\hbar })Y_{lm}(\theta ,{\phi })\,\frac{1}{r\sqrt{F}}u(r).$$ Then the radial wave function satisfies the equation $$\label{RadEq} \frac{d^{2}u}{dr^{2}}+\frac{1}{F}\left[ \frac{1}{F}\left( \omega^2-\frac{Q^2 -M^{2}}{r^{4}}\right) -m^2 -\frac{l(l+1)}{r^{2}}\right] u=0$$ where $l=0,1,2,...$ is the orbital quantum number. In principle, this can be reduced to the confluent Heun equation [@Ronveaux], upon substituting $u(r)=\exp(\sqrt{m^2-\omega^2}r)(M-\sqrt{M^2-Q^2}-r)^\alpha(M+\sqrt{M^2-Q^2}-r)^\beta y(r)$ with $\alpha$ and $\beta$ given by bulky expressions in terms of $Q$, $M$ and $m$. The so-obtained confluent Heun equation has singularities at $r=M\pm \sqrt{M^2-Q^2}$ (black-hole horizons) and $r=\infty$. However, in the super-extreme case under consideration these singularity are shifted away from the real line because $M^2-Q^2<0$. As a consequence, the reduction to the confluent Heun form does not simplify the problem. It is more useful to note that the radial equation (\[RadEq\]) holds invariant under rescaling $$\label{rescaling} Q\rightarrow \alpha Q, \quad M\rightarrow \alpha M, \quad m\rightarrow \frac{1}{\alpha}m, \quad \omega\rightarrow \frac{1}{\alpha}\omega, \quad R\rightarrow \alpha r$$ and thus one of the three parameters $Q$, $M$ and $m$ can be fixed arbitrarily. Close to the naked singularity at $r=0$, the radial equation simplifies $$\label{RadEq0} \frac{d^{2}u}{dr^{2}}-\frac{[1+l(l+1)]Q^2 - M^{2}}{Q^{4}}\, u=0, \qquad r\rightarrow 0.$$ If we now adopt the leading WKB approximation, choosing the radial wave function in the form $u(r)\propto\exp (iS(r))$, then in the super-extreme case we have $S'^{2}(r)<0$ for small $r$. Hence, the naked singularity is located in the classically forbidden region. As for the bound states we are interested in, this means that the wave function asymptotics is dominated by the term which decreases exponentially as $r$ approaches $0$. Note that for small $r$ the condition of the WKB-method applicability $|S''(r)/S'^{2}(r)|\ll 1$ is still satisfied. Nevertheless, the exact behavior of the wave function at $r\rightarrow 0$ remains undetermined. The reason is that in a pair of linearly independent solutions to Eq. (\[RadEq\]) both solutions are locally normalized at $r\rightarrow 0$ and thus none of them can be discarded. As seen from Eq. (\[RadEq0\]), such a pair may easily be composed by taking a solution with $u(0)=0$ and another one with $u'(r)|_{r=0}=0$. In the general case, one should consider a linear combination of these solutions and employ the mixed boundary condition $(u'(r)+a\,u(r))|_{r=0}=0$ with an arbitrary real parameter $a$. Physically, the dependence of the dynamics on this parameter indicates that the naked singularity carries some additional degrees of freedom beyond those contained in the metric. It is said [@Ishibashi99] that the RN singularity has “hair”. Mathematically, the ambiguity in choosing the boundary condition stems from the non-uniqueness of the self-adjoint extension to the wave operator on the naked RN background [@Ishibashi99; @Ishibashi03; @Stalker]. As a possible resolution, the Friedrichs extension was suggested which implies the choice $a=0$ and is naturally connected with the quadratic form of energy [@Ishibashi03; @Stalker]. It should be stressed that the application of the WKB method will permit us to bypass this problem at all because the method does not uses the value of the wave function in the origin. In the end of this Section it is worthy to compare the above picture in terms of the areal radial coordinate $r$ with that based on the usual tortoise coordinate $x$ defined by $dr=Fdx$. Adjusting the integration constant, one can always put the limit of $r\rightarrow 0$ in correspondence with $x\rightarrow 0$. Then in place of the radial equation (\[RadEq\]) one obtains $$\label{RadEqX} \frac{d^{2}\phi}{dx^{2}}+\left[\omega^2 -\left(m^2+ \frac{l(l+1)}{r^2}+\frac{2M}{r^3}-\frac{2Q^2}{r^4}\right)F \right] \, \phi=0$$ where $\phi(x)=u(r)/\sqrt{F(r)}$. In the limit of $x\rightarrow 0$ one has $x\sim r^3/(3Q^2)$ and this equation reduces to the Schrödinger-type one with the inverse quadratic potential $$\label{RadEq1} \frac{d^{2}\phi}{dx^{2}}+\frac{2}{9x^{2}}\, \phi=0.$$ Then the textbook analysis [@Landau] shows that the particle cannot fall on the origin because the numerical coefficient of the potential in Eq. (\[RadEq1\]) does not exceed its critical value 1/4. This is in agreement with our previous conclusion that the naked singularity is located in the classically forbidden region. However, Eqs. (\[RadEqX\]) and (\[RadEq1\]) contain the singularity at $x=0$ such that the condition of the WKB-method applicability breaks down for small $x$. Therefore, to develop the WKB approximation in the subsequent sections, we employ the initial radial equation (\[RadEq\]) in terms of the areal radial coordinate $r$. Effective potential {#part3} =================== Now we shall establish possible types of the particle bound states on the naked RN background. To that end, let us analyze the shape of the effective potential $V(r)$ introduced upon rewriting the radial equation (\[RadEq\]) in shorthand notations $$\label{SecEq} \frac{d^{2}u}{dr^{2}}+\frac{1}{F^{2}}\left(\omega^{2}-V(r)\right)u=0,$$ where $$\begin{aligned} \label{Veff} V(r)&=&\left(m^2+\frac{l(l+1)}{r^{2}}\right)F+\frac{Q^{2}-M^2}{r^{4}} \nonumber \\ &=&{m}^{2}-{\frac {2M{m}^{2}}{r}}+{\frac {{Q}^{2}{m}^{2}+\Lambda}{{r}^{2 }}}-{\frac {2\Lambda M}{{r}^{3}}}+{\frac {{Q}^{2}\Lambda-{M}^{2}+{Q}^{ 2}}{{r}^{4}}}\end{aligned}$$ and we designated $\Lambda=l(l+1)$. Note that $V(r)$ is positive-defined because $F>0$ for all positive $r$. For a given particle energy $\omega$, the type of the bound state is regulated by the pattern of classically allowed and forbidden regions. From Eq. (\[SecEq\]) one concludes that the classically allowed (forbidden) region is to be defined as a region in which the condition $\omega^{2}-V(r)>0$ ($\omega^{2}-V(r)<0$) holds. Instead of solving those inequalities explicitly, we are looking for local extrema of $V(r)$. If $V(r)$ has a minimum at $r=r_{min}$ and no other extrema, then for $\omega$ close enough to $\sqrt{V(r_{min})}$ the classically allowed region consists of a single line segment and there stable bound states may exist. In this case we call $V(r)$ the single-well potential. If $V(r)$ has two or more extrema, then for certain $\omega$ the classically allowed region must be split into two subregions separated by the classically forbidden region. Because of the tunneling between the subregions, the bound states are now metastable, *i.e.* decaying. The corresponding potential will be referred to as the barrier-shaped one. We are now in the position to deduce what type of the bound states occurs depending on the values of the particle mass $m$ and the singularity parameters $Q$ and $M$. Let us first study the simpler case of the massless particle and then turn to the massive case. Case of $m=0$ ------------- In this case the effective potential simplifies $$\begin{aligned} \label{Veffm0} V(r)={\frac {\Lambda}{{r}^{2 }}}-{\frac {2\Lambda M}{{r}^{3}}}+{\frac {{Q}^{2}\Lambda-{M}^{2}+{Q}^{ 2}}{{r}^{4}}}.\end{aligned}$$ Notice that for the S-waves ($l=0 \Rightarrow \Lambda=l(l+1)=0$) it is monotonically decreasing and thus only scattering states are present in the spectrum. For $l\neq 0$ the formula $$\begin{aligned} \label{crit-m0} \left(\frac{Q}{M}\right)_{cr}^2 = \frac{8+9\Lambda}{8+8\Lambda}\end{aligned}$$ defines a critical charge-to-mass ratio such that for $Q/M>(Q/M)_{cr}$ the effective potential $V(r)$ has no extrema. If $1<Q/M<(Q/M)_{cr}$, there must be one local minimum and one local maximum, so that the potential is barrier-shaped and there emerge the metastable bound states. Noticeably, in the limiting case of the extreme black hole, $Q/M\rightarrow 1$, the local minimum transforms in an event horizon. From Eq. (\[crit-m0\]), one sees that the value $(Q/M)_{cr}$ increases with the growth of $\Lambda$, though not exceeding $\sqrt{9/8}\approx 1.0607$. Thus the metastable bound states may only exist in the narrow range $1<Q/M<(Q/M)_{cr}$. The plots of $V(r)$ for $m=0$, obtained with the values $Q/M=1.01$ and $1.1$ inside and outside this range, respectively, are shown in Fig. 1. ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------  1, 2$. ](reQ11M1m0.eps "fig:") (a) (b) ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- \[fig1\] Case of $m\neq 0$ ----------------- We shall perform the same analysis as in the previous subsection but for the massive particle. For brevity, we set $M=1$ in this subsection; the formulae with arbitrary $M$ can readily be obtained by rescaling (\[rescaling\]). To study the shape of $V(r)$, we find its extrema that amounts to solving the cubic equation $$\label{cubic1} \frac{dV}{dr}=\frac{2m^2}{r^5}(r^{3}+ar^{2}+br+c)=0$$ with the coefficients $$\label{cubic-coef} a=-Q^{2}-\frac{\Lambda}{m^2},\quad b=\frac{3\Lambda}{m^2},\quad c=-\frac{2[(1+\Lambda)Q^{2}-1]}{m^2}.$$ The number of real roots to this cubic equation is controlled by the sign of its determinant defined by $$\label{cubic-D} D=\left(\frac{p}{3}\right)^3+\left(\frac{q}{2}\right)^2$$ where $$\label{cubic-pq} p=-\frac{a^2}{3}+b,\quad q=2\left(\frac{a}{3}\right)^3-\frac{ab}{3}+c.$$ There must be one, two and three real roots when $D>0$, $D=0$ and $D<0$ respectively. Since the signs of the coefficients alternate, all these roots are positive and thus all local extrema of $V(r)$ are of physical meaning. It is convenient to rewrite $D$ in terms of the deviation from the extreme charge value $\delta=Q^2-1$ $$\begin{aligned} \label{Discr} D&=&\frac{1}{108m^8}\left[ \left( 8\,\delta-1 \right) {\Lambda}^{4}+ \left( -78\,{m}^{2}\delta+6\,{m}^{2}+24\,{m}^{2}{\delta}^{2}+8\,\delta \right) {\Lambda}^{3} \right. \nonumber \\ &+& \left( 24\,{m}^{4}{\delta}^{3}+15\,{m}^{4}-84\,{m}^{2}\delta+24\,{m}^ {2}{\delta}^{2}+63\,{m}^{4}{\delta}^{2}+54\,{m}^{4}\delta \right) { \Lambda}^{2} \nonumber \\ &+& \left( 8\,{m}^{6}{\delta}^{4}+32\,{m}^{6}\delta+48\,{m}^ {6}{\delta}^{2}+32\,{m}^{6}{\delta}^{3}+8\,{m}^{6}+24\,{m}^{4}{\delta} ^{3}+156\,{m}^{4}{\delta}^{2}\right. \nonumber \\ &+&\left.\left. 132\,{m}^{4}\delta \right)\Lambda+8\,{m }^{6}\delta+24\,{m}^{6}{\delta}^{2}+24\,{m}^{6}{\delta}^{3}+8\,{m}^{6} {\delta}^{4}+108\,{m}^{4}{\delta}^{2}\right].\end{aligned}$$ For the super-extreme charge under consideration, we have $\delta>0$. Then it becomes evident from (\[Discr\]) that the condition $D>0$ always holds true for the S-states ($\Lambda=0$) and, hence, the corresponding effective potential has one local extremum (minimum), thereby being single-well. Thus, in contrast to the case of $m=0$, the particle with $m\neq 0$ may form stable bound S-states. As an illustration, in Fig. 2 we offer plots of $V(r)$ obtained with $m=0.1$, using $Q/M=1.01$ and $1.1$. In these plots the potential well where the bound states emerge lies below the rest energy level $\omega^2=m^2$ depicted by the dotted line. ----------------------------------------------------------------------------------- ----------------------------------------------------------------------------------   (a) (b) ----------------------------------------------------------------------------------- ---------------------------------------------------------------------------------- \[fig2\] As for the spectrum with $l\neq 0$, we have learnt that in the case of $m=0$ it contains the metastable bound states provided the charge-to-mass ratio is close enough to its extreme value $Q/M=1$. We now argue that in the case of $m\neq 0$ the situation is essentially the same. Indeed, for small $\delta$ the asymptotics of $D$ is given by $$\label{cubic-D1} D=\,{\frac {\Lambda\, \left( 8\,{m}^{2}-\Lambda \right) \left( \Lambda+{m}^{2} \right) ^{2}}{108{m}^{8}}+O(\delta)},$$ so that $D<0$ if $\Lambda>8m^2$. In practice, the last condition holds for all $l\neq 0$, because $m\ll M=1$ (the particle mass is supposed to be much less than the central object mass). The inequality $D<0$ means that $dV/dr$ has three zeros and, as a consequence, $V(r)$ is the barrier-shaped potential with two minima and one maximum. This subcase may include the metastable bound states because one of the minima lies above the level of $\omega^2=m^2$ (see Fig. 2a). On the other hand, if $Q/M$ is far from being extreme, the metastable bound states are excluded. In the case of $m=0$, the critical value (\[crit-m0\]) that provides the necessary and sufficient condition for their exclusion was derived. Now, for $m\neq 0$, we are able to deduce the sufficient but not necessary condition. To do this, we should establish for which values of $\delta$ the condition $D>0$ holds, which guarantees that the effective potential is single-well. Upon demanding that the coefficients in front of powers of $\Lambda$ in (\[Discr\]) be positive, one obtains several restrictions. The strongest one comes from the coefficient of the ${\Lambda}^{2}$-term which becomes positive-defined when $\delta>7/2$. Thus we conclude that the condition $D>0$ holds and, hence, the effective potential is single-well and the metastable bound states are excluded, provided $$\label{Qcrit} \frac{Q^2-M^2}{M^2} >{\frac {7}{2}}, \quad \frac{Q}{M} >\frac {3\sqrt {2}}{2}\approx 2.121...$$ It should be stressed that this is not an exact critical value and, in fact, $D>0$ may hold at lesser values of $Q/M$ when $l$ is small enough. For example, if $Q/M=1.1<2.121$ the effective potential turns out to be single-well for $l=0,1,2$. In the case of $l=0$ it can be seen from Fig. 2b whereas for $l=1,2$ the bottom of the well is located in the region of large $r$ outside the plot. Let us summarize our findings. For $l=0$, the spectrum contains only the scattering states, if the particle is massless, and in addition the stable bound states, if $m\neq 0$. For $l\neq 0$, the metastable bound states emerge if the central object charge-to-mass ratio is close to its extreme value $Q/M=1$. When $Q/M$ exceeds a certain critical value, the metastable bound states disappear and one observes the scattering states and the stable bound states. Energy spectrum {#part4} =============== In this Section we derive the energy spectrum of the particle (= test scalar field perturbation) in the leading WKB approximation. Since the background geometry is naked rather than the black-hole one, our calculation will differ essentially from typical computations of the black-hole quasinormal modes. The latter imply that the solutions to the wave equation go as plane waves close to the horizon. Instead, in our case the solutions have to decrease exponentially deep into the classically forbidden region in the vicinity of the origin. Actually, the barrier-shaped potentials that lead to the metastable bound states in our treatment (see Figs. 1a and 2a) resemble those of Gamow’s theory for $\alpha$-decay (Fig. 3). In turn, the single-well potential we obtained with $m\neq 0$ and $l=0$ resembles the attractive Coulomb potential everywhere except for a narrow region close to the origin. Therefore we shall adopt the ordinary flat-space WKB formulae [@Landau] with the Langer modification for the centrifugal term, $\Lambda=l(l+1)\rightarrow(l+1/2)^2$, which is usual in the Coulomb-like problems [@Heading; @Berry].  \[fig3\] Decaying bound states --------------------- Let us start with the metastable bound states. From the wave equation (\[SecEq\]), it can be seen that the quasiclassical momenta $k(r)$ and $\kappa(r)$ for the classically allowed and forbidden regions respectively should be defined by $$\label{quasimom} k(r)=\frac{1}{F}\sqrt{\omega^2-V(r)}, \qquad \kappa(r)=\frac{1}{F}\sqrt{V(r)-\omega^2}.$$ For the metastable bound states the energy $\omega=\omega_R+i\omega_I$ is complex. In the lowest WKB approximation its real part is determined by the Bohr-Sommerfeld rule [@Landau] $$\label{BSrule} \int_{r_1}^{r_2} \frac{1}{F}\sqrt{\omega_R^2-V(r)}dr=\pi (n+1/2)$$ where $r_{1,2}$ are the turning points at the ends of the potential well (see Fig. 3) and $n$ denotes the excitation number. Further, we derive a Gamow-type formula for the imaginary part of $\omega$. Our derivation follows the conventional procedure [@Mur]. First, multiplying the wave equation (\[SecEq\]) by the complex conjugate wave function, we compose the expression $u''\bar{u}-\bar{u}''u = -(\omega^2-\bar{\omega}^2)F^{-2}|u|^2$ and then integrate it to obtain $$\label{eq-conj} \left.\left(u'\bar{u}-\bar{u}'u\right)\right|_{r\rightarrow\infty} = -4i\omega_R \omega_I\int_{0}^{\infty}F^{-2} |u|^2 dr.$$ Here we took into account that $(u'\bar{u}-\bar{u}'u)$ vanishes at $r=0$ because $u$ must satisfy the boundary condition $u'(0)+au(0)=0$ with real $a$ in order that the Klein-Gordon operator be symmetric. Next, we resort to the well-known WKB formulae for the wave function asymptotics in the classically allowed region [@Landau] $$\label{WKB-wf1} u\sim\frac{C_1}{2\sqrt{k(r)}}\exp\left(i\int_{r_3}^{r}k(r)dr-i\pi/4 \right),\quad r>r_3,$$ $$\label{WKB-wf2} u\sim\frac{C_2}{\sqrt{k(r)}}\cos\left(\int_{r_1}^{r}k(r)dr-\pi/4 \right),\quad r_1<r<r_2.$$ Using the first of these formulae, the left-hand side of (\[eq-conj\]) is estimated to be equal to $i|C_1|^2/2$. The main contribution to the right-hand side of (\[eq-conj\]) comes from the region $r_1<r<r_2$. Approximating the squared cosine in this region by $1/2$, we rewrite (\[eq-conj\]) as $$\frac{i|C_1|^2}{2} = -4i\omega_R \omega_I|C_2|^2\int_{r_1}^{r_2}\frac{dr}{2F^2 k(r)}.$$ Applying the connection formula [@Landau] $C_2=C_1\exp(\int_{r_2}^{r_3}\kappa(r)dr)$, we end up with the requisite expression for the imaginary part of $\omega$ $$\label{Width} \omega_I={\displaystyle -\exp\left(-2 \int_{r_2}^{r_3} \kappa(r)dr \right)}\left[\displaystyle 4\omega_R\int_{r_1}^{r_2} \frac{dr}{F^2 k(r)} \right]^{-1}.$$ This formula as well as the WKB asymptotics (\[WKB-wf1\]) and (\[WKB-wf2\]) is valid if the barrier is nearly impenetrable. It means that the calculated value of $\omega_I$, which determines the decay rate, has to obey the condition $|\omega_I|\ll \omega_R$. Stable bound states ------------------- Now let us turn to the stable bound states occurring in the case of the massive particle. If $m\neq 0$, the effective potential (\[Veff\]) has the Coulomb tail ($V\propto -1/r$), so that a Balmer-type approximate formula for the real bound-state energies $\omega=\omega_R$ can be obtained. To do this, we first compare the wave equation (\[SecEq\]) for our effective potential and the ordinary Schrödinger equation for the Coulomb potential $$\label{CoulEq} \frac{d^{2}\psi}{dr^{2}}+2m\left(E+\frac{Ze^2}{r}-\frac{l(l+1)}{2mr^2} \right)\psi=0.$$ Since at large $r$ the factor $F$ of the RN metrics (\[RNmetric\]) tends to 1, it becomes evident that the term $2Mm^2/r$ in Eq. (\[SecEq\]) corresponds to $2mZe^2/r$ in Eq. (\[CoulEq\]). Then we can define the characteristic radius for our system, $r_B=1/(m^2 M)$, corresponding to the Bohr radius, $1/(mZe^2)$, for Eq. (\[CoulEq\]). Introducing now the dimensionless coordinate $\rho=r/r_B$, the effective potential (\[Veff\]) with the Langer modification is rewritten as $$\begin{aligned} \label{VeffX} V(\rho)&=&{m}^{2}-{\frac {2M^2{m}^{4}}{\rho}}+{\frac {M^2{m}^{4}[{Q}^{2}{m}^{2}+(l+1/2)^2]}{{\rho}^{2 }}}-{\frac {2(l+1/2)^2 M^4 m^6}{{\rho}^{3}}} \nonumber \\ &+&{\frac {M^4 m^8\{{Q}^{2}[1+(l+1/2)^2]-{M}^{2}\}}{{\rho}^{4}}}.\end{aligned}$$ The last two terms in this expression contain higher powers of the particle mass $m$ which is much less than the central object mass $M$. Hence, we may neglect these terms provided that $M$ is not much larger than the Planck mass (equal to 1 in our units). Thus we obtain the truncated potential $$\begin{aligned} \label{VeffTr} V_{trunc}(\rho)&=&{m}^{2}-{\frac {2M^2{m}^{4}}{\rho}}+{\frac {M^2{m}^{4}[{Q}^{2}{m}^{2}+(l+1/2)^2]}{{\rho}^{2 }}}\end{aligned}$$ serving as a good approximation to $V(\rho)$ in the classically allowed region. Using the same reasoning, we approximate $F=1-2M^2 m^2/\rho+M^2 Q^2 m^4/\rho^2$ by $1$. Within the above approximation, the Bohr-Sommerfeld integral (\[BSrule\]) reduces to $$\label{BSruleTr} \int_{\rho_1}^{\rho_2} \sqrt{\omega^2-{m}^{2}+{\frac {2M^2{m}^{4}}{\rho}}-{\frac {M^2{m}^{4}[{Q}^{2}{m}^{2}+(l+1/2)^2]}{{\rho}^{2 }}}} \frac{d\rho}{m^2 M}=\pi (n+1/2)$$ where turning points $\rho_{1,2}$ are zeros of the expression under the square root. This integral is easily calculated, by applying the formula $$\label{BSruleEval} \int_{\rho_1}^{\rho_2} \frac{\sqrt{(\rho-\rho_1)(\rho_2-\rho)}}{\rho}\, d\rho=\pi \left(\frac{\rho_1+\rho_2}{2}-\sqrt{\rho_1 \rho_2} \right).$$ As a result, we get the explicit expression for the stable bound-state energies $$\label{WKBenergies} \omega=m\left[ 1-\frac{m^2 M^2}{\left(n+1/2+\sqrt{(l+1/2)^2+Q^2 m^2}\right)^2}\right]^{1/2}.$$ It has essentially the same structure as the Balmer formula in the Coulomb problem, assuming that the values of the quantum numbers are high, $n, l\gg 1$, as usual in the WKB method. The only difference is that the coupling constant is now given by the product of masses, but not charges. Note that the quantization procedure that starts from the classical particle Hamiltonian has lead us to the same expression [@Kulikov]. Numerical results {#part5} ================= First we examine the spectrum of the metastable bound states for the massless neutral scalar particle in the field of naked RN singularity. It should be noticed that since the depth and the width of the corresponding potential well are finite, the number of the metastable bound states is limited. The energies of all the existing states with $l=1, 2, 3$ calculated according to the WKB formulae (\[BSrule\]) and (\[Width\]) with $Q=1.01$, $M=1$ are presented in Table 1. That the states with larger excitation numbers $n$ do not exist was fixed by observing that the Bohr-Sommerfeld integral (\[BSrule\]) cannot be saturated to its value $\pi (n+1/2)$ even at the maximal allowed bound-state energy $\omega^2$ equal to the peak of the effective potential. As seen from Table 1, for higher $l$ the number of the existing bound states and their half-lives increase as the peak of the effective potential grows with $l$ (see Fig. 1a). --------------------------------- ------------------------------------ ------------------ ------------------ $\phantom{\displaystyle{\sum}}$ $\omega_{WKB}$ $\sqrt{V_{min}}$ $\sqrt{V_{max}}$ $n=0$ $0.294636-i\,0.263\times 10^{-3}$ 0.241152 0.380747 $\phantom{\displaystyle{\sum}}$ $\omega_{WKB}$ $\sqrt{V_{min}}$ $\sqrt{V_{max}}$ $n=0$ $0.423194-i\,0.164\times 10^{-5}$ 0.364334 0.632606 $n=1$ $0.537046-i\,0.227\times 10^{-3}$ $\phantom{\displaystyle{\sum}}$ $\omega_{WKB}$ $\sqrt{V_{min}}$ $\sqrt{V_{max}}$ $n=0$ $0.555002-i\,0.969\times 10^{-10}$ 0.494094 0.884899 $n=1$ $0.673735-i\,0.236\times 10^{-5}$ $n=2$ $0.786160-i\,0.211\times 10^{-3}$ $n=3$ $0.884745-i\,0.523\times 10^{-2}$ --------------------------------- ------------------------------------ ------------------ ------------------ : Energies of the metastable bound states on the naked RN background calculated with $Q=1.01$, $M=1$ and $m=0$. \[ta1\] --------------------------------- ---------------------- ------------------ ------------------ $\phantom{\displaystyle{\sum}}$ $\omega_{WKB}$ $\sqrt{V_{min}}$ $\sqrt{V_{max}}$ $n=0$ $0.37764-i\,0.08936$ 0 0.375 $n=1$ $0.34392-i\,0.27828$ $n=2$ $0.29661-i\,0.48145$ $\phantom{\displaystyle{\sum}}$ $\omega_{WKB}$ $\sqrt{V_{min}}$ $\sqrt{V_{max}}$ $n=0$ $0.62609-i\,0.08873$ 0 0.625 $n=1$ $0.60677-i\,0.26944$ $n=2$ $0.57254-i\,0.45750$ --------------------------------- ---------------------- ------------------ ------------------ : Quasinormal energies of the extreme RN black hole with $Q=M=1$ and $m=0$ taken from Ref. [@Onozawa] \[ta2\] For comparison, in Table 2 we list the first three quasinormal energies (frequencies) of the scalar field on the extreme RN black-hole background with $Q=M=1$ that were computed in Ref. [@Onozawa]. The striking difference between the results in Tables 1 and 2 can be ascribed to the fact that the event horizon is absent in the naked singularity case and thus the metastable bound states and the quasinormal modes are of different nature. As was discussed in the previous Section, the setup for the naked RN bound states resembles that of Gamow’s theory for $\alpha$-decay. In this setup the most longliving states are concentrated near the bottom of the effective potential well $\omega^2\simeq \mathrm{min}[V(r)]$. This can be readily checked by comparing the calculated energies $\omega$ with the values $V_{\mathrm{min}}= \mathrm{min}[V(r)]$ which are also shown in Table 1. In turn, the black-hole quasinormal modes are, roughly speaking, the scattering resonances travelling both to $r=0$ and to $r=\infty$. Thus they have much smaller half-lives and survive for energies close to the peak of the effective potential $\omega^2\simeq \mathrm{max}[V(r)]$ (see Table 2). For the naked RN singularity, the quasinormal modes of the latter type were calculated in the recent work [@Chirenti]. These authors postulate the Dirichlet boundary condition that rules out one of the linearly independent solutions to the Klein-Gordon equation. In the present work we do not specify the boundary condition because it is not needed in the WKB approximation. Moreover, for the metastable bound states we consider the point $r=0$ is located deep into the classically forbidden region and thus the boundary condition cannot affect these states substantially. --------------------------------- ------------------------------------ ------------------ ------------------ $\phantom{\displaystyle{\sum}}$ $\omega_{WKB}$ $\sqrt{V_{min}}$ $\sqrt{V_{max}}$ $n=0$ $0.295156-i\,0.221\times 10^{-3}$ 0.241591 0.383910 $\phantom{\displaystyle{\sum}}$ $\omega_{WKB}$ $\sqrt{V_{min}}$ $\sqrt{V_{max}}$ $n=0$ $0.423496-i\,0.142\times 10^{-5}$ 0.364614 0.634529 $n=1$ $0.537417-i\,0.209\times 10^{-3}$ $\phantom{\displaystyle{\sum}}$ $\omega_{WKB}$ $\sqrt{V_{min}}$ $\sqrt{V_{max}}$ $n=0$ $0.555216-i\,0.869\times 10^{-10}$ 0.494299 0.886277 $n=1$ $0.673975-i\,0.219\times 10^{-5}$ $n=2$ $0.786450-i\,0.200\times 10^{-3}$ $n=3$ $0.885377-i\,0.569\times 10^{-2}$ --------------------------------- ------------------------------------ ------------------ ------------------ : Energies of the metastable bound states on the naked RN background calculated with $Q=1.01$, $M=1$ and $m=0.1$. \[ta3\] Next we investigate the case of the massive particle. In Table 3 the energies of all the states with $l=1, 2, 3$ computed using $Q=1.01$, $M=1$ and $m=0.1$ are listed. Also, in Fig. 4 the imaginary part of $\omega$ is plotted versus the real part of $\omega$ for different values of the particle mass $m$. From Tables 1, 3 and Fig. 4, we see that the real part of $\omega$, *i.e.* the oscillation frequency, grows with increase of $m$, while the imaginary part of $\omega$, representing the decay rate, falls down. Interestingly, the same behavior was observed for the quasinormal modes of the scalar field on the sub-extreme RN black-hole background in Ref. [@Konoplya]. --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------   (a) (b) --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- \[fig4\] Now let analyze the spectrum of the stable bound states. In Table 4 we list the energies $(\omega/m)_{WKB}$ of the ground and the first two excited S-states ($l = 0$) computed according to the WKB formula (\[WKBenergies\]) and also the energies $(\omega/m)_{num}$ obtained by direct numerical integration of the Klein-Gordon equation (\[RadEq\]) with the Dirichlet boundary condition $u(0)=0$. The calculation was made using $Q=1.1$, $M=1$ and various $m$. From Table 4 we see that the binding energy $\omega_{bind}=-(\omega-m)$ increases as the particle mass $m$ increases. However, this binding energy is more than two orders of magnitude smaller than the metastable bound-state energy $\omega_R$ calculated with the same values of $m$ and $M$ (see Table 3). It means that transitions between levels of the stable states due to an external perturbation would have much lower frequencies than the those due to the decay of the metastable bound states. ----------------- -------------------- -------------------- -------------------- -------------------- -------------------- -------------------- $\phantom{n=0}$ $(\omega/m)_{WKB}$ $(\omega/m)_{num}$ $(\omega/m)_{WKB}$ $(\omega/m)_{num}$ $(\omega/m)_{WKB}$ $(\omega/m)_{num}$ $n=0$ 0.998757 0.998733 0.995105 0.994710 0.989266 0.987068 $n=1$ 0.999688 0.998685 0.998764 0.998701 0.997257 0.996906 $n=2$ 0.999861 0.999860 0.999449 0.999429 0.998771 0.998661 ----------------- -------------------- -------------------- -------------------- -------------------- -------------------- -------------------- : Energies of the stable bound S-states on the naked RN background with $Q=1.1$, $M=1$. \[ta4\] Conclusion {#part6} ========== In this work we have studied the states of neutral scalar particles in the field of the naked RN singularity. It has been established that their energy spectrum is qualitatively different from the quasinormal mode spectrum for the RN black hole. The conditions for the bound states to be formed have been found and the possible types of these states have been examined. The first possible type is the metastable, *i.e.* decaying bound states. They occur provided that the charge-to-mass ratio for the central object, $Q/M$, is close to its extreme value $Q/M=1$. If the particle is massless, this is expressed by the inequality $1<(Q/M)^2<[8+9l(l+1)]/[8+8l(l+1)]$ where $l$ is the orbital number. Using the WKB method, we have calculated the metastable bound-state spectrum which shows that there is no continuous transition in energies between the RN naked singularity and the extremal RN black hole cases. The metastable bound states on the naked RN background with $Q/M>1$ have much larger live-times than the quasinormal modes for $Q/M=1$. This is not surprising because for $Q/M>1$ there exists no event horizon to fall on, so that the metastable states can decay only at the expense of the particle escape to spatial infinity upon tunneling through the wide potential barrier. As the second possible type of particle states, we identify the stable bound states. It should be stressed that these states have no analog in the case of the RN black hole and can only be formed by massive particles in the field of the central object with the sufficiently high charge-to-mass ratio. Their energy spectrum, calculated by means of the WKB method, has the structure similar to that of the Coulomb spectrum. Energy gaps between the stable bound states, *i.e.* transition frequencies, are several orders of magnitude smaller than the frequencies for the metastable bound states obtained with the same masses of the particle and the central object. Thus the latter states are more favored to be ever found in experiment. Nevertheless, the stable bound states may manifest themselves in a different way, by making up a scalar condensate around the central object. Then the question arises as to whether such a condensate can provide mass needed to screen the RN singularity. Answering this question requires a self-consistent model for interacting scalar and gravitational fields which may be constructed in a future work. Acknowledgments {#acknowledgments .unnumbered} =============== We thank the anonymous referee for a useful suggestion aimed at improving the paper. This work was supported by the grant under the Cosmomicrophysics program for the Physics and Astronomy Division of the National Academy of Sciences of Ukraine. [00]{} S. Chandrasekhar, *The mathematical theory of black holes*, Clarendon Press, Oxford (1983). J. M. Cohen, R. Gautreau, [*Phys. Rev. D*]{} **19** (1979) 2273. R. M. Wald [*J. Math. Phys.*]{} **21** (1980) 2802. G. T. Horowitz, D. Marolf, [*Phys. Rev. D*]{} **52** (1995) 5670. A. 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--- abstract: 'This survey paper reports on work of Birkar, who confirmed a long-standing conjecture of Alexeev and Borisov-Borisov: Fano varieties with mild singularities form a bounded family once their dimension is fixed. Following Prokhorov-Shramov, we explain how this boundedness result implies that birational automorphism groups of projective spaces satisfy the Jordan property, answering a question of Serre in the positive.' address: - | Mathematisches Institut\ Albert-Ludwigs-Universität Freiburg\ Ernst-Zermelo-Straße 1\ 79104 Freiburg im Breisgau, Germany\  \ Freiburg Institute for Advanced Studies (FRIAS)\ Freiburg im Breisgau, Germany - 'Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, Ernst-Zermelo-Straße 1, 79104 Freiburg im Breisgau, Germany & Freiburg Institute for Advanced Studies (FRIAS), Freiburg im Breisgau, Germany' author: - Stefan Kebekus date: - Janvier 2019 - Janvier 2019 - subtitle: - 'following Birkar and Prokhorov-Shramov' - - title: 'Boundedness results for singular Fano varieties, and applications to Cremona groups' --- Main results ============ Throughout this paper, we work over the field of complex numbers. Boundedness of singular Fano varieties -------------------------------------- A normal, projective variety $X$ is called *Fano* if a negative multiple of its canonical divisor class is Cartier and if the associated line bundle is ample. Fano varieties appear throughout geometry and have been studied intensely, in many contexts. For the purposes of this talk, we remark that Fanos with sufficiently mild singularities constitute one of the fundamental variety classes in birational geometry. In fact, given any projective manifold $X$, the Minimal Model Programme (MMP) predicts the existence of a sequence of rather special birational transformations, known as “divisorial contractions” and “flips”, as follows, $$\xymatrix{ X = X^{(0)} \ar@{-->}[rr]^{α^{(1)}}_{\text{birational}} && X^{(1)} \ar@{-->}[rr]^{α^{(2)}}_{\text{birational}} && ⋯ \ar@{-->}[rr]^{α^{(n)}}_{\text{birational}} && X^{(n)}. }$$ The resulting variety $X^{(n)}$ is either canonically polarised (which is to say that a suitable power of its canonical sheaf is ample), or it has the structure of a fibre space whose general fibres are either Fano or have numerically trivial canonical class. The study of (families of) Fano varieties is thus one of the most fundamental problems in birational geometry. Even though the starting variety $X$ is a manifold by assumption, it is well understood that we cannot expect the varieties $X^{(•)}$ to be smooth. Instead, they exhibit mild singularities, known as “terminal” or “canonical” — we refer the reader to [@KM98 Sect. 2.3] or [@MR3057950 Sect. 2] for a discussion and for references. If $X^{(n)}$ admits the structure of a fibre space, its general fibres will also have terminal or canonical singularities. Even if one is primarily interested in the geometry of *manifolds*, it is therefore necessary to include families of *singular* Fanos in the discussion. In a series of two fundamental papers, [@Bir16a; @Bir16b], Birkar confirmed a long-standing conjecture of Alexeev and Borisov-Borisov, [@MR1298994; @MR1166957], asserting that for every $d ∈ ℕ$, the family of $d$-dimensional Fano varieties with terminal singularities is bounded: there exists a proper morphism of quasi-projective schemes over the complex numbers, $u : 𝕏 → Y$, and for every $d$-dimensional Fano $X$ with terminal singularities a closed point $y ∈ Y$ such that $X$ is isomorphic to the fibre $𝕏_y$. In fact, a much more general statement holds true. \[thm:BAB\] Given $d ∈ ℕ$ and $ε ∈ ℝ^+$, let $\mathcal X_{d,ε}$ be the family of projective varieties $X$ with dimension $\dim_ℂ X = d$ that admit an $ℝ$-divisor $B ∈ ℝ\operatorname{Div}(X)$ such that the following holds true. 1. \[il:BAB1\] The tuple $(X,B)$ forms a pair. In other words: $X$ is normal, the coefficients of $B$ are contained in the interval $[0,1]$ and $K_X+B$ is $ℝ$-Cartier. 2. \[il:BAB2\] The pair $(X,B)$ is $ε$-lc. In other words, the total log discrepancy of $(X,B)$ is greater than or equal to $ε$. 3. \[il:BAB3\] The $ℝ$-Cartier divisor $-(K_X+B)$ is nef and big. Then, the family $\mathcal X_{d,ε}$ is bounded. If $X$ has terminal singularities, then $(X,0)$ is $1$-lc. We refer to Section \[sec:singofpairs\], to Birkar’s original papers, or to [@MR3224718 Sect. 3.1] for the relevant definitions concerning more general classes of singularities. For his proof of the boundedness of Fano varieties and for his contributions to the Minimal Model Programme, Caucher Birkar was awarded with the Fields Medal at the ICM 2018 in Rio de Janeiro. ### Where does boundedness come from? {#ssec:1-1-1v2} The brief answer is: “From boundedness of volumes!” In fact, if $(X_t, A_t)_{t ∈ T}$ is a family of tuples where the $X_t$ are normal, projective varieties of fixed dimension $d$ and $A_t ∈ \operatorname{Div}(X_t)$ are very ample, and if there exists a number $v ∈ ℕ$ such that $$\operatorname{vol}(A_t) := \limsup_{n→∞} \frac{d!·h⁰\bigl( X_t,\, 𝒪_{X_t}(n·A_t) \bigr)}{n^d} < v$$ for all $t ∈ T$, then elementary arguments using Hilbert schemes show that the family $(X_t, A_t)_{t ∈ T}$ is bounded. For the application that we have in mind, the varieties $X_t$ are the Fano varieties whose boundedness we would like to show and the divisors $A_t$ will be chosen as fixed multiples of their anticanonical classes. To obtain boundedness results in this setting, Birkar needs to show that there exists one number $m$ that makes all $A_t := -m·K_{X_t}$ very ample, or (more modestly) ensures that the linear systems $|-m·K_{X_t}|$ define birational maps. Volume bounds for these divisors need to be established, and the singularities of the linear systems need to be controlled. ### Earlier results, related results {#ssec:1-1-2v2} Boundedness results have a long history, which we cannot cover with any pretence of completeness. Boundedness of smooth Fano surfaces and threefolds follows from their classification. Boundedness of Fano *manifolds* of arbitrary dimension was shown in the early 1990s, in an influential paper of Kollár, Miyaoka and Mori, [@KMM92], by studying their geometry as rationally connected manifolds. Around the same time, Borisov-Borisov were able to handle the toric case using combinatorial methods, [@MR1166957]. For (singular) surfaces, Theorem \[thm:BAB\] is due to Alexeev, [@MR1298994]. Among the newer results, we will only mention the work of Hacon-McKernan-Xu. Using methods that are similar to those discussed here, but without the results on “boundedness of complements” ($→$ Section \[sec:bcomp\]), they are able to bound the volumes of klt pairs $(X, Δ)$, where $X$ is projective of fixed dimension, $K_X + Δ$ is numerically trivial and the coefficients of $Δ$ come from a fixed DCC set, [@MR3224718 Thm. B]. Boundedness of Fanos with klt singularities and fixed Cartier index follows, [@MR3224718 Cor. 1.8]. In a subsequent paper [@MR3507257] these results are extended to give the boundedness result that we quote in Theorem \[thm:boundednessCriterion\], and that Birkar builds on. We conclude with a reference to [@Jiang17; @Chen18] for current results involving $K$-stability and $α$-invariants. The surveys [@MR2827803; @MR3821154] give a more complete overview. ### Positive characteristic Apart from the above-mentioned results of Alexeev, [@MR1298994], which hold over algebraically closed field of arbitrary characteristic, little is known in case where the characteristic of the base field is positive. Applications {#ssec:1-2} ------------ As we will see in Section \[sec:jordan\] below, boundedness of Fanos can be used to prove the existence of fixed points for actions of finite groups on Fanos, or more generally rationally connected varieties. Recall that a variety $X$ is *rationally connected* if every two points are connected by an irreducible, rational curve contained in $X$. This allows us to apply Theorem \[thm:BAB\] in the study of finite subgroups of birational automorphism groups. ### The Jordan property of Cremona groups Even before Theorem \[thm:BAB\] was known, it had been realised by Prokhorov and Shramov, [@MR3483470], that boundedness of Fano varieties with terminal singularities would imply that the birational automorphism groups of projective spaces (= Cremona groups, $\operatorname{Bir}(ℙ^d)$) satisfy the *Jordan property*. Recall that a group $Γ$ is said to *have the Jordan property* if there exists a number $j ∈ ℕ$ such that every finite subgroup $G ⊂ Γ$ contains a normal, Abelian subgroup $A ⊂ G$ of index $|G:A| ≤ j$. In fact, a stronger result holds. \[thm:jordan\] Given any number $d ∈ ℕ$, there exists $j ∈ ℕ$ such that for every complex, projective, rationally connected variety $X$ of dimension $\dim_{ℂ} X = d$, every finite subgroup $G ⊂ \operatorname{Bir}(X)$ contains a normal, Abelian subgroup $A ⊆ G$ of index $|G:A| ≤ j$. Theorem \[thm:jordan\] answers a question of Serre [@MR2567402 6.1] in the positive. A more detailed analysis establishes the Jordan property more generally for all varieties of vanishing irregularity, [@MR3292293 Thm. 1.8]. Theorem \[thm:jordan\] ties in with the general philosophy that finite subgroups of $\operatorname{Bir}(ℙ^d)$ should in many ways be similar to finite linear groups, where the property has been established by Jordan more then a century ago. \[thm:jordan-lin\] Given any number $d ∈ ℕ$, there exists $j^{\operatorname{Jordan}}_d ∈ ℕ$ such that every finite subgroup $G ⊂ \operatorname{GL}_d(ℂ)$ contains a normal, Abelian subgroup $A ⊆ G$ of index $|G:A| ≤ j^{\operatorname{Jordan}}_d$. For further information on Cremona groups and their subgroups, we refer the reader to the surveys [@MR3229352; @MR3821147] and to the recent research paper [@Popov18]. For the maximally connected components of automorphism groups of projective varieties (rather than the full group of birational automorphisms), the Jordan property has recently been established by Meng and Zhang without any assumption on the nature of the varieties, [@MZ18 Thm. 1.4]; their proof uses group-theoretic methods rather than birational geometry. For related results (also in positive characteristic), see [@Hu18; @MR3830471; @SV18] and references there. ### Boundedness of finite subgroups in birational transformation groups Following similar lines of thought, Prokhorov and Shramov also deduce boundedness of finite subgroups in birational transformation groups, for arbitrary varieties defined over a finite field extension of $ℚ$. Let $k$ be a finitely generated field over $ℚ$. Let $X$ be a variety over $k$, and let $\operatorname{Bir}(X)$ denote the group of birational automorphisms of $X$ over $\operatorname{Spec}k$. Then, there exists $b ∈ ℕ$ such that any finite subgroup $G ⊂ \operatorname{Bir}(X)$ has order $|G| ≤ b$. As an immediate corollary, they answer another question of Serre[^1], pertaining to finite subgroups in the automorphism group of a field. Let $k$ be a finitely generated field over $ℚ$. Then, there exists $b ∈ ℕ$ such that any finite subgroup $G ⊂ \operatorname{Aut}(k)$ has order $|G| ≤ b$. ### Boundedness of links, quotients of the Cremona group Birkar’s result has further applications within birational geometry. Combined with work of Choi-Shokurov, it implies the boundedness of Sarkisov links in any given dimension, cf. [@MR2784026 Cor. 7.1]. In [@BLZ], Blanc-Lamy-Zimmermann use Birkar’s result to prove the existence of many quotients of the Cremona groups of dimension three or more. In particular, they show that these groups are not perfect and thus not simple. Outline of this paper --------------------- Paraphrasing [@Bir16a p. 6], the main tools used in Birkar’s work include the Minimal Model Programme [@KM98; @BCHM10], the theory of complements [@MR1892905; @MR2448282; @MR1794169], the technique of creating families of non-klt centres using volumes [@MR3224718; @MR3034294] and [@KollarSingsOfPairs Sect. 6], and the theory of generalised polarised pairs [@MR3502099]. In fact, given the scope and difficulty of Birkar’s work, and given the large number of technical concepts involved, it does not seem realistic to give more than a panoramic presentation of Birkar’s proof here. Largely ignoring all technicalities, Sections \[sec:bcomp\]–\[sec:lcthres\] highlight four core results, each of independent interest. We explain the statements in brief, sketch some ideas of proof and indicate how the results might fit together to give the desired boundedness result. Finally, Section \[sec:jordan\] discusses the application to the Jordan property in some detail. Acknowledgements ---------------- The author would like to thank Florin Ambro, Serge Cantat, Enrica Floris, Christopher Hacon, Vladimir Lazić, Benjamin McDonnell, Vladimir Popov, Thomas Preu, Yuri Prokhorov, Vyacheslav Shokurov, Chenyang Xu and one anonymous reader, who answered my questions and/or suggested improvements. Yanning Xu was kind enough to visit Freiburg and patiently explain large parts of the material to me. He helped me out more than just once. His paper [@YXu18], which summarises Birkar’s results, has been helpful in preparing these notes. Even though our point of view is perhaps a little different, it goes without saying that this paper has substantial overlap with Birkar’s own survey [@Bir18]. Notation, standard facts and known results ========================================== Varieties, divisors and pairs ----------------------------- We follow standard conventions concerning varieties, divisors and pairs. In particular, the following notation will be used. If $X$ is a normal, quasi-projective variety and $B ∈ ℝ\operatorname{Div}(X)$ an $ℝ$-divisor on $X$, we write $⌊ B ⌋$, $⌈ B ⌉$ for the round-down and round-up of $B$, respectively. The divisor $\{B\} := B - ⌊ B ⌋$ is called *fractional part of $B$*. A *pair* is a tuple $(X, B)$ consisting of a normal, quasi-projective variety $X$ and an effective $ℝ$-divisor $B$ such that $K_X + B$ is $ℝ$-Cartier. A *couple* is a tuple $(X, B)$ consisting of a normal, projective variety $X$ and a divisor $B ∈ \operatorname{Div}(X)$ whose coefficients are all equal to one. The couple is called *log-smooth* if $X$ is smooth and if $B$ has simple normal crossings support. $ℝ$-divisors ------------ While divisors with real coefficients had sporadically appeared in birational geometry for a long time, the importance of allowing real (rather than rational) coefficients was highlighted in the seminal paper [@BCHM10], where continuity- and compactness arguments for spaces of divisors were used in an essential manner. Almost all standard definitions for divisors have analogues for $ℝ$-divisors, but the generalised definitions are perhaps not always obvious. For the reader’s convenience, we recall a few of the more important notions here. Let $X$ be a normal, projective variety. A divisor $B ∈ ℝ\operatorname{Div}(X)$, which need not be $ℝ$-Cartier, is called *big* if there exists an an ample $H ∈ ℝ\operatorname{Div}(X)$, and effective $D ∈ ℝ\operatorname{Div}(X)$ and an $ℝ$-linear equivalence $B \sim_ℝ H + D$. Let $X$ be a normal, projective variety of dimension $d$. The *volume* of an $ℝ$-divisor $D ∈ ℝ\operatorname{Div}(X)$ is defined as $$\operatorname{vol}(D) := \limsup_{m→∞} \frac{d!·h⁰\bigl( X,\, 𝒪_X(⌊mD⌋) \bigr)}{m^d}.$$ Let $X$ be a normal, quasi-projective variety and let $M ∈ ℝ\operatorname{Div}(X)$. The *$ℝ$-linear system* $|M|$ is defined as $$|M|_ℝ := \{ D ∈ ℝ\operatorname{Div}(X) \,|\, D \text{ is effective and } D \sim_{ℝ} M \}.$$ Invariants of varieties and pairs {#sec:singofpairs} --------------------------------- We briefly recall a number of standard definitions concerning singularities. In brief, if $X$ is smooth, and if $π : {\widetilde}{X} → X$ is any birational morphism, where ${\widetilde}{X}$ it smooth, then any top-form $σ ∈ H⁰ \bigl( X,\, ω_X \bigr)$ pulls back to a holomorphic differential form $τ ∈ H⁰ \bigl( {\widetilde}{X},\, ω_{{\widetilde}{X}} \bigr)$, with zeros along the positive-dimensional fibres of $π$. However, if $X$ is singular, if $π : {\widetilde}{X} → X$ is a resolution of singularities and if $σ ∈ H⁰ \bigl( X,\, ω_X \bigr)$ is any section in the (pre-)dualising sheaf, then the pull-back of $σ$ will only be a rational differential form on ${\widetilde}{X}$ which might well have poles along the positive-dimensional fibres of $π$. The idea in the definition of “log discrepancy” is to use this pole order to measure the “badness” of the singularities on $X$. We refer the reader to one of the standard references [@KM98 Sect. 2.3] and [@MR3057950 Sect. 2] for an-depth discussion of these ideas and of the singularities of the Minimal Model Programme. Since the notation is not uniform across the literature[^2], we spend a few lines to fix notation and briefly recall the central definitions of the field. \[not:logdiscrep\] Let $(X,B)$ a pair and let $π : {\widetilde}{X} → X$ be a log resolution of singularities, with exceptional divisors $(E_i)_{1 ≤ i ≤ n}$. Since $K_X+B$ is $ℝ$-Cartier by assumptions, there exists a well-defined notion of pull-back, and a unique divisor $B_{{\widetilde}{X}} ∈ ℝ\operatorname{Div}({\widetilde}{X})$ such that $K_{{\widetilde}{X}} + B_{{\widetilde}{X}} = π^* (K_X+B)$ in $ℝ\operatorname{Div}({\widetilde}{X})$. If $D$ is any prime divisor on ${\widetilde}{X}$, we consider the *log discrepancy* $$a_{\log}(D, X, B) := 1 - \operatorname{mult}_D B_{{\widetilde}{X}}.$$ The infimum over all such numbers, $$a_{\log}(X, B) := \inf \{ a_{\log}(D, X, B) \:|\: π:{\widetilde}{X} → X \text{ a log resolution and } D ∈ \operatorname{Div}({\widetilde}{X}) \text{ prime}\}$$ is called the *total log discrepancy of the pair $(X,B)$*. The total log discrepancy measures how bad the singularities are: the smaller $a_{\log}(X, B)$ is, the worse the singularities are. Table  lists the classes of singularities will be relevant in the sequel. In addition, $(X, B)$ is called *plt* if $a_{\log}(D, X, B) > 0$ for every resolution $π : {\widetilde}{X} → X$ and every *exceptional* divisor $D$ on ${\widetilde}{X}$. The class of $ε$-lc singularities, which is perhaps the most relevant for our purposes, was introduced by Alexeev. If …, then $(X,B)$ is called “…” ---------------------- --- ------------------------------------ $a_{\log}(X, B) ≥ 0$ … *log canonical (or “lc”)* $a_{\log}(X, B) > 0$ … *Kawamata log terminal (or “klt”)* $a_{\log}(X, B) ≥ ε$ … *$ε$-log canonical (or “$ε$-lc”)* $a_{\log}(X, B) ≥ 1$ … *canonical* $a_{\log}(X, B) > 1$ … *terminal* : Singularities of the Minimal Model Programme[]{data-label="tab:x1"} ### Places and centres The divisors $D$ that appear in the definition log discrepancy deserve special attention, in particular if $a_{\log}(D, X, B) ≤ 0$. Let $(X,B)$ a pair. A *non-klt place* of $(X, B)$ is a prime divisor $D$ on birational models of $X$ such that $a_{\log}(D, X, B) ≤ 0$. A *non-klt centre* is the image on $X$ of a non-klt place. When $(X, B)$ is lc, a non-klt centre is also called an *lc centre*. ### Thresholds Suppose that $(X,B)$ is a klt pair, and that $D$ is an effective divisor on $X$. The pair $(X, B+t·D)$ will then be log-canonical for sufficiently small numbers $t$, but cannot be klt when $t$ is large. The critical value of $t$ is called the *log-canonical threshold*. \[def:lct\] Let $(X,B)$ be a klt pair. If $D ∈ ℝ\operatorname{Div}(X)$ is effective, one defines the *lc threshold of $D$ with respect to $(X,B)$* as $$\begin{aligned} \operatorname{lct}\bigl(X,\, B,\, D \bigr) & := \sup \bigl\{ t ∈ ℝ \:\bigl|\: (X,B+t·D) \text{ is lc} \bigr\}. \\ \intertext{If $Δ ∈ ℝ\operatorname{Div}(X)$ is $ℝ$-Cartier with non-empty $ℝ$-linear system (but not necessarily effective itself), one defines \emph{lc threshold of $|Δ|_{ℝ}$ with respect to $(X,B)$} as} \operatorname{lct}\bigl(X,\, B,\, |Δ|_ℝ \bigr) & := \inf \bigl\{ \operatorname{lct}(X,B,D) \:\bigl|\: D ∈ |Δ|_ℝ \bigr\}. \end{aligned}$$ \[rem:sflct\] In the setting of Definition \[def:lct\], it is a standard fact that $$\operatorname{lct}\bigl(X,\, B,\, |Δ|_ℝ \bigr) = \sup \bigl\{ t ∈ ℝ \:\bigl|\: (X, B+t·D) \text{ is lc for every } D ∈ |Δ|_{ℝ} \bigr\}.$$ In particular, if $(X,B)$ is klt, then $(X, B+t'·D)$ is lc for every $D ∈ |Δ|_{ℝ}$ and every $0 < t' < t$. If $B = 0$, we omit it from the notation and write $\operatorname{lct}\bigl(X,\, |Δ|_ℝ \bigr)$ and $\operatorname{lct}\bigl(X,\, D \bigr)$ in short. Fano varieties and pairs ------------------------ Fano varieties come in many variants. For the purposes of this overview, the following classes of varieties will be the most relevant.   - A projective pair $(X, B)$ is called *log Fano* if $(X, B)$ is lc and if $-(K_X+B)$ is ample. If $B = 0$, we just say that $X$ is Fano. - A projective pair $(X, B)$ is called is called *weak log Fano* if $(X, B)$ is lc and $-(K_X + B)$ is nef and big. If $B = 0$, we just say that $X$ is weak Fano. There exist relative versions of the notions discussed above. If $(X, B)$ is any quasi-projective pair, if $Z$ is normal and if $X → Z$ is surjective, projective and with connected fibres, we say $(X, B)$ is log Fano over $Z$ if it is lc and if $-(K_X + B)$ is relatively ample over $Z$. Ditto with “weak log Fano”. Varieties of Fano type ---------------------- Varieties $X$ that *admit* a boundary $B$ that makes $(X,B)$ a Fano pair are said to be of *Fano type*. This notion was introduced by Prokhorov and Shokurov in [@MR2448282]. We refer to that paper for basic properties of varieties of Fano type. A normal, projective variety $X$ is said to be *of Fano type* if there exists an effective, $ℚ$-divisor $B$ such that $(X,B)$ is klt and weak log Fano pair. Equivalently: there exists a big $ℚ$-divisor $B$ such that $K_X + B \sim_ℚ 0$ and such that $(X, B)$ is a klt pair. \[rem:MoriDream\] If $X$ is of Fano type, recall from [@BCHM10 Sect. 1.3] that $X$ is a “Mori dream space”. Given any $ℝ$-Cartier divisor $D ∈ ℝ\operatorname{Div}(X)$, we can then run the $D$-Minimal Model Programme and obtain a sequence of extremal contractions and flips, $X \dasharrow Y$. If the push-forward of $D_Y$ of $Y$ is nef over, we call $Y$ a minimal model for $D$. Otherwise, there exists a $D_Y$-negative extremal contraction $Y → T$ with $\dim Y > \dim T$, and we call $Y$ a Mori fibre space for $D$. As before, there exists an obvious relative version of the notion “Fano type”. Remark \[rem:MoriDream\] generalises to this relative setting. Varieties of Fano type come in two flavours that often need to be treated differently. The following notion, which we recall for later use, has been introduced by Shokurov. \[def:exceptional\] Let $(X, B)$ be a projective pair, and assume that there exists an effective $P ∈ ℝ\operatorname{Div}(X)$ such that $K_X + B + P \sim_{ℝ} 0$. We say $(X, B)$ is *non-exceptional* if we can choose $P$ so that $(X, B+P)$ is *not* klt. We say that $(X, B)$ is *exceptional* if $(X, B+P)$ is klt for every choice of $P$. b-Divisors and generalised pairs ================================ In addition to the classical notions for singularities of pairs that we recalled in Section \[sec:singofpairs\] above, much of Birkar’s work uses the notion of *generalised polarised pairs*. The additional flexibility of this notion allows for inductive proofs, but adds substantial technical difficulties. Generalised pairs were introduced by Birkar and Zhang in [@MR3502099]. ### Disclaimer {#disclaimer .unnumbered} The notion of generalised polarised pairs features prominently in Birkar’s work, and should be presented in an adequate manner. The technical complications arising from this notion are however substantial and cannot be explained within a few pages. As a compromise, this section briefly explains what generalised pairs are, and how they come about in relevant settings. Section \[ssec:BC\] pinpoints one place in Birkar’s inductive scheme of proof where generalised pairs appear naturally, and explains why most (if not all) of the material presented in this survey should in fact be formulated and proven for generalised pairs. For the purpose of exposition, we will however ignore this difficulty and discuss the classical case only. Definition of generalised pairs ------------------------------- To begin, we only recall a minimal subset of the relevant definitions, and refer to [@Bir16a Sect. 2] and to [@MR3502099 Sect. 4] for more details. We start with the notion of b-divisors, as introduced by Shokurov in [@MR1420223], in the simplest case. \[def:1v2\] Let $X$ be a variety. We consider projective, birational morphisms $Y → X$ from normal varieties $Y$, and for each $Y$ a divisor $M_Y ∈ ℝ\operatorname{Div}(Y)$. The collection $M := (M_Y)_Y$ is called *$b$-divisor* if for any morphism $f : Y' → Y$ of birational models over $X$, we have $M_Y = f_* (M_{Y'})$. Setting as in Definition \[def:1v2\]. A b-divisor $M$ is called *b-$ℝ$-Cartier* if there exists one $Y$ such that $M_Y$ is $ℝ$-Cartier and such that for any morphism $f : Y' → Y$ of birational models over $X$, we have $M_{Y'} = f^* (M_Y)$. Ditto for *b-Cartier* b-divisors. \[def:gpp\] Let $Z$ be a variety. A *generalised polarised pair over $Z$* is a tuple consisting of the following data: 1. a normal variety $X$ equipped with a projective morphism $X → Z$, 2. an effective $ℝ$-divisor $B ∈ ℝ\operatorname{Div}(X)$, and 3. a b-$ℝ$-Cartier b-divisor over $X$ represented as $(φ: X' → X, M')$, where $M' ∈ ℝ\operatorname{Div}(X')$ is nef over $Z$, and where $K_X + B + φ_* M'$ is $ℝ$-Cartier. In the setup of Definition \[def:gpp\], we usually write $M := φ_* M'$ and say that $(X, B+M)$ is a generalised pair with data $X' \overset{φ}{→} X → Z$ and $M'$. In contexts where $Z$ is not relevant, we usually drop it from the notation: in this case one can just assume $X → Z$ is the identity. When $Z$ is a point we also drop it but say the pair is projective. Following [@MR3502099 p. 286] we remark that Definition \[def:gpp\] is flexible with respect to $X'$ and $M'$. To be more precise, if $g : X'' → X'$ is a projective birational morphism from a normal variety, then there is no harm in replacing $X'$ with $X''$ and replacing $M'$ with $g^* M'$. Singularities of generalised pairs ---------------------------------- All notions introduced in Section \[sec:singofpairs\] have analogues in the setting of generalised pairs. Again, we cover only the most basic definition here. Consider a generalised polarised pair $(X, B+M)$ with data $X' \overset{φ}{→} X → Z$ and $M'$, where $φ$ is a log resolution of $(X, B)$. Then, there exists a uniquely determined divisor $B'$ on $X'$ such that $$K_{X'} + B' + M' = φ^* (K_X + B + M)$$ If $D ∈ \operatorname{Div}(X')$ is any prime divisor, the *generalised log discrepancy* is defined to be $$a_{\log}(D, X, B+M) := 1 - \operatorname{mult}_D B'.$$ As before, we define the *generalised total log discrepancy* $a_{\log}(X, B+M)$ by taking the infimum over all $D$ and all resolutions. In analogy to the definitions of Table \[tab:x1\], we say that the generalised polarised pair is *generalised lc* if $a_{\log}(X, B+M) ≥ 0$. Ditto for all the other definitions. Example: Fibrations and the canonical bundle formula {#ssec:gpfib} ---------------------------------------------------- We discuss a setting where generalised pairs appear naturally. Let $Y$ be a normal pair variety, and let $f : Y → X$ be a fibration: the space $X$ is projective, normal and of positive dimension, the morphism $f$ is surjective with connected fibres. Also, assume that $K_Y$ is $ℚ$-linearly equivalent to zero over $X$, so that there exists $L_X ∈ ℚ\operatorname{Div}(X)$ with $K_Y \sim_ℚ f^* L_X$. Ideally, one might hope that it would be possible to choose $L_X = K_X$, but this is almost always wrong — compare Kodaira’s formula for the canonical bundle of an elliptic fibration, [@HBPV Sect. V.12]. To fix this issue, we define a first correction term $B ∈ ℚ\operatorname{Div}(X)$ as $$B := \sum_{\mathclap{\substack{D ∈ \operatorname{Div}(X)\\\text{prime}}}} (1-t_D)·D \quad \text{where} \quad t_D := \operatorname{lct}° \bigl(Y,\, Δ_Y,\, f^*D \bigr)$$ The symbol $\operatorname{lct}°$ denotes a variant of the lc threshold introduced in Definition \[def:lct\], which measures the singularities of $\bigl(Y, f^*D \bigr)$ only over the generic point of $D$. Since $X$ is smooth in codimension one, this also solves the problem of defining $f^* D$. Finally, one chooses $M ∈ ℚ\operatorname{Div}(X)$ such that $K_X + B + M$ is $ℚ$-Cartier and such that the desired $ℚ$-linear equivalence holds, $$K_Y \sim_ℚ f^* ( K_X + B + M ).$$ The divisor $B$ is usually called the “discriminant part” of the correction term. It detects singularities of the fibration, such as multiple or otherwise singular fibres, over codimension one points of $X$. The divisor $M$ is called the “moduli part”. It is harder to describe. While we have defined it only up to $ℚ$-linear equivalence, a more involved construction can be used to define it as an honest divisor. Conjecturally, the moduli part carries information on the birational variation of the fibres of $f$, [@MR1646046]. We refer to [@MR2359346] and to the introduction of the recent research paper [@FL18] for an overview, but see also [@MR3329677]. ### Behaviour under birational modifications We ask how the moduli part of the correction term behaves under birational modification. To this end, let $φ : X' → X$ be a birational morphism of normal, projective varieties. Choosing a resolution $Y'$ of $Y ⨯_X X'$, we find a diagram as follows, $$\xymatrix{ Y' \ar[rr]^{Φ \text{, birational}} \ar[d]_{f'} && Y \ar[d]^f \\ X' \ar[rr]_{φ \text{, birational}} && X. }$$ Set $Δ_{Y'} := Φ^* K_Y - K_{Y'}$. Generalising the definition of $\operatorname{lct}°$ a little to allow for negative coefficients in $Δ_{Y'}$, one can then define $B'$ similarly to the construction above, $$B' := \sum_{\mathclap{\substack{D ∈ \operatorname{Div}(X')\\\text{prime}}}} (1-t'_D)·D \quad \text{where} \quad t'_D := \operatorname{lct}° \bigl(Y',\, Δ_{Y'},\, (f')^*D \bigr).$$ Finally, one may then choose $M' ∈ ℚ\operatorname{Div}(X')$ such that $$\begin{aligned} K_{Y'} + Δ_{Y'} & \sim_ℚ (f')^* (K_{X'} + B' + M' ), \\ K_{X'} + B' + M' & = φ^*(K_X+B+M)\end{aligned}$$ and $B = φ_* B'$ as well as $M = φ_* M'$. ### Relation to generalised pairs Now assume that $Y$ is lc. The divisor $B$ will then be effective. However, much more is true: after passing to a certain birational model $X'$ of $X$, the divisor $M_{X'}$ is nef and for any higher birational model $X'' → X'$, the induced $M_{X''}$ on $X''$ is the pullback of $M_{X''}$, [@MR1646046; @MR2153078; @MR2359346] and summarised in [@Bir16a Thm. 3.6]. In other words, going to a sufficiently high birational model of $X'$ of $X$, the moduli parts $M'$ define an b-$ℝ$-Cartier b-divisor. Moreover, this b-divisor is b-nef. We obtain a generalised polarised pair $(X, B+M)$ with data $X' \overset{φ}{→} X → \operatorname{Spec}ℂ$ and $M'$. This generalised pair is generalised lc by definition. A famous conjecture of Prokhorov and Shokurov [@MR2448282 Conj. 7.13] asserts that the moduli divisor $M_{X''}$ is semiample, on any sufficiently high birational model $X''$ of $X$. More precisely, it is expected that a number $m$ exists that depends only on the general fibre of $f$ such that all divisors $m·M_{X''}$ are basepoint free. If this conjecture was solved, it is conceivable that Birkar’s work could perhaps be rewritten in a manner that avoids the notion of generalised pairs. The construction outlined in this section is used in the proof of “Boundedness of complements”, as sketched in Section \[ssec:BC\] below. It generalises fairly directly to pairs $(Y, Δ_Y)$, and even to tuples where $Δ_Y$ is not necessarily effective, [@Bir16a Sect. 3.4]. Boundedness of complements {#sec:bcomp} ========================== Statement of result ------------------- One of the central concepts in Birkar’s papers [@Bir16a; @Bir16b] is that of a *complement*. The notion of a “complement” is an ingenious concept of Shokurov that was introduced in his investigation of threefold flips, [@zbMATH00146455 Sect. 5]. We recall the definition in brief. \[defn:comple\] Let $(X, B)$ be a projective pair and $m ∈ ℕ$. An *$m$-complement of $K_X + B$* is a $ℚ$-divisor $B^+$ with the following properties. 1. \[il:c1\] The tuple $(X, B^+)$ is an lc pair. 2. \[il:c2\] The divisor $m·(K_X + B^+)$ is linearly equivalent to $0$. In particular, $m·B^+$ is integral. 3. \[il:c3\] We have $m·B^+ ≥ m·⌊B⌋ +⌊(m+1)·\{B\}⌋$. \[rem:3-2\] Setting as in Definition \[defn:comple\]. If $m$ can be chosen such that $m·⌊B⌋ +⌊(m+1)·\{B\}⌋ ≥ m·B$, then Item \[il:c2\] guarantees that $-m·(K_X+B)$ is linearly equivalent to the effective divisor $m·(B^+-B)$. In particular, the sheaf $𝒪_X\bigl(-m·(K_X+B)\bigr)$ admits a global section. In view of Item \[il:c2\], Shokurov considers complements as divisors that make the lc pair $(X, B^+)$ “Calabi-Yau”, hence “flat”. The following result, which asserts the existence of complements with bounded $m$, is one of the core results in Birkar’s paper [@Bir16a]. A proof of Theorem \[thm:boundOfCompl\] is sketched in Section . \[thm:boundOfCompl\] Given $d ∈ ℕ$ and a finite set $\mathcal{R} ⊂ [0,1] ∩ ℚ$, there exists $m ∈ ℕ$ with the following property. If $(X,B)$ is any log canonical, projective pair, where 1. \[il:r1\] $X$ is of Fano type and $\dim X = d$, 2. the coefficients of $B$ are of the form $\frac{l-r}{l}$, for $r ∈ \mathcal{R}$ and $l ∈ ℕ$, 3. $-(K_X+B)$ is nef, then there exists an $m$-complement $B^+$ of $K_X+B$ that satisfies $B^+ ≥ B$. The divisor $B^+$ is also an $(m·l)$-complement, for every $l ∈ ℕ$. \[rem:3-4\] Given a pair $(X,B)$ as in Theorem \[thm:boundOfCompl\] and a number $l ∈ ℕ$ such that $(ml)·B$ is integral, then $ml·⌊B⌋ +⌊(ml+1)·\{B\}⌋ ≥ ml·B$, and Remark \[rem:3-2\] implies that $h⁰ \bigl( X,\, 𝒪_X(-ml·(K_X+B)) \bigr) > 0$. Idea of application ------------------- We aim to show Theorem \[thm:BAB\]: under suitable assumptions on the singularities the family of Fano varieties is bounded. The proof relies on the following boundedness criterion of Hacon and Xu that we quote without proof (but see Sections \[ssec:1-1-1v2\] and \[ssec:1-1-2v2\] for a brief discussion). Recall that a set of numbers is DCC if every strictly descending sequence of elements eventually terminates. \[thm:boundednessCriterion\] Given $d ∈ ℕ$ and a $\operatorname{DCC}$ set $I ⊂ [0,1] ∩ ℚ$, let $\mathcal{Y}_{d,I}$ be the family of pairs $(X,B)$ such that the following holds true. 1. The pair $(X, B)$ is projective, klt, and of dimension $\dim_ℂ X = d$. 2. The coefficients of $B$ are contained in $I$. The divisor $B$ is big and $K_X + B \sim_ℚ 0$. Then, the family $\mathcal{Y}_{d,I}$ is bounded. With the boundedness criterion in place, the following observation relates “boundedness of complements” to “boundedness of Fanos” and explains what pieces are missing in order to obtain a full proof. \[obs:1\] Given $d ∈ ℕ$ and $ε ∈ ℝ^+$, Theorem \[thm:boundOfCompl\] gives a number $m ∈ ℕ$ such that every $ε$-lc Fano variety $X$ with $-K_X$ nef admits an effective complement $B^+$ of $K_X = K_X+0$, with coefficients in the set $\{\frac{1}{m}, \frac{2}{m}, …, \frac{m}{m}\}$. If one could in addition always choose $B^+$ so that $(X,B^+)$ was klt rather than merely lc, then Theorem \[thm:boundednessCriterion\] would immediately apply to show that the family of $ε$-lc Fano varieties with $-K_X$ nef is bounded. As an important step towards boundedness of $ε$-lc Fanos, we will see in Section \[sec:EB\] how the theorem on “effective birationality” together with Theorem \[thm:boundednessCriterion\] and Observation \[obs:1\] can be used to find a boundedness criterion (=Proposition ) that applies to a relevant class of klt, weak Fano varieties. Variants and generalisations {#sec:vandg} ---------------------------- Theorem \[thm:boundOfCompl\] is in fact part of a much larger package, including boundedness of complements in the relative setting, [@Bir16a Thm. 1.8], and boundedness of complements for generalised polarised pairs, [@Bir16a Thm. 1.10]. To keep this survey reasonably short, we do not discuss these results here, even though they are of independent interest, and play a role in the proofs of Theorems \[thm:boundOfCompl\] and \[thm:BAB\]. Idea of proof for Theorem \[thm:boundOfCompl\] {#ssec:BC} ---------------------------------------------- We sketch a proof of “boundedness of complements”, following [@Bir16a p. 6ff] in broad strokes, and filling in some details now and then. In essence, the proof works by induction over the dimension, so assume that $d$ is given and that everything was already shown for varieties of lower dimension. ### Simplification {#simplification .unnumbered} Theorem \[thm:boundOfCompl\] considers a finite set ${\mathcal R}⊂ [0,1] ∩ ℚ$, and log canonical pairs $(X,B)$, where the coefficients of $B$ are contained in the set $$Φ({\mathcal R}) := \bigl\{ \textstyle{\frac{l-r}{l}} \,|\, r ∈ \mathcal{R} \text{ and } l ∈ ℕ \bigr\}.$$ The set $Φ({\mathcal R})$ is infinite, and has $1 ∈ ℚ$ as its only accumulation point. Birkar shows that it suffices to treat the case where the coefficient set is finite. To this end, he constructs in [@Bir16a Prop. 2.49 and Constr. 6.13] a number $ε' ≪ 1$ and shows that it suffices to consider pairs with coefficients in the finite set $Φ({\mathcal R}) ∩ [0,1-ε'] ∪ \{1\}$. In fact, given any $(X,B)$, he considers the divisor $B'$ obtained by replacing those coefficients on $B$ that lie in the range $(1-ε',1)$ with $1$. Next, he constructs a birational model $(X'', B'')$ of $(X, B')$ that satisfies all assumptions Theorem \[thm:boundOfCompl\]. His construction guarantees that to find an $n$-complement for $(X,B)$ it is equivalent to find an $n$-complement for $(X'', B'')$. Among other things, the proof involves carefully constructed runs of the Minimal Model Programme, Hacon-McKernan-Xu’s local and global ACC for log canonical thresholds [@MR3224718 Thms. 1.1 and 1.5], and the extension of these results to generalised pairs [@MR3502099 Thm. 1.5 and 1.6]. Recall from Remark \[rem:MoriDream\] that Assumption \[il:r1\] (“$X$ is of Fano type”) allows us to run Minimal Model Programmes on arbitrary divisors. Along similar lines, Birkar is able to modify $(X'', B'')$ by further birational transformation, and eventually proves that it suffices to show boundedness of complements for pairs that satisfy the following additional assumptions. The coefficient set of $(X, B)$ is contained in ${\mathcal R}$ rather than in $Φ({\mathcal R})$, and one of the following holds true. 1. \[il:sc1\] The divisor $-(K_X+B)$ is nef and big, and $B$ has a component $S$ with coefficient $1$ that is of Fano type. 2. \[il:sc2\] There exists a fibration $f \colon X → T$ and $K_X+B\equiv 0$ along that fibration. 3. \[il:sc3\] The pair $(X,B)$ is exceptional. The main distinction is between Case \[il:sc3\] and Case \[il:sc1\]. In fact, if $(X,B)$ is not exceptional, recall from Definition \[def:exceptional\] that there exists an effective $P ∈ ℝ\operatorname{Div}(X)$ such that $K_X + B + P \sim_{ℝ} 0$ and such that $(X, B+P)$ is *not* klt. This allows us to find a birational model whose boundary contains a divisor with multiplicity one. Case \[il:sc2\] comes up if the runs of the Minimal Model Programmes used in the construction of birational models terminates with a Kodaira fibre space. The three cases \[il:sc1\]–\[il:sc3\] require very different inductive treatments. ### Case \[il:sc1\] {#caseilsc1 .unnumbered} We consider only the simple case where $S = ⌊ B ⌋$ is a normal prime divisor, where $(X,B)$ is plt near $S$ and where $-(K_X+B)$ is ample. Setting $B_S := (K_X+B)|_S - K_S$, the coefficients are contained in a finite set ${\mathcal R}'$ of rational numbers that depends only on ${\mathcal R}$ and on $d$. In summary, the pair $(S, B_S)$ reproduces the assumptions of Theorem \[thm:boundOfCompl\], and by induction we obtain a number $n ∈ ℕ$ that depends only on ${\mathcal R}$ and $d$, such that 1. \[il:i1\] the divisor $n·B_S$ is integral, and 2. \[il:i2\] there exists an $n$-complement $B^+_S$ of $K_S+B_S$. Following [@Bir16a Prop. 6.7], we aim to extend $B^+_S$ from $S$ to a complement $B^+$ of $K_X+B$ on $X$. As we saw in in Remark \[rem:3-4\], Item \[il:i1\] guarantees that $n·(B^+_S-B_S)$ is effective, so that the complement $B^+_S$ gives rise to a section in $$H⁰ \bigl( S,\, n·(B^+_S-B_S) \bigr) = H⁰ \bigl( S,\, -n·(K_S+B_S) \bigr)$$ But then, looking at the cohomology of the standard ideal sheaf sequence, $$H⁰ \bigl( X,\, -n·(K_X+B) \bigr) → \underbrace{H⁰ \bigl( S,\, -n·(K_X+B)|_S \bigr)}_{\not = 0 \text{ by Rem.~\ref{rem:3-4}}} → \underbrace{H¹ \bigl( X,\, -n·(K_X+B) - S \bigr) }_{= 0 \text{ by Kawamata-Viehweg vanishing}}$$ we find that the section extends to $X$ and defines an associated divisor $B^+ ∈ \lvert-(K_X+B)\rvert_ℚ$. Using the connectedness principle for non-klt centres[^3], one argues that $B^+$ is the desired complement. ### Case \[il:sc2\] {#ssect:sc2 .unnumbered} Given a fibration $f: X → T$, we apply the construction of Section \[ssec:gpfib\], in order to equip the base variety $T$ with the structure of a generalised polarised pair $(T, B+M)$, with data $T' \overset{φ}{→} T → \operatorname{Spec}ℂ$ and $M'$. Adding to the results explained in Section \[ssec:gpfib\], Birkar shows that the coefficients of $B$ and $M$ are not arbitrary. The coefficients of $B$ are in $Φ({\mathcal S})$ for some fixed finite set ${\mathcal S}$ of rational numbers that depends only on ${\mathcal R}$ and $d$. Along similar lines, there exists a bounded number $p ∈ ℕ$ such that $p·M$ is integral. The plan is now to use induction to find a bounded complement for $K_T + B + M$ and pull it back to $X$. This plan works out well, but requires us to formulate and prove all results pertaining to boundedness of complements in the setting of generalised polarised pairs. All the arguments sketched here continue to work, *mutatis mutandis*, but the level of technical difficulty increases substantially. ### Case \[il:sc3\] {#caseilsc3 .unnumbered} There is little that we can say in brief about this case. Still, assume for simplicity that $B=0$ and that $X$ is a Fano variety. If we could show that $X$ belongs to a bounded family, then we would be done. Actually we need something weaker: effective birationality. Assume we have already proved Theorem \[thm:effBir\]. Then there is a bounded number $m ∈ ℕ$ such that $|-mK_X|$ defines a birational map. Pick $M∈ |-mK_X|$ and let $B^+ := \frac{1}{m}·M$. Since $X$ is exceptional, $(X, B^+)$ is automatically klt, hence $K_X+B^+$ is an $m$-complement. Although this gives some idea of how one may get a bounded complement but in practice we cannot give a complete proof of Theorem \[thm:effBir\] before proving Theorem \[thm:boundOfCompl\]. Contrary to the exposition of this survey paper, where “boundedness of complements” and “effective birationality” are treated as if they were separate, the proofs of the two theorems are in fact much intertwined, and this is one of the main points where they come together. Many of the results discussed in this overview (“Bound on anti-canonical volumes”, “Bound on lc thresholds”) have separate proofs in the exceptional case. Effective birationality {#sec:EB} ======================= Statement of result ------------------- The second main ingredient in Birkar’s proof of boundedness is the following result. A proof is sketched in Section . \[thm:effBir\] Given $d ∈ ℕ$ and $ε ∈ ℝ^+$, there exists $m ∈ ℕ$ with the following property. If $X$ is any $ε$-lc weak Fano variety of dimension $d$, then $\lvert -m·K_X \rvert$ defines a birational map. The divisors $m·K_X$ in Theorem \[thm:effBir\] need not be Cartier. The linear system $\lvert -m·K_X \rvert$ is the space of effective Weil divisors on $X$ that are linearly equivalent to $-m·K_X$. Idea of application {#ssec:EBA} ------------------- In the framework of [@Bir16a], effective birationality is used to improve the boundedness criterion spelled out in Theorem \[thm:boundednessCriterion\] above. \[prop:4.3a\] Let $d, v ∈ ℕ$ and let $(t_ℓ)_{ℓ ∈ ℕ}$ be a sequence of positive real numbers. Let $\mathcal{X}$ be the family of projective varieties $X$ with the following properties. 1. The variety $X$ is a klt weak Fano variety of dimension $d$. 2. \[il:x3\] The volume of the canonical class is bounded, $\operatorname{vol}(-K_X) ≤ v$. 3. \[il:x4\] For every $ℓ ∈ ℕ$ and every $L ∈ \lvert -ℓ·K_X \rvert$, the pair $(X, t_ℓ·L)$ is klt. Then, $\mathcal{X}$ is a bounded family. The formulation of Proposition \[prop:4.3a\] is meant to illustrate the application of Theorem \[thm:effBir\] to the boundedness problem. It is a simplified version of Birkar’s formulation and defies the logic of his work. While we present Proposition \[prop:4.3a\] as a corollary to Theorem \[thm:effBir\], and to all the results mentioned in Section \[sec:bcomp\], Birkar uses [@Bir16a Prop. 7.13] as one step in the inductive proof of “boundedness of complements” and “effective birationality”. That requires him to explicitly list partial cases of “boundedness of complements” and “effective birationality” as assumptions to the proposition, and makes the formulation more involved. Proposition \[prop:4.3a\] reduces the boundedness problem to solving the following two problems. - Boundedness of volumes, as required in \[il:x3\]. This is covered in the subsequent Section \[sec:volumes\]. - Existence of numbers $t_ℓ$, as required in \[il:x4\]. This amounts to bounding “lc thresholds” and is covered in Section \[sec:lcthres\]. To prove Proposition \[prop:4.3a\], Birkar uses effective birationality in the following form, as a log birational boundedness result. \[thm:lbb\] Given $d,v ∈ ℕ$ and $ε ∈ ℝ^+$. Then, there exists $c ∈ ℝ^+$ and a bounded family ${\mathcal P}$ of couples with the following property. If $X$ is a normal projective variety of dimension $d$ and if $B ∈ ℝ\operatorname{Div}(X)$ and $M ∈ ℚ\operatorname{Div}(X)$ are divisors such that the following holds, 1. the divisor $B$ is effective, with coefficients in $\{0\} ∪ [ε,∞)$, 2. the divisor $M$ is effective, nef and $|M|$ defines a birational map, 3. the difference $M-(K_X+B)$ is pseudo-effective, 4. the volume of $M$ is bounded, $\operatorname{vol}(M) < v$, 5. if $D$ is any component of $M$, then $\operatorname{mult}_D (B+M) ≥ 1$, then there exists a log smooth couple $(X', Σ) ∈ {\mathcal P}$, a rational map $\overline{X} \dasharrow X$ and a resolution of singularities $r : {\widetilde}{X} → X$, with the following properties. 1. \[il:y1\] The divisor $Σ$ contains the birational transform on $M$, as well as the exceptional divisor of the birational map $β$. 2. \[il:y2\] The movable part $A_{{\widetilde}{X}}$ of $r^* M$ is basepoint free. 3. \[il:y3\] If ${\widetilde}{X}'$ is any resolution of $X$ that factors via $X'$ and ${\widetilde}{X}$, $$\xymatrix{ {\widetilde}{X}' \ar[d]_{s\text{, resolution}} \ar[rr]^{{\widetilde}{β}\text{, birational}} && {\widetilde}{X} \ar[d]^{r\text{, resolution}} \\ X' \ar@{-->}[rr]_{β\text{, birational}} && X }$$ then the coefficients of the $ℚ$-divisor $s_* (r ◦ {\widetilde}{β})^* M$ are at most $c$ and ${\widetilde}{β}^* A_{{\widetilde}{X}}$ is linearly equivalent to zero relative to $X'$. Since $|M|$ defines a birational map, there exists a resolution $r : {\widetilde}{X} → X$ such that $r^* M$ decomposes as the sum of a base point free movable part $A_{{\widetilde}{X}}$ and fixed part $R_{{\widetilde}{X}}$. The contraction $X → X''$ defined by $A_{{\widetilde}{X}}$ is birational. Since $\operatorname{vol}(M)$ is bounded, the varieties $X''$ obtained in this way are all members of one bounded family ${\mathcal P}'$. The family ${\mathcal P}'$ is however not yet the desired family ${\mathcal P}$, and the varieties in ${\mathcal P}'$ are not yet equipped with an appropriate boundary. To this end, one needs to invoke a criterion of Hacon-McKernan-Xu for “log birationally boundedness”, [@MR3034294 Lem. 2.4.2(4)], and take an appropriate resolution of the elements in ${\mathcal P}'$. Applying Theorems \[thm:boundOfCompl\] (“Boundedness of complements”) and \[thm:effBir\] (“Effective birationality”), we find a number $m ∈ ℕ$ such that every $X ∈ \mathcal{X}$ admits an $m$-complement for $K_X$ and that $\lvert -m·K_X \rvert$ defines a birational map. If $m$-complements $B^+$ of $K_X$ could always be chosen such that $(X,B^+)$ were klt, we have seen in Observation \[obs:1\] that $\mathcal{X}$ is bounded. However, Theorems \[thm:boundOfCompl\] guarantees only the existence of an $m$-complement $B^+$ of $K_X$ where $(X,B^+)$ is lc. Using the bounded family ${\mathcal P}$ obtained when applying Proposition \[thm:lbb\] with $M = -m·K_X$ and $B = 0$, we aim to find a universal constant $ℓ$ and a finite set $\mathcal{R}$, and then perturb any given $(X,B^+)$ in order to find a boundary $B^{++}$ with coefficients in $\mathcal{R}$ that is $ℚ$-linearly equivalent to $-K_X$ and makes $(X, B^{++})$ klt. Boundedness will then again follow from Theorem \[thm:boundednessCriterion\]. To spell out a few more details of the proof use boundedness of the family ${\mathcal P}$ to infer the existence of a universal constant $ℓ$ with the following property. > If $(X', Σ) ∈ {\mathcal P}$ and if $A_{X'} ∈ \operatorname{Div}(X')$ is contained in $Σ$ with coefficients bounded by $c$, and if $|A_{X'}|$ is basepoint free and defines a birational morphism, then there exists $G_{X'} ∈ |ℓ·A_{X'}|$ whose support contains $Σ$. Now assume that one $X ∈ \mathcal{X}$ is given. It suffices to consider the case where $X$ is $ℚ$-factorial and admits an $m$-complement of the form $B^+ = \frac{1}{m}·M$, for general $M ∈ \lvert -m·K_X \rvert$. To make use of $ℓ$, consider a diagram as discussed in Item \[il:y3\] of Proposition \[thm:lbb\] above and decompose $r^*M = A_{{\widetilde}{X}} + R_{{\widetilde}{X}}$ into its moving and its fixed part. Write $A := r_* A_{{\widetilde}{X}}$ and $R := r_* R_{{\widetilde}{X}}$. Item \[il:y1\] of Proposition \[thm:lbb\] implies that the divisor $A_{X'} := s_* {\widetilde}{β}^* A_{{\widetilde}{X}}$ is then contained in $Σ$, and Item \[il:y3\] asserts that it is basepoint free, defines a birational morphism. So, we find $G_{X'} ∈ |ℓ·A_{X'}|$ as above. Writing $G := r_* {\widetilde}{β}_*s^* G_{X'}$, we find that $G+ℓ·R ∈ |- mℓ·K_X|$, so that $(X, t_{mℓ} G)$ is klt by assumption. We may assume that $t_{mℓ}$ is rational and $t_{mℓ} < \frac{1}{mℓ}$. If $(X, \frac{1}{mℓ}(G+ℓ·R))$ is lc, then set $B' := \frac{1}{mℓ}(G+ℓ·R)$. Otherwise, one needs to use the lower-dimensional versions of the variants and generalisations of boundedness of complements that we discussed in Section \[sec:vandg\] above. To be more precise, using 1. boundedness of complements for generalised polarised pairs for varieties of dimension $≤ d-1$, and 2. boundedness of complements in the relative setting for varieties of dimension $d$, one can always find a universal number $n$ and $B' ≥ t_{mℓ}·(G+ℓ·R)$ where $(X,B')$ is lc and $n·(K_X+B') \sim 0$. Finally, set $$B^{++} := \frac{1}{2}·B^+ + \frac{t}{2m}·A - \frac{t}{2mℓ}·G + \frac{1}{2}·B'$$ and then show by direct computation that all required properties hold. Preparation for the proof of Theorem \[thm:effBir\] --------------------------------------------------- We prepare for the proof with the following proposition. In essence, it asserts that effective divisors with “degree” bounded from above cannot have too small lc thresholds, under appropriate assumptions. Since this proposition may look plausible, we do not go into details of the proof. Further below, Proposition \[prop:lctbd\] gives a substantially stronger result whose proof is sketched in some detail. \[p-non-term-places\] Given $ε' ∈ ℝ^+$ and given a bounded family ${\mathcal P}$ of couples, there exists a number $δ ∈ ℝ^{>0}$ such that the following holds. Given the the following data, 1. an $ε'$-lc, projective pair $({\widehat}{G}, {\widehat}{B})$, 2. a reduced divisor $T ∈ \operatorname{Div}({\widehat}{G})$ such that $\bigl( {\widehat}{G}, \operatorname{supp}({\widehat}{B}+T) \bigr) ∈ {\mathcal P}$, and 3. an $ℝ$-divisor ${\widehat}{N}$ whose support is contained in $T$, and whose coefficients have absolute values $≤ δ$, then $({\widehat}{G}, {\widehat}{B}+{\widehat}{L})$ is klt, for all ${\widehat}{L} ∈ |{\widehat}{N}|_ℝ$. Sketch of proof of Theorem \[thm:effBir\] ----------------------------------------- Assume that numbers $d$ and $ε$ are given. Given an $ε$-lc Fano variety $X$ of dimension $d$, we will be interested in the following two main invariants, $$\begin{aligned} m_X & := \min \{ \: m' ∈ ℕ \,|\, \text{the linear system } \lvert -m'·K_X \rvert \text{ defines a birational map }\} \\ n_X & := \min \{ \: n' ∈ ℕ\; \,|\, \operatorname{vol}(-n'·K_X) ≥ (2d)^d \:\}\end{aligned}$$ Eventually, it will turn out that both numbers are bounded from above. Our aim here is to bound the numbers $m_X$ by a constant that depends only on $d$ and $ε$. Bounding the quotient {#bounding-the-quotient .unnumbered} --------------------- Following [@Bir16a], we will first find an upper bound for the quotients $m_X/n_X$ by a number that depends only on $d$ and $ε$. ### Construction of non-klt centres In the situation at hand, a standard method (“tie breaking”) allows us to find dominating families of non-klt centres; we refer to [@KollarSingsOfPairs Sect. 6] for an elementary discussion, but see also [@Bir16a Sect. 2.31]. Given an $ε$-lc Fano variety $X$ of dimension $d$, and using the assumption that $\operatorname{vol}(-n_X·K_X) ≥ (2d)^d$, the following has been shown by Hacon, McKernan and Xu. \[claim:dfnkc\] Given any $ε$-lc Fano variety $X$, there exists a dominating family ${\mathcal G}_X$ of subvarieties in $X$ with the following property. If $(x, y) ∈ X ⨯ X$ is any general tuple of points, then there exists a divisor $Δ ∈ |-(n_X+1)·K_X|_ℝ$ such that the following holds. 1. \[il:o1\] The pair $(X,Δ)$ is not klt at $y$. 2. \[il:o2\] The pair $(X,Δ)$ is lc near $x$ with a unique non-klt place. The associated non-klt centre is a subvariety of the family ${\mathcal G}_X$. Given $X$, we may assume that the members of the families ${\mathcal G}_X$ all have the same dimension, and that this dimension is minimal among all families of subvarieties that satisfy \[il:o1\] and \[il:o2\]. ### The case of isolated centres If $X$ is given such that the members of ${\mathcal G}_X$ are points, then the elements are isolated non-klt centres. Given $G ∈ {\mathcal G}_X$, standard vanishing theorems for multiplier ideals will then show surjectivity of the restriction maps $$H⁰ \bigl( X,\, 𝒪_X(K_X+Δ) \bigr) → \underbrace{H⁰ \bigl( G,\, 𝒪_X(K_X+Δ)|_G \bigr)}_{≅ ℂ}.$$ In particular, we find that $𝒪_X(K_X+Δ) ≅ 𝒪_X(-n_X·K_X)$ has non-trivial sections. Further investigation reveals that a bounded multiple of $-n_X·K_X$ will in fact give a birational map. ### Non-isolated centres It remains to consider varieties $X$ where the members of ${\mathcal G}_X$ are positive-dimensional. Following [@Bir16a proofs of Prop. 4.6 and 4.8], we trace the arguments for that case in *very* rough strokes, ignoring all of the (many) subtleties along the way. The main observation to handle this case is the following volume bound. \[claim:vB\] There exists a number $v ∈ ℝ^+$ that depends only on $d$ and $ε$, such that for all $X$ and all positive-dimensional $G ∈ {\mathcal G}_X$, we have $\operatorname{vol}(-m_X·K_X|_G) < v$. Going back and looking at the construction of non-klt centres (that is, the detailed proof of Claim \[claim:dfnkc\]), one finds that the construction can be improved to provide families of lower-dimension centres if only the volumes are big enough. But this collides with our assumption that the varieties in ${\mathcal G}_X$ were of minimal dimension.  (Claim \[claim:vB\]) To make use of Claim \[claim:vB\], look at one $X$ where the members of ${\mathcal G}_X$ are positive-dimensional. Choose a general divisor[^4] $M ∈ \lvert -m_X·K_X \rvert$, and let $(x,y) ∈ X ⨯ X$ be a general tuple of points with associated centre $G ∈ {\mathcal G}_X$. Since $G$ is a non-klt centre that has a unique place over it, adjunction (and inversion of adjunction) works rather well. Together with the bound on volumes, this allows us to define a natural boundary ${\widehat}{B}$ on a suitable birational modification ${\widehat}{G}$ of the normalisation of $G$, such that the following holds. 1. The pair $({\widehat}{G}, {\widehat}{B})$ is $ε'$-lc, for some controllable number $ε'$. 2. Writing $E$ for the exceptional divisor of ${\widehat}{G} → G$ and $T := ({\widehat}{B}+E)_{\operatorname{red}}$, the couple $\bigl({\widehat}{G}, \operatorname{supp}({\widehat}{B} + T) \bigr)$ belongs to a bounded family ${\mathcal P}$ that in turn depends only on the numbers $d$ and $ε$. 3. The pull-back of $M$ to ${\widehat}{G}$ has support in $\operatorname{supp}({\widehat}{B} + T)$. ### End of proof The idea now is of course to apply Proposition \[p-non-term-places\], using the family ${\mathcal P}$. Arguing by contradiction, we assume that the numbers $m_X/n_X$ are unbounded. We can then find one $X$ where $n_X/m_X$ is really quite small when compared to the number $δ$ given by Proposition \[p-non-term-places\]. In fact, taking ${\widehat}{N}$ as the pull-back of $\frac{n_X}{m_X}·M$, it is possible to guarantee that the coefficients of ${\widehat}{N}$ are smaller than $δ$. Intertwining this proof with the proof of “boundedness of complements”, we may use a partial result from that proof, and find $L ∈ |-n_X·K_X|_{ℚ}$, whose coefficients are $≥ 1$. Since the points $(x,y) ∈ X ⨯ X$ were chosen generically, the pull-back ${\widehat}{L}$ of $L$ to ${\widehat}{G}$ has coefficients $≥ 1$, and can therefore never appear in the boundary of a klt pair. But then, ${\widehat}{L} ∈ |{\widehat}{N}|_ℝ$, which contradicts Proposition \[p-non-term-places\] and ends the proof. In summary, we were able to bound the quotient $m_X/n_X$ by a constant that depends only on $d$ and $ε$. Bounding the numbers $m_X$ {#bounding-the-numbers-m_x .unnumbered} -------------------------- Finally, we still need to bound $m_X$. This can be done by arguing that the volumes $\operatorname{vol}(-m_X·K_X)$ are bounded from above, and then use the same set of ideas discussed above, using $X$ instead of a birational model ${\widehat}{G}$ of its subvariety $G$. Since some of the core ideas that go into boundedness of volumes are discussed in more detail in the following Section \[sec:volumes\] below, we do not go into any details here. Bounds for volumes {#sec:volumes} ================== Statement of result ------------------- Once Theorem \[thm:BAB\] (“Boundedness of Fanos”) is shown, the volumes of anticanonical divisors of $ε$-lc Fano varieties of any given dimension will clearly be bounded. Here, we discuss a weaker result, proving boundedness of volumes for Fanos of dimension $d$, assuming boundedness of Fanos in dimension $d-1$. \[thm:bvol\] Given $d ∈ ℕ$ and $ε ∈ ℝ^+$, if the $ε$-lc Fano varieties of dimension $d-1$ form a bounded family, then there is a number $v$ such that $\operatorname{vol}(-K_X) ≤ v$, for all $ε$-lc weak Fano varieties $X$ of dimension $d$ Idea of application {#idea-of-application-1} ------------------- We have seen in Section \[ssec:EBA\] how to obtain boundedness criteria for families of varieties from boundedness of volumes. This makes Theorem \[thm:bvol\] a key step in the inductive proof of Theorem \[thm:BAB\]. Idea of proof for boundedness of volumes, following [@Bir16a Sect. 9] --------------------------------------------------------------------- To illustrate the core idea of proof, we consider only the simplest cases and make numerous simplifying assumptions, no matter how unrealistic. The assumption that $ε$-lc Fano varieties of dimension $d-1$ form a bounded family will be used in the following form. \[lem:5-2\] There exists a finite set $I ⊂ ℝ$ with the following property. If $X$ is an $ε$-lc Fano variety of dimension $d-1$, if $r ∈ ℝ^{≥ 1}$ and if $D$ is any non-zero integral divisor on $X$ such that $K_X + r·D \equiv 0$, then $r ∈ I$. We argue by contradiction and assume that there exists a sequence $(X_i)_{i ∈ ℕ}$ of $ε$-lc weak Fanos of dimension $d$ such that the sequence of volumes is strictly increasing, with $\lim \operatorname{vol}(X_i) = ∞$. For simplicity of the argument, assume that all $X_i$ are Fanos rather than weak Fanos, and that they are $ℚ$-factorial. For the general case, one needs to consider the maps defined by multiples of $-K_X$ and take small $ℚ$-factorialisations. Choose a rational $ε'$ in the interval $(0,ε)$. Using explicit discrepancy computations of boundaries of the form $\frac{1}{N}·B'_i$, for $B'_i ∈ \lvert -N·K_{X_i} \rvert$ general, [@KM98 Cor. 2.32], we find a decreasing sequence $(a_i)_{i ∈ ℕ}$ of rationals, with $\lim a_i = 0$, and boundaries $B_i ∈ ℚ\operatorname{Div}(X_i)$ with the following properties. 1. For each $i$, the divisor $B_i$ is $ℚ$-linearly equivalent to $-a_i·K_{X_i}$. 2. The volumes of the $B_i$ are bounded from below, $(2d)^d < \operatorname{vol}(B_i)$. 3. \[il:5-1-3\] The pair $(X_i, B_i)$ has total log discrepancy equal to $ε'$. Passing to a subsequence, we may assume that $a_i < 1$ for every $i$. Again, discrepancy computation show that this allows us to find sufficiently general, ample $H_i ∈ ℚ\operatorname{Div}(X_i)$ that are $ℚ$-linearly equivalent to $-(1-a_i)·K_{X_i}$ and have the property that $(X, B_i+H_i)$ are still $ε'$-lc. Given any index $i$, Item \[il:5-1-3\] implies that there exists a prime divisor $D'_i$ on a birational model $X'_i$ that realises the total log discrepancy. For simplicity, consider only the case where one can choose $X_i = X'_i$ for every $i$, and therefore find prime divisors $D_i$ on $X_i$ that appear in $B_i$ with multiplicity $1-ε'$. Without that simplifying assumption one needs to invoke [@BCHM10 Cor. 1.4.3], in order to replace the variety $X_i$ by a model that “extracts” the divisor $D'_i$. In summary, we can write $$\label{eq:5-2-4} -K_{X_i} \sim_{ℚ} \frac{1}{a_i}·B_i = \frac{1-ε'}{a_i}·D_i + (\text{effective}).$$ As a next step, recall from Remark \[rem:MoriDream\] that the $X_i$ are Mori dream spaces. Given any $i$, we can therefore run the $-D_i$-MMP, which terminates with a Mori fibre space on which the push-forward of $D_i$ is relatively ample. Again, we ignore all technical difficulties and assume that $X_i$ itself is the Mori fibre space, and therefore admits a fibration $X_i → Z_i$ with relative Picard number $ρ(X_i/Z_i) = 1$ such that $D_i$ is relatively ample. Let $F_i ⊆ X_i$ be a general fibre. Adjunction and standard inequalities for discrepancies imply that $F_i$ is again $ε$-lc and Fano. The statement about the relative Picard number implies that any effective divisor on $X_i$ is either trivial or ample on $F_i$. In particular, Equation \[eq:5-2-4\] implies that $-K_{F_i} \equiv s_i·D_i$, where $s_i ≥ \frac{1-ε'}{a_i}$ goes to infinity. If $\dim F_i = d-1$, or more generally if $\dim F_i < d$ for infinitely many indices $i$, this contradicts Lemma \[lem:5-2\] and therefore proves Theorem \[thm:bvol\]. It remains to consider the case where the $Z_i$ are points. Birkar’s proof in this case is similar in spirit to the argumentation above, but technically much *more* demanding. He creates a covering family of non-klt centres, uses adjunction on these centres and the assumption that $ε$-lc Fano varieties of dimension $d-1$ form a bounded family to obtain a contradiction. Bounds for lc thresholds {#sec:lcthres} ======================== The last of Birkar’s core results presented here pertains to log canonical thresholds of anti-canonical systems; this is the main result of Birkar’s second paper [@Bir16b]. It gives a positive answer to a well-known conjecture of Ambro [@MR3556423 p. 4419]. With the notation introduced in Section \[sec:singofpairs\], the result is formulated as follows. \[thm:lct\] Given $d ∈ ℕ$ and $ε ∈ ℝ^+$, there exists $t ∈ ℝ^+$ with the following property. If $(X,B)$ is any projective $ε$-lc pair of dimension $d$ and if $Δ := -(K_X+B)$ is nef and big, then $\operatorname{lct}\bigl(X,\, B,\, |Δ|_ℝ \bigr) ≥ t$. Though this is not exactly obvious, Theorem \[thm:lct\] can be derived from boundedness of $ε$-lc Fanos, Theorem \[thm:BAB\]. One of the core ideas in Birkar’s paper [@Bir16b] is to go the other way and prove Theorem \[thm:lct\] using boundedness, but only for *toric* Fano varieties, where the result has been established by Borisov-Borisov in [@MR1166957]. Idea of application {#idea-of-application-2} ------------------- As pointed out in Section \[ssec:EBA\], bounding lc thresholds from below immediately applies to the boundedness problem. To illustration the application, consider the following corollary, which proves Theorem \[thm:BAB\] in part. \[cor:bab\] Given $d ∈ ℕ$ and $ε ∈ ℝ^+$, the family $\mathcal{X}^{\operatorname{Fano}}_{d,ε}$ of $ε$-lc Fanos of dimension $d$ is bounded. We aim to apply Proposition \[prop:4.3a\] to the family $\mathcal{X}^{\operatorname{Fano}}_{d,ε}$. With Theorem \[thm:bvol\] (“Bound on volumes”) in place, it remains to satisfy Condition \[il:x4\] of Proposition \[prop:4.3a\]: we need a sequence $(t_{ℓ})_{ℓ ∈ ℕ}$ such that the following holds. > For every $ℓ ∈ ℕ$, for every $X ∈ \mathcal{X}^{\operatorname{Fano}}_{d,ε}$ and every $L ∈ \lvert -ℓ·K_X \rvert$, the pair $(X, t_ℓ·L)$ is klt. But this is not so hard anymore. Let $t ∈ ℝ^+$ be the number obtained by applying Theorem \[thm:lct\]. Given a number $ℓ ∈ ℕ$, a variety $X ∈ \mathcal{X}^{\operatorname{Fano}}_{d,ε}$ and a divisor $L ∈ |-ℓ·K_X|$, observe that $\frac{1}{ℓ}·L ∈ |-K_X|_{ℝ}$ and recall from Remark  that $(X, \frac{t}{2ℓ}·L)$ is klt. We can thus set $t_{ℓ} := \frac{t}{2ℓ}$. Preparation for the proof of Theorem \[thm:lct\]: $ℝ$-linear systems of bounded degrees {#ssec:lctbd} --------------------------------------------------------------------------------------- To prepare for the proof of Theorem \[thm:lct\], we begin with a seemingly weaker result that provides bounds for lc thresholds, but only for $ℝ$-linear systems of bounded degrees. This result will be used in Section \[ssec:soft\] to prove Theorem \[thm:lct\] in an inductive manner. \[prop:lctbd\] Given $d$, $r ∈ ℕ$ and $ε ∈ ℝ^+$, there exists $t ∈ ℝ^+$ with the following property. If $(X,B)$ is any projective, $ε$-lc pair of dimension $d$, if $A ∈ \operatorname{Div}(X)$ is very ample with $A-B$ ample and $[A]^d ≤ r$, then $\operatorname{lct}\bigl(X,\, B,\, |A|_ℝ \bigr) ≥ t$. \[rem:6-4\] The condition on the intersection number, $[A]^d ≤ r$ implies that $X$ belongs to a bounded family of varieties. More generally, if we choose $A$ general in its linear system, then $(X,A)$ belongs to a bounded family of pairs. The proof of Proposition \[prop:lctbd\] is sketched below. It relies on two core ingredients. Because of their independent interest, we formulate them separately. \[set:6-5\] Given $d$, $r ∈ ℕ$ and $ε ∈ ℝ^+$, we consider projective, $ε$-lc pairs $(X,B)$ of dimension $d$ where $X$ is $ℚ$-factorial, equipped with the following additional data. 1. A very ample divisor $A ∈ \operatorname{Div}(X)$, with $A-B$ ample and $[A]^d ≤ r$. 2. An effective divisor $L ∈ ℝ\operatorname{Div}(X)$, with $A-L$ ample. 3. A birational morphism $ν : Y → X$ of normal projective varieties, and a prime divisor $T ∈ \operatorname{Div}(Y)$ whose image is a point $x ∈ X$. \[lem:A6-6\] Given $d$, $r ∈ ℕ$ and $ε ∈ ℝ^+$, assume that Proposition \[prop:lctbd\] holds for varieties of dimension $d-1$. Then, there exist integers $n$, $m ∈ ℕ$ and a real number $0 < ε' < ε$, with the following property. Whenever we are in Setting \[set:6-5\], and whenever there exists a number $t < r$ such that 1. the pair $(X, B+t·L)$ is $ε'$-lc, and 2. the log discrepancy is realised by $T$, that is $a_{\log}(T,X,B+t·L) = ε'$, Then there exists an effective divisor $Λ ∈ ℚ\operatorname{Div}(X)$ such that 1. the divisor $n·Λ$ is integral, 2. the tuple $(X,Λ)$ is lc near $x$, and $T$ is an lc place of $(X,Λ)$, and 3. the divisor $m·A-Λ$ is ample. Lemma \[lem:A6-6\] is another existence-and-boundedness result for complements, very much in the spirit of what we have seen in Section \[sec:bcomp\]. The relation to complements is made precise in [@Bir16b Thm. 1.7], which is a core ingredient in Birkar’s proof. In fact, after some birational modification of $Y$, Birkar finds a divisor $Λ_Y ∈ \operatorname{Div}(Y)$ such that $(Y, Λ_Y)$ is lc near $T$ and such that $n·(K_Y + Λ_Y)$ is linearly equivalent to $0$, relative to $X$ and for some bounded number $n ∈ ℕ$. As Birkar points out in [@Bir18 p. 16], one can think of $K_Y + Λ_Y$ as a local-global type of complement. He then takes $Λ$ to be the push-forward of $Λ_Y$ and proves all required properties. \[lem:A6-7\] Given $d$, $r$ and $n ∈ ℕ$ and $ε ∈ ℝ^+$, assume that Proposition \[prop:lctbd\] holds for varieties of dimension $\le d-1$. Then, there exists $q ∈ ℝ^+$, with the following property. Whenever we are in Setting \[set:6-5\], whenever $a(T,X,B)\le 1$, and whenever a divisor $Λ ∈ ℚ\operatorname{Div}(X)$ is given that satisfies the following conditions, 1. $Λ$ is effective and $n·Λ$ is integral, 2. $A-Λ$ is ample, 3. $(X, Λ)$ is lc near $x$, and $T$ is an lc place of $(X,Λ)$, then $T$ appears in the divisor $ν^* L$ with multiplicity $\operatorname{mult}_T ν^*L \le q$. Lemma \[lem:A6-7\] is perhaps the core of Birkar’s paper [@Bir16b]. To begin, one needs to realise that the couples $\bigl( X, \operatorname{supp}(Λ) \bigr)$ that appear in Lemma \[lem:A6-7\] come from a bounded family. This allows us to consider common resolution, and eventually to assume from the outset that $(X, Λ)$ is a log-smooth couple. In particular, $(X, Λ)$ is toroidal, and $T$ can be obtained by a sequence of blowing ups that are toroidal with respect to $(X , Λ)$. Given that toroidal blow-ups are rather well understood, Birkar finds that to bound the multiplicity $\operatorname{mult}_T ν^*L$, it suffices to bound the number of blowups involved. Bounding the number of blowups is hard, and the next few sentences simplify a very complicated argument to the extreme[^5]. Birkar establishes a Noether-normalisation theorem, showing that he may replace the couple $(X, Λ)$, which is log-smooth, by a pair of the form $(ℙ^d, \text{union of hyperplanes})$, which is toric rather than toroidal. Better still, applying surgery coming from the Minimal Model Programme, he is then able to replace $Y$ by a toric, Fano, $ε$-lc variety. But the family of such $Y$ is bounded by the classic result of Borisov-Borisov, [@MR1166957], and a bound for the number of blowups follows. The proof of Proposition \[prop:lctbd\] proceeds by induction, so assume that $d$, $r$, and $ε$ are given and that everything was already shown in lower dimensions. Now, given a $d$-dimensional pair $(X, B)$ and a very ample $A ∈ \operatorname{Div}(X)$ as in Proposition \[prop:lctbd\], we aim to apply Lemma \[lem:A6-6\] and \[lem:A6-7\]. This is, however, not immediately possible because $X$ need not be $ℚ$-factorial. We know from minimal model theory that there exists a small $ℚ$-factorialisation, say $X' → X$, but then we need to compare lc thresholds of $X'$ and $X$, and show that the difference is bounded. To this end, recall from Remark \[rem:6-4\] that the family of all possible $X$ is bounded, which allows us to construct simultaneous $ℚ$-factorialisations in stratified families, and hence gives the desired bound for the differences. Bottom line: we may assume that $X$ is $ℚ$-factorial. Let $ε'$ be the number given by Lemma \[lem:A6-6\]. Next, given any divisor $L ∈ |A|_ℝ$, look at $$s := \sup \{ s' ∈ ℝ \,|\, (X, B+s'·L) \text{ is $ε'$-lc} \}.$$ Following Remark \[rem:sflct\], we would be done if we could bound $s$ from below, independently of $X$, $B$, $A$ and $L$. To this end, choose a resolution of singularities, $ν : Y → X$ and a prime divisor $T ∈ \operatorname{Div}(Y)$ such that $a_{\log}(T,X,B+s·L) = ε'$. For simplicity, we will only consider the case where $ν(T)$ is a point, say $x ∈ X$ — if $ν(T)$ is not a point, Birkar cuts down with general hyperplanes from $|A|$, uses inversion of adjunction and invokes the induction hypothesis in order to proceed. In summary, we are now in a situation where we may apply Lemma \[lem:A6-6\] (“Existence of complements”) to find a divisor $Λ$ and then Lemma \[lem:A6-7\] (“Bound on multiplicity at an lc place”) to bound the multiplicity $\operatorname{mult}_T ν^*L$ from above, independently of $X$, $B$, $A$ and $L$. But then, a look at Definition \[not:logdiscrep\] (“log discrepancy”) shows that this already gives the desired bound on $s$. Preparation for the proof of Theorem \[thm:lct\]: varieties of Picard-number one -------------------------------------------------------------------------------- The second main ingredient in the proof of Theorem \[thm:lct\] is the following result, which essentially proves Theorem \[thm:lct\] in one special case. Its proof, which we do not cover in detail, combines all results discussed in the previous Sections \[sec:bcomp\]–\[sec:volumes\]: boundedness of complements, effective birationality and bounds for volumes. \[p-bnd-lct-global-weak\] Given $d ∈ ℕ$ and $ε ∈ ℝ^+$, assume that Proposition \[prop:lctbd\] (“LC thresholds for $ℝ$-linear systems of bounded degrees”) holds in dimension $≤ d$ and that Theorem \[thm:BAB\] (“Boundedness of $ε$-lc Fanos”) holds in dimension $≤ d-1$. Then, there exists $v ∈ ℝ^+$ such that the following holds. If $X$ is any $ℚ$-factorial, $ε$-lc Fano variety of dimension $d$ of Picard number one, and if $L ∈ ℝ\operatorname{Div}(X)$ is effective with $L \sim_{ℝ} -K_X$, then each coefficient of $L$ is less than or equal to $v$. Sketch of proof of Theorem \[thm:lct\] {#ssec:soft} -------------------------------------- Like other statements, Theorem \[thm:lct\] is shown using induction over the dimension. The following key lemma provides the induction step. \[l-local-lct-bab-to-global-lct\] Given $d ∈ ℕ$, assume that Proposition \[prop:lctbd\] (“LC thresholds for $ℝ$-linear systems of bounded degrees”) holds in dimension $≤ d$ and that Theorem \[thm:BAB\] (“Boundedness of $ε$-lc Fanos”) holds in dimension $≤ d-1$. Then, Theorem \[thm:lct\] (“Lower bound for lc thresholds”) holds in dimension $d$. The first steps in the proof are similar to the proof of Proposition \[prop:lctbd\]. Choose any number $ε'∈ (0,ε)$. Given any projective, $d$-dimension, $ε$-lc pair $(X,B)$ be as in Theorem \[thm:lct\] in dimension $d$ and any divisor $L ∈ |Δ|_{ℝ}$, let $s$ be the largest number such that $(X,B+s·L)$ is $ε'$-lc. We need to show $s$ is bounded from below away from zero. In particular, we may assume that $s<1$. As in the proof of Proposition \[prop:lctbd\], we may also assume $X$ is $ℚ$-factorial. There is a birational modification $φ : Y → X$ and a prime divisor $T ∈ \operatorname{Div}(Y)$ with log discrepancy $$\label{eq:ghjfg} a_{\log}(T,X,B+s·L)=ε'.$$ Techniques of [@BCHM10] (“extracting a divisor”) allow us to assume that $φ$ is either the identity, or that the $φ$-exceptional set equals $T$ precisely. The assumption that $X$ is $ℚ$-factorial allows us to pull back divisors. Let $$B_Y := φ^*(K_X+B)-K_Y \quad\text{and}\quad L_Y := φ^*L.$$ Using the definition of log discrepancy, Definition , the assumption that $(X,B)$ is $ε$-lc and Equation  are formulated in terms of divisor multiplicities as $$\operatorname{mult}_T B_Y ≤ 1-ε \quad\text{and}\quad \operatorname{mult}_T(B_Y+s·L_Y) = 1-ε',$$ hence $\operatorname{mult}_T (s·L_Y) ≥ ε-ε'$. The pair $(Y, B_Y+ s·L_Y)$ is klt and weak log Fano, which implies that $Y$ is Fano type. Recalling from Remark  that $Y$ is thus a Mori dream space, we may run a $(-T)$-Minimal Model Programme and obtain rational maps, $$\xymatrix{ Y \ar@{-->}[rrrr]^{α\text{, extr.\ contractions and flips}} &&&& Y' \ar[rrrr]^{β\text{, Mori fibre space}} &&&& Z', }$$ where $-T$ is ample when restricted to general fibres of $β$. We write $B_{Y'} := α_* B_Y$ and $L_{Y'} := α_* L_Y$ and note that $$-(K_{Y'}+B_{Y'}+s·L_{Y'}) \overset{\text{def.\ of $L$}}{\sim_ℝ} (1-s)L_{Y'} \overset{s < 1}{≥} 0.$$ Moreover, an explicit discrepancy computation along the lines of [@KM98 Cor. 2.32] shows that $(Y', B_{Y'}+s·L_{Y'})$ is $ε'$-lc, because $(Y,B_Y+s·L_Y)$ is $ε'$-lc and because $-(K_Y+B_Y+s·L_Y)$ is semiample. There are two cases now. If $\dim Z' > 0$, then restricting to a general fibre of $Y' → Z'$ and applying Proposition \[prop:lctbd\] (“LC thresholds for $ℝ$-linear systems of bounded degrees”) in lower dimension[^6] shows that the coefficients of those components of $(1-s)·L_{Y'}$ that dominate $Z'$ components of are bounded from above. In particular, $\operatorname{mult}_{T'}(1-s)·L_{Y'}$ is bounded from above. Thus from the inequality $$\operatorname{mult}_{T'}(1-s)·L_{Y'} ≥ \frac{(1-s)·(ε-ε')}{s},$$ we deduce that $s$ is bounded from below away from zero. If $Z'$ is a point, then $Y'$ is a Fano variety with Picard number one. Now $$-K_{Y'} \sim_ℝ (1-s)·L_{Y'} + B_{Y'} + s·L_{Y'}≥ (1-s)·L_{Y'},$$ so by Proposition \[p-bnd-lct-global-weak\], $\operatorname{mult}_{T'}(1-s)·L_{Y'}$ is bounded from above which again gives a lower bound for $s$ as before. Application to the Jordan property {#sec:jordan} ================================== We explain in this section how the boundedness result for Fano varieties applies to the study of birational automorphism groups, and how it can be used to prove the Jordan property. Several of the core ideas presented here go back to work of Serre, who solved the two dimensional case, [@MR2567402 Thm. 5.3] but see also [@MR2648675 Thm. 3.1]. If one is only interested in the three-dimensional case, where birational geometry is particularly well-understood, most arguments presented here can be simplified. Existence of subgroups with fixed points ---------------------------------------- If $X$ is any rationally connected variety, Theorem \[thm:jordan\] (“Jordan property of Cremona groups”) asks for the existence of finite Abelian groups in the Cremona groups $\operatorname{Bir}(X)$. As we will see in the proof, this is almost equivalent to asking for finite groups of automorphisms that admit fixed points, and boundedness of Fanos is the key tool used to find such groups. The following lemma is the simplest result in this direction. Here, boundedness enters in a particularly transparent way. \[lem:j1\] Given $d ∈ ℕ$, there exists a number $j^{\operatorname{Fano}}_d ∈ ℕ$ such that for any $d$-dimensional Fano variety $X$ with canonical singularities and any finite subgroup $G ⊆ \operatorname{Aut}(X)$, there exists a subgroup $F ⊆ G$ of index $|G:F| ≤ j^{\operatorname{Fano}}_d$ acting on $X$ with a fixed point. To keep notation simple, Lemma \[lem:j1\] is formulated for Fanos with canonical singularities, which is the relevant case for our application. In fact, it suffices to consider Fanos that are $ε$-lc. As before, write $\mathcal{X}^{\operatorname{Fano}}_{d,0}$ for the $d$-dimensional Fano variety $X$ with canonical singularities. It follows from boundedness, Theorem \[thm:BAB\] or Corollary \[cor:bab\], that there exist numbers $m$, $v ∈ ℕ$ such that the following holds for every $X ∈ \mathcal{X}^{\operatorname{Fano}}_{d,0}$. 1. The divisor $-m·K_X$ is Cartier and very ample. 2. The self-intersection number of $-m·K_X$ is bounded by $v$. More precisely, $$-[m·K_X]^{d} ≤ v.$$ Given $X$, observe that the associated line bundles $𝒪_X(-m·K_X)$ are $\operatorname{Aut}(X)$-linearised. Accordingly, there exists a number $N ∈ ℕ$, such that every $X ∈ \mathcal{X}^{\operatorname{Fano}}_{d,0}$ admits an $\operatorname{Aut}(X)$-equivariant embedding $X ↪ ℙ^N$. Let $j^{\operatorname{Jordan}}_{N+1}$ be the number obtained by applying the classical result of Jordan, Theorem \[thm:jordan-lin\], to $\operatorname{GL}_{N+1}(ℂ)$, and set $j^{\operatorname{Fano}}_d := j^{\operatorname{Jordan}}_{N+1}·v$. Now, given any $X ∈ \mathcal{X}^{\operatorname{Fano}}_{d,0}$ and any finite subgroup $G ⊆ \operatorname{Aut}(X)$, the $G$ action extends to $ℙ^N$. The action is thus induced by a representation of a finite linear group $Γ$, say $$\xymatrix{ Γ \ar@{^(->}[r] \ar@{->>}[d] & \operatorname{GL}_{N+1}(ℂ) \ar@{->>}[d] \\ G \ar@{^(->}[r] & \operatorname{ℙ\operatorname{GL}}_{N+1}(ℂ). \\ }$$ By Theorem \[thm:jordan-lin\], the classic result of Jordan, we find a finite Abelian subgroup $Φ ⊆ Γ$ of index $|Φ:Γ| ≤ j^{\operatorname{Jordan}}_{N+1}$. Since $Φ$ is Abelian, the $Φ$-representation space $ℂ^{N+1}$ is a direct sum of one-dimensional representations. Equivalently, we find $N+1$ linearly independent, $Φ$-invariant, linear hyperplanes $H_i ⊂ ℙ^{N+1}$. The intersection of suitably chosen $H_i$ with $X$ is then a finite, $Φ$-invariant subset $\{x_1, …, x_r\} ⊂ X$, of cardinality $r ≤ v$. The stabiliser of $x_1 ∈ X$ is a subgroup $Φ_{x_1} ⊂ Φ$ of index $|Φ:Φ_{x_1}| ≤ v$. Taking $F$ as the image of $Φ_{x_1} → G$, we obtain the claim. The proof of Lemma \[lem:j1\] shows that the groups $G$ are close to Abelian. It also gives an estimate for $j^{\operatorname{Fano}}_d$ in terms of the volume bound (“$v$”) and the classical Jordan constant $j^{\operatorname{Fano}}_d$. As a next step, we aim to generalise the results of Lemma \[lem:j1\] to varieties that are rationally connected, but not necessarily Fano. The following result makes this possible. \[lem:xfgs\] Let $X$ be a projective variety with an action of a finite group $G$. Suppose that $X$ is klt, with $Gℚ$-factorial singularities and let $f : X \dasharrow Y$ be a birational map obtained by running a $G$-Minimal Model Programs. Suppose that there exists a subgroup $F ⊂ G$ and an $F$-invariant, rationally connected subvariety $T ⊊ Y$. Then, there exists an $F$-invariant rationally connected subvariety $Z ⊊ X$. Since we are mainly interested to see how boundedness applies to birational transformation groups, we will not explain the proof of Lemma \[lem:xfgs\] in detail. Instead, we merely list a few of the core ingredients, which all come from minimal model theory and birational geometry. - Hacon-McKernan’s solution [@HMcK07] to Shokurov’s “rational connectedness conjecture”, which guarantees in essence that the fibres of all morphisms appearing in the MMP are rationally chain connected. - A fundamental result of Graber-Harris-Starr, [@GHS03], which implies that if $f : X → Y$ is any dominant morphism of proper varieties, where both the target $Y$ and a general fibre is rationally connected, then $X$ is also rationally connected. - Log-canonical centre techniques, in particular a relative version of Kawamata’s subadjunction formula, [@MR3483470 Lem. 2.5]. These results identify general fibres of minimal log-canonical centres under contraction morphisms as rationally connected varieties of Fano type. \[prop:j2\] Given $d ∈ ℕ$, there exists a number $j^{rc}_d ∈ ℕ$ such that for any $d$-dimensional, rationally connected projective variety $X$ and any finite subgroup $G ⊆ \operatorname{Aut}(X)$, there exists a subgroup $F ⊆ G$ of index $|G:F| ≤ j^{rc}_d$ acting on $X$ with a fixed point. We argue by induction on the dimension. Since the case $d=1$ is trivial, assume that $d > 1$ is given, and that numbers $j^{rc}_1, …, j^{rc}_{d-1}$ have been found. Set $$j_d := \max \{j^{rc}_1, …, j^{rc}_{d-1}, j^{\operatorname{Fano}}_d \} \quad\text{and}\quad j^{rc}_d := (j_d)².$$ Assume that a $d$-dimensional, rationally connected projective variety $X$ and a finite subgroup $G ⊆ \operatorname{Aut}(X)$ are given. By induction hypothesis, it suffices to find a subgroup $G' ⊆ G$ of index $|G:G'| ≤ j_d$ and a $G'$-invariant, rationally connected, proper subvariety $X' ⊊ X$. If ${\widetilde}{X} → X$ is the canonical resolution of singularities, as in [@BM96], then ${\widetilde}{X}$ is likewise rationally connected, $G$ acts on ${\widetilde}{X}$ and the resolution morphism is equivariant. Since images of rationally connected, invariant subvarieties are rationally connected and invariant, we may assume from the outset that $X$ is smooth. But then we can run a $G$-equivariant Minimal Model Programme[^7] terminating with a $G$-Mori fibre space, $$\xymatrix{ X \ar@{-->}[rrr]^{G\text{-equivariant MMP}} &&& X' \ar[rrr]^{G\text{-Mori fibre space}} &&& Y. }$$ In the situation at hand, Lemma \[lem:xfgs\] claims that to find proper, invariant, rationally connected varieties on $X$, it is equivalent to find them on $X'$. The fibre structure, however, makes that feasible. Indeed, if the base $Y$ of the fibration happens to be a point, then $X'$ is Fano with terminal singularities, and Lemma \[lem:j1\] applies. Otherwise, let $G_Y$ be the image of $G$ in $\operatorname{Aut}(Y)$, let $G_{X'/Y} ⊆ G$ be the ineffectivity of the $G$-action on $Y$, and consider the exact sequence $$1 → G_{X'/Y} → G → G_Y → 1.$$ As the image of the rationally connected variety $X'$, the base $Y$ is itself rationally connected. By induction hypothesis, using that $\dim Y < \dim X$, there exists a subgroup $F'_Y ⊆ G_Y$ of index $|G_Y:F'_Y| < j_d$ that acts on $Y$ with a fixed point, say $y ∈ Y$. Let $G' ⊂ G$ be the preimage of $G'_Y$. The fibre $X_y$ is then invariant with respect to the action of $G'$ and rationally chain connected by [@HMcK07 Cor. 1.3]. Better still, Prokhorov and Shramov show that it contains a rationally connected, $G'$-invariant subvariety. The induction applies. Proof of Theorem \[thm:jordan\] (“Jordan property of Cremona groups”) {#ssec:prof} --------------------------------------------------------------------- Given a number $d ∈ ℕ$, we claim that the number $j := j^{rc}_d·j^{\operatorname{Jordan}}_d$ will work for us, where $j^{rc}_d$ is the number found in Proposition \[prop:j2\], and $j^{\operatorname{Jordan}}_d$ comes from Jordan’s Theorem \[thm:jordan-lin\]. To this end, let $X$ be any rationally connected variety of dimension $d$, and let $G ⊆ \operatorname{Bir}(X)$ be any finite group. Blowing up the indeterminacy loci of the birational transformations $g ∈ G$ in an appropriate manner, we find a birational, $G$-equivariant morphism ${\widetilde}{X} → X$ where the action of $G$ in ${\widetilde}{X}$ is regular rather than merely birational, see [@MR0337963 Thm. 3]. Combining with the canonical resolution of singularities, we may assume that ${\widetilde}{X}$ is smooth. Proposition \[prop:j2\] will then guarantee the existence of a subgroup $G' ⊆ G$ of index $|G:G'| ≤ j^{rc}_d$ acting on ${\widetilde}{X}$ with a fixed point ${\widetilde}{x}$. Standard arguments (“linearisation at a fixed point”) that go back to Minkowski show that the induced action of $G'$ on the Zariski tangent space $T_{{\widetilde}{x}}({\widetilde}{X})$ is faithful, so that Jordan’s Theorem \[thm:jordan-lin\] applies. In fact, assuming that there exists an element $g ∈ G' ∖ \{e\}$ with $Dg|_{{\widetilde}{x}} = \operatorname{Id}_{T_{{\widetilde}{x}}({\widetilde}{X})}$, choose coordinates and use a Taylor series expansion to write $$g(\vec{x}) = \vec{x} + A_k(\vec{x}) + A_{k+1}(\vec{x}) + …$$ where each $A_m(\vec{x})$ is homogeneous of degree $m$, and $A_k$ is non-zero. Given any number $n$, observe that $$g^n(\vec{x}) = \vec{x} + n·A_k(\vec{x}) + \text{(higher order terms)}.$$ Since the base field has characteristic zero, this contradicts the finite order of $g$. 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Available at <http://www.mi.ras.ru/~prokhoro/preprints/edi.pdf>. Serre’s question is found on page 7. [^2]: The papers [@Bir16a; @Bir16b; @BCHM10] denote the log discrepancy by $a(D, X, B)$, while the standard reference books [@KM98; @MR3057950] write $a(D, X, B)$ for the standard (= “non-log”) discrepancies. [^3]: For generalised pairs, this is [@Bir16a Lem. 2.14] [^4]: the divisor $M$ should really be taken as the movable part, but we ignore this detail. [^5]: see [@Bir18 p. 16f] and [@YXu18 Sect. 10] for a more realistic account of all that is involved. [^6]: or applying Theorem \[thm:BAB\] (“Boundedness of $ε$-lc Fanos”) [^7]: The existence of an MMP terminating with a fibre space is [@BCHM10 Cor. 1.3.3], which we have quoted before. The fact that the MMP can be chosen in an equivariant manner is not explicitly stated there, but follows without much pain.

--- abstract: 'In organic bulk heterojunction solar cells, the donor/acceptor interfacial energy offset ($\Delta E$) is found to provide the driving force for efficient charge separation which gives rise to high short circuit current density ($J_\mathrm{sc}$), but a high $\Delta E$ inevitably undermines the open circuit voltage ($V_\mathrm{oc}$). In this paper, employing the device model method we calculated the steady state current density-voltage ($J-V$) and the $J_\mathrm{sc}-\Delta E$ curves under two different charge separation mechanisms to investigate the optimum driving force required for achieving sizable $V_\mathrm{oc}$ and $J_\mathrm{sc}$ simultaneously. Under the Marcus charge transfer mechanism, with the increased $\Delta E$ the Jsc increases rapidly for $\Delta E\leq 0.2$ eV, and then maintains a nearly constant value before decreasing at the Marcus inverted region, which is due to the accumulation of undissociated excitons within their lifetime and is beneficial for obtaining a sizable $J_\mathrm{sc}$ under a $\Delta E$ much smaller than the reorganization energy $\lambda$. With inclusion of both the electron and the hole transfer pathways of different respective $\lambda$’s into the device model, the experimentally measured $J-V$ curves for donor/acceptor blend with different $\Delta E$’s can be reproduced. For the coherent charge transfer mechanism in which the driving force act as the energy window of accessible charge separated states, with two typical types of density of states for the charge transfer excitons, it is shown that the highest $J_\mathrm{sc}$ can also be achieved under a small $\Delta E$ of 0.2eV if the high-lying delocalized states are harvested in high proportion. This work demonstrates the existence of the optimum driving force of 0.2eV and provides some guidelines for engineering the interfacial energetics to achieve the high balanced $J_\mathrm{sc}$ and $V_\mathrm{oc}$.' author: - Wenchao Yang bibliography: - 'dfpaper.bib' title: 'Achieving balanced open circuit voltage and short circuit current by tuning the interfacial energetics in organic bulk heterojunction solar cells: A drift-diffusion simulation' --- Introduction ============ In organic bulk heterojunction solar cells, the randomly oriented donor/acceptor (D/A) interfaces are generally employed for converting the photogenerated excitons into free charge carriers[@deibel; @hains; @clark; @gunes; @hedley]. There have been plenty of experimental and theoretical works devoted to the investigation of the various interfacial properties, which demonstrated that the performance and stability of the devices are largely determined by the interfacial donor/acceptor morphology[@bavel; @huang; @szarko; @nuzzo], the interfacial molecular orientation and aggregation[@barry; @miller; @fu; @chen; @ryno; @ran; @ndjawa], since they have significant impacts on the interfacial energetics[@poelking; @guo; @fazzi]. In particular, the interfacial energetics plays the central role on the exciton dissociation and charge generation[@clark; @hedley], and thus attracted most of the research attention among all the interfacial properties. It is expected that through investigating the working principles of the interfacial energetics people can optimize it and fabricate high efficiency photovoltaic devices. However, the interfacial energetics is rather complicated, which not only involves the molecular frontier orbitals of the donor and acceptor materials, but also the tightly-bounded singlet excitonic states, the loosely-bounded charge transfer (CT) states and the charge separated (CS) states upon photo-excitation[@ohkita; @tvingstedt; @deibel2; @arndt; @grancini; @jailaubekov; @nuzzo]. Moreover, each type of these excited states consists of many energy levels forming a manifold[@clark; @barker; @gerhard; @ti]. Up to now, the intricate interactions among the states and their effects on the charge separation processes remain hotly debated. Macroscopically, the interfacial energetics is revealed to have direct impacts on both the short circuit current density ($J_{sc}$) and the open circuit voltage ($V_{oc}$) of the devices. For the $V_{oc}$, its upper limit is basically determined by the CT states energy and their disordered effect[@veldman; @vandewal; @burke; @collins; @zou; @guan]; while for the $J_{sc}$, it is experimentally demonstrated that a finite lowest unoccupied molecular orbital (LUMO) or highest occupied molecular orbital (HOMO) level offset across the interface is indispensable for obtaining sizable photocurrent[@ohkita; @dnuzzo; @hoke; @hendriks]. The measured current density-voltage ($J-V$) characteristics for devices with a fixed donor material and different fullerene acceptor materials suggest that, as the acceptor with higher LUMO level is employed, the $V_{oc}$ becomes larger due to the increased effective band gap, whereas the $J_{sc}$ decreases significantly. Especially for the recently popular fullerene-based acceptor of ICBA, when it is blended with P3HT as the photoactive layer, the interfacial LUMO offset is smaller than 0.05eV, and the corresponding $J-V$ curve exhibits a $V_{oc}$ of over 1V but an extremely small $J_{sc}$, representing poor charge generation efficiency[@dnuzzo; @hoke]. For the hole transfer processes, the required HOMO energy offset is found to be even 0.3eV higher than the driving force for electron transfer[@hendriks]. Thus, the interfacial LUMO (HOMO) offset is believed to play important roles on charge separation and is usually referred as the driving force for charge transfer and separation in literature[@rand; @clark; @ohkita; @shoaee; @ward; @coffey; @dimitrov; @wright; @jakowetz]. More rigorously, the driving force can be defined as the difference between the effective band gap and the CT state energy[@jakowetz]. According to the different scenarios proposed to explain how the exciton dissociation and charge separation process proceed at the donor/acceptor interface, the possible roles of the driving force could be the following three folds. First of all, the charge transfer at the interface may be a non-adiabatic process which involves a relatively large reorganization energy. The energy level offset provides the free energy for carriers to reach the intersection of potential energy surfaces and achieve resonant charge transfer, as described by the traditional Marcus theory[@ljakoster; @zhang; @wright; @volpi]. This mechanism has been demonstrated by the measurement of photo-carrier yield for a series of acceptors[@coffey]. Secondly, the driving force may provide the kinetic energy required for the electron-hole polaron pairs to escape from their mutual Coulomb attractive potential, which is the so-called hot CT state dissociation[@clark; @ohkita; @shoaee; @murthy; @grancini; @jailaubekov; @schulze; @fuzzi]. Thirdly, employing the pump-push-photocurrent measurements on the free carrier generation efficiency, Bakulin et al found that there exists a band of delocalized high-lying CT states or some vibrational modes which can facilitate the coherent transport of charge carriers on these states to achieve full separation, while those charge carriers on the low-lying CT states generally recombine geminately and do not contribute to the photocurrent[@bakulin1; @bakulin2; @jakowetz]. Due to this coherent charge separation mechanism the finite driving force seems unnecessary[@chner; @kaake; @whaley]. However, with the increased energy level offset, more delocalized states become accessible for the ballistic or coherent transfer of charge carriers, so that the thus measured charge generation rate still exhibits a weak dependence on the LUMO energy level offset[@jakowetz]. On the other hand, the impacts of donor: acceptor ratio is much stronger in this case, because the high proportion of fullerene acceptor material will spontaneously aggregates and forms crystalline phases, which give rise to much more delocalized electronic states[@jakowetz; @savoire; @tamura; @nan]. Each of the charge generation mechanisms can partly explain the experimental phenomena and it is highly controversial that which one is the dominant. Since most of the measurements are done under the transient pulsed luminescence conditions, it remains unclear to what extent the steady state performance of devices is limited by the charge generation rates under the incoherent (Marcus) or coherent mechanisms. Moreover, as mentioned above, there is always a tradeoff between the $J_{sc}$ and $V_{oc}$ under a specific value of the driving force, and increasing the driving force to boost the free carrier generation will inevitably lead to the decreased $V_{oc}$[@rand; @ohkita]. Thus people need to find the minimum driving force required for efficient charge separation to avoid sacrificing the $V_{oc}$ too much. Actually this has been realized in some non-fullerene acceptor solar cells, where the high and balanced $J_{sc}$ and $V_{oc}$ can be reached simultaneously[@liu; @baran]. But the theoretical explanation for this desirable effect is still lacking. In this paper, we employ the macroscopic device model simulation to investigate the effect of the driving force on the final device performance, especially the $J_{sc}$ which was less intensively studied than the properties of $V_{oc}$ in literature. The interfacial energy offset are incorporated into the device model both through its impacts on the effective band gap and on the exciton dissociation rate or proportion to calculate the $J-V$ curves under different driving forces. The investigations are done on the theoretical frameworks of the incoherent and the coherent charge separation mechanisms. It is found that both of them can give rise to $J-V$ curves similar to the experimentally measured ones. Moreover, there indeed exists an optimum driving force of 0.2eV or so for obtaining balanced $J_{sc}$’s and $V_{oc}$’s. Under the steady state, with the incoherent dissociation mechanism the relatively large $J_{sc}$ can be achieved in a broad range of the interfacial LUMO offset so that it could be restricted to a much less value than the reorganization energy; while with the coherent mechanism, the denser is the distribution of the delocalized CT states above the acceptor LUMO level, the higher is the $J_{sc}$ under small driving forces. The results are consistent with the finding that the efficient charge separation can be achieved under small driving forces[@lee; @heeger], and may also provide clues for the design and preparation of the organic donor/acceptor with optimized interfacial energetics to fabricate devices of high power conversion efficiency. In Sec.\[method\], we describe the model we used in simulation, and in Sec.\[results\], the simulated J-V curves for the incoherent and the coherent dissociation mechanisms are shown, respectively, and the variation of $J_{sc}$ under different driving forces is discussed in detail. Finally, we give the conclusions in Sec.\[conclusion\]. Theoretical device modeling method {#method} ================================== The one dimensional device model provide a straightforward method to calculate the device operating parameters under the influences of various microscopic electronic processes[@smith; @blom]. For the bulk heterojunction devices, the active layer in which the donor and the acceptor phases interpenetrate with each other and form percolating pathways for charge transport is considered as a homogeneous medium. Although the interfacial morphology cannot be taken into account in the model, for finely-mixed donor/acceptor phases this assumption is valid from a macroscopic point of view. In order to produce free charge carriers, the photo-generated singlet excitons must experience two successive dissociation steps. In the first one, the exciton diffuses to the donor/acceptor interfaces and transfer their electrons from the donor phase to the acceptor phase while leaving the holes in the donor phase, forming CT states on the interfaces[@clark]. The exciton dynamics is described by the following continuity equation $$\label{exciton} \frac{\partial X}{\partial t}=D_X\frac{\partial^2 X}{\partial x^2}-\frac{X}{\tau}-k_\mathrm{PET} X + G,$$ where $X$ is the exciton concentration. The terms on the right-hand side of Eq. (\[exciton\]) represents the diffusion, the radiative and non-radiative decay, the dissociation and the photo-generation processes, respectively, with $D_X$ the diffusion coefficient, $\tau$ the lifetime, $k_\mathrm{PET}$ the photo-electron transfer rate and $G$ the optical generation rate. This ultrafast electron transfer process is nonadiabatic and its rate is given by the Marcus theory:[@clark] $$\label{ket} k_\mathrm{PET}=\frac{2\pi}{\hbar\sqrt{4\pi \lambda kT}}V^2\exp\left(-\frac{(\triangle G+\lambda)^2}{4\lambda kT}\right),$$ in which the $V$ stands for the electronic coupling between the donor and acceptor molecules; the $\lambda$ represents the reorganization energy; and the $\triangle G$ is the free energy. In the context of charge transfer at the donor/acceptor heterojunction, the $\triangle G$ is actually equal to the interfacial energy offset. The value of $k_\mathrm{PET}$ is mainly dominated by the exponential factor on the right side of Eq. (\[ket\]), and the corresponding prefactor is assumed to be a constant of $k_0$, which may also represents the coherent (ballistic) charge transfer rate. The coherent charge transfer mechanism arises from the delocalized CT states and its modeling method is postponed to the Sec.\[results\] for compactness. In the incoherent charge separation mechanism, only certain proportion of the thus produced CT states can dissociate and generate free charge carriers, while the others recombine geminately to the ground state. According to the Onsager-Braun theory, the proportion of the successfully dissociated CT states $P(E)$ is mainly dependent on temperature, the electric field strength, and the CT states binding energy. With the approximate form of[@clark] $$\label{pe} P(E)=\exp\left(-\frac{e^2}{4\pi\varepsilon_0\varepsilon kT a}\right)\left(1+\frac{e^3}{8\pi\varepsilon_0\varepsilon (kT)^2}E\right),$$ it is incorporated into the free carrier generation rate. Now the continuity equations for electrons and holes can be written as: $$\begin{aligned} % \nonumber to remove numbering (before each equation) \frac{\partial p}{\partial t} &=& -\frac{1}{e}\frac{\partial J_p}{\partial x}+P(E)k_\mathrm{PET} X-R, \label{pt} \\ \frac{\partial n}{\partial t} &=& \frac{1}{e}\frac{\partial J_n}{\partial x}+P(E)k_\mathrm{PET} X-R. \label{nt} \end{aligned}$$ The electron (hole) current $J_n$ ($J_p$) has the common drift-diffusion form[@blom], with the Einstein’s relation being assumed. At the two ends of the device, the $J_n$ and $J_p$ are defined as the respective net surface recombination currents[@sandberg], which consist of the boundary conditions for Eqs (\[pt\],\[nt\]). The bimolecular recombination rate $$R=\zeta \frac{e(\mu_n+\mu_p)}{\varepsilon_0\varepsilon}(np-n_i^2)$$ where $\zeta$ is the reduction factor with respect to the Langevin bimolecular recombination rate[@burke]. The internal electric field $E(x)$ obeys the ordinary Poisson’s equation $$\label{poisson} \frac{\partial E}{\partial x}=\frac{e}{\varepsilon_0\varepsilon}(p-n)$$ with the constraint that $$\label{cons} \int_0^L E(x)dx=V_\mathrm{ext}-(E_g-\phi_p-\phi_n)/e ,$$ in which $V_\mathrm{ext}$ is the externally applied bias voltage, $E_g$ is the effective band gap, and $\phi_p(\phi_n)$ is the hole (electron) injection barrier at the anode (cathode), namely the effective voltage drop across the device is equal to the applied voltage subtracted by the built-in voltage. Using the equilibrium concentrations of $n(x), p(x)$ and the equilibrium internal field strength $E(x)$ as the initial conditions, the continuity equations and the Poisson’s equation are evolved together under the constant illumination condition to reach the steady state solutions, from which the $J-V$ curves are plotted. The simulation parameters are presented in Table. \[para\] except being noted otherwise. Parameter Symbol Value -------------------------------------------- ------------------ ----------------------------------------------- Donor (Acceptor) band gap $E_g$ 1.8eV Injection barriers $\phi_n, \phi_p$ 0.2eV Relative permitivity $\varepsilon$ 3.5 Active layer thickness $L$ 200nm Effective density of states $N_C, N_V$ $10^{21} \mbox{cm}^{-3}$ Charge carrier Mobilities $\mu_n, \mu_p$ $0.1\,\mbox{cm}^2/\mbox{Vs}$ CT generation rate $G$ $3\times 10^{21} \mbox{cm}^{-3}\mbox{s}^{-1}$ CT state lifetime $\tau$ 100ns CT state radius $a$ 2.25nm Coherent CT rate $k_0$ 0.1$\mbox{ns}^{-1}$ Bimolecular recombination reduction factor $\zeta$ 0.1 Reorganization energy $\lambda$ 0.5eV : The parameters used in the device model simulation\[para\] Results and discussion {#results} ====================== Charge separation through the incoherent Marcus mechanism --------------------------------------------------------- Based on the assumption that the major role of the driving force is to predominantly determine the charge transfer rate as described by the Marcus theory, we calculated the $J-V$ curves under a set of LUMO level offsets $\Delta E_\mathrm{L}$’s to examine their effects on the device performance, which are shown in Fig. \[jv1\]. It is observed that the calculated curves reflect well some features of the experimentally measured $J-V$ curves for the polymer/fullerene bulk heterojunction solar cells with a fixed donor material and varied acceptor materials, that is if a $J-V$ curve shows a high $V_\mathrm{oc}$ the corresponding $J_\mathrm{sc}$ is relatively small, and vice versa[@dnuzzo; @hoke]. Therefore, it is demonstrated that the excess free energy required for achieving a sufficiently high nonadiabatic charge transfer rate $k_\mathrm{PET}$ could be the probable origin of the observed tradeoff between the $J_\mathrm{sc}$ and $V_\mathrm{oc}$ in these devices. Generally, as the driving force increases by 0.1eV each time, the $V_\mathrm{oc}$ decreases exactly by 0.1V, which is simply due to the consequent decreasing of the effective band gap. In the following we will mainly focus on the behavior of $J_\mathrm{sc}$ with the varying driving forces. It can be observed that the $J_\mathrm{sc}$ increases significantly with the increasing $\Delta E_\mathrm{L}$ when $\Delta E_\mathrm{L}$ is as small as 0.1 or 0.2eV. In this case the enhanced electron transfer rate $k_\mathrm{PET}$ produces high concentration of CT states, whose subsequent dissociation leads to the increased photocurrent. As the $\Delta E_\mathrm{L}$ approaches 0.5eV, the $J_\mathrm{sc}$ reaches its maximum and then decreases. ![The calculated $J-V$ curves for different interfacial LUMO level offsets (the driving forces) under the Marcus charge separation mechanisms. The reorganization energy $\lambda$ is set to 0.5eV. In the calculation the Onsager-Braun theory for CT state dissociation is taken into account. []{data-label="jv1"}](JVcurve1){width="8cm"} In order to reveal quantitatively the relationship between the steady state photocurrent and the driving force, we calculated more $J_\mathrm{sc}$’s under different $\Delta E_\mathrm{L}$’s and plotted them in Fig. \[jsc\](a), where the effect of temperature is also examined considering the strong temperature-dependence of $k_\mathrm{PET}$ and $P(E)$. At the room temperature (RT) of 300K, as the $\Delta E_\mathrm{L}$ increases the $J_\mathrm{sc}$ quickly rises to over $8\,\mbox{mA}/\mbox{cm}^2$, and keeps this high and approximately constant value in the wide range from 0.2 and 0.7eV, beyond which the Marcus inverted region emerges and the $J_\mathrm{sc}$ becomes smaller. With the decreasing temperature, the $J_\mathrm{sc}$ reduces greatly due to the reduction of the dissociation proportion $P(E)$ of the CT states. Moreover, for the curves of low temperature, the high-and-flat region shown in the RT curve disappears, and the $J_\mathrm{sc}$ begins to decrease slowly as soon as it reaches its maximum at $\Delta E_\mathrm{L}=0.3$eV. This is because with the increasing $\Delta E_\mathrm{L}$, the built-in field is greatly weakened so that the $P(E)$ decreases as the result of the reduced internal field, leading to inefficient charge extraction and smaller $J_\mathrm{sc}$’s. Therefore, at low temperatures the free charge generation is strongly restricted by the small field dependent CT state dissociation rate. The sole effect of the driving force can be observed in Fig. \[jsc\](b), where we plotted the $J_\mathrm{sc}-\Delta E_\mathrm{L}$ curves calculated by setting $P(E)=1$. In this case all of the $J_\mathrm{sc}-\Delta E_\mathrm{L}$ curves have the high-and-flat region, even though the region gradually shrinks with the decreasing temperatures. In addition, the curves are basically symmetric with respect to the vertical line of $\Delta E_\mathrm{L}=0.5$eV, which is the feature of the Marcus charge transfer rate. Nevertheless, they display large discrepancy with the corresponding $k_\mathrm{PET}-\Delta E_\mathrm{L}$ curves, for the latter have prominent peaks when the $\Delta E_\mathrm{L}=\lambda$ and reduces much more rapidly when the $\Delta E_\mathrm{L}$ deviates from the $\lambda$. This result is in contradictory with Coffey et al’s finding that the photo-carrier relative yield data measured with the time-resolved microwave photoconductivity (TRMC) method for blends of fixed acceptor and different donors can be well fitted by the $k_\mathrm{PET}-\Delta E_\mathrm{L}$ curve[@coffey]. To find the underlying reason of the discrepancy, we calculated the steady state exciton concentration $X$ under the short circuit condition for different $\Delta E_\mathrm{L}$’s and temperatures, which are averaged over the whole active layer thickness, as shown in Fig. \[exdis\]. It is seen that the variation of the exciton concentration $X$ with respect to $\Delta E_\mathrm{L}$ is just opposite to that of $k_\mathrm{PET}$, i.e. the curve has a deep valley precisely at the point of the reorganization energy $\lambda$ of 0.5eV, and this feature does not change at different temperatures. Therefore, the product of $k_\mathrm{PET}X$ which is the CT states generation rate is approximately a constant for a wide range of $\Delta E_L$, which is the origin of the high-and-flat region for the $J_\mathrm{sc}-\Delta E_\mathrm{L}$ curves. Especially for small $\Delta E_\mathrm{L}$’s, under the constant illumination condition in steady state the induced small $k_\mathrm{PET}$ may give rise to high concentration of unquenched excitons, and many of which can dissociate within their lifetime to contribute to the photocurrent. On the contrary, under the pulsed illumination condition in the TRMC experiments, since the supply of excitons is limited by the light-pulse duration for each type of polymer:fullerene blend, only the $k_\mathrm{PET}$ basically governs the variational tendency of the photo-carrier yield with respect to the $\Delta E_\mathrm{L}$, such that a bell-like curve for the photo-carrier yield emerges[@coffey]. The occurrence of high $J_\mathrm{sc}$ under small $\Delta E_\mathrm{L}$ in steady state suggests that it is unnecessary to employ pairs of donor and acceptor materials with their $\Delta E_\mathrm{L}$ approaching the reorganization energy $\lambda$ to achieve the maximum photocurrent. According to Fig. 2, at the RT with $\lambda=0.5$eV, a moderate $\Delta E_\mathrm{L}$ of 0.2-0.3eV can provide sufficient driving force for charge separation at the D/A interface. Thus in principle given a donor material, much $V_\mathrm{oc}$ loss due to the interfacial energy level offset could be saved by employing acceptors with higher LUMO levels to form blend with the donor. However, in some polymer/fullerene blended systems the required driving force for achieving sizable photocurrent is still as high as 0.5eV[@wright; @hendriks]. This may be due to the fact that both of the electron transfer and the hole transfer processes contribute to the photocurrent, and the driving force for the latter is experimentally revealed to be 0.3eV higher than the former[@hendriks]. Consequently, if a significant proportion of the excitons are dissociated through transferring their holes from the acceptor to the donor, a relatively large HOMO offset $\Delta E_\mathrm{H}$ is essential for achieving a sufficiently high hole transfer rate $k_\mathrm{HT}$ and thus the high $J_\mathrm{sc}$. The situation may occur in devices made of non-fullerene acceptors, in which the photo-absorption of acceptors contribute greatly to the exciton formation[@stoltzfus]. Moreover, it is reported that even for fullerene-based acceptors such as ICBA, a large proportion of excitons generated in the polymers can diffuse into them through the F$\ddot{\mbox{o}}$rster resonant energy transfer process, and these excitons can only dissociate through the hole transfer pathway, and the inefficient hole transfer process in polymer-ICBA devices should be responsible for their low $J_\mathrm{sc}$[@dnuzzo; @hoke]. Here we denoted the respective proportions of excitons dissociating through the two pathways as $P_e$ and $P_h=1-P_e$, and calculated the $J_\mathrm{sc}-\Delta E$ curves with the varied $P_e$ to evaluate the combined roles of the electron and hole transfer pathways on the $J_\mathrm{sc}$. Since we assumed the same band gaps for the donor and acceptor materials, the HOMO offset $\Delta E_\mathrm{H}$ is equal to the LUMO offset $\Delta E_\mathrm{L}$; and the reorganization energy for hole transfer is 0.3eV higher than that of the electron transfer. The calculated results are shown in Fig. \[twopathway\]. It is observed that as more excitons are generated in or transferred into the acceptor, the high-and-flat region for $J_\mathrm{sc}$ is maintained, but its onset is shifted to the higher $\Delta E$, which suggests that the driving force required for achieving sizable photocurrent becomes larger if the hole transfer reaction, being of a higher reorganization energy, plays a significant role on exciton dissociation. In addition, the slow decreasing of $J_\mathrm{sc}$’s in the high $\Delta E$ regime is mainly caused by the reduced built-in field rather than the Marcus inverted effect, which is different from the sole electron transfer case (the red dashed line). Based on the above understanding, we calculated the RT $J-V$ curves with the $\Delta E$’s derived from real materials, which are some types of fullerene-based acceptors blended with the donor of PF10TBT, as shown in Fig. \[jvicba\]. The effective energy gap is set to 1.66eV, and the electron transfer driving forces $\Delta E_\mathrm{L}$ are set to 0.18, 0.08, 0.05, 0.03 and 0.01eV, corresponding to the acceptors of PCBM, $\mbox{t}_2$-bis-PCBM, bis-PCBM, si-bis-PCBM and ICBA, respectively. Without taking into account the hole transfer pathway, the high $J_\mathrm{sc}$ exhibited by the curve of PCBM further suggest that only a small driving force of about 0.2eV is required for obtaining sufficiently high photocurrent. Compared with the experimentally measured curves in Ref.[@dnuzzo], the curve of ICBA exhibits a much larger $J_\mathrm{sc}$ of 3.56$\mbox{mA/cm}^2$. So we speculate that when examining the performance of the ICBA-acceptor devices, it is important to incorporate the effects of the high proportion of excitons diffusing into the ICBA and the inefficient hole transfer pathway. With the $P_e=0.2$ and the hole transfer reorganization energy $\lambda_\mathrm{H}=0.8$eV, the recalculated $J-V$ curve (the dashed line) of ICBA gives rise to a relatively realistic $J_\mathrm{sc}$ of 1.25$\mbox{mA/cm}^2$. It is noticed that in this case the $V_\mathrm{oc}$ also becomes smaller because of the reduced photo-carrier density under the open circuit condition. ![The calculated short circuit current density versus the interfacial LUMO offset ($J_\mathrm{sc}-\Delta E_\mathrm{L}$) under the Marcus charge transfer mechanism at different temperatures, with the field and temperature dependent Onsager-Braun CT state dissociation probability $P(E)$ being taken into account (a) or neglected (b) in the calculation. For comparison, the corresponding $k_\mathrm{PET}-\Delta E_\mathrm{L}$ curves with their maximum being scaled to the value of $J_\mathrm{sc}$ at $\Delta E_\mathrm{L}=\lambda=0.5$eV are also plotted in (b). []{data-label="jsc"}](JscDF){width="16cm"} ![The calculated exciton density versus the interfacial LUMO offset under the Marcus charge transfer mechanism at different temperatures. Each point for density corresponds to the average value over the whole active layer in the device. The reorganization energy $\lambda$ is set to 0.5eV[]{data-label="exdis"}](excitondensity){width="8cm"} ![The calculated $J_\mathrm{sc}-\Delta E_\mathrm{L}$ curves under the Marcus charge transfer mechanism for a set of different proportions $P_e/(1-P_e)$ of the electron/hole transfer pathways. The reorganization energy $\lambda_\mathrm{L}, \lambda_\mathrm{H}$ for the electron and hole transfer pathways are set to 0.5 and 0.8eV, respectively. []{data-label="twopathway"}](twopath){width="8cm"} ![The calculated $J-V$ curves for devices with the donor of PF10TBT blended with different acceptors. The effective energy gap is set to 1.66eV; the reorganization energy is set to 0.5eV; and the electron transfer driving forces $\Delta E_\mathrm{L}$ are set to 0.18, 0.08, 0.05, 0.03 and 0.01eV, corresponding to the acceptors of PCBM, $\mbox{t}_2$-bis-PCBM, bis-PCBM, si-bis-PCBM and ICBA, respectively. The dashed line is calculated for the ICBA acceptor by taking into account the hole transfer pathway with $P_e=0.2$ and $\lambda_\mathrm{H}=0.8$eV, which can better fit the experimentally measured $J-V$ curve for the ICBA. []{data-label="jvicba"}](jvicba){width="8cm"} Charge separation through the coherent/ballistic dynamics --------------------------------------------------------- Next we examine the role of the driving force in coherent mechanism for charge separation. In contrast to the Onsager-Braun theory, for the coherent mechanism the charge transfer process is ballistic and band-like rather than diffusive, and results in complete charge separation. The relevant exciton dissociation rate can be assumed to be of a constant rate of $k_0$ which is independent of the ambient temperature and the electric field. But as have been demonstrated by lots of experiments like the PPP, only the proportion of excitons that are energetically resonant with the delocalized CT states (thus they can be deemed as the same entity) in the CT states manifold can participate the ballistic transfer process[@bakulin1; @bakulin2]. To make the picture of the involved energy levels simple, it is assumed that the delocalized CT states accessible for charge separation are those states which are energetically higher than the LUMO level of the acceptor, and the CT states below the acceptor LUMO level are relaxed and localized so that they cannot dissociate successfully but decay to the ground state through geminate recombination. Since the coherent charge transfer process is ultrafast, which occurs within 100fs upon photo-excitation and is prior to the relaxation of the hot CT states[@grancini; @jailaubekov; @whaley; @yao], initially the population in each level of the CT states manifolds does not obey the equilibrium distribution and is considered to be evenly distributed. In particular, the population of the in-gap CT states arises from the low energy such as the near infrared photo-absorption[@troisi], as has been verified by some external quantum efficiency (EQE) measurements[@vandewal]. According to the above considerations, if the total number of states in the intermediate CT states manifold is fixed to be $N$, the corresponding density of states (DOS) $g(E)$ solely determines the proportion of the delocalized CT states $P_\mathrm{band}$ in the whole manifold, thus we have $$\label{pband} P_\mathrm{band}(\Delta)=\frac{1}{N}\int_{E^\prime-\Delta}^{E^\prime} g(E) dE,$$ where $E^\prime$ is the upper limit of the CT states manifold and $\Delta$ is the width of the energy window of the delocalized electronic states, i.e. the interfacial LUMO level offset. To investigate the dynamics of free charge carrier, only the delocalized CT states should be taken into account. Whereas the low-lying ones do not contribute to the photocurrent and are just wasted. Thus the continuity equations of excitons should be modified into the form of $$\label{exciton2} \frac{\partial X}{\partial t}=D_X\frac{\partial^2 X}{\partial x^2}-\frac{X}{\tau}-k_0 X + P_\mathrm{band}G.$$ Solving Eq. (\[exciton2\]) together with other device model equations, the $J-V$ curves for the coherent charge separation mechanism can be obtained. To obtain the actual $P_\mathrm{band}$, the $g(E)$ must be explicitly given. However, for real materials there are many complicated effects impacting the CT DOS, such as the energetic disorder[@mcmahon], the entropic effect due to the different dimensionality of the donor and acceptor molecules[@bregg], the aggregation effects of the fullerene-based acceptors and the image charge effect at the donor/acceptor interface[@savoire; @xyz]. Currently the $g(E)$ can only be calculated using first principle or molecular dynamics methods for specific donor-acceptor material systems[@tamura; @chner; @nan; @savoire; @belj], and there lacks an analytical expression for it. For the convenience of the present phenomenological investigation, we assumed two simplified analytical expressions of $g(E)$ which may approximate the real DOS of the CT states manifold to some extent. ### Hydrogen-atom-like density of states Firstly, considering the CT excitons as bounded polaron pairs, their energy levels resemble those of a hydrogen atom so that the $g(E)$ has the hydrogen-atom DOS like expression[@xyz]. In Fig. \[schematic\](a) we schematically depicted the interfacial energetics, where the zero point of the CT state energy is set to be the acceptor LUMO level, i.e. $E_\mathrm{ct}$ is transformed to $E_\mathrm{ct}-E_g$. A cutoff energy level $E_c$ slightly above the acceptor LUMO level separates the high-lying continuous energy spectrum from the low-lying discrete one. Then for $E\geq E_c$, $g(E)$ equals to a constant of $\alpha (E_a-E_c)^{-3/2}$; while for $E<E_c$, $g(E)=\alpha (E_a-E)^{-3/2}$, in which the parameter $E_a$ is a small positive energy used to avoid the singularity in $g(E)$, and the prefactor $\alpha$ is determined by the normalization condition of $$\label{normalization} N=\int_{E^\prime-W}^{E^\prime} g(E) dE$$ with $W$ the width of the CT state manifold. Based on the Eqs. (\[pband\],\[normalization\]), the proportion $P_\mathrm{band}$ is deduced to be of the form of $$\label{pbandhy} P_\mathrm{band}(\Delta)=\frac{(\Delta-E_c)/(E_a-E_c)+2\left[1-\sqrt{(E_a-E_c)/E_c}\right]}{(\Delta-E_c)/(E_a-E_c)+2\left[1-\sqrt{(E_a-E_c)/(E_c+W-\Delta)}\right]}.$$ Substituting the $P_\mathrm{band}$ into Eq. (\[exciton2\]) with $E_c=0.05$eV, $E_a=0.1$eV and $W=1$eV, we calculated the $J-V$ curves with a set of different driving force $\Delta$’s, which are plotted in Fig. \[jvpband\]. It is observed that the curves exhibit the exactly same behavior as those calculated for the Marcus incoherent charge transfer mechanism (Fig. \[jv1\]). That is with the increased $\Delta$, the $J_\mathrm{sc}$ increases whereas the $V_\mathrm{oc}$ decreases evenly. Thus it is not plausible to gain some clues concerning which charge separation mechanism is the dominant one just from the variation of $J-V$ curves with respect to the driving force. For $\Delta=0.2$eV, the optimum device performance is obtained with $J_\mathrm{sc}=7.14\,\mbox{mA/cm}^2$ and $V_\mathrm{oc}=0.94$V, such that a driving force of 0.2eV is sufficient for achieving balanced $J_\mathrm{sc}$ and $V_\mathrm{oc}$ in devices where the coherent exciton dissociation mechanism plays the major role on charge generation. The quantitative relationship between the $J_\mathrm{sc}$ and $\Delta$ are calculated and presented with the varying cutoff energy $E_c$ and the fixed $E_a=0.1$eV, as shown in Fig. 8(a). For all the curves, the $J_\mathrm{sc}$ increases very rapidly with $\Delta$ when the latter is smaller than 0.2eV, because in the energy window of 0-0.2eV the CT states form a quasi-continuum band with high DOS $g(E)$, and a small increase of $\Delta$ can induce high extra population of the CT states to participate the coherent (ballistic) charge transfer and separation. For $E_c=0.08$eV, the $J_\mathrm{sc}$ reaches an approximately constant high value beyond $\Delta=0.2$eV, being similar to the high-and-flat region appearing in the $J_\mathrm{sc}-\Delta E_L$ curve under the incoherent charge transfer mechanism. On the other hand, with the reduced $E_c$ the $J_\mathrm{sc}$ becomes smaller and remain increases slowly in the relatively high $\Delta$ regime, which is due to the fact that the in-gap CT states close to the acceptor LUMO level are of a high DOS and thus need to be harvested by enhancing the $\Delta$ in order to reach a sufficiently high $J_\mathrm{sc}$. The decreasing of $J_\mathrm{sc}$ around $\Delta=0.8$eV is only caused by the reduction of the built-in electric field and the lowered charge extraction efficiency, rather than the Marcus inverted region in Fig. 2. We also included a curve with the varied $E_a$ of 0.2eV and $E_c=0.08$eV(the dashed line), where compared to the corresponding curve with $E_a=0.1$eV (the red line), the $J_\mathrm{sc}$ reduces significantly as a result of the reduced DOS for the high-lying levels in CT state manifold. The CT states lying in the energy window $\Delta$ can also possibly consist of purely discrete spectrum, namely the $E_c$ level is equal to or above the donor LUMO level, as shown in Fig. \[schematic\](b). In this case we assumed the $E_c$ and $E_a$ to be the same. Then the delocalized CT states proportion $P_{band}$ is modified to $$\label{pband2} P^{\prime}_\mathrm{band}(\Delta)=\frac{1-(1+\Delta/E_c)^{-1/2}}{1-(1+W/E_c)^{-1/2}}.$$ Inserting the $P'_\mathrm{band}$ into the device model, the $J_\mathrm{sc}-\Delta$ curves are calculated and presented in Fig. 8(b). The shape of the curves does not exhibit observable change as compared to that in Fig. 8(a), but it becomes more difficult to achieve a sizable $J_\mathrm{sc}$ for the small $\Delta$. Generally the $J_\mathrm{sc}$ increases with the decreasing $E_c$. Therefore the $E_c$ level should be tuned as close to the donor LUMO level as possible. Combined with the results in Fig. 8(a), we conclude that the criterion for good interfacial energetics being able to facilitate significant coherent exciton dissociation is that, the CT states manifold is low and the energy window $\Delta$ is resonant with at least part of the continuous spectrum and the high-lying discrete spectrum, so that plenty of hot excitons can be harvested. ![The schematic illustration of the donor/acceptor interfacial energetics which contains the delocalized CT states and may facilitate the coherent (ballistic) charge separation. The energy levels of donor are on the left side and those of acceptor are on the right side. The Charge transfer (CT) states manifold (marked in red) is of the width $W$, in which the levels above the acceptor LUMO level are delocalized and resonant with the charge separated (CS) states (marked in green), forming an energy window of $\Delta$ in which the ballistic charge transfer can take place. The energy parameter $E_c$ represents a critical energy level on which the continuous and the discrete spectra in the CT states manifold meet. The $E_c$ level may be in the energy window (a), or above it (b). The case of (b) may occur in the hydrogen-atom-like DOS of the CT states manifold. The specific forms of the CT states DOS are described in the text. []{data-label="schematic"}](schematic){width="16cm"} ![The calculated $J-V$ curves for different driving force $\Delta$’s under the coherent charge separation mechanism. The CT states manifold is of a hydrogen-atom-like DOS, with the energy parameter $E_c=0.05$eV, $E_a=0.1$eV and $W=1$eV. []{data-label="jvpband"}](jvpband){width="8cm"} ; (b) or the manifold is of the purely discrete energy levels below the LUMO level of the donor, corresponding to the DOS schematically illustrated in Fig. \[schematic\](b). []{data-label="jscecea"}](jscecea){width="16cm"} ### Exponential density of states Secondly, motivated by the mobility edge model in organic semiconductors[@neher], we considered CT states manifold to be of the exponential type of DOS. The interfacial energetic levels can be still basically schematically illustrated by the Fig. \[schematic\](a), with $E_c$ representing the cutoff energy level separating the continuous band from the discrete levels. For $E\geq E_c$, the DOS $g(E)=\alpha/E_a$; while for $E<E_c$, $g(E)=\alpha/E_a \exp[(E-E_c)]/E_a$, where $E_a$ is a parameter characterizing the width of the exponential states, and the prefactor $\alpha$ is also determined by the normalization condition Eq. (\[normalization\]). Now it can be deduced that the proportion of the CT states lying in the energy window of the driving force $\Delta$ is $$\label{pbandexp} P^{\prime\prime}_\mathrm{band}(\Delta)=\frac{(\Delta-E_c)+E_a[1-\exp(-E_c/E_a)]}{(\Delta-E_c)+E_a\left[1-\exp\left(\frac{\Delta-W-E_c}{E_a}\right)\right]}.$$ With the energy parameters $E_c, E_a$ and $W$ being set to 0.05, 0.1 and 1eV, we substituted the above $P^{\prime\prime}_\mathrm{band}(\Delta)$ into Eq. (\[exciton2\]) and calculated the $J-V$ curves for different driving force $\Delta$’s, as shown in Fig. \[expjv\](a). It is obvious that the curves show almost the same features with respect to those for hydrogen-atom like CT state DOS (see Fig.\[jvpband\]), suggesting that the device performance is not very sensitive to the specific form of the DOS as long as the number of in-gap levels decreases quickly with the decreasing energy. In Fig. \[expjv\](b) we present the calculated $J_\mathrm{sc}-\Delta$ curves for the varying $E_c$ and $E_a$. Similar to the behaviors exhibited by the curves for hydrogen-atom like DOS, the relatively larger $E_c$ more or less give rises to the higher $J_\mathrm{sc}$, because of the inclusion of the dense high-lying discrete levels in the energy window for ballistic charge transfer. On the other hand, the $E_a$ plays a much more important role on determining the photocurrent. With the increasing of $E_a$ from 0.1eV to 0.2eV, the $J_\mathrm{sc}$ decreases by nearly 2$\mbox{mA/cm}^2$ under the $\Delta$ of 0.2eV. In order to obtain a sizable $J_\mathrm{sc}$ under a small driving force, the width of the in-gap states in the CT manifold should be restricted to a value at least being smaller than 0.2eV. Therefore, the optimization of the DOS for the CT sates manifold is important, which could be realized by changing donor:acceptor ratio of the blend to enhance the fullerene aggregation and crystallization so that more delocalized CT states may be formed. . []{data-label="expjv"}](exponential){width="16cm"} Conclusion and outlook {#conclusion} ====================== In this work, employing the phenomenological device model method we investigated the impacts of the charge separation driving force, which is defined as the donor/acceptor interfacial energy level offsets on the device performance of organic bulk heterojunction solar cells. The driving force $\Delta$ may either provide the free energy required for the incoherent Marcus charge transfer processes to happen or form an energy window where the delocalized CT states reside and facilitate the coherent charge transfer processes. Both of the two kinds of charge separation mechanisms probably play important roles and thus were studied independently by calculating the corresponding $J-V$ and $J_\mathrm{sc}-\Delta$ curves. Generally the $V_\mathrm{oc}$ reduces evenly with the increased $\Delta$, forming a significant $V_\mathrm{oc}$ loss pathway. For the Marcus charge transfer mechanism, with the increasing of $\Delta$ from 0eV, the $J_\mathrm{sc}$ initially increases extremely rapidly and begin to saturate under a small delta of 0.2eV or so; then the $J_\mathrm{sc}$ maintains a high and nearly constant value until the Marcus inverted effect emerges under too high $\Delta$’s, exhibiting a behavior which is largely different from that of the Marcus charge transfer rate $k_\mathrm{PET}$. The underlying reason is found that the reduced $k_\mathrm{PET}$ under a $\Delta$ deviating from the reorganization energy $\lambda$ is precisely compensated by the enhanced density of the accumulated exciton within their lifetime, such that the overall free charge generation rate changes very slowly. When the hole transfer pathway plays innegligible roles on charge separation, the required $\Delta$ for obtaining a sizable $J_\mathrm{sc}$ may become higher due to the relatively larger reorganization energy on the acceptor side, such as the case for the ICBA acceptor based devices. For the coherent mechanism, when calculating the $J-V$ and $J_\mathrm{sc}-\Delta$ curves we assumed the hydrogen-atom-like DOS and the exponential DOS for the interfacial CT states manifold, respectively. The results show similar behaviors and suggest that as long as the energy window formed by the interfacial energy offset (or the driving force) contains part of the continuous spectrum and the dense high-lying discrete levels in the CT state manifold while the low-lying in-gap levels are rare, a great proportion of the CT states can be converted into the fully separated charge carriers and consequently the high $J_\mathrm{sc}$ is obtained under a small $\Delta$ of about 0.2eV, which is consistent with the behavior of $J_\mathrm{sc}$ calculated for the incoherent mechanism. Therefore, regardless of the charge separation mechanism, people can obtain the relatively high $J_\mathrm{sc}$ and $V_\mathrm{oc}$ simultaneously without sacrificing one for the other, which may be hopefully realized in the recently popular non-fullerene acceptor solar cells. In addition, concerning the concrete charge separation mechanism in the actual donor/acceptor blended systems, the coherent and incoherent mechanisms may coexist, which is probably the reason that up to now, in different experiments people have observed that the photocurrent generation follows both the Marcus-type behavior with respect to the driving force and the composition dependence on the donor:acceptor blend ratio. It is demonstrated in our simulation that with a moderate driving force, there is no obvious feature on the $J-V$ curves and the $J_\mathrm{sc}-\Delta$ curves that can identify which one is the dominant mechanism. However, the incoherent mechanism induces strongly temperature-dependent effects for the photocurrent and thus can be singled out through observing the behavior of $J_\mathrm{sc}$ at the lowered ambient temperature. Also, future works on the DOS of CT states may be helpful for acquiring the high $J_\mathrm{sc}$ under the smaller driving forces. The authors would like to thank Professor R. $\ddot{\mbox{O}}$sterbacka for the fruitful discussion and his insightful comments. This work is supported by the National Natural Science Foundation of China under the Contract No. 11604280 and 51602276.

--- abstract: 'An attractive explanation for non-zero neutrino masses and small matter antimatter asymmetry of the present Universe lies in “leptogenesis". At present the [*size*]{} of the lepton asymmetry is precisely known, while the [*sign*]{} is not known yet. In this work we determine the sign of this asymmetry in the framework of two right handed neutrino models by relating the leptogenesis phase(s) with the low energy CP violating phases appearing in the leptonic mixing matrix. It is shown that the knowledge of low energy lepton number violating re-phasing invariants can indeed determine the sign of the present matter antimatter asymmetry of the Universe and hence indirectly probing the light physical neutrinos to be Majorana type.' author: - Kaushik Bhattacharya - Narendra Sahu - Utpal Sarkar - 'Santosh K. Singh' title: '**Leptogenesis and low energy CP phases with two heavy neutrinos**' --- Introduction {#intr} ============ Within the Standard Model (SM) the neutrinos are massless and hence there is no CP violation in the lepton (L) sector. The current evidence [@solar-expt; @atmos-expt; @kamland] from the neutrino oscillation experiments, on the other hand, suggest that neutrinos are massive, however small, and they mix up. The goal of the present neutrino oscillation experiments is to determine the nine degrees of freedom in the low energy neutrino mass matrix. They are parametrized by three masses, three mixing angles and three CP violating phases out of which two are Majorana and one is Dirac. At present the neutrino oscillation experiments able to measure the two mass square differences, the solar and the atmospheric, and three mixing angles with varying degrees of precision, while there is no information about the phases. Assuming that the neutrinos are of Majorana type the small masses of the physical left handed neutrinos can be explained by the elegant seesaw mechanism [@seesawgroup] which involves singlet right-handed neutrinos (type-I seesaw) or triplet Higgs (type-II seesaw) or can be both (hybrid seesaw). In the present article we limit ourselves to the case of type-I seesaw models. Although we call them right-handed neutrinos, in the extensions of the SM they are just singlet fermions that transform trivially under the SM gauge group. So, there is no apparent reasons for the number of heavy singlet neutrinos to be same as the number of left-handed neutrinos. So, for the main part of our discussions we restrict ourselves to only two right-handed neutrinos. These results will also be true when there are three right-handed neutrinos, but the third right-handed neutrino do not mix with the other two neutrinos. We start with three right-handed neutrinos and after some general comments work mostly with two right-handed neutrinos. While there is no information about the absolute mass scales of the physical neutrinos, the currently discovered tiny mass scales; the atmospheric neutrino mass ($\Delta_{atm}=\sqrt{|m_3^2-m_2^2|}$) in the $\nu_{\mu}-\nu_{\tau}$ oscillation and the solar neutrino mass ($\Delta_{\odot}=\sqrt{m_2^2-m_1^2}$) in the $\nu_e-\nu_{\mu}$ oscillation, can be explained by adding at least two right handed neutrinos to the SM Lagrangian. However, with two right-handed neutrinos the seesaw mechanism predicts one of the physical light neutrino mass to be exactly zero which is permissible within the current knowledge of neutrino masses and mixings. The Majorana mass of the right-handed neutrino violates $L$-number and hence is a natural source of L-asymmetry in the early Universe [@fukugita.86; @baryo_lepto_group]. A partial L-asymmetry is then converted to baryon (B) asymmetry through the non-perturbative sphaleron processes, unsuppressed above the electroweak phase transition. Currently the $B$-asymmetry has been measured precisely by the Wilkinson Microwave Anisotropy Probe (WMAP)[@spergel.03] and is given by $$\left( \frac{(n_B-n_{\bar{B}})}{n_\gamma} \right)_0 \equiv \left( \frac{n_B}{n_\gamma} \right)_0=\left(6.1^{+0.3}_{-0.2}\right) \times 10^{-10}. \label{asymmetry}$$ It is legitimate to ask if there are any connecting links between leptogenesis and the CP violation in the low energy leptonic sector, in particular neutrino oscillation and neutrinoless double beta decay. In the context of three right-handed neutrino models several attempts have been taken in the literature to connect the CP violation in leptogenesis and neutrino oscillations [@3RH-models]. It is found that there are almost no links between these two phenomena unless one considers special assumptions [@scpv_models]. In fact it is shown that leptogenesis can be possible irrespective of the CP violation at low energy [@rebelo_prd]. On the other hand, in the two right-handed neutrino models there is a ray of hope connecting leptogenesis with the CP violation in neutrino oscillation [@2RH-models] and neutrinoless double beta decay processes. While the magnitude of CP violation is fairly known in the quark sector, it is completely shaded in the leptonic sector of the SM. Therefore, searching for CP violation in the leptonic sector is of great interest in the present days. It has been pointed out that the Dirac phase, being involved in the L-number conserving processes, can be measured in the long baseline neutrino oscillation experiments [@dirac_phase_group], while the Majorana phase, being involved in the L-number violating processes, can be investigated in the neutrinoless double beta decay [@majorana_phase_group] processes. At present the magnitude of B-asymmetry is precisely known, while the sign of this asymmetry is not known yet. However, by knowing the CP violating phases in the leptonic mixing matrix one can determine the sign of the B-asymmetry. This is the study taken up in this work. We consider a minimal extension of the SM by including two singlet right-handed neutrinos which are sufficient to explain the present knowledge of neutrino masses and mixings. We adopt a general parameterization of the neutrino Dirac Yukawa coupling and give the possible links between the CP violation in leptogenesis and neutrino oscillation, CP violation in neutrinoless double beta decay and leptogenesis. It is shown that the knowledge of low energy CP violating re-phasing invariants can indeed determine the sign of the B-asymmetry since the size of this asymmetry is known precisely. Rest of the manuscript is arranged as follows. In section II we elucidate the canonical seesaw in the framework of three right handed neutrinos. We then display the possible links between leptogenesis and the low energy CP-violating phases appearing in the leptonic mixing matrix in certain special circumstances. It is found that there are almost no links between these two phenomena occurring at two different energy scales. Therefore, in section III we give a parameterization of $m_D$ in the two right-handed neutrino models. In section IV we calculate the neutrino masses and mixings by using the parameterization of $m_D$ given in section III. In section V we estimate the CP violation in leptogenesis. In section VI we consider the re-phasing invariant formalism to study the possible links between the CP violating phases responsible for leptogenesis and the CP violation at low energy phenomena. First we calculate th CP violation in neutrino oscillation and then elucidate its link to leptogenesis. After that we calculate the CP violation in low energy lepton number violating process, i.e., the neutrinoless double beta decay, and then elucidate its link to leptogenesis. We conclude in section VII. Canonical seesaw and parameter counting {#sec2} ======================================= To account for the small neutrino masses we extend the SM by including right-handed neutrinos. The corresponding leptonic Lagrangian is given by $$\begin{aligned} \mathcal{L} &=& \overline{\ell_{Li}}i\gamma^\mu D_\mu\ell_{Li}+\overline{ \ell_{Ri}}i\gamma^\mu \partial_\mu\ell_{Ri}+\overline{N_{R\alpha}}i\gamma^\mu \partial_\mu N_{R\alpha}\nonumber\\ &-&\left( {1\over 2}\overline{(N_{R\alpha})^c}(M_R)_{\alpha \beta}N_{R\beta}+ \overline{\ell_{Li}} \phi (Y_e)_{ij}\ell_{Rj}+\overline{\ell_{Li}}\tilde{\phi} (Y_\nu)_{i\alpha} N_{R\alpha}+H.C.\right)\,, \label{lagrangian}\end{aligned}$$ where $\tilde{\phi}=i\tau^2\phi$ and $i$ runs from 1 to 3, representing the left-handed fields. $\alpha$ represent the right handed neutrino indices. $\ell_{Li}$ represents the ${\rm SU(2)}_L \times {\rm U(1)}_Y$ doublets, $\ell_{Ri}$ and $N_{R\alpha}$ are right-handed singlets of the theory. After the electroweak symmetry breaking the canonical seesaw [@seesawgroup] gives the effective neutrino mass matrix $$\begin{aligned} m_\nu=-m_D M_R^{-1} m_D^T\,, \label{seesaw}\end{aligned}$$ where $m_D=Y_\nu v$ is the Dirac mass matrix of the neutrinos with $v$ is the vev of SM Higgs and that of $M_R$ is the mass matrix of right handed neutrinos. Without loss of generality we consider $M_R$ to be diagonal and in this basis $m_D$ contains rest of the physical parameters that appears in $m_\nu$. The diagonalization of $m_{\nu}$, through the lepton flavor mixing matrix $U_{PMNS}$ [@pmns-matrix], gives us three masses of the physical neutrinos. Its eigenvalues are given by $$\begin{aligned} D_m\equiv {\rm diag.}(m_{1}, m_{2}, m_{3})=U_{PMNS}^\dagger m_{\nu} U_{PMNS}^*\,, \label{diag}\end{aligned}$$ where the masses $m_i$ are real and positive. The standard PDG parametrization [@pdg] of the PMNS matrix reads: $$\begin{aligned} U_{PMNS}= \left( \begin{array}{ccccc} c_{12}c_{13} & & s_{12}c_{13} & &s_{13}e^{-i\delta_{13}}\cr\\ -s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta_{13}} & & c_{12}c_{23}- s_{12}s_{23}s_{13}e^{i\delta_{13}} & & s_{23}c_{13}\cr\\ s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta_{13}} & & -c_{12}s_{23}-s_{12} c_{23}s_{13}e^{i\delta_{13}} & & c_{23}c_{13} \end{array} \right)\,.\,U_{ph} \label{mns-matrix}\end{aligned}$$ where $U_{ph}={\rm diag.}\,(1, e^{i\eta}, e^{i(\xi +\delta_{13})})$ and $c_{ij}$, $s_{ij}$ stands for $\cos \theta_{ij}$ and $\sin \theta_{ij}$ respectively. The two physical phases $\eta$ and $\xi$ associated with the Majorana character of neutrinos are not relevant for neutrino oscillations. Thus we see that there are three phases in the low energy effective theory responsible for CP violation. However, these phases may not give rise to CP violation at high energy regime, in particular, leptogenesis to our interest. In the following we study this in the framework of three and than two right-handed neutrino models. In general if $n$ and $n'$ are the number of generations of the left- and right-handed neutrinos that take part in the seesaw then the total number of parameters in the effective theory is estimated to be [@broncano_plb.03] $$\begin{aligned} N_{\rm moduli}&=&n+n'+nn'\,, \label{mod}\\ N_{\rm phase}&=&n(n'-1)\,. \label{phs}\end{aligned}$$ For $n=3$ and $n^\prime = 3$, $N_{\rm moduli} = 15$ and $N_{\rm phase}=6$, which in the effective theory manifests as three masses of charged leptons, three masses of right-handed neutrinos and remaining 15 parameters including nine moduli and six phases in the Dirac mass matrix $m_D$ in a basis where the charged lepton mass matrix is real and diagonal. In the bi-unitary parameterization the mass matrix $m_D$ can be given as $$m_D=U_L^\dagger m_D^{diag} U_R\,, \label{para_step1}$$ where $U_L$ and $U_R$ are $3\times 3$ unitary matrices. $U_L$ diagonalizes the left-handed sector while $U_R$ is the diagonalizing matrix of $m_D^\dagger m_D$. Any arbitrary $3 \times 3$ unitary matrix $U$ can be written as $$\begin{aligned} U=e^{i\varphi} P_1 {\widetilde U} P_2\,, \label{udef}\end{aligned}$$ where $\varphi$ is an overall phase and $$\begin{aligned} P_1 &=& {\rm diag.}(1, e^{-i\alpha_1}, e^{-i\alpha_2})\,, \label{p1}\\ P_2 &=& {\rm diag.}(1, e^{-i\beta_1}, e^{-i\beta_2})\,, \label{p2}\end{aligned}$$ are phase matrices. ${\widetilde U}$ is a CKM like matrix parametrized by three angles and one embedded phase. Now using Eq. (\[udef\]) in Eq. (\[para\_step1\]) we get m\_D=e\^[i(-\_L+\_R)]{}P\_[2L]{}\^\^ P\_[1L]{}\^m\_D\^[diag]{} P\_[1R]{}[U\_R]{}P\_[2R]{}. \[para\_step2\] Without loss of generality three of the left phases can be absorbed in the redefinition of charged lepton fields. As a result the effective Dirac mass matrix turns out to be m\_D=[U\_L]{}\^P\_3 m\_D\^[diag]{} [U\_R]{}P\_[2R]{}, \[para\_step3\] where $P_3=P_{1L}^\dagger P_{1R}$ is an effective phase matrix. Thus in the models with three right-handed neutrinos $m_D$ contains 15 parameters. In leptogenesis, the CP asymmetry comes in a form $m_D^\dagger m_D$, which contains $P_{2R}$ and ${\widetilde U_R}$, [*i.e.*]{}, m\_D\^m\_D=P\_[2R]{}\^\^(m\_D\^[diag]{})\^2 [U\_R]{}P\_[2R]{}, and hence is independent of $P_{3}$ and ${\widetilde U_L}$. Although it would be good to know the exact relationship of the phases in $P_{2R}$ and ${\widetilde U_R}$ with the phases appearing in the $U_{PMNS}$ matrix but that is not possible. So, we try with some special cases. [**Case-I:**]{} Let us first consider the case, when ${\widetilde U_R}$ is a diagonal matrix. This is the case when the right-handed neutrino Majorana mass matrix is diagonal to start with. The mass matrix can still contain Majorana phases. In that case, ${\widetilde U_R}$ and $m_D^{diag}$ will commute and hence $m_D^\dagger m_D$ will be real and there will not be any leptogenesis. This already tells us that the phases in leptogenesis crucially depends on the mixing of the right-handed physical neutrinos. Even in this case there will be CP violation at low energy as we shall see below. The light neutrino mass matrix is given by $$m_\nu = -{\widetilde U_L}^\dagger (P_3)^2({\widetilde U_R})^2 (P_{2R})^2 (m_D^{diag})^2 M_R^{-1} {\widetilde U_L}^*$$ so that the PMNS matrix will become $$U_{PMNS} = {\widetilde U_L}^\dagger P_3P_{2R}\,.$$ Thus both the Dirac and Majorana phases at low energy are non-vanishing. [**Case-II:**]{} We shall now consider another special case when there is no leptogenesis. If the diagonal Dirac neutrino mass matrix is proportional to a unit matrix, i.e., $m_D = m \cdot I$ ($I$ is the identity matrix), again there is no leptogenesis, $$m_D^\dagger m_D= m^2 \cdot I \,.$$ In this case the light neutrino mass matrix becomes $$m_\nu = -{\widetilde U_L}^\dagger P_3 {\widetilde U_R} P_{2R} m^2 M_R^{-1}P_{2R}{\widetilde U_R}^T P_3 {\widetilde U_L}^*\,,$$ so that the PMNS matrix can be read off to be $$U_{PMNS} = {\widetilde U_L}^\dagger P_3 {\widetilde U_R} P_{2R}\,.$$ Even in this case the Dirac and Majorana phases are present. Thus in both these examples, even if CP violation is observed at low energy neutrino experiments, this CP violation may not be related to leptogenesis. Since it is not possible to make any further progress with three heavy neutrinos, we shall now restrict ourselves to models with two heavy neutrinos. Parameterization of $m_D$ in 2RH Neutrino models {#sec3} ================================================ From now on we shall work with only two right-handed (2RH) neutrinos. This result will be applicable when there are only two heavy neutrinos or when there are three heavy neutrinos but one of them do not mix with others and heavier than the other right-handed neutrinos and hence its contribution to the light neutrinos is also negligible. In the present case where we have $n=3$ and $n'=2$, from Eq. (\[mod\]) and (\[phs\]), we get $N_{\rm moduli}=11$ and $N_{\rm phase}=3$. The 14 parameters in the effective theory manifest them as three masses of charged leptons, two masses of right handed neutrinos and remaining nine parameters including six moduli and three phases appear in the Dirac mass matrix $m_D$. There are various textures and their phenomenological implications of $m_D$ in the 2RH neutrino models that have been considered in the literature [@2RH-textures]. In this article a general parametrization of the $3\times 2$ mass matrix of the Dirac neutrinos is considered. This is given by $$\begin{aligned} m_D=v Y_\nu=v U Y_{2RH}\,, \label{trip}\end{aligned}$$ where $U$ is an arbitrary Unitary matrix and the Yukawa coupling of the two RH neutrino model is given as $$\begin{aligned} Y_{2RH}= \left( \begin{array}{ccc} 0 & & x\\ z && y e^{-i\theta}\\ 0 & & 0 \end{array}\right)\,. \label{tri1}\end{aligned}$$ A derivation of Eq. (\[tri1\]) is given in the appendix \[appA\]. However, we declare that the texture of $Y_{2RH}$ is not unique. By choosing appropriately the $U$ matrix one can place $x,y,z$ at different positions so as to get the different textures of $Y_{2RH}$ as shown in appendix \[appB\]. Using (\[udef\]) in Eq. (\[trip\]) we get $$\begin{aligned} m_D = v{\widetilde U} P_2 Y_{2RH}\,, \label{emd}\end{aligned}$$ where ${\widetilde U}$ contains four parameters including three moduli and one phase, $P_2$ contains two phases and $Y_{2RH}$ contains four parameters including three moduli and one phase which all together makes ten parameters in $m_D$. However, by multiplying the phase matrix $P_2$ with $Y_{2RH}$ one can see that one of the phases in the phase matrix $P_2$, i.e., $\beta_2$ becomes redundant and can be dropped without loss of generality. As a result the total number of effective parameters is actually nine and hence consistent with our counting. Substituting $m_D$, given by Eq. (\[emd\]), in Eq. (\[seesaw\]) we can calculate the effective neutrino mass matrix, $m_\nu$. The diagonalization of $m_{\nu}$, through the lepton flavor mixing matrix $U_{PMNS}$ [@pmns-matrix], then gives us two non-zero masses of the physical neutrinos while setting one of the mass to be exactly zero as shown in the following section. Neutrino masses and mixings in 2RH neutrino models {#sec4} ================================================== The unitary matrix ${\widetilde U}$, appearing in Eq. (\[emd\]), can be parameterized as [^1] $$\begin{aligned} {\widetilde U} = R_{23}(\Theta_{23})R_{13}(\Theta_{13}, \delta'_{13}) R_{12}(\Theta_{12})\,. \label{up}\end{aligned}$$ It turns out that this parameterization is useful in determining the leptonic mixing matrix in 2RH neutrino models. Now from Eqs. (\[seesaw\]) and (\[emd\]) we get the effective neutrino mass matrix to be $$\begin{aligned} m_\nu &=& -v^2 {\widetilde U} P_2 Y_{2RH} M_R^{-1} Y^T_{2RH} P_2 {\widetilde U}^T\,\nonumber\\ &=& - v^2 {\widetilde U}P_2 X P_2 {\widetilde U}^T\,, \label{ss1}\end{aligned}$$ where $$\begin{aligned} X= Y_{2RH} M_R^{-1} Y^T_{2RH}\,. \label{xdef}\end{aligned}$$ For simplicity of the calculation let us take $e^{-i\theta}$ common from 2nd row of $Y_{2RH}$ matrix given by Eq. (\[tri1\]) and absorb it in $P_{2}$ by redefining $\beta_{1}$ as $(\beta_{1}+\theta)\rightarrow \beta_{1}$. As a result opposite phase will reappear with $z$. Then the matrix $Y_{2RH}$ turns out to be $$\begin{aligned} Y_{2RH}=\left(\begin{array}{ccc} 0 & & x\\ z e^{i\theta}&& y\\ 0 & & 0\end{array}\right)\,. \label{tri2}\end{aligned}$$ Using Eq. (\[tri2\]) in the above Eq. (\[xdef\]) we get $$\begin{aligned} X=\left(\begin{array}{ccc} {x^{2}\over M_{2}} & {xy\over M_{2}} & 0\\ {xy\over M_{2}} & {y^{2}\over M_2} +{z^{2}e^{2i\theta}\over M_{1}} & 0\\ 0 & 0 & 0\end{array}\right)\,. \label{x}\end{aligned}$$ In writing the above equation we have used a diagonal basis of the RH neutrinos where $M_R = {\rm diag.}(M_1, M_2)$. For simplicity, we absorb $M_{1}$ and $M_{2}$ in $x, y$ and $z$ as $\frac{x}{\sqrt{M_2}} \rightarrow a, $ $\frac{y}{\sqrt{M_2}}\rightarrow b$ and $\frac{z} {\sqrt{M_1}}\rightarrow c$. So $X$ can be rewritten as: $$\begin{aligned} X=\left(\begin{array}{ccc} a^{2} & ab & 0\\ ab & b^{2}+c^{2}e^{2i\theta} & 0\\ 0 & 0 & 0\end{array}\right)\,. \label{nx} \end{aligned}$$ Looking to the effective neutrino mass matrix as given by Eq. (\[ss1\]) we can guess that the diagonalizing matrix would be of the form $$\begin{aligned} U_{PMNS}={\widetilde U} K\,, \label{uk}\end{aligned}$$ where $K$ is an unitary matrix. Using Eqs. (\[diag\]) and (\[uk\]) in Eq. (\[ss1\]) we see that $$\begin{aligned} D_m = - K^{\dagger}P_{2} X P_{2}K^*\,, \label{dmk}\end{aligned}$$ which implies that $K$ would diagonalize the matrix $P_{2} X P_{2}$. From the structure of $X$ it is clear that one of the light physical neutrinos must be massless. The matrix $K$ can be parameterized as $$\begin{aligned} K=P_2\,R_{12}(\omega,\phi)\,P\,, \label{kpar}\end{aligned}$$ where $P={\rm diag.}(e^{i\eta_{1}/2},e^{i\eta_{2}/2},1)$ and $$\begin{aligned} R_{12}(\omega,\phi)= \left(\begin{array}{ccc} \cos \omega & e^{i\phi}\sin \omega & 0\\ -e^{-i\phi}\sin \omega & \cos \omega & 0\\ 0 & 0 & 1\end{array} \right)\,, \label{rpar}\end{aligned}$$ with $$\begin{aligned} \tan 2\omega &=&\left[\frac{2ab\left(a^4 + b^4+ c^4+ 2a^2b^2 + 2b^2c^2 \cos 2\theta+ 2c^2a^2 cos 2\theta\right)^{1/2}} {\left( -a^{4}+b^{4}+c^{4}+2b^{2}c^{2}\cos 2\theta \right)}\right]\,, \label{omeg}\end{aligned}$$ and $$\begin{aligned} \tan \phi&=&\left[\frac{-c^{2}\sin 2\theta}{a^{2}+b^{2}+c^{2}\cos 2\theta} \right]\,. \label{ph}\end{aligned}$$ Since $R_{12}(\omega,\phi)$ diagonalizes the matrix $X$ the resulting diagonal matrix will have complex eigenvalues in general. However, by choosing appropriately the phases of $P$ one can make the eigenvalues of $X$ real. Using Eqs. (\[omeg\]) and (\[ph\]) we get the eigenvalues $\{\lambda_1, \lambda_2, \lambda_3\}$ of $X$ to be $$\lambda_1=a^2-ab e^{i\phi}\tan \omega\,,~~\lambda_2=e^{-2i\phi}(a^2+ ab e^{i\phi}\cot\omega) ~~~{\rm and}~~~ \lambda_3=0 \label{lambda}$$ The absolute masses of the physical neutrinos are then given by $\{ m_1=v^2 |\lambda_1|, \, m_2=v^2 |\lambda_2|,\, m_3=0\}$. The MSW effect in the solar neutrino oscillation experiments indicates that $m_2 > m_1$. The corresponding mass scale, giving rise to the $\nu_e - \nu_\mu$ oscillation, is given by $$\Delta m^2_\odot\equiv m_2^2-m_1^2=v^4( |\lambda_2|^2-|\lambda_1|^2)\,. \label{solar-mass}$$ Using Eq. (\[lambda\]) in the above equation we get the solar neutrino mass scale to be $$\begin{aligned} \Delta m^2_\odot &=&v^4 \left\{ \left[ (a^2+b^2)^2+c^4+2b^2c^2\cos 2\theta \right]^2 - 4 a^4 c^4 \right\}^{1/2} \nonumber \\ &\simeq& 8\times 10^{-5} eV^2\,. \label{sol-mass}\end{aligned}$$ The atmospheric mass scale, on the other hand, is given by $$\Delta m^2_{atm} \equiv |m_2^2 - m_3^2|=v^4( |\lambda_2|^2-|\lambda_3|^2)\,.$$ Now using Eq. (\[lambda\]) in the above equation we get the atmospheric mass scale to be $$\begin{aligned} \Delta m^2_{atm} &=& \frac{v^{4}}{2}\left((a^{2}+b^{2})^{2} +c^{4}+2b^{2}c^{2}\cos2\theta \right. \nonumber \\ && +\left. \left\{\left((a^{2}+b^{2})^{2}+ c^{4}+2b^{2}c^{2}\cos2\theta\right)^{2}-4a^{4}c^{4}\right\}^{1/2} \right)\,,\nonumber\\ &\simeq & 2\times 10^{-3} eV^2\,. \label{atmos-mass}\end{aligned}$$ These equations may be inverted to obtain $$\begin{aligned} v^4 \left((a^{2}+b^{2})^{2} +c^{4}+2b^{2}c^{2}\cos2\theta \right) &=& 2 \Delta m^2_{atm} -\Delta m^2_\odot \nonumber \\ a^4 c^4 v^8 &=& \Delta m^2_{atm} (\Delta m^2_{atm}-\Delta m^2_\odot).\end{aligned}$$ Now using Eqs. (\[p2\]) and (\[rpar\]) in Eq. (\[kpar\]) we can rewrite the matrix $K$ as $$\begin{aligned} K &=& R_{12}(\omega,\phi+\beta_1)P'\nonumber\\ &=& \left(\begin{array}{ccc} \cos\omega & e^{i(\phi+\beta_{1})}\sin \omega & 0\\ -e^{-i(\phi+\beta_{1})}\sin \omega & \cos \omega & 0\\ 0 & 0 & 1\end{array}\right) \left(\begin{array}{ccc} e^{i\eta_{1}/2} & 0 & 0\\ 0 & e^{i(\eta_{2}/2-\beta_{1})} & 0\\ 0 & 0 & e^{-i{\beta_2}}\end{array} \right)\,. \label{krlab}\end{aligned}$$ Thus using Eqs. (\[krlab\]) and (\[up\]) in Eq. (\[uk\]) the PMNS matrix $U_{PMNS}$ is given as $$U_{PMNS}=R_{23}(\Theta_{23})R_{13}(\Theta_{13},\delta'_{13})R_{12} (\Theta_{12}) R_{12}(w, \phi+\beta_1)P'\,, \label{eff-pmns}$$ where $$\begin{aligned} &&R_{12}(\Theta_{12})R_{12}(\omega,\phi+\beta_{1})=\left(\begin{array}{ccc} \cos\Theta_{12}'e^{i\rho_{1}} & \sin\Theta_{12}'e^{i\rho_{2}} & 0\\ -\sin \Theta_{12}'e^{-i\rho_{2}} & \cos\Theta_{12}'e^{-i\rho_{1}} & 0\\ 0 & 0 & 1\end{array}\right)\nonumber\\ &=& \left(\begin{array}{lll} e^{i\left(\frac{\rho_{1}+\rho_{2}}{2}\right)} & 0 & 0\\ 0 & e^{-i\left(\frac{\rho_{1}+\rho_{2}}{2}\right)} & 0\\ 0 & 0 & 1\end{array} \right)\left(\begin{array}{lll} \cos\Theta_{12}' & \sin \Theta_{12}' & 0\\ -\sin \Theta_{12}' & \cos \Theta_{12}' & 0\\ 0 & 0 & 1\end{array}\right) \left(\begin{array}{lll} e^{i\left(\frac{\rho_{1}-\rho_{2}}{2}\right)} & 0 & 0\\ 0 & e^{-i\left(\frac{\rho_{1}-\rho_{2}}{2}\right)} & 0\\ 0 & 0 & 1\end{array}\right)\,. \label{r12k}\end{aligned}$$ In the above equation we have $$\begin{aligned} \cos 2\Theta_{12}'&=&\cos 2\omega \cos 2\Theta_{12} - \cos(\phi+\beta_{1}) \sin 2\omega \sin 2\Theta_{12}\,, \label{thetap}\\ \sin(\rho_{2}-\rho_{1})&=&\sin(\phi+\beta_{1})\tan \omega \left[ \cot 2\Theta_{12}'+\frac{\cos 2\Theta_{12}}{\sin 2\Theta_{12}' }\right]\,, \label{2-1}\\ \sin(\rho_{1}+\rho_{2})&=&\frac{\sin 2\omega \sin(\phi+\beta_{1})} {\sin 2\Theta_{12}'}\,. \label{2+1}\end{aligned}$$ For further simplification of the PMNS matrix (\[eff-pmns\]) we now compute the matrix product $R_{12}(\Theta_{12})K=R_{12}(\Theta_{12})R_{12} (\omega,\phi+\beta_{1})P'$ which is given as $$\begin{aligned} & &R_{12}(\Theta_{12})R_{12}(\omega,\phi+\beta_{1})P'= e^{i({\eta_1\over 2}-\rho_2)}\nonumber\\ & &\left(\begin{array}{ccc} e^{i(\rho_{1}+\rho_{2})} & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\end{array}\right) \left(\begin{array}{ccc} \cos\Theta_{12}' & \sin\Theta_{12}' & 0\\ -\sin\Theta_{12}' & \cos\Theta_{12}' & 0\\ 0 & 0 & 1\end{array}\right) \left(\begin{array}{ccc} 1 & 0 & 0\\ 0 & e^{i(\rho_{2}-\rho_{1}+(\eta_{2}-\eta_{1})/2-\beta_{1})} & 0\\ 0 & 0 & e^{-i(\beta_2-\rho_2+{\eta_1\over 2})}\end{array} \right)\,. \label{rkmult}\end{aligned}$$ Now using Eq. (\[rkmult\]) in Eq. (\[eff-pmns\]) the $U_{PMNS}$ matrix can be rewritten as: $$\begin{aligned} U_{PMNS}&=&\tilde{U}K\nonumber\\ &=& R_{23}(\Theta_{23})R_{13}(\Theta_{13},\delta_{13})R_{12} (\Theta_{12}')\, \nonumber\\ &&{\rm diag.}(1,e^{i(\rho_{2}-\rho_{1}+ (\eta_{2}-\eta_{1})/2-\beta_{1})}, e^{-i(\beta_2-\rho_2+{\eta_1\over 2})}) \nonumber\\ &=&V\,.\,V_{ph}\,, \label{rdfu}\end{aligned}$$ where $V$ is the CKM like matrix and $V_{ph}$ is the Majorana phase matrix. The effective $CP$ violating phase in the $V$ matrix is given by $$\begin{aligned} \delta_{13}=\delta'_{13}+(\rho_{1}+\rho_{2})\,. \label{dcp}\end{aligned}$$ Note that in writing Eq. (\[rdfu\]) the overall phase $e^{i({\eta_1\over 2}-\rho_2)}$ has been taken out. Moreover, we absorb the unphysical phase matrix ${\rm diag.}(1, e^{-(\rho_1+\rho_2)}, e^{-(\rho_1+\rho_2)})$ into the redefinition of charged lepton fields. From Eqs. (\[mns-matrix\]), (\[up\]) and (\[dcp\]) we see that, for the chosen parameterization of $Y_{2RH}$, two of the mixing angles $\Theta_{23}$ and $\Theta_{13}$ remains same as of the $(2-3)$ and $(1-3)$ mixing angles in PDG parameterization of the leptonic mixing matrix. Thus we can write $\Theta_{23}\equiv \theta_{23}$ and $\Theta_{13}\equiv\theta_{13}$. While $\Theta_{12}$ gets modified to $\Theta'_{12}$ and is given by Eq. (\[thetap\]), the modified CP violating phase $\delta_{13}$ is given by Eq. (\[dcp\]). At present the best fit value of $\Theta_{23}$ is given to be $45^\circ$, while the best fit value with $1\sigma$ error the value of $\Theta'_{12}$ is given to be $33.9^\circ \pm 1.6^\circ$ [@mohapatra_review.06]. The CHOOZ experiment gives a bound on $\Theta_{13}$. Currently the most conservative upper bound on $\Theta_{13}$ at the $3\sigma$ confidence level is given to be [@fogli_group.05] $$sin^2\Theta_{13}<0.048\,,$$ which gives $\Theta_{13}<13^\circ$. Leptogenesis in 2RH neutrino models {#sec5} =================================== The Majorana mass of the RH neutrino violates L-number and hence is considered to be a natural source of L-asymmetry in the early Universe [@fukugita.86] provided its decay violates CP symmetry, a necessary criteria of Sakharov [@sakharov.67]. In a mass basis where the RH neutrinos are real and diagonal the Majorana neutrinos are defined as $N_i={1\over \sqrt{2}} (N_{Ri}\pm N_{Ri}^c)$. In this basis the CP asymmetry is given by $$\epsilon_i=\frac{\Gamma_i-\bar{\Gamma_i}}{\Gamma_i+\bar{\Gamma_i}}\,,$$ where $\Gamma_i$ is the decay rate of $N_i$. If we assume a normal mass hierarchy ($M_1<<M_2$) in the RH neutrino sector then the final L-asymmetry is given by the decay of the lighter RH neutrino, $N_1$. The CP asymmetry parameter, arising from the decay of $N_1$, is then given by $$\epsilon_1={-3\over 16 \pi v^2}\left( {M_1\over M_2} \right) \frac{Im[ (m_D^\dagger m_D)_{12}]^2}{ (m_D^\dagger m_D)_{11} }\,. \label{cpasym}$$ Using Eqs. (\[emd\]) and (\[tri1\]) in the above Eq. (\[cpasym\]) we get $$\epsilon_1={-3\over 16 \pi} \left( {M_1\over M_2}\right) y^2 \sin 2\theta\,. \label{cpasym-1}$$ From the above Eq. (\[cpasym-1\]) it is clear that if $\theta=0$ then there is no CP violation in leptogenesis. Therefore, $\theta$ can be thought of the phase associated with $M_i$ in a basis where $M_i$’s are complex. Moreover, $\theta$ always hangs around $y$. So $y=0$ implies no leptogenesis. We will discuss more about it in sec.VI while we compare the CP violation in leptogenesis, neutrino oscillation and neutrinoless double beta decay processes. We now estimate the magnitude of $L$-asymmetry. A net L-asymmetry arises when $\Gamma_1$ fails to compete with the Hubble expansion parameter, $H=1.67g_*^{1/2}(T^2/M_{pl})$, where $g_*$ is the number of relativistic degrees of freedom at the epoch of temperature $T$. In a comoving volume the $L$-asymmetry is defined as $$Y_L=\epsilon_1 Y_{N_1} d\,, \label{YL-def}$$ where $d$ is the dilution factor arises due to the competitions between $\Gamma_1$ and $H$ at $T\simeq M_1$. Now using Eq. (\[cpasym-1\]) in the above Eq. (\[YL-def\]) we get $$Y_L=-5.97 \times 10^{-5}{M_1\over M_2} \left( \frac{Y_{N_1}d}{10^{-3}}\right) y^2 \sin 2\theta\,. \label{cal-YL}$$ A part of the L-asymmetry is then transferred to the B-asymmetry via the sphaleron processes which are unsuppressed above the electroweak phase transition. Taking into account the particle content in the $SM$ the B- and L-asymmetries are related as $$B=\frac{p}{p-1}L\simeq -0.55 L\,, \label{BL-rel}$$ where $p=28/79$ appropriate for the particle content in the $SM$. As a result we get the net B-asymmetry per comoving volume to be $$Y_B\simeq 3.28 \times 10^{-5}{M_1\over M_2} \left( \frac{Y_{N_1}d}{10^{-3}}\right) y^2 \sin 2\theta\,. \label{com-asy}$$ The observed B-asymmetry, on the other hand, is given by $$\left( \frac{n_B}{n_\gamma} \right) = 7 Y_B = 2.3\times 10^{-4} {M_1\over M_2} \left( \frac{Y_{N_1}d}{10^{-3}}\right) y^2 \sin 2\theta \,. \label{final-asy}$$ Comparing the above Eq. (\[final-asy\]) with the observed matter antimatter asymmetry, given by Eq.(\[asymmetry\]), we get $$y^2 \sin 2\theta=(2.57~~-~~2.78)\times 10^{-6}{M_2\over M_1} \left( \frac{10^{-3}}{Y_{N_1}d} \right)\,. \label{theory-expt}$$ We have shown the allowed values of $y$ in fig. (\[ytheta\]), using $(Y_{N1}d)=10^{-3}$, for hierarchical RH neutrinos in the $y-\theta$ plane. It is shown in fig. \[ytheta\](a) that for $(M_1/M_2)=0.1$ the minimum allowed value of $y$ is $5\times 10^{-3}$. However, this value is lifted up to $1.7\times 10^{-2}$ for $(M_1/M_2)=0.01$ as shown in fig. \[ytheta\](b). CP violation in re-phasing invariant formalism ============================================== It is convenient to study CP violation in a re-phasing invariant formalism. In particular, for the CP violation in the leptonic sector the latter makes it very interesting. The CP violation in any lepton number conserving processes comes out to be of the form [@jarlskog] $$J_{abij} = {\rm Im} [ V_{ai} V_{bj} V^\ast_{aj} V^\ast_{bi} ],$$ where $V$ is the CKM like matrix in the lepton sector. On the other hand, CP violation in any lepton number violating processes will be of the form [@Nieves:1987pp] $$\label{x1} t_{aij} = {\rm Im} [ V_{ai} V^\ast_{aj} (V_{ph})^\ast_{ii} (V_{ph})_{jj}]\,.$$ Now one can have as many independent re-phasing invariant measures $t$ as many independent Majorana CP phases. For three generations there are two independent $t$’s (denoted as $J_1$ and $J_2$) and one J (denoted as $J_{CP}$). For example, in the neutrinoless double beta decay the following re-phasing invariant will appear $$\label{x2} T = {\rm Im} [ V_{ai} V_{aj} V^\ast_{bi} V^\ast_{bj} ] \sim t_{aij} t^\ast_{bij}.$$ It has been shown that the re-phasing invariant CP violating quantity $J_{CP}$ only appears in the neutrino oscillations and that of $J_1$, $J_2$ appears in the neutrinoless double beta decay processes which may be observed in the next generation experiments. CP-violation in leptogenesis and neutrino oscillation ----------------------------------------------------- It has been pointed out that the Dirac phase $\delta_{13}$ can be measured in the long baseline neutrino oscillation experiments [@dirac_phase_group]. In that case the CP violation arises from the difference of transition probability $\Delta P= P_{\nu_e\rightarrow \nu_\mu}-P_{\bar{\nu}_e\rightarrow \bar{\nu}_\mu}$. It can be shown that the transition probability $\Delta P$ is proportional to the leptonic Jarlskog invariant $$J_{CP}=Im [V_{e1}V^*_{e2}V^*_{\mu 1}V_{\mu 2}]\,. \label{jcp}$$ Using Eq. (\[rdfu\]) the re-phasing invariant $J_{CP}$ can be rewritten as $$J_{CP}=\frac{1}{8}\sin 2\Theta'_{12}\sin 2\Theta_{23}\sin 2\Theta_{13} \cos \Theta_{13} \sin (\delta'_{13}+\rho_1+\rho_2)\,. \label{jcp-1}$$ Now using Eqs. (\[omeg\]), (\[ph\]), (\[thetap\]) and (\[2+1\]) in the above Eq. (\[jcp-1\]) we get $$\begin{aligned} J_{CP} &=& \frac{1}{8}\frac{\sin 2\Theta_{23}\sin 2\Theta_{13}\cos \Theta_{13}} {\sqrt{\left[(a^2+b^2)^2+c^4+2b^2c^2\cos 2\theta\right]^2-4a^4c^4}}\nonumber\\ &\times& \left[ 2ab \cos\delta'_{13}\{ -c^2 \sin 2\theta \cos\beta_1+(a^2+b^2+c^2\cos2\theta)\sin\beta_1\}\right.\nonumber\\ &+&\left. 2ab\cos 2\Theta_{12}\sin\delta'_{13}\{ (a^2+b^2+c^2\cos 2\theta)\cos\beta_1+c^2\sin 2\theta \sin\beta_1\}\right.\nonumber\\ &+& \left.\sin \delta'_{13}\sin 2\Theta_{12}(-a^4+b^4+c^4+2b^2 c^2 \cos 2\theta) \right]\,. \label{jcp-2}\end{aligned}$$ From the above Eq. (\[jcp-2\]) it is obvious that $J_{CP}=0$ only if both $\sin \delta'_{13}=0$ and $b=0$, while only $b=0$ (equivalently $y=0$) implies the condition for “no leptogenesis". This indicates that there is no one-to-one correspondence between the CP violation in neutrino oscillation and the CP violation in leptogenesis, even in the 2RH neutrino models. However, it is interesting to see the common regions in the plane of $(n_B/n_\gamma)$ versus $J_{CP}$. This is shown in fig. (\[jcp-fig\]) by taking a typical set of parameters. The main aim is to illustrate the maximal contrast between the positive and negative values of $n_B/n_\gamma$ for a given set of values of $J_{CP}$. This helps us in determining the sign of the asymmetry by knowing the size of the asymmetry. From the fig. (\[jcp-fig\]) it is obvious that for the given set of parameters the positive sign of the asymmetry allows the values of $J_{CP}$ in the range $0.049 - 0.0495$ for $(M_1/M_2)=0.1$. However, this range is significantly reduced for $(M_1/M_2)=0.01$. On the other hand, the negative sign of the asymmetry allows the values of $J_{CP}$ in the range $0.0465 - 0.047$ for $(M_1/M_2)=0.1$ which is further reduced for $(M_1/M_2)=0.01$. In this figure the value of $\Theta_{12}$ is used from fig. (\[theta12-theta\]) where we have shown the allowed values of $\Theta_{12}$ as $\theta$ varies from $0$ to $\pi$. Note that the above results are true for a non-zero $\Theta_{13}$. Consequently the allowed range of values of $J_{CP}$ may vary depending on the values of $\Theta_{13}$. Thus we anticipate that in the 2RH neutrino models a knowledge of $J_{CP}$ can predict the sign of matter antimatter asymmetry of the Universe. We should note that the predictive power of the model depends on the CP violating phases $\beta_1$ and $\delta'_{13}$. This can be visible from fig. (\[jcpvalues-fig\]) where we have shown the variation of $n_B/n_\gamma$ with $J_{CP}$ for different values of $\beta_1$ and $\delta'_{13}$. In particular, for the choice ($\beta_1=\pi/2$, $\delta'_{13}=0$) and ($\beta_1=0$, $\delta'_{13}=\pi/2$), the contrast between the positive and negative values of $n_B/n_\gamma$ is almost zero for a given set of values of $J_{CP}$. On the other hand, for the choice ($\beta_1=\pi/2$, $\delta'_{13}=\pi/2$) and ($\beta_1=0$, $\delta'_{13}=0$), the contrast between the positive and negative values of $n_B/n_\gamma$ is maximal and can be chosen for the present purpose. CP violation in leptogenesis and neutrinoless double-beta decay --------------------------------------------------------------- The observation of the neutrinoless double beta decay would provide direct evidence for the violation of total $L$-number in the low energy effective theory and hence probing the left-handed physical neutrinos to be Majorana type. Note that the $L$-number violation at high energy scale is a necessary criteria for leptogenesis. In the canonical seesaw models this is conspired by assuming that the RH neutrinos are Majorana in nature. However, this assumption doesn’t ensure that the left-handed physical neutrinos are Majorana type. Assuming that the physical neutrinos are of Majorana type we investigate the connecting links between the two $L$-number violating phenomena occurring at two different energy scales. In the low energy effective theory with three generations of left-handed fermions, apart from the $J_{CP}$, one can write two more re-phasing invariants $J_1$ and $J_2$ which designates lepton number violation and CP violation [@Nieves:1987pp]. However, in the models with two RH neutrinos one of the eigen values of the physical light neutrino mass matrix is exactly zero. Therefore, the corresponding phase in the Majorana phase matrix can always be chosen so as to set one of the lepton number violating CP violating re-phasing invariant to zero. In the present case $m_3=0$ and hence the corresponding phase is arbitrary. This is ensured through $\beta_2$ which is redundant and pointed out in Eq. (\[emd\]). Therefore, from Eq. (\[rdfu\]) we can write the only $L$-number violating CP violating re-phasing invariant as: $$\begin{aligned} J&=& Im \left[ V_{e1}V_{e2}^*(V_{ph})_{11}^* (V_{ph})_{22}\right]\nonumber\\ &=& -\frac{1}{2}\sin 2\Theta'_{12} \cos^2\Theta_{13} \sin(\rho_2-\rho_1+ {(\eta_2-\eta_1)\over 2}-\beta_1)\,. \label{J_lep}\end{aligned}$$ Using Eq. (\[rkmult\]) the above Eq. (\[J\_lep\]) can be rewritten as $$\begin{aligned} J &=&-\frac{\cos^2 \Theta_{13}}{2}\frac{1} {\left[ (a^2+b^2)^2+c^4+2b^2c^2 \cos 2\theta+2c^2a^2 \cos 2\theta \right]}\nonumber\\ &\times& \left[ \sin 2\Theta_{12} \cos \theta \{-c^2 \sin 2\theta \cos \beta_1+(a^2+b^2+c^2 \cos 2\theta) \sin \beta_1 \}\right.\nonumber\\ &&\left. \times \sqrt{(a^2+b^2)^2+c^4+2c^2a^2+2b^2c^2 \cos 2\theta}\right. \nonumber\\ &+& \left. \sin 2\Theta_{12} \sin \theta \{c^2 \sin 2\theta \sin \beta_1+ (a^2+b^2+c^2 \cos 2\theta) \cos \beta_1 \}\right.\nonumber\\ && \left. \times \frac{(-a^4+b^4+c^4+2 b^2c^2 \cos 2\theta)}{\sqrt{(a^2+b^2)^2 +c^4+2 c^2a^2+2b^2c^2 \cos 2\theta}}\right.\nonumber\\ &+&\left. \cos 2\Theta_{12}\sin \theta \frac{2ab\{(a^2+b^2)^2+c^4+2b^2c^2 \cos 2\theta +2c^2a^2 \cos 2\theta \}}{\sqrt{(a^2+b^2)^2+c^4+2c^2a^2+ 2b^2c^2 \cos 2\theta}} \right]\,. \label{J-2-final}\end{aligned}$$ In the above Eq. (\[J-2-final\]) the allowed values of $\Theta_{12}$ is obtained from $$\begin{aligned} \cos \Theta'_{12} &=& \left[\frac{1}{2}\left[1+\frac{\left(-a^{4}+b^{4}+ c^{4}+2b^{2}c^{2}\cos2\theta\right)\cos 2\Theta_{12}}{\sqrt{((a^2+b^2)^2+c^4+ 2b^2c^2 \cos 2\theta)^2-4a^4c^4}}\right.\right.\nonumber\\ &&-\left.\left.\sin2\Theta_{12} \frac{2ab\{ c^{2}\sin2\theta\sin\beta_{1} +(a^{2}+b^{2}+c^{2}\cos2\theta)\cos\beta_{1}\} }{\sqrt{((a^2+b^2)^2+c^4+ 2b^2c^2 \cos 2\theta)^2-4a^4c^4}}\right] \right]^{1/2}\,, \label{theta12_prime}\end{aligned}$$ by fixing $\Theta'_{12}=(33.9\pm 1.6)^\circ$. This is shown in fig. (\[theta12-theta\]). From Eq. (\[J-2-final\]) one can see that $J \neq 0$ as $\theta \rightarrow 0$ which is the condition for “no leptogenesis". Thus we see that there is no one-to-one correspondence between the two $L$-number violating processes occurring at two different energy scales. However, it is always interesting to see the overlapping regions in the plane of $\frac{n_B}{n_\gamma}$ versus $J$ as $\theta$ varies from $0$ to $\pi$. This is shown in fig. (\[j2-asy\]) for a typical set of parameters. From fig. (\[j2-asy\]) one can see that for positive sign of the $B$-asymmetry the values of $J$ lie in between $-0.45$ to $-0.1$ for $(M_1/M_2)=0.1$. This range is further reduced to $(-0.4 - -0.15)$ for $(M_1/M_2)=0.01$. On the other hand, for the negative sign of the $B$-asymmetry the values of $J$ lie in the range $(0.05 - 0.45)$ for $(M_1/M_2)=0.1$ and in the range $(0.15 - 0.4)$ for $(M_1/M_2)=0.01$. Thus we see that within the allowed range of parameters the contrast between the positive and negative values of $\frac{n_B}{n_\gamma}$ is maximum for a given set of values of $J$. Therefore, we expect a knowledge of $J$ can precisely determine the sign of $B$-asymmetry since the value of $n_B/n_\gamma$ is known. Finally we note that, unlike $J_{CP}$, $J$ remains non-vanishing even if $\Theta_{13}=0$ [^2]. Now the remaining question to be addressed is how $n_B/n_\gamma$ varies with respect to $J$ for different values of $\beta_1$. This is shown in fig. (\[jvalues-asy\]) for a given set of parameters. One can see that for $\beta_1=0$ and $\beta_1=\pi$ both positive and negative values of $n_B/n_\gamma$ correspond to the same set of values of $J$ which is unwelcome for determination of sign of the asymmetry. On the other hand for $\beta_1\neq 0,\pi$ one can have maximal contrast between the positive and negative values of $n_B/n_\gamma$ for the given set of values of $J$ and hence can be chosen for the present purpose. Conclusions =========== We have studied the connecting links between the CP violating phase(s) giving rise to leptogenesis, occurring at a high energy scale, and the CP violating phases appearing in the low energy phenomena, i.e., neutrino oscillation and neutrinoless double beta decay processes. This is studied in the framework of two right-handed neutrino models. The low energy leptonic CP violation is studied in a re-phasing invariant formalism. It is shown that there are only two re-phasing invariants; (1) The lepton number conserving CP violating re-phasing invariant $J_{CP}$ which can be determined in the future long-baseline neutrino oscillation experiments, (2) The lepton number violating CP violating re-phasing invariant $J$ which can be determined in the neutrinoless double beta decay experiments. It is found that there is no one-to-one correspondence between these two CP violating phenomena, occurring at two different energy scales, even though the number of parameters involving in the seesaw is exactly same as the number of low energy observable parameters. However, in a suitable parameter space we have shown that the overlapping regions in the plane of $n_B/n_\gamma$ versus $J_{CP}$ and $n_B/n_\gamma$ versus $J$ can indeed determine the [*sign*]{} of the matter antimatter asymmetry of the present Universe assuming that the [*size*]{} of the asymmetry is precisely known. Parameterization of $Y_{2RH}$ {#appA} ============================= To parameterize the neutrino Dirac Yukawa coupling in two right-handed neutrino models we follow the same procedure adopted in Ref. [@morozumietal]. Let ${\bf u_1}$, ${\bf u_2}$, ${\bf u_3}$ be three orthonormal 3 dimensional vectors. Using these basis vectors we can write the most general unitary matrix $U$ as: $$\begin{aligned} U = ({\bf u_1}\,\,{\bf u_2}\,\,{\bf u_3})\,. \label{udef1}\end{aligned}$$ Let us consider an arbitrary $3 \times 2$ matrix $Y$ which in terms of the 3-dimensional vectors ${\bf y_1}$ and ${\bf y_2}$ can be written as: $$\begin{aligned} Y = ({\bf y_1}\,\,{\bf y_2})\,. \label{ydef}\end{aligned}$$ Without loss of generality we choose ${\bf u_2}=\frac{{\bf y_1}} {|{\bf y_1}|}$. As a result we get $$\begin{aligned} U^\dagger Y = \left( \begin{array}{cccc} 0& & \alpha_{12}\\ |y_1| & & \alpha_{22}\\ 0& & \alpha_{32} \end{array} \right)\,, \label{udagy}\end{aligned}$$ where $\alpha_{ij}={\bf u_i}^\dagger \cdot {\bf y_j}$. Let $V$ be another unitary matrix which we choose to be of the form: $$\begin{aligned} V= \left( \begin{array}{ccccc} \frac{\alpha_{12}}{\sqrt{|\alpha_{12}|^2+|\alpha_{32}|^2}} & & 0 & & \beta_{13}\\ 0 & & 1 & & 0 \\ \frac{\alpha_{32}}{\sqrt{|\alpha_{12}|^2+|\alpha_{32}|^2}} & & 0 & & \beta_{33} \end{array} \right)\,, \label{vmat}\end{aligned}$$ where $\beta_{ij}$ must follow $\alpha_{12} \beta^*_{13}+\alpha_{32} \beta^*_{33}=0$ and $|\beta_{13}|^2+|\beta_{33}|^2=1$. Consequently we have $$\begin{aligned} V^\dagger U^\dagger Y = \left( \begin{array}{cccc} 0& & \sqrt{|\alpha_{12}|^2 + |\alpha_{32}|^2}\\ |y_1| & & \alpha_{22}\\ 0 & & 0 \end{array} \right)\,,\end{aligned}$$ where we set $V_{32}=0$ by imposing the unitarity condition of $V$. This implies that we can always write any arbitrary $3\times 2$ matrix $$\begin{aligned} Y = W Y_{2RH}\,,\end{aligned}$$ where $W=VU$ is an unitary matrix and the texture of $Y_{2RH}$, the Yukawa coupling in the two right handed neutrino mass models, is given as $$\begin{aligned} Y_{2RH}= \left( \begin{array}{cccc} 0 & & x\\ z& & y e^{-i\theta}\\ 0 & & 0 \end{array}\right)\,,\end{aligned}$$ where $x$, $y$, $z$ and $\theta$ are real numbers. Note that by appropriately choosing the $U$ and $V$ matrices one can construct the $Y_{2RH}$ matrix in twelve possible ways. Possible textures of $Y_{2RH}$ and neutrino mixings {#appB} =================================================== In this appendix we specify the various possible textures of $Y_{2RH}$. One of the particular texture of $Y_{2RH}$ has been used in section \[sec2\] for our work. In the table-I we write all the possible textures of $Y_{2RH}$. [|c|c|c|]{}\ I & $\begin{pmatrix} 0 & 0\cr z & y e^{-i\theta}\cr 0 & x \end{pmatrix}$ $\begin{pmatrix} 0 & 0\cr 0 & x\cr z & ye^{-i\theta} \end{pmatrix}$ & $\begin{pmatrix} 0 & 0\cr x & 0 \cr y e^{-i\theta} & z\end{pmatrix}$ $\begin{pmatrix} 0 & 0\cr y e^{-i\theta} & z\cr x & 0 \end{pmatrix}$\ \ II & $\begin{pmatrix} z & y ^{-i\theta}\cr 0 & 0\cr 0 & x \end{pmatrix}$ $\begin{pmatrix} 0 & x\cr 0 & 0\cr z & y e^{-i\theta}\end{pmatrix}$ & $\begin{pmatrix}y ^{-i\theta} & z\cr 0 & 0\cr x & 0 \end{pmatrix}$ $\begin{pmatrix} x & 0\cr 0 & 0\cr y e^{-i\theta} & z \end{pmatrix}$\ \ III & $\begin{pmatrix} z & y e^{-i\theta}\cr 0 & x\cr 0 & 0 \end{pmatrix} $ $\begin{pmatrix} 0 & x\cr z & y e^{-i\theta}\cr 0 & 0\end{pmatrix} $ & $\begin{pmatrix} y e^{-i\theta} & z\cr x & 0\cr 0 & 0 \end{pmatrix}$ $\begin{pmatrix} x & 0\cr y e^{-i\theta} & z\cr 0 & 0 \end{pmatrix}$\ Each possible $Y_{2RH}$ in table-I will lead to various forms of $X$, apparent from Eq. (\[xdef\]). 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--- abstract: 'A certain Diophantine problem and 2D crystallography are linked through the notion of [*standard realizations*]{} which was introduced originally in the study of random walks. In the discussion, a complex projective quadric defined over $\mathbb{Q}$ is associated with a finite graph. “Rational points" on this quadric turns out to be related to standard realizations of 2D crystal structures. In the last section, it is observed that the number of rational points which correspond to periodic tilings is finite.' address: ' Department of Mathematics, Meiji University, Higashimita 1-1-1, Tama-ku, Kawasaki, 214-8571 Japan ' author: - Toshikazu Sunada title: Standard 2D Crystalline Patterns and Rational Points in Complex Quadrics --- Introduction ============ The standard crystalline pattern is a synonym of the [*standard realization*]{} (or canonical placement) of a crystal structure introduced in [@sk2] which gives the most symmetric crystalline shape among all possible realizations, and is characterized uniquely (up to similar transformations) as a minimizer of a certain energy functional. ![Typical 2D standard realizations[]{data-label="fig:classical"}](classical.eps){width=".6\linewidth"} Figure \[fig:classical\] exhibits some of 2D examples which are familiar to scientists; the square lattice, honeycomb (regular hexagonal lattice), equilateral triangular lattice, and regular kagome lattice. In the 3D case, Diamond, Lonsdaleite[^1], and the nets associated with the face-centered and body-centered lattices give standard crystalline patterns. This notion, via the elementary theory of homology and covering spaces, offers an effective method in a systematic enumeration and design of crystal structures (see [@su4])[^2]. Remarkably, it shows up in asymptotic behaviors of simple random walks ([@sk1]), and is also related to a combinatorial analogue of Abel-Jacobi maps ([@su5]). The aim of this paper is to observe another interesting aspect of standard realizations; specifically we establish a link between standard 2-dimensional crystalline patterns and “rational points" of certain complex quadrics defined over $\mathbb{Q}$. A rational point we mean here is a point in a complex projective space each of whose homogeneous coordinate is represented by a number in an [*imaginary quadratic field*]{}. For instance, the regular kagome lattice (the lower right in Fig. \[fig:classical\]) corresponds to the $\mathbb{Q}(\sqrt{-3})$-rational point $$\Big[\frac{1+\sqrt{-3})}{2}, \frac{1-\sqrt{-3}}{2}, -1, \frac{1+\sqrt{-3}}{2}, \frac{1-\sqrt{-3}}{2}, -1\Big]$$ (or its complex conjugate) of the 2-dimensional projective quadric defined over $\mathbb{Q}$ $$\begin{aligned} &&\{[z_1,z_2,z_3, z_4,z_5,z_6]\in P^5(\mathbb{C});~z_1{}^2+\cdots+z_6{}^2=0,\\ && \qquad z_1+z_6=z_3+z_4,~ z_2+z_4=z_1+z_5,~z_3+z_5=z_2+z_6\},\end{aligned}$$ which is biregular (over $\mathbb{Q}$) to $$\begin{aligned} && \{[w_1,w_2,w_3,w_4]\in P^3(\mathbb{C})|~3w_1{}^2+2w_2{}^2+2w_3{}^2+3w_4{}^2\\ && \qquad \qquad -2w_1w_2-4w_1w_4-2w_1w_3 +2w_2w_4+2w_3w_4=0\}.\end{aligned}$$ The way to give this correspondence is not [*ad hoc*]{}. Indeed there is a systematic procedure to associate a complex projective quadrics ${\bf Q}(X_0)$ defined over $\mathbb{Q}$ with a [*finite graph*]{} $X_0$, which is an interesting object for its own sake. In our setting, $X_0$ turns up as the quotient graph by the translational action of a lattice group on the crystal structure. The point corresponding to the standard crystalline pattern is obtained by taking the intersection of this quadric and a certain (projective) line. It is also observed that every rational point on the quadric yields a standard 2D crystalline pattern. Incidentally all the examples given in Fig. \[fig:classical\] happen to be derived from [*tilings*]{} of plane. As a matter of fact, it is rather rare to find tilings among standard 2D crystalline patterns; namely as will be proved in the last section, there are only finitely many rational points on ${\bf Q}(X_0)$ which correspond to tilings. Standard realizations {#sec:standard} ===================== We first review some basic results about standard realizations (see [@su5] for the terminology used in the present article). A [*graph*]{} is represented by an ordered pair $X = (V,E)$ of the set of [*vertices*]{} $V$ and the set of all [*directed edges*]{} $E$ (note that each edge has just two directions, which are to be expressed by arrows). For an directed edge $e$, we denote by ${\it o}(e)$ the [*origin*]{}, and by ${\it t}(e)$ the [*terminus*]{}. The inversed edge of $e$ is denoted by $\overline{e}$. With these notations, we have $o(\overline{e})=t(e)$, $t(\overline{e})=o(e)$. We also use the notation $E_x$ for the set of directed edges $e$ with $o(e)=x$. Throughout, the degree ${\rm deg}~\!x=|E_x|$ is assumed to be greater than or equal to three for every vertex $x$. The network associated with a $d$-dimensional crystal is identified with a periodic realization of a $d$-dimensional [*topological crystal*]{} (an infinite-fold covering graph) $X=(V,E)$ over a finite graph $X_0=(V_0,E_0)$ whose covering transformation group is a free abelian group $L$ of rank $d$ ($d=2$ or $3$ when we are handling a network associated with a real crystal). Theory of covering spaces tells us that there is a subgroup $H$ (called a [*vanishing subgroup*]{}) such that $H_1(X_0,\mathbb{Z})/H=L$ (note that $H$ is a direct summand of $H_1(X_0,\mathbb{Z})$). Actually the topological crystal $X$ is the quotient graph of the maximal abelian covering graph $X_0^{\rm ab}$ over $X_0$ modulo $H$. We call $X_0^{\rm ab}$ the [*maximal topological crystal*]{} over $X_0$. We denote by $\mu:H_1(X_0,\mathbb{Z})\longrightarrow L$ the canonical homomorphism. Precisely speaking, a periodic realization is a piecewise linear map $\varPhi:X\longrightarrow \mathbb{R}^d$ satisfying $$\varPhi(\sigma x)=\varPhi(x)+\rho(\sigma)\qquad (\sigma \in L),$$ where $\rho:L\longrightarrow \mathbb{R}^d$ is an injective homomorphism whose image is a lattice in $\mathbb{R}^d$.[^3] We call $\rho$ (resp. $\rho(L)$) the [*period homomorphism*]{} (resp. the [*period lattice*]{}) for $\varPhi$. By putting ${\bf v}(e)=\varPhi\big(t(e)\big)-\varPhi\big(o(e)\big)$  $(e\in E)$, we obtain a $L$-invariant function ${\bf v}$ on $E$ which we may identify with a 1-cochain ${\bf v}\in C^1(X,\mathbb{R}^d)$ with values in $\mathbb{R}^d$. Since ${\bf v}$ determines completely $\varPhi$ (up to parallel translations), we shall call ${\bf v}$ the [*building cochain*]{}[^4] of $\varPhi$. One can check that if we identify the cohomology class $[{\bf v}]\in H^1(X_0,\mathbb{R}^d)$ with a homomorphism of $H_1(X_0,\mathbb{Z})$ into $\mathbb{R}^d$ (the [*duality of cohomology and homology*]{}), then $[{\bf v}]=\rho\circ \mu$. In particular, ${\rm Ker}~\![{\bf v}]=H$ and ${\rm Image}~\![{\bf v}]=\rho(L)$. [([@su4])]{} Giving a periodic realization of a topological crystal over $X_0$ is equivalent to giving a 1-cochain ${\bf v}\in C^1(X_0,\mathbb{R}^d)$ such that the image of the homomorphism $[{\bf v}]:H_1(X_0,\mathbb{Z})\longrightarrow \mathbb{R}^d$ is a lattice in $\mathbb{R}^d$. Among all periodic realizations of $X$, there is a “standard" one which is characterized uniquely (up to similar transformations) by the following two conditions: \(1) ([**Harmonicity**]{}) $$\label{eq:harmonic} \displaystyle\sum_{e\in E_{0x}}{\bf v}(e)={\bf 0} \quad (x\in V_0),$$ \(2) ([**Tight-frame condition**]{}[^5]) There exists a positive constant $c$ such that $$\label{eq:standard} \sum_{e\in E_0} \langle{\bf x},{\bf v}(e) \rangle {\bf v}(e)=c{\bf x}\quad ({\bf x}\in \mathbb{R}^d).$$ In the coordinate form, (\[eq:standard\]) is written as $$\sum_{e\in E_0}v_i(e)v_j(e)=c\delta_{ij},$$ where ${\bf v}={}^t\big(v_1(e),\ldots,v_d(e)\big)$. In particular $$\label{eq:cd} \sum_{e\in E_0}\|{\bf v}(e)\|^2=cd.$$ (\[eq:standard\]) is also equivalent to $$\sum_{e\in E_0} \langle{\bf x},{\bf v}(e) \rangle^2=c\|{\bf x}\|^2.$$ The constant $c$ does not play a role since we are considering similarity classes of realizations; later we handle the case $c=2$. [([@sk2], [@sk11])]{} [(1)]{} The standard realization is the unique minimizer, up to similar transformations, of the energy[^6] $$\mathcal{E}(\varPhi)={\rm vol}\big(\mathbb{R}^d/\rho(L)\big)^{-2/d}\sum_{e\in E_0}\|{\bf v}(e)\|^2.$$ [(2)]{} Let $\varPhi:X\longrightarrow \mathbb{R}^d$ be the standard realization. Then there exists a homomorphism $\kappa$ of the automorphism group ${\rm Aut}(X)$ of $X$ into the congruence group $M(d)$ of $\mathbb{R}^d$ such that [(a)]{} when we write $\kappa(g)=\big(A(g),b(g)\big)\in O(d)\times \mathbb{R}^d$, we have $$\varPhi(gx)=A(g)\varPhi(x)+b(g) \qquad (x\in V),$$ [(b)]{} the image $\kappa\big({\rm Aut(X)}\big)$ is a crystallographic group. [**Remark**]{}  Equation (\[eq:harmonic\]) says that the cochain ${\bf v}$ is “harmonic" in the sense that $\delta{\bf v}=0$ where $\delta:C^1(X_0,\mathbb{R}^d)\longrightarrow C^1(X_0,\mathbb{R}^d)$ is the adjoint of the coboundary operator $d:C^0(X_0,\mathbb{R}^d)\longrightarrow C^1(X_0,\mathbb{R}^d)$ with respect to the natural inner products in $C^i(X_0,\mathbb{R}^d)$. Using a discrete analogue of the Hodge–Kodaira theorem, one can prove that the correspondence ${\bf v}\in {\rm Ker}~\!\delta\mapsto [{\bf v}]\in H^1(X_0,\mathbb{R}^d)$ is a linear isomorphism (hence ${\rm dim}~\!{\rm Ker}~\!\delta=db_1(X_0)$ where $b_1(X_0)$ is the first Betti number of $X_0$). Thus given $\rho$, there is a unique harmonic cochain ${\bf v}$ with $[{\bf v}]=\rho\circ \mu$. A realization satisfying (\[eq:harmonic\]) is said to be a [*harmonic realization*]{} [@sk2] (or an equilibrium placement [@d-1]), which is characterized as a minimizer of $\mathcal{E}$ when $\rho$ is fixed. Albanese tori {#sec:alb} ============= The building cochain ${\bf v}$ for the standard realization of $X$ with $c=2$ in Eq. (\[eq:standard\]) is explicitly constructed in the following way ([@su5]). First we provide $H_1(X_0,\mathbb{R})$ with a natural inner product (which allows us to identify $H_1(X_0,\mathbb{R})$ with the Euclidean space $\mathbb{R}^b$, $b=b_1(X_0)$). For this sake, we start with an inner product on $C_1(X_0,\mathbb{R})$. For $e,e'\in E_0$, we set $$\langle e, e'\rangle=\begin{cases} 1 & (e'=e)\\ -1 & (e'=\overline{e})\\ 0 & (\text{otherwise}) \end{cases},$$ which extends to an inner product on $C_1(X_0,\mathbb{R})$ in a natural manner. Restricting this inner product to the subspace $H_1(X_0,\mathbb{R})~(={\rm Ker}~\big(\partial: C_1(X_0,$ $\mathbb{R})\longrightarrow C_0(X_0,$ $\mathbb{R})\big))$, we get an Euclidean structure on $H_1(X_0,\mathbb{R})$. Let $P_{\rm ab}:C_1(X_0,\mathbb{R})\longrightarrow H_1(X_0,\mathbb{R})$ be the orthogonal projection, and put ${\bf v}_{\rm ab}(e)=P_{\rm ab}(e)$, regarding each edge as a $1$-chain. Note that $[{\bf v}^{\rm ab}]:H_1(X_0,\mathbb{Z})\longrightarrow H_1(X_0,\mathbb{R})$ coincides with the injection. One can check that ${\bf v}_{\rm ab}$ is the building cochain of the standard realization of the maximal topological crystal $X_0^{\rm ab}$ (with $c=2$). Let $X$ be a topological crystal over $X_0$ corresponding to a vanishing subgroup $H$ of $H_1(X_0,\mathbb{Z})$. Let $H_{\mathbb{R}}$ be the subspace of $H_1(X_0,\mathbb{R})$ spanned by $H$, and $H_{\mathbb{R}}^{\perp}$ the orthogonal complement of $H_{\mathbb{R}}$ in $H_1(X_0,\mathbb{R})$: $$H_1(X_0,\mathbb{R})=H_{\mathbb{R}}\oplus H_{\mathbb{R}}^{\perp}.$$ Then ${\rm dim}~\hspace{-0.05cm}H_{\mathbb{R}}^{\perp}={\rm rank}~\hspace{-0.05cm}L=d$. By choosing an orthonormal basis of $H_{\mathbb{R}}^{\perp}$, we identify $H_{\mathbb{R}}^{\perp}$ with the Euclidean space $\mathbb{R}^d$. Let $P:H_1(X_0,\mathbb{R})\longrightarrow H_{\mathbb{R}}^{\perp}=\mathbb{R}^d$ be the orthogonal projection. If we put ${\bf v}(e)=P\big({\bf v}_{\rm ab}(e)\big)$, then we find that ${\bf v}$ gives the building cochain of the standard realization of $X$ (with $c=2$). As shown in [@su5], one may reduce the construction of ${\bf v}$ to an elementary computation of matrices. We now consider two flat tori $$\begin{aligned} &&A(X_0)=H_1(X_0,\mathbb{R})/H_1(X_0,\mathbb{Z}),\\ &&A(X_0,H)=\mathbb{R}^d/{\rm Image}~\![{\bf v}],\end{aligned}$$ which, in view of analogy with classical algebraic geometry, are called the [*Albanese torus*]{} of $X_0$ and the [*generalized Albanese torus*]{} of $(X_0,H)$, respectively. The projection $P$ induces the exact sequence $$\label{exactexact} 0\longrightarrow H_{\mathbb{R}}/H\longrightarrow A(X_0)\stackrel{p}{\longrightarrow}A(X_0,H)\longrightarrow 0,$$ In the following proposition, $\kappa(X_0)$ denotes the [*tree number*]{} of $X_0$, the number of spanning trees in $X_0$. [(1) ([@sk12])]{} $ {\rm vol}\big(A(X_0)\big)=\kappa(X_0)^{1/2}. $ $(2)$ ${\rm vol}\big(A(X_0,H)\big)=\kappa(X_0)^{1/2}/{\rm vol}(H_{\mathbb{R}}/H).$ The second claim is a consequence of the exact sequence (\[exactexact\]). The volume ${\rm vol}(H_{\mathbb{R}}/H)$ is computed as $${\rm vol}(H_{\mathbb{R}}/H)=\det (\langle \alpha_i, \alpha_j\rangle)^{1/2},$$ where $\{\alpha_1,\ldots, \alpha_{b-d}\}$ is a $\mathbb{Z}$-basis of $H$. Putting $I(H)=\det (\alpha_i\cdot \alpha_j)$, we shall call $I(H)$ the [*intersection determinant*]{} for $H$, which is evidently a positive integer. We thus have the following formula. $${\rm vol}\big(A(X_0,H)\big)=\kappa(X_0)^{1/2}I(H)^{-1/2},$$ and hence, appealing to (\[eq:cd\]), we obtain $$\label{eq:energymini} \min_{\varPhi}~\!\mathcal{E}(\varPhi)=2d\kappa(X_0)^{-1/d}I(H)^{1/d},$$ where $\varPhi$ runs over all periodic realizations of $X$. Complex quadrics associated with finite\ graphs ======================================== We now embark on a new enterprise. We confine ourselves to 2D standard realizations. Let $X_0=(V_0,E_0)$ be a finite connected graph such that $b_1(X_0)$ is greater than or equal to 2. Put $$\mathbb{H}=\{{\bf z}\in C^1(X_0,\mathbb{C})|~\sum_{e\in E_{0x}}{\bf z}(e)=0~ (x\in V_0)\}.$$ This is nothing but the space of harmonic cochains (we identify $\mathbb{R}^2$ with $\mathbb{C}$), so we find $${\rm dim}_{\mathbb{C}}\mathbb{H}=b_1(X_0)$$ (see Remark at the end of Sect. \[sec:standard\]). We denote by $\mathbb{P}(\mathbb{H})$ the projective space associated with $\mathbb{H}$; that is, $\mathbb{P}(\mathbb{H})$ is the orbit space $(\mathbb{H}\backslash\{{\bf 0}\})/\mathbb{C}^{\times}$ by the natural action of the multiplicative group $\mathbb{C}^{\times}=\mathbb{C}\backslash\{0\}$. For ${\bf z}\neq {\bf 0} (\in \mathbb{H})$, we use the notation $\langle{\bf z}\rangle\in \mathbb{P}(\mathbb{H})$ for the orbit containing ${\bf z}$. It should be noted that an orientation-preserving similar transformation in $\mathbb{R}^2$ is identified with the multiplication by a non-zero complex number. A cochain ${\bf z}\in C^1(X_0,\mathbb{C})=C^1(X_0,\mathbb{R}^2)$ satisfies the tight-frame condition if and only if $\displaystyle\sum_{e\in E_0}{\bf z}(e)^2=0$. [*Proof*]{}  Put ${\bf z}(e)=a(e)+b(e)\sqrt{-1}$, and ${\bf x}={}^t(x,y)$. Then $$\sum_{e\in E_0}\langle {\bf z}(e),{\bf x}\rangle^2= \Big(\sum_{e\in E_0}a(e)^2\Big)x^2+2\Big(\sum_{e\in E_0}a(e)b(e)\Big)xy+ \Big(\sum_{e\in E_0}b(e)^2\Big)y^2.$$ Thus ${\bf z}$ satisfies the tight-frame condition if and only if $$\sum_{e\in E_0}a(e)^2=\sum_{e\in E_0}b(e)^2, \quad \sum_{e\in E_0}a(e)b(e)=0.$$ Since $$\sum_{e\in E_0}{\bf z}(e)^2=\sum_{e\in E_0}\big(a(e)^2-b(e)^2+2\sqrt{-1}a(e)b(e)\big),$$ we get the claim.$\Box$ In view of this lemma, it is natural to consider the quadric defined by $${\bf Q}(X_0)=\Big\{ \langle{\bf z}\rangle\in {\bf P}(\mathbb{H})\big|~\sum_{e\in E_0}{\bf z}(e)^2=0 \Big\}$$ Let $H (\subset H_1(X_0,\mathbb{Z}))$ be a vanishing subgroup such that ${\rm rank}~\!H_1(X_0,\mathbb{Z})/H=2$. We consider the subspace of $\mathbb{H}$ defined by $$\mathbb{W}_H=\{{\bf z}\in \mathbb{H}|~[{\bf z}](\alpha)=0~ (\alpha\in H)\}$$ (remember that the cohomology class $[{\bf z}]$ is identified with a homomorphism of $H_1(X_0,\mathbb{Z})$ into $\mathbb{C}$). Clearly ${\rm dim}_{\mathbb{C}}\mathbb{W}=2$ so that $\mathbb{L}_H={\bf P}(\mathbb{W}_H)$ is a (projective) line in ${\bf P}(\mathbb{H})$. Let ${\bf z}_H\in \mathbb{H}$ be the building cochain of the standard realization of $X$ corresponding to $H$ which was constructed in the previous section. We fix a complex structure on $H_{\mathbb{R}}^{\perp}$, and identify $H_{\mathbb{R}}^{\perp}$ with $\mathbb{C}$. It is evident that both ${\bf z}_H$ and its complex conjugate $\overline{{\bf z}_H}$ belong to $\mathbb{W}_H$. $\mathbb{W}_H$ is spanned by ${\bf z}_H$ and $\overline{{\bf z}_H}$. [*Proof*]{}  It is enough to check that ${\bf z}_H$ and $\overline{{\bf z}_H}$ are linearly independent over $\mathbb{C}$. If $\overline{{\bf z}_H}=w{\bf z}_H$ for some $w\in \mathbb{C}$, then $$\sum_{e\in E_0}|{\bf z}_H(e)|^2=w\sum_{e\in E_0}{\bf z}_{H}(e)^2=0$$ so that ${\bf z}_H={\bf 0}$, thereby a contradiction. $\Box$ The following theorem tells us that the standard realization of the 2D topological crystal over $X_0$ associated with the vanishing group is obtained as intersection of the quadric ${\bf Q}(X_0)$ and the line $\mathbb{L}_H$. ${\bf Q}(X_0)\cap \mathbb{L}_H=\{\langle {\bf z}_H\rangle, \langle \overline{{\bf z}_H}\rangle\}$. ![Quadric and line[]{data-label="fig:inter"}](inter.eps){width=".37\linewidth"} [*Proof*]{}  We only have to check that the line $\mathbb{L}_H$ is not contained in ${\bf Q}(X_0)$. But this follows immediately from $\sum_{e\in E_0}|{\bf z}_H(e)|^2\neq 0$ and $$\sum_{e\in E_0}\big(a{\bf z}_H(e)+b\overline{{\bf z}_H(e)}\big)^2 =2ab\sum_{e\in E_0}|{\bf z}_H(e)|^2\quad (a,b\in\mathbb{C}).$$ $\Box$ To give a coordinate form of ${\bf Q}(X_0)$, choose an orientation $E_0^o=(e_1,\ldots,e_N)$ of $X_0$, and put $$\begin{aligned} %&&E_{0x}=\{e\in E_0|~o(e)=x\},\\ &&E_{0x}^{\rm in}=\{e\in E_0^0|~t(e)=x\},\\ &&E_{0x}^{\rm out}=\{e\in E_0^0|~o(e)=x\}.\end{aligned}$$ Then ${\bf Q}(X_0)$ is identified with $$\begin{aligned} &&\Big\{[z_1,\ldots,z_N]\in P^{N-1}(\mathbb{C})|~z_1{}^2+\cdots+z_N^{2}=0,\\ &&\qquad \qquad \qquad \sum_{i;e_i\in E_{0x}^{\rm in}}z_i=\sum_{j;e_j\in E_{0x}^{\rm out}}z_j ~~(x\in V_0) \Big\}.\end{aligned}$$ via the correspondence ${\bf z}\mapsto (z_1,\ldots,z_N)$ given by $z_i={\bf z}(e_i)$. Note the equation $$\label{eq:kir} \sum_{i;e_i\in E_{0x}^{\rm in}}z_i=\sum_{j;e_j\in E_{0x}^{\rm out}}z_j$$ is equivalent to $\displaystyle\sum_{e\in E_{0x}}{\bf z}(e)=0$, and is a complex version of [*Kirchhoff’s law*]{} in the theory of electric circuits stating that the amount of in-coming currents at a node is equal to the amount of out-going current. ${\bf Q}(X_0)$ is (biregular over $\mathbb{Q}$ to) a non-singular quadric defined over $\mathbb{Q}$ of dimension $b_1(X_0)-2$. [*Proof*]{}  Put $$W=\Big\{{}^t(z_1,\ldots,z_N)|~\sum_{i;e_i\in E_{0x}^{\rm in}}z_i=\sum_{j;e_j\in E_{0x}^{\rm out}}z_j ~~(x\in V_0)\Big\},$$ which is a coordinate form of $\mathbb{H}$. One can find a $N\times b$ matrix $A=(a_{ij})$ with rational entries such that the correspondence ${}^t(w_1,\ldots,w_b)\mapsto {}^t(z_1, \ldots,z_N)$ given by $z_i=\sum_{j=1}^b a_{ij}w_j$ is a linear isomorphism of $\mathbb{C}^b$ onto $W$. Then $$F(w_1,\ldots,w_b)=\sum_{i=1}^N\Big(\sum_{j=1}^ba_{ij}w_j\Big)^2$$ is obviously a positive definite quadratic form. Thus the quadric $$Q=\{[w_1,\ldots,w_b]\in P^{b-1}(\mathbb{C})|~F(w_1,\ldots,w_b)=0\}$$ is non-singular, and is biregular (over $\mathbb{Q}$) to ${\bf Q}(X_0)$. $\Box$ To describe $\{\langle{\bf z}_H\rangle, \langle\overline{{\bf z}_H}\rangle\}$ in the coordinate form, let $\alpha_1,\ldots,\alpha_{b-2}$ be a free $\mathbb{Z}$-basis of the vanishing subgroup $H$. Write $$\alpha_i=\sum_{j=1}^Na_{ij}e_j \quad (i=1,\ldots,b-2).$$ with $a_{ij}\in \mathbb{Z}$. Then $$\mathbb{W}_H=\Big\{{}^t(z_1,\ldots,z_N)\big|~\sum_{j=1}^Na_{ij}z_j=0 ~ (i=1,\ldots,b-2) \Big\}.$$ We thus have $$\begin{aligned} &&\{\langle {\bf z}_H\rangle, \langle\overline{{\bf z}_H}\rangle\}=\Big\{[z_1,\ldots,z_N]\in P^{N-1}(\mathbb{C})|~z_1{}^2+\cdots+z_N^{2}=0,\\ &&\qquad \qquad \qquad\quad \sum_{i;e_i\in E_{0x}^{\rm in}}z_i=\sum_{j;e_j\in E_{0x}^{\rm out}}z_j ~~(x\in V_0),\\ &&\qquad \qquad \qquad\qquad \sum_{j=1}^Na_{ij}z_j=0 ~ (i=1,\ldots,b-2) \Big\}.\end{aligned}$$ Rational points =============== From the previous theorem, it follows that ${\bf z}_H$ and $\overline{{\bf z}_H}$ are obtained by solving a quadratic equation of the form $az_1{}^2+bz_1z_2+cz_2{}^2=0$  ($a,b,c\in \mathbb{Q}$). Hence we conclude that there exists a positive square free integer $D$ such that $\{\langle {\bf z}_H\rangle, \langle\overline{{\bf z}_H}\rangle\}\subset P^{N-1}\big(\mathbb{Q}(\sqrt{-D})\big)$. The following lemma will give another proof for this fact. Suppose that $\{z_1,\ldots,z_N\}$ satisfies $z_1{}^2+\cdots+z_N^{2}=0$. Then $\{z_1,\ldots,z_N\}$ generates a lattice in $\mathbb{C}$ if and only if there exists a positive square free integer $D$ such that $[z_1,\ldots,z_N] \in P^{N-1}\big(\mathbb{Q}(\sqrt{-D})\big)$. [*Proof*]{}  The proof is fairly elementary. Suppose that $\{z_1,\ldots,z_N\}$ generates a lattice whose $\mathbb{Z}$-basis $w_1,w_2$. Then there exist integers $a_i,b_i$ such that $z_i=a_iw_1+b_iw_2$. Substituting this for $z_1{}^2+\cdots+z_N^{2}=0$, we get $$\Big(\sum_{i=1}^Na_i^2\Big)w_1^2+2\Big(\sum_{i=1}^Na_ib_i\Big)w_1w_2+\Big(\sum_{i=1}^Nb_i^2\Big)w_2^2=0$$ so that $$\frac{w_2}{w_1}=\frac{-\sum_{i=1}^Na_ib_i\pm \sqrt{\Big(\sum_{i=1}^Na_ib_i\Big)^2-\Big(\sum_{i=1}^Na_i^2\Big)\Big(\sum_{i=1}^Nb_i^2\Big)}}{\sum_{i=1}^Nb_i^2}.$$ If we denote by $D$ the square free part of the positive integer $$\Big(\sum_{i=1}^Na_i^2\Big)\Big(\sum_{i=1}^Nb_i^2\Big)- \Big(\sum_{i=1}^Na_ib_i\Big)^2,$$ then $x:=w_2/w_1\in \mathbb{Q}(\sqrt{-D})$. Thus (if $z_1\neq 0$), $$\frac{z_i}{z_1}=\frac{a_i+b_ix}{a_1+b_1x}\in \mathbb{Q}(\sqrt{-D}).$$ Therefore $$[z_1,\ldots,z_N]=[1,z_2/z_1,\ldots, z_N/z_1]\in P^{N-1}\big(\mathbb{Q}(\sqrt{-D})\big).$$ Conversely suppose that $[z_1,\ldots,z_N]\in P^{N-1}\big(\mathbb{Q}(\sqrt{-D})\big)$. One may assume $z_i\in \mathbb{Q}(\sqrt{-D})$ without loss of generality. One may also assume $z_1,z_2$ are linearly independent over $\mathbb{R}$; otherwise every $z_i$ is a real scalar multiple of some $z$; say, $z_i=c_iz,~ c_i\in \mathbb{R}$, and hence $0=(c_1^2+\cdots+c_N^2)|z|^2$; thereby leading to a contradiction. Obviously there are rational numbers $\alpha_i,\beta_i$ such that $z_i=\alpha_iz_1+\beta_iz_2 $  $(i=3,\ldots, N)$. This implies that $\{z_1,\ldots,z_N\}$ generates a lattice in $\mathbb{C}$ (which is commensurable to the lattice generated by $z_1,z_2$. $\Box$ [(1)]{} Let ${\bf z}\in C^1(X_0,\mathbb{C})$ be the building cochain of the standard realization of a 2D topological crystal over $X_0$. Then $[z_1, \ldots,z_N]\in {\bf Q}(X_0)\cap P^{N-1}\big(\mathbb{Q}(\sqrt{-D})\big)$. Conversely, for $[z_1, \ldots,z_N]\in {\bf Q}(X_0)\cap P^{N-1}\big(\mathbb{Q}(\sqrt{-D})\big)$, put $${\bf z}(e)=\begin{cases} z_i & (e=e_i)\\ -z_i & (e=\overline{e_i}). \end{cases}$$ Then ${\bf z}$ is the building cochain of the standard realization of a 2D topological crystal. [(2)]{} The set $${\bf Q}(X_0)\cap \bigcup_{D}P^{N-1}\big(\mathbb{Q}(\sqrt{-D})\big)$$ is identified with the family of all similarity classes of standard realizations of 2-dimensional topological crystals over $X_0$. It suffices to prove that for $$[z_1,\ldots,z_N]\in {\bf Q}(X_0)\cap P^{N-1}\big(\mathbb{Q}(\sqrt{-D})\big),$$ the rank of the image of $[{\bf z}]:H_1(X_0,\mathbb{Z})\longrightarrow \mathbb{C}$ is equal to two (note that ${\rm Image}~\![{\bf z}]$ is a subgroup of the lattice generated by $\{z_1,\ldots,z_N\}$). If the rank is one, then $\{z_1,\ldots,z_N\}$ gives rise to 1-dimensional (harmonic) realization, and hence every $z_i$ must be a real scalar multiple of some $z$; thereby a contradiction. $\Box$ We have interest in the description of $D$ in terms of $X_0$ and $H$. Consider the standard realization associated with $$[z_1,\ldots,z_N]\in {\bf Q}(X_0)\cap P^{N-1}\big(\mathbb{Q}(\sqrt{-D})\big).$$ One may assume $z_i\in \mathbb{Q}(\sqrt{-D})$. Let $T=\mathbb{Z}w_1+\mathbb{Z}w_2$ be the period lattice. Writing $w_i=a_i+b_i\sqrt{-D}\in \mathbb{Q}(\sqrt{-D})$, we observe that the energy of the standard realization associated with $[z_1,\ldots,z_N]$ is computed as $$\begin{aligned} &&2{\rm vol}(\mathbb{C}/T)^{-1}(|z_1|^2+\cdots +|z_N|^2)=\frac{4}{|w_1\overline{w_2}-w_2\overline{w_1}|}(|z_1|^2+\cdots +|z_N|^2)\\ &&= \frac{2}{|a_1b_2-a_2b_1|\sqrt{D}}(|z_1|^2+\cdots +|z_N|^2).\end{aligned}$$ On the other hand, in view of (\[eq:energymini\]), this is equal to $$4\kappa(X_0)^{-1/2}I(H)^{1/2}.$$ Since $|z_1|^2+\cdots +|z_N|^2\in \mathbb{Q}$, we have $D$ is equal to the square free part of $\kappa(X_0)I(H)$. The following proposition is shown in a routine way. If ${\bf Q}(X_0)$ has a $\mathbb{Q}(\sqrt{-D})$-rational point, then ${\bf Q}(X_0)\cap P^{N-1}\big(\mathbb{Q}(\sqrt{-D})\big)$ is dense in ${\bf Q}(X_0)$. Examples {#sec:example} ======== We shall illustrate several examples. \(1) The square lattice and the honeycomb have a special feature in the sense that they correspond to rational points of “0-dimensional" quadrics; say: In the case of the square lattice $$\{[z_1,z_2]\in P^1(\mathbb{C});~z_1{}^2+z_2{}^2=0\} =[1,\pm\sqrt{-1}].$$ In the case of the honeycomb $$\begin{aligned} &&\{[z_1,z_2,z_3]\in P^2(\mathbb{C});~z_1{}^2+z_2{}^2+z_3{}^2=0,~z_1+z_2+z_3=0\}\\ &&=\Big[1, \frac{-1\pm\sqrt{-3}}{2}, \frac{-1\mp\sqrt{-3}}{2}\Big].\end{aligned}$$ \(2) The regular kagome lattice, which is, as an abstract graph, a topological crystal over the graph depicted in Fig. \[fig:kagome\], corresponds to the $\mathbb{Q}(\sqrt{-3})$-rational points $$\Big[\frac{1\pm\sqrt{-3})}{2}, \frac{1\mp\sqrt{-3}}{2}, -1, \frac{1\pm\sqrt{-3}}{2}, \frac{1\mp\sqrt{-3}}{2}, -1\Big]$$ of the 2-dimensional projective quadric defined over $\mathbb{Q}$ $$\begin{aligned} &&\{[z_1,z_2,z_3, z_4,z_5,z_6]\in P^5(\mathbb{C});~z_1{}^2+\cdots+z_6{}^2=0,\\ && \qquad z_1+z_6=z_3+z_4,~ z_2+z_4=z_1+z_5,~z_3+z_5=z_2+z_6\}\end{aligned}$$ (the vanishing subgroup is $H=\mathbb{Z}(e_1+e_2+e_3)+\mathbb{Z}(e_4+e_5+e_6)$). ![Example[]{data-label="fig:kagome"}](kagome.eps){width=".3\linewidth"} \(3) The equilateral triangular lattice, which is, as an abstract graph, a topological crystal over the 3-bouquet graph $B_3$, corresponds to the $\mathbb{Q}(\sqrt{-3})$-rational points $$\Big[1, \frac{-1\pm\sqrt{-3}}{2}, \frac{-1\mp\sqrt{-3}}{2}\Big]$$ of the quadric $$\{[z_1,z_2,z_3]\in P^2(\mathbb{C});~z_1{}^2+z_2{}^2+z_3{}^2=0\}$$ (the vanishing group is $\mathbb{Z}(e_1+e_2+e_3)$; the orientation being illustrated in Fig. \[fig:example1\]). \(4) The 2D crystal lattice in Fig. \[fig:example1\], which is also a topological crystal over $B_3$ as an abstract graph, corresponds to $\mathbb{Q}(\sqrt{-6})$-rational points $$[3\pm\sqrt{-6}, -3\pm\sqrt{-6}, \mp\sqrt{-6}]$$ of the quadric $$\{[z_1,z_2,z_3]\in P^2(\mathbb{C})|~z_1{}^2+z_2{}^2+z_3{}^2=0\}$$ (the vanishing groups is $H=\mathbb{Z}(e_1+e_2+2e_3)$). ![Example[]{data-label="fig:example1"}](lattice4.eps){width=".6\linewidth"} \(5) The 2D crystal lattice in Fig. \[fig:8-4\] corresponds to $\mathbb{Q}(\sqrt{-1})$-rational points $$\Big[ 1, \frac{-1\pm\sqrt{-1}}{2}, -\frac{1\pm\sqrt{-1}}{2}, \pm\sqrt{-1}, \frac{-1\pm\sqrt{-1}}{2}, \frac{1\pm\sqrt{-1}}{2} \Big]$$ of the quadric $$\begin{aligned} &&\{[z_1,\ldots,z_6]\in P^5(\mathbb{C});~z_1{}^2+\cdots+z_6{}^2=0,~z_1+z_2+z_3=0,\\ &&\qquad\qquad z_3+z_4=z_5, z_1+z_5=z_6, z_2+z_6=z_4\}\end{aligned}$$ (the vanishing group is $H=\mathbb{Z}(-e_2+e_3+e_5+e_6)$). ![Example[]{data-label="fig:8-4"}](8-4lattice.eps){width=".7\linewidth"} This crystalline pattern is observed when we look at the $K_4$ crystal (diamond twin) toward an suitable direction (see [@su2], [@su5]). \(6) Figure \[fig:dice\] is the so-called [*dice lattice*]{}. This corresponds to $\mathbb{Q}(\sqrt{-3})$-rational points $$\Big[1, \frac{-1\pm\sqrt{-3}}{2}, \frac{-1\mp\sqrt{-3}}{2}, -1, \frac{1\pm\sqrt{-3}}{2}, \frac{1\mp\sqrt{-3}}{2}\Big]$$ of the quadric $$\{[z_1,\ldots,z_6]\in P^5(\mathbb{C})|~z_1{}^2+\cdots+z_6{}^2=0,~ z_1+z_2+z_3=0,~z_4+z_5+z_6=0\}$$ (the vanishing groups is $H=\mathbb{Z}(e_1+\overline{e_3}+e_4+\overline{e_5}) +\mathbb{Z}(e_5+\overline{e_6}+e_3+\overline{e_2})$). ![Example[]{data-label="fig:dice"}](dice.eps){width=".8\linewidth"} \(7) Figure \[fig:cairo1\] is a tiling of pentagons with picturesque properties that has become known as the [*Cairo pentagon*]{}[^7]. Its 1-skeleton is the standard realization of a topological crystal over the finite graph drawn on the right, which corresponds to $\mathbb{Q}(\sqrt{-1}))$-rational points $$\begin{aligned} &&\Big[ -1\mp\frac{\sqrt{-1}}{2}, \frac{1}{2}\mp\sqrt{-1}, 1\pm\frac{\sqrt{-1}}{2}, -\frac{1}{2}\pm\sqrt{-1}, \mp\sqrt{-1},\\ &&\qquad 1, -\frac{1}{2}\mp\sqrt{-1}, -1\pm\frac{\sqrt{-1}}{2}, -1\pm\frac{\sqrt{-1}}{2}, -\frac{1}{2}\mp\frac{\sqrt{-1}}{2} \Big]\end{aligned}$$ of the quadric $$\begin{aligned} &&\{[z_1,\ldots,z_{10}]\in P^{9}(\mathbb{C})|~ z_1{}^2+\cdots+z_{10}{}^2=0,~ z_1=z_5+z_9,~z_2=z_6+z_{10},\\ &&\qquad\qquad \qquad z_1+z_2+z_3+z_4=0,~z_4+z_6+z_7=0,~ z_3+z_5+z_8=0,\\ &&\qquad\qquad \qquad z_9+z_{10}=z_7+z_8 \}.\end{aligned}$$ ![Example[]{data-label="fig:cairo1"}](cairo2.eps){width=".65\linewidth"} Tilings ======= In general, a [*periodic tiling*]{}, symbolically written as $(T,L)$, is a [*tessellation*]{} of figures (tiles) in the plane $\mathbb{R}^2$ which is periodic with respect to the translational action by a lattice group $L$. Two tiles $D$ and $D'$ in $(T,L)$ are said to be [*equivalent*]{} if $D'=D+\sigma$ for some $\sigma\in L$. We denote by $f_{T,L}$ the number of equivalence classes of tiles, and let $D_1,\ldots,D_{f_{T,L}}$ be representatives of equivalence classes, which we call [*fundamental tiles*]{}[^8] of $(T,L)$. Figure \[fig:fund\] is the picture of the tiling we find on a street pavement in Zakopane, Poland[^9]. This tiling has three fundamental tiles depicted on the right. ![A tiling and its fundamental tiles[]{data-label="fig:fund"}](tile1.eps){width=".5\linewidth"} As for topological shapes of tilings, we have \[thm1\] For any natural number $f$, there are only finitely many homeomorphic classes of tilings $(T,L)$ such that $f_{T,L}=f$. Here we say that two tilings $(T_1,L_1)$ and $(T_2,L_2)$ are [*homemorphic*]{} if there is a homemorphism $\varphi:\mathbb{R}^2\longrightarrow \mathbb{R}^2$ and an isomorphism $\psi:L_1\longrightarrow L_2$ satisfying \(1) $\varphi (x+\sigma)=\varphi(x)+\psi (\sigma), \quad (x\in \mathbb{R}^2,~\sigma \in L_1)$, \(2) $\varphi(T_1)=T_2$. The following theorem is deduced from the proof of this theorem. \[thm2\] There are only finitely many 2D topological crystals over $X_0$ whose standard realizations yield tilings. In other words, there are only finitely many rational points in ${\bf Q}(X_0)$ which correspond to tilings. Let $X$ be the 1-skeleton of a given periodic tiling $(T,L)$, and consider the quotient graph $X_0=X/L$. Then $X$ as an abstract graph is a 2D topological crystal over $X_0$ (the net $X$ is not necessarily a crystal net since the edges are allowed to be curved). We shall say that $(T,L)$ is a [*tiling with the base graph*]{} $X_0$. Denote by $\omega:X\longrightarrow X_0$ the covering map. Note that the degree of any vertex (node) $x$ in $X_0$ is greater than or equal to 3. The covering map $\pi$ of $\mathbb{R}^2$ onto the the 2-dimensional torus $\mathbb{R}^2/L$ induces a cellular decomposition of $\mathbb{R}^2/L$. The 1-skeleton of this cellular decomposition is just the finite graph $X_0$ realized in $\mathbb{R}^2/L$ (conversely, a cellular decomposition of the torus yields a tiling). The proof of Theorem \[thm1\] is divided into 4 steps each of which is fairly elementary. \(1) The number of 2-cells in $\mathbb{R}^2/L$ is equal to $f=f_{T,L}$, which coincides with $v-e~(=b_1(X_0)-1)$, where $v$ (resp. $e$) is the number of vertices (resp. edges) in $X_0$. This is because $v-e+f$ is the Euler number of the torus so that $v-e+f=0$. \(2) Let $D_1,\ldots,D_{b-1}$ $(b=b_1(X_0))$ be fundamental tiles. Then $\pi(D_1),\ldots,$ $\pi(D_{b-1})$ are all 2-cells in the torus. Topologically $D_i$ is identified with a polygon with $k_i$ sides where $k_i$ is the number of vertices on $D_i$. If we put $N_{T,L}=\max\{k_1,\ldots,k_{b-1}\}$, then $N_{T,L}\leq 2e$. To show this, let $D_i$ is a tile having $N_{T,L}$ edges on $\partial D_i$. If $N_{T,L}>2e$, then there exist at least three edges in $\partial D_i$ which are mapped to an edge in $X_0$ by $\omega$. This should not happen since the torus is non-singular. Therefore we get the claim. \(3) For a fixed integer $b\geq 0$, there are only finitely many finite graphs $X_0$ $($up to isomorphisms$)$ such that ${\rm deg}~\hspace{-0.04cm}x\geq 3$ for all vertex $x$ and $b_1(X_0)=b$. Indeed, $$3v\leq \sum_{x}{\rm deg}~\!x=2e,$$ so $3v\leq 2e$ and $v\leq 2(b-1)$ (use $v-e=1-b$), and also $e\leq 3b-3$. Therefore the number of vertices and edges is bounded. \(4) Given a finite graph $X_0$ and polygonal 2-cells $D_i$ with $k_i$ sides ($i=1,\ldots,f$, there are only finitely many ways to attach $D_i$’s to $X_0$ to make a cell complex whose underlying space is the torus (more generally a closed surface). Putting altogether, we complete the proof of Theorem \[thm1\]. Theorem \[thm2\] is also a consequence of (1), (2), (4) since if a 2D topological crystal $X$ over $X_0$ yields a tiling $(T,L)$, then $f_{T,L}=b_1(X_0)-1$ and $N_{T,L}\leq 6\big(b_1(X_0)-1\big)$. We shall go a bit further. Let $H$ be the vanishing group for the 2D topological crystal $X$ over $X_0$ which corresponds to a tiling $(T,L)$. If we put $c_i=\partial D_i$ and give the counter-clockwise orientation on $c_i$, then $\omega(c_{b-1})=-(\omega(c_1)+\cdots+\omega(c_{b-2}))$, and $\omega(c_1),\ldots,\omega(c_{b-2})$ form a $\mathbb{Z}$-basis of $H$. To prove the last assertion, let $c$ be a closed path in $X_0$ whose homology class is in $H$. The (any) lifting $\widehat{c}$ of $c$ in $X$ is also closed. As a 1-chain, we may write $$\widehat{c}=\sum_{i=1}^{b-1}n_i\partial D_i,$$ where $n_i$ denotes the winding number of $\widehat{c}$ around a interior point of $D_i$. Thus $$c=\omega(\widehat{c})=\sum_{i=1}^{b-1}n_i\omega(c_i),$$ which implies that $H$ is generated by $\omega(c_1),\ldots,\omega(c_{b-2})$. Next suppose that $$\sum_{i=1}^{b-2}m_i\omega(c_i)=0\qquad (m_i\in \mathbb{Z}).$$ Expressing the left hand side as a 1-chain, and taking a look at the coefficients of directed edges in $X_0$, we find that if $\omega(c_i)$ and $\omega(c_j)$ share an edge, then $m_i=m_j$, and that if $\omega(c_i)$ and $\omega(c_{b-1})$ share an edge, then $m_i=0$, from which it follows that $m_1=\cdots=m_{b-2}=0$. Thus we conclude that $\omega(c_1),\ldots,\omega(c_{b-2})$ comprise a $\mathbb{Z}$-basis of $H$. It is an interesting problem to list all homeomorphic classes of tilings $(T,L)$ with $f_{T,L}=f$. To handle this problem (not yet having been solved completely), it may be useful to introduce the notion of [*height*]{} of vanishing groups as follows[^10]. Take an orientation of $X_0$ and define the norm $\|\alpha\|_1$ of a $1$-chain $\alpha=\displaystyle\sum_{e\in E_0^o}a_ee$ by setting $$\|\alpha\|_1=\sum_{e\in E_0^o}|a_e|,$$ where $E_0^o$ is the set of directed edges for the orientation (it should be noted that $\|\alpha\|_1$ does not depend on the choice of an orientation). For a subset $S=\{\alpha_{1},\ldots, \alpha_{b-2}\}$ of $H_1(X_0,\mathbb{Z})$ which forms a $\mathbb{Z}$-basis of a vanishing subgroup, we put $$h(S)=\max (\|\alpha_{1}\|_1,\ldots, \|\alpha_{b-2}\|_1),$$ and define the [*height*]{} of $H$ by $ h(H)=\min_{S}h(S), $ where $S$ runs over all $\mathbb{Z}$-bases of $H$. Consider two sets $$R_1=\{S|~h(S)\leq h\},\quad R_2=\{H|~h(H)\leq h\}.$$ Certainly $R_1$ is a finite set. The correspondence $ S\mapsto H~(\text{generated by S}) $ yields a surjective map of $R_1$ onto $R_2$. Thus we get: \[thm:finiteness\] There are only a finite number of vanishing subgroups $H$ of the homology group $H_1(X_0,\mathbb{Z})$ such that $(1)$ ${\rm rank}~\hspace{-0.05cm}H_1(X_0,\mathbb{Z})/H=2$, $(2)$ $h(H)\leq h$. Suppose now that $X_0$ is the base graph for a tiling $(T,L)$, and let $H$ be the vanishing group corresponding to the topological crystal $X$ associated with $(T,L)$. Then, using the notations above, we have $$\|\omega(c_i)\|_1\leq k_i\leq N_{T,L}$$ so that $h(H)\leq N_{T,L}\leq 6\big(b_1(X_0)-1\big)$ (this gives another proof of Theorem \[thm2\]). Thus in order to list homeomorphic classes of tiling $(T,L)$ with $f_{T,L}=f$, we first enumerate finite graphs $X_0$ with $b_1(X_0)=f+1$, and then for such $X_0$, we enumerate vanishing groups $H\subset H_1(X_0,\mathbb{Z})$ such that $h(H)\leq 6f$ and check whether $X=X_{0}^{\rm ab}/H$ gives a tiling or not. [999]{} S. J. Chung, T. Hahn, and W. E. Klee, *Nomenclature and generation of three-periodic nets: the vector method*, Acta. Cryst., **A40** (1984), 42–50. O. Delgado-Friedrichs and M. O’Keeffe, *Identification of and symmetry computation for crystal nets*, Acta Cryst., **A59** (2003), 351–360. O. Delgado-Friedrichs, *Barycentric drawings of periodic graphs*, LNCS **2912** (2004), 178–189. J-G. Eon, *Archetypes and other embeddings of periodic nets generated by orthogonal projection*, J. Solid State Chem. **147** (1999), 429–437. J-G. Eon, *Euclidean embeddings of periodic nets: definition of a topologically induced complete set of geometric descriptors for crystal structures*, Acta Cryst. **A67** (2011), 68–86. M. Kotani and T. Sunada, *Standard realizations of crystal lattices via harmonic maps*, Trans. Amer. Math. Soc., **353** (2000), 1–20. M. Kotani and T. Sunada, *Jacobian tori associated with a finite graph and its abelian covering graphs*, Advances in Apply. Math., **24** (2000), 89–110. M. Kotani and T. Sunada, *Albanese maps and off diagonal long time asymptotics for the heat kernel*, Comm. Math. Phys., **209** (2000), 633–670. M. Kotani and T. Sunada, *Spectral geometry of crystal lattices*, Contemporary Math., **338** (2003), 271–305. T. Sunada, *Crystals that nature might miss creating*, Notices Amer. Math. Soc., **55** (2008), 208–215. T. Sunada, *Discrete geometric analysis*, Proceedings of Symposia in Pure Mathematics, (ed. by P. Exner, J. P. Keating, P. Kuchment, T. Sunada, A. Teplyaev), **77** (2008), 51–86. T. Sunada, *Lecture on topological crystallography*, Japan. J. Math. **7** (2012), 1–39. T. Sunada, *Topological crystallography —With a View Towards Discrete Geometric Analysis—*, Surveys and Tutorials in the Applied Mathematical Sciences, Vol. 6, Springer, 2012. [^1]: This carbon allotrope, formed when meteorites containing graphite strike the earth, is named in honor of crystallographer Kathleen Lonsdale, also referred to as the [*hexagonal diamond*]{}. [^2]: Crystallographers proposed a similar idea in [@d-1], [@d1], [@eon], [@eon1] [^3]: The network constructed in this way could be “degenerate" in the sense that different vertices of $X$ are realized as one points, or different edges overlap in $\mathbb{R}^d$. But we shall not exclude these possibilities. [^4]: In [@su4], [@su5], the term “building block" is used. The idea to describe crystal structures by using finite graphs together with vector labeling is due to [@chung] [^5]: The term “tight frame" is the terminology in wavelet analysis. [^6]: The energy defined here is similarity-invariant. [^7]: It is given its name because several streets in Cairo are paved in this design (strictly speaking, it is a bit distorted). This is also called Macmahon’s net, [**mcm**]{}, and a 4-fold pentille (J. H. Conway). [^8]: Some of fundamental tiles can be congruent. For instance, fundamental tiles of the equilateral lattice with respect to the maximal periodic lattice are two congruent equilateral triangles. [^9]: From Wikipedia. [^10]: This notion is introduced for topological crystal of arbitrary dimension ([@su5]).

--- abstract: 'In this paper, we study the codes over the matrix ring over $\mathbb{Z}_4$, which is perhaps the first time the ring structure $M_2(\mathbb{Z}_4)$ is considered as a code alphabet. This ring is isomorphic to $\mathbb{Z}_4[w]+U\mathbb{Z}_4[w]$, where $w$ is a root of the irreducible polynomial $x^2+x+1 \in \mathbb{Z}_2[x]$ and $U\equiv$ ${11}\choose{11}$. We first discuss the structure of the ring $M_2(\mathbb{Z}_4)$ and then focus on algebraic structure of cyclic codes and self-dual cyclic codes over $M_2(\mathbb{Z}_4)$. We obtain the generators of the cyclic codes and their dual codes. Few examples are given at the end of the paper.' address: - ' Department of Mathematics, National Institute of Technology Durgapur, Durgapur, INDIA' - 'Department of Mathematics, Dr. SPM International Institute of Information Technology, Naya Raipur, INDIA' author: - Sanjit Bhowmick - Satya Bagchi - Ramakrishna Bandi title: 'Self-dual cyclic codes over $M_2(\mathbb{Z}_4)$' --- Codes over $\mathbb{Z}_4+u\mathbb{Z}_4$ ,Gray map ,Lee weight ,Self-dual codes : 94B05 ,94B15 Introduction ============ Codes over finite rings are very old on the other hand their applications in digital communication is rather young. Codes over rings have got the attention of researchers only after Hammons et. al. [@Hammons1994] in which they have shown an interesting relation between popular non-linear codes and linear codes over integer residue rings modulo 4, via a map called Gray map. This attracted the researchers to focus on codes over rings and their applications. As a result so many new ring structures have been considered as code alphabets. However most of the study was restricted to codes over commutative rings [@Hammons1994; @Pless1997; @Pless1996]. Recently, codes over a non-commutative ring, the matrix ring over $\mathbb{Z}_2$, i.e., $M_2(\mathbb{Z}_2)$ has been considered as a code alphabet to study space time codes [@Oggier2012]. The advantage of this type of matrix rings is that they are non-commutative, which allows a quotient ring to have wide left and right ideals (cyclic codes). This is also true in the case of skew polynomial rings, however, the polynomial factorization is a big hurdle in the construction of codes over skew polynomial rings, which is not the case with codes over matrix rings. A notable work on cyclic codes over non-commutative finite rings is [@Bachoc1997; @Greferath1999; @Wisbauer1991]. In [@Greferath1999], the authors have studied cyclic codes over $M_2(\mathbb{Z}_2)$ and obtained some optimal codes over the same. Inspired by this work Luo and Uday [@Luo2017] have obtained the structure of cyclic codes over $M_2(\mathbb{Z}_2+u\mathbb{Z}_2)$ and found some optimal cyclic codes. Motivated by this, in this paper, we explored the construction of codes over $M_2(\mathbb{Z}_4)$. The reason for choosing $\mathbb{Z}_4$ is that $\mathbb{Z}_4$ is best suited for the construction of modular lattices and also the relation established by Hammons et. al. [@Hammons1994] between binary non-linear codes and linear codes over $\mathbb{Z}_4$. The approach which is being used in this paper to cyclic codes over $M_2(\mathbb{Z}_4)$ is same as that of cyclic codes over $M_2(\mathbb{Z}_2)$, however, it is not straight forward as it can be seen later in the paper. The paper is organised as follows: In Section 2, we describe the structure of $M_2(\mathbb{Z}_4)$ and show that $M_2(\mathbb{Z}_4)$ is isomorphic to $\mathbb{Z}_4[w]+U\mathbb{Z}_4[w]$, where $w$ is a root of the polynomial $x^2+x+1$ and $U\equiv$ ${11}\choose{11}$. We define a Gray map on $\mathbb{Z}_4[w]+U\mathbb{Z}_4[w]$ to $\mathbb{F}^4_4$ which preserves the Lee weight in $\mathbb{Z}_4[w]+U\mathbb{Z}_4[w]$ and Hamming weight in $\mathbb{F}^4_4$. In section 3, we discuss the structure of cyclic codes and prove some results on the dimension of cyclic codes. In section 4, we obtain the structure of dual cyclic codes and also self-dual codes. Structure of $M_2(\mathbb{Z}_4)$ ================================ Let us denote $\mathcal{R}=M_2(\mathbb{Z}_4)$. $\mathcal{R}$ is a non-commutative ring of matrices of order 2 over $\mathbb{Z}_4$. The set $\mathbb{Z}_4+X\mathbb{Z}_4+Y\mathbb{Z}_4+YX\mathbb{Z}_4$ forms a non-commutative finite ring with respect to component wise addition and multiplication defined in Table \[p1\_table1\]. $\mathcal{R}$ $\cdot$ 1 X Y YX ----------------------- --- ---- --- ---- 1 1 X 0 0 X 0 0 1 X Y Y YX 0 0 YX 0 0 Y YX : Multiplication rule of $\mathcal{R}$[]{data-label="p1_table1"} \[p1\_lemma1\] The ring $M_2(\mathbb{Z}_4)$ is isomorphic to the ring $\mathbb{Z}_4+X\mathbb{Z}_4+Y\mathbb{Z}_4 + YX\mathbb{Z}_4$, i.e., $M_2(\mathbb{Z}_4) \equiv \mathbb{Z}_4+X\mathbb{Z}_4+Y\mathbb{Z}_4+YX\mathbb{Z}_4$. We define a mapping $f: M_2(\mathbb{Z}_4) \longrightarrow \mathbb{Z}_4+X\mathbb{Z}_4+Y\mathbb{Z}_4+YX\mathbb{Z}_4$ such that $f(A) = a+Xb+Yc+YXd$, where $A=\left(\begin{array}{cc} a & b \\ c & d \end{array}\right) \in M_2(\mathbb{Z}_4)$. It is easy to see that $f(A+B)=f(A)+f(B)$. Now we show that $f(AB)=f(A)f(B)$. Let $A=\left(\begin{array}{cc} a & b \\ c & d \end{array}\right)$ and $B=\left(\begin{array}{cc} a_1 & b_1 \\ c_1 & d_1 \end{array}\right)$. Then $AB=\left(\begin{array}{cc} aa_1+bc_1 & ab_1+bd_1 \\ ca_1+dc_1 & cb_1+dd_1 \end{array}\right)$. So $f(AB)=(aa_1+bc_1) +X(ab_1+bd_1) +Y(ca_1+dc_1) +YX(cb_1+dd_1)$. Now, $$\begin{array}{rcl} f(A)f(B) & = & (a +Xb +Yc +YXd)(a_1 +Xb_1 +Yc_1 +YXd_1)\\ & = & (aa_1+bc_1) +X(ab_1+bd_1) +Y(ca_1+dc_1) +YX(cb_1+dd_1 ) \\ & = & f(AB). \end{array}$$ It is easy to see that $f$ is one-one and onto. Hence $M_2(\mathbb{Z}_4) \equiv \mathbb{Z}_4+X\mathbb{Z}_4+Y\mathbb{Z}_4+YX\mathbb{Z}_4$. We consider a subset of $\mathbb{Z}_4 + X\mathbb{Z}_4 + Y\mathbb{Z}_4$ $+YX\mathbb{Z}_4$, namely $W = \lbrace 0$, $X+Y3+YX3$, $X2+Y2+YX2$, $X3+Y+YX$, $1+X+YX3$, $2+X2+YX2$, $3+X3+YX$, $1+YX$, $2+YX2$, $3+YX3$, $1+X2+Y2+YX3$, $3+X2+Y2+YX$, $1+X3+Y+YX2$, $2+X3+Y+YX3$, $2+X+Y3+YX$, $3+X+Y3+YX3 \rbrace$. \[p1\_lemma2\] The subset $W$ forms a commutative ring with respect to component wise addition and multiplication defined on $\mathbb{Z}_4+X\mathbb{Z}_4+Y\mathbb{Z}_4+YX\mathbb{Z}_4$. One can easily verify that $W$ is an abelian group under component wise addition. The other criteria of a ring can be verified using the Table \[p1\_table2\]. For simplicity, we use the following notation:\ $\begin{array}{cccc} a_0=0, & a_{1}= X+Y3+YX3, & a_{2}= X2+Y2+YX2, & a_{3}= X3+Y+YX, \\ a_{4}=1+X+YX3, & a_{5}=2+X2+YX2, & a_{6}= 3+X3+YX, & a_{7}= 1+YX, \\ a_{8}=2+YX2, & a_{9}= 3+YX3, & a_{10}= 1+X2+Y2+YX3, & a_{11}=3+X2+Y2+YX, \\ a_{12}=1+X3+Y+YX2, & a_{13}=2+X3+Y+YX3, & a_{14}=2+X+Y3+YX, & a_{15}=3+X+Y3+YX3. \end{array}$ $\cdot$ 0 $a_1$ $a_2$ $a_3$ $a_4$ $a_5$ $a_6$ $a_7$ $a_8$ $a_9$ $a_{10}$ $a_{11}$ $a_{12}$ $a_{13}$ $a_{14}$ $a_{15}$ ---------- --- ---------- ------- ---------- ---------- ------- ---------- ---------- ------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 $a_1$ 0 $a_6$ $a_5$ $a_4$ $a_9$ $a_8$ $a_7$ $a_1$ $a_2$ $a_3$ $a_{13}$ $a_{14}$ $a_{10}$ $a_{12}$ $a_{15}$ $a_{11}$ $a_2$ 0 $a_5$ 0 $a_5$ $a_8$ 0 $a_8$ $a_2$ 0 $a_2$ $a_2$ $a_2$ $a_{8}$ $a_5$ $a_5$ $a_8$ $a_3$ 0 $a_4$ $a_5$ $a_6$ $a_7$ $a_8$ $a_9$ $a_3$ $a_2$ $a_1$ $a_{14}$ $a_{13}$ $a_{11}$ $a_{15}$ $a_{12}$ $a_{10}$ $a_4$ 0 $a_9$ $a_8$ $a_7$ $a_1$ $a_2$ $a_3$ $a_4$ $a_5$ $a_6$ $a_{15}$ $a_{12}$ $a_{14}$ $a_{11}$ $a_{10}$ $a_{13}$ $a_5$ 0 $a_8$ 0 $a_8$ $a_2$ 0 $a_2$ $a_5$ 0 $a_5$ $a_5$ $a_5$ $a_5$ $a_8$ $a_8$ $a_2$ $a_6$ 0 $a_7$ $a_8$ $a_9$ $a_3$ $a_2$ $a_1$ $a_6$ $a_5$ $a_4$ $a_{12}$ $a_{15}$ $a_{13}$ $a_{10}$ $a_{11}$ $a_{14}$ $a_7$ 0 $a_1$ $a_2$ $a_3$ $a_4$ $a_5$ $a_6$ $a_7$ $a_8$ $a_9$ $a_{10}$ $a_{11}$ $a_{12}$ $a_{13}$ $a_{14}$ $a_{15}$ $a_8$ 0 $a_2$ 0 $a_2$ $a_5$ 0 $a_5$ $a_8$ 0 $a_8$ $a_8$ $a_8$ $a_5$ $a_2$ $a_2$ $a_5$ $a_9$ 0 $a_3$ $a_2$ $a_1$ $a_6$ $a_5$ $a_4$ $a_9$ $a_8$ $a_7$ $a_{11}$ $a_{10}$ $a_{15}$ $a_{14}$ $a_{13}$ $a_{12}$ $a_{10}$ 0 $a_{13}$ $a_2$ $a_{14}$ $a_{15}$ $a_5$ $a_{12}$ $a_{10}$ $a_8$ $a_{11}$ $a_7$ $a_9$ $a_6$ $a_1$ $a_3$ $a_4$ $a_{11}$ 0 $a_{14}$ $a_2$ $a_{13}$ $a_{12}$ $a_5$ $a_{15}$ $a_{11}$ $a_8$ $a_{10}$ $a_9$ $a_7$ $a_4$ $a_3$ $a_1$ $a_6$ $a_{12}$ 0 $a_{10}$ $a_8$ $a_{11}$ $a_{14}$ $a_5$ $a_{13}$ $a_{12}$ $a_5$ $a_{15}$ $a_6$ $a_4$ $a_1$ $a_7$ $a_9$ $a_3$ $a_{13}$ 0 $a_{12}$ $a_5$ $a_{15}$ $a_{11}$ $a_8$ $a_{10}$ $a_{13}$ $a_2$ $a_{14}$ $a_1$ $a_3$ $a_7$ $a_6$ $a_4$ $a_9$ $a_{14}$ 0 $a_{15}$ $a_5$ $a_{12}$ $a_{10}$ $a_8$ $a_{11}$ $a_{14}$ $a_2$ $a_{13}$ $a_3$ $a_1$ $a_9$ $a_4$ $a_6$ $a_7$ $a_{15}$ 0 $a_{11}$ $a_8$ $a_{10}$ $a_{13}$ $a_2$ $a_{14}$ $a_{15}$ $a_5$ $a_{12}$ $a_4$ $a_6$ $a_3$ $a_9$ $a_7$ $a_1$ : Multiplication table of $W$[]{data-label="p1_table2"} We choose the element $1+X+Y+YX$ from $\mathbb{Z}_4+X\mathbb{Z}_4+Y\mathbb{Z}_4+YX\mathbb{Z}_4$ and denote it by $U$, i.e., $U = 1+X+Y+YX$. So $UW = \lbrace 0$, $3+Y3$, $2+Y2$, $1+Y$, $X+YX$, $X2+YX2$, $X3+YX3$, $1+X+Y+YX$, $2+X2+Y2+YX2$, $3+X3+Y3+YX3$, $3+X+Y3+YX$, $1+X3+Y+YX3$, $2+X+Y2+YX$, $3+X2+Y3+YX2$, $1+X2+Y+YX2$, $2+X3+Y2+YX3\rbrace$. This implies that $W \cap UW=\lbrace 0 \rbrace$, which inturn implies that $W+UW=\mathbb{Z}_4+X\mathbb{Z}_4+Y\mathbb{Z}_4+YX\mathbb{Z}_4$ as $W+UW$ is sub-ring of $\mathbb{Z}_4+X\mathbb{Z}_4+Y\mathbb{Z}_4+YX\mathbb{Z}_4$ and $\mid W+UW \mid =256$. Therefore $\mathbb{Z}_4+X\mathbb{Z}_4+Y\mathbb{Z}_4+YX\mathbb{Z}_4 = W+UW$. Let $x^2+x+1$ be a basic irreducible polynomial over $\mathbb{Z}_4$. Then $\dfrac{\mathbb{Z}_4[x]}{\langle x^2+x+1 \rangle}$ is called the Galois extension ring of $\mathbb{Z}_4$ and is denoted by GR(4,2). If $w$ is a root of $x^2+x+1$ then $\dfrac{\mathbb{Z}_4[x]}{\langle x^2+x+1 \rangle} = GR(4,2) \cong \mathbb{Z}_4[w]$. \[p1\_lemma4\] The ring $W$ is isomorphic to the ring $\mathbb{Z}_4[w]$, i.e., $W\cong\mathbb{Z}_4[w]$. We consider the mapping as follows: $0\longmapsto 0$, $X+Y3+YX3\longmapsto w$, $X2+Y2+YX2\longmapsto 2w$, $X3+Y+YX\longmapsto 3w$, $1+X+Y3\longmapsto 3w^2$, $2+X2+Y2\longmapsto 2w^2$, $3+X3+Y\longmapsto w^2$, $1+YX\longmapsto 1$, $2+YX2\longmapsto 2$, $3+YX3\longmapsto 3$, $1+X2+Y2+YX3\longmapsto 2w+1$, $3+X2+Y2+YX\longmapsto 2w+3$, $1+X3+Y+YX2\longmapsto 3w+1$, $2+X3+Y+YX3\longmapsto 3w+2$, $2+X+Y3+YX\longmapsto w+2$, $3+X+Y3+YX2\longmapsto w+3$. It is clear from Table \[p1\_table2\] that this map is a ring isomorphism. Therefore $W\cong\mathbb{Z}_4[w]$. \[p1\_thm1\] The ring $M_2(\mathbb{Z}_4)$ is isomorphic to the ring $\mathbb{Z}_4[w]+U\mathbb{Z}_4[w]$, i.e., $M_2(\mathbb{Z}_4) \cong\mathbb{Z}_4[w]+U\mathbb{Z}_4[w]$. We have $$\begin{aligned} M_2(\mathbb{Z}_4) & \cong & \mathbb{Z}_4+X\mathbb{Z}_4+Y\mathbb{Z}_4+YX\mathbb{Z}_4 ~~\mbox{from Lemma}~ \ref{p1_lemma1}\\ &\cong & W+UW \\ & \cong & \mathbb{Z}_4[w]+U\mathbb{Z}_4[w] ~~\mbox{from Lemma}~ \ref{p1_lemma4}\end{aligned}$$ Therefore $M_2(\mathbb{Z}_4) \cong\mathbb{Z}_4[w]+U\mathbb{Z}_4[w]$. We notice here that the rings $W$ and $\mathbb{Z}_4[w]$ are commutative, however, their extensions, both $W+UW$ and $\mathbb{Z}_4[w]+U\mathbb{Z}_4[w]$ are non-commutative. Summarising the above discussion, we have $\mathcal{R} \cong \mathbb{Z}_4[w]+U\mathbb{Z}_4[w]$, where $U^2=2U$, $U^3=0$, $2U=U2$ and $2U^2=0$. We know that each element of $\mathbb{Z}_4$ has 2-adic representation $a+2b$, where $a,~b \in \mathbb{Z}_2$, so is $\mathbb{Z}_4[w]$. Now we define a Gray map on $\mathcal{R}$. For this, first we define a mapping $\mathcal{R}$ to $\mathbb{Z}^2_4[w]$, and then define a mapping $\mathbb{Z}^2_4[w]$ to $\mathbb{F}^4_4$ so that the Gray map is $$\begin{array}{cccc} \Phi: & \mathcal{R} & \longrightarrow & \mathbb{F}_4^4\\ & a+2b+Uc+U2d & \longmapsto & (d,~ c+d,~ b+d,~ a+b+c+d), \end{array}$$ where $a, b, c, d\in \mathbb{F}_4$. This map can easily be extended to $\mathcal{R}^n$ component wise. The Hamming weight $w_H$ of $x \in \mathbb{F}_4^n$ is defined as the number of non-zero coordinates of $x$. For $x=a+2b+Uc+U2d \in \mathcal{R}^n$, we define the Lee weight of $x$, denoted by $w_L(x)$, as $$w_L(x)=w_H(d)+w_H(d+c) + w_H(d+b)+w_H(a+b+c+d).$$ For any $x$, $y \in \mathcal{R}^n$, the Lee distance $d_L(x, y)$ between $x$ and $y$ is the Lee weight of $x-y$, i.e., $d_L(x,y)=w_L(x-y)$. A linear code $C$ of length $n$ over $\mathcal{R}$ is an $\mathcal{R}$-submodule of $\mathcal{R}^n$. $C$ is said to be a [*[free code]{}*]{} if $C$ has a $\mathcal{R}$-basis. We define the *rank* of a code $C$ as the minimum number of generators for $C$. The Lee distance of $C$ is denoted by $d_L(C)$ and is defined by $d_L(C)=min \lbrace w_L(c)=\sum^{n-1}_{i=0 }w_L(c_i)\vert c=(c_0, c_1,\dots, c_{n-1})\in C\rbrace$. From the above discussion, we can easily verify the following theorem. \[p1\_thm2\] If $C$ is a linear code over $\mathcal{R}$ of length $n$, size $M$ with Lee distance $d_L$, then $\Phi(C)$ is a code of length $4n$ over $\mathbb{F}_4$, size $M$. Cyclic codes over $M_2(\mathbb{Z}_4)$ ===================================== Let $\tau$ be the standard cyclic shift operator on $\mathcal{R}^n$. A linear code $C$ of length $n$ over $\mathcal{R}$ is cyclic if $\tau(c) \in C$ whenever $c \in C$, *i.e.,* if $(c_0, c_1, \ldots, c_{n-1}) \in C$, then $(c_{n-1}, c_0, c_1, \ldots, c_{n-2}) \in C$. As usual, in the polynomial representation, a cyclic code of length $n$ over $\mathcal{R}$ is an ideal of $\frac{\mathcal{R}[x]}{\left\langle x^n-1\right\rangle}$. Note here that $\frac{\mathcal{R}[x]}{\left\langle x^n-1\right\rangle}$ is a ring.\ \[thm2.2\] A linear code $C = C_1 + UC_2$ of length $n$ over $\mathcal{R}$ is cyclic if and only if $C_1$, $C_2$ are cyclic codes of length $n$ over $\mathbb{Z}_4[w]$. Let $c_1+Uc_2 \in C$, where $c_1 \in C_1$ and $c_2 \in C_2$. Then $\tau(c_1+Uc_2) = \tau(c_1) +U\tau(c_2) \in C$, since $C$ is cyclic and $\tau$ is a linear map. So, $\tau(c_1) \in C_1$ and $\tau(c_2) \in C_2$. Therefore $C_1, C_2$ are cyclic codes. Conversely if $C_1$, $C_2$ are cyclic codes, then for any $c_1+Uc_2 \in C$, where $c_1 \in C_1$ and $c_2 \in C_2$, we have $\tau(c_1) \in C_1$ and $\tau(c_2) \in C_2$, and so, $\tau(c_1+Uc_2) = \tau(c_1) +U\tau(c_2) \in C$. Hence $C$ is cyclic. We assume that $n$ is odd for the rest of this paper. Let $\mathcal{R}[x]$ be the ring of polynomials over the ring $\mathcal{R}$. We define a mapping $$\begin{array}{cccc} \mu: & \mathcal{R}[x] & \longrightarrow & \mathbb{F}_4[x] \\ & \sum_{i=0}^n a_ix^i & \longmapsto & \sum_{i=0}^n \mu(a_i)x^i, \end{array}$$ where $\mu(a_i)$ denote reduction of modulo $2$ and $U$. A polynomial $f \in \mathcal{R}[x]$ is called [*[basic irreducible polynomial]{}*]{} if $\mu(f)$ is irreducible over $\mathbb{F}_4$. Two polynomials $f(x), g(x) \in \mathcal{R}[x]$ are said to be *coprime* if there exist $a(x), b(x) \in \mathcal{R}[x]$ such that $$a(x)f(x) + b(x)g(x) = 1~.$$ The polynomial $x^n-1$ factorizes uniquely into pairwise coprime irreducible polynomials over $\mathbb{F}_4$. Let $x^n-1=f_1f_2f_3\cdots f_m$, where $f_i$’s are irreducible polynomials over $\mathbb{F}_4$. Let $f_i$ be a basic irreducible polynomial over $\mathcal{R}$. Then $\dfrac{\mathcal{R}[x]}{\langle f_i \rangle}$ is not a ring but a right module over $\mathcal{R}$. Since $\langle f_i \rangle$ is not two sided ideal of $\mathcal{R}[x]$ so $\dfrac{\mathcal{R}[x]}{\langle f_i \rangle }$ is not a ring for $1 \leq i \leq m$. Then each $\dfrac{\mathcal{R}[x]}{\langle f_i \rangle }$ is a right $\mathcal{R}$-module. We need a non-commutative analogue of the Chinese Remainder Theorem for modules. Let $n$ be an odd integer. Then $$\dfrac{\mathcal{R}[x]}{\langle x^n-1 \rangle} = \bigoplus^m_1 \dfrac{\mathcal{R}[x]}{\langle f_i \rangle}.$$ The proof follows from [@Oggier2012], [@Wisbauer1991]. \[p1\_thm3\] If $f$ be an irreducible polynomial in $\mathbb{F}_4[x]$, then the right $\mathcal{R}$-modules of $\dfrac{\mathcal{R}[x]}{\langle f \rangle}$ are $$\langle 0 \rangle, \langle 1+\langle f \rangle \rangle, \langle U+\langle f \rangle \rangle, \langle 2U+\langle f \rangle \rangle, \langle (2+Um_{f}) + \langle f \rangle \rangle, \langle 2+\langle f \rangle \rangle, \langle \langle 2, U \rangle+\langle f \rangle \rangle,$$ where $m_{f}$ is an unit in $\dfrac{\mathbb{F}_4[x]}{\langle f \rangle}$. Let $I$ be a non-zero right sub-module of $\dfrac{\mathcal{R}[x]}{\langle f \rangle}$ and $g(x) \in \mathcal{R}[x]$ such that $g(x) + \langle f \rangle \in I$ but $g(x)\notin \langle f \rangle$. If $gcd(\mu(g(x))$, $f(x))=1$ then $g$ is invertible $\pmod f$. So $I=\langle 1 + \langle f \rangle \rangle =\dfrac{\mathcal{R}[x]}{\langle f \rangle}$. If $gcd(\mu(g(x)), f(x)) = f(x)$ then there exit $g_1(x)$, $g_2(x)$, $g_3(x)$, $g_4(x)\in\mathbb{F}_4[x]$ such that $g(x) = g_1(x) + Ug_2(x) + 2g_3(x) + 2Ug_4(x)$ with $gcd((g_1(x)), f(x))=f(x)$ then $g(x) +\langle f \rangle = Ug_2(x)+2g_3(x)+2Ug_4(x)+\langle f \rangle$. If $gcd(g_2(x), f(x))=f(x)$ then $g(x)+\langle f \rangle =2g_3(x)+2Ug_4(x)+\langle f \rangle$. It follows that $I=\langle 2+\langle f \rangle \rangle$. If $gcd(g_3(x), f(x))=f(x)$, implies that $I=\langle 2U+\langle f \rangle \rangle$. Also if $gcd(g_2(x), f(x))=1$, then there exits $g^{-1}_2(x)\in \mathbb{F}_4[x]$ such that $g_2(x)g^{-1}_2(x)\equiv 1 \pmod f$. Therefore $2U=2g(x)g^{-1}_2(x)$. Hence $2U+\langle f \rangle =2g(x)g^{-1}_2(x)+\langle f \rangle\in I$. It follows that $ Ug_2(x)+2g_3(x)+\langle f \rangle = g(x) + 2Ug_4(x)+\langle f \rangle\in I$. If $gcd(g_3(x), f(x))=f(x), $ then $I=\langle U+\langle f \rangle \rangle$. Otherwise $gcd(g_3(x), f(x))=1$, then $g^{-1}_3(x)\in \mathbb{F}_4[x]$ such that $g_3(x)g^{-1}_3(x)\equiv 1 \pmod f$. Hence $2+Ug_2(x)g^{-1}_3(x)+\langle f \rangle \in I$, i.e., $\langle 2+Um_{f}+\langle f \rangle \rangle =I$, where $m_{f}=g_2(x)g_3^{-1}(x)$ is an unit in $\dfrac{\mathbb{F}_4[x]}{\langle f \rangle}$. Since $gcd(g_2(x), f(x))=1,~ gcd(g_3(x), f(x))=1$ then there exit $a_1(x)$, $a_2(x)$, $b_1(x)$, $b_2(x)\in\mathbb{F}_4[x]$ such that $g_2(x)a_1(x)+f(x)a_2(x)=1$, $g_3(x)b_1(x)+f(x)b_2(x)=1$. Therefore $$Ub_1(x)+\langle f \rangle = (Ug_2(x) + \langle f \rangle)(a_1(x)b_1(x)+\langle f \rangle)$$ $$2a_1(x)+\langle f \rangle=(Ug_3(x)+\langle f \rangle)(a_1(x)b_1(x)+\langle f \rangle)$$ $$Ub_1(x)+2a_1(x)+\langle f \rangle=(Ug_2(x)+2g_3(x)\langle f \rangle)(a_1(x)b_1(x)+\langle f \rangle)$$ It follows that $I=\langle \langle U, 2 \rangle + \langle f \rangle \rangle$. \[p1\_thm4\] Let $x^n-1=f_1f_2f_3 \cdots f_m$, where $f_i$’s are monic basic irreducible pairwise coprime polynomials in $\mathcal{R}[x]$. Let $\hat{f}_i=\dfrac{x^n-1}{f_i}$. Then any ideal in $\dfrac{\mathcal{R}[X]}{\left\langle x^n-1\right\rangle}$ is the sum of the right sub-modules: $\left\langle \hat{f}_i+\left\langle x^n-1\right\rangle \right\rangle$, $\left\langle 2\hat{f}_i+\left\langle x^n-1\right\rangle \right\rangle$, $\left\langle U\hat{f}_i+\left\langle x^n-1\right\rangle \right\rangle$, $\left\langle 2U\hat{f}_i+\left\langle x^n-1\right\rangle \right\rangle$, $\left\langle (2+Um_f)\hat{f}_i+\left\langle x^n-1\right\rangle \right\rangle$, $\left\langle \left\langle2, U \right\rangle \hat{f}_i+\left\langle x^n-1\right\rangle \right\rangle $, where $m_{f}$ is an unit in $\dfrac{\mathbb{F}_4[x]}{\left\langle f \right\rangle}$. It follows from the Chinese Remainder Theorem for modules and the right $\mathcal{R}$-modules of the $\dfrac{\mathcal{R}[x]}{\left\langle f\right\rangle}$. \[p1\_thm5\] Let $C$ be a cyclic code of length $n$ over $\mathcal{R}$. Then there exists a family of pairwise monic polynomials $F_0, F_1, \dots, F_6\in \mathbb{F}_4[x]$ such that $F_0F_1\cdots F_6=x^n-1$ and $C=\left\langle \hat{F}_1 \right\rangle$ $\oplus$ $\left\langle U\hat{F}_2 \right\rangle $ $\oplus $ $ \left\langle 2\hat{F}_3 \right\rangle $ $\oplus $ $\left\langle 2U\hat{F}_4 \right\rangle $ $\oplus $ $\left\langle (2+Um_f)\hat{F}_5 \right\rangle $ $\oplus $ $ \left\langle \left\langle 2, U\right\rangle \hat{F}_6 \right\rangle$, where $m_{f}$ is an unit in $\dfrac{\mathbb{F}_4[x]}{\left\langle f \right\rangle}$. Moreover $\vert C \vert = 4^\alpha$, where $\alpha = 4 degF_1+2 degF_2+2 degF_3+degF_4+2 degF_5+3 degF_6$. First part follows from a similar argument of Theorem 2 in [@Luo2017]. Now we compute $|C|$. We know that $C=\left\langle \hat{F}_1 \right\rangle $ $\oplus $ $\left\langle U\hat{F}_2 \right\rangle $ $\oplus $ $ \left\langle 2\hat{F}_3 \right\rangle $ $\oplus $ $\left\langle 2U\hat{F}_4 \right\rangle $ $\oplus $ $\left\langle (2+Um_f)\hat{F}_5 \right\rangle $ $\oplus $ $ \left\langle \left\langle 2,U\right\rangle \hat{F}_6 \right\rangle$, which implies that $\vert C \vert = \mid\hat{F}_{1}\mid \cdot \mid U\hat{F}_{2}\mid \cdot \mid 2\hat{F}_{3}\mid \cdot \mid 2U\hat{F}_{4}\mid \cdot \mid (2+Um_f)\hat{F}_{5}\mid \cdot \mid \langle2, U\rangle\hat{F}_{6}\mid$. The rest follows from the fact that $\mid\hat{F}_{1}\mid=4^{4degF_1}$, $|U\hat{F}_{2}|=4^{2degF_2}$, $ \mid 2\hat{F}_{3}\mid=4^{2degF_3}$, $\mid 2U\hat{F}_{4}\mid=4^{degF_4}$, $\mid (2+Um_f)\hat{F}_{5}\mid=4^{2degF_5}$, $\mid \langle2, U\rangle\hat{F}_{6}\mid=4^{3degF_6}$. \[p1\_thm6\] Let $C$ be a cyclic code of length $n$ over $\mathcal{R}$ with $C=\left\langle \hat{F}_1 \right\rangle \oplus \left\langle U\hat{F}_2 \right\rangle \oplus \left\langle 2\hat{F}_3 \right\rangle \oplus \left\langle 2U\hat{F}_4 \right\rangle \oplus \left\langle (2+Um_f)\hat{F}_5 \right\rangle \oplus \left\langle \left\langle 2, U\right\rangle \hat{F}_6 \right\rangle$, where $m_{f}$ is an unit in $\dfrac{\mathbb{F}_4[x]}{\left\langle f \right\rangle}$ and $F=\hat{F}_1+U\hat{F}_2+2\hat{F}_3+2U\hat{F}_4+(2+Um_f)\hat{F}_5+\left\langle 2, U\right\rangle \hat{F}_6$. Then $C=\left\langle F \right\rangle$. For any two distinct $i, j$, $0 \leq i, j \leq 6$, we have $(x^n-1)\vert \hat{F}_i\hat{F}_j$. So $\hat{F}_i\hat{F}_j=0$. Also for any $i$ with $0\leq i \leq 6$, $ F_i ,\hat{F}_i $ are coprime and $ F_i \hat{F}_i=0$. Since $ F_i, \hat{F}_i $ are coprime, there exist $a_i, b_i$ such that $(a_1F_1+b_1\hat{F}_1)(a_2F_2+b_2\hat{F}_2)(a_3F_3+b_3\hat{F}_3)(a_4F_4+b_4\hat{F}_4)(a_5F_5+b_5\hat{F}_5)=1$. This implies that $a_1F_1a_2F_2a_3F_3a_4F_4a_5F_5 + b_1\hat{F}_1a_2F_2a_3F_3a_4F_4a_5F_5 + a_1F_1b_2\hat{F}_2a_3F_3a_4F_4a_5F_5 + a_1F_1a_2F_2b_3\hat{F}_3a_4F_4a_5F_5 +a_1F_1a_2F_2a_3F_3b_4\hat{F}_4a_5F_5 +a_1F_1a_2F_2a_3F_3a_4F_4b_5\hat{F}_5 =1$. On multiplying both side by $\hat{F}_6$, we obtain $\hat{F}_6a_1F_1a_2F_2a_3F_3a_4F_4a_5F_5=\hat{F}_6$. We have $F=\hat{F}_1+U\hat{F}_2+2\hat{F}_3+2U\hat{F}_4+(2+Um_f)\hat{F}_5+\left\langle 2, U\right\rangle \hat{F}_6$. It follows that $Fa_1F_1a_2F_2a_3F_3a_4F_4a_5F_5 =\left\langle 2, U\right\rangle \hat{F}_6a_1F_1a_2F_2a_3F_3a_4F_4a_5F_5,$ which inturn implies that $Fa_1F_1a_2F_2a_3F_3a_4F_4a_5F_5=\langle 2, U\rangle \hat{F}_6$. Hence $ \langle 2, U\rangle \hat{F}_6\in \langle F\rangle$. Continuing this process, we obtain $\hat{F}_1,~ U\hat{F}_2,~ 2\hat{F}_3,~ 2U\hat{F}_4, ~(2+Um_f)\hat{F}_5,~ \left\langle 2, U\right\rangle \hat{F}_6\in \langle F \rangle$. Consequently $C=\langle F \rangle$. Let us denote $R=\dfrac{\mathbb{F}_4[x]}{\left\langle x^n-1 \right\rangle}$. \[p1\_thm7\] Let $C$ be a cyclic code of length $n$ over $\mathcal{R}$. Then there exists a family of polynomials $F, G, H, Q, T\in \mathbb{F}_4[x] $ which are divisors of $x^n-1$ such that $C=\left\langle F \right\rangle_{R} \oplus U\left\langle G \right\rangle_{R} \oplus 2\left\langle H \right\rangle_{R}\oplus 2U\left\langle Q \right\rangle_{R}\oplus (2+Um_f) \left\langle T \right\rangle_{R}$, where $m_{f}$ is an unit in $\dfrac{\mathbb{F}_4[x]}{\left\langle f \right\rangle}$. Moreover $\vert C \vert=4^{5n-(degF+degG+degH+degQ+degT)}$. A similar argument as in [@Luo2017]. Self-dual cyclic codes over $M_2(\mathbb{Z}_4)$ =============================================== For given $\textbf{x}=(x_1, x_2, \dots, x_n)$, $\textbf{y}=(y_1, y_2, \dots, y_n)\in \mathcal{R}^n$, the Euclidean scalar product (or dot product) of $\textbf{x, y}$ is $\textbf{x}\cdot \textbf{y} = x_1y_1+x_2y_2+ \cdots+x_ny_n \pmod 4$. Two vectors $\textbf{x}$ and $\textbf{y}$ in $\mathcal{R}^n$ are called orthogonal if $\textbf{x}\cdot \textbf{y}=0$. For a linear code $C $ over $\mathcal{R}$, its dual code $C^\perp$ is the set of words over $\mathcal{R}$ that are orthogonal to all codewords of $C$, i.e., $C^\perp=\left\lbrace \textbf{x} \in \mathcal{R}^n \mid \textbf{x} \cdot \textbf{y}=0, \forall y\in C \right\rbrace$. A code $C$ is called self-orthogonal if $C \subset C^{\perp}$ and self-dual if $C=C^{\perp}$. Let $f(x)=a_0+a_1x+ \cdots +a_{k-1}x^{k-1} + a_{k}x^{k}$ be a polynomial of degree $k$ with $a_{k}\neq 0$, $a_{0}\neq 0$. The reciprocal $f^\ast(x)$ of $f(x)$ is defined by $$f^\ast(x)=a_0^{-1}x^kf(x^{-1}).$$ \[p1\_thm8\] Let $C$ be a cyclic code of length $n$ over $\mathcal{R}$ with $C=\left\langle \hat{F}_1 \right\rangle $ $\oplus $ $\left\langle U\hat{F}_2 \right\rangle $ $\oplus $ $ \left\langle 2\hat{F}_3 \right\rangle $ $\oplus $ $\left\langle 2U\hat{F}_4 \right\rangle $ $\oplus $ $\left\langle (2+Um_f)\hat{F}_5 \right\rangle $ $\oplus $ $ \left\langle \left\langle 2, U\right\rangle \hat{F}_6 \right\rangle$, where $m_{f}$ is an unit in $\dfrac{\mathbb{F}_4[x]}{\left\langle f \right\rangle}$. Then $C^\perp=\left\langle \hat{F}^\ast_0 \right\rangle $ $\oplus $ $\left\langle U\hat{F}^\ast_2 \right\rangle $ $\oplus $ $ \left\langle 2\hat{F}^\ast_3 \right\rangle $ $\oplus $ $\left\langle 2U\hat{F}^\ast_6 \right\rangle $ $\oplus $ $\left\langle (2+Um_f)\hat{F}^\ast_5 \right\rangle $ $\oplus $ $ \left\langle \left\langle 2, U\right\rangle \hat{F}^\ast_4 \right\rangle$ and $\mid C^\perp\mid=4^{4degF_0+2degF_2+2degF_3+3degF_4+2degF_5+degF_6}$. From the Theorem \[p1\_thm5\], $\mid C \mid = 4^{4degF_1 + 2degF_2 + 2degF_3 + degF_4 + 2degF_5 + 3degF_6}$. Since $\mid C \mid \mid C^\perp \mid=4^{4n}$ and $n=degF_1+degF_2+degF_3+degF_4+degF_5+degF_6$, so $\mid C^\perp \mid=4^{4degF_0+2degF_2+2degF_3+3degF_4+2degF_5+degF_6}$. We denote $C^\ast=\left\langle \hat{F}^\ast_0 \right\rangle $ $\oplus $ $\left\langle U\hat{F}^\ast_2 \right\rangle $ $\oplus $ $ \left\langle 2\hat{F}^\ast_3 \right\rangle $ $\oplus $ $\left\langle 2U\hat{F}^\ast_6 \right\rangle $ $\oplus $ $\left\langle (2+Um_f)\hat{F}^\ast_5 \right\rangle $ $\oplus $ $ \left\langle \left\langle 2, U\right\rangle \hat{F}^\ast_4 \right\rangle $. For $i, j$, $0 \leq i, j \leq 6$, if $i+1=7-j+1$, i.e., $i=7-j,$ we can see that $\hat{F}_{i+1}\hat{F}^\ast_{7-i+1}=0$. If $i+1\neq7-j+1$, i.e., $i\neq7-j,$ then we have $ x^n-1\mid\hat{F}_{i+1}\hat{F}^\ast_{7-i+1}$. Then it follows that $\hat{F}_{i+1}\hat{F}^\ast_{7-i+1}=0$. Therefore $C^\ast \subseteq C^\perp$. Note that $\mid\hat{F}^\ast_{0}\mid=4^{4degF_0}$, $\mid U\hat{F}^\ast_{2}\mid=4^{2degF_2}$, $\mid 2\hat{F}^\ast_{3}\mid=4^{2degF_3}$, $\mid 2U\hat{F}^\ast_{6}\mid=4^{degF_6}$, $\mid (2+Um_f)\hat{F}^\ast_{5}\mid=4^{2degF_5}$, $\mid \langle2, U\rangle\hat{F}^\ast_{4}\mid=4^{degF_4}$. Hence $|C^\ast| = 4^{4degF_0+2degF_2+2degF_3+3degF_4+2degF_5+degF_6} = |C^\perp|$. Consequently $C^\ast = C^\perp$. \[p1\_thm9\] Let $C$ be a cyclic code of length $n$ over $\mathcal{R}$ with $C$ and $C^\perp$ defined as in Theorem \[p1\_thm8\], and $F^\ast=\hat{F}^\ast_0+U\hat{F}^\ast_2+2\hat{F}_3+2U\hat{F}^\ast_6+(2+Um_f)\hat{F}^\ast_5+\left\langle 2, U\right\rangle \hat{F}^\ast_4$. Then $C^\perp=\left\langle F^\ast \right\rangle$. The result follows from a similar argument as in the proof of Theorem \[p1\_thm6\] as $\hat{F}^\ast_i\hat{F}^\ast_j=0$ and $\hat{F}^\ast_i, F^\ast_j$ are coprime for any $i, j$, $0 \leq i,j \leq 6$. \[p1\_thm10\] Let $C$ be a cyclic code of length $n$ over $\mathcal{R}$. Then there exists a family of polynomials $F^\ast, G^\ast, H^\ast, Q^\ast, T^\ast\in \mathbb{F}_4[x] $ which are divisors of $x^n-1$ such that $C^\perp=\left\langle F^\ast \right\rangle_{R} \oplus U\left\langle G^\ast \right\rangle_{R} \oplus 2\left\langle H^\ast \right\rangle_{R}\oplus 2U\left\langle Q^\ast \right\rangle_{R}\oplus (2+Um_f) \left\langle T^\ast \right\rangle_{R}$, where $m_{f}$ is an unit in $\dfrac{\mathbb{F}_4[x]}{\left\langle f \right\rangle}$. Moreover $\vert C^\perp \vert = 4^{5n-(degF^\ast+degG^\ast+degH^\ast+degQ^\ast+degT^\ast)}$. Follows fromTheorem \[p1\_thm7\]. We now prove the main result of this section, a condition for a cyclic code $C$ over $\mathcal{R}$ to be self-dual. From Theorem \[p1\_thm6\] and Theorem \[p1\_thm9\], we can see that a cyclic codes $C$ is self-dual if and only if $F=F^\ast$. This implies that $$\hat{F}_1=\hat{F}^\ast_0, ~~ \hat{F}_2=\hat{F}^\ast_2,~~ \hat{F}_3=\hat{F}^\ast_3,~~\hat{F}_4=\hat{F}^\ast_6,~~\hat{F}_5=\hat{F}^\ast_5,~~ \hat{F}_6=\hat{F}^\ast_4.$$ Again since $\hat{F}_i=\dfrac{x^n-1}{F_i}$, $\hat{F}^\ast_j=\dfrac{x^n-1}{F^\ast_j}$ and $\hat{F}_i=\hat{F}^\ast_j$, we have $F_i=F^\ast_j$. Hence proved the following results. \[p1\_thm11\] Let $C$ be a cyclic code of length $n$ over $\mathcal{R}$ with $C=\left\langle \hat{F}_1 \right\rangle $ $\oplus $ $\left\langle U\hat{F}_2 \right\rangle $ $\oplus $ $ \left\langle 2\hat{F}_3 \right\rangle $ $\oplus $ $\left\langle 2U\hat{F}_4 \right\rangle $ $\oplus $ $\left\langle (2+Um_f)\hat{F}_5 \right\rangle $ $\oplus $ $ \left\langle \left\langle 2 ,U\right\rangle \hat{F}_6 \right\rangle$, where $m_{f}$ is an unit in $\dfrac{\mathbb{F}_4[x]}{\left\langle f \right\rangle}$. Then $C$ is self-dual code if and only if $F_1=F^\ast_0,~~ F_2=F^\ast_2,~~ F_3=F^\ast_3,~~ F_4=F^\ast_6,~~ F_5=F^\ast_5.$ \[p1\_thm12\] Let $C$ be a cyclic code of length $n$ over $\mathcal{R}$ with $C=\left\langle F \right\rangle_{R} \oplus U\left\langle F \right\rangle_{R} \oplus 2\left\langle F \right\rangle_{R}\oplus 2U\left\langle F \right\rangle_{R}\oplus (2+Um_f) \left\langle F \right\rangle_{R},$ where $m_{f}$ is an unit in $\dfrac{\mathbb{F}_4[x]}{\left\langle f \right\rangle}$. Then $C$ is self-dual code if and only if $F=F^\ast,~~ G=G^\ast,~~ H=H^\ast,~~ Q=Q^\ast,~~ T=T^\ast.$ Hermitian Self-dual cyclic codes over $M_2(\mathbb{Z}_4)$ ========================================================= For any two codewords $\textbf{x}=(x_1, x_2, \dots, x_n)$, $\textbf{y}=(y_1, y_2, \dots, y_n)\in \mathcal{R}^n$, the Hermitian inner product is defined as $$\langle\textbf{x},\textbf{y}\rangle=\textbf{x} \cdot {\bar{\textbf{y}}}=x_1\bar{y_1}+x_2\bar{y_2}+ \cdots+x_n\bar{y_n},$$ where “ $\bar{~}$ " called conjugation, for example, $\bar{0}=0$, $\bar{1}=1$, $\bar{w}=w^2$, $\bar{w^2}=w$. The Hermitian dual of $C$, denoted by $C^{\perp_{H}}$, is define as $$C^{\perp_{H}}=\left\lbrace\textbf{x} \in \mathcal{R}^n \mid \langle\textbf{x},\textbf{y}\rangle=0,\forall y\in C \right\rbrace .$$ We can see that $\bar{C}^\perp=C^{\perp_{H}}$. As usual $C$ is called Hermitian self-orthogonal and Hermitian self-dual if $C\subseteq C^{\perp_{H}}$ and $C=C^{\perp_{H}}$, respectively. Let $f(x)=a_0+a_1x+ \cdots +a_{k-1}x^{k-1} + a_{k}x^{k}$ be a polynomial of degree $k$ with $a_{k}\neq 0$, $a_{0}\neq 0$. The reciprocal $f^\ast(x)$ of $f(x)$ is defined by $$f^\ast(x)=a_0^{-1}x^kf(x^{-1}).$$ Denote $\bar{f}(x)=a^2_0+a^2_1x+ \cdots +a^2_{k-1}x^{k-1} + a^2_{k}x^{k}$. It is easy to check that two operations $\ast$ and $\bar{•}$ are commutative, i.e., $\bar{(f^\ast)}(x)=(\bar{f})^\ast(x)$. All the theorems proved in previous section are true with respect to Hermitian inner product as well. So state them here without proofs. \[p1\_thm13\] Let $C$ be a cyclic code of length $n$ over $\mathcal{R}$ with $C=\left\langle \hat{F}_1 \right\rangle $ $\oplus $ $\left\langle U\hat{F}_2 \right\rangle $ $\oplus $ $ \left\langle 2\hat{F}_3 \right\rangle $ $\oplus $ $\left\langle 2U\hat{F}_4 \right\rangle $ $\oplus $ $\left\langle (2+Um_f)\hat{F}_5 \right\rangle $ $\oplus $ $ \left\langle \left\langle 2, U\right\rangle \hat{F}_6 \right\rangle$, where $m_{f}$ is an unit in $\dfrac{\mathbb{F}_4[x]}{\left\langle f \right\rangle}$. Then $C^{\perp_{H}}=\left\langle \hat{\bar{F}}^\ast_0 \right\rangle $ $\oplus $ $\left\langle U\hat{\bar{F}}^\ast_2 \right\rangle $ $\oplus $ $ \left\langle 2\hat{\bar{F}}^\ast_3 \right\rangle $ $\oplus $ $\left\langle 2U\hat{\bar{F}}^\ast_6 \right\rangle $ $\oplus $ $\left\langle (2+Um_f)\hat{\bar{F}}^\ast_5 \right\rangle $ $\oplus $ $ \left\langle \left\langle 2, U\right\rangle \hat{\bar{F}}^\ast_4 \right\rangle$ and $\mid C^{\perp_{H}}\mid=4^{4degF_0+2degF_2+2degF_3+3degF_4+2degF_5+degF_6}$. \[p1\_thm14\] Let $C$ be a cyclic code of length $n$ over $\mathcal{R}$ with $C$ and $C^\perp$ defined as in Theorem \[p1\_thm13\], and $\bar{F}^\ast=\hat{\bar{F}}^\ast_0+U\hat{\bar{F}}^\ast_2+2\hat{\bar{F}}_3+2U\hat{\bar{F}}^\ast_6+(2+Um_f)\hat{\bar{F}}^\ast_5+\left\langle 2, U\right\rangle \hat{\bar{F}}^\ast_4$. Then $C^{\perp_{H}}=\left\langle \bar{F}^\ast \right\rangle$. \[p1\_thm14 a\] Let $C$ be a cyclic code of length $n$ over $\mathcal{R}$. Then there exists a family of polynomials $\bar{F}^\ast, \bar{G}^\ast, \bar{H}^\ast, \bar{Q}^\ast, \bar{T}^\ast\in \mathbb{F}_4[x] $ which are divisors of $x^n-1$ such that $C^{\perp_{H}}=\left\langle \bar{F}^\ast \right\rangle_{R} \oplus U\left\langle \bar{G}^\ast \right\rangle_{R} \oplus 2\left\langle \bar{H}^\ast \right\rangle_{R}\oplus 2U\left\langle \bar{Q}^\ast \right\rangle_{R}\oplus (2+Um_f) \left\langle \bar{T}^\ast \right\rangle_{R}$, where $m_{f}$ is an unit in $\dfrac{\mathbb{F}_4[x]}{\left\langle f \right\rangle}$. Moreover $\vert C^{\perp_{H}} \vert = 4^{5n-(degF^\ast+degG^\ast+degH^\ast+degQ^\ast+degT^\ast)}$. In [@Alahmadi2013 Theoem 2], the authors have claimed that Hermitian Self-dual codes do not exist over $M_2(\mathbb{Z}_2)$ but which is not true. In this paper, we present a condition for a cyclic code over $M_2(\mathbb{Z}_2)$ to be Hermitian Self-dual and also demonstrate the same with an example. We also generalise the same to cyclic codes over $M_2(\mathbb{Z}_4)$. \[p1\_thm15\] Let $\mathcal{C}=<fh,~ ufg>$ be a cyclic code of length $n$ over $M_2(\mathbb{Z}_2)$ with $x^n-1=fgh$. Then $\mathcal{C}$ is Hermitian self-dual code if and only if $f=\hat{\bar{g}}^\ast$ and $h=\hat{\bar{h}}^\ast$. In [@Alahmadi2013 $\mathsection$6], authors have claimed that there does not exist a nontrivial self-dual cyclic codes of length $5$. A contrary example is the following: The factorization of $x^5-1$ is $(x-1)(x^2+wx+1)(x^2+w^2x+1)$ over $\mathbb{F}_4$. Let $f_1=(x-1)$, $f_2=(x^2+wx+1)$ and $f_3=(x^2+w^2x+1)$, then $f_1=\bar{f}^\ast_1$, $f_2=\bar{f}^\ast_3$ and $f_3=\bar{f}^\ast_2$. The following cyclic codes of length $5$ over $M_2(\mathbb{Z}_2)$ are self-dual (Hermitian) codes and their Gray image $\Phi(C)$ has parameters $[10,5,4]$ over $\mathbb{F}_4$. $$\langle f_1f_2, \quad uf_2f_3 \rangle, \qquad \langle f_1f_3, \quad uf_2f_3 \rangle.$$ \[p1\_thm15\] Let $C$ be a cyclic code of length $n$ over $\mathcal{R}$ with $C=\left\langle \hat{F}_1 \right\rangle $ $\oplus $ $\left\langle U\hat{F}_2 \right\rangle $ $\oplus $ $ \left\langle 2\hat{F}_3 \right\rangle $ $\oplus $ $\left\langle 2U\hat{F}_4 \right\rangle $ $\oplus $ $\left\langle (2+Um_f)\hat{F}_5 \right\rangle $ $\oplus $ $ \left\langle \left\langle 2 ,U\right\rangle \hat{F}_6 \right\rangle$, where $m_{f}$ is an unit in $\dfrac{\mathbb{F}_4[x]}{\left\langle f \right\rangle}$. Then $C$ is Hermitian self-dual code if and only if $$\hat{F}_1=\hat{\bar{F}}^\ast_0,~~\hat{F}_2=\hat{\bar{F}}^\ast_2,~~\hat{F}_3=\hat{\bar{F}}^\ast_3,~~\hat{F}_4=\hat{\bar{F}}^\ast_6,~~\hat{F}_5=\hat{\bar{F}}^\ast_5,~~ \hat{F}_6=\hat{\bar{F}}^\ast_4.$$ \[p1\_thm16\] Let $C$ be a cyclic code of length $n$ over $\mathcal{R}$ with $C=\left\langle F \right\rangle_{R} \oplus U\left\langle F \right\rangle_{R} \oplus 2\left\langle F \right\rangle_{R}\oplus 2U\left\langle F \right\rangle_{R}\oplus (2+Um_f) \left\langle F \right\rangle_{R},$ where $m_{f}$ is an unit in $\dfrac{\mathbb{F}_4[x]}{\left\langle f \right\rangle}$. Then $C$ is the Hermitian self-dual code if and only if $$F=\bar{F}^\ast,~~ G=\bar{G}^\ast,~~ H=\bar{H}^\ast,~~ Q=\bar{Q}^\ast,~~ T=\bar{T}^\ast.$$ The factorization of $x^7-1$ is $(x-1)(x^3+x+1)(x^3+x^2+1)$ over $\mathbb{F}_4$. Let $f_1=(x-1)$, $f_2=(x^3+x+1)$ and $f_3=(x^3+x^2+1)$, then $f_1=f^\ast_1$, $f_2=f^\ast_3$ and $f_3=f^\ast_2$. The following cyclic codes of length $7$ over $\mathcal{R}$ are self-dual (Euclidean) codes and their Gray image $\Phi(C)$ has parameters $[28,14,4]$ over $\mathbb{F}_4$.\ $$\langle f_1f_2, \quad rf_2f_3 \rangle, \qquad \langle f_1f_3, \quad rf_2f_3 \rangle, \qquad \mbox{where}~ r\in\{U,2,2+U\},$$ $$\langle 2Uf_1f_3, \quad 2f_1f_2, \quad Uf_1f_3, \quad sf_2f_3 \rangle, \qquad \mbox{where}~ s\in\{2,2+U\},$$ $$\langle 2Uf_1f_2, \quad 2f_1f_3, \quad Uf_1f_3, \quad tf_2f_3 \rangle, \qquad \mbox{where}~ t\in\{2,2+U\}.$$ The factorization of $x^5-1$ is $(x-1)(x^2+wx+1)(x^2+w^2x+1)$ over $\mathbb{F}_4$. Let $f_1=(x-1)$, $f_2=(x^2+wx+1)$ and $f_3=(x^2+w^2x+1)$, then $f_1=\bar{f}^\ast_1$, $f_2=\bar{f}^\ast_3$ and $f_3=\bar{f}^\ast_2$. The following cyclic codes of length $5$ over $\mathcal{R}$ are self-dual (Hermitian) codes and their Gray image $\Phi(C)$ has parameters $[20,10,4]$ over $\mathbb{F}_4$.\ $$\langle f_1f_2, \quad rf_2f_3 \rangle, \qquad \langle f_1f_3, \quad rf_2f_3 \rangle, \qquad \mbox{where}~ r\in\{U,2,2+U\},$$ $$\langle 2Uf_1f_3, \quad 2f_1f_2, \quad Uf_1f_3, \quad sf_2f_3 \rangle, \qquad \mbox{where}~ s\in\{2,2+U\},$$ $$\langle 2Uf_1f_2, \quad 2f_1f_3, \quad Uf_1f_3, \quad tf_2f_3 \rangle, \qquad \mbox{where}~ t\in\{2,2+U\}.$$ Conclusion ========== In 2013, Alahmadi et al. developed cyclic codes over finite matrix ring $M_2(\mathbb{F}_2)$ and their duals as right ideals in terms of two generators. Also cyclic codes over $M_2(\mathbb{F}_2)$ were made the existence of infinitely many nontrivial cyclic codes for Euclidean product. All this was derived of odd length code. In this paper, we constructed the structure of $M_2(\mathbb{Z}_4)$ and developed cyclic dual codes and cyclic self-dual codes over $M_2(\mathbb{Z}_4)$ which is even length codes over $M_2(\mathbb{F}_2)$. In [@Alahmadi2013] [@Luo2017], it is not possible to construct negacyclic code, since the characteristic of structure of the ring is $2$. But in our construction one can form negacyclic code. Welcome to the reader to construct even length codes over $M_2(\mathbb{Z}_4)$. Another useful direction for further study would be to consider LCD codes over $M_2(\mathbb{Z}_4)$. Acknowledgements {#acknowledgements .unnumbered} ================ The author Sanjit Bhowmick is thankful to MHRD for financial support. [1]{} A. Alahmadi, H. Sboui, P. Sol$\acute{\mathrm{e}}$, O. Yemen, Cyclic codes over $M_2(\mathbb{F}_2)$, Journal of the Franklin Institute, 350 (9) (2013) 2837 –2847. C. Bachoc, Applications of coding theory to the construction of modular lattices, Journal of Combinatorial Theory A 78-1 (1997) 92 – 119. M. Greferath, S.E. Schmidt, Linear codes and rings of matrices, Proceedings of AAECC 13, Hawaii, Springer LNCS 1719 (1999) 160 –169. A.R. Hammons Jr., P. Vijay Kumar, A.R. Calderbank, N.J.A. Sloane, P. Sol$\acute{\mathrm{e}}$, The $\mathbb{Z}_4$-linearity of Kerdock, Preparata, Goethals and related codes, IEEE Transactions on Information Theory, IT-40 (1994) 301 – 319. R. Luo, U. Parampalli, Cyclic codes over $M_2(\mathbb{F}_2+u\mathbb{F}_2)$, Cryptography and Communications, (2017) https://doi.org/10.1007/s12095-017-0266-1. F. Oggier, P. Sol$\acute{\mathrm{e}}$, J.-C. Belfiore, Codes over matrix rings for space-time coded modulations, IEEE Transactions on Information Theory IT-58 (2012) 734 –746. V. Pless, P. Sol$\acute{\mathrm{e}}$, Z. Qian, Cyclic self-dual $\mathbb{Z}_4$-codes, Finite Fields and their Applications 3 (1997) 48 –69. V. Pless, Z. Qian, Cyclic codes and quadratic residue codes over $\mathbb{Z}_4$, IEEE Transactions on Information Theory 42 (5) (1996) 1594 – 1600. R. Wisbauer, Foundations of Module and Ring Theory, Gordon and Breach, 1991.

--- abstract: 'The pion-photon Transition Distribution Amplitudes (TDAs) are studied, treating the pion as a bound state in the sense of Bethe-Salpeter, in the formalism of the NJL model. The results obtained explicitly verify support, sum rules and polynomiality conditions. The role of PCAC is highlighted.' author: - | A. Courtoy $^{1}$ and S. Noguera $^{1}$\ $^{1} $ [Departamento de Fisica Teorica and Instituto de Física Corpuscular,]{}\ [Universidad de Valencia-CSIC, E-46100 Burjassot (Valencia), Spain.]{} title: ' Pion-Photon TDAs in the NJL Model [^1] ' --- Hard reactions provide important information for unveiling the structure of hadrons. The large virtuality, $Q^{2}$, involved in the processes allows the factorization of the hard (perturbative) and soft (non-perturbative) contributions in their amplitudes. In recent years a large variety of processes governed by the Generalized Parton Distributions (GPDs), like the Deeply Virtual Compton Scattering, has been considered. A generalization of GPDs to non-diagonal transitions has been proposed in [@Pire:2004ie]. In particular, the easiest case to consider is the pion-photon TDA, governing processes like $\pi^{+}\pi^{-}\rightarrow\gamma^{\ast}\gamma$ or $\gamma ^{\ast}\pi^{+}\rightarrow\gamma\pi^{+}$ in the kinematical regime where the virtual photon is highly virtual but with small momentum transfer. At leading-twist, the vector and axial TDAs, respectively $V(x,\xi,t)$ and $A(x,\xi,t)$, are defined as [@Courtoy:2007vy] [$$\begin{aligned} \int\frac{dz^{-}}{2\pi}e^{ixP^{+}z^{-}}\left. \left\langle \gamma\left( p^{\prime}\right) \right\vert \bar{q}\left( -\frac{z}{2}\right) \gamma ^{+}\hspace*{-0.05cm}\tau^{-}q\left( \frac{z}{2}\right) \left\vert \pi^+\left( p\right) \right\rangle \right\vert _{z^{+}=z^{\bot}=0} & =i\,e\,\varepsilon_{\nu}\,\epsilon^{+\nu\rho\sigma}\,P_{\rho}\,\Delta_{\sigma }\,\frac{V^{\pi^{+}}\left( x,\xi,t\right) }{\sqrt{2}f_{\pi}}~,\label{vectcurr}\\ \int\frac{dz^{-}}{2\pi}e^{ixP^{+}z^{-}}\left. \left\langle \gamma\left( p^{\prime}\right) \right\vert \bar{q}\left( -\frac{z}{2}\right) \gamma ^{+}\hspace*{-0.05cm}\gamma_{5}\tau^{-}q\left( \frac{z}{2}\right) \left\vert \pi^+\left( p\right) \right\rangle \right\vert _{z^{+}=z^{\bot}=0} & =e\,\left( \vec{\varepsilon}^{\bot}\cdot\vec{\Delta}^{\bot}\right) \frac{A^{\pi^{+}}\left( x,\xi,t\right) }{\sqrt{2}f_{\pi}}\nonumber\\ & +e\,\left( \varepsilon\cdot\Delta\right) \frac{2\sqrt{2}f_{\pi}}{m_{\pi }^{2}-t}~\epsilon\left( \xi\right) ~\phi\left( \frac{x+\xi}{2\xi}\right) \quad, \label{axcurr}$$ ]{}where $t=\Delta^{2}=(p^{\prime}-p)^{2}$, $P=\left( p+p^{\prime}\right) /2$, $\xi=\left( p-p^{\prime}\right) ^{+}/2P^{+}$, $\epsilon\left( \xi\right) =1$ for $\xi>0$ and $-1$ for $\xi<0$ and where $f_{\pi}=93$ MeV. For any four-vector $v^{\mu},$ we have the light-cone coordinates $v^{\pm}=\left( v^{0}\pm v^{3}\right) /\sqrt{2}$ and the transverse components $\vec{v}^{\bot}=\left( v^{1},v^{2}\right) .$ Finally, $\phi\left( x\right) $ is the pion distribution amplitude (PDA). Apart from the axial TDA $A(x,\xi,t)$, the axial current Eq.(\[axcurr\]) contains a pion pole contribution, which can be understood as a consequence of PCAC because the axial current must be coupled to the pion. This second term has been isolated in a model independent way. Therefore, all the structure of the incoming pion remains in $A\left( x,\xi,t\right) $. The pion pole term is not a peculiarity of the pion-photon TDAs: a similar contribution would be present in the Lorentz decomposition, in terms of distribution amplitudes, of the axial current for any pair of external particles. This term is only non-vanishing in the ERBL region, i.e. the $x\in\left[ -\xi,\xi\right] $ region, whose kinematics allow the emission or absorption of a pion from the initial state, which is described through the PDA. The $\pi$-$\gamma$ TDAs are related to the vector and axial transition form factors through the sum rules$$\int_{-1}^{1}dx~V^{\pi^{+}}\left( x,\xi,t\right) =\frac{\sqrt{2}f_{\pi}}{m_{\pi}}F_{V}\left( t\right) ~,~~~~~\int_{-1}^{1}dx~A^{\pi^{+}}\left( x,\xi,t\right) =\frac{\sqrt{2}f_{\pi}}{m_{\pi}}F_{A}\left( t\right) \quad. \label{sumerule}$$ As usual, we consider that the currents present in Eqs. (\[vectcurr\]) and (\[axcurr\]) are dominated by the handbag diagram. The method of calculation developed in [@Theussl:2002xp] is here applied. The pion is treated as a bound-state in a fully covariant manner using the Bethe-Salpeter equation and solving it in the NJL model. Gauge invariance is ensured by using the Pauli-Villars regularization scheme. All the invariances of the problem are then preserved. As a consequence, the correct support is obtained, i.e. $x\in\lbrack-1,1],$ vector and axial TDAs obey the sum rules, Eq.(\[sumerule\]), and the polynomiality expansion is recovered in both cases. Moreover, for the DGLAP region, we have obtained the isospin relations $$V\left( -x,\xi,t\right) =-2V\left( x,\xi,t\right) ,~~~~A\left( -x,\xi,t\right) =2A\left( x,\xi,t\right) ,~~~~~~~~~~~\left\vert \xi\right\vert <x<1\quad.\label{isospin}$$ In the figures are depicted both the vector and axial TDAs, which explicit expression are given in [@Courtoy:2007vy], for $m_{\pi}=140$ MeV, $t=-0.5$ GeV$^{2}$ and different values of $\xi$ ranging between $t/(2m_{\pi}^{2}-t)<\xi<1$: the process here does not constrain the skewness variable to be positive. The vector TDA is mainly a function of $\xi^{2}$ and we have depicted only positive values of $\xi.$ For the axial TDA, two quite different behaviours are observed according to the sign of $\xi$. The value we numerically obtain for $F_{V}\left( 0\right) $ is in agreement with [@Yao:2006px], while the one we obtain for $F_{A}\left( 0\right) $ is twice the expected value [@Yao:2006px]. {width="6.5cm"} {width="6.5cm"} Previous studies of the pion-photon TDAs have been released [@Tiburzi:2005nj]. Since both these studies parametrize TDAs by means of double distributions, Ref. [@Courtoy:2007vy] is the first study of the polynomiality property of TDAs. Moreover, in Ref. [@Courtoy:2007vy], the support, sum rules and polynomiality expansion are results (and not inputs) of the calculation. The study of TDAs should lead to interesting estimates of cross-sections for exclusive meson pair production in $\gamma\gamma^{\ast}$ scattering [@Pire:2004ie]. In particular, a deeper study of the pion pole contribution should allow us to give a cross-section estimate for the $\pi\pi$ pair case. [9]{} -2pt B. Pire and L. Szymanowski, Phys. Rev. D **71** (2005) 111501. J. P. Lansberg *et al.*, Phys. Rev. D **73** (2006) 074014. A. Courtoy and S. Noguera, Phys. Rev.  D [**76**]{} (2007) 094026 \[arXiv:0707.3366 \[hep-ph\]\]. S. Noguera, L. Theussl and V. Vento, Eur. Phys. J. A **20** (2004) 483. W. M. Yao *et al.* \[Particle Data Group\], J. Phys. G **33** (2006) 1. B. C. Tiburzi, Phys. Rev. D **72** (2005) 094001. W. Broniowski and E. R. Arriola, Phys. Lett. B **649** (2007) 49. [^1]: This work has been supported by the 6th Framework Program of the European Commission No. 506078 and MEC (FPA 2007-65748-C02-01 and AP2005-5331)

--- abstract: 'After introducing the Szekeres and Lemaître–Tolman cosmological models, the real-time cosmology program is briefly mentioned. Then, a few widespread misconceptions about the cosmological models are pointed out and corrected. Investigation of null geodesic equations in the Szekeres models shows that observers in favourable positions would see galaxies drift across the sky at a rate of up to $10^{-6}$ arc seconds per year. Such a drift would be possible to measure using devices that are under construction; the required time of monitoring would be $\approx10$ years. This effect is zero in the FLRW models, so it provides a measure of inhomogeneity of the Universe. In the Szekeres models, the condition for zero drift is zero shear. But in the shearfree normal models, the condition for zero drift is that, in the comoving coordinates, the time dependence of the metric completely factors out.' author: - 'Andrzej Krasiński$^*$' - 'Krzysztof Bolejko$^*$' title: Exact inhomogeneous models and the drift of light rays induced by nonsymmetric flow of the cosmic medium --- Inhomogeneous models in astrophysics$^1$ ======================================== \[introduction\] Just as was the case with our earlier reviews [@Kras1997; @BKHC2010; @BCKr2011], we define inhomogeneous cosmological models as those exact solutions of Einstein’s equations that contain at least a subclass of nonvacuum and nonstatic Friedmann – Lemaître – Robertson – Walker (FLRW) solutions as a limit. The reason for this choice is that such FLRW models are generally considered to be a good first approximation to a description of our real Universe, so it makes sense to consider only those other models that have a chance to be a still better approximation. Models that do not include an FLRW limit would not easily fulfil this condition. This is a live topic, with new contributions appearing frequently, so any review intended to be complete would become obsolete rather soon. Ref. 1 is complete until 1994, Ref. 2 is a selective update until 2009 and Ref. 3 is a more selective update until the end of 2010. The criterion of the selection in the updates was the usefulness of the chosen papers for solving problems of observational cosmology. In the very brief overview given here we put emphasis on pointing out and correcting the erroneous results that exist in the literature and are being taken as proven truths. Some of them have evolved to become research paradigms, with many followers, some others proceed in this direction. We hope to stop this process, which disturbs and slows down the recognition of the inhomogeneous models as useful devices for understanding the observed Universe. We first present the two classes of inhomogeneous models that, so far, proved most fruitful in their astrophysical application: the Szekeres model [@Szek1975], and its spherically symmetric limit, the Lemaître [@Lema1933] – Tolman [@Tolm1934] (L–T) model. Then we briefly mention the real-time cosmology program, and we give an overview of the erroneous ideas. Finally, we present an effect newly calculated in a few classes of models that is possible to observe and can become a test of homogeneity of the Universe: the drift of light rays induced by nonsymmetric flow of the cosmic medium The Szekeres solution {#Szek} ===================== The (quasi-spherical) Szekeres solution [@Szek1975; @PlKr2006] is, in comoving coordinates $${\rm d} s^2 = {\rm d} t^2 - \frac {{\cal E}^2 {(\Phi / {\cal E}),_r}^2} {1 + 2E(r)} {\rm d} r^2 - \frac {\Phi^2} {{\cal E}^2} \left({\rm d} x^2 + {\rm d} y^2\right),$$ $${\cal E} {\ {\overset {\rm def} =}\ }\frac {(x - {P})^2} {2{S}} + \frac {(y - {Q})^2} {2{S}} + \frac {{S}} 2,\label{2.1}$$ where $E(r)$, $M(r)$, ${P}{(r)}$, ${Q}{(r)}$ and ${S}{(r)}$ are arbitrary functions and $\Phi(t,r)$ obeys $$\label{2.2} {\Phi,_t}^2 = 2E(r) + \frac {2 {M}{(r)}} {\Phi} + \frac 1 3 \Lambda \Phi^2.$$ The source in the Einstein equations is dust, whose mass density in energy units is $$\label{2.3} \kappa \rho = \frac {2 \left(M / {\cal E}^3\right),_r} {(\Phi / {\cal E})^2 \left(\Phi / {\cal E}\right),_r}.$$ Eq. (\[2.2\]) implies that the bang time is in general position-dependent: $$\label{2.4} \int\limits_0^{\Phi}\frac{{\rm d} \widetilde{\Phi}}{\sqrt{2E + 2M / \widetilde{\Phi} + \frac 1 3 \Lambda \widetilde{\Phi}^2}} = t - {t_B(r)}.$$ The general Szekeres metric has no symmetry. It contains the spherically symmetric Lemaître [@Lema1933] – Tolman [@Tolm1934] (L–T) model as the limit of $(P, Q, S)$ being all constant. The latter is usually used with a different parametrisation of the spheres of constant $(t, r)$, namely $$\label{2.5} {\rm d}s^2 = {\rm d}t^2 - \frac{R,_r^2}{1 + 2E} {\rm d}r^2 - R^2(t,r) \left({\rm d}\vartheta^2 + \sin^2 \vartheta {\rm d}\varphi^2 \right),$$ where $R \equiv \Phi$, still obeying (\[2.2\]), and (\[2.3\]) simplifies to $$\label{2.6} \kappa \rho = \frac {2 M,_r} {R^2 R,_r}.$$ The Friedmann limit follows when, in addition, $\Phi (t,r) = r S(t)$, $2E = - k r^2$ where $k =$ const is the FLRW curvature index, and $t_B$ is constant. Real-time cosmology =================== There are ways in which the expansion of the Universe might be directly observed. The authors of Refs. [@QQAm2009; @QABC2012] composed them into a paradigm termed [***real-time cosmology***]{}. One of them is the [***redshift drift***]{}: the change of redshift with time for a fixed light source, induced by the expansion of the Universe. Consider, as an example, an L–T model with $\Lambda$, given by (\[2.5\]) with $R$ obeying (\[2.2\]). Along a single radial null geodesic, directed toward the observer, $t = T(t_o,r)$ (where $t_o$ is the instant of observation) the redshift is [@Bond1947] $$\label{3.1} 1 + z(t_o,r) = {\rm exp} \left[\int_{r_{\rm em}}^{r_{\rm obs}} \frac {R,_{tr}(T(t_o,r), r)} {\sqrt{1 + 2E(r)}} {\rm d} r\right].$$ For any fixed source (i.e. constant $r$), eq. (\[2.2\]) defines a different expansion velocity $R,_t$ for $\Lambda = 0$ and for $\Lambda \neq 0$, and thus allows us to calculate the contribution of $\Lambda$ to $z$ via (\[3.1\]). With the evolution type known, the expansion velocity depends on $r$, and so allows us to infer the distribution of mass along the past light cone. According to the authors of [@QQAm2009; @QABC2012], the “European Extremely Large Telescope” (E-ELT)[^1] could detect the redshift drift during less than 10 years of monitoring a given light source. The Gaia observatory[^2] could achieve this during about 30 years. For more on redshift drift see the contribution by P. Mishra, M.-N. Célérier and T. Singh in these Proceedings. The drift of light rays described in Sec. \[Szredshift\] and following is another real-time cosmology effect. Erroneous ideas and paradigms {#errideas} ============================= The L–T model was noticed in the astrophysics community – but: 1\. Many astrophysicists treat it as an enemy to kill rather than as a useful new device. (Citation from Ref. [@QABC2012]: The Gaia or E-ELT projects could distinguish FLRW from L–T “possibly eliminating an [*exotic alternative explanation to dark energy*]{}”). 2\. Some astrophysicists practise a loose approach to mathematics. An extreme example is to take for granted every equation found in any paper, without attention being paid to the assumptions under which it was derived. Papers written in such a style planted errors in the literature, which then came to be taken as established facts. In this section a few characteristic errors are presented (marked by [**[$\bullet$]{}**]{}) together with their explanations (marked by [**[$*$]{}**]{}). [**[$\bullet$]{}**]{} The accelerating expansion of the Universe is an observationally established fact (many refs., the Nobel Committee among them). [**[$*$]{}**]{} The established fact is the [*smaller than expected observed luminosity of the SNIa supernovae*]{} (but even this is obtained assuming that FLRW is the right cosmological model). [***The accelerating expansion is an element of theoretical explanation of this observation.***]{} When the SNIa observations are interpreted against the background of a suitably adjusted L–T model, they can be explained by matter inhomogeneities along the line of sight, with decelerating expansion [@KHBC2010; @INNa2002]. [**[$\bullet$]{}**]{} Positive sign of the redshift drift is a direct confirmation of accelerated expansion of the space. [**[$*$]{}**]{} The sign of redshift drift is only related to acceleration when homogeneity is assumed[@YKN2011] (see also Mishra, Célérier, and Singh in these Proceedings). [**[$\bullet$]{}**]{} $H_0$, supernovae and the cosmic microwave background radiation (i.e. the size of the sound horizon and the location of the acoustic peaks) are sufficient to rule out inhomogeneous L–T models. [**[$*$]{}**]{} These observations only depend on $D(z)$ and $\rho(z)$, and thus can be accommodated by the L–T model, which is specified by 2 arbitrary functions. Examples of such constructions are given in [@CBKr2010; @INNa2002]. In fact, as follows from the Sachs equations, in the approximation of small null shear (which for most L–T models works quite well[@BoFe2012]), there is a relation between $D(z)$, $\rho(z)$ and $H(z)$, meaning that these 3 observables are not independent[@BoFe2012], and thus allow the L–T model to accommodate more data on the past null cone. [**[$\bullet$]{}**]{} The gravitational potential of a typical structure in the Universe is of the order of $10^{-5}$ and thus, by writing the metric in the conformal Newtonian gauge, one immediately shows that inhomogeneities can only introduce minute ($\approx 10^{-5}$) deviations from the RW geometry. This also means that the evolution of the Universe must be Friedmannian (common argument among cosmologists). [**[$*$]{}**]{} Even if the gravitational potential remains small, its spatial derivatives do not, and thus the model has completely different optical properties than the Friedmann models. This was shown by rewriting one of the L–T Gpc-scale inhomogeneous models in the conformal Newtonian coordinates [@EMR2009]. The gravitational potential of this model remains small yet the distance–redshift relation deviates strongly from that for the background model. The question remains whether small-scale fluctuations (of the order of tens of Mpc) could also modify optical properties and evolution of the Universe. The problem is complicated as it requires solving the Einstein equations and null geodesics for a general matter distribution, which, with current technology, is not possible to do numerically. Therefore, the problem has been addressed in a number of approximations and toy models. Recent studies showed that the optical properties along a single line of sight can be significantly different than in the Friedmann model. Yet, if averaged over all directions, the average distance–redshift relation closely follows that of the model, which describes the evolution of the average density and expansion rate[@BoFe2012]. Thus, the problem reduces to the following one: do the average density and expansion rate follow the evolution of the homogeneous model, i.e. is the evolution of the background affected by small-scale inhomogeneities and does it deviate from the Friedmannian evolution? Some authors claim that matter inhomogeneities cannot affect the background and that the Universe must have Friedmannian properties[@IsWa2006; @GrWa2011], while others argue for strong deviation from the Friedmannian evolution[@Wilt2011; @RBCO2011]. Studies of this problem within the exact models, like L–T, proved that under certain conditions the back-reaction can be large, while under others it remains quite small [@Suss2011], leaving the problem unsolved. For an informative description of the problem and techniques used to address it see Ref. [@BuRa2012; @CELU2011]. [**[$\bullet$]{}**]{} Fitting an L–T model to number counts or the $D_L(z)$ relation results in predicting a huge void, several hundred Mpc in radius, around the centre (too many papers to be cited, literature still growing). Measurements of the dipole component of the CMB radiation then imply that our Galaxy should be very close to the center of this void, which contradicts the “cosmological principle”. [**[$*$]{}**]{} , for example constant $t_B$. When the model is employed at full generality, the giant void is not implied [@CBKr2010]. [**[$*$]{}**]{} , it cannot say which model is “right” and which is “wrong”. [**[$\bullet$]{}**]{} The bang time function must be constant, otherwise the decaying mode of density perturbation is nonzero, which implies large inhomogeneities in the early universe (see, for example, Ref. [@ZMSc2008]). [**[$*$]{}**]{} The bang time function describes the differences in the age between different regions of the Universe. In the L–T and Szekeres models it is also related to the amplitude of the decaying mode. However, these models describe the evolution of dust and therefore cannot be extended to times before the recombination, when the Universe was in a turbulent state: rotation, plasma, pressure gradients all did affect the proper time of an observer (${\rm d} \tau = {\rm d} t~ \sqrt{g_{00} (t,x^i)}$). Eventually, even if the Universe started with a simultaneous big bang, by the time of recombination, due to the standard physical processes, the age of the Universe would have been different at different spatial positions, giving rise to non-constant $t_B(r)$ of the dust L–T or Szekeres model that takes over there. Moreover, the relation between the nonsimultaneous big bang and the decaying mode was established only for the L–T [@Silk1977; @PlKr2006] and Szekeres [@GoWa1982] models. For more general models, not yet explicitly known as solutions of Einstein’s equations, like the ones mentioned above, the connection may be more complicated and indirect. Thus, citing this relation for such a general situation is an illegitimate stretching of a theorem beyond the domain of its assumptions (see also the next entry below). [**[$\bullet$]{}**]{} The L–T models used to explain away dark energy must have their bang-time function constant, or else they “can be ruled out on the basis of the expected cosmic microwave background spectral distortion” [@Zibi2011]. [**[$*$]{}**]{} . This sets them on the wrong track from the beginning. [**[$*$]{}**]{} , [***one must apply it at every step of analysis***]{} of the observational data. To do so, would require a re-analysis of a huge pool of data. C. Hellaby with coworkers [@McHe2008] is working on such a program applied to the L–T model, but the work is far from being completed. Lacking any better chance, we currently use observations interpreted in the FLRW framework to infer about the $M(r)$ and $t_B(r)$ functions in the L–T model. This is justified as long as we intend to point out possibilities, under the tacit assumption that these results will be verified in the future within a complete revision of the observational material on the basis of the L–T model. However, putting “precise” bounds on the L–T model functions using the self-inconsistent mixture of FLRW/L–T data available today is a self-delusion. An example: the spatial distribution of galaxies and voids is inferred from the luminosity distance vs. redshift relation that applies [*only*]{} in the FLRW models. Without assuming the FLRW background, we know nothing about this distribution until we reconstruct it using the L–T model from the beginning. The L–T and Szekeres models cannot be treated as exact models of the Universe, to be taken literally in all their aspects. They are [*exact as solutions of Einstein’s equations*]{}, but when applied in cosmology, they are merely [*the next step of approximation after FLRW*]{}. If the FLRW approximation is good for some purposes, then a more detailed model, [*when applied in a situation, in which its assumptions are fulfilled*]{}, can only be better. The redshift equations in the Szekeres models {#Szredshift} ============================================= Consider two light rays, the second one following the first after a short time-interval $\tau$, both emitted by the same source and arriving at the same observer. The trajectory of the first ray is given by $$\label{5.1} (t, x, y) = (T(r), X(r), Y(r)),$$ the corresponding equation for the second ray is $$\label{5.2} (t, x, y) = (T(r) + \tau(r), X(r) + \zeta(r), Y(r) + \psi(r)).$$ This means that while the first ray intersects a hypersurface $r = r_0$ at $(t, x, y) = (T, X, Y)$, the second ray intersects the same hypersurface not only later, but, in general, at a different comoving location. [***$\Longrightarrow$ In general the two rays will intersect different sequences of intermediate matter worldlines***]{}. The same is true for nonradial rays in the L–T model. Consequently, the second ray is emitted in a different direction and is received from a different direction by the observer. Thus, a typical observer in a Szekeres spacetime should see each light source slowly [***drift across the sky***]{}. How slowly will be estimated further on. As will be seen from the following, [***the absence of this drift is a property of exceptionally simple geometries***]{} (or exceptional directions in more general geometries). We assume that $(\zeta, \psi)$ and $({{{\rm d} {}} / {{\rm d} {r}}}) (\tau, \zeta, \psi)$ are small of the same order as $\tau$, so we neglect all terms nonlinear in any of them and terms involving their products. For any function $f(t, r, x, y)$ the symbol $\Delta f$ will denote $$\label{5.3} f(t + \tau, r, x + \zeta, y + \psi) - f(t, r, x, y)$$ [***linearized in $(\tau, \zeta, \psi)$.***]{} Note: [***the difference is taken at the same value of $r$***]{}. Applying $\Delta$ to the null geodesic equations parametrised by $r$ we obtain the equations of propagation of $(\tau, \zeta, \psi)$ and $(\xi, \eta) {\ {\overset {\rm def} =}\ }({{{\rm d} {}} / {{\rm d} {r}}})$ $(\zeta, \psi)$ along a null geodesic – see both sets of equations fully displayed in Ref. [@KrBo2011]. Repeatable light paths {#repeat} ====================== There will be no drift when, for a given source–observer pair, each light ray will proceed through the same intermediate sequence of matter world lines. Rays having this property will be called [***repeatable light paths (RLP)***]{}. For a RLP we have $$\label{6.1} \zeta = \psi = \xi = \eta = 0$$ all along the ray. The equations of propagation of $(\tau, \zeta, \psi, \xi, \eta)$ become then overdetermined (3 equations to determine the propagation of $\tau$ along a null geodesic), and imply limitations on the metric components. They can be used in 2 ways: 1\. As the condition (on the metric) for [***all***]{} null geodesics to be RLPs. 2\. As the conditions under which special null geodesics are RLPs in subcases of the Szekeres spacetime. In the first interpretation, the equations of propagation should be identities in the components of ${{{\rm d} {x^{\alpha}}} / {{\rm d} {r}}}$, and this happens when $$\label{6.2} \Psi {\ {\overset {\rm def} =}\ }\Phi,_{tr} - \Phi,_t \Phi,_r/\Phi = 0.$$ This means zero shear, i.e. the Friedmann limit. Thus, we have the following [**Corollary:**]{} [***The only spacetimes in the Szekeres family in which [all]{} null geodesics have repeatable paths are the Friedmann models.***]{} In fact, we proved something stronger. Since the drift vanishes in the Friedmann models, we have: [**Corollary 2:**]{} [***The presence of the drift would be an observational evidence for the Universe to be inhomogeneous on large scales.***]{} In the second interpretation, there are only 2 nontrivial (i.e. non-Friedmannian) cases: A. When the Szekeres spacetime is axially symmetric ($P$ and $Q$ are constant). In this case, the RLPs are those null geodesics that stay on the axis of symmetry in each 3-space of constant $t$. B. When the Szekeres spacetime is spherically symmetric ($P, Q, S$ are all constant) – then it reduces to the L–T model. In this case, the radial null geodesics are [*the only*]{} RLPs that exist. A formal proof of this statement is highly complicated [@KrBo2011]. For non-radial rays in the L–T model, the non-RLP phenomenon was predicted, by a different method (and under the name of [***cosmic parallax***]{}), by Quercellini [*et al.*]{} [@QQAm2009; @QABC2012]. Their review contains a broad presentation of the real-time cosmology paradigm, oriented toward observational possibilities.  Examples of non-RLPs in the L–T model {#numex} ===================================== The examples will show the non-RLP effect for nonradial null geodesics in two configurations of the L–T model, shown in Fig. \[densprof\], for different positions of the observers with respect to the center of symmetry. In Example 1 (see Fig. \[example1\]) we use Profile 1, the observer and the light source at 3.5 Gpc from the center of the void, the directions to them at the angle 1.8 rad, and [@BoWy2008] $t_B =0$ – simultaneous Big Bang, $\rho(t_0,r) = \rho_0 \left[ 1 + \delta - \delta \exp \left( - {r^2}/{\sigma^2} \right) \right]$ – the density profile at the current instant, $r {\ {\overset {\rm def} =}\ }R(t_0,r)$ – the radial coordinate, $\rho_0 {\ {\overset {\rm def} =}\ }\rho(t_0,0) = 0.3 \times (3H_0^2)/(8 \pi G) \equiv 0.3 \rho_{\rm critical}$ – the present density at the center of the void, $H_0 = 72$ km s$^{-1}$ Mpc$^{-1}$ – the present value of the Hubble parameter, $\delta = 4.05$, $\sigma = 2.96$ Gpc. The source in Fig. \[example1\] sends three light rays to the same observer, the first of which was received by the observer $5 \times 10^9$ years ago, the second is being received right now, and the third one will be received $5 \times 10^9$ years in the future. Fig. \[fig1\] shows these rays projected on the space $t =$ now along the flow lines of the L–T dust. The source is at the upper right corner of the graph and the observer is at the upper left corner. ![The configuration of the light source and the observer in example 1.[]{data-label="example1"}](example1.ps)  The [***time-averaged***]{} rate of change of the position of the source in the sky, seen by the observer is $$\begin{aligned} \label{7.1} && \dot{\gamma} = \frac {\rm angle\ between\ the\ earlier\ and\ the\ later\ ray} {\rm time\ interval = 5 \times 10^9\ years} \nonumber \\ && \sim 10^{-7}\ \frac {\rm arc sec} {\rm year}.\end{aligned}$$ The rate of drift in the next figures is calculated in the same way. In Examples 2, 3 and 4 (Fig. \[fig2\]), the observer O is at $R_0$ from the center; the angle between the direction toward the galaxy ([$_*$]{}) and toward the origin is $\gamma$. For each $\gamma$ we calculated the rate of change $\dot \gamma$ by eq. (\[7.1\]), and the graphs in Fig. \[fig3\] show $\dot{\gamma}$ as a function of $\gamma$. All the examples have $d = 1$ Gly $\approx 306.6$ Mpc. Example 2 (solid line) has $R_0 = 3$ Gpc and Profile 1; Example 3 (dashed line) has $R_0 = 1$ Gpc and Profile 1; Example 4 (dotted line) has $R_0 = 1$ Gpc and Profile 2 (for which $\delta = 10.0$ instead of $\delta = 4.05$; this is a deeper void in a higher-density background). The amplitude is $\sim 10^{-7}$ for (3) and $\sim 10^{-6}$ for (2) and (4). With the Gaia accuracy of $5-20 \times 10^{-6}$ arcsec, we would need a few years to detect this effect. ![The configuration for Examples 2, 3, 4.[]{data-label="fig2"}](fig2.eps)  RLPs in shearfree normal models {#BarnesRLP} =============================== In the Szekeres models, the condition for all null geodesics to be RLPs was the vanishing of shear. This suggests that the cause of the non-RLP phenomenon might be shear in the cosmic flow. To test this supposition, the existence of RLPs was investigated in those cosmological models in which shear is zero [@Kras2011] – the shearfree normal models found by Barnes [@Barn1973]. They obey the Einstein equations with a perfect fluid source and contain, as the acceleration-free limit, the whole FLRW family. There are four classes of them: the Petrov type D metrics that are spherically, plane and hyperbolically symmetric, and the conformally flat metric found earlier by Stephani [@Step1967]. In the Petrov type D case, the metric in comoving coordinates is $${\rm d} s^2 = \left(\frac {F V,_t} V\right)^2 {\rm d} t^2 - \frac 1 {V^2} \left({\rm d} x^2 + {\rm d} y^2 + {\rm d} z^2\right), \label{8.1}$$ where $F(t)$ is an arbitrary function, related to the expansion scalar $\theta$ by $\theta = 3 / F$. The Einstein equations reduce to the single equation: $$w,_{uu} /w^{2} = f(u), \label{8.2}$$ where $f(u)$ is an arbitrary function, while $u$ and $w$ are related to $(x, y, z)$ and to $V(t,x,y,z)$ differently in each subfamily. We have $$(u, w) = \left\{ \begin{array}{ll} (r^{2}, V) & \mbox{with spherical symmetry}, \\ \ & \ r^2 {\ {\overset {\rm def} =}\ }x^2 + y^2 + z^2;\\ (z, V) & \mbox{with plane symmetry};\\ (x/y, V/y) & \mbox{with hyperbolic symmetry}.\end{array} \right. \label{8.3}$$ The FLRW limit follows when $f = 0$ and $V = R(t) g(x,y,z)$. The conformally flat Stephani solution [@Step1967; @Kras1997] has the metric given by (\[8.1\]), the coordinates are comoving, and $V(t, x, y, z)$ is given by $$\begin{aligned} &&V = \frac 1 R \left\{1 + \frac 1 4 k(t) \left[\left(x - x_{0}(t)\right)^{2} + \left(y - y_{0}(t)\right)^{2} \right.\right. \nonumber \\ &&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ + \left.\left.\left(z - z_{0}(t)\right)^{2}\right]\right\}, \label{8.4}\end{aligned}$$ where $(R, k, x_0, y_0, z_0)$ are arbitrary functions of $t$. This a generalisation of the whole FLRW class, which results when $(k, x_0, y_0, z_0)$ are all constant. In general, (\[8.4\]) has no symmetry. In these models, in the most general cases, generic null geodesics are not RLPs. Consequently, [***it is not shear that causes the non-RLP property***]{}.[^3] In the general type D shearfree normal models, the only RLPs are radial null geodesics in the spherical case and their analogues in the other two cases. In the most general Stephani spacetime, RLPs do not exist. In the axially symmetric subcase of the Stephani solution the RLPs are those geodesics that intersect the axis of symmetry in every space of constant time. In the spherically-, plane- and hyperbolically symmetric subcases, the RLPs are the radial geodesics. The completely drift-free subcases are conformally flat, but more general than FLRW. Their defining property is that their time-dependence in the comoving coordinates can be factored out, and the cofactor metric is static. The FLRW models have the same property. For example, in the drift-free spherically symmetric type D case: $$\begin{aligned} {\rm d} s^2 = \frac 1 {V^2} && \hspace{-3mm} \left\{\left[\left(A_1 + A_2 r^2\right) \left(F S,_t {\rm d} t\right)\right]^2 - {\rm d} r^2\right. \nonumber \\ &&- \left.r^2 \left({\rm d} \vartheta^2 + r^2 \sin^2 \vartheta {\rm d} \varphi^2\right)\right\}, \label{8.5}\end{aligned}$$ the whole non-staticity is contained in $V$: $$\label{8.6} V = B_1 + B_2 r^2 + \left(A_1 + A_2 r^2\right) S(t).$$ The $(A_1, A_2, B_1, B_2)$ are arbitrary constants and $S(t)$ is an arbitrary function. This model is more general than FLRW because the pressure in it is spatially inhomogeneous. The FLRW limit follows when $A_1 \neq 0$ and $B_2 = (A_2/A_1) B_1$. Dependence of RLPs on the observer congruence ============================================= The RLPs are defined relative to the congruence of worldlines of the observers and light sources. So far, we have considered observers and light sources attached to the particles of the cosmic medium, whose velocity field is defined by the spacetime geometry via the Einstein equations. But we could as well consider other timelike congruences, or spacetimes in which no preferred timelike congruence exists, for example Minkowski. It turns out that even in the Minkowski spacetime one can devise a timelike congruence that will display the non-RLP property [@Kras2012]. Take the Minkowski metric in the spherical coordinates $$\label{9.1} {\rm d} s^2 = {\rm d} {t'}^2 - {\rm d} {r'}^2 - {r'}^2 \left({\rm d} \vartheta^2 + \sin^2 \vartheta {\rm d} \varphi^2\right),$$ and carry out the following transformation on it: $$\label{9.2} t' = (r - t)^2 + 1 / (r + t)^2, \qquad r' = (r - t)^2 - 1 / (r + t)^2.$$ The result is the metric $$\begin{aligned} && {\rm d} s^2 = \frac 1 {(r + t)^4}\left\{16 u \left({\rm d} t^2 - {\rm d} r^2\right)\right. \\ && \ \ \left.- \left(u^2 - 1\right)^2 \left({\rm d} \vartheta^2 + \sin^2 \vartheta {\rm d} \varphi^2\right)\right\}, \qquad u {\ {\overset {\rm def} =}\ }r^2 - t^2. \nonumber \label{9.3}\end{aligned}$$ Now we assume that the curves with the unit tangent vector field $u^{\alpha} = \left[(r + t)^2 / \left(4 \sqrt{u}\right)\right] {\delta^{\alpha}}_0$ are world lines of test observers and test light sources. 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--- abstract: 'We present an algorithm to compute the annihilator of (i.e., the linear differential equations for) the multi-valued analytic function $f^\lambda(\log f)^m$ in the Weyl algebra $D_n$ for a given non-constant polynomial $f$, a non-negative integer $m$, and a complex number $\lambda$. This algorithm essentially consists in the differentiation with respect to $s$ of the annihilator of $f^s$ in the ring $D_n[s]$ and ideal quotient computation in $D_n$. The obtained differential equations constitute what is called a holonomic system in $D$-module theory. Hence combined with the integration algorithm for $D$-modules, this enables us to compute a holonomic system for the integral of a function involving the logarithm of a polynomial with respect to some variables.' author: - | Toshinori Oaku\ Department of Mathematics, Tokyo Woman’s Christian University date: 'January 12, 2012' title: An algorithm to compute the differential equations for the logarithm of a polynomial --- Introduction ============ For a given function $u$, it is an interesting problem both in theory and in practice to determine the differential equations which $u$ satisfies. Let us restrict our attention to linear differential equations with polynomial coefficients. Then our problem can be formulated as follows: Let $D_n$ be the Weyl algebra, i.e., the ring of differential operators with polynomial coefficients in the variables $x = (x_1,\dots,x_n)$. An element $P$ of $D_n$ is expressed as a finite sum $$\label{eq:anOperator} P = \sum_{\alpha,\beta \in {{\mathbb N}}^n} a_{\alpha,\beta}x^\alpha\partial^\beta,$$ with $x^\alpha = x_1^{\alpha_1}\cdots x_n^{\alpha_n}$, $\partial^\beta = \partial_1^{\beta_1}\cdots\partial_n^{\beta_n}$ and $a_{\alpha,\beta}\in {{\mathbb C}}$, where $\alpha = (\alpha_1,\dots,\alpha_n),\,\,\beta = (\beta_1,\dots,\beta_n) \in {{\mathbb N}}^n$ are multi-indices with ${{\mathbb N}}= \{0,1,2,\dots\}$ and $\partial_i = \partial/\partial x_i$ ($i=1,\dots,n$) denote derivations. The [*annihilator*]{} of $u$ (in $D_n$) is defined to be $${\mbox{{\rm Ann}}}_{D_n}u = \{P \in D_n \mid Pu=0\},$$ which is a left ideal of $D_n$. Since $D_n$ is a non-commutative Noetherian ring, there exist a finite number of operators $P_1,\dots,P_N \in D_n$ which generate ${\mbox{{\rm Ann}}}_{D_n}u$ as left ideal. Thus we can regard the system $$P_1 u = \cdots = P_Nu = 0$$ of linear (partial or ordinary) differential equations as a maximal one that $u$ satisfies. As to systems of linear differential equations, there is a notion of holonomicity, or being [*holonomic*]{}, which plays a central role in $D$-module theory. See Appendix for a precise definition. A holonomic system of linear differential equations admits only a finite number of linearly independent solutions although it is not a sufficient condition for holonomicity. A [*holonomic function*]{} is by definition a function which satisfies a holonomic system. The importance of the holonomicity lies in, in addition to the finiteness property above, the fact the it is preserved under basic operations on functions such as sum, product, restriction and integration. Hence starting from some basic holonomic functions we can construct various holonomic functions by using such operations. As one of basic holonomic functions, let us consider $f^\lambda$ with a non-constant polynomial $f$ in $x = (x_1,\dots,x_n)$ and a complex number $\lambda$. Then the function $f^\lambda$ is holonomic and there is an algorithm to compute its annihilator strictly ([@OakuJPAA1997],[@BM],[@SST]). Our purpose is to give an algorithm to compute the annihilator of $f^\lambda (\log f)^m$ with a positive integer $m$ and to prove that it is a holonomic function. This is achieved by differentiation with respect to the parameter $s$ of the annihilator of $f^s$ in $D_n[s]$. This method can be extended to functions of the form $f_1^{\lambda_1}\cdots f_N^{\lambda_N} (\log f_1)^{m_1}\cdots(\log f_N)^{m_N}$ for polynomials $f_k$, complex numbers $\lambda_k$ and nonnegative integers $m_k$. Since the algorithm yields a holonomic system, we can apply the integration algorithm for $D$-modules (see [@OTdeRham], [@SST]) to get a holonomic system for the integral of a function involving the logarithm of a polynomial. Annihilators with a parameter ============================= Let $f$ be a non-constant polynomial in $n$ variables $x = (x_1,\dots,x_n)$ with coefficients in the field ${{\mathbb C}}$ of the complex numbers. From an algorithmic viewpoint, we assume that the coefficients of $f$ belong to a computable field. First, we consider formal functions of the form $f^s(\log f)^k$ with an indeterminate $s$. More precisely, for a non-negative integer $m$, we introduce the module $${{\mathcal L}}(f,m) := \bigoplus_{k=0}^m {{\mathbb C}}[x,f^{-1},s]f^s(\log f)^k,$$ of which $f^s(\log f)^k$ are regarded as a free basis over ${{\mathbb C}}[x,f^{-1},s]$. Then ${{\mathcal L}}(f,m)$ has a natural structure of left $D_n[s]$-module, which is induced by the action of the derivation $\partial_j = \partial/\partial x_j$ defined by, for $a \in {{\mathbb C}}[x,f^{-1},s]$, $$\partial_j\{af^s(\log f)^k\} = \left(\frac{\partial a}{\partial x_j} + sa f^{-1}\frac{\partial f}{\partial x_j}\right)f^s(\log f)^k + k a f^{-1}\frac{\partial f}{\partial x_j}f^s(\log f)^{k-1} \quad (j=1,\dots,n)$$ if $k \geq 1$ and $$\partial_j(af^s) = \left(\frac{\partial a}{\partial x_j} + sa f^{-1}\frac{\partial f}{\partial x_j}\right)f^s \quad (j=1,\dots,n).$$ In view of this action, it is easy to see that ${{\mathcal L}}(f,m)/{{\mathcal L}}(f,m-1)$ is isomorphic to ${{\mathcal L}}(f,0) = {{\mathbb C}}[x,f^{-1},s]f^s$ as a left $D_n[s]$-module. Now consider the left $D_n[s]$-submodule $${{\mathcal P}}(f,m) := D_n[s]f^s + \cdots + D_n[s](f^s(\log f)^m)$$ of ${{\mathcal L}}(f,m)$. Our purpose is to determine the annihilator module $${\mbox{{\rm Ann}}}_{D_n[s]}(f^s,\dots,f^s(\log f)^m) := \left\{P = (P_0,P_1,\dots,P_m) \in D_n[s]^{m+1} \mid \sum_{k=0}^m P_k(f^s(\log f)^k) = 0 \right\}$$ and the annihilator ideal $${\mbox{{\rm Ann}}}_{D_n[s]}f^s(\log f)^m := \left\{P \in D_n[s] \mid P(f^s(\log f)^m) = 0 \right\}.$$ Note that there are isomorphisms $$\begin{aligned} {{\mathcal P}}(f,m) &\simeq D_n[s]^{m+1}/{\mbox{{\rm Ann}}}_{D_n[s]}(f^s,\dots,f^s(\log f)^m), \nonumber\\ D_n[s](f^s(\log f)^m) &\simeq D_n[s]/{\mbox{{\rm Ann}}}_{D_n[s]}(f^s(\log f)^m). \nonumber\end{aligned}$$ Now let us regard $f^s\log f$ as a multi-valued analytic function in $(x,s)$ on $\{(x,s)\in {{\mathbb C}}^{n+1} \mid f(x) \neq 0 \}$. \[lemma:independence\] Let $f \in {{\mathbb C}}[x]$ be a non-constant polynomial. Then for $a_i(x) \in {{\mathbb C}}[x]$, $$\sum_{i=0}^m a_i(x)(\log f)^i = 0$$ holds as analytic function if and only if $a_i(x,s) = 0$ for all $i$. We argue by induction on $m$. Let $x_0 \in {{\mathbb C}}^n$ be a non-singular point of the hypersurface $f(x) = 0$, i.e, assume $$f(x_0) = 0,\quad \frac{\partial f}{\partial x_i}(x_0) \neq 0 \quad \mbox{for some $i$ with $1 \leq i \leq n$}.$$ In view of the uniqueness of analytic continuation, we have only to show that each $a_i(x)$ vanishes near $x_0$. Hence we may suppose that $a_i(x)$ are analytic near $x_0 = 0$ and $f(x) = x_1$. That is, $$\label{eq:identity} a_0(x) + a_1(x)\log x_1 + \cdots + a_m(x)(\log x_1)^m = 0$$ holds on a neighborhood $U$ of $0$. Fix a point $x = (x_1,\dots,x_n)$ in $U$ such that $x_1 \neq 0$. By analytic continuation along a circle $(e^{\sqrt{-1}t}x_1,x_2,\dots,x_n)$ with $0 \leq t \leq 2\pi$, the identity (\[eq:identity\]) is transformed to $$a_0(x) + a_1(x)(\log x_1 + 2\pi\sqrt{-1}) + \cdots + a_m(x)(\log x_1 + 2\pi\sqrt{-1})^m = 0.$$ By subtraction, we get an identity of the form $$b_0(x) + b_1(x)\log x_1 + \cdots b_{m-1}(x)(\log x_1)^{m-1} = 0$$ with $$b_{m-1}(x) = 2m\pi\sqrt{-1}a_m(x).$$ From the induction hypothesis it follows that $b_0(x) = \cdots = b_{m-1}(x)=0$, which implies $a_m(x)=0$. We are done by induction on $m$. Computation of the annihilator ============================== Now let us describe an algorithm for computing the annihilator of $f^s(\log f)^m$. \[alg:annfslog\] Input: a non-constant polynomial $f$ in the variables $x = (x_1,\dots,x_n)$ with coefficients in a computable subfield of ${{\mathbb C}}$, a non-negative integer $m$. 1. Let $G = \{P_1(s),\dots,P_k(s)\}$ be a generating set of the left ideal ${\mbox{{\rm Ann}}}_{D_n[s]}f^s := \{P(s) \in D_n[s] \mid P(s)f^s = 0\}$ by using an algorithm of [@OakuJPAA1997] or [@BM] (see also [@LM1]). 2. Let $e_0 = (1,0,\dots,0),\dots,e_{m} = (0,\dots,0,1)$ be the canonical unit vectors of ${{\mathbb C}}^{m+1}$. For each $i = 1,\dots,k$ and $j = 0,1,\dots,m$, set $$P_{i}(s)^{(j)} := \sum_{\nu = 0}^j { j \choose \nu} \frac{\partial^{j-\nu}P_i(s)}{\partial s^{j-\nu}}e_\nu.$$ Output: $G' := \{P_{i}(s)^{(j)} \mid 1\leq i \leq k,\,0\leq j \leq m\}$ generates ${\mbox{{\rm Ann}}}_{D_n[s]}(f^s,\dots,f^s(\log f)^m)$. Input: a non-constant polynomial $f$ in the variables $x = (x_1,\dots,x_n)$ with coefficients in a computable subfield of ${{\mathbb C}}$, a non-negative integer $m$. 1. Let $G'$ be the output of Algorithm \[alg:annfslog\]. 2. Compute a Gröbner base $G''$ of the module generated by $G'$ with respect to a term order $\prec$ for $(D_n[s])^{m+1}$ such that $Me_j \prec M'e_k$ for any monomial $M$ and $M'$ if $k < j$. Let $G_0$ be the set of the last component of each element of $G''$. Output: $G_0$ generates ${\mbox{{\rm Ann}}}_{D_n[s]} f^s(\log f)^m$. \[lemma:Ps\] Let $I$ be a left ideal of $D_n[s]$ generated by $\{P_1(s),\dots,P_k(s)\}$. For $P(s) \in D_n[s]$, and $j \in {{\mathbb N}}$, set $$P(s)^{(j)} := \sum_{\nu = 0}^j {j \choose \nu}\frac{\partial^{j-\nu}P(s)}{\partial s^{j-\nu}}e_\nu.$$ Then the left $D_n[s]$-submodule of $(D_n[s])^{m+1}$ which is generated by $\{P(s)^{(j)} \mid P(s) \in I,\,0 \leq j \leq m\}$ coincides with the one which is generated by $\{P_i(s)^{(j)} \mid 0 \leq i \leq k,\,0 \leq j \leq m\}$ for any integer $m \geq 0$. Let ${{\mathcal N}}$ be the left $D_n[s]$-module generated by $\{P_i(s)^{(j)} \mid 0 \leq i \leq k,\,0 \leq j \leq m\}$ and $P(s)$ be a nonzero element of $I$. Then there exist $$Q_i(s) = \sum_{l=0}^{m_i}Q_{il}s^l \quad (Q_{il} \in D_n)$$ such that $P(s) = \sum_{i=1}^k Q_i(s)P_i(s)$. Then we have $$P(s)^{(j)} = \sum_{i=1}^k\sum_{l=0}^{m_i}Q_{il}(s^lP_i(s))^{(j)}.$$ Hence we have only to show that $(s^lP_i(s))^{(j)}$ belongs to ${{\mathcal N}}$. This can be done as follows: $$\begin{aligned} (s^lP_i(s))^{(j)} &= \sum_{\nu=0}^j {j \choose \nu}\left(\frac{\partial}{\partial s}\right)^{j-\nu} (s^lP_i(s))e_\nu \nonumber \\ &= \sum_{\nu=0}^j {j \choose \nu}\sum_{\mu=0}^{\min\{j-\nu,l\}} {j-\nu \choose \mu}(l)_\mu s^{l-\mu} \left(\frac{\partial}{\partial s}\right)^{j-\nu-\mu}P_i(s)e_\nu \nonumber\\ &= \sum_{\mu=0}^{\min\{j,l\}}{j \choose \mu} (l)_\mu s^{l-\mu} \sum_{\nu=0}^{j-\mu}{j-\mu \choose \nu} \left(\frac{\partial}{\partial s}\right)^{j-\mu-\nu}P_i(s)e_\nu \nonumber\\ &= \sum_{\mu=0}^{\min\{j,l\}}{j \choose \mu} (l)_\mu s^{l-\mu}P_i(s)^{(j-\mu)}, \nonumber\end{aligned}$$ where $(l)_\mu := l(l-1)\cdots(l-\mu+1)$. The output of Algorithm \[alg:annfslog\] coincides with ${\mbox{{\rm Ann}}}_{D_n[s]}(f^s,\dots,f^s(\log f)^m)$. Let $P(s)$ belong to ${\mbox{{\rm Ann}}}_{D_n[s]}f^s$. Differentiating the equation $P(s)f^s = 0$ with respect to $s$, we get $$\sum_{\nu=0}^j { j \choose \nu}\frac{\partial^{j-\nu}P_i(s)}{\partial s^{j-\nu}} (f^s(\log f)^\nu) = 0$$ for $0 \leq j \leq m$. This shows that each $P_{i}(s)^{(j)}$ annihilates $(f^s,\dots,f^s(\log f)^m)$. Set ${{\mathcal M}}:= {\mbox{{\rm Ann}}}_{D_n[s]}(f^s,\dots,f^s(\log f)^m)$. Let ${{\mathcal N}}$ be the left $D_n[s]$-module generated by the output $G'$ of Algorithm \[alg:annfslog\]. The argument above shows that ${{\mathcal N}}$ is a left $D_n$-submodule of ${{\mathcal M}}$. Hence we have only to prove ${{\mathcal N}}= {{\mathcal M}}$. For this purpose let ${{\mathcal N}}_j$ be the left $D_n[s]$-module generated by $\{P_i(s)^{(\nu)} \mid 1 \leq i \leq k,\, 0 \leq \nu \leq j\}$ and set $$\begin{aligned} {{\mathcal M}}_j &:= \{(Q_0,Q_1,\dots,Q_m) \in {{\mathcal M}}\mid Q_\nu = 0 \mbox{ if } \nu > j\}. \nonumber\end{aligned}$$ Let $Q(s) = (Q_0(s),\dots,Q_j(s),0,\dots,0)$ be an element of ${{\mathcal M}}_j$. Then $$\sum_{\nu=0}^j Q_\nu(s)(f^s(\log f)^\nu) = 0$$ holds. In view of the action of $D_n[s]$ on ${{\mathcal L}}(f,j)$ noted in Section 1, this implies $Q_j(s)f^s = 0$. Hence $Q_j(s)^{(j)}$ belongs to ${{\mathcal N}}_j$ by Lemma \[lemma:Ps\]. It is easy to see that $Q(s) - Q_j(s)^{(j)}$ belongs to ${{\mathcal M}}_{j-1}$. This means ${{\mathcal M}}_j = {{\mathcal N}}_j + {{\mathcal M}}_{j-1}$ for $1 \leq j \leq m$. Then we can show that ${{\mathcal N}}_j = {{\mathcal M}}_j$ holds for $1 \leq j \leq m$ by induction on $m$ noting ${{\mathcal N}}_0 = {{\mathcal M}}_0$. If $f$ is weighted homogeneous, i.e., if there exist rational numbers $w_i$ such that $\sum_{i=1}^n w_i x_i\partial_i(f) = f$, then $D_n[s]f^s(\log f)^m$ is isomorphic to ${{\mathcal L}}(f,m)$ as left $D_n[s]$-module. That is, we have an isomorphism $$D_n[s]/{\mbox{{\rm Ann}}}_{D_n[s]} f^s(\log f)^m \simeq (D_n[s])^{m+1}/{\mbox{{\rm Ann}}}_{D_n[s]}(f^s,\dots,f^s(\log f)^m).$$ of left $D_n[s]$-module In fact, this follows from the relations $$\left(\sum_{i=1}^n w_i x_i\partial_i - s\right)(f^s(\log f)^k) = k f^s(\log f)^{k-1} \qquad (k \geq 1).$$ Specialization of the parameter =============================== Let us fix a complex number $\lambda$. (From an algorithmic view point, we assume $\lambda$ lies in a computable subfield of the field ${{\mathbb C}}$.) We set $${{\mathcal L}}(f,m,\lambda) := \bigoplus_{k=0}^m {{\mathbb C}}[x,f^{-1}]f^\lambda(\log f)^k,$$ where $f^\lambda(\log f)^k$ are regarded as a free basis over ${{\mathbb C}}[x,f^{-1}]$. Substituting $\lambda$ for $s$ gives ${{\mathcal L}}(f,m,\lambda)$ a natural structure of left $D_n$-module. In fact, one has $$\partial_j\{af^\lambda(\log f)^k\} \left(\frac{\partial a}{\partial x_j} + \lambda a f^{-1}\frac{\partial f}{\partial x_j}\right)f^\lambda(\log f)^k + k a f^{-1}\frac{\partial f}{\partial x_j}f^\lambda(\log f)^{k-1} \quad (j=1,\dots,n)$$ for $k \geq 1$ and $$\partial_j(af^\lambda) = \left(\frac{\partial a}{\partial x_j} + \lambda a f^{-1}\frac{\partial f}{\partial x_j}\right)f^\lambda \quad (j=1,\dots,n)$$ with $a \in {{\mathbb C}}[x,f^{-1}]$. This implies that ${{\mathcal L}}(f,m,\lambda)/{{\mathcal L}}(f,m-1,\lambda)$ is isomorphic to ${{\mathcal L}}(f,0,\lambda) = D_nf^\lambda$ as a left $D_n$-module. It follows that ${{\mathcal L}}(f,m,\lambda)$ is holonomic since so is $D_nf^\lambda$ as was proved by Bernstein [@Bernstein]. Set $${{\mathcal P}}(f,m,\lambda) := D_nf^\lambda + \cdots + D_n(f^\lambda(\log f)^m)$$ We define the annihilators of $(f^s,\dots,f^\lambda(\log f)^m)$ and of $f^\lambda(\log f)^m$ to be $$\begin{aligned} {\mbox{{\rm Ann}}}_{D_n}(f^\lambda,\dots,f^\lambda(\log f)^k) &:= \{P = (P_0,P_1,\dots,P_m) \in (D_n)^{m+1} \mid \sum_{k=0}^m P_k(f^\lambda(\log f)^k) = 0 \}, \nonumber \\ {\mbox{{\rm Ann}}}_{D_n}f^\lambda(\log f)^m &:= \{P \in D_n \mid P(f^\lambda(\log f)^m) = 0 \} \nonumber\end{aligned}$$ respectively. Then we have isomorphisms $$\begin{aligned} {{\mathcal P}}(f,m,\lambda) &\simeq (D_n)^{m+1} /{\mbox{{\rm Ann}}}_{D_n}(f^\lambda,\dots,f^\lambda(\log f)^k) \nonumber\\ D_n(f^\lambda(\log f)^m) &\simeq D_n/{\mbox{{\rm Ann}}}_{D_n}(f^\lambda(\log f)^m). \nonumber\end{aligned}$$ In the sequel, we need information on the integral roots of the *Bernstein-Sato polynomial* or the *$b$-function* of $f$, which is, by definition, the monic polynomial $b_{f}(s)$ of the least degree such that a formal functional equation $$\label{eq:fseq} P(s)f^{s+1} = b_{f}(s)f^s$$ holds with some $P(s) \in D_n[s]$. The existence of such a functional equations was proved by Bernstein [@Bernstein]. It was proved by Kashiwara [@Kashiwara] that the roots of $b_f(s)=0$ are negative rational numbers. An algorithm to compute $b_f(s)$ and an associated operator $P(s)$ was given in [@OakuDuke]. The following proposition generalizes a result of Kashiwara [@Kashiwara Proposition 6.2]: \[prop:specialization\] Let $b_f(s)$ be the Bernstein-Sato polynomial of $f$, i.e., a polynomial in $s$ of the least degree such that $P(s)f^{s+1} = b_f(s)f^s$ holds with a $P(s) \in D_n[s]$. Let $\lambda$ be a complex number such that $b_f(\lambda-\nu) \neq 0$ for any positive integer $\nu$. Then we have $$\begin{aligned} {\mbox{{\rm Ann}}}_{D_n}(f^\lambda,\dots,f^\lambda(\log f)^k) &= \{P(\lambda) \mid P(s) \in {\mbox{{\rm Ann}}}_{D_n[s]}(f^s,\dots,f^s(\log f)^k), \nonumber \\ {\mbox{{\rm Ann}}}_{D_n}f^\lambda(\log f)^m &= \{ P(\lambda) \mid P(s) \in {\mbox{{\rm Ann}}}_{D_n[s]}f^s(\log f)^m\}. \nonumber\end{aligned}$$ We have only to show the first equality. Assume that $\sum_{k=0}^m P_k (f^\lambda(\log f)^k) = 0$ holds with $P_k \in D_n$. Then there exist non-negative integer $l \geq 0$ and polynomials $a_k(x,s) \in {{\mathbb C}}[x,s]$ such that $$\sum_{k=0}^m P_k (f^s(\log f)^k) = (s-\lambda)\sum_{k=0}^m a_k(x,s)f^{s-l}(\log f)^k.$$ By using the functional equation (\[eq:fseq\]), we can find an operator $Q(s) \in D_n[s]$ such that $$b_f(s-1)\cdots b_f(s-l)f^{s-l} = Q(s)f^s.$$ In view of the action of $D_n[s]$ on ${{\mathcal L}}(f,m)$, there exist $a'_k(x,s) \in {{\mathbb C}}[x,s]$ and a non-negative integers $l_1$ such that $$\begin{aligned} b_f(s-1)\cdots b_f(s-l)f^{s-l}(\log f)^m &= Q(s)\{f^s(\log f)^m\} + \sum_{k=0}^{m-1} a'_k(x,s)f^{s-l_1}(\log f)^k. \nonumber\end{aligned}$$ Proceeding inductively, we conclude that there exist a polynomial $b(s) \in {{\mathbb C}}[s]$ which is a product (possibly with multiplicities) of $b_f(s-j)$ with $j\geq 1$ and operators $\widetilde Q_k(s) \in D_n[s]$ such that $$b(s) \sum_{k=0}^m a_k(x,s)f^{s-l}(\log f)^k = \sum_{k=0}^m \widetilde Q_k(s)\{f^s(\log f)^k\}.$$ Hence $$\widetilde P(s) := b(s)\sum_{k=0}^m P_ke_k - (s-\lambda)\sum_{k=0}^m \widetilde Q_k(s)e_k$$ belongs to ${\mbox{{\rm Ann}}}_{D_n[s]}(f^s,\dots,f^s(\log f)^k)$ and $b(\lambda)\sum_{k=0}^m P_ke_k = \widetilde{P}(\lambda)$. This completes the proof since $b(\lambda) \neq 0$ by the assumption. If $b_f(\lambda-\nu) = 0$ for some positive integer $\nu$, then set $\nu_0 := \max\{\nu \in {{\mathbb Z}}\mid b_f(\lambda-\nu) = 0\}$ and $\lambda_0 := \lambda-\nu_0$. Then $\lambda_0$ satisfies the condition of Theorem \[prop:specialization\]. Then for $(P_0,\dots,P_m) \in (D_n)^{m+1}$, we have $$\begin{aligned} & (P_0,\dots,P_m) \in {\mbox{{\rm Ann}}}_{D_n}(f^\lambda,\dots,f^\lambda(\log f)^k) \nonumber\\ &\Leftrightarrow\quad (P_0f^{\nu_0},\dots,P_mf^{\nu_0}) \in {\mbox{{\rm Ann}}}_{D_n}(f^{\lambda_0},\dots,f^{\lambda_0}(\log f)^k) \nonumber\\ &\Leftrightarrow\quad (P_0,\dots,P_m) \in {\mbox{{\rm Ann}}}_{D_n}(f^{\lambda_0},\dots,f^{\lambda_0}(\log f)^k) : f^{\nu_0}. \nonumber\end{aligned}$$ The module quotient in the last line can be obtained by computing the module intersection or else by syzygy computation. Now let us describe two algorithms for module quotient in general. First, let us define the componentwise product of two elements $P = (P_0,\dots,P_m)$ and $Q = (Q_0,\dots,Q_m)$ of $(D_n)^{m+1}$ to be $PQ := (P_0Q_0,\dots,P_mQ_m)$. Let $N$ be a left $D_n$-submodule of $(D_n)^{m+1}$ and $P$ be a nonzero element of $(D_n)^{m+1}$. Then the module quotient $N:P$ is defined to be $$N:P := \{Q \in (D_n)^{m+1} \mid QP \in N\},$$ which is a left $D_n$-submodule of $(D_n)^{m+1}$. \[alg:quotient\]Input: A set $G_1$ of generators of a left $D_n$-submodule $N$ of $(D_n)^{m+1}$ and a non-zero element $P = (P_0,P_1,\dots,P_m)$ of $(D_n)^{m+1}$. 1. Introducing a new variable $t$, compute a Gröbner base $G_2$ of the left $D_n[t]$-module of $(D_n[t])^{m+1}$ which is generated by $\{(1-t)P_ke_k \mid 0 \leq k \leq m \} \cup \{tQ \mid Q \in G_1 \}$ with respect to a term order $\prec$ such that $x^\alpha\partial^\beta e_j \prec te_k$ for any $j,k \in \{0,1,\dots,m\}$ and $\alpha,\beta \in {{\mathbb N}}^n$. 2. $G_3 := G_2 \cap (D_n)^{m+1}$. 3. $G_4 := \{ Q/P \mid Q \in G_3\}$, where $Q/P$ denotes the element in $(D_n)^{m+1}$ such that $(Q/P)P = Q$ in the sense of compnentwise product. Output: $G_4$ generates the module quotient $N:P$. In fact, we can show in the same way as in the commutative case that $G_3$ generates the left module $N \cap (D_n)^{m+1}P$. In particular, for each $Q \in G_3$, there exists $Q' \in (D_n)^{m+1}$ such that $Q = Q'P$. Let us denote this $Q'$ by $Q/P$. Then $Q'$ belongs to the quotient module $N : P$. Conversely, if $Q'$ belongs to $N : P$, then $Q'P$ belongs to $N \cap (D_n)^{m+1}P$. Hence $Q'$ belongs to the module generated by $G_4$. The correctness of the following algorithm should be clear: Input: A set $G_1 = \{Q_1,\dots,Q_k\}$ of generators of a left $D_n$-submodule $N$ of $(D_n)^{m+1}$ and a non-zero element $P=(P_0,\dots,P_m)$ of $(D_n)^{m+1}$. 1. Compute a set $G_2$ of generators of the syzygy module $${{\mathcal S}}:= \{(S_0,S_1,\dots,S_m,S_{m+1},\dots,S_{m+k}) \in (D_n)^{m+k+1} \mid \sum_{j=0}^m S_jP_je_j + \sum_{j=m+1}^{m+k}S_{m+j}Q_j = 0\}$$ via a Groebner base of the module generated by $P_je_j$ ($0 \leq j \leq m$) and $Q_j$ ($1 \leq j \leq k$). 2. Let $G_3$ be the set of the first $m+1$ components of the elements of $G_2$. Output: $G_3$ generates the module quotient $N:P$. Summed up, the annihilators for $(f^{\lambda}((\log f)^k)_{0 \leq k \leq m}$ and $(f^{\lambda}(\log f)^m$ are computed as follows: \[alg:specialization\]Input: a non-constant polynomial $f$ in the variables $x = (x_1,\dots,x_n)$ with coefficients in a computable subfield of ${{\mathbb C}}$, a number $\lambda$ which belongs to a computable subfield of ${{\mathbb C}}$, a non-negative integer $m$. 1. Compute a set $G_1$ of generators of ${\mbox{{\rm Ann}}}_{D_n[s]}(f^s,\dots,f^s((\log f)^k)$ by Algorithm \[alg:annfslog\]. 2. Compute the (global) Bernstein-Sato polynomial $b_f(s)$ of $f$ by using one of the algorithms in [@OakuDuke], [@OakuJPAA1997], [@BM] or their modifications. 3. Let $\nu_0$ be the largest positive integer $\nu$ such that $b_f(\lambda-\nu) = 0$ if there are any such $\nu$. If there are no positive integer $\nu$ such that $b_f(\lambda-\nu)=0$, then set $\nu_0=0$. 4. Set $\lambda_0 := \lambda - \nu_0$ and $G_2 := G_1|_{s=\lambda_0}$ (substitute $\lambda_0$ for $s$ in each element of $G_1$). 5. If $\nu_0 > 0$, then let $G_3$ be a set of generators of the module quotient $\langle G_2\rangle : f^{\nu_0} = \langle G_2\rangle : (f^{\nu_0},\dots,f^{\nu_0})$, where $\langle G_2\rangle$ denotes the left module generated by $G_2$. 6. If $\nu_0 = 0$, then set $G_3 := G_2$. 7. Compute a Gröbner base $G_4$ of the module generated by $G_3$ with respect to a term order $\prec$ for $(D_n)^{m+1}$ such that $Me_j \prec M'e_k$ for any monomial $M$ and $M'$ if $k < j$. Let $G_5$ be the set of the last component of each element of $G_4$. Output: $G_3$ generates ${\mbox{{\rm Ann}}}_{D_n}(f^\lambda,\dots,f^{\lambda}(\log f)^m)$; $G_5$ generates ${\mbox{{\rm Ann}}}_{D_n}f^{\lambda}(\log f)^m$. In step (3) of the algorithm above, we need only integer roots of the $b$-function. Hence one can employ a method described in [@LM2] to determine all the integer roots of the $b$-function efficiently without computing the whole $b$-function. Implementation and examples =========================== We have implemented the algorithms in a computer algebra system Risa/Asir [@asir], which is capable of Groebner base computation of modules over the ring of differential operators as well as over the ring of polynomials. (one dimensional case) Let $f$ be a square-free polynomial in one variable $x$ with complex coefficients. Since ${\mbox{{\rm Ann}}}_{D_1[s]}f^s$ is generated by $f\partial -sf'$, the annihilator module ${\mbox{{\rm Ann}}}_{D_1[s]}(f^s,\dots,f^s(\log f)^m)$ is generated by $m+1$ elements $$(f\partial_x-sf',0,\cdots,0),\quad (-f',f\partial_x-sf',0,\cdots,0),\quad\cdots,\quad (0,\cdots,0,-mf',f\partial_x-sf')$$ with $\partial_x = d/dx$ and $f' = \partial(f)$. Since the Bernstein-Sato polynomial of $f$ is $s+1$, the substitution $s = \lambda$ gives generators of ${\mbox{{\rm Ann}}}_{D_1}(f^\lambda,\dots,f^\lambda(\log f)^m)$ if $\lambda \neq 0,1,2,\dots$. In particular, ${\mbox{{\rm Ann}}}_{D_1}(f^{-1},\dots,f^{-1}(\log f)^m)$ is generated by $$(\partial_x f,0,\cdots,0),\quad (-f',\partial_x f,0,\cdots,0),\quad\cdots,\quad (0,\cdots,0,-mf',\partial_x f).$$ In view of Algorithm \[alg:specialization\], we can verify that ${\mbox{{\rm Ann}}}_{D_1}(1,\dots,(\log f)^m) = {\mbox{{\rm Ann}}}_{D_1}(f^{-1},\dots,f^{-1}(\log f)^m) : f$ is generated by $$(\partial_x,0,\cdots,0),\quad (-f',f\partial_x,0,\cdots,0),\quad\cdots,\quad (0,\cdots,0,-mf',f\partial_x).$$ Explicit generators of ${\mbox{{\rm Ann}}}_{D_1}(\log f)^m$ for $m \geq 1$ would be complicated: For example, if $f = x^3-x$ and $m=1$, Algorithm \[alg:specialization\] gives generators $$\begin{array}{l} (3x^5-4x^3+x)\partial_x^2+(3x^4+1)\partial_x, \\ (x^3-x)\partial_x^3+(-3x^4+9x^2-2)\partial_x^2+(-3x^3+3x)\partial_x \end{array}$$ of ${\mbox{{\rm Ann}}}_{D_1}\log f$, which is not generated by a single element. Set $f = x^2y^2+z^2$ with $n=2$ and $(x_1,x_2,x_3) = (x,y,z)$, $\partial_x = \partial/\partial x$ and so on. First ${\mbox{{\rm Ann}}}_{D_3[s]}f^s$ is generated by $$\begin{array}{l} -x \partial_x+y \partial_y, \quad y \partial_y+z \partial_z-2 s, \\ z \partial_x-y^2 x \partial_z, \quad z \partial_y-y x^2 \partial_z, \\ -z \partial_x^2+y^3 \partial_z \partial_y+y^2 \partial_z. \end{array}$$ Since the Bernstein-Sato polynomial of $f$ is $b_f(s) = (s+1)^3(2s+3)$, the substitution $s=-1$ gives a set of generators of ${\mbox{{\rm Ann}}}_{D_3}f^{-1}\log f$. Then by ideal quotient computation we get a set of generators $$\begin{array}{l} -x \partial_x+y \partial_y, \quad -z \partial_x+y^2 x \partial_z, \\ \partial_y^2 + x^2 \partial_z^2, \quad \partial_x^2 + y^2 \partial_z^2, \\ -z \partial_y+y x^2 \partial_z, \quad \partial_y \partial_x^2-z y \partial_z^3, \\ -\partial_y^2 \partial_x+z x \partial_z^3, \quad y\partial_y\partial_x + z\partial_z\partial_x \\ y \partial_z \partial_y+z \partial_z^2+\partial_z, \\ y \partial_y^2 +z \partial_z \partial_y + \partial_y, \\ z \partial_y \partial_x+z y x \partial_z^2-y x \partial_z, \\ \partial_y^2 \partial_x^2 + z^2 \partial_z^4 + 2 z \partial_z^3 \end{array}$$ of ${\mbox{{\rm Ann}}}_{D_3}\log f$. Let us consider the integral $$u(t) := \int_{{{\mathbb R}}^3} e^{-t(x^2+y^2+z^2)}\log(x^2y^2+z^2)\,dxdydz,$$ which is well-defined for $t > 0$. Then $u(t)$ satisfies ordinary differential equations $$P_1 u(t) = P_2u(t)=0$$ with $$\begin{aligned} P_1 &=t^3 \partial_t^5+(2 t^4+17 t^2) \partial_t^4+(32 t^3+80 t) \partial_t^3 \nonumber\\& +(-4 t^4+144 t^2+100) \partial_t^2+(-28 t^3+192 t) \partial_t-36 t^2+48, \nonumber\\ P_2&= t^3 \partial_t^4+(3 t^4+14 t^2) \partial_t^3+(2 t^5+35 t^3+52 t) \partial_t^2 \nonumber\\& +(14 t^4+102 t^2+48) \partial_t+18 t^3+66 t. \nonumber\end{aligned}$$ (Appendix) Holonomic systems ============================ Let us present a precise definition of holonomicity. We define the total or the $({{\bf 1}},{{\bf 1}})$-order of nonzero $P \in D_n$ to be $${\mbox{{\rm ord}}}_{({{\bf 1}},{{\bf 1}})}(P) := \max\{ |\alpha|+|\beta| = \alpha_1 + \cdots +\alpha_n + \beta_1 + \cdots + \beta_n \mid a_{\alpha,\beta} \neq 0\}.$$ We set ${\mbox{{\rm ord}}}_w(0) := -\infty$. This induces the filtration $$F_k(D_n) := \{ P \in D_n \mid {\mbox{{\rm ord}}}_{({{\bf 1}},{{\bf 1}})}(P) \leq k\} \quad (k \in {{\mathbb Z}})$$ on the ring $D_n$. Let $M$ be a left $D_n$-module and $\{F_k(M)\}_{k \in {{\mathbb Z}}}$ be a good $({{\bf 1}},{{\bf 1}})$-filtration. This means the following properties: 1. every $F_k(M)$ is a finite dimensional vector space over ${{\mathbb C}}$; 2. $F_k(M) \subset F_{k+1}(M) \quad \mbox{for all $k \in {{\mathbb Z}}$}$; 3. $\displaystyle \bigcup_{k\in{{\mathbb Z}}}F_k(M) = M$; 4. $F_i(D_n)F_k(M) \subset F_{i+k}(M) \quad \mbox{for all $i,k\in {{\mathbb Z}}$}$; 5. there exists $k_1 \in {{\mathbb Z}}$ such that $F_k(M) = 0$ for $k \leq k_1$; 6. there exists $k_2 \in {{\mathbb Z}}$ such that $F_i(D_n)F_k(M) = F_{i+k}(M)$ for $k \geq k_2$. Then there exists a polynomial in $k$ such that $\dim_{{{\mathbb C}}}F_k(M) = p(k)$ for sufficiently large $k$. The degree of $p(k)$ does not depend on the choice of a good $({{\bf 1}},{{\bf 1}})$-filtration of $M$ and is called the dimension of the module $M$, which we denote by $d(M)$. It was proved by Bernstein [@Bernstein] that $d(M) \geq n$ if $M \neq 0$. The following definition is due to Bernstein [@Bernstein]: A finitely generated left $D_n$-module $M$ is called a [*holonomic system*]{} if $d(M) \leq n$. We also call a left ideal $I$ of $D_n$ to be a [*holonomic ideal*]{}, by abuse of terminology, if the left $D_n$-module $D_n/I$ is holonomic. Note that $d(M) \leq n$ is equivalent to $d(M) = n$ or $M=0$ in view of the Bernstein inequality stated above. The dimension $d(M)$ can be computed as the degree of the Hilbert function from a Gröbner base with respect to a term order which is compatible with the total degree. Holonomicity is preserved by operations such as sum, product, restriction to affine subvarieties, and integration with respect to some of the variables (cf. [@Bernstein], [@Bjork]) and they are computable (see e.g., [@OakuIntegral]). Let $R_n := {{\mathbb C}}(x)\langle\partial_1,\dots,\partial_n\rangle$ be the ring of differential operators with rational function coefficients. A $D_n$-module $M$ is said to be of finite rank and the dimension is called the rank of $M$, if $R_nM$ is a finite dimensional vector space over ${{\mathbb C}}(x)$. A holonomic $D_n$-module $M$ is of finite rank but the converse is not true in general. Note that there is an algorithm for a given $D_n$-module $M$ of finite rank to construct a holonomic $D_n$-module $\widetilde M$ and a surjective $D_n$-homomorphism of $M$ to $\widetilde M$ ([@OTW],[@Tsai]). If $M$ is a system of differential equations of finite rank for an analytic function $u$, then we have an isomorphism $\widetilde \ D_n/{\mbox{{\rm Ann}}}_{D_n}u$. Set $f = x^2y^2+z^2$ and consider the function $f^{-1}$. It is easy to see that the operators $$\begin{aligned} f\partial_x + \frac{\partial f}{\partial x} &= (x^2y^2+z^2)\partial_x + 2xy^2, \nonumber \\ f\partial_y + \frac{\partial f}{\partial y} &= (x^2y^2+z^2)\partial_y + 2x^2y, \nonumber \\ f\partial_z + \frac{\partial f}{\partial z} &= (x^2y^2+z^2)\partial_z + 2z \nonumber \end{aligned}$$ annihilate $f^{-1}$. Let $J$ be the left ideal generated by these three operators, a ‘naive’ annihilator. Then the Hilbert function of the $D_3/J$ is $$\frac{1}{30} k^5+ \frac14 k^4 + \frac76 x^3 + \frac54 x^2 , + \frac{43}{10}k$$ which means that the degree of $D_3/J$ is 5 and hence $D_3/J$ is not holonomic although it is of rank one. The true annihilator $I$ of $f^{-1}$ is generated by $$\begin{array}{l} 3 z^2 \partial_x^2-2 y^3 \partial_z \partial_y-2 y^2 \partial_z, \\ 3 z^2 \partial_y-2 y x^2 \partial_z, \\ 3 z^2 \partial_x-2 y^2 x \partial_z, \\ 3 y \partial_y+2 z \partial_z+6, \\ -x \partial_x+y \partial_y \end{array}$$ and the Hilbert function of $D_3/I$ is $$\frac73k^3-\frac32k^2+\frac{43}{6}k-1,$$ which implies that $D_3/I$ is holonomic. The Hilbert function of $D_3/{\mbox{{\rm Ann}}}_{D_3}\log f$ is $$2k^3+\frac32k^2+\frac52k-1.$$ [99]{} Bernstein, I.N., The analytic continuation of generalized functions with respect to a parameter, Functional Analysis and its Applications [**5**]{} (1972), 1–16. Björk, J.-E., Rings of Differential Operators. North-Holland, 1979. Briançon, J., Maisonnobe, Ph., Remarques sur l’idéal de Bernstein associé à des polynômes, Preprint Université de Nice Sophia-Antipolis, no. 650. Kashiwara, M., $B$-functions and holonomic systems — Rationality of roots of $b$-functions, Invent. Math. [**38**]{} (1976), 33–53. Levandovskyy, V. and Morales, J., Computational [$D$]{}-module theory with [<span style="font-variant:small-caps;">Singular</span>]{}, Comparison with other systems and two new algorithms, Proc. of International Symposium on Symbolic and Algebraic Computation (ISSAC’08), ACM Press, 2008, pp. 173–180. Levandovskyy, V. and Morales, J., Algorithms for checking rational roots of $b$-functions and their applications, arXiv:1003.3785. Noro, M., Takayama, N., Nakayama, H., Nishiyama, K., Ohara, K, Risa/Asir: a computer algebra system. [http://www.math.kobe-u.ac.jp/Asir/asir.html]{}, 2011. Oaku, T., An algorithm of computing $b$-functions, Duke Math. J. [**87**]{} (1997), 115–132. Oaku, T., Algorithms for the $b$-function and $D$-modules associated with a polynomial, J. Pure Appl. Algebra, [**117 & 118**]{} (1997), 495–518. Oaku, T., Algorithms for integrals of holonomic functions over domains defined by polynomial inequalities, arXiv:1108.4853v2. Oaku, T., Takayama, N., An algorithm for de Rham cohomology groups of the complement of an affine variety. J.  Pure Appl. Algebra [**139**]{} (1999), 201–233. Oaku, T., Takayama, N., Walther, U., A localization algorithm for D-modules, J. Symbolic Computation 29 (2000), 721-728. Saito, M., Sturmfels, B., Takayama, N., Gröbner Deformations of Hypergeometric Differential Equations. Springer Verlag, 2000. Tsai, H., Weyl closure, torsion, and local cohomology of D-modules, 2000.

--- author: - | Eloy Ayón–Beato\ Departamento de Física, CINVESTAV–IPN,\ Apdo. Postal 14–740, 07000, México D.F., México.\ E-mail: - | Alan Garbarz\ Departamento de Física, Universidad de Buenos Aires FCEN - UBA,\ Ciudad Universitaria, Pabellón 1, 1428, Buenos Aires, Argentina.\ E-mail: - | Gaston Giribet\ Departamento de Física, Universidad de Buenos Aires FCEN - UBA and CONICET,\ Ciudad Universitaria, Pabellón 1, 1428, Buenos Aires, Argentina.\ E-mail: - | Mokhtar Hassaïne\ Instituto de Matemática y Física, Universidad de Talca,\ Casilla 747, Talca, Chile.\ E-mail: title: Analytic Lifshitz black holes in higher dimensions --- Introduction ============ In the last years, promising attempts to extend the AdS/CFT correspondence to other areas of physics have attracted much attention. Within the context of non-relativistic physics, holographic techniques were recently considered with remarkable success. Pioneer work on this matter, where gravity duals to non-relativistic systems were proposed, has been done in Refs. [@Son] and [@McGreevy]. Of particular importance is also the construction of Ref. [@Kachru], where the authors proposed new gravitational duals to scale invariant Lifshitz fixed points with no Galilean invariance. These gravity backgrounds have the form $$ds^{2}=-\frac{r^{2z}}{l^{2z}}dt^{2}+\frac{l^{2}}{r^{2}}dr^{2} +\frac{r^{2}}{l^{2}}d\vec{x}^2, \label{V}$$ where $\vec{x}$ is a $(D-2)$-dimensional vector. These geometries are usually called Lifshitz spacetimes, and admit the following anisotropic scaling symmetry $$\label{U} t\mapsto\lambda^{z}t,\qquad r\mapsto\lambda^{-1}r,\qquad \vec{x}\mapsto\lambda\vec{x},$$ as part of their isometry group. This is the geometric realization of the scale invariance exhibited by their non-relativistic dual systems, which are thought to be formulated on the $(D-1)$-dimensional space located at infinite $r$. In this sense, this picture completely mimics the prescription of the standard AdS/CFT correspondence. The natural extension of the construction of [@Kachru] is to look for black hole configurations that asymptote the Lifshitz spacetimes (\[V\]). Holographically, they should describe the finite temperature behavior of the non-relativistic theories. Black hole solutions of this type are known with the name of “Lifshitz black holes", and the quest for such solutions has received much attention recently. Analytic Lifshitz black hole solutions are scarce and they are actually hard to be found. In spite of the fact their metrics are not of a particularly abstruse form, these are reluctant to appear as exact solutions of theories of gravity with physically sensible matter sources. The main obstacle for these spacetimes to exist are the Birkhoff theorems, which happen to hold for generic models and restrict the subspace of static solutions in a strong way. Nevertheless, some few examples of black hole solutions that are asymptotically Lifshitz spaces were recently found in the literature. One of the first analytic examples was reported in Ref. [@Taylor] for a sort of higher-dimensional dilaton gravity without restricting the value of the dynamical exponent $z$. In Ref. [@Mann], a topological black hole solution which happens to be asymptotically Lifshitz with $z=2$ was found. An example with $z=4$ and with spherical topology was given in Ref. [@BertoldiI]. Numerical solutions for more general values of $z$ were explored in Refs. [@DanielssonI; @Mann; @BertoldiI; @BertoldiII]. More examples of analytic Lifshitz black holes were studied in Refs. [@DanielssonII; @Balasubramanian:2009rx; @charged], and the solution found in [@Balasubramanian:2009rx] is particularly interesting as it corresponds to a remarkably simple analytic example with $z=2$ in $D=4$ dimensions. The difficulty of embedding Lifshitz black holes in string theory was also investigated in Refs. [@TakayanagiI; @TakayanagiII; @Tong]. The holographic description of asymptotically Lifshitz spacetimes was studied in [@Ross]. More recent investigations related to Lifshitz black holes can be found in Refs. [@Sin:2009wi; @elnuestro; @Pang:2009wa; @Otro; @Otro2]. In [@AyonBeato:2009nh] a remarkably simple solution with $z=3$ in absence of matter fields was found for the New Massive Gravity theory [@Bergshoeff:2009hq], which consists of special square-curvature corrections to three-dimensional gravity. Previously, in Ref. [@Adams:2008zk], it was shown that square-curvature corrections to gravity generically can support the Lifshitz spacetimes (\[V\]). The example of Ref. [@AyonBeato:2009nh] is the first to show that these theories also allows the existence of Lifshitz black holes. Another example with $z=3/2$ was subsequently found for a four-dimensional theory with $R^2$–corrections in Ref. [@Cai:2009ac]. Inspired in the results of [@AyonBeato:2009nh] and [@Cai:2009ac], we will investigate in this paper how the introduction of higher-curvature corrections to the Einstein-Hilbert action leads to find a large zoo of analytic Lifshitz black hole solutions in $D$ dimensions. We will begin our search of Lifshitz black holes by considering the simplest example of quadratic corrections. Already in the simplest case we will find interesting solutions, provided a suitable parameterization of the coupling constants, and which hold for generic $z$ in $D$ dimensions. Interestingly enough, we will also exhibit an extremal Lifshitz black hole and an asymptotically Lifshitz black hole with logarithmic decay at infinity. Motivated by the richness of examples we find in the simplest case, we will then consider the most general square-curvature corrections, and we will present several classes of analytic Lifshitz black hole families of solutions in $D\geq 5$ dimensions. Curiously, one of these higher-dimensional families leads, through some particular limiting procedure, to the three-dimensional $z=3$ Lifshitz black hole of [@AyonBeato:2009nh] as well as to a new solution in $D=4$ with critical exponent $z=6$. $R^2$–corrected Lifshitz black holes for any dimension ====================================================== We first consider a gravity theory with $R^2$–corrections $$\label{eq:SquadR2} S[g_{\mu\nu}]=\int{d}^Dx\sqrt{-g} \left(R-2\lambda+\beta_1{R}^2\right),$$ giving the following field equations $$\label{eq:eqR2} G_{\mu\nu}+\lambda{g}_{\mu\nu}+2\beta_1g_{\mu\nu}\square{R} -2\beta_1\nabla_\mu\nabla_\nu{R}+2\beta_1RR_{\mu\nu} -\frac12\beta_1{R}^2g_{\mu\nu}=0.$$ These equations allow Lifshitz spacetimes (\[V\]) as solutions for a generic value of the dynamical exponent $z$ in any dimension, provided a suitable choice of the cosmological constant $\lambda$ and the coupling constant $\beta_1$ that is given by \[eq:LparamR2\] $$\begin{aligned} \lambda &=& -\frac{2z^2+(D-2)(2z+D-1)}{4l^2}, \label{eq:Lparam_lambdaR2}\\ \beta_1 &=& -\frac{1}{8\lambda}.\label{eq:Lparam_R2}\end{aligned}$$ It is well known that this kind of theory (\[eq:SquadR2\]) can be generically mapped into scalar-tensor theories through a conformal transformation of the metric with conformal factor $\Omega^2=1+2\beta_1R$. However, for the particular choice of the coupling constant (\[eq:LparamR2\]), this trick does not work since $R=4\lambda$ for Lifshitz spacetimes. In this sense, the model corresponds to a genuine pure gravity theory. The black hole solutions we will derive below present the same degeneracy for the conformal transformation, and thus have no scalar-tensor counterpart. Our purpose is to explore whether there exist some black hole solutions which asymptote the Lifshitz spacetimes (\[V\]). This analysis is motivated by the existence of a four-dimensional Lifshitz black hole solution for these theories with a specific value of the dynamical exponent $z=3/2$ found in Ref. [@Cai:2009ac]. In fact, we can show that for any dimension $D$, there exists a two-parametric family of solutions given by \[eq:F1\] $$\begin{aligned} ds^2&=&-\frac{r^{2z}}{l^{2z}} \left(1-\frac{M^{-}l^{\alpha_-}}{r^{\alpha_-}} +\frac{M^{+}l^{\alpha_+}}{r^{\alpha_+}}\right)dt^2 +\frac{l^2}{r^2} \left(1-\frac{M^-l^{\alpha_{-}}}{r^{\alpha_{-}}} +\frac{M^+l^{\alpha_{+}}}{r^{\alpha_{+}}}\right)^{-1}dr^2\nonumber\\ &&{}+\frac{r^2}{l^2}d\vec{x}^2,\label{eq:gF1}\\ &&\nonumber\\ \alpha_{\pm}&=&\frac{3z+2(D-2)\pm\sqrt{z^2+4(D-2)(z-1)}}2, \label{eq:alphaF1}\end{aligned}$$ for which the coupling constants are the same as in the purely Lifshitz case (\[eq:LparamR2\]). It is important to mention that this family of geometries has the same constant scalar curvature of the Lifshitz spacetimes. First, it is clear from the expression of $\alpha_{\pm}$, given by (\[eq:alphaF1\]), that the dynamical exponent may take the values $z\in(-\infty,z_{-}]\cup[z_{+},\infty)$ where $$\label{eq:zpm} z_{\pm}=4-2D\pm 2\sqrt{(D-1)(D-2)}.$$ On the other hand, the solution (\[eq:F1\]) represents an asymptotically Lifshitz black hole for $\alpha_\pm>0$, and this occurs for $z\geq z_+$. It is easy to see that the four-dimensional solution of reference [@Cai:2009ac] corresponds to the particular case $M^+=0$ with $z=3/2$, that is $\lambda=-33/8l^2$ and $\alpha_-=3$. For a specific relation between the constants $M^{\pm}$ given by $$M^{+}=\alpha_-(\alpha_+-\alpha_-)^{\frac{\alpha_+-\alpha_-}{\alpha_-}} \left(\frac{M^-}{\alpha_+}\right)^{\frac{\alpha_+}{\alpha_-}},$$ the solution (\[eq:F1\]) has zero temperature, i.e. it is an extremal black hole $$\begin{aligned} ds^2&=&-\frac{r^{2z}}{l^{2z}}\left[1-\frac{\alpha_+}{\alpha_+-\alpha_-} \left(\frac{r_e}{r}\right)^{\alpha_-}+ \frac{\alpha_-}{\alpha_+-\alpha_-} \left(\frac{r_e}{r}\right)^{\alpha_+}\right]dt^2\nonumber\\ &&{} +\frac{l^2}{r^2} \left[1-\frac{\alpha_+}{\alpha_+-\alpha_-} \left(\frac{r_e}{r}\right)^{\alpha_-}+ \frac{\alpha_-}{\alpha_+-\alpha_-} \left(\frac{r_e}{r}\right)^{\alpha_+}\right]^{-1}dr^2+ \frac{r^2}{l^2}d\vec{x}^2,\label{eq:extremal}\end{aligned}$$ where the extremal radius $r_e$ is expressed as $$r_e=l\left(\frac{\alpha_+-\alpha_-}{\alpha_+}M^-\right)^{1/\alpha_-}.$$ The interest on solution (\[eq:F1\]) increases once one notices that when the dynamical exponent approach the value $z=z_+$, defined in (\[eq:zpm\]), there exists an additional solution which asymptotes the Lifshitz spacetime in a much slower way $$\begin{aligned} ds^2&=&-\frac{r^{2z_+}}{l^{2z_+}} \left\{1-\frac{l^{\alpha_0}}{r^{\alpha_0}} \left[M_1+M_2\ln{\left(\frac{r}{l}\right)}\right]\right\}dt^2 +\frac{l^2}{r^2} \left\{1-\frac{l^{\alpha_0}}{r^{\alpha_0}} \left[M_1+M_2\ln{\left(\frac{r}{l}\right)}\right]\right\}^{-1}dr^2 \nonumber\\ &&{}+\frac{r^2}{l^2}d\vec{x}^2,\label{eq:log}\end{aligned}$$ where the parameter $\alpha_0$ is given by $$\label{eq:alpha0} \alpha_0=3\sqrt{(D-1)(D-2)}-2(D-2).$$ The fact of having a weakened (logarithmic) fall-off as a next-to-leading contribution in the asymptotic behavior is well known in the standard AdS/CFT correspondence. In particular, this was one of the key points in recent discussions on three-dimensional massive gravity (see [@Strominger] and reference therein). The extremal version of the logarithmic black hole (\[eq:log\]) is found for $$M_1=\frac{M_2}{\alpha_0}\left[1-\ln\left(\frac{M_2}{\alpha_0}\right) \right],$$ and then the corresponding spacetime geometry reads $$\begin{aligned} ds^2&=&-\frac{r^{2z_+}}{l^{2z_+}} \left\{1-\frac{{r_e}^{\alpha_0}}{r^{\alpha_0}} \left[1+\alpha_0\ln{\left(\frac{r}{r_e}\right)}\right]\right\}dt^2 +\frac{l^2}{r^2} \left\{1-\frac{{r_e}^{\alpha_0}}{r^{\alpha_0}} \left[1+\alpha_0\ln{\left(\frac{r}{r_e}\right)}\right]\right\}^{-1} dr^2\nonumber\\ &&{}+\frac{r^2}{l^2}d\vec{x}^2,\label{eq:logext}\end{aligned}$$ where the extremal radius is defined by $$r_e=l\left(\frac{M_2}{\alpha_0}\right)^{\frac{1}{\alpha_0}}.$$ Let us stress that spacetimes (\[eq:extremal\]) and (\[eq:logext\]) are the first examples of asymptotically Lifshitz black hole solutions with an extremal horizon. Remarkably, the solution (\[eq:log\]) is additionally the first solution with a logarithmic decay. In the list of curiosities, we can also mention that the family of solutions (\[eq:F1\]) contains an asymptotically conformal limit at $z=1$; namely $$\label{eq:STAdS} ds^2=-\left(\frac{r^2}{l^2} -\frac{M^-l^{D-2}}{r^{D-2}}+\frac{M^+l^{D-3}}{r^{D-3}}\right)dt^2 +\left(\frac{r^2}{l^2}-\frac{M^-l^{D-2}}{r^{D-2}} +\frac{M^+l^{D-3}}{r^{D-3}}\right)^{-1}dr^2 +\frac{r^2}{l^2}d\vec{x}^2.$$ For $M^-=0$ ($M^+<0$), the resulting spacetime is nothing but the Schwarzschild-Tangherlini-AdS topological black hole with toroidal horizon ($k=0$ in standard notation) for $\lambda=-D(D-1)/(4l^2)$. For $M^+=0$, the solution corresponds to a different asymptotically AdS toroidal black hole with a faster decay. As an appealing remark, for $D=4$, the solution (\[eq:STAdS\]) is precisely the Reissner-Nordstrom-AdS topological black hole with $k=0$. As a final comment, we would like to point out that the Lagrangian of the gravity action (\[eq:SquadR2\]) for the coupling constant given by (\[eq:Lparam\_R2\]) can be written as a perfect square, $$\label{eq:LF1} R-2\lambda-\frac1{8\lambda}R^2= -\frac{1}{8\lambda}\left(R-4\lambda\right)^2.$$ Therefore, for this choice of the coupling constant, the gravity action is definite positive and reaches its minimal (vanishing) value for $R=4\lambda$, which is precisely the case of the solutions (\[eq:F1\]). Then, this solution (or its Euclidean continuation) can be seen as a sort of gravitational instanton. This has to do with the degeneracy of the field equations (\[eq:eqR2\]) in the following sense: for the case of constant scalar curvature solutions, equations (\[eq:eqR2\]) become $$f^{\prime}(R)R_{\mu\nu}-\frac{1}{2}f(R)g_{\mu\nu}=0,$$ where $f(R)$ is the Lagrangian expressed in terms of the scalar curvature. Apart from considering $f^{\prime}(R)\not=0$, which yields Einstein equations with an effective cosmological constant and hence no Lifshitz configurations, the only option for constant scalar curvature solutions to exist is that the value of the scalar curvature be a double root of the Lagrangian $f(R)$. Then, it is clear that the family of black holes obtained in (\[eq:F1\]) will be solutions of any gravity theory with Lagrangian $f(R)=(R-4\lambda)^2H(R)$ where $H$ is a function regular at $R=4\lambda$. More general quadratic corrections ================================== Due to the new and interesting results of the $R^2$–corrected theory (\[eq:SquadR2\]) presented above, it is natural to extend the analysis and explore the existence of asymptotically Lifshitz black hole configurations with the most general quadratic corrections. We will proceed in the same way as before, by first establishing the purely Lifshitz configurations and then by presenting three different classes of asymptotically Lifshitz black hole solutions which correspond to different ranges of the dynamical exponent $z$. As shown below, for any value of the dynamical exponent $z$ at least one family of black hole solutions exists. We consider now the gravity action that includes the most general quadratic-curvature corrections in $D$-dimensions; namely $$\begin{aligned} S[g_{\mu\nu}]&=&\int{d}^Dx\sqrt{-g} \left(R-2\lambda+\beta_1{R}^2 +\beta_2{R}_{\alpha\beta}{R}^{\alpha\beta} +\beta_3{R}_{\alpha\beta\mu\nu}{R}^{\alpha\beta\mu\nu} \right). \label{eq:Squad}\end{aligned}$$ It follows from the Gauss-Bonnet theorem in four dimensions and the vanishing of the Gauss-Bonnet term in three dimensions that in the case $D < 5$, it is sufficient to consider only two of the three quadratic invariants in the Lagrangian. This makes necessary to split the analysis in two parts by first considering the higher-dimensional cases and then to analyze separately the lower dimensional ones, $D=3$ and $D=4$. The action (\[eq:Squad\]) gives rise to the following field equations $$\begin{aligned} G_{\mu\nu}+\lambda{g}_{\mu\nu} +\left(\beta_2+4\beta_3\right)\square{R}_{\mu\nu} +\frac12\left(4\beta_1+\beta_2\right)g_{\mu\nu}\square{R} -\left(2\beta_1+\beta_2+2\beta_3\right)\nabla_\mu\nabla_\nu{R} \nonumber\\ \nonumber\\ {}+2\beta_3R_{\mu\gamma\alpha\beta}R_{\nu}^{~\gamma\alpha\beta} +2\left(\beta_2+2\beta_3\right)R_{\mu\alpha\nu\beta}R^{\alpha\beta} {}-4\beta_3R_{\mu\alpha}R_{\nu}^{~\alpha}+2\beta_1RR_{\mu\nu} \nonumber\\ \nonumber\\ {}-\frac12\left(\beta_1{R}^2+\beta_2{R}_{\alpha\beta}{R}^{\alpha\beta} +\beta_3{R}_{\alpha\beta\gamma\delta}{R}^{\alpha\beta\gamma\delta} \right)g_{\mu\nu}&=&0.\qquad \label{eq:squareGrav}\end{aligned}$$ As in the purely $R^2-$case, Lifshitz spacetimes (\[V\]) are solutions of these field equations for a generic value of the dynamical exponent $z$ in any dimension, provided that \[eq:Lparam\] $$\begin{aligned} \lambda &=& -\frac1{4l^2}\bigg(2z^2+(D-2)(2z+D-1)-\frac{4(D-3)(D-4)z(z+D-2)\beta_3}{l^2}\bigg), \label{eq:Lparam_lambda}\\\nonumber\\ \beta_2 &=& \frac{l^2-2\left[2z^2+(D-2)(2z+D-1)\right]\beta_1 -4\left[z^2-(D-2)z+1\right]\beta_3} {2(z^2+D-2)}.\label{eq:Lparam_beta}\end{aligned}$$ Notice that the above parameterizations coincide with the values previously found for the New Massive Gravity in $D=3$ [@AyonBeato:2009nh], where $\beta_3=0$ and $\beta_2=-(8/3)\beta_1=-1/m^2$. In turn, this generalizes the previous authors’ result. We shall now proceed to present three more different Lifshitz black hole families in $D\geq5$ dimensions. An asymptotically Lifshitz black hole family for $z>2-D$ -------------------------------------------------------- The first family of solutions we present here is described by the following line element \[eq:F2\] $$\label{eq:gF2} ds^2=-\frac{r^{2z}}{l^{2z}} \left(1-\frac{Ml^{(z+D-2)/2}}{r^{(z+D-2)/2}}\right)dt^2+\frac{l^2}{r^2} \left(1-\frac{Ml^{(z+D-2)/2}}{r^{(z+D-2)/2}}\right)^{-1}dr^2 +\frac{r^2}{l^2}d\vec{x}^2,$$ and represents an asymptotically Lifshitz black hole solution of the field equations (\[eq:squareGrav\]) for the dynamical exponent $z>2-D$. The coupling constants allowing the existence of the solution (\[eq:gF2\]) are parameterized in term of the dynamical exponent $z$ by $$\begin{aligned} \lambda&=&\frac{(D-2)}{4l^2}{\Big\{}(197D-389)z^4 +4(19D^2-200D+325)z^3+(D-2)\big[2(5D^2-73D+356)z^2 \nonumber\\\nonumber\\ &&{}+4(D^3-2D^2+15D-62)z+(D+2)(D-1)(D-2)^2\big]{\Big\}} \bigg/ P_4(z),\label{eq:lambdaF2}\\\nonumber\\ \beta_1&=&l^2{\Big\{}27z^6-18(3D-4)z^5 +3(19D^2-168D+356)z^4-12(11D^3-84D^2+196D-120)z^3 \nonumber\\\nonumber\\ &&{}-(D-2)\big[(19D^3-330D^2+2052D-3640)z^2 +2(3D^4-30D^3+124D^2-536D+1024)z\nonumber\\\nonumber\\ &&{}+(D+2)(D-2)^2(D^2-4D+36) \big]{\Big\}} \bigg/{\Big(2(D-3)(D-4)(z+D-2)^2 P_4(z)\Big)}, \label{eq:beta1F2}\\\nonumber\\ \beta_2&=&-2l^2\big[3z^2+(D+2)(D-2)\big] {\Big\{}9z^4-6(3D-4)z^3-8(D^2-10)z^2+2(D^3-4D^2+32D-80)z \nonumber\\\nonumber\\ &&{}-(D-2)\big[D^3+2D^2-12(D-2)\big]{\Big\}} \bigg/{\Big((D-3)(D-4)(z+D-2)^2 P_4(z)\Big)}, \label{eq:beta2F2}\\\nonumber\\ \beta_3&=&l^2\big[3z^2+(D+2)(D-2)\big] {\Big\{}9z^3-3(9D-14)z^2 -(D-2)\big[(5D-62)z+D^2-4D+36\big]{\Big\}} \nonumber\\\nonumber\\ &&{}\bigg/{\Big(2(D-3)(D-4)(z+D-2) P_4(z)\Big)}, \label{eq:beta3F2}\end{aligned}$$ where $P_4$ is a polynomial of degree four in the dynamical exponent $z$ given by $$P_4(z)=27z^4-4(27D-45)z^3 -(D-2)\big[2(5D-116)z^2+4(D^2-D+30)z+(D+2)(D-2)^2\big].$$ It is clear from the expressions of the $\beta_i$ that the solution is defined only for higher dimensions $D\ge5$. The analysis of the lower dimensional cases will be done in the next section. As in the purely $R^2$–case, there exists a conformal limit $z=1$ of the family (\[eq:F2\]) which is given by the following asymptotically AdS black hole $$\label{eq:gz=1F2} ds^2=-\left(\frac{r^2}{l^2} -\frac{Ml^{(D-5)/2}}{r^{(D-5)/2}}\right)dt^2+\left(\frac{r^2}{l^2} -\frac{Ml^{(D-5)/2}}{r^{(D-5)/2}}\right)^{-1}dr^2 +\frac{r^2}{l^2}d\vec{x}^2.$$ More precisely, this spacetime (\[eq:gz=1F2\]) is a solution of the Einstein-Gauss-Bonnet gravity with a fine-tuned coupling constant, since for $z=1$ the parameterizations (\[eq:lambdaF2\])-(\[eq:beta3F2\]) become $$\begin{aligned} \lambda&=&-\frac{(D-1)(D-2)}{4l^2},\label{eq:lambdaz=1F2}\\ \beta_1=-\frac14\beta_2=\beta_3&=&\frac{l^2}{2(D-3)(D-4)}, \label{eq:betasz=1F2}\end{aligned}$$ yielding to the Lagrangian $$\label{eq:L_iCS} R-2\lambda -\frac{(D-1)(D-2)}{8(D-3)(D-4)\lambda}{\cal L}_{\mathrm{GB}},$$ where ${\cal L}_{\mathrm{GB}}=R^2 -4R_{\alpha\beta}{R}^{\alpha\beta} +{R}_{\alpha\beta\mu\nu}{R}^{\alpha\beta\mu\nu}$ is the Gauss-Bonnet term. Note that for $D=5$, the above black hole becomes diffeomorphic to a warped product having as base AdS$_3$ with a 2-plane fiber. Moreover this case precisely corresponds to the Chern-Simons gravity in $D=5$. An asymptotically Lifshitz black hole family for $z>1$ ------------------------------------------------------ The second family of asymptotically Lifshitz black holes we find is valid for $z>1$, \[eq:F3\] $$\begin{aligned} ds^2&=&-\frac{r^{2z}}{l^{2z}} \left(1-\frac{Ml^{2(z-1)}}{r^{2(z-1)}}\right)dt^2+\frac{l^2}{r^2} \left(1-\frac{Ml^{2(z-1)}}{r^{2(z-1)}}\right)^{-1}dr^2 +\frac{r^2}{l^2}d\vec{x}^2,\label{eq:gF3}\end{aligned}$$ and it exists for the following choice of coupling constants $$\begin{aligned} \lambda&=&-\frac{(z-1)\left[z^2-Dz-(D-1)(D-2)\right]}{2l^2(z-D)}, \label{eq:lambdaF3}\\\nonumber\\ \beta_1&=&l^2\Big[3(D-1)(D-2)z^3 -(2D^3-2D^2-11D+20)z^2+(3D^3-14D^2+19D+10)z \nonumber\\\nonumber\\ &&{}+(D+2)(D-4)\Big]\Big/{\Big[}2(D-2)(D-3)(D-4)z(z-1)(z-D)(3z+D-4) {\Big]}, \label{eq:beta1F3}\\\nonumber\\ \beta_2&=&-l^2(D-1)(2z-D-2) {\Big[}6(D-2)z^2-(D^2-3D+8)z-2(D-4) {\Big]}\nonumber\\\nonumber\\ &&{}\Big/{\Big[}2(D-2)(D-3)(D-4)z(z-1)(z-D)(3z+D-4) {\Big]}, \label{eq:beta2F3}\\\nonumber\\ \beta_3&=&\frac{l^2(D-1)(2z-D-2)}{4(D-3)(D-4)z(z-D)}. \label{eq:beta3F3}\end{aligned}$$ As for the $z>2-D$ family, the lower dimensions $D=3$ and $D=4$ are also forbidden here. Clearly, there is no conformal limit $z=1$ for this family. An asymptotically Lifshitz black hole family for $z<0$ ------------------------------------------------------ The last family of Lifshitz black holes we describe is characterized by a negative dynamical critical exponent $z=-|z|$, whose metric reads \[eq:F4\] $$\begin{aligned} ds^2&=&-\frac{l^{2|z|}}{r^{2|z|}} \left(1-\frac{Ml^{|z|}}{r^{|z|}}\right)dt^2%\nonumber\\&&{} +\frac{l^2}{r^2} \left(1-\frac{Ml^{|z|}}{r^{|z|}}\right)^{-1}dr^2 +\frac{r^2}{l^2}d\vec{x}^2,\label{eq:gF4}\end{aligned}$$ while the corresponding coupling constants are parameterized as $$\begin{aligned} \lambda&=&\frac{|z|\left[2|z|^2-4(D-2)|z|+(D-2)(D-3)\right]} {4l^2(2|z|-D+2)},\qquad~ \label{eq:lambdaF4}\\\nonumber\\ \beta_1&=&\frac{l^2\left[3|z|^2-2(D-2)|z|+2D-5\right]} {2(D-3)(D-4)|z|(2|z|-D+2)}, \label{eq:beta1F4}\\\nonumber\\ \beta_2&=&-4\beta_3,\\\nonumber\\ \beta_3&=&\frac{l^2\left[6|z|^2-4(D-2)|z|+(D-1)(D-2)\right]} {4(D-3)(D-4)|z|(2|z|-D+2)}.\qquad~ \label{eq:beta23F4}\end{aligned}$$ The associated Lagrangian describes a fine-tuned $R^2$–corrected Einstein-Gauss-Bonnet theory $$\label{eq:L_z<0} R-2\lambda-\beta_3{\cal L}_{\mathrm{GB}} -\frac{l^2R^2}{4|z|(2|z|-D+2)}.$$ This family of black holes is again defined only in higher dimensions $D\ge5$. Being defined only for negative dynamical critical exponents, it has no conformal analog $z=1$. In the next section, we analyze the lower-dimensional cases $D=3$ and $D=4$. Critical lower dimensional Lifshitz black holes =============================================== The families of Lifshitz black holes given by (\[eq:F2\]), (\[eq:F3\]) and (\[eq:F4\]) are generically forbidden in dimensions lower than $5$. This is due to the fact that the use of a nontrivial value for the coupling constant $\beta_3$ is artificial in these dimensions. Concretely, if one consider theories with $\beta_3\neq0$, and due to the fact that the Gauss-Bonnet combination ${\cal L}_{\mathrm{GB}}$ vanishes in $D=3$ and is a total derivative in $D=4$, it turns out that it is always possible to shift the coupling constants and to end with $\beta_3=0$. This shifting reads $$\label{eq:beta_shift} (\beta_1,\beta_2,\beta_3)\mapsto(\beta_1-\beta_3,\beta_2+4\beta_3,0).$$ Despite families (\[eq:F2\]), (\[eq:F3\]) and (\[eq:F4\]) are formally defined for higher-dimensions, the possibility that new critical solutions exist in lower dimensions for some particular values of the dynamical exponent $z$ is not excluded. A natural way to explore this possibility is to consider a dimensional continuation of the $D$-dimensional expressions and study whether some potential cancellation of the divergences of the coupling constants appears when one expands around $D=3$ and $D=4$. That is what we will do in this section. Using the results of this analysis as an indication, we will explicitly confirm the existence of critical solutions that, indeed, represent Lifshitz black holes in $D=3$ and in $D=4$. The $z=3$ three-dimensional asymptotically Lifshitz black hole -------------------------------------------------------------- Let us start with the dimensional continuation and expansion of the coupling constants of the family (\[eq:F2\]) around $D=3$. The cosmological constant is regular, $\lambda=O\left((D-3)^0\right)$, but the coupling constants exhibit the following singular behavior $$\begin{aligned} \label{eq:betasF2D=3} \beta_1=-\frac14\beta_2=\beta_3&=& -\frac{(z-3)(3z^2+5)(9z^2-12z+11)l^2} {2(z+1)(27z^4-144z^3+202z^2-144z-5)}\times\frac1{D-3}\nonumber\\ \nonumber\\ &&{}+O\left((D-3)^0\right).\end{aligned}$$ This indicates that the only possibility for having a potentially regular behavior for this family at $D=3$ appears for $z=3$. Considering that there is in fact no continuity in the number of dimensions, one can chose the element $z=3$ of the family (\[eq:F2\]). Evaluating after that for $D=3$, the resulting Lifshitz black hole is \[eq:NMG\] $$\label{eq:gNMG} ds^2=-\frac{r^{6}}{l^{6}} \left(1-\frac{Ml^{2}}{r^{2}}\right)dt^2+\frac{l^2}{r^2} \left(1-\frac{Ml^{2}}{r^{2}}\right)^{-1}dr^2 +\frac{r^2}{l^2}d{x}^2,$$ with $\lambda=-13/(2l^2)$, and surprisingly the meaningful coupling constants (i.e. after the shifting (\[eq:beta\_shift\])) are those of New Massive Gravity [@Bergshoeff:2009hq] $$\label{eq:bNMG} \beta_2=-(8/3)\beta_1=2l^2,$$ which gives rise to the three-dimensional Lifshitz black hole previously found by the authors in [@AyonBeato:2009nh]. A similar analysis can be done for the family (\[eq:F3\]); the potential regular behavior occurs in this case for $z=5/2$. However, the resulting solution is not new but corresponds to the case $z=5/2$, $M^+=0$ ($\alpha_-=3$) of the family (\[eq:F1\]) valid in generic dimension $D$. The family (\[eq:F4\]) has no regular limit in $D=3$. The $z=6$ four-dimensional asymptotically Lifshitz black hole ------------------------------------------------------------- In four dimensions, we proceed in a similar way, by doing a dimensional continuation and expanding the coupling constants of the family (\[eq:F2\]) around $D=4$. Again, the cosmological constant is regular, $\lambda=O\left((D-4)^0\right)$, and the coupling constants exhibit singular behavior $$\begin{aligned} \label{eq:betasF2D=4} \beta_1=-\frac14\beta_2=\beta_3&=& \frac{3(z-6)(z^2+4)(3z^2-4z+4)l^2} {2(z+2)(9z^4-84z^3+128z^2-112z-16)}\times\frac1{D-4}\nonumber\\ \nonumber\\ &&{}+O\left((D-4)^0\right).\end{aligned}$$ The indication here is that the only possibility potentially occurs for $z=6$. The $z=6$ element of the family (\[eq:F2\]), when is evaluated in $D=4$, indeed gives rise to a new Lifshitz black hole \[eq:LbhD=4\] $$\label{eq:gLbhD=4} ds^2=-\frac{r^{12}}{l^{12}} \left(1-\frac{Ml^{4}}{r^{4}}\right)dt^2+\frac{l^2}{r^2} \left(1-\frac{Ml^{4}}{r^{4}}\right)^{-1}dr^2 +\frac{r^2}{l^2}(d{x}^2+d{y}^2),$$ with $\lambda=-51/(2l^2)$ and meaningful coupling constants given by $$\label{eq:bLbhD=4} \beta_2=-(25/9)\beta_1=25l^2/64.$$ The corresponding analysis for the family (\[eq:F3\]) in $D=4$ singles out the value $z=3$. Again, the resulting solution is not new but corresponds to the case $z=3$, $M^+=0$ ($\alpha_-=4$) of the family (\[eq:F1\]), which is valid in four dimensions. As before, family (\[eq:F4\]) has no regular limit in $D=4$. Conclusions and open problems ============================= In this paper we have found analytic Lifshitz black hole solutions for gravity with square-curvature corrections in arbitrary dimension. Some open questions remain: - The computation of conserved charges of the asymptotically Lifshitz black holes of higher-curvature gravity would be needed to fully understand the thermodynamical properties of both the gravitational backgrounds and the dual systems. Some important advances in this direction have been done recently in [@Hohm]. - Stability of Lifshitz black hole solutions is another question it would be interesting to address. - Among the family of black holes we exhibited here there are extremal solutions, see (\[eq:extremal\]) and (\[eq:logext\]). An interesting question is that of studying the causal structure of these spacetimes. - Last, the condensed matter interpretation of these backgrounds within the holographic proposal of [@Kachru] deserves to be matter for further study. The authors thank the organizers and participants of the CECS Summer Workshop on Theoretical Physics. The work of A.G. is supported by University of Buenos Aires, Argentina. 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--- abstract: 'The slow decay of charge carriers in polymer–fullerene blends measured in transient studies has raised a number of questions about the mechanisms of nongeminate recombination in these systems. In an attempt to understand this behavior, we have applied a combination of steady-state and transient photoinduced absorption measurements to compare nongeminate recombination behavior in films of neat [poly(3-hexyl thiophene)]{}(P3HT) and P3HT blended with [\[6,6\]-phenyl-C$_{61}$ butyric acid methyl ester]{}(PCBM). Transient measurements show that carrier recombination in the neat P3HT film exhibits second-order decay with a recombination rate coefficient that is similar to that predicted by Langevin theory. In addition, temperature dependent measurements indicate that neat films exhibit recombination behavior consistent with the Gaussian disorder model. In contrast, the P3HT:PCBM blend films are characterized by a strongly reduced recombination rate and an apparent recombination order greater than two. We then assess a number of previously proposed explanations for this behavior, including phase separation, carrier concentration dependent mobility, non-encounter limited recombination, and interfacial states. In the end, we propose a model in which pure domains with a Gaussian density of states are separated by a mixed phase with an exponential density of states. We find that such a model can explain both the reduced magnitude of the recombination rate and the high order recombination kinetics and, based on the current state of knowledge, is the most consistent with experimental observations.' author: - Julien Gorenflot - 'Michael C. Heiber' - Andreas Baumann - Jens Lorrmann - Matthias Gunz - Andreas Kämpgen - Vladimir Dyakonov - Carsten Deibel bibliography: - 'Papers.bib' title: 'Nongeminate recombination in neat P3HT and P3HT:PCBM blend films' --- Introduction ============ Organic solar cells based on polymer–fullerene blends have recently reached power conversion efficiencies as high as 10%. [@green2012review] Paradoxically, those high performances are achieved due to an only marginally understood peculiarity: the inefficiency of nongeminate charge carrier recombination. Such inefficient recombination has been observed in a number of polymer–fullerene blends.[@pivrikas2005a; @parkinson2008; @clarke2011; @zusan2014] The nongeminate recombination of two oppositely charged particles is predicted by the Langevin theory.[@langevin1903] This theory states that the recombination rate of electrons and holes ($R_\text{L}$) is governed by a second-order process, $$R_\text{L} = k_\text{L}np. \label{eqn:bimol}$$ where $n$ is the electron concentration, $p$ is the hole concentration, and $k_\text{L}$ is the Langevin rate constant. This rate constant is defined by assuming that the recombination event is much faster than the rate at which electrons and holes encounter one another. As a result, the Langevin rate constant depends on the charge carrier mobility of each species, $$k_{\text{L}} = \frac{e}{\varepsilon} (\mu_e + \mu_h) . \label{eqn:prefact}$$ where $e$ is the elementary charge, $\epsilon$ the dielectric constant, $\mu_e$ is the electron mobility, and $\mu_h$ is the hole mobility. This implies that for equal densities of electrons and holes, $R \propto n^2$, and an initial carrier density ($n_0$) will decay as $n(t) = n_0/(1+k_\text{L}n_0t)\propto t^{-1}$ in the absence of further photogeneration. For conditions typical of an organic solar cell, with an initial polaron density of 10$^{18}$ cm$^{-3}$, a mobility of 10$^{-4}$ cm$^2$/Vs, and a relative dielectric constant of 3.5, Langevin theory predicts that almost 85$\%$ of charge carriers should recombine nongeminately within 100 ns, which is the minimum time required for the charge extraction.[@guo2010a; @deibel2010review] Yet, external quantum efficiencies of 80$\%$ and higher have been reported under appropriate illumination in polymer–fullerene blends. [@deibel2010review] In [poly(3-hexyl thiophene)]{}(P3HT):[\[6,6\]-phenyl-C$_{61}$ butyric acid methyl ester]{}(PCBM) blends, the actual recombination rate is found to be up to 10$^{4}$ times slower than predicted by the Langevin theory,[@pivrikas2005a; @deibel2009] resulting in the characterization of a reduction factor ($\zeta$),[@deibel2008b] $$\zeta = \frac{R_\text{exp}}{R_\text{L}}. \label{eqn:zeta}$$ Furthermore, a variety of experimental methods have shown that the recombination rate in polymer–fullerene blends does not have the expected second-order kinetics. Instead, higher orders between 2.3 and 2.8 have been found at room temperature.[@nelson2003; @guo2010a; @foertig2009; @shuttle2010] Although under these conditions it is not possible to describe the higher order decay by reduced Langevin recombination based on Eqn. (\[eqn:bimol\]), such reduction factors are still being reported.[@kniepert2011; @mingebach2012; @wetzelaer2013] This discrepancy needs to be resolved in order to understand the detailed processes involved in nongeminate recombination. In this paper, we compare the dynamics of nongeminate recombination in a polymer–fullerene blend (P3HT:PCBM) to those in the neat polymer (P3HT) in order to better understand the origins of the reduced recombination rate and super-second order kinetics. Using pump-probe transient absorption spectroscopy (TA), we measure the polaron decay dynamics from the 10 ns to the 100 s timescale from 59-300 K. In spite of numerous studies concerning charge generation in neat P3HT and P3HT:PCBM blends, as well as charge recombination in P3HT:PCBM blends,[@piris2009; @guo2010a; @grzegorczyk2010; @howard2010; @ohkita2008] TA studies of the nongeminate recombination dynamics in neat P3HT are so far missing. Based on these results, we discuss the feasibility of several proposed models for nongeminate recombination in polymer–fullerene blends. Experimental Methods ==================== The experimental setup for steady state photo-induced absorption (PIA), as well as sample preparation, have been described elsewhere. [@delgado2009] P3HT was purchased from BASF (Sepiolid P200) and PCBM from Solenne. All materials were used without further purification. Solar cells prepared with these batches typically reach efficiencies over 3$\%$. [@schafferhans2010] All materials were dissolved in chlorobenzene at a concentration of 20 mg/ml. The films were deposited onto sapphire substrates by spin-coating and annealed at 140[[$^{\circ}$]{}]{}C for 10 min. Blends with a 1:1 weight ratio were studied. Films were prepared under a nitrogen atmosphere in a glove box. The thickness of the TA films was measured at approximately 300 nm by a profilometer. For TA experiments, samples were excited by a 5 ns pulse of a nitrogen/dye laser at a wavelength of 500 nm with a pulse energy of 25 J/cm$^2$. The generated polarons were probed using their characteristic absorption at 980 nm by an 80 mW cw laser. The decay of this absorption was measured using a FEMTO HCA-S-400M-IN preamplified InGaAs photodiode and recorded by a Tektronix oscilloscope.[@deibel2010review; @guo2010a] The change in optical density ($\Delta OD$) was computed from the transient signal and is directly related to the density of the absorbing species by the absorption cross section.[@kroeze2004] The hole mobility of neat P3HT films was also measured using the charge carrier extraction technique, OTRACE, as described in more detail elsewhere.[@baumann2012] OTRACE samples were prepared by spin-coating a solution of P3HT dissolved in chlorobenzene (30 mg/ml), resulting in 200 nm films as measured by a profilometer. For OTRACE measurements, the sample was directly transfered to a closed-cycle He cryostat without any exposure to air. A pulsed 10 W neutral white Rebel-LED was then used to generate charge carriers in the bulk of the P3HT film. The waveform was applied by an Agilent A81150A waveform generator, and the current transients were amplified by a FEMTO DHPCA-100 current amplifier and then recorded using an Agilent DSO90254A digital storage oscilloscope. Experimental Results ==================== Steady state photoinduced absorption ------------------------------------ In neat P3HT films, several species can coexist due to the lower efficiency of charge carrier photogeneration.[@deibel2010] At low temperatures, PIA spectra indeed exhibits bands due to several species as shown in Fig. \[fig:PIA\]. In addition to the polaronic features visible in the P3HT:PCBM blend films, there is a peak at 1170 nm (1.06 eV) that has been attributed to neutral species in P3HT.[@jiang2002a; @vanhal1999] At 30 K, the tail of this peak is overlapping with the *P2* polaron peak (see Fig. \[fig:PIA\] inset). Yet, at room temperature, the lifetime of those excitonic species becomes too short to contribute to PIA after 1 ns.[@piris2009] We find that their contribution to the absorption at 980 nm continuously decreases when increasing the temperature over 30 K and vanishes above 142 K (Fig. \[fig:PIA\]). ![(Color online) Steady-state PIA spectra of neat P3HT (normalized to absorption at 1.2 eV). The blue line indicates the probe wavelength used for TA; above 140 K, absorption at this wavelength is proportional to the density of polarons in both P3HT and P3HT:PCBM. Inset: Steady-state PIA spectra of neat P3HT (continuous line) and P3HT:PCBM blend (broken line) at a temperature of 30 K. \[fig:PIA\]](PIA.pdf){width="8cm"} In contrast, the density of the neutral species in P3HT:PCBM blends decays on the sub-nanosecond scale even at low temperatures.[@grzegorczyk2010] This decay is explained by efficient charge carrier photogeneration as revealed by the high external quantum efficiency measurements in solar cells based on this blend.[@brabec2001] It is therefore safe to assume that polarons are the only absorbing species at 980 nm in the blend over the time range $10^{-8}$ to $10^{-4}$ s. We conclude that, above 140 K, the absorption at 980 nm is representative of the polaron density in both the neat and blend films. Transient absorption: absorption cross section and photogeneration ------------------------------------------------------------------ TA was probed at 980 nm, which corresponds to the maximum of the so-called *P2* peak (see inset of Fig. \[fig:PIA\]). A number of spectroscopic studies including spin-sensitive methods have assigned this peak to P3HT polaron absorption.[@jiang2002a; @brown2001] The corresponding absorption cross section was determined as follows. For a P3HT:PCBM blend film excited by a 25 J/cm$^2$ pulse of 500 nm light, 10.3$\%$ of the incident light was reflected by the sample and 3.6$\%$ was transmitted, yielding an upper limit for the generated exciton density of 1.8$\times 10^{18}$ cm$^{-3}$. By solving the rate equations described by @howard2010,[@howard2010] including the nonlinear losses due to polaron–exciton annihilation at high excitation intensity, [@ferguson2011] and implementing a Gaussian exciton generation term to represent the laser pulse, we estimated the number of generated positive polarons to be (7.8$\pm$2.0)$\times 10^{17}$ cm$^{-3}$. We also obtained a polaron–exciton annihilation rate of (1.7$\pm$0.7)$\times 10^{-7}$ cm$^3$/s from the excitation intensity dependence of the initial change of optical density ($\Delta OD_0$). From the corresponding change in the optical density ($\Delta OD$), the absorption cross section of P3HT polarons at 980 nm in blend films was evaluated at (1.9$\pm$0.5)$\times 10^{-16}$ cm$^2$, which is in a similar range as recently reported for neat P3HT films.[@leijtens2013] The transient decays of the change in optical density for the neat P3HT and P3HT:PCBM blend films are shown in Fig. \[fig:transients\]. The initial change of optical density at 10 ns ($\Delta OD_0$), which includes geminate recombination and exciton–polaron annihilation, [@howard2010] is virtually temperature independent and is only slightly lowered at temperatures approaching 300 K due to a faster onset of nongeminate losses at higher temperatures. This finding is consistent with earlier reports of temperature independent charge carrier photogeneration in polymer–fullerene blends.[@pensack2009; @grzegorczyk2010; @street2010a] For both neat and blend films, the polaron decay beyond 10 ns is due to nongeminate recombination, with an increasing recombination rate for higher temperatures. In the following sections, we will focus on these nongeminate losses. ![(Color online) (a) Transient absorption decays in neat P3HT for different temperatures (solid lines). The dashed lines show fits including only second-order recombination and the dotted lines show fits accounting for contributions by both first and second-order decay. The asymptotes corresponding to a purely second-order decay are shown for comparison for 111 K, 91 K and 59 K (dashed lines). (b) Transient absorption decays in P3HT:PCBM (solid lines) and power law fits (dashed lines). Typical shape of first and second-order decays (dotted lines) are shown for comparison. \[fig:transients\]](transients.pdf){width="8cm"} Transient absorption: neat P3HT {#sec:exp-p3ht} ------------------------------- In order to gain a deeper understanding of the polaron dynamics, we compare the experimental decays to analytical models based on continuity equations. In the absence of any external contributions (injection or photoexcitation from the ground state) after exciton generation by the laser pulse at $t=0$, the continuity equation describing the total polaron density ($n$) is $$\frac{dn}{dt} = -R \label{eqn:continuity}$$ for $t>0$, where $R$ is the recombination rate. Although Langevin recombination (Eqn. (\[eqn:bimol\])) is the expected loss mechanism for separated polarons, we also consider a first order decay.[@deibel2010review] The sum of a first and a second-order term is able to perfectly fit the decays observed in neat P3HT at temperatures below 140 K, whereas at higher temperatures, the decays are found to be purely second-order. These findings are in agreement with the assignment of the absorption signal in neat P3HT at 980 nm to polarons at temperatures above 140 K and to a sum of contributions from both polarons and neutral species at lower temperatures. These neutral species could be triplet excitons,[@vanhal1999] interchain singlet excitons,[@jiang2002a] or polaron pairs [@howard2010] and are outside the scope of this article. As predicted by Langevin theory, the recombination of polarons in neat P3HT exhibits second-order kinetics as shown in Fig. \[fig:rec\_order\]b. In addition, the temperature dependence of the second-order recombination coefficient ($k_\text{br}$) obtained from the fits in Fig. \[fig:transients\]a is compared to Langevin recombination coefficients ($k_\text{L}$) calculated from temperature dependent mobility measurements in Fig. \[fig:rec\_order\]a. Assuming equal electron and hole mobilities and a dielectric constant of 3.5, Eqn. (\[eqn:prefact\]) was used to calculate the Langevin recombination rate coefficient from several different mobility measurements. The recombination coefficient derived from the transients is very similar to Langevin theory at 250 K when compared to coefficients derived from our OTRACE experiments and from previous CELIV measurements.[@mozer2005a] ToF measurements also indicate similar magnitudes and temperature dependencies as the OTRACE and CELIV measurements shown here.[@mozer2005a; @mauer2010] However, the rate coefficients determined from the transients demonstrate a much weaker temperature dependence than observed in mobility measurements. At 150 K the measured recombination coefficient is more than one order of magnitude greater than expected from Langevin theory, although it remains far below the calculated Langevin recombination rate using the temperature independent local mobility determined by time-resolved microwave conductivity (TRMC).[@grzegorczyk2010] ![(Color online) (a) Recombination coefficients extracted from neat P3HT transients (Fig. \[fig:transients\]) compared to Langevin coefficients calculated from experimental mobility measurements using Eqn. (\[eqn:prefact\]) (b) Apparent recombination order as function of temperature for neat P3HT and P3HT:PCBM blend films.[]{data-label="fig:rec_order"}](recombination_analysis.pdf) Transient absorption: P3HT:PCBM blend ------------------------------------- While the dynamics of charge recombination in neat P3HT appear Langevin-like, the transient absorption signal in the P3HT:PCBM blend exhibits a much slower decay (Fig. \[fig:transients\]b), which is similar to previous reports. [@guo2010a; @foertig2009; @shuttle2010] It is characterized by a reduced recombination rate that does, however, depend on time (or carrier concentration). The recombination order exceeds 2 already at room temperature, increasing to about 7 at 30 K as shown in Fig. \[fig:rec\_order\]b. Since the recombination mechanism is still assumed to be between one electron and one hole, the resulting experimentally determined nongeminate recombination rate is expressed as $$\begin{aligned} R_\text{exp} = k_{\text{exp}}(n) n^2 \propto n^{\lambda+1}, \label{eqn:R_exp}\end{aligned}$$ where $k_\text{exp}$(n) is the carrier concentration dependent rate coefficient and $\lambda+1$ is the recombination order. This form is convenient because the slope of the polaron decay on the log-log plots shown in Fig. \[fig:transients\] is equal to $-1/\lambda$. To produce a recombination order of $\lambda+1$, the recombination rate coefficient must then take on the form $$\begin{aligned} k_\text{exp}(n) \propto n^{\lambda-1}. \label{eqn:k_exp}\end{aligned}$$ Discussion ========== Multiple Trapping and Release (MTR) Model ----------------------------------------- It is well known that charge carrier trapping plays a significant role in P3HT and other organic semiconductors.[@schafferhans2008] Therefore, we approach the interpretation of our experimental data using the multiple-trapping-and-release (MTR) model.[@schmidlin1977; @noolandi1977; @oelerich2012] With this model, the overall charge carrier density ($n$) is split into two populations, free carriers ($n_c$) and trapped carriers ($n_t$). The free carriers are assumed to move at a speed defined by the free carrier mobility ($\mu_c$), and the trapped carriers are assumed to be immobile. Over time, trapped carriers are thermally excited to become free carriers, and free carriers relax into trap states. However, under steady state conditions, the ratio of free to trapped carriers is assumed to be constant. To start our analysis, we assume that there is a very low concentration of intrinsic dark carriers, that the concentrations of electrons and holes are equal due to the symmetric nature of photogeneration, and that mobilities of electrons and holes are equal. Neat P3HT: Langevin-like recombination -------------------------------------- To analyze the second-order decay observed in the neat P3HT measurements, it is then assumed that carrier recombination can only occur between free electrons and free holes, free electrons and trapped holes, or trapped electrons and free holes.[@kirchartz2011; @foertig2012; @deibel2013] As a result, the recombination rate is given by $$\begin{aligned} R \approx \frac{e}{\varepsilon} (2 \mu_{c} n_c^2 + 2 \mu_{c} n_c n_t) = \frac{e}{\varepsilon} 2 \mu_{c} n_c (n_c + n_t) \label{eqn:R_neat}\end{aligned}$$ We then define the fraction of free charge carriers, $\Theta$, where $\Theta=n_c/n$, resulting in a final recombination rate $$\begin{aligned} R \approx 2\frac{e}{\varepsilon}\mu_c \Theta n^2. \label{eqn:R_theta}\end{aligned}$$ As a result, the effective macroscopic mobility ($\mu$) is governed by the free carrier mobility and the fraction of free carriers, $\mu = \Theta\mu_c$. Under these conditions, Eqn. (\[eqn:R\_theta\]) is equivalent to Eqn. (\[eqn:bimol\]), and within the MTR model, $k_\text{L}$ can then be expressed as $$k_\text{L} = 2 \frac{e}{\epsilon}\mu_c \Theta. \label{eqn:MTRprefact}$$ Within this framework, we note that if $\Theta(n)$ is not constant, the macroscopic mobility would be expected to be carrier concentration dependent, resulting in super-second-order recombination. However, if $\Theta(n)$ is constant, the macroscopic mobility would be independent of the charge carrier concentration, and second-order recombination is expected. Considering the charge carrier dynamics measured here, we observe second-order decay between 140 and 300 K (Fig. \[fig:rec\_order\]), implying that neat P3HT can be well described by Eqn. (\[eqn:bimol\]) and that both the macroscopic mobility ($\mu$) and $\Theta$ are independent of the carrier concentration. We assume for now, in accordance with the findings of Oelerich et al.,[@oelerich2012] that P3HT has a Gaussian density of states (DOS) distribution. Then, within the framework of the Gaussian disorder model (GDM), in which $\mu(T) \propto \exp( -( 2\sigma/3k_BT)^2)$,[@bassler1993] we determined the standard deviation of the DOS ($\sigma$) to be 37 meV. Previously, a value of 56 meV was found for holes by photocurrent transient measurements on much thicker samples (c.f. Fig. \[fig:rec\_order\]a).[@mauer2010] Using the GDM, the macroscopic mobility in neat disordered materials has been considered previously by hopping master equation[@pasveer2005] and the MTR model.[@oelerich2012] For $\sigma=$37 meV and temperatures between 140 and 300 K, the mobility depends on the carrier concentration only if more than $10^{-3}$ of the states are occupied.[@pasveer2005] Below that limit, the mobility is predicted to be independent of the carrier concentration. In contrast, if neat P3HT has an exponential DOS, the mobility would be expected to depend on the carrier concentration in all regimes.[@oelerich2012] Therefore, the observed second-order decay implies a carrier concentration independent mobility that is consistent with a Gaussian DOS but not with an exponential DOS. This finding is consistent with the analysis of mobility measurements by Oelerich et al.[@oelerich2012] However, as shown in Fig. \[fig:rec\_order\], we find that the measured recombination coefficient has a weaker temperature dependence than expected from OTRACE and CELIV hole mobility measurements, although not temperature independent as shown in the TRMC measurements. TRMC measures the high frequency photoconductivity, which is assumed to result from the motion of free carriers and to be proportional to $n_c \mu_c$, in contrast to the macroscopic mobility ($\mu$) measured by the other techniques, which calculate the mobility based on long-range charge transport of all carriers. As a result, mobility values derived from TRMC are usually much higher, but it is unclear why the temperature dependence is so weak. We would expect that the free carrier concentration ($n_c$) should be temperature dependent and cause the TRMC signal to have a stronger temperature dependence. This discrepancy makes it difficult to rely on mobility measurements derived from TRMC experiments when describing mechanisms that require longer range charge transport until further studies clarify the nature of the mobility measured by TRMC. In any case, the experimentally determined rate coefficients are greater than the Langevin rates derived from macroscopic mobility measurements at lower temperatures and have a weaker temperature dependence. It is plausible that the mobility that characterizes the charge motion required for recombination is different than that for long range macroscopic charge transport. For example, charge trapping may be effectively shallower for transport on the [.17ex]{}10 nm length scale than on the [.17ex]{}100 nm length scale due to spatial homogeneity arising from the presence of crystalline and amorphous domains. Another possibility is that the electron and hole mobilities have different temperature dependencies. However, further detailed studies are needed to test these concepts. As a result, while Langevin theory works fairly well to describe nongeminate recombination near room temperature, questions remain as to why the temperature dependence is weaker than expected from macroscopic mobility measurements. Nevertheless, the recombination rate has no major reduction factors and is actually slightly greater than expected at lower temperatures. This implies that P3HT is sufficiently homogeneous such that mobile charge carriers can reach their recombination partners everywhere. Comparing to previous recombination measurements on neat P3HT films, our observation of second-order recombination dynamics in neat P3HT films is in contrast to the first-order decays observed in TRMC measurements by Ferguson et al.[@ferguson2011] In their study, they attribute the first-order decay to the presence of a significant dark carrier concentration ([.17ex]{}$10^{19}$ cm$^{-3})$. If the observed behavior is indeed due to the presence of dark carriers, our measurements suggest that the neat P3HT samples that we have tested have a significantly lower dark carrier concentration. Our observation of second-order kinetics indicates that the dark carrier concentration in our samples is less than the range of photogenerated carrier concentrations tested. For an estimated initial exciton concentration of [.17ex]{}$10^{18}$ cm$^{-3}$ due to the excitation laser pulse and an upper bound of [.17ex]{}10% carrier yield, we estimate the photogenerated carrier concentration tested here is in the range of [.17ex]{}$10^{15}$ cm$^{-3}$ to [.17ex]{}$10^{17}$ cm$^{-3}$. As a result, we estimate that the dark carrier concentration in our samples is less than [.17ex]{}$10^{15}$ cm$^{-3}$. P3HT:PCBM blends: Langevin recombination? ----------------------------------------- In contrast to neat P3HT, the P3HT:PCBM blend films clearly display slower recombination and super-second-order decay (Fig. \[fig:rec\_order\]b). Under these conditions, the reduction factor defined in Eqn. \[eqn:zeta\] and calculated using Eqn. \[eqn:bimol\] and Eqn. \[eqn:R\_exp\] becomes carrier concentration dependent with the form $$\zeta = \frac{R_\text{exp}}{R_\text{L}} = \frac{k_\text{exp}(n)}{k_\text{L}} \propto n^{\lambda-1}. \label{eqn:zeta2}$$ To understand the origin of this reduction factor, we discuss several previously proposed hypotheses. Based on the measurements presented here and in previous studies in the literature, we attempt to eliminate those that are inconsistent with the current state of experimental knowledge, highlight those that are still feasible, and finally direct researchers to areas where further measurements are needed. We emphasize that any well-suited model must be able to account for both the magnitude and the carrier concentration dependence of the reduction factor. Previously, it has been suggested that due to the presence of lamellar crystals in P3HT domains, which may promote two-dimensional transport, the resulting recombination behavior in P3HT:PCBM blends is more accurately represented by a two-dimensional Langevin recombination model.[@juska2009] However, our observation that the neat P3HT films, which also have lamellar crystalline domains, do not demonstrate these same characteristics suggests that two-dimensional transport is not a dominant factor and that a two-dimensional Langevin recombination model is not appropriate. In addition, kinetic Monte Carlo simulations implementing anisotropic mobility also conclude that the anisotropy effect is likely to be too weak to be the dominant factor in P3HT:PCBM blends.[@groves2008b] Another previously suggested explanation for the super-second order recombination kinetics is the presence of carrier concentration gradients near the electrodes in operational devices.[@deibel2009] While carrier concentration gradients may enhance this effect in devices, our observation here of similar kinetics on samples without electrodes suggests that is not likely be the dominant cause. With these hypotheses ruled out, we move now to a more detailed discussion of the remaining concepts in the following subsections. ### Effect of phase separation One important difference between traditional Langevin theory and the P3HT:PCBM blend system is the presence of a complex nanoscale phase separated morphology[@erb2005] that spatially limits the motion of the electrons and holes and limits the possible places where recombination can occur. The reduction factor has been previously attributed to the presence of phase separation.[@koster2006; @baumann2011] To assess the effect of phase separation, we derive and compare the expected recombination rate equations for a homogeneous blend and a phase separated blend with pure domains. For a homogeneous blend, similar to neat P3HT, in a system with charge carrier trapping due to energetic disorder,[@schafferhans2010; @foertig2012] the recombination rate can be approximated by Eqn. (\[eqn:R\_theta\]) and, equivalently, by Eqn. (\[eqn:bimol\]). As a result, the standard Langevin rate equation would be expected in a homogeneous blend. Moving now to a phase separated blend, carriers trapped in the interior of the domains are unable to undergo recombination, which should reduce the overall recombination rate. If we assume that very few charge carriers are trapped close to the interface, the recombination rate should be dominated by reactions between free electrons and free holes. In this framework, the resulting recombination rate is $$R \approx 2\frac{e}{\varepsilon} \mu_{c} n_c^2 \approx \Theta k_\text{L} n^2. \label{eqn:R_ps}$$ Here, the trapped charge carriers in the bulk of the domains are protected from recombination as long as they are trapped,[@baumann2011] lowering the overall recombination rate by a factor of $\Theta$. However, given the nanoscale dimensions of the domains ($\approx15$ nm),[@vanbavel2009; @pfannmoeller2011] there can actually be a large fraction of the P3HT volume near the interface. As a simple example, given a spherical domain with a 15 nm diameter, 35% of the volume is within 1 nm of the interface, and it can be expected that 35% of the trapped carriers can participate in recombination. Therefore, the amount of carriers trapped close to interface is, in fact, not likely to be negligible. By including recombination between free carriers and these carriers trapped near the interface, the recombination rate equation becomes $$R \approx \Theta k_\text{L} n^2 + \chi (1-\Theta) k_\text{L} n^2, \label{eqn:R_phi_i}$$ where $\chi$ is the interfacial volume fraction, the fraction of the donor and acceptor volume at the interface with respect to the total volume. Furthermore, if we assume that most carriers at any given time are trapped ($\Theta\ll1$) and that $\Theta\ll\chi$, then $$R \approx \chi k_\text{L} n^2. \label{eqn:R_phi}$$ As a result, this scenario predicts that the reduction factor is approximately equal to $\chi$. However, the magnitude of $\chi$ expected in a nanostructured morphology ($>10^{-1}$) is closer to unity than previously measured reduction factors ([.17ex]{}$10^{-3}$) at room temperature.[@deibel2008b] In addition, a simple phase separation model has no way of explaining the super-second order kinetics. As a result, we find it unlikely that phase separation inherently causes the apparent major deviations from Langevin theory. ### Carrier concentration dependence of mobility Assuming that carrier recombination is still encounter-limited as assumed in Langevin theory, the recombination rate should still depend on the carrier mobility. Up to now, we have assumed that the carrier mobility is independent of the carrier concentration, but a more complex carrier concentration dependence must be considered. @shuttle2010 have attempted to explain the super-second order decay by assuming a carrier concentration dependent mobility in which $\mu(n) \propto n^{\lambda-1}$.[@shuttle2010] Such behavior would only be expected if the materials were to have an exponential DOS. However, we have recently shown that this explanation may not generally hold,[@rauh2012] but we point out here that neither of these studies used a method that probes the charge carrier mobility directly. Therefore, these previous conclusions need to be verified. In addition, @savenije2011 compared TRMC and TA measurements on P3HT:PCBM thin films and concluded that the mobility in P3HT:PCBM blends is time independent on the timescale of tens of nanoseconds onwards, indicating a carrier concentration independent mobility for the range tested.[@savenije2011] However, as discussed in section IV.B on neat P3HT, TRMC mobility measurements may probe behavior that is significantly different than the more macroscopic mobility important for describing nongeminate recombination. As a result, further concentration dependent mobility studies are needed to completely rule out carrier concentration dependent mobility as a main cause of the super-second order recombination kinetics. If a carrier concentration dependent mobility is to be the dominant cause of the observed recombination kinetics, this relationship should be proportional to the concentration dependence of the experimental recombination prefactor, $k_\text{exp}$, shown in Fig. \[fig:k\_vs\_n\]. However, given that neat P3HT mobility is only weakly carrier concentration dependent, consistent with a Gaussian DOS, it is a reasonable assumption that at least the pure P3HT domains should demonstrate similar behavior. But even if the P3HT domains do have a Gaussian DOS, another possibility is that the PCBM domains have an exponential DOS and a mobility that dominates the recombination rate. However, space-charge limited current (SCLC) measurements on PCBM have indicated behavior consistent with a Gaussian DOS,[@mihailetchi2003] but SCLC measurements are also performed at much higher carrier concentrations than are present in working solar cells. To clarify this behavior further, carrier concentration dependent mobility measurements on neat PCBM films are needed as well. ![(Color online) Predicted carrier concentration dependence of the mobility when assuming super-second order recombination is caused only by a carrier concentration dependent mobility. \[fig:k\_vs\_n\]](k_vs_n.pdf) ### Effect of non-encounter limited recombination It has also been proposed that nongeminate recombination is not encounter limited as assumed in Langevin theory.[@hilczer2010; @ferguson2011] In this case, if the actual recombination mechanism itself is slow, the decay of charge carriers in the blend does not depend solely on the encounter probability. Instead of the electron and hole recombining immediately when reaching each other, they form an intermediate polaron pair state that can recombine after some time but can also re-dissociate into free charges. The experimental recombination rate is then defined $$R \approx -\frac{dn}{dt} = k_L n^2 - k_d [PP], \label{eqn:non-encounter}$$ where $k_d$ is the polaron pair dissociation rate and $[PP]$ is the polaron pair concentration. For this to have a major effect, the second term must be much larger than the Langevin term, which means the polaron concentration must persist over long timescales and the polaron pair dissociation rate must be fairly fast. To determine the conditions in which this would occur, we first describe the polaron pair rate equation as $$\frac{d[PP]}{dt} = k_L n^2 - k_r [PP] - k_d [PP], \label{eqn:polaron-pair_rate}$$ where $k_r$ is the rate of the final polaron pair recombination event by which the electron finally returns to the ground state. For polaron pairs to persist in the system, $d[PP]/dt$ must not be much less than zero, and in a special case, when the polaron pair concentration is constant ($d[PP]/dt=0$) and the reduction factor ($\zeta$) is very small, $$\zeta \approx \frac{k_r}{k_d} \label{eqn:zeta_non-encounter}$$ However, if the polaron pair concentration is changing over time, a more complicated expression is necessary. Nonetheless, given the highly efficient charge separation that occurs in P3HT:PCBM blends, it is likely that the dissociation rate is significantly faster than the recombination rate. As a result, while a significant reduction in the observed polaron decay rate could be expected, this model still does not explain the origins of the super-second-order kinetics. ### Effect of interfacial states Another important aspect to consider is the presence of a third mixed phase with unique materials properties compared to the pure phases. P3HT:PCBM blends have been shown to have a more complex morphology than simply pure donor and pure acceptor domains,[@collins2010] and the presence of a mixed amorphous phase has been clearly identified.[@pfannmoeller2011] In addition, the presence of deep states that go beyond a superposition of the tail states of the separate pure materials has been experimentally measured.[@vandewal2009a] These deeper states originate from the close interaction of donor and acceptor molecules at the heterointerface. Taking this into account, Street et al. have proposed recombination via interfacial states.[@street2010; @deibel2010b; @street2010b] With this in mind, we consider a model in which pure domains are separated by an interfacial mixed region that contains a DOS that is different from those present in either of the pure domains. We note that implementing interfacial mixing without a separate DOS simply increases $\chi$, as previously defined in subsection 1, and cannot explain the observed behavior. With separate DOS distributions, however, it is plausible that the macroscopic mobility would be dominated more so by charge transport within the pure domains containing a Gaussian DOS, as indicated by our measurements on neat P3HT, but that the actual recombination event is governed mainly by the spatial and energetic properties of the interfacial regions containing an exponential DOS, which has been indicated by defect spectroscopy.[@foertig2012; @presselt2012] To determine the expected recombination rate in this more complex scenario, we need to consider two separate contributions to the recombination rate, the behavior of the carriers in the pure domains and in the interfacial regions. First, we assume that the majority of the carriers are trapped and that the contribution from free–free recombination is negligible. As a result, the dominant recombination mechanism occurs when free carriers from the pure domains ($n_{c,p}$) are transported to the interfacial regions and recombine with carriers that are already present within the interfacial regions ($n_i$). In this case, the resulting recombination rate becomes $$R \approx 2\frac{e}{\varepsilon}\mu_{c,p} n_{c,p} n_i, \label{eqn:R_interfacial}$$ where $\mu_{c,p}$ is the mobility of the free carriers in the pure phases. Rewriting this in terms of the overall carrier concentration ($n$), where $n=n_{c,p}+n_{t,p}+n_{c,i}+n_{t,i}$, the recombination rate becomes $$R \approx \Phi (1-\Phi) k_L n^2, \label{eqn:R_interfacial_n}$$ where $\Phi$ is the fraction of carriers in the pure phase with respect to all carriers, $n_p/(n_p+n_i)$, and $k_L$ is derived from the mobility of the pure phases. This model can explain both the magnitude and the carrier concentration dependence of the reduction factor when $\Phi$ is large and carrier concentration dependent. Given studies that have indicated an energetic driving force for carriers to diffuse from amorphous mixed regions to more ordered pure domains,[@mcmahon2011; @jamieson2012] it is probable that $\Phi$ would be large, and given different DOS distributions, it is possible that the density of occupied states would be populated in different proportions at different overall carrier concentrations. Here, to give super-second order kinetics, interfacial states would have to fill up proportionally faster than the states in the pure phases when increasing the overall carrier concentration. Further theoretical and experimental studies are needed to test this model in more detail, but given the current state of knowledge and the critical analysis presented here, we find it to provide the most complete explanation to date. Conclusions =========== To conclude, we have used transient absorption spectroscopy to monitor the nongeminate polaron decay in neat P3HT and P3HT:PCBM blend films. In the neat polymer, we observed Langevin-like recombination at temperatures above 140 K with second-order kinetics and a recombination coefficient that is slightly less temperature dependent than macroscopic mobility measurements. To analyze the results, we have used a multiple trapping and release (MTR) model to derive the expected recombination rate equations. For neat materials, the MTR model predicts recombination dynamics consistent with Langevin theory, and the neat P3HT measurements appear to be mostly consistent with this model, aside from the weaker temperature dependence. Most importantly, though, no significant reduction factor was observed in neat P3HT, and dark carriers were not found to play a role in the recombination kinetics. In contrast, the recombination dynamics in the blend films were characterized by a reduced recombination rate and super-second-order recombination kinetics. To narrow down the possible explanations for this behavior, we first eliminated several different models previously proposed, including a two-dimensional Langevin model and carrier concentration gradients. To understand the effect of phase separation, the MTR model was used to derive the expected recombination rate for a homogeneous and phase separated blend. However, phase separation alone was shown to only slightly reduce the recombination rate. In addition, we argued that the mobility is not likely to be strongly carrier concentration dependent but have identified that further measurements are needed to rule it out as the sole contributor to the higher order recombination kinetics. We then considered the idea that nongeminate recombination is not encounter limited as assumed by Langevin theory and found that the recombination rate could indeed be significantly reduced from the rate predicted by Langevin theory. However this would still be unable to explain the origins of the recombination order. Finally, we considered the effects of interfacial states, which have been previously identified and proposed to play a significant role in the recombination behavior. Using the MTR model, we then derived the recombination rate expected when there are pure domains with a Gaussian density of states that are separated by a mixed interfacial phase with an exponential density of states. This scenario is expected to produce both a reduced recombination rate and super-second-order recombination kinetics. While still a qualitative model, we propose that it is most consistent with the available experimental data to date. The current work is supported by the Bundesministerium f[ü]{}r Bildung und Forschung in the framework of the GREKOS project (contract no. 03SF0356B) and the European Commission through the Human Potential Program (Marie-Curie RTN SolarNType contract no. MRTN-CT-2006-035533) and the Deutsche Forschungsgemeinschaft, DFG under the contract INST 93/623-1 FUGG. C.D. gratefully acknowledges the support of the Bavarian Academy of Sciences and Humanities.

--- author: - 'R. T. Garrod' - 'E. Herbst' bibliography: - 'robbib\_2006.bib' date: 'Received xxx 00, 0000; accepted xxx 00, 0000' nocite: - '[@vandertak03a]' - '[@viti06a]' title: 'Formation of methyl formate and other organic species in the warm-up phase of hot molecular cores' --- [xxx]{} [The production of saturated organic molecules in hot cores and corinos is not well understood. The standard approach is to assume that, as temperatures heat up during star formation, methanol and other species evaporate from grain surfaces and undergo a warm gas-phase chemistry at 100 K or greater to produce species such as methyl formate, dimethyl ether, and others. But a series of laboratory results shows that protonated ions, typical precursors to final products in ion-molecule schemes, tend to fragment upon dissociative recombination with electrons rather than just ejecting a hydrogen atom. Moreover, the specific proposed reaction to produce protonated methyl formate is now known not to occur at all. ]{} [We utilize a gas-grain chemical network to probe the chemistry of the relatively ignored stage of hot core evolution during which the protostar switches on and the temperature of the surrounding gas and dust rises from 10 K to over 100 K. During this stage, surface chemistry involving heavy radicals becomes more important as surface hydrogen atoms tend to evaporate rather than react. ]{} [Our results show that complex species such as methyl formate, formic acid, and dimethyl ether can be produced in large abundance during the protostellar switch-on phase, but that both grain-surface and gas-phase processes help to produce most species. The longer the timescale for protostellar switch-on, the more important the surface processes.]{} [xxx]{} Introduction ============ Various large molecules including methyl formate (HCOOCH$_3$) have been detected in a number of hot molecular cores and corinos [@blake87a; @hatchell98a; @nummelin00a; @cazaux03a; @bottinelli04a]. These are chemically rich regions in dense interstellar clouds which are warmed by an associated protostar and have typical densities of 10$^6$ – 10$^8$ cm$^{-3}$ and temperatures on the order of 100 K. Observations suggest methyl formate abundances can be as high as $\sim$10$^{-8} \times n$(H$_2$) in these regions. The standard view of the complex chemistry that pertains in hot cores requires that dust grains build up icy mantles at early times, when the temperature is low ($\sim$10 K) and the core is in a state of collapse [@brown88a]. Later, the formation of a nearby protostar quickly warms the gas and dust, re-injecting the grain mantle material into the gas phase, and stimulating a rich chemistry. Various hot core models [@millar91b; @charnley92b; @caselli93a; @charnley95a] have shown that large oxygen-bearing molecules such as methyl formate may be formed during the “hot” stage when large amounts of methanol are released from the grain mantles. Methanol is easily protonated by H$_3^+$ ions, and the resulting protonated methanol was thought to react with formaldehyde to produce protonated methyl formate. Subsequent dissociative recombination with electrons would then complete the process. In this way, large amounts of methyl formate could be produced on timescales of 10$^4$ – 10$^5$ yr after the evaporation of the ices. However, this gas-phase mechanism has now become rather more contentious. Recent quantum chemical calculations carried out by [@horn04a] have shown that the first stage of the process – reaction between H$_2$CO and CH$_3$OH$_2^+$ – is highly inefficient at producing protonated methyl formate. This is due to a large potential barrier between isomeric complexes, resulting in an activation energy barrier for the reaction on the order of $\sim$15,000 K. Horn et al. suggested other gas-phase routes to produce protonated methyl formate as well as two other complexes, adopting moderate efficiencies for each to recombine with electrons to produce HCOOCH$_3$. Their chemical models (at 100 K) showed that even with optimistic rates, these reactions were unable to reproduce observed abundances. This considerable difficulty is compounded by increasing experimental evidence regarding the efficiencies of the dissociative recombinations of large saturated ions. [@geppert05a; @geppert06a] have investigated the branching ratios of the recombination of protonated methanol, finding that the fraction of recombinations producing methanol is considerably lower than the 50 – 100 % typically assumed in chemical models [see e.g. @leteuff00a]. [@geppert06a] find that just $3 \pm 2$ % of recombinations result in methanol, with more than 80 % resulting in separate C and O groups. This work is in agreement with many previous storage-ring experiments on the products of the dissociative recombination reactions involving other H-rich species, which are dominated by three-body channels. The strong implication for such larger molecules as methyl formate and dimethyl ether, which were hitherto expected to form from their protonated ions, is that their production via recombination must be, at best, similarly inefficient. Therefore, the gas-phase routes to such molecules must be considered incapable of reproducing astronomically observed abundances at temperatures of $\sim$100 K. Numerous allusions have been made [e.g. @charnley95a; @ehrenfreund00a; @cazaux03a; @peeters06a] to the possibility that methyl formate is produced on dust-grain surfaces at some point in the evolution of a hot core when dust temperatures are relatively low. The reaction set of [@allen77a], a list of exothermic and plausible grain-surface reactions, includes one such association between the surface radicals HCO and CH$_3$O, producing HCOOCH$_3$. Many of the reactions listed in this set have been incorporated into the gas-grain chemical model of The Ohio State University (OSU) [@hasegawa92a], but only very few of those involve more than one large radical. This is because at the low temperatures of dense interstellar clouds ($\sim$10 K), such large radicals would be effectively stationary on the grain surfaces, prohibiting their reaction with anything but small, relatively mobile species like atomic hydrogen. At the high temperatures encountered in hot molecular cores, these radicals would no longer be present on grain surfaces at all, having thermally evaporated. However, in the light of the failure of gas-phase chemistry to reproduce methyl formate observations, we must ask how these radicals should behave at intermediate temperatures. Clearly such a physical state must exist in the timeline of hot core evolution, but is it long enough to significantly influence the behaviour of the chemistry? Since most chemical models involve only a rudimentary treatment of grain surfaces, and implement only a gas-phase chemistry, few investigations of the warm-up phase of hot cores have been carried out. [l c]{} Species & $E_D$ (K)\ H & 450\ H$_2$ & 430\ OH & 2850\ H$_2$O & 5700\ N$_2$ & 1000\ CO & 1150\ CH$_4$ & 1300\ HCO & 1600\ H$_2$CO & 2050\ CH$_3$O & 5080\ CH$_3$OH & 5530\ HCOOH & 5570\ HCOOCH$_3$ & 6300\ CH$_3$OCH$_3$ & 3150\ [@viti99a] explored the effects of the selective, time-dependent re-injection of grain mantle-borne species, using the same thermal evaporation mechanism as employed in the OSU gas-grain code [@hasegawa92a]. They used a linear growth in the temperature of the hot core, starting from 10 K, over periods corresponding to the switch-on times for hydrogen burning of variously sized protostars, as determined by [@bernasconi96a]. They investigated the effect of selective re-injection of mantle species on gas-phase chemical evolution. The follow-up work of [@viti04a] used results from temperature programmed desorption (TPD) laboratory experiments to determine temperatures at which particular species should desorb, taking into account phase changes in the icy mantles. Whilst these investigations shed new light on the dependence of the chemistry on (currently ill-defined) physical conditions, they necessarily ignored the effects of the grain-surface chemistry prior to re-injection. In this study, we use the OSU gas-grain code to model the warm-up phase of hot-core evolution, with the primary aim of reproducing observed HCOOCH$_3$ abundances. The code was previously used to model hot cores by [@caselli93a], requiring constant grain temperatures. The code is now capable of dealing with the variable temperatures we expect in the warm-up phase. The grain chemistry allows for time-dependent accretion onto and evaporation (thermal and cosmic ray-induced) from the grain surfaces. Surface-bound species may react together – the majority of reactions are hydrogenations but reactions may take place between other atoms, and smaller radicals such as OH and CO, at rates determined by the grain temperature and diffusion barriers. Following the work of [@ruffle01a] we allow surface-bound molecules to be photodissociated by the (heavily extinguished) interstellar radiation field, and by the cosmic-ray induced field of [@prasad83a]. ------------------------------------------------------------------------------------- ----------------------- -------------------------- ----------------------- Reaction $k(300$ K$)$ $T$-dependence, $\alpha$ $k(10$ K$)$ (cm$^3$ s$^{-1}$) (cm$^3$ s$^{-1}$) \(4) CH$_3$OH$_{2}^{+}$ + H$_2$CO $\rightarrow$ CH$_3$OH$_2$OCH$_{2}^{+}$ + $h \nu$ $3.1 \times 10^{-15}$ -3 $8.4 \times 10^{-11}$ \(5) H$_2$COH$^+$ + H$_2$CO $\rightarrow$ H$_2$COHOCH$_{2}^{+}$ + $h \nu$ $8.1 \times 10^{-15}$ -3 $2.2 \times 10^{-10}$ \(6) CH$_{3}^{+}$ + HCOOH $\rightarrow$ HC(OH)OCH$_3^{+}$ + $h \nu$ $1.0 \times 10^{-11}$ -1.5 $1.6 \times 10^{-9}$ $^{a}$ $k(T) = k(300$ K$) \times (T/300)^{\alpha}$ ------------------------------------------------------------------------------------- ----------------------- -------------------------- ----------------------- \ To this basic chemical model we add three new reactions from [@allen77a], allowing for the formation of methyl formate, dimethyl ether (CH$_3$OCH$_3$) and formic acid (HCOOH) from grain-surface radicals. We employ a new set of desorption and diffusion barriers representing water ice. We also investigate the effects of the warm-up on the gas-phase reactions set out in [@horn04a]. Those reactions show a strong inverse temperature dependence, implying much larger rates at temperatures less than 100 K. Model ===== Physical Model -------------- We adopt a two-stage hot-core model: in stage 1, the gas collapses from a comparatively diffuse state of $n_H = 3 \times 10^{3}$ cm$^{-3}$, according to the isothermal free-fall collapse mechanism outlined in [@rawlings02a] and derived from [@spitzer78a]. This process takes just under 10$^6$ yr to reach the final density of $10^{7}$ cm$^{-3}$. In stage 2 the collapse is halted, and a gradual increase in dust and gas temperatures begins. We base this phase of our physical model on the approach of [@viti04a]. They used the observed protostellar luminosity function of [@molinari00a] to derive effective temperatures throughout the accretion phase of a central protostar, up until it reaches the zero-age main sequence (ZAMS), at which hydrogen burning takes place. They approximated the effective temperature profile as a power law with respect to the age of the protostar, then assumed that the temperature of the nearby hot core material should follow the same functional form. The contraction times determined by [@bernasconi96a] were used to provide a timescale for this process, on the order of 10$^4$ to 10$^6$ yr for stars of 60 to 5 solar masses, respectively. Here, we adopt power law temperature profiles for the hot core material, with a power index of either 1 (linear time-dependence) or 2. We allow the temperature to increase from 10 – 200 K over three representative timescales: $5 \times 10^{4}$, $2 \times 10^{5}$ and $1 \times 10^{6}$ yr, approximating high-mass, intermediate, and low-mass star-formation, respectively. The high gas density of 10$^7$ cm$^{-3}$ implies a strong collisional coupling between the dust and gas temperatures, hence we assign them identical values at all times. Surface Chemistry ----------------- For the grain-surface chemistry, we employ a new set of diffusion energy barriers and desorption energies for each species. Because the diffusion barriers determine the rate at which a species can hop between adjacent binding sites on the surface, the sum of these rates for two reaction partners determines the reaction rate [@hasegawa92a]. Similarly, desorption energies determine the rate of thermal and cosmic ray heating-induced desorption [see @hasegawa93a]. Diffusion barriers are set as a fixed fraction of the desorption energy of each species, as in previous work [e.g. @hasegawa92a; @ruffle00a]. Both the diffusion rates and thermal desorption (evaporation) rates are exponentially dependent on the grain temperature, according to a Boltzmann factor. Of the desorption energies that have been used in previous applications of the gas-grain code, some (including those for C, N, O, and S) correspond to a water ice surface, but most are given for carbonaceous or silicate surfaces. Here we construct a set of values corresponding to water ice for all the surface species in the gas-grain code, following [@cuppen06a]. These authors analysed various experimental data to obtain best values for H, H$_2$, O$_2$ and H$_2$O on non-porous ice. We use these values and add to them data provided by Dr. M. P. Collings (private communication), which correspond to the TPD results obtained by [@collings04a]. This gives us values for CO, CO$_2$, CH$_3$OH, C$_2$H$_2$, HCOOH, N$_2$, NH$_3$, CH$_3$CN, SO, SO$_2$, H$_2$S, and OCS. [We ignore the value for CH$_4$, which was subject to experimental problems – see @collings04a]. We additionally assume that the OH value is half that for H$_2$O, to take account of hydrogen bonding. We use these skeletal values (along with unchanged atomic values, except for H) and extrapolate to other species by addition or subtraction of energies from members of the set, making the assumption that the energies are the sum of the fragments that compose the molecules. The net result is that most species possess desorption barriers around 1.5 times larger than in our previous models, whilst those species that should exhibit hydrogen bonding to the surface can have barriers as much as five times as large. Table \[tab1\] shows a selection of important surface species and their desorption energies, $E_D$. A complete table of energy parameters is available from the authors on request. In previous models, diffusion barriers were set to either 0.3 or 0.77 of the desorption energies for most species (not atomic or molecular hydrogen); see [@ruffle00a]. Here we adopt a blanket value for all species of 0.5 $E_D$, based on an assortment of data on icy surfaces and our best estimate of the role of surface roughness and porosity. ------------------------------------ ------------------------- Species $i$ $n_{i}/n_{H}$ $^{\dag}$ H$_2$ $0.5$ He $0.09$ C $1.4(-4)$ N $7.5(-5)$ O $3.2(-4)$ S $1.5(-6)$ Na $2.0(-8)$ Mg $2.55(-6)$ Si $1.95(-6)$ P $2.3(-8)$ Cl $1.4(-8)$ Fe $7.4(-7)$ $^{\dag}$ $a(b) = a \times 10^{b}$ ------------------------------------ ------------------------- : Initial abundances of H$_2$ and elements.[]{data-label="tab3"} The reaction set we employ derives from that used by [@ruffle01a], which includes grain-surface photodissociation. To that set we add the following three surface reactions from [@allen77a]: $$\mbox{HCO} + \mbox{CH}_{3}\mbox{O} \rightarrow \mbox{HCOOCH}_{3},$$ $$\mbox{CH}_{3} + \mbox{CH}_{3}\mbox{O} \rightarrow \mbox{CH}_{3}\mbox{OCH}_{3},$$ $$\mbox{HCO} + \mbox{OH} \rightarrow \mbox{HCOOH}.$$ These reactions we deem to be the most likely surface routes to each of the three molecules. During the cold phase, the radicals HCO and CH$_3$O are formed by successive hydrogenation of CO, which mostly comes directly from accretion from the gas phase. The reactions H + CO $\rightarrow$ HCO and H + H$_2$CO $\rightarrow$ CH$_3$O each assume an activation energy of 2500 K [@ruffle01a]. Large reserves of H$_2$CO and CH$_3$OH may build up through the hydrogenation process. However, at warmer temperatures such processes should be less efficient, due to the faster thermal evaporation of atomic hydrogen. Under these conditions, processes which break down H$_2$CO and CH$_3$OH may become important. Modified Rates -------------- Applications of the gas-grain code to cold dark clouds in previous publications have made use of “modified” rates in certain situations; specifically, problems arose when solving for species with average grain-surface abundances on the order of unity or below [@caselli98a]. Comparison with results from simple Monte Carlo techniques [@tielens82a] indicated that atomic hydrogen in particular reacted too quickly when using the rate-equation method employed in the gas-grain code. In effect, the hydrogen was reacting more quickly than it was actually landing on the grain surfaces. This problem was fixed by limiting the rate of diffusion of the hydrogen to no more than the rate of accretion. Additionally, when in the “evaporation limit” (i.e. when the rate of evaporation exceeds the rate of accretion), the diffusion rate was limited to the rate of H evaporation instead, since in this case it is the evaporation rate which is the main determinant of surface hydrogen abundance. Whilst [@shalabiea98a] applied the modified rate technique to all species, the effect was found to be small except for atomic H, at the low temperatures of dark clouds ($\sim$$10$ K). The analysis by [@katz99a] of earlier experimental work subsequently showed that surface hydrogen hops between binding sites thermally, rather than via quantum tunnelling, indicating a much reduced diffusion rate for atomic hydrogen. The use of this result by [@ruffle00a; @ruffle01a] showed that the need for modified rates even for atomic hydrogen was marginal. {width="8.5cm"} {width="8.5cm"} However, in this work, we deal with much higher temperatures – at which even the heaviest species can become highly mobile. This means that any species could react at a faster rate than it is accreted or evaporated when some critical temperature is reached. But this does not necessarily mean that modified rates should be invoked for all species and reactions. Many of the heavy radicals in which we are interested here are formed mainly by surface reactions, and are therefore not governed by accretion in the way that many species are whilst at low temperatures. Also, some species can be produced by cosmic ray photodissociation of an abundant surface molecule. These complications suggest that any need to apply the modified rates is highly specific, both to individual species and to particular brief periods for those species. Added to this is the fact that the modified rates are untested against Monte Carlo codes at temperatures much greater than 20 K, meaning that there is no certain need to modify the rates at all in those cases. Hence, we modify only reactions involving atomic hydrogen. It is the only species that requires modification at less than 20 K, and is easily evaporated above 20 K, making it less important in the surface chemistry at such temperatures. As a result, the modified rates mainly come into play in the low-temperature collapse phase. The rates of surface reactions involving atomic hydrogen are modified in the manner outlined by [@ruffle00a]. Additionally, we modify the rates of hydrogen reactions that have activation energy barriers, following the method of [@aikawa05a]. This addition is necessary because at the high densities involved, the quantities of reactant that build up can become large enough that, even with the presence of the activation energy barrier, the rate at which hydrogen reacts is greater than the rate at which it is being deposited on the surfaces. To avoid such a run-away hydrogen chemistry, if the product of the activation barrier term, the surface abundance of the (non-hydrogenic) reactant, and the diffusion rate, is greater than the faster of the rates of accretion and evaporation of atomic hydrogen, then that product is replaced in the rate equation with the accretion or evaporation rate. For a full discussion of this issue see [@caselli02a]. Gas-Phase Chemistry ------------------- Table \[tab2\] shows reactions previously included in [@horn04a] that lead, without activation energy, to protonated methyl formate and to two other methyl formate precursors. Two of the ionic products require significant structural re-arrangement in a subsequent dissociative recombination reaction to produce methyl formate. The rate coefficient given to the first reaction in the table (labelled 4) is unchanged; to the other two (5 and 6) we attribute somewhat weaker temperature dependences due either to competition or a better estimate. At a temperature of 10 K, all three reactions are significantly faster than at the 100 K at which they were tested by Horn et al.; reaction (6) reaches approximately the collisional (i.e. maximum) rate coefficient at this temperature. Hence, we expect contributions from these reactions to be strong during the early stages of the warm-up phase, if the abundances of the reactants are significant. We allow the products of reaction (6) to recombine with electrons to form methyl formate in 5 % of reactions, two times lower than assumed by Horn et al. The protonated products of reactions (4) and (5) are allowed to form methyl formate in just 1 % of recombinations, because each of these channels would require a more drastic structural change within the complex. We also include approximately 200 gas-phase reactions from the UMIST ratefile [@leteuff00a] which were not present in the (predominantly low temperature) OSU reaction set. These reactions have activation energies, but the greater temperatures seen in hot cores justify their inclusion here. The initial gas-phase matter is assumed to consist of atoms with the exception of molecular hydrogen. Shown in table \[tab3\], these values derive from those selected by [@wakelam06a] from the most recent observational elemental determinations for diffuse clouds. Since the high temperatures attained in hot cores are capable of driving off the entirety of the granular icy mantles, we choose relatively undepleted (i.e. diffuse cloud) values for the lighter elements, in contrast with previous applications of the gas-grain code [most pertinently, @caselli93a]. We allow heavier elements to be depleted by an order of magnitude, on the assumption that the remainder is bound within the grain nuclei. This also ensures that fractional ionizations stay at appropriate levels, defined by [@mckee89a] as $X(e) \simeq 10^{-5} n($H$_{2})^{-1/2}$. The behaviour of sulphur on grain surfaces is not fully understood, and the form it takes on/in the grains is not known; observations have so far failed to detect H$_2$S on grains, and some authors suggest OCS may be the dominant form [@vandertak03a]. Moreover, the levels of sulphur-bearing species detected in the gas phase of hot cores are typically not as high as the levels detected in diffuse clouds [see e.g. @wakelam04a]. We therefore deplete sulphur by an order of magnitude from the values of [@wakelam06a]. Results ======= Stage 1: Collapse ----------------- All models investigated here use the same stage 1 collapse model, hence the initial conditions for the warm-up phase are the same in all cases. Figure \[figure1\] shows the evolution of some important gas-phase and grain-surface species, with respect to $n_H$, during the collapse from $3 \times 10^{3}$ – $10^{7}$ cm$^{-3}$. ![Temperature profiles for a protostellar switch-on time of $t=2 \times 10^{5}$ yr, adopting linear ($T_1$) and $t^2$ time dependences ($T_2$).[]{data-label="figure2"}](5560f2.eps){width="8.5cm"} Initially, the collapse is slow, and most of the significant increase in density takes place during the final $5 \times 10^{5}$ yr. Over this period, we see that gas-phase reactions become much faster due to the greater density, whilst accretion rates also increase, allowing for an accelerated surface chemistry. Large amounts of H$_2$O, CO, CH$_4$, H$_2$CO, CH$_3$OH, H$_2$S and NH$_3$ have built up on the grain surfaces by the end of the collapse. Besides water ice, NH$_3$ and CH$_4$ make up a large and fairly constant proportion of the ice. CH$_4$ ranges from $\sim$10 – 24 % of total ice throughout the collapse phase; NH$_3$ is steady at around 15 %. Water, methane and ammonia ices are formed by successive hydrogenation of O, C and N respectively. Carbon monoxide is predominantly formed in the gas phase and then accreted onto surfaces. At the very end of the collapse, CO deposition outpaces H$_2$O formation, due to the shortened supply of atomic oxygen. The model suggests that the outer 50 or so monolayers of ice should have a CO:H$_2$O ratio of approximately 6:4. Formaldehyde, which is formed by hydrogenation of CO, should comprise around 5 – 10 % of these outer layers. The calculated abundances of ices at the end of the collapse are in reasonable agreement with IR observations [e.g. @nummelin01a] except for CO$_2$, which is greatly underproduced at 10 K. Solutions to this problem have been advanced by [@ruffle01b]. Observed CO$_2$ percentages (in dark cloud ices) are typically the same as or a little less than CO levels. Hence, even with efficient CO$_2$ production, there should not be a drastic effect on CO, H$_2$CO, or CH$_3$OH ice abundances in the cold phase. Once formed, CO$_2$ does not take part in further reactions, but may be photodissociated. Stage 2: Warm-up Phase ---------------------- We first explore the behaviour of a “standard” physical and chemical model, before discussing other variations and free parameters. Figure \[figure2\] shows two temperature profiles used in the warm-up phase of the model; our “standard” model uses the $\Delta T_2$ profile with a protostellar switch-on time of $2 \times 10^{5}$ yr, corresponding to the final time acheived in the model, $t_f$. All of reactions (1) – (6) are enabled. Figure \[figure3\] shows plots of pertinent gas-phase and grain-surface species over the final “decade” of evolution; i.e. $2 \times 10^{4}$ – $2 \times 10^{5}$ yr. Over this period, temperatures begin to rise significantly above 10 K, and ultimately reach the final value of 200 K. {width="8.5cm"} {width="8.5cm"} {width="8.5cm"} {width="8.5cm"} ### Methyl Formate, Formic Acid, and Dimethyl Ether Methyl formate is formed in significant quantities, both in the gas phase and on grain surfaces. Figure \[figure3\]a shows the gas-phase abundance at late times – a double-peaked structure is apparent. The first, lesser, peak comes about solely from gas-phase formation via reaction (5). Formaldehyde injected into the gas phase from the grain surfaces at this time is protonated by reaction with H$_3$$^+$ and HCO$^+$ ions. At $t=8 \times 10^{4}$ yr, a temperature of $\sim$40 K obtains, giving a rate coefficient for reaction (5) of $3.4 \times 10^{-12}$ cm$^3$ s$^{-1}$. This is more than an order of magnitude larger than the rate at 100 K [cf. @horn04a], a temperature typical of hot core chemical models. These factors combine to produce just under 10$^{-8}$ with respect to H$_2$ of methyl formate; however, this abundance of HCOOCH$_3$ does not manifest itself fully in the gas phase, since most of it is quickly accreted onto the grains. (The high density produces an accretion timescale of $\sim$2000 yr). This causes the sharp fall-off in gas-phase abundance. Reactions (4) and (6) (see table \[tab2\]) do not contribute significantly to methyl formate production at any stage, firstly because their reactants are not typically present in large quantities at the same times; and secondly, because when the reactants H$_2$CO, HCOOH, and the methanol required as a precursor for reaction (4) are all abundant, temperatures are much higher (around 100 K), and the association rates are consequently much lower. The second, final, peak in the gas-phase methyl formate abundance arises from the evaporation of the surface species. Figure \[figure3\]b plots the grain-surface abundance of HCOOCH$_3$. Although not perceptible from the plot, there is a subtle shift in the formation route as the methyl formate abundance grows. The initial rise at around $5 - 6 \times 10^{4}$ yr is brought about by the surface reaction (1) between the radicals HCO and CH$_3$O. This rise takes place when CO has strongly evaporated, whilst the elevation of the grain-surface temperature increases the mobility of OH and H$_2$CO, causing the rate of the reaction OH + H$_{2}$CO $\rightarrow$ HCO + H$_{2}$O to greatly increase. (The OH radical is produced by the photodissociation of water ice, initiated by cosmic ray-induced photons). The mobility of the resultant HCO has also greatly increased. At these warmer temperatures, the hydrogenation of CO has ceased to be important to HCO formation, due to the lower abundance of atomic H (caused by its faster thermal evaporation). However, when CO strongly evaporates at $\sim 20$ K, the reaction CO + OH $\rightarrow$ CO$_2$ + H – the major destruction route of OH – is removed. This greatly increases the OH abundance, allowing much HCO to be formed. At this point in time, although atomic H is more scarce than at lower temperatures, the reaction H + H$_{2}$CO $\rightarrow$ CH$_{3}$O, is still the dominant mechanism for CH$_{3}$O formation. H$_2$CO is abundant on the grains, having built up during the collapse phase. When significant amounts of H$_2$CO begin to evaporate, at $\sim$$8 \times 10^{4}$ yr ($T \simeq 40$ K), HCO and CH$_{3}$O production is reduced and surface formation of HCOOCH$_3$ is inhibited. At this point, reaction (5) starts to have a strong effect due to the H$_2$CO now present in the gas phase, and much of the methyl formate produced is then accreted. From this time until HCOOCH$_3$ evaporates, the two processes operate in tandem, with the gas-phase/accretion mechanism dominant but gradually decreasing in influence. The gas-phase/accretion route is responsible for the formation of around 25 % of the total HCOOCH$_3$ present on the grain surfaces before evaporation. {width="8.5cm"} {width="8.5cm"} At early times, HCOOH is formed in the gas phase from the dissociative recombination of HCOOH$_2^+$, which forms by radiative association of HCO$^+$ and H$_2$O. However, when CH$_4$ evaporates at around $5 - 6 \times 10^{4}$ yr (see figure \[figure3\]c for the gas-phase profile), it can react with O$_2^+$ ions, forming HCOOH$_2^+$, which then dissociatively recombines with electrons. This process results in the rise in the gas-phase formic acid abundance beginning at $6 \times 10^{4}$ yr. At the same time, as with methyl formate, HCOOH surface formation benefits from the increased formation, and diffusion rate, of HCO, causing the first rise in surface abundance. The next rise in the gas-phase HCOOH abundance comes about when formaldehyde evaporates, which facilitates more OH formation in the gas phase via reaction with atomic oxygen. The OH radical and H$_2$CO then react in the gas phase to form HCOOH via the process OH + H$_{2}$CO $\rightarrow$ HCOOH + H. Both of these gas-phase mechanisms increase the grain-surface HCOOH abundance by subsequent accretion of the product. On the grain surfaces, reaction (3) becomes most important when formaldehyde has evaporated significantly, because previously most OH was destroyed by reaction with H$_2$CO. After its evaporation, formaldehyde may still accrete and then (if it does not immediately evaporate again) it may react with OH to form HCO. Sufficient OH is still available to further react with HCO via reaction (3) because the surface abundance of H$_2$CO is comparatively small. At this stage, the dominant OH formation mechanism is photodissociation of H$_2$O by cosmic ray photons. The post-evaporation surface abundance of formaldehyde is small compared to its pre-evaporation level, but the gas-phase abundance remains high, and so accretion is fast and remains steady. Hence, we find that the formation of formic acid on surfaces is optimised at a temperature where the formaldehyde does not stick to the grains ($\sim$40 K), but resides in the gas phase and accretes and re-evaporates quickly. This allows it to react with OH to form enough HCO for reaction (3), whilst not dominating the OH chemistry to such an extent that reaction (3) suffers. Reaction (3) requires HCO to be formed at a significant rate, since HCO quickly evaporates at such temperatures. This optimum state for reaction (3) ends when the temperature increases to the point where OH evaporates ($\sim$70 K). However, the grain-surface formation mechanism is never dominant; the gas-phase synthesis (OH + H$_2$CO) is still around ten times faster at this point, due to high formaldehyde abundances. The final peak in surface HCOOH abundance and the small blip before the final gas-phase peak are due to increased gas-phase formation as OH evaporates from the grains. {width="8.5cm"} {width="8.5cm"} Finally, the surface-bound HCOOH evaporates, leaving a high gas-phase abundance of $\sim 10^{-7}$ with respect to H$_2$. The majority of HCOOH is formed by the gas-phase reaction. Dimethyl ether (CH$_3$OCH$_3$) is formed initially on grain surfaces by reaction (2). The strong rise up to peak surface abundance begins when the surface HNO abundance falls. This species is until then the main reaction partner of CH$_3$, forming CH$_4$ and NO. CH$_3$ is formed on the surfaces by the hydrogenation of CH$_2$ following the photodissociation of CH$_4$ by cosmic ray-induced photons. The fall in HNO allows CH$_3$ abundances to rise, fuelling reaction (2). Dimethyl ether continues to be formed by the grain-surface mechanism until it evaporates from the grains. Reaction (2) does not contribute much dimethyl ether abundance compared to the final level, and this molecule evaporates earlier than methyl formate or formic acid, at a time of $\sim 10^{5}$ yr. The main formation mechanism is therefore the gas-phase route, which consists of the reaction between methanol and protonated methanol, which is fast when methanol finally evaporates, followed by dissociative recombination. We use a recombination efficiency for producing dimethyl ether of $\sim$1.5 %. ### Other Species The gas-phase profile for H$_2$CO (figure \[figure3\]a) also shows a complicated structure. Again, this is due to the time dependence of the evaporation of different species. The first dip (at $\sim 4.5 \times 10^{4}$ yr) is due to N$_2$ evaporation, which contributes strongly to the NH$_3$ gas-phase abundance; the ammonia competes with atomic oxygen to react with CH$_3$, which is required to form H$_2$CO in the gas phase at that time. The subsequent evaporations of CO and CH$_4$ then push up formaldehyde abundance strongly. The final rise in gas-phase abundance takes place as formaldehyde itself is desorbed from the grain surfaces. The profiles of sulphur-bearing species resulting from this treatment show some features of interest. Importantly, H$_2$CS is the most abundant sulphur-bearing species at late times, in spite of the dominance of H$_2$S on the grain surfaces before this. When H$_2$S evaporates, it is broken down to atomic sulphur by cosmic ray-induced photons. Meanwhile, CH$_3$ is abundant, both as a result of the high CH$_4$ level in the gas phase, and the cosmic ray-induced photodissociation of grain-surface methanol followed by the fast evaporation of the resultant CH$_3$. Atomic S and CH$_3$ then react in the gas phase to produce H$_2$CS. When methanol evaporates, CH$_3$ may still be formed by cosmic ray-induced photodissociation, but also by dissociative recombination of CH$_3$OHCH$_3^+$. The final H$_2$CS abundance is rather larger than observations might suggest, and larger than other models predict. Van der Tak et al. (2003) suggest values of 10$^{-9}$, using radiative transfer models with observations of high mass star-forming regions. Typical H$_2$CS:H$_2$S values are on the order of a few. However, that study suggests SO and SO$_2$ should be of the same order as H$_2$S and H$_2$CS, which is not the case here, except at times of around $8 \times 10^{4}$ yr. The observational study of [@wakelam04b] provides SO, SO$_2$ and H$_2$S values on the order of 10$^{-6}$. However, the sulphur chemistry is complex, and is not the main focus of this paper. We direct the reader to Viti, Garrod & Herbst (2006, in prep.), which builds on the sulphur chemistry analysis of [@viti04a], and includes grain-surface chemistry. ---------------------------------- ---------------------------- ---------------------------- ---------------------------- ---------------------------- Species $t_{f}=5 \times 10^{4}$ yr $t_{f}=2 \times 10^{5}$ yr $t_{f}=1 \times 10^{6}$ yr $t_{f}=2 \times 10^{5}$ yr $T=T_2(t)$ $T=T_2(t)$ $T=T_2(t)$ $T=T_1(t)$ HCOOCH$_3$ (all reactions) 1.9$(-9)$ 7.8$(-9)$ 1.9$(-8)$ 5.3$(-9)$ HCOOCH$_3$ (no gas-phase routes) 1.0$(-10)$ 5.5$(-9)$ 1.9$(-8)$ 3.2$(-9)$ HCOOH 5.5$(-8)$ 1.1$(-7)$ 1.2$(-7)$ 1.2$(-7)$ CH$_3$OCH$_3$ 6.1$(-9)$ 1.4$(-8)$ 1.1$(-9)$ 2.2$(-8)$ H$_2$CO 2.3$(-5)$ 2.3$(-7)$ 3.9$(-10)$ 1.5$(-8)$ CH$_3$OH 3.2$(-5)$ 1.7$(-5)$ 5.9$(-7)$ 1.8$(-5)$ $^{a}$ $a(b) = a \times 10^{b}$ ---------------------------------- ---------------------------- ---------------------------- ---------------------------- ---------------------------- Timescale and Temperature Profile Variations -------------------------------------------- In addition to the standard switch-on time of $2 \times 10^{5}$ yr, we ran additional models with heat-up times of $5 \times 10^{4}$ and $1 \times 10^{6}$ yr. Figures \[figure4\] and \[figure5\] show gas-phase and grain-surface abundances of HCOOCH$_3$ and other molecules, for the shorter and longer times, respectively. With the shorter time to reach a temperature of 200 K, the gas-phase routes to HCOOCH$_3$ become dominant over the surface reactions, whilst for the longer timescale, the surface route dominates even more strongly, and also produces the most methyl formate of any of the models. The major gas-phase route that forms methyl formate, reaction (5), is strongly dependent on the H$_2$CO abundance. With the shorter timescale, $t_{f} = 5 \times 10^{4}$ yr, formaldehyde is abundant until the end of the model. This is not the case with the longer timescale, $t_{f} = 1 \times 10^{6}$ yr, and here the gas-phase route acts for only a short time in comparison to the length of time that the HCOOCH$_3$ resides on the grain surface, gradually being destroyed by cosmic ray-induced photodissociation. However, the productivity of the grain-surface mechanism is [*enhanced*]{} by the longer timescale, because there is a longer period between the time when HCO becomes both abundant and highly mobile (i.e. following CO evaporation), and the time when formaldehyde evaporates – the requirements for reaction (1) to be effective. Since the surface formaldehyde abundance is steady until it ultimately evaporates, and the OH required to form HCO (via OH + H$_2$CO $\rightarrow$ HCO + H$_2$O) is provided steadily by photodissociation of water ice, the amount of methyl formate formed by surface reactions is directly proportional to the timescale. For the shorter timescale, the comparatively long lifetime of H$_2$CO in the gas phase means that most HCOOH is formed in the gas phase. Hence, shorter timescales favour the gas-phase routes, whereas longer timescales favour the grain-surface mechanism. The final CH$_3$OCH$_3$ abundance suffers with the longer timescale, as photodissociation reduces the surface-based methanol abundance before it can evaporate and form dimethyl ether. However, the early peak in CH$_3$OCH$_3$ abundance caused by the evaporation of the surface species is almost as strong as the later peak induced by the gas-phase chemistry. Surface production is increased for similar reasons as stated for HCOOH and HCOOCH$_3$. We also ran models using the linear temperature dependence, $\Delta T_1$, shown in figure \[figure2\], with the standard switch-on time. The difference between the two profiles is that for $\Delta T_2$, the increase in temperature is slower at early times, but then much faster at later times. Hence, with $\Delta T_1$, the hot core spends less time at lower temperatures and more time at higher temperatures. Because the surface reactions (1) – (3) are disfavored for shorter timescales, and because these reactions are most active from $\sim$20 – 60 K, the $\Delta T_1$ profile results in less surface formation since it increases more rapidly in this range of temperatures. Discussion ========== Our results show that significant abundances of saturated organic molecules are produced in hot cores during the protostellar warm-up phase. The results also demonstrate that the gas-phase and grain-surface chemistry during this period are strongly coupled. The large abundances which build up on grain surfaces have a large impact on gas-phase chemistry when they evaporate. But, also, the products of that chemistry can re-accrete, changing the surface chemistry. In addition, the sudden absence of certain species from the grains, following evaporation, can have a strong effect on the surface chemistry. Since hydrogenation is inefficient at warmer temperatures, the abundance of heavy radicals on the grain surfaces is mainly due to the photodissociation of ices such as H$_2$O, and to a lesser extent CH$_3$OH, due to cosmic ray-induced photons. The gas-phase and grain-surface chemistries act in different, complementary ways, and the strongly coupled system yields results which only a full gas-grain treatment can produce. Table \[tab4\] lists the gas-phase abundances obtained with various model parameters for methyl formate, formic acid, dimethyl ether, formaldehyde, and methanol at the time of maximum gaseous methyl formate abundance, which occurs at or near the end of the warm-up phase. These abundances derive from both grain-surface and gas-phase reactions. We have also listed the methyl formate abundances obtained without the gas-phase route, since reaction (5) is problematic. If we look at the first three columns of the table, obtained with varying times for the warm-up phase but all with our standard $\Delta T_{2}$ profile, we see that different species show different sensitivities to the timescale, which is related to the mass of the protostar. Methyl formate increases one order of magnitude in abundance from the shortest to the longest timescale, with a stronger dependence if the gas-phase synthesis is removed. On the other hand, formic acid has an abundance that does not change much. Dimethyl ether shows a rather strange dependence, first increasing then decreasing as the time interval increases. The species with the strongest time dependence is formaldehyde, the abundance of which decreases by almost five orders of magnitude from the shortest to the longest time interval considered. The abundance of methanol also decreases with increasing time interval, although not to as great an extent as that of formaldehyde. [l c c c c c]{} Species && Orion Compact Ridge$^{b}$ & Orion Hot Core$^{b}$ & IRAS 16293$^{c}$ & IRAS 4A$^{d}$\ HCOOCH$_3$-A && 3$(-8)$ & 1.4$(-8)$ & $1.7 \pm 0.7(-7)$ & $3.4 \pm 1.7(-8)$\ HCOOCH$_3$-E && & & $2.3 \pm 0.8(-7)$ & $3.6 \pm 1.7(-8)$\ HCOOH && 1.4$(-9)$ & 8$(-10)$ & $\sim$$6.2(-8)$ & $4.6 \pm 7.9(-9)$\ CH$_3$OCH$_3$ && 1.9$(-8)$ & 8$(-9)$ & $2.4 \pm 3.7(-7)$ & $\leq$$2.8(-8)$\ H$_2$CO && 4$(-8)$ & 7$(-9)$ & $6.0(-8)$$^{e}$ & $2(-8)$$^{f}$\ CH$_3$OH && 4$(-7)$ & 1.4$(-7)$ & $3.0(-7)$$^{e}$ & $\leq$$7(-9)$\ \ \ \ \ \ \ Let us now consider the agreement between our new approach to hot core chemistry and observations of organic molecules in these sources. Table \[tab5\] shows observational results for two hot cores and two hot corinos – IRAS 16293 and IRAS 4A – associated with low-mass protostars rather than the high-mass protostars typical of hot cores [@bottinelli04a]. One can see immediately that observed abundances for IRAS 16293 are larger than for the other cores. However, there is some disagreement over the H$_2$ column densities associated with this source [see e.g. @peeters06a], and this may help to explain the discrepancy. Indeed, the sizes and morphologies of the emitting regions of hot cores are not well defined, and therefore observationally determined abundances may be strongly dependent on beam size. For the Orion Compact Ridge and Hot Core, the observed abundances are best fit by a heat-up interval of between $2 \times 10^{5}$ yr and $1 \times 10^{6}$ yr. Note, though, that the abundances listed in table \[tab4\] refer to times somewhat less than the full interval. The worst disagreement occurs with formic acid, where the calculated abundance is two orders of magnitude too high. One explanation for this discrepancy lies in the gas-phase neutral-neutral reaction (OH + H$_{2}$CO) leading to HCOOH; there is only a small amount of evidence for this channel (see, e.g., http://kinetics.nist.gov/index.php). Methanol is best fit at time intervals near the upper value while the calculated abundances of formaldehyde and dimethyl ether are too low at this time, and methyl formate is not very sensitive to changes of time in the interval range considered. The hot corino IRAS 4A is somewhat richer in methyl formate and formic acid than the hot cores, but has less methanol. Nevertheless, the level of agreement is not very different in the range of heat-up intervals considered. For the case of IRAS 16293, the situation is rather unique. Here the formic acid is in agreement with our large calculated abundance but the methyl formate abundance is more than an order of magnitude higher than our largest value and the dimethyl ether abundance is similarly large. The formaldehyde and methanol abundances are not exceptionally large, and so rule out a very small heat-up interval. In other words, both the hot cores and hot corinos in our sample are fit best by heat-up intervals characteristic of low-mass and intermediate-mass stars, although we cannot distinguish between the two. Our results, while not in excellent agreement with observation and while only focusing on abundances of selected species, add strength to the case that hot cores develop over long timescales, spending significant periods at low to intermediate temperatures before reaching the $\sim$100 – 300 K typical of observed objects. Whilst our suggested heat-up timescales of $2 \times 10^{5}$ – $1 \times 10^{6}$ yr are quite long, our model suggests that observed hot cores could have comparatively short post-evaporation ages, where we define the “evaporation stage” as that at which CH$_3$OH, HCOOH, HCOOCH$_3$, H$_2$O and NH$_3$ evaporate, approximately at the same time (at around 100 K). Note that this post-evaporation era is, according to previous models, when gas-phase chemistry is supposed to produce the complex species considered here [see e.g. @millar91b; @charnley92b; @caselli93a]. Our standard and longer timescale models, $t_f = 2 \times 10^{5}$ and $t_f = 1 \times 10^{6}$ yr, spend approximately $3 \times 10^{4}$ and $3 \times 10^{5}$ yr, respectively, between the evaporation of the large molecules and the end of the model. This corresponds to a change in temperature from $\sim$100 K up to 200 K. The models largely produce acceptable agreement with observed abundances throughout this period, for the species which we investigate here, although for the longer timescale model methyl formate abundance has begun to depreciate seriously after around $10^{5}$ yr post-evaporation. The standard model displays acceptable HCOOCH$_3$ levels right to the end of the run, corresponding to $\sim$$3 \times 10^{4}$ yr post-evaporation. Our results (see figures \[figure3\] and \[figure5\]) therefore suggest that hot cores which show HCOOCH$_3$ abundances of around 10$^{-8}$ with respect to H$_2$ could have post-evaporation ages of anywhere between zero and at least $3 \times 10^{4}$ yr (in the case of cores associated with intermediate-mass protostellar sources), and $1 - 2 \times 10^{5}$ yr (in the case of cores associated with low-mass protostellar sources). Our model suggests that grain-surface formation of HCOOCH$_3$ occurs within a temperature range of $\sim 20$ – $40$ K. The lower bound in this range corresponds to the evaporation of CO, which leaves the OH radical free to react with formaldehyde to form HCO. The upper bound corresponds to the evaporation of H$_2$CO, which is crucial to the formation of both of the radicals required by reaction (1). Hence, the binding energies of these two species are especially important in determining the time period available for reaction (1) to take place. The binding energy of HCO is much less important, since it is too reactive to build up on the grains; its abundance is determined by its rate of reaction with other species. Although the strength of its diffusion barrier is related to the binding energy ($E_{D} = 0.5 E_{B}$), we do not expect the [*final*]{} amount of HCOOCH$_3$ produced to be exceptionally sensitive to this value. Following the experimental evidence of [@collings04a] for other species, [@viti04a] suggested that the evaporation of H$_2$CO is determined by co-desorption with water ice. In this model we allow thermal evaporation of H$_2$CO according to its desorption energy, less than half that of water. However, our model indicates that most formaldehyde should be formed at very late times in the collapse phase, so that H$_2$CO is unlikely to be trapped in the water ices below. Using our model to explicitly track the deposition of monolayers would allow us to investigate this more fully. Conclusions =========== Using a gas-grain network of reactions, we have investigated the formation of methyl formate in hot cores, along with the related species dimethyl ether and formic acid, and to a lesser extent formaldehyde and methanol, during the protostellar warm-up and evaporative phase. We undertook this work because standard models of hot-core chemistry, which rely on gas-phase chemistry at a temperature of 100 K or so to produce complex molecules from precursor methanol, have severe failings [see e.g. @horn04a]. The chemistry during the warm-up period occurs both on grain surfaces and in the gas phase. We find that the formation of HCOOCH$_3$, by either mechanism, requires a gradual warm-up of temperatures, rather than an immediate jump from 10 K to 100 K, as is often assumed. In the case of the grain-surface route, reaction (1), this is so that radicals may become mobile enough to react, without quickly desorbing. Indeed, whilst the time-dependent temperature profiles we use are quite specific, grain-surface formation of methyl formate should be viable as long as temperatures remain in the 20 – 40 K range for a long enough period of time. In the case of the only viable gas-phase route, reaction (5), it is because the reaction rate is strongly temperature dependent – being fastest at low temperatures – but requiring the evaporation of H$_2$CO at intermediate temperatures to become important. The plausibility of the gas-phase route for methyl formate identified in this work is a matter for further study. During the warm-up phase, atomic hydrogen evaporates very quickly and hydrogenation is much less important than during the cold phase. The production of surface radicals is largely determined by cosmic ray-induced photodissociation of ices. However, in some cases, the low surface abundance of atomic H is still sufficient to produce radicals, when the other reactant is very abundant, e.g. formaldehyde. Our results for a variety of species show that both gas-phase and grain-surface reactions are strongly coupled. Such a strong interaction allows processes to take place that could not happen on either the grain surfaces or in the gas phase alone. The results of our calculations show that towards the end of the warm-up period, large abundances of a variety of organic molecules are now present in the gas, in agreement with observation. The observed rich, complex chemistries of hot cores may thus be representative of objects in which complex molecules have very recently evaporated (at around 100 K), contrary to the predictions of models that utilise only a gas-phase chemistry to reproduce all abundances [e.g. @rodgers01a]. The physical model of the warm-up phase, which we base on the analysis of Viti et al. (2004, and references therein), assumes a finite time period for the central protostellar heat source to reach its switch-on time, and therefore for the surrounding gas and dust to vary in temperature gradually. This means that the temperature increases should propagate outwards, and that the chemistries of the inner regions should be further advanced than the outer. Therefore, the typical hot core chemistry itself should propagate outwards. The radial size of regions which exhibit significant complex molecule abundances may then indicate the speed at which the temperature gradient propagates. We hope soon to extend our model to include a much larger network of surface reactions and species. This set will include other astronomically detected molecules such as the isomers of methyl formate and dimethyl ether, as well as some postulated and (so far) undetected species. Surface reaction of heavy radicals will also be extended to allow the formation of other large molecules already present in the gas-grain code. We thank the National Science Foundation (US) for support of the Ohio State University astrochemistry program. Helpful discussions with S. Viti, on the subject of the warm-up phase, with M. Collings, on the subject of thermal desorption of grain mantles, with V. Wakelam, on the subject of atomic abundances, and with S. Widicus Weaver, on the subject of complex molecule formation, are acknowledged.

--- abstract: '3D object detection from raw and sparse point clouds has been far less treated to date, compared with its 2D counterpart. In this paper, we propose a novel framework called FVNet for 3D front-view proposal generation and object detection from point clouds. It consists of two stages: generation of front-view proposals and estimation of 3D bounding box parameters. Instead of generating proposals from camera images or bird’s-eye-view maps, we first project point clouds onto a cylindrical surface to generate front-view feature maps which retains rich information. We then introduce a proposal generation network to predict 3D region proposals from the generated maps and further extrude objects of interest from the whole point cloud. Finally, we present another network to extract the point-wise features from the extruded object points and regress the final 3D bounding box parameters in the canonical coordinates. Our framework achieves real-time performance with 12ms per point cloud sample. Extensive experiments on the 3D detection benchmark KITTI show that the proposed architecture outperforms state-of-the-art techniques which take either camera images or point clouds as input, in terms of accuracy and inference time.' author: - bibliography: - 'references.bib' title: | FVNet: 3D Front-View Proposal Generation for\ Real-Time Object Detection from Point Clouds --- *3D object detection, Point clouds, Real-time.* Acknowledgment {#acknowledgment .unnumbered} ============== Thanks to the National Natural Science Foundation of China (No. 61972157), Science and Technology Commission of Shanghai Municipality Program (No. 18D1205903) and the National Social Science Foundation of China (No. 18ZD22) for funding.

--- abstract: 'String theory can accommodate black holes with the black hole parameters related to string moduli. It is a well known but remarkable feature that the near horizon geometry of a large class of black holes arising from string theory contains a BTZ part. A mathematical theorem (Sullivan’s Theorem) relates the three dimensional geometry of the BTZ metric to the conformal structures of a two dimensional space, thus providing a precise kinematic statement of holography. Using this theorem it is possible to argue that the string moduli space in this region has to have negative curvature from the BTZ part of the associated spacetime. This is consistent with a recent conjecture of Ooguri and Vafa on string moduli space.' --- -1.5cm -0.5cm \ **Black Holes, Holography and Moduli Space Metric** Kumar S. Gupta[^1]\ \ Siddhartha Sen[^2]\ \ [*UCD, Belfield, Dublin 4, Ireland*]{}\ \ \ [*Indian Association for the Cultivation of Science*]{}\ [*Calcutta - 700032, India*]{}\ .5cm October 2006 PACS : 04.70.-s\ String theory suggests that a consistent quantum theory of gravity should admit a holographic description. While a full understanding of quantum gravity is yet to emerge, black hole physics provides a glimpse into various quantum aspects of gravity. It is thus natural to ask what, if any, conclusions can be drawn regarding the metric of string moduli space from the holographic property of black holes. This is the issue we address in this Letter. We would argue that in certain regions of the moduli space of string theories, which corresponds to a large class of black holes, there exists a precise kinematical statement of holography. Our strategy relies on the observation that in certain parts of the string moduli space, the near horizon geometry contains a part with the causal structure of a BTZ black hole [@banados]. This observation has been very successful in the microscopic derivation of the Bekenstein-Hawking entropy for a large class of black holes [@maldastrom; @physrept; @peet], including that for four and five dimensional non-supersymmetric backgrounds [@cvetic; @skenderis], e.g. Schwarzschild black hole. For all these backgrounds, which includes black holes of astrophysical interests, there exists a mathematical theorem (Sullivan’s theorem) [@sull] that provides a precise kinematical statement of holography. Using this theorem we can argue that the metric of the string moduli space corresponding to the BTZ part of the spacetime has to have negative curvature. This precise result is consistent with a recent conjecture of Ooguri and Vafa about the curvature of the string moduli space [@vafa]. We start by briefly recalling the string theories where the near horizon region has the causal structure of a BTZ black hole. The classic example is provided by the ten dimensional type IIB string theory compactified on $AdS_3 \times S^3 \times M^4$ [@maldastrom]. Using the notation of [@physrept], the near horizon geometry of the solutions associated to $Q_1$ $D1$-branes and $Q_5$ $D5$-branes in the string frame is given by = (-dt\^2 + (dx\_5)\^2 ) + g\_6 + g\_6 d \_3\^2 where $g_6$ is the six dimensional string coupling, $(2 \pi \alpha^{\prime})^{-1}$ is the string tensions and $l_s$ is the string length with $l_s^2 = \alpha^{\prime}$. In the near horizon limit $U = \frac{r}{\alpha^{\prime}}$ is kept fixed as $\alpha^{\prime} \rightarrow 0$. The string moduli are fixed by the charges $Q_1$ and $Q_5$. The full geometry contains a contribution from the $M^4$ factor which has not been written above. The metric in (1) represents $AdS_3 \times S^3$ with $SL(2,R) \times SL(2,R)$ as isometry group. The BTZ metric is obtained from $ADS_3$ with a global identification by a fixed element of the isometry group [@banados; @maldastrom] and the BTZ black hole parameters are identified in terms of the moduli. The above solution is supersymmetric in nature. Another class of examples involve considering near extremal four dimensional rotating black hole solutions of toroidally compactified string theory, which admits five dimensional embeddings as rotating black strings [@cvetic]. The near extremal Kerr-Newman black hole falls in this category. The near-horizon geometry in this case is $AdS_3 \times S^2$, and the BTZ black hole is obtained with the usual global identifications of $Ads_3$. The third class of examples include non-extremal and non-supersymmetric black holes in four and five dimensions which admit embedding into M-theory, and a series of U-dualities are then used to map these into configurations containing a BTZ black hole [@skenderis]. This includes the interesting example of four dimensional Schwarzschild black hole. The point of view here is that the U-duality transformations are analogous to gauge transformations which do not change the physical content of the system. By using the U-duality, the four and five dimensional black holes considered here are analyzed in the “BTZ gauge". Finally, it may be noted that in certain regions of the moduli space, the BTZ black hole arises directly as a solution of string theory [@horo]. Thus, by restricting attention to certain specific regions of the string moduli space and by use of dualities, the near-horizon geometry of a large class of black holes, including the ones of astrophysical interest, can be shown to contain a BTZ part. All the above examples were discussed in the literature mainly in the context of the calculation of black hole entropy. In fact, most of the effective string models used to calculate black hole entropy relied on having a BTZ geometry present in the system [@peet]. In all these cases, the BTZ black hole parameters, including the horizon radii, are determined from the string moduli. As a next step we recall the key points which relates the BTZ black hole to a precise kinematical notion of holography. The basic feature of the BTZ black hole which allows this to happen is that the Euclidean BTZ is a locally isomorphic to the hyperbolic 3-manifold $H^3$ which is geometrically finite [@sen]. The three dimensional hyperbolic structures for such a manifold, according to Sullivan’s theorem [@sull], are in 1-1 correspondence with the two dimensional conformal structures of its boundary. More precisely, if $K$ is a geometrically finite hyperbolic 3-manifold with boundary then Sullivan’s theorem states that as long as $K$ admits one hyperbolic realization, there is a 1-1 correspondence between hyperbolic structures on $K$ and conformal structures on its boundary $\partial K$, the latter being the Teichmuller space of the boundary $\partial K$. This is nothing but a precise mathematical statement of holography for the BTZ black hole [@sen; @man]. It is thus evident that for the very large class of black holes discussed above, which contain a BTZ geometry in the near-horizon region, the Sullivan’s theorem would ensure this precise kinematical notion of holography would hold. The notion of holography here is kinematical as no detailed dynamical information about the theory is required to establish this correspondence. This is reminiscent of the “kinematical” nature of the entropy [@skenderis] calculated for the same class of black holes where the knowledge of the underlying CFT is enough to produce the Bekenstein-Hawking formula, without precise knowledge of the dynamics of the associated degrees of freedom [@ent]. In order to proceed, note that the boundary of the BTZ black hole has the topology of $T^2$ and the corresponding Teichmuller space is given by a the fundamental region of the complex variable $\tau$. In other words, two Teichmuller parameters $\tau$ and $\tau^{\prime}$ are equivalent if \^ = ,    ad - bc = 1 and $a,b,c,d \in Z$. The transformation in (2) is nothing but the action of the modular group on $\tau$ generated by the operations S &:& -\ T &:& + 1. In terms of the horizon radii $r_+$ and $r_-$ of the Euclidean BTZ black hole, the effective action of the modular group can be written as S &:& r\_+ r\_-\ T &:& r\_+ r\_+,    r\_- r\_- + r\_+. This establishes the fact the that Riemann surface describing the boundary of the Euclidean BTZ has analytic properties. As mentioned before, the BTZ black hole was appears as a part of the near horizon geometry only in specific regions of the string moduli space. Indeed, the radii $r_+$ and $r_-$ of the BTZ black hole, as well as the corresponding Teichmuller parameter $\tau$, are fixed in terms of the string moduli. The tori obtained under the action of $S$ and $T$ in (4) are equivalent as Riemann surfaces. This implies that the metric in the region(s) of the moduli space where the BTZ black hole appears must also be invariant under $SL(2,Z)$. The moduli space metric with such an $SL(2,Z)$ invariance can be obtained by first constructing an $SL(2,R)$ invariant metric on which the $SL(2,Z)$ invariance can be imposed. To do this first note that under the SL(2,R) transformation , where $\alpha \delta - \beta \gamma =1$ and $ \alpha, \beta, \gamma, \delta \in R$, we have d . It is then easy to check that metric ds\^2 = is $SL(2,R)$ invariant. This can be interpreted as the metric in the region of the string moduli space for which the corresponding near horizon geometry contains a BTZ part. It is interesting to note that this metric has negative scalar curvature [@borel]. In summary, based on the observation that in certain regions of the string moduli space the near-horizon geometry contains a BTZ part, and using a precise notion of holography for BTZ black holes following from Sullivan’s theorem, we have argued that the corresponding regions of the string moduli space must admit a metric with negative curvature. 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Welch, “Exact three-dimensional black holes in string theory", Phys. Rev. Lett. [**71**]{}, 331 (1993) \[arXiv:hep-th/9302126\]; D. Birmingham, I. Sachs and Siddhartha Sen, “Entropy of three-dimensional black holes in string theory", Phys. Lett. B424, 275 (1998) \[arXiv:hep-th/9801019\]. D. Birmingham, C. Kennedy, Siddhartha Sen and A. Wilkins, “Geometrical finiteness, holography, and the BTZ black hole", Phys. Rev. Lett. [**82**]{}, 4164 (1999) \[arXiv:hep-th/9812206\]; D. Birmingham, I. Sachs, S. Sen, “ Exact results for the BTZ black hole", Int. Jour. Mod. Phys. [**D10**]{}, (2001) 833 \[arXiv:hep-th/0102155\]. Y. I. Manin and M. Marcolli, “ Holography principle and arithmetic of algebraic curves", Adv. Theor. Math. Phys. [**5**]{}, (2002) 617 \[arXiv:hep-th/0201036\]. A. Strominger, “Black hole entropy from near horizon microstates", JHEP [**9802**]{}, 009 (1998) \[arXiv:hep-th/9712251\]; S. Carlip, “Black hole entropy from conformal field theory in any dimension", Phys. Rev. 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--- abstract: 'Given a triangulation of a closed surface, we consider a cross ratio system that assigns a complex number to every edge satisfying certain polynomial equations per vertex. Every cross ratio system induces a complex projective structure together with a circle pattern on the closed surface. In particular, there is an associated conformal structure. We show that for any triangulated torus, the projection from the space of cross ratio systems with prescribed Delaunay angles to the Teichmüller space is a covering map with at most one branch point. Our approach is based on a notion of discrete holomorphic quadratic differentials.' address: 'Mathematics Research Unit, Université du Luxembourg, L-4364 Esch-sur-Alzette' author: - Wai Yeung Lam bibliography: - 'holomorphicquad.bib' title: Quadratic differentials and circle patterns on complex projective tori --- Introduction {#sec:introduction} ============ Discrete differential geometry concerns structure-preserving discretizations in differential geometry [@Bobenko2008]. Its goal is to establish a discrete theory with rich mathematical structures such that the smooth theory arises in the limit of refinement. It has stimulated applications in computational architecture and computer graphics. To obtain a structured discrete theory, a main challenge is to decide on the right properties to be preserved under discretization. A prominent example in discrete conformal geometry is Thurston’s circle packing [@Stephenson2005]. In the classical theory, holomorphic functions are conformal, mapping infinitesimal circles to themselves. Instead of infinitesimal size, a circle packing is a configuration of finite-size circles where certain pairs are mutually tangent. Thurston proposed regarding the map induced from two circle packings with the same tangency pattern as a discrete holomorphic function. Using machinery from hyperbolic 3-manifolds, a discrete analogue of the Riemann mapping is deduced from Koebe-Andreev-Thurston theorem. Rodin and Sullivan [@Rodin1987] showed that it converges to the classical Riemann mapping in the case of hexagonal circle packings as the mesh size tends to zero. By considering general combinatorics instead of restricting to the hexagonal mesh, circle packings can approximate quasi-conformal maps proving the measurable Riemann mapping theorem [@He1990; @Williams2019]. A natural question is how this theory can be extended to Riemann surfaces and how the convergence depends on the combinatorics. On the other hand, computer scientists have been using circle packings to approximate conformal maps between surfaces for years [@Boris2006]. Among many numerical schemes, circle packings have the advantage of its discrete nature ready for numerical computation and its effective visualization of conformal stretching. It is believed that in the limit of a suitable refinement, the map induced from circle packings would converge to a classical conformal map. It motivates a systematic study of the interplay between circle packings, combinatorics and conformal structures on surfaces. From the viewpoint of discrete differential geometry, it is advantageous to develop a structured discrete theory that relates successful examples like circle packings. In the previous works [@Lam2016; @Lam2017; @Lam2015a], we developed a notion of discrete holomorphic quadratic differentials from circle packings and connected it to several discrete theories: discrete harmonic functions, discrete integrable systems and discrete minimal surfaces in space. It is intriguing if this notion could be related to the classical Teichmüller theory as like as its smooth counterpart, e.g. as a parametrization of the Teichmüller space. To address these questions, this article investigates circle patterns on surfaces with complex projective structures. A circle pattern in the plane is a realization of a planar graph such that each face has a circumcircle passing through the vertices. For circle patterns on surfaces, they can be formulated in terms of an algebraic system as follows: We denote $M=(V, E, F)$ a triangulation of an oriented closed surface where $V$, $E$ and $F$ are the sets of vertices, edges and faces respectively. Vertices are denoted by $i,j,k$. An unoriented edge is denoted by $\{ij\}=\{ji\}$ indicating its end points are vertices $i$ and $ j$, where $i=j$ is allowed and in that case the edge has to form a non-contractable loop on the surface. Given a realization $z:V \to \mathbb{C}\cup\{\infty\}$ on the Riemann sphere, we associate a complex cross ratio to every common edge $\{ij\}$ shared by triangles $\{ijk\}$ and $\{jil\}$: $$cr_{ij} := -\frac{(z_k - z_i)(z_l -z_j)}{(z_i - z_l)(z_j - z_k)}$$ which encodes how the circumdisk of triangle $z_iz_jz_k$ is glued to that of $z_jz_iz_l$ (see Section \[sec:delaunay\])). It defines a function $\operatorname{cr}: E \to \mathbb{C}$ satisfying certain algebraic equations: \[def:crsys\] Suppose $M=(V,E,F)$ is a triangulation of a closed oriented surface. A cross ratio system on $M$ is an assignment $\operatorname{cr}:E \to \mathbb{C}$ such that for every vertex $i$ with adjacent vertices numbered as $1$, $2$, ..., $n$ in the clockwise order counted from the link of $i$, $$\begin{gathered} \Pi_{j=1}^n \operatorname{cr}_{ij} =1 \label{eq:crproduct}\\ \operatorname{cr}_{i1} + \operatorname{cr}_{i1} \operatorname{cr}_{i2} + \operatorname{cr}_{i1}\operatorname{cr}_{i2}\operatorname{cr}_{i3} + \dots + \operatorname{cr}_{i1}\operatorname{cr}_{i2}\dots\operatorname{cr}_{in} =0 \label{eq:crsum} \end{gathered}$$ where $\operatorname{cr}_{ij} = \operatorname{cr}_{ji}$. Equivalently, $$\Pi_{j=1}^n \left( \begin{array}{cc} \operatorname{cr}_{ij} & 1 \\ 0 & 1 \end{array} \right) = \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right).$$ The cross ratio system is called if \(a) the argument of the cross ratios $\operatorname{Arg}\operatorname{cr}$ takes value in $[0, \pi )$, \(b) the graph $(V, E_+)$ is the 1-skeleton of a CW decomposition of $M$ where $$E_{+}=\{ ij \in E| \operatorname{Arg}\operatorname{cr}_{ij} \neq 0 \}.$$ Furthermore the ramification index $s:V\to \mathbb{N}$ is defined by $$2\pi s_i = \sum_j \operatorname{Arg}\operatorname{cr}_{ij}.$$ Without further notice, we focus on cross ratio systems without branch points (i.e. $s\equiv 1$). A cross ratio system provides a recipe to glue neighboring circumdisks, which resembles to Thurston’s equations for gluing ideal hyperbolic polyhedra [@Thurston1982]. In particular, Equations and have been considered by Fock [@Fock1993]. Since cross ratios are invariant under complex projective transformations (i.e. Möbius transformations), the gluing construction is compatible with the complex projective geometry. One can deduce that every Delaunay cross ratio system defines a complex projective structure on the closed surface $M$ (see Section \[sec:cp1\]. It yields a forgetful map $f: \mathcal{D} \to P(M)$ from the space of all Delaunay cross ratio systems $\mathcal{D}$ to the space of all marked complex projective structures $P(M)$. In this article, we are interested in a subset of cross ratio systems that have prescribed arguments. These correspond to circle patterns where neighbouring circles intersect with prescribed angles, which is a natural generalization of circle packings [@Lam2017]. Suppose $M$ is a closed triangulated surface. A Delaunay angle structure is an assignment of angles $\Theta:E \to [0,\pi)$ with $\Theta_{ij}=\Theta_{ji}$ satisfying the following: (i) For every vertex $i$, $$\sum_j \Theta_{ij} = 2\pi$$ where the sum is taken over the neighboring vertices of $i$ on the universal cover. (ii) For any collection of edges $(e_0,e_1,e_2,\dots,e_n=e_0)$ whose dual edges form a simple closed contractable path on the surface, then $$\sum_{i=1}^{n} \Theta_{e_i} > 2\pi$$ unless the path encloses exactly one primal vertex. We write $P(\Theta)\subset \mathcal{D}$ the space of all Delaunay cross ratio systems $\operatorname{cr}$ with $\operatorname{Arg}\operatorname{cr}\equiv \Theta$. It is known that $P(\Theta)$ is nonempty. By the discrete uniformization theorem [@Bobenko2004; @Rivin], $P(\Theta)$ contains exactly one cross ratio system whose underlying complex projective structure can be reduced to an Euclidean structure if $g=1$ and a hyperbolic structure if $g>1$. Every Delaunay cross ratio system also induces a developing map of the universal cover to the Riemann sphere. Its holonomy yields a representation of the fundamental group in $PSL(2,\mathbb{C})$ up to conjugation. Though the holonomy representation of the space of complex projective structures $P(M)$ is known to be non-injective, its restriction to $P(\Theta)$ is injective: \[thm:holo\] For a Delaunay angle structure $\Theta$ on a torus, $P(\Theta)$ is a real analytic surface homeomorphic to $\mathbb{R}^2$. Furthermore, the holonomy map to the character variety $$\operatorname{hol}: P(\Theta) \to \operatorname{Hom}( \pi_1(M), PSL(2,\mathbb{C}))/PSL(2,\mathbb{C})$$ is an embedding which passes through the slice of Euclidean tori exactly once. Particularly the restriction of the forgetful map $f|_{P(\Theta)}$ is an embedding into the space of marked complex projective structures $P(M)$. We denote $\mathcal{T}(M)$ the Teichmüller space, i.e. the space of marked conformal structures. If $M$ is a torus, $\mathcal{T}(M)$ is a manifold of real dimension 2. Recall that $f:P(\Theta)\to P(M)$ sends a Delaunay cross system to its marked complex projective structure. We further write $\pi: P(M) \to \mathcal{T}(M)$ from the space of complex projective structures to the Teichmüller space. \[thm:covering\] For a Delaunay angle structure $\Theta$ on a torus, the projection $$\pi\circ f:P(\Theta)/\{\operatorname{cr}_0\} \to \mathcal{T}(M)/\{\tau_0\}$$ is a finite-sheet covering map where $\operatorname{cr}_0$ is the unique cross ratio system in $P(\Theta)$ that induces an Euclidean torus and $\tau_0$ is the associated conformal structure. In particular, the projection $$\pi\circ f:P(\Theta)\to \mathcal{T}(M)$$ is a covering map with at most one branch point. Both the proofs of Theorems \[thm:holo\] and \[thm:covering\] rely on discrete holomorphic quadratic differentials (Section \[sec:hqd\]) and Rivin’s results in [@Rivin] (see Section \[sec:cp1tori\]). It remains a conjecture whether $\pi\circ f|_{P(\Theta)}$ is a diffeomorphism for the torus. In particular, the analogues of Theorems \[thm:holo\] and \[thm:covering\] for surfaces with genus $g>1$ are still open. Circle patterns on tori have appeared in various forms. Doyle’s spiral circle packings [@Beardon1994] as a discrete analogue of exponential functions can be seen as a developing map of a circle packing on an affine torus, which is further extended to discretize Painlevé equations from integrable systems [@Agafonov2000]. On the other hand, we established a correspondence between circle patterns on tori and the dimer models from statistical mechanics recently [@fKen2018]. It is intriguing how they would interact with the underlying conformal structures. See Figure \[fig:circlepatterns\] for examples of circle patterns on the torus. Our result also indicates that circle patterns on surfaces play a role of discrete quasi-conformal map. Fixing two conformal structures on the torus and Delaunay angles, Theorem \[thm:covering\] indicates that there always exists circle patterns on complex projective tori with the prescribed conformal structures. Under a suitable refinement of the triangulation, the map induced by such circle patterns could converge to a quasi-conformal map. In particular it would be interesting to investigate how the triangulations affect the Beltrami coefficient, which measure the conformal distortion of the resulting quasi-conformal map. The organization of the paper is as follows: In Section \[sec:background\], we explain the derivation of cross ratio systems and the Delaunay condition. We further introduce discrete holomorphic quadratic differentials and recall their connections to discrete harmonic functions from previous results. In Section \[sec:cp1tori\], we focus on the torus. Using complex affine developing maps, we simplify the equations for cross ratio systems. We then apply discrete harmonic functions and Rivin’s results to deduce Theorems \[thm:holo\] and \[thm:covering\]. In Section \[sec:examples\], we explore a couple of examples and show how the above theorems fail for non-Delaunay cross ratio systems. In Section \[sec:discussion\], we discuss some open problems. Related work ------------ Our work is closely related to Kojima, Mizushima and Tan [@Kojima2003] who proposed to consider circle packings on surfaces with complex projective structures. They conjectured that this configuration space is a manifold whose projection to the Teichmüller space is a diffeomorphism. Here we outline the progress so far: (i) Torus ($g=1$): Mizushima [@Mizushima2000] proved that for the one-vertex triangulation, the projection of the space of circles packings on complex projective tori to the Teichmüller space is a diffeomorphism by explicit computation. Kojima, Mizushima and Tan [@Kojima2003] showed that in the space of circle packings on any given triangulation, the circle packing on an Euclidean tori is contained in a neighborhood homeomorphic to $\mathbb{R}^2$. However, little is known about the global structure of the configuration space and its projection to the Teichmüller space. (ii) Surfaces with genus $g>1$: Similarly, Kojima, Mizushima and Tan [@Mizushima2000] showed that in the space of circle packings on any given triangulation, the circle packing on a hyperbolic surface is contained in a neighborhood homeomorphic to $\mathbb{R}^{6g-6}$. Recently, Schlenker and Yarmola [@Schlenker2018] showed that in the setting of circle patterns, the projection to the Teichmüller space is a proper map. However it remains unknown whether the space of circle patterns with fixed intersection angles is a manifold and whether its projection to the Teichüller space is an immersion. Our contribution is to study the projection from the space of Delaunay circle patterns to the Teichmüller space in the case of tori with arbitrary triangulations. We show that it is proper everywhere and is an immersion almost everywhere. Our use of discrete holomorphic quadratic differentials is novel, which could be applicable to surfaces with genus $g>1$. ![Circle patterns with Delaunay intersection angle $\Theta \equiv \pi/3$ on a triangulated torus. Vertex positions $z$ and combinatorics are shown on the left. Their circumcircles are shown on the right. Notice that not all intersection points are the vertices under consideration. The three tori have different conformal structures. []{data-label="fig:circlepatterns"}](circlea0){width="100.00000%"} ![Circle patterns with Delaunay intersection angle $\Theta \equiv \pi/3$ on a triangulated torus. Vertex positions $z$ and combinatorics are shown on the left. Their circumcircles are shown on the right. Notice that not all intersection points are the vertices under consideration. The three tori have different conformal structures. []{data-label="fig:circlepatterns"}](circleb0){width="100.00000%"} ![Circle patterns with Delaunay intersection angle $\Theta \equiv \pi/3$ on a triangulated torus. Vertex positions $z$ and combinatorics are shown on the left. Their circumcircles are shown on the right. Notice that not all intersection points are the vertices under consideration. The three tori have different conformal structures. []{data-label="fig:circlepatterns"}](circlea1){width="100.00000%"} ![Circle patterns with Delaunay intersection angle $\Theta \equiv \pi/3$ on a triangulated torus. Vertex positions $z$ and combinatorics are shown on the left. Their circumcircles are shown on the right. Notice that not all intersection points are the vertices under consideration. The three tori have different conformal structures. []{data-label="fig:circlepatterns"}](circleb1){width="100.00000%"} ![Circle patterns with Delaunay intersection angle $\Theta \equiv \pi/3$ on a triangulated torus. Vertex positions $z$ and combinatorics are shown on the left. Their circumcircles are shown on the right. Notice that not all intersection points are the vertices under consideration. The three tori have different conformal structures. []{data-label="fig:circlepatterns"}](circlea2){width="100.00000%"} ![Circle patterns with Delaunay intersection angle $\Theta \equiv \pi/3$ on a triangulated torus. Vertex positions $z$ and combinatorics are shown on the left. Their circumcircles are shown on the right. Notice that not all intersection points are the vertices under consideration. The three tori have different conformal structures. []{data-label="fig:circlepatterns"}](circleb2){width="100.00000%"} Background {#sec:background} ========== Möbius transformations {#sec:mobius} ---------------------- Without further notice, we only consider orientation-preserving Möbius transformations on the Riemann sphere. They are in the form $z \mapsto \frac{a z + b}{c z +d}$ for some $a,b,c,d \in \mathbb{C}$ such that $ad - bc =1$ and are also called complex projective transformations. These transformation are generated by Euclidean motions and inversion $z \mapsto 1/z$. In particular, they are holomorphic, map circles to circles and preserve cross ratios. Developing map and holonomy {#sec:develop} --------------------------- Given a cross ratio system $\operatorname{cr}:E \to \mathbb{C}$ on a triangulated surface $M$, we denote $\hat{M} = (\hat{V},\hat{E},\hat{F})$ the universal cover of $M$ with the pull back triangulation and $p$ the covering map. Then there exists a *developing map* $z:\hat{V} \to \mathbb{C}\cup \{ \infty \}$ such that for every edge $ij \in \hat{E}$ $$\label{eq:developingmap} \operatorname{cr}_{p(ij)} = -\frac{(z_k - z_i)(z_l -z_j)}{(z_i - z_l)(z_j - z_k)}.$$ The developing map is unique up to complex projective transformations and has an equivariance property with respect to the fundamental group $\pi_1(M)$: For any $\gamma \in \pi_1(M)$, there exists a complex projective transformation $\rho_{\gamma} \in PSL_2(\mathbb{C})$ such that $$z \circ \gamma = \rho_{\gamma} \circ z.$$ The map $\gamma \mapsto \rho_{\gamma}$ defines a holonomy representation $\operatorname{hol}$ of the cross ratio system unique up to conjugation by elements in $PSL_2(\mathbb{C})$, i.e. $\operatorname{hol}(\operatorname{cr}) \in \operatorname{Hom}( \pi_1(M), PSL(2,\mathbb{C}))/PSL(2,\mathbb{C}))$ represents a point in the so-called character variety.. Conversely, every mapping into the the extended complex plane with the equivariance property induces a cross ratio system on $M$ via . To obtain a developing map, we pick a face $\{ijk\}$ of $\hat{M}$ and assign the vertices to three distinct points $z_i,z_j,z_k$. If $\{jil\}$ is a neighboring face (see Fig. \[fig:delaunay\] left), then $z_l$ is uniquely determined by $ \operatorname{cr}_{p(ij)} = -\frac{(z_k - z_i)(z_l -z_j)}{(z_j - z_k)(z_i - z_l)}$. In this way, nearby triangles are laid out one by one. However, for the developing to be well defined, we have to make sure such construction is independent of the order of the triangles chosen. Indeed by Lemma \[lem:cross2\] below, the conditions in Definition \[def:crsys\] yield that the holonomy of the developing map is trivial around each vertex. The developing map $z$ is then uniquely determined as long as some triangle $z_i,z_j,z_k$ is prescribed. If the three vertices take some other distinct values $\tilde{z}_i,\tilde{z}_j, \tilde{z}_k$, then there exists a unique complex projective transformation $\rho$ mapping $z_i,z_j,z_k$ to $\tilde{z}_i,\tilde{z}_j, \tilde{z}_k$ and the developing maps are related by $\tilde{z} = \rho \circ z$. Conversely, Lemma \[lem:cross1\] implies that a mapping $z:\hat{V} \to \mathbb{C}\cup \{\infty\}$ with the equivariance property induces a cross ratio system on $M$. \[lem:cross1\] Given a point $u$ and a sequence of numbers $z_0, z_1, ...., z_k, ...$ in $\C$ such that $u, z_i, z_{i-1}, z_{i+1}$ are distinct for all $i$, we define the cross ratios for $j=1,2,...$ by $$\operatorname{cr}_{uj} = -\frac{(u-z_{j-1})(z_{j}-z_{j+1})}{(z_{j+1}-u)(z_{j-1}-z_j)}$$ Then $$\prod_{j=1}^n \operatorname{cr}_{uj} =1, \quad \sum_{k=1}^n \prod_{j=1}^k \operatorname{cr}_{uj} =0.$$ Under the above assumption, for $k=1,2, ...$ $$\prod_{j=1}^k \operatorname{cr}_{uj} =\frac{(z_{0}-u)(z_1-u)}{z_0-z_1} \cdot \frac{z_k-z_{k+1}}{(z_k-u)(z_{k+1}-u)}= \frac{(z_{0}-u)(z_1-u)}{z_0-z_1} (\frac{1}{z_k-u} - \frac{1}{z_{k+1}-u})$$ $$\sum_{k=1}^m \prod_{j=1}^k \operatorname{cr}_{uj} =\frac{(z_{0}-u)(z_1-u)}{z_0-z_1} (\frac{1}{z_1-u} - \frac{1}{z_{m+1}-u})$$ In the special case that $z_0=z_n$ and $z_1=z_{n+1}$, we obtain the claim. \[lem:cross2\]Suppose $\operatorname{cr}_1, ..., \operatorname{cr}_n \in \C$ such that $$\prod_{j=1}^n \operatorname{cr}_{uj}= 1, \quad \text{and} \quad \sum_{k=1}^n \prod_{j=1}^k \operatorname{cr}_{uj} =0.$$ Then there exist $u, z_1, ..., z_n =z_0, z_{n+1}=z_1$ $\in \C \cup \{\infty\}$ such that $$\operatorname{cr}_{uj} = -\frac{(u-z_{j-1})(z_{j}-z_{j+1})}{(z_{j+1}-u)(z_{j-1}-z_j)}$$ for all $j$. The vector $(u, z_1, ..., z_n)$ is unique up to $PSL(2, \C)$ action. Once we show that there is a unique configuration with $u=\infty$, $z_1=1$ and $z_0=0$, the uniqueness follows since cross ratios are Möbius invariant and the fact that there always exists a unique Möbius transformations mapping any three distinct points to any three distinct points. Taking $u=\infty$, $z_1=1$ and $z_0=0$, we have $$\operatorname{cr}_{uj}=\frac{z_{j+1}-z_j}{z_j- z_{j-1}}$$ Hence using $z_0, z_1$, we can define inductively that $$z_{k+1}=z_k+\prod_{j=1}^k \operatorname{cr}_{uj}.$$ We claim that $z_{n+1}=1=z_1$ and $z_n=0=z_0$. Indeed, consider the summation, $$\sum_{k=1}^n z_{k+1}=\sum_{k=1}^n (z_k+\prod_{j=1}^k \operatorname{cr}_{uj}).$$ We obtain that $z_{n+1}=z_1$ by the assumption that $\sum_{k=1}^n \prod_{j=1}^k (\operatorname{cr}_{uj})=0$. Now consider $1=z_{n+1}=z_n+\prod_{j=1}^n (\operatorname{cr}_{uj})= z_n+1$. It follows that $z_n=0=z_0$. Therefore the claim holds. Delaunay condition {#sec:delaunay} ------------------ Since the surface $M$ is assumed to be oriented, every triangular face of $M$ is equipped with an orientation. Using this orientation together with a realization of the vertices on the Riemann sphere, we associate a circumdisk to every face. We write $\{ijk\}$ a face with vertices $i,j,k$ that is oriented, i.e. $\{ijk\}=\{jki\}=\{kij\}$ but $\{ijk\}\neq\{jik\}$. Suppose $z_i,z_j,z_k\in \mathbb{C}\cup\{\infty\}$ are distinct. Then there is an oriented circle $C_{ijk}$ passing through $z_i,z_j,z_k$ in cyclic order. It bounds two disks on the Riemann sphere. We denote $D_{ijk}$ the open disk with $C_{ijk}$ as the boundary in positive orientation and call $D_{ijk}$ the circumdisk of $\{ijk\}$ under $z$. ![Two possible configurations for Delaunay cross ratio $\operatorname{Arg}\operatorname{cr}_{ij} \in [0,\pi)$. Left: Two bounded disks. Right: One bounded disk and one unbounded disk,. Both figures satisfy the empty circle condition. The circular arc from $i$ to $j$ of the circle through $\{ijk\}$ is contained in the neighboring disk of $\{jil\}$.[]{data-label="fig:delaunay"}](fig7){width="100.00000%"} ![Two possible configurations for Delaunay cross ratio $\operatorname{Arg}\operatorname{cr}_{ij} \in [0,\pi)$. Left: Two bounded disks. Right: One bounded disk and one unbounded disk,. Both figures satisfy the empty circle condition. The circular arc from $i$ to $j$ of the circle through $\{ijk\}$ is contained in the neighboring disk of $\{jil\}$.[]{data-label="fig:delaunay"}](fig8){width="85.00000%"} Suppose a Delaunay cross ratio system is given and $z:V \to \mathbb{C}\cup\{\infty\}$ is a developing map of the universal cover. Then the cross ratio system satisfies $\operatorname{Arg}\operatorname{cr}\in [0,\pi)$ if and only if the local empty disk condition is satisfied, i.e. for any two neighbouring faces $\{ijk\}$ and $\{jil\}$ (See Fig. \[fig:delaunay\]), they satisfy (i) $z_k \notin D_{jil}$ and (ii) $z_l \notin D_{ijk}$ and (iii) $D_{jil} \cap D_{ijk} \neq \emptyset$ Mapping $z_i$ to infinity by inversion, we have $\operatorname{cr}_{ij} = \frac{z_k-z_j}{z_j-z_l}$ and the claim follows immediately. In particular, $$D_{jil} \cap D_{ijk} = \emptyset$$ if and only if $\operatorname{Arg}\operatorname{cr}_{ij} = \pi$. Complex projective structures {#sec:cp1} ----------------------------- Every Delaunay cross ratio system induces a complex projective structure on the closed surface together with a circle pattern. We recall the definition of a complex protective structure on a surface. A complex projective structure on a surface $M$ is a maximal atlas of charts from open subsets of $M$ to the Riemann sphere such that the transition functions are restrictions of Möbius transformations. These charts are called projective charts. Two complex projective structures are marked isomorphic if there is a diffeomorphism homotopic to identity mapping projective charts to projective charts. We denote $P(M)$ the space of marked complex projective structures. Intuitively, a cross ratio system gives a recipe to glue circumdisks with transition functions as Möbius transformations. For every face $\tau=\{ijk\}$, one assigns the vertices to some distinct points $p_i,p_j,p_k$ in the plane to obtain a circumdisk $D_{\tau}$. Consider a neighboring face $\sigma=\{jil\}$, its circumdisk $D_{\sigma}$ has vertices at $q_j, q_i,q_l$. For the common edge $\{ij\}$, pick four points $z_i,z_j,z_k,z_l$ in the plane such that $\operatorname{cr}_{ij}=-\frac{(z_k-z_i)(z_l-z_j)}{(z_j-z_k)(z_i-z_l)}$. Then there are unique Möbius transformations $\phi_{\tau,ij}$ mapping $p_i,p_j,p_k$ to $z_i,z_j,z_k$ and $\phi_{\sigma,ij}$ mapping $q_i,q_j,q_l$ to $z_i,z_j,z_l$. We define the transition map $\phi_{\tau}^{\sigma}:= \phi^{-1}_{\sigma,ij} \circ \phi_{\tau,ij} $ which is independent of the choice of $z$. Having the cocycle condition into account, one can show that the disjoint union $\sqcup D_{ijk}$ with a quotient relation $x \sim y$ if $y =\phi_{\tau}^{\sigma}(x)$ induces a surface with a complex projective structure. However, it should be careful that in general the resulting surface is not homeomorphic to $M$ unless the Delaunay condition is assumed (See section \[sec:one-vertex\] for a counterexample). [@Fock1993] \[prop:cp1\] Every Delaunay cross ratio system defines a complex projective structure on $M$ together with a circle pattern. It yields a forgetful map $f: \mathcal{D} \to P(M)$ from the space of all Delaunay cross ratio systems $\mathcal{D}$ to the space of all marked complex projective structures $P(M)$. The holonomy representation of a Delaunay cross ratio system is identical to that of the underlying complex projective structure. Discrete holomorphic quadratic differentials {#sec:hqd} -------------------------------------------- We rephrase the equations for cross ratio systems in Definition \[def:crsys\] in a slightly different way. We define $\Phi:\mathbb{C}^{E} \to \mathbb{C}^{2V}$ as follows: for every vertex $i\in V$ $$\begin{gathered} \Phi(\operatorname{cr})_{i,1} := \Pi_{j=1}^m \operatorname{cr}_{ij} -1 \\ \Phi(\operatorname{cr})_{i,2}:= \operatorname{cr}_{i1} + \operatorname{cr}_{i1} \operatorname{cr}_{i2} + \operatorname{cr}_{i1}\operatorname{cr}_{i2}\operatorname{cr}_{i3} + \dots + \operatorname{cr}_{i1}\operatorname{cr}_{i2}\dots\operatorname{cr}_{im} \end{gathered}$$ Furthermore, given an angle structure $\Theta$, we define the exponential maps (i) $\exp:\mathbb{C}^{E} \to \mathbb{C}^{E} $ by $\exp(X)_{ij}:= e^{X_{ij}}$ and (ii) $\exp_{\Theta}: \mathbb{R}^{E} \to \mathbb{C}^{E} $ by $\exp_{\Theta}(X)_{ij}:= e^{X_{ij}+ \mathbf{i} \Theta_{ij}}$ where $\mathbf{i}= \sqrt{-1}$. Notice that $\exp_{\Theta}$ is diffeomorphic to its image and the space of cross ratio systems $P(\Theta)$ can be identified with the zero set $(\Phi\circ \exp_{\Theta})^{-1} \{0\}$. To show that it is a manifold, we need to show that the Jacobian $D(\Phi\circ \exp_{\Theta})$ has constant rank along the zero set. The following is intermediate: \[prop:jacobian\] Suppose $X:E\to \mathbb{C}$ satisfies $\Phi\circ \exp(X)=0$. We write $\operatorname{cr}:=\exp(X)$. Then $q:E\to \mathbb{C}$ is in the kernel of the Jacobian $D(\Phi\circ \exp)_X$ if and only if for every vertex $i$ $$\begin{aligned} 0=&D(\Phi\circ \exp_{\Theta})_X(q)_{i,1}= \sum_j q_{ij} \\ 0=&D(\Phi\circ \exp_{\Theta})_X(q)_{i,2} = q_{i1} \operatorname{cr}_{i1} + (q_{i1} + q_{i2})\operatorname{cr}_{i1} \operatorname{cr}_{i2} + \dots + (q_{i1}+ \dots + q_{im}) \operatorname{cr}_{i1}\operatorname{cr}_{i2}\dots\operatorname{cr}_{im} \end{aligned}$$ We write $Q^{\mathbb{C}}(\operatorname{cr}):=\operatorname{Ker}(D(\Phi\circ \exp)_X)$ which is a complex vector space. In the smooth theory, a holomorphic quadratic differential can be obtained via Schwarzian derivative, which measures the degree how much a holomorphic map fails to be a Möbius transformation. Here we have a discrete analog: A real-valued function $q$ in the Jacobian $D(\Phi\circ\exp)_X$ describes a first-order deformation of the cross system $\operatorname{cr}:=\exp(X)$, which represents an infinitesimal deformation of the underlying circle pattern preserving the intersection angles. Such a deformation is induced by a Möbius transformation if and only if $q\equiv0$. Using the notation of Proposition \[prop:jacobian\], a real-valued function $q:E\to \mathbb{R}$ in the kernel of $D(\Phi\circ \exp)$ is called a discrete holomorphic quadratic differential. These discrete holomorphic quadratic differentials forms a real vector space denoted by $Q^{\mathbb{R}}(\operatorname{cr})$. \[prop:lowerbound\] For every cross ratio system $\operatorname{cr}$ on a closed surface with genus $g$ $$\begin{aligned} \dim_{\mathbb{R}} Q^{\mathbb{R}}(\operatorname{cr})&\geq 6g-6 \\ \dim_{\mathbb{C}} Q^{\mathbb{C}}(\operatorname{cr})&\geq 6g-6 + |V| \end{aligned}$$ It follows by counting the number of variables and linear constraints. An inequality is obtained since the constraints might be linearly dependent. For $Q^{\mathbb{R}}(\operatorname{cr})$, there are $|E|$ real variables. For every vertex, there is one real equation and one complex equation. Hence $$\dim_{\mathbb{R}} Q^{\mathbb{R}}(\operatorname{cr})\geq |E|-|V|-2|V|=6g-6.$$ For $Q^{\mathbb{C}}(\operatorname{cr})$, there are $|E|$ complex variables. For every vertex, there are two complex equations. Hence $$\dim_{\mathbb{C}} Q^{\mathbb{C}}(\operatorname{cr})\geq |E|-|V|-|V|=6g-6+|V|.$$ By expressing the cross ratios in terms of the developing map $z$, we obtain the original formulation of discrete holomorphic quadratic differentials introduced in [@Lam2015a]. Suppose a cross ratio system is given on $M=(V,E,F)$. We denote its developing map $z: \hat{V} \to \mathbb{C}\cup\{\infty\}$ of the universal cover $\hat{M}=(\hat{V},\hat{E},\hat{F})$. Then $q:E \to \mathbb{R}$ is a holomorphic quadratic differential if and only if for every vertex $i \in \hat{V}$ $$\begin{aligned} \sum_j \hat{q}_{ij} &=0 \label{qsum} \\ \sum_j \hat{q}_{ij}/(z_j - z_i) &=0 \label{zsum} \end{aligned}$$ where $\hat{q}$ is the lift of $q$ to the universal cover $\hat{M}$ $$\ \hat{q}_{ij} := q_{p(ij)}.$$ In the smooth theory, a holomorphic quadratic differential differential $q$ on a Riemann surface locally is of the form $q= f(z) \, dz^2$ where $f$ is a holomorphic function. Heuristically, equation mimics the property in the smooth theory that $$q(X)+q(JX)+q(J^2 X) + q(J^3 X) =0$$ where $X$ is any tangent vector and $J$ is the 90-degree rotation. Particularly $q(X)=q(-X)$. The holomorphicity is encoded in Equation : $f$ being holomorphic is equivalent to the 1-form $$q/dz = f(z) \, dz$$ being closed, which holds if and only if $\oint q/dz=0$ over any contractible loop. Discrete harmonic functions: cotangent Laplacian {#sec:harmonic} ------------------------------------------------ We recall some facts about a particular graph Laplacian, whose edge weights depend on a realization of the graph into the plane (See [@Lam2015a] for references). These are essential tools to study the Jacobian of $\Phi\circ \exp_{\Theta}$ and the projection to the Teichmüller space. \[def:cotlap\] Suppose $z:V \to \mathbb{C}$ is a realization of a triangulated surface. Then a function $u:V \to \mathbb{R}$ is harmonic in the sense of the cotangent Laplacian if for every vertex $i$ $$\sum_j c_{ij} (u_j - u_i) = 0.$$ where $c_{ij}=c_{ji}:=\cot \angle jki + \cot \angle ilj $ is called the cotangent weight (See Figure \[fig:delaunay\] left). Here $\angle jki \in (-\pi,\pi)$ and it is positive if $z_j,z_k,z_i$ are in counterclockwise order and is negative otherwise. There is a conjugate harmonic function for every discrete harmonic function as in the smooth theory. On a simply connected domain, a function $u:V \to \mathbb{R}$ is harmonic if and only if there exists $u^*:F \to \mathbb{R}$ unique up to an additive constant such that $$u^*_{ijk} - u^*_{jil} = \frac{1}{2}(\cot \angle jki + \cot \angle ilj) (u_j-u_i)$$ where $\{ijk\}, \{jil\}$ are the left and the right faces of the oriented edge pointing from $i$ to $j$. As in the smooth theory, linear functions are discrete harmonic. For any two complex numbers $a,b$, we write their inner product $ \langle a,b \rangle = {\operatorname{Re}}( a \bar{b})$. Given a realization $z:V \to \mathbb{C}$ of a triangulated surface, a function $u:V \to \mathbb{R}$ is called linear if there exists $a \in \mathbb{C}, b \in \mathbb{R}$ such that $$u_i = \langle a,z_i \rangle + b.$$ It is harmonic with respect to the cotangent Laplacian and its conjugate harmonic function $u^*:F \to \mathbb{R}$ is of the form $$u^*_{ijk} = \langle a, -\mathbf{i} z^*_{ijk} \rangle + d$$ for some $d \in \mathbb{R}$ where $z^*_{ijk}$ is the circumcenter of triangle $\{ijk\}$ and $\mathbf{i}= \sqrt{-1}$. Notice that for any two neighboring triangles $\{ijk\},\{jil\}$, the difference between the circumcenters can be written as $$z^*_{ijk} - z^*_{jil} = \frac{\mathbf{i}}{2}(\cot \angle jki + \cot \angle ilj) (z_j -z_i).$$ Suppose $u$ is linear and of the form $u= \langle a,z \rangle + b$, then its conjugate harmonic function satisfies $$u^*_{ijk} - u^*_{jil} = \frac{1}{2}(\cot \angle jki + \cot \angle ilj) ( \langle a,z_j\rangle -\langle a, z_i \rangle ) = \langle a, -\mathbf{i} z^*_{ijk} \rangle - \langle a, -\mathbf{i} z^*_{jil} \rangle$$ and the claim follows. It is known that there is a correspondence between discrete holomorphic quadratic differentials and discrete harmonic functions with respect to the cotangent Laplacian. We outline here the results from [@Lam2015a]. \[prop:hqdharmonic\] Suppose $z:V \to \mathbb{C}$ is a realization of a simply connected triangular mesh. Then there is a one-to-one correspondence between the following: (i) a discrete holomorphic quadratic differential $q:E \to \mathbb{R}$; (ii) an infinitesimal deformation of the developing map $\dot{z}:V \to \mathbb{C}$ such that the argument of the cross ratios $\operatorname{Im}\log \operatorname{cr}$ is preserved. This $\dot{z}$ is unique up to an infinitesimal Möbius transformation; (iii) a discrete harmonic function $u:V\to \mathbb{R}$ unique up to a linear function. These correspondence has been proved in [@Lam2015a]. Since we shall use them in Section \[sec:upperbound\] and \[sec:immersion\], we sketch the relations here. $(1)\leftrightarrow(2)$: Given a first order deformation $\dot{z}:V \to \mathbb{C}$, we compute the logarithmic change in the cross ratio which yields a holomorphic quadratic differential (see Fig \[fig:delaunay\] left for the notation) $$q_{ij}:= \frac{\dot{z}_i-\dot{z}_k}{z_i-z_k}-\frac{\dot{z}_l-\dot{z}_i}{z_l-z_i}+\frac{\dot{z}_j-\dot{z}_l}{z_j-z_l}-\frac{\dot{z}_k-\dot{z}_j}{z_k-z_j} \left( =\frac{\dot{\operatorname{cr}}_{ij}}{\operatorname{cr}_{ij}} \right)$$ and conversely $\dot{z}$ is determined by $q$ whenever $\dot{z}_i,\dot{z}_j,\dot{z}_k$ is prescribed for some face $\{ijk\}$. Furthermore $q$ is real-valued if and only if $\dot{z}$ preserves $\operatorname{Im}\log \operatorname{cr}$. $(2)\leftrightarrow (3)$: $\dot{z}$ preserve $\operatorname{Im}\log \operatorname{cr}$ if and only if there exists a function $u:V\to\mathbb{R}$ such that for every edge $\{ij\}$ $$\operatorname{Im}\frac{\dot{z}_j-\dot{z}_i}{z_j-z_i} = \frac{u_i + u_j}{2}.$$ The function $u$ is uniquely determined from $\dot{z}$ as follows: Pick a face $\{ijk\}$ then $$u_i := \operatorname{Im}( \frac{\dot{z}_j-\dot{z}_i}{z_j-z_i}+\frac{\dot{z}_i-\dot{z}_k}{z_i-z_k}- \frac{\dot{z}_k-\dot{z}_j}{z_k-z_j})$$ is independent of the choice of faces containing vertex $i$. It turns out that $u$ is a discrete harmonic function with respect to the cotangent Laplacian. It is a linear function if and only if $\dot{z}$ is induced by an infinitesimal Möbius transformation. The conjugate harmonic function $v:F \to \mathbb{R}$ similarly satisfies $$\begin{aligned} \label{eq:conjharmonic} \operatorname{Re}\frac{\dot{z}_j-\dot{z}_i}{z_j-z_i} = -\frac{v_{ijk} \cot \angle ilj + v_{ilj} \cot \angle jki }{\cot \angle ilj + \cot \angle jki }. \end{aligned}$$ Circle patterns on complex projective tori {#sec:cp1tori} ========================================== Complex affine torus and holonomy {#subsec:holonomy} --------------------------------- For the torus, it is known that every complex projective structure can be reduced to an affine structure. A complex affine structure on a surface $M$ is a maximal atlas of charts from open subsets of $M$ to $\mathbb{C}$ such that the transition functions are restrictions of complex affine transformations $z \mapsto a z + b$ for some $a,b \in \mathbb{C}$ with $a \neq 0$. It is elementary to characterize the holonomy representation of the fundamental group of the torus in $PSL(2,\mathbb{C})$ up to conjugation. Assume $\gamma_1,\gamma_2$ are generators of the fundamental group of the torus, which satisfy $\gamma_1 \gamma_2 = \gamma_2 \gamma_1$. Pick a developing map $z: \hat{M} \to \mathbb{C} \cup \{\infty\}$ for the complex projective structure. The corresponding holonomy $\rho_1,\rho_2$ in $PGL(2,\mathbb{C})$ satisfy $\rho_1 \rho_2 = \rho_2 \rho_1$. If $\rho_1$ is not the identity, there are three cases: (I) $\rho_1$ has only one fixed point. Then it is also the only fixed point of $\rho_2$. We can normalize the developing map by composing a Möbius transformation such that the fixed point is at infinity and the holonomy $\rho_1,\rho_2$ are translation, i.e. $\exists \beta_j \in \mathbb{C}$ such that $$(z\circ \gamma_j)_i = \rho_j (z_i) = z_i + \beta_j \quad \forall \, i \in \hat{V}.$$ (II) $\rho_1$ has two fixed points and $\rho_2$ does not exchange them. Then they are fixed points of $\rho_2$ as well. We can normalize the developing map such that the fixed points are at the origin and infinity. The holonomy becomes stretched rotation, i.e. $\exists \alpha_j \in \mathbb{C}$ such that $$(z\circ \gamma_j)_i = \rho_j (z_i) = \alpha_j z_i \quad \forall \, i \in \hat{V}.$$ (III) $\rho_1$ has two fixed points and $\rho_2$ does exchange them. We can normalize the developing map such that the fixed points of $\rho_1$ are the origin and infinity. Then the holonomy is of the form $$\rho_1 (z_i) = - z_i \quad \rho_2 (z_i) = \alpha_2 /z_i \quad \forall \, i \in \hat{V}.$$ In the case where $\rho_1$ is the identity, then we either have case (I) or (II) depending on the number of fixed points of $\rho_2$. Notice that the holonomy in case (I) and (II) becomes affine transformations. Complex projective structures on the torus are induced by affine structures (See [@Gunning1966 Ch.9, p.189-192] and [@Loray2009]). More details can be found in the survey [@Baues2014]. \[prop:gunning\] Every complex projective structure on a torus can be reduced to an affine structure. \[cor:affinedev\] Every developing map of a Delaunay cross ratio system on a torus can be normalized to have holonomy as complex affine maps (type I or II). In particular, all circumdisks are bounded and the Euclidean image of any two neighboring triangles are locally embedded, i.e. there is no fold. Every Delaunay cross ratio system induces a complex projective structure on the torus and thus an affine structure by Proposition \[prop:gunning\]. The developing map for the affine structure has image in $\mathbb{C}$. Hence no circumdisk under this developing map passes through infinity and thus every circumdisk is bounded. It implies the Euclidean images of any two neighboring triangles have no fold, i.e. Figure \[fig:delaunay\] (right) does not occur. The holonomy of type III does occur for some non-Delaunay cross ratio systems. See \[sec:one-vertex\] for examples. Delaunay cross ratio systems on tori {#sec:delcrstori} ------------------------------------ Making use of the affine structures, in this subsection we aim to prove that \[prop:kernel\] The space $Q^{\mathbb{R}}(\operatorname{cr})$ of discrete holomorphic quadratic differentials has dimension $2$ for any Delaunay cross ratio system $\operatorname{cr}$ on the torus. It follows from Lemma \[lem:g1lower\] that $\dim_{\mathbb{R}} Q^{\mathbb{R}}(\operatorname{cr}) \geq 2$ and Lemma \[lem:g1upper\] that $\dim_{\mathbb{R}} Q^{\mathbb{R}}(\operatorname{cr}) \leq 2$. Before discussing the proofs of the lemmas, we mention here an immediate consequence: \[cor:realsurface\] The space of cross ratio systems with prescribed Delaunay angle structure $\Theta$ is a real analytic surface in the algebraic variety $\Phi^{-1}\{0\}$. Recall that $Q^{\mathbb{R}}(\operatorname{cr})$ is precisely the kernel of the Jacobian $D(\Phi\circ \exp_{\Theta})$ at $\exp^{-1}(\operatorname{cr})$. Theorem \[prop:kernel\] implies that the Jacobian $D(\Phi\circ \exp_{\Theta})$ has constant rank $|E|-2$ along $(\Phi\circ \exp_{\theta})^{-1}\{0\} \subset \mathbb{R}^{E}$. The constant rank theorem yields that $(\Phi\circ \exp_{\theta})^{-1}\{0\} \subset \mathbb{R}^{E} \subset \mathbb{C}^{E}$ is a real analytic surface. The corollary also follows from Rivin’s variational approach using hyperbolic volume (see Section \[sec:Rivin\]). In the following two subsections, we provide an alternative proof in order to introduce discrete harmonic functions, which is an essential tool to study the projection from the space of Delaunay cross ratio systems to the Teichmüller space. ### Lower bound for $\dim_{\mathbb{R}} Q^{\mathbb{R}}(\operatorname{cr})$ {#sec:lower} Notice that the lower bound in Proposition \[prop:lowerbound\] does not provide any information for $\dim_{\mathbb{R}} Q^{\mathbb{R}}(\operatorname{cr})$ since $6g-6=0$ for the torus ($g=1$). \[lem:g1lower\] For a Delaunay cross ratio system $\operatorname{cr}$ on the torus, we have $\dim_{\mathbb{R}} Q^{\mathbb{R}}(\operatorname{cr}) \geq 2$. We show that there is linearly dependence in the equations for discrete holomorphic quadratic differentials using the affine developing map $z: \hat{V} \to \mathbb{C}$ in Corollary \[cor:affinedev\]. There are two cases: Case (1): the holonomy consists of translations (type I). Pick an arbitrary function $q:E \to \mathbb{R}$. Then the complex number $q_{ij}/(z_j-z_i)$ is well defined on the oriented edges $e_{ij}$ of $M$. Namely, we have $$q_{ij}/(\rho_k(z_j)-\rho_k(z_i)) = q_{ij}/(z_j-z_i)$$ for $k=1,2$. Thus $$\sum_i \sum_j q_{ij}/(z_j-z_i) = 0$$ where the sum on the right is over the neighboring vertices $j$ of $i$ on the universal cover and the sum on the left is over all the vertices $i$ in a fundamental domain. Case (2): the holonomy consists of stretched rotations (type II). Pick an arbitrary $q:E\to \mathbb{R}$ satisfying $\sum_j q_{ij}=0$. The complex number $z_i q_{ij}/(z_j-z_i)$ is well defined on the oriented edges $e_{ij}$ of $M$. Namely, we have $$\rho_k(z_i) q_{ij}/(\rho_k(z_j)-\rho_k(z_i)) = z_i q_{ij}/(z_j-z_i)$$ for $k=1,2$. Thus we always have $$\sum_i \sum_j z_i \frac{q_{ij}}{(z_j-z_i)} = \frac{1}{2}\sum_i \sum_j \frac{q_{ij} (z_i -z_j)}{(z_j-z_i)} = -\frac{1}{2}\sum_i \sum_j q_{ij} =0$$ where the sum on the right is over the neighboring vertices $j$ of $i$ on the universal cover and the sum on the left is over all the vertices $i$ in a fundamental domain. In both cases, the complex constraints $$\sum_j q_{ij}/(z_j -z_i) =0 \quad \forall i \in V$$ are linearly dependent. Hence $$\dim_{\mathbb{R}} Q^{\mathbb{R}}(\operatorname{cr}) \geq |E| - 3|V| + 2 = 2.$$ ### Upper bound for $\dim_{\mathbb{R}} Q^{\mathbb{R}}(\operatorname{cr})$ {#sec:upperbound} We make use of the maximum principle for the cotangent Laplacian and the correspondence in Proposition \[prop:hqdharmonic\] in order to obtain an upper bound for $\dim_{\mathbb{R}} Q^{\mathbb{R}}(\operatorname{cr})$. Suppose $z:\tilde{V}\to \mathbb{C}$ is a developing map with affine holonomy induced from a Delaunay cross ratio system on a triangulated torus $M=(V,E,F)$. Then the cotangent Laplacian has the following property: (i) the cotangent weights are invariant under deck transformations, i.e. $c_{\gamma(ij)}=c_{ij}$ for any deck transformation $\gamma$ on the universal cover; (ii) the cotangent weights are non-negative; (iii) the maximum principle holds: a discrete harmonic function $u:\tilde{V} \to \mathbb{R}$ achieves a local minimum or maximum at an interior vertex must be constant. Firstly, the cotangent weights depend on the angles of triangles which are preserved by the affine holonomy. Hence the weights are invariant under deck transformations. Secondly, by Corollary \[cor:affinedev\], the circumdisks are bounded and hence all faces under $z$ are counterclockwisely oriented. By Definition \[def:cotlap\], all the angles within faces take positive values and $$c_{ij} = \cot \angle jki + \cot \angle ilj = \frac{\sin(\angle jki + \angle ilj)}{\cos \angle jki \, \cos jki} = \frac{\sin(\operatorname{Arg}\operatorname{cr}_{ij}) }{\cos \angle jki \, \cos jki} \geq 0$$ Thirdly, since the cotangent weights are non-negative, the proof for the maximum principle is a standard argument for graph Laplacian: Suppose $u$ is a discrete harmonic function with a local minimum at an interior vertex $i$ then $$0= \sum_j c_{ij} (u_j-u_i) > 0$$ unless $u_j = u_i$ for all $\{ij\} \in E$ with $c_{ij}\neq0$. Because the surface is connected, it implies $u$ is constant. Under an affine developing map, we can characterize discrete harmonic functions on the universal cover that correspond to discrete holomorphic quadratic differentials on the torus ascertained by Proposition \[prop:hqdharmonic\]. \[lem:periodicu\] Suppose $z$ is a developing map with affine holonomy by $\rho_j = \alpha_j z_j + \beta_j$ for $\alpha_j, \beta_j \in \mathbb{C}$ and $k=1,2$. Let $q:\hat{E} \to \mathbb{R}$ be a lift of a holomorphic quadratic differential from $M$, which satisfies $ (q\circ \gamma)_{ij} = q_{ij}$ and $u$ be one of its corresponding harmonic functions. Then there exists $a_j \in \mathbb{C}, b_j \in \mathbb{R}$ such that $$(u \circ \gamma_j)_i - u_i = \langle a_j, z_i \rangle + b_j$$ with $$\begin{aligned} a_1 \bar{\beta}_2 &= a_2 \bar{\beta}_1 \\ a_2 (\bar{\alpha}_1 - 1) &= a_1 (\bar{\alpha}_2 - 1) . \end{aligned}$$ We define $v_{j,i}:= (u \circ \gamma_j)_i - u_i $ for every vertex $i$ and $j=1,2$. Then it is again a discrete harmonic function and the corresponding holomorphic quadratic differential of $v$ is identically zero. Hence $v$ is a linear function, i.e. there exists $a_j \in \mathbb{C}$ and $b_j \in \mathbb{R}$ such that for every $i$ $$v_{j,i} = (u \circ \gamma_j)_i - u_i= \langle a_j, z_i \rangle + b_j.$$ Since $\rho_1,\rho_2$ commute, we have $$(u\circ \gamma_2 \circ \gamma_1) - u = (u\circ \gamma_1 \circ \gamma_2)- u$$ which implies for all $i$ $$\langle a_2, \alpha_1 z_i + \beta_1 \rangle + b_2 + \langle a_1,z_i \rangle + b_1 = \langle a_1, \alpha_2 z_i + \beta_2 \rangle + b_1 + \langle a_2,z_i \rangle + b_2$$ and thus $$\begin{aligned} \langle a_2, \beta_1 \rangle = \langle a_1, \beta_2 \rangle &\implies {\operatorname{Re}}( a_1 \bar{\beta}_2) = {\operatorname{Re}}(a_2 \bar{\beta}_1 )\\ a_2 (\bar{\alpha}_1 - 1) &= a_1 (\bar{\alpha}_2 - 1) \end{aligned}$$ We consider the conjugate harmonic function $v^*_j$ of $v_j$. They are of the form $$\begin{aligned} v^*_{j,klm} = \langle a_j, -\mathbf{i} z^*_{klm} \rangle + \tilde{b}_j . \end{aligned}$$ for every face $\{klm\}$. Applying a similar argument to $v^*_j$, we can deduce $${\operatorname{Re}}( -\mathbf{i} a_1 \bar{\beta}_2) = {\operatorname{Re}}(-\mathbf{i} a_2 \bar{\beta}_1 )$$ Thus $a_1 \bar{\beta}_2 = a_2 \bar{\beta}_1$. We apply the above lemma to two cases: (i) the affine holonomy has only one fixed point at infinity where we have $\alpha_1=\alpha_2=1$ and (ii) the affine holonomy shares an additional fixed point at the origin where we have $\beta_1=\beta_2=0$. \[lem:dimU\] Suppose the developing map $z$ is a Delaunay triangulation with the holonomy as affine transformations. We consider harmonic functions $u:\tilde{V}\to \mathbb{R}$ in two cases: (i) $\rho_j(z)=z+\beta_j$ for some $\beta_j \in \mathbb{C}$. We define $$\mathcal{U}:=\{ u \text{ harmonic}| \exists a_j \in \mathbb{C}, b_j \in \mathbb{R} \text{ s.t.} (u \circ \gamma_j)_i - u_i = \langle a_j, z_i \rangle + b_j \, \& \, a_1 \bar{\beta}_2 = a_2 \bar{\beta}_1 \}.$$ (ii) $\rho_j(z)=\alpha_i z$ for some $\alpha_j \in \mathbb{C}$. We define $$\mathcal{U}:=\{ u \text{ harmonic}| \exists a_j \in \mathbb{C}, b_j \in \mathbb{R} \text{ s.t.} (u \circ \gamma_j)_i - u_i = \langle a_j, z_i \rangle + b_j \, \& \, a_2 (\bar{\alpha}_1 - 1) = a_1 (\bar{\alpha}_2 - 1) \}.$$ Then in both cases $$\dim \mathcal{U} \leq 5.$$ If $u \in \mathcal{U}$ with $a_1=a_2=b_1=b_2=0$, then $u$ is a bounded harmonic function and hence $u$ is constant as a result of the maximum principle. It implies $\dim \mathcal{U} \leq 5$. \[lem:g1upper\] For a Delaunay cross ratio system $\operatorname{cr}$ on a triangulated torus, we have $$\dim_{\mathbb{R}} Q^{\mathbb{R}}(\operatorname{cr}) \leq 2$$ The set $\mathcal{U}$ contains a 3-dimensional subspace of linear functions which correspond to trivial discrete holomorphic quadratic differentials by Proposition \[lem:dimU\]. It implies the space of holomorphic quadratic differentials is of dimension $\leq \dim \mathcal{U} -3=2$. Rivin’s variational approach {#sec:Rivin} ---------------------------- [@Rivin Theorem 7.2] \[prop:rivin\] Let $\Theta$ a Delaunay angle structure on a triangulated torus and $A_1,A_2 \in \mathbb{R} $. Then there exists a unique affine structure on the torus with affine holonomy $\rho_j(z)=\alpha_j z + \beta_j$ such that $\log |\alpha_j|= A_j$ and the induced cross ratio $\operatorname{cr}$ satisfies $\operatorname{Arg}\operatorname{cr}\equiv \Theta$. The map $z$ is unique up to a global affine transformation. Two different affine structures might correspond to the same complex projective structure. The affine holonomy always have a common fixed point at $\infty$. As long as it shares another fixed point in $\mathbb{C}$, we can apply an inversion to interchange the two fixed points and obtain a new affine structure, but the underlying complex projective structure is the same since the inversion is a Möbius transformation. In the notation of Proposition \[prop:rivin\], an affine holonomy shares two fixed points if and only if $(A_1,A_2)\neq0$. Firstly, $P(\Theta)$ being a real analytic surface in the algebraic variety is asserted in Corollary \[cor:realsurface\]. Proposition \[prop:rivin\] implies that the affine tori with $\operatorname{Arg}\operatorname{cr}\equiv \Theta$ are parameterized by two real numbers $(A_1,A_2)$. By uniqueness, two different affine tori $(A_1,A_2)$ and $(\tilde{A}_1,\tilde{A}_2)$ share the same complex projective structure if and only if $(\tilde{A}_1,\tilde{A}_2)=(-A_1,-A_2)$. Thus the space of affine tori with $\operatorname{Arg}\operatorname{cr}\equiv \Theta$ is a 2-to-1 covering of $P(\Theta)$ branched at the Euclidean torus $(A_1,A_2)=(0,0)$. Since the space of affine tori with $\operatorname{Arg}\operatorname{cr}\equiv \Theta$ is homeomorphic to $\mathbb{R}^2$ (parameterized by $(A_1,A_2)$), we deduce that $P(\Theta)$ is homeomorphic to $\mathbb{R}^2$. Secondly, we claim that the holonomy representation is an embedding. Suppose two Delaunay cross ratio systems represent the same point in the character variety. Since Delaunay cross ratio systems induce complex projective structures and thus affine structures on the torus, their holonomy representation can be reduced to affine transformations. By interchanging the fixed points, the affine holonomy of the Delaunay cross ratio systems coincide. By the uniqueness in \[prop:rivin\], the two Delaunay cross ratio systems must be identical. Thirdly, if two Delaunay cross ratio systems induce the same complex projective structures, then they represent the same point in the character variety. Hence the cross ratio systems are the same again. Projection to the Teichmüller space of the torus {#sec:projection} ================================================ Given a Delaunay cross ratio system on the torus, it induces a complex projective structure and thus a conformal structure. The underlying conformal structure can be read off easily from the affine developing map. Recall that we can parameterize the affine structures with a fixed underlying conformal structure on the torus by a complex parameter $c$ as follows: Start with a Euclidean torus obtained by gluing the opposite sides of a parallelogram spanned by complex numbers $1$ and $\tau$ in the upper half plane. The parameter $\tau$ represent a marked conformal structure in the Teichmmüller space. Let $d$ be a developing map of an affine structure with the same marked conformal structure. Its holonomy satisfies $d(z+1)= \alpha_1 d(z) + \beta_1$ and $d(z+\tau) = \alpha_2 d(z) + \beta_2$. Notice that $d$ is holomorphic and $d' \neq 0$. We have $d''/d'$ holomorphic and periodic on the torus. Thus $d''/d' = c$ for some constant $c\in \mathbb{C}$. In the case $c=0$, $d(z) =az + b$ and we get a Euclidean torus. In the case $c\neq 0$, we have $d(z) = a e^{c z} + b$ for some constants $a,b$, which can be normalized to $d(z)= e^{cz}$ by translation and scaling. Its developing map satisfies $d(z+1) = e^c d(z)$ and $d(z+\tau)= e^{c\tau} d(z)$. Thus the holonomy is generated by $$\begin{gathered} z\circ \gamma_1 = \rho_1(z) = e^{c}z \\ z\circ \gamma_2= \rho_2(z) = e^{c \tau}z\end{gathered}$$ Given an affine developing map $z:\tilde{V} \to \mathbb{C}$, we are going to find its underlying conformal structure. Notice that it is incorrect to conclude that $c$ and $c \tau$ are respectively $\log \frac{(z\circ \gamma_1)_j}{z_j}$ and $\log \frac{(z\circ \gamma_2)_j}{z_j}$ since the branch for the imaginary part of $\log$ is unclear. However, it could be fixed as follows: Associating every face (i.e. dual vertex) with the circumcenter defines $z^{*}:F \to \mathbb{C}$ which has the same affine holonomy representation as $z$. Pick a simple path on the universal cover that project to a loop homotopic to $\gamma_r$ on the torus. Under $z^*$, the path becomes a polygonal curve with vertices $(z^{*}_0,z^{*}_1,\dots,z^{*}_n=\rho_1(z^{*}_0),z^{*}_{n+1}=\rho_1(z^{*}_{1}))$ and we define $$\begin{aligned} \label{eq:hhol} h_r:= \sum_{i=1}^{n} \log \frac{z^*_{i+1}-z^*_{i}}{z^*_{i}-z^*_{i-1}}\end{aligned}$$ where the imaginary of the logarithmic for each term take values in $(-\pi,\pi)$, which is positive if the curve is turning right while negative otherwise. Notice that the $|\operatorname{Im}\log \frac{z^*_{i+1}-z^*_{i}}{z^*_{i}-z^*_{i-1}}|$ is the corner angle of the triangle opposite to $\angle z^*_{i-1}z^*_{i}z^*_{i+1}$. It can verified that the complex parameter $\tau$ and the affine parameter $c$ can be determined by $$c= h_1 \quad \text{and} \quad c\, \tau= h_2.$$ \[lem:bounded\] For any Delaunay cross ratio system, both $\operatorname{Im}c$ and $\operatorname{Im}c \tau$ are bounded by a constant depending only on the triangulation. Equation implies that $$|\operatorname{Im}c| = |\sum_{i=1}^{n} \operatorname{Im}\log \frac{z^*_{i+1}-z^*_{i}}{z^*_{i}-z^*_{i-1}}| \leq n \pi$$ where $n$ depends on the triangulation and is independent of the cross ratio systems. The same argument applies to $\operatorname{Im}c \tau$. Properness {#sec:properness} ---------- \[prop:proper\] Suppose $\Theta$ is a Delaunay angle structure on the torus. Then the projection from the space of Delaunay cross ratios $P(\Theta)$ to the Teichmüller space of the torus is a proper map. Suppose a sequence of Delaunay cross ratio systems $\{\operatorname{cr}^{(k)}\}_{k\in \mathbb{N}}$ in $P(\Theta)$ is given such that their underlying marked conformal structures $\{\tau^{(k)}\}_{k\in \mathbb{N}}$ converge to some conformal structure $\tau^{(\infty)}$ in the upper half plane. Proposition \[prop:rivin\] implies that the sequence of Delaunay cross ratio systems can be represented, up to a sign, by the sequence $\{(A_1^{(k)},A_2^{(k)})\}_{k\in \mathbb{N}}$ in $\mathbb{R}^2$ such that under an affine developing map, these numbers represent the scaling part of the holonomy. Namely, for $r=1,2$ and $k \in \mathbb{N}$ $$A_r^{(k)} = {\operatorname{Re}}h^{(k)}_r$$ where $h$ is defined in . We claim that $\{(A_1^{(k)},A_2^{(k)})\}_{k\in \mathbb{N}}$ has a convergent subsequence. Notice that $$\begin{aligned} \label{eq:sequh} h^{(k)}_2= c^{(k)} \tau^{(k)} = \tau^{(k)} h^{(k)}_1 \end{aligned}$$ Taking the imaginary of both sides yields for $k\in \mathbb{N}$ $${\operatorname{Im}}h^{(k)}_2 = \operatorname{Im}\tau^{(k)} \operatorname{Re}h^{(k)}_1 +\operatorname{Re}\tau^{(k)} \operatorname{Im}h^{(k)}_1 =A^{(k)}_1 \operatorname{Im}\tau^{(k)} +\operatorname{Re}\tau^{(k)} \operatorname{Im}h^{(k)}_1$$ Recall from Lemma \[lem:bounded\] that both $\{\operatorname{Im}h^{(k)}_1\}_{k\in \mathbb{N}}$ and $\{\operatorname{Im}h^{(k)}_2\}_{k\in \mathbb{N}}$ are bounded. Since $\{\tau^{(k)}\}_{k\in \mathbb{N}}$ converges to $\tau^{(\infty)}$ in the upper half plane, we can assume $\{\tau^{(k)}\}_{k\in \mathbb{N}}$ is bounded by considering a subsequence. Thus $\{A^{(k)}_1\}_{k\in \mathbb{N}}$ is bounded and hence has a converging subsequence with the limit denoted as $A^{(\infty)}_1$. Similarly, taking the real part of yields that there is a subsequence of $\{A^{(k)}_2\}_{k\in \mathbb{N}}$ converging to some number $A^{(\infty)}_2$ in $\mathbb{R}$. By Rivin’s variational argument (Proposition \[prop:rivin\]), there is a Delaunay cross ratio system $\operatorname{cr}^{(\infty)}$ corresponding to $(A^{(\infty)}_1,A^{(\infty)}_2)$. Furthermore, one can show that both $\operatorname{cr}$ and $(h_1,h_2) \in \mathbb{C}^2$ depends continuously on the parameters $(A_1,A_2)$ for the affine structures. Hence we deduce that $\{\operatorname{cr}^{(k)}\}_{k\in \mathbb{N}}$ has a subsequence converging to $\operatorname{cr}^{(\infty)}$ and its underlying conformal structure is $\tau^{(\infty)}$ by taking the limit of Equation as $k\to \infty$. Immersion around non-Euclidean tori {#sec:immersion} ----------------------------------- Under an infinitesimal change of the complex projective structure with the conformal structure fixed, the change in the affine holonomy is of the form $$\begin{aligned} (\dot{\rho}_1(z),\dot{\rho}_2(z)) = (\dot{c} e^{c}z, \dot{c}\tau e^{c\tau}z).\end{aligned}$$ \[prop:injective\] Suppose a Delaunay cross ratio system induces a non-Euclidean affine torus ($c\neq0$). Then a discrete holomorphic quadratic differential $q$ yields an infinitesimal change in the complex projective structure while preserving the conformal structure if and only if $q \equiv 0$. We prove by contradiction: Suppose we have a Delaunay cross ratio and we have a developing map whose holonomy has fixed points at the origin and infinity. We write its holonomy as $\rho_1(z) = e^{c}z$ and $\rho_2(z) = e^{c \tau}z$. Let $q$ be a discrete holomorphic quadratic differential as in the assumption and $\dot{z}$ be an infinitesimal deformation of the developing map given by Proposition \[prop:hqdharmonic\]. Since $z$ is equivariant with respect to the fundamental group, we have for $$\begin{aligned} (\dot{z}\circ \gamma_1)_i &= d\rho_1 (\dot{z}_i) + \dot{\rho}_1 (z_i)= e^{c}\dot{z}_i+ \dot{c}e^{c} z_i \\ (\dot{z}\circ \gamma_2)_i &= d\rho_2 (\dot{z}_i) + \dot{\rho}_2 (z_i)= e^{c\tau}\dot{z}_i+ \dot{c}\tau e^{c\tau} z_i\end{aligned}$$ By Proposition \[prop:hqdharmonic\], it determines a discrete harmonic function and it turns out to be a multi-valued function on the torus with constant periods: for every triangle $\{ijk\}$ and $r=1,2$ $$\begin{aligned} (u\circ \gamma_r)_i &= \operatorname{Im}\left( \frac{(\dot{z}\circ \gamma_r)_j - (\dot{z}\circ \gamma_r)_i}{(z\circ \gamma_r)_j - (z\circ \gamma_r)_i} + \frac{(\dot{z}\circ \gamma_r)_k - (\dot{z}\circ \gamma_r)_i}{(z\circ \gamma_r)_k - (z\circ \gamma_r)_i} - \frac{(\dot{z}\circ \gamma_r)_j - (\dot{z}\circ \gamma_r)_k}{(z\circ \gamma_r)_j - (z\circ \gamma_r)_k} \right) \\ &= u_i + \operatorname{Im}( \dot{c} \tau^{r-1})\end{aligned}$$ It induces that its conjugate harmonic function $v:\hat{F} \to \mathbb{R}$ must have constant periods as well. We write for $r=1,2$ $$\begin{aligned} (v \circ \gamma_r)_{ijk} = v_{ijk} + \delta_r\end{aligned}$$ where $\delta_r \in \mathbb{R}$. Using , we deduce that for $r=1,2$ $$\delta_r = -\operatorname{Re}( \dot{c} \tau^{r-1})$$ Though $u$ is a multi-valued function on the torus, its difference $u_j- u_i$ is well defined and we can compute its Dirichlet energy: $$\mathcal{E}_{T}(u):= \frac{1}{2} \sum_{ij \in E} (\cot \angle jkl + \cot \angle jil) (u_j - u_i)^2 = \frac{1}{2} \sum_{ij \in E} (v_{ijk} - v_{jil}) (u_j - u_i)$$ where the sum is over the edges of the torus. Analogous to the Riemann Bilinear Identity in the classical theory, Bobenko and Skopenkov [@Bobenko2016] showed that the Dirichlet energy of such a multi-valued discrete harmonic function is determined by its periods: if $u:\hat{V} \to \mathbb{R}$ is a multivalued discrete harmonic function on a torus and $v:\hat{F} \to \mathbb{R}$ is its conjugate harmonic function with constant periods $A_1,A_2 \in \mathbb{C}$ where ${\operatorname{Re}}A_r = u \circ \gamma_r - u $ and ${\operatorname{Im}}A_r = v \circ \gamma_r - v $ for $r=1,2$. Then the Dirichlet energy is given by the periods $$\mathcal{E}_T(u) = -\operatorname{Im}(A_1 \bar{A}_2)$$ Substituting $A_1= -\mathbf{i}\dot{c}$ and $A_2= -\mathbf{i}\dot{c}\tau$ where $\mathbf{i}=\sqrt{-1}$ for our case, our harmonic function $u$ has energy $$\mathcal{E}_T(u) = -\operatorname{Im}(|\dot{c}|^2 \bar{\tau}) = |\dot{c}|^2 \operatorname{Im}( \tau) \geq 0$$ We claim that this energy is not achievable by a discrete harmonic function unless $\dot{c}=0$. Notice that $u$ is defined on the vertices of a triangulation. We can extend it piecewisely over faces to obtain a piecewise linear function $\tilde{u}: \tilde{M} \to \mathbb{R}$, which has the same periods as $u$. Its Dirichlet energy as in the classical theory is given by: $$\mathcal{E}(\tilde{u}) := \iint_M |\operatorname{grad}\tilde{u}|^2 dA$$ Using the property of the cotangent Laplacian [@Pinkall1993], we have $$\mathcal{E}(\tilde{u}) = \mathcal{E}_T(u) =|\dot{c}|^2 \operatorname{Im}( \tau)$$ We compare this energy with a smooth multi-valued harmonic function on the torus: Consider the function $u^{\dagger}:={\operatorname{Re}}(-\mathbf{i} \frac{\dot{c}}{c} \log z)$ defined on the universal cover of $\mathbb{C}-{0}$. Pulled back by the developing map, it defines a harmonic function on the universal cover of the torus. Indeed it is a smooth multi-valued harmonic function on the torus with constant periods: $$\begin{aligned} u^{\dagger} \circ \gamma_1(z) - u^{\dagger}(z) &= {\operatorname{Re}}(-\mathbf{i}\frac{\dot{c}}{c} \log (e^c z)) - {\operatorname{Re}}(-\mathbf{i}\frac{\dot{c}}{c} \log z) = {\operatorname{Re}}(-\mathbf{i}\dot{c}) = \operatorname{Im}(\dot{c}) \\ u^{\dagger} \circ \gamma_2(z) - u^{\dagger}(z) &= {\operatorname{Re}}(-\mathbf{i}\frac{\dot{c}}{c} \log (e^{c\tau} z)) - {\operatorname{Re}}(-\mathbf{i}\frac{\dot{c}}{c} \log z) = {\operatorname{Re}}(-\mathbf{i}\dot{c}\tau)=\operatorname{Im}(\dot{c} \tau) \end{aligned}$$ whose conjugate harmonic function $u^{\dagger}:= \operatorname{Im}(-\mathbf{i}\frac{\dot{c}}{c} \log z)$ has periods $-\operatorname{Re}(\dot{c})$ and $-\operatorname{Re}(\dot{c}\tau) $ similarly. Using the Riemann bilinear identity from the smooth theory, the Dirichlet energy of $u^{\dagger}$ on the torus equals to $\mathcal{E}(u^{\dagger}) = |\dot{c}|^2 \operatorname{Im}( \tau)$. However, notice that $u^{\dagger}$ is the unique minimizer (up to a constant) of the Dirichlet energy among multi-valued piecewise smooth functions with periods $\operatorname{Im}(\dot{c})$ and $\operatorname{Im}(\dot{c}\tau)$, we have if $\dot{c} \neq 0$ $$|\dot{c}|^2 \operatorname{Im}( \tau) = \mathcal{E}(u^{\dagger}) < \mathcal{E}(\tilde{u}) = |\dot{c}|^2 \operatorname{Im}( \tau)$$ which is a contradiction. It implies $\dot{c}=0$ and the discrete harmonic function $u$ is constant by the maximum principle. Using Proposition \[prop:hqdharmonic\], we have $q\equiv0$. For a Delaunay angle structure $\Theta$ on the torus, Proposition \[prop:injective\] shows that $d(\pi\circ f)$ is injective over $P(\Theta)/\{\operatorname{cr}_0\} $ and hence $ \pi\circ f:P(\Theta)/\{\operatorname{cr}_0\} \to \mathcal{T}(M)/\{\tau_0\}$ is a local homeomorphism. On the other hand, Proposition \[prop:proper\] implies the map is proper. Thus $\pi\circ f:P(\Theta)/\{\operatorname{cr}_0\} \to \mathcal{T}(M)/\{\tau_0\}$ is a finite-sheet covering. Examples {#sec:examples} ======== Non-Delaunay cross ratio systems: Jessen’s orthogonal icosahedron {#sec:jessen} ----------------------------------------------------------------- Jessen’s icosahedron [@Jessen1967] is combinatorially a regular icosahedron with some edges flipped (see Fig. \[fig:Jessen\] left). Its vertices lie on a sphere and all dihedral angles are either $\pi/2$ or $3\pi/2$. It is a non-convex triangulated sphere that is infinitesimally flexible, i.e. it admits an infinitesimal deformation of vertices such that edge lengths are preserved. It is known in [@Lam2015] that the non-trivial infinitesimal isometric deformation induces a non-vanishing holomorphic quadratic differential on its stereographic image (see Fig. \[fig:Jessen\] right). Notice that the orthogonal icosahedron consists two kind of edges: Every vertex is connected to four short edges and one long edge. We define $q_{ij}=1$ on the short edges and $q_{ij}=-4$ on the long edges. One immediately have $\sum_j q_{ij}=0$ for every vertex $i$. Denoting $z$ the stereographic image of the vertices, one can check $\sum_j q/(z_j-z_i) =0$ around every vertex and $q$ is a holomorphic quadratic differential. We thus obtain a non-Delaunay triangulation of a sphere that carries a non-trivial holomorphic quadratic differential. \(A) at (1,-2,0); 1 (E) at (-1,-2,0);2 (D) at (2,0,1);3 (H) at (2,0,-1);4 (B) at (0,-1,2);5 (I) at (0,1,2);6 (F) at (-2,0,1);7 (G) at (-2,0,-1);8 (bA) at (1,2,0);9 (bE) at (-1,2,0);10 (C) at (0,-1,-2);11 (bI) at (0,1,-2);12 (A)–(B)–(C)–(A); (B)–(E)–(C); (A)–(D)–(B); (B)–(F)–(E); (E)–(G)–(C); (A)–(H)–(C); (D)–(I)–(F); (F)–(D); (A)–(bA)–(D); (H)–(bA); (E)–(bE); (I)–(bA); (I)–(bE); (I)–(bI); (bI)–(H); (bI)–(G); (bI)–(bA); (bI)–(bE); (H)–(G); (G)–(bE); (F)–(bE); ![Jessen’s orthogonal icosahedron (left) and its stereographic projection (right).[]{data-label="fig:Jessen"}](jessens.png){width="70.00000%"} Jessen’s orthogonal icosahedron provides a construction to obtain a non-Delaunay cross ratio system on a given surface with the dimension of discrete holomorphic quadratic differentials as large as one desires. To see this, suppose a cross ration system $\operatorname{cr}$ is given on a triangulation $T$ of a surface and consider its developing map. Pick a face $\phi$ of $T$. Then we attach a Möbius image of Jessen’s icosahedron to all the lifts of $\phi$ on the universal cover under the developing map. In this way, we obtain a new triangulation $\tilde{T}$ of the given surface together with a new cross ratio system $\tilde{cr}$. It can be verified that $\dim Q^{\mathbb{R}}(\tilde{\operatorname{cr}}) = \dim Q^{\mathbb{R}}(\operatorname{cr}) +1$. Cross ratio systems on one-vertex triangulated torus {#sec:one-vertex} ---------------------------------------------------- We consider the one-vertex triangulation of the torus, which has three edges and two faces. On the universal cover, each vertex is connected to six edges and the equations in Definition \[def:crsys\] becomes: $$\begin{aligned} 1&= (\operatorname{cr}_1 \operatorname{cr}_2 \operatorname{cr}_3)^2 \\ 0&= (\operatorname{cr}_1 + \operatorname{cr}_1 \operatorname{cr}_2+ \operatorname{cr}_1 \operatorname{cr}_2 \operatorname{cr}_3) ( 1 + \operatorname{cr}_1 \operatorname{cr}_2 \operatorname{cr}_3)\end{aligned}$$ There are two cases for the solutions. **Case (a): $\operatorname{cr}_1 \operatorname{cr}_2 \operatorname{cr}_3=-1$**. One can verify that the holonomy $\rho_1$, $\rho_2$ share the same fixed points and can be normalized as complex affine transformations. On the other hand, $q_1,q_2,q_3$ on the edges yields a holomorphic quadratic differential if and only if $q_1+q_2+q_3=0$. It implies $ \dim_{\mathbb{C}} Q^{\mathbb{C}}=2$ and $ \dim_{\mathbb{R}} Q^{\mathbb{R}}=2$. In particular, the space of the solution with prescribed $| \operatorname{cr}|$ or $\operatorname{Arg}\operatorname{cr}$ is a manifold of real dimension $2$. **Case (b): $\operatorname{cr}_1 \operatorname{cr}_2 \operatorname{cr}_3=1$ and $\operatorname{cr}_1 + \operatorname{cr}_1 \operatorname{cr}_2+ \operatorname{cr}_1 \operatorname{cr}_2 \operatorname{cr}_3=0$** . One can show that $\operatorname{cr}_1, \operatorname{cr}_2, \operatorname{cr}_3$ are of the form $b, -(b+1)/b, -1/(1+b)$ for some complex number $b$ and it is non-Delaunay for any choice of $b$. To see this, pick a vertex $z_0$ and denote its neighboring vertices in $\hat{M}$ as $z_1,z_2,z_3,z_{\tilde{1}},z_{\tilde{2}},z_{\tilde{3}}$ in counterclockwise orientation. On can show that $z_i=z_{\tilde{i}}$ for $i=1,2,3$ and furthermore the holonomy exchange the fixed points of each other and is of type (III) in Section \[subsec:holonomy\], which does not appear for the torus in the smooth theory. In fact, the gluing construction in Proposition \[prop:cp1\] yields a surface with boundary in this case. On the other hand, $\{q_1,q_2,q_3\}$ is in the kernel of $D(\Phi\circ \exp)$ (see Section \[sec:hqd\]) if it is in the form of $$\begin{aligned} q_1 &= (b-1) q_2 \\ q_3 &= -b \, q_2\end{aligned}$$ up to scaling. It implies $\dim_{\mathbb{C}} Q^{\mathbb{C}} =1$. For quadratic differentials, since $q$ is required be to real-valued, we have $$\begin{aligned} \dim_{\mathbb{R}} Q^{\mathbb{R}} &= 1 \quad \text{if } b\in \mathbb{R} \\ \dim_{\mathbb{R}} Q^{\mathbb{R}} &= 0 \quad \text{otherwise}\end{aligned}$$ Discussion and open questions {#sec:discussion} ============================= In this section, we discuss some open questions that extend the conjecture of Kojima, Mizushima and Tan [@Kojima2003]. Surfaces with genus $g$ ----------------------- Theorem \[thm:holo\] and \[thm:covering\] should have counterparts for surfaces other than the torus. The first question is whether $P(\Theta)$ is a manifold for a given Delaunay angle structures. It is equivalently to asking if $Q^{\mathbb{R}}(\operatorname{cr})$ has the right dimension as the Teichmüller space. Suppose $\Theta$ is a Delaunay angle structure on a triangulated sphere ($g=0$). Then along $P(\Theta)$, we have $$\dim_{\mathbb{R}} Q^{\mathbb{R}}(\operatorname{cr}) =0.$$ and $P(\Theta)=\{\operatorname{cr}_0\}$ consists of only one element. The fact that $P(\Theta)=\{\operatorname{cr}_0\}$ consists of only one element is asserted by Rivin’s variational method [@Rivin]. To see $\dim_{\mathbb{R}} Q^{\mathbb{R}}(\operatorname{cr}) =0$, we can use discrete harmonic functions again. Consider a developing map of the underlying complex projective structure, it covers the sphere exactly once since there is only one complex projective structure on the sphere. We normalize the developing map such that only one circumdisk contains infinity and remove the corresponding face. What remains is an embedded triangulated disk in the plane with three boundary vertices. Notice that by Dirichlet’s principle, any discrete harmonic function on a domain with three boundary vertices must be a linear function. Thus, suppose $q \in Q^{\mathbb{R}}(\operatorname{cr})$. Then Proposition \[prop:hqdharmonic\] implies there is a corresponding discrete harmonic function on the triangulated disk which is a linear function. Hence $q\equiv 0$. The space of cross ratio systems (not necessary to be Delaunay) on a triangulated sphere $M=(V,E,F)$ is a manifold of complex dimension $|V|-3$. We can put the vertices on the sphere arbitrarily as long as the endpoints of every edge are distinct. Thus, the configuration space modulo Möbius transformations has complex dimension $|V|-3$. Similarly, one can assign a vector field to each vertex arbitrarily for a given developing map. Modulo infinitesimal Möbius transformations, it implies $\dim_{\mathbb{C}} Q^{\mathbb{C}}(\operatorname{cr})=|V|-3$. \[thm:dim\] For a Delaunay cross ratio system $\operatorname{cr}$ on a closed triangulated surface with genus $g$, we have $$\begin{aligned} \dim_{\mathbb{C}} Q^{\mathbb{C}}(\operatorname{cr}) = \begin{cases} |V| + 1 \quad &\text{if } g =1 \\ |V| + 6g- 6 \quad &\text{if } g \geq 2 \end{cases} \end{aligned}$$ and $$\begin{aligned} \dim_{\mathbb{R}} Q^{\mathbb{R}}(\operatorname{cr}) = 6g - 6 \quad \text{if } g \geq 2 \end{aligned}$$ \[conj:full\] For any Delaunay angle structure $\Theta$ on a closed triangulated surface with genus $g>1$, the holonomy map $$\operatorname{hol}: P(\Theta) \to \operatorname{Hom}( \pi_1(M), PSL(2,\mathbb{C}))/PSL(2,\mathbb{C}))$$ is an embedding of a manifold homeomorphic to $\mathbb{R}^{6g-6}$. Furthermore its projection to the Teichmüller space is diffeomorphic. Prescribed hyperbolic metrics ----------------------------- Indeed, every Delaunay cross ratio system is associated with a locally convex pleated surface in hyperbolic 3-space, where the shearing coordinate between neighboring facets is ${\operatorname{Re}}\log cr$ and the dihedral angle is captured by $\operatorname{Arg}\operatorname{cr}= {\operatorname{Im}}\log \operatorname{cr}$ (See [@Bobenko2010] for the construction). The Delaunay condition is equivalent to surface being locally convex. The space $P(\Theta)$ describe all these pleated surfaces with prescribed dihedral angles. In contrast, one can consider those locally convex pleated surfaces with a prescribed hyperbolic metric $d$. Whenever a triangulation $\mathcal{T}$ of the surface is fixed, the hyperbolic metric $d$ is described by the shear coordinates $X_{d,\mathcal{T}}:E \to \mathbb{R}$. Thus the space of Delaunay cross ratio systems that induce pleated surfaces with hyperbolic metric $d$ can be written as $$P(d) = \cup_{\mathcal{T}} P(X_{d,\mathcal{T}})$$ where $P(X_{d,\mathcal{T}})$ consists of all the Delaunay cross ratio systems $\operatorname{cr}$ on $\mathcal{T}$ with $\log |\operatorname{cr}|\equiv X_{d,\mathcal{T}}$ and the union is taken over all triangulations of the surface with the same vertex set [@Bobenko2010; @Gu20182]. This space is non-empty by the discrete uniformization theorem. If the underlying surface is a torus, then the argument in Section \[sec:cp1tori\] and \[sec:immersion\] yields that $P(d)$ is a surface and its projection to the Teichmüller space is an immersion. It is interesting to consider Conjecture \[conj:full\] with prescribed hyperbolic metrics in place of prescribed dihedral angles. Acknowledgment {#acknowledgment .unnumbered} ============== The author would like to thank Feng Luo and Richard Schwartz for fruitful discussions and Masashi Yasumoto for comments on the draft. Rigidity on complex projective structures ========================================= Here we provide an alternative proof that two Delaunay cross ratio systems inducing the same complex projective structure on the torus must be identical. Though this result can be deduced from Rivin’s variational method (Section \[sec:Rivin\]), the following proof involves the maximum principle which might be inspiring for surfaces with high genus. Let $cr, \tilde{cr}:E \to \mathbb{C}$ be two Delaunay cross ratios system on a triangulated torus with $\operatorname{Arg}(cr) = \operatorname{Arg}(\tilde{cr})$. Suppose they admit developing map $z, \tilde{z}$ with affine holonomy $\rho_j(z)=\alpha_j z + \beta_j$ and $\tilde{\rho}_j(z)=\tilde{\alpha}_j z + \tilde{\beta}_j$ such that $|\alpha_j|=|\tilde{\alpha}_j|$ for $j=1,2$. Then the developing maps $z,\tilde{z}$ differ by a similarity and we have $\operatorname{cr}= \tilde{\operatorname{cr}}$. We claim that the corresponding circumscribed circles differ by a global scaling. It follows from the maximum principle as follows: Whenever there is an edge with $\operatorname{Arg}\operatorname{cr}=0$, we remove the edge and merge the neighboring faces since the corresponding circumcircles coincide. By the definition of Delaunay cross ratio systems, we obtain a cell decomposition $(V,E,F)$ of $M$. The Delaunay condition further implies that under the developing map each face is a convex polygon. Indeed, each face is obtained by merging triangles and so we can argue by induction: If the face consists of one triangle, then it is convex obviously. If there are two triangles $\{ijk\},\{ilj\}$, then $\operatorname{Arg}\operatorname{cr}_{ij} = 0$ implies they form a convex quadrilateral. Suppose we have a convex cyclic k-gon and attach a new triangle $\{jil\}$ to an edge $\{ij\}$ of the k-gon, then the new vertex $z_l$ must lie on the arc between $z_i,z_j$ not containing the other vertices since $\operatorname{Arg}\operatorname{cr}_{ij} = 0$. Hence the resulting $(k+1)$-gon is convex cyclic. We denote the $R:\hat{F}\to \mathbb{R}_{>0}$ and $\tilde{R}:\hat{F}\to \mathbb{R}_{>0}$ the radii of the circumcircles under the developing map $z$ and $\tilde{z}$ of the universal cover $\tilde{M}$. We consider the ratio of the radii $\sigma:= \tilde{R}/R$. Since the holonomy is affine, the ratio $\sigma$ is periodic. Hence, $\sigma$ has a local maximum on a face $f_0$. We denote the neighboring faces as $f_1,f_2,\dots, f_k$ and the intersection angles of the circumcircles as $\phi_1,\phi_2, \dots, \phi_k$. Note $\phi_i = \operatorname{Arg}\operatorname{cr}_i \in (0,\pi)$. We focus on the developing map $z$. Suppose $z_i z_{i+1}$ is the common chord shared by the circles at $f_0$ and $f_i$. We write the circumcenter of $f_0$ as $O$ and denote $2 \alpha_i$ the angle at the center from $Oz_i$ to $Oz_{i+1}$ in counterclockwise orientation (See figure \[fig:centralangles\]). Then $$R_0 \sin \alpha_i = R_i \sin (\phi_i - \alpha_i) = R_i( \sin \phi_i \cos \alpha_i - \cos \phi_i \sin \alpha_i)$$ Hence $$\cot \alpha_i = ( \frac{1}{\sin \phi_i}(\frac{R_0}{R_i} + \cos \phi_i))$$ Since $\sigma_0 \geq \sigma_i $, we have $$\cot \alpha_i = \frac{1}{\sin \phi_i}( \frac{\sigma_i \tilde{R}_0}{\sigma_0 \tilde{R}_i} + \cos \phi_i) \leq \frac{1}{\sin \phi_i}( \frac{\tilde{R}_0}{\tilde{R}_i} + \cos \phi_i) = \cot \tilde{\alpha}$$ Note $\cot(\cdot)$ is monotone decreasing on $(0, \pi)$. Hence $\alpha_i \geq \tilde{\alpha}_i$. ![Central angles. Left: Center within the polygon. Right: Center outside the polygon.[]{data-label="fig:centralangles"}](fig5){width="100.00000%"} ![Central angles. Left: Center within the polygon. Right: Center outside the polygon.[]{data-label="fig:centralangles"}](fig6){width="90.00000%"} On the other hand, because the vertices of the face $f_0$ are in cyclic order on the circle, the sum of the central angles are equal to $2\pi$, we have $$2\pi = \sum 2 \alpha_i \geq \sum 2 \tilde{\alpha}_i = 2 \pi.$$ Thus all the equalities should hold and $\sigma_i = \sigma_0$ for $i=1,2,\dots,k$. Hence the maximum principle holds and $\sigma$ must be constant.

--- author: - 'Eshan D. Mitra' - 'William S. Hlavacek [^1]' bibliography: - 'references.bib' title: Bayesian Uncertainty Quantification for Systems Biology Models Parameterized Using Qualitative Data --- Abstract {#abstract .unnumbered} ======== **Motivation:** Recent work has demonstrated the feasibility of using non-numerical, qualitative data to parameterize mathematical models. However, uncertainty quantification (UQ) of such parameterized models has remained challenging because of a lack of a statistical interpretation of the objective functions used in optimization.\ **Results:** We formulated likelihood functions suitable for performing Bayesian UQ using qualitative data or a combination of qualitative and quantitative data. To demonstrate the resulting UQ capabilities, we analyzed a published model for IgE receptor signaling using synthetic qualitative and quantitative datasets. Remarkably, estimates of parameter values derived from the qualitative data were nearly as consistent with the assumed ground-truth parameter values as estimates derived from the lower throughput quantitative data. These results provide further motivation for leveraging qualitative data in biological modeling.\ **Availability:** The likelihood functions presented here are implemented in a new release of PyBioNetFit, an open-source application for analyzing SBML- and BNGL-formatted models, available online at [www.github.com/lanl/PyBNF](www.github.com/lanl/PyBNF).\ Introduction ============ Mathematical models of the dynamics of cellular networks, such as those defined using BioNetGen Language (BNGL) [@Faeder2009] or Systems Biology Markup Language (SBML) [@Hucka2003], require parameterization for consistency with experimental data. Conventional approaches use quantitative data such as time courses and dose-response curves to parameterize models. We and others have demonstrated that it is also possible to use non-numerical, qualitative data in automated model parameterization [@Oguz2013; @Pargett2013; @Pargett2014; @Mitra2018a]. Our demonstration [@Mitra2018a] used qualitative data in combination with quantitative data. In the method of [@Mitra2018a], the available qualitative data are used to formulate inequality constraints on outputs of a model. Parameterization is performed by minimizing a sum of static penalty functions [@Smith1997] derived from the inequalities. Given a list of $n$ inequalities of the form $g_i<0$ for $i=1,...,n$, where the $g_i$ are functions of model outputs, the objective function is defined as $$\sum_{i=1}^n C_i\cdot\max(0, g_i) \label{eq:static}$$ Static penalty functions have long been used in the field of constrained optimization [@Smith1997]. Each violated inequality contributes to the objective function a quantity equal to a distance from constraint satisfaction (e.g., the absolute difference between the left-hand side and right-hand side of the inequality), multiplied by a problem-specific constant weight $C_i$. The objective function of Equation \[eq:static\] becomes smaller as inequalities move closer to satisfaction, thus guiding an optimization algorithm toward a solution satisfying more of the inequalities. In the study of [@Mitra2018a], the approach proved effective in obtaining a reasonable point estimate for the parameters of a 153-parameter model of yeast cell cycle control developed by Tyson and coworkers [@Chen2000; @Chen2004; @Csikasz-Nagy2006; @Oguz2013; @Kraikivski2015], which had previously been parameterized by hand tuning. The static penalty function approach has limitations. Most notably, the approach requires choosing problem-specific weights $C_i$ for the objective function. Although heuristics exist to make reasonable choices for the weights [@Mitra2018a], there is no rigorous method to do so. A related challenge in using qualitative data is performing uncertainty quantification (UQ). Bayesian UQ (described in many studies, such as [@Kozer2013] and [@Klinke2009]) is a valuable approach that generates the multivariate posterior probability distribution of model parameters given data. This distribution can be used for several types of analyses. 1) The marginal distribution of each parameter can be examined to find the most likely value of that parameter and a credible interval. 2) Marginal distributions of pairs of parameters can be examined to determine which parameters are correlated. 3) Prediction uncertainty can be quantified by running simulations using parameter sets drawn from the distribution. Unfortunately, meaningful Bayesian UQ cannot be performed for models parameterized using qualitative data and penalty function-based optimization, because the penalty functions are heuristics. They are not grounded in statistical modeling. Here, we present likelihood functions that can be used in parameterization and UQ problems incorporating both qualitative and quantitative data. We first present a likelihood function that can be used with binary categorical data, and then a more general form to use with ordinal data comprising three or more categories. We implemented the option to use these likelihood functions in fitting and in Bayesian UQ in our software PyBioNetFit [@Mitra2019a]. We built on existing PyBioNetFit support for qualitative data, which previously allowed only the static penalty function approach. In the first section of Results, we derive the new likelihood functions, which have similarities to both the chi squared likelihood function commonly used in curve fitting with quantitative data, and the logistic function commonly used to model classification error in machine learning. In the second section, we describe how we have added support for the new likelihood functions in PyBioNetFit and provide a guide to using them in optimization and UQ. In the third section, we provide an example application of the new software features. This example shows that qualitative datasets are potentially valuable resources for biological modeling. Methods ======= Likelihood functions presented in Results were implemented as options in PyBioNetFit v1.1.0, available online at <https://github.com/lanl/pybnf>. PyBioNetFit supersedes the earlier BioNetFit [@Thomas2016; @Hlavacek2018]. To illustrate use of the new functionality, we configured and solved an example UQ problem (described in Section \[sec:application\]) using PyBioNetFit v1.1.0. Configuration, model, and synthetic data files used for this example are available online (<https://github.com/RuleWorld/RuleHub/tree/2019Aug27/Contributed/Mitra2019Likelihood>). The model that we used has been published in BNGL format [@Faeder2009] in earlier work [@Harmon2017]. We took the published parameterization to be the ground truth. We adapted the simulation commands included in the BNGL file to produce degranulation outputs for specific conditions, as appropriate for our synthetic datasets described below. We considered 11 instances of the problem using different qualitative and quantitative datasets. To generate synthetic quantitative data, we simulated the model with the assumed ground-truth parameterization, and added Gaussian noise to the desired degranulation outputs. To generate synthetic two-category qualitative data, we performed the same procedure, but recorded only whether the noise-corrupted primary degranulation response was greater or less than the noise-corrupted secondary degranulation response. To generate synthetic three-category qualitative data, we followed the same procedure, but recorded that the primary and secondary responses were approximately equal if the difference between the two responses was less than a designated threshold, which was set at $4.2\times 10^4$ arbitrary units. We performed MCMC sampling using PyBioNetFit’s parallel tempering algorithm. For each dataset considered, we performed four independent runs and combined all samples obtained. Each run consisted of four Markov chains for each of nine temperatures, for a total of 36 chains, with samples saved from the four chains at temperature 1, run for a total of 50,000 steps including an unsampled 10,000-step burn-in period. Each run was performed using all 36 cores of a single Intel Broadwell E5-2695 v4 cluster node. Complete configuration settings are provided in the PyBioNetFit configuration file online. Results ======= Mathematical derivation ----------------------- ### Notation {#sec:problem} By way of introduction to our newly proposed likelihood function for qualitative data, we begin by reviewing Bayesian UQ and its associated likelihood function with a more conventional quantitative dataset. We are given an experimental dataset $\mathbf{y}=\{y_1,...,y_n\}$ and a model $f$. There is no restriction on what type of numerical measurement each $y_i$ represents; for example, it could represent a single data point of a time course, a sample mean of several independent and identically distributed measurements, or an arbitrary function of multiple measured quantities. Within a Bayesian framework, the $y_i$ are taken to be samples from the random variables $\{Y_1,...,Y_n\}$. The model $f$ takes as input a parameter vector ${\boldsymbol{\theta} }$ to predict the expected value of each data point $Y_i$, that is, $f_i({\boldsymbol{\theta} }) = E(Y_i)$. ${\boldsymbol{\theta} }$ is the realization of the random variable $\mathbf{\Theta}$. $f$ is assumed to be deterministic (e.g., an ODE model). Stochastic models would require additional treatment that is beyond the intended scope of this study. In Bayesian UQ, parameter uncertainty is quantified by the posterior probability distribution $P({\boldsymbol{\theta} }|\mathbf{y})$, the probability of a particular parameter set given the data. Markov chain Monte Carlo (MCMC) algorithms can be used to sample the posterior distribution using the fact that, by Bayes’ law, $P({\boldsymbol{\theta} }|\mathbf{y}) \propto P(\mathbf{y}|{\boldsymbol{\theta} })P({\boldsymbol{\theta} })$. The change in the value of $P(\mathbf{y}|{\boldsymbol{\theta} })P({\boldsymbol{\theta} })$ is used to determine whether a proposed move by the MCMC algorithm is accepted. $P({\boldsymbol{\theta} })$ is a user-specified distribution representing prior knowledge about the parameters. Therefore, an important prerequisite for performing Bayesian UQ is an expression for the *likelihood*, $P(\mathbf{y}|{\boldsymbol{\theta} })$. ### Chi squared likelihood function When performing conventional Bayesian UQ using only quantitative data, a common choice of likelihood function (e.g., see [@Kozer2013] and [@Harmon2017]) is the chi squared function. $$-\log P(\mathbf{y}|{\boldsymbol{\theta} }) \propto \chi^2({\boldsymbol{\theta} }) = \sum_{i=1}^n \frac{(y_i-f_i({\boldsymbol{\theta} }))^2}{2\sigma_i^2} \label{eq:chisq}$$ Here $\sigma_i$ is the standard deviation of the measurement $y_i$. If $y_i$ represents the sample mean of several independent trials, it is common to estimate $\sigma_i$ as the standard error of the mean. This likelihood function has a strong theoretical motivation. The underlying assumption is that each $Y_i$ has an independent Gaussian distribution with mean $f_i({\boldsymbol{\theta} })$ and standard deviation $\sigma_i$. Then the probability of a single data point $y_i$ given ${\boldsymbol{\theta} }$ is $$P(y_i|{\boldsymbol{\theta} }) = \frac{1}{\sqrt{2\pi}\sigma_i} \exp(\frac{-(y_i-f_i({\boldsymbol{\theta} }))^2}{2\sigma_i^2})$$ Given that the $Y_i$ are independent, the probability of the complete dataset $\mathbf{y}$ given ${\boldsymbol{\theta} }$ is given by the product $$P(\mathbf{y}|{\boldsymbol{\theta} }) = \prod_{i=1}^n \frac{1}{\sqrt{2\pi}\sigma_i} \exp(\frac{-(y_i-f_i({\boldsymbol{\theta} }))^2}{2\sigma_i^2}) \label{eq:product}$$ When performing MCMC sampling, we typically only need a value *proportional to* $P(\mathbf{y}|{\boldsymbol{\theta} })$ to calculate the ratio $P(\mathbf{y}|{\boldsymbol{\theta} }_1) / P(\mathbf{y}|{\boldsymbol{\theta} }_2)$ for two parameter sets ${\boldsymbol{\theta} }_1$ and ${\boldsymbol{\theta} }_2$. This ratio is used to determine, for example, the probability of transitioning from ${\boldsymbol{\theta} }_1$ to ${\boldsymbol{\theta} }_2$ in the Metropolis-Hastings algorithm. We therefore can ignore proportionality constants in Equation \[eq:product\] that are independent of ${\boldsymbol{\theta} }$. $$P(\mathbf{y}|{\boldsymbol{\theta} }) \propto \prod_{i=1}^n \exp(\frac{-(y_i-f_i({\boldsymbol{\theta} }))^2}{2\sigma_i^2}) \label{eq:propproduct}$$ Taking the negative logarithm of Equation \[eq:propproduct\] results in the conventional chi squared function (Equation \[eq:chisq\]). Therefore, under the assumptions stated in this section, the chi squared function represents the kernel of the negative log likelihood and can be rigorously used in Bayesian UQ algorithms. ### Likelihood function for qualitative data We now consider the situation in which the experimental data are qualitative. By qualitative data, we specifically mean observations that can be expressed as inequality constraints to be enforced on outputs of a model. Our problem statement is nearly identical to that presented in Section \[sec:problem\], except we are no longer given the dataset $\mathbf{y}$. Instead, for each $Y_i$, we are given a constant $c_i$, and told whether $y_i < c_i$ or $y_i > c_i$ was observed. $y_i$ is the sample generated from $Y_i$ and is never observed. $y_i < c_i$ (or $y_i > c_i$) is the observation, which has two possible outcomes. We explicitly write down the procedure to generate these qualitative observations from the $Y_i$, which we refer to as our *sampling model*: To generate observation $i$, sample $y_i$ from $Y_i$ and report whether $y_i < c_i$ or $y_i > c_i$. Without loss of generality, we assume all given observations have the form $y_i < c_i$. If some quantity $A$ yielded an observation $a > k$, we could set $Y_i=-A$ and $c_i=-k$. This form also supports the case of an inequality $A<B$ between two measured quantities, as we could set $Y_i=A-B$ and $c_i=0$. To perform Bayesian analysis, we require an expression for the probability of observing $y_i < c_i$ for all $i$ (rather than observing $y_i > c_i$ for some $i$), given a parameter set ${\boldsymbol{\theta} }$. As shorthand, we will write this as $P(\mathbf{y}<\mathbf{c}|{\boldsymbol{\theta} })$, where $\mathbf{y}$ is a vector of the $y_i$ and $\mathbf{c}$ is a vector of the $c_i$. Following the example of the chi squared likelihood function, we assume each $Y_i$ has a Gaussian distribution with a known standard deviation $\sigma_i$. The mean of the distribution is, as before, taken to be given by the model prediction $f_i({\boldsymbol{\theta} })$. With this distribution, the probability of observing $y_i < c_i$ is, by definition, given by the Gaussian cumulative distribution function (CDF). We will write the CDF of a Gaussian distribution with mean $\mu$ and standard deviation $\sigma$ evaluated at a point $x$ as $\textrm{cdf}(\mu,\sigma,x)$. The conditional probability of interest is as follows: $$P(y_i<c_i|{\boldsymbol{\theta} }) = \textrm{cdf}(f_i({\boldsymbol{\theta} }),\sigma_i,c_i) \label{eq:qualsingle}$$ We note that for ease of implementation, $\textrm{cdf}(\mu,\sigma,x)$ can be written in terms of the error function $\textrm{erf}(x)$, which is implemented in many standard libraries, including the Python and C++ standard libraries. $$\textrm{cdf}(\mu,\sigma,x) = \mu + \frac{1 + \textrm{erf}(\frac{x}{\sigma \sqrt{2}})}{2}$$ As shown in Figure \[fig:logistic\], Equation \[eq:qualsingle\] is intuitively reasonable. If the true mean value of $Y_i$ is much smaller than $c_i$ (relative to the scale of $\sigma_i$), we are very likely to observe $y_i<c_i$, whereas if the mean of $Y_i$ is much larger than $c_i$, we are very unlikely to observe $y_i<c_i$. If the true mean of $Y_i$ is close to $c_i$, we are uncertain whether the observation will be $y_i<c_i$ or $y_i>c_i$ in the face of measurement noise. We note that this function has a similar appearance to the logistic function, which is commonly used to model binary categorization in machine learning. Assuming independence of the $Y_i$, the probability of the entire dataset is given by the product. $$P(\mathbf{y}<\mathbf{c}|{\boldsymbol{\theta} }) = \prod_{i=1}^n \textrm{cdf}(f_i({\boldsymbol{\theta} }),\sigma_i,c_i) \label{eq:qualproduct}$$ Finally, we take the negative logarithm to obtain $$-\log P(\mathbf{y}<\mathbf{c}|{\boldsymbol{\theta} }) = \sum_{i=1}^n -\log \textrm{cdf}(f_i({\boldsymbol{\theta} }),\sigma_i,c_i) \label{eq:qualobj}$$ This function can be used for Bayesian UQ when considering qualitative data in an equivalent way to how the chi squared likelihood function is used when considering quantitative data. ![The proposed form for $P(y_i<c_i|{\boldsymbol{\theta} })$ (Equation \[eq:qualsingle\]). []{data-label="fig:logistic"}](Fig1.eps) ### Likelihood function for qualitative data with model discrepancy {#sec:twocat} The likelihood function in Equation \[eq:qualobj\] has a remaining limitation when it comes to real-world experimental data. To illustrate this concern, we point to the model developed by Tyson and co-workers of yeast cell cycle control [@Chen2000; @Chen2004; @Csikasz-Nagy2006; @Oguz2013; @Kraikivski2015]. Several versions of this model have been parameterized using qualitative data (viability status of yeast mutants) by hand-tuning [@Chen2000; @Chen2004; @Csikasz-Nagy2006; @Kraikivski2015] and with optimization algorithms [@Oguz2013; @Mitra2018a]. In all of these parameterization studies, most but not all of the qualitative observations were satisfied by the reported best-fit parameterization. A few of the observations, however, were different from the model predictions. Due to such anomalous observations, a likelihood model as we have described could give the dataset a very low likelihood given the model and parameters, even though there intuitively is good agreement between the parameterized model and dataset. How can we reconcile anomalous observations? An explanation given by Tyson and co-workers is that a model has a limited amount of detail, which is unable to capture every qualitative observation in the data [@Chen2004]. This explanation suggests using a statistical approach known as model discrepancy or model inadequacy [@Kennedy2001]. The principle of model discrepancy is that when calculating the likelihood of a dataset, one should take into account the difference between the model and reality. Although many statistical studies ignore model discrepancy, it has been shown to be important for performing effective statistical inference for certain problems [@Brynjarsdottir2014]. Given that qualitative data may be generated by high-throughput screening that could easily step outside the scope of a particular model, we believe model discrepancy is an especially important consideration for our applications. Existing treatments of model discrepancy often describe discrepancy with its own probability distribution, such as a Gaussian distribution that is autocorrelated in time [@Brynjarsdottir2014]. Such an approach, which uses an assumption that model discrepancy is correlated for similar observations, is hard to apply to our problem formulation in which the $Y_i$ are taken to be independent (possibly coming from different model outputs). Thus, we take a more generic approach of expressing model discrepancy as a constant probability $\epsilon_i$ for each qualitative observation. $\epsilon_i$ relates to the probability that a given observation is outside the scope of the model. We say that when an observation is made, there is a probability $\epsilon_i$ that $y_i < c_i$ is reported regardless of the expected value of $Y_i$ given by the model. Likewise, there is also a probability $\epsilon_i$ that $y_i > c_i$ reported regardless of $Y_i$. These statements can be formalized as part of our sampling model: To generate observation $i$, make a weighted random choice of one of the following possibilities: - With probability $1-2\epsilon_i$, sample $y_i$ from $Y_i$ and report whether $y_i<c_i$ or $y_i>c_i$ - With probability $\epsilon_i$, report $y_i<c_i$ - With probability $\epsilon_i$, report $y_i>c_i$ With this modification, we have the probability distribution $$P(y_i<c_i|{\boldsymbol{\theta} }) = \epsilon_i + (1-2\epsilon_i) \textrm{cdf}(f_i({\boldsymbol{\theta} }),\sigma_i,c_i)$$ and the likelihood function $$-\log P(\mathbf{y}<\mathbf{c}|{\boldsymbol{\theta} }) = \sum_{i=1}^n -\log (\epsilon_i + (1-2\epsilon_i) \textrm{cdf}(f_i({\boldsymbol{\theta} }),\sigma_i,c_i)) \label{eq:qualobjfinal}$$ Equation \[eq:qualobjfinal\] gives our recommended form for a likelihood function incorporating qualitative data with two possible categorical outcomes ($y_i<c_i$ or $y_i > c_i$). We will refer to this function as the *two-category likelihood function*. Note that although we introduced $\epsilon_i$ for dealing with model structure problems, it could also represent a shortcoming of our postulated Gaussian error model. For example, if an experimental instrument had some probability of reporting a false positive or negative, regardless of whether the mean of $Y_i$ is close to the threshold $c_i$, this non-Gaussian error could be accounted for by increasing the value of $\epsilon_i$. ### Likelihood function for ordinal data with more than two categories {#sec:threecat} We next derive a likelihood function for ordinal categorical data with more than two categories. For simplicity, we suppose an observation has three possible outcomes: $y_i<c_{i,1}$, $c_{i,1}<y_i<c_{i,2}$, and $y_i>c_{i,2}$, for constants $c_{i,1}$ and $c_{i,2}$. An example would be if we were making an ordinary qualitative observation ($y_i<c_i$ or $y_i>c_i$), but another possible outcome of the experiment is $y_i=c_i$ to within the experimental error. Then the cutoffs $c_{i,1}$ and $c_{i,2}$ could be chosen on either side of $c_i$ such that the outcome $c_{i,1}<y_i<c_{i,2}$ corresponds to $y_i$ within measurement error. From the definition of the Gaussian CDF we have $$\label{eq:yltc1} P(y_i<c_{i,1}|{\boldsymbol{\theta} }) = 1-\textrm{cdf}(f_i({\boldsymbol{\theta} }),\sigma, c_{i,1})$$ $$P(c_{i,1}<y_i<c_{i,2}|{\boldsymbol{\theta} }) = \textrm{cdf}(f_i({\boldsymbol{\theta} }),\sigma_i, c_{i,1}) - \textrm{cdf}(f_i({\boldsymbol{\theta} }),\sigma_i, c_{i,2}) \label{eq:pmiddle}$$ $$\label{eq:ygtc2} P(y_i>c_{i,2}|{\boldsymbol{\theta} }) = \textrm{cdf}(f_i({\boldsymbol{\theta} }),\sigma_i, c_{i,2})$$ A simplification is possible under the assumption that $c_{i,1}$ and $c_{i,2}$ are far enough separated that for any $E(Y_i)$, at most two of the three categories have non-negligible probability. That is, if $E(Y_i)$ is close enough to $c_{i,2}$ that observing $y_i>c_{i,2}$ is a probable outcome, $E(Y_i)$ is also high enough above $c_{i,1}$ that observing $y_i<c_{i,1}$ has a probability close to zero. Thus, we assume that for all ${\boldsymbol{\theta} }$, either $\textrm{cdf}(f_i({\boldsymbol{\theta} }),\sigma_i, c_{i,1}) = 1$ or $\textrm{cdf}(f_i({\boldsymbol{\theta} }),\sigma_i, c_{i,2}) = 0$. This assumption is reasonable because if it were false, it would mean the experiment cannot reliably distinguish between the three categories, and so the data would be better analyzed as two-category data. With this assumption, Equation \[eq:pmiddle\] can be rewritten as $$P(c_{i,1}<y_i<c_{i,2}|{\boldsymbol{\theta} }) = \textrm{cdf}(f_i({\boldsymbol{\theta} }),\sigma_i, c_{i,1}) * (1 - \textrm{cdf}(f_i({\boldsymbol{\theta} }),\sigma_i, c_{i,2})) \label{eq:pmiddle2}$$ Note that Equation \[eq:pmiddle2\] is equivalent to Equation \[eq:qualproduct\] for two independent constraints $c_{i,1}<y_i$ and $y_i<c_{i,2}$ arising from two-category observations. This makes for a convenient implementation: rather than explicitly considering the two-sided observation $c_{i,1}<y_i<c_{i,2}$, we can rewrite the observation as two independent one-sided observations $c_{i,1}<y_i$ and $y_i<c_{i,2}$ described by Equation \[eq:qualproduct\]. A modification to the two-category case is necessary when model discrepancy is included as in Equation \[eq:qualobjfinal\]. Here, care must be taken to ensure that in the sampling model the probability of all possible outcomes sums to 1. For example, a reasonable sampling model for a three-category observation would be the following: To generate observation $i$, make a weighted random choice of one of the following possibilities: - With probability $1-3\epsilon_i$, sample $y_i$ from $Y_i$ and report whether $y_i<c_{i,1}$ or $c_{i,1}<y_i<c_{i,2}$ or $c_{i,2}<y_i$ - With probability $\epsilon_i$, report $y_i<c_{i,1}$ - With probability $\epsilon_i$, report $c_{i,1}<y_i<c_{i,2}$ - With probability $\epsilon_i$, report $c_{i,2}<y_i$ Recall that in Equation \[eq:qualobjfinal\], in the case of model discrepancy, the observation is equally likely to be $y_i>c_i$ or $y_i<c_i$ (each of these events is assumed to have probability $\epsilon_i$). In contrast, using the above sampling model, it is half as likely to report $y_i<c_{i,1}$ (probability $\epsilon_i$) as to report $y_i>c_{i,1}$ (probability $2\epsilon_i$). We generalize Equation \[eq:qualobjfinal\] to account for the case of three-category observations by allowing for two separate parameters. We define the positive discrepancy rate $\epsilon_i^+$ as the probability that a constraint in the data is satisfied regardless of $Y_i$, and the negative discrepancy rate $\epsilon_i^-$ as the probability a constraint is violated regardless of $Y_i$. For example, with the above sampling model, for the observation $c_{i,2}<y_i$, we would use $\epsilon_i^+ = \epsilon_i$ and $\epsilon_i^- = 2\epsilon_i$ Our modified likelihood function is $$-\log P(\mathbf{y}<\mathbf{c}|{\boldsymbol{\theta} }) = \sum_{i=1}^n -\log (\epsilon_i^+ + (1-\epsilon_i^+-\epsilon_i^-) \textrm{cdf}(f_i({\boldsymbol{\theta} }),\sigma_i,c_i)) \label{eq:qualobjfinalplus}$$ We will refer to this function as the *many-category likelihood function*. The same formulation can be extended to allow for an arbitrary number of ordinal categories. For example, with four categories defined by the thresholds $c_{i,1}$, $c_{i,2}$, and $c_{i,3}$, we could write expressions analogous to Equations \[eq:yltc1\]-\[eq:ygtc2\] for $P(y_i<c_{i,1}|{\boldsymbol{\theta} })$, $P(c_{i,1}<y_i<c_{i,2}|{\boldsymbol{\theta} })$, $P(c_{i,2}<y_i<c_{i,3}|{\boldsymbol{\theta} })$, and $P(y_i>c_{i,3}|{\boldsymbol{\theta} })$. We illustrate the use of Equation \[eq:qualobjfinalplus\] with a concrete example. Suppose we have a quantity of interest with the corresponding random variable $A$, and we make a qualitative observation with three possible outcomes: $a<100$, $a\approx100$, or $a>100$. Suppose also that based on the sensitivity of the assay, we know that any value of $a$ in the range 85–115 would be reported as “$a\approx100$.” Given this knowledge of the assay sensitivity, we take the standard deviation of $A$ to be 5, that is, we can only confidently report $a<100$ if $a$ is 3 standard deviations below the threshold of 100. We choose the sampling model shown in Figure \[fig:ex3\]A, giving a base probability of 0.03 to each possible outcome due to model discrepancy. Note that this sampling model follows the requirement that the probabilities of all possible outcomes sum to 1. We then formulate the constraint(s) as shown in Figure \[fig:ex3\]B, depending on whether the actual observation is $a<100$, $a\approx100$, or $a>100$. The resulting probabilities are shown in Figure \[fig:ex3\]C as a function of the expected value of $A$ predicted by the model. When using the many-category likelihood function, it is important to consider the underlying sampling model, and choose $\epsilon_i^+$ and $\epsilon_i^-$ in a way such that the probabilities in the sampling model sum to 1. An example of how to correctly choose $\epsilon_i^+$ and $\epsilon_i^-$ given a sampling model is presented in Section \[sec:application\]. ![Example constraints and probabilities arising from a qualitative observation with three possible categorical outcomes. (A) The sampling model associated with the observation. (B) Inequalities and $\epsilon^{+}$ and $\epsilon^{-}$ values associated with each possible observation outcome (C) Plots and equations giving the probability of each possible observation outcome as a function of the expected value of model output $A$. []{data-label="fig:ex3"}](Fig2.eps) ### Combined likelihood function If independent quantitative and qualitative data are available, it is straightforward to combine the chi squared likelihood function for quantitative data with one of the newly presented likelihood functions for qualitative data. One would simply sum Equations \[eq:chisq\] and \[eq:qualobjfinal\] (or \[eq:qualobjfinalplus\]) to obtain the kernel of the negative log likelihood for the combined dataset. The relative weighting of the two datasets is determined by the standard deviations for the quantitative data points and the values of $\sigma_i$ and $\epsilon_i$ for the qualitative observations. Software implementation ----------------------- We implemented the likelihood functions described in the previous section in PyBioNetFit v1.1.0. PyBioNetFit supports both the two-category (Equation \[eq:qualobjfinal\]) and many-category (Equation \[eq:qualobjfinalplus\]) likelihood functions for qualitative data, and supports combining these functions with the chi squared likelihood function for quantitative data. The new options were added via an extension of the Biological Property Specification Language (BPSL) supported by PyBioNetFit. As previously described [@Mitra2019a], a BPSL statement consists of an inequality, followed by an enforcement condition, followed by a weight. For example, in the statement $$\texttt{A<4 at time=1 weight 2}$$ the inequality is `A<4` (referring to some modeled quantity $A$), the enforcement condition is `time=1` (referring to time 1 in a time course), and the weight is declared by `weight 2`. This weight declaration refers to $C_i$ in the previously described static penalty function (Equation \[eq:static\]). Using this formulation, the term added to an objective function for this constraint would be $2 \cdot \max(0,A(1)-4)$, where $A(1)$ is model output $A$ evaluated at time = 1. In PyBioNetFit v1.1.0, we added an alternative to the weight clause to specify parameters of the new likelihood functions. As described in Section \[sec:twocat\], for each inequality in the data, the two-category likelihood function has two user-configurable parameters: the probability $\epsilon_i$ of measuring $y_i<c_i$ regardless of the distribution of $Y_i$, and the standard deviation $\sigma_i$ of the quantity $Y_i$. The value of $1-2\epsilon_i$ (i.e., the probability that the distribution of $Y_i$ is relevant to the experimental result) is supplied to PyBioNetFit with the `confidence` keyword. $\sigma_i$ is supplied to PyBioNetFit with the `tolerance` keyword. Therefore, an example BPSL statement using the two-category likelihood function is $$\texttt{A<4 at time=1 confidence 0.98 tolerance 0.5}$$ This statement would result in using the likelihood function of Equation \[eq:qualobjfinal\] with $\epsilon_i=0.01$, $\sigma_i=0.5$, $c_i=4$, and $Y_i=A(1)$. The resulting term added to the likelihood function is $-\textrm{log}(0.01+0.98\cdot\textrm{cdf}(A(1),0.5,4))$. PyBioNetFit also supports the use of the many-category likelihood function (Equation \[eq:qualobjfinalplus\]) through the specification of separate positive and negative discrepancy rates. In this case, the `confidence` keyword is replaced with the keywords `pmin` to specify $\epsilon_i^-$ (i.e., the minimum value of $P(y_i<c_i|{\boldsymbol{\theta} })$) and `pmax` to specify $1-\epsilon_i^+$ (i.e., the maximum value of $P(y_i<c_i|{\boldsymbol{\theta} })$). For example, the BPSL statement $$\texttt{A<4 at time=1 pmin 0.01 pmax 0.98 tolerance 0.5}$$ would use Equation \[eq:qualobjfinalplus\] with $\epsilon_i^+=0.02$, $\epsilon_i^-=0.01$, $\sigma_i=0.5$, $c_i=4$, and $Y_i=A(1)$. The resulting term added to the likelihood function is $-\textrm{log}(0.01+0.97\cdot\textrm{cdf}(A(1),0.5,4))$. When writing these statements in BPSL, care must be taken to ensure that results are statistically valid. First, note that the `tolerance` specifies the standard deviation of the final random variable $Y_i$ used to sample $y_i$ in Equation \[eq:qualobjfinal\]. For example in the above statement, it refers to the standard deviation of $A(1)$. In the statement `A>B at time=5 confidence 0.98 tolerance 0.5`, `tolerance` refers to the standard deviation of $A(5)-B(5)$, i.e., the sum of the standard deviations of $A(5)$ and $B(5)$. In the statement `A>4 always confidence 0.98 tolerance 0.5`, `tolerance` refers to the standard deviation of $\min(A(t))$, rather than the value of $A$ at any particular time. Second, it is important to keep in mind the underlying sampling model to correctly set `confidence` or `pmin` and `pmax`. For example, in the sampling model of Fig \[fig:ex3\]A, there are three possible constraints each with probability 0.03 to be satisfied due to model discrepancy and probability 0.06 to be violated due to model discrepancy. Therefore, the correct setting is `pmin 0.03 pmax 0.94`. Third, when using PyBioNetFit’s enforcement keywords `always`, `once`, and `between`, it is important to be sure the possible categories in the sampling model are mutually exclusive and cover all possible outcomes. For example, if one of two possible categorical outcomes is `A>4 always`, the other must be `A<4 once` (not `A<4 always`). Likewise, if one category is `A>4 between time=5,time=10`, its negation is `A<4 once between time=5,time=10`. We note that the `once between` enforcement condition used here is a new feature of BPSL introduced in PyBioNetFit v1.1.1. The sampling model is never explicitly input into PyBioNetFit, as equations \[eq:qualobjfinal\] and \[eq:qualobjfinalplus\] are defined regardless of whether the sampling model is well-defined. It is the user’s responsibility to choose a well-defined sampling model and specify constraints accordingly to obtain meaningful results. ![Configuration of the example problem in BPSL. As described in the text, we considered the problem assuming either (A) two possible observation categories or (B) three possible categories. The left column shows an example BPSL statement for each possible category. In these BPSL statements, `p1` refers to the primary degranulation and `p3_`$<t>$ refers to the secondary degranulation after a delay of $t$ minutes. Note that in the three-category case, the middle category requires two separate BPSL statements. The right column shows simulated trajectories of the primary (left) and secondary (right) degranulation responses that are consistent with the BPSL statement. For the three-category case, `degrHigh` and `degrLow` are functions defined in the BNGL model file for use in the BPSL statements.[]{data-label="fig:setup"}](Fig3.eps) Example application {#sec:application} ------------------- To demonstrate the use of qualitative likelihood functions in PyBioNetFit, we performed Bayesian UQ on a synthetic example problem based on the study of [@Harmon2017]. The model of [@Harmon2017] describes the degranulation of mast cells in response to two consecutive stimuli with multivalent antigen. In the original study, it was found that depending on the time delay between the two stimuli, the secondary response could be either stronger or weaker than the primary response. The original data consisted of quantitative degranulation measurements for six different time delays. In our synthetic problem, we suppose that the experimental data took a different form. Rather than quantitative measurements, we assume that it is only possible to measure whether the secondary degranulation is higher or lower than the primary degranulation. These measurements can be seen as case-control comparisons between several conditions of interest (secondary degranulation at various time delays) and a control (primary degranulation). We assume that these measurements can be made at a larger number of time delays than were used in the original study (i.e., we have a less precise but higher throughput instrument than in the actual study). We generated synthetic data of this form using the published parameter values of the model as ground truth. For each time delay in the data, we ran a simulation, and added Gaussian noise to the primary and secondary degranulation outputs before recording whether the primary or secondary was higher. We generated datasets ranging from 4 to 64 time delays. The resulting datasets were implemented in BPSL as illustrated in Figure \[fig:setup\]A. Note that we set the `confidence` to 0.98, allowing for a 0.02 chance of model discrepancy (although there is no true model discrepancy in this synthetic problem). We set the `tolerance` to $1.4 \times 10^4$, which is the standard deviation of the difference between the primary and secondary degranulation values (i.e., twice the standard deviation of the added noise for each individual degranulation value). We configured PyBioNetFit jobs to perform Bayesian UQ by parallel tempering for each dataset. The results are shown in Figure \[fig:bayes\]A-D and Figures S1–S5. Not surprisingly, as the number of qualitative observations increases, we obtain a narrower distribution of parameter values, and these narrower distributions include the ground truth parameter values. This result demonstrates that with a sufficient amount of qualitative data, it is possible to find nontrivial credible intervals for parameter values.  To demonstrate the use of the many-category likelihood function (Equation \[eq:qualobjfinalplus\]), we repeated the analysis using three-category synthetic data. Our three categories allow the secondary degranulation to be measured as smaller, larger, or within error of the primary degranulation. The three-category dataset was declared in BPSL as illustrated in Figure \[fig:setup\]B. Compared to the two-category synthetic data, modifications were required as described in Section \[sec:threecat\]. The assumed sampling model used for the constraints in Figure \[fig:setup\]B is the following, where $Y_i$ represents the primary degranulation minus the secondary degranulation: To generate observation $i$, make a weighted random choice of one of the following possibilities: - With probability 0.97, sample $y_i$ from $Y_i$ and report whether $y_i<-4.2\times 10^4$ or $-4.2\times 10^4<y_i<4.2\times 10^4$ or $4.2\times 10^4<y_i$ - With probability 0.01, report $y_i<-4.2\times 10^4$ - With probability 0.01, report $-4.2\times 10^4<y_i<4.2\times 10^4$ - With probability 0.01, report $4.2\times 10^4<y_i$ We have chosen a threshold of $4.2\times 10^4$ for the difference between primary and secondary degranulation that qualifies as “within error.” This value is three times the standard deviation of $Y_i$, giving the separation of categories required in Section \[sec:threecat\] (i.e., any sampled $y_i$ is consistent with at most two possible categories). This condition allows us to define the middle category ($-4.2\times 10^4<y_i<4.2\times 10^4$) using two independent BPSL statements. The choice of threshold is reflected in the BPSL by the use of the model outputs referred to as `degrHigh` and `degrLow`. Based on the sampling model, each category has a minimum probability of 0.01 due to model discrepancy, and a maximum probability of 0.98 (because the other two categories each have a minimum of 0.01). Therefore, we set `pmin` to 0.01 and `pmax` to 0.98 instead of using the `confidence` keyword. Finally, the `tolerance` is set to $1.4\times 10^4$, the same as for the two-category dataset. The results of parallel tempering using this dataset are illustrated in Figure \[fig:bayes\]E and Figures S6–S10. As expected, compared to the results with two-category dataset of the same size, some parameters are bounded more tightly around their ground truth values. For comparison, we also performed the analysis using synthetic quantitative data generated at the same time delays as in the original study (Figure \[fig:bayes\]F and Figure S11). The quantitative dataset produced distributions even tighter than those of the three-category qualitative data. It is notable how close we can get to the results with quantitative data by using purely qualitative data. Discussion ========== Here we have presented a new statistical framework for using qualitative data in conjunction with Bayesian UQ for biological models. In these models, unidentifiable parameters are common, but Bayesian analysis can determine which parameters and correlations are identifiable, and to what extent the model has predictive value despite unidentifiable parameters. We see this framework as a more statistically rigorous improvement upon our previously described static penalty function approach [@Mitra2018a] (Equation \[eq:static\]). Our new framework can be used for statistical analysis, whereas the previous formulation was simply a heuristic for finding a single reasonable parameter set. Our new likelihood function has applications beyond Bayesian UQ. It can also, like the static penalty function, be used with optimization algorithms to find a point estimate of the best parameters. In such a problem, the global minimum (assuming it can be found by an optimization algorithm) is the maximum likelihood estimate, i.e., the maximum of the posterior distribution. The new likelihood function may also be used for UQ by profile likelihood analysis [@Kreutz2013]. The static penalty function may remain more efficient at point estimation. The cdf-based likelihood function has the limitation that when far from constraint satisfaction, its gradient is near zero, and so it cannot effectively guide the optimization algorithm toward constraint satisfaction. In contrast, the static penalty function provides useful information for optimization at any distance from constraint satisfaction. One potential workflow could be to use the static penalty function for initial optimization, followed by the likelihood function for refinement and evaluation of the best fit. We note that under our new framework, each constraint now has two adjustable settings: $\epsilon_i$ and $\sigma_i$. This may appear worse than the single weight parameter $C_i$ in the static penalty formulation, but the advantage is that both of these parameters have a statistical interpretation. $\epsilon_i$ represents the probability of model discrepancy resulting in a qualitative observation that occurs regardless of the model and its predicted mean. $\sigma_i$ represents the standard deviation of the quantity considered in the constraint. This value might seem challenging to estimate, given we may not even be able to quantitatively measure the quantity of interest. However, much of the same intuition holds as when dealing with Gaussian-distributed quantitative data. In particular, if there is a difference of $2\sigma_i$ between a threshold and the mean, we can be reasonably confident (probability 97.7%) that an observation would yield the correct result (greater or less than the threshold). With a difference of $3\sigma_i$, we can be extremely confident (probability 99.87%). To choose $\sigma_i$, a reasonable thought process would be to ask, “How large of a difference would there have to be for the experiment to be sure to detect the difference?”, and set $\sigma_i$ equal to one third of that difference. Both parameters can be seen as optional. If we don’t expect a scenario in which a constraint is impossible to reconcile with our model, we can set $\epsilon_i=0$, ignoring this aspect of the likelihood function. Likewise, if we have no way to estimate the standard deviation of the measured quantity, we could set $\sigma_i=0$ and use $\epsilon_i$ to set a fixed probability of satisfying the constraint. Thus, the two adjustable constants should be seen as an opportunity to provide all available information about a qualitative observation of interest, rather than as a burden for manual adjustment. We expect that our new formulation of a likelihood function derived from qualitative data will be useful in future modeling studies and will help facilitate the wider adoption of qualitative data as a data source for model parameterization. Acknowledgements {#acknowledgements .unnumbered} ================ We acknowledge computational resources provided by the Institutional Computing program at Los Alamos National Laboratory, which is operated by Triad National Security, LLC for the NNSA of DOE under contract 9233218CNA000001. We thank Steven Sanche for useful discussions. Funding {#funding .unnumbered} ======= This work has been supported by NIH/NIGMS grant R01GM111510. [^1]: Corresponding author. wish@lanl.gov

--- abstract: 'We combine experiments with simulations to investigate the fluid-structure interaction of a flexible helical rod rotating in a viscous fluid, under low Reynolds number conditions. Our analysis takes into account the coupling between the geometrically nonlinear behavior of the elastic rod with a non-local hydrodynamic model for the fluid loading. We quantify the resulting propulsive force, as well as the buckling instability of the originally helical filament that occurs above a critical rotation velocity. A scaling analysis is performed to rationalize the onset of this instability. A universal phase diagram is constructed to map out the region of successful propulsion and the corresponding boundary of stability are established. Comparing our results with data for flagellated bacteria suggests that this instability may be exploited in nature for physiological purposes.' author: - 'M.K. Jawed$^{1}$' - 'N.K. Khouri$^{2}$' - 'F. Da$^{3}$' - 'E. Grinspun$^{3}$' - 'P.M. Reis$^{1,2,}$' bibliography: - 'flagella.bib' title: Propulsion and instability of a flexible helical rod rotating in a viscous fluid --- Bacteria often rely on the deformation of filamentary helical structures, called flagella, for locomotion [@purcellthe1997; @*silverman1977bacterial]. The propulsion arises from a complex fluid-structure interaction (FSI), between the structural flexibility of the flagellum and the viscous forces generated by the flow. This FSI may lead to geometrically nonlinear deformations [@turner2000real; @*berg2013cell; @vogel2012motor], which in turn can be exploited for turning [@son2013bacteria], tumbling  [@macnab1977normal], bundle formation [@brown2012flagellar] and polymorphic transformations [@calladineconstruction1975; @darnton2007force]. Resistive force theories (RFT) [@gray1955propulsion; @lighthill1976flagellar] are often used to model the role of viscous forces on flexible filaments [@machinwave1958; @*takano2003numerical; @*takano2003analysis], at low Reynolds number. These simplify the viscous loading by introducing local geometry-dependent drag coefficients. More sophisticated descriptions consider non-local hydrodynamic effects, albeit typically assuming that the filament is rigid such that elastic forces are ignored [@rodenborn2013propulsion; @spagnoliecomparative2011]. The few studies that have coupled long-range hydrodynamics with elasticity either assume small deflections [@kim2005deformation] or approximate the filament as a network of springs [@flores2005study; @olsonmodeling2013], thereby oversimplifying the mechanics of the problem. One exception is the study of buckling of a straight elastic filament loaded by viscous stresses [@wigginsflexive1998; @*coqrotational2008; @*qianshape2008; @*manghipropulsion2006; @*wolgemuth2000twirling]. More recently, a systematic computational study has been performed on a discretized model based on Kirchhoff’s theory for elastic rods (in the form of a chain of connected spheres), coupled with RFT [@vogel2012motor]. This study was significant in that it was the first, to the best of our knowledge, to report a series of buckling instabilities of the flagellum that arise during locomotion and suggested its relevance to the biological system. Moreover, it addressed the important rotation-translation coupling. However, recent experiments [@rodenborn2013propulsion; @chattopadhyay2009effect; @*jung2007rotational] and simulations [@spagnoliecomparative2011] have pointed to the oversimplifying nature of RFT to model propulsion in a quantitatively predictive manner. Therefore, there is a timely need for a description that fully couples a geometrically nonlinear elastic model of the filament [@kirchhoff1992uber] with long-range hydrodynamic interactions [@lighthill1976flagellar; @*lighthillhelical1996; @*johnson1980improved], along with precision experiments for detailed validation. ![(a) Experimental apparatus: a helical rod (1), is rotated by a motor (2), inside a glycerin bath (3), that is enclosed by an external water tank (4) for temperature control. Two orthogonal video cameras (5,6) record the rod. (b) Schematic diagram of the rod. (c-d) Examples of the deformed rod at $\omega=0.6$ rad/s, from both experiments and simulations: (c1-c2) helical and (d1-d2) buckled configurations (movie in [@SM]). See text for the properties of rod and fluid. []{data-label="fig:fig1"}](Fig1){width="\columnwidth"} Here, motivated by the locomotion of uniflagellated bacteria, we perform a combined experimental and numerical investigation of the dynamics of a helical elastic filament rotated in a viscous fluid. Our goal is to predictively understand the underlying mechanical instabilities. In our precision model experiments, we reproduce and systematically quantify the dynamics of the filament, as a function of the control and physical parameters of the system. In parallel, we perform numerical simulations that model the elastic rod using the Discrete Elastic Rods (DER) method [@bergou2008discrete; @*bergou2010discrete], coupled to a viscous drag described by Lighthill’s slender body theory (LSBT) [@lighthill1976flagellar]. After validating the numerics against experiments, we quantify the steady state configurations of the filament and explore the multi-dimensional phase space of the resulting propulsive force. Existing data on the physical properties of bacterial flagella is sparse [@hoshikawaelastic1983; @*hoshikawaelastic1985; @*trachtenberg1992rigidity; @*flynntheoretical2004; @*kim2005deformation], given the experimental challenges associated with their measurement. As such, we seek a dimensionless description that encompasses the geometric parameters of natural flagella, with an emphasis on the propulsive force and onset of buckling. The phase boundary for this instability is mapped out, onto which we locate a number of natural bacterial systems. These results motivate us to speculate on the potential biological relevance of the mechanical instabilities of rotating flagella. In Fig. \[fig:fig1\]a, we provide a photograph of our apparatus. As a model for flagella, we cast a series of elastomeric rods with vinylpolysiloxane [@miller2014shapes; @*lazarus2013contorting; @*lazarus2013continuation] and independently varied each of the geometric parameters (axial length, $l$, or contour length, $L$, helix radius, $R$, pitch, $\lambda$, and cross-sectional radius, $r_0$, or area moment of inertia, $I=\pi r_0^4/4$) and material properties (the Young’s modulus, $E$ of the rod was determined by analyzing the shape of a suspended annulus [@adami2013elasto]). The rod was assumed to be incompressible (Poisson’s ratio, $\nu \approx 0.5$). During fabrication, a polyvinyl chloride tube was wrapped around a cylindrical object along a helical geometry, and was used as a mold for the rods. The density of the rod was adjusted by adding iron filings (Dowling Magnets) to the polymer, prior to casting. Once cured and demolded, the filament was clamped at one end, immersed in a bath of glycerin (20$\times$20$\times$30$\,\mathrm{cm^3}$), and rotated using a stepper motor [@SM]. Using digital imaging, we reconstructed the deformed configurations of the filament and quantified its dynamics. To ensure constant and reproducible values for the fluid viscosity, the glycerin bath was inserted within an external water tank to accurately control the temperature within $\pm0.5\mathrm{^\circ C}$ (Brinkmann Lauda RC6). By tuning the temperature, 7.6$\le\theta[\mathrm{^\circ C}]\le$ 32.4, we varied the viscosity of glycerin in the range 0.50 $\le \mu[\mathrm{Pa\cdot s}] \le$4.45 ($\pm0.05\,\mathrm{Pa\cdot s}$). The density of glycerin is $\rho_m = 1.24~\mathrm{g/cm^3}$, and despite our best effort for density matching, our rods had a slightly higher value ($\lesssim5\%$) than glycerin, which is however included in the numerics. For our simulations, we combined DER [@bergou2008discrete; @*bergou2010discrete], a robust and efficient computational tool for the mechanics of rods, and LSBT [@lighthill1976flagellar], a viscous force model that accounts for non-local hydrodynamics. Both DER and LSBT were independently validated against precision experiments in Refs. [@jawed2014coiling] and [@rodenborn2013propulsion], respectively. The helical rod is described by its centerline, parameterized by the arc-length, $s$ (Fig. \[fig:fig1\]b). For the fluid loading, LSBT is used to relate [@lighthill1976flagellar] the local velocity, $\mathbf{u}(s)$, and the force per unit length, $\mathbf{f}(s)$, at each point on the rod centerline: $$\mathbf{u}(s) = \frac {\mathbf{f}_{\perp} (s) } { 4 \pi \mu } + \int_{|\mathbf{r}(s', s) | > \delta } \mathbf{f} (s') \cdot \mathbb{J} (\mathbf{r}) \mathrm{d}s\rq{}, \label{eq:LighthillSBT}$$ where $\mathbf{f}_{\perp} (s) = \mathbf{f}(s) \cdot \left( \mathbb{I} - \mathbf{t}(s) \otimes \mathbf{t}(s) \right)$ is the component of $\mathbf f$ in the plane perpendicular to the tangent, $\mathbf{t}(s)$, $\mathbf{r}(s^\prime, s)$ is the position vector from $s^\prime$ to $s$, $\delta = \frac { r_0 \sqrt e} {2}$ is the natural cutoff length, and $\mathbb J (\mathbf r) = \frac {1} {8 \pi \mu} \left( \frac {\mathbb I} {| \mathbf r|} + \frac {\mathbf r \otimes \mathbf r} {| \mathbf r|^3 } \right)$ is the Oseen tensor. Eq. (\[eq:LighthillSBT\]) is then discretized and cast into a $3N$ sized linear system of the form $\mathbf{U} = \mathbf{A F}$, where $N$ is the number of nodes of the discretized rod [@SM]. At each time step in DER, the viscous forces, $\mathbf F$, are evaluated from the velocities, $\mathbf U$, and the matrix $\mathbf A$ that only depends on the geometric configuration of the rod. To advance in time, we apply this external force together with elastic forces, update the rod configuration and iterate. Self-contact, possible only after buckling, is neglected throughout, although this does not compromise the agreement with experiments. We first establish a connection with existing literature for a naturally straight filament rotating in a viscous fluid [@coqrotational2008; @*qianshape2008] and then consider naturally curved rods. In Fig. \[fig:fig2\]a, we present experimental photographs of undeformed (top) and deformed (bottom) configurations, for three representative cases of decreasing the natural radius of curvature of the rod, $R$, while fixing its contour length at $L=12.00 \pm 0.05\,\mathrm{cm}$. These three cases are: i) straight rod (clamped at an angle of $\alpha = 15^\circ$, for consistency with Ref. [@coqrotational2008]), ii) moderately curved rod ($R/L=0.56$, $\alpha=0$) and iii) highly curved rod ($R/L=0.29$, $\alpha=0$). All other parameters for this part of the study were kept fixed: $r_0 = 1.58 \pm 0.02~\mathrm{mm}$, rod density, $\rho_r =1.306 \pm 0.002~\mathrm{g/cm^3}$, $E=1255\pm 49~\mathrm{kPa}$, and $\mu = 1.32 \pm 0.05~\mathrm{Pa\cdot s}$, which ensured a Reynolds number $< 10^{-1}$. The resulting configurations (after initial transients) for these three cases are found to vary dramatically with $R/L$. ![(a) Experimental images of (a1) straight rod, (a2) $R/L=0.56$, and (a3) $R/L=0.29$ in their undeformed (top row) and deformed state (bottom row) at $\omega=3.14$ rad/s. (b) Normalized suspended height, $\bar h$, versus angular velocity, $\omega$, for experiments and simulations. (c) Simulation data of normalized propulsive force, $\bar F_p$, versus $\omega$. []{data-label="fig:fig2"}](Fig2_Resubmit){width="\columnwidth"} The role of natural curvature is quantified further in Fig. \[fig:fig2\]b, where we plot the steady state suspended height (vertical distance from the clamp to the bottom of the rod, $h$) normalized by the height in the non-rotating case, *i.e. $\bar h=h/h_0$*, as a function of the imposed angular velocity, $\omega$. Excellent quantitative agreement is found between experiments and simulations, with no fitting parameters. Given that the value of the propulsion force at the clamp is too low to be measured experimentally, we extract it from the simulations at each time step as $F_p = - \int_0^L (\mathbf{f}\cdot\mathbf{e}_z) \mathrm{ds}$, where $\mathbf f$ is obtained from Eq. (\[eq:LighthillSBT\]). In Fig. \[fig:fig2\]c, we normalize the propulsive force, $\bar{F}_p=F_p L^2/(EI)$, by the characteristic bending force in the rod and plot $\bar{F}_p$ versus $\omega$. Qualitative and quantitative differences are observed between the three cases: straight, moderately curved, and highly curved rods. The first two undergo a shape transition at $\omega \approx 0.2\,$ rad/s. However, the propulsion force of the straight rod is always positive, whereas it is negative for the moderately curved rod. By contrast, the highly curved rod exhibits a non-monotonic behavior: $\bar{h}$ first increases to reach a maximum value at $\omega \approx 1.0$ rad/s, where buckling occurs, and eventually $\bar{h} \approx 1$. Since the propulsion depends on the deformed configuration, the resulting $F_p$ vs. $\omega$ relation is markedly different from the previous two cases; $F_p$ first changes sign from negative to positive at $\omega \approx 0.3\,$ rad/s, reaches a maximum at $\omega \approx 2.4\,$ rad/s and then changes sign again at $\omega \approx 2.6\,$rad/s. The coupled effect of curvature, flexibility and fluid forces can thus produce nontrivial behavior in both geometry and propulsion. Reassured by the quantitative agreement between numerics and experiments, we turn to the dynamics of helical filaments as macroscopic analogues of bacterial flagella [@fujii2008polar; @spagnoliecomparative2011]. These rods were rotated in the glycerin bath ($\mu = 1.6 \pm 0.05~\mathrm{Pa\cdot s}$) at angular velocities in the range $0<\omega\,[\mathrm{rpm}]\leq8\,$. For now, we focus on a case with: $E=1255\pm 49~\mathrm{kPa}$, $\rho_r = 1.273 \pm 0.022\,\mathrm{g/cm^3}$, $l=20 \pm 0.5\,\mathrm{cm}$, $\lambda=5 \pm 0.5\,\mathrm{cm}$, $R=1.59 \pm 0.1\,\mathrm{cm}$, and $r_0 = 1.58 \pm 0.02\,\mathrm{mm}$. Under these conditions, the Reynolds number always remains smaller than $10^{-2}$ and the Stokes flow assumption is appropriate throughout. ![ (a) Sequence of experimental images at $\omega = 0.6$ rad/s. Material properties are provided in the text (movie in  [@SM]). Time series of (b) normalized height, $\bar h (t)$, and (c) normalized propulsive force, $\bar F_p (t)$. (d) Normalized height, $\bar h$, in steady state ($t>360$ s) versus $\omega$. (e) Normalized propulsive force, $\bar F_p$, in steady state versus $\omega$, with the shaded region representing the standard deviation.[]{data-label="fig:fig3"}](Fig3){width="\columnwidth"} In Fig. \[fig:fig3\]a, we present a sequence of experimental photographs for our representative helical rod rotated at $\omega = 0.6\,$rad/s, starting from rest. The corresponding time series for the normalized suspended height, $\bar{h}$, is plotted in Fig. \[fig:fig3\]b for both experiments (solid line) and simulations (dashed line), with good agreement between the two. Any mismatch arises primarily from self-contact that is neglected in the simulations. Initially ($t\lesssim 100\,$s), $\bar{h}\sim 1$ but the configuration eventually becomes increasingly distorted due to the appearance of regions of chiral inversion, even if the axis of the helix remains vertical. At later times, the rod bundles and the suspended height reaches an approximate steady state, with $\bar{h} \sim 0.6$. The time series of the normalized propulsive force, $\bar{F}_p = F_p l^2 / (EI)$, calculated from the simulations, is plotted in Fig. \[fig:fig3\]c. Concurrently with the drop in $\bar h$ at $t \approx 150\,$s, $\bar{F}_p$ becomes increasingly unsteady, which we will show arises through a buckling instability. {width="100.00000%"} In Fig. \[fig:fig3\]d, we plot the late time average of $\bar{h}$ over $740$s past the initial transients ($t>360\,$s) versus $\omega$. We find that $\bar{h}\sim 1$ up to $\omega_b=0.51\,$rad/s, after which it sharply drops. Hereafter, we shall refer to $\omega_b$ as the *critical buckling velocity*, above which fluid loading arising due to the rotation causes the helical filament to buckle. The corresponding $\bar{F}_p$ is plotted in Fig. \[fig:fig3\]e as a function of $\omega$ and we find that it increases monotonically up to $\omega_b$. Note that a rigid helix would yield a linear dependence between $\bar{F}_p$ and $\omega$ [@rodenborn2013propulsion] (dashed line in Fig. \[fig:fig3\]e). For $\omega<\omega_b$, flexibility of the helical filament leads to a sub-linear $\bar{F}_p$, when compared to the rigid case. For $\omega>\omega_b$, the average value of $\bar{F}_p$ drops sharply, albeit with significant fluctuations (the shaded region in Fig. \[fig:fig3\]e correspond to the standard deviation of the averaged force value). Similar to $F_p$, we can also measure the input torque, $T_p$, necessary to sustain rotation, and this allows us to quantify the propulsive efficiency, which we define as $\eta = F_p l / T_p$. The efficiency remains almost constant as a function of angular velocity for $\omega < \omega_b$, but drops to a lower value upon buckling [@SM]. Also, see Ref. [@vogel2012motor] for a characterization of propulsion as a function of input torque. We proceed to rationalize the dependence of the buckling velocity on the physical parameters of our system. The viscous force acting on the helical filament scales as $F_v \sim \mu \omega l^2$. Regarding the helix as an effective beam allows us to estimate its critical buckling load as $F_c \sim EI/l^2$. Instability is expected to occur when $F_v \approx F_c$, which yields $\omega_b = \bar{\omega}_b E I / (\mu l^4)$, where $\bar \omega_b = \hat{\bar{\omega}} (\lambda/l, R/\lambda)$ is a dimensionless function that depends on the geometry of the helix alone. To systematically investigate the dependence of $\omega_b$ on the various parameters, we start with the values for the rod used in Figs. \[fig:fig3\] and \[fig:fig4\]a, and plot $\omega_b$ versus $Er_0^4/ (\mu l^4)$, for a given geometry ($\lambda/l=4$ and $R/\lambda=0.32$). In both experiments (filled symbols) and simulations (open symbols), each one of the four parameters $\{ E, r_0, \mu, l\}$ is independently varied, while fixing the other three. We find that the data collapse onto a straight line, thereby supporting our scaling analysis. To characterize the effect of the geometry $\{\lambda/l, R/\lambda\}$, we independently vary the helix radius, $R$, and pitch, $\lambda$, while fixing $E$, $\mu$, $r_0$ and $l$. In Fig. \[fig:fig4\]b, we plot $\omega_b$ as a function of the normalized pitch, $\lambda/l$ at fixed $R = 1.59\,$cm. The dependence of $\omega_b$ on the normalized radius, $R/\lambda$, at $\lambda=5\,$cm is shown in Fig. \[fig:fig4\]c. From both of these plots, we conclude that $\omega_b$ varies nonlinearly with $\lambda/l$ and $R/\lambda$. These parameters must be taken into account when mapping the results from our model system to a regime that is relevant to bacterial locomotion, which is address next. Finally, we take advantage of the efficiency of our algorithm to provide a biologically relevant description of $\bar{\omega}_b$. We use the parameters of the rod used in Fig. \[fig:fig3\], except that the fluid and flagellum are assumed to be density matched and the axial length is increased to $l=0.4\,$m, so that $r_0 \ll \{\lambda, R, l\}$. Supported by the data (see [@SM]), we approximate $\bar \omega_b = \hat M (R/\lambda) \, \hat N (\lambda / l)$ by $\hat M (R/\lambda) = (R / \lambda) ^{-m}$ with $m = 1.96 \pm 0.05$ (see  [@SM] for details). Using this result, in Fig. \[fig:fig4\]d, we construct a phase diagram for the propulsive force, versus both $\bar \omega \cdot (R / \lambda)^m$ and $\lambda / l $, where $\bar \omega = \omega \mu l^4 / (EI)$ is the normalized angular velocity. In Fig. \[fig:fig4\]d, we also superpose the parameter values corresponding to bacterial flagella of specific organisms (see caption), which are estimated by taking the characteristic orders of magnitude values for $\mu = 10^{-3}\,\mathrm{Pa.s}$, $EI = 10^{-23}\,\mathrm{N m^2}$ [@takano2003analysis] (this estimate ranges from $10^{-24}$ [@fujime1972flexural] to $10^{-22}$ [@hoshikawaelastic1985]), and $\omega = 10^2 - 10^3~\mathrm{Hz}$ [@chattopadhyay2009effect; @magariyama1995simultaneous; @son2013bacteria]. Moreover, the geometric parameters $\{\lambda/l, R/\lambda\}$ for some common bacteria (see caption of Fig. \[fig:fig4\]) are taken from Refs. [@rodenborn2013propulsion; @chattopadhyay2009effect]. The data suggests that natural flagella rotate at a rate approximately within one order of magnitude of $\omega_b$, where we have taken into account the estimated range of $\omega$ (errorbars) and the known uncertainty in $EI$ (rectangles). Note that for simplicity and generality, we ignored the role of the cell body, and focused on a single helical filament, even though a number of the bacteria considered here are multi-flagellated. Our results raise the hypothesis that the flexibility of flagella imposes an upper bound on propulsive force through $\omega_b$, above which buckling occurs. Moreover, in addition to the localized bucking that can happen at the hook of the flagellum [@son2013bacteria], the reconfigurations that arise in the post-buckling regime of the flagellum suggest the possibility of a novel functional turning mechanism. This remains an open question, however, given the current uncertainty on the known properties of flagella. As such, our investigation calls for additional experimental work to more precisely measure the properties of natural bacterial flagella, and more accurately image their dynamics. We thank Roman Stocker for enlightening discussions and we are grateful to the National Science Foundation (CMMI-1129894) for financial support.

--- abstract: 'Through the development of neural machine translation, the quality of machine translation systems has been improved significantly. By exploiting advancements in deep learning, systems are now able to better approximate the complex mapping from source sentences to target sentences. But with this ability, new challenges also arise. An example is the translation of partial sentences in low-latency speech translation. Since the model has only seen complete sentences in training, it will always try to generate a complete sentence, though the input may only be a partial sentence. We show that NMT systems can be adapted to scenarios where no task-specific training data is available. Furthermore, this is possible without losing performance on the original training data. We achieve this by creating artificial data and by using multi-task learning. After adaptation, we are able to reduce the number of corrections displayed during incremental output construction by 45%, without a decrease in translation quality.' address: | Institute for Anthropomatics and Robotics\ KIT - Karlsruhe Institute of Technology, Germany bibliography: - 'mybib.bib' title: 'Low-Latency Neural Speech Translation' --- **Index Terms**: speech translation, low-latency Introduction ============ Neural machine translation (NMT) is currently the state-of-the-art in machine translation, significantly improving translation quality in text translation [@WMT] as well as in speech translation [@IWSLT], where the translation input is the output from a speech recognizer. The main strength of neural machine translation is improved output fluency compared to traditional approaches, such as rule-based or statistical machine translation. However, while the model is able to capture more complex dependencies between the source and target languages, it relies heavily on training data examples to do so. As a consequence, the model lacks the robustness at test time to handle data that is fundamentally different from what was seen in training. There are several scenarios where this can be observed. For example, if the input is incorrectly cased, or if a different dialect is presented at test time which has different spelling or phrasings. In this work, we will concentrate on a speech translation use case in which the translation system is required to provide an initial translation in real time, before the complete sentence has been spoken. To this end, [@Niehues2016LT] presented an approach where partial sentences are translated and later replaced if necessary with the translations of the complete sentences. While we focus on this use case, the results of this work can be easily adapted to other use cases where there are differences between the training and testing scenarios. When applying partial sentence translation to neural machine translation systems, we encounter the problem that the MT system has only been trained on complete sentences, and thus the decoder is biased to generate complete target sentences. When receiving inputs which are partial sentences, the translation outputs are not guaranteed to exactly match with the input content, which can be seen in Example \[motivationExample\]. We observe that the translation is often “fantasized” by the model to be a full sentence, as would have occurred in the training data. In the example below, although the English input ends with *‘all of’*, the system generates the translation *‘todo el mundo’* (Engl. *‘all of the world’*). In other cases, the decoder can fall into an over-generation state and repeat the last word several times (eg. ‘debería, debería, debería’). \[motivationExample\] Examples of challenges in using NMT to translate spoken utterances. ---------- ------------------------------------- English: I encourage all of Spanish: yo animo a todo el mundo . English: now , I should Spanish: ahora debería , debería , debería . ---------- ------------------------------------- In this work, we aim to remedy the problem of partial sentence translation in NMT. Ideally, we want a model that is able to generate appropriate translations for incomplete sentences, without any compromise during other translation use cases. Our approach involves using multi-task learning and the automatic generation of parallel corpora in which both the source and the target sentences are incomplete sentences. Related Work ============ The main topic of our work is adapting to different types of inputs for neural machine translation. Previous works have focused on domain mismatch between the training data and test data [@Kobus2017]. In the case of speech translation, the model may only be exposed to specific issues arising from speech recognition outputs during test time. Since speech input can carry over errors from the ASR system to translation, it is necessary to adapt the model to noisier circumstances. To handle this scenario, previous work has proposed introducing artificially corrupted inputs at training time [@Sperber2017a] or direct training on lattices produced by the speech recognition system [@Sperber2017]. Multi-task learning has commonly been used in various NLP problems to jointly train a single model for several well-established NLP tasks, reducing overhead and improving performance. Such implementations can be seen using the encoder-decoder model with attention mechanism [@Luong2015], in which a single model is trained for part-of-speech tagging, named-entity tagging and machine translation simultaneously [@Niehues2017]. Regarding low-latency speech translation, various approaches to translating small text segments exist using statistical phrase-based models [@sridhar2013segmentation; @oda2014optimizing; @shavarani2015learning] or neural networks [@Gu2017]. Due to the fact that the whole input sentence is not available, it is necessary to find a compromise between translation quality and latency. The decoding process of the neural models also needs to be changed to deal with a stream of inputs, which is non-negligible. It is also possible to use the revision strategy to update the partial translations, which has been implemented in practical systems [@Niehues2016LT]. Low-Latency Speech Translation ============================== In the practice of simultaneous speech translation, translation quality is not the only criterion; it is also important to produce a translation for a spoken utterance in real-time and at low latency. Since a speaker’s utterance can be arbitrarily long, it is necessary for the translation system to start operating before the speaker stops, in which case the system input will consist of incomplete spoken segments instead of full sentences. We explore the translation revision method of [@Niehues2016LT], which has been successfully applied to statistical translation systems. The key idea of this method is that the system iteratively revises translations by re-translating new messages sent by the speech recognition component. These newly sent messages are either replacements of or concatenations to previous ones. As a result, the user sees the translation continually updated in the interface. For example, for the sentence *‘I encourage all of you’*, the system first receives only the beginning of the sentence *‘I’*, with the intermediate translation being *‘yo’*. Afterwards, it receives an update which is the continuation of the previous one: *‘I encourage all of’*. The resulting translation from a typical neural model would be *‘yo animo a todo el mundo’*, hypothesizes a final word. Finally, the whole source sentence is available, and the MT system will update the translation of the sentence to *‘yo animo a todos ustedes’*. As can be seen from the above example, in the last translation step the interface has to update the words *‘todo el mundo’* for *‘todos ustedes’*, which was generated only when the full sentence was available. As a result, we experience a delay which comes from the second to last translation step, which is longer than necessary. The interface also suffers from the update, since nearly half of the sentence needs to be replaced. Despite the fact that the final translation quality in the end does not depend on the processing of each segment, the intermediate translation outputs may change drastically due to source sequence updates. The problem is exacerbated by the fact that a neural machine translation model trained with normal parallel corpora is not able to flexibly generate translation for partial input segments, which were not available during training. The aim of our work is to build an online machine translation models which minimizes the number of words which need to be corrected until the full sentence has been seen. We aim to minimizing such criteria while maintaining translation quality for the complete utterance. Partial Translation =================== As motivated in the introduction, an out-of-the-box NMT system struggles with partial input sentences. In order to improve the flexibility of the model, we investigate generating parallel corpora in which the input and output are also partial sentences. Subsequently, we adjust the training process to make use of the data in order to build a single system that is proficient at translating partial as well as full sentences. Generating Partial Parallel Corpora {#task} ----------------------------------- In order to build a system that is good at translating partial sentences, we need to build partial sentence training data. Since such data is not available, we investigated methods to build an artificial training set from standard parallel data. This has the advantage that the methods can be easily applied to any language pair and domain and no new data has to be collected. Creating the source data is straightforward. Given a source sentence $S=s_1 \ldots s_I$, we can generate $I$ input samples $S^{(i)}=s_1 \ldots s_{i}$ by selecting the first $i$ words. The challenge arises from defining the correct translation for this source string. Since we are using this data in a low-latency speech translation system, the translation of the partial sentence should meet several conditions. First, it should be as long as possible in order to minimize the latency of the system. If we always used only the first word, we would not improve latency over a system that waits until the sentence is finished. Furthermore, to minimize the number of corrections, the translation of $S^{(i)}$ should be a substring of $S^{(i')}$ for all $i' > i$. Thus the translation of $S^{(i)}$ should be a substring of the reference translations. One possible solution is to take the reference translation of the whole sentence. But, it is unrealistic to be able to generate the whole target sentence from only a single word in the source string. Therefore, we investigated two methods to select a reasonable substrings from the reference translation. The first method is motivated by the idea that the translation should constantly generate longer target sequences when receiving longer source segments. Furthermore, word reordering may exist between two languages, for many languages sentence structure is similar. Consequently, a first approximation is to use the same proportion of words from the reference translation as we have from the source sentence. One problem in this case is that we introduce additional noise. If the word order is different, we force the system to guess the words coming next in the source sentence. To avoid this problem, we first generate a word alignment using Giza++ [@Och2003] between the source and target sentences. Then, we select the longest prefix of the reference so that no target words in the prefix are aligned to source words that are not in the partial sentences: $$T^{(i)}=\operatorname*{arg\,max}_{j \in J}\{t_1\ldots t_j | \vee j' \le j: a(j') \le i\}$$ Training Process ---------------- #### Multi-task training {#multi-task-training .unnumbered} Given the artificially produced training data, a first step is to train a model on the newly created partial sentence data and use it for speech translation only. Since both tasks are very similar, we first pre-train a standard NMT system and then fine-tune the system to translate partial sentences. The disadvantage of this approach is that the performance on complete sentences might drop, since NMT models tend to rapidly forget what they have learned before. In order to have a system that is able to generate high quality translations of both complete sentences and partial sentences, we opt to use multi-task learning, treating these as two separate tasks. In our approach, we randomly subsample the partial sentence training data to make it the same size as the original training data, so that the model can put equal emphasis on both tasks. The mixed training data then has twice as many sentences as the baseline system, but significantly less than the system using all partial sentences. Then, we fine-tune the NMT system on both tasks: translating complete sentences as well as the partial counterparts. #### Sequence level optimization {#sequence-level-optimization .unnumbered} Beside multi-task learning, we can also guide the search operation of the model so that the generated output is better matched to the source input. We use reinforcement learning with policy gradient methods [@williams1992simple; @ranzato2015sequence] to train the model to maximize the GLEU score [@wu2016google], which is the combination of $n$-gram precision and recall. This reward function restricts the model from generating sentences that are too long. Since this method is known to have high variance gradients, we follow the method in [@rennie2016self], which estimates a baseline using greedy search to reduce the variance. Experimental Results ==================== ------------- ----------------- ---------------- ------- ----------------- ------- --------- ----------- System Valid (tst2011) Test (tst2012) BLEU BLEU BLEU length (tokens) BLEU Word Up Mssg. Up. Baseline 36.86 31.33 26.66 509K 25.97 182K 15.0K Partial 35.45 30.29 29.48 375K 25.54 98K 11.8K Multi-task 37.05 31.27 30.09 376K 26.00 101K 12.0K Align. ref. 37.13 31.06 30.29 371K 26.30 98K 11.5K RL 37.21 31.25 30.08 540K 26.61 179K 15.1K RL + Multi 37.50 31.21 30.31 377K 26.77 82K 11.5K ------------- ----------------- ---------------- ------- ----------------- ------- --------- ----------- We evaluate the method on three different languages pairs: English-Spanish, English-French and German-English. System description ------------------ For all experiments, we trained systems on the Europarl [@Koehn2005] and the WIT-TED corpora [@Cettolo2012WIT] and tested on test sets from the IWSLT evaluation campaign. All systems were adapted to the TED domain by fine-tuning on the in-domain TED data. For the English$\rightarrow$Spanish and English$\rightarrow$French directions, we also optimize the models towards GLEU scores [@wu2016google] using reinforcement learning (RL). For partial sentence translation (both data generation and training), we utilize only the TED corpus. We used the OpenNMT-py toolkit [@opennmt] to train the systems. For each language pair, we jointly trained BPE [@Sennrich2016] for the source and target languages. ----------- ------- ------- ------- --------- ----------- System Final Mix BLEU Word Up Mssg. Up. Baseline 34.11 31.18 23.84 216K 16.3K Multi 34.40 34.71 23.83 128K 13.5K RL 35.08 34.09 24.31 140K 15.0K RL +Multi 34.84 42.51 24.23 99K 12.1K ----------- ------- ------- ------- --------- ----------- : \[result:enfr\] Results for English to French Evaluation metrics ------------------ We evaluated the translation quality using the BLEU score [@Papineni2002]. Since the ASR output uses automatic sentence segmentation, we need to re-segment the translation to fit the reference translations. Therefore, we used the method described in [@Matusov2005], where the automatic translation is re-segmented in a way that minimizes the word error rate to the reference. In addition, we also need to measure the extent to which we are able to reduce the number of corrections in the spoken language translation (SLT) system. To do so, we roll out all updates from the ASR system and translate each. For each updated translation, we measure the overlap between pairs of consecutive updates $s_t$ and $s_{t+1}$ and calculate the amount of re-writing necessary to produce $s_{t+1}$ after $s_t$. Specifically, the number of corrected words is calculated by the length of the translation of $s_t$, minus the length of the common prefix both translations. As illustrated in the example in Section 3, the final update would lead to $3$ corrected words (Word Up). Since an intermediate word change will force the user to reread all following words, our metric also counts all words following the first corrected word as corrected. We also report the number of messages where at least one word is corrected (Messg. Up.). Experiments ----------- #### Initial results {#initial-results .unnumbered} Our initial results on the English$\rightarrow$Spanish translation task are shown in Table \[result:enes\]. We report results in BLEU on the test and validation set with full sentence translation. Next, we report results on the test data with all possible prefixes, and finally, results on the ASR output. In the initial experiments, we use length ratio to determine the length of the reference for the partial translations as described in Section \[task\]. The baseline system is only trained on complete sentences. All other systems use the baseline system, and continue training using different strategies. The system “Partial” is fine-tuned on all partial sentences. As shown in the first two lines of Table \[result:enes\], the final translation quality drops significantly by ${\sim}1$ BLEU point. On the other hand, the BLEU score calculated on only partial sentences improves by ${\sim}3$ BLEU points. As shown by the number of tokens, the length of translations is reduced by 25%. So, a major problem of the baseline system is that it generates translations that are too long for the partial sentences. When testing on the ASR output in the last two columns, we see that the translation quality of the final hypothesis also drops, but the number of words which are updated is reduced by 45%. Also, the number of messages where at least one word changed is reduced by 20%. The system in the third line uses multi-task learning. In this case, the system is trained to perform both tasks: translation of partial and full sentences. Using this technique, we can combine the advantages of both models and maximize the translation quality of the final hypothesis, while minimizing the number of updates. The system has the same translation quality as the baseline system, with the same reduction in updates as the partial system. #### Performance w.r.t the artificial data {#performance-w.r.t-the-artificial-data .unnumbered} In the second set of experiments, we analyzed the use of the artificial data. We used the alignment-based method to generate the references for the partial sentences. In this case, we again fine-tuned used multi-task learning. As shown in the results in Table \[result:enes\], there is no clear performance difference between the two approaches. When translating text input, the system using length-ratio references is better, while the system using alignment-based methods is better on partial sentence and speech translation. Since the length-ratio based method is simpler, we used this approach for the remainder of this work. #### Sequence-level optimization {#sequence-level-optimization-1 .unnumbered} Finally, we also used reinforcement learning (RL) to optimize the performance of the system directly towards BLEU. These systems are first trained using cross-entropy and continue training using reinforcement learning. Here again, we have a baseline system trained only on the full sentences, and a multi-task system trained on both the full and partial sentences (final two lines of Table \[result:enes\]). As above, we observe that with multi-task learning, we do not lose performance on full sentences, while we can significantly reduce the number of updates. In this case, the number of words updated is further reduced, reaching more than 50% less than the baseline. #### English$\rightarrow$French {#englishrightarrowfrench .unnumbered} We also performed experiments using two English to French systems, which are summarized in Table \[result:enfr\]. We again have a baseline system trained with cross-entropy, and a baseline system where training is continued with RL. The models were evaluated on the full sentences and the mixed set of complete and partial sentences, as well as on the ASR output. Similar to the other translation directions, we improved translation performance on partial sentences and reduced the number of rewritten tokens for the SLT output by using multi-task learning. Interestingly, using reinforcement learning also helped us improve the performance on partial sentences. The RL criteria evaluates the n-gram precision as well as the recall of the translation, which is punished when the generated output is too long. Both methods can be combined to achieve the overall best performance, which reduced the rewritten tokens by up to 50% without compromising translation performance. #### German to English {#german-to-english .unnumbered} Finally, we also performed experiments on English to German as shown in Table \[result:deen\]. Again, we can improve the number of rewrites needed to produce the final output. For this language pair, however, the number of updates messages is only slightly reduced. The reason for this might be the larger reordering needed between the two language pairs. ------------ ------- ----------- ----------- System BLEU Words Up. Messg. Up Baseline 15.52 246K 23.6K Multi-task 15.64 172K 23.1K ------------ ------- ----------- ----------- : \[result:deen\] Results for German to English Examples -------- In addition to the evaluation in the last section, improvements using our approach can be seen through examples for English-Spanish and German-English, shown in Table \[example:deen\]. In both cases, we see that the baseline system is not able to generate translations for very short sequences. In this case, the last word is repeated several times. In addition, since the NMT system is tested on input it is not accustomed to, we see that the NMT decoder relies more heavily on language modeling information and completes the sentence in a way that is typical in the target language, regardless of the source input. For example, we see the added *and so on* in the first message for the German to English system. The multi-task system, however, has been trained to handle partial sentences and is therefore able to generate a correct translation. [ll]{}\ Input: & now,\ Baseline: & ahora ,\ Multi-task: & ahora ,\ Input : & now, I should\ Baseline: & ahora debería , debería , debería .\ Multi-task: & ahora debería\ Input : & now, I should men\ Baseline: & ahora debería hombres hombres .\ Multi-task: & ahora debería\ Input : & now, I should mention that this\ Baseline: & ahora debería mencionar esto .\ Multi-task: & ahora , debo mencionarlo .\ \ Input: & Und\ Baseline: & And and and and and so on.\ Multi-task: & And\ Input : & Und ich habe\ Baseline: & And I have\ Multi-task: & And I have\ Input : & Und ich empfehle Ihnen\ Baseline: & And I recommend you to you\ Multi-task: & And I recommend you\ Input : & Und ich empfehle Ihnen .\ Baseline: & And I recommend you .\ Multi-task: & And I recommend you .\ Finally, another interesting point is how the systems handle punctuation. The baseline model for German to English is only able to generate the correct translation if the sentences ends with a punctuation mark. This can be seen in the two last examples, which contain the same words, but only the second has punctuation. The multi-task system, in contrast, is able to generate the correct translation before the input is correctly punctuated. While most errors happen in very short (one or two words) partial sentences, longer partial sentences can also be problematic because of issues like punctuation, which suggests that ignoring short sentences is not a proper solution. Conclusion ========== Low latency translation is important for real-time speech translation systems. To address this challenge, we improve upon a mechanism to translate partial speech input and make updates in real-time. Our main contribution is to propose a simple method to deal with scenarios where data at inference time is different from the training data, which can be resolved with adaptation. We first showed that using simple techniques to generate artificial data are effective to get more fluent output with less correction. We also illustrated that multi-task learning can help adapt the model to the new inference condition, without losing the original capability to translate full sentences. Combining these two ideas, we are able to maintain high quality translation at low latency, minimizing the number of corrected words by 45%, which significantly improves user experience for practical applications. Acknowledgements ================ This work was supported by the Carl-Zeiss-Stiftung.

--- abstract: 'We have performed a comprehensive investigation of the global integrated flux density of M33 from radio to ultraviolet wavelengths, finding that the data between $\sim$100GHz and 3THz are accurately described by a single modified blackbody curve with a dust temperature of $T_\mathrm{dust}$ = 21.67 $\pm$ 0.30K and an effective dust emissivity index of $\beta_\mathrm{eff}$ = 1.35 $\pm$ 0.10, with no indication of an excess of emission at millimeter/sub-millimeter wavelengths. However, sub-dividing M33 into three radial annuli, we found that the global emission curve is highly degenerate with the constituent curves representing the sub-regions of M33. We also found gradients in $T_\mathrm{dust}$ and $\beta_\mathrm{eff}$ across the disk of M33, with both quantities decreasing with increasing radius. Comparing the M33 dust emissivity with that of other Local Group members, we find that M33 resembles the Magellanic Clouds rather than the larger galaxies, i.e., the Milky Way and M31. In the Local Group sample, we find a clear correlation between global dust emissivity and metallicity, with dust emissivity increasing with metallicity. A major aspect of this analysis is the investigation into the impact of fluctuations in the Cosmic Microwave Background (CMB) on the integrated flux density spectrum of M33. We found that failing to account for these CMB fluctuations would result in a significant over-estimate of $T_\mathrm{dust}$ by $\sim$5K and an under-estimate of $\beta_\mathrm{eff}$ by $\sim$0.4.' author: - 'C.T. Tibbs$^{1}$[^1], F.P. Israel$^{2}$[^2], R.J. Laureijs$^{1}$, J.A. Tauber$^{1}$, B. Partridge$^{3}$, M.W. Peel$^{4}$,' - | L. Fauvet$^{5}$\ $^{1}$Scientific Support Office, Directorate of Science, European Space Research and Technology Centre (ESA/ESTEC),\ Keplerlaan 1, 2201 AZ, Noordwijk, The Netherlands\ $^{2}$Leiden Observatory, Leiden University, P.O. Box 9513, 2300 RA, Leiden, The Netherlands\ $^{3}$Department of Astronomy, Haverford College, Haverford, PA 19041, USA\ $^{4}$Departamento de Física Matematica, Instituto de Física, Universidade de São Paulo, São Paulo, Brazil\ $^{5}$ARGANS Limited, Tamar Science Park, Plymouth, PL6 8BX, UK title: Planck Observations of M33 --- \[firstpage\] galaxies: individual: M33 – galaxies: ISM – galaxies: photometry – infrared: galaxies – submillimetre: galaxies – radio continuum: galaxies Introduction {#Sec:Intro} ============ In the region between high-frequency radio waves ($\nu$$\gtrsim$10GHz) and long-wavelength infrared (IR) emission ($\lambda$$\gtrsim$100$\mu$m), thermal radiation from interstellar dust and ionized gas, as well as non-thermal synchrotron radiation, all contribute to the emission from cosmic objects. By unravelling the various contributions, we may obtain information on the ionising stars and the properties of interstellar dust in a variety of galactic environments. The observations provided by the *Planck* mission [@Tauber:10] allow us to sample the poorly observed far-IR to millimetre (mm) gap in the continuum emission spectrum of objects such as entire galaxies. In the past, attempts have been made to extrapolate the incomplete IR continuum flux density spectrum (frequently, but incorrectly, referred to as a spectral energy distribution or SED[^3]) cutting off somewhere between 100 and 160$\mu$m (*IRAS, Spitzer Space Telescope*) by assuming a single effective integrated dust emissivity index of $\beta_\mathrm{eff}$ = 2 for the Rayleigh-Jeans extrapolation. Often, the values actually measured at wavelengths around 1mm significantly exceed such extrapolated flux densities. This so-called “millimetre excess” was readily interpreted as evidence for a large mass of colder dust [see, for instance, @Galliano:05]. However, both the increased far-IR wavelength coverage (up to 500$\mu$m) of the *Herschel Space Observatory* and the results of terrestrial laboratory experiments have subsequently indicated that the actual value of $\beta_\mathrm{eff}$ is generally $<$2. As a consequence, both the historic millimetre excess and the implied large mass of colder dust, can be reduced to an artefact of interpretation, and effectively disappear. Thus far, reliable and complete continuum flux density spectra ranging from the radio to the mid-IR or even optical wavelengths, well-sampling the mm to far-IR range, have been published for a variety of Milky Way sources, but only for a few galaxies beyond. The wavelength coverage of *Planck* renders extrapolation superfluous; the value of $\beta_\mathrm{eff}$ can be measured directly. This measurement is complicated by the degeneracy between $\beta_\mathrm{eff}$ and the dust temperature, $T_\mathrm{dust}$, derived from flux densities that have finite instrumental noise: the two parameters are inversely correlated [e.g., @Shetty:09; @Juvela:12a; @Juvela:12b]. Nevertheless, the *Planck* spectra of the Local Group galaxies, the Large and Small Magellanic Clouds [LMC and SMC; @Planck_Early_Results_XVII:11], and M31 [@Planck_Intermediate_Results_XXV:15], imply galaxy-wide effective dust emissivities well below two. Similar emissivities have been found for other nearby galaxies [@Planck_Early_Results_XVI:11]. Surprisingly, the complete flux density spectra of the LMC and SMC, incorporating *WMAP* and *COBE* data, published by @Israel:10 and interpreted by @Bot:10, do show a pronounced excess of emission at mm to cm wavelengths. This “new” excess emission is not to be confused with the apparent historical millimetre excess discussed earlier as it does not result from an arbitrary assumption on the dust emissivity, but is a well-sampled spectral feature. Its existence was confirmed by @Planck_Early_Results_XVII:11, who explained the observed excess in the LMC as a fluctuation of the Cosmic Microwave Background (CMB), but admitted to the presence of a significant intrinsic excess in the SMC. @Draine:12 proposed that this intrinsic excess in the SMC could be explained if the interstellar dust includes magnetic nanoparticles, emitting magnetic dipole radiation resulting from the thermal fluctuations in the magnetisation. Perhaps relatedly, an excess of emission at longer (cm) wavelengths has also been observed in many environments within the Milky Way [see @Planck_Intermediate_Results_XV:14 and references within]. This cm excess, more commonly known as anomalous microwave emission (AME), is typically observed at frequencies around 30GHz (or wavelengths of 1cm), is observed to be highly correlated with the IR dust emission [e.g., @Casassus:06; @Tibbs:10; @Tibbs:13; @Planck_Intermediate_Results_XV:14], and is believed to be due to electric dipole radiation from very small rapidly spinning dust grains [@Draine:98]. In this paper we present a study of the small Local Group spiral galaxy M33, using the most recent *Planck* data along with data from the literature, to produce a comprehensive continuum flux density spectrum from radio to ultraviolet (UV) wavelengths. We profit from the fact that, due to its proximity [$d$ = 840 kpc, @Freedman:91] and modest dimensions (approximately 70arcmin $\times$ 40arcmin - see Fig. \[Fig:M33\_PLCK857\]), M33 is an exceedingly well-studied object. In this analysis we will specifically address: (a) the shape of the Rayleigh-Jeans spectrum, (b) the magnitude of the effective dust emissivity spectral index, $\beta_\mathrm{eff}$, and (c) possible differences between the inner and outer regions of M33. Since the flux density spectra of both individual interstellar clouds and entire galaxies have a minimum close to the peak of the CMB, at these frequencies the CMB emission typically exceeds the interstellar contribution. Thus, our results depend critically on the reliability of the CMB subtraction. For this reason, we will pay special attention to an analysis of the CMB fluctuations, as these dominate the M33 spectrum at mm wavelengths. This paper is organised as follows. In Section \[Sec:Data\] we describe the data used in this analysis, while in Section \[Sec:Analysis\] we produce a global continuum flux density spectrum for M33, accounting for contributions from both CMB fluctuations and CO line emission. We also spatially decompose M33 into three annuli, producing a flux density spectrum for each. In Section \[Sec:Discussion\] we discuss the results of our work, and we present our conclusions in Section \[Sec:Conclusions\]. Data {#Sec:Data} ==== [lcccc]{} Telescope/Instrument & $\nu_\mathrm{ref}$ & $\theta$ & $\epsilon_\mathrm{phot}$\ & (GHz) & (arcmin) &\ ***Planck*** & & &\ LFI030 & 28.4 & 32.3 & 1$\%$\ LFI044 & 44.1 & 27.1 & 1$\%$\ LFI070 & 70.4 & 13.3 & 1$\%$\ HFI100 & 100 & 9.7 & 1$\%$\ HFI143 & 143 & 7.3 & 1$\%$\ HFI217 & 217 & 5.0 & 1$\%$\ HFI353 & 353 & 4.9 & 1$\%$\ HFI545 & 545 & 4.8 & 7$\%$\ HFI857 & 857 & 4.6 & 7$\%$\ \ ***Herschel*** & & &\ SPIRE 500$\mu$m & 600 & 0.60 & 10$\%$\ SPIRE 350$\mu$m & 857 & 0.41 & 10$\%$\ SPIRE 250$\mu$m & 1200 & 0.30 & 10$\%$\ PACS 160$\mu$m & 1870 & 0.19 & 15$\%$\ PACS 100$\mu$m & 3000 & 0.12 & 15$\%$\ PACS 70$\mu$m & 4280 & 0.09 & 15$\%$\ \ ***IRAS*/IRIS** & & &\ 100$\mu$m & 3000 & 4.3 & 13.5$\%$\ 60$\mu$m & 5000 & 4.0 & 10.4$\%$\ 25$\mu$m & 12000 & 3.8 & 15.1$\%$\ 12$\mu$m & 25000 & 3.8 & 5.1$\%$\ \ ***Spitzer*** & & &\ MIPS 24$\mu$m & 12500 & 0.10 & 10$\%$\ IRAC 8$\mu$m & 37500 & 0.03 & 10$\%$\ IRAC 5.8$\mu$m & 51700 & 0.03 & 10$\%$\ IRAC 4.5$\mu$m & 66600 & 0.03 & 10$\%$\ IRAC 3.6$\mu$m & 83300 & 0.03 & 10$\%$\ \[Table:Data\]   Planck {#Subsec:Planck} ------ The *Planck* mission [@Tauber:10; @Planck_Early_Results_I:11] was the third cosmological satellite mission to observe the entire sky in a series of wide spectral passbands ($\Delta \nu/\nu\sim$0.3–0.6) designed to sample the CMB. It measured the emission from the sky with the Low Frequency Instrument (LFI) at 28.4, 44.1, and 70.4GHz (1.0–0.4cm) with amplifiers cooled to 20K between 2009 and 2013, and the High Frequency Instrument (HFI) at 100, 143, 217, 353, 545, and 857GHz (3.0–0.35mm), with bolometers cooled to 0.1K between 2009 and early 2012 (cf., Table \[Table:Data\]). In this paper, we use the most recent *Planck* 2015 “full” data release [@Planck_2015_Results_I:16]. These data cover the full mission from 12 August 2009 to 23 October 2013 and are available to download from the *Planck* Legacy Archive.[^4] The full LFI/HFI data processing and calibration procedures are described in @Planck_2015_Results_II:16 [@Planck_2015_Results_III:16; @Planck_2015_Results_IV:16; @Planck_2015_Results_V:16; @Planck_2015_Results_VI:16; @Planck_2015_Results_VII:16; @Planck_2015_Results_VIII:16] with an overview provided in @Planck_2015_Results_I:16. The *Planck* full-sky maps are provided in <span style="font-variant:small-caps;">HEALPix</span> format [@Gorski:05], but for this analysis we extracted 2D projected maps centred on M33 using the <span style="font-variant:small-caps;">Gnomdrizz</span> package [@Paradis:12], which accurately conserves the photometry during the data re-pixelization. After performing this extraction for each of the nine *Planck* frequency maps, the maps were converted from units of CMB temperature to MJysr$^{-1}$ using the coefficients described in @Planck_2013_Results_IX:14. The nine *Planck* frequency maps are displayed in the left column of Fig. \[Fig:Planck\_maps\_all\]. The 2015 *Planck* data have been calibrated on the orbital modulation of the “cosmological dipole”, resulting in extremely high (sub-percent) calibration accuracy [see table 1 from @Planck_2015_Results_I:16]. However, it is important to note that the quoted accuracies are appropriate for diffuse emission at large angular scales, where the calibration signal appears. Additional uncertainties apply at smaller angular scales. For relatively compact sources such as M33, the main additional contributors are related to colour correction and to beam uncertainty. Colour correction uncertainties depend on the spectral shape of the source [@Planck_2015_Results_II:16; @Planck_2015_Results_VII:16], while the beam uncertainties depend on angular scale [@Planck_2015_Results_IV:16; @Planck_2015_Results_VII:16]; for M33 we conservatively assume the entire solid angle uncertainty to be applicable. Combining these uncertainties in quadrature, we conservatively assume a photometric uncertainty of 7% for the 545 and 857GHz bands, and 1% for the other seven bands. Ultimately, the uncertainty on the flux determination of compact sources is limited by fluctuations in both the physical backgrounds and foregrounds rather than photometry errors. Herschel {#Subsec:Herschel} -------- M33 was mapped with *Herschel* within the framework of the open-time key programme as part of the HerM33es KPOT\_ckrame01\_1 [@Kramer:10] and OT2\_mboquien\_4 [@Boquien:15] proposals. The @Kramer:10 observations (observation ID 1342189079 and 1342189080) were performed simultaneously with the PACS (100 and 160$\mu$m) and SPIRE (250, 350, and 500$\mu$m) instruments in parallel mode in two orthogonal directions to map a region of approximately 90arcmin $\times$ 90arcmin. The @Boquien:15 observations (observation ID 1342247408 and 1342247409) were performed solely with the PACS instrument in two orthogonal directions at 70 and 160$\mu$m, and covered a smaller area of approximately 50arcmin $\times$ 50arcmin. These maps have spatial resolutions ranging from approximately 6 to 11arcsec for the PACS maps and approximately 18 to 37arcsec for the SPIRE maps. The fully reduced maps were made publicly available by the HerM33es team as *Herschel* User Provided Data Products, and we downloaded these data from the *Herschel* Science Archive.[^5] Full details of the PACS and SPIRE data reduction and map-making are described in detail by @Boquien:11 [@Boquien:15] and @Xilouris:12, respectively. Following @Boquien:11 and @Xilouris:12, we assume a 15 and 10% photometric uncertainty on the extended emission in these PACS and SPIRE maps, respectively. IRAS {#Subsec:IRAS} ---- The original *IRAS* measurements of M33 were presented and discussed by @Rice:90. In this analysis we use the Improved Reprocessing of the *IRAS* Survey [IRIS; @Miville-Deschenes:05] data for all four *IRAS* bands at 12, 25, 60, and 100$\mu$m. These data have been reprocessed resulting in an improvement in the zodiacal light subtraction, destriping, and calibration. We use the photometric uncertainties estimated by @Miville-Deschenes:05 of 5.1, 15.1, 10.4, and 13.5% for the 12, 25, 60, and 100$\mu$m bands, respectively. Spitzer {#Subsec:Spitzer} ------- *Spitzer* mapped M33 as part of the Guaranteed Time Observations (PID 5, PI. R. Gehrz) and we use the MIPS 24$\mu$m data along with the IRAC 8.0, 5.8, 4.5, and 3.6$\mu$m data. The IRAC observations have spatial resolutions of $\sim$2.0arcsec, while the MIPS 24$\mu$m map has a resolution of 6arcsec. A detailed description of the *Spitzer* photometry of M33 is provided by @Verley:07 [@Verley:09]. For this analysis, we downloaded the data from the *Spitzer* Heritage Archive[^6] and reprocessed the data by subtracting the contribution from the zodiacal light, applying the extended emission correction, mosaicking the data, performing an overlap correction, and subtracting the brightest point sources. This processing was performed using <span style="font-variant:small-caps;">mopex</span> and <span style="font-variant:small-caps;">apex</span> in a similar manner to that discussed in @Tibbs:11, and we assume a calibration uncertainty of 10$\%$ on these maps. Analysis And Results {#Sec:Analysis} ==================== ![All of the *Planck* maps used in this analysis. The first column shows the nine standard *Planck* frequency maps, which contain a contribution from the CMB, while the second, third, fourth, and fifth columns show the CMB map, the CMB mask, and the nine corresponding frequency maps which have had the CMB contribution subtracted using the `SMICA`, `NILC`, `SEVEM`, and `Commander` methods, respectively. Each individual map is 2.5$^\circ$ across and orientated in the Galactic coordinate system as defined in Fig. \[Fig:M33\_PLCK857\].[]{data-label="Fig:Planck_maps_all"}](Fig2.pdf) Aperture photometry {#Sec:Photometry} ------------------- In order to determine the integrated flux densities for M33 we applied aperture photometry to the datasets summarised in Section \[Sec:Data\]. For our aperture photometry analysis we used an elliptical aperture with a semi-major axis of 45arcmin (11kpc at the distance of M33), a semi-major to semi-minor axis ratio of $10^{0.23}$ [@Paturel:03], and a position angle of 22.5degrees with respect to an equatorial reference frame [@Kramer:10], centred on M33 ($\ell$ = 133.60$^{\circ}$, $b$ = $-$31.34$^{\circ}$) as indicated in Fig. \[Fig:M33\_PLCK857\]. The size of the aperture was selected based on computing the integrated flux density in apertures of increasing size to determine when the computed flux density stopped growing. The unrelated background and foreground emission was estimated within an elliptical annulus with inner and outer semi-major axes of 1.15 and 1.50 times the aperture semi-major axis, respectively. The estimated uncertainty on the computed flux densities is a combination of the photometric uncertainty, $\epsilon_\mathrm{phot}$, for which we have adopted the values listed in Table \[Table:Data\], and the flux measurement uncertainty, $\epsilon_\mathrm{bg}$, which contains two terms: the first term is the variance in the aperture flux, and the second term is due to the background/foreground subtraction. Both terms are estimated from the variance in the annulus surrounding the source, computed as $$\epsilon_\mathrm{bg} = \sigma_\mathrm{bg} \left[ N_\mathrm{aper} + \frac{\pi N_\mathrm{aper}^{2}}{2 N_\mathrm{bg}} \right]^{0.5} , \label{equ:unc1}$$ [see also @Laher:12; @Hermelo:16], where $\sigma_\mathrm{bg}$ is the standard deviation of the pixels within the annulus, and $N_\mathrm{aper}$ and $N_\mathrm{bg}$ are the number of pixels within the aperture and the annulus, respectively. The total uncertainty on the computed flux densities is then estimated to be $$\epsilon = \sqrt{\epsilon_\mathrm{phot}^{2} + \epsilon_\mathrm{bg}^{2}} . \label{equ:unc3}$$ The aperture photometry described above takes into account the *average* CMB contribution, but not the effect of the CMB fluctuations, which may have a considerable impact on estimates of the M33 integrated flux density. Therefore, before determining the intrinsic M33 flux density spectrum, we need to consider the effect of the CMB on the shape of the spectrum. CMB contribution {#Subsec:CMB_Contribution} ----------------  At the galactic latitude of M33, Milky Way foreground emission is relatively weak, even at the higher frequencies, and the main background affecting the source flux measurements is the CMB itself, especially at the lower observed frequencies. As can be seen from the left column in Fig. \[Fig:Planck\_maps\_all\], M33 only starts to become clearly visible above the CMB at frequencies $\gtrsim$143GHz. *Planck* has extracted the CMB over the whole sky using four different methods: `SMICA`, a non-parametric method that computes the CMB map by linearly combining all of the *Planck* maps with weights that vary with multipole in the spherical harmonic domain; Needlet Internal Linear Combination (`NILC`), which produces a CMB map using the *Planck* maps between 44 and 857GHz by applying the Internal Linear Combination technique in the needlet (wavelet) domain; `SEVEM`, which estimates a CMB map based on linear template fitting in the map domain using internal templates constructed using the *Planck* data; and `Commander`, which is a Bayesian parametric method that models all of the astrophysical signals in the map domain [@Planck_2015_Results_IX:16]. It is important to note that in these CMB estimations, bright sources in the input maps are masked and the resulting CMB maps contain “inpainted” values within the masked areas. These inpainted values are good estimations of the CMB near the edge of the masked area (to preserve continuity), but are only statistically representative of the CMB within the mask. For this reason, the CMB masks employed by each of the four CMB separation techniques, along with the resulting CMB maps for the vicinity of M33, are displayed in Fig. \[Fig:Planck\_maps\_all\], where it is apparent that M33 was only masked for the `SMICA` analysis. In addition to providing maps of the CMB, the *Planck* Legacy Archive also provide maps containing only foreground emission (i.e., the frequency maps with the CMB contribution subtracted). Since there are four different CMB maps, this produces four sets of CMB-subtracted maps. As recommended by @Planck_2015_Results_IX:16, it is not advisable to produce an analysis that depends solely on a single component separation algorithm, and therefore we investigate the impact of the CMB by incorporating all four of the CMB-subtracted datasets in our analysis. As for the standard *Planck* maps discussed in Section \[Subsec:Planck\], we used <span style="font-variant:small-caps;">Gnomdrizz</span> to extract 2D projected maps of the CMB-subtracted maps and subsequently converted them into MJysr$^{-1}$. These CMB-subtracted maps are displayed in Fig. \[Fig:Planck\_maps\_all\], where it is clear to see variations between the maps, reflecting the different approaches used to estimate the CMB. The differences are most obvious at the lower frequencies $\lesssim$200GHz, where the CMB and its fluctuations are strongest. The *Planck* consortium recommends using the `SMICA` CMB map at small angular scales, as it results in the lowest foreground residuals. However, in this case, since M33 has been masked, it is the least reliable of the four CMB-subtraction techniques. Nonetheless, since all four of the CMB-subtracted maps are in principle statistically indistinguishable, we keep all four of them in our analysis. In order to quantify the effect of the CMB as a contaminant of the integrated emission within M33, we compared the flux densities (estimated using aperture photometry as described in Section \[Sec:Photometry\]) in each of the nine *Planck* bands for the five different datasets (the standard *Planck* frequency maps and the `SMICA`, `NILC`, `SEVEM`, and `Commander` CMB-subtracted maps), which we plot in Fig. \[Fig:CMB\_Flux\_Dens\]. These plots clearly illustrate how the CMB impacts the estimate of the integrated flux density. Not only are there differences between the flux density computed using the standard (non CMB-subtracted) maps, but there are also variations between the different CMB estimation methods. These variations are systematic as they only depend on the map frequency and input CMB amplitude, with `SMICA` and `SEVEM` producing the highest and lowest flux densities, respectively, and they also reflect that M33 is masked for the `SMICA` analysis, but not for the other three methods. Fig. \[Fig:CMB\_Flux\_Dens\] also quantitatively shows what can be seen in Fig. \[Fig:Planck\_maps\_all\], which is that the CMB contamination is non-negligible at $\nu$$\lesssim$217GHz, while at higher frequencies the emission from M33 is dominant and hence the CMB contribution becomes increasing negligible. Contamination by CO line emission {#Subsec:CO_Contribution} --------------------------------- The wide passbands of the *Planck* instruments cover the rest frequencies of various molecular lines. Emission from these lines will contaminate the measured broadband continuum flux densities, causing the latter to be overestimated. As discussed by @Planck_2013_Results_XIII:14, Galactic line emission from carbon monoxide (CO) is strongly detected in the *Planck* bands. Specifically, only the $J$=1-0, $J$=2-1, and $J$=3-2 transitions of $^{12}$CO need to be considered, as only they are strong enough to have a significant effect on the *Planck* HFI100, HFI217, and HFI353 bands, respectively. Although the *Planck* data themselves have been used to produce maps of the $^{12}$CO emission [@Planck_2015_Results_X:16], these maps were produced for the velocity range of the Milky Way, which is substantially different to that of M33. We have inspected these maps and determined that they are largely unsuitable for this analysis. However, we can still investigate the magnitude of possible CO contamination by using the $^{12}$CO maps obtained with ground-based telescopes. M33 has been observed by @Heyer:04, who used the Five College Radio Astronomy Observatory (FCRAO) 14m telescope to map the $J $=1-0 CO emission, while @Druard:14 used the $J$=2-1 CO maps observed using the Institut de Radioastronomie Millimétrique (IRAM) 30m telescope. Based on these data, we computed a flux density of the $J$=1-0 CO emission in the HFI100 band to be $\sim$0.1Jy, which when compared to the total flux densities estimated in the Planck 100GHz band accounts for $\lesssim$9% of the emission. Likewise, from the IRAM data [see also @Hermelo:16] we estimated a $J$=2-1 CO flux density in the HFI217 band of $\sim$0.7Jy, which accounts for $\lesssim$4% of the emission in that band. Finally, the compilation of CO line ratios in spiral galaxies (Israel et al., in prep.) suggests that the integrated brightness temperature ratio between the $J$=3-2 and $J$=1-0 lines is $\sim$0.7, predicting a corresponding $J$=3-2 CO flux density of $\sim$1.9Jy, which is $\lesssim$3% of the emission in the HFI353 band. Therefore, we use these flux densities to correct for the contribution from CO line emission. Throughout the rest of this analysis, we include small flux density corrections by subtracting the estimated CO flux density from the observed flux densities at 100, 217, and 353GHz in order to determine the intrinsic M33 continuum flux density spectrum as accurately as possible. Global continuum flux density spectrum and spectral energy distribution {#Subsec:Full_SED} ----------------------------------------------------------------------- To produce the global continuum flux density spectrum for M33, we performed aperture photometry on the *Planck*, *Herschel*, IRIS, and *Spitzer* maps at full angular resolution. The resulting flux densities are listed in Table \[Table:M33\_Flux\_Dens\]. As discussed in Section \[Subsec:CMB\_Contribution\], to account for the effect of the CMB, we computed the flux densities for the standard *Planck* maps along with the CMB-subtracted *Planck* maps. The flux densities listed in Table \[Table:M33\_Flux\_Dens\] have also been corrected for CO contamination as described in Section \[Subsec:CO\_Contribution\]. For a proper analysis of the M33 continuum spectrum, we expand its frequency (wavelength) range by adding results available in the literature, in addition to those listed in Table \[Table:M33\_Flux\_Dens\]. These include the radio flux densities from the compilation by @Israel:92 and from @Tabatabaei:07a, as well as the 2MASS $J$, $H$, and $K_{S}$ band flux densities by @Jarrett:03, the $U$, $B$, $V$ values from @DeVaucouleurs:91, and the *GALEX* far- and near-UV flux densities by @Lee:11. The resulting flux density spectra, with and without CMB-subtraction, are displayed in Fig. \[Fig:M33\_Full\_CFD\], while the corresponding SEDs, obtained by multiplying each flux density by its frequency, are shown in Fig. \[Fig:M33\_Full\_SED\]. --------------------------- ----------- ------------ ------------------ ------------------ ------------------ ------------------ ------------------ -- Instrument Frequency Wavelength Flux Density `SMICA` `NILC` `SEVEM` `Commander` \[GHz\] \[mm\] \[Jy\] LFI030 28.4 10.6 0.46 $\pm$ 0.05 0.55 $\pm$ 0.03 0.51 $\pm$ 0.03 0.47 $\pm$ 0.03 0.49 $\pm$ 0.03 LFI044 44.1 6.80 0.61 $\pm$ 0.12 0.77 $\pm$ 0.08 0.63 $\pm$ 0.08 0.57 $\pm$ 0.08 0.64 $\pm$ 0.08 LFI070 70.4 4.26 1.83 $\pm$ 0.12 1.42 $\pm$ 0.14 1.07 $\pm$ 0.14 0.79 $\pm$ 0.14 0.99 $\pm$ 0.14 HFI100 100 3.00 2.93 $\pm$ 0.17 2.25 $\pm$ 0.09 1.54 $\pm$ 0.09 1.10 $\pm$ 0.08 1.45 $\pm$ 0.09 HFI143 143 2.10 7.41 $\pm$ 0.26 5.99 $\pm$ 0.08 4.86 $\pm$ 0.08 4.07 $\pm$ 0.07 4.64 $\pm$ 0.08 HFI217 217 1.38 21.1 $\pm$ 0.4 18.7 $\pm$ 0.2 17.2 $\pm$ 0.2 16.2 $\pm$ 0.2 17.0 $\pm$ 0.2 HFI353 353 0.849 76.6 $\pm$ 0.8 75.2 $\pm$ 0.8 74.0 $\pm$ 0.8 73.1 $\pm$ 0.8 74.0 $\pm$ 0.8 HFI545 545 0.550 241.0 $\pm$ 16.9 241.0 $\pm$ 16.9 239.0 $\pm$ 16.7 238.0 $\pm$ 16.7 239.0 $\pm$ 16.7 SPIRE 500$\mu$m 600 0.500 345.0 $\pm$ 34.5 344.0 $\pm$ 34.4 342.0 $\pm$ 34.2 341.0 $\pm$ 34.1 341.0 $\pm$ 34.1 SPIRE 350$\mu$m 857 0.350 768.0 $\pm$ 76.8 768.0 $\pm$ 76.8 766.0 $\pm$ 76.6 764.0 $\pm$ 76.5 766.0 $\pm$ 76.6 HFI857 857 0.350 693.0 $\pm$ 48.6 694.0 $\pm$ 48.7 693.0 $\pm$ 48.5 692.0 $\pm$ 48.5 693.0 $\pm$ 48.6 SPIRE 250$\mu$m 1200 0.250 1500 $\pm$ 150 1500 $\pm$ 150 1500 $\pm$ 150 1500 $\pm$ 150 1500 $\pm$ 150 PACS 160$\mu$m 1870 0.160 2230 $\pm$ 334 2250 $\pm$ 337 2250 $\pm$ 338 2250 $\pm$ 338 2260 $\pm$ 338 PACS 100$\mu$m 3000 0.100 1400 $\pm$ 210 1380 $\pm$ 208 1380 $\pm$ 207 1380 $\pm$ 208 1380 $\pm$ 207 IRIS 100$\mu$m 3000 0.100 1340 $\pm$ 181 1330 $\pm$ 179 1330 $\pm$ 179 1330 $\pm$ 179 1330 $\pm$ 179 PACS 70$\mu$m 4280 0.0700 535.0 $\pm$ 80.3 531.0 $\pm$ 79.7 529.0 $\pm$ 79.4 530.0 $\pm$ 79.6 527.0 $\pm$ 79.1 IRIS 60$\mu$m 5000 0.0600 464.0 $\pm$ 48.2 465.0 $\pm$ 48.3 464.0 $\pm$ 48.2 468.0 $\pm$ 48.7 460.0 $\pm$ 47.9 IRIS 25$\mu$m 12000 0.0250 53.0 $\pm$ 8.0 51.0 $\pm$ 7.7 50.6 $\pm$ 7.7 50.0 $\pm$ 7.6 50.8 $\pm$ 7.7 MIPS 24$\mu$m 12500 0.0240 53.7 $\pm$ 5.4 52.6 $\pm$ 5.3 52.4 $\pm$ 5.2 52.1 $\pm$ 5.2 52.5 $\pm$ 5.3 IRIS 12$\mu$m 25000 0.0120 43.5 $\pm$ 2.3 43.5 $\pm$ 2.3 43.5 $\pm$ 2.3 43.5 $\pm$ 2.3 43.5 $\pm$ 2.3 IRAC 8$\mu$m 37500 0.00800 77.0 $\pm$ 7.7 77.0 $\pm$ 7.7 77.0 $\pm$ 7.7 77.0 $\pm$ 7.7 77.0 $\pm$ 7.7 IRAC 5.8$\mu$m 51700 0.00580 62.5 $\pm$ 6.3 62.5 $\pm$ 6.3 62.5 $\pm$ 6.3 62.5 $\pm$ 6.3 62.5 $\pm$ 6.3 IRAC 4.5$\mu$m 66600 0.00450 14.2 $\pm$ 1.4 14.2 $\pm$ 1.4 14.2 $\pm$ 1.4 14.2 $\pm$ 1.4 14.2 $\pm$ 1.4 IRIS 3.6$\mu$m 83300 0.00360 18.8 $\pm$ 1.9 18.8 $\pm$ 1.9 18.8 $\pm$ 1.9 18.8 $\pm$ 1.9 18.8 $\pm$ 1.9 \[Table:M33\_Flux\_Dens\] --------------------------- ----------- ------------ ------------------ ------------------ ------------------ ------------------ ------------------ -- Decomposition of the continuum flux density spectrum {#Subsec:Decomposition} ---------------------------------------------------- In order to quantify the resulting flux density spectra, we fitted each of them with a model simultaneously fitting contributions representing thermal dust emission, as a combination of two modified blackbodies at different temperatures (the use of two modified blackbodies is to insure an accurate fit to the peak of the cold dust component), but with identical dust emissivity indices, $$S_{\nu}^\mathrm{dust} = \sum\limits_{i=1}^{2} C_\mathrm{dust,i} \left( \frac{\nu}{\nu_{0}} \right)^{\beta_\mathrm{eff}} B_{\nu}(T_\mathrm{dust,i}) , \label{equ:S_Td}$$ non-thermal synchrotron emission, $$S_{\nu}^\mathrm{sync} = C_\mathrm{sync} \left( \frac{\nu}{\nu_{0}} \right)^{\alpha_\mathrm{sync}} , \label{equ:S_sync}$$ free-free emission, $$S_{\nu}^\mathrm{ff} = C_\mathrm{ff} \left( \frac{\nu}{\nu_{0}} \right)^{\alpha_\mathrm{ff}} , \label{equ:S_ff}$$ and AME, assuming that it is due to spinning dust emission, $$S_{\nu}^\mathrm{AME} = C_\mathrm{AME} j_{\nu} , \label{equ:S_AME}$$ where $j_{\nu}$ is the spinning dust emissivity for the warm ionised medium computed using the spinning dust model, <span style="font-variant:small-caps;">spdust</span> [@Ali-Haimoud:09; @Silsbee:11]. For each flux density spectrum we used the IDL fitting routine <span style="font-variant:small-caps;">mpfit</span> [@Markwardt:09], which employs the Levenberg-Marquardt least-squares minimisation technique, to fit the data between 1.4GHz and 24$\mu$m for $C_\mathrm{dust,i}$, $T_\mathrm{dust,i}$, $\beta_\mathrm{eff}$, $C_\mathrm{sync}$, $\alpha_\mathrm{sync}$, $C_\mathrm{ff}$, $\alpha_\mathrm{ff}$, and $C_\mathrm{AME}$. During the fitting process, $C_\mathrm{dust,i}$, $T_\mathrm{dust,i}$, $C_\mathrm{sync}$, $C_\mathrm{AME}$ were constrained to be physically realistic (i.e., $\ge$0), $\beta_\mathrm{eff}$ and $\alpha_\mathrm{sync}$ were unconstrained, while $\alpha_\mathrm{ff}$ was fixed to $-$0.1 and $C_\mathrm{ff}$, which as discussed below, was constrained based on additional observations. The estimated uncertainties on these fitted parameters were computed from the resulting covariance matrix.   [lccccc]{} Data & $\beta_\mathrm{eff}$ & $T_\mathrm{dust}$ & $\alpha_\mathrm{sync}$ & $\alpha_\mathrm{ff}$ & $\chi^{2}_\mathrm{red}$\ & & \[K\]\ standard & 0.93 $\pm$ 0.01 & 26.33 $\pm$ 0.18 & $-$1.14 $\pm$ 0.08 & $-$0.1 (fixed) & 3.00\ `SMICA` & 1.20 $\pm$ 0.01 & 22.16 $\pm$ 0.17 & $-$1.05 $\pm$ 0.06 & $-$0.1 (fixed) & 2.38\ `NILC` & 1.36 $\pm$ 0.01 & 21.59 $\pm$ 0.14 & $-$1.01 $\pm$ 0.05 & $-$0.1 (fixed) & 1.46\ `SEVEM` & 1.44 $\pm$ 0.01 & 21.60 $\pm$ 0.13 & $-$1.07 $\pm$ 0.06 & $-$0.1 (fixed) & 3.78\ `Commander` & 1.39 $\pm$ 0.01 & 21.51 $\pm$ 0.14 & $-$1.02 $\pm$ 0.05 & $-$0.1 (fixed) & 1.62\ \ Mean of CMB-subtracted & 1.35 $\pm$ 0.10 & 21.67 $\pm$ 0.30 & $-$1.03 $\pm$ 0.03 & $-$0.1 (fixed) & -\ \[Table:Fitted\_Parameters\] In order to derive reliable thermal dust parameters, the unrelated contributions of the thermal (free-free) and non-thermal (synchrotron) emission components of the gas must be accurately determined. Unfortunately, the decomposition of the low-frequency radio continuum of galaxy flux density spectra is usually not clear-cut because of the degeneracy between the free-free and synchrotron contributions, especially since the intrinsic spectral index of the synchrotron emission is not known. This degeneracy specifically hampers the determination of any AME contribution to the observed emission spectrum and additional constraints are desirable. In the case of M33, such constraints exist. The sum of the directly measured H<span style="font-variant:small-caps;">ii</span> region flux densities corresponds to 235mJy at 10GHz [@Israel:74; @Israel:80]. Unfortunately, this does not include any contribution from the diffuse emission and the tally of H<span style="font-variant:small-caps;">ii</span> regions is incomplete, especially at large radii. It thus provides us only with a useful lower limit. However, a more accurate estimate of the total thermal radio emission may be obtained from the integrated H$\alpha$ emission ($I_\mathrm{H\alpha}$ = 3.6$\times$10$^{-13}$Wm$^{-2}$) measured by @Hoopes:00, after first correcting for global extinction. The M33 SED shown in Fig. \[Fig:M33\_Full\_SED\] exhibits a peak at optical wavelengths ($\sim$5$\times$10$^{5}$GHz) representing the directly observed integrated starlight, and another peak in the far-IR ($\sim$2$\times$10$^{3}$GHz) representing absorbed and re-emitted starlight. Since the second peak is less than the first one, a relatively minor fraction of all starlight is intercepted by dust, and the (small) global extinction can be estimated from the ratio of the peak fluxes. The M33 SED resembles those of the SMC and, in particular, the LMC [@Israel:10], with an optical luminosity exceeding the IR luminosity by a factor of $\sim$2.5. From the luminosity ratio of the far-IR and optical peaks in Fig. \[Fig:M33\_Full\_SED\], we estimate a visual extinction A$_{V}$ = 0.4 $\pm$ 0.1mag, of which 0.1mag is due to the Milky Way foreground [@DeVaucouleurs:91]. Assuming A$_\mathrm{H\alpha}$ = 0.81A$_{V}$, which is a typical Milky Way extinction curve [@Fitzpatrick:07], and at these wavelengths is very similar to typical SMC and LMC extinction curves [@Gordon:03], this corresponds to an H$\alpha$ extinction A$_\mathrm{H\alpha}$ = 0.24 $\pm$ 0.08mag *internal* to M33. Hence, the corrected H$\alpha$ flux is (4.5 $\pm$ 0.5) $\times$ 10$^{-13}$Wm$^{-2}$. Using $$\frac{S^\mathrm{ff}_{\nu}}{I_\mathrm{H\alpha}} = 1.15\times10^{-14} [1-0.21\times\mathrm{log}\left(\frac{\nu}{\mathrm{GHz}}\right)]~\mathrm{Hz^{-1}} , \label{equ:S_Halpha}$$ we find that the free-free flux density at 10GHz is $S^\mathrm{ff}_\mathrm{10\,GHz}$ = 410 $\pm$ 45mJy. This is higher than the value of 280mJy inferred from the work by @Buczilowski:88, but in agreement with the value of 400mJy that follows from the determination by @Tabatabaei:07a [@Tabatabaei:07b]. Therefore, during the fitting process we constrained $C_\mathrm{ff}$ to be 410 $\pm$ 45mJy at 10GHz, and fixed $\alpha_\mathrm{ff}$ = $-$0.1. This is an important element in the decomposition of the observed continuum spectrum, essential for a reliable evaluation of the AME contribution in the 5–50GHz frequency range. The full results of our fitting analysis are shown in Fig. \[Fig:M33\_Full\_CFD\], including the fitted parameters (which are also listed in Table \[Table:Fitted\_Parameters\]), the individual fitted components, the overall flux density spectrum fit, and the normalise residuals of the fits. It is clear that there is significant difference between the fit to the standard *Planck* data compared to the CMB-subtracted *Planck* data. Although there is a spread in the fitted $T_\mathrm{dust}$ and $\beta_\mathrm{eff}$ values estimated from the CMB-subtracted data (blue, pink, red, and green curves in Fig. \[Fig:M33\_Full\_CFD\]), the fit to the standard data (black curve in Fig. \[Fig:M33\_Full\_CFD\]) is significantly outside this range. Focusing solely on the fits to the CMB-subtracted data, and combing these four fits, we find that the dust emission spectrum of M33 between $\sim$100GHz and 3THz is adequately described by a single modified blackbody curve, with a peak of $\sim$2000Jy, a mean dust temperature $T_\mathrm{dust}$ = 21.67 $\pm$ 0.30K, and a mean effective dust emissivity $\beta_\mathrm{eff}$ = 1.35 $\pm$ 0.10. There is also a warm dust component with a mean temperature of 61.89 $\pm$ 0.67K that was forced to have the same effective dust emissivity as the cold dust component. Since this warm component was simply included to insure an accurate fit to the peak of the cold dust component, we do not interpret this any further. The mean synchrotron radio continuum spectral index is $\alpha_\mathrm{sync}$ = $-$1.03 $\pm$ 0.03. The relevant mean values are also listed in Table \[Table:Fitted\_Parameters\]. For the individual entries we list the internal errors, whereas the errors given for the mean values reflect the rather larger dispersion of the individual values. Comparing the mean values to the standard values (i.e., comparing the first and last rows in Table \[Table:Fitted\_Parameters\]) we find that not correcting for the CMB contribution would result in a significant over-estimate of $T_\mathrm{dust}$ (by $\sim$5K) and under-estimate of $\beta_\mathrm{eff}$ (by $\sim$0.4), clearly highlighting the importance of correcting for the CMB. We computed the mean fraction of thermal radio emission at 20cm and 3.6cm to be $\sim$16% and $\sim$49%, respectively, which are consistent with the estimates from @Tabatabaei:07b, confirming that our estimate of the thermal emission from the H$\alpha$ emission is reasonable. Not surprisingly, our estimated synchrotron spectral index is also consistent with the results obtained by @Tabatabaei:07b. Although our fitted free-free emission amplitudes match the limits of the estimate obtained from the H$\alpha$ observations, we confirmed that even without imposing this constraint on the free-free emission, we find consistent results, with a mean dust temperature of $T_\mathrm{dust}$ = 22.36 $\pm$ 0.69K and a mean spectral index of $\beta_\mathrm{eff}$ = 1.31 $\pm$ 0.10, but the fraction of free-free emission is decreased to $\sim$11% and $\sim$34% at 20cm and 3.6cm, respectively. Ignoring the radio data, and simply fitting the data at frequencies $\ge$100GHz, we find a slightly lower value of $\beta_\mathrm{eff}$ = 1.28 $\pm$ 0.10. The fact that these different approaches all yield consistent results confirms that our fit is not biased by the radio data, the decomposition between the thermal and non-thermal radio emission, nor the degeneracy between the free-free emission and the AME. ![SEDs for M33 for the five datasets (standard *Planck* data, along with the `SMICA`, `NILC`, `SEVEM`, and `Commander` CMB-subtracted data).[]{data-label="Fig:M33_Full_SED"}](Fig5.pdf) We find that the AME in M33 is at best a minor component, both in an absolute sense and when compared to the free-free and synchrotron emission. Even though we modelled the AME using a spinning dust model for the warm ionised medium (as was used by @Planck_Intermediate_Results_XXV:15 for their M31 analysis), we obtained consistent results using the cold neutral medium, warm neutral medium, or molecular cloud spinning dust models. Based on observations of AME in the Milky Way, the ratio between the AME emission at 30GHz and the thermal dust emission at 100$\mu$m (often incorrectly referred to as an AME emissivity) is of the order of $\sim$2$\times$10$^{-4}$ [@Todorovic:10; @Planck_Intermediate_Results_XV:14]. Therefore, since we find a 100$\mu$m flux density of $\sim$1350Jy, this would lead us to expect $\sim$0.3Jy of AME at 30GHz, while we only estimate an AME flux density of $\lesssim$0.04Jy. Although this ratio between the AME and thermal dust emission is sensitive to dust temperature [as discussed by @Tibbs:12b], these results indicate that there is significantly less AME in M33 compared to our own Galaxy, which is consistent with what has been observed in M82, NGC253, and NGC4945 [@Peel:11]. On the other hand, in M31 the AME appears to be much more prominent, with a tentative detection that is comparable to what would be expected based on the AME level observed in our own Galaxy [@Planck_Intermediate_Results_XXV:15]. However, we note that for M31, the lack of observations between $\sim$1 and 20GHz could bias the fit, which is not the case for M33, M82, NGC253, and NGC4945, where the wavelength coverage is more complete. Finally, we emphasise that a single dust temperature and a single effective emissivity index, i.e., a single curve, provides a reasonable fit to the observed data between $\sim$100GHz and 3THz. There is no spectral break, and there is no indication of an “excess” of any kind in the global (spatially integrated) flux density spectrum of M33. ![*Planck* 857GHz map convolved to 10arcmin angular resolution. Superimposed are the three independent elliptical apertures (white ellipses with semi-major axes, $a$ = 8, 5.33, and 2.67kpc) and the background annulus (green ellipses). The semi-major to semi-minor axis ratio was fixed to 10$^{0.23}$.[]{data-label="Fig:M33_PLCK857_10arcmin"}](Fig6_adobe.pdf)  \  \  \ Radial variations {#Subsec:Annuli} ----------------- In view of the radial changes in the brightness of the M33 disk, it is of interest to establish whether or not the flux density spectrum changes with radial distance from the center of M33. For this purpose, we now use the *Planck*, *Herschel*, and *IRAS* maps at the best common angular resolution (10arcmin), ignoring the three *Planck* LFI bands, which have beams larger than 10arcmin. Even so, the limited extent of M33 allows only three fully independent concentric ellipses to be constructed with semi-major axes of 8, 5.33, and 2.67kpc. As before, the background/foreground emission was estimated within an elliptical annulus with inner and outer semi-major axes of 1.15 and 1.50 times the semi-major axis of the 8kpc aperture, respectively (see Fig. \[Fig:M33\_PLCK857\_10arcmin\]). @Tabatabaei:07a [@Tabatabaei:07b] have shown that across the disk of M33, the radial profiles of the far-IR and radio emission are very similar. Therefore, in addition to fitting the far-IR/sub-mm emission within each annulus, we also approximated the radio emission by scaling the curves from Fig. \[Fig:M33\_Full\_CFD\]. To do this, we computed the mean ratio between the 160$\mu$m flux density and the 4.8GHz flux density from the analysis in Section \[Subsec:Full\_SED\], and assuming that this is constant, we estimated the level of the radio emission within each annulus. We also assume that the fraction of free-free emission at 4.8GHz is fixed across M33, and fit the total radio emission with a synchrotron spectral index identical to the mean found for the entire galaxy. As before, we fit for synchrotron, free-free, AME, and thermal dust emissions. However, for this analysis we combine the four flux density measurements at each wavelength using the scatter as a measure of the uncertainty and perform a single fit, rather than a separate fit for each of the four CMB-subtracted maps. The resulting flux density spectra for each of the three elliptical annuli representing the inner, middle, and outer regions of M33 are displayed in Fig. \[Fig:M33\_radial\], along with the corresponding normalised residuals of the fits. The radial dependence of both the dust temperature, $T_\mathrm{dust}$, and the effective emissivity, $\beta_\mathrm{eff}$, can be inferred from the plots, and it is clear that *both* dust temperature and the effective dust emissivity decrease with increasing radius. In the center of M33, $T_\mathrm{dust}$ = 22.36 $\pm$ 0.16K and $\beta_\mathrm{eff}$ = 1.53 $\pm$ 0.01. As the radius increases, both decrease to $T_\mathrm{dust}$ = 21.42 $\pm$ 0.16K and $\beta_\mathrm{eff}$ = 1.38 $\pm$ 0.01 in the middle annulus (semi-major axis between 2.67 and 5.33kpc) and to $T_\mathrm{dust}$ = 19.24 $\pm$ 0.15K and $\beta_\mathrm{eff}$ = 1.29 $\pm$ 0.02 in the outer annulus (semi-major axis between 5.33 and 8.00kpc). We emphasize that this result cannot be caused by the $T_\mathrm{dust}$, $\beta_\mathrm{eff}$ degeneracy referred to earlier as that would require $\beta_\mathrm{eff}$ to increase as $T_\mathrm{dust}$ decreases. Our finding that both $T_\mathrm{dust}$ and $\beta_\mathrm{eff}$ decrease with radius is qualitatively similar to the result obtained by @Tabatabaei:14, who used Monte Carlo simulations on more limited *Herschel* flux density spectra between 100 and 500$\mu$m to deduce a simultaneous decrease of dust temperature (from $\sim$24K to $\sim$18K) and emissivity (from $\sim$1.8 to $\sim$1.2) going from the center of M33 out to a radius of 6kpc. The more extended spectral coverage presented here allow quantitatively more robust results even though the spatial resolution is lower. Discussion {#Sec:Discussion} ========== The *global* continuum flux density spectrum of M33 is characterised by an overall emissivity $\beta_\mathrm{eff}$ = 1.35 $\pm$ 0.10, which is below the value of 1.5 estimated from combined *Herschel* and *Spitzer* observations down to 600GHz (500$\mu$m) by @Xilouris:12. The difference illustrates the bias introduced by the lack of low-frequency flux densities that most tightly constrain the Rayleigh-Jeans slope of the flux density spectrum. Even though we can fit the Rayleigh-Jeans part of the M33 global flux density spectrum with a single modified blackbody, it is a priori not likely that all of the dust in M33 radiates at a single temperature. However, a superposition of modified blackbodies representing grains with emissivities, $\beta_\mathrm{g}$, radiating at a range of temperatures may create a profile that is observationally hard to distinguish from a single-temperature modified blackbody profile with an apparent emissivity $\beta_\mathrm{eff}\leq\beta_\mathrm{g}$, especially when dust temperature and emissivity are negatively correlated as originally suggested by @Dupac:03 and @Desert:08, and later confirmed by @Planck_2013_Results_XI:14. Our analysis clearly shows that this is the case. The global flux density spectrum, whose Rayleigh-Jeans part is well-defined by a *single* modified blackbody is shown to be the sum of at least three different flux density spectra representing the inner, middle, and outer regions of M33, each with Rayleigh-Jeans sections equally well fitted by a single modified blackbody. As the number of sub-spectra is only limited by the available angular resolution, we expect that each of these in turn could be decomposed further. Dust mass {#Subsec:Mass} --------- Using $$M_\mathrm{dust} = \frac{S_{\nu} d^{2}}{\kappa_{\nu} B_{\nu}(T_\mathrm{dust})} , \label{equ:Dust_Mass}$$ where $\kappa_{\nu}$ is the dust opacity, we estimated the global dust mass of M33, along with the dust mass in each of the three annuli. It is known that values of the dust opacity in the literature can vary by orders of magnitude [see @Clark:16], and in this work we adopt a value of $\kappa_{\nu}$ = 1.4m$^{2}$kg$^{-1}$ at 160$\mu$m taken from the “standard model” dust properties from @Galliano:11. Incorporating our results from Sections \[Subsec:Decomposition\] and \[Subsec:Annuli\] into Equation \[equ:Dust\_Mass\], we estimated a global dust mass for M33 of (2.3$\pm$0.4$)\times$10$^{6}$ M$_{\odot}$, and (0.8$\pm$0.1)$\times$10$^{6}$ M$_{\odot}$, (1.1$\pm$0.2)$\times$10$^{6}$ M$_{\odot}$, and (0.7$\pm$0.1)$\times$10$^{6}$ M$_{\odot}$ for the inner, middle, and outer regions of M33, respectively. We find that our global dust mass estimated assuming a single modified blackbody is consistent, within the uncertainties, with the sum of the three dust masses estimated for the sub-regions, suggesting that fitting the entirety of M33 is degenerate with fitting the three sub-regions. Local Group sample {#Subsec:Local_Group} ------------------ The global dust emissivity, $\beta_\mathrm{eff}$ = 1.35 $\pm$ 0.10, for M33 may also be compared to those derived from *Planck* observations of other Local Group galaxies: $\beta_{eff}$ = 1.62 $\pm$ 0.10, 1.62 $\pm$ 0.11, 1.48 $\pm$ 0.25, and 1.21 $\pm$ 0.27 for the Milky Way [@Planck_2013_Results_XI:14], M31 [@Planck_Intermediate_Results_XXV:15], the LMC [@Planck_Early_Results_XVII:11], and the SMC [@Planck_Early_Results_XVII:11], respectively. We find that the M33 emissivity is significantly lower than that observed in the Milky Way and M31, and is more consistent with the values found in the Magellanic Clouds, with M33 actually falling between the LMC and the SMC values. Interestingly, these dust emissivities closely follow the mean metallicities of the Local Group galaxies: 12+log\[O/H\] = 8.32 $\pm$ 0.16, 8.67 $\pm$ 0.04, 8.72 $\pm$ 0.19, 8.43 $\pm$ 0.05, and 8.11 $\pm$ 0.03, for M33, the Milky Way, M31, the LMC, and the SMC, respectively [@Pagel:03; @Toribio:16]. To illustrate this, in Fig. \[Fig:M33\_beta\_z\] we plot the dust emissivity as a function of metallicity for these 5 galaxies (filled symbols), which clearly shows that the dust emissivity increases with increasing metallicity. This trend is also observed across the M33 disk itself, where our observed emissivity gradient follows the metallicity gradient [@Toribio:16], as can be seen when we plot the results from our three annuli within M33 (open squares) in Fig. \[Fig:M33\_beta\_z\]. $T_\mathrm{dust}$ and $\beta_\mathrm{eff}$ radial variations {#Subsec:Radial_Variations} ------------------------------------------------------------ The apparent decrease in both $T_\mathrm{dust}$ and $\beta_\mathrm{eff}$ with increasing M33 radius was discussed in some detail by @Tabatabaei:14. Without fully subscribing to their conclusion, we note that there are, in principle, two possible physical explanations for the observed radial decreases. The first involves dust grain composition and di-electric properties. For instance, the dust emissivity may decrease with the average interstellar energy density. Mechanical and radiative erosion of dust grains should be stronger in the more energetic inner regions than in the more quiescent outer regions. This would favour more delicate carbon/ice dust grains in the outer regions and more robust silicate-rich grains in the inner regions. The intrinsic dust grain composition may also undergo radial changes following radial gradients in the population of stellar dust producers. The second explanation involves large-scale dust cloud properties. Dust cloud heating and effective emissivity may decrease with the average radiation field, more specifically the mix of dust cloud temperatures within a specific temperature range many change as a function of irradiation. For instance, consider the possibility that each of the profiles in Fig. \[Fig:M33\_radial\] actually represent a collection of *dust clouds and filaments* with identical emissivities but different temperatures. In a radially decreasing *average* radiation field, clouds with temperatures at the high end would occur less frequently and have a smaller filling factor, resulting in a more skewed composite profile with a downward shift of apparent mean temperature and a consequent flattening of the Rayleigh-Jeans slope. However, the results presented in this analysis do not allow us to distinguish between these possibilities. ![The global effective emissivity index, $\beta_\mathrm{eff}$, of Local Group galaxies as a function of metallicity, 12+log\[O/H\]. Filled symbols mark the global values of the identified galaxies, while the open squares mark the three independent regions (inner, middle, and outer) within M33.[]{data-label="Fig:M33_beta_z"}](Fig8.pdf) Comparison to previous studies {#Subsec:Hermelo} ------------------------------ Our analysis is not the first to produce a full flux density spectrum of M33, as @Hermelo:16 used the *Planck* 2013 “nominal” mission data along with a single CMB-subtraction method (`SMICA`) to derive the full flux density spectrum for M33. Using complex models [@Groves:08; @Popescu:11], they fitted their M33 flux density spectrum, deriving an excess of emission at mm/sub-mm wavelengths. However, in this analysis, we not only use the most recent *Planck* 2015 “full” mission data, but we also incorporate and evaluate four different CMB-subtraction techniques. As we have discussed in some depth, the contribution from the CMB fluctuations is significant and must be accurately accounted for. From Fig. \[Fig:M33\_Full\_CFD\] and Table \[Table:Fitted\_Parameters\] it is clear that there is scatter in both $T_\mathrm{dust}$ and $\beta_\mathrm{eff}$ between each of the four CMB-subtraction methods, highlighting the dangers of adopting a single method. In this analysis, we have chosen to fit our M33 flux density spectrum with a relatively simple model (i.e., a modified blackbody) and find no indication of any emission excess. A major advantage of fitting a modified blackbody to the data, rather than a complex model, is that this approach has been adopted by previous *Planck* analyses [e.g., @Planck_Early_Results_XVII:11; @Planck_2013_Results_XI:14; @Planck_Intermediate_Results_XXV:15], allowing direct comparisons to be made with other galaxies. Conclusions {#Sec:Conclusions} =========== We have performed a comprehensive analysis of the global continuum flux density spectrum of M33 over a very large wavelength range from radio to UV wavelengths. In the course of this analysis, we have demonstrated the importance of accurately accounting for the contribution of CMB fluctuations to the flux density spectrum, which if neglected, results in an over-estimate of $T_\mathrm{dust}$ of $\sim$5K and and under-estimate of $\beta_\mathrm{eff}$ of $\sim$0.4. Surprisingly, we find that the global integrated emission of M33 between $\sim$100GHz and 3THz is adequately described by a single modified blackbody curve, with a mean dust temperature $T_{dust}$ = 21.67 $\pm$ 0.30K and a mean effective dust emissivity $\beta_{eff}$ = 1.35 $\pm$ 0.10, even though such constancy of emission over all of the galaxy and throughout the line of sight is physically unlikely. In order to investigate this, we split M33 into the three independent annuli that the available resolution allows. We find that *both* $T_\mathrm{dust}$ and $\beta_\mathrm{eff}$ decrease from the centre to the outskirts of M33. This correlation is not due to any observational effect and it confirms in a direct manner an earlier conclusion reached by @Tabatabaei:14. The dust emission spectrum between $\sim$100GHz and 3THz of each of the three sub-regions can be fitted with a single (but not identical) modified blackbody curve, and as the sum of the three curves (the global emission curve) itself is well-fitted with a single modified blackbody curve, we conclude that the combination of individual flux density spectra representing different parts of M33 is highly degenerate with the mean flux density spectrum. Comparing the global far-IR emission of M33 to that of the other Local Group galaxies for which coverage is available, we find that M33 resembles the Magellanic Clouds rather than the larger spirals, the Milky Way and M31. Within this limited sample, there is a good correlation between the observed Rayleigh-Jeans slope, $\beta_\mathrm{eff}$, and the metallicity of these galaxies, with $\beta_\mathrm{eff}$ increasing with increasing metallicity. This global correlation for the Local Group galaxies is further strengthened by the finding that it also holds *within* M33. The internal M33 $\beta_\mathrm{eff}$ gradient follows its metallicity gradient, with the inner part of M33 much like the LMC, and the outer part much like the lower-metallicity SMC. Acknowledgments {#acknowledgments .unnumbered} =============== We thank the anonymous referee for providing detailed comments that have improved the content of this paper. MWP acknowledges grant \#2015/19936-1, São Paulo Research Foundation (FAPESP). 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[^4]: http://pla.esac.esa.int/pla/ [^5]: http://archives.esac.esa.int/hsa/whsa/ [^6]: http://sha.ipac.caltech.edu/applications/Spitzer/SHA/

--- abstract: 'Since the launch of the Fermi Gamma-ray Space Telescope on 11 June 2008, significant detections of high energy emission have been reported only in six Gamma-ray Bursts (GRBs) until now. In this work we show that the lack of detection of a GeV spectrum excess in almost all GRBs, though somewhat surprisingly, can be well understood within the standard internal shock model and several alternatives like the photosphere-internal shock (gradual magnetic dissipation) model and the magnetized internal shock model. The delay of the arrival of the $>100$ MeV photons from some Fermi bursts can be interpreted too. We then show that with the polarimetry of prompt emission these models may be distinguishable. In the magnetized internal shock model, high linear polarization level should be typical. In the standard internal shock model, high linear polarization level is still possible but much less frequent. In the photosphere-internal shock model, the linear polarization degree is expected to be roughly anti-correlated with the weight of the photosphere/thermal component, which may be a unique signature of this kind of model. We also briefly discuss the implication of the current Fermi GRB data on the detection prospect of the prompt PeV neutrinos. The influences of the intrinsic proton spectrum and the enhancement of the neutrino number at some specific energies, due to the cooling of pions (muons), are outlined.' author: - | Yi-Zhong Fan$^{1,2}$ [^1]\ $^{1}$[Niels Bohr International Academy, Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen, Denmark]{}\ $^{2}$[Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210008, China]{} title: 'Interpretation and implication of the non-detection of GeV spectrum excess by Fermi $\gamma$-ray Space Telescope in most GRBs' --- \[firstpage\] gamma rays: bursts $-$ polarization $-$ radiation mechanism: nonthermal $-$ acceleration of particles $-$ elementary particles: neutrino Introduction ============ Gamma-ray Bursts (GRBs) are the most extreme explosion discovered so far in the universe. With the discovery of the afterglows and then the measurement of the redshifts in 1997 [see @wijers00 for a review], the cosmological origin of GRBs has been firmly established. The modeling of the late ($t>10^{4}$ s) afterglow data favors the external forward shock model [see @piran99; @mesz02; @zm04 for reviews]. The radiation mechanisms employed in the modeling are synchrotron radiation and synchrotron self-Compton (SSC) scattering. In the early time the prolonged activity of the central engine plays an important role in producing afterglow emission too, particularly in X-ray band [e.g., @Katz98; @FW05; @Nousek06; @Zhang06]. The radiation mechanisms, remaining unclear, are assumed to be the same as those of the prompt soft gamma-ray emission. It is expected that in the Fermi era the origin of the prompt emission can be better understood. This is because the Large Area Telescope (LAT) and the Gamma-ray Burst Monitor (GBM) onboard Fermi satellite (http://fermi.gsfc.nasa.gov/) can measure the spectrum in a very wide energy band (from 8 keV to more than 300 GeV), with which some models may be well distinguished. For example, in the standard internal shock model the SSC radiation can give rise to a distinct GeV excess while in the magnetized outflow model no GeV excess is expected. Motivated by the detection of some $>100$ MeV photons from quite a few GRBs by the Compton Gamma Ray Observatory satellite in 1991$-$2000 [e.g., @Hurley94; @fm95; @Gonz03], the prompt high energy emission has been extensively investigated and most calculations are within the framework of the standard internal shocks [e.g., @Pilla98; @pw04; @gz07; @Bosnjak09 cf. Giannios 2007]. The detection prospect for LAT seems very promising [see @fp08 for a recent review]. Since the launch of Fermi satellite on 11 June 2008, significant detections of prompt high energy emission from GRBs have been only reported in GRB 080825C [@Bouvier08], GRB 080916C [@Abdo09], GRB 081024B [@Omodei08], GRB 090323 [@Ohno09b], GRB 090328 [@Cutini09], possibly and GRB 090217 [@Ohno09] until now (5 May 2009). Though the detection of 3 prompt photons above $10$ GeV from GRB 080916C at redshift $z\sim 4.5$ [@Abdo09; @Greiner09] is amazing and may imply a very high initial Lorentz factor of the outflow $\Gamma_{\rm i}>1800$ and an efficient acceleration of particles to very high energy [@Zou09], the non-detection of a significant $>100$ MeV emission from most GRBs may be a better clue of the underlying physics. A delay in the onset of the $>100$ MeV emission with respect to the soft gamma-rays, as detected in GRB 080825C, GRB 080916C, GRB 081024B and GRB 090323, may be the other clue of the GRB physics [@Abdo09; @Ohno09b]. In this work we focus on these two novel observational features. This work is structured as the following. In section 2 we discuss the constraint of the current Fermi GRB data on the standard internal shock model and several alternatives. In section 3 we look for distinguished signals in linear polarization of the prompt emission. In section 4 we briefly discuss the implication of the current Fermi results on the detection prospect of PeV neutrinos from GRBs. We summarize our results in section 5. Interpreting the lack of GeV excess in most GRBs and the delay of the arrival of the $>100$ MeV photons ======================================================================================================= In the leading [*internal shock model*]{} for the prompt emission [@npp92; @px94; @rm94; @dm98], the ultra-relativistic outflows are highly variable. The faster shells ejected at late times catch up with the slower ones ejected earlier and then power energetic forward/reverse shocks at a radius $R_{\rm int} \sim 5\times 10^{13}~(\Gamma_{\rm i}/300)^2 (\delta t_{\rm v}/{\rm 10~ms})$ cm, where $\Gamma_{\rm i}$ is the initial Lorentz factor of the outflow and $\delta t_{\rm v}$ is the intrinsic variability timescale. Part of the shock energy has been used to accelerate electrons and part has been given to the magnetic field. If the outflow is magnetized [@usov92; @dt92; @lb03; @gs05], we call the shocks generated in the collisions within the outflow the [*magnetized internal shocks*]{} [@Spruit01; @fan04]. The synchrotron radiation of the shock-accelerated electrons may peak in soft gamma-ray band and then account for the observed prompt emission. This model has been widely accepted for the following good reasons: (1) For an ultra-relativistic outflow moving with an initial Lorentz factor $\Gamma_{\rm i}$, the velocity (in units of $c$) is $\beta_{\rm i}=\sqrt{1-1/\Gamma_{\rm i}^2}$. A small velocity dispersion $\delta \beta_{\rm i}\sim \beta_{\rm i}/(2\Gamma_{\rm i}^2)$ will yield a very different Lorentz factor. As a result, internal shocks within the GRB outflow seem inevitable. (2) In the numerical simulation of the collapsar launching relativistic outflow, people found highly variable energy deposition in the polar regions in a timescale as short as $\sim 50$ ms [@mw99]. (3) This model can naturally account for the variability that is well detected in prompt gamma-ray emission [@kps97]. On the other hand, this model usually predicts a fast cooling spectrum $F_\nu \propto \nu^{-1/2}$ in the X-ray band. However, the data analysis finds a typical X-ray spectrum $F_\nu \propto \nu^{0}$ [@preece00; @band93]. Such a divergence between the model and the observation data is the so-called “the low energy spectral index crisis" [@gc99]. Another potential disadvantage of the internal shock model is its low efficiency of converting the kinetic energy of the outflow into prompt emission [e.g., @Kumar99]. Among the various solutions put forward, a plausible scenario is the [*photosphere-internal shock model*]{} [e.g., @rm05; @peer06; @tmr07]. The idea is that the thermal emission leaking from the photosphere is the dominant component of the prompt sub-MeV photons [@tho94; @mr00; @Ryde05; @ioka07]. The nonthermal high energy emission is likely the external inverse Compton (EIC) radiation of the internal shock-accelerated electrons cooled by the thermal photons from the photosphere [@rm05; @peer05; @peer06; @tmr07]. If the electrons are accelerated by gradual magnetic energy dissipation rather than by internal shocks, it is called the [*photosphere-gradual magnetic dissipation model*]{} [@gian07]. There is an increasing interest in these two kinds of models since: (1) In the spectrum analysis people did find evidences for a thermal emission component in dozens of bright GRBs [@Ryde05; @Ryde06; @RF09; @McGl09]. (2) The emission from the photosphere can naturally account for the temporal behaviors of the temperature and flux of these thermal radiation [@peer08]. (3) The overall spectrum of the prompt emission can be reasonably interpreted [e.g., @peer06; @gian07]. (4) The GRB efficiency can be much higher than that of the internal shock model [see @RF09 and the references therein]. In this section we test these four models with the current Fermi GRB data. It is somewhat surprisingly to see that none of these models have been ruled out. Explaining the lack of GeV spectrum excess in most GRBs {#sec:GeV-excess} ------------------------------------------------------- ### The standard internal shock model In this model, the outflow is baryonic and the thermal emission during the initial acceleration of the outflow is ignorable. The prompt emission is powered by energetic internal shocks. There are three basic assumptions. (i) $\epsilon_{\rm e}$, $\epsilon_{\rm B}$, $\epsilon_{\rm p}$ fractions of shock energy have been given to electrons, magnetic field and protons, respectively (note that $\epsilon_{\rm e}+\epsilon_{\rm B}+\epsilon_{\rm p}=1$). (ii) The energy distribution of the shock-accelerated electrons is a single power-law. (iii) The prompt soft gamma-ray emission is attributed to the synchrotron radiation of the shocked electrons. For internal shocks generating at $R_{\rm int}$, the typical random Lorentz factor of the electrons can be estimated as (see section 4.1.1 of @fp08 for details) $$\gamma'_{\rm e, m} \sim 760~(1+Y_{\rm ssc})^{1/4}L_{\rm syn,52}^{-1/4} R_{\rm int,13}^{1/2}(1+z)^{1/2}(\varepsilon_{\rm p}/300~{\rm keV})^{1/2}, \label{eq:gamma_em}$$ where $\varepsilon_{\rm p}=h\nu_{\rm m}$ is the observed peak energy of the synchrotron-radiation spectrum ($\nu F_\nu$), $h$ is the Planck’s constant, $L_{\rm syn}$ is the synchrotron-radiation luminosity of the internal shock emission, $Y_{\rm ssc} \sim {[-1+\sqrt{1+4\epsilon_{\rm e}/(1+g^{2})\epsilon_{\rm B}}]}/2$ is the regular SSC parameter [@se01; @fp08; @Tsvi09], and $g\sim \gamma'_{\rm e,m} \varepsilon_{\rm p}/\Gamma_{\rm i} m_{\rm e}c^2$. The SSC in the extreme Klein-Nishina regime ($g\gg 1$) is very inefficient. If that happens the non-detection of GeV spectrum excess in most Fermi GRBs can be naturally explained. With the typical parameters adopted in eq.(\[eq:gamma\_em\]) we have $g\sim 1$, for which the SSC may still be important (i.e., $Y_{\rm ssc}\geq 1$). In this work the convenience $Q_{\rm x}=Q/10^{\rm x}$ has been adopted in cgs units except for some specific notations. The SSC radiation will peak at $$\begin{aligned} h\nu_{\rm m, ssc} &\sim & {2{\gamma'}_{\rm e, m}^2 \varepsilon_{\rm p} \over 1+2g} \sim 220 {\rm GeV}~(1+2g)^{-1}(1+Y_{\rm ssc})^{1/2}\nonumber\\ && L_{\rm syn,52}^{-1/2} R_{\rm int,13}(1+z)(\varepsilon_{\rm p}/300~{\rm keV})^{2}. \label{eq:SSC_flare}\end{aligned}$$ Taking into account the energy loss of the electrons via inverse Compton scattering on prompt soft gamma-rays, the cooling Lorentz factor can be roughly estimated as $$\gamma'_{\rm e,c} \sim 0.03 L_{\rm syn,52}^{-1}R_{\rm int, 13}\Gamma_{2.5}^3.$$ In reality $\gamma'_{\rm e,c}$ is always larger than 1. The derived $\gamma'_{\rm e,c}<1$ just means that the electrons have lost almost all their energies and are sub-relativistic. Prompt high energy photons above the cut-off frequency $\nu_{\rm cut}$ will produce pairs by interacting with softer photons and will not escape from the fireball. Following @ls01 and @fp08, we have $$\begin{aligned} h\nu_{\rm cut} &\approx & 2~{\rm GeV}~(1+z)^{-1}(\varepsilon_{\rm p}/300~{\rm keV})^{(2-p)/p}L_{\rm syn,52}^{-2/p}\nonumber\\ && \delta t_{\rm v,-2}^{\rm 2/p}\Gamma_{\rm i,2.5}^{(2p+8)/p}. \label{eq:nu_cut}\end{aligned}$$ The SSC radiation spectra can be approximated by $F_{\nu_{\rm ssc}} \propto \nu^{-1/2}$ for $\nu_{\rm m}<\nu<\nu_{\rm m,ssc}$, and $F_{\nu_{\rm ssc}} \propto \nu^{-p/2}$ ($F_{\nu_{\rm ssc}} \propto \nu^{-p}$) for $\nu>\nu_{\rm m,ssc}$ and $g \leq 1$ ($g\gg 1$). The energy ratio of the SSC radiation emitted below $\nu_{\rm cut}$ to the synchrotron radiation in the energy range $\nu_{\rm m}<\nu<\max\{\nu_{\rm M},~\nu_{\rm cut}\}$ can be estimated as $$\begin{aligned} {\cal R} &\sim & {Y_{\rm ssc} \nu_{\rm m}^{(2-p)/2} \int^{\nu_{\rm cut}}_{\nu_{\rm m}}\nu^{-1/2}d\nu \over \nu_{\rm m,ssc}^{1/2} \int^{\max\{\nu_{\rm M},~\nu_{\rm cut}\}}_{\nu_{\rm m}}\nu^{-p/2}d\nu } \nonumber\\ &\approx & (p-2)(\nu_{\rm cut}/\nu_{\rm m,ssc})^{1/2}Y_{\rm ssc},\end{aligned}$$ where $\nu_{\rm M}\approx 30~\Gamma_{\rm i}(1+z)^{-1}~$ MeV is the maximal synchrotron radiation frequency of the shocked electrons [@cw96]. For $Y_{\rm ssc} \sim 1$, $p\sim 2.5$ (corresponding to the typical $\gamma-$ray spectrum $F_\nu \propto \nu^{-1.25}$ for $h\nu>\varepsilon_{\rm p}$) and $\nu_{\rm cut}\ll \nu_{\rm m,ssc}$, we have ${\cal R}\ll 1$. Therefore there is no GeV excess in the spectrum, in agreement with the data. In other words, the non-detection of a significant high energy component is due to a too large $h\nu_{\rm m,ssc}\sim$ TeV and a relative low $h\nu_{\rm cut}\sim$ GeV (see Fig.\[fig:Schematic-1\] for a schematic plot). The other possibility is that $(1+g^{2})\varepsilon_{\rm B}>4\epsilon_{\rm e}$, for which $Y_{\rm ssc}\sim {\cal O}(1)$, i.e., the SSC radiation is unimportant and can be ignored. ![A possible interpretation of the non-detection of the SSC component by Fermi satellite in the internal shock model. The high energy photons with an energy $>h\nu_{\rm cut}$ have been absorbed by the soft gamma-rays and energetic $e^{\pm}$ pairs are formed. The pairs will lost their energy through synchrotron radiation and/or inverse Compton scattering and then produce soft gamma-rays.[]{data-label="fig:Schematic-1"}](Fig1.eps) ### The photosphere$-$internal shock model @tho94 proposed the first photosphere model for the prompt gamma-ray emission, in which the nonthermal X-ray and gamma-ray emission are attributed to the Compton upscattering of the thermal emission by the mildly relativistic Alfvén turbulence. The spectra of GRBs can be nicely reproduced [see also @peer06; @gian07]. It is, however, difficult to explain the energy dependence of the width of the gamma-ray pulse [@feni95; @norr96]. As shown in @tmr07, such a puzzle may be solved in the photosphere$-$ internal shocks model, in which the sub-MeV emission is dominated by the thermal emission of the fireball and the nonthermal tail is the EIC radiation of the electrons accelerated in internal shocks at a radius $R_{\rm int}\sim 10^{14}$ cm [e.g., @mr00; @rm05; @peer06]. In this model the shock accelerated-electrons take a power-law energy distribution $dn/d\gamma'_{\rm e} \propto {\gamma'}_{\rm e}^{-p}$ for $\gamma'_{\rm e}\geq \gamma'_{\rm e,m} \sim 1$. Such an initial distribution can not keep if the electrons cool down rapidly. As shown in @SPN98, in the presence of steady injection of electrons, the energy distribution can be approximated by $N_{\gamma_{\rm e}} \propto {\gamma'}_{\rm e}^{-(p+1)}$ for $\gamma'_{\rm e}>\max\{\gamma'_{\rm e,c}, ~\gamma'_{\rm e,m}\}$ and $N_{\gamma_{\rm e}} \propto {\gamma'}_{\rm e}^{-2}$ for $\gamma'_{\rm e,c}< \gamma'_{\rm e}<\gamma'_{\rm e,m}$. Under what conditions can shock acceleration generate a particle distribution with $\gamma'_{\rm e,m} \sim 1$, with a significant fraction of the outflow energy deposited in the nonthermal particles? There could be two ways. One is to assume that electron/positron pair creation in the outflow is so significant that the resulting pairs are much more than the electrons associated with the protons. As a result, the fraction of shock energy given to each electron/positron will be much smaller than that in the case of a pair-free outflow and a $\gamma'_{\rm e,m} \sim 1$ is achievable [@tmr07]. The other way is to assume that the particle heating is continuous. In the internal shock scenario, this could happen if an outflow shell consists of many sub-shells and the weak interaction between these sub-shells may be able to produce multiple shocks that can accelerate electrons continually with a very small $\gamma'_{\rm e,m}$. Since both $\gamma'_{\rm e,m}$ and $\gamma'_{\rm e,c}$ are $\sim 1$, the EIC spectrum should be $F_\nu \propto \nu^{-p/2}$ in the MeV-TeV energy range and there is no GeV excess for $p\sim 2.5$, consistent with the Fermi data. ### The magnetized internal shock model {#sec:mag} If the unsteady GRB outflow carries a moderate/small fraction of magnetic field, the collision between the fast and slow parts will generate strong internal shocks and then produce energetic soft gamma-ray emission. As usual, the ratio between the magnetic energy density and the particle energy density is denoted as $\sigma$. In the ideal MHD limit, for $\sigma\gg 1$ just a very small fraction of the upstream energy can be converted into the downstream thermal energy. Therefore the GRB efficiency is very low[^2]. That’s why people concentrate on the internal shocks with a magnetization $\sigma\leq 1$ [@Spruit01; @fan04]. For the magnetized internal shocks with $\sigma\geq 0.1$, no significant high energy emission is expected since: (a) The SSC emission of the internal shocks is weak. Therefore there is no distinct GeV excess in the spectrum. (b) The synchrotron-radiation spectrum may be very soft, which renders the detection of GeV photons from GRBs more difficult. The reason is the following. For an isotropic diffusion and a relativistic shock, the electron energy distribution index can be estimated by [@KW05] $$p \sim (3\beta_{\rm u}-2\beta_{\rm u}\beta_{\rm d}^2+\beta_{\rm d}^3)/(\beta_{\rm u}-\beta_{\rm d})-2. \label{eq:KW05}$$ However, in the presence of a large scale coherent magnetic field, the diffusion is highly anisotropic rather than isotropic [@Morl07a]. There are thus corrections to eq.(\[eq:KW05\]). But as long as the scattering is not very forward- or backward-peaked, these corrections are small [@Keshet06]. Taking into account the anisotropic correction, @Morl07 found a spectrum steep to $p\sim 3$ for $\sigma \sim 0.05$. In the ion-electron shock simulation, the acceleration of particles at the un-magnetized shock front is a lot more efficient than that with a $\sigma\sim 0.1$ [@Spit06]. Motivated by these two possible evidences, we adopt eq.(\[eq:KW05\]) to estimate the spectral slope of accelerated particles at the magnetized shock fronts. The validity of our approach can be tested by the advanced numerical simulations in the future. In the case of $\sigma=0$, for an ultra-relativistic shock, $\beta_{\rm u} \rightarrow 1$ and $\beta_{\rm d} \rightarrow 1/3$, we have $p \rightarrow 2.22$. But for an ultra-relativistic magnetized shock, $\beta_{\rm u} \rightarrow 1$ and [e.g., @fan04] $$\beta_{\rm d} \approx {1\over 6}(1+\chi+\sqrt{1+14\chi+\chi^2}), \label{eq:fan04}$$ where $\chi\equiv \sigma/(1+\sigma)$, please note that $\sigma$ is measured in the upstream. Note in this work we just discuss the ideal MHD limit, i.e., there is no magnetic energy dissipation at the shock front. For $0<\sigma\ll 1$, we have $p\sim (4.22-2\sigma)/(1-2\sigma)-2>2.22$. For $\sigma\gg1$, we have $\beta_{\rm d} \rightarrow 1-1/2\sigma$ and $p \sim 4\sigma-1 \gg 2.22$. Correspondingly, the electron spectrum is very soft or even thermal-like. Adopting $\beta_{\rm u}\sim 1$ and substituting $\sigma \sim (1,~0.5,~0.1,~0.01)$ into eqs.(\[eq:KW05\]-\[eq:fan04\]), we have $p\sim (6.6,~4.5,~2.7,~2.3)$. The (very) soft high energy spectra of some GRBs [e.g., @preece00 see also our Fig.\[fig:beta\] for the Fermi GRBs] may be interpreted in this way. (0,180) (0,0) If the weak prompt high energy emission of GRBs is indeed attributed to the magnetization of the outflow, one can expect that the smaller the $p$, the stronger the high energy emission. The ongoing analysis of the LAT data will test such a correlation. ### The photosphere$-$gradual magnetic dissipation model @gian07 calculated the emission of a Poynting-flux-dominated GRB outflow with gradual magnetic energy dissipation (reconnection). In his scenario, the energy of the radiating electrons is determined by heating and cooling balance. The mildly relativistic electrons stay thermal throughout the dissipation region because of Coulomb collisions (Thomson thick part of the flow) and exchange of synchrotron photons (Thomson thin part). Rather similar to @tho94, the resulting spectrum naturally explains the observed sub-MeV break of the GRB emission and the spectral slopes. In this scenario, different from the magnetized internal shock model, the higher the initial $\sigma$, the harder the spectrum (see the Fig.2 of @gian07 for illustration). For an initial $\sigma\leq 40$ (corresponding to the baryon loading $L/\dot{M}c^2 \sim \sigma^{3/2} \leq 250$, where $\dot{M}$ is the mass loading rate), the resulting $>10$ MeV spectrum is very soft [see also @Drenkhahn02; @DS02], accounting for the failed detection of the GeV spectrum excess in most GRBs. Interpreting the delay of the arrival of the $>100$ MeV photons {#sec:GeV-delay} --------------------------------------------------------------- In both the collapsar and the compact star merger models for GRBs [see @piran99; @mesz02; @zm04 for reviews], the early outflow may suffer more serious baryon pollution and thus have a smaller $\Gamma_{\rm i}$ than the late ejecta [@ZhangW04]. This may explain the delay of the arrival of the $>100$ MeV emission since as long as $$\Gamma_{\rm i}\leq \Gamma_{\rm i,c}=180~(1+z)^{p \over 2p+8}({h\nu_{\rm cut} \over 100~{\rm MeV}})^{p\over 2p+8}({\varepsilon_{\rm p}\over 300~{\rm keV}})^{p-2 \over 2p+8}L_{\rm syn,52}^{1 \over 4+p}\delta t_{\rm v,-2}^{-{1\over 4+p}},$$ [*the $>100$ MeV photons can not escape from the emitting region freely and thus can not be detected.*]{} In the photosphere-gradual magnetic dissipation model, a small $\Gamma_{\rm i}$ implies a low initial magnetization of the outflow, for which the high energy spectrum can be very soft [@DS02; @gian07]. In the magnetized internal shock model, the delay of the onset of the LAT observation indicates a larger magnetization of the early internal shocks if $\Gamma_{\rm i}>\Gamma_{\rm i,c}$. In the collapsar scenario, before the breakout, the initial outflow is choked by the envelope material of the massive star (Zhang et al. 2004). The ultra-relativistic reverse shock may be able to smooth out the velocity/energy-density dispersion of the initial ejecta. So the internal shocks generated within the early/breakout outflow may be too weak to produce a significant non-thermal radiation component. The early emission is then dominated by the thermal component from the photosphere and may last a few seconds (provided that the chocked material has a width comparable to that of the envelope of the progenitor). The outflow launched after the breakout of the early ejecta can escape from the progenitor freely and the consequent internal shocks can be strong enough to produce energetic non-thermal radiation. The photosphere-internal shock model therefore might be able to naturally account for the delay in the onset of the LAT observation. In summary, before and after the onset of the $>100$ MeV emission, it seems the physical properties of the outflow have changed. The linear polarization signal of the prompt $\gamma-$ray emission ================================================================== As discussed in section 2, the failed detection of the GeV spectrum excess in most GRBs can be understood in either the standard internal shock model or several alternatives. Therefore we need independent probes to distinguish between these scenarios. Our current purpose is to see whether the polarimetry in gamma-ray band can achieve such a goal. In this section, we firstly investigate the linear polarization property of the photosphere-internal shock model ([*the results may apply to the photosphere-gradual magnetic dissipation model as well*]{}) since it has not been reported by others yet[^3]. We then briefly discuss the linear polarization signals expected in the magnetized internal shock model and in the standard internal shock model since they have been extensively discussed in the literature [e.g., @Lyu03; @Granot03; @Waxman03; @NPW03; @FXW08; @Toma09]. Linear polarization signal of the photosphere-internal shock model {#sec:Lin-thermal} ------------------------------------------------------------------ ![The coordinates in the comoving frame of the emitting region. The polarization vector $\hat{\Pi}'$ is along the ${\rm X'}-$direction for a positive $P$ given in eq.(\[eq:AA81-b\]) otherwise it would be along $\hat{\rm X}'\times \hat{k}'_{\rm o}$. []{data-label="fig:coordinate-2"}](Fig3.eps) In this work we assume an uniform outflow. At any point in the outflow there is a preferred direction, the radial direction, in which the fluid moves. We choose the ${\rm Z}'$-direction of the fluid local frame coordinate to be in that direction. The ${\rm Y}'-$direction is chosen to be within the place containing the line of sight (i.e., the scattered photon $k'_{\rm o}$) and the ${\rm Z}'-$axis (see Fig.\[fig:coordinate-2\]). In this frame, the incident photons ($k'$) are along the ${\rm Z}'-$direction. As usual, we assume that the electrons are isotropic in the comoving frame of the emitting region[^4]. The incident photons are the thermal emission from the photosphere and are unpolarized. The energy of the incident and the scattered photons are $h\nu'_{\rm se}$ and $h\nu'$, respectively. As shown in @AA81 and @Bere82: (I) In the cloud of the isotropic electrons of energies $\gamma'_{\rm e}m_{\rm e}c^2$, the spectrum of photons, upscattered at the angle $\theta'$ relative to the direction of the seed photon beam, can be approximated by $$\begin{aligned} {d N_\gamma \over dt d\nu'd\Omega'}& \approx &{3\sigma_T c \over 16\pi {\gamma'}_e^2}{n_{\nu'_{\rm se}} d\nu'_{\rm se} \over \nu'_{\rm se}} {x\over 2 \beta'_{\rm e}Q} (4A_0^2-4A_0+B_0), \label{eq:AA81-a}\end{aligned}$$ where $\beta'_{\rm e}=(1-1/{\gamma'}_{\rm e}^2)^{-1/2}$ and $$\begin{aligned} A_0 &=& {1\over 2\delta {\gamma'}_{\rm e}^2 x}({1\over {\cal T}_1}-{1\over {\cal T}_2}), \nonumber\\ B_0 &=& {{\cal T}_2\over {\cal T}_1}+{{\cal T}_1\over {\cal T}_2},\nonumber\\ {\cal T}_1 &=& (1-\cos \theta'){1+x+x\delta \cos \theta'-\delta x^2 \over Q^2},\nonumber\\ {\cal T}_2 &=& (1-\cos \theta'){1+x-x\delta \cos \theta'+\delta \over Q^2},\nonumber\\ Q &\equiv & \sqrt{1+x^{2}-2x\cos \theta'},\nonumber\\ \delta &\equiv & h \nu'_{\rm se}/({\gamma'}_{\rm e} m_{\rm e} c^{2}),~~x\equiv \nu'/\nu'_{\rm se}.\end{aligned}$$ Please note that $x$ ranges from $x_{\rm m}$ to $x_{_{\rm M}}$ that are given by $$\begin{aligned} x_{\rm m,_{\rm M}}=1+{\{{\cal A} \mp {\gamma'}_{\rm e}{\beta'}_{\rm e}\sqrt{{\gamma'}_{\rm e}^2(1+\delta)^{2}(1-\cos \theta')^{2}+\sin^{2}\theta'}\}\over 1+2\delta {\gamma'}_{\rm e}^2 (1-\cos \theta')+{\gamma'}_{\rm e}^2\delta^2(1-\cos \theta')^{2}},\end{aligned}$$ where ${\cal A}\equiv (1-\cos \theta'){\gamma'}_{\rm e}^2[{\beta'}_{\rm e}^2-\delta -\delta^2 (1-\cos \theta')]$. \(II) The polarization degree is $$P \approx {4(A_0-A_0^2) \over B_0-4A_0+4A_0^2}. \label{eq:AA81-b}$$ For the relativistic electrons (i.e., $\beta_{\rm e}\rightarrow 1$), eq.(\[eq:AA81-a\]) and eq.(\[eq:AA81-b\]) take the simplified forms $$\begin{aligned} {d N_\gamma \over dt d\nu' d\Omega'} &\approx & {3\sigma_T c \over 16\pi {\gamma'}_e^2}{n_{\nu'_{\rm se}} d\nu'_{\rm se} \over \nu'_{\rm se}} [1+ {\xi^2 \over 2(1-{\xi})}-{2\xi \over b_\theta (1-\xi)}\nonumber\\ &+&{2\xi^2 \over b_\theta^2 (1-\xi)^2}], \label{eq:AA81-a1}\end{aligned}$$ $$P\approx {{2\xi \over b_\theta (1-\xi)}-{2\xi^2 \over b_\theta^2 (1-\xi)^2}\over 1+ {\xi^2 \over 2(1-{\xi})}-{2\xi \over b_\theta (1-\xi)}+{2\xi^2 \over b_\theta^2 (1-\xi)^2}}, \label{eq:AA81-b1}$$ where $\xi \equiv h\nu'/({\gamma'}_{\rm e} m_{\rm e} c^2)$, $b_\theta=2(1-\cos \theta'){\gamma'}_{\rm e} h\nu'_{\rm se}/(m_{\rm e} c^2)$, and $h\nu'_{\rm se}\ll h\nu' \leq {\gamma'}_{\rm e} m_{\rm e} c^2 b_\theta /(1+b_\theta)$. Since the emitting region is moving relativistically, the angle $\theta'$ (in Fig.\[fig:coordinate-2\]) corresponding to the line of sight (L.o.S) is given by $$\cos \theta'=(\cos \theta-\beta_{\rm i})/(1-\beta_{\rm i} \cos \theta),$$ where $\theta$ is the angle between the line of sight and the emitting point (measured in the observer’s frame). The azimuthal angle $\phi$ varying from $0$ to $2\pi$ is defined in Fig.\[fig:Cartoon\]. The polar angle $\theta$ ranges from $0$ to $\theta_{\rm v}+\theta_{\rm j}$. The angle between the vector $(\sin \theta \cos \phi,~\sin \theta \sin \phi,~\cos \theta)$ and the central axis of the ejecta (C.A. in Fig.\[fig:Cartoon\]) is denoted as $\Theta$ and is given by $$\cos \Theta =-\sin \theta_{\rm v}\sin \theta \sin \phi+\cos \theta_{\rm v}\cos \theta.$$ Please [*bear in mind*]{} that in the following radiation calculation, the flux is set to be zero if $\cos \Theta<\cos \theta_{\rm j}$ because these points $(\theta,\phi)$ are outside of the cone of the ejecta. ![Sketch of the geometrical set-up used to compute the polarization signal. We take the L.o.S as the $z-$axis. The ${\rm y}$-axis ($x-$axis) is within (perpendicular to) the plane containing the line of sight and central axis of the ejecta.[]{data-label="fig:Cartoon"}](Fig4.eps) The EIC radiation flux in the observer frame is $$F_{\nu,\rm EIC}\propto %={(1+z)\over 4 \pi D_L^2} \int{ {\cal D}^3 h\nu'{d N_\gamma \over dt d\nu'd\Omega'} N_{\gamma_{\rm e}}d{\gamma'}_{\rm e} d\Omega},$$ where $\nu={\cal D}\nu'/(1+z)$, ${\cal D}=[\Gamma_{\rm i}(1-\beta_{\rm i} \cos \theta)]^{-1}$ is the Doppler factor, and $\Omega$ is the solid angle satisfying $d\Omega=\sin \theta d\theta d\phi$. The polarized radiation flux is $$Q_{\nu,\rm EIC}\propto %={(1+z)\over 4 \pi D_L^2} \int{ {\cal D}^3 h\nu' P \cos 2 \phi {d N_\gamma \over dt d\nu'd\Omega'} N_{\gamma_{\rm e}}d{\gamma'}_{\rm e} d\Omega}.$$ The polarization degree of the EIC emission is $$P_{\rm \nu, EIC}=|Q_{\rm \nu, EIC}|/F_{\nu,\rm EIC}.$$ One can see that a non-zero net polarization is expected as long as $\theta_{\rm v}>0$. In the numerical example, we assume that the seed photons have a thermal spectrum (as suggested in the photosphere model) $$n_{\nu'_{\rm se}}\propto {(h\nu'_{\rm se})^{2} \over e^{h\nu'_{\rm se}/kT'}-1},$$ where $kT'\approx kT/2\Gamma_{\rm i}$ is the temperature (measured in the rest frame of the emitting region) of the thermal emission. In the calculation we take $\Gamma_{\rm i} \sim 300$ and $kT \sim 100$ keV. The electron distribution is taken as $N_{\gamma_{\rm e}} \propto {\gamma'}_{\rm e}^{-(1+p)} \propto {\gamma'}_{\rm e}^{-3.5}$ for ${\gamma'}_{\rm e}>2$, otherwise $N_{\gamma_{\rm e}} =0$. For comparison purpose we also consider the case of $N_{\gamma_{\rm e}} \propto {\gamma'}_{\rm e}^{-2}$. The numerical results are presented in Fig.\[fig:Pol-1\]. One can see that the polarization degrees expected in these two representative cases are only slightly different. We also find that a moderate linear polarization level ($P_{\rm \nu,EIC}>10\%$) is achievable only for $\theta_{\rm v}\gtrsim \theta_{\rm j}+1/(3\Gamma_{\rm i})$. (0,200) (0,0) (0,200) (0,0) Currently the prompt emission consists of a thermal and a non-thermal components. The thermal component with the flux $F_{\rm \nu,th}$ is expected to be unpolarized while the nonthermal EIC component may have a high linear polarization level. The observed polarization degree $$P_{\rm \nu,obs}={|Q_{\rm \nu,EIC}|\over F_{\rm \nu,EIC}+F_{\rm \nu,th}}$$ should be strongly frequency-dependent. Roughly speaking, the linear polarization degree is anti-correlated with the weight of the thermal component. With an energy $\sim kT$, the emission is dominated by the thermal component and $P_{\rm kT,obs}$ is low. For $h\nu \gg kT$, the emission is dominated by the EIC component and $P_{\nu,\rm obs} \sim P_{\nu,\rm EIC}$, as illustrated in Fig.\[fig:Pol-2\]. This unique behavior[^5] can help us to distinguish it from other models. The probability of detecting a moderate/high linear polarization degree (${\cal R}_{\rm pol}$), however, is not high (Please note that we do not take into account the weak events for which a reliable polarimetry is impossible). On the one hand, a high linear polarization level is achievable only for $\theta_{\rm v} \geq \theta_{\rm j}+1/3\Gamma_{\rm i}$. On the other hand, $\theta_{\rm v}-\theta_{\rm j}\lesssim 1/\Gamma_{\rm i}$ is needed otherwise the burst will be too weak to perform the gamma-ray polarimetry. For $\Gamma_{\rm i}\theta_{\rm j}\gg 1$, we have $${\cal R}_{\rm pol} \sim 4 /(3\Gamma_{\rm i} \theta_{\rm j}) \approx 5\% ~\Gamma_{\rm i,2.5}^{-1}\theta_{\rm j,-1}^{-1}. \label{eq:R_pol}$$ During the revision of this work, @McGl09 reported their analysis on the spectrum and the polarization properties of GRB 061122. They found out that the spectrum was better fitted by the superposition of a thermal and a non-thermal components and the photons in the “thermal" emission dominated energy range had a (much) lower polarization level than those in the higher energy band. These two characters are in agreement with the photosphere-internal shock model (or the photosphere-gradual magnetic dissipation model). Linear polarization level expected in the standard internal shock model ----------------------------------------------------------------------- In the standard internal shock model, the polarization of the synchrotron radiation depends on both the [*poorly known*]{} configuration of magnetic field generated in internal shocks and the geometry of the visible emitting region. Assuming a random magnetic field that remains planar in the plane of the shock, @Waxman03 and @NPW03 showed that a high linear polarization level can be obtained when a narrow jet is observed from the edge, like in the photosphere-internal shock model. For the jets on-axis ($\theta_{\rm v}\leq \theta_{\rm j}$), the linear polarization degree is low (see also Gruzinov 1999 and Toma et al. 2009). The detection probability of a moderate/high linear polarization degree can also be estimated by eq.(\[eq:R\_pol\]). High linear polarization degree expected in the magnetized internal shock model ------------------------------------------------------------------------------- For the magnetized internal shock model, the prompt soft $\gamma-$ray emission is attributed to the synchrotron radiation of the electrons in ordered magnetic field and a high linear polarization level is expected [@Lyu03; @Granot03]. The physical reason is the following. The magnetic fields from the central engine are likely frozen in the expanding shells. The toroidal magnetic field component decreases as $R^{-1}$, while the poloidal magnetic field component decreases as $R^{-2}$. At the radius of the “internal” energy dissipation (or the reverse shock emission), the frozen-in field is dominated by the toroidal component. For an ultra-relativistic outflow, due to the relativistic beaming effect, only the radiation from a very narrow cone (with the half-opening angle $\leq 1/\Gamma_{\rm i}$) around the line of sight can be detected. As long as the line of sight is off the symmetric axis of the toroidal magnetic field, the orientation of the viewed magnetic field is nearly the same within the field of view. The synchrotron emission from such an ordered magnetic field therefore has a preferred polarization orientation (i.e. perpendicular to the direction of the toroidal field and the line of sight). Consequently, the linear polarization of the synchrotron emission of each electrons could not be effectively averaged out and the net emission should be highly polarized. The detection prospect of a high linear polarization degree is very promising (i.e., ${\cal R}_{\rm pol}\sim 100\%$). The above argument applies to the reverse shock emission as well if the outflow is magnetized [@fan04a].\ [l|c|c|c|c|c]{} model & unique polarization property & ${\cal R}_{\rm pol}$\ standard internal shocks & & $\lesssim$ 10%\ photosphere-internal shocks$^\dagger$ & strongly frequency-dependent$^\ddagger$ & $\lesssim$ 10%\ magnetized internal shocks & & $\sim$ 100%\ \ \ \[tab:stat\] As summarized in Tab.\[tab:stat\], for the magnetized internal shock model, a high linear polarization level should be typical, while for two other models a moderate/high linear polarization degree is still possible but much less frequent. So the statistical analysis of the GRB polarimetry results may be able to distinguish the magnetized internal shock model from the others [see aslo @Toma09]. In the photosphere-internal shock model the polarization degree is expected to be strongly frequency-dependent. Such a remarkable behavior, if detected, labels its physical origin. Indeed there were some claims of the detection of high linear polarization degree in the soft $\gamma-$ray emission of GRB 021206 [@CB03 however see Rutledge & Fox 2004], GRB 930131, GRB 960924 [@Willis05], GRB 041219A [@McGl07; @Gotz09], and GRB 061122 [@McGl09]. These results are consistent with each other as the errors are very large. The situation is inconclusive and additional data is needed to test these results. Measuring polarization is of growing interest in high energy astronomy. New technologies are being invented, and several polarimeter projects are proposed, such as, in the gamma-ray band, there are the Advanced Compton Telescope Mission [@Boggs06], POET [@Hill08] and others [see @Toma09 for a summary]. So in the next decade reliable polarimetry of GRBs in gamma-ray band may be realized and we can impose tight constraint on the models. At present, the most reliable polarimetry is in UV/optical band [e.g., @Covi99; @Wijers99]. The optical polarimetry of the prompt emission and the reverse shock emission require a quick response of the telescope to the GRB alert. This is very challenging. Mundell et al. (2007) reported the optical polarization of the afterglow, at 203 sec after the initial burst of $\gamma-$rays from GRB 060418, using a ring polarimeter on the robotic Liverpool Telescope. Their robust ($90\%$ confidence level) upper limit on the percentage of polarization, less than $8\%$, coincides with the fireball deceleration time at the onset of the afterglow. Such a null detection is, however, not a surprise because for this particular burst the reverse shock emission is too weak to outshine the unpolarized forward shock emission [@JF07]. Quite recently, the robotic Liverpool Telescope performed the polarimetry measurement of the reverse shock emission of GRB 090102 (Kobayashi 2009, private communication). Following @fan02, @zhang03 and @KP03, it is straightforward to show that the reverse shock of GRB 090102 is magnetized. Consequently the optical flash is expected to be highly polarized. If confirmed in the ongoing data analysis, the magnetized outflow model for some GRBs will be favored. Implication on the detection prospect of PeV neutrino emission ============================================================== The site of the prompt $\gamma-$ray emission may be an ideal place accelerating protons to ultra-high energy [@Vietri95; @Waxman95]. These energetic protons can produce high-energy neutrinos via photomeson interaction, mainly through $\Delta$-resonance [@WB97]. The resulting neutrinos have a typical energy $E_{\rm \nu,obs} \sim 5\times 10^{14}~{\rm eV}~\Gamma_{2.5}^2 [(1+z)^2\epsilon_{\gamma, \rm obs}/1~{\rm MeV}]^{-1}$. Significant detections are expected if GRBs are the main source of ultra-high energy cosmic rays [@WB97]. The underlying assumption is that the proton spectrum is not significantly softer than $dN/dE\propto E^{-2}$. The current Fermi observations do not provide an observational evidence for such a flat particle spectrum. Below we discuss the detection prospect of PeV neutrinos implicated by the non-detection of high energy emission from most GRBs. In the magnetized outflow model, the acceleration of a significant part of protons to energies $\geq 10^{16}$ eV is highly questionable because of the resulting soft proton spectrum. In the photosphere-internal shock model, $\gamma'_{\rm e,m}\sim 1$ is needed [@tmr07]. The efficiency of accelerating protons to very high energy depends on the mechanism of the particle heating. For example, in the case of multiple internal shocks, each pair of internal shocks are expected to be very weak since $\Gamma_{\rm sh}-1 \sim 0.04 (\gamma'_{\rm e,m}/5)(\epsilon_{\rm e}/0.2)^{-1}[3(p-2)/(p-1)]^{-1}$, where $\Gamma_{\rm sh}$ is the Lorentz factor representing the strength of the shock ($\Gamma_{\rm sh}\sim 1$ for Newtonian shocks). So the acceleration of the protons to ultra-high energy is less efficient than the standard internal shocks. This is particularly the case if the acceleration is mainly via second-order Fermi process, in which the acceleration of particles depends on the shock velocity sensitively. Even in the standard internal shock model, the generation of $10^{20}$ eV protons and the production of PeV-EeV neutrinos may be not as promising as that claimed in most literature adopting a proton spectrum $dN/dE \propto E^{-2}$. Such a flat spectrum is predicted for [*Newtonian*]{} shocks and has been confirmed by the supernova remnant observations. But the typical MeV spectrum $F_\nu \propto \nu^{-1.25}$ [@preece00] of GRBs suggests $dN/dE\propto E^{-2.5}$, supposing the accelerated protons and electrons have the same spectrum. The non-detection of $>100$ MeV photon emission from most GRBs implies soft electron (possibly) and proton spectra. Given a proton spectrum $dN/dE\propto E^{-2.22}$ that is predicted in the relativistic shock acceleration model (the First-order Fermi mechanism), the kinetic energy of the ejecta needs to be $\sim 100$ times of the $\gamma-$ray radiation energy if GRBs are indeed the main source of the observed $\sim 10^{20}$ eV cosmic rays [see @Dermer08 and the reference therein]. In other words, the GRB efficiency should be as low as $\sim 1\%$. If correct, the number of protons at $E\sim 10^{16}$ eV will be quite a few times what assumed in @Guetta04. Correspondingly the PeV neutrino flux will be higher. However, the current afterglow modeling usually yields a typical GRB efficiency $\sim 10\%$ [e.g., @FP06; @Zhang07] or larger [e.g., @PK01; @Granot06]. Below we discuss a new possibility–The proton spectrum is curved. In the “low-energy" part, the spectrum may be steepened significantly by the leakage of the very high energy cosmic rays from the ejecta [see @Hillas05 and the references therein]. The “high-energy" spectrum part may be a lot flatter. For example, in the numerical simulation of cosmic rays accelerated in some supernova remnants, a spectrum $dN/dE \propto E^{-1.7}$ at the high energy part is obtained [e.g., @Volk02; @Bere03]. If holding for GRBs as well and GRBs are the main source of the $10^{20}$ eV cosmic rays, the PeV neutrino spectrum will be harder than that predicted in @Guetta04. For instance, the neutron spectra $\varepsilon_{\rm \nu}^2dN/d\varepsilon_{\nu} \propto \varepsilon_{\nu}^0$ and $\varepsilon_{\rm \nu}^2dN/d\varepsilon_{\nu} \propto \varepsilon_{\nu}^{-2}$ in their Fig.3 will be hardened by a factor of $\varepsilon_{\nu}^{0.3}$. But the total flux may be just $\sim 10\%$ times that predicted in @Guetta04 because in this scenario the protons are not as many as that suggested in a flat spectrum $dN/dE\propto E^{-2}$ for $E\ll 10^{20}$ eV. There is a process, ignored in some previous works, that can enhance the detection prospect a little bit. After the pions (muons) are generated, the high energy pions (muons) will lose energy via synchrotron radiation before decaying, thus reducing the energy of the decay neutrinos (e.g. Guetta et al. 2004). As a result, above $\varepsilon_{\nu_{\mu}}^{\rm c} \sim{10^{17} \over 1+z}\epsilon_{\rm e}^{1/2}\epsilon_{\rm B}^{-1/2}L_{\gamma,52}^{-1/2}\Gamma_{\rm i, 2.5}^{4}\delta t_{\rm v,-2}~{\rm eV}$ ($\varepsilon_{\bar{\nu}_{\mu},\nu_{\rm e}}^{\rm c} \sim \varepsilon_{\nu_{\mu}}^{\rm c}/10$), the slope of the corresponding neutrino spectrum steepens by 2, where $L_{\gamma}$ is the luminosity of the $\gamma-$ray emission [@Guetta04]. However we do not suggest a smooth spectral transition around $\varepsilon_{\nu_{\mu}}^{\rm c}$ or $\varepsilon_{\bar{\nu}_{\mu},\nu_{\rm e}}^{\rm c}$ because the cooling of pions (muons) will cause a pile of particles at these energies (see also Murase & Nagataki 2006). A simple estimate suggests that the number of neutrinos in the energy range $(0.5,~1)\varepsilon_{\nu_{\mu}}^{\rm c}$ should be enhanced by a factor of $\sim 3$. A schematic plot of the muon neutrino spectrum in the standard internal shock model is shown in Fig.\[fig:Schematic\]. ![The schematic plot of the PeV muon neutrino spectrum in the standard internal shock model.[]{data-label="fig:Schematic"}](Fig7.eps) Conclusion ========== In the pre-Fermi era, it is widely expected that significant GeV emission will be detected in a good fraction of bright GRBs if they are powered by un-magnetized internal shocks [e.g., @Pilla98; @pw04; @gz07; @fp08]. The detection of a distinct excess at GeV-TeV energies, the SSC radiation component of such shocks, will be a crucial evidence for the standard fireball model. The non-detection of the GeV spectrum excess in almost all Fermi bursts [@Abdo09] is a surprise but does not impose a tight constraint on the models. For example, in the standard internal shock model, the non-detection can be attributed to a too large $h\nu_{\rm m,ssc} \sim $ TeV and a relative low $h\nu_{\rm cut} \sim$ GeV. Some alternatives, such as the photosphere-internal shock model, the magnetized internal shock model and the photosphere-gradual magnetic dissipation model, can be in agreement with the data, too (see Tab.\[tab:sum\] for a summary). We attribute the delay in the onset of LAT detection in quite a few Fermi bursts to the unfavorable condition for GeV emission of the early outflow (see section \[sec:GeV-delay\] for details). model the physical reason ----------------------------- -------------------------------------------------------- -- -- -- standard internal shocks $h(\nu_{\rm m,ssc},~\nu_{\rm cut})\sim (\rm TeV,~GeV)$ photosphere-internal shocks very small $\gamma'_{\rm e,m}$ and $\gamma'_{\rm e,c}$ magnetized internal shocks soft synchrotron spectrum and weak SSC photosphere-gradual magnetic dissipation very small $\gamma'_{\rm e,m}$ and $\gamma'_{\rm e,c}$ : The physical reasons for the lack of GeV spectrum excess in most GRBs. \[tab:sum\] With the polarimetry of GRBs people can potentially distinguish between some prompt emission models (see Tab.\[tab:stat\] for a summary; see also Toma et al. 2009). We show in section \[sec:Lin-thermal\] that in the photosphere-internal shock model the linear polarization degree is roughly anti-correlated with the weight of the thermal component and will be highly frequency-dependent. Such a unique behavior, if detected, labels its physical origin. However, a moderate/high linear polarization level is expected only when the line of sight is outside of the cone of the ejecta (i.e., $\theta_{\rm v}>\theta_{\rm j}$). In addition, $\theta_{\rm v}-\theta_{\rm j}\lesssim 1/\Gamma_{\rm i}$ is needed otherwise the burst will be too weak to perform the gamma-ray polarimetry. Consequently the detection prospect is not very promising. In this work we have also briefly discussed the detection prospect of prompt PeV neutrinos from GRBs. The roles of the intrinsic spectrum of the protons and the cooling of pions (muons) have been outlined. The latter always increases the neutrino numbers at the energies $\varepsilon_{\nu_{\mu}}^{\rm c}$ or $\varepsilon_{\bar{\nu}_{\mu},\nu_{\rm e}}^{\rm c}$ by a factor of 3. The former, however, is uncertain. If the protons have an intrinsic spectrum $dN/dE\propto E^{-2.22}$ and have a total energy about tens times that emitted in gamma-rays, the detection prospect would be as good as, or even better than that presented in @Guetta04. If the proton spectrum traces that of the electrons, i.e., typically $dN/dE\propto E^{-2.5}$, the detection prospect would be discouraging. Acknowledgments {#acknowledgments .unnumbered} =============== We thank the anonymous referee for very helpful suggestions/comments and Drs. X. F. Wu, K. Toma, and Y. C. Zou for communication. 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The particles are accelerated and the synchrotron radiation of the ultra-relativistic electrons peaks at $\nu_{\rm m}\sim 5\times10^{19}\sigma_{2}^{3}(\epsilon_{\rm e}/0.2)^2[3(p-2)/(p-1)]^{2} \Gamma_{2.5} {t_{v,m}}_{-3}~{\rm Hz}$  [@fzp05], provided that a significant part of the magnetic energy has been converted to the thermal energy of the particles. In this scenario, the SSC radiation is expected to be very weak because in the rest frame of the electrons having $\gamma'_{\rm e,m} \sim 10^{4}(\epsilon_{\rm e}/0.2)[3(p-2)/(p-1)]\sigma_{2}$  [@fzp05], the soft gamma-rays have an energy $\gamma'_{\rm e,m}h\nu_{\rm m}/\Gamma_{\rm i}\gg m_{\rm e}c^2$, i.e., the SSC is in the extreme Klein-Nishina regime and is very inefficient. The non-detection of high energy emission from most GRBs is, of course, consistent with this model. [^3]: In the final stage of preparing the manuscript, the author was informed by K. Toma and X. F. 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[^5]: For the synchrotron radiation of the electrons moving in a random magnetic field (the standard internal shock model) or in an ordered magnetic filed (e.g., the magnetized internal shock model), before and after the peak of the spectrum, the polarization degree changes because the polarization properties depend on the profile of the spectrum. However, such a dependence is weak, as shown in @Granot03.

--- abstract: 'The development of robot control programs is a complex task. Many robots are different in their electrical and mechanical structure which is also reflected in the software. Specific robot software environments support the program development, but are mainly text-based and usually applied by experts in the field with profound knowledge of the target robot. This paper presents a graphical programming environment which aims to ease the development of robot control programs. In contrast to existing graphical robot programming environments, our approach focuses on the composition of parallel action sequences. The developed environment allows to schedule independent robot actions on parallel execution lines and provides mechanism to avoid side-effects of parallel actions. The developed environment is platform-independent and based on the model-driven paradigm. The feasibility of our approach is shown by the application of the sequencer to a simulated service robot and a robot for educational purpose.' author: - title: '**A Platform-independent Programming Environment for Robot Control** ' --- INTRODUCTION {#sec:intro} ============ The development of robot control programs is a difficult, error-prone and complex exercise. At first, the developer needs a good understanding about the task which the robot should fulfill. This includes knowledge about the environment in which the robot is embedded and an idea of the desired behavior of the robot. Second, platform-specific knowledge about the robot platform itself is needed. For instance, a vacuum cleaning robot is small and usually equipped with a cleaning device, whereas a service robot is tall and equipped with a manipulator to reach high cupboards and to grasp objects. This diversity in the mechanical and electrical structure reflects not only in the appearance and hardware design, but also in the software system. A robot application developer must be aware of this diversity in order to develop control programs in an optimal way. However, such a robot platform-oriented development of robot control programs leads (very often) to non-reusable control programs. Though, reuse of robot control programs is desirable, because the control program describes the logic of a specific task (e.g. an office delivery task) which could be applied on different robot platforms and reused (or partly reused) in similar tasks. To overcome the dependency of a specific platform (e.g. operating systems or even programming languages) the Software Engineering community introduced the concept of Platform-independent Models (PIM) and Platform-specific Models (PSM) within the model-driven software development paradigm. Here, models play a vital role to capture and abstract generic aspects from domain-specific aspects. Recently, the model-driven paradigm has been adopted by the robotics community. In [@c3] Baer et al. introduced a platform independent modeling language providing modeling elements for communication and collaboration infrastructures. The approach has been applied successfully for cooperative soccer playing robots. In Schlegel et al. [@c4] a component-based robot software development approach has been applied to the model-driven paradigm in order to generate a wide variety of target-specific source code (e.g. for embedded controllers) without modifying the platform-independent model. Another model-driven approach is described in [@c7]. Here, Alonso et al. introduced a platform-independent component-model which captures certain aspects of a robotic software system in three distinct views, namely behavioral, structural, and component. The approach has been applied to generate Ada source-code for a Cartesian robot.\ However, to the best of our knowledge, the model-driven paradigm has been so far not applied to the development of platform-independent robot control programs, even so this domain could benefit from it. In particular, the development of robot control programs where concurrency is crucial (e.g. in behavior-based and hybrid control approaches as [@c8]) is an error-prone task, because actions that may technically run in parallel can have side-effects. For instance, a manipulator can grasp an object while a camera processes images to detect objects. In such a situation the manipulator may block the field of view and the camera may not find the appropriate object. These pitfalls make the entry-level in robot programming very high and complicates the fast creation of robot control programs. Therefore, supporting tools and frameworks to ease the development are needed.\ One promising graphical programming environment with parallelism support is the Urbi Studio by the company Gostai [@d14]. Urbi Studio provides a graphical state-machine based programming environment which is bound to their custom programming language urbiscript. In addition to state-machine based programming Urbi Studio provides a time-based graphical sequencer to initialize and set variables over time – a specialty provided by the framework. However, the environment is tightly bound to the Urbi execution engine which allows to monitor the program execution in the graphical environment. The company aims to make all robots compatible, which leads into a concrete implementation and C++ wrapping of the robot framework to the Urbi framework. Another graphical programming environment which also provides the possibility to describe parallel sequences is the Nao Choregraphe from Aldebaran Robotics [@d15]. It is delivered with their Nao robot. This environment allows flow diagram based programming of Nao robots and provides a time-based sequencer to create motor control commands. The environment is able to execute code on the Nao robot either in Urbi or the Python language.\ In this paper we present a programming environment for graphical sequencing of robot actions. The approach is based on the model-driven paradigm. Exchangeable domain-specific languages are used to generate platform-specific source code for a wide variety of robots without changing the control logic. The paper is structured as follows. In Section \[sec:problemstatement\] the robot control approach, namely the sequencing of concurrent actions is introduced. Our model-driven approach to achieve platform-independence is described in Section \[sec:approach\]. The application of the graphical sequencer is shown in Section \[sec:usecases\] by the application to a simulated service robot and a a robot for educational purpose. In Section \[sec:conclusion\] our approach will be discussed and we conclude with some lessons learned. ACTION SEQUENCING FOR ROBOT CONTROL {#sec:problemstatement} =================================== We are interested in the programming of robot tasks. For instance, the autonomous delivery of documents in an office environment. A task $\mathbf{T} = \{\mathbf{a_{1}}, \mathbf{...}, \mathbf{a_{n}}\}$ is composed of a set of finite actions (e.g. grasping an object, following a person, or moving to a location) from the set of actions $\mathbf{A}$ and $\mathbf{T} \subseteq \mathbf{A}$. Each action $\mathbf{a} \in \mathbf{A}$ is a tuple of the following form $$\mathbf{a} = \langle \mathbf{n}, \mathbf{C} \rangle$$ where $\mathbf{n}$ is an unique name of the action and $\mathbf{C}$ is a set of execution constraints $\mathbf{e}$. Furthermore, a global data space $\mathbf{D}$ is used to store global variables which are needed, modified and shared by the actions. An execution constraint is formally defined as $$\mathbf{e} = \langle \mathbf{\Lambda}, \mathbf{a} \rangle \quad \mathbf{a} \in \mathbf{A}$$ where $\mathbf{\Lambda}$ is an execution operator which constraints the execution of the action $\mathbf{a}$ in a temporal manner. In our case $\mathbf{\Lambda}$ describes the order of actions. Hence, our robot control approach may be described as a dependency graph [@d16] with nodes as actions and edges as execution constraints describing the predecessor and successor relationship between actions. Figure 1 shows an example of such an execution order. Here, the action **A**, **B**, and **C** are executed initially, since no constraints are defined. Action **D** is constrained. **D** must only start as soon as the predecessors **A** and **B** finished. Furthermore, the action **E** is constrained. **E** is the last action and must only be executed, if all predecessors (namely **A**, **C** and **D**) finished. =\[fill=blue,draw=none,text=white\] \(C) [$\mathbf{C}$]{}; (B) [$\mathbf{B}$]{}; (C) \[below of=B\] [$\mathbf{C}$]{}; (B) \[above of=C\] [$\mathbf{B}$]{}; (A) \[above of=B\] [$\mathbf{A}$]{}; (D) \[right of=B\] [$\mathbf{D}$]{}; (E) \[right of=D\] [$\mathbf{E}$]{}; \(A) edge node (E) (A) edge node (D) (B) edge node (D) (C) edge node (E) (D) edge node (E); \[fig:graph\] APPROACH {#sec:approach} ======== In this section we will describe our approach that is based on a model driven paradigm. We show how the concepts of meta-modeling and domain-specific languages are applied to develop a platform and robot system independent programming environment which focuses on parallel action sequences. Requirements ------------ The programming environment shall assist robot programmers to develop robot control sequences as those described in Section \[sec:problemstatement\]. Thereby it shall fulfill two main requirements. First, a simplification of action sequences, so that no experts in the field are required. In order to achieve this, an intuitive graphical program editor must be provided. Besides the simplification of action sequence creation the environment must assist the developer to reduce failures that occur due to parallelism in such sequences, which could be e.g. a manipulator that grasps an object and blocks the field of view of a camera recording images. Secondly, it is required that the environment is independent of the target robot. It must be possible to describe robot action sequences for any robot platform. Thereby, different robot classes can have different programming elements e.g. a vacuum cleaning robot may not have a manipulator and a programmer should not have the possibility to use a grasp action on these robot types. The programming elements must be adaptable and even the same control programs shall be applicable to different robots. Meta-model ---------- The programming model is structured by a meta-model which describes the structure of the introduced action sequences, namely the abstract syntax and semantics. The concept follows the scheme of Model-Driven Software Development (MDSD) defined by Stahl and Völte [@c9]. The meta-model describes the basic programming elements ‘Action’, ‘Resource Component’, and ‘Variable’. See Figure \[fig:PaperMetaModel\]. - **Action:** An action is a certain thing that a robot can do, like ‘move to a position’, ‘set motor speeds’ or ‘capture an image’. Actions can work on a set of predefined global variables which are used as parameter or return values, comparable to function calls. Actions can be put into an execution sequence as defined in Section \[sec:problemstatement\]. Further, to assist concurrent programming, every action has a list of concrete actions, which are not allowed to be scheduled simultaneously. Every action belongs to a resource component. - **Resource Component:** A resource component enables the usage of a certain action set. For instance, the definition of a concrete resource component ‘manipulator’ enables the programmer to use the actions ‘move arm to pose’ and ‘grasp’. If a robot does not have a manipulator, it also should not enable the programmer to use those actions. A resource component can schedule a single action at a time in a serial manner. Only the use of multiple resource components allows the creation of parallel action sequences. So, if a robot has only a single resource component ‘manipulator’ it shall not be able to execute multiple ‘grasp’ actions simultaneously. An additional resource component like human machine interface would allow simultaneous speech output to grasping actions. The term resource component refers not necessarily to a specific hardware device. It may also represent a computational unit like a functional library for planning, numerical computation, mapping or navigation. - **Variable:** The meta-model further contains the structure for variables that are globally defined. Variables are either simple or arbitrarily cascaded like structs in the C language. Domain-specific language ------------------------ The meta-model defines abstract constructs and does not define concrete functionality. Only the abstract elements action, resource component and variable are described. The concrete instances which are available to the programmer are formalized by the Domain Specific Language (DSL). The DSL can be described for different types of robot classes. Every DSL has to define which concrete resource components are provided by a specific robot class e.g. vacuum cleaners. For every resource component it must be defined which specific types of actions are provided by the resource. Those actions must each explicitly declare which other actions are not allowed to run simultaneously. Further, the DSL specifies the possible variables that a programmer can use. As mentioned, those can be basic variables like integer or float, or more complex structures build from other defined variables like the structure of an address. An example of a concrete DSL is sketched in Figure \[fig:PaperExampleDSL\]. It shows an excerpt of a DSL for a vacuum cleaning robot and defines the resource components *DriveBase* with the actions *MoveFwd* and *Stop* and the resource component *CleaningDevice* with the action *Discharge*. The shown DSL has adds a parallelism constraint. It describes that the action *moveFwd* may never run simultaneously to the action *discharge*. This removes the semantical error that a vacuum cleaning robot could move while distributing all collected dirt on the floor. Finally, to get running robot code, a template-based code generator is used. The code generator is the first place in the process where a concrete robot systems is targeted since the program sequence and DSL can be defined for a complete robot class. Every resource component and every action can have an arbitrary number of code templates. Those templates must be written based on the robot framework and programming language. They must implement the action sequences for the robot. This makes it possible to generate code for different robot systems from the same sequence. Further, an arbitrary number of main templates can be specified. This may be necessary for robot code in e.g. C++ where multiple code files are required (source files and headers). Further, it enables the definition of code templates in different languages or frameworks for the same robot types but different concrete instances. An overview of the system in the MDSD terminology is shown in Figure \[fig:SystemConceptMDSD\]. The figure shows the relation between the action sequence programmer, the programming environment and configuration as well as the domain expert that describes the domain specific elements. - A **programmer** can create a concrete program model respective formal model with the programming environment which is an action sequence for a robot class. It is based on a DSL that contains the description of the concrete programming elements for the robot class. This DSL is based on a meta-model which provides structure and is the same for any formal model and robot. - The **domain expert** specifies the DSL via a configuration file. Profound knowledge in programming a certain robot respective robot class is needed. Moreover, the domain expert has to provide the code templates for a specific robot to enable the translation from formal model to specific robot code. Implementation -------------- The system is developed with Java and the Eclipse Rich Client Platform RCP [@c13] and is operating system independent. The graphical editor is implemented with the Eclipse Graphical Editing Framework [@c11] and allows the creation of the described action sequences. The meta-model is formalized as a set of Java classes while the DSL configuration can be separately loaded as XML files. Moreover, the existing sequences can be robot independent serialized to XML. The editor validates the generated action sequences on completeness, e.g. are all names unique, variables instantiated and parameters set and verifies if the defined parallelism constraints are not violated. For code generation the Apache Velocity Engine [@c12] is integrated. This code genaration library was chosen because of its very clear and powerful template language. The amount, location and names of the templates are also configurable with XML. USE CASES {#sec:usecases} ========= Service robotics use case ------------------------- In the first use case a simulated service robot system with a holonomic drive system and two manipulators has been integrated in the environment. An example sequence which makes the robot move to a certain position and grasp an object is shown in Figure \[fig:sampleMRDS\]. For this, a minimalistic DSL containing the resource components for each gripper and the drive base has been designed with actions for closing a gripper, moving the arm to certain positions and moving the robots base. An excerpt of the DSL is given in Listing \[list:dslconfigscheduler\]. {} <ResourceComponent type="Manipulator"> <Action returnType="String" actionIdentifier="MoveManipulator"> <ParameterList> <Parameter type="Vector3" name="targetPose"> </Parameter> <Parameter type="Vector3"name="orientation"> </Parameter> </ParameterList> <NotAllowedSimultaneousActionTypes> <NotAllowedSimultaneousAction type="MoveTo"> </NotAllowedSimultaneousAction> </NotAllowedSimultaneousActionTypes> </Action> </ResourceComponent> To generate code, every action and resource component has its custom code template. An excerpt of the moveManipulator action is given in Listing \[list:MRDSTemplateMoveManipulator\]. The template uses keywords of the velocity engine which are indicated by a ’\#’ for control structures and ’\$’ for data access from the formal model. Listing \[list:MRDSGeneratedMoveManipulator\] shows the generated code that results for this template. {} //Create list of parameters parameters = new List<ParameterVariable>(); //fill list of parameters #foreach($Parameter in $Action.getParameters()) //Add previous initialized variables parameters.Add(getVariable( "$Parameter.getVariable().getName()")); #end //Create robot specific action ExecutionElement $Action.getName() = new ExecElement(MOVE_MANIPULATOR, parameters)); {} //Create list of parameters parameters = new List<ParameterVariable>(); //Add previous initialized variables parameters.Add(getVariable("targetPose")); parameters.Add(getVariable("orientation")); //Create robot system specific action ExecutionElement MoveMani = new ExecElement(MOVE_MANIPULATOR, parameters)); Educational use case -------------------- A further use case based on a different configured DSL has been implemented with the same programming environment. This DSL configures the environment to program a Lego Mindstorm NXT educational robot. The robot was constructed with a left and right ultra sonic sensor, that are oriented at the robot front. The robot is actuated by a differential drive base. This use case shows the implementation of a Braitenbergs vehicles [@c10] so that the robot avoids obstacles. The program sequence is shown in Figure \[fig:sampleLegoObstacle\]. In Braitenberg vehicles the sensor readings are directly mapped to actuators which can result in complex reactive behaviors. For the use case the robots left sonar is mapped to the left motor and the right sonar is mapped to the right motor. The closer the robot senses an object on its right sensor, the higher the sensor value respective right motor speed which results in a turning behavior away from the obstacle. In the sequence shown in the figure, the sensors are read simultaneously and their results are passed to the left and right motors. This is done by two global variables storing the left and right ultra sonic sensor readings. After reading, these variables are used by the motors to set the speed. The left sensor shares a variable with the right motor and the right sensor shares a variable with the left motor. If an obstacle is detected in the left sensor, the motor on the right gets slower since it acts directly on the sensor reading that gets smaller by closer obstacle distance. This results in a turn away from the obstacle. The sequence is translated to NXC code that must be compiled and transfered to the robot. CONCLUSION {#sec:conclusion} ========== This paper presented a programming environment that allows the graphical programming of robot action sequences comparable to dependency graphs. Main focus of the environment is the simplification of robot programming especially in the sense of parallel program development. The designed system is based on the model-driven software development aspects which makes it possible to re-use it with any robot system or even transform the same described program into code for different robot platforms. In order to achieve this, a meta-model has been designed while robot specific DSLs and code generation settings can be loaded via XML configuration files. Two exemplary use-cases were described in which a a service robot and a robot for educational purpose were programmed with the presented environment. The environment is successfully used in lectures to teach students the programming of Lego NXT Robots. There it has been proven as a valuable tool to teach parallel programming on real robots. In the future we will develop domain-specific languages and code generation templates for different robot platforms so for example our service robot Johnny Jackanapes [@c14] which is participating in the service robot competition RoboCup@Home. Further, the programming model will be enhanced by decisional elements to allow a state machine oriented programming. ACKNOWLEDGMENTS =============== The research leading to these results has received funding from the European Community’s Seventh Framework Programme (FP7/2007-2013) under grant agreement no. FP7-ICT-231940-BRICS (Best Practice in Robotics). [99]{} P. A. Baer, R. Reichle, and K. Geihs. The Spica Development Framework – Model-Driven Software Development for Autonomous Mobile Robots. In [*IAS-10, pp. 211–220*]{}. 2008. Christian Schlegel, Thomas Ha"sler, Alex Lotz and Andreas Steck. Robotic Software Systems: From Code-Driven to Model-Driven Designs. In [*Proc. 14th Int. Conf. on Advanced Robotics (ICAR), Munich*]{}. 2009. Diego Alonso, Cristina Vincente-Chicote, Francisco Ortiz, Juan Pastor, Barbara Alvarez. V3CMM: A 3-View Component Meta-Model for Model-Driven Robotic Software Development. In [*Journal of Software Engineering for Robotics*]{}. 2010 Joachim Hertzberg and Frank Schönherr. Concurrency in the DD&P Robot Control Architecture. In [*Proc. of The Int. NAISO Congress on Information Science Innovations (ISIÕ2001)*]{}. 2001. Gostai. Urbi Studio, [*\[Online\] http://www.gostai.com/products/studio/*]{}, May 2010. Aldebaran Robotics. Nao Choregraphe, [*\[Online\] http://www.aldebaran-robotics.com/en*]{}, July 2010. Susan Horwitz and Thomas Reps. The use of program dependence graphs in software engineering, [*Proceedings of the 14th international conference on Software engineering*]{}, p.392-411, May 1992. Thomas Stahl and Markus Völter. Model-Driven Software Development. [*Wiley & Sons*]{}, 2006, Jeff McAffer, Jean-Michel Lemieux and Chris Aniszczyk. Eclipse Rich Client Platform - Designing, Coding and Packaging Java Applications, [*Addison-Wesley, the eclipse series*]{}, 2006. The Eclipse Foundation. The Graphical Editing Framework, [*\[Online\] http://www.eclipse.org/gef*]{}, July 2010. Apache Software Foundation. The Apache Velocity Framework, [*\[Online\] http://velocity.apache.org/*]{}, July 2010. Valentin Braitenberg. Vehicles: Experiments in synthetic psychology, [*MIT Press*]{}, 1984. Thomas Breuer, Geovanny Giorgana, Frederik Hegger, Christian Müller, Zha Jin, Michael Reckhaus, Jan Paulus, Nico Hochgeschwender, Ronny Hartanto, Paul Ploeger and Gerhard Kraetzschmar. The b-it-bots RoboCup@Home 2010 Team Description Paper, RoboCup-2010, Singapore, 2010.

Time-stamp: &lt;\#1 \#2&gt;[\#2]{} In January 2019 the journal *Nature* reported on an exciting development in Machine Learning: the very first issue of journal *Nature Machine Intelligence* contains a paper that describes a learning problem whose solvability is neither provable nor refutable on the basis of the standard ${{\mathsf}{ZFC}}$ axioms of Set Theory. In this note I describe what the fuss is all about and indicate that maybe the problem is not so undecidable after all. Introduction {#introduction .unnumbered} ============ In the paper [@machlearnCH], in *Nature Machine Intelligence*, its authors exhibit an abstract machine-learning situation where the learnability is actually neither provable nor refutable on the basis of the axioms of ${{\mathsf}{ZFC}}$. This was deemed so exciting that the mother journal *Nature* actually devoted two commentaries to this: see [@Reyzin] and [@Castelvecchi]. The first of these, [@Reyzin], is rather matter-of-fact in its description of the problem but the second manages, in just a few lines, to mix up Gödel’s Incompleteness Theorems and the undecidability of the Continuum Hypothesis. It misstates the former — “Gödel discovered logical paradoxes” — and misinterprets the latter: “a paradox known as the Continuum Hypothesis”. In popular parlance one can say that Gödel *employed* the Liar’s Paradox in his proof of his incompleteness theorems but those are not paradoxes, they are, as the name indicates, *theorems*. And the Continuum Hypothesis is not a paradox; it is ‘simply’ a statement that cannot be proved nor refuted on the basis of the usual axioms of Set Theory. In the first part of this note I will explain what the set theory behind the paper is and where the undecidability comes from. In the second part I will show why I think that the problem is not undecidable at all: there is no algorithm that solves this particular learning problem. The amount of Set Theory needed to appreciate the arguments in this paper is not too large. We shall meet the cardinal numbers $\aleph_k$ for $k\le\omega$ as well as the cardinality of the continuum: $2^{\aleph_0}$. We shall also use the ordinals $\omega_k$ as ‘typical’ well-ordered sets of cardinality $\aleph_k$. The first chapter of Kunen’s book [@MR597342] more than suffices for our purposes. The Learning Problem ==================== The following is a summary of the parts of [@machlearnCH] that lead to the undecidability result. The authors start with the following real-life situation as an instance of their general learning problem. A website has a collection of advertisements that it can show to its visitors; each advertisement, $A$, comes with a set, $F_A$, of visitors for whom it is of interest: say if $A$ advertises running shoes then $F_A$ contains avid runners (or people who just like snazzy shoes). Choosing the optimal advertisement to display amounts to choosing a finite set from a population while maximizing the probability that the visitor is actually in that set. The problem is that the probabilty distribution is unknown. Rather than dwell on this particular example the authors make an abstraction: Given a set $X$ and a family ${\mathcal{F}}$ of subsets of $X$ find a member of ${\mathcal{F}}$ whose measure with respect to an unknown probability distribution is close to maximal. This should be done based on a finite sample generated i.i.d. from the unknown distribution. The undecidability manifests itself when we let $X$ be the unit interval $\I$ and ${\mathcal{F}}$ the family ${\operatorname{fin}}\I$ of finite subsets of $\I$. Learning functions ------------------ In the general situation the abstract problem described above is made more explicit and quantitative as follows. For the unknown probability distribution $P$ on $X$ find $F\in{\mathcal{F}}$ such that $E_P(F)$ is quite close to ${\operatorname{Opt}}(P)$, which is defined to be $\sup_{Y\in{\mathcal{F}}}E_P(Y)$. To quantify this further a *learning function* for a ${\mathcal{F}}$ is defined to be a function $$G:\bigcup_{k\in\N}X^k\to{\mathcal{F}}$$ with certain desirable properties. In this case the desirable properties are captured in the following definition of an *$(\epsilon,\delta)$-EMX learner* for ${\mathcal{F}}$. This is a function $G$ as above such that for some $d\in\N$, depending on $\epsilon$ and $\delta$, the following inequality holds $$\Pr\limits_{S\sim P^d} \left[E_P\bigl(G(S)\bigr)\le{\operatorname{Opt}}(P)-\varepsilon\right]\le\delta$$ for all distributions $P$ with finite support. The letters EMX abbreviate ‘estimating the maximum’. A combinatorial translation =========================== The first step in [@machlearnCH] is to translate the existence of a suitable function $G$ into a statement that is a bit more amenable to set-theoretic treatment. This translation involves what the author call monotone compression schemes. Here and later we use $[X]^n$ to denote the family of $n$-element subsets of $X$. \[def.scheme\] Let $m$ and $d$ be two natural numbers with $m>d$. An *$m\to d$ monotone compression scheme* for a family ${\mathcal{F}}$ of finite subsets of a set $X$ is a function $\eta:[X]^d\to{\mathcal{F}}$ such that whenever $A$ is an $m$-element subset of $X$ it has a $d$-element subset $B$ such that $A\subseteq \eta(B)$, where we identify $B$ with a point in $X$ that enumerates it. This is slightly different from the formulation of Definition 2 in [@machlearnCH], which leaves open the possibility that ${\mathopen|{A}\mathclose|}<m$ and that ${\mathopen|{B}\mathclose|}<d$, as it uses indexed sets. It is clear from the results and their proofs that our definition captures the essence of the notion. There is a second unnamed function implicit in Definition \[def.scheme\]: the choice of the subset $B$ of $A$, we call this function $\sigma$. So our schemes consist of a pair of functions: $\sigma:[X]^m\to[X]^d$ and $\eta:[X]^d\to{\mathcal{F}}$; they should satisfy $A\subseteq (\eta\circ\sigma)(A)$ for all $A$. Also, in the cases that we are interested in the set $X$ comes with a linear order $\prec$; in that case we can identify $[X]^m$, the family of $m$-element subsets with a subset of the product $X^m$. Every set corresponds to its monotone enumeration: $[X]^m=\{x\in X^m:(i<j<m)\to(x_i\prec x_j)\}$. The translation is now as follows. For an upward-directed family ${\mathcal{F}}$ of finite sets the existence of a $(\frac13,\frac13)$-EMX learning function is equivalent to the existence of a natural number $m$ and an $(m+1)\to m$ monotone compression scheme for ${\mathcal{F}}$. The proof of necessity takes the natural number $d$ in the learning function and produces a monotone $(m+1)\to m$ compression scheme with $m=\lceil\frac32d\rceil$. At this point the authors turn to the special case of the unit interval $\I$ and its family ${\operatorname{fin}}\I$ of finite subsets and prove the following. \[thm.weakCH\] There is a monotone $(m+1)\to m$ compression scheme for ${\operatorname{fin}}\I$ for some $m\in\N$ if and only if $2^{\aleph_0}<\aleph_\omega$. As the inequality $2^{\aleph_0}<\aleph_\omega$ is both consistent with and independent of the axioms of ${{\mathsf}{ZFC}}$ the same holds for the existence of a compression scheme and for the existence of a $(\frac13,\frac13)$-EMX learning function. Theorem \[thm.weakCH\] is an immediate consequence of the set of equivalences in the following theorem. \[Theorem-1\] Let $k\in\N$ and let $X$ be a set. Then there is a $(k+2)\to(k+1)$ monotone compression scheme for the finite subsets of $X$ if and only ${\mathopen|{X}\mathclose|}\le\aleph_k$. Indeed, $2^{\aleph_0}<\aleph_\omega$ if and only if ${\mathopen|{\I}\mathclose|}=\aleph_k$ for some $k\in\N$. In the next section we take a closer look at monotone compression schemes and point out a connection with an old result of Kuratowski’s. On compression schemes and decompositions ========================================= We begin by giving an equivalent description of monotone compression schemes that does not mention the function $\eta$. This shows that it is $\sigma$ that is doing the compressing. Let $m$ and $d$ be natural numbers and let $X$ be a set. There is an $m\to d$ monotone compression scheme for the finite subsets of $X$ if and only if there is a finite-to-one function $\sigma:[X]^m\to[X]^d$ such that $\sigma(x)\subseteq x$ for all $x$. If the pair ${\langle{\eta},{\sigma}\rangle}$ determines an $m\to d$ monotone compression scheme then $\sigma$ is finite-to-one. For let $y\in[X]^d$ then $\sigma(x)=y$ implies $x\subseteq\eta(y)$, hence there are at most $\binom{M}{m}$ such $x$, where $M={\bigl|{\eta(y)}\bigr|}$. Conversely, if $\sigma$ is as in the statement of the proposition then we can let $\eta(y)=\bigcup\{x:\sigma(x)=y\}$. Kuratowski’s decompositions --------------------------- The following theorem, proved by Kuratowski in [@MR0048518] provides one direction in his characterization of when a set has cardinality at most $\aleph_k$. \[thm.Kuratowski\] The power $\omega_k^{k+2}$ can be written as the union of $k+2$ sets, $\{A_i:i<k+2\}$, such that for every $i<k+2$ and every point $\langle x_j:j<k+2\rangle$ in $\omega_k^{k+2}$ the set of points $y$ in $A_i$ that satisfy $y_j=x_j$ for $j\neq i$ is finite. In Kuratowski’s words “$A_i$ is finite in the direction of the $i$th axis”. The case $k=0$ is easy: let $A_0=\{{\langle{m},{n}\rangle}:m\le n\}$ and $A_1=\{{\langle{m},{n}\rangle}:m>n\}$. The rest of the proof proceeds by induction on $k$. We give the step from $k=0$ to $k=1$ in some detail and leave the other steps to the reader. To decompose $\omega_1^3$ into three sets $A_0$, $A_1$ and $A_2$ we apply the Axiom of Choice to choose (simultaneously) for each infinite ordinal $\alpha$ in $\omega_1$ a decomposition $\{X(\alpha,0),X(\alpha,1)\}$ of $(\alpha+1)^2$, say by choosing well-orders of type $\omega$ and then using the decomposition obtained for $k=0$. - One puts $\langle\alpha,\beta,\gamma\rangle$ into $A_0$ if $\beta$ is the largest coordinate and ${\langle{\alpha},{\gamma}\rangle}\in X(\beta,0)$ or if $\gamma$ is the largest coordinate and ${\langle{\alpha},{\beta}\rangle}\in X(\gamma,0)$. - One puts $\langle\alpha,\beta,\gamma\rangle$ into $A_1$ if $\alpha$ is the largest coordinate and ${\langle{\beta},{\gamma}\rangle}\in X(\alpha,0)$ or if $\gamma$ is the largest coordinate and ${\langle{\alpha},{\beta}\rangle}\in X(\gamma,1)$. - One puts $\langle\alpha,\beta,\gamma\rangle$ into $A_2$ if $\alpha$ is the largest coordinate and ${\langle{\beta},{\gamma}\rangle}\in X(\alpha,1)$ or if $\beta$ is the largest coordinate and ${\langle{\alpha},{\gamma}\rangle}\in X(\gamma,0)$. To see that $A_0$ is finite in the direction of the $0$th coordinate take ${\langle{\beta},{\gamma}\rangle}\in\omega_1^2$, then $\langle\alpha,\beta,\gamma\rangle\in A_0$ implies $\beta$ is largest and ${\langle{\alpha},{\gamma}\rangle}\in X(\beta,0)$, or $\gamma$ is largest and ${\langle{\alpha},{\beta}\rangle}\in X(\gamma,0)$; in either case $\alpha$ belongs to a finite set. A similar argument works for $A_1$ and $A_2$ of course. The inductive steps for larger $k$ are modelled on this step. We now show how Theorem \[thm.Kuratowski\] can be used to prove sufficiency in Theorem \[Theorem-1\]. From a decomposition as in Theorem \[thm.Kuratowski\] we construct a finite-to-one function $\sigma:[\omega_k]^{k+2}\to[\omega_k]^{k+1}$ such that $\sigma(x)\subseteq x$ for all $x$. We assume, without loss of generality, that the sets $A_i$ are disjoint. Let $x\in[\omega_k]^{k+2}$ (so $i<j<k+2$ implies $x_i<x_j$). Take (the unique) $i$ such that $x\in A_i$ and let $\sigma(x)$ be the point in $\omega_k^{k+1}$ that is $x$ but without its coordinate $x_i$. In terms of sets we would have set $\sigma(x)=x\setminus\{x_i\}$. This function is finite-to-one: if $y\in[\omega_k]^{k+1}$ then for each $i<k+2$ there are only finitely many $x$ in $A_i$ with $y=\sigma(x)$. As mentioned above Kuratowski’s result works both ways: if $X^{k+2}$ admits a decomposition as above for $\omega_k^{k+2}$ then ${\mathopen|{X}\mathclose|}\le\aleph_k$. This suggests that the necessity in Theorem \[Theorem-1\] is related to the converse of Theorem \[thm.Kuratowski\]. This is indeed the case: one can construct a Kuratowski-type decomposition from a compression scheme, but because of our definition of the schemes we only get a decomposition of the subset $[\omega_k]^{k+2}$ of the whole power. This can be turned into one for the whole power but the process is a bit messy so we leave it be. The proof of necessity from [@machlearnCH] closes the circle of implications that proves the following. For a set $X$ and a natural number $k$ the following are equivalent: 1. ${\mathopen|{X}\mathclose|}\le\aleph_k$, 2. $X^{k+2}$ admits a Kuratowski-type decomposition into $k+2$ sets, 3. there is a $(k+2)\to(k+1)$ monotone compression scheme for the finite subsets of $X$. We sketch the proof of that last implication for completeness sake. Both it and Kuratowski’s necessity proof use a form of the following lemma. Let $k$, $l$, and $m$ be natural numbers with $m>l$. Assume $\sigma:[\omega_{k+1}]^{m+1}\to[\omega_{k+1}]^{l+1}$ determines an $(m+1)\to(l+1)$ monotone compression scheme. Then there is an $m\to l$ monotone compression scheme for $\omega_k$. We start by determining an ordinal $\delta$ as follows. Let $\delta_0=\omega_k$. Given $\delta_n$ use the fact that $\sigma$ is finite-to-one to find an ordinal $\delta_{n+1}>\delta_n$ such that every $x\in[\omega_{k+1}]^{m+1}$ that satisfies $\sigma(x)\in[\delta_n]^{l+1}$ is in $[\delta_{n+1}]^{m+1}$. In the end let $\delta=\sup_n\delta_n$. Then $\delta$ satisfies: every $x\in[\omega_{k+1}]^{m+1}$ that satisfies $\sigma(x)\in[\delta]^{l+1}$ is in $[\delta]^{m+1}$. We define an $m\to l$ monotone compression scheme for $\delta$. If $x\in[\delta]^m$ then $y=x\cup\{\delta\}$ is in $[\omega_{k+1}]^{m+1}$ and so $\sigma(y)\subseteq y$. It is not possible that $\sigma(y)\subseteq x$ by the choice of $\delta$, hence $\delta\in\sigma(y)$ and so setting $\varsigma(x)=\sigma(y)\setminus\{\delta\}$ defines a map $\varsigma:[\delta]^m\to[\delta]^l$. This map is finite-to-one and satisfies $\varsigma(x)\subseteq x$ for all $x$. To finish the proof of necessity we argue by induction and contradiction. If ${\mathopen|{X}\mathclose|}=\aleph_{k+1}$ and there is a finite-to-one $\sigma:[X]^{k+2}\to[X]^{k+1}$ with $\sigma(x)\subseteq x$ for all $x$ then there is a subset $Y$ of $X$ with ${\mathopen|{Y}\mathclose|}=\aleph_k$ and a finite-to-one $\varsigma:[Y]^{k+1}\to[X]^k$ with $\varsigma(x)\subseteq x$ for all $x$. This would contradict the obvious inductive assumption. We leave it as an exercise to the reader to ponder what absurdity would arise in the case $k=0$. Algorithmic considerations ========================== In this section we address a point already raised by the authors in [@machlearnCH]: the functions that are used in the previous sections are quite arbitrary and not related to any recognizable algorithm. Indeed, the constructions of the compression schemes for uncountable sets blatantly applied the Axiom of Choice: once by assuming that the underlying sets were well-ordered and again when in every step of the induction a choice of well-orders of type $\omega_k$ needed to be made. One may therefore wonder what happens if we impose some structure on the maps in question. One possible way of separating out ‘algorithmic’ functions is by requiring them to have nice descriptive properties. If ‘nice’ is taken to mean ‘Borel measurable’ then the desired functions do not exist. Continuity and Borel measurability ---------------------------------- Here we show, for arbitray $m\in\N$, that there does not exist an $(m+1)\to m$ monotone compression scheme for the finite subsets of $\I$ where the function $\sigma$ is Borel measurable. To this end let $m$ be a natural number and let $\sigma:[\I]^{m+1}\to[\I]^m$ be a function such that $\sigma(x)\subseteq x$ for all $x$. One can apply [@MR0006493]\*[Theorem VI.7]{} and deduce that there is a point $y$ such that the fiber $\sigma{^\gets}(y)$ is one-dimensional, but in this case there is an elementary and more informative argument. To this end let $x\in[\I]^{m+1}$ and assume for notational convenience that $\sigma(x)=\langle x_i:i<m\rangle$, i.e., that the coordinate $x_m$ is left out of $x$ when forming $\sigma(x)$. Let $\varepsilon=\frac13\min\{x_{i+1}-x_i:i<m\}$ and let $\delta>0$ be such that $\delta\le\varepsilon$ and for all $y\in[\I]^{m+1}$ with ${\mathopen\|{y-x}\mathclose\|}<\delta$ we have ${\mathopen\|{\sigma(y)-\sigma(x)}\mathclose\|}<\varepsilon$. Now if $y\in[\I]^{m+1}$ and ${\mathopen\|{y-x}\mathclose\|}<\delta$ then $\abs{y_i-x_i}<\varepsilon$ for all $i\le m$. Also, when $i<j$ we have $x_j-x_i>3\epsilon$. It follows that $y_m-x_i>\epsilon$ for all $i<m$. This implies that $\sigma(y)=\langle y_i:i<m\rangle$ for all $y$ with ${\mathopen\|{y-x}\mathclose\|}<\delta$. This shows that for every $i$ the set $O_i=\bigl\{x\in[\I]^{m+1}:\sigma(x)=x\setminus\{x_i\}\bigr\}$ is open. Because $[\I]^{m+1}$ is connected there is one $i$ such that $O_i=[\I]^{m+1}$. This shows that $\sigma$ cannot be finite-to-one. The above proof can be used/adapted to show that if $\sigma$ is Borel measurable it is not finite-to-one either. \[If $\sigma$ is Borel measurable then $\sigma$ is not finite-to-one\] There is a dense $G_\delta$-set $G$ in $[\I]^{m+1}$ such that the restriction of $\sigma$ to $G$ is continuous, see [@MR0217751]\*[§31 II]{}. Let $x\in G$. As in the previous proof we assume $\sigma(x)=\langle x_i:i<m\rangle$ and we obtain a $\delta>0$ such that $\sigma(y)=\langle y_i:i<m\rangle$ for all $y\in G$ that satisfy ${\mathopen\|{y-x}\mathclose\|}<\delta$. By the Kuratowski-Ulam theorem, [@KurUlam], we can find a point $y$ in $G$ with ${\mathopen\|{y-x}\mathclose\|}<\delta$ such that the set of points $t$ in the interval $(x_m-\delta,x_m+\delta)$ for which $y_t=\sigma(y)*\langle t\rangle$ belongs to $G$ is co-meager. But for every such point we have $\sigma(y_t)=\sigma(y)$ and this shows that $\sigma$ is not finite-to-one. EMX learning is impossible -------------------------- As we saw above a learning function is a function $G$ from the union $\bigcup_{k\in\N}\I^k$ to the family of finite subsets of $\I$. We can call such a function continuous or Borel measurable if its restriction to each individual power is. In the construction of an $(m+1)\to m$ compression scheme from a learning function the authors use its restriction to just one of these powers $\I^d$, where $d\le m$. The definition of $\eta(S)$ involves taking the union of $G(T)$ for all $d$-element subsets $T$ of $S$, hence a union of $\binom md$ many sets. The definition of $\sigma$ involves choosing one $m$-element subset with a certain property from of a given $m+1$-element set. The latter choice can be made explicit using a Borel linear order on the family of all finite subsets of $\I$, or just $[\I]^m$. An analysis of this procedure shows that if $G$ is Borel measurable then so are $\sigma$ and $\eta$. The results of this section then imply that a Borel measurable learning function does not exist. In this author’s opinion that means that the title of [@machlearnCH] should be emended to “EMX-learning is impossible”. On the other hand … ------------------- One may argue that the choice of the unit interval in [@machlearnCH] is a bit of a red herring. None of the arguments in the paper use the structure of $\I$ in any significant way. In the step from the problem of the advertisements to the more abstract problem there is no real need to go to the unit interval. One may equally well use the set of rational numbers to code or rank the elements of the learning set. In that case there is, as we have seen, a $2\to1$ monotone compression scheme for the finite subsets of $\N$: simply let $\sigma(x)=\max x$; the corresponding function $\eta$ is defined by $\eta(n)=\{i:i\le n\}$. It is an easy matter to transfer this scheme to the family of finite subsets of the rational numbers. Whether this scheme gives rise to a useful EMX learning function remains to be seen.

--- abstract: 'We present measurements of the branching fraction  and longitudinal polarization fraction  for  decays, with . The data sample, collected with the  detector at the SLAC National Accelerator Laboratory, represents $465 \times 10^6$ produced  pairs. We measure ${\mbox{${\mbox{${\cal B}$}}({\mbox{$B^0 \ra {\mbox{$a_1(1260)^+ $}}\ {\mbox{$a_1(1260)^- $}}\ $}})$}}\times [{\mbox{${\mbox{${\cal B}$}}({\mbox{${\mbox{$a_1(1260)^+ $}}\ra \pim \pip \pip $}})$}}\ ]^2 = {\ensuremath{{(11.8 \pm 2.6 \pm 1.6)}}\times 10^{-6}}$ and ${\mbox{$f_L$}}= 0.31 \pm 0.22 \pm 0.10$, where the first uncertainty is statistical and the second systematic. The decay mode is measured with a significance of 5.0 standard deviations including systematic uncertainties.' title: '**Observation and Polarization Measurement of $B^0$  Decay**' ---  \  \  \ -[PUB]{}-[09]{}/[011]{}\ SLAC-[PUB]{}-[13709]{}\ authors\_apr2009\_bad2144.tex Charmless $B$ decays to final states involving two axial-vector mesons (AA) have received considerable theoretical attention in the last few years [@Cheng; @Calderon]. Using QCD factorization, the branching fractions of several  decay modes have been calculated. Predictions for the branching fraction of the $B^0 \ra {\mbox{$a_1(1260)^+ $}}\ {\mbox{$a_1(1260)^- $}}$ decay mode vary between $37.4 \times 10^{-6} $ [@Cheng] and $6.4 \times 10^{-6} $ [@Calderon]. Branching fractions at this level should be observable with the  data sample, which can be used to discriminate between the predictions. The predicted value of the longitudinal polarization fraction ${\mbox{$f_L$}}$ is 0.64 [@Cheng]. The only available experimental information on this $B$ decay mode is the branching fraction upper limit (UL) of $2.8 \times 10^{-3}$ at 90% confidence level (CL) measured by CLEO [@a1a1CLEO]. The measured value ${\mbox{$f_L$}}\sim 0.5$ in penguin-dominated $B \ra \phi K^*$ decays [@PhiKst] is in contrast with naive standard model (SM) calculations predicting a dominant longitudinal polarization (${\mbox{$f_L$}}\sim 1$) in $B$ decays to vector-vector (VV) final states. The naive SM expectation is confirmed in the tree-dominated $B \ra \rho \rho$ [@RhoRho] and $B^+ \ra \omega \rho^+$ [@OmegaRho] decays. A value of ${\mbox{$f_L$}}\ \sim 1$ is found in vector-tensor $B \ra \phi K^*_2(1430)$ decays [@Kst2], while ${\mbox{$f_L$}}\ \sim 0.5$ is found in $B \ra \omega K^*_2(1430)$ decays [@OmegaRho] (see Ref [@PDGReview] for further discussion). The small value of  observed in $B \ra \phi K^*$ decays has stimulated theoretical effort, such as the introduction of non-factorizable terms and penguin-annihilation amplitudes [@SMCal]. Other explanations invoke new physics [@NP]. Measurement of  in  [@notazione] decays will provide additional information. We present the first measurements of the branching fraction and polarization in  decays, with  [@CC]. We do not separate the P-wave $(\pi\pi)_{\rho}$ and the S-wave $(\pi\pi)_{\sigma}$ components in the $a_1 \rightarrow 3\pi$ decay; a systematic uncertainty is estimated due to the difference in the selection efficiencies [@A1Pi]. Due to the limited number of signal events expected in the data sample, we do not perform a full angular analysis. Using helicity formalism, and after integration over the azimuthal angle between the decay planes of the two mesons, the predicted angular distribution $d\Gamma/d\cos{\theta} $ is: $$\begin{aligned} \frac{1}{\Gamma}\frac{d\Gamma}{d\cos\theta} \propto {\mbox{$f_L$}}(1 - \cos^2{\theta}) + \frac{1}{2} {\mbox{$f_T$}}(1+\cos^2{\theta}),\end{aligned}$$ where ${\mbox{$f_T$}}= 1 - {\mbox{$f_L$}}$ and $\theta$ is the angle between the normal to the decay plane of the three pions of one   and the flight direction of the other , both calculated in the rest frame of the first . The results presented here are based on data collected with the  detector [@BABARNIM] at the PEP-II asymmetric-energy $e^+e^-$ collider [@pep] located at the SLAC National Accelerator Laboratory. The analysis uses an integrated luminosity of 423.0 fb$^{-1}$, corresponding to $(465 \pm 5) \times 10^6$  pairs, recorded at the $\Upsilon (4S)$ resonance at a center-of-mass energy of $\sqrt{s}=10.58\ {\mbox{$\textrm{GeV}$}}$. An additional 43.9 fb$^{-1}$, taken about 40  below this energy (off-resonance data), is used for the study of  continuum background ($\epem\ra\qqbar$, with $q = u, d, s, c$). Charged particles are detected, and their momenta measured, by a combination of a vertex tracker (SVT) consisting of five layers of double-sided silicon microstrip detectors, and a 40-layer central drift chamber (DCH), both operating in the 1.5 T magnetic field of a superconducting solenoid. The tracking system covers 92% of the solid angle in the center-of-mass frame. We identify photons and electrons using a CsI(Tl) electromagnetic calorimeter (EMC). Further charged-particle identification is provided by the specific energy loss ([$\mathrm{d}\hspace{-0.1em}E/\mathrm{d}x$]{}) in the tracking devices and by an internally reflecting ring-imaging Cherenkov detector (DIRC) covering the central region. A $K/\pi$ separation of better than four standard deviations is achieved for momenta below 3 , decreasing to 2.5 $\sigma$ at the highest momenta in the $B$ decay final states. A more detailed description of the reconstruction of charged tracks in  can be found elsewhere [@Aux]. Monte Carlo (MC) simulations of the signal decay mode, continuum, backgrounds and detector response [@Geant] are used to establish the event selection criteria. The MC signal events are simulated as decays to ${\mbox{$a_1^+ $}}\ {\mbox{$a_1^- $}}$ with ${\mbox{$a_1 $}}\ra \rho(770) \pi$. The  meson parameters in the simulation are: mass $m_0 = 1230$  and width $\Gamma_0 = 400$  [@PDG2008; @EvtGen]. We reconstruct the decay of  into three charged pions. Two pion candidates are combined to form a $\rho^0$ candidate. Candidates with an invariant mass between $0.51$ and $ 1.10$ are combined with a third pion to form an $a_1$ candidate. The $a_1$ candidate is required to have a mass between $0.87$ and $1.75$ . We impose several particle identification requirements to ensure the identity of the signal pions. We also require the $\chi^2$ probability of the $B$ vertex fit to be greater than 0.01 and the number of charged tracks in the event to be greater or equal to seven. A $B$ meson candidate is kinematically characterized by the energy-substituted mass $\mes \equiv \sqrt{(s/2 + {{\bf p}}_0\cdot {{\bf p}}_B)^2/E_0^2 - {{\bf p}}_B^2}$ and energy difference ${\ensuremath{\Delta E}}\equiv E_B^*-\sqrt{s}/2$, where the subscripts $0$ and $B$ refer to the initial  and the $B$ candidate in the laboratory frame, respectively, and the asterisk denotes the  frame. The resolutions in and [$\Delta E$]{} are about $3.0$  and $20$ , respectively. We require candidates to satisfy $5.27 \le \mes \le 5.29$  and $ -90 < {\ensuremath{\Delta E}}< 70$ . Background arises primarily from random track combinations in continuum events. We reduce this background by using the angle [$\theta_{\rm T}$]{} between the thrust axis of the $B$ candidate and the thrust axis of the rest of the event (ROE), evaluated in the  rest frame. The distribution of $|{\ensuremath{\cos{\ensuremath{\theta_{\rm T}}}}}|$ is sharply peaked near $1$ for combinations drawn from jet-like continuum events and is nearly uniform for  events; for this reason, we require $|{\ensuremath{\cos{\ensuremath{\theta_{\rm T}}}}}|<0.65$. Background can also arise from  events, especially events containing a charmed meson (these are mostly events with five pions and a mis-identified kaon in the final state). The charmed background includes peaking modes, with structures in  and [$\Delta E$]{} that mimic signal events, and non-peaking “generic" modes. To suppress the charm background, we reconstruct $D$ and $D^*$ mesons. Events are vetoed if they contain $D$ or $D^*$ candidates with reconstructed masses within 20 MeV/$c^2$  (window size of about $\pm$2$\sigma$) of the nominal charmed meson masses [@PDG2008]. The mean number of $B$ candidates per event is 2.9. If an event has multiple $B$ candidates, we select the candidate with the highest $B$ vertex $\chi^2$ probability. From MC simulation, we find that this algorithm selects the correct candidate 90% of the time in signal events while inducing negligible bias. Using MC simulation of signal events with longitudinal (transverse) polarization, signal events are divided in two categories: correctly reconstructed signal (CR), where all candidate particles come from the correct signal , and self-cross feed (SCF) signal, where candidate particles are exchanged with a ROE particle. The fraction of SCF candidates is $31.8\pm3.2$ ($19.4\pm1.9$)%. We determine the number of signal events (the signal yield) from an unbinned extended maximum-likelihood (ML) fit. The seven input observables are [$\Delta E$]{}, , a Fisher discriminant  [@Aux], the two  masses and the two ${\ensuremath{{\cal H}}}=|\cos{\theta}|$. The Fisher discriminant  combines four variables calculated in the  frame: the absolute values of the cosines of the angles with respect to the beam axis of the $B$ momentum and the thrust axis of the $B$ decay products, and the zeroth and second angular Legendre moments $L_{0,2}$ of the momentum flow about the $B$ thrust axis. The Legendre moments are defined by $ L_k = \sum_m p_m \left|\cos\theta_m\right|^k$, where $\theta_m$ is the angle with respect to the $B$ thrust axis of a track or neutral cluster $m$, $p_m$ is its momentum, and the sum includes the ROE particles only. There are five hypotheses in the likelihood model: signal, continuum, and three  components, which take into account charmless, generic charm and peaking charm backgrounds. The likelihood function is: $${\cal L}= e^{-\left(\sum_{j=1}^5 n_j\right)} \prod_{i=1}^N \left[\sum_{j=1}^5 n_j {\cal P}_j ({\bf x}_i)\right],$$ where $N$ is the number of input events, $n_j$ is the number of events for hypothesis $j$ and ${\cal P}_j ({\bf x}_i)$ is the corresponding probability density function (PDF), evaluated with the observables ${\bf x}_i$ of the $i$th event. Since correlations among the observables are small ($<10$%), we take each ${\cal P}$ as the product of the PDFs for the separate variables. The signal includes both CR and SCF signal components with the SCF fraction fixed in the fit to the value estimated from MC simulation. Both CR and SCF signals are used to measure the branching fraction and polarization. The PDF of the signal takes the form: $$\begin{aligned} P_{sig} & = & {\mbox{$f_L$}}\left(1- g_L^{SCF}\right) {\cal P}_{CR,L} + {\mbox{$f_L$}}g_L^{SCF} {\cal P}_{SCF,L} \\ & + & {\mbox{$f_T$}}\left(1- g_T^{SCF}\right) {\cal P}_{CR,T} + {\mbox{$f_T$}}g_T^{SCF} {\cal P}_{SCF,T} \, \nonumber\end{aligned}$$ where $g_L^{SCF}$ ($g_T^{SCF}$) is the fraction of SCF in longitudinal (transverse) polarized signal events and ${\cal P}_{CR,L}$, ${\cal P}_{SCF,L}$ (${\cal P}_{CR,T}$, ${\cal P}_{SCF,T}$) are the signal PDFs of CR and SCF signal components for longitudinal (transverse) polarization. We determine the PDF parameters from Monte Carlo simulation for the signal and  backgrounds and from off-resonance data for the continuum background. We parameterize  and [$\Delta E$]{} using a Gaussian function with exponential tails [@cruj] for the CR signal and charmless components, and using polynomials for all other components, except for the  distribution for continuum events which is described by the ARGUS empirical phase space function [@argus] $x\sqrt{1-x^2}\exp{\left[-\xi(1-x^2)\right]}$, where $x\equiv2\mes/\sqrt{s}$ and $\xi$ is a parameter. The  mass is described by a relativistic Breit-Wigner function for the CR signal component, an asymmetric Gaussian plus a linear polynomial for the SCF signal component, and polynomials for the remaining components. The Fisher variable is parametrized with an asymmetric Gaussian plus a linear polynomial in all cases. The $\cal H$ variables are parametrized with a Gaussian plus a linear polynomial for the charm peaking component and with a polynomial in all other cases. The parameters left free in the fit are the signal, continuum, and three  component yields, and . We also float some of the parameters of the continuum PDFs: the three parameters of the asymmetric Gaussian part of , and one parameter each for the [${\cal H}$]{}, the $a_1$ masses and [$\Delta E$]{}. Large data samples of $B$ decays to charmed final states ($B^0 \rightarrow {D^*}^- a_1^+$), which have similar topology to the signal, are used to verify the simulated resolutions in  and [$\Delta E$]{}. Where the data samples reveal differences from the Monte Carlo we shift or scale the resolution function used in the likelihood fits. Any bias in the fit, which arises mainly from neglecting the small correlations among the discriminating observables, is determined from a large set of simulated experiments for which the continuum background is generated from the PDFs, and into which we have embedded the expected number of   background, signal and SCF events chosen randomly from fully simulated Monte Carlo samples. ----------------------------------------- --------------------------- Signal yield $545 \pm 118$ Signal yield bias $ +14$ bias $ -0.06$ $\epsilon_L$ (%) $9.0$ $\epsilon_T$ (%) $10.0$ $S$ $(\sigma)$ $5.0$ ${\mbox{${\cal B}$}}\ (\times 10^{-6})$ $11.8 \pm 2.6 \pm 1.6$ $0.31 \pm 0.22 \pm 0.10 $ ----------------------------------------- --------------------------- : Fitted signal yield and yield bias (in events), bias on , detection efficiencies $\epsilon_L $ and $\epsilon_T $ for events with longitudinal and transversal polarization, respectively, significance $S$ (including systematic uncertainties), measured branching fraction  and fraction of longitudinal polarization  with statistical and systematic uncertainties.[]{data-label="tab:results"} The fit results are presented in Table \[tab:results\]. The detection efficiencies are calculated as the ratio of the number of signal MC events passing all the cuts to the total number generated. We compute the branching fraction by subtracting the fit bias from the measured yield, and dividing the result by the number of produced pairs times the product of the daughter branching fractions and the detection efficiency. We assume that the branching fractions of the  to  and  are each 50%. The branching fraction and are corrected for the slightly different reconstruction efficiencies in longitudinal and transversal polarizations. The statistical uncertainty on the signal yield is taken as the change in the central value when the quantity $-2\ln{\cal L}$ increases by one unit from its minimum value. The significance is the square root of the difference between the value of $-2\ln{\cal L}$ (with systematic uncertainties included) for zero signal and the value at its minimum. In this calculation we have taken into account the fact that the floating ${\mbox{$f_L$}}$ parameter is not defined in the zero signal hypothesis. Figure \[fig:projections\] shows the projections of  , [$\Delta E$]{} , the  invariant mass,  and ${\ensuremath{{\cal H}}}$ for a subset of the data for which the ratio of the signal likelihood to the total likelihood (computed without using the variable plotted) exceeds a threshold that optimizes the sensitivity. A systematic uncertainty of 38 events on the signal yield due to the PDF parametrization is estimated by varying the signal PDF parameters within their uncertainties, obtained through comparison of MC and data in control samples. The uncertainty from the fit bias (7 events) is taken as half the correction itself. Uncertainty from lack of knowledge of the  meson parameters is 31 events. We vary the SCF fractions by their uncertainties and estimate a systematic uncertainty of 12 events. A systematic uncertainty of 19 events from possible contamination by $B^0 \ra {\mbox{$a_1(1260)^+ $}}\ a_2(1320)^-$ background events is estimated with simulated MC experiments. The uncertainty due to cross feed between the signal and non-resonant backgrounds, evaluated with MC events, is 10 events. Uncertainties of 1.4% and 3.6% are associated with the track efficiency and particle identification, respectively. Differences between data and simulation for the [$\cos{\ensuremath{\theta_{\rm T}}}$]{} variable lead to a systematic uncertainty of 2.5%. Assuming that 20% of  decays proceed through the S-wave $(\pi\pi)_\sigma$ channel [@PDG2008], we estimate a systematic uncertainty of 6.8% from the difference in reconstruction efficiency between the P-wave $(\pi \pi)_{\rho}$ and S-wave components. The uncertainty in the total number of  pairs in the data sample is 1.1%. The total systematic uncertainty, obtained by adding the individual terms in quadrature, is 12.9%. The main systematic uncertainties on   arise from the fit bias ($0.03$), the variation of PDF parameters ($0.08$), the  parametrization ($0.04$) and the non-resonant background (0.02). In conclusion, we have measured the branching fraction: ${\mbox{${\mbox{${\cal B}$}}({\mbox{$B^0 \ra {\mbox{$a_1^+ $}}\ {\mbox{$a_1^- $}}\ $}})$}}\times [{\mbox{${\cal B}$}}( {\mbox{${\mbox{$a_1^+ $}}\ra (3 \pi)^+ $}})]^2 = {\ensuremath{{(11.8 \pm 2.6 \pm 1.6)}}\times 10^{-6}}$ and the fraction of longitudinal polarization ${\mbox{$f_L$}}= 0.31 \pm 0.22 \pm 0.10$. Assuming that  is equal to , and that  is equal to 100% [@PDG2008], we obtain ${\mbox{${\mbox{${\cal B}$}}({\mbox{$B^0 \ra {\mbox{$a_1^+ $}}\ {\mbox{$a_1^- $}}\ $}})$}}= {\ensuremath{{(47.3 \pm 10.5 \pm 6.3)}}\times 10^{-6}}$. The decay mode is seen with a significance of 5.0 $\sigma$ including systematic uncertainties. 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--- abstract: 'We prove two dichotomy results for detecting long paths as patterns in a given graph. The [[NP]{}]{}-hard problem [Longest Induced Path]{} is to determine the longest induced path in a graph. The [[NP]{}]{}-hard problem [Longest Path Contractibility]{} is to determine the longest path to which a graph can be contracted to. By combining known results with new results we completely classify the computational complexity of both problems for $H$-free graphs. Our main focus is on the second problem, for which we design a general contractibility technique that enables us to reduce the problem to a matching problem.' author: - Walter Kern - 'Daniël Paulusma[^1]' title: 'Contracting to a Longest Path in $H$-Free Graphs' --- Introduction {#s-intro} ============ The [Hamiltonian Path]{} problem, which is to decide if a graph has a hamiltonian path, is one of the best-known problems in Computer Science and Mathematics. A more general variant of this problem is that of determining the length of a longest path in a graph. Its decision version [Longest Path]{} is equivalent to deciding if a graph can be modified into the $k$-vertex path $P_k$ for some given integer $k$ by using vertex and edge deletions. Note that an alternative formulation of [Hamilton Path]{} is that of deciding if a graph can be modified into a path (which must be $P_n$) by using only edge deletions. As such, these problems belong to a wide range of graph modification problems where we seek to modify a given graph $G$ into some graph $F$ from some specified family of graphs ${\cal F}$ by using some prescribed set of graph operations. As [Hamiltonian Path]{} is [[NP]{}]{}-complete (see [@GJ79]), [Longest Path]{} is [[NP]{}]{}-complete as well. The same holds for the problem [Longest Induced Path]{} [@GJ79], which is to decide if a graph $G$ contains an induced path of length at least $k$, that is, if $G$ can be modified into a path $P_k$ for some given integer $k$ by using only vertex deletions. Here we mainly focus on the variant of the above two problems corresponding to another central graph operation, namely edge contraction. This variant plays a role in many graph-theoretic problems, in particular [Hamilton Path]{} [@HV78; @HV81]. The *contraction* of an edge $uv$ of a graph $G$ deletes the vertices $u$ and $v$ and replaces them by a new vertex made adjacent to precisely those vertices that were adjacent to $u$ or $v$ in $G$ (without introducing self-loops or multiple edges). A graph $G$ contains a graph $G'$ as a [*contraction*]{} if $G$ can be modified into $G'$ by a sequence of edge contractions. [.99]{} <span style="font-variant:small-caps;">[Longest Path Contractibility]{}</span>\ ----------------- ------------------------------------------------------- *    Instance:* [a connected graph $G$ and a positive integer $k$.]{} *Question:* [does $G$ contain $P_k$ as a contraction?]{} ----------------- ------------------------------------------------------- The [Longest Path Contractibility]{} problem is [[NP]{}]{}-complete as well [@BV87]. Due to the computational hardness of [Longest Path]{}, [Longest Induced Path]{} and [Longest Path Contractibility]{} it is natural to restrict the input to special graph classes. We briefly discuss some known complexity results for the three problems under input restrictions. A common property of most of the studied graph classes is that they are [*hereditary*]{}, that is, they are closed under vertex deletion. As such, they can be characterized by a family of forbidden induced subgraphs. In particular, a graph is [*$H$-free*]{} if it does not contain a graph $H$ as an induced subgraph, and a graph class is [*monogenic*]{} if it consists of all $H$-free graphs for some graph $H$. Hereditary graph classes defined by a small number of forbidden induced subgraphs, such as monogenic graph classes, are well studied, as evidenced by studies on (algorithmic and structural) decomposition theorems (e.g. for bull-free graphs [@Ch12] or claw-free graphs [@CS05; @HMLW11]) and surveys for specific graph problems (e.g. for [Colouring]{} [@GJPS17; @RS04]). All the known [[NP]{}]{}-hardness results for [Hamiltonian Path]{} carry over to [Longest Path]{}. For instance, it is known that [Hamiltonian Path]{} is [[NP]{}]{}-complete for chordal bipartite graphs and strongly chordal split graphs [@Mu96], line graphs [@Be81] and planar graphs [@GJT76]. Unlike for [Hamiltonian Path]{}, there are only a few hereditary graph classes for which the [Longest Path]{} problem is known to be polynomial-time solvable; see, for example [@UU07]. In particular, [Longest Path]{} is polynomial-time solvable for circular-arc graphs [@MB14], distance-hereditary graphs [@GHK13], and cocomparability graphs [@IN13; @MC12]. The latter result generalized the corresponding results for bipartite permutation graphs [@UV07] and interval graphs [@IMN11]. The few graph classes for which the [Longest Induced Path]{} problem is known to be polynomial-time solvable include the classes of $k$-chordal graphs [@Ga02; @IOY08], AT-free graphs [@KMT03], graphs of bounded clique-width [@CMR00] (see also [@KMT03]) and graphs of bounded mim-width [@JKT17]. Finding a longest induced path in an $n$-dimensional hypercube is known as the [Snake-in-the-Box]{} problem [@Ka58], which has been well studied.[^2] Unlike the [Longest Path]{} and [Longest Induced Path]{} problems, [Longest Path Contractibility]{} is [[NP]{}]{}-complete even for [*fixed*]{} $k$ (that is, $k$ is not part of the input). In order to explain this, let $F$-[Contractibility]{} be the problem of deciding if a graph $G$ contains some fixed graph $F$ as a contraction. The complexity classification of $F$-[Contractibility]{} is still open (see [@BV87; @LPW08; @LPW08b; @HKPST12]), but Brouwer and Veldman [@BV87] showed that already $P_4$-[Contractibility]{} and $C_4$-[Contractibility]{} are [[NP]{}]{}-complete (where $C_k$ denotes the $k$-vertex cycle). In fact, $P_4$-[Contractibility]{} problem is [[NP]{}]{}-complete even for $P_6$-free graphs [@HPW09], whereas Heggernes et al. [@HHLP14] showed that $P_6$-[Contractibility]{} is [[NP]{}]{}-complete for bipartite graphs.[^3] The latter result was improved to $k=5$ in [@DP17]. Moreover, $P_7$-[Contractibility]{} is [[NP]{}]{}-complete for line graphs [@FKP13]. Hence, [Longest Path Contractibility]{} is [[NP]{}]{}-complete for all these graph classes as well. On the positive side, [Longest Path Contractibility]{} is polynomial-time solvable for $P_5$-free graphs [@HPW09]. Our interest in the [Longest Induced Path]{} problem also stems from a close relationship to a vertex partition problem, which played a central role in the graph minor project of Robertson and Seymour [@RS95], as we will explain. Our Results {#our-results .unnumbered} ----------- We first give a dichotomy for [Longest Induced Path]{} using known results for [Hamiltonian Path]{} and some straightforward observations (see Section \[s-pre\] for a proof). Our main result is a dichotomy for [Longest Path Contractibility]{}. We use ‘+’ to denote the disjoint union of two graphs, and a [*linear forest*]{} is the disjoint union of one or more paths. \[t-main0\] Let $H$ be a graph. If $H$ is a linear forest, then [Longest Induced Path]{} restricted to $H$-free graphs is polynomial-time solvable; otherwise it is [[NP]{}]{}-complete. \[t-main\] Let $H$ be a graph. If $H$ is an induced subgraph of $P_2+P_4$, $P_1+P_2+P_3$, $P_1+P_5$ or $sP_1+P_4$ for some $s\geq 0$, then [Longest Path Contractibility]{} restricted to $H$-free graphs is polynomial-time solvable; otherwise it is [[NP]{}]{}-complete. Theorem \[t-main\] shows that [Longest Path Contractibility]{} is polynomial-time solvable for $H$-free graphs only for some specific linear forests $H$. This is in contrast to the situation for [Longest Induced Path]{}, as shown by Theorem \[t-main0\]. To extend the aforementioned results from [@DP17; @FKP13; @HHLP14; @HPW09] for [Longest Path Contractibility]{} to the full classification given in Theorem \[t-main\] we do as follows. First, in Section \[s-poly\], we prove the four new polynomial-time solvable cases of Theorem \[t-main\]. In each of these cases $H$ is a linear forest, and proving these cases requires the most of our analysis.[^4] Every linear forest $H$ is $P_r$-free for some suitable value of $r$ and $P_r$-free graphs do not contain $P_r$ as a contraction. Hence, it suffices to prove that for each $1\leq k\leq r-1$, the $P_k$-[Contractibility]{} problem is polynomial-time solvable for $H$-free graphs for each of the four linear forests listed in Theorem \[t-main\]. In fact, as $P_3$-[Contractibility]{} is trivial, we only have to consider the cases where $4\leq k\leq r-1$. Our general technique for doing this is based on transforming an instance of $P_k$-[Contractibility]{} for $k\geq 5$ into a polynomial number of instances of $P_{k-1}$-[Contractibility]{} until $k=4$. For $k=4$ we cannot apply this transformation, as this case - as we outline below - is closely related to the $2$-[Disjoint Connected Subgraphs]{} problem. This problem takes as input a triple $(G,Z_1,Z_2)$, where $G$ is a graph with two disjoint subsets $Z_1$ and $Z_2$ of $V(G)$. It asks if $V(G)\setminus (Z_1\cup Z_2)$ has a partition into sets $S_1$ and $S_2$, such that $Z_1\cup S_1$ and $Z_2\cup S_2$ induce connected subgraphs of $G$. Robertson and Seymour [@RS95] proved that the more general problem $k$-[Disjoint Connected Subgraphs]{} (for $k$ subsets $Z_i$) is polynomial-time solvable as long as the union of the sets $Z_i$ has constant size.[^5] However, in our context, $Z_1$ and $Z_2$ may have arbitrarily large size. In that case, $2$-[Disjoint Connected Subgraphs]{} is [[NP]{}]{}-complete even if $|Z_1|=2$ (and only $Z_2$ is large) [@HPW09]. To work around this obstacle, we use the fact [@HPW09] that the two outer vertices of the $P_4$, to which the input graph $G$ must be contracted, may correspond to single vertices $u$ and $v$ of $G$. We then “guess” $u$ and $v$ to obtain an instance $(G-\{u,v\},N(u),N(v))$ of $2$-[Disjoint Subgraphs]{}. That is, we seek for a partition of $(V(G)\setminus \{u,v\})\setminus ((N_u)\cup N(v))$ into sets $S_u$ and $S_v$, such that $N(u)\cup S_u$ and $N(v)\cup S_v$ are connected. The latter implies that we can contract these two sets to single vertices corresponding to the two middle vertices of the $P_4$. After guessing $u$ and $v$ we exploit their presence, together with the $H$-freeness of $G$, for an extensive analysis of the structure of $S_u$ and $S_v$ of a potential solution $(S_u,S_v)$. To this end we introduce in Section \[s-p4\] some general terminology and first show how to check in general for solutions in which the part of $S_u$ or $S_v$ that ensures connectivity of $N(u)\cup S_u$ or $N(v)\cup S_v$, respectively, has bounded size. We call such solutions constant. If we do not find a constant solution, then we exploit their absence. For the more involved cases we show that in this way we can branch to a polynomial number of instances of a standard matching problem. In Section \[s-hard\] we prove the new [[NP]{}]{}-completeness results. In particular, we prove that $P_k$-[Contractibility]{}, for some suitable value of $k$, is [[NP]{}]{}-complete for bipartite graphs of large girth, strengthening the known result for bipartite graphs of [@HHLP14]. In Section \[s-classification\] we show how to combine our new polynomial-time and [[NP]{}]{}-hardness results with the known [[NP]{}]{}-completeness results for $K_{1,3}$-free graphs [@FKP13] and $P_6$-free graphs [@HPW09] in order to obtain Theorem \[t-main\]. In Section \[s-cycle\], we briefly discuss the cycle variant of our problem, called the [Longest Cycle Contractibility]{} problem [@Bl82; @Ha99; @Ha02]. Its complexity classification for $H$-free graphs is still incomplete, but we show that it differs from the classification of [Longest Path Contractibility]{} for $H$-free graphs. In Section \[s-con\] we pose some open problems. In particular, the complexity classification of [Longest Path]{} is still open for $H$-free graphs, and we describe the state-of-art for this problem. Preliminaries {#s-pre} ============= In Section \[s-gt\] we give some general graph-theoretic terminology and a helpful lemma for $P_4$-free graphs. In Section \[s-t1\] we give a short proof of Theorem \[t-main0\]. In Section \[l-et\] we give some terminology related to edge contractions. General Terminology and a Lemma for $P_4$-Free Graphs {#s-gt} ----------------------------------------------------- We consider finite undirected graphs with no self-loops. Let $G=(V,E$ be a graph. Let $S\subseteq V$. Then $G[S]=(S,\{uv\in E\; |\; u,v\in S\})$ denotes the subgraph of $G$ [*induced*]{} by $S$. We say that $S$ is [*connected*]{} if $G[S]$ is connected. We may write $G-S=G[V\setminus S]$. The *neighbourhood* of $v\in V$ is the set $N(v)=\{u\; |\; uv\in E\}$ and the [*closed neighbourhood*]{} is $N[v]=N(v)\cup \{v\}$. The [*length*]{} of a path $P$ is its number of edges. The [*distance*]{} $\operatorname{dist}_G(u,v)$ between vertices $u$ and $v$ is the length of a shortest path between them. Two disjoint sets $S, T\subset V$ are [*adjacent*]{} if there is at least one edge between them; $S$ and $T$ are [*(anti)complete*]{} to each other if every vertex of $S$ is (non)adjacent to every vertex of $T$. The set $S$ [*covers*]{} $T$ if every vertex of $T$ has a neighbour in $S$. The [*subdivision*]{} of an edge $e=uv$ in $G$ replaces $e$ by a new vertex $w$ and two new edges $uw$ and $wv$. A graph $G$ is [*$H$-free*]{} for some other graph $H$ if $G$ does not contain $H$ as an induced subgraph. For a set $H_1,\ldots,H_p$ of graphs, $G$ is [*$(H_1,\ldots,H_p)$-free*]{} if $G$ is $H_i$-free for $i=1,\ldots,p$. A graph is [*complete bipartite*]{} if it consists of a single vertex or its vertex set can be partitioned into two independent sets $A$ and $B$ that are complete to each other. The [*claw*]{} $K_{1,3}$ is the complete bipartite graph with $|A|=1$ and $|B|=3$. The graph $K_n$ is the complete graph on $n$ vertices. The [*disjoint union*]{} $G_1+\nobreak G_2$ of two vertex-disjoint graphs $G_1$ and $G_2$ is the graph $(V(G_1)\cup V(G_2), E(G_1)\cup E(G_2))$; the disjoint union of $r$ copies of a graph $G$ is denoted $rG$. A [*forest*]{} is a graph with no cycles. A [*linear forest*]{} is a forest of maximum degree at most 2, that is, a disjoint union of one or more paths. The [*join*]{} operation $\times$ adds an edge between every vertex of $G_1$ and every vertex of $G_2$. A graph $G$ is a [*cograph*]{} if $G$ can be generated from $K_1$ by a sequence of join and disjoint union operations. A graph is a cograph if and only if it is $P_4$-free (see, e.g., [@BLS99]). The following well-known lemma follows from this fact and the definition of a cograph. In particular, to prove that a connected $P_4$-free graph $G$ has a spanning complete bipartite graph with partition classes $A$ and $B$, we can do as follows: take the complement $\overline{G}=(V,\{uv\; |\; uv\not \in E\; \mbox{and}\; u\neq v\}$ of $G$ and put the vertex set of one connected component of $\overline{G}$ in $A$ and all the other vertices of $\overline{G}$ in $B$. \[l-p4\] Every connected $P_4$-free graph on at least two vertices has a spanning complete bipartite subgraph, which can be found in polynomial time. We remind the reader of the following notions.The [*girth*]{} of a graph $G$ that is not a forest is the number of vertices in a shortest induced cycle of $G$. The [*line graph*]{} $L(G)$ of a graph $G=(V,E)$ has $E$ as vertex set and there is an edge between two vertices $e_1$ and $e_2$ of $L(G)$ if and only if $e_1$ and $e_2$ have a common end-vertex in $G$. Every line graph is readily seen to be $K_{1,3}$-free. The Proof of Theorem \[t-main0\] {#s-t1} -------------------------------- We now present a short proof for Theorem \[t-main0\]. We start with the following lemma. \[l-girthpath\] Let $p\geq 3$ be some constant. Then [Longest Induced Path]{} is [[NP]{}]{}-complete for graphs of girth at least $p$. We reduce from [Hamiltonian Path]{}. Let $G$ be a graph on $n$ vertices. We subdivide each edge $e$ of $G$ exactly once and denote the set of new vertices $v_e$ by $V'$. We denote the resulting graph by $G'$ and note that $G'$ is bipartite with partition classes $V$ and $V'$. We claim that $G$ has a Hamiltonian path if and only if $G'$ has an induced path of length $2n-2$. First suppose that $G$ has a Hamiltonian path $u_1u_2\cdots u_n$. Then the path on vertices $u_1, v_{u_1u_2}, u_2, \ldots, v_{u_{n-1}u_n}, u_n$ is an induced path of length $2n-2$ in $G'$. Now suppose that $G'$ has an induced path $P'$ of length $2n-2$. Then either $P'$ starts and finished with a vertex of $V$, or $P'$ starts and finishes with a vertex of $V'$. In the first case $P'$ contains $n$ vertices of $G$, so $P$ contains all vertices $u_1,\ldots,u_n$ of $G$, say in this order. Then $u_1u_2\cdots u_n$ is a Hamiltonian path of $G$. In the second case $P'$ contains $n-1$ vertices of $V$, say vertices $u_1,\ldots,u_{n-1}$ in that order. As $P'$ is an induced path and vertices of $V'$ are only adjacent to vertices of $V$, this means that the end-vertices of $P'$ are both adjacent to $u_n$. Hence, we find that $u_1u_2\cdots u_n$ is a Hamiltonian path of $G$ (and the same holds for $u_nu_1\cdots u_{n-1}$). We note that the girth of $G'$ is twice the girth of $G$. Hence, we obtain the result by applying this trick sufficiently many times. We also need the following lemma. \[l-line\] The [Longest Induced Path]{} problem is [[NP]{}]{}-complete for line graphs. We reduce from [Hamiltonian Path]{}. Let $G=(V,E)$ be a graph on $n$ vertices. We construct the line graph $L(G)$ of $G$. We claim that $G$ has a Hamiltonian path if and only if $L(G)$ has an induced path on $n-1$ vertices. First suppose that $P$ is a Hamiltonian path in $G$. Then the edges of $P$ form an induced path of length $n-1$ in $L(G)$. Now suppose that $L(G)$ has an induced path $\tilde{P}$ on $n-1$ vertices. Let $e_1, \dots, e_{n-1}$ be the $n-1$ edges of $\tilde{P}$ in that order. As $\tilde{P}$ is induced in $L(G)$, no two edges $e_i$ and $e_j$ with $i<j$ have a vertex $v \in V$ in common unless $j=i+1$. Hence, $P=\{e_1, \dots, e_{n-1}\}$ must be a Hamiltonian path in $G$. We are now ready to prove Theorem \[t-main0\]. [**Theorem \[t-main0\]. (restated)**]{} [*Let $H$ be a graph. If $H$ is a linear forest, then [Longest Induced Path]{} restricted to $H$-free graphs is polynomial-time solvable; otherwise it is [[NP]{}]{}-complete.*]{} Let $G$ be an $H$-free graph. First suppose that $H$ is a linear forest. Then there exists a constant $k$ such that $H$ is an induced subgraph of $P_k$. This means that the length of a longest induced path of $G$ is at most $k-1$. Hence, we can determine a longest path in $G$ in $O(n^{k-1})$ time by brute force. Now suppose that $H$ is not a linear forest. First assume that $H$ contains a cycle. Let $g$ be the girth of $H$. We set $p=g+1$. Then the class of $H$-free graphs contains the class of graphs of girth at least $p$. Hence, we can use Lemma \[l-girthpath\] to find that [Longest Induced Path]{} is [[NP]{}]{}-complete for $H$-free graphs. Now assume that $H$ contains no cycle. As $H$ is not a linear forest, $H$ must be a forest with at least one vertex of degree at least 3. Then the class of $H$-free graphs contains the class of $K_{1,3}$-free graphs. Recall that every line graph is $K_{1,3}$-free. Hence, the class of line graphs is contained in the class of $H$-free graphs. Then we can use Lemma \[l-line\] to find that [Longest Induced Path]{} is [[NP]{}]{}-complete for $H$-free graphs. Terminology Related to Edge Contractions {#l-et} ---------------------------------------- Recall that the contraction of an edge $uv$ of a graph $G$ is the operation that deletes $u$ and $v$ from $G$ and replaces them by a new vertex made adjacent to precisely those vertices that were adjacent to $u$ or $v$ in $G$ (without introducing self-loops or multiple edges). We denote the graph obtained from a graph $G$ by contracting $e=uv$ by $G/e$. We may denote the resulting vertex by $u$ (or $v$) again and say that we [*contracted $e$*]{} on $u$ (or $e$ on $v$). Recall also that a graph $G$ contains a graph $H$ as a contraction if $G$ can be modified into $H$ via a sequence of edge contractions. Alternatively, a graph $G$ contains a graph $H$ as a contraction if and only if for every vertex $x\in V(H)$ there exists a nonempty subset $W(x)\subseteq V(G)$ of vertices in $G$ such that: - $W(x)$ is connected; - the set ${\cal W}=\{W(x)\; |\; x\in V_H\}$ is a partition of $V(G)$; and - for every $x_i,x_j\in V(H)$, $W(x_i)$ and $W(x_j)$ are adjacent in $G$ if and only if $x_i$ and $x_j$ are adjacent in $H$. By contracting the vertices in each $W(x)$ to a single vertex we obtain the graph $H$. The set $W(x)$ is called an $H$-[*witness bag*]{} of $G$ for $x$. The set ${\cal W}$ is called an [*$H$-witness structure*]{} of $G$ (which does not have to be unique). A pair of (non-adjacent) vertices $(u,v)$ of a graph $G$ is $P_k$-*suitable* for some integer $k\geq 3$ if and only if $G$ has a $P_k$-witness structure ${\cal W}$ with $W(p_1)=\{u\}$ and $W(p_k)=\{v\}$, where $P_k=p_1\dots p_k$; see Figure \[f-p4witness\] for an example. ![Two $P_4$-witness structures of a graph; the grey vertices form a $P_4$-suitable pair [@HPW09].[]{data-label="f-p4witness"}](p4witness.pdf) The following known lemma shows why $P_k$-suitable pairs are of importance. \[l-outer\] For $k\geq 3$, a graph $G$ contains $P_k$ as a contraction if and only if $G$ has a $P_k$-suitable pair. Lemma \[l-outer\] leads to the following auxiliary problem, where $k\geq 3$ is a fixed integer, that is, $k$ is not part of the input. [.99]{} <span style="font-variant:small-caps;">[$P_k$-Suitability]{}</span>\ ----------------- ---------------------------------------------------------------- *    Instance:* [a connected graph $G$ and two non-adjacent vertices $u,v$.]{} *Question:* [is $(u,v)$ a $P_k$-suitable pair?]{} ----------------- ---------------------------------------------------------------- The next, known observation follows from the fact that $P_k$-[Contractibility]{} is trivial for $k\leq 2$, whereas for $k=3$ we can use Lemma \[l-outer\] combined with the observation that $P_3$-[Suitability]{} is polynomial-time solvable (two non-adjacent vertices $u$, $v$ form a $P_3$-suitable pair in a connected graph $G$ if and only if $G-\{u,v\}$ is connected). \[l-trivial\] For $k\leq 3$, [$P_k$-Contractibility]{} can be solved in polynomial time. We now show the following lemma, which will be helpful for proving our results. \[l-reduce\] Let $k\geq 4$ and let $(G,u,v)$ be an instance of $P_k$-[Suitability]{} with $u$ and $v$ at distance $d > k$. Let $P$ be a shortest path from $u$ to $v$. Then $(G,u,v)$ can be reduced in polynomial time to $d-2$ instances $(G/e,u,v)$, one for each edge $e\in E(P)$ that is not incident to $u$ and $v$, with $\operatorname{dist}(u,v) =d-1$, such that $(G,u,v)$ is a yes-instance if and only if at least one of the new instances $(G/e,u,v)$ is a yes-instance of $P_k$-[Suitability]{}. First suppose that $(G,u,v)$ is a yes-instance of $P_k$-[Suitability]{}. Then $G$ has a $P_k$-witness structure ${\cal W}$ with $W(p_1)=\{u\}$ and $W(p_k)=\{v\}$. As $d\geq k$, at least one bag of ${\cal W}$ will contain both end-vertices of an edge $e$ of $P$. Then contracting $e$ yields a $P_{k}$-witness structure ${\cal W}'$ for $(G/e,u,v)$ with $W'(p_1)=\{u\}$ and $W'(p_k)=\{v\}$. As $W(p_1)$ and $W(p_k)$ only contain $u$ and $v$, respectively, the end-vertices of $e$ belong to some bag $W(p_i)$ with $2\leq i \leq k-1$. Hence, $e$ is not incident to $u$ and $v$. Now suppose that $P$ contains an edge $e$ not incident to $u$ and $v$ such that $(G/e,u,v)$ is a yes-instance of $P_k$-[Suitability]{}. Then $G/e$ has a $P_k$-witness structure ${\cal W}'$ with $W'(p_1)=\{u\}$ and $W'(p_k)=\{v\}$. Let $e=st$ and say that we contracted $e$ on $s$. As $e$ is not incident to $u$ and $v$, we find that $\{s,t\}\cap \{u,v\}=\emptyset$. Hence, $s$ belongs to some bag $W'(p_i)$ with $2\leq i \leq k-1$. Then in $W'(p_i)$ we uncontract $e$ (so the new bag will contain both $s$ and $t$). This yields a $P_k$-witness structure ${\cal W}$ of $G$ with $W(p_1)=\{u\}$ and $W(p_k)=\{v\}$. In our polynomial-time algorithms for constructing $P_k$-witness structures (to prove Theorem \[t-main\]) we put vertices in certain sets that we then try to extend to $P_k$-witness bags (possibly via branching) and we will often apply the following rule: [Contraction Rule.]{} If two adjacent vertices $s$ and $t$ end up in the same bag of some potential $P_k$-witness structure, then contract the edge $st$. For a graph $G=(V,E)$, we say that we [*apply the [Contraction Rule]{} on some set*]{} $U\subseteq V$ if we contract every edge in $G[U]$. The advantage of applying this rule is that we obtain a smaller instance and that we can exploit the fact that the resulting set $G[U]$ has become independent. It is easy to construct examples that show that a class of $H$-free graphs is not closed under contraction if $H$ contains a vertex of degree at least 3 or a cycle. However, all polynomial-time solvable cases of Theorem \[t-main\] involve forbidding a linear forest $H$. The following known lemma, which is readily seen, shows that the [Contraction Rule]{} does preserve $H$-freeness as long as $H$ is a linear forest. Hence, we can safely apply the rule in our proofs of the polynomial-time solvable cases of Theorem \[t-main\]. \[l-contract\] Let $H$ be a linear forest and let $G$ be an $H$-free graph. Then the graph obtained from $G$ after contracting an edge is also $H$-free. The Polynomial-Time Solvable Cases of Theorem \[t-main\] {#s-poly} ======================================================== In this section we prove that [Longest Path Contractibility]{} is polynomial-time solvable for $H$-free graphs if $H=P_2+P_4$ (Section \[s-p2p4\]), $H=P_1+P_2+P_3$ (Section \[s-p1p2p3\]), $H=P_1+P_5$ (Section \[s-p1p5\]) and $H=sP_1+P_4$ for every integer $s\geq 0$ (Section \[s-sp1p4\]). To solve [Longest Path Contractibility]{} in each of these cases we will eventually check if the input graph can be contracted to $P_4$. This turns out to be the hardest situation to deal with in our proofs. Due to Lemma \[l-outer\], we can solve it by checking for each pair of distinct vertices $u$, $v$ with $N(u)\cap N(v)=\emptyset$ if $(G,u,v)$ is a yes-instance of $P_4$-[Suitability]{} (note that for any other pair $u$, $v$, we have that $(G,u,v)$ is a no-instance of $P_4$-[Suitability]{}). In Section \[s-p4\] we first provide a general framework by introducing some additional terminology and one general result for solving $P_4$-[Suitability]{}. On Contracting a Graph to $\mathbf{P_4}$ {#s-p4} ---------------------------------------- Let $(G,u,v)$ be an instance of $P_4$-[Suitability]{}. For every $P_4$-witness structure of $G$ with $W(p_1)=\{u\}$ and $W(p_4)=\{v\}$ (if it exists), every neighbour of $u$ belongs to $W(p_2)$ and every neighbour of $v$ belongs to $W(p_3)$. Throughout our proofs we let $T(u,v)=V(G)\setminus (N[u]\cup N[v])$ denote the set of remaining vertices of $G$, which still need to be placed in either $W(p_2)$ or $W(p_3)$. We write $T=T(u,v)$ if no confusion is possible. We say that a partition $(S_u,S_v)$ of $T$ is a [*solution*]{} for $(G,u,v)$ if $N(u)\cup S_u$ and $N(v)\cup S_v$ are both connected. Hence, a solution $(S_u,S_v)$ for $(G,u,v)$ corresponds to a $P_4$-witness structure ${\cal W}$ of $G$, where $W(p_1)=\{u\}$, $W(p_2)=N(u)\cup S_u$, $W(p_3)=N(v)\cup S_v$ and $W(p_4)=\{p_4\}$. A solution $(S_u,S_v)$ for $(G,u,v)$ is [*$\alpha$-constant*]{} for some constant $\alpha\geq 0$ if the following holds: either $S_u$ contains a set $S_u'$ of size $|S_u'|\leq\alpha$ such that $N(u)\cup S_u'$ is connected, or $S_v$ contains a set $S_v'$ of size $|S_v'|\leq\alpha$ such that $N(v)\cup S_v'$ is connected. We prove the following lemma. \[l-constant\] Let $(G,u,v)$ be an instance of $P_4$-[Suitability]{}. For every constant $\alpha\geq 0$, it is possible to check in $O(n^{\alpha+2})$ time if $(G,u,v)$ has an $\alpha$-constant solution. We first do the following check for vertex $u$. For each set $S$ of size $|S|\leq \alpha$ we check if $N(u)\cup S$ is connected and if every vertex of $N(v)$ is in the same connected component $D$ of the subgraph of $G$ induced by $(T\setminus S)\cup N(v)$. If so, then we put all vertices of $T \setminus V(D)$ in $S_u$ and all vertices of $T\cap V(D)$ in $S_v$. As $G$ is connected, this yields a solution $(S_u,S_v)$ for $(G,u,v)$. This takes $O(n^2)$ time for each set $S$. As the number of sets $S$ is $O(n^\alpha)$, the total running time is $O(n^{\alpha+2})$. We can do the same check in $O(n^{\alpha+2})$ time for vertex $v$. This proves the lemma. Let $(S_u,S_v)$ be a solution for an instance $(G,u,v)$ of $P_4$-[Suitability]{} that is not $7$-constant (the value $\alpha=7$ comes from our proofs). If $G[S_u]$ and $G[S_v]$ each contain at least one edge, then $(S_u,S_v)$ is [*double-sided*]{}. If exactly one of $G[S_u]$, $G[S_v]$ contains an edge, then $(S_u,S_v)$ is [*single-sided*]{}. If both $S_u$ and $S_v$ are independent sets, then $(S_u,S_v)$ is [*independent*]{}. The Case $\mathbf{H=P_2+P_4}$ {#s-p2p4} ----------------------------- We now show that [Longest Path Contractibility]{} is polynomial-time solvable for $(P_2+P_4)$-free graphs. As mentioned, we will do so via the auxiliary problem $P_k$-[Suitability]{}. We first give, in Lemma \[l-top4\], a polynomial-time algorithm for $P_4$-[Suitability]{} for $(P_2+P_4)$-free graphs. This is the most involved part of our algorithm. As such, we start with an outline of this algorithm. [*Outline of the $P_4$-Suitability Algorithm for $(P_2+P_4)$-free graphs.*]{}\ We first observe that for an instance $(G,u,v)$, we may assume that $u$ and $v$ are of distance at least 3, and consequently, $N(u)\cap N(v)=\emptyset$, and moreover we may assume that $N(u)$ and $N(v)$ are independent. Recall that $T=V(G)\setminus (N[u]\cup N[v])$. To get a handle on the adjacencies between $T$ and $V(G)\setminus T$ we will apply a (constant) number of branching procedures. For example, we will prove in this way that $G[T]$ may be assumed to be $P_4$-free. Each time we branch we obtain, in polynomial time, a polynomial number of new, smaller instances of [$P_4$-Suitability]{} satisfying additional helpful constraints, such that the original instance is a yes-instance if and only if at least one of the new instances is a yes-instance. We then consider each new instance separately. That is, we either solve, in polynomial time, the problem for each new instance or create a polynomial number of new and even smaller instances via some further branching. Our first goal is to check if $(G,u,v)$ has an $7$-constant solution. If so then we are done. Otherwise we prove that the absence of $7$-constant solutions implies that $(G,u,v)$ has no double-sided solution either. Hence, it remains to test if $(G,u,v)$ has a single-sided solution or an independent solution. We check single-sidedness with respect to $u$ and $v$ independently. We show that in both cases this leads either to a solution or to a polynomial number of smaller instances, for which we only need to check if they have an independent solution. This will enable us to branch in such a way that afterwards we may assume that $T$ is an independent set and that the solution we are looking for is equivalent to finding a star cover of $N(u)$ and $N(v)$ with centers in $T$. The latter problem reduces to a matching problem, which we can solve in polynomial time. \[l-top4\] $P_4$-[Suitability]{} can be solved in polynomial time for $(P_2+P_4)$-free graphs. Let $(G,u,v)$ be an instance of $P_4$-[Suitability]{}, where $G$ is a connected $(P_2+P_4)$-free graph. We may assume without loss of generality that $u$ and $v$ are of distance at least 3, that is, $u$ and $v$ are non-adjacent and $N(u)\cap N(v)=\emptyset$; otherwise $(G,u,v)$ is a no-instance. Recall that $T=V(G)\setminus (N[u]\cup N[v])$. Recall also that we are looking for a partition $(S_u,S_v)$ of $T$ that is a solution for $(G,u,v)$, that is, $N(u)\cup S_u$ and $N(v)\cup S_v$ must both be connected. In order to do so we will construct partial solutions $(S_u',S_v')$, which we try to extend to a solution $(S_u,S_v)$ for $(G,u,v)$. We use the [Contraction Rule]{} from Section \[s-pre\] on $N(u)\cup S_u'$ and $N(v)\cup S_v'$, so that these two sets will become independent. By Lemma \[l-contract\], the resulting graph will always be $(P_2+P_4)$-free. For simplicity, we will denote the resulting instance by $(G,u,v)$ again. After applying the [Contraction Rule]{} the size of the set $T$ will be reduced if a vertex $t\in T$ was involved in an edge contraction with a vertex from $N(u)$ or $N(v)$. In that case we say that we [*contracted $t$ away*]{}. At the beginning of our algorithm, $S_u'=S_v'=\emptyset$, and we start by applying the [Contraction Rule]{} on $N(u)$ and $N(v)$. This leads to the following claim. [[*\[c-ind\] $N(u)$ and $N(v)$ are independent sets.*]{}\ ]{}\ [**Phase 1: Exploiting the structure of $\mathbf{G[T]}$**]{} In the first phase of our algorithm, we will look into the structure of $G[T]$. Suppose $G[T]$ contains an induced $P_4$ on vertices $a_1$, $a_2$, $a_3$, $a_4$. If there exists a vertex $t\in N(u)$ not adjacent to any vertex of $\{a_1,a_2,a_3,a_4\}$, then $\{u,t\}\cup \{a_1,a_2,a_3,a_4\}$ induces a $P_2+P_4$ in $G$, a contradiction. Hence, $\{a_1,a_2,a_3,a_4\}$ must cover $N(u)$. Similarly, $\{a_1,a_2,a_3,a_4\}$ must cover $N(v)$. Suppose $G[T]$ has another induced $P_4$ on vertices $\{b_1,b_2,b_3,b_4\}$ such that $\{a_1,a_2,a_3,a_4\}\cap \{b_1,b_2,b_3,b_4\}=\emptyset$. By the same arguments, $\{b_1,b_2,b_3,b_4\}$ also covers $N(u)$ and $N(v)$. This means that $N(u)\cup \{a_1,a_2,a_3,a_4\}$ and $N(v)\cup \{b_1,b_2,b_3,b_4\}$ are both connected. We put each remaining vertex of $T$ into either $S_u$ or $S_v$ (which is possible, as $G$ is connected). This yields a ($4$-constant) solution for $(G,u,v)$. From now on, assume that $G[T]$ contains no induced copy of $P_4$ that is vertex-disjoint from $a_1a_2a_3a_4$ (so, every other induced $P_4$ in $G[T]$ contains at least one vertex of $\{a_1,a_2,a_3,a_4\}$). Below we will branch into $O(n^{16})$ smaller instances in which $G[T]$ is $P_4$-free, such that $(G,u,v)$ has a solution if and only if at least one of these new instances has a solution. [**Branching I**]{} ($O(n^{16})$ branches)\ We branch by considering every possibility for each $a_i$ $(1\leq i\leq 4$) to go into either $S_u$ or $S_v$ for some solution $(S_u,S_v)$ of $(G,u,v)$ (if it exists). We do this vertex by vertex leading to a total of $2^4$ branches. Suppose we decide to put $a_i$ in $S_u$. If $a_i$ is adjacent to a vertex of $N(u)$, then we apply the [Contraction Rule]{} on $N(u)\cup \{a_i\}$ to contract $a_i$ away. If $a_i$ is not adjacent to any vertex of $N(u)$, then we do as follows. For each solution $(S_u,S_v)$ with $a_i\in S_u$, there must exist a shortest path $P_i$ in $G[N(u)\cup S_u]$ from $a_i$ to a vertex of $N(u)$ (as $N(u)\cup S_u$ is connected). As $G$ is $(P_2+P_4)$-free, $G$ is $P_7$-free. Hence, $P_i$ must have at most six vertices and thus at most four inner vertices. We consider all possibilities of choosing at most four vertices of $T$ to belong to $S_u$ as inner vertices of $P_i$. As we may need to do this for $i=1,\ldots,4$, the above leads to a total of $O(n^{16})$ additional branches. For each branch we do as follows. For $i=1,\ldots,4$ we apply the [Contraction Rule]{} on $N(u)\cup \{a_i\} \cup V(P_i)$ to contract $a_i$ and the vertices of $V(P_i)$ away. We denote the resulting instance by $(G,u,v)$ again. Note that the property (Claim \[c-ind\]) that $N(u)$ and $N(v)$ are independent sets is maintained. Moreover, as every induced $P_4$ in $G[T]$ contained at least one vertex of $\{a_1,\ldots,a_4\}$, the following claim holds now as well. [[*\[c-p4free\] $G[T]$ is $P_4$-free.*]{}\ ]{}\ We now prove the following claim. [[*\[c-either\] Let $(S_u,S_v)$ be a solution for $(G,u,v)$ that is not $7$-constant. Let $t,x_1,x_2$ be three vertices of $T$ with $tx_1\notin E(G)$, $tx_2\notin E(G)$ and $x_1x_2\in E(G)$. If $t,x_1,x_2$ are in $S_u$, then every neighbour of $t$ in $N(u)$ is adjacent to at least one of $x_1,x_2$. If $t,x_1,x_2$ are in $S_v$, then every neighbour of $t$ in $N(v)$ is adjacent to at least one of $x_1,x_2$.*]{}\ ]{}\ [*Proof of Claim \[c-either\].*]{} We assume without loss of generality that $t,x_1,x_2$ belong to $S_u$. Suppose $t$ has a neighbour $w\in N(u)$ that is not adjacent to $x_1$ and $x_2$. Suppose there exists a vertex $w' \in N(u)$ not adjacent to any of $t,x_1,x_2$. Then, as $N(u)$ is independent by Claim \[c-ind\], $\{x_1,x_2\}\cup \{w',u,w,t\}$ is an induced $P_2+P_4$, a contradiction. Hence, $\{t,x_1,x_2\}$ covers $N(u)$. As $N(u)\cup S_u$ is connected, $G[N(u)\cup S_u]$ contains a shortest path $P$ from $t$ to $x_1$. As $G$ is $(P_2+P_4)$-free, $G$ is $P_7$-free. Hence, $P$ has at most four inner vertices (possibly including $x_2$). As $V(P)\cup \{x_2\} \cup N(u)$ is connected and $|V(P) \cup \{x_2\}|\leq 7$, we find that $(S_u,S_v)$ is a $7$-constant solution, a contradiction. This proves the claim. We will use the above claim at several places in our proof, including in the next stage. [**Phase 2: Excluding 7-constant solutions and double-sided solutions**]{} We first show that we may exclude double-sided solutions if we have no $7$-constant solutions. [[*\[c-double\]If $(G,u,v)$ has a double-sided solution, then $(G,u,v)$ also has a $7$-constant solution.*]{}\ ]{}\ [*Proof of Claim \[c-double\].*]{} For contradiction, assume that $(G,u,v)$ has a double-sided solution $(S_u,S_v)$ but no $7$-constant solution. By definition, $G[S_u]$ and $G[S_v]$ contain some edges $x_1x_2$ and $x_1'x_2'$, respectively. Then $N(u)$ must contain a vertex $w$ that is not adjacent to $x_1$ and $x_2$; otherwise the two vertices $x_1,x_2$, which belong to $S_u$, cover $N(u)$ and this would imply that $(S_u,S_v)$ is a $2$-constant solution, and thus also a $7$-constant solution. As $N(u)\cup S_u$ is connected and $N(u)$ is an independent set by Claim \[c-ind\], set $S_u$ must contain a vertex $t$ that is adjacent to $w$. Then, by Claim \[c-either\], vertex $t$ must be adjacent to at least one of $x_1,x_2$, say $x_1$. For the same reason, $S_v$ contains a vertex $t'$ that is adjacent to at least one of $x_1',x_2'$, say $x_1'$, and to some vertex $w'\in N(v)$ that is not adjacent to $x_1'$ and $x_2'$. Let $y\in N(v)$. If no vertex of $\{t,x_1,x_2\}$ is adjacent to $y$, then $\{v,y\}\cup \{u,w,t,x_1\}$ is an induced $P_2+P_4$ in $G$, unless $wy\in E(G)$. However, in that case $\{x_1,x_2\}\cup \{u,w,y,v\}$ is an induced $P_2+P_4$, a contradiction. Hence, $\{t,x_1,x_2\}$ covers $N(v)$. For the same reason we find that $\{t',x_1',x_2'\}$ covers $N(u)$. Then $(G,u,v)$ has a $3$-constant solution $(S_u^*,S_v^*)$ (which is $7$-constant by definition) with $\{t',x_1',x_2'\}\subseteq S_u^*$ and $\{t,x_1,x_2\}\subseteq S_v^*$, a contradiction. This proves the claim. Recall that a solution $(S_u,S_v)$ for $(G,u,v)$ is single-sided if exactly one of $G[S_u]$, $G[S_v]$ contains an edge and independent if $S_u,S_v$ are both independent sets. We now do as follows. First we check in polynomial time if $(G,u,v)$ has a $7$-constant solution by using Lemma \[l-constant\]. If so, then we are done. From now on assume that $(G,u,v)$ has no $7$-constant solution. Then, by Claim \[c-double\] it follows that $(G,u,v)$ has no double-sided solution. From the above, it remains to check if $(G,u,v)$ has a single-sided solution or an independent solution. If $(G,u,v)$ has a single-sided solution $(S_u,S_v)$ that is not independent, then either $S_u$ or $S_v$ is independent. Our algorithm will first look for a solution $(S_u,S_v)$ where $S_u$ is independent. We say that it is doing a [*$u$-feasibility check*]{}. If afterwards we have not found a solution $(S_u,S_v)$ where $S_u$ is independent, then our algorithm will repeat the same steps but now under the assumption that the set $S_v$ is independent. That is, in that case our algorithm will perform a [*$v$-feasibility check*]{}. [**Phase 3: Doing a $\mathbf{u}$-feasibility check**]{} We start by exploring the structure of a solution $(S_u,S_v)$ that is either single-sided or independent, and where $S_u$ is an independent set. As $S_u$ and $N(u)$ are both independent sets, $G[N(u)\cup S_u]$ is a connected bipartite graph. Hence, $S_u$ contains a set $S_u^*$, such that $S_u^*$ covers $N(u)$. We assume that $S_u^*$ has minimum size. Then each $s\in S_u^*$ has a nonempty set $Q(s)$ of neighbours in $N(u)$ that are not adjacent to any vertex in $S_u^*\setminus \{s\}$; otherwise we can remove $s$ from $S_u^*$, contradicting our assumption that $S_u^*$ has minimum size. We call the vertices of $Q(s)$ the [*private*]{} neighbours of $s$ with respect to $S_u^*$. We note that $N(u)\cup S_u^*$ does not have to be connected. However, as $(G,u,v)$ has no $7$-constant solution, and thus no $1$-constant solution, we find that $S_u^*$ has size at least 2. We may assume that there is no vertex $t\in S_u\setminus S_u^*$, such that $N(t)\cap N(u)$ strictly contains $N(s)\cap N(u)$ for some $s\in S_u^*$ (otherwise we put $t$ in $S_u^*$ instead of $s$). Let $Q_u$ be the union of all private neighbour sets $Q(s)$ ($s\in S_u^*$). As $|S_u^*|\geq 2$, we observe that $G[Q_u\cup S_u^*]$ is the disjoint union of a set of at least two stars whose centers belong to $S_u^*$. First suppose that $N(u)\setminus Q_u=\emptyset$. As $N(u)\cup S_u$ is connected and $G[Q_u\cup S_u^*]$ is the disjoint union of at least two stars, there exists a vertex $t\in S_u\setminus S_u^*$ that is adjacent to vertices $z\in Q(s)$ and $z'\in Q(s')$ for two distinct vertices $s,s'\in S_u^*$. As $N(u)\setminus Q_u=\emptyset$, we find that $Q(s)=N(s)\cap N(u)$. By our choice of $S_u^*$, this means that $Q(s)$ contains at least one vertex $w$ that is not adjacent to $t$. Similarly, $Q(s')$ contains a vertex $w'$ that is not adjacent to $t$. By the definition of $Q(s)$ and $Q(s')$, we find that $w$ and $z$ are not adjacent to $s'$, and $w'$ is not adjacent to $s$. Then $\{w',s'\}\cup \{w,s,z,t\}$ is an induced $P_2+P_4$ of $G$, a contradiction. From the above we find that $N(u)\setminus Q_u\neq \emptyset$. Let $y\in N(u)\setminus Q_u$. As $S_u^*$ covers $N(u)$ and $y\notin Q_u$, we find that $y$ must be adjacent to at least two vertices $s,s'\in S_u^*$. Suppose $y$ is not adjacent to some vertex $s^*\in S_u^*$. Let $z\in Q(s)$ and $z^*\in Q(s^*)$. By the definition of the sets $Q(s)$ and $Q(s^*)$, we find that $z$ is not adjacent to $s'$ and $s^*$ and that $z^*$ is not adjacent to $s$ and $s'$. In particular it holds that $z\neq z^*$. Then $\{s^*,z^*\}\cup \{z,s,y,s'\}$ is an induced $P_2+P_4$ in $G$, a contradiction. Hence, $y$ must be adjacent to all of $S_u^*$, that is, $N(u)\setminus Q_u$ must be complete to $S_u^*$. Note that this implies that $N(u)\cup S_u^*$ is connected. To summarize, if $(G,u,v)$ has a solution $(S_u,S_v)$ in which $S_u$ is an independent set, then the following holds for such a solution $(S_u,S_v)$: - The set $S_u$ contains a subset $S_u^*$ of size at least 2 that covers $N(u)$, such that each vertex in $S_u^*$ has a nonempty set $Q(s)$ of private neighbours with respect to $S_u^*$, and moreover, the set $N(u)\setminus Q_u$, where $Q_u=\bigcup_{s\in S_u^*}Q(s)$, is nonempty and complete to $S_u^*$. *Remark.* We emphasize that $S_u^*$ is unknown to the algorithm, as we constructed it from the unknown $S_u$, and consequently, our algorithm does not know (yet) the sets $Q(s)$. [**Phase 3a: Reducing $\mathbf{N(u)\setminus Q_u}$ to a single vertex $\mathbf{w_u}$**]{} We will now branch into a polynomial number of smaller instances, in which $N(u)\setminus Q_u$ consists of just one single vertex $w_u$. As we will show below, we can even identify $w_u$ and $Q_u$ for each of these new instances. Again, we will ensure that if one of these new instances has a solution, then $(G,u,v)$ has as solution. If none of these new instances has a solution, then $(G,u,v)$ may still have a solution $(S_u,S_v)$. However, in that case $S_u$ is not an independent set, while $S_v$ must be an independent set. As mentioned, we will check this by doing a $v$-feasibility check as soon as we have finished the $u$-feasibility check. [**Branching II**]{} ($O(n^4)$ branches)\ We will determine exactly those vertices of $N(u)$ that belong to $Q_u$ via some branching, under the assumption that $(G,u,v)$ has a solution $(S_u,S_v)$, where $S_u$ is independent, that satisfies (P). By (P), $S_u^*$ consists of at least two (non-adjacent) vertices $s$ and $s'$. Let $w\in Q(s)$ and $w'\in Q(s')$. We branch by considering all possible choices of choosing these four vertices. This leads to $O(n^4)$ branches, which we each process in the way described below. If we selected $s$ and $s'$ correctly, then $s,s'$ belong to an independent set $S_u$ that together with $S_v=T\setminus S_u$ forms a solution for $(G,u,v)$ that is not $7$-constant. This implies that $\{s,s'\}$ does not cover $N(u)$. Hence, we can pick a vertex $w^*\in N(u)\setminus \{w,w'\}$. If $w^*$ is adjacent to both $s$ and $s'$, then $w^*$ must belong to $N(u)\setminus Q_u$. In the other case, that is, if $w^*$ is adjacent to at most one of $s,s'$, then $w^*$ must belong to $Q_u$. Hence, we have identified in polynomial time the (potential) sets $Q_u$ and $N(u)\setminus Q_u$. Moreover, by applying the [Contraction Rule]{} on $N(u)\cup \{s,s'\}$ we can contract $s$ and $s'$ away. This also contracts all of $N(u)\setminus Q_u$ into a single vertex which, as we mentioned above, we denote by  $w_u$. Thus $w_u$ is complete to $S_u^*$. We denote the resulting instance by $(G,u,v)$ again. We also let $T_1=N(w_u)\cap T$ and $T_2=T\setminus T_1$. Note that $S_u^*\subseteq T_1$. As $S_u^*$ covers $N(u)$ and every vertex of $S_u^*$ is adjacent to $w_u$, we find that $N(u) \cup S_u^*$ is connected. Due to the latter and because every vertex of $T_2$ is not in $S_u^*$, we may put without loss of generality every vertex $t\in T_2$ with a neighbour in $N(v)$ in $S_v$. That is, we may contract such a vertex $t$ away by applying the [Contraction Rule]{} on $N(v)\cup \{t\}$. By the same reason, we may contract every edge between two vertices in $T_2$. Hence, we have proven the following claim. [[*\[c-t2\]$T_2$ is an independent set that is anticomplete to $N(v)$.*]{}\ ]{}\ Note that by definition, no vertex of $T_2$ is adjacent to $w_u$ either. In a later stage we will modify $T_2$ and this property may no longer hold. However, we will always maintain the properties that $T_2$ is independent and anticomplete to $N(v)$. By Lemma \[l-constant\] we check in polynomial time if $(G,u,v)$ has a $7$-constant solution. If so, then we are done. From now on suppose that $(G,u,v)$ has no $7$-constant solution. Recall that we are still looking for a single-sided or independent solution $(S_u,S_v)$, where $S_u$ is an independent set. We first show that we can modify $G$ in polynomial time such that afterwards $G[T]$ is $(K_3+P_1)$-free. Suppose $G[T]$ contains an induced $K_3+P_1$, say with vertices $x_1,x_2,x_3,y$ and edges $x_1x_2$, $x_2x_3$, $x_3x_1$. Consider a solution $(S_u,S_v)$ for $(G,u,v)$, where $S_u$ is an independent set. Recall that we already checked on $7$-constant solutions. Hence, $(S_u,S_v)$ is not $7$-constant. As $S_u$ is an independent set, at least two of $x_1,x_2,x_3$, say $x_1,x_2$, must belong to $S_v$. Then $(S_v \cap (N[x_1]\cup N[x_2]))\cup N(v)$ is connected; otherwise, as $S_v\cup N(v)$ is connected by definition, there would exist a vertex $t\in S_v\setminus (N[x_1]\cup N[x_2])$ with a neighbour in $N(v)$ that is not adjacent to $x_1$ and $x_2$, contradicting Claim \[c-either\]. As $y$ does not belong to $N[x_1]\cup N[x_2]$, this means that $y$ is not needed for $S_v$. From the above we can do as follows. If $y$ has a neighbour in $N(u)$, then we contract $y$ away by applying the [Contraction Rule]{} on $N(u)\cup \{y\}$. Otherwise, if $y$ has no neighbour in $N(u)$, then $y\in T_2$. As $N(u)\cup S_u^*$ is connected for some set $S^*_u \subseteq S_u\cap T_1$, this means that $y$ is not needed for $S_u$ either. Hence, we may contract the edge between $y$ and an arbitrary neighbour of $y$ (as $G$ is connected, $y$ has at least one such neighbour). We apply this rule, in polynomial time, for every induced copy of $K_3+P_1$ in $G[T]$. Note that Claim \[c-t2\] is maintained and that in the end the following claim holds. [[*\[c-t1\]$G[T]$ is $(K_3+P_1)$-free.*]{}\ ]{}\ We will now do some further branching to obtain $O(n)$ smaller instances in which $G[T_1]$ is $K_3$-free, such that the following holds. If one of these new instances has a solution, then $(G,u,v)$ has as solution. If none of these new instances has a solution, then $(G,u,v)$ may still have a solution $(S_u,S_v)$, but in that case $S_u$ is not an independent set while $S_v$ must be an independent set; this will be verified when we do the $v$-feasibility check. [**Branching III**]{} ($O(n)$ branches)\ We consider all possibilities of putting one vertex $t\in T_1$ in $S_u$. This leads to $O(n)$ branches. For each branch we do as follows. As $t$ is adjacent to $w_u$ (because $t\in T_1$), we can contract $t$ away using the [Contraction Rule]{} on $N(u)\cup \{t\}$. As $S_u$ is independent, every neighbour $t'$ of $t$ in $T_1$ must go to $S_v$. If such a neighbour $t'$ is adjacent to a vertex of $N(v)$, this means that we may contract $t'$ away by using the [Contraction Rule]{} on $N(v)\cup \{t'\}$. If $t'$ has no neighbour in $N(v)$, then we put $t' $ in $T_2$. By the [Contraction Rule]{} we may contract all edges between $t'$ and its neighbours in $T_2$, such that $T_2$ is an independent set again that is anticomplete to $N(v)$, so Claim \[c-t2\] is still valid (but $T_2$ may now contain vertices adjacent to $w_u$). We denote the resulting instance by $(G,u,v)$ again. As $G[T]$, and consequently, $G[T_1]$ is $(K_3+P_1)$-free due to Claim \[c-t1\], we find afterwards that the following holds for each branch. [[*\[c-t1b\]$G[T_1]$ is $K_3$-free.*]{}\ ]{}\ By Lemma \[l-constant\] we check in polynomial time if $(G,u,v)$ has an $7$-constant solution. If so, then we are done. From now on assume that $(G,u,v)$ has no $7$-constant solution. Note that $(G,u,v)$ has no double-sided solution either, as then the original instance has a double-sided solution, which we already ruled out (alternatively, apply Claim \[c-double\]). We will focus on the following task (recall that a solution $(S_u,S_v)$ for $(G,u,v)$ is independent if both $S_u$ and $S_v$ are independent sets). [**Phase 3b: Looking for independent solutions after branching**]{} We will now branch to $O(n^5)$ smaller instances for which the goal is to find an independent solution. As before, if one of the newly created instances has a solution, then $(G,u,v)$ has as solution. If none of these new instances has a solution, then $(G,u,v)$ may still have a solution $(S_u,S_v)$. However, in that case $S_u$ is not independent and $S_v$ must be an independent set. This will be verified when we do the $v$-feasibility check. We say that an instance $(G,u,v)$ satisfies the $(*)$-property if the following holds: $(*)$ If $(G,u,v)$ has a solution $(S_u,S_v)$ where $S_u$ is an independent set, then $(G,u,v)$ has an independent solution. Let $D_1,\ldots,D_q$ be the connected components of $G[T]$ for some $q\geq 1$. First suppose that every $D_i$ consists of a single vertex. Then $G[T]$ is an independent set. Hence, any solution for $(G,u,v)$ will be independent. We conclude that $(*)$ holds already. Now suppose that at least one of $D_1,\ldots,D_q$, say $D_1$, has more than one vertex. We first consider the case where another $D_i$, say $D_2$, also has more than one vertex. We claim that $(*)$ is again satisfied already. In order to see this, assume that $(G,u,v)$ has a solution $(S_u,S_v)$, where $S_u$ is an independent set, but $S_v$ contains two adjacent vertices $x_1$ and $x_2$. We assume without loss of generality that $x_1$ and $x_2$ belong to $D_2$. Hence, $V(D_1)$ is anticomplete to $\{x_1,x_2\}$. Suppose $D_1\cap S_v\neq \emptyset$. Let $t\in D_1\cap S_v$. Recall that $(S_u,S_v)$ is not a $7$-constant solution, as $(G,u,v)$ does not have such solutions. Then, by Claim \[c-either\], we find that $S_v$ has a set $S_v'$ that contains $x_1,x_2$ but not $t$, such that every vertex of $N(v)$ is adjacent to a vertex of $S_v'$ and $S_v'\cup N(v)$ is connected. Hence, we may put $t$ into $S_u$. Similarly, we may put every other vertex of $D_1\cap S_v$ into $S_u$. As $T_2$ is an independent set by Claim \[c-t2\], at least one vertex of $D_1$ belongs to $T_1$ and is thus adjacent to $w_u\in N(v)$ by definition. This means that $(S_u\cup (V(D_1)\cap S_v),S_v\setminus V(D_1))$ is another solution for $(G,u,v)$. However, this solution is double-sided, a contradiction. So, from now on, we assume that $D_1$ contains more than one vertex and that $D_2,\ldots,D_q$ each have exactly one vertex. Recall that $T_2$ is an independent set that is anticomplete to $N(v)$ due to Claim \[c-t2\]. Suppose $t\in T_2$ does not belong to $D_1$. Then $t$ is an isolated vertex of $G[T]$ that is not adjacent to any vertex of $N(v)$. As $G$ is connected, $t$ is adjacent to at least one vertex of $N(u)$. We apply the [Contraction Rule]{} on $N(u)\cup \{t\}$ to contract $t$ away. Afterwards, we find that every vertex of $T_2$ must belong to $D_1$. Let $B_1,\ldots,B_p$ be the connected components of $G[T_1\cap V(D_1)]$ for some $p\geq 1$. By Claim \[c-p4free\], $G[T]$, and thus $G[T_1\cap V(D_1)]$, is $P_4$-free (note that we only contracted edges during the branching and thus maintained $P_4$-freeness due to Lemma \[l-contract\]). As $G[T_1]$ is also $K_3$-free by Claim \[c-t1\], each $B_i$ is a complete bipartite graph on one or more vertices due to Lemma \[l-p4\]. First suppose that $p=1$. Recall that $T_2$ is an independent set by Claim \[c-t2\] that belongs to $D_1$. In this case we show how to branch into $O(n^2)$ new and smaller instances, such that $(G,u,v))$ has a solution $(S_u,S_v)$, in which $S_u$ is an independent set, if and only if one of these new instances has such a solution. Moreover, each new instance will have the property that either $(*)$ has been obtained or $p\geq 2$ holds. [**Branching IV**]{} ($O(n^2)$ branches)\ We consider each possibility of choosing one vertex $t \in B_1$ to be placed in $S_u$. This leads to $O(n)$ branches. In each branch we contract $t$ away by the [Contraction Rule]{} on $N(u)\cup \{t\}$ (note that $tw_u\in E(G)$, as $t\in T_1$). Since $S_u$ is an independent set, we must place all neighbours of $t$ in $S_v$. In order to contract these neighbours away using the [Contraction Rule]{} we may need to branch once more by considering every vertex that is in $B_1$ and that has at least one neighbour in $N(v)$. This leads to $O(n)$ additional branches. Hence, the total number of branches for this stage is $O(n^2)$. For each branch we observe that $T_1$ has become an independent set (as the vertices in the components $D_2,\ldots, D_q$ form an independent set as well). By applying the [Contraction Rule]{} we ensure that $T_2$ is an independent set that remains anticomplete to $N(v)$. First suppose that $T_1\cap V(D_1)$ consists of a single vertex $t^*$. If $T_2\neq \emptyset$, then we do as follows. Recall that we are looking for a solution $(S_u,S_v)$ with $T_2\subseteq S_v$. As $N(v)\cup S_v$ must be connected but $T_2$ is anticomplete to $N(v)$ by Claim \[c-t2\], vertex $t^*$ must be placed into $S_v$. Hence, if $t^*$ is not adjacent to a vertex in $N(v)$, we discard the branch. Otherwise we contract $T_2\cup \{t^*\}$ away by applying the [Contraction Rule]{} on $N(v)\cup T_2\cup \{t^*\}$. Hence, we obtained $T_2=\emptyset$. As $T_1$ is an independent set, this means that $(*)$ holds. Now suppose that $T_1\cap V(D_1)$ consists of more than one vertex. As $T_1$ is an independent set, this means that $G[T_1\cap V(D_1)]$ has $p\geq 2$ connected components. Hence, we have arrived in the case where $p\geq 2$. We denote the resulting instance by $(G,u,v,)$ again, and we let also $B_1,\ldots,B_p$ denote the connected components of $G[T_1\cap V(D_1)]$ again. From the above we are now in the situation where $(G,u,v)$ is an instance for which $p\geq 2$ holds. By Lemma \[l-p4\] and because $D_1$ is connected and $P_4$-free, $D_1$ has a spanning complete bipartite graph $B^*$. As $p\geq 2$, all vertices of $V(B_1)\cup \cdots \cup V(B_p)$ belong to the same partition class of $B^*$. By definition, these vertices are in $T_1$. Hence, as $T_2$ is an independent set in $D_1$, all vertices of $T_2$ form the other bipartition class of $B^*$. Consequently, $T_2$ is complete to $T_1\cap V(D_1)$. We will do some branching. [**Branching V**]{} ($O(n)$ branches)\ Every vertex of $T_2$ will belong to $S_v$ in any solution $(S_u,S_v)$ where $S_u$ is an independent set, but without having any neighbours in $N(v)$ due to Claim \[c-t2\]. This means that $S_v$ contains at least one vertex $t$ of $V(D_1)\cap T_1$. We branch by considering all possibilities of choosing this vertex $t$. Indeed, as $T_2$ is complete to $T_1$, it suffices to check single vertices $t\in T_1$ that have a neighbour in $N(v)$. This leads to $O(n)$ branches. For each branch we do as follows. We contract the vertices of $T_2\cup \{t\}$ away using the [Contraction Rule]{} on $N(v)\cup T_2\cup \{t\}$. We denote the resulting instance by $(G,u,v)$ and observe that $T_2=\emptyset$, so $T=T_1$. Note that $G[T]=G[T_1]$ now consists of connected components $B_1',\ldots,B_{p'}'$ for some $p'\geq 1$, where each $B_i'$ is a complete bipartite graph. If every $B_i'$ consists of a single vertex, then $G[T]$ is an independent set. Hence, any solution for $(G,u,v)$ will be independent. We conclude that $(*)$ holds. Now suppose that at least one of $B_1',\ldots,B_{p'}'$, say $B_1'$, has more than one vertex. If another $B_i'$, say $B_2'$, also has more than one vertex, then $(*)$ is also satisfied already. We can show this in the same way as before, namely when we proved this for the sets $D_1,\ldots,D_q$. From now on we may assume that $B_1'$ consists of more than one vertex and that $B_2',\ldots,B_{p'}'$ have only one vertex. So, in particular, $B_1'$ is a complete bipartite graph on at least two vertices. We will do some branching. [**Branching VI**]{} ($O(n^2)$ branches)\ We consider each possibility of choosing one vertex $t \in B_1'$ to be placed in $S_u$. This leads to $O(n)$ branches. In each branch we contract $t$ away by applying the [Contraction Rule]{} on $N(u)\cup \{t\}$ (note that $tw_u\in E(G)$, as $t\in T_1$, so we can indeed do this). Since $S_u$ is an independent set, we must place all neighbours of $t$ in $S_v$. In order to contract these neighbours away using the [Contraction Rule]{} we proceed as follows. All neighbours of $t$ that are adjacent to $N(v)$ we can contract away by applying the [Contraction Rule]{} on $N(v)\cup \{t\}$. If all neighbours of $t$ disappeared this way, this yields $T=T_1$, an independent set as required. Otherwise, we need to include another vertex of $B'_1$ into $S_v$. So we branch on the $O(n)$ vertices $t' \in B_1'$ that are adjacent to $v$. Contracting such a $t'$ away makes all other neighbours of $t$ adjacent to $N(v)$ and we can contract them away. In any case, eventually we will end up with $T=T_1$ being independent. Consequently, $S_v$ must be an independent set for any solution $(S_u,S_v)$ where $S_u$ is an independent set. This means that we achieved $(*)$. If we have not yet found a solution, then by achieving $(*)$, as shown above, we have reduced the problem to $O(n^5)$ instances, for which we search for an independent solution. We consider these new instances one by one. For simplicity, we denote the instance under consideration by $(G,u,v)$ again. [**Phase 3c: Searching for private solutions**]{} In this phase we introduce a new type of independent solution that we call private. In order to define this notion, we first describe our branching procedure which will get us to this new notion. [**Branching VII.**]{} ($O(n^4)$ branches)\ First we process $N(v)$ in the same way as we did for $N(u)$ in Branching II. That is, in polynomial time via $O(n^4)$ branches, we find a partition of $N(v)$ into a set $Q_v$ of private neighbours and a vertex $w_v$ that will be complete to $S_v$. To be more specific, if $(G,u,v)$ has a solution $(S_u,S_v)$ in which $S_u$ and $S_v$ are independent sets, then the following holds for such a solution $(S_u,S_v)$: - The independent set $S_u$ contains a subset $S_u^*$ of size at least 2 that covers $N(u)$, such that each $s\in S_u^*$ has a nonempty set $Q_u(s)$ of private neighbours with respect to $S_u^*$, and moreover, the set $N(u)\setminus Q_u$, where $Q_u=\bigcup Q_u(s)$, consists of a single vertex $w_u$ that is complete to $S_u^*$. - The independent set $S_v$ contains a subset $S_v^*$ of size at least 2 that covers $N(v)$, such that each $s\in S_v^*$ has a nonempty set $Q_v(s)$ of private neighbours with respect to $S_v^*$, and moreover, the set $N(v)\setminus Q_v$, where $Q_v=\bigcup Q_v(s)$, consists of a single vertex $w_v$ that is complete to $S_v^*$. We call an independent solution $(S_u,S_v)$ satisfying (P1) and (P2) a *private solution*. We emphasize that by now all branches are guaranteed to have private solutions or no solutions at all. Thus in what follows we will only search for private solutions. While doing this we may modify the instance $(G,u,v)$, but we will always ensure that private solutions are pertained. In particular, if we contract a vertex $t \in S_u^*$ to $w_u$ using the [Contraction Rule]{} on $N(u)\cup \{t\}$, then this leads to a private solution $(S_u,S_v)$ with $t \notin S_u^*$. Then all private neighbours of $t$ become adjacent to $w_u$ and, by the [Contraction Rule]{}, they get contracted to $w_u$ as well. However, if $t \notin S_u^*$, then contracting $t$ to $w_u$ will make the neighbours of $t$ in $N(u)$ adjacent to $w_u$ and the [Contraction Rule]{} contracts these to $w_u$. As a consequence, some vertices in $S_u^*$ may have no private neighbours in $N(u)$ and hence leave $S_u^*$. If this reduces $|S_u^*|$ to $1$, then we will notice this by checking for $1$-constant solutions, which takes polynomial time due to Lemma \[l-constant\]. If we find a $1$-constant solution, then we stop and conclude that our original instance is a yes-instance. Otherwise, we know that $|S_u^*| \ge 2$, and hence private solutions pertain (should such solutions exist at all). In the remainder, we will perform this test implicitly whenever we apply the [Contraction Rule]{}. We now prove the following two claims. [[*\[c-both\]Every vertex of $T$ is adjacent to both $w_u$ and $w_v$.*]{}\ ]{}\ *Proof of Claim \[c-both\].* Consider a vertex $t\in T$. First suppose that $t\in T$ is neither adjacent to $w_u$ nor to $w_v$. As $G$ is connected, $t$ will be adjacent to some other vertex in $S_u$ or $S_v$ in every solution $(S_u,S_v)$. Hence, $(G,u,v)$ has no independent solutions, and thus no private solutions, and we can discard the branch. From now on assume that every vertex in $T$ is adjacent to at least one of $w_u$, $w_v$. If $t\in T$ is adjacent to only $w_u$ and not to $w_v$, then by the same argument we must apply the [Contraction Rule]{} on $N(u)\cup \{t\}$. Similarly, if $t\in T$ is adjacent to only $w_v$ and not to $w_u$, then we must apply the [Contraction Rule]{} on $N(v)\cup \{t\}$. We discard a branch whenever two adjacent vertices in $T$ were involved in an edge contraction with some neighbour in $N(u)$, or with some neighbour in $N(v)$. [[*\[c-bipartite\] If $(G,u,v)$ has a private solution, then $G[T]$ must be the disjoint union of one or more complete bipartite graphs.*]{}\ ]{}\ [*Proof of Claim \[c-bipartite\].*]{} If $G[T]$ is not bipartite, then $(G,u,v)$ has no independent solution $(S_u,S_v)$, as $T=S_u\cup S_v$. Hence, $(G,u,v)$ has no private solution. Assume that $G[T]$ is bipartite. By Claim \[c-p4free\], $G[T]$ is $P_4$-free. Then the claim follows by Lemma \[l-p4\]. By Claim \[c-bipartite\] we may assume that $G[T]$ is the disjoint union of one or more complete bipartite graphs; otherwise we discard the branch. We now prove that $T$ can be changed into an independent set via some branching. Suppose $T$ is not an independent set yet. Let $B_1,\ldots,B_r$, for some $r\geq 1$, denote the connected components of $G[T]$ that have at least one edge (note that $G[T]$ may also contain some isolated vertices). By Claim \[c-bipartite\], every $B_i$ is complete bipartite. [[*\[c-4\] If $(G,u,v)$ has a private solution, then $r\leq 3$.*]{}\ ]{}\ *Proof of Claim \[c-4\].* Assume that $r\geq 4$. We will prove that $(G,u,v)$ has no private solution. Suppose that $T$ contains four connected components with edges, say $B_1,\ldots, B_4$, for which the following holds: $V(B_1)$ covers some subset $A_1^u\subseteq N(u)$ and $V(B_2)$ covers some subset $A_2^u\subseteq N(u)$, such that $A_1^u\setminus A_2^u \neq \emptyset$, whereas $V(B_3)$ covers some subset $A_3^v\subseteq N(v)$ and $V(B_4)$ covers some subset $A_4^v\subseteq N(v)$, such that $A_3^v\setminus A_4^v \neq \emptyset$. Let $w\in A_1^u\setminus A_2^u$, say $w$ is adjacent to vertex $s$ of $B_1$ (and not to any vertex of $B_2$). Let $x_1$ and $x_2$ be two adjacent vertices of $B_2$, which exist as $B_2$ contains an edge. Suppose $V(B_1)\cup V(B_2)$ does not cover $N(u)$. Then there exists a vertex $w'$ that has no neighbour in $V(B_1)\cup V(B_2)$. However, then $\{x_1,x_2\}\cup \{s,w,u,w'\}$ is an induced $P_2+P_4$ of $G$, a contradiction. Hence, $V(B_1) \cup V(B_2)$ covers $N(u)$. Similarly, we find that $V(B_3)\cup V(B_4)$ covers $N(v)$. Then, as each vertex of $T$ is adjacent to both $w_u$ and $w_v$, we find that $G[N(u)\cup V(B_1)\cup V(B_2)]$ and $G[N(u)\cup V(B_3)\cup V(B_4)]$ are connected. This is not possible, as then the original instance has a double-sided solution, which we already ruled out after Claim \[c-double\]. If two sets from $V(B_1),\ldots,V(B_r)$, say $V(B_1)$ and $V(B_2)$, cover the same subset of $N(u)$ *and* the same subset of $N(v)$, then we can apply the [Contraction Rule]{} on $N(u)\cup V(B_1)$ and on $N(v)\cup V(B_2)$ to find that the original instance has a double-sided solution if it has a solution. However, as we already ruled this out, this is not possible either. Now consider the sets $B_1$ and $B_2$. From the above, we deduce the following. We may assume without loss of generality that $V(B_1)$ and $V(B_2)$ cover different subsets of $N(u)$. This implies that $V(B_3),\ldots, V(B_r)$ all cover the same subset $A$ of $N(v)$. We can also apply the above on $B_1$ and $B_3$ to find that $B_1$ and $B_3$ must either cover different subsets of $N(u)$ or different subsets of $N(v)$. Suppose $B_1$ and $B_3$ cover different subsets of $N(v)$. Then, again from the above, $B_2, B_4,\ldots, B_r$ must cover the same subset of $N(u)$. As $B_1$ and $B_2$ cover different subsets of $N(u)$, this means that $B_1$ and $B_4$ cover different subsets of $N(u)$. This implies that $B_2$ must cover the same set $A$ as $B_4,\ldots,B_r$. As $B_3$ covers $A$ as well, this means that $B_2$ and $B_3$ cover the same subset of $N(v)$. Hence, they must cover different subsets of $N(u)$. However, the latter implies that $B_1$ and $B_4$ cover the same subset of $N(v)$. As $B_4$ covers $A$, just like $B_3$, we find that $B_1$ and $B_3$ cover the same subset $A$ of $N(v)$, a contradiction. Hence, $B_1$ and $B_3$ cover the same subset of $N(v)$, namely $A$, and by symmetry the same holds for $B_2$. As $V(B_1)$ covers $A_1^u$ and $V(B_2)$ covers $A_2^u$ such that $A_1^u\setminus A_2^u \neq \emptyset$, we can use the same arguments as before to deduce that $V(B_1)$ and $V(B_2)$ must cover $N(u)$. We put the vertices of $B_1$ and $B_2$ into $S_u$ and the vertices of $B_i$ for $i\geq 3$ plus all other (isolated) vertices of $T$ into $S_v$. If $(S_u,S_v)$ is a solution for $(G,u,v)$, then the original solution has a double-sided solution, which we already ruled out. Hence, there exists a vertex $z$ of $N(v)$ that is not adjacent to any vertex of $T\setminus (V(B_1)\cup V(B_2))$. However, as every $V(B_i)$ covers the same subset $A$ of $N(v)$, no vertex of $B_1$ and $B_2$ is adjacent to $z$ either. This implies that $(G,u,v)$ is a no-instance, meaning that we can discard this branch. By Claim \[c-4\] we may assume that $r\leq 3$, that is, $G[T]$ has at most three connected components $B_i$ with an edge; otherwise we discard the branch. As $r\leq 3$, we can now do some branching to obtain $O(1)$ smaller instances in which $T$ is an independent set, such that $(G,u,v)$ has a private solution if and only if at least one of these new instances has a private solution. [**Branching VIII**]{} ($O(1)$ branches)\ For $i=1,\ldots, r$, let $Y_i$ and $Z_i$ be the bipartition classes of $B_i$. Let $1\leq i\leq r$. As $S_u$ and $S_v$ must be independent sets and every $B_i$ is complete bipartite, either $Y_i$ belongs to $S_u$ and $Z_i$ belongs to $S_v$, or the other way around. We branch by considering both possibilities. We do this for each $i\in \{1,\ldots, r\}$. This leads to $2^r\leq 2^3$ branches, as $r\leq 3$ due to Claim \[c-4\]. In each branch we apply the [Contraction Rule]{} to contract $Y_i$ and $Z_i$ away (note that here our remark about pertaining private solutions applies). We consider every resulting instance separately. We denote such an instance again by $(G,u,v)$, for which we have proven the following claim. [[*\[c-indep\] $T$ is an independent set.*]{}\ ]{}\ We now continue as follows. As $T$ is an independent set by Claim \[c-indep\], the sets $S_u$ and $S_v$ of any solution $(S_u,S_v)$ will be independent (should $(G,u,v)$ have a solution). Recall also that $\{w_u,w_v\}$ is complete to $T$ by Claim \[c-both\]. We are looking for a private solution $(S_u,S_v)$, which we recall is an independent solution for which sets $S_u^*$ and $S_v^*$ exist so that (P1) and (P2) are satisfied. We make the following observation. Let $R=T\setminus (S_u^*\cup S_v^*)$ be the set of all other vertices of $T$. Consider a vertex $z\in R$. We note that if $z\in S_u$, then $(S_u\setminus \{z\},S_v\cup \{z\})$ is also a solution for $(G,u,v)$; this follows from (P1) and (P2) and the fact that $w_v$ is adjacent to $z\in T$. Similarly, if $z\in S_v$, then $(S_u\cup \{z\},S_v\setminus \{z\})$ is a solution as well. We prove the following four claims. [[*\[c-atleast\] Let $w\in N(u)\cup N(v)$. Then we may assume without loss of generality that $w$ is adjacent to at least two vertices of $T$.*]{}\ ]{}\ [*Proof of Claim \[c-atleast\].*]{} Suppose $N(u)$ or $N(v)$, say $N(u)$, contains a vertex $w$ that is adjacent to at most one vertex of $T$. If $w$ has no neighbours in $T$, then $(G,u,v)$ has no solution and we discard the branch. Suppose $w$ has exactly one neighbour $z\in T$. Then $z$ belongs to $S_u^*$ for every (private) solution $(S_u,S_v)$ of $(G,u,v)$ (assuming $(G,u,v)$ is a yes-instance). Hence, we may apply the [Contraction Rule]{} on $N(u)\cup \{z\}$. We apply this operation exhaustively, while pertaining private solutions as before. It may happen that in this process it turns out that two vertices $z,z'$ both belong to $S_u^*$ for every (private) solution $(S_u,S_v)$ of $(G,u,v)$, while they share a neighbour in $N(u)\setminus \{w_u\}$. This contradicts (P1). Hence, in this case we find that $(G,u,v)$ does not have a private solution and we may discard the branch. Otherwise, in the end, we have obtained in polynomial time an instance with the desired property. As we ensure that private solutions pertain, the size of $S_u^*$ remains at least $2$. [[*\[c-notall\] Let $z\in T$. Then we may assume without loss of generality that $z$ is non-adjacent to at least one vertex of $N(u)$ and to at least one vertex of $N(v)$.*]{}\ ]{}\ [*Proof of Claim \[c-notall\].*]{} Suppose $z\in T$ is adjacent to all vertices of $N(u)$ or to all vertices of $N(v)$, say to all vertices of $N(u)$. Then we can check in polynomial time if $T\setminus \{z\}$ covers $N(v)$. If so, then $\{z,T\setminus \{z\}$ is a ($1$-constant) solution of $(G,u,v)$ and we can stop. Otherwise $z$ must belong to $S_v$ for any solution $(S_u,S_v)$ of $(G,u,v)$. In that case we may apply the [Contraction Rule]{} on $N(v)\cup \{z\}$. We apply this operation exhaustively (we again recall that we ensure that private solutions pertain by checking for $1$-constant solutions). Moreover, it may happen that during this process two vertices $z,z'$ will end up in the same set $S_u$ or $S_v$ for any private solution $(S_u,S_v)$, while sharing a neighbour in $N(u)\setminus \{w_u\}$. As the sets $S_u$ and $S_v$ are independent in a private solution $(S_u,S_v)$, this means that $(G,u,v)$ does not have a private solution and we may discard the branch. Otherwise, in the end, we have obtained in polynomial time an instance with the desired property. [[*\[c-three\] Let $s$ and $t$ be any two distinct vertices of $T$. Then we may assume without loss of generality that either $N(u)\cap N(s)\cap N(t)=\{w_u\}$; or $N(u)\cap N(s)=N(u)\cap N(t)$; or $\{s,t\}$ covers $N(u)$. Similarly, we may assume without loss of generality that either $N(v)\cap N(s)\cap N(t)=\{w_v\}$; or $N(v)\cap N(s)=N(v)\cap N(t)$; $\{s,t\}$ covers $N(v)$.*]{}\ ]{}\ [*Proof of Claim \[c-three\].*]{} By symmetry it suffices to prove only the first statement. Assume $T$ contains two vertices $s$ and $t$, for which there exist distinct vertices $w\in (N(u)\setminus \{w_u\})\cap N(s)\cap N(t)$; $w'\in (N(u)\cap N(s))\setminus N(t)$ and $w''\in N(u)\setminus (N(s)\cup N(t))$. Note that $w_u\notin \{w,w',w''\}$. Recall that in this stage we are looking for private solutions for $(G,u,v)$. Consider an arbitrary private solution $(S_u,S_v)$ (if it exists). Then $w''\in Q_u(z)$ for some $z\in S_u^*$. Note that $z\notin \{s,t\}$, as neither $s$ nor $t$ is adjacent to $w''$. The above means that $z$ must be adjacent to at least one of $w,w'$, as otherwise the set $\{w'',z\}\cup \{w',s,w,t\}$ induces a $P_2+P_4$ in $G$, which is not possible. Hence, at least one of $w$ or $w'$ will be a private neighbour of $z$, that is, will belong to $Q_u(z)$. As $s$ is adjacent to both $w$ and $w'$ and $N(u)\setminus Q_u=\{w_u\}$ (see property (P1) of the definition of a private solution), this means that $s$ does not belong to $S_u^*$. We conclude that $s$ belongs to $R=T\setminus (S_u^*\cup S_v^*)$ or to $S_v^*$ for any private solution $(S_u,S_v)$ of $(G,u,v)$. As $s$ is adjacent to $w_v\in N(v)$, we may therefore apply the [Contraction Rule]{} on $N(v)\cup \{s\}$, ensuring persistence of private solutions (in case there are any) in the usual way. We do this exhaustively, and in the end we find that the claim holds. Note that we obtained this situation in polynomial time. [[*\[c-wu\] Let $s$ and $t$ be any two distinct vertices of $T$ that together cover $N(u)$. Then there exists a nonempty set $A(v)\subseteq N(v)$ that is complete to $\{s,t\}$ and anticomplete to $T\setminus \{s,t\}$, or $(G,u,v)$ has a $2$-constant solution. The same holds for $u$ and $v$ interchanged.*]{}\ ]{}\ [*Proof of Claim \[c-wu\].*]{} Assume without loss of generality that $\{s,t\}$ covers $N(u)$. Then we find that $(\{s,t\}, T\setminus \{s,t\})$ is a $2$-constant solution unless $N(v)$ contains a nonempty set $A(v)$ that is anticomplete to $T\setminus \{s,t\}$). By Claim \[c-atleast\] we find that $A(v)$ is complete to $\{s,t\}$. We will use Claims \[c-atleast\]–\[c-wu\] to prove the following claim. [[*\[c-notboth\] Let $s$ and $t$ be two distinct vertices in $T$ such that $\{s,t\}$ covers $N(u) \cup N(v)$. Then $(G,u,v)$ has a $2$-constant solution.*]{}\ ]{}\ [*Proof of Claim \[c-notboth\].*]{} Assume that $(G,u,v)$ has no $2$-constant solution. Then by Claim \[c-wu\], there is a nonempty set $A(u)\subseteq N(u)$ that is complete to $\{s,t\}$ and anticomplete to $T\setminus \{s,t\}$. Similarly, there exists a nonempty set $A(v)\subseteq N(v)$ that is complete to $\{s,t\}$ and anticomplete to $T\setminus \{s,t\}$. By Claim \[c-notall\] we find that $N(u)$ contains a vertex $w$ that is not adjacent to $s$. As $\{s,t\}$ covers $N(u)$, this means that $w$ is adjacent to $t$. By Claim \[c-atleast\] we find that $w$ is adjacent to some vertex $s'\in T\setminus \{s,t\}$. As $s'$ is anticomplete to $A(u)$, Claim \[c-three\] tells us that $\{s',t\}$ covers $N(u)$. By the same argument, there exists a vertex $t'$ such that $\{s,t'\}$ covers $N(v)$. Putting $s',t$ in $S_u$ and $s,t'$ in $S_v$ (together with all other vertices of $T$) yields a $2$-constant solution $(S_u,S_v)$ of $(G,u,v)$. This is a contradiction. We continue as follows. By Lemma \[l-constant\] we check in polynomial time if $(G,u,v)$ has a $2$-constant solution. If so, then we are done. Otherwise, we obtain the following claim, which immediately follows from Claim \[c-notboth\] and the fact that if one pair of vertices of $T$ covers $N(u)$ and another pair covers $N(v)$, then we obtained a $2$-constant solution. [[*\[c-onlyr\] We may assume without loss of generality that every pair of (distinct) vertices $\{s,t\}$ in $T$ does not cover $N(u)$; hence, $\{s,t\}$ may only cover $N(v)$.*]{}\ ]{}\ We call a pair of vertices $s,t$ of $T$ a [*$2$-pair*]{} if $\{s,t\}$ covers $N(v)$. Let $T_v$ be the set of vertices of $T$ involved in a 2-pair. We continue by proving the following claim. [[*\[c-2pair\] Every vertex of $T_v$ belongs to exactly one $2$-pair.*]{}\ ]{}\ [*Proof of Claim \[c-2pair\].*]{} Let $s\in T_v$. By definition, $s$ belongs to at least one $2$-pair. For contradiction, suppose that $s$ belongs to more than one $2$-pair. Then there exist vertices $t_1$, $t_2$ in $T_v$, such that $\{s,t_1\}$ and $\{s,t_2\}$ both cover $N(v)$. As $(G,u,v)$ has no $2$-constant solution, $N(u)$ contains a nonempty set $A_1(u)\subseteq N(u)$ that is complete to $\{s,t_1\}$ and anticomplete to $T\setminus \{s,t_1\}$ due to Claim \[c-wu\]. By the same claim, $N(u)$ contains a nonempty set $A_2(u)\subseteq N(u)$ that is complete to $\{s,t_2\}$ and anticomplete to $T\setminus \{s,t_2\}$. Let $w_1\in A_1(u)$ and $w_2\in A_2(u)$; note that $w_2\neq w_u$. Then $w_1$ is adjacent to $s$ but not to $t_2$, whereas $w_2\neq w_u$ is a common neighbour of $s$ and $t_2$. As $\{s,t_2\}$ does not cover $N(u)$ due to Claim \[c-onlyr\], this contradicts Claim \[c-three\]. We next prove that actually $T_v = \emptyset$. Suppose that $T_v\neq \emptyset$. Let $(s,t) \in T_v$. By Claim \[c-wu\], there exists a nonempty subset $A(u)$ of $N(u)$ that is complete to $\{s,t\}$ and anticomplete to $T\setminus \{s,t\}$. As $(G,u,v)$ has no $2$-constant solution, $s$ and $t$ do not cover all of $N(u)$. By Claim \[c-notall\], we find that $s$ is not adjacent to some vertex $w\in N(v)$. As $(s,t)$ is a 2-pair, $t$ is adjacent to $w$. By Claim \[c-atleast\], we find that $w$ is adjacent to a vertex $z\in T\setminus \{s,t\}$. From Claim \[c-2pair\] it follows that $(t,z)$ is not a 2-pair, so $t$ and $z$ do not cover all of $N(v)$. By Claim \[c-three\] and the fact that $t$ and $z$ have a common neighbour different from $w_v$, namely $w$, this means that $t$ and $z$ are adjacent to the same neighbours in $N(v)$. However, then $(s,z)$ is 2-pair, contradicting Claim \[c-2pair\]. This means that we have indeed proven the following claim. [[*\[c-tempty\] $T_v=\emptyset$.*]{}\ ]{}\ [**Phase 3d: Translating the problem into a matching problem**]{} We are now ready to translate the instance $(G,u,v)$ into an instance of a matching problem. Recall that $w_u$ and $w_v$ are the vertices in $N(u)$ and $N(v)$ that are complete to $T$. By Claims \[c-three\] and \[c-tempty\] we can partition $N(u)\setminus \{w_u\}$ into sets $N_1(u)\cup \dots \cup N_q(u)$ for some $q\geq 1$ such that two vertices of $N(u)$ have the same set of neighbours in $T$ if and only if they both belong to $N_i(u)$ for some $i\in \{1,\ldots,q\}$. Similarly, we can partition $N(v)\setminus \{w_v\}$ into sets $N_1(v)\cup \dots \cup N_r(v)$ for some $r\geq 1$ such that two vertices of $N(v)$ have the same set of neighbours in $T$ if and only if they both belong to $N_i(v)$ for some $i\in \{1,\ldots,r\}$. We may remove all but one vertex of each $N_h(u)$ and each $N_i(v)$ to obtain an equivalent instance, which we denote by $(G,u,v)$ again. Let $G'$ be the graph obtained from $G$ after removing the vertices $u,v,w_u,w_v$ and every edge between a vertex of $N(u)$ and a vertex of $N(v)$. Note that $G'$ is bipartite with partition classes $(N(u)\setminus \{w_u\})\cup (N(v)\setminus \{w_v\})$ and $T$. It remains to compute a maximum matching $M$ in $G'$. We can do this by using the Hopcroft-Karp algorithm, which runs in $O(m\sqrt{n})$-time on bipartite graphs with $n$ vertices and $m$ edges. If $|M|=|N(u)|+|N(v)|-2$, then each vertex in $(N(u)\setminus \{w_u\})\cup (N(v)\setminus \{w_v\})$ is incident to an edge of $M$, and hence, we found a (private) solution for $(G,u,v)$. If $|M|<|N(u)|+|N(v)|-2$, then $(G,u,v)$ has no (private) solution, and we discard the branch. The above concludes the description of the $u$-feasibility check. If we found a branch with a solution, then we translate it in polynomial time to a solution for the original instance. Otherwise we perform Phase 4. [**Phase 4: Doing a $\mathbf{v}$-feasibility check**]{} As mentioned, our algorithm now does a $v$-feasibility check, that is, it checks for the existence of a solution $(S_u,S_v)$, where $S_v$ is an independent set and $G[S_u]$ may contain edges. As we can repeat exactly the same steps as in Phase 3, this phase takes polynomial time as well. This concludes the description of our algorithm. The correctness of our algorithm follows from the above description. We now analyze its run-time. The branching is done in eight stages, namely Branching I-VIII and yields a total number of $O(n^{30})$ branches. As explained in each step above, processing each branch created in Branching I-VI until we start branching again takes polynomial time. Checking for $1$-constant solutions to ensure survival of private solutions takes constant time as well. Moreover, processing each of the branches created in Branch VII takes polynomial time as well. We conclude that the total running time of our algorithm is polynomial. Via Lemma \[l-reduce\] and a reduction to $P_4$-[Suitability]{} we obtain: \[l-top5\]$P_5$-[Suitability]{} can be solved in polynomial time for $(P_2+P_4)$-free graphs. Let $(G,u,v)$ be an instance of $P_5$-[Suitability]{}, where $G$ is a connected $(P_2+P_4)$-free graph. We may assume without loss of generality that $u$ and $v$ are of distance at least 4 from each other, as otherwise $(G,u,v)$ is a no-instance. By the [Contraction Rule]{} and Lemma \[l-contract\] we may also assume without loss of generality that $N(u)$ and $N(v)$ are both independent sets; otherwise if, say, $G[N(u)]$ contains an edge $e$, then we contract $e$ to obtain an equivalent but smaller instance $(G',u,v)$, where $G'$ is also $(P_2+P_4)$-free due to Lemma \[l-contract\]. First suppose $|N(u)|=1$, say $N(u)=\{u'\}$ for some $u'\in V(G)$. Then we solve $P_4$-[Suitability]{} on instance $(G-u,u',v)$. We can do this in polynomial time due to Lemma \[l-top4\]. Now suppose $|N(u)|\geq 2$. Note that $\operatorname{dist}(u,v) \leq 5$, as $G$ is $P_7$-free. By Lemma \[l-reduce\] we may assume that $\operatorname{dist}(u,v)=4$. We will explore the structure of the $P_5$-witness bags $W(p_2)$ and $W(p_3)$ should they exist. Let $Z$ be the set that consists of all vertices $z$ with $\operatorname{dist}(u,z)=\operatorname{dist}(z,v)=2$. Then $Z$ must be a subset of $W(p_3)$. As $N(u)$ is not connected, $W(p_2)$ must contain at least one other vertex $s$ adjacent to some vertex $t \in N(u)$. Suppose $s$ is non-adjacent to some other vertex $t'\in N(u)$. Let $w$ be a neighbour of $v$. As $s \in W(p_2)$ and $w \in W(p_4)$, we find that $s$ and $w$ are not adjacent. Then the set $\{v,w\}\cup \{s,t,u,t'\}$ induces a $P_2+P_4$ in $G$, a contradiction. Hence, $s$ is adjacent to every vertex of $N(u)$. We consider all possibilities of choosing vertex $s$ from the set $V(G)\setminus (N[u]\cup N[v]\cup Z)$. This leads to $O(n)$ branches. In each branch we contract the set $N(u)\cup \{s\}$ to a single vertex $u'$. Let $G'$ be the resulting graph. Then we solve $P_4$-[Suitability]{} on instance $(G',u',v)$. As $G'$ is $(P_2+P_4)$-free by Lemma \[l-contract\], we can do this in polynomial time due to Lemma \[l-top4\]. From the above we conclude that we can check in polynomial time if $(u,v)$ is a $P_5$-suitable pair of $G$. We use Lemma \[l-top5\] to prove Lemma \[l-top6\]. \[l-top6\]$P_6$-[Suitability]{} can be solved in polynomial time for $(P_2+P_4)$-free graphs. Let $(G,u,v)$ be an instance of $P_6$-[Suitability]{}, where $G$ is a connected $(P_2+P_4)$-free graph. We may assume without loss of generality that $u$ and $v$ are of distance at least 5 from each other, as otherwise $(G,u,v)$ is a no-instance. We may also assume without loss of generality that $N(u)$ and $N(v)$ are both independent sets; otherwise if, say, $G[N(u)]$ contains an edge $e$, then we contract $e$ to obtain an equivalent but smaller instance $(G',u,v)$, where $G'$ is also $(P_2+P_4)$-free due to Lemma \[l-contract\]. First suppose $|N(u)|=1$, say $N(u)=\{u'\}$ for some $u'\in V(G)$. Then we solve $P_5$-[Suitability]{} on instance $(G-u,u',v)$. We can do this in polynomial time due to Lemma \[l-top5\]. Now suppose $|N(u)|\geq 2$. We assume $W(p_1)=\{u\}$ and we will explore the structure of the $P_6$-witness bag $W(p_2)$ should it exist. As $N(u)$ is not connected, $W(p_2)$ must contain at least one other vertex $s$. Suppose that $s$ is adjacent to some vertex $t\in N(u)$ and non-adjacent to some other vertex $t'\in N(u)$. Let $w$ be a neighbour of $v$. Then the set $\{v,w\}\cup \{s,t,u,t'\}$ induces a $P_2+P_4$ in $G$, a contradiction. Hence, $s$ is adjacent to every vertex of $N(u)$. We consider all possibilities of choosing vertex $s$ from the set $V(G)\setminus (N[u]\cup N[v])$. This leads to $O(n)$ branches. In each branch we contract the set $N(u)\cup \{s\}$ to a single vertex $u'$. Let $G'$ be the resulting graph. Then we solve $P_5$-[Suitability]{} on instance $(G',u',v)$. As $G'$ is $(P_2+P_4)$-free by Lemma \[l-contract\], we can do this in polynomial time due to Lemma \[l-top5\]. From the above we conclude that we can check in polynomial time if $(u,v)$ is a $P_6$-suitable pair of $G$. We now combine Lemmas \[l-outer\] and \[l-trivial\] with Lemmas \[l-top4\]–\[l-top6\] to obtain the following theorem. \[t-p2p4\] The [Longest Path Contractibility]{} problem is polynomial-time solvable for $(P_2+P_4)$-free graphs. Let $G$ be a connected $(P_2+P_4)$-free graph. We may assume without loss of generality that $G$ has at least one edge. Note that $G$ is $P_7$-free. Hence, $G$ does not contain $P_7$ as a a contraction. By combining Lemmas \[l-top4\]–\[l-top6\] with Lemma \[l-outer\] we can check in polynomial time if $G$ contains $P_k$ as a contraction for $k=6,5,4$. If not, then we check if $G$ contains $P_3$ as a contraction by using Lemma \[l-trivial\] combined with Lemma \[l-outer\]. If not then, as $G$ has an edge, $P_2$ is the longest path to which $G$ can be contracted to. The Case $\mathbf{H=P_1+P_2+P_3}$ {#s-p1p2p3} --------------------------------- We will prove that [Longest Path Contractibility]{} is polynomial-time solvable for $(P_1+P_2+P_3)$-free graphs. We will start by showing that $P_4$-[Suitability]{} is polynomial-time solvable for $(P_1+P_2+P_3)$-free graphs. The proof of this result uses similar but more simple arguments than the proof of Lemma \[l-top4\]. \[l-top4c\] The $P_4$-[Suitability]{} problem can be solved in polynomial time for $(P_1+P_2+P_3)$-free graphs. Let $(G,u,v)$ be an instance of $P_4$-[Suitability]{}, where $G$ is a connected $(P_1+P_2+P_3)$-free graph. We may assume without loss of generality that $u$ and $v$ are of distance at least 3, that is, $u$ and $v$ are non-adjacent and $N(u)\cap N(v)=\emptyset$; otherwise $(G,u,v)$ is a no-instance. Recall that $T=V(G)\setminus (N[u]\cup N[v])$. Recall also that we are looking for a partition $(S_u,S_v)$ of $T$ that is a solution for $(G,u,v)$, that is, $N(u)\cup S_u$ and $N(v)\cup S_v$ must both be connected. In order to do so we will construct partial solutions $(S_u',S_v')$, which we try to extend to a solution $(S_u,S_v)$ for $(G,u,v)$. We use the [Contraction Rule]{} from Section \[s-pre\] on $S_u'$ and $S_v'$, so that these sets will become independent. By Lemma \[l-contract\], the resulting graph will always be $(P_1+P_2+P_3)$-free. For simplicity, we denote the resulting instance by $(G,u,v)$ again. After applying the [Contraction Rule]{} the size of the set $T$ may be reduced by at least one. As before, if $t\in T$ was involved in an edge contraction with a vertex from $N(u)$ or $N(v)$ when applying the rule, then we say that we contracted $t$ away. We start by applying the [Contraction Rule]{} on $N(u)$ and $N(v)$. This leads to the following claim. [[*\[c-ind2\] $N(u)$ and $N(v)$ are independent sets.*]{}\ ]{}\ We now check if $(G,u,v)$ has an $8$-constant solution, which we can check in polynomial time due to Lemma \[l-constant\]. If so, then $(G,u,v)$ is a yes-answer and we stop. From now on suppose that $(G,u,v)$ has no $8$-constant solution. Then we prove the following claim (recall that a solution $(S_u,S_v)$ is independent if $S_u$ and $S_v$ are independent sets). [[*\[c-ind3\] Every solution of $(G,u,v)$ is independent (if $(G,u,v)$ has solutions).*]{}\ ]{}\ [*Proof of Claim \[c-ind3\].*]{} Let $(S_u,S_v)$ be a solution for $(G,u,v)$ that is not independent, say $s,t$ belong to $S_u$ with $st\in E(G)$. If $\{s,t\}$ is anticomplete to a set of two neighbours $w,w'$ of $u$, then $\{v\}\cup \{s,t\}\cup \{w,u,w'\}$ is an induced $P_1+P_2+P_3$ of $G$, a contradiction. Hence, $\{s,t\}$ covers all but at most one vertex of $N(u)$. Suppose that $\{s,t\}$ covers $N(u)$, Then, as $s$ and $t$ are adjacent in $G$, we find that $(S_u,S_v)$ is a $2$-constant solution and thus a $8$-constant solution, which is not possible. Hence, $N(u)$ contains a unique vertex $w$ that is not adjacent to $s$ and $t$, but that is adjacent to some $z\in T\setminus \{s,t\}$. As $G$ is $(P_1+P_2+P_3)$-free, $G$ is $P_8$-free. Then $G[N(u)\cup S_u]$ contains a path $P$ on at most seven vertices from $s$ to $z$. The path $P$, together with vertex $t$ that may not be on $P$, shows that $(S_u,S_v)$ is a $8$-constant solution, a contradiction. We will now analyze the structure of an independent solution $(S_u,S_v)$. As $S_u$ and $N(u)$ are both independent sets, $G[N(u)\cup S_u]$ is a connected bipartite graph. Hence, $S_u$ contains a set $S_u^*$, such that $S_u^*$ covers $N(u)$. We assume that $S_u^*$ has minimum size. Then each $s\in S_u^*$ has a nonempty set $Q(s)$ of vertices in $N(u)$ that are not adjacent to any vertex in $S_u^*\setminus \{s\}$; otherwise we can remove $s$ from $S_u^*$, contradicting our assumption that $S_u^*$ has minimum size. We call the vertices of $Q(s)$ the [*private*]{} neighbours of $s\in S_u^*$ with respect to $S_u^*$. As $(G,u,v)$ has no $8$-constant solution, and thus no $1$-constant solution, we find that $S_u^*$ has size at least 2. Suppose $Q(s)$ contains at least two private neighbours $w_1,w_2$ of some vertex $s\in S_u^*$. As $|S_u^*|\geq 2$, there exists a vertex $s'\in S_u^*$ with $s'\neq s$. Let $w_3\in Q(s')$. Then $\{v\}\cup \{w_3,s'\}\cup \{w_1,s,w_2\}$ is an induced $P_1+P_2+P_3$ of $G$, a contradiction. Hence, each set $Q(s)$ has size 1. We denote the unique vertex of $Q(s)$ by $w_u^s$. So, $w_u^s$ is adjacent to $s$ but not to any other vertex from $S_u^*$. Let $Q_u$ be the set of all vertices $w_u^s$. Then $G[Q_u\cup S_u^*]$ is the disjoint union of $|S_u^*|$ edges. We claim that the set $N(u)\setminus Q_u$ is complete to $S_u^*$. In order to see this, let $w\in N(u)\setminus Q_u$. By definition, $w$ is adjacent to at least two vertices $s_1,s_2$ of $S_u^*$. For contradiction, assume that $w$ is not adjacent to some vertex $s_3\in S_u^*$. Then $\{v\} \cup \{s_3,w_u^{s_3}\}\cup \{s_1,w,s_2\}$ induces a $P_1+P_2+P_3$ in $G$, which is not possible. As $G[N(u)\cup S_u]$ is connected as well and $S_u$ is an independent set, every vertex $t\in S_u\setminus S_u^*$ must be adjacent to at least one vertex of $N(u)$. However, we claim that every vertex of $S_u\setminus S_u^*$ is adjacent to at most one vertex of $Q_u$. For contradiction, assume that $S_u\setminus S_u^*$ contains a vertex $t$ that is adjacent to two vertices of $Q_u$, say to $w_u^s$ and $w_u^{s'}$ for some $s,s'\in S_u^*$ with $s\neq s'$. Recall that $S_u$ is independent. Consequently, if $t$ is non-adjacent to $w_u^{s''}$ for some $s''\in S_u^*\setminus \{s,s'\}$, then $G$ contains an induced $P_1+P_2+P_3$ with vertex set $\{v\} \cup \{w_u^{s''},s''\} \cup \{w_u^s,t,w_u^{s'}\}$, a contradiction. Hence, $t$ is adjacent to every vertex of $Q_u$. If $t$ is adjacent to every vertex of $N(u)$, then $(G,u,v)$ has a $1$-constant solution, and this an $8$-constant solution, which we ruled out already. Hence, the set $N(u)\setminus N(t)$ is nonempty. As $t$ is adjacent to every vertex of $Q_u$, the set $N(u)\setminus N(t)$ is a subset of $N(u)\setminus Q_u$. Recall that $N(u)\setminus Q_u$ is complete to $S_u^*$. Hence, $N(u)\setminus N(t)$ is complete to $S_u^*$. Let $s\in S_u^*$. Then $\{s,t\}$ covers $N(u)$, and moreover $G[N(u)\cup \{s,t\}]$ is connected. This means that $(G,u,v)$ has a $2$-constant solution and thus an $8$-constant solution, which is not possible. We conclude that every vertex of $S_u\setminus S_u^*$ is adjacent to at most one vertex of $Q_u$. Finally, we prove that $N(u)\setminus Q_u$ is nonempty. For contradiction, assume that $N(u)\setminus Q_u$ is empty. Then $N(u)=Q_u$. As $G[Q_u\cup S_u^*]$ is the disjoint union of a number of edges, and $G[N(u)\cup S_u]$ is connected, there must exist a vertex $t\in S_u\setminus S_u^*$ that is adjacent to at least two vertices of $Q_u$. However, we proved above that this is not possible. We conclude that $N(u)\setminus Q_u$ is nonempty. We can deduce all the claims above with respect to $v$ as well. To summarize, any independent solution $(S_u,S_v)$ for $(G,u,v)$ satisfies the following two properties: - The independent set $S_u$ contains a subset $S_u^*$ of size at least 2 that covers $N(u)$, such that each vertex $s\in S_u^*$ has exactly one private neighbour $w_u^s$ in $N(u)$ with respect to $S_u^*$, and moreover, the set $N(u)\setminus Q_u$, where $Q_u=\{w_u^s\; |\; s\in S_u^*\}$, is nonempty and complete to $S_u^*$, and every vertex of $S_u\setminus S_u^*$ is adjacent to at most one vertex of $Q_u$ and to at least one vertex of $N(u)\setminus Q_u$. - The independent set $S_v$ contains a subset $S_v^*$ of size at least 2 that covers $N(v)$, such that each vertex in $s\in S_v^*$ has exactly one private neighbour $w_v^s$ in $N(v)$ with respect to $S_v^*$, and moreover, the set $N(v)\setminus Q_v$, where $Q_v=\{w_v^s\; |\; s\in S_v^*\}$, is nonempty and complete to $S_v^*$, and every vertex of $S_v\setminus S_v^*$ is adjacent to at most one vertex of $Q_v$ and to at least one vertex of $N(u)\setminus Q_v$. *Remark.* We emphasize that $S_u^*$ and $S_v^*$ are unknown to the algorithm, as we constructed it from the unknown sets $S_u$ and $S_v$, and consequently our algorithm does not know (yet) the sets $Q_u$ and $Q_v$. We will now branch into $O(n^8)$ smaller instances in which $N(u)\setminus Q_u$ and $N(u)\setminus Q_v$ consist of just one single vertex $w_u$ and $w_v$, respectively, such that $(G,u,v)$ has an independent solution if and only if at least one of the new instances has an independent solution. Moreover, we will be able to identify $w_u$ and $w_v$, and consequently, the sets $Q_u$ and $Q_v$, in polynomial time. [**Branching**]{} ($O(n^8)$ branches)\ We will determine exactly those vertices of $N(u)$ that belong to $Q_u$ via some branching, under the assumption that $(G,u,v)$ has an independent solution $(S_u,S_v)$ that satisfies (P1) and (P2). By (P1), $S_u^*$ consists of at least two (non-adjacent) vertices $s$ and $s'$. By (P2), $S_v^*$ consists of at least two (non-adjacent) vertices $t$ and $t'$. We branch by considering all possible choices of choosing these four vertices together with their private neighbours $w_u^s$, $w_u^{s'}$, $w_v^{t}$, $w_v^{t'}$ (which are unique by (P1) and (P2)). This leads to $O(n^8)$ branches. For each branch we do as follows. We discard the branch in which $G[\{s,s',w_u^s,w_u^{s'}\}]$ and $G[\{t,t',w_v^t,w_v^{t'}\}]$ are not both isomorphic to $2P_2$. We put a vertex $y\in N(u)$ in $N(u)\setminus Q_u$ if and only if $y$ is a common neighbour of $s$ and $s'$. This gives us the set $Q_u$. We obtain the set $Q_v$ in the same way. If there exists a vertex in $Q_u\setminus \{w_u^s,w_u^{s'}\}$ that is adjacent to one of $s,s'$, then we discard the branch. We also discard the branch if there exists a vertex in $Q_v\setminus \{w_v^t,w_v^{t'}\}$ that is adjacent to one of $t,t'$. Moreover, by applying the [Contraction Rule]{} on $N(u)\cup \{s,s'\}$ we can contract $s$ and $s'$ away. This contracts all vertices of $N(u)\setminus Q_u$ into a single vertex which we denote by $w_u$ due to (P1). Similarly, we branch $t$ and $t'$ away and this leads to the contraction of $N(v)\setminus Q_v$ into a single vertex $w_v$ due to (P2). Note that we have identified $w_u$ and $w_v$ in polynomial time. We denote the resulting instance by $(G,u,v)$ again. Consider a vertex $z\in T$. Firs suppose that $z$ is not adjacent to $w_u$. Then $z$ does not belong to $S_u$ in any independent solution $(S_u,S_v)$ for $(G,u,v)$ by (P1).Hence $z$ must belong to $S_v$ for any independent solution $(S_u,S_v)$ for $(G,u,v)$. However, (P2) tells us that If $z$ is not adjacent to $w_v$, then $z$ cannot belong to the set $S_v$ of any independent solution $(S_u,S_v)$ for $(G,u,v)$. Hence, in that case we must discard the branch. Otherwise, that is, if $z$ is adjacent to $w_v$, then we check the following. If $z$ has two neighbours in $N(v)\setminus \{w_v\}$, then $z$ does not belong to $S_v$ in any independent solution $(S_u,S_v)$ for $(G,u,v)$ due to (P2). Hence, we will discard the branch. If $z$ is adjacent to at most one vertex of $N_v\setminus \{w_v\}$, then we apply the [Contraction Rule]{} on $N(v)\cup \{z\}$ to contract $z$ away. As a side effect, the possible neighbour of $z$ in $N_v\setminus \{w_v\}$ will be contracted away as well. Now suppose that $z$ is not adjacent to $w_v$. Then we perform the same operation with respect to $u$. We apply this operation exhaustively on both $u$ and $v$. This takes polynomial time. In the end we either discarded the branch or have found a new instance, which we also denote by $(G,u,v)$ again, in which every vertex of $T$ is adjacent to $w_u$ and to $w_v$. Consider again a vertex $z\in T$. If $z$ is adjacent to only $w_u$ and $w_v$ and to at most one other vertex $w$ in $N(u)\cup N(v)$, then we apply the [Contraction Rule]{} on $G[N(u)\cup \{z\}]$ (if $w\in N(u)$) or $G[N(v)\cup \{z\}]$ (in the other two cases) in order to contract $z$ away. As a side effect, the possible other neighbour of $z$ in $(N(u)\cup N(v)) \setminus \{w_u,w_v\}$ will be contracted away as well. If $z$ is adjacent to more than one vertex of $N(u)\setminus \{w_u\}$, then $z$ does not belong to $S_u$ in any independent solution $(S_u,S_v)$ for $(G,u,v)$. We check if $z$ is adjacent to more than one vertex of $N(v)\setminus \{w_v\}$. If so, then $z$ does not belong to $S_v$ in any independent solution $(S_u,S_v)$ for $(G,u,v)$. In that case we will discard the branch. Otherwise we will apply the [Contraction Rule]{} on $N(v)\cup \{z\}$ to contract $z$ away. Again, as a side effect, the possible neighbour of $z$ in $N(v)\setminus \{w_v\}$ will be contracted away as well. If $z$ is adjacent to more than one vertex of $N(u)\setminus \{w_v\}$, we perform a similar operation with respect to $u$. We apply this rule exhaustively. This takes polynomial time. In the end we find that every vertex of $T$ is adjacent to $w_u$ and $w_v$ and to exactly one vertex of $Q_u$ and to exactly one vertex of $Q_v$. We now remove all edges of $G[T]$. We also remove $w_u$ and $w_v$ from the graph. This yields a bipartite graph $G'$ with partition classes $N(u)\cup N(v)\setminus \{w_u,w_v\}$ and $T$. It remains to compute a maximum matching $M$ in $G'$. We can do this by using the Hopcroft-Karp algorithm [@HK73], which runs in $O(m\sqrt{n})$-time on bipartite graphs with $n$ vertices and $m$ edges. If $|M|=|N(u)|+|N(v)|-2$ then we found a solution for $(G,u,v)$; otherwise we discard the branch. Note that we did not explicitly forbid that two adjacent vertices of $T$ ended up in $S_u$ or two adjacent vertices of $T$ ended up in $S_v$: we have ruled out the existence of such solutions already (but they would still be perfectly acceptable if they did exist). As mentioned, we translate a solution found for some branch into a solution for the original instance. We can do so in polynomial time. If we find no yes-answer for the instance of any branch, then we conclude that the original instance has no solution. The correctness of our algorithm follows from the above description. We now analyze its run-time. There is only one branching procedure, which yields a total number of $O(n^{8})$ branches. As explained above, processing each branch takes polynomial time. In particular, checking for $8$-constant solutions takes polynomial time due to Lemma \[l-constant\]. We conclude that the total running time of our algorithm is polynomial. We proceed in the same way as in the case where $H=P_2+P_4$. That is, we will use Lemma \[l-top4c\] to prove Lemma \[l-top5c\]. Then we use Lemma \[l-top5c\] to prove Lemma \[l-top6c\], and we use Lemma \[l-top6c\] to prove Lemma \[l-top7c\]. \[l-top5c\] The $P_5$-[Suitability]{} problem can be solved in polynomial time for $(P_1+P_2+P_3)$-free graphs. Let $(G,u,v)$ be an instance of $P_5$-[Suitability]{}, where $G$ is a connected $(P_1+P_2+P_3)$-free graph. We may assume without loss of generality that $u$ and $v$ are of distance at least 4 from each other, as otherwise $(G,u,v)$ is a no-instance. We may also assume without loss of generality that $N(u)$ and $N(v)$ are independent sets; otherwise, say $N(u)$ contains an edge, we apply the [Contraction Rule]{} on $N(u)$ to obtain an equivalent but smaller instance $(G',u,v)$, where $G'$ is also $(P_1+P_2+P_3)$-free due to Lemma \[l-contract\]. If $N(u)$ consists of exactly one vertex $u'$, then we can instead solve $P_5$-[Suitability]{} on instance $(G-u,u',v)$. By Lemma \[l-top5c\] this takes polynomial time. Hence, we may assume that $N(u)$, and for the same reason, $N(v)$ have size at least 2. By Lemma \[l-reduce\] we may assume that $\operatorname{dist}(u,v)=4$. Let $M$ consist of all vertices of $G$ that are of distance 2 from $u$ and of distance 2 from $v$. Note that $M\neq \emptyset$, as $\operatorname{dist}(u,v)=4$. Moreover, if $G$ has a $P_5$-witness structure ${\cal W}$ with $W(p_1)=\{u\}$ and $W(p_5)=\{v\}$, then $M\subseteq W(p_3)$ must hold. Let $z,z'$ be two vertices in $N(v)$. Suppose $x\notin M\cup \{u\}$ is adjacent to $w\in N(u)$ but not to $w'\in N(u)$. As $x$ is not in $M$ and adjacent to $w\in N(u)$, we find that $x$ is not adjacent to $z$ and $z'$. However, then $\{w'\}\cup \{w,x\}\cup \{z,v,z'\}$ induces a $P_1+P_2+P_3$ in $G$, a contradiction. Hence, every vertex not in $M\cup \{u\}$ is either complete to $N(u)$ or anticomplete to $N(u)$. This means that if $G$ has a $P_5$-witness structure ${\cal W}$ with $W(p_1)=\{u\}$ and $W(p_5)=\{v\}$, then the following holds: $W(p_2)\setminus N(u)$ contains a vertex $s$, such that $N(u)\cup \{s\}$ is connected. We now branch by considering all possibilities of choosing this vertex $s$; note that we only have to consider vertices of $G$ that are of distance 2 from $u$ and that are not in $M$. This leads to $O(n)$ branches. We consider each branch separately, as follows. First we contract all edges in $G[N(u)\cup \{s\}]$. If this does not yield a single vertex $u'$, then we discard the branch. Otherwise we let $G'$ be the resulting graph. The graph $G'-u'$ consists of at least two connected components, one of which consists of vertex $u$, and the other one contains $v$ and $N(v)$. We contract away the vertices of any other connected component $D$ of $G'-u'$ by applying the [Contraction Rule]{} on $\{u'\}\cup V(D)$. It remains to check if $(G'-u,u',v)$ is a yes-instance of $P_4$-[Suitability]{}. We can do this in polynomial time via Lemma \[l-top4c\]. As there are $O(n)$ branches, the total running time of our algorithm is polynomial. \[l-top6c\] The $P_6$-[Suitability]{} problem can be solved in polynomial time for $(P_1+P_2+P_3)$-free graphs. Let $(G,u,v)$ be an instance of $P_6$-[Suitability]{}, where $G$ is a connected $(P_1+P_2+P_3)$-free graph. We may assume without loss of generality that $u$ and $v$ are of distance at least 5 from each other, as otherwise $(G,u,v)$ is a no-instance. We may also assume without loss of generality that $N(u)$ and $N(v)$ are independent sets; otherwise, say $N(u)$ contains an edge, we apply the [Contraction Rule]{} on $N(u)$ to obtain an equivalent but smaller instance $(G',u,v)$, where $G'$ is also $(P_1+P_2+P_3)$-free due to Lemma \[l-contract\]. If $N(u)$ consist of exactly one vertex $u'$, then we can instead solve $P_5$-[Suitability]{} on instance $(G-u,u',v)$. By Lemma \[l-top5c\] this takes polynomial time. Hence, we may assume that $N(u)$, and for the same reason, $N(v)$ are independent sets of size at least 2. Let $z,z'$ be two vertices in $N(v)$. Suppose $x\notin N(u)\cup \{u\}$ is adjacent to $w\in N(u)$ but not to $w'\in N(u)$. Then $\{w'\}\cup \{w,x\}\cup \{z,v,z'\}$ induces a $P_1+P_2+P_3$ in $G$, a contradiction. Hence, every vertex not in $N(u)\cup \{u\}$ is either complete to $N(u)$ or anticomplete to $N(u)$. This means that if $G$ has a $P_5$-witness structure ${\cal W}$ with $W(p_1)=\{u\}$ and $W(p_5)=\{v\}$, then the following holds: $W(p_2)\setminus N(u)$ contains a vertex $s$, such that $N(u)\cup \{s\}$ is connected. We now branch by considering all possibilities of choosing this vertex $s$. This leads to $O(n)$ branches. We consider each branch separately, as follows. First we contract all edges in $G[N(u)\cup \{s\}]$. If this does not yield a single vertex $u'$, then we discard the branch. Otherwise we let $G'$ be the resulting graph. The graph $G'-u'$ consists of at least two connected components, one of which consists of vertex $u$, and the other one contains $v$ and $N(v)$. We contract away the vertices of any other connected component $D$ of $G'-u'$ by applying the [Contraction Rule]{} on $\{u'\}\cup V(D)$. It remains to check if $(G'-u,u',v)$ is a yes-instance of $P_4$-[Suitability]{}. We can do this in polynomial time via Lemma \[l-top5c\]. As there are $O(n)$ branches, the total running time of our algorithm is polynomial. \[l-top7c\] The $P_7$-[Suitability]{} problem can be solved in polynomial time for $(P_1+P_2+P_3)$-free graphs. Let $(G,u,v)$ be an instance of $P_7$-[Suitability]{}, where $G$ is a connected $(P_1+P_2+P_3)$-free graph. We may assume without loss of generality that $u$ and $v$ are of distance at least 6 from each other, as otherwise $(G,u,v)$ is a no-instance. Note that in fact $u$ and $v$ are of distance exactly 6 from each other, as otherwise $G$ contains an induced $P_1+P_2+P_3$. We may also assume without loss of generality that $N(u)$ is an independent set; otherwise we apply the [Contraction Rule]{} on $N(u)$ to obtain an equivalent but smaller instance $(G',u,v)$, where $G'$ is also $(P_1+P_2+P_3)$-free due to Lemma \[l-contract\]. Suppose $N(u)$ contains two vertices $w$ and $w'$. As $u$ and $v$ are of distance 6 from each other, there exists a vertex $y$ with $\operatorname{dist}(u,y)=\operatorname{dist}(v,y)=3$. Let $z\in N(v)$. Then the set $\{y\}\cup \{v,z\}\cup \{w,u,w'\}$ induces a $P_1+P_2+P_3$ in $G$, a contradiction. Hence, $N(u)$ consist of exactly one vertex $u'$. We can therefore solve $P_6$-[Suitability]{} on instance $(G-u,u',v)$. By Lemma \[l-top6c\] this takes polynomial time. We are now ready to prove the main result of Section \[s-p1p2p3\]. \[t-p1p2p3\] The [Longest Path Contractibility]{} problem is polynomial-time solvable for $(P_1+P_2+P_3)$-free graphs. Let $G$ be a connected $(P_1+P_2+P_3)$-free graph. We may assume without loss of generality that $G$ has at least one edge. Then $G$ is $P_8$-free. Hence, $G$ does not contain $P_8$ as a a contraction. By combining Lemmas \[l-top4c\]–\[l-top7c\] with Lemma \[l-outer\] we can check in polynomial time if $G$ contains $P_k$ as a contraction for $k=7,6,5,4$. If not, then we check if $G$ contains $P_3$ as a contraction by using Lemma \[l-trivial\] combined with Lemma \[l-outer\]. If not then, as $G$ has an edge, $P_2$ is the longest path to which $G$ can be contracted to. The Case $\mathbf{H=P_1+P_5}$ {#s-p1p5} ----------------------------- We will prove that [Longest Path Contractibility]{} is polynomial-time solvable for $(P_1+P_5)$-free graphs. This result extends a corresponding result of [@HPW09] for $P_5$-free graphs. Its proof is based on the same but slightly generalized arguments as the result for $P_5$-free graphs and comes down to the following lemma. \[l-all\] Let $k\geq 4$ and let $G$ be a $(P_1+P_5)$-free graph with a $P_k$-suitable pair $(u,v)$ such that $N(u)$ is an independent set. Then $G$ has a $P_k$-witness structure ${\cal W}$ with $W(p_1)=\{u\}$ and $W(p_k)=\{v\}$, for which the following holds: $W(p_2)\setminus N(u)$ contains a set $S$ of size at most $2$ such that $N(u)\cup S$ is connected. As $(u,v)$ is a $P_k$-suitable pair, $G$ has a $P_k$-witness structure ${\cal W}$ with $W(p_1)=\{u\}$ and $W(p_k)=\{v\}$. For contradiction, assume that $W(p_2)\setminus N(u)$ contains no set $S$ of size at most 2 such that $N(u)\cup S$ is connected. Then $W(p_2)\setminus N(u)$ contains at least three vertices $x_1$, $x_2$, $x_3$ such that one of the following holds: - for $i=1,2,3$, vertex $x_i$ is adjacent to some vertex $w_i\in N(u)$ with $w_i\notin N(x_h)\cup N(x_j)$, where $\{h,i,j\}=\{1,2,3\}$; or - $N(u)\subseteq N(x_1)\cup N(x_2)$, but $G[N(u)\cup \{x_1\}\cup \{x_2\}]$ is not connected. First assume that (i) holds. Recall that $N(u)$ is an independent set. Then $x_1x_2 \in E(G)$, as otherwise the set $\{v\}\cup \{x_1,w_1,u,w_2,x_2\}$ induces a $P_1+P_5$ in $G$, which is not possible. However, now the set $\{v\}\cup \{w_3,u,w_2,x_2,x_1\}$ induces a $P_1+P_5$ in $G$, a contradiction. Hence, (i) cannot hold. Now assume that (ii) holds. As $(G,u,v)$ has no $1$-constant solution, $x_1$ has a neighbour $w_1\in N(u)$ not adjacent to $x_2$ and $x_2$ has a neighbour $w_2\in N(u)$ not adjacent to $x_1$. As $G[N(u)\cup \{x_1\}\cup \{x_2\}]$ is not connected but $N(u)\subseteq N(x_1)\cup N(x_2)$, we have that $x_1x_2\notin E(G)$ However, then the set $\{v\}\cup \{x_1,w_1,u,w_2,x_2\}$ induces a $P_1+P_5$ in $G$, a contradiction. Hence (ii) does not hold either, a contradiction. As a consequence of Lemma \[l-all\], we get that $P_4$-[Suitability]{} is easy and that $P_k$-[Suitability]{} reduces to $P_4$-[Suitability]{}, as we will see. \[l-top4b\] The $P_4$-[Suitability]{} problem can be solved in polynomial time for $(P_1+P_5)$-free graphs. Let $(G,u,v)$ be an instance of $P_4$-[Suitability]{}, where $G$ is a connected $(P_1+P_5)$-free graph. We may assume without loss of generality that $u$ and $v$ are of distance at least 3 from each other, as otherwise $(G,u,v)$ is a no-instance. We may also assume without loss of generality that $N(u)$ is an independent set; otherwise we apply the [Contraction Rule]{} on $N(u)$ to obtain an equivalent but smaller instance $(G',u,v)$, where $G'$ is also $(P_1+P_5)$-free due to Lemma \[l-contract\]. By Lemma \[l-all\] we find that if $(G,u,v)$ has a solution, then $G$ has a $2$-constant solution. We can check the latter in $O(n^4)$ time by Lemma \[l-constant\]. \[l-top5b\] The $P_5$-[Suitability]{} problem can be solved in $O(n^6)$ time for $(P_1+P_5)$-free graphs. Let $(G,u,v)$ be an instance of $P_5$-[Suitability]{}, where $G$ is a connected $(P_1+P_5)$-free graph. We may assume without loss of generality that $u$ and $v$ are of distance at least 4 from each other, as otherwise $(G,u,v)$ is a no-instance. We may also assume without loss of generality that $N(u)$ is an independent set; otherwise we apply the [Contraction Rule]{} on $N(u)$ to obtain an equivalent but smaller instance $(G',u,v)$, where $G'$ is also $(P_1+P_5)$-free due to Lemma \[l-contract\]. If $(u,v)$ is a $P_5$-suitable pair, then by Lemma \[l-all\], $G$ has a $P_5$-witness structure ${\cal W}$ with $W(p_1)=\{u\}$ and $W(p_5)=\{v\}$, for which the following holds: $W(p_2)\setminus N(u)$ contains a set $S$ of size at most $2$, such that $N(u)\cup S$ is connected. We now branch by considering all possibilities of choosing this set $S$. This leads to $O(n^2)$ branches. We consider each branch separately, as follows. First we contract all edges in $G[N(u)\cup S]$. If this does not yield a single vertex $u'$, then we discard the branch. Otherwise we let $G'$ be the resulting graph. The graph $G'-u'$ consists of at least two connected components, one of which consists of vertex $u$, and the other one contains $v$ and $N(v)$. If there are more components in $G'-u'$ than these two, we contract each such component $D$ to $u'$ by applying the [Contraction Rule]{} on $\{u'\}\cup V(D)$. It remains to check if $(G'-u,u',v)$ is a yes-instance of $P_4$-[Suitability]{}. We can do this in $O(n^4)$ time via Lemma \[l-top4b\]. As there are $O(n^2)$ branches, the total running time of our algorithm is $O(n^6)$. \[l-top6b\] The $P_6$-[Suitability]{} problem can be solved in $O(n^8)$ time for $(P_1+P_5)$-free graphs. We reduce $P_6$-[Suitability]{} to $P_5$-[Suitability]{} in exactly the same way we reduced $P_5$-[Suitability]{} to $P_4$-[Suitability]{} in the proof of Lemma \[l-top5b\]. This leads to $O(n^2)$ branches. For each branch we apply Lemma \[l-top5b\], which takes $O(n^6)$ time. Hence the total running time of $O(n^8)$. We are now ready to prove the main result of Section \[s-p1p5\]. \[t-p1p5\] The [Longest Path Contractibility]{} problem is polynomial-time solvable for $(P_1+P_5)$-free graphs. Let $G$ be a connected $(P_1+P_5)$-free graph. We may assume without loss of generality that $G$ has at least one edge. Note that $G$ is $P_7$-free. Hence, $G$ does not contain $P_7$ as a a contraction. By combining Lemmas \[l-top4b\]–\[l-top6b\] with Lemma \[l-outer\] we can check in polynomial time if $G$ contains $P_k$ as a contraction for $k=6,5,4$. If not, then we check if $G$ contains $P_3$ as a contraction by using Lemma \[l-trivial\] combined with Lemma \[l-outer\]. If not then, as $G$ has an edge, $P_2$ is the longest path to which $G$ can be contracted to. The Case $\mathbf{H=sP_1+P_4}$ {#s-sp1p4} ------------------------------ We adopt/extend the notation from Section \[s-p4\]. Let $(G,u,v)$ be an instance of $P_k$-[Suitability]{} with $k \ge 4$. A *solution* is a witness structure $\mathcal{W}=\{W(p_1), \dots, W(p_k)\}$ with $W(p_1)=\{u\}, W(p_2)=N(u) \cup S_u , W(p_{k-1}) = N(v) \cup S_v$ and $W(p_k)=\{v\}$. We let $T:= V\setminus (N[u] \cup N[v])$. Thus $S_u$ and $S_v$ are disjoint subsets of $T$ such that $N(u) \cup S_u$ and $N(v) \cup S_v$ are connected. As in Section \[s-p4\], we call a solution $\alpha$-constant if there exists a subset $S_u'\subseteq S_u$ with $N(u) \cup S_u'$ connected and $|S_u'| \le \alpha$, or there exists $S_v'\subseteq S_v$ with $N(v)\cup S_v'$ connected and $|S_v' |\le \alpha$. Let $(\{u\}, N(u) \cup S_u, \dots)$ be a solution and $S'_u \subseteq S_u$ such that $N(u) \cup S'_u$ is connected. We define the *closure* $\overline{S'_u}$ of $S'_u$ as the set of all vertices in $S_u$ that are connected to $v$ in $G$ only via $N(u) \cup S'_u$. \[l-clos\] Let $(\{u\},N(u) \cup S_u, W(p_3), \dots, W(p_{k-1}),\{v\})$ be a solution for an instance $(G,u,v)$ of $P_k$-[Suitability]{} for some $k\geq 4$. If $S_u' \subseteq S_u$ such that $N(u) \cup S'_u$ is connected, then $(\{u\},N(u) \cup \overline{S'_u}, W(p_3)\cup S_u \setminus \overline{S'_u}, \dots,W(p_{k-1}),\{v\})$ is also a solution for $(G,u,v)$. We check the three properties for witness structures. All bags in the new partition are mutually disjoint. Connectedness of $W(p_3)\cup S_u \setminus \overline{S'_u}$: Any $s \in S_u\setminus\overline{S'_u}$ is joined to $v$ by a path $P$ that does not pass through $N(u) \cup S'_u$ (by definition). Moreover, $P$ does not hit $\overline{S'_u}$ since from there; by definition, we cannot reach $v$ without passing through $N(u) \cup S'_u$. Since vertices in $W(p_2)=N(u)\cup S_u$ are only adjacent to vertices in $W(p_1) \cup W(p_2) \cup W(p_3)$, we find that $P$ must be contained in $S_u \setminus \overline{S'_u}$ until it eventually reaches $W(p_3)$. Connectedness of $W(p_3) \cup S_u \setminus \overline{S'_u}$ follows. As it turns out, it suffices to search for $\alpha$-constant solutions: \[l-constps\] An instance $(G,u,v)$ of $P_k$-[Suitability]{}, where $G$ is $(sP_1+P_4)$-free, has a solution if and only if it has an $\alpha$-constant solution, where $\alpha=(s+2)(2s+4)$. We may, as usual, assume that $N(u)$ is independent (otherwise we apply the [Contraction Rule]{} to $N(u)$ without any effect on $S_u$ in solutions $(\{u\}, N(u) \cup S_u, \dots)$). First suppose that $(G,u,v)$ has an $\alpha$-constant solution. Then obviously $(G,u,v)$ has a solution. Now suppose that $(G,u,v)$ has a solution $(\{u\}, N(u) \cup S_u, \dots)$. Let $t \in S_u$ and let $S_u^* \subseteq S_u$ be a minimum size subset that covers (e.g., dominates) $N(u)$. Each $z \in S_u^*$ is connected to $t$ by some path $P_z \subseteq S_u$. Since $G$ is $(sP_1+P_4)$-free, $P_z$ has at most $2s+4$ vertices. Hence, $S_v':= \bigcup_{z \in S_u^*} P_z$ has size at most $|S_u^*|(2s+4)$ and is connected and covers $N(u)$ (as $S_u^*$ does). Then $(G,u,v)$ is an $\alpha$-constant solution, unless $|S_u'| > \alpha$. From now on suppose that $|S_u'| > \alpha$, so in particular $|S_u^*| > s+2$. We show that $S_u^*$ is independent. For contradiction, assume that $z,z'\in S_u^*$ are adjacent. Let $w,w'\in N(u)$ be private neighbours of $z,z'$, resp. Then $wzz'w'$ induces a $P_4$. Since $G$ is $(sP_1+P_4)$-free, $\{z,z'\}$ must cover almost all vertices in $N(u)$ (which may be assumed independent) except at most $s-1$ vertices, say, $w_1, \dots, w_{s-1}$. Thus a minimum size cover $S_u^*$ of $N(u)$ has at most $s+1$ vertices ($z,z'$ and at most $s-1$ others covering $w_1, \dots, w_{s-1}$), contradicting the fact that $|S_u^*| > s+2$. Next we prove that any two vertices $z,z'\in S_u^*$ cover disjoint sets in $N(u)$. For contradiction, assume that $z,z'\in S_u^*$ have a common neighbour $w \in N(u)$. Since $z\in S_u^*$ also has a private neighbour $w' \in N(u)$, we find an induced $P_4=w'zwz'$ and conclude that $\{z, z'\}$ must cover all but at most $s-1$ vertices in $N(u)$, a contradiction again. From the above we conclude that $S_u^* \cup N(u)$ is a disjoint union of stars. Recall that $|S_u^*| > s+2> 1$. Therefore, to be connected, $S_u$ must contain a vertex $t \in S_u \setminus S_u^*$ connecting two vertices $z,z'\in S_u^*$. Let again $w$ be a private neighbour of $z$ in $N(u)$. Then $wztz'$ is a $P_4$, implying that $\{t,z,z'\}$ must cover all but $s-1$ vertices in $N(u)$, leading to a contradiction as before. Summarizing, we have shown that $(S_u, \dots)$ is an $\alpha$-constant solution. Combining Lemmas \[l-clos\] and \[l-constps\] gives the desired result: \[t-sp1p4\] For every constant $s\geq 0$, the [Longest Path Contractibility]{} problem is polynomial-time solvable for $(sP_1+P_4)$-free graphs. By Lemma \[l-constps\] we may focus on $\alpha$-constant solutions, If $(G,u,v)$ has an $\alpha$-constant solution $(\{u\}, N(u) \cup S_u, \dots)$ with $S_u'\subseteq S_u$ of size at most $\alpha$, we may guess this set $S_u'$ and extend it to its closure $\overline{S'_u}$ (by adding all vertices that are connected to the rest of the graph only through $N(u)\cup S_u'$ using Lemma \[l-clos\]) in time $O(n^{\alpha+2})$. We may then contract $\{u\} \cup \overline{S'_u}$, thereby reducing $P_k$-[Suitability]{} to $P_{k-1}$-[Suitability]{}. Since $G$ is $(sP_1+P_4)$-free, we may assume that $k \le 2s+4$. (If $k\geq 2s+5$, then every instance $(G,u,v)$ where $G$ is $(sP_1+P_4)$-free is a no-instance of $P_k$-[Suitability]{}). Thus only $2s$ such reductions are required. The NP-Complete Cases of Theorem \[t-main\] {#s-hard} =========================================== In this section we prove the new [[NP]{}]{}-complete cases of Theorem \[t-main\]. A [*hypergraph*]{} ${\cal H}$ is a pair $(Q,{\mathcal S})$, where $Q=\{q_1,\ldots,q_m\}$ is a set of $m$ [*elements*]{} and ${\mathcal S}=\{S_1,\ldots,S_n\}$ is a set of $n$ [*hyperedges*]{}, which are subsets of $Q$. A [*$2$-colouring*]{} of ${\cal H}$ is a partition of $Q$ into two (nonempty) sets $Q_1$ and $Q_2$ with $Q_1\cap S_j \ne\emptyset$ and $Q_2 \cap S_j \ne\emptyset$ for each $S_j$. This leads to the following decision problem. [.99]{} <span style="font-variant:small-caps;">[Hypergraph 2-Colourability]{}</span>\ ----------------- ----------------------------------------- *    Instance:* [a hypergraph ${\cal H}$.]{} *Question:* [does ${\cal H}$ have a 2-colouring?]{} ----------------- ----------------------------------------- Note that [Hypergraph 2-Colourability]{} is [[NP]{}]{}-complete even for hypergraphs ${\cal H}$ with $S_i\ne \emptyset$ for $1\leq i\leq n$ and $S_n=Q$. By a reduction from [Hypergraph 2-Colourability]{}, Brouwer and Veldman [@BV87] proved that $P_4$-[Contractibility]{} is [[NP]{}]{}-complete. That is, from a hypergraph ${\cal H}$ they built a graph $G_{\cal H}$, such that ${\cal H}$ has a $2$-colouring if and only if $G_{\cal H}$ has $P_4$ as a contraction. We first recall the graph $G_{\cal H}$ from [@BV87], which was obtained from a hypergraph ${\cal H}$ with $S_i\ne \emptyset$ for $1\leq i\leq n$ and $S_n=Q$ (see Figure \[f-hypergraph\] for an example). - Construct the [*incidence graph*]{} of $(Q,{\mathcal S})$, which is the bipartite graph with partition classes $Q$ and ${\cal S}$ and an edge between two vertices $q_i$ and $S_j$ if and only if $q_i\in S_j$. - Add a set ${\mathcal S}'=\{S_1',\ldots,S_n'\}$ of $n$ new vertices, where we call $S_j'$ the [*copy*]{} of $S_j$. - For $i=1,\ldots m$ and $j=1,\ldots, n$, add an edge between $q_i$ and $S_j'$ if and only if $q_i\in S_j$. - For $j=1,\ldots, n$ and $\ell=1,\ldots,n$, add an edge between $S_j$ and $S_\ell'$, so the subgraph induced by ${\cal S}\cup {\cal S}'$ will be complete bipartite. - For $h=1,\ldots, m$ and $i=1,\ldots,m$, add an edge between $q_h$ and $q_i$, so $Q$ will be a clique. - Add two new vertices $t_1$ and $t_2$. - For $j=1,\ldots,n$, add an edge between $t_1$ and $S_j$, and between $t_2$ and $S_j'$. ![An example of a graph $G_{\cal H}$ for some hypergraph ${\cal H}$ [@LPW08].[]{data-label="f-hypergraph"}](hypergraph.pdf) As mentioned, Brouwer and Veldman [@BV87] proved the following. \[l-bv87\] A hypergraph ${\cal H}$ has a $2$-colouring if and only if $G_{\cal H}$ has $P_4$ as a contraction. A [*split graph*]{} is a graph whose vertex set can be partitioned into two (possibly empty) sets $K$ and $I$, where $K$ is a clique and $I$ is an independent set. It is well known that a graph is split if and only if it is $(2P_2,C_4,C_5)$-free [@FH77]. Let ${\cal H}$ be a hypergraph. Observe that the subgraphs of $G_{\cal H}$ induced by $Q\cup {\cal S}$ and $Q\cup {\cal S}'$, respectively, are split graphs. Hence, we make the following observation. \[l-split\] Let ${\cal H}$ be a hypergraph. Then the subgraphs of $G_{\cal H}$ induced by $Q\cup {\cal S}$ and $Q\cup {\cal S}'$, respectively, are $(2P_2,C_4,C_5)$-free. We will need the following known lemma from [@HPW09]. \[l-p6\] Let ${\cal H}$ be a hypergraph. Then the graph $G_{\cal H}$ is $P_6$-free. We complement Lemma \[l-p6\] with the following lemma. \[l-freefree\] Let ${\cal H}$ be a hypergraph. Then the graph $G_{\cal H}$ is $(2P_1+2P_2,3P_2, 2P_3)$-free. We will prove that $G=G_{\cal H}$ is $(2P_1+2P_2,3P_2, 2P_3)$-free by considering each graph in $\{2P_1+2P_2,3P_2, 2P_3\}$ separately. [**$\mathbf{(2P_1+2P_2)}$-freeness.**]{} For contradiction, assume that $G$ contains a subgraph $H$ isomorphic to $2P_1+2P_2$. Let $D_1$ and $D_2$ be the two connected components of $H$ that contain an edge. Let $x$ and $y$ denote the two isolated vertices of $H$. First suppose that one of $x,y$, say $x$, belongs to ${\cal S}\cup {\cal S}'$, say to ${\cal S}$. Then $D_1$ and $D_2$ do not contain $t_1$ and also do not contain any vertex from ${\cal S}'$. The latter implies that $D_1$ and $D_2$ cannot contain vertex $t_2$ either. Hence, $D_1$ and $D_2$ only contain vertices from ${\cal S}\cup Q$, contradicting Lemma \[l-split\]. Hence, $x$ and $y$ must both belong to $Q\cup \{t_1,t_2\}$. Suppose one of them, say $x$, belongs to $Q$. Then $D_1$ and $D_2$ do not contain any vertices from $Q$ and thus only contain vertices from ${\cal S}\cup {\cal S}'\cup \{t_1,t_2\}$. However, $G[{\cal S}\cup {\cal S}'\cup \{t_1,t_2\}]$ is complete bipartite, and thus $2P_2$-free, a contradiction. We thus found that $\{x,y\}=\{t_1,t_2\}$. Then $D_1$ and $D_2$ may not contain any vertices from ${\cal S}\cup {\cal S}'$. Consequently, $D_1$ and $D_2$ only contain vertices from $Q$. This is not possible, as $Q$ is a clique. We conclude that $G$ is $(2P_1+2P_2)$-free. [**$\mathbf{3P_2}$-freeness.**]{} For contradiction, assume that $G$ contains a subgraph $H$ isomorphic to $3P_2$. Let $D_1,D_2,D_3$ be the three connected components of $H$. Suppose one of $D_1,D_2,D_3$, say $D_1$, contains a vertex from ${\cal S}\cup {\cal S}'$, say $D_1$ contains a vertex from ${\cal S}$. Then $D_2$ and $D_3$ do not contain $t_1$ and also do not contain any vertex from ${\cal S}'$. The latter implies that $D_2$ and $D_3$ cannot contain vertex $t_2$ either. Hence, $D_2$ and $D_3$ only contain vertices from ${\cal S}\cup Q$, contradicting Lemma \[l-split\]. This means that $H$ contains no vertex from ${\cal S}\cup {\cal S}'$. Consequently, $H$ does not contain $t_1$ and $t_2$ either. However, then $H$ consists of vertices from $Q$ only. This is not possible, as $Q$ is a clique. We conclude that $G$ is $3P_2$-free. [**$\mathbf{2P_3}$-freeness.**]{} For contradiction, assume that $G$ contains a subgraph $H$ isomorphic to $2P_3$. Let $D_1$ and $D_2$ be the two connected components of $H$. Suppose one of $D_1,D_2$, say $D_1$, contains a vertex from $Q$. As $Q$ is a clique, this means that $D_1$ must contain at least one vertex of ${\cal S}\cup {\cal S}'$, say $D_1$ contains a vertex of ${\cal S}$. Then $D_2$ cannot contain any vertex from $Q\cup \{t_1\}$ or from ${\cal S}'$. The latter implies that $D_2$ does not contain $t_2$ either. Hence, $D_2$ only contains vertices from ${\cal S}$. This is not possible, as ${\cal S}$ is an independent set. We conclude that neither $D_1$ nor $D_2$ contains a vertex from $Q$. Hence, $H$ only contains vertices from ${\cal S}\cup {\cal S}'\cup \{t_1,t_2\}$. However, $G[{\cal S}\cup {\cal S}'\cup \{t_1, t_2\}]$ is complete bipartite, and thus $2P_3$-free, a contradiction. We conclude that $G$ is $2P_3$-free. It is readily seen that $P_4$-[Contractibility]{} belongs to [[NP]{}]{}. Hence, we obtain the following result from Lemmas \[l-bv87\], \[l-p6\], and \[l-freefree\]. \[t-hard\] [$P_4$-Contractibility]{} is [[NP]{}]{}-complete for $(2P_1+2P_2,3P_2, 2P_3,P_6)$-free graphs. By modifying the graph $G_{\cal H}$ we prove the next theorem. \[t-girth\] Let $p\geq 4$ be some constant. Then [$P_{2p}$-Contractibility]{} is [[NP]{}]{}-complete for bipartite graphs of girth at least $p$. We assume without loss of generality that $p$ is even. We reduce again from [Hypergraph $2$-Colouring]{}, using a suitable subdivision of the graph $G_{\mathcal{H}}$ in order to satisfy the bipartiteness and girth constraints. Let $\mathcal{H}$ be a hypergraph. We first construct $G_{\mathcal{H}}$. We then subdivide edges in $G_{\mathcal{H}}$ as follows. The edge $S_nS'_n$ is *not* subdivided. All other edges $S_iS'_j$ are subdivided by an *even* number of vertices, namely by $p-2$, each. All edges joining $Q$ to $\mathcal{S'}$ are also subdivided $p-2$ times. So each of these edges becomes a path of odd length ($p-1$). Edges joining $Q$ to itself or to $\mathcal{S}$ are subdivided by an *odd* number of vertices, namely $p-1$, each. So each of these edges becomes a path of even length ($p$). In addition, we attach paths of length $p-2$, one to each of $t_1$ and $t_2$. Denote these paths by $P_i$ with end-vertices $t_i$ and, say, $\bar{t}_i, i=1,2$. Call the resulting graph $\bar{G}_{\mathcal{H}}$. In what follows we will denote the paths of length $p-1$ or $p$ obtained by subdividing an edge $xy$ in $G_{\mathcal{H}}$ by $\overline{xy}$. The distance between $\bar{t}_1$ and $\bar{t}_2$ in $\bar{G}_{\mathcal{H}}$ equals $2(p-2)+3=2p-1$. The unique shortest path is given by $P=(P_1,S_n,S'_n,P_2)$. No other pair of vertices in $\bar{G}_{\mathcal{H}}$ is this far apart. (For example, the distance between $S_i \in \mathcal{S}$ and $q \in Q\backslash S_i$ equals $p$ or $2+p$, the length of the path via $t_1$ and $S_n$.) It is straightforward to check that $\bar{G}_{\mathcal{H}}$ is bipartite: Any path joining $\mathcal{S}$ to $\mathcal{S'}$ has odd length. Hence, there are no odd cycles that hit both $\mathcal{S}$ and $\mathcal{S'}$. Similarly, all paths joining $Q$ and $\mathcal{S}'$ have odd length. So there cannot be any odd cycle in the subgraph induced by $Q\cup \mathcal{S}'$. The same argument applies to cycles in the subgraph induced by $Q \cup \mathcal{S}$. Here, again, all paths between $Q$ and $\mathcal{S}$ have the same parity (this time even). This shows that $\bar{G}_{\mathcal{H}}$ is indeed bipartite. The graph $\bar{G}_{\mathcal{H}}$ also has girth at least $p$. (Recall that any subdivided edge became a path of length at least $p-1$.) It remains to prove that ${\cal H}$ has a 2-colouring if and only if $\bar{G}_{\mathcal{H}}$ contains $P_{2p}$ as a contraction. First suppose that ${\cal H}$ has a 2-colouring $(Q_1,Q_2)$. Then define a contraction of $\bar{G}_{\mathcal{H}}$ to $P$ with corresponding witness structure $\{W(x), x \in P\}$ as follows. For each $(i,j) \neq (n,n)$ pick any subdivision vertex $v_{ij} \in \overline{S_iS'_j}$ and let $P_{ij}$ denote the (vertices of the) subpath of $\overline{S_iS_j'}$ from $S_i$ to $v_{ij}$. Similarly, let $P'_{ij}$ denote the (vertices of) $\overline{S_iS_j'}\backslash P_{ij}$, the “other half” of the path from $S_i$ to $S_j'$. Now the witnesses can be defined as follows: $$\begin{aligned} \nonumber W(t)&= \{t\} ~\text{for} ~t \in P_1 \\ \nonumber W(S_n)& = \bigcup_i \{\overline{S_iq}~|~q \in S_i\cap Q_1\} \cup \bigcup_{q,q'\in Q_1}\overline{qq'} \cup \bigcup_{(i,j)\neq(n,n)} P_{ij}\\ \nonumber W(S'_n)& = \bigcup_i \{\overline{S'_iq}~|~q \in S'_i\cap Q_2\} \cup \bigcup_{q,q'\in Q_2}\overline{qq'} \cup \bigcup_{(i,j)\neq(n,n)} P'_{ij}\\ \nonumber W(t)&= \{t\} ~\text{for} ~t \in P_2 \\ \nonumber\end{aligned}$$ To check correctness, we verify the three conditions for witnesses (observing that disjointness of the bags $W(x)$ is obvious). - $W(S_n)$ is connected: Indeed, all of $P_{ij}$ is connected to $S_i$ and this (as we assume $Q=Q_1\cup Q_2$ is a $2$-colouring of $\mathcal{H}$) contains some $q \in Q_1$, so $\overline{S_iq}$ joins $S_i$ to $q$. The latter, in turn, is joined to $S_n$. The same arguments apply to $W(S'_n)$. - Any two consecutive bags $W(x)$ and $W(y)$ (that is, when $x$ and $y$ are neighbours in $P$) are adjacent: Indeed, $t_1$ is adjacent to $S_n$, $S_n$ is adjacent to $S'_n$, and $S'_n$ is adjacent to $t_2$. - If $x$ and $y$ are non-adjacent in $P$, then $W(x)$ and $W(y)$ are non-adjacent in $\bar{G}_{\mathcal{H}}$: Indeed, $t_1$ is only adjacent to $W(S_n)$ and this in turn is only adjacent to $W(S'_n)$ and $t_1$. Thus, indeed, $\bar{G}_{\mathcal{H}}$ contains $P_{2p}$ as a contraction. Now suppose that $\bar{G}_{\mathcal{H}}$ contains $P_{2p}$ as a contraction. Since $\bar{t}_1$ and $\bar{t}_2$ are the only vertices at distance $2p-1$ in $\bar{G}_{\mathcal{H}}$, the only possibility is that $\bar{G}_{\mathcal{H}}$ contracts to $P=(P_1,S_n, S'_n, P_2)$. Let $\{W(x), x\in P\}$ be a corresponding witness structure. *Claim 1:*\ (i) $S_i \in W(t_1)\cup W(S_n)$ and $S'_i \in W(t_2)\cup W(S'_n)$ for $i=1, \dots, n$.\ (ii) $q \in W(t_1) \cup W(S_n)\cup W(S'_n) \cup W(t_2)$ for all $q \in Q$.\ *Proof of Claim 1.* In order to have all $W(x), x \in P$ connected, the subdivision vertices on $\overline{S_iS'_j}$ must belong to the same bags $W(x)$ as either $S_i$ or $S'_j$. The vertices $S_i$ and $S'_j$, however, must be in different (adjacent) bags: Indeed, $S_i, S'_j\in W(x)$ would imply that both $t_1$ and $t_2$ were adjacent to (or contained in) $W(x)$, contradicting the third condition for witness structures. The same argument shows that $S_i$ must either be in $W(t_1)$ or an adjacent bag, that is, in $W(S_n)$ or in $W(t)$, where $t$ is the unique neighbour of $t_1$ in $P_1$. The latter, however, is impossible: If $S_i \in W(t)$, then $S_i$ must be connected to $t$ within $W(t)$. But the only path joining $S_i$ to $t$ in $\bar{G}_{\mathcal{H}}$ runs through $t_1$, which does not belong to $W(t)$. Thus, indeed, (i) follows. Part (ii) can be proved in the same way: If $q\in W(t)$ with $t \in P_1\backslash \{t_1\}$, then $q$ should be connected to $t$ within $W(t)$. But, again, the only path connecting $q$ and $t$ runs through $t_1$, a contradiction. We claim that the partition $Q=Q_1 \cup Q_2$ given by $Q_1= Q \cap (W(t_1)\cup W(S_n))$ and $Q_2:= Q \cap (W(t_2) \cup W(S'_n))$ is a $2$-colouring of ${\cal H}$. That is, we will show that each $S_i \in \mathcal{S}$ contains some $q \in Q_1$ and, similarly, each $S'_i \in \mathcal{S'}$ contains some $q \in Q_2$. Let $S_i \in \mathcal{S}$. From Claim 1 it follows that $S_i \in W(t_1)\cup W(S_n)$. For each $q \in S_i$ we follow the path $\overline{S_iq}$ from $S_i$ to $q$ in $\bar{G}_{\mathcal{H}}$. Let $v$ be the last vertex on this path that belongs to $W(t_1)\cup W(S_n)$. If $v=q$, then $q \in Q \cap (W(t_1)\cup W(S_n)) =Q_1$ and we are done. Hence, assume $v\neq q$. Then, in particular, $q \notin W(t_1)\cup W(S_n)$. From Claim 1 we know that $q \in W(t_2) \cup W(S'_n)$. The path $\overline{S_iq}$ starts in $S_i \in W(t_1) \cup W(S_n)$ and ends in $q \in W(t_2)\cup W(S'_n)$. Since only $W(S_n)$ and $W(S'_n)$ are adjacent, this path must eventually pass from $W(S_n)$ to $W(S'_n)$ for the last time. Hence, $v\in W(S_n)$ and, therefore, must be connected to $S_n$ within $W(S_n)$. As $v$ is a subdivision vertex on $\overline{S_iq}$, this connection can only be via $S_i$ or $q$. But $q$ is not in $W(S_n)$, so the connection must be via $S_i$ and we conclude that $S_i \in W(S_n)$. Hence, $S_i$ must be connected to $S_n$ within $W(S_n)$. The only paths in $\bar{G}_{\mathcal{H}}$ connecting $S_i$ to $S_n$ run through either $t_1$ (which does *not* belong to $W(S_n)$) or some $S'_j$ (which also does not belong to $W(S_n)$) or - the last possibility - some $\tilde{q} \in S_i$. Hence, indeed, at least one such $\tilde{q} \in S_i$ must belong to $W(S_n)$. But then $\tilde{q} \in Q_1$ (by definition of $Q_1$), as required. As a consequence of Theorem \[t-girth\], [Longest Path Contractibility]{} is [[NP]{}]{}-complete for bipartite graphs of arbitrarily large girth. This strengthens the corresponding result for bipartite graphs, which following from a result of [@HHLP14]. For our dichotomy result we need the following consequence of Theorem \[t-girth\]. \[c-girth\] Let $H$ be a graph that has a cycle. Then [Longest Path Contractibility]{} is [[NP]{}]{}-complete for $H$-free graphs. Let $g$ be the girth of $H$. We set $p=g+1$ and note that the class of $H$-free graphs contains the class of graphs of girth at least $p$. Hence, we can apply Theorem \[t-girth\]. The Proof of Theorem \[t-main\] {#s-classification} =============================== We will use the following result from [@FKP13] as a lemma (in fact this result holds even for line graphs which form a subclass of the class of $K_{1,3}$-free graphs). \[l-claw\] The $P_7$-[Contractibility]{} problem is [[NP]{}]{}-complete for $K_{1,3}$-free graphs. By using the results from the previous sections and the above result we can now prove our classification theorem. [**Theorem \[t-main\]. (restated)**]{} [*Let $H$ be a graph. If $H$ is an induced subgraph of $P_1+P_5$, $P_1+P_2+P_3$, $P_2+P_4$ or $sP_1+P_4$ for some $s\geq 0$, then [Longest Path Contractibility]{} restricted to $H$-free graphs is polynomial-time solvable; otherwise it is [[NP]{}]{}-complete.*]{} If $H$ is an induced subgraph of $P_1+P_5$, $P_1+P_2+P_3$, $P_2+P_4$ or $sP_1+P_4$ for some $s\geq 0$, then we use Theorems \[t-p2p4\]–\[t-sp1p4\] to find that [Longest Path Contractibility]{} is polynomial-time solvable for $H$-free graphs. From now on suppose $H$ is not of this form. Below we will prove that in that case [Longest Contractibility]{} is [[NP]{}]{}-complete for $H$-free graphs. If $H$ contains a cycle, then we apply Corollary \[c-girth\] to prove that [Longest Path Contractibility]{} is [[NP]{}]{}-complete for $H$-free graphs. Assume that $H$ is a forest. If $H$ has a vertex of degree at least 3, then the class of $H$-free graphs contains the class of $K_{1,3}$-free graphs. Hence, we can apply Lemma \[l-claw\] to find that [Longest Path Contractibility]{} is [[NP]{}]{}-complete for $H$-free graphs. From now on we assume that $H$ is a linear forest. As $H$ is not an induced subgraph of $sP_1+P_4$, we find that $H$ contains at least one edge. We distinguish three cases. [**Case 1.**]{} The number of connected components of $H$ is at least 3.\ First suppose that at least three connected components of $H$ contain an edge. Then $H$ contains an induced $3P_2$. This means we may apply Theorem \[t-hard\] to find that [Longest Path Contractibility]{} is [[NP]{}]{}-complete for $H$-free graphs. Now suppose that exactly two connected components of $H$ contain an edge. If $H$ contains at least four connected components, then $H$ contains an induced $2P_1+2P_2$. This means we may apply Theorem \[t-hard\] to find that [Longest Path Contractibility]{} is [[NP]{}]{}-complete for $H$-free graphs. Hence, $H=P_1+P_r+P_s$ for some $2\leq r\leq s$. If $s\geq 4$, then $H$ contains an induced $2P_1+2P_2$. This means we may apply Theorem \[t-hard\] to find that [Longest Path Contractibility]{} is [[NP]{}]{}-complete for $H$-free graphs. If $s=3$ and $r=2$, then $H=P_1+P_2+P_3$, a contradiction. If $s=3$ and $r=3$, then $H$ contains an induced $2P_3$. This means we may apply Theorem \[t-hard\] to find that [Longest Path Contractibility]{} is [[NP]{}]{}-complete for $H$-free graphs. Hence, $s=2$, and thus $r=2$. Then $H=P_1+2P_2$ is an induced subgraph of $P_1+P_5$, a contradiction. Finally suppose that exactly one connected component of $H$ contains an edge. Then $H=sP_1+P_r$ for some $r\geq 2$. As $H$ is not an induced subgraph of $sP_1+P_4$, we find that $r\geq 5$. If $r\geq 6$, then $H$ contains an induced $P_6$. This means we may apply Theorem \[t-hard\] to find that [Longest Path Contractibility]{} is [[NP]{}]{}-complete for $H$-free graphs. Hence, $r=5$. As $H\neq P_1+P_5$, we find that $s\geq 2$. Then $H=sP_1+P_5$ contains an induced $2P_1+2P_2$. This means we may apply Theorem \[t-hard\] to find that [Longest Path Contractibility]{} is [[NP]{}]{}-complete for $H$-free graphs. [**Case 2.**]{} The number of connected components of $H$ is exactly 2.\ Then $H=P_r+P_s$ for some $r$ and $s$ with $1\leq r\leq s$. If $r\geq 3$ then $H$ contains an induced $2P_3$, and if $s\geq 6$ then $H$ contains an induced $P_6$. In both cases we apply Theorem \[t-hard\] to find that [Longest Path Contractibility]{} is [[NP]{}]{}-complete for $H$-free graphs. From now on assume that $r\leq 2$ and $s\leq 5$. If $s\leq 4$, then $H$ is an induced subgraph of $P_2+P_4$, a contradiction. Hence, $s=5$. If $r=1$, then $H=P_1+P_5$, a contradiction. Thus $r=2$. Then $H$ contains an induced $3P_2$. This means we may apply Theorem \[t-hard\] to find that [Longest Path Contractibility]{} is [[NP]{}]{}-complete for $H$-free graphs. [**Case 3.**]{} The number of connected components of $H$ is exactly 1.\ If $H=P_r$ for some $r\leq 5$, then we use Theorem \[t-p1p5\]. Otherwise $P_6$ is an induced subgraph of $H$, and we use Theorem \[t-hard\]. Longest Cycle Contractibility {#s-cycle} ============================= The length of of a longest cycle a graph $G$ can be contracted to is called the [*co-circularity*]{} [@Bl82] or [*cyclicity*]{} [@Ha99] of $G$. This leads to the following decision problem. [.99]{} <span style="font-variant:small-caps;">[Longest Cycle Contractibility]{}</span>\ ----------------- ----------------------------------------------- *    Instance:* [a connected graph $G$ and an integer $k$.]{} *Question:* [does $G$ contain $C_k$ as a contraction?]{} ----------------- ----------------------------------------------- Hammack proved that [Longest Cycle Contractibility]{} is [[NP]{}]{}-complete for general graphs [@Ha02] but polynomial-time solvable for planar graphs [@Ha99]. It is also known that $C_6$-[Contractibility]{}, and thus [Longest Cycle Contractibility]{}, is [[NP]{}]{}-complete for $K_{1,3}$-free graphs [@FKP13] and bipartite graphs [@DP17], and thus for $C_r$-free graphs if $r$ is odd. The purpose of this section is to show that the complexities of [Longest Cycle Contractibility]{} and [Longest Path Contractibility]{} may not coincide on $H$-free graphs. For a given hypergraph ${\cal H}=(Q,{\cal S})$ we first construct the graph $G_{\cal H}$ as before. We then add an edge between vertices $t_1$ and $t_2$. This yields the graph $G_{\cal H}'$. We need the following result from [@BV87]. \[l-bv87b\] A hypergraph ${\cal H}$ has a $2$-colouring if and only if $G_{\cal H}'$ has $C_4$ as a contraction. We now prove the following lemma. \[l-p2p4again\] Let ${\cal H}$ be a hypergraph. Then the graph $G_{\cal H}'$ is $(P_2+P_4)$-free. For contradiction, assume that $G_{\cal H}'$ contains a subgraph $H$ isomorphic to $P_2+P_4$. Let $D_1$ be the connected component of $H$ on two vertices, and let $D_2$ be the connected components of $H$ on four vertices. As the subgraph of $G_{\cal H}'$ induced by ${\cal S}\cup {\cal S}'\cup \{t_1,t_2\}$ is complete bipartite and thus $P_4$-free, we find that $D_2$ must contains a vertex of $Q$. As $Q$ is a clique, $D_2$ must also contain at least two vertices from ${\cal S}\cup {\cal S}'$. We may without loss of generality assume that $D_2$ contains a vertex from ${\cal S}$. This means that $D_1$ contains no vertex from $Q\cup {\cal S}'\cup \{t_1\}$. Hence, $D_1$ only contains vertices of ${\cal S}\cup \{t_2\}$. As the latter set is independent, this is not possible. We conclude that $G_{\cal H}'$ is $(P_2+P_4)$-free. We note that, in line with our polynomial-time result of [Longest Path Contractibility]{} for $(P_2+P_4)$-free graphs (Theorem \[t-p2p4\]), the graph $G_{\cal H}$ may not be $(P_2+P_4)$-free: as $t_1$ and $t_2$ are not adjacent in $G_{\cal H}$, two vertices of $Q$ together with vertices $t_1,S_j,S_\ell',t_2$ may form an induced $P_2+P_4$ in $G_{\cal H}$. It is readily seen that $C_4$-[Contractibility]{} belongs to [[NP]{}]{}. Hence, we obtain the following result from Lemmas \[l-bv87b\] and \[l-p2p4again\]. \[t-cycle\] [$C_4$-Contractibility]{} is [[NP]{}]{}-complete for $(P_2+P_4)$-free graphs. Theorem \[t-cycle\] has the following consequence. \[c-cycle\] [Longest Cycle Contractibility]{} is [[NP]{}]{}-complete for $(P_2+P_4)$-free graphs. Recall that [Longest Path Contractibility]{} is polynomial-time solvable for $(P_2+P_4)$-free graphs by Theorem \[t-main\]. Hence, combining this result with Corollary \[c-cycle\] shows that the two problems behave differently on $(P_2+P_4)$-free graphs. Conclusions {#s-con} =========== We completely classified the complexities of [Longest Induced Path]{} and [Longest Path Contractibility]{} problem for $H$-free graphs. Such a classification is still open for [Longest Path]{} and below we briefly present the state of art. A graph is [*chordal bipartite*]{} if it is bipartite and every induced cycle has length 4. In other words, a graph is chordal bipartite if and only if it is $(C_3,C_5,C_6,\ldots)$-free. A direct consequence of the [[NP]{}]{}-hardness of [Hamiltonian Path]{} for chordal bipartite graphs and strongly chordal split graphs [@Mu96], or equivalently, strongly chordal $(2P_2,C_4,C_5)$-free graphs [@FH77] is that [Hamiltonian Path]{}, and therefore, [Longest Path]{} is [[NP]{}]{}-complete for $H$-free graphs if $H$ has a cycle or contains an induced $2P_2$. The [[NP]{}]{}-hardness of [Hamiltonian Path]{} for line graphs [@Be81], and thus for $K_{1,3}$-free graphs, implies the same result for $H$-free graphs if $H$ is a forest with a vertex of degree at least 3. On the positive side, [Longest Path]{} is polynomial-time solvable for $P_4$-free graphs due to the corresponding result for its superclass of cocomparability graphs [@IN13; @MC12]. This leaves open the following cases. Determine the computational complexity of [Longest Path]{} for $H$-free graphs when: - $H=sP_1+P_r$ for $3\leq r\leq 4$ and $s\geq 1$ - $H=sP_1+P_2$ for $s\geq 2$ - $H=sP_1$ for $s\geq 3$. We showed that the complexities of [Longest Cycle Contractibility]{} and [Longest Path Contractibility]{} do not coincide for $H$-free graphs. However, the complexity of [Longest Cycle Contractibility]{} for $H$-free graphs has not been settled yet. For instance, if $H$ is a cycle, the cases $H=C_4$ and $H=C_6$ are still open. 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P. van ’t Hof, M. Kaminski, D. Paulusma, S. Szeider, and D. M. Thilikos. On graph contractions and induced minors. , 160(6):799–809, 2012. P. van ’t Hof, D. Paulusma, and G. J. Woeginger. Partitioning graphs into connected parts. , 410(47-49):4834–4843, 2009. T. Watanabe, T. Ae, and A. Nakamura. On the removal of forbidden graphs by edge-deletion or by edge-contraction. , 3(2):151–153, 1981. T. Watanabe, T. Ae, and A. Nakamura. On the [N]{}[P]{}-hardness of edge-deletion and -contraction problems. , 6(1):63–78, 1983. [^1]: Supported by Research Project Grant RPG-2016-258 from the Leverhulme Trust. [^2]: The complexity status of [Snake-in-the Box]{} is still open. A table of world records for small values of $n$ can be found at http://ai1.ai.uga.edu/sib/sibwiki/doku.php/records. [^3]: In [@HHLP14], the problem is equivalently formulated as a graph modification problem: can a graph $G$ be modified into a graph $F$ from some specified family ${\cal F}$ by using at most $\ell$ edge contractions for some given integer $\ell\geq 0$? We refer to for instance [@ALSZ17; @AST17; @AH83; @BGHP14; @CG13; @Ep09; @GHP13; @GM13; @HHLLP14; @HHLP13; @LMS13; @MOR13; @WAN81; @WAN83], for both classical and fixed-parameter tractibility results for various families ${\cal F}$ including the family of paths, which form the focus in this paper. [^4]: This is in line with research for other graph problems restricted to $H$-free graphs. In fact, classes of $H$-free graphs, where $H$ is a linear forest are still poorly understood. There is a whole range of graph problems, e.g. [Independent Set]{}, [$3$-Colouring]{}, [Feedback Vertex Set]{}, [Odd Cycle Transversal]{}, and [Dominating Induced Matching]{}, for which it is not known if they are [[NP]{}]{}-complete on $P_k$-free graphs for some integer $k$, such that they are [[NP]{}]{}-complete on $P_k$-free graphs (see [@BDFJP]). [^5]: If every $Z_i$ has size 2, then we obtain the well-known [$k$-Disjoint Paths]{} problem.

--- abstract: 'We show how the LHC potential to detect a rather light CP-even Higgs boson of the NMSSM, $H_1$ or $H_2$, decaying into CP-odd Higgs states, $A_1A_1$, can be improved if Higgs-strahlung off $W$ bosons and (more marginally) off top-antitop pairs are employed alongside vector boson fusion as production modes. Our results should help extracting at least one Higgs boson signal over the NMSSM parameter space.' --- preprint SHEP-06-21\ [**Another step towards a no-lose theorem\ for NMSSM Higgs discovery at the LHC**]{}\ \ The Minimal Supersymmetric Standard Model (MSSM) is affected by the so-called ‘$\mu$-problem’. Its Superpotential contains a dimensionful parameter, $\mu$, that, upon Electro-Weak Symmetry Breaking (EWSB), provides a contribution to the masses of both Higgs bosons and Higgsino fermions. Furthermore, the associated soft Supersymmetry (SUSY) breaking term mixes the two Higgs doublets. Now, the presence of $\mu$ in the Superpotential before EWSB indicates that its natural value would be either 0 or the Planck mass $M_P$. On the one hand, $\mu = 0$ would mean that no mixing is actually generated between Higgs doublets at any scale and the minimum of the Higgs potential occurs for $< H_d > = 0$, so that one would have in turn massless down-type fermions and leptons after SU(2) symmetry breaking. On the other hand, $\mu \approx M_P$ would reintroduce a ‘fine-tuning problem’ in the MSSM since the Higgs scalars would acquire a huge contribution $\sim\mu^2$ to their squared masses (thus spoiling the effects of SUSY, which effectively removes otherwise quadratically divergent contributions to the Higgs mass from SM particles). Therefore, the values of this (arbitrary) parameter $\mu$ are phenomenologically constrained to be close to $M_{\rm SUSY}$ or $M_W$. The most elegant solution to the $\mu$-problem is to introduce a new singlet scalar field $S$ into the theory and replace the $\mu$-term in the MSSM Superpotential by an interaction term[^1] $\sim \hat S \hat H_u \hat H_d$. At the same time, also the soft term $B\mu H_u H_d$ is replaced by the dimension-4 term $\sim A_{\lambda} S H_u H_d$. When the extra scalar field $S$ acquires a Vacuum Expectation Value (VEV), an effective $\mu$ term, naturally of the EW scale, is generated automatically. This idea has been implemented in the Next-to-Minimal Supersymmetric Standard Model (NMSSM) [@NMSSM], described by the Superpotential $$\label{WNMSSM} W_{\rm{NMSSM}} = {\hat{Q}}{\hat{H}_u}{\bf{h_u}}{\hat{U}^C} + {\hat{H}_d}{\hat{Q}}{\bf{h_d}}{\hat{D}^C} + {\hat{H}_d}{\hat{L}}{\bf{h_e}}{\hat{E}^C} + \lambda\hat{S}(\hat{H}_u\hat{H}_d)+\frac{1}{3}\kappa\hat{S}^3,$$ where $\hat{S}$ is an extra Higgs iso-singlet Superfield, $\lambda$ and $\kappa$ are dimensionless couplings and the last ($Z_3$ invariant) term is required to explicitly break the dangerous Peccei-Quinn (PQ) U(1) symmetry [@PQ][^2]. (See Ref. [@slightly] for NMSSM Higgs sector phenomenology with an exact or slightly broken PQ symmetry.) However, due to its $Z_3$ symmetry, the NMSSM has a domain wall problem, as discussed in the last few references in [@DW]. This is to be solved by additional terms that break $Z_3$ explicitely. Although the latter can generate dangerous tadpole diagrams, as discussed in the first few references in [@DW], scenarios that solve both problems simultaneously are proposed in [@KT]. (Alternative formulations to the NMSSM – known as the Minimal Non-minimal Supersymmetric Standard Model (MNSSM) and new Minimally-extended Supersymmetric Standard Model or nearly-Minimal Supersymmetric Standard Model (nMSSM) – exist [@other-non-minimal].) Another positive feature of all these non-minimal SUSY models is that they predict the existence of a (quasi-)stable singlet-type neutralino (the singlino) that could be responsible for the Dark Matter (DM) of the universe, albeit this occurs only in limited regions of parameter space [@DM]. Finally, in these extended SUSY models, the singlet Superfield $\hat{S}$ has no SM gauge group charge (so that MSSM gauge coupling unification is preserved) and one can comfortably explain the baryon asymmetry of the Universe by means of a strong first order EW phase transition [@baryon1] (unlike the MSSM, which requires a light top squark and Higgs boson barely compatible with experimental bounds [@baryon2]). Clearly, in eq. (\[WNMSSM\]), upon EWSB, a VEV will be generated for the real scalar component of $\hat S$ (the singlet Higgs field), $<S>$, alongside those of the two doublets $<H_u>$ and $<H_d>$ (related by the parameter $\tan\beta= <H_u>/<H_d>$). In the absence of fine-tuning, one should expect these three VEVs to be of the order of $M_{\rm{SUSY}}$ or $M_W$, so that now one has an ‘effective $\mu$-parameter’, $\mu_{\rm{eff}}=\lambda <S>$, of the required size, thus effectively solving the $\mu$-problem. In the end, in the NMSSM, the soft SUSY-breaking Higgs sector is described by the Lagrangian contribution $$V_{\rm NMSSM} = m_{H_u}^2|H_u|^2+m_{H_d}^2|H_d|^2+m_{S}^2|S|^2 + \left(\lambda A_\lambda S H_u H_d + \frac{1}{3}\kappa A_\kappa S^3 + {\rm h.c.}\right),$$ with $A_\lambda$ and $A_\kappa$ dimensionful parameters of ${\cal O}(M_{\rm{SUSY}})$. As a result of the introduction of an extra complex singlet scalar field, which only couples to the two MSSM-type Higgs doublets, the Higgs sector of the NMSSM comprises of a total of seven mass eigenstates: a charged pair $H^\pm$, three CP-even Higgses $H_{1,2,3}$ ($M_{H_1}<M_{H_2}<M_{H_3}$) and two CP-odd Higgses $A_{1,2}$ ($M_{A_1}<M_{A_2}$). Consequently, Higgs phenomenology in the NMSSM may plausibly be different from that of the MSSM. In view of the upcoming CERN Large Hadron Collider (LHC), quite some work has been dedicated to probing the Next-to-Minimal Supersymmetric Standard Model (NMSSM) [@NMSSM] Higgs sector over recent years. Primarily, there have been attempts to extend the so-called ‘no-lose theorem’ of the MSSM [@Dai:1993at] to the case of the NMSSM [@NoLoseNMSSM1a; @NoLoseNMSSM1b][^3]. From this perspective, it was realised that at least one NMSSM Higgs boson should remain observable at the LHC over the NMSSM parameter space that does not allow any Higgs-to-Higgs decay. However, when the only light non-singlet (and, therefore, potentially visible) CP-even Higgs boson, $H_1$ or $H_2$, decays mainly to two very light CP-odd Higgs bosons, $A_1 A_1$, one may not have a Higgs signal of statistical significance at the LHC. From the preliminary studies in Ref. [@NoLoseNMSSM1b] though, it appeared that using the $qq\to qq W^+W^-,qqZZ\to qq H_{1,2} \to qq A_1A_1$ detection mode, i.e., via Vector Boson Fusion (VBF), may lead to the possibility of establishing a no-lose theorem in the NMSSM, particularly if the lightest CP-odd Higgs mass is such that there can happen abundant $A_1A_1\to b\bar b\tau^+\tau^-$ decays, with both $\tau$-leptons being detected via their $e,\mu$ leptonic decays[^4]. At high luminosity, this signal may be detectable at the LHC as a bump in the tail of a rapidly falling mass distribution. However, this procedure relies on the background shape to be accurately predictable. These analyses were based on Monte Carlo (MC) event generation (chiefly, via the SUSY routines of the [HERWIG]{} v6.4 code [@SHERWIG]) and a toy detector simulation ([GETJET]{}, based on UA1 software). Further analyses based on [PYTHIA]{} v6.2 [@Sjostrand:2001yu] and a more proper ATLAS detector simulation ([ATLFAST]{}) [@Baffioni:2004gdr] found that the original selection procedures may need improvement in order to extract a signal [@Orsay]. While the jury is still out on this particular analysis, we would like here to advertise the possibilities offered by exploiting Higgs-strahlung (HS) off gauge bosons ($q\bar q'\to W^{\pm*}\to W^\pm H_{1,2}$, with a subleading component from $q\bar q\to Z^{0*}\to Z^0 H_{1,2}$) and, more marginally, off heavy quark pairs (chiefly top quarks, $q\bar q,gg\to t\bar t H$, because of the small $\tan\beta$ values involved in the scenarios outlined in [@NoLoseNMSSM1b]) as the underlying Higgs production modes, in place of or – better – alongside VBF. In fact, for the $H_{1,2}$ masses of relevance to the above analyses, say, 50 to 120 GeV, Higgs-strahlung gives cross sections comparable to VBF, if not larger for smaller $M_{H_{1,2}}$ values. However, we will not be performing here a detector analysis, including parton shower and hadronisation effects, as in [@NoLoseNMSSM1b; @Baffioni:2004gdr]. Rather, in this brief report, we will limit ourselves to proving that, after enforcing standard LHC triggers (at partonic level) on $W$ decays in Higgs-strahlung and on forward/backward jets in VBF, there are regions of NMSSM parameter space were the yield of the former is of the same size as that of the latter, no matter what the $A_{1}A_1$ decay pattern may be. Therefore, we conclude that our results are encouraging in an attempt to establish the aforementioned NMSSM no-lose theorem at the LHC. For a general study of the NMSSM Higgs sector (without any assumption on the underlying SUSY-breaking mechanism) we used here the [NMHDECAY]{} code (version 1.1) [@Ellwanger:2004xm]. (We have verified that the pattern described below does not change if one adopts the newest version [@NMHDECAY2].) This program computes the masses, couplings and decay Branching Ratios (BRs) of all NMSSM Higgs bosons in terms of the model parameters taken at the EW scale. The computation of the spectrum includes leading two-loop terms, EW corrections and propagator corrections. [NMHDECAY]{} also takes into account theoretical as well as experimental constraints from negative Higgs searches at collider experiments. For our purpose, instead of postulating unification, we fixed the soft SUSY breaking terms to a very high value, so that they have little or no contribution to the outputs of the parameter scans. Consequently, we are left with six free parameters: the usual tan$\beta$, the Yukawa couplings $\lambda$ and $\kappa$, the soft trilinear terms $A_\lambda$ and $A_\kappa$ plus $\mu_{\rm eff} = \lambda\langle S\rangle$. We have used [NMHDECAY]{} to scan over the NMSSM parameter space defined in [@Moretti:2006sv] (borrowed from [@Ellwanger:2005uu]), with the aforementioned six parameters taken in the following intervals[^5]: $\lambda$ : 0.0001 – 0.75, $\kappa$ : $-$0.65 – +0.65, $\tan\beta$ : 1.6 – 54,\ $\mu$, $A_{\lambda}$, $A_{\kappa}$ : $-$1000 – +1000 GeV.\ Remaining soft terms which are fixed in the scan include:\ $\bullet\phantom{a}m_{Q_3} = m_{U_3} = m_{D_3} = m_{L_3} = m_{E_3} = 2$ TeV,\ $\bullet\phantom{a}A_{U_3} = A_{D_3} = A_{E_3} = 1.5$ TeV,\ $\bullet\phantom{a}m_Q = m_U = m_D = m_L = m_E = 2$ TeV,\ $\bullet\phantom{a} M_1 = M_2 = M_3 = 3$ TeV.\ The allowed decay modes for neutral NMSSM Higgs bosons are into any SM particle, plus into any final state involving all possible combinations of two Higgs bosons (neutral and/or charged) or of one Higgs boson and a gauge vector as well as into all possible sparticles. We have performed our scan over several millions of randomly selected points in the specified parameter space. The data points surviving all constraints are then used to determine the cross sections for NMSSM Higgs hadro-production. As the SUSY mass scales have been set well above the EW one, the production modes exploitable in simulations at the LHC are the usual ones, the so-called ‘direct’ Higgs production modes of [@Kunszt:1996yp]. As we are aiming at comparing the yield of VBF ([$qq\to qqH$]{}) against HS off $W$ bosons (W-HS) ([$qq\to WH$]{}) and off $t\bar t$ pairs (tt-HS) ([$gg\to ttH$]{}), it is of relevance to study in Fig. \[fig:XsectSM\] the light Higgs, $H$, hadro-production cross sections at the LHC in the SM, as the NMSSM rates would be obtained from these (for a given Higgs mass) by rescaling the $VVH$ and $ttH$ couplings. We see that in the SM W-HS dominates for Higgs masses below 80 GeV while VBF becomes the leading channel above such a value (in the NMSSM these two processes are rescaled by the same amount). The case tt-HS is generally subleading (even in presence of appropriate NMSSM couplings), but not negligible at low Higgs masses. Besides, as intimated earlier, notice that HS off $Z$ boson is always very small, so we will ignore it in the remainder of the paper. It is also worth recalling that gluon-gluon fusion ([$gg\to H$]{}), despite being the mode with largest production rates, plays no role in our case, as $H_{1,2}\to A_1A_1$ decay channels would not be extractable in this case from the background. (Notice in the figure the normalisation via NLO QCD throughout.) As a second step we computed the NMSSM total cross section times BR into $A_1A_1$ pairs for VBF and W-HS + tt-HS for each of the two lightest neutral Higgs bosons, $H_1$ and $H_2$. We display these rates in Fig. \[fig:XsectBRH\] as a function of $M_{H_1}$ and $M_{H_2}$. Here, one can appreciate that there exist more possibilities of establishing a $H_1$ signal than one due to $H_2$. Whereas the potential to detect the heavier of these two Higgs states is confined to masses above 115 GeV or so and probably below 140 GeV, where VBF is largely dominant with respect to W-HS + tt-HS, in the case of the light state there exists a low mass window where production rates via the latter two processes combined are comparable to those from the former, most often within 10–20% from each other. In fact, at times, W-HS + tt-HS rates are larger than those for VBF, the more so the lower the $H_1$ mass. (Recall that all parameter points examined here are compliant with collider bounds, even those at very low Higgs mass, as these correspond to reduced Higgs couplings to gauge bosons.) Now, one should bear in mind that the rates in Fig. \[fig:XsectBRH\] do not include yet the efficiency to trigger on the signal. In the case of VBF, one triggers on one forward and one backward jet, with $p_T > 20$ GeV, $|\eta| < 5$ and $\eta({\rm fwd})\cdot\eta({\rm bwd}) < 0$. The efficiency is here about 60%. In the case of W-HS, one triggers on a high transverse momentum lepton (electron or muon), with $p_T > 20$ GeV and $|\eta| < 2.5$. In this case the efficiency is lower, about 19%, primarily due to the fact that a $W$ boson decays into electron/muons only about 20% of the times. [The efficiency for tt-HS is 14%, as one top is required to decay hadronically and the other leptonically.]{} (Note that the efficiency values quoted are basically independent of the Higgs mass.) Even so, the W-HS component, aided by the tt-HS one, would make a sizable addition to the production rates of VBF. As we expect the efficiency of extracting whichever $H_{1,2}\to A_1A_1$ decays to be the same in both processes[^6], we see a potential in improving the signal yield by using all mentioned channels, beyond what achieved by using VBF alone. --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![Cross section times BR of $H_1$ (left) and $H_2$ (right) plotted against their respective masses. The symbol ‘$\cdot$’ refers to VBF while ‘$+$’ to W-HS + tt-HS.[]{data-label="fig:XsectBRH"}](H1_mass-bit.eps "fig:") ![Cross section times BR of $H_1$ (left) and $H_2$ (right) plotted against their respective masses. The symbol ‘$\cdot$’ refers to VBF while ‘$+$’ to W-HS + tt-HS.[]{data-label="fig:XsectBRH"}](H2_A1A1_mass-bit.eps "fig:") --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- By recalling that the efficiency to trigger on VBF is at least three times the one to isolate W-HS + tt-HS, it is of particular interest to estimate the proportion of points where the latter gives more cross section than the former. Despite we found that W-HS + tt-HS very rarely exceeds VBF by more than a factor of three, there are clear zones of NMSSM parameters space where W-HS + tt-HS is consistently larger than VBF, those producing $M_{H_1}$ values below 80 GeV, indeed the SM crossing point seen in Fig. \[fig:XsectSM\]. Evidently, this mass range is of relevance to $H_1\to A_1A_1$ decays only, see Fig. \[fig:XsectBRH\]. In fact, for the case of $H_2\to A_1A_1$, cross sections are much smaller in comparison and VBF is always very dominant, as – for potentially detectable rates – $M_{H_2}$ is above $\approx115$ GeV and below $\approx140$ GeV. Finally, notice that $H_2\to H_1H_1$ decays very often compete with $H_2\to A_1A_1$ [@Ellwanger:2005uu]. In fact the former occur almost as often as the latter over the NMSSM parameter space investigated here. To make use of this channel too, a slight modification of the procedures advocated in [@NoLoseNMSSM1b] would be required. Even after accounting for the trigger efficiencies, the VBF cross sections plotted in Fig. \[fig:XsectBRH\] are in the same range as those probed in [@NoLoseNMSSM1b][^7], so that, for similar $M_{H_1}$ and $M_{H_2}$ masses, we would expect to obtain the same overall detection efficiencies seen back then also for all our points falling in the mass range, say, 50 to 120 GeV. Crucially, NMSSM parameter points giving the highest cross sections for VBF are the same yielding the largest rates for W-HS + tt-HS. More in general, from Figs. \[fig:paramsH\]a–b, one can also gather where the regions of highest cross sections, for both channels (VBF and W-HS + tt-HS) and Higgs flavours ($H_1$ and $H_2$), lie in the NMSSM parameter space. In particular, their distribution is quite homogeneous as they are not located in some specific areas (i.e., in a sense, not ‘fine-tuned’). Altogether, the proportion of parameter space where the two production modes yield potentially detectable Higgs signals (at least according to the analysis in [@NoLoseNMSSM1b]), say, above 1–2 pb (prior to including tagging efficiencies and $A_1$ decay rates), is 0.21% for VBF and 0.13% for W-HS + tt-HS. However, if production cross sections of 4 pb or upwards are required to render the $H_{1}\to A_1A_1$ signal visible, then the rates reduce to $0.096\%$ and $0.0019\%$, respectively. For the case of $H_{2}\to A_1A_1$, the numbers are typically 20 and 10 times smaller, for the case of VBF and W-HS + tt-HS, respectively. Clearly, while the production cross sections (after triggering), the selection procedures and efficiencies to extract the Higgs decays may well be the same in both samples, the background will differ. In fact, whilst in the case of VBF the latter is dominated by top-antitop pair production and decay for V-HS and tt-HS we expect that (more manageable) $WZ$ + jets events will be the largest noise, assuming the most promising Higgs signature discussed above (i.e., $b\bar b\tau^+\tau^-$). A detailed phenomenological study, based upon parton shower, hadronisation and detector simulation (like in Refs. [@NoLoseNMSSM1b; @Baffioni:2004gdr]), is obviously in order before drawing any firm conclusions from our very preliminary study. (In this respect, it is also interesting to see how the mass of the decaying Higgs bosons, $H_1$ and $H_2$, relates to that of the light $A_1$ state: this is illustrated in Fig. \[fig:mass\].) Nonetheless, we thought it worthwhile to alert the LHC experiments to the possibility of supplementing the search for $H_{1,2}\to A_1A_1$ signals via VBF with that through W-HS + tt-HS, as such Higgs decays are relevant in a region of NMSSM parameter space where the two production modes are competitive. Whilst the efficiency of tagging two forward/backward jets in VBF is three times higher than that to trigger on a high transverse momentum electron/muon in W-HS + tt-HS (mainly in virtue of the leptonic BR suppression in the second case), the combination of the latter two remains competitive with the former over the Higgs mass range relevant to these decays, 50 to 120 GeV or so, the more so the lighter the mass of the decaying Higgs state. (Notice that such a low mass scenario is one alleviating the so-called ‘little fine-tuning problem’ of the MSSM, resulting in LEP failing to detect a light CP-even Higgs boson, predicted over most of the MSSM parameter space, as in the NMSSM the mixing among more numerous CP-even or CP-odd Higgs fields enables light mass states being produced at LEP yet they can remain undetected because of their reduced couplings to $Z$ bosons.) Thus, the chances of establishing a no-lose theorem in the NMSSM at the LHC via the aforementioned Higgs-to-Higgs decay mode might improve considerably if the Higgs state strongly coupled to gauge bosons is the lightest one. Our analysis was based on a fairly extensive scan of the NMSSM parameter space incorporating the latest experimental constraints. Detailed MC event generation studies will be available soon. Acknowledgements {#acknowledgements .unnumbered} ---------------- SM thanks Cyril Hugonie for discussions. 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Z. Kunszt, S. Moretti and W.J. Stirling, Z. Phys. C [**74**]{} (1997) 479. ------------------------- ------------------------------       ------------------------- ------------------------------ (a) ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![Cross section times BR of $H_1$ (left) and $H_2$ (right) when potentially visible, i.e., limited to those NMSSM parameter points for which both cross sections times BRs are larger than 2(1) pb for $H_1(H_2)$, plotted against the following parameters: (a) $\tan\beta$, $\lambda$, $\kappa$; (b) $A_\lambda$, $A_\kappa$ and $\mu_{\rm{eff}}$.[]{data-label="fig:paramsH"}](alambda_H1.eps "fig:") ![Cross section times BR of $H_1$ (left) and $H_2$ (right) when potentially visible, i.e., limited to those NMSSM parameter points for which both cross sections times BRs are larger than 2(1) pb for $H_1(H_2)$, plotted against the following parameters: (a) $\tan\beta$, $\lambda$, $\kappa$; (b) $A_\lambda$, $A_\kappa$ and $\mu_{\rm{eff}}$.[]{data-label="fig:paramsH"}](alambda_H2_A1A1.eps "fig:") ![Cross section times BR of $H_1$ (left) and $H_2$ (right) when potentially visible, i.e., limited to those NMSSM parameter points for which both cross sections times BRs are larger than 2(1) pb for $H_1(H_2)$, plotted against the following parameters: (a) $\tan\beta$, $\lambda$, $\kappa$; (b) $A_\lambda$, $A_\kappa$ and $\mu_{\rm{eff}}$.[]{data-label="fig:paramsH"}](akappa_H1.eps "fig:") ![Cross section times BR of $H_1$ (left) and $H_2$ (right) when potentially visible, i.e., limited to those NMSSM parameter points for which both cross sections times BRs are larger than 2(1) pb for $H_1(H_2)$, plotted against the following parameters: (a) $\tan\beta$, $\lambda$, $\kappa$; (b) $A_\lambda$, $A_\kappa$ and $\mu_{\rm{eff}}$.[]{data-label="fig:paramsH"}](akappa_H2_A1A1.eps "fig:") ![Cross section times BR of $H_1$ (left) and $H_2$ (right) when potentially visible, i.e., limited to those NMSSM parameter points for which both cross sections times BRs are larger than 2(1) pb for $H_1(H_2)$, plotted against the following parameters: (a) $\tan\beta$, $\lambda$, $\kappa$; (b) $A_\lambda$, $A_\kappa$ and $\mu_{\rm{eff}}$.[]{data-label="fig:paramsH"}](mu_H1.eps "fig:") ![Cross section times BR of $H_1$ (left) and $H_2$ (right) when potentially visible, i.e., limited to those NMSSM parameter points for which both cross sections times BRs are larger than 2(1) pb for $H_1(H_2)$, plotted against the following parameters: (a) $\tan\beta$, $\lambda$, $\kappa$; (b) $A_\lambda$, $A_\kappa$ and $\mu_{\rm{eff}}$.[]{data-label="fig:paramsH"}](mu_H2_A1A1.eps "fig:") ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- (b) ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![Distribution of the $H_1$ (left) and $H_2$ (right) masses with respect to that of $A_1$, when VBF (top) and W-HS + tt-HS (bottom) are potentially visible, i.e., limited to those NMSSM parameter points for which both cross sections times BRs are larger than 2(1) pb for $H_1(H_2)$.[]{data-label="fig:mass"}](mH1_mA1_vv.eps "fig:") ![Distribution of the $H_1$ (left) and $H_2$ (right) masses with respect to that of $A_1$, when VBF (top) and W-HS + tt-HS (bottom) are potentially visible, i.e., limited to those NMSSM parameter points for which both cross sections times BRs are larger than 2(1) pb for $H_1(H_2)$.[]{data-label="fig:mass"}](mH2_mA1_vv.eps "fig:") ![Distribution of the $H_1$ (left) and $H_2$ (right) masses with respect to that of $A_1$, when VBF (top) and W-HS + tt-HS (bottom) are potentially visible, i.e., limited to those NMSSM parameter points for which both cross sections times BRs are larger than 2(1) pb for $H_1(H_2)$.[]{data-label="fig:mass"}](mH1_mA1_tt+w.eps "fig:") ![Distribution of the $H_1$ (left) and $H_2$ (right) masses with respect to that of $A_1$, when VBF (top) and W-HS + tt-HS (bottom) are potentially visible, i.e., limited to those NMSSM parameter points for which both cross sections times BRs are larger than 2(1) pb for $H_1(H_2)$.[]{data-label="fig:mass"}](mH2_mA1_tt+w.eps "fig:") ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- [^1]: Hereafter, hatted variables describe Superfields while un-hatted ones stand for the corresponding scalar components. [^2]: One could also gauge the U(1)$_{\rm PQ}$ group, so that the $Z_3$ symmetry is embedded in the local gauge symmetry [@Z3]. [^3]: See Refs. [@Erice]–[@Moretti:2006sv] for a complementary approach, named ‘more-to-gain theorem’, attempting to define regions of the NMSSM parameter space where more Higgs states are visible at the LHC than those available within the MSSM. [^4]: The scope of other decays, $A_1A_1\to jjjj$, $A_1A_1\to jj\tau^+\tau^-$ (where $j$ represents a light quark jet) or $A_1A_1\to \tau^+\tau^-\tau^+\tau^-$ is very much reduced in comparison. [^5]: Notice that a top quark pole mass of $m_t=175$ GeV was used as default, though we have verified that values within current error bands (see [@topmass]) have a numerically small impact on our analysis, thus leaving the main conclusions of the paper unchanged. [^6]: If anything, since no actual $b$-tagging was enforced in the analyses of Refs. [@NoLoseNMSSM1b; @Baffioni:2004gdr], whenever $A_1A_1$ hadronic decays are present, we would expect the efficiency to worsen for the case of VBF, because of jet combinatorics. [^7]: We have in fact been able to reproduce most of the points discussed therein.

--- abstract: 'We introduce a model independent method for the determination of the hadronic contribution to the QED running coupling, $\Delta\alpha_{\text{HAD}}(M_{Z}^{2})$, requiring no $e^+e^-$ annihilation data as input. This is achieved by calculating the heavy-quark contributions entirely in perturbative QCD, whilst the light-quark resonance piece is determined using available lattice QCD results. Future reduction in the current uncertainties in the latter shall turn this method into a valuable alternative to the standard approach. Subsequently, we find that the precision of current determinations of $\Delta\alpha_{\text{HAD}}(M_{Z}^{2})$ can be improved by some $20\%$ by computing the heavy-quark pieces in PQCD, whilst using $e^+e^-$ data only for the low-energy light-quark sector. We obtain in this case $\Delta\alpha_{\text{HAD}}(M_{Z}^{2})=275.7(0.8)\, \times 10^{-4}$, which currently is the most precise value of $\Delta\alpha_{\text{HAD}}(M_{Z}^{2})$.' author: - 'S. Bodenstein' - 'C. A. Dominguez' - 'K. Schilcher' - 'H. Spiesberger' title: 'Hadronic contribution to the QED running coupling $\alpha(M_{Z}^2)$' --- INTRODUCTION ============ Of the subset of three parameters that enter the electroweak sector of the Standard Model (SM) of particle physics, $G_F, M_Z$ and $\alpha(M_{Z}^{2})$, the least precisely known is the electromagnetic coupling at the $Z$ boson mass, $\alpha(M_{Z}^{2})$. This is primarily due to hadronic contributions which are not calculable using perturbative QCD (PQCD). Increasing the precision of $\alpha(M_{Z}^{2})$ is important for, amongst other things, obtaining a Standard Model fit of the Higgs mass. Currently, there is a minor tension between the recently measured mass of a potential Higgs boson, $M_H= 126.0 (0.4)(0.4)\,\text{GeV}$ [@higgsm1], or $M_H= 125.3 (0.4)(0.5)\,\text{GeV}$ [@higgsm2], and a mass of $91^{+30}_{-23}\,\text{GeV}$ (at the 68% confidence level) obtained from global SM fits to electroweak precision data [@davier2011].\ The running QED coupling $\alpha$ can be parameterized as $$\alpha(s)=\frac{\alpha(0)}{1-\Delta\alpha_\text{L}(s)-\Delta\alpha_\text{HAD}(s)}\;,$$ where $\Delta\alpha_\text{L}$ is the leptonic contribution, which can be determined with high precision in perturbation theory, and $\Delta\alpha_\text{HAD}$ is the hadronic term. Of particular interest is the QED coupling at the scale $M_Z$. Denoting $\alpha\equiv \alpha(0)$ in the sequel, $\Delta\alpha_{\text{HAD}}(M_{Z}^{2})$ can be written as $$\label{eq:alpha1} \Delta\alpha_{\text{HAD}}(M_{Z}^{2})=4 \,\pi \, \alpha\left\{\Pi(0) - Re \,[\Pi(M_{Z}^{2})]\right\}\;,$$ where $\Pi(s)$ is the standard electromagnetic current correlator $$\begin{aligned} \Pi_{\mu\nu} (q^2) &=& i \int d^4x\, e^{iqx} \langle 0|\, T \left(j^{\text{\,EM}}_{\mu}(x), j^{\text{\,EM}}_{\nu}(0) \right)|0\rangle \nonumber\\ &=& (q_\mu q_\nu - q^2 g_{\mu\nu}) \Pi(q^2)\;,\end{aligned}$$ with $j^{\text{EM}}_\mu(x)=\sum_f Q_f \bar{f}(x) \gamma_\mu f(x)$, and the sum is over all quark flavors $f=\{u,d,s,c,b,t\}$, with charges $Q_f$. Invoking analyticity and unitarity for $\Pi(s)$, and using the optical theorem, i.e. $R(s)=12\pi\, \text{Im}\,\Pi(s)$, where $R(s)$ is the normalized $e^+e^-$ cross-section, one can write Eq. as a dispersion integral [@cabibbo1961] $$\label{EQ:dispersion} \Delta\alpha_{\text{HAD}}(M_{Z}^{2})=\frac{\alpha \, M_{Z}^{2}}{3\, \pi}P\int^{\infty}_{4m_{\pi}^{2}}\frac{R(s)}{s(M_{Z}^{2}-s)}ds \;,$$ where $P$ denotes the principal part of the integral. This dispersion relation is useful as it only requires knowledge of $R(s)$, which can be determined experimentally. The standard approach to determining $\Delta\alpha_{\text{HAD}}(M_{Z}^{2})$ is to evaluate Eq. making use of $e^+e^-$ annihilation data for $R(s)$ in the resonance regions, and either use the PQCD prediction for $R(s)$ above these regions (see e.g. [@davier2011]), or make use of all the available $e^+e^-$ data and fill in the gaps using the PQCD prediction (see e.g. [@hagiwara2011; @Actis]). Since the use of data is the primary source of uncertainty, other analyses have attempted to reduce the dependence of $\Delta\alpha_{\text{HAD}}(M_{Z}^{2})$ on $e^+e^-$ data by a variety of methods that place a greater emphasis on PQCD. One approach in this direction is to subtract a polynomial from the weight function in Eq. to reduce the impact of the data contribution. In order to compensate, this polynomial weighted integral is added to the right hand side of Eq. and evaluated in PQCD (plus non-perturbative corrections given in the framework of the Operator Product Expansion) using a circular contour integral (see e.g. [@groote1998; @kuhn1998]). Another approach is to first calculate $\Delta\alpha_{\text{HAD}}(-s_0)$ ($s_0>0$ with $s_0$ large enough for PQCD to be valid), whose weight function deemphasizes the low-energy region. Subsequently, $\Delta\alpha_{\text{HAD}}(-s_0)$ is run to $\Delta\alpha_{\text{HAD}}(M_{Z}^2)$ using the PQCD prediction of the Adler function [@jegerlehner2008].\ The purpose of this paper is two-fold. First, to calculate the complete heavy quark (charm, bottom, and top) contributions to $\Delta\alpha_{\text{HAD}}(M_{Z}^2)$ using only PQCD, which to our knowledge has not been done before. Interestingly, this will significantly reduce the total uncertainty in $\Delta\alpha_{\text{HAD}}(M_{Z}^{2})$, as the use of $e^+e^-$ data in the charm-quark region leads to an error equivalent to that from the use of $e^+e^-$ data in the light-quark resonance region. Second, to show how existing Lattice QCD (LQCD) calculations involved in the evaluation of the hadronic contribution to $g-2$ of the muon can be used to calculate the light-quark contribution to $\Delta\alpha_{\text{HAD}}(M_{Z}^{2})$ entirely from theory. This will allow for the first model-independent determination of $\Delta\alpha_{\text{HAD}}(M_{Z}^{2})$ that makes no use at all of $e^+e^-$ cross-section data. At present, though, current precision of LQCD results do not allow this method to compete with the standard approach.\ ![[]{data-label="fig:qed"}](feynQED.eps){height="0.8in" width="3.3in"} We begin by considering the heavy quark contribution to Eq., which can be written as $$\label{eq:AlphaHeavy} \Delta\alpha^{(f)}_{\text{HAD}}(M_{Z}^{2})=4\pi\alpha\left(\Pi^{(f)}(0)- Re \,[\Pi^{(f)}(M_{Z}^{2})]\right) \;,$$ where $f=\{c,b,t\}$ are the heavy quark flavors, and in the sequel it should be understood that it is the real part of the correlators that enters in the time-like region. It should be noticed that one only needs knowledge of the correlator at $s=0$ and at $s=M_{Z}^{2}$. The latter scale is way above either the charm- or the bottom-quark pair production resonance region, so that one can safely use the high-energy expansion of the heavy quark correlator. This is known to $\mathcal{O}(\alpha_{s}^{3})$ (with partial results at $\mathcal{O}(\alpha_{s}^{4})$). In addition, $\Pi^{(f)}(0)$ has also been calculated in PQCD to $\mathcal{O}(\alpha_{s}^{3})$. The other key inputs are the recent high precision bottom- and charm-quark masses obtained from LQCD [@lattice] in the $\overline{\text{MS}}$-scheme. This scheme will be used here in all PQCD calculations. Prior to these LQCD determinations, the charm- and bottom-quark masses were obtained using $e^+e^-$ data, a procedure we wish to avoid as we aim at an entirely theoretical determination of $\Delta\alpha^{(f)}_{\text{HAD}}(M_{Z}^{2})$. As explained later, it turns out that Eq. is problematic for the charm-quark contribution (but not for the bottom- or the top-quark). The reason being its strong dependence on the renormalization scale, which must be the same for both $\Pi^{(c)}(0)$ and $\Pi^{(c)}(M_{Z}^2)$. Therefore, we introduce two additional approaches in the charm-quark sector which are significantly less sensitive to this problem. The first is inspired by the Adler function approach of [@jegerlehner2008], to wit. We note that $$\label{eq:ADLER} \frac{d}{ds}\Delta\alpha^{(c)}_{\text{HAD}}(s)= - 4\pi \alpha \frac{d}{ds}\Pi^{(c)}(s)=\frac{\alpha}{3 \,\pi} \frac{D^{(c)}(s)}{s} \;,$$ where the real part is understood, and $D^{(c)}(s)$ is the Adler function in the charm-quark channel. Integrating Eq. gives $$\begin{aligned} \Delta\alpha^{(c)}_{\text{HAD}}(M^{2}_{z})&\equiv&\bigl[\Delta\alpha^{(c)}_{\text{HAD}}(M^{2}_{z})-\Delta\alpha^{(c)}_{\text{HAD}}(s_0)\bigr]\nonumber\\ &+&\Delta\alpha^{(c)}_{\text{HAD}}(s_0)=\frac{\alpha}{3\pi} \int^{M_{Z}^{2}}_{s_0}\frac{D^{(c)}(s)}{s}ds \nonumber\\ &+& 4 \pi \alpha \left(\Pi^{(c)}(0)-\Pi^{(c)}(s_0)\right) \;.\label{eq:adler}\end{aligned}$$ We choose $s_0$ large enough so that PQCD is valid, but still $s_0\ll M_{Z}^{2}$. One can then use one scale for the second term on the right hand side above, whilst another scale for integrating over the Adler function (one could also use a running scale, e.g. $\mu^2=s$). The second, but similar approach, is to use Cauchy’s residue theorem to rewrite the dispersion relation Eq. to obtain $$\begin{aligned} &&\Delta\alpha^{(c)}_{\text{HAD}}(M^{2}_{z})=4\, \alpha \,M_{Z}^{2}\Bigl[ \frac{i}{2}\oint_{|s|=s_0}ds\,\frac{\Pi^{(c)}(s)}{s(M_{Z}^{2}-s)}\nonumber \\ &&+\pi \,\frac{\Pi^{(c)}(0)}{M_Z^2}\Bigr]+\frac{\alpha \, M_{Z}^{2}}{3 \,\pi}P\int^{\infty}_{s_0}\frac{R^{(c)}(s)}{s(M_{Z}^{2}-s)}ds \;, \label{eq:FESR}\end{aligned}$$ which is only valid for $s_0<M_{Z}^{2}$. As usual, $s_0$ will be taken large enough so that PQCD is valid. Once again, this allows for the use of more than one scale appropriate for the different regions. In addition, $R^{(c)}(s)$ is known partially up to $\mathcal{O}(\alpha_{s}^{4})$. The final virtue of this approach is that it will allow a careful region-by-region comparison with the standard approach based on Eq.. The sum rule Eq. is very similar to the FESR used in precision charm- and bottom-quark mass determinations employing experimental data on $R(s)$ [@bodenstein2011; @bodenstein2012]. Hence, to determine $\Delta\alpha^{(f)}_{\text{HAD}}(M^{2}_{z})$ entirely from theory it is essential to use a non-QCD sum rule determination of the charm- and bottom-quark masses, such as e.g. that from LQCD. The procedure just outlined for determining $\Delta\alpha^{(f)}_{\text{HAD}}(M^{2}_{z})$ is not necessary for the bottom- and top-quark counterparts, as Eq. for $f=b,t$ gives results that are essentially renormalization scale independent. However, we have checked that the FESR and the Adler function approaches give the same result as that using Eq. for the charm-quark contribution, although the latter has a much larger error. ![[]{data-label="fig:convergence"}](Fig.2.eps){height="2.8in" width="3.6in"} VECTOR CURRENT CORRELATOR IN QCD ================================ We provide in this section a summary of the available theoretical information on the vector current correlator in QCD. The flavor $f$-quark current correlator can be split as $$\label{eq:total} \Pi^{(f)}(s)=\Pi^{(f)}_{\text{PQCD}}(s)+\Pi^{(f)}_{\text{NP}}(s)+\Pi^{(f)}_{\text{QED}}(s) \;,$$ where $f\in\{uds,c,b,t\}$, $\Pi^{(f)}_{\text{PQCD}}$ is the PQCD contribution, $\Pi^{(f)}_{\text{NP}}$ is the contribution from non-perturbative power corrections given in the framework of the Operator Product Expansion (OPE), and $\Pi^{(f)}_{\text{QED}}$ are QED corrections as shown in Fig. \[fig:qed\]. The dominant contribution to Eq. is the perturbative part $\Pi^{(f)}_{\text{PQCD}}$. In the massless case, appropriate for the $u,d,s$ quarks, the high energy limit of $\Pi^{(f)}_{\text{PQCD}}$ is known exactly up to order $\mathcal{O}(\alpha_{s}^{3})$, and up to a real constant to order $\mathcal{O}(\alpha_{s}^{4})$ (the full result is given in computer readable form in [@QCD0]). In the heavy-quark case, though, one needs both the low- and the high-energy expansions of the correlator. In the low-energy limit, with a single heavy quark and $n_f$ active flavors, the vector correlator can be written as $$\Pi_f(s)=\frac{3 Q^{2}_{f}}{16\, \pi^2}\sum_{i=0}^{\infty}\bar{C}_i z^i \;, \label{eq:LE}$$ where $z\equiv s/(4\bar{m}_{f}^{2})$, and $\bar{m}_{f}$ is the mass of the quark of flavor-$f$ in the $\overline{\text{MS}}$ scheme at the scale $\mu$. The coefficients $\bar{C}_0$ and $\bar{C}_1$ were determined up to $\mathcal{O}(\alpha_{s}^{3})$ in [@QCD1; @QCD2], $\bar{C}_2$ in [@C2], and $\bar{C}_3$ in [@C3]. In the high-energy limit the heavy quark correlator is written as the massless one with added quark-mass corrections $$\Pi(s)=Q_{f}^{2}\sum_{n=0}^{\infty}\left(\frac{\alpha_s(\mu^2)}{\pi}\right)^n\Pi^{(n)}(s)\;,$$ where $$\Pi^{(n)}(s)=\sum_{i=0}^{\infty}\left(\frac{\bar{m}^{2}_{f}(\mu)}{s}\right)^i\Pi^{(n)}_i \;.$$ ![[]{data-label="Fig:data"}](Fig3.eps){height="3.0in" width="3.5in"} [lccccccc]{} &\ & $[0-3.7\,\text{GeV}]_{uds}$ & $[3.7-9.3\,\text{GeV}]_{udsc}$ & $[9.3-40\,\text{GeV}]_{udscb}$ & $[>40\,\text{GeV}]_{udscb} $ & $[>40\,\text{GeV}]_t$ & Total\ Standard approach [@davier2011] & $79.29\pm 0.69$ & $60.21\pm 0.51$ & $93.50\pm 0.16$ & $42.70\pm 0.06$ & $-0.72\pm 0.01$ & $275.0\pm 1.0$\ \ This work & $79.39\pm 0.68$ & $60.46\pm 0.33$ & $93.82\pm 0.14$ & $42.76\pm 0.06$ & $-0.76\pm 0.03$ & $275.7\pm 0.8$\ The terms $\Pi^{(0)}$ and $\Pi^{(1)}$ above are known exactly, whilst $\Pi^{(2)}$ is for all practical purposes also known exactly, i.e. the mass corrections up to $\mathcal{O}(m^{60})$ are given in [@QCD2; @Pi2a; @Pi2b; @QCD3]. At order $\mathcal{O}(\alpha_{s}^{3})$, there is only partial information on the mass corrections. The functions $\Pi^{(3)}_0$ and $\Pi^{(3)}_1$ are known exactly [@QCD4], whilst the logarithmic terms of $\Pi^{(3)}_1$ are given in [@QCD5]. The constant term has been estimated using Pade approximants [@QCD6], but it was found to be negligible. At order $\mathcal{O}(\alpha_{s}^{4})$ the logarithmic terms in $\Pi^{(4)}_0$ and $\Pi^{(4)}_1$ are known [@QCD7; @QCD8], but not the constant terms. Hence, for the heavy quarks no $\mathcal{O}(\alpha_{s}^{4})$ terms will be included in any contour integral. However, $\Pi^{(4)}_0$ and $\Pi^{(4)}_1$ will be included when integrating over $\text{Im}(\Pi(s))$, as the unknown constant terms do not contribute in this case. At $\mathcal{O}(\alpha_{s}^{3})$, there are singlet diagrams contributing for the first time (Fig. \[fig:qed\]). The logarithmic terms are known in the high energy case [@QCD5], and we will use these in Eq. to estimate the singlet contribution. An example of a lowest order contribution to $\Pi^{(f)}_{\text{QED}}(s)$ is shown in Fig. \[fig:qed\], after substituting gluons by photons. Finally, the leading-order non-perturbative contribution to $\Pi^{(f)}_{\text{NP}}(s)$ for heavy quarks is from the gluon condensate $\left<(\alpha_s/\pi)G^2\right>$. This has been determined from data on $\tau$-decays [@G2tau], and with a very large uncertainty from data on $e^+ e^-$ annihilation into hadrons [@G2R]. The conservative value $\left<(\alpha_s/\pi)G^2\right>=(0.006\pm 0.012)\,\text{GeV}^4$ will be used in the sequel. To estimate the error arising from the incomplete knowledge of the correlator in PQCD (truncation error), we take the difference between the $\mathcal{O}(\alpha_{s}^n)$ and $\mathcal{O}(\alpha_{s}^{n-1})$ results, where $n$ is the highest available order. We will check this by also varying the scale $\mu$. Finally, many errors will be 100% correlated or anti-correlated between different regions, such as e.g. the error in $\alpha_s$.\ As input, the PDG value of the Z-mass will be used, i.e. $M_Z=91.1876(21)$ [@PDG]. For the strong coupling we use the result from the determination of Davier [*et al.*]{} [@davier2011] $$\label{eq:coupling} \alpha_{s}(M_{Z}^{2})=0.1193(28) \;,$$ in order to facilitate the comparison of the final result for $\Delta\alpha_{\text{HAD}}(M_{Z}^{2})$. This value has a far more conservative error than the PDG value of $\alpha_{s}(M_{Z}^{2})=0.1184(7)$ [@PDG]. However, this will hardly matter as our results turn out to be largely independent of the strong coupling. For the charm- and bottom-quark masses, we use the most recent LQCD determination [@lattice], $\bar{m}_{c}^{(4)}(3\,\text{GeV})=0.986(6)\,\text{GeV}$ and $\bar{m}_{b}^{(5)}(10\,\text{GeV})=3.617(25)\,\text{GeV}$. These values are in very good agreement with QCD sum rule determinations [@bodenstein2011; @bodenstein2012; @kuhn2007]. For the top-quark mass, we use $\bar{m}_{t}^{(5)}(\bar{m}_{t})=160.0(3.5)\,\text{GeV}$ [@topmass]. If one employs either the Adler function or the FESR approach, the coupling and quark masses need to be run across flavor thresholds. This will be done using the Mathematica program `RunDec` [@rundec].\ CHARM-QUARK CONTRIBUTION ======================== We consider first the evaluation of $\Delta\alpha_{\text{HAD}}^{(c)}(M_{Z}^{2})$ directly from Eq. . As mentioned previously, there is a problem with the strong dependence of $\Delta\alpha_{\text{HAD}}^{(c)}(M_{Z}^{2})$ on the renormalization scale $\mu$. In fact, if one varies $\mu$ in the interval $\mu = 2\,\text{GeV} - M_{Z}$ then $\Delta\alpha^{(c)}_{\text{HAD}}(M_{Z}^{2})$ changes by $0.77\times 10^{-4}$, which is a large variation in the context of the current precision of $\Delta\alpha_{\text{HAD}}(M_{Z}^{2})$. We choose instead a value of $\mu$ leading to a good convergence of the perturbative series. Figure \[fig:convergence\] shows the difference between the order $\mathcal{O}(\alpha_{s}^{3})$ and the order $\mathcal{O}(\alpha_{s}^{2})$ results for $4\pi\alpha(\Pi^{(c)}(0)-\text{Re}[\Pi^{(c)}(M_{Z}^{2})])$, i.e. the truncation error $\Delta \text{tr}$, as a function of the renormalization scale $\mu$. A value of $\mu \simeq 3\,\text{GeV}$ ensures good convergence. Using $n_f=5$, this leads to $$\begin{aligned} \Delta\alpha^{(c)}_{\text{HAD}}(M_{Z}^{2})&=&4\pi\alpha\left(\Pi^{(c)}(0)- \,[\Pi^{(c)}(M_{Z}^{2})]\right)\nonumber\\ &=& (79.19\pm 0.13_{\Delta \text{tr}}\pm 0.03_{\Delta \alpha_s} \nonumber\\ &\pm& 0.01_{\Delta\langle G^2\rangle} \pm 0.11_{\Delta\bar{m}_c})\times 10^{-4},\end{aligned}$$ where the errors are due to truncation ($\Delta \text{tr}$), and to uncertainties in $\alpha_s$ ($\Delta \alpha_s$), in the gluon condensate ($\Delta \langle G^2 \rangle$), and in the charm-quark mass ($\Delta \bar{m}_c$). Interestingly, this value corresponds to a global minimum of $\Delta\alpha^{(c)}_{\text{HAD}}(M_{Z}^{2})$ as a function of $\mu$. In any case, given this strong $\mu$-dependence we shall not use this method to determine the charm-quark contribution. We consider instead the Adler function approach using Eq. which involves two terms, a high energy and a low energy contribution. Starting with the low energy part, and using $n_f=4$, $s_0=(9.3\,\text{GeV})^2$ (which is below the bottom-quark threshold), and an initial value $\mu=5\,\text{GeV}$ (to be made to vary in a wide range later), we find $$\begin{aligned} &&4 \,\pi \,\alpha \,(\Pi^{(c)}(0)- \,\Pi^{(c)}(s_0))=(29.57\pm 0.25_{\Delta \text{tr}}\nonumber\\ &&\pm \,0.05_{\Delta \alpha_s} \pm\, 0.01_{\Delta\langle G^2\rangle} \pm \, 0.12_{\Delta\bar{m}_c})\times 10^{-4}\;. \label{eq:adlercharm1}\end{aligned}$$ Allowing now $\mu$ to vary in the wide range $\mu = (2\,- 9.3)\,\text{GeV}$, results in a variation of the central value above in the range $[29.31 - 29.62]\times 10^{-4}$, which is within the truncation error. For the high energy term, we use $n_f=5$ and an initial value $\mu=\frac{1}{2}M_{Z}^{2}$ to obtain $$\begin{aligned} \frac{\alpha}{3\pi} \int^{M_{Z}^{2}}_{s_0}\frac{D^{(c)}(s)}{s}ds &=&(49.91\pm 0.05_{\Delta \text{tr}} \pm \, 0.04_{\Delta \alpha_s} \nonumber\\ &\mp& 0.01_{\Delta\bar{m}_c})\times 10^{-4} \;. \label{eq:adlercharm2}\end{aligned}$$ Varying $\mu$ in the interval $\mu = 9.3\,\text{GeV} - M_Z$, produces a variation of the central value above in the range $[49.82 - 49.95] \times 10^{-4}$. Adding both contributions, Eqs. and , the total result from this approach is $$\begin{aligned} \Delta\alpha^{(c)}_{\text{HAD}}(M_{Z}^{2})&=&(79.49\pm 0.30_{\Delta \text{tr}}\pm 0.09_{\Delta \alpha_s}\nonumber\\ &\pm& 0.01_{\Delta\langle G^2\rangle} \pm 0.11_{\Delta\bar{m}_c})\times 10^{-4} \;,\end{aligned}$$ where we notice that the errors $\Delta\bar{m}_c$ in Eqs. and are anti-correlated. The third method consists in using the FESR Eq.. The values $n_f=4$, and $\mu=5\,\text{GeV}$, have been used both in the contour integral, with $s_0 = (9.3\,\text{GeV})^2$, as well as in the residue $\Pi^{(c)}(0)$. For the integral involving $R^{(c)}(s)$ we use $n_f=5$ (the contribution above the top threshold is not numerically important), and obtain $$\begin{aligned} \label{eq:charmFESR} \Delta\alpha^{(c)}_{\text{HAD}}(M_{Z}^{2})&&=(79.34\pm 0.26_{\Delta \text{tr}}\pm 0.04_{\Delta \alpha_s}\nonumber\\ && \pm 0.01_{\Delta\langle G^2\rangle} \pm 0.11_{\Delta\bar{m}_c})\times 10^{-4} \;.\end{aligned}$$ Varying $\mu$ in the respective regions as done previously produces changes in the central value of $\Delta\alpha^{(c)}_{\text{HAD}}(M_{Z}^{2})$ well within the truncation error. This result will be adopted for the charm-quark contribution as it is has a much smaller uncertainty due to $\alpha_s$, and a slightly smaller truncation error. It should be mentioned that at $s=0$ the low energy expansion, Eq., is a well convergent power series expansion in the strong coupling. In the vicinity of $s=0$, Eq. also converges well, even if the charm-quark is at the borderline between light and heavy quarks. This is a standard procedure in the determinations of the charm-quark mass from QCD sum rules or from LQCD.\ BOTTOM- AND TOP-QUARK CONTRIBUTIONS =================================== In this case all three methods give essentially the same result for $\Delta\alpha^{(b)}_{\text{HAD}}(M_{Z}^{2})$. Using e.g. Eq. with $n_f=5$ and $\mu=10\,\text{GeV}$ gives $$\begin{aligned} \label{eq:beauty} \Delta\alpha^{(b)}_{\text{HAD}}(M_{Z}^{2})&=& 4\pi\alpha\left(\Pi^{(b)}(0)- \,[\Pi^{(b)}(M_{Z}^{2})]\right)\nonumber\\ &=&(12.79\pm 0.06_{\Delta \text{tr}}\pm 0.009_{\Delta \alpha_s} \nonumber\\ &\pm& 0.03_{\Delta\bar{m}_b})\times 10^{-4}\;.\end{aligned}$$ Varying $\mu$ in the very wide range $\mu = 10\,\text{GeV} - 10 M_Z$ only changes this result by $0.04\times 10^{-4}$, which shows a remarkable scale independence.\ For the top quark, one can use the low energy expansion to calculate both $\Pi^{(t)}(0)$ and $\Pi^{(t)}(M_{Z}^{2})$. Up to $\mathcal{O}(\alpha_s)$ the full analytic correlator is known. We have verified that there is no appreciable difference between results using the low-energy expansion of the correlator or using the full expression up to this order to determine $\Pi^{(t)}(M_{Z}^{2})$. At higher orders, one can reconstruct the full analytic behavior of the correlator using Pade approximants, as done in [@QCD6] at order $\mathcal{O}(\alpha^{3}_{s})$. Using these results we find that it is perfectly safe to use the low energy expansion of the correlator. With $\mu=\bar{m}_{t}$ and $n_f=6$, we find $$\begin{aligned} \label{eq:truetop} \Delta\alpha^{(t)}_{\text{HAD}}(M_{Z}^{2})&=& 4 \pi \alpha\left(\Pi^{(t)}(0)- \, \Pi^{(t)}(M_{Z}^{2})\right)\nonumber\\ &=&(-0.76\pm 0.03_{\Delta\bar{m}_t})\times 10^{-4} \;,\end{aligned}$$ where only the uncertainty in the top-quark mass produces a non-negligible uncertainty in $\Delta\alpha^{(t)}_{\text{HAD}}(M_{Z}^{2})$. Authors $\Delta\alpha^{(5)}_{\text{HAD}}(M_{Z}^{2}) ({\mbox{in units of}}\, 10^{-4})$ Method -------------------------------------------- ------------------------------------------------------------------------------- --------------------------------------------------------------------------------- Groote *et al.* (1998) [@groote1998] 277.6(4.1) PQCD driven (using polynomial weights and global duality) Kühn *et al.* (1998) [@kuhn1998] 277.5(1.7) PQCD driven (using polynomial weights and global duality) Burkhardt *et al.* (2011) [@burkhardt2011] 275.0(3.3) Data driven Trocóniz *et al.* (2005) [@troconiz2005] 274.9(1.2) Data driven Jegerlehner (2008) [@jegerlehner2008] 275.94(2.19) Data driven Jegerlehner (2011) [@jegerlehner2011] 274.98(1.35) Adler function approach Hagiwara *et al.* (2011) [@hagiwara2011] 276.26(1.38) Data driven Davier *et al.* (2011) [@davier2011] 275.7(1.0) PQCD in range $1.8<\sqrt{s}<3.7\,\text{GeV}$, and for $\sqrt{s}>5\,\text{GeV}$, otherwise data. This work 276.5(0.8) Data for $\sqrt{s}<1.8\,\text{GeV}$, and PQCD for $\sqrt{s}>1.8\,\text{GeV}$. This work $\sim 273$ LQCD + PQCD LIGHT-QUARK CONTRIBUTION ======================== In contrast to the heavy quark contributions, one cannot use PQCD to determine the light-quark correlator at low energies. There are two approaches to achieve this, i.e. using $e^+e^-$ data for $R(s)$, or LQCD determinations of $\Pi(s)$ (in the space-like region), with both being used in the sequel. For the $e^+e^-$ data approach, and below the onset of PQCD, we use the integrated result of [@davier2011] to avoid the complicated task of dealing with the vast amount of $e^+e^-$ data available (for a recent independent analysis see [@G2R]). The result of [@davier2011], integrated up to the PQCD threshold $\sqrt{s}=1.8\,\text{GeV}$ is $$\begin{aligned} \label{eq:light} \frac{\alpha M_{Z}^{2}}{3 \pi} & \int^{(1.8 \text{GeV})^2}_{0}\frac{R^{(uds)}_{\text{data}}(s)}{s(M_{Z}^{2}-s)}ds \nonumber\\ &= (55.02\pm 0.66)\times 10^{-4} \;.\end{aligned}$$ Above the PQCD threshold, and using the massless order $\mathcal{O}(\alpha_{s}^{4})$ PQCD expression for $R^{(uds)}(s)$ we find $$\begin{aligned} &&\frac{\alpha M_{Z}^{2}}{3\pi}\int^{\infty}_{(1.8\,\text{GeV})^2}\frac{R^{(uds)}_{\text{PQCD}}(s)}{s(M_{Z}^{2}-s)}ds \nonumber\\ &=& (129.26\pm 0.16_{\Delta \text{tr}} \pm 0.29_{\Delta \alpha_s})\times 10^{-4}, \end{aligned}$$ which added to Eq. gives the total light-quark contribution $$\begin{aligned} \label{eq:totaluds} \Delta\alpha^{(uds)}_{\text{HAD}}(M_{Z}^{2})&=&(184.28\pm 0.66_{\text{data}}\pm 0.16_{\Delta \text{tr}}\nonumber\\ &\pm& 0.29_{\Delta \alpha_s})\times 10^{-4}\;.\end{aligned}$$ A large effort is currently underway to determine $\Pi^{(uds)}(s)$ in the space-like region using LQCD. A key aim is to provide a first-principles determination of the hadronic contribution to the $g-2$ of the muon [@Blum; @Aubin; @boyle2011; @renner2011; @wittig2012]. Here we describe two methods for obtaining $\Delta\alpha^{(uds)}_{\text{HAD}}(M_{Z}^{2})$, entirely from theory, from a combination of LQCD and PQCD. This is inspired by [@g-2BOD], where an entirely theoretical determination of $g-2$ was proposed. It must be emphasized, though, that LQCD results are currently not precise enough to compete with the $e^+e^-$ approach. For instance, one source of uncertainty arises from disconnected Feynman diagrams, which are currently not included in LQCD calculations, and which lead to an estimated 10% systematic uncertainty [@boyle2011].\ The first method is based on the FESR Eq., with $\Pi^{(uds)}(0)$ determined from LQCD, and the two integrals computed in PQCD. In LQCD it is not possible to calculate directly $\Pi^{(uds)}(0)$. Instead, the correlator is computed for values very close to $s=0$, and then these results are fitted and extrapolated to the origin to obtain $\Pi_{uds}(0)$. For instance, the phenomenologically inspired fitting function used in [@boyle2011] is of the form $$\label{eq:fit} \Pi_{uds}(s)=A-\frac{F_{1}^{2}}{m_{1}^{2}-s}-\frac{F_{2}^{2}}{m_{2}^{2}-s} \;,$$ with fit parameters given in [@boyle2011]. An alternative, model independent approach to extrapolating LQCD data is based on Pade approximants [@Aubin2]. This approach would be appropriate in future precision determinations based on improved LQCD data. An important observation is that neither $\Pi_{uds}(0)$ nor the contour integral in Eq. are observable quantities. Therefore, it is essential to compute both of these quantities in the same renormalization scheme, and at the same scale, so that the observable difference between $\Pi_{uds}(0)$ and the contour integral is scheme-independent. A problem arises because PQCD schemes, such as $\overline{\text{MS}}$, are not easy to relate to LQCD renormalization schemes. The latter lead to a prediction of $\Pi^{(uds)}(s)$ which differs from the $\overline{\text{MS}}$ results by the constant $\Pi^{(uds)}(0)$, which is precisely what is needed in Eq.. The standard approach to fix this constant is to impose agreement between PQCD and LQCD results at some value $s = -s^*$ where PQCD is expected to be valid. This procedure would then allow for a determination of $\Pi^{(uds)}(0)$ from LQCD. Above $s \simeq - 2 \, {\mbox{GeV}}^2$ LQCD results already are in agreement with PQCD, so to determine $\Pi^{(uds)}(0)$ we choose $s^*= - 3.5\,\text{GeV}^2$ to be on the safe side, together with $s_0=(3.72\,\text{GeV})^2$ which corresponds to the onset of the charm-quark region, and $n_f=3$. The renormalization scale was varied in the wide range between the $\tau$-lepton mass and the charm threshold, i.e. $\mu=1.77 - 3.7\,\text{GeV}$. This produces a negligible change in $\Delta\alpha^{(uds)}_{\text{HAD}}(M_{Z}^{2})$ of $0.05 \times 10^{-4}$. There are eight available LQCD fits [@boyle2011] for different pion masses, with each giving a result for $\Pi^{(uds)}(0)$ in need of extrapolation to the actual physical value of the pion mass. Using a simple linear extrapolation fit gives the results shown in Fig. \[Fig:data\], leading to $$\Pi^{(uds)}(0)\sim 0.08758 \ \ (\mu=2\,\text{GeV}) \;.$$ Using this value in Eq., together with a PQCD evaluation of the integrals, gives $$\label{eq:uds} \Delta\alpha^{(uds)}_{\text{HAD}}(M_{Z}^{2})\sim 181 \times 10^{-4} \;.$$ No error is given above in view of the current uncertainties in LQCD.\ The second method to determine $\Delta\alpha^{(uds)}_{\text{HAD}}(M_{Z}^{2})$ entirely from theory makes use of the Adler function. One important advantage over the FESR approach is that there is no need to enforce the matching between PQCD and LQCD at any point. In order to involve the Adler function we write $\Delta\alpha^{(uds)}_{\text{HAD}}(M_{Z}^{2})$ as $$\begin{aligned} \label{eq:last} \Delta\alpha^{(uds)}_{\text{HAD}}(M_{Z}^{2})&\equiv&\Delta\alpha^{(uds)}_{\text{HAD}}(-s_0)+\bigl[\Delta\alpha^{(uds)}_{\text{HAD}}(s_0) - \Delta\alpha^{(uds)}_{\text{HAD}}(-s_0)\bigr] +\bigl[\Delta\alpha^{(uds)}_{\text{HAD}}(M_z^2)-\Delta\alpha^{(uds)}_{\text{HAD}}(s_0)\bigr]\nonumber\\ &=& 4\pi\alpha \;\left[\Pi^{(uds)}_{\text{LQCD}}(0)-\Pi^{(uds)}_{\text{LQCD}}(-s_0)\right] +\frac{\alpha}{3\pi} \int^{s_0}_{-s_0}\frac{D^{(uds)}_{\text{PQCD}}(s)}{s}ds +\frac{\alpha}{3\pi} \int^{M_{Z}^{2}}_{s_0}\frac{D^{(uds)}_{\text{PQCD}}(s)}{s}ds ,\end{aligned}$$ where $s_0$ is large enough for PQCD be valid, and the real part of the expression in the last line is to be understood. Notice that the line integral in the interval $(-s_0, + s_0)$ is well defined. In fact, since $D(s)$ is an analytic function the integration was performed on a semi-circular contour of radius $|s_0|$, avoiding the origin. Evaluating each of the three terms on the right hand side of Eq., with $s_0 = s^* = - 3.5 \; {\mbox{GeV}}^2$, we find $$\label{eq:LQCDuds} 4\pi\alpha \;\left[\Pi^{(uds)}_{\text{LQCD}}(0)-\Pi^{(uds)}_{\text{LQCD}}(-s_0)\right] \simeq 52.32 \, \times 10^{-4}\,,$$ $$\frac{\alpha}{3\pi} \int^{s_0}_{-s_0}\frac{D^{(uds)}_{\text{PQCD}}(s)}{s}ds = 2.56 \, \times 10^{-4} \,,$$ $$\frac{\alpha}{3\pi} \int^{M_{Z}^{2}}_{s_0}\frac{D^{(uds)}_{\text{PQCD}}(s)}{s}ds = 125.95 \, \times 10^{-4} \,,$$ which add up to $$\label{eq:total2} \Delta\alpha^{(uds)}_{\text{HAD}}(M_{Z}^{2}) = 181 \times 10^{-4}\,,$$ as already given in Eq.. The errors in the total PQCD contributions are $$\begin{aligned} \label{eq:errorPQCD} \frac{\alpha}{3\pi}& \int^{s_0}_{-s_0}\frac{D^{(uds)}_{\text{PQCD}}(s)}{s}ds + \frac{\alpha}{3\pi} \int^{M_{Z}^{2}}_{s_0}\frac{D^{(uds)}_{\text{PQCD}}(s)}{s}ds \nonumber\\ &= (128.5 \pm 0.2_{\Delta \text{tr}} \pm 0.3 _{\Delta \alpha_s}) \times 10^{-4} \,.\end{aligned}$$ The contribution of the gluon condensate is at the level of one order of magnitude smaller than the uncertainty in $\alpha_s$, hence it can be neglected. Once LQCD determinations of Eq. achieve enough accuracy, it would become possible to determine $\Delta\alpha_{\text{HAD}}(M_{Z}^{2})$ entirely from theory, after adding to the LQCD light-quark contribution the heavy-quark results Eqs., , and . CONCLUSIONS =========== Adding up all of the contributions, i.e. Eqs., , , and , the final result for $\Delta\alpha_{\text{HAD}}(M_{Z}^{2})$ is $$\begin{aligned} \label{eq:finalf} \Delta\alpha_{\text{HAD}}(M_{Z}^{2})&=& \left(275.7 \pm 0.66_{\Delta \text{data}} \pm 0.44_{\Delta \text{tr}} \pm 0.26_{\Delta \alpha} \right.\nonumber \\ &\pm& \left. 0.11_{\Delta \bar{m}_c}\right)\times 10^{-4} \nonumber\\ &=& (275.7\pm 0.8)\times 10^{-4} \;,\end{aligned}$$ where $n_f = 6$ has been used, and the uncertainties due to the bottom- and the top-quark masses, and due to the gluon condensate, are negligible. This result can be compared with $\Delta\alpha_{\text{HAD}}(M_{Z}^{2})= (275.0\pm 1.0)\times 10^{-4}$ from [@davier2011] (for $n_f=6$)(other five quark flavor results are listed in Table II). The primary reason for this $20\%$ reduction in uncertainty is our PQCD calculation of the contribution of the charm-quark resonance region, which is given in Table \[Tab:results\].\ We comment in closing on the relation between $\Delta\alpha_{\text{HAD}}(M_{Z}^{2})$ and the value of the Higgs mass. To begin with, as is well known both $\Delta\alpha_{\text{HAD}}(M_{Z}^{2})$ and the QCD strong coupling $\alpha_s(M_Z^2)$ enter into the global SM fit to electroweak precision data. At first sight one could suspect that the central role played by PQCD in our approach could lead to a stronger correlation between $\Delta\alpha_{\text{HAD}}(M_{Z}^{2})$ and $\alpha_s(M_Z^2)$. However, this is not the case. In fact, quite the contrary, i.e. in comparison with the standard approach our FESR method reduces the overall dependence of $\Delta\alpha_{\text{HAD}}(M_{Z}^{2})$ on $\alpha_s(M_Z^2)$. This happens because the $\alpha_s(M_Z)$ dependence of the combination of contour integral plus Cauchy residue in Eq. is anti-correlated with the $\alpha_s(M_Z)$ dependence of the integral involving $R(s)$. Hence, there is some cancellation of the $\alpha_s(M_Z)$ dependence, which does not take place in the standard approach which uses data to determine the heavy-quark resonance contribution. Quantitatively, the functional dependence of the central value of $\Delta\alpha_{\text{HAD}}(M_{Z}^{2})$ on the value of $\alpha_s(M_Z)$ in the standard approach [@davier2011] is approximately $132.1 [\alpha_s(M_Z) - 0.1193] \times 10^{-4}$, while in our method it becomes approximately $107.7 [\alpha_s(M_Z) - 0.1193] \times 10^{-4}$.\ The correlation between $\Delta\alpha_{\text{HAD}}^{(5)}(M_{Z}^{2})$ and the logarithm of the Higgs mass, $\ln{M_H}$, was found in [@HM] to be $-0.395$ (for an earlier determination see [@HM0]). Using our result for $\Delta\alpha_{\text{HAD}}^{(5)}(M_{Z}^{2})$ in Table II would lead to a Higgs mass $M_H \simeq 87 \, {\mbox{GeV}}$, somewhat lower than the value from [@davier2011] $M_H \simeq 91^{+30}_{-23} \, {\mbox{GeV}}$, thus increasing the tension between a possible Higgs of mas $M_H \simeq 126 \, {\mbox{GeV}}$, and the fitted Higgs mass. ACKNOWLEDGMENTS =============== This work was supported in part by the National Research Foundation (South Africa) and by the Alexander von Humboldt Foundation (Germany). The authors thank A. Denig, M. Fritsch, and F. Jegerlehner for helpful discussions. [99]{} S. Chatrchyan [*et al.*]{}, CMS Coll., Phys.Lett. B [**716**]{}, 30 (2012). G. Aad [*et al.*]{}, ATLAS Coll., Phys.Lett. B [**716**]{}, 1 (2012). M. Davier, A. Hoecker, B. Malaescu and Z. Zhang, Eur. Phys. J. C [**71**]{}, 1515 (2011). N. Cabibbo and R. Gatto, Phys. Rev. [**124**]{}, 1577 (1961). K. Hagiwara, A. Hoecker, R. Liao, A. Martin, D. Nomura and T. Teubner, J. Phys. 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=6.0in =8.5in =.1in 0.2in 0.2in 0.2in Abstract 0.2in The production of single photons in relativistic heavy ion collisions at CERN SPS, BNL RHIC and CERN LHC energies is re-examined in view of the recent studies of Aurenche et al which show that the rate of photon production from quark gluon plasma, evaluated at the order of two loops far exceeds the rates evaluated at one-loop level which have formed the basis of all the estimates of photons so far. We find that the production of photons from quark matter could easily out-shine those from the hadronic matter in certain ideal conditions. Single photons can be counted among the first signatures [@first] which were proposed to verify the formation of deconfined strongly interacting matter- namely the quark gluon plasma (QGP). Along with dileptons- which will have similar origins, they constitute electro-magnetic probes which are believed to reveal the history of evolution of the plasma, through a (likely) mixed phase and the hadronic phase, as they do not re-scatter once produced and their production cross section is a strongly increasing function of temperature. During the QGP phase, the single photons are believed to originate from Compton ($q\,(\overline{q})\,g\,\rightarrow\,q\,(\overline{q})\,\gamma$) and annihilation ($q\,\overline{q}\,\rightarrow\,g\,\gamma$) processes [@joe; @rolf] as well as bremsstrahlung processes ($q\,q\,(g)\,\rightarrow\,q\,q\,(g)\,\gamma$). Recently in the first evaluation of single photons within a parton cascade model [@pcm], it was shown [@pcmphot] that the fragmentation of time-like quarks ($q\,\rightarrow\,q\,\gamma$) produced in (semi)hard multiple scatterings during the pre-equilibrium phase of the collision leads to a substantial production of photons (flash of photons!), whose $p_T$ is decided by the $Q^2$ of the scatterings and not the temperature as in the above mentioned calculations. The upper limit for production of single photons in $S+Au$ collisions at SPS energies [@wa80] has been used by several authors to rule out simple hadronic equations of states [@prl] and the final results for the $Pb+Pb$ collisions at SPS energies are eagerly awaited. In a significant development Aurenche et al [@pat] have recently evaluated the production of photons in a QGP up to two loops and shown that the bremsstrahlung process gives a contribution which is similar in magnitude to the Compton and annihilation contributions evaluated up to the order of one loop earlier [@joe; @rolf]. This is in contrast to the ‘expectations’ that the bremsstrahlung contributions drop rapidly with energy (see Ref. [@kryz; @dipali] for estimates within a soft photon approximation). They also reported an entirely new mechanism for the production of hard photons through the annihilation of an off-mass shell quark and an anti-quark, where the off-mass shell quark is a product of scattering with another quark or gluon and which completely dominates the emission of hard photons. This process is similar to the annihilation of quarks in the presence of the chromo-electric field which may develop when two nuclei pass through each other due to colour exchange, and which can absorb the unbalanced energy and momentum to ensure the feasibility of the process which is absent in vacuum [@avijit]. If confirmed, this has far reaching consequences for the search of single photons from the relativistic heavy ion collisions. The rate for the production of hard photons evaluated to one loop order using the effective theory based on resummation of hard thermal loops is given by [@joe; @rolf]: $$E\frac{dN}{d^4x\,d^3k}=\frac{1}{2\pi^2}\,\alpha\alpha_s\, \left(\sum_f e_f^2\right)\, T^2\, e^{-E/T}\,\ln(\frac {cE}{\alpha_s T})$$ where the constant $c\approx$ 0.23. The summation runs over the the flavours of the quarks and $e_f$ is the electric charge of the quarks in units of charge of the electron. The rate of production of photons due to the bremsstrahlung processes evaluated by Aurenche et al is given by: $$E\frac{dN}{d^4x\,d^3k}=\frac{8}{\pi^5}\,\alpha\alpha_s\, \left(\sum_f e_f^2\right)\, \frac{T^4}{E^2}\, e^{-E/T}\,(J_T-J_L)\,I(E,T)$$ where $J_T\approx$ 4.45 and $J_L\approx - $4.26 for 2 flavours and 3 colour of quarks. For 3 flavour of quarks, $J_T\approx$ 4.80 and $J_L\approx - $4.52. $I(E,T)$ stands for; $$\begin{aligned} I(E,T)&=&\left[ 3\zeta(3)+\frac{\pi^2}{6}\frac{E}{T}+ \left(\frac{E}{T}\right)^2\ln(2)\right.\nonumber\\ & &+4\,Li_3(-e^{-|E|/T})+2\,Li_2(-e^{-|E|/T})\nonumber\\ & &\left. -\left(\frac{E}{T}\right)^2\,\ln(1+e^{-|E|/T})\right]~,\end{aligned}$$ and the poly-logarith functions $Li$ are given by; $$Li_a(z)=\sum_{n=1}^{+\infty}\frac{z^n}{n^a}~~.$$ And finally the contribution of the $q\overline{q}$ annihilation with scattering obtained by them is given by: $$E\frac{dN}{d^4x\,d^3k}=\frac{8}{3\pi^5}\,\alpha\alpha_s\, \left(\sum_f e_f^2\right)\, ET \, e^{-E/T}\,(J_T-J_L)$$ We plot these rates of emission of photons from a QGP at $T=$ 250 MeV (Fig. 1) for an easy comparison. The dashed curve gives the contribution of the Compton and annihilation processes evaluated to the order of one loop by Kapusta et al [@joe], the dot-dashed curve gives the bremsstrahlung contribution evaluated to two-loops by Aurenche et al [@pat] while the solid curve gives the results for the annihilation with scattering evaluated by the same authors. The dotted curve gives the results for the bremsstrahlung contribution evaluated within a soft-photon approximation (and using thermal mass for quarks and gluons) obtained by Pal et al [@dipali]. We see that at larger energies the annihilation of quarks with scattering really dominates over the rest of the contributions by more than a order of magnitude. How much of this dominance does survive when we integrate the radiation of photons over the history of evolution of the system, specially as the QGP phase occurring in the early stages of the evolution necessarily occupies smaller four-volume compared the hadronic matter, which is known to have an emission rate similar to the quark matter at a given temperature [@joe] at least when only the Compton and the annihilation terms are used? In order to ascertain this we consider central collision of lead nuclei at SPS, RHIC and LHC energies. We assume that a chemically and thermally equilibrated quark-gluon plasma is formed at $\tau_0=$ 1 fm/$c$ at SPS and at 0.5 fm/$c$ at RHIC and LHC energies. While there are indications that the plasma produced at the energies under consideration may indeed attain thermal equilibrium at around $\tau_0$ chosen here [@pcm; @therm], it is not quite definite that it may be chemically equilibrated. It may be recalled that the parton cascade model which properly accounts for multiple scatterings uses a cut-off in momentum transfer and virtuality to regulate the divergences in the scattering and the branching amplitudes for partons. This could underestimate the extent of chemical equilibration, by a cessation of interactions when the energy of the partons is still large which would not be the case if the screening of the partonic interactions could be accounted for. The self-screened parton cascade [@sspc] on the other hand attempts to remove these cut-offs by estimating the screening offered by the partons which have larger $p_T$ (and hence materialize earlier) to the partons which have smaller $p_T$ (and hence materialize later). However it does not explicitly account for multiple scattering except for what is contained in the Glauber approximation utilized there. In these exploratory calculations we assume a chemical equilibration at the time $\tau_0$ such that the initial temperature is obtained from the Bjorken condition [@bj]; $$\frac{2\pi^4}{45\zeta(3)}\,\frac{1}{\pi R_T^2}\frac{dN}{dy}=4 a T_0^3\tau_0$$ where we have chosen the particle rapidity densities as 825, 1734, and 5625 respectively at SPS, RHIC, and LHC energies for central collision of lead nuclei [@kms] and taken $a=47.5\pi^2/90$ for a plasma of mass-less quarks (u, d, and s) and gluons. We assume the phase transition to take place at $T=$ 160 MeV, and the freeze-out to take place at 100 MeV. We use a hadronic equation of state consisting of all the hadrons and resonances from the particle data table which have a mass less then 2.5 GeV [@jean]. The rates for the hadronic matter have been obtained [@joe] from a two loop approximation of the photon self energy using a model where $\pi-\rho$ interactions have been included. The contribution of the $A_1$ resonance is also included according to the suggestions of Xiong et al [@li]. The relevant hydrodynamic equations are solved using the procedure [@hydro] discussed earlier and a integration over history of evolution is performed [@jean]. In Fig. 2 we show our results for central collision of lead nuclei at energies which are reached at CERN SPS. We give the contribution of the quark matter (from the QGP phase and the mixed phase) labeled as QM and that of the hadronic matter (from the mixed phase and the hadronic phase) separately. We see that if we use the rates obtained earlier by Kapusta et al, there is no window when the radiations from the quark-matter could shine above the contributions from the hadronic matter. However, once the newly obtained rates are used we see that the quark matter may indeed out-shine the hadronic matter up to $p_T=$ 2 GeV, from these contributions alone. Note that by tracking the history from $\tau_0$= 1 fm/$c$ onward, we have not included the pre-equilibrium contributions [@pcmphot] which will make a large contribution at higher momenta. The contribution of hard QCD photons [@qcd] obtained by scaling the results for $pp$ collisions by the nuclear thickness. The results for RHIC energies (Fig. 3) are quite interesting as now the window over which the quark matter out-shines the hadronic contributions stretches to almost 3 GeV. Once again the addition of the pre-equilibrium contributions at larger $p_T$ would substantially widen this window. At LHC energies this window extends to beyond 4 GeV, and considering that perhaps the local thermalization at LHC (and also at RHIC) could be attained earlier than what is definitely a very conservative value here, these results provide the exciting possibility that if these conditions are met the quark matter may emit photons which may be almost an order of magnitude larger than those coming from the hadronic matter over a fairly wide window. As mentioned earlier, the pre-equilibrium contribution (due to the very larger initial energy) should be much larger here and we may have the exciting possibility that the quark matter may out-shine the hadronic matter over a very large window indeed. How will the results change if the QGP is not in chemical equilibrium? While it is not easy to perform the estimates similar to the one done by Aurenche et al for a chemically non-equilibrated plasma, it is reasonable to assume that the rates will fall simply because then the number of quarks and gluons will be smaller. Some of this short-fall will be off-set by the much larger temperatures which the parton cascade models predict. If one considers a chemically equilibrating plasma [@smm] then the quark and gluon fugacities will increase with time and at least the contributions from the latter stages will not be strongly suppressed. It is still felt that the loss of production of high $p_T$ (from early times) photons due to chemical non-equilibration would be more than off-set by the increased temperature and the pre-equibrium contribution, which can be quite large. We conclude that the newly obtained rates for emission of photons from QGP (evaluated to the order of two loops) suggest that if chemically equilibrated plasma is produced then there will exist a fairly wide window where the photons from quark matter may outshine the photons from hadronic matter. Even in the absence of chemical equilibration these results indicate an enhanced radiation from the quark matter which is of considerable interest. Acknowledgments {#acknowledgments .unnumbered} =============== The author gratefully acknowledges the hospitality of University of Bielefeld where part of this work was done. He would also like to acknowledge useful discussions with Jean Cleymans and Francois Gelis. He is especially grateful to Haitham Zaraket for suggesting that the exact expression for the bremsstrahlung contribution be used. [37]{} E. L. Feinberg, Nuovo Cim. A [**34**]{}, 391 (1976); E. V. Shuryak, Phys. Lett. [**78**]{}, 150 (1978). J. Kapusta, P. Lichard, and D. 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--- abstract: | The effect of one-gluon-exchange (OGE) pair-currents on the ratio $\mu_p G_E^p/G_M^p$ for the proton is investigated within a nonrelativistic constituent quark model (CQM) starting from $SU(6) \times O(3)$ nucleon wave functions, but with relativistic corrections. We found that the OGE pair-currents are important to reproduce well the ratio $\mu_p G_E^p/G_M^p$. With the assumption that the OGE pair-currents are the driving mechanism for the violation of the scaling law we give a prediction for the ratio $\mu_n G_E^n/G_M^n$ of the neutron. author: - 'Murat M. Kaskulov' - Peter Grabmayr title: 'Effect of gluon-exchange pair-currents on the ratio $\mu_p G_E^p/G_M^p$' --- [*Introduction:*]{}    Recently the ratio $\mu_p G_E^p/G_M^p$ between the electric $G^p_E(Q^2)$ and magnetic $G^p_M(Q^2)$ form factors of the proton has been extracted from experimental data on the recoil proton polarization in elastic electron scattering with polarized electrons up to $Q^2\sim$5 GeV$^2$ [@Milbrath1999; @Jones2000; @Gayou2001; @Gayou2002]. These experiments are of importance because they are direct measurements of the form factor ratio, and the present results are in contradiction to previous analyses [@Milbrath1999; @Jones2000; @Gayou2001; @Gayou2002; @Brash2002]. Historically, the determination of the electric and magnetic form factors up to several GeV were based on the Rosenbluth separation, and they were found compatible with the scaling laws: $$\label{scaling} G_E^p(Q^2) = G_M^p(Q^2)/{\mu_p} = G_{D}(Q^2)\ \ .$$ where $G_{D}(Q^2)$ represents the dipole form factor. The form factors and particularly the ratio give insight to the main features of the dynamical processes and are very useful for a test of the nucleon models [@Thomas_book]. The remarkable feature of the new experimental data is that they show a decrease of the ratio $\mu_{p} G_E^p/G_M^p$ from unity, indicating a significant deviation from this simple scaling law, but also from the simple constituent quark model. Within different hadronic models the calculations for the proton ratio $\mu_p G_E^p/G_M^p$ became available, with Ref. [@Frank2] presenting one of the earliest. We will restrict this discussion to the most recent calculations which agree reasonably well with the trend of the experimental data and which will allow to make predictions at higher $Q^2$ than the present data. In the cloudy bag model (CBM) [@Thomas], the pion field required by chiral symmetry is quantized and coupled to the MIT bag [@MIT1]. Addition of the pion cloud improves the MIT bag model results [@Lu1], in which the decrease of $\mu_p G_E^p/G_M^p$ is an inherent property. It was shown for a CBM formulated on the light cone [@Miller], that the combination of Poincaré invariance and pion effects is sufficient to describe $\mu_p G_E^p/G_M^p$. Several groups have studied different effects within CQMs. In the Goldstone boson exchange CQM [@GlozmanRiska] the baryon is considered as a system of three constituent quarks with an effective $qq$ hyperfine interaction mediated by the octet of pseudoscalar mesons. This model together with the point-form spectator approximation [@GlozmanBoffi], which provides a covariant framework, leads to a rather close description of the nucleon form factors and the available $\mu_p G_E^p/G_M^p$ data. Calculations of Ref. [@Cardarelli] performed within CQM and light-front formalism, showed that a suppression of the ratio can be expected in the CQM, if the relativistic effects generated by kinematical $SU(6)$ breaking due to the Melosh rotation of the constituent spins are taken into account. Finally, the most recent calculations based on relativistic quark models are from Ref. [@MillerFrank], where the hadron helicity nonconservation induced by the Melosh transformation was recognised to affect the ratio. The implementation of relativity is an common feature of all these works and all emphasize the necessity of both kinematical and dynamical relativistic corrections for the interpretations of the decrease of the ratio $\mu_p G_E^p/G_M^p$. In the non-relativistic constituent quark model (NRCQM) [@Isgur], the effective degrees of freedom are the massive quarks moving in a self-consistent potential whose specific form is dictated by considerations of QCD. Other degrees of freedom like Goldstone bosons or gluons are not considered in the original version and effectively absorbed into the constituent quarks. Theoretically, the explicit introduction of the additional degrees of freedom in the nucleon structure will change its properties compared to expectations based on simple quark models in which the baryon is described as a three-quark state only. Among different improvements to the naive CQM which could be essential for dynamical properties of the nucleons, the most important ones are relativistic kinematical corrections, the introduction of a mesonic cloud via pion-loop corrections, and dynamical corrections due to the interaction currents and to the creation of quark-antiquark ($q \bar q$) pairs. For low momentum transfer, $q \bar q$ pairs (sea-quarks) are dominant and the mesonic degrees of freedom become increasingly important. However, in a recent study [@Geiger] on “un-quenching” the quark model, strong cancellations between the hadronic components of the $q \bar q$ sea were found which tend to make the nucleon transparent to photons. These studies provide a natural way of understanding the success of the valence quark model even though the $q\bar q$ sea is very strong. At higher momentum transfer and in the presence of residual $qq$ interaction, the e.m. operators must be supplemented by the two-body exchange currents. The inclusion of two-body terms leads beyond the single-quark impulse approximation, and in dependence on the model for the $qq$ interaction effectively represents the gluonic or mesonic exchange degrees of freedom in the e.m. current operator. In this sense the physical picture should be similar to nuclear physics, where at low momentum transfer the nucleons are reasonable degrees of freedom, but at higher momentum transfer the meson-exchange currents play a prominent role [@Kaskulov:2002mc]. In this work we continue our studies [@Grabmayr1] of the possible role of interaction currents, in particular OGE pair-currents, for the e.m. properties of the nucleon. We use the NRCQM with relativistic corrections, coming from the Lorentz boost of the nucleon wave function, together with gluonic corrections for the calculation of the proton ratio $\mu_{p} G_E^p/G_M^p$ at momentum transfers beyond 1 GeV$^2$, where effects of the soft pionic cloud should be less important. We show that gluonic corrections to the CQM are important, and that the ratio $\mu_{p} G_E^p/G_M^p$ is well reproduced by the $SU(6) \times O(3)$ wave function of the nonrelativistic quark model. [*The nucleon in the NRCQM:*]{}    In the quark model, baryons are considered as three-quark configurations. The ground state has positive parity with all three quarks in their lowest state, and the total angular momentum (isospin) of baryons is obtained by appropriately combining the quark spins (isospins). In the NRCQM [@Isgur] a baryon is treated as a non-relativistic three-quark system, and in the simplest case of equal quark masses $m_q$ it is described by the Hamiltonian: $$\begin{aligned} \label{H3q} \mathcal{H}_{3q} &=&~~{\displaystyle}\sum_{i=1}^{3} \Big(m_q + \frac{{\displaystyle}{\bf p}_i^2}{{\displaystyle}2 m_q} \Big) - \ \frac{{\displaystyle}{\bf P}^2}{{\displaystyle}6 m_q} \nonumber \\ && + {\displaystyle}\sum_{i<j}^{3} V^{(conf)}({\bf r}_i,{\bf r}_j) + {\displaystyle}\sum_{i<j}^{3} V^{(res)}({\bf r}_i,{\bf r}_j) \end{aligned}$$ where ${\bf r}_i$, ${\bf p}_i$ are the spatial and momentum coordinates of the $i$-th quark, respectively, and [**P**]{} is the centre-of-mass momentum. The Hamiltonian ${\cal H}_{3q}$ consists of the nonrelativistic kinetic energy, a confinement potential $V^{(conf)}$, and a residual interaction $V^{(res)}$. Here, we take a two-body harmonic oscillator (h.o.) confinement potential: $V^{(conf)}({\bf r}_i,{\bf r}_j) ~\sim~ {\bf \lambda}_i \cdot {\bf \lambda}_j ( {\bf r}_i - {\bf r}_j)^2,~ $where ${\bf \lambda}_i$ are the Gell-Mann colour matrices of the $i$-th quark, with $\left<{\bf \lambda}_i \cdot {\bf \lambda}_j\right> = -8/3$ for a $qq$ pair in a baryon. The phenomenological residual interaction $V^{(res)}$ can be based on various $qq$ potentials [@GlozmanRiska; @Isgur], which reflect the symmetries and properties of QCD. Up to now, its dynamical origin is rather uncertain. We use a standard OGE interaction, the strength of which is determined by the strong coupling constant $\alpha_s$. However, unlike perturbative QCD, where the strong coupling constant $\alpha_s$ goes to zero at large inter-quark momenta, we take $\alpha_s$ of the NRCQM as an effective momentum independent constant. We start from the simplest form of the NRCQM, i.e. without configuration mixing, in which the nucleon $| N \rangle$ is described by the lowest h.o. three quark configurations $(0s)^3[3]_X$ in the translationally-invariant shell model (TISM): $$| N \rangle = \Big|(0s)^3 [3]_X L=0, ST = \frac{1}{2} \frac{1}{2} [3]_{ST},~ J^P = \frac{1}{2}^{+} \Big\rangle$$ where the colour part is omitted. After having removed the centre-of-mass coordinate ${\bf R}$ from the TISM configuration, the ground state eigenfunction depends only on the Jacobi relative coordinates ${\bf \rho}_1$ and ${\bf \rho}_2$ of the quarks: $$| (0s)^3 ({\bf \rho}_1,{\bf \rho}_2) \rangle \sim \exp \left( - \frac{1}{4 b^2} {\bf \rho}^2_1 - \frac{1}{3 b^2} {\bf \rho}^2_2 \right)$$ where the constant $b$ determines the average hadronic size of the baryon. Note that the elimination of ${\bf R}$ is crucial for correctly counting the baryonic states. This is one reason why the nonrelativistic approach is so successful in spectroscopy. [*The nucleon e.m. Sachs form factors:*]{}    The nucleon e.m. form factors are functions of the square of the momentum transfer in the scattering process ${Q}^2 = -q^{\mu} q_{\mu}$. The Sachs form factors, $G_{E(M)}$, fully characterize the charge and current distributions inside the nucleon [@Sachs1] and can be written in terms of Dirac and Pauli form factors $\mathcal{F}_1$ and $\mathcal{F}_2$, respectively. The most general form of the nucleon e.m. operator $J^{\mu}_{em}(x)$, which defines $\mathcal{F}_1$ and $\mathcal{F}_2$, satisfies the requirements of relativistic covariance and the condition of gauge invariance; it is of the form $$\begin{aligned} \langle N(p',s') | J^{\mu}_{em}(0) | N(p,s) \rangle &=& \\ \nonumber \bar{u}({\bf p}',s') \Big[ \gamma^{\mu} \mathcal{F}_{1}(Q^2) &+& i\frac{\sigma^{\mu\nu}q_{\nu}}{2 M_N}\mathcal{F}_{2}(Q^2) \Big] u({\bf p}, s) ,\end{aligned}$$ with $q^{\nu} = p'^{\nu} - p^{\nu}$. The Breit frame, where the incoming momentum ${\bf p} = - {\bf q}/2$ is scattered to the momentum ${\bf p}' = {\bf q}/2$, is characterized by ${Q}^2 = {\bf q}^2$. In this frame the nucleon electric $G_E$ and magnetic $G_M$ form factors can be interpreted as Fourier transforms of the distributions of charge and magnetization, respectively: $$\begin{aligned} \Big< N_{s'}(\frac{{\bf q}}{2})\Big|~{\bf J}_{em}(0)~ \Big| N_s(-\frac{{\bf q}}{2}) \Big> &=&\chi^{\dagger}_{s'}\frac{i{\bf\sigma}\times{\bf q}}{2 M_N} \chi_s G_M(q^2) ~~~~\\ \Big< N_{s'}(\frac{{\bf q}}{2}) \Big| ~J^{0}_{em}(0)~ \Big| N_s(-\frac{{\bf q}}{2}) \Big> &=& \chi^{\dagger}_{s'} \chi_s G_E(q^2) \ \\end{aligned}$$ where $\chi^{\dagger}_{s'}$ and $\chi_s$ are Pauli spinors for the initial and final nucleons. Starting from the rest frame, the spherical nucleon is expected to undergo a Lorentz contraction along the direction of motion. Results of previous studies suggest that the consistent treatment of the form factors should be supplemented by the relativistic boost [@Wagenbrunn]. But a complete solution of a covariant many-body problem is difficult; the use of the light-cone dynamics [@Chung] for constituent quarks leads to the introduction of additional parameters. Thus, a semiclassical prescription proposed in Ref. [@Licht] and successfully applied in a CBM [@Lu1] is used here. Thereby, the relativistic form factors can be derived in the Breit frame from the corresponding nonrelativistic ones by a simple substitution: $$\label{Lboost} G_{E(M)}(Q^2) \to \eta G_{E(M)}(\eta Q^2),$$ where $ {\displaystyle}\eta = {M^2_N}/{E^2_N} $ and $ E_N^2 = M_N^2 + {{\bf q}^2}/{4}$. The scaling factor $\eta$ in the argument of $G_{E(M)}$ arises from the coordinate transformation of the struck quark, and the pre-factor in Eq.(\[Lboost\]) comes from the reduction of the integral measure of the two spectator quarks in the Breit frame. This simple boost together with the NRCQM nucleon wave function does not addmix configurations with nonzero orbital angular momentum; it leads to the hadron helicity conserving solution. Note, that imposing Poincaré invariance in a relativistic CQM causes substantial violation of the helicity conservation rule [@MillerFrank], and results in an asymptotic behaviour of form factors which differs from that as expected in pQCD [@Lepage]. We first consider the nucleon single-quark current ${j}_{q_i}^{\mu}(x)$ contribution:$J^{\mu}_{em}(x) = \sum_{i=1}^{3} {j}_{q_i}^{\mu}(x).$ In the CQM the e.m. vertex of the internal quarks should be assumed to have a spatially extended structure that may be described by a form factor $F_{q}({\bf q}^2)$. The most general form for the covariant e.m. current operator of the constituent quarks is written as [@Gross]: $$\label{J_3q_Modif} {j}_{q_i}^{\mu}(x) = \mathcal{Q}_i \bar{q}_i(x) \Big\{ \gamma^{\mu} + \Big(F_q({\bf q}^2) - 1 \Big) \Big[\gamma^{\mu} - \frac{\gamma \cdot q q^{\mu}}{q^2} \Big] \Big\} q_i(x),$$ where $q_i(x)$ is the quark field operator, $\mathcal{Q}_i$ is its charge in units of $e$:  $\mathcal{Q}_i = 1/2 \left[ 1/3 + \tau^3_i \right].$ This vertex, in which the first term corresponds to pointlike quarks, maintains the requirement of current conservation, as the form factor modification appears only in a purely transverse term. The nonrelativistic reduction of Eq.(\[J\_3q\_Modif\]) for pointlike quarks, $F_{q}({\bf q}^2)=1$, leads to the standard one-body e.m. current operators: $\hat{\rho}_{3q}({\bf q}) = \sum_{i=1}^{3} \mathcal{Q}_i e^{i \bf{q} \cdot {\bf r}_i}$ and $\hat{\bf j}_{3q}({\bf q}) = \frac{1}{2 m_q} \sum_{i=1}^{3} \mathcal{Q}_i e^{i \bf{q} \cdot {\bf r}_i} \Bigl( {\bf p}'_i + {\bf p}_i + i {\bf \sigma}_i \times {\bf q} \Bigr), ~ $ where we have retained only the lowest order contributions. This is in spirit of a NRCQM, where the main contribution to the e.m. moments is expected to come from the non-relativistic single quark currents, which by the choice of the effective quark mass already incorporates substantial relativistic corrections [@Buchmann]. It follows that one should not use next-to-leading order relativistic corrections proportional to $\sim {\bf q}^2/8 m_q^2 $ in the charge operator $\hat{\rho}_{3q}({\bf q})$, for example the Darwin-Foldy term, if one ignores them in the kinetic energy. The naive CQM results in the following nucleon e.m. form factors $G^{(3q)}_E$ and $G^{(3q)}_M$: $$\begin{aligned} \label{OB_E} G^{(3q)}_E ({\bf q}^2) &=& e_N \exp \left(-{\bf q}^2 b^2/6 \right) \\ \label{OB_M} G^{(3q)}_M ({\bf q}^2) &=& \frac{M_N}{m_q} \ \mu_N \exp \left(-{\bf q}^2 b^2/6\right) \end{aligned}$$ where $e_N$ and $\mu_N$ are the charge and CQM magnetic moment of the nucleon: $e_N = \frac{1}{2} \langle N |(1 + \tau_3) | N \rangle, ~\mu_N = \frac{1}{6} \langle N | ( 1 + 5 \tau_3 ) | N \rangle .$ Due to the same momentum dependence, Eqs.(\[OB\_E\]) and (\[OB\_M\]) lead to the scaling law noted in Eq.(\[scaling\]); a ratio of unity is obtained as presented by the long dashed line in Fig. \[fig:gegmprot4\]. Clearly, the scaling law is in contradiction with the recent proton experiments [@Jones2000; @Milbrath1999; @Gayou2001; @Gayou2002]. [*The OGE pair-current:*]{}   In the presence of residual OGE interactions between the quarks the total current operator of the hadron cannot simply be a sum of free quark currents, but must be supplemented by two-body currents. These two-body currents are closely related to the $qq$ potential from which they can be derived by minimal substitution. Since the effect of the residual $qq$ potential is clearly seen in the excited spectra of hadrons, one expects the corresponding two-body currents to play an important role in various e.m. properties of hadrons. Both, the photon and the gluons interacting with quarks can produce $q\bar q$ pairs leading to pair-current contributions to e.m. quark current as provided by OGE. The two-body terms we consider are depicted in Fig. \[OGE\]. The nonrelativistic reduction of these diagrams leads to the following configuration space e.m. current operators [@Grabmayr1; @Sanctis]: $$\begin{aligned} \label{rhooge} \rho_{3q}^{(OGE)} = - i \frac{ \alpha_s}{16 m_q^3} \sum_{i < j} {\bf \lambda}_i \cdot {\bf \lambda}_j \frac{ \mathcal{Q}_i}{r^3_{ij}} \left[ e^{i {\bf q}\cdot {\bf r}_i} \Big({\bf q} \cdot ({\bf r}_{i} - {\bf r}_{j}) \right. \hspace{0.cm} \nonumber \\ + \left. \Big[{\bf \sigma}_i \times {\bf q}\Big] \Big[{\bf \sigma}_j \times ({\bf r}_{i}-{\bf r}_j)\Big] \Big) + (i \leftrightarrow j) \right] \hspace{0.8cm} \\ \label{joge} {\bf j}_{3q}^{(OGE)} = - \frac{\alpha_s}{8 m_q^2} \sum_{i < j} {\bf \lambda}_i \cdot {\bf \lambda}_j \frac{ \mathcal{Q}_i}{r^3_{ij}} \hspace{3.3cm} \nonumber \\ \times ~ \Big[ e^{i {\bf q} \cdot {\bf r}_i} \Big[({\bf \sigma}_i + {\bf \sigma}_j ) \times ({\bf r}_{i}-{\bf r}_j)\Big] + (i \leftrightarrow j) \Big] \hspace{0.2cm}\end{aligned}$$ These OGE pair-currents describe a $q\bar q$ pair creation process induced by the external photon with subsequent annihilation of the $q\bar q$ pair into a gluon, which is then absorbed by an another quark. These currents are of relativistic origin as reflected in the higher powers of $1/m_{q}$ as compared to the one-body e.m. current operators. Because the gluon does not carry any isospin the OGE pair-current has the same isospin structure as the one-body currents. Eqs.(\[rhooge\]) and (\[joge\]) result in the following electric $G^{(OGE)}_{E}$ and magnetic $G^{(OGE)}_{M}$ form factors: $$\begin{aligned} \label{OGE_E} \left\{ \begin{array}{r} G^{(OGE)}_{E_p} \\ G^{(OGE)}_{E_n} \end{array} \right\} &=& -\frac{\alpha_s}{m_q^3} ~ q ~ e^{-q^2 b^2 /24} \left\{ \begin{array}{r} 1/3 \\ - 2/9 \end{array} \right\} \mathcal{K}(q) ~~~~~~\\ \label{OGE_M} \left\{ \begin{array}{r} G^{(OGE)}_{M_p} \\ G^{(OGE)}_{M_n} \end{array} \right\} &=& \frac{\alpha_s}{m_q^2} ~ \frac{M_N}{q} ~ e^{-q^2 b^2 /24} \left\{ \begin{array}{r} 2/3 \\ - 2/9 \end{array} \right\} \mathcal{K}(q) ~~~~~~\end{aligned}$$ The function $\mathcal{K}$ in the above expressions is: $$\mathcal{K}(q) = 4 \pi \Big( \frac{1}{2 \pi b^2} \Big)^{3/2} \int_{0}^{\infty} d r ~ e^{-r^2/(2 b^2)} j_1({q r}/{2})$$ where $j_1({q r}/{2})$ is the spherical Bessel function. The interaction of the incoming photon with a $q\bar q$ pair can be considered as a point-like interaction or as being dominated by intermediate vector mesons. The latter leads to an additional dipole form factor, $ {\displaystyle}F_{\gamma q \bar q}({\bf q}^2) = \Lambda_{\gamma q \bar q}^2/ \left(\Lambda_{\gamma q \bar q}^2 + {\bf q}^2\right)$, reflecting the extended structure of the $\gamma q \bar q$ vertex. $\Lambda_{\gamma q \bar q}$ can be considered as a free parameter or simply can be taken equal to the $\rho$-meson mass. [*Results:*]{}   In this work we consider the effect of the OGE pair-current corrections to the NRCQM nucleon e.m. form factors, particularly for the ratio $\mu_{p} G^p_{E}/G^p_{M}$. The ratio is calculated for a quark mass of $m_q$=400 MeV and the respective quark core radius of $b$=0.5 fm. In Fig. \[fig:gegmprot4\] calculations with different $\alpha_s$ are shown to indicate the sensitivity. In the insert of Fig. \[fig:gegmprot4\] we show results towards higher values of $Q^2$ for the best description of the present data by $\alpha_s$ = 0.4 with (solid curve) and without Lorentz boost (dashed curve). Our results indicate that the  $\mu_{p} G^p_{E}/G^p_{M}$ continues to decrease and that it will cross zero at $Q^2\sim$8.1 GeV$^2$. From this, a negative value of the ratio must be expected for the planed measurements in JLAB at $Q^2\sim$9 GeV$^2$. Deviations could be explained due to an extended $\gamma q \bar q$ vertex as demonstrated by using a $\Lambda_{\gamma q \bar q}$=770 MeV (dotted line). The introduction of such states does not affect very much our results up to $\sim$10 GeV$^2$, but strongly influences the behaviour of $\mu_{p} G^p_{E}/G^p_{M}$ for higher $Q^2$. For quark masses in the range $m_q\sim313\div400$ MeV and bag radii of $b\sim0.4\div0.6$ fm one can find also a good description of the data with reasonable values for $\alpha_s\sim0.2\div0.6$ [@Sanctis]. However, these are not able to reproduce the $N-\Delta$ mass splitting in the case of pure OGE. It seems likely that the observed mass splitting is the result of a linear combination of the pion-loop contributions and OGE [@Thomas_book]. In this sense pionic contributions could produce the desirable effect of reducing the size of the strong coupling constant $\alpha_s$, needed for the reproduction of the  $\mu_{p} G^p_{E}/G^p_{M}$. The ratio $\mathcal{F}^p_2/\mathcal{F}^p_{1}$ can be directly derived from $G^p_{E}/G^p_{M}$. It is predicted in Ref. [@Miller] to be constant for values of $Q^2$ up to 20 GeV$^2$ and it is understood as a result of the Melosh transformation, which reflects relativistic effects. Our results are shown in Fig. \[fig:qf2f1-highQ\]. The “kinematical” background formed by the naive CQM results (dot-dashed curve), $\mathcal{F}^p_2/\mathcal{F}^p_{1} = 1/(1+ \kappa_p Q^2/4M^2_M)$, underestimates the data and is not affected by the Lorentz boost, a failure which is overcompensated when adding the OGE pair-currents (dashed curve). It is due to the Lorentz boost (solid curve) acting on the OGE currents to reproduce the flattening in $Q\mathcal{F}^p_2/\mathcal{F}^p_{1}$. Following Ref. [@MillerFrank], we also study the high $Q^2$-behavior. The ratio falls for asymptotic values of $Q^2$ as $Q\mathcal{F}^p_2/\mathcal{F}^p_{1} \sim 1/Q$, and allows to make a smooth transition to the scaling behavior as expected from pQCD [@Lepage]. In Ref. [@MillerFrank] the ratio $Q \mathcal{F}^p_2/\mathcal{F}^p_{1}$ falls less quickly as in our case and in pQCD, both stated a notion of the hadron helicity conservation. We also confirm the statement of Ref. [@MillerFrank], that a plateau seen in Fig. \[fig:qf2f1-highQ\] is the result of a broad maximum occuring near $Q^2\sim10$ GeV$^2$. Recent experimental progress in using polarized nuclear targets will allow to obtain the neutron ratio $G_E^n/G_M^n$. As well known in the $SU(6)$ limit $G_E^n(Q^2)$ is zero [@IsgurNeutron]. We can treat $G_E^n(Q^2)$ as a result of the residual OGE-force in the form of gluonic currents and with the assumption that the OGE pair-currents are the driving mechanism of the scaling law violation we can calculate the neutron ratio $G_E^n/G_M^n$ using the best results for the proton. The results are shown in Fig. \[fig:gegmneut\]. Recombining Eqs. \[OB\_E\], \[OB\_M\], \[OGE\_E\] and \[OGE\_M\] leads to a simple approximate result in analytic form between $G_E^n/G_M^n$ and that of the proton,  $G_E^p/G_M^p$: $$\label{eq:approx} \mu_n G_E^n/G_M^n \simeq \frac{2}{3} ~(1 - \mu_p G_E^p/G_M^p),$$ which works remarkably well from low up to very high $Q^2$, and actually insensitive to the choice of the parameters (insert Fig. \[fig:gegmneut\]). In conclusion, we would like to mention that the internal dynamics of the nucleon are much more complex than we have presented in this work. First of all it is interesting to examine the effect of nonvalence Fock states [@KG]: $$\Psi_N = \left( \begin{array}{c} \Psi(3q) \\ \Psi(3q + q \bar q) \end{array} \right)$$ reflecting $q \bar q$ fluctuations of the constituent quarks. This question is closely related to the possible role of the mesonic cloud. Very useful discussions with V.I. Kukulin are gratefully acknowledged. 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--- abstract: 'Using the Riemann-Hilbert approach, we explicitly construct the asymptotic $\Psi$-function corresponding to the solution $y\sim\pm\sqrt{-x/2}$ as $|x|\to\infty$ to the second Painlevé equation $y_{xx}=2y^3+xy-\alpha$. We precisely describe the exponentially small jump in the dominant solution and the coefficient asymptotics in its power-like expansion.' address: 'St Petersburg Department of Steklov Mathematical Institute, Fontanka 27, St Petersburg 191011, Russia' author: - 'A.A. Kapaev' title: 'Quasi-linear Stokes phenomenon for the Hastings-M[c]{}Leod solution of the second Painlevé equation' --- Introduction {#intro} ============ The second Painlevé equation, $$\label{p2}\tag{$P_2$} y_{xx}=2y^3+xy-\alpha,\quad \alpha=const,$$ was introduced more than a century ago in a classification of the second order ODEs $y_{xx}=R(x,y,y_x)$ with the Painlevé property [@ince]. To this date, equation $P_2$ has found interesting and important applications in the modern theory of nonlinear waves [@abl_seg; @hast_mcleod], plasma physics [@ZKM], bifurcation theory [@maree], random matrices and combinatorics [@TW; @TW2], theory of semi-classical orthogonal polynomials [@BI] and others. Among its various solutions, we distinguish those with a [*monotonic*]{} asymptotic behavior as $x\to\pm\infty$ (look for complete list of asymptotics and connection formulae to transcendent solutions of $P_2$ in [@kapaev4]). For instance, if $\alpha=0$ and $x\to+\infty$, there exists a 1-parameter family of solutions approximated with exponential accuracy by decreasing solutions of the Airy equation, $y_{xx}=xy$, see [@abl_seg], $$\label{Airy_sol} y\simeq a\mbox{\rm Ai}(x)= \frac{a}{2\sqrt\pi}x^{-1/4}e^{-\frac{2}{3}x^{3/2}} \bigl(1+{\cal O}(x^{-3/2})\bigr),\quad a=const.$$ If in the above asymptotics $0\leq a<1$, then the Painlevé function is bounded and oscillates as $x\to-\infty$, $$y\simeq b(-x)^{-1/4}\sin\bigl(\tfrac{2}{3}(-x)^{3/2}-\tfrac{3}{4}b^2\ln(-x) +\phi\bigr),$$ with real constants $b$ and $\phi$ determined by $a$, cf. [@abl_seg]. If $a>1$, then the solution blows up at a finite point. If $a=1$, then the solution of $P_2(\alpha=0)$ grows like a square root as $x\to-\infty$, $y\simeq\sqrt{-x/2}$, see [@hast_mcleod]. For the first time, the asymptotic as $|x|\to\infty$ behavior of the Painlevé transcendents in the complex domain was studied by Boutroux [@boutroux]. Generically, the Painlevé asymptotics within the sectors $\arg x\in\bigl(\frac{\pi}{3}n,\frac{\pi}{3}(n+1)\bigr)$, $n\in{\Bbb Z}$ is described by a modulated elliptic sine which trigonometrically degenerates along the directions $\arg x=\frac{\pi}{3}n$. Besides generic solutions, Boutroux described 1- and 0-parameter reductions of the trigonometric asymptotic solutions which admit analytic continuation from the ray $\arg x=\frac{\pi}{3}n$ into an adjacent complex sector. As $x\to\infty$ in the interior of the relevant complex sector, such solution is represented in the leading order by a power sum of $x$ and of a [*trans-series*]{} which is a sum of exponentially small terms, $$\label{gen_struct} y=\hbox{(power series)}+\hbox{(exponential terms)}.$$ Expansions of such kind can be obtained using a conventional perturbation analysis. In [@kapaev:P1; @kapaev:P2], it was observed that asymptotic solutions of the form (\[gen\_struct\]) exhibit a [*quasi-linear Stokes phenomenon*]{}, i.e. a discontinuity in a minor term with respect to $\arg x$. The first rigorous study of the quasi-linear Stokes phenomenon associated to solutions $y\sim\alpha/x$ as $x\to\infty$ for $P_2$ and $y\sim\sqrt{-x/6}$ as $x\to\infty$ for $P_1$ is presented in [@its_kapaev1; @kapaev2], where the reader can find a discussion of other approaches to the same problem. Below, we study a quasi-linear Stokes phenomenon for the second Painlevé transcendent with the monotonic asymptotic behavior $y\sim\sqrt{-x/2}$ as $x\to-\infty$. Our main tool is the isomonodromy deformation method [@JMU; @FN; @its_nov] in the form of the Riemann-Hilbert problem approach via the nonlinear steepest descent method [@DZ]. As in [@its_kapaev1; @kapaev2], we pursue a two-fold goal, i.e.  (a) we bring an exact meaning to the formal expression (\[gen\_struct\]), and (b) we evaluate the asymptotics of the coefficients of the leading power series in (\[gen\_struct\]). Riemann-Hilbert problem for $P_2$ {#RH_p2} ================================= The inverse problem method in the form of the Riemann-Hilbert (RH) problem was first applied to an asymptotic study of $P_2$ in [@FN]. Further study of the RH problem can be found in [@DZ; @FA; @FZ; @IFK; @its_kapaev3]. According to [@FN], the set of generic Painlevé functions is parameterized by two of the Stokes multipliers of the associated linear system denoted below by the symbols $s_k$, $k=0,1,2$. As it is shown in [@its_kapaev3], if $s_0(1+s_0s_1)\neq0$ then the asymptotic solution of the RH problem within the sector $\arg x\in(\frac{2\pi}{3},\pi)$ can be expressed in terms of elliptic $\theta$-functions which in turn yields an elliptic asymptotics to the Painlevé function itself. Assuming that the condition $s_0=0$ holds true, we arrive to an RH problem which leads to a decreasing asymptotics $y\sim\alpha/x$, see [@its_kapaev1]. In the present paper, we first construct an asymptotic solution to the relevant RH problem as $|x|\to\infty$, $\arg x\in(\frac{2\pi}{3},\pi)$ assuming that $$\label{Stokes_cond} 1+s_0s_1=0.$$ Below, we use the RH problem of Flaschka and Newell [@FN] modified as it is proposed in [@its_kapaev1]. This RH problem comes in a standard way from a collection of properly normalized solutions of the Lax pair for $P_2$, \[Lax\_pair\_p2\] $$\begin{aligned} \label{Lax_pair_p2_lambda} &\frac{\partial\Psi}{\partial\lambda}\Psi^{-1}= -i\bigl(4\lambda^2+x+2y^2\bigr)\sigma_3 -\bigl(4y\lambda+\frac{\alpha}{\lambda}\bigr)\sigma_2 -2y_x\sigma_1, \\\label{Lax_pair_p2_x} &\frac{\partial\Psi}{\partial x}\Psi^{-1}= -i\lambda\sigma_3-y\sigma_2,\end{aligned}$$ where $\sigma_3=\bigl(\begin{smallmatrix}1&0\\0&-1\end{smallmatrix}\bigr)$, $\sigma_2=\bigl(\begin{smallmatrix}0&-i\\i&0\end{smallmatrix}\bigr)$, $\sigma_1=\bigl(\begin{smallmatrix}0&1\\1&0\end{smallmatrix}\bigr)$. Let us introduce the piece-wise oriented contour $\gamma=C\cup\rho_+\cup\rho_-\cup_{k=0}^5\gamma_k$ which is the union of the rays $\gamma_k=\{\lambda\in{\Bbb C}\colon\ |\lambda|>r,\ \arg\lambda=\frac{\pi}{6}+\frac{\pi}{3}(k-1)\}$, $k=0,1,\dots,5$, oriented toward infinity, the clock-wise oriented circle $C=\{\lambda\in{\Bbb C}\colon\ |\lambda|=r\}$, and of two vertical radiuses $\rho_+=\{\lambda\in{\Bbb C}\colon\ |\lambda|<r,\ \arg\lambda=\frac{\pi}{2}\}$ and $\rho_-=\{\lambda\in{\Bbb C}\colon\ |\lambda|<r,\ \arg\lambda=-\frac{\pi}{2}\}$ oriented to the origin. The contour $\gamma$ divides the complex $\lambda$-plane into 8 regions $\Omega_k$, $k\in\{\mbox{\rm left},\mbox{\rm right},0,1,\dots,5\}$. $\Omega_{\mbox{\rm\tiny left}}$ and $\Omega_{\mbox{\rm\tiny right}}$ are left and right halves of the interior of the circle $C$ deprived the radiuses $\rho_+$, $\rho_-$. The regions $\Omega_k$, $k=0,1,\dots,5$, are the sectors between the rays $\gamma_k$ and $\gamma_{k-1}$ outside the circle, see Figure \[fig1\]. Let each of the regions $\Omega_k$, $k=\mbox{\rm right},0,1,2$, be a domain for a holomorphic $2\times2$ matrix function $\Psi_k(\lambda)$. Denote the collection of $\Psi_k(\lambda)$ by $\Psi(\lambda)$, $$\label{Psi_collect} \Psi(\lambda)\bigr|_{\lambda\in\Omega_k}=\Psi_k(\lambda),\qquad \Psi(\rme^{\rmi\pi}\lambda)=\sigma_2\Psi(\lambda)\sigma_2.$$ Let $\Psi_+(\lambda)$ and $\Psi_-(\lambda)$ be the limits of $\Psi(\lambda)$ on $\gamma$ to the left and to the right, respectively. Let us also introduce two matrices $\sigma_+=\bigl(\begin{smallmatrix}0&1\\0&0\end{smallmatrix}\bigr)$, $\sigma_-=\bigl(\begin{smallmatrix}0&0\\1&0\end{smallmatrix}\bigr)$. The RH problem we talk about is the following one: 1. Find a piece-wise holomorphic $2\times2$ matrix function $\Psi(\lambda)$ such that $$\label{Psi_at_infty} \Psi(\lambda)e^{\theta\sigma_3}\to I,\qquad \lambda\to\infty,\qquad \theta=i\bigl(\tfrac{4}{3}\lambda^3+x\lambda\bigr),$$ and $$\label{Psi_at_0} \|\Psi_{\mbox{\rm\tiny right}}(\lambda)\lambda^{-\alpha\sigma_3}\|\leq \mbox{\rm const},\qquad \lambda\to0;$$ 2. on the contour $\gamma$, the jump condition holds $$\label{jump_cond} \Psi_+(\lambda)=\Psi_-(\lambda)S(\lambda),$$ where the piece-wise constant matrix $S(\lambda)$ is given by equations: – on the rays $\gamma_k$, $$\label{p23} S(\lambda)\bigr|_{\gamma_k}=S_k,\qquad S_{2k-1}=I+s_{2k-1}\sigma_-,\qquad S_{2k}=I+s_{2k}\sigma_+,$$ with the constants $s_k$ satisfying the constraints $$\label{s_constraints} s_{k+3}=-s_k,\qquad s_1-s_2+s_3+s_1s_2s_3=-2\sin\pi\alpha;$$ – on the radiuses $\rho_{\pm}$, $S(\lambda)$ is specified by the equations $$\begin{gathered} \label{M_rels} \lambda\in\rho_-\colon\ \Psi_{\mbox{\rm\tiny left}}(\rme^{2\pi i}\lambda)= \Psi_{\mbox{\rm\tiny right}}(\lambda)M, \\ \shoveleft{ \lambda\in\rho_+\colon\ \Psi_{\mbox{\rm\tiny right}}(\lambda)= \Psi_{\mbox{\rm\tiny left}}(\lambda)\sigma_2M\sigma_2, }\hfill\end{gathered}$$ where $$\begin{gathered} \label{M} M=\Bigl(e^{i\pi(\alpha-\frac{1}{2})\sigma_3} +J_{\pm}\sigma_{\pm}\Bigr)(-i\sigma_2), \\ \shoveleft{\text{with}} \\ \shoveleft{ J_+=0\quad\hbox{ if }\quad \tfrac{1}{2}+\alpha\notin{\Bbb N},\quad \hbox{i.e.}\quad \alpha\notin\bigl\{\tfrac{1}{2},\tfrac{3}{2},\dots\bigr\}, } \\ \shoveleft{ J_-=0\quad\hbox{ if }\quad \tfrac{1}{2}-\alpha\notin {\Bbb N}, \quad\hbox{i.e.}\quad \alpha\notin\bigl\{-\tfrac{1}{2},-\tfrac{3}{2},\dots\bigr\}; }\hfill\end{gathered}$$ – on the circle $C$, the piece-wise constant jump matrix $S(\lambda)$ is defined by the equations $$\begin{gathered} \label{E_rels} \Psi_0(\lambda)=\Psi_{\mbox{\rm\tiny right}}(\lambda)ES_0^{-1},\qquad \Psi_1(\lambda)=\Psi_{\mbox{\rm\tiny right}}(\lambda)E,\qquad \Psi_2(\lambda)=\Psi_{\mbox{\rm\tiny right}}(\lambda)ES_1, \\ \shoveleft{ \Psi_3(\lambda)= \Psi_{\mbox{\rm\tiny left}}(\lambda)\sigma_2E\sigma_2S_3^{-1},\qquad \Psi_4(\lambda)=\Psi_{\mbox{\rm\tiny left}}(\lambda)\sigma_2E\sigma_2, } \\ \shoveleft{ \Psi_5(\lambda)=\Psi_{\mbox{\rm\tiny left}}(\lambda)\sigma_2E\sigma_2S_4. }\hfill\end{gathered}$$ The connection matrix $E$, the Stokes matrices $S_k$ and the monodromy matrix $M$ satisfy the cyclic relation, $$\label{E_conditions} ES_1S_2S_3=\sigma_2M^{-1}E\sigma_2.$$ A solution of the RH problem (\[Psi\_at\_infty\])–(\[E\_conditions\]), if exists, is unique. \[resonant\] For the $\lambda$-equation associated with PII (\[Lax\_pair\_p2\_lambda\]) as well as for the above RH problem, the values $\alpha=\frac{1}{2}+n$, $n\in{\Bbb Z}$, are called [*resonant*]{} since the corresponding $\Psi$-function may have a logarithmic singularity at the origin. It is a quite common fallacy, that the RH problem is [*not*]{} uniquely solvable for resonant values of such parameters. As a matter of facts, monodromy data form a locally smooth complex surface (\[s\_constraints\]) with the special points $\alpha=\frac{1}{2}+n$, $n\in{\Bbb Z}$, $s_1=-s_2=s_3=(-1)^{n+1}$. To the latter, one has to attach a copy of ${\Bbb CP}^1$. A complex parameter describing the attached space ${\Bbb CP}^1$ can be interpreted as the ratio of a column entries of the connection matrix $E$ [@FN]. Thus neglecting the attached space ${\Bbb CP}^1$ may lead to the loss of uniqueness in the RH problem solution. The asymptotics of $\Psi(\lambda)$ as $\lambda\to\infty$ is given by $$\label{Y_expansion} Y(\lambda):=\Psi(\lambda)e^{\theta\sigma_3}= I+\lambda^{-1}\bigl(-i{\Eu H}\sigma_3+\frac{y}{2}\sigma_1\bigr)+ {\cal O}(\lambda^{-2}),$$ where $$\label{H_def} {\Eu H}=\tfrac{1}{2}y_x^2-\tfrac{1}{2}y^4-\tfrac{1}{2}xy^2+\alpha y$$ is the Hamiltonian for the second Painlevé equation. Thus $y(x)$ can be extracted from the “residue" of $Y(\lambda)$ at infinity, $$\label{y_from_Y} y=2\lim_{\lambda\to\infty}\lambda Y_{12}(\lambda)= 2\lim_{\lambda\to\infty}\lambda Y_{21}(\lambda).$$ Equation (\[y\_from\_Y\]) specifies the Painlevé transcendent as the function $y=f(x,\alpha,\{s_k\})$ of the independent variable $x$, of the parameter $\alpha$ in the equation and of the Stokes multipliers $s_k$ (generically, the connection matrix $E$ can be expressed via $s_k$ using equation (\[E\_conditions\]) modulo arbitrary left diagonal (or triangular for half-integer $\alpha$) multiplier; at the special points of the monodromy surface (\[s\_constraints\]) $s_1=-s_2=s_3=(-1)^{n+1}$, $\alpha=\frac{1}{2}+n$, $n\in{\Bbb Z}$, the connection matrix $E$ contains a parameter $r\in{\Bbb C}P^1$ which specifies a relevant classical solution of $P_2$). Using the solution $y=f(x,\alpha,\{s_k\})$ and the symmetries of the Stokes multipliers described in [@kapaev:ell], we obtain further solutions of $P_2$: $$\begin{gathered} \label{P_symmetries} y=-f(x,-\alpha,\{-s_k\}),\qquad y=\overline{f(\bar x,\bar\alpha,\{\overline{s_{4-k}}\})}, \\ \shoveleft{ y=\rme^{\rmi \frac{2\pi}{3}n} f(\rme^{\rmi \frac{2\pi}{3}n}x,\alpha,\{s_{k+2n}\}), }\hfill\end{gathered}$$ where the bar means the complex conjugation. Riemann-Hilbert problem for $1+s_0s_1=0$ ======================================== First of all observe that our RH problem can be transformed to an equivalent RH problem with the jump contour consisting of three branches, see Figure \[fig2\] (the method of transformation of Riemann-Hilbert graphs is explained in detail in [@its_kapaev3; @its_kapaev1]). Let us assume now that $$\label{1+s0s1=0} 1+s_0s_1=0.$$ Constraints (\[s\_constraints\]) imply that $s_1-s_0=-2\sin\pi\alpha$ as well, so that $$\label{s13} s_1=e^{-i\pi\sigma(\alpha+\frac{1}{2})},\quad s_0=-1/s_1=e^{i\pi\sigma(\alpha-\frac{1}{2})},\quad \sigma^2=1,$$ while the parameter $s_2$ remains arbitrary. Using (\[1+s0s1=0\]), it is straightforward to check that the jump matrices in the transformed jump graph in Figure \[fig2\] are as follows, $$\begin{gathered} \label{S1S0S1} (S_1S_0S_1)^{-1}= \begin{pmatrix} 0&1/s_1\\-s_1&0 \end{pmatrix},\quad \sigma_2(S_1S_0S_1)^{-1}\sigma_2= \begin{pmatrix} 0&s_1\\-1/s_1&0 \end{pmatrix}, \\ \shoveleft{ S_1S_0S_1S_2\sigma_2S_1^{-1}\sigma_2= \begin{pmatrix} 0&-1/s_1\\s_1&s_1(s_1+s_2) \end{pmatrix}= \begin{pmatrix} 0&-1/s_1\\ s_1&0 \end{pmatrix} \begin{pmatrix} 1&s_1+s_2\\ 0&1 \end{pmatrix} }, \\ \sigma_2S_1S_0S_1S_2\sigma_2S_1^{-1}= \begin{pmatrix} s_1(s_1+s_2)&-s_1\\1/s_1&0 \end{pmatrix}= \begin{pmatrix} 0&-s_1\\ 1/s_1&0 \end{pmatrix} \begin{pmatrix} 1&0\\ -s_1-s_2&1 \end{pmatrix}.\end{gathered}$$ For the connection matrix $E$ we have: \[hatE\] $$\begin{gathered} \label{hatEa} \alpha-\tfrac{1}{2}\notin{\Bbb Z},\quad J_+=J_-=0,\quad M=e^{i\pi\alpha\sigma_3}i\sigma_1 \colon \\ \begin{cases} \sigma=-1,\quad s_1=-e^{i\pi(\alpha-\frac{1}{2})}\colon\quad ES_0^{-1}S_1^{-1}=p^{\sigma_3} \begin{pmatrix} 1&0\\ \frac{s_1^2(s_1+s_2)}{1-s_1^2}&1 \end{pmatrix}, \\ \sigma=+1,\quad s_1=-e^{-i\pi(\alpha-\frac{1}{2})}\colon\quad ES_0^{-1}S_1^{-1}=p^{\sigma_3} i\sigma_2 \begin{pmatrix} 1&0\\ \frac{s_1^2(s_1+s_2)}{1-s_1^2}&1 \end{pmatrix}, \end{cases}\end{gathered}$$ where $p$ is constant; $$\begin{gathered} \label{hatEb} \alpha-\tfrac{1}{2}=n\in{\Bbb Z},\quad s_1=-s_0=(-1)^{n+1},\quad M\,i\sigma_2=(-1)^nI +J_{\pm}\sigma_{\pm} \colon\quad \\ \begin{cases} n\in{\Bbb Z}_{<0},\quad J_+=0,\quad J_-\neq0\colon\quad ES_0^{-1}S_1^{-1}=p^{\sigma_3} \begin{pmatrix} 1&0\\ q&1 \end{pmatrix}, \\ n\in{\Bbb Z}_{\geq0},\quad J_+\neq0,\quad J_-=0\colon\quad ES_0^{-1}S_1^{-1}=p^{\sigma_3}i\sigma_2 \begin{pmatrix} 1&0\\ q&1 \end{pmatrix}, \end{cases}\end{gathered}$$ where $p$ and $q$ are constants. If $\alpha-\frac{1}{2}=n\in{\Bbb Z}$ and $s_1=-s_0=(-1)^{n+1}$ then the condition $J_+=J_-=0$ is equivalent to the equation $s_2=(-1)^n$. On the one hand, these points have important geometrical meaning as being special points of the monodromy surface (\[s\_constraints\]). On the other hand, in this case, equation (\[E\_conditions\]) does not provide any restriction to the connection matrix $E$ which becomes arbitrary. Taking into account that, for $\alpha-\frac{1}{2}=n\in{\Bbb Z}$, the matrix $E$ is determined up to an upper or lower in dependence on the sign of $n$ triangular left multiplier, it contains one essential parameter which parameterizes a family of the (classical) Painlevé functions. Case $\alpha-\frac{1}{2}\not\in{\Bbb Z}$ and $1+s_0s_1=0$ {#case_a-1/2_not_Z} ========================================================= In the case of the non-special point of the monodromy surface, even for a half-integer $\alpha$, the entries of the connection matrix $E$ do not affect the Painlevé function, and thus the jump graph can be deformed to one depicted in Figure \[fig3\]. Here, $$\label{Sigma_def} \Sigma= \begin{pmatrix} 0&1/s_1\\ -s_1&0 \end{pmatrix},\quad S_2S_4^{-1}= \begin{pmatrix} 1&s_1+s_2\\ 0&1 \end{pmatrix},$$ and other jump matrices are as above. For our convenience, we put the node points of the jump graph to the points $\lambda_{1,2}=\pm\sqrt{-x/2}$, $\arg x\in\bigl[\frac{2\pi}{3},\pi\bigr]$. It is worth to note that the jump graph consists of the level lines $\Im g(\lambda)=const$, $\Re g(\lambda)=const$ emanating from the critical points $\lambda=\pm\sqrt{-x/2}$ and $\lambda=0$ for the function $$\label{g-8_def} g(\lambda)=i\tfrac{4}{3}\bigl(\lambda^2+\tfrac{x}{2}\bigr)^{3/2}.$$ The function $\Psi(\lambda)$ is normalized at infinity by the asymptotic condition (\[Psi\_at\_infty\]). Since we eliminate from the jump graph the circle around the origin, the asymptotics for $\Psi(\lambda)$ at the origin (\[Psi\_at\_0\]) has to be replaced by the condition $$\label{Psi_at_0_non_spec} \bigl\|\Psi(\lambda)S_1S_0E^{-1} \lambda^{-\alpha\sigma_3}\bigr\|\leq \mbox{\rm const},\quad \lambda\to+0.$$ The RH problem depends on the free parameter $s_2$ due to a jump across the imaginary axis. Using a quadratic change $\lambda^2=\xi$ supplemented by a gauge transformation of the $\Psi$-function, it is possible to obtain a disjoint jump graph. Unfortunately, unlike the cases considered in [@its_kapaev1; @kapaev2], in this case, one meets a difficulty with the normalization of the transformed RH problem which results in ambiguity in the asymptotics of the Painlevé transcendent. Thus we prefer to deal with the connected jump graph shown in Figure \[fig3\]. Reduced RH problem with $\bf 1+s_0s_1=s_1+s_2=0$. ------------------------------------------------- Consider the non-special RH problem for $P_2$ corresponding to $1+s_0s_1=s_1+s_2=0$. Then the non-specialty assumption implies that $\alpha-\frac{1}{2}$ is not integer. The RH problem jump graph coincides with one depicted in Figure \[fig3\] except for the jump across the curve lines emanating from the origin and approaching the vertical direction since now $$S_2S_4^{-1}=\sigma_2S_2S_4^{-1}\sigma_2=I.$$ The reduced RH problem is formulated as follows: find a piece-wise holomorphic function $\hat\Psi(\lambda)$ such that 1. $\hat\Psi(\lambda)e^{\theta\sigma_3}\to I,\quad \lambda\to\infty$; 2. $\bigl\|\hat\Psi_-(\lambda)S_1S_0E_0^{-1} \lambda^{-\alpha\sigma_3}\bigr\|\leq \mbox{\rm const},\quad \lambda\to0$,where $E_0$ is the connection matrix $E$ defined in (\[hatE\]) corresponding to $s_1+s_2=0$; 3. $\hat\Psi(\lambda)$ is discontinuous across the contour shown in Figure \[fig3\] (with the trivial jumps across the lines emanating from the origin and approaching the vertical directions). \[McLeod1\] Let $\arg x\in[\frac{2\pi}{3},\pi]$, $\alpha-\frac{1}{2}\notin{\Bbb Z}$ and $1+s_0s_1=s_1+s_2=0$. Then, for large enough $|x|$, there exists a unique solution to the reduced RH problem. If additionally the parameter $\sigma\in\{+1,-1\}$ is defined in such a way that $s_1=e^{-i\pi\sigma(\alpha+\frac{1}{2})}$ then the corresponding Painlevé function has the asymptotics $$\label{y1_def} y(x)=y_1(x,\alpha,\sigma)= \sigma\sqrt{e^{-i\pi}\tfrac{x}{2}}+{\cal O}(x^{-2/5}),\quad |x|\to\infty,\quad \arg x\in\bigl[\tfrac{2\pi}{3},\pi\bigr].$$ Uniqueness. Let the reduced RH problem admit two solutions $\hat\Psi_1(\lambda)$ and $\hat\Psi_2(\lambda)$. Since all the jump matrices are unimodular, their determinants $\det\hat\Psi_j(\lambda)$ are continues across the jump contour, bounded at the origin and therefor are entire functions. Using the Liouville theorem and the normalization of $\hat\Psi(\lambda)$ we conclude that $\det\hat\Psi_j(\lambda)\equiv1$. Therefore there exists the ratio $\chi(\lambda)=\hat\Psi_1(\lambda)\hat\Psi_2^{-1}(\lambda)$ which is continuous across the jump contour for $\hat\Psi_j(\lambda)$, remains bounded as $\lambda\to0$ and is normalized to the unit matrix as $\lambda\to\infty$. Therefore, by the Liouville theorem, $\chi(\lambda)\equiv I$. Existence. Consider an RH problem on the curve line segment $[-\sqrt{-x/2},0)\cup(0,\sqrt{-x/2}]$, where $\arg\sqrt{-x/2}\in\bigl[-\frac{\pi}{6},0\bigr]$, see Figure \[fig3a\] (because $\Psi$-function can be analytically continued to ${\Bbb C}\backslash\{0\}$, the jump contours in the RH problem can be bent in any convenient way and even “kiss" the infinity), with the quasi-permutation jump matrices: [**RH model problem 1.**]{} 1. $\Phi^0(\lambda)e^{\theta\sigma_3}\to I,\quad \lambda\to\infty$; 2. $\bigl\|\Phi^0_-(\lambda)S_1S_0E_0^{-1} \lambda^{-\alpha\sigma_3}\bigr\|\leq \mbox{\rm const},\quad \lambda\to0$; 3. $$\begin{aligned} &\Phi^0_+(\lambda)=\Phi^0_-(\lambda)\sigma_2\Sigma\sigma_2,\quad \lambda\in\bigl(-\sqrt{-x/2},0\bigr), \\ &\Phi^0_+(\lambda)=\Phi^0_-(\lambda)\Sigma^{-1},\quad \lambda\in\bigl(0,\sqrt{-x/2}\bigr), \\ &\sigma_2\Sigma\sigma_2= \begin{pmatrix} 0&s_1\\ -1/s_1&0 \end{pmatrix},\quad \Sigma^{-1}= \begin{pmatrix} 0&-1/s_1\\ s_1&0 \end{pmatrix}.\end{aligned}$$ A solution to this RH problem is found explicitly in [@IFK] in some different notations. Namely, on the complex $\lambda$-plane cut along $[-\sqrt{-x/2},\sqrt{-x/2}]$, define the following scalar and matrix functions $\beta,Y,\delta$: $$\label{beta_def} \beta(\lambda)=\Bigl( \frac{\lambda-\sqrt{-x/2}}{\lambda+\sqrt{-x/2}} \Bigr)^{1/4},$$ whose branch is fixed by the condition $$\beta\to1\quad\text{as}\quad \lambda\to\infty;$$ $$\label{Y_def} Y(\lambda)=\frac{1}{2} \begin{pmatrix} \beta+\beta^{-1}&-\sigma(\beta-\beta^{-1})\\ -\sigma(\beta-\beta^{-1})&\beta+\beta^{-1} \end{pmatrix}= \tfrac{1}{\sqrt2} \begin{pmatrix} 1&1\\1&-1 \end{pmatrix} \beta^{-\sigma\sigma_3} \tfrac{1}{\sqrt2} \begin{pmatrix} 1&1\\1&-1 \end{pmatrix},$$ $$\label{delta_def} \delta(\lambda)=\Bigl( \frac{\sqrt{\lambda^2+\frac{x}{2}}-i\sqrt{-x/2}} {\sqrt{\lambda^2+\frac{x}{2}}+i\sqrt{-x/2}} \Bigr)^{\nu},\quad \nu=-\frac{1}{2\pi i}\ln(i\sigma s_1)=\frac{\sigma\alpha}{2},$$ where the branch of the square root is fixed by its asymptotics $\sqrt{\lambda^2+\frac{x}{2}}=\lambda+{\cal O}(\lambda^{-1})$ as $\lambda\to+\infty$, and $z^{\nu}$ is defined on the plane cut along the negative part of the real axis and its main branch is chosen. As easy to see, $$\label{delta_symm} \delta(-\lambda)\delta(\lambda)=1.$$ Furthermore, the functions introduced above enjoy the following jump properties: $$\begin{aligned} \label{Y_jump} &Y_+(\lambda)=Y_-(\lambda)(-i\sigma)\sigma_1,\quad \lambda\in\bigl(-\sqrt{-x/2},\sqrt{-x/2}\bigr), \\\label{delta_jump} &\delta_+(\lambda)\delta_-(\lambda)=e^{2\pi i\nu},\quad \lambda\in\bigl(-\sqrt{-x/2},0\bigr), \\ &\delta_+(\lambda)\delta_-(\lambda)=e^{-2\pi i\nu},\quad \lambda\in\bigl(0,\sqrt{-x/2}\bigr), \notag \\\label{g_jump} &g_+(\lambda)=-g_-(\lambda),\quad \lambda\in\bigl(-\sqrt{-x/2},\sqrt{-x/2}\bigr).\end{aligned}$$ Therefore the function $$\label{Phi0_def} \Phi^0=Y\delta^{\sigma_3}e^{-g\sigma_3}$$ satisfies the jump conditions of model RH problem 1. Because of the asymptotics as $\lambda\to\infty$ $$\begin{aligned} \label{beta_Y_delta_g_as8} &\beta(\lambda)=1-\frac{1}{2\lambda}\sqrt{-x/2}+{\cal O}(\lambda^{-2}), \notag \\ &Y(\lambda)=I+\frac{1}{2\lambda}\sigma\sqrt{-\tfrac{x}{2}}\,\sigma_1 +{\cal O}(\lambda^{-2}), \notag \\ &\delta(\lambda)=1+{\cal O}(\lambda^{-2}), \notag \\ &g(\lambda)=i\bigl( \tfrac{4}{3}\lambda^3+x\lambda\bigr) +i\tfrac{x^2}{8\lambda} +{\cal O}(\lambda^{-3}),\end{aligned}$$ we find the asymptotics as $\lambda\to\infty$ of $\Phi^0(\lambda)$, $$\label{Phi0_as8} \Phi^0(\lambda)=\Bigl( I+\tfrac{1}{2\lambda} \bigl(-\tfrac{ix^2}{4}\sigma_3+\sigma\sqrt{-\tfrac{x}{2}}\,\sigma_1\bigr) +{\cal O}(\lambda^{-2})\Bigr) e^{-\theta\sigma_3}.$$ Using asymptotics as $\lambda\to0$, $$\begin{aligned} \label{beta_Y_delta_g_as0} &\beta_-(\lambda)=e^{-i\pi/4}+{\cal O}(\lambda), \notag \\ &Y_-(\lambda)=\frac{1}{\sqrt2} \Bigl(I+i\sigma\sigma_1+{\cal O}(\lambda)\Bigr), \notag \\ &\delta_-(\lambda)=e^{-i\pi\nu}(-2x)^{\nu}\lambda^{-2\nu} \bigl(1+{\cal O}(\lambda^2)\bigr), \notag \\ &g_-(\lambda)=-\tfrac{\sqrt2}{3}(-x)^{3/2}+\sqrt2(-x)^{1/2}\lambda^2 +{\cal O}(\lambda^4x^{-1/2}),\end{aligned}$$ it is easy to see that $$\label{Phi0_as0} \Phi^0_-(\lambda)=C\bigl(I+{\cal O}(\lambda)\bigr) \lambda^{-\sigma\alpha\sigma_3}$$ with some constant matrix $C$, therefore, using (\[hatEa\]), $\bigl\|\Phi^0_-(\lambda)S_1S_0E_0^{-1} \lambda^{-\alpha\sigma_3}\bigr\|\leq\mbox{\rm const}$. Thus the function $\Phi^0(\lambda)$ (\[Phi0\_def\]) solves model RH problem 1. \[quasi\_perm\] A similar quasi-permutation RH problem in an elliptic case is solved in [@its_kapaev3]. More general quasi-permutation RH problem is solved in [@korotkin]. For the subsequent discussion it is also worth to note that $$\label{Phi0_sym} \sigma_2\Phi^0(-\lambda)\sigma_2=\Phi^0(\lambda).$$ We also point out that $\Phi^0(\lambda)$ is singular at $\lambda=\pm\sqrt{-x/2}$. To “smoothen" this singularity, let us consider the following [**RH model problem 2.**]{} Find a piece-wise holomorphic function $\Phi(\lambda)$ with the jumps indicated in Figure \[fig3b\]. Since we do not normalize $\Phi(\lambda)$, it is determined up to a left multiplication in an entire matrix function. If we will use this solution in a domain different from ${\Bbb C}$, then $\Phi(\lambda)$ is determined up to a left multiplication in a matrix function holomorphic in this domain. It is possible to construct a solution $\Phi(\lambda)$ of this model problem using the classical Airy functions. The Airy function $\Ai(z)$ can be defined using the Taylor expansion [@BE; @olver], $$\label{Ai} \Ai(z)=\frac{1}{3^{2/3}\Gamma(\frac{2}{3})} \sum_{k=0}^{\infty} \frac{3^k\Gamma(k+\frac{1}{3})z^{3k}}{\Gamma(\frac{1}{3})(3k)!} -\frac{1}{3^{1/3}\Gamma(\frac{1}{3})} \sum_{k=0}^{\infty} \frac{3^k\Gamma(k+\frac{2}{3})z^{3k+1}}{\Gamma(\frac{2}{3})(3k+1)!}.$$ Asymptotics at infinity of this function and its derivative are as follows, $$\begin{gathered} \label{Ai_as} \Ai(z)=\tfrac{z^{-1/4}}{2\sqrt\pi}e^{-\frac{2}{3}z^{3/2}} \Bigl\{\sum_{n=0}^N(-1)^n3^{-2n} \frac{\Gamma(3n+\frac{1}{2})}{\Gamma(\frac{1}{2})(2n)!}z^{-\frac{3n}{2}} +{\cal O}\bigl(z^{-\frac{3}{2}(N+1)}\bigr)\Bigr\}, \\ \Ai'(z)=-\tfrac{z^{1/4}}{2\sqrt\pi}e^{-\frac{2}{3}z^{3/2}} \Bigl\{\sum_{n=0}^N(-1)^{n+1}3^{-2n}\bigl(3n+\tfrac{1}{2}\bigr) \frac{\Gamma(3n-\frac{1}{2})}{\Gamma(\frac{1}{2})(2n)!}z^{-\frac{3n}{2}} +{\cal O}\bigl(z^{-\frac{3}{2}(N+1)}\bigr) \Bigr\}, \\ \text{as}\quad z\to\infty,\quad \arg z\in(-\pi,\pi).\end{gathered}$$ Introduce the matrix function $Z_0(\lambda)$, $$\label{Z0_def} Z_0(\lambda)=\sqrt{2\pi}\,e^{-i\pi/4} \begin{pmatrix} v_2(z)&v_1(z)\\ \frac{d}{dz}v_2(z)&\frac{d}{dz}v_1(z) \end{pmatrix}e^{-i\frac{\pi}{4}\sigma_3},$$ where $$\label{Airy_notations} v_2(z)=e^{i2\pi/3}Ai(e^{i2\pi/3}z),\quad v_1(z)=Ai(z).$$ As $|z|\to\infty$, $\arg z\in\bigl(-\pi,\frac{\pi}{3}\bigr)$, this function has the asymptotics $$\label{Z_0_as} Z_0(z)=z^{-\sigma_3/4}\frac{1}{\sqrt2} \begin{pmatrix} 1&1\\ 1&-1 \end{pmatrix} \Bigl(I+{\cal O}(z^{-3/2})\Bigr) e^{\frac{2}{3}z^{3/2}\sigma_3}.$$ Also, introduce the matrix functions $$\begin{gathered} \label{Airies} Z_1(z):=Z_0(z)G_0,\quad Z_2(z):=Z_1(z)G_1,\quad Z_3(z):=Z_2(z)G_2, \\ \shoveleft{ G_0=G_2= \begin{pmatrix} 1&0\\ -i&1 \end{pmatrix},\quad G_1= \begin{pmatrix} 1&-i \\ 0&1 \end{pmatrix}. }\hfill\end{gathered}$$ By the properties of the Airy functions [@BE; @olver], $Z_j(z)$ have the asymptotics (\[Z\_0\_as\]) as $z\to\infty$ and $\arg z\in\bigl(-\pi+\frac{2\pi}{3}j,\frac{\pi}{3}+\frac{2\pi}{3}j\bigr)$, $j=0,1,2,3$. Define a piece-wise holomorphic function $Z^{RH}(z)$, $$\label{RH_Airy} Z^{RH}(z)=\begin{cases} Z_0(z)(is_1)^{\sigma_3/2},\quad \arg z\in\bigl(-\frac{\pi}{3},0),\\ Z_j(z)(is_1)^{\sigma_3/2},\quad \arg z\in\bigl(-\frac{2\pi}{3}+\frac{2\pi}{3}j,\frac{2\pi}{3}j\bigr), \quad j=1,2,\\ Z_3(z)(is_1)^{\sigma_3/2},\quad \arg z\in\bigl(\frac{4\pi}{3},\frac{5\pi}{3}). \end{cases}$$ By construction, $Z^{RH}(z)$ has the uniform asymptotics $$\label{Z_RH_as} Z^{RH}(z)=z^{-\sigma_3/4}\frac{1}{\sqrt2} \begin{pmatrix} 1&1\\ 1&-1 \end{pmatrix} \Bigl(I+{\cal O}(z^{-3/2})\Bigr) e^{\frac{2}{3}z^{3/2}\sigma_3} (is_1)^{\sigma_3/2},\quad z\to\infty,$$ and jump properties indicated in Figure \[fig5\]. Using the change of the independent variable $$\label{z(lambda)} z=e^{i\pi}2^{2/3}\bigl(\lambda^2+\frac{x}{2}\bigr),$$ and the notation $$\label{zeta_def} \zeta:=\lambda-\sqrt{-x/2},$$ we compute the “ratio" of $\Phi^0$ and $Z^{RH}$ in the annulus $c_1|x|^{-\frac{1}{2}+\epsilon}\leq|\zeta|\leq c_2|x|^{-\frac{1}{2}+\epsilon}$, where $0<c_1<c_2$ and $\frac{1}{4}<\epsilon<\frac{1}{2}$ are some constants, $$\begin{gathered} \label{Phi0_Z_ratio} \Phi^0(\lambda)\bigl(Z^{RH}(z)\bigr)^{-1}= \Bigl(I+{\cal O}\bigl(x^{-1/4}\zeta^{1/2}\bigr) +{\cal O}\bigl(x^{-1/2}\zeta^{-2}\bigr)\Bigr) \times \\ \times\tfrac{1}{\sqrt2} \begin{pmatrix} 1&1\\1&-1 \end{pmatrix} \Bigl(\frac{1-\sigma}{2}i\sigma_1 +\frac{1+\sigma}{2}I\Bigr) e^{i\frac{\pi}{4}\sigma_3} 2^{\frac{2}{3}\sigma_3} (-x/2)^{\sigma_3/4}.\end{gathered}$$ Define the matrix function $\Phi_r(\lambda)$, $$\label{Phir_def} \Phi_r(\lambda)= \tfrac{1}{\sqrt2} \begin{pmatrix} 1&1\\1&-1 \end{pmatrix} \Bigl(\frac{1-\sigma}{2}i\sigma_1 +\frac{1+\sigma}{2}I\Bigr) e^{i\frac{\pi}{4}\sigma_3} 2^{\frac{2}{3}\sigma_3} (-x/2)^{\sigma_3/4} Z^{RH}\bigl(z(\lambda)\bigr).$$ Let us introduce the piece-wise holomorphic function $\Phi(\lambda)$, $$\label{Phi_def} \Phi(\lambda)= \begin{cases} \Phi_r(\lambda),\quad \bigl|\lambda-\sqrt{-x/2}\bigr|<R,\\ \sigma_2\Phi_r(e^{-i\pi}\lambda)\sigma_2,\quad \bigl|\lambda+\sqrt{-x/2}\bigr|<R,\\ \Phi^0(\lambda),\quad \bigl|\lambda\pm\sqrt{-x/2}\bigr|>R, \end{cases}$$ where $R=c|x|^{-\frac{1}{2}+\epsilon}$ for a constant $c>0$ and $\frac{1}{4}<\epsilon<\frac{1}{2}$. We look for the exact solution of the reduced RH problem in the form of the product $$\label{chi_red_def} \hat\Psi(\lambda)=\chi(\lambda)\Phi(\lambda).$$ The correction function $\chi(\lambda)$ solves the RH problem whose jump graph is shown in Figure \[fig6\]. The jump matrix across the circle centered at $\lambda=\sqrt{-x/2}$ is given by the ratio $\Phi^0(\lambda)\Phi_r^{-1}(\lambda)$ which, due to (\[Phi0\_Z\_ratio\]), (\[Phir\_def\]) and choosing $\epsilon=2/5$, satisfies the estimate $$\label{jump_r_circle} \bigl\|\Phi^0(\lambda)\Phi_r^{-1}(\lambda)-I\bigr\|\leq c|x|^{-3/10}.$$ The jump matrices across the exterior parts of infinite contours approach the unit matrices exponentially fast, $$\label{jump_tails} \bigl\|\Phi^0(\lambda)S_j\bigl(\Phi^0(\lambda)\bigr)^{-1}-I\bigr\|\leq C|x|^{3/20}e^{-\frac{2^{15/4}}{3}|x|^{3/4}|\zeta|^{3/2}}.$$ Now, the solvability of the reduced RH problem is straightforward. Indeed, consider the system of singular integral equations $\chi_-=I+K\chi_-$ equivalent to the RH problem for $\chi(\lambda)$, $$\label{chi_singular_integral} \chi_-(\lambda)=I+\frac{1}{2\pi i}\int_{\gamma}\chi_-(\xi) \bigl(G(\xi)-I\bigr)\frac{d\xi}{\xi-\lambda_-},$$ where $\gamma$ is the contour shown in Figure \[fig6\] and $G(\lambda)=\Phi_-(\lambda)S_j\Phi_+^{-1}(\lambda)$ is the relevant jump matrix. Using the estimates (\[jump\_r\_circle\]), (\[jump\_tails\]) and the boundedness of the Cauchy operator in $L^2(\gamma)$, the singular integral operator $K$, which is a superposition of the operator of the right multiplication in $G(\lambda)-I$ and of the Cauchy operator $C_-$, satisfies the estimate, $\bigl\|K\bigr\|_{L^2(\gamma)}\leq c|x|^{-2/5}$, where the precise value of the positive constant $c$ is not important for us. Thus $K$ is a contracting operator in $L^2(\gamma)$ for large enough $|x|$. Since $KI$ is a square integrable function on $\gamma$, and observing that $\rho=\chi_--I$ satisfies the singular integral equation $\rho=KI+K\rho$, we find $\rho\in L^2(\gamma)$ and therefore $\chi_-=I+\rho$ by iterations. The solution found by iterations implies the asymptotics of $\chi(\lambda)$ as $\lambda\to\infty$, $$\label{chi_as} \chi(\lambda)=I+\frac{1}{\lambda}{\cal O}(x^{-2/5}) +{\cal O}(\lambda^{-2}).$$ Using (\[chi\_as\]) and (\[Phi0\_as8\]) in (\[chi\_red\_def\]) and taking into account the expansion (\[Y\_expansion\]), we see that the Painlevé function corresponding to the reduced RH problem has the asymptotics as $|x|\to\infty$, $\arg x\in\bigl[\frac{2\pi}{3},\pi\bigr]$, $$\label{y_as} y(x)=\sigma\sqrt{e^{-i\pi}\frac{x}{2}}+{\cal O}(x^{-2/5}).$$ This completes the proof. RH problem with $\bf 1+s_0s_1=0$ and $\bf s_1+s_2\neq0$. -------------------------------------------------------- Let us go to the case of $1+s_0s_1=0$ and arbitrary $s_1+s_2\neq0$, see Figure \[fig3\]. We look for the solution $\Psi(\lambda)$ in the form of the product $$\label{hat_chi_def} \Psi(\lambda)=\hat\chi(\lambda)\hat\Psi(\lambda),$$ where $\hat\Psi(\lambda)$ is the solution of the reduced RH problem, i.e.with $s_1+s_2=0$. The correction function $\hat\chi(\lambda)$ satisfies the RH problem on two level lines $\Im g(\lambda)=const$ emanating from the origin and approaching the vertical direction, see Figure \[fig7\], 1. $\hat\chi(\lambda)\to I,\quad \lambda\to\infty$; 2. $\bigl\|\hat\chi(\lambda)\hat\Psi(\lambda)S_1S_0E^{-1} \lambda^{-\alpha\sigma_3}\bigr\|\leq\mbox{\rm const},\quad \lambda\to+0$; 3. $\hat\chi_+(\lambda)=\hat\chi_-(\lambda)\hat G(\lambda)$, $\hat G(\lambda):=\hat G_1(\lambda)= \hat\Psi_-(\lambda)S_2S_4^{-1}\hat\Psi_-^{-1}(\lambda),\quad \lambda\in(0,+i\infty)$, $\hat G(\lambda):=\hat G_2(\lambda)= \hat\Psi_-(\lambda)\sigma_2S_2S_4^{-1}\sigma_2\hat\Psi_-^{-1}(\lambda),\quad \lambda\in(0,-i\infty)$, $\sigma_2\hat G_1(-\lambda)\sigma_2=\hat G_2(\lambda)$. \[McLeod2\] If $\alpha-\frac{1}{2}\notin{\Bbb Z}$, $1+s_0s_1=0$, $\arg x\in[\frac{2\pi}{3},\pi]$ and $|x|$ is large enough, then the RH problem (\[Psi\_at\_infty\])–(\[E\_conditions\]) is uniquely solvable. The asymptotics of the corresponding Painlevé function as $|x|\to\infty$ in the indicated sector is given by $$\begin{gathered} \label{mcleod-_as} y=y_1(x,\alpha,\sigma)- \\ -\frac{s_1+s_2}{\pi} 2^{-\frac{5}{2}\sigma\alpha-\frac{7}{4}} \Gamma(\tfrac{1}{2}+\sigma\alpha) (e^{-i\pi}x)^{-\frac{3}{2}\sigma\alpha-\frac{1}{4}} e^{-\frac{2\sqrt2}{3}(e^{-i\pi}x)^{3/2}} \bigl(1+{\cal O}(x^{-1/4})\bigr),\end{gathered}$$ where $y_1(x,\alpha,\sigma)\sim\sigma\sqrt{e^{-i\pi}x/2}$ is the solution of the Painlevé equation corresponding to $1+s_0s_1=s_1+s_2=0$, while the parameter $\sigma\in\{+1,-1\}$ is defined by the use of the equation $s_1=e^{-i\pi\sigma(\alpha+\frac{1}{2})}$. It is enough to prove the solvability of the RH problem for $\hat\chi(\lambda)$ above. In a neighborhood of the imaginary axis, $\hat\Psi(\lambda)=\chi(\lambda)\Phi^0(\lambda)$, see (\[chi\_red\_def\]) and (\[Phi\_def\]). Hence, using (\[Phi0\_def\]), we have the following expression for the jump matrix across $(0,+i\infty)$, \[hat\_G12\] $$\begin{aligned} \label{hat_G1} &\hat G_1(\lambda)=I+(s_1+s_2)\delta^2e^{-2g}\chi Y\sigma_+Y^{-1}\chi^{-1}, \\\label{hat_G2} &\hat G_2(\lambda)=I-(s_1+s_2)\delta^{-2}e^{2g}\chi Y\sigma_-Y^{-1}\chi^{-1}.\end{aligned}$$ Because the jump contour coincides with the level line $\Im g(\lambda)=const$, the jump matrix $\hat G_1(\lambda)$ exponentially approaches the unit matrix as $\lambda\to+i\infty$. Similarly, $\hat G_2(\lambda)-I$ decreases exponentially as $\lambda\to-i\infty$. For the exponential $e^{g(\lambda)}$, due to (\[beta\_Y\_delta\_g\_as0\]), the origin is the saddle point. The jump matrices have algebraic singularity at $\lambda=0$, \[hat\_G12\_at0\] $$\begin{aligned} \label{hat_G1_at0} &\hat G_1(\lambda)=I+(s_1+s_2)e^{-2\pi i\nu} (-2x)^{-2\nu}\lambda^{4\nu} e^{-\frac{2\sqrt2}{3}(-x)^{3/2}+2\sqrt2(-x)^{1/2}\lambda^2}\times \notag \\ &\times \tfrac{1}{2}(i\sigma\sigma_3+\sigma_1) \bigl(1+{\cal O}(\lambda)+{\cal O}(x^{-9/10})\bigr), \\\label{hat_G2_at0} &\hat G_2(\lambda)=I-(s_1+s_2) e^{2i\pi\nu}(-2x)^{-2\nu}\lambda^{4\nu} e^{-\tfrac{2\sqrt2}{3}(-x)^{3/2}+2\sqrt2(-x)^{1/2}\lambda^2)}\times \notag \\ &\times \tfrac{1}{2}(i\sigma\sigma_3+\sigma_1) \bigl(1+{\cal O}(\lambda)+{\cal O}(x^{-9/10})\bigr).\end{aligned}$$ Consider the model RH problem with the jump matrices $\tilde G_j(\lambda)$, $j=1,2$, of the form (\[hat\_G12\]) but with the matrix $\chi(\lambda)Y(\lambda)$ replaced by its asymptotics at $\lambda=0$, i.e. \[tilde\_G12\] $$\begin{aligned} \label{tilde_G1} &\tilde G_1(\lambda)=I+(s_1+s_2)\delta^2e^{-2g} \tfrac{1}{2}(i\sigma\sigma_3+\sigma_1), \\\label{tilde_G2} &\tilde G_2(\lambda)=I-(s_1+s_2)\delta^{-2}e^{2g} \tfrac{1}{2}(i\sigma\sigma_3+\sigma_1).\end{aligned}$$ Assume for a moment that $$\label{nu>=0} \Re\nu\geq0,\quad \text{i.e.}\quad \Re(\sigma\alpha)\geq0.$$ Since $\tilde G_j(\lambda)-I$ are constant nilpotent matrices multiplied in scalar functions, solution of the jump problem is given by the Cauchy integral, $$\begin{gathered} \label{tilde_chi_sol} \tilde\chi(\lambda)=I+\frac{s_1+s_2}{2\pi i} \Bigl\{ \int_0^{+i\infty} \delta^2e^{-2g} \frac{d\zeta}{\zeta-\lambda} +\int_{-i\infty}^0 \delta^{-2}e^{2g} \frac{d\zeta}{\zeta-\lambda} \Bigr\} \tfrac{1}{2}(i\sigma\sigma_3+\sigma_1)= \\ =I+\frac{s_1+s_2}{2\pi i} \int_0^{+i\infty} \delta^2e^{-2g} \frac{\lambda\,d\zeta} {\zeta^2-\lambda^2} (i\sigma\sigma_3+\sigma_1).\end{gathered}$$ The condition (\[nu&gt;=0\]) ensures the convergence of the above integrals. Obviously, $\tilde\chi(\lambda)\to I$ as $\lambda\to\infty$. As to asymptotics at the origin, the same condition (\[nu&gt;=0\]) ensures that $\bigl\|\tilde\chi(\lambda)\bigr\|\leq\mbox{\rm const}$, and $\bigl\|\tilde\chi(\lambda)\hat\Psi(\lambda)S_1S_0E^{-1} \lambda^{-\alpha\sigma_3}\bigr\|\leq\mbox{\rm const}$. We will look for $\hat\chi(\lambda)$ in the form of the product $$\label{X_def} \hat\chi(\lambda)=X(\lambda)\tilde\chi(\lambda).$$ For the correction function $X(\lambda)$, we have the RH problem with the jump contour shown in Figure \[fig7\], but with different jump matrices: 1. $X(\lambda)\to I,\quad \lambda\to\infty$; 2. $\bigl\|X(\lambda)\bigr\|\leq\mbox{\rm const},\quad \lambda\to+0$; 3. $X_+(\lambda)=X_-(\lambda)H(\lambda)$,$H(\lambda)=\tilde\chi_-(\lambda)\hat G(\lambda)\tilde G^{-1}(\lambda)\tilde\chi_-^{-1}(\lambda)$. In the equivalent singular integral equation, $$\label{X_sing_eq} X_-(\lambda)=I+\frac{1}{2\pi i}\int_{\gamma} X_-(\zeta)\bigl(H(\zeta)-I\bigr)\frac{d\zeta}{\zeta-\lambda_-},$$ where $\gamma$ is the jump graph in Figure \[fig7\] and $\lambda_-$ is the right limit of $\lambda$ on the jump contour, or, in the symbolic form, $X_-=I+KX_-$, the operator $K$ is the superposition of the right multiplication in $H-I$ and of the Cauchy operator $C_-$. Using the boundedness of $C_-$ in $L^2(\gamma)$, we estimate the norm $$\label{K_H-I_norm} \bigl\|K\bigr\|_{L^2(\gamma)}\leq c\bigl\|H-I\bigr\|_{L^2(\gamma)}\leq c'|x|^{-(5\nu+1)/2}e^{-\frac{8}{3}|x|^{3/2}\cos\frac{3}{2}(\arg x-\pi)}\leq c''|x|^{-1/2},$$ where $c$, $c'$ and $c''$ are positive constants whose precise value is not important for us. Taking into account the assumption $\Re\nu\geq0$ (\[nu&gt;=0\]), the singular integral operator $K$ is contracting for large enough $|x|$, $\arg x\in\bigl[\frac{2\pi}{3},\pi\bigr]$, and therefore equation (\[X\_sing\_eq\]) is solvable by iterations. To find the asymptotics of the relevant Painlevé function, it is enough to find asymptotics of $\tilde\chi(\lambda)$ (\[tilde\_chi\_sol\]) as $\lambda\to\infty$, $$\begin{gathered} \label{tilde_chi_as} \tilde\chi(\lambda)=I-\frac{s_1+s_2}{2\pi i\lambda} \int_0^{+i\infty}\delta^2e^{-2g}\,d\zeta \bigl(1+{\cal O}(\lambda^{-2})\bigr) (i\sigma\sigma_3+\sigma_1)= \\ =I+\lambda^{-1}h(i\sigma\sigma_3+\sigma_1)+{\cal O}(\lambda^{-3}), \\ h=-\frac{s_1+s_2}{2\pi} 2^{-5\nu-\frac{7}{4}} \Gamma(2\nu+\tfrac{1}{2}) (-x)^{-3\nu-\frac{1}{4}} e^{-\frac{2\sqrt2}{3}(-x)^{3/2}} \bigl(1+{\cal O}(x^{-1/2})\bigr).\end{gathered}$$ Thus the asymptotics of $\Psi(\lambda)$ as $\lambda\to\infty$ (\[Y\_expansion\]) is given by $$\begin{gathered} \label{Psi_as_s1+s2ne0} \Psi(\lambda)e^{\theta\sigma_3}=X\tilde\chi\hat\Psi e^{\theta\sigma_3}= I+\lambda^{-1}\bigl(-i{\Eu H}\sigma_3+\frac{y}{2}\sigma_1\bigr)+ {\cal O}(\lambda^{-2})= \\ = I+\lambda^{-1}\Bigl( -i({\Eu H}_2-\sigma\hat h_3)\sigma_3 +\frac{y_1+2\hat h_1}{2}\sigma_1 \Bigr)+ {\cal O}(\lambda^{-2}),\end{gathered}$$ where $\hat h_j=h\bigl(1+{\cal O}(x^{-2\nu-1/4})\bigr)$, $j=3,1$, involves the contribution of the correction function $X(\lambda)$. Comparison of two lines in (\[Psi\_as\_s1+s2ne0\]) yields the asymptotics of the Painlevé function, $y=y_1+2\hat h_1$, which turns into (\[mcleod-\_as\]) for $\Re(\sigma\alpha)\geq0$, $\alpha-\frac{1}{2}\notin{\Bbb Z}$ substituting $\nu=\sigma\alpha/2$. Validity of the asymptotic formula (\[mcleod-\_as\]) without the restriction (\[nu&gt;=0\]) follows from the observation that (\[mcleod-\_as\]) is invariant with respect to Bäcklund transformations. Indeed, the Schlesinger transformation of the $\Psi$-function, $$\begin{gathered} \label{Schlesinger} \tilde\Psi(\lambda)=R(\lambda)\Psi(\lambda), \\ R(\lambda)=I -\frac{q_{\epsilon}}{2i\lambda}(\sigma_3-i\epsilon\sigma_1),\quad q_{\epsilon}= \frac{\alpha+\frac{\epsilon}{2}}{y_x-\epsilon(y^2+\frac{x}{2})},\quad \epsilon\in\{+1,-1\},\end{gathered}$$ yields the $\Psi$-function, associated to a new Painlevé transcendent $$\label{Backlund} \tilde y=y+\epsilon q_{\epsilon},\quad \tilde y_x=y_x+\epsilon(\tilde y^2-y^2),\quad \tilde\alpha=-\alpha-\epsilon,$$ characterized however by the same Stokes multipliers. The latter with the definition of the parameter $\sigma$ via $s_1=e^{-i\pi\sigma(\alpha+\frac{1}{2})}= \tilde s_1=e^{-i\pi\tilde\sigma(\tilde\alpha+\frac{1}{2})}$ implies the transformation of the parameter $\sigma$, $\tilde\sigma=-\sigma$ for either value of $\epsilon=\pm1$. If both $\Re(\sigma\alpha)\geq0$ and $\Re(\tilde\sigma\tilde\alpha)=\Re(\sigma\alpha)+\sigma\epsilon\geq0$, then the asymptotics of both $y$ and $\tilde y$ is described by (\[mcleod-\_as\]) (supplemented by tilde over $y$, $\sigma$ and $\alpha$ if necessary). This observation constitutes the invariance of the asymptotic formula with respect to the Bäcklund transformation. Since the latter transformation is a bi-rational transformation in the space with coordinates $(y,y_x)$, the above invariance can be confirmed algebraically. Therefore, if $\Re(\tilde\sigma\tilde\alpha)<0$ while $\Re(\sigma\alpha)\geq0$, the asymptotics of $\tilde y$ and $y$ are described by the formula (\[mcleod-\_as\]) (supplemented by tilde if necessary). The iterated use of the Bäcklund transformations completes the proof. The increasing degenerate Painlevé functions -------------------------------------------- Applying the second of the symmetries (\[P\_symmetries\]) to (\[mcleod-\_as\]) and changing the argument of $x$ in $2\pi$, we obtain \[McLeod3\] If $\alpha-\frac{1}{2}\notin{\Bbb Z}$, $1+s_0s_1=0$, $\arg x\in[\pi,\frac{4\pi}{3}]$ and $|x|$ is large enough, then the asymptotics of the Painlevé function is given by $$\begin{gathered} \label{mcleod-bar_as} y=y_0(x,\alpha,\sigma)- \\ -\frac{s_2-s_0}{\pi} 2^{-\frac{5}{2}\sigma\alpha-\frac{7}{4}} \Gamma(\tfrac{1}{2}+\sigma\alpha) (e^{-i\pi}x)^{-\frac{3}{2}\sigma\alpha-\frac{1}{4}} e^{-\frac{2\sqrt2}{3}(e^{-i\pi}x)^{3/2}} \bigl(1+{\cal O}(x^{-1/4})\bigr),\end{gathered}$$ where $y_0(x,\alpha,\sigma)=\overline{y_1(\bar x,\bar\alpha,\sigma)} \sim\sigma\sqrt{e^{-i\pi}x/2}$ is the solution of the Painlevé equation corresponding to $1+s_0s_1=s_2-s_0=0$, while the parameter $\sigma\in\{+1,-1\}$ is defined by the use of the equation $s_1=e^{-i\pi\sigma(\alpha+\frac{1}{2})}$. The solutions $y_1(x,\alpha,\sigma)$ and $y_0(x,\alpha,\sigma)=\overline{y_1(\bar x,\bar\alpha,\sigma)}$ are meromorphic functions of $x\in{\Bbb C}$ and thus can be continued beyond the sectors indicated in Theorems \[McLeod2\] and \[McLeod3\]. To find the asymptotics of the solution $y_0(x,\alpha,\sigma)$ in the interior of the sector $\arg x\in[\frac{2\pi}{3},\pi]$, we apply (\[mcleod-\_as\]) with $s_2=s_0$. Similarly, we find the asymptotics of the solution $y_1(x,\alpha,\sigma)$ in the interior of the sector $\arg x\in[\pi,\frac{4\pi}{3}]$ using (\[mcleod-bar\_as\]) with $s_2=-s_1$. Either expression implies that, if $|x|\to\infty$, $\arg x\in[\frac{2\pi}{3},\frac{4\pi}{3}]$, $$\begin{gathered} \label{y0-y1} y_0(x,\alpha,\sigma)-y_1(x,\alpha,\sigma)= \\ =i\sigma \frac{2^{-\frac{5}{2}\sigma\alpha-\frac{3}{4}}} {\Gamma(\tfrac{1}{2}-\sigma\alpha)} (e^{-i\pi}x)^{-\frac{3}{2}\sigma\alpha-\frac{1}{4}} e^{-\frac{2\sqrt2}{3}(e^{-i\pi}x)^{3/2}} \bigl(1+{\cal O}(x^{-1/4})\bigr),\end{gathered}$$ where we have used the relation $$(s_0+s_1)\Gamma\bigl(\tfrac{1}{2}+\sigma\alpha\bigr)= -\frac{2i\pi\sigma}{\Gamma\bigl(\tfrac{1}{2}-\sigma\alpha\bigr)}.$$ The formula (\[y0-y1\]) constitutes the quasi-linear Stokes phenomenon for the increasing degenerate asymptotic solution of $P_2$. Due to exponential decay of the difference (\[y0-y1\]) in the interior of the sector $\arg x\in\bigl(\frac{2\pi}{3},\frac{4\pi}{3}\bigr)$, we have solutions of $P_2$, $$\begin{gathered} \label{y0_ext1} y=y_1(x,\alpha,\sigma)\simeq\sigma\sqrt{e^{-i\pi}x/2},\quad |x|\to\infty,\quad \arg x\in\bigl[\tfrac{2\pi}{3},\tfrac{4\pi}{3}\bigr), \\ \text{for}\quad s_0=-e^{i\pi\sigma(\alpha+\frac{1}{2})},\quad s_1=e^{-i\pi\sigma(\alpha+\frac{1}{2})},\quad s_2=-e^{-i\pi\sigma(\alpha+\frac{1}{2})},\end{gathered}$$ and $$\begin{gathered} \label{y1_ext0} y=y_0(x,\alpha,\sigma)\simeq\sigma\sqrt{e^{-i\pi}x/2},\quad |x|\to\infty,\quad \arg x\in\bigl(\tfrac{2\pi}{3},\tfrac{4\pi}{3}\bigr], \\ \text{for}\quad s_0=-e^{i\pi\sigma(\alpha+\frac{1}{2})},\quad s_1=e^{-i\pi\sigma(\alpha+\frac{1}{2})},\quad s_2=-e^{i\pi\sigma(\alpha+\frac{1}{2})}.\end{gathered}$$ Applying the rotational symmetry of (\[P\_symmetries\]) to $y_1$ and $y_0$, we find solutions $$\begin{gathered} \label{y_2n+1_def} y=y_{2n+1}(x,\alpha,\sigma):= e^{i\frac{2\pi}{3}n}y_1(e^{i\frac{2\pi}{3}n}x,\alpha,\sigma)\simeq \sigma(-1)^n\sqrt{e^{-i\pi}x/2}, \\ |x|\to\infty,\quad \arg x\in\bigl[\tfrac{2\pi}{3}-\tfrac{2\pi}{3}n, \tfrac{4\pi}{3}-\tfrac{2\pi}{3}n\bigr), \\ \text{for}\quad s_{2n}=-e^{i\pi\sigma(\alpha+\frac{1}{2})},\quad s_{2n+1}=e^{-i\pi\sigma(\alpha+\frac{1}{2})},\quad s_{2n+2}=-e^{-i\pi\sigma(\alpha+\frac{1}{2})},\end{gathered}$$ and $$\begin{gathered} \label{y_2n_def} y=y_{2n}(x,\alpha,\sigma):= e^{i\frac{2\pi}{3}n}y_0(e^{i\frac{2\pi}{3}n}x,\alpha,\sigma)\simeq \sigma(-1)^n\sqrt{e^{-i\pi}x/2}, \\ |x|\to\infty,\quad \arg x\in\bigl(\tfrac{2\pi}{3}-\tfrac{2\pi}{3}n, \tfrac{4\pi}{3}-\tfrac{2\pi}{3}n\bigr], \\ \text{for}\quad s_{2n}=-e^{i\pi\sigma(\alpha+\frac{1}{2})},\quad s_{2n+1}=e^{-i\pi\sigma(\alpha+\frac{1}{2})},\quad s_{2n+2}=-e^{i\pi\sigma(\alpha+\frac{1}{2})}.\end{gathered}$$ Comparing the Stokes multipliers, we observe the symmetries $$\label{y_n_symm} y_n(x,\alpha,\sigma)=y_{n+6}(x,\alpha,\sigma),\quad y_{2n-1}(x,\alpha,\sigma)=y_{2n}(x,\alpha,-\sigma).$$ Equation (\[y0-y1\]) and definitions (\[y\_2n+1\_def\]) and (\[y\_2n\_def\]) imply the differences take place, $$\begin{gathered} \label{y_2n+1-y_2n} y_{2n}(x,\alpha,\sigma)-y_{2n+1}(x,\alpha,\sigma)= \\ =i\sigma \frac{2^{-\frac{5}{2}\sigma\alpha-\frac{3}{4}}} {\Gamma(\tfrac{1}{2}-\sigma\alpha)} e^{i\frac{2\pi}{3}n} (e^{-i\pi}e^{i\frac{2\pi}{3}n}x)^{-\frac{3}{2}\sigma\alpha-\frac{1}{4}} e^{-\frac{2\sqrt2}{3}(e^{-i\pi}e^{i\frac{2\pi}{3}n}x)^{3/2}} \bigl(1+{\cal O}(x^{-1/4})\bigr), \\ |x|\to\infty,\quad \arg x\in\bigl[\tfrac{2\pi}{3}-\tfrac{2\pi}{3}n, \tfrac{4\pi}{3}-\tfrac{2\pi}{3}n\bigr],\end{gathered}$$ which are exponentially small in the interior of the indicated sectors. Using the second of the equations (\[y\_n\_symm\]), we establish the analytic continuation of the asymptotics as $|x|\to\infty$, $$\begin{gathered} \label{max_as_sect} y_{2n-1}(x,\alpha,(-1)^{n-1}\sigma)=y_{2n}(x,\alpha,(-1)^n\sigma)\simeq \sigma\sqrt{e^{-i\pi}x/2}, \\ |x|\to\infty,\quad \arg x\in\bigl(\tfrac{2\pi}{3}(1-n),\tfrac{2\pi}{3}(3-n)\bigr).\end{gathered}$$ Coefficient asymptotics for $\alpha-\frac{1}{2}\not\in{\Bbb Z}$ =============================================================== An elementary investigation shows that the equation $P_2$, $y_{xx}=2y^2+xy-\alpha$, can be satisfied by the formal series depending on $\alpha$ and a parameter $\sigma=\pm1$, $$\label{mcleod-formal} y_f(x,\alpha,\sigma)= \sigma\sqrt{-x/2}\sum_{n=0}^{\infty}b_n(-x)^{-3n/2} +{\cal O}(x^{-\infty}).$$ Given $\sigma$, the series (\[mcleod-formal\]) is determined uniquely since its coefficients $b_n$ are determined by the recurrence relation, $$\begin{gathered} \label{bn_recurrence} b_0=1,\quad b_1=\frac{\sigma\alpha}{\sqrt2}, \\ b_{n+2}=\frac{9n^2-1}{8}b_{n} -\sum_{m=1}^{n+1}b_mb_{n+2-m} -\frac{1}{2}\sum_{l=1}^{n+1}\sum_{m=1}^{n+2-l}b_lb_mb_{n+2-l-m}.\end{gathered}$$ Several initial terms of the expansion are given by $$\begin{gathered} \label{mcleod_expansion} y_f(x,\alpha,\sigma)=\sigma\sqrt{-x/2}\Bigl\{1+ \frac{\sigma\alpha}{\sqrt2(-x)^{3/2}} -\frac{1+6\alpha^2}{8(-x)^{3}} +\frac{\sigma\alpha(11+16\alpha^2)}{8\sqrt2(-x)^{9/2}}- \\ -\frac{73+708\alpha^2+420\alpha^4}{128(-x)^{6}} +\frac{\sigma\alpha(1021+2504\alpha^2+768\alpha^4)} {64\sqrt2(-x)^{15/2}}- \\ -\frac{10657+129918\alpha^2+132060\alpha^4+24024\alpha^6} {1024(-x)^{9}}+ \\ +\frac{\sigma\alpha(248831+786304\alpha^2+416400\alpha^4+49152\alpha^6)} {512\sqrt2(-x)^{21/2}} +{\cal O}\bigl(x^{-12}\bigr)\Bigr\}.\end{gathered}$$ To find the asymptotics of the coefficients $b_n$ in (\[mcleod-formal\]) as $n\to\infty$, let us construct a sectorial analytic function $\hat y(t)$, $$\begin{gathered} \label{hat_mcleod} \hat y(t)=y_{2n-1}(e^{i\pi}t^2,\alpha,(-1)^{n-1}\sigma) =y_{2n}(e^{i\pi}t^2,\alpha,(-1)^n\sigma), \\ |t|\to\infty,\quad \arg t\in\bigl(-\tfrac{\pi}{3}n, \tfrac{\pi}{3}-\tfrac{\pi}{3}n\bigr).\end{gathered}$$ The function $\hat y(x)$ has a finite number of simple poles and, by construction, has the uniform asymptotics near infinity, $\hat y(t)\simeq\sigma t/\sqrt2$ as $|t|\to\infty$. Using (\[mcleod-formal\]), it has the following formal series expansion, $$\label{mcleod_hat_y_expansion} \hat y(t)= \sigma\frac{t}{\sqrt2}\sum_{n=0}^{\infty}b_nt^{-3n}.$$ Let $y^{(N)}(x)$ be a partial sum $$\label{mcleod_yN} y^{(N)}(t)=\sigma\frac{t}{\sqrt2}\sum_{n=0}^{N-1}b_nt^{-3n},$$ and $v^{(N)}(t)$ be a product $$\label{mcleod_vn} v^{(N)}(t)=t^{3N-2}(\hat y(t)-y^{(N)}(t))= \frac{\sigma}{t\sqrt2}\sum_{n=0}^{\infty}b_{n+N}t^{-3n}.$$ Because $t^{3N-2}y^{(N)}(t)$ is polynomial and $\hat y(t)/t$ is bounded as $|t|\geq\rho$, the integral of $v^{(N)}(t)$ along the circle of the radius $|t|=\rho$ containing all the pole singularities of $\hat y(t)$ satisfies the estimate $$\label{mcleod_vN_integral} \Bigl|\oint_{|t|=\rho}v^{(N)}(t)\,dt\Bigr|\leq \rho^{3N-2}\oint_{|t|=\rho}|\hat y(t)|\,dl\leq 2\pi\rho^{3N}\max_{|t|=\rho}|\hat y(t)/t|=C\rho^{3N}.$$ On the other hand, inflating the sectorial arcs of the circle $|t|=\rho$, we find that $$\label{mcleod_vN_inflation} \oint_{|t|=\rho}v^{(N)}(t)\,dx=\oint_{|t|=R}v^{(N)}(t)\,dt +\sum_{n=-3}^2\int_{e^{i\frac{\pi}{3}n}(\rho,R)}\bigl( v_+^{(N)}(t)-v_-^{(N)}(t)\bigr)\,dt.$$ Since $v^{(N)}(t)=\frac{\sigma}{t\sqrt2}b_N+{\cal O}(t^{-3})$, the first of the integrals in the r.h.s. of (\[mcleod\_vN\_inflation\]) is $$\label{bN_int} \oint_{|x|=R}v^{(N)}(t)\,dt=\pi i\sigma\sqrt{2}\,b_N+{\cal O}(R^{-2}).$$ Last six integrals in (\[mcleod\_vN\_inflation\]) are computed using (\[hat\_mcleod\]), (\[y\_2n+1\_def\]), (\[y\_2n\_def\]), (\[y\_n\_symm\]) and (\[y0-y1\]), $$\begin{gathered} \label{bN_eval} \sum_{n=-3}^2\int_{e^{i\frac{\pi}{3}n}(\rho,R)}\bigl( v_+^{(N)}(t)-v_-^{(N)}(t)\bigr)\,dt= \\ =3\int_{\rho}^R t^{3N-2}\bigl( y_0(e^{i\pi}t^2,\alpha,\sigma) -y_1(e^{i\pi}t^2,\alpha,\sigma)\bigr)\,dt+ \\ +3(-1)^{N-1}\int_{\rho}^R t^{3N-2}\bigl( y_0(e^{i\pi}t^2,\alpha,-\sigma) -y_1(e^{i\pi}t^2,\alpha,-\sigma)\bigr)\,dt= \\ =3i\sigma \frac{2^{-\frac{5}{2}\sigma\alpha-\frac{3}{4}}} {\Gamma(\tfrac{1}{2}-\sigma\alpha)} \int_{\rho}^R t^{3(N-\sigma\alpha)-\frac{5}{2}} e^{-\frac{2\sqrt2}{3}t^3} \bigl(1+{\cal O}(t^{-1/2})\bigr)\,dt+ \\ +3(-1)^{N}i\sigma \frac{2^{\frac{5}{2}\sigma\alpha-\frac{3}{4}}} {\Gamma(\tfrac{1}{2}+\sigma\alpha)} \int_{\rho}^R t^{3(N+\sigma\alpha)-\frac{5}{2}} e^{-\frac{2\sqrt2}{3}t^3} \bigl(1+{\cal O}(t^{-1/2})\bigr)\,dt= \\ =i\sigma 2^{-\frac{3}{2}N} 3^{N-\frac{1}{2}} \Bigl\{ 6^{-\sigma\alpha} \frac{\Gamma(N-\sigma\alpha-\tfrac{1}{2})} {\Gamma(\tfrac{1}{2}-\sigma\alpha)} \bigl(1+{\cal O}(N^{-1/6})\bigr)+ \\ +(-1)^N 6^{\sigma\alpha} \frac{\Gamma(N+\sigma\alpha-\tfrac{1}{2})} {\Gamma(\tfrac{1}{2}+\sigma\alpha)} \bigl(1+{\cal O}(N^{-1/6})\bigr) \Bigr\} + \\ +{\cal O}(\rho^{3(N+|\mbox{\tiny\rm Re}\alpha|)-\frac{3}{2}}) +{\cal O}\bigl(e^{-\frac{2\sqrt2}{3}R^3} R^{3(N+|\mbox{\tiny\rm Re}\alpha|)-\frac{9}{2}}\bigr).\end{gathered}$$ Thus, letting $R=\infty$, we find the asymptotics as $N\to\infty$ of the coefficient $b_N$ in (\[mcleod-formal\]), $$\begin{gathered} \label{bn_fin} b_N=-\frac{1}{\pi\sqrt{6}} \Bigl(\frac{3}{2\sqrt2}\Bigr)^N \Bigl\{ 6^{-\sigma\alpha} \frac{\Gamma(N-\sigma\alpha-\tfrac{1}{2})} {\Gamma(\tfrac{1}{2}-\sigma\alpha)} \bigl(1+{\cal O}(N^{-1/6})\bigr)+ \\ +(-1)^N 6^{\sigma\alpha} \frac{\Gamma(N+\sigma\alpha-\tfrac{1}{2})} {\Gamma(\tfrac{1}{2}+\sigma\alpha)} \bigl(1+{\cal O}(N^{-1/6})\bigr) \Bigr\} +{\cal O}(\rho^{3N}).\end{gathered}$$ In particular, the asymptotics (\[bn\_fin\]) is consistent with the elementary observation that, for $\alpha=0$, all odd coefficients vanish, $b_{2n-1}=0$; for even coefficients, the leading order asymptotics reduces to the one-term expression, i.e. $$\begin{gathered} \label{bn_fin_a=0} \text{if}\quad \alpha=0\colon\quad b_{2N-1}=0, \\ b_{2N}=-\frac{\sqrt2}{\pi^{3/2}\sqrt{3}} \Bigl(\frac{3}{2\sqrt2}\Bigr)^{2N} \Gamma(2N-\tfrac{1}{2}) \bigl(1+{\cal O}(N^{-1/6})\bigr) +{\cal O}(\rho^{3N}).\end{gathered}$$ \[mcleod\_err\_bound\] The error bound in (\[mcleod-\_as\]), (\[mcleod-bar\_as\]) and (\[y0-y1\]) can be improved to ${\cal O}(x^{-3/2})$ which implies the error bound for (\[bn\_fin\]) ${\cal O}(N^{-1})$. \[alpha=1/2+n\] It is possible to prove that the asymptotic formula (\[bn\_fin\]) remains valid for half-integer values of $\alpha$ as well. 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--- author: - | \ \ $^1$ School of Physics, University of Melbourne, Parksville, VIC 3010, Australia\ $^2$ ICRAR, University of Western Australia, 35 Stirling Highway, Crawley, Western Australia 6009, Australia\ $^3$ Centre for Astrophysics & Supercomputing, Swinburne University of Technology, P.O. Box 218, Hawthorn, VIC 3122, Australia\ $^4$ Scuola Normale Superiore, Piazza dei Cavalieri 7, I-56126 Pisa, Italy bibliography: - 'main.bib' date: draft version title: | Dark-ages Reionization $\&$ Galaxy Formation Simulation I:\ The dynamical lives of high redshift galaxies --- \[firstpage\] cosmology: dark ages, reionization, first stars – cosmology: early Universe – cosmology: theory – galaxies: formation – galaxies: high redshift Introduction {#sec-intro} ============ \[sec-introduction\] Simulations {#sec-simulations} =========== Analysis {#sec-analysis} ======== Summary and conclusions {#sec-summary} ======================= Acknowledgements {#acknowledgements .unnumbered} ================ Mass function fitting {#sec-appendix} =====================

--- abstract: 'We show that finite-dimensional order unit spaces equipped with a continuous sequential product as defined by Gudder and Greechie are homogeneous and self-dual. As a consequence of the Koecher-Vinberg theorem these spaces therefore correspond to Euclidean Jordan algebras. We remark on the significance of this result in the context of reconstructions of quantum theory. In particular, we show that sequential product spaces that have locally tomographic tensor products, i.e. their vector space tensor products are also sequential product spaces, must be C$^*$ algebras. Finally we remark on a couple of ways these results can be extended to the infinite-dimensional setting of JB- and JBW-algebras and how changing the axioms of the sequential product might lead to a new characterisation of homogeneous cones.' author: - | John van de Wetering\ Radboud University Nijmegen, Netherlands\ `john@vdwetering.name` bibliography: - '../bibliography.bib' title: Sequential Product Spaces are Jordan Algebras --- Introduction ============ The set of observables of a quantum system can be represented as the space of self-adjoint operators on a complex Hilbert space $B(H)^{\text{sa}}$. This space has a variety of algebra-like structures that can be associated to it, the most well-known of which is the *Jordan product* $a*b := \frac12(ab+ba)$. In the 30’s Jordan, von Neumann and Wigner hoped to find generalisations of the quantum mechanical formalism by considering general spaces equipped with an axiomatisation of this algebraic structure. They however found that the resulting *Euclidean Jordan algebras* (EJAs) have a strikingly simple classification [@jordan1993algebraic], and hence that this algebraic approach does not allow you to go far beyond quantum theory. The significance of EJAs was further established by the Koecher-Vinberg theorem that states that any homogeneous and self-dual ordered vector space is a Euclidean Jordan algebra [@koecher1957positivitatsbereiche]. These two results, the Koecher-Vinberg theorem and the classification by Jordan, von Neumann and Wigner, lie at the heart of many *reconstructions of quantum theory* where intuitively sensible axioms from which quantum theory can be derived are sought [@hardy2011reformulating; @wetering2018reconstruction; @selby2018reconstructing; @barnum2014higher; @gunson1967algebraic] (although it should be noted that these theorems are not used directly in all approaches [@tull2016reconstruction; @hohn2017toolbox; @chiribella2011informational]). Other algebraic structures on $B(H)^{sa}$ studied in an axiomatic way are the *quadratic Jordan algebras* that axiomatize the map $(a,b)\mapsto aba$ or the more general *triple product* $(a,b,c)\mapsto \frac{1}{2}(abc+cba)$. The definitions of the Jordan product and the triple product don’t have a particularly compelling physical motivation: the product does not correspond to any type of physical process. In this paper we will look at a different structure that *does* follow naturally from physical processes. Given two positive operators $a$ and $b$ we define their *sequential product* as $a{\,\&\,}b := \sqrt{a}b\sqrt{a}$. When $a$ and $b$ represent *effects*, i.e. possible outcomes in a measurement, then the sequential product models the act of first getting the outcome $a$ and then the outcome $b$, hence the name *sequential* product (although this composition is also known as a *Lüders process*). It is important to note that this product is only defined for positive operators (since otherwise the square root wouldn’t be defined), and that this operation is not bilinear, associative or commutative. Gudder and Greechie introduced the concept of a *sequential effect algebra* to study the sequential product in a more abstract setting [@gudder2002sequential]. While they studied the structure of the sequential product on the very general structure of *effect algebras*, we will restrict ourselves to the more concrete setting of *order unit spaces*: An *order unit space* $(V, \leq, 1)$ is an ordered real vector space with the additional property that $1$ is a *strong Archimedean unit*: 1. Strong unit: For all $a\in V$ we can find $n\in {\mathbb{N}}$ such that $-n1\leq a \leq n$. 2. Archimedean: For $a\in V$ when $a\leq \frac1n 1$ for all $n\in{\mathbb{N}}_{>0}$, then $a\leq 0$. We call the elements $a\in V$ with $0\leq a \leq 1$ the *effects* of $V$ which we will denote by $[0,1]_V$. The *states* of $V$ are positive linear maps $\omega:V\rightarrow {\mathbb{R}}$ such that $\omega(1) = 1$. Ordered vector spaces with a strong unit represent the most general kinds of systems allowed in causal *generalised probabilistic theories* [@barrett2007information] and hence they form a suitable background to studying models of general physical theories. The Archimedity condition states intuitively that there are no effects that cannot be distinguished by a state. More precisely, order unit spaces are precisely the ordered vector spaces where the states order-separate the elements: if $\omega(v)\leq \omega(w)$ for all states $\omega$ then $v\leq w$. Note that an order unit space has a norm induced by the order in the following way: ${\norm{a} := \inf\{r\in {\mathbb{R}}_{\geq 0}~;~ -r1\leq a\leq r1\}}$. Whenever we refer to continuity in the context of order unit spaces it should be understood to refer to this norm. The object of study in this paper is an order unit space with an operation modelled after the sequential product on $B(H)^{\text{sa}}$. To be specific: \[def:seqprod\] Let $(V, \leq, 1, \&)$ be an order unit space equipped with a binary operation\ ${\&:[0,1]_V\times [0,1]_V \rightarrow [0,1]_V}$. We write $a{\,\lvert\,}b$ and say $a$ and $b$ are *compatible* when $a{\,\&\,}b = b{\,\&\,}a$. We call $V$ a *sequential product space* and $\&$ a *sequential product* when $\&$ satisfies the following properties for all $a,b,c \in [0,1]_V$: 1. \[ax:add\] Additivity: $a{\,\&\,}(b+c) = a{\,\&\,}b+ a{\,\&\,}c$. 2. \[ax:cont\] Continuity: The map $a\mapsto a{\,\&\,}b$ is continuous in the norm. 3. \[ax:unit\] Unitality: $1{\,\&\,}a = a$. 4. \[ax:orth\] Compatibility of orthogonal effects: If $a{\,\&\,}b = 0$ then also $b{\,\&\,}a =0$. 5. \[ax:assoc\] Associativity of compatible effects: If $a{\,\lvert\,}b$ then $a{\,\&\,}(b{\,\&\,}c) = (a{\,\&\,}b){\,\&\,}c$. 6. \[ax:compadd\] Additivity of compatible effects: If $a{\,\lvert\,}b$ then $a {\,\lvert\,}1-b$, and if also $a{\,\lvert\,}c$ then $a{\,\lvert\,}(b+c)$. 7. \[ax:compmult\] Multiplicativity of compatible effects: If $a{\,\lvert\,}b$ and $a{\,\lvert\,}c$ then $a{\,\lvert\,}(b{\,\&\,}c)$. The properties we require of $\&$ are the same as that of a sequential product in a sequential effect algebra [@gudder2002sequential] except for condition \[ax:cont\] which is new. It should be noted that the standard sequential product $a{\,\&\,}b = \sqrt{a}b\sqrt{a}$ on $B(H)^{\text{sa}}$ is not fully characterised by these axioms, as there are multiple binary operations that satisfy these axioms [@weihua2009uniqueness]. It is possible however to characterise the standard sequential product using related sets of axioms [@gudder2008characterization; @westerbaan2016universal; @wetering2018characterisation]. It has been established in the authors previous work [@wetering2018characterisation] that Euclidean Jordan algebras allow a binary operation satisfying these properties and hence are examples of sequential product spaces. The main purpose of this paper is to establish the converse: Let $V$ be a finite-dimensional sequential product space, then $V$ is order-isomorphic to a Euclidean Jordan algebra. Since EJAs are very well understood and in particular classified we can use this theorem to prove additional results. In particular, using the classification of Jordan algebras and a property called *local tomography* we can infer when sequential product spaces are C$^*$-algebras. Let $V$ and $W$ be finite-dimensional sequential product spaces. We say that they have a *locally tomographic composite* when their vector space tensor product $V\otimes W$ is also a sequential product space with $(a_1\otimes b_1){\,\&\,}(a_2\otimes b_2) = (a_1{\,\&\,}a_2)\otimes (b_1{\,\&\,}b_2)$ for all effects $a_i$ in $V$ and $b_i$ in $W$. In the context of generalised probabilistic theories, the property of local tomography states that local measurements on each of the subsystems is enough to fully characterise bipartite states. It is a property that holds for regular quantum theory, but fails for, for instance, quantum theory over the real numbers. Let $V$ be a finite-dimensional sequential product space that has a locally tomographic composite with itself, then there exists a C$^*$-algebra $A$ such that $V$ is isomorphic to $A^{\text{sa}}$ as a Jordan algebra. Note that C$^*$-algebras being singled out among all the EJAs by local tomography is not surprising as similar results were obtained in [@barnum2014local; @selby2018reconstructing; @masanes2014entanglement], but in combination with the results regarding the sequential product it does give a novel understanding of the mathematical structure of quantum theory: > A causal probabilistic physical theory that satisfies local tomography and that has a well-behaved notion of sequential measurement must be modelled by C$^*$-algebras. There have been quite a few characterisations of quantum theory using operational axioms [@barnum2014higher; @selby2018reconstructing; @masanes2014entanglement; @chiribella2011informational; @tull2016reconstruction; @wetering2018reconstruction; @wilce2016royal; @hohn2017toolbox; @hardy2001quantum], but the one presented above is different in a couple of ways. First of all, other characterisations and reconstructions have their axioms refer to a multitude of structures, like the existence of certain systems, transformations and pure states, instead of focusing on a single aspect, which this characterisation does with regards to sequential measurement. Second, all reconstructions of quantum theory that the author is aware of have axioms ensuring the existence of suitable reversible (i.e. invertible) dynamics in the theory. In contrast, this characterisation of C$^*$-algebras doesn’t directly say anything about the existence of reversible maps. In addition to the above theorems, we will also establish a couple of infinite-dimensional versions of the first theorem relating infinite-dimensional sequential product spaces to JB- and JBW-algebras. We refer to section \[sec:infiniterank\] for the details. The structure of the paper is as follows: the main theorem that sequential product spaces are EJAs will be proved using the Koecher-Vinberg theorem which requires us to show that the space is both *homogeneous* and *self-dual*. In section \[sec:prelim\] we will cover known results originally presented in [@gudder2002sequential; @wetering2018characterisation] regarding sequential product spaces, culminating in a proof of a spectral theorem and a proof of the homogeneity of the space. Then in section \[sec:selfdual\] we will prove self-duality of the space, using results regarding lattices of projections of Alfsen and Schultz [@alfsen2012state; @alfsen2012geometry] and a characterisation result concerning low rank homogeneous spaces of Ito and Louren[ç]{}o [@ito2017p]. At this point sequential product spaces have been established to be EJAs, but only in a rather indirect way. In section \[sec:jordanproduct\] we directly construct the Jordan product using the sequential product. In section \[sec:loctom\] we show how the additional requirement of local tomography forces the sequential product space to be a C$^*$-algebra. Section \[sec:infiniterank\] discusses infinite dimensional generalisations of the main theorem, while in section \[sec:axioms\] we discuss how changing the axioms of a sequential product impacts the results of this paper. *Acknowledgements.*—The author would like to thank Bas and Bram Westerbaan for all the useful and insightful conversations regarding effect algebras and order unit spaces and Alex Kolmus and Ema Alsina for suggestions to improve the manuscript. This work is supported by the ERC under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC grant n$^\text{o}$ 320571. Preliminaries {#sec:prelim} ============= As mentioned in the introduction, our main goal is to show that a sequential product space is homogeneous and self-dual, let us start therefore with the definition of these properties. Let $V$ be an order unit space. An *order isomorphism* is a linear map $\Phi: V\rightarrow V$ such that $\Phi(a)\geq 0 \iff a\geq 0$ for all $a\in V$. Denote the interior of the positive cone of $V$ by $C$, i.e. $a\in C \iff \exists \epsilon\in{\mathbb{R}}_{>0}: \epsilon 1 \leq a$. We call $V$ *homogeneous* when for all $a,b \in C$ there exists an order isomorphism $\Phi$ such that $\Phi(a) = b$. Let $V$ be an order unit space. We call $V$ *self-dual* when there exists an inner product $\inn{\cdot,\cdot}$ such that for all $a\in V$: $a\geq 0$ if and only if $\inn{a,b} \geq 0$ for all $b\geq 0$. To give a complete picture of the theory of sequential product spaces we will repeat some of the known basic results regarding sequential products and sequential product spaces that can be found in for instance [@gudder2002sequential; @gudder2005uniqueness; @wetering2018characterisation]. This section on preliminaries will end with the existence proofs of spectral decompositions of effects and the corollary of homogeneity that follows from it originally shown in [@wetering2018characterisation]. Unless otherwise stated, we will let $V$ denote a finite-dimensional sequential product space, ${E=[0,1]_V}$ its set of effects and $\&: E\times E\rightarrow E$ a sequential product. For $a\in E$ we let $a^\perp=1-a$ denote its *complement* which by virtue of $a$ lying in the unit interval of $V$ is also an effect. \[prop:basic\] Let $a,b,c \in E$. 1. \[prop:unitzero\] $a{\,\&\,}0 = 0{\,\&\,}a = 0$ and $a{\,\&\,}1=1{\,\&\,}a = a$. 2. \[prop:decreasing\] $a{\,\&\,}b \leq a$. 3. \[prop:orderpreserve\] If $a\leq b$ then $c{\,\&\,}a\leq c{\,\&\,}b$. Originally proved in Ref. [@gudder2002sequential]. 1. We of course have $a{\,\lvert\,}a$ and by \[ax:compadd\] we have $a{\,\lvert\,}a^\perp$. Using \[ax:compadd\] again we then see that $a{\,\lvert\,}a+a^\perp = 1$ so that by \[ax:unit\] $1{\,\&\,}a = a{\,\&\,}1 = a$. Using \[ax:compadd\] again we also have $a{\,\lvert\,}1^\perp = 0$ so that it remains to show that $a{\,\&\,}0 = 0$. By \[ax:add\] we get $a{\,\&\,}0 = a{\,\&\,}(0+0) = a{\,\&\,}0 + a{\,\&\,}0$ so that indeed $a{\,\&\,}0 = 0$. 2. By the previous point and \[ax:add\] $a = a{\,\&\,}1 = a{\,\&\,}(b+(1-b)) = a{\,\&\,}b + a{\,\&\,}(1-b)$ so that indeed $a{\,\&\,}b \leq a$, as $a{\,\&\,}(1-b) \geq 0$. 3. Since $a\leq b$ we have $b-a \geq 0$ so that using \[ax:add\] we have $c{\,\&\,}b = c{\,\&\,}(b-a + a) = c{\,\&\,}(b-a) + c{\,\&\,}a$, from which we derive $c{\,\&\,}(b-a) = c{\,\&\,}b - c{\,\&\,}a$. Since the lefthandside is greater than zero, the righthandside must be as well. \[prop:linearity\] Let $a,b\in E$ and let $q$ be any rational number between zero and one, and $\lambda$ any real number between zero and one. 1. $a{\,\&\,}(qb) = q(a{\,\&\,}b)$. 2. $a{\,\&\,}(\lambda b) = \lambda(a{\,\&\,}b)$. 3. $(\lambda a){\,\&\,}b = a{\,\&\,}(\lambda b) = \lambda(a{\,\&\,}b)$. 4. \[prop:commumult\] If $a{\,\lvert\,}b$ then $a{\,\lvert\,}\lambda b$. Originally proved in Ref. [@wetering2018characterisation]. 1. Of course $a{\,\&\,}b = a{\,\&\,}(n \frac1n b) = n a{\,\&\,}(\frac1n b)$ by \[ax:add\]. Dividing by $n$ gives $a{\,\&\,}(\frac1n b) = \frac1n (a{\,\&\,}b)$. By summing this equation multiple times we see that we get $a{\,\&\,}(q b) = q(a{\,\&\,}b)$ for any rational $0\leq q\leq 1$. 2. Let $q_i$ be an increasing sequence of positive rational numbers that converges to $\lambda$. Using the norm of the order unit space we compute $$\begin{aligned} \norm{\lambda (a{\,\&\,}b) - a{\,\&\,}(\lambda b)} &= \norm{(\lambda - q_i)(a{\,\&\,}b) + q_i(a{\,\&\,}b) - a{\,\&\,}(\lambda b)} \\ &= \norm{(\lambda-q_i)(a{\,\&\,}b) - a{\,\&\,}((\lambda - q_i)b)}. \end{aligned}$$ Because $(\lambda - q_i)b\leq (\lambda-q_i)\norm{b}1$ and using proposition \[prop:orderpreserve\] we have $\norm{a{\,\&\,}((\lambda - q_i)b)} \leq \norm{a}\norm{(\lambda -q_i)b} = (\lambda - q_i)\norm{a}\norm{b}$. Then $\norm{\lambda (a{\,\&\,}b) - a{\,\&\,}(\lambda b)} \leq 2(\lambda - q_i)\norm{a}\norm{b}$. This expression indeed goes to zero as $i$ increases so that indeed $\lambda (a{\,\&\,}b) = a{\,\&\,}(\lambda b)$. 3. Clearly $\frac{1}{n}a{\,\lvert\,}\frac{1}{n}a$ so that by \[ax:compadd\] $\frac{1}{n}a{\,\lvert\,}a$. In the same way we also get $qa{\,\lvert\,}a$ and $qa^\perp {\,\lvert\,}a^\perp$ for any rational $0\leq q\leq 1$. Using the rule $a{\,\lvert\,}b \implies a{\,\lvert\,}b^\perp$ from \[ax:compadd\] we then also get $qa^\perp {\,\lvert\,}a$ so that $a{\,\lvert\,}(qa+qa^\perp)=q1$. Then indeed $(q1){\,\&\,}a = a{\,\&\,}(q1) = q(a{\,\&\,}1) = qa$ so that also $(qa){\,\&\,}b = (a{\,\&\,}(q1)){\,\&\,}b = a{\,\&\,}((q1){\,\&\,}b)) = a{\,\&\,}qb = q(a{\,\&\,}b)$. Now let $\lambda\in[0,1]$ be a real number and let $q_i$ be a sequence of rational numbers converging to $\lambda$ so that also $q_i a \rightarrow \lambda a$ and $q_i(a{\,\&\,}b) \rightarrow \lambda(a{\,\&\,}b)$. Then $q_i(a{\,\&\,}b) = (q_ia){\,\&\,}b \rightarrow (\lambda a){\,\&\,}b$ by \[ax:cont\]. We conclude that $(\lambda a){\,\&\,}b = \lambda(a{\,\&\,}b) = a{\,\&\,}(\lambda b)$. 4. Suppose $a{\,\lvert\,}b$, then using the previous point $a{\,\&\,}(\lambda b) = \lambda (a{\,\&\,}b) = \lambda (b{\,\&\,}a) = (\lambda b){\,\&\,}a$. As a result of this proposition, the *left-multiplication map* $L_a:E\rightarrow E$ for $a\in E$ given by $L_a(b) = a{\,\&\,}b$ can be extended by linearity to the entirety of $V$ by $L_a(b-c) = L_a(b) - L_a(c)$. Similarly we can define the sequential product for any element in the positive cone of $V$ (not necessarily below the identity) by rescaling: $a{\,\&\,}b := \norm{a} ((\frac{1}{\norm{a}} a){\,\&\,}b)$. An effect $p\in E$ is called *sharp* when the only effect below both $p$ and $p^\perp$ is the zero effect, i.e when the following implication holds: $b\leq p$ and $b\leq p^\perp$ implies $b=0$. When $V=B(H)^{\text{sa}}$ the sharp effects are precisely the projections. This should be clear considering the following proposition: \[prop:sharpness\] Let $a\in E$ be an effect, $a$ is sharp if and only if $a{\,\&\,}a^\perp = 0$ if and only if $a{\,\&\,}a = a$. Originally proved in Ref. [@gudder2002sequential]. The equivalence of $a{\,\&\,}a^\perp =0$ and $a{\,\&\,}a = a$ follows straightforwardly by \[ax:add\] and proposition \[prop:unitzero\]: $a = a{\,\&\,}1 = a{\,\&\,}(a+a^\perp) = a{\,\&\,}a + a{\,\&\,}a^\perp$. So let us assume $a$ is sharp. By \[ax:compadd\] we have $a{\,\lvert\,}a^\perp$ so that $a{\,\&\,}a^\perp = a^\perp {\,\&\,}a$. By \[prop:decreasing\] we have $a{\,\&\,}a^\perp \leq a$ and similarly $a^\perp {\,\&\,}a \leq a^\perp$, the expression $a{\,\&\,}a^\perp$ is therefore below both $a$ and $a^\perp$ and by sharpness of $a$ has to be zero. Now suppose $a{\,\&\,}a^\perp = 0$ and let $b\leq a$ and $b\leq a^\perp$. If $b\leq a^\perp$ then by \[prop:decreasing\] we get $a{\,\&\,}b \leq a{\,\&\,}a^\perp =0$, and similarly we get $a^\perp {\,\&\,}b = 0$. By \[ax:orth\] we have $b{\,\lvert\,}a$ and $b{\,\lvert\,}a^\perp$ so that $b = b{\,\&\,}1 = b{\,\&\,}(a+a^\perp) = b{\,\&\,}a + b{\,\&\,}a^\perp = a{\,\&\,}b + a^\perp {\,\&\,}b = 0 + 0 = 0$. We call two sharp effects $p$ and $q$ *orthogonal* when $p{\,\&\,}q = 0$. Of course by \[ax:orth\] orthogonality is a symmetric relation, and we note that therefore orthogonal effects are also compatible. Let $a\in E$ be an effect. We define the powers of $a$ inductively to be $a^0 := 1$ and $a^n := a{\,\&\,}a^{n-1}$. We define the *classical algebra of $a$* to be the linear space $C(a)$ spanned by all the powers of $a$ and $a^\perp$. \[prop:polyspace\] Let $a\in E$ be an effect. $C(a)$ is a commutative sequential product space. $C(a)$ inherits the order structure from $V$ in the obvious way. Of course $a{\,\lvert\,}a^\perp$ and because of \[ax:compmult\] we have $a^n {\,\lvert\,}(a^\perp)^m$ for all $n$ and $m$. Because of \[ax:compadd\] and proposition \[prop:commumult\] linear combinations of compatible effects are also compatible and hence all effects of $C(a)$ are compatible. We are now in a position to use the seminal representation theorem from Kadison: [@kadison1951representation] Let $V$ be a complete order unit space with a bilinear operation $\circ$ that preserves positivity: $a\circ b \geq 0$ when $a,b\geq 0$. Then there exists a compact Haussdorff space $X$ such that $V\cong C(X)$, the space of continuous real-valued functions on $X$. This isomorphism is both an order- and algebra-isomorphism. Let $a\in E$ be an effect. $C(a)\cong {\mathbb{R}}^n$ for some $n\in {\mathbb{N}}$. The sequential product is linear in the second argument. Since $C(a)$ is a commutative sequential product space by proposition \[prop:polyspace\], its product is also linear in the first argument, and hence this operation is bilinear. It obviously preserves positivity by definition, so that Kadisons theorem applies and $C(a) \cong C(X)$. Since $V$ is finite-dimensional, $C(a)$ has to be so as well, and hence $C(X)$ is finite-dimensional. Of course $C(X)$ is finite-dimensional only when $X$ is finite so that $X$ is necessarily discrete. We conclude that indeed $C(X) \cong {\mathbb{R}}^n$. Let $a\in E$ be an effect. There exists a set of orthogonal sharp effects $p_i$ compatible with $a$ and positive scalars $\lambda_i$ such that $a = \sum_i \lambda_i p_i$. By the previous proposition $C(a) \cong {\mathbb{R}}^n$ and this space is obviously spanned by orthogonal sharp effects, hence we can find the desired $p_i$ and $\lambda_i$. By construction $p_i\in C(a)$ so that they are compatible with $a$. We will refer to a decomposition of $a$ in the above sense as a *spectral decomposition* of $a$. The existence of these decomposition is already enough to show that the space must be homogeneous: \[prop:homogen\] Let $C$ denote the cone of *strictly positive* elements in $V$, i.e the elements $v\in V$ such that $\exists \epsilon>0$ with $\epsilon 1\leq v$. The cone $C$ is homogeneous, i.e. for every $v,w\in C$ there exists an order isomorphism $\Phi: V\rightarrow V$ such that $\Phi(v)=w$. Originally proved in Ref. [@wetering2018characterisation]. For an arbitrary positive element $a$ we can find a spectral decomposition $a=\sum_i \lambda_i p_i$ such that $\lambda_i > 0$, i.e. we don’t write the zero ‘eigenvalues’. It is then straightforward to check that $a$ lies in the interior of the positive cone if and only if $\sum_i p_i=1$. In that case we define its *inverse* $a^{-1}=\sum_i \lambda_i^{-1} p_i$. Since the $p_i$ are all compatible and $a$ and $a^{-1}$ are linear combinations of these effects, they are also compatible and we calculate $a{\,\&\,}a^{-1} = \sum_{i,j} \lambda_i \lambda_j^{-1} p_i {\,\&\,}p_j = \sum_i \lambda_i \lambda_i^{-1} p_i = \sum_i p_i = 1$ so that $a^{-1}$ is indeed the inverse of $a$ with respect to the sequential product. The multiplication map $L_a(b) := a{\,\&\,}b$ is positive and has a positive inverse $L_{a^{-1}}$ due to \[ax:assoc\]: $a^{-1}{\,\&\,}(a{\,\&\,}b) = (a^{-1}{\,\&\,}a){\,\&\,}b = 1{\,\&\,}b = b$. The map $L_a$ is therefore an order isomorphism when $a$ is strictly positive. Now, for $a$ and $b$ strictly positive and hence invertible, define $\Phi: V\rightarrow V$ by $\Phi = L_bL_{a^{-1}}$. As this is a composition of order isomorphisms, it is also an order isomorphism and of course $\Phi(a) = b{\,\&\,}(a^{-1}{\,\&\,}a) = b{\,\&\,}1 = b$ as desired. Proof of self-duality {#sec:selfdual} ===================== With homogeneity of $V$ now established, we set our sights on proving self-duality. We will do this by first showing that the sharp effects form an atomic lattice with the *covering property* as defined in [@alfsen2012state]. The covering property states that for every sharp effect $p$ there is a unique number $r$ called the *rank* of $p$ such that we can write $p=\sum_{i=1}^r p_i$ where the $p_i$ are atomic and orthogonal. Using this definition we can define the rank of a space as equal to the rank of the unit effect. The existence of well-defined ranks of sharp effects allows us to reduce the question of self-duality to that of self-duality in rank 2 spaces. This problem is in turn solved by appealing to the classification result of Ref. [@ito2017p] that homogeneous spaces of rank 2 are always self-dual. The lattice of sharp effects ---------------------------- \[prop:sharpprop\] Let $a\in E$ be any effect and let $p\in E$ be sharp. 1. \[prop:belowsharp\] $a\leq p$ if and only if $p{\,\&\,}a = a{\,\&\,}p = a$ if and only if $p^\perp {\,\&\,}a = 0$. 2. \[prop:abovesharp\] $p\leq a$ if and only if $p{\,\&\,}a = a{\,\&\,}p = p$.   1. Suppose $a\leq p$ with $p$ sharp. Then $p^\perp {\,\&\,}a \leq p^\perp {\,\&\,}p = 0$ by proposition \[prop:sharpness\] and \[prop:decreasing\]. Hence $a{\,\lvert\,}p^\perp$ and $a{\,\lvert\,}p$ so that $a = a{\,\&\,}(p+p^\perp) = a{\,\&\,}p + a{\,\&\,}p^\perp = a{\,\&\,}p = p{\,\&\,}a$. For the other direction we note that $a=p{\,\&\,}a \leq p$ by \[prop:decreasing\]. 2. Suppose $p\leq a$ with $p$ sharp, then $a^\perp \leq p^\perp$ with $p^\perp$ sharp so that by the previous point $a{\,\lvert\,}p$ and $p{\,\&\,}a^\perp = 0$ so that $p=p{\,\&\,}(a+a^\perp) = p{\,\&\,}a$. For an effect $a\in E$ we let $\ceil{a}$ denote the smallest sharp element above $a$ (when it exists) and $\floor{a}$ the largest sharp element below $a$ (when it exists). The ceiling and the floor exist for any $a$. Moreover, writing $a=\sum_i \lambda_i p_i$ with $1\geq \lambda_i> 0$ and the $p_i$ sharp and orthogonal, then $\ceil{a} = \sum_i p_i$ and $\floor{a}=\ceil{a^\perp}^\perp$. Write $a=\sum_i \lambda_i p_i$. Of course $\sum_i p_i$ is an upper bound of $a$. Suppose $a\leq r$ for some sharp $r$. Then $\lambda_i p_i \leq r$, so by proposition \[prop:belowsharp\] $r{\,\&\,}(\lambda_i p_i) = \lambda_i p_i$. Using linearity we can rewrite this expression to $\lambda_i (r{\,\&\,}p_i) = \lambda_i p_i$ so that $r{\,\&\,}p_i = p_i$. So again by \[prop:belowsharp\] $p_i\leq r$ and $p_i{\,\lvert\,}r$ from which we get $r{\,\&\,}\sum_i p_i = \sum_i r{\,\&\,}p_i = \sum_i p_i$ so that also $\sum_i p_i\leq r$ which proves that it is the least upper bound. The other statement now follows because $a\leq b \iff b^\perp \leq a^\perp$. As a corollary of the above we also see that $\ceil{\lambda a} = \ceil{a}$ when $1\geq \lambda > 0$ and that $a$ is sharp if and only if $\ceil{a}=a$ or $\floor{a}=a$. We also note that $a\leq b$ implies that $\ceil{a}\leq \ceil{b}$. \[prop:lattice\] The sharp effects form a lattice: for two sharp effects $q$ and $p$, their least upper bound $p\vee q$ and greatest lower bound $p\wedge q$ exist. Furthermore the following relation holds between them: $(p\vee q)^\perp = p^\perp \wedge q^\perp$. We claim that $p\vee q = \ceil{\frac12(p+q)}$. Note that $p\leq p+q$ and thus that $\frac12 p \leq \frac12(p+q)$ so that $p = \ceil{p} = \ceil{\frac12p} \leq \ceil{\frac12(p+q)}$. Similarly we also have $q\leq \ceil{\frac12(p+q)}$ and thus this is an upper bound. Suppose now that $p\leq a$ and $q\leq a$ for some sharp $a$. Then of course also $\frac12(p+q)\leq \frac12(a+a) = a$. Taking the ceiling on both sides then shows that indeed $p\vee q = \ceil{\frac12(p+q)}$. To find $p\wedge q$ we note that $(\cdot)^\perp$ is an order-antiautomorphism, and thus that it interchanges joins with meets: $(p\vee q)^\perp = p^\perp\wedge q^\perp$. \[prop:sharplattice\] Let $a\in E$ be any effect and let $p\in E$ be sharp. 1. \[prop:meetsharp\] $p{\,\&\,}a = 0$ if and only if $p+a\leq 1$ in which case it is the least upper bound of the two. When $p{\,\&\,}a = 0$ their sum $p+a$ is sharp if and only if $a$ is also sharp. 2. \[prop:joinsharp\] If both $a$ and $p$ are sharp and $a$ and $p$ are compatible then $p{\,\&\,}a$ is sharp and equal to their join: $p\wedge a = p{\,\&\,}a$. Originally proved in Ref. [@gudder2002sequential]. 1. $p{\,\&\,}a = 0$ if and only if $p^\perp{\,\&\,}a = a$ which by proposition \[prop:abovesharp\] is true if and only if $a\leq p^\perp = 1-p$ so that indeed $p+a\leq 1$. That $p+a$ is an upper bound of $p$ and $a$ is obvious. Suppose now that $b$ is also an upper bound so that $p\leq b$ and $a\leq b$. We then calculate using proposition \[prop:abovesharp\] $p = p{\,\&\,}b = p{\,\&\,}(b-a + a) = p{\,\&\,}(b-a) + p{\,\&\,}a = p{\,\&\,}(b-a)$ so that $p\leq b-a$ again by \[prop:abovesharp\]. This gives $p+a \leq b$ so that $p+a$ is indeed the least upper bound. Now to show $p+a$ is sharp if and only if bot $p$ and $a$ are sharp: since $p{\,\&\,}a = 0$ we have $p{\,\lvert\,}a$ and thus also $p{\,\lvert\,}p+a$ and $a{\,\lvert\,}p+a$ by \[ax:compadd\]. We calculate $(p+a){\,\&\,}(p+a) = p{\,\&\,}p + 2 p{\,\&\,}a + a{\,\&\,}a = p + a{\,\&\,}a = (p+a) + (a-a{\,\&\,}a)$. We therefore have $(p+a){\,\&\,}(p+a) = p+a$ if and only if $a-a{\,\&\,}a = 0$ which proves the result by proposition \[prop:sharpness\]. 2. Suppose both $p$ and $a$ are sharp and that $p{\,\lvert\,}a$. We calculate: $$(p{\,\&\,}a){\,\&\,}(p{\,\&\,}a) = (p{\,\&\,}a){\,\&\,}(a{\,\&\,}p) =p{\,\&\,}(a{\,\&\,}(a{\,\&\,}p)) = p{\,\&\,}(a{\,\&\,}p) = p{\,\&\,}(p{\,\&\,}a) = p{\,\&\,}a$$ where we have used that $a{\,\lvert\,}a{\,\&\,}p$ and $p{\,\lvert\,}a{\,\&\,}p$ by \[ax:compmult\]. Hence $p{\,\&\,}a$ is sharp. It is a lower bound of $p$ and $a$ by \[prop:decreasing\]. Suppose $b\leq p, a$ is also a lower bound. We calculate $p{\,\&\,}a = p{\,\&\,}(a-b + b) = p{\,\&\,}(a-b) + p{\,\&\,}b = p{\,\&\,}(a-b) + b \geq b$, where we have used that $p{\,\&\,}b = b$ as a consequence of proposition \[prop:belowsharp\]. \[lem:ceilzero\] Let $a,b\in E$. If $b{\,\&\,}a = 0$ then $b{\,\&\,}\ceil{a} = 0$. Write $a= \sum_i \lambda_i p_i$. If $b{\,\&\,}a = 0 = \sum_i \lambda_i b{\,\&\,}p_i$, then we must have $b{\,\&\,}p_i=0$ for all $p_i$. Since $\ceil{a}=\sum_i p_i$ the claim follows. \[lem:ceilceil\] Let $p\in E$ be sharp and $a\in E$ arbitrary, then $\ceil{p{\,\&\,}a} = \ceil{p{\,\&\,}\ceil{a}}$. First of all $p{\,\&\,}a \leq p{\,\&\,}\ceil{a}$ so that $\ceil{p{\,\&\,}a}\leq \ceil{p{\,\&\,}\ceil{a}}$. It suffices therefore to prove the other inequality. Because $p{\,\&\,}a \leq p{\,\&\,}1 = p$ we also have $\ceil{p{\,\&\,}a}\leq \ceil{p}=p$ implying $\ceil{p{\,\&\,}a}^\perp{\,\&\,}p =0$ so that $\ceil{p{\,\&\,}a}^\perp$ and $p$ are compatible. Now because $p{\,\&\,}a \leq \ceil{p{\,\&\,}a}$ we can use proposition \[prop:belowsharp\] to write $0=\ceil{p{\,\&\,}a}^\perp {\,\&\,}(p{\,\&\,}a) = (\ceil{p{\,\&\,}a}^\perp{\,\&\,}p){\,\&\,}a = (\ceil{p{\,\&\,}a}^\perp {\,\&\,}p){\,\&\,}\ceil{a} = \ceil{p{\,\&\,}a}^\perp{\,\&\,}(p{\,\&\,}\ceil{a})$ where we have used lemma \[lem:ceilzero\] to replace $a$ with $\ceil{a}$. Since $\ceil{p{\,\&\,}a}^\perp{\,\&\,}(p{\,\&\,}\ceil{a}) = 0$ we use \[prop:belowsharp\] again to conclude $p{\,\&\,}\ceil{a}\leq \ceil{p{\,\&\,}a}$ so that indeed $\ceil{p{\,\&\,}\ceil{a}}\leq \ceil{p{\,\&\,}a}$. Atomic effects {#sec:atomeffect} -------------- We call a nonzero sharp effect $p\in E$ *atomic* if for all $a\in E$ with $a\leq p$ we have $a=\lambda p$ for some $\lambda\in[0,1]$. Every sharp effect can be written as a sum of orthogonal atomic effects. Let $p$ be sharp and suppose it is not atomic, then we can find $0\leq a\leq p$ such that $a\neq \lambda p$ for any $\lambda\in [0,1]$. Write $a= \sum_i \lambda_i q_i$ where the $q_i$ are sharp and orthogonal. Then $\lambda_i q_i\leq p$ and thus also $\ceil{\lambda_i q_i} = q_i\leq \ceil{p}=p$. If all the $q_i$ are equal to $p$, then $a$ is a multiple of $p$, so at least one of the $q_i$ is strictly smaller than $p$. We can repeat this process for $q_i$ and $p-q_i$, getting a sequence of nonzero orthogonal sharp effects that sum up to $p$. Since the space is finite-dimensional and orthogonal effects are linearly independent this process must stop after a finite amount of steps in which case we are left with atomic effects. \[cor:spectralatomic\] Every $a\in V$ can be written as $a=\sum_i \lambda_i p_i$ where the $p_i$ are orthogonal sharp atomic effects. For every $a\in V$ we can find a spectral decomposition in terms of orthogonal sharp effects. The previous proposition shows that these sharp effects can be further decomposed into atomic effects. Recall that the norm in an order unit space is defined as ${\norm{a}:=\inf\{r~;~-r 1 \leq a \leq r1\}}$. \[lem:atomicnorm\] A non-zero effect $p$ is atomic if and only if we have $p{\,\&\,}a = \norm{p{\,\&\,}a}p$ for all $a\in E$. First we establish that the norm of any non-zero sharp effect is equal to $1$. Let $q$ be sharp. We see that $q = q{\,\&\,}q \leq q {\,\&\,}(\norm{q} 1) = \norm{q} q{\,\&\,}1 = \norm{q} q$ so that $\norm{q}\geq 1$. But since $q\leq 1$ we also have $\norm{q}\leq 1$. Suppose $p$ is atomic. Because $0\leq p{\,\&\,}a \leq p$ we must have $p{\,\&\,}a = \lambda p$ for some $0\leq \lambda \leq 1$ so that $\norm{p{\,\&\,}a} = \lambda \norm{p} = \lambda$ because $p$ is sharp. For the other direction first note that $p= p{\,\&\,}\ceil{p} = \norm{p{\,\&\,}\ceil{p}} p = \norm{p}p$ so that necessarily $\norm{p}=1$ (since $p\neq 0$). By writing $p$ as a linear combination of sharp effects we see that then also $\norm{p^2}=1$. Now $p{\,\&\,}p = \norm{p^2} p = p$ so that $p$ is sharp. Let $q\leq p$ be non-zero sharp. Then $q=p{\,\&\,}q = \norm{p{\,\&\,}q}p=p$ (using again that $\norm{q}=1$ because $q$ is sharp) so there are no non-trivial sharp effects below $p$. Now if $a=\sum_i \lambda_i q_i$ lies below $p$ we see that $\lambda_iq_i\leq p$ so that $\ceil{\lambda_i q_i} = q_i\leq \ceil{p}=p$ so that $q_i=p$ and thus $a=\lambda p$. Since all $a\in E$ can be written in this way we conclude that this holds for all $a\leq p$, so that $p$ is indeed atomic. The set of atomic effects is closed in the norm topology. Let $p_n\rightarrow p$ be a norm converging set of atomic effects $p_n$. We need to show that $p$ is also atomic. As a result of the previous lemma we have $p_n{\,\&\,}a = \norm{p_n {\,\&\,}a} p_n$ for all effects $a$. By continuity of $\&$ (i.e. axiom \[ax:cont\]) we have $p_n{\,\&\,}a \rightarrow p{\,\&\,}a$ so that $p{\,\&\,}a = \lim p_n {\,\&\,}a = \lim \norm{p_n{\,\&\,}a}p_n = \norm{p{\,\&\,}a} p$. Using the previous lemma again we conclude that $p$ is indeed atomic. \[prop:atompreservation\] Let $a\in E$ be arbitrary and $p\in E$ be atomic, then $a{\,\&\,}p$ is proportional to an atomic effect. The property that $0\leq a\leq p \implies a=\lambda p$ is determined by the order, so any order isomorphism preserves it. If $a$ is invertible then $L_a: V\rightarrow V$ given by $L_a(b):= a{\,\&\,}b$ is an order isomorphism, so that $L_a(p)$ must be proportional to an atomic effect. For non-invertible $a$ we write $a_n = a+\frac{1}{n}$, so that $a_n$ is invertible and the sequence $a_n$ converges to $a$. Let $q_n = (a_n{\,\&\,}p)/\norm{a_n {\,\&\,}p}$, then all the $q_n$ are atomic. By the continuity condition \[ax:cont\] we get $a_n{\,\&\,}p \rightarrow a{\,\&\,}p$ so that also $\norm{a_n {\,\&\,}p}\rightarrow \norm{a{\,\&\,}p}$. The sequence $q_n$ is therefore also convergent and since the set of atomic effects is closed by the previous corollary we conclude that $q_n\rightarrow q=(a{\,\&\,}p)/\norm{a{\,\&\,}p}$ is atomic. The Covering Property {#sec:coverprop} --------------------- At this point we know that the set of sharp effects forms an atomic lattice, but in fact it has the much stronger *covering property* that allows us to attach a *rank* to each sharp effect: the amount of atomic effects needed to make the effect. To show this we need some results from Alfsen and Schultz [@alfsen2012geometry] that unfortunately were proven in a slightly different setting. We will repeat these results with very similar proofs, but adapted to work in the setting of sequential product spaces. \[lem:wedgeminus\] [@alfsen2012geometry Lemma 8.9] Let $q$ and $p$ be sharp with $q\leq p$, then $p-q = p\wedge q^\perp$. Let $q$ and $p$ be sharp with $q\leq p$, then $p-q$ is sharp and $p{\,\lvert\,}q$, so also $p{\,\lvert\,}q^\perp$ so that $p-q = p{\,\&\,}q^\perp = p\wedge q^\perp$ by \[prop:joinsharp\]. [@alfsen2012geometry Theorem 8.32]: Let $p$ be sharp and $a$ arbitrary, then $\ceil{p{\,\&\,}a} = (\ceil{a}\vee p^\perp)\wedge p$. Because $\ceil{p{\,\&\,}a} = \ceil{p{\,\&\,}\ceil{a}}$ by lemma \[lem:ceilceil\] it suffices to prove this for sharp $a$. We prove the equality by showing that an inequality holds in both directions. Since $p^\perp \leq a\vee p^\perp$ we have $p^\perp {\,\lvert\,}(a\vee p^\perp)$ by \[prop:belowsharp\] so that in turn $p{\,\lvert\,}(a\vee p^\perp)$ by \[ax:compadd\]. We proceed by using \[ax:assoc\]: $(a\vee p^\perp){\,\&\,}(p{\,\&\,}a) = ((a\vee p^\perp){\,\&\,}p){\,\&\,}a = p{\,\&\,}((a\vee p^\perp){\,\&\,}a) = p{\,\&\,}a$ because $a\vee p^\perp \geq a$. Therefore $p{\,\&\,}a \leq a\vee p^\perp$ which implies that $\ceil{p{\,\&\,}a} \leq a\vee p^\perp$. Since also $p{\,\&\,}a\leq p$ and therefore $\ceil{p{\,\&\,}a}\leq p$ we conclude that $\ceil{p{\,\&\,}a}\leq (a\vee p^\perp)\wedge p = (\ceil{a}\vee p^\perp)\wedge p$. Now for the other direction: we obviously have $p^\perp{\,\&\,}(p{\,\&\,}a) = (p^\perp {\,\&\,}p){\,\&\,}a = 0$ by \[ax:compadd\] and \[ax:assoc\] so that by lemma \[lem:ceilzero\] $p^\perp{\,\&\,}\ceil{p{\,\&\,}a}=0$. Using \[prop:belowsharp\] we see then that $p{\,\lvert\,}\ceil{p{\,\&\,}a}^\perp$ and therefore by \[prop:joinsharp\] that $\ceil{p{\,\&\,}a}^\perp{\,\&\,}p = \ceil{p{\,\&\,}a}^\perp\wedge p$. Since $p{\,\&\,}a \leq \ceil{p{\,\&\,}a}$ we calculate using \[prop:belowsharp\]: $0=\ceil{p{\,\&\,}a}^\perp{\,\&\,}(p{\,\&\,}a) = (\ceil{p{\,\&\,}a}^\perp {\,\&\,}p){\,\&\,}a = (\ceil{p{\,\&\,}a}^\perp \wedge p){\,\&\,}a$ so that $a\leq (\ceil{p{\,\&\,}a}^\perp\wedge p)^\perp = \ceil{p{\,\&\,}a}\vee p^\perp$ by proposition \[prop:lattice\]. Then of course also $a\vee p^\perp \leq \ceil{p{\,\&\,}a}\vee p^\perp$ and by noting that $\ceil{p{\,\&\,}a}$ and $p^\perp$ are orthogonal and using \[prop:meetsharp\]: $\ceil{p{\,\&\,}a}\vee p^\perp = \ceil{p{\,\&\,}a}+p^\perp$. Bringing the $p^\perp$ to the other side and using lemma \[lem:wedgeminus\] (which applies because $p^\perp\leq a\vee p^\perp$) then gives $(a\vee p^\perp)\wedge p = a\vee p^\perp - p^\perp \leq \ceil{p{\,\&\,}a}$. \[prop:coverprop\] [@alfsen2012geometry Proposition 9.7]: The lattice of sharp effects has the covering property: for $q$ atomic, the expression $(q\vee p)\wedge p^\perp = (q\vee p)-p$ is either zero or atomic. In other words: when $q$ does not lie below $p$ then there is no sharp effect lying strictly between $p$ and $q\vee p$. By the previous lemma $(q\vee p)\wedge p^\perp = \ceil{p^\perp {\,\&\,}q}$. Since $p^\perp{\,\&\,}q$ is proportional to an atom by proposition \[prop:atompreservation\], it is either zero (when $q\leq p$) in which case we are done, or non-zero in which case $\ceil{p^\perp {\,\&\,}u}$ is an atom, which also proves the statement. The equality $(q\vee p)\wedge p^\perp = (q\vee p)-p$ follows directly from lemma \[lem:wedgeminus\]. The last observation is proven as follows. Suppose $p< r < q\vee p$. Subtract $p$ to get $0< r-p < q\vee p - p$. As $r-p$ is sharp and $q\vee p -p$ has been established to be atomic, this is not possible. Let $p$ be sharp and let $p_i$ be a collection of atomic orthogonal effects such that $p=\sum_i^n p_i$. The minimal size of such a collection is called the *rank* of $p$. The rank of a sequential product space is defined to be the rank of the unit effect. With the covering property proven we can finally prove the following ‘dimension’ theorem: [@alfsen2012state Proposition 1.66] Write $p=\sum_i^n p_i$ where the $p_i$ are orthogonal and atomic, then $n={\text{rnk}\xspace}~ p$, i.e. all ways of writing $p$ as a sum of atomic effects require an equal amount of atomic effects. Furthermore, suppose $q\leq p$ then ${\text{rnk}\xspace}~ q \leq {\text{rnk}\xspace}~ p$ and if also ${\text{rnk}\xspace}~ q = {\text{rnk}\xspace}~ p$ then necessarily $q=p$. Let $p^\prime = p_1\vee\ldots\vee p_{n-1}$. Then $p^\prime\vee p_n = p$ and by the covering property (proposition \[prop:coverprop\]) there is no sharp effect strictly between $p^\prime$ and $p$. Suppose now $q\leq p$ is atomic and suppose that $q$ is not below $p^\prime$. Then $p^\prime \vee q$ must be strictly greater than $p^\prime$, but since this must also lie below $p$ we conclude that $p^\prime \vee q = p$. Let $p = \sum_j^r q_j = q_1\vee\ldots\vee q_r$ where $r:={\text{rnk}\xspace}~ p$ is the minimal amount of terms needed to write $p$ as a sum of atomic effects. We must then of course have $r\leq n$. Let $q=q_2\vee\ldots\vee q_r$, then $q$ must lie strictly below $p$ since $q_1\leq p$ but not $q_1\leq q$. It then follows that there must be a $p_i$ such that $p_i$ does not lie below $q$ as well, since otherwise $p = p_1\vee\ldots\vee p_n \leq q < p$. Without loss of generality let this $p_i$ be $p_1$. By the previous paragraph we must have $p_1\vee q = p_1\vee q_2\ldots\vee q_r = p$. This procedure can be repeated with $q_2,\ldots, q_r$ until we are left with the equation $p_1\vee \ldots\vee p_r = p$. Suppose $n>r$, then because $p_n$ is orthogonal to all the other $p_i$’s we have in particular $p_n\leq p_1^\perp \wedge \ldots \wedge p_r^\perp = (p_1\vee \ldots \vee p_r)^\perp = p^\perp$. Since also $p_n\leq p$ we get $p_n=0$ by sharpness which is a contradiction. We therefore have $n=r$. Now suppose $q=\sum_j^s q_j\leq p=\sum_i^r p_i$. Where $s={\text{rnk}\xspace}~ q$. Since $p-q$ is sharp we can write $p-q = \sum_k^t v_k$. Then because $p = \sum_j^s q_j + \sum_k^t v_k$ we must by the previous points have $s+t = r$ so that indeed ${\text{rnk}\xspace}~ q\leq {\text{rnk}\xspace}~ p$. When these ranks are equal we must have $t=0$ so that indeed $p-q = 0$. \[cor:rank\] Let $p\neq q$ be two atomic sharp effects and suppose $0\leq a\leq p\vee q$, then $a=\lambda_1 r_1 + \lambda_2 r_2$ where the $r_i$ are orthogonal and atomic and $r_1 + r_2 = p\vee q$. By proposition \[prop:coverprop\] $(p\vee q) - p$ is atomic so that $p\vee q$ can be written as the sum of two atomic sharp effects so that indeed ${\text{rnk}\xspace}~ p\vee q = 2$. Suppose $0\leq a \leq p\vee q$. Let $a=\sum_i^n \lambda_i r_i$ be a spectral decomposition of $a$ with the $r_i$ orthogonal and atomic. Of course $\ceil{a} \leq p\vee q$ so that by the previous proposition we must have ${\text{rnk}\xspace}\ceil{a} \leq 2$. Since also by the previous proposition ${\text{rnk}\xspace}~\sum_i^n r_i = n$ we see that we must have $n=2$ and thus that $a$ is as desired. Self-duality {#sec:subselfdual} ------------ The important concept of this section will be that of *strict convexity* of a cone, since this is related to a characterisation theorem for homogeneous spaces. Let $C$ be a positive cone of an order unit space $V$. We call $F\subseteq C$ a *face* of $C$ if $F$ is a convex set such that whenever $\lambda a+\lambda^\perp b \in F$ with $0<\lambda<1$ then $a,b \in F$. The face $\{\lambda p~;~ \lambda \in {\mathbb{R}}_{\geq 0}\}$ of $C$ defined by an extreme point $p\in C$ is called an *extreme ray*. A face is called *proper* when it is non-empty and not equal to $C$. If the only proper faces of a cone are extreme rays the cone is *strictly convex*. \[prop:itochar\] [@ito2017p] Let $V$ be a finite-dimensional ordered vector space with a strictly convex homogeneous positive cone, then $V$ is order isomorphic to a spin-factor, i.e. $V\cong H\oplus {\mathbb{R}}$ where $H$ is a real finite-dimensional Hilbert space with the order on $H\oplus {\mathbb{R}}$ given by $(v,t)\geq 0 \iff t\geq \norm{v}_2$. \[def:orderideal\] Let $p$ and $q$ be two unequal atomic effects. We define the *order ideal* generated by $p$ and $q$ as $V_{p,q}:= \{v\in V~;~\exists n: -n~p\vee q \leq v \leq n ~p\vee q\}$. $V_{p,q}$ is an order unit space with unit $p\vee q$. If we have $a,b\in [0,1]_{V_{p,q}}$ then $a{\,\&\,}b \leq a \leq p\vee q$ so that the sequential product restricts to this space. We therefore conclude by proposition \[prop:homogen\] that this space has a homogeneous positive cone. Let $p$ and $q$ be two unequal atomic effects. The positive cone of $V_{p,q}$ is strictly convex. Let $F$ be a proper face of the positive cone of $V_{p,q}$. Let $a\in F$ and write $a=\lambda (\lambda^{-1} a) + \lambda^\perp 0$, so that $\lambda^{-1} a \in F$. We see that $F$ is closed under positive scalar multiplication and thus that we can restrict ourselves to effects. Let $a\in F$ be an effect. By corollary \[cor:rank\] we can write $a=\lambda r + \mu r^\perp$ for some $\lambda,\mu\geq 0$ and $r$ atomic. Suppose both $\lambda,\mu >0$, then because $F$ is a face $r,r^\perp \in F$ so that $\frac{1}{2}1 = \frac{1}{2}(r+r^\perp)\in F$. But then since $1=s+s^\perp$ for any atomic $s$ we see that $F$ has to be the entire positive cone. We conclude that we must have had $a=\lambda r$ for some atomic $r$. If there were some other atomic $s \in F$, then we can consider $a = \frac{1}{2}(r + s)$. We know that $a$ can’t be atomic so we can write it as $a=\lambda r + \mu r^\perp$ with $\lambda, \mu > 0$ which is a contradiction. We conclude that $F$ is an extreme ray and thus that the positive cone of $V_{p,q}$ is strictly convex. Let $p$ and $q$ be two different atomic effects, then $V_{p,q}$ is isomorphic to a spin-factor. Follows directly from the previous lemma and proposition \[prop:itochar\]. Recall that a *state* on an order unit space is a positive linear map $\omega: V\rightarrow {\mathbb{R}}$ such that $\omega(1) = 1$. For an atomic effect $p$ in a spin-factor there exists a unique state $\omega_p$ such that $\omega_p(p)=1$. A spin-factor has *symmetry of transition probabilities* [@alfsen2012geometry]: $\omega_p(q)=\omega_q(p)$ for any two atomic effects $p$ and $q$. We can use the previous results to prove that symmetry of transition probabilities also holds for arbitrary (finite-dimensional) sequential product spaces. \[prop:uniquestate\] Let $p,q \in E$ be atomic effects. There exist unique pure states $\omega_p$ and $\omega_q$ such that $\omega_p(p)=1$ and $\omega_q(q)=1$. Furthermore for these states we have $\omega_p(q)=\omega_q(p)$. The states separate the points of an order unit space [@alfsen2012state Corollary 1.27] so that for $p$ we can find a state $\omega$ such that $\omega(p)\neq 0$. Define $\omega_p(a) := \omega(p{\,\&\,}a)/(\omega(p))$, then $\omega_p$ is a state and $\omega_p(p)=1$. Suppose there is another state $\omega^\prime$ such that $\omega^\prime(p)=1$. Let $q\neq p$ be any other atomic effect (if there is no atomic $q\neq p$ then $V\cong {\mathbb{R}}$ and we are already done) and look at the restrictions of the states $\omega_p$ and $\omega^\prime$ to the space $V_{p,q}$. These restriction maps are still states as $\omega_p(p\vee q)\geq \omega_p(p)=1$ (and similarly for $\omega^\prime$). Because states with this property are unique on spin-factors we can conclude that these restricted states are equal on this subspace and in particular $\omega_p(q)=\omega^\prime(q)$. Since $q$ was arbitrary and the atomic effects span $V$ we conclude that $\omega_p=\omega^\prime$ so that $\omega_p$ is indeed unique. For any two atomic $p$ and $q$ we can look at their unique pure states $\omega_p$ and $\omega_q$ as restricted to $V_{p,q}$ for which we know that $\omega_p(q)=\omega_q(p)$ which finishes the proof. \[prop:atomicprod\] Let $p$ and $q$ be atomic sharp effects. $p$ and $q$ are orthogonal, i.e. $p{\,\&\,}q = q{\,\&\,}p = 0$ if and only if $\omega_p(q) = \omega_q(p) = 0$. Furthermore, $p{\,\&\,}q = \omega_p(q) p$. Note that if $q{\,\&\,}p =0$ then by proposition \[prop:meetsharp\] $q+p\leq 1$ so that $1=\omega(p)\leq\omega_p(q+p)\leq \omega(1)=1$ from which we conclude that $\omega_p(q)=0$. So if $p$ and $q$ are orthogonal then $\omega_p(q)=\omega_q(p)=0$. For the converse we will show that $p{\,\&\,}q = \omega_p(q) p$, from which it directly follows that $\omega_p(q) = 0 \implies p{\,\&\,}q = 0$. Since $p$ is atomic we of course have $p{\,\&\,}q = \lambda p$ for some $\lambda \geq 0$. Let $\omega^\prime(a) = \omega_p(p{\,\&\,}a)$, then $\omega^\prime(p) = \omega_p(p{\,\&\,}p) = \omega_p(p) = 1$, so that by the uniqueness of $\omega_p$ we have $\omega^\prime = \omega_p$. We then see that $\omega_p(q) = \omega^\prime(q) = \omega_p(p{\,\&\,}q) = \omega_p(\lambda p) = \lambda \omega_p(p) = \lambda$. There exists an inner product on $V$ such that the positive cone is self-dual with respect to this inner product. For atomic $p$ and $q$ we let $\inn{p,q}:= \omega_p(q)=\omega_q(p)=\inn{q,p}$. We can then extend it by linearity to arbitrary $a=\sum_i \lambda_i p_i$ and $b=\sum_j \mu_j q_j$ by $\inn{a,b}:= \sum_{i,j} \lambda_i\mu_j \inn{p_i,q_i}$. This is well-defined since $\inn{a,b} = \sum_i \lambda_i \omega_{p_i}( \sum_j \mu_j q_j) = \sum_i \lambda_i \omega_{p_i}(b) = \sum_j \mu_j \omega_{q_j}(a)$ so that this is independent of the representation of $a$ and $b$ in as linear combinations of atomic effects. Now $\inn{a,a} = \sum_{i,j} \lambda_i \lambda_j \omega_{p_i}(p_j) = \sum_i \lambda_i^2$ since $p_i$ and $p_j$ are orthogonal when $i\neq j$ and $\omega_{p_i}(p_i)=1$. We conclude that $\inn{a,a}\geq 0$ and that it is only equal to zero when $a=0$ so that $\inn{\cdot,\cdot}$ indeed is an inner product. If $a$ and $b$ are positive elements then we can write them as $a=\sum_i \lambda_i p_i$ and $b=\sum_j \mu_j q_j$ where all the $\lambda_i$ and $\mu_j$ are greater than zero. It then easily follows that $\inn{a,b}\geq 0$ because $\omega_{p_i}(q_j)\geq 0$. Conversely if we have $a=\sum_i \lambda_i p_i$ with $\lambda_i$ not necessarily positive with $\inn{a,b}\geq 0$ for all $b\geq 0$, then we can in particular take $b=p_j$ to see that $0\leq \inn{a,p_j} = \lambda_j$ from which we conclude that indeed $a\geq 0$. A finite-dimensional sequential product space is isomorphic to a Euclidean Jordan algebra. By proposition \[prop:homogen\] the space is homogeneous, and by the previous proposition it is self-dual. The Koecher-Vinberg theorem [@koecher1957positivitatsbereiche] states that any homogeneous self-dual ordered vector space is order-isomorphic to a Euclidean Jordan algebra. The Jordan product from a sequential product {#sec:jordanproduct} ============================================ The Koecher-Vinberg theorem is a rather indirect way of establishing the Jordan algebra structure of the space. Since we don’t have just a homogeneous self-dual space, but we also have access to the sequential product we can in fact construct the Jordan product directly. That is what we will strive for in this section. We will use the construction of the Jordan product from the work of Alfsen and Schultz [@alfsen2012geometry], but then adapted to our setting. We call a real vector space $V$ a *Jordan algebra* when it has a bilinear commutative operation $*$ that satisfies the *Jordan identity*: $a*(b*(a*a)) = (a*b)*(a*a)$. We call $V$ a *Euclidean* Jordan algebra when it is furthermore a Hilbert space with inner product $\inn{\cdot, \cdot}$ such that $\inn{a*b,c} = \inn{b,a*c}$. Note: By writing the Jordan product operator $T_a(b) := a*b$ and using the commutativity of the product we can also write the Jordan identity as $T_aT_{a*a} = T_{a*a}T_a$. Let $p$ be an atomic sharp effect and let $b\in V$ be arbitrary. We define their *Jordan product* as $p*b = \frac{1}{2}({\text{id}}+ L_p - L_{p^\perp}) b$. \[lem:restrict\] Let $p$ and $q$ be atomic sharp effects, then $p^\perp {\,\&\,}q = p^\prime {\,\&\,}q$ where $p^\prime = p\vee q - p$. First note that $p^\perp = 1-p = 1-p\vee q + p\vee q -p = (p\vee q)^\perp + p^\prime$ and hence that $p^\prime \leq p^\perp$ so that $p^\prime {\,\lvert\,}p^\perp$ by proposition \[prop:belowsharp\]. We then also have $p^\perp{\,\&\,}(p\vee q) = (p\vee q){\,\&\,}p^\perp = (p\vee q) {\,\&\,}((p\vee q)^\perp + p^\prime) = p^\prime$. Now using the fact that we are working with compatible effects and that $q\leq p\vee q$ we calculate $p^\perp {\,\&\,}q = p^\perp {\,\&\,}((p\vee q){\,\&\,}q) = (p^\perp {\,\&\,}(p\vee q)){\,\&\,}q = p^\prime {\,\&\,}q$. \[lem:atomiccommute\] [@alfsen2012geometry Lemma 9.29]: Let $p$ and $q$ be sharp atomic effects. 1. $p*q = q*p$. 2. When $p{\,\&\,}q = 0$ we have $p*q=0$ and in that case for any $b\in V$: $p*(q*b)=q*(p*b)$. 3. $p*p = p$.   1. If $p=q$ this is trivial, so assume that $p\neq q$. Let us denote $p^\prime = p\vee q - p$. By proposition \[prop:coverprop\] $p^\prime$ is atomic. By proposition \[prop:atomicprod\] we have $p{\,\&\,}q = \omega_p(q) p = \inn{p,q} p$ and similarly $p^\prime {\,\&\,}q = \inn{p^\prime ,q} p^\prime$. Expanding the definition of $p*q$ and using lemma \[lem:restrict\] to write $p^\perp {\,\&\,}q = p^\prime {\,\&\,}q$ where $p^\prime = p\vee q - p$ we calculate $$\begin{aligned} 2 (p*q) &= q + \inn{p,q}p - \inn{p^\prime, q} p^\prime \\ &= q + \inn{p,q}p - \inn{p^\prime, q} (p\vee q - p) \\ &= q + (\inn{p,q}p + \inn{p^\prime, q})p - \inn{p^\prime, q} (p\vee q) \\ &= q + \inn{p\vee q,q}p + \inn{p\vee q - p, q} (p\vee q) \\ &= q+p + (1-\inn{p,q})(p\vee q) \end{aligned}$$ which is indeed symmetric in $p$ and $q$. 2. When $p{\,\&\,}q = 0$ we have $q\leq p^\perp$ so that $p^\perp {\,\&\,}q = q$ which indeed gives $p*q = {\frac12(q + p{\,\&\,}q - p^\perp {\,\&\,}q)} = \frac12(q-q) = 0$. For the second point we note that because $p{\,\&\,}q = 0$ we have $p{\,\lvert\,}q, q^\perp$ and $q{\,\lvert\,}p^\perp$, and hence that the maps $L_p, L_{p^\perp}, L_q$ and $L_{q^\perp}$ commute so that the maps $b\mapsto p*b$ and $b\mapsto q*b$ will commute as well. 3. Follows immediately from $p{\,\&\,}p = p$ and $p^\perp {\,\&\,}p =0$. As a result of this lemma we can extend the Jordan product by linearity to the entirety of the space. Let $a,b\in V$ be arbitrary. Let $a=\sum_i \lambda_i p_i$ and $b=\sum_j \mu_j q_j$ be spectral decompositions with the $p_i$ and $q_j$ atomic. Define their Jordan product as $a*b = \sum_{i,j} \lambda_i \mu_j p_i*q_j$. We write $T_a:V\rightarrow V$ for the operator that sends $b$ to $a*b$. \[prop:jordanproduct\] The Jordan product is well-defined, bilinear, commutative and furthermore 1. If $a{\,\lvert\,}b$ then $T_aT_b = T_bT_a$. 2. \[prop:jordancomm\] If $a{\,\lvert\,}b$ then $T_a b = a^+{\,\&\,}b - a^- {\,\&\,}b$ where $a^+$ and $a^-$ are the unique orthogonal positive elements such that $a=a^+ - a^-$. We first note that we of course have $a*b = \sum_i \lambda_i p_i*b$ so that the definition is independent of how $b$ is represented as a sum of atomic sharp effects. By the previous lemma the product is commutative and therefore we see it is bilinear and well-defined. 1. Suppose $a{\,\lvert\,}b$. By considering the classical algebra spanned by both $a$ and $b$ we can find an orthogonal set of atomic sharp effects $p_i$ such that $a=\sum_i \lambda_i p_i$ and $b=\sum_i \mu_i p_i$. Since $p_i*p_j=0$ for $i\neq j$ we have $p_i*(c*p_j) = p_j*(c*p_i)$ by lemma \[lem:atomiccommute\] so that we can then write $$T_bT_a c = b*(a*c) = \sum_{i,j} \mu_i\lambda_j p_i*(c*p_j) = \sum_{i,j} \mu_i \lambda_j p_j*(c*p_i) = a*(c*b) = T_aT_b c$$ which holds for all $c$. We conclude that $T_bT_a=T_aT_b$. 2. If atomic $p_i$ commutes with $b$, then $p_i*b = \frac{1}{2}({\text{id}}+ U_{p_i} - U_{p_i^\perp})b = \frac{1}{2}(b + p_i{\,\&\,}b - p_i^\perp b) = \frac{1}{2}((p_i+p_i^\perp){\,\&\,}b + p_i{\,\&\,}b - p_i^\perp b) = p_i{\,\&\,}b$. So by writing $a=\sum_i \lambda_i p_i = \sum_{i, \lambda_i>0} \lambda p_i - \sum_{i, \lambda_i<0}\lvert \lambda_i\rvert p_i = a^+ - a^-$ with $p_i{\,\lvert\,}b$ the desired result follows by linearity. Let $V$ be a finite-dimensional sequential product space, then it is a Euclidean Jordan algebra with the Jordan product as defined above. We have already established that the product is bilinear and commutative. Note that since $a{\,\lvert\,}a$ we get $a*a = a^+{\,\&\,}a - a^-{\,\&\,}a = (a^+)^2 + (a^-)^2 = a^2$ using proposition \[prop:jordancomm\]. Because of course $a{\,\lvert\,}a^2$ we get $T_aT_{a*a} = T_{a*a}T_a$ as a consequence of the previous proposition so that the Jordan identity holds. Since $a*a = a^2\geq 0$ we also see that the algebra is *formally real*: if $\sum_i a_i*a_i = 0$ then for all $i$: $a_i=0$. It is a well-known result (see for instance [@faraut1994analysis Proposition VIII.4.2]) that if a Jordan product is formally real, that the algebra is Euclidean, with the product being symmetric with regards to the (essentially unique) self-dual inner product. Note: It is also possible to show in a more direct manner that the Jordan product is symmetric with respect to the inner product. We sketch here how to do so. First it must be established that $L_a$ for invertible $a$ commute with their adjoints $L_a^*$ by exploiting the fact that $\Theta = L_{a^{-1}}L_a^*$ must be a unital order-isomorphism necessarily satisfying $\Theta^{-1} = \Theta^*$. Because the mapping $a\mapsto L_a$ is continuous, the result extends to all $a$ and hence it holds in particular for $L_p$ with $p$ sharp. It is a standard result that an idempotent map that commutes with its adjoint is in fact self-adjoint. Since the Jordan product is defined as a linear combination of product maps of sharp effects this indeed establishes the desired result. Local Tomography and C\*-algebras {#sec:loctom} ================================= In this section we will let $V$ and $W$ be finite-dimensional sequential product spaces, and hence by the previous sections Euclidean Jordan algebras. For the duration of this section we will assume that their linear algebraic tensor product $V\otimes W$ is also a sequential product space and that the sequential product satisfies $$(a_1\otimes b_1){\,\&\,}(a_2\otimes b_2) = (a_1{\,\&\,}a_2) \otimes (b_1{\,\&\,}b_2).$$ Note that by definition of the tensor product any element of $V\otimes W$ can be written as $\sum_i \lambda_i a_i\otimes b_i$ where $a_i\in V$ and $b_i\in W$. \[prop:atomictensor\] Let $p\in V$ and $q\in W$ be atomic, then $p\otimes q\in V\otimes W$ is also atomic. Because $(p\otimes q){\,\&\,}(p\otimes q) = (p{\,\&\,}p)\otimes (q{\,\&\,}q) = p\otimes q$ it is sharp. Let $c = \sum_i \lambda_i a_i\otimes b_i$ be an arbitrary element of $V\otimes W$, then using lemma \[lem:atomicnorm\] $(p\otimes q){\,\&\,}c = \sum_i \lambda_i (p{\,\&\,}a_i)\otimes (q{\,\&\,}b_i) = \sum_i \lambda_i \norm{p{\,\&\,}a_i}\norm{q{\,\&\,}b_i} (p\otimes q) = \mu (p\otimes q)$ for some $\mu\in {\mathbb{R}}$. Since $c$ was arbitrary we conclude that $p\otimes q$ is atomic as a result of lemma \[lem:atomicnorm\]. Let $V$ be a sequential product space with effects $E$. We call $c\in V$ *classical* when it is compatible with all other effects: $a{\,\lvert\,}c$ for all $a\in E$. We will call a classical effect *minimal* when there is no non-zero classical effect strictly below it. \[prop:classicaltensor\] Let $c\in V$ and $d\in W$ be classical, then $c\otimes d$ is classical in $V\otimes W$. $c{\,\lvert\,}a$ for all $a \in V$ and $d{\,\lvert\,}b$ for all $b\in W$, therefore $c\otimes d{\,\lvert\,}a\otimes b$ and the same holds for linear combinations of these elements which span the entirety of $V\otimes W$. We call an EJA *simple* if it contains no non-trivial classical effects. We can write any EJA uniquely as $E_1\oplus\ldots \oplus E_k$ where the $E_i$ are simple EJAs, which we will refer to as the *summands* of the EJA. An EJA with $k$ summands has exactly $k$ minimal classical effects, corresponding to the units of each of the summands. Each other sharp classical effect is a sum of these minimal ones. \[lem:atomicoverlap\] Let $p$ and $q$ be atomic effects in an EJA. If $q{\,\&\,}p \neq 0$, then $p$ and $q$ belong to the same simple summand. Suppose $c$ is a sharp classical effect. Then $c{\,\&\,}p = p{\,\&\,}c = \lambda p$, for some $\lambda\in[0,1]$. But also $\lambda p = c{\,\&\,}p=(c^2){\,\&\,}p = c{\,\&\,}(c{\,\&\,}p) = \lambda (c{\,\&\,}p) = \lambda^2 p$, so that $\lambda = 1$ or $\lambda = 0$. So either $p\leq c$ or $p$ and $c$ are orthogonal. Suppose $p\leq c$, and that $q{\,\&\,}p \neq 0$, then $0\neq q{\,\&\,}p \leq q{\,\&\,}c =\lambda q$, so that $\lambda\neq 0$ and thus $q\leq c$. If we let $c$ be the identity of the summand that $p$ belongs to we see that the desired property follows. Let $p_1,\ldots,p_r$ be a maximal collection of orthogonal non-zero atomic effects in a simple EJA, then there exists an atomic effect $q$ such that $q{\,\&\,}p_i\neq 0$ for all $i$. We do this by case distinction using the classification of simple EJAs [@jordan1993algebraic]. Either the space is a spin-factor, in which case a maximal collection is always given by a sharp atomic $p$ and its complement $p^\perp$. Any $q\neq p,p^\perp$ cannot be orthogonal to them, because the space is of rank 2, so that indeed $q{\,\&\,}p\neq 0$ and $q{\,\&\,}p^\perp \neq 0$. If the space is not a spin-factor, then it must be of the form $B(H)^{\text{sa}}$ for a real, complex, quaternionic or octonion finite-dimensional Hilbert space $H$ (in the case of the octonions we must have $\dim H=3$). For such a space the atomic idempotents correspond to unit vectors of the underlying Hilbert space: $p_i = \lvert v_i\rangle\langle v_i \rvert$ where $v_i\in H$ is some unit vector. We can then take $q=\lvert w\rangle\langle w \rvert$ with $w=\frac{1}{\sqrt{r}}\sum_{i=1}^r v_i$. It should then be clear that $\inn{q,p_i}\neq 0$ and hence $q{\,\&\,}p_i \neq 0$. \[prop:tensordirectsum\] Let $V = E_1\oplus\ldots \oplus E_m$ and $W = F_1\oplus \ldots \oplus F_n$ with the $E_i$ and $F_j$ being simple EJAs. Let $1\leq k \leq n$ and $1\leq l\leq m$. Let $p_1,\ldots, p_r$ be a maximal collection of orthogonal non-zero atomic effects in $E_k$, and let $q_1,\ldots, q_s$ be such a maximal collection in $F_l$. Then $(p_i\otimes q_j)_{i=1,j=1}^{r,s}$ belong to the same simple summand in $V\otimes W$ and they form a maximal collection of orthogonal non-zero atomic effects in this summand. We let $p$ be atomic such that $p{\,\&\,}p_i\neq 0$ for all $i$ which exists by the previous lemma and similarly we let $q$ be atomic such that $q{\,\&\,}q_j \neq 0$ for all $j$. By proposition \[prop:atomictensor\] $p\otimes q$ and $p_i\otimes q_j$ will be atomic for all $i$ and $j$. By construction we of course have $0\neq (p{\,\&\,}p_i)\otimes (q{\,\&\,}q_j)=(p\otimes q){\,\&\,}(p_i\otimes q_j)$ and by lemma \[lem:atomicoverlap\] the $p_i\otimes q_j$ must then belong to the summand of $p\otimes q$ for all $i$ and $j$. Since $\sum_i p_i=1_{E_k}$, this sum is a classical effect. The same holds for $\sum_j q_j=1_{F_l}$. Their tensor product $1_{E_k}\otimes 1_{F_l} = \sum_{i,j} p_i\otimes q_j$ is then also classical by proposition \[prop:classicaltensor\]. Since the only nonzero classical effect in a simple summand is the identity this expresion must be equal to the identity of this summand. As a result the set $(p_i\otimes q_j)_{i,j}$ is indeed maximal. Using this proposition we conclude that for each of the summands $E$ of $V$ and $F$ of $W$ there must exist a summand in $V\otimes W$ which has rank ${\text{rnk}\xspace}~ E~ {\text{rnk}\xspace}~ F$. Because the tensor product map is obviously injective this factor must have dimension at least $\dim E~ \dim F$, and then because of local tomography the dimension must be strictly equal. Now let $V$ be a sequential product space for which the tensor product $V\otimes V$ exists so that the above must in particular be true when $E=F$, i.e. if $E$ is a simple factor of $V$ then there must exist a simple factor with rank $({\text{rnk}\xspace}E)^2$ and dimension $(\dim E)^2$. \[prop:simpleEJAsquare\] Let $E$ be a simple Euclidean Jordan algebra of rank $r> 1$ and dimension $N$. There exists a simple Euclidean Jordan algebra of rank $r^2$ and dimension $N^2$ if and only if $E=B(H)^{\text{sa}}$ where $H$ is a complex finite-dimensional Hilbert space. If $E=B(H)^{\text{sa}}$ is a complex matrix algebra the property is obviously true by considering $B(H\otimes H)^{\text{sa}}$ as the simple EJA of rank $r^2$ and dimension $N^2$. We simply check that every other simple EJA is not a possibility. If $E= B(H)^{\text{sa}}$ where $H$ is the 3-dimensional octonion Hilbert space, then $r=3$ and $N=27$. The highest dimensional simple EJA of rank $9$ is the quaternionic system which has dimension $9*(2*9-1)=153<27^2 = 729$ so that this is not possible. If $E=B(H)^{\text{sa}}$ with $H$ quaternionic, then $N = r(2r-1)$. The highest dimensional simple EJA of rank $r^2$ is also quaternionic so that its dimension is $r^2(2r^2-1)$. It is easy to check that $N^2 = r^2(2r-1)^2 > r^2(2r^2-1)$ when $r>1$ so that again, it cannot be this space. If $E=B(H)^{\text{sa}}$ where $H$ is real, then by dimension counting we can again see that there does not exist an EJA with rank $r^2$ and dimension $N^2$. A spin factor always has rank 2. The rank 4 EJAs have dimension 10, 16 and 28. The only one of these which is a square is 16. The 4 dimensional spin-factor corresponds to the qubit which is indeed $B(H)^{\text{sa}}$ with $H$ a 2-dimensional complex Hilbert space. Suppose $V$ is a finite-dimensional sequential product space for which the linear algebraic tensor product space $V\otimes V$ is also a sequential product space with product satisfying $(a\otimes b){\,\&\,}(c\otimes d) = (a{\,\&\,}c)\otimes (b{\,\&\,}d)$ for all $a,b,c,d \in V$, then there exists a C$^*$-algebra $A$ such that $V\cong A^{\text{sa}}$. As established, $V$ is a Euclidean Jordan algebra. As a result of proposition \[prop:tensordirectsum\] for each summand of $V = E_1\oplus\ldots\oplus E_n$ there must exist a simple EJA of rank $({\text{rnk}\xspace}E_i)^2$ and dimension $(\dim E_i)^2$. By proposition \[prop:simpleEJAsquare\] this is only possible if $E_i=B(H)^{\text{sa}}$ where $H$ is a complex Hilbert space. Therefore $V$ is a direct sum of complex matrix algebras which means it is the set of self-adjoint elements of a C$^*$-algebra. Infinite-dimensional sequential product spaces {#sec:infiniterank} ============================================== In order to state the following infinite-dimensional generalisations of the main theorem we must first give an appropriate definition of infinite-dimensional Jordan algebras. We call an order unit space $V$ a *JB-algebra* when it is complete in its norm topology and it has a Jordan product $*$ such that $\norm{a*a} = \norm{a}^2$ and $\norm{a*b} \leq \norm{a}\norm{b}$ for all $a,b \in V$. Let $V$ be an order unit space. A subset $S\subseteq V$ is called *bounded* when there exists $r\in {\mathbb{R}}$ such that $\norm{a}\leq r$ for all $a\in S$. It is called *directed* when for all $s_1,s_2 \in S$ we can find $s\in S$ such that $s_1,s_2 \leq s$. We call $V$ *bounded directed complete* when for any bounded directed set $S$ we can find a least upper bound, i.e. an element $t\in V$ such that for all $s\leq t$ for all $s\in S$. We call a state $\omega: V\rightarrow {\mathbb{R}}$ on an order unit space *normal* when it preserves suprema of bounded directed sets: $\omega(\bigvee S) = \bigvee \omega(S)$. We say that $V$ has *enough normal states* when the normal states order-separate the elements, i.e. when $\omega(v) \leq \omega(w)$ for all normal states $\omega$ implies that $v\leq w$. A *JBW-algebra* is a JB-algebra that is bounded directed complete and has enough normal states. When thinking about these algebras it is helpful to keep in mind the following analogy: JB-algebras are to C$^*$-algebras as JBW-algebras are to von Neumann algebras. As shown in the authors previous work [@wetering2018characterisation], we can still derive a spectral decomposition theorem and homogeneity for an infinite-dimensional bounded direct complete sequential product space when we also require the sequential product to be normal. In order to use homogeneity as in proposition \[prop:itochar\] however we need atomic effects. In the main theorem, the finite-dimensionality of the space ensured the existence of atomic effects, but in infinite dimension, atomic effects don’t have to exist. For example, if $V$ would be the space of self-adjoint elements of a type II von Neumann algebra then it does not contain a single non-zero atomic effect. We see therefore that if we want to work with atomic effects, that we must require them explicitly: Let $V$ be a sequential product space. We call $V$ *atomic* when below every sharp effect we can find a non-zero atomic sharp effect. Unfortunately, at this point we still cannot use the characterisation of rank 2 spaces from [@ito2017p] as this classification only holds for finite-dimensional spaces. It is currently unclear whether the generalisation of this theorem to infinite-dimensional spaces holds, although it seems reasonable. At this point we therefore have to add one more condition: We say that an atomic sequential product space $V$ has *finite bits* when the spaces $V_{p,q}$ (see definition \[def:orderideal\]) generated by sharp atomic effects $p$ and $q$ are finite-dimensional. Note that there do exist sequential product spaces that do not have finite bits, namely spin-factors of infinite-dimension. But the author is not aware of any space of rank greater than 2 that does not contain a spin-factor as a subsystem, that doesn’t have finite bits. Let $V$ be an atomic bounded directed complete sequential product space with finite bits where the sequential product is normal, then $V$ is a JB-algebra. As shown in [@wetering2018characterisation] a directed complete sequential product space with normal sequential product is homogeneous. The necessary propositions presented in sections \[sec:atomeffect\], \[sec:coverprop\] and \[sec:subselfdual\] still hold and since the spaces $V_{p,q}$ are required to be finite-dimensional, the characterisation theorem of [@ito2017p] applies so that we indeed have symmetry of transition probabilities. Since the atomic effects lie dense in the space (see also [@wetering2018characterisation]) we have then satisfied all the conditions of [@alfsen2012geometry Theorem 9.38] so that $V$ is indeed a JB-algebra. This theorem can with a few extra requirements be lifted to a more specific statement: Let $V$ be an atomic bounded directed complete sequential product space with finite bits and a normal sequential product and enough normal states, then $V$ is a JBW-algebra. If furthermore $V$ is of infinite rank and the only classical effects are $0$ and $1$ then $V$ is isomorphic to $B(H)^{\text{sa}}$ where $H$ is an infinite-dimensional real, complex or quaternionic Hilbert space. It has already been established that a $V$ with these properties is a JB-algebra. It is also a JBW-algebra, since we explicitly require bounded directed completeness and enough normal states. Because $V$ contains no non-trivial classical effects it must be a factor. A JBW factor of infinite rank such that there is an atomic effect below each sharp effect is isomorphic to $B(H)^{\text{sa}}$ where $H$ is a real, complex or quaternionic Hilbert space [@hanche1984jordan Theorem 7.5.11]. The statement of this theorem in its current form is quite hampered. It seems reasonable that this theorem should be able to be generalised. In particular, the requirement of enough normal states is probably unnecessary: there is a theorem stating that type I AW$^*$-algebra factors correspond exactly to type I von Neumann algebra factors. A similar theorem seems reasonable to hold for type I JBW-algebras and the ‘JBAW’-algebras. Furthermore, it seems reasonable that the characterisation of rank 2 homogeneous spaces of [@ito2017p] holds for infinite-dimensional spaces, in which case the condition on finite bits can be dropped. A much larger class of spaces could be covered if the atomicity requirement could somehow be dropped. In fact, we conjecture the following: > **Conjecture:** Let $V$ be a bounded directed complete sequential product space with a normal sequential product and enough normal states, then $V$ is a JBW-algebra. It is currently not clear whether the tools we have now are adequate to prove this statement. We will however show that using an additional assumption on the sharp effects we can prove a similar theorem: Let $V$ be a bounded directed complete sequential product space with a normal sequential product and enough normal states, such that for all sharp effects $p$ and $q$ the relation $ p{\,\&\,}q + p^\perp{\,\&\,}q^\perp = q{\,\&\,}p + q^\perp {\,\&\,}p^\perp$ holds, then $V$ is a JBW-algebra. When $V$ allows spectral decompositions, is bounded directed complete and has enough normal states, a sufficient condition for it being a JBW-algebra is that the relation defined above for sharp effects holds true for so called *compressions* (Corollary 9.45 and Theorem 9.43 of [@alfsen2012geometry]). As shown in [@wetering2018characterisation] we know that $V$ has spectral decompositions. It is however a priori not clear whether the sequential product map of a sharp effect $L_p(a) = p{\,\&\,}a$ is indeed a compression as defined by Alfsen and Schultz. It is true though that these maps are *complemented*, so that the proof of the relevant theorem in [@alfsen2012geometry] still carries trough when the compressions are replaced with the maps $L_p$. Minimality of axioms {#sec:axioms} ==================== In this section we will discuss the minimality of the conditions and the axioms needed to show that finite-dimensional sequential product spaces are Euclidean Jordan algebras. For easy reference we copy definition \[def:seqprod\] here: A map ${\&:[0,1]_V\times [0,1]_V \rightarrow [0,1]_V}$ is called a sequential product when it satisfies the following properties for all $a,b,c \in [0,1]_V$: 1. Additivity: $a{\,\&\,}(b+c) = a{\,\&\,}b+ a{\,\&\,}c$. 2. Continuity: The map $a\mapsto a{\,\&\,}b$ is continuous in the norm. 3. Unitality: $1{\,\&\,}a = a$. 4. Compatibility of orthogonal effects: If $a{\,\&\,}b = 0$ then also $b{\,\&\,}a =0$. 5. Associativity of compatible effects: If $a{\,\lvert\,}b$ then $a{\,\&\,}(b{\,\&\,}c) = (a{\,\&\,}b){\,\&\,}c$. 6. Additivity of compatible effects: If $a{\,\lvert\,}b$ then $a {\,\lvert\,}1-b$, and if also $a{\,\lvert\,}c$ then $a{\,\lvert\,}(b+c)$. 7. Multiplicativity of compatible effects: If $a{\,\lvert\,}b$ and $a{\,\lvert\,}c$ then $a{\,\lvert\,}(b{\,\&\,}c)$. First of all, for the proofs of the main theorems, axiom \[ax:compmult\] is actually not needed since the following weaker version is sufficient: Suppose $a{\,\lvert\,}b,c$ and that $b{\,\lvert\,}c$, then $a{\,\lvert\,}(b{\,\&\,}c)$. Using axiom \[ax:assoc\] repeatedly: $a{\,\&\,}(b{\,\&\,}c) = (a{\,\&\,}b){\,\&\,}c = (b{\,\&\,}a){\,\&\,}c = b{\,\&\,}(a{\,\&\,}c) = b{\,\&\,}(c{\,\&\,}a) = (b{\,\&\,}c){\,\&\,}a$. The reason we included axiom \[ax:compmult\] is because it is part of the definition of a sequential effect algebra, and because when defining the classical algebra of an element when working in infinite dimension, it is needed to show that the algebra is closed under multiplication. Of the other axioms, the ones that seem less essential are \[ax:orth\] and \[ax:cont\], so it would be interesting to see what can be done without them. To define and study the classical algebra of an effect, \[ax:orth\] is not needed and \[ax:cont\] is only needed to show that $(\lambda 1){\,\&\,}a = \lambda a$. The spectral theorem and the homogeneity of the space can thus be proven without using these axioms if \[ax:unit\] is changed to $(\lambda 1){\,\&\,}a = \lambda a$. When restricting to rank 2 spaces, axioms \[ax:add\], \[ax:unit\], \[ax:assoc\] and \[ax:compadd\] are then sufficient to prove that the space is a Euclidean Jordan algebra (or specifically, a spin-factor). Since the spectral theorem is also what is needed to show that $L_a$ for $a$ invertible is an order-isomorphism, it should be clear that on an EJA this restricted set of axioms already greatly reduces the possible sequential-product-like maps. Note also that using the T-algebra formalism of Vinberg [@vinberg1967theory] it is possible to find an associative binary operation (see the beginning of section 4 of [@chua2003relating] for this operation) for the positive elements in any finite-dimensional homogeneous space that satisfies axioms \[ax:add\], \[ax:cont\], \[ax:unit\], \[ax:assoc\] (and by associativity also \[ax:compmult\]) but this product does not satisfy axioms \[ax:orth\] and \[ax:compadd\]. This actually leads to an interesting observation: if either the proof of homogeneity in section \[sec:prelim\] can be shown to hold without use of axiom \[ax:compadd\] or if a binary product on homogeneous spaces can be found that also satisfies \[ax:compadd\], then this would give us a new characterisation of homogeneous spaces. In particular, in the second case, it would show that \[ax:orth\] is the key to establishing self-duality. When one considers more general ordered vector spaces than order unit spaces, one can find non-trivial totally ordered vector spaces that allow a commutative bilinear product, and hence a sequential product [@basmaster]. These spaces are pathological in the sense that they have ‘infinitesimal’ effects, i.e. effects than cannot be distinguished using states.

--- abstract: 'We discuss the well-known phenomenon of spontaneous symmetry breaking for a linear sigma model for scalar and pseudoscalar mesons based on the meson composite structure and the normalization of the quantum states. To test our formulation and validate our approach we give another proof of the Goldstone theorem and derive the corresponding mass eigenstates of the theory. We briefly describe the possible wave function of a meson that leads to the adequate mass eigenstates.' author: - 'Amir H. Fariborz $^{\it \bf a}$ ' - 'Renata Jora $^{\it \bf b}$ ' title: | \ [Spontaneous symmetry breaking and Goldstone theorem for composite states revisited]{} --- Introduction ============ It is now well known that QCD is the theory of the strong interaction. At higher energies the coupling constant is small and there is an adequate perturbative description of the theory in terms of quarks and gluons. At low energies the coupling constant is large, the theory confines and one needs to introduce new dynamical degrees of freedom, the hadrons. The low energy hadron spectrum includes among others three very light particles, the pions. In order to explain the smallness of their masses and other properties of the hadron spectra one assumes that the corresponding meson Lagrangian is endowed with a chiral flavor symmetry: $SU(2)_L\times SU(2)_R$ (we consider only two flavors). This symmetry is both spontaneously broken by the formation of the chiral condensate as well as explicitly broken in the presence of the quark masses. The phenomenon of spontaneous chiral symmetry breaking is evident in a linear sigma model that includes both scalars and pseudoscalars and it is a particular example of spontaneous symmetry breaking of a global or gauge group which was described in a series of pioneering works [@Nambu]-[@Hagen]. The aim of our work is to analyze spontaneous symmetry breaking from the point of view of the composite structure of the meson states. Although determining the exact substructure of the mesons in terms of the constituent quarks can be quite nontrivial, some general features may be considered and some important properties may be derived based on those. In section II we will show what one can learn about the meson states and spontaneous symmetry breaking for composite theories by considering the meson quark substructure. In essence we give another proof of the Goldstone theorem based on the normalization of the quantum meson states. In section III we compute the mass eigenstates of the hamiltonian after spontaneous symmetry breaking. Section IV contains a description of a possible meson wave function that leads to the correct mass spectrum. Section V is dedicated to the conclusions. Composite states and the goldstone theorem ========================================== We consider a model that contains composite scalars and pseudoscalars, the most relevant example being the $SU(2)_L \times SU(2)_R$ linear sigma model for the mesons. The corresponding meson states have the structure [@Schechter]: $$\begin{aligned} M_{ij}\propto\bar{\Psi}_i^A(\frac{1+\gamma^5}{2})\Psi^A_j, \label{mestr45546}\end{aligned}$$ where $i$, $j$ denote the flavor and $A$ represents the color index. The model displays spontaneous symmetry breaking down to $SU(2)_V$ as soon as the vacuum condensates $\bar{\Psi}_i^A\Psi_i^A$ forms. In a linear sigma model the particles arrange themselves in triplets and singlets of the flavor group according to the representation: $$\begin{aligned} &&\pi^+\propto(-i)\bar{d}\gamma^5u \nonumber\\ &&\pi^0\propto i\frac{1}{\sqrt{2}}[\bar{d}\gamma^5d -\bar{u}\gamma^5u] \nonumber\\ &&\eta\propto -i\frac{1}{\sqrt{2}}[\bar{d}\gamma^5u+\bar{u}\gamma^5d] \nonumber\\ &&a^+\propto \bar{d}u \nonumber\\ &&a^0\propto\frac{1}{\sqrt{2}}[\bar{u}u-\bar{d}d] \nonumber\\ &&\sigma\propto\frac{1}{\sqrt{2}}[\bar{u}u+\bar{d}d]. \label{mes55466}\end{aligned}$$ Here $\pi^{\pm}$, $\pi^0$ and $\eta$ correspond to the pseudoscalar states whereas $a^{\pm}$, $a^0$ and $\sigma$ to the scalar ones. We are only considering a $\bar{q}q$ model to illustrate the connection but in reality it is known that the light scalars are considerably more complex and contain large four-quark component and glue component. If one considers a simple Lagrangian that contains only the kinetic terms and a mass term then both the meson states in Eq. (\[mes55466\]) and the quantities $\bar{\Psi}_i(\frac{1+\gamma^5}{2})\Psi_j$ are eigenstates of the Hamiltonian with the same mass. Although our arguments could extend easily to the model above in what follows we shall consider a simplified version of it with a single quark, assuming this to be $u$. The model has a simple $U(1)_L \times U(1)_R$ symmetry and contains a scalar, $\Phi_0\propto\frac{1}{\sqrt{2}}\bar{u}u$ and a pseudoscalar $\Phi_1\propto-i\frac{1}{\sqrt{2}}\bar{u}\gamma^5u$. The Lagrangian of interest is given simply by: $$\begin{aligned} {\cal L}=\partial^{\mu}(\Phi_0-i\Phi_1)\partial_{\mu}(\Phi_0+i\Phi_1) -m^2(\Phi_0^2+\Phi_1^2)=\frac{1}{2}\partial^{\mu}M^{\dagger}\partial_{\mu}M-\frac{1}{2}m^2M^{\dagger}M. \label{lagr45534}\end{aligned}$$ Note that both the fields $\Phi_0$ and $\Phi_1$ or $u_R^{\dagger}u_L$ and $u_L^{\dagger}u_R$ may be considered eigenstates of the Hamiltonian. In terms of the latter fields the Lagrangian becomes: $$\begin{aligned} {\cal L}\propto\frac{1}{2}[\partial^{\mu}(u_L^{\dagger}u_R)\partial_{\mu}(u_R^{\dagger}u_L)-m^2(u_L^{\dagger}u_R)(u_R^{\dagger}u_L)], \label{lagr45546}\end{aligned}$$ since $\Phi_0+i\Phi_1\propto \frac{1}{\sqrt{2}}u_R^{\dagger}u_L$. Assume that a vacuum condensate of $\Phi_0$ forms: $$\begin{aligned} \langle|\Phi_0(x)|\rangle=\langle|\Phi_0(0)|\rangle=v \label{v45546}\end{aligned}$$ Although we do not know exactly the exact structure of the scalars in terms of the up quark fields we can write quite safely the analogue of Eq.(\[v45546\]) as function of the individual quark states: $$\begin{aligned} \int \frac{d^3p}{(2\pi)^3}\int \frac{d^3q}{(2\pi)^3}f(\vec{p},\vec{q})\sum_{s,r}\langle 0|b^s_p\bar{v}^s(p)v^r(q)b_q^{r\dagger}|0\rangle\neq 0, \label{resttrrtt6}\end{aligned}$$ where $b^s_p$ and $b_q^{r\dagger}$ are operators of annihilation and creation, $v$ is the Dirac spinor solution of the equation of motion and $f(\vec{p},\vec{q})$ is a function that encapsulates our lack of knowledge with regard to the scalar structure in terms of the constituents. The result will contain a delta function $ \delta(\vec{p}-\vec{q})$ and the element $\bar{v}^s(p)v^r(q)=-2m\delta_{rs}$, where $m$ is the mass of an individual quark. It can be observed that the matrix element in Eq. (\[resttrrtt6\]) is in general different than zero if the quark has a nonzero mass. Let us consider the wave packets associated to the fields $\Phi_0$ and $\Phi_1$: $$\begin{aligned} &&|\Phi_0\rangle=\int \frac{d^3\vec{k}}{(2\pi)^3}\frac{1}{\sqrt{2E_k}}\Phi_0(\vec{k})|\vec{k}\rangle \nonumber\\ &&|\Phi_1\rangle=\int \frac{d^3\vec{k}}{(2\pi)^3}\frac{1}{\sqrt{2E_k}}\Phi_1(\vec{k})|\vec{k}\rangle, \label{res553443}\end{aligned}$$ where $\Phi_0(\vec{k})$ and $\Phi_1(\vec{k})$ are the Fourier transforms of the respective spatial wave function. Since $\Phi_0$ and $\Phi_1$ are composite they can be expressed quite generically in terms of the constituent quarks as: $$\begin{aligned} &&|\Phi_0\rangle=\int \frac{d^3\vec{k}}{(2\pi)^3}\frac{d^3\vec{p}}{(2\pi)^3}f(\vec{p},\vec{k})\sum_{r,s}\bar{u}^s(k)v^r(p)|\vec{k},s;\vec{p},r\rangle \nonumber\\ &&|\Phi_1\rangle=\int \frac{d^3\vec{k}}{(2\pi)^3}\frac{d^3\vec{p}}{(2\pi)^3}f(\vec{p},\vec{k})\sum_{r,s}\bar{u}^s(k)\gamma^5v^r(p)|\vec{k},s;\vec{p},r\rangle. \label{const5546}\end{aligned}$$ Here the function $f(\vec{k},\vec{p})$ is as before and $u$, $v$ are the standard spinor solutions of the Dirac equation of motion. It is necessary to have the same function $f(\vec{k},\vec{p})$ for both the scalar and the pseudoscalar in order to have the correct representation in terms of the quarks: $$\begin{aligned} &&\Phi_0+i\Phi_1\propto u_R^{\dagger}u_L \nonumber\\ &&\Phi_0-i\Phi_1\propto u_L^{\dagger}u_R. \label{res3324567}\end{aligned}$$ Note that here as opposed to the Eq. (\[const5546\]) $u$ designates the up quark. Having the generic expressions both for $|\Phi_0\rangle$ and $|\Phi_1\rangle$ we shall compute the normalization of states. Thus, $$\begin{aligned} &&\langle \Phi_0|\Phi_0\rangle=\int \frac{d^3\vec{k}}{(2\pi)^3}\frac{d^3\vec{p}}{(2\pi)^3}\frac{d^3\vec{q}}{(2\pi)^3}\frac{d^3\vec{w}}{(2\pi)^3}|f(\vec{k},\vec{p})|^2\times \nonumber\\ &&\sum_{t,m}\sum_{s,r}\bar{v}^t(q)u^m(w)\bar{u}^s(k)v^r(p)\langle \vec{q},t;\vec{w},m|\vec{k},s;\vec{p},r\rangle. \label{res546645}\end{aligned}$$ Then, $$\begin{aligned} \langle \vec{q},t;\vec{w},m|\vec{k},s;\vec{p},r\rangle=2E_p2E_k\delta_{t,r}\delta_{m,s}(2\pi)^3\delta(\vec{k}-\vec{w})(2\pi)^3\delta(\vec{p}-\vec{q}), \label{form66577}\end{aligned}$$ such that Eq. (\[res546645\]) becomes: $$\begin{aligned} \langle \Phi_0|\Phi_0\rangle=\int \frac{d^3\vec{k}}{(2\pi)^3}\frac{d^3\vec{p}}{(2\pi)^3}|f(\vec{k},\vec{p})|^2\times\sum_{r,s}\bar{v}^r(p)u^s(k)\bar{u}^s(k)v^r(p). \label{res221332}\end{aligned}$$ Using (we use the conventions in [@Srednicki] with $g^{\mu\nu}=(-1,1,1,1)$): $$\begin{aligned} &&\sum_su^s(k)\bar{u}^s(k)=-\gamma^{\mu}k_{\mu}+m \nonumber\\ &&\sum_r\bar{v}^r(p)\gamma^{\mu}v^r(p)=2p^{\mu}\times 4 \nonumber\\ &&\sum_r\bar{v}^r(p)v^r(p)=(-2m)\times 4 \label{usef554664}\end{aligned}$$ we obtain: $$\begin{aligned} \langle \Phi_0|\Phi_0\rangle=\int \frac{d^3\vec{k}}{(2\pi)^3}\frac{d^3\vec{p}}{(2\pi)^3}|f(\vec{k},\vec{p})|^2(4)(-1)[2p^{\mu}k_{\mu}+2m^2]. \label{finalres553663}\end{aligned}$$ Similarly one finds: $$\begin{aligned} &&\langle \Phi_1|\Phi_1\rangle=\int \frac{d^3\vec{k}}{(2\pi)^3}\frac{d^3\vec{p}}{(2\pi)^3}|f(\vec{k},\vec{p})|^2\times (-1)\sum_{r,s}\bar{v}^r(p)\gamma^5u^s(k)\bar{u}^s\gamma^5(k)v^r(p)= \nonumber\\ &&=\int \frac{d^3\vec{k}}{(2\pi)^3}\frac{d^3\vec{p}}{(2\pi)^3}|f(\vec{k},\vec{p})|^2(4)(-1)[2p^{\mu}k_{\mu}-2m^2]. \label{res221332}\end{aligned}$$ The two expressions for the fields $\Phi_0$ and $\Phi_1$ are completely identical apart from the factors: $$\begin{aligned} &&A=[2p^{\mu}k_{\mu}+2m^2] \nonumber\\ &&B=[2p^{\mu}k_{\mu}-2m^2] \label{factors55343}\end{aligned}$$ where $A$ corresponds to $\Phi_0$ and B to $\Phi_1$. One can further write: $$\begin{aligned} &&A=2p^{\mu}k_{\mu}+2m^2=(p+k)^2-p^2-k^2+2m^2=(p+k)^2+4m^2 \nonumber\\ &&B=2p^{\mu}k_{\mu}-2m^2=(p+k)^2-p^2-k^2-2m^2=(p+k)^2, \label{calc66453}\end{aligned}$$ where we used the fact that for the spinor solutions of the equation of motion $p^2=-m^2$ and $k^2=-m^2$ where $m$ is the mass of the quark. Since generically one can consider $p+k$ the momentum of the scalar and pseudoscalar state (it is sufficient to consider the wave function for the composite field to determine that) one can make a change of variable $p+k=q$, $k=q-p$ to find the Fourier modes of the scalar and pseudoscalar fields. In order to have a meaningful theory with a Lagrangian and a Hamiltonian we should be able to normalize simultaneously the states $\Phi_0$ and $\Phi_1$. Note that if $m=0$ the normalization of the states is the same and we are able to normalize them simultaneously. As $m$ becomes different than zero the vacuum condensate forms and the normalization of the states differs showing that the $\Phi_0$ and $\Phi_1$ have actually different masses. Then a necessary condition for the theory to make sense is to have: 1)Case I $$\begin{aligned} &&A=(2p^{\mu}k_{\mu})_1+2m^2=x^2 \nonumber\\ &&B=(2p^{\mu}k_{\mu})_2-2m^2=x^2 \label{case235}\end{aligned}$$ or, 2\) Case II $$\begin{aligned} &&A=(2p^{\mu}k_{\mu})_1+2m^2=-x^2 \nonumber\\ &&B=(2p^{\mu}k_{\mu})_2-2m^2=-x^2. \label{case233}\end{aligned}$$ where we added subscript $1$ and $2$ to show that the overall energy and mass for the quadrivector $p+k$ is different for the two fields. Both conditions for each case must be satisfied simultaneously. Then for case I we pick $(2p^{\mu}k_{\mu})_1=x^2-2m^2$ and for case II we pick $(2p^{\mu}k_{\mu})_2=-x^2+2m^2$ to determine that in either of these situations: $$\begin{aligned} |2p^{\mu}k_{\mu}|=|2m^2-x^2|\leq 2m^2. \label{res44356}\end{aligned}$$ Then using the fact that: $$\begin{aligned} |2p^{\mu}k_{\mu}|=\Bigg|2\sqrt{m^2+\vec{p}^2}\sqrt{m^2+\vec{k}^2}-2\vec{p}\cdot\vec{k}\Bigg|\geq 2m^2 \label{res32678}\end{aligned}$$ we determine that $|2p^{\mu}k_{\mu}|=2m^2$ and consequently $x^2=0$ or $x^2=4m^2$. We can choose both possibilities but the second one would lead to zero mass for the scalar and non zero for the pseudoscalar. We thus pick the first choice as corresponding to the reality. However in the case where one considers two quarks one with the constituent mass $m$ and the other with the constituent mass $-m$ the second choice is the valid one and leads to a zero mass for the pseudoscalar. The second choice is also equivalent to considering both quarks with the same imaginary mass $im$. We shall discuss the second choice from a different point of view in more details in section IV. Then from either of the Eq. (\[case235\]) and (\[case233\]) we determine: $$\begin{aligned} &&A=(2p^{\mu}k_{\mu})_1+2m^2=0=(p+k)^2+4m^2. \nonumber\\ &&B=(2p^{\mu}k_{\mu})_2-2m^2=0=(p+k)^2. \label{case23}\end{aligned}$$ Since $(p+k)$ represents the four momentum of the scalar and pseudoscalar states we obtain that after spontaneous symmetry breaking the mass of the scalar $\Phi_0$ will become $4m^2$ whereas the pseudoscalar will be massless. This is quite a generic result which depends little or at all on the exact structure of the scalars and pseudoscalars in terms of the constituent quarks. Note that the factors of zero in the wave packet should be compensated by additional factors to be introduced in the denominators, more exactly in the structure function $f(\vec{k},\vec{p})$. Then the wave function for the composite scalars and pseudoscalars is constructed regularly afterwards with the new energy factors just determined through this method. In summary we start with a picture where the fields $\Phi_0$ and $\Phi_1$ have the same mass be that zero to find out that the apparition of a quark condensate and the normalization of states leads to a change of picture where the scalar gains a different mass and the pseudoscalar becomes massless. In consequence we gave another proof of the Goldstone theorem applicable to composite theories. A different perspective on the mass eigenstates =============================================== Assume instead of considering the initial Lagrangian in terms of the fields $\Phi_0$ and $\Phi_1$ we consider it in terms of the eigenstates (of the initial Lagrangian) $S_1\propto u_R^{\dagger}u_L$ and $S_2=S_1^{\dagger}\propto u_L^{\dagger}u_R$. Initially we have: $$\begin{aligned} &&\langle S_1^{\dagger}|{\cal H}|S_1\rangle=\vec{p^2}+m^2 \nonumber\\ &&\langle S_2^{\dagger}|{\cal H}|S_2\rangle=\vec{p^2}+m^2 \label{res665890}\end{aligned}$$ with all the other matrix elements zero (here we took into account the normalization of states in order to get energy squared on the right hand side). Thus the hamiltonian is diagonal on the states $S_1$ and $S_2$. Assume that the quarks get a mass different than zero. Then a vacuum condensate forms because the element, $$\begin{aligned} \sum_{s,r}\bar{v}^r(q)(\frac{1-\gamma^5}{2})u^s(k)\bar{u}^s(k)(\frac{1-\gamma^5}{2})v^r(q)\approx (-8)m^2 \label{r665487}\end{aligned}$$ is different than zero. Then the matrix elements in Eq. (\[res665890\]) remain the same and new matrix elements different than zero emerge: $\langle S_2^{\dagger}|{\cal H}|S_1\rangle=\langle S_1^{\dagger}|{\cal H}|S_2\rangle=a^2$. We do not know exactly the value of $a$ but we can diagonalize the hamiltonian to find that there are two eigenvalues: $$\begin{aligned} &&\lambda_1=\vec{p}^2+m^2-a^2 \nonumber\\ &&\lambda_2=\vec{p}^2+m^2+a^2. \label{res546788}\end{aligned}$$ We ask for the first eigenvalue to have the mass zero according to the Goldstone theorem. We obtain $a^2=m^2$ and $\lambda_2=p^2=\vec{p}^2+2m^2$. The eigenstates in this case are exactly $\Phi_1=\frac{S_1-S_2}{\sqrt{2}}$ with the mass zero and $\Phi_2=\frac{S_1+S_2}{\sqrt{2}}$ with the mass $2m^2$. Note that this is exactly the result that one would have obtained from a linear sigma model with a wrong sign mass term and a $\Phi^4$ interaction. In this case however we did not depart from the purely kinetic plus mass term. The scalar and pseudoscalar wave functions ========================================== Let us consider again the $SU(2)_L \times SU(2)_R$ model mentioned in section II after the dynamical breaking of the symmetry through the formation of a vacuum condensate. For illustration we just pick two states $\pi^+\propto \bar{d}\gamma^5u$ and $a^+\propto\bar{d}u$. We know that after symmetry breaking the pion should be massless so the question that arises is: what kind of quark substructure should $\pi^+$ and $a^+$ have to lead naturally to the desired mass spectrum? An example can be the following: $$\begin{aligned} &&\pi^+\approx \int d^4 y \bar{d}(x+y)\gamma^5u(x-y) \nonumber\\ &&a^+\approx \int d^4 y \bar{d}(x+y)u(x-y) \label{states4342}\end{aligned}$$ Note that we shall consider one quark with constituent mass $m$, the other with constituent mass $-m$ to account for usual mechanism of spontaneous symmetry breaking where the particles have the wrong mass term in the Lagrangian. We consider that both quark states satisfy the Dirac equation with the mentioned masses. Then we apply the operator $\partial^{\mu}\partial_{\mu}$ to the pion (here we use $g^{\mu\nu}=(1,-1,-1,-1))$: $$\begin{aligned} &&\partial^{\mu}\partial_{\mu}\pi^+\approx \nonumber\\ &&\approx \int d^4 y [(\partial^{\mu}\partial_{\mu})\bar{d}(x+y)\gamma^5u(x-y)+ \bar{d}(x+y)\gamma^5\partial^{\mu}\partial_{\mu}u(x-y)]+ \nonumber\\ &&\int d^4 y [2(\partial^{\mu}\bar{d})(x+y)\gamma^5\partial_{\mu}u(x-y)]= \nonumber\\ &&-2m^2\int d^4 y \bar{d}(x+y)\gamma^5u(x-y)-\int d^4 y [2(\partial^{\mu}\bar{d})(x+y)\gamma^5\partial_{\mu}u(x-y)]. \label{res9973546}\end{aligned}$$ We will treat the last term on the right hand side of Eq. (\[res9973546\]) separately. We use: $$\begin{aligned} 2g^{\mu\nu}=2\gamma^{\mu}\gamma^{\nu}-[\gamma^{\mu},\gamma^{\nu}] \label{g54664}\end{aligned}$$ to write, $$\begin{aligned} &&\int d^4 y [2(\partial^{\mu}\bar{d})(x+y)\gamma^5\partial_{\mu}u(x-y)]= \nonumber\\ &&\int d^4 y [2g^{\mu\nu}(\partial_{\mu}\bar{d})(x+y)\gamma^5\partial_{\nu}u(x-y)]= \nonumber\\ &&\int d^4 y [2(\partial_{\mu}\bar{d})(x+y)\gamma^5\gamma^{\mu}\gamma^{\nu}\partial_{\nu}u(x-y)]- \nonumber\\ &&\int d^4 y [(\partial_{\mu}\bar{d})(x+y)\gamma^5[\gamma^{\mu},\gamma^{\nu}]\partial_{\nu}u(x-y)]= \nonumber\\ &&+2m^2\int d^4 y \bar{d}(x+y)\gamma^5u(x-y)-\int d^4 y [(\partial_{\mu}\bar{d})(x+y)\gamma^5[\gamma^{\mu},\gamma^{\nu}]\partial_{\nu}u(x-y)]. \label{res539076}\end{aligned}$$ Here the minus sign in the first term on the right hand side of the equation comes from the anticommutator of $\gamma^{\mu}$ with $\gamma^5$. The last term on the right hand side of Eq, (\[res539076\]) is: $$\begin{aligned} &&\int d^4 y [(\partial_{\mu}\bar{d})(x+y)\gamma^5[\gamma^{\mu},\gamma^{\nu}]\partial_{\nu}u(x-y)]= \nonumber\\ &&\int d^4 y \int \frac{d^4p}{(2\pi)^4} \frac{d^4k}{(2\pi)^4}\exp[ip(x+y)]\exp[ik(x-y)]\bar{d}(p)\gamma^5[\gamma^{\mu},\gamma^{\nu}]u(k)p_{\mu}k_{\nu}= \nonumber\\ &&\int d^4 y \int \frac{d^4p}{(2\pi)^4} \frac{d^4k}{(2\pi)^4}\delta(p+k)(-p_{\mu}p_{\nu})\bar{d}(p)\gamma^5[\gamma^{\mu},\gamma^{\nu}]u(k)=0, \label{proof657}\end{aligned}$$ so it cancels. By adding the results in Eqs. (\[res9973546\]) and (\[res539076\]) we obtain: $$\begin{aligned} \partial^{\mu}\partial{\mu}\pi^+=0 \label{res4343}\end{aligned}$$ showing that the pseudoscalar is massless. Similarly one can show for $a^+$: $$\begin{aligned} \partial^{\mu}\partial_{\mu}a^+=-4m^2a^+, \label{res4343}\end{aligned}$$ signaling that the mass of $a^+$ is twice the absolute mass of an individual quark. Conclusions =========== In this paper we discussed the formation of a vacuum condensate and spontaneous symmetry breaking from the point of view of the meson substructure in terms of the constituent quarks. The main result of the paper is to give another proof of the Goldstone theorem for composite states based on the normalization of the wave packets associated to the mesons. Although we considered a simple $U(1)_L\times U(1)_R$ meson model our arguments can be extended easily to more flavors and larger symmetries by simply computing matrix elements corresponding to the meson states. We were thus able to confirm once more the standard features of spontaneous symmetry breaking. One important remark is in order. In our work we showed that spontaneous symmetry breaking occurs as soon as the quarks or mesons gain constituent masses. However the order in which this happens is not clear cut. In our simple model we showed that it is impossible for massive mesons to survive at any scale without spontaneous symmetry breaking. Consequently the picture in which confinement and chiral symmetry breaking occur at the same scale might be the correct one. We relied on the quantum properties of the meson states in the theory and on the corresponding symmetry and we did not use the $\Phi^4$ term present in the linear sigma model Lagrangian. Of course in order to give an accurate description of the reality one would need to take into account also the explicit symmetry breaking terms. We also discussed the mass eigenstates of the Hamiltonian after spontaneous symmetry breaking and briefly described a possible meson wave function that leads to the correct spectrum of states. Acknowledgments {#acknowledgments .unnumbered} =============== -.5cm The work of R. J. was supported by a grant of the Ministry of National Education, CNCS-UEFISCDI, project number PN-II-ID-PCE-2012-4-0078. 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--- author: - 'Yang Li, John Klingner, and Nikolaus Correll' bibliography: - 'reference.bib' title: Distributed Camouflage for Swarm Robotics and Smart Materials --- Introduction ============ We wish to design artificial camouflage systems that can quickly adapt to a large variety of environments. Inspired by the capabilities of cephalopods, which tightly integrate sensing, actuation (color change), neural computation and communication, we are interested in a distributed artificial approach that mimics this tight integration [@yu2014adaptive; @mcevoy2015materials]. While animals employ camouflage mostly for escaping predators, camouflage in an engineering context is typically motivated by clandestine military operations. More broadly, everything from small robots to buildings could use these techniques to more seamlessly be a part of their environment. Nature employs a large variety of techniques to achieve these goals. For example, moths mimic patterns that they would expect in their environments, sea animals use mottle patterns to soften their contours, and other animals decorate their body with artifacts from the environment [@stevens2009animal]. Two animals with notable camouflage abilities are the cuttlefish and octopus, who can dramatically alter the coloration and patterning of their skin and switch between different environments in a matter of seconds [@hanlon1988adaptive]. These creature’s camouflage behavior is not only driven by the animals’ visual system (which is color-blind [@messenger1977evidence]) or brain [@messenger2001cephalopod], but has also been shown to rely on local sensing and control [@ramirez2015eye]. There have been multiple attempts to achieve active camouflage using a combination of cameras and projection [@inami2003optical; @lin2009framework]. Although such systems provide “perfect” camouflage, they are highly dependent on the observer’s viewpoint. Mimicking the background exactly is rarely employed in the animal kingdom, where a few simple families of patterns — mottled, striped, or simply uniform [@hanlon2007cephalopod] — dominate. Creating such patterns requires only local coordination [@meinhardt1982models], suggesting a combination of high-level selection of appropriate motor programs [@messenger2001cephalopod] and self-organization [@meinhardt1982models]. Here, we are not concerned with perfectly matching the background, but rather aim to replicate the pattern matching ability of natural systems, which are able to fool sophisticated predators. Distributing the sensing and actuation for camouflage generation makes an implementation scalable for a variety of factors, such as resolution of the camouflage pattern, the size of the area being camouflaged, and robustness against the failure of individual units. Further, a distributed camouflage system could respond to local changes in the environment, in particular when deployed on non-trivial 3D surfaces. In this paper, we present a fully distributed approach, which we implement on a swarm of Droplets [@farrow2014miniature], each equipped with the ability to sense and emit color as well as communicate with its local neighbors. Although there exist multiple attempts to design artificial chromatophores, most work focuses on component technology, i.e. the ability to color change in a soft substrate [@rossiter2012biomimetic; @morin2012camouflage], but very few works articulate the systems challenges that require not only local color changes, but also local sensing and computation [@yu2014adaptive], or investigate the ability to co-locate simple signal processing with the sensors themselves [@fekete2009distributed]. Our algorithm can be broken into three phases, each described in detail below. First, we estimate a color and gradient histogram with a consensus algorithm among the particles. This information is then used to determine the parameters of a pattern formation algorithm. Finally, the pattern is formed using a reaction-diffusion process. The “background” in to which the swarm is trying to camouflage is projected on to the particles from above, requiring them to have color sensors. To simplify the color-identification process, the paper focuses on two-tone patterns. Finally, we arrange the particles in a grid pattern, which allows us to implement a discrete convolution operation and simplifies debugging the pattern at the low resolution that swarms in the order of tens of particles can afford. Distributed Camouflage Algorithm {#sec:camouflage} ================================ In this section, we describe the distributed camouflage algorithm. Fig. \[fig:pipeline\] illustrates the steps of the algorithm in broad strokes. First, each robot measures the color projected on it. Then, it exchanges the measured color with neighboring robots. Once received, neighbors’ color information is used to compute an estimated probability for the various pattern types based on local information (Section \[sec:descriptor\]). Next, the swarm communicates their local pattern probabilities, using a weighted-average consensus algorithm to compute the most likely global pattern (Section \[sec:consensus\]). Once consensus has been achieved, the swarm reproduces the pattern collaboratively with a reaction-diffusion process (Section \[sec:generator\]). ![Pipeline of the Distributed Camouflage Algorithm []{data-label="fig:pipeline"}](pics/pipeline.png){width="\columnwidth"} Pattern Descriptor {#sec:descriptor} ------------------ Once each robot has measured the local environment’s color, and communicated that information, they apply a filter mask (see Fig. \[fig:filter\]) to compute a discrete approximation of the second-order color derivative in both the horizontal and vertical directions. This is quite similar in concept to kernels used in edge detection and other computer-vision tasks [@dalal2005histograms]. Indeed, if the grid of robots is viewed as an image with each robot a pixel, these two pattern descriptors are simply the value the pixel would have after each of the two convolutions. These second-order derivatives are the *Pattern Descriptors* – denoted $P_x$ and $P_y$ for the horizontal and vertical directions respectively – and are used to calculate the most probable local pattern. ![Illustration of applying the two second order derivative masks.[]{data-label="fig:filter"}](pics/filter.png){width="0.5\columnwidth"} To be specific, with $M$ denoting my local color and $T$, $R$, $B$, and $L$ denoting the color of my top, right, bottom, and left neighbors, $P_x$ and $P_y$ are given by: $$\begin{aligned} P_x &= L + R - 2M \\ P_y &= T + B - 2M\end{aligned}$$ A pattern-probability array $p = [p_h, p_v, p_m]$ is used to record each robot’s pattern, where $p_h$ represents the probability of a pattern-type is selected and given a probability of $1$ based on our local *Pattern Descriptors*, and the other probabilities are all 0. This is shown in the equation below, where $T$ is some threshold value and $\left|val\right|$ is used to indicate $\texttt{abs}\left(val\right)$. $$\begin{aligned} \label{eq:pattern_descriptor} p = [p_h, p_v, p_m] = \begin{cases} [1, 0, 0] & \text{if } \left|P_y\right|-\left|P_x\right|>T \\ [0, 1, 0] & \text{if } \left|P_x\right|-\left|P_y\right|>T \\ [0, 0, 1] & \text{otherwise} \end{cases}\end{aligned}$$ Note that a grid representation has only been chosen for the simplicity of performing (and explaining) the mathematical operations, but one could equally well perform the described convolutions using continuous representations and local range and bearing information. Distributed Average Consensus Scheme {#sec:consensus} ------------------------------------ Once each robot has computed the most likely local pattern (ie, computed $p = [p_h, p_v, p_m]$), they need to achieve consensus on the global pattern. We use the distributed average consensus scheme [@xiao2005scheme] for this purpose. In each step of this scheme, the robot updates its local $p$ to be a weighted average of its own and its neighbors’. This step is repeated many times, allowing information to diffuse through the swarm. Since the weighted average just uses local information, each step takes the same amount of time regardless of the number of robots in the swarm. The number of steps needed was determined experimentally. The weighted-average calculation uses Metropolis weights, defined as: $$\label{eq: weights} W_{i,j} = \left\{ \begin{array}{ll} \frac{1}{1+max\{ d_i, d_j \}}& \text{if } (i, j) \in E,\\ 1-\sum_{(i,k) \in E} {W_{i,k}}& \text{if } i = j,\\ 0& \text{ otherwise.} \end{array} \right.$$ The Metropolis weights are well-suited for distributed algorithms, since weight-calculation requires only local knowledge. Further, it is proven in [@xiao2005scheme] that Metropolis weights guarantee convergence of the average consensus provided that the infinitely occurring communication graphs are jointly connected. Once the robots have converged, the largest value in $p$ represents the most likely global pattern. For example, $p_h > p_v$ and $p_h > p_m$ indicates that the most likely global pattern is horizontal stripes. Pattern Generator {#sec:generator} ----------------- In this section, we describe the distributed pattern formation algorithm to generate a proper pattern to match the environment. ![Illustration of local activator-inhibitor model: on the left, the activation region (orange) is defined by $A_x$ and $A_y$ while the inhibition region (gray) is defined by $I_x$ and $I_y$; on the right, $W_1$ and $W_2$ are the two field values. $R1$ is related to $A_x$ and $A_y$ and $R2$ is related to $I_x$ and $I_y$, []{data-label="fig:morphogen"}](pics/morphogen.png){width="0.8\columnwidth"} Now that a global pattern has been selected, the robots next need to generate an appropriate camouflage pattern. We use the pattern-formation algorithm presented by Young [@young1984local]: a local activator-inhibitor model. In this model, each cell (robot) is either ‘on’ or ‘off’, and can generate two kinds of morphogens: activator morphogen and inhibitor morphogen. Together, these form a “morphogenetic field”. Note that the activator should be inside of the inhibitor (see left of Figure \[fig:morphogen\]). The cells (robots) in the activator morphogen contribute to stimulate change for nearby ‘on’ cells, and cells in the inhibitor morphogen contribute to stimulate change for nearby ‘off’ cells. ![Activator (orange) and Inhibitor (gray) Regions for each of the three patterns.[]{data-label="fig:droplet_morphogen"}](pics/droplet_morphogen.png){width="0.95\columnwidth"} During each step of this algorithm, each cell changes its ‘off’/‘on’ status based on the combined effect of all nearby morphogenetic fields. More specifically, a ‘strength’ is calculated with each ‘on’ robot contributing a positive value ($W_1$) if in the activator region or contributing a negative value ($W_2$) if outside the activator region and inside the inhibitor region. The robot then changes its state to ‘on’ if the strength is greater than 0, and to ‘off’ otherwise. This step is repeated until the states converge to a stable pattern. In [@young1984local] the author observes that convergence typically takes around five steps. This was consistent with our observations. In this framework, the different types of patterns are represented with differently shaped activator and inhibitor regions. The regions for each pattern are shown in Figure \[fig:droplet\_morphogen\]. Note that the region sizes mean that each robot only requires information from robots within two hops of it. Simulated Results {#sec:simulation} ================= We implemented the algorithm introduced above on a centralized system for testing. By presenting some simulated results here, we hope to demonstrate the algorithm’s functionality and add clarity to the explanation above. We run these tests with three images, one for each of the pattern types. Each image is $128 \times 128$ pixels, and gray scale. We simulate $64$ ($8 \times 8$) robots. Note that this grid of $8 \times 8$ robots is in many ways analogous to the sensor of a digital camera, albeit a camera with only $8 \times 8$ sensors and thus with very low resolution. If you were to recapture our test images with such a low resolution camera, many different pixels in the test image would contribute to the camera’s output, resulting in a very blurry image. We therefore downsample the input image by taking the average of $16 \times 16$ pixel blocks. This blurred image is used as the color sensed by each robot for selecting the most likely pattern. For pattern generation, the initial on/off state is determined by making the blurred image binary (ie. white and black). Figure \[fig:8simResults\] shows the entire process for each of the three input images. ; Once the Droplets calculate the local pattern based on their sensed color and that of their neighbors, they need to achieve consensus on the global pattern. As has been discussed in Section \[sec:consensus\], convergence of this value is guaranteed. Next, the pattern generator described above is used (Section \[sec:generator\]) with the activator and inhibitor regions seen in Figure \[fig:droplet\_morphogen\]. The activator field value of $W_1 = 1$ was used, as suggested in paper [@young1984local]. The inhibitor field value, $W_2$, is a parameter which gives rough control over what proportion of the robots are ‘on’ in the final pattern. We found that $W_2 = -0.75$ gave qualitatively good results for all three of the pattern generators. Finally, we start pattern generation with each robot’s initial state to be ‘on’ or ‘off’ status based by the sensed value. If the value is less than $127$ we set it black, otherwise we set it white. The pattern generator runs for ten iterations. Robots on the image boundary use a reflection of their neighbors. A robot on the top row, for example, would count its bottom neighbor twice, as the top row is empty. To further test the simulated algorithm, we added a simple noise model. For measurement error, instead of always assigning the appropriate color to a robot based on its position, we assign a uniformly random color with probability $\rho_{meas}$. For communication error, at each step in the algorithm where information from a neighbor is shared with a robot for a calculation (including the step where a robot’s neighbors are calculated in the first place), are robot does not share this information with probability $\rho_{comm}$. ![The y-axis is the pixel difference from the ‘correct’ pattern and the x-axis is the error probability. The red line shows the effect of measurement errors ($\rho_{meas}$). The blue line shows the effect of communication errors ($\rho_{comm}$). The green line shows the effect of both measurement and communication errors ($\rho_{comm}=\rho_{meas}$). Each data point reflects the mean result over 10 trials of the forest image.[]{data-label="fig:errorAnalysis"}](pics/error_analysis.png){width="0.6\linewidth"} For a quantitative measure of the effects of error, we calculated the total absolute difference between the final generated pattern in the presence of error, and the final generated pattern without any error (as visible in the bottom row of Figure \[fig:8simResults\]). These results are charted in Figure \[fig:errorAnalysis\]. Note that, with the $8x8$ images used, a purely random image should give us a difference of $32$, on average. The algorithm seems quite robust to errors of up to $0.15--0.2$. After these thresholds, the error increases sharply. (Results shown here are for the forest image, with other images yielding similar results.) Qualitatively, we observe that even as errors started to appear, many of the resulting patterns still looked ‘good’, i.e., still had prominent vertical stripes. The main determining factor as the probability of error increased seemed to be in the global pattern detection. If the correct pattern (vertical stripes in this case) is selected, the resulting pattern will fit well even with large errors. Correct pattern selection grows increasingly infrequent, however. Hardware Implementation {#sec:hardware_imp} ======================= To validate the proposed algorithm and to understand the sorts of errors that real hardware introduces, we implemented the algorithm described above on a swarm of “Droplets”  [@farrow2014miniature; @klingner2014]. The Droplets are an open-source platform, with source code and manufacturing information available online[^1]. Each Droplet is roughly cylindrical with a radius of $2.2\,\mathrm{cm}$ and a height of $2.4\,\mathrm{cm}$. The Droplets use an Atmel xMega128A3U micro-controller, and receive power via their legs through a floor with alternating strips of $+5V$ and $GND$. Each Droplet has six infrared emitters, receivers, and sensors, which are used for communication and for the range and bearing system [@farrow2014miniature]. The top of each board has sensors to detect the color and brightness of ambient light, and an RGB LED. Each droplet has a 16-bit unique ID. In our implementation, each Droplet maintains an array of neighbor’s IDs. Messages are labeled with phase flags and attached with Droplets’ IDs. The Droplets are synchronized using a firefly synchronization algorithm [@mirollo1990synchronization; @werner2005firefly]. A simple TDMA protocol is used with $37$ slots, each $350$ms long. Each frame is thus $12.95$s long. Each robot is assigned a slot based on its unique id modulo 37. The number of slots (37) was chosen to be large enough that the probability of two adjacent robots sharing a slot is low, but small enough that the algorithm runs quickly. ![Neighbor array. The orange neighbors(0-3) are used for pattern recognition; the green neighbors(4-7) are used in addition to the orange for pattern consensus.All pictured neighbors (orange, green and gray) are used for pattern formation.[]{data-label="fig:droplet_neighbors"}](pics/droplet_neighbors.png){width="0.4\columnwidth"} In Phase 0 (neighbor identification), we initialize and configure the neighbor ID arrays which store neighboring Droplets’ IDs. Range and bearing information is used to calculate positions for each Droplet’s immediate neighbors, and neighbors-of-neighbors are learned by listening to the messages sent by Droplets each slot, which contain that Droplet’s neighbors. The positions of Droplets and their indices in the array are illustrated in Fig. \[fig:droplet\_neighbors\]. We allot $20$ frames for Phase 0, since the neighbor information is critical to the three phases. In Phase 1 (color sensing and recognition), each Droplet communicates the color it senses, and stores the colors its neighbors sense, as learned through communications. Once this is complete, each (non-boundary) Droplet should know the ID and position of 12 neighbors, as well as those neighbor’s sensed colors. With this information, the Droplets calculate an pattern probability array $p$ as described in Section \[sec:descriptor\]. This phase is allotted $10$ frames. In Phase 2 (pattern consensus), each Droplet communicates its pattern probability array $p$ and receives pattern probability arrays from its neighbors. At the end of each frame, each Droplet updates its pattern probability array according to the weighted-average consensus algorithm, as described in Section \[sec:consensus\]. Each ‘step’ of the consensus algorithm spans one frame. This phase is allotted $35$ frames. In Phase 3 (pattern formation), each Droplet communicates its intended color for the generated pattern, and receives that information from neighboring Droplets. At the end of each frame, each Droplet updates its color in the generated pattern from corresponding Droplets. Each Droplet exchanges pattern color message with neighbors. At the end of each frame, each Droplet updates its pattern color according to the pattern generation algorithm described in Section \[sec:generator\]. This phase is allotted $20$ frames. Hardware Results {#sec:hardware_results} ================ ![Initial condition (left), final pattern with projected pattern (middle) and final pattern (right) for camouflaging the tiger stripe pattern[]{data-label="fig:t_7_8_11"}](pics/t_7_8_11_init.png "fig:"){width="0.3\columnwidth"} ![Initial condition (left), final pattern with projected pattern (middle) and final pattern (right) for camouflaging the tiger stripe pattern[]{data-label="fig:t_7_8_11"}](pics/t_7_8_11_final_patternOn.png "fig:"){width="0.3\columnwidth"} ![Initial condition (left), final pattern with projected pattern (middle) and final pattern (right) for camouflaging the tiger stripe pattern[]{data-label="fig:t_7_8_11"}](pics/t_7_8_11_final.png "fig:"){width="0.3\columnwidth"} ![Convergence of pattern probabilities of randomly chosen Droplets camouflaging tiger stripe pattern[]{data-label="fig:t_7_8_11_pp"}](pics/t_7_8_11_pp.png){width="0.6\columnwidth"} A hardware implementation of the swarm camouflage algorithm is shown in Figure \[fig:t\_7\_8\_11\]. For this test, the projected image for the Droplets to sense is a tiger stripe pattern. The results of this test are interesting because a striped pattern is maintained, despite the failure of two units. This, in addition to the more-difficult-to-count failures in communication and color sensing. Figure \[fig:t\_7\_8\_11\_pp\] shows the pattern probability convergence for a random sampling of Droplets, when run with a simple horizontal stripe pattern projected on them. The swarm reaches consensus on a horizontal pattern, converging to $p_h=0.61$. Conclusion ========== We present a distributed camouflage system in which a robot swarm can sense the environment color, recognize the local pattern, achieve consensus on the global pattern, and generate a camouflage pattern consistent with the environment the robots are in. In our design, pattern descriptors are proposed for recognizing local patterns. A weighted-average consensus scheme is then utilized, allowing the swarm to converge to a common global pattern. Finally, a pattern formation model is applied to each robot which generates a pattern appropriate for the background. This is accomplished using local communication and simple mathematical operations. We simulated the proposed algorithm on a couple of patterns from nature: a desert, a forest, and leopard skin. After going through all the phases in the algorithm, and successfully agreeing on a global pattern, the simulation results show that robots with wrong color reading can correct themselves to match the global pattern. This is especially obvious for the horizontal and vertical patterns. We also tried to test the distributed algorithm by applying it on the Droplet swarm robotics platform. The results from the Droplets is promising since the robots can agree on the global pattern and show a proper matching pattern even if individual Droplets stop working. As communication on the Droplets is not perfectly reliable, the resultant patterns exhibit some random variations; they do not perfectly match simulation. Even these variations, however, will roughly follow the desired background pattern, seeming to bend or twist around the erroneous robot. In the future, we wish to test the algorithm on a more purpose-built hardware platform which would allow for higher-resolution patterns, and extend the algorithm to include consensus on the dominant colors and patterns consisting of more than two colors. This research has been supported by NSF grant \#1150223. [^1]: <http://github.com/correlllab/cu-droplet>

--- abstract: | Drawdowns measuring the decline in value from [the]{} historical running maxima over a given period of time, are considered as extremal events from the standpoint of risk management. To date, research on the topic has mainly focus on the side of severity by studying the first drawdown over certain [pre-specified]{} size. In this paper, we extend the discussion by investigating the frequency of drawdowns, and some of their inherent characteristics. We [consider]{} two types of drawdown time sequences depending on whether a historical running maximum [is reset or not]{}. For each type, we study the frequency rate of drawdowns, the Laplace transform of the $n$-th drawdown time, the distribution of the running maximum and the value process at the $n$-th drawdown time, as well as some other quantities of interest. Interesting relationships between these two drawdown time sequences are also established. Finally, insurance policies protecting against the risk of frequent drawdowns are also proposed and priced. *Keywords*: Drawdown; Frequency; Brownian motion *MSC*(2000): Primary 60G40; Secondary 60J65 91B2[4]{} author: - 'David Landriault[^1]' - 'Bin Li[^2]' - 'Hongzhong Zhang[^3]' title: On the Frequency of Drawdowns for Brownian Motion Processes --- 15.5pt Introduction ============ We consider a drifted Brownian motion $X=\{X_{t},t\geq0\}$, defined on a filtered probability space $(\Omega,\{\mathcal{F}_{t},t\geq0\},\mathbb{P})$, with dynamics$$X_{t}=x_{0}+\mu t+\sigma W_{t},$$ where $x_{0}\in\mathbb{R} $ is the initial value, $\mu\in\mathbb{R} $, $\sigma>0$, and $\{W_{t},t\geq0\}$ is a standard Brownian motion. The time of the first drawdown over size $a>0$ is denoted by $$\tau_{a}:=\inf\{t>0:M_{t}-X_{t}\geq a\}, \label{tau a}$$ where $M=\left\{ M_{t},t\geq0\right\} $ with $M_{t}:=\sup_{s\in\lbrack 0,t]}X_{t}$ is the running maximum process of $X$. Here and henceforth, we follow the convention that $\inf{\emptyset}=\infty$ and $\sup{\emptyset}=0$. Drawdown is one of the most frequently quoted path-dependent risk indicators for mutual funds and commodity trading advisers (see, e.g., Burghardt et al. [@DrawdownRisk]). From a risk management standpoint, large drawdowns should be considered as extreme events of which both the severity and the frequency need to be investigated. Considerable attention has been paid to the severity aspect of the problem by [pre-specifying]{} a threshold, namely $a>0$, of the size of drawdowns, and subsequently studying various properties associated to the first drawdown time $\tau_{a}$. In this paper, we extend the discussion by investigating the frequency of drawdowns. To this end, we derive the joint distribution of the $n$-th drawdown time, the running maximum, and the value process at the drawdown time for a drifted Brownian motion. Using the general theory on renewal process, we proceed to characterize the behavior of the frequency of drawdown episodes in a long time-horizon. Finally, we introduce some insurance policies which protect against the risk associated with frequent drawdowns. These policies are similar to the sequential barrier options in over-the-counter (OTC) market (see, e.g., Pfeffer [@Pfef01]). Through Carr’s randomization of maturities, we provide closed-form pricing formulas by making use of the main theoretical results of the paper. Literature review ----------------- The first drawdown time $\tau_{a}$ is the first passage time of the drawdown process $\left\{ M_{t}-X_{t},t\geq0\right\} $ to level $a$ or above. It has been extensively studied in the literature of applied probability. The joint Laplace transform of $\tau_{a}$ and $M_{\tau_{a}}$ was first derived by Taylor [@Taylor75] for a drifted Brownian motion. Lehoczky [@Lehoczky77] extended the results to a general time-homogeneous diffusion by a perturbation approximation approach. An infinite series expansion of the distribution of $\tau_{a}$ was derived by Douady et al. [@DSY00] for a standard Brownian motion and the results were generalized to a drifted Brownian motion by Magdon et al. [@MDD04]. [The dual of drawdown, known as drawup, measures the increase in value from the historical running minimum over a given period of time. ]{}The probability that a drawdown precedes a drawup is subsequently studied by Hadjiliadis and Vecer [@HadjVece] and Pospisil et al. [@PospVeceHadj] under the drifted Brownian motion and the general time-homogeneous diffusion process, respectively. Mijatovic and Pistorius [@MijatovicPistorius] derived the joint Laplace transform of $\tau_{a}$ and the last passage time at level $M_{\tau_{a}}$ prior to $\tau_{a}$, associated with the joint distribution of the running maximum, the running minimum, and the overshoot at $\tau_{a}$ for a spectrally negative Lévy process. The probability that a drawdown precedes a drawup in a finite time-horizon is studied under drifted Brownian motions and simple random walks in [@ZhanHadj]. [More recently, [@Zhang13; @ZhanHadj12] studied Laplace transforms of the drawdown time, the so-called speed of market crash, and various occupation times at the first exit and the drawdown time for a general time-homogeneous diffusion process.]{} In quantitative risk management, drawdowns and its descendants have become an increasingly popular and relevant class of path-dependent risk indicators. A portfolio optimization problem with constraints on drawdowns was explicitly solved by Grossman and Zhou [@GrosZhou] in a Black-Scholes framework. Hamelink and Hoesli [@HameHoes04] used the relative drawdown as a performance measure in optimization of real estate portfolios. Chekhlov et al. [@ChekUryaZaba] proposed a new family of risk measures called conditional drawdown and studied parameter selection techniques and portfolio optimization under constraints on conditional drawdown. Some [novel]{} financial derivatives were introduced by Vecer [@Vece06] to hedge maximum drawdown risk. Pospisil and Vecer [@PospVece] invented a class of Greeks to study the sensitivity of investment portfolios to running maxima and drawdowns. Later, Carr et al. [@CarrZhanHaji] introduced a class of European-style digital drawdown insurances and proposed semi-static hedging strategies using barrier options and vanilla options. The swap type insurances and cancelable insurances against drawdowns were studied in Zhang et al. [@ZhanLeunHadj]. Definitions ----------- [While sustaining downside risk can be appropriately characterized using the drawdown process and the first drawdown time, economic turmoil and volatile market fluctuations are better described by quantities containing more path-wise information, such as the frequency of drawdowns. ]{}The existing knowledge about the first drawdown time $\tau_{a}$ provides only limited and implicit information about the frequency of drawdowns. For the purpose of tackling the problem of frequency directly and systematically, we define below two types of drawdown time sequences depending on whether the last running maximum needs to be recovered or not. The first sequence $\{\tilde{\tau}_{a}^{n},n\in\mathbb{N} \}$ is called the *drawdown times with recovery*, defined recursively as$$\tilde{\tau}_{a}^{n}:=\inf\{t>\tilde{\tau}_{a}^{n-1}:M_{t}-X_{t}\geq a,M_{t}>M_{\tilde{\tau}_{a}^{n-1}}\}, \label{tau til}$$ where $\tilde{\tau}_{a}^{0}=0$. Note that, after each $\tilde{\tau}_{a}^{n-1}$, the corresponding running maximum $M_{\tilde{\tau}_{a}^{n-1}}$ must be recovered before the next drawdown time $\tilde{\tau}_{a}^{n}$. In other words, the running maximum is reset and updated only when the previous one is revisited. Since the sample paths of $X$ are almost surely (a.s.) continuous, we have that $M_{\tilde{\tau}_{a}^{n}}-X_{\tilde{\tau}_{a}^{n}}=a$ a.s. if $\tilde{\tau}_{a}^{n}<\infty$. The second sequence $\{\tau_{a}^{n},n\in\mathbb{N} \}$ is called the *drawdown times without recovery*, defined recursively as$$\tau_{a}^{n}:=\inf\{t>\tau_{a}^{n-1}:M_{[\tau_{a}^{n-1},t]}-X_{t}\geq a\}, \label{tau}$$ where $\tau_{a}^{0}:=0$ and $M_{[s,t]}:=\sup_{s\leq u\leq t}X_{u}$. From definition (\[tau\]), it is implicitly assumed that the running maximum $M_{\tau_{a}^{n}}$ is reset to $X_{\tau_{a}^{n}}$ at the drawdown time $\tau_{a}^{n}$. In fact, $\tau_{a}^{n}$ is the so-called iterated stopping times associated with $\tau_{a}$ defined as$$\tau_{a}^{n}=\left\{ \begin{array} [c]{ll}\tau_{a}^{n-1}+\tau_{a}\circ\theta_{\tau_{a}^{n-1}}, & \text{when }\tau _{a}^{n-1}\text{ and }\tau_{a}\circ\theta_{\tau_{a}^{n-1}}\text{ are finite,}\\ \infty, & \text{otherwise,}\end{array} \right. \label{iterated}$$ where $\theta$ is the Markov shift operator such that $X_{t}\circ\theta _{s}=X_{s+t}$ for $s,t\geq0$. Note that both $\tau_{a}^{n}$ and $\tilde{\tau}_{a}^{n}$ are independent of the initial value $x_{0}$ for not only the drifted Brownian motion $X$, but also a general Lévy process. In view of definitions (\[tau\]) and (\[tau til\]), it is clear that the following inclusive relation of the two types of drawdown times holds:$$\{\tilde{\tau}_{a}^{n},n\in\mathbb{N} \}\subset\{\tau_{a}^{n},n\in\mathbb{N} \}.$$ In other words, for each $n\in\mathbb{N} $, there exists a unique positive integer $m\geq n$ such that $\tilde{\tau }_{a}^{n}=\tau_{a}^{m}$ (if $\tilde{\tau}_{a}^{n}<\infty$). Our motivation for introducing the two drawdown time sequences are as follows. The drawdown times with recovery $\{\tilde{\tau}_{a}^{n},n\in\mathbb{N} \}$ are easy to identify from the sample paths of $X$ by searching the running maxima. Moreover, they are consistent with definition (\[tau a\]) of the first drawdown $\tau_{a}$ in the sense that a drawdown can be considered as incomplete if the running maximum has not been revisited. However, there are also some crucial drawbacks of $\{\tilde{\tau}_{a}^{n},n\in\mathbb{N} \}$ which motivate us to introduce the drawdown times without recovery $\{\tau_{a}^{n},n\in\mathbb{N} \}$. First, the downside risk during recovering periods is neglected. One or more larger drawdowns may occur in a recovering period. Second, the threshold $a$ needs to be adjusted to gain a more integrated understanding about the severity of drawdowns. In other words, the selection of $a$ becomes tricky. Third, the requirement of recovery is too strong. In real world, a historical high water mark may never be recovered again, [as in the case of a financial bubble [@Bubble11].]{} The rest of the paper is organized as follows. In Section 2, some preliminaries on exit times and the first drawdown time $\tau_{a}$ of the drifted Brownian motion $X$ are presented. In Section 3, the frequency rate of drawdowns, and the Laplace transform of $\tilde{\tau}_{a}^{n}$ associated with the distribution of $M_{\tilde{\tau}_{a}^{n}}$ and/or $X_{\tilde{\tau}_{a}^{n}}$ are derived. Section 4 is parallel to Section 3 but studies the drawdown times without recovery $\{\tau_{a}^{n},n\in\mathbb{N} \}$. Interesting connections between the two drawdown time sequences are established. In Section 5, some insurance contracts are introduced to insure against the risk of frequent drawdowns. Preliminaries ============= Henceforth, for ease of notation, we write $\mathbb{E}_{x_{0}}[\,\cdot \,]=\mathbb{E}[\left. \cdot\,\right\vert X_{0}=x_{0}]$ for the conditional expectation, $\mathbb{P}_{x_{0}}\{\,\cdot\,\}$ for the corresponding probability and $\mathbb{E}_{x_{0}}[\,\cdot\,;U]=\mathbb{E}_{x_{0}}[\,\cdot\,{1}_{U}]$ with ${1}_{U}$ denoting the indicator function of a set $U\subset\Omega$. In particular, when $x_{0}=0$, we drop the subscript $x_{0}$ from the conditional expectation and probability. For $x\in\mathbb{R} $, let $T_{x}^{+}=\inf\left\{ t\geq0:X_{t}>x\right\} $ and$\ T_{x}^{-}=\inf\left\{ t\geq0:X_{t}<x\right\} $ be the first passage times of $X$ to levels in $\left[ x,\infty\right) $ and $\left( -\infty,x\right] $, respectively. For $a<x<b$ and $\lambda>0$, it is known that$$\mathbb{E}_{x}[\mathrm{e}^{-\lambda T_{a}^{-}}]=\mathrm{e}^{\beta_{\lambda }^{-}(x-a)}\qquad\text{and}\qquad\mathbb{E}_{x}[\mathrm{e}^{-\lambda T_{b}^{+}}]=\mathrm{e}^{\beta_{\lambda}^{+}(x-b)}, \label{one-sided L}$$ where $\beta_{\lambda}^{\pm}=\frac{-\mu\pm\sqrt{\mu^{2}+2\lambda\sigma^{2}}}{\sigma^{2}}$ (see, e.g., formula 2.0.1 on Page 295 of Borodin and Salminen [@borodin2002handbook]). By letting $\lambda\rightarrow0+$ in (\[one-sided L\]), we have$$\mathbb{P}_{x}\left\{ T_{b}^{+}<\infty\right\} =\mathrm{e}^{\frac{-\mu +|\mu|}{\sigma^{2}}(x-b)}\qquad\text{and}\qquad\mathbb{P}_{x}\left\{ T_{a}^{-}<\infty\right\} =\mathrm{e}^{\frac{-\mu-|\mu|}{\sigma^{2}}(x-a)}. \label{one-sided P}$$ From Taylor [@Taylor75] or Equation (17) of Lehoczky [@Lehoczky77], we have the following joint Laplace transform of the first drawdown time $\tau_{a}$ and its running maximum $M_{\tau_{a}}$. \[lem 1\]For $\lambda,s>0$, we have$$\mathbb{E}\left[ \mathrm{e}^{-\lambda\tau_{a}-sM_{\tau_{a}}}\right] =\frac{c_{\lambda}}{b_{\lambda}+s} \label{JL}$$ where $b_{\lambda}=\frac{\beta_{\lambda}^{+}\mathrm{e}^{-\beta_{\lambda}^{-}a}-\beta_{\lambda}^{-}\mathrm{e}^{-\beta_{\lambda}^{+}a}}{\mathrm{e}^{-\beta_{\lambda}^{-}a}-\mathrm{e}^{-\beta_{\lambda}^{+}a}}$ and $c_{\lambda }=\frac{\beta_{\lambda}^{+}-\beta_{\lambda}^{-}}{\mathrm{e}^{-\beta_{\lambda }^{-}a}-\mathrm{e}^{-\beta_{\lambda}^{+}a}}$. A Laplace inversion of (\[JL\]) with respect to $s$ results in $$\mathbb{E}[\mathrm{e}^{-\lambda\tau_{a}};M_{\tau_{a}}>x]=\frac{c_{\lambda}}{b_{\lambda}}\mathrm{e}^{-b_{\lambda}x},\label{L1}$$ for $x>0$. Furthermore, letting $x\rightarrow0+$ in (\[L1\]), we immediately have $$\mathbb{E}[\mathrm{e}^{-\lambda\tau_{a}}]=c_{\lambda}/b_{\lambda}.\label{lap}$$ A numerical evaluation of the distribution function of $\tau_{a}$ (and more generally $\tau_{a}^{n}$ and $\tilde{\tau}_{a}^{n}$) by an inverse Laplace transform method will be given at the end of Section 4. Other forms of infinite series expansion of the distribution of $\tau_{a}$ were derived by Douady et al. [@DSY00] and Magdon et al. [@MDD04] for a standard Brownian motion and a drifted Brownian motion, respectively. By taking the derivative with respect to $\lambda$ in (\[lap\]) and letting $\lambda \rightarrow0+$, we have $$\mathbb{E}[\tau_{a}]=\frac{\sigma^{2}\mathrm{e}^{2\mu a/\sigma^{2}}-\sigma ^{2}-2\mu a}{2\mu^{2}}.$$ It is straightforward to check that $$\lim_{\lambda\rightarrow0+}b_{\lambda}=\lim_{\lambda\rightarrow0+}c_{\lambda }=\frac{\gamma}{\mathrm{e}^{\gamma a}-1},\label{bc}$$ where $\gamma=\frac{2\mu}{\sigma^{2}}$. In the risk theory literature, the constant $\gamma$ is known as the *adjustment coefficient*. In particular, when $\mu=0$, the quantity $\frac{\gamma}{\mathrm{e}^{\gamma a}-1}$ is understood as $\lim_{\gamma\rightarrow0}\frac{\gamma}{\mathrm{e}^{\gamma a}-1}=\frac{1}{a}$. It follows from (\[lap\]) and (\[bc\]) that $$\mathbb{P}\left\{ \tau_{a}<\infty\right\} =\lim_{\lambda\rightarrow 0+}\mathbb{E}\left[ \mathrm{e}^{-\lambda\tau_{a}}\right] =1.$$ Furthermore, we have $$\mathbb{P}\left\{ M_{\tau_{a}}\geq x\right\} =\mathbb{P}\left\{ M_{\tau _{a}}\geq x,\tau_{a}<\infty\right\} =\lim_{\lambda\rightarrow0+}\mathbb{E}\left[ \mathrm{e}^{-\lambda\tau_{a}};M_{\tau_{a}}\geq x\right] =\mathrm{e}^{-\frac{\gamma x}{\mathrm{e}^{\gamma a}-1}}.\label{M}$$ which implies that the running maximum at the first drawdown time $M_{\tau _{a}}$ follows an exponential distribution with mean $\left( \mathrm{e}^{\gamma a}-1\right) /\gamma$ (see, e.g., Lehoczky [@Lehoczky77]). The drawdown times with recovery ================================ We begin our analysis with the drawdown times with recovery $\{\tilde{\tau }_{a}^{n},n\in\mathbb{N} \}$ given that their structure leads to a simpler analysis than their counterpart ones without recovery. We first consider the asymptotic behavior of the frequency rate of drawdowns with recovery. Let $\tilde{N}_{t}^{a}=\sum\nolimits_{n=1}^{\infty}1_{\left\{ \tilde{\tau}_{a}^{n}\leq t\right\} }$ be the number of drawdowns with recovery observed by time $t\geq0$, and define $\tilde{N}_{t}^{a}/t$ to be the frequency rate of drawdowns. It is clear that $\left\{ \tilde{N}_{t}^{a},t\geq0\right\} $ is a delayed renewal process where the first drawdown time is distributed as $\tau_{a}$, while the subsequent inter-drawdown times are independent and identically distributed as $T_{X_{\tau_{a}}+a}^{+}\circ\tau_{a}$. From Theorem 6.1.1 of Rolski et al. [@Rolskibook], it follows that, with probability one, $$\lim_{t\rightarrow\infty}\frac{\tilde{N}_{t}^{a}}{t}=\left\{ \begin{array} [c]{lc}\frac{1}{\mathbb{E}[\tau_{a}]+\mathbb{E}[T_{a}^{+}]}=\frac{2\mu^{2}}{\sigma^{2}\left( \mathrm{e}^{2\mu a/\sigma^{2}}-1\right) }, & \text{if }\mu>0,\\ 0, & \text{if }\mu\leq0. \end{array} \right.$$ Moreover, one could easily obtain some central limit theorems for $\tilde {N}_{t}^{a}$ by Theorem 6.1.2 of Rolski et al. [@Rolskibook]. Next, we study the joint Laplace transform of $\tilde{\tau}_{a}^{n}$ and $M_{\tilde{\tau}_{a}^{n}}$. Note that $X_{\tilde{\tau}_{a}^{n}}=M_{\tilde {\tau}_{a}^{n}}-a$ a.s. whenever $\tilde{\tau}_{a}^{n}<\infty$, and thus the following theorem is sufficient to characterize the triplet $\left( \tilde{\tau}_{a}^{n},M_{\tilde{\tau}_{a}^{n}},X_{\tilde{\tau}_{a}^{n}}\right) $. \[thm M til L\]For $n\in\mathbb{N} $ and $\lambda,x\geq0$, we have$$\mathbb{E}\left[ \mathrm{e}^{-\lambda\tilde{\tau}_{a}^{n}};M_{\tilde{\tau }_{a}^{n}}>x\right] =\left( \frac{c_{\lambda}}{b_{\lambda}}\right) ^{n}\mathrm{e}^{-(n-1)\beta_{\lambda}^{+}a}\sum_{m=0}^{n-1}\frac{(b_{\lambda }x)^{m}}{m!}\mathrm{e}^{-b_{\lambda}x}. \label{M tau til L}$$ To prove this result, we first condition on the first drawdown time $\tau_{a}$ and subsequently on the time for the process $X$ to recover its running maximum. Using the strong Markov property of $X$ and (\[JL\]), it is clear that $$\begin{aligned} \mathbb{E}\left[ \mathrm{e}^{-\lambda\tilde{\tau}_{a}^{n}-sM_{\tilde{\tau }_{a}^{n}}}\right] & =\mathbb{E}\left[ \mathrm{e}^{-\lambda\tilde{\tau }_{a}^{n}-sM_{\tilde{\tau}_{a}^{n}}};\tilde{\tau}_{a}^{n}<\infty\right] \nonumber\\ & =\mathbb{E}\left[ \mathrm{e}^{-\lambda\tau_{a}-sM_{\tau_{a}}}\right] \mathbb{E}\left[ \mathrm{e}^{-T_{a}^{+}}\right] \mathbb{E}\left[ \mathrm{e}^{-\lambda\tilde{\tau}_{a}^{n-1}-sM\tilde{\tau}_{a}^{n-1}}\right] \nonumber\\ & =\frac{c_{\lambda}}{b_{\lambda}+s}\mathrm{e}^{-\beta_{\lambda}^{+}a}\mathbb{E}\left[ \mathrm{e}^{-\lambda\tilde{\tau}_{a}^{n-1}-sM\tilde{\tau }_{a}^{n-1}}\right] \nonumber\\ & =\left( \frac{c_{\lambda}}{b_{\lambda}+s}\right) ^{n-1}\mathrm{e}^{-(n-1)\beta_{\lambda}^{+}a}\mathbb{E}\left[ \mathrm{e}^{-\lambda\tau _{a}-sM_{\tau_{a}}}\right] \nonumber\\ & =\left( \frac{c_{\lambda}}{b_{\lambda}+s}\right) ^{n}\mathrm{e}^{-(n-1)\beta_{\lambda}^{+}a}. \label{babu}$$ Given that $\left( b_{\lambda}/\left( b_{\lambda}+s\right) \right) ^{n}$ is the Laplace transform of an Erlang random variable (rv) with mean $n/b_{\lambda}$ and variance $n/\left( b_{\lambda}\right) ^{2}$, a tail inversion of (\[babu\]) wrt $s$ yields (\[M tau til L\]). In particular, letting $x\rightarrow0+$, we have$$\mathbb{E}\left[ \mathrm{e}^{-\lambda\tilde{\tau}_{a}^{n}}\right] =\left( c_{\lambda}/b_{\lambda}\right) ^{n}\mathrm{e}^{-(n-1)\beta_{\lambda}^{+}a}, \label{L tau til}$$ for $n\in\mathbb{N} $. Furthermore, letting $\lambda\rightarrow0+$ in (\[L tau til\]), together with (\[bc\]) and $\lim_{\lambda\rightarrow0+}\beta_{\lambda}^{+}=\frac {-\mu+|\mu|}{\sigma^{2}}$, we have $$\mathbb{P}\left\{ \tilde{\tau}_{a}^{n}<\infty\right\} =\left\{ \begin{array} [c]{lc}1, & \text{if }\mu\geq0,\\ \mathrm{e}^{(n-1)\gamma a}, & \text{if }\mu<0. \end{array} \right. \label{tau til P}$$ In other words, a historical running maximum may never be recovered if the drift $\mu<0$. For $n\in\mathbb{N} $ and $x>0$, we have$$\mathbb{P}\left\{ M_{\tilde{\tau}_{a}^{n}}>x,\tilde{\tau}_{a}^{n}<\infty\right\} =\left\{ \begin{array} [c]{lc}\mathrm{e}^{-\frac{\gamma x}{\mathrm{e}^{\gamma a}-1}}\sum_{m=0}^{n-1}\frac {1}{m!}\left( \frac{\gamma x}{\mathrm{e}^{\gamma a}-1}\right) ^{m}, & \text{if }\mu\geq0,\\ \mathrm{e}^{(n-1)\gamma a}\mathrm{e}^{-\frac{\gamma x}{\mathrm{e}^{\gamma a}-1}}\sum_{m=0}^{n-1}\frac{1}{m!}\left( \frac{\gamma x}{\mathrm{e}^{\gamma a}-1}\right) ^{m}, & \text{if }\mu<0. \end{array} \right. \text{.} \label{M tau til P}$$ Substituting (\[L tau til\]) into (\[M tau til L\]) yields $$\mathbb{E}\left[ \mathrm{e}^{-\lambda\tilde{\tau}_{a}^{n}};M_{\tilde{\tau }_{a}^{n}}>x\right] =\mathbb{E}\left[ \mathrm{e}^{-\lambda\tilde{\tau}_{a}^{n}}\right] \sum_{m=0}^{n-1}\frac{(b_{\lambda}x)^{m}}{m!}\mathrm{e}^{-b_{\lambda}x}. \label{pep}$$ Taking the limit when $\lambda\rightarrow0+$ in (\[pep\]), and then using (\[bc\]), one arrives at$$\mathbb{P}\left\{ M_{\tilde{\tau}_{a}^{n}}>x,\tilde{\tau}_{a}^{n}<\infty\right\} =\mathbb{P}\left\{ \tilde{\tau}_{a}^{n}<\infty\right\} \sum_{m=0}^{n-1}\frac{(\frac{\gamma x}{\mathrm{e}^{\gamma a}-1})^{m}}{m!}\mathrm{e}^{-\frac{\gamma x}{\mathrm{e}^{\gamma a}-1}}\text{.} \label{pep1}$$ Substituting (\[tau til P\]) into (\[pep1\]) results in (\[M tau til P\]). Note that (\[pep1\]) indicates $$\mathbb{P}\left\{ M_{\tilde{\tau}_{a}^{n}}>x\left\vert \tilde{\tau}_{a}^{n}<\infty\right. \right\} =\sum_{m=0}^{n-1}\frac{1}{m!}\left( \frac{\gamma x}{\mathrm{e}^{\gamma a}-1}\right) ^{m}\mathrm{e}^{-\frac{\gamma x}{\mathrm{e}^{\gamma a}-1}}\text{,} \label{abc}$$ for all $\mu\in\mathbb{R}$. This result can be interpreted probabilistically. Indeed, when $\tilde{\tau}_{a}^{n}<\infty$, $M_{\tilde{\tau}_{a}^{m}}-M_{\tilde{\tau}_{a}^{m-1}}$ follows an exponential distribution with mean $\left( \mathrm{e}^{\gamma a}-1\right) /\gamma$ for $m=1,2,...,n$. From the strong Markov property, the rv’s $M_{\tilde{\tau}_{a}^{m}}-M_{\tilde{\tau}_{a}^{m-1}}$ for all $m=1,2,...,n$ are all independent, and thus $M_{\tilde{\tau}_{a}^{n}}=\sum_{m=1}^{n}\left( M_{\tilde{\tau}_{a}^{m}}-M_{\tilde{\tau}_{a}^{m-1}}\right) $ is an Erlang rv with survival function (\[abc\]). In particular, when $n\rightarrow\infty$, it is easy to check that $\lim_{n\rightarrow\infty}\mathbb{P}\left\{ M_{\tilde{\tau}_{a}^{n}}>x\right\} =\mathbb{P}\left\{ T_{x}^{+}<\infty\right\} $ which agrees with (\[one-sided P\]). For completeness, we conclude this section with a result that is immediate from (\[M tau til L\]) and the fact that $M_{\tilde{\tau }_{a}^{n}}-X_{\tilde{\tau}_{a}^{n}}=a$ a.s. whenever $\tilde{\tau}_{a}^{n}<\infty$. For $n\in\mathbb{N} $ and $x\geq-a$, we have$$\mathbb{E}\left[ \mathrm{e}^{-\lambda\tilde{\tau}_{a}^{n}};X_{\tilde{\tau }_{a}^{n}}>x\right] =\left( \frac{c_{\lambda}}{b_{\lambda}}\right) ^{n}\mathrm{e}^{-(n-1)\beta_{\lambda}^{+}a}\sum_{m=0}^{n-1}\frac{\left( b_{\lambda}(x+a)\right) ^{m}}{m!}\mathrm{e}^{-b_{\lambda}(x+a)}.$$ Drawdown times without recovery =============================== In this section, we focus on the drawdown times without recovery which are more challenging to analyze than their counterparts with recovery. Let $N_{t}^{a}=\sum\nolimits_{n=1}^{\infty}1_{\left\{ \tau_{a}^{n}\leq t\right\} }$ be the number of drawdowns without recovery by time $t\geq0$. Clearly, $\left\{ N_{t}^{a},t\geq0\right\} $ is a renewal process with independent inter-drawdown times, all distributed as $\tau_{a}$. By Theorem 6.1.1 of Rolski et al. [@Rolskibook], it follows that, with probability one,$$\lim_{t\to\infty}\frac{N_{t}^{a}}{t}=\frac{1}{\mathbb{E}\left[ \tau _{a}\right] }=\frac{2\mu^{2}}{\sigma^{2}\mathrm{e}^{2\mu a/\sigma^{2}}-\sigma^{2}-2\mu a}\text{,}$$ which is consistent with our intuition based on (\[iterated\]). Here again, one can also obtain some central limit theorems for $N_{t}^{a}$ by an application of Theorem 6.1.2 of Rolski et al. [@Rolskibook]. Next, we characterize the joint distribution of $\left( \tau_{a}^{n},X_{\tau_{a}^{n}}\right) $ by deriving an explicit expression for $\mathbb{E}[\mathrm{e}^{-\lambda\tau_{a}^{n}};X_{\tau_{a}^{n}}>x]$. \[yyz\]For $n\in\mathbb{N} $ and $\lambda,x>0$, the joint distribution of $\left( \tau_{a}^{n},X_{\tau_{a}^{n}}\right) $ satisfies $$\mathbb{E}[\mathrm{e}^{-\lambda\tau_{a}^{n}};X_{\tau_{a}^{n}}>x]=\left( \frac{c_{\lambda}}{b_{\lambda}}\right) ^{n}\mathrm{e}^{-b_{\lambda}(x+na)}\sum_{m=0}^{n-1}\frac{\left( b_{\lambda}(x+na)\right) ^{m}}{m!}. \label{L tau x}$$ Given that $X_{\tau_{a}^{n}}+na$ is a positive rv (and $X_{\tau_{a}^{n}}$ is not), we prove (\[L tau x\]) by first deriving an expression for the joint Laplace transform of $\left( \tau_{a}^{n},X_{\tau_{a}^{n}}+na\right) $. By conditioning on the first drawdown time and its associated value process, and by making use of the strong Markov property and (\[JL\]), it is clear that for all $s\ge0$, $$\begin{aligned} \mathbb{E}\left[ \mathrm{e}^{-\lambda\tau_{a}^{n}-s\left( X_{\tau_{a}^{n}}+na\right) }\right] & =\mathbb{E}\left[ \mathrm{e}^{-\lambda\tau _{a}-s\left( X_{\tau_{a}}+a\right) }\right] \mathbb{E}\left[ \mathrm{e}^{-\lambda\tau_{a}^{n-1}-s\left( X_{\tau_{a}^{n-1}}+\left( n-1\right) a\right) }\right] \nonumber\\ & =\mathbb{E}\left[ \mathrm{e}^{-\lambda\tau_{a}-sM_{\tau_{a}}}\right] \mathbb{E}\left[ \mathrm{e}^{-\lambda\tau_{a}^{n-1}-s\left( X_{\tau _{a}^{n-1}}+\left( n-1\right) a\right) }\right] \nonumber\\ & =\frac{c_{\lambda}}{b_{\lambda}+s}\mathbb{E}\left[ \mathrm{e}^{-\lambda\tau_{a}^{n-1}-s\left( X_{\tau_{a}^{n-1}}+\left( n-1\right) a\right) }\right] \nonumber\\ & =\left( \frac{c_{\lambda}}{b_{\lambda}+s}\right) ^{n}\text{.} \label{aaa}$$ The Laplace transform inversion of (\[aaa\]) with respect to $s$ results in$$\mathbb{E}\left[ \mathrm{e}^{-\lambda\tau_{a}^{n}};\left( X_{\tau_{a}^{n}}+na\right) \in\mathrm{d} y\right] =\left( c_{\lambda}\right) ^{n}\frac{y^{n-1}\mathrm{e}^{-b_{\lambda}y}}{\left( n-1\right) !}\mathrm{d} y\text{,} \label{aaa1}$$ for $y\geq0$. Integrating (\[aaa1\]) over $y$ from $x+na$ to $\infty$ yields (\[L tau x\]). Letting $s\rightarrow0+$ in (\[aaa\]), it follows that $$\mathbb{E}[\mathrm{e}^{-\lambda\tau_{a}^{n}}]=\left( c_{\lambda}/b_{\lambda }\right) ^{n}=\left( \mathbb{E}[\mathrm{e}^{-\lambda\tau_{a}}]\right) ^{n}. \label{L tau}$$ Note that (\[L tau\]) and (\[bc\]) implies that$$\mathbb{P}\left\{ \tau_{a}^{n}<\infty\right\} =1.$$ It is worth pointing out that the relation $\mathbb{E}\left[ \mathrm{e}^{-\lambda\tau_{a}^{n}}\right] =\left( \mathbb{E}\left[ \mathrm{e}^{-\lambda\tau_{a}}\right] \right) ^{n}$ holds more generally for $X$ a general Lévy process or a renewal risk process (also known as the Sparre Andersen risk model [@AndersenRiskmodel]) given that the inter-drawdown times $\tau_{a}^{1}$, and $\left\{ \tau_{a}^{n}-\tau_{a}^{n-1}\right\} _{n\geq2}$ form a sequence of i.i.d. rvs. Similarly, letting $\lambda\rightarrow0+$ in (\[L tau x\]), it follows that $$\mathbb{P}\left\{ X_{\tau_{a}^{n}}\geq x\right\} =\mathrm{e}^{-\frac {\gamma(x+na)}{\mathrm{e}^{\gamma a}-1}}\sum_{m=0}^{n-1}\frac{\left( \frac{\gamma(x+na)}{\mathrm{e}^{\gamma a}-1}\right) ^{m}}{m!}, \label{X tau P}$$ for $n\in\mathbb{N} $ and $x\geq-na$. As expected, (\[X tau P\]) is the survival function of an Erlang rv with mean $n\left( \mathrm{e}^{\gamma a}-1\right) /\gamma$ and variance $n\left( \left( \mathrm{e}^{\gamma a}-1\right) /\gamma\right) ^{2}$, later translated by $-na$ units. Our objective is now to include $M_{\tau_{a}^{n}}$ in the analysis of the $n$-th drawdown time. A result particularly useful to do so is provided in Lemma \[constLT\] which consider a specific constrained Laplace transform of the first passage time to level $x$. \[constLT\]For $n\in\mathbb{N} $ and $x>0$, the constrained Laplace transform of $T_{x}^{+}$ together with this first passage time occurring before $\tau_{a}^{n}$ is given by$$\mathbb{E}\left[ \mathrm{e}^{-\lambda T_{x}^{+}};T_{x}^{+}<\tau_{a}^{n}\right] =\mathrm{e}^{-b_{\lambda}x}\sum_{j=0}^{n-1}\left( c_{\lambda }\mathrm{e}^{-b_{\lambda}a}\right) ^{j}\frac{x(x+ja)^{j-1}}{j!}\text{.} \label{cLT}$$ We prove this result by induction on $n$. For $n=1$, we have$$\begin{aligned} \mathbb{E}\left[ \mathrm{e}^{-\lambda T_{x}^{+}};T_{x}^{+}<\tau_{a}^{1}\right] & =\mathbb{E}\left[ e^{-\lambda T_{x}^{+}}\right] -\mathbb{E}\left[ e^{-\lambda T_{x}^{+}};T_{x}^{+}>\tau_{a}^{1}\right] \\ & =\mathrm{e}^{-\beta_{\lambda}^{+}x}-\int_{0}^{x}\mathbb{E}\left[ \mathrm{e}^{-\lambda\tau_{a}^{1}}; M_{\tau_{a}^{1}}\in\mathrm{d} y\right] \,\mathbb{E}_{y-a}\left[ \mathrm{e}^{-\lambda T_{x}^{+}}\right] \\ & =\mathrm{e}^{-\beta_{\lambda}^{+}x}-\int_{0}^{x}c_{\lambda}\mathrm{e}^{-b_{\lambda}y}\,\mathrm{e}^{-\beta_{\lambda}^{+}\left( x-y+a\right) }\mathrm{d} y\\ & =\mathrm{e}^{-\beta_{\lambda}^{+}x}-c_{\lambda}\mathrm{e}^{-\beta_{\lambda }^{+}a}\frac{\mathrm{e}^{-\beta_{\lambda}^{+}x}-\mathrm{e}^{-b_{\lambda}x}}{b_{\lambda}-\beta_{\lambda}^{+}}\text{,}$$ where we used in the third equality. On the other hand, using the fact that $c_{\lambda}\mathrm{e}^{-\beta _{\lambda}^{+}a}=b_{\lambda}-\beta_{\lambda}^{+}$, we have $$\mathbb{E}\left[ \mathrm{e}^{-\lambda T_{x}^{+}};T_{x}^{+}<\tau_{a}^{1}\right] =\mathrm{e}^{-b_{\lambda}x}\text{.}$$ We now assume that (\[cLT\]) holds for $n=1,2,...,k-1$ and shows that (\[cLT\]) also holds for $n=k$. Indeed, by the total probability formula,$$\begin{aligned} \mathbb{E}\left[ \mathrm{e}^{-\lambda T_{x}^{+}};T_{x}^{+}<\tau_{a}^{k}\right] & =\mathbb{E}\left[ \mathrm{e}^{-\lambda T_{x}^{+}};T_{x}^{+}<\tau_{a}^{1}\right] +\mathbb{E}\left[ \mathrm{e}^{-\lambda T_{x}^{+}};\tau_{a}^{1}<T_{x}^{+}<\tau_{a}^{k}\right] \nonumber\\ & =\mathrm{e}^{-b_{\lambda}x}+\int_{0}^{x}\mathbb{E}\left[ \mathrm{e}^{-\lambda\tau_{a}};M_{\tau_{a}}\in\mathrm{d}y\right] \mathbb{E}_{y-a}\left[ \mathrm{e}^{-\lambda T_{x}^{+}};T_{x}^{+}<\tau_{a}^{k-1}\right] \mathrm{d}y\nonumber\\ & =\mathrm{e}^{-b_{\lambda}x}+\int_{0}^{x}c_{\lambda}\mathrm{e}^{-b_{\lambda }y}\,\mathbb{E}\left[ \mathrm{e}^{-\lambda T_{x-y+a}^{+}};T_{x-y+a}^{+}<\tau_{a}^{k-1}\right] \mathrm{d}y\text{.} \label{parta}$$ Substituting (\[cLT\]) at $n=k-1$ into (\[parta\]) yields$$\begin{aligned} & \mathbb{E}\left[ \mathrm{e}^{-\lambda T_{x}^{+}};T_{x}^{+}<\tau_{a}^{k}\right] \\ & =\mathrm{e}^{-b_{\lambda}x}+c_{\lambda}\mathrm{e}^{-b_{\lambda}\left( x+a\right) }\sum_{j=0}^{k-2}\int_{0}^{x}\left( c_{\lambda}\mathrm{e}^{-b_{\lambda}a}\right) ^{j}\frac{\left( x-y+a\right) (x-y+\left( j+1\right) a)^{j-1}}{j!}\mathrm{d}y\\ & =\mathrm{e}^{-b_{\lambda}x}+c_{\lambda}\mathrm{e}^{-b_{\lambda}\left( x+a\right) }\left( x+\sum_{j=1}^{k-2}\left( c_{\lambda}\mathrm{e}^{-b_{\lambda}a}\right) ^{j}\int_{0}^{x}\left( \frac{\left( y+\left( j+1\right) a\right) ^{j}}{j!}-a\frac{\left( y+\left( j+1\right) a\right) ^{j-1}}{\left( j-1\right) !}\right) \mathrm{d}y\right) \\ & =\mathrm{e}^{-b_{\lambda}x}\left( 1+c_{\lambda}\mathrm{e}^{-b_{\lambda}a}x+\sum_{j=2}^{k-1}\left( c_{\lambda}\mathrm{e}^{-b_{\lambda}a}\right) ^{j}\frac{x\left( x+ja\right) ^{j-1}}{j!}\right) \\ & =\mathrm{e}^{-b_{\lambda}x}\sum_{j=0}^{k-1}\left( c_{\lambda}\mathrm{e}^{-b_{\lambda}a}\right) ^{j}\frac{x(x+ja)^{j-1}}{j!}\text{.}$$ This completes the proof. In the next theorem, we provide a distributional characterization of the $n$-th drawdown time $\tau_{a}^{n}$ with respect to both $M_{\tau_{a}^{n}}$ and $X_{\tau_{a}^{n}}$. \[jointd\]For $n\in\mathbb{N} $ and $x>0$, we have $$\begin{aligned} & \mathbb{E}\left[ \mathrm{e}^{-\lambda\tau_{a}^{n}};M_{\tau_{a}^{n}}>x,X_{\tau_{a}^{n}}\in\mathrm{d}y\right] \nonumber\\ & =\left( c_{\lambda}\right) ^{n}\mathrm{e}^{-b_{\lambda}(y+na)}\sum _{m=0}^{n-1}\frac{x(x+ma)^{m-1}(y-x+(n-m)a))^{n-1-m}\mathrm{1}_{\left\{ y-x+(n-m)a\geq0\right\} }}{m!(n-m-1)!}\mathrm{d}y\text{.} \label{JJ}$$ By conditioning on the drawdown episode during which the drifted Brownian motion process $X$ reaches level $x$ for the first time and subsequently using the strong Markov property, we have $$\begin{aligned} & \mathbb{E}\left[ \mathrm{e}^{-\lambda\tau_{a}^{n}};M_{\tau_{a}^{n}}>x,X_{\tau_{a}^{n}}\in\mathrm{d}y\right] \nonumber\\ & =\sum_{m=0}^{n-1}\mathbb{E}\left[ \mathrm{e}^{-\lambda\tau_{a}^{n}};M_{\tau_{a}^{n}}>x,X_{\tau_{a}^{n}}\in\mathrm{d}y,\tau_{a}^{m}<T_{x}^{+}<\tau_{a}^{m+1}\right] \nonumber\\ & =\sum_{m=0}^{n-1}\mathbb{E}\left[ \mathrm{e}^{-\lambda T_{x}^{+}};\tau _{a}^{m}<T_{x}^{+}<\tau_{a}^{m+1}\right] \mathbb{E}_{x}\left[ \mathrm{e}^{-\lambda\tau_{a}^{n-m}};X_{\tau_{a}^{n-m}}\in\mathrm{d}y\right] \label{J1}$$ From Lemma \[constLT\], we know that$$\begin{aligned} \mathbb{E}\left[ \mathrm{e}^{-\lambda T_{x}^{+}};\tau_{a}^{m}<T_{x}^{+}<\tau_{a}^{m+1}\right] & =\mathbb{E}\left[ \mathrm{e}^{-\lambda T_{x}^{+}};\tau_{a}^{m}<T_{x}^{+}\right] -\mathbb{E}\left[ \mathrm{e}^{-\lambda T_{x}^{+}};\tau_{a}^{m+1}<T_{x}^{+}\right] \nonumber\\ & =\left( c_{\lambda}\right) ^{m}\frac{x(x+ma)^{m-1}}{m!}\mathrm{e}^{-b_{\lambda}\left( x+ma\right) }\text{.} \label{J2}$$ By Theorem \[yyz\], we have $$\begin{aligned} & \mathbb{E}_{x}\left[ \mathrm{e}^{-\lambda\tau_{a}^{n-m}};X_{\tau_{a}^{n-m}}\in\mathrm{d}y\right] \nonumber\\ & =\frac{\left( c_{\lambda}\right) ^{n-m}(y-x+(n-m)a)^{n-m-1}\mathrm{e}^{-b_{\lambda}(y-x+(n-m)a)}1_{\left\{ y-x+(n-m)a\geq0\right\} }}{(n-m-1)!}\mathrm{d}y. \label{J3}$$ Substituting (\[J2\]) and (\[J3\]) into (\[J1\]) and simplifying, one easily obtains (\[JJ\]). Recall that $\tau_{a}^{1}=\tilde{\tau}_{a}^{1}=\tau_{a}$ and $X_{\tau_{a}}=M_{\tau_{a}}-a$ a.s.. Therefore, by letting $\lambda\rightarrow0+$ and $x=a$ in (\[J2\]), it follows that, for $m=0,1,2,\cdots$,$$\begin{aligned} \mathbb{P}\left\{ \tilde{\tau}_{a}^{2}=\tau_{a}^{2+m}\right\} & =\mathbb{P}\{\tau_{a}^{m}<T_{a}^{+}<\tau_{a}^{m+1}\}\nonumber\\ & =\frac{(m+1)^{m-1}}{m!}\left( \frac{\gamma a}{\mathrm{e}^{\gamma a}-1}\right) ^{m}\mathrm{e}^{-\frac{\left( m+1\right) \gamma a}{\mathrm{e}^{\gamma a}-1}}, \label{eq:f2}$$ which is the probability mass function of a generalized Poisson rv (see, e.g., Equation (9.1) of Consul and Famoye [@Lagrangian2006] with $\theta =\lambda=\gamma a/(\mathrm{e}^{\gamma a}-1)$). For completeness, a rv $Y$ has a generalized Poisson$\left( \theta,\lambda\right) $ distribution if its probability mass function $p_{Y}$ is given by $$p_{Y}\left( m\right) =\frac{\theta\left( \theta+\lambda m\right) ^{m-1}e^{-\theta-\lambda m}}{m!}\text{,\qquad}m=0,1,2,...\text{,}$$ when both $\theta,\lambda>0$. Note that a generalization of (\[eq:f2\]) will be proposed in Theorem \[thm tt\]. \[rk dd\]Equation (\[eq:f2\]) can be interpreted as follows: the number of drawdowns **without** recovery between two successive drawdowns with recovery follows a generalized Poisson distribution with $\theta =\lambda=\gamma a/(\mathrm{e}^{\gamma a}-1)$. The following result connecting the two drawdown time sequences is provided. It should be noted that the rv $N_{\tilde{\tau}_{a}^{k}}^{a}-k$ represents the number of drawdowns without recovery over the first $k$ drawdowns with recovery. When $k=2$, (\[allo\]) coincides with (\[eq:f2\]). \[thm tt\]For any $k\in\mathbb{N} $, $N_{\tilde{\tau}_{a}^{k}}^{a}-k$ follows a generalized Poisson distribution with parameters $\theta=(k-1)\gamma a/($$^{\gamma a}-1)$ and $\lambda=\gamma a/(\mathrm{e}^{\gamma a}-1)$, i.e., for $m=0,1,2,\ldots,$ we have$$\mathbb{P}\left\{ \tilde{\tau}_{a}^{k}=\tau_{a}^{k+m}\right\} =\mathbb{P}\left\{ N_{\tilde{\tau}_{a}^{k}}^{a}=k+m\right\} =\frac{k-1}{m+k-1}\frac{\left( \frac{\left( m+k-1\right) \gamma a}{\mathrm{e}^{\gamma a}-1}\right) ^{m}}{m!}\mathrm{e}^{-\frac{\left( m+k-1\right) \gamma a}{\mathrm{e}^{\gamma a}-1}}. \label{allo}$$ It is clear that $\left\{ \tilde{\tau}_{a}^{k}=\tau_{a}^{k+m}\right\} \ $corresponds to the event that $m$ drawdowns without recovery will occur over the first $k$ drawdowns with recovery, i.e. $$\left\{ \tilde{\tau}_{a}^{k}=\tau_{a}^{k+m}\right\} =\left\{ N_{\tilde {\tau}_{a}^{k}}^{a}=k+m\right\} \text{.}$$ Next we prove $N_{\tilde{\tau}_{a}^{k}}^{a}-k$ follows a generalized Poisson distribution. By Remark \[rk dd\] and the strong Markov property of $X$, we know that the numbers of drawdowns without recovery between any two successive drawdowns with recovery are i.i.d. and follow a generalized Poisson distribution with $\theta=\lambda=\gamma a/(\mathrm{e}^{\gamma a}-1)$. Thus, $$N_{\tilde{\tau}_{a}^{k}}^{a}-k=\sum_{i=2}^{k}\left( N_{\tilde{\tau}_{a}^{i}}^{a}-N_{\tilde{\tau}_{a}^{i-1}}^{a}-1\right) \text{,}$$ corresponds to a sum of i.i.d. rv’s with a generalized Poisson distribution $\theta=\lambda=\gamma a/(\mathrm{e}^{\gamma a}-1)$. Using Theorem 9.1 of Consul and Famoye [@Lagrangian2006], we have that $N_{\tilde{\tau}_{a}^{k}}^{a}-k$ follows a generalized Poisson distribution with parameters $\theta=(k-1)\gamma a/($$^{\gamma a}-1)$ and $\lambda=\gamma a/(\mathrm{e}^{\gamma a}-1)$. Next, we propose the following corollary which can be viewed as an extension to Taylor [@Taylor75] and Lehoczky [@Lehoczky77] from the first drawdown case to the $n$-th drawdown without recovery. \[aab\]For $n\in\mathbb{N} $ and $x>0$, we have$$\mathbb{E}\left[ \mathrm{e}^{-\lambda\tau_{a}^{n}};M_{\tau_{a}^{n}}>x\right] =\left( \frac{c_{\lambda}}{b_{\lambda}}\right) ^{n}\sum_{m=0}^{n-1}\frac{x(x+ma)^{m-1}b_{\lambda}^{m}}{m!}\mathrm{e}^{-b_{\lambda}\left( ma+x\right) }.$$ Taking the integral of (\[JJ\]) with respect to $y$ in $(-na,\infty)$, we have$$\begin{aligned} \mathbb{E}\left[ \mathrm{e}^{-\lambda\tau_{a}^{n}};M_{\tau_{a}^{n}}>x\right] & =(c_{\lambda})^{n}\sum_{m=0}^{n-1}\frac{x(x+ma)^{m-1}}{m!(n-m-1)!}\int_{x-(n-m)a}^{\infty}\mathrm{e}^{-b_{\lambda}(y+na)}(y-x+(n-m)a)^{n-m-1}\mathrm{d}y\\ & =(c_{\lambda})^{n}\sum_{m=0}^{n-1}\frac{x(x+ma)^{m-1}}{m!(n-m-1)!}\int _{0}^{\infty}\mathrm{e}^{-b_{\lambda}(z+x+ma)}z^{n-m-1}\mathrm{d}z\\ & =(c_{\lambda})^{n}\sum_{m=0}^{n-1}\frac{x(x+ma)^{m-1}}{m!(n-m-1)!}\mathrm{e}^{-b_{\lambda}(x+ma)}\int_{0}^{\infty}\mathrm{e}^{-b_{\lambda}z}z^{n-m-1}\mathrm{d}z\\ & =(c_{\lambda})^{n}\sum_{m=0}^{n-1}\frac{x(x+ma)^{m-1}}{m!b_{\lambda}^{n-m}}\mathrm{e}^{-b_{\lambda}(x+ma)}.\end{aligned}$$ which completes the proof. The marginal distribution of $M_{\tau_{a}^{n}}$ can easily be obtained from Corollary \[aab\] by letting $\lambda\rightarrow0+$ and subsequently making use of (\[bc\]). Indeed, $$\mathbb{P}\left\{ M_{\tau_{a}^{n}}>x\right\} =\sum_{m=0}^{n-1}\frac{x(x+ma)^{m-1}\left( \frac{\gamma}{\mathrm{e}^{\gamma a}-1}\right) ^{m}}{m!}\mathrm{e}^{-\frac{\gamma(ma+x)}{\mathrm{e}^{\gamma a}-1}}\text{.} \label{PP}$$ Rearrangements of (\[PP\]) yields$$\mathbb{P}\left\{ M_{\tau_{a}^{n}}>x\right\} =\sum_{k=0}^{n-1}D_{k,n}\frac{\left( \frac{\gamma x}{\mathrm{e}^{\gamma a}-1}\right) ^{k}}{k!}\mathrm{e}^{-\frac{\gamma x}{\mathrm{e}^{\gamma a}-1}}\text{,} \label{survivalM}$$ where $D_{0,n}=1$, and$$D_{k,n}=\sum_{m=k}^{n-1}\frac{k\left( \frac{m\gamma a}{\mathrm{e}^{\gamma a}-1}\right) ^{m-k}}{m\left( m-k\right) !}\mathrm{e}^{-\frac{m\gamma }{\mathrm{e}^{\gamma a}-1}a}=\sum_{m=0}^{n-1-k}\frac{k\left( \frac{\left( m+k\right) \gamma a}{\mathrm{e}^{\gamma a}-1}\right) ^{m}}{(m+k)m!}\mathrm{e}^{-\frac{\left( m+k\right) \gamma a}{\mathrm{e}^{\gamma a}-1}}\text{,} \label{D1}$$ for $k=1,2,...,n-1$. Note that by substituting $k$ by $k+1$ in (\[allo\]), it follows that (\[D1\]) can be rewritten as$$D_{k,n}=\sum_{m=0}^{n-1-k}\mathbb{P}\left\{ \tilde{\tau}_{a}^{k+1}=\tau _{a}^{k+1+m}\right\} \text{,}$$ which is equivalent to $$D_{k,n}=\mathbb{P}\left\{ \tilde{\tau}_{a}^{k+1}\leq\tau_{a}^{n}\right\} =\mathbb{P}\left\{ \tilde{N}_{\tau_{a}^{n}}^{a}>k\right\} \text{.}$$ Then, $$\mathbb{P}\left\{ M_{\tau_{a}^{n}}\in\mathrm{d}y\right\} =\sum_{k=1}^{n}d_{k,n}\frac{\left( \frac{\gamma a}{\mathrm{e}^{\gamma a}-1}\right) ^{k}y^{k-1}e^{-\frac{\gamma a}{\mathrm{e}^{\gamma a}-1}y}}{\left( k-1\right) !}\mathrm{d}y\text{,}$$ where $\left\{ d_{k,n}\right\} _{k=1}^{n}$ are given by$$\begin{aligned} d_{k,n} & \equiv D_{k-1,n}-D_{k,n}\\ & =\sum_{j=k}^{n}\frac{k-1}{j-1}\frac{\left( \frac{\left( j-1\right) \gamma a}{\mathrm{e}^{\gamma a}-1}\right) ^{j-k}}{\left( j-k\right) !}\mathrm{e}^{-\frac{\left( j-1\right) \gamma a}{\mathrm{e}^{\gamma a}-1}}\left( 1-\sum_{m=0}^{n-j-1}\frac{(m+1)^{m-1}}{m!}\left( \frac{\gamma a}{\mathrm{e}^{\gamma a}-1}\right) ^{m}\mathrm{e}^{-\frac{\left( m+1\right) \gamma a}{\mathrm{e}^{\gamma a}-1}}\right) \text{.}$$ In conclusion, $M_{\tau_{a}^{n}}$ follows a mixed-Erlang distribution which is an important class of distribution in risk management (see, e.g., Willmot and Lin [@WillLin] for an extensive review of mixed Erlang distributions). \[mE\]Note that the distribution of $M_{\tau_{a}^{n}}$ does not come as a surprise. Indeed, one can obtain the structural form of the distribution of $M_{\tau_{a}^{n}}$ by conditioning on $\tilde{N}_{\tau_{a}^{n}}^{a}$, namely the number of drawdowns with recovery over the first $n$ drawdowns (without recovery). Using the strong Markov property of the process $X$ and Equation (\[M\]), it follows that $M_{\tau_{a}^{n}}\left\vert \tilde{N}_{\tau_{a}^{n}}^{a}=m\right. $ is an Erlang rv with mean $m\frac{\mathrm{e}^{\gamma a}-1}{\gamma}$ and variance $m\left( \frac{\mathrm{e}^{\gamma a}-1}{\gamma }\right) ^{2}$ for $m=1,2,...,n$. Thus, in (\[survivalM\]), $D_{k,n}$ can be interpreted as the survival function of $\tilde{N}_{\tau_{a}^{n}}^{a}$, i.e. $$D_{k,n}=\mathbb{P}\left\{ \tilde{N}_{\tau_{a}^{n}}^{a}>k\right\} =\mathbb{P}\left\{ \tilde{\tau}_{a}^{k+1}\leq\tau_{a}^{n}\right\} .$$ The next corollary investigates the actual drawdown $M_{t}-X_{t}$ at $t=\tau_{a}^{n}$. For $a\leq x\leq na$, we have $$\begin{aligned} & \mathbb{E}\left[ \mathrm{e}^{-\lambda\tau_{a}^{n}};M_{\tau_{a}^{n}}-X_{\tau_{a}^{n}}\leq x\right] \\ & =(c_{\lambda})^{n}\mathrm{e}^{-b_{\lambda}(na-x)}\sum_{m=0}^{n-1}\left( \frac{(na-x)^{m}}{b_{\lambda}^{n-m}m!}-\frac{\mathrm{1}_{\left\{ x\leq(n-m)a\right\} }((n-m)a-x)^{n-m-1}\int_{0}^{\infty}\mathrm{e}^{-b_{\lambda}y}y(y+ma)^{m-1}\mathrm{d}y}{m!(n-m-1)!}\right) .\end{aligned}$$ We have$$\begin{aligned} & \mathbb{E}\left[ \mathrm{e}^{-\lambda\tau_{a}^{n}};M_{\tau_{a}^{n}}-X_{\tau_{a}^{n}}>x\right] \nonumber\\ & =\int_{-x}^{\infty}\mathbb{E}\left[ \mathrm{e}^{-\lambda\tau_{a}^{n}};M_{\tau_{a}^{n}}-X_{\tau_{a}^{n}}>x,X_{\tau_{a}^{n}}\in\mathrm{d}y\right] +\mathbb{E}\left[ \mathrm{e}^{-\lambda\tau_{a}^{n}};M_{\tau_{a}^{n}}-X_{\tau_{a}^{n}}>x,X_{\tau_{a}^{n}}\leq-x\right] \nonumber\\ & =\int_{-x}^{\infty}\mathbb{E}\left[ \mathrm{e}^{-\lambda\tau_{a}^{n}};M_{\tau_{a}^{n}}>x+y,X_{\tau_{a}^{n}}\in\mathrm{d}y\right] +\mathbb{E}\left[ \mathrm{e}^{-\lambda\tau_{a}^{n}};X_{\tau_{a}^{n}}\leq-x\right] \nonumber\\ & =\int_{-x}^{\infty}\mathbb{E}\left[ \mathrm{e}^{-\lambda\tau_{a}^{n}};M_{\tau_{a}^{n}}>x+y,X_{\tau_{a}^{n}}\in\mathrm{d}y\right] +\left( c_{\lambda}/b_{\lambda}\right) ^{n}\left( 1-\mathrm{e}^{-b_{\lambda}(na-x)}\sum_{m=0}^{n-1}\frac{\left( b_{\lambda}(na-x)\right) ^{m}}{m!}\right) , \label{MX}$$ where the last step is due to (\[L tau x\]). Moreover, by Theorem \[jointd\], the first term of (\[MX\])$$\begin{aligned} & \int_{-x}^{\infty}\mathbb{E}\left[ \mathrm{e}^{-\lambda\tau_{a}^{n}};M_{\tau_{a}^{n}}>x+y,X_{\tau_{a}^{n}}\in\mathrm{d}y\right] \\ & =(c_{\lambda})^{n}\sum_{m=0}^{n-1}\frac{((n-m)a-x)^{n-m-1}\mathrm{1}_{\left\{ -x+(n-m)a\geq0\right\} }}{m!(n-m-1)!}\int_{-x}^{\infty}\mathrm{e}^{-b_{\lambda}(y+na)}(x+y)(x+y+ma)^{m-1}\mathrm{d}y\\ & =(c_{\lambda})^{n}\sum_{m=0}^{n-1}\frac{((n-m)a-x)^{n-m-1}\mathrm{1}_{\left\{ x\leq(n-m)a\right\} }}{m!(n-m-1)!}\int_{0}^{\infty}\mathrm{e}^{-b_{\lambda}(z-x+na)}z(z+ma)^{m-1}\mathrm{d}z\\ & =(c_{\lambda})^{n}\mathrm{e}^{-b_{\lambda}\left( na-x\right) }\sum _{m=0}^{n-1}\frac{((n-m)a-x)^{n-m-1}\mathrm{1}_{\left\{ x\leq(n-m)a\right\} }}{m!(n-m-1)!}\int_{0}^{\infty}\mathrm{e}^{-b_{\lambda}z}z(z+ma)^{m-1}\mathrm{d}z.\end{aligned}$$ Substituting this back into (\[MX\]), we complete the proof. To complete the section, we consider a numerical example to compare the distribution of the $n$-th drawdown times $\tilde{\tau}_{a}^{n}$ and $\tau _{a}^{n}$ whose Laplace transforms are given in (\[L tau til\]) and (\[L tau\]), respectively. We implement a numerical inverse Laplace transform approach proposed by Abate and Whitt [@AbatWhit06]. For ease of notation, we denote the cumulative distribution functions of $\tau_{a}^{n}$ and $\tilde{\tau}_{a}^{n}$ by $F_{n}$ and $\tilde{F}_{n}$, respectively. **Table 4.1** Distribution of the $n$-th drawdown times when $a=0.1$ and $\sigma=0.2$$$\begin{tabular} [c]{c|c|c|c}\hline & $\mu=0.1$ & $\mu=0$ & $\mu=-0.1$\\\hline \multicolumn{1}{l|}{\begin{tabular} [c]{l}\\ $n=1$\\ $n=2$\\ $n=3$\\ $n=4$\\ $n=5$\\ $n=6$\end{tabular} } & \multicolumn{1}{|l|}{\begin{tabular} [c]{ll}$F_{n}(1)$ & $\tilde{F}_{n}(1)$\\ 0.9779 & 0.9779\\ 0.8759 & 0.4865\\ 0.6651 & 0.1024\\ 0.4060 & 0.0082\\ 0.1942 & 0.0002\\ 0.0721 & 0.0000 \end{tabular} } & \multicolumn{1}{|l|}{\begin{tabular} [c]{ll}$F_{n}(1)$ & $\tilde{F}_{n}(1)$\\ 0.9908 & 0.9908\\ 0.9366 & 0.4406\\ 0.7926 & 0.0885\\ 0.5652 & 0.0070\\ 0.3262 & 0.0002\\ 0.1492 & 0.0000 \end{tabular} } & \multicolumn{1}{|l}{\begin{tabular} [c]{ll}$F_{n}(1)$ & $\tilde{F}_{n}(1)$\\ 0.9967 & 0.9967\\ 0.9719 & 0.3636\\ 0.8874 & 0.0663\\ 0.7166 & 0.0050\\ 0.4871 & 0.0001\\ 0.2696 & 0.0000 \end{tabular} }\\\hline \end{tabular} $$ Table 4.1 presents the probabilities that at least $n$ drawdowns with or without recovery occurs before time $1$ for different values of the drift $\mu$. We observe that $F_{n}(1)>\tilde{F}_{n}(1)$ for $n\geq2$ due to the relation between $\tau_{a}^{n}$ and $\tilde{\tau}_{a}^{n}$ given in (\[allo\]). In addition, it shows that $F_{n}(1)$ increases as $\mu$ decreases. However, we observe the opposite trend for $\tilde{F}_{n}(1)$ when $n\geq2$. This is because the previous running maximum is less likely to be revisited for a smaller $\mu$. Since the drawdown risk is in principle a type of downside risk, we think smaller $\mu$ should lead to higher downside risks. In this sense, we suggest that the drawdown times without recovery are better to capture the essence of drawdown risks. **Table 4.2** Distribution of drawdown times when $a=0.1$ and $\sigma=0.12$$$\begin{tabular} [c]{c|c|c|c}\hline & $\mu=0.1$ & $\mu=0$ & $\mu=-0.1$\\\hline \multicolumn{1}{l|}{\begin{tabular} [c]{l}\\ $n=1$\\ $n=2$\\ $n=3$\\ $n=4$\\ $n=5$\\ $n=6$\end{tabular} } & \multicolumn{1}{|l|}{\begin{tabular} [c]{ll}$F_{n}(1)$ & $\tilde{F}_{n}(1)$\\ 0.5663 & 0.5663\\ 0.1592 & 0.0339\\ 0.0225 & 0.0002\\ 0.0016 & 0.0000\\ 0.0001 & 0.0000\\ 0.0000 & 0.0000 \end{tabular} } & \multicolumn{1}{|l|}{\begin{tabular} [c]{ll}$F_{n}(1)$ & $\tilde{F}_{n}(1)$\\ 0.7845 & 0.7845\\ 0.3755 & 0.0494\\ 0.0986 & 0.0002\\ 0.0137 & 0.0000\\ 0.0010 & 0.0000\\ 0.0000 & 0.0000 \end{tabular} } & \multicolumn{1}{|l}{\begin{tabular} [c]{ll}$F_{n}(1)$ & $\tilde{F}_{n}(1)$\\ 0.9257 & 0.9257\\ 0.6509 & 0.0463\\ 0.2891 & 0.0002\\ 0.0730 & 0.0000\\ 0.0099 & 0.0000\\ 0.0007 & 0.0000 \end{tabular} }\\\hline \end{tabular} $$ Table 4.2 is the equivalent of Table 4.1 with a lower volatility $\sigma =0.12$. We notice that $F_{n}(1)$ and $\tilde{F}_{n}(1)$ decrease as $\sigma$ decreases. We also have an interesting observation that the trend of $\tilde{F}_{2}(1)$ is not monotone in $\mu$. Again, this is because the occurrence of $\tilde{\tau}_{a}^{n}$ for $n\geq2$ necessitates a recovery for the previous running maximum. Smaller drift does imply higher drawdown risk, meanwhile the recovery becomes more difficult. Insurance of frequent relative drawdowns ======================================== In this section, we consider insurance policies protecting against the risk of frequent drawdowns. We denote the price of an underlying asset by $S=\{S_{t},t\geq0\}$, with dynamics $$\mathrm{d}S_{t}=rS_{t}\mathrm{d}t+\sigma S_{t}\mathrm{d}W_{t}^{\mathbb{Q} }\text{,\qquad}S_{0}=s_{0}\text{,}$$ where $r>0$ is the risk-free rate, $\sigma>0$ and $\{W_{t}^{\mathbb{Q} },t\geq0\}$ is a standard Brownian motion under a risk-neutral measure $\mathbb{Q} $. It is well known that $$S_{t}=s_{0}\mathrm{e}^{X_{t}}, \label{SX}$$ where $X_{t}=(r-\frac{1}{2}\sigma^{2})t+\sigma W_{t}^{\mathbb{Q} }$. In practice, drawdowns are often quoted in percentage. For fixed $0<\alpha<1$, we denote the time of the first relative drawdown over size $\alpha$ by $$\eta_{\alpha}(S)=\inf\left\{ t\geq0:M_{t}^{S}-S_{t}\geq\alpha M_{t}^{S}\right\} ,$$ where $M_{t}^{S}=\sup_{0\leq u\leq t}S_{u}$ represents the running maximum of $S$ by time $t$. By (\[SX\]), it is easy to see that the relative drawdown of the geometric Brownian motion $S$ corresponds to the actual drawdown of a drifted Brownian motion $X$, namely$$\eta_{\alpha}(S)=\inf\left\{ t\geq0:M_{t}^{X}-X_{t}\geq-\log(1-\alpha )\right\} =\tau_{\bar{\alpha}}(X),$$ where $\bar{\alpha}=-\log(1-\alpha)$. Similarly, we denote the relative drawdown times with and without recovery by$$\tilde{\eta}_{\alpha}^{n}(S)=\inf\{t>\tilde{\eta}_{\alpha}^{n-1}(S):M_{t}^{S}-S_{t}\geq\alpha M_{t}^{S},M_{t}^{S}>M_{\tilde{\eta}_{\alpha}^{n-1}(S)}^{S}\},$$ and $$\eta_{\alpha}^{n}(S)=\inf\{t>\eta_{\alpha}^{n-1}(S):M_{[\eta_{\alpha}^{n-1}(S),t]}^{S}-S_{t}\geq\alpha M_{[\eta_{\alpha}^{n-1}(S),t]}^{S}\}\text{,}$$ respectively. Therefore, we have $$\tilde{\eta}_{\alpha}^{n}(S)=\tilde{\tau}_{\bar{\alpha}}^{n}(X)\qquad \text{and\qquad}\eta_{\alpha}^{n}(S)=\tau_{\bar{\alpha}}^{n}(X). \label{it}$$ Next, we consider two types of insurance policies offering a protection against relative drawdowns. For the first one, we assume that the seller pays the buyer $\$k$ at time $T$ if a total of $k$ relative drawdowns over size $0<\alpha<1$ occurred prior to time $T$ (for all $k$). For the relative drawdown times with and without recovery, by (\[it\]), the risk-neutral prices are given by $$\tilde{V}_{1}(T)=\mathrm{e}^{-rT}\sum_{k=1}^{\infty}k\mathbb{Q} \left\{ \tilde{N}_{T}^{\bar{\alpha}}(X)=k\right\} =\mathrm{e}^{-rT}\mathbb{E}^{\mathbb{Q} }[\tilde{N}_{T}^{\bar{\alpha}}(X)],$$ and $$V_{1}(T)=\mathrm{e}^{-rT}\sum_{k=1}^{\infty}k\mathbb{Q} \left\{ N_{T}^{\bar{\alpha}}(X)=k\right\} =\mathrm{e}^{-rT}\mathbb{E}^{\mathbb{Q} }[N_{T}^{\bar{\alpha}}(X)],$$ respectively. For the second type of policies, the seller pays the buyer $\$1$ at the time of each relative drawdown time as long as it occurs before maturity $T$. Hence, their risk-neutral prices are $$\tilde{V}_{2}(T)=\sum_{k=1}^{\infty}\mathbb{E}^{\mathbb{Q} }[\mathrm{e}^{-r\tilde{\tau}_{\bar{\alpha}}^{k}(X)};\tilde{\tau}_{\bar{\alpha }}^{k}(X)\leq T],$$ and $$V_{2}(T)=\sum_{k=1}^{\infty}\mathbb{E}^{\mathbb{Q} }[\mathrm{e}^{-r\tau_{\bar{\alpha}}^{k}(X)};\tau_{\bar{\alpha}}^{k}(X)\leq T]\text{,}$$ respectively. \[price\] For $\lambda>0$, we have$$\begin{array} [c]{ll}\int_{0}^{\infty}\mathrm{e}^{-\lambda T}V_{1}(T)\mathrm{d}T=\frac{1}{\lambda+r}\frac{\bar{c}_{\lambda+r}/\bar{b}_{\lambda+r}}{1-\bar{c}_{\lambda+r}/\bar{b}_{\lambda+r}}, & \int_{0}^{\infty}\mathrm{e}^{-\lambda T}\tilde{V}_{1}(T)\mathrm{d}T=\frac{1}{\lambda+r}\frac{\bar{c}_{\lambda +r}/\bar{b}_{\lambda+r}}{1-\mathrm{e}^{-\bar{\beta}_{\lambda+r}^{+}a}\bar {c}_{\lambda+r}/\bar{b}_{\lambda+r}},\\ \int_{0}^{\infty}\mathrm{e}^{-\lambda T}V_{2}(T)\mathrm{d}T=\frac{1}{\lambda }\frac{\bar{c}_{\lambda+r}/\bar{b}_{\lambda+r}}{1-\bar{c}_{\lambda+r}/\bar {b}_{\lambda+r}}, & \int_{0}^{\infty}\mathrm{e}^{-\lambda T}\tilde{V}_{2}(T)\mathrm{d}T=\frac{1}{\lambda}\frac{\bar{c}_{\lambda+r}/\bar{b}_{\lambda+r}}{1-\mathrm{e}^{-\bar{\beta}_{\lambda+r}^{+}a}\bar{c}_{\lambda +r}/\bar{b}_{\lambda+r}}, \end{array}$$ where $\bar{b}_{\lambda}=\frac{\bar{\beta}_{\lambda}^{+}\mathrm{e}^{-\bar{\beta}_{\lambda}^{-}\bar{\alpha}}-\bar{\beta}_{\lambda}^{-}\mathrm{e}^{-\bar{\beta}_{\lambda}^{+}\bar{\alpha}}}{\mathrm{e}^{-\bar{\beta }_{\lambda}^{-}\bar{\alpha}}-\mathrm{e}^{-\bar{\beta}_{\lambda}^{+}\bar {\alpha}}}$, $\bar{c}_{\lambda}=\frac{\bar{\beta}_{\lambda}^{+}-\bar{\beta }_{\lambda}^{-}}{\mathrm{e}^{-\bar{\beta}_{\lambda}^{-}\bar{\alpha}}-\mathrm{e}^{-\bar{\beta}_{\lambda}^{+}\bar{\alpha}}}$ and $\bar{\beta }_{\lambda}^{\pm}=\frac{-r+\frac{1}{2}\sigma^{2}\pm\sqrt{(r-\frac{1}{2}\sigma^{2})^{2}+2\lambda\sigma^{2}}}{\sigma^{2}}$. We provide the proof for $\int_{0}^{\infty}V_{1}(T)\mathrm{e}^{-\lambda T}\mathrm{d}T$ and $\int_{0}^{\infty}V_{2}(T)\mathrm{e}^{-\lambda T}\mathrm{d}T$ only. The other two results can be derived in a similar fashion. From the definition of $N_{T}^{\bar{\alpha}}(X)$, we have the following relation $$\mathbb{E}^{\mathbb{Q} }\left[ N_{T}^{\bar{\alpha}}(X)\right] =\sum_{k=1}^{\infty}\mathbb{Q}\left\{ N_{T}^{\bar{\alpha}}(X)\geq k\right\} =\sum_{k=1}^{\infty}\mathbb{Q}\left\{ \tau_{\bar{\alpha}}^{k}(X)\leq T\right\} .$$ By (\[L tau\]), it follows that$$\begin{aligned} \int_{0}^{\infty}V_{1}(T)\mathrm{e}^{-\lambda T}\mathrm{d}T & =\int _{0}^{\infty}\mathrm{e}^{-(\lambda+r)T}\mathbb{E}^{\mathbb{Q} }[N_{T}^{\bar{\alpha}}(X)]\mathrm{d}T\\ & =\sum_{k=1}^{\infty}\int_{0}^{\infty}\mathrm{e}^{-(\lambda+r)T}\mathbb{Q}\left\{ \tau_{\bar{\alpha}}^{k}(X)\leq T\right\} \mathrm{d}T\\ & =\frac{1}{\lambda+r}\sum_{k=1}^{\infty}\mathbb{E}^{\mathbb{Q} }[\mathrm{e}^{-(\lambda+r)\tau_{\bar{\alpha}}^{k}(X)}]\\ & =\frac{1}{\lambda+r}\sum_{k=1}^{\infty}\left( \frac{\bar{c}_{\lambda+r}}{\bar{b}_{\lambda+r}}\right) ^{n}\\ & =\frac{1}{\lambda+r}\frac{\bar{c}_{\lambda+r}/\bar{b}_{\lambda+r}}{1-\bar{c}_{\lambda+r}/\bar{b}_{\lambda+r}}.\end{aligned}$$ For $\int_{0}^{\infty}V_{2}(T)\mathrm{e}^{-\lambda T}\mathrm{d}T$, by Fubini’s theorem and (\[L tau\]), we have$$\begin{aligned} \int_{0}^{\infty}V_{2}(T)\mathrm{e}^{-\lambda T}\mathrm{d}T & =\sum _{k=1}^{\infty}\int_{0}^{\infty}\mathbb{E}^{\mathbb{Q} }[\mathrm{e}^{-r\tau_{\bar{\alpha}}^{k}(X)};\tau_{\bar{\alpha}}^{k}(X)\leq T]\mathrm{e}^{-\lambda T}\mathrm{d}T\\ & =\sum_{k=1}^{\infty}\int_{0}^{\infty}\int_{0}^{T}\mathrm{e}^{-rt}\mathbb{Q} \left\{ \tau_{\bar{\alpha}}^{k}(X)\in\mathrm{d}t\right\} \mathrm{e}^{-\lambda T}\mathrm{d}T\\ & =\sum_{k=1}^{\infty}\frac{1}{\lambda}\int_{0}^{\infty}\mathrm{e}^{-(\lambda+r)t}\mathbb{Q} \left\{ \tau_{\bar{\alpha}}^{n}(X)\in\mathrm{d}t\right\} \\ & =\sum_{k=1}^{\infty}\frac{1}{\lambda}\left( \frac{\bar{c}_{\lambda+r}}{\bar{b}_{\lambda+r}}\right) ^{n}\\ & =\frac{1}{\lambda}\frac{\bar{c}_{\lambda+r}/\bar{b}_{\lambda+r}}{1-\bar {c}_{\lambda+r}/\bar{b}_{\lambda+r}}.\end{aligned}$$ This completes the proof. It is worth pointing out that, through expansion of the randomized prices in Corollary \[price\] in terms of exponentials, it is possible to obtain semi-static hedging portfolios as in [@CarrZhanHaji]. Moreover, capped insurance contracts against frequency of drawdowns can also be formulated and priced using Theorems \[thm M til L\], \[yyz\], and Corollary \[aab\]. To conclude, we consider a pricing example for the four types of insurance contracts proposed earlier. The same numerical Laplace transform approach as in the last section is applied. **Table 5.1** Insurance contracts prices when $\alpha=15\%$ and $r=5\%$$$\begin{tabular} [c]{ll|l|l|l|l}\hline & & $V_{1}(T)$ & $\tilde{V}_{1}(T)$ & $V_{2}(T)$ & $\tilde{V}_{2}(T)$\\\hline $T=1$ & $\sigma=0.1$ & 0.1102 & 0.1091 & 0.1120 & 0.1108\\ $T=2$ & $\sigma=0.1$ & 0.3011 & 0.2769 & 0.3131 & 0.2885\\ $T=3$ & $\sigma=0.1$ & 0.4743 & 0.4031 & 0.5058 & 0.4318\\\hline $T=1$ & $\sigma=0.2$ & 1.1777 & 0.7873 & 1.2043 & 0.8081\\ $T=2$ & $\sigma=0.2$ & 2.3815 & 1.1842 & 2.4977 & 1.2550\\ $T=3$ & $\sigma=0.2$ & 3.4651 & 1.4519 & 3.7279 & 1.5890\\\hline \end{tabular} $$ As expected, Table 5.1 shows that type 2 contracts have higher prices than type 1 contracts because of earlier payments (at the moment of each drawdown time instead of the maturity $T$). It also shows that $\tilde{V}_{1}(T)$ and $\tilde{V}_{2}(T)$ are respectively lower than $V_{1}(T)$ and $V_{2}(T)$ due to $\tau_{a}^{n}\leq\tilde{\tau}_{a}^{n}$. All the prices increase as $T$ increases or $\sigma$ increases. Moreover, we can expect that the prices will decrease as $\alpha$ or $r$ increases. The latter is due to a higher discount rate which is the risk-free rate under the risk-neutral measure $\mathbb{Q} $. 0.5cm **Acknowledgments.** The authors would like to thank Professor Gord Willmot and an anonymous referee for their helpful remarks and suggestions. Support for David Landriault from a grant from the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged, as is support for Bin Li from a start-up grant from the University of Waterloo. [99]{} (2006). A unified framework for numerically inverting [L]{}aplace transforms. 408–421. (1957). On the collective theory of risk in case of contagion between claims. 104–125. (2002). 2nd ed. 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[^1]: Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, ON, N2L 3G1, Canada (dlandria@uwaterloo.ca) [^2]: Corresponding Author: Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, ON, N2L 3G1, Canada (bin.li@uwaterloo.ca) [^3]: Department of Statistics, Columbia University, New York, NY, 10027, USA (hzhang@stat.columbia.edu)

--- author: - Dhrubojyoti Roy - Mukundan Sridharan - Satyajeet Deshpande - Anish Arora bibliography: - 'dtnbib.bib' title: 'Achieving Throughput via Fine-Grained Path Planning in Small World DTNs' ---

KANAZAWA-17-06\ September, 2017 [**Dark matter stability and one-loop neutrino mass generation based on Peccei-Quinn symmetry**]{} [Daijiro Suematsu]{}\ [*Institute for Theoretical Physics, Kanazawa University, Kanazawa 920-1192, Japan*]{} [**Abstract**]{}\ We propose a model which is a simple extension of the KSVZ invisible axion model with an inert doublet scalar. Peccei-Quinn symmetry forbids tree-level neutrino mass generation and its remnant $Z_2$ symmetry guarantees dark matter stability. The neutrino masses are generated by one-loop effects as a result of the breaking of Peccei-Quinn symmetry through a nonrenormalizable interaction. Although the low energy effective model coincides with an original scotogenic model which contains right-handed neutrinos with large masses, it is free from the strong $CP$ problem. Introduction ============ The standard model (SM) has been confirmed by the discovery of the Higgs scalar [@higgs]. However, it is now considered to be extended to explain several experimental and observational data such as neutrino masses and mixings [@nexp; @t13], and dark matter (DM) [@dm]. Strong $CP$ problem is also one of such problems suggested by an experimental bound of the electric dipole moment of a neutron [@strongcp]. Invisible axion models are known to give a simple and interesting solution to it [@ksvz; @dfsz]. The KSVZ model, which is one of such realizations, is an extension of the SM by a complex singlet scalar and a pair of colored fermions. It has a global $U(1)$ symmetry, which is violated only by the QCD anomaly and plays a role of Peccei-Quinn (PQ) symmetry [@pq]. If the spontaneous breaking of this $U(1)_{PQ}$ symmetry occurs, a pseudo Nambu-Goldstone boson associated to this breaking called axion appears to solve the strong $CP$ problem [@axion]. If the axion decay constant $f_a$ is large enough such as $10^9~{\rm GeV}<f_a<10^{12}~{\rm GeV}$ due to a vacuum expectation value (VEV) of the singlet scalar, the axion mass is very small and its coupling is extremely weak so as not to cause any contradiction with experiments and astrophysical observations [@fa]. On the other hand, the $U(1)_{PQ}$ breaking is known to cause $N$ degenerate minima for the axion potential due to the QCD anomaly depending on both the field contents and the PQ charge assignment for them. As a result, the model is generally annoyed by the dangerous production of topologically stable domain walls [@dw]. It can be escapable only for $N=1$ unless one consider the domain wall free universe brought about by inflation. If a certain subgroup of $U(1)_{\rm PQ}$ remains as a discrete symmetry broken only by the QCD anomaly in a model with $N=1$, it could present an interesting scenario in relation to the DM physics at the low energy regions.[^1] In this paper, we consider such a possibility in an extension of the KSVZ model, in which an inert doublet scalar and three right-handed neutrinos are added. The low energy effective model obtained from it after the breakdown of the $U(1)_{PQ}$ symmetry is reduced to the original scotogenic neutrino mass model with an effective $Z_2$ symmetry [@ma]. This $Z_2$ symmetry could guarantee the stability of a lightest neutral component of the inert doublet scalar to give a DM candidate. The neutrino masses are generated through a one-loop effect as a result of the $U(1)_{PQ}$ breaking. The relevant diagram is caused by both right-handed neutrinos and a nonrenormalizable interaction between the inert doublet scalar and the ordinary Higgs doublet. The model might be recognized as a well motivated simple framework at high energy regions for the original scotogenic model. The remaining parts are organized as follows. In the next section, we introduce a model by fixing charge assignment of $U(1)_{PQ}$ to the field contents. We discuss basic features of the model such as remnant effective symmetry, scalar mass spectrum, vacuum stability and so on. In section 3, phenomenological features such as neutrino mass generation, leptogenesis and DM abundance in this model are discussed. The consistency of the scenario is also studied from a viewpoint of the vacuum stability and a cut-off scale of the model. We summarize the paper in section 4. An extension of the KSVZ model ============================== The KSVZ model is constructed by introducing a singlet complex scalar $S$ and a vector-like colored fermions $(D_L, D_R)$ to the SM [@ksvz]. We assume $D_{L,R}$ as triplets of the color $SU(3)$. Although they are $SU(2)_L$ singlets, they could have a suitable weak hypercharge $Y$, in general. This point is crucial for phenomenological consistency of the model as discussed below. The model has a global $U(1)_{PQ}$ symmetry and its charge is assigned to $S$ and $D_{L,R}$, but it is not assigned to the SM contents. We assume the existence of a gauge invariant Yukawa coupling $y_DS\bar D_LD_R$ so that the PQ mechanism could work to solve the strong $CP$ problem. This requires that the PQ charge $X$ of these new ingredients should satisfy $X_S=X_{D_L}-X_{D_R}$. On the other hand, this symmetry should be chiral to have the QCD anomaly and $X_{D_L}\not=X_{D_R}$ is satisfied. Thus, this $U(1)_{PQ}$ is spontaneously broken through the VEV of $S$. The $U(1)_{PQ}$ transformation $D_{L,R}\rightarrow e^{iX_{D_{L,R}}\alpha}D_{L,R}$ for the colored fermions $D_{L,R}$ shifts the QCD $\theta$ parameter through the anomaly as [@strongcp; @dw] $$\theta_{\rm QCD}\rightarrow \theta_{\rm QCD}-\frac{1}{2}(X_{D_R}-X_{D_L})\alpha.$$ Since $\theta_{\rm QCD}$ has a period $2\pi$, the model is invariant for $\alpha=\frac{2\pi k}{N}$ where $N\equiv\frac{1}{2}|X_{D_R}-X_{D_L}|$ is an integer and $k=0,1,\cdots,N-1$. This means that the model could have a discrete symmetry $Z_N$ after taking account of the QCD anomaly.[^2] If we assign the $U(1)_{PQ}$ charge $S$ as $X_S=2$, the model has $N=1$ and no degenerate minima in the axion potential. Thus, the model has no domain wall problem as is well known.[^3] Here, we note that an effective $Z_2$ symmetry could remain after the symmetry breaking due to $\langle S\rangle\not=0$ although it is violated by the QCD anomaly. Since the SM contents are supposed to have no PQ charge, it could play an important role in the leptonic sector of the model to guarantee the stability of the lightest $Z_2$ odd field in that sector, which could be DM. If both $D_L$ and $D_R$ cannot couple with quarks, which occurs in case $Y(D_{L,R})=0$ for example, they are stable and then its relic abundance has to be smaller than the DM abundance [@lmn]. Even if its relic abundance satisfies such a condition, the existence of the fractionally charged $D$ hadrons is generally forbidden by the present bound obtained from the search of fractionally charged states. On the other hand, if we assign $Y=-\frac{1}{3}$ or $\frac{2}{3}$ to $D_{L,R}$, all the $D$ hadrons can have integer charge. In that case, the $D$ relic abundance will restrict the $D$ mass into a narrow range such as $m_D~{^>_\sim}~1$ TeV [@lmn]. Moreover, they are allowed to couple with quarks through a renormalizable Yukawa interaction as long as their PQ charge is zero. For example, using the left handed quark doublet $q_L$ and the Higgs doublet $\phi$ or $\tilde\phi(\equiv i\tau_2\phi^\ast)$, the coupling $\tilde\phi\bar q_LD_R$ is allowed for $D_R$ with $X=0$ and $Y=-\frac{1}{3}$ and also $\phi\bar q_L D_R$ for $D_R$ with $X=0$ and $Y=\frac{2}{3}$. In these cases, $D_R$ decays to the SM fields through these couplings. $D_L$ can also decay via the mass mixing with $D_R$ induced by the coupling $y_DS\bar D_LD_R$ through $\langle S\rangle\not=0$. As a result, the mass $m_D$ has no constraint other than the bound obtained through the accelerator experiments. Anyway, in the model where the PQ charge is assigned as discussed above, the strong $CP$ problem could be solved without inducing any cosmological and astrophysical difficulty, as long as the symmetry breaking scale satisfies $10^9~{\rm GeV}<\langle S\rangle<10^{12}~{\rm GeV}$. Now, we consider a modification of this model by introducing an inert doublet scalar $\eta$ and three right-handed neutrinos $N_i$. The PQ charge assignment of the fields contained in the model is shown in Table 1. Invariant terms under the assumed symmetry for the Yukawa couplings and the scalar potential of the relevant fields are summarized as $$\begin{aligned} -{\cal L}_y&=&y_DS\bar D_LD_R + h_D\bar q_L\tilde\phi D_R + y_iS\bar N^c_iN_i +h_{\alpha i}\bar \ell_\alpha\eta N_i +{\rm h.c.}, \nonumber \\ V&=&m_S^2S^\dagger S+\kappa_1(S^\dagger S)^2+\kappa_2(S^\dagger S)(\phi^\dagger\phi) +\kappa_3(S^\dagger S)(\eta^\dagger\eta) \nonumber \\ &+&m_\eta^2\eta^\dagger\eta +m_\phi^2\phi^\dagger\phi +\lambda_1(\phi^\dagger\phi)^2 +\lambda_2(\eta^\dagger\eta)^2 +\lambda_3(\phi^\dagger\phi)(\eta^\dagger\eta) +\lambda_4(\phi^\dagger\eta)(\eta^\dagger\phi) \nonumber \\ &+&\frac{\lambda_5}{2}\left[\frac{S}{M_\ast}(\eta^\dagger\phi)^2 +{\rm h.c.}\right], \label{smodel}\end{aligned}$$ where $\lambda_5$ is taken to be real and $M_\ast$ is a cut-off scale of the model. The quark generation index is abbreviated in the Yukawa coupling $h_D$. We find that $V$ given in eq. (\[smodel\]) is the most general scalar potential up to the dimension 5. $D_L$ $D_R$ $S$ $\eta$ $N_i$ ------- ---------------- ---------------- ----- ---------------- ------- -- -- -- -- -- $Y$ $-\frac{1}{3}$ $-\frac{1}{3}$ 0 $-\frac{1}{2}$ 0 $X$ 2 0 2 1 $-1$ $Z_2$ $+$ $+$ $+$ $-$ $-$ [   The hypercharge $Y$ and the $U(1)_{\rm PQ}$ charge $X$ of new fields in the model. The SM contents are assumed to have no PQ charge. Parity for the effective symmetry $Z_2$ which remains after the $U(1)_{PQ}$ breaking is also listed.]{} After the symmetry breaking due to $\langle S\rangle\not=0$, $D_{L,R}$, $N_i$ and $S$ are found to get masses such as $m_D=y_D\langle S\rangle$, $M_i=y_i\langle S\rangle$ and $M_S^2=4\kappa_1\langle S\rangle^2$, respectively. Since $D_{L,R}$ can decay to the SM fields through the second term in ${\cal L}_y$ as discussed above, there is no thermal relic of $D_{L,R}$ in the present Universe. The effective model at the scale below $M_S$ could be obtained by integrating out $S$ [@stab]. This can be done by using the equation of motion for $S$. As its result, we obtain the corresponding effective model whose scalar potential of the light scalars can be written as $$\begin{aligned} V_{\rm eff}&=&\tilde m_\phi^2(\phi^\dagger\phi)+\tilde m_\eta^2(\eta^\dagger\eta) +\tilde\lambda_1(\phi^\dagger\phi)^2 +\tilde\lambda_2(\eta^\dagger\eta)^2 +\tilde\lambda_3(\phi^\dagger\phi)(\eta^\dagger\eta) +\lambda_4(\phi^\dagger\eta)(\eta^\dagger\phi) \nonumber \\ &+&\frac{\tilde\lambda_5}{2}\left[(\phi^\dagger\eta)^2 +{\rm h.c.}\right], \label{effpot}\end{aligned}$$ where we use the shifted parameters which are defined as $$\begin{aligned} &&\tilde\lambda_1=\lambda_1-\frac{\kappa_2^2}{4\kappa_1}, \qquad \tilde\lambda_2=\lambda_2-\frac{\kappa_3^2}{4\kappa_1}, \qquad \tilde\lambda_3=\lambda_3-\frac{\kappa_2\kappa_3}{2\kappa_1}, \nonumber\\ &&\tilde\lambda_5=\lambda_5\frac{\langle S\rangle}{M_\ast}, \qquad \tilde m_\phi^2=m_\phi^2+\kappa_2\langle S\rangle^2, \qquad \tilde m_\eta^2=m_\eta^2+\kappa_3\langle S\rangle^2. \label{gcoupl}\end{aligned}$$ We note that the model contains the neutrino Yukawa couplings between heavy right-handed neutrinos and the inert doublet scalar as shown in the above ${\cal L}_y$. Vacuum stability condition for the scalar potential $V_{\rm eff}$ in eq. (\[effpot\]) is known to be given as [@pstab] $$\tilde\lambda_1>0, \qquad \tilde\lambda_2>0, \qquad \tilde\lambda_3>-2\sqrt{\tilde\lambda_1\tilde\lambda_2}, \qquad \tilde\lambda_3+\lambda_4-|\tilde\lambda_5|> -2\sqrt{\tilde\lambda_1\tilde\lambda_2}, \label{instab}$$ and these should be satisfied at the energy region $\mu<M_S$. On the other hand, at $M_S<\mu<M_\ast$, both the same conditions for $\lambda_{1,2,3}$ as eq. (\[instab\]) except for the last one and new conditions $$\kappa_1>0, \qquad \kappa_2>-2\sqrt{\lambda_1\kappa_1}, \qquad \kappa_3>-2\sqrt{\lambda_2\kappa_1}, \label{stability2}$$ should be satisfied. The couplings in both regions should be connected through eq. (\[gcoupl\]). We can examine whether these conditions could be satisfied or not by using one-loop renormalization group equations (RGEs). This is the subject studied later. This effective model obtained after the spontaneous breaking of $U(1)_{PQ}$ is just the original scotogenic model [@ma].[^4] This model connects the neutrino mass generation with the DM existence. It has been extensively studied from various phenomenological view points [@radnm; @tribi; @stabf1; @ks; @infl]. In the present case, the right-handed neutrinos do not have their masses in a TeV region but they are considered to be much heavier. The coupling $\tilde\lambda_5$ which is crucial for the one-loop neutrino mass generation is derived from a nonrenormalizable term as a result of the PQ symmetry breaking. The model contains the inert doublet scalar $\eta$ which has odd parity of the remnant effective $Z_2$. It has charged components $\eta^\pm$ and two neutral components $\eta_{R,I}$. Their mass eigenvalues can be expressed as $$M_{\eta^\pm}^2=\tilde m_\eta^2 +\tilde\lambda_3\langle\phi\rangle^2, \qquad M_{\eta_{R,I}}^2=\tilde m_\eta^2+\left(\tilde\lambda_3+\lambda_4 \pm\tilde\lambda_5\right)\langle\phi\rangle^2. \label{mscalar}$$ We suppose $\tilde m_\eta=O(1)$ TeV although it requires fine tuning because of $|\langle S\rangle|\gg |\langle\phi\rangle|$. As a result of the effective $Z_2$ symmetry, the lightest one among the components of $\eta$ is stable to be a DM candidate if it is neutral. If it is supposed to be $\eta_R$, we find that this requires $\lambda_4<0$ and $\tilde\lambda_5<0$ as long as $|\tilde\lambda_5|\ll |\lambda_4|$ is satisfied. On the other hand, since $\tilde m_\eta^2\gg \langle\phi\rangle^2$ is satisfied in eq. (\[mscalar\]), the mass eigenvalues of the components $\eta$ are found to be degenerate enough so that the coannihilation processes among them are expected to be effective. This observation suggests that the abundance of $\eta_R$ could be suitably suppressed and then it could be a good DM candidate as the ordinary inert doublet model [@inert1; @inert2]. The charged states with the mass of $O(1)$ TeV are also expected to be detected in the accelerator experiments. Phenomenological features ========================= Neutrino mass, leptogenesis and DM relic abundance -------------------------------------------------- In this model, neutrino masses are forbidden at tree-level. However, since both the right-handed neutrino masses and the mass difference between $\eta_R$ and $\eta_I$ are induced after the $U(1)_{PQ}$ breaking, the small neutrino masses can be generated radiatively through one-loop diagrams in the same way as the original scotogenic model. Since $M_{\eta_{R,I}}^2\gg |M_{\eta_R}^2-M_{\eta_I}^2|$ is satisfied, the neutrino mass formula can be approximately written as $${\cal M}_{\alpha\beta}=\sum_i h_{\alpha i}h_{\beta i}\Lambda_i, \qquad \Lambda_i\simeq \frac{\tilde\lambda_5\langle\phi\rangle^2}{8\pi^2M_i} \ln\frac{M_i^2}{\bar M_\eta^2}, \label{lnmass}$$ where $\bar M_\eta^2= \tilde m_\eta^2 +\left(\tilde\lambda_3+\lambda_4\right)\langle\phi\rangle^2$. In order to take account of the constraints from the neutrino oscillation data in the analysis, we may fix the flavor structure of neutrino Yukawa couplings $h_{\alpha i}$ at the one which induces the tri-bimaximal mixing [@tribi][^5] $$h_{ej}=0, \quad h_{\mu j}=h_{\tau j}\equiv h_j \quad (j=1,2); \qquad h_{e3}=h_{\mu 3}=-h_{\tau 3}\equiv h_3, \label{flavor}$$ where the charged lepton mass matrix is assumed to be diagonal. In that case, the mass eigenvalues are estimated as $$\begin{aligned} &&m_1=0, \qquad m_2= 3|h_3|^2\Lambda_3, \nonumber \\ &&m_3=2\left[|h_1|^4\Lambda_1^2+|h_2|^4\Lambda_2^2+ 2|h_1|^2|h_2|^2\Lambda_1\Lambda_2\cos 2(\theta_1-\theta_2) \right]^{1/2}, \label{nmass}\end{aligned}$$ where $\theta_j={\rm arg}(h_j)$. As is known generally and found also from this mass formula, neutrino masses could be determined only by two right-handed neutrinos. It means that the mass and neutrino Yukawa couplings of a remaining right-handed neutrino could be free from the neutrino oscillation data as long as its contribution to the neutrino mass is negligible. In eq. (\[nmass\]), such a situation can be realized for $|h_1|^2\Lambda_1 \ll |h_2|^2\Lambda_2$. This is good for the thermal leptogenesis [@leptg] since a sufficiently small neutrino Yukawa coupling $h_1$ makes the out-of-equilibrium decay of the right-handed neutrino $N_1$ possible.[^6] We find that the squared mass differences required by the neutrino oscillation data could be explained if we fix the parameters relevant to the neutrino masses, for example, as $$\begin{aligned} &&M_1= 10^8~{\rm GeV}, \qquad M_2=4\times 10^8~{\rm GeV}, \qquad M_3= 10^9{\rm GeV}, \nonumber \\ &&|h_1|= 10^{-4.5}, \qquad |h_2|\simeq 7.2\times 10^{-4}\tilde\lambda_5^{-0.5}, \qquad |h_3|\simeq 3.1\times 10^{-4}\tilde\lambda_5^{-0.5}, \label{yukawa}\end{aligned}$$ for $\tilde m_\eta=1$ TeV. Using these values, we can estimate the expected baryon number asymmetry through the out-of-equilibrium decay of the thermal $N_1$ by solving the Boltzmann equation as done in [@ks]. The numerical analysis shows that the required baryon number asymmetry could be generated for $M_1~{^>_\sim}~10^8$ GeV, which is somewhat smaller than the Davidson-Ibarra bound [@di] in the ordinary thermal leptogenesis. In case of the parameter set given in (\[yukawa\]), we find $Y_B\left(\equiv\frac{n_B}{s}\right) =4.0\times 10^{-10}$ if we assume $\tilde\lambda_5=2.5\times 10^{-3}$ and a maximal $CP$ phase in the $CP$ violation parameter $\varepsilon$. In Fig. 1, we plot $Y_B$ as a function of $\tilde\lambda_5$. Its feature can be easily understood by taking account of eq. (\[yukawa\]). If $\tilde\lambda_5$ takes larger values, the neutrino Yukawa couplings become smaller to make the $CP$ violation $\varepsilon$ in the $N_1$ decay smaller but also the washout of the generated lepton number asymmetry smaller. On the other hand, if $\tilde\lambda_5$ takes smaller values, the neutrino Yukawa couplings become larger to induce the reverse effects. This makes the required baryon number asymmetry be generated only for the $\tilde\lambda_5$ in the limited regions as found in this figure. epsf =7.5cm [  Baryon number asymmetry $Y_B$ generated through the out-of-equilibrium decay of $N_1$. $Y_B$ is plotted as a function of $\tilde\lambda_5$ for the parameter set shown in eq. (\[yukawa\]), which can explain the neutrino mass differences required by the neutrino oscillation data. Horizontal dotted lines show the required value for $Y_B$. ]{} The relic abundance of $\eta_R$ is tuned to the observed value if the couplings $\tilde\lambda_3$ and $\lambda_4$ take suitable values. In fact, since $\tilde m_\eta$ is assumed to be of $O(1)$ TeV in this scenario, the mass of each component of $\eta$ could be degenerate enough for wide range values of $\tilde\lambda_3$ and $\lambda_4$ as remarked at eq. (\[mscalar\]). This makes the coannihilation among them effective enough to reduce the $\eta_R$ abundance [@ks]. We search the region of $\tilde\lambda_3$ and $\lambda_4$, which realizes the required DM abundance as the $\eta_R$ relic abundance by taking the values of $\tilde m_\eta$ and $\tilde\lambda_5$ as the ones given below eq. (\[yukawa\]). They are suitable for the explanation of the neutrino oscillation data and the cosmological baryon number asymmetry. In the estimation of the DM relic abundance, we follow the procedure given in [@gs] where the coannihilation effects are taken into account. We present a brief review of the procedure adopted here. The $\eta_R$ relic abundance is estimated as $$\Omega h^2\simeq \frac{1.07\times 10^9~{\rm GeV}^{-1}}{J(x_F)g_\ast^{1/2}m_{\rm pl}},$$ where $g_\ast$ is the relativistic degrees of freedom. The freeze-out temperature $T_F(\equiv\frac{M_{\eta_R}}{x_F})$ of $\eta_R$ and $J(x_F)$ are defined as $$x_F=\ln\frac{0.038m_{\rm pl}g_{\rm eff}M_{\eta_R}\langle\sigma_{\rm eff}v\rangle} {(g_\ast x_F)^{1/2}}, \qquad J(x_F)=\int^\infty_{x_F}\frac{\langle\sigma_{\rm eff}v\rangle}{x^2}dx.$$ In these formulas, the effective annihilation cross section $\langle\sigma_{\rm eff}v\rangle$ and the effective degrees of freedom $g_{\rm eff}$ are expressed as[^7] $$\langle\sigma_{\rm eff}v\rangle=\frac{1}{g_{\rm eff}^2}\sum^4_{i,j=1} \langle\sigma_{ij}v\rangle\frac{n^{\rm eq}_i}{n^{\rm eq}_1} \frac{n^{\rm eq}_j}{n^{\rm eq}_1}, \qquad g_{\rm eff}=\sum_{i=1}^4\frac{n^{\rm eq}_i}{n^{\rm eq}_1},$$ where $\langle\sigma_{ij}v\rangle$ is the thermally averaged (co)annihilation cross section and $n^{\rm eq}_i$ is the thermal equilibrium number density of $\eta_i$. If the former is expanded by the thermally averaged relative velocity $\langle v^2\rangle$ as $\langle\sigma_{ij}v\rangle = a_{ij}+b_{ij}\langle v^2\rangle$, it could be approximated only by $a_{ij}$ since $\langle v^2\rangle\ll1$ is satisfied for the cold DM. Final states of the relevant (co)annihilation are composed only of the SM contents. The corresponding $a_{\rm eff}$ can be approximately calculated as [@ks; @inert2] $$\begin{aligned} &&a_{\rm eff}=\frac{(1+2c_w^4)g^4}{128\pi c_w^4M_{\eta_1}^2}\left(N_{11}+N_{22}+ 2N_{34}\right) +\frac{s_w^2g^4}{32\pi c_w^2M_{\eta_1}^2} \left(N_{13}+N_{14}+N_{23}+N_{24}\right) \nonumber \\ &&+\frac{1}{64\pi M_{\eta_1}^2}\left[ \left(\tilde\lambda_+^2+\tilde\lambda_-^2 +2\tilde\lambda_3^2\right)(N_{11}+N_{22}) +(\tilde\lambda_+-\tilde\lambda_-)^2(N_{33}+N_{44}+ N_{12})\right. \nonumber \\ &&+\left\{(\tilde\lambda_+-\tilde\lambda_3)^2 +(\tilde\lambda_--\tilde\lambda_3)^2\right\}(N_{13}+N_{14}+N_{23}+N_{24}) \nonumber \\ &&+\left.\left\{(\tilde\lambda_+ +\tilde\lambda_-)^2 +4\tilde\lambda_3^2\right\}N_{34}\right], \label{cross}\end{aligned}$$ where $\tilde\lambda_\pm=\tilde\lambda_3+\lambda_4\pm\tilde\lambda_5$ and $N_{ij}$ is defined by using $M_{\eta_i}$ given in eq. (\[mscalar\]) as $$N_{ij}\equiv\frac{1}{g_{\rm eff}^2} \frac{n_i^{\rm eq}}{n_1^{\rm eq}}\frac{n_j^{\rm eq}} {n_1^{\rm eq}} =\frac{1}{g_{\rm eff}^2} \left(\frac{M_{\eta_i}M_{\eta_j}}{M_{\eta_1}^2}\right)^{3/2} \exp\left[-\frac{M_{\eta_i}+M_{\eta_j}-2M_{\eta_1}}{T}\right]. \label{eqfactor}$$ =7.5cm [   Points plotted by a red solid line in the $(\tilde\lambda_3,~\lambda_4)$ plane can realize the required DM relic abundance $\Omega h^2=0.12$ as the relic $\eta_R$ abundance. The last condition in eq.(\[instab\]) is satisfied at a region above the straight line which represents $\tilde\lambda_3+\lambda_4=|\tilde\lambda_5|- 2\sqrt{\tilde\lambda_1\tilde\lambda_2}$ for a fixed $\tilde\lambda_2$.]{} We use this procedure to find the points in the $(\tilde\lambda_3,~\lambda_4)$ plane, where the required DM abundance $\Omega_{\rm DM}h^2=0.12$ is realized by $\eta_R$. In Fig. 2, we plot such points by a red solid line for $\tilde m_\eta=1$ TeV and $\tilde\lambda_5=2.5\times 10^{-3}$ which are used in the previous part. In this figure, we take account of the condition $\lambda_4<0$ which has been already discussed in relation to eq. (\[mscalar\]). Moreover, if we use the Higgs mass formula $m_{h^0}^2=4\tilde\lambda_1\langle\phi\rangle^2$, we find $\tilde\lambda_1\simeq 0.13$ for $m_{h^0}=125$ GeV and then the last condition in eq. (\[instab\]) can be also plotted for a fixed $\tilde\lambda_2$ in the same plane.[^8] An allowed points are contained in the region above a straight line which is fixed by an assumed value of $\tilde\lambda_2$. We give two examples here. Although the DM abundance can be satisfied for the negative value of $\tilde\lambda_3$, we find that such cases contradict with the vacuum stability condition for $\tilde\lambda_3$ given in eq. (\[instab\]). The figure shows that $\tilde\lambda_3$ and/or $|\lambda_4|$ are required to take rather large values for realization of the DM abundance. This suggests that the RG evolution of the scalar quartic couplings $\tilde\lambda_i$ could be largely affected if they are used as initial values at the weak scale. In that case, vacuum stability and perturbativity of the model could give constraints on the model. In the next part, we focus our study on this point. Before proceeding to this subject, we comment on the contribution of the axion to the DM abundance and also a possible violation of $U(1)_{PQ}$ by the quantum gravity effect. In this model, the axion could also contribute to the DM abundance through the misalignment mechanism. If the initial misalignment of the axion is written as $\langle\theta_i\rangle$, the axion contribution to the present energy density is estimated as [@strongcp] $$\Omega_ah^2=2\times 10^4\left(\frac{\langle S\rangle} {10^{16}~{\rm GeV}}\right)^{7/6}\langle\theta_i^2\rangle.$$ The axion contribution to the DM abundance crucially depends on the scale of $\langle S\rangle$ and $\langle\theta_i\rangle$. This estimation shows that it could be too small to give the required value $\Omega_{\rm DM} h^2=0.12$ for $\langle S\rangle< 10^{11}$ GeV even if we assume $\langle\theta_i\rangle=O(1)$.[^9] Thus, the axion contribution to the DM abundance is sub-dominant or negligible for $\langle S\rangle< 10^{11}$ GeV. In this region of $\langle S\rangle$, the result obtained for $(\tilde\lambda_3,\lambda_4)$ through the above study can be still applicable even if the axion contribution to the DM abundance is taken into account. Although we assume that $U(1)_{PQ}$ is exact in this study, continuous global symmetry is suggested to be violated by the quantum gravity. This possible effect on the PQ mechanism has been studied [@gravity]. If the $U(1)_{\rm PQ}$ symmetry is violated by the gravity induced effective interaction which is suppressed by the Planck scale such as $$\frac{|S|^{n+3}}{M_{\rm pl}^n}\left(gS+g^\ast S\right),$$ it has been shown that $n\ge 6$ should be satisfied for the PQ mechanism to give a solution to the strong $CP$ problem in case that $|g|$ is of $O(1)$. If accidental appearance of global $U(1)$ happens due to some discrete or continuous gauge symmetry [@disgauge], it might protect the PQ symmetry up to sufficiently higher order operators. The same breaking effect could also affect the axion CDM abundance [@gravity]. If the contribution to the axion mass due to the quantum gravity is small compared to the one due to the QCD anomaly, $\langle S\rangle\simeq 10^{11}$ GeV is required for saturating $\Omega h^2=0.12$ by the axion contribution. Even if its contribution to the axion mass is larger than the one from the QCD anomaly within the bound which is required so as not to disturb the PQ mechanism, $\langle S\rangle\simeq 10^{11}$ GeV is required again for saturating $\Omega h^2=0.12$. Thus, $\eta_R$ could play a dominant role in the DM abundance as long as $\langle S\rangle$ is smaller than $10^{11}$ GeV. The stability for $\eta_R$ could be also violated through the same effect. The most effective processes for the $\eta_R$ decay are induced by nonrenormalizable Yukawa couplings such as $$\frac{S^{n}}{M_{\rm pl}^{n}}\left(h_u\bar q_Lu_R\eta+h_d\bar q_Ld_R\tilde\eta +h_e\bar \ell_Le_R\tilde\eta\right). \label{decay}$$ If the allowed dimension for these kind of operators is the same as the one which guarantees the PQ mechanism to work, the lifetime of $\eta_R$ could be longer than the age of our universe in case $m_\eta=O(1)$ TeV and $h_{u,d,e}=O(1)$ as long as we take $\langle S\rangle\simeq 10^{10}$ GeV. If the lower dimension operators such as $n< 6$ are allowed, its lifetime cannot be long enough to be the DM at the present universe. Consistency of the scenario with a cut-off scale of the model ------------------------------------------------------------- It is crucial to check what kind of values of the right-handed neutrino mass $M_i$ and $\tilde\lambda_5$ could be consistent with a value of $\langle S\rangle$ which is restricted by the axion physics. In this model, DM is identified with $\eta_R$ whose mass is of $O(1)$ TeV. In such a mass region, we find that its abundance is determined by the values of the scalar quartic couplings $\tilde\lambda_3$ and $\lambda_4$. On the other hand, these couplings could affect the vacuum stability and also the perturbativity of the model through the radiative effects on the scalar quartic couplings $\tilde\lambda_i$. Here, we examine the consistency of the values of $\tilde\lambda_3$ and $\lambda_4$ required to realize of the DM abundance with these issues.[^10] Since the breaking of the perturbativity is considered to be relevant to a scale for the applicability of the model, we could obtain an information for the cut-off scale $M_\ast$. It allows us to judge whether the required value for $\tilde\lambda_5$ by the neutrino masses and the leptogenesis could be induced through the VEV of $S$. The one-loop $\beta$-functions for the scalar quartic couplings in the effective model at energy regions below $M_S$ are given as follows [@rge], $$\begin{aligned} \beta_{\tilde\lambda_1}&=&24\tilde\lambda_1^2 +\tilde\lambda_3^2+(\tilde\lambda_3+\lambda_4)^2 +\tilde\lambda_5^2 \nonumber\\ &+&\frac{3}{8}\left(3g^4+g^{\prime 4}+2g^2g^{\prime 2}\right) -3\tilde\lambda_1\left(3g^2+g^{\prime 2}-4h_t^2\right)-6h_t^4, \nonumber \\ \beta_{\tilde\lambda_2}&=&24\tilde\lambda_2^2+\tilde\lambda_3^2 +(\tilde\lambda_3+\lambda_4)^2+\tilde\lambda_5^2 \nonumber\\ &+&\frac{3}{8}\left(3g^4+g^{\prime 4}+2g^2g^{\prime 2}\right) -3\tilde\lambda_2\left(3g^2+g^{\prime 2}\right), \nonumber \\ \beta_{\tilde\lambda_3}&=&2(\tilde\lambda_1+\tilde\lambda_2) (6\tilde\lambda_3+2\lambda_4) +4\tilde\lambda_3^2+2\lambda_4^2+2\tilde\lambda_5^2 \nonumber\\ &+&\frac{3}{4}\left(3g^4+g^{\prime 4}-2g^2g^{\prime 2}\right) -3\tilde\lambda_3\left(3g^2+g^{\prime 2}-2h_t^2\right), \nonumber \\ \beta_{\lambda_4}&=&4(\tilde\lambda_1+\tilde\lambda_2)\lambda_4 +8\tilde\lambda_3\lambda_4+4\lambda_4^2 +8\tilde\lambda_5^2+3g^2g^{\prime 2} -3\lambda_4\left(3g^2+g^{\prime 2}-2h_t^2\right), \nonumber\\ \beta_{\tilde\lambda_5}&=&4(\tilde\lambda_1+\tilde\lambda_2)\tilde\lambda_5 +8\tilde\lambda_3\tilde\lambda_5+12\lambda_4\tilde\lambda_5 -3\tilde\lambda_5\left(3g^2+g^{\prime 2}-2h_t^2\right), \end{aligned}$$ where $\beta_\lambda$ is defined as $\beta_\lambda=16\pi^2\mu\frac{d\lambda}{d\mu}$. In these equations, we can expect that the positive contributions of $\tilde\lambda_3$ and $\lambda_4$ to the $\beta$-functions of $\tilde\lambda_{1,2}$ tend to save the model from violating the first two vacuum stability conditions in eq. (\[instab\]). On the other hand, the same contributions of $\tilde\lambda_3$ and $\lambda_4$ could induce the breaking of the perturbativity of the model at a rather low energy scale since they could give large positive contributions to $\beta_{\tilde\lambda_1}$, $\beta_{\tilde\lambda_2}$ and $\beta_{\tilde\lambda_3}$. Here, we identify a cut-off scale $M_\ast$ of the model with a scale where any of the perturbativity conditions $\lambda_i(M_\ast)<4\pi$ and $\kappa_i(M_\ast)<4\pi$ is violated.[^11] In this case, $M_\ast >|\langle S\rangle|$ should be satisfied. If $M_\ast$ is smaller than $\langle S\rangle$, the consistency of the scenario is lost. We analyze this issue by solving the above one-loop RGEs at $\mu<M_S$ and also the ones at $\mu>M_S$, which are given in Appendix. The quartic couplings $\tilde\lambda_i$ in the tree-level potential at the energy scale $\mu<M_S$ are connected with the ones $\lambda_i$ at $\mu>M_S$ through eq. (\[gcoupl\]). Since the masses of the right-handed neutrinos $N_i$ are considered to be heavy in the present model, they decouple at the scale $\mu<M_i~^<_\sim~O(M_S)$ to be irrelevant to the RGEs there. On the other hand, the mass of the colored fields $D_{L,R}$ can take any values larger than 1 TeV as discussed before, they can contribute to the RGEs at larger scales than their mass. In the present study, we assume that $D_{L,R}$ is light of $O(1)$ TeV but its Yukawa coupling $h_D$ with the ordinary quarks is small enough.[^12] Thus, they are considered to contribute substantially only to the $\beta$-functions of the gauge couplings. In this study, we take its hypercharge as $Y=-\frac{1}{3}$ as shown in Table 1. The free parameters in the scalar potential of the effective model (\[effpot\]) are $\tilde\lambda_1,~\tilde\lambda_2,~\tilde\lambda_3,~\lambda_4$ and $\tilde\lambda_5$ at $M_Z$ as long as we assume $\tilde m_\eta=1$ TeV.[^13] Among them, we should fix $\tilde\lambda_5$ at a value used in the discussion of the neutrino mass and the leptogenesis. Both $\tilde\lambda_3$ and $\lambda_4$ are fixed at values determined through the DM relic abundance as shown in Fig. 2. We also have $\tilde\lambda_1\simeq 0.13$ from the Higgs mass. From this point of view, $\tilde\lambda_2$ is an only remaining parameter. Thus, if we solve the RGEs varying the value of $\tilde\lambda_2$ for other fixed parameters, we can find $M_\ast$ checking the vacuum stability for each $\tilde\lambda_2$. =7.5cm =7.5cm [   Left panel: running of the scalar quartic couplings for $t=\ln\frac{\mu}{M_Z}$. $\tilde\lambda_2=0.23$, $\tilde\lambda_3=0.65$ and $\lambda_4=-0.806$ are used as the initial values at $\mu=M_Z$. A vertical line corresponds to $t=\ln\left(\frac{M_S}{M_Z}\right)$. The running of the SM Higgs quartic coupling $\lambda$ is also plotted as a reference. Right panel: the cut-off scale $M_\ast$ as a function of $\tilde\lambda_2$ which is fixed as a value at $M_Z$ for four points marked by the black bulbs in Fig. 2 where $\Omega h^2=0.12$ is satisfied.]{} In the left panel of Fig. 3, as an example, we present the running of the scalar quartic couplings $\tilde\lambda_{1,2,3}$ for the initial values $\tilde\lambda_2=0.23$, $\tilde\lambda_3=0.65$ and $\lambda_4=-0.806$ at $M_Z$ by assuming the $U(1)_{PQ}$ breaking scale as $\langle S\rangle=M_S=10^{10}$ GeV. In the same panel, we also plot the value of $\tilde\lambda_3+\lambda_4-|\tilde\lambda_5| +2\sqrt{\tilde\lambda_1\tilde\lambda_2}$ as $C[\lambda_{34}]$, which corresponds to the last one in eq. (\[instab\]). In this example, we can see that the vacuum stability is kept until the cut-off scale $M_\ast \simeq 1.54\times 10^{13}$ GeV. These values of $\langle S\rangle$ and $M_\ast$ can naturally realize the assumed value for $\tilde\lambda_5$ through the relation given in eq.(\[gcoupl\]) just by taking $\lambda_5$ as a value of $O(1)$. This feature can be verified for other allowed values of $\tilde\lambda_3$ and $\lambda_4$. Here, we note that the axion contribution to the DM abundance can be neglected for a value such as $\langle S\rangle<10^{11}$ GeV. In the right panel of Fig. 3, we plot $M_\ast$ as a function of $\tilde\lambda_2$ for four sets of $(\tilde\lambda_3,~\lambda_4)$ which are shown by black bulbs in Fig. 2. End points found in the two lines represent the value of $\tilde\lambda_2$ for which the vacuum stability is violated before reaching $M_\ast$. This figure shows that $\tilde\lambda_2$ which is restricted to a rather narrow region can make $M_\ast$ appropriate values in order to realize a required value of $\tilde\lambda_5$ for $\langle S\rangle<10^{11}$ GeV. This study suggests that the scenario could work well without strict tuning of the relevant parameters. As found from the above study, the simultaneous explanation of the neutrino masses and the DM abundance could be preserved in this extended model in the same way as in the original scotogenic model. We should stress that no other additional constraint from the DM physics and the neutrino physics is brought about by taking the present scenario. The cosmological baryon number asymmetry is expected to be explained through the out-of-equilibrium decay of the lightest right-handed neutrino. The required right-handed neutrino mass could be smaller compared with the Davidson-Ibarra bound in the ordinary thermal leptogenesis [@di]. This is consistent with the result in [@ks] where the mass bound of the right-handed neutrino for the successful leptogenesis is shown to be relaxed in the radiative neutrino mass model in comparison with the ordinary seesaw model . Finally, we give brief comments on possible experimental signatures of the model. The present model might be examined through (i) the search of the $\eta_R$ DM and the charged scalars $\eta^\pm$ through the DM direct detection experiments and the accelerator experiments, (ii) the search of the mixing of $D$ with the ordinary quarks although it could be observed only in the light $D$ case, and (iii) the search of the axion whose coupling with photon is characterized by $g_{a\gamma\gamma}=\frac{m_a}{\rm eV} \frac{2.0}{10^{10}{\rm GeV}}(6 Y^2-1.92)$, where $Y$ is the hypercharge of $D$ [@lmn]. Summary ======= We have proposed an extension of the KSVZ invisible axion model so as to include a DM candidate and explain the small neutrino masses. An extra inert doublet scalar $\eta$ and three right-handed neutrinos $N_i$ are introduced as new ingredients. After the $U(1)_{PQ}$ symmetry breaking, its subgroup $Z_2$ could remain as a remnant effective symmetry, which is violated through the QCD anomaly but it can play the same role as the $Z_2$ in the scotogenic neutrino mass model. Since only the new ones $\eta$ and $N_i$ have its odd parity, the model reduces to the scotogenic model which has $Z_2$ in the leptonic sector. The neutrino masses are generated at one-loop level and the DM abundance can be explained by the thermal relics of the neutral component of $\eta$. The cosmological baryon number asymmetry could be generated through the out-of-equilibrium decay of a right-handed neutrino in the same way as the ordinary thermal leptogenesis in the tree-level seesaw model. However, the bound for the right-handed neutrino mass can be relaxed in this model. Since this simple extension can relate the strong $CP$ problem to the origin of neutrino masses and DM, it may be a promising extension of both the KSVZ model and the scotogenic model. Appendix {#appendix .unnumbered} ======== The $\beta$-function for the scalar quartic couplings at $\mu>M_S$ are given as $$\begin{aligned} \beta_{\lambda_1}&=&24\lambda_1^2 +\lambda_3^2+(\lambda_3+\lambda_4)^2 + \kappa_2^2 +\frac{3}{8}\left(3g^4+g^{\prime 4}+2g^2g^{\prime 2}\right) \nonumber \\ &-&3\lambda_1\left(3g^2+g^{\prime 2}-4h_t^2\right)-6h_t^4, \nonumber \\ \beta_{\lambda_2}&=&24\lambda_2^2+\lambda_3^2 +(\lambda_3+\lambda_4)^2 +\kappa_3^2 +\frac{3}{8}\left(3g^4+g^{\prime 4}+2g^2g^{\prime 2}\right) -3\lambda_2\left(3g^2+g^{\prime 2}\right) \nonumber \\ &+&4\lambda_2\left[2(h_1^2+h_2^2)+3h_3^2\right] -8(h_1^2+h_2^2)^2-18h_3^4, \nonumber \\ \beta_{\lambda_3}&=&2(\lambda_1+\lambda_2) (6\lambda_3+2\lambda_4) +4\lambda_3^2+2\lambda_4^2 +2\kappa_2\kappa_3 +\frac{3}{4}\left(3g^4+g^{\prime 4}-2g^2g^{\prime 2}\right) \nonumber \\ &-&3\lambda_3\left(3g^2+g^{\prime 2}-2h_t^2\right) +2\lambda_3\left[2(h_1^2+h_2^2)+3h_3^2\right], \nonumber \\ \beta_{\lambda_4}&=&4(\lambda_1+\lambda_2)\lambda_4 +8\lambda_3\lambda_4+4\lambda_4^2 +3g^2g^{\prime 2}-3\lambda_4\left(3g^2+g^{\prime 2}-2h_t^2\right) \nonumber \\ &+&2\lambda_4\left[2(h_1^2+h_2^2)+3h_3^2\right], \nonumber\\ \beta_{\kappa_1}&=& 20\kappa_1^2+2\kappa_2^2+2\kappa_3^2 +4\kappa_1\left(3y_D^2+\sum_iy_i^2\right) -2\left(3y_D^4+\sum_iy_i^4\right), \nonumber \\ \beta_{\kappa_2}&=& 4\kappa_2^2+2\kappa_2(6\lambda_1+4\kappa_1) +2\kappa_3(2\lambda_3+\lambda_4)+ 2\kappa_2\left(3y_D^2+\sum_iy_i^2\right) \nonumber \\ &-&\frac{3}{2}\kappa_2(3g^2+g^{\prime 2}-4h_t^2), \nonumber \\ \beta_{\kappa_3}&=&4\kappa_3^2+2\kappa_3(6\lambda_2+4\kappa_1) +2\kappa_2(2\lambda_3+\lambda_4)+ 2\kappa_3\left(3y_D^2+\sum_iy_i^2\right) \nonumber \\ &-&\frac{3}{2}\kappa_3\left[3g^2+g^{\prime 2}- \frac{4}{3}\left(2(h_1^2+h_2^2)+3h_3^2\right)\right], \end{aligned}$$ where eq. (\[flavor\]) is assumed for the flavor structure of neutrino Yukawa couplings. The $\beta$-functions for the gauge couplings and the Yukawa couplings for top, $D$ and neutrinos are given as $$\begin{aligned} &&\beta_{g_s}=-11+\frac{2}{3}(6+\delta)g_s^3, \quad \beta_g= -3g^3, \quad \beta_{g^\prime}=(7+4Y^2\delta)g^{\prime 3}, \nonumber \\ &&\beta_{h_t}=h_t\left(\frac{9}{2}h_t^2-8g_s^2-\frac{9}{4}g^2 -\frac{17}{12}g^{\prime 2}\right), \quad \beta_{y_k}=y_k\left(y_k^2+3y_D^2+\sum_iy_i^2\right),\nonumber \\ &&\beta_{y_D}=y_D\left(-8g_3^3-6Y^2\delta g^{\prime 2}+4y_D^2+\sum_iy_i^2 \right), \nonumber \\ &&\beta_{h_{1,2}}=h_{1,2}\left[-\frac{9}{4}g^2- \frac{3}{4}g^{\prime 2}+5(h_1^2+h_2^2)+3h_3^2+\frac{1}{2}\sum_iy_i^2\right], \nonumber \\ &&\beta_{h_3}=h_3\left[-\frac{9}{4}g^2- \frac{3}{4}g^{\prime 2}+2(h_1^2+h_2^2)+6h_3^2+\frac{1}{2}\sum_iy_i^2\right],\end{aligned}$$ where $\delta$ stands for the number of extra color triplets $D_{L,R}$. Since $D_{L,R}$ is assumed to be light in this study, $\delta$ is treated as 1. 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However, it is beyond the scope of the present study and we do not discuss it here. [^13]: Quartic couplings $\kappa_i$ for $S$ are fixed as $\kappa_1=\frac{M_S^2}{4\langle S\rangle^2}$ and $\kappa_{2,3}=0.1$ at $M_S$ in the present study. As easily found from RGEs, larger values of $\kappa_{2,3}$ make $M_\ast$ smaller.

--- abstract: 'In this paper we propose a generalization of deep neural networks called deep function machines (DFMs). DFMs act on vector spaces of arbitrary (possibly infinite) dimension and we show that a family of DFMs are invariant to the dimension of input data; that is, the parameterization of the model does not directly hinge on the quality of the input (eg. high resolution images). Using this generalization we provide a new theory of universal approximation of bounded non-linear operators between function spaces. We then suggest that DFMs provide an expressive framework for designing new neural network layer types with topological considerations in mind. Finally, we introduce a novel architecture, RippLeNet, for resolution invariant computer vision, which empirically achieves state of the art invariance.' author: - | William H. Guss\ Machine Learning at Berkeley\ University of California, Berkeley\ `wguss@ml.berkeley.edu` bibliography: - 'dfm.bib' title: 'Deep Function Machines: Generalized Neural Networks for Topological Layer Expression' ---

--- abstract: 'Defects in silicon carbide are of intense and increasing interest for quantum-based applications due to this material’s properties and technological maturity. We calculate the multi-particle symmetry adapted wave functions of the negatively charged silicon vacancy defect in hexagonal silicon carbide via use of group theory and density functional theory and find the effects of spin-orbit and spin-spin interactions on these states. Although we focused on $\textrm{V}_{\textrm{Si}}^-$ in 4H-SiC, because of its unique fine structure due to odd number of active electrons, our methods can be easily applied to other defect centers of different polytpes, especially to the 6H-SiC. Based on these results we identify the mechanism that polarizes the spin under optical drive, obtain the ordering of its dark doublet states, point out a path for electric field or strain sensing, and find the theoretical value of its ground-state zero field splitting to be 68 MHz, in good agreement with experiment. Moreover, we present two distinct protocols of a spin-photon interface based on this defect. Our results pave the way toward novel quantum information and quantum metrology applications with silicon carbide.' author: - 'Ö. O. Soykal' - Pratibha Dev - 'Sophia E. Economou' title: 'Silicon vacancy center in 4H-SiC: Electronic structure and spin-photon interfaces' --- Over the last several years, deep-center defects in solids have been intensely researched for applications in quantum information [@Togan_nature11; @Bernien_Nature13], quantum sensing and nanoscale imaging [@Grinolds_NP13] including bioimaging [@Balasubramanian_Nature08; @Shi_science15]. Their success and popularity stem from their unique properties, combining advantages from atomic and solid state systems—-most notably long spin coherence times even at room temperature and integrability into a solid state matrix. The NV center in diamond is the most studied defect for quantum technologies, so that its properties, strengths and limitations are by now very well understood. Deep defect centers in silicon carbide (SiC) have emerged as strong contenders due to this material’s significantly lower cost, availability of mature microfabrication technologies [@Song_OptExp11; @Maboudian_JVSci13], and favorable optical emission wavelengths [@Baranov_prb11]. Some of the stable defects in SiC have the same structure as the NV center in diamond in terms of symmetry and the number of active electrons and, as a result, spin and electronic structure. Such defects include the silicon-carbon divacancy, which has been investigated over the last several years [@Son_PRL06; @Koehl_nature11; @Falk_natcom13; @Christle_nature14]. Experiments [@Mizuochi2002; @Baranov_prb11; @Riedel_PRL12; @Kraus_NP14; @Kraus_scirep14; @Widmann_nmat15; @Carter_PRBRC15] on the Si *monovacancy* ($\textrm{V}_{\textrm{Si}}^-$) have shown that this is a distinct defect in terms of electronic and spin structure. It features a ground state with total spin $3/2$ [@Mizuochi2002; @Kraus_NP14], offering both quantitative improvements and qualitatively new capabilities [@Kraus_scirep14] compared to NV-like defects. To date, room temperature spin polarization and coherent control of $\textrm{V}_{\textrm{Si}}^-$ have been implemented via electron spin resonance [@Soltamov_PRL12; @Widmann_nmat15] and optically detected magnetic resonance (ODMR) [@Sorman_PRB2000; @Baranov_prb11; @Kraus_NP14; @Carter_PRBRC15]. Unlike the well-studied NV center in diamond [@Lenef_PRB96; @maze_njp11; @doherty_njp11], theoretical studies of the $\textrm{V}_{\textrm{Si}}^-$ in SiC have been mostly limited to finding single-particle levels and their energies via density functional theory (DFT) [@janzen_physicab09; @Weber_PNAS10; @gali_JMatR12]. While such DFT calculations are an important first step, it is of crucial importance to obtain the multi-particle electronic structure to understand the properties of this defect and take full advantage of the novel opportunities it affords. ![(color online) $\textrm{V}_{\textrm{Si}}^-$ in 4H-SiC: (a) $C_{3\nu}$-structure of the defect, and the optically-active orbitals of $\textrm{V}_{\textrm{Si}}^-$ using DFT: (b) $\bar{u}$ ($A_1$ symmetry), (c) $\bar{v}$ ($A_1$), and (d) $\bar{e}_{x,y}$ ($E$). Only carbons near the $\textrm{V}_{\textrm{Si}}^-$ are shown for clarity.[]{data-label="Fig1"}](Figure1.pdf){width="8.0cm"} [|c||c|c|c|c|l|c|]{} Orbital & $S$ & $m_s$ & $\Gamma$ & $\Gamma_o\otimes\Gamma_s$ & & $\textrm{Label}$\ & & & & & & $\Psi_\textrm{g}^1$\ & & & & & & $\Psi_\textrm{g}^2$\ & & & & & $\begin{array}{l}||ve_x\bar{e}_y+v\bar{e}_xe_y+\bar{v}e_xe_y\rangle/\sqrt{3}\end{array}$ & $\Psi_\textrm{g}^3$\ & & & & & $\begin{array}{l}||\bar{v}\bar{e}_xe_y+\bar{v}e_x\bar{e}_y+v\bar{e}_x\bar{e}_y\rangle/\sqrt{3}\end{array}$ & $\Psi_\textrm{g}^4$\ & & & $E_{1/2}^+$ & & & $\Psi_{\textrm{d}1}^1$\ & & & $E_{1/2}^-$ & & & $\Psi_{\textrm{d}1}^2$\ & & & & & &\ & & & & & &\ & & & $E_{1/2}^+$ & & $\begin{array}{l}||ve_x\bar{e}_y+v\bar{e}_xe_y-2\bar{v}e_xe_y\rangle/\sqrt{6}\end{array}$ & $\Psi_{\textrm{d}2}^1$\ & & & $E_{1/2}^-$ & & $\begin{array}{l}||\bar{v}\bar{e}_xe_y+\bar{v}e_x\bar{e}_y-2v\bar{e}_x\bar{e}_y\rangle/\sqrt{6}\end{array}$ & $\Psi_{\textrm{d}2}^2$\ & & & & & &\ & & & & & &\ & & & & & &\ & & & & & &\ & & & & & &\ & & & & & &\ & & & $E_{1/2}^+$ & & $\begin{array}{l}||ve_x\bar{e}_x+ve_y\bar{e}_y\rangle/\sqrt{2}\end{array}$ & $\Psi_{\textrm{d}4}^1$\ & & & $E_{1/2}^-$ & & $\begin{array}{l}||\bar{v}\bar{e}_xe_x+\bar{v}\bar{e}_ye_y\rangle/\sqrt{2}\end{array}$ & $\Psi_{\textrm{d}4}^2$\ & & & $E_{1/2}^+$ & & & $\Psi_{\textrm{d}5}^1$\ & & & $E_{1/2}^-$ & & & $\Psi_{\textrm{d}5}^2$\ & & & & & &\ & & & & & &\ & & & $E_{1/2}$ & $E\otimes \prescript{1}{}E_{3/2}$ & $\begin{array}{l}||uve_x\rangle\end{array}$, $\begin{array}{l}||uve_y\rangle\end{array}$ & $\Psi_{\textrm{q}2}^1$, $\Psi_{\textrm{q}2}^2$\ & & & $E_{1/2}$ & $E\otimes \prescript{2}{}E_{3/2}$ & $\begin{array}{l}||\bar{u}\bar{v}\bar{e}_x\rangle\end{array}$, $\begin{array}{l}||\bar{u}\bar{v}\bar{e}_y\rangle\end{array}$ & $\Psi_{\textrm{q}2}^3$, $\Psi_{\textrm{q}2}^4$\ & & & & & &\ & & & & & &\ & & & & & &\ & & & & & &\ & & & & & &\ & & & & & &\ In this Letter we address this need by calculating the multi-particle wave functions of $\textrm{V}_{\textrm{Si}}^-$ through a combination of group theory and DFT. We explicitly find the ground states as well as the excited state manifolds, considering both the orbital and the spin degrees of freedom. Furthermore, we investigate the effects of spin-orbit and spin-spin interactions. Based on these results we (i) explain quantitatively the spin polarization mechanism in experiments, (ii) find the zero-field splitting, in good agreement with experiment, (iii) present a mechanism that allows this defect to be used for electric field or strain sensing, and (iv) propose two spin-photon interface protocols enabled by the rich electronic structure of this defect, including the generation of strings of entangled photons and the creation of a Lambda system with potential applications in quantum technologies. The $C_{6\nu}$ symmetry of bulk 4H-SiC is lowered to the $C_{3\nu}$ point group in the presence of $\textrm{V}_{\textrm{Si}}^-$. The local geometry of $\textrm{V}_{\textrm{Si}}^-$ is shown in Fig.\[Fig1\](a), where the missing silicon leaves four dangling bonds ($sp^3$-orbitals) on the surrounding carbons. Single electron molecular orbitals (MO) can be constructed from symmetry-adapted linear combinations of the three equivalent $sp^3$-orbitals ($a$, $b$ and $c$) from the basal-plane carbons and the $sp^3$-orbital, $d$, belonging to the carbon atom on the $C_3$-axis that coincides with the crystalline $c$-axis. Using the standard projection operator technique [@Tinkham2003] and our DFT results as a guide \[Fig.\[Fig1\](b)-(d)\], we obtain the following MOs of the defect center: $u{=}\alpha_u (a+b+c){+}\beta_u d$, $v{=}\alpha_v (a+b+c){+}\beta_v d$, $e_x{=}\alpha_x (2c-a-b)$, and $e_y{=}\alpha_y (a-b)$, where the coefficients are given in [@supplement]. The orbitals, as calculated by DFT, are shown in Fig.\[Fig1\]. The functions $u$ and $v$ transform as $A_1$, $e_X$ and $e_Y$ transform as the $x$ and $y$ components of the $E$ representation respectively and the states are listed in order of increasing energy according to our DFT calculations. The electronic configuration of this defect is modeled by three holes, a simpler but equivalent picture to that of five active electrons. Then, the three-hole lowest energy quartet configurations are identified as $ve_xe_y$, $ue_xe_y$, and $uve_x$ (or $uve_y$), respectively, increasing in energy [@supplement]. The tensor products of $u$, $v$, and $e_{x,y}$ states with the total spin eigenstates comprise our basis set, from which we calculate the multi-particle symmetry-adapted states compatible with $C_{3\nu}$. The odd number of particles here results in a much more complicated structure compared to NV centers in diamond and divacancies in SiC. Thus, we obtain the multi-particle wave functions systematically by use of the projection operator on the basis states for both the orbital and the spin degree of freedom: $$\mathcal{P}^{(j)}=(I_j/h)\sum_R\chi^{(j)}(R)^*\Gamma^{(j)}(R),\label{2}$$ where, $\chi^{(j)}(R)$ is the character of operation $R$ in the $j^{\text{th}}$ irreducible representation [@supplement], and $\Gamma$ is the irreducible matrix representation for the $R$ symmetry operator (tensor product of the three-particle orbital and spin operators [@supplement]). The resulting symmetry adapted states are shown in Table\[Table3\], and are characterized by the total spin $S$, the orbital and spin symmetry, as well as their overall symmetry. These classifications are of key importance in understanding the nature of these states, their additional interactions, as well as the allowed optical or spin-orbit assisted transitions and selection rules. The ground state manifold has $S{=}3/2$ (quartet), while there are nearby additional manifolds (each a doublet, $S{=}1/2$) with some having the same orbital composition as the ground state and split from each other only due to Coulomb interactions (see Fig.\[Fig3\] and [@supplement]). The states are split and mixed further by spin-orbit (SO) and spin-spin interactions. The SO coupling is $$\mathcal{H}_{SO}=\sum_j \bm{\ell}_j\cdot \bm{s}_j,\label{3}$$ where $\ell_j$ and $s_j$ are orbital and spin angular momentum operators belonging to the $j^{\mathrm{th}}$ hole. The former is defined as $(\ell_j)_i=\epsilon_{ikl}[\nabla V(\bm{r}_j)]_k[\bm{p}_j]_l/2m^2 c^2$ where the $V(\bm{r_j})$ is the local potential, $\bm{p_j}$ is the hole momentum operator with coordinate indices $i,k,l$. The components of both $\mathbf{\ell}$ and $\mathbf{s}$ transform as the $(E_Y,E_X,A_2)$ representation and the $H_{SO}$ Hamiltonian itself transforms as $A_1$. With these symmetry classifications we see that the diagonal part of $H_{SO}$, $\sum_j \ell_{j,z} s_{j,z}$, will only couple states of the same $L$ and $S$ and of orbital symmetry $E$ (since $A_1{\subset}E{\otimes} A_2{\otimes}E$). Thus, the ground states do not split due to this term, while states $\{\Psi^j_\textrm{d}\}$ and $\{\Psi^j_\textrm{q2}\}$ shift and/or mix within their manifolds, as shown in Fig.\[Fig2\] by $\Delta_{\textrm{d}}{=}\langle\phi^E_\xi||L_z^{A_2}||\phi^E_\xi\rangle/(2\sqrt{2})$ and $\Delta_{\textrm{q}}{=}\langle\phi^E_{uve}||L_z^{A_2}||\phi^E_{uve}\rangle/(2\sqrt{2})$ respectively (given in terms of reduced matrix elements and $\xi=\{e^3, v^2e\}$). Note that the total orbital angular momentum operator is used here, which is equivalent to using Eq. \[3\] for matrix elements between states of the same total $S$ and $L$ [@Tinkham2003]. The transverse parts of the SO interaction, $\sum_j \ell_{j,\bot} s_{j,\bot}$, couple states of different total spin and orbital character $\{u,v\}$ to both $e_x$ and $e_y$ at single particle level. Hence the ground states will couple to $\{\Psi^j_\textrm{d1}\}$ (defined in Table\[Table3\]) via these transverse SO terms. This coupling is crucial both in explaining existing experiments and in designing future applications. The key is to notice that ground states and q1 excited states with $|S_z|{=}3/2$ couple more strongly to excited $\{\Psi^j_\textrm{d1}\}$ ($e^3$) states compared to the states with $|S_z|{=}1/2$. In fact using the states of Table\[Table3\] we can show that the ratio of the matrix elements is $\sqrt{3}$. From this we identify the dominant intersystem crossing channel that constitutes the spin polarization mechanism seen in recent experiments at the single-spin level [@Widmann_nmat15] with h-site ($V_2$) defects, where optical driving polarizes the system into the $|S_z|{=}3/2$ states. This mechanism, shown in Fig.\[Fig3\], also successfully predicts the recently seen increase in the ODMR photo-luminescence intensity with microwave drive [@Sorman_PRB2000; @Baranov_prb11; @Kraus_NP14; @Widmann_nmat15; @Carter_PRBRC15]. We can also consider first-order perturbing corrections to the ground state wave functions from the excited dark doublet states through spin-orbit coupling (see Fig.\[Fig3\]). The different strength of the SO matrix elements (e.g., the extra involvement of $l_{j,z}s_{j,z}$ with $m_s=\pm 1/2$ states only) will cause a different degree of admixture of excited states to the $|S_z|{=}3/2$ and $|S_z|{=}1/2$ ground states, which in turn will allow an electric field [@Dolde_NP11], strain and mechanical motion [@Soykal_PRL11; @Grinolds_NanoLett12; @MacQuarrie_PRL13] to couple ground states with different $|S_z|$ projections. This paves the way toward unexplored SiC-based applications in sensing. ![(color online) Electronic configuration of $\textrm{V}_{\textrm{Si}}^-$, shown in terms of the wave functions given in Table\[Table3\]. The splittings are shown explicitly for the SO and spin-spin interactions. The spin quartets are grouped on the left half whereas the metastable doublets are on the right. The states with subscript q and d denote excited quartet and doublet states, respectively. The dashed (green) arrows indicate the mixing due to spin-spin interactions. []{data-label="Fig2"}](Figure2.pdf){width="8.5"} ![(color online) Spin polarization channel of $\textrm{V}_{\textrm{Si}}^-$ through the spin-orbit assisted dominant intersystem crossing $\prescript{4}{}A_{2}(ue^2){\rightarrow}\prescript{2}{}E(e^3){\rightarrow}\prescript{4}{}A_{2}(ve^2)$ and all other allowed channels are shown in dashed lines. Thicker lines of blue and green indicate 3$\times$ faster transition rate from or to $m_s=\pm 3/2$ states by the transverse component of spin-orbit $\lambda_\bot$ whereas orange represents a channel via the longitudinal $\lambda_z$ component. Energies of the doublets are ordered in terms of the one-particle Coulomb Hamiltonian $\chi=\langle\phi|\sum h_i|\phi\rangle$ and leading many-particle direct integrals, i.e. $j^0=\int\rho_{aa}(r_1) V_{R}(r_1,r_2)\rho_{aa}(r_2)d^3r_1 d^3r_2$, of Coulomb repulsion [@supplement]. []{data-label="Fig3"}](Figure3.pdf){width="8.5"} Next we consider the spin-spin interaction between the holes. The Hamiltonian is $$\mathcal{H}_{S}=\frac{\mu_0 g^2 \mu_B^2}{4\pi}\sum_{i>j}\frac{1}{r_{ij}^3}\left\{\bm{s}_i\cdot \bm{s}_j-3\left(\bm{s}_i\cdot \bm{\hat{r}}_{ij}\right)\left(\bm{s}_j\cdot \bm{\hat{r}}_{ij}\right)\right\},\label{4}$$ where $g$ is the electron g-factor, $\mu_0$ is the vacuum permeability, and $\mu_B$ is the Bohr magneton. The spin operator of each hole, the distance to each other and its unit vector are $s_{i}$, $r_{ij}$ and $\bm{\hat{r}}_{ij}$, respectively. The spin-spin splittings of the quartets and doublets are shown in Fig.\[Fig2\] in terms of the splitting parameters defined as $\gamma_\textrm{g}{=}\gamma_0 \langle\phi^{A_2}_{ve^2}||I_2||\phi^{A_2}_{ve^2}\rangle/\sqrt{10}$, $\gamma_{\textrm{q1}}{=}\gamma_0 \langle\phi^{A_2}_{ue^2}||I_2||\phi^{A_2}_{ue^2}\rangle/\sqrt{10}$, $\gamma_\textrm{d}{=}\gamma_0\langle\phi^{E}_{\xi}||I_2||\phi^{E}_{\xi}\rangle/(6\sqrt{10})$, $\gamma_{\textrm{q2}}^1{=}\gamma_0\langle\phi^{E}_{uve}||I_2||\phi^{E}_{uve}\rangle/\allowbreak(2\sqrt{10})$ and $\gamma_{\textrm{q2}}^2{=}\gamma_{\textrm{q2}}^1(1-1.028\zeta)$, where $I_2$ is an irregular solid harmonic of second rank, i.e. $I_l^m{=}\sqrt{4\pi/(2l+1)}Y_l^m/r^{l+1}$, $\gamma_0 {=} \mu_0g^2\mu_B^2/4\pi$, and $\zeta{=}\langle\phi^{E}_{uve}||I_2||\phi^{E}_{uve}\rangle/\Delta_{\textrm{q}}{\approx} 0$, see [@supplement]. Using in these expressions the calculated bond lengths $d{=}2.058$ Å, $a{=}2.055$ Å, and $\theta_0{=}35.26^\circ$ from our DFT results, we estimate the zero field splitting (ZFS) to be $2|\gamma_\textrm{g}| {=} 68$ MHz, in good agreement with experiments [@Mizuochi2002; @Kraus_NP14; @Carter_PRBRC15; @Sorman_PRB2000]. However, we found a negative $D$ for the ground state, i.e. $\mathcal{H}_S{\simeq}D[S_z^2-S(S+1)/3]$, causing $m_s=\pm 1/2$ to be energetically higher than the $m_s=\pm 3/2$ states contrary to the some assumptions of $D{>}0$ in literature. In the limit of perfect tetrahedral ($T_d$) symmetry, our calculation also leads to a vanishing ZFS (0 MHz) consistent with the lack of any ZFS with $\textrm{V}_{\textrm{Si}}^-$ centers in 3C-SiC. Based on Table\[Table3\], the rich structure of the various transitions and immunity to all local perturbing electric and strain fields (Kramer’s degeneracy) enable the design of a spin-photon interface for applications in quantum computing and quantum communications. Below we propose two such protocols. First consider the ground states with $|S_z|{=}3/2$, split by a B-field along the $C_3$ axis, $\Psi_g^{\pm}{=}\Psi_\textrm{g}^1\pm \Psi_\textrm{g}^2$. The excited states of interest are $\Psi_e^+{=}\Psi^2_\textrm{q2}{-}i\Psi^1_\textrm{q2}$ and $\Psi_e^-{=}\Psi^4_\textrm{q2}{+}i\Psi^3_\textrm{q2}$, which are degenerate energy eigenstates after SO has been included (Fig.\[Fig2\]); these states have $|S_z|{=}3/2$, and since the g-factor is the same in ground and excited states [@janzen_physicab09] they split by the same amount as the lower levels. They are also the only states which are not coupled to the states of $\prescript{4}{}A_2$ q1 manifold via $\sum_j \ell_{j,\bot} s_{j,\bot}$ terms. The allowed optical transitions between these sets of states are $\Psi_g^+ {\leftrightarrow} \Psi_e^+$ and $\Psi_g^- {\leftrightarrow} \Psi_e^-$ with right and left circularly polarized light respectively, Fig.\[Fig4\](a). A coherently excited superposition of the two excited states decays to an entangled spin-photon state, $|\Psi_g^+\rangle |\sigma +\rangle {+} |\Psi_g^-\rangle |\sigma -\rangle$. Repeating this process produces additional photons, all entangled with the spin and each other, resulting in a multiphoton Greenberger-Horne-Zeilinger state. Augmenting the optical protocol with microwaves can couple the ground states and allow the production of a cluster state [@raussendorf_PRL01], similarly to a proposal for quantum dots [@Lindner_PRL09; @Economou_PRL10]. Next we consider a B-field perpendicular to the $C_3$ axis. This mixes all four ground states, and from these we select $\Psi_g^\alpha$ and $\Psi_g^\beta$, along with the excited state $\Psi_e^\gamma$ (all of them given in [@supplement] in terms of the states of Table\[Table3\]). Then a $\Lambda$-system can be formed, Fig.\[Fig4\](b). This three-level system can be used in numerous quantum applications and demonstrations, including coherent population trapping [@santori_optexp06], optical spin qubit rotations [@Economou_PRL07; @Yale_PNAS10] and generation of spin-photon entanglement [@Togan_science10; @Bernien_Nature13] with applications in quantum repeaters [@briegel_PRL98]. ![(color online) (a) A B-field parallel to the $C_3$ axis enables the creation of two two-level systems with the same transition frequency but orthogonal polarizations. Periodic coherent pumping followed by spontaneous emission leads to strings of entangled photons. (b) A B-field perpendicular to the $C_3$ axis allows for the creation of a Lambda system. []{data-label="Fig4"}](figure4.pdf){width="7."} In summary, we addressed the crucial need of calculating the multi-particle fine structure of the silicon vacancy defect in SiC. Based on the resulting spectrum we identified the intersystem crossing channel that polarizes the system, found a mechanism to enable quantum sensing applications, and proposed two spin-photon interface protocols. Our work opens further opportunities in understanding these defects and in implementing novel quantum technological applications. This work was supported in part by ONR. Computer resources were provided by the DoD HPCMP. Ö.O.S. and P.D. acknowledge the NRL-NRC Research Associateship Program. We thank S. Carter, Sang-Yun Lee, and Amrit De for comments on the manuscript. 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--- abstract: 'We study interaction-induced broken symmetry phases that can arise in metallic or semimetallic band structures with two nested Weyl or Dirac loops. The odered phases can be of the charge or (pseudo)spin density wave type, or superconductivity from interloop pairing. A general analysis for two types of Weyl loops is given, according to whether a local reflection symmetry in momentum space exists or not, for Hamiltonians having a global PT symmetry. The resulting density-wave phases always have lower total energy, and can be metallic, insulating, or semimetallic (with nodal loops), depending on both the reflection symmetry of the loops and the symmetry transformation that maps one loop onto the other. We extend this study to nested $\mathbb{Z}_2$ nodal lines, for which the ordered phases include also nodal point and nodal chain semimetals, and to spinful Dirac nodal lines. Superconductivity from interloop pairing can be fully gapped only if the initial double loop system is semimetallic.' author: - 'Miguel A. N. Araújo$^{1,2}$, and Linhu Li$^3$' title: 'Broken-symmetry phases of interacting nested Weyl and Dirac loops ' --- Introduction ============ Since the discovery of topological insulators, band structures of fermionic systems with non-trivial momentum-space topology have received much attention in modern condensed matter physics. Their low energy description involves Dirac-like band dispersions, which in some cases imply gapless band structures characterized by the presence of nodal points or lines. Among these, are the nodal line semimetals (NLSMs) [@mullen; @hklattice]. A NLSM has valence and conduction bands touching along one-dimensional (1D) lines in the three-dimensional (3D) momentum space, and feature two-dimensional (2D) “drumhead" surface states surrounded by the nodal lines [@Burkov; @Yang2014; @Chen2015; @Weng2015; @Zhang2016]. Contrary to the well-studied topological insulating phases and nodal point semimetals, the 1D nodal lines of NLSMs provide rich topological structures such as links and knots [@Links_knots1; @Links_knots2; @Links_knots3; @Links_knots4; @Links_knots5], which cannot be described unambiguously by a single sign (e.g. the $Z_2$ number) or a integer (e.g. the Chern number) [@multi_links_Li]. On the other hand, a variety of gapped and gapless topological phases have been predicted in NLSMs (while possibly breaking certain symmetries). For example, a spin-orbit interaction can induce 3D Dirac semimetals from a NLSM [@SOC1; @SOC2], and periodical driving such as linear or circular polarized light may induce different types of nodal points [@light_driven1; @light_driven2; @light_driven3; @light_driven4; @light_driven5; @light_driven6]. By introducing various types of extra gapped terms, a NLSM can also be driven into several different types of topological insulators, including the recently discovered high-order topological insulators [@TI_NL1; @TI_NL2]. Spontaneous symmetry breaking from interactions in three dimensional systems with Weyl/Dirac nodal points or lines have also been addressed. For single nodal loop (NL) systems, superconducting and charge (or spin) density wave instabilities have been investigated using renormalization of fermionic interactions[@Nandkishore2], including also the mean-field description of the ordered phases[@Nandkishore1; @roy; @ryu]. Certain symmetries, such as spatial inversion or time-reversal, imply that Weyl nodes must occur at an even number of Brillouin Zone (BZ) points. Charge and spin density waves, as well as superconducting phases, which arise from nested spherical Fermi surfaces (FSs) in doped (or uncompensated) Weyl/Dirac points have been discussed[@wangye]. Weyl or Dirac NLs, on the other hand, do not necessarily have to exist in pairs. Although two-loop semimetals have not yet been found in nature, pairs of linked NLs (or Hopf-link structures) have been theoretically proposed[@Links_knots3; @Links_knots4; @Links_knots6]. Furthermore, a class of NLs protected by a combination of inversion and time-reversal has recently been discussed[@Z2_loop], which carry $\mathbb{Z}_2$ monopole charges, and must therefore be created or annihilated in pairs. This has motivated us to address the spontaneous symmetry breaking from short range interactions in two-loop band structures, when the NLs are related through a nesting vector ${{\bf Q}}$ in the BZ. We describe the density wave and superconducting phases, which can be metallic, semimetallic (with double NLs) or fully gapped. A systematic analysis for two-band NL models is given, where the NLs can either satisfy a local reflection symmetry in the loop plane or not. If a global PT symmetry exists, then a symmetry operation can relate the two NLs. These properties combined determine the nature of the ordered phases. We also study specific four-band models that have appeared in the recent literature, such as the $\mathbb{Z}_2$ NLs, and NLs arising from perturbed Dirac points. Superconducting phases arising from pairing of fermions in different loops are also considered for all cases of singlet and triplet gap functions in loop space as well as (pseuso)spin space. But we have restricted our search to order parameters with time-reversal symmetry (TRS) and fully gapped phases, because the latter are expected to be more stable. The possibility of gap functions with a winding number, which break TRS, is not addressed here. The structure of the paper is as follows. In Section \[modelsec\] we introduce the local $k\cdot p$ Hamiltonians for two-band Weyl NLs, which can either satisfy a local reflection symmetry in the NL plane, or not. The Hubbard interaction and the density wave order parameters associated with the NL nesting vector are also introduced. The density wave phases are described in Section \[densec\], and Section \[examples\] is devoted to example models and to a four-band system that was not included in the general analysis of the previous Sections: the nested $\mathbb{Z}_2$ NLs. In Section \[Diracsec\], we study spin degenerate Dirac NLs and also NLs arising from perturbed Dirac points. The superconducting pairing between nested NLs is studied in Section \[supersec\]. The analysis is focused on interloop pairing and time-reversal symmetric order parameters. In Section \[concsec\] we present our conclusions. Model {#modelsec} ===== We consider spinless fermions and let $\boldsymbol\tau$ denote the Pauli matrices acting on the pseudo-spin (orbital) degree of freedom. We assume that the band structure has two degenerate loops. If the Hamiltonian has PT symmetry, both loops involve the same Pauli matrices, $\tau_a\,,\tau_b$, so that each one can be locally described by $k\cdot p$ Hamiltonians: $$\begin{aligned} H_0({{\bf k}})&=& v_1\left( p_{\parallel} - p_o \right) \tau_a + v_2 p_\perp \tau_b \,, \\ H_0({{\bf k}}+\ {{\bf Q}})&=& g_1v_1\left( p_{\parallel} - p_o \right) \tau_a +g_2 v_2 p_\perp \tau_b\,,\end{aligned}$$ \[loop1\] Here, and throughout the paper, $\bf p=\hbar{{\bf k}}$ and the subscripts $\perp$($\parallel$) refer to the components perpendicular (parallel) to the loop plane, and $g_{1(2)}$ are + or - signs. The loops are nested by the vector ${{{\bf Q}}}$. We shall refer to the NLs in Eq. (\[loop1\]) as “model-1” loops. Such NLs are protected by a local reflection[@bian] in the loop plane, ${\cal R}=(p_\perp\rightarrow -p_\perp)\otimes \tau_a$. Such a NL can be topologically characterized by a $\pi$ Berry phase along a trajectory enclosing the NL [@Burkov; @Zhang2016]. At zero chemical potential, the system is a semimetal and the FS consists of the two nested NLs. We shall also take non-degeneracy into account by considering an energy offset $\delta$ between the loops and make the replacement $H_0({{\bf k}})\rightarrow H_0({{\bf k}}) -\delta$, $H_0({{\bf k}}+\ {{\bf Q}})\rightarrow H_0({{\bf k}}+\ {{\bf Q}}) +\delta$. For positive $\delta$ and zero chemical potential, the FSs are torus-shaped, the one from $H_0({{\bf k}}+\ {{\bf Q}}) $ is in the lower (hole) band, while the FS from $H_0({{\bf k}})$ is in the upper (electron) band. However, NLs are not necessarily protected by reflection symmetry. Here we also consider a more general model of nested NLs without reflection symmetry. The Hamiltonian reads $$\begin{aligned} H_0({{\bf k}})&=& \left[v_{1\parallel}\left( p_{\parallel} - p_o \right) + v_{1\perp} p_\perp \right] \tau_a + v_2 p_\perp \tau_b\,, \\ H_0({{\bf k}}+\ {{\bf Q}})&=& \left[g_1'v_{1\parallel}\left( p_{\parallel} - p_o\right)+ g_2'v_{1\perp}p_\perp \right]\tau_a + g_2 v_2 p_\perp \tau_b\,, \nonumber\\\end{aligned}$$ \[loop2\] where $g_{1(2)}'$ are + or - signs. We refer to these as “model-2” loops. The extra $p_{\parallel}$ in the $\tau_a$ term changes the pseudospin texture near a NL, but does not affect the topological properties associated with the Berry phase. Examples of both types of NLs will be given in Section \[modelsec\]. The above two types of loops respond differently to the interactions, as shown in the following sections. Normaly, one should expect that a perturbation arises that will lift the degeneracy between nested FSs. The perturbation may result from interactions and, in a normal system, usually takes the form of some charge or spin wave with the wave vector ${{{\bf Q}}}$. Also, superconducting pairing between fermions in different NLs will be considered. In the rest of the paper we shall set to unity the velocity prefactors in the Hamiltonians (\[loop1\]) and (\[loop2\]), as they are not really necessary for the analysis that follows. Density wave phases {#densec} =================== Interaction and mean-field theory --------------------------------- We introduce a Hubbard interaction, $$\begin{aligned} \hat U &=& U\sum_{{\bf r}}\hat n_1({{\bf r}})\hat n_2({{\bf r}})\,, \label{localU}\end{aligned}$$ where the indices $1,2$ refer to the orbital degree of freedom. Doing a mean field theory decoupling, the interaction takes the form $$\begin{aligned} \hat U_{MF} = U \sum_{{\bf r}}\left[ \langle n_1({{\bf r}}) \rangle \hat n_2({{\bf r}}) + \hat n_1({{\bf r}})\langle n_2({{\bf r}})\rangle - \langle n_1({{\bf r}}) \rangle \langle n_2({{\bf r}})\rangle \right]\,. \label{mfloop}\end{aligned}$$ A pseudospin density wave (PSDW) phase with the same nesting wave vector ${{{\bf Q}}}$ is characterized by $$\begin{aligned} \langle n_j({{\bf r}})\rangle = \frac 1 2 n + \bar m (-1)^j\cos({{{\bf Q}}}\cdot{{\bf r}})\,,\end{aligned}$$ where $\bar m$ is the amplitude and $j=1,2$. Although this type of ordering describes an imbalance in orbital occupation, it is not a charge density wave (CDW) because the charge at site ${{\bf r}}$ is spatially constant, $n$. Omitting the factor $(-1)^j$, then a true CDW is obtained. We introduce the annihilation operator $\hat\psi_j({{\bf r}}) $, at point ${{\bf r}}$, with pseudospin index $j$. Then, Eq. (\[mfloop\]) can be rewritten as: $$\begin{aligned} &&\hat U_{eff} = \frac{Un}{2} \sum_{{\bf r}}\left( \hat\psi_1^\dagger({{\bf r}}) \hat\psi_2^\dagger({{\bf r}}) \right) \tau_0 \left( \begin{array}{c} \hat\psi_1({{\bf r}}) \\ \hat\psi_2({{\bf r}}) \end{array} \right)\nonumber\\ &&+\ U \bar m \sum_{{\bf r}}\cos({\boldsymbol Q}\cdot{{\bf r}}) \left( \hat\psi_1^\dagger({{\bf r}}) \hat\psi_2^\dagger({{\bf r}}) \right) \tau_3 \left( \begin{array}{c} \hat\psi_1({{\bf r}}) \\ \hat\psi_2({{\bf r}}) \end{array} \right)\nonumber\\ &=& \frac{Un}{2} \sum_{{\bf k}}\left( \hat c_{{{\bf k}},1}^\dagger \hat c_{{{\bf k}},2}^\dagger \right) \tau_0 \left( \begin{array}{c} \hat c_{{{\bf k}},1}\\ \hat c_{{{\bf k}},2} \end{array} \right) \nonumber\\ &+& \frac{U \bar m}{2} \sum_{{\bf k}}\left[ \left( \hat c_{{{\bf k}}+{{\bf Q}},1}^\dagger \hat c_{{{\bf k}}+{{\bf Q}},2}^\dagger \right) \tau_3 \left( \begin{array}{c} \hat c_{{{\bf k}},1}\\ \hat c_{{{\bf k}},2} \end{array} \right) + \left( \hat c_{{{\bf k}},1}^\dagger \hat c_{{{\bf k}},2}^\dagger \right) \tau_3 \left( \begin{array}{c} \hat c_{{{\bf k}}+{{\bf Q}},1}\\ \hat c_{{{\bf k}}+{{\bf Q}},2} \end{array} \right)\right]\nonumber\\ \label{UEff}\end{aligned}$$ Replacing $\tau_3\rightarrow\tau_0$ in Eq. (\[UEff\]), we can describe a true CDW. We write the effective Hamiltonian $H_{eff}({{\bf k}})$ matrix in operator basis $( \hat c_{{{\bf k}},1}, \hat c_{{{\bf k}},2} \hat c_{{{\bf k}}+{{\bf Q}},1}, \hat c_{{{\bf k}}+{{\bf Q}},2}) \equiv( \boldsymbol c_{{{\bf k}}} \ \boldsymbol c_{{{\bf k}}+{{\bf Q}}} )$ and introduce a factor $\frac 1 2$ to avoid double counting of momenta in the BZ: $$\begin{aligned} \hat H_{eff}&=& \frac 1 2 \sum_{{\bf k}}\left( \boldsymbol c^\dagger_{{{\bf k}}}\ \boldsymbol c^\dagger_{{{\bf k}}+{{\bf Q}}} \right) H_{eff}({{\bf k}}) \left( \begin{array}{c} \boldsymbol c_{{{\bf k}}} \\ \boldsymbol c_{{{\bf k}}+{{\bf Q}}} \end{array}\right)\,, \\ H_{eff}({{\bf k}}) &=& \left( \begin{array}{cc} H_0({{\bf k}}) & U\bar m \tau_\alpha\\ U\bar m \tau_\alpha & H_0({{\bf k}}+{{\bf Q}}) \end{array}\right) \,, \label{martizHloops} \end{aligned}$$ where $\alpha=0$ for CDW, or $\alpha=3$ for PSDW. The mean field equations for this Hamiltonian are derived in Appendix \[APMF\]. The effective Hamiltonian Eq. (\[martizHloops\]) is by no means restricted to the case of a local interaction as in Eq. (\[localU\]). A nearest-neighbor interaction, for instance, would produce an effective Hamiltonian of the same form, but where the bare interaction parameter would be multiplied by a ${{\bf Q}}$-dependent form factor which could still be denoted by “$U$”. The nesting property of the Fermi surface leads to a divergence of the susceptibilities in momentum space at the nesting wavevector[@gruner]. This always leads to a density wave with momentum $\bf Q$, described by the mean field couplings $<\hat c_{{{\bf k}}}^\dagger\hat c_{{{\bf k}}+{{\bf Q}}}>$, and the relevant interaction “$U$” would be the Fourier component of the interaction for the nesting wavevector. model-1 loops {#model1_general} ------------- We now introduce the Pauli matrices $t_\mu$ operating in loop space $({{\bf k}},{{\bf k}}+{{\bf Q}})$. For type-1 models the unperturbed double loop Hamiltonian has the form $$\begin{aligned} H_{eff}^0({{\bf k}}) = \left( p_{\parallel} - p_o \right) t_i\tau_a+ p_\perp t_j\tau_b -\delta\ t_3 \label{H0eff}\end{aligned}$$ where $i,j$ can only take values 0 or 3. The effective Hamiltonian (\[martizHloops\]) can be written as $$H_{eff}({{\bf k}})= H_{eff}^0({{\bf k}}) + U\bar m\ t_1 \tau_\alpha \label{umgap}$$ Supose that $\delta=0$, that is, degenerate loops at perfect compensation. It is clear that if the perturbing term $U\bar m t_1 \tau_\alpha$ anticommutes with only one term of $H_{eff}^0({{\bf k}}) $ then the resulting system still is a double NL semimetal. On the other hand, if $U\bar mt_1\tau_\alpha$ anticommutes with $H_{eff}^0({{\bf k}})$ then the resulting system is a gapped insulator, and if $\left[U\bar mt_1 \tau_\alpha, H_{eff}^0({{\bf k}})\right]=0$ then the original loops are shifted and a metallic phase arises with torus-shaped FSs, one of them hole-like, and the other electron-like. Next we establish a criterion based on how a unitary transformation maps one loop onto the other. If the Hamiltonian has PT symmetry, one can always find a rotation through a Pauli matrix $\tau_\beta$ that maps one model-1 loop at ${{\bf k}}$ into the other at ${{\bf k}}+{{\bf Q}}$: $$\begin{aligned} H_0({{\bf k}}) = \tau_{\beta} H_0({{\bf k}}+\ {{\bf Q}}) \tau_{\beta} \,. \label{rotj} \end{aligned}$$ It is then convenient to apply a unitary transformation to the effective Hamiltonian in Eq. (\[martizHloops\]) according to $$\begin{aligned} A H_{eff}({{\bf k}}) A^\dagger &=& \left( \begin{array}{cc} H_0({{\bf k}}) -\delta & U\bar m \tau_\alpha\tau_{\beta}\\ U \bar m \tau_{\beta}\tau_\alpha & H_0({{\bf k}}) +\delta \end{array} \right) \label{rotatedloops}\\ A&=&\left( \begin{array}{cc} 1 & 0 \\ 0 &\tau_{\beta} \end{array} \right)\,.\label{Amatrix} \end{aligned}$$ The energy spectrum can be obtained by performing appropriate rotations on the matrix (\[rotatedloops\]), as shown in Appendix \[appa\]. We list all the four possibilities as follows. 1. If $\tau_\alpha \tau_{\beta}=1$, the spectrum reads: $$\begin{aligned} E=\pm \sqrt{ \left( p_{\parallel} - p_o \right)^2+p_\perp^2} \pm \sqrt{U^2\bar m^2 + \delta^2} \,,\label{CDWk0}\end{aligned}$$ (uncorrelated $\pm$ signs). In this case, $H_0({{\bf k}})=\tau_\alpha H_0({{\bf k}}+{{\bf Q}})\tau_\alpha$. The density wave produces a “level repulsion” effect by introducing (increasing) an energy splitting between the degenerate (non-degenerate) NLs. The density wave phase has two toroidal FSs, one hole-like and one electron-like. 2. In the case where $\tau_\alpha \tau_{\beta}\propto\tau_a$, the energy spectrum is: $$\begin{aligned} E^2 &=& \left( p_{\parallel} - p_o \right)^2 + p_\perp^2+ U^2\bar m^2 +\delta^2 \nonumber\\ &\pm& 2\sqrt{\left( p_{\parallel} - p_o \right)^2(U^2\bar m^2 + \delta^2)+ p_\perp^2\delta^2}\,,\label{CDWka}\end{aligned}$$ which yields two NLs at $p_\perp=0$, $p_{\parallel}=p_o\pm\sqrt{U^2\bar m^2 + \delta^2}$. As $p_\parallel$ can only take a positive value, the loop with the minus sign will shrink into a point and vanish when $\sqrt{U^2\bar m^2 + \delta^2}$ becomes larger then $p_o$. 3. For $\tau_\alpha \tau_{\beta}\propto\tau_b$ we get: $$\begin{aligned} E^2 &=& \left( p_{\parallel} - p_o \right)^2 + p_\perp^2+ U^2\bar m^2 +\delta^2 \nonumber\\ &\pm& 2\sqrt{ p_\perp^2(U^2\bar m^2 + \delta^2)+ \left( p_{\parallel} - p_o \right)^2\delta^2}\,.\label{CDWkb}\end{aligned}$$ This spectrum also gives two NLs, given by $p_\parallel = p_o$, $p_\perp=\pm \sqrt{U^2\bar m^2 + \delta^2}$. Unlike the previous case, these two loops only move along the $p_\perp$ direction when tuning $U$ or $\delta$, while their radii remain unchanged. 4. For the case $\tau_\alpha \tau_{\beta}\propto\tau_{c(\neq a,b)}$ we have $$\begin{aligned} E^2 &=& U^2\bar m^2 + \left[ \sqrt{ \left( p_{\parallel} - p_o \right)^2+p_\perp^2} \pm \delta \right]^2 \,,\label{CDWkc}\end{aligned}$$ which is then fully gapped. This is the case where $H_0({{\bf k}})=-\tau_\alpha H_0({{\bf k}}+{{\bf Q}})\tau_\alpha$. At half filling (zero chemical potential), all the above energy dispersions lead to a lowering of the energy for $U\bar m\neq 0$, so, the density wave phase is energetically favorable. In a single NL, the density of states vanishes linearly at the chemical potential, so the broken symmetry phase appears only for $U$ above a finite critical value, $U_{cr}$. In any case if the phase transition is second order, then $U\rightarrow U_{cr}^+ \Rightarrow\ \bar m\rightarrow 0$. The PSDW cases occur for $U>U_{cr}>0$ but the CDW ordering requires negative $U<U_{cr}<0$, hence an attractive interaction (see Appendices \[APMF\] and \[MFloop\]). From Eq. (\[rotj\]) we see that $\tau_\alpha H_0({{\bf k}}) \tau_\alpha = \tau_\alpha\tau_{\beta} H_0({{\bf k}}+\ {{\bf Q}}) \tau_{\beta}\tau_\alpha$ and therefore, the density-wave phase is a nodal line semi-metal if $$H_0({{\bf k}}) \neq \pm \tau_\alpha H_0({{\bf k}}+\ {{\bf Q}})\tau_\alpha \,,\label{critCDW}$$ where $\alpha=0$ for CDW, or $\alpha=3$ for PSDW. For model-1 loops we can make the following observations regarding symmetry. The Hamiltonian (\[rotatedloops\]) for $\delta=0$ is chiral as it anti-commutes with the operator $\tau_{c\neq a,b}$, and contains the degenerate NLs. This chiral symmetry can be broken by a term of the form: [*(i)*]{} $t_\mu\tau_c$ which may fully gap the spectrum, or yield a semimetal, depending on its exact form; [*(ii)*]{} $t_\mu\tau_0$ which shifts the original loops and leads to a metallic spectrum. On the other hand, a term of the form $t_j\tau_a$ or $t_j\tau_b$ ($j=1,2,3$), preserves the chiral symmetry and yields the NL semimetal even if $\delta t_3 \neq 0$ is already present, as shown in Appendix \[appa\]. model-2 loops {#model2_general} ------------- As we shall see, the criteria (\[critCDW\]) do not always apply to model-2 loops. The type-2 loop Hamiltonian at $\mathbf{k}$ is (omitting velocity prefactors): $$\begin{aligned} H_0({{\bf k}})&=& \left( p_{\parallel} - p_o + p_\perp\right)\tau_a + p_\perp \tau_b\,,\end{aligned}$$ and the loop at ${{\bf k}}+ {{\bf Q}}$ can always be related to that in ${{\bf k}}$ by either: [*(i)*]{} a rotation through a Pauli matrix if $g'_1g'_2=1$ in Eq. (\[loop2\]); or [*(ii)* ]{} a reflection in the loop plane if $g'_1g_2'=-1$. If [*(i)*]{} holds, then one can again rotate the effective Hamiltonian according to equations (\[rotatedloops\])-(\[Amatrix\]), and the resulting spectra can be obtained from Eqs (\[CDWk0\])-(\[CDWkc\]) for model-1 loops, with the replacement $p_{\parallel} - p_o\rightarrow p_{\parallel} - p_o+p_\perp$. But in case [*(ii)*]{} the two NLs can be related through a reflection in the NL’s plane, $$\begin{aligned} H_0({{\bf k}}) &=& {\cal R} H_0({{\bf k}}+\ {{\bf Q}}) {\cal R}^\dagger\,,\\ {\cal R} &=& (p_\perp\rightarrow - p_\perp) \tau_{\beta}\,, \end{aligned}$$ \[reflection\] which corresponds to the following cases, depending on $\tau_{\beta}$: $$\begin{aligned} \begin{aligned} & H_0({{\bf k}}+\ {{\bf Q}})=\\ &= \left( p_{\parallel} - p_o - p_\perp\right)\tau_a - p_\perp \tau_b\equiv H_0(-k_\perp) &\beta&=0;\\ &= \left( p_{\parallel} - p_o - p_\perp\right)\tau_a + p_\perp \tau_b &\beta&=a;\\ &= \left[ -\left( p_{\parallel} - p_o\right) + p_\perp\right]\tau_a - p_\perp \tau_b & \beta&=b;\\ &= \left[ -\left( p_{\parallel} - p_o\right) + p_\perp\right]\tau_a + p_\perp \tau_b & \beta&=c\neq a,b\,. \end{aligned}\end{aligned}$$ We apply the same rotation to the effective Hamiltonian, using Eqs. (\[rotatedloops\])-(\[Amatrix\]): $$\begin{aligned} A H_{eff}({{\bf k}}) A^\dagger &=& \left( \begin{array}{cc} H_0({{\bf k}}) & U\bar m \tau_\alpha\tau_{\beta} \\ U \bar m \tau_{\beta}\tau_\alpha & H_0(-k_\perp) \end{array} \right)\label{difficult}\,. \end{aligned}$$ For finite $\delta$ one cannot write the energy dispersion in closed form. We analytically deal with the degenerate case at perfect compensation, $\delta=0$, below, and show numerical results for nonzero $\delta$ in Fig.\[fig1\]. The figure shows the two inner bands of Hamiltonians (\[mode2\_case1\]), (\[mode2\_case2\]), (\[mode2\_case3\]), and (\[mode2\_case4\]) in the two-dimensional space of $(p_{\parallel} ,p_\perp)$. A NL for $U\bar m = \delta=0$ then looks like a Dirac cone at point $(p_o ,0)$. The splitting of the original NLs can be seen as the appearance of two Dirac cones in the plot. For finite $\delta$, the Dirac cone axis is tilted. Similarly to the discussion for model-1 loops, we also list all the four possibilities. 1. If $\tau_\alpha\tau_{\beta}=1$: $$\begin{aligned} A H_{eff}({{\bf k}}) A^\dagger &=& \left( p_{\parallel} - p_o \right)\tau_a + p_\perp t_3\tau_a + p_\perp t_3\tau_b\nonumber\\ &+& U\bar m \ t_1 - \delta t_3\,,\label{mode2_case1} \end{aligned}$$ For $\delta=0$ (perfect compensation) the spectrum obeys $$\begin{aligned} E^2 = p_\perp^2 + \left[ \sqrt{p_\perp^2 + (U\bar m)^2} \pm \left( p_{\parallel} - p_o\right) \right]^2 \end{aligned}$$ which has two nodal lines for $p_\perp=0$ and $p_{\parallel} - p_o=\pm U\bar m$. By turning on $\delta$, the two NLs are tilted along the $p_{\perp}$ direction, and move along the $p_{\parallel}$ direction, as shown in Fig. \[fig1\](a). It can also be seen from Eq. (\[mode2\_case1\]) that if $p_\perp=0$ the dispersion relation has two nodal lines: $|p_{\parallel} - p_o|=\sqrt{U^2\bar m^2+\delta^2}$. Therefore, one of the loops will shrink into the origin as $p_{\parallel}\rightarrow 0$ when $\delta^2+U^2\overline{m}^2\rightarrow p_0^2$, and become gapped for larger $\delta$. 2. If $\tau_\alpha\tau_{\beta}\propto\tau_a$: $$\begin{aligned} A H_{eff}({{\bf k}}) A^\dagger &=& \left( p_{\parallel} - p_o \right)\tau_a + p_\perp t_3\tau_a + p_\perp t_3\tau_b\nonumber\\ &+& \epsilon_{k\alpha a}U\bar m \ t_2\tau_a - \delta t_3\,,\label{mode2_case2} \end{aligned}$$ For $\delta=0$ (perfect compensation) the spectrum obeys $$\begin{aligned} E^2 &=& \left( p_{\parallel} - p_o\right)^2 + 2 p_\perp^2 +(U\bar m)^2 \nonumber\\ & \pm & 2 \sqrt{ \left( p_{\parallel} - p_o\right)^2 \left( p_\perp^2 +U^2\bar m^2\right) + p_\perp^2 U^2\bar m^2 } \end{aligned}$$ which, for $\delta=0$, has the same two nodal lines as in the previous case, and behavior of these lines with nonzero $\delta$ is also identical. However, these loops show a quadratic dispersion along the $p_\perp$ direction, as shown in Fig. \[fig1\](b). 3. If $\tau_\alpha\tau_{\beta}\propto\tau_b$: $$\begin{aligned} A H_{eff}({{\bf k}}) A^\dagger &=& \left( p_{\parallel} - p_o \right)\tau_a + p_\perp t_3\tau_a + p_\perp t_3\tau_b\nonumber\\ &+& \epsilon_{k\alpha b}U\bar m \ t_2\tau_b - \delta t_3\,, \label{mode2_case3}\\ E^2 &=& p_\perp^2 + \left( \sqrt{ \left( p_{\parallel} - p_o\right)^2 + U^2\bar m^2} \pm p_\perp \right)^2 \end{aligned}$$ which is a fully gapped insulator, and remains gapped with nonzero $\delta$ \[Fig. \[fig1\](c)\]. 4. If $\tau_\alpha\tau_{\beta}\propto\tau_c(\neq a,b)$: $$\begin{aligned} A H_{eff}({{\bf k}}) A^\dagger &=& \left( p_{\parallel} - p_o \right)\tau_a + p_\perp t_3\tau_a + p_\perp t_3\tau_b\nonumber\\ &+& \epsilon_{k\alpha c}U\bar m \ t_2\tau_c - \delta t_3\,,\label{mode2_case4}\end{aligned}$$ which for $\delta=0$ has the spectrum $$\begin{aligned} E^2 &=& \left( p_{\parallel} - p_o\right)^2 + 2 p_\perp^2 + U^2\bar m^2 \nonumber\\ &\pm& 2p_\perp\sqrt{ \left( p_{\parallel} - p_o\right)^2 + 2U^2\bar m^2 }\end{aligned}$$ with two nodal lines at $p_{\parallel} - p_o=0$ and $\sqrt{2}p_\perp=\pm U\bar m$. Unlike the nodal lines in the first two cases, these two are tilted along $p_{\parallel}$ and move away from each other along $p_{\perp}$ for $\delta\neq 0$. ![The energy dispersion of the two inner bands of the spectrum. Panels (a)-(d) correspond to the four cases in Eqs. (\[mode2\_case1\]), (\[mode2\_case2\]), (\[mode2\_case3\]), and (\[mode2\_case4\]) respectively. The energy offset $\delta$ is: (a1)-(d1) $\delta=0$; (a2)-(d2) $\delta=0.5$; and (a3)-(d3) $\delta=1$. The other parameters are: $p_o=1$, and $U\bar m =0.5$. []{data-label="fig1"}](fig1a.pdf "fig:"){width="1\linewidth"} ![The energy dispersion of the two inner bands of the spectrum. Panels (a)-(d) correspond to the four cases in Eqs. (\[mode2\_case1\]), (\[mode2\_case2\]), (\[mode2\_case3\]), and (\[mode2\_case4\]) respectively. The energy offset $\delta$ is: (a1)-(d1) $\delta=0$; (a2)-(d2) $\delta=0.5$; and (a3)-(d3) $\delta=1$. The other parameters are: $p_o=1$, and $U\bar m =0.5$. []{data-label="fig1"}](fig1b.pdf "fig:"){width="1\linewidth"} ![The energy dispersion of the two inner bands of the spectrum. Panels (a)-(d) correspond to the four cases in Eqs. (\[mode2\_case1\]), (\[mode2\_case2\]), (\[mode2\_case3\]), and (\[mode2\_case4\]) respectively. The energy offset $\delta$ is: (a1)-(d1) $\delta=0$; (a2)-(d2) $\delta=0.5$; and (a3)-(d3) $\delta=1$. The other parameters are: $p_o=1$, and $U\bar m =0.5$. []{data-label="fig1"}](fig1c.pdf "fig:"){width="1\linewidth"} ![The energy dispersion of the two inner bands of the spectrum. Panels (a)-(d) correspond to the four cases in Eqs. (\[mode2\_case1\]), (\[mode2\_case2\]), (\[mode2\_case3\]), and (\[mode2\_case4\]) respectively. The energy offset $\delta$ is: (a1)-(d1) $\delta=0$; (a2)-(d2) $\delta=0.5$; and (a3)-(d3) $\delta=1$. The other parameters are: $p_o=1$, and $U\bar m =0.5$. []{data-label="fig1"}](fig1d.pdf "fig:"){width="1\linewidth"} Example models {#examples} ============== In this section we investigate the density wave phases in several specific lattice models which have been widely studied in the literature. The examples in Secs. \[model1\_example\] and \[model2\_example\] fall into the two types of NLs discussed in previous sections. The remainder of this section will be devoted to model beyond the simplest two-band cases. A model-1 loop example {#model1_example} ---------------------- A lattice model with two “model-1" loops can be described by the Hamiltonian, $$\begin{aligned} H_0({{\bf k}})&=& \left( \cos k_x + \cos k_y - m \right)\tau_a + \cos{k_z} \tau_b\label{model1}\,.\end{aligned}$$ When $|m|<2$, the system has two NLs at $\cos{k_x}+\cos{k_y}-m=0$, in two parallel planes $k_z=\pm \pi/2$. They are nested by the vector $\mathbf{Q}=(0,0,\pi)$, and can be mapped to each other as $$\begin{aligned} H_0({{\bf k}})=\tau_aH_0({{\bf k}}+{\bf Q})\tau_a\,.\end{aligned}$$ Introducing a Hubbard interaction, the effective Hamiltonian can be written as $$\begin{aligned} H_{eff}({{\bf k}}) &=& \left( \cos k_x + \cos k_y - m \right)t_0\tau_a + \cos{k_z} t_3\tau_b \nonumber\\ &+& U\bar m\ t_1\tau_{\alpha}\,.\end{aligned}$$ We analyze first the CDW case ($\tau_\alpha=\tau_0$). Following the discussion in the previous Section, this condition corresponds to the second case in Sec. \[model1\_general\], where each NL gets split into two. Indeed, from the commuting relations between different terms, we can obtain the energy dispersion as $$E_{CDW}= \pm \sqrt{\cos^2 {k_z} + \left[ \cos k_x + \cos k_y - m \pm U\bar m \right]^2 }\,,$$ and the split NLs are given by $$\begin{aligned} \cos{k_z} = 0\,,\mbox{and}\ \cos k_x + \cos k_y = m \pm U\bar m.\end{aligned}$$ In the PSDW case ($\tau_\alpha=\tau_3$), different choices of $a$ and $b$ in Eq. (\[model1\]) will lead to different phases of the system. In such case, the effective Hamiltonian reads $$\begin{aligned} H_{eff}({{\bf k}}) &=& \left( \cos k_x + \cos k_y - m \right)t_0\tau_a + \cos{k_z} t_3\tau_b \nonumber\\ &+& U\bar m\ t_1\tau_3,\end{aligned}$$ and the possibilities are summarized in Table \[tab1\], and listed explicitly as follows. a$\backslash$ b 1 2 3 ----------------- ------- ------- -------- 1 X loops gapped 2 loops X gapped 3 metal metal X : Possibilites for $\tau_a$, $\tau_b$ matrices in the loop model (\[model1\]). And the resulting different outcomes of the PSDW phase.[]{data-label="tab1"} In the case of $\tau_a=\tau_3$, the energy spectrum reads $$\begin{aligned} E_{PSDW}&=&\pm\sqrt{ \left( \cos k_x + \cos k_y - m \right)^2 + \cos {k_z} ^2 }\pm U\bar m \nonumber\\\end{aligned}$$ with uncorrelated $\pm$ signs. This is the first case in Sec. \[model1\_general\], and the ordered phase is metallic. If $a,b\neq 3$, the spectrum reads $$\begin{aligned} E_{PSDW}^2&=& \left( \cos k_x + \cos k_y - m \right)^2 + \left( \cos{k_z} \pm U\bar m \right)^2\,,\end{aligned}$$ where each original NL splits into two with different $k_z$, as in the third cases in Sec. \[model1\_general\]. Finally, when $\tau_b=\tau_3$, the spectrum takes the form, $$\begin{aligned} E^2&=& \left( \cos k_x + \cos k_y - m \right)^2 + \cos {k_z} ^2 + U^2\bar m^2\,.\end{aligned}$$ Thus the system is fully gapped by a nonzero $U \bar m$ and becomes an insulator, as in the fourth case in Sec. \[model1\_general\]. A model-2 loop example {#model2_example} ---------------------- By including $\cos{k_z}$ in the $\tau_a$ term in Eq. (\[model1\]), we obtain a system with “model-2" loops, described by $$\begin{aligned} H_0({{\bf k}})= \left( \cos k_x + \cos k_y + \cos k_z- m \right)\tau_a + \cos k_z \tau_b.\label{model2}\end{aligned}$$ This system has two parallel NLs with $k_z=\pm\pi/2$ when $-2<m<2$. The continuous approximation for these two loops, as in Eqs. (\[loop2\]), satisfies $g'_1=1$, and $g_2=g'_2=-1$. Thus, the effective Hamiltonian including the Hubbard interaction is given by $$\begin{aligned} H_{eff} &=& \left( \cos k_x + \cos k_y - m \right)t_0\tau_a + \cos k_z t_3\tau_a\nonumber\\ &+& \cos k_z t_3\tau_b + U\bar m t_1\tau_{\alpha}\,. \label{16}\end{aligned}$$ For the CDW case ($\tau_{\alpha}=\tau_0$), the spectrum reads $$\begin{aligned} E^2_{CDW}&=& \left[ \cos k_x + \cos k_y - m \pm \sqrt{U^2\bar m^2+\cos^2{k_z}} \right]^2\nonumber\\ &&+\cos {k_z}^2 \,,\end{aligned}$$ thus each of the original NLs splits into two. The new NLs are given by $$\begin{aligned} \cos{k_z} = 0\,, \hspace{0.2cm} \cos k_x + \cos k_y = m \pm U\bar m.\end{aligned}$$ In the case of PSDW ordering ($\tau_{\alpha}=\tau_3$), there are three possible outcomes as summarized in Table \[tab2\]. In the case of $\tau_a=\tau_3$, the energy spectrum reads $$E^2_{PSDW}=(\cos k_x+\cos k_y-m)^2+2\cos^2 k_z+U^2\bar m^2$$ $$\begin{aligned} \pm2\sqrt{(\cos k_x+\cos k_y-m)^2\left[ \cos^2 k_z + U^2\bar m^2 \right] + (U\bar m\cos k_z)^2 }\,,\nonumber\\\end{aligned}$$ where each NL splits into two as in the second case in Sec. \[model2\_general\]. The condition of the NLs is the same as for the CDW case; however, as discussed in the previous section \[Fig. \[fig1\](b)\], these NLs have a quadratic dispersion along $k_z$. a$\backslash$ b 1 2 3 ----------------- ------- ------- -------- 1 X loops gapped 2 loops X gapped 3 loops loops X : Possibilites for $\tau_a$, $\tau_b$ matrices in the loop model (\[model2\]). And the resulting different outcomes of the PSDW phase.[]{data-label="tab2"} If $\tau_b=\tau_3$ then the spectrum reads $$\begin{aligned} E_{PSDW}^2&=& \left[ \cos k_z\pm \sqrt{ U^2\bar m^2+ \left( \cos k_x + \cos k_y -m\right)^2} \right]^2\nonumber\\ &+& \cos^2k_z \,,\end{aligned}$$ which is an insulating phase as in the third case in Sec. \[model2\_general\]. The remaining possibility, $a,b\neq 3$, where the spectrum reads $$\begin{aligned} E^2_{PSDW}&=&(\cos k_x+\cos k_y-m)^2+2\cos^2 k_z+U^2\bar m^2\nonumber\\ &&\pm2\cos k_z\sqrt{ (\cos k_x+\cos k_y-m)^2 +2U^2\bar m^2 }\,,\nonumber\\\end{aligned}$$ yields two NLs given by $$\begin{aligned} \cos{k_z} = \pm U\bar m /\sqrt{2}\,, \ \cos k_x + \cos k_y = m\,.\end{aligned}$$ This is the splitting along $k_z$, as in the fourth case in Sec. \[model2\_general\]. Finally, if we add an extra term $H_{\delta}=\delta\sin{k_z}\tau_0$ to the original Hamiltonian of Eq. (\[model2\]), we can induce a energy offset $\delta$ of the two original NLs, and tilt the resulting NLs after the Hubbard interaction is introduced. Nested $\mathbb{Z}_2$ NLs {#sec:Z2} -------------------------- We have hitherto considered examples of two-band NLSMs, which verify our results for general two-band models. If an extra degree of freedom, say, a (pseudo)spin-1/2 subspace, is introduced, the Hilbert space is enlarged and the possibility of nodal lines carrying a $\mathbb{Z}_2$ monopole charge arises, which must be created in pairs[@Z2_loop]. It is beyond the scope of this paper to extend the general analysis of Sec III. Instead, we explicitly consider a recent four-band model for $Z_2$ NLs [@Z2_loop] and study density-wave order due to nesting. The model reads $$\begin{aligned} H_0({{\bf k}}) &=& \sin k_x \tau_0\sigma_1+ \sin k_y \tau_2\sigma_2+ \sin k_z \tau_0\sigma_3 + m \tau_1\sigma_1\,,\nonumber\\ \label{Z2_H}\end{aligned}$$ and has the spectrum $$\begin{aligned} E_0^2({{\bf k}})&=& \left( \sqrt{\sin^2k_x + \sin^2k_y}\pm m \right)^2 + \sin^2k_z\,.\label{spectrum_Z2}\end{aligned}$$ There are eight NLs, centered at momenta ${{\bf k}}$ with Cartesian components $k_i=0,\pi$ for $i=x,y,z$. We introduce a Hubbard interaction in four-dimensional space assuming that the repulsion exists between the two orbitals in subspace of $\boldsymbol\sigma$: $$\begin{aligned} \hat U&=& U \sum_{{{\bf r}}, \nu}n_{\nu 1}({{\bf r}}) n_{\nu 2}({{\bf r}}) = \frac{U}{2} \sum_{{{\bf r}}} n({{\bf r}})\tau_0\sigma_1 n({{\bf r}})\,,\end{aligned}$$ so that the remaining index $\nu=1,2$ for the two components in the subspace of $\boldsymbol\tau$ produces a two-fold degeneracy. In the mean-field approximation, the interaction reads (apart from unimportant constants) $$\begin{aligned} \hat U_{eff}&=& U\bar m \sum_{{{\bf k}}} \hat c_{{{\bf k}}+{{\bf Q}}}^\dagger \tau_0\sigma_z \hat c_{{{\bf k}}} \,,\end{aligned}$$ where we defined the PSDW as: $$\langle n_{\nu j}({{\bf r}})\rangle = \frac{n}{2} + \bar m (-1)^j\cos({{\bf Q}}\cdot{{\bf r}}) \,,$$ The effective 8x8 Hamiltonian has the same form as that in Eq. (\[martizHloops\]), but the anti-diagonal blocks are now written as $U\bar m \tau_0\sigma_3$. Two of the original $\mathbb{Z}_2$ loops are now coupled by the interaction. Such coupling provides an extra pseudospin-1/2 subspace, and the interaction term breaks the SU(2) symmetry in this space. As a result, the pair of NLs may either survive, shrink to point nodes, or be gapped out by the PSDW. Depending on vector ${{\bf Q}}$, the effective Hamiltonian takes the form of $$\begin{aligned} H_{eff} &=& \sin k_x t_{\alpha_x} \tau_0\sigma_1 + \sin k_y t_{\alpha_y} \tau_2\sigma_2 + \sin k_z t_{\alpha_z} \tau_0\sigma_3 \nonumber\\ &+&m t_0 \tau_1\sigma_1 + U\bar m t_1 \tau_0\sigma_3\,,\end{aligned}$$ where the Pauli matrix $t_{\alpha_i}$ equals $t_0$ ($t_3$) for $Q_i=0$ ($Q_i=\pi$). Thus, different choices of $Q_i$ determine whether each term commutes or anticommutes with the interaction term $U\bar m t_1 \tau_0\sigma_3$. The possibilities with different choices of ${{\bf Q}}$ are summarized as follows. ![The effect of a PSDW on the $\mathbb{Z}_2$ loops for different nesting vectors ${{\bf Q}}$. (a) to (d) show four different cases of ${{\bf Q}}$ where the resulting system is still a semimetal. The red lines are the original NLs of Hamiltonian $H_0({{\bf k}})$ in Eq. (\[Z2\_H\]), whereas the blue lines and stars are the nodal lines or points in the PSDW phase. The parameters are: $m=\sqrt{2}/2$ and $U\bar m=0.5$. We only plot for $k_i\in [-\pi/2,\pi/2]$, as the graph repeats itself with a period of $\pi$.\[fig2\]](fig2.pdf){width="1\linewidth"} 1. ${{\bf Q}}=(1,1,1)\pi$: $$\begin{aligned} H_{eff} &=& \sin k_x t_3 \tau_0\sigma_1 + \sin k_y t_3 \tau_2\sigma_2 + \sin k_z t_3 \tau_0\sigma_3 \nonumber\\ &+& m t_0 \tau_1\sigma_1 + U\bar m t_1 \tau_0\sigma_3\,, \\ E^2 &=& \left( \sqrt{\sin^2 k_x + \sin^2k_y} \pm \sqrt{m^2 +U^2\bar m^2}\right)^2 + \sin^2k_z\,. \nonumber\\ \label{E111}\end{aligned}$$ The spectrum is composed of four two-fold degenerate bands, and has the same NLs as the original $H_0$, albeit with enlarged radius \[Fig. \[fig2\](a)\]: $$\begin{aligned} k_z=0,\pi; \ \sqrt{\sin^2 k_x + \sin^2k_y} = \sqrt{m^2 +U^2\bar m^2}\,. \label{eq56}\end{aligned}$$ Since the energy dispersions in Eq. (\[E111\]) are two-fold degenerate, the NLs are four-fold degenerate. 2. ${{\bf Q}}=(1,1,0)\pi$: $$\begin{aligned} H_{eff} &=& \sin k_x t_3 \tau_0\sigma_1 + \sin k_y t_3 \tau_2\sigma_2 + \sin k_z t_0 \tau_0\sigma_3 \nonumber\\ &+& m t_0 \tau_1\sigma_1 + U\bar m t_1 \tau_0\sigma_3\,,\end{aligned}$$ which has the same NLs as in Eq. (\[eq56\]), and as shown in FIg. \[fig2\](b), despite some minor variation of the spectrum near the NLs. 3. ${{\bf Q}}=(1,0,1)\pi$: $$\begin{aligned} H_{eff} &=& \sin k_x t_3 \tau_0\sigma_1 + \sin k_y t_0 \tau_2\sigma_2 + \sin k_z t_3 \tau_0\sigma_3 \nonumber\\ &+& m t_0 \tau_1\sigma_1 + U\bar m t_1 \tau_0\sigma_3\,.\end{aligned}$$ The full spectrum also has the two-fold degeneracy. while the NLs are gapped in most regions, leaving only pairs of four-fold degenerate points at $$\begin{aligned} \sin k_y = \sin k_z=0; \ \sin k_x = \pm \sqrt{m^2 +U^2\bar m^2}\,,\end{aligned}$$ as shown in Fig. \[fig2\](c). We note that the case ${{\bf Q}}=(0,1,1)\pi$ has a similar spectrum, which can be obtained from the above by performing the substitution $k_x\leftrightarrow k_y$. 4. ${{\bf Q}}=(1,0,0)\pi$: $$\begin{aligned} H_{eff} &=& \sin k_x t_3 \tau_0\sigma_1 + \sin k_y t_0 \tau_2\sigma_2 + \sin k_z t_0 \tau_0\sigma_3 \nonumber\\ &+& m t_0 \tau_1\sigma_1 + U\bar m t_1 \tau_0\sigma_3\,.\end{aligned}$$ Energy zeros are obtained if two conditions are simultaneously satisfied: $$\begin{aligned} \sum_{i=1}^3 \sin^2k_i = U^2\bar m^2 + m^2\\ m\sin k_z=\pm U\bar m \sin k_y\,.\end{aligned}$$ Geometrically, one can think of the first condition as defining a spherical surface, for small ${{\bf k}}$, and the second one as two planes. The intersection between the surface and the planes yields two NLs, obtained by rotating the original loops (in the XY plane) around the $x$ axis in opposite directions. These two NLs are given by different pairs of energy bands, and thus form a nodal chain \[Fig. \[fig2\](d)\]. We note that spectrum for the case ${{\bf Q}}=(0,1,0)\pi$ is similar to the case ${{\bf Q}}=(\pi,0,0)$, and differs only in the interchange $k_x\leftrightarrow k_y$. The nodal chain is obtained from the two original NLs by a rotation around the $y$ axis. 5. ${{\bf Q}}=(0,0,1)\pi$ $$\begin{aligned} H_{eff} &=& \sin k_x t_0 \tau_0\sigma_1 + \sin k_y t_0 \tau_2\sigma_2 + \sin k_z t_3 \tau_0\sigma_3 \nonumber\\ &+& m t_0 \tau_1\sigma_1 + U\bar m t_1 \tau_0\sigma_3\,.\end{aligned}$$ In this case the system is a fully gapped insulator. Nested NLs in Dirac systems {#Diracsec} =========================== In this Section we study density wave phases in four-band spinfull Hamiltonians with NLs. The spin degree of freedom allows us to distinguish two types of ordered phases: true and hidden spin density waves (SDWs). In subsection \[Diracperturb\] we consider NLs obtained from a perturbed[@Burkov; @mitschell] Dirac Hamiltonian. Spin degenerate loops --------------------- The simplest way to go from a Weyl to a Dirac loop is to introduce spin degeneracy, $$H_0({{\bf k}})\ \rightarrow\ H_0({{\bf k}}) \sigma_0$$ where $\sigma_\mu$ acts in spin space. We assume Hubbard repulsion between two fermions having opposite spins in the same orbital (labeled by the index $j$): $$\begin{aligned} \hat U = \sum_{{{\bf r}}, j=1,2} \hat n_{j{\uparrow}}({{\bf r}})\hat n_{j{\downarrow}}({{\bf r}})\,. \label{Usigma}\end{aligned}$$ In principle one could have ordered phases with ferromagnetic (FM) or antiferromagnetic (or SDW) configurations of the spin: $$\begin{aligned} \langle n_{j\sigma}\rangle &=& \frac 1 4 n + \bar m \sigma \qquad \sigma=\pm 1\,,\qquad\mbox{Stoner FM}\\ \langle n_{j\sigma}\rangle &=& \frac 1 4 n + \bar m \sigma (-1)^j \qquad\mbox{hidden Stoner FM}\\ \langle n_{j\sigma}\rangle &=& \frac 1 4 n + \bar m \sigma \cos({{\bf Q}}\cdot{{\bf r}}) \qquad\mbox{true SDW}\\ \langle n_{j\sigma}\rangle &=& \frac 1 4 n + \bar m \sigma (-1)^j \cos({{\bf Q}}\cdot{{\bf r}}) \qquad\mbox{hidden SDW} \label{hiddenSDW} \end{aligned}$$ Because a NL’s density of states vanishes linearly with energy, the Stoner criterion precludes the FM orderings for weak interactions[@roy], and they are not related to the nesting ${{\bf Q}}$. In the following, we shall concentrate on SDW phases. Considering the hidden SDW, Eq. (\[hiddenSDW\]), the effective interaction reads: $$\begin{aligned} \hat U_{eff} &=& -U\sum_{{{\bf r}},j} \left[ \left( \frac n 4 \right)^2 - \bar m^2 \cos^2({{\bf Q}}\cdot{{\bf r}}) - \frac n 4 \sum_{\sigma} \hat\psi^\dagger_{j\sigma}({{\bf r}}) \hat\psi_{j\sigma}({{\bf r}}) \right]\nonumber\\ &+&U\bar m\sum_{{{\bf r}},j,j',\sigma\sigma'} \hat\psi^\dagger_{j\sigma}({{\bf r}})\ \tau^{jj'}_3\sigma^{\sigma\sigma'}_3 \cos({{\bf Q}}\cdot{{\bf r}})\ \hat\psi_{j'\sigma'}({{\bf r}}) \,,\label{tSDW}\end{aligned}$$ where the field operator $\hat\psi_{j\sigma}({{\bf r}})$ now includes the spin index $\sigma$. Similarly, the field operator in momentum space $\boldsymbol c_{{\bf k}}$ now denotes $ c_{{{\bf k}},j,\sigma}$ for all $j,\sigma$. The Hamiltonian matrix in $\left( \boldsymbol c^\dagger_{{{\bf k}}}\ \boldsymbol c^\dagger_{{{\bf k}}+{{\bf Q}}} \right)$ space reads (apart from unimportant constants), $$\begin{aligned} H_{eff}({{\bf k}}) &=& \left( \begin{array}{cc} H_0({{\bf k}}) & U\bar m \tau_\alpha\sigma_3 \\ U\bar m \tau_\alpha\sigma_3 & H_0({{\bf k}}+{{\bf Q}}) \end{array}\right)\,,\label{SDWDirac}\end{aligned}$$ where $\alpha=3$ describes a hidden SDW \[as in Eq. (\[tSDW\])\], and $\alpha=0$ describes a true SDW. The off-diagonal block $U\bar m t_1\tau_3\sigma_3$ has exactly the same (anti)commutation relations with the other Hamiltonian terms, as in the Weyl case of Sections \[model1\_general\] and \[model2\_general\] . All the criteria and spectra established for the Weyl case still hold, if one just replaces $\bar m\rightarrow\bar m \sigma_3$. Since this term appears as $(U\bar m)^2$ in the dispersion relations, there is no spin splitting in the spectra. We note that single antiferromagnetic NLs have been discussed in the literature[@jingwang]. Two perturbed Dirac points {#Diracperturb} -------------------------- A nodal line Dirac semimetal can be obtained starting from a pristine 3D Dirac semimetal[@mitschell] of the form $H_D({{\bf k}}) = -\tau_3{{\bf p}}\cdot\boldsymbol\sigma$ and perturbing it with terms of the form $a_\mu\tau_\mu\otimes b_\nu\sigma_\nu$. Suppose, for instance, $$H_0({{\bf k}}) = -\tau_3{{\bf p}}\cdot\boldsymbol\sigma + \tau_1\boldsymbol b\cdot\boldsymbol\sigma$$ Without loss of generality assume $\boldsymbol b \parallel \hat z$. The term $p_z\tau_3\sigma_3$ anticommutes with the others, so: $$E^2 =p_z^2 + \left( \sqrt{p_x^2+p_y^2}\pm b \right)^2 \,.$$ We note that this dispersion relation is very similar to that in Eq. (\[spectrum\_Z2\]) for the $\mathbb{Z}_2$ loops. However, the effect of the Hubbard interaction is different, as these two systems couple the two sets of (pseudo)spin-1/2 subspace in different ways. On the other hand, Dirac points described by $H_D$ above do not exist alone if an additional symmetry, such as time-reversal or inversion, is present. For instance, a time-reversal symmetry (TRS) relates two Dirac points at $-\frac 1 2 {{\bf Q}}$ and $+\frac 1 2 {{\bf Q}}$ in such a way that: $$\begin{aligned} \sigma_2H_0^*\left(\frac {{\bf Q}}2 -{{\bf k}}\right)\sigma_2 = H_0\left(-\frac {{\bf Q}}2 +{{\bf k}}\right)\,,\end{aligned}$$ it then follows that, for $\boldsymbol b=0$, $H_0\left(- {{\bf Q}}/2 +{{\bf k}}\right) =H_0\left( {{\bf Q}}/2 +{{\bf k}}\right) =H_D({{\bf k}})$. Therefore, the two unperturbed Dirac points have the same $k\cdot p$ Hamiltonian. Including the $\tau_1\boldsymbol b\cdot\boldsymbol\sigma$ term, which breaks TRS, we obtain the model, $$\begin{aligned} H_0\left(-\frac {{\bf Q}}2 +{{\bf k}}\right) &=& -\tau_3{{\bf p}}\cdot\boldsymbol\sigma +\tau_1\boldsymbol b\cdot\boldsymbol\sigma\,,\\ H_0\left(\frac {{\bf Q}}2 +{{\bf k}}\right) &=& -\tau_3{{\bf p}}\cdot\boldsymbol\sigma +\tau_1\boldsymbol b\cdot\boldsymbol\sigma\,,\end{aligned}$$ \[Tpartial\] and the effective Hamiltonian has equal diagonal blocks. A different version of the above model that would preserve TRS symmetry reads: $$\begin{aligned} H_0\left(-\frac {{\bf Q}}2 +{{\bf k}}\right) &=& -\tau_3{{\bf p}}\cdot\boldsymbol\sigma +\tau_1\boldsymbol b\cdot\boldsymbol\sigma\,,\\ H_0\left(\frac {{\bf Q}}2 +{{\bf k}}\right) &=& -\tau_3{{\bf p}}\cdot\boldsymbol\sigma -\tau_1\boldsymbol b\cdot\boldsymbol\sigma\,.\end{aligned}$$ \[Tcase\] If we now consider the role of inversion symmetry ${{\bf k}}\rightarrow -{{\bf k}}$, the two Dirac points are related by $$\begin{aligned} H_0\left(\frac {{\bf Q}}2 -{{\bf k}}\right) = H_0\left(-\frac {{\bf Q}}2 +{{\bf k}}\right) &=& -\tau_3{{\bf p}}\cdot\boldsymbol\sigma + \tau_1\boldsymbol b\cdot\boldsymbol\sigma\,,\hspace{1cm} \\ \Rightarrow H_0\left(\frac {{\bf Q}}2 +{{\bf k}}\right) &=& \tau_3{{\bf p}}\cdot\boldsymbol\sigma +\tau_1\boldsymbol b\cdot\boldsymbol\sigma\,, \label{Pcase}\end{aligned}$$ therefore, the two Dirac points have different $k\cdot p$ Hamiltonians. Next we study the effects of a hidden SDW and a true SDW for these different cases, still assuming $\boldsymbol b \parallel \hat z$. For a hidden SDW, the effective Hamiltonian for the TRS breaking model in Eq. (\[Tpartial\]) is then $$\begin{aligned} \hat H_{eff} = t_0 \left[-\tau_3{{\bf p}}\cdot\boldsymbol\sigma + b\tau_1 \sigma_3\right] + U\bar m t_1\tau_3\sigma_3\,,\end{aligned}$$ which, by inspection produces the eight band spectrum: $$E^2 = \left( - p_z\pm U\bar m \right)^2 + \left( \sqrt{p_x^2+p_y^2}\pm b \right)^2$$ where the $\pm$ signs are uncorrelated. This corresponds to splitting each loop along the $k_z$ direction. If one considers, instead, a true SDW phase, $$\begin{aligned} \hat H_{eff} = t_0 \left[-\tau_3{{\bf p}}\cdot\boldsymbol\sigma + b\tau_1\sigma_3\right]+ U\bar m t_1\tau_0\sigma_3\,,\end{aligned}$$ then there are four doubly degenerate bands: $$\begin{aligned} E^2&=& b^2 + U^2\bar m^2+ {{\bf p}}^2\nonumber\\ &\pm& 2\sqrt{ b^2\left( U^2\bar m^2 + p_x^2 + p_y^2 \right) + U^2\bar m^2p_z^2 } \label{mathspec}\end{aligned}$$ with nodal lines given by $ p_z=0\,, p_x^2 + p_y^2 = b^2 - U^2\bar m^2$. So, the initial two loops still exist but their radius shrinks. In the TRS model, Eq. (\[Tcase\]), the hidden SDW phase is described by the effective Hamiltonian: $$\begin{aligned} \hat H_{eff}&=& -t_0\tau_3{{\bf p}}\cdot\boldsymbol\sigma + bt_3\tau_1\sigma_3 + U\bar m t_1\tau_3\sigma_3\,.\end{aligned}$$ The spectrum is the same as in Eq. (\[mathspec\]). So, the initial two loops still exist but their radius shrinks. And a true SDW phase is described by the effective Hamiltonian: $$\begin{aligned} \hat H_{eff} &=& -t_0 \tau_3( p_x\sigma_1 + p_y\sigma_2) - p_z t_0\tau_3\sigma_3 + b\ t_3\tau_1\sigma_3 \nonumber\\ &+& U\bar m t_1\tau_0\sigma_3\,,\end{aligned}$$ which produces the spectrum with eight bands: $$E^2 =\left( \sqrt{p_x^2+p_y^2} \pm b \right)^2 + \left( p_z \pm U\bar m \right)^2\,,$$ where the $\pm$ signs are uncorrelated. This corresponds to splitting each loop along $p_z=\pm U\bar m$. For the case with inversion symmetry, in Eq. (\[Pcase\]), a hidden SDW phase is described by the effective Hamiltonian: $$\begin{aligned} H_{eff} ({{\bf k}})= -t_3 \tau_3{{\bf p}}\cdot\boldsymbol\sigma + b \ t_0\tau_1\sigma_3 + U\bar m t_1\tau_3\sigma_3\,, \label{PhiddenSDW}\end{aligned}$$ which produces the eight band spectrum: $$E^2 =p_z^2 + \left( \pm \sqrt{b^2 + U^2\bar m^2} \pm \sqrt{p_x^2+p_y^2} \right)^2\,,$$ (uncorrelated$\pm$ signs). This corresponds to splitting each nodal line by changing its radius. A true SDW is obtained by changing $\tau_3\rightarrow\tau_0$ in the last term of Eq. (\[PhiddenSDW\]). The resulting spectrum, $$E^2 =p_z^2 + \left( \sqrt{p_x^2+p_y^2} \pm b \pm U\bar m \right)^2\,,$$ (with uncorrelated $\pm$ signs) also has NLs given by $p_z=0$, $\sqrt{p_x^2+p_y^2} =|b\pm U\bar m|$. In the remaining case, where $H_0(-{{\bf Q}}/2 + {{\bf k}})=-H_0({{\bf Q}}/2 + {{\bf k}})$, a hidden SDW phase is described by the effective Hamiltonian: $$\begin{aligned} H_{eff} ({{\bf k}})= t_3 \left[ -\tau_3{{\bf p}}\cdot\boldsymbol\sigma + b \tau_1\sigma_3 \right]+ U\bar m t_1\tau_3\sigma_3\,,\end{aligned}$$ which, by inspection produces the eight band spectrum: $$E^2 =p_z^2 + \left( \sqrt{p_x^2+p_y^2} \pm b \pm U\bar m \right)^2\,,$$ with uncorrelated$\pm$ signs. Therefore, each loop splits in the radial direction. The true SDW is described by the effective Hamiltonian: $$\begin{aligned} \hat H_{eff} = t_3 \left[ -\tau_3{{\bf p}}\cdot\boldsymbol\sigma + b \tau_1\sigma_3 \right]+ U\bar m t_1\tau_0\sigma_3\,,\end{aligned}$$ which, by inspection produces the eight band spectrum: $$E^2 = p_z^2 + \left( \pm \sqrt{p_x^2+p_y^2} \pm \sqrt{ b^2 + U^2\bar m^2} \right)^2 \,,$$ where the $\pm$ signs are uncorrelated. Again, this corresponds to splitting each nodal loop in the radial direction. Superconductivity {#supersec} ================= When considering a single Weyl NL, the pairing block of the Bogolyubov-deGennes (BdG) matrix in the particle-hole basis[@sacramento1; @sacramento2; @beri], takes the form $$\hat \Delta({{\bf k}}) = \left[ d_0({{\bf k}})\tau_0+ \boldsymbol d({{\bf k}})\cdot\boldsymbol\tau \right] i\tau_2 \,,$$ and fermionic statistics imposes that $\hat \Delta({{\bf k}}) = \hat \Delta^T(-{{\bf k}})$. Close to the nodal lines the 3D momentum, ${{\bf p}}=\hbar{{\bf k}}$, can be parametrized as $$\begin{aligned} p_x &=& (p_0 + \tilde p\cos\phi) \cos\theta\\ p_y &=& (p_0 + \tilde p\cos\phi) \sin\theta \,,\\ p_z&=&p_\perp=\tilde p\sin\phi\,, \end{aligned}$$ \[pcoordinates\] which is to be inserted in the $k\cdot p$ loops models. Here, $\theta$ is the azimuthal angle along the loop, $\tilde p$ is the radius of a torus involving the NL, and the angle $\phi$ wraps around the latter[@Nandkishore1]. Note that, according to Eq. (\[pcoordinates\]), momentum inversion ${{\bf p}}\rightarrow -{{\bf p}}$ is equivalent to $\theta\rightarrow\theta+\pi$ and $\phi\rightarrow -\phi$, while reflection in the loop’s plane, $p_z\rightarrow -p_z$, is equivalent to $\phi\rightarrow -\phi$. In the semimetal case (undoped, or compensated case) the FS reduces to the NL and ${{\bf p}}$ reduces to the angle $\theta$ on the loop. In the doped case, any point on the torus shaped FS can be labeled by two angles, $\theta,\phi$. The functions $ d_0({{\bf k}})$ and $\boldsymbol d({{\bf k}})$ describe (pseudo-spin) singlet and triplet pairing, respectively. One can expand the singlet pairing function quite generally as $$d_0({{\bf k}}) = \sum_{l_1,l_2} e^{il_1\theta}\left[ \Delta_{l_1l_2}\cos(l_2\phi) + \tilde \Delta_{l_1l_2}\sin(l_2\phi) \right]\,.$$ An analogous expansion can be written for $\boldsymbol d({{\bf k}})$. If there are two nested Weyl loops, then an additional loop label must be introduced and the Pauli matrix $t_\mu$ operates in the two-dimensional loop space. For a two-loop system then, we write the pairing matrix as $$\hat \Delta({{\bf k}}) = \left[ d_0({{\bf k}})\tau_0+ \boldsymbol d({{\bf k}})\cdot\boldsymbol\tau \right] i\tau_2 t_\mu\,.$$ The BdG Hamiltonian matrix in the particle-hole basis has the form $$H({{\bf k}})= \left( \begin{array}{cc} \hat\Xi ({{\bf k}})& \hat \Delta ({{\bf k}})\\ \hat\Delta^\dagger ({{\bf k}}) & -\hat\Xi^T(-{{\bf k}}) \end{array}\right) \label{BdGmatrix}$$ with $\hat\Xi= diag(H_1, H_2)$ . The Hamiltonians $H_{1(2)}$ are the $k\cdot p$ Weyl NL models. The total Hamiltonian is then $$\begin{aligned} \hat H=\frac 1 2 \sum_{{{\bf k}}}{\boldsymbol c}^\dagger H({{\bf k}}) {\boldsymbol c}\,,\end{aligned}$$ where $ {\boldsymbol c}=(\hat {\boldsymbol c}_{{{\bf k}},1},\ \hat {\boldsymbol c}_{{{\bf k}},2},\ \hat {\boldsymbol c}_{-{{\bf k}},1}^\dagger,\ {\boldsymbol c}_{-{{\bf k}},2}^\dagger )^T$. If the two NLs are centered at BZ points $\pm{{\bf Q}}/2$ respectively, then the inter-NL pairing is the “usual” pairing between opposite momenta, and we shall take this to be the case. If not, then the Cooper pair would have a finite quasi-momentum (a Fulde-Ferrel-Larkin-Ovchinnikov state[@LOFF1; @LOFF2; @LOFF3]). The cases $\mu=0,1,3$, are different from the case $\mu=2$ regarding the parity of the functions $ d_0({{\bf k}})$ and $\boldsymbol d({{\bf k}})$. In the cases $\mu=1,2$, electrons on different loops are being paired: an electron $({{\bf k}},1)$ is being paired with another $(-{{\bf k}},2)$. The cases $\mu=0,3$ describe intra-NL pairing, where the scattering of two particles from one NL into the other may be included, and $id_0\tau_2t_3$ would describe sign-reversed s-wave pairing, analogous to that in pnictide superconductors[@BangChoi]. Inter-NL pairing with $\mu=1$ (interloop triplet pairing), requires $d_0$ to be even and ${\boldsymbol d}$ to be odd function of ${{\bf k}}$; if $\mu=2$ (interloop singlet), then $d_0$ and ${\boldsymbol d}$ have the opposite parities. The BdG matrix decouples into two blocks each associated with the vector spaces $(\hat c_{{{\bf k}},1}, \hat c_{-{{\bf k}},2}^\dagger)^T$ and $(\hat c_{{{\bf k}},2}, \hat c_{-{{\bf k}},1}^\dagger)^T$, respectively. Since we expect a fully gapped excitation spectrum to have higher condensation energy than a nodal spectrum, we shall examine the cases where $d_0$ and $\boldsymbol d$ are constant on the FS (in the $\mu=1$ and $\mu=2$ cases, respectively). If TRS holds, then these order parameters must also be real. Model-1 loops {#model-1-loops} ------------- Assuming a positive energy offset, $\delta$, the interband pairing occurs between the electronic toroidal FS from the $H_1-\delta$ loop, and the hole-like FS from the $H_2+\delta$ loop. As in previous literature, this is best done by considering projective form factors[@wangye; @Nandkishorepro] onto the conduction or valence band. Let $U_{1(2)}$ be the unitary matrices which diagonalize $H_{1(2)}$, so that $U_sH_sU_s^\dagger=\sqrt{\left( |{{\bf p}}_\parallel| - p_0 \right)^2 + p_\perp^2}\ \tau_3\equiv \tilde p\tau_3$ for $s=1,2$. The positive and the negative branches are the conduction and valence bands, respectively. Because for model-1 loops there is always a Pauli matrix $\tau_\beta$ such that $H_1=\tau_\beta H_2\tau_\beta$, it then follows that $U_2=U_1\tau_\beta$. We can apply this same unitary transformation to the BdG matrix in Eq. (\[BdGmatrix\]) as: $$\begin{aligned} \left( \begin{array}{cc}\Lambda & 0\\ 0 & \Lambda^*(-{{\bf k}})\end{array}\right)H({{\bf k}}) \left( \begin{array}{cc}\Lambda^\dagger & 0\\ 0 &\Lambda^T(-{{\bf k}}) \end{array}\right) \nonumber\\ = \left( \begin{array}{cc} \tilde p\tau_3t_0 -\delta\tau_0 t_3 & \Lambda \hat\Delta \Lambda^T(-{{\bf k}}) \\ \Lambda^*(-{{\bf k}}) \hat\Delta^\dagger \Lambda^\dagger& -\tilde p\tau_3t_0+\delta \tau_0 t_3 \end{array}\right) \label{UBdGU}\end{aligned}$$ where $\Lambda=diag(U_1, U_2)$. The off-diagonal pairing block is then $\Lambda({{\bf k}}) \hat\Delta({{\bf k}}) \Lambda^T(-{{\bf k}})$. For $\delta>0$, only the pairing between the conduction band of $H_1$ and the valence band of $H_2$ is considered. From the BdG matrix in Eq. (\[UBdGU\]) we obtain the submatrix operating in this two-fold subspace as: $$\begin{aligned} H^{FS}=\left( \begin{array}{cc} \tilde p -\delta & \Delta_{FS}({{\bf k}}) \\ \Delta_{FS}^*({{\bf k}}) & \tilde p -\delta \end{array}\right) \label{condval1}\end{aligned}$$ where $\Delta_{FS}({{\bf k}})$ is the pairing function on the FS which, from Eq. (\[UBdGU\]) and for $\mu=1$ reads: $$\begin{aligned} \Delta_{FS}({{\bf k}}) = \left[ U_1({{\bf k}}) \left(d_0+{\boldsymbol d}\cdot{\boldsymbol\tau}\right) i\tau_2U_2^T(-{{\bf k}})\right]_{12} \label{pairingfunction}\end{aligned}$$ It is then clear from Eq. (\[condval1\]) that the spectrum is $E=\tilde p -\delta \pm |\Delta_{FS}({{\bf k}})|$, and is gapless. At finite doping, no gapped state is to be expected from FS interloop pairing between non-degenerate model-1 Weyl loops. The situation is different for the degenerate ($\delta=0$) case, however, where the FS is composed of two nodal lines. From Eq. (\[UBdGU\]) and $t_\mu=t_1$ we obtain a BdG matrix restricted to the subspace $\left( U_1({{\bf k}}) \hat{\boldsymbol c}_{{{\bf k}},1} \,, U_2^*(-{{\bf k}}) \hat{\boldsymbol c}_{-{{\bf k}}, 2}^\dagger \right)$, as $$\begin{aligned} H_{12}'=\left( \begin{array}{cc} \tilde p\tau_3 & U_1(d_0 + {\boldsymbol d}\cdot{\boldsymbol \tau})i\tau_2 U_2^T \\ -iU_2^*\tau_2 (d_0 + {\boldsymbol d}\cdot{\boldsymbol \tau}) U_1^\dagger & -\tilde p\tau_3 \end{array}\right) \label{UBU2}\end{aligned}$$ For the sake of definiteness we consider the NL models with $\tau_a=\tau_1,\tau_b=\tau_2$, so that $$\begin{aligned} H_1(\phi) &=& \tilde p\left( \cos\phi\tau_1 + \sin\phi\tau_2\right)\,,\label{h1phi}\\ U_1({{\bf k}})&=& \frac{1}{\sqrt{2}}\left( \begin{array}{cc} 1 & e^{-i\phi}\\ -1 & e^{-i\phi} \end{array}\right)\,.\label{u1phi}\\end{aligned}$$ We note that all the other $(\tau_a,\tau_b)$ cases can be related to this through a suitable rotation in pseudo-spin space. From Eq. (\[u1phi\]), one can see that $U_1^\dagger({{\bf k}}) =U_1^T(-{{\bf k}}) $. Interloop pairing is described by the off-diagonal block in Eq (\[UBU2\]): $$\begin{aligned} && U_1({{\bf k}}) \left(d_0+{\boldsymbol d}\cdot{\boldsymbol\tau}\right) i\tau_2\tau_\beta^TU_1^T(-{{\bf k}})=\nonumber\\ &=& \left(\begin{array}{cc} id_0\sin\phi + id_2+d_3\cos\phi & d_0\cos\phi + d_1 + id_3\sin\phi\\ -d_0\cos\phi + d_1 - id_3\sin\phi & - id_0\sin\phi + id_2-d_3\cos\phi \end{array}\right) \,, \nonumber \\ &=& \left(\begin{array}{cc} d_3 +id_2\cos\phi- id_1\sin\phi & -d_0 - d_1\cos\phi- d_2\sin\phi \\ -d_0 + d_1\cos\phi + d_2\sin\phi & d_3 -id_2\cos\phi+ id_1\sin\phi \end{array}\right) \,, \nonumber\\ &=& \left(\begin{array}{cc} -id_0 -id_1\cos\phi-id_2\sin\phi & d_1\sin\phi - d_2\cos\phi +id_3\\ - d_1\sin\phi + d_2\cos\phi +id_3 & -id_0 +id_1\cos\phi+id_2\sin\phi \end{array}\right)\,, \nonumber\\ &=& \left(\begin{array}{cc} -d_0\cos\phi - d_1-id_3\sin\phi & -id_0\sin\phi - id_2 -d_3\cos\phi\\ id_0\sin\phi - id_2 +d_3\cos\phi & d_0\cos\phi - d_1+id_3\sin\phi \end{array}\right)\,,\nonumber\\ \label{udelu}\end{aligned}$$ for the cases $\beta=0,1,2,3$, respectively. Note that for the case $t_\mu=t_2$, we simply have to multiply both sides of Eq. (\[udelu\]) by $-i$. For $\mu=1$ a fully gapped FS can only happen for constant $d_0$ because $\boldsymbol d$ is an odd function and must have nodes on the NLs. In this case, only for $\beta=2$ a gapped spectrum is obtained: $E^2 = \tilde p^2 + d_0^2$. For $\mu=2$ (interloop singlet) and constant real $\boldsymbol d$ there are more possibilities. If $\beta=0$ a fully gapped spectrum $E^2 = \tilde p^2 + d_2^2$; if $\beta=1$ a fully gapped spectrum $E^2 = \tilde p^2 + d_3^2$; for $\beta=3$ the fully gapped spectrum $E^2 = \tilde p^2 + d_1^2$. Gapped spectra result from intraband pairing. Interband pairing leads to nodal spectra for the same reason as in the $\delta>0$ case. Model-2 loops {#model-2-loops} ------------- In a model-2 loop we replace Eqs. (\[h1phi\])-(\[u1phi\]) with $$\begin{aligned} H_1(\phi) &=& \tilde p\left[ (\cos\phi+ \sin\phi)\tau_1 + \sin\phi\tau_2\right]\,,\label{h21phi}\\ U_1({{\bf k}})&=& \frac{1}{\sqrt{2}}\left( \begin{array}{cc} e^{i \omega} & 1\\ e^{-i \omega}& -1 \end{array}\right)\,,\label{u21phi}\end{aligned}$$ where $\omega=arg(e^{i\phi}+\sin\phi)$. If $H_1 = \tau_\beta H_2\tau_\beta$, then the conclusions are the same as for model-1 loops, with the replacement $\tilde p\rightarrow \tilde p|e^{i\phi}+\sin\phi|$. We now consider the case where the two loops are related through the reflection operation in Eq. (\[reflection\]). Because the reflection implies $\phi\rightarrow -\phi$, the energy dispersions are different for $H_1$ and $H_2$. In the non-degenerate case ($\delta>0$), $H^{FS}$ now takes the form $$\begin{aligned} H^{FS}=\left( \begin{array}{cc} \tilde p |e^{i\phi}+\sin\phi|-\delta & \Delta_{FS}({{\bf k}}) \\ \Delta_{FS}^*({{\bf k}}) & \tilde p|e^{i\phi}-\sin\phi| -\delta \end{array}\right)\end{aligned}$$ and the resulting spectrum allows gapless excitations, as was the case for model-1 loops. In the degenerate case, we find it more convenient not to perform the rotation in Eq. (\[UBdGU\]), and diagonalize the original BdG matrix restricted to the subspace $\left( \hat{\boldsymbol c}_{{{\bf k}},1} \,, \hat{\boldsymbol c}_{-{{\bf k}}, 2}^\dagger \right)$, instead. In this subspace, the two diagonal blocks of the BdG matrix, which follow from Eq. (\[BdGmatrix\]), are $ H_1(\phi)$ and $$\begin{aligned} -H_2^T(-\phi)&=&- \tau_\beta H_1^T(\phi)\tau_\beta\,, \label{H1H2phi}\end{aligned}$$ which follows from (\[h21phi\]) and the reflection operation that relates both loops: $H_2(\phi)=\tau_\beta H_1(-\phi)\tau_\beta$. For $t_\mu=t_1$ (interloop triplet) the BdG reads: $$\begin{aligned} H_{12}'=\left( \begin{array}{cc} H_1\phi) & (d_0 + {\boldsymbol d}\cdot{\boldsymbol \tau})i\tau_2 \\ -i\tau_2 (d_0 + {\boldsymbol d}\cdot{\boldsymbol \tau}) & - \tau_\beta H_1^T(\phi)\tau_\beta \end{array}\right) \label{UBU3}\end{aligned}$$ We identify the TRS cases where the excitation spectrum is fully gapped. For constant $d_0$ and ${\boldsymbol d}=0$, the gapped spectra are obtained for $\beta=1$ and $\beta=3$, respectively: $$\begin{aligned} \beta=1: E^2 &=&d_0^2 + {\tilde p}^2 \sin^2\phi + \left( d_0\pm \tilde p| \sin\phi+\cos\phi | \right)^2\,,\nonumber \\ \\ \beta=3: E^2 &=&d_0^2 + {\tilde p}^2 \left[ \sin^2\phi + (\sin\phi+\cos\phi )^2\right]\,.\end{aligned}$$ In the case of interloop singlet $t_\mu=t_2$, we consider $d_0=0$ and constant $\boldsymbol d$. Gapped spectra exist for: $\beta=0$ and nonzero $d_3$; $\beta=1$ and nonzero $d_2$; $\beta=2$ and nonzero $d_1$. All these cases have similar spectra: $$\begin{aligned} \beta=0: E^2 &=&d_3^2 + {\tilde p}^2 \left[ \sin^2\phi + (\sin\phi+\cos\phi )^2\right] \,,\ \ \\ \beta=1: E^2 &=&d_2^2 + {\tilde p}^2 \left[ \sin^2\phi + (\sin\phi+\cos\phi )^2\right]\,,\ \ \\ \beta=2: E^2 &=&d_1^2 + {\tilde p}^2 \left[ \sin^2\phi + (\sin\phi+\cos\phi )^2\right]\,.\ \\end{aligned}$$ Pairing between Dirac loops --------------------------- Including the spin degree of freedom, we may discuss the pairing between spin degenerate loops $H_1\otimes\sigma_0$ and $H_2\otimes\sigma_0$ described by the BdG matrix: $$\begin{aligned} H_{12,s}'=\left( \begin{array}{cc} H_1\otimes\sigma_0 & (d_0 + {\boldsymbol d}\cdot{\boldsymbol \tau})i\tau_2 (t_\mu)_{12}\sigma_s \\ -i\sigma_s\tau_2 (d_0 + {\boldsymbol d}\cdot{\boldsymbol \tau})(t_\mu)_{21} & -H_2^T \otimes\sigma_0 \end{array}\right)\nonumber\\\end{aligned}$$ Whatever the choice for $s=1,2,3$, $H_{12,s}'$ decouples in subblocks for which the results obtained above for Weyl systems can be applied. The (anti)symmetric property of the matrices $t_\mu$, $\sigma_s$ will determine whether the functions $d_0$,$\boldsymbol d$ should be odd of even: if for instance, $s=1,3$ then the parity of $d_0$,$\boldsymbol d$ is as in the Weyl case; if, however, $s=2$ (spin singlet), then the parities should be reversed. Summary and Conclusions {#concsec} ======================= We have described broken symmetry phases of nested Weyl and Dirac NLs that are induced by a short range interaction. We made a systematic analysis for two-band Hamiltonians with PT symmetry, where the two nested Weyl NLs can be mapped onto each other through a rotation or reflection operator. Charge and (pseudo)spin density waves always lower the energy and the broken symmetry phase can be metallic, semimetallic or insulating, depending on the operator that maps the the initial NLs onto each other, and on whether they enjoy a local reflection symmetry in the loop plane. This outcome does not depend on whether the initial system is semimetallic or metallic (when the initial FS is composed of two toroidal FSs, one hole- and one electron-like). If the initial system is semimetallic, spontaneous symmetry breaking requires a finite interaction which is attractive for CDWs and repulsive for PSDWs. We have also studied specific four-band models, including the $\mathbb{Z}_2$ NLs, spin degenerate Dirac systems, and NL’s derived from perturbed spinful Dirac nodal points. The PSDW phases from $\mathbb{Z}_2$ NLs include nodal point and nodal chain semimetals. Fully gapped superconducting phases from electron pairing in different NLs (interloop pairing), with TRS, have been found. They include all possibilities of triplet and singlet pairing in loop space and spin space. There has recently been an intensive search for topological semimetal materials. Given that point nodes tend to appear in pairs for symmetry reasons, it is conceivable that suitable engineering can produce double NLs. Indeed, a recent proposal for realizing point nodes (Dirac or Weyl), and pairs of NLs by strain engineering in SnTe and GeTe is relevant here[@LauOrtix]. Another recent proposal concerns layered ferromagnetic rare-earth-metal monohalides [*LnX*]{} ([*Ln*]{}=La, Gd; [*X*]{}=Cl, Br) and a pair of mirror-symmetry protected nodal lines in La[*X*]{} and Gd[*X*]{}[@Nie]. Also, splitting of Dirac rings into pairs of Weyl rings by spin-orbit interaction in InNbS$_2$ has been proposed[@Du]. Two groups of Dirac nodal rings have been experimentally detected[@Lou] in ZrB$_2$. However, the detection of pairs of NLs at the Fermi level is presently still lacking. We have not addressed the competition between different orderings or interactions, but such an extension of our work might be relevant to real materials. We have also neglected the effect of the long-ranged tail of the Coulomb interaction which could be present if the starting system is a NL semimetal with the screening radius diverging near the Fermi level. In this respect, the study in Ref\[[@roy]\] for a single NL suggests that the critical interaction strength for orderings where a fully gapped spectrum arises could be lowered. Ordered phases[@gruner], such as orbital and/or spin density waves, can be detected through neutron scattering, or resonant soft x-ray scattering[@xray]. The band structure itself may be studied with angle-resolved photoemission spectroscopy. Acknowledgments {#acknowledgments .unnumbered} =============== M.A.N.A. acknowledges partial support from Fundação para a Ciência e Tecnologia (Portugal) through Grant No. UID/CTM/04540/2013, and the hospitality of Computational Science Research Center, Beijing, China, where this work was initiated. M.A.N.A. would like to thank Vítor R. Vieira, Bruno Mera, and Tilen Cadez for a discussion. Energy spectrum for Hamiltonian (\[rotatedloops\]) {#appa} ================================================== Taking $\alpha=0$, (CDW case), for instance, and $(a,b)=(1,2)$, the rotated effective Hamiltonian in Eq. (\[rotatedloops\]) then reads: $$A H_{eff}({{\bf k}}) A^\dagger = \left( p_{\parallel} - p_o \right) t_0\tau_1+ p_\perp t_0\tau_2 +U\bar m\ t_1 \tau_{\beta}- \delta\ t_3\tau_0\,. \label{AHAdelta}$$ We perform a suitable rotation on the Hamiltonian $AH_{eff}A^\dagger$ in Eq. (\[AHAdelta\]) so that its energy spectrum can be written down by inspecting the (anti)commutation relations among its terms. If $\beta=1$ or 2, we introduce a SU(2) rotation in $t_\mu$ space so that in the end, only the matrix $t_3$ appears. The required rotation is $$W=\cos\frac \theta 2 -it_2\tau_{\beta} \sin\frac \theta 2 \,,$$ with the rotation angle $\theta$ given by $$\begin{aligned} \sin\theta &=& \frac{U\bar m}{\sqrt{U^2\bar m^2 + \delta^2}}\,,\\ \cos\theta &=& \frac{\delta}{\sqrt{U^2\bar m^2 + \delta^2}}\,,\end{aligned}$$ so that the rotated Hamiltonian for $\beta=1$ reads $$\begin{aligned} WAH_{eff}A^\dagger W^\dagger &=& \left( p_{\parallel} - p_o \right)\tau_1 + p_\perp\cos\theta\ \tau_2 \nonumber\\ &+& p_\perp\sin\theta\ t_2\tau_3\nonumber\\ &-& \sqrt{U^2\bar m^2 + \delta^2}\ t_3\,,\end{aligned}$$ and the energy spectrum obeys $$\begin{aligned} E^2&=& \left[ \sqrt{ \left( p_{\parallel} - p_o \right)^2 + p_\perp^2\cos^2\theta}\pm \sqrt{U^2\bar m^2 + \delta^2} \right]^2 \nonumber\\ &+& p_\perp^2\sin^2\theta\,,\end{aligned}$$ which is equivalent to Eq. (\[CDWka\]). For $\beta=2$ we have $$\begin{aligned} WAH_{eff}A^\dagger W^\dagger &=& \left( p_{\parallel} - p_o \right)\cos\theta\ \tau_1 + p_\perp\tau_2 \nonumber\\ &-& \left( p_{\parallel} - p_o \right)\sin\theta\ t_2\tau_3\nonumber\\ &-& \sqrt{U^2\bar m^2 + \delta^2}\ t_3\,,\end{aligned}$$ and the energy spectrum obeys $$\begin{aligned} E^2&=& \left[ \sqrt{ \left( p_{\parallel} - p_o \right)^2 \cos^2\theta+ p_\perp^2}\pm \sqrt{U^2\bar m^2 + \delta^2} \right]^2 \nonumber\\ &+& \left( p_{\parallel} - p_o \right)^2\sin^2\theta\,,\end{aligned}$$ which is equivalent to Eq. (\[CDWkb\]). In the case $\beta=3$, it is preferable to perform a SU(2) rotation in $\tau$ space in order to eliminate one of the Pauli matrices $\tau$. To this aim, we introduce $$R_c=\cos\frac \theta 2 -i\tau_3\sin\frac \theta 2 \,,$$ with $$\begin{aligned} \sin\theta &=& \frac{ p_{\parallel} - p_o }{\sqrt{\left( p_{\parallel} - p_o \right)^2 + \delta^2}} \,,\\ \cos\theta &=& \frac{p_\perp}{\sqrt{ \left( p_{\parallel} - p_o \right)^2 + \delta^2}}\,,\end{aligned}$$ so that the rotation of the Hamiltonian now works out as, $$\begin{aligned} R_cAH_{eff}A^\dagger R_c^\dagger &=& {\sqrt{ \left( p_{\parallel} - p_o \right)^2 + \delta^2}}\ \tau_2 + U\bar m \ t_1\tau_3-\delta \ t_3\,.\nonumber\\\end{aligned}$$ More generally, if the product $\tau_\alpha\tau_{\beta}=i\epsilon_{\alpha\beta j}\tau_j$, then the above results for the energy still hold, because the appearance of the factor $i$ in the $U\bar m$ term would lead to the replacement $t_1\rightarrow t_2$ in (\[AHAdelta\]), which does not change the (anti)commutation relations among the Hamiltonian terms. Mean field treatment of PSDW/CDW {#APMF} ================================ Given the order parameter for a PSDW, $ \langle n_s({{\bf r}})\rangle = \frac 1 2 n + \bar m (-1)^s\cos({{{\bf Q}}}\cdot{{\bf r}})$, one may transform to Fourier space as $$\begin{aligned} \langle n_s({{\bf r}})\rangle =\frac 1 N\sum_{{{\bf q}}} \langle c_{{{\bf q}}s}^\dagger c_{{{\bf q}}s} \rangle + e^{i{{\bf Q}}\cdot{{\bf r}}}\langle c_{{{\bf q}}s}^\dagger c_{{{\bf q}}+{{\bf Q}}s} \rangle + e^{-i{{\bf Q}}\cdot{{\bf r}}}\langle c_{{{\bf q}}+{{\bf Q}}s}^\dagger c_{{{\bf q}}s} \rangle\,.\nonumber\\\end{aligned}$$ Using $\hat\psi_s({{\bf r}}) =\sum_{{{\bf q}}} e^{i{{\bf q}}\cdot{{\bf r}}} c_{{{\bf q}}s}/\sqrt{N}$, where $N$ is the number of momentum values in the summation, we see that the above $\langle n_s({{\bf r}})\rangle$ is obtained if $$\begin{aligned} \frac 1 N\sum_{{{\bf q}}} \langle c_{{{\bf q}}s}^\dagger c_{{{\bf q}}s} \rangle&=& \frac 1 2 n\,,\\ \frac 1 N\sum_{{{\bf q}}} \langle c_{{{\bf q}}+{{\bf Q}}s}^\dagger c_{{{\bf q}}s} \rangle = \frac 1 N\sum_{{{\bf q}}} \langle c_{{{\bf q}}s}^\dagger c_{{{\bf q}}+{{\bf Q}}s} \rangle &=& \frac 1 2 \bar m (-1)^s\,, \label{MFQ}\end{aligned}$$ for $\alpha=3$ (PSDW). If $\alpha=0$ (CDW) then the factor $(-1)^s$ should be omitted. The Hamiltonian is given by $$\begin{aligned} \hat H_{eff} = \frac 1 2\sum_{{{\bf k}}} (\hat {\boldsymbol c}_{{{\bf k}}}^\dagger \hat {\boldsymbol c}_{{{\bf k}}+{{\bf Q}}}^\dagger) H_{eff}({{\bf k}}) \left( \begin{array}{c}\hat {\boldsymbol c}_{{{\bf k}}}\\ \hat {\boldsymbol c}_{{{\bf k}}+{{\bf Q}}} \end{array} \right)\end{aligned}$$ where $\hat{\boldsymbol c}_{{{\bf k}}} = (\hat c_{{{\bf k}},1} \hat c_{{{\bf k}},2})$. We assume that $H_{eff}({{\bf k}})$ is diagonalized by a unitary matrix, $S$, so that $SH_{eff}({{\bf k}})S^\dagger$ is the diagonal matrix composed of the eigenenergies. Then, the operators $\hat{\boldsymbol\gamma}$ which destroy the elementary excitations are given by $$\begin{aligned} \hat{\boldsymbol\gamma} = S \left( \begin{array}{c}\hat {\boldsymbol c}_{{{\bf k}}}\\ \hat {\boldsymbol c}_{{{\bf k}}+{{\bf Q}}} \end{array} \right)\,. \label{gammaS}\end{aligned}$$ Following Eq. (\[MFQ\]) we can see that $$\begin{aligned} \frac 1 N\sum_{{{\bf k}}} \langle (\hat {\boldsymbol c}_{{{\bf k}}}^\dagger \hat {\boldsymbol c}_{{{\bf k}}+{{\bf Q}}}^\dagger) \left( \begin{array}{cc} 0 & \tau_\alpha \\ \tau_\alpha & 0 \end{array} \right) \left( \begin{array}{c}\hat {\boldsymbol c}_{{{\bf k}}}\\ \hat {\boldsymbol c}_{{{\bf k}}+{{\bf Q}}} \end{array} \right) \rangle = \mp 2\bar m \left\{ \begin{array}{c} \alpha=3\\ \alpha=0\end{array} \right.\end{aligned}$$ or, in the eigenbasis using (\[gammaS\]), $$\begin{aligned} &&\frac 1 N\sum_{{{\bf k}}} \langle \hat{\boldsymbol\gamma}^\dagger S \left( \begin{array}{cc} 0 & \tau_\alpha \\ \tau_\alpha & 0 \end{array} \right) S^\dagger\hat{\boldsymbol\gamma} \rangle\ =\mp 2\bar m \left\{ \begin{array}{c} \alpha=3\\ \alpha=0\end{array} \right. \nonumber\\ &=& \frac 1 N\sum_{{{\bf k}}} \sum_j \left[ S \left( \begin{array}{cc} 0 & \tau_\alpha \\ \tau_\alpha & 0 \end{array} \right) S^\dagger \right]_{jj} f(E_j({{\bf k}})) \,, \label{SalphaS}\end{aligned}$$ where $f(x)=1/(1+e^{x/T})$ denotes the Fermi-Dirac distribution function, and $j=1,...,4$ denotes a band index. The energy dispersions, $E_j({{\bf k}})$, are given in the main text. However, it is more convenient to work with the transformed Hamiltonian $A H_{eff}A^\dagger$ as in the main text, which implies that all operators are similarly rotated and $S\rightarrow SA^\dagger$ above. Then, Eq. (\[SalphaS\]) can be written in the form, $$\begin{aligned} &&\frac 1 N\sum_{{{\bf k}}} \sum_j \left[ SA^\dagger \left( \begin{array}{cc} 0 & \tau_\alpha\tau_{\beta} \\ \tau_{\beta}\tau_\alpha & 0 \end{array} \right) AS^\dagger \right]_{jj} f(E_j({{\bf k}}))\nonumber\\ &=& \mp 2\bar m \left\{ \begin{array}{c} \alpha=3\\ \alpha=0\end{array} \right. \,.\nonumber\\ \label{APMFeq}\end{aligned}$$ Critical interaction $U_{cr}$ for degenerate NLs {#MFloop} ================================================ We consider a circular nodal line and use the momentum parametrization in Eq. (\[pcoordinates\]). We linearize the theory in a toroidal region surrounding the NL up to a momentum cutoff: $0<\tilde p<\tilde p_c$, $0<\theta,\phi<2\pi$. The volume element is $d^3p=(p_0 + \tilde p\cos\phi)\tilde p\cdot d\tilde p d\theta d\phi$. The number of ${{\bf k}}$ terms in the toroidal region around the NL is then given by $$\begin{aligned} N=\frac{1}{(2\pi\hbar)^3}\int d^3p &=& \frac{p_0\tilde p_c^2}{4\pi\hbar^3}\,.\label{N}\end{aligned}$$ In order to simplify the calculations, it is assumed that the dispersion relation has the same velocity, $v$, in the NL plane and perpendicular to it. Then, using $p_\parallel - p_0 = \tilde p\cos\phi$, the model-1 NL Eq. (\[loop1\].a) reads $$\begin{aligned} H_0({{\bf k}}) = v\tilde p \left( \cos\phi\tau_1 + \sin\phi\tau_2\right)\,,\end{aligned}$$ where $v=v_1=v_2$ and we shall take $\delta=0$. As before, we proceed considering the velocity $v=1$ and but shall restore it in the final result for $U_{cr}$. In the cases where $\tau_\alpha \tau_{\beta}\propto\tau_1$ and $\tau_\alpha\tau_{\beta}\propto\tau_2$ the ordered phases are semimetalic and yield similar mean field equations. In the case where $\tau_\alpha\tau_{\beta}\propto\tau_2$, for instance, the negative energy bands are $$\begin{aligned} E_{1(2)} &=&-\sqrt{ \tilde p^2\pm 2\tilde pU\bar m\sin\phi + U^2\bar m^2}\end{aligned}$$ and the l.h.s. of Eq. (\[APMFeq\]) takes the form, $$\begin{aligned} &&\frac 1 N\sum_{{{\bf k}}} \sum_j \left[ SA^\dagger \left( \begin{array}{cc} 0 & \tau_\alpha\tau_{\beta} \\ \tau_{\beta}\tau_\alpha & 0 \end{array} \right) AS^\dagger \right]_{jj} f(E_j({{\bf k}})) \nonumber\\ &=& \frac 1 N\ \int \frac{2(\tilde p\sin\phi-U\bar m) (p_0 + \tilde p\cos\phi) d^3p }{(2\pi\hbar)^3\sqrt{ \tilde p^2- 2\tilde pU\bar m\sin\phi + U^2\bar m^2}}\,.\end{aligned}$$ In the limit $U\bar m\rightarrow 0$ one can Taylor expand the integrand to first order. 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--- abstract: 'Passive scalar equation is considered in a turbulent homogeneous incompressible Gaussian velocity field. The turbulent nature of the field results in non-smooth coefficients in the equation. A strong, in the stochastic sense, solution of the equation is constructed using the Wiener Chaos, and the properties of the solution are studied. The results apply to both viscous and conservative motions.' author: - 'S. V. Lototsky' - 'B. L. Rozovskii' title: Time Evolution of a Passive Scalar in a Turbulent Incompressible Gaussian Velocity Field --- [^1] [^2] Passive Scalar in a Gaussian Field ================================== We consider the following transport equation to describe the evolution of a passive scalar $\theta$ in a random velocity field $\bv$: $$\label{eq:ps} \dot{\theta}(t,x)=0.5\nu\Delta \theta(t,x) - {\mathbf v}(t,x) \cdot \nabla \theta(t,x) +f(t,x);\ x\in \bR^d,\ d>1.$$ Our interest in this equation is motivated by the on-going progress in the study of the turbulent transport problem (E and Vanden Eijnden [@EVE], Gawȩdzki and Kupiainen [@GK], Gawȩdzki and Vergasola [@GV], Kraichnan [@Krch], etc.) We assume in (\[eq:ps\]) that $\bv=\bv(t,x)\in \bR^d$, $d\geq 2$, is an isotropic Gaussian vector field with zero mean and covariance $$E(v^i(t,x)v^j(s,y))=\delta(t-s)C^{ij}(x-y)$$ with some matrix-valued function $C=(C^{ij}(x), i,j=1, \ldots, d)$. It is well-known (see, for example, LeJan [@LeJan]) that in the physically interesting models the matrix-valued function $C=C(x)$ has the Fourier transform $\hat{C}=\hat{C}(z)$ given by $$\hat{C}(z)=\frac{A_0}{(1+|z|^2)^{(d+\alpha)/2}}\left(a\frac{zz^*}{|z|^2}+\frac{b}{d-1} \left(I-\frac{zz^T}{|z|^2}\right) \right),$$ where $z^*$ is the row vector $(z_1, \ldots, z_d)$, $z$ is the corresponding column vector, $|z|^2=z^*z$, $I$ is the identity matrix; $\alpha>0,a\geq 0,b\geq 0,A_0>0$ are real numbers. Similar to [@LeJan], we assume that $0<\alpha<2$. By direct computation (cf. [@BH]), the vector field $\bv=(v^1, \ldots, v^d)$ can be written as $$\label{eq:v} v^i(t,x)=\sum_{k\geq 0} \sigma^i_k(x)\dot{w}_k(t),$$ where $\dot{w}_k(t)$, $k\geq 1$, are independent standard Gaussian white noises and $\{\sigma_k, \ k\geq 1\}$ is a CONS in the space $H_C$, the reproducing kernel Hilbert space corresponding to the kernel function $C$. The space $H_C$ is all or part of the Sobolev space $H^{(d+\alpha)/2}(\bR^d; \bR^d)$. It follows from (\[eq:v\]) that $\sum_k \sigma_{k}^i(x)\sigma_{k}^j(y)=C^{ij}(x-y)$ for all $x,y$; in particular, $\sigma_k^i(x)\sigma_k^j(x)=C^{ij}(0)$ for all $x$. If $a>0$ and $ b>0$, then the matrix $\hat{C}$ is invertible and $$H_C=\{ f\in \bR^d: \int_{\bR^d} \hat{f}^*(z)\hat{C}^{-1}(z)\hat{f}(z)dz < \infty \} = H^{(d+\alpha)/2}(\bR^d; \bR^d),$$ because $\|\hat{C}(z)\| \sim (1+|z|^2)^{-(d+\alpha)/2}$. If $a>0$ and $b=0$, then $$H_C=\left\{ f\in \bR^d: \int_{\bR^d} |\hat{f}(z)|^2(1+|z|^2)^{(d+\alpha)/2}dz < \infty; \ zz^*\hat{f}(z)=|z|^2\hat{f}(z) \right\},$$ the subset of gradient fields in $H^{(d+\alpha)/2}(\bR^d; \bR^d)$ (those are vector fields $f$ for which $\hat{f}(z)=z\hat{F}(z)$ for some scalar $F \in H^{(d+\alpha+1)/2}$). If $a=0$ and $b>0$, then $$H_C=\left\{ f\in \bR^d: \int_{\bR^d} |\hat{f}(z)|^2(1+|z|^2)^{(d+\alpha)/2}dz < \infty; \ z^*\hat{f}(z)=0 \right\},$$ the subset of divergence free fields in $H^{(d+\alpha)/2}(\bR^d; \bR^d)$. By the embedding theorems, each $\sigma_k^i$ is a bounded continuous function on $\bR^d$; in fact, every $\sigma_k^i$ is Hölder continuous of order $\alpha/2$. In addition, being an element of the corresponding space $H_C$, each $\sigma_k$ is a gradient field if $b=0$ and is divergence free if $a=0$. To simplify the further presentation and to make the model (\[eq:ps\]) more physically relevant, we consider the divergence-free velocity field and assume that the stochastic integration is in the sense of Stratonovich. Under these assumptions, equation (\[eq:ps\]) becomes $$\label{eq:ps2s} d{\theta}(t,x)=0.5\nu\Delta \theta(t,x)dt- \sum_{k}\sigma_{k}(x)\cdot \nabla\theta(t,x) \circ dw_k(t).$$ With divergence-free functions $\sigma_k$, the equivalent Ito formulation is $$\label{eq:ps2i} d{\theta}(t,x)=0.5(\nu\Delta \theta(t,x)+C^{ij}(0)D_iD_j\theta(t,x))dt- \sigma_{k}^i(x)D_i\theta(t,x) dw_k(t).$$ In what follows, we construct a solution of (\[eq:ps2i\]) using Wiener Chaos. A Review of the Wiener Chaos ============================ Let $\mbF=(\Omega, \cF, \{\cF_t\}_{t\geq 0}, \mbP)$ be a stochastic basis with the usual assumptions. On $\mbF$ consider a collection $(w_k(t),k\geq 1,t\geq 0)$ of independent standard Wiener processes. For a fixed $0<T<\infty$, let $\cF^W_T$ be the sigma-algebra generated by $w_k(t),\ k\geq 1, \ 0<t<T$, and $L_2(\cF^W_T)$ the collection of $\cF^W_T$-measurable square integrable random variables. For the Fourier cosine basis $\{m_k,\ k\geq 1,\}$ in $L_2((0,T))$ with $$\label{eq:cosbas} m_1(t)=\frac{1}{\sqrt{T}}, \ m_k(t)=\sqrt{\frac{2}{T}}\cos\left(\frac{\pi(k-1)t}{T}\right),\ k\geq 2,$$ define the independent standard Gaussian random variables $$\xi_{ik}=\int_0^T m_i(s)dw_k(s).$$ Consider the collection of multi-indices $$\cJ=\Big\{ \alpha =(\alpha_i^k,\ i,k\geq 1),\ \alpha_{i}^k\in \{0,1,2,\ldots\},\ \sum_{i,k} \alpha_i^k<\infty \Big\}.$$ The set $\cJ$ is countable, and, for every $\alpha \in \cJ$, only finitely many of $\alpha_i^k$ are not equal to zero. For $\alpha \in {\mathcal J}$, define $$|\alpha|=\sum_{i,k} \alpha_i^k, \ \alpha!=\prod_{i,k}\alpha_i^k!,$$ and $$\xi_{\alpha}=\frac{1}{\sqrt{\alpha!}}\prod_{i,k}H_{\alpha_{i}^{k}}(\xi_{ik}),$$ where $$H_{n}(t)=e^{t^{2}/2} \frac{d^{n}}{dt^{n}}e^{-t^{2}/2}$$ is $n$-th Hermite polynomial. In particular, if $\alpha\in\cJ$ is such that $\alpha_i^k=1$ if $i=j$ and $k=l$, and $\alpha^k_i=0$ otherwise, then $\xi_{\alpha}=\xi_{jl}$. The space $L_2(\cF^W_T)$ is called the Wiener Chaos space. The $N$-th Wiener Chaos is the linear subspace of $L_2(\cF^W_T)$, generated by $\xi_{\alpha},\ |\alpha|=N$. The following is a classical results of Cameron and Martin [@CM]. \[th:CM\] The collection $\{ \xi_{\alpha},\ \alpha \in \cJ\}$ is an orthonormal basis in the space $L_2(\cF^W_T)$. In addition to the original source [@CM], the proof of this theorem can be found in many other places, for example, in [@HKPS]. By Theorem \[th:CM\] every element $v$ of $L_2(\cF^W_T)$ can be written as $$v=\sum_{\alpha \in \cJ} v_{\alpha}\xi_{\alpha},$$ where $v_{\alpha}=\mbE (v\xi_{\alpha})$. The Wiener Chaos Solution of the Passive Scalar Equation ======================================================== Using the summation convention, define the operators $\cA=0.5(\nu\Delta+C^{ij}(0)D_iD_j)$ and $\cM_k=\sigma_k^iD_i$. Assume that $\theta_0 \in L_2(\bR^d)$ and define $\theta_{\alpha}(t,x)$ by $$\label{eq:def2} \theta_{\alpha}(t)=\theta_{0}I(|\alpha|=0) + \int_0^t \cA \theta_{\alpha}(s,x)ds+ \int_0^t\sum_{i,k}\sqrt{\alpha_i^k}\cM_k \theta_{\alpha^-(i,k)}(s)m_i(s)ds.$$ Notice that for every function $f \in H^1_2(\bR^d)$, $$\sum_{k\geq 1} \|\cM_k f\|^2_{L_2(\bR^d)}=(\sigma_k^j\sigma_k^iD_if, D_jf)= (C^{ij}(0)D_if, D_jf),$$ where $(\cdot, \cdot)$ is the inner product in $L_2(\bR^d)$. Since the matrix $(C^{ij}(0), i,j=1, \ldots, d)$ is positive definite, we conclude that there exist positive numbers $c_1, c_2$ so that, for every function $f \in H^1_2(\bR^d)$, $$\label{eq:PSnorm} c_1\|\nabla f\|_{L_2(\bR^d)}^2 \leq\sum_{k\geq 1} \|\cM_k f\|^2_{L_2(\bR^d)} \leq c_2 \|\nabla f\|_{L_2(\bR^d)}^2.$$ \[th:main\] 1. For every $\nu\geq 0$ and every $t\in [0,T]$, the series $$\label{eq:PSsum} \sum_{\alpha \in \cJ} \theta_{\alpha}(t,x)\xi_{\alpha}$$ converges in $L_2(\Omega; L_2(\bR^d))$ to a process $\theta=\theta(t,x)$. 2. If $\nu>0$, then, for every $\varphi\in C^{\infty}_0(\bR^d)$, the process $\theta(t,x)$ satisfies $$\label{eq:PSsol1} \begin{split} (\theta,\varphi)(t)&=(\theta_0, \varphi)-0.5\nu\int_0^t (\nabla \theta,\nabla \varphi)(s)ds- 0.5\int_0^tC^{ij}(0)(D_i\theta, D_j\varphi)(s)ds\\ &-\int_0^t(\sigma^i_kD_i\theta,\varphi)dw_k(s) \end{split}$$ with probability one for all $t \in [0,T]$ at once, where $(\cdot, \cdot)$ is the inner product in $L_2(\bR^d)$. Also, $$\label{eq:PSnorm1} \mbE \|\theta\|_{L_2(\bR^d)}^2(t) + \nu \int_0^t\mbE\|\nabla\theta\|_{L_2(\bR^d)}^2(s)ds = \|\theta_0\|_{L_2(\bR^d)}^2.$$ 3. If $\nu=0$, then, for every $\varphi\in C^{\infty}_0(\bR^d)$, the process $\theta(t,x)$ satisfies $$\label{eq:PSsol0} (\theta,\varphi)(t)=(\theta_0, \varphi)+0.5\int_0^t C^{ij}(0)(\theta, D_iD_j\varphi)(s)ds +\int_0^t(\theta,\sigma^i_kD_i\varphi)dw_k(s)$$ with probability one for all $t \in [0,T]$ at once. Also, $$\label{eq:PSnorm0} \mbE \|\theta\|_{L_2(\bR^d)}^2(t) \leq \|\theta_0\|_{L_2(\bR^d)}^2.$$ Equalities (\[eq:PSsol1\]) and (\[eq:PSsol0\]) mean that $\theta=\theta(t,x)$ is the solution of the transport equation in the traditional sense of theory of stochastic partial differential equations, that is, it is a strong solution in the stochastic sense, satisfying the corresponding equation in the generalized function sense. The solution is also unique in the class of $L_2((0,T)\times \Omega; L_2(\bR^d)) $ random functions, because any other solution will automatically have the same Wiener Chaos expansion. The uniqueness can, in fact, be established in a much wider class of generalized random functions. The proof of Theorem \[th:main\] is based on the following lemmas. \[lm:LMR\] The system of equations (\[eq:def2\]) has a unique solution so that every $\theta_{\alpha}$ is a smooth bounded function of $x$ for $t>0$ and, if $T_t, \ t\geq 0,$ is the heat semigroup generated by the operator $0.5(\nu\Delta+C^{ij}(0)D_iD_j)$, then, for every $N\geq 0$, $$\label{eq:L2norm} \begin{split} &\sum_{|\alpha|=N} |\theta_\alpha(t,x)|^2 \\ &=\sum_{k_1, \ldots, k_N=1}^{\infty} \int_0^t \int_0^{s_N}\ldots\int_0^{s_2} |T_{t-s_N}\cM_{k_N} \ldots T_{s_2-s_1}\cM_{k_1}T_{s_1}\theta_0(x)|^2ds_1\ldots ds_N. \end{split}$$ and $$\label{eq:L2gnorm} \begin{split} &\sum_{|\alpha|=N} |\nabla\theta_\alpha(t,x)|^2 \\ &=\sum_{k_1, \ldots, k_N=1}^{\infty} \int_0^t \int_0^{s_N}\ldots\int_0^{s_2} |\nabla T_{t-s_N}\cM_{k_N} \ldots T_{s_2-s_1}\cM_{k_1}T_{s_1}\theta_0(x)|^2ds_1\ldots ds_N. \end{split}$$ [**Proof.**]{} See Proposition A.1 in [@LMR]. [$\Box$0.15in]{} \[lm:tail1\] Assume that $\nu\geq 0$. Define $\theta_N(t,x)=\sum_{n=0}^N \sum_{|\alpha|=n}\theta_{\alpha}(t,x)\xi_{\alpha}$. Then, for all $t \in [0,T]$, $$\label{eq:PStr} \begin{split} &\mbE \|\theta_N\|^2_{L_2(\bR^d)}(t)=\|\theta_0\|_{L_2(\bR^d)}^2- \nu\sum_{n=0}^N \sum_{|\alpha|=n} \int_0^t\|\nabla \theta_{\alpha}\|_{L_2(\bR^d)}^2(s)ds \\ &-\sum_{k_1, \ldots, k_{N+1}} \int_0^t\ldots\int_0^{s_2} \|\cM_{k_{N+1}}T_{s-s_N}\cM_{k_N} \ldots T_{s_2-s_1}\cM_{k_1}T_{s_1}\theta_0\|_{L_2(\bR^d)}^2ds^Nds. \end{split}$$ [**Proof.**]{} By Lemma \[lm:LMR\], after integration with respect to $x$, $$\label{eq:PSL2norm} \begin{split} &\sum_{|\alpha|=N} \|\theta_\alpha\|_{L_2(\bR^d)}^2(t) \\ &=\sum_{k_1, \ldots, k_{N}=1}^{\infty} \int_0^t \int_0^{s_N}\ldots\int_0^{s_2} \|T_{t-s_N}\cM_{k_N} \ldots T_{s_2-s_1}\cM_{k_1} T_{s_1}\theta_0\|_{L_2(\bR^d)}^2ds_1\ldots ds_N. \end{split}$$ If $F_N(t)=\sum_{|\alpha|=N} \|\theta_\alpha(t)\|_{L_2(\bR)}^2$, then $$\begin{split} &\frac{d}{dt}F_N(t) \\ &=\sum_{k_1, \ldots, k_{N}} \int_0^t \int_0^{s_{N-1}}\ldots\int_0^{s_2} \|\cM_{k_N} T_{t-s_{N-1}}\cM_{k_{N-1}} \ldots T_{s_2-s_1}\cM_{k_1} T_{s_1}u_0\|_{L_2(\bR)}^2ds^{N-1} \\ &+2\sum_{k_1, \ldots, k_{N}} \int_0^t\ldots\int_0^{s_2} \left(\cA T_{t-s_N}\cM \ldots T_{s_1}u_0, T_{t-s_N}\cM \ldots T_{s_2-s_1}\cM_{k_N} T_{s_1}u_0\right) ds^N. \end{split}$$ It remains to notice that, for every smooth function $f=f(x)$, $$2(\cA f,f)=-\nu\|\nabla f\|_{L_2(\bR)}^2- \sum_{k\geq1}\|\cM_k f\|_{L_2(\bR)}^2.$$ Equality (\[eq:PStr\]) now follows. [$\Box$0.15in]{}Notice that (\[eq:PStr\]) implies both the $L_2(\Omega;L_2(\bR^d))$ convergence of the series $$\sum_{\alpha}\theta_{\alpha}(t,x)\xi_{\alpha}$$ for every $t\in [0,T]$ and inequality (\[eq:PSnorm0\]). \[lm:tail2\] If $\nu>0$, then, for every $t\in [0,T]$, $$\label{eq:PStail} \lim_{N\to \infty} \sum_{k_1, \ldots, k_{N+1}} \int_0^t\ldots\int_0^{s_2} \|\cM_{k_{N+1}}T_{s-s_N}\cM_{k_N} \ldots T_{s_2-s_1}\cM_{k_1}T_{s_1}\theta_0\|_{L_2(\bR^d)}^2 ds^Nds=0.$$ [**Proof.**]{} Define $$\label{eq:PStail1} \begin{split} &F_N(t) \\ &= \sum_{k_1, \ldots, k_{N+1}} \int_0^t\ldots\int_0^{s_2} \|\cM_{k_{N+1}}T_{s-s_N}\cM_{k_N} \ldots T_{s_2-s_1}\cM_{k_1}T_{s_1}\theta_0\|_{L_2(\bR^d)}^2 ds^Nds. \end{split}$$ By (\[eq:PSnorm\]) and Lemma \[lm:LMR\], $F_N(t)\leq c_2 \sum_{|\alpha|=N} \int_0^t \|\nabla \theta_{\alpha}\|_{L_2(\bR^d)}^2(s)ds$. Lemma \[lm:tail1\] then implies that the series $\sum_{N\geq 0} F_N(t)$ converges for all $t \in [0,T]$. Therefore, $\lim_{N\to \infty} F_N(t)=0$ and the statement of the lemma follows. [$\Box$0.15in]{} Since (\[eq:PStr\]) and (\[eq:PStail\]) imply (\[eq:PSnorm1\]), to complete the proof of the theorem it remains to establish (\[eq:PSsol1\]) and (\[eq:PSsol0\]). The necessary arguments are similar to the proof of Theorem 3.5 in [@MR.gs] [10]{} P. Baxendale and T. E. Harris, *[Isotropic Stochastic Flows]{}*, Ann. Probab. **14** (1986), no. 4, 1155–1179. R. H. Cameron and W. T. Martin, *[ The Orthogonal Development of Nonlinear Functionals in a Series of Fourier-Hermite Functions]{}*, Ann. Math. **48** (1947), no. 2, 385–392. W. E and E. Vanden Eijden, *[Generalized Flows, Intrinsic Stochasticity, and Turbulent Transport]{}*, Proc. Nat. Acad. Sci. **97** (2000), no. 15, 8200–8205. K. Gawȩdzki and A. Kupiainen, *[Universality in Turbulence: An Exactly Solvable Model]{}*, Low-Dimensional Models in Statistical Physics and Quantum Field Theory (H. Grosse and L. Pittner, eds.), Springer, Berlin, 1996, pp. 71–105. K. Gawȩdzki and M. Vergassola, *[Phase Transition in the Passive Scalar Advection]{}*, Physica D **138** (2000), 63–90. T. Hida, H-H. Kuo, J. Potthoff, and L. Sreit, *[White Noise]{}*, Kluwer Academic Publishers, Boston, 1993. R. H. Kraichnan, *[Small-Scale Structure of a Scalar Field Convected by Turbulence]{}*, Phys. Fluids **11** (1968), 945–963. Y. LeJan and O. Raimond, *[Integration of Brownian Vector Fields]{}*, Ann. Probab. **30** (2002), no. 2, 826–873. S. Lototsky, R. Mikulevicius, and B. L. Rozovskii, *[Nonlinear Filtering Revisited: A Spectral Approach]{}*, SIAM Journal on Control and Optimization **35** (1997), no. 2, 435–461. R. Mikulevicius and B. L. Rozovskii, *[Global $L_2$-solutions of Stochasic Navier-Stokes Equations]{}*, (2003), To appear. [^1]: The work was partially supported by the Sloan Research Fellowship and by the ARO Grant DAAD19-02-1-0374. [^2]: The work was partially supported by the ARO Grant DAAD19-02-1-0374.

--- author: - 'E. Piceno' - 'A. Rosado' - 'E. Sadurní' title: 'Fundamental constraints on two-time physics' --- Introduction ============ Two time physics has been a tantalizing possibility of dimensionally-extended descriptions of our world [@Scherk:1979zr; @Rubakov:1983bz; @Weinberg:1983xy; @Alvarez:1983kt; @Barrow:1987sr], in particular for those extensions related to extra-dimensional high-energy physics and their inherent compactification mechanisms [@Hosotani:1983xw; @Cremmer:1976zc; @Flacke:2005hb; @Kong:2005hn]. The focus of attention in such theoretical endeavours is the existence of phenomenological as well as consistency constraints that allow to answer the simplest, yet most profound question on the nature of space-time. Undoubtedly, all areas of physics are affected by its outcome. Noteworthy attempts [@Petriello:2002uu; @Appelquist:2000nn; @Haisch:2007vb; @Goertz:2011hj; @ADDM2002; @DDG1999] have directed their efforts to distill numerical bounds on physical quantities – i.e. energies or masses – using a theoretical apparatus related to extended-metric versions of classical and quantum field theories. Posed in this language –i.e. a problem of group representations on dimensionally extended manifolds– the challenge seems fit for particle physicists, who are well versed in mechanisms [@Giardino:2011zz; @Allahverdi:2003aq] that either hide or expose measurable quantities in the domain of current experimental capabilities. We believe, however, that even more fundamental restrictions may arise from the point of view of dynamical systems when their evolution is extended to two parameters. At such an elementary level of formality, we encounter already a number of compromising questions. Indeed, we may formulate initially symmetric two-time dynamics in compliance with Newton’s second law and discover, with little effort, that any $1+2$ dimensional extension can be integrated under reasonable smoothness conditions on the functions involved. The main result is that the dynamics occur on a restricted surface and therefore only one time. This would be a surprising result in principle, since typical integrability conditions rest on the existence of integrals of the motion that match the system’s dimensionality. Our results show that in the presence of two times, not only the integrability conditions must be rethought, but also that $1+2$ extensions render simple solutions with a preferred time axis at the level of classical mechanics. It is the resulting motion and not the shape of the equations what yields an effective one-time description. We can formulate the $2+2$ and $3+2$ dimensional extensions in the same manner, therefore these extensions yield similar results. In these cases, two-time evolution takes place only in a dimensionally reduced surface of the force space. Thus we still have one preferred time direction for the case of a general force, and two times for very special cases. Less restrictive conditions can be derived in quantum mechanical settings, either following the approach of dimensionally extended wave equations, or by extending the number of evolution generators that ensure unitary dynamics. In regard with the former, the conservation of probability is at stake: dimensionally extended continuity equations yield, upon integration, a conserved probability along a single time direction. In the present work we show that circumventing such a drawback via unitary evolution leads, after dynamical analysis, to classical limits constrained again by Newton’s second law for two times –now at the level of quantum averages. However, our probability-conserving approach also shows that a full quantum-mechanical regime allows a small window of opportunity determined by a generalized uncertainty relation in energies and times. Furthermore, such a relation provides bounds on the observability of two-time physics in terms of fluctuations in level spacings, Planck’s constant and the norm of elapsed times. We present our discussion starting with the simplest classical example in section \[s1\], and we gradually move to an extended Heisenberg picture in section \[s2\]. We conclude in section \[s3\]. Fundamental aspects and motivation ================================== String theory and Kaluza-Klein compactification [@bailin1987] rely on dimensional extensions that are normally proposed in the space-like part of the Minkowski metric. Why not time? The old idea of introducing stationary modes for extra dimensions in mass operators (e.g. in dimensionally extended Klein-Gordon equations) does produce effective masses related to compactification scales. In the time-like part, these extensions would obviously enter with the opposite sign: if $\Box_{d+2}$ is the $d+2$-dimensional D’Alembert operator, one has that the wavefunction $\phi$ for a scalar particle of mass $m$ satisfies = 0, \[add1\] where \_[d+2]{}= + - - . \[add1.1\] The introduction of additional times would correct such a mass; if we set $\phi(\v x, t_1,t_2) = e^{-i\omega t_2}\tilde \phi(\v x, t_1)$, leads to = 0. \[add2\] The new mass for this mode is then $\tilde m^2 c^4 = m^2 c^4 - \hbar^2 \omega^2$. Strong implications on the values of these masses would arise due to extra time axes. Usually, it is said [@bailin1987] that such extensions should not be considered due to an imaginary effective mass, but here we note that previously existing masses with the value $m$ (coming either from compactified space-like coordinates or other fundamental mechanisms) still allow a real positive $\tilde m$, up to a certain frequency cut off $\omega_{max}$ determined by $\tilde m^2 > 0$. This entails the existence of a fundamental period $\tau = 2\pi / \omega$ that should be compared with a fundamental length $R = \hbar/mc$. The condition that avoids tachyonic –imaginary mass– representations of the Poincaré group acting on $d+1$-dimensional Minkowski space is the inequality $c \tau > R$, while the equality does not produce mass. Other important observations on the existence of more times are related to the proper formulation of dynamical systems for two parameters, which is the main purpose of this paper. We admit that nature is described by integro-differential equations that govern physical quantities, e.g. the position of a particle in space, the value of a classical field, or the space dependence of probability densities in the quantum world. Regardless of the specific form of such equations, the evolution of a system is produced by nature once a set of initial data is provided. This should also apply for more than one time axis; therefore, a consistent description of this type of systems should be given. For this reason, and in the case of classical mechanics, it seems mandatory to address the problem from the point of view of Newton’s second law, with a properly specified force and some initial conditions –in general, Hamiltonian systems of two times. In connection with experimental aspects, the situation must be described as purely observational. This is a common denominator in every extra-dimensional theory, both for space and time. One looks for quantites that can be produced or influenced by extra-dimensional physics –the typical example being the aforementioned masses– and their values are detected in our $3+1$-dimensional world. Indeed, we shall see that this is not limited to the alluded masses, but also includes corrections to the time-energy uncertainty principle arising in quantum-mechanical transition probabilities and quantum mechanical fluctuations; specifically, in section \[S2.3\] we shall analyze the standard deviation of a particle’s position, leading to the conclusion that its value depends strongly on time-energy fluctuations satisfying a new inequality (\[27\]). If the new inequality is confirmed, it would give a hint for new physics. It should be clear though, that many effects could be falsified by other (new) physical considerations; the confirmation of a corrected mass or a new uncertainty principle is not an undeniable proof of extra dimensions, neither spatial nor temporal. Any experimental apparatus made in our world measures the effects produced in it, but the apparatus could hardly venture into other dimensions merely by its motion. In the same manner, we may touch our shadow projected on a wall, but our shadow cannot stick out its arm and shake our hand. Classical dynamics of two times \[s1\] ====================================== We start by allowing Newton’s second law to accomodate two times via a function $x^î(t_1,t_2)$ representing position coordinates. Then, adopting units where mass can be ignored, there must be two differential and non-preferential equations of the form p\^[i]{}\_j = + A\^[i]{}\_[j]{}, i=1,2,,d j=1,2\[1\] = F\^[i]{}\_[kj]{} \[2\] where, for convenience, space indices are placed above quantities, and time indices below. Eq. (\[1\]) shows the necessity of considering two velocities and therefore two momenta, perhaps with a connection $A^{i}_j$ coming from gauge interactions. The partial derivatives with respect to times 1 and 2 are now required. Eq. (\[2\]) expresses an extension of Newton’s second law whenever $F^i_{jk}$ is a specified force – now a tensor in time indices. As in any specified force governing the dynamics of a system, we have a generic dependence $F^i_{jk}(t_1,t_2; \v x(t_1,t_2))$. In addition, if $F$ depends explictly on the two times, there is little we could do to rule out the relevance of a second time, which compels to consider $F^i_{jk}(\v x(t_1,t_2))$ as a specific force law (autonomous differential equations). Now we may proceed to analyze the dynamical consequences of such a force field: combining (\[1\]) and (\[2\]) and abbreviating with $\partial_j$ the time derivatives, we obtain the following antisymmetrization in time indices \_ = \_ x\^[i]{} + \_. \[3\] If $x^{i}$ is a sufficiently differentiable function (a reasonable requirement at least in some domain of the variables $t_1, t_2$) we may eliminate the first term and retain the antisymmetrization due to all other contributions to the momenta: F\^[i]{}\_ = \_, \[4\] as expected. It is striking to see that any attempt at generalizing physics in this way is compromised by space dimensionality, as we now discuss. One spatial dimension --------------------- If $d=1$, the equations of motion and the chain rule can be used to prove the following relations \_1 \_2 p\_k = (p\_1 - A\_1) , \[5\] \_2 \_1 p\_k = (p\_2 - A\_2) . \[6\] As before, we admit that each function $p_k(t_1,t_2)$ is continuously differentiable and compare (\[5\]) and (\[6\]) for $k=1,2$ (p\_1-A\_1) = (p\_2-A\_2), \[7\] (p\_1-A\_1) = (p\_2-A\_2). \[8\] Surprisingly, we can establish now an equation for ’orbits’ relating $p_1, p_2$ and the elements of the force tensor as functions of $x$. This is done by solving the previous system of equations to find ()\^2 = (x), \[9\] F’\_[11]{} F’\_[22]{} = F’\_[12]{} F’\_[21]{}, \[10\] {width="8.6cm"} where we denote the derivative with respect to $x$ by a prime. These orbits are indeed surfaces that we describe in figure \[fig:1\]. The surface is valid for any initial condition, excluding the possibility of a full foliation of the extended phase space $x,p_1,p_2$. This means that the initial value problem is well posed if the second velocity is constrained by the dynamics in the $(x,p_1)$ plane. Another way to look at this comes from a convenient definition of a field $\v \fcal =(\sqrt{F'_{22}F'_{21}},-\sqrt{F'_{11}F'_{12}})$, showing thus that all the evolution happens under the restriction v\_[12]{} x(t\_1,t\_2) = 0. \[11\] As a consequence, the function $x(t_1,t_2)$ is constant in the direction parallel to $\fcal$. Regardless of the explicit form of $\fcal$, there is a family of non-intersecting curves in $(t_1,t_2)$ defining the evolution, which picks only one direction –but not the arrow of time– as a result of consistent dynamics. It is also possible to find a restriction on the field $\fcal$ with the aim of supressing solutions with closed curves in the plane $t_1,t_2$ – a most desirable condition, in view of causality. This type of analysis shall become important in the higher dimensional cases discussed below. Two and three spatial coordinates --------------------------------- The number of constraints emerging in the higher dimensional cases can be determined in a straightforward way. It is advantegeous to reproduce the one-dimensional calculation in (\[7\]) and (\[8\]) in order to find the corresponding dimensionally-extended expressions. The smoothnes condition on the functions $p^i_{j}$ (i.e. commutability of time derivatives) is expressed as \_l F\_[jk]{}\^i = \_j F\_[lk]{}\^i. \[1.1\] Using the chain rule again and ignoring any explicit time dependence of $F$ (we are analyzing autonomous systems!), this equality acquires the form \_[m=1,...,d]{} \_l x\^m = \_[m=1,...,d]{} \_j x\^m \[1.2\] where $d$ is the spatial dimension. For simplicity, we consider now $A=0$ (symmetric $F$) and write (\[1.2\]) as \_[m=1,...,d]{} = 0. \[1.3\] For $d=2$ this condition comprises in fact four equations that can be cast as a homogenous linear system with four velocities as variables: $$\label{1.4} \left(\begin{array}{cccc}F^1_{21,x}&-F^1_{11,x}&F^1_{21,y}&-F^1_{11,y}\\ F^1_{22,x}&-F^1_{12,x}&F^1_{22,y}&-F^1_{12,y}\\ F^2_{21,x}&-F^2_{11,x}&F^2_{21,y}&-F^2_{11,y}\\ F^2_{22,x}&-F^2_{12,x}&F^2_{22,y}&-F^2_{12,y}\end{array}\right) \left( \begin{array}{c} p_1^1 \\ p_2^1 \\ p_1^2 \\ p_2^2 \end{array} \right) =0.,$$ where $$\label{defpar} F_{jk,x^m}^i\equiv\frac{\partial F_{jk}^i}{\partial x^m} ,$$ and $x\equiv x^1$, $y\equiv x^2$. The analogue of (\[10\]) now shows in the form of a determinant, i.e. any non-trivial force field allowing non-zero velocities must satisfy $$\label{1.5} \det\left(\begin{array}{cccc}F^1_{21,x}&-F^1_{11,x}&F^1_{21,y}&-F^1_{11,y}\\ F^1_{22,x}&-F^1_{12,x}&F^1_{22,y}&-F^1_{12,y}\\ F^2_{21,x}&-F^2_{11,x}&F^2_{21,y}&-F^2_{11,y}\\ F^2_{22,x}&-F^2_{12,x}&F^2_{22,y}&-F^2_{12,y}\end{array}\right)=0.$$ With this condition and the general solution of (\[1.4\]) we find once more a parallel restriction on the evolution of each coordinate: \_[12]{} x(t\_1,t\_2) = 0, \_[12]{} y(t\_1,t\_2) = 0 \[1.6\] where the fields $\ccal$ and $\dcal$ are defined in terms of derivatives of the force, given in \[app1\]. As noted in the previous section, more constraints can be imposed by means of physical considerations. For example, ruling out closed curves in $t_1, t_2$ for each function $x$ and $y$ entails the elimination of the curl of each parallel field \_[12]{} = 0, \_[12]{} = 0. \[1.7\] since each curl contains only one component, (\[1.7\]) provides two additional restrictions. The number of conditions in our treatment beg for a careful count of restrictions and free variables. We may have effective one-time dynamics if the two vector fields $\ccal$, $\dcal$ turn out to be parallel; fortunately this constitutes a single restriction ( [cc]{} \_1 & \_2\ \_1 & \_2 ) = 0 \[1.8\] which does not change the dimensionality of the original space. We have in principle 8 independent functions for the entries of the force tensor, but they reduce to 6 when symmetry is invoked. Eqns. (\[1.7\]) reduce the dimensionality by two, but the most important condition for a non-trivial solution is (\[1.5\]), which subtracts yet another dimension. In total, we are left with 3 free functions for a space of 8 components, and (\[1.8\]) represents only a two-dimensional exclusion in such a space. In conclusion, non-trivial dynamics (i.e. with two times) is attainable for a reduced space of possible forces: in this sense, two-time physics must be driven by a special kind of force. The three dimensional case leads to more intricate expressions, which however follow the same logic as in the previous discussion. A similar procedure using (\[1.3\]) with $d=3$ leads to the restriction $$\det \left( \begin{array}{cccccc} F_{21,x}^1&-F_{11,x}^1&F_{21,y}^1&-F_{11,y}^1&F_{21,z}^1&-F_{11,z}^1\\ F_{21,x}^2&-F_{11,x}^2&F_{21,y}^2&-F_{11,y}^2&F_{21,z}^2&-F_{11,z}^2\\ F_{21,x}^3&-F_{11,x}^3&F_{21,y}^3&-F_{11,y}^3&F_{21,z}^3&-F_{11,z}^3\\ F_{22,x}^1&-F_{12,x}^1&F_{22,y}^1&-F_{12,y}^1&F_{22,z}^1&-F_{12,z}^1\\ F_{22,x}^2&-F_{12,x}^2&F_{22,y}^2&-F_{12,y}^2&F_{22,z}^2&-F_{12,z}^2\\ F_{22,x}^3&-F_{12,x}^3&F_{22,y}^3&-F_{12,y}^3&F_{22,z}^3&-F_{12,z}^3\end{array} \right) =0. \label{1.9}$$ As before, the linear system can be expressed as a number of orthogonality conditions for some vector fields $\ccal_1$, $\ccal_2$ and $\ccal_3$ (the procedure to obtain their expressions is indicated in \[app2\]): \_i \_[12]{} x\^[i]{} = 0, \_[12]{} \_i = 0, i=1,2,3. \[1.10\] Finally, the $3+2$ dimensional evolution is governed by tensors which are dimensionally restricted. We have $12$ original components, $3$ symmetry conditions, $3$ irrotational conditions, and a vanishing determinant, i.e. $12-3-3-1=5$ dimensions. An accidental case inside such a space is still possible, since $\ccal_1 \, || \, \ccal_2 \, || \, \ccal_3$ yields effective one-time evolution, but this does not reduce the dimensionality any further. We reach thus the same conclusion as in the $d=2$ case. Remarks on classical results ---------------------------- So far, we have found the necessary constraints [*on the force fields and space dimensions *]{}to produce variations of coordinates in two times. These constraints are conditions that should not be confused with the well-known theory of constraints proposed by Dirac [@Dirac1964]. However, there seems to be an important analogy with this interesting topic when dealing with special relativity of many bodies and it deserves a few comments. In this setting, multiple times emerge from various four-vectors $x_{\mu}^{(i)}$ for each particle, where $i$ is the particle index, and attached to each vector there is a seemingly independent time variable $x_{0}^{(i)}$. The constraints that emerge from this theory correspond to the invariance of each rest mass $m_i$, and they have the form $p^{(i)\mu}_ip^{(i)}_{\mu}=m_i^2c^2$, i.e. on-shell. Since each $p^{(i)0}_i$ is determined by momenta and masses through an equality, we are in the presence of a surface constraint. It must be stressed, however, that the apparent dependence on multiple times has been reduced in the past to single-time dynamics by Barut and coworkers [@barut], as well as Moshinsky and Nikitin [@moshinsky1] and Moshinsky and Sadurni [@moshsadurni], where it was possible to describe the dynamics of a composite system with a single Hamiltonian corresponding to the time-like component of the total four-momentum. Moreover, interactions between particles could be introduced using perpendicular (frame-dependent) four-vectors together with relative Jacobi coordinates generalized to Minkowski space – here one should note the contrast with common criticisms related to the inconsistency of certain types of relativistic interactions [@lienert1]. Some nice results regarding this topic were summarized in [@moshinsky2](in particular, Chapter XII, 315-316), especially those connected with the many-body Dirac oscillator. For more recent applications to spectroscopy of relativistic composites, see [@sadurni1; @sadurni2], where similar ideas came into play for strongly interacting relativistic particles. The treatment in [@sadurni1], eqns. (6)-(11) is of special interest for this topic. In all, two-time physics of a single particle is not a particular case of typical relativistic multi-body physics, but it constitutes a parallel line of study for such problems. Furthermore, one should not deny the possibility that our result of a single surface in 3D phase space (\[10\]), (\[11\]) could be better formulated using Hamiltonian constraints. Also, a cautionary remark is in order. Although the bridge connecting the classical and the quantum world could be built based on classical constraints [@komar] through weak equalities and the Hilbert space obeying them, it is important to recognize that the quantization of classical constraints is not equivalent to the constraint of quantum systems. This fact was first noted by Jensen [@jensen], further elaborated by daCosta [@dacosta] in connection with curvature, and it is of utmost importance to our endeavours. Very specific effects emerging from this difference can be found in [@sadurni3; @sadurni4], e.g. bound states with no classical counterpart. For this reason and in the following sections, the quantum side of the two-time problem will be tackled from scratch; this will ensure generality regarding constraints. Quantum mechanics of two times \[s2\] ===================================== There is more than one approach to extend wave equations to include two time-like axes. The use of augmented metrics has been proposed [@Hosotani:1983xw; @Bonezzi:2010prd; @SS1979; @Bars2001; @BarsT2010; @BarsK1997B; @Bars2008; @BarsDM1999; @VNRS2005; @BarsK1997D], which can be traced back to the least action principle where densities are defined in an extended Minkowski space. In connection with the many body problem in relativity, we have already discussed some similarities and differences with our topic. It should be stressed now that, although such theories exist, the triumphant view of our physical world is the one provided by Relativistic Quantum Field Theory [@ryder]. As explained in many textbooks, only one set of coordinates is needed and many-particle states are represented in second quantization with Fock states. One could reproduce here the first attempts to describe Quantum Electrodynamics using multiple sets of coordinates for $N$ charged fermions with $N$ arbitrary but fixed. It is interesting that the first physical theory that necessitated multiple times was proposed by the old masters of quantum theory, precisely in this direction: Dirac [@Dirac2], Bloch [@bloch] and Tomonaga [@tomonaga]. Additional efforts in the study of multi-particle physics with many times were made in more recent years, e.g. [@lienert1; @crater; @sazdjian]. Our quantum mechanical study involves two lines of reasoning that are different, in principle. In our first approach we shall explore the behavior of a quantum field in the context of augmented metrics, leading to very stringent conditions on the evolution. In our second approach, we shall derive conditions for arbitrary Hamiltonians and states, without alluding to coordinates at the level of wavefunctions. This treament is an extension of our previous premises for the classical case: two dynamical non-preferential equations. We shall see a clear physical difference between the two approaches, the second being the most general. Augmented metrics imply extensions of a single wave equation, while unitary evolution of two times demands the existence of two Schrödinger equations. As a consequence, the second approach is more permissive, providing less restrictive conditions for the possibility of including the variation of physical quantities in two times. In any case, the general meaning of a wavefunction is not affected by our considerations; let us spell out its interpretation in the context of two times. If $|\psi\>$ is a physical state and $|x\>$ represents a state of coordinates $x$, then the probability amplitude of finding a physical system[^1] between $x$ and $x + dx$ is simply $\<x|\psi\>$. At later times, because of the superposition principle –we have a Hilbert space– there is a linear operator that transforms the state $|\psi\>$ to $|\psi (t_1, t_2)\>$, dropping the initial times, and therefore the amplitude evolves to $\<x|\psi(t_1,t_2)\>$ – note that the evolution operator may arise from a prescribed differential equation, such as Klein-Gordon, Dirac, etc. The density $\rho$ is thus the probability of finding the physical system in the coordinates between $x$ and $x+dx$ at the values $t_1, t_2$ of the time parameters, and it is given by $\rho = |\<x|\psi(t_1,t_2)\>|^2$. With appropriate modifications, this is even the case for Klein-Gordon equations –second order in time– as shown, for example, in [@mostafazadeh2006]. Extended wave equations \[s2.1\] -------------------------------- Quantum physics rests on the assumption that wavefunctions and probabilites are related. In any theory of dimensionally extended wave equations, it is compulsory to include a conserved quantity that can be identified with total probability (with the proper normalization $P=1$). As in any field theory, the alluded conservation law is ensured by Nöther’s theorem and therefore by the presence of symmetry. Taking this as a starting point, leads us to consider field equations with gauge symmetries, such that quantum-mechanical waves describing the evolution of matter can be related to a continuity equation satisfied by the corresponding Nöther currents. A specific example with the $1+2$ dimensional Dirac equation is discussed in appendix \[app3\]. In the general case, we only need to analyze the continuity equation for extended dimensionality. The idea is to find a conserved probability along two times, avoiding the case in which this occurs only for one time. The latter obviously implies that there can be only one-time dynamics, while the former still makes a case for non-trivial physics. Later, we shall find that in this general scenario, the probability density must be very special. We start with the extended continuity equation \_ j\^ &=& 0, g\_ = {+,+,-,,- },\ , &=& 1,, d+2. \[2.1\] In order to obtain the conserved ’charge’ (in the Nöther sense), we must split this equation into time-like and spce-like components, with the aim of performing space integration under some specific boundary conditions. In our previous notation, we write \_[12]{} ǰ\_ - \_[x]{} ǰ\_= 0, \[2.2\] with ǰ\_ &=& (j\_1,j\_2),\ ǰ\_ &=& (j\_3,, j\_[d+2]{})=(j\_x,j\_y, j\_z, ),\ \_[12]{} &=& (\_1,\_2),\ \_[x]{} &=& ( ,, ). \[2.3\] Now we are ready to integrate over a $d+1$ dimensional space in order to obtain the total derivative of some quantity; this can be done in two different ways, over volumes $V_1$ and $V_2$: { \_[V\_1]{} dt\_2 dx j\_1 } = \_[V\_1]{} dt\_2 d x ( \_2, \_[x]{} ) ( -j\_2, ǰ\_ )\ \[2.4\] { \_[V\_2]{} dt\_1 dx j\_2 } = \_[V\_2]{} dt\_1 d x ( \_1, \_[x]{} ) ( -j\_1, ǰ\_ ).\ \[2.5\] but now we find two conserved quantities after applying the divergence theorem to the r.h.s. and imposing vanishing integrals on the boundary (which is always necessary, even in single-time dynamics)[^2]. This leads to &=& \_[S\_i]{} d š ( -j\_k, ǰ\_ ) = 0,\ i&=&1,2, k=1,2, k i\ Q\_i (t\_i) && \_[V\_i]{} dt\_k dx j\_i \[2.6\] From here we see that there are two densities $\rho_i = \int dt_k j_i$ corresponding to each conserved charge. Our task now is to find a conserved quantity in both times $Q(t_1,t_2)$ using (\[2.6\]). As a consequence, we shall see that any probability density such that d x (x; t\_1, t\_2) = Q(t\_1,t\_2) = 1 \[2.7\] must be a separable function of $t_1,t_2$. In order to obtain an expression for $\rho$ in terms of the aforementioned densities $\rho_i$, let us consider a functional $\rho \left[ \rho_1, \rho_2 \right]$ and replace it back in (\[2.7\]). Computing the time derivatives \_i d x &=& \_i Q(t\_1,t\_2) = 0\ &=& d x { \_i \_1 + \_i \_2 }\ &=& d x \_i \_i \[2.8\] where the last line follows from $\partial_1 \rho_2 = \partial_2 \rho_1 =0$. Moreover, since this integral vanishes, the result must be independent of $t_1$ and $t_2$. If it is independent of $t_1$ for $i=2$, then $\partial \rho / \partial \rho_2$ cannot depend on $\rho_1$ (unless $\rho_1$ is trivially a constant) and a similar argument yields that $\partial \rho / \partial \rho_1$ cannot depend on $\rho_2$. This means that $\rho$ must be separable $\rho = \alpha g_1(\rho_1) + \beta g_2(\rho_2)$ with $\alpha, \beta$ two constants. Now we consider $i=2$ when the integral does not depend on $t_2$, which leads to d x g\_2’(\_2) \_2 \_2 = 0 = \_2 Q\_2 = d x \_2 \_2, \[2.9\] and finally the desired result is $\beta g_2' = 1$, i.e. $\rho$ is linear in $\rho_1, \rho_2$. This is indeed a very strong result: = \_1 + \_2. \[2.10\] As a consequence, the total charge is also separable $Q = \alpha Q_1 + \beta Q_2$, and the constants $\alpha, \beta$ can be chosen in terms of the normalization condition. We finish our discussion by recognizing that separability has a restrictive effect on averages, when computed with our $\rho$. The average of any function of the position (say $A(\v x)$) is separable as well A = d x A \_1 + d x A \_2. \[2.11\] A dynamical consequence of this property is that $\<\v x \>$ itself is separable, i.e. $\< x^i \> = f^i_1(t_1) + f^i_2(t_2)$. According to the Ehrenfest’s theorem, in the classical limit (for any [*bona fide *]{}quantization scheme) one must recover Hamilton’s equations of motion for these averages. The force (as treated in our previous discussions) should act then in such a way that the evolution is separable, leading to F\_[12]{}\^i = \_1 \_2 x\^i = 0 \[2.12\] but in general we have a non-vanishing diagonal F\_[11]{}\^i = \_1\^2 x\^i 0, F\_[22]{}\^i = \_2\^2 x\^i 0. \[2.13\] When the dynamical equations (\[2.13\]) or their classical counterparts (\[2\]) are decoupled in this way, one finds a strong consequence: F\_[jj]{}\^i ( x ) &=& F\_[jj]{}\^i ( f\_1(t\_1) + f\_2(t\_2) ) = \_j\^2\ &=& \_j\^2 f\^i\_j(t\_j) \[2.14\] so now $F^i_{11}$ may depend only on $t_1$ and $F^i_{22}$ may depend only on $t_2$. For this reason, either $f^i_1 =$ constant or $f^i_2=$ constant. In this way we have shown that in the classical limit, this scheme forces a single time dependence of the average position. The external forces are also constrained, in the sense that any external stimulus must produce evolution in only one parameter. This argument is valid for $i=1,\cdots,d$, but it is possible to analyze simple cases such as $d=1$, where the force tensor must have only one non-vanishing component. Once more, parallels can be drawn regarding multi-particle theories. Tensor conservation laws have been formulated with the aim of including every particle current in a vanishing integral [@sazdjian]. It seems that this approach has been very helpful in ensuring conservation of probabilities where times are necessarily attached to particle labels. However, such treatments do not include two times for a single particle – they do not reduce to such a case. A vector current (as opposed to tensor) for a single particle in $d+2$ dimensions obeys a different conservation law, since a partition of the remaining $d$ spatial coordinates in 3d spaces is not necessary, nor the imposition of a vanishing integral of a tensor for surfaces covering each particle’s coordinates. With this, we conclude that the separability results of the density in this section are quite restrictive. Unitary evolution with two times \[s2.2\] ----------------------------------------- It is important to stress that the issue of probability conservation has been raised regarding extended differential equations using the first approach. For this reason, in our second approach we postulate a Hilbert space of physical states $|a\>$ evolving under two parameters by means of a unitary transformation $U(t_1,t_2)$ such that U(t\_1,t\_2; t\_[01]{}, t\_[02]{}) |a = |a(t\_1,t\_2), UU\^=U\^U=1.\ \[12\] Let us drop the initial time label $ t_{01}, t_{02}$ to simplify the notation. An infinitesimal expansion of $U$ reveals a Hermitean generator $H(t_1,t_2)$ which is in fact composed by two operators in each time direction $H_1, H_2$. By a standard procedure, we can show that each evolution leads to a Schrödinger equation for the operator $U$ in terms of the corresponding generator (note that this is deduced and not postulated): H\_i(t\_1,t\_2) U(t\_1,t\_2) = i \_i U(t\_1,t\_2), i=1,2. \[13\] This is equivalent to a unitary group with two parameters, but in order to elucidate its structure we must resort once more to smoothness conditions on $U$, at least in some region of the space $t_1,t_2$. Commutability of derivatives implies \_1(H\_2 U) = \_2 (H\_1 U), \[14\] or multiplying by $U^{\dagger}$ from the right and using (\[13\]) i \_ = , \[15\] which tell us that the generators obey fundamental non-commutability conditions in terms of their derivatives. Finally, (\[13\]) can be used to establish two Schrödinger equations when acting on a specific state. Similar conditions were derived in [@petrat]. In order to work out that the result is, in fact, independent of any interpretational relation to multi-particle physics, as well as to show that the multiple times do not necessarily need to be attached to particles, we have presented the result in a different way. In addition, we would like to mimic here the treatment used in the classical case by analyzing the evolution of observables. The Heisenberg picture can be extended naturally in this framework if $A^{\mbox{\scriptsize H}}\equiv U^{\dagger}AU$, leading to i \_i A\^ = , i=1,2. \[16\] For any observable, in particular for position $x$, we have again two velocities [^3]: v\_i = - \[17\] and four possible accelerations: a\_[ij]{} = \_i v\_j = - F\_[ij]{}. \[18\] This is in fact Newton’s second law if we admit that the dynamics is governed by previous specification of the operator $F_{ij}$ [^4]. With these relations we may now establish a set of dynamical equations for averages using a fixed state $|\psi\>$: a\_[ij]{} = \^2\_[ij]{}x =F\_[ij]{}. \[19\] This is our two-time version of Ehrenfest’s theorem; a good classical limit is ensured when $x_{\mbox{\scriptsize Classical}}=\<x\>$ and $F_{ij}(x_{\mbox{\scriptsize Classical}}) = \<F_{ij}\>$. The second condition clearly depends on the state $|\psi\>$ we choose, but imposing such a limit leads directly to our previously obtained conditions, i.e. the evolution of averages must pick one direction in the plane $t_1,t_2$. Fluctuations and fundamental constraints \[S2.3\] ------------------------------------------------- In a full quantum-mechanical regime, we may analyze the possibility of ’leakage’ of the evolution along other directions. In a general framework, a reasonable assumption comes into play: If physics is to be invariant under translations of the origin of time, then $\partial_i H_j =0$ and consequently $\left[H_i,H_j\right]=0$ [^5]. Therefore, the Hamiltonian generators share an eigenbasis and we may analyze the dynamics in a suitable matrix representation where $x^{mn}=\<m|x|n\>, H_i |n\>=E_i^n |n\>$. Computing the matrix elements of both sides of (\[18\]) and defining $\Delta^{nm}_i \equiv E^n_i-E^m_i$ yields \_i \_j x\^[nm]{}= x\^[nm]{}\^[nm]{}\_i \^[nm]{}\_j, \[20\] which is symmetric in $i,j$, as expected. Once more, velocities $v_i$ and their matrix elements $v_i^{nm}=i\Delta^{nm}_i x^{nm}/\hbar$ can be assumed to be smooth functions, rendering the following conditions \^[nm]{}\_1 \_2 x\^[nm]{} - \^[nm]{}\_2\_1 x\^[nm]{} = 0. \[21\] The evolution of each matrix element is thus restricted by a geometric condition of the type v\^[nm]{} \_[12]{} x\^[nm]{}=0 \[22\] where the field $\v \fcal^{nm}=(\Delta^{nm}_2,-\Delta^{nm}_1)$ now depends on energy quanta $n,m$, in contrast with its classical counterpart (\[11\]). A convenient change of variables in the two-time plane helps us to understand the nature of each direction; we propose a rotation of variable angle \^[nm]{}\_1 &=& (\^[nm]{}) t\_1 + ( \^[nm]{}) t\_2\ \^[nm]{}\_2 &=& -(\^[nm]{}) t\_1 + (\^[nm]{}) t\_2 \[23\] with (\^[nm]{}) &=& =,\ (\^[nm]{}) &=& =. \[24\] The geometric restriction (\[22\]) now implies that $x^{nm}$ depends only on the first variable $\tau^{nm}_1$ and not on $\tau^{nm}_2$. In other words, each element ’chooses’ its own time direction as a function of the energies. When $n=m$ we recover the classical limit, but in the full quantum picture the single-time evolution is applicable only to individual processes. The possibility of having an overall two-time physics may take place, provided that fluctuations in $x$ or any other observable are non-vanishing: \^2 &=&x\_\^2-x\^2\_\ &=& x\_\^2 - \_[nm]{} \_m\^\* \_n |x\^[nm]{}(0)|\^2 ()\ \[25\] where $\psi_n = \<n|\psi\>$ and $\<x\>$ is now replaced by its limit $x_{\mbox{\scriptsize Classical}}$. The average $\<x^2\>$ is what we must analyze in our search for residual effects. Although many elaborate procedures can be employed in the study of quantum evolution and decoherence –e.g. density operators and master equations [@Zurek:2003; @Schlosshauer:2005; @Isar:1994]– the expression in (\[25\]) serves well our purposes; the more terms involved in $\<x^2\>$, the more plausible is to have superpositions along many time directions $\tau_1^{nm}$. Quantumness thus appears as a bound to such many possible choices of single-time dynamics. A second effect deserves a mention: the effective time of the evolution becomes one and only one when the angle is fixed $\theta^{nm}=\theta$, which is tantamount to saying that a single direction has been picked by the system rather than the elimination of $t_2$ in favor of $t_1$. A possible scenario occurs when a proportionality $E_1^{n}-E_1^{m} = \epsilon (E_2^{n}-E_2^{m}) $ holds, i.e. when the generators coincide in their shape $H_1 = \epsilon H_2 + E_0$. In our attempt to explore new physics, it is worth noting that the ressemblance between the spectra of $H_1$ and $H_2$ leads to a small probability of finding two-time dynamics in the fluctuations of any observable. Extended uncertainty principle \[S2.3\] --------------------------------------- As a continuation of our analysis, we note now that (\[25\]) would contain strong oscillations – therefore vanishing contributions – unless the argument of the exponential satisfies a new uncertainty relation: (E\^n\_1-E\^m\_1) t\_1 + (E\^n\_2-E\^m\_2) t\_2 \~\[26\] or in compact notation E\_1 t\_1 + E\_2 t\_2 \~. \[27\] The inequality is also attainable, but if the argument of the exponential is too small, there will be no chance to perform at least one oscillation and the evolution would not have produced significant variations on $\<x^2\>$. As a final task, let us estimate the variation in the angle of the trajectories around some average value. We define a length in time variables $t = \sqrt{t_1^2+t_2^2}$ and take $\phi$ as the angle between the vectors $(t_1,t_2)$ and $(\Delta_1^{nm},\Delta_2^{nm})=(\Delta E_1, \Delta E_2)$. We have, by virtue of (\[27\]) = 1. \[28\] The effects in quantum fluctuations, as we have seen, come from variations of energies as functions of $n,m$. In order to estimate the fluctuations on $\phi$, we obtain the differential $\delta \phi$ by computing the derivative of $\cos \phi$ in both sides of (\[28\]). This allows to express $\delta \phi$ in terms of increments of energy spacings $\delta(\Delta E_i)$: = ,\ \[29\] and if $\hbar$ is retained to lowest order, we have the lowest possible estimate: \~. \[30\] Our chances to detect this small width $\delta \phi$ from our new uncertainty principle are greatly bound by level spacings of both generators of the evolution. The inequality in (\[28\]) can be applied to find , \[31\] which is now independent of physical constants. Although this bound does not guarantee the existence of two time axes, we can be sure that the particular details of Hamiltonians $H_1,H_2$ – in other words, the model of our physical system – are enough to constrain observability. From the denominator of (\[31\]) we can see that large spacings destroy any possibility of a finite width. We must stress though that the inequality is controlled also by the fluctuations $\delta(\Delta E)$, alluding now to the composition of wavepackets comprised of many energy scales. The energy differences can indeed fluctuate if $|\psi\>$ contains both a fine and a coarse structure of levels. We have reached thus the conclusion that the richness in composition as a whole gives the opportunity of observing sustained two-time evolution. A two-level system –the most quantum mechanical example that we know– does not suffice, nor an experiment of very massive particles isolated in the scale $100$ GeV. Conclusion \[s3\] ================= The introduction of two times through non-preferential and probability-conserving laws leads to strong bounds. We have seen how quantum mechanics may be less restrictive, where the roles of Planck’s constant and the level spacing fluctuations are equally important. Violations to our fundamental constraints would imply severe deviations from known physics, such as an invalid Newton’s second law or higher order differential equations. In the quantum domain, probability conservation is attached to the very notion of probability and we do not believe that such a strong hypothesis could be dropped. Financial support from CONACyT under project CB 2012-180585 is acknowledged. Parallel vector fields ====================== Fields in 2+2 dimensions \[app1\] --------------------------------- The components of the fields that restrict the dynamics for the $2+2$ dimensional case are \_1 = [|cc|]{}&\ & , \_2 = [|cc|]{}’&’\ ’& ’ ,\ \_1 = [|cc|]{}&\ & , \_2= [|cc|]{}’&’\ ’&’ , \[a1.7\] with the quantities = [|cc|]{}F\_[21,x]{}\^x&F\_[21,y]{}\^x\ F\_[21,x]{}\^y&F\_[21,y]{}\^y ,= [|cc|]{}F\_[11,x]{}\^x&F\_[12,x]{}\^x\ F\_[11,x]{}\^y&F\_[21,x]{}\^y ,\ = [|cc|]{}F\_[21,y]{}\^x&F\_[22,y]{}\^x\ F\_[21,x]{}\^x&F\_[22,x]{}\^x ,= [|cc|]{}F\_[21,x]{}\^x&F\_[12,x]{}\^x\ F\_[21,x]{}\^x&F\_[22,x]{}\^x ,\ \[a1.8\] =-,= [|cc|]{}F\_[11,y]{}\^x&F\_[21,y]{}\^x\ F\_[11,y]{}\^y&F\_[21,y]{}\^y ,=-,= [|cc|]{}F\_[11,y]{}\^x&F\_[12,y]{}\^x\ F\_[21,y]{}\^x&F\_[22,y]{}\^x , \[a1.9\] ’= [|cc|]{}F\_[21,x]{}\^x&F\_[11,y]{}\^x\ F\_[21,x]{}\^y&F\_[11,y]{}\^y ,’=,’= [|cc|]{}F\_[11,y]{}\^x&F\_[12,y]{}\^x\ F\_[21,x]{}\^x&F\_[22,x]{}\^x ,’=, \[a1.10\] = [|cc|]{}F\_[21,y]{}\^y&F\_[11,x]{}\^y\ F\_[21,y]{}\^x&F\_[11,x]{}\^x ,=,= [|cc|]{}F\_[22,y]{}\^x&F\_[12,x]{}\^x\ F\_[21,y]{}\^x&F\_[11,x]{}\^x ,=. \[a1.11\] Fields in 3+2 dimensions \[app2\] --------------------------------- The smoothness of the functions $p^{i}_j$ and the chain rule applied to the derivative of the force lead, in this case, to the following linear system $$\left(\begin{array}{cccccc} F_{21,x}^1&-F_{11,x}^1&F_{21,y}^1&-F_{11,y}^1&F_{21,z}^1&-F_{11,z}^1\\ F_{21,x}^2&-F_{11,x}^2&F_{21,y}^2&-F_{11,y}^2&F_{21,z}^2&-F_{11,z}^2\\ F_{21,x}^3&-F_{11,x}^3&F_{21,y}^3&-F_{11,y}^3&F_{21,z}^3&-F_{11,z}^3\\ F_{22,x}^1&-F_{12,x}^1&F_{22,y}^1&-F_{12,y}^1&F_{22,z}^1&-F_{12,z}^1\\ F_{22,x}^2&-F_{12,x}^2&F_{22,y}^2&-F_{12,y}^2&F_{22,z}^2&-F_{12,z}^2\\ F_{22,x}^3&-F_{12,x}^3&F_{22,y}^3&-F_{12,y}^3&F_{22,z}^3&-F_{12,z}^3\end{array}\right) \left(\begin{array}{c} p^{1}_1 \\ p^{1}_2 \\ p^{2}_1 \\ p^{2}_2 \\ p^{3}_1 \\ p^{3}_2 \end{array} \right)=0. \label{a1.12}$$ From here, and the condition of a vanishing determinant (\[1.9\]), we may eliminate successively the variables $p_2^3$ in terms of other $p'$s, $p_1^3$ in terms of the remaining $p'$s and so forth. All the resulting relations are linear in $p$, but not in $F$. The final substitution is equivalent to the first condition in (\[1.10\]): \_[1,1]{} p\_1\^1 + \_[1,2]{} p\_2\^1 =0 = \_1 \_[12]{} x. \[a1.13\] Once this expression has been established, we may solve for other $p$’s in order to obtain the remaining two \_[2,1]{} p\_1\^2 + \_[2,2]{} p\_2\^2 =0 = \_2 \_[12]{} y,\ \_[3,1]{} p\_1\^3 + \_[3,2]{} p\_2\^3 =0 = \_1 \_[12]{} z, \[a1.14\] where the components of the three vectors $\ccal_i, i=1,2,3$ can be solved completely in terms of $F_{mn,k}^{i}$. This completes the procedure, but as an example we provide $\ccal_1$ explicitly (the other three can be obtained by trivial permutations): $$\ccal_{1,j} = \det\left(\begin{array}{cc} A^3_2(j)&A^3_2(3)\\ A^3_3(j)&A^3_3(3)\end{array}\right) ,\qquad j=1,2$$ where $$A^3_m(n)=\det\left(\begin{array}{cc} A^2_m(n)&A^2_m(4)\\ A^2_4(n)&A^2_4(4) \end{array}\right), \qquad m,n = 1, \cdots, 3.$$ $$A^2_m(n)=\det\left(\begin{array}{cc} A_m^1(n)&A^1_m(5)\\ A^1_5(n)&A^1_5(5) \end{array}\right), \qquad m,n = 1, \cdots, 4.$$ $$A^1_m(n)=\det\left(\begin{array}{cc} c_{mn}&c_{m1}\\ c_{6n}&c_{61} \end{array}\right), \qquad m,n = 1, \cdots, 5.$$ and the matrix $c_{mn}$ is given in terms of the force tensor $$(c_{mn})= \left(\begin{array}{cccccc} F_{21,x}^1&-F_{11,x}^1&F_{21,y}^1&-F_{11,y}^1&F_{21,z}^1&-F_{11,z}^1\\ F_{21,x}^2&-F_{11,x}^2&F_{21,y}^2&-F_{11,y}^2&F_{21,z}^2&-F_{11,z}^2\\ F_{21,x}^3&-F_{11,x}^3&F_{21,y}^3&-F_{11,y}^3&F_{21,z}^3&-F_{11,z}^3\\ F_{22,x}^1&-F_{12,x}^1&F_{22,y}^1&-F_{12,y}^1&F_{22,z}^1&-F_{12,z}^1\\ F_{22,x}^2&-F_{12,x}^2&F_{22,y}^2&-F_{12,y}^2&F_{22,z}^2&-F_{12,z}^2\\ F_{22,x}^3&-F_{12,x}^3&F_{22,y}^3&-F_{12,y}^3&F_{22,z}^3&-F_{12,z}^3\end{array}\right).$$ Continuity in the $1+2$ Dirac equation {#app3} ====================================== The usual $d+1$ dimensional Dirac equation has an associated conserved current which is well understood. We proceed to extend the Dirac equation to $1+2$ dimensions and obtain an appropiate conserved current, and therefore a suitable probability density. We find that the derived conserved current is consistent with section \[s2.1\]. Thus, we end up with the same conclusions, and we have a classical limit with a single time evolution. With the energy-momentum conservation equation ($\hbar=c=1$) $$\begin{aligned} \left\{\Box_{1+2}+m^2\right\}\Psi=0 \label{A2.1}\end{aligned}$$ as the starting point, we want to obtain an equation of order 1 in the two times. In order to do so, a set of Dirac $\gamma$ matrices is needed. Denoting time components by $\mu = 1,2$ and the space component by $\mu=3$, we write the square root of (\[A2.1\]) as $$\begin{aligned} \left\{i\gamma_\mu\partial^\mu-m\right\}\Psi(\v x)=0, \label{A2.2}\end{aligned}$$ where we use Clifford’s condition $$\begin{aligned} \left\{\gamma_\mu,\gamma_\nu\right\}=2g_{\mu\nu} \v 1, \label{A2.3}\end{aligned}$$ and $g_{\mu\nu}$ is given by (\[2.1\]); we find that a $2\times 2$ representation is enough, and the $\gamma$ matrices may be given by $$\begin{aligned} \gamma_3 = \gamma_x=i\sigma_3,\quad \gamma_1=\sigma_1,\quad \gamma_2=\sigma_2. \label{A2.4}\end{aligned}$$ With the extended Dirac equation (\[A2.2\]), now we proceed to find a conserved current $j^\mu$, $$\begin{aligned} \partial_\mu j^\mu=0, \label{A2.5}\end{aligned}$$ ensured by Nöether’s theorem. The usual way (in $3+1$ dimensions) to find such a conserved current does not work well in this case, as a consequence of the effective hamiltonian not being hermitean: $$\begin{aligned} i\partial_1\Psi=-i\partial_2[(\gamma_1\gamma_2)\Psi]+i\frac{\partial}{\partial x}[(\gamma_1\gamma_3)\Psi]+m\gamma_1\Psi. \label{A2.6}\end{aligned}$$ This further implies that the field describing the antiparticle cannot be defined as $\Psi^\dagger\gamma_3$. However, by making a coordinate inversion while considering the hermitean conjugate of Dirac equation $$\begin{aligned} \Psi^\dagger(-x_\nu)\left\{+i\gamma^\dagger_\mu\overleftarrow{\partial}^\mu-m\right\}=0, \label{A2.7}\end{aligned}$$ we are led to a conservation equation of the form $$\begin{aligned} \partial^\mu\left[i\Psi^\dagger(-x_\nu)\gamma_3\gamma_\mu\Psi(x_\nu)\right]=0. \label{A2.8}\end{aligned}$$ For the current, we consider only the hermitean part of the previous quantity $$\begin{aligned} j_\mu=\frac{1}{2}i\Psi^\dagger(-x_\nu)\gamma_3\gamma_\mu\Psi(x_\nu)+ \mbox{h.c.} \label{A2.9}\end{aligned}$$ and the $\bar{\Psi}$ describing antiparticles must be defined as $$\begin{aligned} \bar{\Psi}(x_\nu)=\Psi^\dagger(-x_\nu)\gamma_3. \label{A2.10}\end{aligned}$$ With this conserved current we can obtain the probability density and total charge as done in section \[s2.1\]: $$\begin{aligned} \rho({\bf x} ;t_1,t_2)=\alpha\int dt_2j_1+\beta\int dt_1j_2 \nonumber \\ =\alpha\rho_1({\bf x},t_1)+\beta\rho_2({\bf x},t_2) \label{A2.11}\end{aligned}$$ and $$\begin{aligned} P=\int d{\bf x}\rho({\bf x};t_1,t_2)=\alpha P_1(t_1)+\beta P_2(t_2); \label{A2.12}\end{aligned}$$ which, once again, are separable. The conclusions in section \[s2.1\] apply, and then, in the classical limit, the average position evolves as a function of only one time. Positivity of densities ----------------------- Probability densities must be positive quantities in all space. Although the Klein-Gordon current corresponding to (\[A2.1\]) could be used as a conserved current, textbook observations on such currents reveal that only positive energy components can be related to positive densities; the conserved quantity here is the [*charge. *]{} For Dirac currents, it is left to prove that the $1+2$ theory allows positive densities. If we go back to (\[A2.11\]), we will note that the positivity is a property that corresponds to $\rho$ in general. In particular, if $j_1$ and $j_2$ are positive, then the result should follow, but these are not the most general conditions. It seems appropriate, though, to find the restrictions in the solutions of the two-time Dirac’s equation that allow such a scenario. Compare with [@Lienert2015] and the structure of the resulting theory. We start by noting that either the imaginary part or the real part of (\[A2.9\]) can be used as a valid current. As long as the sign of these components does not change, we may define $j_1,j_2$ up to a sign. In order to compute $j$’s, we write the solutions of (\[A2.2\]) as linear combinations of on-shell waves = e\^[i k\_ x\^]{} \_[+]{}(0) + e\^[-i k\_ x\^]{} \_[-]{}(0). \[app2.1.1\] Substitution of this Dirac spinor on Dirac’s equation yields the conditions {-\_ k\^ - m } \_[+]{}(0) = 0, \[app2.1.2\] {\_ k\^ - m } \_[-]{}(0) = 0, \[app2.1.3\] while the vanishing determinant of each linear operator is simply the on-shell condition, which is independent of the sign of $k_{\mu}$. Now, in terms of components, the spinors are \_(0) = ( [c]{} C\_\^1\ C\_\^2 ) \[app2.1.4\] and substitution of (\[app2.1.4\]) in the imaginary part $\Im \left[\cdot \right]$ of (\[A2.9\]) gives j\_1 = 2 (2k\_x\^)\ + 2 , \[app2.1.5\] j\_2 = 2 (2k\_x\^)\ + 2 . \[app2.1.6\] Had we used the real part $\Re \left[\cdot \right]$ of (\[A2.9\]), an overall factor $\sin(2k_{\mu}x^{\mu})$ would have spoilt the result, since this factor changes its sign in space and time. Now, (\[app2.1.5\]) and (\[app2.1.6\]) preserve their sign (positive) if the following conditions are met || ||, \[app2.1.7\] || ||. \[app2.1.8\] Since (\[app2.1.2\]) and (\[app2.1.3\]) relate $C_{\pm}^{1}$ with $C_{\pm}^{2}$, the positivity conditions (\[app2.1.7\]) and (\[app2.1.8\]) constrain the normalizations of $\Psi_{+}(0)$, $\Psi_{-}(0)$, together with their relative phase. With these conditions, we have reached positivity. 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The latter indicate that our extended wavefunctions do not necessarily vanish for $t_i \rightarrow \pm \infty$, i.e. no localization in time is required. [^3]: Here we do not need to impose any representation on momenta, nor a canonical commutator with $x$. However, note that the prescription $p=-i\hbar \partial/\partial x$ must be modified to accomodate two times. [^4]: Equations of a finite order in time are sufficient. Quantum mechanical versions of the Abraham-Lorentz force could be proposed, but this would make our treatment lengthier. [^5]: In the general case $\partial_i H_j \neq 0$, we can prove that smoothness of $x(t_1,t_2)$ leads in fact to $\left[x, \left[H_i, H_j \right] \right]=0$, i.e. $i \left[H_i, H_j \right]$ must be compatible with $x$ and it may contain other physical observables such as spin, color, flavor, etc.

--- abstract: 'In the last several years, remote sensing technology has opened up the possibility of performing large scale building detection from satellite imagery. Our work is some of the first to create population density maps from building detection on a large scale. The scale of our work on population density estimation via high resolution satellite images raises many issues, that we will address in this paper. The first was data acquisition. Labeling buildings from satellite images is a hard problem, one where we found our labelers to only be about 85% accurate at. There is a tradeoff of quantity vs. quality of labels, so we designed two separate policies for labels meant for training sets and those meant for test sets, since our requirements of the two set types are quite different. We also trained weakly supervised footprint detection models with the classification labels, and semi-supervised approaches with a small number of pixel-level labels, which are very expensive to procure.' author: - | Amy Zhang[^1]\ Facebook\ `amyzhang@fb.com`\ Xianming Liu\ Facebook\ `xmliu@fb.com`\ Andreas Gros\ Facebook\ `andreasg@fb.com`\ Tobias Tiecke\ Facebook\ `ttiecke@fb.com`\ bibliography: - 'egbib.bib' title: Building Detection from Satellite Images on a Global Scale --- Introduction ============ With the recent improvements in remote sensing technology, there has been a lot of work in building detection and classification from high resolution satellite imagery. However, we are the first to implement a system on a global scale. Other work uses handpicked features to define buildings [@DBLP:journals/corr/Cohen0KCD16] [@autorooftop] which would not scale well across countries with very different styles of buildings. The work closest to ours is done by Yuan [@DBLP:journals/corr/Yuan16], which also uses pixel level convolutional neural networks for building detection, but is only validated on a handful of cities in the US and likely would not transfer well to smaller settlements or other countries. In order to speed up our pipeline we need a fast bounding box proposal algorithm to limit the number of images that need to be run through our convolutional neural network. To maintain high recall, however, we need to be careful to not filter out too many candidates. We used a naive bounding box proposal algorithm, by performing straight edge detection to extract smaller masks to run through our classification network. This reduced the amount of landmass to process by 50%. The distribution of buildings is still very negatively skewed, where only 2% of proposals are positive. This also means we need to sample a large number of masks in order to have confident precision and recall numbers by country. We also use a weak building classifier to filter masks with over 0.3 IoU (intersection over union) by choosing the mask with the highest probability of containing a building in the center, since these overlapping masks are likely to contain the same building. Discovering systematic issues with our models is also a slow, manual problem that requires visualization of .kmz files, pinpointing large numbers of false positive or false negative areas, and debugging the causes. The problems encountered included noise, contrast issues, cloud cover, or just deficiencies in the model, and we set up a feedback loop to fix those problems. We will be open sourcing our population density results as well as our labeled dataset as a benchmark for future efforts. Dataset Collection Issues ========================= We have two goals for data collection, obtaining labels for training, and accuracy numbers on a country level. Obtaining accuracy numbers of the entire pipeline for a single country requires randomly sampling from all possible 64x64 masks. That distribution is incredibly skewed, and randomly sampling enough masks to obtain a reasonable confidence interval on accuracy is expensive. Instead, we measure how well our neural network performs building classification by randomly sampling from the distribution of masks generated by our bounding box proposal algorithm. The assumption is that the bounding box proposal algorithm only eliminates clear negatives, so reduces skew on the underlying distribution without affecting recall of the overall pipeline. This drops the number of labels we need by a factor of 10, because our new distribution now is 2%-5% positive. Collecting a training set went through several iterations because we want a more balanced dataset for training so the model can get enough samples of both the background and the building classes. We also employ simple active learning techniques by sampling from masks the network was “less sure” about, where the probability was closer to the threshold. Generalizing a Global Model =========================== Training a global building classification model has trade-offs. Buildings can look very different across different countries, but there is still a lot of information that can be transferred from country to country. We initially started with a model trained only on Tanzania, which when applied to a new country had a large drop in accuracy. However, we found that as we labeled data in more countries and re-trained our model with the new data, our new global model performed better on Tanzania than a Tanzania specific model. The generalizations learned from other countries made the model more robust. Another argument for training a global model is that building a large training set takes time, and the amount of data required to train a model from scratch for each country was prohibitive. The trade-off is that the global model doesn’t work equally well on all countries, and we found it necessary to perform some amount of model specialization. We fine-tuned the global model with the same samples it had seen from the initial training, but only from a handful of countries that we wanted it to improve upon. We saw gains of 20-40% in precision and recall on the validation set using the extra fine-tuning step, but noticed there were trade-offs. The training and validation sets gave no evidence of overfitting, but we saw an increase in systematic false positives when visualizing the results on a country level, in certain countries. Building Classification Model ----------------------------- The classification model we trained was a weakly supervised version of SegNet [@DBLP:journals/corr/BadrinarayananK15], which is a fast yet accurate pixel classification network that uses deconvolution layers. We trained with weak “pixel level” labels, and generate a mask level probability using global average pooling on the final pixel level probabilities over the 64x64 mask. We have 500TB of satellite imagery, and being able to run the model over all these countries (multiple times) is crucial for fast iteration. It was a non-trivial task to develop a model that was large enough to capture the complex idea of what defines a building, while also being small enough to run quickly during inference time. SegNet performed well on this by saving the indices from the max pooling layers to perform non-linear upsampling in the deconvolution layers. Building Segmentation Model --------------------------- ![Semantic segmentation results using weakly-supervised model.[]{data-label="fig:segmentation"}](./footprints.png){width="50.00000%"} Finely pixel-wise labeled data is extremely time consuming to acquire, and errors will accumulate especially for small foreground objects. Instead of utilizing fully supervised semantic segmentation method such as FCN [@long2015fully], we investigated weakly supervised segmentation models relying on feedback neural network [@cao2015look], which utilizes the large amount of “cheap” weakly-supervised training data. Notably, to increase the efficiency of semantic segmentation, the classification model is composed to filter out negative candidate regions. By combining results from both models, the segmentation model successfully suppress false positives and generate best results, with an example shown in Figure \[fig:segmentation\] Dealing with Systematic Errors ============================== Finding Systematic Errors ------------------------- The precision and recall numbers we measure by randomly sampling from the mask candidates do not account for systematic errors arising from varying satellite image quality. To discover those systematic errors, we adopt both visually inspection and evaluation using external data. Intuitively, we visualize our results by construction *KMZ* files and overlaying with Google Earth to manually pinpoint areas of concern. We also use this strategy to sample *ambiguous* training data to fine-tune our model to reduce the chance of further systematic errors. Moreover, we also quantitatively measure systematic errors at a coarser scale by comparing our results with external datasets on those areas with adequate data coverage. However, it is still an open question to discover systematic errors on large scale with less manual work. Data Quality ------------ One of the reasons for systematic errors is also issues with data quality. The satellite images are taken at various times of day, and pre-processed across multiple layers for the highest quality image. However, areas with a lot of cloud cover tend to have much fewer clear images taken, and so quality suffers. This has an impact on our model, since most of the data is randomly or semi-randomly sampled, and so it does not get a lot of exposure to these poorer quality images during training. We use geographical meta-information to further detect the cloud occlusion during deploying stage. Another key factor of low data quality comes from noise, which are introduced in either imaging or image enhancing phases. Traditional image denoising approach such as BM3D [@dabov2006image] is computationally expensive in handling large imagery files, and can only work for limited type of noises, such as white noise. To this end, we train a shallow neural network end-to-end by mimicking several kinds of noise existed in satellite images. The trained denoising model is appended as a transformer before imagery is fed to the classification network. Comparison of classification results of the same low data quality area before and after denoising is shown in Figure \[fig:denoising\]. Results ======= Overall the SegNet model by itself achieves a precision and recall of $pr=0.9$, $re=0.89$ on a global dataset where the imbalance is such that $93\%$ of the randomly sampled testing data is not a building. Below we have some heat maps generated of building density in three countries: Mozambique, Madagascar, and India. So far we have released datasets for 5 countries: Haiti, Malawi, Ghana, South Africa, and Sri Lanka. The rest are pending validation with third party groups. Below we show precision recall curves and best F-score with confidence intervals for each of the countries released. The estimation of population density via settlement buildings as a proxy results in significant improvement compared with previous efforts. Figure \[fig:stateofart\] shows the comparison of previous highest resolution estimation from Galantis and our own results. This gives a totally new perspective to various social / economic research. ![Comparison of Galantis and our results[]{data-label="fig:stateofart"}](./stateofart.png){width="50.00000%"} Conclusion ========== We have built one of the first building detection systems that can be deployed at a global scale. Future work includes reducing the amount of iteration required to achieve a robust model as we roll out to more countries, the biggest problem of which is detecting systematic errors. Detecting and solving these systematic issues in classification is still a work in progress. We are still looking into ways to automate the data validation process and data collection methods further, which will also shorten the length of each iteration required to improve our dataset accuracy. [^1]: Authors are of equal contribution

--- abstract: 'In this paper we address the problem of pool based active learning, and provide an algorithm, called UPAL, that works by minimizing the unbiased estimator of the risk of a hypothesis in a given hypothesis space. For the space of linear classifiers and the squared loss we show that UPAL is equivalent to an exponentially weighted average forecaster. Exploiting some recent results regarding the spectra of random matrices allows us to establish consistency of UPAL when the true hypothesis is a linear hypothesis. Empirical comparison with an active learner implementation in Vowpal Wabbit, and a previously proposed pool based active learner implementation show good empirical performance and better scalability.' author: - | Ravi Ganti, Alexander Gray\ School of Computational Science & Engineering, Georgia Tech\ gmravi2003@gatech.edu, agray@cc.gatech.edu bibliography: - 'upal\_arxiv.bib' title: 'UPAL: Unbiased Pool Based Active Learning' --- Introduction ============ In the problem of binary classification one has a distribution $\cD$ on the domain $\cX\times\cY\subseteq \bbR^d\times\{-1,+1\}$, and access to a sampling oracle, which provides us i.i.d. labeled samples $\cS=\{(x_1,y_1),\ldots,(x_n,y_n)\}$. The task is to learn a classifier $h$, which predicts well on unseen points. For certain problems the cost of obtaining labeled samples can be quite expensive. For instance consider the task of speech recognition. Labeling of speech utterances needs trained linguists, and can be a fairly tedious task. Similarly in information extraction, and in natural language processing one needs expert annotators to obtain labeled data, and gathering huge amounts of labeled data is not only tedious for the experts but also expensive. In such cases it is of interest to design learning algorithms, which need only a few labeled examples for training, and also guarantee good performance on unseen data. Suppose we are given a labeling oracle $\cO$, which when queried with an unlabeled point $x$ returns the label $y$ of $x$. Active learning algorithms query this oracle as few times as possible and learn a provably good hypothesis from these labeled samples. Broadly speaking active learning (AL) algorithms can be classified into three kinds, namely membership query (MQ) based algorithms, stream based algorithms and pool based algorithms. All these three kinds of AL algorithms query the oracle $\cO$ for the label of the point, but differ in the nature of the queries. In MQ based algorithms the active learner can query for the label of a point in the input space $\cX$, but this query might not necessarily be from the support of the marginal distribution $\cD_{\cX}$. With human annotators MQ algorithms might work poorly as was demonstrated by Lang and Baum in the case of handwritten digit recognition [-@baum1992query], where the annotators were faced with the awkward situation of labeling semantically meaningless images. Stream based AL algorithms [@cohn1994improving; @chu2011unbiased] sample a point $x$ from the marginal distribution $\cD_{\cX}$, and decide on the fly whether to query $\cO$ for the label of $x$? Stream based AL algorithms tend to be computationally efficient, and most appropriate when the underlying distribution changes with time. Pool based AL algorithms assume that one has access to a large pool $\cP=\{x_1,\ldots,x_n\}$ of unlabeled i.i.d. examples sampled from $\cD_{\cX}$, and given budget constraints $B$, the maximum number of points they are allowed to query, query the most informative set of points. Both pool based AL algorithms, and stream based AL algorithms overcome the problem of awkward queries, which MQ based algorithms face. However in our experiments we discovered that stream based AL algorithms tend to query more points than necessary, and have poorer learning rates when compared to pool based AL algorithms. Contributions. -------------- 1. In this paper we propose a pool based active learning algorithm called UPAL, which given a hypothesis space $\cH$, and a margin based loss function $\phi(\cdot)$ minimizes a provably unbiased estimator of the risk $\bbE[\phi(y h(x))]$. While unbiased estimators of risk have been used in stream based AL algorithms, no such estimators have been introduced for pool based AL algorithms. We do this by using the idea of importance weights introduced for AL in Beygelzimer et al. [-@beygelzimer2009importance]. Roughly speaking UPAL proceeds in rounds and in each round puts a probability distribution over the entire pool, and samples a point from the pool. It then queries for the label of the point. The probability distribution in each round is determined by the current active learner obtained by minimizing the importance weighted risk over $\cH$. Specifically in this paper we shall be concerned with linear hypothesis spaces, i.e. $\cH=\bbR^d$. 2. In theorem \[thm:ewa\] (Section \[sec:ewa\]) we show that for the squared loss UPAL is equivalent to an exponentially weighted average (EWA) forecaster commonly used in the problem of learning with expert advice [@cesa2006prediction]. Precisely we show that if each hypothesis $h\in\cH$ is considered to be an expert and the importance weighted loss on the currently labeled part of the pool is used as an estimator of the risk of $h\in\cH$, then the hypothesis learned by UPAL is the same as an EWA forecaster. Hence UPAL can be seen as pruning the hypothesis space, in a soft manner, by placing a probability distribution that is determined by the importance weighted loss of each classifier on the currently labeled part of the pool. 3. In section \[sec:consistency\] we prove consistency of UPAL with the squared loss, when the true underlying hypothesis is a linear hypothesis. Our proof employs some elegant results from random matrix theory regarding eigenvalues of sums of random matrices [@hsu2011analysis; @hsu2011dimension; @tropp2010user]. While it should be possible to improve the constants and exponent of dimensionality involved in $n_{0,\delta},T_{0,\delta},T_{1,\delta}$ used in theorem \[thm:main\], our results qualitatively provide us the insight that the the label complexity with the squared loss will depend on the condition number, and the minimum eigenvalue of the covariance matrix $\Sigma$. This kind of insight, to our knowledge, has not been provided before in the literature of active learning. 4. In section \[sec:expts\] we provide a thorough empirical analysis of UPAL comparing it to the active learner implementation in Vowpal Wabbit (VW) [@langford2010vowpal], and a batch mode active learning algorithm, which we shall call as BMAL [@hoi2006batch]. These experiments demonstrate the positive impact of importance weighting, and the better performance of UPAL over the VW implementation. We also empirically demonstrate the scalability of UPAL over BMAL on the MNIST dataset. When we are required to query a large number of points UPAL is upto 7 times faster than BMAL. Algorithm Design {#sec:alg_design} ================ A good active learning algorithm needs to take into account the fact that the points it has queried might not reflect the true underlying marginal distribution. This problem is similar to the problem of dataset shift [@quinonero2008dataset] where the train and test distributions are potentially different, and the learner needs to take into account this bias during the learning process. One approach to this problem is to use importance weights, where during the training process instead of weighing all the points equally the algorithm weighs the points differently. UPAL proceeds in rounds, where in each round $t$, we put a probability distribution $\{p_i^t\}_{i=1}^n$ on the entire pool $\cP$, and sample one point from this distribution. If the sampled point was queried in one of the previous rounds $1,\ldots,t-1$ then its queried label from the previous round is reused, else the oracle $\cO$ is queried for the label of the point. Denote by $\Qit\in\{0,1\}$ a random variable that takes the value 1 if the point $x_i$ was queried for it’s label in round $t$ and 0 otherwise. In order to guarantee that our estimate of the error rate of a hypothesis $h\in\cH$ is unbiased we use importance weighting, where a point $x_i\in\cP$ in round $t$ gets an importance weight of $\frac{\Qit}{\pit}$. Notice that by definition $\bbE[\Qit|\pit]=1$. We formally prove that importance weighted risk is an unbiased estimator of the true risk. Let $\cDn$ denote a product distribution on ${(x_1,y_1),\ldots,(x_n,y_n)}$. Also denote by $Q_{1:n}^{1:t}$ the collection of random variables $Q_{1}^1,\ldots,Q_{n}^1,\ldots, Q_{n}^t$. Let $\langle \cdot,\cdot \rangle$ denote the inner product. We have the following result. \[thm:unbiased\] Let $\hatLth{\mbox{$\;\stackrel{\mbox{\tiny\rm def}}{=}\;$}}\frac{1}{nt}\sum_{i=1}^n \sum_{\tau=1}^t \frac{\Qitau}{\pitau} \phi(y_i\langle h,x_i\rangle),$ where $p_i^{\tau}>0$ for all $\tau=1,\ldots,t$. Then $$\bbE_{Q_{1}^{1},\ldots,Q_{n}^{t},\cDn} \hatLth=L(h).$$ $$\begin{gathered} \bbE_{Q_{1:n}^{1:t},\cDn}\hatLth =\bbE_{Q_{1:n}^{1:t},\cDn} \frac{1}{nt}\sum_{i=1}^n\sum_{\tau=1}^t \frac{\Qitau}{\pitau} \phi(y_i\langle h,x_i\rangle)\nonumber =\bbE_{Q_{1:n}^{1:t},\cDn} \frac{1}{nt}\sum_{i=1}^n\sum_{\tau=1}^t \bbE_{Q_{i}^\tau|Q_{1:n}^{1:\tau-1},\cDn} \frac{\Qitau}{\pitau} \phi(y_i\langle h,x_i\rangle) =\\\bbE_{\cDn} \frac{1}{nt}\sum_{i=1}^n\sum_{\tau=1}^t \phi(y_i\langle h,x_i\rangle)=L(w).\qedhere \end{gathered}$$ The theorem guarantees that as long as the probability of querying any point in the pool in any round is non-zero $\hat{L}_{t}(h)$, will be an unbiased estimator of $L(h)$. How does one come up with a probability distribution on $\cP$ in round $t$? To solve this problem we resort to probabilistic uncertainty sampling, where the point whose label is most uncertain as per the current hypothesis, $h_{A,t-1}$, gets a higher probability mass. The current hypothesis is simply the minimizer of the importance weighted risk in $\cH$, i.e. $h_{A,t-1}=\arg\min_{h\in\cH} \hat{L}_{t-1}(h)$. For any point $x_i\in\cP$, to calculate the uncertainty of the label $y_i$ of $x_i$, we first estimate $\eta(x_i){\mbox{$\;\stackrel{\mbox{\tiny\rm def}}{=}\;$}}\bbP[y_i=1|x_i]$ using $h_{A,t-1}$, and then use the entropy of the label distribution of $x_i$ to calculate the probability of querying $x_i$. The estimate of $\eta(\cdot)$ in round $t$ depends both on the current active learner $h_{A,t-1}$, and the loss function. In general it is not possible to estimate $\eta(\cdot)$ with arbitrary convex loss functions. However it has been shown by Zhang [-@zhang2004statistical] that the squared, logistic and exponential losses tend to estimate the underlying conditional distribution $\eta(\cdot)$. Steps 4, 11 of algorithm \[alg:poolal\] depend on the loss function $\phi(\cdot)$ being used. If we use the logistic loss i.e $\phi(yz)=\ln(1+\exp(-yz))$ then $\hat{\eta_t}(x)=\frac{1}{1+\exp(-yh_{A,t-1}^Tx)}$. In case of squared loss $\hat{\eta_t}(x)=\min\{\max\{0,w_{A,t-1}^Tx\},1\}$. Since the loss function is convex, and the constraint set $\cH$ is convex, the minimization problem in step 11 of the algorithm is a convex optimization problem. 1\. Set num\_unique\_queries=0, $h_{A,0}=0$, $t=1$. 2. Set $\Qit=0$ for all $i=1,\ldots,n$. 3. Set $p_{\text{min}}^{t}=\frac{1}{nt^{1/4}}$. 4. Calculate $\hat{\eta_t}(x_i)=\bbP[y=+1|x_i,h_{A,{t-1}}]$. 5. Assign $\pit=p_{\text{min}}^{t}+(1-np_{\text{min}}^t)\frac{\hat{\eta}_t(x_i)\ln(1/\hat{\eta}_t(x))+(1-\hat{\eta}_t(x_i))\ln(1/(1-\hat{\eta}_t(x_i)))}{\sum_{j=1}^n\hat{\eta}_t(x_j)\ln(1/\hat{\eta}_t(x_j))+(1-\hat{\eta}_t(x_j))\ln(1/(1-\hat{\eta}_t(x_j)))}$. 6. Sample a point (say $x_j$) from $p^t(\cdot)$. 7. Reuse its previously queried label $y_j$. 8. Query oracle $\cO$ for its label $y_j$. 9. . 10. Set $Q_j^t=1$. 11. Solve the optimization problem: $h_{A,t}=\arg\min_{h\in \cH} \sum_{i=1}^n\sum_{\tau=1}^t \frac{Q_{i}^{\tau}}{p_i^{\tau}}\phi(y_ih^Tx_i)$. 12. $t\leftarrow t+1$. 13. Return $h_{A}{\mbox{$\;\stackrel{\mbox{\tiny\rm def}}{=}\;$}}h_{A,t}$ By design UPAL might requery points. An alternate strategy is to not allow requerying of points. However the importance weighted risk may not be an unbiased estimator of the true risk in such a case. Hence in order to retain the unbiasedness property we allow requerying in UPAL. The case of squared loss {#sec:ewa} ------------------------ It is interesting to look at the behaviour of UPAL in the case of squared loss where $\phi(yh^Tx)=(1-yh^Tx)^2$. For the rest of the paper we shall denote by $\ha$ the hypothesis returned by UPAL at the end of $T$ rounds. We now show that the prediction of $\ha$ on any $x$ is simply the exponentially weighted average of predictions of all $h$ in $\cH$. \[thm:ewa\] Let $$\begin{aligned} z_i{\mbox{$\;\stackrel{\mbox{\tiny\rm def}}{=}\;$}}&\sum_{t=1}^T\frac{\Qit}{\pit} &\Sighz&{\mbox{$\;\stackrel{\mbox{\tiny\rm def}}{=}\;$}}\sum_{i=1}^n z_ix_ix_i^T\\ v_z{\mbox{$\;\stackrel{\mbox{\tiny\rm def}}{=}\;$}}& \sum_{i=1}^{n} z_iy_ix_i &c{\mbox{$\;\stackrel{\mbox{\tiny\rm def}}{=}\;$}}& \sum_{i=1}^n z_i. \end{aligned}$$ Define $w\in\bbR^d$ as $$\label{eqn:w} w=\frac{\int_{\bbR^d} \exp(-\hat{L}_{T}(h))h~\mathrm{d}h}{\int_{\bbR^d}\exp(-\hat{L}_{T}(h))~\mathrm{d}h}.$$ Assuming $\Sighz$ is invertible we have for any $x_0\in \bbR^d$, $w^Tx_0=h_A^Tx_0$. By elementary linear algebra one can establish that $$\begin{aligned} \ha&=\Sighzi v_z\label{eqn:ha}\\ \hat{L}_{T}(h)&=(h-\Sighzi v_z)\Sighz(h-\Sighzi v-z). \end{aligned}$$ Using standard integrals we get $$Z{\mbox{$\;\stackrel{\mbox{\tiny\rm def}}{=}\;$}}\int_{\bbR^d}\exp(-\hat{L}_{T}(h))~\mathrm{d}h= \exp(-c-v_z^T\Sighzi v_z)\sqrt{\pi^d}\sqrt{\det(\Sighzi)}.$$ In order to calculate $w^Tx_0$, it is now enough to calculate the integral $$I{\mbox{$\;\stackrel{\mbox{\tiny\rm def}}{=}\;$}}\int_{\bbR^d} \exp(-\hat{L}_{T}(h))~h^Tx_0~\mathrm{d}w.$$ To solve this integral we proceed as follows. Define $I_1=\int_{\bbR^d}\exp(-\hat{L}_{T}(h))~ h^Tx_0~\mathrm{d}h$. By simple algebra we get $$\begin{aligned} I&=\int_{\bbR^d} \exp(-w^T\Sighz w+2w^Tv_z-c)~ w^Tx_0~\mathrm{d}w\\ &=\exp(-c-v_z^T\Sighzi v_z)I_1.\label{eqn:eqn_I} \end{aligned}$$ Let $a=h-\Sighzi v_z$. We then get $$\begin{aligned} I_1&=\int_{\bbR^d}h^Tx_0\exp\left(-(h-\Sighzi v_z)\Sighz(h-\Sighzi v_z)\right)~\mathrm{d}h\nonumber\\ &=\int_{\bbR^d} (a^Tx_0+v_z^T\Sighzi x_0)\exp(-a^T\Sighz a)~\mathrm{d}a\nonumber\\ &=\underbrace{\int_{\bbR^d} (a^Tx_0)\exp(-a^T\Sighz a)~\mathrm{d}a}_{I_2}+\nonumber \underbrace{\int_{\bbR^d} v_z^T\Sighzi x_0\exp(-a^T\Sighz a)~\mathrm{d}a}_{I_3}.\label{eqn:eqn_2}\end{aligned}$$ Clearly $I_2$ being the integrand of an odd function over the entire space calculates to 0. To calculate $I_3$ we shall substitute $\Sighz=SS^T$, where $S\succ 0$. Such a decomposition is possible since $\Sighz\succ 0$. Now define $z=S^Ta$. We get $$\begin{aligned} I_3&=v_z^T\Sighzi x_0\int \exp(-z^Tz)~\det(S^{-1})~\mathrm{d}z\\ &=v_z^T\Sighzi x_0 \det(S^{-1})\sqrt{\pi^{d}}.\label{eqn:eqn_for_I3}\end{aligned}$$ Using equations (\[eqn:eqn\_I\], \[eqn:eqn\_2\], \[eqn:eqn\_for\_I3\]) we get $$\begin{gathered} I=(\sqrt{\pi})^dv_z^T\Sighzi x_0 ~\det(S^{-1})\exp(-c-v_z^T\Sighzi v_z). \end{gathered}$$ Hence we get $$w^Tx_0=v_z^T\Sighzi x_0\frac{\det(S^{-1})}{\sqrt{\det(M^{-1})}}=v_z^T\Sighzi x_0=\ha^Tx_0,$$ where the penultimate equality follows from the fact that $\det(\Sighzi)=1/\det(\Sighz)=1/(\det(SS^T))=1/(\det(S))^2$, and the last equality follows from equation \[eqn:ha\]. Theorem \[thm:ewa\] is instructive. It tells us that assuming that the matrix $\Sighz$ is invertible, $\ha$ is the same as an exponentially weighted average of all the hypothesis in $\cH$. Hence one can view UPAL as learning with expert advice, in the stochastic setting, where each individual hypothesis $h\in\cH$ is an expert, and the exponential of $\hat{L}_{T}$ is used to weigh the hypothesis in $\cH$. Such forecasters have been commonly used in learning with expert advice. This also allows us to interpret UPAL as pruning the hypothesis space in a soft way via exponential weighting, where the hypothesis that has suffered more cumulative loss gets lesser weight. Bounding the excess risk {#sec:consistency} ======================== It is natural to ask if UPAL is consistent? That is will UPAL do as well as the optimal hypothesis in $\cH$ as $n\rightarrow \infty,T\rightarrow \infty$? We answer this question in affirmative. We shall analyze the excess risk of the hypothesis returned by our active learner, denoted as $h_{A}$, after $T$ rounds when the loss function is the squared loss. The prime motivation for using squared loss over other loss functions is that squared losses yield closed form estimators, which can then be elegantly analyzed using results from random matrix theory [@hsu2011analysis; @hsu2011dimension; @tropp2010user]. It should be possible to extend these results to other loss functions such as the logistic loss, or exponential loss using results from empirical process theory [@vandegeer2000empirical]. Main result ----------- \[thm:main\] Let $(x_1,y_1),\ldots (x_n,y_n)$ be sampled i.i.d from a distribution. Suppose assumptions A0-A3 hold. Let $\delta\in(0,1)$, and suppose $n\geq n_{0,\delta},T\geq \max\{T_{0,\delta},T_{1,\delta}\}$. With probability atleast $1-10\delta$ the excess risk of the active learner returned by UPAL after $T$ rounds is $$L(h_A)-L(\beta)= O\left(\frac{1}{n}+\frac{n}{\sqrt{T}}(d+2\sqrt{d\ln(1/\delta)}+2\ln(1/\delta))\right).$$ Assumptions, and Notation. -------------------------- 1. (Invertibility of $\Sigma$) The data covariance matrix $\Sigma$ is invertible. 2. (Statistical leverage condition) There exists a finite $\gamma_0\geq 1$ such that almost surely $$||\Sig^{-1/2}{x}||\leq \gamma_0\sqrt{d}.$$ 3. There exists a finite $\gamma_1\geq 1$ such that $\bbE[\exp(\alpha^Tx)]\leq \exp\left(\frac{||\alpha||^2\gamma_1^2}{2}\right)$. 4. (Linear hypothesis) We shall assume that $y=\beta^Tx+\xi(x)$, where $\xi(x)\in [-2,+2]$ is additive noise with $\bbE[\xi(x)|x]=0$. Assumption A0 is necessary for the problem to be well defined. A1 has been used in recent literature to analyze linear regression under random design and is a Bernstein like condition [@rokhlin2008fast]. A2 can be seen as a softer form of boundedness condtion on the support of the distribution. In particular if the data is bounded in a d-dimensional unit cube then it suffices to take $\gamma_1=1/2$. It may be possible to satisfy A3 by mapping data to kernel spaces. Though popularly used kernels such as Gaussian kernel map the data to infinite dimensional spaces, a finite dimensional approximation of such kernel mappings can be found by the use of random features [@rahimi2007random]. **Notation.** 1. $h_A$ is the active learner outputted by our active learning algorithm at the end of $T$ rounds. 2. $$\begin{aligned} \forall i=1,\ldots,n: z_i{\mbox{$\;\stackrel{\mbox{\tiny\rm def}}{=}\;$}}\sum_{t=1}^T\frac{\Qit}{\pit} &\hspace{30pt} \Sighz{\mbox{$\;\stackrel{\mbox{\tiny\rm def}}{=}\;$}}\sum_{i=1}^n z_ix_ix_i^T\\ \psi_z{\mbox{$\;\stackrel{\mbox{\tiny\rm def}}{=}\;$}}\sum_{i=1}^nz_i\xi(x_i)x_i& \hspace{30pt}\Sigh{\mbox{$\;\stackrel{\mbox{\tiny\rm def}}{=}\;$}}\frac{1}{n}\sum_{i=1}^n x_ix_i^T\\ \Sig{\mbox{$\;\stackrel{\mbox{\tiny\rm def}}{=}\;$}}\bbE[xx^T]& \hspace{30pt}\Sighz{\mbox{$\;\stackrel{\mbox{\tiny\rm def}}{=}\;$}}\sum_{i=1}^n z_ix_ix_i^T\\ n_{0,\delta}{\mbox{$\;\stackrel{\mbox{\tiny\rm def}}{=}\;$}}7200d^2\gamma_0^4(d\ln(5)+\ln(10/\delta))&\hspace{30pt} T_{1,\delta}{\mbox{$\;\stackrel{\mbox{\tiny\rm def}}{=}\;$}}12+512\sqrt{2}d^{8/3}\gamma_0^{16/3}\ln^{4/3}(d/\delta) \end{aligned}$$ $$T_{0,\delta}{\mbox{$\;\stackrel{\mbox{\tiny\rm def}}{=}\;$}}\gamma_1^{16/3}d^{8/3}\ln^{4/3}(d/\delta)\ln^{8/3}(n/\delta)\lambdamin^{8/3}(\Sigma)+4\ln(d/\delta)\frac{\lambdamax(\Sigma)}{\lambdamin(\Sigma)},$$ where $\delta\in(0,1)$. Overview of the proof --------------------- The excess risk of a hypothesis $h\in\cH$ is defined as $L(h)-L(\beta)=\bbE_{x, y\sim\cD} [(y-h^Tx)^2-(y-\beta^Tx)^2]$. Our aim is to provide high probability bounds for the excess risk, where the probability measure is w.r.t the sampled points $(x_1,y_1),\ldots,(x_n,y_n), Q_1^1,\ldots,Q_{n}^T$. The proof proceeds as follows. 1. In lemma \[lem:decompose\], assuming that the matrices $\Sighz,\Sigh$ are invertible we upper bound the excess risk as the product $||\Sig^{1/2}\Sighzi\Sig^{1/2}||^2||\Sig^{-1/2}\Sigh^{1/2}||^2 ||\Sigh^{-1/2}\psi_z||^2$. The prime motivation in doing so is that bounding such “squared norm” terms can be reduced to bounding the maximum eigenvalue of random matrices, which is a well studied problem in random matrix theory. 2. In lemma \[lem:1\] we provide an upper bound for $||\Sig^{-1/2}\Sigh^{1/2}||^2$. To do this we use the simple fact that the matrix 2-norm of a positive semidefinite matrix is nothing but the maximum eigenvalue of the matrix. With this obsercation, and by exploiting the structure of the matrix $\Sigh$, the problem reduces to giving probabilistic upper bounds for maximum eigenvalue of a sum of random rank-1 matrices. Theorem \[thm:litvak\] provides us with a tool to prove such bounds. 3. In lemma \[lem:2\] we bound $||\Sig^{1/2}\Sighzi\Sig^{1/2}||^2$. The proof is in the same spirit as in lemma \[lem:1\], however the resulting probability problem is that of bounding the maximum eigenvalue of a sum of random matrices, which are not necessarily rank-1. Theorem \[thm:mat\_bern\] provides us with Bernstein type bounds to analyze the eigenvalues of sums of random matrices. 4. In lemma \[lem:3\] we bound the quantity $||\Sigh^{-1/2}\psi_z||^2$. Notice that here we are bounding the squared norm of a random vector. Theorem \[thm:quadratic\] provides us with a tool to analyze such quadratic forms under the assumption that the random vector has sub-Gaussian exponential moments behaviour. 5. Finally all the above steps were conditioned on the invertibility of the random matrices $\Sigh,\Sighz$. We provide conditions on $n,T$ (this explains why we defined the quantities $n_{0,\delta},T_{0,\delta},T_{1,\delta}$) which guarantee the invertibility of $\Sigh,\Sighz$. Such problems boil down to calculating lower bounds on the minimum eigenvalue of the random matrices in question, and to establish such lower bounds we once again use theorems \[thm:litvak\], \[thm:mat\_bern\]. Full Proof ---------- We shall now provide a way to bound the excess risk of our active learner hypothesis. Suppose $\ha$ was the hypothesis represented by the active learner at the end of the T rounds. By the definition of our active learner and the definition of $\beta$ we get $$\begin{aligned} \ha&=\arg\min_{h\in\cH} ~\sum_{i=1}^n \sum_{t=1}^T\frac{\Qit}{\pit} (y_i-h^Tx_i)^2=\sum_{i=1}^n z_i (y_i-h^Tx_i)^2=\Sighzi v_z\\ \beta&=\arg\min_{h\in\cH}\bbE(y-\beta^Tx)^2=\Sigi\bbE[yx].\end{aligned}$$ \[lem:decompose\] Asumme $\Sighz,\Sigh$ are both invertible, and assumption A0 applies. Then the excess risk of the classifier after $T$ rounds of our active learning algorithm is given by $$\label{eqn:decompose} L(h_{A})-L(\beta)\leq ||\Sig^{1/2}\Sighzi\Sig^{1/2}||^2||\Sig^{-1/2}\Sigh^{1/2}||^2 ||\Sigh^{-1/2}\psi_z||^2.$$ $$\begin{aligned} L(\ha)-L(\beta)&=\bbE[(y-\ha^Tx)^2-(y-\beta^Tx)^2]\nonumber\\ &=\bbE_{x,y}[\ha^Txx^T\ha-2y\ha^Tx-\beta^Txx^T\beta+2y\beta^Tx]\nonumber\\ &=\ha^T\Sigma \ha-2\ha^T\bbE[xy]-\beta^T\Sigma\beta+2\beta^T\Sigma\beta \text{~[Since $\Sig\beta=\bbE[yx]$]}\nonumber\\ &=\ha^T\Sigma \ha-\beta^T\Sigma\beta-2\ha^T\Sigma\beta+2\beta^T\Sigma\beta \nonumber\\ &=\ha^T\Sigma \ha+\beta^T\Sigma\beta-2\ha^T\Sigma\beta \nonumber\\ &=||\Sigma^{1/2}(\ha-\beta)||^2\label{eqn:exrisk1}. \end{aligned}$$ We shall next bound the quantity $||\ha-\beta||$ which will be used to bound the excess risk in Equation ( \[eqn:exrisk1\]). To do this we shall use assumption A3 along with the definitions of $\ha,\beta$. We have the following chain of inequalities. $$\begin{aligned} \ha&=\Sighzi v_z\nonumber\\ &=\Sighzi\sum_{i=1}^n z_i y_ix_i\nonumber\\ &=\Sighzi\sum_{i=1}^n z_i(\beta^Tx_i+\xi(x_i))x_i\nonumber\\ &=\Sighzi\sum_{i=1}^n z_i x_ix_i^T\beta+z_i\xi(x_i)x_i\nonumber\\ &=\beta+\Sighzi\sum_{i=1}^n z_i\xi(x_i)x_i=\beta+\Sighzi\psi_z.\label{eqn:habd}\end{aligned}$$ Using Equations  \[eqn:exrisk1\],\[eqn:habd\] we get the following series of inequalities for the excess risk bound $$\begin{aligned} L(\ha)-L(\beta)&=||\Sig^{1/2}\Sighzi\psiz||^2\nonumber\\ &=||\Sig^{1/2}\Sighzi\Sigh^{1/2}\Sigh^{-1/2}\psiz||^2\nonumber\\ &=||\Sig^{1/2}\Sighzi\Sig^{1/2}\Sig^{-1/2}\Sigh^{1/2}\Sigh^{-1/2}\psiz||^2\\ &\leq ||\Sig^{1/2}\Sighzi\Sig^{1/2}||^2||\Sig^{-1/2}\Sigh^{1/2}||^2 ||\Sigh^{-1/2}\psi_z||^2.\qedhere\end{aligned}$$ The decomposition in lemma \[lem:decompose\] assumes that both $\Sighz,\Sigh$ are invertible. Before we can establish conditions for the matrices $\Sighz,\Sigh$ to be invertible we need the following elementary result. \[prop:expmoments\_statlev\] For any arbitrary $\alpha\in \bbR^d$, under assumption A1 we have $$\bbE[\exp(\alpha^T\Sig^{-1/2}x)]\leq 5\exp\left(\frac{3d\gamma_0^2||\alpha||^2}{2}\right).$$ From Cauchy-Schwarz inequality and A1 we get $$-||\alpha||\gamma_0\sqrt{d} \leq -||\alpha||~||\Sig^{-1/2}x|| \leq \alpha^T\Sig^{-1/2}x\leq ||\alpha||~||\Sig^{-1/2}x||\leq ||\alpha||\gamma_0\sqrt{d} .$$ Also $\bbE[\alpha^T\Sig^{-1/2}x]\leq ||\alpha||\gamma_0\sqrt{d}$. Using Hoeffding’s lemma we get $$\begin{aligned} \bbE[\exp(\alpha^T\Sig^{-1/2}x)]&\leq \exp\left(||\alpha||\gamma_0\sqrt{d}+\frac{||\alpha||^2d\gamma_0^2}{2}\right)\\ &\leq 5\exp(3||\alpha||^2d\gamma_0^2/2).\qedhere\end{aligned}$$ The following lemma will be useful in bounding the terms $||\Sig^{1/2}\Sighzi\Sig^{1/2}||$, $||\Sig^{-1/2}\Sigh^{1/2}||^2$. \[lem:J\] Let $J{\mbox{$\;\stackrel{\mbox{\tiny\rm def}}{=}\;$}}\sum_{i=1}^n \Sigma^{-1/2}x_ix_i^T\Sigma^{-1/2}$. Let $n\geq n_{0,\delta}$. Then the following inequalities hold separately with probability atleast $1-\delta$ each $$\begin{aligned} \lambdamax(J)\leq n+6dn\gamma_0^2\left[\sqrt{\frac{32(d\ln(5)+\ln(10/\delta))}{n}}+\frac{2(d\ln(5)+\ln(10/\delta))}{n}\right]\leq 3n/2\\ \lambdamin(J)\geq n-6dn\gamma_0^2\left[\sqrt{\frac{32(d\ln(5)+\ln(10/\delta))}{n}}+\frac{2(d\ln(5)+\ln(10/\delta))}{n}\right]\geq n/2. \end{aligned}$$ Notice that $\bbE[\Sig^{-1/2}x_ix_i^T\Sig^{-1/2}]=I.$ From Proposition \[prop:expmoments\_statlev\] we have $\bbE[\exp(\alpha^T\Sig^{-1/2}x)]\leq 5\exp(3||\alpha||^2 d\gamma_0^2/2)$. By using theorem \[thm:litvak\] we get with probability atleast $1-\delta$: $$\lambdamax\left(\frac{1}{n}\sum_{i=1}^n (\Sig^{-1/2}x_i)(\Sig^{-1/2}x_i)^T\right)\leq 1+6d\gamma_0^2\left[\sqrt{\frac{32(d\ln(5)+\ln(2/\delta))}{n}}+\frac{2(d\ln(5)+\ln(2/\delta))}{n}\right].$$ Put $n\geq n_{0,\delta}$ to get the desired result. The lower bound on $\lambdamin$ is also obtained in the same way. \[lem:inv\_sigh\] Let $n\geq n_{0,\delta}$. With probability atleast $1-\delta$ separately we have $\Sigh\succ 0$, $\lambdamin(\Sigh)\geq \frac{1}{2}\lambdamin(\Sigma)$, $\lambdamax(\Sigh)\leq \frac{3}{2}\lambdamax(\Sigma)$. Using lemma \[lem:J\] we get for $n\geq n_{0,\delta}$ with probability atleast $1-\delta$, $\lambdamin(J)\geq 1/2$ and with probability atleast $1-\delta$, $\lambdamax(\Sigma)\leq 3/2$. Finally since $\Sigma^{1/2}J\Sigma^{1/2}=\Sigh$, and $J\succ 0,\Sigma\succ 0$, we get $\Sigh\succ 0$. Further we have the following upper bound with probability atleast $1-\delta$: $$\begin{aligned} \label{eqn:sig_ub} \lambdamax(\Sigh)&=||\Sigma^{1/2}J\Sigma^{1/2}||\\ &\leq ||\Sigma^{1/2}||^2~||J|| \\ &\leq ||\Sigma||~||J||\\ &=\lambdamax(\Sigma)\lambdamax(J)\\ &\leq \frac{3}{2} \lambdamax(\Sigma),\end{aligned}$$ where in the last step we used the upper bound on $\lambdamax(J)$ provided by lemma \[lem:J\]. Similarly we have the following lower bound with probability atleast $1-\delta$ $$\begin{aligned} \label{eqn:sig_lb} \lambdamin(\Sigh)&=\frac{1}{\lambdamax(\Sig^{-1/2}J^{-1}\Sig^{-1/2})}\\ &=\frac{1}{||\Sig^{-1/2}J^{-1}\Sig^{-1/2}||}\\ &\geq \frac{1}{||\Sig^{-1}||~||J^{-1}||~||\Sig^{-1/2}||}\\ &=\lambdamin(\Sigma)\lambdamin(J)\\ &\geq\frac{\lambdamin(\Sigma)}{2},\end{aligned}$$ where in the last step we used the lower bound on $\lambdamin(J)$ provided by lemma \[lem:J\]. The following proposition will be useful in proving lemma \[lem:inv\_sighz\]. \[prop:normxibound\] Let $\delta\in(0,1)$. Under assumption A2, with probability atleast $1-\delta$, $\sum_{i=1}^n||x_i||^4\leq 25\gamma_1^4d^2\ln^2(n/\delta)$ From A2 we have $\bbE[\exp(\alpha^Tx)]\leq \exp(\frac{||\alpha||^2\gamma_1^2}{2})$. Now applying theorem \[thm:quadratic\] with $A=I_{d}$ we get $$\bbP[||x_i||^2\leq d\gamma_1^2+2\gamma_1^2\sqrt{d\ln(1/\delta)}+2\gamma_1^2\ln(1/\delta)]\geq 1-\delta.$$ The result now follows by the union bound. \[lem:inv\_sighz\] Let $\delta\in(0,1)$. For $T\geq T_{0,\delta}$, with probability atleast $1-4\delta$ we have $\lambdamin(\Sighz)\geq \frac{nT\lambdamin(\Sigma)}{4}>0$. Hence $\Sighz$ is invertible. The proof uses theorem \[thm:mat\_bern\]. Let $M_t'{\mbox{$\;\stackrel{\mbox{\tiny\rm def}}{=}\;$}}\sum_{i=1}^n \frac{\Qit}{\pit} x_ix_i^T$, so that $\Sighz=\sum_{t=1}^T M_t'$. Now $\bbE_tM_t'=n\Sigh$. Define $R_t'{\mbox{$\;\stackrel{\mbox{\tiny\rm def}}{=}\;$}}n\Sigh-M_t'$, so that $\bbE_t R_t'=0$. We shall apply theorem \[thm:mat\_bern\] to the random matrix $\sum R_t'$. In order to do so we need upper bounds on $\lambdamax (R_t')$ and $\lambdamax (\frac{1}{T}\sum_{t=1}^T \bbE_t R_t'^2)$. Let $n\geq n_{0,\delta}$. Using lemma \[lem:inv\_sigh\] we get with probability atleast $1-\delta$ $$\begin{aligned} \lambdamax(R_t')=\lambdamax(n\Sigh-M_t')\leq \lambdamax(n\Sigh)\leq \frac{3n\lambdamax(\Sig)}{2}{\mbox{$\;\stackrel{\mbox{\tiny\rm def}}{=}\;$}}b_2.\end{aligned}$$ $$\begin{aligned} \lambdamax\left[\frac{1}{T}\sum_{t=1}^T \bbE_t R_t'^2\right]&=\frac{1}{T}\lambdamax\left[\sum_{t=1}^T \bbE_t(n\Sigh-M_t')^2\right]\label{eqn:jump-2}\\ &=\frac{1}{T}\lambdamax(-n^2T\Sigh^2+\sum_{t=1}^T\bbE_t\sum_{i=1}^n \frac{\Qit}{(\pit)^2}(x_ix_i^T)^2)\label{eqn:jump-1}\\ &=\frac{1}{T}\lambdamax(-n^2T\Sigh^2+\sum_{t=1}^T\sum_{i=1}^n \frac{1}{\pit}(x_ix_i^T)^2)\label{eqn:jump0}\\ &\leq \frac{1}{T}\lambdamax(\sum_{i=1}^n\sum_{t=1}^T\frac{1}{\pit}(x_ix_i^T)^2)-n^2\lambdamin^2(\Sigh)\label{eqn:jump1}\\ &\leq nT^{1/4}\lambdamax(\sum_{i=1}^n (x_ix_i^T)^2)\label{eqn:jump2}\\ &\leq nT^{1/4}\sum_{i=1}^n\lambdamax^2(x_ix_i^T)\label{eqn:jump3}\\ &=nT^{1/4}\sum_{i=1}^n ||x_i||^4\label{eqn:jump4}\\ &\leq 25\gamma_1^4d^2n^2T^{1/4}\ln^2(n/\delta){\mbox{$\;\stackrel{\mbox{\tiny\rm def}}{=}\;$}}\sigma_2^2\label{eqn:jump5}.\end{aligned}$$ Equation \[eqn:jump-1\] follows from Equation \[eqn:jump-2\] by the definition of $M_t'$ and the fact that at any given $t$ only one point is queried i.e. $\Qit Q_{j}^t=0$ for a given $t$. Equation \[eqn:jump0\] follows from equation \[eqn:jump-1\] since $E_{t}\Qit=\pit$. Equation \[eqn:jump1\] follows from Equation \[eqn:jump0\] by Weyl’s inequality. Equation \[eqn:jump2\] follows from Equation \[eqn:jump1\] by substituting $p_{\text{min}}^t$ in place of $\pit$. Equation \[eqn:jump3\] follows from Equation \[eqn:jump2\] by the use of Weyl’s inequality. Equation \[eqn:jump4\] follows from Equation \[eqn:jump3\] by using the fact that if $p$ is a vector then $\lambdamax(pp^T)=||p||^2$. Equation \[eqn:jump5\] follows from Equation \[eqn:jump4\] by the use of proposition \[prop:normxibound\]. Notice that this step is a stochastic inequality and holds with probability atleast $1-\delta$. Finally applying theorem \[thm:mat\_bern\] we have $$\begin{aligned} \bbP\left[\lambdamax(\frac{1}{T}\sum_{t=1}^T R_t')\leq \sqrt{\frac{2\sigma_2^2\ln(d/\delta)}{T}}+\frac{b_2\ln(d/\delta)}{T}\right]\geq 1-\delta\\ \implies \bbP\left[\lambdamax(n\Sigh-\frac{1}{T}\sum_{t=1}^TM_t')\leq \sqrt{\frac{2\sigma_2^2\ln(d/\delta)}{T}}+\frac{b_2\ln(d/\delta)}{T}\right]\geq 1-\delta\\ \implies \bbP\left[\lambdamin(n\Sigh)-\frac{1}{T}\lambdamin\left(\sum_{t=1}^TM_t'\right)\leq \sqrt{\frac{2\sigma_2^2\ln(d/\delta)}{T}}+\frac{b_2\ln(d/\delta)}{T}\right]\geq 1-\delta\end{aligned}$$ Substituting for $\sigma_2,b_2$, rearranging the inequalities, and using lemma \[lem:inv\_sigh\] to lower bound $\lambdamin(\Sigh)$ we get $$\begin{aligned} \bbP\left[\lambdamin(\sum_{t=1}^TM_t')\geq T\lambdamin(n\Sigh)-\sqrt{2T\sigma_2^2\ln(d/\delta)}-b_2\ln(d/\delta)\right]\geq 1-\delta\\ \implies \bbP\left[\lambdamin(\sum_{t=1}^TM_t')\geq \frac{nT\lambdamin(\Sigma)}{2}-\sqrt{2T\sigma_2^2\ln(d/\delta)}-b_2\ln(d/\delta)\right]\geq 1-2\delta\\ \implies \bbP\left[\lambdamin(\sum_{t=1}^TM_t')\geq\frac{nT\lambdamin(\Sigma)}{2}-5\sqrt{2}\gamma_1^2dnT^{5/8}\sqrt{\ln(d/\delta)}\ln(n/\delta)-\frac{n\ln(d/\delta)\lambdamax(\Sigma)}{2}\right]\geq 1-4\delta\end{aligned}$$ For $T\geq T_{0,\delta}$ with probability atleast $1-4\delta$, $\lambdamin\sum_{t=1}^TM_t'=\lambdamin(\Sighz)\geq\frac{nT\lambdamin(\Sigma)}{4}$. \[lem:1\] For $n\geq n_{0,\delta}$ with probability atleast $1-\delta$ over the random sample $x_1,\ldots,x_n$ $$||\Sig^{-1/2}\Sigh^{1/2}||^2\leq 3/2.$$ $$\begin{aligned} ||\Sig^{-1/2}\Sigh^{1/2}||^2&=||\Sigh^{1/2}\Sig^{-1/2}||^2\\ &=\lambdamax(\Sig^{-1/2}\Sigh\Sig^{-1/2})\\ &=\lambdamax\left(\frac{1}{n}\sum_{i=1}^n (\Sig^{-1/2}x_i)(\Sig^{-1/2}x_i)^T\right)\\ &=\lambdamax\left(\frac{J}{n}\right)\\ &\leq 3/2 \end{aligned}$$ where in the first equality we used the fact that $||A||=||A^T||$ for a square matrix $A$, and $||A||^2=\lambdamax(A^TA)$, and in the last step we used lemma \[lem:J\]. \[lem:2\] Suppose $\Sighz$ is invertible. Given $\delta\in (0,1)$, for $n\geq n_{0,\delta}$, and $T\geq \max\{T_{0,\delta}, T_{1,\delta}\}$ with probability atleast $1-3\delta$ over the samples $$||\Sig^{1/2}\Sighzi\Sig^{1/2}||^2\leq\frac{400}{n^2T^2}.$$ The proof of this lemma is very similar to the proof of lemma \[lem:inv\_sighz\]. From lemma \[lem:inv\_sighz\] for $n\geq n_{0,\delta}, T\geq T_{0,\delta}$ with probability atleast $1-\delta$, $\Sighz\succ 0$. Using the assumption that $\Sig\succ 0$, we get $\Sig^{1/2}\Sighzi\Sig^{1/2}\succ 0$. Hence $||\Sig^{1/2}\Sighzi\Sig^{1/2}||=\lambdamax(\Sig^{1/2}\Sighzi\Sig^{1/2})=\frac{1}{\lambdamin(\Sig^{-1/2}\Sighz\Sig^{-1/2})}$. Hence it is enough to provide a lower bound on the smallest eigenvalue of the symmetric positive definite matrix $\Sig^{-1/2}\Sighz\Sig^{-1/2}$. $$\begin{aligned} \lambdamin(\Sig^{-1/2}\Sighz\Sig^{-1/2})&=\lambdamin\left(\sum_{i=1}^n z_i\Sig^{-1/2} x_ix_i^T\Sig^{-1/2}\right)\\ &=\lambdamin(\sum_{t=1}^T\underbrace{\sum_{i=1}^n \frac{\Qit}{\pit}\Sig^{-1/2} x_ix_i^T\Sig^{-1/2}}_{{\mbox{$\;\stackrel{\mbox{\tiny\rm def}}{=}\;$}}M_t})\\ &=\lambdamin\left(\sum_{t=1}^T M_t\right).\end{aligned}$$ Define $R_t{\mbox{$\;\stackrel{\mbox{\tiny\rm def}}{=}\;$}}J-M_t$. Clearly $\bbE_{t}[M_t]=J$, and hence $\bbE[R_t]=0$. From Weyl’s inequality we have $\lambdamin(J)+\lambdamax\left(\frac{-1}{T}\sum_{t=1}^T M_t\right)\leq \lambdamax(\frac{1}{T}\sum_{t=1}^T R_t)$. Now applying theorem \[thm:mat\_bern\] on $\sum R_t$ we get with probability atleast $1-\delta$ $$\label{eqn:matbern} \lambdamin(J)+\lambdamax\left(\frac{-1}{T}\sum_{t=1}^T M_t\right)\leq \lambdamax\left(\frac{1}{T}\sum_{t=1}^T R_t\right)\leq \sqrt{\frac{2\sigma_ 1^2\ln(d/\delta)}{T}}+\frac{b_1\ln(d/\delta)}{3T},$$ where $$\begin{aligned} \lambdamax\left(\frac{1}{T}\sum_{t=1}^T J-M_t\right)\leq b_1\\ \lambdamax\left(\frac{1}{T}\sum_{t=1}^T \bbE_{t}(J-M_t)^2\right)\leq \sigma_1^2\end{aligned}$$ Rearranging Equation (\[eqn:matbern\]) and using the fact that $\lambdamax(-A)=-\lambdamin(A)$ we get with probability atleast $1-\delta$, $$\label{eqn:matbernrear} \lambdamin\left(\sum_{t=1}^T M_t\right)\geq T\lambdamin(J)-\sqrt{2T\sigma_ 1^2\ln(d/\delta)}-\frac{b_1\ln(d/\delta)}{3}.$$ Using Weyl’s inequality [@horn90matrix] we have $\lambdamax(\frac{1}{T}\sum_{t=1}^T J-M_t)\leq \lambdamax(J)\leq \frac{3n}{2}$ with probability atleast $1-\delta$, where in the last step we used lemma (\[lem:J\]). Let $b_1{\mbox{$\;\stackrel{\mbox{\tiny\rm def}}{=}\;$}}\frac{3n}{2}$. To calculate $\sigma_1^2$ we proceed as follows. $$\begin{aligned} \lambdamax\left(\frac{1}{T}\sum_{t=1}^T \bbE_{t}(J-M_t)^2\right)&=\frac{1}{T}\lambdamax\left(\sum_{t=1}^T \bbE_{t}(M_t^2)-J^2\right)\label{eqn:here0}\\ &\leq \frac{1}{T}\lambdamax\left(\sum_{t=1}^T \bbE_{t}M_t^2\right)\label{eqn:here1}\\ &=\frac{1}{T}\lambdamax\left(\sum_{t=1}^T \bbE_t \left(\sum_{i=1}^n \frac{\Qit}{\pit}\Sigma^{-1/2}x_ix_i^T\Sigma^{-1/2}\right)^2\right)\label{eqn:here2}\\ &=\frac{1}{T}\lambdamax\left(\sum_{t=1}^T \bbE_t \sum_{i=1}^n \frac{\Qit}{(\pit)^2}(\Sigma^{-1/2}x_ix_i^T\Sigma^{-1/2})^2\right)\label{eqn:here3}\\ &=\frac{1}{T}\lambdamax\left(\sum_{t=1}^T \sum_{i=1}^n \frac{1}{\pit}(\Sigma^{-1/2}x_ix_i^T\Sigma^{-1/2})^2\right)\label{eqn:here4}\\ &\leq \frac{1}{T}\sum_{t=1}^T\sum_{i=1}^n \frac{1}{\pit}||\Sig^{-1/2}x_i||^4\label{eqn:here5}\\ &\leq \frac{d^2\gamma_0^4}{T}\sum_{i=1}^n\sum_{t=1}^T \frac{1}{\pit}\label{eqn:here6}\\ &\leq \frac{nd^2\gamma_0^4}{T}\sum_{t=1}^T \frac{1}{p_{\text{min}}^t}\label{eqn:here7}\\ &\leq n^2d^2\gamma_0^4 T^{1/4}{\mbox{$\;\stackrel{\mbox{\tiny\rm def}}{=}\;$}}\sigma_1^2\label{eqn:here8}.\end{aligned}$$ Equation \[eqn:here1\] follows from Equation \[eqn:here0\] by using Weyl’s inequality and the fact that $J^2\succeq 0$. Equation  \[eqn:here3\] follows from Equation \[eqn:here2\] since only one point is queried in every round and hence for any given $t,i\neq j$ we have $\Qit Q_{j}^t=0$, and hence all the cross terms disappear when we expand the square. Equation (\[eqn:here4\]) follows from Equation (\[eqn:here3\]) by using the fact that $\bbE_{t}Q_t=p_t$. Equation (\[eqn:here5\]) follows from Equation (\[eqn:here4\]) by Weyl’s inequality and the fact that the maximum eigenvalue of a rank-1 matrix of the form $vv^T$ is $||v||^2$. Equation (\[eqn:here6\]) follows from Equation (\[eqn:here5\]) by using assumption A1. Equation \[eqn:here8\] follows from Equation (\[eqn:here7\]) by our choice of $p_{min}^t=\frac{1}{n\sqrt{t}}$. Substituting the values of $\sigma_1^2, b_1$ in \[eqn:matbernrear\], using lemma \[lem:J\] to lower bound $\lambdamin(J)$, and applying union bound to sum up all the failure probabilities we get for $n\geq n_{0,\delta},T\geq \max\{T_{0,\delta},T_{1,\delta}\}$ with probability atleast $1-3\delta$, $$\begin{gathered} \lambdamin\left(\sum_{t=1}^T M_t\right)\geq T\lambdamin(J)-\sqrt{2T^{5/4}n^2d^2\gamma_0^4\ln(d/\delta)}-3n/2\\ \geq \frac{nT}{2}-\sqrt{2}T^{5/8}nd\gamma_0^2\sqrt{\ln(d/\delta)}-3n/2\geq nT/4.\qedhere\end{gathered}$$ The only missing piece in the proof is an upper bound for the quantity $||\Sigh^{-1/2}\psi_z||^2$. The next lemma provides us with an upper bound for this quantity. \[lem:3\] Suppose $\Sigh$ is invertible. Let $\delta\in (0,1)$. With probability atleast $1-\delta$ we have $$||\Sigh^{-1/2}\psi_z||^2\leq (2nT^2+56n^3T\sqrt{T})(d+2\sqrt{d\ln(1/\delta)}+2\ln(1/\delta)).$$ Define the matrix $A\in \bbR^{d\times n}$ as follows. Let the $i^{\text{th}}$ column of $A$ be the vector $\frac{\Sigh^{-1/2}x_i}{\sqrt{n}}$, so that $AA^T=\frac{1}{n}\Sigh^{-1/2}x_ix_i^T\Sigh^{-1/2}=I_d$. Now $||\Sigh^{-1/2}\psi_z||^2=||\sqrt{n}Ap||^2$, where $p=(p_1,\ldots,p_n)\in \bbR^n$ and $p_i=\xi(x_i)z_i$ for $i=1,\ldots,n$. Using the result for quadratic forms of subgaussian random vectors (threorem \[thm:quadratic\]) we get $$\begin{gathered} \label{eqn:norm_Ap2} ||Ap||^2\leq \sigma^2(\operatorname{tr}(I_d)+2\sqrt{\operatorname{tr}(I_d)\ln(1/\delta)}+2||I_d||\ln(1/\delta))=\sigma^2(d+2\sqrt{d\ln(1/\delta)}+2\ln(1/\delta)), \end{gathered}$$ where for any arbitrary vector $\alpha$, $\bbE[\exp(\alpha^Tp)]\leq \exp(||\alpha||^2\sigma^2)$. Hence all that is left to be done is prove that $\alpha^Tp$ has sub-Gaussian exponential moments. Let $$D_t{\mbox{$\;\stackrel{\mbox{\tiny\rm def}}{=}\;$}}\sum_{i=1}^n \frac{\alpha_i\xi(x_i)\Qit}{\pit}-\alpha^T\xi~~~\forall t=1,\ldots,T.$$ With this definition we have the following series of equalities $$\begin{aligned} \label{eqn:decompose_condexpec} \bbE[\exp(\alpha^Tp)]=\bbE[\exp(\sum D_t+T\alpha^T\xi)]=\bbE\left[\exp(T\alpha^T\xi)\bbE[\exp(\sum D_t)|\cDn]\right].\end{aligned}$$ Conditioned on the data, the sequence $D_1,\ldots,D_T$, forms a martingale difference sequence. Let $\xi=[\xi(x_1),\ldots,\xi(x_n)]$. Notice that $$\label{eqn:bound_Dt} -\alpha^T\xi-\frac{2||\alpha||}{p_{\text{min}}^t}\leq D_t\leq -\alpha^T\xi+\frac{2||\alpha||}{p_{\text{min}}^t}.$$ We shall now bound the probability of large deviations of $D_t$ given history up until time $t$. This allows us to put a bound on the large deviations of the martingale sum $\sum_{t=1}^T D_t$. Let $a\geq 0$. Using Markov’s inequality we get $$\begin{aligned} \bbP[D_t\geq a|Q_{1:n}^{1:t-1},\cDn]&\leq \min_{\gamma>0}~\exp(-\gamma a)\bbE[\gamma D_t|Q_{1:n}^{1:t-1},\cDn]\\ &\leq\min_{\gamma>0}\exp\left(\frac{2\gamma^2||\alpha||^2}{(p_{\text{min}}^t)^2}-\gamma a\right)\\ & \leq \exp\left(\frac{-a^2}{8||\alpha||^2n^2\sqrt{t}}\right).\end{aligned}$$ In the second step we used Hoeffding’s lemma along with the boundedness property of $D_t$ shown in equation \[eqn:bound\_Dt\]. The same upper bound can be shown for the quantity $\bbP[D_t\leq a|Q_{1:n}^{1:t-1},\cDn]$. Applying lemma \[lem:modazuma\] we get with probability atleast $1-\delta$, conditioned on the data, we have $$\frac{1}{T}\sum_{t=1}^T D_t\leq \sqrt{\frac{448||\alpha||^2n^2\ln(1/\delta)}{\sqrt{T}}}\\\implies \sum_{t=1}^T D_t\leq \sqrt{112||\alpha||^2n^2T^{3/2}\ln(1/\delta)}.$$ Hence $\sum_{t=1}^T D_t$, conditioned on data, has sub-Gaussian tails as shown above. This leads to the following conditional exponential moments bound $$\label{eqn:sumdt} \bbE[\exp(\sum_{t=1}^T D_t)|\cD_n]=\exp\left(56||\alpha||^2n^2T\sqrt{T}\ln(1/\delta)\right).$$ Finally putting together equations \[eqn:decompose\_condexpec\], \[eqn:sumdt\] we get $$\bbE[\exp(\alpha^Tp)]\leq \bbE\exp(T\alpha^T\xi)\exp(56||\alpha||^2n^2T\sqrt{T})\leq \exp((2T^2+56n^2T\sqrt{T})||\alpha||^2),$$ In the last step we exploited the fact that $-2 \leq \xi(x_i)\leq 2$, and hence by Hoeffding lemma $\bbE[\exp(\alpha^T\xi)]\leq \exp(2||\alpha||^2)$. This leads us to the choice of $\sigma^2=2T^2+56n^2T\sqrt{T}$. Substituting this value of $\sigma^2$ in equation \[eqn:norm\_Ap2\] we get $$||Ap||^2\leq (2T^2+56n^2T\sqrt{T})(d+2\sqrt{d\ln(1/\delta)}+2\ln(1/\delta)),$$ and hence with probability atleast $1-\delta$, $$||\Sigh^{-1/2}\psi_z ||^2=n||Ap||^2\leq (2nT^2+56n^3T\sqrt{T})(d+2\sqrt{d\ln(1/\delta)}+2\ln(1/\delta)).$$ We are now ready to prove our main result. \[**Proof of theorem \[thm:main\]**\] For $n\geq n_{0,\delta}$ and $T\geq \max\{T_{0,\delta},T_{1,\delta}\}$ from lemma  \[lem:inv\_sigh\], \[lem:inv\_sighz\], both $\Sighz$, and $\Sigh$ are invertible with probability atleast $1-\delta, 1-4\delta$ respectively. Conditioned on the invertibility of $\Sighz,\Sigma$ we get from lemmas \[lem:1\]-\[lem:3\], $||\Sigi\Sigh^{1/2}||^2\leq 3/2$ and $||\Sig^{1/2}\Sighzi\Sig^{1/2}||^2\leq400/n^2T^2$, and $||\Sigh^{-1/2}\psi_z||^2\leq (2nT^2+56n^3T^{3/2})(d+2\sqrt{d\ln(1/\delta)+2\ln(1/\delta)})$ with probability atleast $1-\delta,1-3\delta,1-\delta$ respectively. Using lemma \[lem:decompose\] and the union bound to add up all the failure probabilities we get the desired result. Related Work ============ A variety of pool based AL algorithms have been proposed in the literature employing various query strategies. However, none of them use unbiased estimates of the risk. One of the simplest strategy for AL is uncertainty sampling, where the active learner queries the point whose label it is most uncertain about. This strategy has been popularl in text classification [@lewis1994sequential], and information extraction [@settles2008analysis]. Usually the uncertainty in the label is calculated using certain information-theoretic criteria such as entropy, or variance of the label distribution. While uncertainty sampling has mostly been used in a probabilistic setting, AL algorithms which learn non-probabilistic classifiers using uncertainty sampling have also been proposed. Tong et al. [-@tong2001support] proposed an algorithm in this framework where they query the point closest to the current svm hyperplane. Seung et al. [-@seung1992query] introduced the query-by-committee (QBC) framework where a committee of potential models, which all agree on the currently labeled data is maintained and, the point where most committee members disagree is considered for querying. In order to design a committee in the QBC framework, algorithms such as query-by-boosting, and query-by-bagging in the discriminative setting [@Abe1998query], sampling from a Dirichlet distribution over model parameters in the generative setting [@mccallumzy1998employing] have been proposed. Other frameworks include querying the point, which causes the maximum expected reduction in error [@zhu2003combining; @guo2007optimistic], variance reducing query strategies such as the ones based on optimal design [@flaherty2005robustdesign; @zhang2000value]. A very thorough literature survey of different active learning algorithms has been done by Settles [-@settlestr09]. AL algorithms that are consistent and have provable label complexity have been proposed for the agnostic setting for the 0-1 loss in recent years [@dasgupta2007general; @beygelzimer2009importance]. The IWAL framework introduced in Beygelzimer et al. [-@beygelzimer2009importance] was the first AL algorithm with guarantees for general loss functions. However the authors were unable to provide non-trivial label complexity guarantees for the hinge loss, and the squared loss. UPAL at least for squared losses can be seen as using a QBC based querying strategy where the committee is the entire hypothesis space, and the disagreement among the committee members is calculated using an exponential weighting scheme. However unlike previously proposed committees our committee is an infinite set, and the choice of the point to be queried is randomized. Experimental results {#sec:expts} ==================== We implemented UPAL, along with the standard passive learning (PL) algorithm, and a variant of UPAL called RAL (in short for random active learning), all using logistic loss, in matlab. The choice of logistic loss was motivated by the fact that BMAL was designed for logistic loss. Our matlab codes were vectorized to the maximum possible extent so as to be as efficient as possible. RAL is similar to UPAL, but in each round samples a point uniformly at random from the currently unqueried pool. However it does not use importance weights to calculate an estimate of the risk of the classifier. The purpose of implementing RAL was to demonstrate the potential effect of using unbiased estimators, and to check if the strategy of randomly querying points helps in active learning. We also implemented a batch mode active learning algorithm introduced by Hoi et al. [-@hoi2006batch] which, we shall call as BMAL. Hoi et al. in their paper showed superior empirical performance of BMAL over other competing pool based active learning algorithms, and this is the primary motivation for choosing BMAL as a competitor pool AL algorithm in this paper. BMAL like UPAL also proceeds in rounds and in each iteration selects $k$ examples by minimizing the Fisher information ratio between the current unqueried pool and the queried pool. However a point once queried by BMAL is never requeried. In order to tackle the high computational complexity of optimally choosing a set of $k$ points in each round, the authors suggested a monotonic submodular approximation to the original Fisher ratio objective, which is then optimized by a greedy algorithm. At the start of round $t+1$ when, BMAL has already queried $t$ points in the previous rounds, in order to decide which point to query next, BMAL has to calculate for each potential new query a dot product with all the remaining unqueried points. Such a calculation when done for all possible potential new queries takes $O(n^2t)$ time. Hence if our budget is $B$, then the total computational complexity of BMAL is $O(n^2B^2)$. Note that this calculation does not take into account the complexity of solving an optimization problem in each round after having queried a point. In order to further reduce the computational complexity of BMAL in each round we further restrict our search, for the next query, to a small subsample of the current set of unqueried points. We set the value of $p_{\text{min}}$ in step 3 of algorithm 1 to $\frac{1}{nt}$. In order to avoid numerical problems we implemented a regularized version of UPAL where the term $\lambda||w||^2$ was added to the optimization problem shown in step 11 of Algorithm 1. The value of $\lambda$ is allowed to change as per the current importance weight of the pool. The optimal value of $C$ in VW [^1] was chosen via a 5 fold cross-validation, and by eyeballing for the value of $C$ that gave the best cost-accuracy trade-off. We ran all our experiments on the MNIST dataset(3 Vs 5) [^2], and datasets from UCI repository namely Statlog, Abalone, Whitewine. Figure \[fig:expt\_results\] shows the performance of all the algorithms on the first 300 queried points. Sample size ------------- ------ ------- -------- ------- Time Error Time Error 1200 65 7.27 60 5.67 2400 100 6.25 152 6.05 4800 159 6.83 295 6.25 10000 478 5.85 643.17 5.85 : \[tab:fixed\_B\] Budget Speedup -------- ------ ------- --------- ------- ----- Time Error Time Error 500 859 5.79 1973 5.33 2.3 1000 1919 6.43 7505 5.70 3.9 2000 4676 5.82 32186 5.59 6.9 : \[tab:fixed\_n\] On the MNIST dataset, on an average, the performance of BMAL is very similar to UPAL, and there is a noticeable gap in the performance of BMAL and UPAL over PL, VW and RAL. Similar results were also seen in the case of Statlog dataset, though towards the end the performance of UPAL slightly worsens when compared to BMAL. However UPAL is still better than PL, VW, and RAL. Active learning is not always helpful and the success story of AL depends on the match between the marginal distribution and the hypothesis class. This is clearly reflected in Abalone where the performance of PL is better than UPAL atleast in the initial stages and is never significantly worse. UPAL is uniformly better than BMAL, though the difference in error rates is not significant. However the performance of RAL, VW are significantly worse. Similar results were also seen in the case of Whitewine dataset, where PL outperforms all AL algorithms. UPAL is better than BMAL most of the times. Even here one can witness a huge gap in the performance of VW and RAL over PL, BMAL and UPAL. One can conclude that VW though is computationally efficient has higher error rate for the same number of queries. The uniformly poor performance of RAL signifies that querying uniformly at random does not help. On the whole UPAL and BMAL perform equally well, and we show via our next set of experiments that UPAL has significantly better scalability, especially when one has a relatively large budget $B$. Scalability results ------------------- Each round of UPAL takes $O(n)$ plus the time to solve the optimization problem shown in step 11 in Algorithm 1. A similar optimization problem is also solved in the BMAL problem. If the cost of solving this optimization problem in step $t$ is $c_{opt,t}$, then the complexity of UPAL is $O(nT+\sum_{t=1}^T c_{opt,t})$. While BMAL takes $O(n^2B^2+\sum_{t=1}^Tc'_{t,opt})$ where $c'_{t,opt}$ is the complexity of solving the optimization problem in BMAL in round $t$. For the approximate implementation of BMAL that we described if the subsample size is $|S|$, then the complexity is $O(|S|^2B^2+\sum_{t=1}^Tc'_{t,opt})$. In our first set of experiments we fix the budget $B$ to 300, and calculate the test error and the combined training and testing time of both BMAL and UPAL for varying sizes of the training set. All the experiments were performed on the MNIST dataset. Table \[tab:fixed\_B\] shows that with increasing sample size UPAL tends to be more efficient than BMAL, though the gain in speed that we observed was at most a factor of 1.8. In the second set of scalability experiments we fixed the training set size to 10000, and studied the effect of increasing budget. We found out that with increasing budget size the speedup of UPAL over BMAL increases. In particular when the *budget was 2000, UPAL is arpproximately 7 times faster than BMAL.* All our experiments were run on a dual core machine with 3 GB memory. Conclusions and Discussion ========================== In this paper we proposed the first unbiased pool based active learning algorithm, and showed its good empirical performance and its ability to scale both with higher budget constraints and large dataset sizes. Theoretically we proved that when the true hypothesis is a linear hypothesis, we are able to recover it with high probability. In our view an important extension of this work would be to establish tighter bounds on the excess risk. It should be possible to provide upper bounds on the excess risk in expectation which are much sharper than our current high probability bounds. Another theoretically interesting question is to calculate how many unique queries are made after $T$ rounds of UPAL. This problem is similar to calculating the number of non-empty bins in the balls-and-bins model commonly used in the field of randomized algorithms [@motwani1995ra], when there are $n$ bins and $T$ balls, with the different points in the pool being the bins, and the process of throwing a ball in each round being equivalent to querying a point in each round. However since each round is, unlike standard balls-and-bins, dependent on the previous round we expect the analysis to be more involved than a standard balls-and-bins analysis. Some results from random matrix theory ====================================== \[thm:quadratic\](Quadratic forms of subgaussian random vectors [@litvak2005smallest; @hsu2011analysis]) Let $A\in \bbR^{m\times n}$ be a matrix, and $H{\mbox{$\;\stackrel{\mbox{\tiny\rm def}}{=}\;$}}AA^T$, and $r=(r_1,\ldots,r_n)$ be a random vector such that for some $\sigma\geq0$, $$\bbE[\exp(\alpha^Tr)]\leq \exp\left(\frac{||\alpha||^2\sigma^2}{2}\right)$$ for all $\alpha\in \bbR^n$ almost surely. For all $\delta\in(0,1)$, $$\bbP~\left[||Ar||^2>\sigma^2\operatorname{tr}(H)+2\sigma^2\sqrt{\operatorname{tr}(H^2)}\ln(1/\delta)+2\sigma^2||H||\ln(1/\delta)\right]\leq \delta.$$ The above theorem was first proved without explicit constants by Litvak et al. [@litvak2005smallest] Hsu et al [@hsu2011analysis] established a version of the above theorem with explicit constants. \[thm:litvak\](Eigenvalue bounds of a sum of rank-1 matrices) Let $r_1,\ldots r_n$ be random vectors in $\bbR^d$ such that, for some $\gamma>0$, $$\begin{aligned} \bbE[r_ir_i^T|r_1,\ldots,r_{i-1}]&=I\\ \bbE[\exp(\alpha^Tr_i)|r_1,\ldots,r_{i-1}]&\leq \exp(||\alpha||^2\gamma/2) ~\forall \alpha \in \bbR^d. \end{aligned}$$ For all $\delta \in (0,1)$, $$\bbP\left[\lambdamax\left(\frac{1}{n}\sum_{i=1}^n r_ir_i^T\right)>1+2\epsilon_{\delta,n} \vee \lambdamin\left(\frac{1}{n}\sum_{i=1}^n r_ir_i^T\right)<1-2\epsilon_{\delta,n}\right]\leq \delta,$$ where $$\epsilon_{\delta,n}=\gamma\left(\sqrt{\frac{32(d~\ln(5)+\ln(2/\delta))}{n}}+\frac{2(d\ln(5)+\ln(2/\delta))}{n}\right).$$ We shall use the above theorem in Lemma  \[lem:inv\_sigh\], and lemma \[lem:J\]. \[thm:mat\_bern\](Matrix Bernstein bound) Let $X_1\ldots,X_n$ be symmetric valued random matrices. Suppose there exist $\bar{b},\bar{\sigma}$ such that for all $i=1,\ldots,n$ $$\begin{aligned} \bbE_i[X_i]&=0\\ \lambdamax(X_i)&\leq \bar{b}\\ \lambdamax\left(\frac{1}{n}\sum_{i=1}^n \bbE_{i}[X_i^2]\right)&\leq \bar{\sigma}^2.\end{aligned}$$ almost surely, then $$\begin{aligned} \bbP\left[\lambdamax\left(\frac{1}{n}\sum_{i=1}^n X_i\right)> \sqrt{\frac{2\bar{\sigma}^2\ln(d/\delta)}{n}}+\frac{\bar{b}\ln(d/\delta)}{3n}\right]\leq \delta.\end{aligned}$$ A dimension free version of the above inequality was proved in Hsu et al [@hsu2011dimension]. Such dimension free inequalities are especially useful in infinite dimension spaces. Since we are working in finite dimension spaces, we shall stick to the non-dimension free version.  [@shamir2011variant]\[lem:modazuma\] Let $(Z_1,\cF_1),\ldots,(Z_T,\cF_T)$ be a martingale difference sequence, and suppose there are constants $b\geq 1,c_t>0$ such that for any $t$ and any $a>0$, $$\max\{\bbP[Z_t\geq a|\cF_{t-1}],\bbP[Z_t\leq -a|\cF_{t-1}]\}\leq b\exp(-c_ta^2).$$ Then for any $\delta>0$, with probability atleast $1-\delta$ we have $$\frac{1}{T}\sum_{t=1}^T Z_t\leq \sqrt{\frac{28b\ln(1/\delta)}{\sum_{t=1}^Tc_t}}.$$ The above result was first proved by Shamir [@shamir2011variant]. Shamir proved the result for the case when $c_1=\ldots=c_{T}$. Essentially one can use the same proof with obvious changes to get the above result.  [see @cesa2006prediction page 359] Let $X$ be a random variable with $a\leq X\leq b$. Then for any $s\in\bbR$ $$\bbE[\exp(sX)]\leq \exp\left(s\bbE[X]+\frac{s^2(b-a)^2}{8}\right).$$ Let $A, B$ be positive semidefinite matrices. Then $$\lambdamax(A)+\lambdamin(B)\leq \lambdamax(A+B)\leq \lambdamax(A)+\lambdamax(B).$$ The above inequalities are called as Weyl’s inequalities [see @horn90matrix chap. 3] [^1]: The parameters initial\_t, $l$ were set to a default value of 10 for all of our experiments. [^2]: The dataset can be obtained from <http://cs.nyu.edu/~roweis/data.html>. We first performed PCA to reduce the dimensions to 25 from 784.

--- abstract: '[Liquids generally become more ordered upon cooling. However, it has been a long-standing debate on whether such structural ordering in liquid water takes place continuously or discontinuosly: continuum vs. mixture models. Here, by computer simulations of three popular water models and analysis of recent scattering experiment data, we show that, in the structure factor of water, there are two overlapped peaks hidden in the apparent “first diffraction peak”, one of which corresponds to the neighboring O-O distance as in ordinary liquids and the other to the longest periodicity of density waves in a tetrahedral structure. This unambiguously proves the coexistence of two local structural motifs. Our findings not only provide key clues to settle long-standing controversy on the water structure but also allow experimental access to the degree and range of structural ordering in liquid water. ]{}' author: - Rui Shi - Hajime Tanaka bibliography: - 'fsdpWater.bib' title: | Distinct signature of two local structural motifs of liquid water\ in the scattering function --- Water is ubiquitous in our planet and plays vital roles in many biological, geological, meteorological, and technological processes. Despite its simple molecular structure, water shows many unique thermodynamic and dynamic properties in the liquid state, such as a density maximum at 4 $^\circ$C, a rapid increase of isothermal compressibility and a dynamic fragile-to-strong transition upon cooling [@Debenedetti2003; @gallo2016water]. These unusual properties, which are absent in ordinary liquids, are well-known as “water’s anomalies”. It is widely believed that the anomalies are linked to water’s structural ordering towards tetrahedral structures stabilized by four hydrogen bonds (H-bonds). Even after intensive studies for more than a century, however, how such structural ordering takes place is still a matter of hot debate without convergence. Two conflicting different scenarios have continued to exist until now: ‘continuum models’ based on a broad unimodal distribution of structural components and ‘mixture models’ based on a bimodal distribution of structural components reflecting the coexistence of two (or more) types of local structures [@narten1969observed; @eisenberg2005structure; @handle2017supercooled]. The mixture model dates back to Wilhelm Röntgen, who proposed in 1892 that water can be regarded as ’icebergs’ in a fluid ’sea’ [@rontgen1892ueber]. Later various mixture models have been developed. One famous example is the mixture model of Linus Pauling, who proposed in 1959 that water is mixture of clathrate-like structure and interstitial molecules [@pauling1959structure]. These mixture models, however, have been continuously challenged by the continuum model dating back to John Pople, who proposed in 1951 that water’s structure can be described by a continuously distorted H-bond network [@pople1951molecular]. These two types of models lead to fundamentally different understandings of water structure. Despite such a clear difference in the physical picture, there has been no convergence of this debate over a century. The main reason is the lack of [*experimental*]{} evidence exclusively supporting either of the two models. In this Letter, we provide such clear evidence that liquid water is indeed a mixture of two types of local structural motifs, from simulations of three popular water models and detailed analysis of recent scattering experiments. First we need to explain the precise nature of our two-state model to clarify essential differences from ‘continuum models’ and other types of mixture models that regard water as a mixture of two types of liquids, i.e. low-density (LDL) and high-density liquids (HDL). Our two-state model that regards water as a mixture of ordered ($S$) and less ordered local structural motifs ($\rho$-state) [@Tanaka_review; @Tanaka2000] is characterized by the following five features: (1) $S$- and $\rho$-states in liquid water are defined by [*local structures around a central molecule*]{} and characterized by low and high local symmetry, energy, density, and entropy, respectively. In the one-phase regime of water far from the second-critical point (if it exists), the two structural motifs can have only short coherence lengths. Thus, our model is essentially different from a mixture model of LDL and HDL. We stress that they are macroscopic phases of water that can exist only below the second critical point; (2) Reflecting the presence of the two states, the distribution of a proper structural descriptor should have a bimodal distribution composed of two Gaussians (not necessarily two delta functions; note that there is no unique configuration for each state under thermal fluctuations); (3) The two-state model effectively transforms to a continuum-like model at high temperatures/high pressures where there exists only $\rho$ state, because the ordered $S$-structure can hardly survive due to the entropy/volume penalties; (4) The $T, P$-dependence of the fraction of the two structural motifs should obey the thermodynamic two-state equations [@Tanaka_review]. (5) The existence of a second critical point is a sufficient but not necessary condition for the two-state model. Recently we have shown that we can detect two structure motifs by a microscopic structural descriptor $\zeta$ (see Methods), which characterizes the translational order in the second shell, and confirmed the above five features on a microscopic level by computer simulations of several popular water models [@Russo2014; @shi2018impact; @shi2018Microscopic; @shi2018origin; @shi2018common]. These studies have clearly indicated that water is a dynamic mixture of the two states [@Tanaka_review; @Tanaka2000; @holten2013nature; @Russo2014; @singh2016two; @Biddle2017; @Singh2017; @de2018viscosity; @russo2018water; @shi2018impact; @shi2018common; @shi2018origin]—$S$-state \[locally favored tetrahedral structure (LFTS)\] and $\rho$-state \[disordered normal-liquid structure (DNLS)\]. The former stabilized by four H-bonds has lower symmetry, density, energy and entropy than the latter. A typical snapshot of LFTS and DNLS is shown in Fig. \[fig:fnfsZeta\]a. We have also found that the fraction $s$ of the LFTS, serving as an order parameter characterizing the degree of structural ordering, changes with temperature $T$ and pressure $P$, obeying the prediction of the thermodynamic two-state model [@Tanaka_review; @Tanaka2000; @Singh2017; @de2018viscosity; @russo2018water; @shi2018common; @shi2018origin]. These results provide strong computational support for the two-state model. The shape of the distribution function of a physical quantity is a key to judge which of mixture and continuum models is relevant, because the former predicts a bimodal distribution at a certain range of $T$ and $P$ whereas the latter always predicts a unimodal Gaussian distribution. Our structural descriptor $\zeta$ clearly shows the bimodality. Then, a key question is whether a quantity directly related to local density shows such bimodality or not. The answer is yes. We show the distribution $P(N_\mathrm{fs})$ of the coordination number $N_\mathrm{fs}$ in Fig. \[fig:fnfsZeta\]b. We note that $N_\mathrm{fs}$ is the number of water molecules in the spherical first-shell volume $V_{\rm fs}$ of radius of 3.5 Å, and thus proportional to the local number density, $N_{\rm fs}/V_{\rm fs}$ (see Methods (Characterization of local density) for the relevance of this estimation of local density). We can see that $P(N_\mathrm{fs})$ has clear bimodality. Furthermore, $P(N_\mathrm{fs})$ and $P(\zeta)$ both can be properly characterized by two Gaussian functions (see Methods) with the same fraction $s$ (see Fig. \[fig:fnfsZeta\]b), following the prediction of the thermodynamic two-state model (Fig. \[fig:fnfsZeta\]c-e and Fig. \[fig:fnfsT\]-\[fig:fnfs2Gau-small\]). This clearly indicates the anti-correlation between $\zeta$ and local density \[see the above feature (1)\]. This result strongly contradicts with the prediction of the continuum models that $P(N_\mathrm{fs})$ should be unimodal Gaussian, which is the case for simple liquids such as Lennard-Jones liquids (Fig. \[fig:fnfs-lj\]). We note that three popular water models all show the bimodal distributions of $P(N_\mathrm{fs})$ (Fig. \[fig:fnfsT\]). We can see that $P(N_\mathrm{fs})$ exhibits a unimodal distribution at very high $T$, but it transforms to a bimodal one (composed of two Gaussians) upon cooling for all the three water models. This clearly indicates the failure of continuum models, and supports the two-state model (see the above Features (1)-(4)). ![Structural bimodality in the coordination number distribution of liquid water. [**a**]{}, A snapshot of liquid TIP4P/2005 water at 1 bar and 240 K, where $s\approx 0.5$, i.e., near the Schottky temperature $T_{s=1/2}$. Two structural motifs, LFTS and DNLS, are highlighted by blue and red colors, respectively. LFTS has four H-bonded nearest neighbors with tetrahedral coordination, whereas DNLS typically has five or more nearest neighbors (typically three of them being H-bonded) with broken tetrahedral symmetry. [**b**]{}, Correlation between the number of water molecules in the spherical first-shell volume of radius of 3.5 Å, $N_\mathrm{fs}$ (i.e., the coordination number), and the structural descriptor $\zeta$ at 1 bar and 240 K. The distributions of $N_\mathrm{fs}$ and $\zeta$ are shown in the right and top sides, respectively. Both distributions show clear bimodal features, which can be properly described by the sum of two Gaussian functions (black lines), with blue and red shades corresponding to LFTS and DNLS respectively. [**c**]{}, Distribution of coordination number $N_\mathrm{fs}$ as a function of temperature, $P \left(N_\mathrm{fs}, T\right)$, is shown by colored balls with red and blue for higher and lower temperatures, respectively. The colored surface is the fit to two Gaussian functions by Eqs. \[eq:pnfs\] - \[eq:SDNLS\]. [**d**]{}-[**e**]{}, $T$-dependence of the two Gaussian components of $P \left(N_\mathrm{fs}, T\right)$ in side ([**d**]{}) and top ([**e**]{}) view. One Gaussian component locating at $N_\mathrm{fs} \approx 4$ corresponds to LFTS, whereas the other one at $N_\mathrm{fs} \simeq 5$ corresponds to DNLS, in agreement with the snapshot in [**a**]{}. The fraction of the two Gaussian components is consistent with the fraction determined by $\zeta$ distribution in [**b**]{} and the theoretical two-state model (Eq. \[eq:s\]).[]{data-label="fig:fnfsZeta"}](fnfsZeta-small){width="14.5cm"} It has sometimes been argued that the unimodal Gaussian distribution of density fluctuations is a signature against mixture models. However, we point out that it is not the case: Under thermal fluctuations, any thermodynamic order parameters, e.g. density $\rho$ and local structural order $s$, should have unimodal Gaussian distributions. This is because the free energy of a system, $f(\rho,s)$, can be expressed by quadratic terms in the one-phase homogeneous region (see, e.g., Ref. [@tanaka1998simple]). Although theoretically obvious, we can confirm it from the fact that the macroscopic density distributions in liquid water and other single- or two-component liquids commonly show Gaussian distributions (see the results of Lennard-Jones (LJ) liquid, SiO$_2$, and Cu$_{64}$Zr$_{36}$ in Fig. \[fig:fdensity\]) irrespective of whether the local density distribution is unimodal or bimodal (unimodal for LJ liquid whereas bimodal for H$_2$O, SiO$_2$ and Cu$_{64}$Zr$_{36}$). The same is applied for the distribution of another macroscopic order parameter $s$ (estimated from $\zeta$): $P(s)$ has a unimodal Gaussian distribution, even when the underlying microscopic structural descriptor $\zeta$ has a bimodal distribution composed of two Gaussians (Fig. \[fig:fs\_zeta\]). This difference between macroscopic and microscopic distributions clearly indicates that the bimodal structural ordering in liquid water is highly localized, in agreement with Feature (1) in the introduction. Here we note that a mixture model of LDL and HDL should result in the bimodal distributions of macroscopic order parameters $\rho$ and $s$, contrary to the above results. So far we show that computer simulations of classical water models allow us to directly access the distributions of $\zeta$ and $N_\mathrm{fs}$ and provide strong evidence for the presence of the two types of structural motifs. Unfortunately, however, we cannot access such microscopic molecular-level information by experiments. So an experimentally accessible structural descriptor is highly desirable to close a long-standing debate on the structure of liquid water. The most powerful experimental methods to access the local structures of materials are x-ray and neutron scatterings, by which we can measure the structure factor, i.e. the density-density correlation in reciprocal space: $$S(\bm{k}) = \frac{1}{N}\langle\rho_{\bm{k}} \rho_{-\bm{k}}\rangle \label{eq:sk}$$ where $\langle \cdots \rangle$ denotes the ensemble average, $N$ is the number of particles, $\rho_{\bm{k}} = \sum_{i=1}^{N} \exp \left(-i\bm{k}\cdot \bm{r}_{i}\right)$ is the number density, $\bm{r}_i$ is the position vector of particle $i$, and $\bm{k}$ is the wave vector. In a crystal, the density $\rho_{\bm{k}}$ has components only at particular wave vectors $\bm{k}$’s because of the periodic arrangement of particles, leading to sharp diffraction spots at those wave vectors in the structure factor. These spots provide a complete description of a crystal structure. On the other hand, liquids and amorphous solids do not possess long-range translational order, and, as a result, only board isotropic amorphous halos are usually observed, which makes their structural characterization extremely difficult. This has also been the case for liquid water. So far no evidence of the coexistence of two types of structural motifs has been detected in $S(k)$ ($k=|\bm k|$), which has been a main cause of a continuous doubt on the two-state model. In this Letter, however, we report a new analysis of $S(k)$ focusing on the first few peaks, which provides direct experimental evidence for the coexistence of the two types of structural motifs and thus supports the two-state model. To do so, we focus on the lowest wave number peak in liquid water. In simple liquids such as hard spheres and LJ liquids, the first diffraction peak usually appears at the wave number $k \hat{r}/2\pi \simeq 1$ corresponding to the average nearest neighbor distance $\hat{r}$, or the average interparticle distance. However, it has been reported that $S(k)$ of a wide class of materials has a peak at a lower wave number whose corresponding length is longer than the average nearest neighbor distance [@elliott1991medium]. Such a peak has been widely observed in the so-called tetrahedral liquids such as SiO$_2$, GeO$_2$, BeF$_2$, Si, Ge and C, and widely known as the first sharp diffraction peak (FSDP) [@shi2019distinct]. The emergence of FSDP has been considered as a signature of intermediate-range structural ordering in liquids and amorphous states. Recently we have discovered [@shi2019distinct] that FSDP of these liquids originates from the scattering from the density wave characteristic of a tetrahedral unit in LFTS, which is the fundamental structural motif of tetrahedral materials. More precisely, a density wave whose wave vector corresponds to the height $H$ of the tetrahedral structure (e.g., along the $Z$ direction in Fig. \[fig:sqooTetraZeta\]a) generates a sharp diffraction peak specifically at $k_\mathrm{T1}=k \hat{r}/2\pi \simeq 3/4$, i.e., FSDP. We note that a tetrahedral unit produces four peaks in the range $0.5 \leq k \hat{r}/2\pi \leq 3.0$ with peak wave numbers labelled as $k_{\mathrm{T}i}$ ($i=1\sim4$) from low to high $k$ [@shi2019distinct] (Fig. \[fig:sqooTetraZeta\]b). If the two structural motifs revealed by $\zeta$ for water models are also relevant to real water, there should be the corresponding distinct signatures in the experimentally measured structure factor. Such a signature is indeed seen from the locations of the first diffracton peak in the structure factor of low $T$ and high $P$ water (Fig. \[fig:sqoo\_lowT\_highP\]). We can see a more distinct signature in simulated model waters, for which we are able to access both much lower temperatures (predominantly composed of LFTS), and higher pressures (predominantly composed of DNLS) than for experiments, without suffering from ice crystallisation. Figures \[fig:sqooTetraZeta\]b and c show the partial O-O structure factor of TIP4P/2005 water at low $T$ (LFTS-dominant) and high $P$ (DNLS-dominant), respectively, together with those of typical amorphous tetrahedral materials, C, Si and Ge. We can clearly see that low-$T$ water shows the structure factor very similar to the typical amorphous tetrahedral materials and its FSDP is exactly located at $k_\mathrm{T1}=k \hat{r}/2\pi \simeq 3/4$, as expected [@shi2019distinct]. On the other hand, high-$P$ water has a first diffraction peak at $k_\mathrm{D1}=k \hat{r}/2\pi \simeq 1$, as simple liquids do, reflecting its (partially) disordered nature. In the two-state regime lying between the two extreme conditions, where LFTS and DNLS coexist with comparable populations, distinct signatures from the two structural motifs are expected to appear in the structure factor of liquid water. To reveal local structural characteristics in the wave-number space, we employ what is called “the Debye scattering function” [@debye1915zerstreuung] (see Eqs. \[eq:skdebye\] - \[eq:skzeta\] and Fig. \[fig:sqdebye\]). This allows us to access the correlation between the local structure of each structural motif characterized by $\zeta$ and its local structure factor on the firm theoretical basis. Figures \[fig:sqooTetraZeta\]d and e show the $\zeta$ dependent O-O structure factor $S(k,\zeta)$ of TIP4P/2005 water at 1 bar and 240 K, where water has equal amount of LFTS and DNLS (or, $s=1/2$). We refer this particular temperature to the Schottky temperature [@shi2018common] and denote it as $T_{s=1/2}$. Strikingly, we can see two distinct peaks at $k_\mathrm{T1}$ and $k_\mathrm{D1}$ in different $\zeta$ domains, which are nicely characterized by the two Gaussian components in the distribution of $\zeta$. Thus, we may conclude that the two peaks at $k_\mathrm{T1}$ and $k_\mathrm{D1}$ in the structure factor should correspond to LFTS and DNLS, respectively (Figs. \[fig:sqooTetraZeta\]e and f). We have also confirmed the same feature for TIP5P and ST2 water (Fig. \[fig:sqooZeta\]). Together with the bimodality of the structural descriptor $\zeta$ and coordination number $N_\mathrm{fs}$, this result further supports the two-state model. We emphasize that our finding indicates that we can now access the two-state signature [*experimentally*]{} by analysing the structure factor of real water. Unfortunately, however, because the $k_\mathrm{T1}$ and $k_\mathrm{D1}$ peaks are close to each other, they are heavily overlapped under substantial thermal fluctuations, which makes a clear separation difficult. This difficulty is a source of the long-standing controversy. Thanks to the strong two-state nature in liquid silica—a tetrahedral liquid structurally similar to water [@shi2018impact]—and large scattering cross sections of the atoms, we recently found that the apparent “first diffraction peak” in the Si-Si partial structure factor of silica is indeed a doublet: A Lorentzian peaked at $k_\mathrm{T1}$ and a Gaussian peaked at $k_\mathrm{D1}$ are necessary to properly describe the apparent “first diffraction peak”. Moreover, the integrated intensity of the $k_\mathrm{T1}$ component is proportional to the fraction $s$ of LFTS, which is determined independently from a microscopic structural descriptor $z$ [@shi2018impact]; namely, it obeys the prediction of the thermodynamic two-state model [@shi2019distinct]. ![Structural bimodality in the structure factor of liquid water. [**a**]{}, A schematic representation of a regular tetrahedron formed by five water molecules with the nearest O-O distance $r_{\rm OO}$, height $H$, width $D$, edge length $L$. [**b**]{}, The O-O partial structure factor $S_\mathrm{AA}(k)$ ($\mathrm{AA}=\mathrm{OO}$) of a regular tetrahedron (formed by five oxygen atoms) and simulated TIP4P/2005 liquid water at 1 bar, 194 K, together with the structure factor of typical amorphous tetrahedral materials, C [@gilkes1995], Si [@laaziri1999] and Ge [@etherington1982]. Deeply supercooled liquid water clearly shows four characteristic peaks ($k_{Ti}$ ($i=1 \sim 4$)) common to tetrahedral materials. The peak position of FSDP, $k_{T1}=kr_\mathrm{OO}/2\pi \simeq 3/4 $, indicated by the arrow cooresponds to the height $H$ of an LFTS [@shi2019distinct]. [**c**]{}, The O-O partial structure factor of TIP5P/2005 water at 10000 bar, 250 K (see Fig. \[fig:sqooP\] for results of real and TIP5P water). It shows a characteristic peak of normal disordered systems at $k_{D1}=kr_\mathrm{OO}/2\pi \simeq 1 $ (see the arrow). [**d**]{},[**e**]{}, The $\zeta$-dependent partial O-O structure factor $S(k,\zeta)$ of TIP4P/2005 water at 1 bar, 240 K, calculated by Debye’s scattering equation (Eqs. \[eq:ski\] - \[eq:skzeta\]), in side ([**d**]{}) and top ([**e**]{}) view (see Fig. \[fig:sqooZeta\] for TIP5P and ST2 models). The characteristic peaks, $k_\mathrm{T1}$ and $k_\mathrm{D1}$, are highlighted by blue and red circles, respectively. [**f**]{}, The distribution of $\zeta$ shows two Gaussian components, corresponding to LFTS (blue shade) and DNLS (red shade) respectively. The wave numbers in [**b**]{}, [**c**]{}, [**d**]{}, [**e**]{} are scaled by the nearest neighbor distance $r_\mathrm{AA}$ for all cases (A = C, Si, Ge and O in water).[]{data-label="fig:sqooTetraZeta"}](sqooTetraZeta-small){width="16cm"} ![Analysis of O-O partial structure factors of real water and model waters at ambient pressure. [**a**]{}-[**d**]{}, X-ray scattering data of real water [@skinner2014structure; @pathak2019intermediate]. [**e**]{}-[**h**]{}, TIP4P/2005 model. [**i**]{}-[**l**]{}, TIP5P model. [**m**]{}-[**p**]{}, ST2 model. The O-O partial structure factors are shown by spheres in 3D plots ([**a**]{}, [**e**]{}, [**i**]{}, [**m**]{}) and circles in 2D plots ([**b**]{}, [**f**]{}, [**j**]{}, [**n**]{}) with more blue and red color for lower and higher temperatures respectively. The colored surfaces in ([**a**]{}, [**e**]{}, [**i**]{}, [**m**]{}) are the fits of our model (two Lorentzian (L1 + L2) and two Gaussian (G1 + G2) functions; see Eq. \[eq:sk2\] for the details) to the structure factors. In [**b**]{}, [**f**]{}, [**j**]{}, [**n**]{}, four characteristic peaks obtained from the fit are displayed by broken lines and assigned as indicated by the arrows in [**b**]{}. The colored surfaces in the right two columns show the temperature dependences of the four characteristic peaks from the fit in side view ([**c**]{}, [**g**]{}, [**k**]{}, [**o**]{}) and top view ([**d**]{}, [**h**]{}, [**l**]{}, [**p**]{}). The color bar is shown in [**c**]{} and the number in it denotes the ratio of the peak height to the maximum height of each peak over the temperature and wave number ranges shown in each image. The wave number is scaled by the nearest neighbor O-O distance $r_\mathrm{OO}$.[]{data-label="fig:sqooT"}](sqooT-small){width="16.6cm"} Recent progress of x-ray scattering techniques enables to measure structure factors of liquid water with high accuracy down to 254.1 K [@skinner2014structure; @skinner2013benchmark], which makes a detailed structural analysis possible even for real water, as for silica. Here we analyse the O-O partial structure factors of real water as well as TIP4P/2005, TIP5P, and ST2 waters by using four peak functions for fitting (see Fig. \[fig:sqooT\]). Indeed, we find that the apparent “first diffraction peak” in the O-O partial structure factors of real water as well as model waters can be nicely described by the sum of a Lorentzian (L1) and a Guassian (G1) over a wide temperature range (Figs. \[fig:sqooT\] and \[fig:sqoofit\]). We call this fitting scheme ‘scheme II’ (see Methods). In particular, the Lorentzian and Gaussian functions have peaks at $k_\mathrm{T1}=kr_{\mathrm{OO}}/2\pi \sim 3/4$ and $k_\mathrm{D1}=kr_{\mathrm{OO}}/2\pi \sim 1$, corresponding to LFTS and DNLS respectively, in agreement with the above-mentioned silica case and the Debye scattering function shown in Fig. \[fig:sqooTetraZeta\]. The integrated intensity of the Lorentzian peak follows the prediction of the two-state model and agrees well with the fraction of LFTS, $s$, determined independently by $\zeta$ and $N_\mathrm{fs}$. Here we note that the Lorentzian and Gaussian shapes reflect the different nature of the two structural motifs: LFTS has rather unique local tetrahedral order, whereas DNLS intrinsically has high structural fluctuations. The presence of the bimodality in the experimental structure factor of liquid water (Fig. \[fig:sqooT\]), as well as in $\zeta$ [@Russo2014; @shi2018Microscopic; @shi2018impact; @shi2018common] and $N_\mathrm{fs}$, together with their inter-consistency, unambiguously show the existence of the two types of local structures (LFTS and DNLS) in liquid water and thus support the two-state description of liquid water. Real water TIP4P/2005 TIP5P ST2 ----------------- ------------ ------------ --------- --------- $\Delta E$ (K) -1929.0 -1802.0 -3355.9 -4612.5 $\Delta \sigma$ -8.2845 -7.5779 -13.134 -16.106 $T_{s=1/2}$ 232.85 237.80 255.51 286.39 : Two-state parameters for real water and model waters. \[Tablepara\] Unlike at low $T$, we find that at high $T$ only one Gaussian function is enough to properly describe the apparent “first diffraction peak” in the experimental and simulated O-O structure factors. We call this fitting scheme ‘Scheme I’. One might think that Scheme I might work even at any temperatures, which is expected for continuum models. Thus, to rationalise the relevance of Scheme II at low $T$, or to confirm the bimodality of the apparently first diffraction peak in an unambiguous manner, we show in Fig. \[fig:fitError\] the difference in the mean squared residual, which measures the deviation of the fit from the data, between Schemes I and II as a function of the scaled temperature $T/T_{s=1/2}$. We can clearly see a tendency common to both real water and simulated model waters: At temperatures above 1.1$T_{s=1/2}$ a single Gaussian (Scheme I) can describe the apparent “first diffraction peak” in the structure factor equally well as a Gaussian plus a Lorentzian function (Scheme II). Below 1.1$T_{s=1/2}$, on the other hand, Scheme I starts to seriously fail in describing the data, reflecting the rapid growth of the fraction of LFTS below that temperature. The failure of Scheme I at low temperatures not only supports the emergence of the bimodality in the apparent “first diffraction peak” there, but also explain why the two-state feature can hardly be observed in liquid water at ambient condition [@smith2005unified; @clark2010small; @niskanen2019compatibility]. Moreover, our two-state description (Scheme II) of the structure factor provides a direct experimental access to the degree and range of local structural ordering in real water. The fraction $s$ of LFTS, which is propotional to the integrated intensity of FSDP at $k_\mathrm{T1}$, increases rapidly towards the LDL limit ($s\simeq 1$) upon cooling, as shown in Fig. \[fig:sl\]a. The increase is faster for TIP5P and ST2 water than for TIP4P/2005 and the real water, indicating the “over-structured" tendency in the former two models. In the two-state language, TIP5P and ST2 models overestimate the energy gain and entropy loss upon the formation of LFTS, as shown by the two-state-model parameters in Table \[Tablepara\]. Figure \[fig:sl\]b shows the increase of the coherence length estimated from the width of FSDP (Eq. \[eq:L\]) upon cooling. Below $T_{s=1/2}$, the coherence lengths estimated from the experimental and simulated structure factors increase and converge towards the $s \rightarrow 1$ limit upon cooling. Above $T_{s=1/2}$, on the other hand, the fraction of LFTS, i.e. the integrated intensity of FSDP, is rather small and thus the data suffer from large uncertainty. In any case, the coherence length is very short, bounded between $\sim2$ Å of a single tetrahedron and $\sim6.5$ Å of LDA ice (see Fig. \[fig:sqooTetraLDA\] for the detail), in agreement with the previous measurements of structural correlation length [@xie1993noncritical; @huang2009inhomogeneous; @kim2017maxima] in real water and dynamic correlation length in TIP5P water [@shi2018origin; @shi2018common] \[Feature (1) in the introduction\]. ![Degree and range of local tetrahedral ordering in liquid water at ambient pressure. [**a**]{}, The integrated intensity of FSDP ($k_\mathrm{T1}$ peak in Fig. \[fig:sqooT\]b) as a measure of the degree of local tetrahedral ordering, or the fraction $s$ of LFTS, which monotonically increases upon cooling. The horizantal dash line indicates the upper limit of tetrahedral ordering in liquid water. [**b**]{}, The coherence length of FSDP as a measure of the range of local tetrahedral ordering, which monotonically increases with decreasing temperature. The high and low temperature limits of the coherence length from a single tetrahedron and LDA ice [@mariedahl2018] are shown by violet dash dot line and navy dot line, respectively (see Fig. \[fig:sqooTetraLDA\]). The correlation lengths determined by the Ornstein-Zernike analysis of small-angle X-ray scattering data (typically $k<0.5$ Å$^{-1}$) by different groups [@xie1993noncritical; @huang2009inhomogeneous; @kim2017maxima] are shown by dotted lines. The maximum correlation length of $\xi\simeq 4.1$ Å at $T_{s=1/2} = 229.2$ K estimated from recent small-angle X-ray scattering measurements of liquid water droplets [@kim2017maxima] is shown by the magenta star symbol. []{data-label="fig:sl"}](sl){width="14cm"} We show the first clear experimentally accessible evidence in the structure factor for the dynamical coexistence of the two types of structural motifs, LFTS and DNLS, supporting the two-state description of liquid water. We reveal that liquid water exhibits the so-called FSDP in the structure factor as other tetrahedral liquids do. The FSDP provides crucial information on the fraction of LFTS (degree of structural ordering, i.e., the order parameter of the two-state model (Eq. \[eq:s\])) and its coherence length (range of structural ordering). We hope that these findings will contribute to the convergence of long-standing debates on the structure of water. [**Acknowledgements**]{} The authors are grateful to R. Evans for his valuable suggestion on the analysis regarding the criticality. This study was partly supported by Scientific Research (A) and Specially Promoted Research (KAKENHI Grants No. JP18H03675 and No. JP25000002 respectively) from the Japan Society for the Promotion of Science (JSPS) and the Mitsubishi Foundation. Methods {#methods .unnumbered} ======= Simulation of water {#simulation-of-water .unnumbered} ------------------- Classical molecular dynamics simulations were performed in a periodic cubic box containing 1000 TIP4P/2005 [@abascal2005general] water molecules by using the Gromacs package [@Hess2008] with a time step of 2 fs. Intermolecular van der Waals forces and Coulomb interactions in real space were truncated at 9 Å, and long-range Coulomb interactions were treated by the particle-mesh Ewald method. Long-range dispersion corrections for energy and pressure were applied. All simulations were performed in *NPT* ensemble with temperature and pressure kept constant by Nos[é]{}-Hoover thermostat and Parrinello-Rahman barostat, respectively. All the bonds are constrained by using the LINCS algorithm. Long-time simulations (typically longer than 100 times molecular reorientation time) were performed after equilibration in a wide temperature range from 194 to 300 K at 1 bar, and in a wide pressure range from 1 to 10000 bar at 250 K. At 194, 197 and 200 K, two independent trajectories were generated to enhance the statistics. The simulation times for production runs are summarized in Tables \[TabletimeT\] and \[TabletimeP\]. The simulation details of TIP5P and ST2 model can be found in Refs. [@shi2018origin; @shi2018common]. Ice nucleation has not been observed at any temperature and pressure studied in this work for all the water models. $T$ (K) 194 197 200 210 220 230 240 ---------- --------------------------- --------------------------- --------------------------- ----- ----- ----- ----- $t$ (ns) $\mathbf{2} \times 38000$ $\mathbf{2} \times 41000$ $\mathbf{2} \times 23000$ 600 40 10 5 $T$ (K) 250 260 270 280 290 300 $t$ (ns) 3 2 1.2 1.2 1.0 1.0 : Simulation times $t$ used for production runs for TIP4P/2005 water at 1 bar. The bold multiplers indicate the number of independent runs. \[TabletimeT\] $P$ (bar) 1 1000 1800 4000 6000 8000 10000 ----------- --- ------ ------ ------ ------ ------ ------- $t$ (ns) 3 2 2.6 20 40 80 100 : Simulation times $t$ used for production runs for TIP4P/2005 water at 250 K. \[TabletimeP\] Simulation of silica {#simulation-of-silica .unnumbered} -------------------- A two-component system of 3456 BKS [@Beest1990; @Saika2000] silica (3456 silicon and 6912 oxygen ions) in a periodic cubic box was simulated with a time step of 0.5 fs by using the LAMMPS package [@Plimpton1995]. Short range interactions were truncated at 5.5 Å, and long-range electrostatic interactions were treated by using the particle-particle particle-mesh method. All simulations were performed in *NPT* ensemble with temperature and pressure controlled by Nos[é]{}-Hoover thermostat and barostat, respectively. Production runs were carried out at ambient pressure for 200 ps each at 5500 K and 6000 K, after equilibration runs of the same length. Simulation of metallic glass Cu$_{64}$Zr$_{36}$ {#simulation-of-metallic-glass-cu_64zr_36 .unnumbered} ----------------------------------------------- A binary metallic glass Cu$_{64}$Zr$_{36}$, containing 6400 copper and 3600 zirconium atoms was simulated by using the embedded atom method potential [@mendelev2009]. Molecular dynamics simulations were carried out with a time step of 1.0 fs by using the LAMMPS package [@Plimpton1995]. Periodic boundary condition was applied to all directions of the cubic box. Nos[é]{}-Hoover thermostat and barostat were employed to keep the temperature at 1800 K and pressure at 1 bar, respectively. A production run of 400 ps was performed at 1 bar, 1800 K after a 300 ps equilibration run. Simulation of Lennard-Jones (LJ) liquid {#simulation-of-lennard-jones-lj-liquid .unnumbered} --------------------------------------- A one-component LJ system of 6912 particle interacting via the standard 12-6 LJ potential was simulated with a time step of 0.005 by using the LAMMPS package [@Plimpton1995]. The interaction was truncated and force-shifted at $2.5\sigma$, so that both the potential and force smoothly go to zero at $2.5\sigma$. *NVT* and *NPT* simulations were performed at $\rho=0.7$, $t=0.75$ and $p=0.05$, $t=0.75$ (in reduced unit) for 4000000 steps, respectively. Characterization of local density {#characterization-of-local-density .unnumbered} --------------------------------- In this work the local density is characterized by the coordination number that measures the number of neighboring water molecules in the first coordination shell of a center molecule. The first coordination shell is defined as a sphere with a radius corresponding to the position of first minimum in the oxygen-oxygen radial distribution function, which is typically $\sim3.5$ Å, in comparable with the coherence length for liquid water (Fig. 4b). Other approaches such as Voronoi tessellation [@duboue2015characterization] and density in grids [@soper2010recent; @english2011density] have also been used to measure the local density of liquid water. The former estimates local density by using the Voronoi volume of each molecule, whereas the latter calculates the number of molecules in a small cubic box with different sizes (typically $>9$ Å). Both of these two methods report unimodal density distributions, which have often been taken as direct evidence against the two-state model. However, we argue that neither of the two methods is a proper measure of the local density. For the Voronoi tessellation method, it has been shown that its application to tetrahedral materials such as amorphous silicon suffers from a serious deficiency because of the low coordination number [@tsumuraya1993statistics]. For the grid method, on the other hand, because of the small coherence length of the structural motifs (Fig. 4b), a box of $9 \times 9 \times 9$ Å is too large to detect the local density fluctuation associated with them: For a box significantly larger than the size of the local structural motifs, the density distribution is expected to be unimodal and Gaussian, as shown in Fig. \[fig:fdensity\]. Fitting formula for the coordination number distribution {#fitting-formula-for-the-coordination-number-distribution .unnumbered} -------------------------------------------------------- At high enough temperatures, the distribution of coordination number, $P\left(N_\mathrm{fs}\right)$, of liquid water shows a unimodal Gaussian distribution as simple LJ liquid (see Fig. \[fig:fnfs-lj\] and Fig. \[fig:fnfsT\]a,d,g). However, $P\left(N_\mathrm{fs}\right)$ of liquid water significantly deviates from a single Gaussian distribution and instead displays a bimodal distribution upon cooling, which strongly suggests the development of two structural motifs in supercooled water (see Fig. 1c and Fig. \[fig:fnfs2Gau-small\]). As a result, we find that $P\left(N_\mathrm{fs}\right)$ can be properly described by the sum of two Gaussian functions, whose integrated intensity corresponds to the fraction of LFTS and DNLS respectively, $$\begin{split} P\left(N_\mathrm{fs}\right) & = \frac{s}{\sigma_\mathrm{LFTS} \sqrt{2\pi}} \exp \left[-\frac{\left(N_\mathrm{fs}-N_\mathrm{LFTS}\right)^2}{2\sigma_\mathrm{LFTS}^2}\right] \\ & + \frac{1-s}{\sigma_\mathrm{DNLS} \sqrt{2\pi}} \exp \left[-\frac{\left(N_\mathrm{fs}-N_\mathrm{DNLS}\right)^2}{2\sigma_\mathrm{DNLS}^2}\right] \label{eq:pnfs} \end{split}$$ In this equation, $s$ is defined by Eq. \[eq:s\] and other parameters can be described as follows, $$N_\mathrm{LFTS} = a_{00} + a_{01}T + a_{02}T^2 \label{eq:NLFTS}$$ $$N_\mathrm{DNLS} = a_{10} + a_{11}T + a_{12}T^2 \label{eq:NDNLS}$$ $$\sigma_\mathrm{LFTS} = b_{00} + b_{01}T + b_{02}T^2 \label{eq:SLFTS}$$ $$\sigma_\mathrm{DNLS} = b_{10} + b_{11}T + b_{12}T^2 \label{eq:SDNLS}$$ Calculation of the structure factor {#calculation-of-the-structure-factor .unnumbered} ----------------------------------- The structure factor $S(\bm{k})$, defined as the density-density correlation in reciprocal space, can be obtained by $$S(\bm{k}) = \frac{1}{N}\langle\rho_{\bm{k}} \rho_{-\bm{k}}\rangle \label{eq:skS}$$ where $\langle \cdots \rangle$ represents the ensemble average, $\bm{k}$ is the wave vector, $N$ is the number of particles and $\rho_{\bm{k}}$ is the Fourier component of the number density $\rho$, which is given by $$\rho_{\bm{k}} = \stackrel[i=1]{N}{\sum}\exp \left(-i\bm{k}\cdot \bm{r}_{i}\right) \label{eq:rhok}$$ where $\bm{r}_i$ is the coordinates of atom $i$. For an isotropic system, the structure factor is a function of only the magnitude of the wave vector, $k$: $S(k)$. Debye scattering function {#debye-scattering-function .unnumbered} ------------------------- The structure factor can also be calculated by the Debye scattering function [@debye1915zerstreuung]: $$S(k) = 1 + \frac{1}{N} \stackrel[i=1]{N}{\sum} \stackrel[j\neq i]{N}{\sum} \frac{\sin (kr_{ij})}{kr_{ij}} W(r_{ij}) \label{eq:skdebye}$$ where $W(r_{ij}) = \frac{\sin (\pi r_{ij}/r_{c})}{\pi r_{ij}/r_{c}}$ is the window function [@zhang2019improving] and $r_{c}$ is a cutoff distance. Debye scattering function allows for a local structural characterization by the molecular structure factor: $$S_{i}(k) = 1 + \stackrel[j\neq i]{N}{\sum} \frac{\sin (kr_{ij})}{kr_{ij}} W(r_{ij}) \label{eq:ski}$$ Then, the correlation between molecular structure factor $S_{i}(k)$ and local structure descriptor $\zeta$ can be evaluated by the $\zeta$-dependent structure factor: $$S(k,\zeta) = \frac{\stackrel[i]{N}{\sum} S_{i}(k)\delta(\zeta-\zeta(i))}{\stackrel[i]{N}{\sum} \delta(\zeta-\zeta(i))} \label{eq:skzeta}$$ Fitting formula for the structure factor {#fitting-formula-for-the-structure-factor .unnumbered} ---------------------------------------- The first three peaks, the first of which is actually a doublet, in the structure factor of liquid water can be well described by two Lorentzian and two Gaussian functions as $$\begin{aligned} S\left(k\right) = \frac{f_\mathrm{T1}}{\pi}\frac{\Gamma_\mathrm{T1}}{\left(k-k_\mathrm{T1}\right)^2 + \Gamma_\mathrm{T1}^2} + \frac{f_\mathrm{D1}}{\sigma_\mathrm{D1} \sqrt{2\pi}} \exp \left[-\frac{\left(k-k_\mathrm{D1}\right)^2}{2\sigma_\mathrm{D1}^2}\right] \\+ \frac{f_\mathrm{T2}}{\pi}\frac{\Gamma_\mathrm{T2}}{\left(k-k_\mathrm{T2}\right)^2 + \Gamma_\mathrm{T2}^2} + \frac{f_\mathrm{T3}}{\sigma_\mathrm{T3} \sqrt{2\pi}} \exp \left[-\frac{\left(k-k_\mathrm{T3}\right)^2}{2\sigma_\mathrm{T3}^2}\right] \label{eq:sk2} \end{aligned}$$ where the subscripts denote the peaks in the O-O partial structure factor as shown in Fig. 3b. In this equation, all the parameters depend on temperature and pressure, and therefore a large set of parameters are needed to describe the structure factor of liquid water at different thermodynamic conditions (12 parameters for each temperature and pressure). However, thanks to the weak temperature dependence of the fitting parameters (except for $f_\mathrm{T1}$ and $f_\mathrm{D1}$), we found that they can be well described by a set of polynomial functions up to the second order: $$k_{x} (T) = \tilde{k}_{x0} + \tilde{k}_{x1} \hat{T} + \tilde{k}_{x2} \hat{T}^2 \label{eq:polyk}$$ $$\Gamma_{x} (T) = \tilde{\Gamma}_{x0} + \tilde{\Gamma}_{x1} \hat{T} + \tilde{\Gamma}_{x2} \hat{T}^2 \label{eq:polyg}$$ $$\sigma_{x} (T) = \tilde{\sigma}_{x0} + \tilde{\sigma}_{x1} \hat{T} + \tilde{\sigma}_{x2} \hat{T}^2 \label{eq:polys}$$ $$f_{x} (T) = \tilde{f}_{x0} + \tilde{f}_{x1} \hat{T} + \tilde{f}_{x2} \hat{T}^2 \label{eq:polyf}$$ where the subscript $x$ ($x = \textrm{T1, D1, T2 or T3}$) denotes parameters for each peak and $\hat{T}=T/T_\mathrm{ref}$ with $T_\mathrm{ref} = 373.15$ K. This procedure allows for a simultaneous fitting of a large set of structure factors measured in a wide temperature range, which largely reduces the number of fitting parameters. Moreover, we found in practice that the fitting accuracy will not be affected if we set $\tilde{k}_\mathrm{x2}=0$ (for $x = \textrm{T1, D1, T2 or T3}$), and fix the value of $k_\mathrm{T3}$ and $\tilde{k}_\mathrm{T11}/\tilde{k}_\mathrm{D11}$ properly. Although the $k_\mathrm{T3}$ peak is only partially included in the fitting, its position is read from the data and fixed in the fitting precedure. On the other hand, the intensities of T1 and D1 peaks vary significantly with temperature and pressure, corresponding to the change in the fractions of the two structural motifs. In our previous study [@shi2019distinct], we found that the integrated intensity $f_{T1}$ of FSDP is proportional to the fraction $s$ of LFTS in liquid silica as $$f_\mathrm{T1} (T,P) = a\cdot s \label{eq:ft1}$$ where $a$ is a positive constant. This knowledge is directly applied to liquid water, since they are both characterized by the same two-state features [@shi2019distinct]. Here $s$ can further be described by the two-state model with negligible cooperativity as [@Tanaka_review; @Tanaka2000; @russo2018water; @shi2018impact; @shi2018Microscopic; @shi2018origin; @shi2018common]: $$s(T,P) = \frac{1}{1+\exp \left(\frac{\Delta E - T \Delta \sigma + P \Delta V}{k_\mathrm{B}T}\right)} \label{eq:s}$$ where $k_\mathrm{B}$ is the Boltzmann constant, $\Delta E$, $\Delta \sigma$ and $\Delta V$ are the energy, entropy, and volume differences between LFTS and DNLS in the two-state model. At ambient pressure, the term $P\Delta V$ is negligibly small. The parameters $\Delta E$ and $\Delta \sigma$ for TIP5P and ST2 waters have already been determined in our previous work [@shi2018common]. For TIP4P/2005 and real water, $\Delta E$ and $\Delta \sigma$ were independently determined by applying the two-state model (Eq. \[eq:s\]) to the fraction $s(T)$ of LFTS that can be obtained from the $g_\mathrm{OO}(r)$ by $s(T)=1-g_\mathrm{OO}(r=r_\mathrm{HB})$ [@Russo2014], where $r_\mathrm{HB}=3.5$ Å, according the Luzar-Chandler definition of H-bond [@Luzar1996hydrogen]. The parameters $\Delta E$ and $\Delta \sigma$ for real water, TIP4P/2005, TIP5P and ST2 waters are summarised in Table 1 in the main text. Since $D1$ peak is exclusively from DNLS, whose fraction is given by $(1-s)$, we can formulate the temperature and pressure dependence of $f_{D1}$ as $$f_{D1} (T,P) = b(1-s)=b \left[1- \frac{1}{1+\exp \left( \frac{\Delta E - T \Delta \sigma + P \Delta V}{k_\mathrm{B}T}\right)}\right] \label{eq:fd1}$$ where $b$ is a positive constant. After all the above considerations, only 25 free fitting parameters are necessary to fit O-O partial structure factors of liquid water at all the temperatures studied in this work. The Gaussian function represents the scattering peak coming from the interatomic correlation of DNLS (see main text). Simple liquids such as LJ and hard spheres liquids usually have this peak at the wave number corresponding to the neighboring interactomic distance $r$, and thus we constrained its position $k_{0}r_\mathrm{OO}/2\pi$ to be close to 1. The Fourier transform of a Lorentzian function is an exponentially decaying function in real space. The coherence length $\lambda$ of FSDP, which characterizes the range of coherent tetrahedral ordering, can be estimated by $$\lambda=1/\Gamma_\mathrm{T1} \label{eq:L}$$ where $\Gamma_\mathrm{T1}$ is the half width of FSDP. ![An example of the typical distribution of coordination number $N_\mathrm{fs}$, $P\left(N_\mathrm{fs}\right)$, for a simple liquid. Here we show $P\left(N_\mathrm{fs}\right)$ of Lennard-Jones liquid at $\rho = 0.7$ and $T=0.75$. $P\left(N_\mathrm{fs}\right)$ can be well described by a Gaussian distribution as indicated by the red curve.[]{data-label="fig:fnfs-lj"}](fnfs-lj){width="12cm"} ![The distribution of coordination number $N_\mathrm{fs}$, $P\left(N_\mathrm{fs}\right)$, for simulated water. Here we show $P\left(N_\mathrm{fs}\right)$ for TIP4P/2005 water at 380 K ([**a**]{}), 280 K ([**b**]{}), 240 K ([**c**]{}), for TIP5P water at 360 K ([**d**]{}), 280 K ([**e**]{}), 260 K ([**f**]{}), and for ST2 water at 360 K ([**g**]{}), 300 K ([**h**]{}), 285 K ([**i**]{}), at ambient pressure. At high temperatures, $P\left(N_\mathrm{fs}\right)$ shows a Gaussian distribution as simple Lennard-Jones liquid does (Fig. \[fig:fnfs-lj\]), whereas at low temperatures it changes to bimodal distributions, which can be properly described by the sum of two Gaussian functions (with blue and red shades). The fraction of the two Gaussian components agrees well with the fraction independently determined by $\zeta$ distribution and the prediction of the theoretical two-state model (Eq. \[eq:s\]). One component (with blue shade) corresponds to LFTS, in which the central water typically has $\sim 4$ H-bonded nearest neighbors, whereas the other (with red shade) corresponds to DNLS, in which the central water has $\sim 5$ nearest neighbors (three of them being H-bonded typically) on average at ambient pressure.[]{data-label="fig:fnfsT"}](fnfsT){width="16cm"} ![$T$-dependence of the distribution of coordination number $N_\mathrm{fs}$, $P\left(N_\mathrm{fs}, T\right)$, for three water models. [**a**]{}, [**d**]{}, [**g**]{}, $P \left(N_\mathrm{fs}, T\right)$ for simulated liquid TIP4P/2005 ([**a**]{}), TIP5P ([**d**]{}) and ST2 ([**g**]{}) water at ambient pressure. The colored surfaces in [**a**]{}, [**d**]{}, [**g**]{} are the fits to two Gaussian functions by Eqs. \[eq:pnfs\] - \[eq:SDNLS\]. The two Gaussian components of $P \left(N_\mathrm{fs}, T\right)$ are shown in side ([**b**]{}, [**e**]{}, [**h**]{}) and top ([**c**]{}, [**f**]{}, [**i**]{}) view for TIP4P/2005 ([**b**]{}, [**c**]{}), TIP5P ([**e**]{}, [**f**]{}) and ST2 ([**h**]{}, [**i**]{}) water.[]{data-label="fig:fnfs2Gau-small"}](fnfs2Gau-small){width="16cm"} ![Macroscopic density distribution for single- and two-component systems. Macroscopic density distribution $P(\rho)$ for two-component BKS silica at $T=6000$ K, $P=1$ bar ([**a**]{}), two-component metallic glass Cu$_{64}$Zr$_{36}$ at $T=1800$ K, $P=1$ bar ([**b**]{}), single-component Lennard-Jones liquid at $T=0.75$, $P=0.05$ ([**c**]{}) and single-component TIP4P/2005 water at $T=240$ K, $P=1$ bar ([**d**]{}), TIP5P water at $T=260$ K, $P=1$ bar ([**e**]{}) and ST2 water at $T=285$ K, $P=1$ bar ([**f**]{}). The macroscopic density distributions in panels [**d**]{}, [**e**]{} and [**f**]{} were measured at the same conditions as in panels [**c**]{}, [**f**]{} and [**i**]{} in Fig. \[fig:fnfsT\] for TIP4P/2005, TIP5P and ST2 water, respectively. The macroscopic density distribution always remains unimodal and Gaussian for both single- and two-component liquids, under thermal fluctuations, as it should be.[]{data-label="fig:fdensity"}](fdensity){width="14cm"} ![Distribution of local structural descriptor $\zeta$ and macroscopic order parameter $s$. [**a**]{}-[**c**]{}, Distribution of local structural descriptor $\zeta$, $P(\zeta)$, for TIP4P/2005 ([**a**]{}), TIP5P ([**b**]{}) and ST2 water ([**c**]{}) at ambient pressure. [**d**]{}-[**f**]{}, Distribution of macroscopic structural order parameter $s$, $P(s)$, for TIP4P/2005 ([**d**]{}), TIP5P ([**e**]{}) and ST2 water ([**f**]{}) at ambient pressure. All the distributions are shown at $T\approx T_{s=1/2}$ (240 K for TIP4P/2005 water, 260 K for TIP5P water and 285 K for ST2). The distribution of the macroscopic order parameter, $P(s)$, always remains unimodal and Gaussian for all the models, whereas the distribution of local order, $P(\zeta)$, is clearly bimodal with two Gaussian components (blue and red shades correspond to LFTS and DNLS, respectively). The macroscopic order parameter $s$ is estimated as the fraction of molecules whose $\zeta > \zeta_c$ at each time frame. A threshold value $\zeta_c$ ($\simeq 0.5$ Å) is chosen to satisfy $s\approx 0.5$ after time averaging.[]{data-label="fig:fs_zeta"}](fs_zeta){width="14cm"} ![Partial O-O structure factors of liquid water measured by x-ray scattering experiments at 0.1 MPa, 254.1 K [@skinner2014structure] (blue circle) and 362 MPa, 300 K [@skinner2016structure] (red square). Distinct characteristic peaks $k_\mathrm{T1}$ and $k_\mathrm{D1}$ corresponding to LFTS and DNLS respectively, are clearly located at different wave numbers for these two conditions, whereas the other peaks are located at almost the same positions.[]{data-label="fig:sqoo_lowT_highP"}](sqoo_lowT_highP){width="12cm"} ![The validity of the Debye’s scattering function analysis. The parital O-O structure factors obtained by the two-body density correlation (Eq. \[eq:skS\], black circle) and Debye scattering equation (Eq. \[eq:skdebye\], red curve) for TIP4P/2005 ([**a**]{}), TIP5P ([**b**]{}) and ST2 ([**c**]{}) water at 240 K, 260 K and 285 K, respectively, at ambient pressure. The structure factors calculated from the two methods agree well with each other. A small deviation at the second peak $kr_\mathrm{OO}/2\pi\approx 1.3$ mainly comes from the effect of the Window function we employed (see Methods).[]{data-label="fig:sqdebye"}](sqdebye){width="10cm"} ![The $\zeta$ dependent partial O-O structure factor $S(k,\zeta)$ obtained by Debye’s scattering equation (Eqs. \[eq:ski\] and \[eq:skzeta\]). Side ([**a**]{}, [**d**]{}, [**g**]{}) and top ([**b**]{}, [**e**]{}, [**h**]{}) views for TIP4P/2005 ([**a**]{}, [**b**]{}) TIP5P ([**d**]{}, [**e**]{}) and ST2 ([**g**]{}, [**h**]{}) water at 240 K, 260 K and 285 K, respectively, at ambient pressure. In ([**b**]{}, [**e**]{}, [**h**]{}) the characteristic peaks, $k_\mathrm{T1}$ and $k_\mathrm{D1}$, are highlighted by blue and red circles, respectively. [**c**]{},[**f**]{},[**i**]{}, The distribution of $\zeta$ shows two Gaussian components, corresponding to LFTS (blue shade) and DNLS (red shade) respectively for TIP4P/2005 ([**c**]{}), TIP5P ([**f**]{}) and ST2 ([**i**]{}) water. The wave numbers are scaled by the nearest neighbor O-O distance $r_\mathrm{OO}$.[]{data-label="fig:sqooZeta"}](sqooZeta-small){width="14cm"} ![Partial structure factor $S_\mathrm{AA}(k)$ of simulated water (A=O) and silica (A=Si) at $T\approx T_{s=1/2}$. At ambient pressure, $T_{s=1/2} \approx $ 240, 260, 285 and 5000 K for TIP4P/2005, TIP5P, ST2 water and BKS silica, respectively. Silica shows very similar anomalous behaviors as water does, although no critical point has been found in silica [@lascaris2014search]. Recently strong evidence supporting the existence of two structural motifs—LFTS and DNLS, in liquid silica has been revealed for BKS silica [@shi2018impact; @shi2019distinct]. The similarity between the structure factor of liquid water and liquid silica at the same fraction of LFTS further supports the two-state feature in both liquids.[]{data-label="fig:sqWaterSilicaTw"}](sqWaterSilicaTw){width="12cm"} ![Experimental O-O partial structure factor of water at high pressure. [**a**]{}, X-ray scattering data [@skinner2016structure]. [**b**]{}, TIP4P/2005 model. [**c**]{}, TIP5P model. The arrow denotes the direction of pressure increase. The wave number is scaled by the nearest neighbor O-O distance $r_\mathrm{OO}$.[]{data-label="fig:sqooP"}](sqooP){width="10cm"} ![Decomposition of the O-O partial structure factor of real water and model waters at ambient pressure. [**a**]{}-[**c**]{}, X-ray scattering data [@skinner2014structure; @skinner2013benchmark; @pathak2019intermediate]. [**d**]{}-[**f**]{}, TIP4P/2005 water. [**g**]{}-[**i**]{}, TIP5P water. [**j**]{}-[**l**]{}, ST2 water. The O-O partial structure factors are shown by black squares. The red lines are the cumulative fits of two Lorentzian (L1 + L2) and two Gaussian (G1 + G2) functions to the structure factors. The four characteristic peaks obtained from the fits are displayed by green, blue, orange and magenta curves. The FSDP is highlighted by green shading. The wave number is scaled by the nearest neighbor O-O distance $r_\mathrm{OO}$.[]{data-label="fig:sqoofit"}](sqoofit){width="16cm"} ![The fitting quality of structure factors. Two schemes were applied to the analysis of the partial O-O structure factor $S_\mathrm{OO}(k)$ of liquid water. In Scheme I, the “apparent” first three peaks (from low to high wave number) in $S_\mathrm{OO}(k)$ are fitted to one Gaussian (1st peak), one Lorentizan (2nd peak), and one Gaussian (3rd peak) function, respectively. All the fitting parameters are free, and thus we need a new set of 9 parameters for each temperature. In Scheme II, the “apparent” first three peaks (from low to high wave number) in $S_\mathrm{OO}(k)$ are fitted to one Lorentzian + one Gaussian (1st peak), one Lorentizan (2nd peak), and one Gaussian (3rd peak) function, respectively. We constrain the temperature dependence of the fitting parameters. We need only up to 25 free fitting parameters for simultaneous fitting of $S_\mathrm{OO}(k)$ at all the temperatures (see Methods). Scheme I is chosen for comparison, since it reasonably well describes $S_\mathrm{OO}(k)$ of liquid water at high temperatures with only three functions (8 fitting parameters at each temperature with position of $k_\mathrm{T3}$ peak being fixed). The mean squared residual $\chi^2$ measures the deviation of the fits from the data. The difference in the mean squared residual between Schemes I and II, $\Delta \chi^2 = \chi^2_\mathrm{I} - \chi^2_\mathrm{II}$, is shown for real water, simulated TIP4P/2005, TIP5P and ST2 waters as a function of scaled temperature $T/T_{s=1/2}$. All the curves collapse into a master curve, suggesting a universal behavior independent of models. For all the systems, $\Delta \chi^2$ remains almost zero above $1.1T_{s=1/2}$, whereas sharply increases below $1.1T_{s=1/2}$, reflecting the fast growth of LFTS upon cooling (Fig. 4a). The failure of Scheme I at lower temperatures strongly suggests that the structure of liquid water fundamentally changes with temperature. At low enough temperatures (below $\sim 1.1T_{s=1/2}$), liquid water can no longer be treated to be microscopically homogeneous unlike at high temperatures, and should be regarded as a dynamical mixture of two local structure motifs. The two structure motifs have different contributions to the “apparent” first diffraction peak in $S_\mathrm{OO}(k)$ (Figs. 2-3), leading to the failure of Scheme I that assumes only a single type of contribution to the peak. For real water, there are three data points significantly deviating from the master curve, which are attributed to the experimental errors in the structure factor data (see Figure \[fig:noise\])[]{data-label="fig:fitError"}](fitError){width="12cm"} ![Noise analysis in the experimental structural factor of supercooled water. [**a**]{}, The height of the 1st, 2nd and 3rd peak in the experimental partial O-O structure factor as a function of temperature. The height of the 3rd peak is multiplied by a factor of 1.16 for clarification. [**b**]{}, The height of the 1st and 2nd trough in the experimental partial O-O structure factor as a function of temperature. In [**a**]{} and [**b**]{} the filled and open symbols are data from Ref. [@skinner2014structure] and [@pathak2019intermediate], respectively. [**c**]{}, The experimental partial O-O structure factor of supercooled water at 234.8, 240.9 and 243.8 K (Ref. [@pathak2019intermediate]). [**d**]{}, The experimental partial O-O structure factor of supercooled water at 254.1, 263.1, 264.0, 268.1 and 277.1 K (solid line from Ref. [@skinner2014structure] and dashed line from Ref. [@pathak2019intermediate]). The inset shows the enlarged plot of the 1st peak inside the black square. Clearly, the structure factors measured at 234.8 K, 264.0 K and 268.1 K suffer from large uncertainty, because they significantly deviate from the normal temperature dependence, as highlighted by the black and red circles in [**a**]{} and [**b**]{}. The large errors at these three temperatures result in the three outliers in the $\Delta \chi^2$ value for real water, deviating from the master curve. For example, at 234.8 K the 3rd peak and the 2nd trough of the structure factor is substantially smaller than the expectation from the temperature trend, which is the major source of the deviation of the fits from the data by our fitting scheme II (see Figure \[fig:sqoofit\]a). We note that scheme I independently fits structure factors at different temperatures, whereas scheme II simultaneously fits structure factors at all the temperatures. Because scheme II uses much less fitting parameters, it is more robust, but at the same time, more sensitive to the errors in the data than scheme I.[]{data-label="fig:noise"}](noise){width="16cm"} ![Analysis of the O-O partial structure factor of a single regular tetrahedron and experimental LDA ice. [**a,**]{} Decomposition of the structure factor of a single regular tetrahedron (Fig. 2b) by two Lorentzian (L1 + L2) and one Gaussian (G) functions. [**b,**]{} Decomposition of the structure factor of experimental LDA ice [@mariedahl2018] by the same functions. The three characteristic peaks obtained from the fit are displayed by blue, orange and magenta curves. The coherence lengths of a single regular tetrahedron and experimental LDA ice were thus estimated to be $\sim2$ Å and $\sim6.5$ Å respectively, from the widths of the corresponding FSDP’s (blue curves). The wave number is scaled by the nearest neighbor O-O distance $r_\mathrm{OO}$.[]{data-label="fig:sqooTetraLDA"}](sqooTetraLDA){width="10cm"}

--- author: - 'Clément Dombry[^1] Paul Jung[^2]' title: 'Approximation of stable random measures and applications to linear fractional stable integrals.' --- [**Key words:**]{} stable random measure, moving average, fractional stable motion, Lindeberg-Feller. [**AMS Subject classification:**]{} 60G22, 60G52, 60G57, 60H05 Introduction ============ Stable integration is an important tool in the theory of $\aa$-stable processes. Similar to the theory for Gaussian processes, it is known ([@samorodnitsky1994stable Sec. 13.2]) that all stable processes, satisfying mild conditions, can be constructed from integrals of the form \[stableprocess\] X\_t=\_E f\_t(x) M\_(dx),tT, where $M_\alpha$ is an independently scattered $\alpha$-stable random measure on the measurable space $(E,{{\mathcal E}})$ with control measure $m$ and $(f_t)_{t\in T}$ is a kernel such that $f_t\in L^\alpha(E,{{\mathcal E}},m)$ for all $t\in T$. If $T=\R$ and $f_t=1_{[0,t]}$ then $X_t$ is an $\aa$-stable Levy motion having independent and stationary increments (the symmetric case is the stable analog of Brownian motion). In this work, we approximate the finite-dimensional distributions of using a Riemann sum-type scheme. These approximations are useful for the dual purposes of intuition and simulation of stable processes. The weak convergence of our scheme is facilitated by a Lindeberg-Feller type stable limit theorem, which we have not previously seen in the literature. A couple of different discrete approximations of stable processes have appeared previously in the literature. One approach is Lepage’s series which was improved upon in a series of papers by J. Rosinski (see [@rosinski2001series] and the references therein). In the present paper, we use a lattice approximation of stable integrals which extends, to $f\in L^\aa(\R^d)$, the “moving-average" discrete approximations of L-FSMs in [@davydov; @maejima1983class; @astrauskas1983limit; @davis1985limit] corresponding to the case $f_t=1_{[0,t]}$. The work of [@kasahara1988weighted] improved upon these earlier papers to obtain discrete approximations of slightly more general stable processes, while [@avram1992weak] showed that tightness of discretized L-FSMs cannot be achieved in the $J_1$-Skorokhod topology. In [@kokoszka1995fractional], it was shown that discretized L-FSMs satisfy the fractional ARIMA equations and a closer look at issues concerning absolute convergence was taken. A secondary purpose of this work is to generalize certain Gaussian integrals to the $\aa$-stable case and, as in [@kasahara1988weighted], we then approximate such integrals with the scheme just described. In the past fifteen years or so, there has been an effort to develop stochastic integrals with respect to a broader class of Gaussian processes than just Brownian motion. In particular, consider Gaussian processes with stationary increments, but replace the independent increments condition with the weaker condition of self-similarity. Normalizing the variance at $t=1$ to unity, one gets the single parameter family of fractional Brownian motions (FBM) with Hurst self-similarity parameter $0<H<1$. The theory of integration with respect to FBM is difficult because FBM is not a semi-martingale. Nevertheless, rapid progress has been made using several different approaches (with significant overlap between them). Roughly speaking, they can be categorized into four approaches which use, respectively, fractional derivatives and integrals, Malliavin calculus, fractional white noise theory, and path-wise integration (see [@biagini2008]). In Section \[sec:lfsm\], we consider a generalization of the FBM integral based on fractional integro-differentiation to $\aa$-stable analogs of FBM called the linear fractional stable motions[^3] (L-FSMs). By “$\aa$-stable analog", we mean that a L-FSM is a self-similar, symmetric stable process with stationary increments. Any process with these properties is called a FSM. In contrast to the Gaussian picture, for each admissible $(\aa,H)$ pair, there is not a unique (normalized) FSM, up to finite-dimensional distributions. Moreover, for each $(\aa,H)$ pair with $0<H<1$ and $H\neq 1/\aa$, there are infinitely many L-FSMs. These L-FSMs are represented by where $E=\R$ is equipped with Lebesgue measure, $M_\aa$ is symmetric, and \[LFSMkernel\] &&f\^[a,b]{}\_t(x):=\ &&a$(t-x)_+^{H-1/\aa} - (-x)_+^{H-1/\aa}$ + b$(t-x)_-^{H-1/\aa} - (-x)_-^{H-1/\aa}$ for properly normalized order pairs $(a,b)$ where $a,b\ge 0$ (see [@samorodnitsky1994stable Sec. 7.4] for more details). Here $x_-=|x|$ if $x<0$ and $0$ otherwise (similarly for $x_+$). The family of L-FSMs were the first FSMs to be constructed and studied, and much is known about them. Our motivation comes partly from [@pipiras2000integration] which handles the $\aa=2$ case. As in their work, we restrict ourselves to deterministic integrands, but [@pipiras2000integration] shows that even in the $\aa=2$ case, the theory for deterministic integrands is not completely trivial. An integral with respect to L-FSM will be defined as an integral with respect to a linear fractional stable random measure which we define for $\aa>1$ and all permissable Hurst parameters $0<H<1$. We have recently learned that when $H>1/\aa$, [@maejima2008limit] has developed similar integrals and also discrete approximations for them. However, the convergence results for their approximations concern a strictly smaller class of integrands. In particular, they require bounded integrands which are piece-wise continuous (we require no continuity or boundedness) and which must satisfy a faster tail decay than ours. The rest of the paper is organized as follows. In Section 2, we review the notion of stable random measures and present our result concerning the convergence of discretizations of stable random measures. In Section 3, integrals with respect to L-FSMs are defined, and their approximation by moving averages of i.i.d. random variables are discussed. Section 4 is devoted to the proofs. Discrete approximations of random measures =========================================== A useful viewpoint is that a random measure is a stochastic process: Let $(E,{{\mathcal E}})$ be a measurable space and $V$ be a vector space of measurable functions $f:E\to{\mathbb{R}}$. A random measure on $(E,{{\mathcal E}})$ is a stochastic process $(M[f])_{f\in V}$ satisfying the linearity property: for all $a_1,a_2\in{\mathbb{R}}$ and $f_1,f_2\in V$, $$\label{eq:lin} M[a_1f_1+a_2f_2]=a_1M[f_1]+a_2M[f_2] \quad {\rm almost\ surely.}$$ Let us make a few comments concerning this definition. First of all, the linearity property ensures that the finite-dimensional distributions of the process $(M[f])_{f\in V}$ are determined by its one-dimensional distributions. If ${\mathbf{1}}_A\in V$ for $A\in{{\mathcal E}}$, we note $M(A)=M[{\mathbf{1}}_A]$ which is thought of as the random measure of the set $A$. If $M(A_i)$ are independent for disjoint sets $A_1,\ldots, A_k$, then $M$ is said to be [*independently scattered*]{}. For general $f\in V$, to emphasize the analogy with usual integration, the notation $M[f]=\int_E f(x)M(dx)$ is often used. Finally, if one so pleases, one may also view the random measure $M$ as a random linear functional on the linear space $V$ (see for example [@dudley1969random]). Let ${{\mathcal S}}_\aa(\sigma)$ be the symmetric $\alpha$-stable () law of index $\alpha\in (0,2]$ with $\sigma \geq 0$ being the scale parameter[^4]. We denote the characteristic function of ${{\mathcal S}}_\aa(\sigma)$ by $$\label{eq:fcstable1} \lambda_\alpha(\theta)=\exp\left(-|\sigma\theta|^\alpha \right), \quad \theta\in{\mathbb{R}}.$$ To reduce notation, when $\sigma=1$ we simply write ${{\mathcal S}}_\alpha={{\mathcal S}}_\aa(1)$. We now consider the class of independently scattered random measures, i.e. those where $M[f]$ is for all $f\in V$. Suppose that $(E,{{\mathcal E}},m)$ is a measure space where $m$ is a $\sigma$-finite measure and ${{\mathcal E}}_0$ is the class of measurable sets with finite $m$-measure. Following [@samorodnitsky1994stable Sec. 3.3], we say that the independently scattered random measure $M_\alpha$ has [*control measure*]{} $m$ if $M_\alpha(A)$ has distribution ${{\mathcal S}}_\alpha(m(A)^{1/\aa})$ for all $A\in{{\mathcal E}}_0$. For such random measures, it can be shown that $V=L^\aa(E)$ (see [@samorodnitsky1994stable Ch. 3]) and that the distributions $M_\aa(A), A\in{{\mathcal E}}_0$ uniquely determine the characteristic functions {iM\_å\[f\]}= -\_E |f(x)|\^å m(dx). In the Gaussian case $\alpha=2$, this is just the usual Wiener integral. In the rest of this section we develop a discrete approximation of $M_\aa$ when $E={\mathbb{R}}^d$ with Lebesgue control measure. We begin by recalling that the domain of attraction of ${{\mathcal S}}_\aa $ consists of random variables $\xi$ such that $$\label{eq:xi1} a_n^{-1}\Big(\sum_{k=1}^n \xi_k -b_n\Big) \Longrightarrow {{\mathcal S}}_\aa \quad \mathrm{as}\ n\to\infty,$$ where $a_n>0$ and $b_n\in{\mathbb{R}}$ are normalization constants and the $\xi_k$’s are i.i.d. copies of $\xi$. In the sequel, we will assume $\eta$ is , and $\xi$ is not only in the domain of attraction of $\eta$, but also that the normalization constants are precisely $$\label{eq:xi2} a_n=n^{1/\alpha}\quad \mbox{and}\quad b_n=0,\quad n\geq 1.$$ When $\aa<2$, such distributions are said to be in the domain of [*normal*]{} attraction of ${{\mathcal S}}_\aa$ which is not to be confused with the normal domain of attraction. We propose a discrete approximation of $M_\alpha$ based on the lattice $h{\mathbb{Z}}^d\subset {\mathbb{R}}^d$ with edge length $h$. Let $(\xi_k)_{k\in{\mathbb{Z}}^d}$ be a random field of i.i.d. copies of $\xi$ satisfying and and formally define $$\label{eq:series} M_\aa^h[f]:=\sum_{k\in{\mathbb{Z}}^d} f^h(k)\xi_{k},$$ where for $I^d=[0,1)^d$, $f^h:\Z^d\mapsto \R$ is $$\begin{aligned} \label{def:superh} f^h(k)&:=&\int_{h(k+I^d)} f(x) \,dx, \quad f\in L^1_{\text{loc}}(\R^d).\end{aligned}$$ Note that we have implicitly fixed an enumeration $\{k_n, n\geq 1\}$ of $\Z^d$ and convergence of $\sum_{k\in{\mathbb{Z}}^d}a_k$ really means convergence of $\sum_{n=1}^\infty a_{k_n}$. The discrete random measures $M^h_\aa$ approximate $M_\aa$ in the following sense: \[maineq2prop\] Fix $\aa\in(0,2]$. If $\aa\in[1,2]$, let $f_t\in L^\aa(\R^d)$ for all $t$ in an index set $T$. If $\aa\in(0,1)$, for a fixed $\epsilon>0$ let $f_t\in L^{\aa-\epsilon}\cap L^1(\R^d)$ for all $t\in T$. Then as $h\to 0$\[maineq2\] M\_å\^h\[f\_t\] M\_. The notation $\stackrel{fdd}{\longrightarrow}$ denotes weak convergence of the finite dimensional distributions, i.e., convergence in distribution of $ M_\aa^h[f]$ for all linear combinations $f=\th_1f_{t_1}+\cdots +\th_nf_{t_n}$. When the functions are indexed by one-dimensional time, it was shown in [@avram1992weak], that even for the simple family $f_t=1_{[0,t]}\in L^\aa(\R)$, the above convergence does not hold in the $J_1$-Skorokhod topology[^5]. Theorem \[maineq2prop\] will follow from a Lindeberg-Feller type result for stable distributions which we state in Theorem \[theo:whitecase\] below. Let us make one more remark before stating Theorem \[theo:whitecase\]. One motivation for was to provide a means to simulate a process $X_t=\int_{{\mathbb{R}}^d}f_t(x)M_\alpha(dx)$. For such simulations, it is natural to let the $\xi_k$’s be i.i.d. copies of ${{\mathcal S}}_\aa $ (rather than only in the domain of normal attraction). If one is concerned only with one-dimensional distributions (a single function $f$), then a better approximation is given by replacing $f^h(k)$ in by \[eq:one-dd\] u\_k:=$\int_{h(k+I^d)}f(x)^{<\aa>} \,dx$\^[&lt;1/å&gt;]{} where we have used the notation $x^{<\aa>}:=\mathrm{sign}(x)|x|^\alpha$. In fact, using $u_k$, one can check that the approximation is exact, and the right and left sides of are equal in distribution for every $h>0$. The reason we have not used for the general approximation scheme is due to the fact that is no longer a random measure under because the linearity property does not hold. The analysis of the finite-dimensional distributions then becomes much more difficult. \[theo:whitecase\] Suppose $(\xi_{k,j})_{k,j\in\N}$ is an i.i.d. array of random variables in the domain of normal attraction of ${{\mathcal S}}_\alpha$, $\alpha\in(0,2]$, and $(u^{(j)})_{j\in\N}$ is a sequence of vectors in $\ell^\alpha$, i.e. $u^{(j)}:=(u^{(j)}_k)_{k\in\N}\in \ell^\alpha$ for all $j\in\N$. If 1. $\lim_{j\to\ff} \|u^{(j)}\|_{\ell^\alpha}=\sigma$ and\ 2. $\lim_{j\to\ff} \|u^{(j)}\|_{\ell^\infty}=0$ then $ \sum_{k}u_k^{(j)}\xi_{k,j}<\ff $ a.s. for each $j\in\N$ and $$\sum_{k\in\N} u_k^{(j)}\xi_{k,j} \Longrightarrow {{\mathcal S}}_\alpha(\sigma) \quad \mathrm{as}\ j\to\infty.$$ [**Remarks:**]{} 1. The condition that the $\xi_{k,j}$ be identically distributed can be relaxed slightly to the condition that $\E[ \exp(i\theta \xi_{k,j}) ] = 1-|\theta|^\alpha + o(|\theta|^\alpha)$ holds uniformly in $k,j$ as $\theta\to 0$. For example, they may be chosen from a finite family of distributions in the domain of normal attraction of ${{\mathcal S}}_\aa$. 2. The a.s. convergence $\sum_{k\in\N} u_k^{(j)}\xi_{k,j}<\infty$ in fact occurs if and only if $u=(u_k)_{k\in\N}\in\ell^{\aa}$ as will be seen in Lemma \[lem1\]. 3. Although the series $\sum_{k\in\N} u_k^{(j)}\xi_{k,j}$ may not converge absolutely, switching the order of summation does not change the convergence in distribution to ${{\mathcal S}}_\aa(\sigma)$. This will be apparent in the proof. 4. In the Gaussian case, the result can be seen as a variant of the usual Lindeberg-Feller Theorem by noticing that condition 2, concerning $\ell_\infty$, is equivalent to $$\lim_{j\to\ff}\sum_k 1{\{|u^{(j)}_k|>\eps\}}=0$$ for all $\eps>0$. More generally when , the result is related to Theorem 3.3 of [@petrov1995limit] which gives necessary and sufficient conditions for convergence of sums of independent triangular arrays to a given infinitely divisible distribution. In particular, the conditions of Theorem \[theo:whitecase\] above imply the [*infinite smallness*]{} condition (cf. Eq. (3.2) in [@petrov1995limit]). However, it is unclear how to obtain Theorem \[theo:whitecase\] from [@petrov1995limit Thm 3.3] in a manner simpler than the proof of Theorem \[theo:whitecase\] provided below. Linear fractional stable random measures {#sec:lfsm} ======================================== To simplify matters, in this section we will restrict our attention to the one-dimensional case $E=\R^1$ equipped with Lebesgue measure. For higher dimensions, see the first remark following Corollary \[theo:fraccase\]. Also, in this section we assume that $1<\aa\le 2$. Fractional integro-differentiation and L-FSM integrals ------------------------------------------------------ In this subsection we define the stochastic integration of suitable functions with respect to different L-FSMs in terms of stable random measures which are not independently scattered. This is achieved using fractional integrals and derivatives. The intuition behind our definition is based on two facts. The first is that fractional integrals and derivatives can be realized using convolutions, and the second is that convolutions are moving averages. The practice of using fractional integro-differentiation for analogous integrals with respect to FBM was initiated in [@decreusefond1999stochastic], and was subsequently used in [@pipiras2000integration]. We note that the $M$ operator, which is fundamental in the development of the so-called WIS integral ([@elliot2003]), is simply fractional integro-differentiation in disguise. Before we define our integral, let us review some preliminaries concerning fractional integro-differentiation. The Riemann-Liouville integrals are defined, for $ f \in L^p(\R), 1\le p<1/ \delta$ and $0< \delta<1$, by \[fracintegral\] (I\^ \_[+]{} f )(x)&:=& \_[-]{}\^x dt\ &=& \_ dt\ (I\^ \_[-]{} f )(x)&:=& \_ dt Our notation is consistent with the standard reference on this topic, [@samko1987integrals Sec. 5.1], where some basic properties of the above can be found. For example, if $ f $ is in the Schwartz space and we allow for $ \delta\in\N$, then gives the usual integral, as can be seen by Cauchy’s formula for repeated integration: \_[-]{}\^x\_[-]{}\^[t\_[n]{}]{}\_[-]{}\^[t\_2]{} f (t\_1) dt\_1 dt\_[n-1]{}dt\_n = \_[-]{}\^x(x-t)\^[n-1]{} f (t) dt.Also, the above fractional integrals have the semigroup property for $ \delta,\gamma>0$ and $ \delta+\gamma<1$: I\^ \_I\^\_ f = I\^[ +]{}\_ f . For sufficiently nice $ f $, this semigroup property extends to all $ \delta,\gamma>0$. Suppose $f\in \CC^1 $ and $f'\in L^{1}$. These are sufficient conditions for the following Riemann-Liouville derivatives to exist: \[fracderiv\] (\^\_[+]{} f )(x)&:=& \_ dt\ (\^\_[-]{} f )(x)&:=& \_ dt. If $ f \in L^1$, it is known that the inversion $\DD_{\pm}^\bb I^\bb_\pm f = f $ holds. Bringing the derivative inside the integral in , the Riemann-Liouville integrals and derivatives of $ f $ can be seen as convolutions of $f$ and $ f'$ with the family \[def:w\] w\_[a,b]{}(x)=w\_[a,b]{}\^[()]{}(x):=ax\_-\^[-]{} + bx\_+\^[-]{}, (0,1) where we have set $\beta=1-\delta$. \[maindef\] Fix $1<\aa\le 2$ and $a,b\ge 0$. 1. If $\bb\in(1/\aa,1)$, let $f\in L^1\cap L^\aa $. The [*linear fractional random measure*]{} with [*long range dependence*]{} is defined by \[posintegral\] M\_[,H]{} \[f\]&:=& M\_å\[aI\^\_- f+b[I\^\_+ f]{}\]=M\_\[fw\_[a,b]{}\^[()]{}\] where the Hurst parameter is given by $H=1+1/\aa-\bb$. 2. If $\bb\in(0,1/\aa)$, let $f\in \CC^1 $ and $f'\in L^1\cap L^\alpha $. The [*linear fractional random measure*]{} with [*anti-persistence*]{} is defined by \[negintegral\] M\_[,H]{} \[f\]&:=& M\_å\[a\^\_- f+b[\^\_+ f]{}\]=M\_\[(fw\_[a,b]{}\^[()]{})\] where $H=1/\aa-\bb$. It is not hard to check that $I^\bb_\pm f$ and $\DD^\bb_\pm f$ are in $L^\aa$ so that and are well-defined: to see this, split $w_{a,b}^{(\bb)}$ into an $L^1$ and $L^\aa$ function using $1_{[-\epsilon,\epsilon]} +1_{[-\epsilon,\epsilon]^c}$ and apply Young’s convolution inequality, \[Young\] fg\_r f\_p g\_q +=+1. In fact, one can slightly improve the condition for to $f\in L^{\alpha(1+\alpha(1-\bb)+\epsilon)^{-1}}\cap L^{\alpha(1+\alpha(1-\bb)-\epsilon)^{-1}}$ for some $\eps>0$, and a similar condition can be found for and $f'$. However, in the interest of simple notation, we will not utilize these meager improvements in the sequel. Let us remark that the fact that is well-defined coincides with Proposition 3.2 in [@pipiras2000integration] for the Gaussian case. By the linearity of convolutions, it follows that $M_{\alpha,H}$ is a random measure. Also note that $M_{\alpha,H} [f]$ can be interpreted as the integral of $f$ with respect to a L-FSM in which case we write M\_[,H]{} \[f\]\_ f dL\_[å,H]{}. To check consistency with , we see that &&M\_\[1\_[\[0,t\]]{}w\_[a,b]{}\]\ &=&\_(\_1\_[\[0,t\]]{}(y)$a(x-y)_-^{-\bb}+b(x-y)_+^{-\bb}$dy)M\_(dx)\ &=&\_(\_1\_[\[0,t\]]{}(y)$a(y-x)_+^{-\bb}+b(y-x)_-^{-\bb}$dy)M\_(dx)\ &=& \_ f\_t\^[a,b]{}(x) M\_(dx) and &&M\_\[(1\_[\[0,t\]]{}w\_[a,b]{})(x)\]\ &=&\_\_1\_[\[0,t\]]{}(y)$a(x-y)_-^{-\bb}+b(x-y)_+^{-\bb}$dyM\_(dx)\ &=& \_ f\_t\^[a,b]{}(x) M\_(dx). When $f\in \CC^1 $ and that $f'\in L^{1}$, one can rewrite as && \_ dt\ &=& \_0\^f’ (x-t) \_t\^ ds\ &=& \_0\^ ds.The right-hand side above is slightly more general then and is called the [*Marchaud derivative*]{}. This is the fractional derivative used in [@pipiras2000integration], however, to keep a unified notation in our approximations of the next subsection, we will continue with the Riemann-Liouville derivative. Discrete approximations of linear fractional stable measures ------------------------------------------------------------ Let $1<\alpha\leq 2$, and consider the stationary moving average process $(\hat\xi_k)_{k\in{\mathbb{Z}}}$ obtained by “linearly filtering" an i.i.d. sequence $(\xi_l)_{l\in{\mathbb{Z}}}$ in the domain of normal attraction of ${{\mathcal S}}_\aa$: $$\label{eq:defhatxi} \hat\xi_k:=\sum_{l\in{\mathbb{Z}}}\B_{k-l}\xi_l.$$ Lemma \[lem1\] shows that if $\B\in\ell^\alpha$, the series converges almost surely. Recall the definition of $f^h$ from and denote the inversion of a sequence by $\check\B_k:=\B_{-k}$ A first stab at approximating a L-FSM integral of $f$, as defined in the previous subsection, might be to mimic and look at $\sum_{k\in{\mathbb{Z}}} f^h_k\hat\xi_{k}$ for appropriate filters $v$ (which would also depend on $h$). This is, for example, the approach of [@kasahara1988weighted] and [@maejima2008limit]. Then formally, $$\begin{aligned} \label{convolution convergence} \sum_{k\in{\mathbb{Z}}} f^h_k\hat\xi_{k}&=&\sum_{k,l\in{\mathbb{Z}}} f^h_k\B_{k-l}\xi_l \\&=&\sum_{l\in{\mathbb{Z}}} \(f^h\ast\check\B\)_l\xi_l <\ff.\nn\end{aligned}$$ However, in view of the right-hand side above, it is easier and perhaps more natural to first convolve $f$ with $w_{a,b}=w_{a,b}^{(\beta)}$ and then approximate the convolution on a lattice with side-length $h$. In particular, for $w_{a,b}$ corresponding to $H\in(1/\aa,1)$, define $$\label{eq:series2} M^h_{\alpha,H}[f]:=\sum_{k\in{\mathbb{Z}}} \(f\ast w_{a,b}\)^h_k \xi_{k},\quad \quad f\in L^1\cap L^{\aa}(\R)$$ where the sequence $\(f\ast w_{a,b}\)^h$ is defined according to . Alternatively, for $w_{a,b}$ corresponding to $H\in(0,1/\aa)$, define $$\label{eq:series2n} M^h_{\alpha,H}[f]:=\sum_{k\in{\mathbb{Z}}} \(f'\ast w_{a,b}\)^h_k\xi_{k},\quad \quad f\in \CC^1(\R), f'\in L^1\cap L^{\aa}(\R).$$ By and the remark above it, $f\ast w_{a,b}\in L^\aa(\R)$. Thus one obtains, from a direct application of Theorem \[maineq2prop\], the following corollary: \[theo:fraccase\] Fix $1<\alpha\leq 2$. Suppose that for all $t$ in an index set $T$, $f_t\in L^1\cap L^\aa$ when $H\in(1/\alpha,1)$ or $f_t\in \CC^1, f'_t\in L^1\cap L^\aa$ when $H\in(0, 1/\alpha)$. Then as $h\to 0$: \[theo:fraccase equation\] M\^h\_[,H]{}\[f\_t\]M\_[,H]{} \[f\_t\]. [**Remarks:**]{} 1. It is not hard to extend the $H>1/\aa$ case to stable random measures on $\R^d$ by generalizing the two fixed values $a,b\ge 0$ (representing the negative and positive directions) to a function on the unit sphere $S^{d-1}\subset \R^d$. However, one then has to specify what is meant by “stationary increments” as there are different possibilities for $d>1$. 2. Extending the $H<1/\aa$ case to higher dimensions is more difficult. One possibility is to consider the Marchaud derivative in place of the Riemann-Liouville derivative (see also the next remark). 3. When $H>1/\aa$, Eq. has been shown by various authors in the case where $f_t=1_{[0,t]}$ (see [@kasahara1988weighted] and its references). However, when $H<1/\aa$, to our knowledge, even the case $f_t=1_{[0,t]}$ has not appeared in the literature. It is, however, related to the normalization suggested in Theorem 5.2 of [@kasahara1988weighted] which can be thought of as a discrete Marchaud derivative in the case where $f_t=1_{[0,t]}$. Proofs {#sec:proof} ====== Before delving into the proofs, let us recall some facts about the domain of attraction of a stable distribution. We write $f(x)\sim g(x)$ as $x\to c$ if $\lim_{x\to c}f(x)/g(x)=1$. For $\alpha\in(0,2)$, the following statements are equivalent (see [@geluk1997stable Theorem 1] with $p=1/2$): - $\xi$ is in the domain of attraction of ${{\mathcal S}}_\alpha$ (i.e. Eq. holds); - the tail function $t\mapsto {\mathbb{P}}(|\xi|\geq t)$ is regularly varying at infinity with index $-\alpha$ and ${\mathbb{P}}(\xi\leq -t)\sim {\mathbb{P}}(\xi\geq t)$ as $t\to \infty$; - the characteristic function $\lambda(\theta)={\mathbb{E}}\left[e^{i\theta\xi}\right]$ satisfies - $\theta\mapsto 1- \mathrm{Re}(\lambda(\theta))$ is regularly varying at $0$ with index $\alpha$, - for all $x\neq 0$, $$\lim_{\theta\to 0} \frac{x\mathrm{Im}(\lambda(\theta x))-\mathrm{Im}(\lambda(\theta ))}{1- \mathrm{Re}(\lambda(\theta))}=0.$$ Moreover, if conditions (i)-(iii) hold, then $$1- \mathrm{Re}(\lambda(\theta))\sim c_\alpha {\mathbb{P}}(|\xi|\geq 1/\theta)\quad \mbox{with}\ c_\alpha=\int_0^\infty x^{-\alpha}\sin x\, dx$$ as $\theta\to 0$ and also $$\mathrm{Im}(\lambda(\theta))=\theta \int_0^{1/\theta}({\mathbb{P}}(\xi\geq s)-{\mathbb{P}}(\xi\leq-s))ds+ o({\mathbb{P}}(|\xi|\geq 1/\theta)).$$ Also, Remark 3 of [@geluk1997stable] shows that one may choose the normalization constants so that $$\lim_{n\to \infty} n\Big(1- \mathrm{Re}(\lambda(1/a_n))\Big) =1\quad \mbox{and}\quad b_n=n\mathrm{Im}(\lambda(1/a_n)).$$ Recall from that in the present framework, we have assumed $a_n=n^{1/\alpha}$ and $b_n=0$. Thus, $$\label{eq:heavy-tail} {\mathbb{P}}( \xi\geq t) \sim {\mathbb{P}}( \xi\leq -t)\sim \frac{1}{2c_\alpha} t^{-\alpha} \quad \mbox{as}\ \theta\to 0.$$ and $$\label{eq:fcstable2} \lambda(\theta)=1-|\theta|^\alpha+o(|\theta|^\alpha)=\lambda_\alpha(\theta)+o(|\theta|^\alpha)\quad \mbox{as}\ \theta\to 0.$$ where $\lambda_\alpha$ is defined in Eq. . Furthermore, (\[eq:heavy-tail\]) implies that there exists $C>0$ such that for any $s>0$ $$\label{eq:stablestim2} {\rm Var}[\xi{\mathbf{1}}_{\{|\xi|\leq s\}}]\leq Cs^{2-\alpha}\quad {\rm and} \quad \E[|\xi|{\mathbf{1}}_{\{|\xi|\leq s\}}]\leq Cs^{1-\alpha}.$$ Proof of Theorem \[theo:whitecase\] ----------------------------------- We begin with a lemma which shows the $\ell^\alpha$ is the right space for the sequence $u$. \[lem1\] If $(\xi_k)_{k\in\N}$ is an i.i.d. sequence in the domain of normal attraction of ${{\mathcal S}}_\aa $, then $\sum_{k}u_k\xi_{k}<\ff$ if and only if $u\in\ell^\alpha$. The case $\alpha=2$ is standard and omitted. Consider $\alpha\in (0,2)$. Recall Kolmogorov’s Three-series Theorem: $\sum_{k} u_k\xi_k$ converges a.s. if and only if for any $s>0$, the following three series converge $$\sum_{k\in \N} {\mathbb{P}}\left[|u_k\xi_k |>s\right],\quad \sum_{k\in \N} {\rm Var}\left [u_k\xi_k {\mathbf{1}}_{\{|u_k\xi_k |\leq s\}}\right], \quad \sum_{k\in \N} \E\left[u_k\xi_k{\mathbf{1}}_{\{|u_k\xi_k |\leq s\}}\right] .$$ Eq. (\[eq:heavy-tail\]) implies $$\nn {\mathbb{P}}\left[|u_k\xi_k |>s\right]\sim C|u_k|^\alpha s^{-\alpha}$$ and hence the first series converges if and only if $u\in \ell^\alpha$. If $u\in\ell^\alpha$, then (\[eq:stablestim2\]) implies the convergence of the third series since $$|u_k| \E[|\xi_k| 1_{\{|\xi_k|<s/|u_k|\}}] \leq |u_k| C(s/|u_k|)^{\alpha-1} = Cs^{\alpha-1} |u_k|^\alpha.$$ The convergence of the second series is obvious. If $\eta_{k,j}$ are i.i.d. ${{\mathcal S}}_\aa$ random variables, then for any fixed $j$ the above lemma allows us to write \[eq:fc-stableintegral\] {i\_[k]{} u\^[(j)]{}\_k\_[k,j]{}}=\_[k]{} \_å( u\^[(j)]{}\_k)= -|u\^[(j)]{}\_[\_å]{}|\^åand $$\label{eq:fc-noise} \E\exp\{i\theta \sum_{k\in\N} u^{(j)}_k\xi_{k,j}\}=\prod_{k\in\N} \lambda\left( u^{(j)}_k\th \right).$$ Note that since $\|u\|_{\ell_\aa}^\aa:=\sum_{k\in\N} |u(k)|^\aa$ absolutely converges, the order in which the summation and products above are taken is irrelevant. It suffices to show that as $ j\to\ff$, $$\label{eq:diff} \prod_{k\in\N} \lambda\left( u^{(j)}_k\th \right)=\prod_{k\in\N} \lambda_\aa\left( u^{(j)}_k\th \right)+o(1).$$ We fix $j$ and estimate the difference of the above products using the following fact : if $(z_i)_{i\in I}$ and $(z_i')_{i\in I}$ are two families of complex numbers with moduli no greater than $1$ and such that the products $\prod_{i\in I}z_i$ and $\prod_{i\in I}z'_i$ converge, then \[standardinequality\] |\_[iI]{}z’\_i -\_[iI]{}z\_i| \_[iI]{} |z’\_i - z\_i|. We therefore have $$\begin{aligned} \left|\prod_{k\in\N} \lambda\left( u^{(j)}_k\th \right)- \prod_{k\in\N} \lambda_\aa\left( u^{(j)}_k\th \right)\ \right| \leq \ \sum_{k\in\N} \ \left|\ \lambda \left(u^{(j)}_k\th \right)-\lambda_\alpha \left(u^{(j)}_k\th \right)\ \right|\label{eq:diff1}.\end{aligned}$$ Equation (\[eq:fcstable2\]) implies[^6] that the function $g$ defined by $g(0)=0$ and $$g(u)=|u|^{-\alpha}\left|\lambda(u)-\lambda_\aa(u)\right|\ \ ,\ \ u\neq 0,$$ is continuous and bounded and for any $k\in\N$, we have $$\left| \lambda \left( u^{(j)}_k\th \right)- \lambda_\aa \left( u^{(j)}_k\th \right) \right| = g( u^{(j)}_k\th )| u^{(j)}_k\th |^\alpha .$$ In order to obtain a uniform estimate on the above, define the function $\tilde g:\R^+\to\R^+$ by $$\tilde g(v) :=\sup_{|u|\leq v} |g(u)|.$$ Note that $\tilde g$ is continuous, bounded and vanishes at $0$, and that for any $k\in \N$ such that $| u^{(j)}_k\th |\leq \varepsilon$, $$\label{eq:diff2} \left| \lambda \left( u^{(j)}_k\th \right)- \lambda_\aa \left( u^{(j)}_k\th \right) \right|\leq \tilde g(\varepsilon)| u^{(j)}_k\th |^\alpha .$$ Let $\varepsilon>0$. Equations (\[eq:diff1\]) and (\[eq:diff2\]) together yield $$\begin{aligned} & &\left|\prod_{k\in\N} \lambda\left( u^{(j)}_k\th \right)- \prod_{k\in\N} \lambda_\aa\left( u^{(j)}_k\th \right)\ \right|\\ &\leq& \ \tilde g(\varepsilon)\sum_{k\in\N}\left| u^{(j)}_k\th \right|^\alpha{\mathbf{1}}_{\{| u^{(j)}_k\th |\leq\varepsilon\}} + 2\sum_{k\in\N}{\mathbf{1}}_{\{| u^{(j)}_k\th |>\varepsilon\}}.\end{aligned}$$ Now, by the continuity of $\tilde g$ at $0$, $\tilde g(\varepsilon)$ is small when $\varepsilon$ is small. Eq. follows since $\lim_{j\to\ff} \|u^{(j)}\|_{\ell^\infty}=0$ implies $\sum_{k\in\N}{\mathbf{1}}_{\{| u^{(j)}_k\th |>\varepsilon\}}\to 0$ as $j\to\ff$. Proof of Theorem \[maineq2prop\] -------------------------------- Let $\lfloor\cdot\rfloor$ denote the floor function applied to each coordinate of $\R^d$. Define $\fh:\R^d\mapsto\R$ to be a piece-wise constant function approximating $f\in L^1_{\text{loc}}(\R^d)$: \[def:fh\] (x)&:=& \_[h(h\^[-1]{}x+I\^d)]{}h\^[-d]{} f(y) dy\ &=&\_[h(k+I\^d)]{} h\^[-d]{} f(y) dy,   xh(k+I\^d)\ &=& h\^[-d]{} f\^h(k), xh(k+I\^d).Note that \[normsequal\] f\^h\_[\^]{}=\_[L\^]{}. \[laa lemma\]For $\aa\in[1,2]$, suppose $f\in L^\aa(\R^d)$. Then as $h\to 0$, $$\lim_{h\to 0}\| \fh-f\|_{L^\aa}=0.$$ To reduce notation we assume $d=1$, but the proof holds for general $d$. Fix $k\in\Z$ and consider the sequence of $h$’s such that $h=2^{-j}$ for $j\in\N$. We will exploit the fact that $\fh1_{[k,k+1)}$ is a martingale (in time $j$) with respect to Lebesgue measure on $[k,k+1)$ and with respect to the $\sigma$-fields generated by the sets $2^{-j}[i,i+1), i\in\Z$. For $\aa\ge 1$, $|\fh|^\aa1_{[k,k+1)}$ is a submartingale which, by the martingale convergence theorem, converges a.s. to $|f|^\aa1_{[k,k+1)}$. Thus, Fatou’s lemma gives \[eq:aag1\] 1\_[\[k,k+1)]{}\_[L\^å]{}\^åf1\_[\[k,k+1)]{}\_[L\^å]{}\^å. Since $\fh1_{[k,k+1)}$ converges a.s. and the $L^\aa$-norms converge, we have convergence in $L^\aa(\R)$ of $\fh1_{[k,k+1)}$ and also for $\fh1_{[-N,N)}$ for any $N\in\N$. For $f\in L^\aa(\R)$ without compact support, simply choose $N$ so that $$\|f1_{[-N,N)^c}\|_{L^\aa}^\aa<\epsilon.$$ Since $|\fh|^\aa1_{[k,k+1)}$ is a submartingale, we also have uniformly in $h$. Finally, to extend the above to general $h\to 0$. Note that all we really require is a sequence of lattices such that finer lattices are sublattices of prior ones and that the mesh size goes to zero. But any such sequence has the same limit in $L^\aa(\R)$, thus we conclude that the only real requirement is that the mesh size goes to zero. By the Crámer-Wold device, we must show that for all $\theta_1,\ldots,\theta_n\in{\mathbb{R}}$ and $f_1,\ldots,f_n\in L^\aa(\R^d)$, $$\sum_{i=1}^n \theta_iM_\alpha^h[f_i]\Longrightarrow \sum_{i=1}^n \theta_iM_\alpha[f_i]\quad \mbox{as}\ h\to 0.$$ Our proof uses Theorem \[theo:whitecase\]. First note that the comment following shows that switching the order of summation in the series $M_\alpha^h[f_i]$ does not affect its distribution. This, together with the linearity of $M_\alpha$ and $M_\alpha^h$, allows us to reduce the above to verifying $$\nn M_\alpha^h[f]\Longrightarrow M_\alpha[f]\quad \mbox{as}\ h\to 0$$ for a single $f\in L^\aa(\R^d)$. This will follow from Theorem \[theo:whitecase\] provided we check the two conditions $$\label{eq:cond1} \lim_{h\to 0}\|f^h\|_{\ell^\alpha}=\|f\|_{L^\alpha}$$ and \[eq:cond2\] \_[h0]{}f\^h\_[\^]{}=0. We consider $\aa\in[1,2]$ first. Condition easily follows from and Lemma \[laa lemma\]. For , note that convergence of the $L^1$ norms of $|\fh|^\aa$, coupled with a.e. convergence, shows that the family $\{|\fh|^\aa\}_{h\in\N}$ is uniformly integrable. For $\aa\in(0,1)$, we first consider the sequence of $h$’s such that $h=2^{-j}$ for $j\in\N$. By uniform integrability and the martingale convergence theorem (see the proof of Lemma \[laa lemma\]), we see that $\fh1_{[k,k+1)}$ converges in $L^1$ to $f1_{[k,k+1)}$. The final comment in the proof of Lemma \[laa lemma\] shows the convergence also holds for arbitrary $h\to 0$. Next, note that contains $L^\aa([k,k+1))$ and that the endomorphism on $L^1([k,k+1))$ which maps $$f1_{[k,k+1)}\ \mapsto \ |f|^\aa1_{[k,k+1)}$$ is continuous. Thus Eq. holds for $\aa\in(0,1)$. Since $f\in L^\aa$ we can choose $N_1$ so that $\|f1_{[-N_1,N_1)^c}\|_{L^\aa}^\aa$ is small. However, to uniformly bound the tails of the $f_h$, we will use the stronger condition of $f\in L^{\aa-\epsilon}$. In particular, there exist $N_2>0$, $C>0$ and $\delta>\alpha^{-1}$ such that $|x|\ge N_2$ implies $|f(x)|\leq C|x|^{-\delta}$. We have for $|x|\geq N_2+{h}$ that $${h}(\lfloor{h}^{-1}x\rfloor+I)\subset (-N_2,N_2)^c$$ and $$\label{eq:maj-unif} |\fh(x)|^\alpha=\left|{h}^{-1}\int_{{h}(\lfloor{h}^{-1}x\rfloor+I)}f(y) \, dy \right|^\alpha\leq C^\aa{h}^{1-\alpha}(|x|-{h})^{-\alpha\delta}.$$ Since $\alpha\delta>1$, follows from . Finally, as before, we see that along with a.e. convergence gives for $\aa\in(0,1)$. Acknowledgements {#acknowledgements .unnumbered} ================ We are grateful to Gennady Samorodnitsky for helpful correspondence. [EVDH03]{} A. Astrauskas. Limit theorems for sums of linearly generated random variables. , 23(2):127–134, 1983. F. Avram and M.S. Taqqu. Weak convergence of sums of moving averages in the $\alpha$-stable domain of attraction. , pages 483–503, 1992. F. Biagini, Y. Hu, B. [Ø]{}ksendal, and T. Zhang. . Springer Verlag, 2008. S. Cambanis and M. Maejima. Two classes of self-similar stable processes with stationary increments. , 32(2):305–329, 1989. Y.A. Davydov. . , 15(3):498–509, 1970. R. Davis and S. Resnick. Limit theory for moving averages of random variables with regularly varying tail probabilities. , pages 179–195, 1985. L. Decreusefond and A.S. [Ü]{}st[ü]{}nel. . , 10(2):177–214, 1999. RM Dudley. Random linear functionals. , 136:1–24, 1969. R.J. Elliott and J. Van Der Hoek. . , 13(2):301–330, 2003. J.L. Geluk and L.F.M. Haan. Stable probability distributions and their domains of attraction. Technical report, Tinbergen Institute, 1997. Y. Kasahara and M. Maejima. Weighted sums of iid random variables attracted to integrals of stable processes. , 78(1):75–96, 1988. P.S. Kokoszka and M.S. Taqqu. Fractional arima with stable innovations. , 60(1):19–47, 1995. M. Maejima. On a class of self-similar processes. , 62(2):235–245, 1983. M. Maejima and S. Suzuki. Limit theorems for weighted sums of infinite variance random variables attracted to integrals of linear fractional stable motions. , 31(2):259–271, 2008. V.V. Petrov. . Oxford Science Publications, 1995. V. Pipiras and M.S. Taqqu. . , 118(2):251–291, 2000. J. Rosinski. . , page 401, 2001. S.G. Samko, A.A. Kilbas, and O.I. Marichev. . London: Gordon and Breach, 1987. G. Samorodnitsky and M.S. Taqqu. . Chapman & Hall/CRC, 1994. [^1]: Laboratoire LMA, CNRS UMR 7348, Université de Poitiers, Téléport 2, BP 30179, F-86962 Futuroscope-Chasseneuil cedex, France. Email: clement.dombry@math.univ-poitiers.fr [^2]: Department of Mathematics, University of Alabama Birmingham, USA. Email: pjung@uab.edu [^3]: The term [*linear fractional stable motion*]{} was introduced in [@cambanis1989two] due to its close relation to linear time series (moving average processes). [^4]: Our approximation in Thm \[maineq2prop\], as well as the Lindeberg-Feller result, can be extended to stable distributions with skewness $\nu\neq 0$, however, to simplify calculations and notation we have assumed symmetry. [^5]: In [@avram1992weak], it was also shown that under the right conditions, convergence does occur in Skorokhod’s $M_1$ topology. [^6]: We have assume $\alpha\in(0,2)$ for Eq. , but for $\alpha=2$ it is well-known.

--- abstract: | There exist different kinds of averaging of the differences of the energy-momentum and angular momentum in normal coordinates [**NC(P)**]{} which give tensorial quantities. The obtained averaged quantities are equivalent mathematically because they differ only by constant scalar dimensional factors. One of these averaging was used in our papers \[1-8\] giving the [*canonical superenergy and angular supermomentum tensors*]{}. In this paper we present another averaging of the differences of the energy-momentum and angular momentum which gives tensorial quantities with proper dimensions of the energy-momentum and angular momentum densities. But these averaged relative energy-momentum and angular momentum tensors, closely related to the canonical superenergy and angular supermomentum tensors, [*depend on some fundamental length $L>0$*]{}. The averaged relative energy-momentum and angular momentum tensors of the gravitational field obtained in the paper can be applied, like the canonical superenergy and angular supermomentum tensors, to [*coordinate independent*]{} analysis (local and in special cases also global) of this field. We have applied the averaged relative energy-momentum tensors to analyze vacuum gravitational energy and momentum and to analyze energy and momentum of the Friedman (and also more general) universes. The obtained results are interesting, e.g., the averaged relative energy density is [*positive definite*]{} for the all Friedman universes. address: 'Institute of Physics, University of Szczecin, Wielkopolska 15; 70-451 Szczecin, POLAND[^1]' author: - Janusz Garecki title: 'The averaged tensors of the relative energy-momentum and angular momentum in general relativity and some of their applications' --- The averaged relative energy-momentum and angular momentum tensors in general relativity ======================================================================================== In the papers \[1-8\] we have defined the canonical superenergy and angular supermomentum tensors, matter and gravitation, in general relativity ([**GR**]{}) and studied their properties and physical applications. In the case of the gravitational field these tensors gave us some substitutes of the non-existing gravitational energy-momentum and gravitational angular momentum tensors. The canonical superenergy and angular supermomentum tensors were obtained pointwise as a result of some special averaging of the differences of the energy-momentum and angular momentum in normal coordinates [**NC(P)**]{}. The role of the normal coordinates [**NC(P)**]{} is, of course, auxilliary, only to extract tensorial quantities even from pseudotensorial ones. The dimensions of the components of the canonical superenergy and angular supermomentum tensors can be written down as:\[the dimensions of the components of an energy-momentum or angular momentum tensor (or pseudotensor)\]$\times m^{-2}$. In this paper we propose a new averaging of the energy-momentum and angular momentum differences in [**NC(P)**]{} which is very like to the averaging used in \[1-8\] and which gives the averaged quantities with proper dimensionality of the energy-momentum and angular momentum densities. Namely, we propose the following general definition of the averaged tensor (or pseudotensor) $T_a^b$ $$<{\it T_a^{~b}}(P)> := \displaystyle\lim_{\varepsilon\to 0}{\int\limits_{\Omega}{\bigl[{\it T_{(a)}^{~~~(b)}}(y) - {\it T_{(a)}^{~~~(b)}}(P)\bigr] d\Omega}\over\varepsilon^2/2\int\limits_{\Omega}d\Omega},$$ where $${\it T_{(a)}^{~~~(b)}}(y) := T_i^{~k}(y){}e^i_{~(a)}(y){}e_k^{~(b)}(y),$$ $${\it T_{(a)}^{~~~(b)}}(P):= T_i^{~k}(P){}e^i_{~(a)}(P){}e_k^{~(b)}(P) = {\it T_a^{~b}}(P)$$ are the tetrad (or physical) components of a tensor or a pseudotensor $T_i^{~k}(y)$ which describes an energy-momentum distribution, $y$ is the collection of normal coordinates [**NC(P)**]{} at a given point [**P**]{}, $e^i_{~(a)}(y),~e_k^{~(b)}(y)$ denote an orthonormal tetrad field and its dual, respectively, $$e^i_{~(a)}(P) = \delta^i_a,~e_k^{~(a)}(P) =\delta^a_k,~e^i_{~(a)}(y)e_i^{~(b)}(y) =\delta_a^b,$$ and they are parallelly propagated along geodesics through [**P**]{}. For a sufficiently small domain $\Omega$ which surrounds [**P**]{} we require $$\int\limits_{\Omega}{y^id\Omega} = 0,~~\int\limits_{\Omega}{y^iy^kd\Omega} = \delta^{ik} M,$$ where $$M = \int\limits_{\Omega}{(y^0)^2 d\Omega} = \int\limits_{\Omega} {(y^1)^2 d\Omega} = \int\limits_{\Omega}{(y^2)^2 d\Omega} = \int\limits_{\Omega}{(y^3)^2 d\Omega},$$ is a common value of the moments of inertia of the domain $\Omega$ with respect to the subspaces $y^i = 0, ~~(i = 0,1,2,3)$. The procedure of averaging of an energy-momentum tensor or an energy-momentum pseudotensor given in (1) is a four-dimensional modification of the proposition by Mashhoon \[9-12\]. Let us choose $\Omega$ as a small analytic ball defined by $$(y^0)^2 + (y^1)^2 + (y^2)^2 + (y^3)^2 \leq R^2 = \varepsilon^2 L^2,$$ which can be described in a covariant way in terms of the auxiliary positive-definite metric $h^{ik} := 2v^iv^k - g^{ik}$, where $v^i$ are the components of the four-velocity of an observer [**O**]{} at rest at [**P**]{} (see, e.g., \[1-8\]). $\varepsilon$ means a small parameter: $\varepsilon\in (0;1)$ and $L >0$ is a fundamental length. Since at [**P**]{} the tetrad and normal components are equal, from now on we will write the components of any quantity at [**P**]{} without (tetrad) brackets, e.g., $T_a^{~b}(P)$ instead of $T_{(a)}^{~~~(b)}(P)$ and so on. Let us now make the following expansions for the energy-momentum tensor of matter $T_i^{~k}(y)$ and for $e^i_{~(a)}(y), e_k^{~(b)}(y)$ \[13\] $$\begin{aligned} T_i^{~k}(y) &=& {\hat T}_i^{~k} + \nabla_l {\hat T}_i^{~k}y^l +1/2{{\hat T}_i^{~k}}{}_{,lm} y^ly^m + R_3\nonumber\\ &=& {\hat T}_i^{~k} +\nabla_l{\hat T}_i^{~k} y^l +1/2\biggl[\nabla_{(l}\nabla_{m)}{\hat T}_i^{~k}\nonumber\\ &-&1/3{\hat R}^c_{~(l\vert i\vert m)}{}{\hat T}_c^{~k} +1/3 {\hat R}^k_{~(l\vert c\vert m)}{} {\hat T}_i^{~c}\biggr] y^ly^m + R_3,\end{aligned}$$ $$e^i_{~(a)}(y) = {\hat e}^i_{~(a)} + 1/6{\hat R}^i_{~lkm}{\hat e}^k_{~(a)} y^ly^m + R_3,$$ $$e_k^{~(b)}(y) = {\hat e}_k^{~(b)} -1/6{\hat R}^p_{~lkm}{\hat e}_p^{~(b)} y^ly^m + R_3,$$ which give (1) in the form $$<_m T_a^{~b}(P)> =\displaystyle\lim_{\varepsilon\to 0}{\int\limits_{\Omega}\bigl[\nabla_l{\hat T}_a^{~b} y^l + 1/2\nabla_{(l}\nabla_{m)}{\hat T}_a^{~b}y^ly^m + THO\bigr]d\Omega\over \varepsilon^2/2\int\limits_{\Omega}d\Omega},$$ where $THO$ means the terms of higher order in the expansion of the differences $T_{(a)}^{~~~(b)}(y) - T_{(a)}^{~~~(b)}(P) = T_{(a)}^{~~~(b)}(y) - T_a^{~b}(P)$; $R_3$ is the remainder of the third order and $\nabla$ denotes covariant differentiation. Hat denotes the value of an object at [**P**]{} and the round brackets denote symmetrization from which the indices inside vertical lines, e.g., $(a\vert c\vert b)$ are excluded. The first and $THO$ terms in the numerator of (11) do not contribute to $<_m T_a^{~b}(P)>$. Hence, we finally get from (11) $$<_m T_a^{~b}(P)> = _m S_a^{~b}(P) {L^2\over 6},$$ where $$_m S_a^{~b}(P) := \delta^{lm}\nabla_{(l}\nabla_{m)}{\hat T}_a^{~b}$$ is the [*canonical superenergy tensor of matter*]{} \[1-8\]. By introducing the four velocity ${\hat v}^l \dot = \delta^l_0,~v^lv_l =1$ of an observer [**O**]{} at rest at [**P**]{} and the local metric ${\hat g}^{ab}\dot = \eta^{ab}$, where $\eta^{ab}$ is the inverse Minkowski metric, one can write (13) in a covariant way as $$_m S_a^{~b} (P;v^l) = \bigl(2{\hat v}^l{\hat v}^m - {\hat g}^{lm}\bigr)\nabla_{(l}\nabla_{m)} {\hat T}_a^{~b}.$$ The sign $\dot =$ means that an equality is valid only in some special coordinates. The matter superenergy tensor $_m S_a^{~b}(P;v^l)$ [*is symmetric*]{}. As a result of an averaging the tensor $_m S_a^{~b}(P;v^l)$, and in consequence the averaged tensor $<_m T_a^{~b}(P;v^l)>$, do not satisfy any local conservation laws in general relativity. However, these tensors satisfy trivial local conservation laws[^2] in special relativity (see, e.g., \[1-8\]). Now let us take the gravitational field and make the expansion $$\begin{aligned} _E t_i^{~k}(y) &=& {\alpha\over 9}\biggl[{\hat B}^k_{~ilm} + {\hat P}^k_{~ilm}\nonumber\\ &-& {\delta_i^k\over 2}{\hat R}^{abc}_{~~~l}\bigl({\hat R}_{abcm} + {\hat R}_{acbm}\bigr) + 2\beta^2\delta_i^k{\hat E}_{(l\vert g} {\hat E}^g_{~\vert m)}\nonumber\\ &-& 3\beta^2{\hat E}_{i(l\vert}{\hat E}^k_{~\vert m)} + 2\beta{\hat R}^k_{~(gi)(l\vert}{\hat E}^g_{~\vert m)}\biggr] y^ly^m + R_3.\end{aligned}$$ Here $_E t_i^{~k}$ mean the components of the canonical Einstein energy-momentum pseudotensor of the gravitational field. In a holonomic frame we have $$\begin{aligned} _E t_i^{~k}&=&\alpha\Bigl\{\delta^k_i g^{ms}\bigl(\Gamma^l_{mr}\Gamma^r_{sl}-\Gamma^r_{ms}\Gamma^l_{rl}\bigr)\nonumber\\ &+& g^{ms}_{~~,i}\bigl[\Gamma^k_{ms}-1/2\bigl(\Gamma^k_{tp} g^{tp}- \Gamma^l_{tl} g^{kt}\bigr)g_{ms}\nonumber\\ & -&1/2\bigl(\delta^k_s \Gamma^l_{ml} + \delta^k_m \Gamma^l_{sl}\bigr)\bigr]\Bigr\}.\end{aligned}$$ $$\alpha = {c^4\over 16\pi G} = {1\over 2\beta},~E_i^{~k} := T_i^{~k} - 1/2\delta_i^k T,$$ and, in any frame, $$B^b_{~alm} := 2R^{bik}_{~~~(l\vert}{}R_{aik\vert m)} -1/2\delta^b_a{}R^{ijk}_{~~~l} R_{ijkm},$$ is the [*Bel-Robinson tensor*]{}, while the tensor $$P^b_{~alm} := 2 R^{bik}_{~~~(l\vert}{}R_{aki\vert m)} -1/2\delta_a^b{} R^{ijk}_{~~~l} R_{ikjm}$$ is very closely related to the former[^3]. The expansion (15) with the help of (9) and (10) gives the following averaged gravitational relative energy-momentum tensor $$<_g t_a^{~b}(P;v^l)> = _g S_a^{~b}(P;v^l){L^2\over 6},$$ where the tensor $_g S_a^{~b}(P;v^l)$ is the [*canonical superenergy tensor*]{} for the gravitational field \[1-8\]. We have \[1-8\] $$\begin{aligned} _g S_a^{~b}(P;v^l) &=& {2\alpha\over 9}\bigl(2{\hat v}^l{\hat v}^m - {\hat g}^{lm}\bigr)\biggl[{\hat B}^b_{~alm} + {\hat P}^b_{~alm}\nonumber\\ &-& 1/2\delta_a^b{\hat R}^{ijk}_{~~~m}\bigl({\hat R}_{ijkl} + {\hat R}_{ikjl}\bigr) + 2\beta^2\delta_a^b{\hat E}_{(l\vert g}{\hat E}^g_{~\vert m)}\nonumber\\ &-& 3\beta^2{\hat E}_{a(l\vert}{\hat E}^b_{~\vert m)} + 2\beta{\hat R}^b_{~(ag)(l\vert} {\hat E}^g_{~\vert m)}\biggr].\end{aligned}$$ In vacuum the tensor $_g S_a^{~b}(P;v^l)$ reduces to the simpler form $$_g S_a^{~b} (P;v^l) = {8\alpha\over 9}\bigl(2{\hat v}^l{\hat v}^m - {\hat g}^{lm}\bigr)\biggl[{\hat R}^{b(ik)}_{~~~~~(l\vert}{\hat R}_{aik\vert m)} -1/2\delta_a^b{\hat R}^{i(kp)}_{~~~~~(l\vert}{\hat R}_{ikp\vert m)}\biggr],$$ which is symmetric and the quadratic form $_g S_{ab}(P;v^l){\hat v}^a{\hat v}^b$ is [*positive-definite*]{}. In vacuum we also have the [*local conservation laws*]{} $$\nabla_b~{_g {\hat S}_a^{~b}} = 0.$$ and the analogous laws satisfied by the averaged tensor $<_g t_a^{~b}(P;v^l)>$. The averaged energy-momentum tensors $<_m T_a^{~b}(P;v^l)>$ and $<_g t_a^{~b}(P;v^l)>$ can be considered as the [*averaged tensors of the relative energy-momentum*]{}. They can also be interpreted as the [*fluxes*]{} of the appropriate canonical superenergy. It is easily seen from the formulas (12) and (20). Now let us consider the [*averaged angular momentum tensors*]{} in [**GR**]{}. The constructive definition of these tensors, in analogy to the definition of the averaged energy-momentum tensors, is as follows. In normal coordinates [**NC(P)**]{} we define $$<M^{(a)(b)(c)}(P)> = <M^{abc}(P)> :=\displaystyle\lim_{\varepsilon\to 0}{ \int\limits_{\Omega}{\bigl[M^{(a)(b)(c)}(y) - M^{(a)(b)(c)}(P)\bigr]d\Omega}\over\varepsilon^2/2\int\limits_{\Omega}d\Omega},$$ where $$M^{(a)(b)(c)}(y) := M^{ikl}(y){}e_i^{~(a)}(y){}e_k^{~(b)}(y){}e_l^{~(c)}(y),$$ $$M^{(a)(b)(c)}(P) := M^{ikl}(P) e_i^{~(a)}(P){}e_k^{~(b)}(P){}e_l^{~(c)}(P) = M^{ikl}(P)\delta_i^a\delta_k^b\delta_l^c = M^{abc}(P),$$ are the [*physical*]{} (or tetrad) components of the field $M^{ikl}(y)= (-)M^{kil}(y)$ which describes the angular momentum densities [^4]. As in (2) and (3) , $e^i_{~(a)}(y),~~e_k^{~(b)}(y)$ denote mutually dual orthonormal tetrads parallelly propagated along geodesics through [**P**]{} such that $e^i_{~(a)}(P) = \delta^i_a,~~e_k^{~(b)}(P) = \delta_k^b$. The compact four-dimensional domain $\Omega$ is defined in the same way as in the formula (1) and we will again take $\Omega$ as a sufficiently small four-dimensional ball with centre at [**P**]{} and with radius $R = \varepsilon L$. At [**P**]{} the tetrad and normal components of an object are equal. We apply this once more and omit tetrad brackets for the indices of any quantity attached to the point [**P**]{}; for example, we write $M^{abc}(P)$ instead of $M^{(a)(b)(c)}(P)$ and so on. For matter as $M^{ikl}(y)$ we take $$_m M^{ikl}(y) =\sqrt{\vert g\vert}\bigl[y^i T^{kl}(y) - y^k T^{il}(y)\bigr],$$ where $T^{ik}(y) = T^{ki}(y)$ are the components of a symmetric energy-momentum tensor of matter and $y^i$ denote the normal coordinates [**NC(P)**]{}. The formula (27) gives us the total angular momentum densities, orbital and spinorial, because the symmetric energy-momentum tensor of matter $T^{ik} = T^{ki}$ comes from the canonical one by using the [*Belinfante-Rosenfeld*]{} symmetrization procedure and, therefore, includes the canonical spin of matter \[14\]. For the gravitational field we take the gravitational angular momentum pseudotensor proposed by Bergmann and Thomson \[14,18\] which in a [**NC(P)**]{} (and in any other holonomic frame) reads $$_g M^{ikl}(y) = _F U^{i[kl]}(y) - _F U^{k[il]}(y)+\sqrt{\vert g\vert}\bigl(y^i _{BT} t^{kl} - y^k _{BT} t^{il}\bigr),$$ where, in a holonomic frame, $$_F U^{i[kl]}:= g^{im}{}_F U_m^{~[kl]} =\alpha g^{im}{g_{ma}\over\sqrt{\vert g\vert}}\biggl[(-g)\bigl(g^{ka} g^{lb} - g^{la} g^{kb}\bigr)\biggr]_{,b}$$ are [*Freud’s superpotentials*]{} with the first index raised and $$_{BT} t^{kl}:= g^{ki}{} _Et_i^{~l} + {g^{mk}_{~~,p}\over\sqrt{\vert g\vert}} {} _F U_m^{~[lp]}$$ are the components of the [*Bergmann-Thomson*]{} gravitational energy-momentum pseudotensor \[14,18\]. $_E t_i^{~k}$ mean the components of the [*Einstein canonical gravitational energy-momentum pseudotensor*]{} of the gravitational field. The Bergmann-Thomson gravitational angular pseudotensor is most closely related to the Einstein canonical energy-momentum complex $_E K_i^{~k}:= \sqrt{\vert g\vert}\bigl(T_i^{~k} + _E t_i^{~k}\bigr)$, matter and gravitation, and it has better physical and transformational properties than the famous gravitational angular momentum pseudotensor proposed by Landau and Lifschitz \[15-17\]. This is why we apply it here. One can interpret the Bergmann-Thomson gravitational angular momentum pseudotensor as the sum of the [*spinorial part*]{} $$S^{ikl} := _F U^{i[kl]} - _F U^{k[il]}$$ and the [*orbital part*]{} $$O^{ikl} := \sqrt{\vert g\vert}\bigl(y^i{}_{BT} t^{kl} - y^k{}_{BT} t^{il}\bigr)$$ of the gravitational angular momentum “densities”. Substitution of (27) and (28) (expanded up to third order), (9),(10) and the expansion $$\sqrt{\vert g\vert} = 1 -1/6{\hat R}_{ab}y^ay^b + R_3 = 1-1/6\beta{\hat E}_{ab}y^ay^b + R_3,$$ into (24) gives us the following [*averaged angular momentum tensors*]{} for matter and gravitation respectively $$<_m M^{abc}(P;v^l)> = _m S^{abc}(P;v^l){L^2\over 6},$$ $$<_g M^{abc}(P;v^l)> = _g S^{abc}(P;v^l){L^2\over 6}.$$ Here $$_m S^{abc}(P;v^l) = 2\bigl[\bigl(2{\hat v}^a{\hat v}^p - {\hat g}^{ap}\bigr)\nabla_p{\hat T}^{bc} - \bigl(2{\hat v}^b{\hat v}^p - {\hat g}^{bp}\bigr) \nabla_p {\hat T}^{ac}\bigr],$$ and $$\begin{aligned} _g S^{abc}(P;v^l) &=& \alpha\bigl(2{\hat v}^p{\hat v}^t - {\hat g}^{pt}\bigr)\biggl[\beta\bigl({\hat g}^{ac}{\hat g}^{br} -{\hat g}^{bc}{\hat g}^{ar}\bigr)\nabla_{(t}{\hat E}_{pr)}\nonumber\\ &+& 2{\hat g}^{ar}\nabla_{(t}{\hat R}^{(b}_{~~p}{}^{c)}_{~~r)} - 2{\hat g}^{br} \nabla_{(t}{\hat R}^{(a}_{~~p}{}^{c)}_{~~r)}\nonumber\\ &+& 2/3{\hat g}^{bc}\bigl(\nabla_r{\hat R}^r_{~(t}{}^{a}_{~p)} -\beta\nabla_{(p} {\hat E}^a_{~t)}\bigr) -2/3{\hat g}^{ac}\bigl(\nabla_r {\hat R}^r_{~(t}{}^b_{~p)} - \beta\nabla_{(p} {\hat E}^b_{~t)}\bigr)\biggr]\end{aligned}$$ are the components of the [*canonical angular supermomentum tensors*]{} for matter and gravitation, respectively \[4,6,8\]. In special relativity the averaged tensor $<_m M^{abc}(P;v^l)>$, and the canonical angular supermomentum tensors for matter $_m S^{abc}(P;v^l)$ satisfy trivial conservation laws \[1-8\]. In the framework of the [**GR**]{} only the tensors $_g S^{abc}(P;v^l)$ and $<_g M^{abc}(P;v^l)>$ satisfy local conservation laws in vacuum. In vacuum, when $T_{ik} = 0 \Longleftrightarrow E_{ik} := T_{ik} -1/2 g_{ik}T = 0$, the canonical gravitational angular supermomentum tensor $_g S^{abc}(P;v^l) = (-) _g S^{bac}(P;v^l)$ given by (37) simplifies to $$_g S^{abc}(P;v^l) =2\alpha\bigl(2{\hat v}^p{\hat v}^t - {\hat g}^{pt}\bigr)\biggl[{\hat g}^{ar}\nabla_{(p}{\hat R}^{(b}_{~~t}{}^{c)}_{~~r)} -{\hat g}^{br}\nabla_{(p} {\hat R}^{(a}_{~~t}{}^{c)}_{~~r)}\biggr].$$ Some remarks are in order: 1. The orbital part $O^{ikl} =\sqrt{\vert g\vert}\bigl(y^i_{BT} t^{kl} - y^k _{BT} t^{il}\bigr)$ of the $_g M^{ikl}$ [*does not contribute*]{} to the tensor $_g S^{abc}(P;v^l)$ and, therefore, also to the tensor $<_g M^{abc}(P;v^l)>$. Only the spinorial part $S^{ikl} = _F U^{i[kl]} - _F U^{k[il]}$ gives nonzero contribution to these tensors. 2. The averaged angular nomentum tensors $<_gM^{abc}(P;v^l)>, ~~<_m M^{abc}(P;v^l)>$, like as the canonical angular supermomentum tensors, [*do not need*]{} any radius-vector for existing. The averaged tensors $< _m M^{abc}(P;v^l)>,~~<_g M^{abc}(P;v^l)>$, likely as the averaged relative energy-momentum tensors, can be interpreted as the [*averaged tensors of the relative angular momentum*]{}[^5] and also as the [*fluxes*]{} of the appropriate angular supermomentum. The formulas (12),(20),(34) and (35) give the direct link beteween the canonical superenergy and angular supermomentum tensors $$_gS_a^{~b}(P;v^l),~_m S_a^{~b}(P;v^l), ~_g S^{abc}(P;v^l), ~~_m S^{abc}(P;v^l)$$ and the averaged relative energy-momentum and angular momentum tensors $$<_g t_a^{~b}(P;v^l)>, <_m T_a^{~b}(P;v^l)>, <_g M^{abc}(P;v^l)>, <_m S^{abc}(P;v^l)>.$$ Namely, it is easily seen from these formulas that the averaged relative energy-momentum and angular momentum tensors [*differ*]{} from the canonical superenergy and angular supermomentum tensors [*only*]{} by the constant scalar multiplicator ${L^2\over 6}$, where $L>0$ means some fundamental length. Thus, from the mathematical point of view, these two kind of tensors are equivalent. Physically they [*are not*]{} because their components have different dimension. Moreover the averaged energy-momentum and angular momentum tensors depend on a fundamental length $L>0$, i.e., they need introduction a supplementary element into [**GR**]{}[^6]. Owing to the last fact and the formulas (12),(20), (34), (35) it seems that the canonical superenergy and angular supermomentum tensors are [*more fundamental*]{} than the averaged energy-momentum and angular momentum tensors. But the averaged energy-momentum and angular momentum tensors have one important superiority over the canonical superenergy and angular supermomentum tensors: their components [*possesse proper dimensions*]{} of the energy-momentum and angular momentum densities. The averaged tensors $$<_g t_a^{~b}(P;v^l)>, ~<_m T_a^{~b}(P;v^l)>,~~ <_g M^{abc}(P;v^l)>, ~<_m M^{abc}(P;v^l)>$$ depend on the four-velocity ${\vec v}$ of a fiducial observer [**O**]{} which is at rest at the beginning [**P**]{} of the normal coordinates [**NC(P)**]{} used for averaging and on some fundamental length $L>0$. After fixing the fundamental length $L$ one can determine univocally these tensors along the world line of an observer [**O**]{}. In general one can [*unambiguously determine*]{} these tensors (after fixing $L$) in the whole spacetime or in some domain $\Omega$ if in the spacetime or in the domain $\Omega$ a geometrically distinguised timelike unit vector field ${\vec v}$ exists. An example of such a kind of the spacetime is given by Friedman universes. One can try to fix[^7] the fundamental length $L$, e.g., by using loop quantum gravity. Namely, one can take as $L$ the smallest length $l$ over which the classical model of the spacetime is admissible. Following loop quantum gravity \[19-29\] one can say about continuous classical differential geometry already just a few orders of magnitude above the Planck scale, e.g., for distances $l\geq 100L_P = 100\sqrt{{G\hbar\over c^3}} \approx 10^{-33}$ m. So, one can take as the fundamental length $L$ the value $L = 100 L_P \approx 10^{-33}$ m.[^8] After fixing the fundamental length $L$ one has the averaged relative energy-momentum and angular momentum tensors as precisely defined as the canonical superenergy and angular supermomentum tensors are. The averaged tensors (with $L$ fixed or no) $$<_mT_a^{~b}(P;v^l)>, <_g t_a^{~b}(P;v^l)>, <_m M^{abc}(P;v^l)>, <_g M^{abc}(P;v^l)>$$ give us as good tool to a local analysis ( and also to global analysis iff in spacetime a privileged global unit timelike vector field exists)) of the gravitational and matter fields as the canonical superenergy and angular supermomentum tensors $$_m S_a^{~b}(P;v^l), ~_g S_a^{~b}(P;v^l), ~_m M^{abc}(P;v^l), ~_g M^{abc}(P;v^l)$$ give. For example, one can apply the averaged energy-momentum and angular momentum tensors to the all problems which have been analyzed in the papers \[1-8\] by using the canonical superenergy and angular supermomentum tensors. Some applications of the averaged relative energy-momentum tensors ================================================================== In this paper we apply the averaged gravitational relative energy-momentum tensor$<_g t_a^{~b}(P;v^l)>$ only to decide if free vacuum gravitational field has energy-momentum; especially, if gravitational waves carry any energy-momentum, and the averaged gravitational and matter relative energy-momentum tensors to analyze the energy and momentum of the Friedman universes. Albrow and Tryon were the first who assumed that the net energy of the closed Friedman universes may be equal to zero \[30-31\]. We will show in this paper that this assumption is, most probably, [*incorrect*]{}. Let us begin from the vacuum gravitational energy and momentum. The problem was revived recently because some authors conjectured \[32-36\], by using coordinate dependent[^9] pseudotensors and double index complexes, that the energy and momentum in general relativity are confined only to the regions of non-vanishing energy-momentum tensor of matter and that the gravitational waves carry no energy and momentum. The argumentation is the following. For some solutions to the Einstein equations and in some special coordinates, e.g., in Bonnor’s spacetime \[37\] in Bonnor’s or in Kerr-Schild coordinates, the Einstein canonical gravitational energy-momentum pseudotensor (and other most frequently used gravitational energy-momentum pseudotensors also) [*globally vanishes*]{} outside of the domain in which $T^{ik}\not= 0$. The analogous global vanishing of the canonical pseudotensor $_E t_a^{~b}$ we have for the plane and for the plane-fronted gravitational waves in, e.g., null coframe \[3,38\]. But one should emphasize that all these results are [*coordinate dependent*]{} \[3,7,38\], i.e., in [*other coordinates*]{} the used gravitational energy-momentum pseudotensors [*do not vanish*]{} in vacuum. Moreover, one should interpret physically the global vanishing of the canonical pseudotensor (and other pseudotensors also) in some coordinates in vacuum as a [*global cancellation*]{} of the energy-momentum of the real gravitational field which has $R_{iklm}\not= 0$ with energy-momentum of the inertial forces field which has $R_{iklm}=0$; [*not as a proof of vanishing of the energy-momentum of the real gravitational field*]{}. It is because the all used pseudotensors were entirely constructed from the Levi-Civita’s connection $\Gamma^i_{~kl} = \Gamma^i_{~lk}$ and from the metric $g_{ik}$ which describe a mixture of the real gravitational field ($R_{iklm}\not= 0$) and an inertial forces field ($R_{iklm}= 0$). In order to get the coordinate independent results about energy-momentum of the [*the real gravitational field*]{} one must use tensorial expressions which depend on curvature tensor, like the averaged gravitational relative energy-momentum tensor $<_g t_a^{~b}(P;v^l)>$. This tensor vanishes iff $R_{iklm}=0$, i.e., iff the spacetime is flat and we have no real gravitational field. When calculated, the averaged gravitational relative energy-momentum tensor$<_g t_a^{~b}(P;v^l)>$ always gives the [*positive-definite*]{} averaged free relative gravitational energy density and, in the case of a gravitational wave, its non-zero flux. It is easily seen from the our papers \[1-8,38\] in which we have used the canonical gravitational superenergy tensor and from the formula (20) of this paper which gives the direct connection between the averaged relative gravitational energy-momentum tensor and the canonical gravitational superenergy tensor. Thus, the conjecture about localization of the gravitational energy only to the regions of the non-vanishing energy-momentum tensor of matter [*is incorrect*]{} for the real gravitational field which has $R_{iklm}\not= 0$. It is interesting that the gravitational angular momentum pseudotensor (28) [*does not vanish*]{} in Bonnor’s spacetime and in Bonnor’s coordinates [*outside*]{} of the domain in which $T^{ik}\not= 0$. This important fact which, as I think, is unknown for the authors of the conjecture, gives other [*direct proof*]{} that this conjecture [*is incorrect*]{}. If the conjecture were correct, then we would have an absurd situation: the energy-momentum density–free vacuum gravitational field has non-vanishing “densities” of the angular momentum. In a similar way as above one can use the averaged gravitational relative angular momentum tensor $<_g M^{abc}(P;v^l)>$ to coordinate independent analysis of the angular momentum of the real gravitational field. Now, let us pass to the problem of the energy and momentum of the Friedman universes. Of course, the problem of the global energy and global linear (or angular) momentum for Friedman universes (and also for more general universes) is not [*well-posed*]{} from the physical point of view because these universes are not asymptotically flat spacetimes \[39\]. Despite this important fact recently many authors concluded \[40-50\] that the energy and momentum of the Friedman universes, flat and closed, are equal to zero locally and globally (flat universes) or only globally (closed universes). Such conclusion, which has a mathematical sense, originated from calculations performed in special comoving coordinates called “Cartesian coordinates” by using [*coordinate dependent*]{} double index energy-momentum complexes, matter and gravitation. One can introduce in [**GR**]{} many different energy-momentum complexes. The six of them are most frequently used: Einstein’s canonical complex, Landau-Lifshitz complex, Bergmann-Thomson complex, Møller complex, Papapetrou complex and Weinberg energy-momentum complex. These all energy-momentum complexes [*are neither geometrical objects nor coordinate independent objects*]{}, e.g., they can vanish in some coordinates locally or globally and in other coordinates they can be different from zero. It results that the double index energy–momentum complexes and the gravitational energy-momentum pseodotensors [*have no physical meaning*]{} to a local analysis of the gravitational field, e.g., to study gravitational energy-density distribution. They can be reasonably used [*only to calculate the global quantities*]{} for the very precisely defined asymptotically flat spacetimes (in spatial or in null direction). The general opinion is that the best one of the all possible double index energy-momentum complexes from physical and geometrical points of view is the canonical Einstein’s double index energy-momentum complex $_E K_i^{~k} = \sqrt{\vert g\vert}\bigl(T_i^{~k} + _E t_i^{~k}\bigr)$. The global results obtained by use of this canonical energy-momentum complex are usually treated as correct and giving some pattern. In fact, the other double index energy-momentum complexes were constructed following the instruction: they should give the same global results as the Einstein energy-momentum complex gives at least in the simplest cases, e.g., in the case of a closed system. That is why we have confined in the paper (and also in the all our previous papers) only to this double index energy-momentum complex. So, let us consider the results of the formal calculations of the global energy and momentum for Friedman universes in the standard comoving coordinates by using canonical Einstein’s double index energy-momentum complex. Any other sensible double index energy-momentum complex gives equivalent results. 1. In the “Cartesian coordinates” $(t,x,y,z)$ in which the line element has the form[^10] $$ds^2 = dt^2 - R^2(t){(dx^2+dy^2+dz^2)\over[1+k/4(x^2+y^2+z^2)]^2},~~~k = 0,^+_-1,$$ we obtain after simple calculations \[1,5\] that for flat universes the global quantities $P_i ~(i = 0,1,2,3)$, where $P_i$ mean the components of the energy-momentum contained inside of a slice $t = const$, are equal to zero. In this case the all integrands (energy and momentum “densities”) in the integrals on $P_i~~(i = 0,1,2,3)$ identically vanish because they are multiplied by the curvature index $k$. So, one can say that for flat Friedman universes the integral quantities $P_i ~~(i=0,1,2,3)$ vanish locally and globally in the “Cartesian” coordinates[^11]. For closed Friedman universes we also get $P_i = 0,~~(i=0,1,2,3)$, but this time the integrands do not vanish. Only after integration one gets that the integrals representing $P_i, (i =0,1,2,3)$ are equal to zero. In the case of the open Friedman universes one gets $E = P_0 = (-)\infty, ~P_1 = P_2 = P_3 = 0$. The integrands also do not vanish in this case. 2. In the coordinates $(t,\chi,\vartheta,\varphi)$ in which the line element reads $$ds^2 = dt^2 -R^2(t)[d\chi^2 + S^2(\chi)(d\vartheta^2 + \sin^2\vartheta d\varphi^2)],$$ where $$S(\chi) =\bigl\{sin\chi~~if~~ k = 1, ~~\chi~~ if~~ k=0,~~ sh\chi~~ if~~ k =-1\bigr\},$$ one gets drastically different results: $E = P_0 =(-)\infty, ~~ P_1 = (-)\infty,~~P_2 = P_3 = 0$ for flat universes; $ E = P_0 = {\pi\over 2} R(t), ~~P_1 = P_2 = P_3 =0$ for closed univeres and $E = P_0 =(-)\infty, ~~ P_1 =(-)\infty, ~~P_2 = P_3 =0$ for open universes. 3. Finally, in the coordinates $(t,r,\vartheta,\varphi)$ in which the line element has the form $$ds^2 = dt^2 - R^2(t)[{dr^2\over(1-kr^2)} + r^2(d\vartheta^2+\sin^2\vartheta d\varphi^2)],~~k=0,^+_-1,$$ we obtain the following results: $E = P_0 =(-)\infty,~~ P_1 = (-)\infty, ~~P_2 = P_3 = 0$ for flat universes; $E = P_0 = {\pi\over 4}R(t),~~P_1 = (-)\infty,~~P_2 = P_3 = 0$ for closed universes and $E= P_0 =(-)\infty,~~ P_1 =(-)\infty,~~P_2 = P_3 =0$ for open Friedman universes. In the all cases in which the integrands (=“densities” of the calculated four-momentum) do not vanish, these integrands go to zero if $R(t)\longrightarrow 0$. So, these integrands (“densities”of the energy-momentum) are not suitable for analysis of the Big-Bang singularity. The authors which assert that the energy and momentum of the Friedman universes, flat and closed, are equal to zero have performed their calculations only in the “Cartesian” comoving coordinates $(t,x,y,z)$ by using coordinate dependent double index energy-momentum complexes and have got zero results. But in the case of the Friedman universes the “Cartesian”coordinates [*are by no means better*]{} than the comoving coordinates $(t,\chi,\vartheta,\varphi)$ or $(t,r,\vartheta,\varphi)$ in which we have obtained non-zero results. Only in a flat and in an asymptotically flat spacetimes one can distinguish in some reasonable way the Cartesian coordinates; but [*not in the case of the Friedman universes*]{}. So, the conclusion of these authors about vanishing of the energy and linear momentum of the Friedman universes, flat and closed, [*cannot be correct*]{}. By using double index energy-momentum complexes one rather should conclude that the energy and momentum of the Friedman universes explicite depend on the used comoving coordinates and, therefore, that [*they are undetermined*]{} locally and globally. This last conclusion is very sensible because [*one cannot measure the global energy and global momentum of the Friedman (and more general) universes*]{}. One can do this only in the case of an isolated system \[39\]. On the other hand the former conclusion directly follows from the coordinate dependence of the energy-momentum complexes. May be one would try to support the [*mathematically sensible*]{} hypothesis which states that energy and momentum of the Friedman universes, flat and closed, disappear by using coordinate independent expressions, like Pirani’s expression on global energy, matter and gravitation, or like single index Komar’s expression (Komar’s single index complex) on global energy-momentum and global angular momentum, matter and gravitation [^12]. The Pirani’s expression (for the energy only, see, eg., \[51\]) is unique and can be applied in a spacetime having a privileged set of observers whose world-lines form a normal congruence. In such spacetime there exists a family of spatial hypersurfaces which are orthogonal to the four-velocities of this set of observers. The Pirani’s expression is coordinate independent but it has two defects: calculated total energy density, matter and gravitation, [*is not positive-definite*]{}, and, if the congruence is geodesic, then the total energy-density [*is identically zero*]{}, and, in consequence, the global energy [*trivially vanishes in the case*]{}. However, this zero values [*are not a property of the gravitational and matter fields*]{}. They are only a property of the geodesics congruence. In Friedman universes does exist privileged set of observers called [*fundamental or isotropic*]{} observers. For these observers the four-velocity ${\vec v}$ has components $v^k = \delta^k_0$ in a comoving coordinates and the family of the spatial hypersurfaces orthogonal to ${\vec v}$ is given by $t = const.$ But, unfortunately, the congruence of the isotropic observers in Friedman universes is geodesic and, therefore, the Pirani’s expression [*fails*]{} in the case [*giving trivially zero*]{}. On the other hand, coordinate independent Komar’s expression (see, e.g., \[51-53\]) [*needs Killing vector fields*]{}: translational timelike Killing vector field as [*energy descriptor*]{}, translational spatial Killing vector fields as [*descriptors of the linear momentum*]{} and rotational spatial Killing vector fields as [*descriptors of the angular momentum*]{}. Friedman universes admit only six linearly independent spatial Killing vector fields, three translational Killing vector fields and three rotational Killing vector fields (see, e.g., \[54\]). So, one can consider in Friedman universes six coordinate independent integrals (scalars) which correctly represent (from mathematical point of view) the components of the global linear momentum and the components of the global angular momentum (see, eg., \[54\]). These integrals [*trivially*]{} vanish for Friedman universes, i.e., integrands in these integrals [*identically*]{} vanish, independently of the curvature index $k=0,^+_-1$. This is very sensible result and it can be interpreted as a mathematically correct proof that the linear and angular momentum for Friedman universes disappear in a comoving coordinates. But we still have a problem with energy of the Fiedman universes [*because we have no energy descriptor*]{}, i.e., translational timelike Killing vector field, in these universes. Therefore, one cannot use the coordinate independent Komar’s expression in order to calculate correctly from the mathematical point of view the energy of the Friedman universes. If one formally uses in Komar’s expression the four-velocity of the privileged set of the isotropic observers as the energy descriptor, then [*one will get identically zero*]{} because for a geodesic timelike congruence the integrand in this expression, like integrand in Pirani’s expression, identically vanishes. But this vanishing is also only a property of the geodesics congruence. It is not a property of the gravitational and matter fields. Resuming, one cannot use the coordinate independent Pirani’s and Komar’s expressions in order to correctly prove[^13] the statement that the energy of the Friedman universes disappears, i.e., that these universes are complete energetic nonentity. For this purpose one cannot also use the coordinate independent KBL bimetric approach \[55\] because the results obtained in this approach depend not only on the used background but also on mapping of the real spacetime onto this background. Therefore, the [*mathematically sensible*]{} statement that the closed and flat Friedman universes have no energetic content [*is still not satisfactory proved*]{}. It is interesting that the using of the coordinate independent averaged relative energy-momentum tensors to analyze the energetic content of the Friedman universes lead us to [*positive-definite results*]{} for the all Friedman universes. Namely, let us apply the averaged relative energy-momentum tensors for gravitation $<_g t_i^{~k}(P;v^l)>$ and for matter $<_m T_i^{~k}(P;v^l)>$ to calculate the averaged relative energy density for Friedman (and more general) universes. With this aim let us define $$_g\epsilon := <_g t_a^{~b}(P;v^l)>v^av_b$$ —– the averaged relative gravitational energy density, $$_m\epsilon:= <_m T_a^{~b}(P;v^l)>v^av_b$$ —– the averaged relative matter energy density, and $$\epsilon:= _g\epsilon + _m \epsilon$$ —– the averaged relative total energy density. Here $v^a$ are the components of the four-velocity of an observer [**O**]{} which is studying gravitational and matter fields. In Friedman universes, if we take as the observers [**O**]{} the globally defined set of the fundamental observers, then we can also define the global averaged total relative energy $E$ of a Friedman universe $$E := \int\limits_{t = const}\epsilon\sqrt{\vert g\vert} d^3v ~{\dot =}\int\limits_{t = const} \bigl[<_g t_i^{~0}> + <_m T_i^{~0}>\bigr] v^i\sqrt{\vert g\vert} d^3v,$$ and, in analogous way, the global averaged relative energy for matter and for gravitation. Here $d^3v$ means the product of the diferentials of the coordinates which parametrize slices $t = const$ of the Friedman universes, e.g., $d^3v = dxdydz$ in the Cartesian comoving coordinates $(t,x,y,z)$. After something tedious but very simple calculations we will obtain for Friedman universes \[1,2,5\][^14]: 1. $_g\epsilon$, $_m\epsilon$ and, in consequence $\epsilon$, are [*positive definite*]{} for the all Friedman universes. 2. $\displaystyle\lim_{R\to 0}{}_g\epsilon = \displaystyle\lim_{R\to 0} {}_m\epsilon =\displaystyle\lim_{R\to 0}{}\epsilon = +\infty, ~~(k = 0,^+_- 1)$. It follows from this that one can use the averaged relative energy densities to study the Big-Bang singularity. 3. $\displaystyle\lim_{R\to\infty}{} _g\epsilon = \displaystyle\lim_{R\to\infty}{} _m \epsilon = \displaystyle\lim_{R\to\infty}{}\epsilon = 0, ~~(k = 0,-1)$. 4. The global averaged relative energies, gravitation, matter and total, are infinite ($+\infty$) for flat and for open Friedman universes and they are finite and positive for closed Friedman universes. Also the other three invariant integrals which formally represent the components $P_{(\alpha)} ~~(\alpha = 1,2,3)$ of the global averaged relative linear momentum for Friedman universes $$P_{(\alpha)}:= \int\limits_{t = const}\bigl\{<_g t_i ^{~0}> + < _m T_i ^{~0}>\bigr\}e^i_{~(\alpha)}\sqrt{\vert g\vert}d^3v,~~(\alpha = 1,2,3),$$ [*vanish trivially*]{} in a comoving coordinates \[1,2,5\] because the integrands in these integrals (densities of the averaged relative linear momentum components) [*identically vanish*]{} \[1,2,5\]. Here $e^i_{~(\alpha)}, ~~(\alpha = 1,2,3)$ mean the components of the three translational spatial Killing vector fields (descriptors of the linear momentum) which exist in the Friedman universes (see, e.g., \[54\]). We would like to emphasize that the integrals (51) and (52) do not depend on the used coordinates. They depend only on a slice $t = const$. The all above results are very sensible and satisfactory from the physical point of view. We will finish this Section with remark that the analogous situation as for flat Friedman universes one has also for the more general, only homogeneous, Kasner vacuum universes \[15\] and Bianchi–type I universes filled with stiff matter (see, e.g., \[44-50, 56-59\]). Namely, the most frequently used double-index energy-momentum complexes, when used in Cartesian comoving coordinates to analyze of these universes, give zero results locally and globally. Of course, in other comoving coordinates, e.g., in the $t,r,\vartheta,\varphi$ comoving coordinates, we have non-zero and globally divergent results. If one applies the averaged relative energy-momentum tensors $<_g t_a^{~b}(P;v^l)>, \\<_m T_a^{~b}(P;v^l)>$ to analyze of a vacuum Kasner universe and a Bianchi–type I universe filled with stiff matter, then one gets the following, coordinate independent results: 1. The averaged relative gravitational energy of a vacuum Kasner universe has [*positive-definite*]{} density and the same limits when $t\longrightarrow 0$ or when $t\longrightarrow +\infty$ as it was in the case of a flat Friedman universe. Also the suitable integral global quantity defined in analogous way as in the case of the Friedman universes is divergent to $+\infty$. 2. For an expanding Bianchi–type I universe filled with stiff matter the averaged relative gravitational energy density and the averaged relative energy-density for matter are still [*positive-definite*]{} and lead to divergent to $+\infty$ global energies. Thus, one can conclude that these two more general, only homogeneous universes, like Friedman flat universes, also [*are not energetic nonentity*]{}. Concerning of the components of the linear momentum for Kasner vacuum universes and for Bianchi–type I universes filled with stiff matter one can easily check that these components, defined in analogous way as in the case of the Friedman universes, [*identically vanish locally and globally*]{} in a comoving coordinates. Conclusion ========== We have introduced in the paper the averaged tensors of the relative energy-momentum and the averaged tensors of the relative angular momentum, for matter and for gravitation. These tensors are very closely related to the canonical superenergy and angular supermomentum tensors and they can be used to analyze the same problems which we have analyzed in the our papers \[1-8\] with the help of the canonical superenergy and angular supermomentum tensors. The superiority of the averaged relative energy-momentum and angular momentum tensors in comparison with the canonical superenergy and angular supermomentum tensors is the following: the averaged tensors have proper dimensionality of the energy-momentum and angular momentum densities. The averaged relative energy-momentum and relative angular momentum tensors of the gravitational field [*refer to the energy-momentum and angular momentum of the real gravitational field*]{} for which we have $R_{iklm}\not= 0$. These tensors vanish iff $R_{iklm} =0$, i.e., iff [*we have no real gravitational field*]{}. In our opinion the all existing (and projected in near future) detectors of the gravitational waves will measure the averaged relative gravitational energy density and its flux; not the gravitational energy defined by pseudotensors. It is easily seen from the fact that the acting of these detectors relies on the equations of the geodesics deviation which explicitly depend on the curvature tensor. In this paper we have applied the averaged relative gravitational energy-momentum tensor to decide if free vacuum gravitational field has energy and momentum and the averaged gravitational and matter relative energy-momentum tensors to analyze energy and momentum of the Friedman universes and also to analyze the Kasner and Bianchi–type I universes. The latter problem is recently very popular despite the fact that the problem of the global quantities for Friedman universes (and for more general cosmological models also) [*is not well-posed from the physical point of view*]{}. The global energy and momentum [*have physical meaning*]{} only when spacetime is asymptotically flat either in spatial or null direction. Of course, this is not a case of the Friedman and Kasner or Bianchi–type I cosmological models. We have obtained the following results: 1. The real vacuum gravitational field for which we have $R_{iklm}\not= 0$ [*always*]{} possesses his own positive-definite averaged relative energy density and in the cases in which the gravitating system is not at rest, the gravitational field possesses also the non-zero averaged relative linear momentum. 2. The coordinate independent averaged relative energy-momentum tensors, gravitation and matter, give positive-definite densities of the averaged relative energy, matter and gravitation, for the all Friedman universes. Therefore, these tensors indicate that the Friedman universes [*are not energetic nonentity*]{}. They [*are not energetic nonentity*]{} in the following sense: one can construct from the canonical energy-momentum complex, matter and gravitation, non-local tensorial, i.e., coordinate-independent expressions with correct dimensions which give positive-definite energy densities for the all Friedman universes. The averaged relative energy-momentum tensors tensors give also zero values of the averaged relative linear momenta for these universes in a comoving coordinates. The above results directly follow from the results obtained in the our previous papers \[1-5\] in which we have used the canonical superenergy (and angular supermomentum) tensors, gravitation and matter, and from the formulas (12) and (20) of this paper which connect the averaged relative energy-momentum tensors with the canonical superenergy tensors. The coordinate independent results presented in this paper for the Friedman universes are very satisfactory from the physical point of view. Much more satisfactory than the strange, coordinate dependent results which one obtains by using gravitational energy-momentum pseudotensors and double index energy-momentum complexes, matter and gravitation. By using of these objects one can only conclude that the energy and momentum of the Friedman universes [*are undetermined*]{} locally and globally. The analogous conclusion as given above for Friedman universes is also correct for the more general Kasner and Bianchi–type I universes. We are planning to use in a future the averaged relative energy-momentum tensors, and also the averaged tensors of the relative angular momentum, to analyze much more general homogeneous universes, like the universes which have been considered in the papers \[44-50, 56-60\]. [\[1\]]{} J. Garecki, [*Rep.Math.Phys.,*]{} [**33**]{} (1993) 57. [\[2\]]{} J. Garecki, [*Int.J.Theor.Phys.*]{}, [**35**]{} (1996) 2195. [\[3\]]{} J. Garecki, [*Rep.Math.Phys*]{}., [**40**]{} (1997) 485. [\[4\]]{} J. Garecki, [*J.Math.Phys*]{}., [**40**]{} (1999) 4035. [\[5\]]{} J. Garecki, [*Rep.Math.Phys*]{}., [**43**]{} (1999) 397. [\[6\]]{} J. Garecki, [*Rep.Math.Phys*]{}., [**44**]{} (1999) 95. [\[7\]]{} J. Garecki, [*Ann. Phys. 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[\[58\]]{} Oktay Aydogdu et al., “Energy Density Associated with the Bianchi Type–II Spacetime”, arXiv:gr-qc/0601133. [\[59\]]{} P. Halpern, “Energy of the Taub cosmological solution”, arXiv:gr-qc/0606095. [\[60\]]{} M.S. Berman, “On the energy of the universe”, arXiv:gr-qc/0605063. [^1]: e-mail: garecki@sus.univ.szczecin.pl [^2]: Trivial local conservation laws because the integral superenergetic quantities or, equivalently, integral averaged relative energy-momentum calculated from them for a closed system in special relativity vanish. [^3]: Very closely related because this tensor has almost the same analytic form as the Bel-Robinson tensor and the same symmetry properties. [^4]: Of course, $M^{abc}(P) = 0$, but we leave $M^{abc}(P)$ in our formulas. [^5]: Of course, the angular momentum is always relative quantity, in principle. Despite that we will keep the term [*relative angular momentum tensors*]{}. [^6]: The fundamental length $L>0$ must be infinitesimally small because its existence violates local Lorentz invariance. It is generally belived that a fundamental length exists in Nature. [^7]: But this is [*not necessary*]{}. One can effectively use the averaged energy-momentum and angular momentum tensors [*without fixing L*]{} explicitly. [^8]: Concerning other propositions fixing of $L$ see, e.g., \[9–12\]. [^9]: By “coordinate dependent” quantity we mean a quantity which is not a tensor (in general–which is not a tensor valued p-form). By “coordinate independent” quantity we mean a tensor quantity (in general – a tensor valued p-form). [^10]: From now on we will use [*geometrized units*]{} in which $G = c =1$. [^11]: It is interesting that the angular momentum “densities” when calculated, e.g., by using Bergmann–Thomson angular momentum complex (28) do not vanish in the case even for flat Friedman universes. [^12]: We would like to remark that the Pirani’s and Komar’s expressions, though coordinate independent, depend (like double index energy-momentum complexes) not only on real gravitational field ($R_{iklm}\not= 0$) but also on inertial forces field ($R_{iklm} = 0$). [^13]: Correctly from the mathematical point of view. [^14]: The results given below are easily seen from the our previous papers \[1,2,5\] and from the formulas (12) and (20) which connect the canonical superenergy tensors used in the papers \[1,2,5\] with the averaged relative energy and momentum tensors which we are using in this paper.

--- abstract: 'We consider the localization game played on graphs, wherein a set of cops attempt to determine the exact location of an invisible robber by exploiting distance probes. The corresponding optimization parameter for a graph $G$ is called the localization number and is written ${\zeta(G)}$. We settle a conjecture of [@nisse1] by providing an upper bound on the chromatic number as a function of the localization number. In particular, we show that every graph with ${\zeta(G)} \le k$ has degeneracy less than $3^k$ and, consequently, satisfies $\chi(G) \le 3^{{\zeta(G)}}$. We show further that this degeneracy bound is tight. We also prove that the localization number is at most 2 in outerplanar graphs, and we determine, up to an additive constant, the localization number of hypercubes.' address: - 'Department of Mathematics, Ryerson University, Toronto, ON, Canada, M5B 2K3' - 'Department of Mathematics, University of Rhode Island, University of Rhode Island, Kingston, RI,USA, 02881' author: - Anthony Bonato - 'William B. Kinnersley' title: Bounds on the localization number --- [Introduction]{} *Graph searching* focuses on the analysis of games and graph processes that model some form of intrusion in a network and efforts to eliminate or contain that intrusion. One of the best known examples of graph searching is the game of [*Cops and Robbers*]{}, wherein a robber is loose on the network and a set of cops attempts to capture the robber. How the players move and the rules of capture depend on which variant is studied. There are many variants of graph searching studied in the literature, which are either motivated by problems in practice or inspired by foundational issues in computer science, discrete mathematics, and artificial intelligence, such as robotics and network security. For a survey of graph searching see [@bp; @by; @fomin], and see [@bonato] for more background on Cops and Robbers. We focus in the present paper on a variant of Cops and Robbers, called the [*localization game*]{}, in which the cops only have partial information on the location of the robber. The variant we discuss is motivated by a real-world tracking problem with mobile receivers and a cell phone user. The receivers are placed in various locations, and the user is in motion and is only detectable by the strength of their signal to the receivers (measured by their distance to the receivers). The receivers, who do not know the user’s location, may appear anywhere and relocate over time. The goal is to uniquely determine the location of the user. See, for example, [@bahl]. The localization game was first introduced for one receiver by Seager [@seager1; @seager2] and was further studied in [@brandt; @car]. In this game, there are two players moving on a connected graph, with one player controlling a set of $k$ *cops*, where $k$ is a positive integer, and the second controlling a single *robber*. Unlike in Cops and Robbers, the cops play with imperfect information: the robber is invisible to the cops during gameplay. The game is played over a sequence of discrete time-steps; a *round* of the game is a move by the cops together with the subsequent move by the robber. The robber occupies a vertex of the graph, and when the robber is ready to move during a round, he may move to a neighboring vertex or remain on his current vertex. A move for the cops is a placement of cops on a set of vertices (note that the cops are not limited to moving to neighboring vertices). At the beginning of the game, the robber chooses his starting vertex. After this, the cops move first, followed by the robber; thereafter, the players move on alternate steps. Observe that any subset of cops may move in a given round. In each round, the cops occupy a set of vertices $u_1, u_2, \ldots , u_k$ and each cop sends out a *cop probe*, which gives their distance $d_i$, where $1\le i \le k$, from $u_i$ to the robber. Hence, in each round, the cops determine a *distance vector* $(d_1, d_2, \ldots , d_k)$ of cop probes, which is unique up to the ordering of the cops. Note that relative to the cops’ position, there may be more than one vertex $x$ with the same distance vector. We refer to such a vertex $x$ as a *candidate*. For example, in an $n$-vertex clique with a single cop, so long as the cop is not on the robber’s vertex, there are $n-1$ many candidates. The cops win if they have a strategy to determine, after finitely many rounds, a unique candidate, at which time we say that the cops [*capture*]{} the robber. If there is no unique candidate in a given round, then the robber may move in the next round and the cops may move to other vertices resulting in an updated distance vector. The robber wins if he is never captured. For a connected graph $G$, define the *localization number* of $G$, written ${\zeta(G)}$, to be the least integer $k$ for which $k$ cops have a winning strategy over any possible strategy of the robber (that is, we consider the worst case that the robber a priori knows the entire strategy of the cops). As placing a cop on each vertex gives a distance vector with unique value of $0$ on the location of the robber, ${\zeta(G)}$ is at most $n$ and hence is well-defined. The localization number is related to the metric dimension of a graph, in a way that is analogous to how the cop number is related to the domination number. The *metric dimension* of a graph $G$, written $\mathrm{dim}(G)$, is the minimum number of cops needed in the localization game so that the cops can win in one round; see [@hm; @slater]. Hence, ${\zeta(G)} \le \mathrm{dim}(G)$, but in many cases this inequality is far from tight. The bound of ${\zeta(G)} \le {\left \lfloor \frac{(\Delta +1)^2}{4} \right \rfloor} +1$, where $\Delta$ is the maximum degree of $G$, was shown in [@has]. In [@nisse1], Bosek et al. showed that ${\zeta(G)}$ is bounded above by the pathwidth of $G$ and that the localization number is unbounded even on graphs obtained by adding a universal vertex to a tree. They also proved that computing ${\zeta(G)}$ is **NP**-hard for graphs with diameter 2, and they studied the localization game for geometric graphs. The *centroidal localization game* was considered in [@nisse2], where it was proved, among other things, that the centroidal localization number (and hence the localization number) of outerplanar graphs is at most 3. In [@dfp], the localization number was studied for binomial random graphs with diameter 2. Bosek et al. conjectured (see [@nisse1], Conjecture 16) that there is a function $f$ such that every graph with ${\zeta(G)} \le k$ satisfies $\chi(G) \le f(k),$ where $\chi(G)$ is the chromatic number of $G$. We settle this conjecture in Corollary \[cor:chromatic\]. In particular, by exploiting a lower bound on the localization number using graph degeneracy, we show that $\chi(G) \le 3^{{\zeta(G)}}$. The degeneracy bound is proven to be tight via a non-trivial example utilizing a graph built from strong powers of cycles. In Theorem \[thm:outerplanar\], we prove that outerplanar graphs have localization number at most 2. We finish by giving an asymptotically tight upper bound on the localization number of the hypercube; in particular, in Theorem \[thm:hypercube\], we show that for all positive integers $n$, ${\zeta(Q_n)} \le {\left \lceil \log_2 (n-1) \right \rceil} + 2$. Throughout, all graphs considered are simple, undirected, connected, and finite. For a reference on graph theory, see [@Wes01]. [Degeneracy and localization]{} Our first result is a general lower bound on the localization number of a graph in terms of its degeneracy. The [*degeneracy*]{} of a graph $G$ is the maximum, over all subgraphs $H$ of $G$, of $\delta(H)$. Note that the degeneracy of any nonempty graph must be a positive integer. For a vertex $u$ in a graph $G$, we define $N_G[u]$ to be the set of neighbors of $u$ along with the vertex $u$ itself. \[thm:degeneracy\] If $G$ is a graph with degeneracy $k$, where $k$ is a positive integer, then ${\zeta(G)} \ge \log_3 (k+1)$. Let $G$ be a graph with degeneracy $k$ and let $H$ be a subgraph of $G$ with $\delta(H) = k$. Suppose we play the localization game on $G$ with $m$ cops. It suffices to show that the robber can win provided that $m < \log_3 (k+1)$. In particular, we show how he can perpetually evade capture while always occupying a vertex of $H$. Toward this end, we claim that for all $v \in V(H)$, and for every cop probe $(u_1, u_2, \dots, u_m)$, there are at least two vertices in $N_H[v]$ sharing the same distance vector. Let $d_i = d_G(u_i,v)$, and note that for all $w \in N_H[v]$ we have $d_G(u_i,w) \in \{d_i-1,d_i,d_i+1\}$. Thus, between them, the vertices of $N_H[v]$ correspond to at most $3^m$ different distance vectors. Since $m < \log_3 (k+1)$, there are at most $k$ distance vectors represented in $N_H[v]$; since ${\left \vert N_H[v] \right \vert} \ge k+1$, by the Pigeonhole Principle some distance vector corresponds to at least two vertices in $N_H[v]$, as claimed. The robber’s strategy is now straightforward. Suppose that, on some robber turn, the robber occupies some vertex $v$ in $H$. If in fact the robber is choosing an initial position, then he instead pretends that he already occupies some arbitrary vertex $v$ of $H$ and wishes to move to some neighbor of $v$. Before making his move, the robber considers the cops’ subsequent probe. He next finds some two vertices in $N_H[v]$, say $w$ and $x$, that share the same distance vector with respect to this probe. The robber moves to $w$; the cops cannot uniquely locate him, since to the best of their knowledge, he could occupy either $w$ or $x$. Thus the game continues. The robber can repeat this strategy indefinitely, thereby forever evading capture. Johnson and Koch [@joh] proved that under a slightly different model of the localization game, if ${\zeta(G)} = 1$, then $\chi(G) \le 4$. In the game they studied, the robber was not allowed to move to a vertex that the cops had just probed. Our model gives the robber slightly more power and thus can slightly lower the localization number. In particular, under our model, if ${\zeta(G)} = 1$, then $\chi(G) \le 3$. Bosek et al. [@nisse1] asked whether $\chi(G)$ is, in general, bounded above by some function of ${\zeta(G)}$. We answer this question in the affirmative; Theorem \[thm:degeneracy\] yields a short proof. \[cor:chromatic\] For every graph $G$, we have $\chi(G) \le 3^{{\zeta(G)}}$. Let $G$ be any graph and let $k$ be its degeneracy. It is well-known that $\chi(G) \le k+1$, which in turn is at most $3^{{\zeta(G)}}$ by Theorem \[thm:degeneracy\]. When $G$ is bipartite, Theorem \[thm:degeneracy\] can be improved. \[thm:degeneracy\_bipartite\] If $G$ is a bipartite graph with degeneracy $k$, where $k$ is a positive integer, then ${\zeta(G)} \ge \log_2 k$. The proof proceeds exactly as with Theorem \[thm:degeneracy\], except that for all $w \in N_H(v)$ we now have $d_G(u_i,w) \in \{d_i-1,d_i+1\}$, since no neighbor of $v$ occupies the same partite set as $v$. Thus the vertices of $N_H(v)$ correspond to at most $2^m$ different distance vectors, so if $m < \log_2 k$, then some distance vector corresponds to more than one vertex in $N_H(v)$. We remark that results analogous to Theorem \[thm:degeneracy\] and Corollary \[cor:chromatic\] are known for metric dimension. Chartrand et al. [@chart] showed that $\mathrm{dim(G)} \ge \log_3(\Delta(G)+1)$, while Chappell et al. [@chap] showed that if $\mathrm{dim}(G) = m$, then $\chi(G) \le 2^m$; both bounds were shown to be tight. We conclude this section by showing that Theorem \[thm:degeneracy\] is tight. To do this we produce, for all $k$, a graph $G_k$ with degeneracy $k$ and localization number $\log_3 (k+1)$. Recall that the [*strong product*]{} of graphs $G$ and $H$ is the graph with vertex set $V(G) \times V(H)$, where $(u,v)$ is adjacent to $(u',v')$ provided that $u$ is adjacent to $u'$ in $G$ and $v=v'$, $u = u'$ and $v$ is adjacent to $v'$ in $H$, or $u$ is adjacent to $u'$ in $G$ and $v$ is adjacent to $v'$ in $H$. We construct $G_k$ as follows. Begin with the $k$-fold strong product of copies of $C_{40}$. We refer to the vertices of this strong product as [*core vertices*]{}, and we represent each one using a $k$-dimensional vector with entries in $\{0,1,\dots, 39\}$; distinct vertices are adjacent provided that they differ by at most 1 (modulo $40$) in every coordinate. In addition to the core vertices, $G_k$ contains $2k$ [*satellite vertices*]{}. For all $i \in \{1,2, \dots, k\}$ and $t \in \{0,10\}$, we add edges joining the satellite vertex $s_{i,t}$ to all core vertices whose $i$th coordinate equals $t$. We then subdivide each of these edges into a path of length $40$; we refer to the paths produced from this subdivision (including the original endpoints of the edge, namely the satellite and core vertex) as [*threads*]{} emanating from the corresponding satellite. We will make repeated use of the following fact: for a core vertex $w$, if $w = (w_1, w_2, \dots, w_{k})$, then $d(s_{i,t},w) = 40 + \min\{{\left \vert w_i-t \right \vert},40-{\left \vert w_i-t \right \vert}\}$. To see this, let $w' = (w_1, w_2, \dots, w_{i-1}, t, w_{i+1}, \dots, w_{k})$; it is clear that some shortest path from $s_{i,t}$ to $w$ contains $w'$, so $d(s_{i,t}, w) = d(s_{i,t},w') + d(w',w) = 40 + \min\{{\left \vert w_i-t \right \vert},40-{\left \vert w_i-t \right \vert}\}$. In particular, $d(s_{i,t},w)$ depends only on the $i$th coordinate of $w$. \[thm:degen\_tight\] For all positive integers $k$, the graph $G_k$ has degeneracy $3^k-1$ and localization number $k$. The $k$-fold strong product of copies of $C_{40}$ is regular of degree $3^k-1$, so clearly the degeneracy of $G_k$ is at least $3^k-1$. By Theorem \[thm:degeneracy\], we now have ${\zeta(G_k)} \ge \log_3(3^k) = k$. To complete the proof, it suffices to show that $k$ cops can locate a robber on $G$ and hence ${\zeta(G_k)} \le k$. Label the cops $1, 2, \dots, k$. Before presenting the full details of the cops’ strategy, we give an overview. In general, on each turn of the game, the robber either occupies some core vertex $(z_1, z_2, \dots, z_k)$ or some vertex on a thread ending at some such core vertex. (It is also possible that the robber could occupy a satellite, but this case will be very easily dispatched.) To locate the robber, the cops need to determine coordinates $z_1, \dots, z_k$. For each $i \in \{1, \dots, k\}$, cop $i$ will attempt to determine $z_i$, which she does by probing either $s_{i,0}$ or $s_{i,10}$. As we will show, it is relatively easy for the cops to locate the robber provided that he begins in the core and never leaves, and it likewise easy for the cops to locate the robber provided that they can be certain he has left the core; the key difficulty is in distinguishing between these two cases. We present the cops’ strategy in three stages. Before presenting the cops’ main strategy, we explain how they can locate the robber if at some point in the game some cop observes a distance smaller than 40 or larger than 60 (which would immediately indicate that the robber has left the core). Next, we give the cops’ main strategy, and we explain how this enables them to locate the robber provided that they can be certain he has never left the core. Finally, we explain how the cops proceed if there is some ambiguity as to whether or not the robber has ever left the core. First suppose that at some point in the game, some cop $c$ observes a distance strictly less than $40$; letting $v_c$ denote the satellite that this cop has just probed, the cops can infer that the robber occupies some thread emanating from $v_c$. Let $z$ be the core vertex at the other end of this thread, and let $z = (z_1, z_2, \dots, z_{k})$. The cops seek to determine the coordinates of $z$, which they can do with their next probe. Say that cop $c$, when probing $v_c$, observed a distance of $40-d$ for some positive integer $d$. If $d=40$, then the robber occupies $v_c$ and the game is over, so suppose otherwise. Cop $c$ has already determined $z_c$: it is $0$ if $v_c = s_{c,0}$ and 10 if $v_c = s_{c,10}$. Likewise, she knows the robber’s distance from $z$ along the thread. At the time of the cops’ first probe, the robber was on an internal vertex in some thread, so with his ensuing move, he can only have moved along the thread. With her next probe, cop $c$ probes $v_c$ again, and again she learns the robber’s distance from $z$ along the thread. Once again we may suppose that the robber does not occupy $v_c$, since otherwise he has been located. Now consider some other cop $i$. Cop $i$ can determine the distance from her first probe to $z$ by taking the distance she just observed and subtracting $d$, since the shortest path from her probe to the robber must pass through $z$, and the robber is $d$ steps from $z$ along the thread. On her next turn, she probes whichever of $s_{i,0}$ and $s_{i,10}$ she did not just probe. As before, she can determine her distance to $z$ using the results of cop $c$’s second probe. At the time of the cops’ first probe, the robber was on an internal vertex in some thread, so with his ensuing move, he can only have moved along the thread. Thus, the coordinates of the endpoint of that thread – that is, $z$ – cannot have changed with his last move. Cop $i$ knows, from her two probes, both $\min\{z_i,40-z_i\}$ and $\min\{{\left \vert z_i-10 \right \vert},40-{\left \vert z_i-10 \right \vert}\}$; using this information, she can uniquely determine $z_i$. Collectively, the cops can uniquely determine $z$, so they know which thread the robber occupies; since they also know the robber’s distance from $z$ along the thread, they have successfully located him. Now suppose instead that at some point, some cop observed a distance of $60+d$ for some positive integer $d$. Once again this indicates that the robber occupies some thread, but this time the cops cannot necessarily determine which satellite that thread emanates from. If any cop observed a distance smaller than 40, then the cops can locate the robber using the strategy above, so suppose otherwise. On the cops’ next turn, each cop $i$ probes whichever of $s_{i,0}$ and $s_{i,10}$ she did not just probe. If the robber still occupies a vertex internal to the thread, then some cop must observe a distance smaller than 40, and once again the cops can locate the robber. Otherwise, the cops know that the robber has just moved into the core; hence, at the time of the cops’ first probe, the robber was exactly one step from the core. Taking this into account, each cop $i$ now has enough information to determine $z_i$ as in the previous paragraph, so once again the cops can locate the robber. We now give the cops’ “main” strategy. If at any point any cop observes a distance smaller than 40 or greater than 60, then the cops can locate the robber as explained above, so we assume throughout that this never happens. The cops will attempt to determine the robber’s location within three rounds. The cops initially operate under the presumption that the robber always remains within the core, but they will remain alert for any indications that this may not be the case. Under this presumption, let $x = (x_1, x_2, \dots, x_{k})$ denote the robber’s position at the time of the cops’ first probe, let $x' = (x'_1, x'_2, \dots, x'_{k})$ denote his position at the time of the second probe, and let $x'' = (x''_1, x''_2, \dots, x''_{k})$ denote his position at the time of the third probe. The cops aim to determine $x''$ and thus win the game with their third probe. Below we describe a strategy for each individual cop. For each $i \in \{1, \dots, k\}$, cop $i$ aims to determine $x''_i$. Depending on the results of her probes, she may detect the possibility that the robber might have entered the interior of a thread emanating from either $s_{i,0}$ or $s_{i,10}$; if this happens, then we say that coordinate $i$ is *critical*. Should any coordinates be deemed critical within the cops’ first three turns, the cops will need additional probes to determine whether or not the robber has, in fact, left the core. On the cops’ first turn, each cop $i$ probes satellite $s_{i,0}$; suppose she observes a distance of $40+d_i$ for some nonnegative integer $d_i$. We consider five possibilities based on the value of $d_i$: - $2 \le d_i \le 8$. In this case, either $2 \le x_i \le 8$ or $32 \le x_i \le 38$; consequently, either $0 \le x''_i \le 10$ or $30 \le x''_i \le 39$. On her second and third turns, cop $i$ probes $s_{i,10}$. She can now uniquely determine $x''_i$, as all 21 possible values for $x''_i$ yield different distances from $s_{i,10}$. - $12 \le d_i \le 20$. In this case, $12 \le x_i \le 28$ so $10 \le x''_i \le 30$. As in Case (1), by probing $s_{i,10}$ on her next two turns, cop $i$ can uniquely determine $x''_i$. - $d_i=1$. In this case, $x_i \in \{39,1\}$. On her second turn, cop $i$ probes $s_{i,10}$; say she observes a distance of $40+d'_i$. If $d'_i \not = 10$, then she can determine $x''_i$ by probing $s_{i,10}$ on her third turn. If instead $d'_i = 10$, then more care is needed. We know that $x'_i = 0$. This is problematic, since between the cops’ second and third probes, the robber could leave the core and enter the interior of a thread emanating from $s_{i,0}$. Regardless, on her third turn, cop $i$ probes $s_{i,10}$. If she observes a distance of 49 then she knows that $x''_i = 1$, and if she observes a distance of 50 then she knows that $x''_i = 0$. If she observes a distance of $51$, then either $x''_i = 39$ or the robber has entered the interior of a thread emanating from $s_{i,0}$, but she cannot determine which; in this case, we deem coordinate $i$ to be critical. - $d_i=0$. Here, we know $x_i = 0$. Again, this indicates that the robber might leave the core and enter a thread emanating from $s_{i,0}$. On her second turn, cop $i$ probes $s_{i,0}$ once again; assuming that she doesn’t observe a distance smaller than 40, we must have $x'_i \in \{39,0,1\}$. On her third turn, she probes $s_{i,10}$. As in Case (3), if she observes any distance other than 51 then she can determine $x''_i$; otherwise, she knows that either $x''_i = 39$ or the robber has entered the interior of a thread emanating from $s_{i,0}$, and again coordinate $i$ is critical. - $9 \le d_i \le 11$. On her second turn, cop $i$ probes $s_{i,10}$; assuming that she does not observe a distance smaller than 40, she can verify that the robber has not yet left the core. On her third turn, she probes $s_{i,0}$. As in Cases (4) and (5), she may be able to conclude that the robber has not left the core, in which case she can determine $x''_i$. Otherwise, she knows only that either the robber has entered the interior of some thread emanating from $s_{i,10}$ or $x''_i = 11$; in this case, once again coordinate $i$ is critical. After the cops’ third probe, if there are no critical coordinates, then the cops can be certain that the robber hasn’t left the core, and thus (as outlined above) they can uniquely determine his position. Suppose instead that at least one coordinate is critical. For each $i \in \{1, \dots, k\}$, let $y_i$ denote cop $i$’s “predicted” value for $x''_i$ – that is, the value of $x''_i$ provided that the robber has not left the core. After the cops’ third probe and the robber’s ensuing turn, let $(z_1, z_2, \dots, z_{k})$ denote either the robber’s current position (if in fact he remains in the core) or the core vertex at the end of the thread on which the robber resides (if he has left the core). The cops play as follows, with each cop $i$’s strategy depending on the value of $y_i$. 1. If $y_i = 39$, then cop $i$ probes $s_{i,0}$. If she observes a distance smaller than 40, then the cops can locate the robber as explained earlier. If she observes a distance of exactly 40, then the robber must be in the core with $z_i = 0$. If she observes a distance of 41, then the robber cannot possibly have just left the interior of a thread emanating from $s_{i,0}$, so $x''_i = y_i = 39$. Consequently, the robber must be in the core and so $z_i = 39$, since if the robber had just entered the interior of a thread emanating from some other satellite, then cop $i$ would have observed a distance of 42. Finally, if she observes a distance of 42, then perhaps the robber was in the core, has just entered the interior of a thread, and $z_i = 39$, or perhaps the robber remains in the core and $z_i = 38$; in this case, coordinate $i$ remains critical after the cops’ turn. Note that the cops can uniquely determine $z_i$ provided that they can, collectively, determine whether or not the robber is currently in the core. 2. If $y_i = 11$, then cop $i$ probes $s_{i,10}$. As usual, if she observes a distance smaller than 40, then the cops can locate the robber. If she observes a distance of 40, then the robber is presently in the core and $z_i = 10$. If she observes a distance of 41, then necessarily $z_i = 11$ and the robber remains in the core. If she observes a distance of 42, then perhaps the robber was in the core, has just entered some thread, and $z_i = 11$, or perhaps he remains in the core and $z_i = 12$; in this last case, coordinate $i$ remains critical. 3. If $1 \le y_i \le 9$, then cop $i$ probes $s_{i,0}$. Suppose she observes a distance of $40+d$ for some nonnegative integer $d$. She now knows that either the robber remains in the core and $z_i = d$ or that the robber has entered some thread and $z_i = d-1$. 4. If $12 \le y_i \le 29$, then cop $i$ probes $s_{i,10}$. As in the previous case, she can determine $z_i$ provided that the cops can deduce whether or not the robber remains in the core. 5. If $30 \le y_i \le 38$, then by probing $s_{i,0}$, cop $i$ can again determine $z_i$ provided that the cops can deduce whether or not the robber remains in the core. 6. If $y_i = 0$, then cop $i$ probes $s_{i,10}$. If she observes a distance of 51, then the robber may have just entered the interior of some thread (possibly emanating from $s_{i,0}$), or it could instead be that the robber remains in the core and $z_i = 39$; in this case, coordinate $i$ remains critical after the cops’ turn. Otherwise, as before, the cop has enough information to determine $z_i$ provided that the cops can determine whether or not the robber remains in the core. 7. If $y_i = 10$, then cop $i$ probes $s_{i,0}$. As in the previous case, if she observes a distance of 51, then the robber may have just entered the interior of some thread (possibly emanating from $s_{i,10}$), or it could instead be that $z_i = 11$; once again, coordinate $i$ remains critical after this round. Otherwise, the cop again has enough information to determine $z_i$ provided that the cops can determine whether or not the robber remains in the core. In each case, if the cops can conclusively determine whether or not the robber is currently in the core, then cop $i$ can determine $z_i$ for all $i \in \{1, \dots, k\}$ and hence the cops can locate the robber. If any cop observes a distance of exactly 40, then the robber must be in the core, so the cops can locate him. If all distances observed exceed 40 but no coordinates are critical after this last round of probes, then again the the robber must be in the core and the cops can locate him. Finally, suppose one or more coordinates are critical after this round, so the cops cannot tell whether or not the robber is presently in the core. By the strategy above, the cops can be certain that the robber does not occupy the endpoint, in the core, of any thread; if he did, then they would have noticed this, concluded that he was in the core, and located him. Thus, if in fact the robber does presently reside in the core, then he cannot possibly move into the interior of a thread with his next move. Consequently, if the cops repeat the above strategy once more on their next turn, then there cannot be any critical coordinates; thus the cops can determine whether or not the robber is now in the core, after which they can locate him. We do not have a construction demonstrating the tightness of Theorem \[thm:degeneracy\_bipartite\]. However, the localization number of the hypercube $Q_{k}$ exceeds the bound in Theorem \[thm:degeneracy\_bipartite\] by no more than 2; see Theorem \[thm:hypercube\]. [Outerplanar graphs]{} Bosek et al. [@nisse1] showed that ${\zeta(G)}$ can be unbounded on the class of planar graphs and asked whether the same is true of outerplanar graphs. They answer this question in the negative in [@nisse2], by showing that ${\zeta(G)} \le 3$ when $G$ is outerplanar. They actually prove $\zeta^*(G)\le 3,$ where $\zeta^*(G)$ is the corresponding parameter in the *centroidal localization game*. In each round of this game (which is similar to the localization game), the cops receive only the relative distances between their location and the robber. More precisely, in this game, if the cops probe $u_1,u_2,\ldots , u_k$ and the robber is on $y$, then for all $1\le i <j \le k$ the cops learn whether $d(u,y)=0$, $d(u_i,y)=d(u_j,y)$, $d(u_i,y)<d(u_j,y),$ or $d(u_i,y)>d(u_j,y).$ Note that for all graphs $G$, we have that ${\zeta(G)} \le \zeta^*(G)$. Bosek et al. [@nisse2] ask whether there exists an outerplanar graph with localization number 3; that is, whether their bound on ${\zeta(G)}$ is tight. We answer this question by showing that in fact ${\zeta(G)} \le 2$ when $G$ is outerplanar. (This bound is clearly tight; for example, ${\zeta(C_3)} = 2$.) Recall that a [*block*]{} of a graph $G$ is a maximal 2-connected subgraph of $G$; every graph is the edge-disjoint union of its blocks. \[thm:outerplanar\] If $G$ is an outerplanar graph, then ${\zeta(G)} \le 2$. We give a strategy for two cops to locate a robber on $G$. Throughout the game, the cops will maintain a set of vertices called the [*cop territory*]{}. The cop territory will be a connected subgraph of $G$, and the cops will distinguish two distinct vertices of the cop territory as the [*endpoints*]{} of the territory. The cops will maintain three invariants: (1) : Immediately after a probe, the cops can be certain that the robber does not occupy any vertex of the cop territory. (2) : No vertex in the cop territory, with the possible exception of the endpoints, is adjacent to any vertex outside the cop territory. (3) : Both endpoints belong to the same block of $G$. We give a strategy for the cops to gradually enlarge the cop territory; since $G$ is finite, this process cannot continue indefinitely, so the cops must eventually locate the robber. Throughout the game, if either cop observes a distance of 0 on her probe, then she has located the robber and the cops have won; thus, in the proof below, we implicitly assume that this has not happened. The cops’ general approach is as follows. The cops will focus on one block of $G$ at a time. Over the course of several turns, they will ensure that the robber does not occupy any vertex of this block and, in the process, expand the cop territory to contain all vertices in the block. They will then move on to a new block that is “closer” to the robber and repeat the process until they have located the robber. Throughout the proof, $B$ will denote the block that the cops are currently probing, and ${v_{L}}$ and ${v_R}$ will denote the endpoints of the cop territory. We sometimes refer to ${v_{L}}$ (respectively ${v_R}$) as the [*left endpoint*]{} (resp. [*right endpoint*]{}) of the cop territory, and we refer to the cop who has most recently probed ${v_{L}}$ (resp. ${v_R}$) as the [*left cop*]{} (resp. [*right cop*]{}). For a vertex $v$ in $B$, we define $G_v$ to be the (possibly empty) subgraph of $G-v$ not containing any vertices of $B$. Informally, $G_v$ is the collection of blocks “attached to” $v$; that is, those blocks on the other side of $v$ from $B$. (See Figure \[fig:outerplanar\_gv\].) In what follows, we will repeatedly use the following observation: for any two distinct vertices $u$ and $v$ in $B$, if the robber occupies $G_v$, then he must be closer to $v$ than to $u$. ![. An outerplanar graph $G$ with subgraphs $G_u$ and $G_v$, the collections of blocks attached to $u$ and to $v$, respectively.[]{data-label="fig:outerplanar_gv"}](fig1.pdf) Initially, the cops choose any block $B$ of $G$, choose adjacent vertices within $B$ to comprise the cop territory, designate these vertices ${v_{L}}$ and ${v_R}$, and probe them. It is evident that all three invariants hold. To show how the cops can enlarge the cop territory, we consider the structure of $B$. Suppose first that $B$ is $K_2$. Since both ${v_{L}}$ and ${v_R}$ are cutvertices (or pendant vertices) in $G$, the robber must be closer to one than to the other; without loss of generality, suppose he is closer to ${v_{L}}$. The cops now know that the robber cannot be in $G_{{v_R}}$, so they may add all vertices of $G_{{v_R}}$ to the cop territory. On their next turn, the cops choose any neighbor of ${v_{L}}$ that is not in the cop territory and designate this vertex to be the new ${v_R}$. They then probe ${v_{L}}$ and ${v_R}$, and they add ${v_R}$ to the cop territory. The robber cannot occupy ${v_{L}}$ or ${v_R}$ (since otherwise the cops would have located him), and he was unable to pass through ${v_{L}}$ with his previous move, so he cannot be in the cop territory. Moreover, it is clear that no vertex in the cop territory aside from the endpoints can have any neighbor outside the cop territory. Finally, both endpoints clearly belong to the same block of $G$ (which the cops now take as the new block $B$). Thus all three invariants have been maintained, and the cops have successfully enlarged the cop territory. Suppose instead that $B$ is not $K_2$. In this case, $B$ must itself be a 2-connected outerplanar graph. Recall that a $2$-connected outerplanar graph can be represented as a Hamiltonian cycle with non-crossing chords drawn inside it. Consider some such representation of $B$, and label its vertices $v_1, v_2, \dots, v_n$ in clockwise cycle order. (For convenience, we may wish to refer to $v_{n+1}$, $v_{n+2}$, etc. later in the proof; indices should be adjusted modulo $n$ where needed.) The intersection of the cop territory with $V(B)$ will consist of vertices $v_{\ell}, v_{\ell+1}, \dots, v_{r}$ for some $\ell$ and $r$; that is, it is an “arc” of the outer cycle. By symmetry, we may suppose at all times that ${v_{L}}= v_{\ell}$ and ${v_R}= v_r$. (Note that this means that whenever the endpoints of the cop territory change, the values of $\ell$ and $r$ change accordingly.) Henceforth, the cops play as follows. The left cop probes ${v_{L}}$, while the right cop probes ${v_R}$. Suppose that the robber was at distance ${d_L}$ from ${v_{L}}$ and distance ${d_R}$ from ${v_R}$.\ [**Case 1**]{}: All vertices of $B$ belong to the cop territory.\ If in fact all of $V(G)$ belongs to the cop territory, then the cops have won, so suppose otherwise. By invariant (2), every vertex outside the cop territory that is adjacent to a vertex inside the cop territory must be adjacent to ${v_{L}}$ or ${v_R}$, so the robber must reside in either $G_{{v_{L}}}$ or $G_{{v_R}}$. Since all vertices of $B$ belong to the cop territory, ${v_{L}}$ and ${v_R}$ are either equal or adjacent along the outer cycle of $B$. If ${v_{L}}= {v_R}$, then $G_{{v_{L}}} = G_{{v_R}}$. If instead ${v_{L}}$ is adjacent to ${v_R}$, then we cannot have ${d_L}= {d_R}$; if ${d_L}< {d_R}$ then the robber occupies a vertex in $G_{{v_{L}}}$, and if ${d_R}< {d_L}$ then the robber occupies a vertex in $G_{{v_R}}$. We assume henceforth that the robber occupies a vertex in $G_{{v_{L}}}$; a symmetric argument suffices for the case where he occupies a vertex in $G_{{v_R}}$. If $G_{{v_{L}}} \not = G_{{v_R}}$, then the cops add all vertices of $G_{{v_R}}$ to the cop territory. To proceed, the cops must determine which component of $G_{{v_{L}}}$ contains the robber. Within $G_{{v_{L}}}$, let $B_1, B_2, \dots, B_m$ be the blocks containing ${v_{L}}$. For $i \in \{1, \dots, m\}$, let $C_i$ be the subgraph of $G_{{v_{L}}}$ induced by ${v_{L}}$ and all vertices in the same component of $G_{{v_{L}}}-{v_{L}}$ as the vertices of $B_i-{v_{L}}$. (Informally, $C_i$ consists of all vertices “on the same side of” ${v_{L}}$ as $B_i$.) Note that any two $C_i$ share only one vertex, namely ${v_{L}}$, and the $C_i$ together contain all vertices in $G_{{v_{L}}}$. The cops aim to determine which of these components the robber occupies. They begin by determining whether or not the robber occupies $C_1$. If $B_1 = K_2$, then they can easily do this by probing both ${v_{L}}$ and the other vertex of $B_1$, so suppose otherwise. Within $B_1$, let $w_1, w_2, \dots, w_k$ be the neighbors of ${v_{L}}$, in clockwise order around the outer cycle of $B_1$. The cops probe ${v_{L}}$ and $w_1$; let ${d_L}$ and $d_1$ denote the robber’s distances from ${v_{L}}$ and $w_1$, respectively. Note that $d_1 \in \{{d_L}-1,{d_L},{d_L}+1\}$. If $d_1 \le {d_L}$, then the robber must be in $C_1$. The cops now take $B_1$ as the new block $B$, take ${v_{L}}$ and $w_1$ as the new left and right endpoints of the cop territory, and add all vertices of $C_2 \cup C_3 \cup \dots \cup C_m$ to the cop territory. Suppose instead that $d_1 = {d_L}+1$. On their next turn, the cops probe ${v_{L}}$ and $w_2$; let ${d_L}'$ and $d_2$, respectively, be the distances observed. Once again, if $d_2 \le {d_L}'$, then the robber must be in $C_1$ and the cops play as outlined in the preceding paragraph. Otherwise, we must have $d_2 = {d_L}'+1$. We claim that for all vertices $u$ in $B_1$ that lie on the clockwise arc from $w_1$ to $w_2$ (inclusive), the robber cannot occupy either $u$ or $G_{u}$. Suppose otherwise, let $z$ denote the robber’s current position, and let $y$ denote the robber’s previous position (that is, his position at the time of the cops’ previous probe). Since $d_2 = {d_L}'+1$, some shortest path from $w_2$ to $z$ passes through ${v_{L}}$, and thus through $w_1$ as well (since $u$ lies on the arc from $w_1$ to $w_2$). Consequently, we have that $d(w_1,z) = d(w_2,z)-2 = d_2-2 = {d_L}'-1$. Because $y$ and $z$ are adjacent, we also have $d_1 = d(w_1,y) \le d(w_1,z)+1 = {d_L}'$. Similarly, $d(w_2,y) = d_1-2 = {d_L}-1$, hence $d_2 = d(w_2,z) \le d(w_2,y)+1 = {d_L}$. Thus ${d_L}\ge d_2 = {d_L}'+1$, and yet ${d_L}' \ge d_1 = {d_L}+1$, so ${d_L}\ge {d_L}'+1 \ge {d_L}+2$, a contradiction. The cops next probe ${v_{L}}$ and $w_3$, use this information to determine whether or not the robber lies between $w_2$ and $w_3$, and proceed in this manner until they either determine that the robber occupies $C_1$ (at which point they proceed as explained earlier) or exhaust all neighbors of ${v_{L}}$ in $B_1$. In the latter case, they repeat the process in $B_2$, then $B_3$, and so forth. Since the cops probe ${v_{L}}$ on every turn, the robber cannot move between the $C_i$, so eventually the cops determine which $C_i$ contains the robber, at which point they enlarge the cop territory and proceed into a new block.\ [**Case 2**]{}: ${d_L}= 1, {d_R}= 1,$ or both.\ If both ${d_L}$ and ${d_R}$ are 1, then the robber’s position is uniquely determined, since ${v_{L}}$ and ${v_R}$ can have at most one common neighbor outside the cop territory. Thus, suppose that ${d_L}= 1$ but ${d_R}> 1$; a symmetric argument suffices when ${d_R}= 1$ and ${d_L}> 1$. Note that since ${d_R}> {d_L}$, the robber cannot occupy $G_{{v_R}}$; if any vertices of $G_{{v_R}}$ do not yet belong to the cop territory, then the cops add them. We consider two cases. (Refer to Figure \[fig:outerplanar\_case2\].) - Suppose ${v_{L}}$ is adjacent to $v_{r+1}$. Since ${d_R}> 1$, the robber cannot enter ${v_R}$ on his ensuing turn. The cops now add $v_{r+1}$ to the cop territory and take ${v_{L}}$ and $v_{r+1}$ as the new endpoints. Due to the presence of edge ${v_{L}}v_{r+1}$, there cannot be any edges joining $v_r$ to vertices of $B$ not in the cop territory, so invariant (2) still holds. The cops have successfully enlarged the cop territory. - Suppose ${v_{L}}$ is not adjacent to $v_{r+1}$. Of all the neighbors of ${v_{L}}$ in $B$ that are outside the cop territory, let $v_{s}$ denote the one furthest counterclockwise. On their next turn, the left cop probes ${v_{L}}$ while the right cop probes $v_{s-1}$. The cops now take ${v_{L}}$ and $v_{s-1}$ to be the left and right endpoints of the cop territory, respectively, and add to the cop territory $v_{r+1}, \dots, v_{s-1}$ along with $G_{v_{r+1}}, \dots, G_{v_{s-1}}$. The robber cannot possibly occupy the cop territory: by choice of $s$ and the fact that ${d_L}= 1$, prior to his last move the robber could not have occupied $v_i$ or $G_{v_i}$ for any $i \in \{r+1, \dots, s-1\}$, and he cannot have reached any of these in just one step – except perhaps for $v_{s-1}$, which the cops have just probed. Thus invariant (1) holds; invariants (2) and (3) clearly hold as well. Finally, since ${v_{L}}$ is not adjacent to $v_{r+1}$, we have $s \ge r+2$. Thus $v_{s-1}$ is further clockwise than ${v_R}$, so the cops have enlarged the cop territory.\ ![. Top: Case 2(a). Bottom: Case 2(b). Filled vertices represent the interior of the cop territory; shaded vertices represent the endpoints; unfilled vertices represent the robber territory. Only block $B$ is pictured.[]{data-label="fig:outerplanar_case2"}](fig2a.pdf) ![. Top: Case 2(a). Bottom: Case 2(b). Filled vertices represent the interior of the cop territory; shaded vertices represent the endpoints; unfilled vertices represent the robber territory. Only block $B$ is pictured.[]{data-label="fig:outerplanar_case2"}](fig2b.pdf) [**Case 3**]{}: ${d_L}> 1, {d_R}> 1,$ and exactly one of ${v_{L}}$ and ${v_R}$ lies on a chord of $B$ joining it to a vertex outside the cop territory.\ Suppose that ${v_{L}}$ lies on such a chord while ${v_R}$ does not; the other case is similar. - If all vertices of $G_{{v_R}}$ belong to the cop territory, then on their next turn the cops add $v_{r+1}$ to the cop territory as the new right endpoint. (Note that since ${d_R}> 1$, the robber could not have entered ${v_R}$ on his last turn, so he cannot be in the cop territory.) - If part of $G_{{v_R}}$ does not belong to the cop territory and ${d_R}\ge {d_L}$, then the robber cannot occupy $G_{{v_R}}$, so the cops may safely add all vertices of $G_{{v_R}}$ to the cop territory. - Suppose part of $G_{{v_R}}$ does not belong to the cop territory and ${d_R}< {d_L}$. If any vertices of $G_{{v_{L}}}$ do not yet belong to the cop territory, then the cops add them now. Out of all neighbors of ${v_{L}}$ in $B$ that do not belong to the cop territory, let $v_s$ be the one furthest counterclockwise. We claim that for all $i \in \{\ell-1, \ell-2, \dots, s\}$, the robber cannot have occupied either $v_i$ or $G_{v_i}$ immediately after the cops’ probe. To see this, note that the shortest path from ${v_R}$ to any such vertex must pass through ${v_{L}}$ or $v_s$, and ${v_{L}}$ is at least as close to both of these vertices as ${v_R}$. Thus on their next turn the cops may take ${v_R}$ and $v_s$ as the new endpoints of the cop territory and add $v_i$ and all vertices of $G_{v_i}$ for all $i \in \{\ell-1, \ell-2, \dots, s\}$.\ [**Case 4**]{}: ${d_L}> 1, {d_R}> 1,$ and both ${v_{L}}$ and ${v_R}$ lie on chords of $B$ joining them to vertices outside the cop territory.\ Of all vertices of $B$ adjacent to ${v_{L}}$, let $v_{s}$ be the farthest counterclockwise; of all vertices of $B$ adjacent to ${v_R}$, let $v_{t}$ be the farthest clockwise. Let $H_L$ denote the subgraph comprised of ${v_{L}}= v_{\ell}, v_{\ell-1}, \dots, v_{s}$ and $G_{v_{\ell}}, G_{v_{\ell-1}}, \dots, G_{v_{s}}$. Likewise, let $H_R$ denote the subgraph comprised of ${v_R}= v_r, v_{r+1}, \dots, v_{t}$ and $G_{v_{r}}, G_{v_{r+1}}, \dots, G_{v_{t}}$. The cops would like to determine which of these subgraphs (if either) the robber presently inhabits. We consider two subcases. - Suppose first that $v_{s} \not = v_{t}$. If the robber is in $H_L$, then ${d_L}< {d_R}$: any path from ${v_R}$ to a vertex in $H_L$ must pass through either ${v_{L}}$ or $v_s$, and ${v_{L}}$ is closer than ${v_R}$ to both of these. Thus, if ${d_L}\ge {d_R}$, then the robber cannot be in $H_L$, so the cops add all vertices of $H_L$ to the cop territory and take $v_{s}$ and ${v_R}$ as the endpoints. Invariant (1) holds since the robber did not occupy $H_L$ before his last move and could only have entered $H_L$ through $v_{s}$; invariant (2) holds by choice of $v_{s}$. Likewise, if ${d_R}> {d_L}$, then the cops add $H_R$ to the cop territory and take ${v_{L}}$ and $v_t$ as the endpoints. \[fig:outerplanar\_case4\]  - Suppose now that $v_{s} = v_{t}$. This time, if the robber occupies $H_L$, we know only that ${d_L}\le {d_R}$ (and likewise if he occupies $H_R$, then ${d_R}\le {d_L}$). If ${d_L}\not = {d_R}$, then the cops proceed as above. Otherwise, more care is needed. In clockwise order, let ${v_R}= w_1, w_2, \dots, w_k$ be the neighbors of $v_{s}$ in $B$ that are counterclockwise from $v_{s}$. For $i \in \{1, \dots, k-1\}$, let [*sector $i$*]{} refer to the arc of the outer cycle of $B$ from $w_i$ to $w_{i+1}$ (inclusive), together with the subgraphs $G_u$ for all vertices $u$ in this arc. The cops aim to determine which sector (if any) the robber occupies. On their next turn, the cops probe $v_{s}$ and $w_2$; let $d'_L$ and $d'_R$ denote the distances observed. If $d'_L \ge d'_R$, then the robber cannot presently reside in $H_L$: every shortest path from $w_2$ to a vertex in $H_L$ must pass through either ${v_{L}}$ or $v_{s}$, and $v_{s}$ is closer to both of these than $w_2$ is. In this case, as before, the cops may add all vertices of $H_L$ to the cop territory and take $v_{s}$ and ${v_R}$ as the endpoints. Thus we may suppose that $d'_{L} < d'_R$; since $v_{s}$ and $w_2$ are adjacent, we must have $d'_R = d'_{L}+1$. We claim that the robber cannot occupy sector 1. Suppose to the contrary that the robber does occupy some vertex $u$ in sector 1, and note that $u \not = w_2$ (since the cops have just probed $w_2$). Since $d'_R = d'_{L}+1$, some shortest path from $w_2$ to the robber passes through $v_{s}$ and, since the robber is in sector 1, through ${v_R}$ as well. Thus, the distance from ${v_R}$ to $u$ is $d'_{L}-1$; since $u$ is adjacent to the robber’s previous position, ${d_R}\le d({v_R},u)+1 = d'_{L} = d'_R-1$. On the cops’ previous turn (when they probed ${v_{L}}$ and ${v_R}$), we had ${d_L}= {d_R}$, so some shortest path from ${v_R}$ to the robber passed through $v_{s}$; since $u$ is in the interior of sector 1, the robber must have been in sector 1 on the previous turn, so this path must also have passed through $w_2$. Thus, the distance from $w_2$ to the robber on that turn was ${d_R}-2$, so $d'_R = d(w_2,u) \le ({d_R}-2)+1 = {d_R}-1$. We now have $${d_R}\le d'_R-1 \le ({d_R}-1)-1 = {d_R}-2,$$ a contradiction. After the cops probe $v_s$ and $w_2$, and after the robber makes his ensuing move, he still cannot have entered the cop territory: since ${d_L}= {d_R}\ge 2$, he cannot have passed through either ${v_{L}}$ or ${v_R}$. Moreover, before the robber’s most recent move, the cops deduced that he was not in sector 1; hence he cannot have entered the interior of sector 1. The cops now repeat this strategy, but with $w_2$ taking the place of ${v_R}$. In particular, on their next turn, they probe ${v_{L}}$ and $w_2$; let ${d_L}$ and ${d_R}$ be the distances observed. If ${d_L}\not = {d_R}$, then they can add either $H_L$ or $H_R$ to the cop territory, as before. If ${d_L}= {d_R}= 1$, then the robber must occupy $v_{s}$. If ${d_L}= {d_R}\ge 2$, then on their next turn the cops probe $v_{s}$ and $w_3$. Depending on the results of that probe, the cops can either add $H_L$ to the cop territory or deduce that the robber is not in sector 2. (Note that he also cannot be in sector 1: he cannot have traveled through $v_{s}$, and since ${d_R}\ge 2$, he cannot have traveled through $w_2$ either.) Repeating this argument, the cops can eventually add either $H_L$ or $H_R$ to the cop territory and proceed.\ [**Case 5**]{}: ${d_L}> 1, {d_R}> 1,$ and neither ${v_{L}}$ nor ${v_R}$ lie on chords of $B$ joining them to vertices outside the cop territory.\ Suppose first that both $G_{{v_{L}}}$ and $G_{{v_R}}$ contain vertices outside the cop territory. If ${d_L}\ge {d_R}$, then the robber cannot inhabit $G_{{v_{L}}}$, so the cops can add all vertices of $G_{{v_{L}}}$ to the cop territory. Otherwise the robber cannot inhabit $G_{{v_R}}$, so the cops can instead add $G_{{v_R}}$ to the cop territory. (In either case, ${v_{L}}$ and ${v_R}$ remain the endpoints.)\ Finally, suppose that $G_{{v_R}}$ contains no vertices outside the cop territory. (The case where $G_{{v_{L}}}$ contains no vertices outside the cop territory is similar.) Vertex ${v_R}$ has only one neighbor outside the cop territory, namely $v_{r+1}$. The robber cannot have been on $v_{r+1}$ last round (since ${d_R}> 1$), so the cops may add $v_{r+1}$ to the cop territory and take it as the new right endpoint. [Hypercubes]{} We conclude the paper by giving an asymptotically tight upper bound on the localization number of the hypercube. \[thm:hypercube\] For all positive integers $n$, we have that ${\zeta(Q_n)} \le {\left \lceil \log_2 n \right \rceil} + 2$. We represent vertices of $Q_n$ using binary ordered $n$-tuples, where two vertices are adjacent provided that the corresponding $n$-tuples differ in exactly one coordinate. For this proof, it will be convenient to index coordinates starting from $0$; that is, our $n$-tuples have coordinates $0$ through $n-1$ (rather than $1$ through $n$). We show how ${\left \lceil \log_2 n \right \rceil} + 2$ cops can locate a robber on $Q_n$. We distinguish two cops, which we refer to as “cop $C_0$” and “cop $C_1$”, and we refer to the rest of the cops as [*maintenance cops*]{}. The cops will locate the robber over the course of $n$ probes. Intuitively, cops $C_0$ and $C_1$ will be in charge of “learning” one coordinate of the robber’s position in each round, while the maintenance cops will be responsible for “updating” any coordinates that may have changed with the robber’s last move. In the first round of the game, the cops aim to determine coordinate $0$ of the robber’s position. Subsequently, for $k \in \{2, \dots, n\}$, we suppose that just before to the cops’ $k$th probe they know coordinates $0$ through $k-2$ of the robber’s position prior to his most recent move, and with their ensuing probe they aim to determine coordinates $0$ through $k-1$ of his current position. On each cop turn, $C_0$ probes the vertex $(0,0,\dots,0)$. This probe will give the cops some insight into which “direction” the robber is moving. In particular, when the robber’s distance to $C_0$ decreases from one round to the next, the cops know that some coordinate of the robber’s position has changed from $1$ to $0$. Likewise, if the robber’s distance to $C_0$ has increased, then some coordinate of his position has changed from $0$ to $1$, and if the distance to $C_0$ remains unchanged, then the robber hasn’t moved. For $k \in \{1, \dots, n\}$, in the $k$th round of the game, cop $C_1$ probes the vertex for which coordinate $k-1$ is 1 and all other coordinates are 0. The results of this probe, in conjunction with the results of $C_0$’s probe, allow the cops to determine coordinate $k-1$ of the robber’s current position. Finally, we explain the maintenance cops’ strategy. Label these cops $0, \dots, {\left \lceil \log_2 n \right \rceil}-1$. Fix $k \in \{1, \dots, n\}$. Recall that for $k \ge 2$, just before the cops’ $k$th probe, we suppose that the cops know coordinates $0$ through $k-2$ of the robber’s position prior to his last move. With this next probe, the cops aim to determine coordinates $0$ through $k-1$ of the robber’s current position. We have already seen how the probes by $C_0$ and $C_1$ let the cops determine coordinate $k-1$ of the robber’s position; it is the maintenance cops’ job to “update” coordinates $0$ through $k-2$ to reflect the robber’s most recent move. To do this, for each $i \in \{0, \dots, {\left \lceil \log_2 n \right \rceil}-1\}$, maintenance cop $i$ probes the vertex of $Q_n$ in which, for all $j \in \{0, \dots, n-1\}$, coordinate $j$ is 1 if and only if the binary representation of $j$ has a 1 in the “$2^i$” bit. (If $k=1$, then there is no need to update any coordinates of the robber’s position; however, the maintenance cops still probe these vertices, since the results will be needed in the next round of the game.) Now suppose $k\ge 2$ and suppose that on the robber’s last turn, coordinate $j$ of his position changed from a 0 to a 1. (The case where some coordinate changes from 1 to 0 is symmetric, and the probe by $C_0$ allows the cops to distinguish between these cases – as well as to detect the case where the robber remains in place.) Those maintenance cops probing a vertex where coordinate $j$ is 1 see that the robber has moved one step closer to their probes, while the others see that he has moved one step farther away. Thus, for each $i \in \{0, 1, \dots, {\left \lceil \log_2 n \right \rceil} - 1\}$, maintenance cop $i$ can determine whether the binary representation of $j$ has a 0 or a 1 in the $2^i$ bit. Between them, the cops have enough information to determine $j$. Since the cops now know which coordinate of the robber’s position has changed, they can update their information about coordinates 0 through $k-2$ of his position (if indeed $0 \le j \le k-2$); in total, the cops now know coordinates $0$ through $k-1$ of the robber’s position, as desired. After their $n$th probe, the cops know all $n$ coordinates of the robber’s position, and so they have located him. The cop strategy used above can actually be applied to a slightly more general class of graphs. Recall that the [*Cartesian product*]{} of graphs $G$ and $H$, written $G {\, \Box \,}H,$ is the graph with vertex set $V(G) \times V(H)$, where $(u,v)$ is adjacent to $(u',v')$ provided that $u$ is adjacent to $u'$ in $G$ and $v=v'$, or $u = u'$ and $v$ is adjacent to $v'$ in $H$. \[thm:hypercube\_generalization\] If $G = G_0 {\, \Box \,}G_1 {\, \Box \,}\dots {\, \Box \,}G_{n-1}$, where each $G_i$ is a path, then ${\zeta(G)} \le {\left \lceil \log_2 n \right \rceil} + 2$. In lieu of a full proof of Theorem \[thm:hypercube\_generalization\], we explain how the strategy from Theorem \[thm:hypercube\] can be adapted. As before, we represent vertices of $G$ as ordered $n$-tuples, but they need no longer be binary $n$-tuples; instead, for $i \in \{0, \dots, n-1\}$, coordinate $i$ can take on any value from $0$ up to ${\left \vert V(G_i) \right \vert}-1$. To locate a robber on $G$, the cops follow the same strategy as in Theorem \[thm:hypercube\], with one change: for all $i \in \{0, \dots, n-1\}$, whenever a probe would have originally had 1 in coordinate $i$, the probe should instead have ${\left \vert V(G_i) \right \vert}-1$ in that coordinate. (All other coordinates remain unchanged.) As before, in the $k$th round of the game, the cops aim to determine the first $k-1$ coordinates of the robber’s position. It is straightforward to verify the following: - In round $k$, cops $C_0$ and $C_1$ can determine coordinate $k-1$ of the robber’s position. - In each round, $C_0$ can determine whether the robber has incremented some coordinate of his position, decremented some coordinate, or remained in place. - In each round, if the robber has changed his position, then the maintenance cops can determine which coordinate has changed. As in the original strategy, when the robber increments some coordinate $j$ of his position, those maintenance cops whose probe has ${\left \vert V(G_j) \right \vert}-1$ in that coordinate will see that the robber has moved closer to them, while the rest will see that he has moved farther away; collectively, the maintenance cops have enough information to determine $j$. (A similar argument works if the robber decrements some coordinate of his position.)\ Theorems \[thm:degeneracy\_bipartite\] and \[thm:hypercube\] together show that ${\left \lceil \log_2 n \right \rceil} \le {\zeta(Q_n)} \le {\left \lceil \log_2 n \right \rceil} + 2$. It is interesting to note that although the localization number and metric dimension are closely connected, we know ${\zeta(Q_n)}$ up to an additive constant, but we know only that $\mathrm{dim}(Q_n) \sim \frac{2n}{\log_2 n}$ (see [@cm; @er; @lind]). 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--- abstract: 'We show that for all $d\in \{3,\ldots,n-1\}$ the size of the largest component of a random $d$-regular graph on $n$ vertices around the percolation threshold $p=1/(d-1)$ is $\Theta(n^{2/3})$, with high probability. This extends known results for fixed $d\geq 3$ and for $d=n-1$, confirming a prediction of Nachmias and Peres on a question of Benjamini. As a corollary, for the largest component of the percolated random $d$-regular graph, we also determine the diameter and the mixing time of the lazy random walk. In contrast to previous approaches, our proof is based on a simple application of the switching method.' author: - Felix Joos - Guillem Perarnau bibliography: - 'crit\_ref.bib' title: Critical percolation on random regular graphs --- Introduction ============ For every $d\in \{3,\ldots,n-1\}$, let ${\mathcal{G}}_{n,d}$ be the set of all simple and vertex-labelled $d$-regular graphs on $n$ vertices and let $G_{n,d}$ be a graph chosen uniformly at random from ${\mathcal{G}}_{n,d}$. For $p\in[0,1]$, let $G_{n,d,p}$ be a graph obtained from $G_{n,d}$ by retaining each edge independently with probability $p$. The goal of this paper is to study the order of the largest component of $G_{n,d,p}$, denoted by $L_1(G_{n,d,p})$, in terms of $n,d$ and $p$. Most of the literature in the area focuses either on fixed $d\geq 3$ or on $d=n-1$. Goerdt [@goerdt2001giant] showed the existence of a critical probability, $p_{{crit}}:=1/(d-1)$, such that for every fixed $d\geq 3$ and every $\epsilon>0$ the following holds with probability $1-o(1)$: if $p\leq (1-\epsilon)p_{{crit}}$, then $L_1(G_{n,d,p})=O(\log{n})$, while if $p\geq (1+\epsilon)p_{{crit}}$, then $L_1(G_{n,d,p})=\Theta(n)$. Similar results were also obtained in a more general setting by Alon, Benjamini and Stacey [@alon2004percolation]. For $d=n-1$, the random graph $G_{n,d,p}$ corresponds to the classic Erdős-Rényi random graph $G_{n,p}$. In their seminal paper [@erd6s1960evolution], Erdős and Rényi proved that for every $\epsilon>0$, the following holds with probability $1-o(1)$: if $p\leq (1-\epsilon)/n$, then the largest component of $G_{n,p}$ has order $O(\log n)$, if $p= 1/n$ (critical probability), then it has order $\Theta(n^{2/3})$, while if $p\geq (1+ \epsilon)/n$, then it has linear order. Both for fixed $d\geq 3$ and for $d=n-1$, the behaviour around the critical probability has attracted a lot of interest. It is well established that the critical window in $G_{n,p}$ around $p=1/n$ is of order $n^{-1/3}$ (see e.g. [@nachmias2010critical2]). More precise estimates can be found in [@LPW94]. Benjamini posed the problem of determining the width of the critical window in $G_{n,d,p}$ around $p_{{crit}}=1/(d-1)$ (see [@nachmias2010critical; @pittel2008edge]). Nachmias and Peres [@nachmias2010critical] and Pittel [@pittel2008edge], independently showed that the critical window exhibits mean-field behaviour for fixed $d\geq 3$, namely, the following holds with probability $1-o(1)$: for every fixed $\lambda\in{\mathbb{R}}$, if $p=\frac{1+\lambda n^{-1/3}}{d-1}$, then $L_1(G_{n,d,p})=\Theta(n^{2/3})$. See also Riordan [@riordan2012phase] for more precise results on $L_1(G_{n,d,p})$ in the critical window. The case when $d$ is an arbitrary function of $n$ is much less understood. It follows from existing results in the literature[^1] that for every $d\in \{3,\ldots,n-1\}$, the critical probability for the existence of a linear order component in $G_{n,d,p}$ is $1/(d-1)$. Results inside the critical window for given $d$-regular graphs have also been obtained in the context of transitive graphs under the finite triangle condition [@borgs2005random] or under certain expansion conditions [@nachmias2009mean]. Finally, similar results have been obtained for irregular degree sequences whenever the average degree is bounded by a constant [@bollobas2015old; @fountoulakis2007percolation; @fountoulakis2016percolation; @janson2008percolation]. In view that both the sparse regime (fixed $d\geq 3$) and the densest one ($d=n-1$) exhibit similar properties, Nachmias and Peres [@nachmias2010critical] suggested that the mean-field behaviour extends to every $d\in \{3,\ldots,n-1\}$. In this paper we confirm this prediction in the critical window and thus answer the question posed by Benjamini for all $d\in \{3,\ldots,n-1\}$. \[thm:main\] Suppose $\lambda\in \mathbb{R}$ and $d,n\in {\mathbb{N}}$ such that $3\leq d \leq n-1$ and $n$ is sufficiently large. Let $p=\frac{1+\lambda n^{-1/3}}{d-1}$. Then for every sufficiently large $A=A(\lambda)$, we have $${\mathbb{P}}[L_1(G_{n,d,p}) \notin [A^{-1}n^{2/3}, An^{2/3}]]\leq 20A^{-1/2}\;.$$ The upper bound in Theorem \[thm:main\] directly follows from the upper bound for $d$-regular graphs in Proposition 1 in [@nachmias2010critical]. The proof of the lower bound is more intricate and we devote the rest of the paper to it. Most of the previous work on the component structure of $G_{n,d,p}$ uses the configuration model introduced by Bollobás in [@bollobas1980probabilistic]. The configuration model, denoted by $G_{n,d}^*$, is a model of random $d$-regular multigraphs on $n$ vertices. Conditional on $G_{n,d}^*$ being simple, one obtains the uniform distribution on ${\mathcal{G}}_{n,d}$. It is well-known (see for example [@wormald1999models]) that $$\begin{aligned} \label{eq:simple} {\mathbb{P}}[G_{n,d}^* \text{ simple}] = e^{-\Omega({d^2})}\;.\end{aligned}$$ While ${\mathbb{P}}[G_{n,d}^* \text{ simple}]$ is constant for fixed $d\geq 3$, it quickly tends to $0$ if $d$ grows with $n$, and new ideas are needed to study $G_{n,d}$. A standard tool to estimate probabilities for $G_{n,d}$ when $d$ grows with $n$ is the switching method, introduced by McKay in [@mckay1985asymptotics]. For instance, this method has been used to estimate  for $d=o(\sqrt{n})$ [@mckay1991asymptotic] or to determine several combinatorial properties of $G_{n,d}$ when $d$ grows with $n$ [@krivelevich2001random]. The proof of the lower bound in Theorem \[thm:main\] is based on the analysis of an exploration process in $G_{n,d,p}$ using the switching method. [The central quantity that we track through the process is the number of edges between the explored and unexplored parts of the graph, denoted by $X_t$. Our proof relies on sharp estimations of the first and second moments of $X_t$.]{} This approach is inspired by recent developments of the switching method for the study of the component structure of random graphs with a given degree sequence [@fountoulakis2016percolation; @joos2016how]. We take this opportunity to illustrate the use of our method with a simple proof that makes no assumptions on $d$. [**The critical window.** Theorem \[thm:main\] shows that the critical window has width $\Omega(n^{-1/3})$. Proposition 1 in [@nachmias2010critical] implies that, as $\lambda\to -\infty$, the typical order of the largest component is $o(n^{2/3})$. Following analogous ideas as the ones used in the proof of Theorem \[thm:main\], one obtains that, as $\lambda\to \infty$, the typical order of the largest component is $\omega(n^{2/3})$. More precisely, there exist constants $c,C>0$ such that for every $3\leq d\leq n-1$ and $\lambda >0$, if $p=\frac{1+\lambda n^{-1/3}}{d-1}$, then $${\mathbb{P}}\left[L_1(G_{n,d,p}) \leq c\cdot \lambda n^{2/3}\right]\leq C\lambda^{-1}\;.$$ The proof of this statement is simpler than the proof of our main theorem, since the assumption $\lambda>0$ implies that $X_t$ has positive drift. In particular, the first part of the exploration process can be analysed using a first moment argument only and for the entire process it suffices to control the variance of $X_t$ from above. It follows that the width of the critical window is $\Theta(n^{-1/3})$. ]{} [In its current form, our method does not give sharp estimates for $L_1(G_{n,d,p})$ in the barely subcritical and barely supercritical regimes. However, we believe that similar estimates as the ones in Lemma \[lem:exp\] hold in general and may be used to extend the results of Nachmias and Peres in [@nachmias2010critical] to all $d\in \{3,\ldots,n-1\}$.]{} [**Diameter and Mixing Time.** We present a consequence]{} of Theorem \[thm:main\]. For a component ${\mathcal{C}}$, let ${\text{\rm diam}}({\mathcal{C}})$ denote its diameter and let $T_{\text{\rm mix}}({\mathcal{C}})$ denote the mixing time of the lazy random walk on ${\mathcal{C}}$. Theorem 1.2 in [@nachmias2008critical] implies the following corollary. \[cor:diam\] Suppose $\lambda\in {\mathbb{R}}$ and $d,n\in {\mathbb{N}}$ such that $3\leq d \leq n-1$ and $n$ is sufficiently large. Let $p=\frac{1+\lambda n^{-1/3}}{d-1}$. Let $\mathcal{C}$ be the largest component of $G_{n,d,p}$. Then, for every $\epsilon>0$, there exists $A= A(\lambda,\epsilon)$ such that $${\mathbb{P}}[{\text{\rm diam}}(\mathcal{C}) \notin [A^{-1}n^{1/3}, An^{1/3}]]< \epsilon\;.$$ and $${\mathbb{P}}[T_{{\text{\rm mix}}}(\mathcal{C}) \notin [A^{-1}n, An]]< \epsilon\;.$$ [**Organisation of the paper.**]{} The paper is organized as follows. In Section \[sec:explo\], we describe our exploration process of $G_{n,d,p}$ and introduce different quantities we will track during the process. In Section \[sec:swi\], we present our main combinatorial tool (switching method) and prove two technical lemmas. In Section \[sec:analy\], we use these lemmas to study a single step of the exploration process. Finally, in Section \[sec:proof\], we conclude with the proof of the lower bound in Theorem \[thm:main\]. The exploration process {#sec:explo} ======================= Before describing the exploration process, we briefly introduce some notation. For a graph $G$, a subset of vertices $X$ of $G$, and a vertex $u$ of $G$, we write $d_G(u)$ for the number of neighbours of $u$ in $G$ and $d_{G,X}(u)$ for the number of neighbours of $u$ in $G$ that belong to $X$. We also write $\Delta(G)$ for the maximum degree of $G$. Finally, for $p\in [0,1]$, we write $G_p$ for the graph where each edge in $G$ is independently retained with probability $p$. We will use an exploration process to reveal the component structure of $G_{n,d,p}$. Let us denote the vertex set by $V$, which we equip with a linear order (from now on $V$ is always a vertex set of size $n$). For technical reasons, we perform our exploration process not on $G_{n,d,p}$, but on what we call an input. An *input* is a tuple $(G,{\mathfrak{S}})$, where $G\in \mathcal{G}_{n,d}$ and ${\mathfrak{S}}=\{\sigma_v\}_{v\in V}$ is a collection of $n$ permutations of length $d$. For each vertex of $G$, arbitrarily label the edges incident to it with distinct elements from $\{1,\ldots, d\}$. Thus every edge receives two labels. In fact, we may think about this as a labelling of the semi-edges of $G$. Let ${\mathcal{I}}$ be the set of all inputs $(G,{\mathfrak{S}})$ where $G\in {\mathcal{G}}_{n,d}$ and ${\mathfrak{S}}$ is a collection of $n$ permutations of length $d$. Observe that every graph in $G\in {\mathcal{G}}_{n,d}$ gives rise to exactly $(d!)^n$ inputs. Thus, choosing an input uniformly at random from ${\mathcal{I}}$ and ignoring the edge-labels is equivalent to choosing $G_{n,d}$. Let ${\mathfrak{S}}_{n,d}$ be a collection of $n$ permutations of length $d$ each chosen independently and uniformly at random. Hence, if an input is chosen uniformly at random from ${\mathcal{I}}$, then this input is distributed as $(G_{n,d},{\mathfrak{S}}_{n,d})$. Next, we describe our exploration process on an input $(G,{\mathfrak{S}})$. First, for every $uv\in E(G)$, we denote by $I(uv)$ the indicator random variable that is $1$ if $uv$ belongs to $G_p$ (it percolates) and $0$ otherwise. If $I(uv)$ is revealed, we say that the edge $uv$ has been exposed. For each integer $t\geq 0$, the set $S_t$ consists of the vertices explored up to time $t$ (with $S_0=\emptyset$); the bipartite graph $F_t$, with bipartition $(S_t,V{\setminus}S_t)$, consists of all edges in $G$ between $S_t$ and $V {\setminus}S_t$ that have been exposed and have failed to percolate; and the graph $H_t$, with vertex set $S_t$, consists of all edges in $G$ within $S_t$, that is, $H_{t}:=G[S_{t}]$. Let ${\mathcal{H}}_t$ be the history of all random choices we make until time $t$ (which we will treat as an event). We now describe how to obtain ${\mathcal{H}}_{t+1}$, given ${\mathcal{H}}_{t}$. Suppose there exists at least one vertex $u\in S_t$ such that $d_{H_t}(u)+d_{F_t}(u)<d$. Among all such vertices $u$, let $v_{t+1}$ be the vertex which comes first in the linear order of $V$. Let $w_{t+1}$ be the vertex $w\in V {\setminus}{S_t}$ with $v_{t+1}w\in E(G){\setminus}E(F_t)$ that minimizes $\sigma_{v_{t+1}}(\ell(w))$, where $\ell(w)$ is the label of the semi-edge incident to $v_{t+1}$ that corresponds to $v_{t+1}w$. Thereafter, we expose $v_{t+1}w_{t+1}$. If $I(v_{t+1}w_{t+1})=0$, then we set $S_{t+1}:=S_t$, [$Y_{t+1}:=0$, $Z_{t+1}:=0$]{} and we let $F_{t+1}$ be the graph obtained from $F_t$ by adding $v_{t+1}w_{t+1}$. If $I(v_{t+1}w_{t+1})=1$, then we set $$\begin{aligned} S_{t+1}:=S_t\cup \{w_{t+1}\},\enspace Y_{t+1}:=d_{F_t}(w_{t+1}) ,\enspace Z_{t+1}:=d_{G,S_t}(w_{t+1})-Y_{t+1}-1,\enspace \end{aligned}$$ and we let $F_{t+1}$ be the graph obtained from $F_t$ by deleting all edges incident to $w_{t+1}$ and moving $w_{t+1}$ to the other side of the bipartition. [Since $H_{t+1}=G[S_{t+1}]$, we also reveal all the edges between $w_{t+1}$ and $S_t$.]{} Observe that $Z_{t+1}$ counts the number of neighbours of $w_{t+1}$ in $S_t{\setminus}\{v_{t+1}\}$ whose corresponding edge has not yet been exposed. If $d_{H_t}(u)+d_{F_t}(u)=d$ for all $u\in S_t$, that is, every edge incident to a vertex in $S_t$ has been exposed, then we pick a vertex $x\in V{\setminus}S_t$ that minimises $d_{F_t}(x)$ and set $w_{t+1}:=x$, $S_{t+1}:=S_t\cup \{w_{t+1}\}$, [$Y_{t+1}:=d_{F_t}(w_{t+1})$, $Z_{t+1}:=0$]{} and we let $F_{t+1}$ be the graph obtained from $F_t$ by deleting all edges incident to $w_{t+1}$ and by moving $w_{t+1}$ to the other side of the bipartition. Observe that, in any of the above-mentioned cases, $|E(F_{t+1})|\leq |E(F_{t})|+1$ and hence $|E(F_{t})|\leq t$. A crucial parameter of our exploration process is the number of edges between $S_t$ and $V{\setminus}S_t$ which have not yet been exposed: $$\begin{aligned} X_t:=\sum_{u\in S_t}(d -d_{H_t}(u)-d_{F_t}(u))\;.\end{aligned}$$ For the sake of simplicity, we define $\eta_{t+1}:=X_{t+1}-X_t$. If $X_t>0$, then $$\begin{aligned} \label{eq:change} \eta_{t+1}&= -(1-I(v_{t+1}w_{t+1}))+I(v_{t+1}w_{t+1})(d-2- Y_{t+1} - 2Z_{t+1}) \;,\end{aligned}$$ and if $X_t=0$, then $$\begin{aligned} \label{eq:change2} \eta_{t+1}&= d-Y_{t+1}\;.\end{aligned}$$ [Note that $Y_{t+1}$ and $Z_{t+1}$ are measurable random variables given ${\mathcal{H}}_t$ and thus $\eta_{t+1}$ is a predictable sequence with respect to ${\mathcal{H}}_t$.]{} The switching method and some applications {#sec:swi} ========================================== In this section we explain the switching method and we present two simple applications. In Lemma \[lem:UB\_prob\] we use the switching method to bound the probability from above that two vertices are adjacent. In Lemma \[lem: back edges\] we provide an upper bound on the expectation of the number of neighbours of a vertex in a specified set of vertices. Let $G$ be a graph and let $x_1,x_2,x_3,x_4$ be distinct vertices of $G$. Suppose $x_1x_2,x_3x_4\in E(G)$ and $x_1x_4,x_2x_3\notin E(G)$. A *switching* on the $4$-cycle $x_1x_2x_3x_4$ transforms $G$ into a graph $G'$ by deleting $x_1x_2,x_3x_4$ and adding $x_1x_4,x_2x_3$. Observe that the degree sequence of $G$ is preserved by the switching. In particular, if $G$ is $d$-regular, then so is $G'$. Moreover, the switching operation is reversible: if $G$ can be transformed into $G'$ by a switching, then $G$ can be also obtained from $G'$ by a switching on the same $4$-cycle. Finally, there is a natural way to extend the notion of a switching from graphs to inputs by simply preserving the labels on each semi-edge. Switchings can be used to obtain bounds on the probability that $G_{n,d}$ satisfies a certain property. Suppose ${\mathcal{A}},{\mathcal{B}}$ are disjoint subsets of ${\mathcal{G}}_{n,d}$. Suppose that for every graph $G\in {\mathcal{A}}$, there are at least $a$ switchings that transform $G$ into a graph in ${\mathcal{B}}$ and for every graph $G'\in {\mathcal{B}}$, there are at most $b$ switchings that transform $G'$ into a graph in ${\mathcal{A}}$. By double-counting the number of switchings between ${\mathcal{A}}$ and ${\mathcal{B}}$, we obtain $a|{\mathcal{A}}| \leq b |{\mathcal{B}}|$. Thus $a{\mathbb{P}}[{\mathcal{A}}]\leq b{\mathbb{P}}[{\mathcal{B}}]$, where we define ${\mathbb{P}}[{\mathcal{S}}]:=|{\mathcal{S}}|/{|\mathcal{G}_{n,d}|}$ for every ${\mathcal{S}}\subseteq {\mathcal{G}}_{n,d}$. \[lem:UB\_prob\] Suppose $d,n\in {\mathbb{N}}$ such that $3\leq d \leq n/4$ and $S\subseteq V$ such that $|S|\leq n/6$. Let $H$ be a graph with vertex set $S$ and let $F$ be a bipartite graph with vertex partition $(S,V{\setminus}S)$ with $\Delta(F\cup H)\leq d$. Let $u\in S$ and $v\in V{\setminus}S$ such that $uv\notin E(F)$. Then $$\begin{aligned} {\mathbb{P}}[uv\in E(G_{n,d})\mid G_{n,d}[S]=H,\, F\subseteq G_{n,d}]\leq \frac{6(d-d_H(u)-d_F(u))}{n}\;.\end{aligned}$$ Let ${\mathcal{F}}^+$ be the set of graphs $G\in {\mathcal{G}}_{n,d}$ such that $G[S]=H$, $F\subseteq G$ and $uv\in E(G)$, and let ${\mathcal{F}}^-$ be the set of graphs $G\in {\mathcal{G}}_{n,d}$ such that $G[S]=H$, $F\subseteq G$ but $uv\notin E(G)$. We will only perform switchings that involve edges and non-edges that are not contained in $E(H)\cup E(F)$. This ensures that the graph $G'$ obtained from a switching also satisfies $G'[S]=H$ and $F\subseteq G'$. Suppose $G\in {\mathcal{F}}^+$. In order to bound the number of switchings from below it suffices to switch on a cycle $uvxy$ that satisfies $xy\in E(G)$, $uy,vx\notin E(G)$, and $x,y\in V{\setminus}S$. There are at least $dn-2d|S|$ ordered edges $xy$ with both endpoints in $V{\setminus}S$. There are at most $d^2$ edges $xy$ such that $x$ is at distance at most $1$ from $v$ and at most $d^2$ edges $xy$ such that $y$ is at distance at most $1$ from $u$. Thus, there are at least $dn-2d|S|-2d^2 \geq dn/6$ switchings that transform $G$ into a graph in ${\mathcal{F}}^-$. Suppose now $G\in {\mathcal{F}}^-$. Then there are clearly at most $d\cdot(d-d_H(u)-d_F(u))$ switchings that transform $G$ into a graph in ${\mathcal{F}}^+$. It follows that $$\begin{aligned} {\mathbb{P}}[uv\in E(G_{n,d}) &\mid G_{n,d}[S]=H,\, F\subseteq G_{n,d}]\\ &\leq \frac{d(d-d_H(u)-d_F(u))}{dn/6}\cdot {\mathbb{P}}[uv\notin E(G_{n,d})\mid G_{n,d}[S]=H,\, F\subseteq G_{n,d}]\\ &\leq \frac{6(d-d_H(u)-d_F(u))}{n}\;. \qedhere\end{aligned}$$ \[lem: back edges\] Suppose $d,n\in {\mathbb{N}}$ such that $3\leq d \leq n/4$ and $S\subseteq V$ such that $|S|\leq n/6$. Let $H$ be a graph with vertex set $S$ and let $F$ be a bipartite graph with vertex partition $(S,V{\setminus}S)$ with $\Delta(F\cup H)\leq d$. Let $v\in V{\setminus}S$. Then $$\begin{aligned} {\mathbb{E}}[d_{G,S}(v)-d_{F}(v) \mid G_{n,d}[S]=H,\; F\subseteq G_{n,d}]\leq 6d|S|/n.\end{aligned}$$ For every $k\geq 0$, let ${\mathcal{F}}_k$ be the set of graphs $G\in {\mathcal{G}}_{n,d}$ such that $G[S]=H$, $F\subseteq G$, and $d_{G,S}(v)-d_{F}(v)=k$. As in Lemma \[lem:UB\_prob\], we will only perform switchings using edges and non-edges that are not contained in $E(H)\cup E(F)$. Consider a graph in ${\mathcal{F}}_k$. There are at most $(d-d_{F}(v))\cdot d|S|\leq d^2|S|$ switchings that lead to a graph in ${\mathcal{F}}_{k+1}$. For every graph in ${\mathcal{F}}_{k+1}$, we can use a switching on a cycle $uvxy$ that satisfies $uv,xy\in E(G){\setminus}E(F)$, $uy,vx\notin E(G)$ and $u\in S$, and $v,x,y\in V{\setminus}S$. There are $k+1$ choices for $uv$ and, for any particular choice of $uv$, there are at least $dn-2d|S|-2d^2\geq dn/6$ choices for the (ordered) edge $xy$. Hence, there are at least $(k+1)dn/6$ switchings that lead to a graph in ${\mathcal{F}}_k$. Thus, for every $k\geq 0$, we obtain $$\begin{aligned} \label{eq:bound} {\mathbb{P}}[{\mathcal{F}}_{k+1}] \leq \frac{6d|S|/n}{(k+1)}\cdot {\mathbb{P}}[{\mathcal{F}}_{k}]\;.\end{aligned}$$ Let $X$ be a Poisson distributed random variable with mean $6d|S|/n$. Lemma 3.4 in [@Mrandom2012] together with  implies that for every $m\geq 0$ $${\mathbb{P}}[d_{G,S}(v)-d_{F}(v)\geq m \mid G_{n,d}[S]=H,\; F\subseteq G_{n,d}] \leq {\mathbb{P}}[X\geq m]\;,$$ which implies the statement of the lemma. Analysis of the exploration process {#sec:analy} =================================== In this section we show how to control the expectation of $\eta_t$ and $\eta_t^2$. We first use Lemmas \[lem:UB\_prob\] and \[lem: back edges\] to bound the expectation of $Y_{t+1}$ and $Z_{t+1}$ from above. \[lem:exp2\] Suppose $d,n\in {\mathbb{N}}$ such that $3\leq d\leq n-1$ and $n$ is sufficiently large. Fix $p\in [0,1]$. Consider the exploration process described above on $(G_{n,d},{\mathfrak{S}}_{n,d})$ with percolation probability $p$ and suppose $t\leq d n^{2/3}$. Conditional on ${\mathcal{H}}_t$ satisfying $|S_t|\leq 5n^{2/3}$, we have $$\begin{aligned} {\mathbb{E}}[Y_{t+1}|{\mathcal{H}}_t]\leq 20dn^{-1/3} \text{\enspace and \enspace } {\mathbb{E}}[Z_{t+1}|{\mathcal{H}}_t]\leq 180dn^{-1/3}\;.\end{aligned}$$ If ${\mathcal{H}}_t$ satisfies $X_t=0$, then $Y_{t+1}\leq t/(n-|S_t|)\leq 2dn^{-1/3}$ by our choice of $w_{t+1}$ (we always choose the vertex $x$ that minimises $d_{F_t}(x)$) and $|E(F_t)|\leq t$. Note that $Z_{t+1}=0$ by definition. Hence we may assume from now on that $X_t>0$. Note that if $d\geq n/4$, then the lemma follows directly from the fact that $Y_{t+1}\leq |S_t|\leq 5n^{2/3} \leq 20dn^{-1/3}$, and similarly for $Z_{t+1}$. Thus, in the following we assume that $d\leq n/4$. Given $w\in V{\setminus}S_t$ such that $v_{t+1}w\notin E(F_t)$, we apply Lemma \[lem:UB\_prob\] with $S=S_t$, $F=F_t$, $H=H_t$, $u=v_{t+1}$ and $v=w$ to obtain $${\mathbb{P}}[v_{t+1}w\in E(G_{n,d}) \mid v_{t+1}w\notin E(F_t),{\mathcal{H}}_t]\leq \frac{6(d-d_{H_t}(v_{t+1})-d_{F_t}(v_{t+1}))}{n}\;.$$ Observe that we run our exploration process on inputs. In order to apply Lemma \[lem:UB\_prob\], we fix the semi-edge labelings and perform switchings on the graphs. Since $\sigma_{v_{t+1}}$ is a random permutation, each edge incident to $v_{t+1}$ that is not contained in $E(F_t)\cup E(H_t)$ is chosen with the same probability to continue the exploration process. Hence, given that $v_{t+1}w\in E(G_{n,d}){\setminus}E(F_t)$, the probability that $w_{t+1}=w$ is precisely $(d-d_{H_t}(v_{t+1})-d_{F_t}(v_{t+1}))^{-1}$. Therefore, $$\begin{aligned} &\quad\,\, {\mathbb{P}}[w_{t+1}=w\mid v_{t+1}w\notin E(F_t),{\mathcal{H}}_t]\nonumber \\ &={\mathbb{P}}[w_{t+1}=w\mid v_{t+1}w\in E(G_{n,d}){\setminus}E(F_t),{\mathcal{H}}_t]\cdot{\mathbb{P}}[v_{t+1}w\in E(G_{n,d}) \mid v_{t+1}w\notin E(F_t),{\mathcal{H}}_t]\nonumber \leq \frac{6}{n}\;.\end{aligned}$$ Since ${\mathbb{P}}[w_{t+1}=w\mid v_{t+1}w\in E(F_t) , {\mathcal{H}}_t]=0$, it follows that for every $w\in V{\setminus}S_t$ $$\begin{aligned} \label{eq:prob} {\mathbb{P}}[w_{t+1}=w\mid {\mathcal{H}}_t]&\leq \frac{6}{n}\;.\end{aligned}$$ Using that $|E(F_t)|\leq t$, we conclude $$\begin{aligned} {\mathbb{E}}[Y_{t+1}|{\mathcal{H}}_t] &= \sum_{w\in V{\setminus}S_t} d_{F_t}(w){\mathbb{P}}[w_{t+1}=w|{\mathcal{H}}_t] \stackrel{\eqref{eq:prob}}{\leq} \frac{6}{n}\sum_{w\in V{\setminus}S_t} d_{F_t}(w) \leq \frac{6}{n}\cdot t\leq 6d n^{-1/3}\;.\end{aligned}$$ We now prove the second statement. Given $w\in V{\setminus}S_t$ with ${\mathbb{P}}[w_{t+1}=w\mid {\mathcal{H}}_t]>0$ (that is, $v_{t+1}w\notin E(F_t)$), we apply Lemma \[lem: back edges\] with $S=S_t$, $F$ obtained from $F_t$ by adding $v_{t+1}w$, $H=H_t$, and $v=w$, to obtain $$\begin{aligned} {\mathbb{E}}[Z_{t+1}|{\mathcal{H}}_t] &= \sum_{w\in V{\setminus}S_t} {\mathbb{E}}[Z_{t+1}|w_{t+1}=w,v_{t+1}w\notin E(F_t),{\mathcal{H}}_t]{\mathbb{P}}[w_{t+1}=w\mid v_{t+1}w\notin E(F_t), {\mathcal{H}}_t] \\ &\stackrel{\eqref{eq:prob}}{\leq} \sum_{w\in V{\setminus}S_t} \frac{6d|S_t|}{n} \cdot \frac{6}{n} \leq 180d n^{-1/3}\;.\qedhere\end{aligned}$$ \[lem:exp\] Suppose $\mu\geq 0$ and $d,n\in {\mathbb{N}}$ such that $3 \leq d\leq n-1$ and $n$ is sufficiently large. Consider the exploration process described above on $(G_{n,d},{\mathfrak{S}}_{n,d})$ with $p=\frac{1-\mu n^{-1/3}}{d-1}$ and suppose $t\leq d n^{2/3}$. Conditional on $|S_t|\leq 5n^{2/3}$, then $$\begin{aligned} {\mathbb{E}}[\eta_{t+1}|{\mathcal{H}}_t]\geq -(570+\mu)n^{-1/3} \enspace \text{ and }\enspace {\mathbb{E}}[\eta_{t+1}^2|{\mathcal{H}}_t] \geq d/4\;.\end{aligned}$$ Moreover, if $X_t>0$, then ${\mathbb{E}}[\eta_{t+1}^2|{\mathcal{H}}_t] \leq d$. First assume that $X_t>0$. Recall that for any ${\mathcal{H}}_t$ [and for any edge $uv$ that has not been exposed yet]{}, we have ${\mathbb{E}}[I(uv)\mid {\mathcal{H}}_t]={p=}(1-\mu n^{-1/3})/(d-1)$. [Recall that $Y_{t+1}$ and $Z_{t+1}$ are measurable with respect to ${\mathcal{H}}_t$.]{} Taking conditional expectations on  and using Lemma \[lem:exp2\], we obtain $$\begin{aligned} {\mathbb{E}}[\eta_{t+1}|{\mathcal{H}}_t] &=-\left(1-\frac{1-\mu n^{-1/3}}{d-1}\right) + \frac{1-\mu n^{-1/3}}{d-1}(d-2 -{\mathbb{E}}[Y_{t+1}|{\mathcal{H}}_t]-2{\mathbb{E}}[Z_{t+1}|{\mathcal{H}}_t])\\ &\geq -\frac{{\mathbb{E}}[Y_{t+1}|{\mathcal{H}}_t]+2{\mathbb{E}}[Z_{t+1}|{\mathcal{H}}_t]}{d-1} -\mu n^{-1/3}\\ &\geq -\frac{380 dn^{-1/3}}{d-1} -\mu n^{-1/3} \geq -(570+\mu)n^{-1/3}\;,\end{aligned}$$ since $d\geq 3$. Again, by Lemma \[lem:exp2\] and , we obtain $$\begin{aligned} {\mathbb{E}}[\eta_{t+1}^2|{\mathcal{H}}_t] &=\left(1-\frac{1-\mu n^{-1/3}}{d-1}\right)(-1)^2+ \frac{1-\mu n^{-1/3}}{d-1}{\mathbb{E}}[(d-2 -Y_{t+1}-2Z_{t+1})^2\mid{\mathcal{H}}_t]\\ &\geq \frac{d-2}{d-1}+ \frac{(1-\mu n^{-1/3})(d-2)^2}{d-1} -\frac{2(d-2)({\mathbb{E}}[Y_{t+1}|{\mathcal{H}}_t]+2{\mathbb{E}}[Z_{t+1}|{\mathcal{H}}_t])}{d-1}\\ &\geq (1-\mu n^{-1/3})(d-2) -2({\mathbb{E}}[Y_{t+1}|{\mathcal{H}}_t]+2{\mathbb{E}}[Z_{t+1}|{\mathcal{H}}_t])\\ &\geq (1-\mu n^{-1/3})(d-2) -760dn^{-1/3}\\ &\geq d/4\;,\end{aligned}$$ where the last inequality holds since $d\geq 3$ and $n$ is sufficiently large. Observe that ${\mathbb{E}}[\eta_{t+1}^2|{\mathcal{H}}_t]\leq d$ follows from a similar argument as $(d-2 -Y_{t+1}-2Z_{t+1})^2\leq (d-2){^2}$. If $X_t=0$, then clearly ${\mathbb{E}}[\eta_{t+1}|{\mathcal{H}}_t]\geq 0$ and, since ${\mathbb{E}}[\eta_{t+1}^2|{\mathcal{H}}_t]={\mathbb{E}}[(d-Y_{t+1})^2|{\mathcal{H}}_t]$, similarly as before, one can prove that ${\mathbb{E}}[\eta_{t+1}^2|{\mathcal{H}}_t]\geq d/4$. \[lem:S\_t\] Suppose $\mu\geq 0$ and $d,n\in {\mathbb{N}}$ such that $3\leq d\leq n-1$ and $n$ is sufficiently large. Consider the exploration process described above on $(G_{n,d},{\mathfrak{S}}_{n,d})$ with $p=\frac{1-\mu n^{-1/3}}{d-1}$. Then, for every fixed $\delta>0$ and all $0\leq t_1\leq t_2 \leq 5dn^{2/3}$, we have $$\begin{aligned} &{\mathbb{P}}\left[ |S_{t_2}{\setminus}S_{t_1}| - \frac{t_2-t_1}{d-1} \geq -\delta n^{2/3} \right]= {1-o(n^{-2})}\quad\text{ and}\\ &{\mathbb{P}}\left[ |S_{t_2}{\setminus}S_{t_1}| - \frac{t_2-t_1}{d-1} - \left\lceil\frac{{t_2}}{5d/6}\right\rceil\leq \delta n^{2/3} \right]= {1-o(n^{-2})}\;.\end{aligned}$$ We add a vertex to $S_t$ either if $I(v_{t+1}w_{t+1})=1$ or if we start exploring a new component of $G_{n,d,p}$ at time $t+1$. Thus, [$|S_{t_2}{\setminus}S_{t_1}|$]{} stochastically dominates a binomial random variable with parameters [$t_2-t_1$]{} and $(1-\mu n^{-1/3})/(d-1)$. A standard application of Chernoff’s inequality implies the first statement. Let $A_t\subseteq S_t$ be the set of vertices that start a new component in $G_{n,d,p}$. [For every $0\leq t\leq 5dn^{2/3}$, let $a_t:=|A_t|$, let $c_t:=|S_t{\setminus}A_t|$ and let $b_t:=|S_t{\setminus}(S_{t_1}\cup A_t)|$]{}. Observe that [$c_t$]{} is stochastically dominated by a binomial random variable with parameters $t$ and $1/(d-1)$. Using Chernoff’s inequality, [we have $ c_t\leq 8 n^{2/3}$]{} with probability [$1-o(n^{-2})$]{} for any $t\leq 5dn^{2/3}$. We claim that for every [$0\leq t\leq 5dn^{2/3}$ and conditional on $c_t\leq 8n^{2/3}$]{}, we have $a_t\leq \lceil\frac{t}{5d/6}\rceil$. Indeed, the claim is true for $t\in \{0,1\}$. Assume that $t\geq 2$ and that the claim holds for every $t'\in \{0,\ldots,t-1\}$. If $X_{t-1}>0$, then $a_t=a_{t-1}$ and we are done. Thus, assume that $X_{t-1}=0$. Let $s$ be the largest integer $s'\in \{0,\ldots,t-2\}$ such that $X_{s'}=0$ (it exists since $X_0=0$ and $t\geq 2$). Recall that $w_{s+1}$ is a vertex $x\in V{\setminus}S_s$ that minimises $d_{F_s}(x)$. It follows that $$d_{F_{s}}(w_{s+1})\leq \frac{|E(F_{s})|}{n-(a_s+{c_s})}\leq \frac{s}{n- \lceil s/(5d/6)\rceil- 8n^{2/3}}\leq \frac{d}{6}\;,$$ provided that $n$ is large enough. Hence, $X_{s+1}\geq 5d/6$ and the process will not start a new component for the next $5d/6$ steps. In particular, $s+ 5d/6\leq t$. This implies $a_{t}= a_s+1\leq \lceil\frac{s}{5d/6}\rceil+1\leq \lceil\frac{t}{5d/6}\rceil$. [ Since $|S_{t_2}{\setminus}S_{t_1}|\leq a_{t_2}+b_{t_2}$, the second part of the lemma now follows from the upper bound on $a_{t_2}$ (which holds as we assume $c_t\leq 8n^{2/3}$) and an upper bound on $b_{t_2}$ obtained by Chernoff’s inequality.]{} Proof of Theorem \[thm:main\] {#sec:proof} ============================= As we mentioned in the introduction, due to the result of Nachmias and Peres, we only need to prove a lower bound. Since it suffices to prove the lower bound of the statement for $\lambda\leq 0$, we use the definition $\mu:=-\lambda$. We now present a brief overview of the proof. In the first phase, we show that with probability at least $1-A^{-1/2}$, the process $X_t$ exceeds $A^{-1/4} dn^{1/3}$ in the first $dn^{2/3}$ steps. In the second phase and conditional on the success of the first phase, we show that $X_t$ stays positive for at least $2A^{-1}dn^{2/3}$ steps with probability at least $1-A^{-1/2}$. From standard concentration inequalities, this gives the existence of a component of order at least $A^{-1}n^{2/3}$, concluding the proof. This proof strategy was introduced by Nachmias and Peres to prove the same statement for fixed $d\geq 3$ [@nachmias2010critical] and for $d=n-1$ [@nachmias2010critical2]. We remark that, in comparison to [@nachmias2010critical], our analysis of the exploration process is simpler, as we do not need to track the number of vertices $x\in V{\setminus}S_t$ which satisfy $d_{F_t}(x)=k$ for $k\in \{0,1,\ldots,d\}$. If $d\geq 3$ is fixed, as in [@nachmias2010critical], almost every vertex $x$ satisfies $d_{F_t}(x)\in \{0,1\}$. However, this is no longer true if $d$ is an arbitrary function of $n$. We avoid the technicalities involved with this issue by averaging over the values of $d_{F_t}(x)$. **First phase:** We start with the definition of a few parameters. Let $h:=A^{-1/4} dn^{1/3}$, $T_1:= 5dn^{2/3}/6$ and $T_2:=2A^{-1} d n^{2/3}$. In addition, we define the following stopping times: $$\begin{aligned} \tau_h&:=\min\{t:\, X_t\geq h\}\wedge T_1\\ \tau_S^1&:=\min\{t:\, |S_t|\geq 3n^{2/3}\}\\ \tau_1&:= \tau_h\wedge \tau_S^1\;.\end{aligned}$$ Recall that $X_{t+1}=\eta_{t+1}+X_t$. Note also that for every $t< \tau_1$, we have $X_t\leq h$ and $|S_t|\leq 5n^{2/3}$. Hence, Lemma \[lem:exp\] implies that $$\begin{aligned} {\mathbb{E}}[X_{t+1}^2-X_t^2|{\mathcal{H}}_t] &\geq {\mathbb{E}}[\eta_{t+1}^2|{\mathcal{H}}_t] +2{\mathbb{E}}[\eta_{t+1} X_{t}|{\mathcal{H}}_t] \geq d/4- 2\cdot(570+\mu) n^{-1/3} h\geq {d/5}\;,\end{aligned}$$ provided that $A$ is large enough with respect to $\mu$ (and thus, with respect to $\lambda$). Hence $X_{t\wedge \tau_1}^2 - (t\wedge \tau_1)d/5$ is a submartingale. By the Optional Stopping theorem for submartingales (see for example [@GrimStirz] p.491), ${\mathbb{E}}[X_{\tau_1}^2 - \frac{d}{5}\tau_1]\geq {\mathbb{E}}[X_{0}^2] = 0$, which implies that ${\mathbb{E}}[\tau_1]\leq \frac{5}{d}{\mathbb{E}}[X_{\tau_1}^2]$. Since $X^2_{\tau_1}\leq (h+d)^2\leq 2 h^2$, we obtain $${\mathbb{P}}[\tau_1=T_1]\leq \frac{{\mathbb{E}}[\tau_1]}{T_1} \leq \frac{5{\mathbb{E}}[X_{\tau_1}^2]}{d T_1}\leq\frac{10 h^2}{dT_1} = 12A^{-1/2}\;.$$ By Lemma \[lem:S\_t\] [with $t_1=0$ and $t_2=T_1$]{}, we have ${\mathbb{P}}[\tau_S^1\leq T_1]=o(1)$. Thus $$\begin{aligned} \label{eq:T1} {\mathbb{P}}[\{\tau_h=T_1\} \cup \{\tau_S^1\leq \tau_h\}] \leq {\mathbb{P}}[\tau_1=T_1]+{\mathbb{P}}[\tau_S^1\leq T_1] \leq 12A^{-1/2}+o(1) \leq 13A^{-1/2}\;.\end{aligned}$$ We conclude that [the event]{} ${\mathcal{E}:=}\{\tau_h<T_1,\tau_h< \tau^1_S\}$ holds with probability at least $1-13A^{-1/2}$. In particular, with probability at least $1-13A^{-1/2}$, the random process $X_t$ exceeds $h$ before time $T_1$. **Second phase:** Write ${\mathbb{P}}_*$ and ${\mathbb{E}}_*$ for the probability and the expectation conditional on ${\mathcal{E}}$. We define $$\begin{aligned} \tau_0:&=\min\{t: X_{\tau_h+t}=0\}\wedge T_2\\ \tau^2_S:&=\min\{t: |S_{\tau_h+t}{\setminus}S_{\tau_h}|\geq {2}n^{2/3}\}\\ \tau_2:&=\tau_0\wedge \tau_S^2\;.\end{aligned}$$ Consider the random variable $$W_t:=h- \min \{h, X_{\tau_h+t}\} \;.$$ Hence $$\begin{aligned} W_{t+1}^2-W_{t}^2 &\leq (h- \min\{h,X_{\tau_h+t}\}-\eta_{\tau_h+t+1})^2-(h- \min\{h,X_{\tau_h+t}\})^2\\ &= \eta_{\tau_h+t+1}^2-2 \eta_{\tau_h+t+1}(h- \min\{h,X_{\tau_h+t}\})\\ &\leq \eta_{\tau_h+t+1}^2 -2 \eta_{\tau_h+t+1}h \;.\end{aligned}$$ If $t<\tau_2$ and $n$ is sufficiently large, we can apply Lemma \[lem:exp\] and this leads to (provided $A$ is sufficiently large with respect to $\mu$) $$\begin{aligned} {\mathbb{E}}_*[W_{t+1}^2-W_{t}^2\mid {\mathcal{H}}_{\tau_h+t}] \leq d+2 \cdot (570+\mu)n^{-1/3} \cdot h \leq 2d\;.\end{aligned}$$ Thus, $W_{t\wedge \tau_2}^2-2d(t\wedge\tau_2)$ is a supermartingale. Similar as before, we use the Optimal Stopping theorem to conclude that $$\begin{aligned} {\mathbb{E}}_*[W_{\tau_2}^2] \leq 2d{\mathbb{E}}_*[\tau_2] \leq 2dT_2 \;.\end{aligned}$$ Thus $$\begin{aligned} {\mathbb{P}}_*[\tau_2<T_2]\notag &= {\mathbb{P}}_*[\tau_0<T_2,\tau^2_S>T_2]+{\mathbb{P}}_*[\tau^2_S\leq T_2]\\ &\leq {\mathbb{P}}_*[W_{\tau_2}\geq h]+{\mathbb{P}}_*[|S_{\tau_h +T_2}{\setminus}S_{\tau_h}|\geq {2}n^{2/3}]\\ &\leq {\mathbb{P}}_*[W^2_{\tau_2}\geq h^2]+o(1)\\ &\leq \frac{{\mathbb{E}}_*[W_{\tau_2}^2]}{h^2}+o(1)\leq 5A^{-1/2}\;,\end{aligned}$$ where we used Lemma \[lem:S\_t\] [with $t_1=\tau_h$ and $t_2=\tau_h+T_2$]{} for the second inequality. [(Observe that we cannot apply Lemma \[lem:S\_t\] directly, because we assume $\mathcal{E}$ holds and $\tau_h$ is a random time. However, as $\tau_h\leq T_1$, a simple union bound with $t_1=k$ and $t_2=k+T_2$ for all $k\leq T_1$ together with the fact that ${\mathbb{P}}[\mathcal{E}]\geq 1-13A^{-1/2}\geq 1/2$, yields the desired result.)]{} It follows that $$\begin{aligned} {\mathbb{P}}[\{\tau_2<T_2\}\cup \{\tau_h=T_1\} \cup \{\tau_S^1\leq \tau_h\}] &\leq {\mathbb{P}}[\{\tau_h=T_1\} \cup \{\tau_S^1\leq \tau_h\}]+ {\mathbb{P}}_*[\tau_2<T_2]\\ &\stackrel{\eqref{eq:T1}}{\leq} 13A^{-1/2} +5A^{-1/2}=18A^{-1/2}\;.\end{aligned}$$ Since all the vertices explored from time $\tau_h$ to $\tau_h+\tau_2$ belong to the same component of $G_{n,d,p}$, there exists a component of size at least $|S_{\tau_h+\tau_2}{\setminus}S_{\tau_h}|$. As $\tau_2=T_2= 2A^{-1} dn^{2/3}$ with probability at least $1-18A^{-1/2}$, by Lemma \[lem:S\_t\] [with $t_1=\tau_h$ and $t_2=\tau_h+T_2$ (as above, strictly speaking, we apply Lemma \[lem:S\_t\] with $t_1=k$ and $t_2=k+T_2$ for all $k\leq T_1$ and use the fact that ${\mathbb{P}}[\mathcal{E}]\geq 1/2$]{}) with probability at least $1-18A^{-1/2}-o(1)\geq 1-19A^{-1/2}$, there exists a component of size at least $A^{-1} n^{2/3}$. **Acknowledgements:** The authors want to thank Nikolaos Fountoulakis, Michael Krivelevich, and Asaf Nachmias for fruitful discussions on the topic [and the anonymous referees for their valuable comments.]{} plus 1fill Version Felix Joos\ \ Guillem Perarnau\ \ School of Mathematics, University of Birmingham, Birmingham\ United Kingdom [^1]: The non-existence of a linear order component when $p\leq (1-\epsilon)p_{{crit}}$ follows from Proposition 1 in [@nachmias2010critical]. The existence of a linear order component when $p\geq (1+\epsilon)p_{{crit}}$ follows from the expansion properties of $G_{n,d}$ (see Corollary 2.8 in [@krivelevich2001random]) and the results on $(n,d,\lambda)-$graphs in [@krivelevich2013phase].

--- abstract: 'We present methods to assess whether gamma-ray excesses towards Milky Way dwarf galaxies can be attributed to astrophysical sources rather than to dark matter annihilation. As a case study we focus on Reticulum II, the dwarf which shows the strongest evidence for a gamma-ray signal in Fermi data. Dark matter models and those with curved energy spectra provide good fits to the data, while a simple power law is ruled out at $97.5\%$ confidence. We compare RetII’s spectrum to known classes of gamma-ray sources and find a useful representation in terms of spectral curvature and the energy at which the spectral energy distribution peaks. In this space the blazar classes appear segregated from the confidence region occupied by RetII. Pulsars have similar gamma-ray spectra to RetII but we show that RetII is unlikely to host a pulsar population detectable in gamma rays. Tensions with astrophysical explanations are stronger when analyzing 6.5 years of Pass 7 than with the same amount of Pass 8 data, where the excess is less significant. These methods are applicable to any dwarf galaxy which is a promising dark matter target and shows signs of gamma-ray emission along its line of sight.' author: - 'Alex Geringer-Sameth' - 'Savvas M. Koushiappas' - 'Matthew G. Walker' - Vincent Bonnivard - Céline Combet - David Maurin bibliography: - 'bibfile.bib' title: Astrophysical explanations of suspected dark matter signals in dwarf galaxies --- Dark matter is the dominant form of mass in the Universe but has, so far, been characterized only through its gravitational effects on astronomical scales. Its microscopic nature, holding fundamental implications for particle physics and cosmology, has yet to be revealed. Astrophysical searches for the high energy particles produced if dark matter annihilates with its antiparticle are a promising way to discover and characterize weakly interacting massive particles with a mass in the GeV–TeV range [see, e.g. @1996PhR...267..195J; @2000RPPh...63..793B; @2005PhR...405..279B]. Dark matter annihilation ought to take place anywhere in the universe where dark matter particles encounter each other with sufficient frequency. The annihilation products typically result in the generation of gamma rays, which suffer little deflection or absorption on their way to Earth. This motivates a great variety of searches for anomalous gamma-ray emission in different targets [e.g. @2005PhR...405..279B], including the Galactic Center [e.g. @2011PhLB..697..412H; @2014PhRvD..90b3526A; @2015PhRvD..91f3003C], the Galactic halo [e.g. @2010NuPhB.840..284C; @2011PhRvD..83l3516B; @2018arXiv180404132C], Milky Way dwarf galaxies [e.g. @1990Natur.346...39L; @2015PhRvD..91h3535G; @2015PhRvL.115w1301A], galaxy groups and clusters [e.g. @2010JCAP...05..025A; @2012JCAP...07..017A; @2018PhRvL.120j1101L], large scale structure [e.g. @2011MNRAS.416.2247X; @2014PhRvD..90b3514A; @2015PhRvL.114x1301R], and in the isotropic gamma-ray background [e.g. @2010JCAP...04..014A; @2010JCAP...11..041A; @2015JCAP...09..008T]. In searches for dark matter annihilation that gives rise to emission over a continuous energy range (as opposed to a monoenergetic gamma-ray line), conventional astrophysical processes produce gamma rays which compete with the (possibly subdominant) dark matter signal. In this regard, Milky Way dwarf spheroidal galaxies are unique as compared with other targets: they are dark matter dominated systems which contain no known sources of astrophysical emission, making them particularly clean laboratories for dark matter annihilation searches. In recent years, dwarf searches have benefited tremendously from full-sky observations of the Large Area Telescope on board the Fermi Gamma-Ray Space Telescope (Fermi LAT) [@2009ApJ...697.1071A], analysis techniques capable of combining observations of many targets [@2011PhRvL.107x1303G; @2011PhRvL.107x1302A; @2012APh....37...26M; @2013JCAP...03..018S; @2014PhRvD..90k2012A; @2014PhRvD..89d2001A; @2015PhRvD..91h3535G; @2015PhRvL.115w1301A; @2016JCAP...02..039M; @2017PhRvD..95h2001A; @2018ApJ...853..154A], the discovery of new Milky Way satellites [e.g. @2010AdAst2010E..21W; @2013NewAR..57..100B; @2015PhRvD..91f3515H], and the characterization of the dark matter distributions within these systems . With the increasing sensitivity of this effort, it is important to consider how evidence of annihilation in dwarfs might first present itself. While the first goal is always to detect any gamma-ray excess, when indications of one appear we must be prepared to rigorously evaluate whether it originates from dark matter annihilation. In this paper we consider such “next steps” that can be taken to test a dark matter hypothesis when a signal presents itself. This work is motivated by the 2015 discovery of Reticulum II (RetII), a nearby Milky Way dwarf galaxy found in photometric data from the Dark Energy Survey (DES) [@2015ApJ...805..130K; @2015ApJ...807...50B] and confirmed spectroscopically as a system dynamically dominated by dark matter [@2015ApJ...808..108W; @2015ApJ...808...95S; @2015ApJ...811...62K]. Intriguingly, analysis of 6.5 years of Fermi data reveals a gamma-ray excess between about 2 and 10 GeV significant at the $p=0.0001$ to $0.01$ level, depending on how the background is modeled [@2015PhRvL.115h1101G] ($p$ being the probability that background processes alone can generate such “signal-like” data). While this finding was confirmed by an independent analysis [@2015JCAP...09..016H] of the same data, known as “Pass 7”, the Fermi-LAT and DES collaborations found a decreased significance, $p=0.05$, for RetII using a reprocessing of the raw LAT data (“Pass 8”) over a similar 6 year baseline [@2015ApJ...809L...4D] (and a background model analogous to the one that yielded $p=0.01$ above). Subsequent analyses, using $\sim6-7$ years of data, also show decreased significance in Pass 8 [@2017ApJ...834..110A; @2018ChPhC..42b5102Z] as compared with Pass 7. Since these last studies were performed, however, the Pass 8 significance has apparently continued to rise. Recently @2018arXiv180506612L presented a 9-year analysis that closely follows the procedure of the 6-year studies [@2015PhRvL.115w1301A; @2015ApJ...809L...4D; @2017ApJ...834..110A]. They find that RetII steadily grows in significance from three to six to nine years in Pass 8. They give an uncalibrated significance of $TS=13.5$ ($TS$ being twice the log-likelihood ratio between the dark matter annihilation and background-only hypotheses). While they do not attempt to quantify the trials factor due to testing multiple dark matter masses, annihilation channels, and halo profiles, the local significance (assuming $\chi_1^2/2$ statistics) is $p \sim 10^{-4}$ (compare with $p\sim 10^{-6} - 10^{-5}$ in Pass 7 [Fig. 2 in @2015PhRvL.115h1101G]; a trials factor of $\sim3 - 10$ usually suffices for testing multiple masses and channels [@2015PhRvD..91h3535G]). We analyze 6.9 and 10 years of Pass 8 data with the identical method described in [@2015PhRvL.115h1101G] and also observe a rise in significance. The reason for the change in significance between Pass 7 and Pass 8 remains unclear (we discuss consistency in Sec. \[sec:discussion\]). Nonetheless, in both data sets RetII possesses the most significant gamma-ray signal of any dwarf galaxy [@2018arXiv180506612L; @2015PhRvL.115h1101G; @2015PhRvD..91h3535G; @2015JCAP...09..016H]. We take RetII as a case study in the first indication of excess gamma rays from the direction of a dwarf galaxy and consider further hurdles that a dark matter interpretation must overcome. That is, we operate under the assumption that there is a gamma ray source along the direction toward RetII and then seek to characterize this source (we discuss the basis of this assumption in Sec. \[sec:significance\]). We emphasize that a dark matter origin cannot be established as long as there is a plausible astrophysical explanation for the gamma-ray excess. The goal of this work is to rigorously address this possibility. One strategy is to observe the dwarf at longer wavelengths to try to identify a possible astrophysical counterpart. In a dedicated radio observation, @2017JCAP...07..025R identified two blazar candidates (BL Lacs) among the sources in RetII’s vicinity. As a population, blazars are often associated with gamma-ray emission. However, the distributions of radio, optical, and X-ray fluxes of gamma-ray loud vs. gamma-ray quiet blazars are highly overlapping [e.g. @2015ApJ...810...14A], making the prediction of any individual blazar’s gamma-ray flux quite challenging. At higher frequencies,  report a detection of 511 keV emission from the direction of RetII, which the authors suggest may be due to an accreting black hole (microquasar) within RetII . Whether or not this speculative scenario entails gamma ray emission remains to be seen. In this work we look deeper into the gamma-ray data to assess astrophysical explanations of the excess. Possibilities include a population of gamma-ray emitting objects in the dwarf galaxy itself or a chance alignment with an unrelated, distant gamma-ray source. We begin by quantifying the goodness of fit of dark matter annihilation and astrophysical spectra to the gamma-ray data (Sec. \[sec:GOF\]). Then we consider each of the astrophysical possibilities: we compare the shape of RetII’s energy spectrum to known classes of gamma-ray sources (Sec. \[sec:3fgl\]) and assess whether RetII hosts one or more gamma-ray emitting pulsars (Sec. \[sec:pulsars\]). In a companion paper we evaluate a dark matter annihilation hypothesis for the signal. Data selection ============== This work considers both Pass 7 and Pass 8 Fermi-LAT data sets. These are two separate reductions of the raw spacecraft data into reconstructed lists of events along with the associated instrument response functions. To facilitate comparison between them we use data collected during the time when both are available: between Fermi mission weeks 9 and 368 (August 4, 2008 to June 24, 2015; 6.9 years). Pass 7 data is not available after this time. For Pass 7, events and instrument response functions are obtained with version `v9r33p0` of the Fermi Science Tools[^1]. We extract Pass 7 Reprocessed SOURCE class events within $15{^\circ}$ of RetII using `gtselect` with the recommended `zmax=100{^\circ}`. Events must be detected within “good time intervals” found using `gtmktime` with recommended filter `DATA_QUAL==1 && LAT_CONFIG==1` and `roicut=no`. The instrument response (PSF and exposure) in the direction of RetII are obtained by running `gtselect` with a radius of $0.5{^\circ}$, `gtmktime` with the above filter but with `roicut=yes`, `gtltcube` with default options, and `gtpsf` with 17 log-spaced energies between 133.3 MeV and 1.333 TeV, `thetamax=10{^\circ}`, and `ntheta=500`. Pass 8 SOURCE events and instrument responses are found using the same procedure except we use version `v10r0p5` of the Science Tools, the recommended `zmax = 90{^\circ}` in `gtselect`, and `DATA_QUAL>0 && LAT_CONFIG==1` in `gtmktime`. Exposures and PSFs agree with those computed with the latest version `v11r5p3` of the Science Tools at the subpercent level. We define a region of interest (ROI) as a circle of radius $0.5{^\circ}$ containing events with energies between 0.5 and 300 GeV. Events in the ROI centered on RetII are used to consider various models for the signal. @2015PhRvL.115h1101G adopted 1 GeV as the lower end of the energy range. This work includes energies down to 0.5 GeV in order to explore a variety of dark matter and astrophysical models. Dark matter annihilation spectra have shapes which may be a good fit to the data between 1 and 300 GeV but not at energies below 1 GeV. Reducing the energy threshold to 0.5 GeV allows us to better evaluate various interpretations of the data. Due to the energy-dependent point spread function (PSF) of the Fermi LAT, lowering the energy threshold potentially allows for contamination of an ROI by gamma rays from nearby point sources. However, in the case of RetII such sources can be safely ignored[^2] Background model and the existence of a source toward RetII =========================================================== Adopted background model \[sec:bgdef\] -------------------------------------- For the gamma-ray background in the direction of RetII we adopt the Poisson background model used in [@2015PhRvL.115h1101G]: the number of background events in the RetII ROI is a Poisson variable; background events are distributed isotropically within the $0.5{^\circ}$ ROI and their energies are independent samples from a given energy spectrum. We adopt the background energy spectrum derived by the Fermi collaboration[^3]. It is the sum of an isotropic component[^4] and a diffuse interstellar component[^5]. The diffuse flux is averaged within a circle of radius $1{^\circ}$ centered on RetII (the effect of changing the size of this region is negligible). As shown in [@2015PhRvL.115h1101G] this model is a very good fit to the average background within $10{^\circ}$ of RetII above 0.2 GeV. We denote the expected background flux $dF_b(E) / dEd\Omega$ (flux of background events per energy per solid angle). Detection significances \[sec:significance\] -------------------------------------------- As mentioned in the introduction, two different ways of modeling the gamma-ray background yield two different detection significances for RetII. We reflect on the meaning of this discrepancy and show that comparing the two background models can yield inferences about the presence of a source in the direction of RetII. Significance represents the degree to which we can reject a “background-only” null hypothesis as an explanation of the data. The $p$ value is the probability of obtaining the observed data (or data more “signal-like”, as quantified by a test statistic) if the null hypothesis were true. For source detection the hypothesis is of the form: The detected events within the RetII ROI are produced by “background processes” only. Under the background model of Sec. \[sec:bgdef\] the significance of RetII using Pass 7 data is $p \approx 10^{-4}$ (see [@2015PhRvL.115h1101G] and Table \[tab:gof78\] of this paper). A second background model and resulting significance test is provided by the “empirical background” technique [@2011PhRvL.107x1303G; @2015PhRvD..91h3535G] (see also [@2014PhRvD..89d2001A; @2015PhRvD..91f1302C; @2015PhRvL.115w1301A; @2017ApJ...834..110A] for an analogous “blank sky locations” method). Here, the same test statistic applied to the RetII ROI is applied to random locations within $10{^\circ}$ of RetII, building up the empirical probability distribution of the test statistic under the background hypothesis. The underlying assumption is that whatever background processes are at work near RetII are also at work in the direction of RetII. Then the $p$ value is the fraction of all sampled background ROIs that look more “signal-like” than the RetII ROI (i.e. have a larger value of the test statistic). @2015PhRvL.115h1101G find $p \approx 0.01$ from the Pass 7 data using this method. If we sharpen up what is meant by “background processes” we will see that the two different significances for RetII come from testing two distinct hypotheses. The first tested hypothesis is that the background model of Sec. \[sec:bgdef\] with no additional contributions can explain the RetII data. The energy spectrum $dF_b(E) / dEd\Omega$ is based mainly on physical models of the Milky Way’s interstellar medium [@2016ApJS..223...26A] (e.g. cosmic ray interactions with gas), diffuse structures like the Fermi bubbles, and isotropic emission over the whole sky (which also accounts for cosmic ray contamination). Such a fundamentally diffuse process of emission is governed by Poisson statistics, with an energy spectrum changing smoothly from place to place. In particular, the model does not include discrete “bright” point sources, i.e. those with fluxes at or above the level of the diffuse processes. Obtaining $p\approx10^{-4}$ for this hypothesis means that the emission towards RetII is either due to a rare statistical fluctuation or to the adoption of an incorrect spectral model (this is the usual issue of statistics vs. systematics). If the spectral model is incorrect it can be for two reasons: the presence of a localized “bright” source of gamma rays or the inadequate modeling of the diffuse physical processes. The last explanation (no additional source but a mismodeling of the diffuse processes) is unlikely for a few reasons. First, the background model is a very good fit overall to the $10{^\circ}$ region surrounding RetII, showing no systematic deviation from the data [@2015PhRvL.115h1101G; @2015ApJ...809L...4D]. Second, the background spectrum has a significantly different shape from the observed RetII spectrum [Fig. 1 of @2015PhRvL.115h1101G], and a good fit cannot be obtained by changing its normalization (e.g. if the amount of gas along the line of sight were underestimated in the model; see also [@Hoof2018]). In fact, there is no place in the sky where the Fermi diffuse model has the shape of the RetII spectrum[^6], which we take as an indication that conventional diffuse processes cannot give rise to such a spectrum. Third, the RetII excess is localized. If the diffuse model were incorrect there would likely be a highly spatially-correlated excess. Furthermore, the diffuse model shows no large variations or complicated behavior near the location of RetII, which is $50{^\circ}$ off the galactic plane. Finally, the RetII signal is undiminished if, instead of SOURCE events, we compute the significance using Pass 7 ULTRACLEAN events, which are a subset of SOURCE events reconstructed with higher quality and suffering a smaller cosmic ray contamination. With cautious confidence in the diffuse model, the remaining possibilities are either a “bright” source toward RetII or a Poisson fluctuation in the detected events. We can use the results of the hypothesis test based on the empirical background model to explore this further. Whereas $p\approx 10^{-4}$ is the probability that diffuse processes such as cosmic ray interactions with gas and extragalactic isotropic emission can explain the RetII data, $p\approx 0.01$ is the probability of an excess from [*all mechanisms*]{} besides emission from a dwarf galaxy (i.e. from any cause other than dark matter annihilation within the dwarf). The additional mechanisms in this more inclusive hypothesis are those stated above: the presence of additional gamma-ray sources along the line of sight and systematic deviations from the assumed diffuse model. That this probability of 0.01 is relatively high by particle physics standards means that, for this set of data [*and*]{} using this particular test statistic, we cannot reject the hypothesis that there is no dark matter annihilation taking place in RetII. However, as we show next, we can use this information in a back of the envelope calculation that shows that the RetII data are much more likely to be due to a source than to a background fluctuation. Consider all the significant-looking sky locations like RetII’s. How many contain “bright” sources (i.e. those with flux above the diffuse level) and how many are Poisson fluctuations of the diffuse processes (assuming they are correctly modeled)? Let ${\mathrm{P}}(F_b)$ be the probability that the brightest source in a random ROI has a flux below the diffuse background level $F_b$. We seek the probability that a given ROI with “RetII-like” gamma-ray data $D$[^7] does not contain a bright source: ${\mathrm{P}}(F_b \mid D)$. This can be rewritten in terms of $p$ values: ${\mathrm{P}}(F_b \mid D) = {\mathrm{P}}(D \mid F_b) {\mathrm{P}}(F_b) / {\mathrm{P}}(D)$, where the two hypothesis tests discussed correspond to ${\mathrm{P}}(D \mid F_b) \approx 10^{-4}$ and ${\mathrm{P}}(D)\approx 0.01$. Since ${\mathrm{P}}(F_b) < 1$ we must have that ${\mathrm{P}}(F_b \mid D) < 0.01$[^8]. In words, the probability of a statistical fluctuation is less than 1% and the probability that there is a source with flux above the diffuse background level is greater than 99%. Given the data, and based only on the observable statistics of the gamma-ray sky, it is over 100 times more likely that the RetII excess arises from an above-background source rather than from a Poisson fluctuation of the diffuse background. This rough calculation has no bearing on whether or not the emission is caused by dark matter annihilation. Rather, it justifies us taking the simple existence of a source as the [*starting point*]{} for our explorations here. We note that commonly used test statistics [e.g. @1996ApJ...461..396M; @2015PhRvD..91h3535G; @2015PhRvL.115w1301A] are designed to be powerful at rejecting the diffuse model as the null hypothesis, not at distinguishing a dark matter signal from a previously unidentified astrophysical point source. This paper focuses on this second question (see also [@2015JCAP...09..016H; @2015PhRvD..91f1302C; @2015JCAP...05..056L; @2017AJ....153..253M] for progress in incorporating unknown source populations at the level of the likelihood). Gamma-ray likelihood ==================== The gamma-ray observable ${\mathbf{X}}_\gamma$ is the list of events $i$ with energies $E_i$ and angular separations $\phi_i$ from the center of the ROI. Dividing the events into bins of energy and angular separation we have $n_j$ events in bin $j$. Model parameters (e.g. dark matter particle properties, energy spectrum shape parameters, or those describing RetII’s dark matter halo) are denoted by $\theta$. The expected number of counts in bin $j$ is $\mu_j(\theta)$. For the adopted Poisson background model, the probability (or likelihood) of observing the set ${\mathbf{n}}\equiv (n_1, n_2, \dots)$ given model parameters $\theta$ is simply the product of Poisson distributions: $${\mathrm{P}}({\mathbf{n}}\mid \theta) = \exp \left( - \sum_j \mu_j(\theta) \right) \prod_j \frac{\mu_j(\theta)^{n_j}}{n_j !}. \label{eqn:likelihoodbinned}$$ The expected counts can be divided into source (signal) and background components: $\mu_j(\theta) = B_j + S_j(\theta)$, where $B_j$ and $S_j(\theta)$ are integrals of the differential expected counts $b(E)$ and $s(E,\phi \mid \theta)$ over the $E$ and $\phi$ range of bin $j$. The differential background $b(E)$ (predicted background events per energy per solid angle) is $$b(E) = \frac{d F_b(E)}{dE d\Omega} \epsilon(E),$$ where $dF_b(E) / dEd\Omega$ is the adopted background flux model of Sec. \[sec:bgdef\] and $\epsilon(E)$ is the Fermi-LAT exposure (effective area $\times$ time) in the direction of RetII. The differential signal $s(E,\phi \mid \theta)$ for a point source depends on energy and angular separation from RetII and on the model parameters $\theta$: $$s(E,\phi \mid \theta) = \frac{d F(E \mid \theta)}{dE} \epsilon(E) {\mathrm{PSF}}(\phi \mid E), \label{eqn:signalfluxpoint}$$ where $dF(E \mid \theta)/dE$ is the source photon flux per energy and the $\phi$ dependence is governed entirely by the instrument’s PSF[^9]. We discuss the choice to model RetII as a point source rather than an extended one in Sec. \[sec:DMdef\]. Our statistical tests will be based on an unbinned likelihood. As the size of the bins in $E$ and $\phi$ shrink to zero Eq. \[eqn:likelihoodbinned\] becomes $${\mathrm{P}}({\mathbf{X}}_\gamma \mid \theta) \propto \exp\left( -\int (s+b) dEd\Omega \right) \prod\limits_i (s_i + b_i), \label{eqn:likelihoodunbinned}$$ where the integral is over the entire ROI (i.e. all energies and angular separations). The product in Eq. \[eqn:likelihoodunbinned\] is over the individual observed events, i.e. $s_i = s(E_i, \phi_i \mid \theta)$. In the limit of small bins the constant of proportionality in Eq. \[eqn:likelihoodunbinned\] goes to zero. It is convenient to normalize the probability by a term which does not depend on the model parameters $\theta$. A likelihood ratio where the denominator is the probability under the background-only model is a convenient choice. Dividing Eq. \[eqn:likelihoodbinned\] by itself but with all $S_j=0$ yields a finite limit as the bins become infinitesimal (cf. Eq. 22 of [@2015PhRvD..91h3535G]): $$\frac{{\mathrm{P}}({\mathbf{X}}_\gamma \mid \theta)}{{\mathrm{P}}({\mathbf{X}}_\gamma \mid s=0)} = \exp\left( -\int s\, dEd\Omega \right) \prod\limits_i \left(1 + \frac{s_i}{b_i} \right). \label{eqn:likelihoodratio}$$ Source models \[sec:sourcemodels\] ================================== We consider two classes of models to describe the gamma ray source toward RetII: phenomenological descriptions of astrophysical sources and dark matter annihilation within RetII. Astrophysical source models --------------------------- We model astrophysical sources as point sources with either power law or curved “log parabola” spectra. These two functional forms are used to describe the vast majority of gamma-ray sources in the Fermi Third Source Catalog (3FGL) [@2015ApJS..218...23A]. In the 3FGL each source (unless it is a pulsar) is fit with both a power law and a log parabola spectrum. If the log parabola spectrum is found to be a significantly better fit (difference in test statistic greater than 16) it is adopted as the “spectral type” in the catalog. Of the 3034 sources in the catalog, 2523 are described by a power law spectrum and 395 are assigned log parabola spectra. The remaining 116 sources are pulsars (and the extremely bright blazar 3C 454.3) and are fit with power laws with exponential or superexponential cutoffs. We consider a pulsar interpretation of the RetII signal in Sec. \[sec:pulsars\]. We note that other spectral shapes (e.g. broken power law) may provide better fits to some sources. However, for the purpose of comparing RetII’s spectrum to those of known gamma ray sources we adopt the same spectral models used in the 3FGL. The power law spectrum has two model parameters, a normalization $F_0$ and a slope $\alpha$, $$\frac{d F(E \mid \theta)}{dE} = F_0 \left( \frac{E}{E_0}\right)^{-\alpha}, \label{eqn:powerlawdef}$$ where $E_0$ is an arbitrary reference energy that we fix to 1 GeV. The log parabola spectrum has an additional curvature parameter $\beta$: $$\frac{d F(E \mid \theta)}{dE} = F_0 \left( \frac{E}{E_0}\right)^{-\alpha - \beta \log(E/E_0)}, \label{eqn:logparaboladef}$$ where $\log$ is the natural logarithm. In the 3FGL the reference energy, called the pivot energy $E_p$, varies from source to source. Changing the reference energy changes the parameter $\alpha$ [@2012ApJS..199...31N]: ${\alpha(E_p) = \alpha(E_0) + 2\beta \log(E_p/E_0)}$. We convert the $\alpha(E_p)$’s given in the 3FGL to ${\alpha(E_0=1\,{\mathrm{GeV}})}$ for this work. Dark matter annihilation \[sec:DMdef\] -------------------------------------- For dark matter annihilation the model parameters are $\theta = (M, {\langle {\sigma}v \rangle}, {\mathrm{ch}}, J)$, with the first three representing the dark matter particle mass, its velocity-averaged annihilation cross section, and the annihilation channel (i.e. Standard Model final state). We treat RetII as a point source of gamma-rays (see below). Therefore, as far as gamma-ray emission is concerned, its dark matter halo is parameterized by a single quantity $J$, the integral over the halo volume of the dark matter density squared divided by the line-of-sight distance squared. The dark matter annihilation flux to be used in Eq. \[eqn:signalfluxpoint\] is given by (e.g. [@2015PhRvD..91h3535G]) $$\frac{d F(E \mid\theta)}{dE} = \frac{{\langle {\sigma}v \rangle}J}{8\pi M^2} \frac{dN_\gamma(E)}{dE}, \label{eqn:signalfluxDM}$$ where $dN_\gamma/dE$ is the number of gamma-rays emitted per annihilation (per energy) for the given final state channel and mass $M$. For $dN_\gamma/dE$ we adopt the spectra computed by @2011JCAP...03..051C, which include electroweak corrections [@2011JCAP...03..019C]. For point-source emission $J$ is exactly degenerate with ${\langle {\sigma}v \rangle}$ in Eq. \[eqn:signalfluxDM\] and we treat ${\langle {\sigma}v \rangle}J$ as a single parameter which normalizes the amplitude of the signal. In this way our results are independent of any particular choice of $J$. Equations \[eqn:signalfluxpoint\] and \[eqn:signalfluxDM\] represent a point source approximation. To be accurate, $J$ must be replaced with the $J$-profile $dJ(\phi)/d\Omega$ (e.g. [@2015ApJ...801...74G]) and convolved with the PSF as described in [@2015PhRvD..91h3535G]. The $J$-profile is the integral of the square of the dark matter density along the line of sight as a function of the angle $\phi$ from the center of the dwarf. The use of Eqs. \[eqn:signalfluxpoint\] and \[eqn:signalfluxDM\] is justified if the $J$-profile is much narrower than Fermi’s PSF. The 68% containment angle of the gamma-ray PSF for the RetII observation is about $0.5{^\circ}$ at 2 GeV and decreases to $0.2{^\circ}$ at 10 GeV. Interestingly, the median posterior estimate of the 68% containment angle for RetII’s $J$-profile, as measured by @2015ApJ...808L..36B, is roughly $1{^\circ}$, with half the sampled $J$-profiles being more extended. However, $0.5{^\circ}$ corresponds to 260 pc at the distance to RetII (30 kpc), while the halflight radius of RetII is only 58 pc [@2018arXiv180408627M] and the outermost spectroscopically confirmed member star is at a projected distance of 90 pc [@2015ApJ...808..108W]. This means that inferences about the extent of RetII’s dark matter halo strongly depend on assumptions about the halo beyond the radius probed by observations. Thus at the present time no firm conclusions can be made about RetII’s dark matter distribution on angular scales of $0.5{^\circ}$. We note that @2015JCAP...09..016H find that the Pass 7 data do not prefer a departure from the point source assumption for RetII, while @2018arXiv180506612L find a a slight ($\Delta \chi^2 = 1.3$) preference for extension in 9 years of Pass 8. Goodness of fit of various spectral models \[sec:GOF\] ====================================================== In this section we consider which spectral models are good fits to the gamma-ray data from RetII and which cannot explain the emission. Method \[sec:GOFmethod\] ------------------------ We use a likelihood ratio to assess the goodness of fit of the various emission models to the RetII data. Under the null hypothesis we wish to test, the emission is governed by a particular spectral model and associated parameters $\theta_0$. For example, $\theta_0$ might be dark matter annihilation with a given mass, channel, and value of ${\langle {\sigma}v \rangle}J$. Or it could be a log parabola model with a specified $F_0$, $\alpha$, and $\beta$. A powerful test of the null hypothesis is performed by comparing $\theta_0$ to plausible alternatives using a likelihood ratio. We take these alternatives to be any of the spectral models described in Sec. \[sec:sourcemodels\]. The test statistic, a function of the gamma-ray data ${\mathbf{X}}_\gamma$, is (e.g. [@Kendall5th; @casella2002statistical]) $$\lambda({\mathbf{X}}_\gamma) = 2 \log \frac{{\mathrm{P}}({\mathbf{X}}_\gamma \mid \hat{\theta})}{{\mathrm{P}}({\mathbf{X}}_\gamma \mid \theta_0)}. \label{eqn:likelihoodratiotest}$$ In this equation, $\hat{\theta}$ is the model which maximizes the likelihood for the given data set ${\mathbf{X}}_\gamma$. The maximization is performed over all spectral types (dark matter, power law, or log parabola) and all parameters within those types (e.g. mass, channel, $F_0$, $\alpha$, $\beta$, etc.). Large values of $\lambda({\mathbf{X}}_\gamma)$ indicate that the hypothesis $\theta_0$ is a poor fit to the data ${\mathbf{X}}_\gamma$. We use Eq. \[eqn:likelihoodratio\] to compute the likelihood ratio. Our set of alternative models are not nested, the true values of $\theta$ may lie beyond the boundaries of the parameter space for the log parabola model (see below), and it is unclear whether the number of events in the $0.5{^\circ}$ ROI is large enough to apply Wilks theorem [@wilks1938; @Kendall5th]. We therefore simulate large numbers of fake ROIs to directly construct the sampling distribution of $\lambda({\mathbf{X}}_\gamma)$ under a given hypothesis $\theta_0$ (see [@2016MNRAS.458L..84A] for a possible alternative). Background events are generated using the model of Sec. \[sec:bgdef\] and signal events using the models of Sec. \[sec:sourcemodels\]. The goodness of fit $p$ value for $\theta_0$ is the fraction of realizations with a larger value of $\lambda({\mathbf{X}}_\gamma)$ than obtained for the observed RetII data. We find that, for the best fitting models $\theta_0$, the distribution of $\lambda({\mathbf{X}}_\gamma)$ is not well described by a $\chi^2$ distribution. However, for the 13 best fitting models (11 dark matter, power law, and log parabola) the PDF of $\lambda({\mathbf{X}}_\gamma)$ is fairly well described by a gamma distribution with shape parameter $k\approx3$ and scale parameter $\theta \approx 1.4$, suggesting that a scaled version of $\lambda({\mathbf{X}}_\gamma)$ may be distributed as a $\chi^2$ variable with 6 degrees of freedom[^10]. The maximization of the likelihood is performed over a grid of model parameters (except for the normalizations ${\langle {\sigma}v \rangle}J$ and $F_0$ which can vary continuously). We have chosen the grid to be fine enough so that the results are not sensitive to the discreteness. For the dark matter models the allowed masses run from the mass of the final state particle up to 1 TeV in log-spaced steps where neighboring masses differ by 2%. For the power law spectrum we consider indices $\alpha$ running from 1 to 3 in steps of 0.01 (the range for 3FGL sources is between 1.1 to 5.7). For the log parabola models, $\alpha$ runs from $-1$ to 5 in steps of 0.05 (the range in the 3FGL is -0.54 to 4.6), and $\beta$ runs from 0.05 to 1 in steps of 0.05 (the range in the 3FGL is 0.03 to 1). Though values of $\beta$ greater than 1 are physical, the upper limit of 1 is imposed in order to more easily compare with the 3FGL, where 1 is the maximum allowable $\beta$. Additionally, the log parabola model is meant to model astrophysical sources and it is appropriate to restrict its parameter space to where such sources are expected to lie. We discuss the relaxation of the $\beta <1$ requirement in Sec. \[sec:3fgl\]. Results ------- We find the best fitting model parameters (those which maximize the likelihood in Eq. \[eqn:likelihoodratio\]) for each spectral class: power law, log parabola, and dark matter for each annihilation channel. We then test whether each of these best fit models is actually a good fit to the RetII data using the likelihood ratio test described above. That is, we set $\theta_0$ to a best fit model, generate fake data ${\mathbf{X}}_\gamma$ under this model to find the distribution of $\lambda({\mathbf{X}}_\gamma)$, and find what fraction of fake data sets have $\lambda({\mathbf{X}}_\gamma)$ higher than that observed for RetII. The resulting $p$ values are shown in Table \[tab:gof78\]. [dddddddddd]{} & & & &\ & & & &\ & & & & & & & & &\ -0.70 & -1.00\^\*& 1.00\^\* & 0.95 & 3.0 &0.80 & 2.5 & 1.3 & 0.73 &0.90\ 2.09 &1.99& & & 10.6 & 2.8 & 9.9 & 4.9 & 0.025 &0.16\ & 19.9 & 7.0 & & 0.027\ \ & &\ & &\ & 6.2 & 8.1 & 2.9 & 1.7 & 0.0 & 0.0 & 1.0 & 1.0\ & 6.2 & 8.1 & 6.5 & 3.5 & 0.8 & 0.46 & 0.93 & 0.97\ & 13.9 & 19.5 & 1.1 & 0.63 & 2.7 & 1.5 & 0.70 & 0.89\ & 77.6 & 120.5 & 3.8 & 2.2 & 5.1 & 3.0 & 0.29 & 0.51\ & 34.8 & 55.1 & 1.7 & 0.96 & 5.4 & 3.2 & 0.24 & 0.44\ & 31.5 & 47.9 & 1.5 & 0.81 & 5.4 & 3.2 & 0.24 & 0.44\ & 32.8 & 48.9 & 1.5 & 0.83 & 5.4 & 3.2 & 0.24 & 0.44\ & 180.0\^\* & 195.0 & 11.1 & 4.4 & 5.5 & 3.0 & 0.23 & 0.48\ & 135.3 & 214.3 & 7.8 & 4.6 & 5.6 & 3.2 & 0.23 & 0.45\ & 90.0\^\* & 99.5 & 6.3 & 2.5 & 5.7 & 3.2 & 0.21 & 0.43\ & 100.0\^\* & 124.6 & 7.0 & 3.2 & 5.7 & 3.3 & 0.21 & 0.42\ The best fits (highest $p$ values) are for a source with a log parabola spectrum or dark matter particles annihilating into leptons, followed by annihilation into quarks and gauge bosons. For every dark matter model there is at least one particle mass for which the fit is acceptable. Power law models, on the other hand, are in tension with the Pass 7 RetII data with $p = 0.025$. Specifically, if there were a power law source in the direction of RetII with spectral index of $\alpha = 2.09$ there is only a 2.5% chance of finding $\lambda({\mathbf{X}}_\gamma)$ as large is it is measured to be. In other words, RetII appears to have a significantly curved spectrum. Note that Table \[tab:gof78\] only shows best fitting models, not uncertainties in model parameters. In the next section we discuss constraints on the log parabola model parameters and a companion paper will explore dark matter parameter space (see also Fig. 4 in [@2015PhRvL.115h1101G]). Figure \[fig:speckstackmodels\] shows the best fitting model spectra and compares them with the observed RetII data. The data points show the empirical spectrum of RetII derived from the observed event counts within $0.25{^\circ}$ of RetII. These are binned in energy (5 bins per decade starting at 0.2 GeV) and error bars show 68% Poisson confidence intervals. The empirical flux is the number of counts divided by the exposure and energy bin width. Model spectra as well as the background spectrum are plotted as curves. Pass 7 and Pass 8 results are shown with their respective best fitting models (we discuss consistency in Sec. \[sec:discussion\]). ![\[fig:speckstackmodels\] Energy spectrum of RetII compared with best fitting models. Data points (same for each row) are the observed spectrum derived from events detected within $0.25{^\circ}$ of RetII along with 68% Poisson error bars. The thin gray curve is the predicted background and dashed curves show the source contribution. Solid colored curves are the sum of background and source. The first three rows are models of dark matter annihilation while the last describes a generic astrophysical source with a curved spectrum. The two columns show results for Pass 7 and Pass 8 data.](specstack2colminimodels_Ret2_eebbtauWWlogparabola_v4.pdf) Figure \[fig:speckstackannuli\] illustrates the fit as a function of energy and angular separation from RetII. The $0.5{^\circ}$ RetII ROI is divided into four annuli with equal solid angle, which are shown as different rows. The best fitting log parabola and power law models are plotted. In Figs. \[fig:speckstackmodels\] and \[fig:speckstackannuli\] the model spectra (Eqs. \[eqn:powerlawdef\], \[eqn:logparaboladef\], and \[eqn:signalfluxDM\]) are scaled by the PSF integrated within the corresponding annulus in order to compare with data. The energy dependence of the PSF explains why, for example, the power law spectrum is not a straight line. ![\[fig:speckstackannuli\] Same as Fig. \[fig:speckstackmodels\] but showing the fits at varying angular separation from RetII. Different rows correspond to spectra constructed from different annuli. The best fit power law and log parabola source models are compared.](specstack2colminiRbins_Ret2_logparabola_powerlaw_v4.pdf) Comparison with astrophysical populations ========================================= \[sec:3fgl\]Sources in the 3FGL ------------------------------- Among the models meant to describe astrophysical sources, the log parabola spectrum is a perfectly acceptable fit to the data ($p=0.73$ in Pass 7). This motivates a comparison with the various source populations present in the 3FGL. Another likelihood ratio test statistic is used to place constraints on the $\alpha$ and $\beta$ parameters of the log parabola spectrum that can describe the RetII emission. The space of alternative hypotheses (the numerator in Eq. \[eqn:likelihoodratiotest\]) is restricted to include only log parabola spectra with $-1 \leq \alpha \leq 5$, $0 \leq \beta \leq 1$, and any value for $F_0$. To test whether a given set $\alpha, \beta$ is an acceptable fit to the data we maximize the likelihood over the normalization $F_0$ while holding $\alpha$ and $\beta$ fixed. This constrained maximum likelihood value is used as the null hypotheses in the denominator of Eq. \[eqn:likelihoodratiotest\]. With fixed numbers of degrees of freedom in the null and alternative hypotheses, we cautiously make use of the $\chi^2$ approximation to the likelihood ratio [e.g. @Kendall5th]. In this case $\lambda({\mathbf{X}}_\gamma)$ should be distributed as $\chi^2$ with 2 degrees of freedom and regions of $\alpha, \beta$ space where $\lambda({\mathbf{X}}_\gamma) > 2.3\, (6.2)$ are ruled out at 68.3% (95.4%) significance. The contours in Fig. \[fig:logparabola3fgl\] show the resulting confidence intervals. The large black cross shows the best fitting parameters for the Pass 7 RetII data, occurring at the edge of the allowable parameter space at $(\alpha, \beta)=(-0.7,1)$. Solid lines show the 68% and 95% confidence regions for Pass 7 (the large black dashed circle and dashed black contour show the best fit and 68% region for Pass 8; the 95% contour includes the entire figure since $\lambda({\mathbf{X}}_\gamma)<6.2$ for the log parabola model in Pass 8). To check the coverage of the confidence intervals we simulated $10^4$ fake data sets for each of 20 points along the 68% and 95% contours (using the best fit $F_0$ at that $(\alpha,\beta)$ value). For each fake data set we find the sampling distribution of $\lambda({\mathbf{X}}_\gamma)$ and directly find the $p$ value for the RetII observation (the fraction of fake data sets with $\lambda({\mathbf{X}}_\gamma)$ larger than the RetII value). This exact $p$ value is compared with the approximate $p$ value obtained using a $\chi^2$ distribution with 2 degrees of freedom. In these experiments we find the actual $p$ to fall between 0.4 and 0.8 times the approximate $p$ value, indicating that the contours are conservative (i.e. the probability they enclose the true value of $\alpha$ and $\beta$ is greater than 68% and 95%). In terms of “sigma values” (where $1\sigma=68.3\%$ and $2\sigma=95.4\%$), the contours correspond to sigma values about $0.1\sigma$ to $0.3\sigma$ higher than stated. This is perhaps expected since the $\chi^2$ approximation should break down when the true parameters are at the boundary of the parameter space (or beyond). Figure \[fig:logparabola3fgl\] also shows the spectral parameters of 298 sources which are assigned a log parabola spectrum in the 3FGL catalog. Different markers denote different source classes, as described in Table 6 of [@2015ApJS..218...23A]. Extragalactic and unassociated sources are listed in the legend on the left and galactic sources on right. Of the 395 curved 3FGL sources 114 have one or more analysis flags set, indicating that some aspect of their analysis is problematic (e.g. detection significance or measured flux unstable to changes in the diffuse model, located near a brighter source, poor quality of spectral fit; see [@2015ApJS..218...23A] for details). The majority of these are unassociated sources, have $\beta > 0.25$, and are located very close to the Galactic plane where the source density is high and the diffuse model more uncertain. We remove the 97 sources with $|b|<5{^\circ}$ that have an analysis flag set (other than the flag indicating $\beta=1$). The remaining 17 flagged sources are shown with faded markers in Fig. \[fig:logparabola3fgl\]. This selection removes sources with likely biased parameters that are anyway unlikely to be counterparts of a source at RetII’s location ($b \approx -50{^\circ}$). Error bars on the individual 3FGL sources are omitted for clarity but we note that the sizes of the errors on $\alpha$ and $\beta$ are each highly correlated with the value of $\beta$. For the sources in each of four $\beta$ bins ($\beta \in [0, 0.25]$, $[0.25,0.5]$, $[0.5,0.75]$, $[0.75,1]$) we show the median uncertainty on $\alpha$ and $\beta$ as a series of error bars running up the right-hand side of the figure.  The RetII contours are quite large compared to the 3FGL error bars because RetII is detected at much lower significance than these sources. There are a number of unassociated 3FGL sources (empty black circles) within the RetII $2\sigma$ Pass 7 contour, and even several BL Lacs (empty blue diamonds) and active galaxies of uncertain type (filled green circles). The vast majority of sources that can be associated with galactic or extragalactic counterparts, however, have significantly different spectral shape than RetII. In particular, the two blazar classes (BL Lacs and flat spectrum radio quasars) that make up the bulk of associated curved sources populate a relatively well-defined region in $\alpha, \beta$ space with $\beta \lesssim 0.3$. Of the 2918 non-pulsar 3FGL sources, 298 (with $|b|>5{^\circ}$ and no analysis flag) have significant curvature and of these only 5 (40) lie within the 68% (95%) Pass 7 RetII contours. Even with the limited photon counts, the data suggest that RetII may have spectral parameters substantially different from almost all other known gamma-ray sources. From Eq. \[eqn:logparaboladef\] we see that the maximum of $E^2 dF/dE$ (the spectral energy distribution) occurs at an energy ${E_\mathrm{peak}}$ where $\log({E_\mathrm{peak}}/E_0) = (1 - \alpha/2)/\beta$. In Fig. \[fig:logparabola3fgl\] contours of constant ${E_\mathrm{peak}}$ (gray dashed lines) are straight lines radiating from $(\alpha,\beta)=(2,0)$, with positive slope if ${E_\mathrm{peak}}< E_0$ and negative slope if ${E_\mathrm{peak}}> E_0$. The degeneracy direction in the $\alpha, \beta$ contours suggests that ${E_\mathrm{peak}}$ is what is actually being measured in the data, rather than $\alpha$ and $\beta$ individually. This is verified in Fig. \[fig:logparabola3fgl\_Epeakbeta\], where we reparameterize the log parabola spectra using ${E_\mathrm{peak}}$ instead of $\alpha$ and find the best fit within the range $10~{\mathrm{MeV}}< {E_\mathrm{peak}}< 1~{\mathrm{TeV}}$ and $0<\beta<1$ . Representative error bars for 3FGL sources are obtained with the same binning procedure used in Fig. \[fig:logparabola3fgl\] (we use simple error propagation to find the errors on $\log({E_\mathrm{peak}}/E_0)$ and note that for some sources with the lowest measured $\beta$’s the errors on ${E_\mathrm{peak}}$ can reach 100%, reflecting the fact that ${E_\mathrm{peak}}$ is ill-defined as $\beta\rightarrow 0$). While the data imply a lower limit on RetII’s curvature parameter, they provide a well-constrained measurement of the peak of its spectral energy distribution as expected from, e.g., Fig. \[fig:speckstackmodels\]. It appears that FSRQs radiate most of their gamma-ray energy in photons of systematically lower energy than RetII does.  It is clear from Fig. \[fig:logparabola3fgl\_Epeakbeta\] that curvatures larger than $\beta=1$ will improve the fit. If we relax the constraint from the 3FGL that $\beta \leq 1$, the best fitting $\beta$ increases from 1 to 3.0 in Pass 7 (3.8 in Pass 8), and $\lambda({\mathbf{X}}_\gamma)$ becomes similar to the best fitting models of Table \[tab:gof78\]. Since ${E_\mathrm{peak}}$ and $\beta$ are uncorrelated we measure them each individually, maximizing the likelihood over the other parameter and $F_0$, and assuming that the likelihood ratio is governed by a $\chi^2$ distribution with 1 degree of freedom. We measure $3.1 < {E_\mathrm{peak}}/{\mathrm{GeV}}< 4.5$ and $1.4 < \beta < 5.9$ at 68.3% confidence for the Pass 7 data. For Pass 8 we find $4.0 < {E_\mathrm{peak}}/{\mathrm{GeV}}< 6.4$ and $\beta > 1.6$. As with Pass 7, the likelihood ratio rises rapidly as $\beta$ decreases from its best fit value, but $\lambda({\mathbf{X}}_\gamma)$ is still less than 1 when $\beta=10$, yielding a one-sided confidence interval on $\beta$. As expected, ${E_\mathrm{peak}}$ is constrained rather precisely while the data essentially provide a lower bound on $\beta$. Pulsars \[sec:pulsars\] ----------------------- Among the source classes in the 3FGL, pulsars are notable for their significantly curved spectra. About 75% of the pulsars in the 3FGL have a curvature significance greater than $4\sigma$ (as compared with FSRQs (17%), BL Lacs (3%), blazars of uncertain type (3%), supernova remnants (57%), globular clusters (40%), and unassociated sources (17%)). If RetII hosts one or more gamma-ray emitting pulsars that may explain its curved spectrum. We make an estimate of the pulsar contribution to RetII’s gamma-ray flux by considering the 15 globular clusters in the 3FGL. The gamma-ray emission from globular clusters is likely powered by populations of millisecond pulsars (MSPs) . For each globular cluster, we scale its gamma-ray flux to what it would be at the distance of RetII. We also scale the gamma-ray flux according to the ratio of V-band luminosity $L_V$ of the globular cluster [@1996AJ....112.1487H 2010 edition] to that of RetII [@2015ApJ...805..130K]. In this way each globular cluster provides an estimate of the pulsar emission which might be expected from RetII. As RetII has an old, metal-poor stellar population, similar to globular clusters, the visual luminosity is a proxy for number of stars. The luminosity scaling assumes that the number of MSPs in a system is proportional to the number of stars. However, MSPs are typically found in binary systems, and the stellar encounter rate in globular clusters correlates with both their abundance of X-ray binaries (possible progenitors of MSPs) and their gamma-ray luminosity . Because of its extremely low stellar density compared to globular clusters (hence low encounter rate), RetII likely harbors a far smaller fraction of binary systems than do globular clusters, and therefore a far fewer number of MSPs per unit luminosity. Furthermore, selection bias leads to the globular clusters in the 3FGL having higher gamma-ray luminosities than expected from the scaling based on the population of Milky Way globular clusters (there are 104 gamma-ray quiet globular clusters in the Harris catalog with greater $L_V/D^2$ than the 3FGL globular cluster with the smallest $L_V/D^2$). For these reasons, our globular cluster scaling is conservative and will tend to overestimate the RetII pulsar flux. ![\[fig:GCs\] The pulsar contribution to RetII’s gamma-ray flux compared with the diffuse background level. Each thin curve shows the spectrum of a gamma-ray detected globular cluster scaled according to the distance and luminosity of the globular cluster relative to RetII. Solid lines correspond to globular clusters with power law spectra in the 3FGL, dashed lines to those with log parabola spectra. The thick curve shows the diffuse background. Fluxes are integrated over a region of radius $0.25{^\circ}$ as in Fig. \[fig:speckstackmodels\].](fig_GCs_vs_bgflux_025deg.pdf) Figure \[fig:GCs\] compares the diffuse background level toward RetII to the estimates of pulsar emission provided by each scaled globular cluster. Fluxes are shown integrated within $0.25{^\circ}$ of RetII (compare with Figs. \[fig:speckstackmodels\] and \[fig:speckstackannuli\]). While the spectral shape of globular cluster emission is often quite similar to what we observe from RetII, the expected flux is too small to explain the RetII signal by over an order of magnitude. We also consider the peak intensity (flux per solid angle) of the scaled globular cluster emission: the maximum value of the PSF multiplied by the point source flux (see Eq. \[eqn:signalfluxpoint\]). Except for Palomar 9, each scaled globular cluster has a gamma-ray intensity an order of magnitude or more below the background estimate in RetII’s direction. Palomar 9’s intensity lies slightly above background at energies above 30 GeV. However, at these energies we expect fewer than a single event to be detected by Fermi. We conclude that it is highly unlikely that a population of MSPs could give rise to an observable gamma-ray signal from RetII. Another way to see the implausibility of the MSP explanation is to note that the estimated number of MSPs in gamma-ray emitting globular clusters range from about ten to at most a few hundred . This relative handful of MSPs occur in densely packed systems of millions of stars. RetII, with about 1000 solar luminosities, is unlikely to possess a single MSP. In fact, using a sample of globular clusters [*not*]{} selected by gamma-ray luminosity, @2016JCAP...08..018H find the occurrence of MSPs in globular clusters to be about 1 per $10^6$ solar luminosities. The results of this section, based on simple scaling arguments, are in agreement with the conclusions of @2016ApJ...832L...6W. In that study, the pulsar contribution to dwarf galaxy gamma-ray fluxes is estimated by constructing a gamma-ray luminosity function for isolated Milky Way MSPs and then scaling the Milky Way population down by the ratio of dwarf galaxy to Milky Way stellar mass. The authors find an expected pulsar contribution 1 to 5 orders of magnitude below the diffuse background gamma-ray flux for all ultrafaint dwarf galaxies. Our arguments do not address the possibility of an MSP unrelated to RetII that happens to lie along the line of sight. The probability of such a coincidence can be estimated with population synthesis simulations . In particular, @2016ApJ...832L...6W predict the flux from foreground pulsars to be similar to the flux from those internal to RetII. Dedicated searches in radio [e.g. @2017JCAP...07..025R] and X-rays may also be able to discover an interloping pulsar. Discussion \[sec:discussion\] ============================= Among the types of background gamma-ray sources that might lie along RetII’s line of sight, blazars are perhaps the most likely candidates. This population makes up the majority of associated 3FGL sources at high galactic latitudes. In addition, a large fraction of the unassociated 3FGL sources likely have blazar counterparts [e.g. @2012ApJ...752...61M; @2012ApJ...753...83A; @2016PhRvL.116o1105A; @2015PhR...598....1F]. Generic radio sources [@2003MNRAS.342.1117M; @2012MNRAS.422.1527M] number in the hundreds of thousands, with only a tiny fraction being associated with a gamma-ray source. In contrast, around 30% of the approximately 3000 known blazar candidates appear in the 3FGL . As we have shown, comparing the gamma-ray spectra of blazars with that of a tentative gamma-ray source offers a way of making a distinction between dark matter annihilation and blazar emission. Taken at face value, the high curvature and spectral energy peak of RetII are markedly different from the two main blazar types. In particular, ${E_\mathrm{peak}}$ is measured more robustly than $\beta$ making the separation between RetII and the FSRQs especially clean. The separation between RetII and the BL Lacs is based on the apparently large curvature of RetII, though this separation is less marked than for the FSRQs. This is particularly important as @2017JCAP...07..025R have identified two BL Lac candidates behind RetII. As Fermi increases its exposure (and if there is in fact a source in RetII’s direction), confidence regions in Fig. \[fig:logparabola3fgl\_Epeakbeta\] will shrink, the catalog of gamma-ray loud blazars will expand, and the comparison between RetII and blazar types will come into sharper focus. In contrast to blazars, our analysis of gamma-ray spectral shape cannot distinguish a dark matter signal from pulsar emission. This spectral similarity has been a central issue in the search for dark matter annihilation at the Galactic Center [e.g. @2011JCAP...03..010A; @2013PhRvD..88h3009H; @2016PhRvL.116e1102B; @2016PhRvL.116e1103L]. Searches in dwarf galaxies appear to avoid the problem (Sec. \[sec:pulsars\], also [@2016ApJ...832L...6W]). Of course, RetII may be a system with a peculiar history [e.g. @2016ApJ...830...93J; @2017arXiv170706871S] and our analogy with globular clusters may break down. Future study of RetII at all wavelengths will help pin down possible gamma-ray sources within (and behind) RetII. Our discussion in Sec. \[sec:significance\] about the relative probabilities of a statistical fluctuation vs. an additional gamma-ray source in RetII’s direction is based on a well known property of the Fermi sky: sampling random sky locations turns up more “high-significance” locations (i.e. hot pixels) than would be expected from the Poisson statistical fluctuations of the diffuse model (see, e.g. [@2015PhRvD..91h3535G; @2014PhRvD..89d2001A; @2015PhRvD..91f1302C; @2015PhRvL.115w1301A; @2017ApJ...834..110A] in the context of dark matter searches). This phenomenon has been invoked to argue for a millisecond pulsar explanation of the Galactic Center gamma-ray excess [@2015JCAP...05..056L; @2016PhRvL.116e1102B; @2016PhRvL.116e1103L] and, at higher latitudes, to constrain source populations [@2010JCAP...01..005F; @2011ApJ...738..181M; @2014ApJ...796...14C; @2016ApJ...826L..31Z; @2016ApJS..225...18Z; @2016PhRvL.116o1105A; @2017ApJ...839....4D; @2016ApJ...832..117L] as well as dark matter annihilation [@2009JCAP...07..007L; @2010PhRvD..82l3511B; @2015JCAP...09..027F; @2017arXiv171001506Z]. In this work, the task is to understand the origin of one particular hot pixel (i.e. RetII) that is known to host a dark matter halo with a large $J$ value. As the origin of the excess high-significance locations becomes better understood (i.e. via “1-point function” analyses) it will be possible to quantify the probability that a RetII-like observation is caused by a particular class of sources. Finally, we return to the differences between Pass 7 and Pass 8. There are differences in detection significance and in the best fitting properties of the RetII source when analyzed with the two data sets. The ultimate solution requires finding the probability of jointly obtaining the Pass 7 and Pass 8 results when there either is or is not a source in RetII’s direction. This is beyond the scope of this work. However, our analysis can partially address the consistency question: is there a single RetII energy spectrum consistent with both the Pass 7 and Pass 8 data sets? Though we analyzed the two data sets independently, in reality they are highly overlapping [e.g. @2015PhRvL.115w1301A]: dividing the events detected within $15{^\circ}$ of RetII into energy bins we find that, over the same 6.9 observation, about 60-70% of events between 1 and 10 GeV found in one data set are also found in the other. We can obtain a necessary condition for consistency by treating the Pass 7 and Pass 8 data sets as two independent observations of the same object. If there is no single spectrum that is a good fit to both data sets when they are treated independently then the data sets will certainly be inconsistent had their dependence been properly included. The best fitting log parabola model to the combined Pass 7/Pass 8 data set (with all three log parabola parameters completely free) is $\alpha=-8.1$, $\beta=3.5$, ${E_\mathrm{peak}}=4.2\,{\mathrm{GeV}}$, and $F_0 = 1.5\times 10^{-13} \,{{\mathrm{cm}}}^{-2} \sec^{-1} {\mathrm{GeV}}^{-1}$. Taking this model as the null model $\theta_0$ (Sec. \[sec:GOFmethod\]) the goodness of fit test statistic is $\lambda({\mathbf{X}}_\gamma)=1.1$ for the Pass 7 data, and $\lambda({\mathbf{X}}_\gamma)=1.0$ for the Pass 8 data. The distribution for $\lambda({\mathbf{X}}_\gamma)$ should be $\chi^2$ distributed with 3 degrees of freedom, giving $p\approx 0.8$ for both Pass 7 and Pass 8. When considered independently, the confidence intervals for the log parabola parameters inferred from Pass 7 and 8 are therefore highly overlapping. Correlations between the two data sets will increase the level tension but we conclude that consistency is plausible. There are also indications of consistency between 6.9 years of Pass 7 the 9-year Pass 8 results of @2018arXiv180506612L. The best fitting dark model reported by [@2018arXiv180506612L] is for dark matter with a mass of 16 GeV annihilating into $\tau$ leptons. While they do not report a best fitting annihilation cross section, this mass fits the 6.9-year data essentially as well as the best fitting masses we find ($M=13.9~{\mathrm{GeV}}$ in Pass 7 and 19.5 GeV in 6.9 years of Pass 8; $\lambda({\mathbf{X}}_\gamma)$ changes by $\sim 0.1$ when shifting these masses to 16 GeV while maximizing with respect to ${\langle {\sigma}v \rangle}J$). As for level of significance, the reported $TS=13.5$ corresponds to $p \approx 10^{-4}$ for the diffuse Poisson background model as mentioned in the introduction. Using Fig. 11 of [@2017ApJ...834..110A], which is based on a blank sky calibration that takes into account a trials factor needed for searching multiple masses, such a $TS$ value corresponds to $p\approx 0.01$. This blank sky method for evaluating significance is approximately analogous to the empirical background sampling method which yielded $p=0.01$ in 6.5 years of Pass 7 data [@2015PhRvL.115h1101G]. Thus the arguments of Sec. \[sec:significance\] may well hold for the 9-year Pass 8 data as well as the 6.5-year Pass 7 data. Conclusions =========== We present a series of analyses to follow up on the detection of a gamma-ray signal from the direction of a dwarf galaxy. Our main focus is on assessing whether there is a plausible astrophysical interpretation for the signal. We first quantify the probability that the excess is due to a Poisson fluctuation of diffuse background processes vs. the existence of a previously unknown point source source along the line of sight. We show that comparing the gamma-ray spectrum of the new source to those of known classes of gamma-ray emitters can help rule out a chance alignment with an unrelated background object. Finally, we estimate the level of emission from a population of pulsars within the dwarf which could mimic a dark matter signal. These analyses are applied to Fermi observations of the Reticulum II dwarf, the most promising dwarf dark matter signal seen so far. We find that a line of sight featuring a gamma-ray excess like RetII’s has high likelihood (probability greater than 99%) of hosting a gamma-ray source with flux above the diffuse background level. We use a simple log parabola parameterization of RetII’s gamma-ray spectrum and compare with known sources in the 3FGL catalog. RetII has a significantly curved energy spectrum, which is a distinctive feature among gamma-ray sources. We find that of the blazar types (which represent the majority of high latitude associated gamma-ray sources), flat spectrum radio quasars emit most of their gamma-ray energy at lower energy (${E_\mathrm{peak}}$) than RetII does. BL Lacs can emit at energies comparable to RetII’s, though they in general have spectral curvatures too low to explain the RetII data. All of these conclusions are stronger when considering 6.9 years of Pass 7 data than the same amount of Pass 8 data, for which the significance of the RetII excess is lower and all confidence regions expand considerably. For any promising dark matter target, not just RetII, these techniques will help to distinguish a dark matter explanation from an astrophysical one. We acknowledge useful discussions with Kev Abazajian, Gordon Blackadder, Ian Dell’Antonio, Raphael Flauger, Sebastien Fromenteau, Rick Gaitskell, Manoj Kaplinghat, François Lanusse, Sandhya Rao, Pat Scott, Sukhdeep Singh, Louie Strigari, Roberto Trotta, and Aaron Vincent. SMK is supported by DE-SC0017993. MGW is supported by NSF grants AST-1313045 and AST-1412999. . [^1]: <http://fermi.gsfc.nasa.gov/ssc/> [^2]: The nearest source in Fermi’s Third Source Catalog (3FGL) [@2015ApJS..218...23A] is a BL Lac blazar, 1RXS J032521.8-563543, located $2.9{^\circ}$ from RetII. Adopting the spectral model from the 3FGL and the Pass 7 PSF and exposure in the direction of RetII, this source is expected to contribute 0.3 events to the RetII ROI (within uncertainties in the spectral model this can rise to 0.36 events). Other nearby sources (at $3.7{^\circ}$ and $4.4{^\circ}$) contribute significantly fewer events. In contrast, Galactic diffuse and isotropic gamma-ray backgrounds are expected to contribute 140 events to the RetII ROI. [^3]: <http://fermi.gsfc.nasa.gov/ssc/data/access/lat/BackgroundModels.html> [^4]: Pass 7: `iso_source_v05.txt`, Pass 8: `iso_P8R2_SOURCE_V6_v06.txt` [^5]: Pass 7: `gll_iem_v05_rev1.fit`, Pass 8: `gll_iem_v06.fits` [^6]: We determined this by extracting the spectrum of the Fermi Pass 7 diffuse interstellar emission model within every $0.125{^\circ}\times 0.125{^\circ}$ pixel covering the whole sky and adding it to the isotropic spectral model. At no location does the spectral energy distribution $E^2 dF/dE$ peak above 1 GeV (compare with $\gtrsim 2~{\mathrm{GeV}}$ for RetII; see Secs. \[sec:GOF\] and \[sec:3fgl\]), with 99.8% of the sky having $E^2 dF/dE$ peak below 0.7 GeV (the Pass 8 model gives similar results). [^7]: More precisely, $D$ is the statement that the test statistic for the ROI is larger than the one observed in RetII. [^8]: Applying this argument to the actual RetII ROI would take us into Bayesian territory. In that case $P(D)$ is greater than 0.01, but to quantify it we would have to assign degrees of belief to the various particle properties of dark matter and the parameters describing RetII’s dark matter halo. [^9]: In this work we do not model the finite energy resolution of the LAT ($\Delta E/E \lesssim 0.1$ for $E \gtrsim 0.5\,{\mathrm{GeV}}$) as the spectra we consider are much broader than this. [^10]: This holds when the true parameters are sufficiently far from the boundary of parameter space. In the case where the null hypothesis is background-only (i.e. $F_0=0$) about 10% the samples have $\lambda({\mathbf{X}}_\gamma)=0$ while the rest are gamma-distributed with $k \approx0.8$ and $\theta \approx 2.3$.

--- abstract: 'In this paper, we will show that $\RT^{2}+\WKLo$ is a $\Pi^{1}_{1}$-conservative extension of $\BN[3]$.' author: - 'Theodore A. Slaman' - Keita Yokoyama bibliography: - 'bib-rt2inf.bib' date: 'July 5, 2018 ' title: | The strength of Ramsey’s theorem for\ pairs and arbitrarily many colors --- Introduction ============ The strength of Ramsey’s theorem is well-studied in the setting of reverse mathematics. In this paper, we will focus on the first-order consequences of Ramsey’s theorem for pairs over the base system $\RCAo$. On the first-order part of Ramsey’s theorem for pairs and two colors ($\RT^{2}_{2}$), Hirst[@Hirst-PhD] showed that it implies $\BN[2]$ and then Cholak/Jockusch/Slaman[@CJS] proved that $\RT^{2}_{2}+\WKLo+\IN[2]$ is a $\Pi^{1}_{1}$-conservative extension of $\IN[2]$. Thus, its first-order part is in between $\BN[2]$ and $\IN[2]$. There are many studies to determine the exact strength, and recently Chong/Slaman/Yang[@CSY2017] showed that $\RCAo+\RT^{2}_{2}$ does not imply $\IN[2]$, and Patey/Yokoyama[@PY2018] showed that $\WKLo+\RT^{2}_{2}$ is a $\Pi^{0}_{3}$-conservative extension of $\BN[2]$, which means that the first-order part of $\RT^{2}_{2}$ is closer to $\BN[2]$. How about the strength of Ramsey’s theorem for pairs and arbitrarily many colors ($\RT^{2}$)? Over $\RCAo$, one may easily see that $\RT^{2}_{k}$ implies $\RT^{2}_{k+1}$, but that does not mean $\RT^{2}_{2}$ implies $\RT^{2}$ since the induction available within $\RCAo$ is not strong enough. Indeed, the case for $\RT^{2}$ is very similar to the case for $\RT^{2}_{2}$ and the following are known. $\RT^{2}+\RCAo$ implies $\BN[3]$. \[thm:CJS-RT2\] $\RT^{2}+\WKLo+\IN[3]$ is a $\Pi^{1}_{1}$-conservative extension of $\IN[3]$. Hence, the first-order part of $\RT^{2}$ is between $\BN[3]$ and $\IN[3]$. Here, we will sharpen the proof of this theorem, and determine the exact first-order part of $\RT^{2}$, namely it is $\BN[3]$. For the basic notions of this area, see [@CJS; @Slicing-the-truth; @SOSOA]. The first-order part of $\RT^{2}$ ================================= Our main conservation theorem is the following. $\RT^{2}+\WKLo$ is a $\Pi^{1}_{1}$-conservative extension of $\BN[3]$. To show the main theorem, we will sharpen the argument from [@CJS], which is used for the proof of Theorem \[thm:CJS-RT2\]. Over $\RCAo$, $\RT^{2}$ is equivalent to $\D^{2}$ plus $\COH$. Here, $\D^{2}$ and $\COH$ are the following statements. - $\D^{2}$: for any $k\in \N$ and any $\Delta^{0}_{2}$-partition $\N=\bigsqcup_{i<k} \mc{A}_{i}$, there exists an infinite set $Z\subseteq \N$ such that $Z\subseteq \mc{A}_{i}$ for some $i<k$, - $\COH$: for any infinite sequence of sets $\langle R_{i}: i\in\N \rangle$, there exists an infinite set $Z\subseteq \N$ such that $(Z\subseteq^{*} R_{i}\vee Z\subseteq^{*} \N\setminus R_{i})$ for any $i\in\N$. (Note that $\N$ denotes the set of all natural numbers within $\RCAo$, *i.e.*, if $\mc M=(M,S)$ is a model of $\RCAo$, $\N^{\mc M}=M$.) Since we already know that $\RCAo+\RT^{2}$ implies $\BN[3]$, we will consider the first-order strength of the above two statements over $\BN[3]$. Note that $\D^{2}$ and $\COH$ are both $\Pi^{1}_{2}$-statements, and $\Pi^{1}_{1}$-conservation results for $\Pi^{1}_{2}$-statements can be amalgamated, *i.e.*, if both of $\RCAo+\BN[3]+\D^{2}$ and $\RCAo+\BN[3]+\COH$ are $\Pi^{1}_{1}$-conservative over $\BN[3]$ then so is $\RCAo+\BN[3]+\D^{2}+\COH$, which is equivalent to $\RCAo+\RT^{2}$ (see [@Y2010]). The strength of $\COH$ (together with weak König’s lemma) over $\BN[3]$ is already known. $\WKLo+\COH+\BN[3]$ is a $\Pi^{1}_{1}$-conservative extension of $\BN[3]$. Thus, what we need is the following. \[thm:main-conservation\] $\RCAo+\D^{2}+\BN[3]$ is a $\Pi^{1}_{1}$-conservative extension of $\BN[3]$. In [@CJS], it is shown by a variant of Mathias forcing that a computable instance of $\D^{2}$ admits a $\low_{2}$-solution. On the other hand, $\low_{2}$-sets preserve $\BN[3]$ since they won’t add any new $\Sigma^{0}_{3}$-sets. Thus, the following theorem is essential for Theorem \[thm:main-conservation\]. \[thm:main-construction\] Let $(M,\{B\})$ be a countable model of $\BN[3]$, and let $M=\bigsqcup_{i<k} \mc{A}_{i}$ be a $\Delta^{B}_{2}$-partition of $M$ for some $k\in M$. Then there exists an unbounded $\Delta^{B}_{3}$-set $G\subseteq M$ such that $G\subseteq \mc{A}_{i}$ for some $i<k$, and any $\Sigma^{B\oplus G}_{3}$ subset of $M$ is already $\Sigma^{B}_{3}$ in $M$. We will prove this theorem in the next section. Assuming this theorem, it is routine work to prove Theorem \[thm:main-conservation\]. Assume that $\BN[3]$ does not prove a $\Pi^{1}_{1}$-sentence $\A X\psi(X)$. Then there exists a countable model $(M,S)\models \BN[3]$ such that $(M,S)\models \neg\psi(B)$ for some $B\in S$. For $X,Y\subseteq M$, $X\le_{T} Y$ means that $X$ is $\Delta^{Y}_{1}$ in $M$. By using Theorem \[thm:main-construction\] repeatedly, one can construct an $\omega$-length sequence of subsets of $M$, $B=B_{0}\le_{T}B_{1}\le_{T} \dots$ so that - for any $m\in\omega$ and $\Delta^{B_{m}}_{2}$-partition $M=\bigsqcup_{i<k} \mc{A}_{i}$, there exist $n\ge m$ and an unbouded set $G\le_{T} B_{n}$ such that $G\subseteq \mc{A}_{i}$ for some $i<k$, and, - any $\Sigma^{B_{m}}_{3}$ subset of $M$ is already $\Sigma^{B}_{3}$ in $M$. Put $\bar S=\{X\subseteq M: X\le_{T} B_{m}, m\in\omega\}$, then $(M,\bar S)\models \RCAo+\D^{2}+\BN[3]$ but $\neg\psi(B)$ is still true in $(M,\bar S)$. Hence $\RCAo+\D^{2}+\BN[3]$ does not prove $\A X\psi(X)$. Construction ============ In this section, we will prove Theorem \[thm:main-construction\]. The main idea is formalizing a computability theoretic construction within a nonstandard model of arithmetic. The following theorem is a basic tool to formalize standard arguments for $\Pi^{0}_{1}$-classes, and we will use it freely throughout this section. \[thm:Pi01-in-WKL\] Let $\varphi(X,A)$ be a $\Pi^{0}_{1}$-formula with exactly displayed the set variables. 1. There exists a $\Pi^{0}_{1}$-formula $\psi(A)$ such that $\WKLo$ proves $\E X\varphi(X,A)\leftrightarrow \psi(A)$. 2. $\WKLo$ proves that $\E X\varphi(X,A)$ is equivalent to the statement that there exists (a $\Delta^{A}_{2}$-code for) a low set $Y$ relative to $A$ such that $\varphi(Y,A)$. 3. For a given $\Delta^{0}_{2}$-definable set $\mc{A}$ (possibly not a second-order object), $\WKLo+\BII$ proves $\E X\varphi(X,\mc{A})\to \E X\E Y\varphi(X,Y)$. Thus, “there exists $\Delta^{0}_{2}$-definable set $\mc{A}$ such that $\E X\varphi(X,\mc A)$” can be described by a $\Pi^{0}_{1}$-formula. 1 is a well-known fact, see, e.g., [@SOSOA Lemma VIII.2.4]. 2 is a low basis theorem for $\Pi^{0}_{1}$-classes which is formalizable within $\II$ [@HK1989]. With $\BII$, one can mimic the proof of 1 for $\Delta^{0}_{2}$-sets, 3 easily follows from that. As we mentioned in the previous section, we want to formalize the second $\low_{2}$-solution construction for $\D^{2}$ from [@CJS] within $\BN[3]$. However, that construction uses $\IN[3]$ in two parts, to find the right color for a solution, and to do $\mathbf{0}''$-primitive recursion. In the following construction, we need to avoid these. To overcome the first problem, we will construct solutions for all possible colors, and see that it works for at least one color in the end. For the second problem, we will still use $\mathbf{0}''$-primitive recursion. In a nonstandard model $(M,S)\models\BN[3]$, $\mathbf{0}''$-primitive recursion might end in nonstandard numbers of steps which form a proper cut of $M$. Thus, we will decide some finite collection of $\Sigma^{0}_{2}$-statements at each step, and finally decide all $\Sigma^{0}_{2}$-statements before $\mathbf{0}''$-primitive recursion ends, adapting Shore’s blocking argument. Now we start the construction. Let $(M,\{B\})$ be a countable model of $\BN[3]$. By the following theorem, we will work within $(M,S)\models \WKLo+\BIII$ with $B\in S$. Let $(M,\{B\})$ be a countable model of $\BN[3]$. Then there exists $S\subseteq \mc{P}(M)$ such that $B\in S$ and $(M,S)\models \WKLo+\BN[3]$. In what follows, we will mimic the “double jump control” method in [@CJS]. Let $\bigsqcup_{i<k}\mc{A}_{i}=M$ be a $\Delta^{B}_{2}$-partition for some $k\in M$ and $B\in S$. A quintuple $p=(\bar{F}, X, \sigma, \ell_{0},\ell_{1})$ is said to be a *pre-condition* if - $\ell_{0},\ell_{1}\in M$, $\sigma:\ell_{0}\times k\to 2$, - $\bar{F}$ is a $k$-tuple of finite sets $\langle F_{i}: i<k \rangle$ such that $F_{i}\subseteq \mc{A}_{i}$, - $X$ is coded by $\ell_{1}$ and (a $\Delta^{B}_{2}$ code for) an infinite $\low^{B}$ set $X_{0}$ as $X=X_{0}\cap(\ell_{1},\infty)$, - $\max \bar{F}\cup \{\ell_{0}\}<\ell_{1}$, and a code for $X_{0}$ is bounded by $\ell_{1}$. Here, we call a pair of $k$-tuple of finite sets and another set $(\bar{F},X)$ with $\min X>\max\bar{F}$ a Mathias pair. (In what follows, we will mainly deal with an infinite Mathias pair, *i.e.*, a Mathias pair with $X$ infinite, but quantification for Mathias pairs ranges over possibly finite Mathias pairs.) For finite sets $E,F$ and another set $X$, we write $E\in (F,X)\leftrightarrow F\subseteq E\subseteq F\cup X$. For two Mathias pairs $(\bar{F},X),(\bar{E}, Y)$, we say that $(\bar{E}, Y)$ *extends* $(\bar{F},X)$ (write $(\bar{F},X)\ge (\bar{E}, Y)$) if $E_{i}\in (F_{i},X)$ for every $i<k$, and $Y\subseteq X$. Next, we define how Mathias pairs force $\Sigma^{0}_{1}$ and $\Sigma^{0}_{2}$-formulas at each color. To control the complexity of forcing formulas, we consider a triple of the form $(\bar{F},X,\ell)$, which is a Mathias pair $(\bar{F},X)$ with a bound $\ell\in M$. Let $\theta(n, G[n])$ be a $\Sigma^{0}_{0}$-formula with a new variable $G$. Then we define strong forcing $\Vdash^{+}$ for a pair of color $i$ and a $\Sigma^{0}_{1}$-formula $\E n\,\theta(n, G[n])$ as $$(\bar{F},X,\ell)\Vdash^{+}\langle i,\E n\,\theta(n, G[n]) \rangle\Leftrightarrow \E n\le \max F_{i}\,\theta (n, F_{i}[n]).$$ Similarly, let $\theta(m,n, G[n])$ be a $\Sigma^{0}_{0}$-formula with a new variable $G$. Then we define forcing $\Vdash$ for a pair of color $i$ and a $\Sigma^{0}_{2}$-formula $\E m\A n\,\theta(m,n, G[n])$ as $$(\bar{F},X,\ell)\Vdash\langle i,\E m\A n\,\theta(m,n,G[n]) \rangle\Leftrightarrow \E m\le \ell\,\A E\in (F_{i},X)\A n\le\max E\,\theta (m,n, E[n]).$$ Let $\pi(e, m, G)\equiv \A n\, \pi_{0}(e,m,n,G[n])$ be a universal $\Pi^{B,G}_{1}$-formula, *i.e.*, a universal $\Pi^{0}_{1}$-formulas with a new set variable $G$ (and a set parameter $B$). For a finite partial function $\sigma\subseteq M\times k\to 2$, we let $$\begin{aligned} \sigma_{+}&:=\{\langle i, \E m\,\pi(e,m,G) \rangle: \sigma(e,i)=1\},\\ \sigma_{+,i,\le \ell}&:=\{\langle i, \E m(\pi(e,m,G)\wedge m\le \ell) \rangle: \sigma(e,i)=1\},\\ \sigma_{-}&:=\{\langle i, \E m\,\pi(e,m,G) \rangle: \sigma(e,i)=0\}.\end{aligned}$$ Let $\sigma$ be a finite partial function $\sigma\subseteq M\times k\to 2$. 1. A Mathias pair $(\bar{F},X)$ is said to be *$\sigma$-large* if for any finite sets of (possibly finite) Mathias pairs $\{(\bar{E}^{t}, Y^{t})\}_{t<s}$ and any bound $\ell'\in M$ such that for all $t<s$ and for all $i<k$, ${E}^{t}_{i}\subseteq \mc{A}_{i}$, $(\bar{E}^{t}, Y^{t})\le (\bar{F}, X)$, $\ell'\ge \max\bar{E}^{t}$, and $X\supseteq \bigsqcup_{t<s}Y^{t}\supseteq X\setminus \ell'$ (*i.e.*, $Y^{t}$’s partition a superset of $X\setminus \ell'$ which is included in $X$), there exists $t<s$ such that $(\bar{E}^{t}, Y^{t},\ell')\not\Vdash\langle i,\psi \rangle$ for any $\langle i,\psi \rangle\in \sigma_{+}$ and $Y^{t}$ is not bounded by $\ell'$. 2. Let $i<k$, $\ell\in M$. Then a Mathias pair $(\bar{F},X\cap \mc{A}_{i})$ is said to be *$\sigma$-large at $i$ up to $\ell$* if the largeness holds for $\sigma_{+,i,\le\ell}$ instead of $\sigma_{+}$ with considering all possible $\Delta^{0}_{2}$-definable sets for $Y^{t}$’s. Formally, $(\bar{F},X\cap \mc{A}_{i})$ is *$\sigma$-large at $i$ up to $\ell$* if for any $\Delta^{0}_{2}$-definable finite sets of Mathias pairs $\{(\bar{E}^{t}, Y^{t})\}_{t<s}$ and any bound $\ell'\in M$ such that for all $t<s$, ${E}^{t}_{i}\subseteq \mc{A}_{i}$, $(\bar{E}^{t}, Y^{t})\le (\bar{F}, X\cap\mc{A}_{i})$, $\ell'\ge \max\bar{E}^{t}$, and $X\cap \mc{A}_{i}\supseteq \bigsqcup_{t<s}Y^{t}\supseteq (X\cap \mc{A}_{i})\setminus \ell'$, there exists $t<s$ such that $(\bar{E}^{t}, Y^{t},\ell')\not\Vdash\langle i,\psi \rangle$ for any $\langle i,\psi \rangle\in \sigma_{+,i,\le\ell}$ and $Y^{t}$ is not bounded by $\ell'$. (Here, we consider all $\Delta^{0}_{2}$-definable sets in $(M,S)$ with any parameters from $S$. Be aware that we do not restrict to $\Delta^{B}_{2}$-sets.) Roughly speaking, $\sigma$-largeness guarantees that one can find an extension without forcing any $\langle i,\psi \rangle\in \sigma_{+}$ in the future construction. \[rem:largeness\] 1. The notion “$(\bar{F},X)$ is $\sigma$-large” won’t be changed whether we consider Mathias pairs $(\bar{E}^{t}, Y^{t})$ with $Y^{t}$ being a set in the structure or a $\Delta^{0}_{2}$-definable set by Theorem \[thm:Pi01-in-WKL\].3, and it is described by a $\Pi^{B}_{2}$-formula. 2. For the case “$(\bar{F},X\cap \mc{A}_{i})$ is $\sigma$-large at $i$ up to $\ell$”, it is essential to consider $\Delta^{0}_{2}$-definable sets, and thus the statement cannot be described by a $\Pi^{B}_{2}$-formula. In the following construction (which will be $B''$-primitive recursive), we will avoid checking this requirement directly. A pre-condition $p=(\bar{F}^{p}, X^{p}, \sigma^{p}, \ell_{0}^{p},\ell_{1}^{p})$ is said to be a *condition* if 1. $(\bar{F}^{p}, X^{p})$ is $\sigma^{p}$-large, 2. $(\bar{F}^{p}, X^{p},\ell_{1}^{p})\Vdash \langle i,\psi \rangle$ for any $\langle i,\psi \rangle\in \sigma^{p}_{-}$, 3. if $(\bar{F}^{p}, X^{p}\cap \mc{A}_{i})$ is $\sigma^{p}$-large at $i$ up to $\ell_{0}^{p}$, then, $\A m\le \ell_{0}^{p}$, $(\bar{F}^{p}, X^{p},\ell_{1}^{p})\Vdash^{+} \langle i, \neg\pi(e,m,G) \rangle$ for any $e\le \ell_{0}^{p}$ with $\sigma^{p}(e,i)=1$. Define $\mathbb{P}$ as the set of all conditions. For given two conditions $p,q\in \mathbb{P}$, $q$ properly extends $p$ ($p\succ q$) if $$(\bar{F}^{p}, X^{p})\ge(\bar{F}^{q}, X^{q})\wedge \ell_{1}^{p}\le \ell_{0}^{q}\wedge \sigma^{p}\subseteq\sigma^{q}.$$ For a given condition $p=(\bar{F}^{p}, X^{p}, \sigma^{p}, \ell_{0}^{p},\ell_{1}^{p})$, we want to find an extension of $p$. For this, we introduce a weaker version of the largeness notion. Let $\sigma$ be a finite partial function $\sigma\subseteq M\times k\to 2$. A Mathias pair $(\bar{F},X)$ is said to be *$\sigma$-fair* if - there exist a finite set of Mathias pairs $\{(\bar{E}^{t}, Y^{t})\}_{t<s}$ and a bound $\ell'\in M$ such that ${E}^{t}_{i}\subseteq \mc{A}_{i}$, $(\bar{E}^{t}, Y^{t})\le (\bar{F}, X)$, $\ell'\ge \max\bar{E}^{t}$, $X\supseteq \bigsqcup_{t<s}Y^{t}\supseteq X\setminus \ell'$ such that for any $t<s$, - if $(\bar{E}^{t}, Y^{t},\ell')\not\Vdash\langle i,\psi \rangle$ for any $\langle i,\psi \rangle\in \sigma_{+}$, then $(\bar{E}^{t}, Y^{t},\ell')\Vdash\langle i,\psi \rangle$ for every $\langle i,\psi \rangle\in \sigma_{-}$, or, - $Y^{t}$ is bounded by $\ell'$, and, - for any finite set of Mathias pairs $\{(\bar{E}^{t}, Y^{t})\}_{t<s}$ and a bound $\ell'\in M$ which witness the condition $(\dag)$, there exists $t<s$ such that $(\bar{E}^{t}, Y^{t},\ell')\not\Vdash\langle i,\psi \rangle$ for any $\langle i,\psi \rangle\in \sigma_{+}$ and $Y^{t}$ is not bounded by $\ell'$. Note that “$(\bar{F},X)$ is $\sigma$-fair” can be described by a boolean combination of $\Sigma^{B}_{2}$ and $\Pi^{B}_{2}$ formulas. \[lem:fair-extension\] Let $p=(\bar{F}^{p}, X^{p}, \sigma^{p}, \ell_{0}^{p},\ell_{1}^{p})$ be a condition, and let $\ell'\ge \ell_{1}^{p}$. Then $(\bar{F}^{p}, X^{p})$ is $\tau$-fair for some $\tau:\ell'\times k\to 2$ extending $\sigma^{p}$. Moreover, one can find a lexicographically maximal such $\tau$. Since $p$ is a condition, $(\bar{F}^{p}, X^{p})$ is $\sigma^{p}$-fair. We will see by $\Sigma^{0}_{2}$-induction that for any finite set $H\subseteq M\times k$, there exists $\tau:\dom(\sigma^{p})\cup H\to 2$ such that $\tau\supseteq \sigma^{p}$ and $(\bar{F}^{p}, X^{p})$ is $\tau$-fair. For this, we only need to see that for any $\sigma'$ extending $\sigma^{p}$ such that $(\bar{F}^{p}, X^{p})$ is $\sigma'$-fair and $(e_{0},i_{0})\in M\times k\setminus \dom(\sigma')$, either $\sigma'\cup\{(e_{0},i_{0},0)\}$ or $\sigma'\cup\{(e_{0},i_{0},1)\}$ satisfies the fairness condition for $(\bar{F}^{p}, X^{p})$. Assume that $(\bar{F}^{p}, X^{p})$ is not $\sigma'\cup\{(e_{0},i_{0},1)\}$-fair. Since any finite set of Mathias pairs and a bound which witness the condition $(\dag)$ for $(\bar{F}^{p}, X^{p})$ to be $\sigma'$-fair actually witness $(\dag)$ for $(\bar{F}^{p}, X^{p})$ to be $\sigma'\cup\{(e,i,1)\}$-fair, the condition $(\dag\dag)$ for $(\bar{F}^{p}, X^{p})$ to be $\sigma'\cup\{(e_{0},i_{0},1)\}$-fair must fail. Thus, there exist a finite set of Mathias pairs $\{(\bar{E}^{t}, Y^{t})\}_{t<s}$ and a bound $\ell'\in M$ which witness the condition $(\dag)$ for $\sigma'\cup\{(e,i,1)\}$ such that for any $t<s$, $(\bar{E}^{t}, Y^{t},\ell')\Vdash\langle i,\psi \rangle$ for some $\langle i,\psi \rangle\in \sigma'_{+}\cup\{\langle i_{0}, \E m\,\pi(e_{0},m,G) \rangle\}$ or $Y^{t}$ is bounded by $\ell'$. Thus, for any $t<s$, if $Y^{t}$ is not bounded by $\ell'$, then $(\bar{E}^{t}, Y^{t},\ell')\not\Vdash\langle i,\psi \rangle$ for any $\langle i,\psi \rangle\in \sigma'_{+}$ implies $(\bar{E}^{t}, Y^{t},\ell')\Vdash\langle i,\psi \rangle$ for any $\langle i,\psi \rangle\in \sigma'_{-}\cup\{\langle i_{0}, \E m\,\pi(e_{0},m,G) \rangle\}$. This means $\{(\bar{E}^{t}, Y^{t})\}_{t<s}$ and $\ell'$ witness the condition $(\dag)$ for $(\bar{F}^{p}, X^{p})$ to be $\sigma'\cup\{(e_{0},i_{0},0)\}$-fair. The condition $(\dag\dag)$ for $\sigma'\cup\{(e_{0},i_{0},0)\}$ is automatically satisfied by the same condition for $\sigma'$. \[lem:left-most-extension\] For any $p\in\mathbb{P}$, there exists $q\in \mathbb{P}$ such that $q\prec p$. Moreover, one can construct such an extension in a “left-most” way, *i.e.*, there is a canonical definable way to choose needed elements in the construction of an extension. For a given condition $p=(\bar{F}^{p}, X^{p}, \sigma^{p}, \ell_{0}^{p},\ell_{1}^{p})\in \mathbb{P}$, put $\ell_{0}=\ell_{1}^{p}$. By Lemma \[lem:fair-extension\], there exists a lexicographically maximal $\tau:\ell_{0}\times k\to 2$ which extends $\sigma^{p}$ such that $(\bar{F}^{p}, X^{p})$ is $\tau$-fair. Then one can find a family of low Mathias pairs $\{(\bar{E}^{t}, Y^{t})\}_{t<s}$ (of smallest index) and a bound $\ell'\in M$ which witness $(\dag)$. By $(\dag\dag)$, pick the smallest $t<s$ such that $(\bar{E}^{t}, Y^{t})$ is $\tau$-large. Such a $t<s$ exists by $\BII$ since for any $\ell''\ge \ell'$ and for any $\{(\bar{D}^{t}, Z^{t})\}_{t<s''}$ which refines $\{(\bar{E}^{t}, Y^{t})\}_{t<s}$, one can apply $(\dag\dag)$ for $\{(\bar{D}^{t}, Z^{t})\}_{t<s''}$ and $\ell''$. Note that $\tau$-largeness implies that $(\bar{E}^{t}, Y^{t},\ell')\not\Vdash\langle i,\psi \rangle$ for any $\langle i,\psi \rangle\in \tau_{+}$ and $Y^{t}$ is infinite, thus, by $(\dag)$, $(\bar{E}^{t}, Y^{t},\ell')\Vdash\langle i,\psi \rangle$ for any $\langle i,\psi \rangle\in \tau_{-}$. Now $(\bar{E}^{t}, Y^{t},\tau,\ell_{0},\ell')$ satisfies the first and second clauses to be a condition. For the third clause, we use the following claims. We say that $(\bar{D}', Z')$ is a finite extension of $(\bar{D}, Z)$ at $i$ if $(\bar{D}', Z')\le(\bar{D}, Z)$, $Z\setminus Z'$ is finite, and $D'_{i'}=D_{i'}$ for any $i'\neq i$. One can observe that finite extensions preserve $\tau$-largeness. Let $(\bar{D}, Z)$ be a finite extension of $(\bar{E}^{t}, Y^{t})$ at $i$. If $(\bar{D}, Z\cap \mc{A}_{i})$ is $\tau$-large at $i$ up to $\ell_{0}$, then for any $e< \ell_{0}$ such that $\tau(e,i)=1$, there exists a finite extension $(\bar{D}', Z')\le(\bar{D}, Z)$ at $i$ such that $D'_{i}\in (D_{i},Z\cap\mc{A}_{i})$ and $(\bar{D}', Z',\max\bar{D}'\cup\{\ell'\})\Vdash^{+} \langle i, \A m\le \ell_{0}\,\neg\pi(e,m,G) \rangle$. If $(\bar{E}^{t}, Y^{t}\cap\mc{A}_{i})$ is $\tau$-large at $i$ up to $\ell_{0}$, then there exists a finite extension $(\bar{D}', Z')\le(\bar{E}^{t}, Y^{t})$ at $i$ such that $D'_{i}\in ({E}^{t}_{i}, Y^{t}\cap\mc{A}_{i})$ and $(\bar{D}', Z',\max\bar{D}'\cup\{\ell'\})\Vdash^{+} \langle i, \A m\le \ell_{0}\,\neg\pi(e,m,G) \rangle$ for all $e<\ell_{0}$ such that $\tau(e,i)=1$. One can easily check the first claim by unfolding the definition of $\tau$-largeness at $i$ up to $\ell_{0}$. Since finite extensions preserve $\tau$-largeness at $i$, the second claim is obtained by applying the first claim repeatedly. (This is possible within $\III$.) Now we define $(\bar{D}^{*},Z^{*})\le (\bar{E}^{t}, Y^{t})$ as follows. For each $i<k$, check whether there exists a finite extension $(\bar{D}', Z')\le(\bar{E}^{t}, Y^{t})$ at $i$ such that $D'_{i}\in ({E}^{t}_{i}, Y^{t}\cap\mc{A}_{i})$ and $(\bar{D}', Z',\max\bar{D}'\cup\{\ell'\})\Vdash^{+} \langle i, \A m\le \ell_{0}\,\neg\pi(e,m,G) \rangle$ for all $e<\ell_{0}$ with $\tau(e,i)=1$. (Note that this condition can be expressed by a $\Sigma^{B}_{2}$-formula.) Put $D^{*}_{i}=D'_{i}$ if such $\bar{D}'$ exists, and put $D^{*}_{i}=E^{t}_{i}$ otherwise. (More precisely, one can pick minimal such $\bar{D}^{*}$ within $\III$.) Put $Z^{*}=Y^{t}\setminus[0,\max \bar{D}^{*}]$. Then, by the second claim, one can observe that for all $i<k$ and $e\le \ell_{0}$, $(\bar{D}^{*}, Z^{*}, \max\bar{D}^{*}\cup\{\ell'\})\Vdash^{+} \langle i, \A m\le \ell_{0}\,\neg\pi(e,m,G) \rangle$ if $(\bar{D}^{*}, Z^{*}\cap \mc{A}_{i})$ is $\tau$-large at $i$ up to $\ell_{0}$ and $\tau(e,i)=1$. Take the minimal $\ell_{1}$ so that $\ell_{1}$ bounds $\max\bar{D}^{*}\cup\{\ell'\}$ and a code for $Z^{*}$. Then $q=(\bar{D}^{*}, Z^{*}, \tau, \ell_{0},\ell_{1})$ is the desired extension. For a given $p\in \mathbb{P}$, the extension constructed in the proof of Lemma \[lem:left-most-extension\] is said to be a *left-most successor of $p$.* Note that “$q$ is a left-most successor of $p$” can be described by a boolean combination of $\Sigma^{0}_{2}$ and $\Pi^{0}_{2}$ formulas. Let $p_{0}\succ p_{1}\succ \dots$ be the left-most path of $\mathbb{P}$, *i.e.*, $p_{i+1}$ is a left-most successor of $p_{i}$. More formally, put $$\begin{aligned} \mc{G}&=\{p_{n}: \E \langle p_{j}\mid j\le n \rangle (p_{0}=(\emptyset, \N, \emptyset, 0,1)\wedge \A j<n(p_{j+1}\mbox{ is a left-most successor of }p_{j}))\},\\ J&=\{n : \E \langle p_{j}\mid j\le n \rangle (p_{0}=(\emptyset, \N, \emptyset, 0,1)\wedge \A j<n(p_{j+1}\mbox{ is a left-most successor of }p_{j}))\}. \end{aligned}$$ Both of $J$ and $\mc{G}$ are $\Sigma^{B}_{3}$. Note that $J$ may form a proper cut of $M$. \[lem:unbdd-G\] $\mc{G}$ is unbounded, *i.e.*, for any $x\in M$, there exists $p_{i}\in \mc{G}$ such that $\ell_{1}^{p_{i}}>x$. Assume that $\mc{G}$ is bounded by some $\bar{\ell}\in M$. Then the first existential quantifier in the definition of $J$ is bounded. Thus it is defined by a boolean combination of $\Sigma^{B}_{2}$ and $\Pi^{B}_{2}$ formulas. Hence $J$ has a maximal element by $\III$, which contradicts Lemma \[lem:left-most-extension\]. Thus, $\mc{G}$ is cofinal in $M$. Our next task is to see that at some $i<k$, the construction of a solution works for any $j\in J$. If we can find such $i<k$, then $\bigcup_{j\in J} F_{i}^{p_{j}}$ is unbounded in $M$. For each $j\in J$, put $$\begin{aligned} \eta^{j}:=\{i<k: \A m\le \ell_{0}^{p_{j}} (\bar{F}^{p_{j}}, X^{p_{j}},\ell_{1}^{p_{j}})\Vdash^{+} \langle i, \neg\pi(e,m,G) \rangle\mbox{ for any }e\le \ell_{0}^{p_{j}}\mbox{ with }\sigma^{p_{j}}(e,i)=1\}. \end{aligned}$$ Here, $i\in\eta^{j}$ means that the construction for color $i$ is sill working at stage $j\in J$. Trivially, $\eta^{j}\supseteq\eta^{j'}$ if $j\le j'$. \[lem:color-survive1\] $\eta^{j}\neq \emptyset$ for any $j\in J$. By the definition of the condition, it is enough to show that $(\bar{F}^{p_{j}}, X^{p_{j}}\cap \mc{A}_{i})$ is $\sigma^{p_{j}}$-large at $i$ up to $\ell_{0}^{p_{j}}$ for some $i<k$. Assume not, then for each $i<k$ there exists a witness $\{(\bar{E}^{t,i}, Y^{t,i})\}_{t<s_{i}}$ so that $(\bar{F}^{p_{j}}, X^{p_{j}}\cap \mc{A}_{i})$ is not $\sigma^{p_{j}}$-large at $i$ up to $\ell_{0}^{p_{j}}$. Then the union $\{(\bar{E}^{t,i}, Y^{t,i})\}_{t<s_{i},i<k}$ indicates that $(\bar{F}^{p_{j}}, X^{p_{j}})$ is not $\sigma^{p_{j}}$-large by Remark \[rem:largeness\].1, which is a contradiction. \[lem:color-survive2\] There exists $i<k$ such that $i\in \eta^{j}$ for any $j\in J$. Assume that such $i<k$ does not exist. Then we have $\A i<k\,\E \bar{\ell}\,\E j\in J(i\notin \eta^{j}\wedge \ell_{1}^{p_{j}}<\bar{\ell})$. Thus, by $\BIII$, there exists $\ell\in\N$ such that $\A i<k\, \E j\in J(i\notin \eta^{j}\wedge \ell_{1}^{p_{j}}<\bar{\ell})$. By Lemma \[lem:unbdd-G\], there exists $p_{j'}\in\mc{G}$ such that $\ell_{1}^{p_{j'}}>\bar{\ell}$. Then $\eta^{j'}=\emptyset$ by the monotonicity of $\eta^{j}$, which contradicts Lemma \[lem:color-survive1\]. By Lemma \[lem:color-survive2\], pick a color $i<k$ such that $i\in \eta^{j}$ for every $j\in J$ and put $G:=\bigcup_{j\in J} F_{i}^{p_{j}}$. Then $G\subseteq \mc{A}_{i}$. Take $e_{\inf}\in \N$ so that $\A m\E n>m(n\in G)\leftrightarrow \A m\neg\pi(e_{\inf},m,G)$. Then, for large enough $j\in J$, $\sigma^{p_{j}}(e_{\inf},i)=1$ since “$G$ is finite” is never forced by an infinite Mathias pair. Thus, $G$ is infinite by the third clause of the definition of conditions. $G$ is $\Delta^{B}_{3}$ since $x\in G\leftrightarrow \E j\in J (\ell_{0}^{p_{j}}>x\wedge x\in F_{i}^{p_{j}})$ and $x\notin G\leftrightarrow \E j\in J (\ell_{0}^{p_{j}}>x\wedge x\notin F_{i}^{p_{j}})$. For any $e\in \N$, $\A m \neg\pi(e,m,G)$ holds if and only if $\E j\in J(\ell_{0}^{p_{j}}>e\wedge \sigma^{p_{j}}(e,i)=1)$. Thus, any $\Pi^{B\oplus G}_{2}$-formula is equivalent to a $\Sigma^{B}_{3}$-formula, and hence any $\Sigma^{B\oplus G}_{3}$-formula is equivalent to a $\Sigma^{B}_{3}$-formula.

--- abstract: 'The modified local spin density functional and the related local potential for excited states is tested by employing the ionization potential theorem. The functional is constructed by splitting $k$-space. Since its functional derivative cannot be obtained easily, the corresponding potential is given by analogy to its ground-state counterpart. Further to calculate the highest occupied orbital energy $\epsilon_{max}$ accurately, the potential is corrected for its asymptotic behavior by employing the van Leeuwen and Baerends correction to it. $\epsilon_{max}$ so obtained is then compared with the $\Delta$SCF ionization energy calculated using the MLSD functional. It is shown that the two match quite accurately.' address: 'Department of Physics, Indian Institute of Technology, Kanpur 208 016, India' author: - 'M. Hemanadhan, Md. Shamim and Manoj K. Harbola' title: ' Testing excited-state energy density functional and potential with the ionization potential theorem ' --- Introduction {#sec:introd} ============ Ground-state density functional theory (gDFT) is the most widely used theory for electronic structure calculations  [@book:Parr-Yang:1989; @book:Dreizler-Gross:1990; @book:March:1992; @book:Engel-Dreizler:2011]. The key to its success has been accurate exchange-correlation functionals $E_{xc}$ developed over the past few decades [@becke:1988; @perdew-burke-ernzerhof:1996; @Tau-Perdew-etal:2003]. The exchange-correlation potential $v_{xc}$ required for the self-consistent calculations (SCF) is then obtained either by taking functional derivative of the $E_{xc}$ or in some cases by using model potentials [@leeuwen-baerends:1994; @Umezawa:2006; @becke-johnson:2006]. It is then natural to ask if the ground-state theory can be extended to study excited-states to perform self-consistent Kohn-Sham calculations for the density and total energy of excited-states. Although time-dependent density functional theory (TDDFT) is now routinely used for calculations of excitation energies and the corresponding oscillator strengths, the theory has its limitations [@book:Ullrich:2012]. On the other hand, the progress of time-independent excited-state DFT (eDFT) has been slow. Some of the earlier work includes the extension of ground-state theory to the lowest energy states of a given symmetry by Gunnarsson and Lundqvist [@gunnarsson-lundqvist:1976; @gunnarsson-lundqvist:1976:err:1977], Ziegler et al. [@ziegler-rauk-baerends:1977] and von Barth [@barth:1979]. Subsequent work are the development of ensemble theory to excited-states by Theophilou [@theophilou:1979], Gross, Oliveira, Kohn [@gross-oliveira-kohn:1988; @oliveira-gross-kohn:1988], and its application to study transition energies of atoms by Nagy [@nagy:1996]. Recently, the work by G[ö]{}rling [@gorling:1999] and Levy and Nagy [@levy-nagy:1999; @nagy-levy:2001], both based on constrained-search approach [@Levy:1979], rekindled interest in eDFT. Following this, Samal and Harbola explored density-functional theory for excited-states further  [@harbola:2002; @harbola:2004; @samal-harbola:2005; @samal-harbola:2006; @samal-harbola:2006b]. A crucial requirement for implementing eDFT is the appropriate functionals for the excited-states. These functionals should be as easy to use as the ground-state functionals and be such that improved functionals can be built upon them. For the ground-states such a functional is provided by the local-density approximation (LDA), which is based on the homogeneous electron gas (HEG). Motivated by this, we have proposed an LDA-like functional for excited-states [@samal-harbola:2005]. This functional is also obtained using the homogeneous electron gas. The spin-generalization of the functional, the modified local spin-density (MLSD) functional has been shown to lead to accurate transition-energies [@samal-harbola:2005]. Encouraged by this, we have been subjecting our method of constructing the functional to more and more severe tests [@hemanadhan-harbola:2010; @hemanadhan-harbola:2012; @shamim-harbola:2010]. With this in mind, we test our method for the satisfaction of the ionization potential (IP) theorem in this paper. According to the ionization potential (IP) theorem for the ground-states [@PhysRevLett.49.1691; @Levy-Perdew-Sahni:1984; @Katriel-Davidson:1980] or an excited-states, the highest occupied Kohn-Sham orbital energy ($\epsilon_{max}$) for a system is equal to the negative of the ionization potential $I$ [@PhysRevLett.49.1691]. Thus $$\epsilon_{max} = -I(N) \equiv E(N) - E(N-1) \label{eq:ip}$$ where $E(N)$, and $E(N-1)$ are the energies corresponding to $N$ and $N-1$ electron systems such that $I(N)$ is smallest. The difference of these energies for the $N$ and $N-1$ electron system calculated self-consistently is referred to as $\Delta$SCF. The relationship of Eq.  arises because the asymptotic decay of the electronic density of a system is related to its ionization potential; on the other hand, for a Kohn-Sham system it is governed by $\epsilon_{max}$, thereby relating the two quantities. Thus, if the exact functionals were known, the corresponding Kohn-Sham calculation will give $\epsilon_{max}$, $E(N)$ and $E(N-1)$ so that Eq.  is satisfied. However, this is not the case when approximate functionals are used. For instance, when ground-state calculations are done using the LDA, the $\Delta$SCF values are accurate, but the $\epsilon_{max}$ are roughly $50\%$ of the $\Delta$SCF energy or the experimental values [@w2012crc]. This is due to the fact that LDA potential decays exponentially rather than correctly as $-1/r$ for $r \rightarrow \infty$. Therefore it is less binding for the outermost electrons. For the ground-states, it is seen that if the asymptotic behavior of the potential is improved, $\epsilon_{max}$ becomes close to $E(N)-E(N-1)$. Two ways of making such a correction are the van Leeuwen and Baerends (LB) [@leeuwen-baerends:1994] method and the range-separated hybrid (RSH) methods [@inbook:savin:chong:1995; @leininger-stoll-werner-savin:1997; @iikura-tsundea-yanai-hirao:2001; @yanai-tew-handy:2004; @baer-neuhauser:2005; @kronik-tamar-abramson-baer:2012]. In the LB method, a correction term is added to the LDA potential to make the effective potential go as $-1/r$ asymptotically, while in the RSH approach the Coulomb term is split into long-range (LR) and short-range (SR) part. Thus, $r^{-1}$ can be written as $ r^{-1} \operatorname{erf}(\gamma r) + r^{-1} \operatorname{erfc}(\gamma r )$ where $\gamma$ is a parameter [@inbook:savin:chong:1995; @leininger-stoll-werner-savin:1997; @iikura-tsundea-yanai-hirao:2001; @yanai-tew-handy:2004; @baer-neuhauser:2005; @kronik-tamar-abramson-baer:2012]. Here the first term is long-range and approaches $2\gamma/\sqrt{\pi}$ as $r\rightarrow 0$, while the second term is close to $\frac{\exp(-\gamma r)}{r})$ [@Bohm-Pines:1953c] and is short range. In the RSH approach, the long-range part is treated exactly and the short-range part within the LDA. Recently, Stein et. al [@stein-eisenberg-kronik-baer:2010] have applied this idea to study the band gaps for a wide range of systems. In their work $\gamma$ is fixed by the satisfaction of the IP theorem. Motivated by their work, we have studied the IP theorem using the LB potential. The line of our investigation is as follows: We first show that the LB potential leads to the satisfaction of the IP theorem for the ground-states to a high degree of accuracy. We then ask: does our approach of constructing excited-state energy functionals give the same level of accuracy for IP theorem for excited-states when applied with the LB potential ? This then provides a test for our approach. The positive results of our calculations point to the correctness of our method of dealing with excited-state functionals. The LB correction to the LDA potential is given as $$- \beta \rho_{\sigma}^{1/3}(\mathbf{r}) \frac{x^2_{\sigma}}{1+3\beta x_{\sigma} \sinh^{-1}(x_{\sigma})} \label{eq:lbgradient}$$ where the parameter $\beta$ is obtained by fitting the LB potential so that it resembles closely to the exact potential for the beryllium atom ($\beta=0.05$), and $x_{\sigma}$ is a dimensionless ratio given by $x_{\sigma} = \frac{|\nabla \rho_{\sigma}|}{\rho_{\sigma}^{4/3}}$. In the present paper, the parameter $\beta$ is chosen to satisfy IP theorem, similar to the work of Stein et al. [@stein-eisenberg-kronik-baer:2010]. The difference with the work of Ref. [@stein-eisenberg-kronik-baer:2010] is that in the present work the potential is given entirely in terms of the density whereas in RSH functional it is written using both the wavefunction and the density. We note that recently the LB potential has also been applied to calculate satisfactorily the band gaps of a wide variety of bulk systems [@Prashant-Harbola-etal:2013]. In the following, we present in Section \[sec:gr-theory\] the results of application of the LB potential to the ground-states of several atoms. It is shown that with the help of parameter $\beta$, the LB potential can be optimized to satisfy IP theorem to a very high degree. The results for the ground-state set up the standard against which the excited-state results are to be judged for the functional and the corresponding potential proposed for the excited-states. After this we study the IP theorem for excited-states using the LB correction in conjunction with the modified LDA potential based on the idea [@inbook:Harbola-etal:Ghosh-Chattaraj:2013] of splitting $k$-space for excited-states. It is shown that the IP theorem is satisfied more accurately with the modified LDA potential in comparison to the ground-state LDA expression for the potential. In addition the modified potential has proper structure at the minimum of radial density in contrast to the ground-state LDA potential that has undesirable features at these points [@cheng-Wu-Voorhis:2008]. Results for the ground-state IP theorem using LB potential {#sec:gr-theory} ========================================================== In this section, we first present the ground-state exchange-only $\epsilon_{max}$ and $\Delta$SCF energies obtained with the LDA and the LB potential for few atoms. Following that, we also present the results with correlation functional included. The results for the ground-states are not entirely new in light of some previous work [@Banerjee-Harbola:1999] but it is necessary to give them here to put the new results of excited-states in proper perspective. The LDA exchange energy functional $E_{x}$ [@dirac30] is given by $$E^{LDA}_{x}[\rho(\mathbf{r})] = -\frac{3}{4} \left(\frac{3}{\pi} \right)^{1/3} \int \rho^{4/3} (\mathbf{r}) d\mathbf{r} \label{eq:Ex-LDA}$$ and the corresponding potential $v^{LDA}_{x}$ required for self-consistency calculations is $$v_{x}^{LDA} = - \left( \frac{6\rho(\mathbf{r})}{\pi} \right)^{1/3} \label{eq:vx-LDA}$$ Spin generalization of the expression of Eq. , the local spin-density approximation (LSD), is obtained by using $$E_x^{LSD} [\rho_{\alpha},\rho_{\beta}] = \frac{1}{2} E_x [2\rho_{\alpha}] + \frac{1}{2} E_x [2\rho_{\beta}].$$ In Table \[tab:gr-x-IP\], the $\epsilon_{max}$ and $\Delta$SCF obtained using the spin-generalized LDA exchange functional Eq.  and its potential Eq.  is shown. As is well-known and noted earlier, the LSD underestimates the highest occupied orbital energy (HO) roughly by $50\%$, due to incorrect asymptotic exponential behavior of the LDA exchange-potential of Eq. . The $\Delta$SCF energies, however, are close to the HF values. As stated in the previous section, $\epsilon_{max}$ and $\Delta$SCF energies become consistent with each other if asymptotically the potential goes correctly as $-1/r$. The van-Leeuwen and Barends (LB) potential does that. The LB potential $v_x^{LB}$ [@leeuwen-baerends:1994], is calculated by including the LB correction of Eq.  to the LSD potential of Eq.  and is given as $$v_{x,\sigma}^{LB}(\mathbf{r}) = v_{x,\sigma}^{LSD} - \beta \rho_{\sigma}^{1/3}(\mathbf{r}) \frac{x^2_{\sigma}}{1+3\beta x_{\sigma} \sinh^{-1}(x_{\sigma})} \label{eq:vx-LB}$$ where $v_{x,\sigma}^{LSD} = \frac{ \delta E_x^{LSD}}{\delta \rho_{\sigma}}$. In the original LB potential, parameter $\beta=0.05$. In the present work, in addition to using this value of $\beta$, we also optimize it by the satisfaction of IP theorem. In the latter calculation, $\beta$ is varied until $\epsilon_{max}$ and $\Delta$SCF energies match, i.e $$\epsilon^{\beta}_{max} = E(N,\beta) - E(N-1,\beta)$$ Here, $\epsilon^{\beta}_{max}$ is the highest-occupied eigen-value for a specific choice of $\beta$. The price for employing the asymptotically corrected model exchange potential is that the corresponding exchange functional is not known. Although in the past Levy-Perdew relation [@levy-perdew:1985] has been used to get the corresponding exchange-energies from the potential [@Banerjee-Harbola:1999], this may not always be correct [@Gaiduk-Chulkov-Staroverov:2009]. In this section we therefore use the potential above in the KS calculations but employ the LSD exchange functional for calculating the energies. Presented in Table \[tab:gr-x-IP\] are the results for $\epsilon_{max}$ and $\Delta$SCF energies using the LSD potential, the LB potential with $\beta=0.05$ and the LB potential with optimized $\beta$. As mentioned above, the exchange energy functional used is the LSD functional itself for all the three potentials. Also shown are the $\Delta$SCF energies obtained from HF calculations. Comparing the results of the LSD and LB calculations with the corresponding numbers in Hartree-Fock theory, it is evident that (i) the $\Delta$SCF values given by the LSD functional are reasonably close to the corresponding HF values, and (ii) by making the potential correct in the asymptotic regions, $\epsilon_{max}$ improves substantially and becomes close to the $\Delta$SCF values. Interestingly, the match between $\epsilon_{max}$ obtained with the LB potential and the $\Delta$SCF values is better than that in the Hartree-Fock theory. Next, motivated by the work of Ref. [@stein-eisenberg-kronik-baer:2010], we tune the parameter $\beta$ in the LB potential so that $\epsilon_{max}$ matches with the $\Delta$SCF energies. The optimized $\beta$ and the corresponding energies are also shown in Table \[tab:gr-x-IP\]. As is evident from Table \[tab:gr-x-IP\], choosing $\beta$ through IP theorem, the highest orbital energy $\epsilon_{max}$ improves. We note that according to Koopmans theorem [@Koopmans:1934], the orbital energy $\epsilon_{i}$ is close to the removal energy of the electron from that orbital. However we find that DFT results are better in this regard. The results of Table \[tab:gr-x-IP\] are depicted in Fig. \[fig:ip-gr-x\] where we have plotted the $\Delta$SCF results against $-\epsilon_{max}$ for LSD, LB and HF theories. We see that the LB results are closest to the $\Delta$SCF$=-\epsilon_{max}$ line. Having presented our results for the exchange-only calculations we next include correlation using the LDA. The correlation functional we use is that parametrized by Vosko, Wilk and Nusair [@vosko-wilk-nusair:1980]. The orbital energies $\epsilon_{max}$ and the $\Delta$SCF energies for the LB and $\beta$ optimized LB are presented in Table \[tab:gr-xc-IP\] in comparison with the experimental results [@w2012crc]. We see from Table \[tab:gr-xc-IP\] that with the asymptotically corrected LB potential, the IP theorem is satisfied remarkably well. The parameter $\beta$ in the LB potential is tuned to satisfy IP theorem (Ref. Eq. ) The $\epsilon_{max}$, so obtained matches with experiments in a much better way. The radial density and the exchange potential for Li ground-state obtained using the LDA and the LB potentials are shown in Fig. \[fig:vx-Li-ground\]. Also shown in Fig. \[fig:vx-Li-ground\] is the KLI potential [@krieger-li-iafrate:1992a], which is essentially the exact exchange potential, for comparison. It is evident that from about $r=0.2 a.u.$ onwards, the LB potentials with both $\beta=0.05$ and the optimized $\beta$ are quite close to the KLI potential. The discrepancy of the LB potential for $r<0.2 a.u$ corresponds to the non-zero $\beta$. Furthermore, all the three potentials go as $-1/r$ in the asymptotic regions. On the other hand, the LSD potential underestimates the exact potential all over. The bump in the potential for Li is at the minimum in the radial densities [@lindgren:1971]. Having given the results for the ground-states, we now turn our attention to excited-states and show that the exchange functional and potential constructed for these states by splitting the $k$-space for HEG give results with similar accuracy. Split $k$-space method for constructing excited-state energy functionals and excited-state potential {#sec:excited} ==================================================================================================== In eDFT, we have put forth the idea that the excited state energies be calculated using the modified local spin density (MLSD) functional developed over the past few years [@samal-harbola:2005; @shamim-harbola:2010; @hemanadhan-harbola:2010; @hemanadhan-harbola:2012]. The basis of the MLSD exchange energy functional is the split $k$-space method [@inbook:Harbola-etal:Ghosh-Chattaraj:2013]. In this method, the $k$-space is split in accordance to the orbital occupation of a given excited-state. In Fig. \[fig:k-space\], we show an excited-state, where some orbitals (core) are occupied, followed by vacant (unocc) orbitals and then again orbitals are occupied (shell). To construct excited-state functionals, the density for each point is mapped onto the $k$-space of an HEG. The corresponding split $k$-space, also shown in Fig. \[fig:k-space\], is constructed according to the orbital occupation i.e. the $k$-space is occupied from $0$ to $k_1$, vacant from $k_1$ to $k_2$ and then again occupied from $k_2$ to $k_3$ where $k_1, k_2, k_3$ are given by $$\begin{aligned} k_{1}^{3}(\mathbf{r}) &= 3\pi^{2}\rho_{c}(\mathbf{r}) \label{eq:k1} \\ k_{2}^{3}(\mathbf{r})-k_{1}^{3}(\mathbf{r}) &= 3\pi^{2}\rho_{v}(\mathbf{r}) \label{eq:k2} \\ k_{3}^{3}(\mathbf{r})-k_{2}^{3}(\mathbf{r}) &= 3\pi^{2}\rho_{s}(\mathbf{r}) \label{eq:k3}\end{aligned}$$ in terms of $\rho_{c}$, $\rho_{v}$ and $\rho_{s}$ corresponding to the electron densities of core, vacant (unoccupied) and the shell orbitals. Further, $$\begin{aligned} \rho_{c}(\mathbf{r}) &= \sum\limits^{n_1}_{i=1} {\left| \phi_{i}^{core}(\mathbf{r})\right|}^{2} \\ \rho_{v}(\mathbf{r}) &= \sum\limits^{n_2}_{i=n_1+1} {\left| \phi_{i}^{unocc}(\mathbf{r})\right|}^{2} \\ \rho_{s}(\mathbf{r}) &= \sum\limits^{n_3}_{i=n_2+1} {\left| \phi_{i}^{shell}(\mathbf{r})\right|}^{2}\end{aligned}$$ where first $n_1$ orbitals are occupied, $n_1+1$ to $n_2$ are vacant followed by occupied orbitals from $n_2+1$ to $n_3$. The total electron density $\rho({\bf r})$ is given as $$\begin{aligned} \rho({\mathbf{r}})=\rho_{c}({\mathbf{r}})+\rho_{s}({\mathbf{r}}) \\ \textrm{or} \hspace{2em} \rho({\mathbf{r}})=\rho_1({\mathbf{r}}) -\rho_2({\mathbf{r}}) + \rho_3({\mathbf{r}})\end{aligned}$$ with $\rho_1=\rho_c,\rho_2=\rho_c+\rho_v $ and $\rho_3=\rho_c+\rho_v+\rho_s$. Using this idea, we have constructed the kinetic [@hemanadhan-harbola:2010] and exchange-energy functionals [@samal-harbola:2005] for excited-states, and shown that these functionals lead to accurate kinetic, exchange, and transition energies. We point out that the application of the ground-state LSD functional generally leads to poor results for excited-states. The generality of this idea to construct energy functionals for other class of systems also leads to accurate energies [@shamim-harbola:2010; @hemanadhan-harbola:2012]. Encouraged by these studies, we now subject this method to test by the IP theorem. For this study, we consider the class of excited-systems as shown in Fig. \[fig:k-space\] for which the MLSD functional is given by [@samal-harbola:2005] $$\begin{aligned} E_X^{MLDA}[\rho] & = \int \rho(\mathbf{r}) \left[ \epsilon(k_3)-\epsilon(k_2)+\epsilon(k_1) \right] d\mathbf{r} + \frac{1}{8\pi^3} \int \left(k_3^2-k_1^2 \right)^2 \ln \left( \frac{k_3+k_1}{k_3-k_1}\right) d\mathbf{r} \nonumber \\ & - \frac{1}{8\pi^3} \int \left(k_3^2-k_2^2 \right)^2 \ln \left( \frac{k_3+k_2}{k_3-k_2}\right) d\mathbf{r} + \frac{1}{8\pi^3} \int \left(k_2^2-k_1^2 \right)^2 \ln \left( \frac{k_2+k_1}{k_2-k_1}\right) d\mathbf{r} \label{eq:xmlda}\end{aligned}$$ where $\epsilon(k_i)=\frac{-3k_i}{4\pi}$ is the exchange energy per particle for the ground state of HEG with Fermi wavevector $k_i$. Like the ground-state functional the modified local spin density (MLSD) functional is given as $$E_X^{MLSD}[\rho] = \frac{1}{2} E_X^{MLDA}[2 \rho_{\alpha}] + \frac{1}{2} E_X^{MLDA}[2 \rho_{\beta}] \label{eq:exmlsd}$$ The corresponding potential $v^{MLSD}_x$ is given as $$v_{x,\sigma}^{MLSD}(\textbf{r})= \frac{\delta E_{X}^{MLSD}[\rho]}{\delta\rho_{\sigma}(\mathbf{r})} \label{xp}$$ However, it has not been possible to get a workable analytical expression for $ v_{x}^{MLSD}({\bf r})$ out of Eqs.  and . Therefore on the basis of arguments based on ground-state theory, we try to model the potential. For completeness we note the earlier attempts to construct accurate excited-state potentials by Gaspar [@gaspar:1974] and Nagy [@nagy:1990]. They have given an ensemble averaged exchange potential for the excited states and using this potential, they calculate excitation energy for single electron excitations. In the next section we propose an excited-state LDA-like exchange potential based on split $k$-space. This potential is similar to its ground-state LDA counterpart. We refer to this as the MLSD potential. We further correct the potential for its asymptotic behavior with the LB correction. With the asymptotically corrected MLSD potential, we show that the IP theorem for excited-states is satisfied to a good accuracy. Generalization of Dirac exchange potential for excited-states using split $k$-space ----------------------------------------------------------------------------------- The Hartree-Fock exchange potential for a system of electrons is given by$$v_{x,i}^{HF}=v_{x}(\phi_{i})=-\sum_{j}\int\frac{\phi^{\ast}_{j}({\bf r'})\phi_{i}({\bf r'})\phi_{j}({\bf r})} {\phi_{i}({\bf r})|{\bf r}- {\bf r'}|}d{\bf r'} \label {hfx}.$$ For homogeneous electron gas, the wavefunction is given by $$\phi_{{\bf k}}({\bf r}) = \frac{1}{{\sqrt V}} e^{ \left (i{\bf k}\cdot {\bf r}\right )} \label {wf}.$$ where $V$ is the volume of the system. Using this form of wavefunction in Eq.  we get an exchange potential for one-gap systems shown in Fig. \[fig:k-space\] to be for $\phi_k(\mathbf{r})$ $$\begin{aligned} v_{x}(k)=&-\frac{1}{\pi} \left [k_{1}-k_{2}+k_{3} + \frac {k_{1}^{2}-k^{2}}{2k}\ln\left|\frac{k+k_{1}}{k-k_{1}}\right| \right. - \left. \frac {k_{2}^{2}-k^{2}}{2k}\ln\left|\frac{k+k_{2}}{k-k_{2}}\right|+\frac {k_{3}^{2}-k^{2}}{2k}\ln\left|\frac{k+k_{3}}{k-k_{3}}\right|\right ] \label{eq:hfxpi}\end{aligned}$$ where $k_1, k_2$, and $k_3$ are given by Eqns , ,  This potential is orbital dependent. To make this potential an orbital independent potential we draw the analogy from the ground state exchange potential, where the exact LDA potential is equal to the HF potential for highest occupied molecular orbital (HOMO). $$v^{HF}_{x,i}(\mathbf{r})|_{i=max} = \frac{\delta E_x^{LDA}[\rho(\mathbf{r})]}{\delta \rho(\mathbf{r})} \label{eq:hflda}$$ Therefore we take the potential for the electron in HOMO as the exchange potential for all the electrons. For this we put $k=k_{3}$ in Eq. , and get the following expression for the MLSD exchange potential $$\begin{aligned} v_{x}^{MLSD}= & -\frac{k_{3}}{\pi}\left [1-x_{2}+x_{1}-\frac{1}{2}(1-x_{1}^{2})\ln\left|\frac{1+x_{1}}{1-x_{1}}\right| \right. \left. +\frac{1}{2}(1-x_{2}^{2})\ln\left|\frac{1+x_{2}}{1-x_{2}}\right|\right ] \label {eq:vxmlsd}\end{aligned}$$ where, $ x_{1}=\frac{k_{1}}{k_{3}}, x_{2}=\frac{k_{2}}{k_{3}} $ The MLSD potential of Eq.  is also obtained by taking the functional derivative of the exchange functional $E_x^{MLDA}$ of Eq.  with respect to $\rho_{3}(\mathbf{r})$, corresponding to the largest wave-vector in the $k$-space. Thus, we reach the same result from two different paths; this in some sense assures us about the correctness of the approach taken. When this potential is corrected for its asymptotic behavior by adding the LB correction, we obtain the modified LB (MLB) potential. In the following Section, we test the MLB potential using the IP theorem for excited-states and show that it satisfies the IP theorem as accurately as the LB potential does for the ground-states. On the other hand, the LB potential does not lead to as accurate as satisfaction of the IP theorem indicating thereby that the potential derived on the basis of splitting $k$-space is more appropriate for the excited-state calculations. Results for excited-states {#sec:ex-result} ========================== The MLSD potential is the ground-state counterpart of the LSD potential. To correct the potential in the asymptotic region, we include the LB gradient term of Eq.  corresponding to largest wave-vector $k_3$ in the MLSD potential and obtain the MLB potential. $$v_{x,\sigma}^{MLB} = v_x^{MLSD} - \beta \rho_{3,\sigma}^{1/3}(\mathbf{r}) \frac{x^2_{3,\sigma}}{1+3\beta x_{3,\sigma} \sinh^{-1}(x_{3,\sigma})} \label{eq:vxmlb}$$ In performing self-consistent calculations, it is this potential that is employed as the exchange potential in the excited-state Kohn-Sham equations. Our calculations are performed using the central-field approximation [@Slater:1929] whereby the potential is taken to be spherically symmetric. Having obtained the orbitals the exchange energy is then calculated using the MLSDSIC functional [@samal-harbola:2005] and is given as $$E_{X}^{MLSDSIC}=E_{X}^{MLSD}-\sum_{i}^{rem}{E_i}^{SIC}-\sum_{i}^{add}{E_i}^{SIC} \label{eq:mlsdsic}$$ where, $$E_{i}^{SIC}\left[\phi_i\right]= \int\int\frac{|\phi_{i}(\mathbf{r}_{1})|^{2}|\phi_{i} ({\bf r}_{2})|^{2}}{|{\bf r}_{1}-{\bf r}_{2}|}d{\bf r}_{1}d{\bf r}_{2} +E^{LSD}_{X}\left[\rho\left(\phi_i\right)\right] \label{eq:sic}$$ where the summation index $i$ in Eq.  runs over the orbitals from which the electrons are removed and create a gap, and to the orbitals to which the electrons are added. $E^{LSD}_{X}\left[\rho\left(\phi_i\right)\right]$ is the exchange energy corresponding to the $\phi_i$ orbital in the LSD approximation. Using the $\Delta$SCF energy obtained from these calculations and the eigenvalues from the Kohn-Sham calculations, we study the IP theorem. For our study, we have considered systems for which both the atomic excited-states and its ionic states can be represented by a single Slater determinant; this is so because LSD/MLSD is accurate for such states  [@barth:1979]. Presented in Table \[tab:ex-x-IP\] are the $\epsilon_{max}$ and $\Delta$SCF energies for different excited-states obtained using the LB potential of Eq. , and the excited-state MLB potential of Eq. . In both the LB and the MLB potentials, we have used $\beta = 0.05$. Further, the energies for both the potentials are calculated using the MLSDSIC exchange energy functional. The HF $\epsilon_{max}$ and $\Delta$SCF are also shown in Table \[tab:ex-x-IP\] for comparison. The results of Table III are shown graphically in Fig. \[fig:ip-ex-x\]. It is evident from the figure that the MLB potential satisfies the IP theorem accurately while the LB and HF both deviate from it. Thus accounting for the occupation of orbitals in the $k$-space gives better results for the theorem. Let us next check how does the MLB potential compare with the KLI potential for excited-states. Plotted in Fig. \[fig:vx-Li-excited-3s1\] are the radial density and the corresponding excited-state exchange potential of Li $(3s^1 \ ^2S)$ within the LB and the MLB approximations. Also shown in the figure the exact exchange potential, obtained through KLI method [@Nagy:1997]. It is clear from the figure that the split $k$-space based MLB potential has a structure resembling the KLI potential for the excited-state: very close to it in the inter-shell region from about 0.1 a.u. onwards and beyond. This is similar to the relation between the LB potential and the KLI potential for the ground-states. The LB potential for the excited-states, on the other hand, is not close to the exact potential and has undesirable features at the minimum of radial density which are not present in the MLB potential. Similar unsmooth behavior is observed [@cheng-Wu-Voorhis:2008] in the LSD potential. In addition, the MLB potential is closer to the KLI potential in the interstitial and the asymptotic region, similar to what the LB potential did for the ground-states. The discrepancy between the potentials near the nucleus that was present in the ground-states is also present here. Nonetheless it is clear that the exchange potentials obtained on the basis of split $k$-space give a much better description of an excited-state than the ground-state LB potential. To sum up, we have shown that excited-state energy functional and its asymptotically corrected potential based on split $k$-space satisfy IP theorem with a great accuracy in the exchange-only limit. This can be improved further by optimizing $\beta$. In Table \[tab:ex-x-IP\], we also present the results obtained by varying the parameter $\beta$ in the excited-state MLB potential until $\epsilon_{max}$ matches with the $\Delta$SCF energies. The $\epsilon_{max}$ so obtained using the excited-state potential is close to the HF values. For $B \ (3p^1 \ ^2P)$ we are unable to tune the $\beta$ using the MLB potential. We now wish to include correlation and compare our results with experiments. The lack of correlation potential for excited-states forces us to rely on the ground-state potential. In Table \[tab:ex-xc-IP\] are the calculations performed using the ground-state VWN potential. It is seen that similar to the ground-state, the $\Delta$SCF energies obtained with the split $k$-space functional are close to the experimental values. Also shown in table are the $\beta$ tuned energies to satisfy IP theorem. By imposing IP theorem, $\epsilon_{max}$ improves over the $\beta=0.05$ values and is closer to the experimental values for all atoms. Concluding Remarks {#sec:conclusion} ================== To conclude we have shown that splitting $k$-space according to the occupation of Kohn-Sham orbitals is a good way of constructing excited-state potential. The potential so constructed, when corrected for its long-range behavior, gives highly accurate eigenvalues for the upper most orbital in the sense of IP theorem: the eigenvalues and the $\Delta$SCF energies obtained from the energy functional by splitting $k$-space agree with one another to a great degree. This shows that split $k$-space method could be the proper path to follow for constructing excited-state energy functionals. Acknowledgments =============== M. Hemanadhan wishes to thank Council of Scientific and Industrial Research (CSIR), New Delhi for financial support. [10]{} R. G. Parr and W. 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Team, NIST Atomic Spectra Database (version 5.0), 2012. ![Plot of $\epsilon_{max}$ vs. $\Delta$SCF energies using different exchange-only potentials.[]{data-label="fig:ip-gr-x"}](fig_IP_ground_x.ps){width="4.5in" height="6.5in"} ![Ground-state radial density and the exchange potential of Li ($2s^1 \ 2S$) for the up spin obtained using different approximations for the potential.[]{data-label="fig:vx-Li-ground"}](pl-gr-Li.ps){height="5.5in" width="3.8in"} ![(a) Orbital and (b) the corresponding $k$-space occupation in the excited state configuration of a homogeneous electron gas (HEG).[]{data-label="fig:k-space"}](kfig.eps){height="3.5in" width="4.5in"} ![Plot of $\epsilon_{max}$ vs. $\Delta$SCF energies using different exchange-only potentials of LB, MLB and HF.[]{data-label="fig:ip-ex-x"}](fig_IP_excited_x.ps){width="4.5in" height="6.5in"} ![Excited-state radial density and the exchange potential of Li ($3s^1 \ 2S$) for the up spin obtained using different approximations for the potential.[]{data-label="fig:vx-Li-excited-3s1"}](pl-ex-Li.ps){height="5.5in" width="3.8in"} =0.11cm ----------------------- ---------------------------- ------------- ---------------------------- ------------- --------- --------------------------------------------------- ---------------------------- ------------- Atoms/ion $-\epsilon_{\textrm{max}}$ $\Delta$SCF $-\epsilon_{\textrm{max}}$ $\Delta$SCF $\beta$ $-\epsilon_{\textrm{max}} (=\Delta \textrm{SCF})$ $-\epsilon_{\textrm{max}}$ $\Delta$SCF He($1s^2 \ ^1S$) 0.517 0.811 0.794 0.810 0.064 0.809 0.918 0.862 Li($2s^1 \ ^2S$) 0.100 0.185 0.175 0.182 0.073 0.182 0.196 0.196 Be($2s^2 \ ^1S$) 0.170 0.281 0.282 0.278 0.043 0.278 0.309 0.296 B($2s^2 2p^1 \ ^2P$) 0.120 0.278 0.263 0.274 0.075 0.273 0.310 0.291 C($2s^2 2p^2 \ ^3P$) 0.196 0.396 0.366 0.394 0.104 0.392 0.433 0.396 N($2s^2 2p^3 \ ^4S$) 0.276 0.515 0.476 0.513 0.112 0.511 0.568 0.513 O($2s^2 2p^4 \ ^3P$) 0.210 0.436 0.448 0.431 0.035 0.432 0.632 0.437 F($2s^2 2p^5 \ ^2P$) 0.326 0.597 0.585 0.594 0.060 0.594 0.730 0.578 Ne($2s^2 2p^6 \ ^1S$) 0.443 0.754 0.724 0.751 0.077 0.749 0.850 0.729 ----------------------- ---------------------------- ------------- ---------------------------- ------------- --------- --------------------------------------------------- ---------------------------- ------------- ----------------------- ---------------------------- ------------- ------------------- --------------------------------------------------- ------- Atom Expt. [@w2012crc] $-\epsilon_{\textrm{max}}$ $\Delta$SCF $\beta$ $-\epsilon_{\textrm{max}} (=\Delta \textrm{SCF})$ He($1s^2 \ ^1S$) 0.851 0.892 0.106 0.890 0.904 Li($2s^1 \ ^2S$) 0.193 0.198 0.066 0.198 0.198 Be($2s^2 \ ^1S$) 0.320 0.329 0.072 0.329 0.342 B($2s^2 2p^1 \ ^2P$) 0.296 0.312 0.086 0.311 0.305 C($2s^2 2p^2 \ ^3P$) 0.401 0.431 0.115 0.430 0.414 N($2s^2 2p^3 \ ^4S$) 0.511 0.550 0.117 0.548 0.534 O($2s^2 2p^4 \ ^3P$) 0.516 0.506 0.041 0.507 0.501 F($2s^2 2p^5 \ ^2P$) 0.647 0.661 0.065 0.660 0.640 Ne($2s^2 2p^6 \ ^1S$) 0.782 0.813 0.082 0.811 0.792 ----------------------- ---------------------------- ------------- ------------------- --------------------------------------------------- ------- ------------------------- ---------------------------- ------------- ---------------------------- ------------- --------- ---------------------------------------------------- ---------------------------- ------------- -- -- Atom $-\epsilon_{\textrm{max}}$ $\Delta$SCF $-\epsilon_{\textrm{max}}$ $\Delta$SCF $\beta$ $-\epsilon_{\textrm{max}}(={\Delta \textrm{SCF})}$ $-\epsilon_{\textrm{max}}$ $\Delta$SCF Li($2p^1 \ ^2P$) 0.117 0.109 0.096 0.114 0.300 0.114 0.129 0.129 B($2s^1 2p^2 \ ^2D$) 0.226 0.175 0.166 0.185 0.120 0.183 0.276 0.192 C($2s^1 2p^3 \ ^3D$) 0.279 0.202 0.200 0.215 0.090 0.213 0.402 0.233 N($2s^1 2p^4 \ ^4P$) 0.328 0.227 0.232 0.242 0.070 0.241 0.522 0.241 O($2s^1 2p^5 \ ^3P$) 0.466 0.362 0.387 0.368 0.035 0.370 0.601 0.364 F($2s^1 2p^6 \ ^2S$) 0.601 0.543 0.533 0.539 0.055 0.538 0.703 0.497 Ne$^+$($2s^12p^6\ ^2S$) 1.429 1.369 1.339 1.370 0.075 1.369 1.553 1.334 Li($3s^1 \ ^2S$) 0.076 0.085 0.069 0.072 0.080 0.073 0.074 0.074 Li($4s^1 \ ^2S$) 0.042 0.052 0.035 0.051 0.122 0.038 0.038 0.038 B($3s^1 \ ^2S$) 0.107 0.139 0.108 0.122 0.200 0.121 0.114 0.114 B($3p^1 \ ^2P$) 0.078 0.096 0.067 0.088 - - 0.079 0.079 Be($2s^1 3s^1 \ ^3S$) 0.095 0.116 0.094 0.101 0.100 0.102 0.100 0.100 ------------------------- ---------------------------- ------------- ---------------------------- ------------- --------- ---------------------------------------------------- ---------------------------- ------------- -- -- --------------------------- ---------------------------- ------------- --------- --------------------------------------------------- --------------- Atom $-\epsilon_{\textrm{max}}$ $\Delta$SCF $\beta$ $-\epsilon_{\textrm{max}} (=\Delta \textrm{SCF})$ Expt. [@NIST] Li($2p^1 \ ^2P$) 0.110 0.128 0.230 0.127 0.130 B($2s^1 2p^2 \ ^2D$) 0.214 0.252 0.600 0.247 0.257 C($2s^1 2p^3 \ ^3D$) 0.262 0.300 0.500 0.295 0.318 N($2s^1 2p^4 \ ^4P$) 0.308 0.344 0.350 0.339 0.348 O($2s^1 2p^5 \ ^3P$) 0.453 0.441 0.040 0.442 0.471 F($2s^1 2p^6 \ ^2S$) 0.594 0.604 0.056 0.600 0.623 Ne$^+$($2s^1 2p^6 \ ^2S$) 1.409 1.444 0.080 1.443 1.442 Li($3s^1 \ ^2S$) 0.079 0.081 0.060 0.081 0.074 Li($4s^1 \ ^2S$) 0.042 0.046 0.122 0.046 0.039 B($3s^1 \ ^2S$) 0.123 0.136 0.200 0.136 0.122 B($3p^1 \ ^2P$) 0.079 0.099 - - 0.083 Be($2s^1 3s^1 \ ^3S$) 0.107 0.112 0.082 0.112 0.105 --------------------------- ---------------------------- ------------- --------- --------------------------------------------------- ---------------

--- abstract: 'The famous two weights problem consists in characterising all possible pairs of weights such that the Hardy projection is bounded between the corresponding weighted $L^2$ spaces. Koosis’ theorem of 1980 gives a way to construct a certain class of pairs of weights. We show that Koosis’ theorem is closely related to (in fact, is a direct consequence of) a spectral perturbation model suggested by de Branges in 1962. Further, we show that de Branges’ model provides an operator-valued version of Koosis’ theorem.' address: - 'Department of Mathematics, King’s College London, Strand, London, WC2R 2LS, U.K.' - 'Department of Mathematics, Michigan State University, East Lansing, MI 48824, U.S.A.' author: - Alexander Pushnitski - Alexander Volberg title: Spectral perturbation theory and the two weights problem --- Introduction and main result {#sec.a} ============================ Introduction {#sec.a1} ------------ Let $P_\pm$ be the Hardy projections in $L^2({{\mathbb T}})$ (${{\mathbb T}}$ is the unit circle parameterised by $(0,2\pi)$): $$(P_\pm f)(e^{i\theta}) = \pm \lim_{r\to1\mp0}\int_0^{2\pi}\frac{f(e^{it})}{1-re^{i(\theta-t)}}\frac{dt}{2\pi}. \label{a1}$$ In its simplest form, the two weights problem consists in the characterisation of all pairs of weights $v_j:{{\mathbb T}}\to [0,\infty)$, $j=0,1$, such that $$P_+: L^2({{\mathbb T}},v_0(e^{it})dt) \mapsto L^2({{\mathbb T}}, v_1(e^{it})dt) \label{a2}$$ is a bounded operator. (Of course, one could equally speak of $P_-$). If $v_1=v_0$, then the characterisation of such weights is given by the celebrated Muckenhoupt condition [@Muck]: $$\sup_\Delta \left( \frac1{{\lvert\Delta\rvert}}\int_\Delta v_0(e^{it})dt \cdot \frac1{{\lvert\Delta\rvert}}\int_\Delta v_0(e^{it})^{-1}dt \right)<\infty,$$ where the supremum is taken over the set of all intervals $\Delta\subset(0,2\pi)$ and ${\lvert\Delta\rvert}$ is the length of the interval $\Delta$. If there is no *a priori* relation between $v_0$ and $v_1$, the two weights problem is open, despite many years of efforts. Some necessary and some sufficient conditions are known but no effective complete description of all pairs of weights $v_0$, $v_1$ was available available till recently. The recent news at the time of writing is that the conjunction of three preprints [@NTV], [@LSSUT], [@L] proved a long-standing conjecture of Nazarov–Treil–Volberg (see [@Vo]), stating that for the Hilbert transform the so-called two-weight $T1$ theorem is valid. However, the conditions of $T1$ theorem are not easily translated (if at all) into conditions on weights. Under these circumstances, any partial information on the problem is valuable. One such piece of information is Koosis’ theorem [@Koosis1]: For every weight $v_0\geq0$ such that $0<v_0(e^{it})<1$ for a.e. $t\in(0,2\pi)$ and $v_0^{-1}\in L^1({{\mathbb T}})$, one can find another weight $v_1$, $0\leq v_1\leq v_0$, such that $\log v_1\in L^1({{\mathbb T}})$ and such that the Hardy projection $P_+$ is bounded between the weighted spaces . Koosis’ proof (see also [@NVY Appendix]) is an ingenious calculation, but one can argue that it has a rather *ad hoc* flavour. The purpose of this note is to point out that Koosis’ theorem follows *naturally* from the formalism of spectral perturbation theory (more precisely, scattering theory) in the form suggested by de Branges in [@deB]. In fact, the statement we get in this way is more general than the original Koosis’ theorem; we obtain an *operator-valued* analogue. That is, our $L^2$ spaces consist of functions on ${{\mathbb T}}$ with values in a Hilbert space ${{\mathcal K}}$ and our weights are functions with values in the Schatten classes of compact operators in ${{\mathcal K}}$. We hope that this note will attract the attention of experts to the connection between the two weights problem and scattering theory. We believe that this connection is yet to be thoroughly explored. Preliminaries {#sec.a1a} ------------- First we would like to rewrite the two weights problem in an equivalent form. Let $f\in L^2({{\mathbb T}},v_0(e^{it})dt)$ and suppose that the weight $v_0$ vanishes on some open set. Then the function $f$ is not defined on this open set, and therefore it is not clear how to define the projections $P_\pm f$ by . This suggests that the integration in the definition of the projections $P_\pm$ should be performed with respect to the weighted measure $v_0(e^{it})dt$. Thus, for a weight $w_0:{{\mathbb T}}\to[0,\infty)$, we define the *weighted Hardy projections $P_\pm^{(w_0)}$* by $$(P_\pm^{(w_0)}f)(e^{i\theta}) = \pm \lim_{r\to1\mp0}\int_0^{2\pi}\frac{w_0(e^{it})f(e^{it})}{1-re^{i(\theta-t)}}\frac{dt}{2\pi}; \label{a3}$$ the existence of the limits will be discussed separately. If $v_0(e^{it})>0$ for a.e.  $t$, then a simple argument with replacing $f$ by $v_0f$ shows that $P_+$ is a bounded operator between the spaces if and only if $$P_+^{(w_0)}: L^2({{\mathbb T}},w_0(e^{it})dt)\to L^2({{\mathbb R}},w_1(e^{it})dt) \label{a4}$$ is bounded, where $w_1=v_1$ and $w_0=v_0^{-1}$. Thus, we obtain For every weight $w_0\geq0$ such that $w_0(e^{it})>0$ for a.e. $t\in(0,2\pi)$ and $w_0\in L^1({{\mathbb T}})$, one can find another weight $w_1\geq0$ with $w_1w_0\leq1$ and $\log w_1\in L^1({{\mathbb T}})$ such that the weighted Hardy projection $P_+^{(w_0)}$ is a bounded operator between the spaces . It is this second version of Koosis’ theorem that we will discuss in this paper. Operator valued functions {#sec.a1b} ------------------------- Let ${{\mathcal K}}$ be a Hilbert space; the case $\dim {{\mathcal K}}<\infty$ is not excluded, neither it is trivial. We denote by $(\cdot,\cdot)$ the inner product in ${{\mathcal K}}$ and by ${\lVert\cdot\rVert}$ the norm in ${{\mathcal K}}$. Notation ${\mathcal{B}}({{\mathcal K}})$ stands for the set of all bounded linear operators on ${{\mathcal K}}$ and ${\mathbf{S}}_p$, $1\leq p<\infty$, denotes the Schatten class of compact operators in ${{\mathcal K}}$; in particular, ${\mathbf{S}}_1$ is the trace class. We denote by ${\lVert\cdot\rVert}_p$ the norm in ${\mathbf{S}}_p$ and by ${\lVert\cdot\rVert}_{\mathcal{B}}$ the norm in ${\mathcal{B}}({{\mathcal K}})$. As usual, for $w\in{\mathcal{B}}({{\mathcal K}})$, notation $w\geq0$ means that $(w\chi,\chi)\geq0$ for all elements $\chi\in{{\mathcal K}}$, and in the same way $w\leq C$, where $C$ is a constant, means $(w\chi,\chi)\leq C{\lVert\chi\rVert}^2$ for all $\chi\in{{\mathcal K}}$. For any $w\in{\mathcal{B}}({{\mathcal K}})$ such that $w\geq0$, the square root $w^{1/2}$ is defined via the functional calculus for self-adjoint operators. Below we work with “nice” ${{\mathcal K}}$-valued functions of the form $$f(\mu)=\sum_i (\mu-z_i)^{-1} \chi_i, \quad \mu\in{{\mathbb T}}, \quad \chi_i\in{{\mathcal K}}, \quad {\lvertz_i\rvert}\not=1, \label{a5}$$ where the sum has finitely many terms. We will denote by ${\mathcal{L}}$ the set of all such “nice” functions $f$. Let $w:{{\mathbb T}}\to{\mathcal{B}}({{\mathcal K}})$ be a Borel measurable function. Suppose that $w$ is non-negative i.e. $w(e^{it})\geq0$ for a.e.  $t\in(0,2\pi)$, and that $w$ satisfies $$\int_0^{2\pi}(w(e^{it})\chi,\chi)\frac{dt}{2\pi} \leq C{\lVert\chi\rVert}^2 \label{a4a}$$ for some constant $C$ and all $\chi\in{{\mathcal K}}$. Then for any $f\in{\mathcal{L}}$ we can define the quasi-norm $${\lVertf\rVert}_{L^2(w)}^2 = \int_0^{2\pi} (w(e^{it})f(e^{it}),f(e^{it}))\frac{dt}{2\pi}. \label{a4b}$$ After taking the quotient over the subspace of functions $f$ with ${\lVertf\rVert}_{L^2(w)}=0$, we obtain a norm on the quotient space; the space obtained by taking the closure is, by definition, the weighted space $L^2(w)$. Thus, by construction, ${\mathcal{L}}$ is dense in $L^2(w)$. Main result and discussion {#sec.a2} -------------------------- Let $1\leq p<\infty$, and let $w_0:{{\mathbb T}}\to{\mathbf{S}}_p$ be a Borel measurable function. We assume that $w_0$ is non-negative and satisfies $$\int_0^{2\pi} {\lVertw_0(e^{it})\rVert}_p \frac{dt}{2\pi}<\infty;$$ this, of course, implies . For convenience, we will assume that $w_0$ is normalised so that the above integral equals one: $$\int_0^{2\pi} {\lVertw_0(e^{it})\rVert}_p \frac{dt}{2\pi}=1. \label{a6}$$ For such weight $w_0$ and for $f\in{\mathcal{L}}$, we define the weighted Hardy projections $P_\pm^{(w_0)}$, as in the scalar case, by . It is clear that for every $r\not=1$, the integrals in converge absolutely in the norm of ${{\mathcal K}}$. \[th.a1\] Let $1\leq p<\infty$, and let $w_0:{{\mathbb T}}\to{\mathbf{S}}_p$ be a Borel measurable non-negative ($w_0\geq0$ a.e.) weight function which satisfies . Then for all $f\in{\mathcal{L}}$ (i.e. for all $f$ of the form ) and for a.e.  $\theta\in(0,2\pi)$, the limits in exist in the norm of ${{\mathcal K}}$. Further, there exists a non-trivial Borel measurable non-negative weight function $w_1:{{\mathbb T}}\to{\mathcal{B}}({{\mathcal K}})$, which satisfies $$\int_0^{2\pi}(w_1(e^{it})\chi,\chi)\frac{dt}{2\pi} \leq {\lVert\chi\rVert}^2, \quad \forall \chi\in{{\mathcal K}},$$ and there exist contractions (i.e. operators of norm $\leq1$) $X$, $Y_+$, $Y_-$, acting from $L^2(w_0)$ to $L^2(w_1)$, such that the weighted Hardy projections $P_\pm^{(w_0)}$ can be represented as $$P_\pm^{(w_0)}=\pm\frac{i}{2}(X-Y_\pm). \label{a7a}$$ In particular, $$P_\pm^{(w_0)}: L^2(w_0)\to L^2(w_1)$$ are contractions. Let us discuss this result. 1\. It is easy to see that the sum $P_+^{(w_0)}+P_-^{(w_0)}$ is simply the operator of multiplication by $w_0$: $$(P_+^{(w_0)}f)(e^{i\theta})+(P_-^{(w_0)}f)(e^{i\theta}) = w_0(e^{i\theta})f(e^{i\theta}).$$ By , it follows that this operator of multiplication has norm $\leq1$. From this it follows that $$w_0(e^{i\theta})^{1/2}w_1(e^{i\theta})w_0(e^{i\theta})^{1/2}\leq 1 \label{a9}$$ for a.e.  $\theta\in(0,2\pi)$; see the end of Section \[sec.d\] for the details of this argument. 2\. In fact, more than is true; we note without proof that the boundedness of $P_\pm^{(w_0)}$ implies that $$(P_r*w_0)^{1/2}(P_r*w_1)(P_r*w_0)^{1/2}\leq C$$ for all $r<1$ with some constant $C$; here $P_r*w_{0,1}$ is the convolution with the Poisson kernel $$P_r(\theta)=\frac{1-r^2}{1+r^2-2r\cos\theta}. \label{a9a}$$ 3\. Of course, the boundedness of $P_\pm^{(w_0)}$ implies that the weighted Hilbert transform $$(H^{(w_0)} f)(e^{i\theta}) = \lim_{r\to1} \int_{0}^{2\pi} w_0(e^{it}) \frac{2\sin(\theta-t)}{1+r^2-2r \cos(\theta-t)} f(e^{it})\frac{dt}{2\pi}$$ is a bounded map from $L^2(w_0)$ to $L^2(w_1)$. 4\. If $p=1$, the weight $w_1$ can be chosen to satisfy $$\int_0^{2\pi} {\lVertw_1(e^{it})\rVert}_1\frac{dt}{2\pi}<\infty; \label{a10}$$ see the end of Section \[sec.d\]. 5\. The weight function $w_1$ constructed in the Koosis theorem is non-degenerate in the sense that $\log w_1\in L^1({{\mathbb T}})$. The weight function $w_1$ that we construct in Theorem \[th.a1\] is also non-degenerate in the following sense. One has $$w_0(\mu) = D_0^+(\mu)^* w_1(\mu) D_0^+(\mu), \quad \text{ a.e.\ $\mu\in{{\mathbb T}}$,}$$ where $D_0^+$ is an operator valued function to be constructed below (see ). The function $D_0^+$ satisfies ${\lVertD_0^+(\cdot)\rVert}_{\mathcal{B}}\in L^{1,\infty}({{\mathbb T}})$ and $D_0^+(\mu)$ has a bounded inverse for a.e.  $\mu\in{{\mathbb T}}$. In particular, $$\operatorname{rank}w_0(\mu)=\operatorname{rank}w_1(\mu), \quad \text{ a.e.\ $\mu\in{{\mathbb T}}$,} \label{a12}$$ and $${\lVertw_1(\mu)\rVert}_{\mathcal{B}}\geq \frac{{\lVertw_0(\mu)\rVert}_{\mathcal{B}}}{{\lVertD_0^+(\mu)\rVert}_{\mathcal{B}}^2}, \quad \text{ a.e.\ $\mu\in{{\mathbb T}}$.} \label{a30}$$ By , we have $$\log {\lVertw_1(\mu)\rVert}_{\mathcal{B}}\geq \log {\lVertw_0(\mu)\rVert}_{\mathcal{B}}- 2\log^+{\lVertD_0^+(\mu)\rVert}_{\mathcal{B}},$$ and ${\lVertD_0^+(\cdot)\rVert}_{\mathcal{B}}\in L^{1,\infty}({{\mathbb T}})$ implies $\log^+{\lVertD_0^+(\cdot)\rVert}_{\mathcal{B}}\in L^p({{\mathbb T}})$ for all $p<\infty$. The outline of the proof {#sec.a3} ------------------------ We consider the absolutely continuous (a.c.) operator valued measure on ${{\mathbb T}}$ given by $$d\nu_0(e^{i\theta})=w_0(e^{i\theta})\frac{d\theta}{2\pi}. \label{a13}$$ For this measure $\nu_0$, we exhibit (see Lemma \[lma.b1\]) a Hilbert space ${{\mathcal H}}$, a unitary operator $U_0$ in ${{\mathcal H}}$ and a contraction $G:{{\mathcal H}}\to{{\mathcal K}}$ such that $$\nu_0(\delta)=GE_{U_0}(\delta)G^*, \quad \delta\subset{{\mathbb T}}, \label{a14}$$ where $E_{U_0}$ is the projection-valued spectral measure of $U_0$, and $\delta\subset{{\mathbb T}}$ is any Borel set. Next, we construct (see ) a unitary operator $U_1$ in ${{\mathcal H}}$ such that the identities $$\begin{aligned} (\alpha+\psi_0(z)) (\alpha-\psi_1(z)) &=I, \label{a15} \\ (\alpha-\psi_1(z)) (\alpha+\psi_0(z)) &=I, \label{a16}\end{aligned}$$ hold true for all ${\lvertz\rvert}\not=1$; here $\alpha$ is the auxiliary bounded self-adjoint operator given by $$\alpha=\sqrt{I-(GG^*)^2}, \label{a17}$$ and $$\psi_j(z)=i G\frac{U_j+z}{U_j-z} G^*. \label{a17a}$$ Further, similarly to , we set $$\nu_1(\delta)=GE_{U_1}(\delta)G^*, \quad \delta\subset {{\mathbb T}}. \label{a18}$$ We will be able to prove (in Lemma \[lma.c2\]) that the a.c. part of the measure $\nu_1$ can be represented as $$d\nu^{\text{\rm (ac)}}_1(e^{i\theta})=w_1(e^{i\theta})\frac{d\theta}{2\pi}$$ with some operator valued non-negative weight function $w_1$. Note that this is not automatic: the Radon-Nikodym theorem for operator valued measures in general fails; to see this, consider the spectral measure of a self-adjoint or unitary operator with a non-trivial a.c. component. Key to our construction is the connection between the weighted Hardy projections $P_\pm^{(w_0)}$ and certain operators appearing in scattering theory for the pair $U_0$, $U_1$. We use the formalism suggested by de Branges [@deB] with some simplifications due to Kuroda [@Ku]. This formalism makes use of the weighted Hilbert spaces $L^2(\nu_j)$, $j=0,1$ of ${{\mathcal K}}$-valued functions on ${{\mathbb T}}$. They are defined, similarly to , starting from the quasi-norm $${\lVertf\rVert}_{L^2(\nu_j)}^2 = \int_0^{2\pi} d(\nu_j(e^{it})f(e^{it}),f(e^{it}))$$ on the set ${\mathcal{L}}$, by taking a quotient and then a closure. We note that $\nu_0=\nu^{\text{\rm (ac)}}_0$ and $$L^2(\nu_1)\subset L^2(\nu_1^{\text{\rm (ac)}}) \quad \text{ and } \quad {\lVertf\rVert}_{L^2(\nu_1^{\text{\rm (ac)}})} \leq {\lVertf\rVert}_{L^2(\nu_1)}. \label{a20}$$ Following de Branges, we define some auxiliary bounded operators $X$, $Y_+$ and $Y_-$ acting from $L^2(\nu_0)$ to $L^2(\nu_1^{{\text{\rm (ac)}}})$. First we denote (cf. , ) $$D_0(z) = \alpha+\psi_0(z), \quad D_1(z) = -\alpha+\psi_1(z). \label{a21}$$ By , we have $$D_0(z)D_1(z)=D_1(z)D_0(z)=-I, \quad {\lvertz\rvert}\not=1. \label{a22}$$ Let $$X: L^2(\nu_0)\to L^2(\nu_1) \label{a23}$$ be the linear operator, defined on the dense set ${\mathcal{L}}$ by $$(Xf)(\mu) = \sum_i (\mu-z_i)^{-1} D_0(z_i)\chi_i, \quad f(\mu)=\sum_i (\mu-z_i)^{-1}\chi_i. \label{a24}$$ It turns out (see Lemma \[lma.b2\]) that $X$ is a unitary operator between the spaces . Moreover, this is true for *any* operators $U_0$, $U_1$, $G$, $\alpha$, related by –; assumption is not relevant here. This fact is part of de Branges’ construction [@deB]. Bearing in mind the embedding , we see that $X$ is a contraction as a map from $L^2(\nu_0)$ to $L^2(\nu_1^{\text{\rm (ac)}})$. Further, by the spectral theorem for the unitary operator $U_0$, we have $$\psi_0(z) = i\int_0^{2\pi}\frac{e^{it}+z}{e^{it}-z}d\nu_0(e^{it}) = i\int_0^{2\pi}\frac{e^{it}+z}{e^{it}-z} w_0(e^{it})\frac{dt}{2\pi}. \label{a24a}$$ Thus, $\psi_0$ is the Cauchy transform of $w_0$. Using assumption on the weight $w_0$ and the *UMD property* (see e.g. [@deF]) of the space ${\mathbf{S}}_p$, $1<p<\infty$, we check (in Lemma \[lma.c1\]) that the limits $$D_0^\pm(e^{i\theta}) = \lim_{r\to1\pm0} D_0(re^{i\theta}), \quad D_1^\pm(e^{i\theta}) = \lim_{r\to1\pm0} D_1(re^{i\theta}) \label{a25}$$ exist for a.e.  $\theta\in(0,2\pi)$ in the operator norm. For $p=1$, this was proven in [@deB]; for $p>1$, this fact is borrowed from from our related work [@PuV]. Again following de Branges, we consider the operators $$Y_\pm: f(\mu)\mapsto D_0^{\pm}(\mu)f(\mu), \quad \mu\in{{\mathbb T}}, \label{a26}$$ defined initially on the set ${\mathcal{L}}$, and show that $Y_\pm$ extend as isometric operators $$Y_\pm: L^2(\nu_0)\to L^2(\nu_1^{\text{\rm (ac)}}).$$ Finally, a simple calculation (see Section \[sec.d\]) shows that $P_\pm^{(w_0)}$, $X$, $Y_\pm$ are related by . We note that $Y_\pm$ are unitarily equivalent to the wave operators $W_\pm(U_1,U_0)$ (see [@Ku]), although we will not need this fact. Identities , and the map $X$ {#sec.b} ============================ The construction of $G$, $U_0$, $U_1$, $\alpha$ {#sec.b1} ------------------------------------------------ Let ${{\mathcal H}}$ be the Hilbert space of all Borel measurable ${{\mathcal K}}$-valued functions on ${{\mathbb T}}$ with the norm $${\lVertf\rVert}_{{\mathcal H}}^2 = \int_0^{2\pi} {\lVertf(e^{it})\rVert}^2 \frac{dt}{2\pi}.$$ Let $U_0$ be the operator of multiplication by $e^{it}$ in ${{\mathcal H}}$. Let $G:{{\mathcal H}}\to{{\mathcal K}}$ be defined by $$Gf=\int_0^{2\pi} w_0(e^{it})^{1/2} f(e^{it})\frac{dt}{2\pi}.$$ Then our assumption implies that $G$ is a contraction: $$\begin{gathered} {\lVertGf\rVert} \leq \left(\int_0^{2\pi}{\lVertw_0(e^{it})^{1/2}\rVert}_{\mathcal{B}}^2\frac{dt}{2\pi}\right)^{1/2} \left(\int_0^{2\pi}{\lVertf(e^{it})\rVert}^2\frac{dt}{2\pi}\right)^{1/2} \\ = \left(\int_0^{2\pi}{\lVertw_0(e^{it})\rVert}_{\mathcal{B}}\frac{dt}{2\pi}\right)^{1/2} {\lVertf\rVert}_{{\mathcal H}}\leq {\lVertf\rVert}_{{\mathcal H}}.\end{gathered}$$ It is clear that setting $\nu_0(\delta)=GE_{U_0}(\delta)G^*$ (see ) yields . Next, let $$\Theta =2\sin^{-1}(G^*G);$$ thus, $\Theta$ is a bounded self-adjoint operator in ${{\mathcal H}}$ with $\sigma(\Theta)\subset [0,\pi)$ and $$G^*G=\sin(\tfrac12 \Theta). \label{b2}$$ Set $$U_1=\exp(\tfrac{i}2 \Theta)U_0\exp(\tfrac{i}2 \Theta) \label{b3}$$ and let $\alpha$ be defined by . \[lma.b1\] Let $U_0$, $U_1$, $G$, $\alpha$ be as described above. Then identities , hold true. The measure $\nu_1$, defined by , satisfies $\nu_1({{\mathbb T}})=\nu_0({{\mathbb T}})$ and $${\lVert\nu_1({{\mathbb T}})\rVert}\leq 1. \label{b3a}$$ Denote $$\beta=\sqrt{I-(G^*G)^2};$$ clearly, we have $$\alpha G=G\beta. \label{b5}$$ Comparing and the definition of $\beta$, we find that $$\beta=\cos(\tfrac12\Theta).$$ Using this and a little algebra, we obtain $$U_1G^*G+G^*GU_0+i(U_1\beta-\beta U_0)=0.$$ From here by straightforward manipulation we obtain the identity $$(U_1-z)G^*G(U_0-z) + i\bigl((U_1+z)\beta(U_0-z)-(U_1-z)\beta(U_0+z)\bigr) - (U_1+z)G^*G(U_0+z)=0$$ for any $z\in{{\mathbb C}}$. Taking ${\lvertz\rvert}\not=1$ and multiplying by $(U_1-z)^{-1}$ on the left and by $(U_0-z)^{-1}$ on the right, we get $$G^*G + i\left( \frac{U_1+z}{U_1-z}\beta-\beta\frac{U_0+z}{U_0-z} \right) - \frac{U_1+z}{U_1-z}G^*G\frac{U_0+z}{U_0-z} =0.$$ Multiplying this by $G$ on the left and by $G^*$ on the right and using that (by ) $$(GG^*)^2=I-\alpha^2,$$ we obtain $$-\alpha^2 + iG\frac{U_1+z}{U_1-z}\beta G^* - i G\beta\frac{U_0+z}{U_0-z} G^* - G\frac{U_1+z}{U_1-z}G^* G\frac{U_0+z}{U_0-z} G^* = -I.$$ Finally, using , this transforms into . The relation is obtained by taking adjoints in and changing $z$ to $\overline{z}^{-1}$. By , , we have $\nu_0({{\mathbb T}})=\nu_1({{\mathbb T}})=GG^*$. The estimate follows from the inequality ${\lVertG\rVert}\leq 1$. In fact, the construction of [@deB; @Ku] allows for a whole family of possible choices for operators $G$, $U_0$, $U_1$, suitable for our argument. For simplicity, we have chosen only one representative of this family. In order to clarify the ideas behind Lemma \[lma.b1\], let us sketch the analogous argument for the case of the weights $w_0$, $w_1$ on the real line. In this case the construction naturally leads to self-adjoint (rather than unitary) operators and the algebra is somewhat more transparent. Let a non-negative weight $w_0:{{\mathbb R}}\to{\mathcal{B}}({{\mathcal K}})$ satisfy $$\int_{{\mathbb R}}{\lVertw_0(t)\rVert}_{{\mathbb R}}dt<\infty.$$ Let ${{\mathcal H}}_{{\mathbb R}}$ be the $L^2$ space of ${{\mathcal K}}$-valued functions on ${{\mathbb R}}$ with the norm $${\lVertf\rVert}_{{{\mathcal H}}_{{\mathbb R}}}^2 = \int_{{\mathbb R}}{\lVertf(t)\rVert}^2dt.$$ Let $A_0$ be the operator of multiplication by the independent variable $t$ in ${{\mathcal H}}_{{\mathbb R}}$ and let $G_{{\mathbb R}}:{{\mathcal H}}_{{\mathbb R}}\to{{\mathcal K}}$ be given by $$G_{{\mathbb R}}f=\int_{{\mathbb R}}w_0(t)^{1/2} f(t)dt.$$ We set $A_1=A_0+G_{{\mathbb R}}^*G_{{\mathbb R}}$. Then from the standard resolvent identity we get $$\begin{gathered} (I+G_{{\mathbb R}}(A_0-z)^{-1}G_{{\mathbb R}}^*) (I-G_{{\mathbb R}}(A_1-z)^{-1}G_{{\mathbb R}}^*) \\ = (I-G_{{\mathbb R}}(A_1-z)^{-1}G_{{\mathbb R}}^*) (I+G_{{\mathbb R}}(A_0-z)^{-1}G_{{\mathbb R}}^*) =I;\end{gathered}$$ this is the analogue of , . One sets $$\nu_j^{{\mathbb R}}(\delta)=G_{{\mathbb R}}E_{A_j}(\delta)G_{{\mathbb R}}^*, \quad j=0,1, \quad \delta\subset {{\mathbb R}},$$ and the rest of the construction is very similar to the case of measures on ${{\mathbb T}}$. The map $X$ {#sec.b2} ----------- Let the map $X$ be defined by , . \[lma.b2\] The map $X$ is unitary between the spaces $L^2(\nu_0)$ and $L^2(\nu_1)$. For $j=0,1$, the functions $\psi_j$ (see ) can be expressed as $$\psi_j(z)=i\int_0^{2\pi}\frac{e^{it}+z}{e^{it}-z}d\nu_j(e^{it}). \label{b6}$$ We note two identities for $\psi_j$: $$\begin{gathered} \frac{\psi_j(z_1)-\psi_j(z_2)^*}{z_1-\overline{z_2}^{-1}} = 2i\int_0^{2\pi} \frac{e^{it}}{(e^{it}-z_1)(e^{it}-\overline{z_2}^{-1})} d\nu_j(e^{it}), \label{b7} \\ \psi_j(z)^*=\psi_j(\overline{z}^{-1}). \label{b8}\end{gathered}$$ Next, using , , we have for ${\lvertz_{1,2}\rvert}\not=1$: $$\begin{gathered} \psi_0(z_1)-\psi_0(z_2)^* = D_0(z_1)-D_0(z_2)^* \\ = -D_0(z_2)^*D_1(z_2)^*D_0(z_1) + D_0(z_2)^*D_1(z_1)D_0(z_1) \\ = D_0(z_2)^*(-\psi_1(z_2)^*+\psi_1(z_2))D_0(z_1).\end{gathered}$$ Combining this with , , we get $$\int_0^{2\pi} \frac{d\nu_0(e^{it})}{(e^{-it}-\overline{z_2})(e^{it}-z_1)} = D_0(z_2)^* \int_0^{2\pi} \frac{d\nu_1(e^{it})}{(e^{-it}-\overline{z_2})(e^{it}-z_1)} D_0(z_1). \label{b9}$$ Now let $$f_1(\mu)=(\mu-z_1)^{-1}\chi_1, \quad f_2(\mu)=(\mu-z_2)^{-1}\chi_2, \quad \mu\in{{\mathbb T}}, \label{b10}$$ where ${\lvertz_{1,2}\rvert}\not=1$ and $\chi_{1,2}\in{{\mathcal K}}$. Then from we get $$(f_1,f_2)_{L^2(\nu_0)} = (Xf_1,Xf_2)_{L^2(\nu_1)}.$$ This extends to all $f_1,f_2\in{\mathcal{L}}$. It follows that $X$ is an isometry. By considering an operator $X_1$ defined in a similar way with $D_1$ instead of $D_0$, and using , we obtain $XX_1=-I$, hence $X$ is a surjection. Thus, $X$ is a unitary operator. The boundary values of $D_0$ and $D_1$ {#sec.c} ====================================== Existence of boundary values of $D_0$ and $D_1$ {#sec.c1} ----------------------------------------------- \[lma.c1\] The limits $D_0^\pm(e^{i\theta})$, $D_1^\pm(e^{i\theta})$ (see ) exist for a.e.  $\theta\in(0,2\pi)$ in the operator norm. For $p=1$, this was proven in [@deB]. 1\. First we consider the limits $D_0^\pm$. We have $$D_0(z)=\alpha+\psi_0(z),$$ where $\psi_0(z)$ is given by . Thus, it suffices to consider the limits of $\psi_0$. By , it suffices to consider the limits as $z$ approaches the unit circle from inside the unit disk. Without loss of generality assume $p>1$ in . In fact, we will prove the existence of the non-tangential limits $$\lim_{\genfrac{}{}{0pt}{}{z\to e^{i\theta}}{z\in S_\theta}} \psi_0(z)$$ in the norm of ${\mathbf{S}}_p$. Here $S_\theta$ is the appropriate sector of opening $\pi/2$ with the vertex at $e^{i\theta}$ (see e.g. [@Koosis2 Section VIII:C3]). The argument below is presented in more detail in our related work [@PuV]. The function $\psi_0$ is the Cauchy transform of the weight function $w_0$ (see ). Consider the non-tangential maximal function $$(T w_0)(e^{i\theta}) = \sup \left\{{\left\lVert\psi_0(z)\right\rVert}_p: z\in S_\theta\right\}.$$ The key fact is that for $1<p<\infty$, the Banach space ${\mathbf{S}}_p$ possesses the UMD property, see [@deF]; that is, the Hilbert transform and many other integral transforms are bounded as operators in $L^2$ spaces of ${\mathbf{S}}_p$-valued functions. Using this, one can prove that the (non-linear) operator $T$ is of the weak 1-1 type, i.e. $T w_0$ belongs to the weak $L^{1,\infty}({{\mathbb T}})$ class. Next, using this fact and repeating the classical construction of Privalov’s uniqueness theorem (see e.g. [@Koosis2 Section III:D]), for any ${\varepsilon}>0$ one constructs a simply connected domain ${{\mathcal D}}$ in the unit disk such that ${\lVert\psi_0\rVert}_p$ is bounded in ${{\mathcal D}}$ and the boundary of ${{\mathcal D}}$ contains the unit circle ${{\mathbb T}}$ up to a set of measure ${\varepsilon}$. Let $\varphi$ be a conformal map of the unit disk onto ${{\mathcal D}}$. Then $F(z)=\psi_0(\varphi(z))$ is a bounded ${\mathbf{S}}_p$-valued analytic function on the unit disk. By standard results on Banach space valued analytic functions (see e.g. [@Bu]), $F(z)$ attains non-tangential boundary values in ${\mathbf{S}}_p$ norm a.e.  on the unit circle. It follows that the function $\psi_0$ attains non-tangential boundary values in ${\mathbf{S}}_p$ norm on the unit circle minus a set of measure ${\varepsilon}$. Sending ${\varepsilon}\to0$, one obtains the desired result. 2\. Let us consider the limits of $D_1$. Since $D_1(z)=-D_0(z)^{-1}$, it suffices to prove that the limiting operators $D_0^\pm(e^{i\theta})$ have bounded inverses for a.e.  $\theta$. We do this by employing an argument from [@Yafaev]. We have $$D_0(z)=D_0(0)\left(I+D_0(0)^{-1}(D_0(z)-D_0(0))\right),$$ and therefore it suffices to check that the operators $$I+D_0(0)^{-1}(D_0^\pm(e^{i\theta})-D_0(0)) \label{c1}$$ have a bounded inverse for a.e.  $\theta$. By , we have $\psi_0(z)\in{\mathbf{S}}_p$ for all ${\lvertz\rvert}\not=1$. Let $q\geq p$ be any integer; consider the regularised determinant $$d(z)=\operatorname{Det}_q(I+D_0(0)^{-1}(\psi_0(z)-\psi_0(0))).$$ The functional $A\mapsto\operatorname{Det}_q(I+A)$ is continuous (in fact, analytic) on ${\mathbf{S}}_q$. Thus, $d(z)$ is analytic in $z$ and by the previous step of the proof, $d(z)$ has non-tangential boundary values a.e.  on the unit circle. Applying Privalov’s uniqueness theorem, we obtain that these boundary values are non-zero a.e.  on the unit circle. Now since $\operatorname{Det}_q(I+A)\not=0$ if and only if $I+A$ has a bounded inverse, we conclude that the operators have bounded inverses for a.e.  $\theta$. The a.c. part of $\nu_1$ {#sec.c2} ------------------------ Taking $z_1=z_2=re^{i\theta}$ in , one obtains $$\psi_j(re^{i\theta})-\psi_j(re^{i\theta})^* = 2i\int_0^{2\pi} P_r(\theta-t)d\nu_j(e^{it}), \label{c2}$$ where $P_r$ is the Poisson kernel on ${{\mathbb T}}$. From the existence of the boundary values of $\psi_j$ on ${{\mathbb T}}$ (see Lemma \[lma.c1\]) it folows that the r.h.s. of attains a limit (in the operator norm) as $r\to1$ for a.e.  $\theta\in(0,2\pi)$. Of course, by the definition of $\nu_0$ we have $$w_0(e^{i\theta}) = \lim_{r\to1} \int_0^{2\pi} P_r(\theta-t)d\nu_0(e^{it}) \label{c7}$$ for a.e.  $\theta$. Similarly, we *define* the weight function $w_1$ by $$w_1(e^{i\theta})= \lim_{r\to1} \int_0^{2\pi} P_r(\theta-t)d\nu_1(e^{it}) \label{c3}$$ for a.e.  $\theta$. In Lemmas \[lma.c2\] and \[lma.c3\], we follow de Branges’ work [@deB]. \[lma.c2\] The a.c. part of the measure $\nu_1$ is given by $$d\nu_1^{\text{\rm (ac)}}(e^{i\theta})=w_1(e^{i\theta})\frac{d\theta}{2\pi}, \quad \text{ a.e.\ $\theta\in(0,2\pi)$.} \label{c4}$$ Of course, in the scalar case $\dim{{\mathcal K}}<\infty$ formula follows directly from ; the point here is to consider the general case. Let $\chi_1,\chi_2\in{{\mathcal K}}$; consider the scalar (complex-valued) measure $(\nu_1(\cdot)\chi_1,\chi_2)$. If $\nu_1^{\text{\rm (ac)}}$ and $\nu_1^{\text{\rm (sing)}}$ are the a.c. and the singular parts of $\nu_1$ with respect to the Lebesgue measure on ${{\mathbb T}}$, then $$(\nu_1(\cdot)\chi_1,\chi_2) = (\nu_1^{\text{\rm (ac)}}(\cdot)\chi_1,\chi_2) + (\nu_1^{\text{\rm (sing)}}(\cdot)\chi_1,\chi_2)$$ gives the unique decomposition of the scalar measure $(\nu_1(\cdot)\chi_1,\chi_2)$ into the a.c. and singular parts. By the scalar theory, we have $$\lim_{r\to1}\int_0^{2\pi} P_r(\theta-t)d(\nu_1^{\text{\rm (sing)}}(e^{it})\chi_1,\chi_2)=0$$ for a.e.  $\theta$. Thus, using , we obtain $$(w_1(e^{i\theta})\chi_1,\chi_2) = \lim_{r\to1} \int_0^{2\pi} P_r(\theta-t)d(\nu_1^{\text{\rm (ac)}}(e^{it})\chi_1,\chi_2). \label{c5}$$ Now take $f_1$, $f_2$ as in ; multiplying by $(e^{i\theta}-z_1)^{-1}(e^{-i\theta}-\overline{z_2})^{-1}$ and integrating, we get $$\int_0^{2\pi} (w_1(e^{i\theta})f_1(e^{i\theta}), f_2(e^{i\theta}))\frac{d\theta}{2\pi} = \int_0^{2\pi} d(\nu_1^{\text{\rm (ac)}}(e^{i\theta})f_1(e^{i\theta}),f_2(e^{i\theta})).$$ By linearity, this extends to all $f_1,f_2\in{\mathcal{L}}$. This yields . The operators $Y_\pm$ {#sec.c3} --------------------- Next, we consider the operators $Y_\pm$ of multiplication by $D_0^\pm$, see . \[lma.c3\] The operators $Y_\pm$ are unitary maps from $L^2(\nu_0)$ to $L^2(\nu_1^{\text{\rm (ac)}})$. Taking $z_1=z_2=re^{i\theta}$ in , we obtain $$\int_0^{2\pi} P_r(\theta-t)d\nu_0(e^{it}) = D_0(re^{i\theta})^* \int_0^{2\pi} P_r(\theta-t)d\nu_1(e^{it}) \ D_0(re^{i\theta}).$$ Taking $r\to1\pm0$ and using Lemma \[lma.c2\], we get $$w_0(\mu) = D_0^\pm(\mu)^*w_1(\mu)D_0^\pm(\mu), \quad \text{ a.e.\ $\mu\in{{\mathbb T}}$.}$$ This shows that $Y_\pm$ are isometries. Considering the operators of multiplication by the boundary values of $D_1$ and using the identity , it is easy to prove that $Y_\pm$ are surjections, so they are unitary operators. The proof of Theorem \[th.a1\] {#sec.d} ============================== The weight function $w_1$ has been defined by . By construction, it is non-negative. It is Borel measurable as a pointwise norm limit of continuous weight functions. Let us prove that the limits in exist in ${{\mathcal K}}$ and $$(P^{(w_0)}_\pm f)(e^{i\theta}) = \pm\frac{i}{2}((Xf)(e^{i\theta})-(Y_\pm f)(e^{i\theta})) \label{d1}$$ for a.e. $\theta$. Take $f(\mu)=(\mu-z)^{-1}\chi$, $\chi\in{{\mathcal K}}$, ${\lvertz\rvert}\not=1$. We have $$D_0(z)=\alpha+i\int_0^{2\pi}d\nu_0(e^{it})\frac{e^{it}+z}{e^{it}-z} = \alpha - i\int_0^{2\pi}d\nu_0(e^{it}) + 2i \int_0^{2\pi}d\nu_0(e^{it})\frac{e^{it}}{e^{it}-z},$$ and therefore, by the definition of $X$, $$(Xf)(e^{i\theta}) = \left(\alpha - i\int_0^{2\pi}d\nu_0(e^{it})\right)f(e^{i\theta}) + 2i \int_0^{2\pi}d\nu_0(e^{it})\frac{e^{it}}{(e^{i\theta}-z)(e^{it}-z)}\chi.$$ For the second term in the above sum, we have $$\begin{gathered} \int_0^{2\pi} d\nu_0(e^{it})\frac{e^{it}}{(e^{i\theta}-z)(e^{it}-z)}\chi = \int_0^{2\pi} d\nu_0(e^{it})\frac{f(e^{i\theta})-f(e^{it})}{e^{it}-e^{i\theta}}e^{it} \\ = \lim_{r\to1} \left\{ \int_0^{2\pi} d\nu_0(e^{it})\frac{f(e^{i\theta})}{1-re^{i(\theta-t)}} - \int_0^{2\pi} d\nu_0(e^{it})\frac{f(e^{it})}{1-re^{i(\theta-t)}} \right\},\end{gathered}$$ where the limits exist in the norm of ${{\mathcal K}}$. Putting this together, after a little algebra we get $$(Xf)(e^{i\theta}) = \lim_{r\to1} \left\{D_0(re^{i\theta})f(e^{i\theta}) -2i \int_0^{2\pi} d\nu_0(e^{it})\frac{f(e^{it})}{1-re^{i(\theta-t)}}\right\}. \label{d3}$$ By Lemma \[lma.c1\] the limits $$\lim_{r\to\pm1} D_0(re^{i\theta})f(e^{i\theta})$$ exist in the norm of ${{\mathcal K}}$. Thus, the limits of the integral in also exist. Recalling the definition of $P_\pm^{(w_0)}$, we obtain . By and , we have $$\begin{gathered} w_0(e^{i\theta}) = \lim_{r\to1}\int_0^{2\pi} P_r(\theta-t)d\nu_0(e^{it}) \\ = \frac1{2i}\lim_{r\to1} (\psi_0(re^{i\theta})-\psi_0(\tfrac1r e^{i\theta})) = \frac1{2i}\lim_{r\to1} (D_0(re^{i\theta})-D_0(\tfrac1r e^{i\theta})).\end{gathered}$$ Thus, if we denote by $Y_0$ the operator of multiplication by $w_0(e^{i\theta})$, acting from $L^2(\nu_0)$ to $L^2(\nu_1^{\text{\rm (ac)}})$, we obtain $$Y_0=\frac1{2i}(Y_+-Y_-),$$ and therefore ${\lVertY_0\rVert}\leq 1$. This yields $$\int_0^{2\pi} (w_1(e^{i\theta})w_0(e^{i\theta})f(e^{i\theta}), w_0(e^{i\theta})f(e^{i\theta}))\frac{d\theta}{2\pi} \leq \int_0^{2\pi} (w_0(e^{i\theta})f(e^{i\theta}),f(e^{i\theta}))\frac{d\theta}{2\pi},$$ which implies . Suppose $p=1$. Then $$\operatorname{Tr}(GG^*) = \int_0^{2\pi} \operatorname{Tr}( w_0(e^{i\theta}))\frac{d\theta}{2\pi} \leq 1,$$ hence $G$ is Hilbert-Schmidt. Then $$\int_0^{2\pi} {\lVertw_1(e^{i\theta})\rVert}_1\frac{d\theta}{2\pi} = \int_0^{2\pi} \operatorname{Tr}(w_1(e^{i\theta}))\frac{d\theta}{2\pi} = \operatorname{Tr}(\nu_1^{\text{\rm (ac)}}({{\mathbb T}})) \leq \operatorname{Tr}(\nu_1({{\mathbb T}})) = \operatorname{Tr}(GG^*)\leq 1,$$ i.e. ${\lVertw_1(\cdot)\rVert}_1\in L^1({{\mathbb T}})$. [12]{} *Perturbations of self-adjoint transformation,* American Journal of Mathematics, **84**, no. 4 (1962), 543–560. *Hardy spaces of vector-valued functions.* (Russian) Investigations on linear operators and theory of functions, VII. Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) **65** (1976), 5–16. *Moyennes quadratiques pondérées de fonctions périodiquess et de leurs conjuguées harmoniques,* C. R. Acad. Sci. Paris, Ser. A, **291** (1980), 255–257. *Introduction to $H_p$ spaces.* Second edition. Cambridge University Press, Cambridge, 1998. *On a stationary approach to scattering problem,* Bull. Amer. Math. Soc. **70** (1964), 556–560. , *Two Weight Inequality for the Hilbert Transform: A Real Variable Characterization, II*, preprint, arXiv1301.4663v3. , *Two weight inequality for the Hilbert transform: a real variable characterization*, preprint, arXiv 1201.4319v6. , *Weighted norm inequalities for the Hardy maximal function,* Trans. AMS **165** (1972), 207–226. *Two weight estimate for the Hilbert transform and Corona decomposition for non-doubling measures*, preprint, arXiv:1003.1596. *Asymptotics of orthogonal polynomials via the Koosis’ theorem,* Math. Res. Lett. **13**, no. 5–6 (2006), 975–983. *Scattering theory and Banach space valued singular integrals,* preprint, arXiv:1211.6694. *Martingale and integral transforms of Banach space valued functions.* Probability and Banach spaces (Zaragoza, 1985), 195–222, Lecture Notes in Math., 1221, Springer, Berlin, 1986. *Mathematical scattering theory. General theory.* American Mathematical Society, Providence, RI, 1992. *Calderón–Zygmund capacities and operators on non-homogeneous spaces*, CBMS Series in Math., v. 100, Amer. Math. Soc., 2003, pp. 1–165.

--- abstract: 'The Cherenkov Telescope Array (CTA) is the major next-generation observatory for ground-based very-high-energy gamma-ray astronomy. It will improve the sensitivity of current ground-based instruments by a factor of five to twenty, depending on the energy, greatly improving both their angular and energy resolutions over four decades in energy (from 20 GeV to 300 TeV). This achievement will be possible by using tens of imaging Cherenkov telescopes of three successive sizes. They will be arranged into two arrays, one per hemisphere, located on the La Palma island (Spain) and in Paranal (Chile). We present here the optimised and final telescope arrays for both CTA sites, as well as their foreseen performance, resulting from the analysis of three different large-scale Monte Carlo productions.' address: - 'Dept. of Physics and Centre for Advanced Instrumentation, Durham University, South Road, Durham DH1 3LE, United Kingdom' - 'Instituto de Astrofísica de Andalucía-CSIC, Glorieta de la Astronomía s/n, E-18008, Granada, Spain' - 'Universidad Nacional Autónoma de México, Delegación Coyoacán, 04510 Ciudad de México, Mexico' - 'Pontificia Universidad Católica de Chile, Avda. Libertador Bernardo O’ Higgins No 340, borough and city of Santiago, Chile' - 'University of Geneva - Département de physique nucléaire et corpusculaire, 24 rue du Général-Dufour, 1211 Genève 4, Switzerland' - 'INFN Dipartimento di Scienze Fisiche e Chimiche - Università degli Studi dell’Aquila and Gran Sasso Science Institute, Via Vetoio 1, Viale Crispi 7, 67100 L’Aquila, Italy' - 'Instituto de Astronomia, Geofísica, e Ciências Atmosféricas - Universidade de São Paulo, Cidade Universitária, R. do Matão, 1226, CEP 05508-090, São Paulo, SP, Brazil' - 'LUTH and GEPI, Observatoire de Paris, CNRS, PSL Research University, 5 place Jules Janssen, 92190, Meudon, France' - 'INAF - Osservatorio di astrofisica e scienza dello spazio di Bologna, Via Piero Gobetti 101, 40129 Bologna, Italy' - 'INAF - Osservatorio Astrofisico di Arcetri, Largo E. Fermi, 5 - 50125 Firenze, Italy' - 'INFN Sezione di Perugia, Via A. Pascoli, 06123 Perugia, Italy' - 'Università degli Studi di Perugia, Via A. Pascoli, 06123 Perugia, Italy' - 'INAF - Osservatorio Astronomico di Roma, Via di Frascati 33, 00040, Monteporzio Catone, Italy' - 'INFN Sezione di Napoli, Via Cintia, ed. G, 80126 Napoli, Italy' - 'University of Oxford, Department of Physics, Denys Wilkinson Building, Keble Road, Oxford OX1 3RH, United Kingdom' - 'EMFTEL department and IPARCOS, Universidad Complutense de Madrid, E-28040 Madrid, Spain' - 'Laboratoire Univers et Particules de Montpellier, Université de Montpellier, CNRS/IN2P3, CC 72, Place Eugène Bataillon, F-34095 Montpellier Cedex 5, France' - 'Institute for Cosmic Ray Research, University of Tokyo, 5-1-5, Kashiwa-no-ha, Kashiwa, Chiba 277-8582, Japan' - 'School of Physics and Astronomy, Monash University, Melbourne, Victoria 3800, Australia' - 'ISDC Data Centre for Astrophysics, Observatory of Geneva, University of Geneva, Chemin d’Ecogia 16, CH-1290 Versoix, Switzerland' - 'INAF - Osservatorio Astronomico di Brera, Via Brera 28, 20121 Milano, Italy' - 'RIKEN, Institute of Physical and Chemical Research, 2-1 Hirosawa, Wako, Saitama, 351-0198, Japan' - 'Centro Brasileiro de Pesquisas Físicas, Rua Xavier Sigaud 150, RJ 22290-180, Rio de Janeiro, Brazil' - 'INFN Sezione di Padova and Università degli Studi di Padova, Via Marzolo 8, 35131 Padova, Italy' - 'Instituto de Astrofísica de Canarias and Departamento de Astrofísica, Universidad de La Laguna, La Laguna, Tenerife, Spain' - 'Institut für Theoretische Physik, Lehrstuhl IV: Weltraum- und Astrophysik, Ruhr-Universität Bochum, Universitätsstraße 150, 44801 Bochum, Germany' - 'Harvard-Smithsonian Center for Astrophysics, 60 Garden St, Cambridge, MA 02180, USA' - 'Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany' - 'INFN Sezione di Torino, Via P. Giuria 1, 10125 Torino, Italy' - 'Pidstryhach Institute for Applied Problems in Mechanics and Mathematics NASU, 3B Naukova Street, Lviv, 79060, Ukraine' - 'Institut de Physique Nucléaire, IN2P3/CNRS, Université Paris-Sud, Université Paris-Saclay, 15 rue Georges Clemenceau, 91406 Orsay, Cedex, France' - 'Department of Physics and Astronomy, University of California, Los Angeles, CA 90095, USA' - 'INFN Sezione di Bari and Politecnico di Bari, via Orabona 4, 70124 Bari, Italy' - 'Institut de Fisica d’Altes Energies (IFAE), The Barcelona Institute of Science and Technology, Campus UAB, 08193 Bellaterra (Barcelona), Spain' - 'Institute of Physics of the Czech Academy of Sciences, Na Slovance 1999/2, 182 21 Praha 8, Czech Republic' - 'INAF - Osservatorio Astrofisico di Catania, Via S. Sofia, 78, 95123 Catania, Italy' - 'Radboud University Nijmegen, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands' - 'INFN and Università degli Studi di Siena, Dipartimento di Scienze Fisiche, della Terra e dell’Ambiente (DSFTA), Sezione di Fisica, Via Roma 56, 53100 Siena, Italy' - 'Departament de Física Quàntica i Astrofísica, Institut de Ciències del Cosmos, Universitat de Barcelona, IEEC-UB, Martí i Franquès, 1, 08028, Barcelona, Spain' - 'Centre for Space Research, North-West University, Potchefstroom Campus, 2531, South Africa' - 'Department of Physics, Columbia University, 538 West 120th Street, New York, NY 10027, USA' - 'Department of Physics, TU Dortmund University, Otto-Hahn-Str. 4, 44221 Dortmund, Germany' - 'Astronomical Observatory, Department of Physics, University of Warsaw, Aleje Ujazdowskie 4, 00478 Warsaw, Poland' - 'Armagh Observatory and Planetarium, College Hill, Armagh BT61 9DG, United Kingdom' - 'Kavli Institute for Particle Astrophysics and Cosmology, Department of Physics and SLAC National Accelerator Laboratory, Stanford University, 2575 Sand Hill Road, Menlo Park, CA 94025, USA' - 'INAF - Istituto di Astrofisica Spaziale e Fisica Cosmica di Palermo, Via U. La Malfa 153, 90146 Palermo, Italy' - 'Universidade Cruzeiro do Sul, Núcleo de Astrofísica Teórica (NAT/UCS), Rua Galvão Bueno 8687, Bloco B, sala 16, Libertade 01506-000 - São Paulo, Brazil' - 'INAF - Istituto di Astrofisica Spaziale e Fisica Cosmica di Milano, Via Bassini 15, 20133 Milano, Italy' - 'INFN Sezione di Pisa, Largo Pontecorvo 3, 56217 Pisa, Italy' - 'The Henryk Niewodniczański Institute of Nuclear Physics, Polish Academy of Sciences, ul. Radzikowskiego 152, 31-342 Cracow, Poland' - 'INAF - Osservatorio Astronomico di Capodimonte, Via Salita Moiariello 16, 80131 Napoli, Italy' - 'Escola de Engenharia de Lorena, Universidade de São Paulo, Área I - Estrada Municipal do Campinho, s/n°, CEP 12602-810, Brazil' - 'INFN Sezione di Trieste and Università degli Studi di Udine, Via delle Scienze 208, 33100 Udine, Italy' - 'AIM, CEA, CNRS, Université Paris Diderot, Sorbonne Paris Cité, Université Paris-Saclay, F-91191 Gif-sur-Yvette, France' - 'University of the Witwatersrand, 1 Jan Smuts Avenue, Braamfontein, 2000 Johannesburg, South Africa' - 'Centre for Astrophysics & Relativity, School of Physical Sciences, Dublin City University, Glasnevin, Dublin 9, Ireland' - 'CIEMAT, Avda. Complutense 40, 28040 Madrid, Spain' - 'Aix Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France, 163 Avenue de Luminy, 13288 Marseille cedex 09, France' - 'INAF - Istituto di Radioastronomia, Via Gobetti 101, 40129 Bologna, Italy' - 'Cherenkov Telescope Array Observatory, Saupfercheckweg 1, 69117 Heidelberg, Germany' - 'Universidade Federal Do Paraná - Setor Palotina, Departamento de Engenharias e Exatas, Rua Pioneiro, 2153, Jardim Dallas, CEP: 85950-000 Palotina, Paraná, Brazil' - 'Laboratoire Leprince-Ringuet, École Polytechnique (UMR 7638, CNRS/IN2P3, Université Paris-Saclay), 91128 Palaiseau, France' - 'Institute of Space Sciences (ICE, CSIC), Campus UAB, Carrer de Magrans s/n, 08193 Barcelona, Spain; Institut d’Estudis Espacials de Catalunya (IEEC), 08034 Barcelona, Spain; and Institució Catalana de Recerca i Estudis Avançats (ICREA) Barcelona, Spain' - 'INFN Sezione di Bari, via Orabona 4, 70126 Bari, Italy' - 'Instituto de Física de São Carlos, Universidade de São Paulo, Av. Trabalhador São-carlense, 400 - CEP 13566-590, São Carlos, SP, Brazil' - 'INFN Sezione di Bari and Università degli Studi di Bari, via Orabona 4, 70124 Bari, Italy' - 'Institut für Astronomie und Astrophysik, Universität Tübingen, Sand 1, 72076 Tübingen, Germany' - 'APC, Univ Paris Diderot, CNRS/IN2P3, CEA/lrfu, Obs de Paris, Sorbonne Paris Cité, France, 10, rue Alice Domon et Léonie Duquet, 75205 Paris Cedex 13, France' - 'University of Rijeka, Department of Physics, Radmile Matejcic 2, 51000 Rijeka, Croatia' - 'Institute for Theoretical Physics and Astrophysics, Universität Würzburg, Campus Hubland Nord, Emil-Fischer-Str. 31, 97074 Würzburg, Germany' - 'Sorbonne Université, Univ Paris Diderot, Sorbonne Paris Cité, CNRS/IN2P3, Laboratoire de Physique Nucléaire et de Hautes Energies, LPNHE, 4 Place Jussieu, F-75005 Paris, France' - 'Escola de Artes, Ciências e Humanidades, Universidade de São Paulo, Rua Arlindo Bettio, 1000 São Paulo, CEP 03828-000, Brazil' - 'Astronomical Observatory of Taras Shevchenko National University of Kyiv, 3 Observatorna Street, Kyiv, 04053, Ukraine' - 'Western Sydney University, Locked Bag 1797, Penrith, NSW 2751, Australia' - 'INAF - Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, 35122 Padova, Italy' - 'Deutsches Elektronen-Synchrotron, Platanenallee 6, 15738 Zeuthen, Germany' - 'Department of Physics, Nagoya University, Chikusa-ku, Nagoya, 464-8602, Japan' - 'GRAPPA, University of Amsterdam, Science Park 904 1098 XH Amsterdam, The Netherlands' - 'Instituto de Física Teórica UAM/CSIC and Departamento de Física Teórica, Campus Cantoblanco, Universidad Autónoma de Madrid, c/ Nicolás Cabrera 13-15, Campus de Cantoblanco UAM, 28049 Madrid, Spain' - 'INFN Sezione di Trieste and Università degli Studi di Trieste, Via Valerio 2 I, 34127 Trieste, Italy' - 'Unitat de Física de les Radiacions, Departament de Física, and CERES-IEEC, Universitat Autònoma de Barcelona, E-08193 Bellaterra, Spain, Edifici C3, Campus UAB, 08193 Bellaterra, Spain' - 'Alikhanyan National Science Laboratory, Yerevan Physics Institute, 2 Alikhanyan Brothers St., 0036, Yerevan, Armenia' - 'IRFU, CEA, Université Paris-Saclay, F-91191 Gif-sur-Yvette, France' - 'Universidad Andrés Bello UNAB, República N° 252, Santiago, Región Metropolitana, Chile' - 'Academic Computer Centre CYFRONET AGH, ul. Nawojki 11, 30-950 Cracow, Poland' - 'Department of Natural Sciences, The Open University of Israel, 1 University Road, POB 808, Raanana 43537, Israel' - 'Max-Planck-Institut für Physik, Föhringer Ring 6, 80805 München, Germany' - 'University of Liverpool, Oliver Lodge Laboratory, Liverpool L69 7ZE, United Kingdom' - 'Univ. Bordeaux, CNRS, IN2P3, CENBG, UMR 5797, F-33175 Gradignan., 19 Chemin du Solarium, CS 10120, F-33175 Gradignan Cedex, France' - 'Department of Physics, Konan University, Kobe, Hyogo, 658-8501, Japan' - 'Santa Cruz Institute for Particle Physics and Department of Physics, University of California, Santa Cruz, 1156 High Street, Santa Cruz, CA 95064, USA' - 'Institute of Particle and Nuclear Studies, KEK (High Energy Accelerator Research Organization), 1-1 Oho, Tsukuba, 305-0801, Japan' - 'Palacky University Olomouc, Faculty of Science, RCPTM, 17. listopadu 1192/12, 771 46 Olomouc, Czech Republic' - 'Josip Juraj Strossmayer University of Osijek, Trg Svetog Trojstva 3, 31000 Osijek, Croatia' - 'ICTP-South American Institute for Fundamental Research - Instítuto de Física Teórica da UNESP, Rua Dr. Bento Teobaldo Ferraz 271, 01140-070 São Paulo, Brazil' - 'INFN Sezione di Roma, Piazza Aldo Moro 5 I, 00185 Roma, Italy' - 'Faculty of Physics, Astronomy and Applied Computer Science, Jagiellonian University, ul. prof. Stanisława Łojasiewicza 11, 30-348 Kraków, Poland' - 'Universität Erlangen-Nürnberg, Physikalisches Institut, Erwin-Rommel-Str. 1, 91058 Erlangen, Germany' - 'Institut de Recherche en Astrophysique et Planétologie, CNRS-INSU, Université Paul Sabatier, 9 avenue Colonel Roche, BP 44346, 31028 Toulouse Cedex 4, France' - 'University of Iowa, Department of Physics and Astronomy, Van Allen Hall, Iowa City, IA 52242, USA' - 'Institut für Astro- und Teilchenphysik, Leopold-Franzens-Universität, Technikerstr. 25/8, 6020 Innsbruck, Austria' - 'Division of Physics and Astronomy, Graduate School of Science, Kyoto University, Sakyo-ku, Kyoto, 606-8502, Japan' - 'Department of Physics, Tokai University, 4-1-1, Kita-Kaname, Hiratsuka, Kanagawa 259-1292, Japan' - 'Dept. of Physics and Astronomy, University of Leicester, Leicester, LE1 7RH, United Kingdom' - 'Centro de Ciências Naturais e Humanas - Universidade Federal do ABC, Rua Santa Adélia, 166. Bairro Bangu. Santo André - SP - Brasil . CEP 09.210-170, Brazil' - 'Tuorla Observatory, Department of Physics and Astronomy, University of Turku, FI-21500 Piikkiő, Finland' - 'Department of Physics, Humboldt University Berlin, Newtonstr. 15, 12489 Berlin, Germany' - 'Escuela Politécnica Superior de Jaén, Universidad de Jaén, Campus Las Lagunillas s/n, Edif. A3, 23071 Jaén, Spain' - 'Saha Institute of Nuclear Physics, Bidhannagar, Kolkata-700 064, India' - 'Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 72 boul. Tsarigradsko chaussee, 1784 Sofia, Bulgaria' - 'University of Białystok, Faculty of Physics, ul. K. Ciołkowskiego 1L, 15-254 Białystok, Poland' - 'School of Physics, University of New South Wales, Sydney NSW 2052, Australia' - 'Physik-Institut, Universität Zürich, Winterthurerstrasse 190, 8057 Zürich, Switzerland' - 'Hiroshima Astrophysical Science Center, Hiroshima University, Higashi-Hiroshima, Hiroshima 739-8526, Japan' - 'Nicolaus Copernicus Astronomical Center, Polish Academy of Sciences, ul. Bartycka 18, 00-716 Warsaw, Poland' - 'INFN Sezione di Roma Tor Vergata, Via della Ricerca Scientifica 1, 00133 Rome, Italy' - 'INAF - Istituto di Astrofisica e Planetologia Spaziali (IAPS), Via del Fosso del Cavaliere 100, 00133 Roma, Italy' - 'Department of Physics, University of Bath, Claverton Down, Bath BA2 7AY, United Kingdom' - 'Graduate School of Science and Engineering, Saitama University, 255 Simo-Ohkubo, Sakura-ku, Saitama city, Saitama 338-8570, Japan' - 'Faculty of Management Information, Yamanashi-Gakuin University, Kofu, Yamanashi 400-8575, Japan' - 'Department of Physics, Yamagata University, Yamagata, Yamagata 990-8560, Japan' - 'Charles University, Institute of Particle & Nuclear Physics, V Holešovičkách 2, 180 00 Prague 8, Czech Republic' - 'Institute for Space-Earth Environmental Research, Nagoya University, Chikusa-ku, Nagoya 464-8601, Japan' - 'Graduate School of Technology, Industrial and Social Sciences, Tokushima University, Tokushima 770-8506, Japan' - 'School of Physics & Center for Relativistic Astrophysics, Georgia Institute of Technology, 837 State Street, Atlanta, Georgia, 30332-0430, USA' - 'Université Grenoble Alpes, CNRS, Institut de Planétologie et d’Astrophysique de Grenoble, 414 rue de la Piscine, Domaine Universitaire, 38041 Grenoble Cedex 9, France' - 'LAPP, Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS-IN2P3, 74000 Annecy, France, 9 Chemin de Bellevue - BP 110, 74941 Annecy Cedex, France' - 'Institut für Physik & Astronomie, Universität Potsdam, Karl-Liebknecht-Strasse 24/25, 14476 Potsdam, Germany' - 'Department of Physics and Electrical Engineering, Linnaeus University, 351 95 Växjö, Sweden' - 'Landessternwarte, Universität Heidelberg, Königstuhl, 69117 Heidelberg, Germany' - 'University of Johannesburg, Department of Physics, University Road, PO Box 524, Auckland Park 2006, South Africa' - 'School of Physical Sciences, University of Adelaide, Adelaide SA 5005, Australia' - 'University of Alabama, Tuscaloosa, Department of Physics and Astronomy, Gallalee Hall, Box 870324 Tuscaloosa, AL 35487-0324, USA' - 'National Astronomical Research Institute of Thailand, 191 Huay Kaew Rd., Suthep, Muang, Chiang Mai, 50200, Thailand' - 'Space Research Centre, Polish Academy of Sciences, ul. Bartycka 18A, 00-716 Warsaw, Poland' - 'The University of Manitoba, Dept of Physics and Astronomy, Winnipeg, Manitoba R3T 2N2, Canada' - 'Toruń Centre for Astronomy, Nicolaus Copernicus University, ul. Grudziądzka 5, 87-100 Toruń, Poland' - 'Center for Astrophysics and Cosmology, University of Nova Gorica, Vipavska 11c, 5270 Ajdovščina, Slovenia' - 'University of Wisconsin, Madison, 500 Lincoln Drive, Madison, WI, 53706, USA' - 'INAF - Osservatorio Astrofisico di Torino, Via Osservatorio 20, 10025 Pino Torinese (TO), Italy' - 'University of Hawai’i at Manoa, 2500 Campus Rd, Honolulu, HI, 96822, USA' - 'Faculty of Science, Ibaraki University, Mito, Ibaraki, 310-8512, Japan' - 'University of Groningen, KVI - Center for Advanced Radiation Technology,Zernikelaan 25, 9747 AA Groningen,The Netherlands' - 'Department of Physics and Astronomy, University of Utah,Salt Lake City, UT 84112-0830, USA' author: - 'A. Acharyya' - 'I. Agudo' - 'E.O. Angüner' - 'R. Alfaro' - 'J. Alfaro' - 'C. Alispach' - 'R. Aloisio' - 'R. Alves Batista' - 'J.-P.Amans' - 'L. Amati' - 'E. Amato' - 'G. Ambrosi' - 'L.A. Antonelli' - 'C. Aramo' - 'T. Armstrong' - 'F. Arqueros' - 'L. Arrabito' - 'K. Asano' - 'H. Ashkar' - 'C. Balazs' - 'M. Balbo' - 'B. Balmaverde' - 'P. Barai' - 'A. Barbano' - 'M. Barkov' - 'U. Barres de Almeida' - 'J.A. Barrio' - 'D. Bastieri' - 'J. Becerra González' - 'J. Becker Tjus' - 'L. Bellizzi' - 'W. Benbow' - 'E. Bernardini' - 'M.I. Bernardos' - 'K. Bernlöhr' - 'A. Berti' - 'M. Berton' - 'B. Bertucci' - 'V. Beshley' - 'B. Biasuzzi' - 'C. Bigongiari' - 'R. Bird' - 'E. Bissaldi' - 'J. Biteau' - 'O. Blanch' - 'J. Blazek' - 'C. Boisson' - 'G. Bonanno' - 'A. Bonardi' - 'C. Bonavolontà' - 'G. Bonnoli' - 'P. Bordas' - 'M. Böttcher' - 'J. Bregeon' - 'A. Brill' - 'A.M. Brown' - 'K. Brügge' - 'P. Brun' - 'P. Bruno' - 'A. Bulgarelli' - 'T. Bulik' - 'M. Burton' - 'A. Burtovoi' - 'G. Busetto' - 'R. Cameron' - 'R. Canestrari' - 'M. Capalbi' - 'A. Caproni' - 'R. Capuzzo-Dolcetta' - 'P. Caraveo' - 'S. Caroff' - 'R. Carosi' - 'S. Casanova' - 'E. Cascone' - 'F. Cassol' - 'F. Catalani' - 'O. Catalano' - 'D. Cauz' - 'M. Cerruti' - 'S. Chaty' - 'A. Chen' - 'M. Chernyakova' - 'G. Chiaro' - 'M. Cieślar' - 'S.M. Colak' - 'V. Conforti' - 'E. Congiu' - 'J.L. Contreras' - 'J. Cortina' - 'A. Costa' - 'H. Costantini' - 'G. Cotter' - 'P. Cristofari' - 'P. Cumani' - 'G. Cusumano' - 'A. D’Aì' - 'F. D’Ammando' - 'L. Dangeon' - 'P. Da Vela' - 'F. Dazzi' - 'A. De Angelis' - 'V. De Caprio' - 'R. de Cássia dos Anjos' - 'F. De Frondat' - 'E.M. de Gouveia Dal Pino' - 'B. De Lotto' - 'D. De Martino' - 'M. de Naurois' - 'E. de Oña Wilhelmi' - 'F. de Palma' - 'V. de Souza' - 'M. Del Santo' - 'C. Delgado' - 'D. della Volpe' - 'T. Di Girolamo' - 'F. Di Pierro' - 'L. Di Venere' - 'C. Díaz' - 'S. Diebold' - 'A. Djannati-Ataï' - 'A. Dmytriiev' - 'D. Dominis Prester' - 'A. Donini' - 'D. Dorner' - 'M. Doro' - 'J.-L. Dournaux' - 'J. Ebr' - 'T.R.N. Ekoume' - 'D. Elsässer' - 'G. Emery' - 'D. Falceta-Goncalves' - 'E. Fedorova' - 'S. Fegan' - 'Q. Feng' - 'G. Ferrand' - 'E. Fiandrini' - 'A. Fiasson' - 'M. Filipovic' - 'V. Fioretti' - 'M. Fiori' - 'S. Flis' - 'M.V. Fonseca' - 'G. Fontaine' - 'L. Freixas Coromina' - 'S. Fukami' - 'Y. Fukui' - 'S. Funk' - 'M. Fü[ß]{}ling' - 'D. Gaggero' - 'G. Galanti' - 'R.J. Garcia López' - 'M. Garczarczyk' - 'D. Gascon' - 'T. Gasparetto' - 'M. Gaug' - 'A. Ghalumyan' - 'F. Gianotti' - 'G. Giavitto' - 'N. Giglietto' - 'F. Giordano' - 'M. Giroletti' - 'J. Gironnet' - 'J.-F. Glicenstein' - 'R. Gnatyk' - 'P. Goldoni' - 'J.M. González' - 'M.M. González' - 'K.N. Gourgouliatos' - 'T. Grabarczyk' - 'J. Granot' - 'D. Green' - 'T. Greenshaw' - 'M.-H. Grondin' - 'O. Gueta' - 'D. Hadasch' - 'T. Hassan' - 'M. Hayashida' - 'M. Heller' - 'O. Hervet' - 'J. Hinton' - 'N. Hiroshima' - 'B. Hnatyk' - 'W. Hofmann' - 'P. Horvath' - 'M. Hrabovsky' - 'D. Hrupec' - 'T.B. Humensky' - 'M. Hütten' - 'T. Inada' - 'F. Iocco' - 'M. Ionica' - 'M. Iori' - 'Y. Iwamura' - 'M. Jamrozy' - 'P. Janecek' - 'D. Jankowsky' - 'P. Jean' - 'L. Jouvin' - 'J. Jurysek' - 'P. Kaaret' - 'L.H.S. Kadowaki' - 'S. Karkar' - 'D. Kerszberg' - 'B. Khélifi' - 'D. Kieda' - 'S. Kimeswenger' - 'W. Kluźniak' - 'J. Knapp' - 'J. Knödlseder' - 'Y. Kobayashi' - 'B. Koch' - 'J. Kocot' - 'N. Komin' - 'A. Kong' - 'G. Kowal' - 'M. Krause' - 'H. Kubo' - 'J. Kushida' - 'P. Kushwaha' - 'V. La Parola' - 'G. La Rosa' - 'M. Lallena Arquillo' - 'R.G. Lang' - 'J. Lapington' - 'O. Le Blanc' - 'J. Lefaucheur' - 'M.A. Leigui de Oliveira' - 'M. Lemoine-Goumard' - 'J.-P. Lenain' - 'G. Leto' - 'R. Lico' - 'E. Lindfors' - 'T. Lohse' - 'S. Lombardi' - 'F. Longo' - 'A. Lopez' - 'M. López' - 'A. Lopez-Oramas' - 'R. López-Coto' - 'S. Loporchio' - 'P.L. Luque-Escamilla' - 'E. Lyard' - 'M.C. Maccarone' - 'E. Mach' - 'C. Maggio' - 'P. Majumdar' - 'G. Malaguti' - 'M. Mallamaci' - 'D. Mandat' - 'G. Maneva' - 'M. Manganaro' - 'S. Mangano' - 'M. Marculewicz' - 'M. Mariotti' - 'J. Martí' - 'M. Martínez' - 'G. Martínez' - 'H. Martínez-Huerta' - 'S. Masuda' - 'N. Maxted' - 'D. Mazin' - 'J.-L. Meunier' - 'M. Meyer' - 'S. Micanovic' - 'R. Millul' - 'I.A. Minaya' - 'A. Mitchell' - 'T. Mizuno' - 'R. Moderski' - 'L. Mohrmann' - 'T. Montaruli' - 'A. Moralejo' - 'D. Morcuende' - 'G. Morlino' - 'A. Morselli' - 'E. Moulin' - 'R. Mukherjee' - 'P. Munar' - 'C. Mundell' - 'T. Murach' - 'A. Nagai' - 'T. Nagayoshi' - 'T. Naito' - 'T. Nakamori' - 'R. Nemmen' - 'J. Niemiec' - 'D. Nieto' - 'M. Nievas Rosillo' - 'M. Nikołajuk' - 'D. Ninci' - 'K. Nishijima' - 'K. Noda' - 'D. Nosek' - 'M. Nöthe' - 'S. Nozaki' - 'M. Ohishi' - 'Y. Ohtani' - 'A. Okumura' - 'R.A. Ong' - 'M. Orienti' - 'R. Orito' - 'M. Ostrowski' - 'N. Otte' - 'Z. Ou' - 'I. Oya' - 'A. Pagliaro' - 'M. Palatiello' - 'M. Palatka' - 'R. Paoletti' - 'J.M. Paredes' - 'G. Pareschi' - 'N. Parmiggiani' - 'R.D. Parsons' - 'B. Patricelli' - 'A. Pe’er' - 'M. Pech' - 'P. Peñil Del Campo' - 'J. Pérez-Romero' - 'M. Perri' - 'M. Persic' - 'P.-O. Petrucci' - 'O. Petruk' - 'K. Pfrang' - 'Q. Piel' - 'E. Pietropaolo' - 'M. Pohl' - 'M. Polo' - 'J. Poutanen' - 'E. Prandini' - 'N. Produit' - 'H. Prokoph' - 'M. Prouza' - 'H. Przybilski' - 'G. Pühlhofer' - 'M. Punch' - 'F. Queiroz' - 'A. Quirrenbach' - 'S. Rainò' - 'R. Rando' - 'S. Razzaque' - 'O. Reimer' - 'N. Renault-Tinacci' - 'Y. Renier' - 'D. Ribeiro' - 'M. Ribó' - 'J. Rico' - 'F. Rieger' - 'V. Rizi' - 'G. Rodriguez Fernandez' - 'J.C. Rodriguez-Ramirez' - 'J.J. Rodríguez Vázquez' - 'P. Romano' - 'G. Romeo' - 'M. Roncadelli' - 'J. Rosado' - 'G. Rowell' - 'B. Rudak' - 'A. Rugliancich' - 'C. Rulten' - 'I. Sadeh' - 'L. Saha' - 'T. Saito' - 'S. Sakurai' - 'F. Salesa Greus' - 'P. Sangiorgi' - 'H. Sano' - 'M. Santander' - 'A. Santangelo' - 'R. Santos-Lima' - 'A. Sanuy' - 'K. Satalecka' - 'F.G. Saturni' - 'U. Sawangwit' - 'S. Schlenstedt' - 'P. Schovanek' - 'F. Schussler' - 'U. Schwanke' - 'E. Sciacca' - 'S. Scuderi' - 'K. Sedlaczek' - 'M. Seglar-Arroyo' - 'O. Sergijenko' - 'K. Seweryn' - 'A. Shalchi' - 'R.C. Shellard' - 'H. Siejkowski' - 'A. Sillanpää' - 'A. Sinha' - 'G. Sironi' - 'V. Sliusar' - 'A. Slowikowska' - 'H. Sol' - 'A. Specovius' - 'S. Spencer' - 'G. Spengler' - 'A. Stamerra' - 'S. Stanič' - 'Ł. Stawarz' - 'S. Stefanik' - 'T. Stolarczyk' - 'U. Straumann' - 'T. Suomijarvi' - 'P. Świerk' - 'T. Szepieniec' - 'G. Tagliaferri' - 'H. Tajima' - 'T. Tam' - 'F. Tavecchio' - 'L. Taylor' - 'L.A. Tejedor' - 'P. Temnikov' - 'T. Terzic' - 'V. Testa' - 'L. Tibaldo' - 'C.J. Todero Peixoto' - 'F. Tokanai' - 'L. Tomankova' - 'D. Tonev' - 'D.F. Torres' - 'G. Tosti' - 'L. Tosti' - 'N. Tothill' - 'F. Toussenel' - 'G. Tovmassian' - 'P. Travnicek' - 'C. Trichard' - 'G. Umana' - 'V. Vagelli' - 'M. Valentino' - 'B. Vallage' - 'P. Vallania' - 'L. Valore' - 'J. Vandenbroucke' - 'G.S. Varner' - 'G. Vasileiadis' - 'V. Vassiliev' - 'M. Vázquez Acosta' - 'M. Vecchi' - 'S. Vercellone' - 'S. Vergani' - 'G.P. Vettolani' - 'A. Viana' - 'C.F. Vigorito' - 'J. Vink' - 'V. Vitale' - 'H. Voelk' - 'A. Vollhardt' - 'S. Vorobiov' - 'S.J. Wagner' - 'R. Walter' - 'F. Werner' - 'R. White' - 'A. Wierzcholska' - 'M. Will' - 'D.A. Williams' - 'R. Wischnewski' - 'L. Yang' - 'T. Yoshida' - 'T. Yoshikoshi' - 'M. Zacharias' - 'L. Zampieri' - 'M. Zavrtanik' - 'D. Zavrtanik' - 'A.A. Zdziarski' - 'A. Zech' - 'H. Zechlin' - 'A. Zenin' - 'V.I. Zhdanov' - 'S. Zimmer' - 'J. Zorn' bibliography: - 'references.bib' title: Monte Carlo studies for the optimisation of the Cherenkov Telescope Array layout --- =10000 =10000 Monte Carlo simulations ,Cherenkov telescopes ,IACT technique ,gamma rays ,cosmic rays Introduction ============ Cosmic rays and very-high-energy (VHE, few tens of GeV and above) gamma rays reaching Earth’s atmosphere produce cascades of subatomic particles called air showers. Ultrarelativistic charged particles generated within these showers produce photons through the Cherenkov effect. Most of this light is emitted at altitudes ranging between 5 to 15 km, and it propagates down to ground level as a quasi-planar, thin disk of Cherenkov photons orthogonal to the shower axis. Imaging atmospheric Cherenkov telescopes (IACTs) are designed to capture images of these very brief optical flashes, generally lasting just a few ns. By placing arrays of IACTs within the projected light pool of these showers and analysing the simultaneous images taken by these telescopes, it is possible to identify the nature of the primary particle and reconstruct its original energy and incoming direction. Building on the experience gained through the operation of the current IACTs (H.E.S.S.[^1], MAGIC[^2], and VERITAS[^3]), the next generation of ground-based very-high-energy gamma-ray telescope is currently under construction. The [^4] [@CTA_concept; @CTAICRC2015] will detect gamma rays in the energy range from 20 GeV to 300 TeV with unprecedented angular and energy resolutions for ground-based facilities, outperforming the sensitivity of present-day instruments by more than an order of magnitude in the multi-TeV range [@CTA_MC_2017]. This improvement will be possible by using larger arrays of telescopes. As a cost-effective solution to improve performance over four decades of energy, telescopes will be built in three different sizes: Large-Sized Telescopes (LSTs) [@LSTgamma], Medium-Sized Telescopes (MSTs) [@MSTgamma; @SCTgamma] and Small-Sized Telescopes (SSTs) [@Montaruli:2015]. To provide full-sky coverage, IACT arrays will be installed in two sites, one in each hemisphere: at Paranal (Chile) and at La Palma (Canary Islands, Spain). Each telescope class will primarily cover a specific energy range: LSTs, with a $\sim370$ m$^2$ reflecting dish and a camera with a field of view (FoV) of $\sim4.3^\circ$, will allow the reconstruction of the faint low-energy showers (below 100 GeV), not detectable by smaller telescopes. In this energy range the rejection of the cosmic-ray background is limited by the modest number of particles created in the air showers. Due to the relatively high flux of low-energy gamma rays and the large associated construction costs, few LSTs will be built at each site. They have been designed for high-speed slewing allowing short repositioning times to catch fast transient phenomena on time scales of minutes to days, such as gamma-ray bursts [@ScienceCTA]. MSTs, with a larger FoV of $\sim7.6^\circ$, will populate the inner part of the array, increasing the number of telescopes simultaneously observing each shower, enhancing the angular and energy resolutions within the core energy range (between 100 GeV and 10 TeV). Two different MST designs have been proposed: the Davies-Cotton MST (DC-MST) and the Schwarzschild-Couder MST (SC-MST) [@MSTgamma; @SCTgamma]. The DC-MST is a 12m-diameter single-mirror IACT built with modified Davies-Cotton optics and a mirror area of $\sim88$ m$^2$. Two different cameras have been prototyped for this telescope: NectarCam and FlashCam [@nectar; @flashcam]. The SC-MST features a two-mirror optical design with a 9.7 m diameter primary mirror and an area of $\sim41$m$^2$. The dual-mirror setup corrects spherical and comatic aberrations, allowing a finer shower image pixelisation, enhancing angular resolution and off-axis performance. Above a few TeV, Cherenkov light from electromagnetic showers becomes significantly brighter, not requiring such large reflecting surfaces for their detection. At the same time, the gamma-ray flux decreases with energy, so in order to detect a sufficient number of these high-energy events, a large ground surface needs to be covered. SSTs, with a mirror area of $\sim$ 8 m$^2$ and a FoV of $>8^\circ$, have been designed with this purpose. A large number of SSTs will populate the outer part of the array covering a total surface area of up to 4.5 km$^2$. Three variants of SSTs have been proposed: two designs of SC-SSTs, the ASTRI and the GCT, both with primary mirror diameters of 4 m, and a DC-SST, the SST-1M, with a single 4 m diameter mirror [@Montaruli:2015]. The northern and southern observatories will make the full VHE gamma-ray sky accessible to CTA. As a cost-effective solution to maximise scientific output, each site will have different telescope layouts. The CTA southern site will be larger to take advantage of its privileged location for observation of the Galactic Center and most of the inner half of the Galactic Plane, regions with a high density of sources with spectra extending beyond 10 TeV. Its baseline design foresees 4 LSTs, 25 MSTs and 70 SSTs. The northern site will be more focused on the study of extragalactic objects and will be composed of 4 LSTs and 15 MSTs. No SSTs are planned to be placed in the northern hemisphere. Detailed simulations are required to estimate the performance of an IACT array [@Konrad:2008; @APP_CTA_MC; @MC_ICRC:2013], which is evaluated by quantities like the minimum detectable flux, sensitive FoV or its angular and energy resolutions. All these estimators are strongly dependent on a set of parameters related to both the telescope design and the array layout (i.e. the arrangement of telescope positions on ground). Other scenarios (e.g. standalone operations of sub-arrays composed of only LSTs, MSTs or SSTs, or short downtime periods of some telescope) need to be also taken into consideration during the layout optimisation phase to ensure that the CTA performance is not critically affected. The objective of this work is to optimise the telescope layout of a given number of telescopes, maximising performance, while complying with all CTA requirements. These requirements were derived as a cost-effective solution to obtain excellent performance over a wide range of very different physics cases [@ScienceCTA], to ensure the scientific impact of the future observatory. Array layout considerations --------------------------- Optimal array layouts are mainly characterised by the configuration of each telescope type and by the number and arrangement of these telescopes. Each telescope type configuration is mainly described by its light collection power, dominated by mirror area, photo sensor efficiency, and camera FoV and pixelation, with optics chosen so that the optical point spread function matches the pixel size. A generic telescope cost model was used with mirror area, FoV and pixel size as primary parameters, so that all proposed array layouts that were compared during these optimisation studies could be considered of approximately equal cost. As a first step, semi-analytical performance estimations were carried out using parameterisations for the responses of each telescope type. These studies allowed us to perform quick estimates of gamma-ray and cosmic-ray detection rates for a wide variety of telescope configurations and arrangements. Simulations of regular square grids of telescopes were performed to quantify the impact of parameters such as mirror area, FoV, pixel size or telescope spacing. To validate and fine tune the optimal telescope configurations calculated with these simplified approaches, a series of large-scale MC simulations were performed sequentially, described in more detail in Section \[sec:mcprod\]. Telescopes are arranged in concentric arrays of different telescope sizes, ordered in light collection power, from a compact low-energy array at the centre to an extended high-energy array, providing an effective area that increases with energy. The light pool size of air showers increases with energy, from a radius of about 120 m for $\sim$ 30 GeV showers to more than 1000 m for multi-TeV showers. In the sub-TeV to TeV domain, telescope spacing of about 100 m to 150 m optimises sensitivity, providing an equilibrium between having more images per air shower and a reasonable collection area. For TeV energies and above, larger distances are preferred to improve the collection area, given that, at these energies, the cosmic-ray background can be rejected almost completely and the achievable sensitivity is photon-rate limited. The baseline design number of telescopes (4 LSTs, 25 MSTs and 70 SSTs for CTA-South and 4 LSTs and 15 MSTs for CTA-North) was fixed after a combined effort involving the production of large-scale MC simulations, evaluation of the performance of very different array layouts [@APP_CTA_MC], and study of the effect of this diverse set of layouts over a large variety of key scientific cases [@DM_2013; @CRs_2013; @AGNs_2013; @surveys_2013; @CTA_GRBs; @pulsars_2013]. This study presents the final baseline arrays for both the northern and southern sites. large-scale productions, described in section \[sec:mcprod\], were used to estimate the performance of a very large variety of layouts. The main considerations taken into account in the performance evaluation are outlined in section \[sec:conlayout\], while the final baseline arrays and their performances are presented in sections \[sec:souths\] and \[sec:norths\] for the southern and northern site, respectively. CTA Monte Carlo production and analysis {#sec:mcprod} ======================================= Given the unprecedented scale of the project, a constant effort has been devoted over the past five years to define and optimise the telescope layouts. Three large-scale productions were conducted and analysed with this purpose [@MC_ICRC:2013; @Hassan-2015; @ICRC2017:Layout]. In addition to the layout optimisation, these productions have been used to: - estimate the expected performance [@CTA_MC_2017; @APP_CTA_MC], - guide the design of the different telescope types and compare their capabilities [@Wood:2014; @Prod2_SCMST; @GCT_MC_2015], - provide input to the site selection process by evaluating the effect of the characteristics of each site on the array performance. Among the considered site attributes there were altitude, geomagnetic field, night-sky background level and aerosol optical depth [@Site_Paper; @MC_ICRC_site:2015; @2013APh....45....1S]. As described in [@APP_CTA_MC; @Site_Paper; @Hassan-2015], each large-scale production requires the definition of a large telescope layout, called the master layout. Each master layout comprises hundreds of telescopes distributed over an area of about 6 km$^2$ and are designed to contain numerous possible layouts of equivalent cost. To identify the optimal arrangement, these plausible layouts are extracted, analysed and their performances are compared with respect to each other. For each MC production, telescope models were sequentially improved, becoming more realistic in each iteration thanks to the increasing input coming from the prototype telescopes. Air showers initiated by gamma rays, cosmic-ray nuclei and electrons are simulated using the package [@corsika]. The telescope response is simulated using [@Konrad:2008], used by the HEGRA and H.E.S.S. experiments. The simulated products generated by these large-scale productions resemble the data that will be supplied by the future CTA hardware and software. The performance of each telescope layout is estimated by analysing these data products using reconstruction methods [@evnDisplay; @MARS], developed for the current generation of IACTs, and adapted for analysis of the arrays, briefly described in section \[sec:anchain\]. The first large-scale production ([`prod1`]{}) covered a wide range of different layouts [@APP_CTA_MC], from very compact ones, focused on low energies, to very extended ones, focused on multi-TeV energies. The evaluation of these layouts, studying their impact on a range of science cases [@DM_2013; @CRs_2013; @AGNs_2013; @surveys_2013; @CTA_GRBs; @pulsars_2013], resulted in a clear preference for intermediate layouts with a balanced performance over a wide energy range. The second large-scale production ([`prod2`]{}) refined the layout optimisation studies [@Hassan-2015] while putting an additional emphasis on assessing the effect of site-related parameters over performance at the proposed sites to host the Observatory [@MC_ICRC_site:2015]. Results from this production concluded that all proposed sites were excellent candidates to host CTA, but that sites at moderate altitudes ($\sim2000$ m) give the best overall performances [@Site_Paper]. Given the wide scope of this production, the layout optimisation performed [@Hassan-2015] is estimated to be $\sim10$% away from the optimum performance, mainly due to the limited number of simulated telescope positions for a given site. The third large-scale production ([`prod3`]{}) was carried out for the primary site candidates, Paranal (Chile) and La Palma (Spain). Telescope design configurations were updated and a significantly larger and more realistic set of available telescope positions were included (see Fig. \[fig:Layout\_ALL\_Prod3\]). The aim of this production was to refine the optimisation, defining the final telescope layout for both arrays by reducing the optimisation uncertainty to the few percent level, while preserving the goal of a balanced intermediate layout fulfilling all performance requirements. To validate the baseline arrays inferred from this work (see section \[sec:souths\]), this production was extended using identical telescope models. Telescope locations were further refined by considering a total of 210 positions for Paranal. All results presented in this paper, unless otherwise stated, refer to this third large-scale production. The optimisation of the arrays required a significant computational effort: the third large-scale production for the Paranal site alone required million HEP-SPEC06 CPU hours[^5] and PB of disk storage. Most of these simulations were carried out on the CTA computing grid, using the European Grid Infrastructure and utilising the DIRAC framework as interware [@dirac-general; @Arrabito-2015], as well as on the computer clusters of the Max-Planck-Institut für Kernphysik. The subsequent analysis was carried out using the DIRAC framework, as well as the computing clusters at the Deutsches Elektronen-Synchrotron and at the Port d’Informació Científica. Simulated telescope layouts {#SimTelLay} --------------------------- ![Simulated telescope positions within the third large-scale MC production (see section \[sec:mcprod\] for details). *Top*: La Palma telescope positions including all radially-scaled MST layouts. The available positions are restricted by the site topography, buildings and roads. *Bottom*: Paranal telescope positions before applying any radially-symmetric transformation (scaling number 1). LST positions are indicated by red circles, MSTs by green triangles, and SSTs by blue squares.[]{data-label="fig:Layout_ALL_Prod3"}](Figures/TelCoordLaPalma.pdf "fig:"){height="0.4\textheight"} ![Simulated telescope positions within the third large-scale MC production (see section \[sec:mcprod\] for details). *Top*: La Palma telescope positions including all radially-scaled MST layouts. The available positions are restricted by the site topography, buildings and roads. *Bottom*: Paranal telescope positions before applying any radially-symmetric transformation (scaling number 1). LST positions are indicated by red circles, MSTs by green triangles, and SSTs by blue squares.[]{data-label="fig:Layout_ALL_Prod3"}](Figures/TelCoordParanal.pdf "fig:"){height="0.4\textheight"} ![*Top-left*: Radially-symmetric distortion factors for the five different scalings applied to the CTA-South layouts, as a function of the radial distance to the centre of the array before the applied transformation. *Top-right to bottom-right*: an example of the five resulting scaled layouts for one of the Paranal site candidates (“S1’’). LST positions are indicated by red circles, MSTs by green triangles, and SSTs by blue squares. Taken from [@ICRC2017:Layout].[]{data-label="fig:distfactor"}](Figures/distfactor.pdf "fig:"){width="44.00000%"} ![*Top-left*: Radially-symmetric distortion factors for the five different scalings applied to the CTA-South layouts, as a function of the radial distance to the centre of the array before the applied transformation. *Top-right to bottom-right*: an example of the five resulting scaled layouts for one of the Paranal site candidates (“S1’’). LST positions are indicated by red circles, MSTs by green triangles, and SSTs by blue squares. Taken from [@ICRC2017:Layout].[]{data-label="fig:distfactor"}](Figures/3HB1-FD-1.pdf "fig:"){width="45.00000%"} ![*Top-left*: Radially-symmetric distortion factors for the five different scalings applied to the CTA-South layouts, as a function of the radial distance to the centre of the array before the applied transformation. *Top-right to bottom-right*: an example of the five resulting scaled layouts for one of the Paranal site candidates (“S1’’). LST positions are indicated by red circles, MSTs by green triangles, and SSTs by blue squares. Taken from [@ICRC2017:Layout].[]{data-label="fig:distfactor"}](Figures/3HB1-FD-2.pdf "fig:"){width="45.00000%"} ![*Top-left*: Radially-symmetric distortion factors for the five different scalings applied to the CTA-South layouts, as a function of the radial distance to the centre of the array before the applied transformation. *Top-right to bottom-right*: an example of the five resulting scaled layouts for one of the Paranal site candidates (“S1’’). LST positions are indicated by red circles, MSTs by green triangles, and SSTs by blue squares. Taken from [@ICRC2017:Layout].[]{data-label="fig:distfactor"}](Figures/3HB1-FD-3.pdf "fig:"){width="45.00000%"} ![*Top-left*: Radially-symmetric distortion factors for the five different scalings applied to the CTA-South layouts, as a function of the radial distance to the centre of the array before the applied transformation. *Top-right to bottom-right*: an example of the five resulting scaled layouts for one of the Paranal site candidates (“S1’’). LST positions are indicated by red circles, MSTs by green triangles, and SSTs by blue squares. Taken from [@ICRC2017:Layout].[]{data-label="fig:distfactor"}](Figures/3HB1-FD-4.pdf "fig:"){width="45.00000%"} ![*Top-left*: Radially-symmetric distortion factors for the five different scalings applied to the CTA-South layouts, as a function of the radial distance to the centre of the array before the applied transformation. *Top-right to bottom-right*: an example of the five resulting scaled layouts for one of the Paranal site candidates (“S1’’). LST positions are indicated by red circles, MSTs by green triangles, and SSTs by blue squares. Taken from [@ICRC2017:Layout].[]{data-label="fig:distfactor"}](Figures/3HB1-FD-5.pdf "fig:"){width="45.00000%"} Layouts with a more compact and denser distribution of telescopes improve the direction and energy reconstruction of showers (the limiting factor for the low/mid-energy range of CTA, between 20 GeV and 5 TeV), while larger and sparser layouts improve the collection area and event statistics (the limiting factor for the highest energies), see also discussion in [@APP_CTA_MC]. To find the most efficient inter-telescope distance for CTA, each layout candidate is modified by applying several radially-symmetric scaling factors (see Fig. \[fig:distfactor\]). On top of that, in order to maintain the radial symmetry of the array in the shower projection for typical observation directions near source culmination, the southern array layouts were stretched by a factor of 1.06 in the north-south direction and compressed by a factor 1/1.06 in the east-west direction. The assumption of an average culmination zenith angle of $z \sim 27^{\circ} \approx \arccos{(1/1.06^2)}$, is based on long-term observation statistics from H.E.S.S., MAGIC, and VERITAS. The simulated telescope positions are shown in Figure \[fig:Layout\_ALL\_Prod3\]. In the case of La Palma, for which a combination of all scaled layouts is shown, these positions were constrained by site topography, as well as by existing buildings and roads. For Paranal, the layout was based on a hexagonal grid[^6] with some additional positions. Five sets of radially-symmetric transformations were applied to the master telescope layout shown at the bottom of Fig. \[fig:Layout\_ALL\_Prod3\], as detailed in [@ICRC2017:Layout]. Changing the scaling, each telescope is moved radially so that its new position ($x$, $y$) satisfies $\sqrt{x^2 + y^2} = r \cdot D(r)$, where $r$ is the distance to the centre of the array before the applied transformation and $D(r)$ is the distortion factor, shown in Fig. \[fig:distfactor\] (top-left). These transformations change the inter-telescope distance from close to optimal for the low/mid energies to increasingly larger separations for the higher energies. As an example, the five resulting scaled arrays for one CTA-South layout are shown in Fig. \[fig:distfactor\]. By studying the performance of each simulated scaling, we attempt to find the optimal layout that balances reconstruction quality and event quantity. At the energy range where the LSTs dominate (below $\sim100$ GeV), the influence of the other telescope types is small, therefore LST spacing optimisation is studied independently and their positions are constant among the five different scalings for both sites. The layout naming convention used throughout the text is the following: All layout names start with either the letter “S’’, for CTA-South candidates, or “N’’, for CTA-North candidates, followed by a number indicating the array variant. When referring to the different scalings of each candidate, an additional number is added after the layout name, e.g. “S2-3’’ indicates the scaling 3 of the layout “S2’’. This scheme has two exceptions: the layout “SI-$N_{scaling}$’’, with an alternative MST distribution shown in Fig. \[fig:Layoutisland\], and layouts “S7’’ and “S8’’, products of the merging between different scalings, shown in Fig. \[fig:3HB9Layout\] and discussed in section \[sec:souths\]. The telescope number and positions of the CTA-South array candidates are shown in Fig. \[fig:naming\]. ![Simulated telescope positions for the different CTA-South array candidates. The positions of each telescope sub-system is shown separately for the arrays “S1’’ to “S4’’. The table shows the number of telescopes per type for all layout candidates.[]{data-label="fig:naming"}](Figures/LSTNaming.pdf){width="\textwidth"} ![Simulated telescope positions for the different CTA-South array candidates. The positions of each telescope sub-system is shown separately for the arrays “S1’’ to “S4’’. The table shows the number of telescopes per type for all layout candidates.[]{data-label="fig:naming"}](Figures/MSTNaming.pdf){width="\textwidth"} \ ![Simulated telescope positions for the different CTA-South array candidates. The positions of each telescope sub-system is shown separately for the arrays “S1’’ to “S4’’. The table shows the number of telescopes per type for all layout candidates.[]{data-label="fig:naming"}](Figures/SSTNaming.pdf){width="\textwidth"} Name LST MST SST ------------------ ----- ----- ----- -- SI-N$_{scaling}$ 4 24 72 S1-N$_{scaling}$ 4 24 73 S2-N$_{scaling}$ 3 24 73 S3-N$_{scaling}$ 3 24 73 S4-N$_{scaling}$ 3 24 73 S7 3 24 73 S8 4 25 70 The total number of simulated unique telescope positions adds up to 892 for the southern site and 99 for the northern site. At the time the layouts were defined, different alternative designs for the medium and small size telescopes were under consideration and the number of telescopes of each design was not yet fixed. To ensure that the layout resulting from the optimisation does not depend on a certain telescope model, all prototype designs and cameras were simulated, resulting in a total of 3092 simulated telescopes. This way, the performance of each proposed baseline array can be studied for all the different combinations of MST/SST models. Analysis and evaluation criteria {#sec:anchain} -------------------------------- In order to perform the telescope layout optimisation, parameters describing the performance of a given layout need to be defined and maximised. As in [@Site_Paper], the primary criteria used in this work to evaluate performance is the differential sensitivity, i.e. the minimum detectable flux from a steady source over a narrow energy range and a fixed observation time. This parameter depends on the collection area, angular resolution and rate of background events, mostly composed by cosmic-ray hadrons and electrons that survive the gamma-ray selection criteria (cuts). The differential sensitivity is calculated by optimising in each energy bin the cuts on the shower arrival direction, background rejection efficiency and minimum telescope event multiplicity[^7]. It is computed by requiring a five standard deviation (5$\sigma$) detection significance in each energy bin (equation 17 from [@LiMa], with an off-source to on-source exposure ratio of five, assuming a power-law spectrum of $E^{-2.6}$), and the signal excess to be at least five times the expected systematic uncertainty in the background estimation (1%), and larger than ten events. The figure of merit used for the evaluation and comparison of the scientific performance of CTA layouts is called the performance per unit time (PPUT). PPUT is the unweighted geometrical mean of the reference point-source flux sensitivity, $F_\mathrm{sens,ref}$, to the achieved sensitivity, $F_\mathrm{sens}$, over a given energy range with $N$ logarithmically uniform bins (five per decade) in energy: $$\mathrm{PPUT} = \left( \prod_{i=1}^{N} \frac{F_\mathrm{sens,ref}(i)}{F_\mathrm{sens}(i)} \right)^{1/N}$$ The reference sensitivity was derived from the analysis of previous simulations carried out by the Consortium, based on initial and conservative assumptions on the telescope parameters (see [@APP_CTA_MC]). These reference values, together with other performance requirements (e.g. minimum angular and energy resolutions), constitute the prime goals of the CTA design concept. PPUT may be calculated for the whole CTA-required energy range to estimate the overall performance, i.e. from 20 GeV up to 300 (50) TeV for CTA-South (North), or for energy sub-ranges, to evaluate specific telescope sub-system capabilities. PPUT is defined such that a larger number corresponds to better performance. Statistical uncertainties of all PPUT values, calculated by propagating the differential sensitivity errors associated with the MC event statistics, are below the 3% level. When comparing PPUT values, these uncertainties are unrealistic given that the performance of all layouts in a given site are calculated from the same set of simulated showers. Statistical uncertainties of PPUT values are therefore not shown in this work. Except if specified differently, all performance curves and PPUT values shown in this work correspond to a differential sensitivity to a point-like source in the centre of the FoV with an observation time of 50 hours. The sensitivity of these layouts to sources located at larger angular distances from the centre of the FoV was also evaluated. All telescope layouts presented here were required to comply with a minimum off-axis performance: the radius of the FoV region in which the point-source sensitivity is within a factor two of the one at the centre must be larger than 1$^\circ$ for the LST sub-system (array composed by all and only LSTs) and larger than 3$^\circ$ for the MST and SST sub-systems. Two fully independent analysis chains, Eventdisplay [@evnDisplay] and MARS [@MARS] (thoroughly tested by the VERITAS and MAGIC collaborations, respectively), have been used to process the full MC production (at 20$^\circ$ zenith angle) for a large number of telescope configurations for both the Paranal and La Palma sites. In addition, the ImPACT analysis [@2014APh....56...26P] was used to produce a cross-check for a small subset of these configurations and the baseline analysis [@APP_CTA_MC] was used to validate some results on same-type telescope sub-systems. Eventdisplay, MARS and the methods of the baseline analysis perform classical analyses based on second moment parameterisation of the Cherenkov images [@hillas], with different choice of algorithms for image cleaning, background suppression (Boosted Decision Trees, Random Forest or Lookup tables) and energy reconstruction (Lookup tables or Random Forest). ImPACT is based on a maximum likelihood fit of shower images to pre-generated MC templates, and has proven effective in the analysis of H.E.S.S. data. In all four cases, background suppression cuts are tuned to achieve the best performance (maximising sensitivity) in each bin of reconstructed energy. See [@Site_Paper; @APP_CTA_MC] for more details on the analysis. Figure \[fig:difanchain\] shows the PPUT values (between 20 GeV to 125 TeV) of the five scalings simulated for a given CTA-South array candidate, analysed with three of the analysis chains described. The results of the different analyses are, in general, fairly consistent. As shown in Fig. \[fig:difanchain\], despite their small differences, the conclusion on the optimal layout is the same regardless of the choice of analysis package. ![Comparison of performance (expressed in terms of PPUT, see text) of a range of simulated array layouts for three different analysis chains, relative to the PPUT value attained by each of them on the “S1-3’’ layout. The five layouts are presented in Fig. \[fig:distfactor\]. The symbols shown in the legend indicate the various analysis chains.[]{data-label="fig:difanchain"}](Figures/PPUT_difanchain.pdf){width="0.9\linewidth"} Telescope Configurations ------------------------ The third large-scale MC production was simulated using the most realistic and detailed modelling of all telescopes and camera types available. Given that the prototype telescopes were in the development stage at the time of the production (summer 2015), some telescope and camera parameters used within these models may be different from the final ones. These differences are expected to have a small effect on single-telescope performance, so all conclusions inferred from this study will still be valid, as long as the -proposed telescopes do not undergo major design changes. SC-MSTs were excluded from this study due to technical limitations. The limited available memory during computation did not allow the production of sufficient event statistics for their performance evaluation. Given the relatively similar mirror area and FoV of DC-MSTs and SC-MSTs, it is unlikely that the replacement of some DC-MSTs with SC-MSTs in the proposed layouts would result in a sub-optimal array layout. As the final configuration of telescope types is not known at this point (e.g. how many SSTs of each design will be constructed), the analysis always considers arrays of a single MST and SST design. All possible combinations between the two DC-MST cameras and the three SST models have been studied to ensure that the layout choice does not depend on specific telescope configurations. Figure \[fig:PPUT\_NFDG\] shows as an example the PPUT values of some CTA-South arrays using different combinations of telescope models: NectarCam/GCT, NectarCam/SST-1M, FlashCam/GCT, and FlashCam/SST-1M. The relative differences of the PPUT values between the different configurations for a given array layout are below 5% and clearly show the same trend upon changes of the array layout and scaling. ![Comparison of performance (expressed in terms of PPUT, see text) of a range of simulated CTA-South array layouts for different combinations of telescope model configurations, each relative to the “S1-3’’ layout. The different “S1’’ layout scalings are pictured in Fig. \[fig:distfactor\], while the “SI’’ layouts are described in section \[sec:MSTSSTpat\]. The symbols shown in the legend indicate the various telescope configurations. []{data-label="fig:PPUT_NFDG"}](Figures/PPUT_NFDG.pdf){width="0.9\linewidth"} Layout Optimisation {#sec:conlayout} =================== The final numbers of telescopes of each type is now fixed for both hemispheres, defined as the most cost-effective solution to maximise performance over the key scientific cases [@ScienceCTA]. The number of telescopes that the baseline arrays will be composed of are 4/25/70 LST/MST/SST for CTA-South and 4/15 LST/MST for CTA-North. With the number of telescopes fixed, the layout optimisation was performed following these considerations (in approximate order of priority): 1. Full system performance requirements. 2. Telescope sub-system performance requirements (e.g. MST-only array performance). 3. Topographical constraints of the selected sites. 4. Shadowing between neighbouring telescopes (i.e. telescopes structure intersecting the FoV of other telescopes during large zenith angle observations). 5. Performance of partially-operating arrays (e.g. resulting from telescope staging or downtime). 6. Impact on the ease of calibration and the likely magnitude of systematic effects. For C1, the main optimisation parameter is the differential sensitivity of the full array, while simultaneously ensuring that the energy resolution, the angular resolution and the FoV requirements are still met. C2 ensures that the system works in a close-to-optimal fashion also when operated as individual (LST, MST or SST) sub-systems. C3 is critical for the northern site (La Palma), but was not needed for the southern site, where no significant constraints are expected. C4 sets a minimum telescope spacing for pairs of each telescope size combination. If possible, without moving significantly away from the optimum performance for the baseline, point C5 was addressed by ensuring that partially completed systems are still close to optimal. In the case of the LSTs, of which only four telescopes will be installed on each site, the effect of telescope downtime was taken into consideration due to the expected occasional maintenance of one of these telescopes. For MSTs and SSTs, a few missing telescopes due to maintenance is not expected to significantly affect the performance. Finally, point C6 was addressed by requiring some overlap between different telescope sub-systems even when the array is partially completed. LST optimal separation {#subsec:lsts} ---------------------- ![Differential sensitivity and differential sensitivity ratio as a function of energy for two configurations of three LSTs with equal area (*bottom*): arranged as half a square of 115 m on a side (*top right*) or an isosceles triangle with two 127 m sides (close to equilateral, *top left*). The layouts are slightly stretched in the north-south direction and compressed in the east-west direction, as explained in section \[SimTelLay\]. The ratio is calculated so that higher values correspond to better sensitivity.[]{data-label="fig:3LST"}](Figures/Tel3LST123.pdf "fig:"){width="0.45\linewidth"} ![Differential sensitivity and differential sensitivity ratio as a function of energy for two configurations of three LSTs with equal area (*bottom*): arranged as half a square of 115 m on a side (*top right*) or an isosceles triangle with two 127 m sides (close to equilateral, *top left*). The layouts are slightly stretched in the north-south direction and compressed in the east-west direction, as explained in section \[SimTelLay\]. The ratio is calculated so that higher values correspond to better sensitivity.[]{data-label="fig:3LST"}](Figures/Tel3LST567.pdf "fig:"){width="0.45\linewidth"} ![Differential sensitivity and differential sensitivity ratio as a function of energy for two configurations of three LSTs with equal area (*bottom*): arranged as half a square of 115 m on a side (*top right*) or an isosceles triangle with two 127 m sides (close to equilateral, *top left*). The layouts are slightly stretched in the north-south direction and compressed in the east-west direction, as explained in section \[SimTelLay\]. The ratio is calculated so that higher values correspond to better sensitivity.[]{data-label="fig:3LST"}](Figures/3LSTs.pdf "fig:"){width="0.9\linewidth"} Below $\sim100$ GeV the LSTs will dominate CTA performance, as these will be the only telescopes with enough reflecting surface to detect the faint low-energy showers. For this reason, the layout of the MST and SST positions have no strong impact in this energy range, therefore their spacing optimisation can be studied independently. These showers are generally triggered within impact distances[^8] below 150 m, i.e. similar to the light pool radius of about 120 m [@Site_Paper]. As the light-pool size increases with the energy of the primary particle, the optimal LST spacing is expected to be smaller than for MSTs or SSTs. The optimal shape of the LST sub-system in the shower-plane projection is expected to be a square for four LSTs and an equilateral triangle for three LSTs. This is confirmed in Figure \[fig:3LST\], which shows the low-energy differential sensitivity of a three LST layout with an isosceles shape, close to equilateral, compared to a three LST layout with a half-square shape. The optimisation of the LST layout beyond these considerations is thus a question of separation only. At too-short separations, the projected lever arm in the stereoscopic shower reconstruction is too small for most events while at too-large separations too few showers are detected simultaneously by three or four LSTs (required for an optimal cosmic-ray background rejection). As described in section \[sec:mcprod\], the second large-scale MC production assessed CTA performance over a wide range of site candidates. Realistic values of the altitude and geomagnetic field strength at each site were used in the shower simulation [@Site_Paper]. Nine different LST positions were included at each site, allowing the analysis of several equivalent layouts (e.g. pairs of two LSTs) with different inter-telescope distances. Archival simulation sets for the following site candidates were available for this analysis (see [@Site_Paper] for details on each site): Aar (near Aus, Namibia) at 1640 m altitude, two sites at Leoncito (Argentina) at 1650 and 2660 m, and SAC (San Antonio de los Cobres, Argentina) at 3600 m altitude. To test the array performance at lower altitudes, an additional hypothetical Aar site was simulated at 500 m altitude. For the SAC site candidate, at whose altitude the Cherenkov light pool is significantly smaller, an additional set of simulations were performed with the telescope spacing reduced by a factor of 0.84, allowing us to test a larger number of telescope distances. ![Performance (expressed in terms of PPUT, see text) of LST squared layouts of different sizes located at different CTA-South candidate sites (*left*: observations towards north, *right*: observations towards south), in the energy range 30 to 300 GeV, using the baseline analysis described in [@APP_CTA_MC].[]{data-label="fig:lstseparationsquare"}](Figures/lstseparationsquare.pdf){width="0.9\linewidth"} For a layout of four LSTs in a square shape, side distances of 71, 100, and 141 m (plus 59, 84, and 119 m only for SAC) were available. Figure \[fig:lstseparationsquare\] shows the dependence of the LST sub-system performance versus telescope separation for all the studied sites. For the Paranal site, with an altitude and geomagnetic field falling between the two simulated Leoncito sites shown in Fig. \[fig:lstseparationsquare\], a separation of about 100 m (square side length) is favoured. For the case of LST pairs, there were nine different distances available between 58 to 255 m. As shown in Figure \[fig:lstseparation\], a rather flat optimum is found at 130 m, with close-to-optimum performance for separations ranging from about 100 m up to 150 m, with no significant change in energy threshold over this range. The optimum separation over the whole LST energy range (more relevant for observation with the LST sub-system only) is not significantly larger than for just the lowest energies (relevant for observations with the full array). ![Performance (expressed in terms of PPUT, see text) and energy threshold of pairs of LSTs as a function of their separation. PPUT values are calculated from the average of the Aar and the two Leoncito site candidates (with an average altitude close to that of the Paranal site) and are also averaged over observations pointing towards north and south. The upper panel shows PPUT values in the energy ranges of 25 GeV to 125 GeV and 25 GeV to 1.25 TeV; the lower panel shows the calculated energy threshold by using the true energy value that leaves 10% of the events below the cut value (after either the trigger or the analysis) [@Colin2009]. The performance is derived from the baseline analysis described in [@APP_CTA_MC].](Figures/LSTseparation.pdf){width="0.9\linewidth"} . \[fig:lstseparation\] Taking all these results into account, a squared layout of four LSTs with an optimised side distance of 115 m to 120 m would provide both full-system and sub-system optimal performance. In order to make sure the rest of the listed considerations, such as geological constrains for the La Palma site or improved staging scenarios for Paranal, are complied with, minor modifications were needed to be applied to these positions. As shown in Figure \[fig:3LST\], such minor modifications of the LST layout are expected to affect the performance at only the few percent level. MST and SST patterns {#sec:MSTSSTpat} -------------------- ![Layouts with different MST patterns: “S1’’ (*top*), with a strictly hexagonal pattern and “SI’’ (*bottom*), with four islands and a hexagonal core. The LST positioning in the two cases is the same, while the SSTs have been rearranged. Both layouts correspond to their scaling 2 variation. The distance of each telescope to its nearest neighbour of the same type is shown on the right.[]{data-label="fig:Layoutisland"}](Figures/3HB1-FD-Scaling-2.pdf "fig:"){width="0.9\linewidth"} ![Layouts with different MST patterns: “S1’’ (*top*), with a strictly hexagonal pattern and “SI’’ (*bottom*), with four islands and a hexagonal core. The LST positioning in the two cases is the same, while the SSTs have been rearranged. Both layouts correspond to their scaling 2 variation. The distance of each telescope to its nearest neighbour of the same type is shown on the right.[]{data-label="fig:Layoutisland"}](Figures/TelCoord3HI1-2.pdf "fig:"){width="0.9\linewidth"} ![Relative PPUT values for different energy ranges for the layout with a hexagonal MST pattern (“S1’’) and a layout with an MST pattern presenting four islands (“SI’’), both for the southern site, relative to “S1-3’’. Open and filled symbols correspond to observation times of 5 h and 50 h, respectively.[]{data-label="fig:PPUTisland"}](Figures/Paranal_PPUT_Paper_scaling_island_v3.pdf){width="0.99\linewidth"} As introduced in section \[sec:mcprod\], the master layout of simulated telescopes used in this work is based on a hexagonal layout to enhance the statistics of showers simultaneously detected by at least three telescopes [@Colin_2009]. From this layout, two different MST patterns were studied: a hexagonal one (as in “S1’’, top of figure \[fig:Layoutisland\]) and one presenting an inner hexagonal core with fewer telescopes and four surrounding islands of three MSTs each (as in “SI’’, bottom of figure \[fig:Layoutisland\]). Because of the repositioning of MSTs, some SSTs have been moved in order to provide uniform coverage. The positions of the LSTs are shared between the two layouts. As shown in Fig. \[fig:PPUTisland\], the two layouts provide comparable overall sensitivity over the whole energy range (20 GeV to 125 TeV). Over the low and medium energy ranges (20 GeV to 1.25 TeV) the hexagonal pattern is preferred, given the higher number of MSTs simultaneously used to reconstruct these contained showers (i.e. showers whose light pools are fully contained inside the area covered by CTA telescopes). Between 1.25 TeV and 12.5 TeV, the island pattern provides better performance due to the improved reconstruction of high-energy showers triggering telescopes near the edge of the array. This improvement fades above 12 TeV, for energies dominated by the SST sub-system. The hexagonal MST pattern was chosen as the preferred option given its improved performance over a wider energy range. Two different observation times were tested in this comparison, 5 and 50 hours, to make sure that the inferred conclusions are not dependent on the observation time. Southern site baseline array {#sec:souths} ============================ The PPUT values for six different energy ranges were calculated for three different CTA-South layout candidates (“S2’’ and “S4’’, calculated with respect to “S3’’) and their five different radial scalings. As shown in Fig. \[fig:PPUTscalings\], more compact arrays improve performance below $\sim1$ TeV, but have poorer performance compared to arrays with larger scalings at higher energies. Taking these results into account, a new layout is defined combining the MST layout with moderate radial scaling (2) and the SST layout with strong scaling (5), labelled as “S7’’. As shown in Fig. \[fig:PPUTscalings\], it is the layout with best overall performance, outperforming most alternatives in every energy range. ![Relative PPUT values for different energy ranges for several CTA-South layout candidates, relative to “S3-3’’. The resulting PPUT values obtained by combining the MST layout with moderate radial scaling (2) and the SST layout with strong scaling (5) are shown labelled as “scaling 2+5’’.[]{data-label="fig:PPUTscalings"}](Figures/Paranal_PPUT_Paper_scaling_v2.pdf){width="0.99\linewidth"} However, minor modifications are still necessary to be applied to “S7’’ for two important reasons: 1) it includes slightly different numbers of telescopes with respect to the defined baseline (4 LSTs, 25 MSTs and 70 SSTs) and 2) the distribution of the SSTs is sub-optimal for independent sub-system operation and complicates cross-calibration. The proposed baseline layout for -South is therefore a slightly modified version of “S7’’, named “S8’’ (both shown in Fig. \[fig:3HB9Layout\]). The performed modifications are discussed below: ![The best performing layouts from Fig. \[fig:PPUTscalings\]: “S7’’, on the left, and the proposed baseline layout for the southern site, “S8’’, on the right.[]{data-label="fig:3HB9Layout"}](Figures/Tel3HB8.pdf "fig:"){width="0.49\linewidth"} ![The best performing layouts from Fig. \[fig:PPUTscalings\]: “S7’’, on the left, and the proposed baseline layout for the southern site, “S8’’, on the right.[]{data-label="fig:3HB9Layout"}](Figures/Tel3HB9.pdf "fig:"){width="0.49\linewidth"} - The LST layout is rather independent of the optimisation of the system as a whole. The proposed four LST layout is an intermediate step between a square and a double-equilateral triangle, with the advantage that it performs significantly better than a square for a three LST stage, without significant degradation of the full system performance. This compromise also works better than the double-equilateral triangle configuration for the situation where one of the east-west pair of telescopes is unavailable (e.g. due to maintenance activities). The east-west pair of telescopes represents the best option for a two LST stage-1[^9], and therefore the chosen telescope separation is close to optimal for a two-telescope system (as shown in Sec. \[subsec:lsts\]). - The MST layout for the proposed array is identical to “S7’’ except for the addition of a central MST. The central MST is particularly useful for MST sub-system operation, surveying performance and LST-MST cross-calibration. - The SST positions are modified from “S7’’ by removing four telescopes (“S7’’ has 74 SSTs) and smoothing their distribution. Four SSTs are moved within the boundary of the dense MST array to enhance the SST-only sub-system performance, to provide better MST-SST cross-calibration and to smooth the performance transition between the MST-dominated to the SST-dominated energy range. After fixing the four inner telescopes and the outer boundary edge of the layout (so that the highest energy performance is not affected), the spacing of the remaining telescopes is adjusted to minimise the inter-telescope distance. As mentioned in section \[sec:mcprod\], some telescope positions within “S8’’ were not available and needed to be added to the third large-scale MC production. This extension was necessary to confirm that these modifications were not strongly affecting performance. As shown in Fig. \[fig:PPUTscalings\], the overall PPUT of “S8’’ matches the one attained by “S7’’. Even if the performance above $\sim1$ TeV is slightly affected by subtracting four SSTs, “S8’’ outperforms most layout alternatives, while taking into account all considerations listed in section \[sec:conlayout\]. For these reasons, “S8’’ is the final telescope layout proposed as the baseline for the CTA southern site. Northern site baseline array {#sec:norths} ============================ As discussed in section \[sec:mcprod\], the available telescope positions of the CTA-North layout were mainly constrained by site topography, buildings and roads. As Figure \[fig:LaPalmaPPUT\] illustrates, the best overall performance from the simulated layouts is achieved by the widest MST spacing considered. This large spacing does not have an impact on the low energy performance while guaranteeing the best sensitivity at higher energies. An even wider spacing, while possible for some of the telescopes, is forbidden by the logistical constraints of the site. The position of the four LSTs was fixed by orography and existing constraints, with LST-1 already under construction. Several solutions are still possible for alternative MST layouts, some of which are shown in Fig. \[fig:AL4\], maintaining the same inter-telescope distance. All these alternative layouts achieve similar performance, as shown in Fig. \[fig:LPPPUT3b\], while complying with the constraints imposed by the site. ![Relative PPUT values for the different scalings of the proposed layout for the northern site, all shown in Fig. \[fig:Layout\_ALL\_Prod3\], relative to the scaling 3.[]{data-label="fig:LaPalmaPPUT"}](Figures/LP_PPUT_Paper_scaling_v2.pdf){width="0.9\linewidth"} ![Several layouts proposed as baseline arrays for the northern site, together with the position of buildings, roads, and the two MAGIC telescopes. The orography constraints are not shown. The layouts share the LST positions and roughly the same inter-telescope distances between MSTs.[]{data-label="fig:AL4"}](Figures/Tel3AL4AN15_roads.pdf "fig:"){width="49.00000%"} ![Several layouts proposed as baseline arrays for the northern site, together with the position of buildings, roads, and the two MAGIC telescopes. The orography constraints are not shown. The layouts share the LST positions and roughly the same inter-telescope distances between MSTs.[]{data-label="fig:AL4"}](Figures/Tel3AL4BN15_roads.pdf "fig:"){width="49.00000%"} ![Several layouts proposed as baseline arrays for the northern site, together with the position of buildings, roads, and the two MAGIC telescopes. The orography constraints are not shown. The layouts share the LST positions and roughly the same inter-telescope distances between MSTs.[]{data-label="fig:AL4"}](Figures/Tel3AL4CN15_roads.pdf "fig:"){width="49.00000%"} ![Several layouts proposed as baseline arrays for the northern site, together with the position of buildings, roads, and the two MAGIC telescopes. The orography constraints are not shown. The layouts share the LST positions and roughly the same inter-telescope distances between MSTs.[]{data-label="fig:AL4"}](Figures/Tel3AL4DN15_roads.pdf "fig:"){width="49.00000%"} ![Several layouts proposed as baseline arrays for the northern site, together with the position of buildings, roads, and the two MAGIC telescopes. The orography constraints are not shown. The layouts share the LST positions and roughly the same inter-telescope distances between MSTs.[]{data-label="fig:AL4"}](Figures/Tel3AL4FN15_roads.pdf "fig:"){width="49.00000%"} ![Several layouts proposed as baseline arrays for the northern site, together with the position of buildings, roads, and the two MAGIC telescopes. The orography constraints are not shown. The layouts share the LST positions and roughly the same inter-telescope distances between MSTs.[]{data-label="fig:AL4"}](Figures/Tel3AL4GN15_roads.pdf "fig:"){width="49.00000%"} ![Relative PPUT values for several different candidates for the northern layout, relative to “N3’’. The differences between the layouts are less than 5%.[]{data-label="fig:LPPPUT3b"}](Figures/LP3b_PPUT_Paper_scaling_v2.pdf){width="0.9\linewidth"} Conclusion ========== ![CTA differential sensitivity (multiplied by energy squared) compared to those of present day instruments (from [@publicCTAsensi]): Fermi-LAT [@LATSensi], MAGIC [@MAGICSensi1; @MAGICSensi2], H.E.S.S. [@HessSensi], VERITAS [@VeritasSensi], and HAWC [@HAWCSensi][]{data-label="fig:AllInstr"}](Figures/CTA-Performance-prod3b-v1.pdf){width="0.95\linewidth"} The Cherenkov Telescope Array will be the next generation gamma-ray instrument in the VHE range. It will be composed of two separate arrays: the southern observatory will be installed at Paranal (Chile). The northern array, the construction of which has already started with LST-1, will be built on the island of La Palma (Spain). These baseline arrays are the result of a concerted effort involving three different large-scale MC productions performed during the last several years. The main purpose of the last large-scale production was to define the final layouts to be constructed in both sites. As a result, a single layout (right of Fig. \[fig:3HB9Layout\]) is proposed for CTA-South. It features a four LST rhombus layout (intermediate step between a square and a double-equilateral triangle), an hexagonal MST layout, and SSTs homogeneously distributed on a circle of about 1.1 km radius. Several similarly performing layouts are instead proposed for CTA-North (Fig. \[fig:AL4\]). Given the nearly identical performance of different layouts for CTA-North, the final layout will be fixed based on ease of construction, once a better understanding on the site constraints is attained. This study shows that the inter-telescope optimum distance of the LSTs is between 100 and 150 m, with a rather flat low-energy performance over these values. The MSTs will provide better performance over the core-energy range of CTA when distributed over a hexagonal grid slightly stretched by applying an azimuthally-symmetric transformation, with inter-telescope distances ranging between 150 and 250 m. The SSTs, present in the southern hemisphere site only, provide better performance in a layout with a strong scaling, with inter-telescope distances ranging between 190 and 300 m. While the main parameter used in the optimisation is differential sensitivity over the different energy ranges, other considerations were also taken into account. Apart from considering the constraints imposed by the characteristics of the selected sites, minor modifications were applied to the baseline arrays to improve the performance of different staging scenarios (slightly modifying the final LST layout), the cross-calibration between different telescope types, and the stand-alone sub-system performance (mainly by adding SSTs in the inner part of the layout). All these layouts comply with the performance requirements imposed by the CTA Consortium for both sites over the full energy range. CTA will outperform present day instruments by more than an order of magnitude in sensitivity in the multi-TeV range, as can be seen in Fig. \[fig:AllInstr\]. The differential sensitivities presented in Fig. \[fig:AllInstr\], together with all the instrument response functions of the proposed baseline arrays, are publicly available [@publicCTAsensi] and they were used in the study of key science projects [@ScienceCTA]. As shown in all the performance comparisons performed throughout this work, the optimisation reaches the few percent level in precision, showing that smaller modifications to these baseline arrays will not lead to significant performance losses. In addition, several different implementations for the SST and MST telescopes were tested and resulted in equivalent conclusions, proving that this optimisation is also valid even if different telescope designs undergo minor modifications. ### Acknowledgments {#acknowledgments .unnumbered} We gratefully acknowledge financial support from the following agencies and organizations: State Committee of Science of Armenia, Armenia; The Australian Research Council, Astronomy Australia Ltd, The University of Adelaide, Australian National University, Monash University, The University of New South Wales, The University of Sydney, Western Sydney University, Australia; Federal Ministry of Science, Research and Economy, and Innsbruck University, Austria; Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro (FAPERJ), Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP), Ministry of Science, Technology, Innovations and Communications (MCTIC), Brasil; Ministry of Education and Science, National RI Roadmap Project DO1-153/28.08.2018, Bulgaria; The Natural Sciences and Engineering Research Council of Canada and the Canadian Space Agency, Canada; CONICYT-Chile grants PFB-06, FB0821, ACT 1406, FONDECYT-Chile grants 3160153, 3150314, 1150411, 1161463, 1170171, Pontificia Universidad Católica de Chile Vice-Rectory of Research internationalization grant under MINEDUC agreement PUC1566, Chile; Croatian Science Foundation, Rudjer Boskovic Institute, University of Osijek, University of Rijeka, University of Split, Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, University of Zagreb, Faculty of Electrical Engineering and Computing, Croatia; Ministry of Education, Youth and Sports, MEYS LM2015046, LTT17006 and EU/MEYS CZ.02.1.01/0.0/0.0/16\_013/0001403, CZ.02.1.01/0.0/0.0/17\_049/0008422, Czech Republic; Ministry of Higher Education and Research, CNRS-INSU and CNRS-IN2P3, CEA-Irfu, ANR, Regional Council Ile de France, Labex ENIGMASS, OSUG2020, P2IO and OCEVU, France; Max Planck Society, BMBF, DESY, Helmholtz Association, Germany; Department of Atomic Energy, Department of Science and Technology, India; Istituto Nazionale di Astrofisica (INAF), Istituto Nazionale di Fisica Nucleare (INFN), MIUR, Istituto Nazionale di Astrofisica (INAF-OABRERA) Grant Fondazione Cariplo/Regione Lombardia ID 2014-1980/RST\_ERC, Italy; ICRR, University of Tokyo, JSPS, MEXT, Japan; Netherlands Research School for Astronomy (NOVA), Netherlands Organization for Scientific Research (NWO), Netherlands; University of Oslo, Norway; Ministry of Science and Higher Education, DIR/WK/2017/12, the National Centre for Research and Development and the National Science Centre, UMO-2016/22/M/ST9/00583, Poland; Slovenian Research Agency, Slovenia, grants P1-0031, P1-0385, I0-0033, J1-9146; South African Department of Science and Technology and National Research Foundation through the South African Gamma-Ray Astronomy Programme, South Africa; MINECO National R+D+I, Severo Ochoa, Maria de Maeztu, CDTI, PAIDI, UJA, FPA2017-90566-REDC, Spain; Swedish Research Council, Royal Physiographic Society of Lund, Royal Swedish Academy of Sciences, The Swedish National Infrastructure for Computing (SNIC) at Lunarc (Lund), Sweden; Swiss National Science Foundation (SNSF), Ernest Boninchi Foundation, Switzerland; Durham University, Leverhulme Trust, Liverpool University, University of Leicester, University of Oxford, Royal Society, Science and Technology Facilities Council, UK; U.S. National Science Foundation, U.S. Department of Energy, Argonne National Laboratory, Barnard College, University of California, University of Chicago, Columbia University, Georgia Institute of Technology, Institute for Nuclear and Particle Astrophysics (INPAC-MRPI program), Iowa State University, the Smithsonian Institution, Washington University McDonnell Center for the Space Sciences, The University of Wisconsin and the Wisconsin Alumni Research Foundation, USA. The research leading to these results has received funding from the European Union’s Seventh Framework Programme (FP7/2007-2013) under grant agreements No 262053 and No 317446. This project is receiving funding from the European Union’s Horizon 2020 research and innovation programs under agreement No 676134. We would like to thank the computing centres that provided resources for the generation of the Instrument Response Functions [@gridAcknowledgement]. [^1]: <https://www.mpi-hd.mpg.de/hfm/HESS/HESS.shtml>. [^2]: <https://magic.mpp.mpg.de/>. [^3]: <https://veritas.sao.arizona.edu/>. [^4]: <http://www.cta-observatory.org/>. [^5]: The HEP-wide benchmark for measuring CPU performance. See specifications in [^6]: As discussed in [@Colin_2009], a square grid is preferred to enhance two telescope events while a hexagonal layout favours the simultaneous detection of showers by three or more telescopes, the latter being more suitable for CTA. [^7]: The event multiplicity is the number of telescopes simultaneously detecting a shower. [^8]: The impact distance is the between the telescope location and the shower axis. [^9]: The east-west telescope pair provides better stereoscopic reconstruction while pointing north/south, the preferred sky directions in which sources culminate.

--- abstract: 'In this paper, we give a simple description of the deformations of a map between two smooth curves with partially prescribed branching, in the cases that both curves are fixed, and that the source is allowed to vary. Both descriptions work equally well in the tame or wild case. We then apply this result to obtain a positive-characteristic Brill-Noether-type result for ramified maps from general curves to the projective line, which even holds for wild ramification indices. Lastly, in the special case of rational functions on the projective line, we examine what we can say as a result about families of wildly ramified maps.' author: - Brian Osserman bibliography: - 'hgen.bib' title: 'Deformations of covers, Brill-Noether theory, and wild ramification' --- [^1] Introduction ============ In studying ramified maps of curves, questions frequently arise which demand fixing ramification on the source, or branching on the target. In the case that the target curve is $\P^1$, the former is treated by the theory of linear series, which naturally works up to automorphism of the image, so we will refer to this as the linear series perspective. In contrast, we will refer to working with fixed branching on the target (and typically allowing the source curve itself to vary) as the branched covers perspective. Often, and particularly in the context of degeneration arguments over $\C$ [@e-h1], these perspectives have been considered more or less interchangeable. However, recently a number of fundamental differences have come to light (see for instance [@os7 Prop. 5.4, Rem. 8.3]), particularly in positive characteristic, and it has also proven fruitful to pass between the perspectives to take advantages of the distinct features of each, perhaps most notably in Tamagawa’s [@ta1]. In this note, we examine the deformation theory of covers with partial branching specified, and then apply it to the perspective of linear series to obtain a ramified Brill-Noether theorem in positive characteristic, in the case of one-dimensional target. Our deformation theory result is straightforward to obtain from extremely well-known results, but does not appear to be stated in the literature. It is the following. \[main-def\] Given $C, D$ smooth curves over a field $k$, and $f:C \rightarrow D$ of degree $d$, together with $k$-valued points $P_1, \dots, P_n$ of $C$ such that $f$ is ramified to order at least $e_i$ at each $P_i$ for some $e_i$, then the space of first-order infinitesmal deformations of $f$ together with the $P_i$ over $k$, such that $f(P_i)$ remains fixed and the $P_i$ remain ramified to order at least $e_i$, is parametrized by $$\label{main-eq} H^0(C, f^* T_D (-\sum_i (e_i-\delta_i)P_i))\oplus k^{\sum_i(1-\delta_i)},$$ where $\delta_i=0$ if $p | e_i$ or $f$ is ramified to order higher than $e_i$ at $P_i$ and is $1$ otherwise. Furthermore, the space of first-order infinitesmal deformations of $C,$ the $P_i$ and $f$, fixing $f(P_i)$ and preserving the ramification condition at each $P_i$, is parametrized by $$\label{main-eq2} \H^1(C, T_C(-\sum_i P_i) \rightarrow f^* T_D(-\sum_i e_i P_i)) \cong k^{d(2-2g_D)-(2-2g_C)-\sum_i (e_i-1)},$$ where the last isomorphism requires also that $f$ be separable. Finally, both statements still hold when some $e_i$ are allowed to be $0$, which we interpret to put no condition on the $P_i$ at all. In the case that $f$ is tame and ramified to order exactly $e_i$ at the $P_i$, this is well-known. The main observation of the theorem, particularly in the first case, is that in order to obtain a useful theory, it is important to also consider the moduli of the ramification points, even when the branch points remain fixed. In the classical setting, this issue does not arise. Next, if one considers the situation with $D=\P^1$, and ramification points specified on $C$, the perspective changes from branched covers to linear series with prescribed ramification. From this point of view, classical Brill-Noether theory gives a lower bound on the dimension of the space of maps as the $P_i$ (and even $C$) are allowed to move. The deformation theory of Theorem \[main-def\] gives the necessary upper bound, and allows us to conclude the following Brill-Noether result, generalizing [@e-h1 Thm. 4.5] to positive characteristic in the case $r=1$. \[brill-noether\] Fix $d, n$ and $e_1, \dots, e_n$, together with $n$ general points $P_i$ on a general curve $C$ of genus $g$. Then the space of separable maps of degree $d$ from $C$ to $\P^1$, ramified to order at least $e_i$ at $P_i$, and taken modulo automorphism of the image space, is pure of dimension $2d-2-g-\sum_i (e_i-1)$. Finally, we note that although from the perspective of branched covers, tame ramification is always well-behaved and wild ramification seems more pathological, the situation is not the same from the perspective of ramified linear series. Indeed, from this perspective a simple dimension count justifies the fact that wildly-ramified maps always come in infinite families. On the other hand, we have examples from [@os7 Prop. 5.4] of cases where tame ramification could only produce infinite families of separable maps with fixed ramification, and as a result of Brill-Noether theorem, can exist only for special configurations of $P_i$. We make some elementary observations that, at least in certain cases when $C=D=\P^1$, wildly ramified linear series are in fact rather well-behaved. One such result is the following. \[wild\] Fix $d,n,m$, together with $n$ general points $P_i$ on $\P^1$, and $e_1, \dots, e_n$, with $e_i$ wild for $i \leq m$, and $e_i$ tame for $i>m$, and satisfying $2d-2 = m+\sum_i (e_i-1)$. Then the dimension of the space of separable maps of degree $d$ from $\P^1$ to $\P^1$, ramified to order exactly $e_i$ at $P_i$ and unramified elsewhere, and taken modulo automorphism of the image space, is exactly $m$. Moreover, if $m=1$, $e_1=p$, and $e_i<p$ for $i>1$, this space is non-empty if and only if the corresponding space is non-empty when one replaces $e_1=p$ with $e_1=p-1$, and considers maps of degree $d-1$. Note that except when explicitly stated otherwise, we make no assertions about the non-emptyness of the space of maps with given ramification. However, the last statement of Theorem \[wild\] certainly produces cases of wild ramification in which for general $P_i$, the space of maps is non-empty of the expected dimension. This is thus better behavior than the pathological tame examples mentioned above. The subject of existence and non-existence will be taken up in [@os12]. Although the proof given here of Theorem \[brill-noether\] is not necessary for [@os12], it does provide the only purely algebraic, instrinsically characteristic-$p$ argument for the existence and non-existence results in question. Acknowledgements {#acknowledgements .unnumbered} ================ I would like to thank Johan de Jong for his tireless and invaluable guidance, and Ravi Vakil for his helpful conversations. Deformation Theory ================== Let $C, D$ be smooth curves over a field $k$, and $f:C \rightarrow D$ a morphism of degree $d>0$. We begin by reviewing some standard deformation theory, so that we can use formal local analysis to obtain Theorem \[main-def\]. It is well-known (see, e.g., [@va2 Appendix]) that the first-order infinitesmal deformations of $f$ are parametrized by $H^0(C, f^*(T_D))$, deformations of a pointed curve $(C,\{P_i\})$ by $H^1(C,T_C(-\sum_i P_i))$, and deformations of $(C,\{P_i\},f)$ by the hypercohomology group $\H^1(C,T_C(-\sum_i P_i)\rightarrow f^*T_D)$. By the same token, deformations of $k$-valued points $P_i$ are parametrized simply by $H^0(\Spec k, f^* (T_C)) \cong k$ (note that this is different from the case of deforming $C$ along with the $P_i$ because in this case, there are no automorphisms of $C$ to mod out by). These may be verified directly on the Cech cocycle level using the facts that $T_C$ is the sheaf of infinitesmal automorphisms of $C$, and that deformations of smooth, pointed curves are always locally trivial. In the case of pointed curves, one trivializes the deformation (including of the points) locally on $C$, and obtains the $1$-cocycle by considering the resulting transition functions, taking values in infinitesmal automorphisms; in order to give well-defined deformations of the $P_i$, these transition functions must vanish along them. To prove Theorem \[main-def\], we therefore simply need to determine the locus inside $H^0(C, f^*(T_D)) \oplus k^n$ corresponding to maps which fix $f(P_i)$ and preserve the ramification at $P_i$, and similarly for $\H^1(C,T_C(-\sum_i P_i)\rightarrow f^*T_D)$. We accomplish this easily by formal local analysis. Since the first statement we are trying to prove gives a self-contained, purely local description of a subspace of $H^0(C, f^*(T_D)) \oplus k^n$, and fixing $f(P_i)$ and the ramification at $P_i$ are likewise purely local conditions, it suffices to check agreement formally locally around each $P_i$ and $f(P_i)$. Accordingly, let $s,t$ be formal coordinates at $P_i$, $f(P_i)$ respectively. We then have $f(s)=\sum_{j \geq 0} a_j s^j$ for some $a_j \in k$ with $a_j=0$ for $j<e_i$. First, a deformation of $f$ will be of the form $\tilde{f}(s)=f(s)+\epsilon \sum_{j \geq 0} b_j s^j$, with the vanishing order of the $b_j$ being the vanishing order of the section of $H^0(C, f^*(T_D))$ inducing the deformation. Since $P_i$ corresponds to $s=0$, a deformation of $P_i$ can be written simply as $\epsilon x$ for $x \in k$. If we fix both $P_i$ and $f(P_i)$, then requiring that ramification of order $e_i$ be preserved is simply equivalent to requiring that $b_j=0$ for $j < e_i$. If we fix $f(P_i)$, but allow $P_i$ to move, the condition that $f(P_i)$ is fixed is simply $\tilde{f}(\epsilon x)=0$, while the condition that $f$ remain ramified to order at least $e_i$ at $P_i$ may be expressed vanishing to order at least $e_i$ of $\tilde{f}$ when expanded around $s-\epsilon x$. Taylor expansion yields $\tilde{f}(s)= \sum_{j\geq 0} (a_j + \epsilon (b_j+(j+1)a_{j+1} x))(s-\epsilon x)^j$. We see that for the first $e_i-1$ terms to vanish, we need $b_j=0$. For the $e_i$th term, which is $j=e_i-1$, we need $b_{e_i-1}+(e_i)a_{e_i} x=0$. The condition that $\tilde{f}(\epsilon x)=0$ may be written $b_0+a_1 x=0$. In the case $e_i>1$, this is automatically satisfied, while for $e_i=1$, it is redundant with the ramification condition. Now, if $e_i a_{e_i}$ is non-zero, we see that $b_{e_i-1}$ may be chosen arbitrarily, and uniquely determines $x$, giving the classical case of the theorem, where $\delta=1$. On the other hand, if $e_i a_{e_i}=0$, then we must have $b_{e_i-1}=0$, but $x$ can be arbitrary, giving the $\delta=0$ case and completing the proof of the theorem. Given the deformation-theoretic machinery we have already recalled, the second statement of the theorem is even easier. Indeed, in our formal-local trivialization, we assume by construction that the sections also correspond to the trivial deformation, so that we are in the case above that we have fixed both $P_i$ and $f(P_i)$, where we already noted we simply find that $b_j=0$ for all $j<e_i$, which is equivalent to saying that our $0$-cochain of $f^*T_D$ must vanish to order $e_i$ at $P_i$, as desired. If $f$ is separable, the map $T_C(-\sum_i P_i) \rightarrow f^* T_D(-\sum_i e_i P_i))$ is injective, with cokernel equal to the skyscraper sheaf of length $\delta_{P_i}-(e_i-1)$ at each $P_i$, where $\delta_{P_i}$ is the order of the different of $f$ at $P_i$. The Riemann-Hurwitz formula then gives the desired value for the dimension of the deformation space. Finally, it is also clear from our construction that both statements work when some $e_i=0$ and no condition is placed on the corresponding $P_i$. Brill-Noether Theory ==================== In this section, we prove Theorem \[brill-noether\], using classical Brill-Noether theory together with Theorem \[main-def\]. Given two smooth curves $C,D$ over a finite-type $k$-scheme $S$, and integers $d,n$ and $e_1, \dots e_n$, we have a moduli scheme $MR:=MR(C,D,d,e_1,\dots e_n)$ parametrizing $(n+1)$-tuples $(f, P_1, \dots P_n)$, where $f:C \rightarrow D$ is a separable morphism of degree $d$, the $P_i$ are distinct sections of $C$, and $f$ is ramified to order at least $e_i$ at $P_i$ (and possibly elsewhere); see [@os7 Appendix]. This comes with natural forgetful morphisms $\ram: MR \rightarrow C^n$ and $\branch: MR \rightarrow D^n$ giving the portion of the ramification and branch loci of the map $f$ which is mandated by the definition of $MR$; that is to say, the $P_i$ and $f(P_i)$ respectively. If we fix a scheme-valued point $b$ in $D^n$, $\branch^{-1}(b)$ is then the locus of maps $f:C \rightarrow D$ with the specified branching above each of the $n$ points corresponding to $b$. Recall that in the case that $D=\P^1$, $MR$ admits a natural quotient scheme $\overline{MR}$ which parametrizes the appropriate linear series on $C$, together with ramification sections; that is to say, $\overline{MR}$ represents the quotient functor obtained simply by modding out by postcomposition with $\Aut(\P^1)$. This may be realized as a classical relative $G^r_d$ scheme over the base $C^n$, with prescribed ramification at the corresponding $n$ sections. Since this group action fixes $\ram$, we have that $\ram$ factors through $\overline{MR}$. We present the $g=0$ case of Theorem \[brill-noether\] first, as a simpler and more direct illustration of the general idea, using the first statement of Theorem \[main-def\] directly. We thus specialize to the case that $C=\P^1$. It is not hard to see that separable maps from $\P^1$ to itself of degree $d$ are parametrized by an open subscheme of the Grassmannian $\G(1,d)$, and a ramification condition of order $e_i$ corresponds to a Schubert cycle of codimension $e_i-1$; see, e.g., [@os7 §2]. Moreover, this description works in the relative setting, so we conclude that $\overline{MR}$ has codimension $\sum_i (e_i-1)$ in the trivial $\G(1,d)$ bundle over $C^n=(\P^1)^n$. With these observations, we may easily prove the theorem. Since $\G(1,d)$ is smooth of relative dimension $2d-2$ over $(\P^1)^n$, $\overline{MR}$ has dimension at least $n+2d-2-\sum_i (e_i-1)$, and it follows that $\dim MR = \dim \overline{MR} + \dim \Aut(\P^1) \geq n+2d+1 -\sum_i(e_i-1)$. On the other hand, by Theorem \[main-def\] if we are given an $f \in MR$ the dimension of the tangent space of its fiber of $\branch$ is $h^0(\P^1, f^* T_{\P^1} (-\sum_i (e_i-\delta_i)))+\sum_i(1-\delta_i)$ where $\delta_i=0$ if $p | e_i$ or $f$ is ramified to order higher than $e_i$ at $P_i$ and is $1$ otherwise. Since $T_{\P^1} \cong \O(2)$, we have $f^* T_{\P^1} (-\sum_i (e_i-\delta_i)) \cong \O(2d- \sum_i(e_i-\delta_i))$. If $2d-\sum_i(e_i-\delta_i)$ is negative, Riemann-Hurwitz implies that $MR$ is empty, and otherwise we find that our $h^0$ is given by $2d+1-\sum_i(e_i-\delta_i)$, and the dimension of our tangent space by $2d+1-\sum_i(e_i-1)$. Thus $MR$ has dimension at most $n+2d+1-\sum_i (e_i-1)$, and this must give its dimension precisely. The theorem then follows. We now consider the higher-genus case, assuming initially that $g \geq 2$. Instead of working over $\Spec k$, we let our base $S$ be a scheme étale over the moduli space $\M_{g,0}$ over $k$, and let $C$ be the corresponding universal curve over $S$. Even in this relative setting, if we twist by a sufficiently ample divisor $D$ on $C$ (since $g \geq 2$, we could use high powers of the anticanonical sheaf), and then impose vanishing along $D$, we can realize $\overline{MR}$ as a closed subscheme of a Grassmannian bundle over $\Pic_S(C)\times _S C^n$, with the ramification conditions corresponding to relative Schubert cycles. This construction is carried out in the more general setting of limit linear series on families of curves of compact type by Eisenbud and Harris in the proof of [@e-h1 Thm. 3.3]. The above classical description again gives a dimensional lower bound for $\overline{MR}$, this time as $(n+3g-3)+(2d-2-g)-(\sum_i(e_i-1))$, giving that $\dim MR \geq n+2g+2d-2-\sum_i(e_i-1)$. On the other hand, we see that the tangent space to a fiber of $MR$ over a point $(\P^1)^n$ (under the branch morphism composed with $S \rightarrow \Spec k$) is precisely a deformation of the corresponding curve $C_0$ with marked points $P_i$, together with the map to $\P^1$, fixing the $f(P_i)$ and the ramification conditions at the $P_i$. Since $g \geq 2$, there are no infinitesmal automorphisms to mod out by in the corresponding deformation theory problem of Theorem \[main-def\], so we find that the tangent space of the fiber is described by that theorem, and thus has dimension $2d-2+2g-\sum_i (e_i-1)$. We find as before that the dimension of $MR$ is at most, hence exactly, $n+2d-2+2g-\sum_i(e_i-1)$, and a fiber of $\overline{MR}$ over a general point of $C^n$ must have dimension $n+2d-2+2g-\sum_i(e_i-1)-(n+3g-3)-3=2d-2-g-\sum_i(e_i-1)$, as desired. We conclude with the $g=1$ case (we could handle the $g=0$ case similarly, but since we have already given a proof in that case, we will not do so). Here, we argue as when $g \geq 2$, but let $S$ be étale over $\M_{1,1}$. The Brill-Noether lower bound works as before to give the relative dimension of $2d+1-g-\sum_i(e_i-1)$ for $MR$ over $C^n$, with the ample divisor in the construction arising from our given section. Because $S$ has dimension $3g-3+1$ in this case, we find we need the fiber of $MR$ over a point $(\P^1)^n$ to have dimension $1$ greater than before. The tangent space of this fiber at a point $(C_0,\{P_i\}_i,f)$ now includes a $P_0$ with $e_0=0$, giving the extra dimension, as desired. One observes in the $g=1$ case above that even if all ramification is specified, the fiber dimension for fixed branching is 1. The reason for this is that the construction of $MR$ doesn’t see the marked point on the genus 1 curve which comes from a point of $S$, and still allows changing the ramification sections $P_1,\dots,P_n$ by automorphism of the underlying curve $C_0$. We remark that although the $g=0$ case of this theorem is extremely easy in characteristic $0$, the situation is more delicate in characteristic $p$. In particular, in the intersection of the ramification Schubert cycles frequently has an excess intersection corresponding to inseparable maps of lower degree. Furthermore, examples such as $x^{p+2}+tx^p+x$ give tamely ramified situations where all non-empty fibers of $\ram$ have greater than the expected dimension; in such situations, $\ram(MR)$ necessarily fails to dominate $(\P^1)^n$ even though the expected dimension is non-negative. However, the argument of Theorem \[brill-noether\] implies that this can never happen for $\branch(MR)$. In the situation of Theorem \[brill-noether\], but allowing the $P_i$ and $C$ to vary as in the proof, if $f\in MR$ is any point with any neighborhood $U \subset MR$, then $U$ dominates $(\P^1)^n$ under the $\branch$ morphism, with all fibers smooth of dimension $2d-2+2g-\sum(e_i-1)+\epsilon_g$, where $\epsilon_g$ is the dimension of the infinitesmal automorphism space of a curve of genus $g$ (i.e., 3 if $g=0$, 1 if $g=1$, and 0 otherwise). We saw in the proof of Theorem \[brill-noether\] that if $MR$ is non-empty, it is pure of dimension $n+2g+2d-2+\epsilon_g-\sum_i(e_i-1)$, and that the tangent space at any point in any fiber of the $\branch$ morphism has dimension precisely $2d-2+2g+\epsilon_g-\sum_i(e_i-1)$. The corollary follows. Wild Ramification ================= We conclude with some largely elementary remarks on wild ramification, again in the situation that $C=D=\P^1$. The first observation is that by Riemann-Hurwitz in characteristic $p$, if any $e_i$ are wild, in order to have separable maps with the desired ramification, we must have $2d-2>\sum_i(e_i-1)$. The codimension count of the previous section then implies that the separable maps with at least the specified ramification will necessarily form an infinite family. Thus, the fact that wildly ramified maps come in infinite families is elementary from the perspective of linear series. With the exception of one application of Theorem \[brill-noether\] to prove Theorem \[wild\], all our observations will be of a completely elementary nature, but we hope they may shed some light on the behavior of wildly ramified maps. The primary assertion follows from Theorem \[brill-noether\] together with the observation that under our hypotheses, the locus of maps $f$ with exactly the specified ramification is open in the locus of maps with at least the specified ramification. Indeed, having ramification exactly $e_i$ at $P_i$ is always open, since ramification can only decrease under deformation. On the other hand, by Riemann-Hurwitz a deformation cannot have additional ramification away from the $P_i$, since the different at each wild $e_i$ is necessarily at least $e_i$. The second assertion will be a special case of the following proposition. Considering Theorem \[brill-noether\] and the preceding argument, one might be led to expect that the space of wildly ramified maps having higher different at the wild points would have higher dimension. However, this is not necessarily the case. In the argument for openness above, we use minimality of the different in a key way. Indeed, if one deforms a map with greater than minimal discriminant, a new ramification point specializing to the wild point can appear, as illustrated (indirectly) by the following two propositions. \[foo\] Let $d,n$ and $e_1, \dots, e_n$ be positive integers, with the $e_i$ less than $p$, and $2d-2=\sum_i (e_i-1)$. Also, let $P_1, \dots, P_n$ be distinct points on $\P^1$. Then there exists a separable map of degree $d$ from $\P^1$ to itself, ramified to order $e_i$ at $P_i$, if and only if there exists a separable map of degree $d+p-e_1$, ramified to order $e_i$ at $P_i$ for $i>1$, and order $p$ at $P_1$. The dimension of the space wild maps in this situation is $1$ more than the dimension of the space of tame ones. We may assume that $P_1 = \infty$, and $f(\infty)=\infty$. Then we can go back and forth between the wild and tame cases simply by adding appropriate multiples of $x^p$, since the fact that $e_i <p$ for $i>1$ implies that the ramification away from $\infty$ will remain unchanged. We also use that the different in the wild case at $\infty$ is less than $2p$, so that if we subtract a multiple of $x^p$ from a wild map, the degree of the numerator cannot drop below the degree of the denominator. The difference in dimension comes from the fact that the multiple of $x^p$ added to obtain a wild map can be arbitrary. Let $d,n$ and $e_1, \dots, e_n$ be positive integers, with $e_1=d=p$, $e_i$ less than $p$ for $i>1$, and $2d-2>\sum_i (e_i-1)$. Also, let $P_1, \dots, P_n$ be distinct points on $\P^1$. Then the space of separable maps of degree $d$ from $\P^1$ to itself, ramified to order $e_i$ at $P_i$ and unramified elsewhere, and taken modulo automorphism of the image, is non-empty of dimension $1$. Without loss of generality, we can assume $P_1=\infty$ and $f(\infty)=\infty$; then $f$ is given by a polynomial of degree $p$. Since $e_i<p$ for $i>1$, the ramification conditions for $i>1$ determine the derivative of $f$ (up to scaling). On the other hand, since $\sum_{i>1}(e_i-1)<p-1$, an $f$ with the desired derivative always exists. The space is $1$-dimensional because the $x^p$ term may be scaled independently from the lower-order terms. Up until now, all of our examples have suggested that the dimension of a wildly ramified family will always be equal to the number of wildly ramified points. However, the following example shows that this is not always the case, even for families existing for general $P_i$. The family $\frac{x^{2p}+t_1x^{p+1}+t_2}{x^p+t_1 x}$ for $t_1,t_2$ non-zero is a two-dimensional family of rational functions (modulo automorphism of the image) ramified to order $p$ at infinity, and unramified elsewhere. One can try to say more about the case with a single wildly ramified point by generalizing the argument of Proposition \[foo\], inductively inverting as necessary and subtracting off inseparable polynomials. However, there are subtleties to be aware for this sort of argument. In particular, neither the tame ramification indices nor the dimension of the tame family obtained in this process will be determined by the ramification indices and degree of the wildly ramified map. Indeed, the maps $\frac{x^5(x^{10}+x^7-2x)+1}{x^{10}+x^7-2x}$ and $\frac{x^5(x^5(x^5+x^4-x^3+2x)+x^2+2x+1)+x^5+x^4-x^3+2x} {x^5(x^5+x^4-x^3+2x)+x^2+2x+1}$ in characteristic $5$ are both of degree $15$, ramified to order $5$ at infinity, and simply ramified at the $6$th roots of unity, but the tame functions they reduce to are $x^7-2x$ and $\frac{x^2+2x+1}{x^5+x^4-x^3+2x}$ respectively; the former moves in a one-dimensional family, while the latter doesn’t. Nonetheless, it is interesting to note that each of the wild maps moves in a $2$-dimensional family. However, even in the tame case the situation of one index being at least $p$ while the others are less than $p$ is pathological, so it is not clear how much general intuition one should attempt to draw from this case. That said, it is interesting that at least for this example, it seems the dimension in the wild case is in fact more uniform than the dimension in the tame case. This suggests that an approach other than reducing to the tame case is likeliest to be productive for analyzing the dimension of families of wildly ramified maps. [^1]: This paper was partially supported by fellowships from the National Science Foundation and the Japan Society for the Promotion of Science.

$\quad$ [**Metallic Edge States in Zig-Zag\ Vertically-Oriented MoS$_2$ Nanowalls**]{}\ M. Tinoco$^{1,2,\dagger}$, L. Maduro$^{1,\dagger}$ and S. Conesa-Boj$^{1,*}$ * $^{1}$ Kavli Institute of Nanoscience, Delft University of Technology,\ 2628CJ Delft, the Netherlands.*  $^{2}$ ICTS – Centro Nacional de Microscopía Electrónica,\ Universidad Complutense, 28040 Madrid, Spain. \ [**Abstract**]{} The remarkable properties of layered materials such as MoS$_2$ strongly depend on their dimensionality. Beyond manipulating their dimensions, it has been predicted that the electronic properties of MoS$_2$ can also be tailored by carefully selecting the type of edge sites exposed. However, achieving full control over the type of exposed edge sites while simultaneously modifying the dimensionality of the nanostructures is highly challenging. Here we adopt a top-down approach based on focus ion beam in order to selectively pattern the exposed edge sites. This strategy allows us to select either the armchair (AC) or the zig-zag (ZZ) edges in the MoS$_2$ nanostructures, as confirmed by high-resolution transmission electron microscopy measurements. The edge-type dependence of the local electronic properties in these MoS$_2$ nanostructures is studied by means of electron energy-loss spectroscopy measurements. This way, we demonstrate that the ZZ-MoS$_2$ nanostructures exhibit clear fingerprints of their predicted metallic character. Our results pave the way towards novel approaches for the design and fabrication of more complex nanostructures based on MoS$_2$ and related layered materials for applications in fields such as electronics, optoelectronics, photovoltaics, and photocatalysts.\  $^{*}$Corresponding author: [s.conesaboj@tudelft.nl ](mailto:s.conesaboj@tudelft.nl )  $^{\dagger}$ Equal contribution. Introduction {#introduction .unnumbered} ============ The ability of crafting new materials in a way that makes possible controlling and enhancing their properties is one of the main requirements of the ongoing nanotechnology revolution [@ref1; @ref2]. In this context, a family of materials that has attracted intense attention recently are 2D layered materials, such as MoS$_2$, which belong to the group of transition metal dichalcogenides (TMDs). These materials have been extensively studied due to their promising electrical and optical properties [@ref3; @ref4; @ref5; @ref6]. A defining feature of TMDs is that they exhibit a lack of inversion symmetry, which leads to the appearance of a variety of different edge structures. The most common of these, consisting on dangling bounds, are the armchair (AC) and the zig-zag (ZZ) edge structures. Of particular relevance in this context, the electronic properties of MoS$_2$ have been predicted to be affected by the presence of the different edge structures in rather different ways. For instance, the AC edges have been predicted to be semiconducting, while the ZZ edges should exhibit instead metallic behavior [@ref7; @ref8; @ref9; @ref10]. Moreover, [*ab-initio*]{} theoretical calculations predict that these metallic states at the edges of MoS$_2$ could lead to the formation of plasmons [@ref11]. Beyond this tuning of electronic properties, other attractive applications of these active edge sites arise in photocatalysis, such as their use in hydrogen evolution reactions (HER) [@ref12; @ref13; @ref14; @ref15]. With these motivations, it is clear that the design and fabrication of MoS$_2$ nanostructures with morphologies that maximize the number of exposed active edge sites is a key aspect for further improvements in terms of applications. In this respect, significant efforts have been pursued to realize the systematic bottom-up growth of vertically-oriented standing MoS$_2$ layers. This configuration leads to the edge sites facing upwards, therefore maximizing the number of exposed edge sites as compared with the more common horizontal configuration, where its basal plane lies parallel to the substrate [@ref16; @ref17; @ref18; @ref19; @ref20; @ref21; @ref22; @ref23]. However, this bottom-up approach is hampered by a lack of reproducibility due to the complexities of the growth mechanism. Another limitation within the bottom-up approach is that the specific type of edges exposed cannot be selectively grown. Ideally, one would like to combine the best of both worlds. On the one hand, it is important to be able to controllably grow MoS$_2$ nanostructures that exhibit the largest possible surface area of edge structures, as it is achieved by the bottom-up strategy summarized above. On the other hand, one would also like to be able to select the specific type of edge sites exposed, in particular, by selecting whether these correspond to AC or to ZZ edges. Therefore, the main goal of this work is to bridge these two requirements by realizing a novel approach to the growth of vertically-oriented standing MoS$_2$ layers with full control on the nature of the exposed edge sites. To achieve this goal, here we adopt a well-stablished top-down approach based on focus ion beam (FIB) in a way that allows us to selectively pattern both types of edges (AC and ZZ) within out-of-plane (vertical) MoS$_2$ nanostructures. In the context of patterning layered materials, the usefulness of FIB has been repeatedly demonstrated. [@ref24; @ref25; @ref26]. By means of this technique, we are able to selectively maximize the density of exposed edge sites while controlling their type. Subsequently, by combining high-resolution transmission electron microscopy (TEM) with electron energy-loss spectroscopy (EELS) measurements, we are able to confirm not only the crystallographic nature of both the AC and ZZ MoS$_2$ surfaces, but also we can demonstrate that, despite the roughness and imperfections induced during the fabrication procedure, the ZZ MoS$_2$ nanostructures clearly exhibit a metallic character, in agreement with the theoretical predictions from [*ab-initio*]{} calculations [@ref11]. The results of this work will open new opportunities for nanoengineering the edge type in MoS$_2$ nanostructures as well as in related layered materials, paving the way towards novel exciting opportunities both for fundamental physics and technological applications in electronics, optoelectronics, photovoltaics, and photocatalysts. Results {#results .unnumbered} ======= From crystal structure considerations, the possible angles between adjacent flat edges within MoS$_2$ flakes should be multiples of 30. Specifically, the expected angles between adjacent AC and ZZ edge structures in a MoS$_2$ flake such as that of [**Fig. \[fig1\]a**]{} should be 30, 90, and 120, as illustrated in [**Fig. \[fig1\]b**]{}. Based on this information, we have designed the orientation of the different areas of the MoS$_2$ flake that subsequently will be patterned. In this way, we can ensure the full control over the resulting specific edge crystallographic orientation. [**Figs. \[fig1\]c**]{} and[** \[fig1\]d**]{} display a scanning electron microscopy (SEM) image of the MoS$_2$ flake that has been used for the fabrication of the nanostructures, taken before and after the milling respectively. Before the milling is performed, a protective metallic layer of tungsten (W) with a thickness of 500 nm was deposited on top of the selected areas of the MoS$_2$ flake. Subsequently, we performed a series of milling and cleaning processes in order to construct the vertically-aligned MoS$_2$ nanostructures. [**Fig. \[fig1\]d**]{} displays three ordered vertically-oriented patterned arrays of MoS$_2$ nanostructures, which in the following are denoted as nanowalls (NWs). Two of these sets of nanowalls are oriented perpendicularly with respect to each other, guaranteeing that this way one of two arrays will correspond to AC (ZZ) NWs while the other array will correspond instead to the complementary ZZ (AC) ones. These NWs are found to exhibit a uniform thickness being ($89 \pm 5$) nm (central array in [**Fig. \[fig1\]d**]{}) and ($68 \pm 5$) nm (rightmost array in [**Fig. \[fig1\]d**]{}). Note that the left-most array was fabricated without the protective metal layer. To further examine the crystallographic nature of the resulting vertical MoS$_2$ nanostructures, transmission electron microscopy (TEM) studies were carried out. For these studies, we lifted out two of the MoS$_2$ NWs from the two different patterned NWs arrays using a micromanipulator. Subsequently, the nanostructure was mounted onto a TEM half-grid. This whole procedure takes place within the FIB chamber. ![(a) Atomic model of a MoS$_2$ flake viewed along the \[0001\] direction, where we indicate the corresponding zig-zag (ZZ) and armchair (AC) edges. (b) From geometric considerations, we can determine the possible values that the angles between adjacent AC and ZZ edges should take; (c) and (d) SEM micrographs of the MoS$_2$ flake used for patterning the nanowalls, taken before and after the milling respectively. In (d), three different set of arrays can be observed. The left-most array was fabricated without the protective metal layer, while the other arrays used instead this protective metal layer. \[fig1\] ](Figure1_Overview_ZZ-AC.pdf){width="80.00000%"} [**Fig. \[fig2\]a**]{} displays a high-angle annular dark-field scanning transmission electron microscopy (HAADF-STEM) image of a selected region of the ZZ MoS$_2$ nanowall, bracketed between the Si substrate and the metallic protective layer. [**Fig. \[fig2\]b**]{} shows the corresponding chemical compositional of this nanowall obtained by means of energy dispersive X-ray (EDS) spectroscopy measurements. From the EDS map, the different chemical components of the NWs can be clearly distinguished: the MoS$_2$ segment, embedded within the protective metal layer tungsten (W), and the silicon (Si) substrate. From the crystalline structure studies carried out by means of high-resolution TEM measurements ([**Fig. \[fig3\]**]{}), we are able to confirm the specific edge site configuration for the two NW arrays. [**Figs. \[fig3\]a**]{} and[** \[fig3\]b**]{} display the results of the TEM measurements on the AC and ZZ MoS$_2$ surfaces respectively. By comparing the two crystallographic orientations, AC and ZZ, we can observe the differences between the atomic arrangement of each surface, which are consequently characterized by different fast Fourier transforms (FFTs) (shown in the insets of [**Figs. \[fig3\]a**]{} and[** \[fig3\]b**]{}). From these results, it is clearly noticeable the excellent agreement between the experimental FFTs obtained from the TEM measurements and the corresponding ones calculated theoretically in terms of the expected atomic configuration (shown in [**Figs. \[fig3\]c**]{} and[** \[fig3\]d**]{}). These results provide direct confirmation that these vertically-oriented MoS$_2$ nanowalls are in fact exposing ZZ and AC edge terminations, therefore validating our fabrication strategy. Fingerprinting the edge-type nature of MoS$_2$ nanowalls {#fingerprinting-the-edge-type-nature-of-mos_2-nanowalls .unnumbered} --------------------------------------------------------- In order to pin down the local electronic properties of the AC and ZZ MoS$_2$ NWs, electron energy-loss spectroscopy (EELS) measurements have been carried out in a scanning transmission electron microscope (STEM). In [**Fig. \[fig4\]**]{} we show the energy-loss spectra corresponding to both the AC and ZZ surfaces, taken at different points along the length of the nanowall. As it can be observed in the two sets of EELS spectra, the MoS$_2$ bulk plasmon signal appears at 23.4 eV in both samples with similar intensities and general shape, in agreement with previous analyses  [@ref27; @ref28]. Nevertheless, the MoS$_2$ surface plasmon peak, present at 15.2 eV, turns out to appear only on a restricted subset of the spectra of the ZZ-nanowalls. Considering that the fabricated AC-nanowalls are thinner than the ZZ-terminated ones, the presence of the surface MoS$_2$ plasmon on the ZZ-nanowalls cannot be attributed to a lower thickness of the sample. Therefore, the origin of this peak should be caused by another phenomenon. In that respect, it is important to notice that the MoS$_2$ surface plasmon peak appears and disappears in a periodic manner, depending on the specific position along the nanowall where the EELS spectrum is collected. It is found that the positions which correspond to local maxima of the intensity associated to this surface plasmon peak are separated by around 12 nm between each other. This behavior can be attributed to the presence of metallic surface plasmon polaritons (SPP), which are planar waves appearing at the interfaces between a metal and a dielectric material under some external excitation, such as an electron beam [@ref29]. That could correspond to the oscillatory character present in our EELS spectra. Therefore, from this analysis, we can conclude that the ZZ MoS$_2$ NWs surfaces present a clear metallic character. On the contrary, the AC MoS$_2$ NWs do not exhibit such metallic behavior. With this result, we can hereby confirm that the ZZ MoS$_2$ NWs are dominantly enclosed by zig-zag edges structures. It is also worth mentioning here that no signal arising from neither the metal layer nor the Ga used for the FIB milling were present at any of the acquired EELS spectra, indicating that the possible contamination from Ga in the nanowalls is non-existing. Discussion {#discussion .unnumbered} ========== In order to further validate the onset of the metallic behavior observed in the ZZ MoS$_2$ nanowalls (NWs), we calculated the corresponding density of states (DOS) by means of [*ab-initio*]{} calculations in the framework of density functional theory (DFT). The van der Waals (vdW) interactions characteristic of MoS$_2$ were incorporated by using the nonlocal vdW functional model [@ref30] as implemented in the WIEN2k code (see Methods for further details). We modeled the ZZ MoS$_2$ nanowall by constructing a 1x3x1 supercell of MoS$_2$, as shown in [**Fig. \[fig5\]a**]{}. In order to minimize the interactions between periodic images due to the 3D boundary conditions, we introduced a vacuum layer such that the distance between periodic images is 17.170 . The resulting calculated total DOS for the ZZ MoS$_2$ NW is displayed in [**Fig. \[fig5\]b**]{}. A clear absence of a gap in the DOS near the Fermi energy is observed, which implies a finite probability (11.65 states/eV) for states just below and above the Fermi energy level being populated, highlighting the metallic behavior of the ZZ MoS$_2$ NWs. [**Fig. \[fig5\]b**]{} (middle panel) also displays the individual contributions of the 4d states of Mo atoms located at the surface of the ZZ MoS$_2$ nanowall. These 4d states of Mo are also observed to cross over the Fermi energy (1.35 states/eV), contributing therefore to the metallic character of the ZZ MoS$_2$ nanowall. The individual contribution of the 3p states of S atoms located at the surface of the NW turns out to be much smaller from the DFT calculation, 0.08 states/eV. Therefore, the dominant contribution to the metallic character of ZZ MoS$_2$ nanowalls can be confidently attributed to Mo-4d states of MoS$_2$. In this work, we have presented a novel approach for the top-down fabrication of ordered vertically-oriented MoS$_2$ nanostructures (denoted as nanowalls) which makes possible to achieve at the same time, a large density of exposed active edge sites while also to controllably select whether these are of the AC or ZZ types. The crystallographic nature of the exposed surfaces has been validated by means of high-resolution TEM measurements. We have also studied the local electronic properties of these NW surfaces by means of EELS, finding direct evidence of the metallic character of the ZZ surfaces as indicated by the presence of MoS$_2$ surface plasmon peak. The metallic nature of the ZZ MoS$_2$ nanowalls can be exploited to open new opportunities for nanoengineering the edge type in MoS$_2$ nanostructures as well as in related layered materials. This would allow new exciting opportunities both for fundamental physics and technological applications in electronics, optoelectronics, photovoltaics, and photocatalysts. Methods {#methods .unnumbered} ------- [*Focus ion beam patterning for the fabrication of edge-controlled MoS$_2$ nanowalls*]{}. MoS$_2$ bulk crystal obtained from Alfa Aesar (99.999% purity) was mechanical exfoliated with Poly-Di-Methyl-Siloxaan (PDMS) and then transferred to a SiO$_2$/Si substrate. The MoS$_2$ nanostructures were milled using a FEI Helios G4 CX focus ion beam. The ion milling procedure was carried out using a very low energy electron beam of 15 kV, and an ion beam of 2 pA. Before the milling procedure was carried out, a protective metal (W) layer of 500 nm of thickness was deposited on top the selected areas.\ [*Characterization techniques*]{}. Transmission Electron Microscopy (TEM) measurements were carried out in a Titan Cube microscope using an acceleration voltage of 300 kV. Its spatial resolution at Scherzer defocus conditions is 0.08 nm in the High-Resolution Transmission Electron Microscopy (HRTEM) mode, whilst the resolution is around 0.19 nm in the HAADF-STEM (High Angle Annular Dark Field – Scanning Transmission Electron Microscopy) mode. Electron Energy Loss Spectroscopy (EELS) experiments were carried out using a Gatan Imaging Filter (GIF) spectrometer, employing a collection semi-angle of 2.95 mrad, a convergence semi-angle of 14 mrad, and an aperture of 2 mm. The energy resolution obtained by using these parameters in EELS was 0.9 eV, with an exposure time of 0.1 s/spectrum and an energy dispersion of 0.1 eV/channel.\ [*First-principle calculations*]{}. The density of states (DOS) calculations were performed using both linearized augmented plane wave (LAPW) and local orbitals (LO) methods implemented in the WIEN2k package [@ref31]. The nonlocal van der Waals [@ref32; @ref33] (vdW) interactions used for the DOS calculations uses optB88 [@ref34] for the exchange term, the local density approximation [@ref35] (LDA) for the correlation term, and the DRSLL kernel for the non-local term [@ref36]. For the non-local vdW integration the cut-off density rc was set to 0.3 bohr$^{-3}$, while the plane wave expansion cut-off Gmax was set to 20 bohr$^{-1}$. No spin polarization was considered. The lattice parameters were found by volume and force optimization of the supercell, such that the force on each atom was less than 1.0 mRy/bohr. The total energy convergence criteria was set to be 0.1 mRy between self-consistent field (SCF) cycles, while the charge convergence criteria was set to 0.001e, with e the elementary unit charge. The core and valence electron states were seperated by an energy gap of -6.0 Ry. Furthermore, the calculations used an R\*kmax of 6.0, where R is the radius of the smallest Muffin Tin sphere, and kmax is the largest k-vector. The first Brillouin zone for the lattice parameter calculations was sampled with 100 k-points using the tetrahedon method of Blöchl et al. [@ref37], which corresponds to 21 k-points in the irreducible Brillouin zone. With the above parameters the optimized lattice parameters were $a = 3.107 {\textup{\AA}}$ and $c = 12.087 {\textup{\AA}}$, which are in good agreement with the experimental values $a = 3.161 {\textup{\AA}}$ and $c = 12.295 {\textup{\AA}}$  [@ref38]. The DOS was calculated with a denser k-point sampling of the Brillouin zone consisting of 1600 k-points, corresponding to 630 k-points in the irreducible Brillouin zone. Acknowledgements {#acknowledgements .unnumbered} ---------------- M. T and L. M. acknowledge support from the Netherlands Organizational for Scientific Research (NWO) through the NanoFront program. S. C.-B. acknowledge financial support from ERC through the Starting Grant “TESLA” grant agreement number 805021. Contributions {#contributions .unnumbered} ------------- M. T. and L. M. prepared the samples. M. T. performed the FIB milling and the TEM measurements. M. T. analyzed the TEM data. M. T. and S. C. B. prepared the figures and the discussion of the results. L. M. performed the structure modeling and DFT calculations. S.C. B. designed and supervised the experiments. All the authors contributed to the writing of the manuscript. Competing interests {#competing-interests .unnumbered} ------------------- The authors declare that they have no competing interests. Corresponding author {#corresponding-author .unnumbered} -------------------- Correspondence about this work should be sent to Sonia Conesa Boj.

--- abstract: 'Quantum interference between one- and two-photon absorption pathways allows coherent control of interband transitions in unbiased bulk semiconductors; carrier population, carrier spin polarization, photocurrent injection, and spin current injection can all be controlled. We calculate injection spectra for these effects using a $14 \times 14$ $\mathbf{k}\cdot \mathbf{p}$ Hamiltonian including remote band effects for five bulk semiconductors of zinc-blende symmetry: InSb, GaSb, InP, GaAs, and ZnSe. Microscopic expressions for spin-current injection and spin control accounting for spin split bands are presented. We also present analytical expressions for the injection spectra derived in the parabolic-band approximation and compare these with the calculation nonperturbative in $\mathbf{k}$.' author: - 'R. D. R. Bhat' - 'J. E. Sipe' date: 'December 28, 2005' title: 'Calculations of two-color interband optical injection and control of carrier population, spin, current, and spin current in bulk semiconductors.' --- Introduction ============ When a bulk semiconductor is simultaneously irradiated by an optical field and its phase-coherent second harmonic, quantum interference between one- and two-photon absorption pathways enables excitation of carrier distributions with interesting properties [@Atanasov96; @Fraser99PRL; @BhatSipe00; @Stevens_pssb]. Such excitation, even without an external bias, can produce ballistic photocurrents [@Hache97], spin-polarized currents [@StevensJAP], and pure spin currents [@StevensPRL03; @HubnerPRL03]. Characteristically of quantum interference, these currents are sensitive to the phases of the two optical fields. In noncentrosymmetric semiconductors, the phases can also be used to control the total population of photoexcited carriers [@Fraser99PRL], and the net carrier spin polarization [@Stevens_pssb; @Stevens05_110]. Which of these effects occur depends on the polarization states of the fields. These are examples of “$n+m$” coherent control schemes, in which a two-color light field controls a physical or chemical process by interference of $n$- and $m$-photon transitions [@Manykin67; @ShapiroBrumer97; @Gordon99]. In semiconductors, “1+2” excitation has been discussed for impurity-band absorption [@Entin89], free carrier absorption [@Baskin88; @Entin89; @Alekseev99], quantum wells [@Dupont95; @Potz98; @Khurgin98; @Najmaie03; @Marti04; @MartiReview04; @RumyantsevIQEC_04; @Najmaie05a; @Duc05], and quantum wires [@Marti05], but our interest here is “1+2” coherent control of interband transitions in unbiased bulk semiconductors [@Atanasov96; @Fraser99PRL; @BhatSipe00; @Stevens_pssb; @vanDrielSipeReview01; @StevensReview04]. Such experiments have been performed with either (a) two fields, typically short pulses, one the generated second harmonic of the other [@Cote99; @Cote03; @Fraser03; @Fraser99PRL; @HubnerPRL03; @Hache97; @HacheIEEE98; @Kerachian04; @Roos03; @StevensReview04; @Stevens_pssb; @StevensJAP; @Stevens05_110; @StevensSST04; @StevensJAP03; @StevensPRL03], or (b) a single ultrashort pulse having at least an octave bandwidth [@Fortier04; @Roos05]. Previous microscopic calculations of “1+2” processes in bulk semiconductors fall into two categories: *ab initio* density functional methods have been used for current injection [@Atanasov96] and population control [@Fraser99PRL], while simple analytical band models perturbative in $\mathbf{k}$ (with at most eight spherical, parabolic bands) have been used for current injection [@Atanasov96; @Sheik-Bahae99; @BhatSipe00; @Kral00; @BhatSipeExcitonic05] and spin-current injection [@BhatSipe00]. The former are best suited for excess energies on the order of eVs, while the latter are only valid for excitation close to the band edge and cannot be applied to population and spin control, which vanish in such centrosymmetric models. In this article, we calculate “1+2” processes using an intermediate model that diagonalizes the $\mathbf{k}\cdot \mathbf{p}$ Hamiltonian in a basis of 14 $\Gamma$-point states with remote band effects included perturbatively. The model contains empirically determined parameters [@PZ96; @WinklerBook]. Fourteen-band models (also called five-level models) have been used to calculate band structures [@Rossler84; @PZ90; @MayerRossler91; @MayerRossler93b; @PZ96], linear [@MayerRossler93; @BhatPRL05] and non-linear [@HW94; @Hutchings95; @Hutchings97; @Bhat_TPS_05] optical properties, and spin decoherence properties [@Lau01; @Lau_condmat] of GaAs and other semiconductors. Winkler has recently reviewed 14-band models [@WinklerBook]. The model is nonperturbative in $\mathbf{k}$ and includes nonparabolicity, warping, spin-splitting, and interband spin-orbit coupling. We apply the 14-band model to the zinc-blende semiconductors InSb, GaSb, InP, GaAs, and ZnSe. We compare these results with analytic expressions derived in the parabolic-band approximation (PBA) based on an expansion in $\mathbf{k}$ about the $\Gamma$ point of $\mathbf{v}_{n,m} (\mathbf{k})$, which is the matrix element governing optical transitions. A one-photon transition is called “allowed” if the zeroth-order term in its expansion is nonzero, and called “forbidden” otherwise. Two-photon transitions have two velocity matrix elements, and thus have a hyphenated label depending on the lowest-order terms in the expansions for each matrix element. For example, if both matrix elements are independent of $\mathbf{k}$ to lowest order, the two-photon transition is called “allowed-allowed”. For current injection and spin-current injection, we use expressions derived previously with an eight band model [@BhatSipe00; @BhatSipeExcitonic05]. For population control and spin control, we derive expressions with the 14-band model. The comparison between the PBA expressions and the numerical calculation establishes an important microscopic difference between current and spin-current control on the one hand, and population and spin control on the other hand. Close to the band-gap, the former result from the interference of allowed one-photon transitions and allowed-forbidden two-photon transitions, whereas the latter result from the interference of allowed one-photon transitions and allowed-allowed two-photon transitions. This difference was posited previously based on heuristic arguments [@Fraser03; @StevensReview04]. Most of the early theory on semiconductor “1+2” processes processes conceptually separated the optical injection of densities and currents from the relaxation and transport of these quantities. We follow this approach, and in this article, focus on microscopic calculations of the optical injection. We note that relaxation and transport have been studied with an effective circuit model [@HacheIEEE98; @Roos05], hydrodynamic equations [@Atanasov96; @Cote03], Boltzmann transport in the relaxation time approximation [@HubnerPRL03], a non-equilibrium Green function formalism [@Kral00], and the semiconductor Bloch equations [@Marti04; @MartiReview04; @RumyantsevIQEC_04; @Duc05]. We model the optical field as a superposition of monochromatic fields of frequency $\omega $ and $2\omega $: $$\mathbf{E}(t) = \mathbf{E}_{\omega} \exp (-i\omega t)+\mathbf{E}_{2\omega}\exp (-i2\omega t)+c.c. \label{eq:Efield}$$ and we sometimes write $\mathbf{E}_{\omega / 2 \omega} = E_{\omega / 2 \omega} \mathbf{e}_{\omega / 2 \omega}$ and $E_{\omega / 2 \omega} = |E_{\omega / 2 \omega}| \exp (i \phi_{\omega / 2 \omega})$. We describe the fourteen-band model in Section \[sec:Model\], and use it to study “1+2” current injection in Section \[sec:Current\], “1+2” spin-current injection in Section \[sec:SpinCurrent\], “1+2” population control in Section \[sec:Population\], and “1+2” spin control in Section \[sec:Spin\]. We calculate the injection of each “1+2” process using microscopic expressions derived using velocity gauge ($\mathbf{A}\cdot\mathbf{v}$) coupling in the long wavelength approximation, treating the field perturbatively in the Fermi’s golden rule limit, and using the independent-particle approximation [@Atanasov96; @Fraser99PRL; @BhatSipe00; @footnoteIPAExcitonic]. For spin-current injection and spin control, we use microscopic expressions that include the coherence between spin-split bands. In Appendix \[App:kdepSO\], we justify the neglect of $\mathbf{k}$-dependent spin-orbit coupling. The parabolic-band approximation results are derived and discussed in Appendix \[App:PBA\_a-a\], and compared with the numerical calculations in Sections \[sec:Current\]–\[sec:Spin\]. We summarize and conclude in Section \[sec:Summary\]. Model\[sec:Model\] ================== The fourteen-band model Hamiltonian, which includes important remote-band effects to order $k^{2}$, and which we denote $H_{14}$, is given explicitly by Pfeffer and Zawadski [@PZ96; @footnotePZtypo]. The fourteen bands (counting one for each spin), are shown in Fig. \[fig:14BandDiagram\]. They comprise six valence bands (two each for split-off, heavy and light holes) and eight conduction bands (the two $s$-like ones at the band edge, and the six next lowest ones which are $p$-like). We now briefly review the derivation of $H_{14}$. ![A schematic diagram of the fourteen-band model, indicating band abbreviations (left), energies (center), symmetry of the $\Gamma$-point states (right), and one- and two-photon transitions. $\Gamma_{6}$, $\Gamma_{7}$, and $\Gamma_{8}$ indicate irreducible representations of the $T_{d}$ double group, whereas $\Gamma_{1}$ and $\Gamma_{4}$ indicate irreducible representations of the $T_{d}$ point group. Note that spin-splitting, nonparabolicity, and warping are not shown in this diagram.[]{data-label="fig:14BandDiagram"}](fbm_14BandDiagram) The one-electron field-free Hamiltonian is $H=H_{0}+H_{SO}$, where $H_{0}=p^{2}/\left( 2m\right) +V$, the potential $V\left( \mathbf{r}\right) $ has the symmetry of the crystal, and the spin-orbit interaction $H_{SO}$ is $$H_{SO}=\frac{\hbar }{4m^{2}c^{2}}\bm{\sigma }\cdot \left( \bm{\nabla } V\times \mathbf{p}\right) ,$$ where $\bm{\sigma }$ is the dimensionless spin operator, $\bm{\sigma }=2 \mathbf{S}/\hbar $. Note that relativistic corrections proportional to $\left| \bm{\sigma }\times \bm{\nabla }V\right| ^{2}$ have been neglected [@LaxBook]. The eigenstates of $H$ are Bloch states $\left| n\mathbf{k}\right\rangle $ with energy $\hbar \omega _{n}\left( \mathbf{k}\right) $. The associated spinor wave function $\phi _{n\mathbf{k}}\left( \mathbf{r}\right) \equiv \langle \mathbf{r} | n\mathbf{k} \rangle $ can be written $\phi _{n\mathbf{k}}\left( \mathbf{r}\right) =u_{n\mathbf{k}}\left( \mathbf{r}\right) \exp \left( i\mathbf{k}\cdot \mathbf{r}\right) $, where the spinor functions $u_{n\mathbf{k}}\left( \mathbf{r}\right) $ have the periodicity of the crystal lattice. We use the notation $\left| \overline{n \mathbf{k}}\right\rangle $ to denote the kets for the $u$-functions; i.e. $ u_{n\mathbf{k}}\left( \mathbf{r}\right) = \langle \mathbf{r} | \overline{n\mathbf{k}} \rangle $. Note that $\left| \overline{n \mathbf{k}}\right\rangle =\exp {\left( -i\mathbf{k}\cdot \mathbf{r}\right) } \left| n\mathbf{k}\right\rangle $. The Hamiltonian for the $u$-function kets, known as the $\mathbf{k}\cdot \mathbf{p}$ Hamiltonian, is [@LaxBook; @YuCardonaChapter2] $$H_{\mathbf{k}}=e^{-i\mathbf{k}\cdot \mathbf{r}}He^{i\mathbf{k}\cdot \mathbf{r}}=H+\frac{\hbar ^{2}k^{2}}{2m}+\hbar \mathbf{k}\cdot \mathbf{v},$$ where the velocity operator $\mathbf{v}\equiv \left( i/\hbar \right) \left[ H,\mathbf{r}\right] $ is $$\mathbf{v}=\frac{1}{m}\mathbf{p}+\frac{\hbar }{4m^{2}c^{2}}\left( \bm{\sigma }\times \bm{\nabla }V\right) . \label{eq:v_operator}$$ The second term in $\mathbf{v}$, the anomalous velocity, which leads to $\mathbf{k}$-dependent spin-orbit coupling in $H_{\mathbf{k}}$, can be neglected for the processes we consider as shown in Appendix \[App:kdepSO\]; in the rest of this article, we assume that it vanishes. The states $\left| n,\mathbf{k=\mathbf{0}}\right\rangle $ are a complete set of eigenstates for the Hamiltonian $H$ on the space of cell-periodic functions. Thus cell-periodic eigenstates of $H_{\mathbf{k}}$ can be expanded in the infinite set of states $\left| n,\mathbf{k=\mathbf{0}}\right\rangle $. The “bare” fourteen-band model truncates this expansion to a set of fourteen states, corresponding to the fourteen bands closest in energy to the fundamental band gap at the $\Gamma$ point [@PZ90]. In a semiconductor of zinc-blende symmetry, the states $\left\{ \left| n, \mathbf{k=\mathbf{0}} \right\rangle |n=1..14\right\}$ are conveniently expanded in the eigenstates of $H_{0}$, $\left\{ \left| S\right\rangle ,\left| X\right\rangle ,\left| Y\right\rangle ,\left| Z\right\rangle ,\left| x\right\rangle ,\left| y\right\rangle ,\left| z\right\rangle \right\} \otimes \left\{ \left| \uparrow \right\rangle ,\left| \downarrow \right\rangle \right\} $, where, under the point group $T_{d}$, $\left| S\right\rangle $ transforms like $\Gamma _{1}$, $\left\{ \left| X\right\rangle ,\left| Y\right\rangle ,\left| Z\right\rangle \right\} $ and $\left\{ \left| x\right\rangle ,\left| y\right\rangle ,\left| z\right\rangle \right\} $ transform like $\Gamma _{4}$ [@YuCardonaChapter2]. The $\left\{ \left| \uparrow \right\rangle ,\left| \downarrow \right\rangle \right\}$ comprises the usual spin $1/2$ states: \[eq:Pauli\] $$\begin{aligned} \left\langle \uparrow \right| \bm{\sigma }\left| \uparrow \right\rangle &=-\left\langle \downarrow \right| \bm{\sigma }\left| \downarrow \right\rangle =\mathbf{\hat{z}} \\ \left\langle \uparrow \right| \bm{\sigma }\left| \downarrow \right\rangle &= \left( \left\langle \downarrow \right| \bm{\sigma }\left| \uparrow \right\rangle \right)^{*}= \mathbf{\hat{x}}-i\mathbf{\hat{y}}.\end{aligned}$$ The non-zero matrix elements of $\left( \bm{\nabla }V\times \mathbf{p}\right) $ are $$\begin{aligned} \left\langle X\right| \left( \bm{\nabla }V\times \mathbf{p}\right) ^{y}\left| Z\right\rangle &\equiv i\frac{4m^{2}c^{2}}{3\hbar }\Delta _{0}, \\ \left\langle x\right| \left( \bm{\nabla }V\times \mathbf{p}\right) ^{y}\left| z\right\rangle &\equiv i\frac{4m^{2}c^{2}}{3\hbar }\Delta _{0}^{\prime}, \\ \left\langle X\right| \left( \bm{\nabla }V\times \mathbf{p}\right) ^{y}\left| z\right\rangle &\equiv i\frac{4m^{2}c^{2}}{3\hbar }\Delta^{-},\end{aligned}$$ cyclic permutations of these \[e.g. $\left\langle x\right| \left( \bm{\nabla }V\times \mathbf{p}\right) ^{y}\left| z\right\rangle =\left\langle z\right| \left( \bm{\nabla }V\times \mathbf{p}\right) ^{x}\left| y\right\rangle =\left\langle y\right| \left( \bm{\nabla }V\times \mathbf{p}\right) ^{z}\left| x\right\rangle $\], and those generated by Hermitian conjugation of these. The above equations define the spin-orbit energies $\Delta _{0}$ and $\Delta _{0}^{\prime}$, and the interband spin-orbit coupling $\Delta^{-}$ [@CPB65; @CCF88]. The fourteen basis states $\left\{ \left| n,\mathbf{k=\mathbf{0}} \right\rangle |n=1..14\right\}$ for $H_{14}$ are \[eq:basis\] $$\begin{aligned} \left| \Gamma _{7v},\pm 1/2\right\rangle &= \pm \frac{1}{\sqrt{3}}\left| Z\right\rangle \left| \alpha _{\pm }\right\rangle +\frac{1}{\sqrt{3}}\left| X\pm iY\right\rangle \left| \alpha _{\mp }\right\rangle \\ \left| \Gamma _{8v},\pm 1/2\right\rangle &= \mp \sqrt{\frac{2}{3}}\left| Z\right\rangle \left| \alpha _{\pm }\right\rangle +\frac{1}{\sqrt{6}}\left| X\pm iY\right\rangle \left| \alpha _{\mp }\right\rangle \\ \left| \Gamma _{8v},\pm 3/2\right\rangle &= \pm \frac{1}{\sqrt{2}}\left| X\pm iY\right\rangle \left| \alpha _{\pm }\right\rangle \\ \left| \Gamma _{6c},\pm 1/2\right\rangle &= i\left| S\right\rangle \left| \alpha _{\pm }\right\rangle \\ \left| \Gamma _{7c},\pm 1/2\right\rangle &= \pm \frac{1}{\sqrt{3}}\left| z\right\rangle \left| \alpha _{\pm }\right\rangle +\frac{1}{\sqrt{3}}\left| x\pm iy \right\rangle \left|\alpha _{\mp}\right\rangle \\ \left| \Gamma _{8c},\pm 1/2\right\rangle &= \mp \sqrt{\frac{2}{3}}\left| z\right\rangle \left| \alpha _{\pm }\right\rangle +\frac{1}{\sqrt{6}}\left| x\pm iy \right\rangle \left|\alpha _{\mp}\right\rangle \\ \left| \Gamma _{8c},\pm 3/2\right\rangle &= \pm \frac{1}{\sqrt{2}}\left| x\pm iy\right\rangle \left| \alpha _{\pm }\right\rangle ,\end{aligned}$$ where $\left| \alpha _{+}\right\rangle =\left| \uparrow \right\rangle $ and $\left| \alpha _{-}\right\rangle =\left| \downarrow \right\rangle $. The states are labeled with their transformation property under the double group for $T_{d}$, and with a pseudo-angular momentum notation. In the basis , $H_{\mathbf{k}=\mathbf{0}}$ is diagonal except for terms proportional to $\Delta ^{-}$. The connection between the eigenvalues of $H_{\mathbf{k}=\mathbf{0}}$ for the $\Gamma $-point eigenstates and the eigenvalues of $H_{0}$ is given by Pfeffer and Zawadski [@PZ90]. The nonzero matrix elements of momentum, which appear in $H_{\mathbf{k}}$, are \[eq:P\_matrix\] $$\begin{aligned} \left\langle S\right| p^{x}\left| X\right\rangle &= \left\langle S\right| p^{y}\left| Y\right\rangle =\left\langle S\right| p^{z}\left| Z\right\rangle \equiv i m P_{0} / \hbar \\ \left\langle S\right| p^{x}\left| x\right\rangle &= \left\langle S\right| p^{y}\left| y\right\rangle =\left\langle S\right| p^{z}\left| z\right\rangle \equiv i m P_{0}^{\prime} / \hbar \\ \begin{split} \left\langle X\right| p^{y}\left| z\right\rangle &= \left\langle Y\right| p^{z}\left| x\right\rangle =\left\langle Z\right| p^{x}\left| y\right\rangle = \left\langle Z\right| p^{y}\left| x\right\rangle \\ &= \left\langle Y\right| p^{x}\left| z\right\rangle =\left\langle X\right| p^{z}\left| y\right\rangle \equiv i m Q / \hbar . \end{split}\end{aligned}$$ Eq.  defines the parameters $P_{0}$, $P_{0}^{\prime}$, and $Q$. They are sometimes expressed as energies $E_{P}$, $E_{P^{\prime}}$, and $E_{Q}$ with the connections $E_{P}= 2 m P_{0}^{2}/ \hbar ^{2}$, etc. The “bare” fourteen-band model has eight empirical parameters $E_{g}$, $\Delta_{0}$, $E_{0}^{\prime}$, $\Delta_{0}^{\prime}$, $\Delta^{-}$, $P_{0}$, $Q$, and $P_{0}^{\prime}$. Its quantitative accuracy is improved by adding important remote band effects to order $k^{2}$ using Löwdin perturbation theory [@Lowdin51], which adds $\mathbf{k}$-dependent terms to the truncated $14\times14$ Hamiltonian so that its solutions better approximate those of the full Hamiltonian [@PZ96]. The remote band effects are governed by the parameters $\gamma _{1}$, $\gamma _{2}$, $\gamma _{3}$, $F$, and $C_{k}$. The parameters $\gamma _{1} $, $\gamma _{2}$, and $\gamma _{3}$ are modified Luttinger parameters that account for remote band effects on the valence bands. They are related to the usual Luttinger parameters $\gamma _{1L}$, $\gamma _{2L}$, and $\gamma _{3L}$ by the couplings with $\Gamma _{6c}$, $\Gamma _{7c}$, and $\Gamma _{8c}$ bands, which are already accounted for in the “bare” fourteen-band model [@PZ96]: $$\begin{aligned} \gamma_{1} &= \gamma_{1L} - \frac{E_{P}}{3 E_{g}} - \frac{E_{Q}}{3 E_{0}^{\prime}} - \frac{E_{Q}}{3 E_{0}^{\prime}+\Delta_{0}^{\prime}}, \\ \gamma_{2} &= \gamma_{2L} - \frac{E_{P}}{6 E_{g}} + \frac{E_{Q}}{6 E_{0}^{\prime}}, \\ \gamma_{3} &= \gamma_{3L} - \frac{E_{P}}{6 E_{g}} - \frac{E_{Q}}{6 E_{0}^{\prime}}.\end{aligned}$$ The parameter $F$ accounts for remote band effects on the lowest conduction band, essentially fixing its effective mass to the experimentally observed value. Finally, the parameter $C_{k}$ is the small $\mathbf{k}$-linear term in the valence bands [@CCF88]. The remote band effects can be removed by setting $\gamma _{1}=-1$ and $\gamma _{2}=\gamma _{3}=F=C_{k}=0$. The model includes neither remote band effects on the $uc$ bands, nor remote band effects on the $\Gamma _{6c}$-$\Gamma _{8v}$ and $\Gamma _{6c}$-$\Gamma _{7v}$ momentum matrix elements, although such terms exist in principle [@WinklerBook]. In summary, $H_{14}$ is a fourteen-band approximation to $H_{\mathbf{k}}$ that incorporates some remote band effects. It can be found in Eq. (5) of Pfeffer and Zawadzki, although with a slightly different notation [@PZ96]. With their notation on the left, and ours on the right: $E_{0}= - E_{g}$, $E_{1}=E_{0}^{\prime}-E_{g}$, $\Delta_{1}=\Delta_{0}^{\prime}$, $\overline{\Delta}=\Delta^{-}$, $P_{1}=P_{0}^{\prime}$. Also, our $\Delta_{0}$ differs from theirs by a minus sign. Other authors have also used different notations [@WinklerBook]. The fourteen bands are shown schematically in Fig. \[fig:14BandDiagram\] along with the symmetry notation of the $\Gamma$-point states, and the notation used to label the bands. Material parameters ------------------- Numerical values for the thirteen parameters of the model are listed in Table \[Table:parameters\] for InSb, GaSb, InP, GaAs, and ZnSe. They are taken from the literature, where they were chosen to fit low-temperature experimental data. Of the two parameter sets discussed by Pfeffer and Zawadzki for GaAs, we use the one corresponding to $\alpha =0.085$ that they find yields better results [@PZ96]. For InP, GaSb, and InSb, we use parameters from Cardona, Christensen and Fasal [@CCF88]. For cubic ZnSe, we use the parameters given by Mayer and Rossler [@MayerRossler93b], we use a calculated value of $C_{k}$ [@CCF88], and we use $\Delta ^{-}=-0.238$ eV to give a $k^{3}$ conduction band spin-splitting that matches the *ab initio* calculation of Cardona, Christensen and Fasal [@CCF88]. Winkler used these same parameters for ZnSe, but took $\Delta ^{-}=0$ [@WinklerBook]. There is more uncertainty in the parameters for ZnSe than in those for the other materials [@MayerRossler93b], but we include it as an example of a semiconductor with a larger band gap. [c|ddddd]{} & & & & &\ $E_{g}$ (eV) & 1.519 & 1.424 & 0.813 & 0.235 & 2.820\ $\Delta_{0}$ (eV) & 0.341 & 0.108 & 0.75 & 0.803 & 0.403\ $E_{0}^{\prime}$ (eV) & 4.488 & 4.6 & 3.3 & 3.39 & 7.330\ $\Delta_{0}^{\prime}$ (eV) & 0.171 & 0.50 & 0.33 & 0.39 & 0.090\ $\Delta^{-}$ (eV) & -0.061 & 0.22 & -0.28 & -0.244 & -0.238\ $P_{0}$ (eVÅ) & 10.30 & 8.65 & 9.50 & 9.51 & 10.628\ $Q$ (eVÅ) & 7.70 & 7.24 & 8.12 & 8.22 & 9.845\ $P_{0}^{\prime}$ (eVÅ) & 3.00 & 4.30 & 3.33 & 3.17 & 9.165\ $\gamma_{1L}$ & 7.797 & 5.05 & 13.2 & 40.1 & 4.30\ $\gamma_{2L}$ & 2.458 & 1.6 & 4.4 & 18.1 & 1.14\ $\gamma_{3L}$ & 3.299 & 1.73 & 5.7 & 19.2 & 1.84\ $F$ & -1.055 & 0 & 0 & 0 & 0\ $C_{k}$ (meVÅ) & -3.4 & -14 & 0.43 & -9.2 & -14 The parabolic-band approximation calculations use parameters from Table \[Table:parameters\], and average effective masses derived from the parameters in Table \[Table:parameters\]. Matrix elements --------------- The relations between matrix elements of the Bloch states and matrrix elements of the $u$-function kets are $$\begin{gathered} \mathbf{v}_{nm}\left(\mathbf{k} \right) \equiv \left\langle n\mathbf{k}\right| \mathbf{v} \left| m\mathbf{k} \right\rangle =\left\langle \overline{n\mathbf{k}}\right| \mathbf{v}\left| \overline{m\mathbf{k}}\right\rangle +\frac{\hbar \mathbf{k}}{m}\delta _{nm}, \label{eq:v Bloch u} \\ \left\langle n\mathbf{k} \right| \mathbf{S} \left| m\mathbf{k}\right\rangle =\left\langle \overline{n\mathbf{k}}\right| \mathbf{S}\left| \overline{m\mathbf{k}}\right\rangle \\ \left\langle n\mathbf{k} \right| v^{i}S^{j} \left| m\mathbf{k} \right\rangle =\left\langle \overline{n\mathbf{k}}\right| v^{i}S^{j}\left| \overline{m\mathbf{k}}\right\rangle +\frac{\hbar k^{i}}{m}\left\langle \overline{n\mathbf{k}}\right| S^{j}\left| \overline{m\mathbf{k}} \right\rangle .\end{gathered}$$ The matrix elements of the velocity operator, $\mathbf{v}$, neglecting the anomalous velocity as discussed in Appendix \[App:kdepSO\], can be calculated using , , and the right side of . The matrix elements of the spin operator $\mathbf{S}$, can be found from Eq. . The matrix elements of $v^{i}S^{j}$ can be similarly found in the basis of eigenstates of $H_{0}$. Each of these can then be rotated to the basis in which the states $\left| \overline{m\mathbf{k}}\right\rangle$ are expanded. It is well known that, in a crystal, $\mathbf{v}_{nn}\left( \mathbf{k}\right) = \nabla _{\mathbf{k}}\omega _{n}\left( \mathbf{k}\right) $. More generally, $$\label{eq:v nablaH identity} \mathbf{v}_{nm}\left( \mathbf{k}\right) = \nabla_{\mathbf{k}} \left\langle n \mathbf{k} \right| H \left| m \mathbf{k}\right\rangle = \left\langle \overline{n \mathbf{k}} \right| \nabla_{\mathbf{k}} H_{\mathbf{k}} \left| \overline{m \mathbf{k}}\right\rangle.$$ These identities can be proven from the definitions $H_{\mathbf{k}}=e^{-i\mathbf{k}\cdot \mathbf{r}}He^{i\mathbf{k}\cdot \mathbf{r}}$ and $\mathbf{v}=\left( i/\hbar \right) \left[ H,\mathbf{r}\right]$, even for a non-local Hamiltonian. But when remote band effects are included in a finite band model, they no longer hold. That is, $\mathbf{v}_{nm}\left( \mathbf{k}\right)$ calculated using and eigenstates of $H_{14}$ is not equal to $\left\langle \overline{n \mathbf{k}} \right| \nabla_{\mathbf{k}} H_{14} \left| \overline{m \mathbf{k}}\right\rangle$. We explicitly restore these identities by using $\left\langle \overline{n \mathbf{k}} \right| \nabla_{\mathbf{k}} H_{14} \left| \overline{m \mathbf{k}}\right\rangle$ to calculate $\mathbf{v}_{nm}\left( \mathbf{k}\right)$. This approach can be described as including remote band effects in the velocity operator. It was used for an eight band calculation of linear absorption by Enders et al [@Enders95]. This step is not critically important for the effects calculated here, since remote band effects are generally small. $\mathbf{k}$-space integration ------------------------------ The optical calculations in this article have the form $\Theta $, where $$\label{eq:GeneralOpticalCalc} \Theta =\sum_{c,v}\int d^{3}kf_{cv}\left( H_{\mathbf{k}}\right) \delta \left( \hbar \omega _{cv}\left( \mathbf{k}\right) -2\hbar\omega\right) ,$$ where $f_{cv}$ depends on matrix elements and energies of eigenstates of $H_{\mathbf{k}}$, and where $\omega _{nm}\left( \mathbf{k}\right) \equiv \omega _{n}\left( \mathbf{k}\right) -\omega _{m}\left( \mathbf{k}\right) $. The integral in is understood to be restricted to the first Brillioun Zone, but we do not actively enforce the restriction, since the photon energies considered here cause transitions well within the first Brillioun Zone. Writing $\mathbf{k}=\left( k_{cv},\theta _{\mathbf{k}},\phi _{\mathbf{k}}\right) $ in spherical coordinates, where $k_{cv}$ is the solution to $$\hbar \omega _{cv}\left( k_{cv},\theta _{\mathbf{k}},\phi _{\mathbf{k}}\right) -2\hbar \omega=0, \label{eqn:RootFinding}$$ we have $$\Theta =8\sum_{c,v}\int_{0}^{\pi /2}\int_{0}^{\pi /2}\frac{k_{cv}^{2}\sin \theta _{\mathbf{k}} f_{cv}\left( H_{\mathbf{k}}\right) }{\left| \hbar \left( \mathbf{v}_{cc}\left( \mathbf{k}\right) -\mathbf{v}_{vv}\left( \mathbf{k} \right) \right) \cdot \mathbf{\hat{k}}\right| }d\phi _{\mathbf{k}}d\theta _{\mathbf{k}} \label{eqn:BZIntegration},$$ where we have used $\bm{\nabla }\omega _{n}\left( \mathbf{k}\right) = \mathbf{v}_{nn}\left( \mathbf{k}\right) $ and the cubic symmetry of the crystal. It is numerically convenient to do the sum over any degenerate bands before the integral over $\theta _{\mathbf{k}}$ and $\phi _{\mathbf{k}}$. Approximations\[approximations\_defined\] ----------------------------------------- The calculations of “1+2” effects in the following sections are primarily labeled by the Hamiltonian used to approximate $H_{\mathbf{k}}$. The complete fourteen-band model is denoted $H_{14}$. The bare fourteen-band model, denoted $H_{14\text{-Bare}}$, is $H_{14}$ without remote band effects. The $8\times 8$ subset of the fourteen band Hamiltonian within the basis $\left\{ \Gamma _{6c},\Gamma _{8v},\Gamma _{7v}\right\} $ is denoted $H_{8}$. The spherical eight-band model, denoted $H_{8\text{Sph}}$, is derived from $H_{8}$ by setting $ C_{k}=0$ and replacing $\gamma _{2}$ and $\gamma _{3}$ by $\tilde{\gamma} \equiv \left( 2\gamma _{2}+3\gamma _{3}\right) /5$;[@Baldereschi73] it is a spherical approximation to the Kane model including remote band effects [@Kane57]. The aforementioned calculations are non-perturbative in $\mathbf{k}$; that is, in each case, the Hamiltonian is solved numerically at each $\mathbf{k}$. The perturbative calculations of Appendix \[App:PBA\_a-a\] are denoted PBA (parabolic-band approximation). The microscopic expression for each of the “1+2” effects contains a sum over intermediate bands, which originates from the two-photon amplitude. Unless otherwise noted, calculations include all possible intermediate bands (eg., $H_{14}$ includes fourteen intermediate bands, and $H_{8\text{Sph}}$ includes eight intermediate bands). Calculations that restrict this sum are secondarily labeled to reflect the restriction. The label “$H_{14}$, no $uc$” uses $H_{14}$, but does not include $uc$ bands as intermediate states. The label “$H_{14}$, no $uc$/$so$” uses $H_{14}$, but includes neither $uc$ nor $so$ bands as intermediate states. The label “$H_{14}$, 2BT” uses $H_{14}$, but only includes two-band terms (terms for which the intermediate band is the same as the initial or final band). Similar labels are used for $H_{8\text{Sph}}$, for example, “$H_{8\text{Sph}}$-PBA, no so” uses the perturbative solution to $H_{8\text{Sph}}$ and does not include $so$ intermediate states. Current\[sec:Current\] ====================== The current injection rate due to the field can be written $$%\begin{split} \dot{J}^{i} = \eta _{\left( 1\right) }^{ijk}E_{2\omega}^{j*} E_{2\omega}^{k} + \dot{J}_{(I)}^{i} + \eta _{\left( 2\right) }^{ijklm}E_{\omega}^{j*}E_{\omega}^{k*}E_{\omega}^{l}E_{\omega}^{m}, %\end{split} \label{eq:phenom_current}$$ where $\mathbf{J}$ is the macroscopic current density, and $$\dot{J}_{(I)}^{i} = \eta_{(I)} ^{ijkl}E_{\omega}^{j*}E_{\omega}^{k*}E_{2\omega}^{l}+c.c. \label{eq:phenom_current_I}$$ The third rank tensor $\eta _{\left( 1\right) }^{ijk}$ describes one-photon current injection (the circular photogalvanic effect [@SturmanFridkin; @GanichevReview]), the fifth rank tensor $\eta _{\left( 2\right) }^{ijklm}$ describes two-photon current injection, and the fourth rank tensor $\eta_{(I)} ^{ijkl}$ describes “1+2” current injection [@Atanasov96]. Aversa and Sipe showed that $\eta_{(I)} ^{ijkl}$ is related to a doubly divergent part of the third-order nonlinear susceptibility $\chi ^{\left( 3\right) }$ [@Aversa96]. In cubic materials with point group symmetry $T_{d}$, $O_{h}$ or $O$, a general fourth rank tensor has four independent components, but due to the intrinsic symmetry $\eta_{(I)} ^{ikjl}=\eta_{(I)} ^{ijkl}$, $\eta_{(I)} $ has only three independent components; there are 21 non-zero components of $\eta_{(I)} $ in the standard cubic basis: $\eta _{(I)}^{aaaa}=\eta_{(I)} ^{bbbb}=\eta_{(I)} ^{cccc}$, $\eta_{(I)} ^{baab}=\eta_{(I)} ^{abba}=\eta_{(I)} ^{caac}=\eta_{(I)} ^{acca}=\eta_{(I)} ^{cbbc}=\eta_{(I)} ^{bccb}$, and $\eta_{(I)} ^{a\left( ab\right) b}=\eta_{(I)} ^{b\left( bc\right) c}=\eta_{(I)} ^{c\left( ca\right) a}=\eta_{(I)} ^{a\left( ac\right) c}=\eta_{(I)} ^{c\left( cb\right) b}=\eta_{(I)} ^{b\left( ba\right) a}$ (the components in parentheses can be exchanged), where $a$, $b$, and $c$ denote components along the principal cubic axes [@Atanasov96]. This can be written $$\eta _{(I)}^{ijkl}=i\frac{\eta _{B1}}{2}\left( \delta ^{ij}\delta ^{kl}+\delta ^{ik}\delta ^{jl}\right) +i\eta _{B2}\delta ^{il}\delta ^{jk} +i\eta _{C}\delta ^{ijkl}, \label{eq:phenom_eta_form}$$ where $\delta ^{ij}$ is a Kronecker delta and the only non-isotropic part is $\delta ^{ijkl}$, which we define in the principal cubic basis as $\delta ^{ijkl}=1$ when $i=j=k=l$ and zero otherwise. The three independent components are $\eta _{B1}\equiv -2i\eta ^{aabb}$, $\eta _{B2}\equiv -i\eta ^{abba}$, and $\eta _{C}\equiv 2i\eta ^{aabb}+i\eta ^{abba}-i\eta ^{aaaa}$. Thus, in a cubic material, $$\begin{split} \dot{J}_{(I)}^{i} =& i\eta _{B1}\left( \mathbf{E}_{\omega}^{*}\cdot \mathbf{E}_{2\omega}\right) E_{\omega}^{i*}+i\eta _{B2}\left( \mathbf{E}_{\omega}\cdot \mathbf{E}_{\omega}\right) ^{*}E_{2\omega}^{i} \\ &+i\eta _{C}\delta ^{ijkl}E_{\omega}^{j*}E_{\omega}^{k*}E_{2\omega}^{l}+c.c. \end{split}$$ This generalizes the notation we used previously for a calculation in the parabolic-band approximation [@BhatSipe00], with the connection $\eta _{B1}=eDB_{1}/\hbar $, and $\eta _{B2}=eDB_{2}/\hbar $. In that, or any other spherical approximation, $\eta _{C}=0$. To calculate $\eta_{(I)} $, we use the microscopic expression first given by Atanasov et al. [@Atanasov96], modified to explicitly include the sum over spin states [@vanDrielSipeReview01; @Najmaie03]. An alternate microscopic expression has been derived in the length gauge [@Aversa96], but it has not yet been used in a calculation. In the independent particle approximation that we employ here, $\eta_{(I)} $ is purely imaginary [@Atanasov96] and hence $\eta _{B1}$, $\eta _{B2}$, and $\eta _{C}$ are real, although they can be complex if excitonic effects are included [@BhatSipeExcitonic05]. ![(color online): Spectra of $\eta _{B1}$ (black lines), $\eta _{B2}$ (red lines), and $\eta _{C}$ (blue lines) for GaAs. Panel (a) shows the contributions from each initial valence band; dashed, dotted, and dashed-dotted lines include only transitions from the $hh$, $lh$, and $so$ bands respectively, while the solid lines include all three transitions. The thin solid, light brown line in (a) is the total $\mathrm{Re} \left( \eta _{(I)}^{aaaa} \right)$. Panel (b) separates the total into electron (dashed) and hole (dotted) contributions.\[fig:currentGaAs\_initial\]](fbm_current_GaAs_initials_and_eh_method1) The spectra of $\eta _{B1}$, $\eta _{B2}$, and $\eta _{C}$, calculated for GaAs, are shown in Fig.\[fig:currentGaAs\_initial\](a) along with the contributions to each tensor component from each possible initial valence band. For a given photon energy, electrons photoexcited from the $hh$ band have higher energies and velocities than electrons photoexcited from the $lh$ band; hence the dominant component $\eta _{B1}$ is larger for $hh$-$c$ transitions than $lh$-$c$ transitions. The smallness of $\eta _{B2}$ is due to contributions from the $hh$-$c$ transitions having opposite sign to the $lh$-$c$ transitions, as shown previously in the PBA [@BhatSipe00]. Figure \[fig:currentGaAs\_initial\](b) separates each tensor component into an electron contribution and a hole contribution (denoted $\eta _{e}$ and $\eta _{h}$ by Atanasov et al [@Atanasov96]). Electrons make a larger contribution to $\eta _{B1}$ than holes, due to the lower effective mass (and hence higher velocity) of an electron than of a hole (much lower, in the case of a heavy hole) with the same crystal momentum. Holes dominate $\eta _{B2}$ at lower photon energies, while electrons dominate $\eta _{B2}$ at higher energies. Both electrons and holes contribute equally to the anisotropic component $\eta _{C}$.  To help in understanding the importance of the various intermediate states, in Fig. \[fig:currentGaAs\_models\] we compare the calculated current injection tensor elements with various degrees of approximation described in Sec. \[approximations\_defined\]. The component $\eta _{B1}$ (and hence $\eta_{(I)} ^{aaaa}$, since $\eta _{B1}$ is larger than $\eta _{B2}+\eta _{C}$) is dominated by two-band terms. Three-band terms cause an increase, by as much as 34%, of $\eta _{B1}$ \[the difference between the dashed and solid black lines in Fig. \[fig:currentGaAs\_models\](a)\]. Although not shown in Fig. \[fig:currentGaAs\_models\], most of the increase is due to three-band terms with the $so$ band as an intermediate state. Terms with the $uc$ bands as intermediate states only cause a small increase to $\eta _{B1}$ (the difference between the dotted and solid black lines). The warping of the bands is clearly not important for $\eta _{B1}$, since the calculation with $H_{8\text{Sph}}$ closely approximates the calculation “$H_{14}$, no $uc$”, which includes the same intermediate states. Surprisingly, the “$H_{8\text{Sph}}$-PBA, 2BT” result [@BhatSipe00; @BhatSipeExcitonic05] closely approximates the complete, non-perturbative fourteen-band calculation, even at excess photon energies for which band nonparabolicity is significant. This is due to a fortuitous compensation between the neglect of nonparabolicity and the neglect of three-band terms. The compensation is not as complete for all materials. The component $\eta _{B2}$, which determines the current due to orthogonal linearly polarized fields, is less forgiving to approximations than the component $\eta _{B1}$. We have already seen in Fig. \[fig:currentGaAs\_initial\] that $\eta _{B2}$ is small due to a near cancellation of $hh$ and $lh$ initial states. Reasonable accuracy on $\eta _{B2}$ thus requires higher accuracy on the contribution from each initial state. In particular, three-band terms must not be neglected. By comparing the dashed-dotted and solid lines in Fig. \[fig:currentGaAs\_models\](b), it can be seen that, whereas the sum of the two-band terms is negative, the sum of the three-band terms is positive and of the same magnitude. It is useful to divide the three-band terms into three groups: those with intermediate state from the $hh$ or $lh$ bands, those with intermediate state from the $so$ band, and those with intermediate state from one of the $uc$ bands. We find that each group contributes roughly the same positive amount to $\eta _{B2}$ for excess photon energies less than $\Delta _{0}$. The groups are added successively to the 2BTs in the dashed, dotted, and solid lines in Fig. \[fig:currentGaAs\_models\](b). Three-band terms with $so$ intermediate states are less important at the higher excess photon energies in Fig. \[fig:currentGaAs\_models\](b). The warping of the bands makes a small but non-negligible contribution to $\eta _{B2}$, as seen in the difference between the dashed-double-dotted and dotted lines of Fig. \[fig:currentGaAs\_models\](b). The solid brown line in Fig. \[fig:currentGaAs\_models\](b) is the “$H_{8\text{Sph}}$-PBA, no $so$” result [@BhatSipe00]. At low excess photon energies, it greatly underestimates $\eta _{B2}$ due to the neglect of $so$ and $uc$ intermediate states, while at excess photon energies greater than 100 meV, this is partly compensated for by the neglect of nonparabolicity. It appears from the difference between “$H_{8\text{Sph}}$-PBA, no $so$” and “$H_{14}$, no $uc$/$so$” in Fig. \[fig:currentGaAs\_models\](b) that nonparabolicity becomes important at energies above 70 meV. The term $\eta _{C}$ is purely due to cubic anisotropy by definition; in any model that is spherically symmetric it is identically zero. There is no cubic anisotropy in the “bare” (i.e. without remote band effects) eight-band model on the set $\left\{ \Gamma _{6c},\Gamma _{8v},\Gamma _{7v}\right\} $. Cubic anisotropy in the fourteen-band model is due to the momentum matrix elements governed by the parameters $E_{Q}$ and $E_{P^{\prime }}$, the interband spin-orbit coupling $\Delta ^{-}$, and remote bands through $(\gamma _{2}-\gamma _{3})$ and $C_{k}$. From Fig.\[fig:currentGaAs\_models\](c), it can be seen that three-band terms are important for $\eta _{C}$. In fact, with only 2BTs included, $\eta _{C}$ is positive for GaAs, whereas it is negative with all terms included. From Fig. \[fig:currentGaAs\_models\](c) it can also be seen that the $so$ band and $uc$ bands are important as intermediate states for $\eta _{C}$. Our calculation of $\eta_{(I)} $ is of the same order of magnitude as the *ab initio* calculation of Atanasov et al.[@Atanasov96], but its spectral dependence is different. In particular, $\eta _{B1}$ agrees more closely with the PBA calculation, as seen in Fig. \[fig:currentGaAs\_models\](a). Atanasov et al. had attributed the difference between their *ab initio* and PBA calculations to the assumption of $\mathbf{k}$-independent velocity matrix elements in the PBA [@Atanasov96]. However, our calculation accounts for the $\mathbf{k}$-dependence of velocity matrix elements and agrees closely (for $\eta _{B1}$ and $\mathrm{Re}\eta ^{aaaa}$) to the PBA. The earlier *ab initio* calculation [@Atanasov96] was, in fact, inaccurate at low photon energies due to various computational issues; an improved *ab initio* calculation agrees with the spectral dependence at low photon energy given here [@NastosPrivateCommunication].  Figure \[fig:current\_4materials\] shows the spectra of $\eta _{B1}$, $\eta _{B2}$, and $\eta _{C}$ calculated with $H_{14}$ for InSb, GaSb, InP, and ZnSe. The dashed black line in Fig.\[fig:current\_4materials\] is the PBA result [@BhatSipe00; @BhatSipeExcitonic05]. The PBA appears to be a reasonable approximation to $\eta _{B1}$ for excess energies less than about $0.2E_{g}$. In each material, $\eta _{B2}\ll \eta _{B1}$, and in each material except for ZnSe, the sign of $\eta _{B2}$ varies as a function of frequency. The component $\eta _{C}$, which arises due to cubic anisotropy, is negative for each material. The cubic anisotropy of current injection due to colinearly polarized fields can be significant enough that it should be measurable. For fields colinearly polarized along $\mathbf{\hat{e}}$, specified by polar angles $\theta $ and $\phi $ relative to the cubic axes, $$\mathbf{\dot{J}}_{(I)} \cdot \mathbf{\hat{e}}=2\mathrm{Im}\left( E_{\omega}^{2}E_{2\omega}^{*}\right) \left[ \eta _{B1}+\eta _{B2}+\eta _{C}-\frac{\eta _{C}}{2}f\left( \theta ,\phi \right) \right] ,$$ where $f\left( \theta ,\phi \right) =\sin ^{2}\left( 2\theta \right) +\sin ^{4}\left( \theta \right) \sin ^{2}\left( 2\phi \right) $. In general, $\mathbf{\dot{J}}_{(I)}$ also has a component perpendicular to $\mathbf{\hat{e}}$ that is proportional to $\eta _{C}$, but it vanishes for $\mathbf{\hat{e}}$ parallel to $\left \langle 001\right\rangle $, $\left \langle 110\right\rangle $, $\left\langle 111\right\rangle $. The field polarization that maximizes the current injection depends on the relative sign of $\eta _{C}$ and $\mathrm{Re}\eta ^{aaaa}=\eta _{B1}+\eta _{B2}+\eta _{C}$. When they have the opposite sign, current injection is a minimum for $\mathbf{\hat{e}\parallel }\left \langle 001\right \rangle $ ($f=0$) and a maximum for $\mathbf{\hat{e}\parallel }\left \langle 111\right \rangle$ ($f=4/3$); for light normally incident on a $\left\{ 001\right\} $ surface, the largest current injection occurs when $\mathbf{\hat{e}\parallel }\left \langle 110\right \rangle$ ($f=1$). When they have the same sign, current-injection is a maximum for $\mathbf{\hat{e}\parallel }\left \langle 001\right \rangle $ and a minimum for $\mathbf{\hat{e}\parallel }\left \langle 111\right \rangle$. From the GaAs results shown in Fig.\[fig:currentGaAs\_initial\](a), the current injection for the three cases $\mathbf{\hat{e}\parallel }\left \langle 001\right \rangle$, $\mathbf{\hat{e}\parallel }\left \langle 110\right \rangle$, and $\mathbf{\hat{e}\parallel }\left \langle 111\right \rangle$ are in the ratio 1 to 1.14 to 1.20 at the band edge, 1 to 1.15 to 1.20 at 200 meV excess photon energy, and 1 to 1.22 to 1.29 at 500 meV excess photon energy. In contrast, the *ab initio* calculation of Atanasov et al. yields larger ratios, for example 1 to 1.32 to 1.43 at 300 meV excess photon energy [@Atanasov96]. This disagreement is consistent with the inaccuracy of the *ab initio* calculation discussed above. Initial experiments with GaAs used $\mathbf{\hat{e}\parallel }\left[ 001\right] $ [@Hache97; @StevensJAP], whereas Roos et al. exploited the larger signal for $\mathbf{\hat{e} \parallel }\left[ 110\right] $ [@Roos03]. For each of the materials shown in Fig.\[fig:current\_4materials\], the minimum current injection is for $\mathbf{\hat{e}\parallel } \left \langle 001\right \rangle$. It is worth noting that two-photon absorption is also a minimum with $\mathbf{\hat{e}\parallel }\left \langle 001\right \rangle$ for many semiconductors [@Dvorak94; @Hutchings94; @Murayama95]. It seems that both “1+2” current injection and two-photon absorption with linearly polarized fields are larger for $\mathbf{\hat{e}}$ directed along the bonds. The cubic anisotropy of “1+2” current injection is pronounced for cross-linearly polarized fields and opposite-circularly polarized fields. For example, for cross-linearly polarized fields normally incident on $\left( 001\right) $ with $\mathbf{\hat{e}}_{\omega} = \mathbf{\hat{a}} \cos \phi + \mathbf{\hat{b}} \sin \phi $ and $\mathbf{\hat{e}}_{2\omega} = - \mathbf{\hat{a} } \sin \phi + \mathbf{\hat{b}} \cos \phi $, $$\begin{split} \mathbf{\dot{J}}_{(I)} =& \mathrm{Im}\left( E_{\omega}^{2}E_{2\omega}^{*}\right) \\ &\times \left[ \left( 2\eta _{B2}+\eta _{C}\sin ^{2}\left( 2\phi \right) \right) \mathbf{\hat{e}}_{2\omega}-\frac{\eta _{C}}{2}\sin \left( 4\phi \right) \mathbf{\hat{e}}_{\omega}\right] . \end{split}$$ For fields with opposite circular polarizations, the current injection is proportional to $\eta _{C}$ and is hence purely anisotropic. The component $\eta _{C}$ causes a type of current injection that has not previously been noted. In all “1+2” experiments considered thus far with light normally incident on a surface, the direction of current injection lies in the plane of the surface. However, with co-linearly polarized light fields normally incident on a $\left( 111\right) $ surface, the current can have a component into (or out of) the surface. The current in this case is $$%\begin{split} \mathbf{\dot{J}}_{(I)} = 2\mathrm{Im} \left( E_{\omega}^{2}E_{2\omega}^{*}\right) \left[ \bar{\eta} \mathbf{\hat{e}} +\frac{\sqrt{2}}{6} \eta _{C}\cos \left( 3\theta \right) \mathbf{\hat{z }} \right], %\end{split}$$ where $\bar{\eta} \equiv \left( \eta _{B1}+\eta _{B2}+\frac{1}{2}\eta _{C}\right)$, $\mathbf{\hat{z}}$ is the $\left[ 111\right] $ direction, and $\theta$ is the angle between $\mathbf{\hat{e}}$ and the $\left[ 2\bar{1}\bar{1}\right] $ direction. Thus, $\eta _{C}$ governs this “surfacing” current. Population control\[sec:Population\] ==================================== The carrier injection rate due to the field can be written $\dot{N}=\dot{N}_{(1)} +\dot{N}_{(I)} +\dot{N}_{(2)}$, where $N$ is the density of electron-hole pairs, $\dot{N}_{(1)}=\xi _{\left( 1\right) }^{ij}E_{2\omega}^{i*}E_{2\omega}^{j}$ is one-photon absorption, $\dot{N}_{(2)}= \xi _{\left( 2\right) }^{ijkl}E_{\omega}^{i*}E_{\omega}^{j*}E_{\omega}^{k}E_{\omega}^{l}$ is two-photon absorption, and $$\dot{N}_{(I)} = \xi _{(I)}^{ijk}E_{\omega}^{*i}E_{\omega}^{*j}E_{2\omega}^{k}+c.c.$$ is “1+2” population control [@Fraser99PRL]. The third-rank tensor $\xi _{(I)}^{ijk}$ has intrinsic symmetry $\xi _{(I)}^{jik}=\xi _{(I)}^{ijk}$. In centrosymmetric materials, such as those with the diamond structure (point group $O_{h}$), $\xi _{(I)}^{ijk}$ is identically zero; hence, population control requires a noncentrosymmetric material. In a material with zinc-blende symmetry (point group $T_{d}$), $\xi _{(I)}^{ijk}$ has only one independent component; in the standard cubic basis, $\xi_{(I)} ^{abc}=\xi_{(I)} ^{cab}=\xi_{(I)} ^{bca}=\xi_{(I)} ^{acb}=\xi_{(I)} ^{bac}=\xi_{(I)}^{cba}$ are the only non-zero components, where $a$, $b$, and $c$ denote components along the principal cubic axes. We calculate $\xi _{(I)}$ with the microscopic expression given by Fraser et al., which was derived in the independent-particle approximation, and is restricted to $\hbar \omega <E_{g}<2\hbar \omega $ [@Fraser99PRL]. Under those conditions, $\xi _{(I)}$ is real and is proportional to the imaginary part of the susceptibility for second harmonic generation (SHG) [@Fraser99PRL; @SipeShkrebtii00]; specifically, (in mks) $$\xi _{(I)}^{abc}=\frac{ 2\varepsilon _{0} }{\hbar }\mathrm{Im}\chi ^{\left( 2\right) cba}\left( -2\omega ;\omega ,\omega \right) \label{zetaI_chi2_relation}.$$ This connection to SHG, which can be derived from considerations of energy transfer and macroscopic electrodynamics [@Fraser99PRL; @BhatSipeExcitonic05], is important because the imaginary part of $\chi ^{\left( 2\right) }\left( -2\omega ;\omega ,\omega \right) $ has sometimes been presented *en route* to a calculation of $\left| \chi ^{(2)}\right| $ [@Bell71; @Moss87; @Ghahramani91; @Huang93; @LewYanVoon94; @HughesSipe96; @Adolph98]. As well, analytic expressions have been derived for the dispersion of SHG by using simple band models, with approximations appropriate for $2\hbar \omega $ near the band gap [@Bell71; @Kelley63a; @Kelley63b; @Rustagi69; @Bell72; @Jha72]. However, these earlier works did not connect $\mathrm{Im}\chi ^{\left( 2\right) }\left( -2\omega ;\omega ,\omega \right) $ with population control, and in fact typically stated that it was not independently observable.  Fig. \[fig:pop\_Imchi2\] shows the calculation of $\mathrm{Im}\chi ^{\left( 2\right) cba}\left( -2\omega ;\omega ,\omega \right) $ for InSb, GaSb, InP, GaAs and ZnSe. Also shown for comparison is the PBA expression (\[eq:popControlPBA\]), derived in Appendix \[App:PBA\_a-a\]. Each spectrum can be divided into roughly three regions. At very low excess photon energies, visible in the log-log plot Fig. \[fig:pop\_Imchi2\](f), the spectrum is roughly independent of $\omega $. This flat part of the spectrum disappears if $C_{k} $ is set to zero; hence, it is due to the $\mathbf{k}$-linear term in the $c$ band spin-splitting. Next higher in photon energy, up to about 100 meV in GaSb, InP, GaAs, and ZnSe (up to about 15 meV in InSb), is a region where the agreement with the analytic expression (\[eq:popControlPBA\]) is best. In this region, the ratio $X_{2}/X_{1}$, defined in Appendix \[App:PBA\_a-a\], is $0.37$ for InSb, $0.30$ for GaSb, $-0.25$ for InP, $0.08$ for GaAs, and $0.07$ for ZnSe. At higher photon energies, the dispersion of $\mathrm{Im}\chi ^{\left( 2\right) cba}\left( -2\omega ;\omega ,\omega \right) $ deviates from the PBAexpression due to band nonparabolicity and warping, $\mathbf{k}$-dependence of matrix elements, and transitions from the split-off band, which are not included in (\[eq:popControlPBA\]). If we remove the two-band transitions $hh$-$\left\{ hh,c\right\}$-$c$, $lh$-$\left\{ lh,c\right\}$-$c$, and $so$-$\left\{ so,c\right\}$-$c$, then the calculation of $\mathrm{Im}\chi^{\left( 2\right)}$ (or $\xi _{(I)}$) is unchanged. This is expected for materials of zinc-blende symmetry [@Kelley63b; @Aspnes72]. Further, many years ago Aspnes argued that the so-called “virtual hole terms” of the form $lh$-$\{so,hh\}$-$c$ and $hh$-$\{so,lh\}$-$c$ make only a small contribution to $\chi ^{\left( 2\right) }\left( 0\right) $ [@Aspnes72]. Such terms have been neglected in some previous calculations of $\chi ^{\left( 2\right) }$ dispersion [@Moss87; @Huang93]. By removing the virtual hole terms, leaving only $\left\{ so,lh,hh \right\}$-$uc$-$c$ transitions, we find $\xi _{(I)}$ is reduced by only 6–10% over the range from the band edge to 500 meV above the gap for GaAs. It is thus clear that inclusion of the $uc$ bands is necessary for a calculation of population control. For some purposes it is also sufficient, since if remote band effects are removed from the model, leaving the “bare” fourteen-band model [@PZ90; @HW94], $\xi_{(I)}$ is decreased by only 7–10% from its full value for GaAs. For most materials, the results in Fig. \[fig:pop\_Imchi2\] are in reasonable agreement with previous calculations of $\mathrm{Im} \chi ^{\left( 2\right) }$ [@Adolph98; @HughesSipe96; @Huang93; @Bell71], although most previous calculations had poor spectral resolution in this energy range. However, for ZnSe, the situation is more complicated. The calculation of Huang and Chin is about an order of magnitude smaller than ours [@Huang93], and that of Ghahramani et al. is about 5 times smaller than ours [@Ghahramani91]. Note also that Huang and Chin calculated $\chi ^{\left( 2\right) }\left( 0\right) $ for ZnSe to be an order of magnitude smaller than experimental results [@Huang93]. Wagner et al. have measured the dispersion of $|\chi ^{\left( 2\right)}|$, which is an upper bound on $\mathrm{Im}\chi ^{\left( 2\right)}$; for ZnSe it is about a factor of two smaller than our calculation of $\mathrm{Im} \chi ^{\left( 2\right) }$ [@Wagner98]. Note that Wagner et al. give a different set of band parameters than we have used here [@Wagner98]. The magnitude of $\xi _{(I)}^{abc}$ determines the magnitude of population control, but in an experiment one is more interested in the depth of the phase-dependent modulation of the carrier absorption, i.e. the control ratio $R$ [@Fraser99PRL]. It is $$R=\frac{\dot{N}_{(I)}}{\dot{N}_{(1)}+\dot{N}_{(2)}}=\frac{\xi _{(I)}^{ijk}E_{2\omega}^{i*}E_{\omega}^{j}E_{\omega}^{k}+c.c.}{\xi _{(1)}^{ij}E_{2\omega}^{i*}E_{2\omega}^{j}+\xi _{(2)}^{ijkl}E_{\omega}^{i*}E_{\omega}^{j*}E_{\omega}^{k}E_{\omega}^{l}}.$$ This ratio is largest for field amplitudes that equalize $\dot{N}_{(1)}$ and $\dot{N}_{(2)}$ [@Fraser03]; in what follows, we assume this condition has been met. The ratio then depends only on $\xi_{(I)}^{ijk}$, $\xi_{(1)}^{ij} $, $\xi_{(2)}^{ijkl}$, and the polarizations of the two fields. For light normally incident on a $\left( 111\right) $ surface, linearly-polarized fields yield $R=\sqrt{2}\xi _{(I)}^{abc}/\sqrt{3 \xi _{(1)}^{aa} \xi _{(2)}^{aaaa} \left( 1 - \sigma /2 \right)}$, while opposite circularly-polarized fields yield $$R=2\xi _{(I)}/\sqrt{3 \xi _{(1)} \xi _{(2)}^{aaaa} \left( 1-\sigma /6 - \delta \right) }, \label{ratio opposite circular}$$ where $\sigma \equiv (\xi _{(2)}^{aaaa} - \xi _{(2)}^{aabb} - 2 \xi _{(2)}^{abab})/ \xi_{(2)}^{aaaa}$ and $\delta \equiv (\xi _{(2)}^{aaaa} + \xi _{(2)}^{aabb} - 2 \xi _{(2)}^{abab})/ (2 \xi_{(2)}^{aaaa})$ are two-photon absorption anisotropy and circular dichroism parameters [@Dvorak94; @HW94]. Stevens et al. found that for light normally incident on a $\left( 111\right) $ surface of GaAs, opposite circularly polarized fields yield the largest ratio [@Stevens_pssb; @StevensReview04]. For light normally incident on a $\left( 110 \right)$ surface, fields linearly polarized along $\left[ 1\bar{1}1 \right]$ yield $$R=2\xi _{(I)}/\sqrt{3\xi _{(1)} \xi _{(2)}^{aaaa} \left( 1-2 \sigma /3 \right) }.\label{e:ratio_111}$$ The polarization configuration that yields a global maximum for the control ratio depends on the material and photon energy; we have found that (\[ratio opposite circular\]) is the maximum except for very close to the band edge, where (\[e:ratio\_111\]) is the maximum. To calculate the population control ratio, it is desirable to use values of $\xi _{(I)}$, $\xi _{(1)}$, and $\xi _{(2)}$ calculated within the same set of approximations. We use microscopic expressions for $\xi _{(1)}$ and $\xi _{(2)}$ in the independent-particle approximation [@Atanasov96], and calculate them within the fourteen-band model. Note that our calculation of two-photon absorption ($\xi _{(2)}$) is similar to that of Hutchings and Wherrett [@HW94], but that our model includes remote band effects. Fig. \[fig:pop\_ratio\] shows the calculated spectra of the population control ratio (\[ratio opposite circular\]) for various semiconductors. For each material, the ratio is close to unity at the band edge, then drops steeply, but flattens out to some non-zero ratio as photon energy is increased. In general, the smaller the band gap (or conduction band effective mass) of the material, the narrower the range over which the ratio drops, and the lower the ratio at higher excess photon energy. Worth noting is the particularly large ratio for ZnSe. Also plotted in Fig.\[fig:pop\_ratio\] is the ratio appropriate for linearly-polarized fields normally incident on a $\left( 111\right) $ surface of GaAs, which was the configuration in the experiment of Fraser et al [@Fraser99PRL]. For all materials, the ratio (\[e:ratio\_111\]) reaches exactly unity at the band edge, in agreement with the PBA calculation (\[popcontrol\_ratio\_bandedge\]) in Appendix \[App:PBA\_a-a\]. ![(color online) Calculated population control ratios appropriate for opposite circularly polarized fields normally incident on a $\left( 111\right) $ surface of InSb, GaSb, InP, GaAs, and ZnSe. The blue, dotted line is the ratio for linearly polarized fields normally incident on a $\left( 111\right) $ surface of GaAs.[]{data-label="fig:pop_ratio"}](fbm_ratio2_labelled) The only previous theoretical calculation of the population-control ratio, which was for GaAs, missed finding the large ratio near the band edge because it was based on *ab initio* calculations of $\xi _{(1)}$, $\xi _{(2)}$ and $\xi _{(I)}$ that had poor spectral resolution near the band edge [@Fraser99PRL]. Over the rest of the spectrum shown in Fig. \[fig:pop\_ratio\], it is about a factor of two smaller than our calculation. This is consistent with the previous calculation being based on a calculation of the two-photon absorption coefficient $\xi _{(2)}$ that is too large by comparison with other calculations [@HW94; @Murayama95]. The population-control ratio has been measured only in GaAs [@Fraser99PRL; @Fraser03; @Stevens_pssb; @StevensReview04; @Stevens05_110]. The measured ratios on $\left( 111\right) $-GaAs, at excess photon energies of 180 meV [@Fraser99PRL; @Fraser03] and 312 meV [@Stevens_pssb; @StevensReview04] were 4 to 5 times smaller than our calculation. Some of the difference can be attributed to phase mismatch and large sample thickness [@Fraser99PRL; @Fraser03; @Stevens_pssb; @StevensReview04]. An experiment on a $\left( 110\right) $-grown multiple quantum well was complicated by an additional cascaded second harmonic effect [@Stevens05_110]. Spin current \[sec:SpinCurrent\] ================================ Spin-current density can be quantified by a second-rank pseudotensor $K^{ij}$ defined as the average value of the product $v^{i}S^{j}$, where $\mathbf{v}$ is the velocity operator and $\mathbf{S}$ is the spin operator [@BhatSipe00]. Note that some authors alternately choose the first index to represent spin and the second index to represent velocity [@Rashba03]. Also, due to the spin-orbit part of the velocity operator—the so-called “anomalous” velocity \[the second term in \]—$\mathbf{v}$ and $\mathbf{S}$ do not commute, and thus $v^{i}S^{j}$ is not Hermitian. Instead, one should take $( v^{i}S^{j} + S^{j}v^{i} ) /2$ as the operator for spin-current. But since we neglect the anomalous velocity (see Appendix \[App:kdepSO\]), this is not necessary. The spin-current injection rate due to the field can be written $$%\begin{split} \dot{K}^{ij}= \mu _{\left( 1\right) }^{ijkl}E_{2\omega}^{k*}E_{2\omega}^{l} + \dot{K}_{(I)}^{ij} +\mu _{\left( 2\right) }^{ijklmn}E_{\omega}^{k*}E_{\omega}^{l*}E_{\omega}^{m}E_{\omega}^{n}, %\end{split} \label{eq:phenom_K}$$ where the pseudotensor $\mu _{\left( 1\right) }^{ijkl}$ describes one-photon spin-current injection [@BhatPRL05], the pseudotensor $\mu _{\left( 2\right) }^{ijklmn}$ describes two-photon spin-current injection, and $$\dot{K}_{(I)}^{ij} = \mu_{(I)} ^{ijklm}E_{\omega}^{*k}E_{\omega}^{*l}E_{2\omega}^{m}+c.c.$$ is “1+2” spin-current injection [@BhatSipe00]. The fifth-rank pseudotensor $\mu_{(I)} ^{ijklm}$ has intrinsic symmetry $\mu_{(I)} ^{ijlkm}=\mu_{(I)} ^{ijklm}$. In an isotropic material, $\mu_{(I)} ^{ijklm}$ has three independent components, while in a cubic material (with $T_{d}$, $O$, or $O_{h}$ symmetry) $\mu_{(I)} ^{ijklm}$ has six independent components. The four parameters $A_{i}$, $i=1$–$4$, that we used previously to describe spin-current injection in an isotropic model [@BhatSipe00] can be reduced to three independent components with identities such as $\varepsilon ^{ijm}\delta ^{kl}-\varepsilon ^{ijk}\delta ^{lm}+\varepsilon ^{jkm}\delta ^{il}-\varepsilon ^{ikm}\delta ^{jl}=0$ [@Kearsley75]. For a cubic material, $\mu_{(I)} ^{ijklm}$ has 54 non-zero elements in the principal cubic basis, and can be written $$\begin{split} \mu_{(I)} ^{ijklm}=& \frac{\mu_{N1}}{2}\left( \varepsilon ^{jml}\delta ^{ik}+\varepsilon ^{jmk}\delta ^{il}\right) +\mu_{N3}\varepsilon ^{ijm}\delta ^{kl} \\ & +\frac{\mu_{N2}}{2}\left( \varepsilon ^{iml}\delta ^{jk}+\varepsilon ^{imk}\delta ^{jl}\right) +\mu_{C1}\delta ^{ikln}\varepsilon ^{njm} \\ &+\mu_{C2}\delta ^{jkln}\varepsilon ^{nim}+\frac{\mu_{C3}}{2}\left( \delta ^{ijkn}\varepsilon ^{nml}+\delta ^{ijln}\varepsilon ^{nmk}\right) , \end{split} \label{eq:phenom_mu_form}$$ where the non-isotropic tensor $\delta ^{ijkl}$ has nonzero components $\delta ^{aaaa}=\delta ^{bbbb}=\delta ^{cccc}=1$, where $a$, $b$, and $c$ denote components along the principal cubic axes. The six independent components are $\mu_{N1}\equiv 2\mu_{(I)}^{acaba}$, $\mu_{N2}\equiv 2\mu_{(I)}^{caaba}$, $\mu_{N3}\equiv \mu_{(I)}^{abccc}$, $\mu_{C1}\equiv \mu_{(I)}^{abaac} -\mu_{N1}-\mu_{N3}$, $ \mu_{C2}\equiv \mu_{(I)}^{baaac} -\mu_{N2}+\mu_{N3}$, and $\mu_{C3}\equiv 2\mu_{(I)}^{aaacb}-\mu_{N1}-\mu_{N2}$. Thus in a cubic material, $$\begin{split} \dot{K}_{(I)}^{ij}=& \mu_{N1}E_{\omega}^{*i}\left( \mathbf{E}_{2\omega}\times \mathbf{E}_{\omega}^{*}\right) ^{j}+\mu_{N2}\left( \mathbf{E}_{2\omega}\times \mathbf{E}_{\omega}^{*}\right) ^{i}E_{\omega}^{*j} \\ & +\mu_{N3}\varepsilon ^{ijk}E_{2\omega}^{k}\left( \mathbf{E}_{\omega}^{*}\cdot \mathbf{E} _{\omega}^{*}\right) +\mu_{C3}\delta ^{ijkl}E_{\omega}^{*k}\left( \mathbf{E}_{2\omega}\times \mathbf{E}_{\omega}^{*}\right) ^{l} \\ & +\left( \mu_{C1}\delta ^{ikln}\varepsilon ^{njm}+\mu_{C2}\delta ^{jkln}\varepsilon ^{nim}\right) E_{\omega}^{*k}E_{\omega}^{*l}E_{2\omega}^{m}+c.c. \end{split}$$ Note that the injection of $\left\langle \mathbf{v\cdot S}\right\rangle $ is zero in a cubic material, i.e., $\dot{K}^{ij}$ is traceless. In an isotropic model, such as the one we used previously [@BhatSipe00], $\mu_{C1}=\mu_{C2}=\mu_{C3}=0$. The connection to our previous notation is $\mu_{N1}=D\left( A_{1}-A_{4}\right) $, $\mu_{N2}=D\left( A_{2}+A_{4}\right) $, and $\mu_{N3}=D\left( A_{3}+A_{4}\right) $ [@BhatSipe00]. The spin-current injection can be divided into a contribution from electrons $\dot{K}^{ij}_{(I;e)}$, and a contribution from holes $\dot{K}^{ij}_{(I;h)}$; that is, $\dot{K}^{ij}_{(I)}=\dot{K}^{ij}_{(I;e)}+\dot{K}^{ij}_{(I;h)}$ (similarly, $\mu^{ijklm}_{(I)}=\mu^{ijklm}_{(I;e)}+\mu^{ijklm}_{(I;h)}$). Expressions in the PBA for both the electron and hole spin current are given elsewhere [@BhatSipe00]; here we focus on the electron spin current, since hole spin relaxation is typically very fast [@Hilton02; @Yu05]. A microscopic expression for the spin-current injection was derived previously in the Fermi’s golden rule (FGR) limit of perturbation theory and applied to a model in which all bands are doubly degenerate [@BhatSipe00]. However, it is unsuitable for a calculation with $H_{14}$, which accounts for the small splitting of the spin degeneracy that occurs in materials of zinc-blende symmetry [@Dresselhaus55; @CCF88; @PikusReview88]. If the spin-split bands were well separated, then the microscopic expression for $\dot{K}^{ij}_{(I;e)}$ would be $$\begin{split} \dot{K}^{ij}_{(I;e)} =& \frac{2\pi }{L^{3}}\sum_{c,v,\mathbf{k}}\left\langle c\mathbf{k} \right| v^{i}S^{j}\left| c\mathbf{k}\right\rangle \\ & \times \left[ \left( \Omega _{c,v,\mathbf{k}}^{\left( 2\right) }\right) ^{*}\Omega _{c,v,\mathbf{k}}^{\left( 1\right) }+c.c.\right] \delta \left( 2\omega -\omega _{cv}\left( \mathbf{k}\right) \right) , \end{split}$$ where $L^{3}$ is a normalization volume; the one-photon amplitude $\Omega _{c,v,\mathbf{k}}^{\left( 1\right) }$ is $$\Omega _{c,v,{\mathbf{k}}}^{\left( 1\right) }=i\frac{e}{2\hbar \omega }\mathbf{E} _{2\omega } \cdot \mathbf{v}_{c,v}\left( \mathbf{k}\right), \label{OPamplitude}$$ where the charge on an electron is $e$ ($e<0$), and the two-photon amplitude $\Omega _{c,v,\mathbf{k}}^{\left( 2\right) }$ is $$\Omega_{c,v,\mathbf{k}}^{(2)} = \left( \frac{e}{\hbar \omega}\right)^{2} \sum_{n}\frac{\left( \mathbf{E}_{\omega }\cdot \mathbf{v}_{c,n}\left( \mathbf{k}\right) \right) \left( \mathbf{E}_{\omega }\cdot \mathbf{v}_{n,v}\left( \mathbf{k}\right) \right) }{\omega_{nv}\left( \mathbf{k}\right) -\omega }. \label{TPamplitude}$$ However, for the photon energies and materials studied here, the spin-splitting is small; it is comparable to the broadening that one would calculate from the scattering time of the states, and also to the laser bandwidth for typical ultrafast experiments. Thus, the spin-split bands should be treated as quasidegenerate in FGR, with the result $$\begin{split} \dot{K}_{(I;e)}^{ij} =& \frac{2\pi }{L^{3}}\sum_{c,c^{\prime }}^{\prime }\sum_{v,\mathbf{k}} \langle c\mathbf{k} | v^{i}S^{j} | c^{\prime }\mathbf{k} \rangle \left( \Omega _{c,v,\mathbf{k}}^{\left( 2\right) }\right) ^{*}\Omega _{c^{\prime },v,\mathbf{k}}^{\left( 1\right) } \\ & \times \frac{1}{2}\left[ \delta \left( 2\omega -\omega _{cv}\left( \mathbf{k} \right) \right) +\delta \left( 2\omega -\omega _{c^{\prime }v}\left( \mathbf{ k}\right) \right) \right] +c.c., \end{split}$$ where the prime on the summation indicates a restriction to pairs $\left( c,c^{\prime }\right) $ for which either $c^{\prime }=c$, or $c$ and $c^{\prime }$ are a quasidegenerate pair. The optical excitation of the coherence between spin-split bands can be justified using the semiconductor optical Bloch equation approach, as was done for the one-photon spin properties [@BhatPRL05]. Note that this issue does not arise for “1+2” current injection or “1+2” population control, since $\langle c \mathbf{k} | \mathbf{v} | c^{\prime} \mathbf{k} \rangle$ and $\langle c \mathbf{k} | c^{\prime} \mathbf{k} \rangle$ vanish between spin-split bands. Using the time-reversal properties of the Bloch functions, we find that $\mu_{(I;e)} $ is real, and can be written as $$\label{eq:mu_micro} \begin{split} \mu _{(I;e)}^{ijklm}=&i\left( \frac{e}{\hbar \omega }\right) ^{3}\frac{\pi }{ 2L^{3}}\sum_{c,c^{\prime }}^{\prime }\sum_{v,\mathbf{k}}\sum_{n} \delta \left( 2\omega -\omega _{cv}\left( \mathbf{k} \right) \right) \\ &\times \mathrm{Re} \left\{ \frac{\left\langle c\mathbf{k}\right| v^{i}S^{j}\left| c^{\prime }\mathbf{k}\right\rangle }{\omega _{nv\mathbf{k}}-\omega }\left[ M_{c,c^{\prime },v}^{klm}-\left( M_{c^{\prime },c,v}^{klm}\right) ^{*}\right] \right\} , \end{split}$$ where $$M_{c,c^{\prime },v}^{klm}\equiv \frac{1}{2} v_{c^{\prime }v}^{m} (\mathbf{k}) \left[ v_{cn}^{k*} (\mathbf{k}) v_{nv}^{l*}(\mathbf{k}) + v_{cn}^{l*} (\mathbf{k}) v_{nv}^{k*} (\mathbf{k}) \right] .\label{eq:Mccpv}$$ That $\mu_{(I;e)}$ in is purely real is a consequence of the independent-particle approximation [@BhatSipeExcitonic05]. ![(color online): Calculated spectra of GaAs spin-current injection components and their contributions from each initial valence band; dashed, dotted, and dashed-dotted lines include only transitions from the $hh$, $lh$, and $so$ bands respectively, while the solid lines include all three transitions. Panel (a) shows $\mu_{N1}$ (black lines), $\mu_{N2}$ (red lines), and $\mu_{N3}$ (blue lines). Panel (b) shows $\mu_{C1}$ (black lines), $\mu_{C2}$ (red lines), and $\mu_{C3}$ (blue lines).[]{data-label="fig:spincurrentGaAs_initial"}](fbm_spincurrent_GaAs_initials) Calculation results ------------------- The spectra of the independent components of $\mu _{(I;e)}$, calculated for GaAs, are shown in Fig.\[fig:spincurrentGaAs\_initial\] and Fig.\[fig:spincurrentGaAs\_models\]. Figure \[fig:spincurrentGaAs\_initial\] also shows contributions from each possible initial valence band. Figure \[fig:spincurrentGaAs\_models\] shows the spin-current injection calculated with various degrees of approximation described in Sec.\[approximations\_defined\]. The only other calculation of “1+2” spin-current injection for bulk GaAs is our earlier calculation, which used a spherical, parabolic-band approximation to the eight-band model and did not include the $so$ band as an intermediate state [@BhatSipe00]; it is shown in Fig.\[fig:spincurrentGaAs\_models\] for $\mu_{N1}$, $\mu_{N2}$, and $\mu_{N3}$.  The term $\mu_{N1}$ has the largest magnitude of the six independent parameters of $\mu_{(I;e)}$. Since it is negative for $hh$ and $lh$ transitions but positive for $so$ transitions, it peaks in magnitude at $2\hbar \omega $ just above $E_{g}+\Delta _{0}$ (the energy at which $so$ transitions become allowed). Two band terms make the largest contribution to $\mu_{N1}$, followed by three-band terms with $hh$ or $lh$ intermediate states. The $so$ and $uc$ intermediate states make a very small contribution to $\mu_{N1}$ for excess energies less than 200 meV. The warping of the bands is not important for $\mu_{N1}$, since the calculation with $H_{8\text{Sph}}$ closely approximates the “$H_{14}$, no $uc$” calculation, which includes the same intermediate states. The “$H_{8\text{Sph}}$-PBA, no $so$” calculation, which we derived previously [@BhatSipe00], is a good approximation to $\mu_{N1}$ at excess energies below 250 meV; nonparabolicity becomes important at higher energies. The $hh$ contribution has a larger magnitude than the $lh$ contribution in part because three-band terms increase the magnitude of the $hh$ contribution, but decrease that of the $lh$ contribution, as expected from the PBA expression . The term $\mu_{N2}$ is negative for $hh$ transitions, positive for $lh$ transitions, and negligible for $so$ transitions. The calculation “$H_{14}$, 2BT” is a good approximation to the calculation $H_{14}$. However, the three-band terms are not small; rather, they nearly cancel. In particular the transition $hh$-$lh$-$c$ makes a large positive contribution to $\mu_{N2}$, while the transition $hh$-$so$-$c$ makes a large negative contribution. Since our earlier PBA calculation included the former but not the latter [@BhatSipe00], it is a poor approximation to $\mu_{N2}$. But by including only 2BTs, it is a fair approximation for excess energies less than 200 meV. This agreement is fortuitous, since the calculation $H_{8\text{Sph}}$ underestimates the magnitude of $\mu_{N2}$, and the PBA leads to an overestimation of the magnitude of $\mu_{N2}$. The term $\mu_{N3}$ is negligible when only 2BTs are included, in agreement with the PBA [@BhatSipe00]. The $hh$-$lh$-$c$ transitions are positive, while the $lh$-$hh$-$c$ transitions are negative; the former is larger, and thus $\mu_{N3}$ is positive when $so$ intermediate states are neglected. Both $lh$-$so$-$c$ and $hh$-$so$-$c$ are negative and substantial enough to make the total $\mu_{N3}$ negative. Consequently, our earlier PBA result [@BhatSipe00], which neglects $so$ intermediate states, is a poor approximation to $\mu_{N3}$. Upper conduction bands make a fairly small contribution to $\mu_{N3}$, and warping does not seem to be important for $\mu_{N3}$ since the calculation with $H_{8\text{Sph}}$ is a good approximation. As expected, the terms $\mu_{C1}$, $\mu_{C2}$, and $\mu_{C3}$ are zero when calculated with $H_{8\text{Sph}}$. The term $\mu_{C1}$ is negligible when only 2BTs are included. Transitions with intermediate states in the set $\left\{ hh,lh,so\right\} $ comprise roughly two-thirds of $\mu_{C1}$. The anisotropy of these transitions is not simply due to the warping of the $hh$ and $lh$ bands, which we have determined by a calculation (not shown) using $H_{8}$ without the remote band contribution to the velocity. Rather, it comes from wave function mixing of the $\Gamma _{8c}$ and $\Gamma _{7c}$ states into the valence and $c$ band states. The cubic anisotropy of two-photon absorption has been attributed to such wave function mixing [@Dvorak94; @HW94]. The other third of the full $\mu_{C1}$ is due to transitions with the $uc$ intermediate state, which would be forbidden close to the $\Gamma $ point if the material were isotropic. We also note that each three-band term makes a positive contribution to $\mu_{C1}$. The term $\mu_{C2}$ is nearly negligible when only 2BTs are included. Transitions from the $hh$ and $lh$ bands have opposite sign, and those from the $so$ band are negligible. About half of $\mu_{C2}$ is due to the transitions $hh$-$lh$-$c$ and $lh$-$hh$-$c$, and the other half is due to transitions with the $uc$ intermediate states. Transitions with $so$ intermediate states are negligible. As with $\mu_{C1}$, the anisotropy of the $hh$-$lh$-$c$ and $lh$-$hh$-$c$ transitions is due to the wave function mixing of the $\Gamma _{8c}$ and $\Gamma _{7c}$ states into the $hh$, $lh$, and $c$ band states. The term $\mu_{C3}$ is positive for $hh$ transitions, negative for $lh$ transitions, and negligible for $so$ transitions. The transitions $hh$-$so$-$c$ and $lh$-$so$-$c$ account for most of the value of $\mu_{C3}$, but 2BTs are not negligible. Transitions with $uc$ intermediate states reduce the value of $\mu_{C3}$ by as much as 10%. Most of $\mu_{C3}$, especially at energies less than 200 meV, is due to the warping of the $hh$ and $lh$ bands. Consistent with this, we find that remote band effects are somewhat important for $\mu_{C3}$; when remote band effects are removed, the calculation of $\mu_{C3}$ is about 25% larger than the full calculation. Note that $\mu_{C3}$ is far more sensitive to remote band effects than any other optical property calculated in this article.  In Fig. \[fig:spincurrent\_4materials\] we plot the spectra of the independent components of the spin current density pseudotensor for InSb, GaSb, InP, and ZnSe. The spin current tensor is largest for InSb in agreement with the PBA expressions in Appendix \[App:PBA\_a-a\]. We also note that $\mu_{N3}$ is positive for InSb and GaSb at low excess photon energy, whereas it is negative for InP, GaAs, and ZnSe. Configurations -------------- Co-circularly polarized fields generate a spin-polarized current, which can be characterized by its degree of spin polarization $f\equiv \left(2e/\hbar \right) \dot{K_{e}^{ij}}\hat{q}^{i}\hat{n}^{j}/|\dot{\mathbf{J}_{e}}|$, where $\hat{\mathbf{n}}$ is a unit vector normal to the polarization plane of the fields, and $\hat{\mathbf{q}}$ is a unit vector in the direction of $\mathbf{J}_{e}$ [@BhatSipe00]. Essentially, $f=\left\langle vS\right\rangle / \left\langle v\right\rangle$. Since this measure aims to characterize the photoexcited distribution of electrons, we neglect holes from both $\dot{K}$ and $\dot{J}$ in this calculation [@footnote2000f]. For fields normally incident on a $\left( 001\right) $ surface (i.e.$\mathbf{e}_{\omega}=\mathbf{e}_{2\omega}=\left( \mathbf{\hat{x}}\pm i\mathbf{\hat{y}}\right) /\sqrt{2}$), the spin current is $$\begin{split} \dot{K}_{(I)}^{ij} =& \mp \sqrt{2}|E_{2\omega}| |E_{\omega}|^{2} \\ & \times \left[ \left( \mu_{N1}+ \frac{\mu_{C1}}{2}\right) \hat{m}_{\pm }^{i}\hat{z}^{j}+\left( \mu_{N2}+\frac{\mu_{C2}}{2}\right) \hat{z}^{i}\hat{m}_{\pm }^{j}\right], \end{split}$$ where $\mathbf{\hat{m}}_{\pm }=\sin \left( 2\phi _{\omega}-\phi _{2\omega}\right) \mathbf{\hat{x}}\pm \cos \left( 2\phi _{\omega}-\phi _{2\omega}\right) \mathbf{\hat{y}}$, the current is $\dot{\mathbf{J}}_{(I)} = \sqrt{2} E_{2\omega} E_{\omega}^{2} \left( \eta _{B1}+\eta _{C} /2 \right) \hat{\mathbf{m}}_{\pm }$, and the degree of spin polarization is $$f=\frac{2e}{\hbar }\frac{\mu_{N1}+\mu_{C1}/2}{\eta _{B1}+\eta _{C}/2}.\label{eq:SC:f:001}$$ For fields normally incident on a $\left( 111\right) $ surface, $\dot{\mathbf{J}}_{(I)}=\sqrt{2}E_{2\omega}E_{\omega}^{2}\left( \eta _{B1}+\eta _{C}/3\right) \hat{\mathbf{m}}_{\pm }$, and $$f=\frac{2e}{\hbar }\frac{\mu_{N1}+\mu_{C1}/3+\mu_{C3}/3}{\eta _{B1}+\eta _{C}/3}.\label{eq:SC:f:111}$$ The degree of spin polarization is plotted for GaAs in Fig.\[fig:SC\_measures\](a). The cubic anisotropy is small, but clearly seen, especially at low excess photon energies. The other materials have very similar degrees of spin polarization. ![(a) Degree of polarization of spin-polarized current due to co-circularly polarized fields. (b) Displacement of spins in pure spin current due to cross-linearly polarized fields.[]{data-label="fig:SC_measures"}](fbm_SC_measures) A pure spin current, without an electrical current, can be generated with cross-linearly polarized fields [@BhatSipe00]. We consider fields polarized in the $\left( 001\right) $ plane, with the $\omega $ field polarized at an angle $\theta $ to the $\mathbf{\hat{x}}$ axis (i.e. $\left[ 100\right] $) and the $2\omega $ field polarized at an angle $\theta $ to the $\mathbf{\hat{y}}$ axis ($\mathbf{e}_{\omega} = \mathbf{\hat{x}} \cos \theta + \mathbf{\hat{y}} \sin \theta$ and $\mathbf{e}_{2\omega} = - \mathbf{\hat{x}} \sin \theta + \mathbf{\hat{y}} \cos \theta$). The spin current is $$\begin{split} \dot{K}_{(I)}^{ij} =& -\frac{1}{2} |E_{2\omega}| |E_{\omega}|^{2}\cos \left( 2\phi _{\omega}-\phi _{2\omega}\right) \left[ \left( 4\mu_{N1}+4\mu_{N3} +3\mu_{C1}+\mu_{C1}\cos \left( 4\theta \right) \right) e_{\omega}^{i}\hat{z}^{j} \right.\\ & \left.-\sin \left( 4\theta \right)\left(\mu_{C1} e_{2\omega}^{i} \hat{z}^{j} +\mu_{C2} \hat{z}^{i}e_{2\omega}^{j}\right) +\left( 4\mu_{N2}-4\mu_{N3}+3\mu_{C2}+\mu_{C2}\cos \left( 4\theta \right) \right) \hat{z} ^{i}e_{\omega}^{j} \right] \end{split}$$ This pure spin current is typically measured by the resulting displacement of up and down spins [@StevensPRL03; @HubnerPRL03]. The finite displacement results from transport and scattering of the electrons. Using the Boltzmann transport equation in the relaxation time approximation with space-charge effects justifiably neglected [@HubnerPRL03], one finds $d^{i}\left( \mathbf{\hat{z}} \right) =\left( 4\tau /\hbar \right) \dot{K}^{ij}\hat{z}^{j}/\left( \dot{N} _{\left( 1\right) }+\dot{N}_{\left( 2\right) }\right) $ [@BhatPRL05]. Here, $\mathbf{d}\left( \mathbf{\hat{z}}\right) $ is the displacement of spins measured with respect to the quantization direction $\mathbf{\hat{z}}$, and $\tau $ is the momentum relaxation time. We assume the field intensities have been chosen to balance one- and two-photon absorption, a condition that is $\theta $-dependent due to the cubic anisotropy of two-photon absorption. Thus, $$\mathbf{d}\left( \mathbf{\hat{z}}\right) \cdot \mathbf{e}_{\omega}=\frac{\tau}{\hbar} \frac{\left( 4\mu_{N1}+4\mu_{N3}+3\mu_{C1}+\mu_{C1}\cos \left( 4\theta \right) \right) }{\sqrt{\xi_{(1)} ^{xx}\xi _{(2)}^{xxxx}\left( 1-\left(\sigma/2\right) \sin^{2} \left( 2\theta \right) \right) }}\label{eq:SC:separationdistance_w}$$ and $$\mathbf{d}\left( \mathbf{\hat{z}}\right) \cdot \mathbf{e}_{2\omega}=\frac{\tau}{\hbar} \frac{\mu_{C1}\sin \left( 4\theta \right) }{\sqrt{\xi_{(1)} ^{xx}\xi _{(2)}^{xxxx}\left( 1-\left(\sigma/2\right) \sin^{2} \left( 2\theta \right) \right) }},\label{eq:SC:separationdistance_2w}$$ where $\sigma $ is the two-photon absorption cubic-anisotropy factor given explicitly in the next section [@Dvorak94; @HW94]. At $\theta =0$ and $\theta =\pi /4$, $\mathbf{d}$ is parallel to $\mathbf{e}_{\omega}$. The spin separation distance is plotted in Fig. \[fig:SC\_measures\](b), where we have assumed a momentum relaxation time of 100 fs for each material. This calculation of the spin separation distance is a significant improvement over our initial calculations [@StevensPRL03; @HubnerPRL03], which used the eight-band PBA and neglected three-band terms from the two-photon amplitude (“$H_{8\text{Sph}}$-PBA, 2BT”). Stevens et al. measured a spin separation distance of 20 nm in a GaAs multiple quantum well at an excess photon energy of 200 meV, and estimated a momentum relaxation time of $\tau=45$ fs [@StevensPRL03]. For $\tau=45$ fs, we calculate a spin separation distance of 20.0 nm for bulk GaAs at 200 meV. Hübner et al. measured a spin separation distance of 24 nm (the photoluminescence spot separation is half this distance) in cubic ZnSe at an excess photon energy of 280 meV, and estimated a momentum relaxation time of $\tau=100$ fs [@HubnerPRL03]. The calculation in Fig. \[fig:SC\_measures\](b) yields $d=23.6$ nm for ZnSe at 280 meV. In both cases, we now find very good agreement with the experiment, whereas the previous model resulted in larger spin separation distances. Of course, this agreement is contingent on the accuracy of the momentum relaxation time estimates. Note that both the degree of spin polarization for co-circularly polarized fields and the spin-separation distance, plotted in Fig.\[fig:SC\_measures\], have a kink at excess photon energy $\Delta_{0}$ and decrease at higher excess photon energies. A similar kink and decrease, due to the onset of transitions from the split-off band, occurs for both one-photon spin injection [@DyakonovOptOrient] and two-photon spin injection [@Bhat_TPS_05]. Spin control\[sec:Spin\] ======================== The spin injection rate due to the field can be written $\dot{\mathbf{S}} = \dot{\mathbf{S}}_{(1)} + \dot{\mathbf{S}}_{(I)} + \dot{\mathbf{S}}_{(2)}$, where $\mathbf{S}$ is the macroscopic spin density, $\dot{S}_{(1)}^{i}= \zeta _{\left( 1\right) }^{ijk}E_{2\omega}^{j*}E_{2\omega}^{k}$ is one-photon spin injection [@DyakonovOptOrient], $\dot{S}_{(2)}^{i} = \zeta _{(2)}^{ijklm}E_{\omega}^{j*}E_{\omega}^{k*}E_{\omega}^{l}E_{\omega}^{m}$ is two-photon spin injection [@Bhat_TPS_05], and $$%\begin{split} \dot{S}_{(I)}^{i}= \zeta_{(I)}^{ijkl} E_{\omega}^{*j} E_{\omega}^{*k} E_{2\omega}^{l}+c.c. %\end{split} \label{eq:macro:spin}$$ is “1+2” spin control [@Stevens_pssb]. In previous sections and in some of the expressions below, we use $\mathbf{S}$ to denote the single-particle spin operator. It should be obvious by context when $\mathbf{S}$ refers to the macroscopic spin density and when it refers to that spin operator. The fourth-rank pseudotensor $\zeta _{(I)}^{ijkl}$ has intrinsic symmetry on the indices $j\leftrightarrow k$. Such a pseudotensor is zero in the presence of inversion symmetry; hence, “1+2” spin control requires materials of lower symmetry. For zinc-blende symmetry (point group $T_{d}$), a general fourth-rank pseudotensor has three independent parameters and 18 non-zero elements in the standard cubic basis; forcing the $j\leftrightarrow k$ symmetry leaves two independent parameters $$\begin{aligned} i\zeta _{IA} & \equiv \zeta _{(I)}^{abba}=\zeta _{(I)}^{caac}=\zeta _{(I)}^{bccb}=-\zeta _{(I)}^{acca}=-\zeta _{(I)}^{cbbc}=-\zeta _{(I)}^{baab} , \label{macro_zetaIA} \\ \begin{split} i\zeta _{IB} & \equiv \zeta _{(I)}^{aabb}=\zeta _{(I)}^{ccaa}=\zeta _{(I)}^{bbcc}=-\zeta _{(I)}^{aacc}=-\zeta _{(I)}^{ccbb}=-\zeta _{(I)}^{bbaa} \\ & =\zeta _{(I)}^{abab}=\zeta _{(I)}^{caca}=\zeta _{(I)}^{bcbc}=-\zeta _{(I)}^{acac}=-\zeta _{(I)}^{cbcb}=-\zeta _{(I)}^{baba} . \label{macro_zetaIB} \end{split}\end{aligned}$$ The spin injection has a contribution from electrons $\mathbf{\dot{S}}_{(I;e)}$, and a contribution from holes $\mathbf{\dot{S}}_{(I;h)}$; that is, $\mathbf{\dot{S}}_{(I)}=\mathbf{\dot{S}}_{(I;e)}+\mathbf{\dot{S}}_{(I;h)}$, and $\zeta_{(I)}=\zeta_{(I;e)} + \zeta_{(I;h)}$. We treat the spin-split bands as quasidegenerate when taking the FGR limit of perturbation theory, as discussed for the spin current in Section \[sec:SpinCurrent\], deriving the microscopic expression $$\begin{split} \mathbf{\dot{S}}_{(I;e)}=&\frac{2\pi }{L^{3}}\sum_{c,c^{\prime }}^{\prime }\sum_{v,\mathbf{k}}\left\langle c\mathbf{k}\right| \mathbf{S}\left| c^{\prime }\mathbf{k}\right\rangle \left( \Omega _{c,v,\mathbf{k}}^{\left( 2\right) }\right) ^{*}\Omega _{c^{\prime },v,\mathbf{k}}^{\left( 1\right) } \\ & \times \frac{1}{2}\left[ \delta \left( 2\omega -\omega _{cv}\left( \mathbf{k} \right) \right) +\delta \left( 2\omega -\omega _{c^{\prime }v}\left( \mathbf{ k}\right) \right) \right] +c.c., \end{split}$$ where the prime on the summation indicates a restriction to pairs $\left( c,c^{\prime }\right) $ for which either $c^{\prime }=c$, or $c$ and $c^{\prime }$ are a quasidegenerate pair. Using the time-reversal properties of the Bloch functions, we find that $\zeta_{(I;e)} $ is purely imaginary and can be written $$\begin{split} \zeta _{(I;e)}^{ijkl}=& i\left( \frac{e}{\hbar \omega }\right) ^{3}\frac{\pi }{ 2L^{3}}\sum_{c,c^{\prime }}^{\prime }\sum_{v,\mathbf{k}}\sum_{n} \delta \left( 2\omega -\omega _{cv}\left( \mathbf{k} \right) \right) \\ & \times \mathrm{Re} \left\{ \frac{\left\langle c\mathbf{k}\right| S^{i}\left| c^{\prime } \mathbf{k}\right\rangle }{\omega _{nv\mathbf{k}}-\omega }\left[ M_{c,c^{\prime },v}^{jkl}+\left( M_{c^{\prime },c,v}^{jkl}\right) ^{*}\right] \right\} ,\label{eq:zeta_micro} \end{split}$$ where $M_{c,c^{\prime },v}^{jkl}$ is given in Eq. (\[eq:Mccpv\]). ![(color online) Spin control pseudotensor components $\zeta_{IA}$ (black lines), $\zeta_{IB}$ (red lines), and $\left( \zeta _{IA}+2\zeta _{IB} \right)$ (blue line) with breakdown into initial states. Dotted lines include transitions from the $lh$ band, dashed lines include transitions from the $hh$ band, dashed-dotted lines include transitions from the $so$ band, and solid lines include all transitions.[]{data-label="fig:spin_GaAs_initials"}](fbm_spin_GaAs_initials_labelled) ![Spin control pseudotensor for GaAs. The calculations are $H_{14}$ (solid black lines), “$H_{14}$, no $uc$” (dotted lines), “$H_{14}$, no $uc$/$so$” (dashed-dotted lines), “$H_{14}$, 2BT” (dashed lines), and “$H_{14}$-PBA” (solid grey lines).[]{data-label="fig:spin_GaAs_models"}](fbm_spin_GaAs_models) The spectra of $\zeta _{IA}$ and $\zeta _{IB}$ for GaAs are shown in Figs. \[fig:spin\_GaAs\_initials\] and \[fig:spin\_GaAs\_models\]. Figure \[fig:spin\_GaAs\_initials\] also shows the contributions from each possible initial valence band. Figure \[fig:spin\_GaAs\_models\] shows the spin control calculated with various degrees of approximation described in Sec.\[approximations\_defined\]. The term $\zeta _{IA}$ decreases from zero at the band edge to a maximum negative value at 40 meV, mostly due to transitions from the $hh$ band, and is positive at higher excess photon energies, mostly due to transitions from the $lh$ band. The low energy behavior is in agreement with the PBA result (\[eq:spin\_IA\_PBA\]), in which the ratio of $hh:lh$ transitions is $\left( m_{c,hh}/m_{c,lh} \right)^{3/2}$. Transitions with $so$ and $uc$ intermediate states dominate the decrease in $\zeta_{IA}$ at low excess photon energies, as seen in Fig. \[fig:spin\_GaAs\_models\](a); they are the only non-zero transitions in the PBA result (\[eq:spin\_IA\_PBA\]). The contribution from $uc$ intermediate states is negative and approximately constant over most of the spectrum, whereas the contribution from $so$ intermediate states changes from negative to positive as transitions from the $so$ band become allowed ($2\hbar \omega >E_{g}+\Delta_{0}$). The contribution from 2BTs, which is zero in the PBA, is positive over the whole spectrum. The breakdown of the PBA is due to the increase in magnitude of the 2BTs. In fact, the sum of the PBA and the 2BTs is a good approximation to the full calculation. We also note that a calculation with $H_{8}$ for $\zeta _{IA}$ yields a nearly negligible result; thus, the contribution from intermediate states within the set $\left\{ so,lh,hh,c\right\} $ (including 2BTs) is due to the mixing of the $\Gamma _{7c}$ and $\Gamma _{8c}$ wavefunctions with these states. The term $\zeta _{IB}$ is larger in magnitude than the term $\zeta _{IA}$ over most of the calculated spectrum. It falls to a maximum negative value at 95 meV, sharply increases when transitions from the $so$ band become allowed, and is positive at higher excess photon energy. At lower photon energies, transitions from the $hh$ band and transitions from the $lh$ band both make negative contributions to $\zeta _{IB}$; in the PBA result (\[eq:spin\_IB\_PBA\]) the ratio of $hh:lh$ transitions is $\left( m_{c,hh}/m_{c,lh} \right)^{3/2}$. Fig.\[fig:spin\_GaAs\_models\](b) reveals that $\zeta _{IB}$ is essentially due to contributions from $uc$ intermediate states, and 2BTs. Over the whole spectrum, the former are negative while the latter are positive. The smallness of the contribution from $so$ intermediate states is also seen in the PBA result (\[eq:spin\_IB\_PBA\]), since $Z_{+} \gg Z_{-}^{\prime}$ in that expression. We also note that a calculation with $H_{8}$ for $\zeta _{IB}$ yields a nearly negligible; thus, the contribution from intermediate states within the set $\left\{ so,lh,hh,c\right\} $ (including 2BTs) is due to the mixing of the $\Gamma _{7c}$ and $\Gamma _{8c}$ wave functions with these states.  We have also calculated the spin-control pseudotensor for the semiconductors InSb, GaSb, InP, and ZnSe. The results are shown in Fig. \[fig:spin\_4materials\] along with the parabolic-band approximations (\[eq:spin\_IA\_PBA\]) and (\[eq:spin\_IB\_PBA\]). The magnitude of spin control is determined by $\zeta_{(I)}$, but in an experiment one is more interested in the depth of the phase-dependent modulation of the spin polarization signal. One possible definition for the signal is the ratio of spin injection measured with both $\omega$ and $2\omega$ fields to the sum of the spin injections measured with circularly polarized fields of each frequency [@Stevens_pssb]. The amplitude of its modulation is $$\frac{\left| \dot{S}_{(I)}^{z} \right|}{\dot{S}_{(1)}^{z}\left( \sigma ^{+}\right) +\dot{S}_{(2)}^{z}\left( \sigma ^{+}\right) }, \label{eq:spin_ratio_spin_norm}$$ where the argument $\left( \sigma ^{+}\right)$ indicates injection with a $\sigma ^{+}$ polarized field. This ratio, which is largest for field amplitudes that equalize $\dot{S}_{(1)}^{z}\left( \sigma ^{+}\right)$ and $\dot{S}_{(2)}^{z}\left( \sigma ^{+}\right)$, was measured by Stevens et al. with excess photon energies of 150 meV and 280 meV [@Stevens_pssb; @Stevens05_110]. The ratio has an undesirable feature: it can exceed unity. Close to the band edge in many semiconductors (at 2 meV in GaAs), there is a photon energy for which $\dot{S}_{(2)}^{z}\left( \sigma ^{+}\right) =0$ [@Bhat_TPS_05]. At that photon energy, it is impossible to choose field amplitudes to balance one- and two-photon spin injection with circular polarized fields \[$\dot{S} _{(1)}^{z}\left( \sigma ^{+}\right) =\dot{S}_{(2)}^{z}\left( \sigma ^{+}\right) $\], and thus the maximum ratio has a singularity. Even if the condition $\dot{S}_{(1)}\left( \sigma ^{+}\right) = \dot{S}_{(2)}\left( \sigma ^{+}\right) $ is relaxed, the ratio can exceed unity. This is because $\dot{S} _{(1)}^{z}\left( \sigma ^{+}\right) $ and $\dot{S}_{(2)}^{z}\left( \sigma ^{+}\right) $ have opposite sign close to the band gap [@Bhat_TPS_05], and thus it is possible, by appropriate choice of field amplitudes, to make the denominator of the ratio arbitrarily small. An alternate ratio to characterize the spin control, which has an upper bound of unity, is $$R_{S}=\frac{2}{\hbar }\frac{\left| \dot{S}_{(I)}^{z} \right|}{\dot{N}_{(1)}+\dot{N}_{(2)}} . \label{eq:spin_ratio_population_norm}$$ It is the amplitude of phase-dependent oscillation of the degree of spin polarization, and it is most useful when there is little or no population control. We assume the fields are chosen to balance one- and two-photon absorption. For most photon energies and materials this is nearly the same as balancing one- and two-photon spin injection. For normal incidence on a $\left( 111\right) $ sample, opposite circularly polarized fields yield $$R_{S}=\frac{2}{\hbar }\frac{\left| \zeta _{IA}+2\zeta _{IB} \right|} {\sqrt{3\xi _{(1)}^{aa}\xi _{(2)}^{aaaa}\left( 1-\sigma /6-\delta \right) }} .$$ For normal incidence on a $\left( 110\right) $ sample, opposite circularly polarized fields $$R_{S}=\frac{2}{\hbar }\frac{3\left| \zeta _{IA}+2\zeta _{IB}\right| }{4\sqrt{ 2\xi _{(1)}^{aa}\xi _{(2)}^{aaaa}\left( 1-\delta -\sigma /8\right) }},$$ and orthogonal linearly polarized fields ($xy$-polarized) yield $$R_{S}\left( \alpha \right) =\frac{2}{\hbar }\frac{\left| \left( \zeta _{IA}+2\zeta _{IB}\right) \left( r+3\sin ^{2}\alpha \right) \cos \alpha \right|} {2\sqrt{\xi _{(1)}^{aa}\xi _{(2)}^{aaaa}\left( 1-\frac{1}{2}\sigma \left( \sin ^{2}\alpha \right) \left( 1+3\cos ^{2}\alpha \right) \right) }},$$ where $r\equiv -2 \zeta_{IA} / (\zeta_{IA} + 2 \zeta_{IB})$ [@Stevens_pssb], and $\alpha$ is the angle between the polarization of the $\omega$ field ($\mathbf{E}_{\omega}$) and the $\left[ 001\right] $ axis, which lies in the $\left( 110\right) $ plane. The angle that maximizes $R_{S}$ depends on photon energy through $r$ and $\sigma $. We determine it numerically. ![(color online) Spin-control ratio normalized by carrier population \[Eq. (\[eq:spin\_ratio\_population\_norm\])\]. In (a), for GaAs, black lines are $\left( 111\right)$-incident, opposite circularly polarized fields; green lines are $\left( 110\right)$-incident, opposite circularly polarized fields; and red lines are $\left( 110\right)$-incident, orthogonal linearly polarized fields. In (b), for InSb, GaSb, InP, GaAs, and ZnSe, solid lines are $\left( 111\right) $-incident, opposite circularly polarized fields. The dotted line in (b) is $\left( 110\right) $-incident, orthogonal linearly polarized fields for InP.[]{data-label="fig:spin_ratio"}](fbm_spinratio) The ratio $R_{S}$ for GaAs is plotted in Fig.\[fig:spin\_ratio\](a). For $\left( 111\right)$-incidence, opposite circularly polarized fields yield the highest ratio over the studied range of photon energies. For $\left( 110\right) $-incidence, opposite circularly polarized fields yield the highest ratio, except for between 190 meV and 415 meV when $xy$-polarized fields the highest ratio. For $xy$-polarized fields, the angle that yields the largest ratio decreases from 0.99 rad to 0.53 rad from the band edge to 320 meV, and is zero for higher excess energies. The ratio $R_{S}$ for the five semiconductors InSb, GaSb, InP, GaAs, and ZnSe are plotted in Fig. \[fig:spin\_ratio\](b). At low photon energy, opposite circularly polarized fields normally incident on $\left( 111\right) $ yield the largest ratio for InSb, GaSb, GaAs, and ZnSe, whereas orthogonal linearly polarized fields normally incident on $\left(110\right)$ yield the largest ratio for InP. Summary\[sec:Summary\] ====================== We have studied the four “1+2” coherent control effects—current injection, spin-current injection, population control, and spin control—in bulk semiconductors having zinc-blende symmetry. We used an empirical, fourteen-band $\mathbf{k} \cdot \mathbf{p}$ Hamiltonian and examined the relative importance to each effect of the possible initial and intermediate states. We have also studied the crystal orientation and polarization dependencies of each effect. Cubic anisotropy is small in some cases, but large in others. We have compared the numerical calculation with analytic expressions, derived in the parabolic-band approximation, to show the value and limitations of the latter. The PBA expressions, where they are accurate, are useful to show how the effects scale in different materials. The comparison between the two approaches establishes that, at low excess photon energies, “1+2” current injection and “1+2” spin-current injection are due to interference of allowed one-photon transitions and allowed-forbidden two-photon transitions, whereas “1+2” population control and “1+2” spin control are due to interference of allowed one-photon transitions and allowed-allowed two photon transitions. It also explains the large population- and spin-control ratios predicted by the fourteen-band calculation close to the band edge, where allowed-allowed two-photon transitions dominate allowed-forbidden two-photon transitions. Neither “1+2” population control, nor “1+2” spin control have yet been experimentally studied in that spectral range. Neglect of the anomalous velocity and $\mathbf{k}$-dependent spin-orbit coupling\[App:kdepSO\] ============================================================================================== The anomalous velocity, i.e. $\mathbf{v}_{A}\equiv \left( \mathbf{v}-\mathbf{p}/m\right) =\hbar \left( \bm{\sigma }\times \bm{\nabla } V\right) /\left( 4m^{2}c^{2}\right) $, which leads to $\mathbf{k}$-dependent spin-orbit coupling in $H_{\mathbf{k}}$ from the term $\hbar \mathbf{k}\cdot \mathbf{v}_{A}$, is often neglected in $\mathbf{k}\cdot \mathbf{p}$ models [@Kane56; @CardonaPollak66; @Rossler84; @PZ90; @PZ96]. Some authors have treated matrix elements of $\bm{\nabla }V$ as additional independent parameters [@Dresselhaus55; @Rustagi69; @Bahder90; @Ostromek96]. For example, Bahder, who gives the matrix for $\hbar \mathbf{k}\cdot \mathbf{v}_{A}$ within the eight-band model, defines the model parameter [@Bahder90; @footnoteC0Ck] $$C_{0} \equiv \frac{1}{\sqrt{3}}\frac{\hbar ^{2}}{4m^{2}c^{2}}\left\langle S\right| \nabla _{x}V\left| X\right\rangle .$$ Ostromek used the value $C_{0}=0.16$ eV [Å]{} to fit the eight-band model to experimental results [@Ostromek96]. We here relate matrix elements of $\bm{\nabla }V$ (and hence matrix elements of $\mathbf{v}_{A}$) to other parameters of the model, thereby demonstrating that they can be neglected for the effects we consider. Bir and Pikus showed that the identity $\left[ H_{0},\mathbf{p}\right] =i\hbar \bm{\nabla }V$ leads to $\left\langle X\right| \nabla _{y}V\left| Z\right\rangle =0$ [@BirPikusBook]. An application of that identity to the remaining nonzero matrix elements yields $$\begin{aligned} \label{eq:nablaV_vc} \left\langle S\right| \nabla _{x}V\left| X\right\rangle & =\frac{mP_{0}}{\hbar ^{2}}\left( E_{S}-E_{X}\right),\\ \left\langle S\right| \nabla _{x}V\left| x\right\rangle & =-\frac{mP_{0}^{\prime }}{\hbar ^{2}}\left( E_{x}-E_{S}\right),\\ \left\langle X\right| \nabla _{y}V\left| z\right\rangle & =\left\langle Z\right| \nabla _{y}V\left| x\right\rangle =-\frac{mQ}{\hbar ^{2}}\left( E_{x}-E_{X}\right) ,\end{aligned}$$ and similar results for cyclic permutations and Hermitian conjugates of these. The energies $E_{S}$, $E_{X}$, and $E_{x}$ are the eigenvalues of $\left| S\right\rangle $, $\left| X\right\rangle $, and $\left| x\right\rangle $ with respect to the Hamiltonian $H_{0}$. Their values are fixed by the requirement that the eigenvalues of $H_{\mathbf{k}=\mathbf{0}}$ yield the parameters $E_{g}$, $E_{0}^{\prime }$, $\Delta _{0}$, and $\Delta _{0}^{\prime }$ [@PZ90]. Neglecting the small contribution from $\Delta ^{-}$, $E_{S}-E_{X}=E_{g}+\Delta _{0}/3$, $E_{x}-E_{S}=E_{0}^{\prime }-E_{g}+2\Delta _{0}^{\prime }/3$, and $E_{x}-E_{X}=E_{0}^{\prime }+2\Delta _{0}^{\prime }/3+\Delta _{0}/3$. Thus, gives matrix elements of $\bm{\nabla }V$ in terms of other model parameters. In particular, with parameters from Table \[Table:parameters\] for GaAs, we find $C_{0}=5\times 10^{-6}$ eV[Å]{}. From the point of view of the theory of invariants [@Suzuki74; @TrebinRossler79; @BirPikusBook; @WinklerBook], $\mathbf{k}$-dependent spin-orbit coupling amounts to using different values of $P_{0}$ for $\Gamma _{8}$ and $\Gamma _{7}$ valence bands (and similar changes for $P_{0}^{\prime }$ coupling and $Q$ coupling) [@WinklerBook]. In terms of $C_{0}$, $P_{0}\rightarrow P_{7}\equiv P_{0}+2\sqrt{3}C_{0}$ for couplings with $\Gamma _{7}$ bands and $P_{0}\rightarrow P_{8}\equiv P_{0}-\sqrt{3}C_{0}$ for couplings with $\Gamma _{8}$ bands. From , $$\frac{P_{7}-P_{8}}{P_{0}}= \frac{\sqrt{3}C_{0}}{P_{0}}=\frac{3\left( E_{S}-E_{X}\right) }{4mc^{2}} \approx \frac{3E_{g}-\Delta_{0} }{4mc^{2}}.$$ This is very small, since the numerator is on the order of eV, whereas $mc^{2}=5.11\times 10^{5}$ eV. And since this relative change in the matrix element depends on the ratio of $C_{0}$ to $P_{0}$, even the overly large coupling value of $C_{0}=0.16$eV[Å]{} has only a small effect on optical properties [@BhatPRL05; @Bhat_TPS_05]. For comparison, consider interband spin-orbit coupling parameterized by $\Delta^{-}$. In the eight-band model, interband spin-orbit coupling is a remote band effect (since it is a coupling with the $uc$ bands), which effectively causes $P_{0}\rightarrow \widetilde{P}_{7}\equiv P_{0} + \left( 2 \Delta^{-} P_{0}^{\prime} \right) / \left[ 3 \left( E_{0}^{\prime} + \Delta_{0}\right)\right]$ and $P_{0}\rightarrow \widetilde{P}_{8}\equiv P_{0} - \left( \Delta^{-} P_{0}^{\prime}\right) /\left[3 \left( E_{0}^{\prime} + \Delta_{0}^{\prime}\right)\right]$. Thus, $$\frac{\widetilde{P}_{7}-\widetilde{P}_{8}}{P_{0}} \approx \frac{\Delta^{-}}{E_{0}^{\prime}}\frac{P_{0}^{\prime}}{P_{0}}.$$ This effect, which is included in our calculation, is small (it is $4\times 10^{-3}$ in GaAs), but it is orders of magnitude larger than the relative change due to $\mathbf{k}$-dependent spin-orbit coupling. The above suggests that $\mathbf{k}$-dependent spin-orbit coupling can be neglected for the processes we consider in bulk, cubic materials. Parabolic band approximations\[App:PBA\_a-a\] ============================================= In this appendix, we discuss parabolic-band approximation (PBA) expressions, which are perturbative in the Bloch wave vector $\mathbf{k}$, for “1+2” coherent control effects. Current\[PBA:PBA:Current\] -------------------------- There have been several different calculations of $\eta $ in the PBA [@Atanasov96; @Sheik-Bahae99; @BhatSipe00; @BhatSipeExcitonic05]. Using a two-band model (one conduction and one valence band), Atanasov *et al*. obtained $\eta _{B1}\propto \left( 2\hbar \omega -E_{g}\right) ^{3/2}$ and $\eta _{B2}=0$ [@Atanasov96]. Using a three-band model, but only accounting for two-band terms, Shiek-Bahae studied the approximate scaling of “1+2” current injection spectra with the band gap $E_{g}$ and concluded that $\eta _{B1}$ and $\eta _{B2}$ are proportional to $E_{g}^{-2}\left( 2x-1\right) ^{3/2}\left( 2x\right) ^{-4}$, where $x\equiv \hbar \omega /E_{g}$ [@Sheik-Bahae99]. Our earlier PBA calculation was based on an 8-band model, included both two- and three-band terms in the two-photon amplitude, but did not include terms with the $so$ band as an intermediate state [@BhatSipe00]. More recently, we included the $so$ band as an intermediate state, but only for two-band terms [@BhatSipeExcitonic05]. The 2BTs in the 8-band model result differ from the 2BTs in the three-band model result of Sheik-Bahae by material independent factors. Spin Current ------------ The spin current PBA result is presented elsewhere [@BhatSipe00]. Here we summarize our earlier result in a new notation. For the electron spin current, $\mu_{C1}=\mu_{C2}=\mu_{C3}=0$, and \[eq:PBA\_SCe\] $$\begin{aligned} \begin{split} \mu_{N1;e} &= D\frac{m}{m_{c}}\left( \frac{m_{c,hh}}{m}\right) ^{3/2}\left( 1+Z_{c}\right) -D\frac{m}{m_{c}}\left( \frac{m_{c,hh}}{m}\right) ^{5/2}\frac{E_{P}}{3E_{g}}\frac{1-Z_{c}}{1+x m_{c,hh}/m_{hh,lh}}\\ & \quad + D\frac{m}{m_{c}}\left(\frac{m_{c,lh}}{m}\right) ^{3/2}\left( \frac{7}{3}-Z_{c}\right) -D\frac{m}{m_{c}}\left( \frac{m_{c,lh}}{m}\right) ^{5/2}\frac{E_{P}}{3E_{g}}\frac{1}{1-x m_{c,lh}/m_{hh,lh}} , \end{split} \label{eq:PBA_N1e}\\ \begin{split} \mu_{N2;e} &= D\frac{m}{m_{c}}\left( \frac{m_{c,hh}}{m}\right) ^{3/2}\left( 1+Z_{c}\right) -3D\frac{m}{m_{c}}\left( \frac{m_{c,hh}}{m}\right) ^{5/2}\frac{E_{P}}{3E_{g}}\frac{1-Z_{c}}{1+x m_{c,hh}/m_{hh,lh}} \\ & \quad -D \frac{m}{m_{c}} \left( \frac{m_{c,lh}}{m}\right)^{3/2} \left( 1-\frac{7}{3}Z_{c}\right)-Z_{c}D\frac{m}{m_{c}} \left( \frac{m_{c,lh}}{m}\right) ^{5/2}\frac{E_{P}}{3E_{g}}\frac{1}{1-x m_{c,lh}/m_{hh,lh}} , \end{split} \label{eq:PBA_N2e} \\ \begin{split} \mu_{N3;e} &= -2D\frac{m}{m_{c}}\left( \frac{m_{c,hh}}{m}\right) ^{5/2}\frac{E_{P} }{3E_{g}}\frac{1-Z_{c}}{1+x m_{c,hh}/m_{hh,lh}} \\ & \quad +2\left( 1-Z_{c}\right) D\frac{m}{m_{c}}\left( \frac{m_{c,lh}}{m}\right) ^{5/2}\frac{ E_{P}}{3E_{g}}\frac{1}{1-x m_{c,lh}/m_{hh,lh}} , \end{split} \label{eq:PBA_N3e}\end{aligned}$$ where $x \equiv \left( 2\hbar \omega -E_{g}\right) /\left( \hbar \omega \right) $, $m_{n,m}^{-1}=m_{n}^{-1}-m_{m}^{-1}$, $D$ is given in Ref. , and $Z_{c} \equiv \frac{1}{3}\frac{E_{P}\Delta_{0} }{E_{g}\left( \Delta_{0} +E_{g}\right) }\frac{m_{c}}{m}$. In and ($\mu_{N1;e}$ and $\mu_{N2;e}$), the first term is from the $hh$-$c$ transition, the second term is from the $hh$-$lh$-$c$ transition, the third term is from the $lh$-$c$ transition, and the fourth term is from the $lh$-$hh$-$c$ transition. In for $\mu_{N3;e}$, the first term is from the $hh$-$lh $-$c$ transition, and the second term is from the $lh$-$hh$-$c$ transition. Note that two-band terms make no contribution to $\mu_{N3;e}$. Spin ---- To calculate optical effects due to the interference of allowed one-photon transitions and allowed-allowed two-photon transitions, we approximate the spin and velocity matrix elements and the energy denominator by their values at the $\Gamma $ point, and approximate the energy bands in the $\delta $-function as spherical and parabolic, neglecting the small $\mathbf{k}$-linear term $C_{k}$ and the small $k^{3}$ spin-splitting. We used this method previously for two-photon spin injection [@Bhat_TPS_05]. Since bands are degenerate at the $\Gamma $ point, the lowest-order approximation to the matrix elements still depends on the direction $\hat{\mathbf{k}}$ [@BirPikusBook]. However, by averaging the microscopic expression over physical systems rotated by each point group operation \[which is equivalent to averaging over each term in Eq. \[macro\_zetaIA\] or Eq. \[macro\_zetaIB\]\], one can make the calculation using $\Gamma$-point states with pseudo-angular momentum quantized along $\hat{\mathbf{z}}$. The integral over $\mathbf{k}$ becomes a straightforward integral over the density of states in this approximation. The $\Gamma$-point basis states are given in . However, all but the $\Gamma _{6c}$ states are not eigenstates at the $\Gamma $ point due to spin-orbit coupling between upper conduction and valence bands parameterized by $\Delta ^{-}$. Using eigenstates to first order in $\Delta ^{-}$ [@Bhat_TPS_05], we find \[eq:spin\_PBA\] $$\begin{aligned} \zeta _{IA} &= -\frac{\left( -e^{3}\right) }{3\pi } \left( \left( \frac{m_{c,hh}}{m}\right) ^{3/2}+\left( \frac{m_{c,lh}}{m}\right) ^{3/2}\right) \frac{\sqrt{2\hbar \omega -E_{g}}}{\left( 2\hbar \omega \right) ^{3}} \sqrt{E_{Q}} \left( Z_{-}+Z_{+}^{\prime }+Z_{-}^{\prime \prime }\right) ,\label{eq:spin_IA_PBA} \\ \zeta _{IB} &= -\frac{\left( -e^{3}\right) }{6\pi } \left( \left( \frac{m_{c,hh}}{m}\right) ^{3/2}+\left( \frac{m_{c,lh}}{m}\right) ^{3/2}\right) \frac{\sqrt{2\hbar \omega -E_{g}}}{\left( 2\hbar \omega \right) ^{3}} \sqrt{E_{Q}} \left( Z_{+}+Z_{-}^{\prime }+Z_{+}^{\prime \prime }\right) ,\label{eq:spin_IB_PBA}\end{aligned}$$ where $$\begin{aligned} Z_{\pm } &= \sqrt{E_{P}E_{P^{\prime }}}\left( \frac{1}{E_{0}^{\prime }-\hbar \omega }\pm \frac{1}{E_{0}^{\prime }+\Delta _{0}^{\prime }-\hbar \omega }\right) \\ %\begin{split} Z_{\pm }^{\prime } &= -\frac{\Delta ^{-}E_{P}}{3}% \left[ \left( \frac{2}{E_{0}^{\prime }+\Delta _{0}}+\frac{1}{E_{0}^{\prime }+\Delta _{0}^{\prime }}\right) \frac{1}{\Delta _{0}+\hbar \omega }+ \frac{2}{E_{0}^{\prime }+\Delta _{0}}\frac{1}{E_{0}^{\prime }-\hbar \omega }\pm \frac{1}{E_{0}^{\prime }+\Delta _{0}^{\prime }}\frac{1}{ E_{0}^{\prime }+\Delta _{0}^{\prime }-\hbar \omega }\right] %\end{split} \\ Z_{\pm }^{\prime \prime } &= -\frac{\Delta ^{-}E_{P^{\prime }}}{3} \frac{1}{E_{0}^{\prime }+\Delta _{0}^{\prime }}\left( \frac{1}{E_{0}^{\prime }-\hbar \omega }\pm \frac{1}{E_{0}^{\prime }+\Delta _{0}^{\prime }-\hbar \omega }\right)\end{aligned}$$ In $Z_{\pm }$, the first term is from intermediate $sc$ states and the second term is from intermediate $lc$ and $hc$ states. In $Z_{\pm }^{\prime } $, the first term is from intermediate $so$ states, the second term is from intermediate $sc$ states, and the third term is from intermediate $lc$ and $hc$ states. In $Z_{\pm }^{\prime \prime }$, the first term is from intermediate $so$ states, and the second term is from intermediate $hc$ and $lc$ states. The term $Z_{\pm }^{\prime \prime }$ can be neglected for typical semiconductors. Note that $\left( \zeta _{IA}+2\zeta _{IB}\right) $ has contributions only from intermediate $so$ and $sc$ states. This only includes transitions from initial $hh$ and $lh$ states; transitions from initial $so$ states, which contribute when $2\hbar \omega >E_{g}+\Delta _{0}$, have been neglected. Population ---------- We derive an expression for population control using the same method used above for spin control. To first order in $\Delta ^{-}$, $$\xi _{(I)}^{abc}=\frac{-e^{3}}{3\pi }\frac{2}{\hbar }\left[ \left( \frac{m_{c,hh}}{m}\right) ^{3/2}+\left( \frac{m_{c,lh}}{m}\right) ^{3/2}\right] \frac{\sqrt{2\hbar \omega -E_{g}}}{\left( 2\hbar \omega \right) ^{3}}\sqrt{E_{Q}}\left( X_{1}+X_{2}+X_{3}\right), \label{eq:popControlPBA}$$ where $$\begin{aligned} X_{1} &=\sqrt{E_{P}E_{P^{\prime }}}\left( \frac{1}{E_{0}^{\prime }-\hbar \omega }+\frac{1}{E_{0}^{\prime }+\Delta _{0}^{\prime }-\hbar \omega } \right) , \\ X_{2} &=-\frac{\Delta ^{-}}{3}E_{P}\left[ \frac{2 \left( E_{0}^{\prime }+\Delta _{0}\right)^{-1}}{E_{0}^{\prime }-\hbar \omega }- \frac{\left( E_{0}^{\prime }+\Delta _{0}^{\prime }\right)^{-1}}{E_{0}^{\prime }+\Delta _{0}^{\prime }-\hbar \omega }+ \frac{2\left( E_{0}^{\prime }+\Delta _{0} \right)^{-1} +\left( E_{0}^{\prime }+\Delta _{0}^{\prime } \right)^{-1}}{\Delta _{0}+\hbar \omega } \right] ,\\ X_{3} &=-\frac{\Delta ^{-}}{3}\frac{E_{P^{\prime }}}{E_{0}^{\prime }+\Delta _{0}^{\prime }}\left( \frac{1}{E_{0}^{\prime }-\hbar \omega }+\frac{1}{E_{0}^{\prime }+\Delta _{0}^{\prime }-\hbar \omega }\right) .\end{aligned}$$ Note that $\left( -e^{3} \right)$ is positive. For typical semiconductors, $X_{3}$ can be neglected and $$\frac{X_{2}}{X_{1}} \approx -\frac{\Delta^{-}}{2\left( \Delta_{0} + \hbar \omega \right)} \sqrt{\frac{E_{P}}{E_{P^\prime}}} .$$ In $X_{2}$, the most important term is the last, which comes from the interference of $\left\{ hh,lh\right\}$-$so$-$c$ two-photon transitions and $\left\{ hh,lh\right\}$-$c$ one-photon transitions. The expression (\[eq:popControlPBA\]) only includes the allowed-allowed transitions from the $hh$ and $lh$ bands. At photon energies for which $2\hbar \omega >E_{g}+\Delta _{0}$, one should add the contribution due to the transition $so$-$uc$-$c$. Because of , is also an analytical expression for $\mathrm{Im}\chi ^{\left( 2\right) abc}\left( -2\omega ;\omega ,\omega \right)$. Jha and Wynne have also used $\mathbf{k}$-independent velocity matrix elements and spherical, parabolic bands to derive an expression for $\chi ^{\left( 2\right) abc}\left( -2\omega ;\omega ,\omega \right)$, but they did not include the interband spin-orbit coupling term $\Delta^{-}$ [@Jha72]. Taking the imaginary part of their Eq.4.4 for $\hbar \omega <E_{g}<2\hbar \omega $, and correcting a factor of $\pi $ error, reproduces the $\mathrm{Im}\chi ^{\left( 2\right) abc}\left( -2\omega ;\omega ,\omega \right)$ one would find from but with $X_{2}=X_{3}=0$. Also, they make the approximation $\hbar \omega \approx E_{g}/2$ in the term $X_{1}$. To get a PBA expression for the population control ratio requires PBA expressions for one- and two-photon absorption. We take the same approach used to derive , but for simplicity, we take $\Delta ^{-}=0$ in the following. In the PBA, at photon energies $2\hbar \omega <E_{g}+\Delta _{0}$, one-photon absorption is $$\xi _{(1)}^{ij}=\frac{e^{2}}{3\pi }\frac{\sqrt{2m}E_{P}}{\hbar ^{2}}\left( \left( \frac{m_{c,lh}}{m}\right) ^{\frac{3}{2}}+\left( \frac{m_{c,hh}}{m} \right) ^{\frac{3}{2}}\right) \frac{\sqrt{2\hbar \omega -E_{g}}}{\left( 2\hbar \omega \right) ^{2}} \delta^{ij}.$$ In a material of cubic symmetry, the two-photon absorption tensor $\xi_{(2)}^{ijkl}$ has three independent components $\xi_{(2)}^{aaaa}$, $\xi_{(2)}^{aabb}$, and $\xi_{(2)}^{abab}$, which are alternately parameterized by the set $\left\{ \xi_{(2)}^{aaaa}, \sigma, \delta \right\}$ (see Sec.\[sec:Population\]). The allowed-forbidden two-photon absorption in the isotropic Kane model, neglecting three- and four-band terms, is $$\xi _{(2)}^{ijkl}=\bar{\xi}_{(2)} \left[ \sqrt{\frac{m_{c,hh}}{m}} \left( \frac{3}{2}\delta ^{ik}\delta ^{jl}+\frac{3}{2}\delta ^{il}\delta ^{jk}-\delta ^{ij}\delta ^{kl}\right) + \sqrt{\frac{m_{c,lh}}{m}} \left( \frac{11}{6}\delta ^{ik}\delta ^{jl}+\frac{11}{6}\delta ^{il}\delta ^{jk}+\delta ^{ij}\delta ^{kl}\right) \right],$$ where $$\bar{\xi}_{(2)} \equiv \frac{64\sqrt{2}}{15\pi }\frac{e^{4}E_{P}}{\sqrt{m}}\frac{ \left( 2\hbar \omega -E_{g}\right) ^{\frac{3}{2}}}{\left( 2\hbar \omega \right) ^{6}} .$$ Note the additional symmetry, $\xi _{(2)}^{aaaa}=2\xi _{(2)}^{abab}+\xi _{(2)}^{aabb}$ in this isotropic model. The allowed-allowed two-photon absorption, neglecting $\Delta _{0}^{\prime }/\left( E_{0}^{\prime }-E_{g}+\hbar \omega \right) $, has $\xi _{(2)}^{aaaa}=\xi _{(2)}^{aabb}=0$ and $$\xi _{(2)}^{abab}=\xi _{(1)}^{aa}\frac{e^{2}}{\omega ^{2}m^{2}}\frac{2m}{E_{P}} \frac{E_{P_{0}^{\prime }}E_{Q}}{\left( E_{0}^{\prime }-E_{g}+\hbar \omega \right) ^{2}},$$ which agrees with Arifzhanov and Ivchenko [@Arifzhanov75]. Thus, at photon energies for which allowed-allowed transitions dominate two-photon absorption, $$R\approx \frac{\xi _{(I)}}{\sqrt{\xi _{(1)}\xi _{(2)}^{abba}}}=1, \label{popcontrol_ratio_bandedge}$$ whereas when allowed-forbidden transitions dominate two-photon absorption, $$R=2\hbar \omega \sqrt{\frac{E_{Q}E_{P^{\prime }}}{E_{P}\left( 2\hbar \omega -E_{g}\right) }}\sqrt{\frac{\left( \frac{m_{c,hh}}{m}\right) ^{3/2}+\left( \frac{m_{c,lh}}{m}\right) ^{3/2}}{\frac{9}{10}\sqrt{\frac{m_{c,hh}}{m}}+% \frac{11}{10}\sqrt{\frac{m_{c,lh}}{m}}}}\left\{ \frac{1}{\Delta _{0}^{\prime }+E_{0}^{\prime }-E_{g}+\hbar \omega }+\frac{1}{E_{0}^{\prime }-E_{g}+\hbar \omega }\right\}$$ This work was financially supported by the Natural Science and Engineering Research Council, Photonics Research Ontario, and the US Defense Advanced Research Projects Agency. We gratefully acknowledge many stimulating discussions with Ali Najmaie, Fred Nastos, Eugene Sherman, Art Smirl, Marty Stevens, and Henry van Driel. 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--- abstract: 'Recent developments on medium modifications of hadron properties in dense and hot hadronic matter are discussed. I will focus in particular on the behavior of spectral functions associated to collective scalar-isoscalar modes, kaons and vector mesons from ordinary nuclear matter to highly excited matter produced in relativistic heavy-ion collisions from SIS to SPS energies. Various theoretical approaches are presented in connection with the interpretation of experimental data. Special emphasis will be put on the role of chiral dynamics and chiral symmetry restoration. I will discuss in particular to which extent the broadening of the rho meson peak signals the onset of chiral symmetry restoration.' author: - | Guy Chanfray\ Institut de Physique Nucléaire de Lyon, IN2P3-CNRS et Université Claude-Bernard Lyon I, 43 Bd du 11 Novembre 1918, 69622 Villeurbanne Cedex, France title: 'Hadrons in dense and hot matter : implications of chiral symmetry restoration' --- 1\. true cm Introduction ============ The problem of the in-medium modifications of hadron properties have motivated numerous works in the recent years both at the experimental and theoretical levels. One major goal of this rapidly developing field is the very fundamental problem of progressive chiral symmetry restoration with increasing temperature and/or baryonic density. In this talk I will try to draw some conclusions on what has been established recently starting from very much discussed examples (rho and sigma mesons, kaons). It will be also emphasized that relevant information can be obtained from variety of sources ranging from intermediate energy physics in the GeV range probing ordinary nuclear matter up to ultrarelativistic heavy-ion collisions probing hadronic matter under extreme conditions. Chiral symmetry : breaking and restoration ========================================== Asymptotic freedom and color confinement are usually considered as the most prominent properties of our theory of strong interaction, Quantum Chromodynamics (QCD). However QCD also possesses an almost exact symmetry, the $SU(2)_L\otimes SU(2)_R$ chiral symmetry which is certainly the most important key for the understanding of many phenomena in low energy hadron physics. This symmetry originates from the fact that the QCD Lagrangian is almost invariant under the separate flavor $SU(2)$ transformations of right-handed $q_R=(u_R,d_R)$ and left-handed $q_L=(u_L,d_L)$ light quark fields $u$ and $d$ : $q_R\to e^{i\vec\tau.\vec\alpha_R/2}\,q_R,\qquad q_L\to e^{i\vec\tau.\vec\alpha_L/2}\,q_L$. The small explicit violation of chiral symmetry is given by the mass term of the QCD Lagrangian which is ${\cal L}_{\chi SB}=-m_q\, (\bar u u+\bar d d)$, neglecting isospin violation. The averaged light quark mass $m_q=(m_u+m_d)/2\le$ 10 MeV, the scale of explicit chiral symmetry breaking, has to be compared with typical hadron masses of order $1$ GeV, indicating that the symmetry is excellent and in the exact chiral limit ($m_q=0$) left-handed and right-handed quarks decouple. From the associated left-handed and right-handed conserved currents, one usually introduces two linear combinations, the vector and axial currents : $${\cal V}^\mu_k=\bar q\,\gamma^\mu\, {\tau_k\over 2}\, q,\qquad {\cal A}^\mu_k=\bar q\,\gamma^\mu\,\gamma_5 \,{\tau_k\over 2}\, q\label{CURR}$$ The corresponding charges $Q_k^V$ and $Q_k^A$ commute with the QCD hamiltonian. However at variance with the vector charges (which actually coincide with the isospin operators) the axial charges of the QCD vacuum are not zero : $Q_k^A |0\rangle \ne 0$. Hence the QCD vacuum does not possess the symmetry of the vacuum [*i.e.*]{} chiral symmetry is spontaneously broken (SCSB). This key property of the QCD vacuum is evidenced by a set of remarkable properties listed below. - The appearance of (nearly) massless goldstone bosons : the pions with extremely small mass compared to other hadrons. - The building-up of a chiral quark condensate : $\langle \bar q q \rangle =\langle \bar u u +\bar d d \rangle/ 2$ which explicitly mixes, in the broken vacuum, left-handed and right-handed quarks ($\langle \bar q q \rangle =\langle \bar q_L q_R + \bar q_R q_L\rangle/2$). Another order parameter at the hadronic scale is the pion decay constant $f_\pi=94$ MeV which is related to the quark condensate by the Gell-Mann-Oakes-Renner relation : $-2 m_q <\bar q q>_{vac}=m^2_\pi f^2_\pi$ valid to leading order in the current quark mass. It leads to a large negative value $<\bar q q>_{vac}\simeq$ -(240 MeV)$^3$ indicating strong dynamical breaking of chiral symmetry. - The absence of parity doublets. The normal Wigner realization of chiral symmetry would imply a doubling of the hadron spectrum. Each hadron would have a “chiral partner” with opposite parity and (nearly) the same mass. This is obviously not the case since the possible chiral partners (such as $\pi(140)-\sigma(400-1200)$, $\rho(770)-a_1(1260)$, $N(940)-N^*(1535)$) show a large mass splitting $\Delta M=500$ MeV. When hadronic matter is heated and compressed, initially confined quarks and gluons start to percolate between the hadrons to be finally liberated. This picture is supported by lattice simulations showing that strongly interacting matter exhibits a sudden change in energy- and entropy-density (possibly constituting a true phase transition) within a narrow temperature window around $T_c=170$ MeV. This transition is accompanied by a sharp decrease of the quark condensate indicating chiral symmetry restoration. However far before the critical region, partial restoration should follow through the simple presence of hadrons. Indeed, inside the hadrons the scalar density, originating either from the valence quarks or from the pion scalar density (the virtual pion cloud), is positive hence decreasing the quark condensate. Said differently, the presence of hadrons locally restores chiral symmetry. This statement can be made quantitative since, to leading order in hadron densities, the condensate evolves according to [@DR90; @CO92]: $$R={\langle\langle \bar q q\rangle\rangle (\rho_h, T)\over \langle \bar q q\rangle_{vac}}=1-\sum_h\,{\rho_h \Sigma_h\over f_\pi^2 m_\pi^2}.\label{DROP}$$ Each hadron species present with scalar density $\rho_h$ contributes to the dropping of the condensate through a characteristic quantity $\Sigma_h$ directly related to the integrated quark scalar density inside the hadron $h$ : $\Sigma_h/m=\int_h\, d{\bf r}\, \langle h| \bar u u+\bar d d |h\rangle$. In nuclear matter the relevant quantity is the nucleon sigma commutator $\Sigma_N\simeq 45$ MeV. Putting the numbers together one finds a $30 \%$ restoration at normal nuclear matter density. The sigma commutator and the dropping of the chiral condensate can be estimated with effective theories in terms of the relevant degrees of freedom. It receives contribution from valence quarks, scalar field and virtual pion cloud. There is strong indication (from model calculations and analysis of photon data) showing that the major part of the nucleon sigma term comes from the pion cloud piece [@JA92; @BI92; @CH99]: $ \Sigma_N^{(\pi)}=\int_N\, d{\bf r} \langle N| m^2_\pi\,\vec\Phi_\pi^2/2 |N\rangle\simeq 30$ MeV. The leading order formula (\[DROP\]), valid only for a non-interacting medium can be promoted to an exact one by replacing in nuclear matter the pion-nucleon sigma term by the full pion-nucleus sigma term per nucleon. Various higher order contributions have been examined. The role of short-range correlations has been found to be weak [@DCE1] but pion exchange contribution ([*i.e.*]{} the modification of the pion scalar-density $\langle \Phi^2_\pi\rangle$ itself related to the full longitudinal spin-isospin response and p-wave collective pionic modes) yields an increase of the pion-nucleon sigma term of about $5$ MeV at normal density [@MA00]. The acceleration of chiral symmetry restoration has not been found in the work of ref.[@LU00] based on an effective chiral lagrangian. In the framework of relativistic theories short range repulsion (omega exchange) also yields a deceleration of symmetry restoration [@BR96]. Despite the fact that the condensate is not an experimental observable, it is hardly conceivable that such a strong modification of the QCD vacuum should not have spectacular consequences on hadronic properties, namely on the hadronic spectral functions. A number of works have been devoted to the possible link between the evolution of the masses and the condensates using various models (sigma models , NJL model,..) most of the time at the mean field level. This activity has culminated with the universal scaling laws proposed by Brown and rho, where hadron masses drop together with quark and gluon condensates [@SCAL]. However, the link between the evolution of the masses and the condensates cannot be an absolute one. For instance the pionic piece of the quark condensate does not contribute to the evolution of the mass [@GU00]. It manifests in a more subtle way through the mixing of the vector and axial-vector correlators. More generally the modification of hadronic spectral functions is certainly not restricted to the shift of centroids of mass distributions. In that respect we will discuss how chiral dynamics may generate a softening/sharpening or a broadening of hadronic spectral functions with some specific and highly debated examples (sigma, kaon, and rho mesons). The key question is the relationship between the observed reshaping and chiral symmetry restoration. One possible strategy to obtain this crucial connection is to make a simultaneous study of the spectral functions associated with chiral partners. A very important example is the rho meson and the axial-vector meson $a_1$ and we will see that there is a mixing of the associated current correlators trough the presence of the pion scalar density as already mentioned just above. Scalar-isoscalar modes in nuclear matter ======================================== There are at least two excellent reasons to study the in-medium modifications of the pion-pion interaction in the scalar-isoscalar channel both being related to fundamental questions in present-day nuclear physics. The first one relies on the binding energy of nuclear matter since a modification of the correlated two-pion exchange may have some deep consequences on the saturation mechanism [@MA99]. The second one is the direct connection with chiral symmetry restoration. Such a restoration implies that there must be a softening of a collective scalar-isoscalar mode, usually called the sigma meson, which becomes degenerate with its chiral partner [*i.e.*]{} the pion at full restoration density, even if this meson is not well identified in the vacuum. This also implies that at some density the sigma meson spectral function should exhibit a spectacular enhancement near the two-pion threshold. This effect can be seen as a precursor effect of chiral symmetry restoration associated with large fluctuations of the quark condensate near phase transition [@HA99]. The medium effect which has been first proposed is a consequence of the modification of the two-pion propagator and unitarized $\pi\pi$ interaction from the softening of the pion dispersion relation by p-wave coupling to $p-h$ and $\Delta-h$ states [@SC88]. The existence of the collective pionic modes produces a strong accumulation of strength near the two-pion threshold in the scalar-isoscalar channel around $\rho_0$. On the experimental side the CHAOS collaboration has measured at TRIUMF the invariant mass distribution of the produced pion pair in $A (\pi^+,\pi^+\pi^-)$ and $A (\pi^+,\pi^+\pi^+)$ for incoming pions of energy $283$ MeV (fig.1) [@BO00]. In the $\pi^+ \pi^-$ channel a strongly A-dependent accumulation of strength is growing up from hydrogen to lead. This effect is not present in the $\pi^+ \pi^+$ channel which is is purely isospin $I=2$ contrary to the $\pi^+ \pi^-$ channel predominantly isoscalar. Since the angular distribution shows that the $\pi^+\pi^-$ pair is almost in a pure s-wave state, this process probes scalar-isoscalar modes inside the nucleus. According to the first realistic calculations [@SC98], this reshaping of the strength coming from the p-wave pionic collective modes may provide a partial explanation of the CHAOS data. These last results have been questioned in another recent paper where it is found that pion absorption forces the reaction to occur at very peripheral density [@VI99]. Hence the effect of chiral symmetry restoration has to be included on top of p-wave pionic effects to reach a better description of the data. This has been achieved in recent works using the linear sigma model implemented by adding a one-parameter form factor $v(k)$ to fit the phase shifts in vacuum once the scattering amplitude is unitarized [@ZH00]. The (in-medium) unitarized scalar-isoscalar $\pi\pi$ T matrix (in the CM frame and at total energy $E$ of the pion pair) has been taken as [@SC98; @ZH00] : $$\langle{\bf k}, -{\bf k}|T(E)|{\bf k}', -{\bf k}'\rangle=v(k) v(k')\, {6 \lambda (E^2-m^2_\pi)\over 1-3\lambda \Sigma(E)} \,\bigg( E^2-m^2_\sigma- {6 \lambda^2 f^2_\pi \Sigma(E)\over 1-3\lambda \Sigma(E)}\bigg)^{-1}\label{TMAT}$$ where $\lambda$ is the $\sigma\pi\pi$ coupling and the last factor in (\[TMAT\]) is nothing but the unitarized sigma meson propagator $D_\sigma(E)$ ([*i.e*]{} with two-pion loop). The p-wave collective effects are embedded in the two-pion loop : $$\Sigma(E)=\int {d{\bf q}\over (2\pi)^3}\, v(q)\, \int{i \,dq_0\over 2\pi}\, D_\pi({\bf q}, q_0)\, D_\pi(-{\bf q},\, E-q_0).$$ The pion propagator $D_\pi({\bf q}, q_0)$ is calculated in a standard nuclear matter approach and incorporates the p-wave coupling of the pion to delta-hole states with short-range screening described by the usual $g'_{\Delta\Delta}=0.5$ parameter. Chiral symmetry restoration can be accounted for by dropping the sigma mass according to : $m_\sigma(\rho)=m_\sigma\,(1-\alpha \,\rho/\rho_0)$. Such a density dependence very naturally arises in this model from the tad-pole graph where the sigma meson directly couples to the nuclear density. In [@HA99] a value of $\alpha$ in the range of 0.2 to 0.3 was found. The result for the invariant-mass distribution $Im D_{\sigma}(E_{\pi\pi})$ is shown in fig.2 at saturation density. One observes a dramatic downward shift of the mass distribution as compared to the vacuum. The low-energy enhancement, already present without sigma-mass modification ($\alpha=0$) and induced by the density dependence of the pion loop, is strongly reinforced as the in-medium $\sigma$-meson mass is included. For $\alpha=0.2$ and $\alpha=0.3$ the peak height is increased by a factor 2 and 4 respectively. Similarly for the T-matrix, a sizable effect can be noticed in its imaginary part which might be sufficient to explain the findings of the CHAOS collaboration. To facilitate the comparison with theoretical calculations the experimental group has presented his data in the form of a composite ratio [@BO00]: $$C^A_{\pi\pi}=\big(M^A_{\pi\pi}/\sigma^A_T\big)\,/\, \big(M^N_{\pi\pi}/\sigma^N_T\big)$$ where $\sigma^A_T$ ($\sigma^N_T$) is the measured total cross section of the $(\pi, 2 \pi)$ process in nuclei (nucleon). As is apparent on fig.3, while a conventional calculation is able to reproduce the $\pi^+\pi^+$ case, chiral symmetry restoration on top of p-wave effects is needed to reach a better agreement with data in the $\pi^+\pi^-$ case. It is therefore very tempting to conclude that a precursor effect of chiral symmetry restoration has been seen in this experiment. However one has to keep in mind that this comparison with theoretical predictions is at best semi-quantitative and a full calculation for the absolute yields incorporating all the complicated reaction dynamics together with the relevant medium effects remains to be done before firm conclusions can be drawn. Nevertheless experimental efforts in that direction has certainly to be encouraged since the still poorly known in-medium scalar field is of utmost importance in present-day nuclear physics. Kaons in dense matter ===================== The strong interest for studying kaon production in relativistic heavy-ion collisions originates from two complementary reasons. The first one is the sensitivity of kaon properties and propagation to the state of the produced matter and the second one is the fact that the basic kaon-nucleon interaction is governed to a large extent by the $SU(3)$ extension of chiral symmetry. Hence kaon production provides a unique opportunity to study chiral dynamics in a dense and hot medium. Recent data taken in particular at SIS by the KaoS and FOPI collaborations have demonstrated sizable medium effects; the details and experimental data can be found in the contribution of P.Senger in this present volume [@SE00]. The absence of in-plan flow and the quite strong azimuthal emission of kaons ($K^+$) together with the unexpected high sub-threshold $K^-/K^+$ ratio favorize a strong in-medium attraction for the anti-kaons ($K^-$) and a more moderate but significant repulsion for the kaon ($K^+$). This effect has been predicted by many theoretical calculations based on chiral symmetric frameworks. From a chiral symmetry lagrangian taken at the mean-field level, one can easily establish the leading order terms modifying kaon and anti-kaon masses at finite proton and neutron densities : $$\begin{aligned} \Delta m_{K^+}&=&+{1\over 2}\,{1\over 4 f^2_\pi} \bigg(3\,(\rho_p+\rho_n)+(\rho_p-\rho_n)\bigg) -\,{\Sigma_{KN}\over 2 m_K\, f^2_\pi}\,\rho \label{NASS}\\ \Delta m_{K^-}&=&-{1\over 2}\,{1\over 4 f^2_\pi} \bigg(3\,(\rho_p+\rho_n)+(\rho_p-\rho_n)\bigg) -\,{\Sigma_{KN}\over 2 m_K\, f^2_\pi}\,\rho\,.\label{MASS}\end{aligned}$$ The first contribution arises from vector current interaction between the pseudo-scalar mesons and the nucleons and is determined by chiral symmetry alone (Weinberg-Tomozawa theorem). It gives the bulk of repulsion for the kaon and attraction for the anti-kaons. The second term of scalar nature is proportional to the not very well-known kaon-nucleon sigma term $\Sigma_{KN}$ and yields attraction in both cases; it is actually to next order in the chiral perturbation expansion and could be balanced by other repulsive terms at this order but without altering too much the strong mass splitting of $S=1$ and $S=-1$ modes in matter. For the specific case of $K^+$, there is a consensus between the various transport code approaches in favor of repulsion to account for the flow variables data [@LI96; @WA00; @CA99]. For what concerns the $K^+$ spectra and excitation functions in the sub-threshold or near-threshold region there is also a consensus for the dominant role played by secondary processes [@CA99; @FU97; @AI00] such as $\pi N\to YK$ or $N\Delta\to NYK$ ($Y=\Lambda, \Sigma$) which are very sensitive to modifications of in-medium masses. Although, there are still uncertainties for the input cross-sections (this is especially important for the $N\Delta$ channel), transport code calculations also favor in-medium repulsion for the $K^+$ [@FU99; @AI00]. The sensitivity to the EOS has also been investigated. It turns out that when in-medium kaon mass is incorporated the sensitivity to the EOS decreases. In the case of soft EOS , the system reaches a higher density yielding a higher $K^+$ production; this effect can be counterbalanced by the larger increase of the $K^+$ mass increasing the threshold production. Nevertheless, according to [@FA00] detailed study of Au-Au versus C-C excitation functions with in-medium repulsion favors a soft EOS with compressibility of the of the order of $200$ MeV. Transport calculations [@LI97; @SI97; @CA99] also show that sub-threshold $K^-$ spectra are better reproduced if a dropping kaon mass of the type given by the mean field approach (\[NASS\],\[MASS\]) is incorporated. It has been realized that the situation is not so simple because the anti-kaon interaction in the medium is governed by a very rich and rather complex chiral dynamics as we will discuss below. To begin with, we have to understand a rather paradoxal situation. On one hand, although the basic $K^- N$ Weinberg-Tomozawa is attractive, the scattering lengths imply a repulsive interaction in vacuum. On the other hand, kaonic atom data are compatible with a strongly attractive anti-kaon optical potential as large as 200 MeV once extrapolated at normal nuclear matter density. The solution of this paradox is the existence of a very peculiar object, the $\Lambda(1405)$ resonance. Following ref. [@WA96; @LUKA; @KOCH; @RA00] one can start with a coupled channel equation for the vacuum $\bar K N$ scattering matrix, schematically written as : $$\langle \bar K N|T|\bar K N\rangle=\langle \bar K N|V_{WT}|\bar K N\rangle +\langle \bar K N|V_{WT}|M B\rangle\, G_{M B}\, \langle M B|T|\bar K N\rangle$$ where $G_{M B}$, with $M(\bar K, \pi)$ and $B(N, \Lambda, \Sigma)$, is the meson-baryon propagator in the intermediate state. It turns out that the attractive Weinberg-Tomozawa interaction ($V_{WT}$) in the isospin $I=0$ channel is sufficiently strong that it generates a pole at 27 MeV below the $K^- p$ threshold. This pole correspond to a quasi-bound state, the $\Lambda(1405)$ which decays into $\pi \Sigma$ with a width of about 50 MeV. This is precisely the existence of this quasi-bound state what makes the $K^-p$ interaction repulsive at threshold, while the Weinberg-Tomozawa amplitude is attractive. Going at finite density Pauli blocking starts to work in the intermediate $\bar K N$ state and the quasi-bound state moves up above threshold. Hence the $\Lambda(1405)$ dissolves already at very small density [@WA96] making the in-medium $K^- N$ interaction attractive as seen in kaonic atom data. However, as pointed out by M. Lutz [@LUKA], a self-consistent incorporation of the in-medium dressed $K^-$ propagator in the intermediate state modifies the picture. The position of the resonance does not really move up but it considerably broadens. The in-medium $K^-$ spectral function exhibits a two-level structure, the lower mode corresponding to the $K^-$ pole branch and the upper to the $\Lambda(1405)$-hole branch with strength decreasing very fast with increasing density (fig.4). Self-consistency makes the two-peak mode barely visible and if the modification of the pion dispersion relation is also included the $K^-$ completely melts with the $\Lambda(1405)-h$ branch even losing its status of quasi-particle [@RA00]. Finally it has been emphasized that the observed enhancement of $K^-$ production has probably little relation with the dropping of the anti-kaon mass or in other words to its optical potential at zero momentum [@KOCH]. Indeed, in the conditions prevealing in heavy-ion collisions, the anti-kaons have a typical momentum of $300$ MeV with respect to the matter rest frame. In that regime a self-consistent calculation shows little attraction if not repulsion and the medium effect might originate from the enhancement of cross-section of important secondary processes such as $\pi \Sigma \to K^- p$ [@KOCH]. Although, there is not yet a quantitative understanding of the $K^-$ production, the physics of the $K^-$ is a beautiful example of the very rich in-medium chiral dynamics. It is also an example of the strong reshaping and broadening of an hadronic spectral function which is also present but for different reasons in the case of the rho meson. Dilepton production and the rho meson ===================================== Low mass dilepton production has been reported as being among the evidences for the formation of a new phase of matter in relativistic heavy-ion collisions at CERN/SPS [@HE00]. In particular the CERES collaboration [@CERES] has observed an important radiation in the invariant mass region $300 -700$ MeV/c beyond what is expected from the conventional sources able to explain the proton-nucleus data (fig.5). Since these conventional sources (the so-called hadronic cocktail) correspond to final state Dalitz decays ($\eta, \eta'\to \gamma e^+ e^-$, $\omega\to \pi^0 e^+ e^-$) and direct vector meson decays ($\rho, \omega, \Phi\to e^+ e^-$), one can conclude that this excess of radiation originates from the interacting fireball before freeze-out. Due to the very large number of produced pions the first candidate is the $\pi^+ \pi^-\to l \bar l$ annihilation process which is dynamically enhanced by the rho meson. Using vacuum meson properties many theoretical groups have included this process within (very) different models for the space-time evolution of $A-A$ reactions. Their results are in reasonable agreement with each other, but in disagreement with the data : the experimental spectra in the mass region $300-600$ MeV/c are significantly underestimated and the rho peak itself has the tendency to be overestimated as seen from fig.6. Thus one came to the conclusion that strong medium effects yielding a flattening of the spectra are needed. This has motivated a considerable theoretical activity that I will now briefly describe. [*Dilepton production from hot and dense matter*]{}. The dilepton production rate (DPR) per unit 4-volume from a hot ($T=1/\beta$) and dense medium is given by : $${dN_{l\bar l}\over d^4x d^4 q }=-{\alpha^2\over 6 \pi^3 M^2}\, {1\over e^{\beta q^0}-1}\,g_{\mu\nu}\,\left(-{1\over \pi}\,Im \Pi_V^{\mu\nu}\right)$$ where $M$ ($M^2=q^2_0-{\bf q}^2$) is the invariant mass of the produced pair. Once the overall thermal factor has been extracted, the DPR is directly proportional to the imaginary part of the current-current correlation function : $$\Pi_V^{\mu\nu} (q)=-i\int\,d^4x\,e^{-iqx}\,\langle\langle J^\mu(x),J^\nu(0) \rangle\rangle(T,\rho_B).$$ For simplicity, we will concentrate on the (prevailing) isospin $I=1$ (isovector) projection of the electromagnetic current : $${\cal V}^\mu_\rho={1\over 2}\,\left(\bar u \gamma^\mu u- \bar d \gamma^\mu d\right)$$ which just coincides with the third component of the conserved vector current of chiral symmetry (see eq.\[CURR\]). We know from the well established Vector Dominance phenomenology (VDM) that the corresponding correlator is accurately saturated by the rho meson. This property is formally incorporated through the famous field-current identity ${\cal V}^\mu_\rho=(m^2_\rho/ g_\rho)\, \rho^\mu$. Hence dilepton production allows to reach the imaginary part of the rho meson propagator, namely the in-medium rho meson spectral function. To have some insight about manifestation of chiral symmetry restoration this vector correlator should be studied simultaneously with the axial-vector correlator in which the properties of the chiral partner of the rho meson, namely the $a_1$ meson, are encoded. In the vacuum the SCSB manifests itself in the marked difference between the $\rho$ and the $a_1$ and the transition from the hadronic to the partonic regime (“duality threshold”) is characterized by the onset of perturbative QCD around $M_{dual}\simeq 1.5$ GeV. In the medium, full chiral symmetry restoration requires the degeneracy of vector and axial correlators over the entire mass range. [*Density expansion.*]{} Several approaches have been put forward to determine the spectral properties of vector mesons in the medium. One method, very usual in nuclear physics, is the low density expansion : $$\Pi^{\mu\nu}(q, T, \mu)=\Pi^{\mu\nu}_{vac}(q)\,+\, \sum_h \rho_h\,\Pi^{\mu\nu}_h(q). \label{VIRIEL}$$ Taking the imaginary part, one obtains the contribution of hadron species $h$ present with density $\rho_h$ to the spectral function. The vacuum piece is extremely well known from $e^+ e^-$ annihilation. The hadronic part is expected to be dominated by the lightest meson ($\pi$) and baryon ($N$). In the chiral reduction formalism [@ZA96], the hadronic matrix elements ($\Pi^{\mu\nu}_h$) can be inferred from a combination of empirical information ($\pi N$, $\rho N$ or $\gamma N$ data...) and chiral Ward identities. Although model independent in spirit, this framework does not allow to perform systematic resummations. Indeed it has the tendency to overestimate the rho meson peak itself because these higher order many-body effects are absent. Nevertheless this approach explicitly contains the already mentioned axial-vector mixing that we will now discuss. [*Axial-Vector mixing.*]{} In the medium the emission and the absorption of thermal (finite temperature) or virtual (finite density) pions is able to transform a vector current into an axial current. In other words, the response of the system to a vectorial probe contains an axial contamination mediated by the pions, the pure vector piece being quenched by the emission and absorption at the same point. Hence increasing temperature or density ([*i.e.*]{} increasing the pion scalar density) makes the axial-vector mixing more and more important until full restoration where axial and vector correlators become identical. This mixing has been formally proven at finite temperature in the chiral limit, using only chiral symmetry. The finite temperature correlators are described to order $T^2$ by the following mixing of zero-temperature correlators [@DE90]: $$\begin{aligned} \Pi^{\mu\nu}_V (q; T)&=&(1-\epsilon)\,\Pi_V^{\mu\nu} (q; T=0) \,+\,\epsilon\,\Pi_A^{\mu\nu} (q; T=0) \label{MIXV}\\ \Pi^{\mu\nu}_A(q; T)&=&(1-\epsilon)\,\Pi_A^{\mu\nu} (q; T=0) \,+\,\epsilon\,\Pi_V^{\mu\nu} (q; T=0)\label{MIXA}\end{aligned}$$ where $\epsilon=T^2/ 6 f^2_\pi$ is directly proportional to the scalar density of the thermal pions. This implies that, to this order, the masses of the $\rho$ and $a_1$ meson do not change although the order parameters (quark condensate and pion decay constant) are modified in contradiction with the BR scaling law. It is amusing to note that full mixing $\epsilon=1/2$ corresponding to full symmetry restoration is realized at $T\simeq 160$ MeV very close to the lattice critical temperature. The above result has been extended beyond chiral limit in the chiral reduction formalism [@ZA96] in which the DPR writes : $$\begin{aligned} {dR\over d^4x d^4q}=& &-{\alpha^2\over \pi^3 \,q^2}\,{1\over e^{\beta q_0}+1} \bigg[Im\,\Pi(q^2)\,-\, {2\over f^2_\pi} \,\int {d{\bf k}\over (2\pi)^3}\, {n(\omega_k)\over\omega_k}\,Im\,\Pi_V(q^2)\nonumber\\ & &+\,{1\over \, f^2_\pi} \,\int {d{\bf k}\over (2\pi)^3}\, {n(\omega_k) \over\omega_k}\,\left(Im\,\Pi_A((q+k)^2)\,+\, Im\,\Pi_A((q-k)^2)\right)\,+....\bigg].\end{aligned}$$ The first term corresponds to the full electromagnetic correlator (with $\rho, \omega$ and $\phi$ pieces) and the second term exhibits the quenching of the (isovector-)vector correlator. The last term represents the axial-vector mixing beyond the soft pion limit ($k \to 0$). The integration over all the pion momenta yields a broadening of the (rho meson) spectrum which has to be understood as an unavoidable consequence of partial chiral symmetry restoration. [*QCD sum rule*]{}. The QCD sum rule approach aims at an understanding of physical current-current correlation functions in terms of QCD by relating the observed (or calculated) hadron spectrum to fundamental condensates $C_n$ (quarks, gluons) [*i.e.*]{} to the non-perturbative QCD vacuum structure. For large space-like momenta ($Q^2=-q^2$), OPE techniques lead to : $$\begin{aligned} {\Pi(q^2=-Q^2)\over Q^2}&=& \int_0^\infty\,{ds\over s}\, {\left(-{1\over\pi}\right)\, Im\,\Pi(s)\over s\,+\,Q^2} \nonumber\\ &=&{d_V\over 12\, \pi^2}\left[-C_0\,ln\left({Q^2\over \mu^2}\right) \,+\,{C_1\over Q^2}\,+\,{C_2\over Q^4}\,+\,{C_3\over Q^6}\,+.....\right] \label{SUMR}\end{aligned}$$ I will not discuss here the technical difficulties and limitations of the method which can be applied both in the vacuum and in the medium. I will only emphasize that it provides a crucial test of consistency between hadronic correlators and fundamental properties such as chiral symmetry and its restoration. Although such a test should be systematically done, the approach is only of little predictive power. It has been found [@KL97; @LEUP] that the generic decrease of the quark and gluon condensates on the right-hand-side is compatible with the phenomenologically left-hand-side if either (a) the vector meson masses decrease (together with small resonance widths as in the $\omega$ meson case) or (b) both width and mass increase (as found in most phenomenological models for the $\rho$ meson). [*Dropping mass scenario.*]{} Early QCD sum rule analysis based on a sharp ansatz for the vector mesons [@HA92] gave a decrease of the vector meson (rho and omega) of about $20\%$ at normal nuclear matter density. At this time this result has been understood as being in favor of the scaling law proposed by Brown and Rho [@SCAL] on the basis of broken scale invariance in QCD : $$\,{f^*_\pi\over f_\pi}\,=\, {m^*_\sigma\over m_\sigma}\,=\,{m^*_\omega\over m_\omega} \,=\,{m_\rho^*\over m_\rho} \,=\,\left({<\bar q q>^*\over <\bar q q>}\right)^{1/3}\,.$$ The rho mass itself plays the role of an order parameter. Although these scaling laws contradict some low density results, chiral symmetry does not forbid the vanishing of the mass at full restoration. But, as already discussed, such a dropping mass scenario is certainly not a necessary consequence of chiral symmetry restoration. Although this scenario yields a reasonable agreement with data [@PREM; @BR97], other hadronic many-body approaches containing other aspects of chiral symmetry restoration without dropping mass are also able to reproduce the data. [*Many-body approaches*]{}. More conservative approaches reside on standard many-body techniques to calculate the self-energy and consequently the rho meson spectral function. They are based on effective hadronic Lagrangians possessing chiral symmetry and incorporating vector dominance. The various parameters (coupling constants and form factor cutoffs) are constrained as much as possible by other data and phenomenology (decay rates, photoabsorption, $\rho N$ scattering [@FRI00],...). One is thus able to evaluate the in-medium modification of the rho meson from its coupling to the various many-body excitations of the dense and hot matter from which one gets the DPR at a given temperature and baryonic chemical potential. As we will see below, once various final state hadronic decays are incorporated on top of the interacting fireball contribution, such an approach is able to reproduce the enhancement of the DPR below the $\rho/\omega$ peak and the apparent depletion of the peak itself. The latter depletion has a pure many-body origin : the propagator formalism generates resummations to all orders which are totally absent in any kind of low-density expansion and/or in incoherent summations of various processes. The total thermal yield in heavy-ion reactions is obtained by a space-time integration over the density-temperature profile for a given collision system modeled within transport or hydrodynamics simulations. Another very successfull attempt is provided by a simple expanding thermal fireball [@CRW2] allowing to incorporate in a rather simple way hadronic many-body effects which are needed to obtain a consistent description of the data. In the most recent calculation [@RAWA] the trajectory starts at $(T,\rho_B)_{in}$=(190 MeV, 2.55$\rho_0$), goes through the experimentally deduced point in hadro-chemical analysis [@STAC] up to thermal freeze out $(T,\rho_B)_{fo}$=(115 MeV, 0.33$\rho_0$). Transport calculations, where no assumption is made about the degree of thermalization, also get better agreement with data once in-medium spectral function is incorporated [@CBRW]. It is convenient at least qualitatively to separate temperature and baryonic density effects. In a hot meson gas the first medium effect is the Bose enhancement of the $\pi\pi$ annihilation or, in terms of the rho meson propagator, the temperature effect affecting the two-pion loop contribution to the rho self-energy. In addition $\rho$-meson scattering, in first rank $\rho \pi\to \omega, a_1$, also significantly contributes to the rho self-energy. It is important to notice that the $a_1$ piece is of axial-vector mixing nature and the corresponding contribution to the DPR can be seen as a consequence of partial chiral symmetry restoration. Historically the first advocated baryonic density effect was the medium modification of the two-pion loop through p-wave coupling of the pion to $\Delta$-h states. This effect, usually referred as the pion cloud contribution, gives a significant enhancement of the DPR below the rho peak [@CHAN] and is mainly related to the coupling of the rho to dressed pions and $\Delta$-h states (the so-called pisobars). Again, the very important point is that it contains an explicit mixing between the vector and the axial baryonic current, as recently demonstrated [@CH99]. Finally, the direct coupling of the rho to baryonic resonances having sizable coupling to the rho has been incorporated. In a many-body language the rho couples to $N^*-h$ excitations building up the so-called rhosobars [@PIRN]. Among them the $N^*(1520)$ plays a prominent role (since the coupling is of s-wave nature) and gives a very important contribution to the low mass enhancement [@PE98]. Contrary to the case of the $a_1$ and pion cloud contributions, the connection with chiral symmetry restoration of the $N^*(1520)$ is not transparent. The resulting full spectrum which incorporates all the above effects within the expanding fireball model nicely accounts for the data (fig.7). A very important observation is that this spectrum is very flat and very close to a pure perturbative quark-gluon spectrum (see lower panel of fig.7). One possible conclusion is that chiral symmetry restoration manifests itself as a lowering of the quark hadron duality threshold from its free space value of 1.5 GeV down to 0.5 GeV near the phase boundary [@RAWA]. Conclusion ========== We have seen the prominent role of in-medium chiral dynamics to generate strong reshaping of hadronic spectral functions. Chiral symmetry restoration itself yields a softening and a sharpening of the scalar-isoscalar modes and the structure seen in ($\pi,\pi\pi$) has been tentatively attributed to precursor effects of the restoration associated to strong fluctuations of the chiral condensate. However, most of the time one observes a broadening and a flattening of hadronic spectral functions. This is in particular true in the rho meson channel and convincing arguments based on detailed calculations yield to the conclusion that dilepton spectra in relativistic heavy-ion collisions probing the phase transition region constitute a possible signature of this restoration associated to a lowering of the quark-hadron duality threshold. Nevertheless it is clear that a strong effort has to be pursued to improve theoretical analysis in connection with present and forthcoming experimental data. For instance, it is not clear that the ($\pi^-,\pi^0 \pi^0$) data [@NI99] can be described within the theoretical framework of section 3. For what concerns dilepton production the connection of the resonance contribution (especially the $N^*(1520)$) with chiral symmetry restoration and axial-vector mixing remains to be fully elucidated although some suggestions have been proposed [@KI00]. 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--- abstract: | Astronomers usually need the highest angular resolution possible when observing celestial objects, but the blurring effect of diffraction imposes a fundamental limit on the image quality from any single telescope. Interferometry allows light collected at widely-separated telescopes to be combined in order to synthesize an aperture much larger than an individual telescope thereby improving angular resolution by orders of magnitude. Because diffraction has the largest effect for long wavelengths, radio and millimeter wave astronomers depend on interferometry to achieve image quality on par with conventional large-aperture visible and infrared telescopes. Interferometers at visible and infrared wavelengths extend angular resolution below the milli-arcsecond level to open up unique research areas in imaging stellar surfaces and circumstellar environments. In this chapter the basic principles of interferometry are reviewed with an emphasis on the common features for radio and optical observing. While many techniques are common to interferometers of all wavelengths, crucial differences are identified that will help new practitioners to avoid unnecessary confusion and common pitfalls. The concepts essential for writing observing proposals and for planning observations are described, depending on the science wavelength, the angular resolution, and the field of view required. Atmospheric and ionospheric turbulence degrades the longest-baseline observations by significantly reducing the stability of interference fringes. Such instabilities represent a persistent challenge, and the basic techniques of phase-referencing and phase closure have been developed to deal with them. Synthesis imaging with large observing datasets has become a routine and straightforward process at radio observatories, but remains challenging for optical facilities. In this context the commonly-used image reconstruction algorithms CLEAN and MEM are presented. Lastly, a concise overview of current facilities is included as an appendix. author: - 'J.D. Monnier & [R.J. Allen ]{}' bibliography: - 'apj-jour.bib' - 'pss\_monnier\_allen\_preprint.bib' title: | Radio & Optical Interferometry:\ Basic Observing Techniques and Data Analysis\ *To be published by Springer in Volume 2 of\ Planets, Stars, & Stellar Systems* --- Interferometry in Astronomy {#sec:introduction} =========================== Introduction ------------ The technique of interferometry is an indispensable tool for modern astronomy. Typically the telescope diameter $D$ limits the angular resolution for an imaging system to $\Theta\approx\frac{\lambda}{D}$ owing to diffraction, but interferometry allows the achievement of angular resolutions $\Theta\approx \frac{\lambda}{B}$ where the baseline $B$ is set by the distance between telescopes. Interferometry has permitted the angular resolution at radio wavelengths to initially reach, and now to significantly surpass, the resolution available with both ground- and space-based optical telescopes. Indeed, radio astronomers routinely create high-quality images with high sensitivity, high angular resolution, and a large field-of-view using arrays of telescopes such as the Very Large Array (VLA), the Combined Array for Research in Millimeter-wave Astronomy (CARMA), and now the Atacama Large Millimeter Array (ALMA). Interferometer arrays are now the instruments of choice for imaging the wide range of spatial structures found both for Galactic and for extragalactic targets at radio wavelengths. At optical wavelengths, interferometry can improve the angular resolution down to the milli-arcsecond level, an order-of-magnitude better than even the Hubble Space Telescope. While atmospheric turbulence limits the sensitivity much more dramatically than for the radio, optical interferometers can nevertheless measure the angular sizes of tens of thousands of nearby Galactic objects and even a growing sample of distant Active Galactic Nuclei (AGN). Recently, optical synthesis imaging of complex objects has been demonstrated with modern arrays of 4–6 telescopes, producing exciting results and opening new avenues for research. Both radio and optical interferometers also excel at precision astrometry, with the potential for [*micro-arcsecond*]{}-level precision for some applications. Currently, it is ground-based radio interferometry (e.g. the VLBA) that provides the highest astrometric performance, although ground-based near-IR interferometers are improving and measure different astronomical phenomena. This chapter will provide an overview of interferometry theory and present some practical guidelines for planning observations and for carrying out data analysis at the premier ground-based radio and optical interferometer facilities currently available for research in astronomy. In this chapter, the term “radio” will be used as a shorthand for the whole class of systems from sub-mm to decametric wavelengths which usually employ coherent high-frequency signal amplification, superheterodyne signal conversion, and digital signal processing, although detailed instrumentation can vary substantially. Likewise, the term “optical” generally describes systems employing direct detection, i.e. the direct combination of the signals from each collector without amplification or mixing with locally-generated signals[^1]. Historically, radio and optical interferometry have usually been discussed and reviewed independently from each other, leaving the student with the impression that there is something fundamentally different between the two regimes of wavelength. Here a different approach is taken, presenting a unified and more wavelength-independent view of interferometry, nonetheless noting important practical differences along the way. This perspective will demystify some of the differing terminology and techniques in a more natural way, and hopefully will be more approachable for a broad readership seeking general knowledge. For a more much detailed treatment of radio interferometry specifically, refer to the classic text by @tms2001 and the series of lectures in the NRAO Summer School on Synthesis Imaging [e.g. @taylor1999]. Optical interferometry basics have been covered in individual reviews by @quirrenbach2001 and by @monnier2003, and a useful collection of course notes can be found in the NASA-Michelson Course Notes [@mss2000] and ESO-VLTI summer school proceedings [@malbet2007]. Recently, a few textbooks have been published on the topic of optical interferometry specifically, including @labeyrie2006, @glindemann2010, and @saha2011. Further technical details can also be found in Chapters 7 and 13 of the first volume in this series. This chapter begins with a brief history of interferometry and its scientific impact on astronomy, a basic scientific context for newcomers that illustrates why the need for better angular resolution has been and continues to be one of the most important drivers for technical innovation in astronomy. Scientific impact ----------------- Using interferometers in a synthesis imaging array allows designers to decouple the diffraction-limited angular resolution of a telescope (which improves linearly with the telescope size) from its collecting area (which, for a filled aperture, grows quadratically with the size). In the middle of the 20th century, radio astronomers faced a challenge in their new science; the newly-discovered “radio stars” were bright enough to be observed with radio telescopes of modest collecting area, but the resolution of conventional “filled aperture” reflecting telescopes was woefully inadequate (by one or more orders of magnitude) to measure the positions and angular sizes of these enigmatic new cosmological objects with a precision sufficient to permit an identification with an optical object. Thus separated-element interferometry, although first applied in astronomy at optical wavelengths [@michelson1921], began to be applied in the radio with revolutionary results. At radio wavelengths, the epoch of rapid technological development began more than 50 years ago, and now interferometry is the “workhorse” technique of choice for most radio astronomers in the world. A steady stream of exciting new results has flowed from these instruments even until today, and a complete census of the major discoveries to date would be very lengthy indeed. Here, our attention is focussed on the earliest historical discoveries that *required* radio interferometers, and we have listed our nominations in Table \[table:discovery\]. In Figure \[fig\_niceimages\]a, the spectacular image of radio jets in quasar 3C175 by the VLA is shown to illustrate the high-fidelity imaging that is possible using today’s radio facilities. [llll]{}\ Solar radio emission from sunspots & 1945-46 & Australia, Sea cliff interferometer & R1\ First Radio Galaxies NGC 4486 & NGC 5128 & 1948 & New Zealand, Sea cliff interferometer & R2\ Identification of Cygnus A& 1951-53 & Cambridge, Würzburg antennas & R3\ Cygnus A double structure & 1953 & Jodrell Bank, Intensity interferometer & R4\ AGN superluminal motions & 1971 & Haystack-Goldstone VLBI & R5\ Dark matter in spiral galaxies & 1972-78 & Caltech interferometer, Westerbork SRT & R6\ Spiral arm structure & kinematics & 1973-80 & Westerbork SRT & R7\ Compact source in Galactic center & 1974 & NRAO Interferometer & R8\ Gravitational lenses & 1979 & Jodrell Bank Mk1 + Mk2 VLBI & R9\ NGC 4258 black hole & 1995 & NRAO VLBA & R10\ \ Physical diameters of hot stars & 1974 & Narrabri Intensity Interferometer& O1\ Empirical effective temperature scale for giants & 1987 & I2T/CERGA & O2\ Survey of IR Dust Shells & 1994 & ISI & O3\ Geometry of Be star disks & 1997 & Mark III & O4\ Near-IR Sizes of YSO disks & 2001 & IOTA & O5\ Pulsating Cepheid $\zeta$ Gem & 2001 & PTI & O6\ Crystalline silicates in inner YSO disks & 2004 & VLTI & O7\ Vega is a rapid rotator & 2006 & NPOI & O8\ Imaging gravity-darkening on Altair & 2007 & CHARA & O9\ Near-IR sizes of AGN & 2009 & Keck-I & O10 Modern long-baseline optical interferometry started approximately 30 years after radio interferometry, following the pioneering experiments and important scientific results with the Narrabri intensity interferometry [e.g., @hb1974] and the heterodyne work of the Townes’ group at Berkeley [@jbt1974]. The first successful direct interference of stellar light beams from separated telescopes was achieved in 1974 [@labeyrie1975] and this was followed by about twenty years of two-telescope (i.e., single baseline) experiments which measured the angular diameters of a variety of objects for the first time. The first imaging arrays with more than two telescopes were constructed in the 1990s, and the COAST interferometer was first to make a true optical synthesis image using techniques familiar to radio astronomers [@baldwin1996]. Keck and VLT interferometers both include 8-m class telescopes, making them the most sensitive facilities in the world. Recently, the CHARA array has produced a large number of new images in the infrared using combinations of 4 telescopes simultaneously. Table \[table:discovery\] lists a few major scientific accomplishments in the history of optical interferometry showing the diversity of contributions in many areas of stellar astronomy and even recent extragalactic observations of active galactic nuclei. With technical and algorithm advances, model-independent imaging has become more powerful and a state-of-the-art image from the CHARA array is presented in Figure \[fig\_niceimages\]b, showing the surface of the rapidly rotating star Alderamin. Interferometry in theory and practice ===================================== Introduction ------------ The most basic interferometer used in observational astronomy consists of two telescopes configured to observe the same object and connected together as a Michelson interferometer. Photons collected at each telescope are brought together to a central location and combined coherently at the “correlator” (radio term) or the “combiner” (optical term). For wavelengths longer than $\sim$0.2 millimeters, the free-space electric field is usually converted into cabled electrical signals and coherently amplified at the focus of each telescope. The celestial signal is then mixed with a local oscillator signal sent to both telescopes from a central location, and the difference frequency transmitted in cables back to the centrally-located correlator. For shorter wavelengths, cable losses increase, and signal transmission moves eventually into free space in a more “optical” mode, using mirrors and long-distance transmission of light beams in (sometimes evacuated) pipes. Depending on the geometry, the light from an astronomical object will in general be received at one telescope before it arrives at the other. If the fractional signal bandwidth $\Delta \nu$ is very narrow (either because of the intrinsic emission properties of the source, e.g. a spectral line, or because of imposed bandwidth limitations in amplifiers and/or filters), then the signal has a high degree of “temporal coherence”, which is to say that the wave packet describing all the photons in the signal is extended in time by $\tau \approx 1/\Delta \nu$ seconds. Expressing the bandwidth in terms of wavelength, $\Delta \nu = (c/\lambda_{0}^{2}) \cdot \Delta \lambda$ where $\lambda_0$ is the band center and $c$ is the speed of light. The coherence time is then $\tau = (1/c) \cdot (\lambda_{0}^{2} / \Delta \lambda)$, and $c \cdot \tau$ is a scale size of the wave packet called the *coherence length*, $L_c = c \cdot \tau = \lambda_{0}^{2} / \Delta \lambda$, [e.g. @hecht2002 Ch. 7]. If the path difference between the two collectors in an interferometer is a significant fraction of $L_c$, an additional time delay must be introduced, otherwise the fringe amplitude will decrease or even disappear. For ground-based systems, the geometry is continually changing for all directions in the sky (except in the directions to the equatorial poles), requiring a continually-changing additional delay to maintain the temporal coherence. The special location on the sky where the adjusted time delay is matched perfectly is often called the “phase center” or point of zero optical path delay (OPD), although such a condition actually defines the locus of a plane passing through the mid-point between the collectors and perpendicular to the baseline, and cutting the celestial sphere in a great circle. Since the telescope optics usually limits the field of view to only a tiny portion of this great circle, adjusting the phase center is the equivalent of “pointing” the interferometer at a given object within that field of view. The final step is to interfere the two beams to measure the *spatial coherence* (often called the mutual coherence) of the electric field as sampled by the two telescopes. If the object observed is much smaller than the angular resolution of the interferometer, then interference is complete and one observes 100% coherence at the correlator/combiner. However, objects that are [ *resolved*]{} (i.e., much larger than the angular resolution of the interferometer) will show less coherence due to the fact that different patches of emission on the object do not interfere at the same time through our system. Figure \[fig\_schematic\] shows two simple cases of an interferometer as a Young’s two-slit experiment to illustrate basic principles. At the left, the interferometer is made up of two slits and the response for a monochromatic point source (i.e., incoming plane waves) is shown. The result should be familiar: an interference fringe modulating the intensity from 100% to 0% with a periodicity that corresponds to a fringe spacing of $\frac{\lambda}{B}$ on the sky. Next to this panel is shown an example of two equal-brightness point sources separated by $\frac{1}{2} \frac{\lambda}{B}$, half the fringe spacing. The location of constructive interference for one point coincides with the location of destructive interference for the other source. Since the two sources are mutually incoherent, the superposition of the two fringe results in an even light distribution, i.e. no fringe at all! In optical interferometry language, the first example fringe has a fringe contrast (or visibility) of $1$ while the second example fringe has a visibility of $0$. Figure \[fig\_interferometers\] contains a schematic of a basic interferometer as typically realized for both radio and optical configurations. While instrumental details vary immensely in how one transmits and interferes the signals for radio, millimeter, infrared, and visible-light interferometers, the basic principles are the same. The foundational theory common to all interferometers will be introduced next. Interferometry in theory {#vcz} ------------------------ The fundamental equation of interferometry is typically derived by introducing the van-Cittert Zernike Theorem and a complete treatment can be found in Chapter 3 of the book by @tms2001. Here the main result will be presented without proof, beginning by defining an interferometric observable called the [*complex visibility*]{}, $\tilde{{\mathcal{V}}}$. The visibility can be derived from the intensity distribution on the sky $I(\vec{\sigma})$ using a given interferometer baseline $\vec{B}$ (which is the separation vector between two telescopes) and the observing wavelength $\lambda$: $$\label{eq:vcz} \tilde{\mathcal{V}} = |\mathcal{V}| e^{i \phi_{\mathcal{V}}} = \int_{\rm sky} A_N (\vec{\sigma}) I(\vec{\sigma}) e^{-\frac{2\pi i}{\lambda} \vec{B}\cdot\vec{\sigma}} d\Omega$$ Here, the $\vec{\sigma}$ represents the vector pointing from the center of the field-of-view (called the “phase center”) to a given location on the celestial sphere using local (East, North) equatorial coordinates and the telescope separation vector $\vec{B}$ also using east and north coordinates. The modulus of the complex visibility $|\mathcal{V}|$ is referred to as the [*fringe amplitude or visibility*]{} while the argument $\phi_{\mathcal{V}}$ is the [ *fringe phase*]{}. $A_N(\vec{\sigma})$ represents the normalized pattern that quantifies how off-axis signals are attenuated as they are received by a given antenna or telescope. In this treatment the astronomical object is assumed to be small in angular size in order to ignore the curvature of the celestial sphere. The physical baseline $\vec{B}$ can be decomposed into components $\vec{u} = (u,v)$ in units of observing wavelength along the east and north directions (respectively) as projected in the direction of our target. The vector $\vec{\sigma}=(l,m) $ also can be represented in rectilinear coordinates on the celestial sphere, where $l$ points along local east and $m$ points north[^2]. Here, $l$ and $m$ both have units of radians. Equation \[eq:vcz\] now becomes: $$\label{eq:vcz2} \tilde{\mathcal{V}}(u,v) = |\mathcal{V}| e^{i \phi_{\mathcal{V}}} = \int_{l,m} A_N (l,m) I(l,m) e^{-2\pi i (u l + v m)} dl dm$$ The fundamental insight from Equation \[eq:vcz2\] is that an interferometer is a Fourier Transform machine – it converts an intensity distribution $I(l,m)$ into measurements of Fourier components $\tilde{\mathcal{V}}(u,v)$ for all the baselines in the array represented by the $(u,v)$ coverage. Since an intensity distribution can be described fully in either image space or Fourier space, the collection of sufficient Fourier components using interferometry allows for an image reconstruction through an inverse Fourier Transform process, although practical limitations lead to compromises in the quality of such images. Interferometry in practice {#interferometry_practice} -------------------------- In this section, the similarities and differences between radio and optical interferometers are summarized along with the reasons for the main differences. Interested readers can find more detailed on specific hardware implementations in Volume I of this series. Modern radio and optical interferometers typically use conventional steerable telescopes to collect photons from the target. In the radio, a telescope is often called an antenna; it is typically a parabolic reflector with a very short focal length (f/D $\approx 0.35$ is common), with signal collection and initial amplification electronics located at the prime focus. Owing to the large value of $\Delta\Theta\sim\frac{\lambda}{{\rm Diameter}}$, the diffraction pattern of the antenna aperture is physically a relatively large region at the prime focus. This fact, coupled with the cost and complexity of duplicating and operating many low-noise receivers in close proximity to each other, has meant that antennas used in radio astronomy typically have only a “single-pixel” signal collection system (a dipole or a “feed horn”) at the prime focus[^3]. Light arriving from various directions on the sky are attenuated depending on the shape of the diffraction pattern, written as $A_N$ in Equation \[eq:vcz2\] and often called the “antenna pattern” or the “primary beam”. The signal collection system may be further limited to a single polarization mode, although systems are common that simultaneously accept both linear (or both circular) polarization states. After initial amplification, the signal is usually mixed with a local oscillator to “down-convert” the high frequencies to lower frequencies that can more easily be amplified and processed further. These lower-frequency signals from the separate telescopes can also be more easily transported over large distances to a common location using e.g. coaxial cable, or by modulating an optical laser and using fiber optics. This common location houses the “back end” of the receiver, where the final steps in signal analysis are carried out including band definition, correlation, digitizing, and spectral analysis. In some cases, the telescope signals are recorded onto magnetic media and correlated at a later time and in a distant location (e.g. the “Very Long Baseline Array” or global VLBI). In the optical, the light from the object is generally focused by the telescope, re-collimated into a compressed beam for free-space transport, and then sent to a central location in a pipe which is typically evacuated to avoid introducing extra air dispersion and ground-level turbulence. In rare cases, the light at the telescope is focused directly into a single-mode fiber, which is the dielectric equivalent to the metallic waveguides used in radio and millimeter receivers. Note that atmospheric seeing is very problematic for even small visible and infrared telescopes while it is usually negligible compared to the diffraction limit for even the largest radio and mm-wave telescopes. Both radio and optical interferometers must delay the signals from some telescopes to match the optical paths. After mixing to a lower frequency, radio interferometers can use switchable lengths of coaxial cable in order to introduce delays. More recently, the electric fields can be directly digitized with bandwidths of $>5$ GHz, and these “bits” can be saved in physical memory and then recalled at a later time. For visible and infrared systems, switchable fiber optics are not practical due to losses and glass dispersion; the only solution is to use an optical “free-space” delay line consisting of a retroreflector moving on a long track and stabilized through laser metrology to compensate for air path disturbances and vibrations in the building. In a radio interferometer, once all the appropriate delays have been introduced the signals from each telescope can be combined. Early radio signal correlators operated in an “optical” mode as simple adding interferometers, running the sum of the signals from the two arms through a square-law detector. The output of such a detector contains the product of the two signals. Unfortunately, the desired correlation product also comes with a large total power component caused by temporally-uncorrelated noise photons contributed (primarily) by the front-end amplifiers in each arm of the interferometer plus atmosphere and ground noise. This large signal demanded excellent DC stability in the subsequent electronics, and it was not long in the history of radio interferometry before engineers found clever switching and signal-combination techniques to suppress the DC component. These days signal combiners deliver only the product of the signals from each arm, and are usually called “correlators”[^4]. Most modern radio/millimeter arrays use digital correlators that introduce time lags between all pairs of telescopes in order to do a full temporal cross-correlation. This allows a detailed wavelength-dependent visibility to be measured, i.e., an interferometric spectrum with $R=\frac{\lambda}{\Delta\lambda}>100000$ if necessary. By most metrics, radio correlators have reached their fundamental limit in terms of extracting full spectral and spatial information and can be fairly sophisticated and complex to configure when correlating multiple bandpasses simultaneously with high spectral resolution[^5]. In the visible and infrared, the electric fields can not be further amplified without strongly degrading the signal-to-noise ratio, and so parsimonious beam combining strategies are common that split the signal using e.g. partly-reflecting mirrors into a small number of pairs or triplets. Furthermore, most optical systems have only modest spectral resolutions of typically $R\sim40$ in order to maintain high signal-to-noise ratio, although a few specialized instruments exist that reach $R>1000$ or even $R>30000$. Signal combination finally takes place simply by mixing the light beams together and modulating the relative optical path difference, either using spatial or temporal encoding. The total power measurement in a visible-light or infrared detector will reveal the interference fringe and a Fourier analysis can be used to extract the complex visibility $\tilde{\mathcal{V}}$. Because the ways of measuring visibilities are quite different, radio and optical interferometrists typically report results in different units. Radio/mm interferometers measure correlated flux density in units of Jansky ($10^{-26}$ W m$^{-2}$ Hz$^{-1}$), just as suggested by Equation \[eq:vcz2\][^6]. In the optical however, interferometers tend to always measure a normalized visibility that varies from $0$ to $1$ – this is simply the correlated signal normalized by the total power. One can convert the latter to correlated flux density by simply multiplying by the known total flux density of the target at the observed wavelengths, or otherwise by carrying out a calibration of the system by a target of known flux density. ### Quantum limits of amplifiers {#quantumnoise} The primary reason why radio and optical interferometers differ so much in their detection scheme is because coherent amplifiers would introduce too much extraneous noise at the high frequencies encountered in the optical and infrared. This difference is fundamental and is explored in more detail in this section. At radio frequencies there are huge numbers of photons in a typical sample of the electromagnetic field, so the net phase of a packet of radio photons (either from the source or from a noisy receiver) is well-defined and amplifiers can operate coherently. The ultimate limits which apply to such amplifiers are dictated by the uncertainty principle as stated by Heisenberg. Beginning with the basic “position - uncertainty” relation $\Delta x \; \Delta p_x \geq h/4 \pi$, it is easy to derive the “energy - time” relation $\Delta E \; \Delta t \geq h/4\pi$. Since the uncertainty in the energy of the $n$ photons in a wave packet can be written as $\Delta E = h \nu \; \Delta n$ and the uncertainty in the phase of the aggregate as $\Delta \phi = 2 \pi \nu \; \Delta t$, this leads to the equivalent uncertainty relation $\Delta \phi \; \Delta n \geq 1/2$. An ideal amplifier which adds no noise to the input photon stream leads to a contradiction of the uncertainty principle. The following argument shows how this happens [adapted from @Heffner1962]: Consider an ideal coherent amplifier of gain $G$ which creates new photons in phase coherence with the input photons, and assume it adds no incoherent photons of its own to the output photon stream. With $n_1$ photons going into such an amplifier, there will be $n_2 = Gn_1$ photons at the output, all with the same phase uncertainty $\Delta \phi_2 = \Delta \phi_1$ with which they went in. In addition, in this model it is expected that $\Delta n_2 = G \Delta n_1$ (no additional “noise” photons unrelated to the signal). But according to the same uncertainty relation, the photon stream coming out of the amplifier must also satisfy $\Delta \phi_2 \; \Delta n_2 \geq 1/2$. This would imply that $\Delta \phi_1 \; \Delta n_1 \geq \frac{1}{2G}$, which for large $G$ says that the input photon number and wave packet phase could be measured with essentially no noise. But this contradicts the same uncertainty relation for the input photon stream, which requires that $\Delta \phi_1 \; \Delta n_1 \geq 1/2$. This contradiction shows that one or more of our assumptions must be wrong. The argument can be saved if the amplifier itself is required to add noise of its own to the photon stream; the following heuristic construction shows how. Using the identity $\Delta n_2 = (G-N) \cdot \Delta n_1 + N\Delta n_1$ at the output (where $N$ is an integer $N \geq 1$), and referring this noise power back to the input by dividing it with the amplifier gain $G$, this leads to $(1 - N/G) \cdot \Delta n_1 + (N/G) \cdot \Delta n_1$ at the input to the amplifier, which for large $G$ is $\Delta n_1$. The smallest possible value of N is $1$. This preserves the uncertainty relation at the expense of an added minimum noise power of $ h \nu$ at the input. @Oliver1965 has elaborated and generalized this argument to include all wavelength regimes, and has shown that the minimum total noise power spectral density $\psi_\nu$ of an ideal amplifier (relative to the input) is $$\psi_\nu = \frac{h \nu}{e^{(h \nu /k T)} - 1} + h \nu \ \ \mbox{Watts/Hz ,}$$ where T is the kinetic temperature that the amplifier input faces in the propagation mode to which the amplifier is sensitive. For $h \nu < kT$ this reduces to $\psi_\nu\approx kT$ Watts/Hz, which can be called the “thermal” regime of radio astronomy. For $h \nu > kT$ this becomes $\psi_\nu \approx h \nu$ Watts/Hz in the “quantum” regime of optical astronomy. The crossover point where the two contributions are equal is where $h \nu / kT = \ln{2}$, or at $\lambda_c \cdot T_c = 20.75$ (mm K). As an illustration of the use of this equation, consider this example: The sensitivity of high-gain radio-frequency amplifiers can usually be improved by reducing their thermodynamic temperatures. However, for instance at a wavelength of 1 mm, it might be unnecessary (depending on details of the signal chain) to aim for a high-gain amplifier design to lower the thermodynamic temperature below about 20K, since at that point the sensitivity is in any case limited by quantum noise. At even shorter wavelengths, the rationale for cooled amplifiers disappears, and at optical wavelengths amplifiers are clearly not useful since the noise is totally dominated by spontaneous emission[^7] and is equivalent to thermal emission temperatures of thousands of degrees. The extremely faint signals common in modern optical observational astronomy translate into very low photon rates, and the addition of such irrelevant photons into the data stream by an amplifier would not be helpful. Atmospheric Turbulence {#turbulence} ---------------------- So far, the analysis of interferometer performance has assumed a perfect atmosphere. However, the electromagnetic signals from cosmic sources are distorted as they pass through the intervening media on the way to the telescopes. These distortions occur first in the interstellar medium, followed by the interplanetary medium in the solar system, then the Earth’s ionosphere, and finally the Earth’s lower atmosphere (the troposphere) extending from an altitude of $\approx 11$ km down to ground level. The media involved in the first three sources of distortion contain ionized gas and magnetic fields, and their effects on signal propagation depend strongly on wavelength (generally as $\propto \lambda^2$) and polarization. At wavelengths shorter than about 10 cm the troposphere begins to dominate. Molecules in the troposphere (especially water vapor) become increasingly troublesome at frequencies above 30 GHz (1 cm wavelength), and the atmosphere is essentially opaque beyond 300 GHz except for two rather narrow (and not very clear) “windows” from 650-700 and 800-900 GHz which are usable only at the highest-altitude sites. The next atmospheric windows appear in the IR at wavelengths less than about 15 microns. The optical window opens around one micron, and closes again for wavelengths shorter than about 350 nm. The behavior of the troposphere is thus of prime importance to ground-based astronomy at wavelengths from the decimeter-radio to the optical. Interferometers are used in the study of structure in the troposphere, and a summary of approaches and results with many additional references is given in @tms2001 [@carilli1999; @sutton1996 Ch. 13]. A discussion oriented towards optical wavelengths can be found in @quirrenbach2000. Since the main focus here is on using interferometers to measure the properties of the cosmic sources themselves, our discussion is limited to some “rules of thumb” for choosing the interferometer baseline length and the time interval between measurements of the source and of a calibrator in order to minimize the deleterious effects of propagation on the fringe amplitudes and (especially) fringe phases. ### Phase fluctuations – length scale Owing to random changes in the refractive index of the atmosphere and the size distribution of these inhomogeneities, the path length for photons will be different along different parallel lines of sight. This fluctuating path length difference grows almost linearly with the separation $d$ of the two lines of sight for separations up to some maximum, called the outer scale length (typically tens to hundreds of meters, with some weak wavelength dependence), and is roughly constant beyond that. Surprisingly, in spite of the differences in the underlying physical processes causing refraction, variations in the index of refraction are quite smooth across the visible and all the way through to the radio. At short radio wavelengths, the fluctuations are dominated by turbulence in the water vapor content; at optical/IR wavelengths, it is temperature and density fluctuations in dry air that dominate. Using a model of fully-developed isotropic Kolmogorov turbulence for the Earth’s atmosphere, the rms path length difference grows according to $\sigma_d \propto d^{5/6}$ for a path separation $d$ [see @tms2001 Ch. 13, for references]. High altitude sites show smaller path length differences as the remaining vertical thickness of the water vapor layer decreases. Relatively large seasonal and diurnal variations also exist at high mountain sites as the atmospheric temperature inversion layer generally rises during the summer and further peaks during mid-day. Variations in $\sigma_d$ by factors of $\sim 10$ are not unusual [see @tms2001 Fig. 13.13], but a rough average value for a good observing site is $\sigma_d \approx 1$ mm for baselines $d \approx 1$ km at millimeter wavelengths, and $\sigma_d \approx 1$ micron for baselines $d \approx 50$ cm at infrared wavelengths. The length scale fluctuations translate into fringe phase fluctuations of $\sigma_\phi = 2\pi\sigma_d/\lambda$ in radians. The [*maximum coherent baseline*]{} $d_0$ is defined as that baseline length for which the rms phase fluctuations reach 1 radian. Using the expressions in the previous paragraph and coefficients suitable for the radio and optical ranges at the better observing sites, two useful approximations are $d_0 \approx 140 \cdot \lambda^{6/5}$ meters for $\lambda$ in millimeters (useful at millimeter radio wavelengths), and $d_0 \approx 10 \cdot \lambda^{6/5}$ centimeters for $\lambda$ in microns (useful at IR wavelengths). These two expressions are in fact quite similar; using the “millimeter expression” to calculate $d_0$ in the IR underestimates the value obtained from the “IR expression” by a factor of 2.8, which is at the level of precision to be expected. At shorter wavelengths (visible and near-infrared), atmospheric turbulence limits even the image quality of small telescopes. This has led to a slightly different perspective for the length scale that characterizes atmospheric turbulence, although it is closely related to the previous description. The Fried length $r_0$ [@fried1965] is the equivalent-sized telescope diameter whose diffraction limit matches the image quality through the atmosphere due to [*seeing*]{}. It turns out that this quantity is proportional to the length scale where the rms phase error over the telescope aperture is $\approx 1$ radian. In other words, apertures with diameters small compared to $r_0$ are approximately diffraction limited, while larger apertures have resolution limited by turbulence to $\approx \lambda / r_0$. It can be shown that, for an atmosphere with fully-developed Kolmogorov turbulence, $r_0 \approx 3.2 d_0$ [@tms2001 Ch. 13]. ### Phase fluctuations – time scale Although fluctuations of order one radian may be no more than a nuisance at centimeter wavelengths, requiring occasional phase calibration (see §\[phasereferencing\]), they will be devastating at IR and visible wavelengths owing to their rapid variations in time. In order to relate the temporal behavior of the turbulence to its spatial structure, a model of the latter is required along with some assumption for how that structure moves over the surface of the Earth. One specific set of assumptions is described in @tms2001 [Ch. 13]; however, for the purposes here it is sufficient to use Taylor’s “frozen atmosphere” model with a nominally-static phase screen that moves across the Earth’s surface with the wind at speed $v_s$. This phase screen traverses the interferometer baseline $d$ in a time $\tau_d = d / v_s$, at the conclusion of which the total path length variation is $\sigma_d$. Taking the critical time scale $\tau_c$ to be when the rms phase error reaches 1 radian, then $\tau_c \approx d_0 / v_s$ with $d_0$ given in the previous paragraph. As an example consider a wind speed of 10 m/s; this leads to $\tau_c \approx 14$ seconds at $\lambda = 1$ mm, and $\approx 10$ milliseconds at $\lambda = 1$ micron. Clearly the techniques required to manage these variations will be very different at the two different wavelength regimes, even though the magnitude of the path length fluctuations (in radians of phase) are similar. Representative values of these quantities are collected in Table \[table:fluctuations\]. ### Calibration – Isoplanatic Angle The routine calibration of interferometer phase and amplitude is usually done by observing a source with known position and intensity inter-leaved in time with the target of interest. At centimeter wavelengths and longer, the discussion in the previous section indicates that such measurements can be done on time scales of minutes to hours, providing ample time to re-position telescopes elsewhere on the sky in order to observe a calibrator. But how close to the target of interest does such a calibrator have to be? Ideally, the calibrator ought to be sufficiently nearby on the celestial sphere that the line of sight traverses a part of the atmosphere with substantially the same phase delay as the line of sight to the target. This angle is called the *isoplanatic angle* $\Theta_{iso}$; it characterizes the angular scale size over which different parts of the incoming wavefront from the target encounter closely similar phase shifts, thereby minimizing the image distortion. The isoplanatic angle can be roughly estimated by calculating the angle subtended by an $r_0$-sized patch at a height $h$ that is characteristic for the main source of turbulence; hence, roughly $\Theta_{iso}\approx\frac{r_0}{h}$. Within a patch on the sky with this angle, the telescope/interferometer PSF remains substantially constant, retaining the convolution relation between the source brightness distribution and the image. Some approximate values are given in Table \[table:fluctuations\] as a guide. At visible and near-IR wavelengths, Table \[table:fluctuations\] shows that the isoplanatic angle is very small, smaller than an arcminute. Unfortunately, the chance of having a suitably bright and point-like object within this small patch of the sky is very low. Even if an object did exist, it would be nearly impossible to repetitively re-position the telescope and delay line at the milli-second level timescale needed to “freeze” the turbulence between target and calibrator measurements. Special techniques to deal with this problem will be discussed further in section \[phasereferencing\]. [ccccc]{} 0.5 $\mu$m (visible) & 4.4 cm & 14 cm & 4.4 ms & $5.5''$\ 2.2 $\mu$m (near-IR) & 26 cm & 83 cm & 26 ms & $33''$\ 1 mm (millimeter) & 140 m & 450 m & 14 sec & $3.5^{\circ}$\ 10 cm (radio) & 35 km & 112 km & 58 min & large\ Planning Interferometer Observations {#observing} ==================================== The issues to consider when writing an interferometer observing proposal or planning the observations themselves include: the desired sensitivity (i.e., the unit telescope collecting area, the number of telescopes to combine at once, the amount of observing time), the required field-of-view and angular resolution (i.e.,the shortest and longest baselines), calibration strategy and expected systematic errors (i.e., choosing phase and amplitude calibrators), the expected complexity in the image (i.e., the completeness of u,v coverage, do science goals demand model-fitting or model-independent imaging), and the spectral resolution (i.e., correlator settings, choice of combiner instrument). Many of these issues are intertwined, and the burden on the aspiring observer to reach a compatible set of parameters can be considerable. Prospective observers planning to use the VLA are fortunate to have a wide variety of software planning tools and user’s guides already at their disposal, but those hoping to use more experimental facilities or equipment which is still in the early phases of commissioning will find their task more challenging. Here, the most common issues encountered during interferometer observations will be introduced. In many ways this is more of a list of things to worry about rather than a compendium of solutions. The basic equations and considerations have been collected in Table \[table\_planning\]. In order to obtain the latest advice on optimizing a request for observing time, or to plan an observing run, observers ought to consult the web sites, software tools, and human assistants available for them at each installation (see Appendix for a list of current facilities). [ll]{} Angular Resolution & $\Theta = \frac{1}{2} \frac{\lambda}{B_{\rm max}}$\ Spectral Resolution & $R=\frac{\lambda}{\Delta \lambda} = \frac{c}{\Delta v}$\ Field-of-View &\ *primary beam & $\Delta\Theta \sim \frac{\lambda}{D_{\rm Telescope}}$\ *bandwidth-smearing & $\Delta\Theta \sim R \cdot \frac{\lambda}{B_{\rm max}}$\ *time-smearing & $\Delta\Theta \sim \frac{230}{\Delta t_{\rm minutes}} \frac{\lambda}{B_{\rm max}}$\ Phase Referencing &\ *Coherence Time & see Table \[table:fluctuations\]\ *Isoplanatic Angle & see Table \[table:fluctuations\]\ ***** Sensitivity ----------- Fortunately modern astronomers can find detailed documentation on the expected sensitivities for most radio and optical interferometers currently available. Indeed, the flexibility of modern instrumentation sometimes defies a back-of-the-envelope estimation for the true signal-to-noise ratio (SNR) expected for a given observation. In order to better understand what limits sensitivity for real systems, the dominant noise sources and the key parameters affecting signal strength are introduced. Most of the focus will be for observations of point sources since resolved sources do not contribute signal to all baselines in an array and this case must be treated with some care. Here, the discussions of the radio and optical cases are separate because of the large differences in the nature of the noise processes (e.g., see §\[quantumnoise\]) and the associated nomenclature. Radio and optical observations lie at the two limits of Bose-Einstein quantum statistics that govern photon arrival rates [e.g., @pathria1972 see §6.3]. At long wavelengths, the occupation numbers are so high that the statistics evolve into the Gaussian limit and where the root-mean-square (rms) fluctuation in the detected power $\Delta$P is proportional to the total power P itself (e.g., $\Delta$Power $\propto$ Power). On the other hand, in the optical limit, the sparse occupation of photon states results in the familiar Poisson statistics where the level of photon fluctuations $\Delta$N is proportional to $\sqrt{N}$. Most of the SNR considerations for interferometers are in common with single-dish radio and standard optical photometry, and so interested readers are referred to the relevant chapters in Volumes 1 and 2 of this series. ### Radio Sensitivity {#radiosensitivity} The signal power spectral density $P_\nu$ received by a radio telescope of effective area $A_e$ (${\rm m}^2$) from a celestial point source of flux density $S_\nu$ (Jansky = Watts/${\rm m}^2$/Hz) is $P_\nu = A_e \cdot S_\nu$ (Watts/Hz). It is common to express this as the power which would be delivered to a radio circuit (wire, coaxial cable, or waveguide) by a matched termination at a physical temperature $T_A$, called the “antenna temperature”, so that $T_A = A_eS_\nu/2k$ (Kelvin) where $k$ = Boltzmann’s constant and the factor 1/2 accounts for the fact that, although the telescope’s reflecting surface concentrates both states of polarization at a focus, the “feed” collects the polarization states separately. As described in section \[quantumnoise\], the amplifier which follows must add noise; this additional noise power (along with small contributions from other extraneous sources in the telescope field of view) $P^{\rm s}_\nu$ can likewise be expressed as $P^{\rm s}_\nu = kT_s/2$, where $T_s$ is the “system temperature.” The rms fluctuations in this noise power will limit the faintest signals that can be distinguished. As mentioned in the previous paragraph, these fluctuations are directly proportional to the receiver noise power itself, so $\Delta T_s \propto T_s$. They will also be inversely proportional to the square root of the number of samples of this noise present in the receiver passband. The coherence time of a signal in a bandwidth $\Delta \nu$ is proportional to $1/\Delta \nu$, so in an integration time $\tau$ there are of order $\tau \Delta \nu$ independent samples of the noise, and the statistical uncertainty will improve as $1/\sqrt{\tau\Delta \nu}$. The ratio of the rms receiver noise power fluctuations to the signal power is therefore: $$\Delta T_s / T_A \propto \frac{2 k T_s}{A_e S \sqrt{\tau \Delta \nu}} \; .$$ The minimum detectable signal $\Delta S$ is defined as the value of $S$ for which this ratio is unity. For this “minimum” value of S the equation becomes: $$\Delta S = \frac{f_c \cdot k T_s}{A_e \sqrt{\tau \Delta \nu}} \; ,$$ The coefficient of proportionality $f_c$ for this equation is of order unity, but the precise value depends on a number of details of how the receiver operates. These details include whether the receiver output contains both polarization states, whether both the in-phase and the quadrature channels of the complex fringe visibility are included, whether the receiver operates in single- or double-sideband mode, and how precisely the noise is quantized if a digital correlator is used. Further discussion of the various possibilities is given in @tms2001 [Chapter 6]. For the present purpose, it suffices to notice that the sensitivity for a specific radio interferometer system improves only slowly with integration time and with further smoothing of the frequency (radial velocity) resolution. The most effective improvements are made by lowering the system temperature and by increasing the collecting area. The point-source sensitivity continues to improve as telescopes are added to an array. An array of $n$ identical telescopes contains $N_b = n(n-1)/2$ distinct baselines. If the signals from each telescope are split into multiple copies, $N_b$ interference pairs can be made. The rms noise in the flux density on a point source including all the data is then $$\Delta S = \frac{f_c \cdot k T_s}{A_e\sqrt{N_b \tau \Delta\nu}} \; .$$ So far the discussion has been made for isolated point sources. Extended sources are physically characterized by their surface brightness power spectral density $B_{\rm surf}(\alpha,\delta,\nu)$ (Jansky/steradian) and by the angular resolution of the observation as expressed by the solid angle $\Omega_b$ of the synthesized beam in steradians (see §\[imaging\]). By analogy with the discussion of rms noise power from thermal sources given earlier, it is usual to express the surface brightness power spectral density for an extended sources in terms of a temperature. This conversion of units to Kelvins is done using the Rayleigh-Jeans approximation to the Planck black-body radiation law, although the radiation observed in the image is only rarely thermally-generated. The conversion from $B_{\rm surf}(\alpha,\delta,\nu)$ (Jansky/steradian) to $T_b$ in Kelvins is $$T_b = \frac{\lambda^2 B_{\rm surf}}{2 k \Omega_b} \; ,$$ which requires ($h\nu/kT << 1$) if the radiation is thermal; otherwise, this conversion can be viewed merely as a convenient change of units. The rms brightness temperature sensitivity in a radio synthesis image from receiver noise alone is then $$\Delta T_b = \frac{f_c \lambda^2 T_s}{2 A_e \Omega_b \sqrt{N_b \tau \Delta \nu}} \; .$$ The final equations above for the sensitivity on synthesis imaging maps shows that the more elements one has, the better the flux density sensitivity will be. For example if one compares an array of $N_b=20$ baselines with an array containing $N_b=10$ baselines, the flux density SNR is improved by a factor $\sqrt{2}$ no matter where the additional 10 baselines are located in the $u,v$ plane. However, the brightness temperature sensitivity does depend critically on the actual distribution of baselines used in the synthesis. For instance, if the same number of telescopes is “stretched out” to double the maximum extent on the ground, the equations above show that the flux density sensitivity $\Delta S$ remains the same, but the brightness temperature sensitivity $\Delta T_b$ is worse by a factor of 4 since the synthesized beam is now 4 times smaller in solid angle. This is a serious limitation for spectral line observations where the source of interest is (at least partially) resolved and where the maximum surface brightness is modest. For instance, clouds of atomic hydrogen in the Galactic ISM never seem to exceed surface brightness temperatures of $\approx 80$ K, so the maximum achievable angular resolution (and hence the maximum useable baseline in the array) is limited by the receiver sensitivity. This can only be improved by lowering the system temperature on each telescope or by increasing the number of interferometer measurements with more telescopes and/or more observing time. A cautionary note is appropriate here. In the case of an optical image of an extended object taken e.g. with charge-coupled device (CCD) camera on a filled aperture telescope, a simple way of improving the SNR is to average neighboring pixels together thereby creating a smoothed image of higher brightness sensitivity. At first sight, the equation for $\Delta T_b$ above suggests that this should also happen with synthesis images, but here the improvement is not as dramatic as it may seem at first sight. The reason is that the action of smoothing is equivalent to discarding the longer baselines in the $u,v$ plane; for instance, reducing the longest baseline used in the synthesis by a factor of 2 would indeed lead to an image with brightness temperature sensitivity which is better by a factor of $2^2$, but the effective reduction of the number of interferometers from $N$ to $N/2$ means that the net improvement is only $2^{1.5}$. A better plan would have been to retain all the interferometers but to shrink the array diameter with the factor 2 by moving the telescopes into a more compact configuration. This is one reason why interferometer arrays are usually constructed to be reconfigurable. ### Visible and Infrared Sensitivity As mentioned earlier, the visible and infrared cases deviate substantially from the radio case. While the sensitivity is still dependent on the collecting area of the telescopes ($A_e$), the dominant noise processes behave quite differently. In the visible and infrared (V/IR), noise is generated by the random arrival times of the photons governed by Poisson statistics $\Delta N = \sqrt{N}$, where $N$ is the mean number of photons expected in a time interval $\tau$ and $\Delta N$ is the rms variation in the actual measured number of photons. Depending on the observing setup (e.g., the observing wavelength, spectral resolution, high visibility case or low visibility case), the dominant noise term can be Poisson noise from the source itself, Poisson noise from possible background radiation, or even detector noise. Because of the centrality of Poisson statistic, it is common to work in units of total detected photo-electrons $N$ within a time interval $\tau$, rather than power spectral density $P_\nu$ or system temperature $T_S$. This conversion is straightforward: $$\begin{aligned} N & = & \eta \frac{P_\nu \Delta\nu }{h \nu} \tau \\ & = & \eta \frac{S_\nu A_e \Delta\nu}{h \nu} \tau \end{aligned}$$ where $\eta$ represents the total system detection efficiency which is the combination of optical transmission of system and the quantum efficiency of the detector and the other variables are the same as for the radio case introduced in the last section. For the optical interferometer, atmospheric turbulence limits the size of the aperture that can be used without adaptive optics (the atmosphere does not limit the useful size of the current generation of single-dish mm-wave and radio telescopes). The Fried parameter $r_0$ sets the coherence length and thus the max($A_e$)$\sim r^2_0$. Likewise without corrective measures, the longest useful integration time is limited to the atmospheric coherence time $\tau\sim\tau_{\rm c}$. There exists a [*coherent volume*]{} of photons that can be used for interferometry, scaling like $r_0 \cdot r_0 \cdot c \tau_c$. As an example, consider the coherent volume of photons for decent seeing conditions in the visible ($r_0 \sim$ 10cm, $\tau_c \sim$ 3.3ms). From this, the limiting magnitude can be estimated by requiring at least 10 photons to be in this coherent volume. Assuming a bandwidth of 100 nm, 10 photons ($\lambda\sim$ 550nm) in the above coherent volume corresponds to a V magnitude of 11.3, which is the best limit one could hope to achieve[^8]. This is more than 14 magnitudes worse than faint sources observed by today’s 8-m class telescopes that can benefit from integration times measured in hours instead of milli-seconds. Because the atmospheric coherence lengths and timescales behave approximately like $\lambda^{\frac{6}{5}}$ for Kolmogorov turbulence, the coherent volume $\propto \lambda^\frac{18}{5}$. Until the deleterious atmospheric effects can be neutralized, ground-based optical interferometers will never compete with even small single-dish telescopes in raw point-source sensitivity. Under the best case the only source of noise is Poisson noise from the object itself. Indeed, this limit is nearly achieved with the best visible-light detectors today that have read-noise of only a few electrons. More commonly, especially in the infrared, detectors introduce the noise that limits sensitivity, typically 10-15 electrons of read-noise in the near-IR for the short exposures required to effectively freeze the atmospheric turbulence. For wavelengths longer than about 2.0$\mu$m (i.e., K, L, M, N bands), Poisson noise from the thermal background begins to dominate over other sources of noise. Highly-sensitive infrared interferometry will require a space platform that will allow long coherence times and low thermal background. Please consult the observer manual for each specific interferometer instrumentation to determine point-source sensitivity. Another important issue to consider is that a low visibility fringe ($\mathcal{V}<<1$) is harder to detect than a strong one. Usually fringe detection sets the limiting magnitude of an interferometer/instrument, and this limit often scales like $N \mathcal{V}$, the number of “coherent” photons. For readnoise or background noise dominant situations (common in NIR), this means that if the point-source ($\mathcal{V}=1$) limiting magnitude is 7.5 then a source with $\mathcal{V}=0.1$ would need to be as bright as magnitude 5.0 to be detected. The magnitude limit worsens even more quickly for low visibility fringes when noise from the source itself dominates, since brighter targets bring along greater noise. Another common expression found in the literature is that the SNR for a visible-light interferometer scales like $N \mathcal{V}^2$. This latter result can be derived by assuming that the “signal” is the average power spectrum $(N \mathcal{V})^2$ and the dominant noise process is photon noise which has a power spectrum that scales like $N$ here. ### Overcoming the Effects of the Atmosphere: Phase Referencing, Adaptive Optics, and Fringe Tracking {#phasereferencing} As discussed above, the limiting magnitude will strongly depend on the maximum coherent integration time that is set by the atmosphere. Indeed, this limitation is very dramatic, restricting visible-light integrations to mere milli-seconds and millimeter radio observations to a few dozen minutes. For mm-wave and radio observations, the large isoplanatic angle and long atmospheric coherence times allow for real-time correction of atmospheric turbulence by using [*phase referencing.*]{} In a phase-referencing observing sequence, the telescopes in the array will alternate between the (faint) science target and a (bright) phase calibrator nearby in the sky. If close enough in angle (within the isoplanatic patch), then the turbulence will be the same between the target and bright calibrator; thus, the high SNR measurement of fringe phase on the calibrator can be used to account for the atmospheric phase changes. Another key aspect is that the switching has to be fast enough that the atmospheric turbulence does not change between the two pointings. With today’s highly-sensitive radio and mm-wave receivers, enough bright targets exist to allow nearly full sky coverage so that most faint radio source will have a suitable phase calibrator nearby.[^9] In essence, phase referencing means that a fringe does not need to be detected within a single coherence time $\tau_c$ but rather one can coherently integrate for as long as necessary with sensitivity improving as $1/\sqrt{t}$. In §\[dataanalysis\] a simple example is presented that demonstrates how phase-referencing works with simulated data. In the visible and infrared, phase referencing by alternate target/calibrator sequences is practically impossible since $\tau_c << 1$ second and $\Theta_{\rm iso}<< 1$ arcminute isoplanatic patch size. In V/IR interferometry, observations still alternate between a target and calibrator in order to calibrate the statistics of the atmospheric turbulence but not for phase referencing. A special case exists for dual-star narrow-angle astrometry [@shao1992] where a “Dual Star” module located at each telescope can send light from two nearby stars down two different beam trains to be interfered simultaneously. At K band, the stars can be as far as $\sim$30$\arcsec$ apart for true phase referencing. This approach is being attempted at the VLT [PRIMA, @delplancke2006] and Keck Interferometers [ASTRA, @woillez2010]. This technique can be applied to only a small fraction of objects owing to the low sky density of bright phase reference calibrators. Adaptive optics (AO) can be used on large visible and infrared telescopes to effectively increase the collecting area $A_e$ term in our signal equation, allowing the full telescope aperture to be used for interferometry. AO on a 10-m class telescope potentially boosts infrared sensitivity by $\times$100 over the seeing limit; however, this method still requires a bright enough AO natural or laser guide star to operate. Currently, only the VLT and Keck Interferometers have adaptive optics implemented for regular use with interferometry. A related technique of [*fringe tracking*]{} is in more widespread use, whereby the interferometer light is split into two channels so that light from one channel is used exclusively for measuring the changing atmospheric turbulence and driving active realtime path length compensation. In the meantime, the other channel is used for longer science integrations (at VLTI, Keck, CHARA). This method improves the limiting magnitude of the system at some wavelengths if the object is substantially brighter at the fringe tracking wavelength, such as for dusty reddened stars. Fringe tracking sometimes can be used for very high spectral observations of stars ordinarily too faint to observe at high dispersion. It is important to mention these other optical interferometer subsystems (e.g., AO, fringe tracker) here because they are crucial for improving sensitivity, but the additional complexities do pose a challenge for observers. Each subsystem has its own sensitivity limit and now multiple wavelengths bands are needed to drive the crucial subsystems. As an extreme example, consider the Keck Interferometer Nuller [@colavita2009]. The R-band light is used for tip-tilt and adaptive optics, the H band is used to correct for turbulence in air-filled Coude path, the K band is used to fringe track and finally the 10$\mu$m light is used for the nulling work. If the object of interest fails to meet the sensitivity limit of any of these subsystems then observations are not possible – most strongly affecting highly reddened sources like young stellar objects and evolved stars. (u,v) Coverage {#uvcoverage} -------------- One central difference between interferometer and conventional single-telescope observations is the concept of (u,v) coverage. Instead of making a direct “image” of the sky at the focal plane of a camera, the individual fringe visibilities for each pair of telescopes are obtained. As discussed in §\[vcz\], each measured complex visibility is a single Fourier component of a portion of the sky. The goal of this subsection is to understand how to estimate (u,v) coverage from the array geometry and which characteristics of Fourier coverage affect the final reconstructed image. For a given layout of telescopes in an interferometer array, the Fourier coefficients can be measured are determined by drawing baselines between each telescope pair. To do this, an (x,y) coordinate system is first constructed to describe the positions of each element of the array; for ground-based arrays in the northern hemisphere, the convention is to orient the +x axis towards the east and the +y axis towards north. The process of determining the complete ensemble of (u,v) points provided by any given array can be laborious for arrays with a large number of elements. A simple method of automating the procedure is as follows. First, construct a distribution in the (x,y) plane of delta functions of unit strength at the positions of all elements. The (u,v) plane coverage can be obtained from the two-dimensional autocorrelation of this distribution, as illustrated in Figure \[fig\_uvcov1\] for four simple layouts of array elements. The delta functions for each array element are shown as dots in the upper row of sketches in this figure, and the corresponding dots in the u,v distributions are shown in the lower row of autocorrelations. Note that each point in the (u,v) plane is repeated on the other side of the origin owing to symmetry; of course the values of amplitude and phase measured on a source at one baseline will be the same whether one thinks of the baseline as extending from telescope 1 to telescope 2, or the converse. For an array of $N$ telescopes, one can measure ${{N}\choose{2}}=\frac{(N)(N-1)}{2}$ independent Fourier components. Sometimes the array geometry may result in the (near-)duplication of baselines in the (u,v) plane. This is the case for array \#2 in the Figure \[fig\_uvcov1\], where the shortest spacing is duplicated 4 times, the next spacing is duplicated 3 times, the following spacing is duplicated twice, and only the longest spacing of this array is unique. While each of these interferometers does contribute statistically independent data as far as the noise is concerned, it is an inefficient use of hardware since the astrophysical information obtained from such redundant baselines is essentially the same. In order to optimize the Fourier coverage for a limited number of telescopes, a layout geometry should be [*non-redundant*]{}, with no baseline appearing more than once, so that the maximum number of Fourier components can be measured for a given array of telescopes. A number of papers have been written on how to optimize the range and uniformity of (u,v) coverage under different assumptions [@golay1971; @keto1997; @holdaway1999]. Note that in the sketches of Figure \[fig\_uvcov1\], array \#4 provides superior coverage in the u,v plane compared to arrays \#3 and \#2 with the same number of array elements. Finally note that the actual (u,v) coverage depends not on the [ *physical baseline separations*]{} of the telescopes but on the [ *projected baseline separations*]{} in the direction of the target. For ground-based observing, a celestial object moves across the sky along a line of constant declination, so the (u,v)-coverage is actually constantly changing with time. This is largely a benefit since earth rotation dramatically increases the (u,v)-coverage without requiring additional telescopes. This type of synthesis imaging is often called [*Earth rotation aperture synthesis*]{}. The details depend on the observatory latitude and the target declination, and a few simple cases are presented in Figure \[fig\_uvcov2\]. In general, sources with declinations very different from the local latitude will never reach a high elevation in the sky, such that the north-south (u,v) coverage will be foreshortened and the angular resolution in that direction correspondingly reduced. Figure \[fig\_uvcov\_compare\] shows the actual Fourier coverage for the 27-telescope Very Large Array (VLA) and for the 6-telescope CHARA Array. For $N=27$, the VLA can measure 351 Fourier components while CHARA ($N=6$) can measure only 15 simultaneously. Notice also in this figure that the ratio between the maximum baseline and the minimum baseline is much larger for the VLA (factor of 50, A array) compared to CHARA (factor of 10). The properties of the (u,v)-coverage can be translated into some rough characteristics of the final reconstructed image. The final image will have an angular resolution of $\sim\frac{\lambda}{B_{\rm max}}$, and note that the angular resolution may not be the same in all directions. It is crucial to match the desired angular resolution with the maximum baseline of the array because longer baselines will over-resolve your target and have very poor (or non-existent) signal-to-noise ratio (see discussion §\[radiosensitivity\]). This functionally reduces the array to a much smaller number of telescopes which dramatically lowers both overall signal-to-noise ratio and the ability to image complex structures. For optical arrays that combine only 3 or 4 telescopes, relatively few (u,v) components are measured concurrently and this limits how much complicated structure can be reconstructed[^10]. From basic information theory under best case conditions, one needs at least as many independent visibility measurements as the number of independent pixels in the final image. For instance, it will take hundreds of components to image a star covered with dozens of spots of various sizes, while only a few data points can be used to measure a binary system with unresolved components. Field-of-view ------------- While the (u,v) coverage determines the angular resolution and quality of image fidelity, the overall imaging field-of-view is constrained by a number of other factors. A common limitation for field-of-view is the primary beam pattern of each individual telescope in the array and this was already discussed in §\[interferometry\_practice\]: $\Delta\Theta\sim\frac{\lambda}{{\rm Diameter}}$. This limit can be addressed by [*mosaicing*]{}, which entails repeated observations over a wide sky area by coordinating multiple telescope pointings within the array and stitching the overlapping regions together into a single wide-field image. This practice is most common in the mm-wave where the shorter wavelengths result in a relatively small primary beam. A useful rule of thumb is that your field-of-view (in units of the fringe spacing) is limited to the ratio of the baseline to the telescope diameter. Most radio and mm-wave imaging is limited by their primary beam, however there is a major push to begin using “array feeds” to allow imaging in multiple primary beams simultaneously. Another limitation to field-of-view is the spectral resolution of the correlator/combiner. The spectral resolution of each channel can be defined as $R=\frac{\lambda}{\Delta\lambda}$. A combiner or correlator can not detect a fringe that is outside the system coherence envelope, which is simply related to the spectral resolution $R$. The maximum observable field of view is $R$ times the finest fringe spacing, or $\Delta\Theta \sim R \cdot \frac{\lambda}{B_{\rm max}}$, often referred to as the [*bandwidth-smearing*]{} limit. Most optical interferometers and also Very Long Baseline Interferometry (VLBI) are limited by bandwidth smearing. A last limitation to field-of-view arises from temporal smearing of data by integrating for too long during an observation. Because the (u,v) coverage is constantly changing due to Earth rotation, time averaging removes information in the (u,v)-plane resulting in reduced field-of-view. A crude field-of-view limit based on this effect is $\Delta\Theta \sim \frac{230}{\Delta t_{\rm minutes}} \frac{\lambda}{B_{\rm max}}$. Both radio and V/IR interferometric data can be limited by temporal-smearing if care is not taken in setting up the data collection, although this limitation is generally avoidable. Spectroscopic Capabilities -------------------------- As for regular radio and optical astronomy, one tries to observe at the crudest spectral resolution that is suitable for the science goal in order to achieve maximum signal-to-noise ratio. However as just discussed, spectral resolution does impact the imaging field-of-view, bringing in another dimension to preparations. While each instrument has unique capabilities that can not be easily generalized, most techniques will require dedicated spectral calibrations as part of observing procedures. “Spectro-interferometry” is an exciting tool in radio and (increasingly) optical interferometry. In this application, the complex visibilities are measured in many spectral channels simultaneously, often across a spectrally-resolved spectral line. This allows the different velocity components to be imaged or modelled independently. For example, this technique can be used for observing emitting molecules in a young stellar object to probe and quantify Keplerian motion around the central mass or for mapping differential rotation on the surface of a rotating star using photospheric absorption lines [e.g., @kraus2008]. Spectro-interferometry is analogous to “integral field spectroscopy” on single aperture telescopes, where each pixel in the image has a corresponding measured spectrum. Another clever example of spectro-interferometry pertains to maser sources in the radio: a single strong maser in one spectral channel can be used as a phase calibrator for the rest of the spectral channels [e.g., @greenhill1998]. Data Analysis Methods {#dataanalysis} ===================== After observations have been completed, the data must be analyzed. Every instrument will have a customized software pipeline to take the recorded electrical signals and transform into useful astronomical quantities. That said, the data reduction process is the similar for most systems and here the basic steps are outlined. Data reduction and calibration overview --------------------------------------- The goal of the data reduction is to produce calibrated complex visibilities and related observables such as closure phases (see §\[closurephase\]). As discussed in §\[observing\], the basic paradigm for interferometric observing is to switch between data of well-known system calibrators and the target of interest. This allows for calibration of changing atmospheric conditions by monitoring the actual phase delay through the atmosphere (in radio) or by statistically correcting for decoherence from turbulence (in optical). One begins by plotting the observed fringe amplitude versus time. Figure \[calsrccal\_a\] shows a schematic example of how data reduction might proceed for the case of high quality radio interferometry observations, such as taken with the EVLA. Here the observed fringe amplitude and phase for a calibrator-target-calibrator sequence is presented. Notice that in this example the fringe amplitude of the calibrator is drifting up with time, as is the observed phase. As long as the switching time between target and calibrator is faster than instrumental gain drifts and atmospheric piston shifts, a simple function can be fitted to the raw calibrator measurements and then interpolated to produce the calibration curves to for the target. Here a 2nd order polynomial has been used to approximate the changing amplitude and phase response. This figure contains an example for only a single baseline, polarization, and spectral channel; there will be hundreds or thousands of panels like this in a dataset taken with an instrument as the EVLA or ALMA. The justification for this fitting procedure can be expressed mathematically. As the wave traverses the atmosphere, telescope, and interferometer beamtrain/waveguides, the electric field can have its phase and amplitude modified.[^11] These effects can be grouped together into a net complex [*gain*]{} of the system for each beam, $\tilde{\mathcal{G}_i}$ – the amplitude of $\tilde{\mathcal{G}_i}$ encodes the net amplification or attenuation of the field strength and the phase term corresponds to combination of time delays in the system and effects from amplifiers in the signal chain. Thus, the measured electric field $\tilde{E}^\prime$ can be written as a product of the original field $\tilde{E}$ times this complex gain: $$\tilde{E}^\prime = \tilde{\mathcal{G}} \tilde{E} \label{monnier_eqn_1}$$ Since the observed complex visibility $\tilde{\mathcal{V}}_{12}$ for a baseline between telescope 1 to telescope 2 is related to the product $\tilde{E}_1 \tilde{E}_2^\ast$, then $$\begin{aligned} \tilde{\mathcal{V}}_{12}^\prime & \propto & \tilde{E}_1^\prime \tilde{E}_2^{\prime\ast} \\ &\propto & \tilde{\mathcal{G}}_1 \tilde{E}_1 \cdot \tilde{\mathcal{G}}_2^\ast \tilde{E}_2^\ast \\ & \propto & \tilde{\mathcal{G}}_1 \tilde{\mathcal{G}}_2^\ast \tilde{\mathcal{V}}_{12} \label{monnier_eqn_2}\end{aligned}$$ Thus the measured complex visibility $\tilde{\mathcal{V}}_{12}^\prime$ is closely related to the true $\tilde{\mathcal{V}}_{12}$, differing only by complex factor $\tilde{\mathcal{G}}_1 \tilde{\mathcal{G}}_2^\ast $. By observing a calibrator with known structure, this gain factor can be measured, even if the calibrator is not a point source for the interferometer. For a radio array, the gain factors are mainly associated with the individual telescope collectors and not the baseline, and so the same gain factors appear in many baselines. This redundancy has led to the development of additional off-line procedures to “self-calibrate” radio imaging data using “closure amplitude” techniques (see §\[closurephase\]). Once the system drifts have been estimated by measurements of the calibrator, this correction can be applied to the whole dataset. Figure \[calsrccal\_b\] shows the calibrated result, where the calibrator flux was assumed to be 30 Jy. In practice, radio phase calibrators are time-variable in flux and so each dataset typically includes an “amplitude calibrator,” a well-studied object with known flux as a reference. These calibrated data can now be averaged and used for further model fitting or synthesis imaging. In the example shown here, both the target and calibrator have reasonable signal-to-noise-ratio. In a more realistic case, the signal-to-noise of the target will be orders of magnitude worse – indeed, in one observing block there may be no discernable signal at all! The calibrator measurements are used to phase up the array and allow for very long phase-coherent integrations (averaging in the complex (u,v) plane). Unfortunately, this “blind” phase referencing can not generally be used in optical interferometry (see §\[phasereferencing\]) where the short atmospheric coherence time and the worse turbulence requires active fringe tracking for both target and calibrator at all times. Note that actual data will not look quite like this simplified schematic. First, raw data might have random data glitches or bad values that need to be flagged. Also, one tends to only observe the calibrator for a short time, just enough to measure the phase. In fact, the time to slew between targets can be similar to the length of time spent integrating on each calibrator. The time spent on the target during each visit is generally as long as possible given the atmospheric coherence time which can vary greatly with baseline length, observing conditions, and wavelength (see §\[turbulence\]). A common complication is that the calibrator may not be an unresolved object nor constant in flux. NRAO maintains a calibrator database that is used to determine the suitability of each calibrator for different situations. As long as the calibrator morphology is known, the observer can apply a visibility amplitude and phase correction to account for the known structure. After this correction, the calibration procedure is the same. For visible and infrared interferometry, the procedure is very similar. In general, optical interferometers measure a time-averaged squared-visibility and not visibility amplitude since the $\mathcal{V}^2$ can be bias-corrected more easily for low signal-to-noise ratio data when observing with no phase referencing [@colavita1999]. As discussed earlier, optical interferometers cannot employ phase referencing between two targets [^12] due to the tiny isoplanatic patch and short temporal coherence times. Instead of averaging fringe phases, closure phases (see §\[closurephase\]) are formed and averaged over longer time frames following a similar interpolation of calibration data. Lastly, calibrators tend to be stars with known or well-estimated diameters. For a given baseline, the observed raw calibrator $\mathcal{V}^2$ are boosted to account for partially resolving them during the observation before the system visibility is estimated. When carrying out spectral line and/or polarization measurements, additional calibrations are required. As for single telescope observations, one must observe a source with known spectrum and/or polarization signature in order to correct for system gains. These procedures add steps to the data reduction but are straightforward. A diversity of packages and data analysis environments are in use for data reduction of interferometer observations. In the radio and mm-wave regime, the most popular packages are AIPS and Miriad. In addition, the CASA package will be used for ALMA and supports many EVLA operations now too. In the visible and infrared, the data reduction packages are usually closely linked to the instrument and are provided by the instrument builders. In most cases there are data analysis “cookbooks” that provide step-by-step examples of how to carry out all steps in the data reduction. Few instruments have complete pipelines that require no user input, although improved scripting is a high priority for future development. A number of summer schools are offered that train new users of interferometer facilities in the details of observation planning and data reduction. The data products from this stage are calibrated complex visibilities and/or closure phases. No astronomical interpretation has occurred yet. The [*de facto*]{} standard data format for radio data is UVFITS. Unfortunately this format is not strictly defined but rather represents the data supported by the NRAO AIPS package and importable into CASA (which uses a new format called the [*Measurement Set*]{}). Recently the VLBA community fully documented and registered the FITS Interferometry Data Interchange (FITS-IDI) format (see [*“FITS-IDI definition document” AIPS Memo 114*]{}). The optical interferometry community saw the problems of radio in having a poorly defined standard and, through IAU-sanctioned activites, crafted a common FITS-based data standard called OIFITS whose specifications were published by @pauls2005. This standard is in wide use by most optical interferometers in the world today. Model-fitting for poor (u,v) coverage ------------------------------------- The ultimate goal of most interferometric observations is to have sufficient data quality and (u,v) coverage to make a synthesis image. With high image fidelity, an astronomer can interact with the image just as one would had it come from a standard telescope imager. Still, cases are plentiful where this ideal situation is not achievable and one will fit a model to the visibility data directly. There are two classes of models: [*geometric* ]{} models and [*physical* ]{} models. [*Geometric models*]{} are simple shapes that describe the emission but without any physics involved. Common examples include Gaussians, uniform disks, binary system of 2 uniform disks, etc. [*Physical models*]{} start with a physical picture including densities, opacities, and sources of energy. Typically a radiative transfer calculation is used to create a synthetic image which can be Fourier-transformed (following Equation \[eq:vcz2\]) to allow fitting to complex visibilities. Geometric models are useful for very simple cases when an object is marginally resolved or when physical models are not available (or not believable!). Physical models are required to connect observations with “real” quantities like densities and temperatures, although size scales can be extracted with either kind of model. In radio and mm-wave, model fitting is now relatively rare[^13] since high-fidelity imaging is often achievable. However in the optical, model fitting is still the most common tool for interpreting interferometry data. In many cases, a simple uniform disk or Gaussian is adequate to express the characteristic size scale of an object. By directly fitting to visibility amplitudes, the data can be optimally used and proper error analysis can be performed. The fitting formulae for the two most common functions can be expressed in closed form as a function of baseline $B$ and wavelength $\lambda$: $$\begin{aligned} |\mathcal{V}| & = & 2 \frac{J_1 (\pi B \Theta_{\rm diameter} / \lambda)}{\pi B \Theta_{\rm diameter} / \lambda} \qquad {\rm case: Uniform~Disk } \\ |\mathcal{V}| & = & e^{-\frac{\pi^2}{4 \ln{2}} (\Theta_{\rm FWHM} B / \lambda)^2} \qquad {\rm case: Gaussian }\end{aligned}$$ Figure \[fitting\_examples\]a illustrates both the model-fitting process and the importance of choosing the most physically-plausible function. Here some simulated visibility data spanning baselines from 0 to 60 m are plotted along with curves for 5 common brightness distributions – a uniform disk, a Gaussian disk, two binary models and a ring model. For the case of marginally resolved objects, only the [*characteristic scale*]{} of a given model can be constrained and there is no way to distinguish [*between models*]{} without longer baseline information [for more elaborate discussion, see @lachaume2003]. Note how all the curves fit the data equally well at short baselines and high visibilities, but that the [*interpretation*]{} of each curve is quite different. Without longer baselines that can clearly distinguish between these models, the observer must rely on theoretical expectations to guide model choice. For example a normal G star should closely resemble a uniform disk in the near-infrared while disk emission from a young stellar object in the sub-mm might be more Gaussian. Despite the uncertainties when fitting to marginally resolved targets, fitting interferometric data allows very precise determinations of model parameters as can be seen in Figure \[fitting\_examples\]b where the CHARA Array was used to monitor the variations in diameter of the Cepheid variable $\delta$ Cep. A recently-published example of model-fitting at longer wavelengths is shown in Figure \[fig\_andrews\]. Here, a semi-analytic physical model (in this case of a circumstellar disk) was used to simultaneously fit the spectral energy distribution along with the visibility data. When realistic physical models are available, multi-wavelength constraints can make a dramatic improvement to the power of high angular resolution data and should be included whenever possible. Synthesis imaging {#imaging} ================= Interferometer data in their raw form are not easy to visualize. Fortunately, as discussed in §\[vcz\], the measured complex visibilities can be transformed into an equivalent brightness distribution on the sky – [*an image*]{}. This procedure is called “aperture synthesis imaging” or more generally “synthesis imaging.” In this section, the critical data analysis steps are described for creating an image in both the ideal case as well as more challenging scenarios when faced with poor (u,v) coverage and phase instability. Ideal case ---------- Under ideal conditions the astronomer will have collected interferometer data with a large number of telescopes including some Earth rotation to fill-in gaps in (u,v) coverage (see §\[uvcoverage\]). In addition, each datum will consist of a fully calibrated complex visibility – both amplitude and phase information. Modern radio arrays such as the VLA and ALMA produce data of this quality when proper phase-referencing procedures (see §\[phasereferencing\]) are employed. Figure \[fig\_imaging1\] depicts the Fourier coverage for a 6 telescope interferometer along with the resulting image from a direct Fourier transform of this coverage for a perfect point source (by construction, the image will be purely real). In this procedure, the values in the (u,v) grid are set to unity where data exists and to zero where no data exists. This resulting image shows artifacts because of the missing (u,v) data that were zeroed out. These artifacts are often called “sidelobes” and show both positive and negative excursions – note that negative flux density is usually a strong sign of a sidelobe since negative values in an image are typically unphysical (except for special applications like absorption line studies or polarization stokes mapping). Note that the central core of the image represents the diffraction-limited angular resolution of this observation. Often in a practical situation, the central core will be elongated because (u,v) coverage is not perfectly symmetric, with longer baselines in some directions than others. The last panel in this figure shows the resulting image for a binary star with 2:1 flux ratio. Notice that the image contains two sources, however both sources show the same pattern of artifacts as the simple point source. Indeed, this supports the previous assertion that missing (u,v) coverage is the main origin of this pattern and suggests that the image quality can be improved by correcting for the sidelobe effect. To proceed, the well-known Convolution Theorem must be introduced: [ *Multiplication in the “(u,v) space” is equivalent to a convolution in “image space.”*]{} Mathematically, this as can be expressed: $$\begin{aligned} {\rm FT} ( \tilde{\mathcal{V}}(u,v) \cdot M(u,v) ) & = & ({\rm FT}~\tilde{\mathcal{V}}) \otimes ({\rm FT}~M) \\ \tilde{\mathcal{V}}(u,v) \cdot M(u,v) & \Leftrightarrow & I(x,y) \otimes B(x,y)\end{aligned}$$ where $\tilde{\mathcal{V}}$ is the full underlying complex visibility that could in principle be measured by the interferometer, $M$ is the (u,v) mask that encodes whether data exists (1) or is missing (0)[^14], FT and $\Leftrightarrow$ denote a Fourier Transform, $I={\rm FT}~\tilde{\mathcal{V}}$ is the true image distribution , and $B={\rm FT}~M$ is called the “Convolving Beam.” Application of the convolution theorem permits an elegant reformulation of the imaging problem into a “deconvolution” problem, where the convolving beam is a complicated function but derived directly from $M$, the observed (u,v) coverage. Since $M$ is exactly known, the convolving beam $B$ is known as well. Note that this deconvolution problem contrasts sharply with the deconvolution problem in adaptive optics imaging where the point source function (PSF) varies in time and is never precisely known. ### CLEAN Algorithm One of the earliest methods developed for deconvolution was the CLEAN algorithm [@hogbom1974], which is still widely used in radio interferometry. In CLEAN, the Fourier transform of the gridded complex visibility data is called the “dirty image” (or sometimes “dirty map”) and the Fourier transform of the (u,v) plane mask is called the “dirty beam.” One first needs to deconvolve the dirty image with the dirty beam. To do this, the true image $I$ is interatively constructed by locating the peak in the dirty image and subtracting from this a scaled version of the dirty beam centered at this location. Here, the scaling of the dirty beam is typically tuned to removed a certain fraction of intensity from the peak, often 5%. One keeps track of how much one removes by collecting the “CLEAN” components in a list. Consider this example. The dirty image has peak of 1.0 Jy at pixel location (3,10). One creates a scaled dirty beam with a peak of 0.05 Jy, shifts the peak to the position (3,10), and subtracts this scaled dirty beam from the dirty image. This CLEAN component is collected and labeled by location (3,10) and flux contribution (0.05 Jy). Continue this procedure, flux is removed from dirty image and CLEAN components are collected. This procedure is halted when the residual dirty image contains only noise. Since the intensity in the true image is expected to be positive-definite, one common criterion for halting the CLEAN cycles is when the largest negative value in the image is comparable to the largest positive value in the image, thereby avoiding any CLEAN components having negative flux. In principle, this collection of delta function point sources (all the CLEAN components) [*is*]{} the best estimate of the image distribution. However, an image of point sources is not visually appealing and a common procedure is to convolve the point source distribution with a “CLEAN” beam, a perfect 2-D Gaussian with a core that matches the FWHM of the dirty beam. Commonly, a filled ellipse will be included in the corner of a CLEANed image showing the 2D FWHM of the restoring beam. Lastly, one adds back the residual image from the dirty map (which should contain only noise) so that the noise level is apparent in the final image and remnant uncorrected sidelobe artifacts, if present, can be readily identified. These steps are illustrated in Figure \[fig\_imaging2\] where the CLEAN procedure has been applied to the examples shown in Figure \[fig\_imaging1\]. The steps of the CLEAN algorithm are summarized below. 1. Create dirty map and beam. 2. Find peak of dirty map. 3. Subtract scaled version of dirty beam from dirty map, removing a small percentage (e.g., $\sim$5%) of peak intensity. Collect CLEAN components. 4. Repeat last step until negative residuals are comparable to positive residuals. 5. Convolve CLEAN components with CLEAN beam. 6. Add back the residuals. While it is beyond the scope to address the weaknesses of the CLEAN algorithm in detail, a few issues are worth mentioning in passing. Typical CLEAN algorithms do not naturally deal with errors unless all the visibility data are of similar quality. The case when each visibility point is weighted equally is called [*natural weighting*]{} (which gives best SNR for detecting faint objects) and the case when each portion of the (u,v) plane is equally weighted, so called [*uniform weighting*]{} (which gives somewhat higher angular resolution but at some loss of sensitivity). @briggs1995 introduced a “ROBUST” parameter than can naturally span these two extremes. Other problems with CLEAN include difficulty reconstructing low surface brightness regions, large scale emission and the fact that the final reconstructed image is of a degraded resolution because the convolving PSF actually suppresses the longest baseline visibilities from the final image. Finally, if the imaging step makes use of the Fast Fourier Transform, several additional artifacts can appear in the image as a consequence of the necessity to grid the input $u,v$ data. ### Maximum Entropy Method (MEM) Another common method for reconstructing images from interferometric data is called the [*Maximum Entropy Method*]{} [@gull1983; @skilling1984]. This approach asks the question: “How to choose which reconstructed image is best, considering that there are an infinite number of images that can fit the interferometer dataset within the statistical uncertainties?” The simple answer here is that the “best” image is the one that both fits the data acceptably and also maximizes the [*Entropy*]{} $S$ defined as: $$S = -\sum_i f_i \ln \frac{f_i}{m_i}$$ where $f_i$ is the (positive-definite) fraction of flux in pixel $i$, and $m_i$ is called the [*image prior*]{} which can encapsulate prior knowledge of the flux distribution (e.g., known from physical considerations or lower resolution observations). @narayan1986 describe general properties of this algorithm in a lucid review article and motivates the above methodology using [ *Bayes’ Theorem*]{}, the cornerstone of so-called [*Bayesian Statistics*]{}. Entropy is an interesting statistic since it quantifies the amount of complexity in a distribution. It is often stated that MEM tries to find the “smoothest image” consistent with the data. This indeed is a highly desirable feature since any structures in the reconstructed image should be based on the data itself and not artifacts from the reconstruction process. However, MEM does not actually select the smoothest image but rather one with “most equal” uniform set of values – MEM does not explicitly take into account spatial structure but only depends on the distribution of pixel values. Indeed with more study, the Maximum Entropy functional has been found not to be that special, except for its privileged appearance in physics. From a broader perspective, maximum entropy can be considered one member of a class of regularizers that allow the inverse problem to be well-defined, and MEM is not necessarily the best nor even suitable for some imaging problems (e.g., other regularizers include total variation or maximum likelihood). MEM performs better for reconstructing smooth large scale emission than CLEAN, although MEM is much more computationally demanding. MEM can naturally deal with heterogeneous data with varying errors since the data is essentially fitted using a $\chi^2$-like statistic. This involves only a “forward” transform from image space to (u,v) space, thus avoiding all the issues of zeros in the (u,v) grid and the need for deconvolution. In addition, MEM images possess [*super-resolution*]{} beyond the traditional $\sim\frac{\lambda}{D}$ diffraction limit since smoothness is introduced in the process only indirectly through the Entropy statistic. For instance, if the object is a point-like object, then the FWHM of the reconstructed MEM image depends on the signal-to-noise of the data, not just the length of the longest baseline. Super-resolution is viewed with some skepticism by practioners of CLEAN because structures beyond the formal diffraction limit may be artifacts of the Entropy functional. MEM has been implemented in many radio interferometry data processing environments, such as AIPS for VLA/VLBA, Caltech VLBI package [@sivia1987], and for optical interferometry [e.g., BSMEM, @buscher1994]. Non-ideal cases for imaging --------------------------- Most modern interferometric arrays in the radio (VLA) and millimeter (ALMA, CARMA) have a sufficient number of elements for good (u,v) coverage and also employ rapid phase referencing for absolute phase calibration. This allows either CLEAN and MEM methods to make imaging possible, although mosaicing large fields can still pose a computational challenge. In optical interferometry and perhaps also at some sub-mm wavelengths, atmospheric turbulence changes the fringes phases so quickly and over such small angular scales that phase referencing is not practical. As discussed earlier, modeling can still be done for the visibility amplitudes but the turbulence scrambles the phase information beyond utility. Fortunately, there is a clever method to recover some of the lost phase information and this is discussed next. ### Closure Phase, the Bispectrum, and Closure Amplitudes {#closurephase} As discussed earlier, phase referencing is used to correct for drifting phases. So what can be done when phase referencing is not possible? Without valid phase information accompanying the visibility amplitude measurements, one cannot carry out the inverse Fourier Transform that lies at the core of synthesis imaging and the CLEAN algorithm specifically. While in some cases the fringe phases do not carry much information (e.g., for symmetrical objects), in general phases carry most of the information for complex scenes. Early in the history of radio interferometry a clever idea, now referred to as “closure phase,” was discovered to recover some level of phase information when observing with three telescopes [@jennison1958]. The method was introduced to partly circumvent the combination of poor receiver stability and variable multi-path propagation in early radio-linked long-baseline ($ \gtrsim 2$ km) interferometer systems at Jodrell Bank. The term “closure phase” itself appeared later on, in the paper by @rogers1974 describing an application at centimeter radio wavelengths using very stable and accurate, but independent, reference oscillators at the three stations in a so-called very long baseline interferometer (VLBI) array. Closure phase was critical for VLBI work in the 1980s although it became less necessary as phase referencing became feasible. Application at optical wavelengths was first mentioned by @rogstad1968, but carried out only much later in the optical range through aperture masking experiments [e.g., @baldwin1986]. By 2006, nearly all separated-element optical arrays with 3 or more elements have succeeded in obtaining closure phase measurements (COAST, NPOI, IOTA, ISI, VLTI, CHARA). An optical observer now can expect closure phases to be a crucial observable for most current instrumentation. The principle behind the power of closure phases is briefly described and the interested reader is referred to @monnier2007 for more detailed information on taking advantage of such phase information in optical interferometry. Consider Figure \[cphasefig\]a in which a time delay is introduced above one slit in a Young’s interferometer. This time delay introduces a phase shift for the detected fringe and the magnitude of the phase shift is independent of the baseline length. For the case of 3 telescopes (see Figure \[cphasefig\]b), a delay above one telescope will introduce phase shifts in [*two fringes*]{}. For instance, a delay above telescope 2 will show up as an equal phase shift for baseline 1-2 and baseline 2-3, but with [*opposite*]{} signs. Hence, the sum of three fringe phases, between 1-2, 2-3, and 3-1, will be insensitive to the phase delay above telescope 2. This argument holds for arbitrary phase delays above any of the three telescopes. In general, the sum of three phases around a closed triangle of baselines, the [*closure phase*]{}, is a good interferometric observable; that is, it is independent of telescope-specific phase shifts induced by the atmosphere or optics. The closure phase $\Phi_{ijk}$ can thus be written in terms of the three telescopes $i$,$j$,$k$ in the triangle: $$\Phi_{ijk} = \phi_{ij} + \phi_{jk} + \phi_{ki}$$ where $\phi_{ij}$ represents the measured Fourier phase for the baseline connecting telescopes $i$,$j$. Alternatively, the closure phase can be written in terms of the ($u_0$,$v_0$,$u_1$,$v_1$) in the Fourier (hyper-)plane where ($u_0$,$v_0$) represents the (u,v) coverage for baseline $i,j$ in the triangle, ($u_1$,$v_1$) represents the (u,v) coverage for baseline $j,k$ in the triangle, and the last leg of the triangle can be calculated from the others since the sum of the 3 baselines must equal zero to be a “closure triangle.” See definition and explanation put forward in documentation of the OI-FITS data format [@pauls2005]. Another method to derive the invariance of the closure phase to telescope-specific phase shifts is through the [*bispectrum*]{}. The bispectrum $\tilde{B}_{ijk}= \tilde{\mathcal V}_{ij} \tilde{\mathcal V}_{jk}\tilde{\mathcal V}_{ki}$ is formed through triple products of the complex visibilities around a closed triangle, where $ijk$ specifies the three telescopes. Using Eq. \[monnier\_eqn\_2\] and using the concept of telescope-specific complex gains $G_i$, it can be seen how the telescope-specific errors affect the measured bispectrum: $$\begin{aligned} \tilde{B}_{ijk} & = & \tilde{\mathcal V}^{\meas}_{ij} \, \tilde{\mathcal V}^{\meas}_{jk}\, \tilde{\mathcal V}^{\meas}_{ki} \\ & = & |G_i| |G_j| e^{i (\Phi^G_i -\Phi^G_j)} \tilde{\mathcal V}^{\true}_{ij} \cdot |G_j| |G_k| e^{i (\Phi^G_j -\Phi^G_k)} \tilde{\mathcal V}^{\true}_{jk} \cdot |G_k| |G_i| e^{i (\Phi^G_k -\Phi^G_i)} \tilde{\mathcal V}^{\true}_{ki} \\ & = & |G_i|^2 |G_j|^2 |G_k|^2 \tilde{\mathcal V}^{\true}_{ij} \cdot \tilde{\mathcal V}^{\true}_{jk} \cdot \tilde{\mathcal V}^{\true}_{ki}\end{aligned}$$ From the above derivation, the bispectrum is a complex quantity whose phase is identical to the closure phase, while the individual telescope gains affect only the bispectrum amplitude. The use of the bispectrum for reconstructing diffraction-limited images from speckle data was developed independently [@weigelt1977] of the closure phase techniques, and the connection between the approaches elucidated only later [@roddier1986; @cornwell1987]. A 3-telescope array with its one triangle can provide a single closure phase measurement, a paltry substitute for the 3 Fourier phases available using phase referencing. However, as one increases the number of elements in the array from 3 telescopes to 7 telescopes, the number of independent closure phases increases dramatically to 15, about 70% of the total 21 Fourier phases available. An array the size of the VLA with 27 antennae is capturing 93% of the phase information. Indeed, imaging of bright objects does not require phase referencing and the VLA can make high quality imaging through closure phases alone. Note that imaging using closure phases alone retains no absolute astrometry information; astrometry requires phase-referencing. A related quantity useful in radio is the [*closure amplitude*]{} (which requires sets of 4 telescopes) and this can be used to compensate for unstable amplifier gains and varying antenna efficiencies [e.g., @readhead1980]. Closure amplitudes are not practical for current optical interferometers partially because most fringe amplitude variations are not caused by telescope-specific gain changes but rather by changing coherence (e.g., due to changing atmosphere). Closure phases (and closure amplitudes) can be introduced into the imaging process in a variety of ways. For the CLEAN algorithm, the closure phases can not be directly used because CLEAN requires estimates of the actual Fourier phases in order to carry out a Fourier transform. A clever iterative scheme known as [*self-calibration*]{} was described by @readhead1978 and @cornwell1981 which alternates between a CLEANing stage and a self-calibration stage that estimates Fourier phases from the closure phases and most recent CLEANed image. As for standard CLEAN itself, self-calibration can not naturally deal with errors in the closure phases and closure amplitudes, and thus is recognized as not optimal. That said, self-calibration is still widely used along with CLEAN even in the case of phase referencing to dramatically improve the imaging dynamic range for imaging of and around bright objects. Closure phases, the bispectrum, and closure amplitudes can be quite naturally incorporated into “forward-transform” image reconstruction schemes such as the Maximum Entropy Method. Recall that MEM basically performs a minimization of a regularizer constrained by some goodness-of-fit to the observed data. Thus, the bispectral quantities can be fitted just like all other observables. That said, the mathematics can be difficult and the program BSMEM [@buscher1994] was one of the first useful software suitable for optical interferometers that successfully solved this problem in practice. Currently, the optical interferometer community have produced several algorithms to solve this problem, including the Building Block Method [@hofmann1993], MACIM [@macim], MIRA [@thiebaut2008], WISARD [@meimon2008], and SQUEEZE [@baron2010]. See @malbet2010 for a description of a recent blind imaging competition between some of these algorithms. Astrometry ---------- Astrometry is still a specialized technique within interferometry and a detailed description is beyond the scope of this chapter. Typically, the precise separation between two objects on the sky, the [*relative astrometry*]{}, is needed to be known for some purpose, such as a parallax measurement. If both objects are known point-like objects with no asymmetric structure, then the precise knowledge of the baseline geometry can be used, along with detailed measurements of interferometric fringe phase, to estimate their angular separation. In general, the astrometric precision $\Delta\Phi$ on the sky is related to the measured fringe phase SNR as follows: $$\Delta\Phi \sim \frac{\lambda}{\rm Baseline} \cdot \frac{1}{\rm SNR}$$ Hence, if one measures a fringe with a signal-to-noise of 100 (which is quite feasible) then one can determine the relative separation of two sources with a precision 100$\times$ smaller than the fringe spacing. Since VLBI and optical interferometers have fringe spacings of $\sim$1 milliarcseconds, this allows for astrometric precision at the 10 [*microarcsecond*]{} level allowing parallax measurements at many kiloparsecs and also allows us to take a close look around the black hole at the center of the Galaxy. Unfortunately, at this precision level many systematic effects become critical and knowledge of the absolute baseline vector between telescopes is crucial and demanding. The interested reader should consult specific instruments and recent results for more information [@reid2009; @muterspaugh2010a for radio and optical respectively]. Concluding remarks ================== In barely more than 50 years, separated-element interferometry has come to dominate radio telescope design as the science has demanded ever-increasing angular resolution. The addition of more elements to an array has provided improvements both in mapping speed and in sensitivity. The early problems with instability in the electronics were solved years ago by improvements in radio and digital electronics, and it is presently not unusual to routinely achieve interferometer phase stability of order $\lambda/1000$ over time spans of many tens of minutes. Sensitivity improvements by further reductions in receiver temperature are reaching a point of diminishing returns as the remaining contributions from local spillover, the Galactic background, and atmospheric losses come to dominate the equation. Advances in the speed and density of digital solid-state devices are presently driving the development of increased sensitivity, with flexible digital signal processors providing wider bands for continuum observations and high frequency resolution for precision spectroscopy. With the commissioning of the billion-dollar international facility ALMA underway and long-term plans for ambitious arrays at longer wavelengths (SKA, LOFAR), the future of “radio” interferometry is bright and astronomers are eager to take advantage of the new capabilities, especially vast improvements to sensitivity. The younger field of optical interferometry is still rapidly developing with innovative beam combination, fringe tracking methods and extensions of adaptive optics promising significant improvements in sensitivity for years to come. In the long-run however, the atmosphere poses a fundamental limit to the ultimate astrometric precision and absolute sensitivity for visible and infrared interferometry that can only be properly overcome by placing the telescopes into space or through exploring exotic sites such as Dome A, Antarctica. Currently however, the emphasis of the majority of the scientific community favors ever-larger collecting area in the near- and mid-IR (e.g. JWST) as astronomers reach to ever greater distances and earlier times, rather than higher angular resolution. Eventually, the limits of diffraction are likely to limit the science return of space telescopes, and the traditional response of building a larger aperture will no longer be affordable. Filled apertures increase in weight and in cost as a power $> 1$ of the diameter [@bely2003]. For ground-based telescopes, cost $\sim D^{2.6}$. In space, the growth is slower, $\sim D^{1.6}$, although the coefficient of proportionality is much larger[^15]. Over the last two decades the prospects have become increasingly bleak for building ever larger filled-aperture telescopes within the anticipated space science budgets. Interferometry permits the connection of individual collectors of modest size and cost into truly gigantic space-based constellations with virtually unlimited angular resolution [@allen2007]. Ultimately, the pressure of discovery is likely to make interferometry in space at optical, UV, and IR wavelengths a necessity, just as it did at radio wavelengths more than half a century earlier. Appendix: Current Facilities {#appendix} ============================ Table \[table:all\_interferometers\] provides a comprehensive list of existing and planned interferometer facilities. These facilities span the range from “private” instruments, currently available only to their developers, to general purpose instruments. An example of the latter is the Square Kilometer Array; this is a major next-generation facility for radio astronomy being planned by an international consortium, with a Project Office at the Jodrell Bank Observatory in the UK (http://www.skatelescope.org/). The location for the SKA has not yet been finalized, but (as of 2011) sites in Australia and South Africa are currently being discussed. Plans for the SKA have spawned several “proof-of-concept” or “pathfinder” instruments which are being designed and built including LOFAR (Europe), MeerKAT (South Africa), ASKAP, MWA, and SKAMP (Australia), and LWA (USA). Table \[table:capabilities\] summarizes the wavelength and angular resolution of currently operating arrays open to the general astronomer. A few of the facilities in this list are still under construction. Figure \[fig\_capabilities\] summarizes the vast wavelength range and angular resolution available using the radio and optical interferometers of the world. Acronym Full Name Lead Institution(s) Location --------- ---------------------------------------------- ------------------------------------------------- ------------------------------------------ ALMA Atacama Large Millimeter Array International/NRAO Chajnantor, Chile ATA Allen Telescope Array SETI Institute Hat Creek Radio Observatory, CA, USA ATCA Australia Telescope Compact Array CSIRO/ATNF Narrabri, Australia CARMA Combined Array for Research Caltech, UCB, UChicago, Big Pine, California USA in Millimeter Astronomy UIUC, UMD CHARA Center for High Angular Resolution Astronomy Georgia State University Mt. Wilson, CA, USA DRAO Dominion Radio Astrophysical Observatory Herzberg Institute of Astrophysics Penticton BC, Canada EVLA Expanded Very Large Array NRAO New Mexico USA EVN The European VLBI Network International Europe, UK, US, S. Africa, China, Russia FASR Frequency Agile Solar Radiotelescope National Radio Astronomy Observatory + 6 others Owens Valley, CA, USA ISI Infrared Spatial Interferometer Univ. California at Berkeley Mt. Wilson, CA, USA Keck-I Keck Interferometer (Keck-I to Keck-II) NASA-JPL Mauna Kea, HI, USA LBTI Large Binocular Telescope Interferometer LBT Consortium Mt. Graham, AZ, USA LOFAR Low Frequency ARray Netherlands Institute Europe for Radio Astronomy - ASTRON LWA Long Wavelength Array US Naval Research Laboratory VLA Site, NM, USA MOST Molonglo Observatory Synthesis Telescope School of Physics, Univ. Sydney Canberra, Australia MRO Magdalena Ridge Observatory Consortium of New Mexico Institutions, Magdalena Ridge, NM, USA MWA Murchison Widefield Array International/MIT-Haystack Western Australia NPOI Navy Prototype Optical Interferometer Naval Research Laboratory/ Flagstaff, AZ, USA U.S. Naval Observatory OHANA Optical Hawaiian Array Consortium (mostly French Institutions), Mauna Kea, HI, USA for Nanoradian Astronomy Mauna Kea Observatories, others PdB Plateau de Bure interferometer International/IRAM Plateau de Bure, FR SKA Square Kilometer Array see Appendix \[appendix\] under study SMA Submillimeter Array Smithsonian Astrophysical Observatory Mauna Kea, HI, USA SUSI Sydney University Stellar Interferometer Sydney University Narrabri, Australia VLBA Very Long Baseline Array NRAO Hawaii to St. Croix, US Virgin Islands VLTI-UT VLT Interferometer (Unit Telescopes) European Southern Observatory Paranal, Chile VLTI-AT VLT Interferometer (Auxiliary Telescopes) European Southern Observatory Paranal, Chile VERA VLBI Exploration of Radio Astrometry Nat. Astron. Obs. Japan (NAOJ) Japan VSOP-2 VLBI Space Observatory Programme-2 Japan Aerospace Exploration Agency Space-Ground VLBI WSRT Westerbork Synthesis Radio Telescope Netherlands Institute for Westerbork, NL Radio Astronomy - ASTRON : Alphabetical list of current and planned interferometer arrays. \[table:all\_interferometers\] ------------ ---------- ------------ -------------------- ------------------------- ------------------------------------------------------------ Maximum Wavelength Observer Resources Acronym Number Size (m) Baseline Coverage web page link SUSI 2 0.14 64 (640$\ast$) m 0.5–1.0$\mu$m http://www.physics.usyd.edu.au/sifa/Main/SUSI NPOI 6 0.12 64 ($>250\ast$) m 0.57–0.85$\mu$m http://www.lowell.edu/npoi/ CHARA 6 1.0 330 m 0.6–2.4$\mu$m http://www.chara.gsu.edu/CHARA/ MRO$\ast$ $\sim$10 $\sim$1.5 $\sim$350 m 0.6–2.5$\mu$m http://www.mro.nmt.edu/Home/index.htm VLTI 4 1.8 – 8.0 130 ($202\ast$) m 1–2.5$\mu$m, 8–13$\mu$m http://www.eso.org/sci/facilities/paranal/telescopes/vlti/ Keck-I 2 10.0 85 m 1.5–4$\mu$m, 8–13$\mu$m Http://planetquest.jpl.nasa.gov/Keck/ LBTI$\ast$ 2 8.4 23$\ast$ m 1–20$\mu$m http://lbti.as.arizona.edu/ ISI 3 1.65 85 ($>100 \ast$) m 8–12$\mu$m http://isi.ssl.berkeley.edu/ ALMA$\ast$ 66 7 – 12 15 km 0.3 mm – 3.6 mm http://science.nrao.edu/alma/index.shtml SMA 8 6 500 m 0.4 mm – 1.7 mm http://www.cfa.harvard.edu/sma/ CARMA 23 3.5 – 10.4 2000 m 1.3, 3, 7 mm http://www.mmarray.org/ PdB 6 15 760 m 1.3, 2, 3 mm http://www.iram-institute.org/EN/ VLBA 10 25 8000 km 3 mm – 28 cm http://science.nrao.edu/vlba/index.shtml eMERLIN 7 25 – 76 217 km 13 mm – 2 m http://www.e-merlin.ac.uk/ EVN 27 14 – 305 $> 10000$ km 7 mm - 90 cm http://www.evlbi.org/ VERA 4 20 2270 km 7 mm – 1.4 cm http://veraserver.mtk.nao.ac.jp/index.html ATCA 6 22 6 km 3 mm – 16 cm http://www.narrabri.atnf.csiro.au/ EVLA 27 25 27 km 6 mm – 30 cm http://science.nrao.edu/evla/index.shtml WSRT 14 25 2700 m 3.5 cm – 2.6 m http://www.astron.nl/radio-observatory /astronomers/wsrt-astronomers DRAO 7 9 600 m 21, 74 cm e-mail: Tom.Landecker@nrc-cnrc.gc.ca GMRT 30 45 25 km 21 cm – 7.9 m http://www.gmrt.ncra.tifr.res.in MOST 64 $\sim 12$ 1600 m 36 cm http://www.physics.usyd.edu.au/sifa/Main/MOST LOFAR many simple 1500 km 1.2 m – 30 m http://www.astron.nl/radio-observatory /astronomers/lofar-astronomers ------------ ---------- ------------ -------------------- ------------------------- ------------------------------------------------------------ : Subset of those interferometer arrays from Table \[table:all\_interferometers\] which are presently open (or will soon be open) for use by qualified researchers from the general astronomy community. “$\ast$” indicates capabilities under development. \[table:capabilities\] We are grateful to the following colleagues for their comments and contributions: W.M. Goss, R.D. Ekers, T.L. Wilson, S. Kraus, M. Zhao, T. ten Brummelaar, A. King, and P. Teuben. {width="6in"} {width="4in"} {width="6in"} {height="3in"} {height="3in"} {width="6in"} {width="6in"} {width="6in"} {width="3in"} {width="3in"} {width="5in"} {width="6in"} {width="6in"} ![(left) Atmospheric turbulence introduces extra path length fluctuations that induce fringe phase shifts. At optical wavelengths, these phase shifts vary by many radians over short time scales ($<<$1 sec) effectively scrambling the Fourier phase information. (right) Phase errors introduced at any telescope causes equal but opposite phase shifts in ajoining baselines, canceling out in the [*closure phase*]{} [see also @readhead1988; @ionic3_2006]. [figures reprinted from @monnier2003] \[cphasefig\] ](JDM_phaseshift.eps "fig:"){height="3in"} ![(left) Atmospheric turbulence introduces extra path length fluctuations that induce fringe phase shifts. At optical wavelengths, these phase shifts vary by many radians over short time scales ($<<$1 sec) effectively scrambling the Fourier phase information. (right) Phase errors introduced at any telescope causes equal but opposite phase shifts in ajoining baselines, canceling out in the [*closure phase*]{} [see also @readhead1988; @ionic3_2006]. [figures reprinted from @monnier2003] \[cphasefig\] ](JDM_cphase2.eps "fig:"){height="3in"} ![Graphical representation of the wavelength coverage and maximum angular resolution available using the radio and optical interferometers of the world. See Table \[table:capabilities\] for more information. \[fig\_capabilities\] ](fig_capabilities.eps){width="6in"} [^1]: It should be emphasized, however, that this distinction is somewhat artificial; the first meter-wave radio interferometers $\approx 65$ years ago were simple Michelson adding interferometers employing direct detection without coherent high-frequency signal amplification. At the other extreme, superheterodyne systems are currently routinely used at wavelengths as short as $10\mu$m, such as the UC Berkeley ISI facility. [^2]: There are several different coordinate systems in use to describe the geometry of ground-based interferometers used in observing the celestial sphere; [see e.g. @tms2001 Chapter 4 and Appendix 4.1]. [^3]: Very recently, several radio observatories have begun to equip their antennas (and at least one entire synthesis telescope) with arrays of such feeds. [^4]: This is not all advantageous; if the data is intended to be used in an imaging synthesis, the absence of the total power component means that the value of the map made from the data will integrate to zero. In other words, without further processing the image will be sitting on a slightly-negative “floor”. If more interferometer spacings around zero are also missing, the floor becomes a “bowl”. All this is colloquially called “the short-spacing problem”, and it adversely affects the photometric accuracy of the image. A significant part of the computer processing “bag of tricks” used to “restore” such images is intended to address this problem, although the only proper way to do that is to obtain the missing data and incorporate it into the synthesis. [^5]: At millimeter and sub-millimeter wavelengths, correlators still do not attain the maximum useful bandwidths for continuum observations [^6]: Recall that an integration of specific intensity over solid angle results in a flux density, often expressed in Jansky. [^7]: Although amplifiers are currently used in the long-distance transmission of near-IR (digital) communication signals in optical fibers, the signal levels are relatively large and low noise is not an important requirement. [^8]: Real interferometers will have a realistic limit about 1-2 orders of magnitude below the theoretical limit due to throughput losses and non-ideal effects such as loss of system visibility. [^9]: At the shortest sub-mm wavelengths, phase-referencing is quite difficult due to strong water vapor turbulence, but can be partially corrected using “water-vapor monitoring” techniques [e.g., @wiedner2001]. [^10]: Fortunately, targets of optical interferometers are generally spatially compact and so sparser (u,v) coverage can often be acceptable. [^11]: In general, also the polarization states and wavefront coherence can also be modified. [^12]: Phase referencing is possible using “Dual Star Feeds” which allows truly simultaneous observing of a pair of objects in the same narrow isoplanatic patch on the sky. This capability has been demonstrated on PTI, Keck, and VLTI. [^13]: Historically, model-fitting was the way data was handled in order to discern source structure in the early days of radio interferometry. A classic example is the model of Cygnus A, which Jennison and Das Gupta fitted to long-baseline intensity-interferometry data at 2.4-m wavelength in 1953 [@JennisonGupta1953]. @Sullivan09 [page 353 et. seq] gives more details of this fascinating story. [^14]: In general, one can consider a full “Spatial Transfer Function” which can have weights between 0 and 1. Here, consider just a simple binary mask in the (u,v) plane for simplicity. [^15]: It’s interesting to note that at some large-enough diameter, ground-based telescopes are likely to become similar in cost to those in space, especially if one considers life-cycle costs.

--- abstract: 'Searches for gravitational waves produced by coalescing black hole binaries with total masses $\gtrsim25\,$M$_\odot$ use matched filtering with templates of short duration. Non-Gaussian noise bursts in gravitational wave detector data can mimic short signals and limit the sensitivity of these searches. Previous searches have relied on empirically designed statistics incorporating signal-to-noise ratio and signal-based vetoes to separate gravitational wave candidates from noise candidates. We report on sensitivity improvements achieved using a multivariate candidate ranking statistic derived from a supervised machine learning algorithm. We apply the random forest of bagged decision trees technique to two separate searches in the high mass $\left( \gtrsim25\,\mathrm{M}_\odot \right)$ parameter space. For a search which is sensitive to gravitational waves from the inspiral, merger, and ringdown (IMR) of binary black holes with total mass between $25\,$M$_\odot$ and $100\,$M$_\odot$, we find sensitive volume improvements as high as $70_{\pm 13}-109_{\pm 11}$% when compared to the previously used ranking statistic. For a ringdown-only search which is sensitive to gravitational waves from the resultant perturbed intermediate mass black hole with mass roughly between $10\,$M$_\odot$ and $600\,$M$_\odot$, we find sensitive volume improvements as high as $61_{\pm 4}-241_{\pm 12}$% when compared to the previously used ranking statistic. We also report how sensitivity improvements can differ depending on mass regime, mass ratio, and available data quality information. Finally, we describe the techniques used to tune and train the random forest classifier that can be generalized to its use in other searches for gravitational waves.' author: - 'Paul T. Baker$^{1}$, Sarah Caudill$^{2}$, Kari A. Hodge$^{3}$, Dipongkar Talukder$^{4}$, Collin Capano$^{5}$, and Neil J. Cornish$^{1}$' date: 'Date: [ JanuaryFebruaryMarchAprilMayJune JulyAugustSeptemberOctoberNovemberDecember ]{}' title: Multivariate Classification with Random Forests for Gravitational Wave Searches of Black Hole Binary Coalescence --- Introduction {#sec:overview} ============ We are rapidly approaching the era of advanced gravitational-wave detectors. Advanced LIGO [@2010aligo] and Advanced Virgo [@2014avirgo] are expected to begin operation in 2015. Within the next decade, these will be joined by the KAGRA [@2014kagra] and LIGO-India [@2014indigo] detectors. The coalescence of compact binaries containing neutron stars and/or stellar mass black holes are expected to be a strong and promising source for the first detection of gravitational waves [@2010rates]. Higher mass sources with total masses $\gtrsim25\,$M$_\odot$ including binary black holes (BBHs) and intermediate mass black holes (IMBHs) are less certain but still potentially strong sources [@2010rates; @2013s6highmass; @2014rdsearch]. Discovery and new science will be possible with detection of gravitational waves from these objects [@1993cbcscience; @2009gwscience]. Measurement of gravitational waves requires exquisitely sensitive detectors as well as advanced data analysis techniques [@2009ligo]. By digging into detector noise for weak signals rather than waiting for a rare, loud event, we increase detection rates. Unfortunately, detector noise can be non-stationary and non-Gaussian, leading to loud, short duration noise transients. Such behavior is particularly troublesome for higher mass searches where the expected in-band signal is of similar duration as noise transients. Traditional searches for compact binary coalescence have utilized multi-detector coincidence, carefully designed ranking statistics, and other data quality methods [@20091styrS5lowmass; @2009s5lowmass186; @2012S6lowmass; @2011s5highmass; @2013s6highmass; @2014rdsearch]. However, in many searches performed to-date over initial LIGO and Virgo data, the sensitivity was limited by an accidental coincidence involving a non-Gaussian transient noise burst [@2011s5highmass; @2013s6highmass; @2014rdsearch]. Only recently have gravitational-wave searches begun to utilize methods that work with the full multidimensional parameter space of classification statistics for candidate events. Previous studies have shown multivariate methods give detection probability improvement over techniques based on single parameter thresholds [@2008cannonbayes], [@2014kari]. Machine learning has a wealth of tools available for the purpose of multivariate statistical classification [@2012amlmla; @2014narskybook]. These include but are not limited to artificial neural networks [@1989nn; @2009nn], support vector machines [@1995svm; @2000svm], and random forests of decision trees [@2001breimanrf]. Such methods have already proven useful in a number of other fields with large data sets and background contamination including optical and radio astronomy  [@2014mlapulsars; @2012ptfbloom; @2013mlabrink] and high energy physics [@2009mlaheph; @2005narskypresent]. Within the field of gravitational wave physics, a search for gravitational-wave bursts associated with gamma ray bursts found a factor of $\sim$3 increase in sensitive volume when using a multivariate analysis with boosted decision trees [@2013mvscburst]. Applications of artificial neural networks to a search for compact binary coalescence signals associated with gamma ray bursts found smaller improvements [@2014ANN]. Machine learning algorithms have successfully been applied to the problem of detector noise artifact classification [@2013mvscglitch]. Additionally, a search for bursts of gravitational waves from cosmic string cusps [@2013cosmicstring] used the multivariate technique described in [@2008cannonbayes]. In this paper, we focus on the development and sensitivity improvements of a multivariate analysis applied to matched filter searches for gravitational waves produced by coalescing black hole binaries with total masses $\gtrsim25\,$M$_\odot$. In particular, we focus on the application to two separate searches in this parameter space. The first, designated the IMR search, looks for gravitational waves from the inspiral, merger, and ringdown of BBHs with total mass between $25\,$M$_\odot$ and $100\,$M$_\odot$. The second, designated the ringdown-only search, looks for gravitational waves from the resultant perturbed IMBH with mass roughly between $10\,$M$_\odot$ and $600\,$M$_\odot$. These investigations are performed over data collected by LIGO and Virgo between 2009 and 2010 so that comparisons can be made with previous IMR and ringdown-only search results [@2013s6highmass; @2014rdsearch]. Using a random forest of bagged decision trees (RFBDT) supervised machine learning algorithm (MLA), we explore sensitivity improvements over each search’s previous classification statistic. Additionally, we describe techniques used to tune and train the RFBDT classifier that can be generalized to its use in other searches for gravitational waves. In Sec. \[sec:detection\], we frame the general detection problem in gravitational-wave data analysis and motivate the need for multivariate classification. In Sec. \[sec:data\], we describe our data set. In Sec. \[sec:detstat\], we explain the method used to classify gravitational-wave candidates in matched-filter searches. In Sec. \[sec:mla\], we review RFBDTs as used in these investigations. In Sec. \[sec:tuning\], we discuss the training set, the multidimensional space used to characterize candidates, and the tunable parameters of the classifier. In Sec. \[sec:results\], we describe the improvement in sensitive volume obtained by the IMR and ringdown-only searches over LIGO and Virgo data from 2009 to 2010 when using RFBDTs. Finally, in Sec. \[sec:summary\] we summarize our results. The Detection Problem {#sec:detection} ===================== Searches for gravitational waves are generally divided based on astrophysical source. The gravitational waveform from compact binary coalescence has a well-defined model [@2006cbcreview; @2012cbcreview]. Thus searches for these types of signals use the method of matched-filtering with a template bank of model waveforms. This is the optimal method for finding modeled signals with known parameters buried in Gaussian noise [@1960matchedfilter; @1962wainstein]. However, if the parameters are not known, matched filtering is not optimal [@2009prix], and additional techniques must be employed to address the extraction of weak and/or rare signals from non-Gaussian, non-stationary detector noise, the elimination or identification of false alarms, and the ranking of gravitational-wave candidates by significance. This paper presents the construction of an [*ad hoc*]{} statistic, automated through machine learning, that can tackle these issues. Searches for compact binary coalescence {#sec:cbc} --------------------------------------- The coalescence of compact binaries generates a gravitational-wave signal composed of inspiral, merger and ringdown phases [@2006cbcreview; @2012cbcreview]. The low frequency inspiral phase marks the period during which the compact objects orbit each other, radiating energy and angular momentum as gravitational waves [@2002inspiralreview]. The signal for low mass systems in the LIGO and Virgo frequency sensitivity bands (i.e., above the steeply rising seismic noise at 40Hz for initial detectors or 10Hz for advanced detectors [@iligoaligonoise]) is dominated by the inspiral phase. Several searches have looked for the inspiral from low mass systems with component masses $>1\,$M$_\odot$ and total mass $< 25\,$M$_\odot$ [@2009s5lowmass186; @20091styrS5lowmass; @2012S6lowmass]. The higher frequency merger phase marks the coalescence of the compact objects and the peak gravitational-wave emission [@2010Rnrreview1; @2009nrreview; @2011nrreview]. Since the merger frequency is inversely proportional to the mass of the binary, the signal for high mass systems in the LIGO and Virgo sensitivity bands could include inspiral, merger and ringdown phases. Searches for high mass signals including all three phases have been performed for systems with total mass between 25$\,$M$_\odot$ and 100$\,$M$_\odot$ [@2011s5highmass; @2013s6highmass]. We designate this as the IMR search. Systems accessible to LIGO and Virgo with even higher total masses will only have a ringdown phase in-band, during which the compact objects have already formed a single perturbed black hole [@1999rdreview; @2007rdwaveform]. Searches for ringdown signals have looked for perturbed black holes with total masses roughly in the range 10$\,$M$_\odot$ to 600$\,$M$_\odot$ and dimensionless spins in the range 0 to 0.99 [@2009s4ringdown; @2014rdsearch]. The dimensionless spin is defined as $\hat{a}=cS/GM^2$ for black hole mass $M$ and spin angular momentum $S$. We designate this as the ringdown-only search. Each of these searches use a matched-filter algorithm with template banks of model waveforms to search data from multiple gravitational-wave detectors. The output is a SNR time series for each detector. We record local maxima, called triggers, in the SNR time series that fall above a predetermined threshold. Low mass searches use template banks of inspiral model waveforms generated at 3.5 post-Newtonian order in the frequency domain [@1995inspiralwaveform; @2004inspiralwaveform]. These waveforms typically remain in the initial LIGO/Virgo frequency sensitivity band for tens of seconds providing a natural defense against triggers arising from short bursts of non-Gaussian noise. The templates for IMR searches include the full inspiral-merger-ringdown waveform, computed analytically and tuned against numerical relativity results. For these investigations, the non-spinning EOBNRv1 family of IMR waveforms was used  [@2007eobnrv1]. The templates, like those for the low mass search, are described by the chirp mass $\mathcal{M}=\eta^{3/5}M$ and symmetric mass ratio $\eta=m_1m_2/M^2$ of the component objects (where $M=m_1+m_2$) [@findchirp]. The duration of high mass waveforms in-band for the initial detectors is much shorter than the duration for low mass waveforms, making the IMR search susceptible to triggers associated with short bursts of non-Gaussian noise. The templates for the search for perturbed black holes, with even higher total mass, is based on black hole perturbation theory and numerical relativity. A perturbed Kerr black hole will emit gravitational waves in a superposition of quasinormal modes of oscillation characterized by a frequency $f_{\ell mn}$ and damping time $\tau_{\ell mn}$ [@1973teukolsky; @1999rdreview]. Numerical simulations have demonstrated that the $\left(\ell, m, n\right)=\left(2,2,0\right)$ dominates the gravitational-wave emission [@2007dominantlmn; @2007rdwaveform]. From here on, we will designate $f_{220}$ as $f_{0}$ and write the damping time $\tau_{220}$ in terms of the quality factor $Q=Q_{220}=\pi f_{220}\tau_{220}$. Ringdown model waveforms decay on the timescale $0.0001\lesssim \tau/\mathrm{s} \lesssim 0.1$, again making this search susceptible to contamination from short noise bursts. The matched-filter algorithms are described in [@findchirp; @2009s4ringdown]. Further details on the templates and template bank construction in the IMR and ringdown-only searches can be found in [@2013s6highmass; @2014rdsearch]. Matched filtering alone cannot completely distinguish triggers caused by gravitational waves from those caused by noise. Thus tools such as data quality vetoes, multi-detector coincidence, and SNR consistency checks are needed [@2010vetoes; @2013ihopepaper]. Additionally, a $\chi^2$ time-frequency signal consistency test augments searches with broadband signal including the IMR search but is less useful for short, quasi-monochromatic ringdown signals [@2005allen]. Finally, each search uses a detection statistic to summarize the separation of signal from background. Details on the construction of a detection statistic are provided in Section \[sec:detstat\]. In general, coincidence tests are applied to single detector triggers to check for multi-detector consistency. The low and high mass searches use an ellipsoidal coincidence test ([*ethinca*]{} [@ethinca]) that requires consistent values of template masses and time of arrival. The ringdown-only search coincidence test similarly calculates the distance $ds^2$ between two triggers by checking simultaneously for time and template coincidence (d$f_0$ and d$Q$) [@2014rdsearch]. When three detectors are operating, if each pair of triggers passes the coincidence test, we store a triple coincidence. We also store double coincidences for particular network configurations as outlined in Section \[sec:data\]. Signal and background {#sec:sigts} --------------------- Evaluating the performance of a detection statistic and training the machine learning classifier require the calculation of detection efficiency at an allowed level of background contamination. In the absence of actual gravitational-wave events, we determine detection efficiency through the use of simulated signals (“injections") added to the detectors’ data streams. To estimate the search background, we generate a set of accidental coincidences using the method of time-shifted data. The simulated signal set is added to the data and a separate search is run. Triggers are recorded corresponding to times when injections were made. The simulated signals are representative of the expected gravitational waveforms detectable by a search. For the IMR and ringdown-only searches, the simulated signals include waveforms from the EOBNRv2 family [@2011eobnr] for systems whose component objects are not spinning and from the IMRPhenomB family [@2011imrphenomb] for systems whose component objects have aligned, anti-aligned, or no spins. Additionally, for the ringdown-only search, we inject ringdown-only waveforms. For a discussion of injection waveform parameters, see Section \[sec:sigtraining\]. Considerations for the injection sets used in training the classifier are discussed in Section \[sec:training\] and in computing search sensitivity are discussed in Section \[sec:results\]. The background rate of accidental trigger coincidence between detectors is evaluated using the method of time-shifted data. We shift the data by intervals of time longer than the light travel time between detectors and then perform a separate search. Any multi-detector coincidence found in the time-shifted search is very likely due to non-Gaussian glitches. We perform searches over 100 sets of time-shifted data and recorded the accidental coincidences found by the algorithm. Details of the method are provided in Section IIIB of [@2013s6highmass] and IIIC of [@2014rdsearch]. For a discussion of the set of accidental coincidences used in training the classifier, see Section \[sec:training\]. Data Set {#sec:data} ======== We performed investigations using data collected by the LIGO and Virgo detectors between July 2009 and October 2010 [@2012s6]. In [@2014rdsearch], we designate this time as Period 2 to distinguish it from the analysis of Period 1 data collected between November 2005 and September 2007. All results reported here consider only Period 2 data. For continuity, we will continue to designate our data analysis time as Period 2. Period 2 covers LIGO’s sixth science run [@2009eLIGO]. During this time, the $4\,$km detectors in Hanford, Washington (H1) and in Livingston, Louisiana (L1) were operating. The $3\,$km Virgo detector (V1) in Cascina, Italy conducted its second and third science run during this time [@2011vsr2]. The investigations were performed using data from the coincident search networks of the H1L1V1, H1L1, H1V1, and L1V1 detectors. Coincidences were stored for all triple and double detector combinations. Data was analyzed separately using the IMR and the ringdown-only search pipelines from the analyses reported in [@2013s6highmass] and [@2014rdsearch]. In order to combat noise transients, three levels of data quality vetoes are applied to remove noise from LIGO-Virgo data when searches are performed. Details of vetoes are provided in [@2010vetoes] and specific descriptions of the use of the vetoes for these Period 2 analyses can be found in [@2013s6highmass; @2014kari; @2014rdsearch].[^1] We analyze the performance of RFBDT classification after the first and second veto levels have been applied and compare this to the performance after the first, second, and third veto levels have been applied. After removal of the first and second veto levels, we were left with 0.48 years of analysis time and after additional removal of the third veto level, we were left with 0.42 years of analysis time. We designate the search over Period 2 after the removal of the first and second veto levels as Veto Level 2 and after the removal of the first, second, and third veto levels as Veto Level 3. In order to capture the variability of detector noise and sensitivity, we divided Period 2 into separate analysis periods of $\sim$1 to 2 months. We then estimated the volume to which each search was sensitive by injecting simulated waveforms into the data and testing our ability to recover them. Details of the method are provided in Section V of [@2013s6highmass; @2014rdsearch]. We repeat this procedure in Section \[sec:results\] in order to quantify the improvement in sensitive volume obtained by each search over LIGO and Virgo data from 2009 to 2010 when using RFBDTs. Detection statistics {#sec:detstat} ==================== We must rank all coincidences based on their likelihood of being a signal. Gravitational-wave data analysis has no dearth of statistics to classify gravitational-wave candidates as signal or background, and often, the ranking statistic will be empirically designed as a composite of other statistics. If the noise in the detector data was Gaussian, a matched-filter SNR would be a sufficient ranking statistic. However, since detector noise is non-Gaussian and non-stationary, we often re-weight the SNR by additional statistics that improve our ability to distinguish signal from background. The exact form will depend on the nature of the signal for which we are searching. A good statistic for differentiating long inspirals may not work well for short ringdowns. Searches for low mass binaries have ranked candidates using matched-filter SNR weighted by the $\chi^2$ signal consistency test value (e.g., [*effective*]{} SNR [@20091styrS5lowmass] and [*new*]{} SNR [@2012S6lowmass]). IMR searches have used similar statistics [@inspiralonlyS4; @S5hm; @2013s6highmass]. Previous ringdown-only searches and studies have used SNR-based statistics to address the non-Gaussianity of the data without the use of additional signal-based waveform consistency tests (e.g., the chopped-L statistic for double [@2009s4ringdown] and triple [@2013talukder; @2014imr] coincident triggers). However, ranking statistics can be constructed using multivariate techniques to incorporate the full discriminatory power of the multidimensional parameter space of gravitational-wave candidates. Several searches have utilized this including [@2013mvscburst; @2013cosmicstring]. In Section \[sec:mla\], we detail the implementation of a multivariate statistical classifier using RFBDTs for the most recent IMR and ringdown-only searches. The final statistic used to rank candidates in order of significance is known as a detection statistic. We combine the ranking statistics for each trigger into a coincident statistic (i.e., a statistic that incorporates information from all detectors’ triggers found in coincidence). This coincident statistic is then used to calculate a combined false alarm rate, the final detection statistic of the search. We determine the coincident statistic $R$ for three different types of coincidences: gravitational-wave search candidates, simulated waveform injection coincidences, and time-shifted coincidences. We then determine a false alarm rate (FAR) for each type of coincidence by counting the number of time-shifted coincidences found in the analysis time $T$ for each of the coincident search networks given in Section \[sec:data\]. For each of the different types of coincidences in each of the search networks, we determine the FAR with the expression: $$\label{eq:far} \mathrm{FAR}=\frac{\sum\limits_{k=1}^{100} N_k(R \ge R^*)}{T}$$ where $N_k$ is the measured number of coincidences with $R \ge R^*$ in the $k^{th}$ shifted analysis for a total of 100 time-shifted analyses. We then rank coincidences by their FARs across all search networks into a combined ranking, known as combined FAR [@2009keppelthesis]. This is the final detection statistic used in these investigations. Machine Learning Algorithm {#sec:mla} ========================== In order to compute the FAR for any candidate, we use the RFBDT algorithm to assign a probability that any candidate is a gravitational-wave signal. This random forest technology is a well-developed improvement over the classical decision tree. Each event is characterized by a feature vector, containing parameters thought to be useful for distinguishing signal from background. A decision tree consists of a series of binary splits on these feature vector parameters. A single decision tree is a weak classifier, as it is susceptible to false minima and over-training [@2005statpattern] and generally performs worse than neural networks [@2005narskypresent]. Random forest technology combats these issues and is considered more stable in the presence of outliers and in very high dimensional parameter spaces than other machine learning algorithms [@2005statpattern; @2014narskybook]. We use the () software package[^2] developed for high energy physics data analysis. This analysis uses the Random Forest of Bootstrap AGGregatED (bagged) Decision Trees algorithm. The method of bootstrap aggregation or “bagging” as described below, tends to perform better in high noise situations than other random forest methods such as boosting [@1999BauerKohaviVoting]. Random forests {#sec:rf} -------------- A random forest of bagged decision trees uses a collection of many decision trees that are built from a set of training data. The training data is composed of the feature vectors of events that we know *a priori* to belong to either the set of signals or background (i.e., coincidences associated with simulated injections and coincidences associated with time-shifted data). Then the decision trees are used to assign a probability that an event that we wish to classify belongs to the class of signal or background. A cartoon diagram is presented in Fig. \[fig:forest\]. To construct a decision tree, we make a series of one-dimensional splits on a random subset of parameters from the feature vector. Each split determines a branching point, or node. Individual splits are chosen to best separate signal and background given the available parameters. There are several methods to determine the optimal parameter threshold at each node. We measure the goodness of a threshold split based on receiver operating characteristic curves (as described in Section \[sec:roc\]). The software package [@2005statpattern] provides several options for optimization criterion. We found that the Gini index [@1912gini] and the negative cross-entropy provided comparable, suitable performance for both searches. Thus we arbitrarily chose the Gini index for the IMR search and the negative cross-entropy for the ringdown-only search. Additional discussion is given in Sect. \[sec:criterion\]. The Gini index is defined by $$G(p)=-2p\bar{p},$$ where $p$ is the fraction of events in a node that are signals and $\bar{p}=1-p$ is the fraction of events in the node that are background. Splits are made only if they will minimize the Gini index. The negative cross-entropy function is defined by $$H(p) = -p \log_2 p - \bar{p} \log_2 \bar{p}.$$ As $H$ is symmetric against the exchange of $p$ and $\bar{p}$, if a node contains as many signal events as background, then $H\left(\frac{1}{2}\right)=1$. A perfectly sorted node has $H(1)=H(0)=0$. By minimizing the negative cross-entropy of our nodes, we find the optimal sort. When no split on a node can reduce the entropy or it contains fewer events than a preset limit, it is no longer divided. At this point, the node becomes a “leaf”. The number of events in the preset limit is known as the “minimum leaf size”. When all nodes have become leaves, we have created a decision tree. This process is repeated to create each decision tree in the forest. Each individual tree is trained on a bootstrap replica, or a resampled set of the original training set, so that each tree will have a different set of training events. Furthermore, a different randomly-chosen subset of feature vector parameters is chosen at each node to attempt the next split. Thus each decision tree in the forest is unique. Results from each tree can be averaged to reduce the variance in the statistical classification. This is the method of bootstrap aggregation or “bagging”. The forest can then be used to classify an event of unknown class. The event is placed in the initial node of each tree and is passed along the various trees’ branches until it arrives at a leaf on each tree. To compute the ranking statistic for an event from a forest of decision trees, we find its final leaf in all trees. The probability that an event is a signal is given by the fraction of signal events in all leaves, $$\label{eq:forest_prob} p_\mathrm{forest}=\frac{\sum s_i}{\sum s_i + b_i} = \frac{1}{N} \sum s_i$$ where $s_i$ and $b_i$ are the number of signal and background events in the $i$th leaf and $N$ is the total number of signal and background events in all final leaves. The final ranking statistic, $M_\mathrm{forest}$, for a forest of decision trees is given by the ratio of the probability that the event is a signal to the probability that the event is background, $$\label{eq:forest_rank} M_\mathrm{forest}=\frac{p_\mathrm{forest}}{1-p_\mathrm{forest}}.$$ This ranking statistic is a probability ratio, indicating how much the signal model is favored over the background model. In order to test the performance of the random forest, we determine how well it sorts events that we know *a priori* are signal or background. Rather than generate a new training set of simulated injections and time-shifted data, we may sort the training set used in construction of the forest. To prevent over-estimation of classifier performance, decision trees cannot be used to classify the same events on which they were trained. Thus we use a round-robin approach to iteratively classify events using a random forest trained on a set excluding those events. We construct ten random forests each using 90% of the training events such that the remaining 10% of events may be safely classified. In this way we can efficiently verify the performance of the random forest using only the original training events. Each forest is trained on and used to evaluate a particular type of double coincidence from the detector network (i.e., H1L1, H1V1, L1V1), as each pair of detectors produces unique statistics. Triple coincidences are split into their respective doubles, as there is not sufficient triple-coincident background to train a separate forest. For a triple coincidence with triggers from detectors $a$, $b$, and $c$, the ranking statistic $M_\mathrm{forest, triple}$ will be the product of $M_\mathrm{forest, double}$ for each pair of triggers, $$\label{eq:triple_forest} M_\mathrm{forest, triple}=\prod_i M_\mathrm{forest, i}$$ where $i \in$ {$ab$, $ac$, $bc$} denotes the possible pairs of double coincident triggers in a triple coincidence. ![A cartoon example of a random forest. There are five decision trees in this random forest. Each was trained on a training set of objects belonging to either the black set or the cyan set. Note that the training set of each decision tree is different from the others. At each numbered node, or split in the tree, a binary decision based on a threshold of a feature vector parameter value is imposed. The decisions imposed at each node will differ for the different trees. When no split on a node can reduce the entropy or it contains fewer events than a preset limit, it is no longer divided and becomes a “leaf”. Consider an object that we wish to classify as black or cyan. Suppose the object ends up in each circled leaf. Then the probability that the object is black is the fraction of black objects in all leaves, $p_\mathrm{forest}=73\%$. []{data-label="fig:forest"}](forest.pdf){width=".45\textwidth"} Tuning {#sec:tuning} ====== There are several issues to consider when optimizing the performance of RFBDT classifiers. Performance of the algorithm is dependent on the quality of the training set (i.e., how well the training data represent the actual population we wish to detect). Additionally, we must select appropriate statistics to include in the feature vector of each coincidence. Finally, RFBDT classifiers have several parameters that must be tuned to optimize the sorting performance. These include number of trees, number of sampled parameters from the feature vector at each node, and minimum leaf size. Improperly choosing these meta parameters will lead to a poorly trained classifier. In general there are two types of mistrained classifiers. An over-trained classifier separates the training data well, but the sort is specific to those data. An over-trained classifier may provide very high or very low $M_\mathrm{forest}$, but these values contain a large systematic bias. Any events that were not well represented by this training set will be misclassified. An under-trained classifier has not gleaned enough information from the training data to sort properly. In this case the classifier is unsure of which set any event belongs, assigning intermediate values of $M_\mathrm{forest}$ to all. Search IMR Ringdown-only ---------------------------- ------- --------------- Number of trees 100 2000 Minimum leaf size 5 65 Total number of parameters 15 24 Number of randomly sampled    parameters per node 6 14 Criterion for optimization Gini- cross- index entropy : Summary of random forest parameters.[]{data-label="tab:summaryrf"} Figure of merit {#sec:roc} --------------- We evaluate the performance of different tunings using receiver operating characteristic (ROC) curves, separately for each search. In general, these curves show detection rate as a function of contamination rate as a discriminant function is varied. For our purposes, thresholds on the combined FAR serve as the varying discriminant function. Thus, both the detection and contamination rates are functions of the combined FAR. Since we seek to improve the sensitivity of our searches, we reject the traditional definition of detection rate and instead define a quantity that depends on sensitive volume. We use a fractional volume computed at each combined FAR threshold, $$\label{eq:fracvol} \frac{V}{V_\mathrm{tot}}=\frac{\sum_i \epsilon_i r_i^3}{\sum_i r_i^3}$$ where $i$ sums over all injections recovered as coincidences by the analysis pipeline, and $r_i$ is the physical distance of the injection. For each combined FAR threshold $\lambda^*$, $\epsilon_i$ counts whether injection $i$ was found with a combined FAR $\lambda_i$ less than or equal to $\lambda^*$, $$\epsilon_i = \left\{ \begin{array}{lr} 1 & : \lambda_i \le \lambda^*\\ 0 & : \lambda_i > \lambda^* \end{array}. \right.$$ In the following sections, we explore tunings and performance for the RFBDT algorithm for different total masses and mass ratios as well as at Veto Level 2 and 3 in order to understand how the application of vetoes affects the RFBDTs. Training set {#sec:training} ------------ The training of the classifier utilizes the signal and background data sets as described in Sect. \[sec:sigts\]. In the following discussions, we consider several issues that arise in the construction of training sets for gravitational-wave classification when using RFBDTs. ### Signal training {#sec:sigtraining} In order to train the classifier on the appearance of signal, we injected sets of simulated waveforms into the data and recorded those found in coincidence by the searches. Both searches injected sets of waveforms from the EOBNRv2 and IMRPhenomB families. The total mass $M$, mass ratio $q$, and component spin $\hat{a}_{1,2}$ distributions of these waveforms are given in Table \[tab:summaryimrts\]. We define the mass ratio as $q=m_>/m_<$ where $m_>=\mathrm{max}(m_1, m_2)$ and $m_<=\mathrm{min}(m_1, m_2)$. The component spins are $\hat{a}_{1,2}=cS_{1,2}/Gm_{1,2}^2$ for the spin angular momenta $S_{1,2}$ and masses $m_{1,2}$ of the two binary components. From this, we define the mass-weighted spin parameter $$\chi_s=\frac{m_1\hat{a}_1+m_2\hat{a}_2}{m_1+m_2}.$$ Additionally, for the ringdown-only search, we injected two sets of ringdown-only waveforms as described in Table \[tab:summaryrdts\]. The two sets gave coverage of the ringdown template bank in $\left(f_0, Q\right)$-space and of the potential $\left(M_f, \hat{a}\right)$-space accessible to the ringdown-only search where $M_f$ is the final black hole mass and $\hat{a}$ is the dimensionless spin parameter. All injections were given isotropically-distributed sky location and source orientation parameters. As described below, only injections that are cleanly found by the search algorithm are used in training the classifiers. For performance investigations in Section \[sec:results\], we determine search sensitivities using all injections found by the searches’ matched filtering pipelines (i.e., not just those that are cleanly found). The IMR search considers full coalescence waveforms from the EOBNRv2 and IMRPhenomB families. The ringdown-only search considers only EOBNRv2 waveforms. These injection sets and their parameters are given in Section \[sec:results\]. ------------------------------ ------------------------------------ ------------------------------------ ------------------------------------ ------------------------------------ Search IMR Ringdown-only IMR Ringdown-only Mass distribution: uniform in $\left(m_1,m_2\right)$ uniform in $\left(M, q\right)$ uniform in $\left(M, q\right)$ uniform in $\left(M, q\right)$ Total mass range: $M/\mathrm{M}_\odot \in [25, 100]$ $M/\mathrm{M}_\odot \in [50, 450]$ $M/\mathrm{M}_\odot \in [25, 100]$ $M/\mathrm{M}_\odot \in [50, 450]$ Mass ratio range: $q \in [1, 10]$ $q \in [1, 10]$ $q \in [1, 10]$ $q \in [1, 10]$ Spin parameter distribution: non-spinning non-spinning uniform in $\chi_s$ uniform in $\chi_s$ Spin parameter range: $\chi_s=0$ $\chi_s=0$ $\chi_s \in [-0.85, 0.85]$ $\chi_s \in [0, 0.85]$ ------------------------------ ------------------------------------ ------------------------------------ ------------------------------------ ------------------------------------ Injection set 1 Injection set 2 --------------- ---------------------------------- ---------------------------------------- Distribution: uniform in $\left(f_0, Q\right)$ uniform in $\left(M_f, \hat{a}\right)$ Parameter 1: $f_0/\mathrm{Hz} \in [50, 2000]$ $M_f/\mathrm{M}_\odot \in [50, 900]$ Parameter 2: $Q \in [2, 20]$ $\hat{a} \in [0, 0.99]$ : Summary of ringdown-only signal training set. Injection set 1 corresponds to coverage of the ringdown template bank in $\left(f_0, Q\right)$-space. Injection set 2 corresponds roughly to the potential $\left(M_f, \hat{a}\right)$-space accessible to the ringdown-only search.[]{data-label="tab:summaryrdts"} To identify triggers associated with simulated waveform injections made into the data, we use a small time window of width $\pm$1.0 second around the injection time. We record the parameters of the trigger with the highest SNR within this time window and associate it with the injection. Unfortunately, when injections are made into real data containing non-Gaussian noise, the injection may occur near a non-Gaussian feature or glitch in the data. In the case where the SNR of the injection trigger is smaller than that of the glitch trigger, the recorded trigger will correspond to the glitch trigger and will not accurately represent the simulated waveform. When using injections to train the classifier on the appearance of gravitational-wave signals, we must be careful to exclude any injections in a window contaminated by a glitch.   Figure \[fig:ifarbump\] demonstrates the issue that can arise when using a contaminated signal training set. These plots show the cumulative distributions of coincident events found as a function of inverse combined false alarm rate for a small chunk of the H1L1V1 network search at Veto Level 3. The ranking statistic used in both Fig. \[fig:ifara\] and Fig. \[fig:ifarb\] is $M_\mathrm{forest}$. However, results for Fig. \[fig:ifara\] were obtained with a RFBDT classifier trained on injections identified using an injection-finding window of width $\pm$1.0 second (i.e., a contaminated injection set). Results for Fig. \[fig:ifarb\] were obtained with a RFBDT classifier trained on injections identified using a narrower injection-finding window of width $\pm$0.01 seconds and after removing any injections made within $\pm$0.5 seconds of a glitch (i.e., a clean injection set). In Fig. \[fig:ifara\], we see that there is an excursion of H1L1V1 gravitational-wave candidate coincidences from the 2$\sigma$ region of the expected background at low values of inverse combined FAR. This excursion for coincidences with low significance is caused by a population of injections that were misidentified because of a nearby glitch in the data. The RFBDT classifier was taught that these glitches designated as injections should be classified as signal. Thus, when similar glitches were found as coincidences in the H1L1V1 network search at Veto Level 3, they were given a boost in their $M_\mathrm{forest}$ rank. However, in Fig. \[fig:ifarb\], we see that by excluding these misidentified injections from the training set, the low significance H1L1V1 coincidences now fall within the 2$\sigma$ region of the expected background. We developed a software tool[^3] for use with the LALSuite gravitational-wave data analysis routines[^4] to construct clean injection sets. Using this tool, we investigated two time window parameters that can be tuned: the width of the injection-finding time window and the width of the injection-removal time window. The injection-finding time window is motivated by the fact that a trigger due to an injection should be found in the data within a few milliseconds of the injection time given the light travel time between detectors. Thus, in Gaussian detector noise, a few millisecond-wide injection-finding window should be sufficient. However, due to non-Gaussian, non-stationary detector noise, the coincidence of triggers associated with an injection could be overshadowed if a loud glitch trigger is nearby. Thus, we allow a much larger window. When conducting searches for gravitational waves, this window is typically set to $\pm$1.0 second from the injection time. However, such a large window results in a contaminated signal training set as we see in Fig. \[fig:ifarbump\]. The injection-removal time window is motivated by the fact that a significant trigger found by the search before injections are performed is a potential contaminating trigger for any injection made similarly in time. A simple time window is used to cross-check whether an injection trigger found by the search could be attributed to a trigger found in detector data before injections were performed. We investigated the performance of the RFBDT classifier for the ringdown-only search separately for detection of $q=1$ and $q=4$ EOBNRv2 simulated waveforms at Veto Level 3. We performed several tuning runs, adjusting the size of the injection-finding and injection-removal windows. We found that an injection-finding window of $\pm$0.01 seconds around an injection and an injection-removal window of $\pm$0.5 seconds around an injection were the most effective at combating the excess of foreground triggers at low significance. These settings were used in designing both the IMR and ringdown-only signal training sets. ### Background training set {#sec:bkgdtraining} The background training set composed of accidental coincidences is not noticeably contaminated by signal. Since this background is constructed by time-shifting the data, it is possible that a real gravitational-wave signal could result in time-shifted triggers contaminating the background training set. However, given the rare detection rates for gravitational waves in the detector data analyzed here, it is unlikely that such a contamination has occurred. However, in the advanced detector era, when gravitational-wave detection is expected to be relatively common, this issue will need to be revisited. An additional issue to consider for the background training set concerns the size of the set. In references [@2013s6highmass; @2014rdsearch] and briefly in Section \[sec:data\], we describe the procedure used to compute the upper limit on BBH and IMBH coalescence rates. The typical procedure involves analyzing data in periods of $\sim$1 to 2 months. For these investigations, we ran the RFBDT classifier for each $\sim$1 to 2 month chunk. However, the size of the background training set for these $\sim$1 to 2 months analyses can be as small as 1% of the total background training set available for the entire Period 2 analysis. Thus, for the ringdown-only search, we took the additional step of examining the performance of the RFBDT classifier in the case where the monthly analyses used background training sets from their respective months and in the case where the monthly analyses used background training sets from the entire Period 2 analysis. As we report in Section \[sec:results\], using a background training set composed of time-shifted coincidences from the entire Period 2 analysis does not result in a clear sensitivity improvement. Feature vector {#sec:featurevect} -------------- A multivariate statistical classifier gives us the ability to use all available gravitational-wave data analysis statistics to calculate a combined FAR. These may include single trigger statistics such as SNR and the $\chi^2$ signal consistency test [@2005allen] as well as empirically-designed composite statistics that were previously used by each search as a classification statistic. The classifier will inherit the distinguishing power of the composite statistics as well as any other information we provide from statistics that may not have been directly folded into the composite statistics. These could include information that highlights inconsistencies in the single triggers’ template parameters or alerts us to the presence of bad data quality. The set of all statistics characterize the feature space and each coincidence identified by the search is described by a feature vector. As explained in Sect. \[sec:rf\], a different subset of feature vector parameters are chosen at each node. Selecting the optimal size of the subset can increase the randomness of the forest and reduce concerns of overfitting. We discuss the tuning of this number in Sect. \[sec:sampled\]. Also, note that the RFBDT algorithm can only make plane cuts through the feature space. It cannot reproduce a statistic that is composed of a non-linear combination of other statistics. As we describe in more detail in Appendix \[sec:appendix1\] and \[sec:appendix2\], if we know [*a priori*]{} a useful functional form for a non-linear composite statistic, we should include that statistic in the feature vector. Such a statistic can only ever be approximated by the plane cuts. Nevertheless, we design feature vectors with a large selection of statistics in the hope that some combination may be useful. Details of the parameters chosen to characterize coincidences with the RFBDT classifier for the IMR and ringdown-only searches are given in Appendices \[sec:appendix1\] and \[sec:appendix2\] and in [@2014kari; @2013bakerthesis]. The parameters are summarized here in Tables \[tab:paramshm\] and \[tab:rdparams\]. These include information from the template parameters and SNR of each single detector trigger as well as composite statistics that combine information from these. Additionally, a few parameters attempt to quantify the quality of the data. One of the data quality parameters for the ringdown-only search is a binary value used to indicate whether a trigger in a coincidence occurred during a time interval flagged for noise transients. The flagged intervals were defined using the hierarchical method for vetoing noise transients known as [*hveto*]{} as described in [@2011hveto]. The LIGO and Virgo gravitational-wave detectors have hundreds of auxiliary channels monitoring local environment and detector subsystems. The [*hveto*]{} algorithm identifies auxiliary channels that exhibit a significant correlation with transient noise present in the gravitational-wave channel and that have a negligible sensitivity to gravitational-waves. If a trigger in the gravitational-wave channel is found to have a statistical relationship with auxiliary channel glitches, a flagged time interval is defined. In Sect. \[sec:rdresults\], we explore the performance of the RFBDT classifier before and after the addition of the [*hveto*]{} parameter to the feature vector. This investigation was done to explore the ability of the classifier to incorporate data quality information. Significant work has been done to identify glitches in the data using multivariate statistical classifiers [@2013mvscglitch] and Bayesian inference [@2014bayeswave]. With more development, this work could be used to provide information to a multivariate classifier used to identify gravitational waves, allowing for powerful background identification and potentially significant improvement to the sensitivity of the search. -------------------------------------- ----------------------------------------------------------------------------------- ------------- Quantity Definition Description $\rho_i$ $\frac{|\left< x_i, h_i\right>|}{\sqrt{\left< h_i, h_i\right>}}$ $\chi^2_i$ $10\left[\sum\limits_{j=0}^{10} (\rho_{j}-\rho_i/10)^2\right]$ $\rho_{\text{eff},i}$ $\frac{\rho_i}{\left[\frac{\chi^2_i}{10}\left(1+\rho_i^2/50\right)\right]^{1/4}}$ $r^2_i$ veto duration $\left\{t:\frac{\chi^2_i(t)}{10+\rho_i(t)^2}>10\right\}$ $\chi^2_{\text{continuous},i}$ $\sum \left| \left<x_i(t)x_i(t+\tau)\right>-\rho_i\right| ^2$ $\rho_{\text{\text{high,combined}}}$ $\sqrt{\sum\limits_i^N \rho^2_{\text{high},i}}$ d$t$ $\left|t_a-t_b\right|$ d$\mathcal{M}_\mathrm{rel}$ $\frac{\left|\mathcal{M}_a-\mathcal{M}_b\right|}{\mathcal{M}_{\text{average}}}$ d$\mathcal{\eta}_\mathrm{rel}$ $\frac{\left|\eta_a-\eta_b\right|}{\eta_\text{average}}$ [*e-thinca*]{} $E$ -------------------------------------- ----------------------------------------------------------------------------------- ------------- ----------------------- -------------------------------------------------------------------- --------------------------------------------------------------------------------------------- Quantity Definition Description $\rho_i$ $\frac{|\left< x_i, h_i\right>|}{\sqrt{\left< h_i, h_i\right>}}$ d$t$ $\left|t_a-t_b\right|$ d$f$ $\left|f_a-f_b\right|$ d$Q$ $\left|Q_a-Q_b\right|$ d$s^2$ $g_{ij}$d$p^i$d$p^j$ $g_{tt}$ $\pi^2 f_0^2\frac{1+4Q^2}{Q^2}$ Metric coefficient in $(t,\,t)$-space $g_{f_0f_0}$ $\frac{1+6Q^2+16Q^4}{4f_0^2\left( 1+2Q^2\right)}$ Metric coefficient in $(f_0,\,f_0)$-space $g_{QQ}$ $\frac{1+28Q^4+128Q^6+64Q^8}{4Q^2\left( 1+6Q^2+8Q^4\right)}$ Metric coefficient in $(Q,\,Q)$-space $g_{tf_0}$ $2\pi Q \frac{1+4Q^2}{1+2Q^2}$ Metric coefficient in $(t,\,f_0)$-space $g_{tQ}$ $2\pi f_0 \frac{1-2Q^2}{\left(1+2Q^2\right)^2}$ Metric coefficient in $(t,\,Q)$-space $g_{f_0Q}$ $\frac{1+2Q^2+8Q^4}{2f_0Q\left( 1+2Q^2\right)^2}$ Metric coefficient in $(f_0,\,Q)$-space $\xi$ max$\left( \frac{\rho_a}{\rho_b},\, \frac{\rho_b}{\rho_a} \right)$ Maximum of the ratio of signal-to-ratios for triggers $a$ to $b$ or $b$ to $a$ ${\rho_N}^2$ $\sum_{i}^N {\rho_i}^2$ Combined network signal-to-noise ratio for $N$ triggers found in coincidence $\rho_\mathrm{S4}$ $\rho_\mathrm{S4,triple},\, \rho_\mathrm{S4,double}$ Detection statistic used in Ref. [@2009s4ringdown]; outlined in Eq.  and (\[eq:s4doub\]) $\rho_\mathrm{S5/S6}$ $\rho_\mathrm{S5/S6,triple},\, \rho_\mathrm{S4,double}$ Detection statistic described in Ref. [@2013talukder]; outlined in Eq.  and (\[eq:s4doub\]) $D_i$ $\frac{\sigma_i}{\rho_i}\left( 1\,\mathrm{Mpc}\right)$ d$D$ $\left|D_a-D_b\right|$ $\kappa$ max$\left( \frac{D_a}{D_b},\, \frac{D_b}{D_a} \right)$ Maximum of the ratio of effective distances for triggers $a$ to $b$ or $b$ to $a$ $n_i$ $n_i\left(\left| t \right| < 0.5\,\mathrm{ms}\right)$ hveto$_i$ $\left\{ \begin{array}{lr} 1 & : \mathrm{{\it hveto}\,\,flag\,\,on} \\ 0 & : \mathrm{{\it hveto}\,\,flag\,\,off} \end{array} \right.$ ----------------------- -------------------------------------------------------------------- --------------------------------------------------------------------------------------------- Random forest parameters ------------------------ A summary of the tunable parameters selected for the RFBDT algorithm for each search is given in Table \[tab:summaryrf\]. ### Number of trees {#sec:trees} We can adjust the number of trees in our forest to provide a more stable $M_\mathrm{forest}$ statistic. Increasing the number of trees results in an increased number of training events folded into the $M_\mathrm{forest}$ statistic calculation. However, the training data contains a finite amount of information and adding a large number of additional trees will ultimately reproduce results found in earlier trees. Furthermore, adding more trees will increase the computational cost of training linearly. In Fig. \[fig:tree1\], we investigate the effect of using a different number of trees for the ringdown-only search on the recovery of $q=1$ EOBNRv2 waveforms at Veto Level 3. We find no significant improvement for using more than 100 trees. Similar results were obtained at Veto Level 2 and for the recovery of $q=4$ EOBNRv2 waveforms. The IMR search trained classifiers with 100 trees in each forest. Initially, for the ringdown-only search, we selected to use 2000 trees in order to offset possible loss in sensitivity due to needing a larger leaf size as described in Sect. \[sec:leaf\]. However, we ultimately found that this did not change the sensitivity. Since computational costs were not high, we left the forest size as 2000 trees for the ringdown-only search. \[fig:b\]![Investigation of the effect of using a different number of trees on the recovery of $q=1$ EOBNRv2 simulated waveforms at Veto Level 3. In general, we find that the use of more than 100 trees gives roughly the same sensitivity regardless of mass ratio or veto level. In this ROC, to adjust for the loss in analysis time in moving from Veto Level 2 to Veto Level 3, we scale the volume fraction in Eq.  by the ratio of analysis times $f=t_\mathrm{VL3}/t_\mathrm{VL2}$. From the analysis times reported in Section \[sec:data\], we find $f=0.88$. []{data-label="fig:tree1"}](trees_cat34_disteff_cfar_test.pdf "fig:"){width=".45\textwidth"} ### Minimum leaf size {#sec:leaf} The minimum leaf size defines the stopping point for the splitting of nodes. We define the minimum number of events allowed in a node before it becomes a leaf. When all nodes become leaves, the recursive splitting of the tree stops. The choice of leaf size is affected by how representative the training data are of actual data and by how many coincident events are in the training data. If the leaf size is too small, the forest will be over-trained. In this case the sort is specific to the training data and may be systematically wrong for anything else. If the leaf size is too large, the forest will be under-trained. The individual decision trees did not have enough chances to make good cuts on the training data. We will be left unsure if any coincident event is signal or background. We are limited by the size of the background training set of time-shifted data. In each monthly analysis, the size of the background training set varied between thousands to hundreds of thousands of coincident events depending on veto level and analyzed networks. A leaf size of 5 worked very well for the IMR search’s trees, but investigations on the ringdown-only search with a leaf size of 5 showed that such a small choice led to an over-trained forest. Some signal *and* background coincidences were given an infinite $M_\mathrm{forest}$ rank (i.e., the classifier was 100% sure that the coincidence was signal). By exploring leaf sizes around 0.1-1% the size of the varied background training sets, we found that a leaf size of 65 eliminated the over-training and also gave good performance for the ringdown-only search. ### Number of sampled parameters {#sec:sampled} At each node, we choose a random subset of parameters to use for splitting. Out of $N_v$ total feature vector parameters, we select $m$ randomly and evaluate the split criteria for each. Thus, a different set of $m$ parameters is available for picking the optimal parameter and its threshold for each branching point. If $m$ is too large, each node will have the same parameters available to make the splits. This can lead to the same few parameters being used over and over again, and the forest will not fully explore the space of possible cuts. Furthermore, because individual trees will be making cuts based on the same parameters, all of the trees in the forest will be very similar. This is an example of over-training. If $m$ is too small, each node would have very few options to make the splits. The classifier would be forced to use poor parameters at some splits, resulting in inefficient cuts. The tree can run up against the leaf size limit before the training events were well-sorted. This is an example of under-training. The classification in this case would be highly dependent on the presence (or lack thereof) of poor parameters. A general rule-of-thumb for a good number of random sampled parameters is $\sim\sqrt{N_v}$ [@2014pyastro]. For the IMR search, of the 15 parameters that make up the feature vector, we empirically found good performance for a selection of 6 randomly chosen parameters at each node. For the ringdown-only search, 14 out of the total 24 feature vector parameters gave good performance. ### Criterion for optimization {#sec:criterion} The optimization criterion is used to select the best thresholds on parameters and proceeds the selection of random sampled parameters for each node. The RFBDT algorithm provides several methods to determine the optimal parameter thresholds. These are grouped by whether the output is composed of a discrete set or a continuous set of $M_\mathrm{forest}$ rankings. While some of the discrete statistics performed well, we preferred to draw rankings from a continuous set. Of the optimization criteria that gave continuous statistics, Gini index [@1912gini] and negative cross-entropy (defined in Sect. \[sec:rf\]) gave good performance and were comparable to each other for both searches. Additionally, in order to obtain a good average separation between signal and background, the suggested optimization criteria are either Gini index or negative cross-entropy [@2005statpattern]. Thus, these two statistics were chosen for the IMR and ringdown-only searches, respectively. The choices were arbitrary in the sense that either optimization criteria would have been suitable for either search. Splits were only made if they minimized the Gini index or the negative cross-entropy. Results {#sec:results} ======= IMR search {#sec:hmresults} ---------- In order to assess the sensitivity improvements of the IMR search to waveforms from BBH coalescing systems with non-spinning components, we use the same set of EOBNRv2 injections used to compute the upper limits on BBH coalescence rates in Sect. VB of [@2013s6highmass]. These injections were distributed approximately uniformly over the component masses $m_1$ and $m_2$ within the ranges $1\le m_i/\mathrm{M}_\odot \le 99$ and $20\le M/\mathrm{M}_\odot \le 109$. Additionally, we use the same set of IMRPhenomB injections used to make statements on sensitivity to spinning and non-spinning BBH coalescences in Sect. VC of [@2013s6highmass]. We use a non-spinning set and a spinning set of IMRPhenomB injections, both uniformly distributed in total mass $25\le M/\mathrm{M}_\odot \le 100$ and uniformly distributed in $q/(q + 1) = m_1/M$ for a given $M$, between the limits $1\le q < 4$. In addition, the spinning injections were assigned (anti-)aligned spin parameter $\chi_s$ uniformly distributed between -0.85 and 0.85. The previous IMR search over Period 2 data [@2013s6highmass] used the combined signal-to-noise and $\chi^2$-based ranking statistic $\rho_{\text{\text{high,combined}}}$ for FAR calculations. For more details on $\rho_{\text{\text{high,combined}}}$, see Table \[tab:paramshm\] and Appendix \[sec:appendix1\]. Here, we report on a re-analysis that replaces $\rho_{\text{\text{high,combined}}}$ with the ranking statistic calculated by the RFBDT, $M_\mathrm{forest}$, as described in Sect. \[sec:rf\]. Additionally, we have chosen a different FAR threshold for calculating sensitivity, rather than the loudest event statistic typically used in calculating upper limits in [@2013s6highmass]. The threshold that we use is the expected loudest FAR, $$\breve{\mathrm{FAR}} = 1/T$$ where $T$ is the total time of the analysis chunk being considered. For a listing of $\breve{\mathrm{FAR}}$ for each analysis chunk and a comparison with the loudest event statistic, see Table 8.1 of [@2014kari]. Improvements in the following section are reported with uncertainties determined using the statistical uncertainty originating from the finite number of injections that we have performed in these investigations. IMR search sensitive $VT$ improvements {#sec:hmsensvt} -------------------------------------- Figure \[fig:hmmtotalvol\] demonstrates the percent improvements in sensitive volume multiplied by analysis time $(VT)$ when using the $M_\mathrm{forest}$ ranking statistic, rather than the $\rho_{\text{\text{high,combined}}}$ ranking statistic. Results are shown at both Veto Levels 2 and 3 for total binary masses from $25\le M/\mathrm{M}_\odot \le 100$ in mass bins of width $12.5\,\mathrm{M}_\odot$. Improvements for EOBNRv2 waveforms are shown in Fig. \[fig:mtotvola\] and for IMRPhenomB are shown in Fig. \[fig:mtotvolb\]. The use of the $M_\mathrm{forest}$ ranking statistic gives improvements in $VT$ over the use of $\rho_{\text{\text{high,combined}}}$ at both Veto Levels 2 and 3. The largest improvements are seen for total masses larger than $50\mathrm{M}_\odot$. The IMR search is more sensitive in these higher mass regions. Thus, larger improvement is found where the search is more sensitive. For EOBNRv2 waveforms, larger improvements are seen at Veto Level 2 than at Veto Level 3. At Veto Level 2, $VT$ improvements ranged from $70_{\pm 13}-109_{\pm 11}$% for EOBNRv2 waveforms and from roughly $9_{\pm 5}-36_{\pm 6}$% for IMRPhenomB waveforms. At Veto Level 3, $VT$ improvements ranged from $10_{\pm 8}-35_{\pm 7}$% for EOBNRv2 waveforms and remained roughly the same for IMRPhenomB waveforms. More investigation is needed to understand why IMRPhenomB improvements are not as strong as EOBNRv2 improvements. One contributing factor could be component spin, which introduces several competing effects on the search including increased horizon distance with positive $\chi_s$, decreased sensitivity due to reduced overlap with EOBNRv1 templates, and higher signal-based $\chi^2$ test values [@2013s6highmass]. It is currently unclear if any of these effects reduce the potential percent improvement seen with the $M_\mathrm{forest}$ ranking statistic. For more detail, Fig. \[fig:hmm1m2vol\] shows the percent improvements in $VT$ for EOBNRv2 waveforms as a function of component masses. At Veto Level 2 in Fig. \[fig:mtotvola\], we see that every mass bin sees a percent improvement in $VT$. At Veto Level 3 in Fig. \[fig:mtotvolb\], again we see that the improvements are smaller than at Veto Level 2. In fact, no improvement is found for the lowest mass bin centered on $\left(5.5, 14.4\right)\,\mathrm{M}_\odot$. In Table \[tab:hmvetoresults\], we explore the percent $VT$ improvements obtained with $M_\mathrm{forest}$ at different veto levels. The improvements reported are made with respect to the sensitive volumes achieved with the $\rho_{\text{\text{high,combined}}}$ ranking statistic at Veto Level 2. These values are presented as a means of comparing sensitivity between Veto Level 2 and Veto Level 3. We see that $M_\mathrm{forest}$ at Veto Level 2 shows greater improvement and hence a more stringent upper limit than $M_\mathrm{forest}$ at Veto Level 3. This is in contrast to the better performance of $\rho_{\text{\text{high,combined}}}$ at Veto Level 3 than at Veto Level 2. For the standard IMR search with the $\rho_{\text{\text{high,combined}}}$ ranking statistic, the additional vetoing of poor quality data at Veto Level 3 was performed with the goal of preventing high SNR noise events from contaminating the list of gravitational-wave candidate events and reducing the sensitivity of the search. However, for the random forest technique, those high SNR noise events are down-weighted in significance due to information contained in other parameters in the feature vector. As a search at Veto Level 2 has more analysis time, it has the potential to have better sensitivity than a search at Veto Level 3. In Table \[tab:hmvetoresults\], we see that the use of the $M_\mathrm{forest}$ ranking statistic for the IMR search has resulted in a better search sensitivity at Veto Level 2. As we discuss in Sect. \[sec:rdresults\], the ringdown-only search did not see the same behavior at Veto Level 2. The information contained in the ringdown-only search’s feature vector may not have had sufficient signal and background separation information to overcome the level of background contamination present at Veto Level 2 as compared to Veto Level 3. -------------------------------------------- -------------------------------------- --------------------- --------------------- $\rho_{\text{\text{high,combined}}}$ $M_\mathrm{forest}$ $M_\mathrm{forest}$ Mass bin $\left( \mathrm{M}_\odot \right)$ Veto Level 3 Veto Level 2 Veto Level 3 25.0 - 37.5 28 $\pm$ 8% 70 $\pm$ 13% 40 $\pm$ 9% 37.5 - 50.0 36 $\pm$ 8% 75 $\pm$ 10% 60 $\pm$ 10% 50.0 - 62.5 35 $\pm$ 7% 109 $\pm$ 11% 82 $\pm$ 10% 62.5 - 75.0 33 $\pm$ 10% 91 $\pm$ 13% 76 $\pm$ 13% 75.0 - 87.5 15 $\pm$ 5% 77 $\pm$ 7% 50 $\pm$ 6% 87.5 - 100.0 41 $\pm$ 7% 108 $\pm$ 9% 86 $\pm$ 9% -------------------------------------------- -------------------------------------- --------------------- --------------------- Ringdown-only search {#sec:rdresults} -------------------- In order to assess the sensitivity improvements of the ringdown-only search to waveforms from binary IMBH coalescing systems with non-spinning components, we use the same set of EOBNRv2 injections used to compute the upper limits on IMBH coalescence rates in Section V of [@2014rdsearch]. Due to the variation in ringdown-only search sensitivity over different mass ratios, we chose to explore sensitivity improvements separately for $q=1$ and $q=4$. This variation occurs because the total ringdown efficiency depends on symmetric mass ratio so that extreme mass ratio systems will not be detectable unless the system is sufficiently close [@2014rdsearch]. The injection sets were distributed uniformly over a total binary mass range from $50 \le M/\mathrm{M}_\odot \le 450$ and upper limits were computed in mass bins of width $50\,$M$_\odot$. The final black hole spins of these injections can be determined from the mass ratios and zero initial component spins [@2009finalspin]. For $q=1$, we find $\hat{a}=0.69$, and for $q=4$, we find $\hat{a}=0.47$. Previous investigations of ranking statistics for the ringdown-only search [@2012talukderthesis; @2012caudillthesis; @2013talukder] found that $\rho_\mathrm{S5/S6}$ provided better sensitivity than the $\rho_\mathrm{S4}$ ranking statistic used as a detection statistic in [@2009s4ringdown]. Thus, here we report on sensitivities based on combined FARs computed using $\rho_\mathrm{S5/S6}$ as a ranking statistic and using $M_\mathrm{forest}$ as a ranking statistic. We follow the same loudest event statistic procedure used in [@2014rdsearch] for calculating upper limits. Improvements in the following section are reported with uncertainties determined using the statistical uncertainty originating from the finite number of injections that we have performed in these investigations. Our complete investigations involve evaluating the performance of the RFBDT classifier for ringdown-only searches over Period 2 data using five separate ranking statistics, described below. Additionally, we explore the improvement separately for recovery of $q=1$ and $q=4$ EOBNRv2 simulated waveforms as well as for Veto Level 2 and Veto Level 3 searches. The first ringdown-only search, to which we will compare performance, utilized the SNR-based statistic $\rho_\mathrm{S5/S6}$ to rank both double and triple coincident events. Details of this ranking statistic are given in Appendix \[sec:appendix2\] and in [@2012talukderthesis; @2012caudillthesis; @2013talukder]. In each of the investigative runs that follow, this statistic becomes a parameter that is added to the feature vector of each coincident event. A summary of the runs is given in Table \[tab:summaryrdruns\]. Run 1 uses a RFBDTs with 2000 trees, a leaf size of 65, and a random selection of 14 parameters out of the 24 total parameters listed in Sect. \[sec:featurevect\] except the [*hveto*]{} parameter. The training set was composed of a clean signal set as outlined in Sect. \[sec:sigtraining\] and background set trained separately for each $\sim$1-2 month chunk of Period 2 as outlined in Sect. \[sec:bkgdtraining\]. Run 2 is identical to Run 1 except that the background training set of the RFBDTs is composed of all Period 2 background coincident events rather than each corresponding $\sim$1-2 month set of background coincident events. We say that the RFBDTs is trained on the “full background set." Run 3 is identical to Run 1 except that the [*hveto*]{} parameter is included in the feature vector of each coincident event. This investigation was done to explore the ability of the RFBDT to incorporate data quality information. Run 4 combines the exceptions of Run 2 and Run 3. Thus, this investigation includes a RFBDT classifier trained on “full background set" and feature vectors that include the [*hveto*]{} parameter. Run Full background training set [*hveto*]{} parameter ----- ------------------------------ ----------------------- 1 No No 2 Yes No 3 No Yes 4 Yes Yes : Summary of ringdown-only search investigations[]{data-label="tab:summaryrdruns"} Ringdown-only sensitive $VT$ improvements {#sec:rdsensvt} ----------------------------------------- Figures \[fig:cat3volrd\] and \[fig:cat4volrd\] demonstrate the percent improvements in sensitive volume multiplied by analysis time $(VT)$ when using the $M_\mathrm{forest}$ ranking statistic, rather than the $\rho_\mathrm{S5/S6}$ ranking statistic at Veto Levels 2 and 3, respectively. Figure \[fig:cat3volrd\] focuses on the comparison of Runs 1-4 over $\rho_\mathrm{S5/S6}$ at Veto Level 2 for each mass ratio. Here we see that all runs perform better than $\rho_\mathrm{S5/S6}$ at Veto Level 2. The largest percent improvements are seen in the lowest and highest mass bins. These are the mass regions where the ringdown-only search is least sensitive. Thus, in these regimes, small changes in $VT$ lead to large percent improvements. This is the reason for the seemingly large percent improvement in Fig. \[fig:cat3volb\] for Run 2. In general, Run 3 and 4 that include the [*hveto*]{} parameter in the feature vector outperform Run 1 and 2 that do not include the [*hveto*]{} parameter. Run 4 most consistently shows the largest $VT$ improvements although the differences are not large at Veto Level 3. At Veto Level 2, $VT$ improvements ranged from $61_{\pm 4}-241_{\pm 12}$% for $q=1$ and from $62_{\pm 6}-236_{\pm 14}$% for $q=4$. We also note in Fig. \[fig:cat3volrd\] that Run 2 is slightly worse than Run 1. This is due to the fact that, generally, it is advantageous to break large analyses up into several smaller chunks to account for sensitivity changes over the run. By training the RFBDTs on the “full background set," we subjected the entire training set to background triggers from the least sensitive times (i.e., times when the background triggers most resembled signal) which resulted in an overall decrease in sensitive volume. In Run 1, these troublesome background triggers would be isolated in the separate training sets for each $\sim$1-2 month chunk of Period 2. However, note that training the RFBDTs on the “full background set" with an [*hveto*]{} data quality parameter in the feature vectors results in Run 4 being more sensitive than Run 3. Figure \[fig:cat4volrd\] focuses on the comparison of Runs 1-4 over $\rho_\mathrm{S5/S6}$ at Veto Level 3 for each mass ratio. Again we see that all runs perform better than $\rho_\mathrm{S5/S6}$ at Veto Level 3 (although percent improvements are not as large as those seen at Veto Level 2), and the largest percent improvements are seen in the lowest and highest mass bins. However, at Veto Level 3, we find that the addition of the [*hveto*]{} data quality parameter in the feature vectors of Run 3 and 4 do not give significant improvements over Run 1 and 2. This fact indicates that the [*hveto*]{} parameter provides no additional information on the most significant glitches for the ringdown-only search that is not already included in the vetoes at Veto Level 3. Although the difference is not large, in general, Run 3 and 4 still outperform Run 1 and 2. At Veto Level 3, $VT$ improvements ranged from $39_{\pm 4}-89_{\pm 8}$% for $q=1$ and from $39_{\pm 5}-111_{\pm 18}$% for $q=4$. In Table \[tab:rdvetoresults\], we explore the percent $VT$ improvements obtained with $M_\mathrm{forest}$ at different veto levels with and without the [*hveto*]{} parameter. The improvements reported are made with respect to the $\rho_\mathrm{S5/S6}$ ranking statistic at Veto Level 2. These values are presented as a means of comparing sensitivity between Veto Level 2 and Veto Level 3. Here we make several observations. First, unlike the behavior we observed for the IMR search, we see that Run 1 at Veto Level 3 shows greater improvement and hence a more stringent upper limit than Run 1 at Veto Level 2. Thus, the removal of poor data quality at Veto Level 3 is an important step for improving the sensitivity of the ringdown-only search. Second, we can compare Run 1 at Veto Level 3 with Run 3 at Veto Level 2. This comparison allows us to gauge whether the gain in analysis time we get by including [*hveto*]{} data quality information in the feature vector at Veto Level 2 outweighs the boost in sensitive volume we gain by removing data flagged by Veto Level 3. We see that Run 1 at Veto Level 3 gives consistently larger percent $VT$ improvements than Run 3 at Veto Level 2. Thus, adding the [*hveto*]{} data quality information in the feature vector does not match the sensitivity improvements from the application of data quality vetoes. However, we note that the [*hveto*]{} data quality information was not specifically tuned for the ringdown-only search nor is it meant to be an exhaustive data quality investigation. Further exploration with more sophisticated data quality information is needed in order to determine whether the classifier can incorporate data quality information and approach the sensitivity achieved by the use of data quality vetoes. -------------------------------------------- ----------------------- --------------- --------------- --------------- --------------- -- $\rho_\mathrm{S5/S6}$ Run 1 Run 1 Run 3 Run 3 Mass bin $\left( \mathrm{M}_\odot \right)$ Veto Level 3 Veto Level 2 Veto Level 3 Veto Level 2 Veto Level 3 50.0 - 100.0 113 $\pm$ 7% 129 $\pm$ 8% 299 $\pm$ 14% 230 $\pm$ 11% 302 $\pm$ 14% 100.0 - 150.0 92 $\pm$ 6% 120 $\pm$ 7% 250 $\pm$ 10% 193 $\pm$ 8% 252 $\pm$ 10% 150.0 - 200.0 88 $\pm$ 5% 95 $\pm$ 6% 196 $\pm$ 8% 143 $\pm$ 7% 201 $\pm$ 8% 200.0 - 250.0 63 $\pm$ 5% 51 $\pm$ 5% 133 $\pm$ 7% 79 $\pm$ 5% 136 $\pm$ 7% 250.0 - 300.0 60 $\pm$ 4% 27 $\pm$ 3% 121 $\pm$ 7% 55 $\pm$ 4% 125 $\pm$ 7% 300.0 - 350.0 66 $\pm$ 6% 47 $\pm$ 5% 139 $\pm$ 8% 82 $\pm$ 6% 143 $\pm$ 8% 350.0 - 400.0 73 $\pm$ 8% 78 $\pm$ 7% 186 $\pm$ 12% 124 $\pm$ 9% 194 $\pm$ 12% 400.0 - 450.0 70 $\pm$ 8% 112 $\pm$ 10% 207 $\pm$ 14% 149 $\pm$ 11% 218 $\pm$ 14% -------------------------------------------- ----------------------- --------------- --------------- --------------- --------------- -- Summary {#sec:summary} ======= This paper presents the development and sensitivity improvements of a multivariate analysis applied to matched filter searches for gravitational waves produced by coalescing black hole binaries with total masses $\gtrsim25\,$M$_\odot$. We focus on the applications to the IMR search which looks for gravitational waves from the inspiral, merger, and ringdown of BBHs with total mass between $25\,$M$_\odot$ and $100\,$M$_\odot$ and to the ringdown-only search which looks for gravitational waves from the resultant perturbed IMBH with mass roughly between $10\,$M$_\odot$ and $600\,$M$_\odot$. These investigations were performed over data collected by LIGO and Virgo between 2009 and 2010 so that comparisons can be made with previous IMR and ringdown-only search results [@2013s6highmass; @2014rdsearch]. We discuss several issues related to tuning RFBDT multivariate classifiers in matched-filter IMR and ringdown-only searches. We determine the sensitivity improvements achieved through the use of a RFBDT-derived ranking statistic over empirical SNR-based ranking statistics while considering the application of data quality vetoes. Additionally, we present results for several modifications on the basic RFBDT implementation including the use of an expansive training set and data quality information. When optimizing the performance of RFBDT classifiers, we found that a RFBDT classifier with 100 trees, a leaf size of 5, and 6 randomly sampled parameters from the feature vector gave good performance for the IMR search while a RFBDT classifier with 2000 trees, a leaf size of 65, and 14 randomly sampled parameters from the feature vector gave good performance for the ringdown-only search. In both cases, we used a training set of “clean“ signal designed to carefully remove contamination from glitches within the software injection-finding time window. This technique eliminated the excursion of gravitational-wave candidate coincidences from the $2\sigma$ region of the expected background at low values of inverse combined FAR as demonstrated in Fig. \[fig:ifarbump\]. Additionally, we examined the performance of the RFBDT classifier in the case where the monthly analyses used background training sets from their respective months and in the case where the monthly analyses used background training sets from the entire Period 2 analysis (i.e., the ”full background set"). We found that using the full background training set does not result in a clear sensitivity improvement unless a data quality [*hveto*]{} parameter is introduced in the feature vector. For the IMR search, we performed a re-analysis replacing $\rho_\mathrm{high,combined}$ with the ranking statistic calculated by the RFBDT, $M_\mathrm{forest}$. Comparisons with $\rho_\mathrm{high,combined}$ were made separately at each veto level. For EOBNRv2 waveforms, the percent improvements in $VT$ were largest at Veto Level 2. Depending on mass bin, the $VT$ improvements ranged from $70_{\pm 13}-109_{\pm 11}$% at Veto Level 2 and from $10_{\pm 8}-35_{\pm 7}$% at Veto Level 3. For IMRPhenomB waveforms, $VT$ improvements ranged from $9_{\pm 5}-36_{\pm 6}$% regardless of veto level. Additionally, we made comparisons across veto levels, using the performance of $\rho_\mathrm{high,combined}$ at Veto Level 2 as the standard. We found that $M_\mathrm{forest}$ at Veto Level 2 shows greater improvement and hence a more stringent upper limit than $M_\mathrm{forest}$ at Veto Level 3. This is in contrast to the better performance of $\rho_\mathrm{high,combined}$ at Veto Level 3 than at Veto Level 2. For the ringdown-only search, we evaluated the performance of the RFBDT classifier using five separate ranking statistics. Comparisons were made with respect to a ringdown-only search that used the $\rho_\mathrm{S5/S6}$ ranking statistic [@2012talukderthesis; @2012caudillthesis; @2013talukder]. The additional four searches used the $M_\mathrm{forest}$ ranking statistic for various instantiations of the RFBDT classifier. Comparisons with $\rho_\mathrm{S5/S6}$ were made separately at each veto level. At Veto Level 2, we found that a RFBDT classifier trained on the full background set and including the data quality [*hveto*]{} parameter in the feature vector resulted in $VT$ improvements in the range $61_{\pm 4}-241_{\pm 12}$% for $q=1$ EOBNRv2 waveforms and in the range $62_{\pm 6}-236_{\pm 14}$% $q=4$ EOBNRv2 waveforms. At Veto Level 3, this same configuration resulted in $VT$ improvements in the range $39_{\pm 4}-89_{\pm 8}$% for $q=1$ EOBNRv2 waveforms and in the range $39_{\pm 5}-111_{\pm 18}$% $q=4$ EOBNRv2 waveforms. Again, we made comparisons across veto levels, using the performance of $\rho_\mathrm{S5/S6}$ at Veto Level 2 as the standard. Unlike the IMR search, we found that $M_\mathrm{forest}$ at Veto Level 3 shows greater improvement and hence a more stringent upper limit than $M_\mathrm{forest}$ at Veto Level 2. Additionally, we found that adding an [*hveto*]{} parameter at Veto Level 2 does not result in the same increase in sensitivity obtained by applying level 3 vetoes to a search using the basic implementation of the RFBDT classifier. With more sophisticated methods for adding data quality information to the feature vector, we may see additional improvements or different behavior. Further exploration is needed. In general, for each search, we found that the RFBDT multivariate classifier results in a considerably more sensitive search than the empirical SNR-based statistic at both veto levels. The software for constructing clean injection sets and the RFBDTs is now implemented in the LALSuite gravitational-wave data analysis routines for use with other matched-filter searches. More investigations will be needed to understand whether lower mass searches for gravitational waves from binary coalescence would benefit from the use of multivariate classification with supervised MLAs. For higher mass searches, particularly those susceptible to contamination from noise transients, RFBDT multivariate classifiers have proven to be a valuable tool for improving search sensitivity. [**Acknowledgments**]{} We gratefully acknowledge the National Science Foundation for funding LIGO, and LIGO Scientific Collaboration and the Virgo Collaboration for access to this data. PTB and NJC were supported by NSF award PHY-1306702. SC was supported by NSF awards PHY-0970074 and PHY-1307429. DT was supported by NSF awards PHY-1205952 and PHY-1307401. CC was partially supported by NSF grants PHY-0903631 and PHY-1208881. This document has been assigned LIGO laboratory document number P1400231. The authors would like to acknowledge Thomas Dent, Chad Hanna, and Kipp Cannon for work during the initial phase of this analysis. The authors would also like to thank Alan Weinstein, Gregory Mendell, and Marco Drago for useful discussion and guidance. [81]{}ifxundefined \[1\][ ifx[\#1]{} ]{}ifnum \[1\][ \#1firstoftwo secondoftwo ]{}ifx \[1\][ \#1firstoftwo secondoftwo ]{}““\#1””@noop \[0\][secondoftwo]{}sanitize@url \[0\][‘\ 12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty [****,  ()](http://stacks.iop.org/0264-9381/27/i=8/a=084006) @noop [ ()]{} [****,  ()](http://stacks.iop.org/0264-9381/29/i=12/a=124007) [ ()](https://dcc.ligo.org/cgi-bin/DocDB/ShowDocument?docid=75988) [****,  ()](http://stacks.iop.org/0264-9381/27/i=17/a=173001) [****,  ()](\doibase 10.1103/PhysRevD.87.022002) [****,  ()](\doibase 10.1103/PhysRevD.89.102006) [****,  ()](\doibase 10.1103/PhysRevLett.70.2984) @noop [****,  ()]{} [****,  ()](http://stacks.iop.org/0034-4885/72/i=7/a=076901) [****,  ()](\doibase 10.1103/PhysRevD.79.122001),  [****,  ()](\doibase 10.1103/PhysRevD.80.047101) [****,  ()](\doibase 10.1103/PhysRevD.85.082002),  [****,  ()](\doibase 10.1103/PhysRevD.83.122005) [****,  ()](http://stacks.iop.org/0264-9381/25/i=10/a=105024) @noop [Ph.D. thesis]{},  () @noop [**]{} (, ) @noop [**]{} (, ) in @noop [**]{}, Vol.  (, , ) pp.  @noop [**]{},  ed. (, , ) [****, ()](\doibase 10.1023/A:1022627411411) @noop [**]{},  ed. (, , ) [****, ()](\doibase 10.1023/A:1010933404324) [****,  ()](\doibase 10.1088/0004-637X/781/2/117),  [****,  ()](\doibase 10.1086/668468),  [****,  ()](\doibase 10.1093/mnras/stt1306),  [****,  ()](\doibase 10.1016/j.engappai.2009.05.004) @noop [ ()]{} [****,  ()](\doibase 10.1103/PhysRevD.88.062006),  @noop [ ]{} [****,  ()](\doibase 10.1103/PhysRevD.88.062003),  [****,  ()](\doibase 10.1103/PhysRevLett.112.131101) [****,  ()](\doibase 10.12942/lrr-2006-6),  [****,  ()](\doibase 10.12942/lrr-2012-8),  [**](http://books.google.com/books?id=RNAyAAAAMAAJ), International series of monographs on electronics and instrumentation (, ) @noop [**]{} (, , ) [****,  ()](\doibase 10.1088/0264-9381/26/20/204013),  @noop [****,  ()]{},  @noop [ ]{} [****,  ()](\doibase 10.1103/RevModPhys.82.3069),  [****,  ()](\doibase 10.1088/0264-9381/26/11/114001),  @noop [ ()]{},  [****,  ()](\doibase 10.12942/lrr-1999-2),  [****,  ()](\doibase 10.1103/PhysRevD.76.064034),  [****, ()](\doibase 10.1103/PhysRevD.80.062001) [****,  ()](\doibase 10.1103/PhysRevLett.74.3515) [****,  ()](\doibase 10.1103/PhysRevLett.93.091101) [****,  ()](\doibase 10.1103/PhysRevD.76.104049),  [****,  ()](\doibase 10.1103/PhysRevD.85.122006) [****,  ()](\doibase 10.1086/152444) [****,  ()](\doibase 10.1103/PhysRevD.75.124018) [****,  ()](http://stacks.iop.org/0264-9381/27/i=16/a=165023) [****,  ()](\doibase 10.1103/PhysRevD.87.024033),  [****, ()](\doibase 10.1103/PhysRevD.71.062001),  @noop [****,  ()]{} [****,  ()](\doibase 10.1103/PhysRevD.84.124052),  [****,  ()](\doibase 10.1103/PhysRevLett.106.241101),  @noop [ ()]{},  [**](https://dcc.ligo.org/LIGO-T060156/public),   (, ) [****,  ()](http://stacks.iop.org/0264-9381/28/i=2/a=025005) [****,  ()](\doibase 10.1103/PhysRevD.77.062002), [****,  ()](\doibase 10.1103/PhysRevD.83.122005), ,  [****,  ()](\doibase 10.1103/PhysRevD.88.122002) @noop [ ()]{},  @noop [Ph.D. thesis]{},  () @noop [ ()]{},  [****, ()](\doibase 10.1023/A:1007515423169) @noop [**]{}, edited by  and  (, ) @noop [Ph.D. thesis]{},  () [****,  ()](http://stacks.iop.org/0264-9381/28/i=23/a=235005) @noop [ ()]{},  @noop [Master’s thesis]{},  () @noop [Ph.D. thesis]{},  () @noop [Ph.D. thesis]{},  () @noop [**]{} (, , ) [****, ()](\doibase 10.1088/0004-637X/704/1/L40) @noop [Ph.D. thesis]{},  () @noop [Ph.D. thesis]{},  () [****,  ()](\doibase 10.1088/0264-9381/28/13/134009),  [****,  ()](\doibase 10.1103/PhysRevD.60.021101) [****, ()](\doibase 10.1103/PhysRevD.68.102003),  IMR search feature vector {#sec:appendix1} ========================= The full list of statistics used for the IMR search’s feature vector are given in Table \[tab:paramshm\]. In the following sections, we provide more detail on the definitions of each statistic. Density distributions of these statistics for this search’s simulated signal and background training sets are shown in Fig. \[fig:hmparams1\] and \[fig:hmparams2\]. Single trigger statistics {#sec:hmsngl_parameters} ------------------------- Single trigger statistics are defined for each individual trigger that makes up each multi-detector coincidence. For the IMR search, single trigger statistics added to the feature vector included the matched-filter SNR from each detector [@findchirp], the $\chi^2$ signal-consistency test for the matched-filter result in a number of frequency bins for each detector [@2005allen], the $r^2$ veto duration for which a weighted $\chi^2$ exceeds a pre-set threshold, and the $\chi_\mathrm{continuous}^2$ quantity for the residual of the SNR and autocorrelation time series in each detector. More details on the matched-filter SNR for the IMR search templates are given in [@2011s5highmass] and a definition is given in Table \[tab:paramshm\]. The $\chi^2$ signal-consistency test, only currently calculated for the low and high mass searches, tests how well the template waveform matches the data in various frequency bands. The bins are constructed so that the matched template contributes an equal amount of SNR to each bin. Then the following quantity is computed, $$\chi^2=10\left[\sum\limits_{i=0}^{10} (\rho_{i}-\rho/10)^2\right]$$ where $\rho_i$ is the SNR contribution from the $i$th bin. The $r^2$ veto duration measures the amount of time that the following quantity is above a threshold of 0.0002, within $6\,$s of the trigger, $$r^2=\frac{\chi^2(t)}{p+\rho(t)^2}$$ where $p=10$ bins are used. This quantity is motivated by the fact that a signal is unlikely to exactly match a template so a non-centrality parameter is introduced to the distribution of the $\chi^2$ signal-consistency test. Thus, rather than thresholding on $\chi^2$, we threshold on $r^2$. The $\chi_\mathrm{continuous}^2$ calculation performs a sum of squares of the residual of the SNR time series and the autocorrelation time series of a single detector trigger. Composite statistics {#sec:hmcomp_parameters} -------------------- Composite statistics are defined by combining single trigger statistics in a meaningful way and are computed once for each coincidence. Although the classifier can approximate such statistics in the multidimensional parameter space (e.g., if they are a combination of the $\rho$ and $\chi^2$), this ability is limited by the tree depth, the number of decision tree cuts before hitting the minimum leaf size. Thus, if we have *a priori* knowledge of a useful functional form for a ranking statistic, we should provide the classifier with this information. By providing this information up front, a classifier can improve upon these good statistics rather than trying to construct them itself. Some of these composite statistics have previously been used as ranking statistics when calculating combined FARs in searches. For the IMR search, we include several previous ranking statistics in the feature vector. The first of these is known as the effective SNR statistic and was used as a ranking statistic in [@2011s5highmass], $$\label{eq:effsnr} \rho_\mathrm{eff}=\frac{\rho}{\left[\frac{\chi^2}{10}\left(1+\rho^2/50\right)\right]^{1/4}}.$$ The second is known as $\rho_\mathrm{high,combined}$, a $\chi^2$-weighted statistic described in detail in [@2013s6highmass] for the IMR search. Due to the different distributions of background triggers over SNR and $\chi^2$ for longer-duration versus shorter-duration templates, a different choice of ranking statistics was selected for each bin in [@2013s6highmass]. For long duration events, the following was used $$\hat{\rho} = \left\{ \begin{array}{lr} \frac{\rho}{\left[ \left(1 + \left(\chi_r^2\right)^3 \right) \right]^{1/6}}& \mathrm{for}\,\,\chi_r^2>1\\ \rho & \mathrm{for}\,\,\chi_r^2 \le1 \end{array} \right.$$ where $\chi_r^2\equiv \chi^2/(2p-2)$ for number of frequency intervals $p=10$. For shorter duration events, Eq. \[eq:effsnr\] was used. Thus, $\rho_\mathrm{high,combined}$ is a piecewise function of $\rho_\mathrm{eff}$ and $\hat{\rho}$ and is combined as a quadrature sum of single-detector statistics. Additionally, we calculate quantities that provide an indication of how close the pair of triggers from different detectors are in the metric space $\left(\mathcal{M},\eta,t\right)$ for the IMR search. These include the difference in arrival time d$t$, the relative difference in chirp mass d$\mathcal{M}_\mathrm{rel}$, the relative difference in the symmetric mass ratio d$\eta_\mathrm{del}$, and a quantity known as the [*e-thinca*]{} test that combines these three by constructing error ellipsoids in time and mass space [@ethinca]. Ringdown-only feature vector {#sec:appendix2} ============================ The full list of statistics used for the ringdown-only search’s feature vector are given in Table \[tab:rdparams\]. In the following sections, we provide more detail on the definitions of each statistic. Density distributions of these statistics for this search’s simulated signal and background training sets are shown in Fig. \[fig:rdparams1\]-\[fig:rdparams4\]. Single trigger statistics {#sec:rdsngl_parameters} ------------------------- For the ringdown-only search, single trigger statistics added to the feature vector included the matched-filter SNR from each detector and the effective distance from each detector, $\mathrm{D}_{\mathrm{eff}}$. More details on the matched-filter SNR, specifically for ringdown templates, are given in [@2009s4ringdown; @2014rdsearch]. The effective distance is equivalent to the distance $r$ to a source that is optimally oriented and located. The theoretical formula for the effective distance is defined in terms of the $F_+$ and $F_\times$ detector antenna pattern functions and the inclination angle $\iota$, $$D_\mathrm{eff}=\frac{r}{\sqrt{F_+^2\left( 1+ \cos^2\iota \right)/4 + F_\times^2\cos^2\iota}}.$$ In practice, however, the effective distance is calculated from the power spectral density of the detector and the matched-filter SNR; see Table \[tab:rdparams\]. Although we did not include them here, additional single trigger statistics may be available to a search (e.g., coherent and null SNRs computed from coherent analyses [@2011coherent; @2013talukder]). Composite statistics {#sec:rdcomp_parameters} -------------------- Composite statistics included in the feature vector for the ringdown-only search include a combined network SNR, a detection statistic used in [@2009s4ringdown], and a ranking statistic detailed in [@2012talukderthesis; @2012caudillthesis; @2013talukder]. The combined network SNR for the $N$ detectors participating in the coincidence, $$\label{eq:ntwksnr} \rho^2_N=\sum\limits_{i}^N\rho_i^2,$$ where $\rho_i$ is the SNR in the $i$th detector, is the optimal ranking statistic for a signal with known parameters in Gaussian noise. In the ringdown-only search in [@2009s4ringdown], due to the dearth of false alarms found in triple coincidence, a suitable statistic for ranking triple coincident events was found to be the the network SNR in Eq. \[eq:ntwksnr\] such that $\rho_\mathrm{S4,triple}=\rho^2_N$. However, [@2009s4ringdown] found a high level of double coincident false alarms, often with very high SNR in only one detector. While it is possible that a real gravitational-wave source could have an orientation that would produce an asymmetric SNR pair, the occurrence is relatively rare in comparison to the occurrence of this feature for false alarms. The network SNR is clearly non-optimal in this case. References [@1999creightonchopL; @2008gogginthesis] found the optimal statistic in such a case to be a “chopped-L" statistic, $$\label{eq:s4doub} \rho_\mathrm{S4,double} = \min \left\{ \begin{array}{c} \rho_{\mathrm{ifo1}}+\rho_{\mathrm{ifo2}} \\ \alpha \rho_{\mathrm{ifo1}}+\beta \\ \alpha \rho_{\mathrm{ifo2}}+\beta \end{array} \right\}$$ where the tunable parameters $\alpha$ and $\beta$ were set to 2 and 2.2, respectively, as described in [@2009s4ringdown]. We include this piece-wise detection statistic composed of $\rho_\mathrm{S4,triple}$ and $\rho_\mathrm{S4,double}$ in the feature vector. For the most recent ringdown-only search [@2014rdsearch], due to a large increase in analysis time and lower SNR thresholds, a significant population of triple coincident false alarms were found. Thus, an additional “chopped-L"-like statistic was developed for triple coincidences, $$\label{eq:s5s6trip} \rho_\mathrm{S5/S6,triple} = \min \left\{ \begin{array}{c} \rho_N \\ \rho_{\mathrm{ifo1}}+\rho_{\mathrm{ifo2}}+\gamma \\ \rho_{\mathrm{ifo2}}+\rho_{\mathrm{ifo3}}+\gamma \\ \rho_{\mathrm{ifo3}}+\rho_{\mathrm{ifo1}}+\gamma \end{array} \right\}$$ where the tunable parameter $\gamma$ was set to 0.75. The development and tuning of this new statistic are described in detail in [@2012talukderthesis; @2012caudillthesis; @2013talukder]. Again, we include this piece-wise detection statistic composed of $\rho_\mathrm{S5/S6,triple}$ and $\rho_\mathrm{S4,double}$ in the feature vector. In addition to these three previous ranking statistics, we include the following simple composite statistics: the maximum of the ratios of the SNRs for triggers in each detector, the difference in recovered effective distances, and the maximum of the ratios of the recovered effective distances. Finally, we calculate quantities that provide an indication of how close the pair of triggers from different detectors are in the metric space $\left(f_0, Q, t\right)$ for the ringdown-only search. These include the difference in arrival time d$t$, the template frequency difference d$f_0$, the template quality factor difference d$Q$, and the 3D-metric distance d$s^2$ between two triggers in $\left(f_0, Q, t\right)$ space [@2014rdsearch; @2003rdds2]. Also included are the 3D-coincidence metric coefficients g$_{tt}$, g$_{f_0f_0}$, g$_{QQ}$, g$_{tf_0}$, g$_{tQ}$, and g$_{f_0Q}$ defined in Table \[tab:rdparams\]. Other parameters {#sec:rdother_parameters} ---------------- Two additional parameters were added to the feature vector for the ringdown-only search in an effort to provide data quality information to the classifier. The first was a binary value used to indicate whether a trigger in a coincidence occurred during a time interval flagged for noise transients. The flagged intervals were defined using the hierarchical method for vetoing noise transients known as [*hveto*]{} as described in [@2011hveto]. The LIGO and Virgo gravitational-wave detectors have hundreds of auxiliary channels monitoring local environment and detector subsystems. The hveto algorithm identifies auxiliary channels that exhibit a significant correlation with transient noise present in the gravitational-wave channel and that have a negligible sensitivity to gravitational-waves. If a trigger in the gravitational-wave channel is found to have a statistical relationship with auxiliary channel glitches, a flagged time interval is defined. The second additional parameter was a count of the number of single detector triggers clustered over a time interval of $0.5\,$ms using a SNR peak-finding algorithm described in [@2008gogginthesis]. The motivation behind this parameter comes from investigations that show that a glitch will be recovered with a different pattern of templates over time than a signal [@2008chadthesis]. Ideally, a $\chi^2$-based statistic could be computed. However, in the absence of this test for the ringdown-only search, we simply provide a count of the number of templates in a small time window around each trigger giving a matched-filter SNR above the threshold. [^1]: The naming convention for veto categorization can vary across searches. We use the convention for Veto Levels 1, 2, and 3 as defined in Section V of [@2010vetoes] [^2]: http://statpatrec.sourceforge.net [^3]: [https://ligo-vcs.phys.uwm.edu/cgit/lalsuite/tree/ pylal/bin/ligolw\_dbinjfind]{} [^4]: [https://www.lsc-group.phys.uwm.edu/daswg/projects/ lalsuite.html]{}

--- author: - bibliography: - 'labels.bib' --- Designers of mathematical models for computational systems need to find appropriate trade-offs between two seemingly contradictory requirements. Automatic verification (and thus usability) typically requires a high level of abstraction whereas prediction accuracy requires a high level of details. &gt;From this perspective, the use of symbolic models for security analysis is particularly delicate since it seems that the inherent high level of abstraction at which such models operate is not able to capture all aspects that are relevant to security. This paper is concerned with one particular such aspect, namely the use of randomization in the construction of cryptosystems [@goldwasser84probabilistic]. A central feature of the computational, complexity-based models is the ability to capture and reason explicitly about the use of randomness. Moreover, randomness is essential to achieve any meaningful notion of security for encryption. In contrast, symbolic models rarely represent randomness directly. For example, a typical representation for the encryption of message $m$ under the public key of entity $B$ is the term ${\{m\}_{\ek(B)}}$. Notice that the symbolic representation does not capture the dependency on the randomness used to generate this ciphertext. While this abstraction may be sufficiently accurate in certain settings [@MW04], in some other settings it is not sufficient. Consider the following flow in some toy protocol: $$\begin{array}{rcl} A& \rightarrow B:& \{m\}_{\ek(B)},\{\{m\}_{\ek(B)}\}_{\ek(B)}\end{array}$$ To implement this flow, each occurrence of ${\{m\}_{\ek(B)}}$ is mapped to a ciphertext. Notice however that the pictorial description does not specify if the two occurrences of ${\{m\}_{\ek(B)}}$ are equal (created with identical coins) or different (created with different coins). In rich enough protocol specification languages disambiguating constructs as above can be easily done. For instance, in a language that has explicit assignments, the two different interpretation for the first message of the protocol can be obtained as $$x:={\{m\}_{\ek(B)}}; \mathsf{send}(x,{\{x\}_{\ek(B)}})\;\;\;\;\mathrm{and}\;\;\;\; \mathsf{send}({\{m\}_{\ek(B)}},{\{{\{m\}_{\ek(B)}}\}_{\ek(B)}})\;\;$$ Here, each distinct occurrence of ${\{m\}_{\ek(B)}}$ is interpreted with different randomness. Other approaches adopt a more direct solution and represent the randomness used for encryption explicitly [@herzog; @abadi01formal; @lowe04analysing; @CortierW-ESOP05]. If we write ${\{m\}_{\ek(B)}}^l$ for the encryption of $m$ under the public key of $B$ with random coins $l$, the two different interpretations of the flow are: $$\mathsf{send}({\{m\}_{\ek(B)}}^{l_1}, {\{{\{m\}_{\ek(B)}}^{l_1}\}_{\ek(B)}}^{l_2})\;\;\;\; \mathrm{and}\;\;\;\; \mathsf{send}({\{m\}_{\ek(B)}}^{l_1}, {\{{\{m\}_{\ek(B)}}^{l_2}\}_{\ek(B)}}^{l_3})$$ A model that employs labels to capture the randomness used in ciphertexts (and signatures) has recently been used to establish soundness of symbolic analysis with respect to computational models [@CortierW-ESOP05]. Their results are based on an emulation lemma: for protocol executions, every computational trace can be mapped to a valid symbolic trace. The mapping is then used to translate security properties that hold in the symbolic model to computational analogues. The next step towards making the soundness result relevant to practice is to carry out the security proofs using some (semi-)automated tools for the symbolic model. However, to the best of our knowledge, none of the popular tools (ProVerif [@blanchet01], CASPER [@casper], Athenta [@athena], AVISPA [@sysdesc-CAV05]), offers capabilities for automatically reasoning in models that use labels. There are at least two solutions to this problem. One possibility is to enhance the symbolic models that underlie existing tools. Unfortunately such a modification would probably require significant effort that involves adapting existing decision procedures, proving their correctness, and verifying and modifying thousands of lines of code. In this paper we put forth and clarify an alternative solution, used implicitly in [@CortierW-ESOP05]. The idea is to keep existing tools unchanged, use their underlying (unlabeled) model to prove security properties, and then show that the results are in fact meaningful for the model with labels. The main result of this paper is to prove that for a large class of security properties the approach that we propose is indeed feasible. We are currently implementing an AVISPA module for computationally sound automatic proofs based on the results of this paper. #### Results. We consider the protocol specification language and the execution model developed in [@CortierW-ESOP05]. The language is for protocols that use random nonces, public key encryption and digital signatures, and uses labels to model the randomness used by these primitives. To each protocol $\Pi$ with labels, we naturally associate a protocol ${\overline{\Pi}}$ obtained by erasing all labels, and extend the transformation to execution traces. To each trace $tr$ of $\Pi$ we associate a trace ${\overline{tr}}$ obtained by erasing labels and we extend this mapping to sets of traces. The first contribution of this paper is a proof that the transformation is sound. More precisely we prove that if $tr$ is a valid trace of $\Pi$ (obtained by Dolev-Yao operations) then ${\overline{tr}}$ is a valid trace of ${\overline{\Pi}}$. Importantly, this result relies on the fact that the specification language that we consider does not allow equality tests between ciphertexts. We believe that a similar result holds for most (if not all) protocol specification languages that satisfy the above condition. The language for specifying protocols (with and without labels) as well as the relation between their associated execution models are in Section \[section-protocol\]. In Section \[section-alogic\] we give two logics, ${\mathcal{L}_1}^l$ and ${\mathcal{L}_1}$, that we use to express security properties for protocols with and without labels, respectively. Informally, the formulas of ${\mathcal{L}_1}$ are obtained by removing the labels from formulas of ${\mathcal{L}_1}^l$. Both logics are quite expressive. For example, it can be used to express standard formulations for secrecy and authenticity properties. Next we focus our attention on translating security properties between the two models. First, notice that the mapping between the model with and that without labels is not faithful since it looses information regarding inequality of ciphertexts. To formalize this intuition we give a protocol $\Pi$ and a formula $\phi$ such that ${\overline{\Pi}}$ satisfies ${\overline{\phi}}$ (the formula that corresponds to $\phi$ in the model without labels), but for which $\Pi$ does not satisfy $\phi$. Anticipating, our example indicates that the source of problems is that $\phi$ may contain equality tests between ciphertexts, and such tests may not be translated faithfully. The counterexample is in Section \[section-main\]. The main result of the paper is a soundness theorem. We show that for a large class of security properties it is possible to carry out the proof in the model without labels and infer security properties in the model with labels. More precisely, we identify ${\mathcal{L}_2}^l$ and ${\mathcal{L}_2}$, fragments of ${\mathcal{L}_1}^l$ and ${\mathcal{L}_1}$ respectively, such that the following theorem holds. Consider an arbitrary protocol $\Pi$ and formula $\phi$ in ${\mathcal{L}_2}^l$. Let ${\overline{\phi}}$ be a formula in ${\mathcal{L}_2}$ obtained by erasing the labels that occur in $\phi$. Then, it holds that: $${\overline{\Pi}}\models{\overline{\phi}} \implies \Pi\models\phi$$ The logics ${\mathcal{L}_2}^l$ and ${\mathcal{L}_2}$ are still expressive enough to contain the secrecy and authentication formulas. The theorem and its proof are in in Section \[section-main\]. {#section-protocol} In this section we provide the syntax of protocols with labels. The presentation is adapted from [@CortierW-ESOP05]. The specification language is similar to the one of Casrul [@RT01]; it allows parties to exchange messages built from identities and randomly generated nonces using public key encryption and digital signatures. Protocols that do not use labels are obtained straightforwardly. Syntax {#section-syntax} ------ Consider an algebraic signature $\Sigma$ with the following sorts. A sort $\id$ for agent identities, sorts $\skey$, $\vkey$, $\ekey$, $\dkey$ containing keys for signing, verifying, encryption, and decryption respectively. The algebraic signature also contains sorts $\nonce$, $\labels$, $\ciphertext$, $\signature$ and $\p$ for nonces, labels, ciphertexts, signatures and pair, respectively. The sort $\labels$ is used in encryption and signatures to distinguish between different encryption/signature of the same plaintext. The sort $\term$ is a supersort containing all other sorts, except $\skey$ and $\dkey$. There are nine operations: the four operations $\ek,\dk,\sk,\vk$ are defined on the sort $\id$ and return the encryption key, decryption key, signing key, and verification key associated to the input identity. The two operations $\mathsf{ag}$ and $\mathsf{adv}$ are defined on natural numbers and return labels. As explained in the introduction, the labels are used to differentiate between different encryptions (and signatures) of the same plaintext, created by the honest agents or the adversary. We distinguish between labels for agents and for the adversary since they do not use the same randomness. The other operations that we consider are pairing, public key encryption, and signing. We also consider sets of sorted variables $\X=\X.n\cup \X.a\cup \X.c\cup \X.s$ and $\X^l=\X\cup \X.l$. Here, $\X.n, \X.a,\X.c, \X.s, \X.l$ are sets of variables of sort nonce, agent, ciphertext, signature and labels, respectively. The sets of variables $\X.a$ and $\X.n$ are as follows. If $k\in{{\mathbb N}}$ is some fixed constant representing the number of protocol participants, w.l.o.g. we fix the set of agent variables to be $\X.a={\{A_1,A_2,\ldots,A_k\}}$, and partition the set of nonce variables, by the party that generates them. Formally: $\X.n=\cup_{A\in \X.a}\X_n(A)^{}$ and $\X_n(A) = \{X^j_A \mid j\in{{\mathbb N}}\}.$ This partition avoids to specify later, for each role, which variables stand for generated nonces and which variables stand for expected nonces. Labeled messages that are sent by participants are specified using terms in ${T}^l$ $$\begin{array}{lll} {{\cal L}}&::= & X.l \mid {\mathsf{ag}(i)} \mid {\mathsf{adv}(j)}\\ {T}^l &::=& \X\mid a \mid \ek(a) \mid \dk(a) \mid \sk(a) \mid \vk(a) \mid n(a,j,s) \mid {\langle {T}^l\; , {T}^l\rangle} \mid {\{{T}^l\}_{\ek(a)}}^{{{\cal L}}} \mid {[{T}^l]_{\sk(a)}}^{{{\cal L}}} \end{array}$$ where $i,j\in{{\mathbb N}}$, $a\in \id$, $j,s\in{{\mathbb N}}$, $a\in \id$. Unlabeled messages are specified similarly as terms in the algebra ${T}$ defined by $$\begin{array}{lll} {T}&::=& \X\mid a \mid \ek(a) \mid \dk(a) \mid \sk(a) \mid \vk(a) \mid n(a,j,s) \mid {\langle {T}\; , {T}\rangle} \mid {\{{T}\}_{\ek(a)}} \mid {[{T}]_{\sk(a)}} \end{array}$$ where $a\in \id$, $j,s\in{{\mathbb N}}$, $a\in \id$. A mapping ${\overline{\cdot}}:{T}^l\rightarrow {T}$ from labeled to unlabeled terms is defined by removing the labels: ${\overline{{\{k\}_{m}}^{l}}} = {\{{\overline{k}}\}_{{\overline{m}}}}$, ${\overline{{[k]_{m}}^{l}}} = {[{\overline{k}}]_{{\overline{m}}}}$, ${\overline{f(t_1,\ldots,t_n)}} = f({\overline{t_1}},\ldots,{\overline{t_n}})$ otherwise. The mapping function is extended to sets of terms as expected. The individual behavior of each protocol participant is defined by a *role* that describes a sequence of message receptions/transmissions. A $k$-party protocol is given by $k$ such roles. The set ${\mathsf{Roles}}^l$ of roles for labeled protocol participants is defined by ${\mathsf{Roles}}^l=(({\{\init\}}\cup{T}^l)\times({T}^l\cup{\{\stop\}}))^*$. A $k$-party labeled protocol is a mapping $\Pi:[k]\to{\mathsf{Roles}}^l$, where $[k]$ denotes the set $\{1,2,\ldots,k\}$. Unlabeled roles and protocols are defined very similarly. The mapping function is extended from labeled protocols to unlabeled protocols as expected. We assume that a protocol specification is such that $\Pi(j)=((l_1^j,r^j_1),(l_2^j,r^j_2),\ldots )$, the $j$’th role in the definition of the protocol being executed by player $A_j$. Each sequence $((l_1,r_1),(l_2,r_2),\ldots )\in{\mathsf{Roles}}^l$ specifies the messages to be sent/received by the party executing the role: at step $i$, the party expects to receive a message conforming to $l_i$ and returns message $r_i$. We wish to emphasize that terms $l_i^j,r_i^j$ are not actual messages, but specify how the message that is received and the message that is output should look like. \[ex\_syntax\] The Needham-Schroeder-Lowe protocol [@lowe96breaking] is specified as follows: there are two roles $\Pi(1)$ and $\Pi(2)$ corresponding to the sender’s and receiver’s role. $$\begin{aligned} A\rightarrow B: & & {\{N_a,A\}_{\ek(B)}}\\ B\rightarrow A: & & {\{N_a,N_b,B\}_{\ek(A)}}\\ A\rightarrow B: & & {\{N_b\}_{\ek(B)}} \end{aligned}$$ $$\begin{aligned} \Pi(1) & = & (\init,{\{X^1_{A_1},A_1\}_{\ek(A_2)}}^{{\mathsf{ag}(1)}}),\;\;\;\; ({\{X^1_{A_1},X^1_{A_2},A_2\}_{\ek(A_1)}}^L, {\{X^1_{A_2}\}_{\ek(A_2)}}^{{\mathsf{ag}(1)}})\\ \Pi(2)& = & ({\{X^1_{A_1},A_1\}_{\ek(A_2)}}^{L_1}, {\{X^1_{A_1},X^1_{A_2},A_2\}_{\ek(A_1)}}^{{\mathsf{ag}(1)}}),\;\;\;\; ({\{X^1_{A_2}\}_{\ek(A_2)}}^{L_2},\stop)\end{aligned}$$ Clearly, not all protocols written using the syntax above are meaningful. In particular, some protocols might be not executable. This is actually not relevant for our result (our theorem also holds for non executable protocols). Execution Model {#section-formal} --------------- We define the execution model only for labeled protocols. The definition of the execution model for unlabeled protocols is then straightforward. If $A$ is a variable or constant of sort agent, we define its knowledge by $\kn(A) = \{\dk(A),\sk(A)\}\cup \X_n(A)$, *i.e.* an agent knows its secret decryption and signing key as well as the nonces it generates during the execution. The formal execution model is a state transition system. A *global state* of the system is given by $(\SId,f,H)$ where $H$ is a set of terms of ${T}^l$ representing the messages sent on the network and $f$ maintains the local states of all session ids $\SId$. We represent session ids as tuples of the form $(n,j,(a_1,a_2,\ldots,a_k))\in({{\mathbb N}}\times{{\mathbb N}}\times {\id}^k)$, where $n\in{{\mathbb N}}$ identifies the session, $a_1,a_2,\ldots,a_k$ are the identities of the parties that are involved in the session and $j$ is the index of the role that is executed in this session. Mathematically, $f$ is a function $f:\SId\to ([\X{\to}{T}^l]\times {{\mathbb N}}\times {{\mathbb N}}),$ where $f(\mathsf{sid})=(\sigma,i,p)$ is the local state of session $\mathsf{sid}$. The function $\sigma$ is a partial instantiation of the variables occurring in role $\Pi(i)$ and $p\in {{\mathbb N}}$ is the control point of the program. Three transitions are allowed. - $(\SId,f,H){\xrightarrow{{\mathbf{corrupt}}(a_1,\ldots,a_l)}} (\SId,f,\cup_{1\leq j\leq l}\kn(a_j)\cup H)$. The adversary corrupts parties by outputting a set of identities. He receives in return the secret keys corresponding to the identities. It happens only once at the beginning of the execution. - The adversary can initiate new sessions: $(\SId,f,H){\xrightarrow{{\mathbf{new}}(i,a_1,\ldots,a_k)}} (\SId',f',H')$ where $H'$, $f'$ and $\SId'$ are defined as follows. Let $s = |\SId| +1$, be the session identifier of the new session, where $|\SId|$ denotes the cardinality of $\SId$. $H'$ is defined by $H'= H$ and $\SId'= \SId\cup\{(s,i,(a_1,\ldots,a_k))\}$. The function $f'$ is defined as follows. - $f'({\mathsf{sid}}) = f({\mathsf{sid}})$ for every ${\mathsf{sid}}\in\SId$. - $f'(s,i,(a_1,\ldots,a_k)) = (\sigma, i, 1)$ where $\sigma$ is a partial function $\sigma:\X{\to}{T}^l$ and: $$\left\{\begin{array}{lll} \sigma(A_j) & = a_j & \quad 1\leq j\leq k \\ \sigma(X^j_{A_i}) & = n(a_i,j,s) & \quad j\in{{\mathbb N}}\end{array}\right.$$ We recall that the principal executing the role $\Pi(i)$ is represented by $A_i$ thus, in that role, every variable of the form $X^j_{A_i}$ represents a nonce generated by $A_i$. - The adversary can send messages: $(\SId,f,H){\xrightarrow{{\mathbf{send}}({\mathsf{sid}},m)}} (\SId,f',H')$ where ${\mathsf{sid}}\in\SId$, $m\in{T}^l$, $H'$, and $f'$ are defined as follows. We define $f'({\mathsf{sid}}') = f({\mathsf{sid}}')$ for every ${\mathsf{sid}}'\neq{\mathsf{sid}}$. We denote $\Pi(j)=((l^j_1,r^j_1),\ldots,(l^j_{k_j},r^j_{k_j}))$. $f({\mathsf{sid}}) = (\sigma, j, p)$ for some $\sigma, j, p$. There are two cases. - Either there exists a least general unifier $\theta$ of $m$ and $l^j_p\sigma$. Then $f'({\mathsf{sid}}) = (\sigma\cup\theta,j,p+1)$ and $H'=H\cup\{r^j_p\sigma\theta\}$. - Or we define $f'({\mathsf{sid}}) = f({\mathsf{sid}})$ and $H'= H$ (the state remains unchanged). If we denote by $\sSId = {{\mathbb N}}\times{{\mathbb N}}\times {\id}^k$ the set of all sessions ids, the set of *symbolic execution traces* is ${\mathsf{SymbTr}}^l \!=\! (\sSId\!\times\! (\sSId\!\to\! ([\X\!{\to}\! {T}^l]\!\times \!{{\mathbb N}}\!\times\! {{\mathbb N}}))\!\times\! 2^{{T}^l})^*$. The set of corresponding unlabeled symbolic execution traces is denoted by ${\mathsf{SymbTr}}$. The mapping function ${\overline{\cdot}}$ is extended as follows: if $tr = (\SId_0,f_0,H_0),\ldots,(\SId_n,f_n,H_n)$ is a trace of ${\mathsf{SymbTr}}^l$, ${\overline{tr}} = ({\overline{\SId_0}},{\overline{f_0}},{\overline{H_0}}),\ldots,({\overline{\SId_n}},{\overline{f_n}},{\overline{H_n}})\in{\mathsf{SymbTr}}$ where ${\overline{\SId_i}}$ simply equal $\SId_i$ and ${\overline{f_i}}:\sSId\!\to\! ([\X\!{\to}\! {T}]\!\times \!{{\mathbb N}}\!\times\! {{\mathbb N}}))$ with ${\overline{f_i}}({\mathsf{sid}}) = ({\overline{\sigma}},i,p)$ if $f_i({\mathsf{sid}}) = (\sigma,i,p)$ and ${\overline{\sigma}}(X) = {\overline{\sigma(X)}}$. [c]{} $$\begin{array}{p{2in}p{2in}p{2in}} \begin{prooftree} \justifies S{\vdash^l}m \using m\in S \end{prooftree} \quad & \begin{prooftree} \justifies S{\vdash^l}b,\ek(b),\vk(b) \using b\in \X.a \end{prooftree} & \mbox{Initial knowledge} \\ \\ \\ \begin{prooftree} S{\vdash^l}m_1 \quad S{\vdash^l}m_2 \justifies S{\vdash^l}{\langle m_1\; , m_2\rangle} \end{prooftree} \quad & \begin{prooftree} S{\vdash^l}{\langle m_1\; , m_2\rangle} \justifies S{\vdash^l}m_i \using i\in \{1,2\} \end{prooftree} & \mbox{Pairing and unpairing} \\ \\ \\ \begin{prooftree} S{\vdash^l}\ek(b) \quad S{\vdash^l}m \justifies S{\vdash^l}{\{m\}_{\ek(b)}}^{{\mathsf{adv}(i)}} \using i\in{{\mathbb N}}\end{prooftree} \quad & \begin{prooftree} S{\vdash^l}{\{m\}_{\ek(b)}}^{l} \quad S{\vdash^l}\dk(b) \justifies S{\vdash^l}m \end{prooftree} & \mbox{Encryption and decryption} \\ \\ \\ \begin{prooftree} S{\vdash^l}\sk(b) \quad S{\vdash^l}m \justifies S{\vdash^l}{[m]_{\sk(b)}}^{{\mathsf{adv}(i)}} \using i\in{{\mathbb N}}\end{prooftree} & \begin{prooftree} S{\vdash^l}{[m]_{\sk(b)}}^{l} \justifies S{\vdash^l}m \end{prooftree} & \mbox{Signature} \end{array}$$ \ \ \ ------------------------------------------------------------------------ \[figure-deducta\] The adversary intercepts messages between honest participants and computes new messages using the deduction relation ${\vdash^l}$ defined in Figure \[figure-deducta\]. Intuitively, $S{\vdash^l}m$ means that the adversary is able to compute the message $m$ from the set of messages $S$. All deduction rules are rather standard with the exception of the last one: The last rule states that the adversary can recover the corresponding message out of a given signature. This rule reflects capabilities that do not contradict the standard computational security definition of digital signatures, may potentially be available to computational adversaries and are important for the soundness result of [@CortierW-ESOP05]. Next, we sketch the execution model for unlabeled protocols. As above, the execution is based on a deduction relation ${\vdash}$ that captures adversarial capabilities. The deduction rules that define ${\vdash}$ are obtained from those of ${\vdash}^l$ (Figure \[figure-deducta\]) as follows. The sets of rules *Initial knowledge* and *Pairing and unpairing* in are kept unchanged (replacing ${\vdash^l}$ by ${\vdash}$, of course). For encryption and signatures we suppress the labels ${\mathsf{adv}(i)}$ and $l$ in the encryption function $\{\_\}^\__\_$ and the signature function ${[\_]_{\_}}^\_$ for rules *Encryption and decryption* and rules *Signature*. That is, the rules for encryption are: $$\begin{prooftree} \quad S{\vdash}\ek(b) \quad S{\vdash}m \justifies S{\vdash}{\{m\}_{\ek(b)}} \end{prooftree} \quad \quad \begin{prooftree} S{\vdash}{\{m\}_{\ek(b)}} \quad S{\vdash}\dk(b) \justifies S{\vdash}m \end{prooftree} \quad$$ and those for signatures are: $$\begin{prooftree} S{\vdash}\sk(b) \quad S{\vdash}m \justifies S{\vdash}{[m]_{\sk(b)}} \end{prooftree} \quad \quad \begin{prooftree} S{\vdash}{[m]_{\sk(b)}} \justifies S{\vdash}m \end{prooftree} \quad$$ We use the deduction relations to characterize the set of valid execution traces. We say that the trace $(\SId_1,f_1,H_1), \ldots, (\SId_n,f_n,H_n)$ is *valid* if the messages sent by the adversary can be computed by Dolev-Yao operations. More precisely, we require that in a valid trace whenever $(\SId_i,f_i,H_i){\xrightarrow{{\mathbf{send}}(s,m)}} (\SId_{i+1},f_{i+1},H_{i+1})$, we have $H_i{\vdash^l}m$. Given a protocol $\Pi$, the set of valid symbolic execution traces is denoted by $\Exec(\Pi)$. The set $\Exec({\overline{\Pi}})$ of execution traces in the model without labels is defined similarly. We thus require that every sent message $m'$ satisfies ${\overline{H_i}} {\vdash}m'$. Playing with the Needham-Schroeder-Lowe protocol described in Example \[ex\_syntax\], an adversary can corrupt an agent $a_3$, start a new session for the second role with players $a_1,a_2$ and send the message ${\{n(a_3,1,1),a_1\}_{\ek(a_2)}}^{{\mathsf{adv}(1)}}$ to the player of the second role. The corresponding valid trace execution is: $$\begin{gathered} \!\!\!\!\! (\emptyset,f_1,\emptyset){\xrightarrow{{\mathbf{corrupt}}(a_3)}}\left(\emptyset,f_1,\kn(a_3)\right) {\xrightarrow{{\mathbf{new}}(2,a_1,a_2)}}\\ \left(\{{\mathsf{sid}}_1\},f_2,\kn(a_3)\right) {\xrightarrow{{\mathbf{send}}({\mathsf{sid}}_1,{\{n_3,a_1\}_{\ek(a_2)}}^{{\mathsf{adv}(1)}})}} \\ \left(\{{\mathsf{sid}}_1\},f_3,\kn(a_3)\cup \{{\{n_3,n_2,a_2\}_{\ek(a_1)}}^{{\mathsf{ag}(1)}}\}\right), \end{gathered}$$ where ${\mathsf{sid}}_1 = (1,2,(a_1,a_2))$, $n_2 = n(a_2,1,1)$, $n_3 = n(a_3,1,1)$, and $f_2,f_3$ are defined as follows: $f_2({\mathsf{sid}}_1) = (\sigma_1,2,1)$, $f_3({\mathsf{sid}}_1) = (\sigma_2,2,2)$ where $\sigma_1(A_1) = a_1$, $\sigma_1(A_2) = a_2$, $\sigma_1(X^1_{A_2}) = n_2$, and $\sigma_2$ extends $\sigma_1$ by $\sigma_2(X^1_{A_1}) = n_3$ and $\sigma_2(L_1) = {\mathsf{adv}(1)}$. Relating the labeled and unlabeled execution models {#sec:relating} --------------------------------------------------- First notice that by induction on the deduction rules, it can be easily shown that whenever a message is deducible, then the corresponding unlabeled message is also deducible. Formally, we have the following lemma. \[lem1\] $S {\vdash^l}m \Rightarrow {\overline{S}} {\vdash}{\overline{m}}$ Note that our main result holds for any deduction rules provided this lemma holds. In particular, the rule $$\begin{prooftree} S{\vdash^l}{[m]_{\sk(b)}}^{l} \justifies S{\vdash^l}m \end{prooftree}$$ and the corresponding unlabeled rule might be removed. Based on the above property we show that whenever a trace corresponds to an execution of a protocol, the corresponding unlabeled trace corresponds also to an execution of the corresponding unlabeled protocol. \[lem4\] $ tr \in \Exec(\Pi) \Rightarrow {\overline{tr}} \in \Exec({\overline{\Pi}}). $ The key argument is that only pattern matching is performed in protocols and when a term with labels matches some pattern, the unlabeled term matches the corresponding unlabeled pattern. The proof is done by induction on the length of the trace. - Let $tr = {\mbox{$(SId_{0}, f_{0}, H_{0})$}}$, where $\SId_0$ and $H_0$ are empty sets. We have ${\overline{H_0}} = H_0$. $f_0$ is defined nowhere, and so is ${\overline{f_0}}$. Clearly, ${\overline{tr}} = (\SId_0, {\overline{f_0}}, {\overline{H_0}})$ is in $\Exec({\overline{\Pi}})$. - Let $tr \in \Exec(\Pi)$, $tr = e_0, ..., e_n = {\mbox{$(SId_{0}, f_{0}, H_{0})$}}, ..., {\mbox{$(SId_{n}, f_{n}, H_{n})$}}$, such that ${\overline{tr}} \in \Exec({\overline{\Pi}})$. We have to show that if $tr' = tr, {\mbox{$(SId_{n+1}, f_{n+1}, H_{n+1})$}} \in \Exec(\Pi)$, then we have ${\overline{tr'}} \in \Exec({\overline{\Pi}})$. There are three possible operations. 1. $corrupt(a_1, ..., a_k)$. It means that $tr = {\mbox{$(SId_{0}, f_{0}, H_{0})$}}, {\mbox{$(SId_{1}, f_{1}, H_{1})$}}$. In this case, we have $\SId_1 = \SId_0 = \emptyset$, $f_1 = f_0$ and $H_1 = H_0 \cup \bigcup_{1 \leq i \leq k} \kn(a_i)$. We can conclude that ${\overline{tr}} = (\SId_0, {\overline{f_0}}, {\overline{H_0}}),(\SId_1, {\overline{f_1}}, {\overline{H_1}})$ is in $\Exec({\overline{\Pi}})$, because there are no labels in $H_1$ and $f_1$ is still not defined. 2. $new(i, a_1, ..., a_k)$. No labels are involved in this operation. The extension made to $f_n$ is the same as is made to ${\overline{f_n}}$. Neither $H_n$ nor ${\overline{H_n}}$ are modified. ${\overline{tr'}} = {\overline{tr}}, (\SId_{n+1}, {\overline{f_{n+1}}}, {\overline{H_{n+1}}})$ is a valid trace. 3. $send(s, m)$. First, we have to be sure that if $m$ can be deduced from $H_n$, then ${\overline{m}}$ can be deduced from ${\overline{H_n}}$. This is Lemma \[lem1\]. Note that $\SId_n = \SId_{n+1}$ thus ${\overline{\SId_n}} = {\overline{\SId_{n+1}}}$. Let $f_n(s)=(\sigma, i, p)$ and $\Pi(i) = (..., (l_p, r_p), ...)$. We have two cases. - Either there is a substitution $\theta$ with $m = l_p\sigma\theta$. Then $f_{n+1}(s) = (\sigma \cup \theta, i, p+1)$. Thus ${\overline{f_n}}(s) = ({\overline{\sigma}}, i, p)$ and ${\overline{f_{n+1}}}(s) = ({\overline{\sigma}} \cup {\overline{\theta}}, i, p+1)$. By induction hypothesis, ${\overline{tr}}$ is a valid trace. From $m = l_p\sigma\theta$ follows ${\overline{m}} = {\overline{l_p}} {\overline{\sigma}}{\overline{\theta}}$. We conclude that ${\overline{tr}}, (\SId_{n+1}, {\overline{f_{n+1}}}, {\overline{H_{n+1}}}) = {\overline{tr'}}$ is a valid trace, thus a member of $\Exec({\overline{\Pi}})$. - Or no substitution $\theta$ with $m = l_p\sigma\theta$ exists. Then $tr' = e_0, ..., e_n, e_{n+1}$ with $e_n = e_{n+1}$. We must show that it is always possible to construct a message $m' \in {T}$, such that there exists no substitution $\theta'$ with $m' = {\overline{l_p}} {\overline{\sigma}} \theta'$. Then, from the validity of $tr'$ and ${\overline{tr}}$ we can deduce the validity of ${\overline{tr'}}$, because ${\overline{e_n}} = {\overline{e_{n+1}}}$. Either there exists no substitution $\theta'$ such that ${\overline{m}} = {\overline{l_p}} {\overline{\sigma}} \theta'$. In that case, we choose $m' = {\overline{m}}$. Or let $\theta'$ be a substitution such that ${\overline{m}} = {\overline{l_p}} {\overline{\sigma}} \theta'$. Then the matching for $m$ fails because of labels. This can be shown by contradiction. Assume $m$ contain no label, *i. e.* $m$ does not contain subterms of the form $\{t\}^l_{\ek(a_i)}$ or $[t]^l_{\sk(a_i)}$, $t \in {T}$. In that case, we have ${\overline{m}} = m$ by definition. &gt;From ${\overline{m}} = {\overline{l_p}} {\overline{\sigma}} \theta'$, we deduce that $m = l_p\sigma\theta'$, contradiction. We deduce that ${\overline{m}}$ contains some subterm of the form $\{t\}_{\ek(a_i)}$ or $[t]_{\sk(a_i)}$. The fact ${\overline{m}} = {\overline{l_p}} {\overline{\sigma}} \theta'$ implies that ${\overline{l_p}}$ has to contain one of the following subterms: $\{t'\}_{\ek(A_i)}$, $[t']_{\sk(A_i)}$ with $t' \in T$ or, a variable of sort ciphertext or signature. Then, we choose $m' = a$ for some agent identity $a\in\X.a$. The term $a$ is deducible from ${\overline{H_n}}$. Now, the matching of $m'$ with ${\overline{l_p}}$ always fails, either because of the encryption or signature occurring in ${\overline{l_p}}$ or because of type mismatch for a variable of type ciphertext or signature in ${\overline{l_p}}$. {#section-alogic} In this section we define a logic for specifying security properties. We then show that the logic is quite expressive and, in particular, it can be used to specify rather standard secrecy and authenticity properties. Preliminary definitions {#section-Preliminary} ----------------------- To a trace $tr = e_1, ..., e_n = {\mbox{$(SId_{1}, f_{1}, H_{1})$}}, ..., {\mbox{$(SId_{n}, f_{n}, H_{n})$}} \in {\mathsf{SymbTr}}$ we associate its set of indices ${\mathcal{I}}(tr) = \{i {\;|\;}e_i \mbox{ appears in the trace } tr \}$. We also define the set of local states ${\mbox{$\mathcal{LS}$}}_{i,p}(tr)$ for role $i$ at step $p$ that appear in trace $tr$ by $ {\mbox{$\mathcal{LS}$}}_{i, p}(tr) = \{(\sigma, i, p) {\;|\;}\exists s \in \SId_k, k \in {\mathcal{I}}(tr), \mbox{ such that } f_k(s) = (\sigma, i, p) \}. $ We assume an infinite set ${\mathit{Sub}}$ of meta-variables for substitutions. We extend the term algebra to allow substitution application. More formally, let ${T_{\mathit{Sub}}}^l$ be the algebra defined by: $$\begin{array}{lll} {{\cal L}}&::= & \varsigma(x_l) \mid {\mathsf{ag}(i)} \mid {\mathsf{adv}(j)} \\ {T_{\mathit{Sub}}}^l &::=& \varsigma(x) \mid a \mid \ek(a) \mid \dk(a) \mid \sk(a) \mid \vk(a) \mid {\langle {T_{\mathit{Sub}}}^l\; , {T_{\mathit{Sub}}}^l\rangle} \mid {\{{T_{\mathit{Sub}}}^l\}_{\ek(a)}}^{{{\cal L}}} \mid {[{T_{\mathit{Sub}}}^l]_{\sk(a)}}^{{{\cal L}}} \end{array}$$ where $x_l\in\X.l$, $\varsigma\in{\mathit{Sub}}$, $i,j\in{{\mathbb N}}$, $x\in\X$, $a\in \id$. The unlabeled algebra ${T_{\mathit{Sub}}}$ is defined similarly. The mapping function between the two algebras is defined by: ${\overline{\varsigma(x)}}= \varsigma(x)$, ${\overline{{\{k\}_{m}}^{l}}} = {\{{\overline{k}}\}_{{\overline{m}}}}$, ${\overline{{[k]_{m}}^{l}}} = {[{\overline{k}}]_{{\overline{m}}}}$, ${\overline{f(t_1,\ldots,t_n)}} = f({\overline{t_1}},\ldots,{\overline{t_n}})$ otherwise. Security Logic {#section-security} -------------- In this section we describe a logic for security properties. Besides standard propositional connectors, the logic has a predicate to specify honest agents, equality tests between terms, and existential and universal quantifiers over the local states of agents. The formulas of the logic ${\mathcal{L}_1}^l$ are defined as follows: $$\begin{array}{lll} F(tr) & ::= & NC(tr, t_1) {\;|\;}(t_1 = t_2) {\;|\;}\neg F(tr) {\;|\;}F(tr) \wedge F(tr) {\;|\;}F(tr) \vee F(tr) {\;|\;}\\ & & {\mbox{$\forall\mathcal{LS}_{i, p}(tr).\varsigma\;$}}F(tr) {\;|\;}{\mbox{$\exists\mathcal{LS}_{i, p}(tr).\varsigma\;$}}F(tr) \end{array}$$ where $tr$ is a parameter of the formula, $i,p\in {{\mathbb N}}$, $\varsigma\in{\mathit{Sub}}$, $t_1$ and $t_2$ are terms of ${T_{\mathit{Sub}}}^l$. Note that formulas are parametrized by a trace $tr$. As usual, we may use $\phi_1\rightarrow\phi_2$ as a shortcut for $\neg\phi_1\vee\phi_2$. We similarly define the corresponding unlabeled logic ${\mathcal{L}_1}$: the tests $(t_1 = t_2)$ are between unlabeled terms $t_1,t_2$ over ${T}_\mathit{sub}$. The mapping function ${\overline{\cdot}}$ is extended as expected. In particular ${\overline{NC(tr, t)}} = NC({\overline{tr}}, {\overline{t}})$, ${\overline{(t_1 = t_2)}} = ({\overline{t_1}} = {\overline{t_2}})$, ${\overline{{\mbox{$\forall\mathcal{LS}_{i, p}(tr).\varsigma\;$}}F(tr)}} = {\mbox{$\forall\mathcal{LS}_{i, p}({\overline{tr}}).\varsigma\;$}}{\overline{F(tr)}}$ and ${\overline{{\mbox{$\exists\mathcal{LS}_{i, p}(tr).\varsigma\;$}}F(tr)}} = {\mbox{$\exists\mathcal{LS}_{i, p}({\overline{tr}}).\varsigma\;$}}{\overline{F(tr)}}$. Here, the predicate $NC(tr, t)$ of arity 2 is used to specify non corrupted agents. The quantifications ${\mbox{$\forall\mathcal{LS}_{i, p}(tr).\varsigma\;$}}$ and ${\mbox{$\exists\mathcal{LS}_{i, p}(tr).\varsigma\;$}}$ are over the local states in the trace that correspond to agent $i$ at control point $p$. The semantics of our logic is defined for *closed* formula as shown in Figure \[interpretation\]. $$\begin{aligned} {\mbox{$[\![ NC(tr, t) ]\!]$}} & = & \left\{ \begin{array}{l l} 1 & \mbox{if } t\in\id \mbox{ and $t$ does not appear in a corrupt action, \textit{i.e.} } \\ & tr=e_1, e_2, ..., e_n \mbox{ and }\\ & \forall a_1,\ldots,a_k \mbox{, s.t. } e_1{\xrightarrow{{\mathbf{corrupt}}(a_1,\ldots,a_k)}}e_2, t\neq a_i \\ 0 & \mbox{otherwise} \end{array} \right. \\ {\mbox{$[\![ (t_1 = t_2) ]\!]$}} & = & \left\{ \begin{array}{l l} 1 & \mbox{if } t_1=t_2 \mbox{ (syntactic equality)}\\ 0 & \mbox{otherwise} \end{array} \right. \\ {\mbox{$[\![ \neg F(tr) ]\!]$}} & = & \neg {\mbox{$[\![ F(tr) ]\!]$}}\\ {\mbox{$[\![ F_1(tr)\wedge F_2(tr) ]\!]$}} & = & {\mbox{$[\![ F_1(tr) ]\!]$}}\wedge {\mbox{$[\![ F_2(tr) ]\!]$}}\\ {\mbox{$[\![ F_1(tr)\vee F_2(tr) ]\!]$}} & = & {\mbox{$[\![ F_1(tr) ]\!]$}}\vee {\mbox{$[\![ F_2(tr) ]\!]$}}\\ {\mbox{$[\![ {\mbox{$\forall\mathcal{LS}_{i, p}(tr).\varsigma\;$}}F(tr) ]\!]$}} & = & \left\{ \begin{array}{l l} 1 & \mbox{if } \forall (\theta, i, p) \in {\mbox{$\mathcal{LS}$}}_{i, p}(tr), \mbox{ we have } {\mbox{$[\![ F(tr)[\theta/\varsigma] ]\!]$}} = 1, \\ 0 & \mbox{otherwise}. \end{array} \right. \\ {\mbox{$[\![ {\mbox{$\exists\mathcal{LS}_{i, p}(tr).\varsigma\;$}}F(tr) ]\!]$}} & = & \left\{ \begin{array}{l l} 1 & \mbox{if } \exists (\theta, i, p) \in {\mbox{$\mathcal{LS}$}}_{i, p}(tr), \mbox{ s.t. } {\mbox{$[\![ F(tr)[\theta/\varsigma] ]\!]$}} = 1, \\ 0 & \mbox{otherwise}. \end{array} \right.\end{aligned}$$ ------------------------------------------------------------------------ \[interpretation\] Next we define when a protocol $\Pi$ satisfies a formula $\phi\in{\mathcal{L}_1}^l$. The definition for the unlabeled execution model is obtained straightforwardly. Informally, a protocol $\Pi$ satisfies $\phi$ if $\phi(tr)$ is true for all traces $tr$ of $\Pi$. Formally: Let $\phi$ be a formula and $\Pi$ be a protocol. We say that $\Pi$ satisfies security property $\phi$, and write $\Pi \models \phi$ if for any trace $tr\in\Exec(\Pi)$, ${\mbox{$[\![ \phi(tr) ]\!]$}}=1$. Abusing notation, we occasionally write $\phi$ for the set $\{tr\mid {\mbox{$[\![ \phi(tr) ]\!]$}}=1\}$. Then, $\Pi\models\phi$ precisely when $\Exec(\Pi)\subseteq\phi$. Examples of security properties {#section-Examples} ------------------------------- In this section we exemplify the use of the logic by specifying secrecy and authenticity properties. ### A secrecy property Let $\Pi(1)$ and $\Pi(2)$ be the sender’s and receiver’s role of a two-party protocol. To specify our secrecy property we use a standard encoding. Namely, we add a third role to the protocol, $\Pi(3) = (X^1_{A_3}, stop)$, which can be seen as some sort of witness. Informally, the definition of the secrecy property $\phi_s$ states that, for two non corrupted agents $A_1$ and $A_2$, where $A_1$ plays role $\Pi(1)$ and $A_2$ plays role $\Pi(2)$, a third agent playing role $\Pi(3)$ cannot gain any knowledge on nonce $X^1_{A_1}$ sent by role $\Pi(1)$. $$\phi_s(tr) ={\mbox{$\forall\mathcal{LS}_{1,1}(tr).\varsigma\;$}} {\mbox{$\forall\mathcal{LS}_{3,2}(tr).\varsigma'\;$}} [ NC(tr, \varsigma(A_1)) \wedge NC(tr, \varsigma(A_2)) \rightarrow \neg (\varsigma'(X^1_{A_3}) = \varsigma(X^1_{A_2})) ]$$ ### An authentication property Consider a two role protocol, such that role 1 finishes its execution after $n$ steps and role 2 finishes its execution after $p$ steps. For this kind of protocols we give a variant of the week agreement property [@lowe97hierarchy]. Informally, this property states that whenever an instantiation of role 2 finishes, there exists an instantiation of role 1 that has finished and they agree on some value for some variable and they have indeed talked to each other. In our example we choose this variable to be $X^1_{A_1}$. Note that we capture that some agent has finished its execution by quantifying appropriately over the local states of that agent. More precisely, we quantify only over the states where it indeed has finished its execution. $$\begin{gathered} \phi_a(tr) = {\mbox{$\forall\mathcal{LS}_{2,p}(tr).\varsigma\;$}} {\mbox{$\exists\mathcal{LS}_{1,n}(tr).\varsigma'\;$}}\\ [NC(tr, \varsigma(A_1))\wedge NC(tr, \varsigma'(A_2)) \rightarrow (\varsigma(X^1_{A_1}) = \varsigma'(X^1_{A_1})) \wedge (\varsigma(A_2) = \varsigma'(A_2)) \wedge (\varsigma(A_1) = \varsigma'(A_1))]\end{gathered}$$ Notice that although in its current version our logic is not powerful enough to specify stronger versions of agreement (like injective or bijective agreement), it could be appropriately extended to deal with this more complex forms of authentication. {#section-main} Recall that our goal is to prove that ${\overline{\Pi}}\models {\overline{\phi}}\Rightarrow \Pi\models \phi$. However, as explained in the introduction this property does not hold in general. The following example sheds some light on the reasons that cause the desired implication to fail. \[ex:counter\] Consider the first step of some protocol where $A$ sends a message to $B$ where some part is intended for some third agent. $$\begin{array}{rcl} A& \rightarrow B:& \{N_a, \{N_a\}_{\ek(C)}, \{N_a\}_{\ek(C)}\}_{\ek(B)} \end{array}$$ The specification of the programs of $A$ and $B$ that corresponds to this first step is as follows (in the definition below $C^1_{A_2}$ and $C^2_{A_2}$ are variables of sort ciphertext). $$\begin{array}{lll} \Pi(1) & = & (\init, {\{\langle X^1_{A_1},\langle {\{X^1_{A_1}\}_{\ek(A_3)}}^{{\mathsf{ag}(1)}}, {\{X^1_{A_1}\}_{\ek(A_3)}}^{{\mathsf{ag}(2)}}\rangle\rangle\}_{\ek(A_2)}}^{{\mathsf{ag}(3)}})\\[2ex] \Pi(2) & = & ({\{\langle X^1_{A_1},\langle C^1_{A_2}, C^2_{A_2}\rangle\rangle\}_{\ek(A_2)}}^{L}, \stop) \end{array}$$ We assume that $A$ generates twice the message $\{N_a\}_{\ek(C)}$. Notice that we stop the execution of $B$ after it receives the first message since this is sufficient for our purpose, but its execution might be continued to form a more realistic example. Consider the security property $\phi_1$ that states that if $A$ and $B$ agree on the nonce $X^1_{A_1}$ then $B$ should have received twice the same ciphertext. $$\begin{gathered} \phi_1(tr) = {\mbox{$\forall\mathcal{LS}_{1,2}(tr).\varsigma\;$}} {\mbox{$\forall\mathcal{LS}_{2,2}(tr).\varsigma'\;$}}\\ NC(tr, \varsigma(A_1)) \wedge NC(tr, \varsigma(A_2)) \wedge (\varsigma(X^1_{A_1}) = \varsigma'(X^1_{A_1})) \rightarrow (\varsigma'(C^1_{A_2}) = \varsigma'(C^2_{A_2}))\end{gathered}$$ This property clearly does not hold for any normal execution of the labeled protocol since $A$ always sends ciphertexts with distinct labels. Thus $\Pi\not\models \phi_1$. On the other hand, one can show that we have ${\overline{\Pi}}\models {\overline{\phi_1}}$ in the unlabeled execution model. Intuitively, this holds because if $A$ and $B$ are honest agents and agree on $X^1_{A_1}$, then the message received by $B$ has been emitted by $A$ and thus should contain identical ciphertexts (after having removed their labels). Logic ${\mathcal{L}_2}^l$ {#section-logic} ------------------------- The counterexample above relies on the fact that two ciphertexts that are equal in the model without labels may have been derived from distinct ciphertexts in the model with labels. Hence, it may be the case that although ${\overline{t_1}}\neq {\overline{t_2}}\Rightarrow t_1\neq t_2$, the contrapositive implication ${\overline{t_1}}={\overline{t_2}}\Rightarrow t_1= t_2$ does not hold, which in turn entails that formulas that contain equality tests between ciphertexts may be true in the model without labels, but false in the model with labels. In this section we identify a fragment of ${\mathcal{L}_1}^l$, which we call ${\mathcal{L}_2}^l$ where such tests are prohibited. Formally, we avoid equality tests between arbitrary terms by forbidding arbitrary negation over formulas and allowing equality tests only between *simple* terms. A term $t$ is said *simple* if $t\in \X.a\cup\X.n$ or $t=a$ for some $a\in \id$ or $t= n(a, j, s)$ for some $a\in \id$, $j,s\in{{\mathbb N}}$. An important observation is that for any simple term $t$ it holds that ${\overline{t}}=t$. The formulas of the logic ${\mathcal{L}_2}^l$ are defined as follows: $$\begin{array}{lll} F(tr) & ::= & NC(tr, t_1) {\;|\;}\neg NC(tr, t_1) \mid F(tr) \wedge F(tr) {\;|\;}F(tr) \vee F(tr) \mid (t_1 \neq t_2) {\;|\;}(u_1 = u_2) \mid\\ & & {\mbox{$\forall\mathcal{LS}_{i,p}(tr).\varsigma\;$}}F(tr) {\;|\;}{\mbox{$\exists\mathcal{LS}_{i,p}(tr).\varsigma\;$}}F(tr), \\ \end{array}$$ where $tr\in {\mathsf{SymbTr}}$ is a parameter, $i,p \in {{\mathbb N}}$, $t_1,t_2\in{T_{\mathit{Sub}}}^l$ and $u_1,u_2$ are simple terms. Since simple terms also belong to ${T_{\mathit{Sub}}}^l$, both equality and inequality tests are allowed between simple terms. The corresponding unlabeled logic ${\mathcal{L}_2}$ is defined as expected. Note that ${\mathcal{L}_2}^l\subset {\mathcal{L}_1}^l$ and ${\mathcal{L}_2}\subset {\mathcal{L}_1}$. Theorem {#section-theorem} ------- Informally, our main theorem says that to verify if a protocol satisfies some security formula $\phi$ in logic ${\mathcal{L}_2}^l$, it is sufficient to verify that the unlabeled version of the protocol satisfies ${\overline{\phi}}$. Let $\Pi$ be a protocol and $\phi\in{\mathcal{L}_2}^l$, then ${\overline{\Pi}}\models{\overline{\phi}} \Rightarrow \Pi\models\phi$. Assume ${\overline{\Pi}}\models{\overline{\phi}}$. We have to show that for any trace $tr\in\Exec(\Pi)$, ${\mbox{$[\![ \phi(tr) ]\!]$}}=1$. &gt;From lemma \[lem4\] it follows that ${\overline{tr}} \in \Exec({\overline{\Pi}})$, thus ${\mbox{$[\![ {\overline{\phi}}({\overline{tr}}) ]\!]$}}=1$, since ${\overline{\Pi}}\models{\overline{\phi}}$. Thus, it is sufficient to show that ${\mbox{$[\![ {\overline{\phi}}({\overline{tr}}) ]\!]$}} \Rightarrow {\mbox{$[\![ \phi(tr) ]\!]$}}$. The following lemma offers the desired property. \[lem3\] Let $\phi(tr)\in{\mathcal{L}_2}^l$ for some $tr\in{\mathsf{SymbTr}}$, ${\mbox{$[\![ {\overline{\phi}}({\overline{tr}}) ]\!]$}}$ implies ${\mbox{$[\![ \phi(tr) ]\!]$}}$. The proof of the lemma is by induction on the structure of $\phi(tr)$. - $\phi(tr) = NC(tr, t)$ or $\phi(tr) = \neg NC(tr, t)$. ${\mbox{$[\![ NC(tr, t) ]\!]$}}=1$, if and only if $t\in \id$ and $t$ does not occur in a ${\mathbf{corrupt}}$ event for the trace $tr$. This is equivalent to ${\overline{t}}\in \id$ and ${\overline{t}}$ does not occur in a ${\mathbf{corrupt}}$ event for the trace ${\overline{tr}}$. Thus ${\mbox{$[\![ NC(tr, t) ]\!]$}}=1$ if and only if ${\mbox{$[\![ {\overline{NC(tr, t)}} ]\!]$}} = {\mbox{$[\![ NC({\overline{tr}}, {\overline{t}}) ]\!]$}}=1$. - $\phi(tr) = (t_1\neq t_2)$. We have that ${\overline{\phi}}({\overline{tr}}) = ({\overline{t_1}} \neq {\overline{t_2}})$ holds. Assume by contradiction that $\phi(tr)$ does not hold, *i.e* $t_1=t_2$. This implies ${\overline{t_1}} = {\overline{t_2}}$, contradiction. - $\phi(tr) = (u_1 = u_2)$ with $u_1,u_2$ simple terms. We have that ${\overline{\phi}}({\overline{tr}}) = ({\overline{u_1}} = {\overline{u_2}})$ holds. Since $u_1$ and $u_2$ are simple terms, we have ${\overline{u_i}} = u_i$, thus $u_1 = u_2$. We conclude that $\phi(tr)$ holds. - The cases $\phi(tr) = \phi_1(tr) \vee \phi_2(tr)$ or $\phi(tr) = \phi_1(tr) \wedge \phi_2(tr)$ are straightforward. - $\phi(tr) = {\mbox{$\forall\mathcal{LS}_{i}(tr).\varsigma\;$}} F(tr)$. If ${\overline{\phi}}({\overline{tr}})$ holds, this means that for all $(\theta, i, p)) \in {\mbox{$\mathcal{LS}$}}_{i, p}({\overline{tr}})$, ${\mbox{$[\![ {\overline{F}}({\overline{tr}})[\theta/\varsigma] ]\!]$}} = 1$. Let $(\theta', i, p) \in {\mbox{$\mathcal{LS}$}}_{i, p}(tr)$. We consider ${\mbox{$[\![ F(tr)[\theta'/\varsigma] ]\!]$}}$. Since $tr \in \Exec(\Pi)$ implies ${\overline{tr}} \in \Exec({\overline{\Pi}})$ (Lemma \[lem4\]), we have $({\overline{\theta'}}, i, p) \in {\mbox{$\mathcal{LS}$}}_{i, p}({\overline{tr}})$. By induction hypothesis, ${\mbox{$[\![ {\overline{F}}({\overline{tr}})[{\overline{\theta'}}/\varsigma] ]\!]$}} = 1$ implies that ${\mbox{$[\![ F(tr)[\theta'/\varsigma] ]\!]$}} = 1$. It follows that $$\forall (\theta', i, p) \in {\mbox{$\mathcal{LS}$}}_{i, p}(tr)\; {\mbox{$[\![ F(tr)[\theta'/\varsigma] ]\!]$}} = 1.$$ Thus, $\phi(tr)$ holds. - $\phi(tr) = {\mbox{$\exists\mathcal{LS}_{i}(tr).\varsigma\;$}} F(tr)$. If ${\overline{\phi}}({\overline{tr}})$ holds, this means that there exists $(\theta, i, p)) \in {\mbox{$\mathcal{LS}$}}_{i, p}({\overline{tr}})$, such that ${\mbox{$[\![ {\overline{F}}({\overline{tr}})[\theta/\varsigma] ]\!]$}} = 1$. By definition of the mapping function, there exists $(\theta', i, p) \in {\mbox{$\mathcal{LS}$}}_{i, p}(tr)$ such that ${\overline{\theta'}}=\theta$. By induction hypothesis, ${\mbox{$[\![ F(tr)[\theta'/\varsigma] ]\!]$}} = 1$. Thus there exists $\theta'$, such that ${\mbox{$[\![ F(tr)[\theta'/\varsigma] ]\!]$}} = 1$. Thus, $\phi(tr)$ holds. {#section-discussion} We conclude with a brief discussion of two interesting aspects of our result. First, as mentioned in the introduction, the only property needed for our main theorem to hold is that the underlying deduction system satisfies the condition in Lemma \[lem1\], that is $S {\vdash^l}m \Rightarrow {\overline{S}} {\vdash}{\overline{m}}$. In fact, an interesting result would be to prove a more abstract and modular version of our theorem. Secondly, a natural question is whether the converse of our main theorem holds. We prove that this is not the case. More precisely, we show that there exists a protocol $\Pi$ and a property $\phi$ such that $\Pi\models \phi$ but ${\overline{\Pi}}\not\models {\overline{\phi}}$. Let $\Pi$ be the protocol defined in Example \[ex:counter\]. Consider a security property $\phi_2$ that states on the contrary that whenever $A$ and $B$ agree on the nonce $X^1_{A_1}$ then $B$ should have received two distinct ciphertexts. Formally: $$\begin{gathered} \phi_2(tr) = {\mbox{$\forall\mathcal{LS}_{1,2}(tr).\varsigma\;$}} {\mbox{$\forall\mathcal{LS}_{2,2}(tr).\varsigma'\;$}}\\ NC(tr, \varsigma(A_1)) \wedge NC(tr, \varsigma(A_2)) \wedge (\varsigma(X^1_{A_1}) = \varsigma'(X^1_{A_1})) \rightarrow (\varsigma'(C^1_{A_2}) \neq \varsigma'(C^2_{A_2}))\end{gathered}$$ where $C^1_{A_2}$ and $C^2_{A_2}$ are variables of sort ciphertext. This property clearly does not hold for any honest execution of the unlabeled protocol since $A$ always sends twice the same ciphertext, and thus ${\overline{\Pi}}\not\models{\overline{\phi_2}}$. On the other hand however, one can show that this property holds for labeled protocols since, if $A$ and $B$ are honest agents and agree on $X^1_{A_1}$, it means that the message received by $B$ has been emitted by $A$ and thus contains two distinct ciphertexts. Thus, $\Pi\models \phi_2$. We conclude that, in general, $\Pi\models \phi$ does not imply ${\overline{\Pi}}\models{\overline{\phi}}$.

--- abstract: 'We briefly review the string technology needed to calculate Yang-Mills amplitudes at two loops, and we apply it to the evaluation of two-loop vacuum diagrams.' address: | NORDITA[^1]\ Blegdamsvej 17, DK–2100 Copenhagen Ø, Denmark\ Dipartimento di Fisica, Politecnico di Torino [^2]\ Corso Duca degli Abruzzi 24, I–10129 Torino, Italy\ author: - 'Lorenzo Magnea$^*$ and Rodolfo Russo$^{\dagger}$' title: 'Two-loop gluon diagrams from string theory' --- Introduction {#introduction .unnumbered} ============ It is well known [@lm:all] that string theory is a powerful calculational tool for the evaluation of tree and one-loop amplitudes in Yang-Mills theory and gravity. During the past two years considerable progress has been made towards the extension of string-inspired techniques to more than one loop [@lm:us]. The calculation of the two-loop Yang-Mills vacuum diagrams, presented here, is the first simple application of this formalism to gauge theories beyond one loop. The general features of string-inspired techniques remain unchanged to all orders in perturbation theory: the field theory limit is obtained by taking the string tension $T = 1/(2\pi\alpha')$ to infinity, decoupling all massive string modes; the corners of string moduli space contributing to the field theory result are those where the integrand of the string amplitude exhibits a singular behavior, and in these regions string moduli are naturally related to Schwinger proper times in field theory; finally, string-derived amplitudes are written in a form which is strongly reminiscent of the world-line formalism in field theory [@lm:sss]. A special feature of calculations done using the bosonic string is the existence of contact terms, corresponding to four-point vertices in field theory, that arise as finite remainders of tachyon exchange. These contributions are present to all orders, however at one loop they can be eliminated by performing a partial integration at the string level. This simplifies calculations, but it obscures the connection between string-derived rules and field theory. At two loops, we find that the same prescription that was used at one loop to handle tachyon exchange (a $\zeta$-function regularization) is sufficient to obtain the correct result. It should be noted that the string derivation is not tied to any special choice of regularization scheme for ultraviolet and infrared divergences. While dimensional regularization is naturally implemented and useful for practical calculations, any regularization scheme that can be applied at the level of Schwinger parameter integrals is just as natural. Thus it is meaningful to calculate vacuum diagrams, although they vanish in dimensional regularization. Multiloop gluon amplitudes {#multiloop-gluon-amplitudes .unnumbered} ========================== Let us begin by recalling the general expression for the color-ordered $h$-loop $M$-gluon planar amplitude in the open bosonic string [@lm:us], A\^[(h)]{}\_P & = & C\_h [N]{}\_0\^M \_h \_[i&lt;j]{} \^[2’ p\_ip\_j]{} \[lm:hmast\]\ & &  , where only terms linear in each polarization should be kept, and we omitted the color factor $N^h\,{\rm Tr}(\lambda^{a_1}\cdots \lambda^{a_M})$. The dimensionless string coupling constant $g_s$ is related to the $d$-dimensional gauge coupling $g_d = g \mu^{(4 - d)/2}$ by g\_s = (2’)\^[1-d/4]{} . \[lm:gs\] The fundamental ingredients of [Eq. (\[lm:hmast\])]{} are the bosonic Green function ${\cal G}^{(h)}(z_i,z_j)$ (the correlator of two scalar fields on the $h$-loop string world sheet), and the measure of integration on moduli space $[dm]_h$. Both these quantities depend only on the genus $h$ of the surface and thus represent the building blocks for the calculation of all diagrams at $h$ loops with an arbitrary number of external states. In particular, the string Green function acts as a generator of the specific world-line Green functions found for particle diagrams of different topology, to which it reduces in the appropriate corners of moduli space. Explicit expressions for the Green function, for the projective transformations $V_i(z)$, which define local coordinate systems around the punctures, and for the normalization constants $C_h$ and ${\cal N}_0$ can be found in Ref. [@lm:us]; here we want to focus on the measure of integration, since it encodes all the information needed for the derivation of the vacuum diagrams. It is given by $$\begin{aligned} [dm]_h & = & \prod_{\mu=1}^{h} \left[ \frac{dk_\mu d \xi_\mu d \eta_\mu}{k_\mu^2 (\xi_\mu - \eta_\mu)^2} ( 1- k_\mu )^2 \right] \label{lm:meas} \left[\det \left( - i \tau_{\mu \nu} \right) \right]^{-d/2} \\ & \times & \prod_{\alpha}\;' \left[ \prod_{n=1}^{\infty} ( 1 - k_{\alpha}^{n})^{-d} \prod_{n=2}^{\infty} ( 1 - k_{\alpha}^{n})^{2} \right]~. \nonumber\end{aligned}$$ The various ingredients of this formula have a geometric intrepretation on a Riemann surface of genus $h$. In particular, $\tau_{\mu \nu}$ is the period matrix, while $k_{\mu}$, $\xi_{\mu}$ and $\eta_{\mu}$ are the moduli of the surface in the Schottky parametrization [@lm:cop]; the primed product over $\alpha$ denotes a product over conjugacy classes of elements of the Schottky group, where only elements that cannot be written as powers of other elements must be included. Three of $2 h + M$ parameters $\xi_\mu$, $\eta_\mu$ and $z_i$ can be fixed using an overall projective invariance of the amplitude. The fixing of this invariance introduces the projective invariant volume element $dV_{abc}$ in [Eq. (\[lm:hmast\])]{}. Including the “multipliers” $k_\mu$, one is left with $3 h - 3 + M$ variables, the correct number of independent moduli for a Riemann surface of genus $h$ with $M$ punctures. In the field theory limit, only the region in moduli space in which the multipliers $k_\mu$ are small gives finite contributions. Yang-Mills vacuum diagrams {#yang-mills-vacuum-diagrams .unnumbered} ========================== Now we turn to the study of vacuum diagrams at two loops, specializing the formulas of the previous section to the case $M=0$, $h=2$. First, we use projective invariance to fix $\xi_1$, $\xi_2$ and $\eta_1$ to $\infty$, $1$ and $0$ respectively. Then we evalute [Eq. (\[lm:meas\])]{} in the field theory limit. To this end, we expand it in power of $k_{\mu}$, and we ignore all terms that are quadratic in one multiplier, since they correspond to the undesired exchange of a massive spin 2 state. In this approximation, [Eq. (\[lm:hmast\])]{} becomes $$\begin{aligned} \label{lm:exp1} A^0_2 & = & N^3 \, C_2 \, (2\pi)^d \int {dk_1\over k_1^2} {dk_2\over k_2^2} {d\eta_2\over (1-\eta_2)^2} \; \\ \nonumber &\times & \left[1+(d-2)(k_1+k_2)+\left((d-2)^2+d\,(1-\eta_2)^2{1+\eta_2^ 2 \over \eta_2^2}\right)k_1 k_2\right] \\ \nonumber &\times & \Bigg[\ln k_1\ln k_2 - \ln^2\eta_2 + {2\,(1-\eta_2)^2\over \eta_2} (k_1\ln k_1 + k_2\ln k_2) + \\ \nonumber & & + {4\,(1-\eta_2)^4\over \eta_2^2}\left(1 + {1+\eta_2 \over 1- \eta_2} \ln\eta_2 \right) k_1 k_2\Bigg]^{-d/2}~~~. \end{aligned}$$ The region of integration can be deduced by studying the Schottky representation of the two–annulus [@lm:scal]. In the small $k$ limit, it is given by $0\leq\sqrt{k_2}\leq \sqrt{k_1}\leq\eta_2\leq 1$. The fixed point $\eta_2$ can be interpreted as the distance between the two loops, so that we can tentatively identify the region $\eta_2\rightarrow 1$ as related to reducible diagrams, and the region $\eta_2\rightarrow 0$ as related to irreducible ones. In the region $\eta_2\rightarrow 1$, in order to isolate the contribution of massless states, we must extract from the integrand of [Eq. (\[lm:exp1\])]{} the term proportional to $k_1^{-1} k_2^{-1} (1-\eta_2)^{-1}$. Then the field theory results are recovered if one introduces the Schwinger proper times $t_i={\alpha}'\ln k_{i}$, $t_3={\alpha}' \ln(1-\eta_2)$, which must remain finite as ${\alpha}'\rightarrow 0$. It can be checked that no terms survive in the limit ${\alpha}'\rightarrow 0$, which is the stringy way to say that the reducible diagram we are considering is zero. There is however a contact interaction leftover from tachyon exchange in the limit $\eta_2\rightarrow 1$. This is obtained by isolating the term in [Eq. (\[lm:exp1\])]{} that is proportional to $k_1^{-1} k_2^{-1}(1-\eta_2)^{-2}$, and requiring that the integrand be independent of $\eta_2$ except for the tachyon double pole. The double pole is then regularized using a $\zeta$-function regularization, as \[lm:reg\] \_0 \~\_0 d x \_[n=1]{}\^n [e]{}\^[- n x]{} \~\_[n=1]{}\^1 \~(0) = -[12]{}  . The field theory contribution to the remaining integral over the proper times $t_1$ and $t_2$ is given by the term that does not depend on ${\alpha}'$, \[lm:tach\] A\^0\_2|\_[\_21]{} = - N\^3 \_0\^dt\_1 dt\_2 (t\_1 t\_2)\^[-d/2]{} . Let us now turn to the region $\eta_2\rightarrow 0$. Here it is convenient to introduce the variables $q_1 = k_2/\eta_2$, $q_2 = k_1/\eta_2$, $q_3 = \eta_2$, which are directly related to the field theory proper times by $t_i={\alpha}'\ln q_{i}$. The region of integration now takes the form $0\leq q_1\leq q_2\leq q_3\leq 1$, and the term that survives in the limit ${\alpha}'\rightarrow 0$ is given by $$\label{lm:asym} A^0_2\big|_{q_3\rightarrow 0} = \frac{g_d^2}{(4\pi)^d} N^3 d (d-2) \int\limits_0^\infty dt_2 \int\limits_0^{t_2} dt_1 \int\limits_0^{t_1} dt_3 {t_1+t_2+2t_3 \over (t_1 t_2+t_1 t_3+t_2 t_3)^{1+d/2}}.$$ The integrand of the last equation is not symmetric in the proper times $t_i$. This prevents us from rewriting the integrations of [Eq. (\[lm:asym\])]{} as indipendent, thus identifying this contribution with that of the irreducible field theory diagram with three propagators. However we expect that also as $\eta_2 \to 0$ there may be contributions with only two propagators, which might, and do, symmetrize [Eq. (\[lm:asym\])]{}. In this case the relevant contribution does not involve the regularization of a tachyon pole, but it simply comes from an integration region in which two moduli are kept very close to each other. In our case the only such region can be parametrized as $q_3 = y q_2$, where we do not associate any proper time to the variable $y$, which is kept finite. This region contributes \[lm:asym2\] A\^0\_2|\_[q\_3q\_2]{} = - N\^3 2 (d-2) \_0\^ dt\_2 \_0\^[t\_2]{} dt\_1 (2 t\_1 t\_2+t\_2\^2)\^[-d/2]{} , where the integrand can be rewritten as \[lm:id\] (2 t\_1 t\_2+t\_2\^2)\^[-d/2]{} = (t\_1 t\_2)\^[-d/2]{} + \_0\^[t\_1]{} dt\_3  . The $(-1)$ factor in [Eq. (\[lm:asym2\])]{} comes from the integration over $y$, where only the contribution from the lower limit of integration is relevant in the field theory limit. [Eq. (\[lm:asym2\])]{} thus symmetrizes [Eq. (\[lm:asym\])]{}, as well as contributing to the contact term. It is easy to check that the sum of Eqs. (\[lm:tach\]), (\[lm:asym\]) and (\[lm:asym2\]) correctly reproduces the sum of the corresponding Feynman diagrams in Yang-Mills theory. In particular, it is amusing to notice that, after doing the $t_3$ integral, the sum of [Eq. (\[lm:asym\])]{} and [Eq. (\[lm:asym2\])]{} vanishes, so that the entire result for the two-loops vacuum diagrams actually comes from the contact term arising from tachyon exchange, given by [Eq. (\[lm:tach\])]{}. Conclusions {#conclusions .unnumbered} =========== We have reviewed some of the results of string perturbation theory that lead to an efficient organization of multiloop Yang-Mills amplitudes, and we have described the simplest application of the method at two loops. The next natural step is the evaluation of the two-loop two-point function, which is expected to contain the two-loop Yang-Mills $\beta$ function, since string amplitudes lead to background field gauges. Work in this direction is in progress. See, for example, M.L. Mangano and S.J. Parke, [*Phys. Rep.*]{} [**200**]{} (1991) 301; and Z. Bern, L. Dixon and D.A. Kosower, [*Ann. Rev. Nucl. Part. Sci.*]{} [**46**]{} (1996) 109, [hep-ph/9602280]{}, and references therein. P. Di Vecchia, A. Lerda, L. Magnea and R. Marotta, [*Phys. Lett.*]{} [**B 351**]{} (1995) 445, [hep-th/9502156]{}. P. Di Vecchia, A. Lerda, L. Magnea, R. Marotta and R. Russo, [*Nucl. Phys.*]{} [**B 469**]{} (1996) 235, [hep-th/9601143]{}; in Erice, [*Theor. Phys.*]{} (1995), [hep-th/9602055]{}; proceedings of the 29th Ahrenshoop Symposium, Buckow, Germany, (August 1995), [hep-th/9602056]{}; proceedings of the Workshop on Gauge Theories, Applied Supersymmetry and Quantum Gravity, London, England, (July 1996), [hep-th 9611023]{}. C. Schubert, [*Acta Phys. Polon.*]{} [**B 27**]{} (1996) 3965, [hep-th/9610108]{}, and references therein . P. Di Vecchia, F. Pezzella, M. Frau, K. Hornfeck, A. Lerda and S. Sciuto, [*Nucl. Phys.*]{} [**B 322**]{} (1989) 317. P. Di Vecchia, A. Lerda, L. Magnea, R. Marotta and R. Russo, [*Phys. Lett.*]{} [**B 388**]{} (1996) 65, [hep-th/9607141]{}; K. Roland, [*Phys. Lett.*]{} [**B 289**]{} (1992) 148. [^1]: On leave from Università di Torino, Italy. [^2]: The work presented here is part of an ongoing collaboration with P. Di Vecchia, A. Lerda and R. Marotta

--- author: - 'Herbert K. Dreiner$^{\ast,}$, Jean-François Fortin$^{\dagger,\$,}$, Jordi Isern$^{\S,\ddag,}$ and Lorenzo Ubaldi$^{\ast,}$' bibliography: - 'Neutralinos\_Draft\_Revised.bib' date: March 2013 title: White Dwarfs constrain Dark Forces --- Introduction {#Intro} ============ White dwarfs (WDs) are simple astrophysical objects whose cooling law is well understood. This fact makes them a good laboratory for testing new models of particle physics. Many such models predict the existence of light bosons or light fermions that interact very weakly with regular matter. If these new particles are produced in a WD, they will typically escape and accelerate the cooling of the star. Thus, determining the cooling law from astrophysical observations can be translated into constraints on particle physics beyond the Standard Model (BSM) [@Raffelt:1996wa]. We first give a brief review of WD cooling. Formally, the cooling evolution of white dwarfs can be written as: \[lwd\] where $L_\gamma$ and $L_\nu$ represent the photon and neutrino luminosities (energy per unit time). The first term on the r.h.s. is the well known contribution of the heat capacity of the star to the total luminosity, the second one represents the contribution of the change of volume. It is in general small since only the thermal part of the electronic pressure, the ideal part of the ions and the Coulomb terms other than the Madelung term contribute [@isern97]. The third term represents the contribution of the latent heat and gravitational readjustement of the white dwarf to the total luminosity at freezing. Finally, $L_x$ and $\dot\epsilon_x$ represent any extra energy sink or source of energy respectively. For many applications, this equation can be easily evaluated assuming an isothermal, almost completely degenerate core containing the bulk of the mass, surrounded by a thin, nondegenerate envelope.[^1] The evolution of white dwarfs can be tested through the luminosity function (LF), $n(l)$, which is defined as the number of white dwarfs of a given luminosity or bolometric magnitude[^2] per unit of magnitude interval and unit volume: \[ewdlf\] where \[bc\] $T_G$ is the age of the Galaxy, $l\equiv-\log(L/{L_{\odot}})$, $M$ is the mass of the parent star (for convenience all white dwarfs are labeled with the mass of the main sequence progenitor), $t_\text{cool}$ is the cooling time down to luminosity $l$, $\tau_\text{cool}=dt/d{M_\text{bol}}$ is the characteristic cooling time, $M_\text{s}$ is the maximum mass of a main sequence star able to produce a white dwarf, and $M_\text{i}$ is the minimum mass of the main sequence stars able to produce a white dwarf of luminosity $l$, and $t_\text{PS}$ is the lifetime of the progenitor of the white dwarf. $\Phi(M)$ is the initial mass function, *i.e.* the number of main sequence stars of mass $M$ that are born per unit mass, and $\Psi(t)$ is the star formation rate, *i.e.* the mass per unit time and volume converted into stars. So the product $\Phi(M)\Psi(\tau)$ is the number of main sequence stars that were born at the right moment to produce a white dwarf of luminosiy $l$ now. Since the total density of white dwarfs is not well known, the computed luminosity function is usually normalized to the bin with the smallest error bar, traditionally the one with $l=3$, in order to compare theory with observations. The star formation rate is not known, but fortunately the bright part of Eq. satisfies [@Isern:2008fs]: If $\Psi$ is a well behaved function and $T_G$ is large enough, the lower limit of the integral is not sensitive to the luminosity, and its value is absorbed by the normalization procedure in such a way that the shape of the luminosity function only depends on the averaged characteristic cooling time of white dwarfs. ![*Luminosity function of white dwarfs.* Red (Harris *et al.* [@Harris:2005gd]) and blue (Krzesinski *et al.* [@Krzesinski:2009]) points represent the luminosity function of all white dwarfs (DA and non-DA families). Magenta points [@DeGennaro:2007yw] represent the luminosity function of the DA white dwarfs alone. Both distributions have been normalized around ${M_\text{bol}}= 13$, see text. The dotted line represents the luminosity function obtained assuming Mestel’s approximation. The continuous lines correspond to full simulations assuming a constant star formation rate and an age of the Galaxy of 13 Gyr for the DA family (black line) and all, DA and non-DA, white dwarfs (blue line).[]{data-label="FigLF"}](figures/wdlf.pdf){width="70.00000%"} It is important to realize that white dwarfs are divided into two broad categories, DA and non-DA. The DA white dwarfs exhibit hydrogen lines in their spectra caused by the presence of an external layer made of almost pure H. This hydrogen layer is absent in the case of non-DAs and, consequently, their spectra is free of the H spectral features. The main result is that the DAs cool down more slowly than the non-DAs [@Althaus:2010pi]. The observed LF is shown in Fig. \[FigLF\] for three different datasets. Note that moving from left to right along the horizontal axis we go from high luminosity (hot, young WDs) to low luminosity (cold, old WDs). The Harris *et al.* [@Harris:2005gd] (red) and the Krzesinski *et al.* [@Krzesinski:2009] (blue) data are representative of all, DAs and non-DAS, white dwarfs. The Harris *et al.* LF has been constructed using the reduced proper motion method which is accurate for cold WDs with ${M_\text{bol}}\gtrsim6$ but not appropriate for hot WDs with ${M_\text{bol}}\lesssim6$, and which have been thus removed from the sample. The Krzesinski *et al.* LF on the other hand has been built employing the UV-excess technique which is accurate for hot WDs with ${M_\text{bol}}\lesssim7$ but inappropriate for the colder ones. Since the datasets overlap and, assuming continuity, it is possible to construct a LF that extends from ${M_\text{bol}}\sim1.5$ to ${M_\text{bol}}\sim16$, although the cool end is affected by severe selection effects. The DeGennaro *et al.* [@DeGennaro:2007yw] sample was also obtained with the proper motion technique, which is why the hot end is not reliable and has been removed. Since the identification of DAs and non-DAs is not clear at low temperatures, the corresponding points of [@DeGennaro:2007yw] have been removed from Figure \[FigLF\]. See Isern *et al.* [@Isern:2012xf] for a detailed discussion. Since ultimately only the slope of the LF is of interest and the total density of WDs is quite uncertain, it is usually more convenient to normalize the LF with respect to one of its values, which is commonly chosen around $\log(L/{L_{\odot}})=-3$. If the cooling were due only to photons and one assumes that Mestel’s approximation [@1952MNRAS.112..583M] holds (*i.e.* ions behave like an ideal gas and the opacity of the radiative envelope follows Kramer’s law), then the LF would be a straight line on this logarithmic plot, which already provides a reasonable fit to the data. Note, however, that the data show a dip for values of ${M_\text{bol}}$ around $6-7$. That is where the neutrinos enter the game: for the hotter WDs (to the left in Fig. \[FigLF\]), neutrino emission becomes more important than photon cooling. When neutrinos are included and the cooling is simulated with a full stellar evolution code the agreement becomes impressive (see the continuous lines of Fig. \[FigLF\]). This agreement can be used to bound the inclusion of new sources or sinks of energy [@Isern:2008fs]. Cooling mechanisms {#Cooling} ================== In this section we review the various cooling mechanisms for WDs. The aim is to provide the reader with a simple understanding of what mechanism dominates in what regime. Photons ------- In WDs the thermal energy is mostly stored in the nuclei which form, to a good approximation, a classical Boltzmann gas. Taking into account the thermal conductance of the surface layers, one can relate the rate of energy loss at the surface to the internal temperature. Using Mestel’s approximation [@1952MNRAS.112..583M] one finds \[EqPhotons\] where $\epsilon_\gamma$ is the energy-loss rate per unit mass and $T_7\equiv\frac{T}{10^7\,\text{K}}$. This constitutes the main cooling for cold (${M_\text{bol}}\gtrsim7$) WDs. Realistic models indicate that $\epsilon_\gamma\propto T^{\beta}$, where $\beta \approx 7/2$, but varies slightly with temperature, chemical composition and mass of the white dwarf. We show in Fig. \[FigLTc\] what this energy loss as a function of the core temperature looks like for a realistic model as opposed to Mestel’s model. ![Luminosity (bolometric magnitude) versus core temperature for a realistic model (continuous line) and for Mestel’s model (dashed line). The photon luminosity $L_\gamma \simeq \epsilon_\gamma M_{\rm WD}$, with $M_{\rm WD}$ the WD mass, is related to ${M_\text{bol}}$ via Eq. (\[EqMbol\]).[]{data-label="FigLTc"}](figures/ltc.pdf){width="60.00000%"} Light bosons vs light fermions in white dwarfs ---------------------------------------------- Additional light bosons and light fermions that interact very weakly can also contribute to the cooling of WDs, but their dominant production mechanisms are usually different. First, consider the DFSZ axion [@Dine:1981rt; @Zhitnitsky:1980tq] as an example of a light boson. It would be mainly produced by the bremsstrahlung process $e+(Z,A)\to e+(Z,A)+a$ as shown in Fig. \[FigDiagrams\] (a). Raffelt gave an intuitive argument [@Raffelt:1985nj] to understand how the corresponding energy emission rate depends on the temperature. It goes as follows: The relevant interaction term in the Lagrangian is $iga\bar{e}\gamma_5e$, where $g=m_e/f_ \text{PQ}$, with $m_e$ the electron mass, $f_\text{PQ}\ge 10^9$ GeV the Peccei-Quinn scale, $a$ the axion field and $e$ the electron field. The axion emission by an electron is analogous to the emission of a photon but, due to the presence of the $\gamma_5$, there is an extra electron spin-flip in the amplitude. Whereas the usual photon bremsstrahlung cross section is proportional to $E_\gamma^{-1}$, the axionic analogue is proportional to $E_a$ due to the extra power $E_a^2$ from the spin-flip nature of the process. For the energy emission rate, we have to multiply the cross section by another factor of $E_a$, which makes it proportional to $E_a^2$. We still have to do the phase space integrals for the initial and final state electrons. Because electrons are degenerate in WDs, these integrals contribute a factor of $T/E_F$ each, with $E_F$ the electron Fermi energy. Combining the factors, the emission rate is proportional to $E_a^2(T/E_F)^2\propto T^4$, given that axion energies will be of the order of the temperature $T$. Next, consider the bremsstrahlung of two fermions $\psi$, as depicted in Fig. \[FigDiagrams\] (b). The intuitive reasoning is analogous to what we just described for axions, but the difference is that the fermions from the electron line are produced in pairs (angular momentum conservation) as opposed to the single axion. This adds an extra factor of the energy ($\sim T$) in the cross section and an extra phase space integral. As a result, we have two more powers of $T$ in the final emission rate, which is therefore proportional to $T^6$. When the calculations are done carefully one gets the following results for the energy-loss rates per unit mass in the two cases [@Raffelt:1996wa] \[EqBrem\] Here, $\alpha_{26}\equiv10^{26}\frac{g^2}{4\pi}$, with $g$ the coupling of the axion to electrons defined above;[^3] $G_\psi$ is the dimensionful coupling for the four-fermion interaction denoted by a red dot in Fig. \[FigDiagrams\], to be compared to the familiar Fermi constant, $G_\text{F}=1.166\times10^{-5}$ GeV$^{-2}$; $C_\psi$ is the effective coupling constant analogous to the effective neutral-current vector coupling constant $C_V=0.964$; $X_j$ is the mass fraction of the element $j$, with nuclear charge $Z_j$ and atomic mass number $A_j$, and the sum runs over the species of nuclei present in the WD. $F_a$ and $F_\psi$ are factors that take into account the effect of screening for Coulomb scattering in a plasma. In WDs $F_a$ and $F_\psi$ are of order one to a good approximation. It is clear from expressions (\[EqPhotons\]) and (\[EqBrem\]) why the bremsstrahlung process for the neutrinos, where $C_\psi G_\psi=C_V G_ \text{F}$, is completely irrelevant in WDs, with internal temperature of the order of $10^7$ K. First, the numerical coefficient in $\epsilon_\psi^\text{brem}$ is suppressed by four orders of magnitude compared to photons (and to axions if we take $\alpha_{26}$ of order one). Second, it has a steeper dependence on the temperature, which makes it less and less relevant as we go to lower temperatures (see Fig. \[FigCompareloss\]). ; ; ; ; Unless we have a model in which $G_\psi$ is significantly bigger than $G_\text{F}$, this contribution is negligible. In fact, the dominant production mechanism of a pair of light fermions in WDs is not bremsstrahlung but is given by the so-called plasmon process [@1963PhRv..129.1383A], which is depicted in Fig. \[FigDiagrams\] (c). That is what we describe next. In vacuum the photon is massless and can not decay into a pair of massive particles, no matter how light they are. But in a medium, as in the interior of a star, the photon dispersion relations are modified and this allows such a decay. What happens is that the photon also acquires a longitudinal polarization and is promoted to the so-called plasmon. One would be tempted to say that the photon becomes massive, but such a statement is strictly speaking incorrect. A better way to think about the plasmon decay, without ever referring to the mass of the photon, is the following: the propagation of an electromagnetic excitation (the plasmon) in the plasma is accompanied by an organized oscillation of the electrons, which in turn serve as a source for emitting a pair of light particles. Figs. \[FigDiagrams\] (c,d) are then understood as follows: the grey blob represents the medium response to the electromagnetic excitation; we can think of the black line outlining the blob as a loop of electrons, with the red dot denoting an effective interaction with the pair of light particles, that can be either fermions or bosons. This is a schematic description. The reader interested in more details is referred to the pedagogic treatment in chapter 6 of Ref. [@Raffelt:1996wa]. The calculation of the plasmon decay [@1963PhRv..129.1383A; @1965NCimA..40..502Z] is quite involved, due to the effects of the medium, and cannot be performed analytically. However, a good approximation, in the case of neutrinos as the products of the decay, was given in Ref. [@Haft:; @1993jt]. The result applies to a wide range of stellar temperatures and densities. Restricting ourselves to WDs, we can write it as \[EqEpsplasmon\] where numerically, to a good approximation \[Eqlambdagamma\] and \[Eqfs\] The plasmon decay depends in a complicated way on the photon dispersion relation in the medium. However its main features can be understood in an approximation where the photons are treated as particles with an effective mass equal to the plasma frequency, $\omega_p$, which in the zero-temperature limit is given by [@Raffelt:1996wa] $\omega_p^2 = 4\pi \alpha n_e/E_F$, with $\alpha$ the fine-structure constant, $n_e$ the electron density and $E_F$ the Fermi energy of the electrons. $\omega_p$ is of the order of a few tens of keV in WDs, slightly higher than the typical WD internal temperature, which is a few keV [@Raffelt:1996wa]. For the plasmon decay to happen, the decay products have to be kinematically accessible. Thus, when we talk about [*new light particles*]{} in this context we mean particles [*lighter than a few tens of keV*]{}. It is not immediately obvious how the energy loss of Eq. compares to the previous ones because of its complicated form, but the differences can be easily visualized in the simple plot in Fig. \[FigCompareloss\]. For WDs whose internal temperature is below $4-5\times 10^7$ K, the cooling is dominated by photons, and perhaps axions. Above that temperature, the plasmon decay into two light particles becomes the main source of energy loss. The contribution from the bremsstrahlung of a pair of fermions is always negligible on the plot. It is useful to translate from temperature to ${M_\text{bol}}$. From Eq. , multiplying by a typical WD mass, $M_{\rm WD}$, that we take to be 0.6 solar masses, we obtain the photon luminosity, $L_\gamma = M_{\rm WD} \epsilon_\gamma$. Plugging it into Eq. (\[EqMbol\]) we obtain an expression that relates the temperature $T_7$ to ${M_\text{bol}}$. Thus, a temperature of $4-5\times 10^7$ K corresponds to values of ${M_\text{bol}}$ between 6 and 7, which is indeed where we see the neutrino dip in Fig. \[FigLF\]. The plasmon decay into a pair of light particles constitutes the dominant cooling mechanism for ${M_\text{bol}}<6-7$, the exact figures depending on the mass of the WD and the properties of the envelope. Note that in models where a light boson couples to the electrons through a Yukawa coupling, the important cooling mechanism is the bremsstrahlung where the boson is produced singly, as in Fig. \[FigDiagrams\] (a). Such is the case for the DFSZ axion. In other models, instead, the light bosons couple indirectly to the electrons through a mediator and can only be produced in pairs, as for example when the bosons are charged under a new symmetry. This is the case for models with a dark sector, for instance, which we study in section \[Examples\]. The dominant production for these bosons is then no longer the bremsstrahlung, but the plasmon decay. A generic constraint on models with new light particles {#Generic} ======================================================= This section describes a generic constraint on models with new light particles obtained from WD cooling and trapping. We also discuss analogous constraints from red giants (RGs) and big bang nucleosynthesis (BBN). White dwarf cooling constraint ------------------------------ In this section we discuss generic constraints from WD cooling due to plasmon decay into new light particles, that can be either fermions or bosons. The only requirement is that they should be lighter than a few tens of keV, for the decay to be kinematically possible. As mentioned at the end of the previous section, such a process affects the LF for values of ${M_\text{bol}}$ below $6-7$. Particle physics models in which new plasmon decay channels are open will potentially be in tension with the data, given the remarkable agreement between standard cooling mechanisms, that include neutrino emission, and the observed LF [@Isern:2012xf]. We want to quantify how much the plasmon decay rate can deviate from the standard one, considering the neutrinos as the only decay products. To achieve this goal, it is useful to introduce a unified formalism reminiscent of the Fermi interactions for fermions. In order to compare with the standard plasmon decay into neutrinos, it is necessary to describe the relevant interaction between neutrinos $\nu$ and electrons $e$. The interaction is given by \[EqLneutrino\] where the contribution from the effective neutral-current axial coupling constant $C_A$ is negligible for our purpose and can be ignored [@Braaten:1993jw]. From this Lagrangian one can compute the plasmon decay rate into two neutrinos. The result is [@1963PhRv..129.1383A; @1965NCimA..; @40..502Z; @1972PhRvD...6..941D] \[EqPlasmneutrino\] where $\alpha$ is the fine-structure constant, $Z_s$ is the plasmon wavefunction renormalization, $\pi_s$ is the effective plasmon mass which enters in the dispersion relation $\omega^2-k^2=\pi_s(\omega,k)$ for a plasmon with frequency $\omega$ and wave vector $k$, and the subscript $s=\{T,L\}$ denotes the plasmon polarizations (transverse and longitudinal, respectively). The explicit forms for $\pi_T$ and $\pi_L$ are involved. They can be found, for example, in Ref. [@Raffelt:1996wa]. We just point out for this discussion that $\pi_s$ is proportional to $\alpha$, so that $\Gamma_{\nu,s}$ goes to zero if we turn off the electromagnetic interaction, as expected. With the standard plasmon decay rate into neutrinos, Eq. , the energy-loss rate per unit mass is given by Eq. with $C_\psi G_\psi=C_VG_\text{F}$, *i.e.* \[Eqplasmons\] and the contribution to the luminosity that appears in Eq. (\[lwd\]) is simply $L_\nu=M_\text{WD}\,\epsilon_\nu^\text{plasmon}$. Let us now turn to new neutrino-like cooling mechanisms for WDs. For BSM models with new light fermions $\psi$, the relevant interactions are given by \[EqLfermions\] which are the appropriate analogs of the four-fermion interaction. The quantities $C_\psi$ and $G_\psi$ have been described above. For new light bosons $\phi$ which must be produced in pairs \[see Fig. \[FigDiagrams\] (d)\], the interaction is \[EqLbosons\] with $\phi^\dagger\overleftrightarrow{\partial}^\mu\phi \equiv \phi^\dagger (\partial^\mu \phi) - (\partial^\mu \phi^\dagger)\phi $. The corresponding quantities are the effective coupling constant $C_\phi$ and the dimensionful parameter $G_\phi$, which is the analog of the Fermi constant. For both interactions the plasmon decay into two new light particles is \[EqPlasmnew\] where $\{C_x,G_x\}$ are given by $\{C_\psi,G_\psi\}$ for new light fermions or $\{C_\phi,G_\phi\}$ for new light bosons. The extra plasmon decay channel will lead to an extra energy-loss rate per unit mass as in Eq. with $C_\psi G_\psi=C_xG_x$, *i.e.* \[Eqplasmonbosons\] and an extra contribution to the total luminosity given by $L_x=M_\text{WD}\,\epsilon_x^\text{plasmon}$, as in Eq. (\[lwd\]). In the following the relevant constants will be denoted simply by $\{C_x,G_x\}$ both for new light fermions and bosons. As already mentioned, in order not to upset the excellent agreement between standard WD cooling mechanisms [@Isern:2012xf], *i.e.* from photon emission and neutrino emission (relevant only for hotter WDs), and observational data, we postulate that plasmon decay into new light particles must not account for more than the plasmon decay into neutrinos. In the massless limit, both for new particles as well as neutrinos, this constraint can be stated simply as \[see Eqs. and \] \[EqConstraint\] In other words, any new sufficiently light particles (*i.e.* which are effectively massless in WDs), that can be produced through plasmon decay in WDs and can escape from WDs, generate extra cooling. This extra cooling must be subdominant compared to standard plasmon decay into neutrinos. To validate this order-one constraint, it is now necessary to properly quantify the agreement between the standard cooling mechanisms and observational data. The standard cooling mechanisms relevant for WDs are photon cooling and plasmon decay into neutrinos. Since we are interested in constraining models which lead to extra neutrino-like cooling, we focus here only on the dataset of DeGennaro *et al.* which covers bolometric magnitudes between $5.5\lesssim{M_\text{bol}}\lesssim12.5$. This range is well understood and clearly exhibits the neutrino dip for ${M_\text{bol}}$ around $6-7$ (see Fig. \[FigLFJordi\]). Moreover, the dataset of DeGennaro *et al.* has the smallest error bars in this range and only contains DA WDs. We start by minimizing the $\chi^2$ for the LF assuming standard cooling mechanisms and Mestel’s approximation. The free parameter is the WD birthrate. The best fit implies a birthrate $\sim 1.6\times 10^{-3}$ pc$^{-3}$ Gyr$^{-1}$, which is a reasonable local WD formation rate [@Liebert:2004bv], for $\chi_\text{min}^2=24.9$. Since there are $N_\text{exp}=18$ data points and $N_\text{th}=1$ free parameters, this provides a decent fit with a reduced chi-square $\chi_\text{red,min}^2=\chi_\text{min}^2/N_\text{dof}=1.47$, where $N_\text{dof}=N_\text{exp}-N_\text{th}=17$ is the total number of degrees of freedom. Next we determine the 90% confidence level exclusion contours for extra cooling from plasmon decay into new light particles, assuming the latter are massless. Since the new plasmon decay channels are reminiscent of the standard plasmon decay into neutrinos, we take here $L_x=S_xL_\nu$, where $S_x$ determines the ratio of the new extra luminosity $L_x$ to the neutrino luminosity $L_\nu $. Now we take $S_x$ as our only free parameter, leaving the WD birthrate fixed to the value determined above. Thus we still have $N_\text{dof}=17$. We then compute the new chi-square, $\chi^2$, including the $L_x$ contribution. From Fig. 36.1 in Ref. [@Nakamura:2010zzi] we find that $\chi^2$ must be such that \[EqDeltachi\] otherwise the extra cooling is excluded at 90% confidence level. Imposing the condition $\Delta\chi^2 <24.8$ translates into the constraint \[EqChiS\] which is equivalent to the one in Eq. , obtained from a simpler and more intuitive physical argument. We provide in Fig. \[FigLFJordi\] three curves for the LF obtained from realistic models. The top one includes only standard cooling (with neutrinos), while the two lower ones include extra neutrino-like contributions with $S_x=0.5$ and $S_x=1$ respectively. ![*Theoretical luminosity function for WDs.* The curves shown include the different contributions to Eq. (\[lwd\]) and correspond to values of $S_x=0, 0.5, 1$ from top to bottom for the $L_x = S_x L_\nu$ contribution. They are superimposed on the data points by DeGennaro *et al.* [@DeGennaro:2007yw]. []{data-label="FigLFJordi"}](figures/wdlfx.pdf){width="70.00000%"} White dwarf trapping constraint ------------------------------- One should also include the effects of trapping. Indeed, as $G_x$ increases the interactions between the new light particles and ordinary matter become stronger. For very large $G_x$ the interactions are too strong and the mean free path of the new light particles is too small for them to escape the WD and thus contribute to its cooling. To make an estimate, we compare the cross section for the scattering of new light particles on ordinary matter, $\sigma_x \propto C_x^2 G_x^2$, with the corresponding one for neutrinos, $\sigma_\nu \propto C_V^2 G_\text{F}^2$. Neutrinos have a mean free path of $\lambda_\nu=(n\sigma_\nu)^{-1}\simeq3000{R_{\odot}}$ in WDs [@Althaus:2010pi]. Requiring that the mean free path of our light particles is bigger than a typical WD radius, $R_\text{WD}\simeq0.019{R_{\odot}}$ [@Raffelt:1996wa], and comparing $\sigma_x$ and $\sigma_\nu$ we find the condition $C_xG_x\lesssim400 \ C_VG_\text{F}$. Combining this with Eq. implies that any new light particles produced in WDs are excluded by cooling considerations if \[EqWDConstraint\] Eq. is the main result of this paper and will be used in section \[Examples\] to constrain BSM models with new light particles. Comparison to constraints from red giants and big bang nucleosynthesis ---------------------------------------------------------------------- In the same line of thoughts, it is possible to obtain cooling constraints from red giants (RGs). Following [@Raffelt:1996wa] the bound from RGs cooling can be translated into $S_x\lesssim2$, which corresponds to $C_xG_x\lesssim1.41C_VG_\text{F}$ and is comparable to, but slightly weaker than what we found in Eq. for WDs. Moreover, since the cores of RGs can be seen as WDs, trapping constraints in RGs will necessary be worse than in WD. Therefore, in this context RGs do not constrain new light particles as well as WDs. Such new light particles could however be very tightly constrained by BBN. Given that we are interested in masses below a few tens of keV, if they were in thermal equilibrium with ordinary matter in the early universe until BBN, that happens at $T\sim 1$ MeV, they would contribute to the number of relativistic degrees of freedom, which is well constrained. To estimate this constraint, we follow [@Steigman:2013yua]. The reactions $e^+ e^- \leftrightarrow \psi \psi$ and $e \psi \leftrightarrow e \psi$, responsible for keeping the light particle, $\psi$, in thermal equilibrium, have a typical cross section $\sigma_x \propto C_x^2G_x^2T^2$, which leads to an interaction rate per particle of $\Gamma_x=n\sigma_x|v|\propto C_x^2G_x^2T^5$, since their number density is $n\propto T^3$. Comparing to the expansion rate $H\propto T^2/M_\text{Pl}$, the decoupling temperature can be estimated as $T_{x,\text{dec}}\propto(C_x^2G_x^2M_\text{Pl})^{-1/3}$, where $M_\text{Pl}$ is the Planck mass. This is completely analogous to the calculation for the neutrinos decoupling temperature, $T_{\nu,\text{dec}}\propto(C_V^2G_\text{F}^2M_\text{Pl})^{-1/3}$. Thus we can write \[EqTdec\] Following [@Steigman:2013yua] the effective number of neutrinos $N_\text{eff}$ is given by \[EqNeff\] where $g_s(T)$ is the ratio of the total entropy density to the photon entropy density and $\Delta N_\nu$ is the number of equivalent neutrinos, *i.e.* $\Delta N_\nu=2\times1$ for a Dirac fermion or $\Delta N_\nu=2\times4/7$ for a complex scalar. Demanding that the number of equivalent neutrinos be smaller than $4$ [@Ade:2013zuv] and taking $T_{\nu,\text{dec}}=3\ \text{MeV}$ [@Steigman:2013yua] leads to the constraint \[EqBBNbound\] which is three (two) orders of magnitude stronger than the WD bound Eq. for new light Dirac fermions (complex scalar bosons). From this analysis it would thus seem that BBN bounds are more competitive than WD bounds in constraining models with new light particles. Note, however, that there are caveats that could invalidate the BBN bounds without modifying the WD constraints. For example, a light \[$\sim\mathscr{O}(\text{MeV})$\] weakly-interacting massive particle (WIMP) whose annihilations heat up the photons but not the neutrinos would result in a lower $N_\text{eff}$ and thus leave more room for extra relativistic degrees of freedom [@Kolb:1986nf; @Serpico:2004nm; @Ho:2012br]. In such a scenario, the bound of Eq. would be relaxed to the extent that the WD constraint would be more competitive. Hence, the WD bound is robust because it is oblivious to possible caveats that would alter BBN considerations. Three examples {#Examples} ============== In this section we consider three examples of BSM scenarios. The first two are supersymmetric extensions of the Standard Model (SM): in the first, the light particle is the neutralino, while in the second, it is the axino. We show that WDs do not put competitive bounds on these models. The situation is different in the third example, where we consider models with a dark sector, in which case the WDs bounds are very competitive. A light neutralino ------------------ The neutralino $\chi_0$ is often the lightest supersymmetric particle in the Minimal Supersymmetric Standard Model. It can be very light, even massless, and still evade all current experimental constraints [@Dreiner:2009ic; @Profumo:2008yg]. For the production of light neutralinos in WDs, that would predominantly occur via plasmon decay, we consider the four-fermion interaction obtained from integrating out the selectron $ \tilde{e}$ (see Fig. \[FigSUSY\])[^4], \[EqLneutralino\] where $G_{\tilde{e}}=\frac{e^2}{4\cos^2\theta_Wm_{\tilde{e}}^2}$ and $C_{\chi_0}=\frac{3}{4}$ [@Dreiner:2003wh], with $e$ the electric charge, $\theta_W$ the weak mixing angle and $m_{\tilde{e}}$ the selectron mass. Since $G_\text{F}=\frac{\sqrt{2}e^2}{8\sin^2\theta_Wm_W^2}$ where $m_W$ is the $W$ gauge boson mass the constraint Eq. can be translated into a lower bound on the selectron mass of \[EqSelectron\] where $m_W=80.4$ GeV and $\sin^2\theta_W=0.23$. Thus, in order to have a significant impact on the LF one needs a selectron lighter than the $W$ gauge boson. The bound in Eq. applies to the case of a massless neutralino. Turning on a small neutralino mass has the effect of pushing the WD bound down to even lower selectron masses. Such light selectrons are already excluded by LEP searches [@Heister:2002jca]. Note that supernovae, contrary to WDs, provide a better arena to constrain the mass of a light neutralino [@Dreiner:2003wh]. Nevertheless, WD cooling bounds do not seem competitive for this process. A light axino ------------- A light axino is in principle very interesting in this context. It has already been argued that the inclusion of an axion gives a better fit to the LF [@Isern:2012ef]. If supersymmetry (SUSY) is realized in nature, the axion would be necessarily accompanied by its fermionic partner, the axino, which could also be very light (see *e.g.* [@Rajagopal:1990yx; @Chun; @Chun199543]). The axino could be pair-produced in the plasmon decay and contribute to the high luminosity part of the LF. When combined with the contribution of the axion one might hope to get an even better fit. Unfortunately, as we explain in the rest of this section, the axino interacts way too weakly so that its contribution to the LF turns out to be completely negligible. Recall that the coupling of axions to electrons is given by $iga\bar{e}\gamma_5e$, with $g=m_e/f_\text{PQ}$. In SUSY there is a corresponding axino-electron-selectron interaction that can be written as $ig\tilde{e}\bar{e}\psi_a$, where $\psi_a$ denotes the axino. If we integrate out the selectron (see Fig. \[FigSUSY\]), the resulting four-fermion interaction between two electrons and two axinos is scalar-like \[*e.g.* $(\bar\psi_a \psi_a) (\bar e e)$\] instead of vector-like \[*e.g.* $(\bar\psi_a \gamma^\mu \psi_a) (\bar e \gamma_\mu e)$\] and thus does not even allow plasmons to decay to pairs of axinos. Being more precise and starting from the derivative interaction between the axion and electrons instead, one obtains higher-dimensional operators after supersymmetrizing and integrating out the selectron, *i.e.* four-fermion interactions between two electrons and two axinos with extra derivatives, which are thus temperature-suppressed compared to the usual plasmon decay. Most importantly however, these interactions are always at least suppressed by $g^2$, which is incredibly tiny for reasonable $f_\text{PQ}\sim10^9-10^{12}$ GeV. Therefore, although the constraint Eq. cannot be applied directly here, the universal suppression just mentioned makes a possible production of axinos absolutely unobservable in WDs. A dark sector ------------- ### The model As seen in the two previous examples, WD cooling might not seem to lead to any strong bounds on new light fermions. The situation is however much more interesting when one considers models of BSM with massive dark photons [@Fayet:1980ad; @Fayet:1990wx]. In these models, which could be of relevance as models of dark matter, a dark sector $\mathscr{L}_\text{D}$ communicates with the SM, $\mathscr{L}_\text{SM}$, solely through kinetic mixing $\mathscr{L}_{\text{SM}\otimes\text{D}}$ [@Holdom:1985ag], *i.e.* \[EqLdark\] Above the electroweak scale the kinetic mixing occurs with strength $\varepsilon_Y$ between the hypercharge gauge group $U(1)_Y$, with the corresponding $F_{\mu\nu}^\text{SM} = \partial_\mu B_\nu - \partial_\nu B_\mu$, and a new Abelian gauge group $U(1)_\text{D}$, with $F_\text{D} ^{\mu \nu} = \partial^\mu A_\text{D}^\nu - \partial^\nu A_\text{D}^\mu$, where $A_\text{D}^\mu$ is the $U(1)_\text{D}$ gauge boson, *i.e.* the dark photon. Below the electroweak scale the mixing involves instead the electromagnetic gauge group, and $\varepsilon=\varepsilon_Y\cos\theta_W$. The dimensionless parameter $\varepsilon$, which should be generated by integrating out massive states charged under both SM and dark gauge groups, is naturally small, $\varepsilon\sim10^{-4}-10^{-3}$. Thus, after rotating the fields appropriately such that gauge bosons have canonically-normalized kinetic terms, the SM fields become millicharged under the [*dark*]{} gauge group [@Cassel:2009pu; @Hook:2010tw], *i.e.* \[EqLSMD\] where $J_\mu^\text{SM}$ is the SM electromagnetic current. Thus, in models with massive dark photons, WD plasmons could decay, through off-shell massive dark photons, to light dark sector particles if they are kinematically available. Note that both dark photon decays to bosons and fermions result in two-particle final states. Thus such plasmon decay through massive dark photons into light dark sector particles is reminiscent of plasmon decay into fermions (*e.g.* neutrinos) irrespective of the spin of the light dark sector particles \[see Fig. \[FigDiagrams\] (c,d)\]. Therefore the relevant constraints for plasmon decay in models with massive dark photons are equivalent to the constraint discussed in section \[Generic\]. We stress the fact that in the scenario we are contemplating, the dark $U(1)_{\rm D}$ gauge group is broken so that the dark photon is massive. Instead, when $U(1)_{\rm D}$ is unbroken, the corresponding gauge boson is commonly referred to as a paraphoton. In this latter case, dark sector particles acquire an electric millicharge, that is a tiny fractional charge under the visible $U(1)_{\rm EM}$, and the constraints are usually shown on the plane given by $\varepsilon$ versus the mass of the dark sector particle [@Davidson:2000hf]. In our case, with the broken dark $U(1)_{\rm D}$, there are no particles with an electric fractional charge. Rather, SM particles have a fractional charge under $U(1)_{\rm D}$, that is quite different. ### Excluded parameter region In order to determine the resulting excluded parameter space it is necessary to integrate out the dark photon, as shown in Fig. \[FigDark\]. This leads to the interaction \[Eqelectronsdark\] where the dark constant is $G_\text{D}=\frac{4\pi\varepsilon\sqrt{\alpha\alpha_\text{D}}}{m_{A_\text{D}}^2}$ and $C_\psi=Q_\psi$, $C_\phi=\frac{Q_ \phi} {2}$. Here $m_{A_\text{D}}$ is the dark photon mass, $\alpha_\text{D}$ is the dark fine-structure constant and $Q_{\psi,\phi}$ are the dark particle charges under the dark gauge group. Note that dark photon decay into a pair of dark gauge bosons is generically not kinematically accessible because the masses of the dark photon and of the other dark gauge bosons are usually of the same order, as, for example, the $Z$ and $W$ gauge bosons in the SM. Comparing with plasmon decay to neutrinos as discussed in section \[Generic\], the constraint Eq. leads to \[EqDark\] where $C_\text{D}=C_{\psi,\phi}$. In Fig. \[FigExclusion\] we show the constraint Eq. and regions in parameter space which have already been explored or will be explored by future experiments [@Bjorken:2009mm], *i.e.* beam dump experiments at SLAC: E137, E141 and E774 [@Riordan:1987aw; @Bross:1989mp; @Andreas:2012mt]; $e^+e^-$ colliding experiments: BaBar [@Aubert:2009au; @Bjorken:; @2009mm] and KLOE [@Archilli:2011zc]; and fixed-target experiments: APEX [@Abrahamyan:2011gv], DarkLight [@Freytsis:2009bh], HPS  [@Boyce:2012ym], MAMI [@Merkel:2011ze] and VEPP-3 [@Wojtsekhowski:2012zq]. 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(0.3,4.3) node; (0.3,7.1) node; (5.3,7.3) node; (5.3,6.5) node\[red\]; (6.6,5.4) node\[purple\]; (7.4,6.7) node\[green\]; (7.4,4.2) node\[red\]; (4.3,5) node\[teal\]; (2.2,3.3) node\[pink\]; (4,4.3) node\[brown\]; (6.2,2.8) node\[brown\]; (7.65,1) node\[rotate=58\]; (6.8,1) node\[rotate=58\]; (5.95,1) node\[rotate=58\]; (0.85,2.8) node\[rotate=37\]; For reasonable dark sector parameters where $\alpha_\text{D}\sim\alpha$, one has $C_\text{D}\sim1$ and $C_\text{D}^2\alpha_\text{D}\sim10^{-3}-10^{-2}$, and thus all but a small fraction of the relevant dark sector parameter space is excluded by WD cooling and the some of the above-mentioned experiments become obsolete *if* dark photons couple to new dark sector fermions and/or bosons which are effectively massless in WDs, *i.e.* lighter than a few keV. In other words, to be viable models of dark photons, any model probed by the above-mentioned experiments cannot have dark sector fermions and/or bosons lighter than a few tens of keV due to WD cooling. Note however that all of the experiments shown in Fig. \[FigExclusion\]—apart from DarkLight, VEPP-3 and the anomalous magnetic moment measurements—assume that dark photons decay predominantly back into the SM. Although this is not possible in WDs (dark photons could only decay back into electron-positron pairs which are not kinematically accessible, the decay to neutrino pairs is negligible), this assumption forbids either light dark sector particles, in which case the WD constraint presented here is irrelevant; or large dark fine-structure constant (relative to $\alpha\varepsilon^2$), for which dark photon decay rate into invisible channel dominates. To investigate this last possibility, we include in Fig. \[FigExclusion\] (see dashed blue line) the WD cooling constraint for which the dark photon decay rate into visible channels dominates over the decay rate into invisible channels, *i.e.* $\Gamma_\text{invisible}\lesssim\Gamma_\text{visible}$ or $C_\text{D}^2\alpha_\text{D}\lesssim\alpha\varepsilon^2$. It is interesting to see that, for very weak dark fine structure constant, the experiments which are sensitive to invisible dark photon decays, *i.e.* DarkLight and VEPP-3, are still constrained by the WD cooling even when dark photons decay predominantly back into the SM. Note that the constraint must be modified for a very light dark photon (again lighter than a few tens of keV), since it could be produced on-shell, which would result in an enhancement of the cooling rate. The resulting constraint would then be even tighter. For such a light dark photon bremsstrahlung might also become important. Finally, it would be of interest to study astrophysical cooling constraints from more energetic objects, like supernovae, to relax the restriction on the masses of the dark particles produced. Discussion and conclusion ========================= We studied constraints from the WD LF on BSM models with new light particles. Whenever these light particles are produced in pairs, whether they are fermions or bosons, the dominant production mechanism in WDs is (usually) given by the plasmon decay. Such a decay is responsible also for the production of neutrino pairs, whose effect is well understood and clearly visible through the dip at ${M_\text{bol}}\sim6-7$ in the LF curve. Adding a significant decay into new light particles would deepen the dip, which would then be in disagreement with the data. This constrains part of the parameter space of these BSM models. More quantitatively, one needs to compare the strength of the interaction between the new light particles and the electrons with the interaction between neutrinos and electrons, *i.e.* the Fermi constant $G_\text{F}$, and require that the former do not exceed the latter. We applied this constraint to three models. We first consider a supersymmetric model with a light neutralino and showed that the WD constraint is not competitive with existing collider bounds. The situation is analogous with an axino, whose interaction is even further suppressed with respect to the neutralino, and does not lead to any interesting constraint. We then explored models with a dark sector, for which the bounds are more relevant. That is due mainly to the fact that the dark photon, that mediates the interaction between the electrons and the light dark sector particles, can be light \[$\sim\mathscr{O}(\text{GeV})$\], which enhances the plasmon decay rate. It turns out that the limits on the dark sector parameter space from energy losses in WDs, as shown in Fig. \[FigExclusion\], are extremely competitive and render some experiments obsolete *if* the dark photon couples to light \[$\sim\mathscr{O}(10\ \text{keV})$\] dark sector particles. Said differently, the dark photon models which are probed by these experiments cannot have light dark sector fermions and/or bosons, due to WD cooling. Such dark sector particles could contribute to the relativistic degrees of freedom, $N_\text{eff}$, in the early universe and alter BBN predictions. BBN bounds can indeed be stronger than those from WD cooling. However, they are subject to caveats and are not as robust. *Note:* During completion of this work An *et al.* [@An:2013yfc] posted a paper on stellar constraints for dark photons. There is no overlap between our work and theirs since they consider dark photons with hard Stückelberg masses. [^1]: The isothermal approximation is not valid when neutrinos are dominant, however the results are still reasonably good and provide a reasonable estimate of the luminosity. The ions do not follow the ideal gas law but the equation of state of a Coulomb plasma—for instance, in the region of interest the specific heat approaches the Dulong–Petit law—and crystallizes at low temperatures, around bolometric magnitude $12-13$, depending on the mass of the star. [^2]: The bolometric magnitude and the luminosity are related through \[EqMbol\] [^3]: For $\alpha_{26}$ of order one, axion cooling becomes comparable to photon cooling and one gets a better fit to the LF [@Isern:2012ef]. This fact can be taken as tentative evidence for the existence of axions, and it explains the choice of the power of 26 in the definition of $\alpha_{26}$. [^4]: A very light neutralino, $m_{\chi_0}\ll 1$ GeV, is almost purely bino and does not couple to the $Z_0$ [@Choudhury].

--- abstract: 'The growing demand of industrial, automotive and service robots presents a challenge to the centralized Cloud Robotics model in terms of privacy, security, latency, bandwidth, and reliability. In this paper, we present a ‘Fog Robotics’ approach to deep robot learning that distributes compute, storage and networking resources between the Cloud and the Edge in a federated manner. Deep models are trained on non-private (public) synthetic images in the Cloud; the models are adapted to the private real images of the environment at the Edge within a trusted network and subsequently, deployed as a service for low-latency and secure inference/prediction for other robots in the network. We apply this approach to surface decluttering, where a mobile robot picks and sorts objects from a cluttered floor by learning a deep object recognition and a grasp planning model. Experiments suggest that Fog Robotics can improve performance by sim-to-real domain adaptation in comparison to exclusively using Cloud or Edge resources, while reducing the inference cycle time by $4\times$ to successfully declutter $86\%$ of objects over $213$ attempts.' author: - 'Ajay Kumar Tanwani, Nitesh Mor, John Kubiatowicz, Joseph E. Gonzalez, Ken Goldberg[^1] [^2]' bibliography: - '0-bibliography-short.bib' title: 'A Fog Robotics Approach to Deep Robot Learning: Application to Object Recognition and Grasp Planning in Surface Decluttering' --- INTRODUCTION ============ The term ‘Cloud Robotics’ describes robots or automation systems that rely on either data or code from the Cloud, i.e. where not all sensing, computation, and memory is integrated into a single standalone system [@Kuffner10; @Kehoe15]. By moving the computational and storage resources to the remote datacenters, Cloud Robotics facilitates sharing of data across applications and users, while reducing the size and the cost of the onboard hardware. Examples of Cloud Robotics platforms include RoboEarth [@Waibel11], KnowRob [@Tenorth13], RoboBrain [@Saxena14], DexNet as a Service [@Tian17; @Li_CASE_18]. Recently, Amazon RoboMaker [@RoboMaker18] and Google Cloud Robotics [@GoogleCR_18] released platforms to develop robotic applications in simulation with their Cloud services. Robots are increasingly linked to the network and thus not limited by onboard resources for computation, memory, or software. Internet of Things (IoT) applications and the volume of sensory data continues to increase, leading to a higher latency, variable timing, limited bandwidth access than deemed feasible for modern robotics applications [@Goldberg19; @Wan16]. Moreover, stability issues arise in handling environmental uncertainty with any loss in network connectivity. Another important factor is the security of the data sent and received from heterogeneous sources over the Internet. The correctness and reliability of information has direct impact on the performance of robots. Robots often collect sensitive information (e.g., images of home, proprietary warehouse and manufacturing data) that needs to be protected. As an example, a number of sensors and actuators using Robot Operating System (ROS) have been exposed to public access and control over the Internet [@DeMarinis18]. **Fog Robotics** is *“an extension of Cloud Robotics that distributes storage, compute and networking resources between the Cloud and the Edge in a federated manner”.* The term Fog Robotics (analogous to Fog Computing[^3] [@Yi15; @Dastjerdi16; @Atlam18]) was first used by Gudi et al. [@Gudi17]. In this paper, we apply Fog Robotics for robot learning and inference of deep neural networks such as object recognition, grasp planning, localization etc. over wireless networks. We address the system level challenges of network limits (high latency, limited bandwidth, variability of connectivity, etc.), security and privacy of data and infrastructure, along with resource allocation and model placement issues. Fog Robotics provides flexibility in addressing these challenges by: 1) sharing of data and distributed learning with the use of resources in close proximity instead of exclusively relying on Cloud resources, 2) security and privacy of data by restricting its access within a trusted infrastructure, and 3) resource allocation for load balancing between the Cloud and the Edge (see Fig. \[Fig: Fogrobotics\_arch\] for an overview and Sec. \[sec: FR\] for details). Shared learning reduces the burden of collecting massive training data for each robot in training deep models, while the models are personalized for each robot at the Edge of the network within a trusted infrastructure. Deploying the deep models at the Edge enables prediction serving at a low-latency of less than $100$ milliseconds. These principles are useful to efficiently train, adapt and deploy massive deep learning models by simulation to reality transfer across a fleet of robots. Surface decluttering is a promising application of service robots in a broad variety of unstructured environments such as home, office and machine shops. Some related examples include cloud-based robot grasping [@Kehoe_2013], grasping and manipulation in home environments [@Ciocarlie_2014], robotic butler with HERB [@Srinivasa09] and PR2 [@Bohren11], combining grasping and pushing primitives in decluttering lego blocks with PR2 [@Gupta15], and robot decluttering in unstructured home environments with low cost robots [@Gupta18]. In this work, we consider decluttering scenarios where a robot learns to pick common machine shop and household objects from the floor, and place them into desired bins. We learn deep object recognition and grasp planning models from synthetic images in the Cloud, adapt the model to the real images of the robot within a trusted infrastructure at the Edge, and subsequently deploy models for low-latency serving in surface decluttering. Contributions ------------- This paper makes four contributions: 1. Motivates and introduces Fog Robotics in the context of deep robot learning. 2. Presents a deep learning based surface decluttering application and demonstrates the use of Fog Robotics compared to the alternatives of exclusive Cloud or exclusive local resources. 3. Presents a domain invariant deep object recognition and grasping model by simulation to real transfer, and evaluates benchmarks for learning and inference with a mobile robot over a wireless network. 4. Surface decluttering experiments with a mobile Toyota HSR robot to grasp $185$ household and machine shop objects over $213$ grasp attempts. Fog Robotics {#sec: FR} ============ While the Cloud can be viewed as a practically infinite pool of homogeneous resources in far away data centers, the Edge of the network is characterized by a limited collection of heterogeneous resources owned by various administrative entities. Resources at the Edge come in various sizes, e.g. content delivery networks, light-weight micro servers, networking devices such as gateways, routers, switches and access points. Fog Robotics explores a continuum between on-board resources on a robot to far away resources in Cloud data centers. The goal is to use the available resources, both at the Edge and in the Cloud, to satisfy the service level objectives including, but not limited to, latency, bandwidth, reliability and privacy. By harnessing the resources close by and not relying exclusively on the Cloud, Fog Robotics provides opportunities such as richer communication among robots for coordination and shared learning, better control over privacy of sensitive data with the use of locally provisioned resources, and flexible allocation of resources based on variability of workload. **Related Work:** Hong et al. proposed ‘Mobile Fog’ to distribute IoT applications from Edge devices to the Cloud in a hierarchical manner [@Hong13]. Aazam and Huh [@Aazam14] presented a resource allocation model for Fog Computing. Bonomi et al. made provision for resource constrained IoT devices in their Fog Computing platform [@Bonomi14]. In [@Yousefpour17], the authors propose a framework to minimize service delays in Fog applications by load sharing. The authors in [@Zao14] use a multi-tier Fog and Cloud computing approach for a pervasive brain monitoring system that can reliably estimate brain states and adapt to track users’ brain dynamics. Lee et al. in [@Lee15] and Alrawais et al. in [@Alrawais17] discuss the security and privacy issues and present solutions for mitigating the security threats. More details of Fog computing are in [@Mouradian18; @Mukherjee18]. Recently, several groups have also advocated the need for Fog Robotics. Katterpur et al. profile the computation times for resource allocation in a fog network of robots [@Kattepur]. Gudi et al. present a Fog Robotics approach for human robot interaction [@Gudi18]. Pop et al. discuss the role of Fog computing in industrial automation via time-sensitive networking [@Pop18]. For more details and updates, see [@Goldberg_fog; @OpenFog; @SongTanwaniGoldberg19]. As an example, a number of battery powered WiFi-enabled mobile robots for surface decluttering can use resources from a close-by fixed infrastructure, such as a relatively powerful smart home gateway, while relying on far away Cloud resources for non-critical tasks. Similar deployments of robots with a fixed infrastructure can be envisioned for industrial warehouses, self-driving cars, flying drones, socially aware cobots and so on. Below, we review the opportunities that Fog Robotics provides for secure and distributed robot learning: Enabling Shared and Distributed Learning ---------------------------------------- Fog Robotics brings computational resources closer to mobile robots that enables access to more data via different sensors on a robot or across multiple robots. Whereas Cloud Robotics assumes the Cloud as a centralized rendezvous point of all information exchange, Fog Robotics enables new communication modalities among robots by finding other optimal paths over the network. Using a Cloud-only approach is inefficient in utilizing the network bandwidth and limits the volume of data that can be shared. Fog Robotics enables computational resources closer to the robots to perform pre-processing, filtering, deep learning, inference, and caching of data to reduce reliance on far away data centers. For example, to support household robots, models trained in the Cloud can be periodically pushed to a smart home gateway instead of directly onto individual robots; such a smart home gateway can act as a cache of local model repository, perform adaptation of a generalized model to the specific household, provide storage of data collected from the household for model adaptation, or even run a shared inference service for local robots to support robots with very limited onboard resources. We demonstrate such an inference service in the context of the surface decluttering application. On a broader scale, resources at a municipal level allow for similar benefits at a geographical level. Such computational resources outside data centers are not merely a vision for the future; they already exist as a part of various projects such as EdgeX Foundry [@edgexfoundry], CloudLab [@Ricci14], EdgeNet [@edgenet], US Ignite [@usignite], PlanetLab [@komosny2015planetlab], PlanetLab Europe [@planetlabeurope], GENI [@berman2014geni], G-Lab [@schwerdel2014future], among others. Security, Privacy, and Control Over Data {#sec: security_privacy} ---------------------------------------- Network connected systems significantly increase the attack surface when compared to standalone infrastructure. Deliberate disruption to wide-area communication (e.g., by targeted Denial of Service (DoS) attacks) is not uncommon [@kolias2017ddos]. The control of data collected by robots and the security of data received from a remote service is a major concern. To this end, a well designed Fog Robotics application can provide a tunable middle ground of reliability, security and privacy between a ‘no information sharing’ approach of standalone isolated deployments and a ‘share everything’ approach of Cloud Robotics. Such control over data, however, is non-trivial as resources at the Edge are partitioned in a number of administrative domains based on resource ownership. Heterogeneity of resources further adds to the security challenge; just keeping various software to the most up-to-date versions is cumbersome. Note that merely encrypting data may not be sufficient. As an example, simple encryption only provides data confidentiality but not data integrity—a clever adversary can make a robot operate on tampered data [@song2001timing]. Moreover, addressing key-management—an integral part of cryptographic solutions—is a challenge in itself [@whitten1999johnny]. Finally, managing the security of ‘live’ data that evolves over time is more challenging than that of a static dump. Data-centric infrastuctures such as Global Data Plane (GDP) [@mor2016toward] can provide a scalable alternative to control the placement and scope of data while providing verifiable security guarantees. GDP uses cryptographically secured data containers called DataCapsules. DataCapsules are analogous to shipping containers that provide certain guarantees on data integrity and confidentiality even when they are handled by various parties during their lifetimes. The integrity and provenance of information in a DataCapsule can be verified by means of small cryptographic proofs [@tamassia2003authenticated]. This allows the owners of data to restrict sensitive information in a DataCapsule to, say, a home or a warehouse. In contrast, existing Cloud storage systems (say Amazon S3) do not provide provable security guarantees and rely solely on the reputation of the Cloud provider to protect the Cloud infrastructure from adversarial infiltration. Similarly, decentralized authorization systems such as WAVE [@Andersen17] can protect the secrecy of data without relying on any central trusted parties. Note that secure execution of data still remains an open challenge. A wider deployment of secure hardware such as Intel’s SGX (Software Guard Extensions) technology [@costan2016intel] has the potential to provide for an end-to-end security. While it is important from an infrastructure viewpoint to maintain control over sensitive data and ensure that it does not leave the boundaries of infrastructure with known security properties, applications also need to be designed around such constraints. We demonstrate such an architecture for privacy preserving Fog Robotics scenario by using synthetic non-private data for training in the Cloud and use real-world private data only for local refinement of models. Flexibility of Resource Placement and Allocation ------------------------------------------------ The Cloud provides seemingly infinite resources for compute and storage, whereas resources at the Edge of the network are limited. Quality of service provisioning depends upon a number of factors such as communication latency, energy constraints, durability, size of the data, model placement over Cloud and/or Edge, computation times for learning and inference of the deep models, etc. This has motivated several models for appropriate resource allocation and service provisioning [@Mukherjee18]. Chinchali et al. use a deep reinforcement learning strategy to offload robot sensing tasks over the network [@Chinchali19]. Nan et al. present a fog robotic system for dynamic visual servoing with an ayschronous heartbeat signal [@Tian18]. Flexibility in placement and usage of resources can give a better overall system design, e.g. offloading computation from the robot not only enables lower unit cost for individual robots but also makes it possible to have longer battery life. Consider, for example, a resource constrained network where GPUs are available on the Cloud and only CPUs are available at the Edge of the network. Even though a GPU provides superior computation capabilities compared to a CPU, the round-trip communication time of using a GPU in the Cloud–coupled with communication latency–vs a CPU locally is application and workload dependent. Note that the capital and operational expense for a CPU is far lower than that of a GPU. Simple application profiling may be used for resource placement in this context [@Kattepur]. However, finding an appropriate balance for performance and cost is challenging when the application demands and the availability of resources keeps changing over time, making continuous re-evaluation necessary [@Xu18]. Deep Learning based Surface Decluttering ======================================== Problem Statement ----------------- We consider a mobile robot equipped with a robotic arm and a two-fingered gripper as the end-effector. The robot observes the state of the floor ${{\boldsymbol{\xi}}}_{t}$ as a RGB image ${{\boldsymbol{I}}}_{t}^{c} \in \mathbb{R}^{640 \times 480 \times 3}$ and a depth image ${{\boldsymbol{I}}}_{t}^{d} \in \mathbb{R}^{640 \times 480}$. The task of the robot is to recognize the objects $\{o_{i}\}_{i=1}^{N}$ as belonging to the object categories $o_{i} \in \{1 \ldots C\}$, and subsequently plan a grasp action ${{\boldsymbol{u}}}_t \in \mathbb{R}^{4}$ corresponding to the $3$D object position and the planar orientation of the most likely recognized object. After grasping an object, the robot places the object into appropriate bins (see Fig. \[Fig: hsr\_setup\] for an overview). In this paper, we learn a deep object recognition and a grasp planning model for surface decluttering with a mobile robot. The object recognition model predicts the bounding boxes of the objects from the RGB image, while the grasp planning model predicts the optimal grasp action from the depth image. We compare the grasp planning approach with a baseline that grasps orthogonal to the centroid of the principal axis of the isolated segmented objects, and uses the depth image to find the height of the object centroid. We are interested in offloading the training and deployment of these models by simulation to reality transfer with Fog Robotics. The deep models are trained with synthetic images in the Cloud, adapted at the Edge with real images of physical objects and then deployed for inference serving to the mobile robot over a wireless network. Simulation and Real Dataset --------------------------- We simulate the decluttering environment in a Pybullet simulator [@Coumans_14]. We collect $770$ 3D object meshes from TurboSquid, KIT and ShapeNet resembling household and machine shop environments, and split them across $12$ categories: screwdriver, wrench, fruit, cup, bottle, assembly part, hammer, scissors, tape, toy, tube, and utility. Camera parameters and viewpoint in the simulator is set according to the real robot facing the floor as shown in Fig. \[Fig: hsr\_setup\]. We randomly drop between $5-25$ objects on the floor from a varying height of $0.2-0.7$ meters, each assigned a random color from a set of $8$ predefined colors. The objects are allowed to settle down before taking the RGB and the depth image and recording the object labels. We generated $20 K$ synthetic images of cluttered objects on the floor following this process. The physical dataset includes $102$ commonly used household and machine shop objects split across $12$ class categories as above (see Fig. \[Fig: obj\_dataset\]). We randomly load $5-25$ objects in a smaller bin without replacement and drop them on $1.2$ sq. meter white tiled floor from different positions. We collected $212$ RGB and depth camera images with an average number of $15.4$ objects per image, and hand label the bounding box and the image categories. Transfer Learning from Simulation to Reality {#sec: obj_rec} -------------------------------------------- We train the deep object recognition model on simulated data and adapt the learned model on real data such that the feature representations of the model are invariant across the simulator and the real images [@Ganin16]. The learning problem considers the synthetic images as belonging to a non-private simulated domain $D_{S}$, and real images belonging to a private real domain $D_{R}$ that is not to be shared with other networks. The simulated and real domain consists of tuples of the form $D_S = \{{{\boldsymbol{\xi}}}_t^{(s)}, {{\boldsymbol{u}}}_t^{(s)}, {{\boldsymbol{y}}}_t^{(s)}\}_{t=1}^{T_S}$ and $D_R = \{{{\boldsymbol{\xi}}}_t^{(r)}, {{\boldsymbol{u}}}_t^{(r)}, {{\boldsymbol{y}}}_t^{(r)}\}_{t=1}^{T_R}$, where ${{\boldsymbol{y}}}_t^{(s)}$ and ${{\boldsymbol{y}}}_t^{(r)}$ correspond to a sequence of bounding boxes of object categories as ground-truth labels for a simulated image and a real image respectively, and $T_S \gg T_R$. The real images and the synthetic images may correspond to different but related randomized environments such as a machine shop and a household environment. For example, we randomize the colors of the 3D object models in the simulated domain, but real world objects have a fixed texture. [  ]{} We use the MobileNet-Single Shot MultiBox Detector (SSD) [@Liu15; @Lin17] algorithm with focal loss and feature pyramids as the base model for object recognition (other well-known models include YOLO, Faster R-CNN; see [@Huang16] for an overview). We modify the base model such that the output of feature representation layer of the model is invariant to the domain of the image, i.e., ${{\boldsymbol{\xi}}}_t \sim D_S \approx {{\boldsymbol{\xi}}}_t \sim D_R$, while minimizing the classification loss $\mathcal{L}_{y_c}$ and the localization loss $\mathcal{L}_{y_l}$ of the model. We add an adversarial discriminator at the output of the feature representation layer that predicts the domain of the image as synthetic or real ${{\boldsymbol{\xi}}}_t \in \{D_S,D_R\}$. The overall model parameters are optimized such that the the object classification loss $\mathcal{L}_{y_c}$ and the localization loss $\mathcal{L}_{y_l}$ is minimized, while the domain classifier loss $\mathcal{L}_{d}$ is maximally confused in predicting the domain of the image [@Goodfellow14; @Ganin16; @Tzeng17]. The trade-off between the loss functions governs the invariance of the model to the domain of the image and the output accuracy of the model. We denote the augmented model as the *domain invariant object recognition* model (DIOR). The domain classification architecture is empirically selected to give better performance with $3$ fully connected layers of $1024, 200$ and $100$ neurons after flattening the output of logits layer. We implement two variations of DIOR: 1) **DIOR\_dann** that shares parameters of the feature representation for both the sim and the real domain [@Ganin16], and 2) **DIOR\_adda** that has separate feature representation layers for the sim and the real domain to allow different feature maps. The parameters of the sim network are pretrained and fixed during the adaptation of the real images in this variant [@Tzeng17]. The cropped depth image from the output bounding box of the object recognition model is fed as input to the Dex-Net grasp planning model adapted from [@mahler2017dex]. The model is retrained on synthetic depth images as seen from the tilted head camera of the robot in simulation. Depth images are readily invariant to the simulator and the real environment. The grasp planning model samples antipodal grasps on the cropped depth image of the object and outputs the top ranked grasp for the robot to pick and place the object into its corresponding bin. Networked System with Execution Environments -------------------------------------------- The overall networked system consists of three modular execution environments (see Fig. \[Fig: docker\_arch\]): 1) the robot environment or its digital twin [@Kirill17] in the simulator that sends images of the environment and receives actuation commands to drive the robot; 2) the control environment responsible for sensing the images, *inferring* the objects and grasp poses from the images using the trained object recognition and grasp planning model, planning the motion of the robot for executing the grasps, and sending the actuation commands to drive the robot; and 3) the learning environment that receives images and labels from the robot or the simulator and splits the data for training and evaluation of the deep models. At the end of the training process, the best performing model on the evaluation set is deployed as an *inference graph* for secured and low-latency prediction serving at the Edge in the *robot-learning-as-a-service* platform. The platform defines a service for robots to easily access various deep learning models remotely over a gRPC server. Note that the robot environment, the control environment and the learning environment are virtual, and their appropriate placement depends on the available storage and compute resources in the network. The software components running in network-connected execution environments are packaged and distributed via Docker images [@Merkel14]. We run an instance of the learning environment to train the deep object recognition model on the Cloud with the non-private synthetic data only, while another instance runs at the Edge of the network that adapts the trained network on real data to extract invariant feature representations from the private (real) and the non-private (synthetic) data. Experiments, Results and Discussion =================================== We now present comparative experiments of deep learning and inference for surface decluttering using: 1) Cloud resources only, 2) Edge resources only, and 3) Fog using resources on both Cloud and Edge. The Edge infrastructure includes a workstation (6-core, 2-thread Intel CPUs, 1.1 TB Hard Disk, with a Titan XP GPU) located in UC Berkeley for Edge computing and storage. We use the Amazon EC$2$ `p2.8xlarge` instance with $8$ Tesla K$80$ GPUs for Cloud compute and use Amazon S3 buckets for Cloud storage. We launch the EC2 instance in two regions: 1) **EC2 (West)** in Oregon (us-west-2), and 2) **EC2 (East)** in Northern Virginia (us-east-1). Sim-to-Real Domain Adaptation over the Network ---------------------------------------------- We divide both the simulated and the real datasets into $60 \%$ training and $40 \%$ evaluation sets: $\{\mathrm{sim\_train, real\_train, sim\_eval, real\_eval}\}$, and estimate the model parameters described in Sec. \[sec: obj\_rec\] under different networks on $\mathrm{real\_eval}$: 1) training in the **Cloud** with only large scale non-private synthetic images $\{\mathrm{sim\_train}\}$, 2) training at the **Edge** with only limited number of private real images $\{\mathrm{real\_train}\}$, and 3) training in the **Fog** with both synthetic and real images on the Edge, using a pretrained model on large scale synthetic data in Cloud, under $3$ baselines: a) **Sim+Real:** training on combined simulation and real data with no domain classifier, b) **DIOR\_dann:** training DIOR with shared parameters for sim and real feature representation, c) **DIOR\_adda:** training DIOR with separate parameters for sim and real feature representations. Results are summarized in Table \[tab\_Cloud\_Fog\_dl\]. We observe that the models give comparable or better mean average precision (mAP) [@Lin14] and classification accuracy on the real images in the Fog in comparison to the models trained exclusively on Cloud or Edge. Naive transfer of model trained on synthetic data does not perform well on the real data with an accuracy of $24.16\%$. Combining sim and real data naively is also suboptimal. The domain invariant object recognition with a few labeled real images provides a trade-off between acquiring a generalized representation versus an accurate adaptation to the real images. The **DIOR\_adda** drastically improves the performance on real domain by partially aligning the feature representation with the sim domain. The **DIOR\_dann** model with shared parameters gives good performance in both domains, which can further be used to update the simulator model in the Cloud [@Chebotar18]. We report the remainder of the results with **DIOR\_dann**. Training time of each model is over $13$ hours on both the Cloud and the Edge(GPU) instances suggesting that the model placement issues are less critical for training. The cropped depth image of the closest object to the robot is fed to the grasp planning model to compute the grasp poses for robot surface decluttering (see Fig. \[Fig: results\_obj\_rec\] for qualitative results of the model on both synthetic and real data). \[tab\_Cloud\_Fog\_dl\] [|c||c|c|c|c|]{} Training Set & mAP & $\mathrm{sim\_eval}$ & $\mathrm{real\_eval}$ & $\mathrm{mix\_eval}$\ \ **Sim** & $0.13$ & $97.97$ & $24.16$ & $55.5$\ \ **Real** & $0.62$ & $24.64$ & $88.1$& $64.92$\ \ **Sim $+$ Real** & $0.33$& $90.40$ & $54.12$ & $69.97$\ **DIOR\_dann** & $0.61$& $96.92$& $86.33$ & ${{\boldsymbol{95.21}}}$\ **DIOR\_adda** & $0.61$& $30.87$& ${{\boldsymbol{90.64}}}$ & $67.82$\ Communication vs Computation Cost for Inference ----------------------------------------------- We deployed the trained models in the robot-learning-as-a-service platform that receives images from the robot as a client, performs inference on a server, and sends back the result to the robot. We measure the round-trip time $t^{\mathrm{(rtt)}}$, i.e., time required for communication to/from the server and the inference time $t^{\mathrm{(inf)}}$. We experiment with four hosts for the inference service in the order of decreasing distance to the robot: EC2 Cloud (West), EC2 Cloud (East), Edge with CPU support only, and Edge with GPU support. Results in Table \[tab\_inference\] show that the communication and not the computation time is the major component in overall cost. Deploying the inference service on the Edge significantly reduces the round-trip inference time and the timing variability in comparison to hosting the service on Cloud, with nearly $4\times$ difference between EC2 Cloud host (East) and Edge host with GPU. Surface Decluttering with the Toyota HSR ---------------------------------------- We test the performance of the trained models on the mobile Toyota HSR robot for surface decluttering. We load $5-25$ objects in a smaller bin from a smaller set of $65$ physical objects and drop them on the floor in front of the robot (see Fig. \[Fig: hsr\_setup\]). The overall accuracy of the domain invariant object recognition and the grasping model on the robot is $90.14\%$ and $86.85\%$, respectively, for a total of decluttering $185$ objects across $213$ grasp attempts. In comparison, grasping orthogonal to the principal axis of the segmented objected resulted in a grasping accuracy of $76.19\%$ only. We found that the grasping results improved substantially by retraining the model with respect to the tilted camera viewpoint of the robot in comparison to the results reported in [@Gupta18]. Note that we remove the pathological objects such as heavy hammers, and objects with very low ground clearance such as wrenches and scissors that the robot is not able to grasp. We observe that the robot performs well in grasping compliant objects and objects with well-defined geometry such as cylinders, screwdrivers, tape, cups, bottles, utilities, and assembly parts (see for video, results and supplementary details). \[tab\_inference\] [|c||c|c|]{} Location & $t^{\mathrm{(inf)}}$ & $t^{\mathrm{(rtt)}}$\ \ **EC2(East)** & $31.93 \pm 1.53$ & $437.63 \pm 100.02$\ **EC2(West)** & ${{\boldsymbol{31.12 \pm 1.28}}}$ & $181.61 \pm 22.71$\ **Edge(CPU)** & $52.34 \pm 4.18$ & $149.32 \pm 21.04$\ **Edge(GPU)** & $33.27 \pm 3.09$ & ${{\boldsymbol{119.40 \pm 12.06}}}$\ \ **EC2(East)** & $1906.59 \pm 224.19$ & $ 4418.34 \pm 1040.59$\ **EC2(West)** & $1880.28 \pm 207.46 $ & $ 2197.76 \pm 199.44$\ **Edge(CPU)** & $ 3590.71 \pm 327.57$ & $ 3710.74 \pm 214.08$\ **Edge(GPU)** & ${{\boldsymbol{1753.65 \pm 201.38}}}$ & ${{\boldsymbol{1873.16 \pm 211.57}}}$\ Conclusions and Future Directions ================================= In this paper, we have introduced a Fog Robotics approach for secure and distributed deep robot learning. Secured compute and storage at the Edge of the network opens up a broad range of possibilities to meet lower-latency requirements, while providing better control over data. Standardizing robot communication with available Edge resources, nonetheless, is challenging for a wider adoption of Fog Robotics. We have presented a surface decluttering application, where non-private (public) synthetic images are used for training of deep models on the Cloud, and real images are used for adapting the learned representations to the real world in a domain invariant manner. Deploying the models on the Edge significantly reduces the round-trip communication time for inference with a mobile robot in the decluttering application. In future work, we plan to deploy various deep models for segmentation, hierarchical task planning etc, for low-latency and secure prediction in a multi-agent distributed environment with a set of robots. ACKNOWLEDGMENT {#acknowledgment .unnumbered} ============== This research was performed at the AUTOLAB at UC Berkeley in affiliation with the Berkeley AI Research (BAIR) Lab, Berkeley Deep Drive (BDD), the Swarm Lab, the Real-Time Intelligent Secure Execution (RISE) Lab, the CITRIS “People and Robots” (CPAR) Initiative, and by the Scalable Collaborative Human-Robot Learning (SCHooL) Project, NSF National Robotics Initiative Award 1734633, and the NSF ECDI Secure Fog Robotics Project Award 1838833. The work was supported in part by donations from Siemens, Google, Amazon Robotics, Toyota Research Institute, Autodesk, ABB, Knapp, Loccioni, Honda, Intel, Comcast, Cisco, Hewlett-Packard and by equipment grants from PhotoNeo, NVidia, and Intuitive Surgical. The authors would like to thank Flavio Bonomi, Moustafa AbdelBaky, Raghav Anand, Richard Liaw, Daniel Seita, Sanjay Krishnan and Michael Laskey for their helpful feedback and suggestions. [^1]: The AUTOLAB at UC Berkeley ([automation.berkeley.edu]{}). [^2]: University of California, Berkeley. [{ajay.tanwani, mor, kubitron, jegonzal, goldberg}@berkeley.edu]{} [^3]: The term “Fog Computing” was introduced by Cisco Systems in $2012$ [@Bonomi12]. Other closely related concepts to Fog Computing are Cloudlets [@Verbelen12] and Mobile Edge Computing [@Roman16].

--- abstract: 'Secondary structure formation of nucleic acids strongly depends on salt concentration and temperature. We develop a theory for RNA folding that correctly accounts for sequence effects, the entropic contributions associated with loop formation, and salt effects. Using an iterative expression for the partition function that neglects pseudoknots, we calculate folding free energies and minimum free energy configurations based on the experimentally derived base pairing free energies. The configurational entropy of loop formation is modeled by the asymptotic expression $ -{c}\ln {m}$, where ${m}$ is the length of the loop and ${c}$ the loop exponent, which is an adjustable constant. Salt effects enter in two ways: first, we derive salt induced modifications of the free energy parameters for describing base pairing and, second, we include the electrostatic free energy for loop formation. Both effects are modeled on the Debye-Hückel level including counterion condensation. We validate our theory for two different RNA sequences: For tRNA-phe, the resultant heat capacity curves for thermal denaturation at various salt concentrations accurately reproduce experimental results. For the P5ab RNA hairpin, we derive the global phase diagram in the three-dimensional space spanned by temperature, stretching force, and salt concentration and obtain good agreement with the experimentally determined critical unfolding force. We show that for a proper description of RNA melting and stretching, both salt and loop entropy effects are needed.' author: - 'Thomas R. Einert' - 'Roland R. Netz' bibliography: - 'RNA\_salt.bib' title: ' Theory for RNA folding, stretching, and melting including loops and salt' --- Introduction {#sec:introduction} ============ Ribonucleic acid (RNA) is one of the key players in molecular biology and has in the past attracted theoretical and experimental physicists because of its intriguing structural and functional properties. RNA has multiple functions: beyond being an information carrier it has regulatory and catalytic abilities [@Gesteland2005]. Comprehending how RNA folds and what influences the folding process are key questions [@Tinoco1971]. Thus, the reliable prediction of RNA structure and stability under various conditions is crucial for our understanding of the functioning of RNA and nucleic acid constructs in general [@Liedl2007; @Dietz2009]. The influence of temperature and solution conditions on RNA folding stays in the interest of experimental groups. Traditionally the thermal melting of RNA was monitored [[via]{} ]{}differential scanning calometry or UV spectroscopy for the bulk ensemble [@Xia1998; @Mathews1999; @Privalov1978; @Vives2002]. More recently, single molecule pulling and unzipping experiments have been used to unveil the influence of different solution conditions and even determine energy parameters [@Liphardt2001; @Vieregg2007; @Huguet2010]. On the theoretical side, RNA denaturation has been modeled on various levels of coarse graining. Focusing on the secondary structure, namely the base pairs (bp), and omitting tertiary interactions, equilibrium folding and unfolding has been modeled very successfully [@McCaskill1990; @Zuker1981; @Markham2005; @Gerland2001; @Bundschuh2005; @Montanari2001; @Mueller2003; @Dimitrov2004; @Imparato2009a; @Einert2008; @Einert2011a]. In the presence of a logarithmic contribution to the loop entropy, it has been shown that homopolymeric RNA, where sequence effects are neglected, features a genuine phase transition, which can be induced by force or temperature [@Mueller2002; @Mueller2003; @Einert2008; @Einert2011a]. However, the specific sequence influences the stretching response of a molecule, which has been shown by @Gerland2001 [@Gerland2003], yet without considering the logarithmic loop entropy. More detailed insights can be obtained by simulations, which are numerically quite costly, though, when compared to models focusing only on secondary structure. Coarse grained, Go-like simulations of short RNA hairpins allowed to analyze the dynamics of the folding and unfolding process [@Hyeon2005; @Hyeon2006]. Ion specific effects have been studied by performing molecular dynamics [@Auffinger2000] or coarse grained simulations [@Tan2006; @Tan2007; @Tan2008; @Jost2009]. Much less is known about the salt dependence of denaturation transitions of RNA. While for DNA numerous corrections of the base pairing free energies due to varying salt concentration exist, see [@Owczarzy2004] and references therein, analogous results for the salt dependence of RNA energy parameters are sparse [@Tan2006]. However, molecular biology and biotechnological applications depend on the reliable prediction of RNA stability for different solution conditions. In this paper we extend these previous works and develop a theory that allows to include all these effects – sequence, salt dependence, logarithmic loop entropy, stretching force – and demonstrate that all are necessary to obtain a complete picture of the thermodynamics of the secondary structure of RNA. Neglecting tertiary interactions, we use a recursion relation, which allows to correctly account for logarithmic and thus non-linear free energy contributions due to the configurational entropy of loops [@Einert2008]. To include the influence of monovalent salt on RNA stability, we model the RNA backbone as a charged polymer interacting [[via]{} ]{}a Debye-Hückel potential and give heuristic formulas for the modification of the loop free energy and the base pairing and stacking free energy parameters. Debye-Hückel is a linear theory, yet we include non-linear effects caused by counterion condensation using Manning’s concept [@Manning1969]. The backbone elasticity of single stranded RNA (ssRNA) is described by the freely jointed chain (FJC) model. Our description allows for a complete description of the behavior of RNA in the three-dimensional phase space spanned by temperature, salt concentration, and external stretching force. We find that for an improved description of RNA melting curves one needs to include both salt effects and loop entropy. Only the combined usage of these two contributions enables to predict the shift of the melting temperature (due to salt) and the cooperativity (due to logarithmic loop entropy), which is illustrated in the case of tRNA-phe. As an independent check we consider the force induced unfolding of the P5ab RNA hairpin and observe good agreement with experimental values with no fitting parameters. The influence of salt is illustrated by melting curves and force extension curves for various salt concentrations. For the P5ab hairpin the phase diagram is determined and slices through the three-dimensional parameter space are shown. Free energy parameterization ============================ RNA folding can be separated into three steps, which occur subsequently and do not influence each other to a fairly good approximation [@Tinoco1999]. The primary structure of RNA is the mere sequence of its four bases cytosine (C), guanine (G), adenine (A), and uracil (U). Due to base pairing, [[i.e. ]{}]{}either the specific interaction of C with G or the interaction of A with U, the secondary structure is formed. Therefore, on an abstract level, the secondary structure is given by the list of all base pairs present in the molecule. Only after the secondary structure has formed, tertiary contacts arise. Pseudoknots [@Richards1969; @Batenburg2000], helix stacking, and base triples [@Higgs2000] as well as the overall three-dimensional arrangement of the molecule are considered as parts of the tertiary structure. The main assumption of hierarchical folding is, that tertiary structure formation operates only on already existing secondary structure elements [@Tinoco1999]. Although cases are known where this approximation breaks down, it generally constitutes a valid starting point [@Cho2009]. In this paper, where the main point is the influence of the loop entropy and the salt concentration on the secondary structure, we therefore neglect tertiary interactions altogether. ![Schematic representation of the secondary structure of an RNA molecule. Dots represent one base, [[i.e. ]{}]{}cytosine, guanine, adenine, or uracil. Solid lines denote the sugar-phosphate backbone bonds, broken lines base pairs, and thick gray lines the non-nested backbone bonds, which are counted by the variable ${M}$, here ${M}=11$. The thick arrows to either side illustrate the force ${F}$ applied to the 5’- and 3’-end.[]{data-label="fig:1"}](049_rna_secondary_structure_scheme_gs_image) Given a set of base pairs, the secondary structure consists of helices and loops as the basic structural units, [[cf. ]{}]{}[fig.]{} \[fig:1\]. Since pseudoknots are neglected, every nucleotide can be attributed unambiguously to exactly one subunit. The free energy of a certain secondary structure is then given by the sum of the free energy contributions of the individual structural subunits, as we will detail now. Free energy of a loop {#sec:free-energy-loop} --------------------- We model the free energy of a loop consisting of $m$ backbone bonds, see [fig.]{} \[fig:1\], with $$\label{eq:1} {{\mathcal G}_{\mathrm{l}}}({m}) = {{\mathcal G}_{\mathrm{l}}^{\mathrm{conf}}}({m}) + {{\mathcal G}_{\mathrm{l}}^{\mathrm{salt}}}({m}) + {{\mathcal G}_{\mathrm{l}}^{\mathrm{init}}}{\;}.$$ The first term is the loop entropy difference between an unconstrained polymer and a ring-like polymer, which is characterized by the loop exponent ${c}$ [@Duplantier1986; @Kafri2000; @Einert2008; @Mueller2002] $$\label{eq:2} {{\mathcal G}_{\mathrm{l}}^{\mathrm{conf}}}({m}) = - {\mathrm{k_B}T}\ln {m}^{-{c}} {\;},$$ with ${\mathrm{k_B}}$ the Boltzmann constant and ${T}$ the absolute temperature. The loop exponent ${c}$ is ${c}_{\mathrm{ideal}}=3/2$ for an ideal polymer and ${c}_{\mathrm{SAW}} = 1.76$ for an isolated self avoiding loop. Helices emerging from the loop limit the configurational space available to the loop and hence increase ${c}$. One obtains ${c}_1=2.06$ for terminal, ${c}_2=2.14$ for internal loops and ${c}_4=2.16$ for a loop with four emerging helices [@Einert2008]. Since the differences between these exponent values are quite small, we assume a constant loop exponent ${c}= 2.1$ in this paper and only compare with the case of vanishing loop entropy characterized by $c=0$. The second term in [eq.]{}  describes the free energy difference between a charged ring of length $m{l_{\mathrm{ss}}}$ and a straight rod of the same length due to electrostatic interactions, with ${l_{\mathrm{ss}}}= \unit{6.4}{\angstrom}$ the length of one ssRNA backbone bond [@Tan2008]. The electrostatics are modeled on the Debye-Hückel level [@Kunze2002] $$\begin{aligned} \label{eq:3} {{\mathcal G}_{\mathrm{l}}^{\mathrm{salt}}}({m}) &= {\mathrm{k_B}T}{l_{\mathrm{B}}}({m}{l_{\mathrm{ss}}}) {{\tau}_{\mathrm{ss}}}^2 \Biggl[ \ln({\kappa}{m}{l_{\mathrm{ss}}}) - \ln(\pi/2) + \gamma - \frac{{\kappa}{m}{l_{\mathrm{ss}}}}{2} \leftidx{_1}{F}{_2}\left( 1/2, \begin{pmatrix} 1\\3/2 \end{pmatrix}, \left(\frac{{\kappa}{m}{l_{\mathrm{ss}}}}{2\pi}\right)^2 \right)\notag\\ & {\quad}{\quad}+ \frac12 \left(\frac{{\kappa}{m}{l_{\mathrm{ss}}}}{\pi}\right)^2 \leftidx{_2}{F}{_3}\left( \begin{pmatrix} 1\\1 \end{pmatrix}, \begin{pmatrix} 3/2\\3/2\\2 \end{pmatrix}, \left(\frac{{\kappa}{m}{l_{\mathrm{ss}}}}{2\pi}\right)^2 \right) \notag\\ & {\quad}{\quad}+ \frac{1}{{\kappa}{m}{l_{\mathrm{ss}}}}\left(1-\exp(-{\kappa}{m}{l_{\mathrm{ss}}}) + {\kappa}{m}{l_{\mathrm{ss}}}\Gamma(0, {\kappa}{m}{l_{\mathrm{ss}}})\right) \Biggr] {\;},\end{aligned}$$ with ${l_{\mathrm{B}}}= {e_0}^2/({\mathrm{k_B}T}4\pi{\varepsilon_0}{\varepsilon_{\mathrm{r}}}) $ the Bjerrum length, which in water has a value of roughly $\unit{7}{\angstrom}$, ${\kappa}^{-1} = \sqrt{{\varepsilon_0}{\varepsilon_{\mathrm{r}}}{\mathrm{k_B}T}/(2{\mathrm{N_A}}{e_0}^2I) }$ the Debye screening length, ${\varepsilon_0}$ the vacuum dielectric constant, ${\varepsilon_{\mathrm{r}}}\approx80$ the relative dielectric constant of water [@Murrell1994], ${I}= 1/2({{\rho}_{\mathrm{a}}}{{z}_{\mathrm{a}}}^2 + {{\rho}_{\mathrm{c}}}{{z}_{\mathrm{c}}}^2)$ the ionic strength, ${{\rho}_{\mathrm{a}}}/{{\rho}_{\mathrm{c}}}$ and ${{z}_{\mathrm{a}}}/{{z}_{\mathrm{c}}}$ the concentration and the valency of the anions/cations, ${\mathrm{N_A}}$ the Avogadro constant, ${e_0}$ the elementary charge, $\gamma\approx0.58$ Euler’s constant, $\Gamma(a,x)$ the incomplete gamma function, and $\leftidx{_p}{F}{_q}$ the generalized hypergeometric functions [@Abramowitz2002]. To account for modifications of the line charge density ${{\tau}_{\mathrm{ss}}}$ due to non-linear electrostatic effects, we employ Manning’s counterion condensation theory [@Manning1969], predicting $${{\tau}_{\mathrm{ss}}}= \min(1/ {l_{\mathrm{ss}}}, 1/({l_{\mathrm{B}}}{{z}_{\mathrm{c}}}))\label{eq:4} {\;}.$$ [Eq.]{}  amounts to a ground state approximation of the electrostatic contribution to the free energy of a loop. This is rationalized by the fact that the electrostatic interaction is screened and decays exponentially over the Debye length, which is roughly ${\kappa}^{-1} = \unit{1}{\nano\meter}$ for salt solution. However, typical distances between bases in a loop are of the order of the helix diameter $d=\unit{2}{\nano\meter}$ or larger. Therefore, we expect electrostatic interactions to be basically independent of the global configuration of a loop, which justifies both the ground state approximation and our additivity approximation, where ion effects and conformational contributions decouple, see [eq.]{} . In the supporting material, see [eq.]{} S4, we give an interpolation formula for [eq.]{}  involving no hypergeometrical functions. The last term in [eq.]{}  is the loop initiation free energy ${{\mathcal G}_{\mathrm{l}}^{\mathrm{init}}}$. As we are employing a logarithmic loop entropy, [eq.]{} , we cannot use the standard value for ${{\mathcal G}_{\mathrm{l}}^{\mathrm{init}}}$, which was extracted from experimental data for a different loop parameterization [@Xia1998; @Mathews1999]. Therefore, a modified value ${{\mathcal G}_{\mathrm{l}}^{\mathrm{init}}}$ is obtained by fitting ${{\mathcal G}_{\mathrm{l}}}({m})$, given by [eq.]{} , to experimental data using ${c}= 2.1$ in ${{\mathcal G}_{\mathrm{l}}^{\mathrm{conf}}}({m})$ and the salt concentration ${\rho}= \unit{1}{\molar}$ in ${{\mathcal G}_{\mathrm{l}}^{\mathrm{salt}}}({m})$, see [fig.]{} \[fig:2\]a. In this figure we show experimentally determined free energies for terminal, internal and bulge loops as a function of the loop size, which exhibit a dependence on the type of the loop. As an approximation, we do not distinguish between those loop types in the theory and consequently fit a single parameter ${{\mathcal G}_{\mathrm{l}}^{\mathrm{init}}}$ to the data, which turns out to be ${{\mathcal G}_{\mathrm{l}}^{\mathrm{init}}}= \unit{1.9}{\kilo\calory\per\mole}$ for ${T}= \unit{300}{\kelvin}$, see supporting material section C. In [fig.]{} \[fig:2\]a the fitted ${{\mathcal G}_{\mathrm{l}}}({m})$ for the loop exponent ${c}=2.1 $ is depicted by the solid line; the other lines illustrate the effect of different loop exponents on the loop free energy according to [eq.]{}  using the same value for ${{\mathcal G}_{\mathrm{l}}^{\mathrm{init}}}$. [fig.]{} \[fig:2\]b illustrates the effect of salt on the loop free energy for a given value of ${c}=2.1$. ![a Free energy of a loop as a function of the number of segments $m$ for different loop exponents ${c}= 0,\,1.5,\,2.1$ (lines) and for NaCl concentration ${\rho}=\unit{1}{\molar} $. Symbols denote experimental values for various types of loops (hairpin, bulge, internal) [@Serra1995; @Tan2008] for ${\rho}= \unit{1}{\molar}$ NaCl. ${{\mathcal G}_{\mathrm{l}}^{\mathrm{init}}}$ is obtained by fitting ${{\mathcal G}_{\mathrm{l}}}({m})$, [eq.]{} , to the experimental data for ${c}= 2.1$ and ${\rho}= \unit{1}{\molar}$. The same salt concentration and the same value for ${{\mathcal G}_{\mathrm{l}}^{\mathrm{init}}}$ is used for plotting the curves with ${c}= 0,\,1.5$. b Salt dependence of the free energy of loops as a function of the number of segments $m$ for different salt concentrations ${\rho}= \unit{1}{\molar},\ \unit{0.1}{\molar},\,\unit{0.01}{\molar}$ and loop exponent ${c}= 2.1$ according to [eq.]{} . []{data-label="fig:2"}](043_bsl_combine_029-030_gs_image.pdf) Free energy of a helix {#sec:free-energy-helix} ---------------------- The free energy of a helix $$\label{eq:5} {{\mathcal G}_{\mathrm{h}}}= {{\mathcal G}_{\mathrm{h}}^{\mathrm{stack}}}+ {{\mathcal G}_{\mathrm{h}}^{\mathrm{init}}}+ {{\mathcal G}_{\mathrm{h}}^{\mathrm{term}}}+ {{\mathcal G}_{\mathrm{h}}^{\mathrm{salt}}}{\;}$$ depends on the sequence $\{b_i\}$, which consists of the four nucleotides $b_i =\mathrm{C,G,A,U}$. The stacking free energy ${{\mathcal G}_{\mathrm{h}}^{\mathrm{stack}}}$ is based on experimentally determined parameters incorporating the base pairing free energy as well as the stacking free energy between neighboring base pairs. In the standard notation, ${{{ g}_{\mathrm{h}}}^{\mathrm{stack}}}[{(b_{i},b_{j})},{(b_{i+1},b_{j-1})}]$ is the contribution of the two neighboring, stacked base pairs ${(b_{i},b_{j})}$ and ${(b_{i+1},b_{j-1})}$ to ${{\mathcal G}_{\mathrm{h}}^{\mathrm{stack}}}$. The explicit values for the enthalpic and entropic parts are given in the supporting material. We use the expanded nearest neighbor model [@Mathews1999; @Xia1998] to calculate the base pairing and stacking contributions of a helical section ranging from base pair ${(i,j)}$ through ${(i+h,j-h)}$ and obtain $$\label{eq:6} {{\mathcal G}_{\mathrm{h}}^{\mathrm{stack}}}= \sum_{h'=1}^h{{{ g}_{\mathrm{h}}}^{\mathrm{stack}}}[(b_{i+h'-1}, b_{j-h'+1}),(b_{i+h'}, b_{j-h'})] {\;}.$$ The initiation and termination free energies in [eq.]{}  take into account weaker pairing energies of AU or GU base pairs at the ends of the helix. We use the standard literature values for ${{\mathcal G}_{\mathrm{h}}^{\mathrm{init}}}$ and ${{\mathcal G}_{\mathrm{h}}^{\mathrm{term}}}$ [@Mathews1999; @Xia1998] and summarize the explicit values in the supporting material. Increasing the salt concentration increases the stability of a helix: First, counterions condense on the negatively charged backbone and reduce the electrostatic repulsion and, second, the diffuse counterion cloud surrounding the charged molecule screens the interaction. We model the two strands of a helix as two parallel rods at distance $d=\unit{2}{\nano\meter}$ interacting [[via]{} ]{}a Debye-Hückel potential characterized by the screening length ${\kappa}^{-1}$. The electrostatic interaction energy per nucleotide with the other strand is given by $$\label{eq:7} {{ g}_{\mathrm{h}}^{\mathrm{DH}}}({\rho})= {\mathrm{k_B}T}{{\tau}_{\mathrm{ds}}}^2{l_{\mathrm{ds}}}^{}{l_{\mathrm{B}}}^{} \int_{-\infty}^\infty \frac{\exp(-{\kappa}\sqrt{d^2+z^2})}{\sqrt{d^2+z^2}} dz = 2 {\mathrm{k_B}T}{{\tau}_{\mathrm{ds}}}^2{l_{\mathrm{ds}}}^{}{l_{\mathrm{B}}}^{} {{\mathrm{K}}_{0}({\kappa}d)} {\;}.$$ ${l_{\mathrm{ds}}}= \unit{3.4}{\angstrom}$ is the helical rise per base pair of double-stranded RNA (dsRNA) and ${{\mathrm{K}}_{0}({\kappa}d)}$ is the zeroth order modified Bessel function of the second kind. Again, we employ Manning’s theory [@Manning1969] to calculate the line charge density ${{\tau}_{\mathrm{ds}}}= \min(1/ {l_{\mathrm{ds}}}, 1/({l_{\mathrm{B}}}{{z}_{\mathrm{c}}}))$. The reference state for the salt correction of the pairing free energy is at temperature ${T}= \unit{300}{\kelvin}$ with monovalent salt concentration ${\rho}= \unit1\molar$, as the experimental pairing free energies ${{{ g}_{\mathrm{h}}}^{\mathrm{stack}}}$ were determined at this concentration. The free energy shift for a helix consisting of $h$ base pairs due to electrostatic interactions is then $$\label{eq:8} {{\mathcal G}_{\mathrm{h}}^{\mathrm{salt}}}= h( {{ g}_{\mathrm{h}}^{\mathrm{DH}}}({\rho}) - {{ g}_{\mathrm{h}}^{\mathrm{DH}}}(\unit{1}{\molar})) {\;}.$$ The use of Debye-Hückel theory to incorporate salt effects enables to include the overall dependence on temperature and salt concentration but involves several approximations. First, we are using Manning’s counterion condensation theory to obtain the actual line charge density of ssRNA and dsRNA [@Manning1969]. However, Manning condensation is known to underestimate the line charge at increasing salt concentration and therefore favors the bound state [@Netz2003]. Second, when calculating the electrostatic energy of a loop we effectively use a ground state approximation and neglect conformational fluctuation effects. Third, when two ssRNA strands come together to form a helix, the line charge density increases since the distance between two bases decreases. The salt dependence of the work to decrease the axial distance between two bases from ${l_{\mathrm{ss}}}= \unit{6.4}{\angstrom}$ to ${l_{\mathrm{ds}}}= \unit{3.4}{\angstrom}$ is neglected. This approximation favors the unbound state. Therefore, it is very important to validate the model we employ, which we do by detailed comparison with experimental data. From the favorable comparison with experiments we tentatively conclude that the various errors partially cancel and the resulting expression for the salt influence is quite accurate. We point out that after determining ${{\mathcal G}_{\mathrm{l}}^{\mathrm{init}}}$ in [eq.]{} , no further fitting is done and only standard literature values are used. Our theory is able to consider variations of the salt concentration as well as of the temperature, which makes it suitable to study RNA melting at various salt concentrations in a consistent way. However, since our approach is solely based on mean field theory, it will become unreliable in the case of multivalent ions, where correlations become important. Also, ion specific effects, which are important for divalent ions such as Mg^2+^ [@Draper2005], are not considered in our approach. Response of the molecule to an external stretching force {#sec:response-molecule-an} -------------------------------------------------------- In atomic force microscope or optical tweezers experiments, it is possible to apply a stretching force ${F}$ to the two terminal bases of the molecule. We model the stretching response of the ${M}$ non-nested backbone bonds, see [fig.]{} \[fig:1\], with the freely jointed chain (FJC) model [@Montanari2001; @Gerland2001; @Gerland2003]. A non-nested bond is defined as a backbone bond, which is neither part of a helix nor part of a loop. It is outside all secondary structure elements and therefore contributes to the end-to-end extension observed in force spectroscopy experiments. The force dependent contribution to the free energy per non-nested monomer is given by $$\label{eq:9} {{{ g}_{\mathrm{f}}}^{\mathrm{FJC}}}= {{{\mathcal G}_{\mathrm{f}}}^{\mathrm{FJC}}}/ {M}= -{\mathrm{k_B}T}\frac{{l_{\mathrm{ss}}}}{{b_{\mathrm{ss}}}} \ln \left(\frac{\sinh(\beta{b_{\mathrm{ss}}}{F})}{\beta{b_{\mathrm{ss}}}{F}}\right) {\;},$$ where $\beta = 1/({\mathrm{k_B}T})$ is the inverse thermal energy and ${b_{\mathrm{ss}}}= \unit{1.9}{\nano\meter}$ is the Kuhn length of ssRNA [@Montanari2001] (we used the Kuhn length of ssDNA as the corresponding ssRNA data is less certain). The stretching response of one non-nested monomer to an external force is then given by $$\label{eq:10} x^{\mathrm{FJC}}({F}) = -\frac{{\mathrm{d}}{{{ g}_{\mathrm{f}}}^{\mathrm{FJC}}}}{{\mathrm{d}}{F}} = {l_{\mathrm{ss}}}{\mathcal L}(\beta{F}{b_{\mathrm{ss}}}) = {l_{\mathrm{ss}}}\left( \coth(\beta{F}{b_{\mathrm{ss}}}) + 1/ (\beta{F}{b_{\mathrm{ss}}})\right) {\;},$$ ${\mathcal L}$ is the Langevin function. Electrostatic effects on the stretching response are considered to be small and hence are neglected [@Marko1995; @Netz2001a]. Calculation of the partition function {#sec:calc-part-funct} ===================================== So far we showed how to calculate the free energy of one given secondary structure. The next step is to enumerate all possible secondary structures and to obtain the partition function, which allows to study the thermodynamics of the system. As we neglect tertiary contacts – and in particular pseudoknots – for any two base pairs ${(i,j)}$ and ${(k,l)}$ with $i<j$, $k<l$, and $i<k$ we have either $i<k<l<j$ or $i<j<k<l$. This allows to derive a recursion relation for the partition function of the secondary structure. In our notation, the canonical partition function ${Q}_{i,j}^{{M}}$ of a sub-strand from base $i$ at the 5’-end through $j$ at the 3’-end depends on the number of non-nested backbone bonds ${M}$ [@Einert2008; @Bundschuh2005; @Mueller2002], see [fig.]{} \[fig:1\]. The recursion relations for ${Q}_{i,j}^{{M}}$ can be written as \[eq:11\] $$\label{eq:9a} {Q}_{i,j+1}^{{M}+1} = \frac{{v_{\mathrm{f}}({M}+1)}}{{v_{\mathrm{f}}({M})}} \left[ {Q}_{i,j}^{{M}} + \sum_{k=i+{M}+1}^{j-N_{\mathrm{loop}}}{Q}_{i,k-1}^{{M}} {Q}_{k, j+1}^{0}\right]$$ and $$\label{eq:9b} {Q}_{k,j+1}^{0}= \sum_{h=1}^{(j-k-N_{\mathrm{loop}})/2} \exp[-\beta{{{\mathcal G}_{\mathrm{h}}}}^{{(k,j+1)}}_{{(k+h,j+1-h)}}]\sum_{m=1}^{j-k-1-2h} {Q}_{k+1+h,j-h}^{m}\frac{\exp[-\beta{{\mathcal G}_{\mathrm{l}}}(m+2)]}{{v_{\mathrm{f}}(m)}} {\;}.$$ [Eq.]{}  describes elongation of an RNA structure by either adding an unpaired base (first term) or by adding an arbitrary sub-strand $ {Q}_{k, j+1}^{0}$ that is terminated by a helix. [Eq.]{}  constructs $ {Q}_{k, j+1}^{0}$ by closing structures with ${m}$ non-nested bonds, summed up in $ {Q}_{k+1+h,j-h}^{{m}}$, by a helix of length $h$. $N_{\mathrm{loop}}=3$ is the minimum number of bases in a terminal loop. ${v_{\mathrm{f}}({M})}$ denotes the number of configurations of a free chain with ${M}$ links and drops out by introducing the rescaled partition function ${\tilde Q}_{i,j}^{{M}}={Q}_{i,j}^{{M}}/{v_{\mathrm{f}}({M})}$ and will not be considered further since its effects on the partition function are negligible. ${{{\mathcal G}_{\mathrm{h}}}}^{{(k,j+1)}}_{{(k+h,j+1-h)}}$ is the free energy of a helix beginning with base pair ${(k,j+1)}$ and ending with base pair ${(k+h,j+1-h)}$ according to [eq.]{} . ${{\mathcal G}_{\mathrm{l}}}(m+2)$ is the free energy of a loop consisting of $m+2$ segments as given by [eq.]{} . ${{\mathcal G}_{\mathrm{l}}}$ and ${{\mathcal G}_{\mathrm{h}}}$ contain all interactions discussed in the previous section. [Eq.]{}  allows to compute the partition function in polynomial time (${\mathcal O}({N}^4)$). Further, our formulation allows to treat non-linear functions for ${{\mathcal G}_{\mathrm{l}}}({m})$ and ${{\mathcal G}_{\mathrm{h}}}(h)$; for instance, ${{\mathcal G}_{\mathrm{l}}}({m})$ is strongly non-linear by virtue of [eqs.]{}  and . The unrestricted partition function of the entire RNA, where the number of non-nested backbone bonds ${M}$ is allowed to fluctuate, is given by $${Z}_{{N}} = \sum_{{M}= 0}^N \exp[-\beta {{{ g}_{\mathrm{f}}}^{\mathrm{FJC}}}{M}]{\tilde Q}_{0,N}^{{M}} \label{eq:12}$$ and contains the influence of force [[via]{} ]{}${{{ g}_{\mathrm{f}}}^{\mathrm{FJC}}}$ defined in [eq.]{} . The partition function ${Z}_{{N}}$ contains all secondary structure interactions, but neglects pseudoknots and other tertiary interactions. As has been argued before, this approximation is known to work very well [@Tinoco1999] and yields reliable predictions for the stability of nucleic acids [@Gruber2008]. Using the same ideas, we determine the minimum free energy (mfe) and the mfe structure. The mfe structure, is defined as the secondary structure, which gives the largest contribution to the partition function. Since it cannot be derived from the partition function itself, it has to be determined from a slightly modified set of recursion relations, see supporting material. Salt dependence of melting curves {#sec:salt-depend-melt} ================================= ![ Melting curve of the 76 bases long tRNA-phe of yeast; the minimum free energy structure at ${\rho}= \unit{1}{\molar}$, ${T}= \unit{300}{\kelvin}$, ${c}= 2.1$ is shown as an inset. Symbols denote experimental melting curves for NaCl concentrations ${\rho}= \unit{20}{\milli\molar}$ (squares) and $\unit{150}{\milli\molar}$ (circles) [@Privalov1978]. Our predictions for different salt concentrations are depicted by the dashed ($\unit{20}{\milli\molar}$), dash-dotted ($\unit{150}{\milli\molar}$), and solid ($\unit{1}{\molar}$) lines. The respective arrows indicate melting temperatures obtained by experiments of another group [@Vives2002] at the same salt concentration ${\rho}=\unit{150}{\milli\molar}$ (left arrow) and ${\rho}=\unit{1}{\molar}$ (right arrow). The dotted line shows our prediction for ${\rho}= \unit{150}{\milli\molar}$ and ${c}= 0$ and exemplifies that a non-zero loop exponent is responsible for rendering the transition more cooperative, in closer agreement with experiment; for $\unit{150}{\milli\molar}$ and ${c}= 0$ the melting temperature is at higher temperatures since the energy parameters are optimized for ${c}= 2.1$. The gray dash-dotted curve is the prediction of the Vienna package, which uses a linearized multi-loop entropy corresponding to ${c}= 0$ and ${\rho}= \unit{1}{\molar}$. This is to be compared to our prediction for ${c}= 2.1$ and ${\rho}= \unit{1}{\molar}$: while the melting temperatures are similar, the cooperativity, [[i.e. ]{}]{}the widths of the peaks are different due to different loop exponents. []{data-label="fig:3"}](031_rna_salt_melting_curve_tRNA-phe_Privalov_gs_image) In this section we calculate melting curves for different salt concentrations by applying [eqs.]{}  and , which include our salt dependent free energy parameterization. In [fig.]{} \[fig:3\] we compare experimental results [@Privalov1978; @Vives2002] with our predictions for the heat capacity of yeast tRNA-phe; the sequence is given in the supporting material section D. The heat capacity is readily obtained by $${C}= {T}\frac{\partial^2{\mathrm{k_B}T}\ln {Z}_N}{\partial{T}^2} {\;},\label{eq:13}$$ where ${Z}_{N}$ is the unrestricted partition function of the RNA at zero force, [eq.]{} . In all our calculations, we use the same literature parameter set for the stacking and pairing free energy ${{{ g}_{\mathrm{h}}}^{\mathrm{stack}}}$. No additional fit parameter enters except the loop initialization free energy ${{\mathcal G}_{\mathrm{l}}^{\mathrm{init}}}$, which is determined in [fig.]{} \[fig:2\]a from a separate experimental data set. The salt dependence of the experimentally observed melting temperatures is reproduced well, compare [fig.]{} \[fig:3\]. The arrows indicate additional experimental results [@Vives2002] for the melting temperature for ${\rho}= \unit{150}{\milli\molar}$ and $\unit{1}{\molar}$, which again coincide with our prediction. We also plot a calculated melting curve for loop exponent ${c}= 0$ and NaCl concentration ${\rho}= \unit{150}{\milli\molar}$, which exhibits a far less cooperative transition than observed in the corresponding curve with ${c}= 2.1$. Finally, we compare our prediction for ${\rho}= \unit{1}{\molar}$ and ${c}= 2.1$ with the prediction of RNAheat in the Vienna Package [@Hofacker1994] for ${\rho}= {1}{\molar}$, which uses a linearized multi-loop entropy amounting to ${c}= 0$ in our framework. The predicted melting temperatures are almost identical. However, the widths of the peaks in both melting curves differ and our melting profile for ${c}= 2.1$ is more peaked. Taking all these observations together leads to the conclusion that only a combined use of logarithmic loop entropy (characterized by a non-zero loop exponent) and salt dependent free energy corrections leads to a correct prediction of melting curves. The additional features in the experimental data, [[e.g. ]{}]{}the shoulder at lower temperatures and the increased width of the experimental curves might be attributed to tertiary structure rearrangements, which are not captured by our approach, or to melting occurring in multiple stages. Salt dependence of stretching curves {#sec:salt-depend-stretch} ==================================== Apart from temperature, force is an important variable to study denaturation of RNA molecules [@Li2008; @Liphardt2001; @Manosas2006; @Seol2007; @Tinoco2004; @Woodside2006; @Woodside2006a; @Hyeon2005; @Gerland2001; @Kumar2008a; @Mueller2002; @Montanari2001]. In [fig.]{} \[fig:4\] we show the salt dependence of stretching curves for yeast tRNA-phe. The stretching curves have been obtained by describing the force response of the ${M}$ non-nested backbone bonds, see [fig.]{} \[fig:1\], with the freely jointed chain (FJC) model, see [eq.]{} , $$\label{eq:14} x({F}) = {\mathrm{k_B}T}\frac{\partial \ln {Z}_N}{\partial{F}} = {\mathrm{k_B}T}\frac{\partial\ln{Z}_N}{\partial{{{ g}_{\mathrm{f}}}^{\mathrm{FJC}}}}\frac{\partial{{{ g}_{\mathrm{f}}}^{\mathrm{FJC}}}}{\partial{F}} = M x^{\mathrm{FJC}}({F}) {\;},$$ where we used the expectation value of the number of non-nested backbone segments $$\label{eq:15} {M}= -{\mathrm{k_B}T}\frac{\partial\ln{Z}_N}{\partial{{{ g}_{\mathrm{f}}}^{\mathrm{FJC}}}} {\;}.$$ ![Salt dependence of stretching curves of tRNA-phe for different salt concentrations ${\rho}= \unit{20}{\milli\molar},\,\unit{150}{\milli\molar},\,\unit{1}{\molar}$. Increasing salt concentration stabilizes the secondary structure due to screening of the electrostatic interaction. The dotted line is the theoretical prediction for the force extension curve of a freely jointed chain, [eq.]{} . The deviation of the predicted curves for RNA from the FJC curve is due to the presence of secondary structure. The observed plateau force is due to the rupture of the secondary structure. We show the force extension curves in a a linear plot and b a double logarithmic plot, indicating that the force extension curve is linear in the low-force regime, before the secondary structure is ruptured apart.[]{data-label="fig:4"}](044_rna_combine_34-35_gs_image) As for the melting curves, one observes that increasing salt concentration stabilizes the structure, leading to higher unfolding forces. All curves converge in the large force limit to a freely jointed chain of the length of the whole RNA molecule $(N-1){l_{\mathrm{ss}}}$, where $N = 76$ is the number of bases in the chain. The deviation for small forces from this theoretical prediction is due to the secondary structure of RNA, which is present at small forces and which becomes disrupted at forces ${F}\gtrsim \unit{\text{3-7}}{\pico\newton}$. In [fig.]{} \[fig:5\]a we show the force extension curve of the P5ab hairpin [@Liphardt2001]; the sequence is given in the supporting material section D. Apart from the salt dependence of the force extension curve, one observes that the unzipping of the helix occurs in two stages. This is seen best by considering the fraction of non-nested segments and its derivative, [fig.]{} \[fig:5\]b. The first stage is a smooth unzipping of the first three base pairs up to the bulge loop visible as a shoulder at ${F}\approx{8}{\pico\newton}$ in the derivative. The second stage is a sharp transition, where the rest of the hairpin unzips. In [fig.]{} \[fig:5\]c we show mfe predictions for the secondary structure at different forces for ${\rho}= \unit{1}{\molar}$ NaCl. For ${F}< \unit{5}{\pico\newton}$, we predict correctly the experimentally observed native state with all base pairs intact [@Liphardt2001]. For forces ${F}\approx\unit{8}{\pico\newton}$, an intermediate state appears, where the first three base pairs are unzipped up to the bulge loop. Denaturation is observed for ${F}\gtrsim \unit{14}{\pico\newton}$. The native structure of the P5ab hairpin contains the stacked pairs ${{\mathrm{GG}}\atop{\mathrm{AA}}}$ – bp[(17,42)]{} and bp[(18,41)]{} [@Liphardt2001]. For this stack, no free energy parameters are available and we use the parameters for the stack ${{\mathrm{GG}}\atop{\mathrm{UU}}}$, instead. However, other parameterizations for this stack work equally well and reproduce the experimental transition force within errors, see [fig.]{} \[fig:9\]. ![a Salt dependence of stretching curves of the 56 bases long RNA hairpin P5ab [@Liphardt2001] for different salt concentrations ${\rho}= \unit{20}{\milli\molar},\,\unit{150}{\milli\molar},\,\unit{1}{\molar}$. Increasing salt concentration stabilizes the secondary structure due to screening of the electrostatic interaction. The dotted line is the force extension curve of an FJC, [eq.]{} . b The fraction of non-nested segments ${M}/{N}$ as a function of force. One observes that the unzipping of the hairpin occurs in two stages, which is visible as a shoulder for $\unit{5}{\pico\newton}\lesssim{F}\lesssim \unit{6-10}{\pico\newton}$ (exact values depend on the salt concentration) and a successive cooperative transition. The inset shows the derivative ${\mathrm{d}}{M}/({N}{\mathrm{d}}{F})$ for ${\rho}= \unit{1}{\molar}$, where the first transition is visible as a shoulder at ${F}\approx\unit{8}{\pico\newton}$. The sharp peak at ${F}=\unit{11}{\pico\newton}$ is the rupture of the complete helix. c Predicted minimum free energy structures of the hairpin P5ab at different forces, see also the supporting material section A. For ${F}< \unit{5}{\pico\newton}$ the hairpin is in the native state with all base pairs intact. At ${F}\approx\unit{8}{\pico\newton}$ the first helix, consisting of three base pairs and bounded by the bulge loop, is ruptured. This causes the first smooth transition. Forces ${F}\gtrsim \unit{14}{\pico\newton}$ lead to the unzipping of the whole hairpin in a very cooperative fashion.[]{data-label="fig:5"}](045_rna_combine_32-33-36_gs_image) ![The effect of different parameterizations for the free energy parameters for ${{\mathrm{GG}}\atop{\mathrm{AA}}}$ on the denaturation curve is marginal. Here, the fraction of non-nested backbone bonds is plotted against the force for the P5ab hairpin and ${T}= \unit{298}{\kelvin}$, ${\rho}= {250}{\milli\molar}$, ${c}= 2.1$. The solid line is obtained with the parameters used for the P5ab hairpin in the rest of this paper ${{{ g}_{\mathrm{h}}}^{\mathrm{stack}}}({{\mathrm{GG}}\atop{\mathrm{AA}}}) = {{{ g}_{\mathrm{h}}}^{\mathrm{stack}}}({{\mathrm{GG}}\atop{\mathrm{UU}}})$. The dotted line is obtained by using ${{{ g}_{\mathrm{h}}}^{\mathrm{stack}}}({{\mathrm{GG}}\atop{\mathrm{AA}}}) = {{{ g}_{\mathrm{h}}}^{\mathrm{stack}}}({{\mathrm{UU}}\atop{\mathrm{AA}}})$, whereas the dashed line is for ${{{ g}_{\mathrm{h}}}^{\mathrm{stack}}}({{\mathrm{GG}}\atop{\mathrm{AA}}}) = 0$. All three curves coincide and differ only slightly at the transition, exhibiting only marginally different transition forces, which all agree with the experimentally observed unfolding force within errors@Liphardt2001. The values of the free energy parameters are given in the supporting material.[]{data-label="fig:9"}](123_rna_salt_compare_different_parameterizations_for_GAGA_in_P5ab_gs_image) Phase diagrams of RNA hairpin P5ab ================================== With the tools established in the previous sections, we are now able to study phase diagrams of RNA. We consider the P5ab hairpin, which is a well studied system [@Liphardt2001; @Gerland2003; @Hyeon2005; @Wen2007; @Cocco2003]. In [fig.]{} \[fig:6\]b the phase diagram in the ${F}$-${\rho}$ plane is shown for ${T}= \unit{298}{\kelvin},\,\unit{300}{\kelvin},\,\unit{320}{\kelvin}$ and ${c}= 2.1$. The phase boundary is defined as the force where half of the helical section is unzipped. For the definition of the phase boundary, we exclude the three unpaired bases at the 5’- and the four bases at the 3’-end, see [fig.]{} \[fig:5\]c, and use the condition ${M}- 7 = ({N}-7)/2$. This threshold value of ${M}/{N}$ is depicted by an arrow in [fig.]{} \[fig:6\]a. Below the phase boundary, the hairpin is stable, above the molecule is denatured. In [fig.]{} \[fig:8\]b we additionally include the experimental results by @Liphardt2001 agreeing nicely with our results. It is important to note, that this transition is not a phase transition in the strict statistical mechanics sense, but just a crossover. A true phase transition is defined as a non-analyticity of the free energy, which can only occur for an infinite system with long-range interactions [@Einert2008]. The three-dimensional phase space we are considering is spanned by temperature, force, and salt concentration. In [figs.]{} \[fig:7\] and \[fig:8\] we show slices in the ${F}$-${T}$ and in the ${T}$-${\rho}$ plane. The phase boundary for the ${F}$-${T}$ plane is determined the same way as in the ${F}$-${\rho}$ plane, yet with varying temperature and fixed salt concentration. The phase boundary in the ${T}$-${\rho}$ plane is determined differently: heat capacity curves as a function of temperature are calculated for different salt concentrations. The position of the peaks in the heat capacity curves (one is depicted by an arrow in [fig.]{} \[fig:8\]a) determine the phase diagram in [fig.]{} \[fig:8\]b. Therefore, slight differences between the phase diagrams in [figs.]{} \[fig:6\], \[fig:7\] on the one hand and [fig.]{} \[fig:8\] on the other hand may arise. We observe that for large salt concentrations, the denaturation forces and temperatures are rather independent of the salt concentration, see [figs.]{} \[fig:6\] and \[fig:8\]. Only when the Debye screening length ${\kappa}^{-1}$ is of the order of the typical length scale of RNA, which is the case for ${\rho}\lesssim\unit{100}{\milli\molar}$, a marked dependence on the salt concentration is observed. ![a Fraction of non-nested segments of the P5ab hairpin as a function of force for different salt concentrations and constant temperature ${T}= \unit{298}{\kelvin}$. The position of the crossover, which is defined as the point where ${M}- 7 = ({N}-7)/2$, [[i.e. ]{}]{} ${M}/{N}=0.56$ (indicated by the arrow), determines the phase diagram. b Phase diagram of the P5ab hairpin in the ${F}$-${\rho}$ plane for different temperatures ${T}= \unit{298}{\kelvin},\,\unit{300}{\kelvin},\,\unit{320}{\kelvin}$. Below the curve the RNA is in the hairpin phase, above the RNA is denatured. The symbol at ${\rho}= \unit{250}{\milli\molar}$, ${F}=\unit{13.3}{\pico\newton}$, and ${T}= \unit{298}{\kelvin}$ denotes the experimental data by @Liphardt2001 and coincides with our prediction.[]{data-label="fig:6"}](046_rna_combine_37-38_gs_image) ![a Fraction of non-nested segments of the P5ab hairpin as a function of force for different temperatures and constant salt concentration ${\rho}= \unit{250}{\milli\molar}$. The position of the crossover (arrow, ${M}/{N}= 0.56$) determines the phase boundary. With increasing temperature a decrease of the denaturation force is observed. Above the melting temperature ${{T}_{\mathrm{m}}}\approx\unit{358}{\kelvin}$ the molecule is always in the denatured state. b Phase diagram of the P5ab hairpin in the ${F}$-${T}$ plane. Below the curve the RNA is in the native hairpin phase, above the RNA is denatured. The symbol denotes experimental values [@Liphardt2001].[]{data-label="fig:7"}](047_rna_combine_39-40_gs_image) ![a Heat capacity curves for different salt concentrations as a function of temperature. The peak position moves to higher temperatures with increasing salt concentration. The positions of the peaks, denoted exemplarily for the curve with ${\rho}=\unit{20}{\milli\molar}$ by the arrow, determine the phase diagram. b Phase diagram of the P5ab hairpin in the ${T}$-${\rho}$ plane. Below the curve the RNA is in the native hairpin phase, above the RNA is denatured.[]{data-label="fig:8"}](048_rna_combine_41-42_gs_image) Conclusions {#sec:conclusions} =========== We construct a theory for RNA folding and melting that includes the effects of monovalent salt, loop entropy, and stretching forces. Our theory is based on salt and temperature dependent modifications of the free energies of RNA helices and loops that include electrostatic interactions on the linear Debye-Hückel level – augmented by Manning condensation – and conformational fluctuation effects [[via]{} ]{}the asymptotic, non-linear expression for the entropy of loop formation. Decreasing salt concentration is shown to generally destabilize RNA folds and to lower denaturation temperatures and forces. The predictions are in good agreement with experimental data as shown for two different scenarios, namely the heat capacity curves for the thermal denaturation of tRNA-phe and the response of the P5ab RNA hairpin to an external pulling force. Due to the usage of the linear Debye-Hückel approximation in conjunction with the Manning condensation concept, our approach is limited to monovalent salt and neglects ion-specific effects. Electrostatic nonlinear and correlation effects could in principle be taken into account by more advanced modeling using variational approaches [@Netz2003], while ion-specific effects could be straightforwardly included using effective interactions between different ions and RNA bases [@Schwierz2010]. More complex phenomena involving multivalent ions such as Mg^2+^ could in principle be modeled by allowing for a few tertiary contacts, which is left for future studies. We find that for a proper description of RNA melting curves, correct modeling of the loop entropy is crucial. A non-zero loop exponent leads to an increased cooperativity of the melting transition and thus makes the heat capacity curve narrower in good agreement with experimental results. We conclude that for a correct description of RNA denaturation thermodynamics, both loop entropy and salt effects are important and should be included in standard structure and melting curve prediction software. Acknowledgements {#sec:acknowledgements} ================ Financial support comes from the DFG [[via]{} ]{}grant NE 810/7. T.R.E. acknowledges support from the Elitenetzwerk Bayern within the framework of CompInt.

--- abstract: | Privacy has been frequently identified as a main concern for system developers while dealing with/managing personal information. Despite this, most existing work on privacy requirements deals with them as a special case of security requirements. Therefore, key aspects of privacy are, usually, overlooked. In this context, wrong design decisions might be made due to insufficient understanding of privacy concerns. In this paper, we address this problem with a systematic literature review whose main purpose is to identify the main concepts/relations for capturing privacy requirements. In addition, the identified concepts/relations are further analyzed to propose a novel privacy ontology to be used by software engineers when dealing with privacy requirements. Keywords {#keywords .unnumbered} ======== Privacy Ontology, Privacy Requirements, Privacy by Design (PbD), Requirements Engineering author: - Mohamad Gharib - Paolo Giorgini - John Mylopoulos title: 'Ontologies for Privacy Requirements Engineering: A Systematic Literature Review' --- Introduction ============ Increasing numbers of today’s systems deal with personal information (e.g., information about citizens, customers, etc.), where such information is protected by privacy laws [@gharibre2016]. Therefore, privacy has become a main concern for system designers. In other words, dealing with privacy related concerns is a must these days because privacy breaches may result in huge costs as well as a long-term consequences [@acquisti2006there; @gellman2002privacy; @hong2004privacy; @camp2002designing; @campbell2003economic]. Privacy breaches might be due lack of appropriate security policies, bad security practices, attacks, data thefts, etc. [@acquisti2006there; @labda2014modeling]. However, most of these breaches can be avoided if privacy requirements of the system-to-be were captured properly during system design (e.g., Privacy by Design (PbD)) [@cavoukian2009privacy; @cavoukian2011privacy; @labda2014modeling], where privacy requirements aim to capture the types and levels of protection necessary to meet the privacy needs of the users. Nevertheless, just few works focused on considering privacy during the system design [@Gurses2011]. More specifically, most existing work on privacy requirements often deal with them either as non-functional requirements (NFRs) with no specific criteria on how such requirements can be met [@anton2002analyzing; @yu2002designing; @mouratidis2007secure], or as a part of security [@zannone2006requirements; @kalloniatis2008addressing], i.e., focusing mainly on confidentiality and overlooking important privacy aspects such as anonymity, pseudonymity, unlinkability, unobservability, etc. On the other hand, privacy is an elusive and vague concept [@solove2002conceptualizing; @solove2006taxonomy; @kalloniatis2008addressing]. Although several efforts have been made to clarify the privacy concept by linking it to more refined concepts such as secrecy, person-hood, control of personal information, etc., there is no consensus on the definition of these concepts or which of them should be used to analyze privacy [@solove2006taxonomy]. This has resulted in a lot of confusion among designers and stakeholders, which has led in turn to wrong design decisions. In this context, a well-defined privacy ontology that captures privacy related concepts along with their interrelations would constitute a great step forward in designing privacy-aware systems. Ontologies have been proven to be a key success factor for eliciting high-quality requirements, and it can facilitate and improve the job of requirements engineers [@souag2012towards; @kaiya2006using; @dzung2009ontology], since it can reduce the conceptual vagueness and terminological confusion by providing a shared understanding of the related concepts between designers and stakeholders [@uschold1996ontologies]. In addition, the ontology should capture privacy requirements in their social and organizational context. Since most complex systems these days (e.g., healthcare systems, smart cities, etc.) are socio-technical systems [@emery1960socio], which consist not only of technical components but also of humans along with their interrelations, where different kinds of vulnerabilities might manifest themselves [@liu2003security; @gharibre2016]. Focusing on the technical aspects and leaving the social and organizational aspects outside the system’s boundary leaves the system open to different kinds of vulnerabilities that might manifest themselves in the social interactions and/or the organizational structure of the system [@liu2003security]. The Flash Crash [@sommerville2012large] and the Allied Irish Bank scandal [@massacci2008detecting] are good examples, where problems were not caused by mere technical failures, but it were also due to several socio-technical related vulnerabilities of the system. This paper applies systematic review techniques to survey available literature to identify the most mature studies that propose privacy ontologies/concepts. In addition, we further analyze the selected privacy related concepts/relations to identify the main ones in order to propose a novel ontology that can be used to capture privacy requirements. This paper is therefore intended to be a starting point to address the problem of identifying a core privacy ontology. The rest of the paper is organized as follows; Section (2) describes the review process and the protocol underlining this systematic review. We present and discuss the review results and findings in Section (3). In Section (4) we propose a novel ontology for privacy requirements engineering. We discuss threats to validity in Section (5). Related work is presented in Section (6), and we conclude and discuss future work in Section (7). Review Process ============== A systematic review can be defined as a systematic process for defining research questions, searching the literature for the best available resources to answer such questions, and collecting available data from the resources for answering the research questions. Following [@kitchenham2004procedures; @keele2007guidelines], the review process (depicted in Figure \[fig:plan\]) consists of three main phases: 1. Planning the review, in which we formulate the research questions and we define the review protocol. 2. Conducting the review, in which we conduct the search process after identifying the search terms and the literature sources, and then we perform the study selection activity. 3. Reporting the results of the review, in which we collect detailed information from the selected studies in order to answer the research questions, and then we use the obtained data to answer the research questions, which we discuss in the following section. Planning the review ------------------- This phase is very important for the success of the review, for it is here that we define the research objectives and the way in which the review will be carried out. This includes two main activities: (1) formulating the research questions that the systematic review will answer; and (2) defining the review protocol that specifies the main procedures to be taken during the review. ### Research questions Formulating the review questions is a very critical activity since these questions are used to derive the entire systematic review methodology [@kitchenham2004procedures]. Therefore, we formulate the following four Research Questions (RQ) to identify the main privacy concepts that have been presented in the literature: RQ1 : What are the privacy concepts/relations that have been used to capture privacy concerns? RQ2 : What are the main concepts/relations that have been used for capturing privacy requirements? RQ3 : Do existing privacy studies cover the main privacy concepts/relations? RQ4 : What are the limitations of existing privacy studies? ### Define the review protocol The review protocol specifies the methods to be followed while conducting the systematic review. Based on [@kitchenham2004procedures; @keele2007guidelines], a review protocol should specify the following: the strategy that will be used to search for primary studies selection; study selection criteria; study quality assessment criteria; data extraction and dissemination strategies. In the rest of this section, we discuss how we specify and perform each of these activities. Conducting the review --------------------- This phase is composed of two main activities: 1- search strategy; and 2- study selection, where each of them is composed of several sub-activities. In what follows, we discuss them. ### Search strategy The search strategy aims to find as many studies relating to the research questions as possible using an objective and repeatable search strategy [@kitchenham2004procedures]. The search activity consists of three main sub-activities: 1- identify the search terms, 2- identify the literature resources, and 3- conduct the search process. **Identify the search terms.** Following [@kitchenham2004procedures; @keele2007guidelines], we derived the main search terms from the research questions. In particular, we used the Boolean AND to link the major terms, and we use the Boolean OR to incorporate alternative synonyms of such terms. The resulting search terms are: (Privacy AND (ontology OR ontologies OR taxonomy OR taxonomies ) OR (Privacy requirements). **Identify the literature resources.** Six electronic database resources were used to primarily extract data for this research. These include: IEEE Xplore, ACM Digital Library, Springer, ACM library, Google Scholar, and Citseerx. **Conduct the search process.** The search process (shown in Figure \[fig:process\]) consists of two main stages: Search stage 1. : We have used the search terms to search the six electronic database sources, and only papers with relevant titles have been selected; Search stage 2. : The reference lists of all primary selected papers were carefully checked, and several relevant papers (25 papers) were identified and added to the list of the primary selected papers. **Study selection.** The selection process (shown in Figure \[fig:process\]) consists of two main stages. Selection stage 1 (primary selection). : Searching the electronic database source returned 240 relevant papers, among which we have identified and removed 33 duplicated papers. Next, we have applied the primary selection criteria on the remaining 207 papers. In particular, we have read the abstract, introduction, and then we skimmed through the rest of paper. We removed all the papers that are not published in the English language, and we excluded all papers that are not related to any of our research questions. Moreover, when we were able to identify multiple version of the same paper, only the most complete one was included. Finally, we excluded any paper that has been published before 1996, since we were not able to find any concrete work related to our research before 1996. The primary selection inclusion and exclusion criteria are shown in Table \[table:inexcriteria\]. The outcome of this selection stage was 107 papers, i.e., we have excluded 100 papers. Selection stage 2 (Quality Assessment (QA)). : At this stage, the QA criteria has been applied to the papers that have resulted from the first selection stage (107 papers) along with the papers that have resulted from the second search stage (25 papers), for a total of 132 papers. In order to identify the most relevant studies that can be used to answer our research questions, we formulated five QA questions (shown in Table \[table:qualityassessment\]) to evaluate the relevance, completeness, and quality of the studies, where each question has only two answers: Yes = 1 or No= 0. The quality score for each study is computed by summing the scores of its QA questions, and the paper is selected only if it scored at least 4. As a result, 98 papers were excluded and 34 studies were selected. The result of the QA of the studies is presented in Table \[table:Quality\] in Appendix A. \[table:inexcriteria\] \[table:qualityassessment\] Reporting the results --------------------- The final phase of the systematic review involves summarizing the results, and it consists of two main activities: 1- data synthesis; and 2- results and discussion. ### Data synthesis In what follows, we describe how data syntheses were executed: Data related to *RQ1* can be extracted directly from the list of selected papers (shown in Table \[table:selected\]). To answer *RQ2*, the contents of the 34 selected studies were further analyzed to identify privacy related concepts along with their interrelations, and list them in a comprehensive table (Table \[table:rq\]). Moreover, we identify the main concepts/relations for capturing privacy requirements based on Table \[table:rq\] & Table \[table:iteration\] that shows the frequency of concepts/relations appearance in the selected studies. To answer *RQ3* data can be derived from Table \[table:limitation\], which summaries the percentage of the main concepts/relations categories that each selected study cover. *RQ4* can be answered by categorizing the studies into four group based on the concepts categories they do not cover. Review results and discussion ============================= This section presents and discusses the findings of this review. First, we start by presenting an overview of the selected studies, and then, we present the findings of this review concerning the research questions. **Overview of selected studies[^1].** 34 studies were selected, where 5 studies were from book chapters, 10 papers were published in journals, 11 papers appeared in conference proceedings, 6 papers came from workshops, and 2 papers were extracted from symposiums. The number of papers by year of publication is presented in Figure \[fig:pubyear\]; while the percentages of the selected studies based on their publishing type are represented in Figure \[fig:pie\]. ![Number of papers by year of publication[]{data-label="fig:pubyear"}](pubyear.eps){width="70.00000%"} **RQ1:** *What are the privacy concepts/relations that have been used to capture privacy concerns?* The review has identified 34 studies that provide concepts and relations that can be used for capturing privacy requirements. The list of the selected studies that answers our first research question (*RQ1*) is presented in Table \[table:selected\], where each paper is described by its identifier, title, author(s), publication year and number of citation. In what follows, we present a summary of the contributions of each selected study. \[table:selected\] **ACM\_03 [@van2004elaborating],** : “*Elaborating Security Requirements by Construction of Intentional Anti-Models*”. Lamsweerde [@van2004elaborating] proposed a goal-oriented approach that extends the KAOS framework for modeling and analyzing security requirements. The framework focus on generating and resolving obstacles/anti-goals to goal satisfaction, i.e., it addresses malicious obstacles/anti-goals (threats) set up by attackers to threaten security goals, and the new security requirements are obtained as countermeasures to resolve these obstacles/anti-goals (threats). The framework adopts several main concepts from KAOS (e.g., agents, goals, etc.) and proposes concepts for building intentional threat models (e.g., obstacles, anti-goal, anti-requirements, attacker, etc.). **ACM\_14 [@labda2014modeling],** : “*Modeling of Privacy-aware Business Processes in BPMN to Protect Personal Data*”. Labda et al. [@labda2014modeling] propose a privacy-aware Business Processes (BP) framework for modeling, reasoning and enforcing privacy constraints. They have identified several privacy-related concepts, including: *Data*, *User*, *Action*, *Purpose*, and *Permissions*. In addition, they identify five concepts that can be used for analysis privacy in BP: (1) *Access control*, (2) *Separation of Tasks (SoT)*, (3) *Binding of Tasks (BoT)*, (4) *User consent*, (5) *Necessity to know (NtK)*. **ACM\_16 [@braghin2008introducing],** : “*Introducing Privacy in a Hospital Information System*”. Braghin et al. [@braghin2008introducing] presented an approach that supports expressing and enforcing privacy-related policies. The approach extends the conceptual model of an open source hospital information system (Care2x) with concepts for role-based privacy management (e.g., subject, processor, and controller), and concepts for supporting the privacy enforcement mechanisms (actions), where such actions can be either inactive or declarative, where the former includes actions that require to access and process data, while the latter includes simple statements representing activities that do not require to interact with the system. **ACM\_35 [@singhal2010ontologies],** : “*Ontologies for Modeling Enterprise Level Security Metrics*”. Singhal and Wijesekera [@singhal2010ontologies] provide a security ontology that supports IT security risk analysis. The ontology identifies which threats endanger which assets and what countermeasures can reduce the probability of the occurrence of a related attack. The concepts of the ontology, includes: *threat*, a potential violation of security, an *attack* exploits vulnerabilities to realize a threat, where *vulnerabilities* are characteristics of target assets that make them prone to attack, and a *risk* is an expectation of loss expressed as a probability that a particular threat will exploit a certain vulnerability, which will result in a harmful result. Finally, *security mechanisms* are designed to prevent threats from happening or mitigating their impact when they occur. **ACM\_40 [@wang2009ovm],** : “*OVM: An Ontology for Vulnerability Management*”. Wang and Guo [@wang2009ovm] propose an ontology for vulnerability management (OVM) that capture the fundamental concepts in information security and their relationship, retrieve vulnerable assets (data) and reason about the cause and impact of such vulnerabilities. The ontology has been built based on the Common Vulnerabilities and Exposures (CVE), Common Weakness Enumeration (CWE), Common Platform Enumeration (CPE), and Common Attack Pattern Enumeration and Classification (CAPEC). The top level concepts of the ontology includes, a *Vulnerability* existing in an *IT\_Product* that can be exploited by an *Attacker* through an *Attack* that compromises the *IT\_Product* and cause *Consequence*. Moreover, *Countermeasures* can be used to protect the *IT\_Product* through mitigating the *Vulnerability*. **CIT\_07 [@velasco2009modelling],** : “*Modeling Reusable Security Requirements Based on an Ontology Framework*”. Velasco et al. [@velasco2009modelling] propose an ontology-based framework for representing and reusing security requirements based on risk analysis. The ontology is based on two ontologies: 1- the risk analysis ontology that is developed based on MAGERIT [@magerit2006methodology], and identifies five types of risk elements: *asset*, *threat*, *safeguard*, *valuation dimension*, *valuation criteria*, and 2- the requirements ontology that models reusable requirements along with their relationships. **CIT\_33 [@liu2003security],** : “*Security and Privacy Requirements Analysis within a Social Setting*”. Liu et al. [@liu2003security] propose a framework for dealing with security and privacy requirements within an agent-oriented modeling framework. They extend *i*\* modeling language to deal with security and privacy requirements, where *i*\* language allows for analyzing security/privacy issues within their social context, which enables for a systematic way of deriving vulnerabilities and threats. Moreover, *i*\* models make it possible to conduct different countermeasure analyses for addressing vulnerabilities and suggesting countermeasures for them. **IEEE\_12 [@souag2013using],** : “*Using Security and Domain ontologies for Security Requirements Analysis*”. Souag et al. [@souag2013using] introduce an ontology-based method for discovering Security Requirements (SR). The process that underlies this method has three main steps, and it starts with the elicitation step that constructs an initial *i*\* requirements model from the stakeholders’ needs/concerns about security. The second step is the SR analysis that depends on production rules to exploit the security-specific ontology to discover threats, vulnerabilities, countermeasures, and resources. These concepts are used to enrich the requirements model by adding new elements (malicious tasks, vulnerability points, etc.). Finally, the domain specific SR analysis step, in which another set of rules explores the domain ontology to improve the requirements model with resources, actors and other concepts that are more specific to the domain at hand. **IEEE\_15 [@tsoumas2006towards],** : “*Towards an Ontology-based Security Management*”. Tsoumas and Gritzalis [@tsoumas2006towards] introduce a security management framework that proposes a Security Ontology (SO), which contains the following concepts, a *stakeholder* possesses an *asset*, which in turn can be compromised by a *vulnerability*. While a *threat* initiated by a *threat agent* targets an *asset* and exploits a *vulnerability* of the asset in order to achieve its goal. Exploitation of a *vulnerability* leads to the realization of an unwanted *incident*, which has a certain *impact*. Furthermore, *countermeasures* reduce the impact of the *threat* with the use of *controls*. Finally, *security policy* formulates the *controls* into a manageable security framework possessed by *stakeholders*. **IEEE\_50 [@Giorgini2005],** : “*Modeling Security Requirements through Ownership, Permission and Delegation*”. Giorgini et al. [@Giorgini2005] introduce Secure Tropos, a formal framework for modeling and analyzing security requirements in their social and organizational context. Secure Tropos proposes several concepts including, an *actor* that covers two concepts (a *role* and an *agent*), a *goal* that can be refined through and/or-decompositions of a root *goal* into finer *sub-goals*, a *task*, and a *resource*. Secure Tropos adopts the notion of *delegation* to model the transfer of objectives (*goals* and *tasks*) from one actor to another, and it adopts *resource provision* among actors. Moreover, it introduces the *ownership* concept that capture the relation between *actors* and *resources* they own. Finally, it provides the *trust* concept to capture the *actors’* expectations in one another concerning their social dependencies, and it introduce the *monitoring* concept to compensate the lack of trust/distrust among *actors* concerning social dependencies. **IEEE\_57 [@kang2013security],** : “*A Security Ontology with MDA for Software Development*”. Kang and Liang [@kang2013security] propose security ontology for software development based on Model Driven Architecture (MDA) paradigm. The ontology includes most popular security concerns mentioned in literature such as *auditing*, *threats*, *accountability*, *non-repudiation*, *risk*, *attacks*, *availability*, *frauds*, *confidentiality*, *asset*, *integrity*, *prevention*, and *Reputation*. **SCH\_18 [@kang2013security],** : “*Eliciting Security Requirements with Misuse Cases*”. Sindre and Opdahl [@kang2013security] present a systematic approach to eliciting security requirements based on *use cases*. They extend the traditional *use case* approach to also consider *misuse cases* that represent unwanted behavior in the system to be developed. In particular, a *use case* diagram contains both, *use cases* and *actors*, as well as *misuse cases* and *misusers*. In addition, *misuse cases* adopts the ordinary *use case* relationships such as *include*, *extend*, and *generalize*. A *use case* is related to a *misuse case* using a directed *association*, which means that a *misuse case* *threatens* the *use case*. Moreover, a use case diagram can contain *security use cases*, which are special *use cases* that can *mitigate* *misuse cases*. In summary, an ordinary *use cases* represent requirements, *security cases* represent security requirements, and *misuse cases* represent security *threats*. **SCH\_24 [@kalloniatis2008addressing],** : “*Addressing Privacy Requirements in System Design. the PriS Method*”. Kalloniatis et al. [@kalloniatis2008addressing] introduce PriS, a security requirements engineering method that consider users’ privacy requirements. PriS considers privacy requirements as business goals and provides a methodological approach for analysing their effect onto the organizational processes. The conceptual model of PriS is based on the Enterprise Knowledge Development (EKD) framework [@loucopoulos1999enterprise], and it includes a set of concepts for modeling privacy requirements, such as: *stakeholders*, *goals* that can be either *strategic goals* or *operational goals*, and *goals* can be *realized* by *processes*. On the other hand, *privacy requirements* are a special type of *goals* (*privacy goals*), which constraint the causal transformation of organizational goals into processes. *Privacy goals* may be decomposed in simpler goals or may *support*/ *conflict* the achievement of other *goals*. Moreover, eight types of *privacy goals* have been identified corresponding to the eight privacy concerns namely, authentication, authorisation, identification, data protection, anonymity, pseydonymity, unlinkability, and unobservability. **SCH\_28 [@mouratidis2007secure],** : “*Secure Tropos: a Security-oriented Extension of the Tropos Methodology*”. Mouratidis and Giorgini [@mouratidis2007secure] introduce extensions to the Tropos methodology [@bresciani2004tropos] to model security concerns throughout the whole development process. Secure Tropos adopts from Tropos methodology concepts for modeling *actors*, *goals*, *resources*, along with their different relations and social dependencies. In addition, it introduces concepts for modeling security requirements, such as a *security constraint* (e.g., privacy, integrity, and availability), which can be decomposed into one or more security sub-constraints. *Security constraint* modeling is divided into *security constraint delegation*, *security constraint assignment*, and *security constraint analysis*. Secure Tropos also introduces *secure entity*, *security features*, *security mechanisms*, a *secure capability*, a *secure dependency*, and the *threat* concept. **SCH\_41 [@solove2006taxonomy],** : “*A Taxonomy of Privacy*”. Solove [@solove2006taxonomy] provides taxonomy for understanding a wide range of privacy related problems. The taxonomy specifies four main groups of possible harmful activities: **(i) information collection**: creates disruption based on the process of data gathering Two sub-classifications of information collection have been identified, *surveillance* and *interrogation*. **(ii) information processing**: refers to the use, storage, and manipulation of data that has been collected. Five different sub-classifications of information processing have been identified: *aggregation*, *identification*, *insecurity*, *secondary use*, and *Exclusion*. **(iii) information dissemination**: in which the data holders transfer the information to others. Seven different sub-classifications of information dissemination have been identified: *breach of confidentiality*, *disclosure*, *exposure*, *increased accessibility*, *blackmail*, *appropriation*, and 7- *distortion*. **(iv) invasion**: involves impingements directly on the individual. Two different sub-classifications of information invasion have been identified: *intrusion* and 2-*decisional interference*. **Spgr\_07 [@massacci2011extended],** : “*An Extended Ontology for Security Requirements*. Massacci et al. [@massacci2011extended] propose ontology for security requirements engineering, the ontology adopts concepts from Secure Tropos methodology [@massacci2007computer], Problem Frame [@haley2008security], and several industrial case studies. The most general concept in the ontology is *Thing*. An *object* is a *thing* that persists, and an *event* is an instantaneous happening that changes some *objects*. The *object* concept can be specialized into *proposition*, *situation*, *entity* and *relationship*. A *proposition* is an *object* representing a true/false statement. A *situation* is a partial world described by a *proposition*. An *entity* is an *object* that has a distinct, separate existence from all other *things*, though that existence need not be material. *Entity* is specialized into *Actor*, *Action*, *Process*, *Resource*, and *Asset*. *Relationship* can be specialized into *do-dependency*, *can-dependency*, *trust-dependency*, *and/or* refinement, *contributes*, *provides*, *uses*. In addition, *damages* is a *relationship* between an *attack* and an *asset*, where the *attack* causes *harm* to the *asset*. *Exploits* is a *relationship* between *attack* and *vulnerability*. *Protects* relates a *security goal* to an *asset*. Finally, *denies* relates an *anti-goal* to a *requirement*. Finally, a *specification* is an *entity* consisting of *actions*, quality propositions, and domain assumptions. *Vulnerability* is a specialization of Situation and is adopted from the Security domain. While a *threat*consists of a situation that includes an attacker and one or more vulnerabilities. **Spgr\_13 [@elahi2009modeling],** : “*A Modeling Ontology for Integrating Vulnerabilities into Security Requirements Conceptual Foundation*”. Elahi et al. [@elahi2009modeling] propose a vulnerability-centric modeling ontology, which integrates empirical knowledge of vulnerabilities into the system development process. They identify a set of core concepts for security requirements elicitation, and they identify another set of concepts for capturing vulnerabilities and their effects on the system. The ontology contains several concepts, including: a *concrete element* that is a tangible entity (e.g., an *activity*, *task*, etc.), and it may *bring* a *vulnerability* into the system. *Exploitation* of *vulnerabilities* can have effects on other elements (*affected elements*), where the *effect* relation is characterized by the *severity* attribute. An *attack* involves the execution of *malicious actions* that one or more *actors* perform to satisfy some *malicious goal*. A *concrete element* may have a *security impact* on *attacks*, which can be interpreted as a *security countermeasure* that can be used to patch *vulnerabilities*. **Spgr\_02\_01 [@asnar2007trust],** : “*From Trust to Dependability Through Risk Analysis*”. Asnar et al. [@asnar2007trust] present an extension of the Tropos Goal-Risk framework. In particular, they introduce an approach to assess risk on the basis of trust relations among actors. In particular, they introduce the notion of trust to extend the risk assessment process. Using this framework, an actor can assess the risk in delegating the fulfillment of his objectives and decide whether or not the risk is acceptable. They also introduce the notion of trust level proposing three trust levels: *Trust*, *Distrust*, and *NTrust* (i.e., neither trust nor distrust), where a low level of trust increases the risk perceived by the depender about the achievement of his objectives. **Spgr\_02\_02 [@asnar2006risk],** : “*Risk Modeling and Reasoning in Goal Models*”. Asnar et al. [@asnar2006risk] propose a goal-oriented approach for modeling and reasoning about risks at requirements level, where risks are introduced and analyzed along the stakeholders’ goals and countermeasures. Their proposed framework is based on the Tropos methodology and extends it with new concepts and qualitative reasoning mechanisms to consider risks since the early phases of the requirements analysis. In their framework, a *risk* is an *event* that has a negative impact on the satisfaction of a *goal*, while a *treatment* is a *countermeasure* that can be adopted in order to mitigate the effects of the *risk*. Moreover, they consider *likelihood* as a property of the *event*, and they capture the *likelihood* by the level of evidence that supports and prevents the occurrence of the *event* (SAT and DEN). On the other hand, *impact* is used to capture the influence of an event to the *goal* fulfillment, and they classify impact under: *strong positive*, *positive*, *negative*, and *strong negative*. **Spgr\_03\_01 [@avizienis2004basic],** : “*Basic Concepts and Taxonomy of Dependable and Secure Computing*”. Avizienis et al. [@avizienis2004basic] propose a new taxonomy for dependable and secure computing based on an extensive analysis of the related literature. The authors provide precise definitions characterizing the various concepts that come into play when addressing the dependability and security of computing and communication systems. The three top-level dimensions of this taxonomy are: *attribute*, *threat*, and *means*. The concept of *attribute* is analyzed in terms of: *availability*; *reliability*; *safety*; *confidentiality*; *integrity*; and *maintainability*. The concept of *threat* is further refined in terms of *fault*, *error*, and *failure*. While the concept of *means* is used to attain the various attributes of dependability and security, where these means can be grouped into four main categories: *fault prevention*; *fault tolerance*; *fault removal*; and *fault forecasting*. **Spgr\_07\_02 [@zannone2006requirements],** : “*A Requirements Engineering Methodology for Trust, Security, and Privacy*”. Zannone [@zannone2006requirements] introduces the Secure i\* (SI\*) methodology that adopts from Secure Tropos the concepts of *actors*, *goals*, *resources*, along with their different relations and social dependencies, and it proposes new relation among roles, namely *supervision*. In SI\*, an *actor* is defined along with a set of *objectives*, *capabilities*, and *entitlements*, which can be modeled through relations between actors and services (goals, tasks, and resources), namely: (1) *require* indicates that an actor intends to achieve a *service*, (2) *be entitled* indicates that an *actor* is the legitimate *owner* of a *service*, and (3) *provide* indicates that the *actor* has the capability to achieve a *service*. The delegation concept is refined in SI\* into: (1) *Delegation of execution (De)*, and (2) *Delegation of permission (De)*. In addition, the trust concept is refined to cope with the refinement of delegation they propose into: (1) *Trust of execution (Te)*, and (2) *Trust of permission (Tp)*. **Spgr\_07\_03 [@lin2003introducing],** : “*Introducing Abuse Frames for Analysing Security Requirements*”. Lin et al. [@lin2003introducing] develop an approach using Problem Frames to analyze security problems in order to determine security vulnerabilities. In particular, they introduce the notion of an anti-requirement as the requirement of a malicious user that can subvert an existing requirement, and they incorporate anti-requirements into abuse frames to represent the notion of a security threat that is imposed by malicious users in a particular problem context. **Spgr\_08\_01 [@mayer2009model],** : “*Model-based Management of Information System Security Risk*”. Mayer [@mayer2009model] proposes ISSRM (Information System Security Risk Management), a security risk management model. The ISSRM reference model addresses risk management at three different levels, combining together *asset*, *risk*, and *risk treatment* views, and it proposes concepts that are ordered in three main groups: **(i) Asset-related concepts** describe what assets are important to protect, and what criteria guarantee asset security; **(ii) Risk-related concepts** present how the risk itself is defined. A *risk* is the combination of a *threat* with one or more *vulnerabilities* leading to a negative impact harming the *assets*; and **(iii) Risk treatment-related concepts** describe what decisions, requirements and controls should be defined and implemented in order to mitigate possible *risks*. **Spgr\_08\_03 [@dritsas2006knowledge],** : “*A Knowledge-based Approach to Security Requirements for E-health Applications*”. Dritsas et al. [@dritsas2006knowledge] propose an ontology that includes the main security related concepts, and use the ontology for designing and developing a set of security patterns that address a subset of these requirements for applications that provide e-health services. The concepts used in the proposed ontology includes: *stakeholder*, *objective*, *threat*, *countermeasure*, *asset*, *vulnerability*, *deliberate attack*, *security pattern* and *security pattern context*. A *security pattern* provides a specific set of *countermeasures*, and a *security pattern context* is defined as a set of *asset*, *vulnerability*, and *deliberate attack* triplets. Therefore, one can start from the generic *security objectives*, find the *security pattern contexts* that match them and choose specific *security pattern*, which ensures that the high level *security objectives* can be fulfilled by implementing the respective *countermeasures*. **Spgr\_13\_01 [@asnar2008risk],** : “*Risk as Dependability Metrics for the Evaluation of Business Solutions: a Model-driven Approach*”. Asnar et al. [@asnar2008risk] adopt and extend the Tropos Goal Model [@asnar2006risk; @asnar2007trust] by considering also the interdependency among the actors within an organization. Through this extension analysts can assess the risk perceived by each actor, taking into account the organizational environment where the actor acts. Based on such analysis, we have provided a method to assist analysts in determining the treatments to be introduced in order to make risks be acceptable by all actors. **Spgr\_13\_02 [@den2003coras],** : “*The CORAS Methodology: Model-based Risk Assessment Using UML and UP*”. Braber [@den2003coras] introduces the CORAS methodology in which the Unified Modeling Language (UML) and Unified Process (UP) are combined to support a model-based risk assessment of security-critical systems. The CORAS ontology propose several concepts, such as *context* that influences the *target*, which contains *assets* and has its *security requirements*. *Security requirements* lead to *security policies*, which protect *assets* by reducing its related *vulnerabilities* that can be exploited by *threats*, which might reduce the *value* of the *asset*. A *Risk* contains an *unwanted incident* that has a certain *consequence* and *frequency* of occurrence. **Spgr\_13\_03 [@elahi2010vulnerability],** : “*A Vulnerability-centric Requirements Engineering Framework: Analyzing Security Attacks, Countermeasures, and Requirements based on Vulnerabilities*. Elahi et al. [@elahi2010vulnerability] adopt and extend their previous work [@elahi2009modeling] by proposing an agent- and goal-oriented framework for eliciting and analyzing security requirements. They refined the goal model evaluation method that helps analysts verifying whether top goals are satisfied with the risk of vulnerabilities and attacks and assess the efficacy of security countermeasures against such risks. More specifically, the evaluation does not only specify if the goals are satisfied, but also makes it possible to understand why and how the goals are satisfied (or denied) by tracing back the evaluation to vulnerabilities, attacks, and countermeasures. **Spgr\_13\_04 [@jurjens2002umlsec],** : “*UMLsec: Extending UML for Secure Systems Development*”. J[ü]{}rjens [@jurjens2002umlsec] proposes UMLsec that is an extension to UML modeling language, which allows for integrating security requirements modeling and analysis within the system development process. UMLsec is able to model security related features such as secrecy, integrity, access control, etc. It represents security feature on UML diagrams by providing several extension mechanisms, namely: (1) stereotypes: a new types of modeling elements that extends the semantics of existing types in the UML meta-model; (2) tagged values: that is used to associate data with model elements and (3) constraints: that are used to define criteria to determine whether requirements are met or not by the system design. In UMLsec, integrity is modeled as a constraint, which can restrict unwanted modification (e.g., insert), but information quality can be affected in several other ways that cannot be captured by this approach. **Spgr\_13\_05 [@matulevivcius2008adapting],** : “*Adapting Secure Tropos for Security Risk Management in the Early Phases of Information Systems Development*”. Matulevi[č]{}ius et al. [@matulevivcius2008adapting] have analyzed how Secure Tropos can be applied to analyze security risks at the early IS development phases. Their analysis suggested a number of improvements for Secure Tropos in order to deal better with security risk management activities. In particular, Secure Tropos could be improved with additional constructs adopted from existing security risk management models (e.g., ISSRM (Information System Security Risk Management)) such as risk, risk treatment, and control. More specifically, among the suggested risk-related concepts is a risk that presents how the risk itself is defined, what are the major principles that should be taken into account when defining the possible risks. The risk is described by the cause of the risk, and the impact of the risk captures the potential negative consequence of the risk, which can be represented by a negative contribution link between the attack and the related security constraint, i.e., the impact negates the security criteria. **Spgr\_13\_07 [@rostad2006extended],** : “*An Extended Misuse Case Notation: Including Vulnerabilities*”. R[ø]{}stad [@rostad2006extended] proposes an extended misuse case notation that includes the ability to represent vulnerabilities and the insider threat. In particular, beside the main concepts of misuse case notation (e.g., *actors*, *use cases*, *misuse cases*, *misusers*, etc.). R[ø]{}stad introduce the *insider* concept to capture inside attackers, since the *misuser* concept in misuse cases was mainly proposed to address outside attackers. More specifically, an *insider* is a *misusers* that is also member of an authorized group for the entity being attacked. In addition, she introduce the *vulnerability* concept that is a weakness in the system, which can be *exploited* by the *insider*. **Spgr\_18\_03 [@fenz2009formalizing],** : “*Formalizing Information Security Knowledge*”. Fenz and Ekelhar [@fenz2009formalizing] introduce security ontology for information security domain knowledge. In their ontology, a *vulnerability* is the absence of a proper safeguard,which could be exploited by a *threat*. A *threat* might threaten an *asset*, and it can be *exploited* by predefined *threat*, and *mitigation* is achieved by the implementation of one or more *control*. In addition, the *severity* of each *vulnerability* is rated by a three-point scale (high, medium, and low). A *threat* has a *source*, and a related *security objectives*. An *asset* is categorized either as a tangible or an intangible asset. While the *data* concept comprises meta-data on the knowledge of an organization. The *person* concept is used to model physical *persons* in the ontology, and the *organization* concept comprises organizations in the broadest sense and assigns *roles* to them. A *role* is a physical person or organization relevant to the organization. Finally, a *location* is used to relate location and threat information in order to assign a priori threat probabilities. **SCH\_24\_02 [@hong2004privacy],** : “*Privacy Risk Models for Designing Privacy-sensitive Ubiquitous Computing Systems*”. Hong et al. [@hong2004privacy] propose a privacy risk model that captures privacy concerns at high abstraction level, and then refining them into concrete issues for specific applications. The privacy risk model consists of two parts: (1) a *privacy risk analysis* that poses a series of questions to help designers think about the social and organizational context in which an application will be used, the technology used to implement that application, and control and feedback mechanisms that end-users will use; and (2) *privacy risk management* that takes the unordered list of privacy risks from the privacy risk analysis, prioritizes them, and helps design teams identify solutions for helping end-users manage those issues. **SCH\_28\_01 [@paja2014sts],** : “*STS-Tool: Security Requirements Engineering for Socio-Technical*”. Paja et al. [@paja2014sts] present the STS-Tool, a modeling and analysis support tool for STS-ml (Socio-Technical Security modeling language), a security requirements modeling language for socio-technical systems. STS-ml consists of three complementary views: 1- The social view, 2- The information view, 3- The authorization view. Through these views, STS-ml supports different types of security needs: (1) *Interaction (security) needs* are security-related constraints on goal delegations and document provisions; (2) *Authorisation needs* determine which information can be used, how, for which purpose, and by whom; (3) *Organisational constraints* constrain the adoption of roles and the uptake of responsibilities. In addition, STS-ml supports the following interaction security needs: 1. Over goal delegations: (a) *No-redelegation*, (b) *Non-repudiation*, (c) *Redundancy*, (d) *Trustworthiness*, and (e) *Availability*. 2. Over-document provisions: (a) *Integrity of transmission*, (b) *Confidentiality of transmission*, (c) *Availability*. 3. From organizational constraints: (a) *Separation of duties (SoD)*, and (b) *Combination of duties (CoD)*. **SCH\_43\_01 [@van2003handbook],** : “*Handbook of Privacy and Privacy-enhancing Technologies*”. Van Blarkom et al. [@van2003handbook] investigate several active areas related to privacy, Privacy-Enhancing Technologies (PET), intelligent software agents, and the inter-relations among these areas. Furthermore, they discussed the concepts of privacy and data protection, the European Directives that rule the protection of personal data and the relevant definitions. In particular, they investigate when personal data items become non-identifiable, the sensitivity of data, automated decisions, privacy preferences, and policies. In addition, they discussed existing technological solutions that offer agent user privacy protection, known under the name Privacy-Enhancing Technologies (PETs), the set of technologies/ principles that underlying PETs, and the legal basis for PET. Moreover, they discussed the Common Criteria for Information Technology Security Evaluation (CC) supplies important information for building privacy secure agents. **RQ2:** *What are the main concepts/relations that have been used for capturing privacy requirements?* Each of the 34 selected studies has been deeply investigated to identify any concept/relation that can be used for capturing privacy requirements. We have focused on identifying any concept/relation that can be used for capturing privacy requirements in their social and organizational context. More specifically, we tried to identify any concept that is related to privacy, social and organizational threats that might threaten privacy needs, treatment/countermeasures that can be used to mitigate threats concerning privacy needs. The result is shown in Table \[table:rq\], which presents the concepts/relations that have been identified in each selected studies. In particular, 55 concepts and relations[^2] have been identified, which have been grouped into four main groups based on their types: [&gt;c c c| p[0.3cm]{} |p[0.3cm]{} |p[0.3cm]{} |p[0.3cm]{} | p[0.3cm]{} |p[0.3cm]{} |p[0.3cm]{} |p[0.3cm]{} |p[0.3cm]{} |p[0.3cm]{} | p[0.3cm]{} |p[0.3cm]{} |p[0.3cm]{} |p[0.3cm]{} |p[0.3cm]{} |p[0.3cm]{} | p[0.3cm]{} |p[0.3cm]{} | p[0.3cm]{} |p[0.3cm]{} |p[0.3cm]{} | p[0.3cm]{} |p[0.3cm]{} | p[0.3cm]{} | p[0.3cm]{} |p[0.3cm]{} | p[0.3cm]{} |p[0.3cm]{} | p[0.3cm]{} | p[0.3cm]{} | p[0.3cm]{} |p[0.3cm]{} | p[0.3cm]{} |p[0.3cm]{} | p[0.3cm]{} |]{} & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & &\ & & & & & [X]{} & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & &\ & & **role** & & & [X]{} & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & &\ & & **agent** & & & [X]{} & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & &\ & & user & & - & - & & & & & & & & & & & & & & & & & & & & & & & & & - & & & & - & & -\ & & stakeholder & & & & & & & & & - & & & - & & & - & & & & & & & & & & & - & & & & & & - & &\ & & person & & & & & & & & & & & & & & & & & & - & & & & & & & & & & & & & & & &\ & & **is\_a** & & & [X]{} & & & & & && & & & & & & & & & & & & & & & & & & & & & & & &\ & & **plays** & & & [X]{} & & & & & && & & & & & & & & & & & & & & & & & & & & & & & &\ & & **goal** & & & [X]{} & & & & & && & & & & & & & & & & & & & & & & & & & & & & & &\ & & objective & & & & & & & & & & & & - & & - & & & & & & & & & & - & & - & & & & & & & &\ & & task & & & & & & & - & - & & - & & & - & & & - & & & - & & - & & - & & & & - & & & - & - & & - &\ & & action & & - & - & & & & - & - & - & & & - & & & & &- & & & & & & & & & & & & & & & & &\ & & **refinement** & & & & & & & & && & & & & & & & & & & & & & & & & & & & & & & & &\ & & asset & - & & & - & & - & & - & - & & - & - & & & & &- & - & & - & & & - & & - & - & & - & & & & & &\ & & **informational** & & & & & & && && & & & & & & & & & & & & & & & & & & & & & & & &\ & & data & & - & - & & & & & & & & & & & & & & - & - & & & & & & & & & & & & & & - & & -\ & & resource & & & & - & & & - & - & & - & & - & - & & & - & & & & & - & & & & & & & & & & & & &\ & & **personal info** & & & & & & &&&& & & & & & & & & & & & & & & & & & & & & & & & &\ & & sensitive info & & & & & & & & & & & & & & & & & - & & & & & & & & & - & & & & & & & & -\ & & **part\_of** & & & & & & &&&&& & & & & & & & & & & & & & & & & & & & & & & &\ & & **own** & & & & & & && && & & & & & & & & & & & & & & & & & & & & & & & &\ & & **obj deleg.** & & && & & & &&& & & & & & & & & & & & & & & & & & & & & & & & &\ & & **perm. deleg.** & & && & & & &&& & & & & & & & & & & & & & & & & & & & & & & & &\ & & **info provision** & & && & & & &&& & & & & & & & & & & & & & & & & & & & & & & & &\ & & **monitor** & & && & & &&&& & & & & & & & & & & & & & & & & & & & & & & & &\ & & **obj trust** & & && & & &&&& & & & & & & & & & & & & & & & & & & & & & & & &\ & & **perm trust** & & && & & &&&& & & & & & & & & & & & & & & & & & & & & & & & &\ & risk & & & & - & & & & & - & & - & & & & & & & & - & - & & & - & & & & & & & - & - & - & - & -\ & **threat** & & & & & & && & & & & & & & & & & & & & & & & & & & & & & & & & &\ & **inten. threat** & & & & & & & && && & & & & & & & & & & & & & & & & & & & & & & &\ & **casual threat** & & & & & & &&& & & & && & & & & & & & & & & & & & & & & & & & &\ & **vulnerability** & & & & & & & & & && & & & & & & & & & & & & & & & & & & & & & & &\ & attack & & & & - & - & & - & & - & & - & - & - & & & - & & & & & - & - & & & & - & & & & & & & &\ & **attacker** & & & & & & & & & && & & & & & & & & & & & & & & & & & & & & & & &\ & **attack method** & & & & & & & & & && & & & & & & & & & & & & & & & & & & & & & & &\ & **impact** & & & & & & & && && & & & & & & & & & & & & & & & & & & & & & & &\ & **threaten** & & & & & & & & & && & & & & & & & & & & & & & & & & & & & & & & &\ & **exploit** & & & & & & & & & && & & & & & & && & & & & & & & & & & & & & & &\ & countermeasure & - & & & & - & - & - & - & - & & - & & & & & & & & - & - & & - & - & - & & - & & & - & & - & & &\ & **mitigate** & & & & & & && & && & & & & & & & & & & & & & & & & & & & & & & &\ & control & & & & & & & & - & - & & & & & & & & & - & & & & & - & & - & & & & & & & & & -\ & treatment & & & & & & & & & & & & & & & & & & & & & & & & - & & & & & & & & & &\ & **s/p goal** & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & &\ & **s/p constraint** & & & & & & & && && & & & & & & & & & & & & & & & & & & & & & & &\ & **s/p policy** & & & & & & & && & & & & & & & & & & & & & & & & & & & & & & & & &\ & **s/p mechanism** & & & & & & & & & && & & & & & & & & & & & & & & & & & & & & & & &\ & **sec/priv req.** & & & & & & & && && & & & & & & & & & & & & & & & & & & & & & & &\ & **confidentiality** & & & & & & & & & && & & & & & & & & & & & & & & & & & & & & & & &\ & integrity & - & & & & & & - & - & - & - & - & & & & & - & & & & & - & - & & & & - & - & & - & & & & - &\ & availability & - & & & & & & - & & - & - & - & & & & & - & & & & & - & - & & & & - & - & & - & & & & - &\ & **non-repudiation** & & & & & & &&& & && & & & & & & & & & & & & & & & & & & & & & &\ & **notice** & & & & & & &&& && & & & & & & & & & & & & & & & & & & & & & & &\ & **anonymity** & & & & & & &&& && & & & & & & & & & & & & & & & & & & & & & & &\ & **transparency** & & & & & & &&& && & & & & & & & & & & & & & & & & & & & & & & &\ & **accountability** & & & & & & &&& && & & & & & & & & & & & & & & & & & & & & & & &\ \[table:rq\] **Organizational.** : 27 concepts and relations have been identified for capturing the agentive entities of the system in terms of their objectives, entitlements, dependencies and their expectations concerning such dependencies. The organizational concepts and relations are further grouped into four sub-categories: **Agentive entities.** : 8 concepts and relations have been identified for capturing the active entities of the system (e.g., actor, user, etc. ). **Intentional entities.** : 5 concepts and relations have been identified for capturing objectives that active entities aim for achieve/want to perform (e.g., goal, task, activity, etc. ). **Informational entities.** : 8 concepts and relations have been identified for capturing informational assets (e.g., data, asset, information, etc.). **Entities interactions.** : 6 concepts and relations have been identified for capturing the entities dependencies and expectations concerning such dependencies (e.g., delegation, dependency, provision, trust, etc. ). <!-- --> **Risk.** : 10 concepts and relations have been identified for capturing risk related aspects (e.g., risk, threat, vulnerabilities, attack, etc.). **Treatment.** : 8 concepts and relations have been identified for capturing treatment related aspects (e.g., treatment, countermeasure, mitigate etc.). **Privacy.** : 9 concepts and relations have been identified for capturing privacy related aspects (e.g., anonymity, confidentiality, etc.). Among the 55 identified concepts and relations, we have selected 38 main concepts and relations that can be used for capturing privacy requirements in their social and organizational context. In particular, these concepts and relations are 17 organizational, 9 risk, 5 treatment, and 7 privacy concepts, and they are shown in **Bold** typeset in Table \[table:rq\]. Each of the selected concepts and relations has been chosen based on the following two criteria: (1) its importance for capturing privacy requirements; and (2) the frequency of its appearance in the selected studies, which is shown in Table \[table:iteration\]. **RQ3:** *Do existing privacy studies cover the main privacy concepts/relations?* We answer *RQ3* by comparing the privacy related concepts/relations presented in each selected study with the main privacy concepts/relations identified while answering *RQ2*. In Table \[table:rq\], we use () when the study presents a main privacy concept/relation, and (-) when the study presents a normal privacy concept/relation. In addition, we use () to mark when a study misses a main concept/relation. Table \[table:limitation\] summarizes the percentage of the main privacy concepts/relations identified in each selected study with respect to the main four categories (organizational, risk, treatment, and privacy). Considering Table \[table:rq\] and Table \[table:limitation\], it is easy to note that most studies miss main privacy related concepts/relations, i.e., none of them cover all the main privacy related concepts/relations. In **RQ4**, we discuss the limitation of each selected study. \[table:iteration\] \[table:limitation\] **RQ4:** *RQ4 What are the limitations of existing privacy studies?* We answer this question by categorizing the studies into four groups (**Group1-4**) [^3] based on the concepts categories (e.g., organizational, risk, treatment, and privacy) that the studies do not appropriately cover: **Group 1,** : contains studies that do not appropriately cover the organizational concepts. In this group, we have identified 25 studies out of the 34 selected ones, including: ACM\_03 Lamsweerde [@van2004elaborating], ACM\_14 Labda et al. [@labda2014modeling], ACM\_16 Braghin et al. [@braghin2008introducing], ACM\_35 Singhal and Wijesekera [@singhal2010ontologies], ACM\_40 Wang and Guo [@wang2009ovm], CIT\_07 Lasheras et al. [@velasco2009modelling], IEEE\_12 Souag et al. [@souag2013using], IEEE\_15 Tsoumas and Gritzalis [@tsoumas2006towards], IEEE\_57 Kang and Liang [@kang2013security], Spgr\_7 Massacci et al. [@massacci2011extended], Spgr\_13 Elahi et al. [@elahi2009modeling], SCH\_18 Sindre and Opdahl [@sindre2005eliciting], SCH\_24 Kalloniatis et al. [@kalloniatis2008addressing], Spgr\_18\_03 Fenz and Ekelhart [@fenz2009formalizing], Spgr\_13\_01 Asnar et al. [@asnar2008risk], Spgr\_13\_02 Braber et al. [@den2003coras], Spgr\_13\_04 Jürjens [@jurjens2002umlsec], Spgr\_13\_07 R[ø]{}stad [@rostad2006extended], Spgr\_08\_01 Mayer [@mayer2009model], Spgr\_08\_03 Dritsas et al. [@dritsas2006knowledge], Spgr\_07\_03 Lin et al. [@lin2003introducing], Spgr\_03\_01 Avi[ž]{}ienis et al. [@avizienis2004basic], Spgr\_02\_01 Asnar et al. [@asnar2007trust], Spgr\_02\_02 Asnar et al. [@asnar2006risk], SCH\_24\_02 Hong et al. [@hong2004privacy], SCH\_43\_01 Blarkom et al. [@van2003handbook]. **Group 2,** : contains studies that do not appropriately cover risk concepts. In this group, we have identified 22 studies out of the 34 selected ones, including: ACM\_14 Labda et al. [@labda2014modeling], ACM\_16 Braghin et al. [@braghin2008introducing], ACM\_35 Singhal and Wijesekera [@singhal2010ontologies], CIT\_07 Lasheras et al. [@velasco2009modelling], IEEE\_50 Giorgini et al. [@Giorgini2005], IEEE\_57 Kang and Liang [@kang2013security], SCH\_18 Sindre and Opdahl [@sindre2005eliciting], SCH\_24 Kalloniatis et al. [@kalloniatis2008addressing], SCH\_28 Mouratidis and Giorgini [@mouratidis2007secure], SCH\_41 Solove [@solove2006taxonomy], Spgr\_13\_01 Asnar et al. [@asnar2008risk], Spgr\_13\_02 Braber et al. [@den2003coras], Spgr\_13\_04 Jürjens [@jurjens2002umlsec], Spgr\_08\_03 Dritsas et al. [@dritsas2006knowledge], Spgr\_07\_02 Zannone [@zannone2006requirements], Spgr\_07\_03 Lin et al. [@lin2003introducing], Spgr\_03\_01 Avi[ž]{}ienis et al. [@avizienis2004basic], Spgr\_02\_01 Asnar et al. [@asnar2007trust], Spgr\_02\_02 Asnar et al. [@asnar2006risk], SCH\_24\_02 Hong et al. [@hong2004privacy], SCH\_28\_01 Paja et al. [@paja2014sts], SCH\_43\_01 Blarkom et al. [@van2003handbook]. **Group 3,** : contains studies that do not appropriately cover treatment concepts. In this group, we have identified 31 studies out of the 34 selected ones, including: ACM\_03 Lamsweerde [@van2004elaborating], ACM\_14 Labda et al. [@labda2014modeling], ACM\_16 Braghin et al. [@braghin2008introducing], ACM\_35 Singhal and Wijesekera [@singhal2010ontologies], ACM\_40 Wang and Guo [@wang2009ovm], CIT\_07 Lasheras et al. [@velasco2009modelling], IEEE\_12 Souag et al. [@souag2013using], IEEE\_50 Giorgini et al. [@Giorgini2005], IEEE\_57 Kang and Liang [@kang2013security], Spgr\_7 Massacci et al. [@massacci2011extended], Spgr\_13 Elahi et al. [@elahi2009modeling], SCH\_18 Sindre and Opdahl [@sindre2005eliciting], SCH\_24 Kalloniatis et al. [@kalloniatis2008addressing], SCH\_41 Solove [@solove2006taxonomy], Spgr\_18\_03 Fenz and Ekelhart [@fenz2009formalizing], Spgr\_13\_01 Asnar et al. [@asnar2008risk], Spgr\_13\_02 Braber et al. [@den2003coras], Spgr\_13\_03 Elahi et al. [@elahi2010vulnerability], Spgr\_13\_04 Jürjens [@jurjens2002umlsec], Spgr\_13\_05 Matulevi[č]{}ius et al. [@matulevivcius2008adapting], Spgr\_13\_07 R[ø]{}stad [@rostad2006extended], Spgr\_08\_01 Mayer [@mayer2009model], Spgr\_08\_03 Dritsas et al. [@dritsas2006knowledge], Spgr\_07\_02 Zannone [@zannone2006requirements], Spgr\_07\_03 Lin et al. [@lin2003introducing], Spgr\_03\_01 Avi[ž]{}ienis et al. [@avizienis2004basic], Spgr\_02\_01 Asnar et al. [@asnar2007trust], Spgr\_02\_02 Asnar et al. [@asnar2006risk], SCH\_24\_02 Hong et al. [@hong2004privacy], SCH\_28\_01 Paja et al. [@paja2014sts], SCH\_43\_01 Blarkom et al. [@van2003handbook]. **Group 4,** : contains studies that do not appropriately cover the privacy concepts. In this group, we have identified 31 studies out of the 34 selected ones, including: ACM\_03 Lamsweerde [@van2004elaborating], ACM\_14 Labda et al. [@labda2014modeling], ACM\_16 Braghin et al. [@braghin2008introducing], ACM\_35 Singhal and Wijesekera [@singhal2010ontologies], ACM\_40 Wang and Guo [@wang2009ovm], CIT\_07 Lasheras et al. [@velasco2009modelling], CIT\_33 Liu et al. [@liu2003security], IEEE\_12 Souag et al. [@souag2013using], IEEE\_15 Tsoumas and Gritzalis [@tsoumas2006towards], IEEE\_50 Giorgini et al. [@Giorgini2005], Spgr\_7 Massacci et al. [@massacci2011extended], Spgr\_13 Elahi et al. [@elahi2009modeling], SCH\_18 Sindre and Opdahl [@sindre2005eliciting], SCH\_24 Kalloniatis et al. [@kalloniatis2008addressing], SCH\_28 Mouratidis and Giorgini [@mouratidis2007secure], SCH\_41 Solove [@solove2006taxonomy], Spgr\_18\_03 Fenz and Ekelhart [@fenz2009formalizing], Spgr\_13\_01 Asnar et al. [@asnar2008risk], Spgr\_13\_02 Braber et al. [@den2003coras], Spgr\_13\_03 Elahi et al. [@elahi2010vulnerability], Spgr\_13\_04 Jürjens [@jurjens2002umlsec], Spgr\_13\_05 Matulevi[č]{}ius et al. [@matulevivcius2008adapting], Spgr\_13\_07 R[ø]{}stad [@rostad2006extended], Spgr\_08\_01 Mayer [@mayer2009model], Spgr\_08\_03 Dritsas et al. [@dritsas2006knowledge], Spgr\_07\_02 Zannone [@zannone2006requirements], Spgr\_07\_03 Lin et al. [@lin2003introducing], Spgr\_03\_01 Avi[ž]{}ienis et al. [@avizienis2004basic], Spgr\_02\_01 Asnar et al. [@asnar2007trust], Spgr\_02\_02 Asnar et al. [@asnar2006risk], SCH\_24\_02 Hong et al. [@hong2004privacy]. Based on the previous categories, we have 15 studies that do not appropriately cover all the four concepts categories, and 13 studies that do not appropriately cover three categories. 5 studies do not appropriately cover two categories, and one study does not appropriately cover only one categories. A detailed description of the concepts and relations that each of these studies does not cover can be obtained from Table \[table:rq\]. Note that most of these studies have not been developed to address privacy related issues. Therefore, it is not a negative thing when they do not cover privacy related concepts. **RQ4** has been considered in this study to assist authors of selected studies, if they aim to extend their frameworks and approaches to cover privacy concerns. A novel privacy ontology ======================== Several resent studies stress the need for addressing privacy concerns during the system design (e.g., Privacy by Design (PbD) [@kalloniatis2008addressing; @labda2014modeling]). Nevertheless, based on the results of this review, it is easy to note that no existing study covers all the main privacy concepts/relations that have been identified in the review, i.e., no existing ontology enables for capturing main privacy aspects and without such ontology it is almost impossible to address privacy concerns during the system design. Therefore, proposing such ontology would be a viable solution for this problem. To this end, we propose a novel privacy ontology based on the main privacy concepts/relations identified in Table \[table:rq\]. The meta-model of our ontology is depicted in Figure \[fig:onto\], and the concepts of the ontology are organized into four main dimensions: **Organizational dimension:** : proposes concepts to capture the social and technical components of the system in terms of their capabilities, objectives, and dependencies. **Risk dimension:** : proposes concepts to capture risks that might endanger privacy needs at the social and organizational levels. **Treatment dimension:** : proposes concepts to capture countermeasure techniques to mitigate risks to privacy needs. **Privacy dimension:** : proposes concepts to capture the stakeholders’ (actors) privacy requirements/needs concerning their personal information. In what follows, we define each of these dimensions in terms of their concepts and relations **(1) Organizational dimension.** Most current complex systems consist of several autonomous components that interact and depend on one another for achieving their objectives. Therefore, this dimension includes the organizational concepts of the system, which have been further organized into several categories, including: intentional entities, entities’ objectives, informational assets, entities interactions, and entities expectations concerning such interactions (social trust). In what follows, we define each of these dimensions along with their concepts and relations. represent the active entities of the system, we have selected three concepts along with two relations: **Actor** : represents an autonomous entity that has intentionality and strategic goals within the system. Actor can be decomposed into sub-units: **Role** : is an abstract characterization of an actor in terms of a set of behaviors and functionalities within some specialized context. A role can be a specialization (**is\_a**) of one another. **Agent** : is an autonomous entity that has a specific manifestation in the system. An agent can **plays** a role or more within the system, i.e., an agent inherits the properties of the roles it plays. the behavior of actors is, usually, determined by the objectives they aim to achieve. Therefore, we adopted the goal concept and and/or decomposition (refinement) relations to represent such objectives. **Goal** : is a state of affairs that an actor intends to achieve. When a goal is too coarse to be achieved, it can be refined through *and/or-decompositions* of a root goal into finer sub-goals. **And-decomposition** : implies that the achievement of the root-goal requires the achievement of all its sub-goals. **Or-decomposition** : is used to provide different alternatives to achieve the root goal, and it implies that the achievement of the root-goal requires the achievement of any of its sub-goals. information is one of the most important concepts when we speak about privacy. Among the available concepts for capturing informational asset, e.g., data [@labda2014modeling], a resource (physical or informational) [@Giorgini2005; @zannone2006requirements; @mouratidis2007secure; @massacci2011extended], asset [@kang2013security; @elahi2010vulnerability], etc., we have adopted the following concepts and relations: **Information** : represents any informational entity without intentionality. Information can be atomic or composite (composed of several parts), and we rely on *part\_of* relation to capture the relation between an information entity and its sub-parts. In the context of this work, we differentiate between two main types of information: **Personal information** : any information that can be *related* (directly or indirectly) to an identified or identifiable legal entity (e.g., names, addresses, medical records, etc.), who has the right to control how such information can be used by others [@braghin2008introducing; @van2003handbook]. **Public information** : any information that cannot be *related* (directly or indirectly) to an identified or identifiable legal entity, or personal information that has been made public by its legal entity [@labda2014modeling]. actors may use information to achieve their goals. Our ontology adopts three relations between goals and information(e.g., *produce*, *read*, and *modify*), where each of these relations can be defined as follows: **Produce** : indicates that information can be created by achieving the goal that is responsible for its production; **Read** : indicates that the goal achievement depends on consuming such information; **Modify** : indicates that the goal achievement depends on modifying such information. as previously mentioned, we differentiate between personal and public information if it can be *related* (directly or indirectly) to an identified or identifiable legal entity. In what follows, we define the *own* concept that relates personal information to its legal entity, and we specify how information owner controls the usage to its personal information. **Own** : indicates that an actor is the legitimate owner of information, where information owner has full control over the use of information it owns. \[fig:onto\] **A permission** : is consent of a particular use of a particular object in a system [@sandhu1996role], i.e., the holder of the permission is allowed to perform some action(s) in the system. Information owner has the authority to control the use of its own information, i.e., the owner can control the delegated permissions over information it owns. In our ontology, information permissions are classified under (P)roduce, (R)ead, (M)odify permissions, which covers the three relations between goals and information that our ontology propose. actors may not have the required capabilities to achieve their own objectives by themselves (e.g., achieve a goal, furnish information, etc.). Therefore, they depend on one another for such objectives. In what follows, we discuss the concepts that are used for capturing the different actors’ social interactions and dependencies. **Information provision** : indicates that an actor has the capability to deliver information to another one, where the source of the provision relation is the provider and the destination is the requester. Information provision has one attribute that describes the provisioning type, which can be either *confidential* or *non-confidential*, where the first guarantee the confidentiality of the transmitted information while the last does not. <!-- --> **Goal delegation** : indicates that one actor delegates the responsibility to achieve a goal to other actors, where the source of delegation called the delegator , the destination is called delegatee, and the subject of delegation is called delegatum. **Permissions delegation** : indicates that an actor delegates the permissions to produce, read and/or modify over a specific information to another actor. the need for trust arises when actors depend on one another for goals or permissions since such dependencies might entail risk [@chopra2003trust; @gharib2015analyzing]. More specifically, a delegator has no warranty that the delegated goal will be achieved or the delegated permissions will not be misused by the delegatee. Therefore, our ontology adopts the notion of trust and distrust to capture the actors’ expectations of one another concerning their delegations: **Trust** : indicates the expectation of trustor that the trustee will behave as expected considering the trustum (e.g., trustee will achieve the delegated goal, or it will not misuse the delegated permission); **Distrust** : indicates the expectation of trustor that the trustee will not behave as expected considering the trustum (e.g., trustee will not achieve the delegated goal, or it will misuse the delegated permission). we rely on monitoring to compensate the lack of trust or distrust in the trustee concerning the trustum [@gans2001modeling; @zannone2006requirements]. **Monitoring** : can be defined as the process of observing and analyzing the performance of an actor in order to detect any undesirable performance [@guessoum2004monitoring], where the source of monitoring is called the monitor and the destination is called monitoree. **(2) Risk dimension.** Risk can be defined as an event that has a negative impact on the system, i.e., it is the possibility that a particular threat will harm one or more asset of a system by exploiting a vulnerability [@kang2013security; @singhal2010ontologies; @mayer2009model; @elahi2009modeling]. In our ontology, risk is not a primitive concept and we do not include it into the ontology, since it can be captured by other concepts such as threat, vulnerabilities, attack, etc. In what follows, we define the risk dimension related concepts along with their interrelations: **A threat** : is a potential incident that *threaten* an asset (personal information) by *exploiting* a *vulnerability* concerning such asset [@mayer2009model; @singhal2010ontologies; @kang2013security]. A *threat* can be either natural (e.g. earthquake, etc.), accidental (e.g. hardware/software failure, etc.), or intentional (e.g. theft of personal information, etc.)[@fenz2009formalizing; @velasco2009modelling; @souag2015security]. Therefore, the ontology differentiates between two types of threat: **Casual threat** : (natural or accidental): a threat that does not require a *threat actor* nor an *attack method*. **Intentional threat** : a threat that require a *threat actor* and a presumed *attack method* [@lin2003introducing; @massacci2011extended]. **Threat actor** : is an actor that aims for achieving the *intentional threat* [@rostad2006extended; @mayer2009model; @elahi2009modeling]. **Attack method** : is a standard means by which a *threat actor* carries out an *intentional threat* [@mayer2009model; @elahi2010vulnerability; @souag2015security]. **Impact** : is the consequence of the *threat* *over* the asset, and it can be characterized by a *severity* attribute that captures the level of the impact (e.g. high, medium or low) [@wang2009ovm; @souag2015security]. **A vulnerability** : is a weakness in the system, asset (personal information), etc. that can be *exploited* by a *threat* [@rostad2006extended; @mayer2009model; @singhal2010ontologies]. **(3) Treatment dimension.** This dimension introduces countermeasure concepts to mitigate risks, we adopted a high abstraction level countermeasure concepts to capture the required protection/treatment level (e.g., privacy goal), which can be refined into concrete protection/treatment constraints (e.g., mechanisms or policies) that can be implemented. The concepts of the treatment dimension are: **A privacy goal** : is an aim to counter threats and prevents harm to personal information by satisfying privacy criteria concerning such information. **A privacy constraint** : is a restriction that is used to realize/satisfy a privacy goal, constraints can be either a privacy policy or privacy mechanism. **A privacy policy** : is a privacy statement that defines the permitted and/or forbidden actions to be carried out by actors of the system toward information. **A privacy mechanism** : is a concrete technique to be implemented for helping towards the satisfaction of privacy goal (attribute). **(4) Privacy dimension.** Introduce concepts to capture the stakeholders’ (actors) privacy requirements/needs concerning their personal information. The concepts of the privacy dimension are: **Privacy requirement** : is used to capture the actors’ (personal information owner/subject) privacy needs at a high abstraction level, and it is specialized from the *privacy goal* concept. Moreover, privacy requirement concept is further specialized into five more refined concepts. **Confidentiality,** : means personal information should be kept secure from any potential leaks and improper access [@solove2006taxonomy; @dritsas2006knowledge; @labda2014modeling]. We rely on the following principles to analyze confidentiality: **Non-disclosure,** : personal information can only be disclosed if the owner’s consent is provided, i.e., the disclosure of the personal information should be under the control of its legitimate owner [@solove2006taxonomy; @dritsas2006knowledge; @braghin2008introducing; @labda2014modeling]. Note that *non-disclosure* also cover information transmission that is why we differentiate between two types of information provision (confidential and non-confidential). **Need to know (NtK),** : an actor should only use information if it is strictly necessary for completing a certain task [@labda2014modeling; @paja2014sts]. **Purpose of use,** : personal information should only be used for specific, explicit, legitimate purposes and not further used in a way that is incompatible with those purposes [@van2003handbook; @solove2006taxonomy; @dritsas2006knowledge]. *Purpose of use* is able to address situations where an actor might be granted a permission to use some personal information for a legitimate purpose, yet after accessing it, he/she might use the information for some other purpose. **Notice,** : the data subject (information owner) should be notified when its information is being collected [@van2003handbook; @solove2006taxonomy; @dritsas2006knowledge]. Notice is considered mainly to address situations where personal information *related* to a legitimate entity (data subject) is being collected without his/her knowledge. **Anonymity,** : the identity of the information owner should not be disclosed unless it is required [@dritsas2006knowledge; @solove2006taxonomy], i.e., the primary/secondary identifiers of the data subject (e.g., name, social security number, address, etc. ) should be removed if they are not required and information still can be used for the same purpose after their removal. We rely on *part\_of* relation to model the internal structure of personal information, i.e., we link the identifiers of the data subject with the rest of the information item by the *part\_of* relation. If the identifiers are not required for the task, they can be easily removed, and information can be used without linking it back to its owner/data subject (unlinkability). **Transparency,** : information owner should be able to know who is using his/her information and for what purposes [@van2003handbook; @dritsas2006knowledge; @kang2013security]. We rely on the following principles to analyze transparency: **Authentication,** : a mechanism that aims at verifying whether actors are who they claim they are [@paja2014sts]. **Authorization,** : a mechanism that aims at verifying whether actors can use information in accordance with their credentials [@dritsas2006knowledge]. **Accountability,** : information owner should have a mechanism available to them to hold information users accountable for their actions concerning information [@dritsas2006knowledge; @kang2013security]. We rely on the following principles to analyze accountability: **Non-repudiation,** : the delegator cannot repudiate he/she delegated; and the delegatee cannot repudiate he/she accepted the delegation [@kang2013security; @paja2014sts]. **Not-re-delegation,** : the delegatee is requested by the delegator not to re-delegate the delegatum, i.e., the re-delegation of a goal/permission is forbidden [@paja2014sts]. Threats to validity =================== After presenting and discussing our systematic literature review, we discuss the threats to its validity in this section. Following Runeson et al. [@runeson2009guidelines], we classify threats to validity under four types: construct, internal, external and reliability: **1- Construct threats:** is concerned with to what extent a test measures what it claims to be measuring [@runeson2009guidelines]. Construct validity is particularly important, since it might influence the internal validity as well [@mackenzie2003dangers]. We have identified the following threats: Poor conceptualization of the construct: : occurs when the predicted outcome of the study is defined too narrowly [@mackenzie2003dangers], i.e., using only one factor to analyze the subject of the study. To avoid this threat, the research objective has been transformed into several research questions and for each of these questions, several factors were specified to evaluate whether they have been properly answered. In addition, we followed the best practices in the area to define the criteria while searching for and selecting the related studies (e.g., inclusion and exclusion criteria, quality assessment criteria, etc.). Systematic error: : may occur while designing and conducting the review. To avoid such threat, the review protocol has been carefully designed based on well-adopted methods, and it has been strictly followed during the different phases of the review. **2- Internal threats:** is concerned with factors that have not been considered in the study, and they could have influenced the investigated factors in the study [@trochim2006research; @runeson2009guidelines]. One internal threat has been identified: Publication bias: : publication bias is a common threat to the validity of systematic reviews, and it refers to a situation where positive research results are more likely to be reported than negatives ones [@keele2007guidelines]. Our review focused on finding privacy related concepts/relations by reviewing the related literature, and there are no positive nor negative research results in such case. Despite this, we have specified very clear inclusion and exclusion criteria, and quality assessment criteria while searching for and selecting the related studies. **3- External threats:** is concerned with to what extent the results of the study can be generalized [@runeson2009guidelines]. One internal threat has been identified: Completeness: : it is almost impossible to capture all related studies, yet our review protocol and search strategy were very carefully designed to cover as much as possible of the related studies. In addition, we might exclude some relevant non-English published studies since we only considered English studies in this review. To mitigate this limitation we performed a manual scan of the references of all the primary selected studies in order to identify those studies that were missed during the first search stage. However, we cannot guarantee that we have identified all the main available studies, which can be used to answer our research questions. **4- Reliability threats:** is concerned with to what extent the study is dependent on the researcher(s), i.e., if another researcher(s) conducted the same study, the result should be the same. The search terms, search sources, inclusion and exclusion criteria, quality assessment questions, etc. are all available, and any researcher can repeat the review and he should get similar results. However, the researcher should take into consideration the time when the studies search process was performed, i.e., the researcher should limit the search time to March 2016. Related work ============ There are few systematic reviews concerning privacy/securities ontologies. For instance, Souag et al. [@souag2012ontologies] performed a systematic review that proposes an analysis and a typology of existing security ontologies. While Blanco et al. [@blanco2008systematic] conducted a systematic review with a main aim for identifying, extracting and analyzing the main proposals for security ontologies. Fabian et al. [@fabian2010comparison] present a conceptual framework for security requirements engineering by mapping the diverse terminologies of different security requirements engineering methods to that framework. Moreover, a security ontology for capturing security requirements have been presented in [@souag2015security]. However, the focus of all the previously mentioned studies was security ontology. On the other hand, Blanco et al. [@blanco2011basis] conduct a systematic review for extracting the key requirements that an integrated and unified security ontology should have. While Mellado et al. [@mellado2010systematic] carried out a systematic review of the existing literature concerning security requirements engineering in order to summarize the current contributions and to provide a road map for future research in this area. Iankoulova and Daneva [@iankoulova2012cloud] performed a systematic review concerning the security requirements of cloud computing. In particular, they have classified the main identified security requirements under nine sub-areas: access control, attack/harm detection, non-repudiation, integrity, security auditing, physical protection, privacy, recovery, and prosecution. Li [@li2011empirical] conducted a systematic review concerning online information privacy concerns, consequences, and moderating effects. Based on the review outcome, he proposed a framework to illustrate the relationships between the previously mentioned factors and to highlight opportunities for further improvement. Finally, Fern[á]{}ndez-Alem[á]{}n et al. [@fernandez2013security] performed systematic literature review for identifying and analyzing critical privacy and security aspects of the electronic health record systems. Conclusions and Future Work =========================== In this paper, we argued that many wrong design decisions might be made due to the insufficient knowledge about the privacy-related concepts, and we advocate that a well-defined privacy ontology that captures the privacy related concepts along with their interrelations can solve this problem. Therefore, we conduct a systematic review concerning the existing privacy/security literature with a main purpose of identifying the main concepts along with their interrelation for capturing privacy requirements. The objectives of the research were considered to have been achieved since the research questions posed have been answered. Moreover, we used the identified concepts/relations for proposing a privacy ontology to be used by software engineers while dealing with privacy requirements. For future work, we aim to develop core privacy ontology to be used by software/security engineers when dealing with privacy requirements. To achieve that, we are planning to contact the authors of the selected studies to get their feedback concerning the proposed privacy ontology. In addition, we will conduct a controlled experiment with software/security engineers to evaluate the usability of the ontology. Finally, we plan to evaluate the completeness and validity of the ontology by deploying it to capture the privacy requirements for two real case studies that belong to different domains (e.g., medical sector and public administration). [100]{} Simont: A security information management ontology framework. In [*Secure and Trust Computing, Data Management and Applications*]{}. Springer, 2011, pp. 201–208. 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Appendix A: Quality assessment application {#appendix-a-quality-assessment-application .unnumbered} ========================================== [ | p[0.6cm]{} | p[2.3cm]{} | p[0.5cm]{} | p[0.5cm]{} | p[0.5cm]{} | p[0.5cm]{} | p[0.5cm]{} | p[0.3cm]{} | p[0.6cm]{} | p[2.3cm]{} | p[0.5cm]{} | p[0.5cm]{} | p[0.5cm]{} | p[0.5cm]{} | p[0.5cm]{} | p[0.3cm]{} | ]{} **N & **ID & **Q1 & **Q2 & **Q3 & **Q4 & **Q5 & **S. & **N & **ID & **Q1 & **Q2 & **Q3 & **Q4 & **Q5 & **S\ ******************************** ****1** & ACM\_02 [@chase2009improving] & ****-** & ****-** & ****-** & ****1** & ****1** & ****2** &************** ****2** & ACM\_03 [@van2004elaborating] & ****1** & ****1** & ****1** & ****1** & ****1** & ****5**\ ****3** & ACM\_04 [@rezgui2002preserving] & ****-** & ****-** & ****-** & ****1** & ****1** & ****2** &**************************** ****4** & ACM\_05 [@kost2012privacy] & ****1** & ****1** & ****-** & ****-** & ****-** & ****2**\ ****5** & ACM\_06 [@gandhi2011discovering] & ****1** & ****-** & ****-** & ****-** & ****1** & ****2** &**************************** ****6** & ACM\_07 [@hinze2009event] & ****1** & ****-** & ****-** & ****1** & ****1** & ****3**\ ****7** & ACM\_08 [@oladimeji2011managing] & ****1** & ****-** & ****-** & ****-** & ****1** & ****2** &**************************** ****8** & ACM\_10 [@srinivasan2014safe] & ****1** & ****-** & ****-** & ****-** & ****-** & ****1**\ ****9** & ACM\_11 [@weber2009mining] & ****-** & ****-** & ****-** & ****-** & ****1** & ****1** &**************************** ****10** & ACM\_13 [@munoz2012surprise] & ****1** & ****-** & ****-** & ****1** & ****-** & ****2**\ ****11** & ACM\_14 [@labda2014modeling] & ****1** & ****1** & ****1** & ****1** & ****-** & ****4** &**************************** ****12** & ACM\_16 [@braghin2008introducing] & ****1** & ****1** & ****1** & ****1** & ****-** & ****4**\ ****13** & ACM\_17 [@compagna2007capture] & ****1** & ****-** & ****-** & ****-** & ****-** & ****1** &**************************** ****14** & ACM\_18 [@yu2008enforcing] & ****1** & ****-** & ****-** & ****-** & ****-** & ****1**\ ****15** & ACM\_19 [@maxwell2010production] & ****1** & ****-** & ****-** & ****-** & ****-** & ****1** &**************************** ****16** & ACM\_22 [@studer2009privacy] & ****1** & ****1** & ****-** & ****1** & ****-** & ****3**\ ****17** & ACM\_23 [@mitra2006privacy] & ****1** & ****-** & ****-** & ****1** & ****1** & ****3** &**************************** ****18** & ACM\_24 [@sullivan2010trust] & ****-** & ****-** & ****-** & ****-** & ****-** & ****0**\ ****19** & ACM\_26 [@mace2010collaborative] & ****1** & ****-** & ****-** & ****-** & ****-** & ****1** &**************************** ****20** & ACM\_28 [@schaefer2009epistemology] & ****-** & ****-** & ****-** & ****-** & ****-** & ****0**\ ****21** & ACM\_30 [@yau2006framework] & ****1** & ****1** & ****-** & ****1** & ****-** & ****3** &**************************** ****22** & ACM\_32 [@alam2006model] & ****1** & ****1** & ****-** & ****-** & ****-** & ****2**\ ****23** & ACM\_34 [@fenz2010ontology] & ****-** & ****-** & ****-** & ****-** & ****-** & ****-** &**************************** ****24** & ACM\_35 [@singhal2010ontologies] & ****1** & ****1** & ****1** & ****1** & ****-** & ****4**\ ****25** & ACM\_36 [@blackwell2010security] & ****-** & ****-** & ****1** & ****-** & ****-** & ****1** &**************************** ****26** & ACM\_37 [@da2007dealing] & ****-** & ****-** & ****-** & ****-** & ****-** & ****0**\ ****27** & ACM\_40 [@wang2009ovm] & ****1** & ****1** & ****1** & ****-** & ****1** & ****4** &**************************** ****28** & IEEE\_03 [@kost2011privacy] & ****1** & ****1** & ****-** & ****-** & ****-** & ****2**\ ****29** & IEEE\_09 [@rahmouni2009privacy] & ****1** & ****-** & ****-** & ****1** & ****-** & ****2** &**************************** ****30** & IEEE\_11 [@liccardo2012ontology] & ****1** & ****-** & ****-** & ****-** & ****-** & ****1**\ ****31** & IEEE\_12 [@souag2013using] & ****1** & ****1** & ****1** & ****1** & ****1** & ****5** &**************************** ****32** & IEEE\_13 [@daramola2012pattern] & ****-** & ****1** & ****-** & ****-** & ****-** & ****1**\ ****33** & IEEE\_14 [@yau2006hierarchical] & ****1** & ****-** & ****-** & ****-** & ****1** & ****2** &**************************** ****34** & IEEE\_15 [@tsoumas2006towards] & ****1** & ****1** & ****1** & ****-** & ****1** & ****4**\ ****35** & IEEE\_18 [@squicciarini2006achieving] & ****1** & ****1** & ****-** & ****1** & ****-** & ****3** &**************************** ****36** & IEEE\_19 [@firesmith2007engineering] & ****1** & ****1** & ****-** & ****-** & ****-** & ****2**\ ****37** & IEEE\_21 [@wang2009environmental] & ****1** & ****-** & ****1** & ****1** & ****-** & ****3** &**************************** ****38** & IEEE\_25 [@chen2004soupa] & ****1** & ****-** & ****-** & ****1** & ****1** & ****3**\ ****39** & IEEE\_26 [@akmayeva2010ontology] & ****-** & ****-** & ****-** & ****-** & ****-** & ****0** &**************************** ****40** & IEEE\_28 [@coma2008context] & ****1** & ****-** & ****-** & ****-** & ****-** & ****1**\ ****41** & IEEE\_30 [@maisonnasse2006detecting] & ****1** & ****-** & ****-** & ****1** & ****-** & ****2** &**************************** ****42** & IEEE\_33 [@gao2010approach] & ****-** & ****-** & ****-** & ****-** & ****-** & ****0**\ ****43** & IEEE\_35 [@chandramouli2013knowledge] & ****1** & ****1** & ****-** & ****1** & ****-** & ****3** &**************************** ****44** & IEEE\_36 [@singh2014comparative] & ****1** & ****-** & ****-** & ****-** & ****-** & ****1**\ ****45** & IEEE\_38 [@lee15ontology] & ****1** & ****-** & ****-** & ****1** & ****-** & ****2** &**************************** ****46** & IEEE\_41 [@garcia2009towards] & ****1** & ****-** & ****-** & ****-** & ****-** & ****1**\ ****47** & IEEE\_42 [@fenz2007information] & ****1** & ****-** & ****-** & ****1** & ****1** & ****3** &**************************** ****48** & IEEE\_48 [@chowdhury2007enabling] & ****1** & ****-** & ****-** & ****1** & ****-** & ****3**\ ****49** & IEEE\_49 [@bishop2003computer] & ****1** & ****-** & ****-** & ****-** & ****1** & ****2** &**************************** ****50** & IEEE\_50 [@Giorgini2005] & ****1** & ****1** & ****1** & ****1** & ****1** & ****5**\ ****51** & IEEE\_51 [@hecker2008privacy] & ****1** & ****1** & ****1** & ****-** & ****-** & ****3** &**************************** ****52** & IEEE\_52 [@hadzic2006use] & ****1** & ****-** & ****-** & ****-** & ****-** & ****1**\ ****53** & IEEE\_54 [@blanquer2009enhancing] & ****1** & ****-** & ****-** & ****1** & ****1** & ****3** &**************************** ****54** & IEEE\_56 [@torrellas2004framework] & ****1** & ****-** & ****-** & ****1** & ****1** & ****3**\ ****55** & IEEE\_57 [@kang2013security] & ****1** & ****1** & ****1** & ****1** & ****-** & ****4** &**************************** ****56** & IEEE\_58 [@vorobiev2006security] & ****1** & ****-** & ****-** & ****1** & ****1** & ****3**\ ****57** & IEEE\_59 [@yan2008ontology] & ****1** & ****-** & ****-** & ****1** & ****-** & ****2** &**************************** ****58** & IEEE\_60 [@ekelhart2007security] & ****1** & ****1** & ****-** & ****1** & ****1** & ****4**\ ****59** & CIT\_01 [@poritz2004property] & ****1** & ****-** & ****-** & ****1** & ****1** & ****3** &**************************** ****60** & CIT\_07 [@velasco2009modelling] & ****1** & ****1** & ****1** & ****1** & ****1** & ****5**\ ****61** & CIT\_09 [@breaux2008analyzing] & ****1** & ****-** & ****-** & ****1** & ****1** & ****3** &**************************** ****62** & CIT\_12 [@chor1998private] & ****1** & ****-** & ****-** & ****1** & ****1** & ****3**\ ****63** & CIT\_13 [@souag2012towards] & ****-** & ****-** & ****-** & ****-** & ****-** & ****-** &**************************** ****64** & CIT\_15 [@ekclhart2007ontological] & ****1** & ****-** & ****-** & ****1** & ****-** & ****2**\ ****65** & CIT\_18 [@Massacci2007] & ****1** & ****1** & ****1** & ****1** & ****1** & ****5** &**************************** ****66** & CIT\_23 [@zhangdeveloping] & ****1** & ****-** & ****-** & ****-** & ****-** & ****1**\ ****67** & CIT\_26 [@langheinrich2001privacy] & ****1** & ****-** & ****-** & ****1** & ****1** & ****3** &**************************** ****68** & CIT\_29 [@vorobiev2008ontology] & ****1** & ****-** & ****-** & ****1** & ****-** & ****2**\ ****69** & CIT\_31 [@ranganathan2003ontologies] & ****1** & ****-** & ****-** & ****-** & ****1** & ****2** &**************************** ****70** & CIT\_33 [@liu2003security] & ****1** & ****1** & ****1** & ****1** & ****1** & ****5**\ ****71** & Spgr\_01 [@kim2005security] & ****1** & ****-** & ****-** & ****-** & ****1** & ****2** &**************************** ****72** & Spgr\_02 [@fabian2010comparison] & ****-** & ****-** & ****-** & ****-** & ****-** & ****-**\ ****73** & Spgr\_03 [@souag2012ontologies] & ****-** & ****-** & ****-** & ****-** & ****-** & ****-** &**************************** ****74** & Spgr\_07 [@massacci2011extended] & ****1** & ****1** & ****1** & ****1** & ****-** & ****4**\ ****75** & Spgr\_08 [@souag2015security] & ****-** & ****-** & ****-** & ****-** & ****-** & ****-** &**************************** ****76** & Spgr\_13 [@elahi2009modeling] & ****1** & ****1** & ****1** & ****1** & ****-** & ****4**\ ****77** & Spgr\_14 [@tsoumas2006security] & ****1** & ****-** & ****-** & ****1** & ****-** & ****2** &**************************** ****78** & Spgr\_18 [@milicevic2010ontology] & ****1** & ****-** & ****-** & ****-** & ****-** & ****1**\ ****79** & Spgr\_19 [@dhiah2006ontology] & ****1** & ****-** & ****-** & ****-** & ****-** & ****1** &**************************** ****80** & Spgr\_20 [@vincent2011privacy] & ****-** & ****-** & ****-** & ****-** & ****-** & ****1**\ ****81** & Spgr\_22 [@gandhi2009ontology] & ****1** & ****-** & ****-** & ****1** & ****-** & ****2** &**************************** ****82** & Spgr\_28 [@heupel2015ontology] & ****1** & ****-** & ****-** & ****1** & ****-** & ****2**\ ****83** & Spgr\_31 [@chen2005soupa] & ****1** & ****-** & ****-** & ****-** & ****1** & ****2** &**************************** ****84** & Spgr\_32 [@mouratidis2003ontology] & ****1** & ****-** & ****-** & ****-** & ****1** & ****2**\ ****85** & Spgr\_34 [@mitre2006legal] & ****1** & ****-** & ****-** & ****-** & ****-** & ****1** &**************************** ****86** & Spgr\_35 [@chowdhury2008capturing] & ****1** & ****-** & ****-** & ****-** & ****-** & ****1**\ ****87** & Spgr\_36 [@pereira2009ontology] & ****1** & ****-** & ****-** & ****-** & ****-** & ****1** &**************************** ****88** & Spgr\_38 [@delgado2003regulatory] & ****1** & ****-** & ****-** & ****-** & ****-** & ****1**\ ************** **N & **ID & **Q1 & **Q2 & **Q3 & **Q4 & **Q5 & **S & **N & **ID & **Q1 & **Q2 & **Q3 & **Q4 & **Q5 & **S\ ******************************** ****89** & Spgr\_41 [@iwaihara2008risk] & ****1** & ****-** & ****-** & ****1** & ****-** & ****2** &************** ****90** & Spgr\_55 [@ekelhart2006security] & ****1** & ****-** & ****-** & ****-** & ****-** & ****1**\ ****91** & Spgr\_56 [@abulaish2011simont] & ****1** & ****-** & ****-** & ****-** & ****1** & ****2** &**************************** ****92** & Spgr\_58 [@ciuciu2011ontology] & ****1** & ****-** & ****-** & ****-** & ****-** & ****1**\ ****93** & Spgr\_60 [@breaux2014eddy] & ****1** & ****-** & ****-** & ****1** & ****1** & ****3** &**************************** ****94** & SCH\_02 [@chen2003ontology] & ****1** & ****1** & ****-** & ****-** & ****1** & ****3**\ ****95** & SCH\_03 [@massacci2005using] & ****1** & ****1** & ****1** & ****1** & ****1** & ****5** &**************************** ****96** & SCH\_06 [@sacco2011privacy] & ****1** & ****-** & ****-** & ****-** & ****1** & ****2**\ ****97** & SCH\_16 [@firesmith2003security] & ****1** & ****-** & ****-** & ****1** & ****1** & ****3** &**************************** ****98** & SCH\_18 [@sindre2005eliciting] & ****1** & ****-** & ****1** & ****1** & ****1** & ****4**\ ****99** & SCH\_20 [@anton2002analyzing] & ****1** & ****1** & ****-** & ****-** & ****1** & ****3** &**************************** ****100** & SCH\_24 [@kalloniatis2008addressing] & ****1** & ****1** & ****1** & ****1** & ****1** & ****5**\ ****101** & SCH\_26 [@haley2008security] & ****1** & ****-** & ****-** & ****1** & ****1** & ****3** &**************************** ****102** & SCH\_27 [@he2003framework] & ****1** & ****-** & ****-** & ****1** & ****1** & ****3**\ ****103** & SCH\_28 [@mouratidis2007secure] & ****1** & ****1** & ****1** & ****1** & ****1** & ****5** &**************************** ****104** & SCH\_32 [@donner2003toward] & ****-** & ****-** & ****-** & ****-** & ****-** & ****0**\ ****105** & SCH\_36 [@alliance2003hipaa] & ****1** & ****-** & ****-** & ****1** & ****-** & ****2** &**************************** ****106** & SCH\_41 [@solove2006taxonomy] & ****1** & ****1** & ****1** & ****1** & ****1** & ****5**\ ****107** & SCH\_43 [@skinner2006information] & ****1** & ****-** & ****-** & ****1** & ****-** & ****2** &**************************** ****108** & Spgr\_18\_01 [@karyda2006ontology] & ****1** & ****-** & ****-** & ****1** & ****-** & ****3**\ ****109** & Spgr\_18\_02 [@raskin2001ontology] & ****-** & ****-** & ****-** & ****-** & ****1** & ****1** &**************************** ****110** & Spgr\_18\_03 [@fenz2009formalizing] & ****1** & ****1** & ****1** & ****-** & ****1** & ****4**\ ****111** & Spgr\_13\_01 [@asnar2008risk] & ****1** & ****1** & ****1** & ****1** & ****1** & ****5** &**************************** ****112** & Spgr\_13\_02 [@den2003coras] & ****1** & ****1** & ****-** & ****1** & ****1** & ****4**\ ****113** & Spgr\_13\_03 [@elahi2010vulnerability] & ****1** & ****1** & ****1** & ****1** & ****1** & ****5** &**************************** ****114** & Spgr\_13\_04 [@jurjens2002umlsec] & ****1** & ****1** & ****-** & ****1** & ****1** & ****4**\ ****115** & Spgr\_13\_05 [@matulevivcius2008adapting] & ****1** & ****1** & ****1** & ****1** & ****1** & ****5** &**************************** ****116** & Spgr\_13\_06 [@mayer2005towards] & ****1** & ****1** & ****1** & ****1** & ****1** & ****5**\ ****117** & Spgr\_13\_07 [@rostad2006extended] & ****1** & ****-** & ****-** & ****1** & ****1** & ****3** &**************************** ****118** & Spgr\_13\_08 [@singh2014revisiting] & ****1** & ****1** & ****-** & ****-** & ****1** & ****3**\ ****117** & Spgr\_13\_07 [@rostad2006extended] & ****1** & ****-** & ****-** & ****1** & ****1** & ****3** &**************************** ****118** & Spgr\_13\_08 [@singh2014revisiting] & ****1** & ****1** & ****-** & ****-** & ****1** & ****3**\ ****119** & Spgr\_08\_01 [@mayer2009model] & ****1** & ****1** & ****1** & ****1** & ****1** & ****5** &**************************** ****120** & Spgr\_08\_02 [@velasco2009modelling] & ****1** & ****1** & ****1** & ****1** & ****-** & ****4**\ ****121** & Spgr\_08\_03 [@dritsas2006knowledge] & ****1** & ****1** & ****1** & ****1** & ****-** & ****4** &**************************** ****122** & Spgr\_07\_01 [@blanco2008systematic] & ****-** & ****-** & ****-** & ****-** & ****-** & ****-**\ ****123** & Spgr\_07\_02 [@zannone2006requirements] & ****1** & ****1** & ****1** & ****1** & ****-** & ****4** &**************************** ****124** & Spgr\_07\_03 [@lin2003introducing] & ****1** & ****-** & ****-** & ****-** & ****1** & ****2**\ ****125** & Spgr\_03\_01 [@avizienis2004basic] & ****1** & ****1** & ****-** & ****1** & ****1** & ****4** &**************************** ****126** & Spgr\_03\_02 [@firesmith2005taxonomy] & ****1** & ****-** & ****-** & ****-** & ****1** & ****2**\ ****127** & Spgr\_02\_01 [@asnar2007trust] & ****1** & ****1** & ****-** & ****1** & ****1** & ****4** &**************************** ****128** & Spgr\_02\_02 [@asnar2006risk] & ****1** & ****1** & ****-** & ****1** & ****1** & ****4**\ ****129** & SCH\_24\_01 [@kalloniatis2005dealing] & ****1** & ****-** & ****-** & ****1** & ****-** & ****2** &**************************** ****130** & SCH\_24\_02 [@hong2004privacy] & ****1** & ****1** & ****1** & ****1** & ****1** & ****5**\ ****131** & SCH\_28\_01 [@paja2014sts] & ****1** & ****1** & ****1** & ****1** & ****-** & ****4** &**************************** ****132** & SCH\_43\_01 [@van2003handbook] & ****1** & ****1** & ****1** & ****1** & ****-** & ****4**\ ************** Appendix B: Overview of all the considered studies {#appendix-b-overview-of-all-the-considered-studies .unnumbered} ================================================== [ | p[0.6cm]{} | p[1.7cm]{} | p[3cm]{} | p[2.9cm]{} | p[1cm]{} | p[0.9cm]{} | p[1.7cm]{} | ]{} ****N** & ****ID** & ****Title** & ****Author(s)** & ****Pub Year** & ****\# Cited** & ****Decision**\ ************** 001 & ACM\_01 [@olivier2002database] & Database Privacy, Balancing Confidentiality, Integrity and Availability & Martin S Olivier & 2002 & 30 & Excluded stage 1\ 002 & ACM\_02 [@chase2009improving] & Improving privacy and security in multi-authority attribute-based encryption & Melissa Chase, Sherman S.M. Chow & 2009 & 375 & Excluded stage 2\ 003 & ACM\_03 [@van2004elaborating] & Elaborating Security Requirements by Construction of Intentional Anti-Models & Axel van Lamsweerde & 2004 & 337 & Selected\ 004 & ACM\_04 [@rezgui2002preserving] & Preserving Privacy in Web Services & Abdelmounaam Rezgui, Mourad Ouzzani, Athman Bouguettaya, Medjahed Brahim & 2002 & 102 & Excluded stage 2\ 005 & ACM\_05 [@kost2012privacy] & Privacy analysis using ontologies & Martin Kost, Johann Christoph Freytag & 2012 & 9 & Excluded stage 2\ 006 & ACM\_06 [@gandhi2011discovering] & Discovering Multidimensional Correlations among Regulatory Requirements to Understand Risk & Robin A. Gandhi, Seok Won Lee & 2011 & 7 & Excluded stage 2\ 007 & ACM\_07 [@hinze2009event] & Event-based applications and enabling technologies & Hinze Annika Kai Sachs, Alejandro Buchmann & 2009 & 110 & Excluded stage 2\ 008 & ACM\_08 [@oladimeji2011managing] & Managing security and privacy in ubiquitous eHealth information interchange & Ebenezer A. Oladimeji, Lawrence Chung, Hyo Taeg Jung, Kim Jaehyoun & 2011 & 12 & Excluded stage 2\ 009 & ACM\_09 [@kumari2012deriving] & Deriving implementation-level policies for usage control enforcement & Prachi Kumari, Alexander Pretschner & 2012 & 18 & Excluded stage 1\ 010 & ACM\_10 [@srinivasan2014safe] & SAFE: Secure and Big Data-Adaptive Framework for Efficient Cross-Domain Communication & Avinash Srinivasan, Wu Jie, Zhu Wen & 2014 & 1 & Excluded stage 2\ 011 & ACM\_11 [@weber2009mining] & Mining and Analysing Security Goal Models in Health Information Systems & Jens H. Weber-Jahnke, Onabajo Adeniyi & 2009 & 6 & Excluded stage 2\ 012 & ACM\_12 [@juan2013decision] & Decision support for partially moving applications to the cloud: the example of business intelligence & Adrian Juan-Verdejo, Henning Baars & 2013 & 10 & Excluded stage 2\ 013 & ACM\_13 [@munoz2012surprise] & Surprise: user-controlled granular privacy and security for personal data in Smarter Context & Juan C. Muñoz, Tamura Gabriel, Norha M. Villegas, and Hausi A. Müller & 2012 & 5 & Excluded stage 2\ 014 & ACM\_14 [@labda2014modeling] & Modeling of privacy-aware business processes in BPMN to protect personal data & Wadha JLabda, Nikolay Mehandjiev, Pedro Sampaio & 2014 & 0 & Selected\ 015 & ACM\_15 [@lammari2011conceptual] & A conceptual meta-model for secured information systems & Nadira Lammari, Jean-Sylvain Bucumi, Jacky Akoka, Isabelle Comyn-Wattiau & 2011 & 2 & Excluded stage 1\ 016 & ACM\_16 [@braghin2008introducing] & Introducing privacy in a hospital information system & Stefano Braghin, Alberto Coen-Porisini, Pietro Colombo, Sabrina Sicari, Alberto Trombetta & 2008 & 9 & Selected\ 017 & ACM\_17 [@compagna2007capture] & How to capture, model, and verify the knowledge of legal, security, and privacy experts: a pattern-based approach & Luca Compagna, Paul El Khoury, Fabio Massacci, Thomas Reshma, Nicola Zannone & 2007 & 29 & Excluded stage 2\ 018 & ACM\_18 [@yu2008enforcing] & Enforcing a Security Pattern in Stakeholder Goal Models & Yijun Yu, Kaiya Haruhiko, Washizaki Hironori, Xiong Yingfei, Hu Zhenjiang, Yoshioka Nobukazu & 2008 & 17 & Excluded stage 2\ 019 & ACM\_19 [@maxwell2010production] & The production rule framework: developing a canonical set of software requirements for compliance with law & Jeremy C. Maxwell, Annie I. Antón & 2010 & 15 & Excluded stage 2\ 020 & ACM\_20 [@tan2006security] & Security issues in a SOA-Based provenance system & Victor Tan, Paul Groth, Simon Miles, Sheng Jiang, Steve Munroe, Sofia Tsasakou, Luc Moreau & 2006 & 73 & Excluded stage 1\ 021 & ACM\_21 [@amagasa2014scheme] & A scheme for privacy-preserving ontology mapping & Toshiyuki Amagasa,Fan Zhang, Jun Sakuma, Hiroyuki Kitagawa & 2014 & 0 & Excluded stage 1\ 022 & ACM\_22 [@studer2009privacy] & Privacy preserving modules for ontologies & Thomas Studer & 2010 & 4 & Excluded stage 2\ 023 & ACM\_23 [@mitra2006privacy] & Privacy-preserving semantic interoperation and access control of heterogeneous databases & Prasenjit Mitra, Chi-Chun Pan, Peng Liu, and Vijayalakshmi Atluri & 2006 & 35 & Excluded stage 2\ 024 & ACM\_24 [@sullivan2010trust] & Trust-terms ontology for defining security requirements and metrics & Kieran Sullivan, Jim Clarke, Barry P. Mulcahy & 2010 & 3 & Excluded stage 2\ 025 & ACM\_25 [@kabir2014user] & User-centric social context information management: an ontology-based approach and platform & Muhammad Ashad Kabir, Jun Han, Jian Yu, Alan Colman & 2014 & 9 & Excluded stage 1\ 026 & ACM\_26 [@mace2010collaborative] & A collaborative ontology development tool for information security managers & John C. Mace, Simon Parkin, Aad van Moorsel & 2010 & 6 & Excluded stage 2\ 027 & ACM\_27 [@bao2007privacy] & Privacy-Preserving Reasoning on the Semantic Web & Jie Bao, Giora Slutzki, Vasant Honavar & 2007 & 37 & Duplicated\ 028 & ACM\_28 [@schaefer2009epistemology] & The Epistemology of Computer Security & Robert Schaefer & 2009 & 6 & Excluded stage 2\ 029 & ACM\_29 [@rodriguez2014survey] & A Survey on Ontologies for Human Behavior Recognition & Rodríguez, Natalia Díaz and Cuéllar, Manuel P Lilius, Johan Calvo-Flores, Miguel Delgado & 2014 & 17 & Excluded stage 1\ 030 & ACM\_30 [@yau2006framework] & A framework for specifying and managing security requirements in collaborative systems & Stephen S. Yau, Chen Zhaoji & 2006 & 17 & Excluded stage 2\ 031 & ACM\_31 [@elcci2014isn] & Isn’t the Time Ripe for a Standard Ontology on Security of Information and Networks? & Atilla El[ç]{}i & 2014 & 3 & Excluded stage 1\ 032 & ACM\_32 [@alam2006model] & Model driven security engineering for the realization of dynamic security requirements in collaborative systems & Muhammad Alam & 2007 & 14 & Excluded stage 1\ 033 & ACM\_33 [@de2008coresec] & CoreSec: an ontology of security applied to the business process of management & Ryan Ribeiro de Azevedo, Fred Freitas, Silas Cardoso de Almeida, Marcelo José SC Almeida, Edson C. de Barros C Filho, Wendell Campos Veras & 2008 & 1 & Excluded stage 1\ 034 & ACM\_34 [@fenz2010ontology] & Ontology-based generation of IT-security metrics & Stefan Fenz & 2010 & 21 & Excluded stage 2\ 035 & ACM\_35 [@singhal2010ontologies] & Ontologies for Modeling Enterprise Level Security Metrics & Singhal Anoop, Wijesekera Duminda & 2010 & 7 & Selected\ 036 & ACM\_36 [@blackwell2010security] & A Security Ontology for Incident Analysis & Clive Blackwell & 2007 & 7 & Excluded stage 2\ 037 & ACM\_37 [@da2007dealing] & Dealing with the formal analysis of Information Security policies through ontologies: a case study & Da Silva, G. M. H., Rademaker, A., Vasconcelos, D. R., Amaral, F. N., Bazílio, C., Costa, V. G., Haeusler, E. H & 2007 & 3 & Excluded stage 2\ 038 & ACM\_38 [@lee2006building] & Building problem domain ontology from security requirements in regulatory documents & Lee, Seok-Won, Robin Gandhi, Divya Muthurajan, Yavagal Deepak. Ahn Gail-Joon & 2006 & 42 & Excluded stage 1\ 039 & ACM\_39 [@vorobiev2010ontology] & Ontology-based analysis of information security standards and capabilities for their harmonization & Vladimir I Vorobiev, Ludmila Fedorchenko, Vadim P Zabolotsky, Alexander V Lyubimov & 2010 & 2 & Excluded stage 1\ 040 & ACM\_40 [@wang2009ovm]& OVM: an ontology for vulnerability management & Ju An Wang, Guo Minzhe & 2009 & 40 & Selected\ 041 & IEEE\_01 [@kagal2004authorization] & Authorization and privacy for semantic Web services & Lalana Kagal, Tim Finin, Massimo Paolucci, Naveen Srinivasan, Katia Sycara, Grit Denker & 2004 & 242 & Excluded stage 1\ 042 & IEEE\_02 [@fernandez2013surveillance] & Surveillance ontology for legal, ethical and privacy protection based on SKOS & Virginia Fernandez Arguedas, Ebroul Izquierdo, Krishna Chandramouli & 2013 & 1 & Excluded stage 1\ 043 & IEEE\_03 [@kost2011privacy] & Privacy Verification Using Ontologies & Martin Kost, Johann-Christoph Freytag, Frank Kargl, Antonio Kung & 2011 & 10 & Excluded stage 2\ 044 & IEEE\_04 [@kayes2013semantic] & A Semantic Policy Framework for Context-Aware Access Control Applications & ASM Kayes, Jun Han, Alan Colman & 2013 & 1 & Excluded stage 1\ 045 & IEEE\_05 [@lioudakis2007proxy] & A Proxy for Privacy: the Discreet Box & Georgios V. Lioudakis, Eleftherios A. Koutsoloukas, Nikolaos Dellas, Sofia Kapellaki, George N. Prezerakos, Dimitra I. Kaklamani, Iakovos S. Venieris & 2007 & 9 & Excluded stage 1\ 046 & IEEE\_06 [@modica2011semantic] & Semantic annotations for security policy matching in WS-Policy & Giuseppe Di Modica, Orazio Tomarchio & 2011 & 1 & Excluded stage 1\ 047 & IEEE\_07 [@modica2011semantic] & Semantic Security Policy Matching in Service Oriented Architectures & Giuseppe Di Modica, Orazio Tomarchio & 2011 & 4 & Excluded stage 1\ 048 & IEEE\_08 [@durbeck2007security] & Security Requirements for a Semantic Service-oriented Architecture & Stefan D[ü]{}rbeck, Rolf Schillinger, Jan Kolter & 2007 & 14 & Excluded stage 1\ 049 & IEEE\_09 [@rahmouni2009privacy] & Privacy compliance in European health grid domains: An ontology-based approach & Hanene Boussi Rahmouni, Tony Solomonides, Marco Casassa Mont, Simon Shiu & 2009 & 9 & Excluded stage 2\ 050 & IEEE\_10 [@chen2011ontological] & An Ontological Study of Data Purpose for Privacy Policy Enforcement & Shan Chen, Mary-Anne Williams & 2011 & 0 & Excluded stage 1\ 051 & IEEE\_11 [@liccardo2012ontology] & Ontology-based Negotiation of Security Requirements in Cloud & Loredana Liccardo, Massimiliano Rak, Giuseppe Di Modica, Orazio Tomarchio & 2012 & 2 & Excluded stage 2\ 052 & IEEE\_12 [@souag2013using] & Using Security and Domain ontologies for Security Requirements Analysis & Amina Souag, Camille Salinesi, Isabelle Wattiau, Haris Mouratidis & 2013 & 4 & Selected\ 053 & IEEE\_13 [@daramola2012pattern] & Pattern-based security requirements specification using ontologies and boilerplates & Olawande Daramola, Guttorm Sindre, Tor Stalhane & 2012 & 4 & Excluded stage 2\ 054 & IEEE\_14 [@yau2006hierarchical] & Hierarchical Situation Modeling and Reasoning for Pervasive Computing & Stephen S Yau, Junwei Liu & 2006 & 83 & Excluded stage 2\ 055 & IEEE\_15 [@tsoumas2006towards] & Towards an Ontology-based Security Management & TSOUMAS Bill, GRITZALIS Dimitris & 2006 & 88 & Selected\ 056 & IEEE\_16 [@khan2012security] & Security oriented service composition: A framework & Khaled M Khan, Abdelkarim Erradi, Saleh Alhazbi, Jun Han & 2012 & 3 & Excluded stage 1\ 057 & IEEE\_17 [@hentea2004multi] & Multi-agent security service architecture for mobile learning & Manana Hentea & 2004 & 6 & Excluded stage 1\ 058 & IEEE\_18 [@squicciarini2006achieving] & Achieving privacy in trust negotiations with an ontology-based approach & A. C. Squicciarini, E. Bertino, E. Ferrari, I. Ray & 2006 & 50 & Excluded stage 2\ 059 & IEEE\_19 [@firesmith2007engineering] & Engineering Safety and Security Related Requirements for Software Intensive Systems & Donald G. Firesmith & 2007 & 30 & Excluded stage 2\ 060 & IEEE\_20 [@chen2004intelligent] & Intelligent agents meet the semantic Web in smart spaces & Harry Chen, Tim Finin, Anupam Joshi, Lalana Kagal, Filip Perich, Dipanjan Chakraborty & 2004 & 277 & Excluded stage 1\ 061 & IEEE\_21 [@wang2009environmental] & Environmental Metrics for Software Security Based on a Vulnerability Ontology & Ju An Wang, Minzhe Guo, Hao Wang, Min Xia, Linfeng Zhou & 2009 & 4 & Excluded stage 2\ 062 & IEEE\_22 [@bouna2011image] & The image protector - A flexible security rule specification toolkit & Bechara Al Bouna, Richard Chbeir, Alban Gabillon & 2011 & 9 & Excluded stage 1\ 063 & IEEE\_23 [@ben2012semantic] & Semantic matching of web services security policies & Monia Ben Brahim, Tarak Chaari, Maher Ben Jemaa, Mohamed Jmaiel & 2012 & 1 & Excluded stage 1\ 064 & IEEE\_24 [@chen2012design] & The design of an ontology-based service-oriented architecture framework for traditional Chinese medicine healthcare & Shih-Wei Chen, Yu-Ting Tseng, Tsai-Ya Lai & 2012 & 0 & Excluded stage 1\ 065 & IEEE\_25 [@chen2004soupa] & Soupa: Standard ontology for ubiquitous and pervasive applications & Harry Chen, Filip Perich, Tim Finin, Anupam Joshi & 2004 & 634 & Excluded stage 2\ 066 & IEEE\_26 [@akmayeva2010ontology] & Ontology of e-Learning security & Galyna Akmayeva, Charles Shoniregun & 2010 & 2 & Excluded stage 2\ 067 & IEEE\_27 [@wei2010design] & The Design and Enforcement of a Rule-based Constraint Policy Language for Service Composition & Wei Wei, Ting Yu & 2010 & 1 & Excluded stage 1\ 068 & IEEE\_28 [@coma2008context] & Context Ontology for Secure Interoperability & Celine Coma, Nora Cuppens-Boulahia1, Frederic Cuppens, Ana Rosa Cavalli & 2008 & 17 & Excluded stage 2\ 069 & IEEE\_29 [@saripalle2015towards] & Towards knowledge level privacy and security using RDF/RDFS and RBAC & Rishi Kanth Saripalle, Alberto De la Rosa Algarin, Timoteus B. Ziminski & 2015 & 0 & Excluded stage 1\ 070 & IEEE\_30 [@maisonnasse2006detecting] & Detecting privacy in attention aware system & Maisonnasse, Jéróme, Nicolas Gourier, Oliver Brdiczka, Patrick Reignier, James L. Crowley & 2006 & 4 & Excluded stage 2\ 071 & IEEE\_31 [@vorobiev2006specifying] & Specifying Dynamic Security Properties of Web Service Based Systems & Artem Vorobiev, Jun Han & 2006 & 17 & Excluded stage 1\ 072 & IEEE\_32 [@elahi2008semantic] & Semantic Access Control in Web Based Communities & Najeeb Elahi, Mohammad MR Chowdhury, Josef Noll & 2008 & 34 & Excluded stage 1\ 073 & IEEE\_33 [@gao2010approach] & An Approach for Privacy Protection Based-On Ontology & Feng Gao, Jingsha He, Shufen Peng, Xu Wu, Xiu Liu & 2010 & 9 & Excluded stage 2\ 074 & IEEE\_34 [@hsieh2011light] & A Light-Weight Ranger Intrusion Detection System on Wireless Sensor Networks & Chia-Fen Hsieh, Yung-Fa Huang, Rung-Ching Chen & 2011 & 8 & Excluded stage 1\ 075 & IEEE\_35 [@chandramouli2013knowledge] & Knowledge modeling for privacy-by-design in smart surveillance solution & Krishna Chandramouli, Virginia Fernandez Arguedas, Ebroul Izquierdo & 2013 & 0 & Excluded stage 2\ 076 & IEEE\_36 [@singh2014comparative] & A comparative study of Cloud Security Ontologies & Vaishali Singh, S.K. Pandey & 2014 & 0 & Excluded stage 2\ 077 & IEEE\_37 [@bao2007privacy] & Privacy-Preserving Reasoning on the Semantic Web & Jie Bao, Giora Slutzki, Vasant Honavar & 2007 & 37 & Excluded stage 1\ 078 & IEEE\_38 [@lee15ontology] & Ontology of Secure Service Level Agreement & Chen-Yu Lee, Krishna M. Kavi, Paul Raymond, Gomathisankaran Mahadevan & 2015 & 0 & Excluded stage 2\ 079 & IEEE\_39 [@koinig2015contrology] & Contrology - An Ontology-Based Cloud Assurance Approach & Ulrich Koinig, Simon Tjoa, Jungwoo Ryoo & 2015 & 0 & Excluded stage 1\ 080 & IEEE\_40 [@sardis2013secure] & Secure Enterprise Interoperability Ontology for Semantic Integration of Business to Business Applications & Emmanuel Sardis, Spyridon V Gogouvitis, Thanassis Bouras, Panagiotis Gouvas, Theodora Varvarigou & 2015 & 0 & Excluded stage 1\ 081 & IEEE\_41 [@garcia2009towards] & Towards a base ontology for privacy protection in service-oriented architecture & Diego Garcia, M. Beatriz F. Toledo, Miriam A. M. Capretz, David S. Allison, Gordon S. Blair, Paul Grace, Carlos Flores & 2009 & 1 & Excluded stage 2\ 082 & IEEE\_42 [@fenz2007information] & Information Security Fortification by Ontological Mapping of the ISO/IEC 27001 Standard & Fenz, Stefan, Gernot Goluch, Andreas Ekelhart, Bernhard Riedl, Edgar Weippl & 2007 & 45 & Excluded stage 2\ 083 & IEEE\_43 [@ahamed2008cctb] & CCTB: Context Correlation for Trust Bootstrapping in Pervasive Environment & Ahamed, Sheikh Monjur, Mehrab, Mohammad Saiful Islam & 2008 & 9 & Excluded stage 1\ 084 & IEEE\_44 [@asim2011interoperable] & An interoperable security framework for connected healthcare & Muhammad Asim, Milan Petkovi/’c, Mike Qu, Changjie Wang & 2011 & 2 & Excluded stage 1\ 085 & IEEE\_45 [@guan2014framework] & A framework for security driven software evolution & Hui Guan, Xuan Wang, Hongj Yang & 2014 & 0 & Excluded stage 1\ 086 & IEEE\_46 [@wei2009research] & Research on Semantic-Based Security Services Model of SOA & Cuncun Wei, Guanghua Chen, Qianqian Ge & 2009 & 0 & Excluded stage 1\ 087 & IEEE\_47 [@ahamed2008cctb] & CCTB: Context Correlation for Trust Bootstrapping in Pervasive Environment & Ahamed, Sheikh I and Monjur, Mehrab and Islam, Mohammad Saiful & 2008 & 9 & Duplicated\ 088 & IEEE\_48 [@chowdhury2007enabling] & Enabling Access Control and Privacy through Ontology & Mohammad M. R. Chowdhury, JosefNoll’ and Juan Miguel Gomez & 2007 & 8 & Excluded stage 2\ 089 & IEEE\_49 [@bishop2003computer] & What is computer security? & Matt Bishop & 2003 & 1916 & Excluded stage 2\ 090 & IEEE\_50 [@Giorgini2005] & Modeling security requirements through ownership, permission and delegation & Paolo Giorgini, Fabio Massacci, John Mylopoulos and Nicola Zannone & 2005 & 198 & Selected\ 091 & IEEE\_51 [@hecker2008privacy] & Privacy Ontology Support for E-Commerce & Michael Hecker, Tharam S. Dillon, and Elizabeth Chang & 2008 & 31 & Excluded stage 2\ 092 & IEEE\_52 [@hadzic2006use] & Use of Ontology Technology for Standardization of Medical Records and Dealing with Associated Privacy Issues & Maja Hadzic, Dillon Tharam, Elizabeth Chang & 2006 & 4 & Excluded stage 2\ 093 & IEEE\_53 [@kanbe2009ontology] & Ontology Alignment in RFID Privacy Protection & Masakazu Kanbe, Shuichiro Yamamoto & 2009 & 2 & Excluded stage 1\ 094 & IEEE\_54 [@blanquer2009enhancing] & Enhancing Privacy and Authorization Control Scalability in the Grid Through Ontologies & Ignacio Blanquer, Hernández Vicente, Segrelles Damiá, Erik Torres & 2009 & 25 & Excluded stage 2\ 095 & IEEE\_55 [@evesti2010ontology] & Ontology-Based Security Adaptation at Run-Time & Antti Evesti, Eila Ovaska & 2010 & 12 & Excluded stage 1\ 096 & IEEE\_56 [@torrellas2004framework] & A framework for multi-agent system engineering using ontology domain modelling for security architecture risk assessment in e-commerce security services & Gustavo A. Santana Torrellas & 2004 & 10 & Excluded stage 2\ 097 & IEEE\_57 [@kang2013security] & A Security Ontology with MDA for Software Development & Wentao Kang, Liang Ying & 2013 & 1 & Selected\ 098 & IEEE\_58 [@vorobiev2006security] & Security Attack Ontology for Web Services & Artem Vorobiev and Jun Han & 2006 & 64 & Excluded stage 2\ 099 & IEEE\_59 [@yan2008ontology] & Ontology-Based Information Content Security Analysis & Pan Yan, Zhao Yanping, Sanxing Cao & 2008 & 7 & Excluded stage 2\ 100 & IEEE\_60 [@ekelhart2007security] & Security Ontologies: Improving Quantitative Risk Analysis & Ekelhart, Andreas, Stefan Fenz, Markus Klemen, Edgar Weippl & 2007 & 88 & Excluded stage 2\ 101 & IEEE\_61 [@d2012ontology]& An Ontology for Run-Time Verification of Security Certificates for SOA & Stefania D’Agostini, Valeria Di Giacomo, Claudia Pandolfo, Domenico Presenza & 2012 & 4 & Excluded stage 1\ 102 & IEEE\_62 [@yau2005adaptable] & An adaptable security framework for service-based systems & Stephen S Yau, Yisheng Yao, Zhaoji Chen, Luping Zhu & 2005 & 13 & Excluded stage 1\ 103 & CIT\_01 [@poritz2004property] & Property attestation—scalable and privacy-friendly security assessment of peer computers & Jonathan Poritz, Matthias Schunter, Els Van Herreweghen, and Michael Waidner & 2004 & 137 & Excluded stage 2\ 104 & CIT\_02 [@guarino1998formal] & Formal Ontology and Information Systems & Nicola Guarino & 1998 & 4406 & Excluded stage 1\ 105 & CIT\_03 [@souag2013using] & Using Security and Domain ontologies for Security Requirements Analysis & Amina Souag, Camille Salinesi, Isabelle Wattiau, Haris Mouratidis & 2013 & 4 & Duplicated\ 106 & CIT\_04 [@kost2012privacy] & Privacy analysis using ontologies & Martin Kost, Johann-Christoph Freytag & 2012 & 9 & Duplicated\ 107 & CIT\_05 [@fenselontologies] & Ontologies: A Silver Bullet for Knowledge Management and Electronic & Dieter Fensel & 2000 & 23 & Excluded stage 1\ 108 & CIT\_06 [@kost2011privacy] & Privacy Verification using Ontologies & Martin Kost, Johann-Christoph Freytag & 2011 & 10 & Duplicated\ 109 & CIT\_07 [@velasco2009modelling] & Modeling Reusable Security Requirements Based on an Ontology Framework & Joaquín Lasheras, Rafael Valencia-García, Jesualdo Tomás Fernández-Breis & 2009 & 30 & Selected\ 110 & CIT\_08 [@foster1998security] & A Security Architecture for Computational Grids & Ian Foster, Carl Kesselman, Gene Tsudik, Steven Tuecke & 1998 & 1765 & Excluded stage 1\ 111 & CIT\_09 [@breaux2008analyzing] & Analyzing regulatory rules for privacy and security requirements & Travis D. Breaux, Annie Antón & 2008 & 251 & Excluded stage 2\ 112 & CIT\_10 [@kim2005security] & Security Ontology for Annotating Resources & Kim Anya, Jim Luo, Myong Kang & 2005 & 151 & Duplicated\ 113 & CIT\_11 [@perrig2002spins]& SPINS: Security Protocols for Sensor Networks & Adrian Perrig, Robert Szewczyk, Justin Douglas Tygar, Victor Wen, David E Culler & 2002 & 4493 & Excluded stage 1\ 114 & CIT\_12 [@chor1998private]& Private Information Retrieval & Benny Chor, Kushilevitz Eyal, Oded Goldreich, Madhu Sudan & 1998 & 1535 & Excluded stage 2\ 115 & CIT\_13 [@souag2012towards] & Towards a new generation of security requirements definition methodology using ontologies & Amina Souag & 2012 & 4 & Excluded stage 2 - Survey paper\ 116 & CIT\_14 [@floridi2005ontological] & The ontological interpretation of informational privacy & Luciano Floridi & 2005 & 109 & Excluded stage 1\ 117 & CIT\_15 [@ekclhart2007ontological] & Ontological mapping of common criteria’s security assurance requirements & Andreas Ekclhart, Stefan Fenz, Gernot Goluch, Edgar Weippl & 2007 & 21 & Excluded stage 2\ 118 & CIT\_16 [@vorobiev2006security] & Security Attack Ontology for Web Services & Artem Vorobiev, Jun Han & 2006 & 57 & Duplicated\ 119 & CIT\_17 [@velasco2009modelling] & An Ontology for Modelling Security: The Tropos Approach & Haralambos Mouratidis, Paolo Giorgini, Gordon Manson & 2003 & 52 & Duplicated\ 120 & CIT\_18 [@Massacci2007] & An Ontology for Secure Socio-Technical Systems & Fabio Massacci, John Mylopoulos, Nicola Zannone & 2007 & 45 & Excluded stage 2 - better version Spgr\_07\_02\ 121 & CIT\_19 [@chen2003ontology] & An Ontology for Context-Aware Pervasive Computing Environments & Harry Chen, Tim Finin, Anupam Joshi & 2003 & 1023 & Duplicated\ 122 & CIT\_20 [@squicciarini2006achieving] & Achieving Privacy in Trust Negotiations with an Ontology-Based Approach & Anna C. Squicciarini, Elisa Bertino, Elena Ferrari, Indrakshi Ray & 2006 & 50 & Duplicated\ 123 & CIT\_21 [@parkin2009information] & An Information Security Ontology Incorporating Human-Behavioral Implications & Simon E Parkin, Aad van Moorsel, Robert Coles & 2009 & 33 & Excluded stage 1\ 124 & CIT\_22 [@vorobiev2007ontological] & An Ontological Approach Applied to Information Security and Trust & Artem Vorobiev, Bekmamedova Nargiza & 2007 & 12 & Excluded stage 1\ 125 & CIT\_23 [@zhangdeveloping] & Developing a privacy ontology for privacy control in context-aware systems & Ni Zhang, Chris Todd & 2005 & 0 & Excluded stage 2\ 126 & CIT\_24 [@mitra2005privacy] & Privacy-preserving ontology matching & Prasenjit Mitra, Peng Liu, Chi-Chun Pan & 2005 & 10 & Excluded stage 1\ 127 & CIT\_25 [@chen2004soupa] & SOUPA: Standard Ontology for Ubiquitous and Pervasive Applications & Harry Chen, Filip Perich, Tim Finin, Anupam Joshi & 2004 & 634 & Duplicated\ 128 & CIT\_26 [@langheinrich2001privacy] & Privacy by Design - Principles of Privacy-Aware Ubiquitous Systems & Marc Langheinrich & 2001 & 769 & Excluded stage 2\ 129 & CIT\_27 [@chen2005soupa] & The SOUPA Ontology for Pervasive Computing & Chen, Harry, Tim Finin, Anupam Joshi & 2005 & 179 & Duplicated\ 130 & CIT\_28 [@hecker2007privacy] & Privacy support and evaluation on an ontological basis & Michael Hecker, Dillon Tharam & 2007 & 5 & Excluded stage 1\ 131 & CIT\_29 [@vorobiev2008ontology] & An ontology framework for managing security attacks and defenses in component based software systems & Artem Vorobiev, Jun Han, Nargiza Bekmamedova & 2008 & 7 & Excluded stage 2\ 132 & CIT\_30 [@gandhi2009ontology] & Ontology Guided Risk Analysis: From Informal Specifications to Formal Metrics & Robin Gandhi, Seok-Won Lee & 2009 & 1 & Duplicated\ 133 & CIT\_31 [@ranganathan2003ontologies] & Ontologies in a pervasive computing environment & Anand Ranganathan, Robert E. McGrath, Roy H. Campbell, Mickunas M. Dennis & 2003 & 65 & Excluded stage 2\ 134 & CIT\_32 [@eiter2006reasoning] & Reasoning with rules and ontologies & Thomas Eiter, Giovambattista Ianni, Axel Polleres, Roman Schindlauer, Hans Tompits & 2006 & 75 & Excluded stage 1\ 135 & CIT\_33 [@liu2003security] & Security and Privacy Requirements Analysis within a Social Setting & Lin Liu, Eric Yu, John Mylopoulos & 2006 & 75 & Selected\ 136 & CIT\_34 [@evfimievski2003limiting] & Limiting Privacy Breaches in Privacy Preserving Data Mining & Alexandre Evfimievski, Johannes Gehrke, Srikant Ramakrishnan & 2003 & 642 & Excluded stage 1\ 137 & CIT\_35 [@mcgrath2003use] & Use of Ontologies in Pervasive Computing Environments & Robert E McGrath, Anand Ranganathan, Roy H Campbell, Mickunas M Dennis & 2003 & 33 & Excluded stage 1\ 138 & Spgr\_01 [@kim2005security] & Security ontology for annotating resources & Kim Anya, Jim Luo, Myong Kang & 2005 & 151 & Excluded stage 2\ 139 & Spgr\_02 [@fabian2010comparison] & A comparison of security requirements engineering methods & Fabian Benjamin, Seda Gurses, Maritta Heisel, Thomas Santen, Holger Schmidt & 2010 & 129 & Excluded stage 2 - Survey paper\ 140 & Spgr\_03 [@souag2012ontologies] & Ontologies for security requirements: A literature survey and classification & Amina Souag, Camille Salinesi, Isabelle Wattiau & 2012 & 41 & Excluded stage 2 - Survey paper\ 141 & Spgr\_04 [@studer2009privacy] & Privacy preserving modules for ontologies & Thomas Studer & 2010 & 4 & Duplicated\ 142 & Spgr\_05 [@yau2006framework] & A framework for specifying and managing security requirements in collaborative systems & Stephen S. Yau, Chen Zhaoji & 2006 & 17 & Duplicated\ 143 & Spgr\_06 [@alam2006model] & Model driven security engineering for the realization of dynamic security requirements in collaborative systems & Muhammad Alam & 2007 & 14 & Duplicated\ 144 & Spgr\_07 [@massacci2011extended] & An Extended Ontology for Security Requirements & Fabio Massacci, John Mylopoulos, Federica Paci, Thein Thun Tun, Yijun Yu & 2011 & 16 & Selected\ 145 & Spgr\_08 [@souag2015security] & A Security Ontology for Security Requirements Elicitation & Amina Souag, Camille Salinesi, Raúl Mazo, Isabelle Comyn-Wattiau & 2015 & 4 & Excluded stage 2 - Survey paper\ 146 & Spgr\_09 [@souag2012ontologies] & Ontologies for Security Requirements: A Literature Survey and Classification & Amina Souag, Camille Salinesi, Isabelle Wattiau & 2012 & 29 & Duplicated\ 147 & Spgr\_10 [@souag2015reusable]& Reusable knowledge in security requirements engineering: a systematic mapping study & Amina Souag, Raúl Mazo, Camille Salinesi, Isabelle Comyn-Wattiau & 2015 & 1 & Excluded stage 1\ 148 & Spgr\_11 [@kim2005security]& Security Ontology for Annotating Resources & Kim Anya, Jim Luo, Myong Kang & 2005 & 151 & Duplicated\ 149 & Spgr\_12 [@tropea2011introducing]& Introducing Privacy Awareness in Network Monitoring Ontologies & Giuseppe Tropea, Georgios V Lioudakis, Nicola Blefari-Melazzi, Dimitra I Kaklamani, Iakovos S Venieris & 2011 & 1 & Excluded stage 1\ 150 & Spgr\_13 [@elahi2009modeling] & A Modeling Ontology for Integrating Vulnerabilities into Security Requirements Conceptual Foundation & Golnaz Elahi, Eric Yu, Nicola Zannone & 2009 & 21 & Selected\ 151 & Spgr\_14 [@tsoumas2006security] & Security-by-Ontology: A Knowledge-Centric Approach & Bill Tsoumas, Panagiotis Papagiannakopoulos, Stelios Dritsas, Dimitris Gritzalis & 2006 & 12 & Excluded stage 2\ 152 & Spgr\_15 [@sicilia2015information] & What are Information Security Ontologies Useful for? & Bill Sicilia, Miguel-Angel García-Barriocanal, Elena Javier Bermejo-Higuera, Salvador Sánchez-Alonso & 2015 & 0 & Excluded stage 1\ 153 & Spgr\_16 [@papagiannakopoulou2014leveraging] & Leveraging Ontologies upon a Holistic Privacy-Aware Access Control Model & Eugenia I Papagiannakopoulou, Maria N Koukovini, Georgios V Lioudakis, Nikolaos Dellas, Joaquin Garcia-Alfaro, Dimitra I Kaklamani, Iakovos S Venieris, Nora Cuppens-Boulahia, Frédéric Cuppens & 2014 & 4 & Excluded stage 1\ 154 & Spgr\_17 [@sure2004towards] & Towards Cross-Domain Security Properties Supported by Ontologies & York Sure, Jochen Haller & 2004 & 5 & Excluded stage 1\ 155 & Spgr\_18 [@milicevic2010ontology] & Ontology-Based Evaluation of ISO 27001. & Danijel Milicevic, Matthias Goeken & 2010 & 3 & Excluded stage 2\ 156 & Spgr\_19 [@dhiah2006ontology] & An Ontology-Based Approach for Managing and Maintaining Privacy in Information Systems & Dhiah el Diehn, Abou-Tair I. Stefan Berlik & 2006 & 5 & Excluded stage 2\ 157 & Spgr\_20 [@vincent2011privacy] & Privacy Protection for Smartphones: An Ontology-Based Firewall & Johann Vincent, Christine Porquet, Maroua Borsali, Harold Leboulanger & 2011 & 8 & Excluded stage 2\ 158 & Spgr\_21 [@ionita2005specifying] & Specifying an Access Control Model for Ontologies for the Semantic Web & Cecilia Ionita, Osborn M, Sylvia L & 2005 & 9 & Excluded stage 1\ 159 & Spgr\_22 [@gandhi2009ontology] & Ontology Guided Risk Analysis: From Informal Specifications to Formal Metrics & Robin Gandhi, Lee Seok-Won & 2009 & 1 & Excluded stage 2\ 160 & Spgr\_23 [@torres2012hit] & HIT Considerations: Informatics and Technology Needs and Considerations & Miguel Humberto Torres-Urquidy, Valerie J. H. Powell, Franklin M. Din, Mark Diehl, Valerie Bertaud-Gounot, W. Ted Klein, Sushma Mishra, Shin-Mey Rose Yin Geist, Monica Chaudhari, Mureen Allen & 2012 & 0 & Excluded stage 1\ 161 & Spgr\_24 [@hoss2007towards] & Towards Combining Ontologies and Model Weaving for the Evolution of Requirements Models & Allyson M Hoss, Doris L Carver & 2008 & 5 & Excluded stage 1\ 162 & Spgr\_25 [@stoica2004ontology] & Ontology Guided XML Security Engine & Andrei Stoica, Csilla Farkas & 2004 & 27 & Excluded stage 1\ 163 & Spgr\_26 [@wangusing] & Using ontologies to perform threat analysis and develop defensive strategies for mobile security & Ping Wang , Kuo-Ming Chao, Chi-Chun Lo, Yu-Shih Wang & 2015 & 1 & Excluded stage 1\ 164 & Spgr\_27 [@weippl2004semanticlife] & SemanticLIFE Collaboration: Security Requirements and Solutions – Security Aspects of Semantic Knowledge Management & Edgar R. Weippl, Alexander Schatten, Shuaib Karim, A. Min Tjoa & 2004 & 13 & Excluded stage 1\ 165 & Spgr\_28 [@heupel2015ontology] & Ontology-Enabled Access Control and Privacy Recommendations & Robin Gandhi, Lee Seok-Won & 2009 & 1 & Excluded stage 2\ 166 & Spgr\_29 [@nissan2012accounting] & Accounting for Social, Spatial, and Textual Interconnections & Ephraim Nissan & 2012 & 0 & Excluded stage 1\ 167 & Spgr\_30 [@hadzic2009case] & Case Study I: Ontology-Based Multi-Agent System for Human Disease Studies & Maja Hadzic, Pornpit Wongthongtham, Tharam Dillon, Elizabeth Chang & 2009 & 0 & Excluded stage 1\ 168 & Spgr\_31 [@chen2005soupa] & The SOUPA Ontology for Pervasive Computing & Harry Chen, Tim Finin, Anupam Joshi & 2015 & 191 & Excluded stage 2\ 169 & Spgr\_32 [@mouratidis2003ontology] & An Ontology for Modelling Security: The Tropos Approach & Haralambos Mouratidis, Paolo Giorgini, Gordon Manson & 2003 & 54 & Excluded stage 2\ 170 & Spgr\_33 [@spyns2008evaluating] & Evaluating Automatically a Text Miner for Ontologies: A Catch-22 Situation? & Peter Spyns & 2008 & 2 & Excluded stage 1\ 171 & Spgr\_34 [@mitre2006legal] & EA Legal Ontology to Support Privacy Preservation in Location-Based Services & Hugo A. Mitre, Ana Isabel González-Tablas, Benjamín Ramos, Arturo Ribagorda & 2006 & 6 & Excluded stage 2\ 172 & Spgr\_35 [@chowdhury2008capturing] & Capturing Semantics for Information Security and Privacy Assurance & Mohammad M. R. Chowdhury, Javier Chamizo, Josef Noll, Juan Miguel Gómez & 2008 & 6 & Excluded stage 2\ 173 & Spgr\_36 [@pereira2009ontology] & An Ontology Based Approach to Information Security & Teresa Pereira, Henrique Santos & 2009 & 10 & Excluded stage 2\ 174 & Spgr\_37 [@panettoefficient] & Efficient Projection of Ontologies & Julius Köpke, Johann Eder, Michaela Schicho & 2013 & 1 & Excluded stage 1\ 175 & Spgr\_38 [@delgado2003regulatory] & Regulatory Ontologies: An Intellectual Property Rights Approach & Haojun Yu, Sun Yuqing, Jinyan Hu & 2012 & 0 & Excluded stage 2\ 176 & Spgr\_39 [@sorli2009ict] & ICT Tools and Systems Supporting Innovation in Product/Process Development & Mikel Sorli, Dragan Stokic & 2009 & 0 & Excluded stage 1\ 177 & Spgr\_40 [@balopoulos2006framework] & A Framework for Exploiting Security Expertise in Application Development & Theodoros Balopoulos, Lazaros Gymnopoulos, Maria Karyda, Spyros Kokolakis, Stefanos Gritzalis, Sokratis Katsikas & 2006 & 0 & Excluded stage 1\ 178 & Spgr\_41 [@iwaihara2008risk] & Risk Evaluation for Personal Identity Management Based on Privacy Attribute Ontology & Mizuho Iwaihara, Murakami Kohei, Ahn Gail-Joon , Masatoshi Yoshikawa & 2008 & 12 & Excluded stage 2\ 179 & Spgr\_42 [@analyti2013framework] & A framework for modular ERDF ontologies & Analyti, Antoniou Anastasia, Grigoris and Damásio, Carlos Viegas, Ioannis Pachoulakis & 2013 & 3 & Excluded stage 1\ 180 & Spgr\_43 [@kayes2013ontology] & An Ontology-Based Approach to Context-Aware Access Control for Software Services & Asm Kayes, Jun Han, Alan Colman & 2013 & 8 & Excluded stage 1\ 181 & Spgr\_44 [@mezgar2007development] & Development of an Ontology-Based Smart Card System Reference Architecture & István Mezgár, Zoltán Kincses & 2007 & 0 & Excluded stage 1\ 182 & Spgr\_45 [@kabilan2007introducing] & Introducing the Common Non-Functional Ontology & Vandana Kabilan, Paul Johannesson, Sini Ruohomaa, Pirjo Moen, Andrea Herrmann, Rose-Mharie Ahlfeldt, Hans Weigand & 2007 & 4 & Excluded stage 1\ 183 & Spgr\_46 [@ryan2003ontology] & Ontology-Based Platform for Trusted Regulatory Compliance Services & Henry Ryan, Peter Spyns, Pieter De Leenheer, Richard Leary & 2003 & 7 & Excluded stage 1\ 184 & Spgr\_47 [@isaza2010intrusion] & Intrusion Correlation Using Ontologies and Multi-agent Systems & Gustavo Isaza, Andrés Castillo, Marcelo López, Luis Castillo, Manuel López & 2010 & 6 & Excluded stage 1\ 185 & Spgr\_48 [@albers2004agent] & Agent Models and Different User Ontologies for an Electronic Market Place & Marcel Albers, Catholijn M Jonker, Mehrzad Karami, Jan Treur, & 2004 & 27 & Excluded stage 1\ 186 & Spgr\_49 [@rajugan2006ontology] & Ontology Views: A Theoretical Perspective & Rajagopal Rajugan, Elizabeth Chang, Tharam S Dillon, & 2006 & 11 & Excluded stage 1\ 187 & Spgr\_50 [@undercoffer2003modeling] & Modeling Computer Attacks: An Ontology for Intrusion Detection & Jeffrey Undercoffer, Anupam Joshi, John Pinkston & 2003 & 136 & Excluded stage 1\ 188 & Spgr\_51 [@denker2003security] & Security for DAML Web Services: Annotation and Matchmaking & Grit Denker, Lalana Kagal, Tim Finin, Massimo Paolucci, Katia Sycara & 2003 & 183 & Excluded stage 1\ 189 & Spgr\_52 [@liu2009ontology] & Ontology-Based Requirements Conflicts Analysis in Activity Diagrams & Chi-Lun Liu & 2009 & 6 & Excluded stage 1\ 190 & Spgr\_53 [@beckers2012ontology] & Ontology-Based Identification of Research Gaps and Immature Research Areas & Kristian Beckers, Stefan Eicker, Stephan Fa[ß]{}bender, Maritta Heisel, Holger Schmidt, Widura Schwittek & 2012 & 4 & Excluded stage 1\ 191 & Spgr\_54 [@ceravolo2003managing] & Managing Identities via Interactions between Ontologies & Paolo Ceravolo & 2003 & 4 & Excluded stage 1\ 192 & Spgr\_55 [@ekelhart2006security] & Security Ontology: Simulating Threats to Corporate Assets & Andreas Ekelhart, Stefan Fenz, Markus D. Klemen, Edgar R. Weippl & 2006 & 31 & Excluded stage 2\ 193 & Spgr\_56 [@abulaish2011simont] & SIMOnt: A Security Information Management Ontology Framework & Muhammad Abulaish, Syed Irfan Nabi, Khaled Alghathbar, Azeddine Chikh & 2011 & 3 & Excluded stage 2\ 194 & Spgr\_57 [@man2005retracted] & Retracted: Shared Ontology for Pervasive Computing & Junfeng Man, Aimin Yang, Xingming Sun & 2005 & 3 & Excluded stage 1\ 195 & Spgr\_58 [@ciuciu2011ontology] & Ontology-Based Matching of Security Attributes for Personal Data Access in e-Health & Ioana Ciuciu, Brecht Claerhout, Louis Schilders, Robert Meersman & 2011 & 5 & Excluded stage 2\ 196 & Spgr\_59 [@blobel2011intelligent] & Intelligent security and privacy solutions for enabling personalized telepathology & Bernd Blobel & 2011 & 5 & Excluded stage 1\ 197 & Spgr\_60 [@breaux2014eddy] & Eddy, a formal language for specifying and analyzing data flow specifications for conflicting privacy & Travis D. Breaux, Hibshi Hanan, Rao Ashwini & 2014 & 16 & Excluded stage 2\ 198 & SCH\_01 [@breaux2008analyzing] & Analyzing regulatory rules for privacy and security requirements & Travis D. Breaux, Annie Antón & 2008 & 230 & Duplicated\ 199 & SCH\_02 [@chen2003ontology] & An Ontology for Context-Aware Pervasive Computing Environments & Harry Chen, Tim Finin, Anupam Joshi & 2003 & 1086 & Excluded stage 2\ 200 & SCH\_03 [@massacci2005using] & Using a security requirements engineering methodology in practice: the compliance with the Italian data protection legislation & Fabio Massacci, Marco Prest, Nicola Zannone & 2005 & 82 & Excluded stage 2 - better version Spgr\_07\_02\ 201 & SCH\_04 [@takabi2010security] & Security and privacy challenges in cloud computing environments & Hassan Takabi, James BD Joshi, Ahn Gail-Joon & 2010 & 660 & Excluded stage 1\ 202 & SCH\_05 [@ashburner2000gene] & Gene Ontology: tool for the unification of biology & Ashburner M, Ball CA, Blake JA, Botstein D, Butler H, Cherry JM, Davis AP, Dolinski K, Dwight SS, Eppig JT, Harris MA, Hill DP, Issel-Tarver L, Kasarskis A, Lewis S, Matese JC, Richardson JE, Ringwald M, Rubin GM, Sherlock G. & 2000 & 15501 & Excluded stage 1\ 203 & SCH\_06 [@sacco2011privacy] & A Privacy Preference Ontology (PPO) for Linked Data & Owen Sacco, Alexandre Passant & 2011 & 52 & Excluded stage 2\ 204 & SCH\_07 [@poritz2004property] & Property attestation—scalable and privacy-friendly security assessment of peer computers & Jonathan Poritz, Matthias Schunter, Els Van Herreweghen, and Michael Waidner & 2004 & 132 & Duplicated\ 205 & SCH\_08 [@chase2009improving] & Improving privacy and security in multi-authority attribute-based encryption & Melissa Chase, Sherman S.M. Chow & 2009 & 375 & Duplicated\ 206 & SCH\_09 [@li2010data] & Data security and privacy in wireless body area networks & Ming Li, Wenjing Lou, Kui Ren & 2010 & 297 & Excluded stage 1\ 207 & SCH\_10 [@ohkubo2003cryptographic] & Cryptographic approach to “privacy-friendly” tags & Miyako Ohkubo, Koutarou Suzuki, Shingo Kinoshita, and others & 2003 & 793 & Excluded stage 1\ 208 & SCH\_11 [@chen2004soupa] & Soupa: Standard ontology for ubiquitous and pervasive applications & Harry Chen, Filip Perich, Tim Finin, Anupam Joshi & 2004 & 634 & Duplicated\ 209 & SCH\_12 [@ferrari2004security] & Security and privacy for web databases and services & Elena Ferrari, Bhavani Thuraisingham & 2004 & 94 & Excluded stage 1\ 210 & SCH\_13 [@weber2010internet] & Internet of Things – New security and privacy challenges & Rolf H Weber & 2010 & 297 & Excluded stage 1\ 211 & SCH\_14 [@el2003privacy] & Privacy and Security in E-Learning & Khalil El-Khatib, Larry Korba, Yuefei Xu, George Yee & 2003 & 86 & Excluded stage 1\ 212 & SCH\_15 [@ferrari2004security] & Security and privacy for web databases and services & Elena Ferrari, Bhavani Thuraisingham & 2004 & 94 & Excluded stage 1\ 213 & SCH\_16 [@firesmith2003security] & Security use cases & Donald G. Firesmith & 2003 & 353 & Excluded stage 2\ 214 & SCH\_17 [@squicciarini2006achieving] & Achieving privacy in trust negotiations with an ontology-based approach & Anna C. Squicciarini, Elisa Bertino, Elena Ferrari, Indrakshi Ray & 2006 & 50 & Duplicated\ 215 & SCH\_18 [@sindre2005eliciting]& Eliciting security requirements with misuse cases & Guttorm Sindre, Andreas L. Opdahl & 2005 & 830 & Selected\ 216 & SCH\_19 [@chen2005soupa] & The SOUPA ontology for pervasive computing & Chen, Harry, Tim Finin, Anupam Joshi & 2005 & 179 & Duplicated\ 217 & SCH\_20 [@anton2002analyzing] & Analyzing website privacy requirements using a privacy goal taxonomy & Annie Antón, Julia B. Earp, Angela Reese & 2002 & 105 & Excluded stage 2\ 218 & SCH\_21 [@jansen2011guidelines] & Guidelines on security and privacy in public cloud computing & Wayne Jansen, Timothy Grance, and others & 2011 & 502 & Duplicated\ 219 & SCH\_22 [@van2004elaborating] & Elaborating security requirements by construction of intentional anti-models & Axel Van Lamsweerde & 2004 & 337 & Duplicated\ 220 & SCH\_23 [@massacci2005using] & Using a security requirements engineering methodology in practice: the compliance with the Italian data protection legislation & Wayne Jansen, Timothy Grance, and others & 2011 & 502 & Excluded stage 1\ 221 & SCH\_24 [@kalloniatis2008addressing] & Addressing privacy requirements in system design: the PriS method & Christos Kalloniatis, Evangelia Kavakli, Stefanos Gritzalis & 2008 & 76 & Selected\ 222 & SCH\_25 [@carminati2006security] & Security conscious web service composition & Barbara Carminati, Elena Ferrari, Patrick CK Hung, & 2006 & 93 & Excluded stage 1\ 223 & SCH\_26 [@haley2008security] & Security requirements engineering: A framework for representation and analysis & Charles B. Haley, Robin Laney, Jonathan D. Moffett, Bashar Nuseibeh & 2008 & 281 & Excluded stage 2\ 224 & SCH\_27 [@he2003framework] & A framework for modeling privacy requirements in role engineering & Qingfeng He, Annie I. Antón & 2003 & 85 & Excluded stage 2\ 225 & SCH\_28 [@mouratidis2007secure] & Secure Tropos: a security-oriented extension of the Tropos methodology & Haralambos Mouratidis, Paolo Giorgini & 2007 & 193 & Selected\ 226 & SCH\_29 [@rezgui2002preserving] & Preserving privacy in web services & Abdelmounaam Rezgui, Mourad Ouzzani, Athman Bouguettaya, Brahim Medjahed & 2002 & 102 & Duplicated\ 227 & SCH\_30 [@gandon2004semantic] & Semantic web technologies to reconcile privacy and context awareness & Fabien L Gandon, Norman M Sadeh & 2004 & 215 & Excluded stage 1\ 228 & SCH\_31 [@hecker2008privacy] & Privacy ontology support for e-commerce & Michael Tharam S. Hecker, Elizabeth Chang Dillon & 2008 & 29 & Duplicated\ 229 & SCH\_32 [@donner2003toward] & Toward a security ontology & Marc Donner & 2003 & 56 & Excluded stage 2\ 230 & SCH\_33 [@gao2010approach] & An approach for privacy protection based-on ontology & Gao Feng, Jingsha He, Shufen Peng, Xu Wu, Xiu Liu & 2010 & 9 & Duplicated\ 231 & SCH\_34 [@kim2005security] & Security ontology for annotating resources & Kim Anya, Jim Luo, Myong Kang & 2005 & 151 & Duplicated\ 232 & SCH\_35 [@brar2004privacy] & Privacy and security in ubiquitous personalized applications & Ajay Brar, Judy Kay & 2004 & 38 & Excluded stage 1\ 233 & SCH\_36 [@alliance2003hipaa] & HIPAA compliance and smart cards: Solutions to privacy and security requirements & Alliance, Smart Card & 2003 & 17 & Excluded stage 2\ 234 & SCH\_37 [@mitra2005privacy] & Privacy-preserving ontology matching & Prasenjit Mitra, Peng Liu, Chi-Chun Pan & 2005 & 10 & Duplicated\ 235 & SCH\_38 [@iwaihara2008risk] & Risk evaluation for personal identity management based on privacy attribute ontology & Mizuho Iwaihara, Murakami Kohei, Gail-Joon Ahn, Masatoshi Yoshikawa & 2008 & 12 & Duplicated\ 236 & SCH\_39 [@kagal2006security] & Security and privacy challenges in open and dynamic environments & Lalana Kagal, Tim Finin, Anupam Joshi, Sol Greenspan, & 2006 & 27 & Excluded stage 1\ 237 & SCH\_40 [@fabian2010comparison] & A comparison of security requirements engineering methods & Fabian Benjamin, Seda Gurses, Maritta Heisel, Thomas Santen, Holger Schmidt & 2010 & 121 & Duplicated\ 238 & SCH\_41 [@solove2006taxonomy] & A taxonomy of privacy & Daniel J Solove & 2006 & 967 & Selected\ 239 & SCH\_42 [@ware1981taxonomy] & A taxonomy for privacy & Willis H Ware & 1981 & 5 & Excluded stage 1\ 240 & SCH\_43 [@skinner2006information] & An information privacy taxonomy for collaborative environments & Geoff Skinner, Song Han, Elizabeth Chang & 2006 & 24 & Excluded stage 2\ \ 241 & Spgr\_18\_01 [@karyda2006ontology] & An ontology for secure e-government applications & M. Karyda, T. Balopoulos, S. Dritsas, L. Gymnopoulos, S. Kokolakis, C. Lambrinoudakis, S. Gritzalis & 2006 & 37 & Excluded stage 2\ 242 & Spgr\_18\_02 [@raskin2001ontology] & Ontology in information security: a useful theoretical foundation and methodological tool & Victor Raskin, Christian F Hempelmann, Katrina E Triezenberg, Sergei Nirenburg & 2001 & 133 & Excluded stage 2\ 243 & Spgr\_18\_03 [@fenz2009formalizing] & Formalizing information security knowledge & Stefan RaskinFenz, Andreas Ekelhart & 2009 & 144 & Selected\ 244 & Spgr\_13\_01 [@asnar2008risk] & Risk as dependability metrics for the evaluation of business solutions: a model-driven approach & Yudistira Asnar, Rocco Moretti, Maurizio Sebastianis, Nicola Zannone & 2008 & 30 & Selected\ 245 & Spgr\_13\_02 [@den2003coras] & The CORAS methodology. model-based risk assessment using UML and UP & Folker den Braber, Theo Dimitrakos, Bjorn A. Gran, Mass S. Lund, Ketil Stolen, Jan O. Aagedal & 2003 & 66 & Selected\ 246 & Spgr\_13\_03 [@elahi2010vulnerability] & A vulnerability-centric requirements engineering framework. analyzing security attacks, countermeasures, and requirements based on vulnerabilities & Golnaz Elahi, Eric Yu, Nicola Zannone & 2010 & 73 & Selected\ 247 & Spgr\_13\_04 [@jurjens2002umlsec] & UMLsec: Extending UML for secure systems development & Jürjens, Jan & 2002 & 583 & Selected\ 248 & Spgr\_13\_05 [@matulevivcius2008adapting] & Adapting secure tropos for security risk management in the early phases of information systems development & Raimundas Matulevi[č]{}ius, Nicolas Mayer, Haralambos Mouratidis, Eric Dubois, Patrick Heymans, Nicolas Genon, & 2008 & 60 & Selected\ 249 & Spgr\_13\_06 [@mayer2005towards] & Towards a risk-based security requirements engineering framework & Nicolas Mayer, André Rifaut, Eric Dubois, and others & 2005 & 52 & Excluded stage 2 - better version Spgr\_08\_01\ 250 & Spgr\_13\_07 [@rostad2006extended] & An extended misuse case notation: Including vulnerabilities and the insider threat & Lillian R[ø]{}stad, & 2006 & 46 & Selected\ 251 & Spgr\_13\_08 [@singh2014revisiting] & Revisiting Security Ontologies & Vaishali Singh, SK Pandey & 2014 & 2 & Excluded stage 2\ 252 & Spgr\_08\_01 [@mayer2009model] & Model-based management of information system security risk & Nicolas Mayer & 2009 & 70 & Selected\ 253 & Spgr\_08\_02 [@velasco2009modelling] & Modelling reusable security requirements based on an ontology framework & Joaquin Velasco, Lasheras Valencia-García, Rafael Fernández-Breis, Tomás Jesualdo, Ambrosio Toval, and others & 2009 & 31 & Excluded stage 2\ 254 & Spgr\_08\_03 [@dritsas2006knowledge] & A knowledge-based approach to security requirements for e-health applications & S. Dritsas, L. Gymnopoulos, M. Karyda, T. Balopoulos, S. Kokolakis, C. Lambrinoudakis, S. Katsikas & 2006 & 17 & Selected\ 255 & Spgr\_07\_01 [@blanco2008systematic] & A systematic review and comparison of security ontologies & Blanco, Carlos and Lasheras, Joaquin and Valencia-García, Rafael and Fernández-Medina, Eduardo and Toval, Ambrosio and Piattini, Mario & 2008 & 79 & Excluded stage 2 - Survey paper\ 256 & Spgr\_07\_02 [@zannone2006requirements] & A requirements engineering methodology for trust, security, and privacy & Nicola Zannone & 2007 & 17 & Selected\ 257 & Spgr\_07\_03 [@lin2003introducing] & Introducing abuse frames for analysing security requirements & Luncheng Lin, Bashar Nuseibeh, Darrel Ince, Michael Jackson, Jonathan Moffett & 2003 & 73 & Excluded stage 2\ 258 & Spgr\_03\_01 [@avizienis2004basic] & Basic Concepts and Taxonomy of Dependable and Secure & Algirdas Avi[ž]{}ienis, Jean-Claude Laprie, Brian Randell, Carl Landwehr & 2004 & 3703 & Selected\ 259 & Spgr\_03\_02 [@firesmith2005taxonomy] & A taxonomy of security-related requirements & Donald G Firesmith & 2005 & 44 & Excluded stage 2\ 260 & Spgr\_02\_01 [@asnar2007trust] & From trust to dependability through risk analysis & Yudistira Asnar, Paolo Giorgini, Fabio Massacci, Nicola Zannone & 2007 & 57 & Selected\ 261 & Spgr\_02\_02 [@asnar2006risk] & Risk modelling and reasoning in goal models & Yudistira Asnar, Paolo Giorgini, John Mylopoulos & 2006 & 17 & Selected\ 262 & SCH\_24\_01 [@kalloniatis2005dealing] & Dealing with privacy issues during the system design process & Christos Kalloniatis, Evangelia Kavakli, Stefanos Gritzalis, & 2005 & 15 & Excluded stage 2\ 263 & SCH\_24\_02 [@hong2004privacy] & Privacy risk models for designing privacy-sensitive ubiquitous computing systems & Jason I Hong, Jennifer Ng, Scott D Lederer, , James A Landay, & 2004 & 218 & Selected\ 264 & SCH\_28\_01 [@paja2014sts] & STS-Tool Security Requirements Engineering for Socio-Technical Systems & Elda Paja, Fabiano Dalpiaz, Paolo Giorgini & 2014 & 2 & Selected\ 265 & SCH\_43\_01 [@van2003handbook] & Handbook of privacy and privacy-enhancing technologies & GW Van Blarkom, JJ Borking, JGE Olk & 2003 & 69 & Selected\ [^1]: An overview of all considered studies is shown in Table \[table:papers\] in Appendix B [^2]: When there are more than one concept with very close meaning, we have chosen the most appropriate one to represent all [^3]: These groups are not mutually exclusive, i.e., a study may belong to all of them

--- author: - 'E. Koulouridis, I. Georgantopoulos, G. Loukaidou, A. Corral, A. Akylas, L. Koutoulidis, E. F. Jiménez-Andrade, J. León Tavares, P. Ranalli' title: The XMM spectral catalog of SDSS optically selected Seyfert 2 galaxies --- Introduction ============ Nearly thirty years ago, the first discovery by Miller & Antonucci (1983) of broad permitted emission lines and a clearly non-stellar continuum in the polarized spectrum of the archetypal Seyfert 2 (Sy2), NGC 1068, was just the beginning of numerous similar observations in a wide variety of galaxies. Ten years later, the unification model of active galactic nuclei (AGN) was formulated upon these observations (Antonucci 1993). According to the unification model, all AGN are intrinsically identical, while the only cause of their different observational features is the orientation of an obscuring torus with respect to our line of sight. In more detail, the AGN type depends on the obscuration of the broad line region (BLR), a small area at close proximity to the SMBH where the broad permitted lines are produced. If the torus happens to be between the observer and the BLR, the optical emission and even the soft X-rays are absorbed. Optical spectropolarimetric observations can reveal the hidden broad line region (HBLR) by highlighting its scattered emission. The observed narrow permitted emission lines are produced at far larger distances from the core, where the torus is irrelevant. As a prediction of this model, the presence and strength of the broad optical emission lines, hence the derived optical spectral type (from type 1 AGN/Sy1s to type 2 AGN/Sy2s, and the intermediate types), should correlate with the amount of intervening material as measured in X-rays. X-ray observations can reveal the exact density of the obscuring torus, even for mildly obscured sources. X-ray surveys with [*Ginga*]{} (Smith and Done 1996) and [*ASCA*]{} (Turner et al. 1997) measured column densities between $10^{22}$ and up to a few times $10^{24}$ $\rm cm^{-2}$ in type 2 AGN samples. More recently, Akylas & Georgantopoulos (2009) and Brightman & Nandra (2011), using [*XMM-Newton*]{}, and Jia et al. (2013, JJ13 hereafter), using [*Chandra,*]{} also studied the obscuration of type 2 X-ray sources in detail (see also Brandt & Alexander (2015) for a recent review). However, even in the hard X-ray band, the X-ray surveys may be missing a fraction of highly obscured sources. These sources are called Compton-thick AGN (see reviews by Comastri 2004 and Georgantopoulos 2013), and they present very high obscuring column densities ($>10^{24} cm^{-2}$ , corresponding to an optical reddening of $A_V>$100). Even though Compton-thick AGN are abundant in the optically selected samples of nearby Seyferts (e.g., Risaliti et al. 1999), only a few tens of Compton-thick sources have been identified from X-ray data. Moreover, Krumpe et al. (2008) found no Compton-thick QSO in their high redshift ($z>0.5$), X-ray selected sample, implying a possible redshift evolution, though this may be due to selection. Although the population of Compton-thick sources remains elusive, there is concrete evidence of its presence. The X-ray background synthesis models can explain the peak of the X-ray background at 30-40 keV, where most of its energy density lies, (Frontera et al. 2007; Churazov et al. 2007) only by invoking a large number of Compton-thick AGN (Gilli et al. 2007). We note, however, that other models (e.g., Treister, Urry & Virani 2009; Akylas et al. 2012) succeed in explaining the X-ray background (XRB) spectrum assuming a lower fraction of CT sources. Additional evidence of a Compton-thick population comes from the directly measured space density of black holes in the local Universe. It is found that this space density could be up to a factor of two higher than predicted from the X-ray luminosity function (Marconi et al. 2004). This immediately suggests that the X-ray luminosity function is missing an appreciable number of obscured AGN. On the other hand, although widely accepted today, the unification model cannot explain a series of observations. For example, Tran et al. (2001) noticed the absence of a HBLR in polarized light in many Sy2 galaxies (non-HBLR Sy2 galaxies), suggesting that there is a class of true Sy2 galaxies that intrinsically lack the broad-line region (see Ho et al. 2008 for a review). Theoretical models attributed the absence of a BLR to either a low Eddington ratio (Nicastro 2000) or to low luminosity (Elitzur & Shlossman 2006). Many studies propose an evolutionary model where a fraction of Sy2 represents the first or the last phase in the life of an AGN (Hunt & Malkan 1999; Dultzin-Hacyan, 1999; Krongold et al. 2002; Levenson et al. 2001; Koulouridis et al. 2006a,b, 2013; Koulouridis 2014, Elitzur, Ho & Trump 2014). This was supported by studies of the local environment of Seyfert galaxies, which showed that Sy2s reside in richer environments compared to Sy1s (e.g., Villaroel & Korn 2014). Unobscured low-luminosity Sy2s were detected via investigation of their X-ray properties (e.g., Pappa et al. 2000, Panessa & Bassani 2002; Akylas & Georgantopoulos 2009). Models of galaxy formation also support this scenario: for example, Hopkins et al. (2008) assert that the AGN is heavily obscured during its birth. During the build-up of its black hole mass, it blows away its cocoon, becoming an unobscured AGN. In this paper, we compile a sample of bona fide optically selected Sy2 galaxies using the SDSS spectra from the data release 10 (DR10). We cross-correlate our sample with the 3XMM/XMMFITCAT spectral catalog (Corral et al. 2015), which contains good quality spectra (at least 50 net counts per XMM detector). We identify a sample of 31 Sy2 galaxies with available X-ray spectra in the redshift range z=0.05-0.3. Our study is complemented by X-ray, mid-IR, and \[OIII\] luminosity ratio diagnostics (Georgantopoulos et al. 2013, Trouille & Barger 2010). This study provides an extension of previous X-ray studies in the local Universe (e.g., Akylas & Georgantopoulos 2009) but also of similar studies at higher redshifts (e.g., JJ13) because of the high S/N X-ray spectra used. We describe our sample selection in §2, the X-ray analysis in \$3, while our results and conclusions are presented in §4 and §5, respectively. Throughout this paper we use $H_0=72$ km/s/Mpc, $\Omega_m=0.27$, and $\Omega_{\Lambda}=0.73$. Sample selection ================ Our sample is composed of Seyfert 2 galaxies with available X-ray spectra within the XMM-Newton Serendipitous Source catalog (Watson et al. 2009, Rosen et al. 2015) and optical spectra within the SDSS-DR10. The names of the sources are taken from the SDSS database. Also, a sequence number is given to each source in the current paper (see Table 1). In the diagrams, interesting sources are followed by their sequence numbers. In the text, the names are followed by the sequence number in parenthesis to make it easier for the reader to trace the sources in the tables and the diagrams. 3 pt ----------- --------------------- ----------- ----------- ----------- ----------- --------------------------- ------------------- -------------- N name obsid ra dec z $\rm N_H (\times10^{22}$) exposure time counts [*(1)*]{} [*(2)*]{} [*(3)*]{} [*(4)*]{} [*(5)*]{} [*(6)*]{} [*(7)*]{} [*(8)*]{} [*(9)*]{} 1 J080429.14+235444.1 504102101 121.1219 23.9127 0.07432 3.18 18300/–/– 86/0/0 2 J080535.00+240950.3 203280201 121.3961 24.1645 0.05971 3.08 5598/8425/– 125/83/0 3 J083139.08+524205.6 92800201 127.9131 52.7016 0.05855 1.16 60280/70920/71790 334/127/166 4 J084002.36+294902.6 504120101 130.0095 29.8175 0.06481 1.83 17870/22630/22640 1072/424/428 5 J085331.05+175339.0 305480301 133.3791 17.8942 0.18659 2.75 34560/–/– 173/0/0 6 J091636.53+301749.3 150620301 139.1524 30.2969 0.12339 1.25 9049/9392/– 431/151/0 7 J100129.41+013633.8 302351001 150.3724 1.6095 0.10423 3.07 31650/42310/42540 310/98/127 8 J101830.79+000504.9 402781401 154.6286 0.0845 0.06233 3.00 15700/20540/20600 753/398/397 9 J103408.58+600152.1 306050701 158.5360 60.0307 0.05101 1.51 8311/–/11420 465/0/133 10 J103456.37+393941.0 506440101 158.7349 39.6614 0.15081 1.96 68400/83070/83850 422/147/120 11 J103515.64+393909.5 506440101 158.8154 39.6527 0.10710 2.03 –/83170/83870 0/112/ 92 12 J104426.70+063753.8 405240901 161.1109 6.6317 0.20991 3.02 24960/–/– 92/0/0 13 J112026.64+431518.4 107860201 170.1109 43.2554 0.14591 1.32 13870/–/– 182/0/0 14 J113549.08+565708.2 504101001 173.9555 56.9522 0.05112 1.07 17490/21310/21320 448/130/127 15 J114826.24+530417.1 204260101 177.1089 53.0717 0.09826 1.23 1701/3632/– 112/94/0 16 J121839.40+470627.6 400560301 184.6649 47.1077 0.09390 1.00 –/37830/37570 0/86/137 17 J123056.11+155212.2 112552101 187.4978 13.5183 0.09816 2.31 8394/–/– 80/0/0 18 J122959.45+133105.7 106061001 187.7338 15.87 0.18768 2.00 4660/–/8979 96/0/98 19 J124214.47+141147.0 504240101 190.5607 14.196 0.15710 2.22 59590/–/80240 1465/0/665 20 J125743.06+273628.2 124710201 194.4296 27.608 0.06839 1.52 30010/–/– 119/0/0 21 J130920.52+212642.7 163560101 197.3359 21.4453 0.27858 1.57 –/28390/28700 0/191/215 22 J131104.66+272807.2 21740201 197.7694 27.469 0.23975 2.06 35000/43000/43100 267/73/75 23 J132525.63+073607.5 200730201 201.3567 7.6022 0.12402 5.02 26900/–/– 174/0/0 24 J134245.85+403913.6 70340701 205.6908 40.6537 0.08926 1.57 26010/35340/35100 393/220/216 25 J135436.29+051524.5 404240101 208.6515 5.2564 0.08152 7.65 11020/–/15780 168/0/101 26 J141602.13+360923.2 14862010 214.0089 36.1567 0.17100 2.56 10910/15780/16140 271/146/145 27 J145720.44–011103.6 502780601 224.3353 –1.1844 0.08735 11.4 7942/–/– 71/0/0 28 J150719.93+002905.0 305750801 226.8330 0.4847 0.18219 10.5 9931/–/– 227/0/0 29 J150754.38+010816.8 402781001 226.9764 1.1381 0.06099 9.81 14330/17900/17890 222/81/81 30 J215649.51–074532.4 654440101 329.2059 –7.7589 0.05541 5.22 42310/73600/75600 134/63/56 31 J224323.18–093105.8 503490201 340.8464 –9.5185 0.14509 2.72 –/113700/114500 0/246/247 ----------- --------------------- ----------- ----------- ----------- ----------- --------------------------- ------------------- -------------- X-ray selection --------------- The XMM-Newton catalog is the largest catalog of X-ray sources ever built. Its current version, 3XMM-DR4 (http://xmmssc-www.star.le.ac.uk/Catalogue/3XMM-DR4/), contains photometric information for half a million source detections, and in addition, spectral and timing data for $\sim$ 120000 of them. The count limit adopted by the 3XMM-DR4 pipeline to derive spectral products is of 100 EPIC net (background subtracted) counts, in order to allow reliable X-ray spectral extraction and analysis. The starting sample was extracted from the XMM-Newton/SDSS-DR7 cross-correlation presented in Georgakakis & Nandra (2011), including more than 40000 X-ray sources. We first selected the sources detected in the X-ray hard band (2-8 keV), a band less affected by obscuration than is the soft one (0.5-2 keV). A total of 1275 sources were found to have available optical spectra within SDSS-DR7. Out of these, 1018 sources had available 3XMM-DR4 spectral data. The corresponding SDSS optical spectra of these 1018 sources were manually examined in order to identify Seyfert 2 galaxies, resulting in our final sample of Sy2s (see next section). It is worth noting that two of these sources have more than one XMM-Newton observation with spectra within 3XMM-DR4, from which we used the longest one. 3 pt -------------- ------------------------------ ----------------------- -------------------------- --------------------------- -------------------- ------------------- ------------- -------------- ------------------- ------------------- ------------------- --------------- N $\rm N_H $ $\Gamma_{soft}$ $\Gamma_{hard}$ EW flux $L_X$ p1/p2 cstat/dof $L_{[OIII]}$ $L_{12}$ $L_{bol}$ log($M_{BH}$) $(\times10^{22}$) $(\times10^{-14})$ $(\times10^{43})$ $(\times10^{42})$ $(\times10^{43})$ $(\times10^{43})$ [*(1)*]{} [*(2)*]{} [*(3)*]{} [*(4)*]{} [*(5)*]{} [*(6)*]{} [*(7)*]{} [*(8)*]{} [*(9)*]{} [*(10)*]{} [*(11)*]{} [*(12)*]{} [*(13)*]{} 1 $18.84_{-5.56}^{+7.68}$ $ 7.6_{-4.6}^{+7.6}$ 1.8$\ddagger$ $ <0.33 $ 22.2 0.27 0.001 105.2/96 0.18 0.4 1.1 6.9 2 $45.83_{-13.94}^{+21.30}$ $ 2.4_{-0.4}^{+0.4}$ 1.8$\ddagger$ $ 0.29_{-0.24}^{+0.71}$ 25.0 0.20 0.015 165.25/210 0.60 0.2 0.3 6.8 3 $23.56_{-3.79}^{+4.00} $ $2.1_{-0.4}^{+0.4}$ $2.1_{-0.4}^{+0.4}$ $ 0.26_{-0.13}^{+0.20}$ 9.3 0.07 0.012 537.63/618 0.17 0.7 1.0 6.3 4 $54.24_{-5.82}^{+5.32} $ $2.8_{-0.1}^{+0.2}$ $ 1.4_{-0.7}^{+0.7}$ $ 0.30_{-0.80}^{+0.80}$ 69.7 0.63 0.024 1088.68/1403 1.11 8.8 19.9 7.6 5 $26.38_{-20.54}^{+27.12} $ $ 2.5_{-0.7}^{+0.7}$ 1.8$\ddagger$ $ <0.17 $ 17.8 1.20 0.293 147.19/189 5.77 11.4 25.9 7.9 6 $<0.06 $ $ <0.74 $ 22.4 0.87 425.68/522 3.47 1.8 2.8 8.2 7 $4.75_{-1.40}^{+1.28} $ $ 1.4_{-0.4}^{+0.4}$ $1.4_{-0.4}^{+0.4}$ $ 0.29_{-0.17}^{+0.20}$ 15.6 0.39 0.018 465.96/529 0.07 1.4 1.9 7.2 8 $2.34_{-0.32}^{+0.38} $ $ <0.10 $ 60.1 0.54 825.83/1078 0.04 0.3 0.4 6.2 9$^\dagger$ $30.32_{-11.62}^{+19.24} $ $ 2.9_{-0.2}^{+0.2} $ 1.8$\ddagger$ $ 1.30_{-0.48}^{+0.80}$ 17.8 0.07 0.293 430.95/474 4.98 7.4 10.3 8.2 10 $64.58_{-23.33}^{+29.99} $ $ 3.0_{-0.2}^{+0.2}$ 1.8$\ddagger$ $ 0.52_{-0.25}^{+0.26}$ 4.1 0.20 0.102 505.46/546 5.37 4.4 6.2 8.2 11 $14.35_{-5.85}^{+6.61}$ $ 6.3_{-1.5}^{+2.1}$ $ 2.0_{-1.1}^{+1.2}$ $ <1.21 $ 9.6 0.25 0.007 189.43/210 0.22 7.4 16.7 7.6 12 $90.26_{-48.60}^{+31.61} $ $ 1.5_{-0.9}^{+0.9}$ $1.5_{-0.9}^{+0.9}$ $ <13.68 $ 11.6 0.92 0.004 82.28/98 1.90 87.9 118.0 8.7 13 $5.49_{-2.51}^{+2.68} $ $ 1.3_{-0.7}^{+0.6}$ $1.3_{-0.7}^{+0.6}$ $ 0.32_{-0.28}^{+0.85}$ 19.1 0.95 0.054 140.57/186 0.23 2.8 6.3 6.0 14 $130.46_{-56.34}^{+69.32}$ $ 2.9_{-0.2}^{+0.2}$ 1.8$\ddagger$ $ <0.65 $ 7.6 0.04 0.006 434.45/462 6.62 22.2 50.8 7.6 15 $1.92_{-0.69}^{+0.88} $ $ <0.48 $ 144 3.37 169.96/204 0.64 0.9 1.2 7.3 16$^\dagger$ $16.07_{-10.34}^{+39.22} $ $ 2.9_{-0.4}^{+0.4}$ 1.8$\ddagger$ $ 0.85_{-0.66}^{+0.75}$ 7.9 0.16 0.173 145.04/155 5.47 9.4 13.1 7.3 17 $4.83_{-2.48}^{+2.80} $ $ 2.3_{-1.1}^{+1.0}$ $2.3_{-1.1}^{+1.0}$ $\star$ 12.6 0.30 0.015 82.11/85 2.06 4.8 6.3 7.4 18 $1.81_{-0.87}^{+1.08} $ $ <0.35 $ 45.2 3.81 183.1/184 0.59 9.3 15.0 6.5 19 $<0.04 $ $ 0.38_{-0.21}^{+0.21}$ 9.9 0.65 748.87/885 1.45 1.4 4.0 8.0 20 $12.62_{-7.20}^{+6.96} $ $1.7_{-1.1}^{+1.0}$ $1.7_{-1.1}^{+1.0}$ $ 0.35_{-0.35}^{+0.51}$ 9.7 0.10 0.052 140.82/167 0.04 0.3 0.4 5.9 21 $<0.08 $ $\star$ 4.7 1.21 214.46/247 0.10 0.5 0.5 6.9 22$^\dagger$ $243.07_{-115.26}^{+303.45}$ $ 2.6_{-0.2}^{+0.2}$ 1.8$\ddagger$ $ 0.65_{-0.60}^{+0.81}$ 4.7 0.37 0.003 296.79/365 3.07 11.7 23.3 7.8 23 $0.39_{-0.22}^{+0.39} $ $ <0.88$ 16.5 0.62 172.56/170 0.13 0.4 1.1 7.4 24 $6.47_{-1.07}^{+1.31} $ $ 2.0_{-0.3}^{+0.4}$ $2.0_{-0.3}^{+0.4}$ $ 0.17_{-0.13}^{+0.17}$ 41.1 0.77 0.009 659.27/707 0.26 2.3 3.4 7.1 25$^\dagger$ $<0.07 $ $<0.70$ 22.7 0.34 274.4/258 0.16 0.9 2.5 6.3 26 $1.98_{-0.47}^{+0.50} $ $ <0.37 $ 46.5 3.47 383.72/507 1.48 42.3 97.8 8.1 27 $5.29_{-3.53}^{+4.69} $ $<0.60$ 13.1 0.23 97.74/80 0.22 1.4 1.9 7.1 28 $28.78_{-12.22}^{+15.82} $ $ 1.7_{-0.7}^{+0.5}$ $1.7_{-0.7}^{+0.5}$ $ <0.46 $ 34.6 2.56 0.045 246.1/310 10.66 2.4 32.3 8.8 29$^\dagger$ $32.18_{-12.68}^{+19.90}$ $ 3.4_{-0.3}^{+0.4}$ 1.8$\ddagger$ $ 1.22_{-0.73}^{+1.89}$ 10.6 0.09 0.155 339.5/354 1.71 1.5 2.1 7.4 30$^\dagger$ $14.95_{-7.28}^{+11.92} $ $3.6_{-0.4}^{+0.5} $ 1.8$\ddagger$ $ 2.08_{-1.24}^{+3.05}$ 3.8 0.03 0.280 266.06/300 0.96 2.7 4.9 7.4 31 $2.61_{-0.68}^{+0.89} $ $ <0.25 $ 10.3 0.51 355.49/417 0.45 0.2 0.4 6.2 -------------- ------------------------------ ----------------------- -------------------------- --------------------------- -------------------- ------------------- ------------- -------------- ------------------- ------------------- ------------------- --------------- Optical selection ----------------- We built the final Sy2 sample based on the emission line properties of their SDSS optical spectra. Initially, we selected only emission line galaxies with redshifts between $z$=0.05 and $z$=0.35. The lower redshift limit excludes all already extensively studied and well-known Seyferts (e.g., Akylas & Georgantopoulos 2009), while the upper limit ensures that the ${\mathrm{H}\alpha}$ and \[NII\] emission lines are within the SDSS spectral range. Furthermore, we excluded all objects where the velocity dispersion of the $\rm H\alpha$ line is greater than 500 km/s, since these objects are certainly broadline AGN. The rest of the objects were placed on a BPT diagram (Baldwin, Phillips, and Terlevich, 1981) and star-forming galaxies, composite galaxies, and LINERS were removed according to the criteria of Kewley et al. (2001) and Schawinski et al. (2007). We used the MPA-JHU emission line fluxes published in DR8 (Brinchmann et al. 2004; Tremonti et al. 2004), although DR10 also contains data from the recent spectroscopic analysis of the Portsmouth Group (Thomas et al. 2013). However, the latter includes only those galaxies from the first two years of observations of the SDSS-III/Baryonic Oscillation Spectroscopic Survey (BOSS) collaboration. We note that a comparison between the two databases by Thomas et al. (2013) has shown that the discrepancy between the calculated emission line fluxes is small. However, the comparison was made after rescaling the Portsmouth values with a factor provided by the “spectofiber” keyword in the MPA-JHU database. This rescaling was originally applied to the MPA-JHU data so that the synthetic r-band magnitude computed from the spectrum matches the r-band fiber magnitude measured by the photometric pipeline. The use of either database does not significantly affect the BPT diagram, since we only need the emission line ratios. We note that in some cases the broadening of the Balmer lines cannot be automatically detected (Seyfert 1.5, 1.8, and especially 1.9), since it only affects the lower part of the lines. As a result, the automated modeling of the line by a single Gaussian may result in lower velocity dispersion values than what is expected from a broad line profile, and the source may be misclassified as a narrow-line AGN. However, since we sought a broad-line-free sample, the spectra of all remaining AGN were eye-inspected with the “interactive spectrum” tool of the SDSS, and all evident intermediate-type Seyferts were removed. After the above filtering, the catalog of Sy2s included 40 objects. Despite the above selection, a number of sources in our sample still have discrepant classifications in the literature; i.e., eight of the sources are listed as Sy1s in Veron-Cetty & Veron (2010, V&V10 hereafter) catalog, plus another one in the NED (NASA extragalactic database). Although none of these objects can actually be a Sy1, we proceeded with our own optical spectrum analysis to determine whether there is any broadening of the permitted emission lines. Optical spectrum analysis ------------------------- The spectra have been retrieved from the SDSS-DR10 and corrected for Galactic extinction using the maps of Schlegel (1998). We use the stellar population synthesis code [^1] to obtain the best fit to an observed spectrum $O_{\lambda}$, taking the corresponding flux error into account. The best fit is a combination of single stellar populations (SSP) from the evolutionary synthesis models of (Bruzual 2003) and a set of power laws to represent the AGN continuum emission. Following the latter approach, several studies have been successful at disentangling the host galaxy and AGN emission components in SDSS spectra (Cid-Fernandes 2011; Tavares 2011). We use a base of 150 SSPs plus six power laws in the form F($\lambda$) = 10$^{20}$($\lambda$ / 4020)$^\beta$, where $\beta$= -0.5, -1, -1.5, -2, -2.5, -3. Each SSP spans six metallicities, Z = 0.005, 0.02, 0.2, 0.4, 1, and 2.5, $Z_{\odot}$, with 25 different ages between 1 Myr and 18 Gyr. Extinction in the galaxy is taken into account in the synthesis, assuming that it arises from a foreground screen with the extinction law of (Cardelli 1989). The code finds the minimum $\chi^{2}$, $$\chi^{2} = \sum_{\lambda} \left( \frac{O_{\lambda}- M_{\lambda}} {\sigma_{obs}}\right) ,$$ where $M_{\lambda}$ is the model spectrum (SSP and power laws), obtaining the corresponding physical parameters of the modeled spectrum: star formation history, $x_{j}$, as a function of a base of SSP models normalized at $\lambda_{0}$, $b_{j,\lambda}$, extinction coefficient of predefined extinction laws, $r_{\lambda}$, and velocity dispersion $\sigma_{\star}$, which obeys the relation $$M_{\lambda}= M_{\lambda 0} \left ( \sum_{j=1}^{N_{SSP}} x_{j}, b_{j, \lambda} r_{\lambda} \right) \otimes G(v_{\star}, \sigma_{\star}) .$$ A detailed description of the code can be found in the publications of the SEAGal collaboration (Cid-Fernandes 2005 ; Cid-Fernandes 2007; Mateus 2006, Asari 2007). In Fig. 1 we present two examples of the spectral decomposition results. After subtracting the stellar background, we use the commercial software [PEAKFIT]{}, by [*Systat Software Inc.*]{}, to model the emission lines. We analyze separately the red ($\rm H\alpha$, N\[II\] and S\[II\] emission lines) and the blue ($\rm H\beta$ and \[OIII\] emission lines) parts of the spectrum. We initially model the emission lines in the blue part, since we are mostly interested in the profile of the \[OIII\]${\lambda5007}$ narrow emission line, with which we also try to fit the lines in the blue part and especially the $\rm H\alpha$. We model the \[OIII\] line with a mixed Gaussian and Lorentzian profile. The contribution of each profile to the fit is a free parameter. If the same profile can also be applied to the red part of the spectrum, we consider this source as a narrow line AGN and keep it in our sample. If there is still a need for an extra broad component to model the $\rm H\alpha$ the source is discarded. In any case, the \[NII\]${\lambda6583}$/\[NII\]${\lambda6548}$ flux ratio should be $\sim3$. We find that seven out of the 40 sources present a broad $\rm H\alpha$ component. Most of these sources belong to the list of ambiguous-type Seyferts that we described in the previous section. Finally, we plot the BPT diagram anew, this time with the line ratios calculated by the above spectral analysis. Although the differences are small, we find that a source that was already close to the AGN-LINER separating line, falls in the LINER region and is therefore excluded. The BPT diagram is plotted in Fig. 2. X-ray spectral fitting ====================== The X-ray data have been obtained with the EPIC (European Photon Imaging Cameras, Strüder et al. 2001; Turner et al. 2001) onboard XMM-Newton. X-ray photons are collected by three detectors (PN, MOS1, and MOS2). All available instrument spectra are modeled simultaneously by using XSPEC, the standard package for X-ray spectral analysis (Arnaud 1996). We used Cash statistics (C-statistics), implemented as cstat in XSPEC to obtain reliable spectral-fitting results even for the lowest quality spectra in our sample. Many of our sources were detected in only one or two of the three detectors (see Table 1). The X-ray spectra of type 2 AGN are usually complicated and consist of multiple components: power-law, thermal, scattering, reflection, and emission lines (see Turner et al. 1997; Risaliti 2002; Ptak et al. 2006; LaMassa et al. 2009). Therefore, no single model could successfully fit the spectra in all cases. We initially tried to model all spectra with a single absorbed power law, but if the fit was not acceptable we added a second power law. Since a strong line is expected in obscured sources, we fit a Gaussian line for the FeK$\alpha$ emission line in both cases. In more detail, this includes: - [Single absorbed power law plus Gaussian FeK$\alpha$ line.]{} We assumed a standard power-law model with two absorption components (wabs\*zwabs\*pow in XSPEC notation) to fit the source continuum emission. The first component models the Galactic absorption. Its fixed values are obtained from Dickey & Lockman (1990) and are listed in Table 1. The second component represents the AGN intrinsic absorption and is left as a free parameter during the modeling procedure. A Gaussian component has also been included to describe the FeK$\alpha$ emission line. We fix the line energy at 6.4 keV in the source rest frame (except in the case of J090036.85+205340.3 (N6) where the line was found at 6.7 keV and implies ionized Fe) and the line width $\sigma$ at 0.01 keV ($\sim$10% of the instrumental line resolution of XMM-Newton). In 12 cases the fitting procedure gives a rejection probability less than 90 per cent and we can accept the model. However, when this simple parametrization is not sufficient to model the whole spectrum, additional components must be included as described in the next paragraph.\ - [Double power law plus Gaussian FeK$\alpha$ line.]{} In the remaining 20 cases, an additional power law was necessary to obtain an acceptable fit (wabs\*(pow+zwabs\*pow), in XSPEC notation). The additional power law is only absorbed by the galactic column density. Initially, the photon indices of the soft (scattered/unabsorbed) and hard (intrinsic/absorbed) power-law components were tied together. However, in 13 cases the value of the hard power-law photon index $\Gamma_{hard}$ was too high (the average photon index of the intrinsic power-law measured in AGN is usually $\sim1.8-2$), and we needed to untie it from the soft one to obtain an acceptable fit. In the cases where the data quality was not high enough to constrain $\Gamma_{hard}$, we fixed it to 1.8 (see Table 2). The X-ray analysis revealed that one of the sources is the brightest galaxy of a contaminating X-ray luminous cluster. We chose to exclude this source from our sample since we cannot provide any reliable X-ray measurements. Our final sample comprises 31 Seyfert 2. In Fig. 3 we present some examples of the X-ray spectra of unobscured ($<10^{22}cm^{-2}$, left panels) and strong FeK$\alpha$-line sources (right panels). Results ======= In the next sections we use various criteria and diagnostic diagrams to investigate the possibility that some objects are more obscured than we can infer from their $\rm N_H$ values and that Compton-thick candidate sources are indeed heavily obscured. Candidate Compton-thick sources ------------------------------- Only two of the sources have $\rm N_H>10^{24}$ cm$^{-2}$, consistent with the high values that define Compton-thick sources. Also, sources (N10) and (N12) are consistent with being CT within the uncertainties. However, except for the column density as a direct indicator of obscuration, there are other criteria, based not only on the X-ray but also on the optical and the infrared emission, that could point to possible Compton-thick sources within our sample. In more detail, a heavily obscured source can have one or more of the following characteristics: 1. Flat X-ray spectrum ($\Gamma<1$). This implies the presence of a strong reflection component that intrinsically flattens the X-ray spectrum at higher energies (e.g., Matt et al. 2000). 2. High equivalent width of the FeK$\alpha$ line ($\sim$1 keV). In this case a Compton-thick nucleus is evident since the line is measured against a heavily obscured continuum (Leahy & Creighton 1993) or only against the reflected component. 3. Low X-ray to mid-infrared ($\rm L_{12}$) luminosity ratio. All Compton-thick sources should have low $\rm L_{2-10 keV}$ to $\rm L_{12}$ ratios, since the mid-IR luminosity of an AGN should be dominated by very hot dust and the X-ray emission should be suppressed by high amounts of absorption (e.g., Lutz et al. 2004; Maiolino et al. 2007). 4. Low X-ray to optical luminosity ratio. The \[OIII\] line emission originates in the narrow line region and is not affected by the circum-nuclear obscuration. Therefore, the ratio between the observed hard X-ray (2-10 keV) and \[OIII\] line luminosity could be used as an indicator of the obscuration of the hard X-ray emission (Mulchaey et al. 1994; Heckman et al. 2005; Panessa et al. 2006; Lamastra et al. 2009; LaMassa et al. 2009; Trouille & Barger2010). ### Flat X-ray spectrum as an indicator of obscuration The first criterion of $\Gamma$ $<1$ is satisfied only by J135436.29+051524.5 (N25). However, this source cannot be included in the Compton-thick candidate sources because there is evidence of partial covering. For more detail see the notes on individual objects in the appendix. ### High equivalent width of the FeK$\alpha$ line as an indicator of obscuration The second criterion of a strong FeK$\alpha$ line is satisfied by four objects (see Table 2). Although the presence of the strong line provides robust evidence of their obscuration, all four exhibit lower $\rm N_H$ values than what is expected by a Compton-thick source. Therefore, we also fit these sources with the model of Brightman & Nandra (2011), which is based on Monte-Carlo simulations. The advantage of this model is that it fits an iron line consistently with the computed $\rm N_H$. Thus, it cannot result in a good fit with a low $\rm N_H$ value and at the same time a high-EW iron line, and vice versa. The fitting confirms that these four sources are indeed Compton-thick. More details can be found in the notes on individual objects in the appendix. Therefore, we do include them in our list of CT sources. Also, we need to examine the X-ray spectra of the unobscured sources carefully for the FeK$\alpha$ line that could give away the presence of obscuration. However, as we can see in Table 2, the line is actually detected only in one out of the five sources, and the equivalent width (EW) is relatively small ($0.38^{+0.21}_{-0.21}$). We do not detect the line in the spectra of any other unobscured source, and the given value of the EW is just the upper limit. Thus, there is no evidence of obscuration based on the the presence of a FeK$\alpha$ line. We note that this criterion is not explicit. High equivalent width lines may also appear in the case of anisotropic distribution of the scattering medium (Ghisellini et al. 1991) or in the case of a time lag between the reprocessed and the direct component (e.g., NGC 2992, Weaver et al. 1996). On the other hand, Compton-thick sources with FeK$\alpha$ EW well below 1 keV have been reported (e.g., Awaki et al. 2000, for Mkn1210). ### The $L_X/L_{12}$ ratio as an indicator of obscuration The detection of a low X-ray to mid-IR luminosity ratio has been widely used as the main instrument for detecting faint Compton-thick AGN, which cannot be easily identified in X-ray wavelengths (e.g., Goulding et al. 2011). This is because the mid-IR luminosity (e.g., 12 $\mu $m or 6 $\mu $m) is a good proxy for the AGN power because it should be dominated by very hot dust that is heated by the AGN (e.g., Lutz et al. 2004; Maiolino et al. 2007). At these wavelengths, the contribution of the stellar light and of colder dust heated by young stars should be small. Gandhi et al. (2009) presented high angular resolution mid-IR (12 $\mu $m) observations of the nuclei of 42 nearby Seyfert galaxies. These observations provide the least contaminated core fluxes of AGN. These authors find a tight correlation between the near-IR fluxes and the intrinsic X-ray luminosity (the Gandhi relation). Spitzer observations do not have the spatial resolution to resolve the core, and the infrared luminosity of an AGN is probably contaminated by the stellar background and the star-forming activity of the galaxy. To obtain an estimate of the purely nuclear 12 $\mu $m infrared luminosity of our sources, we constructed their spectral energy distributions (SED) and computed the various contributions. To model the spectra we used optical data from the SDSS (five optical bands), photometry in the four WISE bands (3.4, 4.6, 12, and 22 $\mu $m) (Wright et al. 2010), and photometry in the three 2MASS bands (J, H, and K) for all sources. Although WISE does include the 12 $\mu $m band, we are only interested in the AGN contribution, so that the construction of the SED and the decomposition of the AGN and host galaxy component is essential. For more details about the code used, the interested reader should refer to Rovilos et al. (2014, Appendix A). In Fig. 4 we present the obscured X-ray luminosities against the 12 $\mu$m luminosities. All our unobscured sources seem to follow the Gandhi-relation closely, and none of them shows unusually high infrared luminosity compared to the X-ray. On the other hand, candidate CT sources are found closer to the dashed line that demarcates the purely CT region. The sources located below this line are all candidate CT according to our analysis. Therefore, it is is unlikely that we are missing any CT candidates among the Sy2 sample. 3 pt ----------- --------------------- ------------------- ------------- ------------------ ---------------------- N name $\rm N_H$ FeK$\alpha$ $\rm L_x/L_{12}$ $\rm L_X/L_{[OIII]}$ [*(1)*]{} [*(2)*]{} [*(3)*]{} [*(4)*]{} [*(5)*]{} [*(6)*]{} 9 J103408.58+600152.1 $>10^{24}\dagger$ x x x 10 J103456.37+393941.0 $>5\times10^{23}$ 12 J104426.70+063753.8 $>9\times10^{23}$ x 14 J113549.08+565708.2 $>10^{24}$ x x 16 J121839.40+470627.6 $>10^{24}\dagger$ x x x 22 J131104.66+272807.2 $>10^{24}$ x 29 J150754.38+010816.8 $>10^{24}\dagger$ x 30 J215649.51–074532.4 $>10^{24}\dagger$ x x x ----------- --------------------- ------------------- ------------- ------------------ ---------------------- : Candidate Compton-thick criteria ### The $\rm N_H$ vs. $L_x/L[OIII]$ ratio as an indicator of obscuration In this section we investigate the possibility that some of the sources are more obscured than we can infer from their column density. In Fig. 5 we plot the column density obtained from the X-ray spectral modeling as a function of the X-ray to optical luminosity ratio. The \[OIII\] luminosities are corrected for reddening using the formula described in Basanni et al. (1999): $\rm L_{[OIII]_{COR}} = L_{[OIII]_{OBS}} [(H\alpha /H\beta)/(H\alpha /H\beta)_o]^{2.94}$, where the intrinsic Balmer decrement $\rm (H\alpha /H\beta )_o$ equals 3. The lower left region in this plot could be possibly occupied by highly obscured or Compton-thick AGN, although their $\rm N_H$ values show the opposite (Akylas & Georgantopoulos 2009). In our case, however, none of the unobscured sources is located in this region, and therefore there is no evidence that their nuclei are heavily obscured. On the other hand, three sources with $\rm N_H>10^{23}$ cm$^{-2}$ are found marginally outside the 3$\sigma$ limit. This implies that they are probably even more obscured than what we calculated by fitting their X-ray spectra. Interestingly, these are the three out of four sources (J103408.58+600152.1 (N9), J121839.40+470627.6 (N16), J215649.51–074532.4 (N30)) for which a high FeK$\alpha$ EW is reported, and they are also found below the CT line in Fig. 4. Therefore, despite the value of the $\rm N_H$, it is evident that the iron line is a robust indicator of obscuration. Once again we can infer that our classification of unobscured and CT sources is valid. Discussion and conclusions ========================== Candidate Compton-thick sources ------------------------------- X-ray spectroscopy shows that the number of Compton-thick AGN in our sample could be as high as eight. N10 was initially included in the CT candidates because it is consistent with being CT within the uncertainties of the calculated column density. However, we chose to exclude this source since it is not confirmed by any of the diagnostics presented in this study (see also LaMassa et al. 2014). Therefore, we are left with seven CT sources, translating to a percentage of $\sim$23%. We find that the number of CT AGN found in our survey agrees with those in other X-ray surveys of optically selected Seyfert galaxies. In more detail, Akylas & Georgantopoulos (2009), using XMM-Newton observations, estimate the number of CT sources among the Seyfert galaxies from the Palomar spectroscopic sample of nearby galaxies (Ho, Filippenko & Sargent 1995). They find a percentage of CT sources of 15-20 %. Since their sample consists of nearby ($<$120 Mpc) Sy2 galaxies, the X-ray observations provide excellent spectra, hence accurate column density measurements classifications of all the AGN in their sample. Also, Malizia et al. (2009) reports that $\sim18\%$ of their hard X-ray selected Sy2 sample is Compton-thick. Nevertheless, considering only the low-redshift sources ($z<0.015$) to remove the selection bias that affects their sample against the detection of CT objects, the percentage becomes $\sim35\%$. They argue that this result is in excellent agreement with the percentage of CT AGN in the optically selected sample of Risaliti, Maiolino & Salvati (1999). We note that because of our sample selection, which requires a sufficient number of photons in order to derive X-ray spectra, we may also be biased against heavily obscured sources. On the other hand, JJ13 in their SDSS optically selected sample of type 2 QSOs, estimate a higher percentage of CT sources that could be as high as 50%, albeit with limited photon statistics. Initially, the percentage they calculate based on the X-ray spectral modeling and the intensity of the FeK$\alpha$ line is significantly lower. However, it reaches 50% after they conclude that at least half of the $\rm N_H$ values of their sources are underestimated, based on their $L_{2-10 keV}/L_{[OIII]}$ ratios. Nevertheless, four out of the seven CT sources in our study are in common with JJ13. Three of them are also reported as CT in JJ13. N12 is not a CT source in JJ13 despite its high $\rm N_H$ and the detection of the FeK$\alpha$ line in their work. A probable reason is that they report an X-ray luminosity that is one order of magnitude higher than the one we measure in the current study. Therefore the $L_{2-10 keV}/L_{[OIII]}$ ratio is higher than their threshold for a CT source. We note that four sources in our sample were initially considered heavily obscured because of the high FeK$\alpha$ EW ($>$1 keV), although their column density was only a few times $10^{23}$ $\rm cm^{-2}$. This suggests that these sources may be attenuated by CT absorbers. Indeed, all CT sources in the local Universe appear to present high EW of the FeK$\alpha$ line (e.g., Fukazawa et al. 2011) owing to suppression of their continuum emission. The discrepancy between the estimated column density and the EW could be attributed to a more complex spectral model that involves a double screen absorber with one of them being CT. In all four cases, by fitting the X-ray spectra with the model of Brightman & Nandra (2011), we confirm that they are indeed heavily obscured ($\rm N_H>10^{24} cm^{-2}$). In addition, according to Table 3, most of them satisfy all our CT criteria. Interestingly, three out of the above four high-EW sources, J103408.58+600152.1 (N9), J215649.51–074532.4 (N30), and J121839.40+470627.6 (N16) lie in the CT regime in the $\rm L_x/L_{12}$ diagram, and (Fig. 4) the same three sources have the lowest $\rm L_X/L_{[OIII]}$ ratios (Fig. 5), again supporting their CT nature. Two of these sources are in common with JJ13 (N9 and N16), and present a high EW in both studies. Unabsorbed Sy2 nuclei --------------------- The X-ray spectral analysis revealed that four[^2] Sy2 galaxies ($\sim$13%) present very low absorption, below $10^{22}$ cm$^{-2}$, in sharp contrast with the unification model of AGN. The percentage of unobscured Sy2 sources varies in the literature, from a few percent ($\sim 3-4\%$) in Risality, Maiolino & Salvati (1999) and in Malizia et al. (2009), to 40% in Page et al. (2006) and 66% in Garcet et al. (2007). Our value is in better agreement with Panessa & Bassani (2002) and Akylas & Georgantopoulos (2009). However, considering that the number of unobscured Sy2s discovered in any of these studies is less than eight, we argue that we roughly agree with all of them, except perhaps with Garcet et al. (2007). Also, we note that our criteria for selecting narrow line AGN are more stringent than in most of the above studies; for example, Risality, Maiolino & Salvati (1999) include Sy1.9 in their sample, and Garcet et al (2007) allow narrow line AGN up to FWHM$_{{\mathrm{H}\alpha}}$=1500 km s$^{-1}$. As we have already discussed, none of our unobscured sources present a low X-ray to \[OIII\] or $\rm L_{12}$ luminosity ratio. They also do not present a strong FeK$\alpha$ line, and therefore we cannot associate them with a highly obscured Compton-thick nucleus. In addition, the FWHM of their ${\mathrm{H}\alpha}$ line is less than 500 km s$^{-1}$, which excludes the possibility of a narrow-line Sy1 classification. Although Tran (2001) argues about the presence of this kind of type 2 AGN in his sample of non-HBLR Sy2s fifteen years ago, their existence is still being strongly debated (see discussion in Antonucci 2012). Below, we summarize important observational and theoretical studies in the field, which attempt to approach this problem from various angles. There is strong evidence that the dusty obscuring torus in low luminosity AGN is absent or is thinner than expected in higher luminosities (e.g., Elitzur & Shlosman 2006; Perlman et al. 2007; van der Wolk et al. 2010). Accordingly, all low luminosity AGN should have been Type 1 sources, which of course is not the case. The only reasonable explanation of this problem is the additional absence of the BLR in such systems. Some authors (e.g., Nicastro 2000; Nicastro, Martocchia & Matt 2003; Bian & Gu. 2007; Marinucci et al. 2012; Elitzur, Ho & Trump 2014) presented arguments that below a specific accretion rate of material into the black hole, and therefore at lower luminosities, the BLR might also be absent. Using data from nearby bright AGN, Elitzur & Ho (2009) conclude that the BLR disappears at bolometric luminosities that are lower than $5 \times 10^{39} (M_{BH}/10^7 M_{\sun})^{2/3} \rm erg\; s^{-1}$, where $M_{BH}$ is the mass of the black hole. They also argue that the quenching of the BLR and the disappearance of the torus can occur either simultaneously or in sequence, with decreasing black hole accretion rate and luminosity. Thus, a possible scenario would be that non-HBLR Sy2 AGN are objects lacking the BLR and possibly the torus. Nicastro, Martocchia & Matt (2003) conclude that the BLR probably does not exist below an accretion rate threshold of $\log(L_{bol}/L_{Edd})=-3$, while Marinucci et al. (2012) argue that true Sy2s can be found below the relatively higher limits of bolometric luminosity $\log L_{bol}=43.9$ and Eddington ratio $\log(L_{bol}/L_{Edd})=-1.9$. Marinucci et al. (2012) derived the bolometric luminosity from the X-ray and the \[OIV\] luminosity and conclude that $L_{[\rm OIII]}$ is not as reliable (see also relevant discussion in Elitzur 2012). We note that Elitzur & Ho (2009) thresholds are relatively low, not only compared to other studies but also for the general Sy2 population (see discussion in the recent review by Netzer 2015). However, the idea that the accretion rate is essential in the formation of the BLR seems to be valid, although the exact limits have not yet been defined and probably also depend on other factors (see discussion in Koulouridis 2014). To evaluate the above limits for our four unobscured sources, we computed their bolometric luminosities from the SED modeling (see §4.2.4). We also calculated their black hole masses using the $M_{BH}-\sigma*$ relation (Tremaine et al. 2002), where $\sigma*$ is the stellar velocity dispersion, calculated from the FWHM of the \[OIII\] emission lines (Greene & Ho 2005). We find that the Elitzur & Ho (2009) limits are very low for our unobscured sources. Nevertheless, all satisfy the bolometric luminosity and Eddington ratio limit of Marrinucci et al. (2012). We note, however, that our Eddington ratios may be overestimated since the Eddington luminosities, derived from the FWHM of the \[OIII\] lines, are probably underestimated (e.g., Bian & Gu 2007). By conducting a two-sample Student’s t-test between the accretion rates of the unobscured and the obscured sources, we conclude that their mean values are significantly different at the 99.9% confidence level. In Fig. 6 we plot the Eddington ratio versus the bolometric luminosity of our Sy2s, but also the discarded intermediate type Seyferts (crosses). We also plot the lines that apparently separate the unobscured sources from the rest of the Sy2 population. These limits are similar to the respective ones found by Marinucci et al. (dashed lines in Fig.6) for HBLR and non-HBLR sources. All four unobscured sources fall into the area where non-HBLR Sy2s are found and the BLR is predicted to not exist. We note that the limits of previous works were based on the differences between HBLR and non-HBLR Sy2s, while our sample is divided into obscured and unobscured sources. The unobscured Sy2s are non-HBLR Sy2s by definition, whereas the obscured sources are not necessarily HBLR Sy2s. Therefore, the presence of obscured Sy2s in the bottom left quarter of the plot may imply the lack of their BLR as well. Interestingly, a number of Compton-thick sources exhibit low accretion rates. This agrees with the evolutionary scheme of AGN proposed by Koulouridis (2014), where a fraction of Compton-thick sources are predicted to emerge shortly after a galaxy interaction or merging event that causes the inflow of gas and dust toward the central region of the galaxy, enhances circumnuclear star formation and triggers the AGN. During this phase the accretion rate is expected to be low and the BLR absent. However, the failure to detect the BLR in CT sources may as well be due to the heavy obscuration and the large covering factor of the nucleus (see next paragraph). We note that the uncertainties that enter the above calculations are large (see Greene & Ho 2005) and our samples fairly small. However, the general tendency of low accretion type-2 AGN to lack any evidence of a BLR is once more evident. An alternative scenario that can explain the lack of detectable BLR in many CT sources is that heavy obscuration does not allow the detection of the BLR even in the polarized spectrum. Marinucci et al. (2012) conclude that 64% of their compton-thick non-HBLR Sy2s exhibit higher accretion rates than the threshold clearly separating the two Sy2 classes. They attributed this discrepancy to heavy absorption along our line of sight, preventing the detection of the actual BLR in their nuclei. Evidently, merging systems constitute a class of extragalactic objects where heavy obscuration occurs (e.g., Hopkins et al. 2008). The merging process may also lead to rapid black hole growth, giving birth to a heavily absorbed and possibly Compton-thick AGN. Thus, we could presume that a number of our non-HBLR mergers, if not all of them, might actually be BLR AGN galaxies, where the high concentration of gas and dust prohibits even the indirect detection of the broad line emission (e.g., Shu et al. 2007). However, other studies have concluded that there is no evidence that non-HBLR Sy2s are more obscured than their HBLR peers (Tran 2003; Yu 2005; Wu 2011), while totally unobscured low-luminosity non-HBLR Sy2s were detected via investigation of their X-ray properties (e.g., Panessa & Bassani 2002; Akylas & Georgantopoulos 2009). The total population of non-HBLR Sy2s is probably a mixture of objects with low accretion rate and/or high obscuration. Koulouridis (2014) argue that both of the above scenarios agree with an AGN evolutionary scheme (Krongold et al. 2002; Koulouridis et al. 2006a, b, 2013), where a low accretion rate is predicted at the beginning and the end of the Seyfert duty cycle, without ruling out the possibility that some HBLR Sy2s could also be created by minor disturbances or even secular processes. Finally, we note that there is always the possibility that the discrepancy between the optical and the X-ray spectra is due to variability, since they were not obtained simultaneously.\ In a nutshell: 1. We found four unobscured sources ($\sim$13%) at odds with the simplest unification scheme. These sources exhibit low accretion rates that agree with previous studies that predict the lack of the BLR in low-accretion-rate AGN. 2. 64% of the Sy2s are obscured with a median column density value of $\rm N_H\sim1.0\times10^{23}cm^{-2}$. 3. The percentage of CT AGN is at $\sim$23%, although direct comparison with previous studies is difficult because of the different selection methodologies. Their heavy obscuration was confirmed using a variety of criteria and diagnostics. We thank the anonymous referee for the insightful comments and suggestions that significantly contributed to improving the quality of the publication. EK acknowledges fellowship funding provided by the Greek General Secretariat of Research and Technology in the framework of the program Support of Postdoctoral Researchers, PE-1145. This work is based on observations obtained with XMM-Newton, an ESA science mission with instruments and contributions directly funded by ESA Member States and the USA (NASA). Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and the U.S. Department of Energy Office of Science. The SDSS-III web site is http://www.sdss3.org/. 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The result of the fit is a high column density, $\rm N_H=220^{+\infty}_{-70}$, characteristic of the CT sources. Other useful values: p1/p2=0.008, cstat/dof=51.9/34, $\Gamma_{soft}=3^{+0.2}_{-0.4}$, $\Gamma_{hard}=1.8$ (fixed).\ - Source 16 - J121839.40+470627.6 Because of the large EW of the FeK$\alpha$ line, but the relatively low $\rm N_H$, we fit the spectrum with the model of Brightman & Nandra (2011). The result of the fit is a high column density, $\rm N_H=2009^{+\infty}_{-135}$, characteristic of the CT sources. Other useful values: p1/p2=0.003, cstat/dof=145/155, $\Gamma_{soft}=3.4^{+0.8}_{-0.6}$, $\Gamma_{hard}=1.8$ (fixed).\ - Source 22 - J131104.66+272807.2 Because of the high $\rm N_H$, but the small EW of the FeK$\alpha$ line, we fit the spectrum with the model of Brightman & Nandra (2011). The result of the fit is a column density value of $\rm N_H=114^{+87}_{-29}\times10^{22}$, which is relatively lower than what is reported in the current study, but again above the limit that characterize CT sources. A strong FeK$\alpha$ line is only present in the pn detector. Other useful values: p1/p2=0.003, cstat/dof=300/366, $\Gamma_{soft}=2.6^{+0.2}_{-0.2}$, $\Gamma_{hard}=1.8$ (fixed).\ - Source 25 - J135436.29+051524.5We chose not to include this source in the unabsorbed list because its photon index $\Gamma$ is extremely flat ($\sim$0.8) if left as a free parameter, and in addition there seems to be a strong FeK$\alpha$ line. It may be a reflection -dominated Compton-thick source, but we cannot confirm this because of the relatively low quality X-ray spectrum. Also, even though the EW seems high, it cannot be considered as a Compton-thick candidate because the scattered percentage is too large ($>30$%) implying partial covering instead of scattered emission.\ - Source 29 - J150754.38+010816.8 Because of the large EW of the FeK$\alpha$ line, but the relatively low $\rm N_H$, we fit the spectrum with the model of Brightman & Nandra (2011). The result of the fit is a high column density, $\rm N_H=211^{+\infty}_{-61}$, characteristic of the CT sources. Other useful values: p1/p2=0.003, cstat/dof=320/355, $\Gamma_{soft}=3.2^{+0.4}_{-0.4}$, $\Gamma_{hard}=1.8$ (fixed).\ - Source 30 - J215649.51–074532.4 Because of the large EW of the FeK$\alpha$ line, but the relatively low $\rm N_H$, we fit the spectrum with the model of Brightman & Nandra (2011). The result of the fit is a high column density, $\rm N_H=1500^{+\infty}_{-1200}$, which is characteristic of the CT sources. Other useful values: p1/p2=0.003, cstat/dof=265/3, $\Gamma_{soft}=3.5^{+0.5}_{-0.5}$, $\Gamma_{hard}=1.8$ (fixed). [^1]: http://www.starlight.ufsc.br/ [^2]: We note that (N25) is a flat X-ray spectrum source with a visible FeK$\alpha$ line, and we have excluded it from the list of unobscured sources (see appendix for more details)