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--- abstract: 'Lanczos-type algorithms are well known for their inherent instability. They typically breakdown when relevant orthogonal polynomials do not exist. Current approaches to avoiding breakdown rely on jumping over the non-existent polynomials to resume computation. This jumping strategy may have to be used many times during the solution process. We suggest an alternative to jumping which consists in switching between different algorithms that have been generated using different recurrence relations between orthogonal polynomials. This approach can be implemented as three different strategies: ST1, ST2, and ST3. We shall briefly recall how Lanczos-type algorithms are derived. Four of the most prominent such algorithms namely $A_4$, $A_{12}$, $A_5/B_{10}$ and $A_5/B_8$ will be presented and then deployed in the switching framework. In this paper, only strategy ST2 will be investigated. Numerical results will be presented.' author: - Muhammad Farooq and Abdellah Salhi bibliography: - 'lanczos1.bib' title: 'A Switching Approach to Avoid Breakdown in Lanczos-type Algorithms' --- 2010 Mathematics Subject Classification: 65F10\ [**Keywords**]{}: Lanczos algorithm; Systems of Linear Equations (SLE’s); Formal Orthogonal Polynomials (FOP’s); Switching; Restarting; Breakdown. Introduction ============ Lanczos-type methods for solving SLE’s are based on the theory of FOP’s. All such methods are implemented via some recurrence relationships between polynomials $P_{k}(x)$ represented by $A_{i}$ or between two adjacent families of orthogonal polynomials $P_{k}(x)$ and $P^{(1)}_{k}(x)$ represented by $A_{i}$ and $B_{j}$ as described in [@94:Baheux; @95:Baheux; @10:Farooq]. The coefficients of the various recurrence relationships between orthogonal polynomials are given as ratios of scalar products. When a scalar product in a denominator vanishes, then a breakdown occurs in the algorithm and the process normally has to be stopped. Equivalently, the breakdown is due to the non-existence of some orthogonal polynomial or polynomials. So, an important issue is how to continue the solution process in such a situation and arrive at a useable result. Several procedures for that purpose appeared in the literature in the last few decades. It has been shown, for instance, that it is possible to jump over non-existing polynomials, [@94:Baheux; @93:Brezinski]; breakdown-free algorithms were thus obtained. The first attempt in this regard was the look-ahead Lanczos algorithm, [@85:Parlett]. Other procedures for avoiding breakdown are also proposed in [@92:Gut; @91:Brezinski; @92:Sadok; @94:Baheux; @93:Brezinski; @99:Brezinski; @97:Graves; @93:Gut; @79:Parlett]. However, they all have their limitations including the possibility of calling the procedure for remedying the breakdown, more than once. In the following, we suggest an alternative to jumping over missing polynomials by switching between different variants of the Lanczos algorithm. The Lanczos approach ==================== We consider a linear system of equations, $$\textit{A}\textbf{x}=\textbf{b},$$ where $\textit{A}\in \textit{R}^{n\times n}$, $\textbf{b}\in \textit{R}^{n}$ and $\textbf{x}\in \textit{R}^{n}$. Let $\textbf{x}_{0}$ and $\textbf{y}$ be two arbitrary vectors in $\textit{R}^{n}$ such that $\textbf{y}\neq0$. The Lanczos method, [@52:Lanczos] consists in constructing a sequence of vectors $\textbf{x}_{k}\in \textit{R}^n$ defined as follows, [@02:Brezinski; @95:Baheux] $$\label{E1}\textbf{x}_{k}-\textbf{x}_{0}\in \textit{K}_{k}(\textit{A}, \textbf{r}_{0}) =span(\textbf{r}_{0}, \textit{A}\textbf{r}_{0},\dots,\textit{A}^{k-1}\textbf{r}_{0}),$$ $$\label{E2}\textbf{r}_{k}=(\textbf{b}-\textit{A}\textbf{x}_{k})\bot\textit{K}_{k}(\textit{A}^{T}, \textbf{y}) =span(\textbf{y}, \textit{A}^T\textbf{y},\dots,\textit{A}^{T^{k-1}}\textbf{y}),$$ where $\textit{A}^T$ denotes the transpose of $\textit{A}$. Equation $(\ref{E1})$ leads to, $$\textbf{x}_{k}-\textbf{x}_{0}=-\alpha_{1}\textbf{r}_{0}-\alpha_{2}\textit{A}\textbf{r}_{0} - \dots -\alpha_{k}\textit{A}^{k-1}\textbf{r}_{0}.$$ Multiplying both sides by $\textit{A}$ and adding and subtracting $\textbf{b}$ on the left hand side gives $$\label{E5} \textbf{r}_{k}=\textbf{r}_{0}+\alpha_{1}\textit{A}\textbf{r}_{0}+\alpha_{2}\textit{A}^2\textbf{r}_{0}+ \dots +\alpha_{k}\textit{A}^{k}\textbf{r}_{0}.$$ From $(\ref{E2})$, the orthogonality condition gives $(\textit{A}^{T^{i}}\textbf{y},\textbf{r}_{k})=0$, for $i=0,\dots,k-1,$ and, by $(\ref{E5})$, we obtain the following system of linear equations $$\begin{aligned} \label{E6}\begin{cases} \alpha_{1}(\textbf{y}, \textit{A}\textbf{r}_{0})+ \dots + \alpha_{k}(\textbf{y}, \textit{A}^k\textbf{r}_{0})= -(\textbf{y}, \textbf{r}_{0}),\\ \vdots\\ \alpha_{1}(\textit{A}^{T^{k-1}}\textbf{y}, \textit{A}\textbf{r}_{0})+ \dots + \alpha_{k}(\textit{A}^{T^{k-1}}\textbf{y}, \textit{A}^k\textbf{r}_{0})= -(\textit{A}^{T^{k-1}}\textbf{y}, \textbf{r}_{0}). \end{cases}\end{aligned}$$ If the determinant of the above system is different from zero then its solution exists and allows to obtain $\textbf{x}_{k}$ and $\textbf{r}_{k}$. Obviously, in practice, solving the above system directly for the increasing value of $k$ is not feasible. We shall now see how to solve this system for increasing values of $k$ recursively. If we set $$\label{E7}P_{k}(x)=1+\alpha_{1}x+\dots+\alpha_{k}x^k,$$ then we can write from $(\ref{E5})$ $$\label{E8} \textbf{r}_{k}=P_{k}(\textit{A})\textbf{r}_{0}.$$ The polynomials $P_{k}$ are commonly known as the residual polynomials, [@93:Brezinski]. Another interpretation of the $P_{k}$ can be found in [@87:Cybenko]. Moreover if we set $c_{i}=(\textit{A}^{T^i}\textbf{y}, \textbf{r}_{0})=(\textbf{y}, \textit{A}^i\textbf{r}_{0})$, $i=0,1,\dots$, and if we define the linear functional $c$ on the space of polynomials by $$c(x^i)=c_{i}, \mbox{ } i=0,1,\dots,$$ $c$ is completely determined by the sequence $\{c_k\}$ and $c_k$ is said to be the moment of order $k$, [@80:Brezinski]. Now, the system (\[E6\]) can be written as $$c(x^iP_{k}(x))=0 \mbox{ for } i=0,\dots,k-1.$$ These conditions show that $P_{k}$ is the polynomial of degree at most $k$, normalized by the condition $P_{k}(0)=1$, belonging to a family of FOP’s with respect to the linear functional $c$, [@92:Brezinski; @80:Brezinski]. Since the constant term of $P_{k}$ in (\[E7\]) is $1$, it can be written as $$P_{k}(x)=1+xR_{k-1}(x)$$ where $R_{k-1}= \alpha_1+ \alpha_2x+...+\alpha_kx^{k-1}$. Replacing $x$ by $\textit{A}$ in the expression of $P_k$ and multiplying both sides by $\textbf{r}_{0}$ and using $(\ref{E8})$, we get $$\textbf{r}_{k}=\textbf{r}_{0}+\textit{A}R_{k-1}(A)\textbf{r}_{0},$$ which can be written as $$b-\textit{A}\textbf{x}_{k}=b-\textit{A}\textbf{x}_{0}+\textit{A}R_{k-1}(A)\textbf{r}_{0},$$ $$-\textit{A}\textbf{x}_{k}=-\textit{A}\textbf{x}_{0}+\textit{A}R_{k-1}(A)\textbf{r}_{0},$$ multiplying both sides by $-\textit{A}^{-1}$, we get $$\textbf{x}_{k}=\textbf{x}_{0}-R_{k-1}(A)\textbf{r}_{0},$$ which shows that $\textbf{x}_{k}$ can be computed from $\textbf{r}_{k}$ without inverting $\textit{A}$. Formal orthogonal polynomials ============================= The orthogonal polynomials $P_{k}$ defined in the previous section are given by the determinantal formula, [@97:Brezinski; @93:Brezinski] $$P_{k}(x)=\frac{\left\vert\begin{array}{cccc} 1 & \cdots & x^k\\ c_{0} & \cdots & c_{k}\\ \vdots & & \vdots\\ c_{k-1} & \cdots & c_{2k-1}\\ \end{array}\right\vert}{\left\vert\begin{array}{cccc} c_{1} & \cdots & c_{k}\\ \vdots & & \vdots\\ c_{k} & \cdots & c_{2k-1}\\ \end{array}\right\vert},$$ where the denominator of this polynomial is $\textit{H}^{(1)}_{k}$, [@93:Brezinski]. Obviously, $P_{k}$ exists if and only if the Hankel determinant $\textit{H}^{(1)}_{k}\neq0.$ Thus, $P_{k+1}$ exists if and only if $\textit{H}^{(1)}_{k+1}\neq0$. We assume that $\forall k$, $\textit{H}^{(1)}_{k}\neq0$. If for some $k$, $\textit{H}^{(1)}_{k}=0$, then $P_{k}$ does not exist and breakdown occurs in the algorithm (in practice the breakdown can occur even if $\textit{H}^{(1)}_{k}\approx0$). Let us now define a linear functional $c^{(1)},$ [@93:Brezinski; @95:Baheux], on the space of real polynomials as $c^{(1)}(x^{i})=c(x^{i+1})=c_{i+1}$ and let $P^{(1)}_{k}$ be a family of orthogonal polynomials with respect to $c^{(1)}$. These polynomials are called monic polynomials, [@93:Brezinski; @95:Baheux], because their highest degree coefficients are always 1, and are given by the following formula $$P^{(1)}_{k}(x)=\frac{\left\vert\begin{array}{cccc} c_{1} & \cdots & c_{k+1}\\ \vdots & & \vdots\\ c_{k} & \cdots & c_{2k}\\ 1 & \cdots & x^k\\ \end{array}\right\vert}{\left\vert\begin{array}{cccc} c_{1} & \cdots & c_{k}\\ \vdots & & \vdots\\ c_{k} & \cdots & c_{2k-1}\\ \end{array}\right\vert}.$$ $P^{(1)}_{k}(x)$ also exists if and only if the Hankel determinant $H^{(1)}_{k}\neq0$, [@93:Brezinski; @95:Baheux], which is also a condition for the existence of $P_{k}(x)$. There exist many recurrence relations between the two adjacent families of polynomials $P_{k}$ and $P^{(1)}_{k}$, [@95:Baheux; @93:Brezinski; @97:Brezinski; @91:Brezinski]. Some of these relations have been reviewed in [@94:Zaglia] and studied in details in [@94:Baheux; @97:Brezinski]. More of these relations have been studied in [@10:Farooq], leading to new Lanczos-type algorithms. A Lanczos-type algorithm consists in computing $P_{k}$ recursively, then $\textbf{r}_{k}$ and finally $\textbf{x}_{k}$ such that $\textbf{r}_{k}=\textbf{b}-\textit{A}\textbf{x}_{k}$, without inverting $A$. In exact arithmetic, this should give the solution to the system $\textit{A}\textbf{x}=\textbf{b}$ in at most $n$ steps [@52:Lanczos; @91:Brezinski], where $n$ is the dimension of the system. For more details, see [@93:Brezinski; @99:Brezinski]. Recalling some existing algorithms ================================== In the following we will recall some of the most recent and efficient Lanczos-type algorithms to be used in the switching framework. The reader should consult the relevant literature for more details. Algorithm $A_{12}$ ------------------ Algorithm $A_{12}$ is based on relation $A_{12}$, [@10:Farooq]. For details on the derivation of the polynomial $A_{12}$, its coefficients and the algorithm itself, please refer to [@10:Farooq]. The pseudo-code of Algorithm $A_{12}$ can be described as follows. Choose $x_{0}$ and $y$ such that $y\neq0$, Choose $\epsilon$ small and positive, as a tolerance, Set $r_{0}=b-Ax_{0}$, $y_{0}=y$, $p=Ar_{0}$, $p_{1}=Ap$, $c_{0}=(y, r_{0})$, $c_{1}=(y, p)$, $c_{2}=(y, p_{1})$, $c_{3}=(y, Ap_{1})$, $\delta=c_{1}c_{3}-c_{2}^2$, $\alpha=\frac{c_{0}c_{3}-c_{1}c_{2}}{\delta}$, $\beta=\frac{c_{0}c_{2}-c_{1}^2}{\delta}$, $r_{1}=r_{0}-\frac{c_{0}}{c_{1}}p$, $x_{1}=x_{0}+\frac{c_{0}}{c_{1}}r_{0}$, $r_{2}=r_{0}-\alpha p+\beta p_{1}$, $x_{2}=x_{0}+\alpha r_{0}-\beta p$, $y_{1}=A^{T}y_{0}$, $y_{2}=A^{T}y_{1}$, $y_{3}=A^{T}y_{2}$. \[k=3\] $y_{k+1}=A^{T}y_{k}$, $q_{1}=Ar_{k-1}$, $q_{2}=Aq_{1}$, $q_{3}=Ar_{k-2}$, $a_{11}=(y_{k-2}, r_{k-2})$, $a_{13}=(y_{k-3}, r_{k-3})$, $a_{21}=(y_{k-1}, r_{k-2})$, $a_{22}=a_{11}$, $a_{23}=(y_{k-2}, r_{k-3})$, $a_{31}=(y_{k}, r_{k-2})$,$a_{32}=a_{21}$, $a_{33}=(y_{k-1}, r_{k-3})$, $s=(y_{k+1}, r_{k-2})$, $t=(y_{k}, r_{k-3})$,$F_{k}=-\frac{a_{11}}{a_{13}}$, $b_{1}=-a_{21}-a_{23}F_{k}$, $b_{2}=-a_{31}-a_{33}F_{k}$, $b_{3}=-s-tF_{k}$, $\Delta_{k}=a_{11}(a_{22}a_{33}-a_{32}a_{23})+a_{13}(a_{21}a_{32}-a_{31}a_{22})$, $B_{k}=\frac{b_{1}(a_{22}a_{33}-a_{32}a_{23})+a_{13}(b_{2}a_{32}-b_{3}a_{22})}{\Delta_{k}}$, $G_{k}=\frac{b_{1}-a_{11}B_{k}}{a_{13}}$, $C_{k}=\frac{b_{2}-a_{21}B_{k}-a_{23}G_{k}}{a_{22}}$, $A_{k}=\frac{1}{C_{k}+G_{k}}$, $r_{k}=A_{k}\{q_{2}+B_{k}q_{1}+C_{k}r_{k-2}+F_{k}q_{3}+G_{k}r_{k-3}\}$, $x_{k}=A_{k}\{C_{k}x_{k-2}+G_{k}x_{k-3}-(q_{1}+B_{k}r_{k-2}+F_{k}r_{k-3})\}$, $x = x_{k}$, Stop. Algorithm $A_{4}$ ----------------- Algorithm $A_{4}$ is based on relation $A_{4}$. Its pseudo-code is as follows. For more details see [@94:Baheux; @95:Baheux]. Choose $x_{0}$ and $y$ such that $y\neq0$, Choose $\epsilon$ small and positive as a tolerance, Set $r_{0}=b-Ax_{0}$, $y_{0} = y$, \[k=0\] $E_{k+1}=-\frac{(y_{k}, r_{k})}{(y_{k-1}, r_{k-1})}$, for $k\geq1$, and $E_{1}=0$, $B_{k+1}=-\frac{(y_{k}, Ar_{k})-E_{k+1}(y_{k}, r_{k-1})}{(y_{k}, r_{k})}$, $A_{k+1}=\frac{1}{B_{k+1}+E_{k+1}}$, $x_{k+1}=A_{k+1}\{B_{k+1}x_{k}+E_{k+1}x_{k-1}-r_{k}\}$, $r_{k+1}=A_{k+1}\{Ar_{k}+B_{k+1}r_{k}+E_{k+1}r_{k-1}\}$. $y_{k+1}=A^Ty_{k}$, Algorithm $A_{5}/B_{10}$ ------------------------ Algorithm $A_{5}/B_{10}$ is based on relations $A_{5}$ and $B_{10}$, first investigated in [@94:Baheux; @95:Baheux]. Its pseudo-code is as follows. Choose $x_{0}$, $y$ and tolerance $\epsilon \geq 0 $; Set $r_{0}=b-Ax_{0}$, $p_{0} = r_{0}$, $y_{0} = y$, $A_{1} = -\frac{(y_{0}, r_{0})}{(y_{0}, Ar_{0})}$, $C^1_{0}= 1$, $r_{1} = r_{0}+A_{1}Ar_{0}$, $x_{1} = x_{0}-A_{1}r_{0}$. $y_{k} = A^Ty_{k-1}$, $D_{k+1} = -\frac{(y_{k}, r_{k})}{C^1_{k-1}(y_{k}, p_{k-1})}$, $p_{k}=r_{k}+D_{k-1}C^1_{k-1}p_{k-1}$ $A_{k+1} = -\frac{(y_{k}, r_{k})}{(y_{k}, Ap_{k})}$, $r_{k+1} = r_{k}+A_{k+1}Ap_{k}$, $x_{k+1} = x_{k}-A_{k+1}p_{k}$. $C^1_{k} = \frac{C^1_{k-1}}{A_{k}}$. Algorithm $A_{8}/B_{10}$ ------------------------ The pseudo-code of $A_{8}/B_{10}$, [@94:Baheux; @95:Baheux], is as follows. Choose $x_{0}$ and $y$ such that $y\neq0$.\ Set $r_0 = b - Ax_0$,\ $z_{0}=r_{0}$,\ $y_{0}=y$,\ $A_{k+1}=-\frac{(\textbf{y}_{k}, \textbf{r}_{k})}{(\textbf{y}_{k}, \textit{A}\textbf{z}_{k})}$,\ $r_{k+1}=r_{k}+A_{k+1}Az_{k}$,\ $x_{k+1}=x_{k}-A_{k+1}z_{k}$.\ $y_{k+1}=A^{T}y_{k}$,\ $C^1_{k+1}=\frac{1}{A_{k+1}}$,\ $B^{1}_{k+1}=-\frac{C^{1}_{k+1}(y_{k+1}, r_{k+1})}{(y_{k}, Az_{k})}$,\ $z_{k+1}=B^{1}_{k+1}z_{k}+C^{1}_{k+1}r_{k+1}$. Switching between algorithms to avoid breakdown =============================================== When a Lanczos-type algorithm fails, this is due to the non-existence of some coefficients of the recurrence relations on which the algorithm is based. The iterate which causes these coefficients not to exist does not cause and should not necessarily cause any problems when used in another Lanczos-type algorithm, based on different recurrence relations. It is therefore obvious that one may consider switching to this other algorithm, when breakdown occurs. This allows the algorithm to work in a Krylov space with a different basis. It is therefore also possible to remedy breakdown by switching. Note that restarting the same algorithm after a pre-set number of iterations works well too, [@12:Farooq] **[Switching strategies]{}** ---------------------------- Different strategies can be adopted for switching between two or more algorithms. These are as follows. 1. **ST1: Switching after breakdown:** Start a particular Lanczos algorithm until a breakdown occurs, then switch to another Lanczos algorithm, initializing the latter with the last iterate of the failed algorithm. We call this strategy ST1. 2. **ST2: Pre-emptive switching:** Run a Lanczos-type algorithm for a fixed number of iterations, halt it and then switch to another Lanczos-type algorithm, initializing it with the last iterate of the first algorithm. Note that there is no way to guarantee that breakdown would not occur before the end of the interval. This strategy is called ST2. 3. **ST3: Breakdown monitoring:** Provided monotonicity of reduction in the absolute value of the denominators in the coefficients of the polynomials involved can be established, breakdown can be monitored as follows. Evaluate regularly those coefficients with denominators that are likely to become zero. Switch to another algorithm when the absolute value of any of these denominators drops below a certain level. This is strategy ST3. **[A generic switching algorithm]{}** ------------------------------------- Suppose we have a set of Lanczos-type algorithms and we want to switch from one algorithm to another using one of the above mentioned strategies ST1, ST2 or ST3. Start the most stable algorithm, if known. Choose a switching strategy from $\{$**[ST1, ST2, ST3]{}$\}$. Continue with current algorithm until it halts; Stop. switch to another algorithm; initialize it with current iterate; Go to 4. Continue with current algorithm for a fixed number of iterations until it stops; Stop. switch to another algorithm, initialize it with the current iterate, Go to 13. Continue with current algorithm and monitor certain parameters for breakdown, until it halts; Stop. switch to another algorithm, initialize it with the current iterate, Go to 22.** However, it is important to mention that we have considered only ST2 in this paper. The convergence tolerance in all of the tests performed is $\epsilon = 1.0e^{-013}$ and the number of iterations per cycle is fixed to $20$. ### **[Switching between algorithms $A_{4}$ and $A_{12}$]{}** In the following, we start with either $A_4$ or $A_{12}$, run it for a fixed number of iterations (cycle) chosen arbitrarily, before switching to the other. The results of this switching algorithm, are compared to those obtained with algorithms $A_{4}$ and $A_{12}$ run individually. We are not changing any of the parameters involved in both algorithms. Details of $A_4$ can be found in [@94:Baheux]. Choose $x_{0}$ and $y$ such that $y\neq0$, set $r_{0}=b-Ax_{0}$, $y_{0}=y$, start either algorithm, run current algorithm for a fixed number of iterations (a cycle) or until it halts; stop; switch to the algorithm not yet run; initialize it with the current iterate; go to 4; [**Remark:**]{} Since restarting can be just as effective as switching, it is easier to implement a random choice between $A_{4}$ and $A_{12}$ at the end of every cycle. Let heads be $A_{4}$ and tails be $A_{12}$. At the toss of a coin, if it shows heads and the algorithm running in the last cycle was $A_{4}$, then the switch is a restart. If the coin shows tails then the switch is a “proper" switch, and $A_{12}$ is called upon. In the numerical results presented below, this is what has been implemented. For more details about restarting see, [@12:Farooq]. ### **[Switching between $A_{4}$ and $A_{5}/B_{10}$ algorithm]{}** Start with $A_{5}/B_{10}$, (details of $A_5/B_{10}$ can be found in [@94:Baheux; @95:Baheux]) do a few iterations and then switch to either $A_{4}$ or $A_{5}/B_{10}$. The procedure is as Algorithm $4$ below. Choose $x_{0}$ and $y$ such that $y \neq 0$; set $r_{0} = b-Ax_{0}$, $y_{0} = y$, $p_{0} = r_{0}$; start with either $A_4$ or $A_{5}/B_{10}$; run it for a fixed number of iterations (cycles) or until it halts stop; switch to either $A_{4}$ or $A_{5}/B_{10}$; initialize it with the last iterate of the algorithm running in the last cycle; go to 4; ### **[Switching between $A_{4}$ and $A_{8}/B_{10}$]{}** Start with either $A_{8}/B_{10}$ (details of $A_8/B_{10}$ can be found in [@94:Baheux; @95:Baheux]) or $A_4$; do a few iterations and then switch to either of them chosen randomly. If the chosen algorithm happens to be the same as the one running in the last cycle, then it is a case of restarting. Otherwise, it is switching. The algorithm is as follows. Choose $x_{0}$ and $y$ such that $y \neq 0$; set $r_{0}=b-Ax_{0}$, $y_{0}=y$, $p_{0}=r_{0}$; start either $A_{4}$ or $A_{8}/B_{10}$; run it for a fixed number of iterations (cycle), or until it halts; stop; switch to either $A_{4}$ or $A_{8}/B_{10}$; initialize it with the iterate of the algorithm run in the last cycle; go to 4. ### **[Switching between $A_{5}/B_{10}$ and $A_{8}/B_{10}$]{}** Here again, switching and restarting are combined in a random way. Start with either $A_{8}/B_{10}$ or $A_{5}/B_{10}$. After a pre-set number of iterations (cycle), switch to either $A_{5}/B_{10}$ or $A_{8}/B_{10}$, randomly chosen. If the chosen algorithm to switch to is the same as the one running in the last cycle then we a have a case of restarting; else it is switching. The algorithm is as follows. Choose $x_{0}$ and $y$ such that $y\neq0$; set $r_{0}=b-Ax_{0}$, $y_{0}=y$, $z_{0}=r_{0}$; start either $A_{8}/B_{10}$ or $A_{5}/B_{10}$; run it for a fixed number of iterations; halt current algorithm; switch to either $A_{5}/B_{10}$ or $A_{8}/B_{10}$; initialize it with the last iterate of the algorithm running in the last cycle;\ go to 4;\ solution found; stop; ### **[Numerical results]{}** Algorithms $1$, $2$, $3$, $4$, [@94:Baheux; @95:Baheux; @10:Farooq] and Algorithms 6, 7, 8 and 9, [@10:Farooq] have been implemented in Matlab and applied to a number of small to medium size problems. The test problems we have used arise in the 5-point discretisation of the operator $-\frac{\partial^{2}}{\partial x^2}-\frac{\partial^{2}}{\partial y^2}+\gamma\frac{\partial}{\partial x}$ on a rectangular region [@94:Baheux; @95:Baheux]. Comparative results are obtained on instances of the problem $\textit{A}\textbf{x}=\textbf{b}$ with $\textit{A}$ and $\textbf{b}$ as below, and with dimensions of $A$ and $\textbf{b}$ ranging from $n=10$ to $n=100$. $$A=\left(\begin{array}{ccccccc} B & -I & \cdots & \cdots & 0\\ -I & B & -I & & \vdots\\ \vdots & \ddots & \ddots & \ddots & \vdots\\ \vdots & & -I & B & -I\\ 0 & \cdots & \cdots & -I & B\\ \end{array} \right),$$with $$B=\left(\begin{array}{ccccccc} 4 & \alpha & \cdots & \cdots & 0\\ \beta & 4 & \alpha & & \vdots\\ \vdots & \ddots & \ddots & \ddots & \vdots\\ \vdots & & \beta & 4 & \alpha\\ 0 & \cdots & & \beta & 4\\ \end{array}\right),$$ and $\alpha=-1+\delta$, $\beta=-1-\delta$. The parameter $\delta$ takes the values $0.0$, $0.2$, $5$ and $8$ respectively. The right hand side $\textbf{b}$ is taken to be $\textbf{b}=\textit{A}\textbf{X}$, where $\textbf{X}=(1, 1, \dots, 1)^{T}$, is the solution of the system. The dimension of $\textit{B}$ is $10$. When $\delta=0$, the coefficient matrix $A$ is symmetric and the problem is easy to solve because the region is a regular mesh, [@06:Gérard]. For all other values of $\delta$, the matrix $A$ is non-symmetric and the problem is comparatively harder to solve as the region is not a regular mesh. ### **[Numerical results]{}** Results obtained with Algorithms 1, 2, 3, 4, and Algorithms 6, 7, 8 and 9 on Baheux-type problems of different dimensions, for different values of $\delta$ are presented in tables 1, 2, 3 and 4, below. Algorithms $1$, $2$, $3$, $4$, executed individually, could only solve problems of dimensions 40 or below. In contrast, the switching algorithms, Algorithms 6, 7, 8 and 9, solved all problems of dimensions up to 4000. These results show that the switching algorithms are far superior to any one of the algorithms considered individually. These echo those obtained by restarting the same algorithm after a predefined number of iterations, [@12:Farooq]. Conclusion ========== We have implemented $A_4$, $A_{12}$, $A_5/B_{10}$ and $A_8/B_{10}$ to solve a number of problems of the type described in Section $5.2.5$ with dimensions ranging from 20 to 4000. The results are compared against those obtained by the switching algorithms, Algorithms 6, 7, 8 and 9 on the same problems. These results show that $A_4$, $A_{12}$, $A_5/B_{10}$ and $A_8/B_{10}$ are not as robust as the switching algorithms. In fact, individual algorithms solved only problems of dimension $n\leq 40$ and that with a poor accuracy. The switching algorithms, however, solved them all with a higher precision. The numerical evidence is strongly in favour of switching. Based on the above results, it is clear that switching is an effective way to deal with the breakdown in Lanczos-type algorithms. It is also clear that the switching algorithms are more efficient particularly for large dimension problems. The cost of switching, in terms of CPU time, in ST2 at least, is not substantial, compared to that of the individual algorithms. It is also quite easy to see that it would not be substantial in ST1 since the cost would be similar to that of ST2. Even in the case of monitoring the coefficients that can vanish, i.e. ST3, the cost should only be that of a test of the form:\ [**if**]{} $|$denominator value$|$ $\leq$ tolerance [**then**]{} stop.\ We have not measured its impact on the overall computing time, but it should not be excessive. This means that switching strategies are worthwhile considering to enhance the efficiency of Lanczos-type algorithms and not just their robustness. Having said that, further research and experimentation are necessary, particularly on the very large scale instances of SLE’s, to establish the superiority of switching algorithms against the state-of-the-art Lanczos-type algorithms with in-built precautions to avoid breakdown such as MRZ and BSMRZ, [@94:Baheux; @91:Brezinski; @na1; @99:Brezinski]. Note that these algorithms are attractive for other reasons too, namely their simplicity and easy implementation. This is the subject of on-going research work. Muhammad Farooq,\ Department of Mathematics,\ University of Peshawar,\ Khyber Pakhtunkhwa, 25120, Pakistan\ mfarooq@upesh.edu.pk, Tel: 00 92 91 9221038, Fax: 00 92 91 9216470\ Abdellah Salhi,\ Department of Mathematical Sciences,\ The University of Essex, Wivenhoe Park,\ Colchester CO4 3SQ, UK\ as@essex.ac.uk, Tel: 00 44 1206 873022, Fax: 00 44 1206 873043\

--- abstract: 'In this article, we perform a 2-d simulation of combustion of neutron star (NS) to hybrid star (HS). We assume that a sudden density fluctuation at the center of the NS initiates a shock discontinuity near the center of the star. This shock discontinuity deconfines NM to 2-f QM, initiating combustion of the star. This combustion front propagates from the center to the surface converting NM to 2-f QM. This combustion stops at a radius of $6 km$ inside the star, as at this density the NM is much stable than QM. Beyond $6 km$ although the combustion stops but the shock wave propagates to the surface. We study the gravitational wave signal for such a PT of NS to HS. We find that such PT has unique GW strain of amplitude $10^{-22}$. These signals last for few tens of $\mu s$ and shows small oscillating behaviour. The power spectrum consists of peaks and at fairly high frequency range. The conversion to NS to HS has a unique signature which would help in defining the PT and the fate of the NS.' author: - R Prasad - Ritam Mallick title: Gravitational waves from phase transition of NS to QS --- Introduction ============ The two recent observation of gravitational wave (GW) from black holes merger (BHM) GW150914 [@abbott] and from a binary neutron star merger (BNSM) GW170817 [@abbott1] has opened a new window towards gravitational wave astronomy and multimessenger astronomy. In the BHM GW is the only signal which otherwise would have left minimal signatures, whereas for BNSM both GW and electromagnetic radiation were detected which opens a whole new window to probe those objects which till now has limited access. The multimessenger detection of BNSM has put a severe constraint on the nature of matter that can occur inside a neutron star (NS) [@annala; @margalit; @radice; @ruiz; @shibata; @most]. The stringent condition has forced to discard many exotic states of nature which were inferred to be at the center of the star. However, there is still a possibility that the interior of the star can have a deconfined state of matter. With more GW detectors coming up shortly, our ability to localize and examine the sources will increase further. Both the ground-based (like LIGO, Virgo, GEO600, TAMA300, Einstein Telescope) and space-based GW detectors (LISA) will have more efficiency with more detection capability (lesser amplitude GW). Also, the detectors would cover a much broader range of frequency range which will help in the detection of more GW sources. So now we are in the age of long-promised GW and multimessenger astronomy. This will help us in tackling problems which are cross-cutting and multidisciplinary. Neutron stars are one of the most exciting subjects of this GW and multimessenger astronomy as had been proved by BNSM GW170817 [@abbott1]. The detection of the BNSM has proved beyond doubt that NSM is one of the best sources of GW. Even rotating NS which have time-varying quadrupole moment can radiate GW and can be one of the sources [@ferrari; @Zimmerman]. Fast rotating protoneutron stars can develop Chandrasekhar-Friedman-Schutz type of instability and can generate enough energy to emit GW which can be detected [@chandrasekhar; @friedman]. Non-radial oscillation can also be a possible mechanism for NS to radiate GWs [@thorne]. Another event in NS that have enough opportunity to develop GW is the phase transition (PT) in NS [@prasad]. All these potential GW detections can open a new window to probe NS interiors and consequently the matter at such nuclear densities. The detection and interpretation of GW170817 have already provided the horizon of maximum mass, the radius and the tidal deformations. The detection of GW from PT in NS is particularly interesting, for it being the signature of PT happening at extreme densities. If such a confined deconfined PT does happen, it will prove the fact that exotic phases of matter do exist at the interior of the stars. We then can have two families of compact stars coexisting. The existence of compact stellar objects with deconfined quark matter was predicted long ago [@Itoh; @Bodmer; @Witten]. Such stars usually originate from NS by the conversion of hadronic matter (HM/NM)) to quark matter (QM). Such a PT can result soon after the birth in a supernovae [@bombaci; @drago; @mintz; @gulminelli] or could happen in cold NS by accreting matter from its companion. The mass accretion would spin up the star (to millisecond pulsars), and there would be a large family of stars with high mass and rotation speed. These are the best candidates to suffer PT at the center of the star which suffers small fluctuations at the center. The PT would likely be first order PT, where the nuclear matter is converted to quark matter (QM). As the density at the center of the star is maximum, the PT would in all probable to start from the center of the star [@prasad]. A combustion front or a shock wave is likely to propagate from the center to the surface of the star. This front converts the NM to QM. The combustion front can spread throughout the star and reach the crust, thereby breaking it and ejecting matter or the front can lose steam inside the star and stop somewhere in between. The former case would result in a strange stable star (SS) whereas the latter would result in a hybrid star (HS). Such exotic stars are more compact than the original NS. Such PT would result in a release in the energy of the order of $10^{52}-10^{53}$ ergs [@bombaci1; @berezhiani; @drago; @sahu]. Some amount of energy will be released in the form of neutrino emission. However, a significant amount of it can go into emitting gravitational waves [@lin; @abdikamalov]. In literature calculattion of gravitational wave emission from phase transition has been obtained via different approaches. In one of the approach, the oscillation modes excited by PT is said to result in GW. The gravitational waves produced due to any given mode is given by, $$h(t)=h_{o} e^{-\frac{t}{\tau}} sin(\omega_{0} t)$$ where $h_{0}$ is amplitude given by $$h_{0} = \frac{4}{\omega_{0} D} \sqrt{\frac{E_{g}}{\tau}}$$ where $E_{g}$ is the energy available to give a oscillation mode, $\omega_{0}$ is frequency of oscillation mode and $\tau$ is damping time scale corresponding to a give mode. The energy released in PT is estimated using the difference in binding energy of NS and HS/QS, given by $$B.E. (r) = M_{g}(r) - M_{B}(r).$$ During phase transition, the energy released is majorly in the form of gamma-ray burst and gravitational waves. Also, there will be other dissipation mechanisms, hence in this approach the energy budget available to gravitational waves is uncertain. In other approach implemented by Lin et al. [@lin] and Admikamalov et al. [@abdikamalov], it is considered that quark matter content when appears inside an NS, due to quark matter EoS having less pressure than nuclear matter the star undergoes a micro-collapse. This results in quadrupole moment change, and the gravitational wave is generated. The calculation of the GW by Lin et al. [@lin] treats the PT to occur instantaneously. The collapse would result in stellar pulsation and using Newtonian gravity they obtained the waveforms of the emitted GW from such collapsar models and found that the strain of the GW can of the order of $10^{-23}$. This calculation was further carried forward by Abdikamalov et al. [@abdikamalov], where they used GR calculation and more realistic EoS models. The PT was assumed to be of finite timescale, but the GW calculation involved were related to the stellar pulsation of different modes. This approach deals mainly with the aftermath of PT. The gravitational waves resulting from the dynamical process of PT has not been investigated. In our recent work [@prasad] (let call it Phase transition in neutron stars (PTNS)), we have simulated the dynamical PT process using a conversion font (shock front) which propagates from the center to the surface and converts quark matter to nuclear matter. In PTNS we have studied the actual dynamical evolution of the combustion front in NS. The conversion of NM to QM takes place at this combustion front, and as the front moves outward to the outer surface of the star the shock intensity decreases and the combustion stops inside the star, however, the velocity of the shock propagation is close to that of light. This indicates a rapid PT, with the timescale, of the order of tens of microseconds. If such a PT happens inside a rapidly rotating neutron star having an axisymmetric shape, the quadrupole moment of the star changes at this small time scale. This can result in a sufficiently strong GW signal, which can be detected by either improved earth based GW interferometers or by space-based GW detectors. In this article, we report the characteristics and the template of such GW signals that are likely to be produced in such PT of NS. In section II we first describe briefly our equation of state (EoS) which we have employed to characterize the QM and HM. Section III is devoted to the description of the numerical model of the star and the combustion of NS to QS. The GW results are given in section IV and finally, in section V we discuss our findings and conclude from them. Equation of State ================= NS matter at the inner core is very dense, and they interact via the strong interaction. In principle the degree of freedom at such densities should be quarks; however, the quarks appear at the core after some fluctuation in the star (after PT). Therefore, to begin with, the degree of freedom is consequently mainly neutron, proton, electron and some other baryons and leptons in a small fraction. The carriers of the nuclear force are assumed to be sigma, omega, and rho. In this calculation we choose PLZ [@reinhard] parameter setting to describe the NM of the star leaving the crust. The EoS is consistent with the recent astrophysical and nuclear constraint and can generate stars more massive than two solar mass. The equatorial radius of the star of mass $1.5$ solar mass is about $15$ Km. After the PT the degree of freedom of the innermost core of the star becomes quarks. The QM is described by the MIT bag model having quark interaction [@chodos]. The QM is composed of only up and down quark. The shock propagation deconfines the hadrons to quarks (2 flavor quark matter). This happens very fast, and this 2 flavor (2-f) matter is metastable. It settles into final stable strange matter (3-f) (with strange quark appearing via weak interaction) at weak interaction time scale. However, this process is much slower than the former deconfinement of quarks [@abhijit]. Therefore, we can treat the two process separately. In this problem, we are dealing with the first process. For the quark matter, the bag value is chosen to be $B^{1/4}=140$ MeV and the quark coupling to be $a_4=0.5$. Fig \[fig-2f\] shows that the 3-f QM (Q-140) is absolutely stable than the HM (PLZ) beyond $p=110$ Mev/fm$^3$. However, the metastable 2-f starts at $p=340$ Mev/fm$^3$ and beyond those densities 2-f QM is stable. In our scenario where the PT happens via a 2-step process, the NM first has to convert to 2-f matter and then to 3-f matter. If the NM is stable than 3-f QM, it will not convert to QM by such a process. The PLZ and 2-f curve (Q-140) cuts at $p=110$ and below those pressure, the 2f QM is metastable and finally has to go to 3-f QM for absolute stability. This, tells us that for a PT taking place from PLZ NS to a quark star (QS) (with Q-140), the final QS is likely to be a hybrid star (HS). Therefore, we only have QM beyond pressure $p=110$ Mev/fm$^3$. In our star, this happens at a distance of $6 km$, and we have the PT or combustion from NM to QM till a distance of $6 km$ and beyond that radial distance we have shock propagation. For the quark matter, the bag value is chosen to be $B^{1/4}=140$ MeV and the quark coupling to be $a_4=0.5$. Hydrodynamic simulation of the combustion of NM to QM ===================================================== This work aims to calculate the strain and power of the GW which would be emitted from such a PT. Shock-induced PT in spherically symmetric neutron star dosen’t lead to GW emission, hence in the present study the neutron star is taken to be axisymmeteric. To solve for an axisymmetric star using the given EoS we employ the rotating neutron star (RNS) code[@komatsu; @Stergioulas]. The structure of the star is described by the Cook-Shapiro-Teukolsky (CST) metric [@cook]. The code can solve for rotating NS using polytropic or tabulated EoS. The code calculates the metric functions of the metric (they are a function of $r$ and $\theta$). It takes the central density and rotational velocities as an input and give back the metric function in terms of $r$ and $\theta$. It also gives the density, pressure as a function of the dependent variables. The total star mass, its equatorial and polar radius, its moment of inertia and its eccentricity are also obtained. Using PLZ EoS neutron star strucutre and properties is obtained for different values of central density keeping the rotational frequency $\omega = 0.3 \times 10^{-4}$ Hz fixed, these different NS models obtained are listed in table 1 and would be used in our calculation of the combustion and also for the GW. The hydrodynamic equation is solved using the GR1D code [@oconnor; @font]. The description of the code can be found in our previous paper, and we only discuss them briefly in this present article [@prasad].The code uses the method of lines for time integration [@hyman]. The spatial discretization is done by finite volume approach [@romero; @font]. All the primitive variables are defined at the cell center and are interpolated at cell interfaces. Piecewise-parabolic method (PPM [@Colella]) is for interpolation of smoothing of fluxes. From the primitive variables the conserved variables are calculated (density, velocity, and energy). The physical fluxes are calculated by HLLE Reimann solver [@font]. Once the conserved variables are calculated the EoS is used to calculate the pressure. So, we have all the thermodynamic variables evolving both spatially and temporally. The solution of the RNS code is used to set the initial configuration at some particular $\theta$. This gives the star profile (density, pressure as a function of radial distance for a specific $\theta$. For our specific problem, we initially give a small density fluctuation near the center of the star (at about $0.5 km$). At an initial time $t=0$, the velocity of both left and right states are zero. As the shock propagates the position of the shock discontinuity concurs with the location of which the speed is maximum. The position of the shock discontinuity can be traced for all time using this method. We ensure that in behind (QM EoS) and front (HM EoS) of the shock the EoS is different, ensuring a simulated PT. In PTNS we have simulated the shock-induced phase transition using one-dimensional general relativistic hydrodynamics code (GR1D). To obtain a PT, we need a significant density fluctuation at the center of the star. The change can be caused due to sudden spin up due to mass accretion. We heuristically assume a density discontinuity at a distance of $0.5$ km from the center of the star. This gives rise to a shock wave. With time the shock wave propagates outward and thereby combusting the NM to QM. The shock wave is evolved according to the solution of the hydrodynamic equation. This is done by the GR1D code. We do a 2-dimensional evolution of the shock profile (in $r$ and $\theta$) solving the GR1D code for different $\theta$ obtained from the RNS code. The initial discontinuity is given for the density, and the pressure discontinuity is obtained from the polytrope. The initial matter velocities at either side of the shock are kept to be zero. We see that with time the discontinuity proceeds towards the periphery of the star from the high-density to the low-density region and its strength gets reduced. To start with, we carry out our study using the PLZ-M1 model having central density $2$ times the nuclear density and $\omega = 0.3 \times 10^{-4}$ Hz, which yields a star of mass $1.5 M_{\odot}$ and radius of $15$ km. We proceed with following steps to ensure that system in hydrodynamics code is a neutron star in 2-dimension and mimics realistic scenario whereas the conversion font propagates the hadronic matter region which it surpasses becomes 2-flavor quark matter. We use RNS code to obtain the profile of neutron star using hadronic matter EoS; it gives us a profile for an axisymmetric neutron star. The $ 0 < \theta < \pi/2 $ is split into 65 values. For a given $\theta$, we obtain $\rho (r, \theta_{fixed})$. These 65 profiles have different values of $r_{max}$ the boundary of the star. The hydrodynamics equations require initial conditions which are the value of density, pressure, and velocity at each point in the system at $t=0$ and the boundary of the star. We evolve this $65$ profiles(each corresponding to a $\theta$) one by one. The overall output of hydrodynamics simulation comes out to be density $\rho (r, \theta_{fixed})$, pressure $p (r, \theta_{fixed})$ and velocity $v (r, \theta_{fixed})$ at each point in the system at any time $t$ along a given direction. By combining all these $65$ profiles along $65$ directions, we get the complete information of star. In fig 2., we show the density variation of the star (before the PT) along the radial distance for PLZ-M1 case. Fig 2 and table 1 are in agreement and shows that the rotation in the star deforms the star. The star is oblate spheroid, with the equatorial radius is elongated than the polar radius. The density (and pressure) fall smoothly from the center to the surface for an unshocked star. ![The velocity of the combustion front and the shock wave is shown as a function of time. Till $25 \mu s$ the combustion wave propagates from the center to a distance of $6$ km. After $6$ km simple shock wave propagates to the surface of the star. The transition from combustion wave to shock wave is seen as a sharp discontinuity in the front velocity. The front velocity is also plotted for the equatorial and polar direction.[]{data-label="vel-f"}](pt_vel.eps){width="3.5in" height="2.8in"} As the shock wave propagates, it converts NM to QM, however after some distance as the density decreases the shock wave starts to lose its strength. Also if there is a formation of the mixed phase, it will dissolve this sharp discontinuity. The smoothing of the discontinuity will depend on how much time needed for the mixed phase region to grow, whereas our propagation of the shock is swift. It is quite possible that it can travel a considerable distance before it smooths out. Therefore, after propagating some distance in the star, the PT will stop and eventually the shock wave may die out. We should mention that we have not done the exact calculation on how long it would take for the sharp discontinuity to dissolve due to the formation of the mixed phase. Such a calculation although may be essential but involves much complexity. For simplicity, we do not consider such a case in our study. The lowest density of the quark matter EoS which we have used is about $1.8$ time the nuclear density, which corresponds to a distance of about $6$ km in the star. Therefore, after a length of $6$ km, the shock evolves without bringing about a PT. We, therefore, have an HS, a star with outer NM and a quark core. [{height="2.95in" width="3.1in"}]{} [{height="2.95in" width="3.1in"}]{} [{height="2.95in" width="3.1in"}]{} Fig 3 shows the density evolution as a function of the radial distance of the star. We have demonstrated this evolution for the equatorial and polar direction. The hydrodynamic equations also solve for the spatial and temporal evolution. Therefore, we have shown $6$ time slice (at four different times including initial configuration) plots of the density and pressure evolution as a function of the radial distance. The EoS ensures that a discontinuity in the density results in a discontinuity in the pressure of the star. In fig 4 we plot the pressure as it evolves both spatially and temporally. The evolution of the pressure strictly follows the development of the density. The hydrodynamic equation ensures that as the combustion starts the matter velocities takes up non zero values. The matter velocities are shown in fig 5. As the frame moves outwards, from the frame of the front, the HM comes towards the front. At the front, it is converted to QM, and beyond the front, the QM goes away from the front. As the star is a closed and a dense system the velocity of the matter peaks up at the surface. This may result in the breaking of the crust and ejection of matter to the star atmosphere. However, gravity may play a crucial role here. In this article we have not taken gravity into account; therefore we refrain from further comments. The density or the pressure plots gives the location of the combustion or the shock front at any given instant of time. Therefore, we can calculate the front velocity by differentiating the shock location with respect to time. In fig 6 we show the shock velocity along the polar and equatorial direction. More or less the nature of the front velocity remains the same. We assume that the combustion wave starts from $0.5$ km and continues till $6$ km. Beyond that only shock wave propagates. Therefore at $6$ km there is a transition from combustion to shock wave. This is seen clearly as a sharp discontinuity of the front velocity. The combustion wave takes around 30 $\mu s$ to travel to $6$ km in the star. This sharp discontinuity has a significant contribution to the GW signal which can be seen later. The 2-dimensional evolution of PT front is shown in fig 7, where for $\phi= \pi/2$ the $\rho=(r,\theta)=\rho(z,y)$ is represented as heatmap for different time instances. Phase transition as source of quadrupole moment change ====================================================== -------- ------------------ --------------- ------------- --------------- ------------- --------------------- ------------ -------------------------------------- Model $\rho_{c}$ $\omega$ $M$ $r_{e}/r_{p}$ $T/W$ I $|h|$ $f_{peaks}$ ($10^{14}$ g/cc) ($10^{4}$ Hz) ($M_{sun}$) ($10^{-2}$) $10^{45}$g cm$^{2}$ kHz PLZ-M1 5.0 0.3 1.50 1.28871 2.73182 2.22 $10^{-22}$ 23.80, 95.23, 166.66, 261.90 PLZ-M2 5.6 0.3 1.82 1.22754 2.44001 2.96 $10^{-22}$ 23.80, 95.23, 166.66, 238.09 PLZ-M3 6.0 0.3 1.99 1.19347 2.29033 3.39 $10^{-22}$ 23.80, 71.42, 142.85, 285.714 PLZ-M4 6.6 0.3 2.2 1.14929 2.10966 3.92 $10^{-22}$ 23.80, 71.42, 119.04, 190.47, 238.09 -------- ------------------ --------------- ------------- --------------- ------------- --------------------- ------------ -------------------------------------- Recent gravitational wave observation of GW170817 [@abbott1] of a neutron star merger has shed new light not only in the field of GW astronomy but also to multimessenger astronomy. The star merger has revealed itself in the form of GW, neutrino production and also in other various spectra of electromagnetic range. With the rejuvenation of GW astronomy and the GW detection, several new gravitational wave detectors are coming up shortly like TAMA300 of Japan, a space-based telescope LISA, Einstein Telescope along with the presently working LIGO, Virgo and GEO600. This detectors and telescopes are much improved and are likely to detect small amplitude GW and with frequency ranging upto KHz. The PT scenario is also a very credible source of generation of GW and the combustion time of the star hints at a very definite template of such a scenario. The small time scale of NM to 2-f QM conversion is quite different from any other astrophysical timescale and therefore if such a process can generate GW it is very likely that the signal of such a process would also be very different. In this section, we calculate the prospect of GW signal due to the combustion of NM to 2-f QM. We have done a relativistic hydrodynamic simulation of the combustion process; however, gravity is not taken into account. The gravity is likely to influence the process, but it is expected that it will not have any considerable effect on the combustion process due to fast burning. The gravity is more likely to have a significant impact on the 2nd step process (2-f QM to 3-F QM) where the star tries to settle down and suffers a gravitational collapse. Such process has been readily studied in the literature [@dimmelheimer; @lin; @abdikamalov] and it suggests that the collapse would last few milliseconds and the GW emission from such a process would have an amplitude of the order of $10^{-23}$. The time frame mentioned in those papers is of the order of 2-f to 3-f (weak interaction) time scale. The possible problem with such a GW signal is that it hints at any type of collapse dynamics. It may come from a PT but even can generate from some reshuffling of the density profile of the star. However, a signal of NM to 2-f QM conversion cannot come from any other process. It also differs from a shock propagation inside an NS (without combustion). For an axisymmetric star, the non-zero components of the gravitational wave field is [@zwerger; @dimmelheimer] $$h_{\theta \theta}^{TT} =\frac{1}{8} \sqrt{\frac{15}{\pi}} \sin^{2}\theta \frac{A_{20}^{E2}}{r}$$ $$h_{\phi \phi}^{TT} =-h_{\theta \theta}^{TT}=h_{+}$$ where $ \theta$ is the angle between the symmetry axis and the line of sight of the observer. $$A_{20}^{E2}= \frac{d^{2}}{dt^{2}} \left( k \int \rho \left( \frac{3}{2} z^{2}-\frac{1}{2} \right)r^{4} dr dz \right)$$ $$z=\cos{\theta}$$ and $$k=\frac{16 \pi^{3/2}}{\sqrt{15}}$$ The above formulae are used for finding the GW emerging from supernova collapse and due to micro-collapse resulting from the aftermath of phase transition. In our recent work, we have modeled the dynamical PT process in NS, and it was observed that the configuration of a neutron star, like the density profile and pressure profiles, exhibited significant change during this process. In this work, we evaluate the quadrupole moment during the dynamical PT process and obtain the GW amplitude and frequency, which eventually sheds light on observation possibility in GW detectors. We present the detailed steps to perform the calculation of $h_{\theta \theta}^{TT}$ due to the phase transition. At $t=0$, we take the originating location of conversion font to be $r_{i}$ a distance very close to the center of the star. Using hydrodynamics simulation after a small time interval (say $dt=10^{-7}$ sec) we evaluate the location ($r_{f}$) reached by conversion font. Once we know $r_{f}$ for the first time step ($dt$), we obtain the density profile generated, wherein star is composed of quark matter EoS and hadronic matter EoS. We integrate the density profile to evaluate the corresponding gravitational wave amplitude $h_{\theta \theta}^{TT}$. Now this $r_{f}$ reached at the end of the first-time step will serve as $r_{i}$ for second-time step; we evaluate the final location reached by conversion font using hydrodynamics simulation at the end second-time step. We repeatedly find the $r_{f}$ and $h_{+}$ for every time step till the conversion font reach the surface of the star. Contrary to PTNS where the 1-dimensional problem was solved, we extended our present calculation to 2-dimensions ($r$, $\theta$) using the RNS code. In fig 8 we plot the quadrupole moment changes as a function of time. We have also shown a curve in which the quadrupole moment of a star changes when a shock propagates through a star. The quadrupole moment rises with time smoothly for a shock wave. However, the situation is quite different for combustion. Initially, it follows the shock curve but starts to deviate after some time. The combustion quadrupole moment curve lies well above the shock curve. Also after $30 \mu s$ (when it reaches a distance of 6 km) the curve first becomes flat and then dips as the shock reaches the outer layer of the star. The combustion stops at the point where the curve becomes flat and only the shock wave propagates out to the surface. Such a change in the quadrupole moment is likely to have some effect on the GW signal and can be very important for the detection of a particular template for such a scenario. Next, in fig 9 we plot the amplitude or the GW strain as a function of time, this involves evaluation double derivative of quadrupole moment and it is sensitive to the choice of time interval $dt$. We use the Savitzky-Golay [@savitzky] filter to perform differentiation, smoothing the data by choosing appropriate window size such that it diminishes variations in data in $10^{-7}$ s, and differentiation is carried out for variations in quadrupole moment which are above $10^{-6}$s. This smoothing of variation suppresses the numerical noise in gravitational wave strain. We have shown curves for the shock propagation and combustion scenario. The shock wave strain first grows with time and then falls smoothly without having any oscillating or periodic behavior. However, the strain amplitude for the combustion of NM to QM is quite different. During the initial burning, the strain increases smoothly; however, at the point of the wave changing from combustion to shock, there is an oscillation of some sort in the signal. This signal is not present in the simple shock propagation and it a well-defined marker for the PT. This marker is particular to the combustion of NM to QM and finally in the formation of a hybrid star. The conversion of the wavefront from combustion to shock is very significant in giving rise to this exact signal. However, if the burning or the PT propagates to the surface of the star such signal would be not present. Such PT scenario is likely to have some different signal. Thereby, looking at the signal coming from an NS, we can likely to tell the difference between whether an NS has converted to a SS or an HS. To obtain the power spectra, we adopt the general procedure to find power spectra of any time-domain signal $h(t)$ with $ 0 \leq t \leq T$. The signal has is a function of time and is descerete, where $n$ values of $h(t)$ is $h_{k}$ at intervals $t = k \times \Delta t$ where $n$ is an integer. The discrete fourier transform of signal is [@press; @dimmelheimer], $$\tilde{h}_{k}= \sum_{n=0}^{N-1} h_{n} \exp{ \left( \frac{2 \pi i n k}{N} \right)}$$ and the corresponding frequency is $$\omega = \frac{2 \pi k}{T} .$$ The power spectrum is given by, $$P(\omega)=\frac{|\tilde{h}_{k}|^{2}}{N^{2}}$$ The amplitude spetrum is given by, $$h(\omega)= \sqrt{P(\omega)}$$. The plot of $P(\omega)$ vs. $\omega$ gives the power spectrum. It contains information about power content in different frequencies of the signal. For better prespective we plot the amplitude spectrum which is square root of power spectrum. We plot the amplitude spectrum of the GW emission of conversion of NS to HS in fig 10. For portion of signal from $t=0\mu s$ to $t=41\mu s$ is considered and with a sampling time of $t=10^{-6}s$, the discrete Fourier transform is carried out. The sampling frequency $10^{6}$ Hz implies that the Nyquist frequency is $5 \times 10^{5}$ Hz, that is characteristic periodicities/ quasi-periodicities upto $5 \times 10^{5}$ Hz can be observed if present in gravitational wave signal. There are three peaks appearing at $20$ KHz, $100$ KHz and $170$ KHz. The peaks appearing at such high frequency is also a unique signal of PT of NS to HS. We have only shown the power spectrum of the combustion scenario and not that of a shock. A pure shock does not show any peak. The peak of the power spectrum at such high frequency is a bit of challenge for GW telescopes. The usual telescopes work at a frequency of few HZ to a thousand Hz (KHz). New telescopes are increasing the frequency range to few KHz. However, such a GW signal has its first peak at tens of KHz, and the other peak occurs at hundred KHz and above. We perform our simulation for different mass models, the different mass models considered and features of GW signature seen for PT in these stars listed in table 1. The gravitational wave strain for different neutron stars undergoing PT is shown in fig. \[strain2\], the PT front gets converted to shock front at 6km which results in a unique signature(a hump) which can be seen. The hump-like signature is seen to shift for different stars, the hump-like signature shifts towards left for massive NS , this is due to PT ending in different stars at different times. The corresponding amplitude spectral density for these signals is represented in figure 11, the peaks indicate the characteristic frequencies present in the GW signals. Neutron star produces gravitational waves in different bands of frequency, some continuous and some short-lived. The continuous signals from neutron star are produced due to its rotation and deformed structure, these signals are usually monochromatic. When PT will happen in such a system two short-lived signals will emerge, one from dynamics of PT and other from micro-collapse. And eventually, the newly formed hybrid star too will emit continuous monochromatic GW. We give the overall picture of the gravitational wave signal coming from an NS, PT and its aftermath in fig 13 and table 2. Initially, the GW amplitude is very low ($10^{-26}$) and is generated by the rotation of the star. The star rotation is kept at $477$ Hz. The rotation of the star is periodic, and we get a sinusoidal curve. The zero of time is arbitrary and can be thought as the time we first start to observe the NS. Then we assume that at $t=2.60$ ms the PT starts, and the NM is combusting to QM. The strain is now of considerable strength ($10^{-22}$). At about 2.63 ms there is a sharp change in the strain (combustion ends, shock continues inside the star). This continues till few more $\mu s$ and then the collapse happens (along with the 2-f to 3-f conversion). The strain is considerable but is weaker than the NM to 2-f combustion signal. This would continue for a few more milliseconds till the central density of the star settles down to a stable HS. Finally, in the 4th panel, the strain (again of amplitude $10^{-26}$) of the HS is shown once the star has settled down. This GW is again coming due to the rotation of the HS. There is a slight change in the amplitude and frequency of the initial NS and final HS as the HS is more compact. The ellipticity of isolated radio pulsars is seen to be between $10^{-4} - 10^{-6}$. The ellipticity of quark/strange stars is thought to be higher than neutron stars, hence we have chosen $10^{-4}$ as ellipticity of neutron star and $10^-{3}$ as ellipticity of hybrid star [@aasiel; @lansky]. ------------- ------------------ --------------- -------------- --------------- ------------- ---------------------- ------------ -- Model $\rho_{c}$ $\omega$ $M$ $r_{e}/r_{p}$ $T/W$ I $\epsilon$ ($10^{14}$ g/cc) ($10^{4}$ Hz) ( $M_{sun}$) ($10^{-2}$) $10^{45}$ g cm$^{2}$ (choosen) NS (PLZ-M1) 5.0 0.300 1.50 1.28871 2.73182 2.22 $10^{-4}$ HS 5.37 0.356 1.43 1.16406 3.33161 1.87882 $10^{-3}$ ------------- ------------------ --------------- -------------- --------------- ------------- ---------------------- ------------ -- Summary and conclusion ====================== In this work, we have done a simulation of an axisymmetric star undergoing combustion from NS to QS. We have mimicked a 2-d simulation by employing the GR1D code in 65 direction of the star (the star is sliced into 65 direction from and the angle between 0 to $\pi/2$. This gives the single quadrant of the star. All the other three quadrants are a replica of this quadrant. Therefore, evolving a single quadrant, we get the information of the whole star. The conversion starts at some point near the center of the star. As the conversion propagates outward, it converts NM to QM. As the star is now axisymmetric and has an oblate spheroid shape, the density distribution along the polar and equatorial direction are not the same. The combustion therefore, proceeds with different velocity along the different direction of the star. We evolve this 65 profiles(each corresponding to a $\theta$) one by one. The output of hydrodynamics simulation gives the value of density $\rho (r, \theta_{fixed})$, pressure $p (r, \theta_{fixed})$ and velocity $v (r, \theta_{fixed})$ at each point in the system at any time $t$ along a given direction. By combining all these 65 profiles along 65 directions, we get the complete information of star. The combustion starts at a distance of $0.5$ km and continues till $6$ km. Till this point the combustion wave continues, and NM suffers a PT. Beyond this distance of the star, the NM is much stable than QM, and there is no PT. Therefore beyond that point, we have a shock wave propagating to the surface of the star. To calculate the prospect of GW signal due to the combustion of NM to 2-f QM, we have done a relativistic hydrodynamic simulation of the combustion process neglecting gravity. The gravity is not likely to influence the process significantly due to fast burning. The gravity is expected to have a significant effect on the 2nd step process (2-f QM to 3-F QM) where the star suffers micro-collapse. As the combustion wave propagates outwards, the internal dynamics of the star changes very rapidly. This changes the overall quadrupole moment of the star which happens very fast.As the quadrupole moment of the star changes, the star emits strong GW emission with the strain of the order of $10^{-22}- 10^{-23}$ depending on the distance of the NS. The Quadrupole moment of the NS changes when the combustion wave ends, and the shock wave begins. In other words, as the PT ceases to take place, there is a change in the quadrupole moment of the star. The gravitational wave amplitude or the strain of the GW depends strongly on the change in quadrupole moment. Therefore, the point where the combustion of NM to QM stops there is a sharp change in the amplitude of the strain. Such a change in quadrupole moment and strain is not seen for ordinary shock wave propagation. This result is very typical of combustion of NM to QM. This marker of a sudden change in the GW strain is very typical of the formation of HS. This type of signal is not present even for the conversion of NS to SS, where the combustion front goes to the surface of the star. This marker can be used to point the difference between an NS to HS conversion and NS to SS conversion. The only problem with the detection of such type of GW signals coming from PT scenario is that the peak in the power spectrum lies in upper KHz to MHz range. The appearance of the peaks at the high-frequency range is also a typical signal of such NS to HS conversion. A simple shock wave propagation does not have such two peaks in the power spectrum. Finally, we have shown the overall picture of such PT from NS to HS. A rotating star emits very low amplitude GW signal beyond our detection capability. As the combustion from NM to QM starts in the star the strain amplitude peaks up and increases continuously. The time when the burning stops a small oscillation in the wave amplitude generates. The combustion lasts only for tens of microseconds. Then the star settles down to a more stable 3-f QM, and the star suffers a collapse. This results in an oscillation of the central density of the star. This process continues for a few milliseconds and the GW strain at this time is of the order of $10^{-23}$. After the HS settles down, it again starts to emit very low amplitude GW signals due to rotation which is hard to detect. In this work, we perform a real dynamical evolution of PT in NS which results in the formation of an HS. We have shown that such PT has a very unique signal which is very different from any other astrophysical signals. Such signals can help us in confirming whether PT in NS is possible or not. It would also help us in determining whether a PT has resulted in the formation of SS or an HS. A hybrid star has features which are very similar to that of NS, and it is very hards to distinguish them. However, we have shown that the PT scenario can distinguish between the formation of SS and HS. We should mention here that we have done the dynamical calculation of the 1st process of PT, that is NM to 2-f QM dynamics. The 2-f to 3-f QM formation dynamics is still to be done. The 2nd process dynamics time scale is similar to the timescale of the star collapse. In our future work, we will be trying to address the dynamics of the second process. We also mention that we have not taken gravity into account as we believe that the effect of gravity in such a fast process is not very huge. However, to be through presently, we are trying to incorporate gravity in our calculation. The author RM is grateful to the SERB, Govt. of India for monetary support in the form of Ramanujan Fellowship (SB/S2/RJN-061/2015) and Early Career Research Award (ECR/2016/000161). RP would like to acknowledge the fianancial support in form of INSPIRE fellowship provided by DST, India. RM and RP would also like to thank IISER Bhopal for providing all the research and infrastructure facilities. [100]{} B. P. Abbott et al. (LIGO Scientific Collaboration and Virgo Collaboration), Phys. Rev. Lett. 116, 061102 (2016) B. P. Abbott et al. (LIGO Scientific Collaboration and Virgo Collaboration), Phys. Rev. Lett. 119, 161101 (2017) E. Annala, T. Gorda, A. Kurkela, and A. Vuorinen, Phys. Rev. Lett. 120, 172703 (2018) B. Margalit and B. D. Metzger, Astrophys. J. Lett. 850, L19 (2017) D. Radice, Astrophys. J. Lett. 838, L2 (2017) M. Ruiz, S. L. Shapiro, and A. Tsokaros, Phys. Rev. D 97, 021501 (2018) M. Shibata, S. Fujibayashi, K. Hotokezaka, K. Kiuchi, K. Kyutoku, Y. Sekiguchi, and M. Tanaka, Phys. Rev. D 96, 123012 (2017) E. R. Most, L. R. Weih, L. Rezzolla, and J. Schaffner-Bielich, Phys. Rev. Lett. 120, 261103 (2018) A. Ferrari and R. Ruffini, Astrophys. J. 158, L71 (1969) M. Zimmerman and E. Szedenits, Phys. Rev. D 20, 351 (1979) S. Chandrasekhar, Phys. Rev. Lett. 24, 611 (1970) J. Friedman and B. Schutz, Astrophys. J. 222, 281 (1978) K. Thorne and A. Campolattaro, Astrophys. J. 149, 591 (1967) R. Prasad & R. Mallick, Astrophys. J. 859, 57 (2018) N. Itoh, Prog. Theor. Phys. 44, 291 (1970) A. R. Bodmer, Phys. Rev. D 4, 1601 (1971) Witten, E., PRD 30, 272 (1984) Bombaci, I. Parenti & I. Vidana, AstroPhys. J. 614, 314 (2004) A. Drago, A. Lavagno & G. Pagliara, Eur. Phys. J. A 19, 197 (2004) B. W. Mintz, E. S. Fraga, G. Pagliara & J. Schaffner-Bielich, Phys. Rev. D 81, 123012 (2010) Gulminelli, F., Raduta, Ad. R., Oertel, M., & Margueron, J., Phys.Rev. C 87, 055809 (2013) I. Bombaci & B. Datta, AstroPhys. J. 530, L69 (2000) Z. Berezhiani, I. Bombaci, A. Drago, F. Frontera & A. Lavagno, AstroPhys. J. 586, 1250 (2003) R. Mallick & P. K. Sahu, Nucl. Phys. A 921, 96 (2014) L. M. Lin, K. S. Cheng, M. C. Chu & W. -M. Suen, Astrophys. J. 639, 382 (2006) E. B. Abdikamalov, H. Dimmelmeier, L. Rezzolla & J. C. Miller, Mon. Not. R. Astron. Soc. 392, 52 (2009) P.-G. Reinhard, Z. Phys. A 329, 257 (1988) A. Chodos, R. L. Jaffe, K. Johnson, C. B. Thorn, and V. F. Weisskopf, Phys. Rev. D 9, 3471 (1974) A. Bhattacharyya, S. K. Ghosh, P. S. Joarder, R. Mallick & S. Raha, Phys. Rev. C 74, 065804 (2006) H. Komatsu, Y. Eriguchi, and I. Hachisu, Mon. Not. R. Astron. Soc. 237, 355 (1989) N. Stergioulas and J. L. Friedman, Astrophys. J. 444, 306 (1995) G. B. Cook, S. L. Shapiro, and S. A. Teukolsky, Astrophys. J. 398, 203 (1992) E. O’Connor and C. D. Ott, Class. Quantum Grav., 27, 114103 (2009) Font J A, Miller M, Suen W-M and Tobias M, Phys. Rev. D. 61, 044011 (2000) Hyman J M, The method of lines solution of partial differential equations Technical Report COO-3077-139 (ERDA Mathematics and Computing Laboratory, Courant Institute of Mathematical Sciences, New York University (1976)) Romero J V, Ibáñez J M, Martı́ J M and Miralles J A, Astrophys. J., 462, 839 (1996) Colella P, & Woodward P R, J. Comput. Phys. 54, 174 (1984) Dimmelmeier H., Font J. A., Mueller E., Astron. & Astrophys. 393, 523 (2002) Zwerger, T. & Mueller, E., Astron. Astrophys., 320, 209 (1997) Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P. 1992, Numerical recipes in C: The art of scientific programming (Cambridge, U. K.: Cambridge University Press) A. Savitzky and M. J. E. Golay, Analytical Chemistry, 36, 8 (1964) J. Aasiel et al., AstroPhys. J. 785, 119 (2014) P. D. Lansky, Pub. Astron. Soc. Aus. (PASA) 32, e034 (2015)

--- abstract: 'We show well-posedness of a diffuse interface model for a two-phase flow of two viscous incompressible fluids with different densities locally in time. The model leads to an inhomogeneous Navier-Stokes/Cahn-Hilliard system with a solenoidal velocity field for the mixture, but a variable density of the fluid mixture in the Navier-Stokes type equation. We prove existence of strong solutions locally in time with the aid of a suitable linearization and a contraction mapping argument. To this end we show maximal $L^2$-regularity for the Stokes part of the linearized system and use maximal $L^p$-regularity for the linearized Cahn-Hilliard system' author: - 'Helmut Abels[^1]   and Josef Weber[^2]' title: 'Local Well-Posedness of a Quasi-Incompressible Two-Phase Flow' --- [**Mathematics Subject Classification (2000):**]{} Primary: 76T99; Secondary: 35Q30, 35Q35, 35R35, 76D05, 76D45\ [**Key words:**]{} Two-phase flow, Navier-Stokes equation, diffuse interface model, mixtures of viscous fluids, Cahn-Hilliard equation Introduction and Main Result ============================ In this contribution we study a thermodynamically consistent, diffuse interface model for two-phase flows of two viscous incompressible system with different densities in a bounded domain in two or three space dimensions. The model was derived by A., Garcke and Grün in [@MR2890451] and leads to the following inhomogeneous Navier-Stokes/Cahn-Hilliard system: $$\begin{aligned} {1} \partial_t (\rho {\textbf{v}}) + & {\text{div}}( \rho {\textbf{v}}\otimes {\textbf{v}}) + {\text{div}}\Big ( {\textbf{v}}\otimes \tfrac{\tilde \rho_1 - \tilde \rho_2}{2} m(\varphi) \nabla (\tfrac{1}{\varepsilon} W'(\varphi) - \varepsilon \Delta \varphi ) \Big ) \nonumber \\ &= {\text{div}}(- \varepsilon \nabla \varphi \otimes \nabla \varphi) + {\text{div}}(2 \eta (\varphi) D{\textbf{v}}) - \nabla q, \label{equation_strong_solutions_1} \\ {\text{div}}{\textbf{v}}&= 0, \\ \partial_t \varphi +{\textbf{v}}\cdot \nabla \varphi &= {\text{div}}( m( \varphi) \nabla \mu ), \\ \mu &= - \varepsilon \Delta \varphi + \frac{1}{\varepsilon} W' (\varphi ) \label{equation_strong_solutions_2} \end{aligned}$$ in $Q_T:= \Omega\times (0,T)$ together with the initial and boundary values $$\begin{aligned} {\textbf{v}}_{|\partial \Omega} = \partial_n \varphi_{|\partial \Omega}= \partial_n \mu_{|\partial \Omega} &= 0 && \text{ on } (0,T) \times \partial \Omega , \\ \varphi (0) = \varphi_0 , {\textbf{v}}(0) &= {\textbf{v}}_0 && \text{ in } \Omega . \label{equation_strong_solutions_inital_data_v}\end{aligned}$$ Here $\Omega \subseteq \mathbb R^d$, $d= 2,3$, is a bounded domain with $C^4$-boundary. In this model the fluids are assumed to be partly miscible and $\varphi\colon \Omega\times (0,T)\to {\mathbb{R}}$ denotes the volume fraction difference of the fluids. ${\textbf{v}}$, $q$, and $\rho$ denote the mean velocity, the pressure and the density of the fluid mixture. It is assumed that the density is a given function of $\varphi$, more precisely $$\rho=\rho(\varphi) = \frac{\tilde{\rho}_1+\tilde{\rho}_2}2 +\frac{\tilde{\rho}_1-\tilde{\rho}_2}2 \varphi \qquad \text{for all }\varphi \in{\mathbb{R}}.$$ where $\tilde{\rho}_1, \tilde{\rho}_2$ are the specific densities of the (non-mixed) fluids. Moreover, $\mu$ is a chemical potential and $W(\varphi)$ is a homogeneous free energy density associated to the fluid mixture, ${\varepsilon}>0$ is a constant related to “thickness” of the diffuse interface, which is described by $\{x\in \Omega: |\varphi(x,t)|<1-\delta\}$ for some (small) $\delta>0$, and $m(\varphi)$ is a mobility coefficient, which controls the strength of the diffusion in the system. Finally $\eta(\varphi)$ is a viscosity coefficient and $D{\textbf{v}}= \frac12(\nabla {\textbf{v}}+ \nabla {\textbf{v}}^T)$. Existence of weak solution for this system globally in time was shown by A., Depner, and Garcke in [@AbelsDepnerGarcke] and [@AbelsDepnerGarckeDegMob] for non-degenerate and degenerate mobility in the case of a singular free energy density $W$. Moreover, Grün showed in [@GruenAGG] convergence (of suitable subsequences) of a fully discrete finite-element scheme for this system to a weak solution in the case of a smooth $W\colon {\mathbb{R}}\to {\mathbb{R}}$ with suitable polynomial growth. In the case of dynamic boundary conditions, which model moving contact lines, existence of weak solutions for this system was shown by Gal, Grasselli, and Wu in [@MR3981392]. In the case of non-Newtonian fluids of suitable $p$-growth existence of weak solutions was proved by A. and Breit [@AbelsBreit]. For the case of a non-local Cahn-Hilliard equation and Newtonian fluids the corresponding results was derived by Frigeri in [@MR3540647] and for a model with surfactants by Garcke and the authors in [@MR3845562]. Recently, Giorgini [@GiorginiPreprint] showed in the two-dimensional case well-posedness locally in time for general bounded domains and globally in time under periodic boundary conditions. In [@AbelsDepnerGarcke] it is shown that the first equation is equivalent to $$\begin{aligned} \nonumber \rho \partial_t {\textbf{v}}+ \Big( \rho {\textbf{v}}+ \tfrac{\tilde \rho_1 - \tilde \rho_2}{2} m(\varphi) & \nabla (\tfrac{1}{\varepsilon} W' (\varphi) - \varepsilon \Delta \varphi ) \Big) \cdot \nabla {\textbf{v}}+ \nabla p - {\text{div}}(2 \eta (\varphi) D{\textbf{v}}) \\\label{eq:equivalent1} &= - \varepsilon \Delta \varphi \nabla \varphi .\end{aligned}$$ This reformulation will be useful in our analysis. For the following we assume: \[strong\_solutions\_general\_assumptions\] 1. Let $\Omega \subseteq \mathbb R^d$ be a bounded domain with $C^4$-boundary and $d = 2,3$. 2. Let $\eta,m \in C^4_b (\mathbb R)$ be such that $\eta (s) \geq \eta_0 > 0$ and $m(s)\geq m_0$ for every $s \in \mathbb R$ and some $\eta_0,m_0 > 0$. 3. The density $\rho\colon {\mathbb{R}}\to {\mathbb{R}}$ is given by $$\begin{aligned} \rho = \rho (\varphi ) = \frac{\tilde \rho_1 + \tilde \rho_2}{2} + \frac{\tilde \rho_2 - \tilde \rho_1}{2} \varphi \qquad \text{ for all } \varphi \in \mathbb R .\end{aligned}$$ 4. $W\colon {\mathbb{R}}\to {\mathbb{R}}$ is twice continuously differentiable. With these assumptions we will show our main existence result on short time existence of strong solutions for -:  \[strong\_solution\_existence\_proof\_of\_strong\_solution\]\ Let $\Omega$, $\eta$, $m$, $\rho$ and $W$ be as in Assumption \[strong\_solutions\_general\_assumptions\]. Moreover, let $\bold v_0 \in H^{1}_0 (\Omega)^d \cap L^2_\sigma (\Omega) $ and $\varphi_0 \in (L^p (\Omega) , W^4_{p,N} (\Omega))_{1- \frac{1}{p}, p}$ be given with $|\varphi_0(x)|\leq 1$ for all $x\in\Omega$ and $4 < p < 6$. Then there exists $T > 0$ such that - has a unique strong solution $$\begin{aligned} \bold v & \in W^1_2 (0,T; L^2_\sigma (\Omega)) \cap L^2 (0,T; H^2 (\Omega)^d \cap H^1_0 (\Omega)^d) , \\ \varphi & \in W^1_p (0,T; L^p (\Omega)) \cap L^p (0,T; W^4_{p,N} (\Omega)),\end{aligned}$$ where $W^4_{p,N}(\Omega)=\{u\in W^4_p(\Omega): \partial_n u|_{\partial\Omega}= \partial_n \Delta u|_{\partial\Omega}=0\}$. We will prove this result with the aid of a contraction mapping argument after a suitable reformulation, similar to [@MR2504845]. But for the present system the linearized system is rather different. The structure of this contribution is as follows: In Section \[sec:prelim\] we introduce some basic notation and recall some results used in the following. The main result is proved in Section \[sec:Main\]. For its proof we use suitable estimates of the non-linear terms, which are shown in Section \[sec:Lipschitz\], and a result on maximal $L^2$-regularity of a Stokes-type system, which is shown in Section \[sec:Linear\]. [**Acknowledgements:**]{} The authors acknowledge support by the SPP 1506 “Transport Processes at Fluidic Interfaces” of the German Science Foundation (DFG) through grant GA695/6-1 and GA695/6-2. The results are part of the second author’s PhD-thesis [@Dissertation_Weber]. Preliminaries {#sec:prelim} ============= For an open set $U\subseteq {\mathbb{R}}^d$, $m\in\mathbb{N}_0$ and $1\leq p \leq\infty$ we denote by $W^m_p(U)$ the $L^p$-Sobolev space of order $m$ and $W^m_p(U;X)$ its $X$-valued variant, where $X$ is a Banach space. In particular, $L^p(U)=W^0_p(U)$ and $L^p(U;X)= W^0_p(U;X)$. Moreover, $B^s_{pq}(\Omega)$ denotes the standard Besov space, where $s\in{\mathbb{R}}$, $1\leq p,q\leq \infty$, and $L^2_\sigma(\Omega)$ is the closure of $C^\infty_{0,\sigma}(\Omega)= \{{\textbf{u}}\in C^\infty_0(\Omega)^d: \operatorname{div} {\textbf{u}}=0\}$ in $L^2(\Omega)^d$ and $\mathbb{P}_\sigma\colon L^2(\Omega)^d\to L^2_\sigma(\Omega)$ the orthogonal projection onto it, i.e., the Helmholtz projection. We will frequently use:  \[theorem\_composition\_sobolev\_functions\]\ Let $\Omega\subseteq {\mathbb{R}}^d$ be a bounded domain with $C^1$-boundary, $m,n\in \mathbb{N}$ and let $1\leq p <\infty$ such that $m - dp > 0$. Then for every $f\in C^m({\mathbb{R}}^N)$ and every $R>0$ there exists a constant $C>0$ such that for all $u\in W^m_p(\Omega)^N$ with $\|u\|_{W^m_p(\Omega)^N}\leq R$, we have $f(u)\in W^m_p(\Omega)$ and $\|f(u)\|_{W^m_p(\Omega)}\leq C$. Moreover, if $f\in C^{m+1}({\mathbb{R}}^N)$, then for all $R>0$ there exists a constant $L>0$ such that $$\|f(u)-f(v)\|_{W^m_p(\Omega)}\leq L \|u-v\|_{W^m_p(\Omega)^N}$$ for all $u, v\in W^m_p(\Omega)^N$ with $\|u\|_{W^m_p(\Omega)^N}, \|v\|_{W^m_p(\Omega)^N}\leq R$. The first part follows from [@RunstSickel Chapter 5, Theorem 1 and Lemma]. The second part can be easily reduced to the first part. In particular we have $uv\in W^m_p(\Omega)$ for all $u,v\in W^m_p(\Omega)$ under the assumptions of the theorem. Let $X_0,X_1$ be Banach spaces such that $X_1\hookrightarrow X_0$ densely. It is well known that $$\label{eq:BUCEmbedding} W^1_p(I;X_0) \cap L^p(I;X_1) \hookrightarrow BUC(I;(X_0,X_1)_{1-\frac1p,p}), \qquad 1\leq p <\infty,$$ continuously for $I=[0,T]$, $0<T<\infty$, and $I=[0,\infty)$, cf. Amann [@Amann Chapter III, Theorem 4.10.2]. Here $(X_0,X_1)_{\theta,p}$ denotes the real interpolation space of $(X_0,X_1)$ with exponent $\theta$ and summation index $p$. Moreover, $BUC(I;X)$ is the space of all bounded and uniformly continuous $f\colon I\to X$ equipped with the supremum norm, where $X$ is a Banach space. Moreover, we will use: \[lemma\_hoelder\_sobolev\_interpolation\_to\_hoelder\] Let $X_0 \subseteq Y \subseteq X_1$ be Banach spaces such that $$\begin{aligned} \|x\|_Y \leq C \|x\|^{1 - \theta}_{X_0} \|x\|^\theta_{X_1}\end{aligned}$$ for every $x \in X_0$ and a constant $C > 0$, where $\theta \in (0,1)$. Then $$\begin{aligned} C^{0, \alpha} ([0,T]; X_1) \cap L^\infty (0,T; X_0) \hookrightarrow C^{0, \alpha \theta} ([0,T] ; Y) . \end{aligned}$$ continuously. The result is well-known and can be proved in a straight forward manner. Proof of the Main Result ========================  \[sec:Main\] We prove the existence of a unique strong solution $({\textbf{v}}, \varphi) \in X_T$ for small $T > 0$, where the space $X_T$ will be specified later. The idea for the proof is to linearize the highest order terms in the equations above at the initial data and then to split the equations in a linear and a nonlinear part such that $$\begin{aligned} \mathcal L ({\textbf{v}}, \varphi ) = \mathcal F ({\textbf{v}}, \varphi ) ,\end{aligned}$$ where we still have to specify in which sense this equation has to hold. To linearize it formally at the initial data we replace ${\textbf{v}}$, $p$ and $\varphi$ by ${\textbf{v}}_0 + \varepsilon {\textbf{v}}$, $p_0 + \varepsilon p$ and $\varphi_0 + \varepsilon \varphi$ and then differentiate with respect to $\varepsilon$ at $\varepsilon = 0$. In (\[equation\_strong\_solutions\_1\]) and the equivalent equation , the highest order terms with respect to $t$ and $x$ are $\rho \partial_t {\textbf{v}}$, $ {\text{div}}(2 \eta (\varphi) D{\textbf{v}})$ and $\nabla p$. Hence the linearizations are given by $$\begin{aligned} \frac{ d}{ \mathit{d \varepsilon}} \left ( \rho (\varphi_0 + \varepsilon \varphi ) \partial_t ({\textbf{v}}_0 + \varepsilon {\textbf{v}}) \right )_{| \varepsilon = 0} &= \rho ' (\varphi_0) \varphi \partial_t {\textbf{v}}_0 + \rho (\varphi_0) \partial_t {\textbf{v}}= \rho_0 \partial_t {\textbf{v}}, \\ \frac{d}{\mathit{d \varepsilon}} \left ( {\text{div}}(2 \eta (\varphi_0 + \varepsilon \varphi) D({\textbf{v}}_0 + \varepsilon {\textbf{v}}) ) \right )_{| \varepsilon = 0} &= {\text{div}}(2 \eta ' (\varphi_0) \varphi D{\textbf{v}}_0) + {\text{div}}(2 \eta (\varphi_0) D{\textbf{v}}) , \\ \frac{d}{\mathit{d \varepsilon}} \nabla (p_0 + \varepsilon p)_{| \varepsilon = 0} &= \nabla p ,\end{aligned}$$ where $\rho_0 := \rho (\varphi_0)$ and $\rho_0 ' := \rho ' (\varphi_0)$. Moreover, we omit the term ${\text{div}}(2 \eta ' (\varphi_0) \varphi D{\textbf{v}}_0) $ in the second linearization since it is of lower order. For the last equation we get the linearization $$\begin{aligned} &\frac{d}{d \tilde \varepsilon} {\text{div}}(m (\varphi_0 + \tilde \varepsilon \varphi) \nabla ( - \varepsilon \Delta (\varphi_0 + \tilde \varepsilon \varphi )))_{|\tilde \varepsilon = 0} &\\ &\quad = - \varepsilon {\text{div}}( m ' (\varphi_0) \varphi \nabla \Delta \varphi_0 ) - \varepsilon {\text{div}}( m (\varphi_0) \nabla \Delta \varphi) .\end{aligned}$$ We can omit the first term since it is of lower order. The second term can formally be reformulated as $$\begin{aligned} - \varepsilon {\text{div}}(m (\varphi_0) \nabla \Delta \varphi) = - \varepsilon m' (\varphi_0) \nabla \varphi_0 \cdot \nabla \Delta \varphi - \varepsilon m (\varphi_0) \Delta ( \Delta \varphi ).\end{aligned}$$ Here the first summand is of lower order again. Hence, the linearization is given by $- \varepsilon m (\varphi_0) \Delta ^2 \varphi$ upto terms of lower order. Due to these linearizations we define the linear operator $\mathcal L \colon X_T \rightarrow Y_T$ by $$\begin{aligned} \mathcal L ({\textbf{v}}, \varphi) = \begin{pmatrix} \mathbb P_\sigma ( \rho_0 \partial_t {\textbf{v}}) - \mathbb P_\sigma ( {\text{div}}(2 \eta (\varphi_0) D{\textbf{v}})) \\ \partial_t \varphi + \varepsilon m(\varphi_0) \Delta^2 \varphi \end{pmatrix} ,\end{aligned}$$ where $\mathcal L$ consists of the principal part of the lionization’s, i.e., of the terms of the highest order. Furthermore, we define the nonlinear operator $\mathcal F \colon X_T \rightarrow Y_T$ by $$\begin{aligned} \mathcal F ( {\textbf{v}}, \varphi) = \begin{pmatrix} \mathbb P_\sigma F_1 ({\textbf{v}}, \varphi) \\ - \nabla \varphi \cdot {\textbf{v}}+ {\text{div}}( \tfrac{1}{\varepsilon} m(\varphi) \nabla W' (\varphi)) + \varepsilon m (\varphi_0) \Delta^2 \varphi - \varepsilon {\text{div}}( m (\varphi) \nabla \Delta \varphi) \end{pmatrix} ,\end{aligned}$$ where $$\begin{aligned} F_1 ({\textbf{v}}, \varphi) = ( & \rho_0 - \rho ) \partial_t {\textbf{v}}- {\text{div}}(2 \eta (\varphi_0) D{\textbf{v}}) + {\text{div}}( 2 \eta (\varphi) D{\textbf{v}}) - \varepsilon \Delta \varphi \nabla \varphi \\ & - \left ( \left ( \rho {\textbf{v}}+ \tfrac{\tilde \rho_1 - \tilde \rho_2}{2} m(\varphi) \nabla (\tfrac{1}{\varepsilon} W' (\varphi) - \varepsilon \Delta \varphi ) \right ) \cdot \nabla \ \right ) {\textbf{v}}.\end{aligned}$$ It still remains to define the spaces $X_T$ and $Y_T$. To this end, we set $$\begin{aligned} Z^1_T &:= L^2 (0,T; H^2 (\Omega)^d \cap H^1_0 (\Omega)^d) \cap W^1_2 (0,T; L^2_\sigma (\Omega)) , \\ Z^2_T &:= L^p (0,T; W^4_{p,N} (\Omega)) \cap W^1_p (0,T; L^p (\Omega))\end{aligned}$$ with $4 < p < 6$, where $$\begin{aligned} W^4 _{p,N} (\Omega) := \{ \varphi \in W^4_p (\Omega) | \ \partial_n \varphi = \partial_n (\Delta \varphi ) = 0 \} .\end{aligned}$$ We equip $Z^1_T$ and $Z^2_T$ with the norms $\|\cdot\|_{Z^1_T} '$ and $\|\cdot\|_{Z^2_T} '$ defined by $$\begin{aligned} \|{\textbf{v}}\|_{Z^1_T} ' & := \|{\textbf{v}}'\|_{L^2 (0,T; L^2 (\Omega))} + \|{\textbf{v}}\|_{L^2 (0,T; H^2 (\Omega))} + \|{\textbf{v}}(0)\|_{(L^2 (\Omega), H^2 (\Omega))_{\frac{1}{2}, 2}} , \nonumber \\ \|\varphi\|_{Z^2_T} ' & := \|\varphi '\|_{L^p (0,T; L^p (\Omega))} + \|\varphi\|_{L^p (0,T; W^4_{p,N} (\Omega))} + \|\varphi (0)\|_{(L^p (\Omega), W^4_p (\Omega))_{1 - \frac{1}{p}, p}} .\nonumber \end{aligned}$$ We use these norms since they guarantee that for all embeddings we will study later the embedding constant $C$ does not depend on $T$, cf. Lemma \[lemma\_embedding\_constant\_does\_not\_depend\_on\_T\]. To this end we use: \[lemma\_embedding\_constant\_does\_not\_depend\_on\_T\] Let $0 < T_0 < \infty$ be given and $X_0$, $X_1$ be some Banach spaces such that $X_1 \hookrightarrow X_0$ densely. For every $0 < T < \frac{ T_0}{2}$ we define $$\begin{aligned} X_T := L^p (0,T; X_1) \cap W^1_p (0,T; X_0),\end{aligned}$$ where $1 \leq p < \infty$, equipped with the norm $$\|u\|_{X_T}:= \|u\|_{L^p(0,T;X_1)}+\|u\|_{W^1_p(0,T;X_0)}+\|u(0)\|_{(X_0,X_1)_{1-\frac1p,p}}.$$ Then there exists an extension operator $E : X_T \rightarrow X_{T_0}$ and some constant $C > 0$ independent of $T$ such that $Eu_{|(0,T)} = u$ in $X_T$ and $$\begin{aligned} \|Eu\|_{X_{T_0}} \leq C \|u\|_{X_T}\end{aligned}$$ for every $u \in X_T$ and every $0 < T < \frac{T_0}{2}$. Moreover, there exists a constant $\tilde C (T_0) > 0$ independent of $T$ such that $$\begin{aligned} \|u\|_{BUC ([0,T]; (X_0, X_1)_{1 - \frac{1}{p},p})} \leq \tilde C (T_0) \|u\|_{X_T}\end{aligned}$$ for every $u \in X_T$ and every $0 <T < \frac{ T_0}{2}$. The result is well-known. In the case $u(0)=0$, one can prove the result with the aid of the extension operator defined by $$\begin{aligned} (Eu) (t) := \begin{cases} u(t) & \text{ if } t \in [0,T], \\ u(2T - t) & \text{ if } t \in (T, 2T], \\ 0 & \text{ if } t \in (2T, T_0] . \end{cases}\end{aligned}$$ The case $u(0)\neq 0$ can be easily reduced to the case $u(0)=0$ by substracting a suitable extension of $u_0$ to $[0,\infty)$. We refer to [@Dissertation_Weber Lemma 5.2] for the details. The last preparation before we can start with the existence proof is the definition of the function spaces $X_T := X_T^1 \times X_T^2$ and $Y_T$ by $$\begin{aligned} X_T^1 & := \{ {\textbf{v}}\in Z^1_T | \ {\textbf{v}}_{|t=0} = {\textbf{v}}_0 \} , \\ X_T^2 & := \{ \varphi \in Z^2_T | \ \varphi_{| t = 0} = \varphi_0 \} , \\ Y_T & := Y^1_T \times Y^2_T := L^2 (0,T; L^2_\sigma (\Omega)) \times L^p (0,T; L^p (\Omega)) ,\end{aligned}$$ where $$\begin{aligned} {\textbf{v}}_0 \in (L^2_\sigma (\Omega), H^2 (\Omega)^d \cap H^1_0 (\Omega)^d \cap L^2_\sigma (\Omega))_{\frac{1}{2}, 2} = H^1_0 (\Omega)^d \cap L^2_\sigma (\Omega)\end{aligned}$$ and $$\begin{aligned} \varphi_0 \in (L^p (\Omega), W^4_{p,N} (\Omega) )_{1 - \frac{1}{p}, p}\end{aligned}$$ are the initial values from (\[equation\_strong\_solutions\_inital\_data\_v\]). Note that in the space $X^2_T$ we have to ensure that $\varphi_{|t=0} = \varphi_0 \in [-1,1]$ since we will use this property to show the Lipschitz continuity of $\mathcal F : X_T \rightarrow Y_T$ in Proposition \[strong\_solution\_proposition\_lipschitz\_continuity\_F\]. Moreover, we note that $X_T$ is not a vector space due to the condition $\varphi_{|t=0} = \varphi_0$. It is only an affine linear subspace of $Z_T := Z^1_T \times Z^2_T$. \[strong\_solution\_proposition\_lipschitz\_continuity\_F\] Let the Assumptions \[strong\_solutions\_general\_assumptions\] hold and $\varphi_0$ be given as in Theorem \[strong\_solution\_existence\_proof\_of\_strong\_solution\]. Then there is a constant $C (T,R) > 0$ such that $$\begin{aligned} \label{strong_solution_estimate_lipschitz_continuity_F} \| \mathcal F (\bold v_1 , \varphi_1 ) - \mathcal F (\bold v_2 , \varphi_2) \| _{Y_T} \leq C (T, R) \|(\bold v_1 - \bold v_2, \varphi_1 - \varphi_2 )\|_{X_T}\end{aligned}$$ for all $(\bold v_i, \varphi_i) \in X_T$ with $\|(\bold v_i, \varphi_i)\|_{X_T} \leq R$ and $i = 1,2$. Moreover, it holds $C(T,R) \rightarrow 0$ as $T \rightarrow 0$. The proposition is proved in Section \[sec:Lipschitz\] below. \[thm:linear\] Let $T>0$ and $\mathcal{L}$, $X_T$ and $Y_T$ be defined as before. Then $\mathcal{L}\colon X_T\to Y_T$ is invertible. Moreover, for every $T_0>0$ there is a constant $C(T_0)>0$ such that $$\|\mathcal{L}^{-1}\|_{\mathcal{L}(Y_T,X_T)}\leq C(T_0)\qquad \text{for all }T\in (0,T_0].$$ This theorem is proved in Section \[sec:Linear\] below. *Proof of Theorem \[strong\_solution\_existence\_proof\_of\_strong\_solution\]:* First of all we note that - is equivalent to $$\begin{aligned} &({\textbf{v}}, \varphi) = \mathcal L^{-1} (\mathcal F ({\textbf{v}}, \varphi)) && \text{in } X_T . \label{strong_solution_fix_point_equation2}\end{aligned}$$ The fact that $\mathcal L$ is invertible will be proven later. Equation (\[strong\_solution\_fix\_point\_equation2\]) implies that we have rewritten the system to a fixed-point equation which we want to solve by using the Banach fixed-point theorem. To this end, we consider some $(\tilde {\textbf{v}}, \tilde \varphi) \in X_T$ and define $$\begin{aligned} M := \| \mathcal L^{-1 } \circ \mathcal F (\tilde {\textbf{v}}, \tilde \varphi)\|_{X_T} < \infty .\end{aligned}$$ Now let $R > 0$ be given such that $(\tilde {\textbf{v}}, \tilde \varphi) \in \overline{B_R^{X_T} (0)}$ and $R > 2M$. Then it follows from Proposition \[strong\_solution\_proposition\_lipschitz\_continuity\_F\] that there exists a constant $C = C(T, R) > 0$ such that $$\begin{aligned} \| \mathcal F ({\textbf{v}}_1 , \varphi_1 ) - \mathcal F ({\textbf{v}}_2 , \varphi_2) \| _{Y_T} \leq C (T, R) \|({\textbf{v}}_1, \varphi_1) - ( {\textbf{v}}_2, \varphi_2)\|_{X_T}\end{aligned}$$ for all $({\textbf{v}}_i, \varphi_i) \in X_T$ with $\|({\textbf{v}}_i, \varphi_i)\|_{X_T} \leq R$, $j = 1,2$, where it holds $C(T,R) \rightarrow 0$ as $T \rightarrow 0$. Furthermore, we choose $T$ so small that $$\begin{aligned} \|\mathcal L^{-1}\|_{\mathcal L (Y_T, X_T)} C (T,R) < \frac{1}{2} .\end{aligned}$$ Here we have to ensure that $\|\mathcal L^{-1}\|_{\mathcal L (Y_T, X_T)} $ does not converge to $+ \infty$ as $T \rightarrow 0$. But since Lemma \[strong\_solution\_lemma\_l\_inverse\_t\_t\_0\_firstpart\] and Lemma \[strong\_solution\_lemma\_l\_inverse\_t\_t\_0\_secondpart\] below yield $\|\mathcal L^{-1}\|_{\mathcal L (Y_T, X_T)} < C (T_0)$ for every $0 < T < T_0$ and for a constant that does not depend on $T$, this is not the case and we can choose $T > 0$ in such a way that the previous estimate holds. Note that $T$ depends on $R$ and in general $T$ has to become smaller the larger we choose $R$.\ Since we want to apply the Banach fixed-point theorem on $\overline{B^{X_T}_R (0)} \subseteq X_T$ as we only consider functions $({\textbf{v}}, \varphi) \in X_T$ which satisfy $\|({\textbf{v}}, \varphi)\|_{X_T} \leq R$, we have to show that $\mathcal L^{-1} \circ \mathcal F$ maps from $\overline{B^{X_T}_R (0)}$ to $\overline{B^{X_T}_R (0)}$. From the considerations above we know that there exists $(\tilde {\textbf{v}}, \tilde \varphi) \in \overline{B^{X_T}_R (0)}$ such that $$\begin{aligned} \label{strong_solution_tilde_v0_tilde_phi0_bounded} \|\mathcal L^{-1} \circ \mathcal F (\tilde {\textbf{v}}, \tilde \varphi) \|_{X_T} = M < \frac{R}{2} .\end{aligned}$$ Then a direct calculation shows $$\begin{aligned} \|\mathcal L ^{-1} \circ \mathcal F ({\textbf{v}}, \varphi) \|_{X_T} & \leq \| \mathcal L ^{-1} \circ \mathcal F ( {\textbf{v}}, \varphi) - \mathcal L ^{-1} \circ \mathcal F (\tilde {\textbf{v}}, \tilde \varphi) \|_{X_T} + \| \mathcal L^{-1} \circ \mathcal F (\tilde {\textbf{v}}, \tilde \varphi) \|_{X_T} \\ & < \|\mathcal L^{-1} \|_{\mathcal L (Y_T, X_T)} \| \mathcal F ({\textbf{v}}, \varphi) - \mathcal F (\tilde {\textbf{v}}, \tilde \varphi) \|_{Y_T} + \frac{R}{2} \\ & \leq \|\mathcal L^{-1} \|_{\mathcal L (Y_T, X_T)} C (R,T) \|({\textbf{v}}, \varphi) - (\tilde {\textbf{v}}, \tilde \varphi)\|_{X_T} + \frac{R}{2} < R \end{aligned}$$ for every $({\textbf{v}}, \varphi) \in \overline{B^{X_T}_R (0)}$, where we used the estimate for the Lipschitz continuity of $\mathcal F$. This shows that $\mathcal L ^{-1} \circ \mathcal F ({\textbf{v}}, \varphi)$ is in $ \overline{B^{X_T}_R (0)}$ for every $({\textbf{v}}, \varphi) \in \overline{ B^{X_T} _R (0)}$, i.e., $$\begin{aligned} \mathcal L^{-1 } \circ \mathcal F : \overline{ B^{X_T}_R (0) } \rightarrow \overline{ B^{X_T} _R (0) } .\end{aligned}$$ For applying the Banach fixed-point theorem it remains to show that the mapping $\mathcal L^{-1 } \circ F \colon B^{X_T}_R (0) \rightarrow B^{X_T} _R (0) $ is a contraction. To this end, let $({\textbf{v}}_i, \varphi_i) \in B^{X_T}_R (0)$ be given for $i = 1,2$. Then it holds $$\begin{aligned} \| \mathcal L^{-1} & \circ \mathcal F ({\textbf{v}}_1, \varphi_1) - \mathcal L^{-1} \circ \mathcal F ({\textbf{v}}_2 , \varphi_2) \|_{X_T} \\ & \leq \| \mathcal L^{-1}\|_{\mathcal L (Y_T, X_T)} C(R,T) \|({\textbf{v}}_1, \varphi_1) - ({\textbf{v}}_2, \varphi_2) \|_{X_T} \\ & < \frac{1}{2} \|({\textbf{v}}_1, \varphi_1) - ({\textbf{v}}_2, \varphi_2) \|_{X_T} ,\end{aligned}$$ which shows the statement. Hence, the Banach fixed-point theorem can be applied and yields some $({\textbf{v}}, \varphi) \in \overline{B^{X_T}_R (0)} \subseteq X_T$ such that the fixed-point equation (\[strong\_solution\_fix\_point\_equation2\]) holds, which implies that $({\textbf{v}}, \varphi)$ is a strong solution for the equations -. Finally, in order to show uniqueness in $X_T$, let $(\hat {\textbf{v}}, \hat \varphi)\in X_T$ be another solution. Choose $\hat{R}\geq R$ such that $(\hat {\textbf{v}}, \hat \varphi)\in \overline{B_{\hat{R}}^{X_T}(0)}$. Then by the previous arguments we can find some $\hat{T}\in (0,T]$ such that (\[strong\_solution\_fix\_point\_equation2\]) has a unique solution. This implies $(\hat {\textbf{v}}, \hat \varphi)|_{[0,\hat{T}]}= ({\textbf{v}}, \varphi)|_{[0,\hat{T}]}$. A standard continuation argument shows that the solutions coincide for all $t\in [0,T]$. Lipschitz Continuity of $\mathcal F$ {#sec:Lipschitz} ==================================== Before we continue we study in which Banach spaces ${\textbf{v}}$, $\varphi$, $\nabla \varphi$, $m (\varphi)$ and so on are bounded. Note that in the definition of $X^2_T$, $p$ has to be larger than $4$ because we will need to estimate terms like $\nabla \Delta \varphi \cdot \nabla {\textbf{v}}$, where $p = 2$ is not sufficient for the analysis and therefore we need to choose $p > 2$. But for most terms in the analysis $p=2$ would be sufficient and $4 < p < 6$ would not be necessary. Nevertheless, for consistency all calculations are done for the case $4 < p < 6$. Due to it holds $$\begin{aligned} \label{strong_solution_v_buc_h1} {\textbf{v}}\in X^1_T \hookrightarrow BUC ([0,T]; B^1_{22} (\Omega)) = BUC ([0,T]; H^1 (\Omega)) ,\end{aligned}$$ where we used $B^s_{22} (\Omega) = H^s_2 (\Omega)$ for every $s \in \mathbb R$. In particular this implies $$\begin{aligned} &\nabla {\textbf{v}}\in L^\infty (0,T; L^2 (\Omega)) \cap L^2 (0,T; L^6 (\Omega)) \hookrightarrow L^{\frac{8}{3}} (0,T; L^4 (\Omega)) , \label{strong_solution_nabla_v_l83_l4} \\ & \nabla {\textbf{v}}\in L^\infty (0,T; L^2 (\Omega)) \cap L^2 (0,T; L^6 (\Omega)) \hookrightarrow L^4 (0,T; L^3 (\Omega)), \label{strong_solution_nabla_v_l4_l3}.\end{aligned}$$ Let $\varphi \in X^2_T$ be given. From it follows $$\begin{aligned} \label{strong_solution_varphi_buc_W4-4p_p} \varphi \in L^p (0,T; W^4_{p,N} (\Omega) ) \cap W^1_p (0,T; L^p (\Omega)) \hookrightarrow BUC ([0,T]; W^{4 - \frac{4}{p}}_p (\Omega)) .\end{aligned}$$ This implies $$\begin{aligned} \label{strong_solution_nabla_delta_phi} \nabla \Delta \varphi \in BUC ([0,T]; W^{1 - \frac{4}{p}}_p (\Omega)) \end{aligned}$$ since $p > 4$. Note that when we write “$\varphi$ is bounded in $Z$" for some function space $Z$, we mean that the set of all functions $\{\varphi \in X_T^2 : \ \|\varphi\|_{X^2_T} \leq R\}$ is bounded in $Z$ in such a way that the upper bound only depends on $R$ and not on $T$, i.e., there exists $C(R) > 0$ such that $\|\varphi\|_{Z} \leq C(R)$ for every $\varphi \in X_T^2$ with $\|\varphi\|_{X^2_T} \leq R$. First of all, we have $$\begin{aligned} \varphi \in W^1_p (0,T; L^p (\Omega)) \hookrightarrow C^{0, 1 - \frac{1}{p}} ([0,T]; L^p (\Omega)) .\end{aligned}$$ Moreover, we already know $\varphi \in BUC([0,T]; W^{4 - \frac{4}{p}}_p (\Omega))$ and we have $$\begin{aligned} (B^{4 - \frac{4}{p}}_{pp} (\Omega), L^p(\Omega))_{\theta, 2} = B^3_{p2} (\Omega) \hookrightarrow W^3_p (\Omega) \end{aligned}$$ together with the estimate $$\begin{aligned} \|\varphi (t)\|_{W^3_p (\Omega)} \leq C \|\varphi (t)\|^{1 - \theta}_{W^{4 - \frac{4}{p}}_p (\Omega)} \|\varphi (t)\|^\theta_{L^p (\Omega)} \end{aligned}$$ for every $t \in [0,T]$. Hence, Lemma \[lemma\_hoelder\_sobolev\_interpolation\_to\_hoelder\] implies $$\begin{aligned} \nonumber \varphi \in &C^{0, 1 - \frac{1}{p}} ([0,T]; L^p (\Omega)) \cap C([0,T]; W^{4 - \frac{4}{p}}_p (\Omega))\\\label{phi_hoelder_continuous_in_time_w3p} &\hookrightarrow C^{0, (1 - \frac{1}{p}) \theta} ([0,T]; W^3_p (\Omega)) .\end{aligned}$$ Because of $W^3_p (\Omega) \hookrightarrow C^2 (\overline \Omega)$ for $d=2,3$ due to $4<p<6$, we obtain that $$\begin{aligned} \label{phi_continuous_in_time_twice_in_omega} \varphi &\text{ is bounded in } C([0,T]; C^2 (\overline \Omega)) .\end{aligned}$$ In the nonlinear operator $\mathcal F \colon X_T \rightarrow Y_T$ the terms $\eta (\varphi)$, $\eta (\varphi_0)$, $m (\varphi)$, $m (\varphi_0)$ and $W' (\varphi)$ appear. Hence, we need to know in which spaces these terms are bounded in the sense that there is a constant $C(R) > 0$, which does not depend on $T$, such that the norms of these terms in a certain Banach space are bounded by $C(R)$ for every $({\textbf{v}}, \varphi) \in X_T$ with $\|({\textbf{v}}, \varphi)\|_{X_T} \leq R$. Due to (\[phi\_hoelder\_continuous\_in\_time\_w3p\]) and because the embedding constant only depends on $R$, it holds $$\begin{aligned} \|\varphi (t)\|_{W^3_p (\Omega)} \leq C(R)\end{aligned}$$ for every $t \in [0,T]$ and $\varphi \in X_T^2$ with $\|\varphi\|_{X^2_T} \leq R$. Hence Theorem \[theorem\_composition\_sobolev\_functions\] yields $$\begin{aligned} \|f(\varphi (t))\|_{W^3_p (\Omega)}, \|f (\varphi_0)\|_{W^3_p (\Omega)}, \|W' (\varphi (t))\|_{W^3_p (\Omega)} \leq C(R)\end{aligned}$$ for every $t \in [0,T]$ and every $\varphi \in X^2_T$ with $\|\varphi\|_{X^2_T} \leq R$, where $ f \in \{\eta, m\}$. Thus $$\begin{aligned} \label{f_of_phi_bounded_in_infty_w3p} f (\varphi), f(\varphi_0), W' (\varphi) \text{ are bounded in } L^\infty (0,T; W^3_p (\Omega)) \text{ for } f \in \{\eta, m\} .\end{aligned}$$ Moreover, Theorem \[theorem\_composition\_sobolev\_functions\] yields the existence of $L > 0$ such that $$\begin{aligned} \label{lipschitz_f_of_phi_w3p} \|f (\varphi_1 (t)) - f (\varphi_2 (t))\|_{W^3_p (\Omega)} & \leq L \|\varphi_1 (t) - \varphi_2 (t)\|_{W^3_p (\Omega)} \end{aligned}$$ for every $t \in [0,T]$, $\varphi_1, \varphi_2 \in X^2_T$ and $f \in \{ \eta, m , W' \}$. In the next step, we want to show that $f(\varphi)$ is bounded in $X_T^2$ and therefore the same embeddings hold as for $\varphi$, where $f \in \{\eta, m , W'\}$. Note that from now on until the end of the proof of the interpolation result for $f(\varphi)$, we always use some general $f \in C^4_b (\mathbb R)$. But all these embeddings are valid for $f \in \{\eta, m, W'\}$. We want to prove that if it holds $\varphi \in X^2_T$ with $\|\varphi\|_{X^2_T} \leq R$, then there exists a constant $C(R) > 0$ such that $\|f(\varphi)\|_{X^2_T} \leq C(R)$. To this end, let $\varphi \in X^2_T$ be given with $\|\varphi\|_{X^2_T} \leq R$. Since we already know $\varphi \in C([0,T]; C^2 (\overline \Omega))$, cf. (\[phi\_continuous\_in\_time\_twice\_in\_omega\]), we can conclude $$\begin{aligned} \|\varphi (t)\|_{C^2 (\overline \Omega)} \leq C (R)\end{aligned}$$ for all $t \in [0,T]$. Hence, it holds $f (\varphi (t)) \in C^2 (\overline \Omega)$ for every $t \in [0,T]$ and $$\begin{aligned} \nabla f ( (\varphi (t)) = f' (\varphi (t)) \nabla \varphi (t) .\end{aligned}$$ Due to (\[f\_of\_phi\_bounded\_in\_infty\_w3p\]), $f ' (\varphi)$ is bounded in $L^\infty (0,T; W^3_p (\Omega))$. In particular, this implies $\|f' (\varphi (t))\|_{W^3_p (\Omega)} \leq C(R)$ for a.e. $t \in (0,T)$ and a constant $C(R) > 0$. Since it holds $\varphi \in L^p (0,T; W^4_p (\Omega))$, it follows $\nabla \varphi (t) \in W^3_p (\Omega)$ for a.e. $t \in (0,T)$. Theorem \[theorem\_composition\_sobolev\_functions\] yields $f' (\varphi (t)) \nabla \varphi (t) \in W^3_p (\Omega)$ for a.e. $t \in (0,T)$ together with the estimate $$\begin{aligned} \|\nabla f (\varphi (t))\|_{W^3_p (\Omega)} = \|f' (\varphi (t)) \nabla \varphi (t)\|_{W^3_p (\Omega)} \leq C \|f ' (\varphi (t))\|_{W^3_p (\Omega)} \|\nabla \varphi (t)\|_{W^3_p (\Omega)}\end{aligned}$$ for a.e. $t \in (0,T)$ and every $\varphi \in X^2_T$ with $\|\varphi\|_{X^2_T} \leq R$. Since $f' (\varphi)$ is bounded in $L^\infty (0,T; W^3_p (\Omega))$ and $\nabla \varphi$ is bounded in $L^p (0,T; W^3_p (\Omega))$, the estimate above implies the boundedness of $\nabla f (\varphi)$ in $L^p (0,T; W^3_p (\Omega))$, i.e., there exists $C(R) > 0$ such that $$\begin{aligned} \|\nabla f (\varphi)\|_{L^p (0,T; W^3_p (\Omega))} \leq C (R) \qquad \text{ for all } \varphi \in X^2_T \text{ with } \|\varphi\|_{X^2_T} \leq R. \end{aligned}$$ Altogether this implies that $$\begin{aligned} f(\varphi) \text{ is bounded in } L^p (0,T; W^4_p (\Omega)) .\end{aligned}$$ Analogously we can conclude from the boundedness of $\varphi $ in $W^1_p (0,T; L^p (\Omega))$ that $f (\varphi) $ is also bounded in $W^1_p (0,T; L^p (\Omega))$ because of $ \frac{d}{dt} f (\varphi (t)) = f' (\varphi (t)) \partial_t \varphi (t), $ where $f' (\varphi ) $ is bounded in $C^0 (\overline Q_T) $. Thus the same interpolation result holds as in (\[phi\_hoelder\_continuous\_in\_time\_w3p\]), i.e., $$\begin{aligned} \label{interpolation_f_of_phi_hoelder_continuous_and_w3p} f(\varphi) \text{ is bounded in } C^{0, ( 1 - \frac{1}{p} ) \theta} ([0,T]; W^3_p (\Omega)) ,\end{aligned}$$ where $\theta := \frac{\frac{4}{p} - 1}{ \frac{4}{p} - 4}$. *Proof of Proposition \[strong\_solution\_proposition\_lipschitz\_continuity\_F\]:* Let $({\textbf{v}}_i, \varphi_i ) \in X_T$ with $\|({\textbf{v}}_i, \varphi_i)\|_{X_T} \leq R$, $i = 1,2$, be given. Then it holds $$\begin{aligned} \label{equation_F_lipschitz} &\| \mathcal F ({\textbf{v}}_1 , \varphi_1 ) - \mathcal F ({\textbf{v}}_2 , \varphi_2) \| _{Y_T} = \| \mathbb P_\sigma ( F_1 ({\textbf{v}}_1, \varphi_1 ) - F_1 ({\textbf{v}}_2, \varphi_2)) \|_{L^2 (Q_T)} \nonumber \\ & \ \ + \|(\nabla \varphi_2 \cdot {\textbf{v}}_2 - \nabla \varphi_1 \cdot {\textbf{v}}_1 ) + \tfrac{1}{\varepsilon} {\text{div}}( m (\varphi_1) \nabla W' (\varphi_1) - m (\varphi_2) \nabla W' (\varphi_2)) \nonumber \\ & \ \ + \varepsilon m (\varphi_0) \Delta^2 (\varphi_1 - \varphi_2) + \varepsilon {\text{div}}( m (\varphi_2) \nabla \Delta \varphi_2 - m (\varphi_1) \nabla \Delta \varphi_2)\|_{L^p (Q_T)} .\end{aligned}$$ For the sake of clarity we study both summands in (\[equation\_F\_lipschitz\]) separately and begin with the first one. Recall that the operator $F_1$ is defined by $$\begin{aligned} F_1 ({\textbf{v}}, \varphi) = & \rho_0 \partial_t {\textbf{v}}- \rho \partial_t {\textbf{v}}- {\text{div}}(2 \eta (\varphi_0) D{\textbf{v}}) + {\text{div}}( 2 \eta (\varphi) D{\textbf{v}}) - \varepsilon \Delta \varphi \nabla \varphi \\ & - \left ( \left ( \rho {\textbf{v}}+ \tfrac{\tilde \rho_1 - \tilde \rho_2}{2} m(\varphi) \nabla (\tfrac{1}{\varepsilon} W' (\varphi) - \varepsilon \Delta \varphi ) \right ) \cdot \nabla \ \right ) {\textbf{v}}\end{aligned}$$ and that it holds $\|\mathbb P_\sigma \|_{\mathcal L (L^2 (\Omega)^d, L^2_\sigma (\Omega))} \leq 1$ for the Helmholtz projection $\mathbb P_\sigma$. We estimate $ \| \mathbb P_\sigma ( F_1 ({\textbf{v}}_1, \varphi_1 ) - F_1 ({\textbf{v}}_2, \varphi_2 ) )\|_{L^2 (Q_T)}$: For the first two terms we can calculate $$\begin{aligned} \|\rho_0 & \partial_t {\textbf{v}}_1 - \rho (\varphi_1) \partial_t {\textbf{v}}_1 - \rho_0 \partial_t {\textbf{v}}_2 + \rho (\varphi_2) \partial_t {\textbf{v}}_2 \|_{L^2(Q_T)} \\ & \leq \| (\rho_0 - \rho (\varphi_1)) \partial_t ({\textbf{v}}_1 - {\textbf{v}}_2) \|_{L^2 (Q_T)} + \| ( \rho (\varphi_1) - \rho (\varphi_2)) \partial_t {\textbf{v}}_2 \|_{L^2 (Q_T)} .\end{aligned}$$ Since it holds $\partial_t {\textbf{v}}_i \in L^2 (0,T; L^2_\sigma (\Omega))$, $i = 1,2$, we need to estimate every $\rho$-term in the $L^\infty$-norm. To this end, we use that $\rho$ is affine linear and $$\begin{aligned} \varphi_i \text{ is bounded in } C^{0, (1 -\frac{1}{p})\theta} ([0,T]; W^3_p (\Omega)) \hookrightarrow C^{0, (1 -\frac{1}{p})\theta} ([0,T]; C^2 ( \overline \Omega)) \end{aligned}$$ for $i = 1,2$ and $\theta = \frac{\frac{4}{p}-1}{\frac{4}{p}-4}$, cf. (\[phi\_hoelder\_continuous\_in\_time\_w3p\]). Then we obtain for the first summand $$\begin{aligned} \| (\rho_0 - \rho (\varphi_1)) \partial_t ({\textbf{v}}_1 - {\textbf{v}}_2) \|_{L^2 (Q_T)} & \leq \|\rho (\varphi_0) - \rho (\varphi_1)\|_{L^\infty (Q_T)} \|\partial_t ({\textbf{v}}_1 - {\textbf{v}}_2)\|_{L^2 (Q_T)} \\ & \leq C \underset{t \in [0,T]}{\sup} \|\varphi_1 (0) - \varphi_1 (t)\|_{L^\infty (\Omega)} \|{\textbf{v}}_1 - {\textbf{v}}_2\|_{X^1_T} \\ & \leq C T^{ (1 - \frac{1}{p} ) \theta} \|\varphi_1\|_{C^{0, (1 - \frac{1}{p}) \theta} ([0,T]; C^2 ( \overline \Omega))} \|{\textbf{v}}_1 - {\textbf{v}}_2\|_{X^1_T} \\ & \leq C R T^{ (1 - \frac{1}{p} ) \theta} \|{\textbf{v}}_1 - {\textbf{v}}_2\|_{X^1_T} .\end{aligned}$$ Analogously the second term can be estimated by $$\begin{aligned} \| & ( \rho (\varphi_1) - \rho (\varphi_2)) \partial_t {\textbf{v}}_2 \|_{L^2 (Q_T)} \leq \|\rho (\varphi_1) - \rho (\varphi_2)\|_{L^\infty (Q_T)} \|{\textbf{v}}_2\|_{X^1_T} \\ & \leq C \underset{t \in [0,T]}{\sup} \| (\varphi_1(t) - \varphi_2 (t)) - (\varphi_1 (0) - \varphi_2 (0))\|_{L^\infty (\Omega)} \|{\textbf{v}}_2\|_{X^1_T} \\ & \leq C R T^{(1 - \frac{1}{p}) \theta} \|\varphi_1 - \varphi_2\|_{C^{0, (1 - \frac{1}{p}) \theta} ([0,T]; C^2 ( \overline \Omega))} \\ & \leq C R T^{ (1 - \frac{1}{p}) \theta} \|\varphi_1 - \varphi_2\|_{X^2_T} .\end{aligned}$$ Here we used the fact that $\varphi_1 (0) = \varphi_0 = \varphi_2 (0)$ for $\varphi_i \in X^2_T$, $i = 1,2$. The next term of $ \| \mathbb P_\sigma ( F_1 ({\textbf{v}}_1, \varphi_1 ) - F_1 ({\textbf{v}}_2, \varphi_2 ) )\|_{L^2 (Q_T)}$ is given by $$\begin{aligned} |&| ( {\text{div}}(2 \eta (\varphi_0) D{\textbf{v}}_2) - {\text{div}}(2 \eta (\varphi_0) D{\textbf{v}}_1 ) ) + ( {\text{div}}(2 \eta (\varphi_1) D{\textbf{v}}_1) - {\text{div}}(2 \eta (\varphi_2) D{\textbf{v}}_2 ) ) \|_{Y^1_T} \\ & \leq \| {\text{div}}(2 (\eta (\varphi_0) - \eta (\varphi_1) ) ( D{\textbf{v}}_2 - D{\textbf{v}}_1 ) ) \|_{Y^1_T} + \| {\text{div}}(2 ((\eta (\varphi_1) - \eta (\varphi_2)) D{\textbf{v}}_2 ) ) \|_{Y^1_T} .\end{aligned}$$ In the next step we apply the divergence on the $\eta(\varphi_i)$- and $D{\textbf{v}}_i$-terms and for the sake of clarity we study both terms in the previous inequality separately. For the first one we use $\eta (\varphi) \in C^{0, (1 -\frac{1}{p}) \theta} ([0,T]; W^3_p (\Omega))$ with $\theta = \frac{\frac{4}{p}-1}{\frac{4}{p} - 4}$, cf. (\[interpolation\_f\_of\_phi\_hoelder\_continuous\_and\_w3p\]), to obtain $$\begin{aligned} \| & {\text{div}}(2 (\eta (\varphi_0) - \eta (\varphi_1) ) ( D{\textbf{v}}_2 - D{\textbf{v}}_1 ) ) \|_{Y^1_T} \\ & \leq \| 2 \nabla (\eta (\varphi_0) - \eta (\varphi_1 ) ) \cdot (D{\textbf{v}}_2 - D{\textbf{v}}_1) \|_{Y^1_T} + \| 2 (\eta (\varphi_0) - \eta (\varphi_1)) \Delta ({\textbf{v}}_2 - {\textbf{v}}_1 ) \|_{Y^1_T} \\ & \leq C \underset{t \in [0,T]}{\sup} \| \nabla \eta (\varphi_1 (0)) - \nabla \eta (\varphi_1 (t)) \|_{C^1 ( \overline \Omega)} \| D{\textbf{v}}_2 - D{\textbf{v}}_1 \|_{L^2 (0,T; H^1 (\Omega))} \\ & \ \ \ + C \underset{t \in (0,T)}{\sup} \| \eta (\varphi_1 (0)) - \eta (\varphi_1 (t)) \|_{C^2 ( \overline \Omega)} \| \Delta ({\textbf{v}}_2 - {\textbf{v}}_1) \|_{L^2 (0,T; L^2 (\Omega))} \\ & \leq C T^{ (1 - \frac{1}{p}) \theta} \|\nabla \eta (\varphi_1)\|_{C^{0, (1 - \frac{1}{p}) \theta } ([0,T]; W^2_p (\Omega))} \|{\textbf{v}}_1 - {\textbf{v}}_2\|_{X^1_T} \\ & \ \ \ + C T^{ (1 - \frac{1}{p}) \theta} \|\eta (\varphi_1)\|_{C^{0, (1 - \frac{1}{p}) \theta} ([0,T]; W^3_p (\Omega))} \|{\textbf{v}}_1 - {\textbf{v}}_2\|_{X^1_T} \\ & \leq C R \left ( T^{ (1 - \frac{1}{p}) \theta}+T^{ (1 - \frac{1}{p}) \theta} \right ) \|{\textbf{v}}_1 - {\textbf{v}}_2\|_{X^1_T} .\end{aligned}$$ Analogously as before we can estimate the second summand by $$\begin{aligned} \| & {\text{div}}(2 ((\eta (\varphi_1) - \eta (\varphi_2)) D{\textbf{v}}_2 ) ) \|_{Y^1_T} \\ & \leq 2 \| \eta ' (\varphi_1) ( \nabla \varphi_1 - \nabla \varphi_2) \cdot D{\textbf{v}}_2 \|_{Y^1_T} + 2 \| ( \eta ' (\varphi_1) - \eta ' (\varphi_2) ) \nabla \varphi_2 \cdot D{\textbf{v}}_2 \|_{Y^1_T} \\ & \ \ \ + 2 \|( \eta (\varphi_1) - \eta (\varphi_2) ) \Delta {\textbf{v}}_2\|_{Y^1_T} .\end{aligned}$$ For the sake of clarity we study these three terms separately again. Firstly, $$\begin{aligned} \|& \eta ' (\varphi_1) ( \nabla \varphi_1 - \nabla \varphi_2) \cdot D{\textbf{v}}_2 \|_{Y^1_T} \leq C (R) \left | \left | \|D{\textbf{v}}_2\|_{L^2 (\Omega)} \|\nabla \varphi_1 - \nabla \varphi_2\|_{C^1 (\overline \Omega)} \right | \right | _{L^2 (0,T)} \\ & \leq C (R) \underset{t \in [0,T]}{\sup} \|\nabla (\varphi_1 (t) - \varphi_2 (t)) - \nabla (\varphi_1 (0) - \varphi_2 (0))\|_{C^1 (\overline \Omega)} \|D{\textbf{v}}_2\|_{L^2 (0,T; L^2 (\Omega))} \\ & \leq C (R) T^{(1 - \frac{1}{p}) \theta} \|\nabla \varphi_1 - \nabla \varphi_2\|_{C^{(1 - \frac{1}{p}) \theta} ([0,T]; W^2_p (\Omega))} \|{\textbf{v}}_2\|_{X^1_T} \\ & \leq C(R) T^{(1 - \frac{1}{p}) \theta} \|\varphi_1 - \varphi_2\|_{X^2_T} ,\end{aligned}$$ where we used in the first step that $\eta ' (\varphi) $ is bounded in $ C([0,T]; C^2 (\overline \Omega))$. Furthermore, (\[lipschitz\_f\_of\_phi\_w3p\]) together with $$\begin{aligned} \varphi \in C^{0, (1 - \frac{1}{p}) \theta} ([0,T]; W^3_p (\Omega)) \hookrightarrow C([0,T]; C^2 (\overline \Omega))\end{aligned}$$ implies $$\begin{aligned} \| & ( \eta ' (\varphi_1) - \eta ' (\varphi_2) ) \nabla \varphi_2 \cdot D{\textbf{v}}_2 \|_{Y^1_T} \\ & \leq \underset{t \in [0,T]}{\sup}\| \eta ' (\varphi_1) - \eta ' (\varphi_2) \|_{W^3_p (\Omega)} \|\nabla \varphi_2\|_{C([0,T];C^1 (\overline \Omega))} \|D{\textbf{v}}_2\|_{L^2 (Q_T)} \\ & \leq C (R) \underset{t \in [0,T]}{\sup} \|\varphi_1 (t) - \varphi_2 (t) \|_{W^3_p (\Omega)} \leq C (R) T^{(1- \frac{1}{p}) \theta} \|\varphi_1 - \varphi_2\|_{X^2_T} \end{aligned}$$ since $\varphi_1(0)-\varphi_2(0)=0$. Analogously to the second summand we can estimate the third one by $$\begin{aligned} \|( \eta (\varphi_1) - \eta (\varphi_2) ) \Delta {\textbf{v}}_2\|_{Y_T} \leq C(R) T^{(1 - \frac{1}{p}) \theta} \|\varphi_1 - \varphi_2\|_{X^2_T} ,\end{aligned}$$ which shows the statement for the second term. For the third term we obtain $$\begin{aligned} \|\rho & (\varphi_2) {\textbf{v}}_2 \cdot \nabla {\textbf{v}}_2 - \rho(\varphi_1) {\textbf{v}}_1 \cdot \nabla {\textbf{v}}_1\|_{Y^1_T} \\ & \leq \|( \rho(\varphi_2) - \rho (\varphi_1)) {\textbf{v}}_2 \cdot \nabla {\textbf{v}}_2 \|_{Y^1_T} + \| \rho(\varphi_1) ({\textbf{v}}_2 - {\textbf{v}}_1) \cdot \nabla {\textbf{v}}_2 \|_{Y^1_T} \\ & \ \ \ + \| \rho (\varphi_1) {\textbf{v}}_1 \cdot (\nabla {\textbf{v}}_2 - \nabla {\textbf{v}}_1 ))\|_{Y^1_T} .\end{aligned}$$ We estimate these three terms separately again. For the first term we use that ${\textbf{v}}_2$ is bounded in $L^\infty (0,T; L^6 (\Omega))$, cf. (\[strong\_solution\_v\_buc\_h1\]), and $\nabla {\textbf{v}}_2$ is bounded in $L^2 (0,T; L^6 (\Omega))$ together with (\[lipschitz\_f\_of\_phi\_w3p\]). Thus $$\begin{aligned} \| & ( \rho(\varphi_2) - \rho (\varphi_1)) {\textbf{v}}_2 \cdot \nabla {\textbf{v}}_2 \|_{Y^1_T} \\ & \leq C (R) T^{(1- \frac{1}{p}) \theta} \|\varphi_2 - \varphi_1 \|_{C^{0, (1 - \frac{1}{p}) \theta} ([0,T]; W^3_p (\Omega))} \|{\textbf{v}}_2\|_{L^\infty (0,T; L^6 (\Omega))} \|\nabla {\textbf{v}}_2\|_{L^2 (0,T; L^6 (\Omega))} \\ & \leq C(R) T^{(1 - \frac{1}{p} ) \theta} \|\varphi_2 - \varphi_1\|_{X^1_T}.\end{aligned}$$ For the second term we use $\rho (\varphi_1) \in C([0,T]; C^2 (\overline \Omega))$, ${\textbf{v}}_i \in L^\infty (0,T; L^6 (\Omega))$ and $\nabla {\textbf{v}}_2 \in L^4 (0,T; L^3 (\Omega))$, cf. (\[strong\_solution\_v\_buc\_h1\]) and (\[strong\_solution\_nabla\_v\_l4\_l3\]), $i = 1,2$. Hence, $$\begin{aligned} \| \rho(\varphi_1) ({\textbf{v}}_2 - {\textbf{v}}_1) \cdot \nabla {\textbf{v}}_2 \|_{Y^1_T} & \leq C (R) T^\frac{1}{4} \|{\textbf{v}}_1 - {\textbf{v}}_2\|_{L^\infty (0,T; L^6 (\Omega))} \| \nabla {\textbf{v}}_2 \|_{L^4 (0,T; L^3 (\Omega))} \\ & \leq C (R) T^\frac{1}{4} \|{\textbf{v}}_1 - {\textbf{v}}_2\|_{X^1_T}.\end{aligned}$$ For the third term we use the same function spaces. This implies $$\begin{aligned} \| \rho (\varphi_1) {\textbf{v}}_1 \cdot (\nabla {\textbf{v}}_2 - \nabla {\textbf{v}}_1 ))\|_{Y_T} &\leq C (R) T^{\frac{1}{4}} \|\nabla {\textbf{v}}_1 - \nabla {\textbf{v}}_2\|_{L^4 (0,T; L^3 (\Omega))} \\ & \leq C (R) T^{\frac{1}{4}} \|{\textbf{v}}_1 - {\textbf{v}}_2\|_{X^1_T} .\end{aligned}$$ Since $ \frac{\tilde \rho_1 - \tilde \rho_2}{2} $ is a constant, we obtain $$\begin{aligned} & \left | \left | \tfrac{\tilde \rho_1 - \tilde \rho_2}{2} m(\varphi_1) \nabla ( \Delta \varphi_1 ) \cdot \nabla {\textbf{v}}_1 - \tfrac{\tilde \rho_1 - \tilde \rho_2}{2} m(\varphi_2) \nabla ( \Delta \varphi_2 ) \cdot \nabla {\textbf{v}}_2 \right | \right |_{Y^1_T} \\ & \ \ \ \leq C \left ( \| m(\varphi_1) \nabla ( \Delta \varphi_1 ) \cdot ( \nabla {\textbf{v}}_1 - \nabla {\textbf{v}}_2 ) \|_{Y^1_T} \right . \\ & \ \ \ \ \ \ + \| m(\varphi_1) ( \nabla (\Delta \varphi_1) - \nabla (\Delta \varphi_2)) \cdot \nabla {\textbf{v}}_2 \|_{Y^1_T} \\ & \ \ \ \ \ \ + \left . \| (m(\varphi_1) - m (\varphi_2) )\nabla (\Delta \varphi_2) \cdot \nabla {\textbf{v}}_2 \|_{Y^1_T} \right ) .\end{aligned}$$ For the sake of clarity we study all three terms separately again. In the following we use $\nabla \Delta \varphi_i \in L^\infty (0,T; L^4 (\Omega))$, cf. (\[phi\_hoelder\_continuous\_in\_time\_w3p\]), $\nabla {\textbf{v}}_i \in L^{\frac{8}{3}} (0,T; L^4 (\Omega))$, cf. (\[strong\_solution\_nabla\_v\_l83\_l4\]), for $i = 1,2$, and $m (\varphi_1) \in C([0,T];C^2 (\overline \Omega))$. Altogether this implies $$\begin{aligned} \| & m(\varphi_1) \nabla ( \Delta \varphi_1 ) \cdot ( \nabla {\textbf{v}}_1 - \nabla {\textbf{v}}_2 ) \|_{Y^1_T} \\ &\leq C T^\frac{1}{8} \|\nabla \Delta \varphi_1\|_{L^\infty (0,T; L^4 (\Omega))} \|\nabla {\textbf{v}}_1 - \nabla {\textbf{v}}_2\|_{L^{\frac{8}{3}} (0,T; L^4 (\Omega))} \\ & \leq C (R) T^\frac{1}{8} \| {\textbf{v}}_1 - {\textbf{v}}_2\|_{X^1_T} .\end{aligned}$$ Analogously the second summand yields $$\begin{aligned} \| m(\varphi_1) ( \nabla (\Delta \varphi_1) - \nabla (\Delta \varphi_2)) \cdot \nabla {\textbf{v}}_2 \|_{Y^1_T} & \leq C (R) T^\frac{1}{8} \|\varphi_1 - \varphi_2\|_{X^2_T} .\end{aligned}$$ For the last term we use $m (\varphi_i) \in C^{0, (1 - \frac{1}{p}) \theta} ([0,T]; W^3_p (\Omega)) \hookrightarrow C^0 ([0,T]; C^2 (\overline \Omega))$ together with (\[lipschitz\_f\_of\_phi\_w3p\]) and obtain $$\begin{aligned} |&| (m(\varphi_1) - m (\varphi_2) )\nabla (\Delta \varphi_2) \cdot \nabla {\textbf{v}}_2 \|_{Y^1_T} \\ & \leq C(R ) T^\frac{1}{8} \|\varphi_1 (t) - \varphi_2 (t)\|_{C^0([0,T]; C^2 (\overline \Omega))} \|\nabla \Delta \varphi_2\|_{L^\infty (0,T; L^4 (\Omega)} \|\nabla {\textbf{v}}_2\|_{L^\frac{8}{3} (0,T; L^4 (\Omega))} \\ & \leq C(R) T^{\frac{1}{8} } \|\varphi_1 - \varphi_2\|_{X^2_T} .\end{aligned}$$ The next term has the same structure as the one before and can be estimated as $$\begin{aligned} \label{strong_solution_nabla_w_prime} & \left | \left | \tfrac{\tilde \rho_1 - \tilde \rho_2}{2} m(\varphi_1) \nabla ( W' (\varphi_1) ) \cdot \nabla {\textbf{v}}_1 - \tfrac{\tilde \rho_1 - \tilde \rho_2}{2} m(\varphi_2) \nabla ( W' (\varphi_2) ) \cdot \nabla {\textbf{v}}_2 \right | \right |_{Y^1_T} \nonumber \\ & \leq C \left ( \| m(\varphi_1) \nabla W' ( \varphi_1) \cdot ( \nabla {\textbf{v}}_1 - \nabla {\textbf{v}}_2 ) \|_{Y^1_T} \right . \nonumber \\ & \ \ \ + \| m(\varphi_1) ( \nabla W'( \varphi_1 ) - \nabla W' ( \varphi_2)) \cdot \nabla {\textbf{v}}_2 \|_{Y^1_T} \nonumber \\ & \ \ \ + \left . \| (m(\varphi_1) - m (\varphi_2) )\nabla W' (\varphi_2 ) \cdot \nabla {\textbf{v}}_2 \|_{Y^1_T} \right ) .\end{aligned}$$ For $\nabla {\textbf{v}}_i$, $i = 1,2$, we use its boundedness in $L^4 (0,T; L^3 (\Omega))$, cf. (\[strong\_solution\_nabla\_v\_l4\_l3\]). Moreover, we know $\nabla W' (\varphi) \in C([0,T]; W^{3 - \frac{4}{p}}_p (\Omega))$ and $m ( \varphi) \in C([0,T]; C^2 (\overline \Omega))$ for $\varphi \in B_R^{X^2_T}$. Using all these bounds we can estimate the three terms in (\[strong\_solution\_nabla\_w\_prime\]) separately. For the first term we obtain $$\begin{aligned} \| m(\varphi_1) \nabla W' ( \varphi_1) \cdot ( \nabla {\textbf{v}}_1 - \nabla {\textbf{v}}_2 ) \|_{Y^1_T} & \leq C (R) T^\frac{1}{4} \|\nabla {\textbf{v}}_1 - \nabla {\textbf{v}}_2\|_{L^4 (0,T; L^3 (\Omega))} \\ & \leq C (R) T^\frac{1}{4} \|{\textbf{v}}_1 - {\textbf{v}}_2\|_{X^1_T} .\end{aligned}$$ For the second summand in (\[strong\_solution\_nabla\_w\_prime\]) we have to estimate the difference $\nabla W' (\varphi_1) - \nabla W' (\varphi_2)$ in an appropriate manner. To this end, we use (\[phi\_hoelder\_continuous\_in\_time\_w3p\]), (\[lipschitz\_f\_of\_phi\_w3p\]) and $W^2_p (\Omega) \hookrightarrow C^1 (\overline \Omega)$. Moreover, we use $\nabla {\textbf{v}}_2 \in L^4 (0,T; L^3 (\Omega))$, cf. (\[strong\_solution\_nabla\_v\_l4\_l3\]), and $m (\varphi ) \in C([0,T]; C^2 (\overline \Omega))$. Then it follows $$\begin{aligned} \| m(\varphi_1) & ( \nabla W'( \varphi_1 ) - \nabla W' ( \varphi_2)) \cdot \nabla {\textbf{v}}_2 \|_{Y^1_T} \\ & \leq C (R) T^\frac{1}{4} \underset{t \in [0,T]}{\sup} \|\nabla W' (\varphi_1 (t) ) - \nabla W' (\varphi_2 (t))\|_{W^2_p (\Omega)} \\ & \leq C (R) T^\frac{1}{4} \underset{t \in [0,T]}{\sup} \|\varphi_1 (t) - \varphi_2 (t)\|_{W^3_p (\Omega)} \\ & \leq C (R) T^{\frac{1}{4} + (1 - \frac{1}{p}) \theta } \| \varphi_1 - \varphi_2 \|_{X^2_T} .\end{aligned}$$ So it remains to estimate the third term of (\[strong\_solution\_nabla\_w\_prime\]). As before we get $$\begin{aligned} \| & (m(\varphi_1) - m (\varphi_2) )\nabla W' (\varphi_2 ) \cdot \nabla {\textbf{v}}_2 \|_{Y^1_T} \\ & \leq C(R) T^{\frac{1}{4} + (1 - \frac{1}{p}) \theta} \|\varphi_1 - \varphi_2\|_{C^{0, (1 - \frac{1}{p} ) \theta} ([0,T]; W^3_p (\Omega))} \\ & \ \ \ \ \|\nabla W' (\varphi_2)\|_{BUC([0,T]; C^1 (\overline \Omega))} \|\nabla {\textbf{v}}_2\|_{L^4 (0,T; L^3 (\Omega))} \\ & \leq C(R) T^{\frac{1}{4} + (1 - \frac{1}{p}) \theta} \|\varphi_1 - \varphi_2\|_{X^2_T} ,\end{aligned}$$ which completes the estimate for (\[strong\_solution\_nabla\_w\_prime\]). Finally, we study the last term of $ \| \mathbb P_\sigma ( F_1 ({\textbf{v}}_1, \varphi_1 ) - F_1 ({\textbf{v}}_2, \varphi_2 ) )\|_{L^2 (Q_T)}$. It holds $$\begin{aligned} \| \Delta \varphi_2 \nabla \varphi_2 - \Delta \varphi_1 \nabla \varphi_1\|_{Y_T} \leq \| \Delta \varphi_2 ( \nabla \varphi_2 - \nabla \varphi_1) \|_{Y_T} + \| ( \Delta \varphi_2 - \Delta \varphi_1) \nabla \varphi_1 \|_{Y_T}.\end{aligned}$$ Using $\Delta \varphi_i \in C([0,T]; C^0 (\overline \Omega))$ and $\nabla \varphi_i \in C^{0, (1- \frac{1}{p} ) \theta} ([0,T]; W^2_p (\Omega))$, $i = 1,2$, cf. (\[phi\_hoelder\_continuous\_in\_time\_w3p\]), the first term can be estimated by $$\begin{aligned} \| \Delta \varphi_2 ( \nabla \varphi_2 - \nabla \varphi_1) \|_{Y^1_T} & \leq C (R) T^{\frac{1}{2} + (1 - \frac{1}{p}) \theta)} \|\nabla \varphi_1 - \nabla \varphi_2\|_{C^{0, ( 1 - \frac{1}{p}) \theta}([0,T]; W^2_p (\Omega))} \\ & \leq C (R) T^{\frac{1}{2} + (1 - \frac{1}{p}) \theta)} \| \varphi_1 - \varphi_2\|_{X^2_T} .\end{aligned}$$ Analogously the second term can be estimated by $$\begin{aligned} \| ( \Delta \varphi_2 - \Delta \varphi_1) \nabla \varphi_1 \|_{Y_T} \leq C (R) T^{\frac{1}{2} + (1 - \frac{1}{p}) \theta)} \| \varphi_1 - \varphi_2\|_{X^2_T} .\end{aligned}$$ Hence, we obtain $$\begin{aligned} \| \mathbb P_\sigma ( F_1 ({\textbf{v}}_1, \varphi_1 ) - F_1 ({\textbf{v}}_2, \varphi_2 ) )\|_{L^2 (Q_T)} & \leq C(R,T) \|({\textbf{v}}_1 - {\textbf{v}}_2), (\varphi_1 - \varphi_2)\|_{X_T}\end{aligned}$$ for a constant $C(R,T) > 0$ such that $C(R,T) \rightarrow 0$ as $T \rightarrow 0$. Remember that we study the nonlinear operator $\mathcal F \colon X_T \rightarrow Y_T$ given by $$\begin{aligned} \mathcal F ({\textbf{v}}, \varphi) = \begin{pmatrix} \mathbb P_\sigma F_1 ({\textbf{v}}, \varphi) \\ - \nabla \varphi \cdot {\textbf{v}}+ {\text{div}}( \frac{1}{\varepsilon} m(\varphi) \nabla W' (\varphi)) + \varepsilon m (\varphi_0) \Delta^2 \varphi - \varepsilon {\text{div}}( m (\varphi) \nabla \Delta \varphi) \end{pmatrix} \end{aligned}$$ and we want to show its Lipschitz continuity such that (\[strong\_solution\_estimate\_lipschitz\_continuity\_F\]) holds. We already showed its Lipschitz continuity for the first part. Now we continue to study the second one. This part has to be estimated in $L^p (0,T; L^p (\Omega))$ for $4 < p < 6$. For the analysis we use the boundedness of $\nabla \varphi $ in $ C([0,T]; C^1 (\overline \Omega))$ and of ${\textbf{v}}$ in $ L^\infty (0,T; L^6 (\Omega))$. Then it holds $$\begin{aligned} \| & (\nabla \varphi_1 \cdot {\textbf{v}}_1 - \nabla \varphi_2 \cdot {\textbf{v}}_2 ) \|_{L^p (Q_T)} \\ & \leq \|\nabla \varphi_1 \cdot ( {\textbf{v}}_1 - {\textbf{v}}_2 )\|_{L^p (Q_T)} + \| ( \nabla \varphi_1 - \nabla \varphi_2 ) \cdot {\textbf{v}}_2\|_{L^p (Q_T)} \\ & \leq T^\frac{1}{p} \|\nabla \varphi_1\|_{L^\infty (Q_T)} \|{\textbf{v}}_1 - {\textbf{v}}_2\|_{L^\infty (0,T; L^6 (\Omega))} \\ & \ \ \ + T^\frac{1}{p} \|\nabla \varphi_1 - \nabla \varphi_2\|_{L^\infty (Q_T)} \|{\textbf{v}}_2\|_{L^\infty (0,T; L^6 (\Omega))} \\ & \leq T^\frac{1}{p} R \|{\textbf{v}}_1 - {\textbf{v}}_2\|_{X^1_T} + T^\frac{1}{p} R \|\varphi_1 - \varphi_2\|_{X^2_T} .\end{aligned}$$ Next we study the term ${\text{div}}(m (\varphi) \nabla W' (\varphi))$. We use the boundedness of $f(\varphi) $ in $C([0,T]; C^2 (\overline \Omega)) \cap C^{0, (1 - \frac{1}{p}) \theta} ([0,T]; W^3_p (\Omega))$ for $f \in \{ m , W' \}$ and $\varphi \in X^2_T $ with $\|\varphi\|_{X^2_T} \leq R$. Then it holds $$\begin{aligned} \|{\text{div}}( & m(\varphi_1) \nabla W' (\varphi_1)) - {\text{div}}( m (\varphi_2) \nabla W' (\varphi_2))\|_{Y^2_T} \\ & \leq C(R) \|m (\varphi_1) \nabla W' (\varphi_1)) - m (\varphi_2) \nabla W' (\varphi_2)\|_{L^p (0,T; W^1_p (\Omega))} \\ & \leq C(R) T^\frac{1}{p} \underset{t \in [0,T]}{\sup} \|m (\varphi_1 (t)) - m (\varphi_2 (t))\|_{W^3_p (\Omega)} \|\nabla W' (\varphi_1)\|_{C([0,T]; C^1 (\overline \Omega))} \\ & \ \ \ + C(R) T^\frac{1}{p} \|m (\varphi_2)\|_{C([0,T]; C^2 (\overline \Omega))} \underset{t \in [0,T]}{\sup} \| W' (\varphi_1 (t)) - W' (\varphi_2 (t)) \|_{W^3_p (\Omega)} \\ & \leq C(R) T^\frac{1}{p} \left ( \underset{t \in [0,T]}{\sup} \|\varphi_1 (t) - \varphi_2 (t)\|_{W^3_p (\Omega)} + \underset{t \in [0,T]}{\sup} \| \varphi_1 (t) - \varphi_2 (t) \|_{W^3_p (\Omega)} \right ) \\ &\leq C(R) T^\frac{1}{p} \underset{t \in [0,T]}{\sup} \|(\varphi_1 (t) - \varphi_2 (t)) - (\varphi_1 (0) - \varphi_2 (0))\|_{W^3_p (\Omega)} \\ & \leq C(R) T^{\frac{1}{p} + (1 - \frac{1}{p} ) \theta} \|\varphi_1 - \varphi_2\|_{C^{0, (1 - \frac{1}{p}) \theta} ([0,T]; W^3_p (\Omega))} \\ & \leq C(R) T^{\frac{1}{p} + (1 - \frac{1}{p} ) \theta} \|\varphi_1 - \varphi_2\|_{X^2_T} .\end{aligned}$$ Here we also used $\varphi_1 (0) = \varphi_2 (0) = \varphi_0$ for $\varphi_1, \varphi_2 \in X^2_T$ and (\[lipschitz\_f\_of\_phi\_w3p\]). Now there remain two terms which we need to study together for the proof of the Lipschitz continuity. Due to the boundedness of $m (\varphi) $ in $ BUC([0,T]; W^3_p (\Omega))$ and of $\nabla \Delta \varphi$ in $ L^p (0,T; W^1_p (\Omega))$, Theorem \[theorem\_composition\_sobolev\_functions\] yields the boundedness of $m (\varphi) \nabla \Delta \varphi$ in $ L^p (0,T; W^1_p (\Omega))$. Hence, this term is well-defined in the $L^p (Q_T)$-norm. We omit the prefactor $\varepsilon $ for both terms again and estimate $$\begin{aligned} &\|m (\varphi_0) \Delta^2 \varphi_1 - m (\varphi_0) \Delta^2 \varphi_2 + {\text{div}}(m (\varphi_2) \nabla \Delta \varphi_2) - {\text{div}}(m (\varphi_1) \nabla \Delta \varphi_1)\|_{L^p (Q_T)} \nonumber \\ & = \| (m(\varphi_0) - m (\varphi_1)) ( \Delta^2 \varphi_1 - \Delta^2 \varphi_2) + m (\varphi_1) \Delta^2 \varphi_1 - m (\varphi_1) \Delta^2 \varphi_2 + \nabla m (\varphi_2) \cdot \nabla \Delta \varphi_2 \nonumber \\ & \ \ \ + m ( \varphi_2) \Delta^2 \varphi_2 - \nabla m (\varphi_1) \cdot \nabla \Delta \varphi_1 - m (\varphi_1) \Delta^2 \varphi_1 \|_{L^p (Q_T)} \nonumber \\ & \leq \| ( m ( \varphi_1 (0) - m (\varphi_1)) (\Delta^2 \varphi_1 - \Delta^2 \varphi_2 )\|_{L^p (Q_T)} + \| (m (\varphi_2) - m (\varphi_1)) \Delta^2 \varphi_2 \|_{L^p (Q_T)} \nonumber \\ & \ \ \ + \| \nabla m (\varphi_2) \cdot \nabla \Delta \varphi_2 - \nabla m ( \varphi_1) \cdot \nabla \Delta \varphi_1 \|_{L^p (Q_T)} \label{complicated_terms_lipschitz_continuity}\end{aligned}$$ For the sake of clarity, we estimate these three terms separately again. Due to the boundedness of $m (\varphi_1)$ in $ C^{0, (1 - \frac{1}{p}) \theta} ([0,T] ; W^3_p (\Omega))$ we obtain for the first term $$\begin{aligned} \| ( m ( & \varphi_1 (0) - m (\varphi_1)) (\Delta^2 \varphi_1 - \Delta^2 \varphi_2 )\|_{L^p (Q_T)} \\ & \leq \underset{t \in (0,T)}{\sup} \| m(\varphi_1 (0)) - m ( \varphi_1 (t)) \|_{C^0 (\overline \Omega)} \|\Delta^2 \varphi_1 - \Delta^2 \varphi_2\|_{L^p (Q_T)} \\ & \leq C(R) T^{(1 - \frac{1}{p}) \theta} \|m (\varphi_1)\|_{C^{0, (1-\frac{1}{p}) \theta } ([0,T]; W^3_p (\Omega))} \|\varphi_1 - \varphi_2\|_{X^2_T} .\end{aligned}$$ Since $m (\varphi_1)$ is bounded in $ C^{0, (1 - \frac{1}{p}) \theta} ([0,T] ; W^3_p (\Omega))$, we can estimate the second term in (\[complicated\_terms\_lipschitz\_continuity\]) by $$\begin{aligned} \| (m (\varphi_2) & - m (\varphi_1)) \Delta^2 \varphi_2 \|_{L^p (Q_T)} \leq \underset{t \in (0,T)}{\sup}\|m (\varphi_2 (t)) - m (\varphi_1 (t))\|_{C^2 (\overline \Omega)} \|\Delta^2 \varphi_2\|_{L^p (Q_T)} \\ & \leq C(R) \underset{t \in (0,T)}{\sup}\|m (\varphi_2 (t)) - m (\varphi_1 (t))\|_{W^3_p (\Omega)} \\ & \leq C(R) \underset{t \in (0,T)}{\sup}\|(\varphi_2 (t) - \varphi_1 (t)) - (\varphi_2(0) - \varphi_1 (0))\|_{W^3_p (\Omega)} \\ & \leq C(R) T^{(1 - \frac{1}{p}) \theta } \|\varphi_1 - \varphi_2\|_{C^{0, ( 1 - \frac{1}{p}) \theta} ([0,T]; W^3_p (\overline \Omega))} ,\end{aligned}$$ where we used (\[lipschitz\_f\_of\_phi\_w3p\]) again in the penultimate step. Finally, we study the last term in (\[complicated\_terms\_lipschitz\_continuity\]). Here we get $$\begin{aligned} \label{strong_solution_lipschitz_continuity_most_complicated_term_} \| & \nabla m (\varphi_2) \cdot \nabla \Delta \varphi_2 - \nabla m ( \varphi_1) \cdot \nabla \Delta \varphi_1 \|_{L^p (Q_T)} \nonumber \\ & \leq \|(\nabla m (\varphi_2) - \nabla m (\varphi_1)) \cdot \nabla \Delta \varphi_2 \|_{L^p (Q_T)} \nonumber \\ & \ \ \ + \|\nabla m (\varphi_1) \cdot ( \nabla \Delta \varphi_2 - \nabla \Delta \varphi_1)\|_{L^p (Q_T)} .\end{aligned}$$ Since $\nabla m (\varphi_1) $ is bounded in $C([0,T]; C^1 (\overline \Omega))$ and $\nabla \Delta \varphi_i$ is bounded in $C([0,T]; L^p (\Omega))$ for $i = 1,2$, we can estimate the second summand by $$\begin{aligned} \|\nabla m ( & \varphi_1) \cdot ( \nabla \Delta \varphi_2 - \nabla \Delta \varphi_1)\|_{L^p (Q_T)} \\ &\leq C(R) T^{\frac{1}{p}} \|\nabla m (\varphi_1)\|_{C ([0,T]; C^1 (\overline \Omega))} \|\nabla \Delta \varphi_1 - \nabla \Delta \varphi_2\|_{C([0,T]; L^p (\Omega))} \\ & \leq C(R) T^\frac{1}{p} \|\varphi_1 - \varphi_2\|_{X^2_T} .\end{aligned}$$ Thus it remains to estimate the first term of (\[strong\_solution\_lipschitz\_continuity\_most\_complicated\_term\_\]). Here we get $$\begin{aligned} \|(\nabla m & (\varphi_2) - \nabla m (\varphi_1)) \cdot \nabla \Delta \varphi_2 \|_{L^p (Q_T)} \\ & \leq C(R) T^\frac{1}{p} \underset{t \in [0,T]}{\sup} \|\nabla m (\varphi_2(t)) - \nabla m (\varphi_1 (t))\|_{C^0 (\overline \Omega)} \|\nabla \Delta \varphi_2\|_{C([0,T]; L^p (\Omega))} \\ & \leq C(R) T^\frac{1}{p} \underset{t \in [0,T]}{\sup} \|m (\varphi_2 (t)) - m (\varphi_1 (t))\|_{W^3_p (\Omega)} \|\varphi_2\|_{C([0,T] ; W^3_p (\Omega))} \\ & \leq C(R) T^\frac{1}{p} \underset{t \in [0,T]}{\sup} \|\varphi_1 (t) - \varphi_2 (t)\|_{W^3_p (\Omega)} \\ & \leq C(R) T^{\frac{1}{p} + (1 - \frac{1}{p}) \theta} \|\varphi_1 - \varphi_2\|_{C^{0, (1 - \frac{1}{p}) \theta} ([0,T] ; W^3_p (\Omega))} .\end{aligned}$$ Hence, (\[strong\_solution\_lipschitz\_continuity\_most\_complicated\_term\_\]) is Lipschitz continuous and therefore also the second part of $\mathcal F$ is Lipschitz continuous. Together with the Lipschitz continuity of the first part of $\mathcal F$ we have shown $$\begin{aligned} \| \mathcal F ({\textbf{v}}_1 , \varphi_1 ) - \mathcal F ({\textbf{v}}_2 , \varphi_2) \| _{Y_T} \leq C (T, R) \|({\textbf{v}}_1 - {\textbf{v}}_2, \varphi_1 - \varphi_2 )\|_{X_T}\end{aligned}$$ for all $({\textbf{v}}_i, \varphi_i) \in X_T$ with $\|({\textbf{v}}_i, \varphi_i)\|_{X_T} \leq R$, $i = 1,2$, and a constant $C(T,R) > 0 $ such that $C (T,R) \rightarrow 0$ as $T \rightarrow 0$. Existence and Continuity of $\mathcal L^{-1}$ {#sec:Linear} ============================================== In the following we need: \[strong\_solution\_showalter\_theorem\_6\_1\] Let the linear, symmetric and monotone operator $\mathcal B$ be given from the real vector space $E$ to its algebraic dual $E'$, and let $E' _b$ be the Hilbert space which is the dual of $E$ with the seminorm $$\begin{aligned} |x|_b = \mathcal B x (x) ^\frac{1}{2}, \qquad x \in E .\end{aligned}$$ Let $A \subseteq E \times E' _b$ be a relation with domain $D = \{ x \in E: \ A(x) \neq \emptyset \}$. Let $A$ be the subdifferential, $\partial \varphi$, of a convex lower-semi-continuous function $\varphi: E_b \rightarrow [0, \infty ]$ with $\varphi (0) = 0$. Then for each $u_0$ in the $E_b$-closure of $\mathrm{dom} (\varphi)$ and each $f \in L^2 (0,T; E' _b)$ there is a solution $u : [0,T] \rightarrow E$ with $\mathcal B u \in C([0,T], E' _b)$ of $$\begin{aligned} \frac{d}{dt} ( \mathcal B u (t) ) + A ( u (t)) \ni f(t) , \qquad 0 < t < T, \end{aligned}$$ with $$\begin{aligned} \varphi \circ u \in L^1 (0,T), \sqrt{t} \frac{d}{dt} \mathcal B u (\cdot ) \in L^2 (0,T; E' _b), u(t) \in D, \text{ a.e. } t \in [0,T] ,\end{aligned}$$ and $\mathcal B u(0) = \mathcal B u_0$. If in addition $u_0 \in \mathrm{dom} (\varphi)$, then $$\begin{aligned} \varphi \circ u \in L^\infty (0,T), \qquad \frac{d}{dt} \mathcal B u \in L^2 (0,T; E' _b) .\end{aligned}$$ The proof of this theorem can be found in [@Showalter Chapter IV, Theorem 6.1]. To prove Theorem \[thm:linear\] we need to show the existence of $(\tilde {\textbf{v}}, \tilde \varphi) \in X_T$ such that (\[strong\_solution\_tilde\_v0\_tilde\_phi0\_bounded\]) holds and to prove that $\mathcal L : X_T \rightarrow Y_T$ is invertible with uniformly bounded inverse, i.e., there exists a constant $C > 0$ which does not depend on $T$ such that $\|\mathcal L^{-1}\|_{\mathcal L (Y_T, X_T)} \leq C$. Recall that the linear operator $\mathcal L \colon X_T \rightarrow Y_T$ is defined by $$\begin{aligned} \mathcal L ({\textbf{v}}, \varphi) = \begin{pmatrix} \mathbb P_\sigma ( \rho_0 \partial_t {\textbf{v}}) - \mathbb P_\sigma ( {\text{div}}(2 \eta (\varphi_0) D{\textbf{v}})) \\ \partial_t \varphi + \varepsilon m (\varphi_0) \Delta^2 \varphi \end{pmatrix} .\end{aligned}$$ We note that the first part only depends on ${\textbf{v}}$ while the second part only depends on $\varphi$. Thus both equations can be solved separately. To show the existence of a unique solution ${\textbf{v}}$ for every right-hand side $\bold f$ in the first equation we use Theorem \[strong\_solution\_showalter\_theorem\_6\_1\]. So we have to specify what $E$, $E ' _b$, $\varphi$ and so on are in the problem we study and show that the conditions of Theorem \[strong\_solution\_showalter\_theorem\_6\_1\] are fulfilled. Then Theorem \[strong\_solution\_showalter\_theorem\_6\_1\] yields the existence of a solution. More precisely, we obtain the following lemma. \[strong\_solution\_existence\_proof\_L\_1\_part1\] Let Assumption \[strong\_solutions\_general\_assumptions\] hold. Then for every $\bold v_0 \in H^1_0 (\Omega)^d \cap L^2_\sigma (\Omega)$, $f \in L^2 (0,T; L^2_\sigma (\Omega))$, $\varphi_0 \in W^{1}_r (\Omega)$, $r > d \geq 2$, and every $0 < T < \infty$ there exists a unique solution $$\begin{aligned} \bold v\in W^1_2 (0,T; L^2_\sigma (\Omega)) \cap L^\infty (0,T; H^1_0 (\Omega)^d)\end{aligned}$$ such that $$\begin{aligned} \mathbb P_\sigma (\rho_0 \partial_t \bold v) - \mathbb P_\sigma (\mathrm{div} (2 \eta (\varphi_0) D \bold v)) &= f && \text{ in } Q_T, \label{strong_solution_L_first_equation} \\ \mathrm{div} (\bold v) &= 0 && \text{ in } Q_T , \\ \bold v_{|\partial \Omega} &= 0 && \text{ on } (0, T) \times \partial \Omega , \\ \bold v (0) &= \bold v_0 && \text{ in } \Omega \label{strong_solution_L_first_equation_initial_data}\end{aligned}$$ for a.e. $(t,x)$ in $(0,T) \times \Omega$, where $\bold v (t) \in H^2 (\Omega)^d$ for a.e. $t \in (0,T)$. Since we want to solve (\[strong\_solution\_L\_first\_equation\])-(\[strong\_solution\_L\_first\_equation\_initial\_data\]) with Theorem \[strong\_solution\_showalter\_theorem\_6\_1\], we define $$\begin{aligned} \mathcal B u := \mathbb P _\sigma ( \rho_0 u) \end{aligned}$$ for $u \in E$, where we still need to specify the real vector space $E$. But as we want to have $\frac{d}{dt} \mathcal B u \in L^2 (0,T; L^2_\sigma (\Omega))$, the dual space $E'_b$ has to coincide with $L^2_\sigma (\Omega)$. But this can be realized by choosing $E = L^2_\sigma (\Omega)$. Then $E_b' \cong L^2_\sigma (\Omega)$ and with the notation in Theorem \[strong\_solution\_showalter\_theorem\_6\_1\] we get the Hilbert space $E'_b$ equipped with the seminorm $$\begin{aligned} |{\textbf{u}}|_b = \mathcal B \bold {\textbf{u}}( {\textbf{u}}) ^\frac{1}{2} &= \left ( {\int \limits_{\Omega}}\mathbb P _\sigma ( \rho_0 {\textbf{u}}) \cdot {\textbf{u}}{\mathit{dx}}\right ) ^\frac{1}{2} = \left ( {\int \limits_{\Omega}}\rho_0 {\textbf{u}}\cdot \mathbb P_\sigma {\textbf{u}}{\mathit{dx}}\right ) ^\frac{1}{2} \\ &= \left ( {\int \limits_{\Omega}}\rho_0 |{\textbf{u}}|^2 {\mathit{dx}}\right ) ^\frac{1}{2} \cong \|{\textbf{u}}\|_{L^2 (\Omega)} .\end{aligned}$$ Thus we obtain $E '_b \cong L^2_\sigma (\Omega) = E_b$. Moreover, we define $A: \mathcal D (A) \rightarrow L^2_\sigma (\Omega) ' \cong L^2_\sigma (\Omega)$ by $$\begin{aligned} \label{strong_solution_definition_A} (A {\textbf{u}})({\textbf{v}}) := \begin{cases} - {\int \limits_{\Omega}}\mathbb P _\sigma {\text{div}}(2 \eta (\varphi_0) D{\textbf{u}}) \cdot {\textbf{v}}{\mathit{dx}}& \text{ if } {\textbf{u}}\in \mathrm{dom} (A) \\ \emptyset & \text{ if } {\textbf{u}}\notin \mathrm{dom} (A) . \end{cases}\end{aligned}$$ for every $ {\textbf{v}}\in L^2_\sigma (\Omega)$ and $\mathcal D (A) = H^2 (\Omega)^d \cap H^1_0 (\Omega)^d \cap L^2_\sigma (\Omega)$. Thus we get for the relation $\mathcal A$ defined by $ \mathcal A := \{ ({\textbf{u}},{\textbf{v}}) : \ {\textbf{v}}= A{\textbf{u}}, \ {\textbf{u}}\in \mathcal D (A) \} $ the following inclusions $$\begin{aligned} \mathcal A = \{ ({\textbf{u}}, - \mathbb P_\sigma {\text{div}}(2 \eta (\varphi_0) D{\textbf{u}}) : {\textbf{u}}\in H^2 (\Omega)^d \cap H^1_0 (\Omega)^d \cap L^2_\sigma (\Omega) \} \subseteq E \times E'_b ,\end{aligned}$$ since the term $ \mathbb P_\sigma {\text{div}}(2 \eta (\varphi_0) D{\textbf{u}})$ is in $L^2 _\sigma (\Omega) ' \cong L^2_\sigma (\Omega)$. Now we define $\psi : L^2_\sigma (\Omega) \rightarrow [0, + \infty]$ by $$\begin{aligned} \label{strong_solution_definition_phi} \psi ({\textbf{u}}) := \begin{cases} {\int \limits_{\Omega}}\eta (\varphi_0) D{\textbf{u}}: D{\textbf{u}}\ dx & \text{ if } {\textbf{u}}\in H^1_0 (\Omega)^d \cap L^2_\sigma (\Omega) = \mathrm{dom} (\psi) , \\ + \infty & \text{ else. } \end{cases}\end{aligned}$$ We note $\psi (0) = 0$ and ${\textbf{v}}_0$ is in the $L^2$-closure of $\mathrm{dom}(\psi)$, i.e., in $L^2_\sigma (\Omega)$. Hence, it remains to show that $\psi$ is convex and lower-semi-continuous and that $A$ is the subdifferential of $\psi$. Then we can apply Theorem \[strong\_solution\_showalter\_theorem\_6\_1\]. But the first two properties are obvious. Thus it remains to show the subdifferential property, which is satisfied by Lemma \[strong\_solution\_A\_coincides\_partial\_phi\] below. Hence, we are able to apply Theorem \[strong\_solution\_showalter\_theorem\_6\_1\] which yields the existence. Moreover, the initial condition is also fulfilled as Theorem \[strong\_solution\_showalter\_theorem\_6\_1\] yields $$\begin{aligned} \mathbb P_\sigma (\rho_0 {\textbf{v}}(0)) = \mathcal B {\textbf{v}}(0) = \mathcal B {\textbf{v}}_0 = \mathbb P_\sigma (\rho_0 {\textbf{v}}_0) \qquad \text{ in } L^2 (\Omega) .\end{aligned}$$ In particular we can conclude $$\begin{aligned} 0 &= \int \limits_\Omega \mathbb P_\sigma ( \rho_0 {\textbf{v}}(0) - \rho_0 {\textbf{v}}_0 ) \cdot \boldsymbol \psi \mathit {dx} = \int \limits_\Omega (\rho_0 {\textbf{v}}(0) - \rho_0 {\textbf{v}}_0) \cdot \boldsymbol \psi \mathit{dx}\end{aligned}$$ for every $\boldsymbol \psi \in C^\infty_{0, \sigma} (\Omega)$. By approximation this identity also holds for $\boldsymbol \psi := {\textbf{v}}(0) - {\textbf{v}}_0 \in L^2_\sigma (\Omega)$ and we get $$\begin{aligned} \int \limits_\Omega \rho_0 |{\textbf{v}}(0) - {\textbf{v}}_0|^2 \mathit{dx} = 0 .\end{aligned}$$ This implies ${\textbf{v}}(0) = {\textbf{v}}_0$ in $L^2_\sigma (\Omega)$. For the uniqueness we consider ${\textbf{v}}_1 , {\textbf{v}}_2 \in W^1_2 (0,T; L^2_\sigma (\Omega)) \cap L^\infty (0,T; H^1_0 (\Omega) \cap L^2_\sigma (\Omega))$ such that (\[strong\_solution\_L\_first\_equation\]) holds for a.e. $(t,x) \in (0,T) \times \Omega$. Then ${\textbf{v}}:= {\textbf{v}}_1 - {\textbf{v}}_2$ solves the homogeneous equation a.e. in $(0,T) \times \Omega$. Testing this homogeneous equation with ${\textbf{v}}$ we get $$\begin{aligned} {\int \limits_{\Omega}}\frac{1}{2} \rho_0 {\textbf{v}}^2_{|t=T} {\mathit{dx}}+ \int \limits_0^T {\int \limits_{\Omega}}2 \eta (\varphi_0) D{\textbf{v}}: D{\textbf{v}}{\mathit{dx}}{\mathit{dt}}= 0.\end{aligned}$$ Hence, it follows ${\textbf{v}}\equiv 0$ and therefore ${\textbf{v}}_1 = {\textbf{v}}_2$, which yields the uniqueness. In the proof above we used that the mapping $A$ coincides with the subdifferential $\partial \varphi$. More precisely, we have the following lemma. \[strong\_solution\_A\_coincides\_partial\_phi\] Let $\Omega \subseteq \mathbb R^d$, $d = 2,3$, be a domain and $\psi\colon L^2_\sigma (\Omega) \rightarrow [0, + \infty]$ be given as in (\[strong\_solution\_definition\_phi\]). Moreover, we consider $A : L^2_\sigma (\Omega) \rightarrow L^2_\sigma (\Omega)$ to be given as in (\[strong\_solution\_definition\_A\]). Then it holds 1. $ \mathcal D (\partial \psi) = \mathcal D (A) . $ 2. $\partial \psi (u) = \{ A u \} \text{ for all } u \in \mathcal D (A) .$ Remember that $$\begin{aligned} \mathcal D (\partial \psi) = \{ {\textbf{v}}\in L^2_\sigma (\Omega) : \ \partial \psi ({\textbf{v}}) \neq \emptyset \}\end{aligned}$$ and $\mathcal D (A) = H^2 (\Omega)^d \cap H^1_0 (\Omega)^d \cap L^2_\sigma (\Omega)$ by definition. **$\bold{1^{st}}$ part:** $\mathcal D (A) \subseteq \mathcal D (\partial \psi)$ and $Au \in \partial \psi (u)$ for every $u \in \mathcal D (A)$.\ To show the first part of the proof let ${\textbf{u}}\in \mathcal D (A)$ be given. If it holds ${\textbf{v}}\in L^2_\sigma (\Omega)$ but ${\textbf{v}}\notin H^1_0 (\Omega)^d$, then the inequality $$\begin{aligned} {\langle A{\textbf{u}}, {\textbf{v}}-{\textbf{u}}\rangle}_{L^2 (\Omega)} \leq \psi ({\textbf{v}}) - \psi ({\textbf{u}})\end{aligned}$$ is satisfied since it holds $\psi ({\textbf{v}}) = + \infty$ in this case by definition.\ So let ${\textbf{v}}\in H^1_0 (\Omega)^d \cap L^2_\sigma (\Omega)$. Then it holds $$\begin{aligned} {\langle A{\textbf{u}}, {\textbf{v}}- {\textbf{u}}\rangle}_{L^2 (\Omega)} &= - {\int \limits_{\Omega}}\mathbb P_\sigma {\text{div}}(2 \eta (\varphi_0) D{\textbf{u}}) \cdot ({\textbf{v}}- {\textbf{u}}) {\mathit{dx}}\\ & = {\int \limits_{\Omega}}2 \eta (\varphi_0) D{\textbf{u}}: D{\textbf{v}}{\mathit{dx}}- {\int \limits_{\Omega}}2 \eta (\varphi_0) D{\textbf{u}}: D {\textbf{u}}{\mathit{dx}}\\ & \leq {\int \limits_{\Omega}}\eta (\varphi_0) |D{\textbf{u}}|^2 {\mathit{dx}}+ {\int \limits_{\Omega}}\eta (\varphi_0) |D{\textbf{v}}|^2 {\mathit{dx}}- 2 {\int \limits_{\Omega}}\eta (\varphi_0) |D{\textbf{u}}|^2 {\mathit{dx}}\\ & = \psi ({\textbf{v}}) - \psi ({\textbf{u}}) \end{aligned}$$ for every ${\textbf{v}}\in H^1_0 (\Omega)^d \cap L^2_\sigma (\Omega)$. This implies that $A{\textbf{u}}$ is a subgradient of $\psi$ at ${\textbf{u}}$, i.e., $ A{\textbf{u}}\in \partial \psi ({\textbf{u}})$, and $\partial \psi ({\textbf{u}}) \neq \emptyset$, i.e., ${\textbf{u}}\in \mathcal D (A) \subseteq \mathcal D (\partial \psi)$. Hence, we have shown the first part of the proof. **$\bold{2^{nd}}$ part:** $\mathcal D (\partial \psi) \subseteq \mathcal D (A) $ and $\partial \psi ({\textbf{u}}) = \{A {\textbf{u}}\}$.\ Let $ {\textbf{u}}\in\mathcal D (\partial \psi) \subseteq \mathrm{dom} (\psi) \subseteq H^1_0 (\Omega)^d \cap L^2_\sigma (\Omega)$ be given. Then by definition of the subgradient there exists $\bold w \in \partial \psi (u) \subseteq \mathcal P (L^2_\sigma (\Omega))$ such that $$\begin{aligned} \label{strong_solution_subdifferential_inequality_phi} \psi ({\textbf{u}}) - \psi ({\textbf{v}}) \leq {\langle \bold w , {\textbf{u}}- {\textbf{v}}\rangle}_{L^2 (\Omega)}\end{aligned}$$ for every ${\textbf{v}}\in L^2_\sigma (\Omega)$. Now we choose ${\textbf{v}}:= {\textbf{u}}+ t \tilde {\textbf{w}}$ for some $\tilde {\textbf{w}}\in H^1_0 (\Omega)^d \cap L^2_\sigma (\Omega)$ and $ t > 0$. Then inequality (\[strong\_solution\_subdifferential\_inequality\_phi\]) yields $$\begin{aligned} \psi ({\textbf{u}}) - \psi ({\textbf{v}}) &= {\int \limits_{\Omega}}\eta (\varphi_0) D{\textbf{u}}: D{\textbf{u}}{\mathit{dx}}- {\int \limits_{\Omega}}\eta (\varphi_0) D({\textbf{u}}+ t \tilde {\textbf{w}}) : D ( {\textbf{u}}+ t \tilde {\textbf{w}}) {\mathit{dx}}\\ &= - 2 t {\int \limits_{\Omega}}\eta (\varphi_0) D{\textbf{u}}: D \tilde {\textbf{w}}{\mathit{dx}}- t^2 {\int \limits_{\Omega}}\eta (\varphi_0) D{\textbf{u}}: D \tilde {\textbf{w}}{\mathit{dx}}\\ & \leq -t {\int \limits_{\Omega}}{\textbf{w}}\cdot \tilde {\textbf{w}}{\mathit{dx}}.\end{aligned}$$ Dividing this inequality by $ -t < 0$ and passing to the limit $t \searrow 0$ yields $$\begin{aligned} {\int \limits_{\Omega}}{\textbf{w}}\cdot \tilde {\textbf{w}}{\mathit{dx}}\leq {\int \limits_{\Omega}}2 \eta (\varphi_0) D{\textbf{u}}: D \tilde {\textbf{w}}{\mathit{dx}}.\end{aligned}$$ When we replace $\tilde {\textbf{w}}$ by $- \tilde {\textbf{w}}$ we can conclude $$\begin{aligned} {\int \limits_{\Omega}}{\textbf{w}}\cdot \tilde {\textbf{w}}{\mathit{dx}}\geq {\int \limits_{\Omega}}2 \eta (\varphi_0) D{\textbf{u}}: D \tilde {\textbf{w}}{\mathit{dx}}.\end{aligned}$$ Thus it follows $$\begin{aligned} \label{strong_solution_left_hand_side_of_lemma_subdifferential} {\int \limits_{\Omega}}{\textbf{w}}\cdot \tilde {\textbf{w}}{\mathit{dx}}= {\int \limits_{\Omega}}2 \eta (\varphi_0) D{\textbf{u}}: D \tilde {\textbf{w}}{\mathit{dx}}\end{aligned}$$ for every $\tilde {\textbf{w}}\in H^1_0 (\Omega)^d \cap L^2_\sigma (\Omega)$. Since we assumed ${\textbf{w}}\in L^2_\sigma (\Omega)$, we can apply Lemma \[strong\_solution\_abels\_rational\_mech\_lemma\_4\] below which yields ${\textbf{u}}\in H^2 (\Omega)^d \cap H^1_0 (\Omega)^d \cap L^2_\sigma (\Omega)$. Using this regularity in (\[strong\_solution\_left\_hand\_side\_of\_lemma\_subdifferential\]) we can conclude $$\begin{aligned} {\int \limits_{\Omega}}{\textbf{w}}\cdot \tilde {\textbf{w}}{\mathit{dx}}= {\int \limits_{\Omega}}2 \eta (\varphi_0) D{\textbf{u}}: D \tilde {\textbf{w}}{\mathit{dx}}= - {\int \limits_{\Omega}}\mathbb P_\sigma {\text{div}}(2 \eta (\varphi_0) D{\textbf{u}}) \cdot \tilde {\textbf{w}}{\mathit{dx}}\end{aligned}$$ for every $\tilde {\textbf{w}}\in H^1_0 (\Omega)^d \cap L^2_\sigma (\Omega)$. Therefore, we obtain ${\textbf{w}}= - \mathbb P _\sigma {\text{div}}(2 \eta (\varphi_0) D{\textbf{u}}) = A{\textbf{u}}$ in $L^2 (\Omega)$, i.e., ${\textbf{u}}\in \mathcal D (A)$ and $\partial \psi ({\textbf{u}}) = \{ A{\textbf{u}}\}$. For the regularity of the Stokes system with variable viscosity we used the following lemma. \[strong\_solution\_abels\_rational\_mech\_lemma\_4\] Let $\eta \in C^2 (\mathbb R)$ be such that $\eta (s) \geq s_0 > 0$ for all $s \in \mathbb R$ and some $s_0 > 0$, $\varphi_0 \in W^{1}_r (\Omega)$, $r > d \geq 2$, with $\|\varphi_0\|_{W^{1}_r (\Omega)} \leq R$, and let $\bold u \in H^1_0 (\Omega)^d \cap L^2_\sigma (\Omega)$ be a solution of $$\begin{aligned} {\langle 2 \eta (\varphi_0) D \bold u, D \boldsymbol{\tilde {\textbf{w}}}\rangle}_{L^2 (\Omega)} = {\langle \bold w , \boldsymbol{\tilde {\textbf{w}}}\rangle}_{L^2 (\Omega)} \qquad \text{ for all }\boldsymbol{\tilde {\textbf{w}}} \in C^\infty_{0, \sigma} (\Omega) ,\end{aligned}$$ where $ \bold w \in L^2 (\Omega)^d$. Then it holds $\bold u \in H^2 (\Omega)^d$ and $$\begin{aligned} \|\bold u\|_{H^2 (\Omega)} \leq C(R) \|\bold w\|_{L^2 (\Omega)} ,\end{aligned}$$ where $C(R)$ only depends on $\Omega$, $\eta$, $r > d$, and $R > 0$. The proof can be found in [@ModelH Lemma 4]. Lemma \[strong\_solution\_existence\_proof\_L\_1\_part1\] implies ${\textbf{v}}\in W_2^1 (0,T; L^2 _\sigma (\Omega)) \cap L^\infty (0,T; H^1_0 (\Omega)^d)$. But as we want to show that ${\textbf{v}}$ is in $X^1_T$, it remains to show ${\textbf{v}}\in L^2 (0,T; H^2 (\Omega)^d)$. To this end, we also use Lemma \[strong\_solution\_abels\_rational\_mech\_lemma\_4\] above. \[strong\_solution\_h2\_regularity\_v\] For the unique solution $\bold v \in W_2^1 (0,T; L^2_\sigma (\Omega)) \cap L^\infty (0,T; H^1_0 (\Omega)^d)$ of (\[strong\_solution\_L\_first\_equation\])-(\[strong\_solution\_L\_first\_equation\_initial\_data\]) from Lemma \[strong\_solution\_existence\_proof\_L\_1\_part1\] it holds $ \bold v \in L^2 (0,T; H^2 (\Omega)^d) . $ Let ${\textbf{v}}\in W_2^1 (0,T; L^2_\sigma (\Omega)) \cap L^\infty (0,T; H^1_0 (\Omega)^d)$ be the unique solution of (\[strong\_solution\_L\_first\_equation\]) from Lemma \[strong\_solution\_existence\_proof\_L\_1\_part1\], i.e., $$\begin{aligned} A ( {\textbf{v}}(t)) &= \bold f (t) - \frac{d}{dt} ( \mathcal B {\textbf{v}}(t)) = \bold f(t) - \mathbb P_\sigma (\rho_0 \partial_t {\textbf{v}}(t)) \quad \text{for all } 0 < t < T .\end{aligned}$$ Since the right-hand side is not the empty set, we get by definition of $A$ $$\begin{aligned} \mathbb P_\sigma ({\text{div}}(2 \eta (\varphi_0) D{\textbf{v}}(t))) = \mathbb P_\sigma (\rho_0 \partial_t {\textbf{v}}(t)) - \bold f (t) \quad \text{for all } 0 < t < T\end{aligned}$$ for given $\bold f \in L^2 (0,T; L^2_\sigma (\Omega))$. From $\partial_t {\textbf{v}}\in L^2 (0,T; L^2_\sigma (\Omega))$ it follows $$\begin{aligned} {\langle 2 \eta (\varphi_0) D{\textbf{v}}(t) , D {\textbf{w}}\rangle}_{L^2 (\Omega)} = {\langle \rho_0 \partial_t {\textbf{v}}(t) - \bold f (t) , {\textbf{w}}\rangle} \qquad \text{ for every } {\textbf{w}}\in C^\infty_{0, \sigma } (\Omega) \end{aligned}$$ and a.e. $t \in (0,T)$. Hence, we can apply Lemma \[strong\_solution\_abels\_rational\_mech\_lemma\_4\] and obtain $$\begin{aligned} \|{\textbf{v}}(t)\|_{H^2 (\Omega)} &\leq C(R) \|\rho_0 \partial_t {\textbf{v}}(t) - \bold f (t) \|_{L^2 (\Omega)} \\ &\leq C(R) \left ( \|\rho_0 \partial_t {\textbf{v}}(t)\|_{L^2 (\Omega)} + \|\bold f (t) \|_{L^2 (\Omega)} \right ) \end{aligned}$$ for a.e. $t \in (0,T)$. Since the right-hand side of this inequality is bounded in $L^2 (0,T)$, this shows the lemma. We still need to ensure that $\|\mathcal L^{-1}\|_{\mathcal L (Y_T, X_T)} $ remains bounded. This is shown in the next lemma. \[strong\_solution\_lemma\_l\_inverse\_t\_t\_0\_firstpart\] Let the assumptions of Lemma \[strong\_solution\_existence\_proof\_L\_1\_part1\] hold and $0 < T_0 < \infty$ be given. Then $$\begin{aligned} \|\mathcal L^{-1}_{1, T}\|_{\mathcal L (Y^1_T, X^1_T)} \leq \|\mathcal L^{-1}_{1, T_0}\|_{\mathcal L (Y^1_{T_0}, X^1_{T_0})} < \infty \qquad \text{ for all } 0 < T < T_0 .\end{aligned}$$ Let $ 0 < T < T_0 $ be given. Lemma \[strong\_solution\_existence\_proof\_L\_1\_part1\] together with Lemma \[strong\_solution\_h2\_regularity\_v\] yields that the operator $\mathcal L_{1, T} : X_T \rightarrow Y_T$ is invertible for every $0 < T < \infty$ and every given $\bold f \in L^2(0,T; L^2_\sigma (\Omega))$, $\varphi_0 \in W^1_r (\Omega)$, ${\textbf{v}}_0 \in H^1_0 (\Omega)^d \cap L^2_\sigma (\Omega)$. Then we define $\bold{ \tilde f } \in L^2 (0,T_0, L^2_\sigma (\Omega))$ by $$\begin{aligned} \bold{\tilde f} (t) := \begin{cases} \bold f(t) & \text{ if } t \in (0,T] , \\ 0 & \text{ if } t \in (T, T_0 ) . \end{cases}\end{aligned}$$ Due to Lemma \[strong\_solution\_existence\_proof\_L\_1\_part1\] together with Lemma \[strong\_solution\_h2\_regularity\_v\] there exists a unique solution $\tilde {\textbf{v}}\in X^1_{T_0}$ of $$\begin{aligned} \mathbb P_\sigma (\rho_0 \partial_t \tilde {\textbf{v}}) - \mathbb P_\sigma ({\text{div}}(2 \eta (\varphi_0) D \tilde {\textbf{v}})) &= \bold{\tilde f} && \text{ in } Q_{T_0}, \\ {\text{div}}( \tilde {\textbf{v}}) &= 0 && \text{ in } Q_{T_0} , \\ \tilde {\textbf{v}}_{|\partial \Omega} &= 0 && \text{ on } (0, T_0) \times \partial \Omega , \\ \tilde {\textbf{v}}(0) &= {\textbf{v}}_0 && \text{ in } \Omega .\end{aligned}$$ So let ${\textbf{v}}\in X^1_T$ be the solution of the previous equations with $T_0$ replaced by $T$. Then $\tilde {\textbf{v}}$ and ${\textbf{v}}$ solve these equations on $(0,T) \times \Omega$. Since the solution is unique, we can deduce $\tilde {\textbf{v}}_{|(0,T)} = {\textbf{v}}$. Hence, $$\begin{aligned} \|\mathcal L^{-1}_{1, T} (\bold f) \|_{X^1_T} &= \| {\textbf{v}}\|_{X^1_{T}} \leq \|\tilde {\textbf{v}}\|_{X^1_{T_0}} = \|\mathcal L^{-1}_{1, T_0} (\bold{\tilde f})\|_{X^1_{T_0}} \\ & \leq \|\mathcal L^{-1}_{1, T_0}\|_{\mathcal L (Y^1_{T_0}, X^1_{T_0})} \|\bold{\tilde f}\|_{Y^1_{T_0}} = \|\mathcal L^{-1}_{1, T_0}\|_{\mathcal L (Y^1_{T_0}, X^1_{T_0})} \| \bold f\|_{Y^1_{T}} ,\end{aligned}$$ which shows the statement since it holds $ \|\mathcal L^{-1}_{1, T_0}\|_{\mathcal L (Y^1_{T_0}, X^1_{T_0})} < \infty$ by the bounded inverse theorem. Finally, we have to show invertibility of the second part of $\mathcal{L}$. \[strong\_solution\_existence\_proof\_L\_1\_part2\] Let Assumption \[strong\_solutions\_general\_assumptions\] hold and $\varphi_0 \in (L^p (\Omega) , W^4_{p,N} (\Omega))_{1 - \frac{1}{p}, p}$, $f \in L^p (0,T; L^p (\Omega)) $ with $4 < p < 6$ be given. Then for every $0 < T < \infty$ there exists $$\begin{aligned} \varphi \in L^p (0,T; W^4_{p,N} (\Omega)) \cap \{ u \in W^1_p(0,T; L^p (\Omega)) : \ u_{|t=0} = \varphi_0 \}\end{aligned}$$ such that $$\begin{aligned} \partial_t \varphi + \varepsilon m (\varphi_0) \Delta^2 \varphi &= f && \text{ in } (0,T) \times \Omega , \label{strong_solution_second_equation_initial_equation} \\ \partial_n \varphi_{|\partial \Omega} &= 0 && \text{ on } (0,T) \times \partial \Omega ,\\ \partial_n \Delta \varphi_{|\partial \Omega} & = 0 && \text{ on } (0,T) \times \partial \Omega , \\ \varphi (0) &= \varphi_0 && \text{ in } \{ 0 \} \times \Omega . \label{strong_solution_second_equation_initial_condition_start}\end{aligned}$$ The result follows from standard results on maximal regularity of parabolic equations, e.g. from [@MR2006641 Theorem 8.2]. Analogously to the previous part we need to ensure that $\|\mathcal L^{-1}\|_{\mathcal L (Y_T, X_T)} $ remains bounded. \[strong\_solution\_lemma\_l\_inverse\_t\_t\_0\_secondpart\] Let the assumptions of Lemma \[strong\_solution\_existence\_proof\_L\_1\_part2\] hold and $0 < T_0 < \infty$ be given. Then $$\begin{aligned} \|\mathcal L^{-1}_{2, T}\|_{\mathcal L (Y^2_T, X^2_T)} \leq \|\mathcal L^{-1}_{2, T_0}\|_{\mathcal L (Y^2_{T_0}, X^2_{T_0})} < \infty \qquad \text{ for all } 0 < T < T_0 .\end{aligned}$$ This lemma can be proven analogously to Lemma \[strong\_solution\_lemma\_l\_inverse\_t\_t\_0\_firstpart\]. From the results of this section Theorem \[thm:linear\] follows immediately. \#1[0=0=0 0 by1pt\#1]{} \[2\][ [\#2](http://www.ams.org/mathscinet-getitem?mr=#1) ]{} \[2\][\#2]{} [10]{} H. Abels, *On a diffuse interface model for two-phase flows of viscous, incompressible fluids with matched densities*, Arch. Rat. Mech. Anal. **194** (2009), no. 2, 463–506. H. Abels, *Existence of weak solutions for a diffuse interface model for viscous, incompressible fluids with general densities*, Comm. Math. Phys. **289** (2009), no. 1, 45–73. H. Abels and D. Breit, *Weak solutions for a non-[N]{}ewtonian diffuse interface model with different densities*, Nonlinearity **29** (2016), no. 11, 3426–3453. H. Abels, D. Depner, and H. Garcke, *Existence of weak solutions for a diffuse interface model for two-phase flows of incompressible fluids with different densities*, J. Math. Fluid Mech. **15** (2013), no. 3, 453–480. , *On an incompressible [N]{}avier-[S]{}tokes/[C]{}ahn-[H]{}illiard system with degenerate mobility*, Ann. Inst. H. Poincaré Anal. Non Linéaire **30** (2013), no. 6, 1175–1190. H. Abels, H. Garcke, and G. Gr[ü]{}n, *Thermodynamically consistent, frame indifferent diffuse interface models for incompressible two-phase flows with different densities*, Math. Models Methods Appl. Sci. **22** (2012), no. 3, 1150013, 40. H. Abels, H. Garcke, and J. Weber, *Existence of weak solutions for a diffuse interface model for two-phase flow with surfactants*, Commun. Pure Appl. Anal. **18** (2019), no. 1, 195–225. H. Amann, *Linear and quasilinear parabolic problems, volume 1: Abstract linear theory*, Birkh[ä]{}user, Basel - Boston - Berlin, 1995. R. Denk, M. Hieber, and J. Pr[ü]{}ss, *R-boundedness, fourier multipliers and problems of elliptic and parabolic type*, Mem. Amer. Math. Soc. **166** (2003), no. 788. S. Frigeri, *Global existence of weak solutions for a nonlocal model for two-phase flows of incompressible fluids with unmatched densities*, Math. Models Methods Appl. Sci. **26** (2016), no. 10, 1955–1993. C. G. Gal, M. Grasselli, and H. Wu, *Global weak solutions to a diffuse interface model for incompressible two-phase flows with moving contact lines and different densities*, Arch. Ration. Mech. Anal. **234** (2019), no. 1, 1–56. A. Giorgini, *Well-posedness of the two-dimensional [A]{}bels-[G]{}arcke-[G]{}rün model for two-phase flows with unmatched densities*, Preprint, arXiv:2006.13060 (2020). G. Grün, *On convergent schemes for diffuse interface models for two-phase flow of incompressible fluids with general mass densities*, SIAM J. Numer. Anal. **51** (2013), no. 6, 3036–3061. T. Runst and W. Sickel, *Sobolev spaces of fractional order, [N]{}emytskij operators, and nonlinear partial differential equations*, De Gruyter Series in Nonlinear Analysis and Applications, vol. 3, Walter de Gruyter & Co., Berlin, 1996. R. E. Showalter, *Monotone operators in [B]{}anach space and nonlinear partial differential equations*, Mathematical Surveys and Monographs, vol. 49, American Mathematical Society, Providence, RI, 1997. J. T. Weber, *[Analysis of diffuse interface models for two-phase flows with and without surfactants]{}*, Ph.D. thesis, University of Regensburg, urn:nbn:de:bvb:355-epub-342471, 2016. [^1]: *Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany* [^2]: *Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany*

--- author: - 'E. Oset' - 'V.K Magas' - 'A. Ramos' - 'H. Toki' date: 'Received: date / Revised version: date' title: Further considerations concerning claims for deeply bound kaon atoms and reply to criticisms --- [leer.eps]{} gsave 72 31 moveto 72 342 lineto 601 342 lineto 601 31 lineto 72 31 lineto showpage grestore Introduction {#intro} ============ A brief story of the recent events around deeply bound kaons atoms could be made as follows: Chiral potentials [@lutz; @angelsself; @schaffner; @cieply; @tolos] provide potentials of depth around 50 MeV attraction at a width around 100 MeV at $\rho =\rho_0$. With these potentials, deeply bound states of binding energy around 30-40 MeV are obtained, but with a width of the order of 100 MeV which would preclude the observation of peaks [@okumura]. A next step of the development appeared with the claims of a very large attractive potential in light nuclei in [@akaishi; @akainew] (AY), around 650 MeV at the center of the nucleus in [@akainew] and with the matter compressed to 10 times nuclear matter density. An experiment was made at KEK with the $K^-$ absorption at rest in $^4He$ [@suzuki] and a peak was seen and attributed first to a strange tribaryon, since its interpretation as a deeply kaon bound state would contradict the predictions of [@akaishi], but afterwords it was reinterpreted as a deeply bound kaon atom since it would match with the corrected version of the potential in [@akainew]. The FINUDA collaboration made the same experiment in different nuclei and found a broad peak in the $\Lambda p$ back to back invariant mass which was attributed to the existence of the $K^- pp$ bound state [@finuda]. With the interpretation of these peaks in clear contradiction with the predictions of the chiral potentials, theoreticians come into the scene: Oset and Toki (OT) [@prc] write a paper indicating that the peak seen at KEK is no proof of a kaon bound state since it can be interpreted in terms of absorption of the $K^-$ by a pair of nucleons going to $\Sigma p$, with the daughter nucleus left as spectator. Parallely, Oset, Magas, Ramos and Toki (MORT) write a paper [@magas] and provide an alternative explanation of the peak seen at FINUDA as coming from $K^- $ absorption in the nucleus going to $\Lambda N$, followed by the rescattering of the $\Lambda$ or the nucleon with the daughter nucleus. After that, a KEK like experiment is made at FINUDA looking at proton spectra following $K^-$ absorption at rest and a peak is indeed found in $^6 Li$ [@newfinuda] which, thanks to the measurement of pions in coincidence, allows the authors to interpret it as coming from $K^-$ absorption by a pair of nucleons going to $\Sigma p$, with the daughter nucleus left as spectator, just the explanation offered by OT in [@prc] for the KEK peak. Incidentally, a second peak seen in the KEK experiment when making a cut of slow pions, and attributed to $K^-$ absorption by a pair of nucleons going to $\Lambda p$ in [@prc] is also seen in [@newfinuda] as a feeble signal and associated to the $\Lambda p$ mechanisms as suggested in [@prc]. This peak is, however, much better seen, as a very narrow peak, in the the $\Lambda p$ back to back invariant mass spectrum of the FINUDA experiment [@finuda]. In between, two novelties have appeared from the japanese side, the experiment has been redone, with an inclusive measurement, without the cuts and acceptance of [@suzuki], and the peaks seem to disappear as reported by Iwasaki in this Conference [@iwasaki]. It was, however, indicated in the discussion that the useful measurement is the one with cuts that reduces background and stresses peaks, which has not been redone [@iwaprivate], and that the the FINUDA data on the proton spectrum showing the KEK like peaks is there to be also seriously considered. The other novelty is the paper by Akaishi and Yamazaki [@criti] criticizing both the approaches of [@prc] and [@magas], and the extra criticism of Akaishi in this Conference criticizing the chiral approach, because of the “unrealistic range” of the interaction used. Actually, no range is used for the interaction because, as we shall see below, all recent versions of the chiral approach rely upon the N/D method and dispersion relations, which only requires the knowledge of the interaction on shell, and the range of the interaction never appears in the formalism. In what follows we show that the recent criticism of Akaishi and Yamazaki, in [@criti] and of Akaishi in his talk have no base. The chiral approach and the N/D method {#sec:1} ====================================== The chiral approach of [@angels], with the on-shell factorization of the potential and the t-matrix, is based on the N/D method. One can find a systematic and easily comprehensible derivation of the ideas of the N/D method applied for the first time to the meson baryon system in [@ulfnsd], which we reproduce here below and which follows closely the similar developments used before in the meson-meson interaction [@ollernsd]. One defines the transition $T-$matrix as $T_{i,j}$ between the coupled channels which couple to certain quantum numbers. For instance in the case of $\bar{K} N$ scattering studied in [@ulfnsd] the channels with zero charge are $K^- p$, $\bar{K^0} n$, $\pi^0 \Sigma^0$,$\pi^+ \Sigma^-$, $\pi^- \Sigma^+$, $\pi^0 \Lambda$, $\eta \Lambda$, $\eta \Sigma^0$, $K^+ \Xi^-$, $K^0 \Xi^0$. Unitarity in coupled channels is written as $$Im T_{i,j} = T_{i,l} \rho_l T^*_{l,j}$$ where $\rho_i \equiv 2M_l q_i/(8\pi W)$, with $q_i$ the modulus of the c.m. three–momentum, and the subscripts $i$ and $j$ refer to the physical channels. This equation is most efficiently written in terms of the inverse amplitude as $$\label{uni} \hbox{Im}~T^{-1}(W)_{ij}=-\rho(W)_i \delta_{ij}~,$$ The unitarity relation in Eq. (\[uni\]) gives rise to a cut in the $T$–matrix of partial wave amplitudes, which is usually called the unitarity or right–hand cut. Hence one can write down a dispersion relation for $T^{-1}(W)$ $$\label{dis} T^{-1}(W)_{ij}=-\delta_{ij}~g(s)_i+ V^{-1}(W)_{ij} ~,$$ with $$\label{g} g(s)_i=\widetilde{a}_i(s_0)+ \frac{s-s_0}{\pi}\int_{s_{i}}^\infty ds' \frac{\rho(s')_i}{(s'-s)(s'-s_0)},$$ where $s_i$ is the value of the $s$ variable at the threshold of channel $i$ and $V^{-1}(W)_{ij}$ indicates other contributions coming from local and pole terms, as well as crossed channel dynamics but [*without*]{} right–hand cut. These extra terms are taken directly from $\chi PT$ after requiring the [*matching*]{} of the general result to the $\chi PT$ expressions. Notice also that $g(s)_i$ is the familiar scalar loop integral of a meson and a baryon propagators. One can simplify the notation by employing a matrix formalism. Introducing the matrices $g(s)={\rm diag}~(g(s)_i)$, $T$ and $V$, the latter defined in terms of the matrix elements $T_{ij}$ and $V_{ij}$, the $T$-matrix can be written as: $$\label{t} T(W)=\left[I-V(W)\cdot g(s) \right]^{-1}\cdot V(W)~.$$ which can be recast in a more familiar form as $$\label{ta} T(W)=V(W)+V(W) g(s) T(W)$$ Now imagine one is taking the lowest order chiral amplitude for the kernel $V$ as done in [@ulfnsd]. Then the former equation is nothing but the Bethe Salpeter equation with the kernel taken from the lowest order Lagrangian and factorized on shell, the same approach followed in [@angels], where different arguments were used to justify the on shell factorization of the kernel. The kernel V plays the role of a potential in ordinary Quantum Mechanics. The on shell factorization of the kernel, justified here with the N/D method, renders the set of coupled Bethe Salpeter integral equations a simple set of algebraic equations. The important thing to note is that both the kernel and the $T$ matrix only appear on shell, for a value of $\sqrt{s}$. The range of the interaction is never used. The loop function is made convergent via a subtraction in the dispersion relation, or equivalently a cut off in the three momentum as used in [@angels], which is proved to be equivalent to the subtraction method in [@ulfnsd]. Akaishi in his talk confuses this cut off in the loop of propagators with the range of the interaction, when they have nothing to do with each other. Even more, the theory must be cut off independent, which means, one can change arbitrarily the cut off by introducing the appropriate higher order counterterms. As a consequence of this, all pathologies of the interaction pointed out by Akaishi in his talk are a pure invention, which has nothing to do with the physics of the problem. Interpretation of the narrow FINUDA peaks and KEK peaks ======================================================= In [@finuda], for absorption in a sample of $^6 Li$, $^7 Li$, $^{12}C$, a narrow peak is seen at $M_I =2340 MeV$ of the back to back $\Lambda p$ system and a wider one at $M_I =2275 MeV$, see Fig.  \[fig:1\]. Let us assume $^7Li$ for simplicity of the discussion. The first thing to recall is the experience of pion absorption that concluded that at low pion energies the absorption was dominated by a direct two body process (even if later on there would be rescattering of the nucleons in the nucleus, giving rise to what was called indirect three body absorption in contrast with the possible direct three body absorption which had a small rate at low energies [@weyer].) We consider the $K^-$ two nucleon absorption mechanism, disregarding the one body mechanisms which do not produce $\Lambda p$ back to back, see Fig. \[fig:1\].\ [-90]{} ![$\Lambda p$ invariant mass distribution of back to back pairs following $K^-$ absorption in a mixture of nuclei, $^6 Li$, $^7 Li$ and $^{12}C$. The inset of the figure shows data corrected for the detector acceptance. From [@finuda].[]{data-label="fig:1"}](finuda.epsi "fig:"){width="30.00000%"} [**Origin of the narrow peak at $M_I =2340 MeV$**]{} : We have the reaction, $$K^- pp + (^5H spectator) \to \Lambda p + (^5H spectator)$$ The kinematics of the reaction is as follows: Let $P$ be the total momentum of the $K¯$-nucleus system, and $p_1$, $p_2$ and $p_3$ the momenta of the $\Lambda$, $p$ and $^5H$ spectator respectively. We have $$(P-p_3)^2= (p_1 +p_2 )^2= {M_{12}}^2$$ from where we deduce that $$\Delta(M_{12})=M(K~Li)\Delta(E_3)/M_{12}$$ This would lead to $\Delta(M_{12})\sim 10 MeV$ for absorption in $^4He$ and $\Delta(M_{12})\sim 1 MeV$ for $^7Li$ if one takes as representative of the Fermi momentum of the quasideuteron or $pp$ pairs $150~ MeV$ for $^4He$ and $50 ~MeV $ for $^7Li$ as suggested in [@criti]. This produces a dispersion of the $p$ momentum in the CM of the same order of magnitude. This quantity is smaller than the main source for $p$ momentum dispersion which is the boost of the proton from the CM of $\Lambda p$ to the frame where the $\Lambda p$ has the Fermi momentum of the initial NN pair, $p_{NN}$. The boost is easily implemented requiring only nonrelativistic kinematics. We have $$\vec{p_p}=\vec{p_{CM}} + m_p~ \vec{V} ;~~~~ \vec{V}=\vec{p_{NN}}/M_{12}$$ $$\Delta(\vec{p_p})^2=m_p^2 ~\vec{V}^2 /3$$ $$\Delta(p_p)= \pm 35 ~MeV/c ~(11~MeV/c)$$ $$~~~~~~~~~~~~~~~~~~for~ p_{NN}=150 ~ MeV/c ~(50 ~MeV/c)$$ Hence we would have a dispersion of proton momentum of $\pm 35 ~MeV$ for $K^-$ absorption in $^4 He$ and $\pm 11 ~MeV$ in the $^7Li$ case. The exercise has been done for $K^- pp \to \Lambda p$ but the results are the same if one has $K^- NN \to \Sigma p$. This latter reaction was the one suggested by OT in [@prc] to explain the KEK peak seen in Fig. \[fig:2\], lower left figure of the panel, around $475 ~ MeV$. The dispersion of Eq. (12) would roughly agree with the peak. [-90]{} ![Proton spectra following $K^-$ absorption in $^4 He$. Lower two figures: left with high pion momentum cut, right with lower pion momentum cut. From [@suzuki].[]{data-label="fig:2"}](kek.epsi "fig:"){width="35.00000%"} We should note that the peaks can be made more narrow, as we have checked numerically by: 1) assuming absorption from a $2p$ orbit of the $K^-$, 2) forcing the $\Lambda p$ pair to go back to back, 3) putting restrictions on the pion momenta. It is interesting to observe in this respect that in the figure of the KEK experiment in the case when the slow pions are selected (lower right figure in the panel ) one can see also a peak in the momentum distribution at $p\sim 545 MeV$, which was identified in [@prc] as coming from $K^-$ absorption going to $\Lambda p$ with the daughter nucleus as a spectator. It is interesting to see that such a signal, “a feeble signal around 580 MeV/c” is seen even in the inclusive spectrum of [@newfinuda], see Fig. \[fig:3\] , and correctly identified there as coming from $K^-$ absorption in $^6Li$ going to $\Lambda p$ (note one has smaller binding of the nucleons here than one has in $^4 He$ and there is no loss of energy as in the case of a thick target of [@suzuki]). [-90]{} ![Proton momentum distribution following the absorption of $K^-$ in $^6 Li$ from [@newfinuda].[]{data-label="fig:3"}](newfinuda.epsi "fig:"){width="30.00000%"} Coming back to the absorption of $K^-$ in $^4He$ it should be noted that the candidate reaction for the peak at 475 MeV/c is the reaction with the rate $$\Sigma^- p~d ~~~~~1.6 ~\%$$ which has been measured by [@katz]. A fraction of this reaction can go with the $d$ as a spectator, and then it is worth mentioning that the fraction of the cross section of this peak is estimated in [@suzuki] at less than 1%, of the order of 0.34 % according to [@thesis]. Interpretation of the wide FINUDA peak ====================================== Next we turn to the wide FINUDA peak in the experiment [@finuda]. This peak was interpreted as naturally coming from the absorption of a $K^-$ from the nucleus going to $\Lambda N$ followed by a rescattering of the nucleon or the Lambda with the remnant nucleus [@magas]. This is the equivalent of the quasielastic peak which appears in all inclusive nuclear reactions with a similar width which is due to the Fermi motion of the nucleons. In [@magas] a calculation was done for the mixture of the different nuclei, as in the experiment and the results are seen in Fig.  \[fig:4\] ![Theoretical calculation of [@magas] versus experiment of the $\Lambda p$ invariant mass distribution of back to back pairs following $K^-$ absorption in a mixture of nuclei, $^6 Li$, $^7 Li$ and $^{12}C$. Histogram theory, bars data from [@finuda].[]{data-label="fig:4"}](chi2.eps){width="50.00000%"} Claims of no peaks in Akaishi and Yamazaki ========================================== Another of the points in the work of [@criti] is that the peaks predicted by OT in [@prc] and MORT in [@magas] are unrealistic and that a proper calculation does not produce any peaks. The curious results are a consequence of a calculation in [@criti] that: 1\) Considers absorption by all four particles at once in the $^4He$ case, disregarding the dynamics found from pion absorption. The spectra essentially reflect phase space with four particles in the final state . 2\) Does not consider absorption by two N with spectator remnant nucleus. 3\) Does not consider angular cuts or particles in coincidence. 4\) Does not consider rescattering of particles.\ And with all this an obvious broad spectrum is obtained. Consequently with their finding the authors of [@criti] write textually: “OT further insist that the same $K^-NN$ absorption mechanism at rest persists in the case of heavier targets as well ($^7$Li and $^{12}$C). However, this proposal contradicts the FINUDA experiment [@finuda], in which they reconstructed an invariant mass spectrum of $M_{\rm inv}(\Lambda p)$. Contrary to the naive expectation of OT, the spectrum shows no such peak at 2340 MeV/$c^2$. ” This assertion could not be more illuminating of the criticism raised. The peak that Akaishi and Yamazaki claim that we predict and does not exist is the one seen exactly at 2340 MeV/$c^2$ in Fig. 3 of [@finuda], which we have reproduced in Fig.  \[fig:1\]. In case there could be some doubts about this peak let us quote textually what the authors of [@finuda] say regarding this peak: “On the other hand, the detector system is very sensitive to the existence of the two nucleon absorption mode $K^- + "pp" \to \Lambda p$ since its invariant mass resolution is 10 MeV/$c^2$ FWHM. The effect of the nuclear binding of two protons is only to move the peak position to the lower mass side of the order of separation energies of two protons ($\sim$  30 MeV), and not to broaden the peak. A sharp spike around 2.34 GeV/$c^2$ may be attributed to this process.” Incidentally this mechanism is the one proposed by [@prc] to explain the peak at 545 MeV/c in the proton spectrum when the slow pion cut is made in [@suzuki]. This peak is also the one that shows in the inclusive momentum spectrum of [@newfinuda] mentioned there as a “feeble signal ” and with the same interpretation. With their claims than no peaks should be seen from these processes AY obviously also contradict the clear peak seen in [@newfinuda] around 500 MeV/c, see Fig. \[fig:3\], which, with the detection in coincidence of the pions coming from $\Sigma \to \pi N$ decay, the authors of [@newfinuda] unmistakeably relate to the $K^- NN \to \Sigma N$ process, the mechanism proposed by OT to explain the lower peak of the spectrum in [@suzuki]. Another of the “proofs” presented in [@criti] is a spectrum of $K^- ~^4He \to \Lambda ~ d~ n$ in which no peak around 560 MeV/c is seen, as one could guess from our interpretation of the process. The comparison is, however, inappropriate. First, the number of counts is of the order of three counts per bin, as average. Second, the rate of $K^-~^4He \to \Sigma^- p~d$ of 1.6  % according to [@katz], and the rate of the peak of the KEK experiment that we attribute to this process, with a value of the order of 0.3 %, indicate that only a fraction of this process will go with the $d$ as a spectator, leading to a peak that can only be seen with far better statistics and resolution than the one in the spectrum of the $K^- ~^4He \to \Lambda ~ d~ n$ experiment [@katz]. Finally, AY present another calculation to prove that the broad FINUDA invariant mass peak requires an explanation based on the $K^- pp$ bound system by 115 MeV. Their results are presented in Fig. 7 of their paper. The calculation made is, however, simply unacceptable. They make the following assumptions: 1\) Calculation in $^4He$ and compare to experiment which is a mixture of $^6Li,~ ^7Li,~ ^{12}C$. 2\) Direct absorption by four nucleons. 3\) No dynamics, just phase space. 4\) Has no rescattering, shown by Magas to be essential to account for the peak. And with this calculation they claim that the $K^- pp$ cluster is bound by 115 MeV !!!. We should also mention here another example of inappropriate comparison. In Fig. 7 of [@criti], which aims at describing the wide FINUDA peak, a vertical line is plotted with the lable OT, presenting this as the position predicted by OT for this FINUDA peak. This comparison is out of place because OT in [@prc] never attempted to predict this broad FINUDA peak. This is done by MORT in [@magas], requiring a different mechanism, the rescattering of the proton or the $\Lambda$ after $K^-$ absorption by two nucleons [@finuda], which automatically produces a peak at lower invariant masses. conclusions =========== - Akaishi and Yamazaki criticisms of Oset Toki and Magas et al, are unfounded. - AY potential with 10 $\rho_0$ compressed matter should not be considered serious. - The claims of KEK and FINUDA for deeply bound kaons were unfounded. - The new FINUDA data on $p$ spectrum following $K^-$ absorption in $^6Li$ has been very clarifying, showing KEK like peaks and interpreting them with the suggestion of Oset and Toki. - The new calculations of Dote and Weise [@weise], and Schevchenko, Gal, Mares [@shevchenko] predicting a bound $K^- pp$ state with 50-70 MeV binding, but more that 100 MeV width, have brought new light to this issue. They do not support the deeply bound narrow $K^- pp$ systems claimed by FINUDA. - The new measurements of $^4He$ X rays by Hayano, Iwasaki et al. [@hayano] are very important to clarify the issue. They clearly contradict predictions of Akaishi based on his potential. - Interesting results from COSY, Buescher et al from $p ~d \to K^+ K^- ~^3He$ in the same direction [@grishina], clearly rejecting such large $K^-~ ^3He$ potentials. Aknowledgments ============== This work is partly supported by DGICYT contract number BFM2003-00856, the Generalitat Valenciana and the E.U. EURIDICE network contract no. HPRN-CT-2002-00311. This research is part of the EU Integrated Infrastructure Initiative Hadron Physics Project under contract number RII3-CT-2004-506078. [99]{} M. Lutz, Phys. Lett. B [**426**]{} (1998) 12 \[arXiv:nucl-th/9709073\]. A. Ramos and E. Oset, Nucl. Phys. A [**671**]{} (2000) 481 \[arXiv:nucl-th/9906016\]. J. Schaffner-Bielich, V. Koch and M. Effenberger, Nucl. Phys. A [**669**]{} (2000) 153 \[arXiv:nucl-th/9907095\]. A. Cieply, E. Friedman, A. Gal and J. Mares, Nucl. Phys. A [**696**]{} (2001) 173 \[arXiv:nucl-th/0104087\]. L. Tolos, A. Ramos and E. Oset, Phys. Rev. C [**74**]{} (2006) 015203 \[arXiv:nucl-th/0603033\]. S. Hirenzaki, Y. Okumura, H. Toki, E. Oset and A. Ramos, Phys. Rev. C [**61**]{} (2000) 055205. Y. Akaishi and T. Yamazaki, Phys. Rev. C [**65**]{} (2002) 044005. Y. Akaishi, A. Dote and T. Yamazaki, Phys. Lett. B [**613**]{} (2005) 140 \[arXiv:nucl-th/0501040\]. T. Suzuki [*et al.*]{}, Phys. Lett. B [**597**]{} (2004) 263. M. Agnello [*et al.*]{} \[FINUDA Collaboration\], Phys. Rev. Lett.  [**94**]{} (2005) 212303. E. Oset and H. Toki, Phys. Rev. C [**74**]{} (2006) 015207 \[arXiv:nucl-th/0509048\]. V. K. Magas, E. Oset, A. Ramos and H. Toki, Phys. Rev. C [**74**]{} (2006) 025206 \[arXiv:nucl-th/0601013\]; arXiv:nucl-th/0611098. M. Agnello [*et al.*]{} \[FINUDA Collaboration\], Nucl. Phys. A [**775**]{} (2006) 35. M. Iwasaki, talk in this Conference. From public discussion and M. Iwasaki, private communication. T. Yamazaki and Y. Akaishi, arXiv:nucl-ex/0609041. E. Oset and A. Ramos, Nucl. Phys. A [**635**]{} (1998) 99 \[arXiv:nucl-th/9711022\]. J. A. Oller and U. G. Meissner, Phys. Lett. B [**500**]{} (2001) 263 \[arXiv:hep-ph/0011146\]. J. A. Oller and E. Oset, Phys. Rev. D [**60**]{} (1999) 074023 \[arXiv:hep-ph/9809337\]. A. Lehmann [*et al.*]{} \[LADS Collaboration\], Phys. Rev. C [**55**]{} (1997) 2931. V. K. Magas, E. Oset and A. Ramos, Phys. Rev. Lett.  [**95**]{} (2005) 052301 \[arXiv:hep-ph/0503043\]. P. A. Katz, K. Bunnell, M. Derrick, T. Fields, L. G. Hyman and G. Keyes, Phys. Rev. D [**1**]{} (1970) 1267. T. Suzuki, PhD Thesis and H. Outa, private communication. W. Weise in this Conference and A. Dote in this Conference. N. V. Shevchenko, A. Gal and J. Mares, arXiv:nucl-th/0610022. R. Hayano, talk in this Conference. V. Y. Grishina, M. Buscher and L. A. Kondratyuk, arXiv:nucl-th/0608072.

--- abstract: 'We introduce and study the notions of hyperbolically embedded and very rotating families of subgroups. The former notion can be thought of as a generalization of peripheral structures of relative hyperbolicity groups, while the later one provides a natural framework for developing a geometric version of small cancellation theory. Examples of such families naturally occur in groups acting on hyperbolic spaces including hyperbolic and relatively hyperbolic groups, mapping class groups, $Out(F_n)$, and the Cremona group. Other examples can be found among groups acting geometrically on $CAT(0)$ spaces, fundamental groups of graphs of groups, etc. Although our technique applies to a wide class of groups, it is capable of producing new results even for well-studied particular classes. For instance, we solve two open problems about mapping class groups, and obtain some results which are new even for relatively hyperbolic groups.' author: - 'F. Dahmani, V. Guirardel, D. Osin' title: Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces --- [[^1]]{} Introduction ============ The notion of a hyperbolic space was introduced by Gromov in his seminal paper [@Gro] and since then hyperbolic geometry has proved itself to be one of the most efficient tools in geometric group theory. Gromov’s philosophy suggests that groups acting “nicely" on hyperbolic spaces have properties similar to those of free groups or fundamental groups of closed hyperbolic manifolds. Of course not all actions, even free ones, are equally good for implementing this idea. Indeed every group $G$ acts freely on the complete graph with $|G|$ vertices, which is a hyperbolic space, so to derive meaningful results one needs to impose some properness conditions on the action. Groups acting on hyperbolic spaces geometrically (i.e., properly and cocompactly) constitute the class of hyperbolic groups. More generally, one can replace properness with its relative analogue modulo a fixed collection of subgroups, which leads to the notion of a relatively hyperbolic group. These classes turned out to be wide enough to encompass many examples of interest, while being restrictive enough to allow building an interesting theory, main directions of which were outlined by Gromov [@Gro]. On the other hand, there are many examples of natural actions of non-relatively hyperbolic groups on hyperbolic spaces: the action of the fundamental group of a graph of groups on the corresponding Bass-Serre tree, the action of the mapping class groups of oriented surfaces on curve complexes, and the action of the outer automorphism groups of free groups on the Bestvina-Feighn complexes, just to name a few. Although these actions are, in general, very far from being proper, they can be used to prove many nontrivial results. The main goal of this paper is to suggest a general approach which allows to study hyperbolic and relatively hyperbolic groups, the examples mentioned in the previous paragraph, and many other classes of groups acting on hyperbolic spaces in a uniform way. The achieve this generality, we have to sacrifice “global properness" (in any reasonable sense). Instead we require the actions to satisfy “local properness", a condition that only applies to selected collections of subgroups. We suggest two ways to formalize this idea. The first way leads to the notion of a *hyperbolically embedded collection of subgroups*, which can be thought of as a generalization of peripheral structures of relatively hyperbolic groups. The other formalization is based on Gromov’s rotating families [@Gro_cat] of special kind, which we call *very rotating families of subgroups*; they provide a suitable framework to study collections of subgroups satisfying small cancellation like properties. At first glance, these two ways seem quite different: the former is purely geometric, while the latter has rather dynamical flavor. However, they turn out to be closely related to each other and many general results can be proved using either of them. On the other hand, each approach has its own advantages and limitations, so they are not completely equivalent. Groups acting on hyperbolic spaces provide the main source of examples in our paper. Loosely speaking, we show that if a group $G$ acts on a hyperbolic space ${\mathbb{X}}$ so that the action of some subgroup $H\le G$ is proper, orbits of $H$ are quasi-convex, and distinct orbits of $H$ quickly diverge, then $H$ is hyperbolically embedded in $G$. If, in addition, we assume that all nontrivial elements of $H$ act on ${\mathbb{X}}$ with large translation length, then the set of conjugates of $H$ form a very rotating family. The main tools used in the proofs of these results are the projection complexes introduced in a recent paper by Bestvina, Bromberg, and Fujiwara [@BBF] and the hyperbolic cone-off construction suggested by Gromov in [@Gro_cat]. This general approach allows us to construct hyperbolically embedded subgroups and very rotating families in many particular classes of groups, e.g., hyperbolic and relatively hyperbolic groups, mapping class groups, $Out(F_n)$, the Cremona group, many fundamental groups of graphs of groups, groups acting properly on proper $CAT(0)$ spaces and containing rank one isometries, etc. Many results previously known for hyperbolic and relatively hyperbolic groups can be uniformly reproved in the general case of groups with hyperbolically embedded subgroups, and very rotating families often provide the most convenient way of doing that. As an illustration of this idea we generalize the group theoretic analogue of Thurston’s hyperbolic Dehn surgery theorem proved for relatively hyperbolic groups in [@Osi07] (see also [@Gr_Ma]). This and other general results from our paper have many particular applications. Despite its generality, our approach is capable of producing new results even for well-studied particular classes of groups. For instance, we answer two well-known questions about normal subgroups of mapping class groups. We also show that the sole existence of non-degenerate (in a certain precise sense) hyperbolically embedded subgroups in a group $G$ has strong implications for the algebraic structure of $G$, complexity of its elementary theory, the structure of the reduced $C^\ast $-algebra of $G$, etc. Note however that the main goal of this paper is to develop a general theory for the future use rather than to prove particular results. Some further applications can be found in [@MacSis; @BW]. The paper is organized as follows. In the next section we provide a detailed outline of the paper and discuss the main definitions and results. We believe it useful to state most results in a simplified form there, as in the main body of the paper we stick to the ultimate generality which makes many statements quite technical. Section 3 establishes the notation and contains some well-known results used throughout the paper. In Sections 4 and 5 we develop a general theory of hyperbolically embedded subgroups and rotating families, respectively. Most examples are collected in Section 6. Section 7 is devoted to the proof of the Dehn filling theorem. Applications can be found in Section 8. Finally we discuss some open questions and directions for the future research in Section 9. #### Acknowledgments. We are grateful to Mladen Bestvina, Brian Bowditch, Montse Casals-Ruiz, Remi Coulon, Thomas Delzant, Pierre de la Harpe, Ilya Kazachkov, Ashot Minasyan, Alexander Olshanskii, and Mark Sapir with whom we discussed various topics related to this paper. We benefited a lot from these discussions. The research of the third author was supported by the NSF grant DMS-1006345 and by the RFBR grant 11-01-00945. **François Dahmani:** Institut Fourier, 100 rue des maths, Université de Grenoble (UJF), BP74. 38 402 Saint Martin d’Hères, Cedex France.\ E-mail: *francois.dahmani@ujf-grenoble.fr* **Vincent Guirardel:** Université de Rennes 1 263 avenue du Général Leclerc, CS 74205. F-35042 RENNES Cedex France.\ E-mail: *vincent.guirardel@univ-rennes1.fr* **Denis Osin:** Department of Mathematics, Vanderbilt University, Nashville 37240, USA.\ E-mail: *denis.osin@gmail.com* [^1]: **2010 Mathematical Subject Classification:** 20F65, 20F67, 20F06, 20E08, 57M27.

--- abstract: 'This work uses Game Theory to study the effectiveness of punishments as an incentive for rational nodes to follow an epidemic dissemination protocol. The dissemination process is modeled as an infinite repetition of a stage game. At the end of each stage, a monitoring mechanism informs each player of the actions of other nodes. The effectiveness of a punishing strategy is measured as the range of values for the benefit-to-cost ratio that sustain cooperation. This paper studies both public and private monitoring. Under public monitoring, we show that direct reciprocity is not an effective incentive, whereas full indirect reciprocity provides a nearly optimal effectiveness. Under private monitoring, we identify necessary conditions regarding the topology of the graph in order for punishments to be effective. When punishments are coordinated, full indirect reciprocity is also effective with private monitoring.' author: - Xavier Vilaça - Luís Rodrigues bibliography: - 'bibfile.bib' title: On the Effectiveness of Punishments in a Repeated Epidemic Dissemination Game --- Introduction ============ Epidemic broadcast protocols are known to be extremely scalable and robust[@Birman:99; @Deshpande:06; @Libo:08]. As a result, they are particularly well suited to support the dissemination of information in large-scale peer-to-peer systems, for instance, to support live streaming[@Li:06; @Li:08]. In such an environment, nodes do not belong to the same administrative domain. On the contrary, many of these systems rely on resources made available by self-interested nodes that are not necessarily obedient to the protocol. In particular, participants may be rational and aim at maximizing their utility, which is a function of the benefits obtained from receiving information and the cost of contributing to its dissemination. Two main incentive mechanisms may be implemented to ensure that rational nodes are not interested in deviating from the protocol: one is to rely on balanced exchanges[@Li:06; @Li:08]; other is to monitor the degree of cooperation of every node and punish misbehavior[@Guerraoui:10]. When balanced exchanges are enforced, in every interaction, nodes must exchange an equivalent amount of messages of interest to each other. This approach has the main disadvantage of requiring symmetric interactions between nodes. In some cases, more efficient protocols may be achieved with asymmetric interactions[@Deshpande:06; @Libo:08; @Guerraoui:10], where balanced exchanges become infeasible. Instead, nodes are expected to forward messages without immediately receiving any benefit in return. Therefore, one must consider repeated interactions for nodes to able to collect information about the behavior of their neighbors, which may be used to detect misbehavior and trigger punishments. Although monitoring has been used to detect and expel free-riders from epidemic dissemination protocols[@Guerraoui:10], no theoretical analysis studied the ability of punishments to sustain cooperation among rational nodes. Therefore, in this paper, we tackle this gap by using Game Theory[@Osborne:94]. The aim is to study the existence of equilibria in an infinitely repeated Epidemic Dissemination game. The stage game consists in a sequence of messages disseminated by the source, which are forwarded by every node $i$ to each neighbor $j$ with an independent probability $p_i[j]$. At the end of each stage, a monitoring mechanism provides information to each node regarding $p_i[j]$. Following work in classical Game Theory that shows that cooperation in repeated games can be sustained using punishing strategies[@Fudenberg:86], we focus on this class of strategies. We assume that there is a pre-defined target for the reliability of the epidemic dissemination process. To achieve this reliability, each node should forward every message to each of its neighbors with a probability higher than some threshold probability $p$, known a priory by the two neighboring nodes. We consider that a player $i$ defects from a neighbor $j$ if it uses a probability lower than $p$ when forwarding information to $j$. Each node $i$ receives a benefit $\beta_i$ per received message, but incurs a cost $\gamma_i$ of forwarding a message to a neighbor. Given this, we are particularly interested in determining the range of values of the ratio benefit-to-cost $(\beta_i/\gamma_i)$ that allows punishing strategies to be equilibria. The wider is this range, the more likely it is for all nodes to cooperate. The main contribution of this paper is a quantification of the effectiveness of different punishing strategies under two types of monitors: public and private. Public monitors inform every node of the actions of every other node with no delays. On the other hand, private monitoring inform only a subset of nodes of the actions of each node, and possibly with some delays. In addition, we study two particular types of punishing strategies: direct and full indirect reciprocity. In the former type, each node is solely responsible for monitoring and punishing each neighbor, individually. The latter type specifies that each misbehaving node should eventually be punished by every neighbor. More precisely, we make the following contributions: - We derive a generic necessary and sufficient condition for a punishing strategy to be a Subgame Perfect Equilibrium under public monitoring. From this condition, we also derive an upper bound for the effectiveness of strategies that use direct reciprocity as an incentive. We observe that this value decreases very quickly with an increasing reliability, in many realistic scenarios. On the other hand, if full indirect reciprocity is used, then this problem can be avoided. We derive a lower bound for the effectiveness of these strategies, which is not only independent from the desired reliability, but also close to the theoretical optimum, under certain circumstances. - Using private monitoring with delays, information collected by each node may be incomplete, even if local monitoring is perfect. We thus consider the alternative solution concept of Sequential Equilibrium, which requires the specification of a belief system that captures the belief held by each player regarding past events of which it has not been informed. For a punishing strategy to be an equilibrium, this belief must be consistent. We provide a definition of consistency that is sufficient to derive the effectiveness of punishing strategies. - Under private monitoring with a consistent belief system, we show that certain topologies are ineffective when monitoring is fully distributed. Then, we prove that, unless full indirect reciprocity is possible, the effectiveness decreases monotonically with the reliability. To avoid this problem, punishments should be coordinated, i.e., punishments applied to a misbehaving node $i$ by every neighbor of $i$ should overlap in time. We derive a lower bound for the effectiveness of full indirect reciprocity strategies with coordinated punishments. The results indicate that the number of stages during which punishments overlap should be at least of the order of the maximum delay of the monitoring mechanism. This suggests that, when implementing a distributed monitoring mechanism, delays should be minimized. The remainder of the paper is structured as follows. Section \[sec:related\] discusses some related work. The general model is provided in Section \[sec:model\]. The analysis of public and private monitoring are given in Sections \[sec:public\] and \[sec:private\], respectively. Section \[sec:conc\] concludes the paper and provides directions of future work. Related Work {#sec:related} ============ There are examples of work that use monitoring to persuade rational nodes to engage in a dissemination protocol. In Equicast[@Keidar:09], the authors perform a Game Theoretical analysis of a multicast protocol where nodes monitor the rate of messages sent by their neighbors and apply punishments whenever the rate drops below a certain threshold. The protocol is shown to be a dominating strategy. Nodes are disposed in an approximately random network, thus, the dissemination process resembles epidemic dissemination. However, given that the network is connected and nodes are expected to forward messages to every neighbor with probability $1$, there is no non-determinism in the delivery of messages. Furthermore, the authors restrict the actions available to each player by assuming that they only adjust the number of neighbors with which they interact and a parameter of the protocol. This contrasts with our analysis, where we consider non-deterministic delivery of messages and a more general set of strategies available to players. Guerraoui et al.[@Guerraoui:10] propose a mechanism that monitors the degree of cooperation of each node in epidemic dissemination protocols. The goal is to detect and expel free-riders. This mechanism performs statistical inferences on the reports provided by every node regarding its neighbors, and estimates the cooperation level of each node. If this cooperation level is lower than a minimum value, then the node is expelled from the network. The authors perform a theoretical and experimental analysis to show that this mechanism guarantees that free-riders only benefit by deviating from the protocol if the degree of deviation is not significantly high. However, no Game Theoretical analysis is performed to determine in what conditions are free-riders willing to abide to the protocol. In [@Li:06; @Li:08], the authors rely on balanced exchanges to provide incentives for nodes to cooperate in dissemination protocols for data streaming. In BAR Gossip[@Li:06], the proposed epidemic dissemination protocol enforces strictly balanced exchanges. This requires the use of a pseudo-random number generator to determine the set of interactions in every round of exchanged updates, and occasionally nodes may have to send garbage as a payment for any unbalance in the amount of information exchanged with a neighbor. A stepwise analysis shows that nodes cannot increase their utility by deviating in any step of the protocol. In FlightPath[@Li:08], the authors remove the need for sending garbage by allowing imbalanced exchanges. By limiting the maximum allowed imbalance between every pair of nodes, the authors show that it is possible for the protocol to be an $1/10$-Nash equilibrium, while still ensuring a streaming service with high quality. Unfortunately, these results might not hold for other dissemination protocols that rely on highly imbalanced exchanges. In these cases, a better alternative might be to rely on a monitoring approach. Other game theoretical analysis have addressed a similar problem, but in different contexts. In particular, the tit-for-tat strategy used in BitTorrent, a P2P file sharing system, has been subjected to a wide variety of Game Theoretical analysis[@Feldman:04; @Rahman:11; @Qiu:04; @Levin:08]. These works consider a set of $n$ nodes deciding with which nodes to cooperate, given a limited number of available connections, with the intent to share content. Therefore, contrary to our analysis, there is no non-determinism in content delivery. Closer to our goal is the trend of work that applies game theory to selfish routing[@Srivastava:06; @Felegyhazi:06; @Ji:06]. In this problem, each node may be a source of messages to be routed along a fixed path of multiple relay nodes to a given destination. The benefit of a node is to have its messages delivered to the destination, while it incurs the costs of forwarding messages as a relay node. This results in a linear relationship between the actions of a player and the utility of other nodes. In our case, that relation is captured by the definition of reliability, which is non-linear. Consequently, the utility functions of an epidemic dissemination and a routing game possess an inherently different structure. Model {#sec:model} ===== We now describe the System and Game Theoretical models, followed by the definition of effectiveness. In Appendix \[sec:epidemic\], we provide a more thorough description of the considered epidemics model and include some auxiliary results that are useful for the analysis. System Model ------------ There is a set of nodes ${\cal N}$ organized into a directed graph $G$. This models a P2P overlay network with a stable membership. Each node has a set of in (${\cal N}_i^{-1}$) and out-neighbors (${\cal N}_i$). Communication channels are assumed to be reliable. We model the generation of messages in this network by considering the existence of a single external source $s$. Its behavior is described by a profile $\vec{p}_s$, which defines for each node $i \in {\cal N}_s$ the probability $p_s[i] \in [0,1)$ of $i$ receiving a message directly from $s$, with the restriction that $p_s[i] > 0$ for some $i$. We consider the graph to be connected from the source $s$, i.e., there exists a path from $s$ to every node $i \in {\cal N}$. Conversely, every node $i$ forwards messages to every neighbor $j \in {\cal N}_i$ with an independent probability $p_{i}[j]$. Provided a profile of probabilities $\vec{p}$, which includes the vector of probabilities $\vec{p}_s$ and $\vec{p}_i$ used by $s$ and by every node $i$, respectively, we can define the reliability of the dissemination protocol as the probability of a node receiving a message. For the analysis, it is convenient to consider the probability of a node $i$ *not* receiving a message, denoted by $q_i[\vec{p}]$. The reliability of the protocol is then defined by $1-q_i[\vec{p}]$. The exact expression of $q_i$ is included in Appendix \[sec:epidemic\]. Monitoring Mechanism. --------------------- The monitoring mechanism emits a signal $s \in {\cal S}$, where every player $i$ may observe a different private signal $s_i \in s$. This signal can take two values for every pair of nodes $j \in {\cal N}$ and $k\in{\cal N}_j$: $s_i[j,k] = \mbox{\emph{cooperate}}$ notifies $i$ that $j$ forwarded messages to $k$ with a probability higher than a specified threshold, and $s_i[j,k] = \mbox{\emph{defect}}$ signals the complementary action. This signal may be public, if all nodes read the same signal, or private, otherwise. Moreover, if the signal is perfectly correlated with the action taken by a node, then monitoring is perfect; otherwise, monitoring is said to be imperfect. We consider that monitoring is performed locally by every node. A possible implementation of such monitoring mechanism in the context of P2P networks can be based on the work of [@Guerraoui:10]. A simpler and cheaper mechanism would instead consist in every out-neighbor $j$ of a given node $i$ recording the fraction of messages sent by $i$ to $j$ during the dissemination of a fixed number $M$ of messages. Then, $j$ may use this information along with an estimate of the reliability of the dissemination of messages to $i$ ($1-q_i$) in order to determine whether $i$ is cooperating or defecting. When a defection is detected, $j$ is disseminates an accusation against $i$ towards other nodes. If $i$ is expected to use $p_i[j]<1$ towards $j$, then monitoring is imperfect. Furthermore, accusations may be blocked, disrupted, or wrongly emitted against one node due to both malicious and rational behavior. However, in this paper, we consider only perfect monitoring, faithful propagation of accusations, and that nodes are rational. Almost perfect monitoring can be achieved with a large $M$. Faithful propagation may be reasonable to assume if the impact of punishments on the reliability of each non-punished node is small and the cost of sending accusations is not significant. We intend to relax these assumptions in future work. In our model, an accusation emitted by a node $j$ against an in-neighbor $i$ may only be received by the nodes that are reachable from $j$ by following paths in the graph. In addition, if we consider the obvious possibility that $i$ might block any accusation emitted by one of its neighbors, then these paths cannot cross $i$. Finally, the number of nodes informed of each defection may be further reduced to minimize the monitoring costs. This restricts the set of in-neighbors of $i$ that may punish $i$ for defecting $j$. In this paper, we consider two alternative models. First, we study public monitoring, where all nodes may be informed about any defection with no delays. Then, we study the private monitoring case, taking into consideration the possible delay of the dissemination of accusations. Game Theoretical Model ---------------------- Our model considers an infinite repetition of a stage game. Each stage consists in the dissemination of a sequence of messages and is interleaved with the execution of the monitoring mechanism, which provides every node with some information regarding the actions taken by other nodes during the stage game. ### Stage Game. The stage game is modeled as a strategic game. An action of a player $i$ is a vector of probabilities $\vec{p}_i \in {\cal P}_i$, such that $p_i[j]>0$ only if $j \in {\cal N}_i$. Thus, $\vec{p}_i$ represents the average probability used by $i$ to forward messages during the stage. It is reasonable to consider that $i$ adheres to $\vec{p}_i$ during the complete stage, since $i$ expects to be monitored by other nodes with regard to a given $\vec{p}_i$. Hence, changing strategy is equivalent to following a different $\vec{p}_i$. Despite $s$ not being a player, for simplicity, we consider that every profile $\vec{p} \in {\cal P}$ implicitly contains $\vec{p}_s$. We can also define a mixed strategy $a_i \in {\cal A}_i$ as a probability distribution over ${\cal P}_i$, and a profile of mixed strategies $\vec{a} \in {\cal A}$ as a vector containing the mixed strategies followed by every player. The utility of a player $i$ is a function of the benefit $\beta_i$ obtained per received message and the cost $\gamma_i$ of forwarding a message to each neighbor. More precisely, this utility is given by the probability of receiving messages ($1-q_i[\vec{p}]$) multiplied by the difference between the benefit per message ($\beta_i$) and the expected cost of forwarding that message to every neighbor ($\gamma_i \sum_{j \in {\cal N}_i} p_{i}[j]$): $$u_i[\vec{p}] = (1-q_i[\vec{p}])(\beta_i - \gamma_i \sum_{j \in {\cal N}_i} p_{i}[j]).$$ If players follow a profile of mixed strategies $\vec{a}$, then the expected utility is denoted by $u_i[\vec{a}]$, which definition depends on the structure of every $a_j$. ### Repeated Game. The repeated game consists in the infinite interleaving between the stage game and the execution of the monitoring mechanism, where future payoffs are discounted by a factor $\omega_i$ for every player $i$. The game is characterized by (possibly infinite) sequences of previously observed signals, named histories. The set of finite histories observed by player $i$ is represented by ${\cal H}_i$ and ${\cal H} = ({\cal H}_i)_{i \in {\cal N}}$ is the set of all histories observed by any player. A pure strategy for the repeated game $\sigma_i \in \Sigma_i$ maps each history to an action $\vec{p}_i$, where $\vec{\sigma} \in \Sigma$ is a profile of strategies. Consequently, $\vec{\sigma}[h]$ specifies for some history $h \in {\cal H}$ the profile of strategies $\vec{p}$ for the stage game to be followed by every node after history $h$ is observed. A behavioral strategy $\sigma_i$ differs from a pure strategy only in that $i$ assigns a probability distribution $a_i \in {\cal A}_i$ over the set of actions for the stage game. For simplicity, we will use the same notation for the two types of strategies. The expected utility of player $i$ after having observed history $h_i$ is given by $\pi_i[\vec{\sigma}|h_i]$. The exact definitions of equilibrium and expected utility depend on the type of monitoring being implemented. Hence, these definitions will be provided in each of the sections regarding public and private monitoring. ### A Brief Note on Notation. Throughout the paper, we will conveniently simplify the notation as follows. Whenever referring to a profile of strategies $\vec{\sigma}$, followed by all nodes except $i$, we will use the notation $\vec{\sigma}_{-i}$. Also, $(\sigma_i,\vec{\sigma}_{-i})$ denotes the composite of a strategy $\sigma_i$ and a profile $\vec{\sigma}_{-i}$. The same reasoning applies to profiles of pure and mixed strategies of the stage game. Finally, we will let $(h,s)$ denote the history that follows $h$ after signal $s$ is observed. Effectiveness ------------- We know from Game Theoretic literature that certain punishing strategies can sustain cooperation if the discount factor $\omega_i$ is sufficiently close to $1$[@Fudenberg:86]. This minimum value is a function of the parameters $\beta_i$ and $\gamma_i$ for every player $i$. More precisely, for larger values of the benefit-to-cost ratio $\beta_i/\gamma_i$, the minimum required value of $\omega_i$ is smaller. In addition, for certain values of the benefit-to-cost ratio, no value of $\omega_i$ can sustain cooperation. Notice that these parameters are specified by the environment and thus cannot be adjusted in the protocol. Thus, a strategy is more effective if it is an equilibrium for wider ranges of $\omega_i$, $\beta_i$, and $\gamma_i$. In this paper, we only measure the effectiveness of a profile of strategies $\vec{\sigma}$ as the allowed range of values for the benefit-to-cost ratio. The effectiveness of a profile $\vec{\sigma} \in \Sigma$ is given by $\psi[\vec{\sigma}] \subseteq [0,\infty)$, such that, if, for every $i \in {\cal N}$, $\frac{\beta_i}{\gamma_i} \in \psi[\vec{\sigma}]$, then there exists $\omega_i \in (0,1)$ for every $i \in {\cal N}$ such that $\vec{\sigma}$ is an equilibrium. Public Monitoring {#sec:public} ================= In this section, we assume that the graph allows public monitoring to be implemented. That is, every node is informed about each defection at the end of the stage when the defection occurred. We can thus simplify the notation by considering only public signals $s \in {\cal S}$ and histories $h \in {\cal H}$. With perfect monitoring, the public signal observed after players follow $\vec{p} \in {\cal P}$ is deterministic. This type of monitoring requires accusations to be broadcast. However, since the dissemination of accusations is interleaved with the dissemination of a sequence of messages, monitoring costs may not be relevant if the size of each accusation is small, compared to the size of messages being disseminated. The section is organized as follows. We start by providing a general definition of punishing strategies and then introduce the definition of expected utility and the solution concept for public monitoring. We then proceed to a Game Theoretical analysis, where we analyze punishing strategies that use direct reciprocity and full indirect reciprocity. Public Signal and Punishing Strategies -------------------------------------- We study a wide variety of punishing strategies, by considering a parameter ${\tau}$ that specifies the duration of punishments. Of particular interest to this analysis is the case where the duration of punishments is infinite, which is known in the Game Theoretical literature as the Grim-trigger strategy. Furthermore, a punishing strategy specifies a Reaction Set ${\mbox{RS}}[i,j] \subseteq {\cal N}$ of nodes that are expected to react to every defection of $i$ from $j$ during ${\tau}$ stages. This set always contains $i$ and $j$, but it may also contain other nodes. In particular, a third node $k \in {\mbox{RS}}[i,j]$ that is an in-neighbor of $i$ ($k \in {\cal N}_i^{-1}$) is expected to stop forwarding any messages to $i$, as a punishment. If $k$ is not a neighbor of $i$, then $k$ may also adapt the probabilities used towards its out-neighbors, for instance, to keep the reliability high for every unpunished node. In order for a node $j \in {\cal N}$ to monitor an in-neighbor $i \in {\cal N}_j^{-1}$, the protocol must define for every history $h \in {\cal H}$ a threshold probability $p_i[j|h]$ with which $i$ should forward messages to $j$. Since $h$ is public, $p_i[j|h]$ is common knowledge between $i$ and $j$, allowing for an accurate monitoring. Given this, the public signal for perfect public monitoring is defined as follows. \[def:pubsig\] For every $h \in {\cal H}$ and $\vec{p}' \in {\cal P}$, let $s = {\mbox{sig}}[\vec{p}'|h]$ be the public signal observed when players follow $\vec{p}'$. For every $i \in {\cal N}$ and $j \in {\cal N}_i$, $s[i,j] = \mbox{\emph{cooperate}}$ if and only if $p_i'[j] \geq p_i[j|h]$. Then, a punishing strategy becomes a set of rules specifying how every $p_i[j|h]$ should be defined. Namely, let $\sigma_i^* \in \Sigma_i$ denote a punishing strategy, which specifies that after a history $h$ every node $i$ should forward messages to a neighbor $j$ with probability $p_i[j|h]$. We will denote by $\vec{\sigma}^* \in \Sigma$ the profile of punishing strategies. The restrictions imposed on every $p_i[j|h]$ can be defined as follows. Every node $i$ evaluates the set of defections observed in a history $h$ by $i$ and every neighbor $j$ to which both nodes should react. Basing on this information, $i$ uses a deterministic function to determine the probability $p_i[j|h]$. For convenience, we will define $p_i[k|h] = 0$ for every node $k \in {\cal N} \setminus {\cal N}_i$ that is not an out-neighbor of $i$. For the precise definition of punishing strategy, we need an additional data structure called Defection Set (${\mbox{DS}}_i[j|h]$) containing the set of defections to which both $i$ and $j$ are expected to react, according to ${\mbox{RS}}$. This information is specified in the form of tuples $(k_1,k_2,r)$ stating that both $i$ and $j$ are expected to react to a defection of $k_1$ from $k_2$ that occurred in the previous $r$-th stage. This way, $p_i[j|h]$ is defined as a function of ${\mbox{DS}}_i[j|h]$. Namely, if ${\mbox{DS}}_i[j|h]$ contains some defection of $j$ from $k$ and $i$ should react to it ($i \in{\mbox{RS}}[j,k]$) or if $i$ defected from $j$, then $p_i[j|h]$ is $0$. Otherwise, $i$ forwards messages to $j$ with any positive probability that is a deterministic function of ${\mbox{DS}}_i[j|h]$. \[def:thr\] Define ${\mbox{DS}}_i[j|h] \subseteq {\cal N} \times {\cal N} \times \mathbb{Z}$ as follows for every $i \in {\cal N}$, $j \in {\cal N}_i$, and $h \in {\cal H}$: - ${\mbox{DS}}_i[j|\emptyset] = \emptyset$. - For $h = (h',s)$, ${\mbox{DS}}_i[j|h] = L_1 \cup L_2$, where: 1. $L_1 = \{(k_1,k_2,r+1) | (k_1,k_2,r) \in {\mbox{DS}}_i[j|h'] \land r +1 < {\tau})\}$. 2. $L_2 = \{(k_1,k_2,0) | k_1, k_2 \in {\cal N} \land i,j \in {\mbox{RS}}[k_1,k_2] \land s[k_1,k_2] = \mbox{\emph{defect}}\}$. For every $h \in {\cal H}$, $i \in {\cal N}$, and $j \in {\cal N}_i$: - If there exists $r <{\tau}$ such that $(i,j,r) \in {\mbox{DS}}_i[j|h]$, then $p_i[j|h] = 0$. - If there exist $r < {\tau}$ and $k \in {\cal N}_j$ such that $(j,k,r) \in {\mbox{DS}}_i[j|h]$, then $p_i[j|h] = 0$. - Otherwise, $p_i[j|h]$ is a positive function of ${\mbox{DS}}_i[j|h]$. We consider that the source $s$ also abides to this strategy. In Section \[sec:pub-evol\], we will show that it follows by construction that if some node $k$ observes a defection of $i$ from a node $j$ and $k \in {\mbox{RS}}[i,j]$, then $k$ reacts to this defection during ${\tau}$ stages, regardless of the ensuing actions of $i$ and the current punishments being applied. In addition, after defecting some neighbor $j$, $i$ does not forward messages to any node of ${\mbox{RS}}[i,j]$ in any of the following ${\tau}$ stages. Expected Utility and Solution Concept ------------------------------------- The expected utility of a profile of pure strategies for every player $i$ and history $h$ is given by: $$\label{eq:pure-util} \pi_i[\vec{\sigma}|h] = u_i[\vec{p}] + \omega_i \pi_i[\vec{\sigma}|(h,{\mbox{sig}}[\vec{p}|h])],$$ where $\vec{p} = \vec{\sigma}[h]$. Conversely, we can define the expected utility for a profile of behavioral strategies as follows: $$\label{eq:behave-util} \pi_i[\vec{\sigma}|h] = u_i[\vec{a}] + \omega_i \sum_{s \in {\cal S}} \pi_i[\vec{\sigma}|(h,s)]pr[s|\vec{a},h],$$ where $\vec{a} = \vec{\sigma}[h]$ and $pr[s|\vec{a},h]$ is defined as $$pr[s|\vec{a},h] = \prod_{j \in {\cal N}}pr_{j}[s|a_j,h],$$ where $pr_{j}[s|a_j,h]$ is the probability of the actions of $j$ in $a_j$ leading to $s$. The considered solution concept for this model is the notion of Subgame Perfect Equilibrium (SPE)[@Osborne:94], which refines the solution concept of Nash Equilibrium (NE) for repeated games. In particular, a profile of strategies is a NE if no player can increase its utility by deviating, given that other players follow the specified strategies. The solution concept of NE is adequate for instance for strategic games, where players choose their actions prior to the execution of the game. However, in repeated games, players have multiple decision points, where they may adapt their actions according to the observed history of signals. In this case, the notion of NE ignores the possibility of players being faced with histories that are not consistent with the defined strategy, e.g., when some defection is observed. This raises the possibility of the equilibrium only being sustained by non-credible threats. To tackle this issue, the notion of SPE was proposed, which requires in addition the defined strategy to be a NE after any history. This intuition is formalized as follows. A profile of strategies $\vec{\sigma}^*$ is a Subgame Perfect Equilibrium if and only if for every player $i \in {\cal N}$, history $h \in {\cal H}$, and strategy $\sigma_i' \in \Sigma_i$, $$\pi_i[\sigma_i^*,\vec{\sigma}_{-i}^*|h] \geq \pi_i[\sigma_i',\vec{\sigma}_{-i}^*|h].$$ While this definition considers variations in strategies, it is possible to analyze only variations in the first strategy for the stage game after any history and holding $\vec{\sigma}^*$ for the remaining stages, as stated by the one-deviation property. For any $a_i' \in {\cal A}_i$ and $\vec{p}_i' \in {\cal P}_i$, let $\sigma_i^*[h|a_i']$ and $\sigma_i^*[h|\vec{p}_i']$ denote the strategies where $i$ always follows $\sigma_i^*$, except after history $h$, when it chooses $a'_i$ and $\vec{p}_i'$ respectively. The same notation will be used for profiles $\vec{a}' \in {\cal A}$ and $\vec{p}' \in {\cal P}$, namely, $\vec{\sigma}^*[h|\vec{a}']$ and $\vec{\sigma}^*[h|\vec{p}']$, respectively. The following property captures the above intuition, which is known to be true from Game Theoretic literature and can be proven in a similar fashion to [@Blackwell:65]: \[prop:one-dev\] **One-deviation.** A profile of strategies $\vec{\sigma}^*$ is a SPE if and only if for every player $i \in {\cal N}$, history $h \in {\cal H}$, and $a_i' \in {\cal A}_i$, $\pi_i[\sigma_i^*,\vec{\sigma}_{-i}^*|h] \geq \pi_i[\sigma_i',\vec{\sigma}_{-i}^*|h]$, where $\sigma_i' = \sigma_i^*[h|a_i']$. Evolution of the Network {#sec:pub-evol} ------------------------ After any history $h \in {\cal H}$, the network induced by $h$ when players follow $\vec{\sigma}^*$ can be characterized by a subgraph, where a link $(i,j)$ is active iff $i$ is not punishing $j$ and $i$ has not defected from $j$ in the last ${\tau}$ stages. All the remaining links are inactive. When considering a profile of punishing strategies $\vec{\sigma}^*$, the evolution of this subgraph over time is deterministic. That is, after a certain number of stages, inactive links become active, such that at most after ${\tau}$ stages we obtain the original graph. Given this, we prove some correctness properties of the punishment strategy, which require the introduction of some auxiliary notation. The complete proofs are in Appendix \[sec:proof-pub-evol\]. All the considered proofs are performed by induction. For any profile of pure strategies $\vec{\sigma} \in \Sigma$ and $h \in {\cal H}$, let ${\mbox{hist}}[h,r|\vec{\sigma}]$ denote the history resulting from players following $\vec{\sigma}$ during $r$ stages, after having observed $h$. That is: $${\mbox{hist}}[h,r|\vec{\sigma}] = (h,(s^{r'})_{r' \in \{1\ldots r\}}),$$ such that $s^1 = {\mbox{sig}}[\vec{\sigma}[h]|h]$ and for every $r' \in \{1\ldots r-1\}$ we have $s^{r'+1} = {\mbox{sig}}[\vec{\sigma}[h']|h']$, where $h' = {\mbox{hist}}[h,r'|\vec{\sigma}]$. Notice that ${\mbox{hist}}[h,0|\vec{\sigma}] = h$ for every $h \in {\cal H}$ and $\vec{\sigma} \in \Sigma$. The following notation will be useful in the analysis, where, for every $r>0$, $h' = {\mbox{hist}}[h,r-1|\vec{\sigma}]$ and $\vec{p}' = \vec{\sigma}[h']$: - $q_i[h,r|\vec{\sigma}] = q_i[\vec{p}']$. - $\bar{p}_i[h,r|\vec{\sigma}] = \sum_{j \in {\cal N}_i} p_i'[j]$. - $u_i[h,r|\vec{\sigma}] = u_i[\vec{p}'] = (1-q_i[h,r|\vec{\sigma}])(\beta_i - \gamma_i \bar{p}_i[h,r|\vec{\sigma}])$. - ${\cal N}_i[h] = \{j \in {\cal N}_i|p_i[j|h]>0\}$. The following lemma characterizes the evolution of the punishments being applied to any pair of nodes. \[lemma:corr-0\] For every $h \in {\cal H}$, $r \in \{1 \ldots {\tau}-1\}$, $i \in {\cal N}$, and $j \in {\cal N}_i$, $${\mbox{DS}}_i[j|h_r^*] = \{(k_1,k_2,r'+r) | (k_1,k_2,r') \in {\mbox{DS}}_i[j|h] \land r' + r < {\tau}\},$$ where $h_r^* = {\mbox{hist}}[h,r|\vec{\sigma}^*]$. The proof is by induction, where for the initial case it follows immediately from the definition of ${\mbox{DS}}_i$ that for every $(k_1,k_2,r' ) \in {\mbox{DS}}_i[j|h]$ such that $r' < 1$, $r'$ is incremented by $1$ in the next stage. Inductively, after $r$ stages, either $r' + r< {\tau}$ and $(k_1,k_2,r' +r) \in {\mbox{DS}}_i[j|h]$ or $(k_1,k_2,r')$ has been removed from ${\mbox{DS}}_i$. (). From this lemma, we obtain the following trivial corollary that simply states that every punishment ends after ${\tau}$ stages. This is true by the fact that for every $h \in {\cal H}$, $i \in {\cal N}$, $j \in {\cal N}_i$, and $(k_1,k_2,r') \in {\mbox{DS}}_i^{h}[j]$, we have $r' < \tau$. \[corollary:corr-0\] For every $h \in {\cal H}$, $r \geq {\tau}$, $i \in {\cal N}$, and $j \in {\cal N}_i$, it holds ${\mbox{DS}}_i[j|h_r^*] = \emptyset$, where $h_r^* = {\mbox{hist}}[h,r|\vec{\sigma}^*]$. The following lemma proves that every node $i$ that defects from a neighbor $j$ expects to be punished exactly during the next ${\tau}$ stages, regardless of the following actions of $i$ or the punishments already being applied to $i$. The auxiliary notation ${\mbox{CD}}_i[\vec{p}'|h]$ is used to denote the characterization of the defections performed by $i$ in $\vec{p}'$ after history $h$. More precisely, for every $i \in {\cal N}$, $${\mbox{CD}}_i[\vec{p}'|h] =\{j \in {\cal N}_i | p_i'[j] < p_i[j|h]\}.$$ \[lemma:corr-1\] For every $h \in {\cal H}$, $\vec{p}' \in {\cal P}$, $r \in \{1 \ldots {\tau}\}$, $i \in {\cal N}$, and $j \in {\cal N}_i$, $$\label{eq:res-corr1} {\mbox{DS}}_i[j|h_r'] = {\mbox{DS}}_i[j|h_r^*] \cup \{(k_1,k_2,r-1) | k_1,k_2 \in {\cal N} \land k_2 \in {\mbox{CD}}_{k_1}[\vec{p}'|h] \land i,j \in {\mbox{RS}}[k_1,k_2]\},$$ where $h_r^* = {\mbox{hist}}[h,r|\vec{\sigma}^*]$, $h_r' = {\mbox{hist}}[h,r| \vec{\sigma}']$, and $\vec{\sigma}' =\vec{\sigma}^*[h|\vec{p}']$ is the profile of strategies where all players follow $\vec{p}'$ in the first stage. By induction, the base case follows from the definition of ${\mbox{DS}}_i[j|h]$ and the fact that $i$ registers every defection of $k_1$ to $k_2$ detected in $\vec{p}'$ by adding $(k_1,k_2,0)$ to ${\mbox{DS}}_i[j|h]$. Inductively, after $r\leq{\tau}$ stages, this pair is transformed into $(k_1,k_2,r-1)$. (). From the previous lemmas, it follows that any punishment ceases after ${\tau}$ stages, which is proven in Lemma \[lemma:corr-2\]. \[lemma:corr-2\] For every $h \in {\cal H}$, $\vec{p}' \in {\cal P}$, $r > {\tau}$, $i \in {\cal N}$, and $j \in {\cal N}_i$, $$\label{eq:res-corr2} {\mbox{DS}}_i[j|h_r'] = {\mbox{DS}}_i[j|h_r^*] = \emptyset,$$ where $h_r^* = {\mbox{hist}}[r|\vec{\sigma}^*]$, $h_r' = {\mbox{hist}}[r| \vec{\sigma}']$, and $\vec{\sigma}' = \vec{\sigma}^*[h|\vec{p}']$. From Corollary \[corollary:corr-0\] and Lemma \[lemma:corr-2\], it follows that after ${\tau}$ stages, every pair $(k_1,k_2,r)$ is removed from ${\mbox{DS}}_i[j|h]$. (). Generic Results {#sec:gen-cond} --------------- This section provides some generic results. Namely, we first derive the theoretically optimal effectiveness, which serves as an upper bound for the effectiveness of any profile of strategies. Then, we derive a simplified generic necessary and sufficient condition for any profile of strategies to be a SPE. The complete proofs are in Appendix \[sec:proof:gen-cond\]. Proposition \[prop:folk\] establishes a minimum necessary benefit-to-cost ratio for any profile of strategies to be a SPE of the repeated Epidemic Dissemination Game. Intuitively, the benefit-to-cost ratio must be greater than the expected costs of forwarding messages to neighbors ($\bar{p}_i = \sum_{j \in {\cal N}_i}p_i[j|\emptyset]$), since otherwise a player has incentives to not forward any messages. This is the minimum benefit-to-cost ratio that provides an enforceable utility as defined by the Folk Theorems[@Fudenberg:86], given that the utility that results from nodes following any profile $\vec{p} \in {\cal P}$ is feasible and the minmax utility is $0$. Consequently, this establishes an upper bound for the effectiveness of any strategy. \[prop:folk\] For every profile of punishing strategies $\vec{\sigma}^*$, if $\vec{\sigma}^*$ is a SPE, then, for every $i \in {\cal N}$, $\frac{\beta_i}{\gamma_i} >\bar{p}_i$. Consequently, $\psi[\vec{\sigma}^*] \subseteq (v,\infty)$, where $v = \max_{i \in {\cal N}} \bar{p}_i$. (). A necessary and sufficient condition for any profile of punishing strategies to be an equilibrium is that no node has incentives to stop forwarding messages to any subset of neighbors, i.e., to drop those neighbors. This condition is named the DC Condition, which is defined as follows: **DC Condition.** \[drop:condition\] For every player $i \in {\cal N}$, history $h \in {\cal H}$, and $D \subseteq {\cal N}_{i}[h]$, $$\label{eq:gen-cond} \sum_{r=0}^{\tau}\omega_i^r (u_i[h,r|\vec{\sigma}^*] - u_i[h,r|\vec{\sigma}']) \geq 0,$$ where $\vec{\sigma}' = (\sigma_i',\vec{\sigma}_{-i}^*)$, $\sigma_i' = \sigma_i^*[h|\vec{p}_i']$, and $\vec{p}_i'$ is defined as: - For every $j \in D$, $p_i'[j] = 0$. - For every $j \in {\cal N}_i \setminus D$, $p_i'[j] = p_i[j|h]$. The following Lemma shows that the DC Condition is necessary. \[lemma:gen-cond-nec\] If $\vec{\sigma}^*$ is a SPE, then the DC Condition is fulfilled. By the One-deviation property, for a profile to be a SPE, a player $i$ must not be able to increase its utility by unilaterally deviating in the first stage. In particular, this is true if $i$ deviates by dropping any subset $D$ of neighbors. Furthermore, since any punishment ends after ${\tau}$ stages, by Lemma \[lemma:corr-2\], we have that if nodes follow the deviating profile $\vec{\sigma}'$, then for every $r > {\tau}$, $$u_i[h,r|\vec{\sigma}^*] =u_i[h,r|\vec{\sigma}'].$$ The DC Condition follows by the One-deviation property and the fact that $$\pi_i[\vec{\sigma}^*|h] - \pi_i[\vec{\sigma}'|h] = \sum_{r = 0}^\infty \omega_i^r(u_i[h,r|\vec{\sigma}^*]-u_i[h,r|\vec{\sigma}']).$$ (). In Lemma \[lemma:gen-cond-suff\], we also show that the DC Condition is sufficient. In order to prove this, we first need to show that every node $i$ cannot increase its utility by not following a pure strategy in every stage game and by not forwarding messages with a probability in $\{0,p_i[j|h]\}$ to every neighbor $j$. This is shown in two steps. First, Lemma \[lemma:best-response1\] proves that any local best response mixed strategy only gives positive probability to an action in $\{0,p_i[j|h]\}$. Second, Lemma \[lemma:best-response2\] proves that there is a pure strategy for the stage game that is a local best response. Define the set of local best response strategies for history $h \in {\cal H}$ and any $i \in {\cal N}$ as: $$BR[\vec{\sigma}_{-i}^*|h] = \{a_i \in {\cal A}_i | \forall_{a_i' \in {\cal A}_i} \pi_i[(\sigma_i^*[h|a_i],\vec{\sigma}_{-i}^*)|h] \geq \pi_i[(\sigma_i^*[h|a_i'],\vec{\sigma}_{-i}^*)|h]\}.$$ Notice that $BR[\vec{\sigma}_{-i}^*|h]$ is not empty. The following lemma first proves that every player $i$ always uses probabilities in $ \{0,p_i[j|h]\}$ towards a neighbor $j$. \[lemma:best-response1\] For every $i \in {\cal N}$, $h \in {\cal H}$, $a_i \in BR[\vec{\sigma}_{-i}^*|h]$, and $\vec{p}_i \in {\cal P}_i$ such that $a_i[\vec{p}_i] > 0$, it is true that for every $j \in {\cal N}_i$ we have $p_i[j] \in \{0,p_i[j|h]\}$. The proof is by contradiction. Namely, assume that for some $\vec{p}_i$ and $a_i$ that is a best response and $a_i[\vec{p}_i] > 0$, we have that $\vec{p}_i$ does not fulfill the restrictions defined above. We can find $a_i'$ and $\vec{p}_i'$ such that: - $\vec{p}_i'$ fulfills the restrictions defined in the lemma. - $a_i'[\vec{p}_i'] = a_i[\vec{p}_i] + a_i[\vec{p}_i']$ and $a_i'[\vec{p}_i] = 0$. By letting $\vec{p}^* = \vec{\sigma}^*[h]$, we have that $${\mbox{sig}}[(\vec{p}_i,\vec{p}^*_{-i})|h] = {\mbox{sig}}[(\vec{p}_i',\vec{p}^*_{-i})|h].$$ Consequently, $$\pi_i[\sigma_i^*[h|a_i],\vec{\sigma}_{-i}^*|h] < \pi_i[\sigma_i^*[h|a_i'],\vec{\sigma}_{-i}^*|h].$$ Thus, $a_i$ cannot be a best response, which is a contradiction. (). We now have to show that there exists a pure strategy in $BR[\vec{\sigma}_{-i}^*|h]$. \[lemma:best-response2\] For every $h \in {\cal H}$ and $i \in {\cal N}$, there exists $a_i \in BR[\vec{\sigma}_{-i}^*| h]$ and $\vec{p}_i \in {\cal P}_i$ such that $a_i[\vec{p}_i] = 1$. First, notice that if only pure strategies are best-responses, then the result follows immediately. If there exists a mixed strategy $a_i$ that is a best response, then $i$ must be indifferent between following any profile $\vec{p}_i$ such that $a_i[\vec{p}_i]>0$. Otherwise, $i$ could find a better strategy $a_i'$. In that case, any such profile $\vec{p}_i$ is a best response. (). Lemma \[lemma:best-response\] is a direct consequence of Lemmas \[lemma:best-response1\] and \[lemma:best-response2\]. \[lemma:best-response\] For every $h \in {\cal H}$ and $i \in {\cal N}$, there exists $\vec{p}_i \in {\cal P}_i$ and a pure strategy $\sigma_i=\sigma_i^*[h|\vec{p}_i]$ such that: 1. For every $j \in {\cal N}_i$, $p_i[j] \in \{0,p_{i}[j|h]\}$. 2. For every $a_i \in {\cal A}_i$, $\pi_i[\sigma_i,\vec{\sigma}_{-i}^*|h] \geq \pi_i[\sigma_i',\vec{\sigma}_{-i}^*|h]$, where $\sigma_i' = \sigma_i^*[h|a_i]$. (). It is now possible to show that the DC Condition is sufficient. \[lemma:gen-cond-suff\] If the DC Condition is fulfilled, then $\vec{\sigma}^*$ is a SPE. If the DC Condition holds, then no player $i$ can increase its utility by dropping any subset of neighbors. By Lemma \[lemma:best-response\], it follows that $i$ cannot increase its utility by following any alternative strategy for the first stage game, which by the One-deviation property implies that the profile $\vec{\sigma}^*$ is a SPE. (). The following theorem merges the results from Lemmas \[lemma:gen-cond-nec\] and \[lemma:gen-cond-suff\]. \[theorem:gen-cond\] $\vec{\sigma}^*$ is a SPE if and only if the DC Condition holds. Direct Reciprocity is not Effective {#sec:direct} ----------------------------------- If $G$ is undirected, then it is possible to use direct reciprocity only, by defining ${\mbox{RS}}[i,j] = \{i,j\}$ for every $i\in{\cal N}$ and $j \in {\cal N}_i$. That is, if $i$ defects from $j$, then only $j$ punishes $i$. Direct reciprocity is the ideal incentive mechanism in a fully distributed environment, since it does not require accusations to be sent by any node. The goal of this section is to show that punishments that use direct reciprocity are not effective, even using public monitoring. To prove this, we first derive a generic necessary benefit-to-cost ratio and then we identify the conditions under which direct reciprocity is ineffective. The complete proofs are included in Appendix \[sec:proof:direct\]. Lemma \[lemma:nec-btc\] derives a minimum benefit-to-cost ratio for direct reciprocity. \[lemma:nec-btc\] If $\vec{\sigma}^*$ is a SPE, then, for every $i \in {\cal N}$ and $j \in {\cal N}_i$, it is true that $q_i' > q_i^*$ and: $$\label{eq:nec-btc} \frac{\beta_i}{\gamma_i} > \bar{p}_i + \frac{p_i[j|\emptyset]}{q_i' - q_i^*}\left(1-q_i' + \frac{1-q_i^*}{{\tau}}\right),$$ where $\vec{p}_i'$ is the strategy where $i$ drops $j$, $\vec{\sigma}' = (\sigma_i^*[\emptyset|\vec{p}_i'],\vec{\sigma}_{-i}^*)$, $q_i'=q_i[\vec{\sigma}'[\emptyset]]$, and $q_i^* = q_i[\vec{\sigma}^*[\emptyset]]$. By the definition of SPE and Theorem \[theorem:gen-cond\], the DC Condition must hold for the initial empty history and every deviation in the first stage where any player $i$ drops an out-neighbor $j$. After some manipulations of the DC Condition for this specific scenario, Inequality \[eq:nec-btc\] is obtained. (). Lemma \[lemma:dir-recip\] also shows that direct reciprocity is not an effective incentive mechanism under certain circumstances. Namely, by letting $q_i^*$ to be the probability of delivery of messages in equilibrium ($q_i[\vec{\sigma}^*[\emptyset]]$), we find that, if $p_i[j|\emptyset] + q_i^* \ll 1$, then the effectiveness is of the order $(1/q_i^*,\infty)$, which decreases to $\emptyset$ very quickly with an increasing reliability. The conditions under which direct reciprocity is ineffective are easily met, for instance, when a node has more neighbors than what is strictly necessary to ensure high reliability. \[lemma:dir-recip\] Suppose that for any $i \in {\cal N}$ and $j \in {\cal N}_i$, $p_i[j|\emptyset] + q_i^* \ll 1$. If $\vec{\sigma}^*$ is a SPE, then: $$\label{eq:dir-necessary} \psi[\vec{\sigma}^*] \subseteq \left(\frac{1}{q_i^*},\infty\right),$$ where $q_i^* = q_i[\vec{\sigma}^*[\emptyset]]$. The proof follows directly from Lemma \[lemma:nec-btc\] and the fact that, as we show in Lemma \[lemma:single-impact\] from Appendix \[sec:epidemic\], we have that if $q_i'$ results from exactly $j$ punishing $i$ for defecting from $j$, then $$q_i' \leq q_i^* \frac{1}{1-p_i[j|\emptyset]}.$$ (). Full Indirect Reciprocity is Sufficient {#sec:indir} --------------------------------------- Unlike direct reciprocity, if full indirect reciprocity is used, then the effectiveness may be independent of the reliability of the dissemination protocol. This consists in the case where for every $i \in {\cal N}$ and $j \in {\cal N}_i$ we have $${\cal N}_i^{-1} \subseteq {\mbox{RS}}[i,j].$$ The goal of this section is to show that, if full indirect reciprocity is used, then the effectiveness is independent of the reliability of the dissemination protocol under certain circumstances. To prove this, we proceed in two steps. First, we conveniently simplify the DC Condition. Then, we derive a sufficient benefit-to-cost ratio for $\vec{\sigma}^*$ to be a SPE. The complete proofs are included in Appendix \[sec:proof:indir\]. The following lemma simplifies the DC Condition for this specific type of punishing strategies. \[lemma:indir-equiv\] The profile of strategies $\vec{\sigma}^*$ is a SPE if and only if for every $h \in {\cal H}$ and $i \in {\cal N}$: $$\label{eq:indir-equiv} \sum_{r=1}^{\tau}(\omega_i^r u_i[h,r|\vec{\sigma}^*]) - (1-q_i[h,0|\vec{\sigma}^*])\gamma_i \bar{p}_i[h,0|\vec{\sigma}^*] \geq 0.$$ This simplification is obtained directly from the DC Condition and the fact that, if a node $i$ has incentives to drop some out-neighbor $j$, then the best response strategy is to drop all out-neighbors. This is proven by defining $\vec{p}''$, where $i$ drops a subset $D$ of out-neighbors, and $\vec{p}'$, where $i$ drops every out-neighbor. We can prove that: - ${\mbox{sig}}[\vec{p}'|h] = {\mbox{sig}}[\vec{p}''|h]$. - $u_i[\vec{p}'] > u_i[\vec{p}'']$. - For any $r>0$, $u_i[h,r|\vec{\sigma}'] = u_i[h,r|\vec{\sigma}'']$, where $\vec{\sigma}'$ and $\vec{\sigma}''$ differ from $\vec{\sigma}^*$ exactly in that $i$ follows $\vec{p}'$ and $\vec{p}''$ in the first stage, respectively. This implies that $\pi_i[\vec{\sigma}'|h] >\pi_i[\vec{\sigma}''|h]$, and therefore the best response for $i$ is to drop all neighbors. (). Theorem \[theorem:indir-suff\] derives a lower bound for the effectiveness of a full indirect reciprocity profile of strategies $\vec{\sigma}^*$. This is done in two steps. First, it is shown that the history $h$ that minimizes the left side of Inequality \[eq:indir-equiv\] results exactly in the same punishments being applied during the first ${\tau}-1$ stages. This is proven in Lemma \[lemma:maxh\]. \[lemma:maxh\] Let $h \in {\cal H}$ be defined such that for every $h' \in {\cal H}$, the value of the left side of Inequality \[eq:indir-equiv\] for $h$ is lower than or equal to the value for $h'$. Then, for every $r \in \{1 \ldots {\tau}-2\}$, $$u_i[h,r|\vec{\sigma}^*] = u_i[h,r+1|\vec{\sigma}^*].$$ The proof is performed by contradiction, where we assume that $h$ minimizes the left side of Inequality \[eq:indir-equiv\], but for some $r \in \{1 \ldots {\tau}- 2\}$ $$u_i[h,r|\vec{\sigma}^*] \neq u_i[h,r+1|\vec{\sigma}^*].$$ This implies that in $h$ a set of punishments ends at the end of stage $r$. We can find $h'$ where those punishments are either postponed or anticipated one stage and such that the left side of Inequality \[eq:indir-equiv\] is lower for $h'$ than for $h$, which is a contradiction. (). It is now possible to derive a sufficient benefit-to-cost ratio for full indirect reciprocity to be a SPE, which constitutes a lower bound for the effectiveness of these strategies. However, this derivation is only valid when the following assumption holds. There must exist a constant $c\geq 1$ such that for every history $h$: $$\label{eq:indir-assum} q_i[h,0|\vec{\sigma}^*] \geq 1- c(1-q_i[h,1|\vec{\sigma}^*]).$$ Intuitively, this states that, after some history $h$, if the value of $q_i$ varies from the first stage to the second due to some punishments being concluded, then this variation is never too large. With this assumption, we can derive a sufficient benefit-to-cost for $\vec{\sigma}^*$ to be a SPE. \[theorem:indir-suff\] If there exists a constant $c \geq 1$ such that, for every $h \in {\cal H}$ and $i \in {\cal N}$, Assumption \[eq:indir-assum\] holds, then $\psi[\vec{\sigma}^*] \supseteq (v,\infty)$, where $$\label{eq:indir-suff} v = \max_{h \in {\cal H}}\max_{i \in {\cal N}} \bar{p}_i[h,0|\vec{\sigma}^*]\left(1 + \frac{c}{{\tau}}\right).$$ We consider the history $h$ that minimizes the left side of Inequality \[eq:indir-equiv\]. Using the result of Lemma \[lemma:maxh\], after some manipulations, we can find that if for every $i \in {\cal N}$, $\beta_i/\gamma_i\in (v,\infty)$, then there exist $\omega_i \in (0,1)$ for every $i \in {\cal N}$ such that Inequality \[eq:indir-equiv\] is true for every history $h'$, which implies by Lemma \[lemma:indir-equiv\] that $\vec{\sigma}^*$ is a SPE. This allows us to conclude that $\psi[\vec{\sigma}^*] \supseteq (v,\infty)$. (). We can then conclude that if $c$ is small or ${\tau}$ is large, and the maximum of $\bar{p}_i[h,0|\vec{\sigma}^*]$ is never much larger than $\bar{p}_i$ for every $i$ and $h$, then the effectiveness of full indirect reciprocity is close to the optimum derived in Proposition \[prop:folk\]. In particular, if Grim-trigger is used ($\tau \to \infty$) and $\bar{p}_i$ is maximal for every $i$, then the effectiveness is optimal. Furthermore, if for any $h$ both $q_i[h,0]$ and $q_i[h,1]$ are small, then the effectiveness of full indirect reciprocity differs from the theoretical optimum only by a factor $1+1/{\tau}$, which is upper bounded by $2$ for any ${\tau}\geq1$. Private Monitoring {#sec:private} ================== When using public monitoring, we make the implicit assumption that the monitoring mechanism is able to provide the same information instantly to every node, which requires the existence of a path from every out-neighbor $j$ of any node $i$ to every node of the graph, that does not cross $i$. In addition, public monitoring is only possible if accusations are broadcast to every node. We now consider private monitoring, where the dissemination of accusations may be restricted by the topology and scalability constraints. However, any node that receives an accusation may react to it. Therefore, the definition of ${\mbox{RS}}$ is no longer necessary. In addition, accusations may be delayed. Private Signals --------------- In private monitoring, signals are determined by the history of previous signals $h$ and the profile $\vec{p}$ followed in the last stage. Namely, ${\mbox{sig}}[\vec{p}|h]$ returns a signal $s$, such that every node $i$ observes only its private signal $s_i \in s$, indicating for every other node $j \in {\cal N}$ whether $j$ cooperated or defected with its out-neighbors in previous stages. The distinction between cooperation and defection is now determined by a threshold probability $p_i[j|h_i]$. If a node $i$ defects an out-neighbor $j$ in stage $r$, then $k$ is informed of this defection with a delay ${d}_k[i,j]$, i.e., $k$ is informed only at the end of stage $r+{d}_k[i,j]$. We only assume that, for every node $i \in {\cal N}$ and $j \in {\cal N}_i$, both $i$ and $j$ are informed instantly of the action of $i$ towards $j$ in the previous stage, i.e.: $${d}_i[i,j] = {d}_j[i,j] = 0.$$ We consider that these delays are common knowledge among players. Moreover, we still assume that monitoring is perfect and that accusations are propagated faithfully. We intend to relax the assumptions in future work. With this in mind, it is possible to provide a precise definition of a private signal. For every player $i \in {\cal N}$ and history $h \in {\cal H}$, we denote by $h_i \in h$ the private history observed by $i$ when all players observe the history $h \in {\cal H}$. If $|h_i| \geq r \geq 1$, then let $h_i^r$ denote the last $r$-th signal observed by $i$, where $h_i^1$ is the last signal. A private signal is defined such that if some node $j$ observes a defection of an in-neighbor $i \in {\cal N}_j^{-1}$, every node $k \in {\cal N}$ such that ${d}_k[i,j]$ is finite (${d}_k[i,j] < \infty$) observes this defection ${d}_k[i,j]$ stages after the end of the stage it occurred. The value ${d}_k[i,j]$ is infinite if and only if accusations emitted by $j$ against $i$ may never reach $k$, either due to every path from $j$ to $k$ crossing $i$ or the monitoring mechanism not disseminating the accusation to $k$. However, if there exists a path from $j$ to $k$ without crossing $i$ and $k \in {\cal N}_i^{-1}$, then ${d}_k[i,j] < \infty$. Formally: \[def:privsig\] For every $i \in {\cal N}$ and $h \in {\cal H}$, let $s_i' \in {\mbox{sig}}[\vec{p}'|h]$ be the private signal observed by $i$ when players follow $\vec{p}' \in {\cal P}$ after having observed $h$. We have: - For every $j \in {\cal N}$ and $k \in {\cal N}_j$ such that ${d}_i[j,k] = 0$, $s_i'[j,k] = \mbox{\emph{cooperate}}$ if and only if $p_j'[k] \geq p_j[k|h_j]$, where $h_j \in h$. - For every $j \in {\cal N}$ and $k \in {\cal N}_j$ such that $0<{d}_i[j,k] < \infty$, $s_i'[j,k] = \mbox{\emph{defect}}$ if and only if: - $|h| \geq {d}_i[j,k]$. - For $h_k \in h$ and $s_k' = h_k^{{d}_i[j,k]}$, $s_k'[j,k] = \mbox{\emph{defect}}$. - For every $j \in {\cal N}$ and $k \in {\cal N}_j$ such that ${d}_i[j,k] = \infty$, $s_i'[j,k] = \mbox{\emph{cooperate}}$. Private Punishments ------------------- In this context, we can define a punishing strategy $\sigma_i^*$ for every node $i$ as a function of the threshold probability $p_i[j|h_i]$ determined by $i$ for every private history $h_i$ and out-neighbor $j$. Notice that in the definition of private signals we assume that an accusation by $j$ is emitted against $i$ iff $i$ uses $p_i[j] < p_i[j|h_i]$ for any private history $h_i$. This was reasonable to assume in public monitoring, where histories were public. Here, the strategy must also specify for every private history $h_j$ the threshold probability $p_i[j|h_j]$, since $h_i$ may differ from $h_j$. In order for $j$ to accurately monitor $i$, for every $h \in {\cal H}$ and $h_i,h_j \in h$, we must have $$\label{eq:common-knowledge} p_i[j|h_i] = p_i[j|h_j].$$ Therefore, both threshold probabilities must be computed as a function of the same set of signals. The only issue with this requirement is that defection signals may arrive at different stages to $i$ and $j$. For instance, if $k_1$ defects from $k_2$ in stage $r$ and ${d}_i[k_1,k_2] < {d}_j[k_1,k_2]$, then $i$ must wait for stage $r+{d}_j[k_1,k_2]$ before taking this defection into consideration in the computation of the threshold probability. Furthermore, as in public monitoring, $i$ and $j$ are expected to react to a given defection for a finite number of stages. However, as we will see later, this number should vary according to the delays in order for punishments to be effective. Thus, for every $k_1 \in {\cal N}$ and $k_2 \in {\cal N}_{k_1}$, we define ${\tau}[k_1,k_2|i,j]$ to be the number of stages during which $i$ and $j$ react to a given defection of $k_1$ from $k_2$. Notice that ${\tau}[k_1,k_2|i,j] = {\tau}[k_1,k_2|j,i]$. This intuition is formalized as follows. As in public monitoring, ${\mbox{DS}}_i[j|h_i]$ denotes the set of defections observed by $i$ and that $j$ *will eventually observe*. This set also contains tuples in the form $(k_1,k_2,r)$. The main difference is that now $i$ may have to wait before considering this tuple in the definition of $p_i[j|h_i]$. We signal this by allowing $r$ to be negative and by using the tuple in the definition of $p_i[j|h_i]$ only when $r \geq0$. When $i$ observes a defection for the first time, it adds $(k_1,k_2,v)$ to ${\mbox{DS}}_i[j|h_i]$, where $$v= \min[{d}_i[k_1,k_2] - {d}_j[k_1,k_2],0].$$ Then, $i$ removes this pair when $r = {\tau}[k_1,k_2|i,j]$. For simplicity, we allow $v$ to take the value $\infty$ when ${d}_j[k_1,k_2] = \infty$, resulting in that $i$ never takes into consideration this defection when determining $p_i[j|h_i]$. This leads to the following definition. \[def:priv-thr\] For every $i \in {\cal N}$, $h_i \in {\cal H}_i$, and $j \in {\cal N}_i \cup {\cal N}_i^{-1}$, define ${\mbox{DS}}_i[j|h_i] \subseteq {\cal N} \times \mathbb{Z}$ as follows: - ${\mbox{DS}}_i[j|\emptyset] = \emptyset$. - For $h_i = (h_i',s_i')$, ${\mbox{DS}}_i[j|h_i] = L_1 \cup L_2$, where: 1. $L_1 = \{(k_1,k_2,r+1) | (k_1,k_2,r) \in {\mbox{DS}}_i[j|h_i'] \land r+1 < {\tau}[k_1,k_2|i,j]\}$. 2. $L_2 = \{(k_1,k_2,v) | k_1,k_2 \in {\cal N} \land s_i'[k_1,k_2] = \mbox{\emph{defect}}\}$, where: $$v= \min[{d}_i[k_1,k_2] - {d}_j[k_1,k_2],0].$$ For every $i \in {\cal N}$, $h_i \in {\cal H}_i$, and $j \in {\cal N}_i$, $\sigma_i^*[h_i] = p_i[j|h_i]$: - Let $K = \{(k_1,k_2,r) \in {\mbox{DS}}_i[j|h_i] | r \geq 0\}$. - If there exists $r \geq 0$ such that $(i,j,r) \in K$, then $p_i[j|h_i] = 0$. - If there exist $r \geq 0$ and $k \in {\cal N}_j$ such that $(j,k,r) \in K$, then $p_i[j|h_i] = 0$. - Otherwise, $p_i[j|h]$ is a positive function of $K$. For every $j \in {\cal N}_i^{-1} \setminus {\cal N}_i$, $\sigma_i^*[h_i] = p_j[i|h_i]$ such that: - Let $K = \{(k_1,k_2,r) \in {\mbox{DS}}_i[j|h_i] | r \geq 0 \}$. - If there exists $r \geq 0$ such that $(j,i,r) \in K$, then $p_j[i|h_i] = 0$. - If there exist $r \geq 0$ and $k \in {\cal N}_i$ such that $(i,k,r) \in K$, then $p_j[i|h_i] = 0$. - Otherwise, $p_j[i|h_i]$ is a positive function of $K$. Expected Utility and Solution Concept ------------------------------------- We now model the interactions as a repeated game with imperfect information and perfect recall, for which the solution concept of Sequential Equilibrium is adequate[@Kreps:85]. Its definition requires the specification of a belief system $\vec{\mu}$. After a player $i$ observes a private history $h_i \in {\cal H}_i$, $i$ must form some expectation regarding the history $h \in {\cal H}$ observed by every player, which must include $h_i$. This is captured by a probability distribution $\mu_i[.|h_i]$ over ${\cal H}$. By defining $\vec{\mu}=(\mu_i)_{i \in {\cal N}}$, we call a pair $(\vec{\sigma},\vec{\mu})$ an assessment, which is assumed to be common knowledge among all players. The expected utility of a profile of strategies $\vec{\sigma}$ is then defined as: $$\label{eq:priv-exp-util} \pi_i[\vec{\sigma}|\vec{\mu},h_i] = \sum_{h \in {\cal H}} \mu_i[h|h_i]\pi_i[\vec{\sigma}|h],$$ where $\pi_i[\vec{\sigma}|h]$ is defined as in the public monitoring case. An assessment $(\vec{\sigma}^*,\vec{\mu}^*)$ is a Sequential Equilibrium if and only if $(\vec{\sigma}^*,\vec{\mu}^*)$ is Sequentially Rational and Consistent. The definition of sequential rationality is identical to that of subgame perfection: An assessment $(\vec{\sigma}^*,\vec{\mu}^*)$ is Sequentially Rational if and only if for every $i \in {\cal N}$, $h_i \in {\cal H}_i$, and $\sigma_i' \in \Sigma_i$, $\pi_i[\vec{\sigma}^*|\vec{\mu}^*,h_i] \geq \pi_i[\sigma_i',\vec{\sigma}^*_{-i}|\vec{\mu}^*,h_i]$. However, defining consistency for an assessment $(\vec{\sigma}^*,\vec{\mu}^*)$ is more intricate. The idea of defining this concept was introduced in [@Kreps:85], intuitively defined as follows in our context. For any profile $\vec{\sigma}^*$, every private history $h_i$ that may be reached with positive probability when players follow $\vec{\sigma}^*$ is said to be consistent with $\vec{\sigma}^*$; otherwise, $h_i$ is inconsistent. For any consistent $h_i$, $\mu_i[h|h_i]$ must be defined using the Bayes rule. The definition of $\mu_i$ for inconsistent private histories varies with the specific definition of Consistent Assessment. It turns out that, in our case, the notion of Preconsistency introduced in [@Hendon:96] is sufficient. We now provide the formal definition of Preconsistency and later provide an interpretation in the context of punishment strategies. Let $pr_i[h'|h,\vec{\sigma}]$ be the probability assigned by $i$ to $h' \in {\cal H}$ being reached from $h \subset h'$ when players follow $\vec{\sigma} \in \Sigma$. Given $h_i \in {\cal H}_i$, we can define $$pr_i[h'|\vec{\mu},h_i,\vec{\sigma}] = \sum_{h \in {\cal H} } \mu_i[h|h_i]pr_i[h'|h,\vec{\sigma}].$$ For $h_i' \in {\cal H}_i$ such that $h_i \subset h_i'$, let: $$pr_i[h_i'|\vec{\mu},h_i,\vec{\sigma}] = \sum_{h' \in {\cal H} : h_i' \in h'} pr_i[h'|\vec{\mu},h_i,\vec{\sigma}].$$ An assessment $(\vec{\sigma}^*,\vec{\mu}^*)$ is Preconsistent if and only if for every $i \in {\cal N}$, $h_i \in {\cal H}_i$, and $h_i' \in {\cal H}_i$ such that $h_i \subset h_i'$, if there exists $\sigma_i' \in \Sigma_i$ such that $pr_i[h_i'|\vec{\mu}^*,h_i,(\sigma_i',\vec{\sigma}^*_{-i})] > 0$, then for every $h' \in {\cal H}$ such that $h_i' \in h'$: $$\mu_i[h'|h_i'] = \frac{pr_i[h'|\vec{\mu}^*,h_i,(\sigma_i',\vec{\sigma}^*_{-i})]}{pr_i[h_i'|\vec{\mu}^*,h_i,(\sigma_i',\vec{\sigma}^*_{-i})]}.$$ The underlying intuition of this definition when considering a profile of punishing strategies $\vec{\sigma}^*$ is as follows. First, notice that a history $h_i$ is consistent with $\vec{\sigma}^*$ if and only if no defections are observed in $h_i$. For any $\sigma_i'$, the profile of strategies $\vec{\sigma}' = (\sigma_i',\vec{\sigma}_{-i}^*)$ may specify non-deterministic actions for the stage game, by only due to $\sigma_i'$. Consider any $h_i'$. If $h_i'$ does not contain any defections, then the only strategies $\sigma_i'$ such that $h_i'$ is consistent with any such $\vec{\sigma}'$ are those where $i$ does not defect any node. Therefore, we can set $h_i$ to the empty history and from the above definition derive the conclusion that $\mu_i[h'|h_i']=1$ if and only if $h'$ does not contain any defections. Similarly, if $h_i'$ only contains defections performed by $i$, then $h_i$ can be the empty set and there must be only one $h'$ such that $\mu_i[h'|h_i'] =1$, which is the history where only defections performed by $i$ are observed by any player. If $h_i'$ contains only defections performed by $i$, then we can use induction on the number of defections committed by other nodes to prove that $\mu_i[h'|h_i'] =1$ if and only if $h'$ contains exactly the defections observed by $i$ in $h_i'$. The base case follows from the two previous scenarios. As for the induction step, there are two hypothesis. If the last defections were performed in the last stage, then there is no $h_i \subset h_i'$ such that $$\label{eq:consistent-aux} pr_i[h_i'|\vec{\mu},h_i,\vec{\sigma}'] > 0.$$ Otherwise, consider that the last defections performed by other nodes occurred in the last $r$-th stage where $r>1$, when $i$ observed $h_i$. In this case, it is true that \[eq:consistent-aux\] holds. Here, there is only one history $h$ such that $h_i \in h$ and $\mu_i[h|h_i] = 1$, which is true by the induction hypothesis. This history contains exactly the defections observed by $i$ in $h_i$, which are also included in $h_i'$. Thus, the only history $h'$ that may follow $h_i$ fulfills the condition that no other defection was performed other than what $i$ observed in $h_i'$. In summary, $\mu[h|h_i] = 1$ if and only if $h$ is the history containing $h_i$ and the set of defections observed by any node $j \in {\cal H}$ in $h_j \in h$ is a subset of the set of defections observed in $h_i$. The importance of this definition of consistency is that in [@Kreps:85] the authors prove that the One-deviation property also holds for Preconsistent assessments, which is sufficient for our analysis. \[prop:priv-one-dev\] **One-deviation.** A Preconsistent assessment $(\vec{\sigma}^*,\vec{\mu}^*)$ is Sequentially Rational if and only if for every player $i \in {\cal N}$, history $h_i \in {\cal H}_i$, and profile $\vec{a}_i' \in {\cal A}_i$, $$\pi_i[\sigma_i^*,\vec{\sigma}_{-i}^*|\vec{\mu}^*,h_i] \geq \pi_i[\sigma_i',\vec{\sigma}_{-i}^*|\vec{\mu}^*,h_i],$$ where $\sigma_i' = \sigma_i^*[h_i|\vec{a}_i']$ is defined as in public monitoring. Evolution of the Network {#sec:priv-evol} ------------------------ When a player $i$ observes a private history $h_i \in {\cal H}_i$, only the histories $h \in {\cal H}$ such that $h_i \in h$ can be observed by other players. Given this, we use the same notation as in public monitoring, when referring to the evolution of the network after a history $h$ is observed. Namely, ${\mbox{hist}}[h,r|\vec{\sigma}]$ is the resulting history starting from the observation of $h$ and when all players follow the pure strategy $\vec{\sigma}$. Therefore, we continue to use the same notation for $q_i$ ($q_i[h,r|\vec{\sigma}]$), $\bar{p}_i$ ($\bar{p}_i[h,r|\vec{\sigma}]$), and $u_i$ ($u_i[h,r|\vec{\sigma}]$). Now, we have $${\cal N}_i[h_i] = \{j \in {\cal N}_i | p_i[j|h_i] > 0\}.$$ The definition of ${\mbox{CD}}_i[\vec{p}|h] \subseteq {\cal N}$ is almost identical to that of the public monitoring case. Namely, for every $i \in {\cal N}$, $h \in {\cal H}$, and $h_i \in h$, $${\mbox{CD}}_i[\vec{p}|h] = \{j \in {\cal N}_i | p_i[j] < p_i[j|h_i]\}.$$ The following lemma proves that every node $k_1$ that defects from an out-neighbor $k_2$ expects $i$ and $j$ to react to this defection during the next ${\tau}[k_1,k_2|i,j]$ stages, regardless of the following actions of $k_1$ or the punishments already being applied to $k_1$. \[lemma:priv-corr-1\] For every $h \in {\cal H}$, $\vec{p}' \in {\cal P}$, $r >0$, $i \in {\cal N}$, and $j \in {\cal N}_i$: $$\label{eq:priv-res-corr1} \begin{array}{ll} {\mbox{DS}}_i[j|h_{i,r}'] =& {\mbox{DS}}_i[j|h_{i,r}^*] \cup \{(k_1,k_2,r -1 - {d}_i[k_1,k_2]+v[k_1,k_2]) | k_1,k_2 \in {\cal N} \land \\ & k_2 \in {\mbox{CD}}_{k_1}[\vec{p}'|h] \land r \in \{{d}_i[k_1,k_2]+1 \ldots {d}_i[k_1,k_2] + {\tau}[k_1,k_2|i,j]-v[k_1,k_2]\}\land\\ & v[k_1,k_2] = \min[{d}_i[k_1,k_2] - {d}_j[k_1,k_2],0]\},\\ \end{array}$$ where $h_{i,r}^* \in {\mbox{hist}}[h,r|\vec{\sigma}^*]$, $h_{i,r}' \in {\mbox{hist}}[h,r| \vec{\sigma}']$, and $\vec{\sigma}' =\vec{\sigma}^*[h|\vec{p}']$ is the profile of strategies where all players follow $\vec{p}'$ in the first stage. By induction, the base case follows from the definition of ${\mbox{DS}}_i$ and the fact that $i$ registers every defection of $k_1$ to $k_2$ in stage ${d}_i[k_1,k_2]$, adding $(k_1,k_2,v[k_1,k_2])$ to ${\mbox{DS}}_i[j|h]$. Inductively, after $r\leq{d}_i[k_1,k_2] +{\tau}[k_1,k_2|i,j] - v[k_1,k_2]$ stages, this pair is transformed into $(k_1,k_2,r-1)$. (). For the sake of completeness, we prove in Lemma \[lemma:priv-history\] that the strategy is well defined, in terms of defining threshold probabilities that are always common knowledge between pairs of players. This supports our assumption in the definition of private signals that an accusation is emitted by $j$ against $i$ iff $i$ uses $p_i[j] < p_i[j|h_j]$ towards $j$. \[lemma:priv-history\] For every $i \in {\cal N}$, $j \in {\cal N}_i$, $h \in {\cal H}$, and $h_i,h_j \in h$: $$p_i[j|h_i] = p_i[j|h_j].$$ It follows from the definition of ${\mbox{DS}}$ that a node $i$ never includes in the set $K$ a tuple $(k_1,k_2,r)$ such that $r<0$, for any out-neighbor $j$. This value is only negative when ${d}_i[k_1,k_2] < {d}_j[k_1,k_2]$, in which case $v$ is set to $-({d}_j[k_1,k_2] - {d}_i[k_1,k_2])$. By Lemma \[lemma:priv-corr-1\], this value only becomes $0$ when $j$ is also informed of this defection, in which case both nodes include the pair in $K$. Consequently, $K$ is always defined identically by $i$ and $j$ after any history $h$, which implies the result. (). Generic Results {#generic-results} --------------- Proposition \[prop:priv-folk\] reestablishes the optimal effectiveness for private monitoring. The proof of this proposition is identical to that of Proposition \[prop:folk\]. The only difference lies in the fact that now the effectiveness of a profile of strategies $\vec{\sigma}^*$ is conditional on a belief system $\vec{\mu}^*$ ($\psi[\vec{\sigma}^*|\vec{\mu}^*]$). \[prop:priv-folk\] For every assessment $(\vec{\sigma}^*,\vec{\mu}^*)$, if $(\vec{\sigma}^*,\vec{\mu}^*)$ is Sequentially Rational, then, for every $i \in {\cal N}$, $\frac{\beta_i}{\gamma_i} \geq \bar{p}_i$. Consequently, $\psi[\vec{\sigma}^*] \subseteq (v,\infty)$, where $v = \max_{i \in {\cal N}} \bar{p}_i$. (). As in the public monitoring case, we can define a necessary and sufficient condition for the defined profile of strategies to be Sequentially Rational, which we name PDC Condition. The proof of both necessity and sufficiency is almost identical to that of public monitoring. **PDC Condition**. For every $i \in {\cal N}$, $h_i \in {\cal H}_i$, and $D \subseteq {\cal N}_i[h_i]$, $$\label{eq:priv-drop} \sum_{h \in {\cal H}}\mu_i^*[h|h_i]\sum_{r=0}^{\infty} \omega_i^r (u_i[h,r|\vec{\sigma}^*] - u_i'[h,r|\vec{\sigma}']) \geq 0,$$ where $\vec{\sigma}' = (\sigma_i^*[h_i|\vec{p}_i'],\vec{\sigma}_{-i}^*)$ and $\vec{p}_i'$ is defined as: - For every $j \in D$, $p_i'[j] = 0$. - For every $j \in {\cal N} \setminus D$, $p_i'[j] = p_i[j|h_i]$. The following corollary captures the fact that the PDC condition is necessary, which follows directly from the One-deviation property and the definition of $\pi_i$ for private monitoring. \[corollary:priv-drop-nec\] If the assessment $(\vec{\sigma}^*,\vec{\mu}^*)$ is Sequentially Rational and Preconsistent, then the PDC Condition is fulfilled. In order to prove that the PDC Condition is sufficient, we proceed in the same fashion to public monitoring, always implicitly assuming that the considered assessment is Preconsistent. Redefine the set of local best responses for every node $i$ and private history $h_i \in {\cal H}_i$ as: $$BR[\vec{\sigma}_{-i}^*|\vec{\mu}^*,h_i] = \{a_i \in {\cal A}_i | \forall_{a_i' \in {\cal A}_i} \pi_i[(\sigma_i^*[h_i|a_i],\vec{\sigma}_{-i}^*)|\vec{\mu}^*,h_i] \geq \pi_i[(\sigma_i^*[h_i|a_i'],\vec{\sigma}_{-i}^*)|\vec{\mu}^*,h_i]\}.$$ \[lemma:priv-best-response1\] For every $i \in {\cal N}$, $h_i \in {\cal H}_i$, $a_i \in BR[\vec{\sigma}_{-i}^*|\vec{\mu}^*,h_i]$, and $\vec{p}_i \in {\cal P}_i$ such that $a_i[\vec{p}_i] > 0$, it is true that for every $j \in {\cal N}_i$ we have $p_i[j] \in \{0,p_i[j|h_i]\}$. (). \[lemma:priv-best-response2\] For every $i \in {\cal N}$ and $h_i \in {\cal H}_i$, there exists $a_i \in BR[\vec{\sigma}_{-i}^*|\vec{\mu}^*,h_i]$ and $\vec{p}_i \in {\cal P}_i$ such that $a_i[\vec{p}_i] = 1$. (). \[lemma:priv-best-response\] For every $i \in {\cal N}$ and $h_i \in {\cal H}_i$, there exists $\vec{p}_i \in {\cal P}_i$ and a pure strategy $\sigma_i=\sigma_i^*[h_i|\vec{p}_i]$ such that: 1. For every $j \in {\cal N}_i$, $p_i[j] \in \{0,p_{i}[j|h_i]\}$. 2. For every $a_i \in {\cal A}_i$, $\pi_i[\sigma_i,\vec{\sigma}_{-i}^*|\vec{\mu}^*,h_i] \geq \pi_i[\sigma_i',\vec{\sigma}_{-i}^*|\vec{\mu}^*,h_i]$, where $\sigma_i' = \sigma_i^*[h_i|a_i]$. (). \[lemma:priv-drop-suff\] If the PDC Condition is fulfilled and $(\vec{\sigma}^*,\vec{\mu}^*)$ is Preconsistent, then $(\vec{\sigma}^*,\vec{\mu}^*)$ is Sequentially Rational. (). The following theorem merges the results from Corollary \[corollary:priv-drop-nec\] and Lemma \[lemma:priv-drop-suff\]. \[theorem:priv-drop\] If $(\vec{\sigma}^*,\vec{\mu}^*)$ is Preconsistent, then $(\vec{\sigma}^*,\vec{\mu}^*)$ is Sequentially Rational if and only if the PDC Condition is fulfilled. Ineffective Topologies ---------------------- An important consequence of Theorem \[theorem:priv-drop\] is that not every topology allows the existence of equilibria for punishing strategies. In fact, if there is some node $i$ and a neighbor $j$ such that every node $k$ that is reachable from $j$ without crossing $i$ is never in between $s$ and $i$, then the impact of the punishments applied to $i$ after defecting from $j$ is null. This intuition is formalized in Lemma \[lemma:paths\], where ${\mbox{PS}}[i,j]$ denotes the set of paths from $i$ to $j$ in $G$. \[lemma:paths\] If the assessment $(\vec{\sigma}^*,\vec{\mu}^*)$ is Preconsistent and Sequentially Rational, then for every $i \in {\cal N}$ and $j \in {\cal N}_i$, there is $k \in {\cal N} \setminus \{i\}$, $x \in {\mbox{PS}}[s,i]$, and $x' \in {\mbox{PS}}[j,k]$, such that $k \in x$ and $i \notin x'$. Assume by contradiction the opposite. Then, nodes can follow $\vec{p}'$ where $i$ drops $j$, such that, if $\vec{p}''$ is the profile resulting from nodes punishing $i$, then by Lemma \[lemma:bottleneck-impact\] from Appendix \[sec:epidemic\], $q_i[\vec{p}''] = q_i[\vec{p}^*]$, where $\vec{p}^* = \vec{\sigma}^*[\emptyset]$. Since $i$ increases its utility in the first stage by deviating, we have $$u_i[\vec{p}''] > u_i[\vec{p}^*].$$ Moreover, by letting $\vec{\sigma}' =\vec{\sigma}^*[\vec{p}']$ to be the profile where exactly $i$ defects $j$, it is true that for every $r>0$ $$u_i[\emptyset,r|\vec{\sigma}'] = u_i[\emptyset,r|\vec{\sigma}^*].$$ This implies that $\pi_i[\vec{\sigma}'|\vec{\mu}^*,\emptyset] > \pi_i[\vec{\sigma}^*|\vec{\mu}^*,\emptyset]$, which is a contradiction. (). The main implication of this result is that many non-redundant topologies, i.e., that do not contain multiple paths between $s$ and every node, are ineffective at sustaining cooperation. This is not entirely surprising, since it was already known that cooperation cannot be sustained using punishments as incentives in non-redundant graphs such as trees[@Ngan:04]. But even slightly redundant structures, such as directed cycles, do not fulfill the necessary condition specified in Lemma \[lemma:paths\]. Although redundancy is desirable to fulfill the above condition, it might decrease the effectiveness of punishments unless full indirect reciprocity may be used, as shown in the following section. Redundancy may decrease Effectiveness ------------------------------------- In addition to the need to fulfill the necessary condition of Lemma \[lemma:paths\], a higher redundancy increases tolerance to failures. We show in Theorem \[theorem:problem\] that if the graph is redundant and it does not allow full indirect reciprocity to be implemented, then the effectiveness decreases monotonically with the increase of the reliability. We consider the graph to be redundant if there are multiple non-overlapping paths from the source to every node. More precisely, if for every $i \in {\cal N}$ and $j \in {\cal N} \setminus \{i\}$, there exists $x \in {\mbox{PS}}[s,i]$ such that $j \notin x$, then $G$ is redundant. The reliability increases as the probabilities $p_i[j|\emptyset]$ approach $1$ for every node $i$ and out-neighbor $j$. This is denoted by $\lim_{\vec{\sigma}^*\to \vec{1}}$. We find that the effectiveness of any punishing strategy that cannot implement full indirect reciprocity converges to $\emptyset$, i.e., no benefit-to-cost ratio can sustain cooperation. This intuition is formalized as follows: $$\label{eq:problem} \lim_{\vec{\sigma}^*\to \vec{1}} \psi[\vec{\sigma}^*|\vec{\mu}^*] = \cdot_{i \in {\cal N},j \in {\cal N}_i} \lim_{p_i[j|\emptyset]\to 1} \psi[\vec{\sigma}^*|\vec{\mu}^*] = \emptyset.$$ Theorem \[theorem:problem\] proves that Equality \[eq:problem\] holds for any graph that does not allow full indirect reciprocity to be implemented, which in our model occurs when there is no path from some $j \in {\cal N}_i$ to some $k \in {\cal N}_i^{-1}$ that does not cross $i$. \[theorem:problem\] If $G$ is redundant and there exist $i \in {\cal N}$, $j \in {\cal N}_i$, and $k \in {\cal N}_i^{-1}$ such that for every $x \in {\mbox{PS}}[j,k]$ we have $i \in x$, then Equality \[eq:problem\] holds. The proof defines a deviating profile of strategies $\vec{\sigma}'$ where exactly $i$ drops $j$. It follows that there is a path $x$ from $s$ to $i$ such that every node $k \in x$ never reacts to this defection. By Definition \[def:priv-thr\], $k$ uses $p_k[l|\emptyset]$ towards every out-neighbor $l$; a value that converges to $1$. It follows from Lemma \[lemma:prob-1\] in Appendix \[sec:epidemic\] that $$\lim_{\vec{\sigma}^* \to 1}( \pi_i[\vec{\sigma}^*|\vec{\mu}^*,\emptyset] - \pi_i[\vec{\sigma}'|\vec{\mu}^*,\emptyset] ) < 0.$$ Therefore, for any $\beta_i$ and $\gamma_i$, the left side of the PDC Condition converges to a negative value. By Theorem \[theorem:priv-drop\], this implies that $\psi[\vec{\sigma}^*|\vec{\mu}^*]$ converges to $\emptyset$. (). Notice that this result does not imply that only full indirect reciprocity is effective at incentivizing rational nodes to cooperate in all scenarios. In fact, in many realistic scenarios, it might suffice for a majority of the in-neighbors of a node $i$ to punish $i$ after any deviation. A more sensible analysis would take into consideration the rate of converge to $\emptyset$ as the reliability increases. However, full indirect reciprocity is necessary in order to achieve an effectiveness fully independent of the desired reliability in any redundant graph. Coordination is Desirable ------------------------- Although full indirect reciprocity is desirable for redundant graphs, we now show that for some definitions of punishing strategies, it might not be sufficient if monitoring incurs large delays. In particular, nodes also need to coordinate the punishments being applied to any node, such that these punishments overlap during at least one stage after the defection, cancelling out any benefit obtained for receiving messages along some redundant path. This intuition is formalized as follows. \[def:coord\] An assessment $(\vec{\sigma}^*,\vec{\mu}^*)$ enforces coordination if and only if for every $i \in {\cal N}$ and $j \in {\cal N}_i$, there exists $r>0$ such that, for every $k \in {\cal N}_i^{-1}$, $$r \in \{{d}_k[i,j] + 1 \ldots {d}_k[i,j] + {\tau}[i,j|k,i]\}.$$ We prove a similar theorem to Theorem \[theorem:problem\], which states that for some definitions of punishing strategies, with a redundant graph, the effectiveness decreases to $\emptyset$ with the reliability. \[theorem:problem-coord\] If the graph is redundant and $\vec{\sigma}^*$ does not enforce coordination, then there is a definition of $\vec{\sigma}^*$ such that Equality \[eq:problem\] holds. Consider the punishing strategy where every node $i$ reacts only to the defections of out-neighbors or to its own defections, and uses $p_i[j|\emptyset]$ in any other case. If $\vec{\sigma}^*$ does not enforce coordination, then for some $i \in {\cal N}$ and $j \in {\cal N}_i$, and for every $r>0$, we can find a path from $s$ to $i$ such that every node $k$ along the path uses the probability $p_k[l|\emptyset]$ towards the next node $l$ in the path. These probabilities converge to $1$ as the reliability increases, which by Lemma \[lemma:prob-1\] implies that $$\lim_{\vec{\sigma}^* \to 1}( \pi_i[\vec{\sigma}^*|\vec{\mu}^*,\emptyset] - \pi_i[\vec{\sigma}'|\vec{\mu}^*,\emptyset] ) < 0,$$ where $\vec{\sigma}'$ is the alternative profile of strategies where exactly $i$ drops $j$. Therefore, for any $\beta_i$ and $\gamma_i$, the left side of the PDC Condition converges to a negative value. By Theorem \[theorem:priv-drop\], this implies that $\psi[\vec{\sigma}^*|\vec{\mu}^*]$ converges to $\emptyset$. (). In order to allow nodes to obtain messages with high probabilities, while keeping the effectiveness independent of the desired reliability, punishments must be coordinated, such that after any deviation any node $i$ expects to be punished by every in-neighbor during ${\tau}>0$ stages. More precisely, for every node $i$, by letting $${\bar{d}}_i = \max_{j \in {\cal N}_i} \max_{k\in{\cal N}_i^{-1}} {d}_k[i,j]$$ to be the maximum delay of accusations against $i$ towards any in-neighbor of $i$, the protocol must define ${\tau}[i,j|k,i]$ for every $k \in {\cal N}_i^{-1}$ and $j \in {\cal N}_i$ in order to fulfill $${\tau}[i,j|k,i] + {d}_k[i,j] \geq {\bar{d}}_i + {\tau}.$$ It is sufficient and convenient for the sake of simplicity to provide a definition of ${\tau}[i,j|k,l]$ for every $k\in{\cal N}$ and $l \in {\cal N}_k$ where ${d}_k[i,j] < \infty$, such that every node stops reacting to a given defection in the same stage. More precisely, \[def:overlap\] For every $k \in {\cal N}$ and $l \in {\cal N}_k$ such that $d_k[i,j] < \infty$, if $k$ and $l$ observe the defection before ${\bar{d}}_i + {\tau}$, i.e., $g = \max[{d}_k[i,j],{d}_l[i,j]] < {\bar{d}}_i + {\tau}$, then $k$ and $l$ react to a defection of $i$ from $j$: $${\tau}[i,j|k,l] = {\bar{d}}_i + {\tau}- g.$$ Otherwise, $${\tau}[i,j|k,l] = 0.$$ This ensures that no node $k$ reacts to a defection of $i$ from $j$ after stage ${\bar{d}}_i + {\tau}$. Coordinated Full Indirect Reciprocity ------------------------------------- We now study the set of punishing strategies that use full indirect reciprocity. This requires the existence of a path from every $j \in {\cal N}_i$ to every $k \in {\cal N}_i^{-1}$, which must not cross $i$. Under some circumstances, the effectiveness of a Preconsistent assessment $(\vec{\sigma}^*,\vec{\mu}^*)$ that uses full indirect reciprocity does not increase with the reliability of the dissemination process. As seen in the previous section, this requires punishments to be coordinated, which we assume to be defined as in Definition \[def:overlap\]. The fact that accusations may be delayed has an impact on the effectiveness, which is quantified in Lemma \[lemma:priv-suff\]. To prove this lemma, we first derive in Lemma \[lemma:delay-equiv\] an intermediate sufficient condition for the PDC Condition to be true. The lemma simplifies the PDC Condition for the worst scenario, where all in-neighbors of a node $i$ begin punishing $i$ for any defection simultaneously. The proofs assume that punishing strategies are defined in a reasonable manner. More precisely, if in reaction to a defection of node $i$ other nodes increase the probabilities used towards out-neighbors other than $i$, then $i$ should never expect a large increase in its reliability during the initial stages, before every in-neighbor starts punishing $i$. This intuition is captured in Assumption \[def:non-neg\]. \[def:non-neg\] (**Assumption**) There exists a constant $\epsilon \in [0,1)$ such that, for every $h\in {\cal H}$, $i \in {\cal N}$, $\vec{p}_i' \in {\cal P}_i$, $\vec{\sigma}' = (\sigma_i^*[h|\vec{p}_i'],\vec{\sigma}_{-i}^*)$, and $r > 0$, $$q_i[h,r|\vec{\sigma}^*] - q_i[h,r|\vec{\sigma}'] < \epsilon.$$ \[lemma:delay-equiv\] If $(\vec{\sigma}^*,\mu^*)$ is Preconsistent, Assumption \[def:non-neg\] holds, and Inequality \[eq:delay-equiv\] is fulfilled for every $i \in {\cal N}$, $h_i \in {\cal H}_i$, and $h \in {\cal H}$ such that $\mu_i^*[h|h_i]>0$, then $(\vec{\sigma}^*,\vec{\mu}^*)$ is Sequentially Rational: $$\label{eq:delay-equiv} - \sum_{r=0}^{{\bar{d}}_i} \omega_i^r ((1-q_i[h,r|\vec{\sigma}^*])\gamma_i \bar{p}_i[h,r|\vec{\sigma}^*] + \epsilon \beta_i)+ \sum_{r={\bar{d}}_i+1}^{{\bar{d}}_i + {\tau}} \omega_i^r u_i[h,r | \vec{\sigma}^*] \geq 0.$$ By the definition of coordinated punishments if $i$ deviates in $\sigma_i'$ by dropping some subset of neighbors such that all players follow $\vec{\sigma}'=(\sigma_i',\vec{\sigma}_{-i}^*)$, then in the worst scenario no node punishes $i$ in any of the first ${\bar{d}}_i$ stages. Therefore, by our assumptions, for every $r \in \{1\ldots {\bar{d}}_i\}$, $$u_i[h,r|\vec{\sigma}^*] - u_i[h,r|\vec{\sigma}'] \geq - \sum_{r=0}^{{\bar{d}}_i} \omega_i^r ((1-q_i[h,r|\vec{\sigma}^*])\gamma_i \bar{p}_i[h,r|\vec{\sigma}^*] + \epsilon \beta_i).$$ Also, for every $r \in \{{\bar{d}}_i+1 \ldots {\bar{d}}_i + {\tau}\}$, every in-neighbor of $i$ punishes $i$, which by Lemma \[lemma:noneib\] from Appendix \[sec:epidemic\] implies $$u_i[h,r|\vec{\sigma}'] = 0.$$ Finally, for every $r \geq {\bar{d}}_i +{\tau}+1$, every node ends its reaction to any defection of $i$ after ${\bar{d}}_i+{\tau}+1$ stages, implying that $$u_i[h,r|\vec{\sigma}^*] = u_i[h,r|\vec{\sigma}'].$$ These three facts have the consequence that if Inequality \[eq:delay-equiv\] is fulfilled, then the PDC Condition holds. Therefore, by Theorem \[theorem:priv-drop\], $(\vec{\sigma}^*,\vec{\mu}^*)$ is Sequentially Rational. (). We can now derive a lower bound for the effectiveness of the considered punishing strategies, in a similar fashion to Theorem \[theorem:indir-suff\]. However, now a stronger assumption is made, defined in \[def:priv-assum\]. The reasoning is similar in that after any history, if a node $i$ defects from some out-neighbor, then the reliability $i$ would obtain during the initial stages when it is not being punished by all in-neighbors is not significantly greater than the reliability $i$ would obtain in the subsequent stages, had $i$ not deviated from the specified strategy. \[def:priv-assum\] (**Assumption**). There exists a constant $c>0$, such that, for every $i \in {\cal N}$, $r \in \{0 \ldots {\bar{d}}_i \}$, and $r' \in \{{\bar{d}}_i+1 \ldots {\bar{d}}_i+ {\tau}\}$, $$q_i[h,r|\vec{\sigma}^*] \geq 1-c(1-q_i[h,r'|\vec{\sigma}^*]).$$ \[lemma:priv-suff\] If $(\vec{\sigma}^*,\vec{\mu}^*)$ is Preconsistent, Assumptions \[def:non-neg\] and \[def:priv-assum\] hold, and Inequality \[eq:priv-suff\] is fulfilled for every $h$, $i \in {\cal N}$, and $r,r' \leq {\bar{d}}_i + {\tau}$ such that $q_i[h,r|\vec{\sigma}^*] <1$ and $q_i[h,r'|\vec{\sigma}^*] <1$, then there exist $\omega_i \in (0,1)$ for every $i \in {\cal N}$ such that $(\vec{\sigma}^*,\vec{\mu}^*)$ is Sequentially Rational: $$\label{eq:priv-suff} \frac{\beta_i}{\gamma_i}> \bar{p}_i[h,r|\vec{\sigma}^*] \frac{1}{A}+\bar{p}_i[h,r'|\vec{\sigma}^*]\frac{1}{B - C},$$ where - $A = 1 - \frac{\epsilon({\bar{d}}_i+1)}{(1-q_i[h,r|\vec{\sigma}^*]){\tau}}$. - $B=\frac{{\tau}}{c}$. - $C=\frac{\epsilon({\bar{d}}_i+1)}{1-q_i[h,r'|\vec{\sigma}^*]}$. The proof considers two histories $h_1$ and $h_2$ that minimize the first and the second factors of Inequality \[eq:delay-equiv\], respectively. Thus, if the following condition is true, then Inequality \[eq:delay-equiv\] is true: $$- \sum_{r=0}^{{\bar{d}}_i} \omega_i^r ((1-q_i^*[h_1,0|\vec{\sigma}^*])\gamma_i \bar{p}_i[h_1,0|\vec{\sigma}^*] + \epsilon \beta_i)+ \sum_{r={\bar{d}}_i+1}^{{\bar{d}}_i+{\tau}+1} \omega_i^r u_i[h_2,0 | \vec{\sigma}^*] \geq 0.$$ After some manipulations, we conclude that the above condition is fulfilled if \[eq:priv-suff\] is true. This implies by Lemma \[lemma:delay-equiv\] that if \[eq:priv-suff\] holds for every $h$, and $r,r' \leq {\bar{d}}_i + {\tau}$, then Inequality \[eq:delay-equiv\] also holds for some $\omega_i \in (0,1)$ and $(\vec{\sigma}^*,\vec{\mu}^*)$ is Sequentially Rational. (). The main conclusion that can be drawn from this lemma is that if we pick the values of ${\tau}$ and $\epsilon$ such that ${\tau}\geq {\bar{d}}+ 1$ and $\epsilon \ll 1$, then we can simplify the above condition to what is expressed in Theorem \[theorem:priv-effect\], where ${\bar{d}}= \max_{i \in {\cal N}} {\bar{d}}_i$ is the maximum delay of the monitoring mechanism. \[theorem:priv-effect\] If $(\vec{\sigma}^*,\vec{\mu}^*)$ is Preconsistent, Assumptions \[def:non-neg\] and \[def:priv-assum\] hold for $\epsilon \ll 1$, and ${\tau}\geq {\bar{d}}+ 1$, then there exists a constant $c>0$ such that $\psi[\vec{\sigma}^*|\vec{\mu}^*] \supseteq (v,\infty)$, where $$v = \max_{i \in {\cal N}} \max_{h \in {\cal H}}\bar{p}_i[h,0|\vec{\sigma}^*](1+c).$$ (). As in public monitoring, the effectiveness is close to optimal only if the initial expected costs of forwarding messages $\bar{p}_i$ are not significantly smaller than the expected costs incurred after any history. Provided this guarantee, if $\epsilon$ is small and ${\tau}$ is chosen to be at least of the order of the maximum delay between out and in-neighbors of any node, then for any other punishing strategy, the effectiveness differs from the optimal by a constant factor. Notice that, although we can adjust the value of ${\tau}$ to compensate for higher delays, it is desirable to have a low maximum delay. First, this is due to the fact that higher values of ${\tau}$ correspond to harsher punishments, which we may want to avoid, especially when monitoring is imperfect and honest nodes may wrongly be accused of deviating. Second, a larger delay decreases the range of values of $\omega_i$ for each benefit-to-cost ratio that ensures that punishing strategies are an equilibrium. In particular, we can derive from the proof of Lemma \[lemma:priv-suff\] the strict minimum $\omega_i$ for Grim-trigger to be an equilibrium. Under our assumptions, it is approximately given by $$\omega_i \geq \sqrt[{\bar{d}}_i]{\frac{\gamma_i \bar{p}_i}{\beta_i}}.$$ For larger values of ${\bar{d}}_i$ and the same benefit-to-cost ratio, the minimum $\omega_i$ is also larger, reducing the likelihood of punishments to persuade rational nodes to not deviate from the specified strategy. Discussion and Future Work {#sec:conc} ========================== From this analysis, we can derive several desirable properties of a fully distributed monitoring mechanism for an epidemic dissemination protocol with asymmetric interactions, which uses punishments as the main incentive. This mechanism is expected to operate on top of an overlay network that provides a stable membership to each node. The results of this paper determine that the overlay should optimally explore the tradeoff between maximal randomization and higher clustering coefficient. The former is ideal for minimizing the latency of the dissemination process and fault tolerance, whereas the latter is necessary to minimize the distances between the neighbors of each node, while maximizing the number of in-neighbors of every node $i$ informed about any defection of $i$. The topology of this overlay should also fulfill the necessary conditions identified in this paper. Furthermore, the analysis of private monitoring shows that each accusation may be disseminated to a subset of nodes close to the accused node, without hindering the effectiveness. As future work, we plan to extend this analysis by considering imperfect monitoring, unreliable dissemination of accusations, malicious behavior, and churn. One possible application of the considered monitoring mechanism would be to sustain cooperation in a P2P news recommendation system such as the one proposed in [@Boutet:13]. Due to the lower rate of arrival of news, a monitoring approach may be better suited in this context. Acknowledgements {#acknowledgements .unnumbered} ================ This work was partially supported by Fundação para a Ciência e Tecnologia (FCT) via the INESC-ID multi-annual funding through the PIDDAC Program fund grant, under project PEst-OE/ EEI/ LA0021/ 2013, and via the project PEPITA (PTDC/EEI-SCR/2776/2012). Epidemic Model {#sec:epidemic} ============== The probability that a node $i$ does not receive a message from $s$ when all players follow $\vec{p}$ can be defined recursively, as follows. \[prop:non-delivery\] Define $\phi$ as follows: i) $\phi[R,\emptyset | \vec{p},L] = 1$ and ii) for $I \neq \emptyset$ and $R' = R \cup I \cup L$: $$\label{eq:prod-non-delivery} \begin{array}{ll} \phi[R,I | \vec{p},L] = \sum_{H \subseteq {\cal N} \setminus R'} (P[I,H|\vec{p}] \cdot Q[{\cal N},R,I,H|\vec{p}]\cdot \phi[R \cup I, H | \vec{p},L]), \end{array}$$ where $$\begin{array}{l} P[I,H|\vec{p}] = \prod_{k \in H} (1 - \prod_{l \in I} (1 - p_l[k])).\\ Q[{\cal N},R,I,H|\vec{p}] = \prod_{k \in {\cal N} \setminus (H \cup R \cup I)} \prod_{l \in I} (1 - p_l[k]). \end{array}$$ Then, $\phi[\emptyset,\{s\}|\vec{p},L]$ is the probability that no node of $L$ receives a message disseminated by $s$. In particular, $q_i[\vec{p}] = \phi[\emptyset,\{s\}|\vec{p},\{i\}]$. (**Justification**) The considered epidemics model is very similar to the Reed-Frost model[@Abbey:52], where dissemination is performed by having nodes forwarding messages with independent probabilities. The main difference is that the probability of forwarding each message is determined by a vector $\vec{p}$, instead of being identical for every node. This implies that the dissemination process can be modeled as a sequence of steps, such that, at every step, there is a set $I$ of nodes infected in the last step, a set $R$ of nodes infected in previous steps other than the last, and a set $S$ of susceptible nodes. Given $R$ and $I$, the probability of the set $H \subseteq {\cal N} \setminus (I \cup R \cup L)$ containing exactly the set of nodes infected at the current step is $$\label{eq:non-delivery} P[I,H|\vec{p}]\cdot Q[{\cal N},I,H|\vec{p}] = \prod_{k \in H} (1 - \prod_{l \in I} (1 - p_l[k])) \cdot \prod_{k \in {\cal N} \setminus (H \cup R \cup I)} \prod_{l \in I} (1 - p_l[k]).$$ That is, every node $i \in H$ is infected with a probability equal to $1$ minus the probability of no node of $I$ choosing $i$, and these probabilities are all independent. Furthermore, all nodes of $I$ do not infect any node of ${\cal N} \setminus (H \cup R \cup I)$. We can characterize $\phi$ with a weighted tree, where nodes correspond to a pair $(R,I)$. Moreover, for every parent node $(R,I)$, each child node corresponds to a pair $(R\cup I,H)$ for every $H \subseteq {\cal N} \setminus (R \cup I \cup L)$. The root node is the pair $(\emptyset,\{s\})$ and leaf nodes are in the form $(R,\emptyset)$ for every $R \subseteq \{s \} \cup {\cal N} \setminus L$. The weight of the transition from $(R,I)$ to $(R\cup I,H)$ is given by \[eq:non-delivery\], which is the probability of exactly the nodes of $H$ being infected among every node of ${\cal N} \setminus (R \cup I \cup L)$. The sum of these factors for any path from $(\emptyset,\{s\})$ to $(R,\emptyset)$ gives the probability of exactly the nodes of $R$ being infected in a specific order. By summing over all leaf nodes in the form $(R,\emptyset)$, we have the total probability of exactly the nodes of $R$ being infected. Finally, by summing over all possible $R \subseteq \{s\} \cup {\cal N} \setminus L$, we obtain the probability of no node in $L$ being infected. In particular, $q_i[\vec{p}] = \phi[\emptyset,\{s\} | \vec{p},\{i\}]$. The following are some useful axioms for the proofs, for any $\vec{p}$, $R$, $I$, $L$, and $R' = R \cup I \cup L$: $$\label{eq:epid-1} \sum_{H \subseteq {\cal N} \setminus R'} P[I,H|\vec{p}]\cdot Q[{\cal N},R,I,H|\vec{p}] = 1.$$ If $(A,B)$ is a partition of ${\cal N}$, then $$\label{eq:epid-2} \begin{array}{ll} &\sum_{H \subseteq {\cal N} \setminus R'} P[I,H|\vec{p}]\cdot Q[{\cal N},R,I,H|\vec{p}]=\\ =&\sum_{H_1 \subseteq A \setminus R'} P[I,H_1|\vec{p}]Q[A,R,I,H_1|\vec{p}] \cdot\\ \cdot&\sum_{H_2 \subseteq B \setminus R'} P[I,H_2|\vec{p}]\cdot Q[B,R,I,H_1|\vec{p}]\\ \end{array}$$ $$\label{eq:epid-3} j \in I \Rightarrow P[I,H|\vec{p}] \leq P[I \setminus \{j\},H|\vec{p}].$$ Deterministic Delivery ---------------------- Lemma \[lemma:prob-1\] proves the straightforward fact that if there is a path from $s$ to some node $i$ where all nodes along the path forward messages with probability $1$, then $q_i=0$. \[lemma:prob-1\] For any $\vec{p} \in {\cal P}$, if there exists $i \in {\cal N}$ and $x \in {\mbox{PS}}[s,i]$ such that for every $r \in \{0\ldots |x|-1\}$ we have $p_{x_r}[x_{r+1}]= 1$, then $q_i[\vec{p}] = 0$. The proof goes by induction on $r$ where the induction hypothesis is that, for every $r \in \{0\ldots |x|-2\}$, $R \subseteq {\cal N} \cup \{s\} \setminus \{i\}$, and $I \subseteq {\cal N} \cup \{s\} \setminus \{i\} \cup R$, such that: - $x_r \in I$, - for every $r' \in \{0\ldots r\}$, $x_{r'} \in R \cup I$, - for every $r' \in \{r+1\ldots |x|-1\}$, $x_{r'} \in {\cal N} \setminus (R \cup I)$, we have $\phi[R,I|\vec{p},\{i\}] = 0$. Consider the base case for $r=|x|-2$ and let $R' = R \cup I \cup \{i\}$. In this case, for $j= x_{r}$, $$\begin{array}{lll} \phi[R,I | \vec{p},\{i\}] &=& \sum_{H \subseteq {\cal N} \setminus R'} (P[I,H|\vec{p}] \cdot Q[{\cal N},R,I,H|\vec{p}] \cdot \phi[R \cup I, H | \vec{p},\{i\}])\\ &\\ &= &\sum_{H \subseteq {\cal N} \setminus R'} (P[,I,H|\vec{p}] \cdot Q[{\cal N}\setminus \{i\},R,I,H|\vec{p}]\\ &&\prod_{l \in I \setminus \{j\}} (1 - p_l[i])(1-p_j[i])\phi[R \cup I, H | \vec{p},\{i\}])\\ &\\ &= &\sum_{H \subseteq {\cal N} \setminus R' } (P[,I,H|\vec{p}] \cdot Q[{\cal N}\setminus \{i\},R,I,H|\vec{p}]\\ & & 0 \cdot \phi[R \cup I, H | \vec{p},\{i\}])\\ &\\ &= &0. \end{array}$$ This proves the base case. Assume now that the hypothesis is true for every $r' \in \{0 \ldots r\}$ and some $r \in \{1 \ldots |x|-2\}$. Let $a = x_{r-1}$ and $b = x_{r}$, let $R_1 = R \cup I$ and $R_2 = R_1 \cup H$. We thus have $$\begin{array}{lll} \phi[R,I | \vec{p},\{i\}] &= &\sum_{H \subseteq {\cal N} \setminus (R_1 \cup \{i\})}(P[I,H|\vec{p}]\cdot Q[{\cal N},R,I,H|\vec{p}]\cdot \phi[R_1, H | \vec{p},\{i\}])\\ &&\\ &= &\sum_{H \subseteq {\cal N} \setminus (R_1 \cup \{i,b\})} (P[I,H|\vec{p}] \cdot Q[{\cal N} \setminus \{b\},R,I,H|\vec{p}]\\ & &\prod_{l \in I \setminus \{a\}} (1 - p_l[b]) (1 - p_a[b]) \phi[R_1, H | \vec{p},\{i\}])\\ &+&\sum_{H \subseteq {\cal N} \setminus (R_1 \cup \{i,b\})} (P[I,H \cup \{b\}|\vec{p}] \cdot Q[{\cal N},R,I,H\cup \{b\}|\vec{p}] \cdot \phi[R_1, H\cup \{b\} | \vec{p},\{i\}])\\ &&\\ &= &\sum_{H \subseteq {\cal N} \setminus (R_1 \cup \{i,b\})} (P[I,H|\vec{p}] \cdot Q[{\cal N} \setminus \{b\},R,I,H|\vec{p}] \cdot 0\cdot \phi[R_1, H | \vec{p},\{i\}])\\ &+&\sum_{H \subseteq {\cal N} \setminus (R_1 \cup \{i,b\})} (P[I,H \cup \{b\}|\vec{p}] \cdot Q[{\cal N},R,I,H\cup \{b\}|\vec{p}] \cdot 0)\\ &&\\ &=&0. \end{array}$$ This proves the induction step for $r-1$. Consequently, since $s = x_0$ and $x_r \in {\cal N}$ for every $r \in \{1 \ldots |x|-1\}$, $$q_i[\vec{p}] = \phi[\emptyset,\{s\}|\vec{p},\{i\}] = 0.$$ Positive Reliability -------------------- Lemma \[lemma:pprob\] shows that if every node forwards messages with a positive probability, then every node of the graph receives a message with positive probability as well. \[lemma:pprob\] For any $\vec{p} \in {\cal P}$, if there exists $i \in {\cal N}$ and $x \in {\mbox{PS}}[s,i]$ such that for every $r \in \{0\ldots |x|-1\}$ we have $p_{x_r}[x_{r+1}]>0$, then $q_i[\vec{p}] <1$. The proof goes by induction on $r$ where the induction hypothesis is that, for every $r \in \{0\ldots |x|-2\}$, $R \subseteq {\cal N} \cup \{s\} \setminus \{i\}$, and $I \subseteq {\cal N} \cup \{s\} \setminus \{i\} \cup R$, such that: - $x_r \in I$, - for every $r' \in \{0\ldots r\}$, $x_{r'} \in R \cup I$, - for every $r' \in \{r+1\ldots |x|-1\}$, $x_{r'} \in {\cal N} \setminus (R \cup I)$, we have $\phi[R,I|\vec{p},\{i\}] < 1$. Consider the base case for $r=|x|-2$ and let $R_1 = R \cup I \cup \{i\}$. In this case, by Axiom \[eq:epid-1\], for $j= x_{r}$, $$\begin{array}{lll} \phi[R,I | \vec{p},\{i\}] &= &\sum_{H \subseteq {\cal N} \setminus R_1} (P[I,H|\vec{p}] \cdot Q[{\cal N},R,I,H|\vec{p}] \cdot \phi[R \cup I, H | \vec{p},\{i\}])\\ & & \\ &= &\sum_{H \subseteq {\cal N} \setminus R_1}(P[I,H|\vec{p}] \cdot Q[{\cal N}\setminus \{i\},R,I,H|\vec{p}] \cdot \\ & & \prod_{l \in I \setminus \{j\}} (1 - p_l[i])(1-p_j[i])\phi[R \cup I, H | \vec{p},\{i\}])\\ &&\\ &< &\sum_{H \subseteq {\cal N} \setminus R_1}P[I,H|\vec{p}] \cdot Q[{\cal N},R,I,H|\vec{p}]\\ &&\\ &=&1. \end{array}$$ This proves the base case. Assume now that the hypothesis is true for every $r' \in \{0 \ldots r\}$ and some $r \in \{1 \ldots |x|-2\}$. Let $a = x_{r-1}$ and $b = x_{r}$. Consider that $R_1 = R \cup I$ and $R_2 = R \cup H \cup I$. We thus have by Axioms \[eq:epid-1\] and \[eq:epid-3\], $$\begin{array}{lll} \phi[R,I | \vec{p},\{i\}] &= &\sum_{H \subseteq {\cal N} \setminus (R_1\cup \{i\})} (P[I,H|\vec{p}] \cdot Q[{\cal N},R,I,H|\vec{p}] \phi[R_1, H | \vec{p},\{i\}])\\ &&\\ &= &\sum_{H \subseteq {\cal N} \setminus (R_1 \cup \{i,b\})} (P[I,H|\vec{p}] \cdot Q[{\cal N} \setminus \{b\},R,I,H|\vec{p}]\\ & & \prod_{l \in I} (1 - p_l[k])\prod_{l \in I \setminus \{a\}} (1 - p_l[k])(1 - p_a[b])\phi[R_1, H | \vec{p},\{i\}])\\ &+&\sum_{H \subseteq {\cal N} \setminus (R_1 \cup \{i,b\})} (P[I,H\cup \{b\}|\vec{p}] \cdot Q[{\cal N},R,I,H \cup \{b\}|\vec{p}] \cdot \phi[R_1, H\cup \{b\} | \vec{p},\{i\}])\\ &&\\ &< &\sum_{H \subseteq {\cal N} \setminus (R_1 \cup \{i,b\})} (P[I,H|\vec{p}] \cdot Q[{\cal N} \setminus \{b\},R,I,H|\vec{p}]\\ & & \prod_{l \in I} (1 - p_l[k])\prod_{l \in I \setminus \{a\}} (1 - p_l[k]) \cdot 1 \cdot \phi[R_1, H | \vec{p},\{i\}])\\ &+&\sum_{H \subseteq {\cal N} \setminus (R_1 \cup \{i,b\})} (P[I \setminus \{a\},H\cup \{b\}|\vec{p}] \cdot Q[{\cal N},R,I \setminus \{a\},H \cup \{b\}|\vec{p}])\\ &&\\ &\leq &\sum_{H \subseteq {\cal N} \setminus (R_1 \cup \{i,b\})} (P[I \setminus \{a\},H|\vec{p}] \cdot Q[{\cal N},R,I \setminus \{a\},H|\vec{p}])\\ &+&\sum_{H \subseteq {\cal N} \setminus (R_1 \cup \{i\})} (P[I \setminus \{a\},H\cup \{b\}|\vec{p}] \cdot Q[{\cal N},R,I \setminus \{a\},H \cup \{b\}|\vec{p}])\\ &&\\ &= &\sum_{H \subseteq {\cal N} \setminus (R_1 \cup \{i\})} (P[I \setminus \{a\},H|\vec{p}] \cdot Q[{\cal N},R,I \setminus \{a\},H|\vec{p}])\\ &&\\ & = &1. \end{array}$$ This proves the induction step for $r-1$. Consequently, since $s = x_0$ and $x_r \in {\cal N}$ for every $r \in \{1 \ldots |x|-1\}$, $$q_i[\vec{p}] = \phi[\emptyset,\{s\}|\vec{p},\{i\}] < 1.$$ Null Reliability ---------------- Lemma \[lemma:noneib\] shows that if every in-neighbor of a node $i$ does not forward messages to $i$, then $q_i = 0$. \[lemma:noneib\] If $\vec{p} \in {\cal P}$ is defined such that for some $i \in {\cal N}$ and for every $j \in {\cal N}_i^{-1}$ $p_j[i] = 0$, then $q_i[\vec{p}] = 1$. The proof goes by induction on $r$ where the induction hypothesis is that for every $r \in \{0\ldots |{\cal N}|\}$, $R \subseteq {\cal N} \cup \{s\} \setminus \{i\}$, and $I \subseteq {\cal N} \cup \{s\} \setminus \{i\} \cup R$ such that $|R|+|I| \leq |{\cal N}|+1-r$, we have $\phi[R,I|\vec{p},\{i\}] = 1$. The base case is for $r=0$, where we have by the definition of $\vec{p}$. $$\phi[R,I|\vec{p},\{i\}] = \prod_{l \in I} (1 - p_l[i]) \phi[R \cup I, \emptyset | \vec{p},\{i\}] = \prod_{l \in I} (1 - 0)\cdot 1 = 1.$$ Assume the induction hypothesis for any $r \in \{0 \ldots|{\cal N}|-1\}$. Consider any two $R$ and $I$ defined as above for $r+1$, such that $|R|+|I| = |{\cal N}|-r$. Let $R_1 = R \cup I \cup \{i\}$ and $R_2 = H \cup R \cup I$. It is true by Axiom \[eq:epid-1\] that: $$\begin{array}{ll} \phi[R,I | \vec{p},\{i\}] = &\sum_{H \subseteq {\cal N} \setminus R_1}P[I,H|\vec{p}] Q[{\cal N},R,I,H|\vec{p}]\phi[R \cup I, H | \vec{p},\{i\}])\\ &\\ &=\sum_{H \subseteq {\cal N} \setminus R_1}P[I,H|\vec{p}] Q[{\cal N},R,I,H|\vec{p}]\cdot 1\\ &\\ & = 1. \end{array}$$ Therefore, for $r=|{\cal N}|$, $q_i[\vec{p}] = \phi[\emptyset,\{s\}|\vec{p},\{i\}]=1$. Uniform Reliability ------------------- Lemma \[lemma:bottleneck-impact\] shows that for some node $i$ and for every path $x$ from $s$ to and $i$, every node of $x$ does not change its probability from $\vec{p}$ to $\vec{p}'$, then $q_i[\vec{p}] = q_i[\vec{p}']$. As an intermediate step, Lemma \[lemma:bt-aux\] proves that the reliability is the same whenever $I$ and $R$ contain the same set of nodes from any path from $s$ to $i$. For any $i \in {\cal N}$, $\vec{p} \in {\cal P}$, and $K \subseteq {\cal N}$, let $$D_i = \{j \in {\cal N} \cup \{s\} \setminus \{i\}| {\mbox{PS}}[j,i] \neq \emptyset\},$$ and: $$p[I,k] = 1 - \prod_{j \in I}(1-p_j[k]).$$ Therefore, it is possible to write: $$\begin{array}{l} P[I,H|\vec{p}] = \prod_{k \in H} p[I,k].\\ Q[{\cal N},R,I,H|\vec{p}] = \prod_{k \in {\cal N} \setminus (R \cup I \cup H)} (1-p[I,k]). \end{array}$$ For any $L_1,L_2 \subseteq {\cal N} \setminus D_i$ such that $L_1 \cap L_2 = \emptyset$, since for every $j \in L$ we have ${\mbox{PS}}[j,i] = \emptyset$ and $p_j[k] =0$ for every $k \in D_i$, and for every $H \subseteq D_i$, $$\label{eq:bta-1} \begin{array}{l} p[I \cup L_1,k] = p[I,k].\\ P[I \cup L_1,H|\vec{p}] = P[I,H|\vec{p}].\\ Q[{\cal N},R \cup L_1,I \cup L_2,H|\vec{p}] = Q[{\cal N},I,H|\vec{p}]. \end{array}$$ \[lemma:bt-aux\] For every $\vec{p} \in {\cal P}$, $i \in {\cal N}$, $R \subseteq {\cal N} \cup \{s\} \setminus \{i\}$, $I \subseteq D_i \setminus R$, and $L_1,L_2 \subseteq {\cal N} \setminus (D_i \cup R)$ such that $L_1 \cap L_2 = \emptyset$, $$\phi[R\cup L_1,I\cup L_2|\vec{p},\{i\}] = \phi[R,I|\vec{p},\{i\}].$$ Fix $\vec{p}$ and $i$. First notice that for every $R \subseteq {\cal N}$ and $L \subseteq {\cal N} \setminus (D_i \cup R)$, $$\label{eq:bta-2} \phi[R,L|\vec{p},\{i\}] = 1.$$ We now prove by induction that for every $R \subseteq {\cal N} \cup \{s\}$, $I \subseteq D_i \setminus R$, and $L_1, L_2\subseteq {\cal N} \setminus (D_i \cup R)$ such that $L_1\cap L_2 = \emptyset$, $$\phi[R, L_1,I\cup L_2|\vec{p},\{i\}] = \phi[R,I|\vec{p},\{i\}].$$ The induction goes on $r \in \{0\ldots |{\cal N}|\}$, where $|R| + |I| + |L| = |{\cal N}| + 1 - r$, where $L = L_1 \cup L_2$. For $r=0$, by Axiom \[eq:epid-1\], \[eq:bta-1\] and \[eq:bta-2\], we can write: $$\label{eq:bta-3} \begin{array}{ll} \phi[R \cup L_1,I\cup L_2 | \vec{p},\{i\}] &= (1 - p[I\cup L_1,i])\phi[R\cup I \cup L,\emptyset | \vec{p},\{i\}]\\ &= (1-p[I,i])\\ &= (1 - p[I,i])\sum_{H \subseteq L } \phi[R \cup I,H|\vec{p},\{i\}]\\ &= (1 - p[I,i])\sum_{H \subseteq {\cal N} \setminus (R \cup I \cup \{i\})}(P[I,H|\vec{p}]\cdot Q[{\cal N},R,I,H|\vec{p}]\\ &\cdot \phi[R \cup I,H|\vec{p},\{i\}])\\ &= \phi[R,I | \vec{p},\{i\}]. \end{array}$$ This proves the base case. Assume the induction hypothesis for every $r' \in \{0 \ldots r\}$ and $r \in \{0\ldots |{\cal N}|-1\}$. By Axioms \[eq:epid-1\] and \[eq:epid-2\], and by \[eq:bta-1\] and the induction hypothesis, we can write: $$\begin{array}{ll} \phi[R \cup L_1,I\cup L_2 | \vec{p},\{i\}] &=\sum_{H \subseteq {\cal N} \setminus (R \cup I \cup L \cup \{i\})}(\\ &P[I \cup L_2,H|\vec{p}] \cdot Q[{\cal N},R \cup L_1,I \cup L_2,H|\vec{p}] \cdot\\ & \phi[R\cup I \cup L,H | \vec{p},\{i\}])\\ &\\ &=\sum_{H_1 \subseteq D_i \setminus (R \cup I \cup \{i\})}(\\ &P[I \cup L_2,H_1|\vec{p}] \cdot Q[D_i,R \cup L_1,I \cup L_2,H_1|\vec{p}]\\ &\sum_{H_2 \subseteq {\cal N} \setminus (R \cup L \cup \{i\} \cup D_i)}(\\ &P[I \cup L_2,H_2|\vec{p}] \cdot Q[{\cal N} \setminus D_i,R \cup L_1,I \cup L_2,H_2|\vec{p}]\\ & \phi[R\cup I \cup L,H_1 \cup H_2 | \vec{p},\{i\}]))\\ &\\ &=\sum_{H_1 \subseteq D_i \setminus (R \cup I \cup L \cup \{i\})}(\phi[R\cup I,H_1| \vec{p},\{i\}]\\ &P[I,H_1|\vec{p}] \cdot Q[D_i,R \cup L_1,I \cup L_2,H_1|\vec{p}]\\ &\sum_{H_2 \subseteq {\cal N} \setminus (R \cup L \cup \{i\} \cup D_i)}(\\ &P[I \cup L_2,H_2|\vec{p}] \cdot Q[{\cal N} \setminus D_i,R \cup L_1,I \cup L_2,H_2|\vec{p}]))\\ &\\ &=\sum_{H_1 \subseteq D_i \setminus (R \cup I \cup L \cup \{i\})}(\phi[R\cup I,H_1| \vec{p},\{i\}]\\ &P[I,H_1|\vec{p}] \cdot Q[D_i,R,I,H_1|\vec{p}])\\ &\\ &=\sum_{H_1 \subseteq D_i \setminus (R \cup I \cup L \cup \{i\})}(\\ &P[I ,H_1|\vec{p}] \cdot Q[D_i,R,I ,H_1|\vec{p}]\\ &\sum_{H_2 \subseteq {\cal N} \setminus (R \cup L \cup \{i\} \cup D_i)}(\\ &P[I,H_2|\vec{p}] \cdot Q[{\cal N} \setminus D_i,R,I,H_2|\vec{p}]\phi[R\cup I,H_1 \cup H_2| \vec{p},\{i\}]))\\ &\\ &=\sum_{H \subseteq {\cal N} \setminus (R \cup I \cup \{i\})}(\\ &P[I,H|\vec{p}] \cdot Q[{\cal N},R,I,H|\vec{p}] \cdot \phi[R \cup I,H | \vec{p},\{i\}])\\ &\\ &=\phi[R,I|\vec{p},\{i\}]. \end{array}$$ This proves the result. \[lemma:bottleneck-impact\] Let $\vec{p},\vec{p}' \in {\cal P}$ be any two profiles of probabilities such that for some $i\in{\cal N}$, for every $x \in {\mbox{PS}}[s,i]$, and for every $j \in x$, $\vec{p}_j = \vec{p}'_j$. Then, $q_i[\vec{p}] = q_i[\vec{p}']$. Assume this to be the case for a fixed $i$, $\vec{p}$, and $\vec{p}'$. Then, for every $x \in {\mbox{PS}}[s,i]$ and $j \in x$, it is true that, for every $k \in {\cal N}_j^{-1}$, there exists $x' \in {\mbox{PS}}[s,i]$ such that $$\label{eq:bi-1} k \in x' \land p_k[j] = p_k'[j].$$ Define $p'[I,k]$ as in \[eq:bta-1\], but for $\vec{p}'$. Condition \[eq:bi-1\] implies that for every $I \subseteq {\cal N} \cup \{s\}$ and $k \in D_i \cup \{i\}$: $$\label{eq:bi-2} p'[I,k] = p[I,k].$$ The rest of the proof is performed by induction on $r$ where the induction hypothesis is that for every $r \in \{0\ldots|{\cal N}|\}$, $R \subseteq {\cal N} \cup \{s\} \setminus \{i\}$, and $I \subseteq {\cal N} \cup \{s\} \setminus \{i\} \cup R$ such that $|R|+|I| \leq |{\cal N}|+1-r$, we have $\phi[R,I|\vec{p},\{i\}] = \phi[R,I|\vec{p}',\{i\}]$. The base case is for $r=0$, where we have by \[eq:bi-2\] and the definition of $\vec{p}$ and $\vec{p}'$: $$\begin{array}{ll} \phi[R,I|\vec{p},\{i\}] & = p[I,i] \phi[R \cup I, \emptyset | \vec{p},\{i\}]\\ &= p'[I,i] \phi[R \cup I, \emptyset | \vec{p}',\{i\}]\\ & = \phi[R,I|\vec{p}',\{i\}]. \end{array}$$ This proves the base case. Now, assume the induction hypothesis for some $r \in \{0\ldots |{\cal N}|-1\}$. It is true by Lemma \[lemma:bt-aux\], by Axioms \[eq:epid-1\] and \[eq:epid-2\], by \[eq:bi-2\], and the induction hypothesis that: $$\begin{array}{ll} \phi[R,I | \vec{p},\{i\}] &=\sum_{H \subseteq {\cal N} \setminus (R \cup I \cup \{i\})}( P[I,H|\vec{p}] \cdot Q[{\cal N},R,I,H|\vec{p}] \cdot\phi[R\cup I,H | \vec{p},\{i\}])\\ &\\ &=\sum_{H_1 \subseteq D_i \setminus (R \cup I \cup \{i\})}(\\ &P[I,H_1|\vec{p}] \cdot Q[D_i,R,I,H_1|\vec{p}]\\ &\sum_{H_2 \subseteq {\cal N} \setminus (R \cup \{i\} \cup D_i)}(\\ &P[I,H_2|\vec{p}] \cdot Q[{\cal N} \setminus D_i,R,I,H_2|\vec{p}]\\ & \phi[R\cup I,H_1 \cup H_2 | \vec{p},\{i\}]))\\ &\\ &=\sum_{H_1 \subseteq D_i \setminus (R \cup I \cup \{i\})}(\phi[R\cup I,H_1| \vec{p},\{i\}]\\ &P[I,H_1|\vec{p}] \cdot Q[D_i,R,I,H_1|\vec{p}]\\ &\sum_{H_2 \subseteq {\cal N} \setminus (R \cup \{i\} \cup D_i)}(\\ &P[I,H_2|\vec{p}] \cdot Q[{\cal N} \setminus D_i,R,I,H_2|\vec{p}]))\\ &\\ &=\sum_{H_1 \subseteq D_i \setminus (R \cup I \cup \{i\})}(\phi[R\cup I,H_1| \vec{p}',\{i\}]\\ &P[I,H_1|\vec{p}'] \cdot Q[D_i,R,I,H_1|\vec{p}'])\\ &\\ &=\sum_{H_1 \subseteq D_i \setminus (R \cup I \cup \{i\})}(\\ &P[I ,H_1|\vec{p}'] \cdot Q[D_i,R,I ,H_1|\vec{p}']\\ &\sum_{H_2 \subseteq {\cal N} \setminus (R \cup L \cup \{i\} \cup D_i)}(\\ &P[I,H_2|\vec{p}'] \cdot Q[{\cal N} \setminus D_i,R,I,H_2|\vec{p}']\phi[R\cup I,H_1 \cup H_2| \vec{p}',\{i\}]))\\ &\\ &=\sum_{H \subseteq {\cal N} \setminus (R \cup I \cup \{i\})}(\\ &P[I,H|\vec{p}'] \cdot Q[{\cal N},R,I,H|\vec{p}'] \cdot \phi[R \cup I,H | \vec{p}',\{i\}])\\ &\\ &=\phi[R,I|\vec{p}',\{i\}]. \end{array}$$ This concludes the proof by induction. Therefore, for $r=|{\cal N}|$, $$q_i[\vec{p}] = \phi[\emptyset,\{s\}|\vec{p},\{i\}]=\phi[\emptyset,\{s\}|\vec{p}',\{i\}] = q_i[\vec{p}'].$$ Single Impact ------------- Lemma \[lemma:single-impact\] provides an upper bound for the impact in the reliability when a single in-neighbor $j$ punishes node $i$. \[lemma:single-impact\] For every $i \in {\cal N}$, $j \in {\cal N}_i^{-1}$, $\vec{p} \in {\cal P}$ such that $p_j[i] < 1$ and $q_i[\vec{p}]>0$, if $\vec{p}'$ is the profile where only $j$ deviates from $p_j[i]$ to $p_j[i]' < p_j[i]$, then $$q_i[\vec{p}'] \leq q_i[\vec{p}] \frac{1-p_{j}[i]'}{1-p_{j}[i]}.$$ Fix $i$, $j$, $\vec{p}$, and $\vec{p}'$. The proof shows by induction that for every $r \in \{0 \ldots |{\cal N}|\}$, $R \subseteq {\cal N} \cup \{s\} \setminus \{i\}$ and $I \subseteq {\cal N} \cup \{s\} \setminus \{i\} \cup R$ such that $|R|+|I|- r =|{\cal N}|-1$: $$\phi[R,I|\vec{p}',\{i\}] \leq \phi[R,I|\vec{p},\{i\}] \frac{1-p_{j}'[i]}{1-p_{j}[i]}.$$ Notice that by the definition of $\phi$, if $j \in R$, then $$\label{eq:si-1} \phi[R,I|\vec{p}',\{i\}] = \phi[R,I|\vec{p},\{i\}].$$ For the base case $r=0$, if $j \in R$, then the result follows immediately. Thus, consider that $j \in I$. We can write $$\begin{array}{ll} \phi[R,I|\vec{p}',\{i\}] & = p[I,i] \phi[R \cup I, \emptyset | \vec{p},\{i\}]\\ &= p[I \setminus \{j\},i] \cdot p'[\{j\},i]\\ &= p[I \setminus \{j\},i] \cdot (1-p_j[i])\frac{1-p_{j}'[i]}{1-p_{j}[i]}\\ &= p[I,i]\frac{1-p_{j}'[i]}{1-p_{j}[i]}\\ & = \phi[R,I|\vec{p},\{i\}]\frac{1-p_{j}'[i]}{1-p_{j}[i]}. \end{array}$$ This proves the induction step for $r=0$. Assume now that the induction hypothesis is true for every $r' \in \{0 \ldots r\}$ and for some $r \in \{0 \ldots|{\cal N}|-1\}$. If $j \notin I$, then by the induction hypothesis and by \[eq:si-1\], $$\begin{array}{ll} \phi[R,I|\vec{p}',\{i\}] &=\sum_{H \subseteq {\cal N} \setminus (R \cup I \cup \{i\})}(P[I,H|\vec{p}']\cdot Q[{\cal N},R,I,H|\vec{p}'] \cdot \phi[R\cup I,H | \vec{p}',\{i\}])\\ &\\ & = \sum_{H \subseteq {\cal N} \setminus (R \cup I \cup \{i\})}(P[I,H|\vec{p}]\cdot Q[{\cal N},R,I,H|\vec{p}] \cdot \phi[R\cup I,H | \vec{p}',\{i\}])\\ &\\ & \leq \sum_{H \subseteq {\cal N} \setminus (R \cup I \cup \{i\})}(P[I,H|\vec{p}]\cdot Q[{\cal N},R,I,H|\vec{p}] \cdot \phi[R\cup I,H | \vec{p},\{i\}] \frac{1-p_{j}'[i]}{1-p_{j}[i]})\\ &\\ & \leq \phi[R,I|\vec{p},\{i\}] \frac{1-p_{j}'[i]}{1-p_{j}[i]}. \end{array}$$ For the final case where $j \in I$, we have by \[eq:si-1\]: $$\begin{array}{ll} \phi[R,I|\vec{p}',\{i\}] &=\sum_{H \subseteq {\cal N} \setminus (R \cup I \cup \{i\})}(P[I,H|\vec{p}']\cdot Q[{\cal N},R,I,H|\vec{p}'] \cdot \phi[R\cup I,H | \vec{p}',\{i\}])\\ &\\ &= \sum_{H \subseteq {\cal N} \setminus (R \cup I \cup \{i\})}(\prod_{k \in H}p[I,k] \\ & \prod_{k \in {\cal N} \setminus (H \cup R \cup I \cup \{i\})} (1- p[I,k])p[I\setminus \{j\},i] (1 - p_j[i])\frac{1-p_{j}'[i]}{1-p_{j}[i]}\phi[R\cup I,H | \vec{p},\{i\}])\\ &\\ & = \phi[R,I|\vec{p},\{i\}] \frac{1-p_{j}'[i]}{1-p_{j}[i]}. \end{array}$$ This concludes the proof. Public Monitoring {#sec:proof:public} ================= Evolution of the Network {#sec:proof-pub-evol} ------------------------ ### Proof of Lemma \[lemma:corr-0\]. {#proof:lemma:corr-0} For every $h \in {\cal H}$, $r \in \{1 \ldots {\tau}-1\}$, $i \in {\cal N}$, and $j \in {\cal N}_i$, $${\mbox{DS}}_i[j|h_r^*] = \{(k_1,k_2,r'+r) | (k_1,k_2,r') \in {\mbox{DS}}_i[j|h] \land r' + r < {\tau}\},$$ where $h_r^* = {\mbox{hist}}[h,r|\vec{\sigma}^*]$. Fix $i$, $h$, and $j$. The proof goes by induction on $r$, where the induction hypothesis is that, for every $r \in \{1 \ldots {\tau}-1\}$, $${\mbox{DS}}_i[j|h_r^*] = \{(k_1,k_2,r'+r) | (k_1,k_2,r') \in {\mbox{DS}}_i[j|h] \land r' + r < {\tau}\}.$$ By Definition \[def:thr\], we have that for every $r \in \{1 \ldots {\tau}-1\}$, $\vec{p}^* = \vec{\sigma}^*[h_r^*]$, and $s^* = {\mbox{sig}}[\vec{p}^*|h]$, $$\label{eq:corr0} {\mbox{DS}}_i[j|h_{r+1}^*] = L_1[r+1|\vec{\sigma}^*] \cup L_2[r+1|\vec{\sigma}^*],$$ where $$\label{eq:corr0-0} \begin{array}{l} L_1[r+1|\vec{\sigma}^*] = \{(k_1,k_2,r'+1)|(k_1,k_2,r') \in {\mbox{DS}}_{i}[j|h_r^*] \land r' +1 < {\tau}\}.\\ L_2[r+1|\vec{\sigma}^*]= \{(k_1,k_2,0) | k_1,k_2 \in {\cal N} \land i,j \in {\mbox{RS}}[k_1,k_2] \land s^*[k_1,k_2] = \mbox{\emph{defect}}\}. \end{array}$$ First, note that by Definition \[def:pubsig\] it holds that $s^*[k_1,k_2]=\mbox{\emph{cooperate}}$ for every $k_1 \in{\cal N}$ and $k_2 \in {\cal N}_{k_1}$. Thus, by \[eq:corr0-0\], for every $r \in \{1\ldots {\tau}-1\}$, $$\label{eq:corr0-1} L_2[r|\vec{\sigma}^*]= \{(k_1,k_2,0) | k_1,k_2 \in {\cal N} \land i,j \in {\mbox{RS}}[k_1,k_2] \land s^*[k_1,k_2] = \mbox{\emph{defect}}\}=\emptyset.$$ By \[eq:corr0-0\], $$L_1[1|\vec{\sigma}^*] = \{(k_1,k_2,r'+1)|(k_1,k_2,r') \in {\mbox{DS}}_{i}[j|h] \land r' +1 < {\tau}\},$$ which, along with \[eq:corr0-1\] and \[eq:corr0\], proves the base case. Now, consider that the induction hypothesis is valid for any $r \in \{1\ldots {\tau}-2\}$. We have by this assumption and by \[eq:corr0-0\] that $$\begin{array}{ll} L_1[r+1|\vec{\sigma}^*] & = \{(k_1,k_2,r'+1)|(k_1,k_2,r') \in {\mbox{DS}}_{i}[j|h_r^*] \land r' +1 < {\tau}\},\\ & = \{(k_1,k_2,r'+1)|(k_1,k_2,r') \in \\ &\{(l_1,l_2,r''+r) | (l_1,l_2,r'') \in {\mbox{DS}}_i[j|h] \land r'' + r< {\tau}\} \land r' + 1 < {\tau}\},\\ & = \{(k_1,k_2,r'+(r+1))|(k_1,k_2,r') \in {\mbox{DS}}_{i}[j|h] \land r' + (r+1) < {\tau}\}. \end{array}$$ This fact, along with \[eq:corr0-1\] and \[eq:corr0\], proves the induction step for $r+1$. ### Proof of Lemma \[lemma:corr-1\]. {#proof:lemma:corr-1} For every $h \in {\cal H}$, $\vec{p}' \in {\cal P}$, $r \in \{1 \ldots {\tau}\}$, $i \in {\cal N}$, and $j \in {\cal N}_i$: $${\mbox{DS}}_i[j|h_r'] = {\mbox{DS}}_i[j|h_r^*] \cup \{(k_1,k_2,r-1) | k_1,k_2 \in {\cal N} \land k_2 \in {\mbox{CD}}_{k_1}[\vec{p}'|h] \land i,j \in {\mbox{RS}}[k_1,k_2]\},$$ where $h_r^* = {\mbox{hist}}[h,r|\vec{\sigma}^*]$, $h_r' = {\mbox{hist}}[h,r| \vec{\sigma}']$, and $\vec{\sigma}' =\vec{\sigma}^*[h|\vec{p}']$ is the profile of strategies where all players follow $\vec{p}'$ in the first stage. Fix $h$, $\vec{p}'$, $i$, and $j$. The proof goes by induction on $r$, where the induction hypothesis is that for every $r \in \{1 \ldots {\tau}\}$, Equality \[eq:res-corr1\] holds. By Definition \[def:thr\], we have that for every $r \leq {\tau}$, $\vec{p}^r = \vec{\sigma}^*[h_r^*]$, and $s^* = {\mbox{sig}}[\vec{p}^r|h_r^*]$: $$\label{eq:corr1} {\mbox{DS}}_i[j|h_{r+1}^*] = L_1[r+1|\vec{\sigma}^*] \cup L_2[r+1|\vec{\sigma}^*],$$ where $$\label{eq:corr1-1} \begin{array}{l} L_1[r+1|\vec{\sigma}^*] = \{(k_1,k_2,r'+1)|(k_1,k_2,r') \in {\mbox{DS}}_{i}[j|h_r^*] \land r' +1 < {\tau}\}.\\ L_2[r+1|\vec{\sigma}^*]= \{(k_1,k_2,0) | k_1,k_2 \in {\cal N} \land i,j \in {\mbox{RS}}[k_1,k_2] \land s^*[k_1,k_2] = \mbox{\emph{defect}}\}. \end{array}$$ Similarly, for every $r \leq {\tau}$, $\vec{p}^r = \vec{\sigma}'[h_r']$, and $s' = {\mbox{sig}}[\vec{p}^r|h_r']$, $$\label{eq:corr1-2} {\mbox{DS}}_i[j|h_{r+1}'] = L_1[r+1|\vec{\sigma}'] \cup L_2[r+1|\vec{\sigma}'],$$ where $$\label{eq:corr1-3} \begin{array}{l} L_1[r+1|\vec{\sigma}'] = \{(k_1,k_2,r'+1)|(k_1,k_2,r') \in {\mbox{DS}}_{i}[j|h_r'] \land r' +1 < {\tau}\}.\\ L_2[r+1|\vec{\sigma}']= \{(k_1,k_2,0) | k_1,k_2 \in {\cal N} \land i,j \in {\mbox{RS}}[k_1,k_2] \land s'[k_1,k_2] = \mbox{\emph{defect}}\}. \end{array}$$ First, note that for every $r \in \{1 \ldots {\tau}\}$ and $\vec{p}'' \in {\cal P}$, such that $\vec{\sigma}'' = \vec{\sigma}^*[h|\vec{p}'']$, we have $\vec{\sigma}''[h'] = \vec{\sigma}^*[h']$ for every $h' \in {\cal H} \setminus \{h\}$. Thus, for $h_r'' = {\mbox{hist}}[h,r|\vec{\sigma}'']$, $\vec{p}^* = \vec{\sigma}''[h_r'']$, and $s^* = {\mbox{sig}}[\vec{p}^*|h_r'']$, we have by Definition \[def:pubsig\] that $s^*[k_1,k_2]=\mbox{\emph{cooperate}}$ for every $k_1 \in{\cal N}$ and $k_2 \in {\cal N}_{k_1}$. Thus, by Definition \[def:thr\], for every $r \in \{1 \ldots {\tau}\}$, $$\label{eq:corr1-4} L_2[r|\vec{\sigma}'']= \{(k_1,k_2,0) | k_1,k_2 \in {\cal N} \land i,j \in {\mbox{RS}}[k_1,k_2] \land s^*[k_1,k_2] = \mbox{\emph{defect}}\} = \emptyset.$$ It follows by \[eq:corr1-1\] and \[eq:corr1-3\] that, for every $r \in \{1 \ldots {\tau}\}$, $$\label{eq:corr1-5} L_2[r|\vec{\sigma}^*] = L_2[r|\vec{\sigma}'] = \emptyset.$$ The base case is when $r=1$. Since $h=h_0' = h_0^*$, by \[eq:corr1-1\] and \[eq:corr1-3\], it is true that: $$\label{eq:corr1-6} L_1[1|\vec{\sigma}'] = L_1[1|\vec{\sigma}^*].$$ Furthermore, if players follow $\vec{p}'$, then for $s' = {\mbox{sig}}[\vec{p}'|h]$, we have $s'[k_1,k_2]=\mbox{\emph{defect}}$ iff $k_2 \in {\mbox{CD}}_{k_1}[\vec{p}'|h]$. Thus: $$\label{eq:corr1-7} \begin{array}{ll} L_2[1|\vec{\sigma}'] & = \{(k_1,k_2,0) | k_1,k_2 \in {\cal N} \land i,j \in {\mbox{RS}}[k_1,k_2] \land s'[k_1,k_2]=\mbox{\emph{defect}}\}\\ & = \{(k_1,k_2,0) | k_1,k_2 \in {\cal N} \land i,j \in {\mbox{RS}}[k_1,k_2] \land k_2 \in {\mbox{CD}}_{k_1}[\vec{p}'|h]\}. \end{array}$$ Consequently, the base case follows from \[eq:corr1\], \[eq:corr1-2\], \[eq:corr1-5\], \[eq:corr1-6\], and \[eq:corr1-7\]. Hence, assume the induction hypothesis for $r \in \{1\ldots {\tau}-1\}$. By the induction hypothesis and by \[eq:corr1-1\], since $r < {\tau}$, it is also true that $$\label{eq:corr1-8} \begin{array}{ll} L_1[r+1|\vec{\sigma}'] & = \{(k_1,k_2,r'+1) | (k_1,k_2,r') \in {\mbox{DS}}_i[j|h_r'] \land r' +1< {\tau}\} \\ &\\ & = \{(k_1,k_2,r'+1) | (k_1,k_2,r') \in {\mbox{DS}}_i[j|h_r^*] \land r' +1 <{\tau}\} \cup \\ & \cup \{(k_1,k_2,r'+1) | (k_1,k_2,r') \in \{(l_1,l_2,r-1) | l_1,l_2 \in {\cal N} \\ &\land l_2 \in {\mbox{CD}}_{l_1}[\vec{p}'|h] \land i,j \in {\mbox{RS}}[l_1,l_2] \} \land r'+1<{\tau}\}\\ &\\ & = L_1[r+1|\vec{\sigma}^*] \cup \{(k_1,k_2,(r+1)-1) |k_1,k_2 \in {\cal N}\\ &\land k_2 \in {\mbox{CD}}_{k_1}[\vec{p}'|h] \land i,j \in {\mbox{RS}}[k_1,k_2]\}. \end{array}$$ The induction hypothesis follows from \[eq:corr1\], \[eq:corr1-2\], \[eq:corr1-5\], and \[eq:corr1-8\] for $r+1$, which proves the result. ### Proof of Lemma \[lemma:corr-2\]. {#proof:lemma:corr-2} For every $h \in {\cal H}$, $\vec{p}' \in {\cal P}$, $r > {\tau}$, $i \in {\cal N}$, and $j \in {\cal N}_i$, $${\mbox{DS}}_i[j|h_r'] = {\mbox{DS}}_i[j|h_r^*] = \emptyset,$$ where $h_r^* = {\mbox{hist}}[r|\vec{\sigma}^*]$, $h_r' = {\mbox{hist}}[r| \vec{\sigma}']$, and $\vec{\sigma}' = \vec{\sigma}^*[h|\vec{p}']$. Fix $h$, $\vec{p}'$, $i$, and $j$. The proof goes by induction on $r$, where the induction hypothesis is that for every $r \in \{1 \ldots {\tau}\}$, Equality \[eq:res-corr2\] holds. By Definition \[def:thr\], we have that for every $r >0$, $\vec{p}^r = \vec{\sigma}^*[h_r^*]$, and $s^* = {\mbox{sig}}[\vec{p}^r|h_r^*]$: $$\label{eq:corr2} {\mbox{DS}}_i[j|h_{r+1}^*] = L_1[r+1|\vec{\sigma}^*] \cup L_2[r+1|\vec{\sigma}^*],$$ where $$\label{eq:corr2-1} \begin{array}{l} L_1[r+1|\vec{\sigma}^*] = \{(k_1,k_2,r'+1)|(k_1,k_2,r') \in {\mbox{DS}}_{i}[j|h_r^*] \land r' +1 < {\tau}\}.\\ L_2[r+1|\vec{\sigma}^*]= \{(k_1,k_2,0) | k_1,k_2 \in {\cal N} \land i,j \in {\mbox{RS}}[k_1,k_2] \land s^*[k_1,k_2] = \mbox{\emph{defect}}\}. \end{array}$$ Similarly, for every $r >0$, $\vec{p}^r = \vec{\sigma}'[h_r']$, and $s' = {\mbox{sig}}[\vec{p}^r|h_r']$, $$\label{eq:corr2-2} {\mbox{DS}}_i[j|h_{r+1}'] = L_1[r+1|\vec{\sigma}'] \cup L_2[r+1|\vec{\sigma}'],$$ where $$\label{eq:corr2-3} \begin{array}{l} L_1[r+1|\vec{\sigma}'] = \{(k_1,k_2,r'+1)|(k_1,k_2,r') \in {\mbox{DS}}_{i}[j|h_r'] \land r' +1 < {\tau}\}.\\ L_2[r+1|\vec{\sigma}']= \{(k_1,k_2,0) | k_1,k_2 \in {\cal N} \land i,j \in {\mbox{RS}}[k_1,k_2] \land s'[k_1,k_2] = \mbox{\emph{defect}}\}. \end{array}$$ First, note that for every $r >0$, $\vec{p}'' \in {\cal P}$, and $\vec{\sigma}'' = \vec{\sigma}^*[h|\vec{p}'']$, we have $\vec{\sigma}''[h'] = \vec{\sigma}^*[h']$ for every $h' \in {\cal H} \setminus \{h\}$. Thus, for $h_r'' = {\mbox{hist}}[r|\vec{\sigma}'']$, $\vec{p}^* = \vec{\sigma}''[h_r'']$, and $s^* = {\mbox{sig}}[\vec{p}^*|h_r'']$, we have by Definition \[def:pubsig\] that $s^*[k_1,k_2]=\mbox{\emph{cooperate}}$ for every $k_1 \in{\cal N}$ and $k_2 \in {\cal N}_{k_1}$. Thus, by Definition \[def:thr\], for every $r>0$, $$\label{eq:corr2-4} L_2[r|\vec{\sigma}'' ]= \{(k_1,k_2,0) | k_1,k_2 \in {\cal N} \land i,j \in {\mbox{RS}}[k_1,k_2] \land s^*[k_1,k_2] = \mbox{\emph{defect}}\} = \emptyset.$$ It follows by \[eq:corr2-1\] and \[eq:corr2-3\] that, for every $r>{\tau}$, $$\label{eq:corr2-5} L_2[r|\vec{\sigma}^*] = L_2[r|\vec{\sigma}'] = \emptyset.$$ By Lemma \[lemma:corr-1\], $${\mbox{DS}}_i[j|h_{{\tau}}'] = {\mbox{DS}}_i[j|h_{{\tau}}^*] \cup \{(k_1,k_2,{\tau}-1)|k_1,k_2 \in {\cal N} \land k_2 \in {\mbox{CD}}_{k_1}[\vec{p}'|h] \land i,j \in {\mbox{RS}}[k_1,k_2]\}.$$ Therefore, by \[eq:corr2-1\] and \[eq:corr2-3\], $$\begin{array}{ll} L_1[{\tau}|\vec{\sigma}'] & = \{(k_1,k_2,r'+1) |(k_1,k_2,r') \in {\mbox{DS}}_i[j|h_{{\tau}}'] \land r'+1<{\tau}\} \\ &\\ & = \{(k_1,k_2,r'+1) | (k_1,k_2,r') \in {\mbox{DS}}_i[j|h_{{\tau}}^*] \land r'+1 <{\tau}\} \cup \\ & \cup \{(k_1,k_2,r'+1) | (k_1,k_2,r') \in \{(l_1,l_2,{\tau}-1)| l_1,l_2 \in {\cal N} \land l_2 \in {\mbox{CD}}_{l_1}[\vec{p}'|h] \\ &\land i,j \in {\mbox{RS}}[l_1,l_2]\} \land r' +1 < {\tau}\}\\ &\\ & = \{(k_1,k_2,r'+1) | (k_1,k_2,r') \in {\mbox{DS}}_i[j|h_{{\tau}}^*] \land r'+1 <{\tau}\} \cup \\ & \cup \{(k_1,k_2,r'+1) | k_1,k_2 \in {\cal N} \land k_2 \in {\mbox{CD}}_{k_1}[\vec{p}'|h] \land i,j \in {\mbox{RS}}[k_1,k_2] \land {\tau}< {\tau}\}\\ &\\ & = \{(k_1,k_2,r'-1) | (k_1,k_2,r') \in {\mbox{DS}}_i[j|h_{{\tau}}^*] \land r' +1 < {\tau}\} \cup \emptyset\\ & = L_1[{\tau}+1|\vec{\sigma}^*]. \end{array}$$ By Corollary \[corollary:corr-0\], $$\label{eq:corr2-6} L_1[{\tau}+1|\vec{\sigma}'] = L_1[{\tau}+1|\vec{\sigma}^*] = \emptyset.$$ By \[eq:corr2\], \[eq:corr2-2\] \[eq:corr2-5\], \[eq:corr2-6\], the base case is true. Now, assume the induction hypothesis for some $r\geq {\tau}+1$. By this assumption, \[eq:corr2-1\], and \[eq:corr2-3\]: $$\label{eq:corr2-7} \begin{array}{ll} L_1[r+1|\vec{\sigma}'] &= \{(k_1,k_2,r'+1) | (k_1,k_2,r') \in {\mbox{DS}}_i[j|h_{r}'] \land r'+1 < {\tau}\}\\ & = \{(k_1,k_2,r'+1) | (k_1,k_2,r') \in \emptyset\} \\ & = \emptyset. \end{array}$$ By Definition \[def:thr\], by \[eq:corr2-2\], and \[eq:corr2-4\], the induction step is true for $r+1$, which proves the result. Generic Results {#sec:proof:gen-cond} --------------- ### Proof of Proposition \[prop:folk\]. {#proof:prop:folk} For every profile of punishing strategies $\vec{\sigma}^*$, if $\vec{\sigma}^*$ is a SPE, then, for every $i \in {\cal N}$, $\frac{\beta_i}{\gamma_i} >\bar{p}_i$. Consequently, $\psi[\vec{\sigma}^*] \subseteq (v,\infty)$, where $v = \max_{i \in {\cal N}} \bar{p}_i$. Let $\vec{p}^* = \vec{\sigma}^*[h]$. The equilibrium utility is $$\pi_i[\vec{\sigma}^*|\emptyset] = \sum_{r=0}^\infty\omega_i^r (1 - q_i[\vec{p}^*])(\beta_i - \gamma_i \bar{p}_i) = \frac{1 - q_i[\vec{p}^*]}{1-\omega_i}(\beta_i - \gamma_i \bar{p}_i).$$ If $\frac{\beta_i}{\gamma_i} \leq \bar{p}_i$, then $$\label{eq:od-1} \pi_i[\vec{\sigma}^*|\emptyset] \leq 0.$$ Let $\sigma_i' \in \Sigma_i$ be a strategy such that, for every $h \in {\cal H}$, $\sigma_i'[h] = \vec{0}$, and let $\vec{\sigma}' = (\sigma_i',\vec{\sigma}_{-i}^*)$, where $\vec{0}=(0)_{j \in {\cal N}_i}$. We have $$\label{eq:od-2} \pi_i[\vec{\sigma}'|\emptyset] = (1-q_i[\vec{p}^*])\beta_i + \pi_i[\vec{\sigma}'|(h,{\mbox{sig}}[\vec{p}'|h])] \geq (1-q_i[\vec{p}^*])\beta_i,$$ where $\vec{p}' = (\vec{0},\vec{p}^*_{-i})$. By Lemma \[lemma:pprob\], $q_i [\vec{p}^*] < 1$. Since $ \pi_i[\vec{\sigma}'|(h,{\mbox{sig}}[\vec{p}'|h])] \geq 0$, it is true that $$\pi_i[\vec{\sigma}^*|\emptyset] \leq 0 < \pi_i[\vec{\sigma}'|\emptyset].$$ This contradicts the assumption that $\vec{\sigma}^*$ is a SPE. ### Proof of Lemma \[lemma:gen-cond-nec\]. {#proof:lemma:gen-cond-nec} If $\vec{\sigma}^*$ is a SPE, then the DC Condition is fulfilled. The proof consists in assuming that $\vec{\sigma}^*$ is a SPE and deriving \[eq:gen-cond\]. By Property \[prop:one-dev\], we must have, for every $h \in {\cal H}$ and $a_i' \in {\cal A}_i$, $$\label{eq:drop-nec} \pi_i[\vec{\sigma}^*|h] - \pi_i[\vec{\sigma}'|h]\geq 0,$$ where $\sigma_i' = \sigma_i^*[h|a_i']$ and $\vec{\sigma}' = (\sigma_i',\vec{\sigma}_{-i}^*)$. This is true for any $\sigma_i'$, where $a_i'[\vec{p}_i'] = 1$ and $\vec{p}_i'$ differs from $\sigma_i^*[h]$ exactly in that $i$ drops the nodes from any set $D \subseteq {\cal N}_{i}[h]$: - For every $j \in D$, $p_i'[j] = 0$. - For every $j \in {\cal N}_i \setminus D$, $p_i'[j] = p_i[j|h]$ For any pure profile of strategies $\vec{\sigma} \in \Sigma$, we can write $$\label{eq:gcn-1} \pi_i[\vec{\sigma}|h] = \sum_{r=0}^{\infty} \omega_i^r u_i[h,r|\vec{\sigma}].$$ By Lemma \[lemma:corr-2\] and Definition \[def:thr\], for every $r > {\tau}$, $j \in {\cal N}$, and $k \in {\cal N}_j$, $$\label{eq:gcn-2} \begin{array}{l} {\mbox{DS}}_j[k|h_r'] = {\mbox{DS}}_j[k|h_r^*] = \emptyset,\\ \\ p_j[k|h_r'] = p_j[k|h_r^*], \end{array}$$ where $h_r' = {\mbox{hist}}[h,r|\vec{\sigma}']$ and $h_r^*={\mbox{hist}}[h,r|\vec{\sigma}^*]$. This implies that for every $r> {\tau}$: $$q_i[h,r|\vec{\sigma}']=q_i[h,r|\vec{\sigma}^*],$$ $$\bar{p}_i[h,r|\vec{\sigma}']=\bar{p}_i[h,r|\vec{\sigma}^*],$$ $$u_i[h,r|\vec{\sigma}'] = u_i[h,r|\vec{\sigma}^*].$$ Thus, \[eq:drop-nec\] and \[eq:gcn-1\] imply \[eq:gen-cond\]. ### Proof of Lemma \[lemma:best-response1\]. {#proof:lemma:best-response1} For every $i \in {\cal N}$, $h \in {\cal H}$, $a_i \in BR[\vec{\sigma}_{-i}^*|h]$, and $\vec{p}_i \in {\cal P}_i$ such that $a_i[\vec{p}_i] > 0$, it is true that for every $j \in {\cal N}_i$ we have $p_i[j] \in \{0,p_i[j|h]\}$. Suppose then that there exist $h \in {\cal H}$, $i \in {\cal N}$, $a_i^1 \in BR[\vec{\sigma}_{-i}^*|h]$, and $\vec{p}^1_i \in {\cal P}_i$ such that $a_i^1[\vec{p}^1_i]>0$ and there exists $j \in {\cal N}_i$ such that $p_i^1[j] \notin \{0,p_i[j|h]\}$. Consider an alternative $a_i^2 \in {\cal A}_i$: - Define $\vec{p}_i^2 \in {\cal P}_i$ such that for every $j \in {\cal N}_i$, if $p_i^1[j] \geq p_i[j|h]$, then $p_i^2[j] = p_i[j|h]$, else, $p_i^2[j]=0$. - Set $a_i^2[\vec{p}_i^2] = a_i^1[\vec{p}_i^1] + a_i^1[\vec{p}_i^2]$ and $a_i^2[\vec{p}_i^1] = 0$. - For every $\vec{p}_i'' \in {\cal P}_i \setminus \{\vec{p}_i^1,\vec{p}_i^2\}$, set $a_i^2[\vec{p}_i''] = a_i^1[\vec{p}_i'']$. Consider the following auxiliary definitions: - $\vec{a}^1 = (a_i^1,\vec{p}_{-i}^*)$ and $\vec{a}^2 = (a_i^2,\vec{p}_{-i}^*)$, where $\vec{p}^* = \vec{\sigma}^*[h]$. - $\sigma_i^1 = \sigma_i^*[h|a_i^1]$ and $\sigma_i^2 = \sigma_i^*[h|a_i^2]$. - $\vec{p}^1 = (\vec{p}_i^1,\vec{p}^*_{-i})$ and $\vec{p}^2 = (\vec{p}_i^2,\vec{p}^*_{-i})$. - $\vec{\sigma}^1 = (\sigma_i^1,\vec{\sigma}^*_{-i})$ and $\vec{\sigma}^2 = (\sigma_i^2,\vec{\sigma}^*_{-i})$. - $s^1= {\mbox{sig}}[\vec{p}^1|h]$ and $s^2={\mbox{sig}}[\vec{p}^2|h]$. Notice that for any $j \in {\cal N}_i$, $p_i^1[j] \geq p_i^2[j]$ and $p_i^1[j] \geq p_i[j|h]$ iff $p_i^2[j] \geq p_i[j|h]$. Thus, by Definition \[def:pubsig\], for any $s \in {\cal S}$, $$\label{eq:br-1} \begin{array}{l} pr_i[s|a_i^1,h] = pr_i[s|a_i^2,h].\\ pr[s|\vec{a}^1,h] = pr[s|\vec{a}^2,h]. \end{array}$$ Moreover, for some $j \in {\cal N}_i$, $p_i^1[j|h] > p_i^2[j|h]$, thus, it is true that $$\label{eq:br-2} u_i[\vec{a}^1] < u_i[\vec{a}^2].$$ Recall that $$\pi_i[\vec{\sigma}^1|h] = u_i[\vec{a}^1] + \omega_i\sum_{s \in {\cal S}} \pi_i[\vec{\sigma}^1|(h,s)]pr[s|\vec{a}^1,h],$$ $$\pi_i[\vec{\sigma}^2|h] = u_i[\vec{a}^2] + \omega_i\sum_{s \in {\cal S}} \pi_i[\vec{\sigma}^2|(h,s)]pr[s|\vec{a}^2,h].$$ By \[eq:br-1\] and the definition of $\vec{\sigma}^1$ and $\vec{\sigma}^2$, $$\sum_{s \in {\cal S}} \pi_i[\vec{\sigma}|(h,s)]pr[s|\vec{a}^1,h] = \sum_{s \in {\cal S}} \pi_i[\vec{\sigma}|(h,s)]pr[s|\vec{a}^2,h].$$ By \[eq:br-2\], $$\pi_i[\vec{\sigma}^1|h] < \pi_i[\vec{\sigma}^2|h].$$ This is a contradiction, since $a_i^1 \in BR[\vec{\sigma}_{-i}^*|h]$ by assumption, concluding the proof. ### Proof of Lemma \[lemma:best-response2\]. {#proof:lemma:best-response2} For every $h \in {\cal H}$ and $i \in {\cal N}$, there exists $a_i \in BR[\vec{\sigma}_{-i}^*| h]$ and $\vec{p}_i \in {\cal P}_i$ such that $a_i[\vec{p}_i] = 1$. For any $h \in {\cal H}$ and $i \in {\cal N}$, if $BR[\vec{\sigma}_{-i}^*|h]$ only contains pure strategies for the stage game, since $BR[\vec{\sigma}_{-i}^*|h]$ is not empty, the result follows. Suppose then that there exists a mixed strategy $a_i^1 \in BR[\vec{\sigma}_{-i}^*|h]$. We know from Lemma \[lemma:best-response1\] that every such $a_i^1$ attributes positive probability to one of two probabilities in $\{0,p_i[j|h]\}$, for every $j \in {\cal N}_i$. Let $\sigma_i^1 = \sigma_i^*[h|a_i^1]$, $\vec{\sigma}^1 = (\sigma_i^1,\vec{\sigma}^*_{-i})$, and denote by ${\cal P}^*[h]$ the finite set of profiles of probabilities that fulfill the condition of Lemma \[lemma:best-response1\], i.e., for every $\vec{p} \in {\cal P}^*[h]$, $j\in{\cal N}$, and $k \in {\cal N}_j$, $p_j[k] \in \{0,p_j[k|h]\}$. We can write $$\label{eq:br2-0} \pi_i[\vec{\sigma}^1|h] = \sum_{\vec{p}_i \in {\cal P}^*_i[h]} (u_i[\vec{p}] + \omega_i \pi_i[\vec{\sigma}^1|(h,{\mbox{sig}}[\vec{p}|h])])a_i^1[\vec{p}_i],$$ where $\vec{p} = (\vec{p}_i,\vec{p}_{-i}^*)$ and $\vec{p}^* = \vec{\sigma}^*[h]$. For any $\vec{p}_i^1 \in {\cal P}_i^*[h]$ such that $a_i^1[\vec{p}_i^1] >0$, let $\vec{p}^* = \vec{\sigma}^*[h]$, $\vec{p}^1 = (\vec{p}_i^1,\vec{p}_{-i}^*)$, $\sigma_i' = \sigma_i^*[h|\vec{p}_i^1]$, and $\vec{\sigma}' = (\sigma_i',\vec{\sigma}^*_{-i})$. There are three possibilities: 1. $\pi_i[\vec{\sigma}^1|h] = \pi_i[\vec{\sigma}'|h]$. 2. $\pi_i[\vec{\sigma}^1|h] < \pi_i[\vec{\sigma}'|h]$. 3. $\pi_i[\vec{\sigma}^1|h] > \pi_i[\vec{\sigma}'|h]$. In possibility 1, it is true that there is $a_i' \in BR[\vec{\sigma}_{-i}^*|h]$ such that $a_i'[\vec{p}_i^1] = 1$ and the result follows. Possibility 2 contradicts the assumption that $a_i^1 \in BR[\vec{\sigma}_{-i}^*|h]$. Finally, consider that possibility 3 is true. Recall that $a_i^1$ being mixed implies $a_i^1[\vec{p}_i^1] < 1$. Thus, there must exist $\vec{p}_i^2 \in {\cal P}_i^*[h] \setminus \{\vec{p}_i^1\}$, $\sigma_i''=\sigma_i^*[h|\vec{p}_i^2]$, and $\vec{\sigma}'' = (\sigma_i',\vec{\sigma}^*_{-i})$, such that $a_i^1[\vec{p}_i^2] > 0$ and $$\label{eq:br2-1} \pi_i[\vec{\sigma}'|h] < \pi_i[\vec{\sigma}''|h].$$ Here, we can define $a_i^2 \in {\cal A}_i$ such that: - $a_i^2[\vec{p}_i^2]=a_i^1[\vec{p}_i^1] + a_i^1[\vec{p}_i^2]$; - $a_i^2[\vec{p}_i^1] = 0$. - For every $\vec{p}_i'' \in {\cal P}^*[h] \setminus \{\vec{p}_i^1,\vec{p}_i^2\}$, $a_i^2[\vec{p}_i''] = a_i^1[\vec{p}_i'']$. Now, let: - $\sigma_i^2 = \sigma_i^*[h|a_i^2]$ and $\vec{\sigma}^2 = (\sigma_i^2,\vec{\sigma}^*_{-i})$. - $\vec{p}^2 = (\vec{p}_i^2,\vec{p}^*_{-i})$. - $\vec{\sigma}' = (\sigma_i[h|\vec{p}_i''],\vec{\sigma}_{-i}^*)$. By \[eq:br2-0\], $$\pi_i[\vec{\sigma}^1|h] = l_1 + \pi_i[\vec{\sigma}'|h]a_i^1[\vec{p}_i^1] + \pi_i[\vec{\sigma}''|h]a_i^1[\vec{p}_i^2],$$ $$\pi_i[\vec{\sigma}^2|h] = l_2 + \pi_i[\vec{\sigma}''|h]a_i^2[\vec{p}_i^2],$$ where $$l_1 = \sum_{\vec{p}'' \in {\cal P}^*[h] \setminus \{\vec{p}_i^1,\vec{p}_i^2\}} (u_i[\vec{p}''] + \omega_i \pi_i[\vec{\sigma}^1|h,s''])a_i^1[\vec{p}_i''],$$ $$l_2 = \sum_{\vec{p}'' \in {\cal P}^*[h] \setminus \{\vec{p}_i^1,\vec{p}_i^2\}} (u_i[\vec{p}''] + \omega_i \pi_i[\vec{\sigma}^2|h,s''])a_i^2[\vec{p}_i''],$$ and $s'' = {\mbox{sig}}[\vec{p}''|h]$. By the definition of $a_i^2$, we have that $l_1 = l_2$. It follows that: $$\begin{array}{ll} \pi_i[\vec{\sigma}^1|h] - \pi_i[\vec{\sigma}^2|h] & = \pi_i[\vec{\sigma}'|h]a_i^1[\vec{p}_i^1] + \pi_i[\vec{\sigma}''|h]a_i^1[\vec{p}_i^2] - \pi_i[\vec{\sigma}''|h](a_i^1[\vec{p}_i^2] + a_i^1[\vec{p}_i^1]) \\ & = (\pi_i[\vec{\sigma}'|h] - \pi_i[\vec{\sigma}''|h])a_i^1[\vec{p}_i^1]. \end{array}$$ If follows from \[eq:br2-1\] that: $$\pi_i[\vec{\sigma}^1|h] < \pi_i[\vec{\sigma}^2|h],$$ contradicting the assumption that $a_i^1 \in BR[\vec{\sigma}_{-i}^*| h]$. This concludes the proof. ### Proof of Lemma \[lemma:best-response\]. {#proof:lemma:best-response} For every $h \in {\cal H}$ and $i \in {\cal N}$, there exists $\vec{p}_i \in {\cal P}_i$ and a pure strategy $\sigma_i=\sigma_i^*[h|\vec{p}_i]$ such that: 1. For every $j \in {\cal N}_i$, $p_i[j] \in \{0,p_{i}[j|h]\}$. 2. For every $a_i \in {\cal A}_i$, $\pi_i[\sigma_i,\vec{\sigma}_{-i}^*|h] \geq \pi_i[\sigma_i',\vec{\sigma}_{-i}^*|h]$, where $\sigma_i' = \sigma_i^*[h|a_i]$. Consider any $h \in {\cal H}$ and $i \in {\cal N}$. From Lemma \[lemma:best-response2\], it follows that there exists $a_i \in BR[\vec{\sigma}_{-i}^*|h]$ and $\vec{p}_i \in {\cal P}_i$ such that $a_i[\vec{p}_i] = 1$. By Lemma \[lemma:best-response1\], every such $a_i$ and $\vec{p}_i$ such that $a_i[\vec{p}_i]=1$ fulfill Condition $1$. Condition $2$ follows from the definition of $BR[\vec{\sigma}_{-i}^*|h]$. ### Proof of Lemma \[lemma:gen-cond-suff\]. {#proof:lemma:gen-cond-suff} If the DC Condition is fulfilled, then $\vec{\sigma}^*$ is a SPE. Assume that Inequality \[eq:gen-cond\] holds for every history $h$ and $D \subseteq {\cal N}_{i}[h]$. In particular, these assumptions imply that, for each $\vec{p}_i \in {\cal P}_i$ such that $p_i[j] \in \{0,p_i[j|h]\}$ for every $j \in {\cal N}_i$, we have $$\label{eq:gcs} \pi_i[\vec{\sigma}^*|h] \geq \pi_i[\sigma_i,\vec{\sigma}^*_{-i}|h],$$ where $\sigma_i = \sigma_i^*[h|\vec{p}_i]$. By Lemma \[lemma:best-response\], there exists one such $\vec{p}_i$ such that $\sigma_i$ is a local best response. Consequently, by \[eq:gcs\], for every $a_i \in {\cal A}_i$ and $\sigma_i'=\sigma_i^*[h|a_i]$, $$\pi_i[\vec{\sigma}^*|h] \geq \pi_i[\sigma_i',\vec{\sigma}^*_{-i}|h].$$ By Property \[prop:one-dev\], $\vec{\sigma}^*$ is a SPE. Direct Reciprocity is not Effective {#sec:proof:direct} ----------------------------------- ### Proof of Lemma \[lemma:nec-btc\]. {#proof:lemma:nec-btc} If $\vec{\sigma}^*$ is a SPE, then, for every $i \in {\cal N}$ and $j \in {\cal N}_i$, it is true that $q_i' > q_i^*$ and: $$\frac{\beta_i}{\gamma_i} > \bar{p}_i + \frac{p_i[j|\emptyset]}{q_i' - q_i^*}\left(1-q_i' + \frac{1-q_i^*}{{\tau}}\right),$$ where $\vec{p}_i'$ is the strategy where $i$ drops $j$, $\vec{\sigma}' = (\sigma_i^*[\emptyset|\vec{p}_i'],\vec{\sigma}_{-i}^*)$, $q_i'=q_i[\vec{\sigma}'[\emptyset]]$, and $q_i^* = q_i[\vec{\sigma}^*[\emptyset]]$. The assumption that $\vec{\sigma}^*$ is a SPE implies by Theorem \[theorem:gen-cond\] that the DC Condition is true for the history $\emptyset$, any node $i \in {\cal N}$, and $D =\{j\}$, where $j \in {\cal N}_i$. Define $\vec{p}_i' \in {\cal P}_i$ as: - $p_i'[j] =0$. - For every $k \in {\cal N}_i \setminus \{j\}$, $p_i'[k] = p_i[k|\emptyset]$. We have that: $$\label{eq:nbtc-0} \begin{array}{ll} \bar{p}_i - \bar{p}_i[\emptyset,0|\vec{\sigma}'] & = \sum_{k \in {\cal N}_i} p_i[k|\emptyset] - \sum_{k \in {\cal N}_i \setminus \{j\}} p_i[k|\emptyset]\\ & = p_i[j|\emptyset]. \end{array}$$ Let $\sigma_i' = \sigma_i^*[\emptyset| \vec{p}_i']$, $\vec{\sigma}' = (\sigma_i',\vec{\sigma}^*_{-i})$, $h_r^* = {\mbox{hist}}[\emptyset,r|\vec{\sigma}^*]$, and $h_r' = {\mbox{hist}}[\emptyset,r|\vec{\sigma}']$. It is true by the definition of $u_i$ and by \[eq:nbtc-0\] that $$\label{eq:nbtc-1} \begin{array}{ll} u_i[\emptyset,0|\vec{\sigma}^*] - u_i[\emptyset,0|\vec{\sigma}'] &= (1-q_i^*)(\beta_i - \gamma_i \bar{p}_i) - (1-q_i^*)(\beta_i - \gamma_i \bar{p}_i[\emptyset,0|\vec{\sigma}'])\\ & = (1-q_i^*)\gamma_i ( \bar{p}_i[\emptyset,0|\vec{\sigma}'] - \bar{p}_i)\\ & = - (1-q_i^*)\gamma_i p_i[j|\emptyset].\\ & = -c, \end{array}$$ where $c = (1-q_i^*)\gamma_i p_i[j|\emptyset]$. Notice that for every $k \in {\cal N} \setminus \{i\}$ $$\label{eq:nb-1} {\mbox{CD}}_k[\vec{p}'|h] = \emptyset,$$ and $$\label{eq:nb-2} {\mbox{CD}}_i[\vec{p}'|h] = \{j\}.$$ From \[eq:nb-1\] and \[eq:nb-2\], and by Lemma \[lemma:corr-1\], and Definition \[def:thr\], for every $r \in \{1 \ldots {\tau}\}$, $$\label{eq:nbtc-2} \begin{array}{ll} {\mbox{DS}}_i[j|h_r'] & = {\mbox{DS}}_i[j|h_r^*] \cup \{(k_1,k_2,r - 1)| k_1,k_2 \in {\cal N} \land i,j \in {\mbox{RS}}[k_1,k_2] \land k_2 \in {\mbox{CD}}_{k_1}[\vec{p}'|h]\}\\ & = {\mbox{DS}}_i[j|h_r^*] \cup \{(i,j,r-1)| i,j \in {\mbox{RS}}[i,j]\}\\ & = \{(i,j,r-1)\}. \end{array}$$ and for every $k \in {\cal N} \setminus \{j\}$ $$\label{eq:nbtc-3} \begin{array}{ll} {\mbox{DS}}_i[k|h_r'] & = {\mbox{DS}}_i[k|h_r^*] \cup \{(k_1,k_2,r-1)| k_1,k_2 \in {\cal N} \land i,k \in {\mbox{RS}}[k_1,k_2] \land k_2 \in {\mbox{CD}}_{k_1}[\vec{p}'|h]\}\\ & = {\mbox{DS}}_i[k|h_r^*] \cup \{(i,k,r-1)| i,k \in {\mbox{RS}}[i,j]\}\\ & = {\mbox{DS}}_i[k|h_r^*] = \emptyset. \end{array}$$ By \[eq:nbtc-2\] and \[eq:nbtc-3\], Definition \[def:thr\], and the definition of $\vec{p}'$, for every $r \in \{1 \ldots {\tau}\}$ $$\label{eq:nbtc-4} \bar{p}_i[\emptyset,r|\vec{\sigma}'] = \sum_{k \in {\cal N}_i \setminus\{j\}} p_i[k|h_r'] =\sum_{k \in {\cal N}_i \setminus\{j\}} p_i[k|\emptyset] = \bar{p}_i - p_i[j|\emptyset].$$ By \[eq:nbtc-2\] and \[eq:nbtc-4\], for every $r \in \{1 \ldots {\tau}\}$, $$\label{eq:nbtc-5} \begin{array}{ll} u_i[\emptyset,r|\vec{\sigma}^*] - u_i[\emptyset,r|\vec{\sigma}'] &= (1-q_i[\emptyset,r|\vec{\sigma}^*])(\beta_i - \gamma_i \bar{p}_i[\emptyset,r|\vec{\sigma}^*]) - \\ &(1-q_i[\emptyset,r|\vec{\sigma}'])(\beta_i - \gamma_i \bar{p}_i[\emptyset,r|\vec{\sigma}'])\\ &\\ & = (1-q_i^*)(\beta_i - \gamma_i \bar{p}_i) - (1-q_i')(\beta_i - \gamma_i \bar{p}_i + p_i[j|\emptyset])\\ & = (q_i' - q_i^*)(\beta_i - \gamma_i \bar{p}_i) - (1-q_i')\gamma_i p_i[j|\emptyset]\\ & = a - b, \end{array}$$ where: - $a = (q_i' - q_i^*)(\beta_i - \gamma_i \bar{p}_i)$. - $b = (1-q_i')\gamma_i p_i[j|\emptyset]$. By Theorem \[theorem:gen-cond\], the assumption that $\vec{\sigma}^*$ is a SPE, \[eq:nbtc-1\], and \[eq:nbtc-5\], $$\label{eq:nbtc-6} \begin{array}{ll} \sum_{r=0}^{{\tau}} \omega_i^r(u_i[h,r|\vec{\sigma}^*] - u_i[h,r|\vec{\sigma}']) &\geq 0\\ -c + \sum_{r=1}^{{\tau}} \omega_i^r (a-b) & \geq 0\\ -c + \frac{\omega_i-\omega_i^{{\tau}+1}}{1-\omega_i}(a-b) & \geq 0\\ -c(1-\omega_i) + (\omega_i-\omega_i^{{\tau}+1})(a-b) & \geq 0\\ \omega_i(a - b + c) - \omega_i^{{\tau}+1}(a-b) -c & \geq 0. \end{array}$$ This is a polynomial function of degree ${\tau}+1 \geq 2$ that has a zero in $\omega_i = 1$ and is negative for $\omega_i = 0$, since, by Lemma \[lemma:pprob\], $c > 0$. For any $\omega_i \in (0,1)$, a solution to \[eq:nbtc-6\] exists only if the polynomial is strictly concave. The second derivative is $$-({\tau}+1){\tau}\omega_i^{{\tau}-1}(a-b),$$ so we must have $a > b$ for this to be true, i.e.: $$\label{eq:nbtc-7} \begin{array}{ll} (q_i' - q_i^*)(\beta_i - \gamma_i \bar{p}_i) &> (1-q_i')\gamma_i p_i[j|\emptyset]\\ \Rightarrow (q_i' - q_i^*)(\beta_i - \gamma_i \bar{p}_i) &> 0. \end{array}$$ By Proposition \[prop:folk\], we know that $\beta_i > \gamma_i \bar{p}_i$. Thus, \[eq:nbtc-7\] implies $q_i' > q_i^*$, which concludes the first part of the proof. Furthermore, a solution to \[eq:nbtc-6\] exists if and only if there is a maximum of the polynomial for $\omega_i \in (0,1)$. We can find the zero of the first derivative in order to $\omega_i$, obtaining $$\label{eq:nbtc-8} \omega_i^{{\tau}} = \frac{a-b+c}{(a-b)({\tau}+1)}.$$ For a solution of \[eq:nbtc-8\] to exist for some $\omega_i \in (0,1)$, it must be true that: 1. $a{\tau}> b{\tau}+ c$. 2. $a - b + c > 0$. Condition 1) yields: $$\label{eq:nbtc-9} \begin{array}{ll} {\tau}(q_i' - q_i^*)(\beta_i - \gamma_i \bar{p}_i) {\tau}& > {\tau}(1-q_i')\gamma_i p_i[j|\emptyset]+ (1-q_i^*)\gamma_i p_i[j|\emptyset]\\ {\tau}(q_i' - q_i^*)\beta_i & > {\tau}(q_i' - q_i^*)\gamma_i \bar{p}_i + {\tau}(1-q_i')\gamma_i p_i[j|\emptyset] + (1-q_i^*)\gamma_i p_i[j|\emptyset]\\ \beta_i & > \gamma_i \bar{p}_i + \frac{1-q_i'}{q_i'-q_i^*}\gamma_i p_i[j|\emptyset] + \frac{1-q_i^*}{{\tau}(q_i' - q_i^*)} \gamma_i p_i[j|\emptyset]\\ \frac{\beta_i}{\gamma_i} & > \bar{p}_i + \frac{1}{q_i'-q_i^*}p_i[j|\emptyset]( 1-q_i' + \frac{1-q_i^*}{{\tau}})\\ \end{array}$$ which is equivalent to \[eq:nec-btc\]. If Condition 1 is true, then: $$a > b + \frac{c}{{\tau}} \Rightarrow a> b - c.$$ That is, Condition 1 implies Condition 2. Therefore, if $\vec{\sigma}^*$ is a SPE for some $\omega_i \in (0,1)$, then \[eq:nec-btc\] must hold, which proves the result. ### Proof of Lemma \[lemma:dir-recip\]. {#proof:lemma:dir-recip} Suppose that for any $i \in {\cal N}$ and $j \in {\cal N}_i$, $p_i[j|\emptyset] + q_i^* \ll 1$. If $\vec{\sigma}^*$ is a SPE, then: $$\psi[\vec{\sigma}^*] \subseteq \left(\frac{1}{q_i^*},\infty\right).$$ Fix $i$ and $j$ for which the assumption holds. Let $\vec{p}_i'$ be defined as: - $p_i'[j] = 0$, - $p_i'[k] = p_i[k|\emptyset]$ for every $k \in {\cal N}_i \setminus \{j\}$. Let $q_i' = q_i[(\vec{p}_i',\vec{p}^*_{-i})]$ and $q_i^* = q_i[\vec{\sigma}^*[\emptyset]]$, where $\vec{p}^* = \vec{\sigma}[\emptyset]$. By Lemma \[lemma:nec-btc\], we must have $$\label{eq:dir-recip-1} \frac{\beta_i}{\gamma_i} > \bar{p}_i + \frac{p_i[j|\emptyset]}{q_i' - q_i^*}\left(1-q_i' + \frac{1-q_i^*}{{\tau}}\right).$$ By Lemma \[lemma:single-impact\] from Appendix \[sec:epidemic\], it is true that $q_i' \leq \frac{q_i^*}{1-p_i[j|\emptyset]}$. By including this fact in \[eq:dir-recip-1\], we obtain: $$\label{eq:dr-5} \begin{array}{ll} \frac{\beta_i}{\gamma_i} &> \bar{p}_i + \frac{p_i[j|\emptyset]}{q_i' - q_i^*}\left(1-q_i' + \frac{1-q_i^*}{{\tau}}\right) > \bar{p}_i + \frac{p_i[j|\emptyset]}{q_i' - q_i^*}(1 - q_i')\\ &\geq \bar{p}_i + \frac{p_i[j|\emptyset](1-p_i[j|\emptyset])}{(1-p_i[j|\emptyset])(q_i^* - q_i^*(1-p_i[j|\emptyset]))}(1- p_i[j|\emptyset] - q_i^*)\\ &= \bar{p}_i + \frac{1}{q_i^*} (1 - p_i[j|\emptyset] - q_i^*)\\ &\approx \bar{p}_i + \frac{1}{q_i^*}\\ &> \frac{1}{q_i^*}. \end{array}$$ The result follows from the fact that if for every $i$ there exists $\omega_i \in (0,1)$ such that $\vec{\sigma}^*$ is a SPE, then $[0,\frac{1}{q_i^*}] \cap \psi[\vec{\sigma}^*] = \emptyset$. Full Indirect Reciprocity is Sufficient {#sec:proof:indir} --------------------------------------- ### Proof of Lemma \[lemma:indir-equiv\]. {#proof:lemma:indir-equiv} The profile of strategies $\vec{\sigma}^*$ is a SPE if and only if for every $h \in {\cal H}$ and $i \in {\cal N}$: $$\sum_{r=1}^{\tau}(\omega_i^r u_i[h,r|\vec{\sigma}^*]) - (1-q_i[h,0|\vec{\sigma}^*])\gamma_i \bar{p}_i[h,0|\vec{\sigma}^*] \geq 0.$$ Using the result from Theorem \[theorem:gen-cond\], it is true that $\vec{\sigma}^*$ is a SPE if and only if the DC Condition is true for every $i \in {\cal N}$, $h \in {\cal H}$, and $D \subseteq {\cal N}_i[h]$. Consider any strategy $\sigma_i'= \sigma_i^*[h|\vec{p}']$ where $\vec{p}' = (\vec{0},\vec{p}_{-i}^*)$ and $\vec{p}^*=\vec{\sigma}^*[h]$. Alternatively, define $\sigma_i''=\sigma_i^*[h|\vec{p}'']$ where $\vec{p}'' = (\vec{p}_i'',\vec{p}^*_{-i})$ such that for some $D \subset {\cal N}_i[h]$: - For every $j \in D$, $p_i''[j] = 0$. - For every $j \in {\cal N}_i \setminus D$, $p_i''[j] = p_i[j|h]$. Define $h_r^* = {\mbox{hist}}[r|\vec{\sigma}^*]$. Let $\vec{\sigma}' = (\sigma_i',\vec{\sigma}_{-i}^*)$, $h_r' = {\mbox{hist}}[r|\vec{\sigma}']$, $\vec{\sigma}'' = (\sigma_i'',\vec{\sigma}_{-i}^*)$, and $h_r'' = {\mbox{hist}}[r|\vec{\sigma}']$. By Lemma \[lemma:corr-1\] and the definition of full indirect reciprocity, for every $r \in \{1 \ldots {\tau}\}$ and $j \in {\cal N}_i^{-1}$, since for every $k \in {\cal N}\setminus \{i\}$, $$\begin{array}{l} {\mbox{CD}}_i[\vec{p}'|h] = {\cal N}_i,\\ {\mbox{CD}}_i[\vec{p}''|h] = D,\\ {\mbox{CD}}_k[\vec{p}'|h] = {\mbox{CD}}_k[\vec{p}''|h] = \emptyset, \end{array}$$ it holds that $$\label{eq:indir-2} \begin{array}{ll} {\mbox{DS}}_j[i|h_r'] & = {\mbox{DS}}_j[i|h_r^*] \cup \{(k_1,k_2,r-1)|k_1,k_2 \in {\cal N} \land k_2 \in {\mbox{CD}}_{k_1}[\vec{p}'|h] \land j,i \in {\mbox{RS}}[k_1,k_2]\}\\ & = {\mbox{DS}}_j[i|h_r^*] \cup \{(i,k,r-1)|k \in {\cal N}_i \land j,i \in {\mbox{RS}}[i,k]\}.\\ & = {\mbox{DS}}_j[i|h_r^*] \cup \{(i,k,r-1)|k \in {\cal N}_i\}. \end{array}$$ and $$\label{eq:indir-3} \begin{array}{ll} {\mbox{DS}}_j[i|h_r''] & = {\mbox{DS}}_j[i|h_r^*] \cup \{(k_1,k_2,r)|k_1,k_2 \in {\cal N} \land k_2 \in {\mbox{CD}}_{k_1}[\vec{p}''|h] \land j,i \in {\mbox{RS}}[k_1,k_2]\}\\ & = {\mbox{DS}}_j[i|h_r^*] \cup \{(i,k,r-1)|k \in D \land j,i \in {\mbox{RS}}[i,k]\}.\\ & = {\mbox{DS}}_j[i|h_r^*] \cup \{(i,k,r-1)|k \in D\}. \end{array}$$ By Definition \[def:thr\], and by \[eq:indir-2\] and \[eq:indir-3\], for every $r \in \{1 \ldots {\tau}\}$ and $j \in {\cal N}_i^{-1}$, $$\label{eq:indir-4} p_j[i|h_r'] = p_j[i|h_r''] = 0.$$ It follows from \[eq:indir-4\] and Lemma \[lemma:noneib\] that for every $r \in \{1 \ldots {\tau}\}$ $$\label{eq:indir-5} \begin{array}{ll} q_i[h,r|\vec{\sigma}'] &= q_i[h,r|\vec{\sigma}''] = 1.\\ u_i[h,r|\vec{\sigma}'] &= u_i[h,r|\vec{\sigma}''] = 0.\\ \end{array}$$ Furthermore, we have $$\bar{p}_i[h,0|\vec{\sigma}'] \leq \bar{p}_i[h,0|\vec{\sigma}''],$$ which implies that $$\label{eq:indir-6} u_i[h,0|\vec{\sigma}'] > u_i[h,0|\vec{\sigma}''].$$ By \[eq:indir-5\] and \[eq:indir-6\], $$\label{eq:indir-7} \sum_{r=0}^{\tau}\omega^r (u_i[h,r|\vec{\sigma}^*] - u_i[h,r|\vec{\sigma}']) < \sum_{r=0}^{\tau}\omega^r(u_i[h,r|\vec{\sigma}^*] - u_i[h,r|\vec{\sigma}'']).$$ Finally, we have $$\label{eq:indir-8} \begin{array}{ll} \sum_{r=0}^{\tau}\omega^r (u_i[h,r|\vec{\sigma}^*] - u_i[h,r|\vec{\sigma}']) &\geq 0\\ (1-q_i[h,0|\vec{\sigma}^*])(\beta_i - \bar{p}_i[h,0|\vec{\sigma}^*]) - (1-q_i[h,0|\vec{\sigma}^*])\beta_i + \sum_{r = 1}^{{\tau}} \omega^r u_i[h,r|\vec{\sigma}^*] & \geq 0\\ \sum_{r = 1}^{{\tau}} \omega^r u_i[h,r|\vec{\sigma}^*] - (1-q_i[h,0|\vec{\sigma}^*])\bar{p}_i[h,0|\vec{\sigma}^*]& \geq 0. \end{array}$$ It is direct to conclude by Theorem \[theorem:gen-cond\] that if $\vec{\sigma}^*$ is a SPE, then DC Condition is fulfilled for $D= {\cal N}_i$. By \[eq:indir-7\], if DC Condition is fulfilled for $D={\cal N}_i$, then it is also fulfilled for every $D \subset {\cal N}_i$, and by Theorem \[theorem:gen-cond\], $\vec{\sigma}^*$ is a SPE. Thus, $\vec{\sigma}^*$ is a SPE iff \[eq:indir-8\] holds. This concludes the proof. ### Proof of Lemma \[lemma:maxh\] {#proof:lemma:maxh} Let $h \in {\cal H}$ be defined such that for every $h' \in {\cal H}$, the left side of Inequality \[eq:indir-equiv\] for $h$ is lower than or equal to the value for $h'$. Then, for every $r \in \{1 \ldots {\tau}-2\}$, $$u_i[h,r|\vec{\sigma}^*] = u_i[h,r+1|\vec{\sigma}^*].$$ The proof goes by contradiction. First, assume that $h$ minimizes the left side of Inequality \[eq:indir-equiv\]: $$\sum_{r=1}^{\tau}(\omega_i^r u_i[h,r|\vec{\sigma}^*]) - (1-q_i[h,0|\vec{\sigma}^*])\gamma_i \bar{p}_i[h,0|\vec{\sigma}^*] \geq 0.$$ This implies that for every $h' \in {\cal H}$, we have: $$\label{eq:mh} \begin{array}{ll} \sum_{r=1}^{\tau}(\omega_i^r u_i[h,r|\vec{\sigma}^*]) - (1-q_i[h,0|\vec{\sigma}^*])\gamma_i \bar{p}_i[h,0|\vec{\sigma}^*]& \\ - \sum_{r=1}^{\tau}(\omega_i^r u_i[h',r|\vec{\sigma}^*]) - (1-q_i[h',0|\vec{\sigma}^*])\gamma_i \bar{p}_i[h',0|\vec{\sigma}^*] & \leq 0\\ \\ \sum_{r=1}^{\tau}\omega_i^r (u_i[h,r|\vec{\sigma}^*] - u_i[h',r|\vec{\sigma}^*]) &\\ + (1-q_i[h,0|\vec{\sigma}^*])\gamma_i (\gamma_i \bar{p}_i[h',0|\vec{\sigma}^*] - \bar{p}_i[h,0|\vec{\sigma}^*]) & \leq 0\\ \end{array}$$ Assume by contradiction that for every $h$ that minimizes the above condition, there is some $r \in \{1 \ldots {\tau}-3\}$, $$u_i[h,r|\vec{\sigma}^*] \neq u_i[h,r+1|\vec{\sigma}^*].$$ Fix $h$. Without loss of generality, suppose $$\label{eq:mh-0} u_i[h,r|\vec{\sigma}^*] < u_i[h,r+1|\vec{\sigma}^*].$$ By Lemma \[lemma:corr-0\] and Definition \[def:thr\], this implies the existence of $D \subset {\cal N}^2$, such that, for every $j \in {\cal N}$ and $k \in {\cal N}_j$, $$\label{eq:mh-1} \begin{array}{l} {\mbox{DS}}_j[k|h_{r+1}^*] = {\mbox{DS}}_j[k|h_{r}^*] \setminus \{(k_1,k_2,{\tau}-1)|(k_1,k_2) \in D \land j,k \in {\mbox{RS}}[k_1,k_2]\},\\ {\mbox{DS}}_j[k|h_{r}^*] = {\mbox{DS}}_j[k|h_{r+1}^*]\cup \{(k_1,k_2,{\tau}-1)|(k_1,k_2) \in D \land j,k \in {\mbox{RS}}[k_1,k_2]\},\\ (k_1,k_2,{\tau}- r-1) \in {\mbox{DS}}_j[k|h], \end{array}$$ where $h_r^* = {\mbox{hist}}[h,r|\vec{\sigma}^*]$, and there exist $j,k$ and $(k_1,k_2) \in D$ such that $i,j \in {\mbox{RS}}[k_1,k_2]$, for instance, $k_1$ and $k_2$. Notice that, if \[eq:mh-0\] is true, then ${\tau}- r -1\geq1$. Thus, we can define $h'$ such that: - $|h'| = |h|$. - ${\mbox{DS}}_j[k|h'] = {\mbox{DS}}_j[k|h] \setminus \{(k_1,k_2,{\tau}- r-1) | (k_1,k_2) \in D\} \cup \{(k_1,k_2,{\tau}- r -2) | (k_1,k_2) \in D\}$. Let $h_r' = {\mbox{hist}}[h',r|\vec{\sigma}^*]$. By Lemma \[lemma:corr-0\], for every $r' \in \{0\ldots r-1\}$, $j \in {\cal N}$, and $k \in {\cal N}_j$, $$\label{eq:mh-2} \begin{array}{ll} {\mbox{DS}}_j[k|h_{r'+1}'] & = \{(l_1,l_2,r''+1) | (l_1,l_2,r'') \in {\mbox{DS}}_j[k|h_{r'}'] \land r'' +1 < {\tau}\}\\ &\\ & = \{(l_1,l_2,r''+r') | (l_1,l_2,r'') \in ({\mbox{DS}}_j[k|h] \setminus \{(k_1,k_2,{\tau}- r- 1) | (k_1,k_2) \in D\}\\ & \cup \{(k_1,k_2,{\tau}- r- 2) | (k_1,k_2) \in D\land j,k \in {\mbox{RS}}[k_1,k_2]\}) \land r'' +r' < {\tau}\}\\ &\\ & = \{(k_1,k_2,r''+r') | (k_1,k_2,r'') \in {\mbox{DS}}_j[k|h] \land (k_1,k_2) \notin D \land r'' +r' <{\tau}\}\\ & \cup \{(k_1,k_2,{\tau}- r+r'-2) | (k_1,k_2) \in D\land j,k \in {\mbox{RS}}[k_1,k_2] \land {\tau}- r - 2 + r' < {\tau}\}\\ &\\ & = {\mbox{DS}}_j[k|h_{r'+1}^*] \setminus \{(k_1,k_2,{\tau}-r+r'-1) | (k_1,k_2) \in D \land j,k \in {\mbox{RS}}[k_1,k_2]\} \\ &\cup \{(k_1,k_2,{\tau}- r + r' - 2) | (k_1,k_2) \in D\land j,k \in {\mbox{RS}}[k_1,k_2] \land r' -2 < r\}.\\ &\\ & = {\mbox{DS}}_j[k|h_{r'+1}^*] \setminus \{(k_1,k_2,{\tau}-r+r'-1) | (k_1,k_2) \in D \land j,k \in {\mbox{RS}}[k_1,k_2]\} \\ &\cup \{(k_1,k_2,{\tau}- r + r' - 2) | (k_1,k_2) \in D\land j,k \in {\mbox{RS}}[k_1,k_2] \land r' -1 < r\}.\\ &\\ & = {\mbox{DS}}_j[k|h_{r'+1}^*] \setminus \{(k_1,k_2,{\tau}-r+r'-1) | (k_1,k_2) \in D \land j,k \in {\mbox{RS}}[k_1,k_2]\} \\ &\cup \{(k_1,k_2,{\tau}- r + r' - 1) | (k_1,k_2) \in D\land j,k \in {\mbox{RS}}[k_1,k_2]\}.\\ &\\ &={\mbox{DS}}_j[k|h_{r'+1}^*]. \end{array}$$ By the definition of ${\mbox{DS}}_j[k|h]$ and ${\mbox{DS}}_j[k|h']$ and by Definition \[def:thr\], for every $r' \in \{0\ldots r\}$, $j \in {\cal N}$, and $k \in {\cal N}_j$, $$\label{eq:mh-3} \begin{array}{l} p_j[k|h_{r'}'] = p_j[k|h_{r'}^*].\\ \bar{p}_i[h',r'|\vec{\sigma}^*] = \bar{p}_i[h,r'|\vec{\sigma}^*].\\ q_i[h',r'|\vec{\sigma}^*] = q_i[h,r'|\vec{\sigma}^*].\\ u_i[h',r'|\vec{\sigma}^*] = u_i[h,r'|\vec{\sigma}^*]. \end{array}$$ Furthermore, by \[eq:mh-1\], $$\label{eq:mh-4} \begin{array}{ll} {\mbox{DS}}_j[k|h_{r+1}'] & = \{(l_1,l_2,r'+1) | (l_1,l_2,r') \in {\mbox{DS}}_j[k|h_{r}'] \land r' +1 <{\tau}\}\\ &\\ & = \{(l_1,l_2,r'+1) | (l_1,l_2,r') \in ({\mbox{DS}}_j[k|h_{r}^*] \cup \{(k_1,k_2,{\tau}-2)|(k_1,k_2) \in D \land j,k \in {\mbox{RS}}[k_1,k_2]\}) \\ &\land r' +1 < {\tau}\})\\ &\\ & = \{(k_1,k_2,r'+1) | (k_1,k_2,r') \in {\mbox{DS}}_j[k|h_{r}^*] \} \cup \\ & \{(k_1,k_2,{\tau}-1)|(k_1,k_2) \in D \land j,k \in {\mbox{RS}}[k_1,k_2] \land {\tau}-1 < {\tau}\}\\ &\\ & = {\mbox{DS}}_j[k|h_{r+1}^*] \cup \{(k_1,k_2,{\tau}- 1)|(k_1,k_2) \in D\land j,k \in {\mbox{RS}}[k_1,k_2]\}\\ & = {\mbox{DS}}_j[k|h_{r}^*]. \end{array}$$ By Definition \[def:thr\] and \[eq:mh-4\], $$\label{eq:mh-5} \begin{array}{l} p_j[k|h_{r+1}'] = p_j[k|{h_{r}^*}].\\ \bar{p}_i[h',r+1|\vec{\sigma}^*] = \bar{p}_i[h,r|\vec{\sigma}^*].\\ q_i[h',r+1|\vec{\sigma}^*] = q_i[h,r|\vec{\sigma}^*].\\ u_i[h',r+1|\vec{\sigma}^*] = u_i[h,r|\vec{\sigma}^*]. \end{array}$$ Finally, by Lemma \[lemma:corr-0\], for every $r'>r$, $j \in {\cal N}$, and $k \in {\cal N}_j$, $$\label{eq:mh-6} \begin{array}{ll} {\mbox{DS}}_j[k|h_{r'+1}'] & = \{(l_1,l_2,r''+1) | (l_1,l_2,r'') \in {\mbox{DS}}_j[k|h_{r'}'] \land r''+1 <{\tau}\}\\ &= \{(l_1,l_2,r'' +1)| (l_1,l_2,r'') \in ({\mbox{DS}}_j[k|h] \setminus \\ &\{(k_1,k_2,{\tau}- r -1) | (k_1,k_2) \in D \land j,k \in {\mbox{RS}}[k_1,k_2]\}\\ &\cup \{(k_1,k_2,{\tau}- r- 2) | (k_1,k_2) \in D \land j,k \in {\mbox{RS}}[k_1,k_2]\}) \land r'' +1 <{\tau}\}\\ &\\ &= \{(l_1,l_2,r''+r'+1)| (l_1,l_2,r'') \in {\mbox{DS}}_j[k|h] \land r'' + r' +1< {\tau}\}\\ & \setminus \{(k_1,k_2,{\tau}- r+r' - 1) | (k_1,k_2) \in D \land j,k \in {\mbox{RS}}[k_1,k_2] \land r'< r\}\\ &\cup \{(k_1,k_2,{\tau}- r+r'-2)| (k_1,k_2) \in D \land j,k \in {\mbox{RS}}[k_1,k_2] \land r'< r+1 \}\\ &\\ &= \{(l_1,l_2,r''+r')| (l_1,l_2,r'') \in {\mbox{DS}}_j[k|h] \land r'' + r' < {\tau}\}\\ & \setminus \{(k_1,k_2,{\tau}- r+r' - 1) | (k_1,k_2) \in D \land j,k \in {\mbox{RS}}[k_1,k_2]\land r'< r\}\\ &\cup \{(k_1,k_2,{\tau}- r+r'-2)| (k_1,k_2) \in D\land j,k \in {\mbox{RS}}[k_1,k_2] \land r'< r\}\\ &\\ & = \{(l_1,l_2,r''-r') | (l_1,l_2,r'') \in {\mbox{DS}}_j[k|h] \land r'' > r' \}\\ & = {\mbox{DS}}_j[k|h_{r'+1}^*]. \end{array}$$ Therefore, by Definition \[def:thr\], for every $r' > r+1$, $j \in {\cal N}$, and $k \in {\cal N}_j$, $$\label{eq:mh-7} \begin{array}{l} p_j[k|h_{r'}'] = p_j[k|h_{r'}^*].\\ \bar{p}_i[h',r'|\vec{\sigma}^*] = \bar{p}_i[h,r'|\vec{\sigma}^*].\\ q_i[h',r'|\vec{\sigma}^*] = q_i[h,r'|\vec{\sigma}^*].\\ u_i[h',r'|\vec{\sigma}^*] = u_i[h,r'|\vec{\sigma}^*]. \end{array}$$ By \[eq:mh-0\], \[eq:mh-3\], \[eq:mh-5\], and \[eq:mh-7\], $$\begin{array}{ll} \sum_{r'=1}^{\tau}\omega_i^{r'}(u_i[h,r'|\vec{\sigma}^*] - u_i[h',r'|\vec{\sigma}^*]) & + (1-q_i[h,0|\vec{\sigma}^*])\gamma_i (\gamma_i \bar{p}_i[h',0|\vec{\sigma}^*] - \bar{p}_i[h,0|\vec{\sigma}^*])\\ = u_i[h,r+1|\vec{\sigma}^*] - u_i[h',r+1|\vec{\sigma}^*] & = u_i[h,r+1|\vec{\sigma}^*] - u_i[h,r|\vec{\sigma}^*] > 0. \end{array}$$ This is a contradiction to \[eq:mh\], which concludes the proof. For the case where $$u_i[h,r|\vec{\sigma}^*] > u_i[h,r+1|\vec{\sigma}^*],$$ the proof is identical, except that $h'$ is defined such that the punishments that end at stage $r$ are anticipated, i.e.: $${\mbox{DS}}_j[k|h'] = {\mbox{DS}}_j[k|h] \setminus \{(k_1,k_2,{\tau}- r-1) | (k_1,k_2) \in D\} \cup \{(k_1,k_2,{\tau}- r ) | (k_1,k_2) \in D\}.$$ ### Proof of Theorem \[theorem:indir-suff\]. {#proof:theorem:indir-suff} If there exists a constant $c \geq 1$ such that, for every $h \in {\cal H}$ and $i \in {\cal N}$, Assumption \[eq:indir-assum\] holds, then $\psi[\vec{\sigma}^*] \supseteq (v,\infty)$, where $$v = \max_{h \in {\cal H}}\max_{i \in {\cal N}} \bar{p}_i[h,0|\vec{\sigma}^*]\left(1 + \frac{c}{{\tau}}\right).$$ Assume by contradiction that, for any player $i \in {\cal N}$, $\vec{\sigma}^*$ is not a SPE for any $\omega_i \in (0,1)$ and that $$\frac{\beta_i}{\gamma_i} > \max_{h \in {\cal H}} \bar{p}_i[h,0|\vec{\sigma}^*]\left(1 + \frac{c}{{\tau}}\right).$$ Fix $i$. The proof considers a history $h$ that minimizes the left side of Inequality \[eq:indir-equiv\]. The reason for this is that, if Inequality \[eq:indir-equiv\] is true for $h$, then it is also true for every other history $h'$. If $\bar{p}_i[h,0|\vec{\sigma}^*] = 0$, then the inequality is trivially fulfilled. Hence, consider that $\bar{p}_i[h,0|\vec{\sigma}^*] > 0$. By Lemma \[lemma:maxh\], for every $r \in \{1 \ldots {\tau}-1\}$, $$u_i[h,r|\vec{\sigma}^*] = u_i[h,r+1|\vec{\sigma}^*].$$ We are left with stage ${\tau}$. Let $u_i^h = u_i[h,1|\vec{\sigma}^*]$. If for every $k \in {\cal N}$ and $l \in {\cal N}_k$ we have ${\mbox{DS}}_k[l|h_{{\tau}}^*] = {\mbox{DS}}_k[l|h_{{\tau}-1}^*]$, where $h_r^* = {\mbox{hist}}[h,r|\vec{\sigma}^*]$, then $$u_i^h = u_i[h,{\tau}|\vec{\sigma}^*].$$ Otherwise, by Lemma \[lemma:maxh\], $$u_i^h \leq u_i[h,{\tau}|\vec{\sigma}^*].$$ Either way, by Lemma \[lemma:maxh\], for every $h' \in {\cal H}$: $$\label{eq:is-1} \begin{array}{ll} -(1-q_i[h,0|\vec{\sigma}^*])\gamma_i \bar{p}_i[h,0|\vec{\sigma}^*] + \sum_{r = 1}^{\tau}\omega_i^r u_i^h & \leq \\ -(1-q_i[h',0|\vec{\sigma}^*])\gamma_i \bar{p}_i[h',0|\vec{\sigma}^*] + \sum_{r=1}^{\tau}\omega_i^r u_i[h',r|\vec{\sigma}^*]. \end{array}$$ We can write: $$\label{eq:is-2} \begin{array}{ll} -(1-q_i[h,0|\vec{\sigma}^*])\gamma_i \bar{p}_i[h,0|\vec{\sigma}^*] + \sum_{r = 1}^{\tau}\omega_i^r u_i^h & \geq 0\\ -a + \frac{\omega_i - \omega_i^{{\tau}+1}}{1 - \omega_i} u_i^h & \geq 0\\ -a + \omega_i(u_i^h + a) - \omega_i^{{\tau}+1} u_i^h & \geq 0, \end{array}$$ where $$a = (1-q_i[h,0|\vec{\sigma}^*])\gamma_i \bar{p}_i[h,0|\vec{\sigma}^*].$$ Again, this Inequality corresponds to a polynomial with degree ${\tau}+1$. If $q_i[h,1|\vec{\sigma}^*] = 1$, then by our assumptions $q_i[h,0|\vec{\sigma}^*] =1$, $a=0$, and the Inequality holds. Suppose then that $$q_i[h,1|\vec{\sigma}^*],q_i[h,0|\vec{\sigma}^*] < 1.$$ The polynomial has a zero in $\omega_i = 1$. If $\bar{p}_i[h,0|\vec{\sigma}^*] = 0$, then $a=0$ and the Inequality holds. Consider, then, that $\bar{p}_i[h,0|\vec{\sigma}^*] > 0$, which implies that $a>0$. In these circumstances, a solution to \[eq:is-2\] exists for $\omega_i \in (0,1)$ iff the polynomial is strictly concave and has another zero in $(0,1)$. This is true iff the polynomial has a maximum in $(0,1)$. The derivatives yield the following conditions: 1. $\exists_{\omega_i \in (0,1)} u_i^h + a - ({\tau}+1) \omega_i u_i^h = 0 \Rightarrow \exists_{\omega_i \in (0,1)} \omega_i = \frac{u_i^h+a}{({\tau}+1)u_i^h}$. 2. $-({\tau}+1){\tau}u_i^h < 0 \Rightarrow u_i^h > 0$. Condition $1$ implies: $$\label{eq:is-3} \begin{array}{ll} u_i^h{\tau}& > a\\ (1-q_i[h,1|\vec{\sigma}^*]) \beta_i {\tau}&> (1-q_i[h,1|\vec{\sigma}^*])\gamma_i\bar{p}_i[h,1|\vec{\sigma}^*] {\tau}+ (1-q_i[h,0|\vec{\sigma}^*])\gamma_i\bar{p}_i[h,0|\vec{\sigma}^*]\\ \frac{\beta_i}{\gamma_i} &> \bar{p}_i[h,1|\vec{\sigma}^*] + \bar{p}_i[h,0|\vec{\sigma}^*]\frac{1}{{\tau}} \frac{1-q_i[h,0|\vec{\sigma}^*]}{1-q_i[h,1|\vec{\sigma}^*]}. \end{array}$$ By the assumption that $q_i[h,0|\vec{\sigma}^*] \leq 1-c(1-q_i[h,1|\vec{\sigma}^*])$, if Inequality \[eq:indir-suff\] is true, then so is \[eq:is-3\]. Furthermore, it also holds that $$\beta_i > \gamma_i \bar{p}_i[h,1|\vec{\sigma}^*] \Rightarrow u_i^h >0,$$ thus, Condition $2$ and \[eq:is-2\] are also true for some $\omega_i \in (0,1)$. By \[eq:is-1\], for every $h' \in {\cal H}$ and $i \in {\cal N}$, Inequality \[eq:indir-equiv\] is true, implying by Lemma \[lemma:indir-equiv\] that $\vec{\sigma}^*$ is a SPE for some value $\omega_i \in (0,1)$. This is a contradiction, proving that if for every $i \in {\cal N}$ we have $\frac{\beta_i}{\gamma_i} \in (v,\infty)$, then $\vec{\sigma}^*$ is a SPE. By the definition of $\psi$, $\psi[\vec{\sigma}^*] \supseteq (v,\infty)$. Private Monitoring {#sec:proof:private} ================== Evolution of the Network {#sec:proof-priv-evol} ------------------------ ### Auxiliary Lemma. \[lemma:pevol-aux\] For every $h \in {\cal H}$, $\vec{p}' \in {\cal P}$, $i \in {\cal N}$, $j \in {\cal N}_i$, $k_1,k_2 \in {\cal N}$, and $r \in \{0 \ldots {d}_i[k_1,k_2] + {\tau}[k_1,k_2|i,j] - v[k_1,k_2] -1\}$, where $v[k_1,k_2] = \min[{d}_i[k_1,k_2] - {d}_j[k_1,k_2],0]$, let $s_i' \in {\mbox{sig}}[\vec{\sigma}'[h_{r}']|h_{r}']$ and $s_i^* \in {\mbox{sig}}[\vec{\sigma}^*[h_{r}^*]|h_{r}^*]$, where $h_r^* = {\mbox{hist}}[h,r|\vec{\sigma}^*]$, $h_r' = {\mbox{hist}}[h,r|\vec{\sigma}']$, and $\vec{\sigma}' = \vec{\sigma}^*[h|\vec{p}']$. Then, we have 1. If $k_2 \notin {\mbox{CD}}_{k_1}[\vec{p}'|h]$ or $k_2 \in {\mbox{CD}}_{k_1}[\vec{p}'|h]$ and ${d}_i[k_1,k_2] > r$, we have $s_i^*[k_1,k_2] = s_i'[k_1,k_2]$. 2. If $k_2 \in {\mbox{CD}}_{k_1}[\vec{p}'|h]$ and ${d}_i[k_1,k_2] = r$, then $s_i^*[k_1,k_2] = \mbox{\emph{cooperate}}$ and $s_i'[k_1,k_2] = \mbox{\emph{defect}}$. 3. Else, $s_i^*[k_1,k_2] = s_i'[k_1,k_2] = \mbox{\emph{cooperate}}$. Fix $i$, $j$, $k_1$, $k_2$, $h$, and $\vec{p}'$. Let $s_{k_2}' \in {\mbox{sig}}[\vec{\sigma}'[h]|h]$ and $s_{k_2}^* \in {\mbox{sig}}[\vec{\sigma}^*[h]|h]$. For $k_2 \notin {\mbox{CD}}_{k_1}[\vec{p}'|h]$ and $h_{k_1} \in h$, $$p_{k_1}'[k_2] = p_{k_1}[k_2|h_{k_1}],$$ and, by Definition \[def:privsig\], $$s_i^*[k_1,k_2] = s_i'[k_1,k_2].$$ If $k_2 \in {\mbox{CD}}_{k_1}[\vec{p}'|h]$ and $r = {d}_i[k_1,k_2]$, then, by the definition of $\vec{\sigma}'$ and $\vec{\sigma}^*$, $$s_{k_2}'[k_1,k_2] = \mbox{\emph{defect}},$$ $$s_{k_2}^*[k_1,k_2] = \mbox{\emph{cooperate}},$$ and, by Definition \[def:privsig\], $$s_i'[k_1,k_2] = \mbox{\emph{defect}}.$$ $$s_i^*[k_1,k_2] = \mbox{\emph{cooperate}}.$$ If $r > {d}_i[k_1,k_2]$, then $$s_i'[k_1,k_2] = s_i^*[k_1,k_2]= \mbox{\emph{cooperate}}.$$ To see this, assume first that $s_i'[k_1,k_2] = \mbox{\emph{defect}}$. Then, by Definition \[def:privsig\], there must exist a round $r'>0$ and history ${\mbox{hist}}[h,r'|\vec{\sigma}']$ after which $k_2$ defects $k_1$. That is, define $r' = r - {d}_i[k_1,k_2]$. For $s_{k_2}'' \in {\mbox{sig}}[\vec{\sigma}'[h_{r'}']|h_{r'}']$, $s_{k_2''}[k_1,k_2] = \mbox{\emph{defect}}$, which is true iff $p_{k_1}''[k_2] < p_{k_1}[k_2|h_{r'}']$. Since $r' > 0$, this contradicts the definition of $\vec{\sigma}'$. Hence, $s_i^*[k_1,k_2] = \mbox{\emph{cooperate}}$. If we assume that $s_i^*[k_1,k_2] = \mbox{\emph{defect}}$, then by Definition \[def:privsig\], there must exist a round $r'>0$ and history ${\mbox{hist}}[h,r'|\vec{\sigma}']$ after which $k_2$ defects $k_1$. As before, since $r'>0$, another contradiction is reached and we can conclude that we must have $s_i^*[k_1,k_2]= \mbox{\emph{cooperate}}$. For $r < {d}_i[k_1,k_2]$,the result holds immediately. This is because by Definition \[def:privsig\], $s_i'[k_1,k_2] = \mbox{\emph{defect}}$ iff $|h| + r\geq {d}_i[k_1,k_2]$ and for $s_{k_2}'' = h_{k_2}^{{d}_i[k_1,k_2]-r}$, $s_{k_2}''[k_1,k_2] = \mbox{\emph{defect}}$. This implies that $$s_{i}^*[k_1,k_2] = \mbox{\emph{defect}}.$$ Similarly, if $s_i'[k_1,k_2] = \mbox{\emph{cooperate}}$, then $s_{k_2}''[k_1,k_2] = \mbox{\emph{cooperate}}$, implying that $$s_{i}^*[k_1,k_2] = \mbox{\emph{cooperate}}.$$ This proves the result. ### Proof of Lemma \[lemma:priv-corr-1\]. {#proof:lemma:priv-corr-1} For every $h \in {\cal H}$, $\vec{p}' \in {\cal P}$, $r >0$, $i \in {\cal N}$, and $j \in {\cal N}_i$: $$\begin{array}{ll} {\mbox{DS}}_i[j|h_{i,r}'] =& {\mbox{DS}}_i[j|h_{i,r}^*] \cup \{(k_1,k_2,r -1 - {d}_i[k_1,k_2]+v[k_1,k_2]) | k_1,k_2 \in {\cal N} \land \\ & k_2 \in {\mbox{CD}}_{k_1}[\vec{p}'|h] \land r \in \{{d}_i[k_1,k_2]+1 \ldots {d}_i[k_1,k_2] + {\tau}[k_1,k_2|i,j]-v[k_1,k_2]\}\land\\ & v[k_1,k_2] = \min[{d}_i[k_1,k_2] - {d}_j[k_1,k_2],0]\},\\ \end{array}$$ where $h_{i,r}^* \in {\mbox{hist}}[h,r|\vec{\sigma}^*]$, $h_{i,r}' \in {\mbox{hist}}[h,r| \vec{\sigma}']$, and $\vec{\sigma}' =\vec{\sigma}^*[h|\vec{p}']$ is the profile of strategies where all players follow $\vec{p}'$ in the first stage. Fix $h$, $\vec{p}'$, $i$, and $j$. The proof goes by induction on $r$, where the induction hypothesis is that for every $r \geq 0$, Equality \[eq:priv-res-corr1\] holds for $r+1$: $$\begin{array}{ll} {\mbox{DS}}_i[j|h_{i,r+1}'] =& {\mbox{DS}}_i[j|h_{i,r+1}^*] \cup \{(k_1,k_2,r - {d}_i[k_1,k_2]+v[k_1,k_2]) | k_1,k_2 \in {\cal N} \land \\ & k_2 \in {\mbox{CD}}_{k_1}[\vec{p}'|h] \land r+1 \in \{{d}_i[k_1,k_2]+1 \ldots {d}_i[k_1,k_2] + {\tau}[k_1,k_2|i,j]-v[k_1,k_2]\}\land\\ & v[k_1,k_2] = \min[{d}_i[k_1,k_2] - {d}_j[k_1,k_2],0]\},\\ \end{array}$$ We will simplify the notation by first dropping the factor $v[k_1,k_2] = \min[{d}_i[k_1,k_2] - {d}_j[k_1,k_2],0]$ whenever possible, and by removing the redundant indexes $k_1,k_2,i,j$, except when distinguishing between different delays. We will also remove the factor $k_1,k_2 \in {\cal N}$. Namely: - ${d}_i[k_1,k_2]= {d}_i$ and ${d}_j[k_1,k_2] = {d}_j$. - $v[k_1,k_2] = v$. - ${\tau}[k_1,k_2|i,j] = {\tau}$. By Definition \[def:priv-thr\], we have that for every $r \geq 0$, $h_r^* = {\mbox{hist}}[h,r|\vec{\sigma}^*]$, and $s_i^* \in {\mbox{sig}}[\vec{\sigma}^*[h_r^*]|h_r^*]$, $$\label{eq:priv-corr1} {\mbox{DS}}_i[j|h_{i,r+1}^*] = L_1[r+1|\vec{\sigma}^*] \cup L_2[r+1|\vec{\sigma}^*],$$ where $$\label{eq:priv-corr1-1} \begin{array}{ll} L_1[r+1|\vec{\sigma}^*] = & \{(k_1,k_2,r'+1)|(k_1,k_2,r') \in {\mbox{DS}}_{i}[j|h_{i,r}^*] \land r' +1 < {\tau}\}.\\ L_2[r+1|\vec{\sigma}^*] = & \{(k_1,k_2,v) | \land s_i^*[k_1,k_2] = \mbox{\emph{defect}}\}. \end{array}$$ Similarly, for every $r\geq0$, $h_r' = {\mbox{hist}}[h,r|\vec{\sigma}']$, and $s_i' \in {\mbox{sig}}[\vec{\sigma}'[h_r']|h_r']$, $$\label{eq:priv-corr1-2} {\mbox{DS}}_i[j|h_{i,r+1}'] = L_1[r+1|\vec{\sigma}'] \cup L_2[r+1|\vec{\sigma}'],$$ where $$\label{eq:priv-corr1-3} \begin{array}{ll} L_1[r+1|\vec{\sigma}'] = & \{(k_1,k_2,r'+1)|(k_1,k_2,r') \in {\mbox{DS}}_{i}[j|h_{i,r}'] \land r' +1 < {\tau}\}.\\ L_2[r+1|\vec{\sigma}'] = & \{(k_1,k_2,v) | s_i'[k_1,k_2] = \mbox{\emph{defect}}\}. \end{array}$$ For any $r \geq 0$, let $s_i' \in {\mbox{sig}}[\vec{\sigma}'[h_{r}']|h_{r}']$ and $s_i^* \in {\mbox{sig}}[\vec{\sigma}^*[h_{r}^*]|h_{r}^*]$, where $h_r^* = {\mbox{hist}}[h,r|\vec{\sigma}^*]$, $h_r' = {\mbox{hist}}[h,r|\vec{\sigma}']$, and $\vec{\sigma}' = \vec{\sigma}^*[h|\vec{p}']$. By Lemma \[lemma:pevol-aux\], we have: 1. $s_i^*[k_1,k_2] = s_i'[k_1,k_2]$ for $k_2 \notin {\mbox{CD}}_{k_1}[\vec{p}'|h]$ and $k_2 \in {\mbox{CD}}_{k_1}[\vec{p}'|h]$ such that ${d}_i> r$. 2. $s_i'[k_1,k_2] = \mbox{\emph{defect}}$ and $s_i^*[k_1,k_2] = \mbox{\emph{cooperate}}$ for $k_2 \in {\mbox{CD}}_{k_1}[\vec{p}'|h]$ such that ${d}_i= r$. 3. $s_i'[k_1,k_2] =s_i^*[k_1,k_2]= \mbox{\emph{cooperate}}$ for $k_2 \in {\mbox{CD}}_{k_1}[\vec{p}'|h]$ such that ${d}_i< r$. By \[eq:priv-corr1-1\], and items 1), 2), and 3) above, $$\label{eq:priv-corr1-4} \begin{array}{ll} L_2[r+1|\vec{\sigma}^*] &= \{(k_1,k_2,v) | s_i^*[k_1,k_2] = \mbox{\emph{defect}}\}\\ &= \{(k_1,k_2,v) | k_2 \notin {\mbox{CD}}_{k_1}[\vec{p}'|h] \land s_i^*[k_1,k_2] = \mbox{\emph{defect}}\}\cup\\ & \{(k_1,k_2,v) | k_2 \in {\mbox{CD}}_{k_1}[\vec{p}'|h] \land {d}_i > r \land s_i^*[k_1,k_2] = \mbox{\emph{defect}}\}, \end{array}$$ Equation \[eq:priv-corr1-4\], 1), 2), and 3) allows us to write: $$\label{eq:priv-corr1-5} \begin{array}{ll} L_2[r+1|\vec{\sigma}'] &= \{(k_1,k_2,v) | s_i'[k_1,k_2] = \mbox{\emph{defect}}\}\\ &\\ &= \{(k_1,k_2,v) | k_2 \notin {\mbox{CD}}_{k_1}[\vec{p}',h] \land s_i'[k_1,k_2] = \mbox{\emph{defect}}\}\\ & \cup \{(k_1,k_2,v) | k_2 \in {\mbox{CD}}_{k_1}[\vec{p}'|h] \land s_i'[k_1,k_2] = \mbox{\emph{defect}}\}\\ &\\ &=\{(k_1,k_2,v) | k_2 \notin {\mbox{CD}}_{k_1}[\vec{p}'|h] \land s_i^*[k_1,k_2] = \mbox{\emph{defect}}\}\cup\\ & \{(k_1,k_2,v) | k_2 \in {\mbox{CD}}_{k_1}[\vec{p}'|h] \land {d}_i > r \land s_i^*[k_1,k_2] = \mbox{\emph{defect}}\} \cup\\ & \{(k_1,k_2,v) | k_2 \in {\mbox{CD}}_{k_1}[\vec{p}'|h] \land {d}_i = r\}\\ &\\ & =L_2[r+1|\vec{\sigma}^*] \cup \{(k_1,k_2,v) | k_2 \in {\mbox{CD}}_{k_1}[\vec{p}'|h] \land {d}_i = r\}. \end{array}$$ Now proceed to the base case, for $r=0$. Since $h={\mbox{hist}}[h,0|\vec{\sigma}^*] = {\mbox{hist}}[h,0|\vec{\sigma}']$, by \[eq:priv-corr1-1\] and \[eq:priv-corr1-3\], it is true that: $$\label{eq:priv-corr1-6} L_1[1|\vec{\sigma}'] = L_1[1|\vec{\sigma}^*].$$ Furthermore, by \[eq:priv-corr1-5\], $$\label{eq:priv-corr1-7} \begin{array}{ll} L_2[r+1|\vec{\sigma}'] &= L_2[r+1|\vec{\sigma}^*] \cup \{(k_1,k_2,v) | k_2 \in {\mbox{CD}}_{k_1}[\vec{p}'|h] \land {d}_i = 0\}\\ &= L_2[r+1|\vec{\sigma}^*] \cup \{(k_1,k_2,r - {d}_i + v) | k_2 \in {\mbox{CD}}_{k_1}[\vec{p}'|h] \land 1 \geq {d}_i +1 \land 1 < {d}_i + {\tau}- v\}\\ &= L_2[r+1|\vec{\sigma}^*] \cup \{(k_1,k_2,r- {d}_i + v) | k_2 \in {\mbox{CD}}_{k_1}[\vec{p}'|h] \land r +1 \in \{{d}_i + 1 \ldots {d}_i + {\tau}- v\}\}. \end{array}$$ The base case is true by \[eq:priv-corr1\], \[eq:priv-corr1-2\], \[eq:priv-corr1-6\] and \[eq:priv-corr1-7\]. Hence, assume the induction hypothesis for some $r \geq 0$ and consider the induction step for $r+1$, which consists in determining the value of ${\mbox{DS}}_i[j|h_{i,r+2}']$. By the induction hypothesis and by \[eq:priv-corr1-1\], $$\label{eq:priv-corr1-8} \begin{array}{ll} L_1[r+2|\vec{\sigma}'] & = \{(k_1,k_2,r'+1)|(k_1,k_2,r') \in (A \cup B)\land r'+1 < {\tau}\}\\ & = \{(k_1,k_2,r'+1)|(k_1,k_2,r') \in A \land r'+1 < {\tau}\} \cup \\ & \{(k_1,k_2,r'+1)|(k_1,k_2,r') \in B \land r'+1 < {\tau}\}, \end{array}$$ where $$\begin{array}{ll} A = & {\mbox{DS}}_{i}[j|h_{i,r+1}^*]\\ B = &\{(k_1,k_2,r - {d}_i+v) |k_2 \in {\mbox{CD}}_{k_1}[\vec{p}'|h] \land r+1 \in \{{d}_i+1 \ldots {d}_i + {\tau}-v\}. \end{array}$$ We have by \[eq:priv-corr1-1\] $$\label{eq:priv-corr1-9} \begin{array}{ll} &\{(k_1,k_2,r'+1)|(k_1,k_2,r') \in A \land r'+1 < {\tau}\}\\ =& \{(k_1,k_2,r'+1)|(k_1,k_2,r') \in {\mbox{DS}}_{i}[j|h_{i,r+1}^*] \land r'+1 < {\tau}\}\\ =L_1[r+2|\vec{\sigma}^*]. \end{array}$$ It is also true that $$\label{eq:priv-corr1-10} \begin{array}{ll} &\{(k_1,k_2,r'+1)|(k_1,k_2,r') \in B \land r'+1 < {\tau}\}\\ =& \{(l_1,l_2,r'+1) | (l_1,l_2,r') \in \{(k_1,k_2,r - {d}_i+v) | k_2 \in {\mbox{CD}}_{k_1}[\vec{p}'|h] \land \\ &r+1 \in \{{d}_i +1\ldots {d}_i + {\tau}-v\}\} \land r'+1<{\tau}\}\\ &\\ = & \{(k_1,k_2,r+1 - {d}_i+v) |k_2 \in {\mbox{CD}}_{k_1}[\vec{p}'|h] \land r+1 \in \{{d}_i+1 \ldots {d}_i + {\tau}-v\} \land r +1 - {d}_i + v < {\tau}\}\\ &\\ =& \{(k_1,k_2,r +1- {d}_i+v) |k_2 \in {\mbox{CD}}_{k_1}[\vec{p}'|h] \land r+1 \in \{{d}_i+1 \ldots {d}_i + {\tau}-v-1\} \}\\ &\\ =& \{(k_1,k_2,r +1- {d}_i+v) |k_2 \in {\mbox{CD}}_{k_1}[\vec{p}'|h] \land r+2 \in \{{d}_i+2 \ldots {d}_i + {\tau}-v \}\} \end{array}$$ By \[eq:priv-corr1\], \[eq:priv-corr1-2\], \[eq:priv-corr1-7\], \[eq:priv-corr1-8\], \[eq:priv-corr1-9\], and \[eq:priv-corr1-10\], $$\begin{array}{ll} {\mbox{DS}}_i[j|h_{i,r+2}'] & = L_1[r+2|\vec{\sigma}^*] \cup\\ & \{(k_1,k_2,r+1 - {d}_i+v) | k_2 \in {\mbox{CD}}_{k_1}[\vec{p}'|h] \land r+2 \in \{{d}_i+2 \ldots {d}_i + {\tau}]-v \} \cup\\ & L_2[r+2|\vec{\sigma}^*] \cup \{(k_1,k_2,v) | k_2 \in {\mbox{CD}}_{k_1}[\vec{p}'|h] \land {d}_i = r+1\}\\ &\\ &= L_1[r+2|\vec{\sigma}^*] \cup L_2[r+2|\vec{\sigma}^*] \cup \\ & \{(k_1,k_2,r+1 - {d}_i+v) | k_2 \in {\mbox{CD}}_{k_1}[\vec{p}'|h] \land r+2 \in \{{d}_i+2 \ldots {d}_i + {\tau}-v \}\cup\\ & \{(k_1,k_2,r +1 - {d}_i + v) | k_2 \in {\mbox{CD}}_{k_1}[\vec{p}'|h] \land {d}_i+1 = r+2\}\\ &\\ &= {\mbox{DS}}_i[j|h_{i,r+2}^*] \cup\\ & \{(k_1,k_2,r+1 - {d}_i+v) | k_2 \in {\mbox{CD}}_{k_1}[\vec{p}'|h] \land r+2 \in \{{d}_i+1 \ldots {d}_i + {\tau}-v\}. \end{array}$$ This proves the induction step and concludes the proof. ### Proof of Lemma \[lemma:priv-history\]. {#proof:lemma:priv-history} For every $i \in {\cal N}$, $j \in {\cal N}_i$, $h \in {\cal H}$, and $h_i,h_j \in h$: $$p_i[j|h_i] = p_i[j|h_j].$$ Fix $i$, $j$, $h$, and $h_i,h_j \in h$. Consider any tuple $(k_1,k_2,r) \in {\mbox{DS}}_i[j|h_i]$ and let $K_i$ and $K_j$ represent the sets used by $i$ and $j$ to compute $p_i[j|h_i]$ and $p_i[j|h_j]$, respectively. By Lemma \[lemma:priv-corr-1\] and Definitions \[def:privsig\] and \[def:priv-thr\], it is true that for some $r' \geq d_i[k_1,k_2]$: $$\label{eq:pvh-0} r=r' - {d}_i[k_1,k_2]+v_i,$$ $|h_i| \geq r'+1$ and, for $s_i = h_i^{r'+1}$, $$\label{eq:pvh-1} s_i[k_1,k_2] = \mbox{\emph{defect}},$$ where $v_i = \min[{d}_i[k_1,k_2] - {d}_j[k_1,k_2],0]$. By Definition \[def:privsig\], this implies that $$|h_i|\geq r' + {d}_i[k_1,k_2]+1,$$ and for $h_{k_2} \in h$ and $s_{k_2} = h_{k_2}^{r' + {d}_i[k_1,k_2]+1}$: $$\label{eq:pvh-2} s_{k_2}[k_1,k_2] = \mbox{\emph{defect}}.$$ Since $r' \geq {d}_i[k_1,k_2]$, if $r <0$, then we have by \[eq:pvh-0\] $$v_i < 0 \Rightarrow {d}_i[k_1,k_2] < {d}_j[k_1,k_2],$$ which implies by Definition \[def:privsig\], that $j$ has yet to observe the defection that caused $i$ to add $(k_1,k_2,v_i)$ to ${\mbox{DS}}_i[j|h_i]$. Consequently, $j$ has not included this tuple in ${\mbox{DS}}_j[i|h_j]$ or in $K_j$. Also, by Definition \[def:priv-thr\], $i$ does not include the tuple in $K_i$, since $r < 0$. Consider, now, that $r \geq0$, where by Definition \[def:priv-thr\] $i$ adds the tuple to $K_i$. By \[eq:pvh-0\], $$\label{eq:pvh-3} r' \geq {d}_i[k_1,k_2] - v_i \geq {d}_i[k_1,k_2] - {d}_i[k_1,k_2] + {d}_j[k_1,k_2] = {d}_j[k_1,k_2].$$ Furthermore, since by Definition \[def:priv-thr\], $r < {\tau}[k_1,k_2|i,j]$, we also have by \[eq:pvh-0\] $$\label{eq:pvh-4} r' < {\tau}[k_1,k_2|i,j] + {d}_i[k_1,k_2] - v_i.$$ If ${d}_i[k_1,k_2] \leq {d}_j[k_1,k_2]$, then $v_i <0$, $v_j = 0$, and by \[eq:pvh-3\] and \[eq:pvh-4\] we have $$r' < {\tau}[k_1,k_2|i,j] + {d}_j[k_1,k_2] - v_j,$$ $$r' +1\in \{{d}_j[k_1,k_2]+1 \ldots {d}_j[k_1,k_2] + {\tau}[k_1,k_2|i,j] -v_j\},$$ where $v_j = \min[{d}_j[k_1,k_2] - {d}_i[k_1,k_2],0]$. By Lemma \[lemma:priv-corr-1\], $j$ adds $(k_1,k_2,r'-{d}_j[k_1,k_2] + v_j)$ to ${\mbox{DS}}_j[i|h_j]$, such that $v_j=0$ and by \[eq:pvh-3\]: $$\label{eq:pvh-5} r' -{d}_j[k_1,k_2] +v_j \geq 0.$$ If ${d}_i[k_1,k_2] > {d}_j[k_1,k_2]$, then $v_i = 0$, $$v_j = -({d}_i[k_1,k_2] - {d}_j[k_1,k_2]),$$ hence, by \[eq:pvh-4\] $${d}_j[k_1,k_2] + {\tau}[k_1,k_2|i,j] -v_j = {\tau}[k_1,k_2|i,j] + {d}_i[k_1,k_2] - v_i> r'.$$ Therefore, by \[eq:pvh-3\], $$r' +1\in \{{d}_j[k_1,k_2]+1 \ldots {d}_j[k_1,k_2] + {\tau}[k_1,k_2|i,j] -v_j\},$$ and by Lemma \[lemma:priv-corr-1\] $j$ adds $(k_1,k_2,r'-{d}_j[k_1,k_2] + v_j)$ to ${\mbox{DS}}_j[i|h_j]$. Again, by \[eq:pvh-0\] and the assumption that $r \geq 0$, $$\label{eq:pvh-6} r'-{d}_j[k_1,k_2] + v_j = r' - {d}_i[k_1,k_2] = r + {d}_i[k_1,k_2] - {d}_i[k_1,k_2]-v_i =r \geq 0.$$ In any case, by \[eq:pvh-5\] and \[eq:pvh-6\], $j$ adds the tuple to $K_j$. This proves that $i$ adds the tuple to $K_i$ iff $j$ adds the tuple to $K_j$, implying that $K_i = K_j$. Since $p_i[j|h_i]$ and $p_i[j|h_j]$ are obtained by applying the same deterministic functions to $K_i$ and $K_j$, respectively, we have $$p_i[j|h_i] = p_i[j|h_j].$$ Generic Results {#sec:proof:priv-drop} --------------- ### Proof of Proposition \[prop:priv-folk\]. {#proof:prop:priv-folk} For every assessment $(\vec{\sigma}^*,\vec{\mu}^*)$, if $(\vec{\sigma}^*,\vec{\mu}^*)$ is Sequentially Rational, then, for every $i \in {\cal N}$, $\frac{\beta_i}{\gamma_i} \geq \bar{p}_i$. Consequently, $\psi[\vec{\sigma}^*] \subseteq (v,\infty)$, where $v = \max_{i \in {\cal N}} \bar{p}_i$. Let $\vec{p}^* = \vec{\sigma}^*[\emptyset]$. For $h_i = \emptyset$, the only history $h$ that fulfills $\mu_i^*[h|h_i]>0$ is $h = \emptyset$. Therefore, the equilibrium utility is also $$\pi_i[\vec{\sigma}^*|\vec{\mu}^*,\emptyset] = \sum_{r=0}^\infty\omega_i^r (1 - q_i[\vec{p}^*])(\beta_i - \gamma_i \bar{p}_i) = \frac{1 - q_i[\vec{p}^*]}{1-\omega_i}(\beta_i - \gamma_i \bar{p}_i).$$ If $\frac{\beta_i}{\gamma_i} \leq \bar{p}_i$, then $$\label{eq:priv-od-1} \pi_i[\vec{\sigma}^*|\vec{\mu}^*,\emptyset] \leq 0.$$ Let $\sigma_i' \in \Sigma_i$ be a strategy such that, for every $h_i \in {\cal H}_i$, $\sigma_i'[h_i] = \vec{0}$, and let $\vec{\sigma}' = (\sigma_i',\vec{\sigma}_{-i}^*)$, where $\vec{0}=(0)_{j \in {\cal N}_i}$. We have $$\label{eq:priv-od-2} \pi_i[\vec{\sigma}'|\vec{\mu}^*,\emptyset] = (1-q_i[\vec{p}^*])\beta_i + \pi_i[\vec{\sigma}'|({\mbox{sig}}[\vec{p}'|\emptyset])] \geq (1-q_i[\vec{p}^*])\beta_i,$$ where $\vec{p}' = (\vec{0},\vec{p}^*_{-i})$. By Lemma \[lemma:pprob\], $q_i [\vec{p}^*] < 1$. Since $ \pi_i[\vec{\sigma}'|{\mbox{sig}}[\vec{p}'|\emptyset]] \geq 0$, it is true that $$\pi_i[\vec{\sigma}^*|\vec{\mu}^*,\emptyset] \leq 0 < \pi_i[\vec{\sigma}'|\vec{\mu}^*,\emptyset].$$ This contradicts the assumption that $\vec{\sigma}^*$ is a SPE. ### Proof of Lemma \[lemma:priv-best-response1\]. {#proof:lemma:priv-best-response1} For every $i \in {\cal N}$, $h_i \in {\cal H}_i$, $a_i \in BR[\vec{\sigma}_{-i}^*|\vec{\mu}^*,h_i]$, and $\vec{p}_i \in {\cal P}_i$ such that $a_i[\vec{p}_i] > 0$, it is true that for every $j \in {\cal N}_i$ we have $p_i[j] \in \{0,p_i[j|h_i]\}$. Suppose then that there exist $i \in {\cal N}$, $h_i \in {\cal H}_i$, $a_i^1 \in BR[\vec{\sigma}_{-i}^*|\vec{\mu}^*,h_i]$, and $\vec{p}^1_i \in {\cal P}_i$ such that $a_i^1[\vec{p}^1_i]>0$ and there exists $j \in {\cal N}_i$ such that $p_i^1[j] \notin \{0,p_i[j|h_i]\}$. Fix any $h$ such that $h_i \in h$ and define an alternative $a_i^2 \in {\cal A}_i$: - $\vec{a}^1 = (a_i^1,\vec{p}_{-i}^*)$ and $\vec{a}^2 = (a_i^2,\vec{p}_{-i}^*)$, where $\vec{p}^* = \vec{\sigma}^*[h]$. - Define $\vec{p}_i^2 \in {\cal P}_i$ such that for every $j \in {\cal N}_i$, if $p_i^1[j] \geq p_i[j|h_i]$, then $p_i^2[j] = p_i[j|h_i]$, else, $p_i^2[j]=0$. - Set $a_i^2[\vec{p}_i^2] = a_i^1[\vec{p}_i^1] + a_i^1[\vec{p}_i^2]$ and $a_i^2[\vec{p}_i^1] = 0$. - For every $\vec{p}_i'' \in {\cal P}_i \setminus \{\vec{p}_i^1,\vec{p}_i^2\}$, set $a_i^2[\vec{p}_i''] = a_i^1[\vec{p}_i'']$. - Define $\sigma_i^1 = \sigma_i^*[h_i|\vec{p}_i^1]$ and $\sigma_i^2 = \sigma_i^*[h_i|\vec{p}_i^2]$. - Set $\vec{\sigma}^1 = (\sigma_i^1,\vec{\sigma}_{-i}^*)$ and $\vec{\sigma}^2 = (\sigma_i^2,\vec{\sigma}_{-i}^*)$. Notice that for any $j \in {\cal N}_i$, $p_i^1[j] \geq p_i^2[j]$ and $p_i^1[j] \geq p_i[j|h_i]$ iff $p_i^2[j] \geq p_i[j|h_i]$. Thus, by Definition \[def:pubsig\], for every $s \in {\cal S}$, $$\label{eq:pvbr-1} \begin{array}{l} pr_i[s|a_i^1,h] = pr_i[s|a_i^2,h].\\ pr[s|\vec{a}^1,h] = pr[s|\vec{a}^2,h]. \end{array}$$ Moreover, for some $j \in {\cal N}_i$, $p_i^1[j|h_i] > p_i^2[j|h_i]$, thus, it is true that $$\label{eq:pvbr-2} u_i[\vec{a}^1] < u_i[\vec{a}^2].$$ Recall that $$\pi_i[\vec{\sigma}^1|h] = u_i[\vec{a}^1] + \omega_i\sum_{s \in {\cal S}} \pi_i[\vec{\sigma}^1|(h,s)]pr[s|\vec{a}^1,h],$$ $$\pi_i[\vec{\sigma}^2|h] = u_i[\vec{a}^2] + \omega_i\sum_{s \in {\cal S}} \pi_i[\vec{\sigma}^2|(h,s)]pr[s|\vec{a}^2,h].$$ By \[eq:pvbr-1\] and the definition of $\vec{\sigma}^1$ and $\vec{\sigma}^2$, $$\sum_{s \in {\cal S}} \pi_i[\vec{\sigma}|(h,s)]pr[s|\vec{a}^1,h] = \sum_{s \in {\cal S}} \pi_i[\vec{\sigma}|(h,s)]pr[s|\vec{a}^2,h].$$ It follows from \[eq:pvbr-2\] that, for every $h \in {\cal H}$ such that $h_i \in h$, $$\pi_i[\vec{\sigma}^1|h] < \pi_i[\vec{\sigma}^2|h].$$ Consequently, $$\pi_i[\vec{\sigma}^1|h_i] < \pi_i[\vec{\sigma}^2|h_i].$$ This is a contradiction, since $a_i^1 \in BR[\vec{\sigma}_{-i}^*|h_i]$ by assumption, concluding the proof. ### Proof of Lemma \[lemma:priv-best-response2\]. {#proof:lemma:priv-best-response2} For every $i \in {\cal N}$ and $h_i \in {\cal H}_i$, there exists $a_i \in BR[\vec{\sigma}_{-i}^*|\vec{\mu}^*,h_i]$ and $\vec{p}_i \in {\cal P}_i$ such that $a_i[\vec{p}_i] = 1$. Fix $i$ and $h_i$. If $BR[\vec{\sigma}_{-i}^*|\vec{\mu}^*,h_i]$ only contains pure strategies for the stage game, since $BR[\vec{\sigma}_{-i}^*|\vec{\mu}^*,h_i]$ is not empty, the result follows. Suppose then that there exists a mixed strategy $a_i^1 \in BR[\vec{\sigma}_{-i}^*|\vec{\mu}^*,h_i]$. We know from Lemma \[lemma:priv-best-response1\] that every such $a_i$ attributes positive probability to one of two probabilities in $\{0,p_i[j|h_i]\}$, for every $j \in {\cal N}_i$. Denote by ${\cal P}_i^*[h_i]$ the finite set of profiles of probabilities that fulfill the condition of Lemma \[lemma:best-response1\], i.e., for every $\vec{p}_i \in {\cal P}_i^*[h_i]$ and $j \in {\cal N}_i$, $p_i[j] \in \{0,p_i[j|h_i]\}$. Define ${\cal P}^*[h]$ similarly for any $h \in {\cal H}$. For any $h \in {\cal H}$ such that $h_i \in h$, we can write $$\label{eq:priv-br2-0} \pi_i[\vec{\sigma}^1|h] = \sum_{\vec{p}_i \in {\cal P}^*_i[h_i]} (u_i[\vec{p}] + \omega_i \pi_i[\vec{\sigma}^1|(h,{\mbox{sig}}[\vec{p}|h])])a_i^1[\vec{p}_i],$$ where $\vec{p} = (\vec{p}_i,\vec{p}_{-i}^*)$ and $\vec{p}^* = \vec{\sigma}^*[h]$. For any $\vec{p}_i^1 \in {\cal P}_i^*[h_i]$ such that $a_i[\vec{p}_i^1] >0$, let $\sigma_i^1 = \sigma_i^*[h_i|a_i^1]$, $\vec{\sigma}^1 = (\sigma_i^1,\vec{\sigma}^*_{-i})$, $\sigma_i' = \sigma_i^*[h_i|\vec{p}_i^1]$, and $\vec{\sigma}' = (\sigma_i',\vec{\sigma}^*_{-i})$. There are three possibilities: 1. $\pi_i[\vec{\sigma}^1|\vec{\mu}^*,h_i] = \pi_i[\vec{\sigma}'|\vec{\mu}^*,h_i]$. 2. $\pi_i[\vec{\sigma}^1|\vec{\mu}^*,h_i] < \pi_i[\vec{\sigma}'|\vec{\mu}^*,h_i]$. 3. $\pi_i[\vec{\sigma}^1|\vec{\mu}^*,h_i] > \pi_i[\vec{\sigma}'|\vec{\mu}^*,h_i]$. In possibility 1, it is true that there exists $a_i' \in BR[\vec{\sigma}_{-i}^*|\vec{\mu}^*,h_i]$ such that $a_i'[\vec{p}_i^1] = 1$ and the result follows. Possibility 2 contradicts the assumption that $a_i^1 \in BR[\vec{\sigma}_{-i}^*|\vec{\mu}^*,h_i]$. Finally, consider that possibility 3 is true. Recall that $a_i^1$ being mixed implies $a_i^1[\vec{p}_i^1] < 1$. Thus, there must exist $\vec{p}_i^2 \in {\cal P}_i^*[h_i]$, $\sigma_i''=\sigma_i^*[h_i|\vec{p}_i^2]$, and $\vec{\sigma}'' = (\sigma_i',\vec{\sigma}^*_{-i})$, such that $a_i^1[\vec{p}_i^2] > 0$ and $$\label{eq:priv-br2-1} \pi_i[\vec{\sigma}'|\vec{\mu}^*,h_i] < \pi_i[\vec{\sigma}''|\vec{\mu}^*,h_i].$$ Here, we can define $a_i^2 \in {\cal A}_i$ such that: - $a_i^2[\vec{p}_i^2]=a_i^1[\vec{p}_i^1] + a_i^1[\vec{p}_i^2]$; - $a_i^2[\vec{p}_i^1] = 0$. - For every $\vec{p}_i'' \in {\cal P}_i^*[h_i] \setminus \{\vec{p}_i^1,\vec{p}_i^2\}$, $a_i^2[\vec{p}_i''] = a_i^1[\vec{p}_i'']$. Now, let $\sigma_i^2 = \sigma_i^*[h_i|a_i^2]$, and $\vec{\sigma}^2 = (\sigma_i^2,\vec{\sigma}^*_{-i})$. By \[eq:priv-br2-0\], it holds that for every $h \in {\cal H}$ such that $h_i \in h$: $$\pi_i[\vec{\sigma}^1|h] = l_1 + \pi_i[\vec{\sigma}'|h]a_i^1[\vec{p}_i^1] + \pi_i[\vec{\sigma}''|h]a_i^1[\vec{p}_i^2],$$ $$\pi_i[\vec{\sigma}^2|h] = l_2 + \pi_i[\vec{\sigma}''|h]a_i^2[\vec{p}_i^2],$$ where $$l_1 = \sum_{\vec{p}'' \in {\cal P}^*[h] \setminus \{\vec{p}_i^1,\vec{p}_i^2\}} (u_i[\vec{p}''] + \omega_i \pi_i[\vec{\sigma}^1|h,s''])a_i^1[\vec{p}_i''],$$ $$l_2 = \sum_{\vec{p}'' \in {\cal P}^*[h] \setminus \{\vec{p}_i^1,\vec{p}_i^2\}} (u_i[\vec{p}''] + \omega_i \pi_i[\vec{\sigma}^2|h,s''])a_i^2[\vec{p}_i''],$$ and $s'' = {\mbox{sig}}[\vec{p}''|h]$. By the definition of $a_i^2$, we have that $l_1 = l_2$. It follows that $$\begin{array}{ll} \pi_i[\vec{\sigma}^1|h] - \pi_i[\vec{\sigma}^2|h] & = \pi_i[\vec{\sigma}'|h]a_i^1[\vec{p}_i^1] + \pi_i[\vec{\sigma}''|h]a_i^1[\vec{p}_i^2] - \pi_i[\vec{\sigma}''|h](a_i^1[\vec{p}_i^2] + a_i^1[\vec{p}_i^1]) \\ & = (\pi_i[\vec{\sigma}'|h] - \pi_i[\vec{\sigma}''|h])a_i^1[\vec{p}_i^1]. \end{array}$$ Consequently, by \[eq:priv-br2-1\], $$\begin{array}{ll} \pi_i[\vec{\sigma}^1|\vec{\mu}^*,h_i] - \pi_i[\vec{\sigma}^2|\vec{\mu}^*,h_i] &= \sum_{h \in {\cal H}}\mu_i^*[h|h_i](\pi_i[\vec{\sigma}^1|h] - \pi_i[\vec{\sigma}^2|h])\\ &=\sum_{h \in {\cal H}}\mu_i^*[h|h_i](\pi_i[\vec{\sigma}'|h] - \pi_i[\vec{\sigma}''|h])a_i^1[\vec{p}_i^1]\\ &=(\pi_i[\vec{\sigma}'|\vec{\mu}^*,h_i] - \pi_i[\vec{\sigma}''|\vec{\mu}^*,h_i])a_i^1[\vec{p}_i^1]\\ &<0. \end{array}$$ Thus, $$\pi_i[\vec{\sigma}^1|\vec{\mu}^*,h_i] < \pi_i[\vec{\sigma}^2|\vec{\mu}^*,h_i],$$ contradicting the assumption that $a_i^1 \in BR[\vec{\sigma}_{-i}^*|\vec{\mu}^*,h_i]$. This concludes the proof. ### Proof of Lemma \[lemma:priv-best-response\]. {#proof:lemma:priv-best-response} For every $i \in {\cal N}$ and $h_i \in {\cal H}_i$, there exists $\vec{p}_i \in {\cal P}_i$ and a pure strategy $\sigma_i=\sigma_i^*[h_i|\vec{p}_i]$ such that: 1. For every $j \in {\cal N}_i$, $p_i[j] \in \{0,p_{i}[j|h_i]\}$. 2. For every $a_i \in {\cal A}_i$, $\pi_i[\sigma_i,\vec{\sigma}_{-i}^*|\vec{\mu}^*,h_i] \geq \pi_i[\sigma_i',\vec{\sigma}_{-i}^*|\vec{\mu}^*,h_i]$, where $\sigma_i' = \sigma_i^*[h_i|a_i]$. Consider any $i \in {\cal N}$ and $h_i \in {\cal H}$. From Lemma \[lemma:priv-best-response2\], it follows that there exists $a_i \in BR[\vec{\sigma}_{-i}^*|\vec{\mu}^*,h_i]$ and $\vec{p}_i \in {\cal P}_i$ such that $a_i[\vec{p}_i] = 1$. By Lemma \[lemma:priv-best-response1\], every such $a_i$ and $\vec{p}_i$ such that $a_i[\vec{p}_i]=1$ fulfill Condition $1$. Condition $2$ follows from the definition of $BR[\vec{\sigma}_{-i}^*|\vec{\mu}^*,h_i]$. ### Proof of Lemma \[lemma:priv-drop-suff\]. {#proof:lemma:priv-drop-suff} If the PDC Condition is fulfilled and $(\vec{\sigma}^*,\vec{\mu}^*)$ is Preconsistent, then $(\vec{\sigma}^*,\vec{\mu}^*)$ is Sequentially Rational. Assume that Inequality \[eq:priv-drop\] holds for every player $i$, history $h_i$ and $D \subseteq {\cal N}_{i}[h_i]$. In particular, these assumptions imply that, for each $\vec{p}_i \in {\cal P}_i$ such that $p_i[j] \in \{0,p_i[j|h_i]\}$ for every $j \in {\cal N}_i$, we have $$\label{eq:priv-gcs} \pi_i[\vec{\sigma}^*|h_i] \geq \pi_i[\sigma_i,\vec{\sigma}^*_{-i}|h_i],$$ where $\sigma_i = \sigma_i^*[h_i|\vec{p}_i]$. By Lemma \[lemma:priv-best-response\], there exists one such $\vec{p}_i$ such that $\sigma_i$ is a local best response. Consequently, by \[eq:priv-gcs\], for every $a_i \in {\cal A}_i$ and $\sigma_i'=\sigma_i^*[h|a_i]$, $$\pi_i[\vec{\sigma}^*|h_i] \geq \pi_i[\sigma_i',\vec{\sigma}^*_{-i}|h_i].$$ By Property \[prop:priv-one-dev\], $(\vec{\sigma}^*,\vec{\mu}^*)$ is Sequentially Rational. ### Proof of Lemma \[lemma:paths\]. {#proof:lemma:paths} If the assessment $(\vec{\sigma}^*,\vec{\mu}^*)$ is Preconsistent and Sequentially Rational and $G$ is redundant, then for every $i \in {\cal N}$ and $j \in {\cal N}_i$, there exists $k \in {\cal N} \setminus \{i\}$, $x \in {\mbox{PS}}[s,i]$, and $x' \in {\mbox{PS}}[j,k]$, such that $k \in x$ and $i \notin x'$. Suppose that there exists a player $i \in {\cal N}$, and a neighbor $j \in {\cal N}_i$ such that for every $k \in {\cal N} \setminus \{i\}$, $x' \in {\mbox{PS}}[j,k]$, and $x \in {\mbox{PS}}[s,i]$, we have $k \notin x$ or $i \in x'$. Define $D_j \subseteq {\cal N}$ and $D \subseteq {\cal N}_i$ as: $$\label{eq:pths1} \begin{array}{l} D_j = \{k \in {\cal N} \setminus \{i\} | \exists_{x \in {\mbox{PS}}[j,k]} i \notin x\}.\\ D = \{j \in {\cal N}_i | \forall_{k \in D_j}\forall_{x \in {\mbox{PS}}[s,i]} \forall_{x' \in {\mbox{PS}}[j,k]} k \notin x \vee i \in x'\}. \end{array}$$ Let ${\mbox{RS}}_D = \cup_{j \in D} D_j$. By our assumptions, $D$ is not empty. Define $\sigma_i' = \sigma_i^*[h_i|\vec{p}_i']$ for every $h_i \in {\cal H}$ such that: - For every $j \in D$, $p_i'[j] = 0$. - For every $j \in {\cal N}_i \setminus D$, $p_i'[j] = p_i[j|\emptyset]$. Let $\vec{\sigma}' = (\sigma_i',\vec{\sigma}_{-i}^*)$. Notice that, for every $k \in {\cal N} \setminus \{i\}$, $$\label{eq:pths4} \begin{array}{ll} {\mbox{CD}}_i[\vec{p}'|h] = D.\\ {\mbox{CD}}_k[\vec{p}'|h] = \emptyset. \end{array}$$ For every $j \in D$ and $k \in {\cal N} \setminus ({\mbox{RS}}_D \cup \{i\})$, we have that ${d}_k[i,j]= \infty$. Therefore, by Lemma \[lemma:priv-corr-1\] and by \[eq:pths4\], for every $l \in {\cal N}_k$, $r\geq0$, $$\label{eq:pths2} {\mbox{DS}}_k[l|h_{k,r}'] = {\mbox{DS}}_k[l|h_{k,r}^*],$$ where $h_{k,r}' \in {\mbox{hist}}[\emptyset,r|\vec{\sigma}']$ and $h_{k,r}^* \in {\mbox{hist}}[\emptyset,r|\vec{\sigma}']$. By Definition \[def:priv-thr\] and \[eq:pths2\], we have that for every $r \geq 0$, $$\label{eq:pths5} \begin{array}{ll} p_k[l|h_{k,r}'] &= p_k[l|h_{l,r}^*]. \end{array}$$ Consequently, by \[eq:pths1\] and \[eq:pths5\], for every $r >0$, there exist $\vec{p}^*= \vec{\sigma}^*[{\mbox{hist}}[\emptyset,r-1|\vec{\sigma}^*]]$ and $\vec{p}' = \vec{\sigma}'[{\mbox{hist}}[\emptyset,r-1|\vec{\sigma}']]$ such that for every $x \in {\mbox{PS}}[s,i]$ and $k \in x \setminus \{i\}$ we have $\vec{p}_k^* = \vec{p}_k'$. It follows from Lemma \[lemma:bottleneck-impact\] of Appendix \[sec:epidemic\] that for every $r \geq 0$ $$\label{eq:ph-2} q_i[\emptyset,r|\vec{\sigma}^*] = q_i[\emptyset,r|\vec{\sigma}'].$$ By the definition of $\vec{p}_i'$, for every $r \geq 0$, $$\label{eq:pths8} \bar{p}_i[\emptyset,r|\vec{\sigma}'] < \bar{p}_i[\emptyset,r|\vec{\sigma}^*].$$ By the definition of $u_i[h,r|\vec{\sigma}]$, from \[eq:ph-2\] and \[eq:pths8\], we have for every $r\geq 0$ $$u_i[\emptyset,r|\vec{\sigma}^*] < u_i[\emptyset,r|\vec{\sigma}'].$$ This implies that $$\sum_{r=0}^{\infty}\omega_i^r (u_i[\emptyset,r|\vec{\sigma}^*] - u_i[\emptyset,r|\vec{\sigma}']) < 0.$$ Since $h=\emptyset$ is the only history such that $\mu_i[h|\emptyset]=1$, by Theorem \[theorem:priv-drop\], $(\vec{\sigma}^*,\vec{\mu}^*)$ cannot be Sequentially Rational, which is a contradiction. Redundancy may Reduce Effectiveness {#sec:proof:coord} ----------------------------------- ### Proof of Theorem \[theorem:problem\]. {#proof:theorem:problem} If $G$ is redundant and there exist $i \in {\cal N}$, $j \in {\cal N}_i$, and $k \in {\cal N}_i^{-1}$ such that for every $x \in {\mbox{PS}}[j,k]$ we have $i \in x$, then Equality \[eq:problem\] holds. Assume that there exist $i \in {\cal N}$, $j \in {\cal N}_i$, and $k \in {\cal N}_i^{-1}$ such that for every $x \in {\mbox{PS}}[j,k]$ we have $i \in x$. This implies by Definition \[def:privsig\] that $$\label{eq:noally} {d}_k[i,j] = \infty.$$ Define $\sigma_i' = \sigma_i^*[h_i|\vec{p}_i']$ for every $h_i \in {\cal H}_i$, such that $p_i'[j] = 0$ and $p_i'[l] = p_i[l|h_i]$ for every $l \in {\cal N}_i \setminus \{j\}$ and let $\vec{\sigma}' = (\sigma_i',\vec{\sigma}_{-i}^*)$. Notice that $$\bar{p}_i[\emptyset,r|\vec{\sigma}'] = \bar{p}_i[\emptyset,r|\vec{\sigma}^*] - p_i[j|\emptyset].$$ By Theorem \[theorem:priv-drop\], if $(\vec{\sigma}^*,\vec{\mu}^*)$ is Sequentially Rational, then, for every $r \geq 0$ and the empty history: $$\label{eq:prb-0} \begin{array}{ll} \sum_{r=0}^{\infty} \omega_i (u_i[\emptyset,r|\vec{\sigma}^*] - u_i[\emptyset,r|\vec{\sigma}']) & \geq 0\\ &\\ \sum_{r=0}^{\infty} \omega_i ((1-q_i[\emptyset,r|\vec{\sigma}^*])(\beta_i - \gamma_i \bar{p}_i[\emptyset,r|\vec{\sigma}^*]) -&\\ (1-q_i[\emptyset,r|\vec{\sigma}'])(\beta_i - \gamma_i \bar{p}_i[\emptyset,r|\vec{\sigma}'])) & \geq 0\\ &\\ \sum_{r=0}^{\infty} \omega_i ((q_i[\emptyset,r|\vec{\sigma}']-q_i[\emptyset,r|\vec{\sigma}^*])(\beta_i - \gamma_i \bar{p}_i[\emptyset,r|\vec{\sigma}^*]) \\ -(1-q_i[\emptyset,r|\vec{\sigma}']) \gamma_i p_i[j|\emptyset] & \geq 0. \end{array}$$ By Lemma \[lemma:prob-1\], since $G$ is connected from $s$ and $q_i$ is continuous in \[0,1\], for every $r \geq 0$: $$\label{eq:prb-1} \lim_{\vec{\sigma}^* \to \vec{1}} q_i[\emptyset,r|\vec{\sigma}^*]= q_i[\vec{1}] = 0.$$ Now, let $\vec{p}^* = \vec{\sigma}^*[\emptyset]$, $\vec{p}' = (\vec{p}_i',\vec{p}_{-i}^*)$, $h_{k,r}' \in {\mbox{hist}}[\emptyset,r|\vec{\sigma}']$, and $h_{k,r}^* \in {\mbox{hist}}[\emptyset,r|\vec{\sigma}^*]$. We have that for every $l \in {\cal N} \setminus \{i\}$: $$\label{eq:prb-2} {\mbox{CD}}_l[\vec{p}'|\emptyset] = \emptyset,$$ and $$\label{eq:prb-3} {\mbox{CD}}_i[\vec{p}'|\emptyset] = \{j\}.$$ It follows immediately by Definition \[def:priv-thr\] and Lemma \[lemma:priv-corr-1\] that, for every $k \in {\cal N}$ and $l \in {\cal N}_k$ such that ${d}_k[i,j]=\infty$, and $r\geq0$, $$\label{eq:prb-4} \begin{array}{l} {\mbox{DS}}_k[l|h_{k,r}'] = {\mbox{DS}}_k[l|h_{k,r}^*].\\ p_{k}[l|h_{k,r}'] = p_{k}[l|h_{k,r}^*]. \end{array}$$ For any $r \geq 0$, let $$\vec{p}'' = \lim_{\vec{\sigma}^* \to \vec{1}} \vec{\sigma}'[h_{r}'].$$ Since $G$ is redundant, there is a path $x \in {\mbox{PS}}[s,k]$ such that $i \notin x$. Furthermore, by \[eq:noally\], for every $l \in x \setminus \{i\}$, $d_l[i,j] = \infty$. By \[eq:prb-4\], this implies that $p_l''[a] = 1$ for every $a \in {\cal N}_l$. Therefore, by Lemma \[lemma:prob-1\] and, since $G$ is connected from $s$ and $q_i$ is continuous in $[0,1]$, for every $r \geq 0$, $$\label{eq:prb-6} \lim_{\vec{\sigma}^* \to \vec{1}} q_i[\emptyset,r|\vec{\sigma}']= q_i[\vec{p}'']= 0.$$ By \[eq:prb-0\], \[eq:prb-1\], and \[eq:prb-6\], $$\label{eq:prb-7} \begin{array}{ll} \lim_{\vec{\sigma}^* \to \vec{1}} \sum_{r=0}^{\infty} \omega_i (u_i[\emptyset,r|\vec{\sigma}^*] - u_i[\emptyset,r|\vec{\sigma}'])) & =\\ &\\ \lim_{\vec{\sigma}^* \to \vec{1}} \sum_{r=0}^{\infty} \omega_i ((q_i[\emptyset,r|\vec{\sigma}']-q_i[\emptyset,r|\vec{\sigma}^*])(\beta_i - \gamma_i \bar{p}_i[\emptyset,r|\vec{\sigma}^*]) &\\ - (1-q_i[\emptyset,r|\vec{\sigma}']) \gamma_i p_i[j|\emptyset] ) &=\\ &\\ \lim_{\vec{\sigma}^* \to \vec{1}} - (1-q_i[\emptyset,r|\vec{\sigma}']) \gamma_i p_i[j|\emptyset] ) &=\\ &\\ -\gamma_i & < 0. \end{array}$$ Therefore, in the limit, the PDC Condition is never fulfilled for any values of $\beta_i$, $\gamma_i$, and $\omega_i \in (0,1)$, which implies by Theorem \[theorem:priv-drop\] that $(\vec{\sigma}^*,\vec{\mu}^*)$ is not Sequentially Rational for an arbitrarily large reliability and $$\lim_{\vec{\sigma}^* \to \vec{1}} \psi[\vec{\sigma}^*|\vec{\mu}^*] = \emptyset.$$ Coordination is Desirable ------------------------- ### Proof of Theorem \[theorem:problem-coord\]. {#proof:theorem:problem-coord} If the graph is redundant and $\vec{\sigma}^*$ does not enforce coordination, then there is a definition of $\vec{\sigma}^*$ such that: $$\lim_{\vec{\sigma}^* \to \vec{1}} \psi[\vec{\sigma}^*|\vec{\mu}^*] = \emptyset.$$ In the aforementioned circumstances, define $\vec{\sigma}^*$ such that for every $i \in {\cal N}$, $j \in {\cal N}_i$, and $h_i \in {\cal H}_i$, $$\label{eq:coo-1} p_i[j|h_i]>0 \equiv p_i[j|h_i] = p_i[j|\emptyset].$$ By the assumption that $\vec{\sigma}^*$ does not enforce coordination, there exists $i \in {\cal N}$ and $j \in {\cal N}_i$ such that for every $r>0$ there is $k \in {\cal N}_i^{-1}$ for which $$\label{eq:coo-2} r \leq {d}_k[i,j] \vee r \geq {d}_k[i,j] + {\tau}[i,j|k,i] + 1.$$ Fix $r$. Let $\sigma_i' = \sigma_i^*[\emptyset|\vec{p}_i']$ and $\vec{\sigma}' = (\sigma_i',\vec{\sigma}_{-i}^*)$, such that - $p_i'[j] = 0$. - For every $k \in {\cal N}_i \setminus \{j\}$, $p_i'[k] = p_i[k|\emptyset]$. For $\vec{p}' = \vec{\sigma}'[\emptyset]$, we have $$\label{eq:coo-3} {\mbox{CD}}_i[\vec{p}'|\emptyset] = \{j\},$$ and for every $k \in {\cal N} \setminus \{i\}$ $$\label{eq:coo-4} {\mbox{CD}}_k[\vec{p}'|\emptyset] = \emptyset.$$ By Lemma \[lemma:priv-corr-1\] and Definition \[def:priv-thr\], and by \[eq:coo-3\], and \[eq:coo-4\], we have for every $a \in {\cal N}$ and $b \in {\cal N}_a$, $h_{a,r}' \in {\mbox{hist}}[\emptyset,r|\vec{\sigma}']$, and $h_{a,r}^* \in {\mbox{hist}}[\emptyset,r|\vec{\sigma}^*]$: $$\label{eq:coo-5} \begin{array}{ll} {\mbox{DS}}_a[b|h_{a,r}'] =& {\mbox{DS}}_a[b|h_{a,r}^*] \cup \{(k_1,k_2,r-1 - {d}_a[k_1,k_2]+v[k_1,k_2]) | k_1,k_2 \in {\cal N} \land \\ & k_2 \in {\mbox{CD}}_{k_1}[\vec{p}'|\emptyset] \land r \in \{{d}_a[k_1,k_2] +1 \ldots {d}_a[k_1,k_2] + {\tau}[k_1,k_2|a,b]-v[k_1,k_2]\}\land\\ & v[k_1,k_2] = \min[{d}_a[k_1,k_2] - {d}_b[k_1,k_2],0]\}\\ &\\ &= \emptyset \cup \{(i,j,r-1 - {d}_a[i,j]+v[i,j]) | v[i,j] = \min[{d}_a[i,j] - {d}_b[i,j],0]\}. \end{array}$$ For ${\mbox{DS}}_k[i|h_{k,r}']$, we have $v[i,j] = 0$, since ${d}_i[i,j] = 0$. Therefore, by \[eq:coo-5\], $$\label{eq:coo-6} p_a[b|h_{a,r}] = p_a[b|\emptyset].$$ Also, for every $a \in {\cal N} \setminus ({\cal N}_i \cup {\cal N}_i^{-1} \cup\{k,i\})$ and $b \in {\cal N}_a$, by Definition \[def:priv-thr\] and the definition of $\vec{\sigma}^*$ in this context, $a$ does not react to a defection of $i$ from $j$, which implies that: $$\label{eq:coo-7} \begin{array}{l} {\mbox{DS}}_a[b|h_{a,r}'] =\{(i,j,r-1 - {d}_a[i,j]+v[i,j]) | v[i,j] = \min[{d}_a[i,j] - {d}_b[i,j],0]\}.\\ p_a[b|h_{a,r}'] = p_a[b|h_{a,r}^*] = p_a[b|\emptyset]. \end{array}$$ Since the graph is redundant, there exists $x \in {\mbox{PS}}[s,k]$ such that, for every $a \in ({\cal N}_i \cup {\cal N}_i^{-1} \cup \{i\}) \setminus \{k\}$, $a \notin x$. Thus, by \[eq:coo-6\] and \[eq:coo-7\], for $$\vec{p}' = \lim_{\vec{\sigma}^* \to \vec{1}} \vec{\sigma}'[{\mbox{hist}}[\emptyset,r|\vec{\sigma}']],$$ there exists a path $x \in {\mbox{PS}}[s,i]$ such that for every $a \in x \setminus \{i\}$ and $b \in {\cal N}_a$ we have $$p_a'[b] = 1.$$ By Lemma \[lemma:prob-1\], given that $q_i$ is continuous in $[0,1]$, for every $r>0$, $$\label{eq:coo-8} \begin{array}{ll} \lim_{\vec{\sigma}^* \to \vec{1}} (q_i[\emptyset,r|\vec{\sigma}^*] - q_i[\emptyset,r|\vec{\sigma}']) &= 0.\\ &\\ \lim_{\vec{\sigma}^* \to \vec{1}} (u_i[\emptyset,r|\vec{\sigma}^*] - u_i[\emptyset,r|\vec{\sigma}']) &=\\ \lim_{\vec{\sigma}^* \to \vec{1}} (1-q_i[\emptyset,r|\vec{\sigma}^*])(\beta_i - \gamma_i \bar{p}_i[\emptyset,r|\vec{\sigma}^*]) - (1-q_i[\emptyset,r|\vec{\sigma}'])(\beta_i - (\gamma_i \bar{p}_i[\emptyset,r|\vec{\sigma}^*] + p_i[j|h_{ir}^*])) &=\\ \lim_{\vec{\sigma}^* \to \vec{1}} -\gamma_i p_i[j|\emptyset] &<0. \end{array}$$ Therefore, in the limit, the PDC Condition is never fulfilled for any values of $\beta_i$ and $\gamma_i$, which implies by Theorem \[theorem:priv-drop\] that $(\vec{\sigma}^*,\vec{\mu}^*)$ is never Sequentially Rational and: $$\lim_{\vec{\sigma}^* \to \vec{1}} \psi[\vec{\sigma}^*|\vec{\mu}^*] = \emptyset.$$ Impact of Delay {#sec:proof:priv-indir} --------------- ### Auxiliary Lemmas. Lemma \[lemma:aux:impdel-1\] shows that the utility obtained by $i$ during the ${\tau}$ stages that follow stage ${\bar{d}}_i$ after any defection by $i$ is null. This is because during that period $i$ is necessarily punished by every in-neighbor. \[lemma:aux:impdel-1\] For every $i \in {\cal N}$, $h \in {\cal H}$ and $h_i \in h$, $D \subseteq {\cal N}_i[h_i]$, and $r \in \{{\bar{d}}_i +1 \ldots {\bar{d}}_i + {\tau}\}$, $u_i[h,r|\vec{\sigma}'] = 0$, where $\vec{\sigma}' = (\sigma_i^*[h_i|\vec{p}_i'],\vec{\sigma}_{-i}^*)$ and $i$ drops every node from $D$ in $\vec{p}_i'$. By Definition \[def:overlap\], since ${d}_i[i,j] = 0$ for every $i$, then, for every $k \in {\cal N}_i^{-1}$, $\max[{d}_k[i,j],{d}_i[i,j]] = {d}_k[i,j]$ and: $$\label{eq:aid1-1} \begin{array}{l} {\tau}[i,j|k,i] \leq {\bar{d}}_i.\\ {\tau}[i,j|k,i] = {\bar{d}}_i - {d}_k[i,j] + {\tau}. \end{array}$$ Notice that for $\vec{p}' = \vec{\sigma}'[h]$ $$\label{eq:aid1-2} {\mbox{CD}}_i[\vec{p}'|h] = D,$$ and for every $j \in {\cal N} \setminus \{i\}$ $$\label{eq:aid1-3} {\mbox{CD}}_j[\vec{p}'|h] = \emptyset.$$ By Lemma \[lemma:priv-corr-1\] and by \[eq:aid1-1\], for every $k \in {\cal N}_i^{-1}$ and $r \in \{{\bar{d}}_i+1 \ldots {\bar{d}}_i + {\tau}\}$: $$\label{eq:aid1-4} \begin{array}{ll} {\mbox{DS}}_k[i|h_{k,r}'] =& {\mbox{DS}}_k[i|h_{k,r}^*] \cup \{(k_1,k_2,r-1 - {d}_k[k_1,k_2]+v[k_1,k_2]) | k_1,k_2 \in {\cal N} \land \\ & k_2 \in {\mbox{CD}}_{k_1}[\vec{p}'|h] \land r \in \{{d}_k[k_1,k_2] + 1\ldots {d}_i[k_1,k_2] + {\tau}[k_1,k_2|k,i]-v[k_1,k_2]\}\land\\ & v[k_1,k_2] = \min[{d}_k[k_1,k_2] - {d}_i[k_1,k_2],0]\},\\ &\\ & ={\mbox{DS}}_k[i|h_{k,r}^*] \cup \{(i,j,r-1 - {d}_k[i,j]+v[i,j]) | j \in D \land \\ & r \in\{{d}_k[i,j]+1 \ldots {d}_k[i,j] + {\tau}[i,j|k,i]-v[i,j]\}\land\\ & v[i,j] = \min[{d}_k[i,j] - {d}_i[i,j],0]\}\\ &\\ & ={\mbox{DS}}_k[i|h_{k,r}^*] \cup \{(i,j,r-1 - {d}_k[i,j]+v[i,j]) | j \in D \land \\ &r \in\{{d}_k[i,j]+1 \ldots {d}_k[i,j] + {\tau}[i,j|k,i] - v[i,j]\} \land v[i,j] = 0\}\\ &\\ &\supseteq {\mbox{DS}}_k[i|h_{k,r}^*] \cup \{(i,j,r-1 - {d}_k[i,j]) | j \in D \land r \in\{{\bar{d}}_i+1 \ldots {\bar{d}}_i + {\tau}\}\}\\ &\\ & = {\mbox{DS}}_k[i|h_{k,r}^*] \cup \{(i,j,r-1 - {d}_k[i,j]+v[i,j])\} \end{array}$$ where $h_{k,r}^* \in {\mbox{hist}}[h,r|\vec{\sigma}^*]$ and $h_{k,r}' \in {\mbox{hist}}[h,r| \vec{\sigma}']$. By \[eq:aid1-4\] and Definition \[def:priv-thr\], it follows that, for every $r \in \{{\bar{d}}_i+1 \ldots {\bar{d}}_i + {\tau}\}$, $$p_k[i|h_{k,r}'] = 0,$$ which by Lemma \[lemma:noneib\] leads to $$\label{eq:aid1-5} \begin{array}{l} q_i[h,r|\vec{\sigma}'] = 1.\\ u_i[h,r|\vec{\sigma}'] = 0. \end{array}$$ This concludes the proof. Lemma \[lemma:aux:impdel-2\] proves that all punishments of a node $i$ are concluded after stage ${\bar{d}}_i + {\tau}$ that follows any defection of $i$. \[lemma:aux:impdel-2\] For every $i \in {\cal N}$, $h \in {\cal H}$ and $h_i \in h$, $D \subseteq {\cal N}_i[h_i]$, and $r > {\bar{d}}_i + {\tau}$, $$u_i[h,r|\vec{\sigma}'] = u_i[h,r|\vec{\sigma}^*],$$ where $\vec{\sigma}' = (\sigma_i^*[h_i|\vec{p}_i'],\vec{\sigma}_{-i}^*)$ and $i$ drops every node from $D$ in $\vec{p}_i'$. Notice that for $\vec{p}' = \vec{\sigma}'[h]$ and $j \in {\cal N} \setminus \{i\}$ $$\label{eq:aid2-1} \begin{array}{l} {\mbox{CD}}_i[\vec{p}'|h] = D.\\ {\mbox{CD}}_j[\vec{p}'|h] = \emptyset. \end{array}$$ By Lemma \[lemma:priv-corr-1\] and by \[eq:aid2-1\], for every $k \in {\cal N}$, $l \in {\cal N}_k$, and $r > {\bar{d}}_i + {\tau}$, $$\label{eq:aid2-2} \begin{array}{ll} {\mbox{DS}}_k[l|h_{k,r}'] =& {\mbox{DS}}_k[l|h_{k,r}^*] \cup \{(k_1,k_2,r-1 - {d}_k[k_1,k_2]+v[k_1,k_2]) | k_1,k_2 \in {\cal N} \land \\ & k_2 \in {\mbox{CD}}_{k_1}[\vec{p}'|h] \land r \in \{{d}_k[k_1,k_2] + 1\ldots {d}_k[k_1,k_2] + {\tau}[k_1,k_2|k,l]-v[k_1,k_2]\}\land\\ & v[k_1,k_2] = \min[{d}_k[k_1,k_2] - {d}_l[k_1,k_2],0]\},\\ &\\ & ={\mbox{DS}}_k[l|h_{k,r}^*] \cup \{(i,j,r-1 - {d}_k[i,j]+v[i,j]) | j \in D \land \\ & r \in\{{d}_k[i,j]+1 \ldots {d}_k[i,j] + {\tau}[i,j|k,l]-v[i,j]\}\land\\ & v[i,j] = \min[{d}_k[i,j] - {d}_l[i,j],0]\}\\ &\\ & ={\mbox{DS}}_k[l|h_{k,r}^*] \cup A\\ \end{array}$$ where $h_{k,r}^* \in {\mbox{hist}}[h,r|\vec{\sigma}^*]$ and $h_{k,r}' \in {\mbox{hist}}[h,r| \vec{\sigma}']$. The goal now is to show that for every $(i,j,r') \in A$, we have $r' < 0$. Fix any $j$. By Definition \[def:overlap\], for every $k \in {\cal N}$ and $l \in {\cal N}_k$ such that $$\label{eq:aid2-4} g=\max[{d}_k[i,j],{d}_l[i,j]] > {\bar{d}}+ {\tau},$$ we have $${\tau}[i,j|k,l] = 0.$$ Thus, by \[eq:aid2-4\], if $g= {d}_k[i,j]$, then we have $v[i,j] = 0$ and $$\begin{array}{l} {d}_k[i,j] + {\tau}[i,j|k,l] - v[i,j] = {d}_k[i,j]. \end{array}$$ Thus, there is no $r$ such that $$r \in\{{d}_k[i,j]+1 \ldots {d}_k[i,j] + {\tau}[i,j|k,l]-v[i,j]\},$$ which implies that $A = \emptyset$ and the result is true. If $g = {d}_l[i,j]$, then we have $v[i,j] = {d}_k[i,j] - {d}_l[i,j]$ and $$\begin{array}{l} {d}_k[i,j] + {\tau}[i,j|k,l] - v[i,j] = {d}_l[i,j].\\ r' = r - 1 - {d}_k[i,j] + v[i,j] = r-1 - {d}_l[i,j]. \end{array}$$ Here, if $r'\geq0$, then ${d}_l[i,j] \leq r-1$ and there is no $r > {\bar{d}}_i + {\tau}$ such that $$r \in\{{d}_k[i,j]+1 \ldots {d}_k[i,j] + {\tau}[i,j|k,l]-v[i,j]\},$$ which implies that $A = \emptyset$. So, either $A= \emptyset$ or $r'<0$, which concludes the step for any $k \in {\cal N}$ and $l \in {\cal N}_k$ that fulfill $\ref{eq:aid2-4}$. Consider now that $$\label{eq:aid2-5} g=\max[{d}_k[i,j],{d}_l[i,j]] < {\bar{d}}_i + {\tau},$$ which results by Definition \[def:overlap\] in $${\tau}[i,j|k,l] = {\bar{d}}_i + {\tau}- g.$$ Notice that $${d}_k[i,j] - v[i,j] = g,$$ which implies that $${d}_k[i,j] + {\tau}[i,j|k,l]-v[i,j] = g + {\bar{d}}_i + {\tau}- g = {\bar{d}}_i + {\tau}.$$ Thus, by \[eq:aid2-4\] there is no $r > {\bar{d}}_i + {\tau}$ such that $$r \in\{{d}_k[i,j]+1 \ldots {d}_k[i,j] + {\tau}[i,j|k,l]-v[i,j]\},$$ which implies that $A = \emptyset$. This allows us to conclude that for every $k \in {\cal N}$, $l \in {\cal N}_k$, and $r > {\bar{d}}_i + {\tau}$, $${\mbox{DS}}_k[l|h_{k,r}'] = {\mbox{DS}}_k[l|h_{k,r}^*] \cup A,$$ where for every $(i,j,r') \in A$ we have $r' < 0$. By Definition \[def:priv-thr\], $i$ adds $(k_1,k_2,r'') \in {\mbox{DS}}_k[l|h_{k,r}^*]$ to $K$ if and only if $i$ adds $(k_1,k_2,r'')$ to ${\mbox{DS}}_k[l|h_{k,r}']$. Therefore, $$\label{eq:aid2-6} \begin{array}{l} p_k[l|h_{k,r}'] = p_k[l|h_{k,r}^*].\\ q_i[h,r|\vec{\sigma}'] = q_i[h,r|\vec{\sigma}^*].\\ u_i[h,r|\vec{\sigma}'] = u_i[h,r|\vec{\sigma}^*]. \end{array}$$ This concludes the proof. ### Proof of Lemma \[lemma:delay-equiv\]. {#proof:lemma:delay-equiv} If $(\vec{\sigma}^*,\mu^*)$ is Preconsistent, Assumption \[def:non-neg\] holds, and Inequality \[eq:delay-equiv\] is fulfilled for every $i \in {\cal N}$, $h_i \in {\cal H}_i$, and $h \in {\cal H}$ such that $\mu_i^*[h|h_i]>0$, then $(\vec{\sigma}^*,\vec{\mu}^*)$ is Sequentially Rational: $$- \sum_{r=0}^{{\bar{d}}_i} \omega_i^r ((1-q_i[h,r|\vec{\sigma}^*])\gamma_i \bar{p}_i[h,r|\vec{\sigma}^*] + \epsilon \beta_i)+ \sum_{r={\bar{d}}_i+1}^{{\bar{d}}_i + {\tau}} \omega_i^r u_i[h,r | \vec{\sigma}^*] \geq 0.$$ Fix $i$, $h_i$, and $h$. Define $\vec{\sigma}' = (\sigma_i^*[h_i|\vec{p}_i'],\vec{\sigma}_{-i}^*)$ for any $D \subseteq {\cal N}_i[h_i]$, where: - For every $j \in D$, $p_i'[j] = 0$. - For every $j \in {\cal N}_i \setminus D$, $p_i'[j] = p_i[j|h_i]$. By Assumption \[def:non-neg\], for every $r \in \{0 \ldots {\bar{d}}_i\}$, $$\label{eq:de-1} \begin{array}{lll} \bar{p}_i[h,r|\vec{\sigma}'] & \geq 0.\\ q_i[h,r|\vec{\sigma}^*] - q_i[h,r|\vec{\sigma}'] &\geq \epsilon.\\ u_i[h,r|\vec{\sigma}^*] - u_i[h,r|\vec{\sigma}'] & = (1-q_i[h,r|\vec{\sigma}^*])(\beta_i -\gamma_i \bar{p}_i[h,r|\vec{\sigma}^*]) -(1-q_i[h,r|\vec{\sigma}'])(\beta_i - \gamma_i \bar{p}_i[h,r|\vec{\sigma}'])\\ & \geq -(1-q_i[h,r|\vec{\sigma}^*])\gamma_i - (q_i[h,r|\vec{\sigma}^*]- q_i[h,r|\vec{\sigma}'])\beta_i\\ & \geq -(1-q_i[h,r|\vec{\sigma}^*])\gamma_i - \epsilon \beta_i. \end{array}$$ By Lemma \[lemma:aux:impdel-1\], for every $r \in \{{\bar{d}}_i+1 \ldots {\bar{d}}_i + {\tau}\}$, $$\label{eq:de-2} u_i[h,r|\vec{\sigma}'] = 0.$$ Finally, by Lemma \[lemma:aux:impdel-2\], for every $r \geq {\bar{d}}_i+{\tau}+1$, $$\label{eq:de-3} \begin{array}{l} u_i[h,r|\vec{\sigma}^*] = u_i[h,r|\vec{\sigma}'].\\ \end{array}$$ It follows from \[eq:de-1\], \[eq:de-2\], and \[eq:de-3\] that: $$\begin{array}{ll} \sum_{r = 0}^{\infty} \omega_i^r(u_i[h,r|\vec{\sigma}^*] - u_i[h,r|\vec{\sigma}']) & \geq\\ - \sum_{r=0}^{{\bar{d}}_i} \omega_i^r ((1-q_i^*[h,r|\vec{\sigma}^*])\gamma_i \bar{p}_i[h,r|\vec{\sigma}^*] + \epsilon \beta_i)+ \sum_{r={\bar{d}}_i+1}^{{\bar{d}}_i + {\tau}} \omega_i^r u_i^*[h,r | \vec{\sigma}^*] . \end{array}$$ Therefore, if Inequality \[eq:delay-equiv\] is fulfilled for every $i \in {\cal N}$, $h_i \in {\cal H}_i$, and $h \in {\cal H}$ such that $\mu_i^*[h|h_i]>0$, then the PDC Condition holds. Consequently, by Theorem \[theorem:priv-drop\], $(\vec{\sigma}^*,\vec{\mu}^*)$ is Sequentially Rational. ### Proof of Lemma \[lemma:priv-suff\]. {#proof:lemma:priv-suff} If $(\vec{\sigma}^*,\vec{\mu}^*)$ is Preconsistent, Assumptions \[def:non-neg\] and \[def:priv-assum\] hold, and Inequality \[eq:priv-suff\] is fulfilled for every $h$, $i \in {\cal N}$, and $r,r' \leq {\bar{d}}_i + {\tau}$ such that $q_i[h,r'|\vec{\sigma}^*] < 1$, then there exist $\omega_i \in (0,1)$ for every $i \in {\cal N}$ such that $(\vec{\sigma}^*,\vec{\mu}^*)$ is Sequentially Rational: $$\frac{\beta_i}{\gamma_i}> \bar{p}_i[h,r|\vec{\sigma}^*] \frac{1}{A}+\bar{p}_i[h,r'|\vec{\sigma}^*]\frac{1}{B - C},$$ where - $A = 1 - \frac{\epsilon({\bar{d}}_i+1)}{(1-q_i[h,r|\vec{\sigma}^*]){\tau}}$. - $B=\frac{{\tau}}{c}$. - $C= \frac{\epsilon({\bar{d}}_i+1)}{1-q_i[h,r'|\vec{\sigma}^*]}$. Consider the above assumptions and assume by contradiction that $(\vec{\sigma}^*,\vec{\mu}^*)$ is not Sequentially Rational. The proof considers history $h_1$ that minimizes the first component of Inequality \[eq:delay-equiv\] and $h_2$ that minimizes the second component, for any history $h \in {\cal H}$. More precisely, fix $h$ and $i$: $$\label{eq:ps-1} \begin{array}{ll} h_1 = \mbox{\emph{argmin}}_{{\mbox{hist}}[h,r|\vec{\sigma}^*]| r \in \{0 \ldots {\bar{d}}_i\}} -((1-q_i[h,r|\vec{\sigma}^*])\gamma_i \bar{p}_i[h,r|\vec{\sigma}^*] + \epsilon \beta_i).\\ h_2 = \mbox{\emph{argmin}}_{{\mbox{hist}}[h,r|\vec{\sigma}^*]| r \in \{{\bar{d}}_i +1 \ldots {\bar{d}}_i + {\tau}\}} u_i[h,r|\vec{\sigma}^*]. \end{array}$$ Let $u_i^{h_2} = u_i[h_2,0|\vec{\sigma}^*]$. We can write: $$\label{eq:ps-2} \begin{array}{ll} - \sum_{r=0}^{{\bar{d}}_i} \omega_i^r ((1-q_i^*[h,r|\vec{\sigma}^*])\gamma_i \bar{p}_i[h,r|\vec{\sigma}^*] + \epsilon \beta_i)+ \sum_{r={\bar{d}}_i+1}^{{\bar{d}}_i + {\tau}} \omega_i^r u_i^*[h,r | \vec{\sigma}^*] & \geq \\ -\sum_{r = 0}^{{\bar{d}}_i} \omega_i((1-q_i[h_1,0|\vec{\sigma}^*])\gamma_i \bar{p}_i[h_1,0|\vec{\sigma}^*] + \epsilon \beta_i) + \sum_{r = 1}^{{\bar{d}}_i + {\tau}} \omega_i^r u_i^{h_2} & =\\ - a \frac{1-\omega_i^{{\bar{d}}_i +1}}{1-\omega_i} + \frac{\omega_i^{{\bar{d}}_i + 1} - \omega_i^{{\bar{d}}_i + {\tau}+1}}{1 - \omega_i} u_i^{h_2}, \end{array}$$ where $$a = (1-q_i[h_1,0|\vec{\sigma}^*])\gamma_i \bar{p}_i[h_1,0|\vec{\sigma}^*] + \epsilon \beta_i.$$ We want to fulfill $$\label{eq:ps-3} \begin{array}{ll} - a \frac{1-\omega_i^{{\bar{d}}_i +1}}{1-\omega_i} + \frac{\omega_i^{{\bar{d}}_i + 1} - \omega_i^{{\bar{d}}_i + {\tau}+1}}{1 - \omega_i} u_i^{h_2} \geq 0\\ -a + \omega_i^{{\bar{d}}_i + 1}(u_i^{h_2} + a) - \omega_i^{ {\bar{d}}_i + {\tau}+1} u_i^{h_2} & \geq 0. \end{array}$$ Again, this inequality corresponds to a polynomial with degree ${\tau}+1$. If $q_i^*[h_1,0|\vec{\sigma}^*] = 1$, then by our assumptions $q_i[h_2,0|\vec{\sigma}^*] =1$, $a=0$, and the Inequality holds. Suppose then that $$q_i[h_1,0|\vec{\sigma}^*], q_i[h_2,0|\vec{\sigma}^*] < 1.$$ The polynomial has a zero in $\omega_i = 1$. If $a=0$, then the Inequality holds. Consider, then, that $a > 0$. In these circumstances, a solution to \[eq:ps-3\] exists for $\omega_i \in (0,1)$ iff the polynomial is strictly concave and has another zero in $(0,1)$. This is true iff the polynomial has a maximum in $(0,1)$. The derivatives yield the following conditions: 1. $\exists_{\omega_i \in (0,1)} ({\bar{d}}_i+1)(u_i^{h_2} + a) - ({\tau}+{\bar{d}}_i + 1) \omega_i u_i^{h_2} = 0 \Rightarrow \exists_{\omega_i \in (0,1)} \omega_i = \frac{({\bar{d}}_i+1)(u_i^{h_2}+a)}{({\tau}+{\bar{d}}_i+1)u_i^{h_2}}$. 2. $-({\tau}+{\bar{d}}_i + 1){\tau}u_i^{h_2} < 0 \Rightarrow u_i^{h_2} > 0$. By our assumptions, Condition $1$ implies that: $$\label{eq:ps-4} \begin{array}{ll} u_i^{h_2}{\tau}& > ({\bar{d}}_i+1)a\\ &\\ (1-q_i[h_2,0|\vec{\sigma}^*]) \beta_i {\tau}&> (1-q_i[h_2,0|\vec{\sigma}^*])\gamma_i\bar{p}_i[h_2,0|\vec{\sigma}^*] {\tau}+ (1-q_i[h_1,0|\vec{\sigma}^*])\gamma_i\bar{p}_i[h_1,0|\vec{\sigma}^*] + \epsilon \beta_i\\ &\\ ((1-q_i[h_2,0|\vec{\sigma}^*]){\tau}- \epsilon({\bar{d}}_i +1)) \beta_i &> (1-q_i[h_2,0|\vec{\sigma}^*])\gamma_i\bar{p}_i[h_2,0|\vec{\sigma}^*] {\tau}+ (1-q_i[h_1,0|\vec{\sigma}^*])\gamma_i\bar{p}_i[h_1,0|\vec{\sigma}^*]\\ &\\ \frac{\beta_i}{\gamma_i} ((1-q_i[h_2,0|\vec{\sigma}^*]){\tau}- \epsilon({\bar{d}}_i +1)) & > (1-q_i[h_2,0|\vec{\sigma}^*])\bar{p}_i[h_2,0|\vec{\sigma}^*] {\tau}+ (1-q_i[h_1,0|\vec{\sigma}^*])\bar{p}_i[h_1,0|\vec{\sigma}^*]. \end{array}$$ Solving in order to the benefit-to-cost ratio, $$\begin{array}{l} \frac{(1-q_i[h_2,0|\vec{\sigma}^*])\bar{p}_i[h_2,0|\vec{\sigma}^*]{\tau}}{(1-q_i[h_2,0|\vec{\sigma}^*]){\tau}- \epsilon({\bar{d}}_i +1)}=\bar{p}_i[h_2,0|\vec{\sigma}^*] \frac{1}{1 - \frac{\epsilon({\bar{d}}_i +1)}{(1-q_i[h_2,0|\vec{\sigma}^*]){\tau}}}=\bar{p}_i[h_2,0|\vec{\sigma}^*] \frac{1}{A}. \end{array}$$ where $A = 1 - \frac{\epsilon({\bar{d}}_i+1)}{(1-q_i[h_2,0|\vec{\sigma}^*]){\tau}}$. Continuing, by Assumption \[def:priv-assum\], it is true that $$\frac{(1-q_i[h_2,0|\vec{\sigma}^*])}{(1-q_i[h_1,0|\vec{\sigma}^*])}\geq \frac{1}{c}.$$ Therefore, $$\begin{array}{l} \frac{(1-q_i[h_1,0|\vec{\sigma}^*])\bar{p}_i[h_1,0|\vec{\sigma}^*]}{(1-q_i[h_2,0|\vec{\sigma}^*]){\tau}- \epsilon({\bar{d}}_i +1)}= \bar{p}_i[h_1,0|\vec{\sigma}^*] \frac{1}{\frac{(1-q_i[h_2,0|\vec{\sigma}^*]){\tau}}{(1-q_i[h_1,0|\vec{\sigma}^*])} - \frac{\epsilon({\bar{d}}_i +1)}{(1-q_i[h_1,0|\vec{\sigma}^*])}}\\ \\ \leq \bar{p}_i[h_1,0|\vec{\sigma}^*] \frac{1}{\frac{{\tau}}{c} - \frac{\epsilon({\bar{d}}_i+1)}{1-q_i[h_1,0|\vec{\sigma}^*]}} = \bar{p}_i[h_1,0|\vec{\sigma}^*] \frac{1}{B -C}. \end{array}$$ where: - $B = \frac{{\tau}}{c}$. - $C= \frac{\epsilon({\bar{d}}_i+1)}{1-q_i[h_1,0|\vec{\sigma}^*]}$. In summary, we have $$\label{eq:ps-5} \begin{array}{ll} \frac{\beta_i}{\gamma_i} &> \bar{p}_i[h_2,0|\vec{\sigma}^*] \frac{1}{1 - A} + \bar{p}_i[h_1,0|\vec{\sigma}^*] \frac{1}{B -C} \Rightarrow\\ \frac{\beta_i}{\gamma_i} ((1-q_i[h_2,0|\vec{\sigma}^*]){\tau}- \epsilon({\bar{d}}_i +1)) & > (1-q_i[h_2,0|\vec{\sigma}^*])\bar{p}_i[h_2,0|\vec{\sigma}^*] {\tau}+ (1-q_i[h_1,0|\vec{\sigma}^*])\bar{p}_i[h_1,0|\vec{\sigma}^*]. \end{array}$$ Consequently, if Inequality \[eq:priv-suff\] is true, then so is \[eq:ps-4\]. Furthermore, it also holds that $$\beta_i > \gamma_i \bar{p}_i[h_1,0|\vec{\sigma}^*] \Rightarrow u_i^h >0.$$ That is, Inequality \[eq:priv-suff\] implies Conditions 1 and 2 of the polynomial for any $h$ and some $\omega_i \in (0,1)$, which by transitivity implies that \[eq:ps-3\] is true. By \[eq:ps-2\], Inequality \[eq:delay-equiv\] is fulfilled for every history $h$. Lemma \[lemma:delay-equiv\], allows us to conclude that $(\vec{\sigma}^*,\vec{\mu}^*)$ is Sequentially Rational. This is a contradiction, proving the result. ### Proof of Theorem \[theorem:priv-effect\]. {#proof:theorem:priv-effect} If $(\vec{\sigma}^*,\vec{\mu}^*)$ is Preconsistent, Assumptions \[def:non-neg\] and \[def:priv-assum\] hold for $\epsilon \ll 1$, and ${\tau}\geq {\bar{d}}+1$, then there exists a constant $c>0$ such that $\psi[\vec{\sigma}^*|\vec{\mu}^*] \supseteq (v,\infty)$, where $$v = \max_{i \in {\cal N}} \max_{h \in {\cal H}}\bar{p}_i[h,0|\vec{\sigma}^*](1+c).$$ The idea is to simplify Inequality \[eq:priv-suff\] for $\epsilon \ll 1$ and ${\tau}\geq {\bar{d}}+ 1$. Recall that $$A = 1 - \frac{\epsilon({\bar{d}}_i+1)}{(1-q_i[h,r|\vec{\sigma}^*]){\tau}}.$$ Thus, this yields $$\frac{1}{A} = \frac{(1-q_i[h,r|\vec{\sigma}^*]) {\tau}}{(1-q_i[h,r|\vec{\sigma}^*]){\tau}(1 - \epsilon \frac{{\bar{d}}_i+1}{{\tau}})} \leq \frac{(1-q_i[h,r|\vec{\sigma}^*]) {\tau}}{(1-q_i[h,r|\vec{\sigma}^*]) {\tau}(1-\epsilon)} \approx 1.$$ Moreover, by Assumption \[def:priv-assum\], $$\begin{array}{l} \frac{1}{B - C} = \frac{1}{\frac{{\tau}}{c} - \frac{\epsilon({\bar{d}}_i+1)}{1-q_i[h,r|\vec{\sigma}^*]}}=\\ \\ \frac{1-q_i[h,r|\vec{\sigma}^*]}{(1-q_i[h,r|\vec{\sigma}^*])\frac{{\tau}}{c} - \epsilon({\bar{d}}_i+1)}\leq\\ \\ \frac{(1-q_i[h,r|\vec{\sigma}^*])}{(1-q_i[h,r+{\tau}|\vec{\sigma}^*]){\tau}- \epsilon({\bar{d}}_i+1)}\leq\\ \\ \frac{(1-q_i[h,r|\vec{\sigma}^*])}{(1-q_i[h,r+{\tau}|\vec{\sigma}^*]- \epsilon)({\bar{d}}_i+1)}\approx\\ \\ \frac{(1-q_i[h,r|\vec{\sigma}^*])}{(1-q_i[h,r+{\tau}|\vec{\sigma}^*])({\bar{d}}_i+1)}\leq\\ \\ \frac{(1-q_i[h,r|\vec{\sigma}^*])}{(1-q_i[h,r+{\tau}|\vec{\sigma}^*]({\bar{d}}_i+1))} \leq\\ \\ \frac{c}{{\bar{d}}_i+1}. \end{array}$$ Thus, for any $r,r' \geq 0$, $$\bar{p}_i[h,r|\vec{\sigma}^*] \frac{1}{A}+\bar{p}_i[h,r'|\vec{\sigma}^*]\frac{1}{B - C} \leq \bar{p}_i[h,r|\vec{\sigma}^*] + \bar{p}_i[h,r'|\vec{\sigma}^*] \frac{c}{{\bar{d}}_i +1}.$$ Thus, there exists a constant $c'= \frac{c}{{\bar{d}}_i + 1}$ such that if for every $i$ we have $$\frac{\beta_i}{\gamma_i} > \max_{h \in {\cal H}} \bar{p}_i[h|\vec{\sigma}^*] (1 + c') \geq \bar{p}_i[h,r|\vec{\sigma}^*] + \bar{p}_i[h,r'|\vec{\sigma}^*] \frac{c}{{\bar{d}}_i +1},$$ then Inequality \[eq:priv-suff\] is fulfilled for every $h$, $r$, and $r'$, and for some $\omega_i \in (0,1)$. By Lemma \[lemma:priv-suff\], this implies $(\vec{\sigma}^*,\vec{\mu}^*)$ is Sequentially Rational and the result follows.

--- abstract: 'We present a theoretical formalism to address the dynamics of textured, noncolliear antiferromagnets subject to spin current injection. We derive sine-Gordon type equations of motion for the antiferromagnets, which are applicable to technologically important antiferromagnets such as Mn$_3$Ir and Mn$_3$Sn, and enables an analytical approach to domain wall dynamics in those materials. We obtain the expression for domain wall velocity, which is estimated to reach $\sim1$ km/s in Mn$_3$Ir by exploiting spin Hall effect with electric current density $\sim10^{11}$ A/m$^2$.' author: - 'Yuta Yamane$^1$, Olena Gomonay$^2$, and Jairo Sinova$^{2,3}$' title: Dynamics of noncollinear antiferromagnetic textures driven by spin current injection --- Since the prediction of staggered magnetic order[@Neel] and its experimental observation in MnO[@Shull], antiferromagnetic (AFM) materials have occupied a central place in the study of magnetism. The absence of macroscopic magnetization in AFMs, however, indicates that they cannot be effectively manipulated and observed by external magnetic field, which has hindered active applications of AFMs in today’s technology. Research in the emergent field of antiferromagnetic spintronics[@Baltz] has shown that electric and spin currents can access AFM dynamics through spin-transfer torques[@Nunez; @Urazhdin; @Xu; @Helen2010; @Hals; @Swaving; @Cheng; @Yamane; @Baldrati] and spin-orbit torques[@Jakob; @Shiino]. Similar to ferromagnets, AFMs can also accommodate topologically nontrivial textures such as domain walls (DWs)[@Baryakhtar1985; @Papanicolau; @Bode] and skyrmions[@Bogdanov; @Barker], which play crucial roles in spintronics applications, e.g., racetrack memories[@Parkin]. The studies on current-driven dynamics of AFM textures have opened an avenue toward AFM-based technologies. Recently, AFMs with noncollinear magnetic configurations are generating increasing attention as they exhibit large magneto-transport and thermomagnetic effects; e.g., anomalous Hall effect[@Chen; @Kubler; @Nakatsuji], anomalous Nernst effect[@Li; @Ikhlas] and magneto-optical Kerr effect[@Feng; @Higo]. These phenomena have their origins in the topological character of the electronic band structures, which in turn are associated with the noncollinear magnetism. To take full advantages of noncollinear AFMs in spintronics applications, it is also important to achieve efficient manipulation of magnetic textures, such as DWs, in those materials. The studies on current-driven dynamics of AFMs, however, have thus far mostly focused on collinear structures. Understanding the effects of electric and spin currents in noncollinear AFMs is being a crucial issue in the community[@Prakhya; @Helen2015; @Liu]. In this paper, we focus on the dynamics of noncollinear AFMs induced by spin current (SC) injection, which may be realized by exploiting spin Hall effect/spin-polarized electric current in an adjacent heavy-metal/ferromagnetic layer. We derive sine-Gordon type equations of motion for the AFMs, including effective forces due to SC injection, external magnetic field, and internal dissipation. Our model can be applied to technologically important triangular AFMs such as Mn$_3$Ir and Mn$_3$Sn. We then study DW dynamics, where an analytical expression for the DW velocity is derived. *Model. —* We consider an antiferromaget (AFM) composed of three equivalent magnetic sublattices (A, B, and C) with constant saturation magnetization $M_{\rm S}$. In our coarse-grained model, the classical vector ${\vec m}_A ({\vec r}, t)$ $(|{\vec m}_A ({\vec r}, t)| = 1)$ is a continuous field that represents the magnetization direction in the sublattice A, with similar definitions for ${\vec m}_B ({\vec r}, t )$ and $ {\vec m}_C ({\vec r}, t )$ (Fig. 1). The magnetic energy density $u$ of the AFM is modeled as follows, $$\begin{aligned} u &=& J_0 \sum_{\langle \zeta \eta \rangle} {\vec m}_\zeta \cdot {\vec m}_\eta + A_1 \sum_{x_i=x,y,z} \sum_{\zeta=A,B,C} \left(\frac{\partial{\vec m}_\zeta}{\partial x_i}\right)^2 \nonumber \\ && - A_2 \sum_{x_i=x,y,z} \sum_{\langle\zeta\eta\rangle} \frac{\partial{\vec m}_\zeta}{\partial x_i} \cdot \frac{\partial {\vec m}_\eta}{\partial x_i} + D_0 {\vec e}_z \cdot \sum_{\langle \zeta \eta \rangle} {\vec m}_\zeta \times {\vec m}_\eta \nonumber \\ && + u_{\rm ani} - \mu_0 M_{\rm S} {\vec H} \cdot \sum_{\zeta=A,B,C} {\vec m}_\zeta , \label{u}\end{aligned}$$ where $J_0$ describes antiferromagnetic exchange coupling between the sublattices, $A_1$ and $A_2$ are the isotropic exchange stiffnesses[@Note], $D_0$ characterizes the homogeneous Dzyaloshinskii-Moriya interaction (DMI), ${\vec H}$ is the external magnetic field, and $\mu_0$ is the vacuum permeability. The symbol $\langle \zeta \eta \rangle$ indicates the sum over the pairs $(\zeta, \eta) = (A,B)$, $(B,C)$, and $(C,A)$. For the anisotropy part $u_{\rm ani}$ we assume $$u_{\rm ani} = - K \sum_{\zeta=A,B,C} \left( {\vec e}_\zeta \cdot {\vec m}_\zeta \right)^2 , \label{u_ani}$$ where $K (>0)$ is the anisotropy constant, and the unit vectors ${\vec e}_\zeta$ indicate the easy axes for ${\vec m}_\zeta$ in the $x$-$y$ plane; ${\vec e}_A = ( - {\vec e}_x + \sqrt{3} {\vec e}_y ) / 2$, ${\vec e}_B = - ( {\vec e}_x + \sqrt{3} {\vec e}_y ) / 2 $, and $ {\vec e}_C = {\vec e}_x $. The magnetic anisotropy of this form applies to triangular AFMs such as the L1$_2$ phase of Mn$_3$Ir[@Ulloa] and the hexagonal phase of Mn$_3$Sn[@Liu], with the (1,1,1) plane of their fcc crystals identified as our $x$-$y$ plane. Although there can also be a smaller out-of-plane anisotropy in realistic materials, Eq. (\[u\_ani\]) suffices for our present purpose of understanding the fundamental response of triangular AFMs to SC injection. ![ Scheme of the studied system; bilayer of noncollinear antiferromagnet (AFM) and nonmagnetic (NM) heavy metal, where the spin current with polarization along ${\vec p}$ is injected into the AFM. Spin current may be created via spin Hall effect (as shown) or by alternative techniques, such as spin-pumping and injection of spin-polarized electric current from a ferromagnetic layer. Domain wall (DW), connecting the all-in (blue) and all-out (red) domains, is driven into motion by the spin current. []{data-label="fig01"}](Fig1.png){width="8cm"} The dynamics of ${\vec m}_\zeta$ ($\zeta =A,B,C$) are assumed to obey the coupled Landau-Lifshitz-Gilbert equations; $$\frac{\partial{\vec m}_\zeta}{\partial t} = - {\vec m}_\zeta \times \gamma {\vec H}_\zeta + \alpha {\vec m}_\zeta \times \frac{\partial{\vec m}_\zeta}{\partial t} - {\vec m}_\zeta \times \left( {\vec m}_\zeta \times {\vec p} \right) , \label{llg}$$ where $\gamma$ and $\alpha$ are the the gyromagnetic ratio and the Gilbert damping constant, respectively, which are assumed for simplicity to be sublattice independent, and $ {\vec H}_\zeta = - (\mu_0 M_{\rm S})^{-1} \delta u / \delta {\vec m}_\zeta$ is the effective magnetic field for the sublattice $\zeta$. The last term in Eq. (\[llg\]) is the Slonczewski-Berger spin-transfer torque[@Slonczewski] due to SC injection. The vector ${\vec p}$ represents the value and polarization of the SC, which depend on the way of SC injection, device materials, geometry, etc. We have assumed that the injected SC transfers the angular momentum equiprobably to each of the sublattices[@Helen2015]. *In-plane triangular approximation. —* We here introduce $$\begin{aligned} {\vec n}_1 &=& \frac{{\vec m}_1 + {\vec m}_2 - 2 {\vec m}_3}{3\sqrt{2}} , \qquad {\vec n}_2 = \frac{- {\vec m}_1 + {\vec m}_2}{\sqrt{6}} , \\ {\vec m} &=& \frac{{\vec m}_1 + {\vec m}_2 + {\vec m}_3}{3} .\end{aligned}$$ Because the AFM exchange coupling responsible for the formation of triangular structure is usually dominant over the other energies, one can safely assume $|{\vec m} ({\vec r}, t)| \ll 1$. The vectors $ {\vec n}_1 $ and ${\vec n}_2$ are then approximated to be orthogonal to each other and have the fixed length as $ | {\vec n}_1 | \simeq | {\vec n}_2 | \simeq 1/ \sqrt{ 2 } $. These two vectors can be regarded order parameters of the AFM[@Helen2015], specifying the particular triangular configuration. We further assume that the in-plane anisotropy is sufficiently large that the triangle is formed in the $x$-$y$ plane with $ | m_\zeta^z | \ll 1$, $\forall\zeta$. This leads to an approximation where only the $x$ and $y$ components of $ {\vec n}_1 $ and $ {\vec n}_2 $ are nonzero (while ${\vec m}$ can still have a finite $z$ component). In this case the orientations of ${\vec n}_1$ and ${\vec n}_2$ in the $x$-$y$ plane can be parameterized by a single azimutal angle $\varphi$[@Andreev; @Helen2015; @Ulloa; @Liu] as $$\begin{aligned} {\vec n}_1 &=& \frac{1}{ \sqrt{ 2 } }\left[ \begin{array}{c} \cos \varphi \\ \sin \varphi \\ 0 \end{array} \right] , \label{n1} \\ {\vec n}_2 &=& {\cal R}_{ \pm \pi / 2 } {\vec n}_1 \equiv \frac{1}{ \sqrt{ 2 } }\left[ \begin{array}{c} \cos ( \varphi \pm \pi / 2 ) \\ \sin ( \varphi \pm \pi / 2 ) \\ 0 \end{array} \right] . \label{n2}\end{aligned}$$ In Eq. (\[n2\]), ${\cal R}_{+\pi/2}$ and ${\cal R}_{-\pi/2}$ select the $+\pi/2$ and $-\pi/2$ rotations of ${\vec n}_2$ against ${\vec n}_1$, respectively, corresponding to the two different chiralities of the triangular structure, defined by ${\rm sgn} \left( {\vec e}_z \cdot {\vec n}_1 \times {\vec n}_2 \right)$; in Fig. 2, four different triangular configurations are shown as examples. Which of ${\cal R}_{+\pi/2}$ and ${\cal R}_{-\pi/2}$ should be chosen is dictated by the DMI and magnetic anisotropy. The DMI favors the ${\cal R}_{+\pi/2}$ (${\cal R}_{-\pi/2}$) chirality if the sign of $D_0$ is negative (positive). The magnetic anisotropy, on the other hand, can never be fully respected by ${\cal R}_{-\pi/2}$ \[Fig. 2. (c) and (d)\], in contrast to ${\cal R}_{+\pi/2}$ where the anisotropy energy is minimized by taking $\varphi=0,\pi$. \[Fig. 2. (a) and (b)\] The ${\cal R}_{-\pi/2}$ chirality is thus favored when the DMI satisfies the condition $2\sqrt{3}D_0 > K$. As a result of the competition between the anisotropy, exchange coupling and DMI, the ${\cal R}_{-\pi/2}$ triangles carry the weak in-plane ferromagnetic moment ${\vec m}$ \[Fig. 2. (c) and (d)\]. Typical materials that host the ${\cal R}_{+\pi/2}$ triangles include the L1$_2$ phase of IrMn$_3$[@Sakuma; @Szunyogh], while the ${\cal R}_{-\pi/2}$ configurations are observed in, e.g., the hexagonal phase of Mn$_3$Z (Z $=$ Sn, Ge, Ga)[@DZhang]. It turns out that, for either chirality, the AFM responds to the injected SC in a similar way. In the following we mostly focus on the ${\cal R}_{+\pi/2}$ case, and later consider configurations with ${\cal R}_{-\pi/2}$. With the parametrization in Eqs. (\[n1\]) and (\[n2\]), the state of an AFM is described by four variables $(\varphi,{\vec m})$. By rewriting Eqs. (\[llg\]) in terms of $(\varphi,{\vec m})$ and assuming $J_0 \gg |D_0| \gg K$, one obtains, up to the first order of ${\vec m}$, the closed equation of motion for $\varphi$, $$c^2 \Box \varphi - \frac{3 \omega_E \omega_K}{2} \sin 2\varphi = 3 \omega_E p_z - \gamma \frac{\partial H_z}{\partial t} + \omega_\alpha \frac{\partial\varphi}{\partial t} , \label{eom}$$ and the explicit expression for ${\vec m}$, $${\vec m} = \frac{1}{ 3\omega_E} \left( - \frac{\partial \varphi}{\partial t} {\vec e}_z + \gamma {\vec H} \right) , \label{m}$$ for the ${\cal R}_{+\pi/2}$ case. We have introduced $ \omega_E = \gamma J_0 / \mu_0 M_{\rm S} $, $\omega_K = 2 \gamma K / \mu_0 M_{\rm S}$, $ \omega_\alpha = 3 \alpha \omega_E $, and $\Box = \nabla^2 - (1/c^2) \partial^2/\partial t^2$ with $ c = \sqrt{3 \omega_E \gamma (2A_1+A_2) / \mu_0 M_{\rm S}}$ the group velocity of spin wave. ![ Triangular magnetic configurations with (a,b) ${\cal R}_{+\pi/2}$ and (c,d) ${\cal R}_{-\pi/2}$ chiralities. They are parametrized by (a,c) $\varphi=0$ and (b,d) $\varphi=\pi$. Dotted lines indicate the easy axes of the magnetic anisotropy. []{data-label="fig02"}](Fig2.png){width="8cm"} Equation (\[m\]) shows that $ {\vec m} $ is expressed in terms of $ \varphi $, and vanishes in the absence of magnetic dynamics ($\partial \varphi/\partial t=0$) and external magnetic field. Equation (\[eom\]) is one of our main results. The rhs of this equation contains the effective forces originating from SC injection, time-varying magnetic field, and internal damping. In the absence of these forces, Eq. (\[eom\]) is reduced to a sine-Gordon equation, consistent with the work in Ref. [@Ulloa]. In the limit of homogeneous systems ($\nabla\varphi=0$) without external magnetic field, Eq. (\[eom\]) then reproduces the result of Ref. [@Helen2015]. Now, our Eq. (\[eom\]) allows one to study inhomogeneous AFM textures under the external driving forces due to SC and magnetic field, and the internal damping. Notice that only the out-of-plane ($z$) component of the SC polarization and of the magnetic field can induce the dynamics of $\varphi$. We should also remark that the DMI does not appear in Eqs. (\[eom\]) and (\[m\]). This is because the DMI energy, which can be written as $3 \sqrt{3} D_0{\vec e}_z \cdot {\vec n}_1\times {\vec n}_2$, is constant within the present approximation, and its contribution to the equations of motion is higher-order. The DMI plays a crucial role, however, in lifting the degeneracy with respect to the chirality (Fig. 2) as discussed above. In the following, we discuss the translational motion of a DW driven by SC (setting ${\vec H}=0$). *Domain wall dynamics. —* The doubly-degenerate ground states for the ${\cal R}_{+\pi/2}$ case are given such that the magnetic anisotropy energy is minimized by $\varphi = 0$ and $\pi$, corresponding to the all-in and all-out configurations, respectively \[Fig. 2 (a) and (b)\]. A DW can be formed as a transition region connecting the two ground states (Fig. 1). Here let us consider a one-dimensional DW extending along the $z$-axis. (Due to the isotropic character of the exchange stiffnesses, our conclusions will be independent of the choice of the direction of DW extension, as long as the SC is polarized along the $z$ axis so defined.) When the rhs of Eq. (\[eom\]) is absent, a standard solution $\varphi_{\rm e} (z)$ for a static DW with the boundary condition $\varphi_{\rm e} (z=\pm\infty) = (0,\pm\pi)$ or $(\pm\pi,0)$ (notice that $\varphi$ is defined in $-\pi\leq\varphi\leq\pi$) is obtained as $ \varphi_{\rm e} (z) = 2 F \tan^{-1} \left[ \exp \left( Q \frac{ z - z_0 }{ \Delta_0 } \right) \right] $; $z_0$ is the coordinate of the DW center, $\Delta_0 = \sqrt{(2A_1+A_2)/2K}$ is the DW width parameter, and $(Q,F)=(\pm1,\pm1)$, satisfying $QF = (1/\pi)\int_{-\infty}^\infty dz (\partial\varphi_{\rm e}/\partial z) $, specifies the boundary condition. To study steady-motion of the DW driven by SC, we employ the following ansatz $$\varphi ( z , t ) = 2 F \tan^{ - 1 } \left[ \exp \left( Q \frac{ z - V t }{ \Delta } \right) \right] , \label{dw}$$ where $V$ is the velocity of DW center and $\Delta$ is the dynamical width parameter. By substituting this ansatz into Eq. (\[eom\]), multiplying the subsequent equation by $\sin\varphi$, and integrating it along the $z$-axis from $z=-\infty$ to $+\infty$, one finds the relation $V = (FQ\pi\Delta/2\alpha) p_z $. In the special case where the DW exhibits an inertial motion in the [*absence*]{} of the rhs of Eq. (\[eom\]), the width parameter $\Delta_{\rm in}$ is given by $$\Delta_{\rm in} = \Delta_0 \sqrt{1 - \frac{V^2}{c^2}} . \label{Delta}$$ Equation (\[Delta\]) implies the Lorentz contraction of the DW, stemming from the Lorentz invariance of the sine-Gordon equation. The rhs of Eq. (\[eom\]) may be regarded perturbation, if $3\omega_E p_z$ and $\omega_\alpha |\partial\varphi/\partial t | \sim \omega_\alpha V/\Delta$ are sufficiently small compared to each term in the lhs. In this perturbative regime one can use the approximation $\Delta = \Delta_{\rm in}$, which leads to $$V = FQ \frac{ \mu p_z }{ \sqrt{ 1 + ( \mu p_z / c )^2 } } , \label{V2}$$ where we have introduced the DW mobility (in the unit of length) $$\mu = \frac{\pi\Delta_0}{2\alpha} . \label{mu}$$ Eq. (\[V2\]) is one of our central results, revealing important natures of the DW dynamics. The sign of $V$ is determined by that of $p_z $, i.e., the polarization of the SC, and the factor $FQ$ that characterizes the DW structure. For $\mu |p_z| / c \ll 1$, the DW velocity depends linearly on $p_z $ as $V \simeq FQ\mu p_z$. Importantly, $V$ monotonically increases with $|p_z|$, in a similar manner as in collinear AFMs[@Shiino]. Our result thus indicates that the absence of the so-called Walker breakdown[@Schryer] is ubiquitous for general AFMs, and a high DW velocity can be achieved by increasing the SC injection. The previous studies showed that noncollinear AFMs have an advantage over collinear ones in the large magneto-transport effects[@Chen; @Kubler; @Nakatsuji; @Feng; @Higo], which provide efficient ways to detect DWs. Now that Eq. (\[V2\]) reveals that the noncollinear AFMs can accommodate DWs moving as fast as in collinear ones, the former are indeed a potential candidate for future spintronics applications. In Fig. 3, Eq. (\[V2\]) is plotted by the solid line as a function of $\mu p_z$ with $(F,Q)=(+1,+1)$. ![ The DW velocity $V$ as a function of SC $\mu p_z$, calculated numerically from Eq. (\[llg\]) (open symbols) and analytically from Eq. (\[V2\]) (solid line). Both $V$ and $\mu p_z$ are measured in the unit of $c$ ($\simeq$16 km/s with the present parameter set shown below). For the simulations we consider a nanowire with dimensions of 1$\times$1 nm$^2\times$20 $\mu$m, dividing it into the unit cells of 1$\times$1$\times$1 nm$^3$. We employ the periodic boundary condition along the $x$ direction, to mimic a thin (in the $y$ direction), wide (in the $x$ direction) nanowire. The parameter values are typical for Mn$_3$Ir[@Sakuma; @Szunyogh]: $J_0 = 2.4\times10^8 $ J/m$^3$, $A_1=0$ (corresponding to taking into only account the nearest-neighbor coupling in the kagome lattice), $A_2 = 2\times10^{-11}$ J/m, $D_0=-2\times10^7$ J/m$^3$, $K = 3\times10^6$ J/m$^3$, $ \mu_0 M_{\rm S} = 1.63 $ T, $ \gamma = 1.76 \times 10^{ 11 } $ Hz/T, and $ \alpha = 10^{ - 2 } $. When the SC is created via spin Hall effect, as in Fig. 1, the electric current density $j_{\rm c}$, corresponding to $\mu p_z = 0.1c$ and $V\simeq4.7$ km/s, is estimated as $j_{\rm c} \simeq 8.5\times10^{11}$ A/m$^2$, using $p_z = (\gamma \hbar / 2eM_{\rm S} ) \theta_{\rm SHE} j_{\rm c} / d$[@Slonczewski] with the spin Hall angle $\theta_{\rm SHE}=0.15$ for NM and the sample thickness $d=1$ nm . ](Fig3.png){width="8cm"} To check the validity of Eq. (\[V2\]), we compute the DW velocity by numerically solving Eq. (\[llg\]), as indicated by the open symbols in Fig. 3. The results of the simulations and the analytical model agree well in the relatively low current regime. The discrepancy starts visibly developing as $\mu p_z$ is increased as large as $\sim c$, where the rhs and lhs of Eq. (\[eom\]) become comparable (with our present choice of parameters) and the perturbative approach is invalid. The deviation of the numerical results from Eq. (\[V2\]) in the high current regime may be attributed to several factors. First, the out-of-plane components of the magnetizations grow with $p_z$, thus reducing the accuracy of the in-plane approximation. Second, the homogeneous SC, represented by the spatial-independent $p_z$ term in Eq. (\[eom\]), acts not only within the DW region, but also on each domain. The SC thus causes the rotation of the domains away from $\varphi=0,\pm\pi$, and the ansatz (\[dw\]) becomes inappropriate. *${\cal R}_{-\pi/2}$ chirality. —* Lastly, we show that qualitatively similar conclusions are obtained for the ${\cal R}_{-\pi/2}$ case. The equations of motion are derived with the same line of approximations used in deriving Eqs. (\[eom\]) and (\[m\]). For the weak ferromagnetic moment ${\vec m}$ one obtains $${\vec m} = \frac{1}{3\omega_E} \left( - \frac{\partial\varphi}{\partial t} {\vec e}_z + \gamma {\vec H} + \frac{\omega_K}{\sqrt{2}} {\cal M} {\vec n} \right) , \label{m2}$$ where the external magnetic field ${\vec H}$ is restored, and ${\cal M} {\vec n} = 2^{-1/2} (-\cos\varphi,\sin\varphi,0)$. Equation (\[m2\]) differs from Eq. (\[m\]) in the third term, which arises from the competition between the magnetic anisotropy, exchange coupling, and DMI, as discussed above. The equation of motion for $\varphi$, up to the first order of ${\vec m}$, is $$\begin{aligned} c^2 \Box \varphi &=& 3 \omega_E p_z - \gamma \frac{\partial H_z}{\partial t} + 3 \alpha \omega_E \frac{\partial\varphi}{\partial t} \nonumber \\ && - \frac{\omega_K \gamma}{2} \left( H_x \sin\varphi + H_y \cos\varphi \right) . \label{eom2}\end{aligned}$$ There are two major differences between the magnetic dynamics for the ${\cal R}_{+\pi/2}$ \[Eq. (\[eom\])\] and ${\cal R}_{-\pi/2}$\[Eq. (\[eom2\])\] cases; First, for ${\cal R}_{-\pi/2}$, in-plane magnetic fields can create additional driving forces \[the last terms in the rhs of Eq. (\[eom2\])\], which originates from the direct Zeeman coupling between the weak ferromagnetic moment \[the last term in Eq. (\[m2\])\] and the magnetic field. Second, the $\sin2\varphi$ term does not appear in Eq. (\[eom2\]), which indicates the absence of effective anisotropy for $\varphi$. In the ${\cal R}_{-\pi/2}$ case, effective anisotropies arise from higher-order terms of ${\vec m}$[@Liu]. For Mn$_3$Sn, indeed, a small anisotropy $\sim10$ J/m$^3$ of the form of $\cos6\varphi$ has been predicted, which leads to formations of 60$^\circ$ DWs[@Liu]. Although a DW is in general not a 180$^\circ$ wall depending on the symmetry of the effective anisotropy, the SC acts on the DW in essentially the same way as on the 180$^\circ$ walls in the ${\cal R}_{+\pi/2}$ case, since the $p_z$ term is identical in Eqs. (\[eom\]) and (\[eom2\]). Most of the conclusions on the DW motion derived before thus hold qualitatively, with renormalization $\varphi \rightarrow \frac{n}{2}\varphi$ for a $n$-fold anisotropy with $n$ an even integer. *Conclusions. —* We have derived sine-Gordon type equations of motion for the noncollinear antiferromagnets, with spin current injection, external magnetic field, and dissipative terms included. We have demonstrated that the injected spin current, when it is polarized perpendicular to the triangular plane, can drive a translational motion of a domain wall. When the spin current is injected by exploiting the spin Hall effect, the domain wall velocity as high as $\sim1$ km/s can be achieved for typical noncollinear antiferromagnets, with realistic electric current density $\sim10^{11}$ A/m$^2$. As the spin current injection into noncollinear antiferromagnets remains to be experimentally demonstrated, our findings provide a guideline for devising future experiments. The authors appreciate Prof. G. Tatara and the members of his group of RIKEN, and Dr. J. Ieda of Japan Atomic Energy Agency, for fruitful discussions and comments on the manuscript. This research was supported by Research Fellowship for Young Scientists from Japan Society for the Promotion of Science, the Alexander von Humboldt Foundation, the Transregional Collaborative Research Center (SFB/TRR) 173 SPIN+X, and Grant Agency of the Czech Republic grant No. 14-37427G. OG also acknowledges the support from DFG (project SHARP 397322108). [99]{} L. Néel, Ann. de physique [**17**]{}, 5 (1932); F. Bitter, Phys. Rev. [**54**]{}, 79 (1938); J. H. Van Vleck, J. Chem. Phys. [**9**]{}, 85 (1941). C. G. Shull and J. S. Smart, Phys. Rev. [**76**]{}, 1256 (1949); C. G. Shull, W. A. Strauser, E. O. Wollan, [*ibid.*]{} [**83**]{}, 333 (1951). V. Baltz, A. Manchon, M. Tsoi, T. Moriyama, T. Ono, and Y. Tserkovnyak, Rev. Mod. Phys. [**90**]{}, 015005 (2018). A. S. Núñez, R. A. Duine, P. Haney, and A. H. MacDonald, Phys. Rev. B [**73**]{}, 214426 (2006); R. A. Duine, P. M. Haney, A. S. Núñez, and A. H. MacDonald, Phys. Rev. B [**75**]{}, 014433 (2007); Z. Wei, A. Sharma, A. S. Núñez, P. M. Haney, R. A. Duine, J. Bass, A. H. MacDonald, and M. Tsoi, Phys. Rev. lett. [**98**]{}, 116603 (2007); P. M. Haney, D. Waldron, R. A. Duine, A. S. Núñez, H. Guo, and A. H. MacDonald, Phys. Rev. B [**75**]{}, 174428 (2007). P. M. Haney and A. H. MacDonald, Phys. Rev. Lett. [**100**]{}, 196801 (2008); S. Urazhdin and N. Anthony, Phys. Rev. Lett. [**99**]{}, 046602 (2007). Y. Xu, S. Wang, and K. Xia, Phys. Rev. Lett. [**100**]{}, 226602 (2008). H. V. Gomonay and V. M. Loktev, Phys. Rev. B [**81**]{}, 144427 (2010); H. V. Gomonay, R. V. Kunitsyn, and V. M. Loktev, [*ibid.*]{} [**85**]{}, 134446 (2012). K. M. D. Hals, Y. Tserkovnyak, and A. Brataas, Phys. Rev. Lett. [**106**]{}, 107206 (2011); E. G. Tveten, A. Qaiumzadeh, O. A. Tretiakov, and A. Brataas, [*ibid.*]{} [**110**]{}, 127208 (2013). A. C. Swaving and R. A. Duine, Phys. Rev. B [**83**]{}, 054428 (2011): J. Phys.: Condens. Matter [**24**]{}, 024223 (2012). R. Cheng and Q. Niu, Phys. Rev. B [**89**]{}, 081105(R) (2014); R. Cheng, J. Xiao, Q. Niu, and A. Brataas, Phys. Rev. Lett. [**113**]{}, 057601 (2014). Y. Yamane, J. Ieda, and J. Sinova, Phys. Rev. B [**94**]{}, 054409 (2016); Y. Yamane, O. Gomonay, H. Velkov, and J. Sinova, [*ibid.*]{} [**96**]{}, 064408 (2017). L. Baldrati, O. Gomonay, A. Ross, M. Filianina, R. Lebrun, R. Ramos, C. Leveille, T. Forrest, F. Maccherozzi, E. Saitoh, J. Sinova, and M. Kläui, arXiv:1810.11326. J. Železný [*et al*]{}, Phys. Rev. Lett. [**113**]{}, 157201 (2014): Phys. Rev. B [**95**]{}, 014403 (2017). T. Shiino, S.-H. Oh, P. M. Haney, S.-W. Lee, G. Go, B.-G. Park, and K.-J. Lee, Phys. Rev. Lett. [**117**]{}, 087203 (2016). V. G. Bar’yakhtar, B. A. Ivanov, and M. V. Chetkin, Sov. Phys. Usp. [[**28**]{}]{}, 7 (1985). N. Papanicolaou, Phys. Rev. B [**51**]{}, 15062 (1995): [*ibid*]{}. [**55**]{}, 12290 (1997). M. Bode, E. Y. Vedmedenko, K. Von Bergmann, A. Kubetzka, P. Ferriani, S. Heinze, and R. Wiesendanger, Nat. Mater. [**5**]{}, 477 (2006). A. N. Bogdanov, U. K. Rö$\beta$ler, M. Wolf, and K.-H. Müller, Phys. Rev. B [**66**]{}, 214410 (2002). J. Barker and O. A. Tretiakov, Phys. Rev. Lett. [**116**]{}, 147203 (2016). S. S. P. Parkin, M. Hayashi, L. Thomas, Science [**320**]{}, 190 (2008). H. Chen, Q. Niu, and A. H. MacDonald, Phys. Rev. Lett. [**112**]{}, 017205 (2014). J. Kübler and C. Felser, Europhys. Lett. [**108**]{}, 67001 (2014). S. Nakatsuji, N. Kiyohara, and T. Higo, Nature [**527**]{}, 212 (2015). X. Li, L. Xu, L. Ding, J. Wang, M. Shen, X. Lu, Z. Zhu, and K. Behnia, Phys. Rev. Lett. [**119**]{}, 056601 (2017). M. Ikhlas, T. Tomita, T. Koretsune, M.-T. Suzuki, D. Nishio-Hamane, R. Arita, Y. Otani, and S. Nakatsuji, Nat. Phys. [**13**]{}, 1085 (2017). W. Feng, G.-Y. Guo, J. Zhou, Y. Yao, and Q. Niu, Phys. Rev. B [**92**]{}, 144426 (2015). T. Higo [*et al.*]{}, Nat. Photonics, [**12**]{}, 73 (2018). K. Prakhya, A. Popescu, and P. M. Haney, Phys. Rev. B [**89**]{}, 054421 (2014). O. V. Gomonay and V. M. Loktev, Low Temp. Phys. [**41**]{}, 698 (2015). J. Liu and L. Balents, Phys. Rev. Lett. [**119**]{}, 087202 (2017). The reader may wonder about the minus sign in front of the $A_2$ term. This minus sign ensures the AFM coupling among the different sublattice moments, as well as the positiveness of the spin wave velocity $c$ defined later. C. Ulloa and A. S. Nunez, Phys. Rev. B [**93**]{}, 134429 (2016). J. C. Slonczewski, J. Magn. Magn. Mat. [**159**]{}, L1 (1996); L. Berger, Phys. Rev. B [**54**]{}, 9353 (1996). A. F. Andreev and V. I. Marchenko, Phys. Usp. [[**23**]{}]{}, 21 (1980). A. Sakuma, K. Fukamichi, K. Sasao, and R. Y. Umetsu, Phys. Rev. B [**67**]{}, 024420 (2003). L. Szunyogh, B. Lazarovits, L. Udvardi, J. Jackson, and U. Nowak, Phys. Rev. B [**79**]{}, 020403(R) (2009). D. Zhang, B. Yan, S.-C. Wu, J. Kübler, G. Kreiner, S. S. P. Parkin, and C. Felser, J. Phys.: Condens. Matter [**25**]{}, 206006 (2013). N. L. Schryer and L. R. Walker, J. Appl. Phys. [**45**]{}, 5406 (1974).

--- abstract: '[The two main features of the Aharonov-Bohm effect are the topological dependence of accumulated phase on the winding number around the magnetic fluxon, and non-locality – local observations at any intermediate point along the trajectories are not affected by the fluxon. The latter property is usually regarded as exclusive to quantum mechanics. Here we show that both the topological and non-local features of the Aharonov-Bohm effect can be manifested in a classical model that incorporates random noise. The model also suggests new types of multi-particle topological non-local effects which have no quantum analog. ]{}' address: | (a) [*[School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel]{}*]{}\ (b) [*[Department of Physics, University of South Carolina, Columbia, SC 29208]{}*]{}\ (c) [*[H.H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol BS8 1TL, UK]{}*]{}\ (d) [*[Hewlett-Packard Laboratories, Stoke Gifford, Bristol BS12 6QZ, UK]{}*]{}\ (e) [*[Department of Physics, Weizmann Institute of Science, Rehovot 76100, Israel]{}*]{} author: - 'Yakir Aharonov$^{a,b}$, Sandu Popescu $^{c,d}$, Benni Reznik$^{a}$ and Ady Stern$^e$ [ ]{}' title: '[**[A classical analog to topological non-local quantum interference effect]{}**]{}' --- In the Aharonov-Bohm (AB) effect[@AB], a phase $$\phi_{AB} =\hbar q\oint \vec A \cdot \vec dl,$$ is accumulated by a charge $q$ upon circulating a solenoid enclosing a magnetic flux. This phase (mod $2\pi$) can be observed in an interference experiment in which the wavefunction of the charge is split into two wavepackets which encircle the solenoid and then interfere when meet together. The phase is [*topological*]{} because it is determined by the number of windings the charge carries out around the solenoid, and is independent of the details of the trajectory. The phase is also [*non-local*]{}: while the magnetic flux in the solenoid clearly affects the resulting interference pattern, it has no local observable consequences along any point on the trajectory. There is no experiment that one can perform anywhere along the trajectory which can tell whether or not there is a magnetic flux inside the solenoid. (In particular, there is no force acting on the charge due to the solenoid.) Various topological analogs of the AB effect have been suggested utilizing light in an optical medium[@milloni], super-fluids[@super] and particles in a gravitational background[@gravity]. However unlike the AB effect, in these analogs one can observe how the global topological effect builds up locally. Hence these models do not reproduce the non-local aspect of the AB effect. (The gravitation analog is an exceptional case. However it requires a non-trivial space-time structure, which is locally flat but globally not equivalent to a Minkowski space-time.) The above analogs suggest that quantum systems differ fundamentally from classical systems as far as non-locality is concerned. In this letter however we show that a classical non-local effect may be constructed without employing a non-trivial space-time structure. The new ingredient which allows for this is the inclusion of a random bath of particles, which “masks" local effects, but does not “screen” the net topological effect. The classical non-local effect we are constructing now is a classical analog of the Aharonov-Casher effect[@AC]. To begin with, consider a particle that is described by the canonical coordinates ${\cal \vec R}$ and ${\cal \vec P}$, and is carrying a magnetic moment $\vec \mu$. Let this particle, (henceforth referred to as the fluxon) interact with the electric field $\vec E$ generated by a homogeneously charged straight wire positioned along the $\hat z$-axis. The non-relativistic Hamiltonian describing this system is the one employed in the study of the Aharonov-Casher effect $$H_{AC} = {\frac{(\vec {{\cal P}} + \frac{1}{c}\vec\mu\times \vec E)^2}{2m}}.$$ For simplicity, in the following we confine the motion of the fluxon to a two dimensional plane orthogonal to the wire and effectively reduce the system to be planar. We denote by $\vec R$, $\vec P$ the two dimensional position and momentum, respectively. We assume that a time dependent scalar potential is applied to the fluxon to make it move along a desired trajectory in the plane, and that this potential does not interact with the magnetic moment. For brevity we do write this potential below, as it does not affect our considerations. Using polar coordinates, the electric field can be written as $\vec E(R) = 2\lambda|\vec \nabla_R\theta| \hat R$, where $\lambda$ is the linear charge density and $\vec\nabla_R$ is the gradient with respect to $\vec R$ and $\hat R$ is the unit vector along $\vec R$. If $\vec \mu$ is aligned in the $\hat z$ direction, the Hamiltonian becomes two-dimensional[@point-Q] $$H_{AC} = {\frac{(\vec P + {\frac{2}{c}}\mu \lambda \vec \nabla_R\theta )^2}{2M}}. \label{hac}$$ Classically, the forces on the particle vanish. However the magnetic field experienced by the fluxon in its rest frame, $\vec B = \frac{\vec v \times \vec E}{c}$ is non-vanishing. Therefore one expects a non-zero torque, $\vec \mu \times \vec B$, and consequently an “internal” precession of the magnetic moment. This latter effect is present both in the quantum and the classical cases [@peshkin-lipkin; @aharonov-reznik]. The precession is not described by the above Hamiltonian because $\mu$ was taken as fixed vector, rather than a degree of freedom. To incorporate the internal precession we next introduce an internal angular momentum variable $\vec L=\hat z L$, with a conjugate internal angular coordinate $\varphi$. We further assume that $\mu\propto L$ (Indeed the magnetic moment of a neutron, for example, $\mu_N = -3.7{\frac{e}{2mc}}{\bf s}$, is proportional to its spin $s$.) Replacing $\mu$ in eq. (\[hac\]) by $L$ we have $$H = {\frac{(\vec P + \xi L \vec\nabla_R\theta )^2}{2m}} \label{hl}$$ where $\xi$ is the resulting net charge/magnetic-moment coupling constant. While $L$ (and the magnitude of $\mu$) are constants of motion, the internal angle $\varphi$, conjugate to $L$, varies in time and satisfies the equation of motion $${\frac{d\varphi}{dt}} = \xi{\frac{d\vec R}{dt}} \cdot\vec\nabla_{R} \theta = \xi{\frac{d\theta}{dt}}. \label{eom}$$ Consequently $\varphi$ is entirely determined by the polar angle $\theta$ of the fluxon relative to the $x$-axis emanating from the position of the charged line[@note1]. Once we fix the initial value $\varphi(\theta_0)$ for a given $\theta_0$, the internal coordinate $\varphi$ at a later time is given by $$\varphi(\theta) = \xi(\theta -\theta_0) + \varphi(\theta_0).$$ Hence as the fluxon moves along a closed loop enclosing the charge, and $\theta$ changes by $2\pi$, the internal angle changes by $2\pi\xi$. Below we refer to the case where $\xi$ is interger as “trivial”, as it leads to an angle winding of a multiple of $2\pi$, and to the case of $\xi$ non-integer as “non-trivial”. This situation is similar to the experimentally observed Aharonov-Casher effect in a Josephson junction array [@Elion]. Up to now we have constructed a model which exhibits a classical analog of a topological effect. However since we can locally observe how the internal angle changes at intermediate points of the trajectory, this model does not capture the non-local feature of an AB-like effect. As we now turn to show, an effectively non-local behavior emerges if we add to the above system two new ingredients. Firstly, we employ two fluxons. Secondly, we consider the interaction of these fluxons with a non-trivial charged particle situated at the origin (i.e. a particle with non-integer $\xi$), and a bath of randomly positioned, moving, charged particles, all leading to trivial angle windings of the fluxon (i.e. particles with integer $\xi$). As each fluxon encircles the origin, the particles of the bath randomize its angle. These particles do not, however, randomize the angle [*difference*]{} between the two fluxons when they coincide in position. Let us denote the coordinates of the two fluxons by $\vec R_k, \vec P_k$, and internal coordinates by $\varphi_k$ and $L_k$. The coupling constants with the particles of the bath are taken to be be “trivial”, i.e., $\xi_{1i}=\xi_{2i}=1$. We denote the coordinates of the bath particles by $\vec r_i=x_i\hat x + y_i\hat y$ and their momenta by $\vec p_i$ (with $i=1...N$). The Hamiltonian of the system becomes, $$\begin{aligned} H &=& \sum_{k=1}^2{{\frac{(\vec P_k + L\vec \nabla_{R_k} [\xi \theta_k +\sum_i \theta_{ki}] )^2 }{2M}}} \nonumber \\ &+&\sum_{i=1}^N {\frac{ (\vec p_i + L\sum_{k=1}^2\vec \nabla_{r_i}\theta_{ik} )^2 }{2m_i}}.\end{aligned}$$ The first term above represents two fluxons, which interact with the charged particle at the origin and with the bath. The second term represents this “bath”. Notice that the kinetic term for the charged particles of the bath includes a vector potential, too[@chargehamiltonian]. The presence of the bath exerts additional vector potential terms $$\vec A_{ki}= L\vec \nabla_{R_k}\theta_{ki}$$ where $\theta_{ik} =\arctan{\frac{y_i-Y_k}{x_i-X_k}}$, is the angle between $\vec r_i -\vec R_k $ and the $x$ axis. As we have seen above (Eq. (\[eom\])) the internal angle changes according to the relative angle between the fluxon and the charged particle. Indeed, the equation of motion for the internal angle is $$\begin{aligned} {\frac{d \varphi_k}{dt}} &=& \xi {\frac{d\vec R_k}{dt}}\cdot \vec\nabla_{R_k}\theta_{k}+ \sum_i\biggl({\frac{d\vec R_k}{dt}}\cdot \vec \nabla_{R_k} + {\frac{d \vec r_i}{dt}} \cdot \vec \nabla_{r_i}\biggr) \theta_{ki} \nonumber \\ &=& \xi {\frac{d\theta_{k}}{dt}} + \sum_i {\frac{d\theta_{ki}}{dt}}.\end{aligned}$$ Clearly, for a sufficiently large number of randomly distributed particles, the effect of the bath on the internal angle becomes chaotic. The time dependence of the $\varphi(t)$ becomes unpredictable. Consider now however the following experiment. We start with the two fluxons situated at the same point. Then one of the fluxons stays fixed while the other moves in a path around the non-trivial charge and returns to its initial point. As noted above, the internal angles of each fluxon change randomly. But consider the the [*relative*]{} internal angle between the fluxons, $$\begin{aligned} \gamma(t) & \equiv &\varphi_2(t)-\varphi_1(t) \\ &=& \xi(\theta_{1}(t)-\theta_{2}(t)) + \sum_{i\in bath} (\theta_{i1}(t) -\theta_{i2}(t)) + constant. \nonumber\end{aligned}$$ We first note that when the two fluxons are located at precisely the same point, $\vec R_1= \vec R_2$ we have $$\theta_{i1}(t) = \theta_{i2}(t).$$ Therefore the random changes induced by the bath in the internal angles of the fluxons are [*identical*]{}, and as long as the fluxons coincide $$\gamma(t)=constant .$$ Once the fluxons move apart, the random time dependence of $\varphi_1(t)$ differs from that of $\varphi_2(t)$, and $\gamma(t)$, the relative internal angle, becomes random. Finally however, when the moving fluxon returns to its original point, after $n$ windings around the origin, and the two fluxons coincide again, $$\gamma_{final} -\gamma_{initial}= 2\pi n\xi + 2\pi N .$$ The first term is the shift caused by the charge at the origin. The second term is due to the bath and $N$ is an integer random number which counts the number of windings of bath particles. The particles of the bath can wind around only one of the fluxons while the fluxons are apart. However when the fluxons coincide, the final relative winding number $N$ is a random integer. More importantly, $(\gamma_{final}-\gamma_{initial}) mod 2\pi$ is unaffected by the bath particles, in sharp contrast to the values of $\varphi_1,\varphi_2,\gamma$ along the trajectory. Thus, upon closing a loop, the random effects due to the bath particles cancel and $(\gamma_{final}-\gamma_{initial}) mod 2\pi$ depends only on the non-trivial charge. In other words, although during the experiment the internal angles change randomly, upon closing a loop and measuring the change of the internal angle of one fluxon with the respect to the other fluxon (which acts as a reference system) we are able to recover information about the non-trivial charge. The effect is [*topological*]{}, because it depends only on the winding numbers and not on the details of the loop. Furthermore, and most important, the effect is [*non-local*]{} because no useful information can be extracted on a local basis (i.e. by monitoring the changes only on parts of the loop); only the closed loop yields information. More generally, we can allow both fluxons to move, starting from the same point and meeting later at some different point, so that the trajectories of the two fluxons form together a closed loop. The non-trivial charge can move as well. In this case $$\gamma_{final} -\gamma_{initial}= 2\pi n\xi + 2\pi N,$$ where $n$ is the winding number of the loop around the non-trivial charge while $N$ represents the winding number of the loop around the bath particles. The result above contains the essence of our effect. We will give a number of generalization later, but first let us make some comments. The key element in our effect is the addition of the random bath of trivial charges. When there are no trivial charges present, the effect is purely local - monitoring the changes of the internal angle we can tell about the presence of the non-trivial charge. The vector-potential generated by the non-trivial charge, $\xi L \vec\nabla_R\theta$ is [*“gauge-invariant"*]{} and [*observable*]{}. As we add more and more trivial charges at random positions and having random motion, the vector-potential generated by the non-trivial charge becomes [*unobservable*]{}. The only gauge invariant quantity becomes the “loop integral", i.e. the change in the relative internal angle over the closed loop. One can see a certain analogy between the observability and non-observability in this case and the gauge independence of the vector potential and its loop interal. The two particle topological effect considered here is, up to a point, analogous to an interference experiment with a single quantum particle such as the Aharonov-Bohm experiment. The relative phase accumulated along two trajectories in the quantum interference effect is hence analogous to the relative internal angle in our case. There are however significant differences. The first, and obvious difference, is that in the AB effect there is a single particle, (whose wave-function is split in two wave-packets), while in the classical analog we have two particles, each following a well-defined classical trajectory. A more subtle difference is the following. In quantum interference one is always sensitive to the relative phase of two wave packets, since the measured quantity is the [*square*]{} of the wave function. In the classical case, in contrast, we are able to generalize our model to a situation where there are three particles, with three internal angles, and where the only observable quantity involves the internal angles of all three particles. To illustrate this, we consider 3 fluxon-like particles which interact with a single non-trivial source with a coupling strength $\xi$ located at the origin, and with 3 [*different*]{} trivial random background charges denoted by $A$, $B$, and $C$. The first fluxon sees particles of type $A$ as positive charges and particles of type $B$ is negative charges. The second fluxon sees type $B$ as a positive charge and and type $C$ as negative charge and the third fluxon sees type $C$ as positive charges and type $A$ as negative charges. The Hamiltonian of the system is then $$\begin{aligned} H_3={\frac{1}{2M_1}}[P_1 +L\nabla_{R_1}(\xi\theta_{1}+\sum_i(\theta_{1i}^A- \theta_{1i}^B))]^2+ \nonumber \\ {\frac{1}{2M_2}}[P_2 +L\nabla_{R_2}(\xi\theta_{2}+ \sum_i(\theta_{2i}^B- \theta_{2i}^C))]^2 + \nonumber \\ {\frac{1}{2M_3}}[P_3 +L\nabla_{R_3}(\xi\theta_{3}+ \sum_i(\theta_{3i}^C- \theta_{3i}^A)]^2 + \nonumber \\ {\frac{1}{2m_A}}\sum_i[p_i^A+L\nabla_{r^A_i}(\theta_{i1}^A-\theta_{i3}^A)]^2+ \nonumber \\ {\frac{1}{m_B}}\sum_i[p_i^B+L\nabla_{r^B_i}(\theta_{i2}^B-\theta_{i1}^B)]^2+ \nonumber \\ {\frac{1}{2m_C}}\sum_i[P_i^C+L\nabla_{r^C_i}(\theta_{i3}^C-\theta_{i2}^C)]^2.\end{aligned}$$ Let $\phi$ be the sum of the three internal angles $\phi=\varphi_1+\varphi_2+\varphi_3$. The change in the sum of the internal angles $\delta\phi$ which in the present model is “shared” by all 3 particles is given by $$\begin{aligned} \delta\phi=\xi(\delta\theta_{1} +\delta\theta_{2} + \delta\theta_{3}) + \sum_i(\delta\theta^A_{1i}-\delta\theta^A_{3i}) + \nonumber \\ \sum_j(\delta\theta^B_{2j}-\delta\theta^B_{1j})+ \sum_k(\delta\theta^C_{3k}-\delta\theta^C_{1k}).\end{aligned}$$ We note that the contribution of the last three sums over $i,j$ and $k$ is random. However when the three fluxons start initially from the same point, and end at the same final point, the random contribution exactly cancel (modulo $2\pi$) and we are left with $$\phi_{final}-\phi_{initial} = 2\pi \xi (n_1+n_2+n_3).$$ Unlike the previous example here the change in $\phi$ yields the sum of the winding numbers of each fluxon $n_1+n_2+n_3$. The effects presented above are classical non-local analogs of quantum vector-potential effects, such as the magnetic A-B effect and the Aharonov-Casher effect. Along the same lines we now present an analog to the scalar A-B effect. This is implemented by the interaction Hamiltonian $$H_{int}=LV(x).$$ In regions where the potential $V(x)$ is constant, this interaction doesn’t generate any [*force*]{}. Indeed, the force due to this interaction term is equal to $F=-L{\frac{{dV}}{{dx}}}$ and it is zero where the potential is constant. On the other hand, the internal angle $\varphi$ is affected: due to the interaction it suffers an additional change of $V\Delta T$, where $\Delta T$ is the time spent in the potential $V$. Again, in the absence of the randomizing charges background, the change of the internal angle due to the potential is observable. However the randomizing background makes the change in the internal angle unobservable. An observable effect can be seen only in a “closed loop" experiment similar to that in the magnetic case. In conclusion, we have described a classical non-local effect, analog to the quantum Aharonov-Bohm and Aharonov-Casher effects. Although many classical analogs to the AB and AC effects are known, they exhibit only the topological character of the AB and AC effects but are local - by local measurements one can see how the topological phase build up gradually. As far as we know, this is the first classical non-local model which does not involve general relativity and non-trivial space-time structures. In our model, although one can measure at any time the internal angle of a “fluxon", the measurement yields no information about a non-trivial charge. Information can be obtained only in experiments in which a loop is closed. The key ingredient which allows us to transform a local topological effect into a non-local one is the addition of random but topologically trivial, noise. A more detailed discussion of the issue of observability versus unobservability in our model and its relations with cryptography are further discussed in [@cryptography]. Y. A., B.R. and A.S. acknowledge support from the Israel Science Foundation, established by the Israel Academy of Sciences and Humanities. Y.A. and B.R. are supported by the Israel MOD Research and Technology Unit. Y.A. acknowledges support and hospitality of the Einstein center at the Weizmann Institute of Science. Y. Aharonov and D. Bohm, Phys. Rev. [**115**]{}, 485 (1959). R. J. Cook, H. Fearn, P. W. Milonni, Am. J. of Phys. [**63**]{}, 705 (1995). F. D. M. Haldane and Y. Wu, Phys. Rev. Lett. 55, 2887-2890 (1985) Ph. Gerbet and R. Jackiw, Commun. Math. Phys. [**123**]{} 229, (1989). B. Reznik Phys. Rev. A [**51**]{}, 1728 (1995). Y. Aharonov and A. Casher, Phys. Rev. Lett. [**53**]{}, 319 (1984). Alternatively, the same effective Hamiltonian can be reached by replacing the line of charge with a point like charged source $Q$ and the point-like magnetic fluxon with line of magnetic fluxons (flux-line) that points in the fixed direction $\hat z$. M. Peshkin and H. J. Lipkin, Phys. Rev. Lett.[**74**]{}, 2847 (1995). quant-ph/9501012. Y. Aharonov and B. Reznik, Phys. Rev. Lett. [**84**]{}, 4790–4793, (2000); P. Hyllus and E. Sjoqvist, Phys. Rev. Lett. [**89**]{}, 198901 (2002); Y. Aharonov and B. Reznik, Phys. Rev. Lett. [**89**]{}, 198902 (2002). In the Hamiltonian (\[hl\]) $\varphi$ is not an independent variable, being fully determined by $\theta$. Employing a Lagrangian analysis necessitates an addition of a kinetic term $L_z^2/2I$, which removes the constraint of $\varphi$ to $\theta$ and leads to the well defined Lagrangian $$L= {\frac{1}{2}} m v^2 + {\frac{1}{2}} I(\dot\varphi - \vec v\cdot\vec \nabla\theta)^2$$ for every finite $I$. Our results are reproduced when the limit $I\rightarrow\infty$ is taken. We note that unlike the usual ${\vec v}\cdot {\vec A}$ coupling, the Lagrangian we obtain has also a quadratic term of $v\cdot A$. WJ Elion et al., Phy. Rev. Lett. [**71**]{}, 2311 (1993). There should be, of course, a kinetic term for the non-trivial charge as well. The term is absent here because, for simplicity, we took this charge to be fixed at the origin i.e. to have infinite mass. Y. Aharonov, S. Popescu and B. Reznik, in preparation.

--- abstract: '**We present the effective Hamiltonian for an electron-nucleus interaction in non-relativistic limit up to the second order in the inverse nucleon mass. This Hamiltonian takes into account the distortion of electron waves and allows to calculate its effect as well as matrix elements for off-shell transitions in $(e,e''p)$ reactions.**' --- 15.5 cm 22.cm -0.005 cm -0.005 cm -0.4 cm [**IFUP-TH 34/94**]{} **The Effective Non-Relativistic Hamiltonian\ for Electron-Nucleus Interaction\ ** **R.Ya.Kezerashvili[^1]** Dipartimento di Fisica, Universita’di Pisa,\ Piazza Torricelli 2, 56100 Pisa, Italy,\ and INFN, Sezione di Pisa,\ Piazza Torricelli 2, 56100 Pisa, Italy The quasielastic $(e,e^{'}p)$ knockout reactions have been the object of a large number of experimental and theoretical studies during the last 20 years. The continuous improvements of the experimental apparatus and the quality of the electron beams have allowed to collect precise information on single-particle properties of nuclei. Together with experimental efforts, a precise theoretical treatment has developed, which correctly takes into account all the main ingredients of this process \[1\]. The basic theoretical treatment used non-relativistic bound nucleon and continuous proton wave functions \[1-7\]. In particular, the single particle wave functions are generated by Hartree-Fock procedure and outgoing distorted proton wave function are taken as the eigenfunction of phenomenological optical potential, obtained through a fit to elastic proton-nucleus scattering. An alternative approach is to use a non-relativistic random phase approximation model to calculate the nuclear states and thereby avoid the use of an optical potential \[8,9\]. On the other hand, in Refs. 10-13 a fully relativistic approach has been adopted. Generally, in a relativistic approach the bound state nucleon wave functions are the solutions of the Dirac equation in the potential wells, derived from a relativistic mean-field Hartree calculation, since the continuous proton wave function is derived using relativistic optical potentials calculated from a Dirac phenomenological global fit to the elastic proton-nucleus scattering observable. The diagram method was suggested in Ref. 14 and it allows consequently to consider many particle short-range correlations in the $(e,e^{'}p)$ reaction. A precise treatment of $(e,e^{'}p)$ reactions needs to include, besides the outgoing proton distortion, also the electron distortion. The interaction of the electrons with the field of the nuclei changes the incoming and outgoing electron wave functions, and more realistic distorted-wave description must be, consequently, introduced in the $(e,e^{'}p)$ reaction. The problem is well known and several efforts have been done to solve it. In Ref. 15 the numerical solution of the Dirac equation for the electron in the nuclear Coulomb field has been obtained with a phase shift analysis based on a partial wave expansion. Assuming that the virtual photon emitted by the electron is absorbed by a single nucleon, the distorted wave functions of the electrons are obtained in ref. 12 by numerically solving the Dirac equation in presence of the static Coulomb potential of the nuclear charge distribution. It is necessary to note that there is disagreement between \[11\] and \[12\], even if they use the same general approach. It is desirable to use an analytic approach to the Coulomb distortion of the incoming and outgoing electron waves in the $(e,e^{'}p)$ reactions because of the rather large and difficult numerical complications. Such approach have been developed and applied in Refs. 4-7. It is based on a high energy approximation for the electron wave function \[16\]. The eikonal electron wave function is expanded up to first \[4\] and second order \[5,6\] in $Z\alpha$ $(\alpha =1/137)$. The cross section was calculated in a non-relativistic approach, based on the impulse approximation. The purpose of the present work is to construct an electron-nucleus interaction Hamiltonian in the non-relativistic limit, taking into account the distortion of the electron waves in the quasielastic electron scattering on nuclei. This Hamiltonian allows for carrying out an analytic calculation of the electron waves distortion effect in the $(e,e^{'}p)$ reaction. We make the following assumptions: i\. the incoming electron wave is distorted by the static Coulomb field of the target nucleus. ii\. the resulting distorted electron wave interacts with a single nucleon inside the nucleus and knocks it out. iii\. the outgoing electron is distorted by the static Coulomb field of the residual nucleus. Thus, our treatment includes distortion of the electron waves due to the static nuclear Coulomb field and we restrict ourself to one photon exchange. Such correction is proportional to $Z/137$ and becomes critical for heavy nuclei. It implies neglecting terms arising from the exchange of two or more photons in the scattering event, terms which are supposed to give a small contribution. The covariant interaction between electrons and relativistic nucleons is, of course, well known and the Dirac equation for a nucleon in an given arbitrary external electromagnetic field is $$\Bigl[\,\gamma_{\mu}\partial_{\mu} + M -ieF_{1}A_{\mu}\gamma_{\mu}+ \frac{e\,K\,F_{2}}{2\,M}\partial_{\nu}A_{\mu}\sigma_{\mu\nu}\, \Bigr]\Psi =0\,,$$ where $$\sigma_{\mu\nu}=-\frac{i}{2}(\gamma_{\mu}\gamma_{\nu}- \gamma_{\nu}\gamma_{\mu}).$$ In formula (1), $F_{1}$ and $F_{2}$ are the electromagnetic nucleon form-factors, having usual normalization, $K$ is the anomalous magnetic moment of the nucleon in nuclear magneton, $M$ is the nucleon mass and the field strength are given through the four-vector potential $A_{\mu}=(\mbox{\boldmath $A$},A_{4})\equiv(\mbox{\boldmath $A$},i\phi)$. If we restrict to the non-relativistic limit, in which nucleons interact like non-relativistic Pauli particles, we need to reduce the Dirac equation (1) to a form involving only two-component spinors for the nucleons. The technique for carrying out the non-relativistic limit of Dirac operators involves the construction of a series of the successive unitary transformations, whose product is known as Foldy-Wouthuysen transformation \[17\]. This transformation decouples equation (1) into two two-component equations, one of which reduces to the non-relativistic description, while the other describes the negative energy states. By following McVoy and Van Hove \[18\], applying the Foldy-Wouthuysen transformation to eq. (1) and retaining terms up to $M^{-2}$ order in the positive energy equation, finally the Hamiltonian $$H=-i\,e\,F_{1}\,A_{4}-\frac{e\,F_{1}}{2\,M} \bigl(\,\mbox{\boldmath $p\cdot A$}+ \mbox{\boldmath $A$}\cdot \mbox{\boldmath $p$}\,\bigr)-$$ $$\frac{e\,(F_{1}+K\,F_{2})}{2\,M}(\mbox{\boldmath $\sigma\cdot$} \mbox {\boldmath $H$})\,+$$ $$\frac{e\,(F_{1}+2\,K\,F_{2})}{8\,M^{2}}\Bigl(\mbox{\boldmath $\sigma \cdot$} \bigl( [\,\mbox{\boldmath $p$}\times \mbox{\boldmath $E$}\,]-[\mbox{\boldmath $E$} \times \mbox{\boldmath $p$}\,]\,\bigr)\Bigr)\,-$$ $$\frac{e\,(F_{1}+2\,K\,F_{2})}{8\,M^{2}} \,div\mbox{\boldmath $E$}$$ is obtained. In the above expression $\mbox{\boldmath $p$}$ and $\mbox{\boldmath $\sigma$}$ are the momentum and Pauli matrices for the nucleon and the fields $\mbox{\boldmath $E$}$ and $\mbox{\boldmath $H$}$ are generated by the four-vector potential $A_{\mu}$ $$\mbox{\boldmath $E$}=i\,grad\,A_{4}-\frac{\partial \mbox{\boldmath $A$}}{\partial t}\,,\quad \mbox{\boldmath $H$}=curl\mbox{\boldmath $A$}\,.$$ The Hamiltonian (3) describe the interaction of the non-relativistic nucleon with an arbitrary electromagnetic field. In the plane wave approximation, when the incoming and the outgoing electron are described by plane waves, $A_{\mu}$ will be the Moller potential \[19\]. By putting the Moller potential into equation (3) and (4), we obtain the McVoy and Van Hove Hamiltonian \[18\]. This Hamiltonian is well known and it has been widely used for investigations of the $(e,e^{'}p)$ processes. What is the effect of the distortion, caused by the interaction of the target and the residual nuclei with the incoming and the outgoing electron waves? Failing a complete calculation, a simple prescription is usually adopted to include a first contribution of electron distortion. The electron plane wave is replaced by $e^{i\mbox{\boldmath $k^{'}r$}}$ \[20\] with $$\mbox{\boldmath $k^{'}$}=\mbox{\boldmath $k$}+ U(0)\mbox{\boldmath $k$}/k \,,$$ where $\mbox{\boldmath $k$}$ is the electron momentum and $U(0)$ is a mean value of the electromagnetic nuclear potential. In Ref. 21 the electron plane wave was replaced by the plane wave with effective momentum $\mbox{\boldmath $k^{'}$}$ and amplitude $k^{'}/k$. This well known result is called the effective momentum approximation. An high energy electron wave function was obtained in Ref. 22 and the final expression for the distorted electron wave is represented by a plane wave with the changed phase and moduli. The corresponding expressions for this function is expanded in powers of $Z\alpha$ and retaining terms up to $Z\alpha$ and $(Z\alpha)^{2}$ are introduced in Refs.4-6 and they represent the plane wave with the changed phase and amplitude too. So, in any case, we can conclude, that from a mathematical point of view, the distortion changes the amplitude and the phase of the plane electron wave. This leads to the corresponding reduced electron density and current. Thus, in general , the electromagnetic potential generated by the distorted electron density and current can be written in the form $$A_{4}=i\frac{4 \pi e\,u^{+}_{f}\,u_{i}}{q_{\mu}^{2}}\,V(\mbox{\boldmath $r$}) \,e^{iq_\mu r_\mu + i\Phi(\mbox{\boldmath $r$})}\,,$$ $$\mbox{\boldmath $A$}= \frac{4\pi e\,u^{+}_{f}\, \mbox{\boldmath $\alpha$}\, u_{i}}{q_{\mu}^{2}}\,V(\mbox{\boldmath $r$})\, e^{iq_\mu r_\mu + i\Phi(\mbox{\boldmath $r$})}\,,$$ where $u$ is the Pauli spinor for the free electron, $\mbox{\boldmath $\alpha$}$ is the Dirac velocity matrix and $q_{\mu}$ is the four-momentum transfer. In formulas (6) $V(\mbox{\boldmath $r$})$ and $\Phi(\mbox{\boldmath $r$})$ are amplitude and phase functions, reproducing the distortion of the incoming and the outgoing electron waves. Those functions have continuous second order derivatives. Putting $V(\mbox{\boldmath $r$})=1$ and $\Phi(\mbox{\boldmath $r$})=0$ into (6), we obtain the Moller potential. Substituting the electromagnetic potential (6), induced by the distorted electron waves, into eqs. (3) and (4), we finally obtain $$H^{'}=\frac{4\,\pi\,e^{2}}{q_{\mu}^{2}} <\,u_{f}|\,F_{1}\,V(\mbox{\boldmath $r$})\, e^{iq_\mu r_\mu + i\Phi(\mbox{\boldmath $r$})}\,-$$ $$\frac{F_{1}}{2\,M}\Bigl(\,(\mbox{\boldmath $p\cdot \alpha$})\, V(\mbox{\boldmath $r$})\,e^{iq_\mu r_\mu + i\Phi(\mbox{\boldmath $r$})}+ V(\mbox{\boldmath $r$})\,e^{iq_\mu r_\mu + i\Phi(\mbox{\boldmath $r$})} (\mbox{\boldmath $p\cdot \alpha$})\Bigr)\,-$$ $$\frac{F_{1}+K\,F_{2}}{2\,M}\,\Bigl\{ i\Bigl(\mbox{\boldmath $\sigma\cdot$} [\, (\, \mbox{\boldmath $q$}+\bigtriangledown \Phi(\mbox{\boldmath $r$})\,) \mbox{\boldmath $\times\alpha$} \,]\,\Bigr)\,V(\mbox{\boldmath $r$})\, e^{iq_\mu r_\mu + i\Phi(\mbox{\boldmath $r$})}\,+$$ $$\Bigl(\mbox{\boldmath $\sigma\cdot$} [\,\bigtriangledown V(\mbox{\boldmath $r$})\mbox{\boldmath $\times\alpha$} \,]\,\Bigr)\,e^{iq_\mu r_\mu + i\Phi(\mbox{\boldmath $r$})}\,\Bigr\}\, +$$ $$\frac{F_{1}+2KF_{2}}{8\,M^{2}}\,\Bigl\{\,i \mbox{\boldmath $\sigma\cdot$}\Bigl( [\,\mbox{\boldmath $p\times$} (\omega \mbox{\boldmath $\alpha$}-\mbox{\boldmath $q$}-\bigtriangledown \Phi(\mbox{\boldmath $r$}) )\,]\, V(\mbox{\boldmath $r$})\,e^{iq_\mu r_\mu + i\Phi(\mbox{\boldmath $r$})}\,-$$ $$V(\mbox{\boldmath $r$})\,e^{iq_\mu r_\mu + i\Phi(\mbox{\boldmath $r$})}\, [\,( \omega \mbox{\boldmath $\alpha$}-\mbox{\boldmath $q$}-\bigtriangledown \Phi(\mbox{\boldmath $r$}))\times \mbox{\boldmath $p$}]\,\Bigr)\,-$$ $$\mbox{\boldmath $\sigma\cdot$}\Bigl( [\,\mbox{\boldmath $p\times$} \bigtriangledown V(\mbox{\boldmath $r$})\,] e^{iq_\mu r_\mu + i\Phi(\mbox{\boldmath $r$})} - e^{iq_\mu r_\mu + i\Phi(\mbox{\boldmath $r$})} [\,\bigtriangledown V(\mbox{\boldmath $r$})\mbox{\boldmath $\times p$}\,]\, \Bigr)\,\,\Bigr\}\,-$$ $$\frac{F_{1}+2K\,F_{2}}{8\,M^{2}}\,\Bigl\{\,(\mbox{\boldmath $q$}+ \bigtriangledown \Phi(\mbox{\boldmath $r$}))^{2}\,V(\mbox{\boldmath $r$})\, e^{iq_\mu r_\mu + i\Phi(\mbox{\boldmath $r$})}\,-$$ $$2ie^{iq_\mu r_\mu + i\Phi(\mbox{\boldmath $r$})} (\mbox{\boldmath $q$}+\bigtriangledown \Phi(\mbox{\boldmath $r$})\,) \mbox{\boldmath $\cdot$}\bigtriangledown V(\mbox{\boldmath $r$})\,-$$ $$iV(\mbox{\boldmath $r$})\, e^{iq_\mu r_\mu + i\Phi(\mbox{\boldmath $r$})} \bigtriangledown ^{2} \Phi(\mbox{\boldmath $r$})- e^{iq_\mu r_\mu + i\Phi(\mbox{\boldmath $r$})} \bigtriangledown ^{2}V(\mbox{\boldmath $r$})\,\Bigr\}\,|\,u_{i}>\,,$$ where $\mbox{\boldmath $\alpha$}$ acts in the space of the electron spinors and $\omega$ is the energy transfer. Eq. (7) introduce the effective Hamiltonian of electron-nucleon interaction in nuclear space, which describes the interaction of electron waves distorted by nuclei with non-relativistic nucleons. When we go over to the consideration of a nucleus, the result of eq. (7) must be summed over the A nucleons. The first three terms of the Hamiltonian (7), which describe the Coulomb, convection current and spin or magnetization current interactions with distorted electron wave are of the order of unity and $M^{-1}$. These terms provide the dominant contributions, especially at lower momentum transfers. The two other terms, of order $M^{-2}$, do contribute since the momentum transfers we deal with may be quite large. They are important at large momentum transfers. If we substitute $V(\mbox{\boldmath $r$})=1$ and $\Phi(\mbox{\boldmath $r$})=0$ into equation (7) the McVoy and Van Hove Hamiltonian \[18\] is obtained. The effective momentum transfer $\mbox{\boldmath $q$}+\bigtriangledown \Phi( \mbox{\boldmath $r$})$ in eq.(7) differs from $\mbox{\boldmath $q$}$ both in magnitude and in direction. When the change of the direction is neglected, the elastic scattering condition is obtained. If we choose $V(\mbox{\boldmath $r$})=1$ and $\Phi(\mbox{\boldmath $r$})=U(0)(\mbox{\boldmath $\hat{q}\cdot r$})$, where $U(0)=3Z\alpha/2R$ with $R=(5/3)^{1/2}<r>^{1/2}$ for the nucleus with charge $Z$ and rms-radius $<r>^{1/2}$, we are considering the distorted electron waves in the approximation used in Ref. 20. In this case, eq. (7) simplifies, because terms containing $\bigtriangledown V(\mbox{\boldmath $r$})$ disappear. The same simplification will happen in the high energy approximation \[18\], but taking the amplitude $k{'}/k$ for the plane wave. The comparison the of equations (6) with the electromagnetic potential obtained in Ref. 4, allows to find corresponding expressions for $V(\mbox{\boldmath $r$})$ and $\Phi(\mbox{\boldmath $r$})$. Using those expressions in the Hamiltonian (7), the distortion of the electron waves will be taken into account, based on the approximation of Ref. 16. In Ref. 3 the non-relativistic Hamiltonian for free electron-nucleon scattering up to the fourth order in the inverse nucleon mass was obtained. Starting from this Hamiltonian and using the same procedure, we obtain the effective non-relativistic Hamiltonian, describing the interaction of the electron wave distorted by nuclei with the nucleon, corrected up to $M^{-4}$. Using the Hamiltonian (7) for the description of $(e,e{'}p)$ reactions, the off-shell transition matrix elements can be calculated and the distortion of electron wave in the initial and final states can be included. I wish to thank Prof. A.Fabrocini for helpful discussion and the Department of Physics of Pisa University for the kind hospitality. 1. S Boffi, C.Giusti and F.D.Pacati. Phys.Rep. [**226**]{} (1993) 1. 2. S Boffi, C.Giusti and F.D.Pacati. Nucl.Phys. [**A 386**]{} (1982) 599. 3. C.Giusti and F.D.Pacati. Nucl.Phys. [**A 336**]{} (1980) 427. 4. C.Giusti and F.D.Pacati. Nucl.Phys. [**A 476**]{} (1987) 717. 5. C.Giusti and F.D.Pacati. Nucl.Phys. [**A 485**]{} (1988) 461. 6. M.Traini, S.Turck-Chièze and A.Zghiche. Phys.Rev. [**C 38**]{} (1988) 2799. 7. M.Traini. Phys.Lett. [**B 213**]{} (1988) 1. 8. J.Ryckebusch, K.Heyde, D.Van Neck and M.waroquier. Phys.Lett. [**B 216**]{} (1989) 252. 9. M.Cavinato, M.Marangoni, and A.M.Saruis. Z.Phys. [**A 335**]{} (1990) 401. 10. A.Picklesimer, J.W.Van Orden and S.J.Wallace. Phys.Rev. [**C 32**]{} (1985) 1312. 11. J.P.McDermott. Phys.Rev.Lett. [**65**]{} (1990) 1991. 12. Yanhe Jin, D.S.Onley and L.E.Wright. Phys.Rev. [**C 45**]{} (1992) 1311. 13. Yanhe Jin, J.K.Zhang, D.S.Onley and L.E.Wright. Phys.Rev. [**C 47**]{} (1993) 2024. 14. R.I.Jibuti and R.Ya.Kezerashvili Yad.Fiz. [**22**]{} (1975) 975 \[Sov.J.Nucl. Phys. [**22**]{} (1975) 508. 15. R.D.Viollier and K.Alder. Helv.Phys.Acta. [**44**]{} (1971) 77. 16. J.Knoll. Nucl.Phys. [**A 223**]{} (1974) 462. 17. L.L.Foldy and S.A.Wouthuysen. Phys.Rev. [**78**]{} (1950) 29. 18. K.W.McVoy and L.Van Hove. Phys.Rev. [**125**]{} (1962) 1034. 19. C.Moller. Z. Phys. [**70**]{} (1931) 786. 20. W.Czyz and K.Gottfried. Ann. of Phys. [**21**]{} (1963) 47. 21. R.Rosenfelder. Ann. of Phys. [**128**]{} (1980) 188. 22. F.Lenz and R.Rosenfelder. Nucl.Phys. [**A 176**]{} (1971) 513. [^1]: Department of Physics, Tbilisi State University, 380028, Tbilisi, Republic of Georgia.

--- abstract: 'Gamma-ray burst (GRB) observations at very high energies (VHE, [*[E]{}*]{} $>$ 100GeV) can impose tight constraints on some GRB emission models. Many GRB afterglow models predict a VHE component similar to that seen in blazars and plerions, in which the GRB spectral energy distribution has a double-peaked shape extending into the VHE regime. VHE emission coincident with delayed X-ray flare emission has also been predicted. GRB follow-up observations have had high priority in the observing program at the Whipple 10m Gamma-ray Telescope and GRBs will continue to be high priority targets as the next generation observatory, VERITAS, comes on-line. Upper limits on the VHE emission, at late times ($>$$\sim$4 hours), from seven GRBs observed with the Whipple Telescope are reported here.' author: - 'D. Horan, R. W. Atkins, H. M. Badran, G. Blaylock, S. M. Bradbury, J. H. Buckley, K. L. Byrum, O. Celik, Y. C. K. Chow, P. Cogan, W. Cui, M. K. Daniel, I. de la Calle Perez, C. Dowdall, A. D. Falcone, D. J. Fegan, S. J. Fegan, J. P. Finley, P. Fortin, L. F. Fortson, G. H. Gillanders, J. Grube, K. J. Gutierrez, J. Hall, D. Hanna, J. Holder, S. B. Hughes, T. B. Humensky, G. E. Kenny, M. Kertzman, D. B. Kieda, J. Kildea, H. Krawczynski, F. Krennrich, M. J. Lang, S. LeBohec, G. Maier, P. Moriarty, T. Nagai, R. A. Ong, J. S. Perkins, D. Petry, J. Quinn, M. Quinn, K. Ragan, P. T. Reynolds, H. J. Rose, M. Schroedter, G. H. Sembroski, D. Steele, S. P. Swordy, J. A. Toner, L. Valcarcel, V. V. Vassiliev, R. G. Wagner, S. P. Wakely, T. C. Weekes, R. J. White, D. A. Williams' title: 'Very High Energy Observations of Gamma-Ray Burst Locations with the Whipple Telescope' --- Introduction {#intro} ============ Since their discovery in 1969 [@Klebesadel:73], gamma-ray bursts (GRBs) have been well studied at many wavelengths. Although various open questions remain on their nature, there is almost universal agreement that the basic mechanism is an expanding relativistic fireball, that the radiation is beamed, that the prompt emission is due to internal shocks and that the afterglow arises from external shocks. It is likely that Lorentz factors of a few hundred are involved, with the radiating particles, either electrons or protons, being accelerated to very high energies. GRBs are sub-classified into two categories, long and short bursts, based on the timescale over which 90% of the prompt gamma-ray emission is detected. Recently, the Swift GRB Explorer [@Gehrels:04] has revealed that many GRBs have associated X-ray flares [@Burrows:05; @Falcone:06a]. These flares have been detected between 10$^2$s and 10$^5$s after the initial prompt emission and have been found to have fluences ranging from a small fraction of, up to a value comparable to, that contained in the prompt GRB emission. This X-ray flare emission has been postulated to arise from a number of different scenarios, including late central engine activity where the GRB progenitor remains active for some time after, or re-activates after, the initial explosion [@Kumar:00; @Zhang:06; @Nousek:06; @Perna:05; @Proga:06; @King:05] and refreshed shocks which occur when slower moving shells ejected by the central engine in the prompt phase catch up with the afterglow shock at late times [@Rees:98; @Sari:00; @Granot:03a; @Guetta:06]. For short GRBs, shock heating of a binary stellar companion has also been proposed [@MacFadyen:05]. It is not yet clear whether the X-ray flares are the result of prolonged central engine activity, refreshed shocks or some other mechanism [@Panaitescu:06]. A very high energy (VHE; E $>$100GeV) component of this X-ray flare emission has also been predicted [@Wang:06]. Within the standard fireball shock scenario [@Rees:92; @Meszaros:93; @Sari:98], many models have been proposed which predict emission at and above GeV energies during both the prompt and afterglow phases of the GRB. These have been summarized by @Zhang:04, and references therein, and include leptonic models in which gamma rays are produced by electron self-inverse-Compton emission from the internal shocks or from the external forward or reverse shocks. Other models predict gamma rays from proton synchrotron emission or photomeson cascade emission in the external shock or from a combination of proton synchrotron emission and photomeson cascade emission from internal shocks. Although GRB observations are an important component of the program at many VHE observatories, correlated observations at these short wavelengths remain sparse even though tantalizing and inherently very important. The sparsity of observations of GRBs at energies above 10MeV is dictated not by lack of interest in such phenomena, or the absence of theoretical predictions that the emission should occur, but by the experimental difficulties. For the observation of photons of energies above 100GeV, only ground-based telescopes are available at present. These ground-based telescopes fall into two broad categories, air shower arrays and atmospheric Cherenkov telescopes (of which the majority are Imaging Atmospheric Cherenkov Telescopes, or IACTs). The air shower arrays, which have wide fields of view making them particularly suitable for GRB searches, are relatively insensitive. There are several reports from these instruments of possible TeV emission. @Padilla:98:Airobicc reported possible VHE emission at [*[E]{}*]{} $>$ 16 TeV from GRB920925c. While finding no individual burst which is statistically significant, the Tibet-AS$\gamma$ Collaboration found an indication of 10TeV emission in a stacked analysis of 57 bursts [@Amenomori:01:TibetGRBs]. The Milagro Collaboration reported on the detection of an excess gamma-ray signal during the prompt phase of GRB970417a with the Milagrito detector [@Atkins00]. In all of these cases however, the statistical significance of the detection is not high enough to be conclusive. In addition to searching the Milagro data for VHE counterparts for 25 satellite-triggered GRBs [@Atkins:05:MilagroGRBCounterparts], the Milagro Collaboration conducted a search for VHE transients of 40 seconds to 3 hours duration in the northern sky [@Atkins:04:MilagroGRB]. No evidence for VHE emission was found from either of these searches and upper limits on the VHE emission from GRBs were derived. Atmospheric Cherenkov telescopes, particularly those that utilize the imaging technique, are inherently more flux-sensitive than air shower arrays and have better energy resolution but are limited by their small fields of view (3-5$^\circ$) and low duty cycle ($\sim$7%). In the Burst And Transient Source Explorer [@Meegan:92] era (1991-2000), attempts at GRB monitoring were limited by slew times and uncertainty in the GRB source position [@Connaughton:97]. Swift, the first of the next generation of gamma-ray satellites which will include AGILE (Astro-rivelatore Gamma a Immagini LEggero) and the Gamma-ray Large Area Space Telescope (GLAST), is beginning to provide arcminute localizations so that IACTs are no longer required to scan a large GRB error box in order to achieve full coverage of the possible emission region. The work in this paper covers the time period prior to the launch of the Swift satellite. The minimum detectable fluence with an IACT, such as the Whipple 10m, in a ten second integration is $<$10$^{-8}$ergcm$^{-2}$ (5 photons of 300GeV in 5x10$^8$cm$^2$ collection area). This is a factor of $>$100 better than GLAST will achieve (3 photons of 10GeV in 10$^4$cm$^2$ collection area). This ignores the large solid angle advantage of a space telescope and the possible steepening of the observable spectrum because of the inherent emission mechanism and the effect of intergalactic absorption by pair production. There have been many predictions of high energy GRB emission in and above the GeV energy range [@Meszaros:94a; @Boettcher:98; @PillaLoeb:98; @Wang:01; @Zhang:01; @Guetta:03b; @Dermer:04; @Fragile:04]; also see @Zhang:04 and references therein. Until AGILE and GLAST are launched, the GRB observations that were made by the EGRET experiment on the Compton Gamma Ray Observatory (CGRO) will remain the most constraining in the energy range from 30MeV to 30GeV. Although EGRET was limited by a small collection area and large dead time for GRB detection, it made sufficient detections to indicate that there is a prompt component with a hard spectrum that extends at least to 100MeV energies. The average spectrum of four bursts detected by EGRET (GRBs910503, 930131, 940217 & 940301) did not show any evidence for a cutoff up to 10 GeV [@Dingus:01]. The relative insensitivity of EGRET was such that it was not possible to eliminate the possibility that all GRBs had hard components [@Dingus98]. EGRET also detected an afterglow component from GRB940217 that extended to 18GeV for at least 1.5 hours after the prompt emission indicating that a high-energy spectral component can extend into the GeV band for a long period of time, at least for some GRBs [@Hurley94]. The spectral slope of this component is sufficiently flat that its detection at still higher energies may be possible [@Mannheim96]. @Meszaros:94b attribute this emission to the combination of prompt MeV radiation from internal shocks with a more prolonged GeV inverse Compton component from external shocks. It is also postulated that this emission could be the result of inverse Compton scattering of X-ray flare photons [@Wang:06]. Although somewhat extreme parameters must be assumed, synchrotron self-Compton emission from the reverse shock is cited as the best candidate for this GeV emission by @Granot:03b, given the spectral slope that was recorded. This requirement of such extreme parameters naturally explains the lack of GRBs for which such a high energy component has been observed. @Guetta:03a postulate that some GRB explosions occur inside pulsar wind bubbles. In such scenarios, afterglow electrons upscatter pulsar wind bubble photons to higher energies during the early afterglow thus producing GeV emission such as that observed in GRB940217. The GRB observational data are extraordinarily complex and there is no complete and definitive explanation for the diversity of properties observed. It is important to establish whether there is, in general, a VHE component of emission present during either the prompt or afterglow phase of the GRB. Understanding the nature of such emission will provide important information about the physical conditions of the emission region. One definitive observation of the prompt or afterglow emission could significantly influence our understanding of the processes at work in GRB emission and its aftermath. In this paper, the GRBs observed with the Whipple 10m Gamma-ray Telescope in response to HETE-2 and INTEGRAL notifications are described. The search for VHE emission is restricted to times on the order of hours after the GRB. In Section \[observe\], the observing strategy, telescope configuration, and data analysis methods used in this paper are described. The properties of the GRBs observed and their observation with the Whipple Telescope are described in Section \[grbs\]. Finally, in Section \[resultsAndDiscussion\], the results are summarized and their implications discussed in the context of some theoretical models that predict VHE emission from GRBs. The sensitivity of future instruments such as VERITAS to GRBs is also discussed. The Gamma-ray Burst Observations {#observe} ================================ Telescope Configuration ----------------------- The observations presented here were made with the 10m Gamma-ray Telescope at the Fred Lawrence Whipple Observatory. Constructed in 1968, the telescope has been operated as an IACT since 1982 [@Kildea:06]. In September 2005, the observing program at the 10m was redefined and the instrument was dedicated solely to the monitoring of TeV blazars and the search for VHE emission from GRBs. Located on Mount Hopkins approximately 40 km south of Tucson in southern Arizona at an altitude of 2300m, the telescope consists of 248 hexagonal mirror facets mounted on a 10m spherical dish with an imaging camera at its focus. The front-aluminized mirrors are mounted using the Davies-Cotton design [@Davies57]. The imaging camera consists of 379 photo-multiplier tubes (PMTs) arranged in a hexagonal pattern. A plate of light-collecting cones is mounted in front of the PMTs to increase their light-collection efficiency. A pattern-sensitive trigger [@Bradbury:02], generates a trigger whenever three adjacent PMTs register a signal above a level preset in the constant fraction discriminators. The PMT signals for each triggering event are read out and digitized using charge-integrating analog to digital converters. In this way, a map of the amount of charge in each PMT across the camera is recorded for each event and stored for offline analysis. The telescope triggers at a rate of $\sim\,25$Hz (including background cosmic ray triggers) when pointing at high ($>$50$^\circ$) elevation. Although sensitive in the energy range from 200GeV to 10TeV, the peak response energy of the telescope to a Crab-like spectrum during the observations reported upon here was approximately 400GeV. This is the energy at which the telescope is most efficient at detecting gamma rays and is subject to a 20% uncertainty. Observing Strategy ------------------ Burst notifications at the Whipple Telescope for the observations described here were received via email from the Global Coordinates Network [@GCN:webpage]. When a notification email arrived, the GRB location and time were extracted and sent to the telescope tracking control computer. An audible alarm sounded to alert the observer of the arrival of a burst notification. If at sufficient elevation, the observer approved the observations and the telescope was commanded to slew immediately to the location of the GRB. The Whipple Telescope slews at a speed of 1$^\circ$s$^{-1}$ and therefore can reach any part of the visible sky within three minutes. Seven different GRB locations were observed with the Whipple 10m Telescope between November 2002 and April 2004. These observations are summarized in Table \[grb\_summary\]. At the time these data were taken, the point spread function of the Whipple Telescope was approximately 0.1$^\circ$ which corresponds to the field of view of one PMT. The positional offsets for the GRB observations (see Table \[observations\_and\_results\]) were all less than this so a conventional “point source” analysis was performed. Data Analysis ------------- The data were analyzed using the imaging technique and analysis procedures pioneered and developed by the Whipple Collaboration [@Reynolds93]. In this method, each image is first cleaned to exclude the signals from any pixels that are most likely the result of noise. The cleaned images are then characterized by calculating and storing the first, second, and third moments of the light distribution in each image. The parameters and this procedure are described elsewhere [@Reynolds93]. Since gamma-ray images are known to be compact and elliptical in shape, while those generated by cosmic ray showers tend to be broader with more fluctuations, cuts can be derived on the above parameters which reject approximately 99.7% of the background images while retaining over 50% of those generated by gamma-ray showers. These cuts are optimized using data taken on the Crab Nebula which is used as the standard candle in the TeV sky. Two different modes of observation are employed at the Whipple Telescope, “[*[On-Off]{}*]{}” and “[*[Tracking]{}*]{}” [@Catanese:98]. The choice of mode depends upon the nature of the target. The GRB data presented here were all taken in the [*[Tracking]{}*]{} mode. Unlike data taken in the [*[On-Off]{}*]{} mode, scans taken in the [*[Tracking]{}*]{} mode do not have independent control data which can be used to establish the background level of gamma-ray like events during the scan. These control data are essential in order to estimate the number of events passing all cuts which would have been detected during the scan in the absence of the candidate gamma-ray source. In order to perform this estimate, a tracking ratio is calculated by analyzing “darkfield data” [[@Horan02]]{}. These consist of [*[Off-source]{}*]{} data taken in the [*[On-Off]{}*]{} mode and of observations of objects found not to be sources of gamma rays. A large database of these scans is analyzed and in this way, the background level of events passing all gamma-ray selection criteria can be characterized as a function of zenith angle. Since the GRB data described in this paper were taken at elevations between 50$^\circ$ and 80$^\circ$, a large sample of darkfield data ($\sim$ 233 hours) spanning a similar zenith angle range was analyzed so that the background during the gamma-ray burst data runs could be estimated. The Gamma-ray Bursts {#grbs} ==================== This paper concentrates on the GRB observations made in response to HETE-2 and INTEGRAL triggers with the Whipple 10m Gamma-ray Telescope; observations made in response to Swift triggers are the subject of a separate paper [@Dowdall]. When the GRB data were filtered to remove observations made at large zenith angles, during inferior weather conditions, and of positions later reported to be the result of false triggers or to have large positional errors, the data from observations of seven GRB locations remained. These GRBs took place between UT dates 021112 and 040422; two have redshifts derived from spectral measurements, one has an estimated redshift and four lie at unknown distances. Five of the sets of GRB follow-up observations were carried out in response to triggers from the high energy transient explorer 2 (HETE-2; @Lamb:00) while two sets of observations were triggered by the international gamma-ray astrophysical laboratory (INTEGRAL; @Winkler:99). In the remainder of this section, the properties of each of the GRBs observed and the results of these observations are presented. A summary of the GRB properties is given in Table \[grb\_summary\] while the observations taken at the Whipple Observatory are summarized in Table \[observations\_and\_results\]. GRB021112 --------- This was a long GRB with a duration of $>$ 5s and a peak flux of $>$3x10$^{-8}$ergcm$^{-2}$s$^{-1}$ in the 8-40keV band [@GCN1682:hete2]. In the 30-400keV energy band, the burst had a peak energy of 57.15keV, a duration of 6.39s and a fluence of 2.1x10$^{-7}$ergcm$^{-2}$ [@hete2webpage]. The triggering instrument was the French Gamma Telescope (FREGATE) instrument on HETE-2. The Milagro data taken during the time of this burst were searched for GeV/TeV gamma-ray emission. No evidence for prompt emission was found and a preliminary analysis, assuming a differential photon spectral index of -2.4, gave an upper limit on the fluence at the 99.9% confidence level of [*[J]{}*]{}(0.2-20TeV) $<$2.6x10$^{-6}$ergcm$^{-2}$ over a 5 second interval [@McEnery:02:GCN1724]. Optical observations with the 0.6-meter Red Buttes Observatory Telescope beginning 1.8 hours after the burst did not show any evidence for an optical counterpart and placed a limiting magnitude of [*[R$_c$]{}*]{}=21.8 (3sigma) on the optical emission: at the time, this was the deepest non-detection of an optical afterglow within 2.6 hours of a GRB [@GCN1776:Optical021112]. Two sets of observations on the location of GRB021112 were made with the Whipple 10m Telescope. The first observations commenced 4.2 hours after the GRB occurred and lasted for 110.6 minutes. Observations were also taken for 55.3 minutes on the following night, 28.6 hours after the GRB occurred. Upper limits (99.7% c.l.) of 0.20 Crab[^1] and 0.30 Crab (E $>$ 400 GeV), respectively, were derived for these observations assuming a Crab-like spectrum (spectral index of -2.49). GRB021204 --------- Little information is available in the literature on this HETE-2 burst. The GRB location was observed with a number of optical telescopes (the RIKEN 0.2m [@Torii:02:GCN1730], the 32 inch Tenagra II [@Nysewander:02:GCN1735], and the 1.05m Schmidt at Kiso Observatory [@GCN1747:RIKEN]) but no optical transient was found to a limiting magnitude of [*[R]{}*]{}=16.5, 2.1 hours after the burst [@Torii:02:GCN1730], and to [*[R]{}*]{}=18.8, 6.2 hours after the burst [@GCN1747:RIKEN]. Whipple observations of this burst location commenced 16.9 hours after the GRB occurred and lasted for 55.3 minutes. An upper limit (99.7% c.l.) of 0.33 Crab was derived for the VHE emission above 400GeV during these observations. GRB021211 --------- This long, bright burst was detected by all three instruments on HETE-2. It had a duration $>$ 5.7s in the 8-40keV band with a fluence of $\sim$ 10$^{-6}$ergcm$^{-2}$ during that interval [@GCN1734:hete2]. The peak flux was $>$8x10$^{-7}$ergcm$^{-2}$s$^{-1}$ (i.e. $>$20 Crab flux) in 5ms [@GCN1734:hete2]. This burst had a peak energy of 45.56keV, a duration of 2.80s, and a fluence of 2.4x10$^{-6}$ergcm$^{-2}$ in the 30-400keV energy band [@hete2webpage]. @Fox:EarlyOptical:03 reported on the early optical, near-infrared, and radio observations of this burst. They identified a break in the optical light curve of the burst at t=0.1-0.2hr, which was interpreted as the signature of a reverse shock. The light curve comprised two distinct phases. The initial steeply-declining flash was followed by emission declining as a typical afterglow with a power-law index close to 1. KAIT observations of the afterglow also detected the steeply declining light curve and evidence for an early break [@Li:EarlyOptical:03]. The optical transient was detected at many observatories [@GCN1736:Super-LOTIS; @GCN1737:KAIT; @GCN1744:ARC-ACO; @GCN1750:MMT; @GCN1751:Magellan-Baade]. The optical transient faded from an R-band magnitude of 18.3, 20.7 minutes after the burst, to an R-band magnitude of 21.1, 5.7 hours after the burst [@Fox:EarlyOptical:03]. @GCN1785:Redshift021211 derived a redshift of 1.006 for this burst based on spectroscopic observations carried out with the European Southern Observatory’s Very Large Telescope (VLT) at Paranal, Chile. Milagro searched for emission at GeV/TeV energies over the burst duration reported by the HETE-2 wide field X-ray monitor. They did not find any evidence for prompt emission and a preliminary analysis, assuming a differential photon spectral index of -2.4, gave an upper limit on the fluence at the 99.9% confidence level of [*[J]{}*]{}(0.2-20TeV)$<$3.8x10$^{-6}$ergcm$^{-2}$ over a 6 second interval [@GCN1740:Milagro]. Whipple observations on this GRB location were initiated 20.7 hours after the GRB and lasted for 82.8 minutes. An upper limit (99.7% c.l.) on the VHE emission of 0.33 Crab (E $>$ 400GeV) was derived from these observations. GRB030329 --------- This GRB is one of the brightest bursts on record. It triggered the FREGATE instrument on HETE-2 in the 6-120keV energy band. It had a duration 22.76 seconds, a fluence of 1.1x10$^{-4}$ergcm$^{-2}$ and a peak energy of 67.86keV in the 30-400keV band [@hete2webpage]. The peak flux over 1.2 seconds was 7x10$^{-6}$ergcm$^{-2}$s$^{-1}$ which is $>$100 times the Crab flux in that energy band [@GCN1997:hete2]. The optical transient was identified by @GCN1985:SSO. Due to its slow decay [@GCN1989:Kyoto] and brightness ([*[R]{}*]{}$\sim$13), extensive photometric observations were possible, making this one of the best-observed GRB afterglows to date. Early observations with the VLT [@GCN2020:VLT] revealed evidence for narrow emission lines from the host galaxy indicating that this GRB occurred at a low redshift of [*[z]{}*]{}=0.1687. Observations of the afterglow continued for many nights as it remained bright with a slow but uneven rate of decline and exhibited some episodes of increasing brightness. These observations are well-documented in the GCN archives. Spectral measurements made on 6 April 2003 by @GCN2107:MMT showed the development of broad peaks in flux, characteristic of a supernova. Over the next few nights, the afterglow emission faded and the features of the supernova became more prominent [@Stanek:03]. These observations provided the first direct spectroscopic evidence that at least a subset of GRBs is associated with supernovae. The afterglow was detected at many other wavelengths. Radio observations with the VLA detected a 3.5mJy source at 8.46GHz. This is the brightest radio afterglow detected to date [@GCN2014:VLA]. The afterglow was also bright at submillimeter [@GCN2088:submm] and near infrared wavelengths [@GCN2040:nir]. The X-ray afterglow was detected by RXTE during a 27-minute observation that began 4 hours 51 minutes after the burst [@GCN1996:rxte]. The flux was $\sim$1.4x10$^{-10}$ergcm$^{-2}$s$^{-1}$ in the 2-10keV band ($\sim$0.007% of the Crab). Whipple observations of the location of GRB030329 commenced 64.6 hours after the prompt emission. In total, 241.4 minutes of observation were taken spanning five nights. The upper limits (99.7% c.l.) from each night of observation are listed in Table \[observations\_and\_results\] and are displayed on the same temporal scale as the optical light curve of the GRB afterglow in Figure \[GRB030329-lightcurve\]. When these data were combined, an upper limit (99.7% c.l.) for the VHE emission above 400GeV of 0.17 Crab was derived. GRB030501 --------- This burst was initially detected by the imager on board the INTEGRAL satellite (IBIS/ISGRI) and was found to have a duration of $\sim$40 seconds [@GCN2183:integral]. The burst was also detected by the Ulysses spacecraft and the spectrometer instrument (SPI-ACS) on INTEGRAL [@GCN2187:ipn]. Triangulation between these two detections allowed a position annulus to be computed for this GRB. As observed by Ulysses, it had a duration of $\sim$75 seconds and had a 25-100keV fluence of approximately 1.1x10$^{-6}$ergcm$^{-2}$ with a peak flux of 4.9x10$^{-7}$ergcm$^{-2}$s$^{-1}$ over 0.25 seconds. Follow-up optical observations with several telescopes did not find evidence for an optical transient [@GCN2201:Wise; @GCN2202:crao; @GCN2224:tarot] to a limiting magnitude of [*[R]{}*]{}=18.0, 0.3-17 minutes after the burst [@GCN2224:tarot] and to a limiting magnitude of [*[R]{}*]{}=20.0, 16.5 hours after the burst [@GCN2201:Wise]. Whipple observations of this burst location commenced 6.6 hours after its occurrence and continued for 83.1 minutes. An upper limit (99.7% c.l.) on the VHE emission (E $>$ 400GeV) during these observations of 0.27 Crab was derived. GRB031026 --------- This burst was located by the FREGATE instrument on HETE-2. It had a duration of 114.2 seconds with a fluence of 2.3x10$^{-6}$ergcm$^{-2}$ in the 25-100keV energy band [@GCN2429:hete2] while in the 30-400keV energy band it had a duration of 31.97s and a fluence of 2.8x10$^{-6}$ergcm$^{-2}$ [@hete2webpage]. Follow-up optical observations were carried out with a number of instruments including the 1.05m Schmidt at the Kiso Observatory [@GCN2427:kiso], the 32 inch Tenagra II Telescope [@GCN2428:tenagra], and the 1.0m Telescope at the Lulin Observatory [@GCN2436:lulin], but no optical transient was found to a limiting magnitude of [*[R]{}*]{}=20.9 for observations taken 6-12 hours after the burst [@GCN2436:lulin] and to [*[I$_c$]{}*]{}=20.4 from observations taken 3.9 and 25.7 hours after the burst [@GCN2433:tenagra]. The 30m IRAM Telescope was used to search the field around the GRB location but did not detect any source with a 250GHz flux density $>$ 16mJy [@GCN2440:iram]. A spectral analysis of the prompt X-ray and gamma-ray emission from this burst revealed it to have a very hard spectrum which is unusual for such a long and relatively faint burst [@GCN2432:hete2]. It was noted that the counts ratio of $>$1.8 between the 7-30keV and 7-80keV FREGATE energy bands was one of the most extreme measured [@GCN2429:hete2]. A “new pseudo-redshift” of 6.67$\pm$2.9 was computed for this burst using the prescription of @Pelangeon:06. Whipple observations of this burst location were initiated 3.7 minutes after receiving the GRB notification. The burst notification however, was not received until more than 3 hours after the prompt GRB emission. Although Whipple observations commenced 3.3 hours after the prompt emission, the first data run is not included here due to inferior weather conditions. The data presented here commenced 3.7 hours after the GRB and continued for 82.7 minutes. An upper limit (99.7% c.l.) for the VHE emission ([*[E]{}*]{}$>$400GeV) of 0.41 Crab was derived. GRB040422 --------- This burst was detected by the imaging instrument (IBIS/ISGRI) on the INTEGRAL satellite in the 15-200 keV energy band. It had a duration of 8 seconds, a peak flux between 20 and 200keV of 2.7photonscm$^{-2}$s$^{-1}$ and a fluence (1s integration time) of 2.5x10$^{-7}$ergcm$^{-2}$ [@GCN2572:integral]. Follow-up observations were carried out by many groups but no optical transient was detected [@GCN2573:eso; @GCN2574:miyazaki; @GCN2576:rotse; @GCN2577:lulin; @GCN2578:loiana; @GCN2580:crao; @GCN2581:xinglong]. The ROTSE-IIIb Telescope at McDonald Observatory began taking unfiltered optical data 22.1s after the GRB. Using the first 110s of data, a limiting magnitude of $\sim$17.5 was placed on the R-band emission from the GRB at this time [@GCN2576:rotse]. Whipple observations of this burst commenced 4.0 hours after the prompt emission and continued for 27.6 minutes. An upper limit (99.7% c.l.) on the VHE emission (E $>$ 400GeV) of 0.62 Crab was derived. Results and Discussion {#resultsAndDiscussion} ====================== Upper limits on the VHE emission from the locations of seven GRBs have been derived over different timescales. For each GRB, a number (1-10) of follow-up 28-minute duration observations were taken with the Whipple 10m Telescope. These GRB data were grouped by UT day and were combined to give one upper limit for each day of observation. The limits range from 20% to 62% of the Crab flux above 400GeV and are presented in Table \[observations\_and\_results\]. In addition to calculating upper limits on the GRB emission for each day, upper limits were calculated for each of the 28-minute scans. These are plotted for each of the GRBs in Figure \[FIG::UPPERLIMITS\]. The usefulness of the upper limits presented here is limited by the fact that five of the GRBs occurred at unmeasured redshifts thus making it impossible to infer the effects of the infrared background light on those observations. In addition to this, the earliest observation was not made at Whipple until 3.68 hours after the prompt GRB emission. Although the Whipple 10m Telescope is capable of beginning GRB observations less than 2 minutes after receiving notification, a number of factors, including notifications arriving during daylight and delays in the distribution of the GRB locations, delayed the commencement of the GRB observations presented here. Although data-taking for GRB031026 began 3.7 minutes after the GRB notification was received, this notification was not distributed by the GCN until 3.3 hours after the GRB had occurred. Thus, the observations presented here cannot be used to place constraints on the VHE component of the initial prompt GRB emission and pertain only to the afterglow emission and delayed prompt emission from GRBs. One of the main obstacles for VHE observations of GRBs is the distance scale. Pair production interactions of gamma rays with the infrared photons of the extragalactic background light attenuate the gamma-ray signal thus limiting the distance over which VHE gamma rays can propagate. Recently however, the H.E.S.S. telescopes have detected the blazar PG1553+113 [@Aharonian06:PG1553]. The redshift of this object is not known but there are strong indications that it lies at [*[z]{}*]{}$>$0.25, possibly as far away as [*[z]{}*]{}=$0.74$. This could represent a large increase in distance to the most distant detected TeV source, revealing more of the universe to be visible to TeV astronomers than was previously thought. Although GRBs lie at cosmological distances, many have been detected at redshifts accessible to VHE observers. Of the GRBs studied here, only 2 had spectroscopic redshifts measured while the redshift of one was estimated by @Pelangeon:06 using an improved version of the redshift estimator of @Atteia03. Since all of the GRBs discussed here were long bursts, it is likely that their redshifts are of order 1. Due to the unknown redshifts of most of the bursts and the uncertainty in the density of the extragalactic background light, the effects of the absorption of VHE gamma rays by the infrared background light have not been included here. @Granot:03a analyzed the late time light curve of GRB030329 and find that the large variability observed at several times (t=1.3-$\sim$1.7 days, $\sim$2.4-2.8 days, $\sim$3.1-3.5 days and at $\sim$4.9-5.7 days) after the burst is most likely the result of refreshed shocks. These time intervals have been highlighted in the top panel of Figure \[GRB030329-lightcurve\] and it can be seen that some of the observations taken at Whipple occurred during these times thus imposing upper limits on the VHE emission during these refreshed shocks. Since GRB030329 occurred at a low redshift ($z\,=\,$0.1685), it is possible that the effects of infrared absorption on any VHE emission component may not have been significant enough to absorb all VHE photons over the energy range to which Whipple is sensitive. Figure \[ulTimeLog\] shows these scan-by-scan upper limits as a function of time since the prompt GRB emission. Also plotted are the predicted fluxes at various times after the GRB by @Zhang:01 and @Peer:SED:04 at $\sim$ 400GeV, and by @Guetta:03a at 250GeV. Although the peak response energy of the Whipple Telescope at the time of these observations was 400GeV, it still had sensitivity, albeit somewhat reduced, at 250GeV. @Razzaque04 predict a delayed GeV component in the GRB afterglow phase from the inverse-Compton up-scattering on external shock electrons. The duration of such a component is predicted to be up to a few hours, softening with time. @Zhang:01 investigated the different radiation mechanisms in GRB afterglows and identified parameter-space regimes in which different spectral components dominate. They found that the inverse-Compton GeV photon component is likely to be significantly more important than a possible proton synchrotron or electron synchrotron component at these high energies. The predictions of @Zhang:01 for VHE emission at different times after a typical “Regime II” burst are shown by squares on Figure \[ulTimeLog\]. Although the observations presented here do not constrain these predictions, the sensitivity is close to that required to detect the emission predicted. A recent analysis of archival data from the EGRET calorimeter has found a multi-MeV spectral component in the prompt phase of GRB941017, that is distinct from the lower energy component [@Gonzalez03]. This high energy component appeared between 10s and 20s after the start of the GRB and had a roughly constant flux with a relatively hard spectral slope for $\sim$200s. This observation is difficult to explain within the standard synchrotron model, thus indicating the existence of new phenomena. @Granot:03b investigated possible scenarios for this high energy spectral component and found that most models fail. They concluded that the best candidate for the emission mechanism is synchrotron self-Compton emission from the reverse shock and predicted that a bright optical transient, similar to that observed in GRB990123, should accompany this high energy component. @Peer:SED:04 explain this high energy tail as emission from the forward shock electrons in the early afterglow phase. These electrons inverse-Compton scatter the optical photons that are emitted by the reverse shock electrons resulting in powerful VHE emission for 100s to 200s after the burst as indicated by the lines on Figure \[ulTimeLog\]. Although the observations presented here did not commence early enough after the prompt GRB emission to constrain such models, the sensitivity of the Whipple Telescope is such that the VHE emission predicted by these models would be easily detectable for low redshift bursts. The prediction of @Guetta:03a for VHE emission 5 x 10$^3$s after the burst from the combination of external Compton emission (the relativistic electrons behind the afterglow shock upscatter the plerion radiation) and synchrotron self-Compton emission (the electrons accelerated in the afterglow emit synchrotron emission and then upscatter this emission to the VHE regime) is indicated by a star on Figure \[ulTimeLog\]. The emission is predicted to have a cutoff at $\sim$250GeV due to pair production of the high energy photons with the radiation field of the pulsar wind bubble. For afterglows with an external density similar to that of the inter-stellar medium, photons of up to 1TeV are possible. It can be seen that, although the upper limits presented here are below the predicted flux from @Guetta:03a, the observations at Whipple took place after this emission was predicted to have occurred. Had data taking at Whipple commenced earlier, the emission predicted by these authors should have been detectable for nearby GRBs. @Razzaque04 investigated the interactions of GeV and higher energy photons in GRB fireballs and their surroundings for the prompt phase of the GRB. They predict that high energy photons escaping from the fireball will interact with infrared and microwave background photons to produce delayed secondary photons in the GeV-TeV range. Although observations of the prompt phase of GRBs are difficult with IACTs since they are pointed instruments with small fields of view which must therefore be slewed to respond to a burst notification, observations in time to detect the delayed emission are possible. There are many emission models which predict significant VHE emission during the afterglow phase of a GRB either related to the afterglow emission itself or as a VHE component of the X-ray flares that have been observed in many Swift bursts. @OBrien:06 analyzed 40 Swift bursts which had narrow-field instrument data within 10 minutes of the trigger and found that $\sim$50% had late (t$>$T90) X-ray flares. If the bulk of the radiation comes via synchrotron radiation as is usually supposed, then by analogy with other systems with similar properties (supernova remnants, active galactic nuclei jets), it is natural to suppose that there must also be an inverse Compton component by which photons are boosted into the GeV-TeV energy range. This process is described by @PillaLoeb:98 who discuss the relationship between the energy at which the high energy cutoff occurs, the bulk Lorentz factor and the size of the emission region. A high energy emission component due to inverse Compton emission has also been considered in detail for GRB afterglows by @SariEsin:01; the predicted flux at GeV-TeV energies is comparable to that near the peak of the radiation in the afterglow synchrotron spectrum. Only direct observations can confirm whether this is so. @Guetta:03b predict that the $\sim$300 GeV photons from the prompt GRB phase will interact with background IR photons, making delayed high energy emission undetectable unless the intergalactic magnetic fields are extremely small. The Swift GRB Explorer has shown that $\sim$50% of GRBs have one or more X-ray flares. These flares have been detected up to 10$^5$s ($\sim$28 hours) after the prompt emission [@Burrows:05]. Indeed, the delayed gamma-ray component detected in BATSE bursts [@Connaughton:02] may also be associated with this phenomenon. Recently, @Wang:06 have predicted VHE emission coincident in time with the X-ray flare photons. In this model, if the X-ray flares are caused by late central engine activity, the VHE photons are produced from inverse Compton scattering of the X-ray flare photons from forward shock electrons. If the X-ray flares originate in the external shock, VHE photons can be produced from synchrotron self-Compton emission of the X-ray flare photons with the electrons which produced them. Should VHE emission be detected from a GRB coincident with X-ray flares, the time profile of the VHE emission could be used to distinguish between these two origins of the X-ray flares. No evidence for delayed VHE gamma-ray emission was seen from any of the GRB locations observed here and upper limits have been placed on the VHE emission at various times after the prompt GRB emission. Although there are no reports of the detection of X-ray flares or delayed X-ray emission from any of these GRBs, it is likely that such emission was present in at least some of them given the frequency with which it has been detected in GRBs observed by Swift. Indeed, the light curve of GRB030329 shows large variability amplitude a few days after the burst and, as shown in Figure \[GRB030329-lightcurve\], Whipple observations were taken during these episodes. Apart from this, a measured redshift is only available for one of the other bursts observed here and it is possible that the remaining five occurred at distances too large to be detectable in the VHE regime. @Soderberg:04 reported on an unusual GRB (GRB031203) that was much less energetic than average. Its similarity, in terms of brightness, to an earlier GRB (GRB980425) suggests that the nearest and most common GRB events have not been detected up until now because GRB detectors were not sensitive enough [@Sazonov04]. Most GRBs that have been studied up until now lie at cosmological distances. They generate a highly collimated beam of gamma rays ensuring that they are powerful enough to be detectable at large distances. Both of the less powerful GRBs detected to date occurred at considerably lower redshifts; GRB980425 at [*[z]{}*]{}=0.0085 and GRB031203 at [*[z]{}*]{}=0.1055. Although @Soderberg:04 conclude that up until now, GRB detectors have only detected the brightest GRBs and that the nearest and most common GRB events have been missed because they are less highly collimated and energetic, @Ramierz:05 argue that the observations of GRB031203 can indeed be the result of off-axis viewing of a typical, powerful GRB with a jet. Should future observations prove there to be a closer, less powerful population of GRBs, these would be prime targets for IACTs. In the past year, the Whipple Observatory 10m Telescope has been used to carry out follow-up observations on a number of GRBs detected by the Swift GRB Explorer. The analysis of these observations will be the subject of a separate paper [@Dowdall]. The Very Energetic Radiation Imaging Telescope Array System (VERITAS) is currently under construction at the Fred Lawrence Whipple Observatory in Southern Arizona. Two of the four telescopes are fully operational and it is anticipated that the four-telescope array will be operational by the end of 2006. GRB observations will receive high priority and, when a GRB notification is received, their rapid follow-up will take precedence over all other observations. The VERITAS Telescopes can slew at 1$^\circ$s$^{-1}$ thus enabling them to reach any part of the visible sky in less than 3 minutes. When an acceptable (i.e. at high enough elevation) GRB notification is received during observing at VERITAS, an alarm sounds to alert the observer that a GRB position has arrived. Upon receiving authorization from the observer, the telescope slews immediately to the position and data-taking begins. Given that the maximum time to slew to a GRB is 3 minutes, and that Swift notifications can arrive within 30s of the GRB, it is possible that VERITAS observations could begin as rapidly as 2-4 minutes after the GRB, depending on its location with respect to the previous VERITAS target. As has been shown above, the Whipple 10m Telescope is sensitive enough to detect the GRB afterglow emission predicted by many authors. With its improved background rejection and greater energy range, VERITAS will be significantly more sensitive for GRB observations than the Whipple 10m Telescope. The VERITAS sensitivity for observations of different durations is shown in Figure \[VERITAS-sensitivity\]. Based on the assumed rate of Swift detections (100 year$^{-1}$), the fraction of sky available to VERITAS, the duty-cycle at its site and the sun avoidance pointing of Swift which maximizes its overlap with nighttime observations, it is anticipated that $\sim$10 Swift GRBs will be observable each year with VERITAS. Acknowledgements ================ The authors would like to thank Emmet Roache, Joe Melnick, Kevin Harris, Edward Little, and all of the staff at the Whipple Observatory for their support. The authors also thank the anonymous referee for his/her comments which were very useful and improved the paper. This research was supported in part by the U. S. Department of Energy, the National Science Foundation, PPARC, and Enterprise Ireland. Extensive use was made of the GCN web pages (http://gcn.gsfc.nasa.gov/). The web pages of Joachim Greiner and Stephen Holland (http://www.mpe.mpg.de/$^\sim$jcg/grbgen.html and\ http://lheawww.gsfc.nasa.gov/$^\sim$sholland/grb/index.html) proved very useful in tracking down references and information related to the GRBs discussed in this paper. Aharonian, F., et al. 2005, Nature (in press) astro-ph/0508073 Aharonian, F., et al. 2006, A&A, 448, L19 Amenomori, M., et al. 2001, AIP Conf. Proc., 558, 844 Atkins, R., et al. 2000, ApJ, 533, L119 Atkins, R., et al. 2004, ApJ, 604, L25 Atkins, R., et al. 2005, ApJ, 630, 996 Atteia, J. L., et al. 2003, astro-ph/0312371 Berger, E., Soderberg, A. M. & Frail, D. A. 2003, GCN Circular 2014 Bertoldi, F., et al. 2003, GCN Circular 2440 Boer, M. & Klotz, A. 2003, GCN Circular 2224 Boettcher, M. & Dermer, C. 1998, ApJ, 499, L131 Bradbury, S. M. & Rose, H. J. 2002, Nucl. Instrum. Meth. A, 481, 521 Burrows, D. N., et al. 2005, Science, 309, 1833 Catanese, M. A., et al. 1998, ApJ, 501, 616 Connaughton, V., et al. 1997, ApJ, 479, 859 Connaughton, V. 2002, ApJ, 567, 1028 Crew, G., et al. 2002, GCN Circular 1734 Davies, J. M. & Cotton, E. S. 1957, Journal of Solar Energy, 1, 16 Dermer, C. & Chiang, J. 1999, Proceedings of GeV-TeV Gamma-Ray Astrophysics Workshop: Toward a Major Atmospheric Cherenkov Telescope, ed. B. L. Dingus (AIP 515, New York, 2000) Dermer, C. & Atoyan, A. 2004, A&A, 418, L5 Dermer, C. 2005, Proceedings of Conference “Towards a Network of Atmospheric Cherenkov Detectors VII,” 27-29 April, 2005, Palaiseau, France Dingus, B. L., Catelli, J. R. & Schneid, E. J. 1998, Proc. 4th Huntsville GRB Symposium, AIP Conf Proc, 428, 349 Dingus, B. L., 2001, in [*[High Energy Gamma Ray Astronomy]{}*]{}, ed. F. A. Aharonian and H. J. Völk, 2001, AIP, 558, 383 Dowdall, C., et al. ([*[in preparation]{}*]{}) Falcone, A. D., et al. 2006a, Proc. of the 16th Annual Astrophysics Conf. in Maryland: Gamma Ray Bursts in the Swift Era, Washington DC (astro-ph/0602135) Fenimore E., et al. 1999, ApJ 512, 683 Fox, D. W., et al 2003, ApJ, 586, L5 Fragile, P. C., Mathews, G. J., Poirier, J. & Totani, T. 2004, Astropart. Phys., 20, 591 Galama, T. J., et al 1998, Nature, 395, 670 Garnavich, P., et al. 2002a, GCN Circular 1750 Garnavich, P. et al. 2002b, GCN Circular 1751 HETE-2 pages at MIT: [*[http://space.mit.edu/HETE/Bursts/Data]{}*]{} Homepage of the Gamma-ray bursts Co-ordinates Network: [*[http://gcn.gsfc.nasa.gov]{}*]{} Gehrels, N., et al. 2004, ApJ, 611, 1005 González, M. M., et al. 2003, Nature, 424, 749 Gotz, D., et al. 2003, GCN Circular 2572 Granot, J., Nakar, E. & Piran, T. 2003, Nature, 426, 138 Granot, J. & Guetta, D. 2003, ApJ, 598, L11 Greiner, J., et al., 2003, GCN Circular 2020 Guetta, D. & Granot, J. 2003a, MNRAS, 340, 115 Guetta, D. & Granot, J. 2003b, ApJ, 585, 885 Guetta, D., et al., 2006, (in press) astro-ph/06062387 Horan, D., et al. 2002, ApJ, 571, 753 Huang, K., et al. 2003a, GCN Circular 2436 Huang, K., et al. 2003b, GCN Circular 2577 Hurley, K., et al. 1994, Nature, 372, 652 Hurley, K., et al., 2003, GCN Circular 2187 Kildea, J., et al., 2006, Astropart. Phys. ([*[in press]{}*]{}) King, A., et al., 2005, ApJ, 630, L113 Klebesadel, R. W., Strong, I. B. & Olson, R. A. 1973, ApJ, 182, L85 Kumar, P. & Piran, T. 2000, ApJ, 532, 286 Lamb, D. Q., et al. 2000, AIP Conf. Proc., 522, 265-268, Ed. S. S. Holt & W. W. Zhang Lamb, D. Q., et al. 2002, GCN Circular 1744 Lamb, D. Q., et al. 2003, GCN Circular 2040 Lazzati, D., et al. 2002, A&A, 396, L5 Lewis, D. A. 1990, Exp. Astronomy, 1, 213 Li, W., et al. 2002, GCN Circular 1737 Li, W., et al 2003, ApJ, 586, L9 Lipkin, Y. M., et al. 2004, ApJ, 606, 381 MacFadyen, A., Ramirez-Ruiz, E. & Zhang, W. 2005, American Astronomical Society Meeting, 207, 151.04 (astro-ph/0510192) Malesani, D., et al. 2003, GCN Circular 2573 Mannheim, K., Hartmann, D. & Funk, B. 1996, ApJ, 467, 532 Marshall, F. E. & Swank, J. H. (2003), GCN Circular 1996 McEnery, J., et al. 2002a, GCN Circular 1724 McEnery, J., et al. 2002b, GCN Circular 1740 Meegan, C. A., et al. 1992, Nature, 355, 143 Mereghetti, S., et al. 2003, GCN Circular 2183 Mészáros, P. & Rees, M. J. 1993, ApJ, 405, 278 Mészáros, P., Rees, M. J. & Papathanassiou, H. 1994, ApJ, 432, 181 Mészáros, P. & Rees, M. J. 1994, MNRAS, 269, L41 Nousek, J. A., et al. 2006, ApJ, 642, 389 Nysewander, M. C., Reichart, D. & Schwartz, M. 2002, GCN Circular 1735 Nysewander, M. C., et al. 2003a, GCN Circular 2428 Nysewander, M. C., et al. 2003b, GCN Circular 2433 O’Brien, P. T., et al. 2006, ApJ (in press), astro-ph/0601125 Ofek, E. O., et al. 2003, GCN Circular 2201 Padilla, L., et al. 1998, A&A, 337, 43 Panateiscu, A., et al. 2006, MNRAS (in press), astro-ph/0508340 Park, H. S., Williams, G. & Barthelmy, S. 2002, GCN Circular 1736 Pe’er, A. & Waxman, E. 2004, ApJ, 603, L1 Pélangeon, A., et al. 2006, Proceedings of the 16th Annual October Astrophysics Conference in Maryland, “Gamma-ray Bursts in the Swift Era”, Washington DC., November 29-December 2, 2005 (astro-ph/0601150) Perna, R., Armitage, P. J. & Zhang, B. 2005, ApJ, 636, L29 Peterson, B. A. & Price, P. A. 2003, GCN Circular 1985 Piccioni, A., et al. 2003, GCN Circular 2578 Pilla, R. P. & Loeb, A. 1998, ApJ, 494, L167 Proga, D. & Zhang, B. 2006, MNRAS, 370, L61 Qui, Y. & Hu, J. 2003, GCN Circular 2581 Ramirez-Ruiz, E., et al. 2005, ApJ, 625, L91 Razzaque, S., Mészáros, P. & Zhang, B. 2004, ApJ, 613, 1072 Rees, M. J. & Mészáros, P. 1998, MNRAS, 248, 41 Rees, M. J. & Mészáros, P. 1998, ApJ, 496, L1 Reynolds, P. T., et al. 1993, ApJ, 404, 206 Ricker, G., et al. 2002, GCN Circular 1682 Ricker, G., et al. 2003a, GCN Circular 2429 Ricker, G., et al. 2003b, GCN Circular 2432 Rumyantsev, V., Pavlenko, E. & Pozanenko, A. 2003a, GCN Circular 2002 Rumyantsev, V., Pavlenko, E. & Pozanenko, A. 2003b, GCN Circular 2580 Rykoff, E. 2003, GCN Circular 2576 Sari, R., Piran, T. & Narayan, R. 1998, ApJ, 497, L17 Sari, R. & Mészáros, P. 2000, ApJ, 535, L33 Sari, R. & Esin, A. A. 2001, ApJ, 548, 787 Sazonov, S. Yu., Lutovinov, L. & Sunyaev, R. A. 2004, Nature, 429, 646 Schaefer, J., et al. 2002, GCN Circular 1776 Soderberg, A. M., et al. 2004, Nature, 429, 648 Sonoda, E., Maeno, S. & Yamauchi, M. 2003, GCN Circular 2574 Smith, I., et al. 2003, GCN Circular 2088 Stanek, K. Z., et al. 2003a, GCN Circular 2107 Stanek, K. Z., et al. 2003b, ApJ, 591, L17 Torii, K., Yamaoka, H., & Kato, Y. 2002, GCN Circular 1730 Uemura, M. 2003, GCN Circular 1989 Urata, Y., et al. 2002, GCN Circular 1747 Urata, Y., et al. 2003, GCN Circular 2427 Vanderspek, R., et al. 2003, GCN Circular 1997 Vreeswijk P., et al. 2002, GCN Circular 1785 Wang, X. Y., Dai, Z., G & Lu, T. 2001, ApJ, 556, 1010 Wang, X. Y., Li, Z. & Mészáros, P. 2006, ApJL (in press), astro-ph/0601229 Winkler, C., et al. 1999, in Proc. 3$^{rd}$ INTEGRAL Workshop - Astro. Lett. and Communications, 29, 309 Zhang, B. & Mészáros, P. 2001, ApJ 559, 110 Zhang, B. & Mészáros, P. 2004, Int. J. Mod. Phys. A19 2385-2472 Zhang, B., et al. 2006, ApJ, 642, 354 [ccccccc]{} 021112 & HETE-2 & 2448 & — & 2.1x10$^{-7}$ & 6.39 & 30-400\ 021204 & HETE-2 & 2486 & — & — & — & —\ 021211 & HETE-2 & 2493 & 1.006 & 2.4x10$^{-6}$ & 2.80 & 30-400\ 030329 & HETE-2 & 2652 & 0.17 & 1.1x10$^{-4}$ & 22.76 & 30-400\ 030501 & INTEGRAL & 596 & — & 1.1x10$^{-6}$ &$\sim$75 & 25-100\ 031026 & HETE-2 & 2882 & 6.67 & 2.8x10$^{-6}$ & 31.97 & 30-400\ 040422 & INTEGRAL & 1758 & — & — & 8 & —\ [cccccc]{} 021112 & 4.24 & 110.56 & 0.013 & 5.1 & $<$0.200\ & 28.63 & 55.28 & 0.013 & 29.0 & $<$0.303\ 021204 & 16.91 & 55.34 & 0.009 & 17.4 & $<$0.331\ 021211 & 20.69 & 82.79 & 0.058 & 21.9 & $<$0.325\ 030329 & 64.55 & 65.21 & 0.060 & 66.2 & $<$0.360\ & 112.58 & 83.17 & 0.022 & 113.8 & $<$0.279\ & 136.23 & 37.55 & 0.022 & 137.0 & $<$0.323\ & 162.14 & 27.74 & 0.022 & 162.4 & $<$0.519\ & 186.16 & 27.73 & 0.022 & 186.4 & $<$0.399\ 030501 & 6.58 & 83.10 & 0.001 & 7.3 & $<$0.265\ 031026 & 3.68 & 82.70 & 0.007 & 4.9 & $<$0.406\ 040422 & 3.99 & 27.63 & 0.062 & 4.2 & $<$0.620\ \ \ \ [^1]: Since the Crab is the standard candle in the VHE regime, it is customary to quote upper limits as a fraction of the Crab flux at the same energy.

--- abstract: 'Recently it was shown that the calculation of quasiparticle energies using the $G_0W_0$ approximation can be performed without computing explicitly any virtual electronic states, by expanding the Green function and screened Coulomb interaction in terms of the eigenstates of the static dielectric matrix. Avoiding the evaluation of virtual electronic states leads to improved efficiency and ease of convergence of $G_0W_0$ calculations. Here we propose a further improvement of the efficiency of these calculations, based on an approximation of density-density response functions of molecules and solids. The approximation relies on the calculation of a subset of eigenvectors of the dielectric matrix using the kinetic operator instead of the full Hamiltonian, and it does not lead to any substantial loss of accuracy for the quasiparticle energies. The computational savings introduced by this approximation depend on the system, and they become more substantial as the number of electrons increases.' author: - Han Yang - Marco Govoni - Giulia Galli bibliography: - 'main.bib' title: 'Improving the efficiency of $G_0W_0$ calculations with approximate spectral decompositions of dielectric matrices' --- \[sec:introduction\]Introduction ================================ Devising accurate and efficient methods to predict the electronic properties of molecules and condensed systems is an active field of research. Density functional theory (DFT) has been widely used for electronic structure calculations.[@DFT1964; @DFT1965; @DFT2004] However, the exact form of the exchange-correlation functional is unknown and therefore DFT results depend on the choice of approximate functionals. Improvement over DFT results may be obtained by using many-body perturbation theory (MBPT).[@ReiningMBPT; @martin2016InteractingElectrons] A practical formulation of MBPT for many electron systems was proposed by Hedin,[@hedin1965] where the self-energy $\Sigma$ is written in terms of the Green’s function $G$ and the screened Coulomb interaction $W$. The $GW$ approximation[@hedin1965; @martin2016InteractingElectrons] has been successful in the description of the electronic properties of several classes of materials and molecules;[@OriginalGW100; @GW100-maggio2017; @Marco2018; @peter2016JCTC; @brawand2016generalization; @golze2019gw-compendium] however the computational cost of $GW$ calculations remains rather demanding and many complex systems cannot yet be studied using MBPT. Hence, algorithmic improvements are required to apply MBPT to realistic systems. One of the most demanding steps of the original implementation of $GW$ calculations[@Louie1985PRL; @Louie1985PRB; @Louie1986; @Strinati1980GWImplementation; @Strinati1982GWImplementation] involves an explicit summation over a large number of unoccupied single particle electronic orbitals, which enter the evaluation of the dielectric matrix $\epsilon$[@adler1962quantum; @wiser1963dielectric] defining the screened Coulomb interaction $W$ ($W=\epsilon^{-1}v_c$, where $v_c$ is the Coulomb interaction). The summation usually converges slowly as a function of the number of virtual orbitals ($N_c$). In recent years, several approaches have been proposed to improve the efficiency of $GW$ calculations. For example, in Ref. it was suggested to replace unoccupied orbitals with approximate physical orbitals (SAPOs); the author of Ref. simply truncated the sum over empty states entering the calculation of the irreducible density-density response function, and assigned the same, average energy to all the empty states higher than a preset value; in a similar fashion, in Ref. an integration over the density of empty states higher than a preset value was used. Other approaches adopted sophisticated algorithms to invert the dielectric matrix, e.g. in Ref. , they employed a Lanczos algorithm. Recently, an implementation of $G_0W_0$ calculations avoiding altogether explicit summations over unoccupied orbitals, as well as the necessity to invert dielectric matrices, has been proposed,[@wilson2008; @wilson2009; @Nguyen2012-GW; @Anh2013-GW; @Marco2015] based on the spectral decomposition of density-density response functions in terms of eigenvectors (also known as projective dielectric eigenpotentials, PDEPs). In spite of the efficiency improvement introduced by such formulation, $G_0W_0$ calculations for large systems remain computationally demanding. In this paper, we propose an approximation to the projective dielectric technique,[@wilson2008] which in many cases leads to computational savings of $G_0W_0$ calculations of 10-50%, without compromising accuracy. The rest of the paper is organized as follows: we describe the proposed methodology in and then we present results for several systems in , followed by our conclusions. \[sec:method\]Methodology ========================= We compute the density-density response function of solids and molecules within the framework of the random phase approximation (RPA), using projective dielectric eigenpotentials (PDEP)[@wilson2008; @wilson2009; @Marco2015]. The accuracy of this approach has been extensively tested for molecules and solids.[@wilson2009] The technique relies on the solution of the Sternheimer’s equation [@Sternheimer1954; @DFT2004; @Galli1993] $$(\hat{H}-\varepsilon_v\hat{I})\lvert\Delta\psi_v\rangle = -\hat{P}_c\Delta \hat{V}\lvert\psi_v\rangle \label{equ:Sternheimer}$$ to obtain the linear variation of the $v$-th occupied electronic orbital, $\lvert\Delta\psi_v\rangle$, induced by the external perturbation $\Delta \hat{V}$. In , $\hat{I}$ is the identity operator, $\hat{P}_c$ is the projector onto the unoccupied states, $\varepsilon_v$ and $\psi_v$ are the $v$-th eigenvalue and eigenvector of the unperturbed Kohn-Sham Hamiltonian $\hat{H}=\hat{K}+\hat{V}_\mathrm{SCF}$, respectively, where $\hat{K}=-\frac{\nabla^2}{2}$ is the kinetic energy, $\hat{V}_\mathrm{SCF}$ is the self-consistent potential. For each perturbation, the first order response of the density $\Delta n$ can be obtained as [@baroni2001phonons] $$\Delta n = 2 \sum_v \psi_v\Delta\psi_v +c.c. \, \label{eq:deltadensity}$$ and \[eq:deltadensity\] can be used to iteratively diagonalize the static symmetrized *irreducible* density-density response, $\tilde{\chi}_0$:[@wilson2008; @wilson2009; @Marco2015] $$\tilde{\chi}_0 = \sum_{i=1}^{N_\mathrm{PDEP}}\left|\xi_i\right\rangle\lambda_i\left\langle\xi_i\right|,\label{equ:chi0}$$ where $\lambda_i$ and $\xi_i$ are eigenvalues and eigenvectors of $\tilde{\chi}_0$ and $N_\mathrm{PDEP}$ is the number of eigenvectors of $\tilde{\chi}_0$, respectively. The eigenvectors $\xi_i$ are called PDEPs: projective dielectric eigenpotentials throughout the manuscript. The symmetrized irreducible density-density response function is defined as $\tilde{\chi}_0 = v_c^{1/2}\chi_0v_c^{1/2}$, where $v_c$ is the Coulomb potential.[@Marco2015] Within the RPA, the symmetrized *reducible* density-density response can be expressed as $\tilde{\chi}=\left(1-\tilde{\chi}_0\right)^{-1}\tilde{\chi}_0$, therefore the $\xi_i$ are also eigenvectors of $\tilde{\chi}$. The projective dielectric technique has also been recently applied beyond the RPA using a finite field method.[@HeMa2018JCTC] When solving the Sternheimer’s equation, it is not necessary to compute explicitly the electronic empty states, because one can write $\hat{P}_c = \hat{I}-\hat{P}_v$, since the eigenvectors of $\hat{H}$ form a complete basis set ($\hat{P}_v$ is the projector onto the occupied states). The use of significantly reduces the cost of $G_0W_0$ calculation from $N_\mathrm{pw}^2N_vN_c$ to $N_\mathrm{PDEP} N_\mathrm{pw}N_v^2$ where $N_v$, $N_c$, $N_\mathrm{PDEP}$, $N_\mathrm{pw}$ are numbers of occupied orbitals (valence bands in solids), virtual orbitals (conduction bands in solids), PDEPs and plane waves, respectively, and importantly $N_\mathrm{PDEP}\ll N_\mathrm{pw}$. The application of the algorithm described above to large systems is hindered by the cost of solving . However, we note that the eigenvalues of $\tilde{\chi}_0$ rapidly converge to zero ,[@Deyu2008PRL; @wilson2008; @wilson2009; @Marco2018] (an example is shown in ). In addition, as shown in Ref. , the eigenvalue spectrum of the dielectric function for eigenvectors higher than the first few, is similar to that of the Lindhard function.[@LindhardFunction1954] Hence we propose to compute the PDEPs of $\tilde{\chi}_0$ corresponding to the lowest eigenvalues using and \[eq:deltadensity\] and to compute the remaining ones with a less costly approach. Inspired by the work of @Rocca2014, we approximate the eigenpotentials corresponding to higher eigenvalues with kinetic eigenpotentials, which are obtained approximating the full Hamiltonian entering with the kinetic operator($\hat{K}$):[@RPA-total-energy-PRL2009; @Rocca2014] $$(\hat{K}-\varepsilon_v\hat{I})\lvert\Delta\psi_v\rangle = -\hat{P}_c\Delta \hat{V}\lvert\psi_v\rangle\label{equ:KineticSternheimer}.$$ We note that the application of the kinetic energy ($\hat{K}$) amounts simply to computing a sum over the plane-wave expansion coefficients multiplied by the square of the $\mathbf{G}$-vectors (we are using a plane-wave basis set). Numerous additional operations involving the solution of a Poisson equation and integrals in real space are necessary to apply the full Hamiltonian in , which includes the Hartree and exchange correlation potentials. Furthermore, the evaluation of the non-local pseudopotentials (not needed when solving ) is an expensive operation in principle of $O(N^3)$, where $N$ is the number of electrons.[@DFT2004; @Galli1993] In the following, we refer to the eigenpotentials from as standard PDEPs (stdPDEP, $\xi_i,\,\,i = 1, \cdots, N_\mathrm{stdPDEP}$) and those from as kinetic PDEPs (kinPDEP, $\eta_i,\,\,i = 1, \cdots, N_\mathrm{kinPDEP}$) and we rewrite the irreducible density-density response function as $$\tilde{\chi}_0 = \sum_{i=1}^{N_\mathrm{stdPDEP}}\left|\xi_i\right\rangle\lambda_i\left\langle\xi_i\right|+\sum_{j=1}^{N_\mathrm{kinPDEP}}\left|\eta_j\right\rangle\mu_j\left\langle\eta_j\right|,$$ where $\xi_i$ and $\eta_j$ are standard and kinetic PDEPs, respectively, and $\lambda_i$ and $\mu_j$ are their corresponding eigenvalues. The procedure to generate stdPDEPs and kinPDEPs is summarized in . We note that during the construction of kinetic PDEPs, the projection operator $\hat{P}=\hat{I}-\sum_{N_\mathrm{stdPDEP}}\left|\xi_i\right\rangle\left\langle\xi_i\right|$ was applied so as to satisfy the orthonormality constraint, $\langle\xi_i|\xi_j\rangle = \delta_{ij}$, $\langle\eta_i|\eta_j\rangle=\delta_{ij}$ and $\langle\xi_i|\eta_j\rangle=0$, $\forall (i,j)$; in addition we applied $v_c^{1/2}$ to perturbations to yield a *symmetrized* irreducible response function $\tilde{\chi}_0$. In our $G_0W_0$ calculations, both the static Green’s function and the statically screened Coulomb interaction are written in the basis of eigenpotentials of the dielectric matrix. Frequency integration is performed using a contour deformation algorithm. A detailed description of the implementation of $G_0W_0$ calculations in the basis of eigenpotentials can be found in Ref. . One should also note that and \[eq:deltadensity\] apply only to semiconductors, but this formalism may be generalized to metallic systems. \[sec:results\]Validation and results ===================================== We now turn to discussing results for molecules and solids obtained by using a combination of standard and kinetic PDEPs. To examine the efficiency and applicability of the approximation proposed in , we performed $G_0W_0$ calculations for a set of closed-shell small molecules, a larger molecule (Buckminsterfullerene $\mathrm{C_{60}}$), and an amorphous silicon nitride/silicon interface ($\mathrm{Si_3N_4/Si}$) with a total of 1152 valence electrons. All Kohn-Sham eigenvalues and eigenvectors were obtained with the QuantumEspresso package,[@QE2009; @QE2017] using the PBE approximation[@PBEfunctional], SG15[@ONCV2015] ONCV[@hamann2013ONCV] pseudopotentials and $G_0W_0$ calculations were carried out with the WEST code.[@Marco2015] We first considered a subset of molecules taken from the G2/97 test set[@G2/97--testset] and calculated their vertical ionization potential (VIP) and electron affinity (EA) using different numbers of stdPDEPs ($N_\mathrm{stdPDEP}$) and kinPDEPs ($N_\mathrm{kinPDEP}$). We chose a plane wave cutoff of 85 Ry and a periodic box of edge 30 Bohr. For all molecules, we included either 20 or 100 stdPDEPs in our calculations, then added 100, 200, 300, 400 kinPDEPs in subsequent calculations, after which an extrapolation was performed ($a+\frac{b}{N_\mathrm{stdPDEP}+N_\mathrm{kinPDEP}}$) to find converged values (one example is shown in ). These results are given in the second (A) and third columns (B) of and .[^1] The reference results reported in the last column (C) of the two tables were obtained with 200, 300, 400, 500 stdPDEPs and an extrapolation was applied. We found that including only 20 stdPDEPs yields quasiparticle energies accurate within $0.1\,\mathrm{eV}$ relative to the reference $G_0W_0$ values obtained using only standard eigenpotentials. (See also Fig. 2 of the Supplementary Information). The two data sets starting from 20 or 100 stdPDEPs enabled us to save 40% and 10% of computer time compared to the time usage needed with only standard eigenpotentials. ![First 500 eigenvalues $\lambda_i$ of the symmetrized irreducible density-density response function $\tilde{\chi}_0$ (see text), for three small molecules: (blue dots), (orange up triangles), and (green down triangles). $N_v$ is the number of occupied orbitals.[]{data-label="fig:wstat-eigens"}](fig/eigenvals-eps-converted-to.pdf){width="0.6\linewidth"} ![The workflow used in this work to generate eigenvectors of the dielectric matrix using the Kohn-Sham Hamiltonian (stdPDEP) and using the kinetic operator (kinPDEP). See text.[]{data-label="fig:workflow"}](fig/workflow-eps-converted-to.pdf){width="0.8\linewidth"} ![Extrapolation of $G_0W_0$ energy of highest occupied orbital of methane with respect to total number of eigenpotential used ($N_\mathrm{stdPDEP}+N_\mathrm{kinPDEP}$). In this plot, $N_\mathrm{stdPDEP} = 20$ and $N_\mathrm{kinPDEP} = 100,\,200,\,300,\,400$ for the four points, respectively.[]{data-label="fig:fitting_example"}](fig/Fitting_example_of_CH4_from120-eps-converted-to.pdf){width="0.6\linewidth"} Molecule A B C -------------------- ------- ------- ------- $\mathrm{C_2H_2}$ 11.07 11.06 11.06 $\mathrm{C_2H_4}$ 10.41 10.40 10.40 $\mathrm{C_4H_4S}$ 8.80 8.77 8.76 $\mathrm{C_6H_6}$ 9.17 9.14 9.13 $\mathrm{CH_3Cl}$ 11.28 11.26 11.25 $\mathrm{CH_3OH}$ 10.58 10.56 10.56 $\mathrm{CH_3SH}$ 9.39 9.36 9.36 $\mathrm{CH_4}$ 14.01 14.01 14.01 $\mathrm{Cl_2}$ 11.51 11.51 11.50 $\mathrm{ClF}$ 12.55 12.55 12.54 $\mathrm{CO}$ 13.51 13.50 13.50 $\mathrm{CO_2}$ 13.32 13.31 13.31 $\mathrm{CS}$ 11.00 10.98 10.98 $\mathrm{F_2}$ 14.99 14.97 14.97 $\mathrm{H_2CO}$ 10.43 10.42 10.42 $\mathrm{H_2O}$ 11.82 11.82 11.81 $\mathrm{H_2O_2}$ 10.87 10.87 10.86 $\mathrm{HCl}$ 12.50 12.50 12.50 $\mathrm{HCN}$ 13.20 13.20 13.20 $\mathrm{Na_2}$ 4.95 4.95 4.95 : Vertical ionization potential (eV) obtained at the $G_0W_0@\mathrm{PBE}$ level of theory with different numbers of standard and kinetic PDEPs. (A) 20 stdPDEPs + up to 400 kinPDEPs and extrapolated; (B) 100 stdPDEPs + up to 400 kinPDEPs and extrapolated; (C) pure stdPDEPs and extrapolated. A detailed discussion of extrapolations of quasiparticle energies can be found in Ref. . See also Fig. 3 of the SI.\[tab:VIP-PBE-small\] Molecule A B C -------------------- ------- ------- ------- $\mathrm{C_2H_2}$ -2.42 -2.41 -2.41 $\mathrm{C_2H_4}$ -1.75 -1.75 -1.75 $\mathrm{C_4H_4S}$ -0.85 -0.81 -0.80 $\mathrm{C_6H_6}$ -1.01 -0.96 -0.96 $\mathrm{CH_3Cl}$ -1.17 -1.16 -1.16 $\mathrm{CH_3OH}$ -0.89 -0.89 -0.89 $\mathrm{CH_3SH}$ -0.88 -0.88 -0.88 $\mathrm{CH_4}$ -0.64 -0.64 -0.64 $\mathrm{Cl_2}$ 1.65 1.64 1.65 $\mathrm{ClF}$ 1.28 1.28 1.28 $\mathrm{CO}$ -1.56 -1.57 -1.57 $\mathrm{CO_2}$ -0.97 -0.97 -0.97 $\mathrm{CS}$ 0.49 0.51 0.51 $\mathrm{F_2}$ 1.16 1.16 1.16 $\mathrm{H_2CO}$ -0.69 -0.68 -0.68 $\mathrm{H_2O}$ -0.90 -0.90 -0.90 $\mathrm{H_2O_2}$ -1.80 -1.79 -1.79 $\mathrm{HCl}$ -1.07 -1.07 -1.07 $\mathrm{HCN}$ -2.08 -2.08 -2.08 $\mathrm{Na_2}$ 0.64 0.63 0.63 : Vertical electron affinity (eV) obtained at the $G_0W_0@\mathrm{PBE}$ level of theory with different numbers of standard and kinetic PDEPs. (A) 20 stdPDEPs + up to 400 kinPDEPs and extrapolated; (B) 100 stdPDEPs + up to 400 kinPDEPs and extrapolated; (C) pure stdPDEPs and extrapolated. We defined the electron affinity as the energy of the first unoccupied state, within the given unit cell used, extrapolated as a function of $N_\mathrm{stdPDEP} + N_\mathrm{kinPDEP}$. See also Ref. and Fig. 3 of the SI.\[tab:EA-PBE-small\] shows our results for the $\mathrm{C_{60}}$ molecule. The structure of $\mathrm{C_{60}}$ (point group $I_h$) was also taken from the NIST computational chemistry database,[@NIST-CCCBDB] (optimized with the $\omega$B97X-D functional and cc-pVTZ basis sets) and no further optimization was carried out. We used the PBE exchange-correlation functional, a plane wave energy cutoff of $\SI{40}{\si{Ry}}$ and cell size of $\SI{40}{\si{bohr}}$, the same as used in Ref. . We performed two groups of calculations starting with 100 and 200 standard eigenpotentials, respectively. For both calculations, we computed quasiparticle energies by adding 100, 200, 300, 400 kinetic eigenpotentials and extrapolation was done in the same manner. As seen in , the results obtained with 200 standard eigenpotentials and additional kinetic eigenpotentials differ at most by from those computed with standard eigenpotentials (extrapolated up to $N_\mathrm{stdPDEP}=2000$). The two sets of calculations starting with $N_\mathrm{stdPDEP}=100$ and $N_\mathrm{kinPDEP}=200$ amounted to savings of 32% and 15%, respectively. We now turn to a more complex system, amorphous silicon nitride interfaced with a silicon surface ($\mathrm{Si_3N_4/Si(100)}$), whose structure was taken from Ref. (See ). This interface is representative of a heterogeneous, low dimensional system. We computed band offsets (BO) by employing two different methods. The first one is based on the calculation of the local density of electronic states (LDOS);[@Anh2013interface; @LDOS-cited-by-Anh] the second one is based on the calculation of the average electrostatic potential which is then used to set a common zero of energy on the two parts of the slab representing the two solids interfaced with each other.[@Walle-alignment-by-potential] The average electrostatic potential was fitted with the method proposed in Ref. . We used a plane wave energy cutoff of $\SI{70}{\si{Ry}}$. We also performed $G_0W_0$@PBE calculations for each bulk system separately and obtained quasiparticle energies. The local density of states is given by: $$D(\varepsilon,z) = 2\sum_i\int\frac{\mathrm{d}x}{L_x}\int\frac{\mathrm{d}y}{L_y} |\psi_i(x,y,z)|^2\delta(\varepsilon-\varepsilon_i),$$ where $z$ is the direction perpendicular to the interface, $\psi_i(x,y,z)$ is the wavefunction, the factor 2 represents spin degeneracy. We computed the variation of the valence band maximum(VBM) and conduction band minimum(CBM) as a function of the direction ($z$) perpendicular to the interface[@Anh2013interface] $$\int_\mathrm{VBM}^{E_F} D(\varepsilon,z)\,\mathrm{d}\varepsilon = \int_{E_F}^\mathrm{CBM}D(\varepsilon,z)\,\mathrm{d}\varepsilon = \Delta \int_{-\infty}^{E_F}D(\varepsilon,z)\,\mathrm{d}\varepsilon$$ where $E_F$ is the Fermi energy and $\Delta$ is an constant that is chosen to be $0.003$.[@Anh2013interface] We follow a common procedure adopted to describe the electronic structure of interfaces described in Ref. and . The band offsets (see ) at the PBE level of theory were determined to be $\SI{0.83}{\si{eV}}$ and $\SI{1.49}{\si{eV}}$ for the valence band and conduction band, respectively, which are in agreement with the results of $\SI{0.8}{\si{eV}}$ and $\SI{1.5}{\si{eV}}$ reported in Ref. . As mentioned above, another method to obtain the valence band offset (VBO) and conduction band offset (CBO) is to align energy levels with respect to electrostatic potentials. Following Ref. , the electrostatic potential was computed as: $$\bar{V}(\mathbf{r}) = V_H(\mathbf{r}) + V_\mathrm{loc}(\mathbf{r}) - \sum_i\bar{V}_\mathrm{at}^{(i)}(|\mathbf{r}-\mathbf{r}_i|)$$ where $\bar{V}_\mathrm{at}^{(i)}$ is the potential near the core region obtained from neutral atom calculations. With this method, VBO and CBO at the PBE level are found to be $\SI{0.89}{\si{eV}}$ and $\SI{1.63}{\si{eV}}$. ![Ball and stick representation of the atomistic structure[@Anh2013interface] of the $\mathrm{Si_3N_4/Si(100)}$ interface used in our study.[]{data-label="fig:interface_structure"}](fig/interface-eps-converted-to.pdf){width="60.00000%"} To compute $G_0W_0$ corrections on band offsets, we performed $G_0W_0$@PBE calculations of bulk silicon and amorphous silicon nitride. In , quasiparticle corrections to Kohn-Sham energies of bulk silicon and amorphous silicon nitride are shown. The second and third columns are computed with 1000 and 2000 standard eigenpotentials. The fitted $G_0W_0$ reference results are extrapolated with 500, 1000, 1500 and 2000 standard eigenpotentials. To test accuracy of kinetic eigenpotentials, we started with 400 stdPDEPs and added 100, 200, 300, 400 kinPDEPs, after which the same extrapolation was applied. We calculated VBO and CBO at the $G_0W_0$ level by applying quasiparticle corrections on PBE results. After applying quasiparticle corrections on LDOS results, VBO and CBO are $\SI{1.41}{\si{eV}}$ and $\SI{1.88}{\si{eV}}$ while the VBO and CBO are found to be $\SI{1.46}{\si{eV}}$ and $\SI{2.02}{\si{eV}}$ after applying corrections to results based on electrostatic potential alignment. Both of them are close to the range of experimental results of $1.5-\SI{1.78}{\si{eV}}$ and $1.82-\SI{2.83}{\si{eV}}$. Time saving when using kinetic eigenpotentials to obtain quasiparticle corrections was approximately $\sim 50\%$. $N_\mathrm{stdPDEP}=1000$ $N_\mathrm{stdPDEP}=2000$ Fit $N_\mathrm{kinPDEP} = 400$ -------------------------- --------------------------- --------------------------- ------- ---------------------------- -- $\mathrm{Si}$ VBM 5.70 5.55 5.45 5.53 $\mathrm{Si}$ CBM 7.03 6.91 6.79 6.82 $a-\mathrm{Si_3N_4}$ VBM 7.14 7.01 7.01 6.99 $a-\mathrm{Si_3N_4}$ CBM 11.99 11.87 11.83 11.83 : Quasiparticle energies of valence band maximum (VBM) and conduction band minimum (CBM) of bulk silicon and amorphous $\mathrm{Si_3N_4}$ computed with standard eigenpotentials and by combining standard and kinetic eigenpotentials. Columns $N_\mathrm{stdPDEP}=1000$ and $N_\mathrm{stdPDEP}=2000$ are calculations done with 1000 and 2000 stdPDEPs; column Fit is extrapolated results; column $N_\mathrm{kinPDEP} = 400$ is calculation with up to 400 kinPDEPs and extrapolated. (See text) \[tab:Bulk-QP\] VBO CBO $E_g^\mathrm{Si}$ $E_g^\mathrm{Si_3N_4}$ -- ----------- ----------- ------------------- ------------------------ ------ LDOS 0.83 1.49 0.67 3.19 Potential 0.89 1.63 0.76 3.19 Ref 0.8 1.5 0.7 3.17 LDOS 1.41 1.88 1.29 4.77 Potential 1.46 2.02 1.29 4.77 Ref 1.5 1.9 1.3 4.87 1.5-1.78 1.82-2.83 1.17 4.5-5.5 : Band gaps of bulk Si, $a-\mathrm{Si_3N_4}$, and band offsets (VBO&CBO) of the interface.(see ) All values are in $\si{eV}$.[]{data-label="tab:bandoffsets"} Ref. ; Ref. ; Estimated by the other three experimental values; Ref. ; Ref. . Conclusion\[sec:conclusion\] ============================ The method introduced in Ref. to compute quasiparticle energies using the $G_0W_0$ approximation avoids the calculation of virtual electronic states and the inversion and storage of large dielectric matrices, thus leading to substantial computational savings. Building on the strategy proposed in Ref. [@wilson2008; @wilson2009] and implemented in the WEST code,[@Marco2015] here we proposed an approximation of the spectral decomposition of dielectric matrices that further improve the efficiency of $G_0W_0$ calculations. In particular we built sets of eigenpotentials used as a basis to expand the Green function and the screened Coulomb interaction by solving two separate Sternheimer equations: one using the Hamiltonian of the system, to obtain the eigenvectors corresponding to the lowest eigenvalues of the response function, and one equation using just the kinetic energy operator to obtain the eigenpotentials corresponding to higher eigenvalues. We showed that without compromising much accuracy, this approximation reduces the cost of $G_0W_0$ calculations by 10%-50%, depending on the system, with the most savings observed for the largest systems studied here. Supplementary Material {#supplementary-material .unnumbered} ====================== See supplementary material for convergence studies of the $G_0W_0$ calculations of the vertical ionization potential of the $\mathrm{CH_4}$ molecule, whcih is taken as a representative example of the molecular systems studied in the main text. We thank He Ma, Ryan L. McAvoy and Ngoc Linh Nguyen for useful discussions. This work was supported by MICCoM, as part of the Computational Materials Sciences Program funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division through Argonne National Laboratory, under contract number DE-AC02-06CH11357. This research used resources of the Argonne Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract DE-AC02-06CH11357, and resources of the University of Chicago Research Computing Center. [^1]: Here we did not attempt to solve the problem on how to accurately compute resonant molecular energy levels: our goal is to compare results obtained with solutions of the Sternheimer equation using the full Hamiltonian () and approximate solutions using only the kinetic operator (). As long as the results obtained with the two procedures agree, we consider the results obtained from as accurate.

--- abstract: 'The aim of this work is the mathematical analysis of the physical time-reversal operator and its definition as a geometrical structure[**,** ]{}in such a way that it could be generalized to the purely mathematical realm. Rigorously, only having such a “time-reversal structure” it can be decided whether a dynamical system is time-symmetric or not.[* *]{}The “time-reversal structures” of several important physical and mathematical examples are presented, showing that there are some mathematical categories whose objects are the (classical or abstract) “time-reversal systems” and whose morphisms generalize the Wigner transformation.' address: - 'I.A.F.E. (Univ. de Bs. As.)' - 'I.F.I.R. / Fac. de Ciencias Exactas, Ing. y Agrim. (Univ. Nac. de Rosario)' - 'Fac. Cs. Exactas, Ing.y Agrim.U.N.R.' author: - 'Mario Alberto Castagnino.' - 'Adolfo Ramón Ordóñez.' - 'Daniela Beatriz Emmanuele.' date: 'December 27th., 2000' title: 'A general mathematical structure for the time-reversal operator.' --- Introduction. ============= The dynamics and the thermodynamics of both, classical and quantum physical systems, are modelized by the mathematical theory of classical and abstract dynamical systems. It is obvious that the [*physical*]{} notion of “time-symmetric (or asymmetric) dynamical systems” requires the definition of a “time-reversal operator”, $K$ [@7]$.$ In fact, every known model of a physical dynamical system [*has*]{} some $K$ operator. E.g., the dynamic of classical physical systems is described in the cotangent fiber bundle $T^{*}(N)$ of its configuration manifold $N$, and therefore the action of $K:T^{*}(N)\rightarrow T^{*}(N)$ is defined as $$p_{q}\mapsto -p_{q} \label{0.1}$$ for any linear functional $p$ on $q\in N,$ or in coordinate’s language: $$(q^{i},p_{i})\mapsto (q^{i},-p_{i}) \label{0.2}$$ in a particular $(q^{i},p_{i})$ coordinate system. In Quantum Mechanics there is the well known Wigner antiunitary operator $K$ defined through the complex conjugation in the position (wave functions) representation: $$\psi (x,t)\mapsto \psi (x,-t)^{*} \label{0.3}$$ In the last few years it was demonstrated that ordinary Quantum Mechanics (with no superselection sectors) can be naturally included in the Hamiltonian formalism of its real Kählerian differentiable manifold of quantum states [@Cire0] [@Cire1] [@Cire2]. The latter one is the real but infinite dimensional manifold of the associated projective space $% {\bf P}({\cal H})$ of its Hilbert states space ${\cal H}$ [^1]. From this point of view, $K:{\bf P}({\cal H})\rightarrow {\bf P}(% {\cal H})$ acts as the cannonical projection to the quotient of (\[0.3\]) $$\lbrack \psi (x,t)]\mapsto \left[ \psi (x,-t)^{*}\right] \label{0.4}$$ Moreover, this result has been generalized to more general quantum systems through its characteristic C\*-algebra $A$, and its pure quantum states space $\partial K(A)$ turns out to be a projective Kähler bundle over the spectrum $\widehat{A},$ whose fiber over the class of a state $\psi ,$ is isomorphic to ${\bf P}({\cal H}_{\psi }),$ being ${\cal H}_{\psi }$ its GNS (Gelfand-Naimark-Segall) representation’s space [@Cire3]. More recently these authors have relaxed the Kählerian structure, showing ”minimal” mathematical structures involved in the quantum principles of superposition and uncertainty, with the aim of considering non linear extensions of quantum mechanics [@Cire4]. But, what happens in more general dynamical systems? Some of them -such as the Bernouilli systems and certain Kolmogorov-systems [@2]- are purely mathematical. Nevertheless, the notion of time-symmetry seems to make sense also for them. So, it would be interesting to know what kind of mathematical structures are involved in these systems. Our aim is to show that: 1.-[*The mere existence of the time reversal operator is a mathematical structure* ]{}consisting of a non trivial involution $K$ of the states space of a general system (with holonomic constraints) $M,$ which splits into a $K$-invariant submanifold (coordinatized by ${\it q}^{i}$) and whose complementary set (the field of the effective action of $K,$ coordinatized by ${\it p}_{i}$) is a manifold with the same dimension of $M.$ This structure is logically independent of the symplectic one [@1], which doesn’t require such a splitting at all. In fact, the essence of symplectic geometry, as its own etymology shows, is the ”common enveloping” of $q$ and $p$, loosing any ”privilege” between them. Actually, [*this* ]{}$K$[*-structure is defined by the action of that part of the complete Galilei group -including the time-reversal- which is allowed to act on the phase space manifold* ]{}$M$ [*by the constraints.* ]{}In fact, only on the homogeneous Euclidean phase space $M^{\prime }={\Bbb R}^{6n},$ it is possible to have the transitive action [@4] of the complete Galilei group[*.* ]{} 2.-[*It is possible to define generalized and purely mathematical “time-reversals”* ]{}allowing a generalization of the notion of “time-symmetry” for a wider class of dynamical systems, including all Bernouilli systems. In fact there are mathematical categories whose objects are the time-reversal (classical or abstract) systems $(M,K)$ and whose morphisms generalize the Wigner transformation [@Wigner][*.* ]{} 3.-When the states space has additional structures, [*there is a possibility of getting a richer time-reversal compatible with these structures.* ]{}For example, in Classical Mechanics the canonical $K$ is a symplectomorphism of phase space, and the Wigner quantum operator is compatible with the Kählerian structure of[* *]{}${\bf P}({\cal H}).$ At first sight (\[0.2\]) is quite similar to (\[0.3\]) and it seems to be some kind of ”complex conjugation” (and the even dimensionality of phase space reinforces this idea). We will prove this fact, namely, the existence of an almost complex structure $J$ with respect to which $K$ is an almost complex time-reversal. This increases the analogy with Quantum Mechanics, where the strong version of the Heisemberg Uncertainty Principle, [@Cire1] [@Cire2] is equivalent to the existence of a complex structure $J,$ by means of which the Wigner time-reversal is defined. 4.-[*It is possible to make a definition of time-reversal systems so general* ]{}that it includes among its examples the real line, the Minkowski space-time, the cotangent fiber bundles, the quantum systems, the classical densities function space, the quantum densities operator space, the Bernouilli systems, etc. The paper is organized as follows: In section 2, the general theory of[* * ]{}[**reversals** ]{}and[** time-reversal systems**]{} is developed. In section 3, the theory of [**abstract**]{} [**reversals** ]{}and [**abstracttime-reversal systems**]{} is given. In section 4, many important examples of time-reversal systems are given[*.*]{} In section 5, we give the abstract time-reversal of Bernouilli systems and we show explicitly the geometrical meaning of our definitions for the Baker’s transformation. Reversals and time-reversal systems =================================== [**Definition:**]{} Let $M$ be a real paracompact, connected, finite or infinite-dimensional differentiable manifold, and let $K:M\rightarrow M$ be a diffeomorphism. We will say that $K$ is a [**reversal**]{} on $M,$ and that $% (M,K)$ is a [**reversal system**]{} if the following conditions are satisfied: (r.1) $K$ is an involution, i.e. $K^2=I_M$ (r.2) The set $N$ of all fixed points of $K$ is an immersed submanifold of $% M,$ such that $M-N$ is a connected or disconnected submanifold of the same dimension of $M$. (In particular, this implies that $M$ is a non trivial involution, i.e. $K\neq I_{M},$ the identity function on $M$) We will say that $N$ is the [**invariant**]{} [**submanifold**]{} of the reversal system. [**Definition:**]{} Let $(M,K)$ be a reversal system. We will say that $M$ is $% K$[**-orientable**]{} if $M-N$ is composed of two diffeomorphic connected components $M_{+}$ y $M_{-}$, and if $$K(M_{+})=M_{-}\;,\;\text{and }K(M_{-})=M_{+} \label{1.0}$$ $M$ is $K$[**-oriented**]{} when -conventionally or arbitrarily- one of these components is selected as [**“positively oriented”**]{}. In that case, $K$ changes the $K$[**-**]{}orientation of $M.$ If there is a complex (or almost complex) structure $J$ on $M$ (and therefore $J^{\prime }=-J$ is another one) and if, in addition, $K$ satisfies: (c.r.) $K$ is complex (or almost complex) as a map from $(M,J)$ to $% (M,J^{\prime })=(M,-J)$, i.e. : $$K_{*}\circ J=-J\circ K_{*} \label{1.1}$$ we will say that $K$ is a [**complex (or almost complex) reversal, or a conjugation.** ]{} Similarly, if a symplectic (or almost symplectic) 2-form $\omega $ is given on $M$ [^2] (and therefore $\omega ^{\prime }=-\omega $ is another one) and if, in addition to (r.1) y (r.2), $K$ satisfies: (s.r.) $K$ is a symplectomorphism from $(M,\omega )$ to $(M,\omega ^{\prime })=(M,-\omega )$, i.e. : $$K^{*}\omega =-\omega \label{1.2}$$ we will say that $K$ is a [**symplectic (or almost symplectic) reversal.** ]{}If we have a symplectic (or almost symplectic) reversal system $% (M,\omega ,K),$ then for every $$m\in M:K_{*}:T_{m}(M)\rightarrow T_{m}(M)$$ is a (toplinear) isomorphism, and if $$i:N\rightarrow M\text{ is the immersion}:i(q)=m$$ and $i_{*}(X_{q})=X_{i(q)}$ is the induced isomorphism, we can define an almost complex structure $J$: $$\begin{aligned} J\left( X_{i(q)}\right) &:&=Y_{m}\Leftrightarrow \omega \left( X_{i(q)},Y_{m}\right) =1 \nonumber \\ J\left( Y_{m}\right) &:&=-X_{i(q)} \label{1.2a}\end{aligned}$$ that is to say, by defining the pairs of “conjugate” vectors (and extending by linearity). Then $$K_{*}\left( X_{i(q)}\right) =X_{i(q)}\;,\;K_{*}\left( Y_{m}\right) =-Y_{m}$$ When $(M,\omega ,J,g)$ is a Kähler (or almost Kähler) manifold, and $% K$ satisfies the properties (r.1), (r.2), (c.r.) and (s.r.), we will say that $K$ is a [**Kählerian (or almost Kählerian) reversal.** ]{}In that case, $K$ is also an isometry $$K^{*}g=g \label{1.3}$$ with respect to the Kähler metric $g$ defined by: $$g(X,Y)=-\omega (X,JY)\text{ for all vector fields }X\text{ and }Y. \label{1.4}$$ [**Definition:**]{} Let $(M,K)$ be a reversal system such that there is a class ${\cal F}$ of flows $\left( S_t\right) _{t\in {\Bbb R}}$ or / and cascades $\left( S_t=S^t\right) _{t\in {\Bbb Z}}$ on $M$ such that, for any $% m\in M,$ and any $t$ in ${\Bbb R}$ (or in ${\Bbb Z}$) satisfies: $$(K\circ S_t\circ K)(m)=S_{-t}(m) \label{1.5}$$ Then we will say that $K$ is a [**time-reversal** ]{}for ${\cal F}$[** **]{}on $% M.$ (In Physics we can take ${\cal F}$ as a class of physical interest. For example, in Classical Mechanics we can take the class of all Hamiltonian flows over a fixed phase space and in Quantum Mechanics the class of solutions of the Schrödinger equation in a fixed states space, etc.) Only having a time-reversal on $M,$ [**time-symmetric**]{} [**(or asymmetric)**]{} dynamical systems $\left( S_{t}\right) $ (flows $\left( S_{t}\right) _{t\in {\Bbb R}}$ ; or cascades $\left( S_{t}=S^{t}\right) _{t\in {\Bbb Z}}$ ) can be defined. In fact, $\left( S_{t}\right) $ will be considered as[** symmetric with respect to the time-reversal** ]{}$K$[**,**]{} if it fulfills $% \forall m\in M$ the condition (\[1.5\]) (or [**asymmetric** ]{}if it doesn’t). When $M$ is orientable (oriented) with respect to a time-reversal $K$, we will say that it is [**time-orientable (oriented).** ]{} [**Definition:** ]{}By a [**morphism**]{} of the reversal system $(M,K)$ into $% (M^{\prime },K^{\prime }),$ we mean a differentiable map $f$ of $M$ into $% M^{\prime }$ such that $$f\circ K=K^{\prime }\circ f \label{1.6}$$ As the composition of two morphisms is a morphism and the identity $I_{M}$ is a morphism, we get a [**category of reversal systems**]{}, whose objects are the reversal systems and whose morphisms are the morphisms of reversal systems. Also we have the subcategories of symplectic, almost complex, Kählerian, etc. reversal systems. Abstract reversals and abstract time-reversal systems ===================================================== [**Definition:**]{} Let $(M,\mu )$, be a measure space, and let $% K:M\rightarrow M$ be an isomorphism (mod 0) [@2]. We will say that $K$ is an [**abstract**]{} [**reversal**]{} on $(M,\mu )$ and that $(M,\mu ,K)$ is an [**abstract**]{} [**reversal system**]{} if the following conditions are satisfied: (a.r.1) $K$ is an involution, i.e. $K^2=I_M$ (mod 0) (a.r.2) The set $N$ of all fixed points of $K$ is a measurable subset of null measure of $M,$ and so $\mu [M-N]=\mu [M]$ (In particular, this implies that $M$ is a non trivial involution, i.e. $K\neq I_{M},$ the identity function on $M$) We will say that $N$ is the [**invariant**]{} [**subset**]{} of the abstract reversal system. [**Definition:**]{} Let $(M,\mu ,K)$ be an abstract reversal system such that there is a class ${\cal F}$ of measure preserving flows $\left( S_{t}\right) _{t\in {\Bbb R}}$ or / and cascades $\left( S_{t}=S^{t}\right) _{t\in {\Bbb Z% }}$ on $M$ such that, for all $m\in M,$ and all $t$ in ${\Bbb R}$ (or in $% {\Bbb Z}$) it satisfies (\[1.5\]). Then, we will say that $K$ is a [**time reversal** ]{}for ${\cal F}$[** **]{}on $(M,\mu )$. Only having an abstract time-reversal on $M,$ [**time-symmetric**]{} [**(or asymmetric)**]{} abstract dynamical systems $\left( S_{t}\right) $ (flows $% \left( S_{t}\right) _{t\in {\Bbb R}}$ ; or cascades $\left( S_{t}=S^{t}\right) _{t\in {\Bbb Z}}$ ) can be defined. In fact, $\left( S_{t}\right) $ will be regarded as[** symmetric with respect to the time-reversal** ]{}$K$[**,**]{} if it fulfills $\forall m\in M$ the condition (\[1.5\]) (or [**asymmetric** ]{}if it doesn’t) [**Definition:** ]{}By a [**morphism**]{} of the abstract reversal system $% (M,\mu ,K)$ into $(M^{\prime },\mu ^{\prime },K^{\prime }),$ we mean a measurable map $f$ of $(M,\mu )$ into $(M^{\prime },\mu ^{\prime })$ such that, $\forall A^{\prime }\subset M^{\prime }$ measurable: $$\mu \left( f^{-1}(A^{\prime })\right) =\mu ^{\prime }\left( A^{\prime }\right) \text{ mod 0} \label{1.7}$$ and $$f\circ K=K^{\prime }\circ f \label{1.8}$$ As the composition of two morphisms is a morphism and the identity $I_{M}$ is a morphism, we get a [**category of abstract reversal systems**]{}, whose objects are the abstract reversal systems and whose morphisms are the morphisms of abstract reversal systems. Examples of time-reversal systems ================================= We will see how the mathematical structure just described can be implemented in all the classical and quantum physical systems, and also generalized to more abstract purely mathematical dynamical systems, as the Bernouilli systems. The real line ------------- Let us consider in the real line ${\Bbb R}$, the mapping $K:{\Bbb R}% \rightarrow {\Bbb R}$ defined by: $$K(t)=-t \label{2.1}$$ Clearly, ${\Bbb R}$ is $K$-orientable, because if $N=\{0\},$ then ${\Bbb R}% -\{0\}={\Bbb R}_{+}\cup {\Bbb R}_{-},$ and $K$ is a canonical time-reversal for the family of translations: for $a\in {\Bbb R}$ fixed, and $t\in {\Bbb R} $, $$\text{ }S_{t}^{a}(x):=x+ta \label{2.2}$$ The Minkowski space-time ------------------------ Let $({\Bbb R}^4,\eta )$ be the Minkowski space-time, with $\eta =$ diag $% (1,-1,-1,-1)$. The invariant submanifold is the spacelike hyperplane $$N=\left\{ (0,x,y,z):x,y,z\in {\Bbb R}\right\}$$ Clearly, fixing $M_{+}$ as the halfspace containing the ”forward” light cone $$\left\{ (ct,x,y,z):c^{2}t^{2}-x^{2}-y^{2}-z^{2}>0\text{ and }t>0\right\}$$ and $M_{-}$ as the halfspace containing the ”backward” light cone $$\left\{ (ct,x,y,z):c^{2}t^{2}-x^{2}-y^{2}-z^{2}>0\text{ and }t<0\right\}$$ and defining $K:{\Bbb R}^{4}\rightarrow {\Bbb R}^{4}$ by: $$K(ct,x,y,z)=(-ct,x,y,z) \label{2.3}$$ we get a canonical $K$-orientation, equivalent to the ussual time-orientation. $K$ is a time-reversal with respect to the temporal translations $$S_{t}^{A}(X):=X+tA=(x^{0}+ta^{0},x^{1},x^{2},x^{3}) \label{2.4}$$ for $A=(a^{0},0,0,0)\in {\Bbb R}^{4}:a^{0}\neq 0$ fixed, and $t\in {\Bbb R}$. [**Remark:** ]{}As an effect of curvature, not every general 4-dimensional Lorentzian manifold $(M,g),$ will be time-orientable [@Licner]. Nevertheless, a time-orientable general space-time is necessary if we are searching for a model of a universe with an arrow of time [@cosmic; @arrow] [@Casta]. In fact, if our universe were represented by a non-time-orientable manifold, it would be impossible to define past and future in a global sense, in contradiction with all our present cosmological observations. Precisely, we know that there are no parts of the universe where the local arrow of time points differently from our own arrow. The cotangent fiber bundle. Classical Mechanics. ------------------------------------------------ A general cotangent fiber bundle needs not to be $K$-orientable. Nevertheless, we have the following result: [**Theorem**]{}: The cotangent fiber bundle (of a finite dimensional differentiable manifold) $\left( T^{*}(N),\pi ,N\right) $ [@1] has a canonical almost Kählerian time-reversal (for the Hamiltonians flows on it). [**Proof:**]{} Let $M$ be the cotangent fiber bundle $T^{*}(N)$ of a real n-dimensional manifold $N.$ In this case $N$ is an embedded submanifold of $% M $, being the embedding $i:N\rightarrow T^{*}(N)$ such that: $$\text{ }i(q)=0_{q}\text{ (the null functional at }q\text{)}$$ Let’s define $$\begin{aligned} K &:&T^{*}(N)\rightarrow T^{*}(N) \nonumber \label{4.1} \\ \forall p_q &\in &T_q^{*}(N):K(p_q)=-p_q \label{2.5}\end{aligned}$$ Because of its definition, this map is obviously an involution, and by its linearity is differentiable and its differential or tangent map $$K_{*}:T\left( T^{*}(N)\right) \rightarrow T\left( T^{*}(N)\right)$$ verifies: $$K_{*}(X_{p_{q}})= %TCIMACRO{ %\QATOPD\{ . {X_{p_{q}}\text{ if }X_{p_{q}}\in i_{*}\left( T_{q}(N)\right) }{-X_{p_{q}}\text{ if }X_{p_{q}}\in T_{p_{q}}\left( \pi ^{-1}(q)\right) } } %BeginExpansion {X_{p_{q}}\text{ if }X_{p_{q}}\in i_{*}\left( T_{q}(N)\right) \atopwithdelims\{. -X_{p_{q}}\text{ if }X_{p_{q}}\in T_{p_{q}}\left( \pi ^{-1}(q)\right) }% %EndExpansion \label{2.6}$$ $X_{p_{q}}\in T_{p_{q}}\left( \pi ^{-1}(q)\right) $ means that it is “vertical” or tangent to the point $p_{q}$ of the fibre in $q$. It must be taken into account that by joining a vertical base with the image of a base in $N$ by the isomorphism $i_{*}$ $,$ we get a base of $T_{p_{q}}\left( T^{*}(N)\right) .$ Let $\omega $ be the canonical symplectic 2-form of the cotangent fiber bundle. As $\omega $ is antisymmetric, in order to evaluate $K^{*}\omega ,$ it is sufficient to consider only three possibilities: $$\begin{aligned} 1)\;\left( X_{p_q},Y_{p_q}\right) &:&X_{p_q}\;,Y_{p_q}\in i_{*}\left( T_q(N)\right) \\ 2)\;\left( X_{p_q},Y_{p_q}\right) &:&X_{p_q}\;,Y_{p_q}\in T_{p_q}\left( \pi ^{-1}(q)\right) \\ 3)\;\left( X_{p_q},Y_{p_q}\right) &:&X_{p_q}\in i_{*}\left( T_q(N)\right) \text{ but }Y_{p_q}\in T_{p_q}\left( \pi ^{-1}(q)\right)\end{aligned}$$ In case 1) $$\omega \left( K_{*}(X_{p_q}),K_{*}(Y_{p_q})\right) =\omega \left( X_{p_q},Y_{p_q}\right) =0 \label{2.7}$$ In case 2) $$\omega \left( K_{*}(X_{p_q}),K_{*}(Y_{p_q})\right) =\omega \left( -X_{p_q},-Y_{p_q}\right) =\omega \left( X_{p_q},Y_{p_q}\right) =0 \label{2.8}$$ In case 3) $$\begin{aligned} \omega \left( K_{*}(X_{p_q}),K_{*}(Y_{p_q})\right) &=&\omega \left( X_{p_q},-Y_{p_q}\right) \nonumber \\ &=&-\omega \left( X_{p_q},Y_{p_q}\right) \label{2.9}\end{aligned}$$ Thus, in any case $$\left( K^{*}\omega \right) \left( X_{p_q},Y_{p_q}\right) =\omega \left( K_{*}(X_{p_q}),K_{*}(Y_{p_q})\right) =-\omega \left( X_{p_q},Y_{p_q}\right) \label{2.10}$$ which proves that $K^{*}\omega =-\omega ,$ the (s.r.) property$.$ Now, let us define $$\begin{aligned} J &:&T\left( T^{*}(N)\right) \rightarrow T\left( T^{*}(N)\right) \nonumber \\ J(X_{p_q}) &=&Y_{p_q}\Leftrightarrow \omega \left( X_{p_q},Y_{p_q}\right) =1 \label{2.10.1}\end{aligned}$$ that is to say, $J(X_{p_q})$ is the canonical conjugate of $X_{p_q}.$ Then, by the antisymmetry of $\omega ,$ clearly $J^{2}=-I.$ In addition $$\begin{aligned} \left( K_{*}\circ J\right) (X_{p_{q}}) &=&K_{*}\left( J(X_{p_{q}})\right) =-J\left( X_{p_{q}}\right) \nonumber \\ &=&-J\left( K_{*}\left( X_{p_{q}}\right) \right) \nonumber \\ &=&\left( -J\circ K_{*}\right) (X_{p_{q}}) \label{2.10.2}\end{aligned}$$ if $X_{p_{q}}\in i_{*}\left( T_{q}(N)\right) $ and $$\begin{aligned} \left( K_{*}\circ J\right) (X_{p_{q}}) &=&K_{*}\left( J(X_{p_{q}})\right) =J(X_{p_{q}}) \nonumber \\ &=&J\left( -K_{*}\left( X_{p_{q}}\right) \right) \nonumber \\ &=&\left( -J\circ K_{*}\right) (X_{p_{q}}) \label{2.10.3}\end{aligned}$$ if $X_{p_{q}}\in T_{p_{q}}\left( \pi ^{-1}(q)\right) .$ So, $K$ preserves both $\omega $ and $J$, and therefore is almost Kählerian. $\Box $ As it is well known, the phase space $M$ of a classical system with a finite number ($n$) of degrees of freedom and holonomic constraints has the particular form $T^{*}(N),$ where $N$ denotes the configuration space of the system. This implies the existence of a privileged submanifold $N$ of $M$. We may enquire: why is this so? The answer is: because every law of Classical Mechanics is invariant with respect to the Galilei group [*which contains all the spatial translations*]{} (and it is itself a contraction of the inhomogeneous Lorentz group [@Hermann]). This forces the configuration space to be a submanifold of some [*homogeneuos*]{} ${\Bbb R}% ^{d}$ space. Now, in this submanifold we also have a privileged system of coordinates: the spatial position coordinates $q_{1}=x_{1},...,q_{d}=x_{d}$, with respect to which the action of the Galilei group has its simplest affine expression. Nevertheless, in general this action will take us away from the configuration manifold $N,$ because it doesn’t fit with the constraints (think for example in the configuration space of a double pendulum -with two united threads- which is a 2-torus, contained in ${\Bbb R}% ^{3}$). Classical Statistical Mechanics ------------------------------- Let us consider the phase space of a classical system $\left( T^{*}(N),\omega \right) $ and take the real Banach space $V=L_{{\Bbb R}% }^1\left( T^{*}(N),\sigma \right) $ containing the probability densities over the phase space, where $\sigma =\omega \wedge ...\wedge \omega $ ($n$ times) is the Liouville measure. $V$ is a real infinite dimensional differentiable manifold modelled by itself. Then, the above defined almost Kählerian time-reversal $K$ on $T^{*}(N)$ induces $\widetilde{K}% :V\rightarrow V$ by: $$\rho \mapsto \widetilde{K}(\rho ):\left( \widetilde{K}(\rho )\right) (m):=\rho \left( K(m)\right) \label{2.11}$$ Clearly, $\widetilde{K}$ is a toplinear involution. Now, let us consider the set $P$ of all (“almost everywhere” equivalent classes of) “$\widetilde{K} $-even” integrable functions $$P=\left\{ \rho \in V:\rho \left( K(m)\right) =\rho (m)\right\} \label{2.12}$$ and the set $I$ of all (“almost everywhere” equivalent classes of) the “$% \widetilde{K}$-odd” integrable functions $$I=\left\{ \rho \in V:\rho \left( K(m)\right) =-\rho (m)\right\} \label{2.13}$$ Trivially, $V=P\oplus I$ , and there are two toplinear projectors mapping any $\rho \in V$ into its “$\widetilde{K}$-even” and “$\widetilde{K}$-odd” parts. $P$ is the invariant subspace of $\widetilde{K}.$ Its complement is the (infinite dimensional) open submanifold of (“almost everywhere” equivalent classes of) integrable functions whose “$\widetilde{% K}$-odd” projection doesn’t vanish$.$ Now, every dynamical system $(S_t)$ (in particular those of the class ${\cal % F}$ of $K$) on $T^{*}(N)$ induces another dynamical system $(U_t)$ on $V:$$$\left( U_t(\rho )\right) (m):=\rho \left( S_{-t}(m)\right) \label{2.14}$$ Considering the class $\widetilde{{\cal F}},$ induced by ${\cal F},$ we conclude that $\widetilde{K}$ is a time-reversal. So we have another example lacking time orientability but having a time-reversal structure . Complex Banach spaces --------------------- A [**complex structure**]{} on a [*real*]{} finite (or infinite) Banach space $% V$ is a linear (toplinear) transformation $J$ of $V$ such that $J^2=-I $ , where $I$ stands for the identity transformation of $V$. [@4] In the case of a [*complex*]{} Banach space $V_{{\Bbb C}}$ , we can consider the associated real vector space (its “realification”) $V=V_{{\Bbb R}}$ composed of the same set of vectors, but with ${\Bbb R}$ instead of ${\Bbb C} $, as the field of its scalars. Then, $J=iI$ is the canonical complex structure of $V_{{\Bbb R}}.$ If $J$ is a complex structure on a finite dimensional real vector space, its dimension must be even. In any case, there exist elements $X_1$, $X_2$,..., $% X_n,...$ of $V$ such that $$\left\{ X_1,...,X_n,...,JX_1,...JX_n,...\right\}$$ is a basis for $V$ [@4]. Let us define $K:V\rightarrow V$ as the “conjugation”, i.e. extending by linearity the assignment $$\forall i=1,2,...:K(X_{i})=X_{i}\;;\;K(JX_{i})=-JX_{i} \label{2.15}$$ Then the real subspace generated by $\{X_{1},X_{2},...\},$ is the subspace of fixed points of $K,$ $N$. So, $(V,J,K)$ is a complex time-reversal system for the class of ”non real translations” $\left( S_{t}^{A}\right) _{t\in {\Bbb R}}$ ($A$ being a linear combination of $JX_{1},...JX_{n},...)$: $$S_{t}^{A}(X)=X+tA\;,\;t\in {\Bbb R} \label{2.16}$$ In fact, $$\begin{aligned} \left( K\circ S_{t}^{A}\circ K\right) (X) &=&K\left( K(X)+tA\right) =X-tA \nonumber \\ &=&S_{-t}^{A}(X) \label{2.17}\end{aligned}$$ Ordinary quantum mechanical systems ----------------------------------- As a particular case of the previous example, let us consider a classical system whose phase space is ${\Bbb R}^{6n},$ and take $V={\cal H}% =L^{2}\left( {\Bbb R}^{3n},\sigma \right) $ (actually, its realification). This choice is motivated by the fact that we want to have a Galilei-invariant Quantum Mechanics, and so [*we must quantify the spatial position coordinates*]{}. There is no cannonical or symplectic symmetry here. Only acting on the wave functions of the position coordinates the Wigner time-reversal operator will be expressed as the complex conjugation. So, we get a complex time-reversal structure. Now, following [@Cire1], let us consider the real but infinite dimensional Kählerian manifold $\left( {\bf P}({\cal H}),\widetilde{J},% \widetilde{\omega },g\right) $ of the associated projective space ${\bf P}(% {\cal H})$ of the Hilbert states space ${\cal H}$ of an ordinary quantum mechanical system. $J$ is the complex structure of ${\cal H}$, and it is the local expresion of $$\widetilde{J}:T\left( {\bf P}({\cal H})\right) \rightarrow T\left( {\bf P}(% {\cal H})\right)$$ $\left( {\bf P}({\cal H}),\widetilde{J},\widetilde{\omega },g\right) $ has a canonical Kählerian time-reversal structure. In fact, we define $K:{\cal % H}\rightarrow {\cal H}$ as in the previous example, and take: $$\widetilde{K}:{\bf P}({\cal H})\rightarrow {\bf P}({\cal H})\text{ by: }% \widetilde{K}\left[ \psi \right] =\left[ K(\psi )\right]$$ then, all the desired properties follows easily. Quantum Statistical Mechanics ----------------------------- Let $V=L^{1}({\cal H})$ denote the complex Banach space generated by all nuclear operators on ${\cal H}$ with the trace norm. This set contains the density operators of Quantum Statistical Mechanics. $V$ is a real infinite dimensional differentiable manifold modelled by itself. Then, the above defined complex time-reversal $K$ on ${\cal H}$ induces $\widehat{K}% :V\rightarrow V$ by: $$\widehat{\rho }\mapsto \widehat{K}(\widehat{\rho }):\left( \widehat{K}(% \widehat{\rho })\right) (\psi ):=\widehat{\rho }\left( K(\psi )\right) \label{2.18}$$ Clearly, $\widehat{K}$ is a toplinear involution. Now, let us consider the set $R$ of all “$\widehat{K}$-real” densities $$R=\left\{ \widehat{\rho }\in V:\widehat{\rho }\left( K(\psi )\right) =% \widehat{\rho }(\psi )\right\} \label{2.19}$$ and the set $I$ of all the “$\widehat{K}$-imaginary” densities $$I=\left\{ \widehat{\rho }\in V:\widehat{\rho }\left( K(\psi )\right) =-% \widehat{\rho }(\psi )\right\} \label{2.20}$$ Trivially, $V=R\oplus I$ , and there are two toplinear projectors mapping any $\rho \in V$ into its “$\widehat{K}$-real” and “$\widehat{K}$-imaginary” parts. $R$ is the invariant subspace of $\widehat{K}.$ Its complement is the (infinite dimensional) open submanifold of (“almost everywhere” equivalent classes of) integrable functions whose “$\widehat{K} $-imaginary” projection is not null$.$ Now, every dynamical system $(S_{t})$ (in particular those of the class $% {\cal F}$ of $K$) on ${\cal H}$ induces another dynamical system $(U_{t})$ on $V$ by setting$:$$$\left( U_{t}(\widehat{\rho })\right) (\psi ):=\widehat{\rho }\left( S_{-t}(\psi )\right) \label{2.21}$$ Considering the class $\widehat{{\cal F}},$ induced by ${\cal F},$ we conclude that $\widehat{K}$ is a complex time-reversal. So we have another example lacking time orientability but having a time-reversal structure. In both Classical and Quantum Statistical Mechanics, we have used the same criterium to choose $N$ and ${\cal H}$ respectively. The densities of the two theories are related by the Wigner integral $W$, which is an essential ingredient in the theory of the classical limit [@Casta-Laura]. In the one dimensional case, it is the mapping $\widehat{\rho }\mapsto \rho =W\left( \widehat{\rho }\right) $ given by: $$\rho (q,p)=\frac{1}{\pi }\int\limits_{-\infty }^{+\infty }\widehat{\rho }% (q-\lambda ,q+\lambda )\,e^{2ip\lambda }\,d\lambda \label{2.21a}$$ where $q$ is the spatial position coordinate [^3], $p$ its conjugate momentum, $\rho (q,p)$ a classical density function, and $$\begin{aligned} \widehat{\rho }(x,x^{\prime }) &=&\left( \sum\limits_{j=1}^{\infty }\rho _{j}\,\overline{\psi _{j}}\otimes \psi _{j}\right) (x,x^{\prime }) \nonumber \\ &=&\sum\limits_{j=1}^{\infty }\rho _{j}\,\overline{\psi _{j}}(x)\psi _{j}(x^{\prime }) \label{2.21b}\end{aligned}$$ is a generic matrix element of a quantum density, being $\left\{ \psi _{j}\right\} _{j=1}^{\infty }$ an orthonormal base of ${\cal H}$, $\rho _{j}\geqslant 0$ and $\sum\limits_{j=1}^{\infty }\rho _{j}=1$. As it is obvious by a simple change of variables, $$W\left[ \widehat{K}\left( \widehat{\rho }\right) \right] \left( q,p\right) =\rho (q,-p)=\rho \left( K(q,p)\right) =\widetilde{K}\left[ W\left( \widehat{% \rho }\right) \right] \left( q,p\right) \label{2.21c}$$ and therefore, $W$ is a morphism between $\left( L^{1}({\cal H}),\widehat{K}% \right) $ and $\left( L_{{\Bbb R}}^{1}\left( T^{*}(N)\right) ,\widetilde{K}% \right) .$ Koopman treatment of Kolmogorov-Systems --------------------------------------- With the definition of time-reversal in the physical examples above, we now face the same definition in purely mathematical dynamical systems. Let $(M,\mu ,S_t)$ be a Kolmogorov system (cascade or flow). As it is well known, this implies that the induced unitary evolution $U_t$ in ${\cal H}% =[1]^{\bot }$ the orthogonal complement of the one dimensional subspace of the classes a. e. of the constant functions in the Hilbert space $L^2(M,\mu ),$ has uniform Lebesgue spectrum of numerable constant multiplicity. This, in turn, implies the existence of a system of imprimitivity $(E_s)_{s\in {\Bbb G}}$ based on ${\Bbb G}$ for the group $(U_t)_{t\in {\Bbb G}}$ , where ${\Bbb G}$ is ${\Bbb Z}$ or ${\Bbb R}$: $$E_{s+t}=U_tE_sU_t^{-1} \label{2.22}$$ Following Misra [@Misra], we define the “Aging” operator $$T=\int\limits_{{\Bbb G}}s\,dE_{s}= %TCIMACRO{ %\QATOPD\{ . {\int\limits_{{\Bbb R}}s\,dE_{s}\text{ for fluxes}}{\sum\limits_{s\in {\Bbb Z}}sE_{s}\text{ for cascades}} } %BeginExpansion {\int\limits_{{\Bbb R}}s\,dE_{s}\text{ for fluxes} \atopwithdelims\{. \sum\limits_{s\in {\Bbb Z}}sE_{s}\text{ for cascades}}% %EndExpansion \label{2.23}$$ Then $$U_{-t}TU_t=T+t \label{2.24}$$ $T$ is selfadjoint in the discrete case, and essentially selfadjoint in the continuous case, and there are eigenvectors in the discrete case, and generalized eigenvectors (antifunctionals) in certain riggings of ${\cal H}$ by a nuclear space $\Phi $ ($\Phi \prec {\cal H}\prec \Phi ^{\times }$) in the continuous case, $\left( \left| \tau ,n\right\rangle \right) _{\tau \in {\Bbb G}}$ , such that: $$\begin{aligned} T\left| \tau ,n\right\rangle &=&\tau \left| \tau ,n\right\rangle \label{2.25} \\ U_{t}\left| \tau ,n\right\rangle &=&(\tau +t)\left| \tau ,n\right\rangle \label{2.26}\end{aligned}$$ Defining $$K\left| \tau ,n\right\rangle =-\left| \tau ,n\right\rangle \label{2.27}$$ It follows easilly that $K$ restricted to ${\cal H}$ is a time-reversal for $% {\cal F}=\{(U_t)\}$ with respect to which $U_t$ is symmetric. Examples of abstract time-reversals =================================== Bernouilli schemes ------------------ Let $M$ be the set $\Sigma ^{{\Bbb Z}}$ of all bilateral sequences (of “bets”) $$m=(a_{j})_{j\in {\Bbb Z}}=(...a_{-2},a_{-1},a_{0},a_{1},a_{2},...) \label{3.1}$$ on a finite set $\Sigma $ with $n$ elements (a “dice” with $n$ faces). Let ${\frak X}$ be the $\sigma $-algebra on $M$ generated by all the subset of the form $$A_{j}^{s}=\{m:a_{j}=s\in \Sigma \} \label{3.2}$$ Clearly, $$M=\bigcup\limits_{s\in \Sigma }A_j^s=\bigcup\limits_{k=1}^nA_j^{s_k} \label{3.3}$$ Let’s define a normalized measure $\mu $ on $M$ by choosing $n$ ordered positive real numbers $p_{1},...,p_{n}$ whose sum is equal to one ($p_{k}$ is the “probability” of getting $s_{k}$ when the “dice” is thrown), and setting: $$\forall k:k=1,...,n\;:p_{k}=\mu (A_{j}^{s_{k}}) \label{3.4}$$ $$\mu \left( A_{j_1}^{s_1}\cap ...\cap A_{j_k}^{s_k}\right) =\mu (A_{j_1}^{s_1})...\mu (A_{j_k}^{s_k}) \label{3.5}$$ where $j_1,...,j_k$ are all different. Let the dynamical authomorphism $S$ be the shift to the right: $$\begin{aligned} S\left( (a_{j})_{j\in {\Bbb Z}}\right) &=&(a_{j}^{\prime })_{j\in {\Bbb Z}} \nonumber \\ \text{where: } &&a_{j}^{\prime }:=a_{j-1} \label{3.6}\end{aligned}$$ The shift preserves $\mu $ because $$\mu \left( S(A_j^{s_k})\right) =\mu (A_{j+1}^{s_k})=p_k \label{3.7}$$ The above abstract dynamical scheme is called a Bernouilli scheme and denoted $B(p_1,...,p_n).$ Let’s define a “cannonical” abstract reversal by: $$\begin{aligned} K\left( (a_{j})_{j\in {\Bbb Z}}\right) &=&(a_{j}^{\prime })_{j\in {\Bbb Z}} \nonumber \\ a_{j}^{\prime } &=&a_{-j+1} \label{3.8}\end{aligned}$$ Clearly, $K$ is an isomorphism, and its invariant set $$N=\bigcap\limits_{j\in {\Bbb Z}}\left\{ \bigcup\limits_{s\in \Sigma }\left( A_j^s\cap A_{-j}^s\right) \right\} \label{3.9}$$ has $\mu $-measure $0.$ In addition $K$ is a time-reversal for the class $% {\cal F}$ of all Bernouilli schemes, because $$K\circ S\circ K=S^{-1} \label{3.10}$$ being $S^{-1}$ the shift to the left. The Baker’s transformation -------------------------- We will show the geometrical meaning of the last two time-reversals for the Baker’s transformation. The measure space is the torus $$M=\left[ 0,1\right] \times \left[ 0,1\right] \;/\sim \;=\left\{ (x,y)% %TCIMACRO{\func{mod}} %BeginExpansion \mathop{\rm mod}% %EndExpansion 1=[x,y]:x,y\in \left[ 0,1\right] \right\}$$ that is to say, $\sim $ is the equivalence relation that identifies the following boundary points: $$(0,x)\sim (1,x)\;\text{and}\;(x,0)\sim (x,1)$$ with its Lebesgue measure. The automorphism $S$ acts as follows: $$S(x,y)= %TCIMACRO{ %\QATOPD\{ . {(2x,\frac 12y)\;\;\;\;\;\;\;\;\;\;if\;0\leq x\leq \frac 12\;,\;0\leq y\leq 1}{(2x-1,\frac 12y+\frac 12)\;if\;\frac 12\leq x\leq 1\;,\;0\leq y\leq 1} } %BeginExpansion {(2x,\frac 12y)\;\;\;\;\;\;\;\;\;\;if\;0\leq x\leq \frac 12\;,\;0\leq y\leq 1 \atopwithdelims\{. (2x-1,\frac 12y+\frac 12)\;if\;\frac 12\leq x\leq 1\;,\;0\leq y\leq 1}% %EndExpansion \label{5.1}$$ It’s clear that $S$ is a non-continuous[** **]{}but measure preserving transformation which involves a contraction in the $y$ direction and a dilatation in the $x$ direction: the contracting and dilating directions at every point $m\in M$ (that is, the vertical and the horizontal lines through each $m$). The torus is a compact, connected Lie group and we can define an involutive automorphism $K$ on $M$ by putting: $$K[x,y]=[y,x]\;;\;x,y\in I=\left[ 0,1\right] \label{5.2}$$ The fixed points of $K$ constitute a submanifold of the torus: the projection of the diagonal $\Delta $ of the unit square $I\times I$ $$N=\Delta \;/\sim =\left\{ [x,x]:x\in I\right\}$$ Then: $$K\circ S^t\circ K=S^{-t},\;\forall t\in {\Bbb Z} \label{5.3}$$ In fact, the first application of $K$ to the generating partition of $S$, rotates the unit square, interchanging the $x$ fibers with the $y$ ones. Then by applicating $S^{t}$ (that is $t$ times $S$) we get a striped pattern of horizontal lines, which is rotated and yields a striped pattern of vertical lines when $K$ is applicated again. The same pattern would be obtained if $S^{-t}$ was used. As it is well known [@2], the Baker transformation is isomorphic to $B(% \frac{1}{2},\frac{1}{2})$. In fact, the map $$(x,y)\mapsto m=(a_{j})_{j\in {\Bbb Z}}\Leftrightarrow x=\sum\limits_{j=0}^{\infty }\frac{a_{-j}}{2^{j+1}}\;\text{and }% y=\sum\limits_{j=1}^{\infty }\frac{a_{j}}{2^{j}} \label{5.4}$$ is an isomorphism (mod 0). Moreover it is an isomorphism of abstract time-reversal systems, because it sends the time-reversal of (\[5.2\]) in the time-reversal of (\[3.8\]). In particular, this implies that the Baker’s map is a Kolmogorov system, and therefore having the corresponding time-reversal for its Koopman treatment. These three reversals are related. Let $\{A,B\}$ be the partition of the unit square into its left and right halves. As it is well known this partition is both independent and generating for the Baker’s map. Let’s define $$\theta _{0}=1-\chi _{A}=% %TCIMACRO{\QATOPD\{ . {\text{\ }1\text{ in }A}{-1\text{ in }B} } %BeginExpansion {\text{\ }1\text{ in }A \atopwithdelims\{. -1\text{ in }B}% %EndExpansion \label{5.5}$$ where $\chi _{A}$ is the characteristic function of the set $A,$ as well as $$\theta _{n}=U^{n}(\theta _{0})=\theta _{0}\circ S^{-n}= %TCIMACRO{ %\QATOPD\{ . {\text{\ }1\text{ in }S^{n}(A)}{-1\text{ in }S^{n}(B)} } %BeginExpansion {\text{\ }1\text{ in }S^{n}(A) \atopwithdelims\{. -1\text{ in }S^{n}(B)}% %EndExpansion \label{5.6}$$ and for any finite set $F=\{n_{1},...,n_{F}\}\subset {\Bbb Z}$, put $$\theta _{F}=\theta _{n_{1}}...\theta _{n_{F}}\;\text{(ordinary product of functions)} \label{5.7}$$ Then, all the eigenvectors of the Aging operator $T$ of $U$ are of the form [@Prigo]: $$T\theta _{F}=n_{m}\theta _{F} \label{5.8}$$ where $n_{m}=\max F.$ Geometrically speaking, $\rho =\theta _{0}$ -that we can identify with $\{A,B\}$- is an eigenvector of age $0,$ and if $U$ acts $% n $ times on it we get an eigenvector of age $n$: $\theta _{n}$ -which can be identified with a set of horizontal fringes-$.$ On the other hand if $% U^{-1}$ acts $n$ times on it we get an eigenvector of age $-n$: $\theta _{-n} $ -which can be identified with a set of vertical fringes-$.$ As expected, the induced action of $K$ sends the “future” horizontal eigenstates of $T$ to the “past” vertical ones, and reciprocally. ACKNOWLEDGMENT The authors wish to express their gratitude to Dr. Sebastiano Sonego for providing an initial and fruitful discussion on the subject of this paper. This work was partially supported by grant PIP 4410 of CONICET (Argentine National Research Council) ARNOLD, V.,[*Mathematical Methods of Classical Mechanics*]{}, Springer-Verlag, New York, 1989. ARNOLD, V. - AVEZ, A., [*Ergodic Problems of Classical Mechanics*]{}, Benjamin, New York, 1968. CASTAGNINO, M., Phys. Rev. D 57, 750 (1998); and [*The global nature of the arrow of time and Bohm-Reichenbach diagram,*]{} in ”Irreversibility and Causality (A selection of articles presented at the 21st International Colloquium on Group Methods in Physics, Goslar, July 1996), A. Bohm et al. Ed. Springer-Verlag, Berlin, page 282 (1998). CASTAGNINO, M., GUNZIG, E., IGURI, S., ORDOÑEZ, A., [*The Kolmogorov-Lax-Phillips systems as branch systems of the Reichemach model*]{}, Proc. 7th. Intrnational Workshop of Instabilities and Non Equilibrium Structures, Valparaíso, Chile, 15/12/1997. CASTAGNINO, M., LAURA, R., [*Functional Approach to Quantum Decoherence and Classical Final Limit*]{}, submitted to Phys. Rev. A (1999). CIRELLI, R., MANIÀ, A., PIZZOCCHERO, L., [*Quantum Phase Space formulation of Schrödinger Mechanics,* ]{}Int. Jour. of Mod. Phys. A, Vol. 6, N 12 (1991), 2133-2146. CIRELLI, R., MANIÀ, A., PIZZOCCHERO, L., [*Quantum mechanics as an infinite-dimensional Hamiltonian system with uncertainty structure: Part I*]{}, J. Math. Phys. 31 (12), Dec. 1990, 2891-2897. CIRELLI, R., MANIÀ, A., PIZZOCCHERO, L., [*Quantum mechanics as an infinite-dimensional Hamiltonian system with uncertainty structure: Part II*]{}, J. Math. Phys. 31 (12), Dec. 1990, 2898-2903. ABBATI, M.C., CIRELLI, R., LANZAVECCHIA, P., MANIÁ, A., [*Pures Quantum-Mechanicals Systems as Kähler Bundles*]{}, Il Nuovo Cimento, Vol. 83 B, N 1, (sett. 1984), 43-60. CIRELLI, R., GATTI, M. ,MANIÀ, A., [*On the nonlinear extension of quantum superposition and uncertainty principles*]{}, Jour. Geom. and Phys. 516 (1998), 1-23. CORNFELD, I.P., FOMIN, S.V., SINAI, Ya.G., [*Ergodic Theory,* ]{}Springer-Verlag, New York, 1982.[* *]{} HELGASON, S., [*Differential Geometry, Lie Groups and Symmetric Spaces*]{}, Acad. Press, New York, 1978. HERMANN, R., [*Lie Groups for Physicists*]{}, Benjamin, New York, 1966. KOBAYASHI, S. AND NOMIZU, K., [*Foundations of Differential Geometry*]{}, Volumes I and II, Interscience Publishers,1969. LASOTA, A. - MACKEY, M., [*Probabilistic properties of deterministic systems*]{}, Cambridge, Univ. Press, 1985. MACKEY, M., [*Time’s Arrow: The origins of Thermodynamic Behaviour*]{}, Springer-Verlag, New York, 1992. MESSIAH, A., [*Mecánica Cuántica*]{}, Ed. Tecnos, Madrid, 1964. LICHNEROWITZ, A., [*Champ de Dirac, champ du neutrino et transformations C, P, T sur un espace-temps courbe*]{}, Ann. Inst. H. Poincaré, Vol. I, n${{}^{o}}$ 3, 233-290, 1964. MISRA, B. , [*Nonequilibrium Entropy, Liapounov Variables and Ergodic Properties of Classical Systems*]{}, Proc. Nat. Acad. Sci. USA, 75, p. 1627, 1978. MISRA, B., PRIGOGINE, I. , COURBAGE, M. , [*From deterministic dynamics to probabilistic descriptions*]{}, Physica 98 A, 1-26, 1979. SPIVAK, M., [*A Comprehensive Introduction to Differential Geometry,*]{} Volume 1, Publish or Perish, Inc., USA, 1979. STERNBERG, S., [*Lectures on Differential Geometry*]{}, Prentice-Hall, New Jersey, 1964. TABOR, M. , [*Chaos and integrability in nonlinear dynamics*]{} , A Wiley-Interscience Publication , John Wiley et Sons. WARNER, F., [*Foundations of Differentiable Manifolds and Lie Groups,*]{} Springer-Verlag, New York, 1983. HILLERY, M. ,O’CONELL, R. F. , SCULLY, M. O. , WIGNER, E.P., [*Distribution Function in Physics: Fundamentals*]{}, Phys. Repp. 106, 3, (1984), 123-167. [^1]: We should remember the fact that ordinary pure quantum states are not [*vectors*]{} $\psi $ (normalized or not) of a Hilbert space ${\cal H}$, but rays or [*projective equivalence classes of vectors*]{} $[\psi ]\in {\bf P}({\cal % H})$. [^2]: In the infinite-dimensional case we require $\omega $ to be [*strongly non-degenerate* ]{}[@Cire4] in the sense that the map $X\mapsto \omega (X,.) $ is a toplinear isomorphism. [^3]: We want to emphasize the necessity of having an homogeneous configuration space (${\Bbb R}$ in the one dimensional case) in order to have the translations $q\mapsto q\pm \lambda $ in $W$ integral.

--- abstract: | CO measurements of 1-4 galaxies have found that their baryonic gas fractions are significantly higher than galaxies at =0, with values ranging from 20-80%. Here, we suggest that the gas fractions inferred from observations of star-forming galaxies at high- are overestimated, owing to the adoption of locally-calibrated CO- conversion factors (). Evidence from both observations and numerical models suggest that  varies smoothly with the physical properties of galaxies, and that  can be parameterised simply as a function of both gas phase metallicity and observed CO surface brightness. When applying this functional form, we find ${\mbox{$f_{\rm gas}$}}\approx 10-40\%$ in galaxies with $M_*=10^{10}-10^{12} \ {\mbox{M$_{\sun}$}}$. Moreover, the scatter in the observed -$M_*$ relation is lowered by a factor of two. The lower inferred gas fractions arise physically because the interstellar media of high- galaxies have higher velocity dispersions and gas temperatures than their local counterparts, which results in an  that is lower than the =0 value for both quiescent discs and starbursts. We further compare these gas fractions to those predicted by cosmological galaxy formation models. We show that while the canonically inferred gas fractions from observations are a factor of 2-3 larger at a given stellar mass than predicted by models, our rederived  values for =1-4 galaxies results in revised gas fractions that agree significantly better with the simulations. author: - | Desika Narayanan$^{1}$[^1][^2], Matt Bothwell$^{1}$, Romeel Davé$^1$\ $^{1}$Steward Observatory, University of Arizona, 933 N Cherry Ave, Tucson, Az, 85721\ date: MNRAS Accepted title: 'Galaxy Gas Fractions at High-Redshift: The Tension between Observations and Cosmological Simulations' --- \[firstpage\] galaxies:formation-galaxies:high-redshift-galaxies:starburst-galaxies:ISM-ISM:molecules Introduction {#section:introduction} ============ Recent technological advances in (sub)millimetre-wave telescope facilities have allowed for the detection of star-forming  gas in large numbers of galaxies at high-redshift via the proxy molecule $^{12}$CO [hereafter, CO; @gre05; @tac06; @cop08; @tac08; @bot09; @dan09; @wag09b; @car10; @dad10a; @dad10b; @gen10; @rie10a; @rie10b; @tac10; @gea11; @cas11; @rie11a; @rie11b; @wan11 see @sol05 for a summary of pre-2005 references]. A major finding from these studies is that, at a given stellar mass, early Universe galaxies tend to be significantly more gas rich than their present-day counterparts, with baryonic gas fractions[^3] (hereafter defined as = $M_{\rm H2}/(M_{\rm H2}+M_*)$) ranging from $\sim 20-80\%$ [e.g. @dad10a; @tac10]. This is consistent with both observational and theoretical results that suggest that even gas rich disc galaxies at 2 are able to form stars rapidly enough that they are comparable to the most extreme merger-driven starburst events in the local Universe [@dad05; @hop10; @dav10]. However, there is a tension between the inferred gas fractions of high- galaxies and galaxy formation models. Hydrodynamic cosmological simulations typically account for the simultaneous growth of galaxies via accretion of gas from the intergalactic medium [e.g. @ker03] as well as the consumption of gas by star formation. A generic feature of these simulations is that the star formation rate of “main sequence galaxies” (galaxies not undergoing a starburst event) is roughly proportional to the accretion rate, and that galaxies tend to have weakly declining gas fractions as their stellar masses increase. Broadly, at a given stellar mass, galaxies in simulations have baryonic gas fractions a factor of 2-3 less than observed gas fractions. This is seen both in hydrodynamic simulations, as well as semi-analytic models [@lag11]. As an example, @dav10 find very few galaxies in a simulated $\sim 150$ Mpc (comoving) volume with stellar mass $M_* \sim 10^{11}$ with gas fractions greater than 30%. This is in contrast to observations which infer gas fractions in comparable mass galaxies up to 80%. One potential solution is that the inferred gas masses from high- galaxies are systematically too large.  masses are typically calculated using the luminosity of the CO (J=1-0) emission line[^4], and then converted to an  mass via a CO- conversion factor [^5]. In the literature,  is typically used bimodally with one value for “quiescent/disc mode” star formation, and a lower value for “starburst/merger mode”. In the Galaxy and Local Group,  is observed to be relatively constant with an average $\approx 6$ [@bli07; @fuk10]. In contrast, dynamical mass modeling of local galaxy mergers suggests that  should be lower in these galaxies by a factor of 2-10 [@dow98; @nar11d]. Despite an observed dispersion in inferred  values from local mergers, a value of $\approx 0.8$ is typically uniformly applied to these starbursts. At higher redshifts, it is more unclear which of the two bimodal values of  to use. For example, for a star-forming disc galaxy that may be undergoing rapid collapse in $\sim$ kiloparsec scale clumps and forming stars at rates $> 100$ (i.e. comparable to local galaxy mergers), is the appropriate  the locally-calibrated “quiescent/disc” value, or the “starburst/merger” value? Similarly, should high-redshift Submillimetre galaxies (SMGs), which are potentially forming stars up to an order of magnitude faster than local mergers, utilise the locally-calibrated “starburst/merger” value? Typically, observational studies use the locally-calibrated quiescent/disc value for high- discs, and the local starburst/merger value for high- SMGs. Recent observational evidence by @tac08 [@bol08; @ler11; @gen12; @sch12] and @pap12, as well as theoretical work by @ost11 [@nar11b; @she11a; @she11b] and @fel12a have suggested that perhaps the picture of a bimodal  is too simplistic, and that  may depend on the physical environment of the interstellar medium (ISM). This picture was expanded upon by @nar12a who developed a functional form for the dependence of  on the CO surface brightness and gas-phase metallicity of a galaxy. When applying this model to observations of high- galaxies, @nar12a found that on average, high- disc galaxies have  values a factor of a few lower than present-epoch discs, and high- SMGs have  values lower than present-day ultraluminous infrared galaxies (ULIRGs), with some dispersion. Physically, this means that for a given observed CO luminosity, the inferred  gas mass should be systematically less than what one would derive using  values calibrated to local galaxies. This owes to warmer and higher velocity dispersion molecular gas in high- galaxies which gives rise to more CO intensity at a given  column density. The model form for  presented by @nar12a finds success in matching local observations of discs and ULIRGs [@nar11b], as well as observed CO- conversion factors for low metallicity systems. Building on these results, in this paper, we reexamine CO detections from high- galaxies utilising the physically motivated functional form for  presented in @nar12a, rather than the traditional bimodal form. We compare our results to those of cosmological hydrodynamic simulations, and show that while the inferred gas fractions derived from the traditional  conversion factor are much larger than those predicted by models, the @nar12a model form for  brings these values down, and in reasonable agreement with simulations. A principle result of the work we will present is that the typical gas fraction of a high- galaxy is typically $\sim 10-40\%$, rather than $\sim40-80\%$ as is inferred when utilising traditional  values. In § \[section:observations\], we describe the literature data utilised here; in § \[section:results\], we present our main results, and in § \[section:summary\], we summarise. ![Histograms of literature values for  for high-  galaxies, as well as those re-derived via Equation \[equation:alphaco\]. The dashed lines denote the literature values, and solid lines the theoretical  values. The black lines are for high- discs, and red for inferred high- mergers (SMGs). On average, the theoretical  values are lower than the locally-calibrated (traditional literature values) for discs and mergers. This owes to higher velocity dispersion and warmer gas in high- galaxies which drives more CO emission per unit  gas mass than in local galaxies.\[figure:alphaco\]](alpha_hist.bothwell.ps)  Methodology {#section:observations} =========== Literature Data {#section:literaturedata} --------------- We examine CO detections of both inferred high- disc galaxies as well as Submillimetre Galaxies with masses ranging from $\sim 10^{10}-10^{12}$ in stellar mass[^6]. A large number of literature SMGs are compiled by @bot12, and include 16 new detections presented in that paper. The compilation by Bothwell et al. includes detections from @ner03 [@gre05; @tac06; @cas09; @bot09] and @eng10. Other SMGs included in our work are compiled in @gen10. The inferred disc galaxies are taken primarily from the compilation of @gen10 and @dad10a. These include galaxies from the SINS sample [@for09], as well as -selected galaxies @dad04. Finally, we include optically-faint radio galaxies (OFRGs) with CO detections from the @cas11 sample. In our sample, a large number of the  and SINS galaxies have been imaged and found to have rotationally dominated gas, consistent with a disc-like morphology. The SMGs are oftentimes assumed to be mergers, though there is some debate over this [@dav10; @nar09; @nar10a; @hay10; @hay11; @hay12a]. The OFRGs are of unknown origin. As we will discuss in § \[section:results\], the global morphology is irrelevant for our model form for , and the general results in this paper. The observational papers that we draw from had to make a number of assumptions. Our philosophy is to simply utilise those assumptions in this paper, and not make any adjustments to assumed numbers. The reason for this is to isolate the effects of applying our model  on the inferred gas fractions. For example, as we will discuss, CO surface brightnesses are required in order to employ the @nar12a model for . When direct measurements are reported, we utilise those. Otherwise, we make the same size assumption that is made in the paper we draw from. Similarly, a number of the detections presented in the aforementioned papers utilised millimetre-wave telescopes, meaning that the observed transition is of higher-lying CO lines in the rest frame. Conversion to the ground state CO (J=1-0) line then occurs via an assumption of CO excitation. Again, we simply utilise the conversion from excited CO lines to CO (J=1-0) as presented in the paper we pull the data from. This said, the assumed excitation ladders in the literature are all relatively similar. It is worth a quick word on the stellar masses of the SMGs in our sample. There is an ongoing literature debate regarding the stellar masses of high-redshift SMGs. Specifically, for the [*same*]{} SMGs, @mic09 and @hai11 find differing stellar masses by up to an order of magnitude (with the Hainline masses being lower). Some attempts to understand the origin of the discrepancy have been reported by @mic12. In this work, we remain agnostic as to which stellar masses are “correct”, and present our results in terms of both sets of observations when relevant. Revised CO- Conversion Factors for Observed Galaxies {#section:alphaco} ---------------------------------------------------- As discussed in § \[section:introduction\], we utilise the functional form of  derived in @nar12a to re-calculate the  gas masses from the high- galaxies in our sample. In this model, the CO- conversion factor can be expressed as $$\label{equation:alphaco} {\mbox{$\alpha_{\rm CO}$}}= \frac{10.7 \times \langle W_{\rm CO}\rangle^{-0.32}}{Z'^{0.65}}$$ where  has units of pc$^{-2}$/K-, $Z'$ is the gas-phase metallicity in units of solar, and $\langle W_{\rm CO} \rangle$ is the luminosity-weighted CO intensity over all GMCs in a galaxy. While $\langle W_{\rm CO} \rangle$ is a difficult quantity to observe, in the limit of uniform distribution of luminosity from the ISM in a galaxy, this reduces to the $L'_{\rm CO}/A$ where $A$ is the area observed ($L'_{\rm CO}/A$ is the CO surface brightness). If the light distribution is actually rather concentrated (and most of the area observed is in dim pixels), the true surface brightness of the pixels which emit most of the light will increase, and the true  will be even lower than what is calculated by Equation \[equation:alphaco\]. This will cause the gas fractions to decrease even further from what utilising the @nar12a model for  derives, thus enhancing our results. The CO surface brightness ($\langle W_{\rm {CO} \rangle}$) serves as a physical parameterisation for the  gas temperature and velocity dispersion, both of which affect the velocity-integrated CO line intensity[^7] at a given  gas mass. The functional form for  also depends on a gas-phase metallicity. Physically,  varies with the gas-phase metallicity due to the growth of CO-dark molecular clouds in low-metallicity gas. In this regime, the required dust to protect CO from photodissociating radiation is not present, but the  is abundant enough to self-shield for survival. For the galaxies in question, metallicity measurements are typically not available. Hence, we assume a solar metallicity ($Z'=1$) for all galaxies. Based on the 2 mass-metallicity relation, galaxies of mass $M_* \sim 10^{11}$ typically have metallicities of order solar [@erb06b]. Thus, an assumption of $Z'=1$ is likely reasonable. We test the validity of this assumption by examining the effect of including a stellar mass-metallicity relation. Following @erb06b, we assume all galaxies above $M_* = 10^{11}$  have $Z'=1$, and that the metallicity evolves as ($M_*$)$^{0.3}$ for lower mass galaxies. The gas fractions in this test do not deviate by more than 10% compared to our assumption that $Z'=1$[^8]. This is because of the weak dependence of metallicity on stellar mass, the weak dependence of  on metallicity, and the fact that gas fractions depend on stellar mass as well as gas mass. It is worth noting that the results in this paper are not entirely dependent on the model for  given in Equation \[equation:alphaco\]. A compilation of observations result in a very similar relation. @ost11 showed that  and $\Sigma_{\rm H2}$ are related via a powerlaw in observed galaxies; a conversion of $\Sigma_{\rm H2}$ to $W_{\rm CO}$ via the relation = $\Sigma_{\rm H2}/W_{\rm CO}$ (and a linear relationship between  and ) gives an -$W_{\rm CO}$ relation that is nearly identical to the model form in Equation \[equation:alphaco\]. Similarly, while we assume $Z'=1$ for the observed galaxies analysed in this work, we note that the model power-law relation between  and $Z'$ is very similar to what has been observed in samples of low-metallicity galaxies [@bol08; @ler11; @gen12]. In this sense, the forthcoming results in this paper could be derived entirely from empirical observational evidence. In Figure \[figure:alphaco\], we plot histograms of the literature  values used in the observations of the galaxies analysed in this work (black), as well as our re-derived  values based on Equation \[equation:alphaco\] (red). We divide the lines into high- mergers (where SMGs are assumed to be mergers in this figure and hereafter) and high- discs so that the reader can see the relative difference between our derived  values and the original ones. While the range of  values is similar in both cases, there is significantly more power toward low  values when utilising our model for both high- discs and high- mergers. ![Difference between observed gas fractions and cosmological simulation gas fractions as a function of stellar mass. The squares are the results when utilising the @mic09 data, and circles from the @hai11 data. The open symbols denote the results when the observed gas fractions are calculated utilising traditional  values, and the filled symbols correspond to values calculated from the @nar12a form for . Our functional form for  results in lower gas fractions and, generally, better agreement between the simulations and observations. \[figure:fgas\_residuals\]](fgas_residuals.ps)  Results and Discussion {#section:results} ====================== We first examine the effect of modifying  from the traditional bimodal values to the @nar12a model on the -$M_*$ relation in high- galaxies. In Figure \[figure:fgas\_mstar\], we show the -$M_*$ relation for all galaxies in our sample utilising both the @mic09 and @hai11 stellar masses. The left panels show the relationship for the observed galaxies when using the traditional bimodal , and the right panels when applying Equation \[equation:alphaco\]. In order to compare with galaxy formation simulations, we overlay the mean -$M^*$ relation from the cosmological hydrodynamic calculations of @dav10 denoted by stars. The error bars in the stars denote the range in possible  values for the simulated galaxies in a given stellar mass bin. The simulated galaxies mostly represent “main-sequence” galaxies which are not typically undergoing a starburst event. When examining the left panels in Figure \[figure:fgas\_mstar\], it is evident that the observed galaxies all have substantially higher gas fractions at a given stellar mass than the simulations. While lower-mass systems can be biased to somewhat higher gas fractions because they are (in part) selected by far-infrared luminosity, we have tried applying similar cuts to simulations and find that this cannot explain the difference. In contrast, when applying a CO- conversion factor which varies smoothly with physical environment, the inferred  gas masses from the CO line measurements drop and come into better agreement with the simulations. Depending on the stellar mass adopted for the SMGs, the gas fractions can drop by up to a factor of 3 for a given galaxy. The reason for the drop in gas fraction when using the @nar12a model for  versus the traditional bimodal value is due to the typical environments of high- galaxies. The gas fractions of high- discs are typically large enough that, absent substantial internal feedback, large $\sim$kpc-scale clumps of gas become unstable and fragment [@spr05a; @cev10; @hop11a]. These clumps can have large internal velocity dispersions ($\sim 50-100 \ {\mbox{km s$^{-1}$}}$), and warm gas temperatures owing to high star formation rates [$\ga 100 \ {\mbox{M$_{\sun}$yr$^{-1}$}}$; @nar11a]. High velocity dispersions and warm gas causes increased CO line luminosity for a given  gas mass, and reduces  [@nar11b]. Because of this, in our model, high- disc galaxies tend to have lower[^9]  values than the traditional present-epoch “quiescent/disc” value (though larger than the traditional present-epoch “starburst/merger” value; Figure \[figure:alphaco\]). The mean derived  for high- discs is 2.5, approximately half that of the @dad10a and @mag11 measurements of high-  galaxies. A similar effect is true for high- starburst galaxies. Owing to extreme star formation rates [potentially up to a thousand ; @nar12b], the gas temperatures and velocity dispersions in violent 2 mergers exceed those of even present-day ULIRGs. Hence, the average  is lower than the average ULIRG value today. Our average derived value for the high- galaxies in our sample is ${\mbox{$\alpha_{\rm CO}$}}\approx 0.5$. @mag11 finds an upper limit of the  of a =4 SMG of 1, and @tac08 finds a reasonable fit to their observed SMGs with an  of unity. The combined effect of our modeling is that  for high- discs is typically lower than that of the traditional =0 “quiescent/disc” value, and  for high- starbursts is lower than that of the traditional =0 “starburst/merger” value (Figure \[figure:alphaco\]). Employing our model  consequently lowers gas masses, and brings gas fractions into better agreement with cosmological galaxy formation models. This is quantitatively shown in Figure \[figure:fgas\_residuals\], where we plot the residuals between the observed data and models for both the traditional , as well as that derived from the @nar12a functional form. The usage of our model form of  reduces the scatter in the -$M_*$ relation in observed galaxies by a factor $\sim 2$ at a given $M_*$. To calculate the reduction in scatter, we compare the standard deviation in galaxy gas fractions within a limited range of stellar masses ($M_* = [5\times10^{10},10^{11}] {\mbox{M$_{\sun}$}}$). Much of the scatter in the original -$M_*$ relation arises from using the bimodal  values. In contrast, our model form of  varies smoothly with the physical conditions in the ISM in a galaxy, and has no knowledge as to whether or not the global morphology of a galaxy is a merger or a disc. So, if some high- disc galaxies actually have physical conditions in their ISM comparable to starbursts, then their  values will be lower than the canonical “quiescent/disc”  (Figure \[figure:alphaco\]). The vice-versa is true for high- SMGs and OFRGs. When accounting for the continuum in physical properties in the ISM of high- galaxies (rather than binning them bimodally), the scatter in the observed -$M_*$ relation reduces substantially. The correlation coefficient between the observed gas fractions and modeled ones increases by $\sim 10\%$ (from $\sim 0.9$ to $0.98$ for both the Michalowski and Hainline masses. The usage of the @mic09 masses result in observed  below the simulations for the highest mass galaxies. This could reflect either an overestimate of masses, or perhaps physical processes that are neglected in the @dav11 simulations where most SMGs (the most massive galaxies) are quiescent, main-sequence objects. Potential neglected physical processes include starbursts which may deplete gas [e.g. @nar09; @nar10b; @nar10a; @hay11], or a stage of gas consumption without replenishment (that ultimately ends in passive galaxies). Utilising our model  additionally results in better agreement between the observed cosmic evolution of the gas fraction of galaxies with redshift and modeled evolution. In Figure \[figure:fgas\_z\], we plot the observed gas fractions of the galaxies in our literature sample against their redshifts. To help guide the eye, we overplot the mean values (with dispersion) in redshift bins of 0.5. We show the predicted values from the analytic model of @dav11 for halos of mass (at =0) $10^{13}$ and $10^{14}$ by the solid and dashed lines respectively, and the simulated points from @dav10 by the stars. When comparing the mean observed values to the predictions from analytic arguments and cosmological simulations, we again see that when using the traditional  calibrated to local values, the inferred gas fractions are significantly higher than the predictions from models. When applying the @nar12a model for , the mean values come into better agreement with simulations. The scatter (as measured as the standard deviation in gas fractions between ${\mbox{$z$}}=0-3$) decreases by $\sim 25\%$ when utilising our model form for  as compared to the traditional values[^10]. It is important to note that it is the normal star-forming galaxies (e.g. the  galaxies represented by the filled blue circles) that come into better agreement with the simulations. On the other hand, SMGs, represented by the open red squares in Figure \[figure:fgas\_z\], have lower gas fractions than the models predict. This is because SMGs are not typical galaxies at high-, but rather rare massive outliers, and therefore not reflected in the predictions for an average galaxy at high-. Comparing the results of this paper to other galaxy formation models is nontrivial. For example, when comparing to galaxies above [L$_{\rm {bol}}$]{}$>10^{11}$ , the gas fractions returned from our model are significantly lower than those predicted by recent semi-analytic models (SAMs). @lag11 utilised the Durham SAM to predict the  content in galaxies over cosmic time. As shown by @bot12, these models substantially over predict the  gas fraction as a function of redshift. However, comparing to galaxies above [L$_{\rm {bol}}$]{}$>10^{12}$  produces better agreement, however (C. Lagos, private communication). @pop12 utilised an indirect methodology to derive the  content in observed galaxies. By inverting the Schmidt relation, and using the @bli06 pressure-based prescription for deriving the /HI ratio, these authours found a gas fraction-stellar mass relation in good agreement with those derived from CO measurements. Implicit in this model, however, is an assumption of an  conversion factor in setting the normalisation of the observed Schmidt relation. In this sense, the measurement is not entirely independent of the methods used in CO-derived gas fractions. Summary {#section:summary} ======= Observed baryonic gas fractions from high-redshift galaxies as inferred from CO measurements are typically higher at a given stellar mass or redshift than cosmological galaxy formation models would predict. These differences can amount to a factor of 2-3 in gas fraction. We suggest that the observed gas fractions are overestimated due to the usage of locally-calibrated CO- conversion factors (). If  scales inversely with the CO surface brightness from a galaxy (as both numerical models and empirical observational evidence suggest), then both high- disc galaxies and high- mergers will have lower average  values than their =0 analogs. This means that for a given CO luminosity, there will be less underlying  gas mass. Applying a functional form for  (Equation \[equation:alphaco\]) decreases the inferred  gas masses by a factor of $\sim 2-3$, and brings them into agreement with cosmological galaxy formation models. Similarly, the usage of our model  reduces the scatter in the observed -$M_*$ relation by a comparable amount. Galaxy gas fractions decrease monotonically with increasing stellar mass, while the average gas fraction of galaxies in a given stellar mass range increases with redshift. Acknowledgements {#acknowledgements .unnumbered} ================ DN acknowledges support from the NSF via grant AST-1009452 and thanks Claudia Lagos and Gergö Popping for helpful conversations. RD was supported by the NSF under grant numbers AST-0847667 and AST-0907998. We additionally thank the anonymous referee for helpful suggestions that improved the presentation of these results. Computing resources were obtained through grant number DMS-0619881 from the National Science Foundation. [65]{} natexlab\#1[\#1]{} , L., [Fukui]{}, Y., [Kawamura]{}, A., [Leroy]{}, A., [Mizuno]{}, N., & [Rosolowsky]{}, E. 2007, in Protostars and Planets V, ed. B. [Reipurth]{}, D. [Jewitt]{}, & K. [Keil]{}, 81–96 , L. & [Rosolowsky]{}, E. 2006, , 650, 933 , A. D., [Leroy]{}, A. K., [Rosolowsky]{}, E., [Walter]{}, F., & [Blitz]{}, L. 2008, , 686, 948 , M. S., [Smail]{}, I., [Chapman]{}, S. C., [Genzel]{}, R., [Ivison]{}, R. J., [Tacconi]{}, L. J., [Alaghband-Zadeh]{}, S., [Bertoldi]{}, F., [Blain]{}, A. W., [Casey]{}, C. M., [Cox]{}, P., [Greve]{}, T. R., [Lutz]{}, D., [Neri]{}, R., [Omont]{}, A., & [Swinbank]{}, A. M. 2012, arXiv/1205.1511 , M. S. [et al.]{} 2010, , 405, 219 , C. L. [et al.]{} 2010, , 714, 1407 , C. M., [Chapman]{}, S. C., [Beswick]{}, R. J., [Biggs]{}, A. D., [Blain]{}, A. W., [Hainline]{}, L. J., [Ivison]{}, R. J., [Muxlow]{}, T. W. B., & [Smail]{}, I. 2009, , 1231 , C. M., [Chapman]{}, S. C., [Neri]{}, R., [Bertoldi]{}, F., [Smail]{}, I., [Coppin]{}, K., [Greve]{}, T. R., [Bothwell]{}, M. S., [Beswick]{}, R. J., [Blain]{}, A. W., [Cox]{}, P., [Genzel]{}, R., [Muxlow]{}, T. W. B., [Omont]{}, A., & [Swinbank]{}, A. M. 2011, , 415, 2723 , D., [Dekel]{}, A., & [Bournaud]{}, F. 2010, , 404, 2151 , K. E. K., [Swinbank]{}, A. M., [Neri]{}, R., [Cox]{}, P., [Alexander]{}, D. M., [Smail]{}, I., [Page]{}, M. J., [Stevens]{}, J. A., [Knudsen]{}, K. K., [Ivison]{}, R. J., [Beelen]{}, A., [Bertoldi]{}, F., & [Omont]{}, A. 2008, , 389, 45 , E. [et al.]{} 2004, , 617, 746 —. 2005, , 631, L13 —. 2010, , 714, L118 —. 2010, , 713, 686 , H., [Daddi]{}, E., [Riechers]{}, D. A., [Walter]{}, F., [Carilli]{}, C. L., [Dickinson]{}, M., [Elbaz]{}, D., & [Morrison]{}, G. E. 2009, , 698, L178 , R., [Finlator]{}, K., & [Oppenheimer]{}, B. D. 2011, arXiv/1108.0426 , R., [Finlator]{}, K., [Oppenheimer]{}, B. D., [Fardal]{}, M., [Katz]{}, N., [Kere[š]{}]{}, D., & [Weinberg]{}, D. H. 2010, , 404, 1355 , D. & [Solomon]{}, P. M. 1998, , 507, 615 , H., [Tacconi]{}, L. J., [Davies]{}, R. I., [Neri]{}, R., [Smail]{}, I., [Chapman]{}, S. C., [Genzel]{}, R., [Cox]{}, P., [Greve]{}, T. R., [Ivison]{}, R. J., [Blain]{}, A., [Bertoldi]{}, F., & [Omont]{}, A. 2010, , 724, 233 , D. K., [Shapley]{}, A. E., [Pettini]{}, M., [Steidel]{}, C. C., [Reddy]{}, N. A., & [Adelberger]{}, K. L. 2006, , 644, 813 , R., [Gnedin]{}, N. Y., & [Kravtsov]{}, A. V. 2012, , 747, 124 , N. M. [et al.]{} 2009, , 706, 1364 , Y. & [Kawamura]{}, A. 2010, , 48, 547 , J. E., [Smail]{}, I., [Moran]{}, S. M., [MacArthur]{}, L. A., [Lagos]{}, C. d. P., & [Edge]{}, A. C. 2011, , 730, L19 , R. [et al.]{} 2010, , 407, 2091 —. 2012, , 746, 69 , T. R. [et al.]{} 2005, , 359, 1165 , L. J., [Blain]{}, A. W., [Smail]{}, I., [Alexander]{}, D. M., [Armus]{}, L., [Chapman]{}, S. C., & [Ivison]{}, R. J. 2011, , 740, 96 , C. C., [Jonsson]{}, P., [Kere[š]{}]{}, D., [Magnelli]{}, B., [Hernquist]{}, L., & [Cox]{}, T. J. 2012, , 424, 951 , C. C., [Kere[š]{}]{}, D., [Jonsson]{}, P., [Narayanan]{}, D., [Cox]{}, T. J., & [Hernquist]{}, L. 2011, , 743, 159 , C. C., [Narayanan]{}, D., [Jonsson]{}, P., [Cox]{}, T. J., [Kere[š]{}]{}, D., [Hopkins]{}, P. F., & [Hernquist]{}, L. 2010, Conference Proceedings for UP2010: Have Observations Revealed a Variable Upper End of the Initial Mass Function? Treyer, Lee, Seibert, Wyder, Neil eds. arXiv/1008.4584 , P. F., [Keres]{}, D., [Murray]{}, N., [Quataert]{}, E., & [Hernquist]{}, L. 2011, arXiv/1111.6591 , P. F., [Younger]{}, J. D., [Hayward]{}, C. C., [Narayanan]{}, D., & [Hernquist]{}, L. 2010, , 402, 1693 , D., [Yun]{}, M. S., & [Young]{}, J. S. 2003, , 582, 659 , C. D. P., [Baugh]{}, C. M., [Lacey]{}, C. G., [Benson]{}, A. J., [Kim]{}, H.-S., & [Power]{}, C. 2011, , 418, 1649 , A. K., [Bolatto]{}, A., [Gordon]{}, K., [Sandstrom]{}, K., [Gratier]{}, P., [Rosolowsky]{}, E., [Engelbracht]{}, C. W., [Mizuno]{}, N., [Corbelli]{}, E., [Fukui]{}, Y., & [Kawamura]{}, A. 2011, , 737, 12 , G. E., [Daddi]{}, E., [Elbaz]{}, D., [Sargent]{}, M., [Dickinson]{}, M., [Dannerbauer]{}, H., [Aussel]{}, H., [Walter]{}, F., [Hwang]{}, H. S., [Charmandaris]{}, V., [Hodge]{}, J., [Riechers]{}, D., [Rigopoulou]{}, D., [Carilli]{}, C., [Pannella]{}, M., [Mullaney]{}, J., [Leiton]{}, R., & [Scott]{}, D. 2011, , 740, L15 , M. J., [Dunlop]{}, J. S., [Cirasuolo]{}, M., [Hjorth]{}, J., [Hayward]{}, C. C., & [Watson]{}, D. 2012, , 541, A85 , M. J., [Hjorth]{}, J., & [Watson]{}, D. 2009, arXiv/0905.4499 , D. 2011, arXiv/1112.1073 , D., [Cox]{}, T. J., [Hayward]{}, C. C., & [Hernquist]{}, L. 2011, , 412, 287 , D., [Cox]{}, T. J., [Hayward]{}, C. C., [Younger]{}, J. D., & [Hernquist]{}, L. 2009, , 400, 1919 , D. & [Dav[é]{}]{}, R. 2012, , 423, 3601 , D., [Dey]{}, A., [Hayward]{}, C. C., [Cox]{}, T. J., [Bussmann]{}, R. S., [Brodwin]{}, M., [Jonsson]{}, P., [Hopkins]{}, P. F., [Groves]{}, B., [Younger]{}, J. D., & [Hernquist]{}, L. 2010, , 407, 1701 , D., [Hayward]{}, C. C., [Cox]{}, T. J., [Hernquist]{}, L., [Jonsson]{}, P., [Younger]{}, J. D., & [Groves]{}, B. 2010, , 401, 1613 , D., [Krumholz]{}, M., [Ostriker]{}, E. C., & [Hernquist]{}, L. 2011, , 418, 664 , D., [Krumholz]{}, M. R., [Ostriker]{}, E. C., & [Hernquist]{}, L. 2012, , 421, 3127 , R. [et al.]{} 2003, , 597, L113 , E. C. & [Shetty]{}, R. 2011, , 731, 41 , P. P., [van der Werf]{}, P., [Xilouris]{}, E., [Isaak]{}, K. G., & [Gao]{}, Y. 2012, , 751, 10 , G., [Caputi]{}, K. I., [Somerville]{}, R. S., & [Trager]{}, S. C. 2012, arXiv/1201.3826 , D. A., [Carilli]{}, C. L., [Walter]{}, F., & [Momjian]{}, E. 2010, , 724, L153 , D. A. [et al.]{} 2011, , 739, L32 —. 2011, , 739, L31 , D. A. o. 2010, , 720, L131 , A., [Leroy]{}, A. K., [Walter]{}, F., [Bigiel]{}, F., [Brinks]{}, E., [de Blok]{}, W. J. G., [Kramer]{}, C., [Rosolowsky]{}, E., [Sandstrom]{}, K., [Schuster]{}, K., [Usero]{}, A., [Weiss]{}, A., & [Wiesemeyer]{}, H. 2012, arXiv/1203.4321 , R. [et al.]{} 2011, , 412, 1686 —. 2011, , 415, 3253 , P. M. & [Vanden Bout]{}, P. A. 2005, , 43, 677 , V., [Di Matteo]{}, T., & [Hernquist]{}, L. 2005, , 361, 776 , L. J. [et al.]{} 2006, , 640, 228 —. 2008, , 680, 246 —. 2010, , 463, 781 , J., [Kanekar]{}, N., & [Carilli]{}, C. L. 2009, , 697, L33 , R. [et al.]{} 2011, , 142, 101 [^1]: Bart J. Bok Fellow [^2]: E-mail: dnarayanan@as.arizona.edu [^3]: This includes a 36% correction for Helium. [^4]: In fact only a few studies directly observe CO (J=1-0) at high-. Typically, higher rotational states are observed, and then down-converted to the ground state via an assumption about the CO excitation. [^5]:  is alternatively monikered , or the $X$-factor. The two are related via  (cm$^{-2} (\rm K-{\mbox{km s$^{-1}$}})^{-1}$)= $6.3 \times 10^{19} \alpha_{\rm CO}$ (${\rm pc}^{-2}$(K-)$^{-1}$). In this paper, we utilise  as notation for the CO-  conversion factor. [^6]: The limits on stellar masses is highly dependent on which literature stellar masses for SMGs we use [@mic09; @hai11]. [^7]: For optically thick gas. [^8]: We note that there is one galaxy which varies by $\sim 22\%$. This is the lowest mass galaxy in the @hai11 stellar mass determinations. [^9]: Increased UV photons produced in high SFR galaxies do have the potential to photodissociate CO. However, these galaxies tend to have large dust to gas ratios. Increased dust columns allow GMCs to reach $A_{\rm V} \approx 1$ quickly, and shield CO from photodissociation throughout the bulk of the GMC [@nar12a]. [^10]: We note that to properly evaluate the evolution of the gas fraction of galaxies with redshift, one would ideally examine the same limited stellar mass range at each redshift interval. Given the limited number of CO detections at high-, however, this is currently infeasible. An examination of the galaxies in Figure \[figure:fgas\_mstar\] shows that the majority of our galaxies reside in a stellar mass range of log$(M_*) = 10.5-11.5$. The lack of clean sample selection is evident in the marginal increase in the already weak correlation coefficients: the correlation coefficient increases from $0.08$ to $0.14$ for the Hainline masses, and remains roughly constant at $\sim 0.15$ for the Michalowski masses. Forthcoming work will address the cosmic evolution of galaxy gas fractions in more detail.

--- abstract: | Given a collection of data points, non-negative matrix factorization (NMF) suggests to express them as convex combinations of a small set of ‘archetypes’ with non-negative entries. This decomposition is unique only if the true archetypes are non-negative and sufficiently sparse (or the weights are sufficiently sparse), a regime that is captured by the separability condition and its generalizations. In this paper, we study an approach to NMF that can be traced back to the work of Cutler and Breiman [@cutler1994archetypal] and does not require the data to be separable, while providing a generally unique decomposition. We optimize the trade-off between two objectives: we minimize the distance of the data points from the convex envelope of the archetypes (which can be interpreted as an empirical risk), while minimizing the distance of the archetypes from the convex envelope of the data (which can be interpreted as a data-dependent regularization). The archetypal analysis method of [@cutler1994archetypal] is recovered as the limiting case in which the last term is given infinite weight. We introduce a ‘uniqueness condition’ on the data which is necessary for exactly recovering the archetypes from noiseless data. We prove that, under uniqueness (plus additional regularity conditions on the geometry of the archetypes), our estimator is robust. While our approach requires solving a non-convex optimization problem, we find that standard optimization methods succeed in finding good solutions both for real and synthetic data. author: - 'Hamid Javadi[^1] and Andrea Montanari[^2]' title: 'Non-negative Matrix Factorization via Archetypal Analysis' --- Introduction ============ Given a set of data points $\bx_1,\bx_2,\cdots,\bx_n\in \reals^d$, it is often useful to represent them as convex combinations of a small set of vectors (the ‘archetypes’ $\bh_1,\dots,\bh_{\ell}$): $$\begin{aligned} \bx_i \approx \sum_{\ell=1}^r w_{i,\ell}\bh_{\ell}\, ,\;\;\;\; w_{i,\ell}\ge 0, \;\; \sum_{\ell=1}^rw_{i,\ell} =1\, . \label{eq:Decomposition} $$ Decompositions of this type have wide ranging applications, from chemometrics [@paatero1994positive] to image processing [@lee1999learning] and topic modeling [@xu2003document]. As an example, Figure \[fig:spectra-no-noise\] displays the infrared reflection spectra[^3] of four molecules (caffeine, sucrose, lactose and trioctanoin) for wavenumber between $1186\; {\rm cm}^{-1}$ and $1530\; {\rm cm}^{-1}$. Each spectrum is a vector $\bh_{0,\ell}\in\reals^d$, with $d=87$ and $\ell\in \{1,\dots, 4\}$. If a mixture of these substances is analyzed, the resulting spectrum will be a convex combination of the spectra of the four analytes. This situation arises in hyperspectral imaging [@ma2014signal], where a main focus is to estimate spatially varying proportions of a certain number of analytes. In order to mimic this setting, we generated $n=250$ synthetic random convex combinations $\bx_1,\dots,\bx_n\in\reals^d$ of the four spectra $\bh_{0,1}$, …, $\bh_{0,4}$, each containing two or more of these four analytes, and tried to reconstruct the pure spectra from the $\bx_i$’s. Each column in Figure \[fig:spectra-no-noise\] displays the reconstruction obtained using a different procedure. We refer to Appendix \[app:Numerical\] for further details. ![Left column: Infrared reflection spectra of four molecules. Subsequent columns: Spectra estimated from $n=250$ spectra of mixtures of the four original substances (synthetic data generated by taking random convex combinations of the pure spectra, see Appendix \[app:Numerical\] for details). Each column reports the results obtained with a different estimator: continuous blue lines correspond to the reconstructed spectra; dashed red lines correspond to the ground truth.[]{data-label="fig:spectra-no-noise"}](recovered_spectra_no_noise-eps-converted-to.pdf){width="120.00000%"} Without further constraints, the decomposition (\[eq:Decomposition\]) is dramatically underdetermined. Given a set of valid archetypes $\{\bh_{0,\ell}\}_{\ell\le r}$, any set $\{\bh_{\ell}\}_{\ell\le r}$ whose convex hull contains the $\{\bh_{0,\ell}\}_{\ell\le r}$ also satisfies Eq. (\[eq:Decomposition\]). For instance, we can set $\bh_\ell=\bh_{0,\ell}$ for $\ell\le r-1$, and $\bh_r= (1+s)\bh_{0,r}-s\bh_{0,1}$ for any $s\ge 0$, and obtain an equally good representation of the data $\{\bx_i\}_{i\le n}$. How should we constrain the decomposition (\[eq:Decomposition\]) in such a way that it is generally unique (up to permutations of the $r$ archetypes)? Since the seminal work of Paatero and Tapper [@paatero1994positive; @paatero1997least], and of Lee and Seung [@lee1999learning; @lee2001algorithms], an overwhelming amount of work has addressed this question by making the assumptions that the archetypes are componentwise non-negative $\bh_{\ell}\ge 0$. Among other applications the non-negativity constraint is justified for chemometrics (reflection or absorption spectra are non-negative), and topic modeling (in this case archetypes correspond to topics, which are represented as probability distributions over words). This formulation has become popular as non-negative matrix factorization (NMF). Under the non-negativity constraint $\bh_{\ell}\ge 0$ the role of weights and archetypes becomes symmetric, and the decomposition (\[eq:Decomposition\]) is unique provided that the archetypes or the weights are sufficiently sparse (without loss of generality one can assume $\sum_{\ell=1}^rh_{\ell,i}=1$). This point was clarified by Donoho and Stodden [@donoho2003does], introduced a separability condition that ensure uniqueness. The non-negative archetypes $\bh_1,\cdots,\bh_{r}$ are separable if, for each $\ell\in [r]$ there exists an index $i(\ell)\in[d]$ such that $(\bh_{\ell})_{i(\ell)}=1$, and $(\bh_{\ell'})_{i(\ell)}=0$ for all $\ell'\neq \ell$. If we exchange the roles of weights $\{w_{i,\ell}\}$ and archetypes $\{h_{\ell,i}\}$, separability requires that $\ell\in [r]$ there exists an index $i(\ell)\in[n]$ such that $w_{i(\ell),\ell}=1$, and $w_{i(\ell),\ell'}=0$ for all $\ell'\neq \ell$ This condition has a simple geometric interpretation: the data are separable if for each archetype $\bh_{\ell}$ there is at least one data point $\bx_i$ such that $\bx_i=\bh_{\ell}$. A copious literature has developed algorithms for non-negative matrix factorization under separability condition or its generalizations [@donoho2003does; @arora2012computing; @recht2012factoring; @arora2013practical; @ge2015intersecting]. Of course this line of work has a drawback: in practice we do not know whether the data are separable. (We refer to the Section \[sec:Discussion\] for a comparison with [@ge2015intersecting], which relaxes the separability assumption.) Further, there are many cases in which the archetypes $\bh_1,\dots,\bh_{\ell}$ are not necessarily non-negative. For instance, in spike sorting, the data are measurements of neural activity ad the archetypes correspond to waveforms associated to different neurons [@roux2009adaptive]. In other applications the archetypes $\bh_{\ell}$ are non-negative, but –in order to reduce complexity– the data $\{\bx_i\}_{i\le n}$ are replaced by a random low-dimensional projection [@kim2008fast; @wang2010efficient]. The projected archetypes loose the non-negativity property. Finally, the decomposition (\[eq:Decomposition\]) is generally non-unique, even under the constraint $\bh_{\ell}\ge 0$. This is illustrated, again, in Figure \[fig:spectra-no-noise\]: all the spectra are strictly positive, and hence we can find archetypes $\bh_1,\dots,\bh_4$ that are still non-negative and whose convex envelope contains $\bh_{0,1},\dots,\bh_{0,4}$. Since NMF is underdetermined, standard methods fail in such applications, as illustrated in Figure \[fig:spectra-no-noise\] . We represent the data as a matrix $\bX\in\reals^{n\times d}$ whose $i$-th row is the vector $\bx_i$, the weights by a matrix $\bW = (w_{i,\ell})_{i\le n, \ell\le r}\in\reals^{n\times d}$ and the prototypes by a matrix $\bH= (h_{\ell,j})_{\ell\le r, j\le d}\in\reals^{r\times d}$. The third column of Figure \[fig:spectra-no-noise\] uses a projected gradient algorithm from [@lin2007projected] to solve the problem $$\begin{aligned} \mbox{minimize}&\;\;\;\;\;\; \|\bX-\bW\bH\|_F^2\, ,\label{eq:StandardNMF}\\ \mbox{subject to}& \;\;\;\;\;\; \bW\ge 0, \;\; \bH\ge 0\, .\nonumber $$ Empirically, projected gradient converges to a point with very small fitting error $\|\bX-\bW\bH\|_F^2$, but the reconstructed spectra (rows of $\bH$) are inaccurate. The second column in the same figure shows the spectra reconstructed using an algorithm from [@arora2013practical], that assumes separability: as expected, the reconstruction is not accurate. In a less widely known paper, Cutler and Breiman [@cutler1994archetypal] addressed the same problem using what they call ‘archetypal analysis.’ Archetypal analysis presents two important differences with respect to standard NMF: $(1)$ The archetypes $\bh_{\ell}$ are not necessarily required to be non-negative (although this constraint can be easily incorporated); $(2)$ The under-determination of the decomposition (\[eq:Decomposition\]) is addressed by requiring that the archetypes belong to the convex hull of the data points: $\bh_{\ell}\in \conv(\{\bx_{i}\}_{i\le n})$. In applications the condition $\bh_{\ell}\in \conv(\{\bx_{i}\}_{i\le n})$ is too strict. This paper builds on the ideas of [@cutler1994archetypal] to propose a formulation of NMF that is uniquely defined (barring degenerate cases) and provides a useful notion of optimality. In particular, we present the following contributions. [**Archetypal reconstruction.**]{} We propose to reconstruct the archetypes $\bh_1,\dots,\bh_{r}$ by optimizing a combination of two objectives. On one hand, we minimize the error in the decomposition (\[eq:Decomposition\]). This amounts to minimizing the distance between the data points and the convex hull of the archetypes. On the other hand, we minimize the distance of the archetypes from the convex hull of data points. This relaxes the original condition imposed in [@cutler1994archetypal] which required the archetypes to lie in $\conv(\{\bx_{i}\})$, and allows to treat non-separable data. [**Robustness guarantee.**]{} We next assume that that the decomposition (\[eq:Decomposition\]) approximately hold for some ‘true’ archetypes $\bh^0_{\ell}$ and weights $w_{i,\ell}^0$, namely $\bx_i = \bx_i^0+\bz_i$, where $\bx^0_i=\sum_{\ell=1}^r w^0_{i,\ell}\bh^0_{\ell}$ and $\bz_i$ captures unexplained effects. We introduce a ‘uniqueness condition’ on the data $\{\bx^0_i\}_{i\le n}$ which is necessary for exactly recovering the archetypes from the noiseless data. We prove that, under uniqueness (plus additional regularity conditions on the geometry of the archetypes), our estimator is robust. Namely it outputs archetypes $\{\hbh_\ell\}_{\ell\le r}$ whose distance from the true ones $\{\bh^0_{\ell}\}_{\ell\le r}$ (in a suitable metric) is controlled by $\sup_{i\le n}\|\bz_i\|_2$. [**Algorithms.**]{} Our approach reconstructs the archetypes $\bh_1,\dots,\bh_r$ by minimizing a non-convex risk function $\cuR_{\lambda}(\bH)$. We propose three descent algorithms that appear to perform well on realistic instances of the problem. In particular, Section \[sec:Algorithm\] introduces a proximal alternating linearized minimization algorithm (PALM) that is guaranteed to converge to critical points of the risk function. Appendix \[app:algo\] discusses two alternative approaches. One possible explanation for the success of such descent algorithms is that reasonably good initializations can be constructed using spectral methods, or approximating the data as separable, cf. Section \[sec:Initialization\]. We defer a study of global convergence of this two-stages approach to future work. An archetypal reconstruction approach ===================================== ![Toy example of archetype reconstruction. Top left: data points (blue) are generated as random linear combinations of $r=3$ archetypes in $d=2$ dimensions (red, see Appendix \[app:Numerical\] for details). Top right: Initialization using the algorithm of [@arora2013practical]. Bottom left: Output of the alternate minimization algorithm of [@cutler1994archetypal] with initialization form the previous frame. Bottom right: Alternate minimization algorithm to compute the estimator (\[eq:Lagrangian\]), with $\lambda = 0.0166$.[]{data-label="fig:example"}](data_example.pdf "fig:"){width="40.00000%"} ![Toy example of archetype reconstruction. Top left: data points (blue) are generated as random linear combinations of $r=3$ archetypes in $d=2$ dimensions (red, see Appendix \[app:Numerical\] for details). Top right: Initialization using the algorithm of [@arora2013practical]. Bottom left: Output of the alternate minimization algorithm of [@cutler1994archetypal] with initialization form the previous frame. Bottom right: Alternate minimization algorithm to compute the estimator (\[eq:Lagrangian\]), with $\lambda = 0.0166$.[]{data-label="fig:example"}](init_example.pdf "fig:"){width="40.00000%"}\ ![Toy example of archetype reconstruction. Top left: data points (blue) are generated as random linear combinations of $r=3$ archetypes in $d=2$ dimensions (red, see Appendix \[app:Numerical\] for details). Top right: Initialization using the algorithm of [@arora2013practical]. Bottom left: Output of the alternate minimization algorithm of [@cutler1994archetypal] with initialization form the previous frame. Bottom right: Alternate minimization algorithm to compute the estimator (\[eq:Lagrangian\]), with $\lambda = 0.0166$.[]{data-label="fig:example"}](linfty_example.pdf "fig:"){width="40.00000%"} ![Toy example of archetype reconstruction. Top left: data points (blue) are generated as random linear combinations of $r=3$ archetypes in $d=2$ dimensions (red, see Appendix \[app:Numerical\] for details). Top right: Initialization using the algorithm of [@arora2013practical]. Bottom left: Output of the alternate minimization algorithm of [@cutler1994archetypal] with initialization form the previous frame. Bottom right: Alternate minimization algorithm to compute the estimator (\[eq:Lagrangian\]), with $\lambda = 0.0166$.[]{data-label="fig:example"}](l1E-4_example.pdf "fig:"){width="40.00000%"} Let $\cQ\subseteq \reals^d$ be a convex set and $D:\cQ\times \cQ\to\reals$, $(\bx,\by)\mapsto D(\bx;\by)$ a loss function on $\cQ$. For a point $\bu\in\cQ$, and a matrix $\bV\in\reals^{m\times d}$, with rows $\bv_1,\dots,\bv_m\in\cQ$, we let $$\begin{aligned} \cuD(\bu;\bV) &\equiv \min \Big\{ D\big(\bu ; \bV^{\sT}\bpi\big)\, :\;\;\;\; \bpi \in \Delta^m\, \Big\} \, ,\\ \Delta^m &\equiv \big\{\bx\in\reals^m_{\ge 0}:\;\; \<\bx,\bfone\> = 1\big\}\, . $$ In other words, denoting by $\conv(\bV) = \conv(\{\bv_1,\dots,\bv_m\})$ the convex hull of the rows of matrix $\bV$, $\cuD(\bu;\bV)$ is the minimum loss between $\bx$ and any point in $\conv(\bV)$. If $\bU\in\reals^{k\times d}$ is a matrix with rows $\bu_{1},\dots, \bu_{k}\in\cQ$, we generalize this definition by letting $$\begin{aligned} \cuD(\bU;\bV) \equiv \sum_{\ell=1}^k \cuD(\bu_{\ell};\bV)\, . $$ While this definition makes sense more generally, we have in mind two specific examples in which $D(\bx;\by)$ is actually separately convex in its arguments $\bx$ and $\by$. (Most of our results will concern the first example.) In this case $\cQ = \reals^d$, and $D(\bx;\by) = \|\bx-\by\|_2^2$. This is the case originally studied by Cutler and Breiman [@cutler1994archetypal]. We take $\cQ = \Delta^d$, the $d$-dimensional simplex, and $D(\bx;\by)$ to be the Kullback-Leibler divergence between probability distributions $\bx$ and $\by$, namely $D(\bx;\by) \equiv \sum_{i=1}^dx_i\log(x_i/y_i)$. Given data $\bx_1,\dots,\bx_n$ organized in the matrix $\bX\in\reals^{n\times d}$, we estimate the archetypes by solving the problem[^4] $$\begin{aligned} \hbH_{\lambda}\in \arg\min\Big\{\cuD(\bX;\bH)+\lambda\, \cuD(\bH;\bX): \;\; \bH\in\cQ^{r}\Big\}\, ,\label{eq:Lagrangian} $$ where we denote by $\cQ^r$ the set of matrices $\bH\in\reals^{r\times d}$ with rows $\bh_1,\dots,\bh_r\in\cQ$. A few values of $\lambda$ are of special significance. If we set $\lambda =0$, and $\cQ =\Delta^d$, we recover the standard NMF objective (\[eq:StandardNMF\]), with a more general distance function $D(\,\cdot\,,\,\cdot\,)$. As pointed out above, in general this objective has no unique minimum. If we let $\lambda\to 0+$ after the minimum is evaluated, $\hbH_{\lambda}$ converges to the minimizer of $\cuD(\bX;\bH)$ which is the ‘closest’ to the convex envelope of the data $\conv(\bX)$ (in the sense of minimizing $\cuD(\bH;\bX)$). Finally as $\lambda\to\infty$, the archetypes $\bh_\ell$ are forced to lie in $\conv(\bX)$ and hence we recover the method of [@cutler1994archetypal]. Figure \[fig:example\] illustrates the advantages of the estimator (\[eq:Lagrangian\]) on a small synthetic example, with $d=2$, $r=3$, $n=500$: in this case the data are non separable. We first use the successive projections algorithm of [@arora2013practical] (that is designed to deal with separable data) in order to estimate the archetypes. As expected, the reconstruction is not accurate because this algorithm assumes separability and hence estimates the archetypes with a subset of the data points. We then use these estimates as initialization in the alternate minimization algorithm of [@cutler1994archetypal], which optimizes the objective (\[eq:Lagrangian\]) with $\lambda=\infty$. The estimates improve but not substantially: they are still constrained to lie in $\conv(\bX)$. A significant improvement is obtained by setting $\lambda$ to a small value. We (approximately) minimize the cost function (\[eq:Lagrangian\]) by generalizing the alternate minimization algorithm, cf. Section \[sec:Algorithm\]. The optimal archetypes are no longer constrained to $\conv(\bX)$, and provide a better estimate of the true archetypes. The last column in Figure \[fig:spectra-no-noise\] uses the same estimator, and approximately solves problem (\[eq:Lagrangian\]) by gradient descent algorithm. In our analysis we will consider a slightly different formulation in which the Lagrangian of Eq. (\[eq:Lagrangian\]) is replaced by a hard constraint: $$\begin{aligned} \mbox{minimize}&\;\;\;\;\; \cuD(\bH;\bX)\, ,\label{eq:HardNoise}\\ \mbox{subject to }&\;\;\;\;\; \cuD(\bx_{i};\bH)\le \delta^2\;\;\;\mbox{ for all } i\in\{1,\dots,n\}\, .\nonumber $$ We will use this version in the analysis presented in the next section, and denote the corresponding estimator by $\hbH$. Robustness ========== In order to analyze the robustness properties of estimator $\hbH$, we assume that there exists an approximate factorization $$\begin{aligned} \bX = \bW_0\bH_0+ \bZ \, ,\label{eq:AssDecomposition} $$ where $\bW_0\in\reals^{n\times r}$ is a matrix of weights (with rows $\bw_{0,i}\in\Delta^r$), $\bH_0\in\reals^{r\times d}$ is a matrix of archetypes (with rows $\bh_{0,\ell}$), and we set $\bX_0 = \bW_0\bH_0$. The deviation $\bZ$ is arbitrary, with rows $\bz_i$ satisfying $\max_{i\le n}\|\bz_{i}\|_2\le \delta$. We will assume throughout $r$ to be known. We will quantify estimation error by the sum of distances between the true archetypes and the closest estimated archetypes $$\begin{aligned} \cuL(\bH_0,\hbH)\equiv \sum_{\ell=1}^r\min_{\ell'\le r} D(\bh_{0,\ell},\hbh_{\ell'})\, . $$ In words, if $\cuL(\bH_0,\hbH)$ is small, then for each true archetype $\bh_{0,\ell}$ there exists an estimated archetype $\hbh_{\ell'}$ that is close to it in $D$-loss. Unless two or more of the true archetypes are close to each other, this means that there is a one-to-one correspondence between estimated archetypes and true archetypes, with small errors. We say that the factorization $\bX_0=\bW_0\bH_0$ satisfies uniqueness with parameter $\alpha>0$ (equivalently, is *$\alpha$-unique*) if for all $\bH\in\cQ^{r}$ with $\conv(\bX_0)\subseteq \conv(\bH)$, we have $$\begin{aligned} \cuD(\bH,\bX_0)^{1/2}\ge \cuD(\bH_0,\bX_0)^{1/2}+ \alpha\,\big\{\cuD(\bH,\bH_0)^{1/2}+\cuD(\bH_0,\bH)^{1/2}\big\}\, . \label{eq:UniquenessAssumption} $$ The rationale for this assumption is quite clear. Assume that the data lie in the convex hull of the true archetypes $\bH_0$, and hence Eq. (\[eq:AssDecomposition\]) holds without error term $\bZ=0$, i.e. $\bX=\bX_0$. We reconstruct the archetypes by demanding $\conv(\bX_0)\subseteq \conv(\bH)$: any such $\bH$ is a plausible explanation of the data. In order to make the problem well specified, we define $\bH_0$ to be the matrix of archetypes that are the closest to $\bX_0$, and hence $\cuD(\bH,\bX_0)\ge \cuD(\bH_0,\bX_0)$ for all $\bH$. In order for the reconstruction to be unique (and hence for the problem to be identifiable) we need to assume $\cuD(\bH,\bX_0) >\cuD(\bH_0,\bX_0)$ strictly for $\bH\neq \bH_0$. The uniqueness assumption provides a quantitative version of this condition. Given $\bX_0$, $\bH_0$, the best constant $\alpha$ such that Eq. (\[eq:UniquenessAssumption\]) holds for all $\bH$ is a geometric property that depend on $\bX_0$ only through $\conv(\bX_0)$. In particular, if $\bX_0= \bW_0\bH_0$ is a separable factorization, then it satisfies uniqueness with parameter $\alpha=1$. Indeed in this case $\conv(\bH_0) = \conv(\bX_0)$, whence $\cuD(\bH,\bX_0) = \cuD(\bH,\bH_0)$ and $\cuD(\bH_0,\bX_0)=\cuD(\bH_0,\bH) = 0$. It is further possible to show that $\alpha\in [0,1]$ for all $\bH_0,\bX_0$. Indeed, we took $\bH_0$ to be the matrix of archetypes that are closest to $\bX_0$. In other words, $\cuD(\bH,\bX_0)\ge \cuD(\bH_0,\bX_0)$ and hence, $\alpha \geq 0$. In addition, since $\conv(\bX_0) \subseteq \conv(\bH_0)$, for $\bh_i$ an arbitrary row of $\bH$ we have $$\begin{aligned} \cuD(\bh_i,\bX_0) \leq \cuD(\bh_i,\bH_0).\end{aligned}$$ Hence, $\cuD(\bH,\bX_0) \leq \cuD(\bH,\bH_0)$ and therefore $$\begin{aligned} \cuD(\bH,\bX_0)^{1/2}\le \cuD(\bH_0,\bX_0)^{1/2}+\big\{\cuD(\bH,\bH_0)^{1/2}+\cuD(\bH_0,\bH)^{1/2}\big\}\, .\end{aligned}$$ Thus, $\alpha \leq 1$. We say that the convex hull $\conv(\bX_0)$ has *internal radius* (at least) $\mu$ if it contains an $r-1$-dimensional ball of radius $\mu$, i.e. if there exists $\bz_0 \in \reals^d$, $\bU\in \reals^{d\times (r-1)}$, with $\bU^\sT\bU = \Id_d$ , such that $\bz_0 + \bU\Ball_{r-1}(\mu)\subseteq \conv(\bX_0)$. We further denote by $\kappa(\bM)$ the condition number of matrix $\bM$. \[thm:Robust\] Assume $\bX = \bW_0\bH_0+ \bZ$ where the factorization $\bX_0=\bW_0\bH_0$ satisfies the uniqueness assumption with parameter $\alpha>0$, and that $\conv(\bX_0)$ has internal radius $\mu>0$. Consider the estimator $\hbH$ defined by Eq. (\[eq:HardNoise\]), with $D(\bx,\by) = \|\bx-\by\|_2^2$ (square loss) and $\delta = \max_{i\le n} \|\bZ_{i,\cdot}\|_2$. If $$\begin{aligned} \max_{i\le n} \|\bZ_{i,\cdot}\|_2\le \frac{\alpha\mu}{30 r^{3/2}}\, , $$ then, we have $$\begin{aligned} \cuL(\bH_0,\hbH)\le \frac{C_*^2\, r^{5}}{\alpha^2} \max_{i\le n} \|\bZ_{i,\cdot}\|^2_2\, , $$ where $C_*$ is a coefficient that depends uniquely on the geometry of $\bH_0$, $\bX_0$, namely $C_* = 120(\sigma_{\rm max}(\bH_0)/\mu)\cdot \max(1, \kappa(\bH_0)/\sqrt{r})$. Algorithms {#sec:Algorithm} ========== While our main focus is on structural properties of non-negative matrix factorization, we provide evidence that the optimization problem we defined can be solved in practical scenarios. A more detailed study is left to future work. From a computational point of view, the Lagrangian formulation is more appealing. For the sake of simplicity, we denote the regularized risk by $$\begin{aligned} \cuR_{\lambda}(\bH) \equiv \cuD(\bX;\bH)+\lambda\, \cuD(\bH;\bX)\, ,\label{eq:RLagrangian} $$ and leave implicit the dependence on the data $\bX$. Notice that this function is non-convex and indeed has multiple global minima: in particular, permuting the rows of a minimizer $\bH$ yields other minimizers. We will describe two greedy optimization algorithms: one based on gradient descent, and one on alternating minimization, which generalizes the algorithm of [@cutler1994archetypal]. In both cases it is helpful to use a good initialization: two initialization methods are introduced in the next section. Initialization {#sec:Initialization} -------------- We experimented with two initialization methods, described below. [*(*1) *Spectral initialization.*]{} Under the assumption that the archetypes $\{\bh_{0,\ell}\}_{\ell\le r}$ are linearly independent (and for non-degenerate weights $\bW$), the ‘noiseless’ matrix $\bX_0$ has rank exactly $r$. This motivates the following approach. We compute the singular value decomposition $\bX =\sum_{i=1}^{n\wedge d} \sigma_i\bu_i\bv_i^{\sT}$, $\sigma_1\ge \sigma_2\ge \dots\ge \sigma_{n\wedge d}$, and initialize $\hbH$ as the matrix $\hbH^{(0)}$ with rows $\hbh^{(0)}_1=\bv_1,\dots,\hbh^{(0)}_r=\bv_r$. [*(*2) *Successive projections initialization.*]{} We initialize $\hbH^{(0)}$ by choosing archetypes $\{\hbh^{(0)}_\ell\}_{1\le \ell\le r}$ that are a subset of the data $\{\bx_i\}_{1\le i\le n}$, selected as follows. The first archetype $\hbh^{(0)}_1$ is the data point which is farthest from the origin. For each subsequent archetype, we choose the point that is farthest from the affine subspace spanned by the previous ones. [ll]{}\ \ \ \ \ 1: & Set $i(1) = \arg\max \{ D(\bx_{i};\bzero):\; i\le n\}$;\ 2: & Set $\hbh^{(0)}_{1}= \bx_{i(1)}$;\ 3: & For $\ell\in \{1,\dots, r\}$\ 4: & Define $V_{\ell}\equiv \aff(\hbh_{1}^{(0)},\hbh_{2}^{(0)},\dots,\hbh_{\ell}^{(0)})$;\ 5: & Set $i(\ell+1) = \arg\max \{\cuD(\bx_{i};V_{\ell})\, :\; i\le n\}$;\ 6: & Set $\hbh^{(0)}_{\ell+1} = \bx_{i(\ell+1)}$;\ 7: & End For;\ 8: & Return $\{\hbh_{\ell}^{(0)}\}_{1\le \ell\le r}$l\ \ This coincides with the successive projections algorithm of [@araujo2001successive], with the minor difference that $V_{\ell}$ is the affine subspace spanned by the first $\ell$ vectors, instead of the linear subspace[^5] This method can be proved to return the exact archetypes if data are separable the archetypes are affine independent [@arora2013practical; @gillis2014fast]. When data are not separable it provides nevertheless a good initial assignment. Proximal alternating linearized minimization {#sec:PALM} -------------------------------------------- The authors of [@bolte2014proximal] develop a proximal alternating linearized minimization algorithm (PALM) to solve the problems of the form $$\begin{aligned} \label{eq:palmform} {\rm{minimize}}\quad\quad \Psi(\bx,\by) = f(\bx) + g(\by) + h(\bx,\by) $$ where $f:\reals^m\to (-\infty,+\infty]$ and $g:\reals^n\to (-\infty,\infty]$ are lower semicontinuous and $h\in C^1(\reals^{m}\times\reals^n)$. PALM is guaranteed to converge to critical points of the function $\Psi$ [@bolte2014proximal]. We apply this algorithm to minimize the cost function (\[eq:RLagrangian\]), with $D(\bx,\by)=\|\bx-\by\|_2^2$ which we write as $$\begin{aligned} \cuR_\lambda(\bH) =\min_{\bW} \Psi (\bH,\bW) = f(\bH) + g(\bW) + h(\bH,\bW)\, . $$ where, $$\begin{aligned} &f(\bH) = \lambda\, \cuD(\bH,\bX),\\ &g(\bW) = \sum_{i=1}^n \Ind\left(\bw_i\in \Delta^r\right),\\ &h(\bH,\bW) = \left\|\bX - \bW\bH\right\|_F^2.\end{aligned}$$ In above equations $\bw_i$ are the rows of $\bW$ and the indicator function $\Ind(\bx\in \Delta^r)$ is equal to zero if $\bx \in \Delta^r$ and is equal to infinity otherwise. By using this decomposition, the iterations of the PALM iteration reads $$\begin{aligned} \label{eq:PALMITER1} &\tbH^{k} = \bH^{k} - \frac{1}{\gamma_1^k}(\bW^k)^\sT\left(\bW^k\bH^k - \bX\right),\\ &\bH^{k+1} = \tbH^{k} - \frac{\lambda}{\lambda + \gamma_1^k}\left(\tbH^{k} - \bPi_{\conv(\bX)}\left(\tbH^{k}\right)\right),\\ &\bW^{k+1} = \bPi_{\Delta^r}\left(\bW^k - \frac{1}{\gamma_2^k}\left(\bW^k\bH^{k+1}-\bX\right)(\bH^{k+1})^\sT\right), \label{eq:PALMITER3}\end{aligned}$$ where $\gamma_1^k$, $\gamma_2^k$ are step sizes and, for $\bM\in\reals^{m_1\times m_2}$, and $\cS\subseteq\reals^{m_2}$ a closed convex set, $\bPi_{\cS}(\bM)$ is the matrix obtained by projecting the rows of $\bM$ onto the simplex $\cS$. \[propo:PALM\] Consider the risk (\[eq:RLagrangian\]), with loss $D(\bx,\by) = \|\bx-\by\|_2^2$, and the corresponding cost function $\Psi(\bH,\bW)$. If the step sizes are chosen such that $\gamma_1^k > \left\|\bW^{k^\sT}\bW^k\right\|_F$, $\gamma_2^k > \max\left\{\left\|\bH^{k+1}\bH^{k+1^\sT}\right\|_F,\eps\right\}$ for some constant $\eps>0$, then $(\bH^k,\bW^k)$ converges to a stationary point of the function $\Psi(\bH,\bW)$. The proof of this statement is deferred to Appendix \[app:PALM\]. It is also useful to notice that the gradient of $\cuR_{\lambda}(\bH)$ can be computed explicitly (this can be useful to devise a stopping criterion). \[propo:Subdiff\] Consider the risk (\[eq:RLagrangian\]), with loss $D(\bx,\by) = \|\bx-\by\|_2^2$, and assume that the rows of $\bH$ are affine independent. Then, $\cuR_{\lambda}$ is differentiable at $\bH$ with gradient $$\begin{aligned} \nabla\cuR_{\lambda}(\bH) &= 2\sum_{i=1}^{n}\balpha^*_{i}\left(\bPi_{\conv(\bH)}(\bx_{i})-\bx_{i}\right) +2\lambda\big(\bH - \bPi_{\conv(\bX)}(\bH)\big)\, ,\\ \balpha_{i}^{*} &= \arg\min_{\balpha \in \Delta^r} \left\| \bH^\sT \balpha - \bx_{i}^\sT\right\|_2\,.\label{eq:Alphai}\end{aligned}$$ where we recall that $\bPi_{\conv(\bX)}(\bH)$ denotes the matrix with rows $\bPi_{\conv(\bX)}(\bH_{1,\cdot}),\dots,\bPi_{\conv(\bX)}(\bH_{r,\cdot})$. The proof of this proposition is given in Appendix \[app:Subdiff\]. Appendix \[app:algo\] also discusses two alternative algorithms. ![Reconstructing infrared spectra of four molecules, from noisy random convex combinations. Noise level $\sigma = 10^{-3}$. Left column: original spectra. The other columns correspond to different reconstruction methods.[]{data-label="fig:projgradsignals_lownoise"}](recovered_signals_sigma=0001-eps-converted-to.pdf){width="120.00000%"} ![As in Figure \[fig:projgradsignals\_lownoise\], with $\sigma = 2\cdot 10^{-3}$ (in blue).[]{data-label="fig:oursignals_highnoise"}](recovered_signals_sigma=0002-eps-converted-to.pdf){width="120.00000%"} ![Risk $\cuL(\bH_0,\hbH)^{1/2}$ vs $\sigma$ for different reconstruction methods. Triangles (blue): anchor words algorithm from [@arora2013practical]. Squares (blue): minimizing the objective function (\[eq:StandardNMF\]) using the projected gradient algorithm of [@lin2007projected]. Circles (red): archetypal reconstruction approach in this paper. Interpolating lines are just guides for the eye. The thick horizontal line corresponds to the trivial estimator $\hbH=0$.[]{data-label="fig:Dvssigma"}](synthetic-data.pdf "fig:"){width="65.00000%"} (-160,0)[$\sigma$]{} (-330,100)[[$\cuL(\bH_0,\hbH)^{1/2}$]{}]{} Numerical experiments --------------------- We implemented both the PALM algorithm described in the previous section, and the two algorithms described in Appendix \[app:algo\]. The outcomes are generally similar. Figures \[fig:projgradsignals\_lownoise\] and \[fig:oursignals\_highnoise\] repeat the experiment already described in the introduction. We generate $n=250$ convex combinations of $r=4$ spectra $\bh_{0,1},\dots,\bh_{0,4}\in\reals^d$, $d=87$, this time adding white Gaussian noise with variance $\sigma^2$. We minimize the Lagrangian $\cuR_{\lambda}(\bH)$, with[^6] $\lambda = 4$ (for Figure \[fig:projgradsignals\_lownoise\]) and $\lambda = 0.8$ (for Figure \[fig:oursignals\_highnoise\]). The reconstructed spectra of the pure analytes appear to be accurate and robust to noise. In Figure \[fig:Dvssigma\] we repeated the same experiment systematically for $10$ noise realizations for each noise level $\sigma$, and report the resulting average loss. Among various reconstruction methods, the approach described in this paper seem to have good robustness to noise and achieves exact reconstruction as $\sigma\to 0$. Discussion {#sec:Discussion} ========== ![Numerical computation of the uniqueness parameter $\alpha$. Left: data geometry. The red hexagon corresponds to $\conv(\bX)$, and the black (equilateral) triangle to the archetypes $\bH_0$, for $L<1/3$. For $L=1/3$ the archetypes are not unique, and for $L\in (1/3,1/2]$, they are given by an equilateral triangle rotated by $\pi/3$ (pointing down). Right: numerical evaluation of the uniqueness constant (red circles). The continuous line corresponds to an analytical upper bound (triangle rotated by $\pi/3$ with respect to $\bH_0$).[]{data-label="fig:Alphanum"}](triangle.pdf "fig:"){width="47.50000%"} ![Numerical computation of the uniqueness parameter $\alpha$. Left: data geometry. The red hexagon corresponds to $\conv(\bX)$, and the black (equilateral) triangle to the archetypes $\bH_0$, for $L<1/3$. For $L=1/3$ the archetypes are not unique, and for $L\in (1/3,1/2]$, they are given by an equilateral triangle rotated by $\pi/3$ (pointing down). Right: numerical evaluation of the uniqueness constant (red circles). The continuous line corresponds to an analytical upper bound (triangle rotated by $\pi/3$ with respect to $\bH_0$).[]{data-label="fig:Alphanum"}](alphanum.pdf "fig:"){width="47.50000%"} We introduced a new optimization formulation of the non-negative matrix factorization problem. In its Lagrangian formulation, our approach consists in minimizing the cost function $\cuR_{\lambda}(\bH)$ defined in Eq. (\[eq:RLagrangian\]). This encompasses applications in which only one of the factors is required to be non-negative. A special case of this formulation ($\lambda\to\infty$) corresponds to the ‘archetypal analysis’ of [@cutler1994archetypal]. In this case, the archetype estimates coincide with a subset of the data points, which is appropriate only under the separability assumption of [@donoho2003does]. Our main technical result (Theorem \[thm:Robust\]) is a robustness guarantee for the reconstructed archetypes. This holds under the uniqueness assumption, which appears to hold for generic geometries of the dataset. In particular, while separability implies uniqueness (with optimal constant $\alpha=1$), uniqueness holds for non-separable data as well. To the best of our knowledge, similar robustness results have been obtained under separability [@recht2012factoring; @arora2013practical; @gillis2014fast; @gillis2015semidefinite] (albeit these works obtain a better dependence on $r$). The only exception is the recent work of Ge and Zou [@ge2015intersecting] who prove robustness under a ‘subset separability’ condition, which provides a significant relaxation of separability. Under this condition, [@ge2015intersecting] develops a polynomial-time algorithm to estimate the archetypes by identifying and intersecting the faces of $\conv(\bH_0)$. However, the algorithm of [@ge2015intersecting] exploits collinearities to identify the faces, and this requires additional ‘genericity’ assumptions. Admittedly, the uniqueness constant $\alpha$ is difficult to evaluate analytically, even for simple geometries of the data. However, by definition it does not vanish except in the case of multiple minimizers, and we expect it typically to be of order one. Figure \[fig:Alphanum\] illustrate this point by computing numerically $\alpha$ for a simple one-parameter family of geometries with $r=3$, $d=2$. The parameter $\alpha$ vanishes at a single point, corresponding to a degenerate problem with multiple solutions. Finally, several earlier works addressed the non-uniqueness problem in classical non-negative matrix factorization. Among others, Miao and Qi [@miao2007endmember] penalize a matrix of archetypes $\bH$ by the corresponding volume. Closely related to our work is the approach of M[ø]{}rup and Hansen [@morup2012archetypal] tha also builds on archetypal analysis. To the best of our knowledge, none of these works establishes robustness of the proposed methods. We conclude by mentioning three important problems that are not addressed by this paper: $(1)$ Are there natural condition under which the risk function $\cuR_{\lambda}(\bH)$ of Eq. (\[eq:RLagrangian\]) can be optimized in polynomial time? We only provided an algorithm that is guaranteed to converge to a critical point. $(2)$ We assumed the rank $r$ to be known. In practice it will need to be estimated from the data. $(3)$ Similarly, the regularization parameter $\lambda$ should be chosen from data. Acknowledgements {#acknowledgements .unnumbered} ================ This work was partially supported by the NSF grant CCF-1319979 and a Stanford Graduate Fellowship. [MBDC[[$^{+}$]{}]{}14]{} Sanjeev Arora, Rong Ge, Yonatan Halpern, David M Mimno, Ankur Moitra, David Sontag, Yichen Wu, and Michael Zhu, *A practical algorithm for topic modeling with provable guarantees.*, ICML (2), 2013, pp. 280–288. Sanjeev Arora, Rong Ge, Ravindran Kannan, and Ankur Moitra, *Computing a nonnegative matrix factorization–provably*, Proceedings of the forty-fourth annual ACM symposium on Theory of computing, ACM, 2012, pp. 145–162. M[á]{}rio C[é]{}sar Ugulino Ara[ú]{}jo, Teresa Cristina Bezerra Saldanha, Roberto Kawakami Harrop Galvao, Takashi Yoneyama, Henrique Caldas Chame, and Valeria Visani, *The successive projections algorithm for variable selection in spectroscopic multicomponent analysis*, Chemometrics and Intelligent Laboratory Systems **57** (2001), no. 2, 65–73. J[é]{}r[ô]{}me Bolte, Shoham Sabach, and Marc Teboulle, *Proximal alternating linearized minimization for nonconvex and nonsmooth problems*, Mathematical Programming **146** (2014), no. 1-2, 459–494. Adele Cutler and Leo Breiman, *Archetypal analysis*, Technometrics **36** (1994), no. 4, 338–347. David L Donoho and Victoria Stodden, *When does non-negative matrix factorization give a correct decomposition into parts?* Nicolas Gillis and Stephen A Vavasis, *Fast and robust recursive algorithmsfor separable nonnegative matrix factorization*, IEEE transactions on pattern analysis and machine intelligence **36** (2014), no. 4, 698–714. [to3em]{}, *Semidefinite programming based preconditioning for more robust near-separable nonnegative matrix factorization*, SIAM Journal on Optimization **25** (2015), no. 1, 677–698. Rong Ge and James Zou, *Intersecting faces: Non-negative matrix factorization with new guarantees*, Proceedings of the 32nd International Conference on Machine Learning (ICML-15), 2015, pp. 2295–2303. Dongmin Kim, Suvrit Sra, and Inderjit S Dhillon, *Fast projection-based methods for the least squares nonnegative matrix approximation problem*, Statistical Analysis and Data Mining **1** (2008), no. 1, 38–51. Chih-Jen Lin, *Projected gradient methods for nonnegative matrix factorization*, Neural computation **19** (2007), no. 10, 2756–2779. P.J. Linstrom and W.G. Mallard (eds.), *Nist chemistry webbook, nist standard reference database number 69*, National Institute of Standards and Technology, Gaithersburg MD, 20899, <http://webbook.nist.gov>, (retrieved January 5, 2017). Daniel D Lee and H Sebastian Seung, *Learning the parts of objects by non-negative matrix factorization*, Nature **401** (1999), no. 6755, 788–791. [to3em]{}, *Algorithms for non-negative matrix factorization*, Advances in neural information processing systems, 2001, pp. 556–562. Wing-Kin Ma, Jos[é]{} M Bioucas-Dias, Tsung-Han Chan, Nicolas Gillis, Paul Gader, Antonio J Plaza, ArulMurugan Ambikapathi, and Chong-Yung Chi, *A signal processing perspective on hyperspectral unmixing: Insights from remote sensing*, IEEE Signal Processing Magazine **31** (2014), no. 1, 67–81. Morten M[ø]{}rup and Lars Kai Hansen, *Archetypal analysis for machine learning and data mining*, Neurocomputing **80** (2012), 54–63. Boris S Mordukhovich and Nguyen Mau Nam, *An easy path to convex analysis and applications*, Synthesis Lectures on Mathematics and Statistics **6** (2013), no. 2, 1–218. Lidan Miao and Hairong Qi, *Endmember extraction from highly mixed data using minimum volume constrained nonnegative matrix factorization*, IEEE Transactions on Geoscience and Remote Sensing **45** (2007), no. 3, 765–777. Pentti Paatero, *Least squares formulation of robust non-negative factor analysis*, Chemometrics and intelligent laboratory systems **37** (1997), no. 1, 23–35. Pentti Paatero and Unto Tapper, *Positive matrix factorization: A non-negative factor model with optimal utilization of error estimates of data values*, Environmetrics **5** (1994), no. 2, 111–126. Jonathan L Roux, Alain D Cheveign[é]{}, and Lucas C Parra, *Adaptive template matching with shift-invariant semi-nmf*, Advances in neural information processing systems, 2009, pp. 921–928. Ben Recht, Christopher Re, Joel Tropp, and Victor Bittorf, *Factoring nonnegative matrices with linear programs*, Advances in Neural Information Processing Systems, 2012, pp. 1214–1222. Fei Wang and Ping Li, *Efficient nonnegative matrix factorization with random projections*, Proceedings of the 2010 SIAM International Conference on Data Mining, SIAM, 2010, pp. 281–292. Wei Xu, Xin Liu, and Yihong Gong, *Document clustering based on non-negative matrix factorization*, Proceedings of the 26th annual international ACM SIGIR conference on Research and development in informaion retrieval, ACM, 2003, pp. 267–273. G[ü]{}nter M Ziegler, *Lectures on polytopes*, vol. 152, Springer Science & Business Media, 2012. Further details on numerical experiments {#app:Numerical} ======================================== The data in Figures \[fig:spectra-no-noise\], \[fig:projgradsignals\_lownoise\], and \[fig:oursignals\_highnoise\] were generated as follows. We retrieved infrared reflection spectra of caffeine, sucrose, lactose and trioctanoin from the NIST Chemistry WebBook dataset [@nist]. We restricted these spectra to the wavenumbers between $1186\; {\rm cm}^{-1}$ and $1530\; {\rm cm}^{-1}$, and denote by $\bh_{0,1},\dots,\bh_{0,4}\in \reals^d$, $d=87$ the vector representations of these spectra. We then generates data $\bx_i\in\reals^d$, $i\le n = 250$ by letting $$\begin{aligned} \bx_i = \sum_{\ell=1}^4 w_{i,\ell}\bh_{\ell}+\bz_i\, , $$ where $\bz_i\sim \normal(0,\sigma^2\id_{d})$ are i.i.d. Gaussian noise vectors. The weights $\bw_{i} =(w_{i,\ell})_{\ell\le 4}$ were generated as follows. The weight vectors $\{\bw_i\}_{1\le i \le 9}$ are generated such that they have $2$ nonzero entries. In other words, $9$ data points are on one dimensional facets of the polytope generated by $\bh_{0,1},\dots,\bh_{0,4}$. In order to randomly generate these weight vectors, for each $1\le i \le 9$, a pair of indices $(\ell_1, \ell_2)$ between $1$ and $4$ is chosen uniformly at random. Then $\{\widetilde\bw\}_{1\le i \le 9}$, $\widetilde\bw\in \reals^2$ are generated as independent Dirichlet random vectors with parameter $(5,5)$. Then we let $w_{i,\ell_1} = \tilde w_{i,1}$ and $w_{i,\ell_2} = \tilde w_{i,2}$ for $1\le i \le 9$. The weight vectors $\{\bw_i\}_{10\le i \le 20}$ each have $3$ nonzero entries. Similar to above, for each of these weight vectors a $3$-tuple of indices $(\ell_1, \ell_2, \ell_3)$ between $1$ and $4$ is chosen uniformly at random. Then we let $w_{i,\ell_1} = \tilde w_{i,1}$, $w_{i,\ell_2} = \tilde w_{i,2}$, $w_{i,\ell_3} = \tilde w_{i,4}$ for $10\le i \le 20$, where $\{\widetilde\bw\}_{10\le i \le 20}$, $\widetilde\bw\in \reals^3$ are i.i.d. Dirichlet random vectors with parameter $(5,5,5)$. The rest of the weight vectors have cardinality equal to $4$. Hence, for $21 \leq i\le 250$, $\bw_i$ are generated as i.i.d. Dirichlet random vectors with parameter $(5,5,5,5)$. Proof of Theorem \[thm:Robust\] {#app:Proof} =============================== In this appendix we prove Theorem \[thm:Robust\]. We start by recalling some notations already defined in the main text, and introducing some new ones. We will then state a stronger form of the theorem (with better dependence on the problem geometry in some regimes). Finally, we will present the actual proof. Throughout this appendix, we assume the square loss $D(\bx,\by) = \|\bx-\by\|_2^2$. Notations and definitions ------------------------- We use bold capital letters (e.g. $\bA$, $\bB$, $\bC$,…) for matrices, bold lower case for vectors (e.g. $\bx$, $\by$, …) and plain lower case for scalars ($aa$, $b$, $c$ and so on). In particular, $\be_i \in \reals^d$ denotes the $i$’th vector in the canonical basis, $E^{r,d} = \{\be_1,\be_2,\dots,\be_r\}$ and for $r\leq d$, $\bE_{r,d}\in \{0,1\}^{r\times d}$ is the matrix whose $i$’th column is $\be_i$, and whose columns after the $r$-th one are equal to $\bzero$. For a matrix $\bX$, $\bX_{i,.}$ and $\bX_{.,i}$ are its $i$’th row and column, respectively. As in the main text, we denote by $\Delta^{m}$ the $m$-dimensional standard simplex, i.e. $\Delta^m = \{\bx\in \reals_{\geq0}^{m}, \langle\bx,\one\rangle = 1\}$, where $\one \in \reals^m$ is the all ones vector. For a matrix $\bH \in \reals^{r\times d}$, we use $\sigma_{\max}(\bH)$, $\sigma_{\min}(\bH)$ to denote its largest and smallest nonzero singular values and $\kappa(\bH)= \sigma_{\max}(\bH)/\sigma_{\min}(\bH)$ to denote its condition number. We denote by $\conv(\bH), \aff(\bH)$ the convex hull and the affine hull of the rows of $\bH$, respectively. In other words, $$\begin{aligned} &\conv(\bH) = \{\bx\in \reals^{d}:\bx = \bH^{\sT}\bpi, \bpi \in \Delta^{r} \},\\ &\aff(\bH) = \{\bx\in \reals^{d}:\bx = \bH^{\sT}\balpha, \langle \one, \balpha \rangle = 1 \}.\end{aligned}$$ We denote by $Q_{r,n}$ is the set of $r$ by $n$ row stochastic matrices. Namely, $$\begin{aligned} Q_{r,n} = \left\{\bPi\in \reals_{\geq 0}^{r\times n}: \langle\bPi_{i,.},\one\rangle = 1\right\}.\end{aligned}$$ with use $Q_r\equiv Q_{r,r}$. Further, $S_r$ is defined as $$\begin{aligned} S_r = \left\{\bPi\in Q_r: \Pi_{i,j}\in \{0,1\}\right\}.\end{aligned}$$ As a consequence, given $\bX\in \reals^{n\times d}$, $\bH_1,\bH_2\in \reals^{r\times d}$, the loss functions $\cuD(\,\cdot\,,\,\cdot\,)$ and $\cuL(\,\cdot\,,\,\cdot\,)$ take the form $$\begin{aligned} \cuD(\bH_1,\bX) &= \min_{\bPi\in Q_{r,n}} \|\bH_1 - \bPi \bX\|^2_F,\\ \cuL(\bH_1,\bH_2) &= \min_{\bPi\in S_r} \|\bH_1 - \bPi \bH_2\|_F^2. $$ We use $\Ball_m(\rho)$ to denote the closed ball with radius $\rho$ in $m$ dimensions, centered at $0$. In addition, for $\bH \in \reals^{m\times d}$ we define the $\rho$-neighborhood of $\conv(\bH)$ as $$\begin{aligned} \Ball_r(\rho;\bH) := \{\bx \in \reals^d: \cuD(\bx,\bH)\leq\rho^2\}.\end{aligned}$$ For a convex set $\mathcal C$ we denote the set of its extremal points by $\ext(\mathcal C)$ and the projection of a point $\bx \in \reals^{d}$ onto $\mathcal C$ by $\bPi_\mathcal C(\bx)$. Namely, $$\begin{aligned} \bPi_{\mathcal C}(\bx) = \arg\min_{\by \in \mathcal C}\|\bx - \by\|_2.\end{aligned}$$ Also, for a matrix $\bX \in \reals^{n\times d}$, and a mapping (not necessarily linear) $\bP:\reals^d \rightarrow \reals^d$, $\bP(\bX) \in \reals^{n\times d}$ is the matrix whose $i$’th row is $\bP(\bX_{i,.})$. Theorem statement ----------------- The statement below provides more detailed result with respect to the one in Theorem \[thm:Robust\]. \[thm:Robust2\] Assume $\bX = \bW_0\bH_0+ \bZ$ where the factorization $\bX_0=\bW_0\bH_0$ satisfies the uniqueness assumption with parameter $\alpha>0$, and that $\conv(\bX_0)$ has internal radius $\mu>0$. Consider the estimator $\hbH$ defined by Eq. (\[eq:HardNoise\]), with $D(\bx,\by) = \|\bx-\by\|_2^2$ (square loss) and $\delta = \max_{i\le n} \|\bZ_{i,\cdot}\|_2$. If $$\begin{aligned} \max_{i\le n} \|\bZ_{i,\cdot}\|_2\le \frac{\alpha\mu}{30 r^{3/2}}\, , $$ then, setting $\delta = \max_{i\le n} \|\bZ_{i,\cdot}\|_2$ in the problem we get $$\begin{aligned} \cuL(\bH_0,\hbH)\le \frac{C^2_*\, r^{5}}{\alpha^2} \max_{i\le n} \|\bZ_{i,\cdot}\|^2_2\, , $$ where $C_*$ is a coefficient that depends uniquely on the geometry of $\bH_0$, $\bX_0$, namely $C_* = 120(\sigma_{\rm max}(\bH_0)/\mu)\cdot \max(1, \kappa(\bH_0)/\sqrt{r})$. Further, if $$\begin{aligned} \max_{i\le n} \|\bZ_{i,\cdot}\|_2\le \frac{\alpha\mu}{330\kappa(\bH_0)r^{5/2}}, $$ then, setting $\delta = \max_{i\le n} \|\bZ_{i,\cdot}\|_2$ in the problem we get $$\begin{aligned} \cuL(\bH_0,\hbH)\le \frac{C_{**}^{2}\,r^4}{\alpha^2} \max_{i\le n} \|\bZ_{i,\cdot}\|^2_2\, , $$ where $C_{**} = 120 \max(\kappa(\bH_0), (\sigma_{\max}(\bH_0)/r+\|\bz_0\|_2)/(\mu r^{1/2}))\cdot\max (1, \kappa(\bH_0)/\sqrt{r})$. Proof ----- ### Lemmas \[lemma:cone\] Let $\mathcal R$ be a convex set and $\mathcal C$ be a convex cone. Define $$\begin{aligned} \gamma_\mathcal C = \max_{\|\bu\|_2=1}\min_{\bv\in \mathcal C, \|\bv\|_2=1}\langle \bu,\bv\rangle.\end{aligned}$$ We have $$\begin{aligned} \min_{\bx\in \mathcal R}\|\bx\|_2 + (1+\gamma_\mathcal C)\max_{x\in \ext(\mathcal R)}\|\bx-\bPi_\mathcal C(\bx)\|_2 \geq \gamma_\mathcal C \min_{\bx \in \ext(\mathcal R)}\|\bx\|_2.\end{aligned}$$ An illustration of this lemma in the case of $\mathcal R \subset \mathcal C$ is given in Figure \[fig:lemma\_cone\]. Note that, $\gamma_{\mathcal C}$ measures the pointedness of the cone $\mathcal C$. Geometrically (for ${\mathcal R}\subseteq {\mathcal C}$) the lemma states that the cosine of the angle between $\arg\min_{\bx\in \mathcal R}\|\bx\|_2$ and $\arg\min_{\bx\in \mathcal \ext(R)}\|\bx\|_2$ is smaller than $\gamma_\mathcal C$. ![Picture of Lemma \[lemma:cone\], in the case, $\mathcal R \subset \mathcal C$.[]{data-label="fig:lemma_cone"}](lemma_cone.pdf){width="70.00000%"} We write $$\begin{aligned} \min_{\bx \in \mathcal R} \|\bx\|_2 = \min_{\bx \in \mathcal R}\max_{\|\bu\|_2 = 1}\left\langle\bu,\bx\right\rangle \geq \max_{\|\bu\|_2=1}\min_{\bx\in \mathcal R}\langle \bu,\bx\rangle = \max_{\|\bu\|_2 = 1}\min_{\bx \in \ext(\mathcal R)}\langle \bu, \bx \rangle.\end{aligned}$$ Replacing $$\begin{aligned} \bx = \bPi_{\mathcal C}(\bx) + \left(\bx - \bPi_{\mathcal C}(\bx)\right),\end{aligned}$$ we get $$\begin{aligned} \min_{\bx \in \mathcal R}\|\bx\|_2 &\geq \max_{\|\bu\|_2 = 1}\min_{\bx\in \ext (\mathcal R)}\left\langle\bu,\bPi_\mathcal C(\bx)+(\bx-\bPi_{\mathcal C}(\bx))\right\rangle \\ &\geq\max_{\|\bu\|_2 = 1}\min_{\bx \in \ext(\mathcal R)}\left\langle\bu,\bPi_{\mathcal C}(\bx)\right\rangle - \max_{\bx \in \ext(\mathcal R)}\|\bx - \bPi_{\mathcal C}(\bx)\|_2.\end{aligned}$$ Hence, using the definition of $\gamma_\mathcal C$, we have $$\begin{aligned} \min_{\bx \in \mathcal R} \|\bx\|_2 \geq \gamma_\mathcal C \min_{\bx\in \ext(\mathcal R)}\|\bPi_\mathcal C(\bx)\|_2-\max_{\bx \in \ext(\mathcal R)}\|\bx - \bPi_\mathcal C(\bx)\|_2. \end{aligned}$$ Note that $$\begin{aligned} \|\bPi_\mathcal C(\bx)\|_2 \geq \|\bx\|_2 - \|\bx-\bPi_\mathcal C(\bx)\|_2.\end{aligned}$$ Therefore, $$\begin{aligned} \min_{\bx \in \mathcal R} \|\bx\|_2 \geq \gamma_{\mathcal C}\min_{\bx \in \ext(\mathcal R)}\|\bx\|_2 - (1+\gamma_\mathcal C)\max_{\bx \in \ext(\mathcal R)}\|\bx-\bPi_\mathcal C(\bx)\|_2,\end{aligned}$$ and this completes the proof. The next lemma is a consequence of Lemma \[lemma:cone\]. \[lemma:dandc\] Let $\bH, \bH_0\in \reals^{r\times d}$, $r\leq d$, be matrices with linearly independent rows. We have $$\begin{aligned} \cuL(\bH_0,\bH)^{1/2}\leq \sqrt{2}\kappa(\bH_0)\cuD(\bH_0,\bH)^{1/2} + (1+\sqrt{2})\sqrt{r}\cuD(\bH,\bH_0)^{1/2}\, . \end{aligned}$$ Consider the cone $\mathcal C_1\subset \reals^d$, generated by vectors $\be_2-\be_1, \dots, \be_r-\be_1\in \reals^d$, i.e, $$\begin{aligned} \mathcal C_1 = \left\{\bv \in \reals^d; \bv = \sum_{i=2}^r{v_i(\be_i - \be_1)}, v_i \geq 0\right\}.\end{aligned}$$ For $\bv \in \mathcal C_1, \|\bv\|_2 = 1$ we have $$\begin{aligned} \bv = \left(-\langle\one,\bx\rangle,\bx,0,0, \dots, 0\right),\end{aligned}$$ where $\bx \in \reals_{\geq 0}^{r-1}$ and $$\begin{aligned} \|\bx\|_2^2 + \left\langle\one,\bx\right\rangle^2 = 1.\end{aligned}$$ Since, $\langle\one,\bx\rangle = \|\bx\|_1 \geq \|\bx\|_2$, we get $\langle \one,\bx\rangle \geq 1/\sqrt{2} $. Thus, for $\bu = -\be_1$, we have $\langle\bu,\bv\rangle \geq 1/\sqrt{2}$. Therefore, for $\gamma_{{\mathcal C}_1}$ defined as in Lemma \[lemma:cone\], we have $\gamma_{\mathcal C_1} \geq 1/\sqrt{2}$. In addition, by symmetry, for $i \in \{1,2,\dots,r\}$, for the cone $\mathcal C_i\subset \reals^d$, generated by vectors $\be_1-\be_i, \be_2-\be_i, \dots, \be_r-\be_i\in \reals^d$ we have $\gamma_{\mathcal C_i} = \gamma \geq 1/\sqrt{2}$. Hence, using Lemma \[lemma:cone\] for $\bH \in \reals^{r\times d}$, $\mathcal R = \conv(\bH)-\be_j$ (the set obtained by translating $\conv(\bH)$ by $-\be_j$), $\mathcal C = \mathcal C_j$ we get for $j = 1,2,\dots, r$ $$\begin{aligned} \min_{\bq\in \Delta^r}\|\be_j - {\bH}^\sT\bq\|_2 &\geq \gamma\min_{\bq\in E^{r,r}}\|\be_j - {\bH}^\sT\bq\|_2 - (1+\gamma)\max_{i \in [r]}\min_{\bq\in \reals_{\geq 0}^r}\|\bH_{i,.}^\sT - \be_j - \bE_{r,d}^\sT\bq + \be_j \langle \one,\bq\rangle\|_2\\ &\geq \gamma\min_{\bq\in E^{r,r}}\|\be_j - {\bH}^\sT\bq\|_2 - (1+\gamma)\max_{i\in [r]}\min_{\bq\in \Delta^r}\|\bH_{i,.}^\sT-\bE_{r,d}^\sT\bq\|_2. $$ Hence, $$\begin{aligned} \sum_{j=1}^r \min_{\bq\in \Delta^r}\|\be_j - {\bH}^\sT\bq\|_2^2 &\geq \gamma^2 \sum_{j=1}^r \min_{\bq\in E^{r,r}}\|\be_j - {\bH}^\sT\bq\|_2^2 + (1+\gamma)^2r\max_{i \in [r]}\min_{\bq\in \Delta^r}\|\bH_{i,.}^\sT-\bE_{r,d}^\sT\bq\|_2^2 \nonumber \\ &- 2\gamma(1+\gamma) \left(\max_{i\in [r]}\min_{\bq\in \Delta^r}\|\bH_{i,.}^\sT-\bE_{r,d}^\sT\bq\|_2\right)\sum_{j=1}^r \min_{\bq\in E^{r,r}}\|\be_j - {\bH}^\sT\bq\|_2\\ &\geq \Bigg[\gamma\bigg(\sum_{j=1}^r \min_{\bq\in E^{r,r}}\|\be_j - {\bH}^\sT\bq\|_2^2 \bigg)^{1/2} \nonumber\\ &- (1+\gamma)\sqrt{r}\bigg(\max_{i \in [r]}\min_{\bq\in \Delta^r}\|\bH_{i,.}^\sT-\bE_{r,d}^\sT\bq\|_2\bigg)\Bigg]^2.\end{aligned}$$ Therefore, $$\begin{aligned} \min_{\bQ\in Q_r}\|\bE_{r,d} - \bQ\bH\|_F \geq \gamma \min_{\bQ\in S_r}\|\bE_{r,d} - \bQ\bH\|_F - (1+\gamma)\sqrt{r}\max_{i \in [r]}\min_{\bq\in \Delta^r}\|\bH_{i,.}^\sT-\bE_{r,d}^\sT\bq\|_2.\end{aligned}$$ Now consider $\bH_0 \in \reals^{r\times d}$ where $\bH_0 = \bE_{r,d}\bM$, $\bH = \bY\bM$, where $\bM\in \reals^{d\times d}$ is invertible. We have $$\begin{aligned} \cuD(\bH_0,\bH)^{1/2} &= \min_{\bQ\in Q_r}\|\bH_0 - \bQ\bH\|_F = \min_{\bQ\in Q_r}\|(\bE_{r,d} - \bQ\bY)\bM\|_F\\ & \geq \sigma_{\min}(\bM)\min_{\bQ\in Q_r}\|\bE_{r,d} - \bQ\bY\|_F \\ & \geq \gamma\sigma_{\min}(\bM)\min_{\bQ\in S_r}\|\bE_{r,d} - \bQ\bY\|_F - \sigma_{\min}(\bM)\sqrt{r}(1+\gamma)\max_{i \in [r]}\min_{\bq\in \Delta^r}\|{\bY_{i,.}^\sT}-\bE_{r,d}^\sT\bq\|_2\\ &= \gamma\sigma_{\min}(\bM)\min_{\bQ\in S_r}\|(\bH_0-\bQ\bH)\bM^{-1}\|_F \nonumber\\ &- \sigma_{\min}(\bM)\sqrt{r}(1+\gamma)\max_{i \in [r]}\min_{\bq\in \Delta^r}\|{(\bM^{-1}})^\sT(\bH_{i,.}^\sT-\bH_0^\sT\bq)\|_2.\end{aligned}$$ Thus, using the fact that $\sigma_{\max}(\bM)/\sigma_{\min}(\bM) = \kappa(\bM) = \kappa(\bH_0)$, $$\begin{aligned} \cuD(\bH_0,\bH)^{1/2} &\geq \frac{\gamma}{\kappa(\bH_0)}\cuL(\bH_0,\bH)^{1/2} - \frac{(1+\gamma)\sqrt{r}}{\kappa(\bH_0)}\max_{i \in [r]}\min_{\bq\in \Delta^r}\|\bH_{i,.}^\sT-\bH_0^\sT\bq\|_2\\ &\geq \frac{\gamma}{\kappa(\bH_0)}\cuL(\bH_0,\bH)^{1/2} - \frac{(1+\gamma)\sqrt{r}}{\kappa(\bH_0)}\cuD(\bH,\bH_0)^{1/2}\, .\end{aligned}$$ Therefore, $$\begin{aligned} \cuL(\bH_0,\bH)^{1/2} \leq \frac{\kappa(\bH_0)}{\gamma}\cuD(\bH_0,\bH)^{1/2} + \frac{(1+\gamma)\sqrt{r}}{\gamma}\cuD(\bH,\bH_0)^{1/2}\, .\end{aligned}$$ Finally, note that the function $f(x) = (1+x)/x$ is monotone decreasing over $\reals_{> 0}$. Hence, for $\gamma \geq 1/\sqrt{2}$, $(1+\gamma)/\gamma \leq 1+\sqrt{2}$. Therefore, we get $$\begin{aligned} \cuL(\bH_0,\bH)^{1/2}\leq \sqrt{2}\kappa(\bH_0)\cuD(\bH_0,\bH)^{1/2} + (1+\sqrt{2})\sqrt{r}\cuD(\bH,\bH_0)^{1/2}\end{aligned}$$ and this completes the proof. We continue with the following lemmas on the condition number of the matrix $\bH$. \[lemma:condition\] Let $\bH_0,\bH\in \reals^{r\times d}$, $r\leq d$,with $\bH$ having full row rank. We have $$\begin{aligned} \sigma_{\max}(\bH) \leq \cuD(\bH,\bH_0)^{1/2}+\sqrt{r}\sigma_{\max}(\bH_0),\label{eq:LemmaCondition1}.\end{aligned}$$ In addition, if $$\begin{aligned} \cuD(\bH_0,\bH)^{1/2} \leq \frac{\sigma_{\min}(\bH_0)}{2},\end{aligned}$$ then $$\begin{aligned} &\kappa(\bH) \leq \frac{2r\sigma_{\max}(\bH_0)+2\cuD(\bH,\bH_0)^{1/2}\sqrt{r}}{\sigma_{\min}(\bH_0)}. \label{eq:LemmaCondition2}\end{aligned}$$ Further, if $$\begin{aligned} \cuD(\bH,\bH_0)^{1/2}+\cuD(\bH_0,\bH)^{1/2} \leq \frac{\sigma_{\min}(\bH_0)}{6\sqrt{r}},\end{aligned}$$ then $$\begin{aligned} &\sigma_{\max}(\bH)\leq 2\sigma_{\max}(\bH_0),\\ &\kappa(\bH) \leq (7/2)\kappa(\bH_0).\end{aligned}$$ For the sake of simplicity, we will write $\cuD_1 = \cuD(\bH,\bH_0)^{1/2}$, $\cuD_2 = \cuD(\bH_0,\bH)^{1/2}$ Note that using the assumptions of Lemma \[lemma:condition\] we have $$\begin{aligned} \label{eq:w,w0} \begin{split} &\bH_0 = \bP\bH + \bA_2; \quad \|\bA_2\|_F= \cuD_2,\\ &\bH = \bR\bH_0 + \bA_1; \quad \|\bA_1\|_F= \cuD_1, \end{split}\end{aligned}$$ where $\bP, \bR \in \reals_{\geq 0}^{r\times r}$ are row-stochastic matrices and $\bA_1, \bA_2 \in \reals^{r\times d}$. Also, $\sigma_{\max}(\bA_1)\leq \|\bA_1\|_F= \cuD_1$, $\sigma_{\max}(\bA_2)\leq \|\bA_2\|_F= \cuD_2$. Therefore, $$\begin{aligned} \sigma_{\max}(\bP)\sigma_{\min}(\bH) \geq \sigma_{\min}(\bP\bH) \geq \sigma_{\min}(\bH_0)-\sigma_{\max}(\bA_2) \geq \sigma_{\min}(\bH_0)-\cuD_2.\end{aligned}$$ In addition, note that for a row stochastic matrix $\bP \in Q_r$, we have $$\begin{aligned} \sigma_{\max}(\bP)\leq \|\bP\|_F = \left(\sum_{i=1}^r\|\bP_{i,.}\|_2^2\right)^{1/2} \leq \left(\sum_{i=1}^r\|\bP_{i,.}\|_1^2\right)^{1/2} \leq \sqrt{r}.\end{aligned}$$ Hence, for $\cuD_2\leq \sigma_{\min}(\bH_0)$ we get $$\begin{aligned} \label{eq:sigma_minH} \sigma_{\min}(\bH)\geq \frac{\sigma_{\min}(\bH_0) - \cuD_2}{\sqrt{r}}\, .\end{aligned}$$ In addition, $$\begin{aligned} \label{eq:sigma_maxH} \sigma_{\max}(\bH) \leq \sigma_{\max}(\bR\bH_0) + \sigma_{\max}(\bA_1) \leq \sigma_{\max}(\bR)\sigma_{\max}(\bH_0) + \cuD_1 \leq \sqrt{r}\sigma_{\max}(\bH_0)+\cuD_1.\end{aligned}$$ Hence, using , , for $\cuD_2\leq \sigma_{\min}(\bH_0)$ we have $$\begin{aligned} \kappa(\bH) \leq \frac{r\sigma_{\max}(\bH_0)+\cuD_1\sqrt{r}}{\sigma_{\min}(\bH_0)-\cuD_2}.\end{aligned}$$ Thus, for $\cuD_2\leq \sigma_{\min}(\bH_0)/2$, we get Eqs. (\[eq:LemmaCondition1\]), (\[eq:LemmaCondition2\]). Now assume that $\cuD_1+\cuD_2 \leq \sigma_{\min}(\bH_0)/(6\sqrt{r})$. In this case, using we have $$\begin{aligned} \bH_0 = \bP(\bR\bH_0 + \bA_1) + \bA_2.\end{aligned}$$ Therefore, $$\begin{aligned} (\Id - \bP\bR)\bH_0 = \bP\bA_1 + \bA_2,\end{aligned}$$ hence, $$\begin{aligned} \Id - \bP\bR = (\bP\bA_1 + \bA_2)\bH_0^{\dagger}\end{aligned}$$ and $$\begin{aligned} \bP\bR = \Id - \bP\bA_1\bH_0^{\dagger} - \bA_2\bH_0^{\dagger}.\end{aligned}$$ where $\bH_0^{\dagger}$ is the right inverse of matrix $\bH_0$. Note that $$\begin{aligned} \sigma_{\max}(\bH_0^{\dagger}) = \sigma_{\min}(\bH_0)^{-1}.\end{aligned}$$ By permuting the rows and columns of $\bH_0$, without loss of generality, we can assume that $R_{ii} = \ \|\bR_{.,i}\|_\infty$. We can write $$\begin{aligned} R_{ii} \geq \left\langle\bP_{i,.},\bR_{.,i}\right\rangle &= 1-(\bP\bA_1\bH_0^{\dagger})_{ii}-(\bA_2\bH_0^{\dagger})_{ii}\\ &\geq 1-\|(\bP\bA_1\bH_0^{\dagger})_{i,.}\|_2 - \|(\bA_2\bH_0^{\dagger})_{i,.}\|_2\\ &\geq 1-\max_{\bu\in \Delta^r}\|\bA_1^\sT \bu\|_2\sigma_{\max}(\bH_0^{\dagger}) - \|(\bA_2)_{i,.}\|_2\sigma_{\max}(\bH_0^{\dagger})\\ &\geq 1-\max_{\bu\in \Delta^r}\|\bu\|_2\sigma_{\max}(\bA_1)\sigma_{\max}(\bH_0^{\dagger}) - \|\bA_2\|_F\sigma_{\max}(\bH_0^{\dagger})\\ &\geq 1- \frac{\cuD_1+\cuD_2}{\sigma_{\min}(\bH_0)}. \label{eq:riibiggerthan}\end{aligned}$$ Hence, for all $i,j\in [r], i\neq j$, since $\bR$ is row-stochastic, $$\begin{aligned} R_{ji} \leq \frac{\cuD_1+\cuD_2}{\sigma_{\min}(\bH_0)}.\end{aligned}$$ Thus, $$\begin{aligned} \left\langle\bP_{i,.},\bR_{.,i}\right\rangle = R_{ii}P_{ii} + \sum_{j\neq i}P_{ij}R_{ji} &\leq R_{ii}P_{ii} + \left(\max_{j\neq i}R_{ji}\right)\sum_{j\neq i}P_{ij}\\ &\leq P_{ii}+\frac{\cuD_1+\cuD_2}{\sigma_{\min}(\bH_0)}(1-P_{ii}).\end{aligned}$$ Therefore, using , $$\begin{aligned} P_{ii} \geq \frac{\sigma_{\min}(\bH_0)-2(\cuD_1+\cuD_2)}{\sigma_{\min}(\bH_0)-(\cuD_1+\cuD_2)}.\end{aligned}$$ Thus, we can write $$\begin{aligned} \bP = \Id + \Delta; \quad \|\Delta_{i,.}\|_1 \leq \frac{2(\cuD_1+\cuD_2)}{\sigma_{\min}(\bH_0)-(\cuD_1 + \cuD_2)}.\end{aligned}$$ Therefore, $$\begin{aligned} \sigma_{\max}(\Delta) \leq \|\Delta\|_F = \left(\sum_{i=1}^r\|\Delta_{i,.}\|_2^2\right)^{1/2} \leq \left(\sum_{i=1}^r\|\Delta_{i,.}\|_1^2\right)^{1/2} \leq \frac{2(\cuD_1+\cuD_2)\sqrt{r}}{\sigma_{\min}(\bH_0)-(\cuD_1 + \cuD_2)}.\end{aligned}$$ Hence, $$\begin{aligned} &\sigma_{\max}(\bP) \leq 1+\frac{2\sqrt{r}(\cuD_1+\cuD_2)}{\sigma_{\min}(\bH_0)-(\cuD_1+\cuD_2)}, &\sigma_{\min}(\bP) \geq 1-\frac{2\sqrt{r}(\cuD_1+\cuD_2)}{\sigma_{\min}(\bH_0)-(\cuD_1+\cuD_2)}.\end{aligned}$$ From we have $\sigma_{\min}(\bP\bH) \geq \sigma_{\min}(\bH_0)-\cuD_2$. Using $\sigma_{\min}(\bP\bH)\leq \sigma_{\max}(\bP)\sigma_{\min}(\bH)$, we get $$\begin{aligned} \sigma_{\min}(\bH) \geq \frac{(\sigma_{\min}(\bH_0)-\cuD_2)(\sigma_{\min}(\bH_0)-(\cuD_1+\cuD_2))}{\sigma_{\min}(\bH_0)-(\cuD_1+\cuD_2)+2\sqrt{r}(\cuD_1+\cuD_2)}.\end{aligned}$$ Further, from we have $\sigma_{\max}(\bP\bH) \leq \sigma_{\max}(\bH_0)+\cuD_2$. Using $\sigma_{\max}(\bP\bH) \geq \sigma_{\min}(\bP)\sigma_{\max}(\bH)$, we get $$\begin{aligned} \sigma_{\max}(\bH) \leq \frac{(\sigma_{\max}(\bH_0)+\cuD_2)(\sigma_{\min}(\bH_0)-(\cuD_1+\cuD_2))}{\sigma_{\min}(\bH_0)-(\cuD_1+\cuD_2)-2\sqrt{r}(\cuD_1+\cuD_2)}.\end{aligned}$$ Hence, for $\cuD_1+\cuD_2\leq \sigma_{\min}(\bH_0)/(6\sqrt{r})$, we have $\sigma_{\max}(\bH)\leq 35\sigma_{\max}(\bH_0)/18 < 2\sigma_{\max}(\bH_0)$. In addition, $$\begin{aligned} \kappa(\bH) &\leq \left(\frac{\sigma_{\max}(\bH_0)+\cuD_2}{\sigma_{\min}(\bH_0)-\cuD_2}\right)\left(1+\frac{4\sqrt{r}(\cuD_1+\cuD_2)}{\sigma_{\min}(\bH_0)-(\cuD_1+\cuD_2)-2\sqrt{r}(\cuD_1+\cuD_2)}\right)\\ &\leq \frac{6\kappa(\bH_0)+1}{5}\left(1+\frac{4}{3}\right) \leq \frac{42\kappa(\bH_0)+7}{15} < \frac{7\kappa(\bH_0)}{2},\end{aligned}$$ and this completes the proof. \[lemma:condition2\] Let $\bX_0 =\bW_0\bH_0\in \reals^{n\times d}$ be such that $\conv(\bX_0)$ has internal radius at least $\mu > 0$, and $\bX=\bX_0+\bZ$ with $ \max_{i\le n}\|\bZ_{i,.}\|_2 \leq \delta$. If $\bH \in \reals^{r\times d}, \bH_{i,.} \in \aff(\bH_0)$ is feasible for problem and has linearly independent rows, then we have $$\begin{aligned} \sigma_{\min}(\bH) \geq \sqrt{2}(\mu-2\delta)\, .\end{aligned}$$ Let $$\begin{aligned} \bX_{i,.}^\prime = \bPi_{\conv(\bH)}(\bX_{i,\cdot}) \equiv \arg\min_{\bx\in \conv(\bH)}\|\bX_{i,\cdot}-\bx\|_2\,.\end{aligned}$$ Note that since $\bH$ is feasible for problem and $\max_{i\le n}\|\bZ_{i,.}\|_2 \leq \delta$ $$\begin{aligned} \|(\bX_0)_{i,.}-\bX^\prime_{i,.}\|_2 \leq \|(\bX_0)_{i,.} - \bX_{i,.}\|_2 + \|\bX_{i,.}-\bX^\prime_{i,.}\|_2 \leq 2\delta.\end{aligned}$$ Therefore, for any $\bx_0 \in \conv(\bX_0)$, writing $\bx_0 = \bX_0^\sT\ba_0$, $\ba_0\in \Delta^n$, we have $$\begin{aligned} \cuD(\bx_0, \bX^\prime)^{1/2} &= \min_{\ba\in \Delta^n}\left\|\bX_0^{\sT}\ba_0- \bX^{\prime \sT}\ba \right\|_2 \leq \left\|\bX_0^{\sT}\ba_0 - \bX^{\prime \sT}\ba_0 \right\|_2\\ &\leq \left(\sum_{i=1}^n (a_0)_i \right)\|(\bX_0)_{i,.}-\bX^\prime_{i,.}\|_2 \leq 2\delta. \label{eq:distx_0convxprime}\end{aligned}$$ Since $\conv(\bX_0)$ has internal radius at least $\mu$, there exists $\bz_0\in\reals^d$, and an orthogonal matrix $\bU\in\reals^{d\times r^\prime}$, $r^\prime = r-1$, such that $\bz_0+\bU\Ball_{r^\prime}(\mu)\subseteq\conv(\bX_0)$. Hence, for every $\bz\in \reals^{r^\prime}$, $\|\bz\|_2 = 1$ there exists $\ba \in \Delta^n$ such that $$\begin{aligned} \mu\bU\bz + \bz_0 = \bX_0^\sT\ba.\end{aligned}$$ Therefore, for any unit vector $\bu$ in column space of $\bU$, for the line segment $$\begin{aligned} \label{eq:lumu} l_{\bu,\mu} = \left\{\bz: \bz = \bz_0 + \alpha \bu, |\alpha| \leq \mu\right\}\subseteq \conv(\bX_0)\, .\end{aligned}$$ Thus, $$\begin{aligned} l_{\bu,\mu} \subseteq \bP_{\bu}(\conv(\bX_0))\end{aligned}$$ where $\bP_{\bu}$ is the orthogonal projection onto the line containing $l_{\bu,\mu}$. Note that using , for any $\bx_0\in \conv(\bX_0)$ we have $$\begin{aligned} \cuD(\bP_{\bu}(\bx_0), \bP_{\bu}(\conv(\bX^\prime)))^{1/2}\leq \cuD(\bx_0,\bX^\prime)^{1/2} \leq 2\delta.\end{aligned}$$ In other words, for any $\bx_0 \in \bP_{\bu}(\conv(\bX_0))$, $D(\bx_0, \bP_{\bu}(\conv(\bX^\prime))) \leq 2\delta$. Therefore, using for any $\bu$ in column space of $\bU$, we have $$\begin{aligned} l_{\bu,\mu-2\delta} \subseteq \bP_{u}(\conv(\bX^\prime)).\end{aligned}$$ This implies that $$\begin{aligned} \bz_0 + \bU\Ball_{r^\prime}(\mu-2\delta)\subseteq \conv(\bX^\prime) \subseteq \conv(\bH).\end{aligned}$$ Hence, for every $\bz\in \reals^{r^\prime}$, $\|\bz\|_2 = 1$ there exists $\ba \in \Delta^r$ such that $$\begin{aligned} (\mu-2\delta)\bU\bz + \bz_0 = \bH^\sT\ba.\end{aligned}$$ Note that $\bH^\sT$ has linearly independent columns. Multiplying the previous equation by $(\bH^\sT)^\dagger$ the left inverse of $\bH^\sT$, we get $$\begin{aligned} (\mu-2\delta)(\bH^\sT)^\dagger\bU\bz + (\bH^\sT)^\dagger\bz_0 = \ba.\end{aligned}$$ Let $$\begin{aligned} &\ba_1 = (\mu-2\delta)(\bH^\sT)^\dagger\bU\bv + (\bH^\sT)^\dagger\bz_0\, ,\\ &\ba_2 = -(\mu-2\delta)(\bH^\sT)^\dagger\bU\bv + (\bH^\sT)^\dagger\bz_0\, ,\end{aligned}$$ where $\bv$ is the right singular vector corresponding to the largest singular value of $(\bH^\sT)^\dagger\bU$. Therefore, we have $$\begin{aligned} &\ba_1 = (\mu - 2\delta)\sigma_{\max}((\bH^\sT)^\dagger\bU)\bv + (\bH^\sT)^\dagger\bz_0,\\ &\ba_2 = -(\mu - 2\delta)\sigma_{\max}((\bH^\sT)^\dagger\bU)\bv + (\bH^\sT)^\dagger\bz_0.\end{aligned}$$ Thus, for $\ba_1, \ba_2 \in \Delta^r$ $$\begin{aligned} \|\ba_1 - \ba_2\|_2 = 2(\mu - 2\delta)\sigma_{\max}((\bH^\sT)^\dagger\bU).\end{aligned}$$ Note that $$\begin{aligned} \|\ba_1 - \ba_2\|_2 \leq {\sqrt{2}}.\end{aligned}$$ Thus, $$\begin{aligned} 2(\mu - 2\delta)\sigma_{\max}((\bH^\sT)^\dagger\bU) = \frac{2(\mu-2\delta)}{\sigma_{\min}(\bH)}\leq \sqrt{2}.\end{aligned}$$ Hence, $$\begin{aligned} \sigma_{\min}(\bH) \geq \sqrt{2}(\mu-2\delta).\end{aligned}$$ The following lemma states an important property of $\hbH$ the optimal solution of problem . \[lemma:optsol\] If $ \max_{i}\|\bZ_{i,.}\|_2 \leq \delta$ and $\hbH$ is the optimal solution of problem , then we have $$\begin{aligned} \cuD(\hbH,\bX_0)^{1/2}\leq \cuD(\bH_0,\bX_0)^{1/2}+3\delta\sqrt{r}.\end{aligned}$$ First note that since $\delta \geq \max_{i}\|\bZ_{i,.}\|_2$, we have $$\begin{aligned} \max_{i\le n}\cuD(\bX_{i,.},\conv(\bH_0))^{1/2} \leq \max_{i\le n}\|\bZ_{i,.}\|_2 \leq \delta.\end{aligned}$$ Hence, $\bH_0$ is a feasible solution for the problem . Therefore, we have $$\begin{aligned} \cuD(\hbH,\bX) \leq \cuD(\bH_0,\bX).\label{eq:Optimality}\end{aligned}$$ Letting $\tilde\balpha_i = \arg\min_{\balpha \in \Delta^n}\|{\hbH_{i,.}}^\sT-\bX^\sT\balpha\|_2$, we have $$\begin{aligned} \cuD(\hbH,\bX) &= \sum_{i=1}^r\min_{\balpha_i\in \Delta^r}\|{\hbH_{i,.}}^\sT- \bX_0^\sT\balpha_i-\bZ^\sT\balpha_i\|_2^2\\ &= \sum_{i=1}^r\min_{\balpha_i\in \Delta^r}\left(\|{\hbH_{i,.}}^\sT- \bX_0^\sT\balpha_i\|_2^2 - 2\left\langle\bZ^\sT\balpha_i,{\hbH_{i,.}}^\sT-\bX_0^\sT\balpha_i\right\rangle + \|\bZ^\sT\balpha_i\|_2^2\right)\\ &= \sum_{i=1}^r\left(\|{\hbH_{i,.}}^\sT- \bX_0^\sT\tilde\balpha_i\|_2^2 - 2\left\langle\bZ^\sT\tilde\balpha_i,{\hbH_{i,.}}^\sT-\bX_0^\sT\tilde\balpha_i\right\rangle + \|\bZ^\sT\tilde\balpha_i\|_2^2\right).\end{aligned}$$ Using the fact that (by triangle inequality) $\|\bZ^\sT\tilde\balpha_i\|_2 \leq \delta$, we have $$\begin{aligned} \cuD(\hbH,\bX) &\geq \sum_{i=1}^r \left(\|{\hbH_{i,.}}^\sT- \bX_0^\sT\tilde\balpha_i\|_2^2 - 2\delta\|{\hbH_{i,.}}^\sT - \bX_0^\sT\tilde\balpha_i\|_2\right)\\ &\geq U^2- 2\delta\sqrt{r} U\end{aligned}$$ where $U^2 = \sum_{i=1}^r\|{\hbH_{i,.}}^\sT- \bX_0^\sT\tilde\balpha_i\|_2^2$. Note that $\cuD(\hbH,\bX)\geq 0$ and for $U\geq 2\delta \sqrt{r}$, the function $U^2-2\delta\sqrt{r}U$ is increasing. Hence, since $$\begin{aligned} U\geq \left(\sum_{i=1}^r\min_{\balpha_i}\|{\hbH_{i,.}}^\sT- \bX_0^\sT\balpha_i\|_2^2\right)^{1/2} = \cuD(\hbH,\bX_0)^{1/2},\end{aligned}$$ we have $$\begin{aligned} \cuD(\hbH,\bX)\geq (U^2-2\delta\sqrt{r}U)\mathbb I_{U\geq 2\delta\sqrt{r}} \geq \cuD(\hbH,\bX_0)-2\delta\sqrt{r}\cuD(\hbH,\bX_0)^{1/2}.\end{aligned}$$ Therefore, $$\begin{aligned} \label{eq:1lemmaoptsol} \cuD(\hbH,\bX)^{1/2} \geq \left(\cuD(\hbH,\bX_0)-2\delta\sqrt{r}\cuD(\hbH,\bX_0)^{1/2}\right)_+^{1/2}\geq \cuD(\hbH,\bX_0)^{1/2}- 2\delta\sqrt{r}.\end{aligned}$$ In addition, $$\begin{aligned} \cuD(\bH_0,\bX) &= \sum_{i=1}^r \min_{\balpha_i\in \Delta^{n}}\|(\bH_0)_{i,.} - \bX_0^\sT\balpha_i-\bZ^\sT\balpha_i\|_2^2\\ &\leq \sum_{i=1}^r \min_{\balpha_i\in \Delta^{n}}\left\{\|(\bH_0)_{i,.} - \bX_0^\sT\balpha_i\|_2 + \|\bZ^\sT\balpha_i\|_2\right\}^2\\ &\leq \sum_{i=1}^r \left\{\min_{\balpha_i \in \Delta^n}\|(\bH_0)_{i,.} - \bX_0^\sT\balpha_i\|_2 + \max_{\balpha_i\in \Delta^n}\|\bZ^\sT\balpha_i\|_2\right\}^2\\ &\leq \left\{\left(\sum_{i=1}^r\min_{\balpha_i\in \Delta^n}\|(\bH_0)_{i,.}- \bX_0^\sT\balpha_i\|_2^2\right)^{1/2} + \delta\sqrt{r} \right\}^2\\ &\leq \left(\cuD(\bH_0,\bX_0)^{1/2} + \delta\sqrt{r}\right)^2.\end{aligned}$$ Hence, $$\begin{aligned} \label{eq:2lemmaoptsol} \cuD(\bH_0,\bX)^{1/2} \leq \cuD(\bH_0,\bX_0)^{1/2} + \delta\sqrt{r}.\end{aligned}$$ Combining equations , , and , we get $$\begin{aligned} \cuD(\hbH,\bX_0)^{1/2}\leq \cuD(\bH_0,\bX_0)^{1/2}+3\delta\sqrt{r}.\end{aligned}$$ This completes the proof of lemma. \[lemma:boundc\] Let $\bX_0$ be such that the uniqueness assumption holds with parameter $\alpha>0$, and $\conv(\bX_0)$ has internal radius at least $\mu>0$. In particular, we have $\bz_0+\bU\Ball_{r-1}(\mu)\subseteq \conv(\bX_0)$ for $\bz_0\in\reals^d$, and an orthogonal matrix $\bU\in\reals^{d\times (r-1)}$. Finally assume $\max_{i\le n}\|\bZ_{i,.}\|_2\leq \delta$. Then for $\hbH$ the optimal solution of problem , we have $$\begin{aligned} \alpha(\cuD(\hbH, \bH_0)^{1/2} + \cuD(\bH_0,\hbH)^{1/2}) \leq 2(1+2\alpha)\left[r^{3/2}\delta\kappa(\bP_0(\hbH)) +\frac{\delta\sqrt{r}}{\mu}\sigma_{\max}(\hbH - \one\bz_0^\sT)\right] +3\delta\sqrt{r}\end{aligned}$$ where $\bP_0:\reals^d\rightarrow \reals^d$ is the orthogonal projector onto $\aff(\bH_0)$ (in particular, $\bP_0$ is an affine map). Let $\tbH$ be such that $\conv(\bX_0) \subseteq \conv(\tbH)$. The uniqueness assumption implies $$\begin{aligned} \cuD(\tbH,\bX_0)^{1/2} \geq \cuD(\bH_0,\bX_0)^{1/2} + \alpha\big(\cuD(\tbH,\bH_0)^{1/2} + \cuD(\bH_0,\tbH)^{1/2}\big).\end{aligned}$$ Note that Lemma \[lemma:optsol\] implies $$\begin{aligned} \cuD(\hbH,\bX_0)^{1/2}\leq \cuD(\bH_0,\bX_0)^{1/2}+3\delta\sqrt{r}.\end{aligned}$$ Therefore, $$\begin{aligned} \cuD(\tbH,\bX_0) ^{1/2}\geq \cuD(\hbH, \bX_0)^{1/2} - 3\delta\sqrt{r} + \alpha\big(\cuD(\tbH,\bH_0)^{1/2} + \cuD(\bH_0,\tbH)^{1/2}\big)\, .\end{aligned}$$ Hence, $$\begin{aligned} \label{eq:alphaCless} \alpha\big(\cuD(\tbH,\bH_0)^{1/2} + \cuD(\bH_0,\tbH)^{1/2}\big) \leq \cuD(\tbH,\bX_0)^{1/2} - \cuD(\hbH,\bX_0)^{1/2} + 3\delta\sqrt{r}.\end{aligned}$$ In addition, for a convex set $S$, by triangle inequality we have $$\begin{aligned} &\left[\sum_{i=1}^n\cuD({\hbH_{i,.}},S)\right]^{1/2} - \left[\sum_{i=1}^n\cuD(\tbH_{i,.},S)\right]^{1/2} \leq \left[\sum_{i=1}^n\left(\cuD({\hbH_{i,.}},S)^{1/2}-\cuD(\tbH_{i,.},S)^{1/2}\right)^2\right]^{1/2}\end{aligned}$$ Therefore, using $$\begin{aligned} | \cuD(\tbH_{i,.},S)^{1/2} - \cuD({\hbH_{i,.}},S)^{1/2}| \leq \|\tbH_{i,.} - \hbH_{i,.}\|_2\end{aligned}$$ we have $$\begin{aligned} \left[\sum_{i=1}^n\cuD({\hbH_{i,.}},S)\right]^{1/2} - \left[\sum_{i=1}^n\cuD(\tbH_{i,.},S)\right]^{1/2} \leq \left[\sum_{i=1}^{n}\|\hbH_{i,.} - \tbH_{i,.}\|_2^2\right]^{1/2} = \|\hbH-\tbH\|_F.\end{aligned}$$ Hence, $$\begin{aligned} \label{eq:CHtildeX0-CHhatX0} |\cuD(\tbH,\bX_0) ^{1/2} - \cuD(\hbH,\bX_0) ^{1/2}| \leq \|\tbH - \hbH\|_F\end{aligned}$$ and $$\begin{aligned} \label{eq:CHtildeH0-CHhatH0} |\cuD(\tbH,\bH_0)^{1/2} - \cuD(\hbH,\bH_0)^{1/2}| \leq \|\tbH - \hbH\|_F.\end{aligned}$$ In addition, similarly to the proof of Lemma \[lemma:optsol\], we can write $$\begin{aligned} \cuD(\bH_0,\tbH) &= \sum_{i=1}^r \min_{\balpha_i\in \Delta^{r}}\|(\bH_0)_{i,.} - {\tbH}^\sT\balpha_i\|_2^2\\ &=\sum_{i=1}^r \min_{\balpha_i\in \Delta^{r}}\|(\bH_0)_{i,.} - {\hbH}^\sT\balpha_i-(\hbH-\tbH)^\sT\balpha_i\|_2^2\\ &\leq \sum_{i=1}^r \min_{\balpha_i\in \Delta^{r}}\left\{\|(\bH_0)_{i,.} - {\hbH}^\sT\balpha_i\|_2 + \|(\hbH-\tbH)^\sT\balpha_i\|_2\right\}^2\\ &\leq \sum_{i=1}^r \left\{\min_{\balpha\in \Delta^r}\|(\bH_0)_{i,.} - {\hbH}^\sT\balpha\|_2 + \max_{\balpha\in \Delta^r}\|(\hbH-\tbH)^\sT\balpha\|_2\right\}^2\\ &\leq \left\{\left(\sum_{i=1}^r\min_{\balpha\in \Delta^r}\|(\bH_0)_{i,.} - {\hbH}^\sT\balpha_i\|_2^2\right)^{1/2} + \sqrt{r}\max_{i\in [r]}\|\hbH_{i,.}-\tbH_{i,.}\|_2 \right\}^2\\ &\leq \left(\cuD(\bH_0,\hbH)^{1/2} + \sqrt{r}\max_{i \in [r]}\|\hbH_{i,.}-\tbH_{i,.}\|_2\right)^2.\end{aligned}$$ Thus, $$\begin{aligned} \label{eq:CHhatH0-CH0Htilde} |\cuD(\bH_0,\tbH)^{1/2} - \cuD(\bH_0,\hbH)^{1/2}| \leq \sqrt{r} \max_{i\in [r]} \|\tbH_{i,.} - \hbH_{i,.}\|_2\end{aligned}$$ Therefore, combining , , , , we get $$\begin{aligned} \label{eq:Cwhatw0} \alpha\big(\cuD(\hbH, \bH_0)^{1/2} + \cuD(\bH_0,\hbH)^{1/2}\big) \leq (1+\alpha)\|\tbH-\hbH\|_F + \alpha\sqrt{r}\max_{i\in [r]}\|\tbH_{i,.} - \hbH_{i,.}\|_2 + 3\delta\sqrt{r}.\end{aligned}$$ Now, we would like to bound the terms $\|\tbH - \hbH\|_F$, $\max_{i\in [r]}\|\tbH_{i,.} - \hbH_{i,.}\|_2$. Note that using the fact that $\hbH$ is feasible for Problem , we have $$\begin{aligned} \cuD(\bX_{i,.}, \hbH) \leq \delta^2\, .\end{aligned}$$ Thus, $$\begin{aligned} \cuD((\bX_0)_{i,.},\hbH)^{1/2} \leq \|\bX_{i,.} - (\bX_0)_{i,.}\|_2 + \cuD(\bX_{i,.},\hbH)^{1/2} \leq 2\delta.\end{aligned}$$ In addition, we know that $(\bX_0)_{i,.} \in \aff(\bH_0)$, where $\aff(\bH_0)$ is a $r-1$ dimensional affine subspace. Therefore, $\conv(\bX_0) \subseteq \aff(\bH_0)$ and, by convexity of $\Ball_{d}(2\delta,\hbH)$, we get $$\begin{aligned} \label{eq:Xi0inballcapaff} \conv(\bX_0) \subseteq \Ball_{d}(2\delta,\hbH)\cap \aff(\bH_0).\end{aligned}$$ First consider the case in which $\hbH_{i,.} \in \aff(\bH_0)$. for all $i\in\{1,2,\dots,r\}$. By a perturbation argument, we can assume that the rows of $\hbH$ are linearly independent, and hence $\aff(\hbH) = \aff(\bH_0)$. Consider $\tilde \bQ\in \reals^{r\times d}$ defined by $$\begin{aligned} &\widetilde Q_{ii} = 1+\xi, \quad\quad \text{if}\;\; i = j\in \{1,2,\dots,r\},\\ &\widetilde Q_{ij} = -\frac{\xi}{r-1}, \;\quad \text{if}\;\; i\neq j\in \{1,2,\dots,r\},\\ &\widetilde Q_{ij} = 0, \quad\quad\quad\quad \text{if}\;\; j\in\{r+1,r+2,\dots,d\}\end{aligned}$$ where $\xi = 2r\delta_0$. Note that for every $\by\in \Ball_d(2\delta_0;\bE_{r,d}) \cap \aff(\bE_{r,d})$, we have $\cuD(\by,\bE_{r,d})^{1/2} \leq 2\delta_0$. In addition, since $\by \in \aff(\bE_{r,d})$, $\langle \by,\one\rangle = 1$. Hence, for $\by\in \Ball_d(2\delta_0;\bE_{r,d}) \cap \aff(\bE_{r,d})$, we can write $$\begin{aligned} \by =\bpi + \bx\end{aligned}$$ where $\bpi \in \conv(\bE_{r,d})$, $\bx\in \reals^d$, $\langle\one,\bx\rangle = 0$, $\|\bx\|_2 \leq 2\delta_0$. It is easy to check that for this $\by$ we have $$\begin{aligned} \by = \sum_{i=1}^{r}\beta_i\widetilde\bQ_{i,.}\end{aligned}$$ where $\bbeta\in \reals^r$ is such that for $i = 1,2,\dots,r$, $$\begin{aligned} \beta_i = \frac{r-1}{r-1+\xi r}(\pi_i + x_i) + \frac{\xi}{r-1+\xi r}.\end{aligned}$$ Further, note that since $\bpi \in \conv(\bE_{r,d})$, $\pi_i\geq 0$ and $x_i \geq -\|\bx\|_2 \geq -2\delta_0$, we have $\pi_i + x_i \geq -2\delta_0$. Hence, for $i\in \{1,2,\dots,r\}$, $$\begin{aligned} \beta_i \geq \frac{-2\delta_0(r-1) + \xi}{r-1+\xi r} = \frac{2\delta_0}{r-1+\xi r} \geq 0.\end{aligned}$$ In addition, $$\begin{aligned} \sum_{i = 1}^r \beta_i = \frac{r\xi}{r-1+\xi r} + \frac{r-1}{r-1+\xi r}\left(\sum_{i=1}^r(\pi_i + x_i) \right)= 1.\end{aligned}$$ Therefore, every $\by\in \Ball_d(2\delta_0;\bE_{r,d}) \cap \aff(\bE_{r,d})$ can be written as a convex combination of the rows of $\widetilde\bQ$. Hence, $$\begin{aligned} \Ball_{d}(2\delta_0;\bE_{r,d})\cap{\aff}(\bE_{r,d})\subseteq\conv(\widetilde\bQ).\end{aligned}$$ Let $\hbH = \bE_{r,d}\bM$, $\bM\in \reals^{d\times d}$. Since $\aff(\hbH) = \aff(\bH_0)$, by taking $\tbH = \widetilde\bQ\bM$, we have $$\begin{aligned} \conv(\tbH) &\supseteq \left[\cup_{\bx\in \conv(\bE_{r,d})}\bM^\sT \Ball_{d}(2\delta_0;\bx)\right] \cap{\aff} (\hbH)\\ &\supseteq \left[\cup_{\bx \in \conv(\hbH)}\Ball_{d}(2\delta_0\sigma_{\min}(\bM);\bx)\right]\cap{\aff} (\hbH)\\ &\supseteq \Ball_{d}(2\delta;\hbH)\cap{\aff}(\bH_0),\end{aligned}$$ provided that $\delta_0 = \delta/\sigma_{\min}(\bM) = \delta/\sigma_{\min}(\hbH)$. Hence, using for this $\delta_0$, $\conv(\bX_0)\subseteq \conv(\tbH)$. Note that for $\widetilde\bQ$, we have $\|\widetilde\bQ_{i,.} - \be_i\|_2 \leq 2r\delta_0$. Thus, $$\begin{aligned} \|\widetilde\bQ- \bE_{r,d}\|_F \leq 2r^{3/2}\delta_0.\end{aligned}$$ Therefore, there exists $\tbH\in \reals^{r\times d}$ such that $\conv(\bX_0) \subseteq \conv(\tbH)$ and $$\begin{aligned} &\|\tbH - \hbH\|_F = \|(\widetilde\bQ - \bE_{r,d})\bM\|_F\leq2r^{3/2}\delta_0\sigma_{\max}(\bM) = 2r^{3/2}\delta_0\sigma_{\max}(\hbH) = 2r^{3/2}\delta\kappa(\hbH),\\ &\max_{i\in [r]}\|\tbH_{i,.} - \hbH_{i,.}\|_2 = \max_{i\in [r]} \|(\widetilde\bQ_{i,.} - \be_i)\bM\|_2 \leq 2r\delta_0\sigma_{\max}(\bM) = 2r\delta_0\sigma_{\max}(\hbH) = 2r\delta\kappa(\hbH).\end{aligned}$$ Now consider the general case in which $\aff(\hbH) \neq \aff(\bH_0)$. Let $\bH^\prime\in \reals^{r\times d}$ be such that $\bH^\prime_{i,.}$ is the projection of $\hbH_{i,.}$ onto $\aff(\bH_0)$. Assuming that the rows of $\bH^\prime$ are linearly independent, $\aff(\bH^\prime) = \aff(\bH_0)$. Note that since $\conv(\bX_0) \in \aff(\bH_0)$, for every point $\bx \in \conv(\bX_0)$, $\cuD(\bx,\bH^\prime)^{1/2}\leq \cuD(\bx,\hbH)^{1/2} \leq 2\delta$. Thus, $$\begin{aligned} (\bX_0)_{i,.} \in \Ball_d(2\delta, \bH^\prime)\cap {\aff}(\bH^\prime).\end{aligned}$$ Therefore, using the above argument for the case where $\aff(\hbH) = \aff(\bH_0)$, we can find $\tbH$ such that $\conv(\bX_0)\subseteq \conv(\tbH)$ and $$\begin{aligned} &\|\tbH - \bH^\prime\|_F \leq 2r^{3/2}\delta\kappa(\bH^\prime),\\ &\max_{i\in [r]}\|\tbH_{i,.} - \bH_{i,.}^\prime\|_2 \leq 2r\delta\kappa(\bH^\prime). \end{aligned}$$ Hence, for every $i=1,2,\dots,r$, $$\begin{aligned} \begin{split} \label{eq:w*-wtilde1} \|\tbH_{i,.} - \hbH_{i,.}\|_2 & \leq \|\tbH_{i,.} -\bH^\prime_{i,.}\|_2 + \|\bH^\prime_{i,.} - \hbH_{i,.}\|_2 \\ & \leq 2r\delta\kappa(\bH^\prime) + \|\bP_0(\hbH_{i,.}) - \hbH_{i,.}\|_2\\ \end{split}\end{aligned}$$ where $\bP_0$ orthogonal projection onto $\aff(\bH_0)$. We next use the assumption on the internal radius of $\conv(\bX_0)$ to upper bound the term $\|\bP_0(\hbH_{i,.}) - \hbH_{i,.}\|_2$. Note that since $\conv(\bX_0) \subseteq \Ball_d(2\delta,\hbH)$, letting $\bar\bH = \hbH - \one\bz_0^\sT$, for some orthogonal matrix $\bU\in\reals^{d\times r^\prime}$, $r^\prime=r-1$, we have $$\begin{aligned} \max_{\|\bz\|_2\leq \mu}\min_{\langle \ba,\one\rangle= 1, \ba\geq 0}\|\bU\bz - \bar\bH^\sT\ba \|_2^2 &= \max_{\|\bz\|_2\leq \mu}\min_{\langle \ba,\one\rangle= 1, \ba\geq 0}\|\bU\bz - (\hbH - \one\bz_0^\sT)^\sT\ba \|_2^2\\ &\leq \max_{\|\bz\|_2\leq \mu}\min_{\langle \ba,\one\rangle= 1, \ba\geq 0}\|\bU\bz + \bz_0 - {\hbH}^\sT\ba\|_2^2 \leq 4\delta^2.\end{aligned}$$ Now, using Cauchy-Schwarz inequality we can write $$\begin{aligned} \label{eq:maxoverzminovera} \max_{\|\bz\|_2\leq \mu}\min_{\|\ba\|_2\leq 1}\|\bU\bz - {\bar\bH}^\sT\ba\|_2^2 \leq \max_{\|\bz\|_2\leq \mu}\min_{\langle \ba,\one\rangle= 1, \ba\geq 0}\|\bU\bz - \bar\bH^\sT\ba \|_2^2 \leq 4\delta^2.\end{aligned}$$ Note that, $$\begin{aligned} \min_{\|\ba\|_2\leq 1}\|\bU\bz - \bar\bH^{\sT}\ba\|_2^2 &=\max_{\rho\geq 0}\min_{\ba}\left\{ \|\bz\|_2^2 - 2\left\langle\bz,\bU^\sT{\bar\bH}^\sT\ba\right\rangle + \left\langle\ba,({\bar\bH}{\bar\bH}^\sT+\rho\Id)\ba\right\rangle - \rho\right\}\\ &= \max_{\rho\geq 0} \left\{\|\bz\|_2^2 - \left\langle{\bar\bH}\bU\bz,({\bar\bH}{\bar\bH}^\sT+\rho\Id)^{-1}{\bar\bH}\bU\bz\right\rangle - \rho\right\}\end{aligned}$$ Hence, using $$\begin{aligned} \mu^2\max_{\rho\geq 0} \Big\{\lambda_{\max}(\Id - \bU^\sT{\bar\bH}^\sT(\bar\bH{\bar\bH}^\sT+\rho\Id)^{-1}\bar\bH\bU)-\rho \leq 4\delta^2\Big\}.\end{aligned}$$ In particular, for $\rho = 0$ we get $$\begin{aligned} \mu^2 \lambda_{\max}(\Id - \bU^\sT{\bar\bH}^\sT(\bar\bH{\bar\bH}^\sT)^{-1}\bar\bH\bU) \leq 4\delta^2.\end{aligned}$$ Taking $\bar\bH = \tilde\bU\bSigma\tilde\bV^\sT$, the singular value decomposition of $\bar\bH$, we have $\sigma_{\max}(\bar\bH) = \sigma_{\max}(\hbH - \one\bz_0^\sT) = \max_{i}\Sigma_{ii}$. Letting $\bU^\sT\tilde\bV = \bQ$, we get $$\begin{aligned} \max_{\rho\geq 0}\lambda_{\max}\left(\Id - \bQ\bQ^\sT\right)\leq \frac{4\delta^2}{\mu^2}.\end{aligned}$$ Letting $q = \sigma_{\min}(\bQ)$, this results in $$\begin{aligned} \label{eq:assumpresult} 1-q^2 \leq \frac{4\delta^2}{\mu^2}.\end{aligned}$$ In addition, note that, by the internal radius assumption, for any $\bz\in \reals^{r^\prime}$, $\bz_0 + \bU\bz \in \aff(\bH_0)$. Further, since $\bz_0 \in \aff(\bH_0)$, $$\begin{aligned} \max_{i \in [r]}\|\bP_0(\hbH_{i,.}) - \hbH_{i,.}\|_2 &= \max_{i \in [r]}\|\bP_\bU(\bar\bH_{i,.}) - \bar\bH_{i,.}\|_2\\ &\leq \max_{\|\ba\|_2\leq 1}\|\bP_\bU(\bar\bH^\sT\ba) - \bar\bH^\sT\ba\|_2\\ &\leq\max_{\|\ba\|_2\leq 1}\|\bP_\bU(\bar\bH^\sT\ba) - \bar\bH^\sT\ba\|_2\\ &\leq\max_{\|\ba\|_2\leq 1}\min_{\bz}\|\bU\bz-{\bar\bH}^\sT\ba\|_2^2\end{aligned}$$ where $\bP_\bU$ is the projector onto the column space of $\bU$. Note that, $$\begin{aligned} \max_{\|\ba\|_2\leq 1}\min_{\bz}\|\bU\bz-{\bar\bH}^\sT\ba\|_2^2 &= \max_{\|\ba\|_2\leq 1}\left\{-\left\langle\ba,\bar\bH\bU\bU^\sT{\bar\bH}^\sT\ba\right\rangle + \left\langle\ba,\bar\bH{\bar\bH}^\sT\ba\right\rangle\right\}\\ &= \lambda_{\max}(\bar\bH{\bar\bH}^\sT - \bar\bH\bU\bU^\sT{\bar\bH}^\sT)\\ &= \lambda_{\max}(\bSigma(\Id-\bQ^\sT\bQ)\bSigma)\\ &\leq \sigma_{\max}(\bar\bH)^2\lambda_{\max}(\Id-\bQ^\sT\bQ) \\ &\leq \sigma_{\max}(\bar\bH)^2(1-q^2)\leq \frac{4\sigma_{\max}(\bar\bH)^2\delta^2}{\mu^2}\end{aligned}$$ where the last inequality follows from . This results in $$\begin{aligned} \max_{i \in [r]}\|\bP_0(\hbH_{i,.}) - \hbH_{i,.}\|_2 \leq \frac{2\sigma_{\max}(\bar\bH)\delta}{\mu} &= \frac{2\sigma_{\max}(\hbH - \one\bz_0^\sT)\delta}{\mu}\end{aligned}$$ Therefore, $\|\bP_0(\hbH) - \hbH\|_F \leq 2\sigma_{\max}(\hbH - \one\bz_0^\sT)\delta\sqrt{r}/\mu$. Hence, using we get $$\begin{aligned} &\max_{i\in[r]}\|\hbH_{i,.} - \tbH_{i,.}\|_2 \leq 2r\delta\kappa(\bP_0(\hbH)) + \frac{2\sigma_{\max}(\hbH - \one\bz_0^\sT)\delta}{\mu},\\ &\|\hbH - \tbH\|_F \leq 2r^{3/2}\delta\kappa(\bP_0(\hbH)) + \frac{2\sigma_{\max}(\hbH - \one\bz_0^\sT)\delta\sqrt{r}}{\mu}.\end{aligned}$$ Replacing this in completes the proof. ### Proof of Theorem \[thm:Robust2\] For simplicity, let $\cuD = \alpha(\cuD(\hbH,\bH_0)^{1/2} + \cuD(\bH_0,\hbH)^{1/2})$. First note that under the assumption of Theorem \[thm:Robust2\] we have $$\begin{aligned} \bz_0 + \bU\Ball_{r^\prime}(\mu)\subseteq \conv(\bX_0) \subseteq \conv(\bH_0).\end{aligned}$$ Therefore, using Lemma \[lemma:condition2\] with $\bH=\bH_0$ and $\delta=0$, we have $$\begin{aligned} \label{eq:sigmaminH0overmu} \mu\sqrt{2} \leq \sigma_{\min}(\bH_0) \leq \sigma_{\max}(\bH_0).\end{aligned}$$ In addition, since $\bz_0 \in \conv(\bH_0)$ we have $\bz_0 = \bH_0^\sT\balpha_0$ for some $\balpha_0 \in \Delta^r$. Therefore, $$\begin{aligned} \label{eq:sigmamaxH0normz0} \|\bz_0\|_2 \leq \sigma_{\max}(\bH_0)\|\balpha_0\|_2 \leq \sigma_{\max}(\bH_0).\end{aligned}$$ Note that $$\begin{aligned} \sigma_{\max}(\hbH - \one\bz_0^\sT) \leq \sigma_{\max}(\hbH) + \sigma_{\max}(\one\bz_0^\sT) = \sigma_{\max}(\hbH) + \sqrt{r}\|\bz_0\|_2.\end{aligned}$$ Therefore, using Lemma \[lemma:boundc\] we have $$\begin{aligned} \label{eq:Clessthan2(1+alpha1+alpha2)} \cuD\leq 2(1+2\alpha) \left(r^{3/2}\delta\kappa(\bP_0(\hbH)) + \frac{\sigma_{\max}(\hbH)\delta r^{1/2}}{\mu} + \frac{r\delta\|\bz_0\|_2}{\mu}\right) + 3\delta r^{1/2}.\end{aligned}$$ In addition, Lemma \[lemma:dandc\] implies that $$\begin{aligned} \label{eq:DH0HhatCH0Hhat} \cuL(\bH_0,\hbH)^{1/2} \leq \frac{1}{\alpha}\max\left\{{(1+\sqrt{2})\sqrt{r}},{\sqrt{2}\kappa(\bH_0)}\right\}\cuD\, .\end{aligned}$$ Further, let $\bP_0$ denote the orthogonal projector on $\aff(\bH_0)$. Hence, $\bP_0$ is a non-expansive mapping: for $\bx, \by \in \reals^d$, $D(\bP_0(\bx),\bP_0(\by))\le D(\bx,\by)$. Therefore, since $\conv(\bH_0) \subset \aff(\bH_0)$, for any $\bh \in \reals^d$ $$\begin{aligned} \cuD (\bP_0(\bh), \bH_0) \le D(\bP_0(\bh), \bP_0(\bPi_{\conv(\bH_0)}(\bh))) \le D(\bh, \bPi_{\conv(\bH_0)}(\bh)) = \cuD(\bh, \bH_0).\end{aligned}$$ Therefore, $$\begin{aligned} \label{eq:cudprojection} \cuD(\bP_0(\hbH),\bH_0) \le \cuD(\hbH,\bH_0).\end{aligned}$$ First consider the case in which $$\begin{aligned} \label{eq:conddeltamain} \delta \leq \frac{\alpha\mu}{30\, r^{3/2}}.\end{aligned}$$ Note that in this case $\delta \leq \mu/2$. Hence, using Lemma \[lemma:condition\] to upper bound $\sigma_{\max}(\hbH)$, $\sigma_{\max}(\bP_0(\hbH))$ and Lemma \[lemma:condition2\] to lower bound $\sigma_{\min}(\bP_0(\hbH))$, by , we get $$\begin{aligned} &\sigma_{\max}(\hbH) \leq \cuD(\hbH,\bH_0)^{1/2} + r^{1/2}\sigma_{\max}(\bH_0) \leq \frac{\cuD}{\alpha} + r^{1/2}\sigma_{\max}(\bH_0),\\ &\kappa(\bP_0(\hbH))= \frac{\sigma_{\max}(\bP_0(\hbH))}{\sigma_{\min}(\bP_0(\hbH))}\leq \frac{\cuD(\bP_0(\hbH),\bH_0)^{1/2} + r^{1/2}\sigma_{\max}(\bH_0)}{\sqrt{2}(\mu - 2\delta)} \nonumber\\ &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\leq\frac{\cuD(\hbH,\bH_0)^{1/2} + r^{1/2}\sigma_{\max}(\bH_0)}{\sqrt{2}(\mu - 2\delta)}\leq \frac{\cuD}{\alpha(\mu - 2\delta)\sqrt{2}} + \frac{r^{1/2}\sigma_{\max}(\bH_0)}{(\mu-2\delta)\sqrt{2}}.\end{aligned}$$ Replacing these in we have $$\begin{aligned} \cuD \leq 2(1+2\alpha)\left[\frac{r^{3/2}\cuD\delta}{\alpha(\mu-2\delta)\sqrt{2}} + \frac{r^2\sigma_{\max}(\bH_0)\delta}{(\mu-2\delta)\sqrt{2}}+\frac{\cuD r^{1/2}\delta}{\alpha\mu} + \frac{r\sigma_{\max}(\bH_0)\delta}{\mu}+ \frac{r\|\bz_0\|_2\delta}{\mu}\right] + 3\delta\sqrt{r}.\end{aligned}$$ Therefore, $$\begin{aligned} \cuD&\left[1-\frac{\sqrt{2}(1+2\alpha)r^{3/2}\delta}{\alpha(\mu-2\delta)}-\frac{2(1+2\alpha) r^{1/2}\delta}{\alpha\mu}\right]\nonumber\\ &\;\;\;\;\;\;\;\leq 2(1+2\alpha)\left[\frac{r^2\sigma_{\max}(\bH_0)\delta}{(\mu-2\delta)\sqrt{2}} + \frac{r\sigma_{\max}(\bH_0)\delta}{\mu} + \frac{r\|\bz_0\|_2\delta}{\mu}\right]+3\delta\sqrt{r}\end{aligned}$$ Notice that condition implies that $\mu-2\delta \geq \mu/2$ and $$\begin{aligned} \frac{\sqrt{2}(1+2\alpha)r^{3/2}\delta}{\alpha(\mu-2\delta)}+\frac{2(1+2\alpha) r^{1/2}\delta}{\alpha\mu} \leq \frac{1}{2}. \end{aligned}$$ Using the previous two equations, under condition we have $$\begin{aligned} \cuD&\leq \frac{4(1+2\alpha)r\delta}{\mu}\left[\frac{5r\sigma_{\max}(\bH_0)}{2} + \|\bz_0\|_2\right] +3\delta\sqrt{r}\nonumber\\ &\leq\frac{4(1+2\alpha)r^2}{\mu}\left[\frac{5\sigma_{\max}(\bH_0)}{2}+\frac{\|\bz_0\|_2}{r} + \frac{3\mu}{4(1+2\alpha)r^{3/2}}\right]\delta. \label{eq:cdeltamain}\end{aligned}$$ Combining this with , and using the fact that $1+2\alpha \leq 3$, we have under condition $$\begin{aligned} \cuL(\bH_0,\hbH)^{1/2}&\le \frac{12r^2}{\mu\alpha}\left(\frac{5\sigma_{\max}(\bH_0)}{2}+\frac{\|\bz_0\|_2}{r} + \frac{3\mu}{4(1+2\alpha)r^{3/2}}\right)\max\left\{(1+\sqrt{2})\sqrt{r}, \sqrt{2}\kappa(\bH_0)\right\}\delta\\ &\le \frac{29\sigma_{\max}(\bH_0)r^{5/2}}{\alpha\mu}\max\left\{1,\frac{\kappa(\bH_0)}{\sqrt{r}}\right\}\left(\frac{5}{2}+\frac{\|\bz_0\|_2}{r\sigma_{\max}(\bH_0)} + \frac{3\mu}{4(1+2\alpha)r^{3/2}\sigma_{\max}(\bH_0)}\right)\delta.\end{aligned}$$ Note that using , and since $\alpha \ge 0$ $$\begin{aligned} \frac{\|\bz_0\|_2}{r\sigma_{\max}(\bH_0)} \leq 1, \quad\quad\quad \frac{3\mu}{4(1+2\alpha)r^{3/2}\sigma_{\max}(\bH_0)} \le \frac{3}{4\sqrt{2}}.\end{aligned}$$ Therefore, $$\begin{aligned} \cuL(\bH_0,\hbH)^{1/2} \leq \frac{120\sigma_{\max}(\bH_0)r^{5/2}}{\alpha\mu}\max\left\{1,\frac{\kappa(\bH_0)}{\sqrt{r}}\right\}\delta.\end{aligned}$$ Thus, $$\begin{aligned} \cuL(\bH_0,\hbH)\le \frac{C_*^2\, r^{5}}{\alpha^2} \max_{i\le n} \|\bZ_{i,\cdot}\|^2_2\, ,\end{aligned}$$ where $C_*$ is defined in Theorem \[thm:Robust2\]. Next, consider the case in which $$\begin{aligned} \label{eq:conddelta2} \delta = \max_{i\le n} \|\bZ_{i,\cdot}\|_2\le \frac{\alpha\mu}{330\kappa(\bH_0)r^{5/2}},\end{aligned}$$ Note that using , and since $1+2\alpha \leq 3$, this condition on $\delta$ implies that $$\begin{aligned} \delta \leq \frac{\alpha\mu\sigma_{\min}(\bH_0)}{12r(1+2\alpha)(5r^{3/2}\sigma_{\max}(\bH_0)+2\|\bz_0\|_2r^{1/2}+3\mu)}.\end{aligned}$$ In particular, condition holds. Hence, using equation we get $$\begin{aligned} \label{eq:cudineqcase2} \cuD \leq \frac{4(1+2\alpha)r^2}{\mu}\left[\frac{5\sigma_{\max}(\bH_0)}{2}+\frac{\|\bz_0\|_2}{r} + \frac{3\mu}{4(1+2\alpha)r^{3/2}}\right]\delta \leq \frac{\alpha\sigma_{\min}(\bH_0)}{6\sqrt{r}}.\end{aligned}$$ Further, note that since $\bP_0$ is a projection onto an affine subspace, for $\bx\in \reals^d$, $\bP_0(\bx) = \widetilde{\bP_0} \bx + \bx_0$ for some $\widetilde{\bP_0} \in \reals^{d \times d}, \bx_0 \in \reals^d$. Hence, for any $\bpi \in \Delta^r$, $\bh = \hbH^\sT \bpi \in \conv(\hbH)$, we have $$\begin{aligned} \bP_0(\bh) = \widetilde{\bP_0}\bh + \bx_0 = \widetilde{\bP_0}\hbH^\sT\bpi + \bx_0 = \sum_{i=1}^r \pi_i\left(\widetilde{\bP_0}\hbH^\sT {\boldsymbol{e}}_i + \bx_0\right) = \sum_{i= 1}^r \pi_i \bP_0(\widehat{\bh_i}) \in \conv(\bP_0(\hbH))\end{aligned}$$ where ${\boldsymbol{e}}_i$ is the $i$’th standard unit vector. Hence, $$\begin{aligned} \bP_0(\conv(\hbH)) \subseteq \conv (\bP_0(\hbH)).\end{aligned}$$ Thus, for $\bh_0 \in \reals^d$ an arbitrary row of $\bH_0$, we have $$\begin{aligned} \cuD(\bh_0,\bP_0(\hbH)) = D(\bh_0,\conv(\bP_0(\hbH))) \leq D(\bh_0, \bP_0(\conv(\hbH))) \leq D(\bh_0, \bP_0(\bPi_{\conv(\hbH)}(\bh_0))).\end{aligned}$$ In addition, using non-expansivity of $\bP_0$, we have $$\begin{aligned} D(\bh_0, \bP_0(\bPi_{\conv(\hbH)}(\bh_0))) \leq D(\bh_0, \bPi_{\conv(\hbH)}(\bh_0)) = D(\bh_0, \conv(\hbH)) = \cuD(\bh_0,\hbH).\end{aligned}$$ This implies that $$\begin{aligned} \label{eq:cudprojection2} \cuD(\bH_0,\bP_0(\hbH))\le \cuD(\bH_0,\hbH).\end{aligned}$$ Therefore, using , and we get $$\begin{aligned} \cuD(\bH_0,\bP_0(\hbH))^{1/2} + \cuD(\bP_0(\hbH),\bH_0)^{1/2} \leq \cuD(\bH_0,\hbH)^{1/2} + \cuD(\hbH,\bH_0)^{1/2} \leq \frac{\cuD}{\alpha}\leq \frac{\sigma_{\min}(\bH_0)}{6\sqrt{r}}.\end{aligned}$$ Hence, in this case Lemma \[lemma:condition\] implies that $$\begin{aligned} &\sigma_{\max}(\hbH) \leq 2\sigma_{\max}(\bH_0),\\ &\kappa(\bP_0(\hbH)) \leq \frac{7\kappa(\bH_0)}{2}.\end{aligned}$$ Replacing this in , we have $$\begin{aligned} \label{eq:hdeltadef} \cuD &\leq (1+2\alpha)r^{1/2}\left(7r\delta\kappa(\bH_0) + \frac{4\sigma_{\max}(\bH_0)\delta + 2\sqrt{r}\|\bz_0\|_2\delta}{\mu}\right) + 3\delta r^{1/2}\\ &\leq 3\delta\sqrt{r}\left(8r\kappa(\bH_0)+ \frac{4\sigma_{\max}(\bH_0) + 2\sqrt{r}\|\bz_0\|_2}{\mu}\right)\end{aligned}$$ Hence, using under assumption , we have $$\begin{aligned} \cuL(\bH_0,\hbH)^{1/2} &\leq 3\sqrt{r}\max\left\{{(1+\sqrt{2})\sqrt{r}},{\sqrt{2}\kappa(\bH_0)}\right\}\left(8r\kappa(\bH_0)+ \frac{4\sigma_{\max}(\bH_0) + 2\sqrt{r}\|\bz_0\|_2}{\mu}\right)\frac{\delta}{\alpha}\\ &\leq 120\max\left\{1,\frac{\kappa(\bH_0)}{\sqrt{r}}\right\}\max\left\{r\kappa(\bH_0), \frac{\sigma_{\max}(\bH_0)+\sqrt{r}\|\bz_0\|_2}{\mu}\right\} \frac{r\delta}{\alpha}.\end{aligned}$$ Hence, for $C_*^{\prime\prime}$ as defined in the statement of the theorem, we get $$\begin{aligned} \cuL(\bH_0,\hbH)^{1/2}\le \frac{C_*^{\prime\prime}\,r}{\alpha} \max_{i\le n} \|\bZ_{i,\cdot}\|_2\, \end{aligned}$$ This completes the proof. Proof of Proposition \[propo:Subdiff\] {#app:Subdiff} ====================================== The proof follows immediately from the following two propositions. \[prop:subgradient1\] Let $\bX\in \reals^{n\times d}$ and $D(\bx,\by) = \|\bx-\by\|_2^2$. Then the gradient of the function $\bu\mapsto \cuD(\bu,\bX)$ is given by $$\begin{aligned} \nabla_{\bu} \cuD(\bu,\bX) = 2(\bu - \bPi_{\conv(\bX)}(\bu)) \, . \label{eq:GradientD}\end{aligned}$$ Note that $\cuD(\bu,\bX)$ is the solution of the following convex optimization problem. $$\begin{aligned} \label{eq:subgradopt} \begin{split} &\mbox{minimize}\quad \left\| \bu - \by\right\|_2^2,\\ &\mbox{subject to}\quad \by = \bX^\sT \bpi,\\ &\quad\quad\quad\quad\quad\;\bpi \geq 0,\\ &\quad\quad\quad\quad\quad\;\langle \bpi, \one \rangle = 1. \end{split}\end{aligned}$$ The Lagrangian for this problem is $$\begin{aligned} \mathcal L (\by,\bpi,\brho,\tilde\rho,\blambda) = \|\bu - \by\|_2^2 + \left\langle \brho,(\by-\bX^\sT\bpi)\right\rangle - \langle \blambda,\bpi\rangle + \tilde\rho(1-\langle \bpi,\one\rangle).\end{aligned}$$ The KKT condition implies that at the minimizer $(\by^*,\bpi^*,\brho^*,\tilde\rho^*,\blambda^*)$, we have $$\begin{aligned} \frac{\partial \mathcal L}{\partial \by} = 0\, ,\end{aligned}$$ and therefore $$\begin{aligned} \label{eq:kkt1} \brho^* = 2(\bu - \by^*)\end{aligned}$$ and the dual of the above optimization problem is $$\begin{aligned} \label{eq:subgraddual1} \begin{split} &\mbox{maximize}\quad -\frac{1}{4}\|\brho\|_2^2 + \left\langle \brho, \bu\right\rangle + \tilde\rho,\\ &\mbox{subject to}\quad \blambda \geq 0,\\ &\quad\quad\quad\quad\quad\; \bX\brho + \tilde\rho\one+\blambda = 0. \end{split}\end{aligned}$$ Note that since is strictly feasible, Slater condition holds and by strong duality the optimal value of is equal to $f(\bu)$. Hence, we have written $f(\bu)$ as pointwise supremum of functions. Therefore, subgradient of $f(\bu)$ can be achieved by taking the derivative of the objective function in at the optimal solution (see Section 2.10 in [@mordukhovich2013easy]). Note that the derivative of this objective function at the optimal solution is equal to $\brho^*= 2(\bu - \by^*) = 2(\bu - \bPi_{\conv(\bX)}(\bu))$ (where we used Eq. ). Since the dual optimum is unique (by strong convexity in $\brho$), the function $\bu\mapsto \cuD(\bu,\bX)$ is differentiable with gradient given by Eq. (\[eq:GradientD\]). \[prop:subgradient2\] Let $\bu\in \reals^{d}$ and $D(\bx,\by) = \|\bx-\by\|_2^2$, and assume that the rows of $\bH_0\in\reals^{r\times d}$ are affine independent. Then the function $\bH\mapsto \cuD(\bu,\bH)$ is differentiable at $\bH_0$ with gradient $$\begin{aligned} \nabla_{\bH}\cuD(\bu,\bH_0) = 2\bpi_0(\bPi_{\conv(\bH_0)}(\bu) - \bu)^{\sT},\;\;\;\;\;\;\; \bpi_0 = \arg\min_{\bpi \in \Delta^r} \left\|\bH_0^\sT\bpi - \bu\right\|^2_2\, . \label{eq:GradientFormula} $$ We will denote by $\bG$ the right hand side of Eq. (\[eq:GradientFormula\]). For $\bV\in\reals^{r\times d}$, we have $$\begin{aligned} \cuD(\bu,\bH_0+\bV) = \min_{\bpi \in \Delta^r} \left\|(\bH_0+\bV)^\sT\bpi - \bu\right\|^2_2\, . $$ Note that $(\bH_0+\bV)$ has affinely independent rows for $\bV$ in a neighborhood of $\bzero$, and hence has a unique minimizer there, that we will denote by $\bpi_{\bV}$. By optimality of $\bpi_{\bV}$, we have $$\begin{aligned} \cuD(\bu,\bH_0+\bV) -\cuD(\bu,\bH_0)& = \left\|(\bH_0+\bV)^\sT\bpi_{\bV} - \bu\right\|^2_2-\left\|(\bH_0+\bV)^\sT\bpi_0 - \bu\right\|^2_2\\ & \le \left\|(\bH_0+\bV)^\sT\bpi_0 - \bu\right\|^2_2-\left\|(\bH_0+\bV)^\sT\bpi_0 - \bu\right\|^2_2\\ & = \<\bG,\bV\>+\|\bV\bpi_0\|_2^2. $$ On the other hand, by optimality of $\bpi_0$, $$\begin{aligned} \cuD(\bu,\bH_0+\bV) -\cuD(\bu,\bH_0) & \ge \left\|(\bH_0+\bV)^\sT\bpi_{\bV} - \bu\right\|^2_2-\left\|(\bH_0+\bV)^\sT\bpi_\bV - \bu\right\|^2_2\\ &= \<2\bpi_\bV(\bPi_{\conv(\bH_0)}(\bu) - \bu)^{\sT},\bV\>+ +\|\bV\bpi_\bV\|_2^2\\ & = \<\bG,\bV\>+2\<(\bpi_\bV-\bpi_0) (\bPi_{\conv(\bH_0)}(\bu) - \bu)^{\sT},\bV\>+\|\bV\bpi_\bV\|_2^2\, . $$ Letting $R(\bV) = |\cuD(\bu,\bH_0+\bV) -\cuD(\bu,\bH_0)-\<\bG,\bV\>|$ denote the residual, we get $$\begin{aligned} \frac{R(\bV)}{\|\bV\|_F}\le \|\bPi_{\conv(\bH_0)}(\bu) - \bu\|_2\|\bpi_{\bV}-\bpi_0\|_2 +\|\bV\|_F(\|\bpi_\bV\|_2+\|\bpi_{0}\|_2)\, . $$ Note that $\bpi_{\bV}$ must converge to $\bpi_0$ as $\bV\to 0$ because $\bpi_0$ is the unique minimizer for $\bV=\bzero$. Hence we get $R(\bV)/\|\bV\|_F\to 0$ as $\|\bV\|_F\to 0$, which proves our claim. Proof of Proposition \[propo:PALM\] {#app:PALM} =================================== We use the results of [@bolte2014proximal] to prove Proposition \[propo:PALM\]. We refer the reader to [@bolte2014proximal] for the definitions of the technical terms in this section. First, consider the function $$\begin{aligned} \label{eq:f(H)definition} f(\bH) = \lambda\cuD(\bH,\bX).\end{aligned}$$ Note that using the main theorem of polytope theory (Theorem 1.1 in [@ziegler2012lectures]), we can write $$\begin{aligned} \conv(\bX) = \left\{\bx \in \reals^d \,|\, \langle \ba_i,\bx\rangle \leq b_i \;\; \text{for} \;\; 1\leq i\leq m\right\}\end{aligned}$$ for some $\ba_i \in \reals^d$, $b_i\in \reals$ and a finite $m$. Hence, using the definition of the semi-algebraic sets (see Definition 5 in [@bolte2014proximal]), the set $\conv(\bX)$ is semi-algebraic. Therefore, the function $f(\bH)$ which is proportional to the sum of squared $\ell_2$ distances of the rows of $\bH$ from a semi-algebraic set, is a semi-algebraic function (See Appendix in [@bolte2014proximal]). Further, the function $$\begin{aligned} \label{eq:g(W)definition} g(\bW) = \sum_{i=1}^n \Ind\left(\bw_i\in \Delta^r\right)\end{aligned}$$ is the sum of indicator functions of semi-algebraic sets (Note that using the same argument used for $\conv(\bX)$, $\Delta^r$ is semi-algebraic). Therefore, the function $g$ is semi-algebraic (See Appendix in [@bolte2014proximal]). In addition, the function $$\begin{aligned} \label{eq:h(H,W)definition} h(\bH,\bW) = \left\|\bX - \bW\bH\right\|_F^2\end{aligned}$$ is a polynomial. Hence, it is semi-algebraic. Therefore, we deduce that the function $$\begin{aligned} \label{eq:Psi(H,W)definition} \Psi (\bH,\bW) = f(\bH) + g(\bW) + h(\bH,\bW)\end{aligned}$$ is semi-algebraic. In addition, since $\Delta^r$ is closed, $\Psi$ is proper and lower semi-continuous. Therefore, $\Psi(\bH,\bW)$ is a KL function (See Theorem 3 in [@bolte2014proximal]). Now, we will show that the Assumptions 1,2 in [@bolte2014proximal] hold for our algorithm. First, note that since $\Delta^r$ is closed, the functions $f(\bH)$ and $g(\bW)$ are proper and lower semi-continuous. Further, $f(\bH)\ge 0,\, g(\bW)\ge 0,\, h(\bH,\bW)\ge 0$ for all $\bH\in \reals^{r\times d},\, \bW\in \reals^{n\times r}$. In addition, the function $h(\bH,\bW)$ is $C^2$. Therefore, it is Lipschitz continuous over the bounded subsets of $\reals^{r\times d}\times\reals^{n\times r}$. Also, the partial derivatives of $h(\bH,\bW)$ are $$\begin{aligned} &\nabla_\bH h(\bH,\bW) = 2\bW^\sT(\bW\bH-\bX),\\ &\nabla_\bW h(\bH,\bW) = 2(\bW\bH - \bX)\bH^\sT.\end{aligned}$$ It can be seen that for any fixed $\bW$, the function $\bH \mapsto \nabla_\bH h(\bH,\bW)$ is Lipschitz continuous with moduli $L_1(\bW) = 2\|\bW^\sT\bW\|_F$. Similarly, for any fixed $\bH$, the function $\bW \mapsto \nabla_\bW h(\bH,\bW)$ is Lipschitz continuous with moduli $L_2(\bH) = 2\|\bH\bH^\sT\|_F$. Note that since in each iteration of the algorithm the rows of $\bW^k$ are in $\Delta^r$. Hence, $$\begin{aligned} \inf\left\{L_1(\bW^k):k\in \mathbb{N}\right\}\ge \lambda_1^-\,\quad\quad \sup\left\{L_1(\bW^k):k\in \mathbb{N}\right\}\le \lambda_1^+\,\end{aligned}$$ for some some positive constants $\lambda_1^-,\, \lambda_1^+$. In addition, note that because the PALM algorithm is a descent algorithm, i.e., $\Psi(\bH^{k}, \bW^{k}) \leq \Psi(\bH^{k-1}, \bW^{k-1})$ for $k \in \mathbb N$, and since $f(\bH)\to \infty$ as $\|\bH\|_F\to \infty$, the value of $L_2(\bH^k) = \|\bH^k\bH^{k^\sT}\|_F$ remains bounded in every iteration. Finally, note that by taking $\gamma_2^k > \max\left\{\left\|\bH^{k+1}\bH^{k+1^\sT}\right\|_F,\eps\right\}$ for some constant $\eps>0$, we make sure that the steps in the PALM algorithm remain well defined (See Remark 3(iii) in [@bolte2014proximal]). Hence, we have shown that the assumptions of Theorem 1 in [@bolte2014proximal] hold. Therefore, using this theorem, the sequence $\left\{\bH^k,\, \bW^k\right\}_{k\in \mathbb N}$ generated by the iterations in - has a finite length and it converges to a stationary point $\left(\bH^*,\bW^*\right)$ of $\Psi$. Other optimization algorithms {#app:algo} ============================= Apart from the proximal alternating linearized minimization discussed in Section \[sec:PALM\], we experimented with two other algorithms, obtaining comparable results. For the sake of completeness, we describe these algorithms here. Stochastic gradient descent {#sec:Proximal} --------------------------- Using any of the initializations discussed in Section \[sec:Initialization\] we iterate $$\begin{aligned} \bH^{(t+1)} = \bH^{(0)} - \gamma_t \bG^{(t)}\, . $$ The step size $\gamma_t$ is selected by backtracking line search. Ideally, the direction $\bG^{(t)}$ can be taken to be equal to $\nabla\cuR_{\lambda}(\bH^{(t)})$. However, for large datasets this is computationally impractical, since it requires to compute the projection of each data point onto the set $\conv(\bH^{(t)})$. In order to reduce the complexity of the direction calculation, we estimate this sum by subsampling. Namely, we draw a uniformly random set $S_t\subseteq [n]$ of fixed size $|S_t|=s\le n$, and compute $$\begin{aligned} \bG^{(t)} &= \frac{2n}{|S_t|}\sum_{i\in S_t} \balpha_{i}^*\left(\bPi_{\conv(\bH)}\left(\bx_{i}\right)-\bx_{i}\right) +2\lambda\left(\bH - \bPi_{\conv(\bX)}\left(\bH\right)\right) \,,\\ \balpha_{i}^{*} &= \arg\min_{\balpha \in \Delta^r} \left\| \bH^\sT \balpha - \bx_{i}^\sT\right\|_2\, .\end{aligned}$$ Alternating minimization ------------------------ This approach generalizes the original algorithm of [@cutler1994archetypal]. We rewrite the objective as a function of $\bW = (bw_i)_{i\le n}$, $\bw_i\in\Delta^r$,$\bH = (\bh_i)_{i\le r}$, $\bh_i\in\reals^d$ and $\bA = (\balpha_{\ell})_{\ell\le r}$, $\balpha_{\ell}\in\Delta^n$ $$\begin{aligned} \cuR_{\lambda}(\bH) &= \min_{\bW,\bA}F(\bH,\bW,\bA)\, ,\\ F(\bH,\bW,\bA) & = \sum_{i=1}^n\Big\|\bx_i-\sum_{\ell=1}^r w_{i\ell}\bh_{\ell}\Big\|_2^3+\lambda \sum_{\ell=1}^r\Big\|\bh_{\ell}-\sum_{i=1}^n\alpha_{\ell,i}\bx_i\Big\|^2_2\, . $$ The algorithm alternates between minimizing with respect to the weights $(\bw_i)_{i\le n}$ (this can be done independently across $i\in\{1,\dots,n\}$) and minimizing over $(\bh_{\ell},\balpha_{\ell})$, which is done sequentially by cycling over $\ell\in\{1,\dots,r\}$. Minimization over $\bw_i$ can be performed by solving a non-negative least squares problem. As shown in [@cutler1994archetypal], minimization over $(\bh_{\ell},\balpha_{\ell})$ is also equivalent to non-negative least squares. Indeed, by a simple calculation $$\begin{aligned} F(\bH,\bW,\bA) & = w^{\rm tot}_{\ell}\big\|\bh_{\ell}-\bv_{\ell}\big\|_2^2 + \lambda\Big\|\bh_{\ell}-\sum_{i=1}^n\alpha_{\ell,i}\bx_i\Big\|^2_2+\widetilde{F}(\bH,\bW,\bA)\\ & = f_{\ell}(\bh_{\ell},\balpha_{\ell};\bH_{\neq \ell},\bW,\bA)+\widetilde{F}(\bH,\bW,\bA)\, . $$ where $\bH_{\neq \ell} = (\bh_i)_{i\neq \ell, i\le r}$, $\widetilde{F}(\bH,\bW,\bA)$ does not depend on $(\bh_{\ell},\balpha_{\ell})$, and we defined $$\begin{aligned} w^{\rm tot}_{\ell} & \equiv \sum_{i=1}^nw^2_{i\ell}\, ,\\ \bv_{\ell} & \equiv \frac{1}{w^{\rm tot}_{\ell}}\,\sum_{i=1}^nw_{i,\ell}\left\{\bx_i-\sum_{j\neq \ell, j\le r} w_{ij}\bh_j\right\}\, . $$ It is therefore sufficient to minimize $f_{\ell}(\bh_{\ell},\balpha_{\ell};\bH_{\neq \ell},\bW,\bA)$ with respect to its first two arguments, which is equivalent to a non-negarive least squares problem. This can be seen by minimizing $f_{\ell}(\cdots )$ explicitly with respect to $\bh_{\ell}$ and writing the resulting objective function. The pseudocode for this algorithm is given below. [ll]{}\ \ \ \ \ 1: & For $\ell\in\{1,\dots,r\}$:\ 2: & Set $\balpha^{(0)}_{\ell} = \arg\min _{\balpha\in\Delta^n}\|\bh^{(0)}_\ell - \bX\balpha_{\ell}\|_2$;\ 3: & For $t\in \{1,\dots, T\}$:\ 4: & Set $\bW^{t} = \arg\min_{\bW}F(\bH^{t-1},\bW,\bA^{t-1})$\ 5: & For $\ell\in\{1,\dots,r\}$:\ 6: & Set $\bh^{(t)}_{\ell},\balpha^{(t)}_{\ell} = \arg\min_{\bh_{\ell},\balpha_{\ell}}f_{\ell}(\bh_{\ell},\balpha_{\ell};\bH_{<\ell}^{t},\bH_{>\ell}^{t-1},\bW^{t},\bA_{<\ell}^{t},\bA_{>\ell}^{t-1})$;\ 7: & End For;\ 8: & Return $\{\hbh_{\ell}^{(T)}\}_{1\le \ell\le r}$\ \ Here $\bH_{< \ell} = (\bh_i)_{ i< \ell}$, $\bH_{> \ell} = (\bh_i)_{\ell< i\le r}$, and similarly for $\bA$. [^1]: Department of Electrical Engineering, Stanford University [^2]: Department of Electrical Engineering and Statistics, Stanford University [^3]: Data were retrieved from the NIST Chemistry WebBook dataset [@nist]. [^4]: This problem can have multiple global minima if $\lambda=0$ or in degenerate settings. One minimizer is selected arbitrarily when this happens. [^5]: The same modification is also used in [@arora2013practical], but we do not apply the full algorithm of this paper. [^6]: These values were chosen as to approximately minimize the estimation error.

--- abstract: 'We present the discovery of ringlike diffuse radio emission structures in the peripheral regions of the Bullet cluster 1E 0657$-$55.8. Ring formations are spanning between 1–3 Mpc away from the center of the cluster, significantly further away from the two already reported relics. Integrated fluxes of four of the sub-regions in the inner ‘ring’ from 4.5 to 10 GHz have also been reported. To understand the possible origin of these structures, here we present a maiden attempt of numerical modelling of a 3D and realistic ‘bullet’ like event in a full cosmological ($\Lambda$CDM) environment with N-body plus hydrodynamics code. We report a simulated ‘bullet’ found inside a (128 Mpc)$^3$ volume simulation with a speed of 2700 km s$^{-1}$, creating a high supersonic bow shock of Mach $M=3.5$ and a clear evidence of temporal separation of dark matter and baryons, assuring no challenge to $\Lambda$CDM cosmology from the bullet event as of now. We are also able to unveil the physics behind the formation of these observed multiple shock structures. Modelled radio emissions in our simulation support a complex combination of merger-associated processes that accelerates and re-accelerates fossil and cosmic-ray electrons. With a time evolution study and the computed radio emissions, we have shown that the ring like formation around the bullet is originated due to the interaction of the strong merger shocks with the accretion shocks at the periphery. The multiple shock structures observed are possibly originated from multiple mergers that have taken place at different times and much before the bullet event.' author: - | Surajit Paul,$^{1}$[^1] Abhirup Datta,$^{2,4}$ Siddharth Malu,$^{2}$ Prateek Gupta,$^{1}$ Reju Sam John,$^{1,3}$ Sergio Colafrancesco$^{5}$\ $^{1}$Department of Physics, SP Pune University, Pune - 411007, India\ $^{2}$Centre of Astronomy, Indian Institute of Technology Indore, Simrol, Khandwa Road, Indore 453552, India\ $^{3}$Department of Physics, Pondicherry Engineering College, Puducherry - 605014, India\ $^{4}$Center for Astrophysics and Space Astronomy, Department of Astrophysical and Planetary Science, University of Colorado,\ Boulder, CO 80309, USA\ $^{5}$School of Physics, University of the Witwatersrand, Private Bag 3, WITS-2050, Johannesburg, South Africa bibliography: - 'bullet9.bib' date: 'Accepted XXX. Received YYY; in original form ZZZ' title: '‘Rings’ of diffuse radio emission surrounding the Bullet cluster' --- =4 \[firstpage\] galaxies: clusters: individual (1E 0657–56, RX J0658–5557) – cosmology: large-scale structure of Universe – Physical Data and Processes: hydrodynamics – radio continuum: general – methods: observational, numerical Introduction {#intro} ============ Galaxy clusters, the most massive gravitationally bound objects in the universe, grow through continuous mergers and accretion processes. Most of the matter in galaxy clusters is in the form of dark matter, and the ordinary matter or baryon is present as a plasma which constitutes the Intra-Cluster Medium (ICM), owing to the high temperature of about 10$^{7}-$10$^8$ K. Mergers and collisions between these galaxy clusters produce shocks and turbulence [@Sarazin_2002ASSL] that accelerate ICM plasma and amplifies magnetic field by compression [@Iapichino_2012MNRAS] and turbulent dynamo [@Subramanian_2006MNRAS] and fills the whole cluster with magnetic fields. Accelerated charged particles in the ICM gyrate in the ambient magnetic fields, give rise to central extended radio synchrotron emission (radio halos) as well as peripheral elongated diffuse emission (radio relics) along the shocks [@Feretti_2012; @Kale_2016JApA]. Large scale structure formation process becomes non-linear and complicated in late times of its evolution with increased rate of mergers [@2000Natur.406..376C; @2015SSRv..188...93P]. This makes it very difficult to formulate any analytical model to understand some of the observed structures and its energetics. Bullet cluster is one of such events. It is an extreme and rare merging event with very complicated physics of structure formation [@Clowe_2006ApJ; @Kraljic_2015JCAP]. Unlike the usual situation where, baryons remain trapped inside the dark matter halos, multi-wavelength observations of bullet cluster (1E0657-56) show an offset of gravitational potential and X-ray peak, indicating a clear separation of baryons from the dark matter (DM) . According to @Lee_2010ApJ [@Kraljic_2015JCAP], this even provides a challenge to the $\Lambda$CDM cosmology. Observations show a shock with Mach number more than $M=3$ ahead of the bullet, indicating extreme high-speed merger. Energetically, bullet cluster has been found to be the hottest (10s of KeV) and most X-ray and SZ bright cluster compared to other similar clusters . These findings suggest that bullet is a violently merging system [@2012PASJ...64...12A]. The energy that release during such mergers (more than $10^{65}$ erg s$^{-1}$) dissipates through shocks that thermalize the ICM and induce large-scale turbulence as well [@Sarazin_2002ASSL; @Paul_2011ApJ; @Iapichino_2010CONF]. Cosmological structure formation shocks are mainly known to be of two kinds [@2003ApJ...593..599R], the first being merger shocks, due to direct interaction of substructures, seen in several cluster mergers by now [@2006Sci...314..791B; @Paul_2014ASInC], most notably in the Bullet cluster [@2015MNRAS.449.1486S]. The shock strength of mergers are usually reported to be $\lesssim$ 3 [@2003ApJ...583..695G]. The second kind, referred to as accretion shocks, occur when gas falls in towards the surroundings of the clusters [@Bagchi_2011ApJ]. These accretion shocks have significantly higher Mach numbers than merger shocks, at $M\gg10$ [@John_2018arXiv; @2000ApJ...542..608M; @2006MNRAS.367..113P; @2008MNRAS.391.1511H], due to the gas in the outskirts of clusters, never being heated before. It has been pointed out that the energy released during mergers create huge pressure in the cluster core and the medium eventually starts expanding supersonically similar to a blast wave [@Ha_2017arXiv; @Paul_2012JoPConS; @Sarazin_2002ASSL] that travels radially as spheroidal wave-front towards the virial radius and moves beyond [@Weeren_2011MNRAS; @Paul_2011ApJ; @Iapichino_2017MNRAS]. So, there could a third situation arise, where, merger shock interacts with the accretion shock and re-energises the ambient particles to much higher energies and can show up in non-thermal emissions with brighter than expected. Clusters are therefore likely to be surrounded by these different kinds of shocks at distances up to a few times the virial radius [@2003ApJ...587..514N; @2008MNRAS.391.1511H] or till the accretion zone. So, the diffuse radio emissions observed at a few Mpc away from the centre of some of the clusters may be associated with these shocks, depending on the merger dynamics and merger activity of the cluster, as evidenced by X-ray morphology, the presence and structure of radio relics and their distances from the merging clusters. In this context we should mention that @2007MNRAS.375...77H have remarked that radio emission resulting from the accretion shocks may also be detected, if confined to a small region. In order to have detectable diffuse emission from accretion shocks or the interacting shocks, a significant amount of diffuse emission, and magnetic field strengths ($\sim\mu$G), combined with sufficient shock acceleration efficiency, are needed [@2008MNRAS.391.1511H; @2001ApJ...563..660F]. Also, depending on the stage of the merger, time elapsed, radio relics may be located $\sim$ Mpcs away from the cluster merger. So, to understand the radio emissions from merger/accretion scenario better, a full cosmological simulation with thermal and non-thermal physics is the set-up that is required. We also need numerical models to compute the possible radio synchrotron emissions from these structures. For this purpose, we have performed a cosmological simulation with the Adaptive Mesh Refinement (AMR), grid-based hybrid (N-body plus hydro-dynamical) code Enzo v. 2.2 [@Bryan_2014ApJS]. And further, computed radio emissions as a post process by implementing electron spectrum from Diffusive Shock Acceleration as well as Turbulent re-acceleration models to recreate the radio map of a bullet like simulated cluster. Radio emission modelling has been done as the post processing numerical analysis by using the yt-tools [@Turk_2011ApJS]. In this paper, we present the observation of unique radio structures in bullet cluster using Australia Telescope Compact Array (ATCA) at 5.5 and 9.0 GHz. Further to understand the observed complex and unusual structures, we present our first ever simulations of a modelled bullet like event in a full cosmological environment. So, after introducing in Section \[intro\], we present the observation, data analysis and reported the findings from observation in Section \[radio\_obs\]. Simulation of a bullet like event and comparison with our observations are presented in Section \[model\]. Finally, discussed the results and concluded in Section \[last\]. Radio Observations & Radio Images {#radio_obs} ================================= \[obs\_journal\] ----------------------------- ------------------------------------------------------------- Co-ordinates (J2000) RA–Dec $06^{\rm h}58^{\rm m}30^{\rm s} -55\degr57\arcmin00\arcsec$ Primary Beam FWHM 8.5$\arcmin$ Synthesized Beam FWHM 35.5$\arcsec\times$13.8$\arcsec$ Natural Resolution (5.5 GHz) Frequency Range 4.5–6.5 GHz & 8–10 GHz Total observing time 14 hours Arrays H168 Amplitude Calibrator PKS B1934$-$638 Phase Calibrator PKS B0823$-$500 Frequency Resolution 1 MHz ----------------------------- ------------------------------------------------------------- : Summary of the ATCA observations[]{data-label="Obs-sum"} Observation details ------------------- The Bullet cluster was observed for a total of 14 hours in the H168 array of the Australia Telescope Compact Array (ATCA) on July 30 & 31, 2010, at 5.5 and 9 GHz. ATCA continuum mode with 1 MHz resolution [@2011MNRAS.416..832W] with all four Stokes parameters and 2048 channels was used for these observations. PKS B1934$-$638 was used as the primary/amplitude calibrator and PKS B0823$-$500 as the secondary/phase calibrator. Observation details has been tabulated in Table \[Obs-sum\]. Data analysis ------------- Data were analyzed using Multichannel Image Reconstruction, Image Analysis and Display (MIRIAD, developed by ATNF [@2011ascl.soft06007S; @1995ASPC...77..433S]). Radio frequency interference induced bad data was excised. The secondary/phase calibrator was observed once every 60 minutes to keep track of variations in phase due to atmospheric effects. A description of our data reduction and calibration method is detailed in . ![5.5 GHz radio image of the Bullet cluster, observed with H168 array of the ATCA, with a synthesized beam 57$\arcsec\times$57$\arcsec$ (indicated as a circle in the bottom left corner). Specific regions are marked as A$-$G and flux densities are given in Table  \[diffsrc\].[]{data-label="cbandimage1"}](5-5ghz_contours_bw01_29mar18.pdf){width="9cm"} ![9 GHz radio image of the Bullet cluster, observed with H168 array of the ATCA, with a synthesized beam 30$\arcsec\times$30$\arcsec$ (indicated as a circle in the bottom left corner).[]{data-label="xbandimage1"}](9ghz_contours_bw01_29mar18.pdf){width="9cm"} The images shown in Figure \[cbandimage1\] and \[xbandimage1\] resulted from the observations presented above. These images were made with antennas 1–5 of the ATCA in the H168 array. Figure \[cbandimage1\] was made using natural weighting. The extent of this image is larger than the figures in @2014MNRAS.440.2901S. Noise rms ($\sigma_{\mathrm{RMS}}$) is 20$\mu$Jy/beam. The 5.5 GHz image has been convolved to 57$\arcsec\times$57$\arcsec$ with a PA of 0$\deg$. The intrinsic resolution in the first two 5.5 GHz images is the same as in . Regions A & B, and five point sources are marked in the figure. Properties of the regions C, D, E, F & G are given in Table  \[diffsrc\]. The two dashed areas indicate roughly the extents of the regions A and B. The dotted line represents the boundary between what we refer to as the peripheral regions or the periphery, and the ‘inner’ part of the Bullet cluster. Figure \[xbandimage1\] was made with uniform weight, and the extent of this image is similar to the figures in @2014MNRAS.440.2901S. Noise rms ($\sigma_{\mathrm{RMS}}$) is 20$\mu$Jy/beam. Results ------- It is evident from Figure \[cbandimage1\], as well as from Figure 1 and 2 of , that there is significant diffuse emission in the periphery of the Bullet cluster, as observed at 5.5 GHz. We point out that the square dotted box in Figure \[cbandimage1\] is the extent of the 2.1 GHz image shown in @2014MNRAS.440.2901S, and, following them, examined diffuse and point sources within the dashed box. We concentrate here on diffuse emission outside this box only. The peripheral diffuse emission has been marked in Figure \[cbandimage1\]. In what follows, we use the coordinates RA: $06^{\rm h}58^{\rm m}30.0^{\rm s}$, DEC: $-55\deg56\arcmin30\arcsec$ as the reference, or the ‘center’: this is the mid–point between the two X-ray peaks of the cluster. Labeling of diffuse emission is done following @2014MNRAS.440.2901S and ; since they label the relics ‘A’ and ‘B’, we label the first diffuse emission region ‘C’ and so on. Because of this labelling, and to avoid confusion, we refer to the relic regions as Relic ‘A’ and Relic ‘B’, henceforth. Outside the dotted box, the diffuse emission has the appearance of ‘rings’ surrounding the central region. These ‘rings’ seem to have three layers, as marked with dotted arcs in Figure \[cbandimage1\]. The northern portions of the diffuse emission have arc-like structures, whereas those in the SE regions have elongated, straight features. Subdividing these regions is difficult, as many of them have ‘bridges’, i.e. the diffuse emission features are connected through small regions of relatively low brightness. However, there is a clear gap between the first and second ‘layers’ of peripheral diffuse emission. We have subdivided only the first ‘layer’ of diffuse emission, and listed their properties in four sub-bands in the 4.5-6.5 GHz range, and two sub-bands in the 8-10 GHz range of frequencies. ![ Contours of the same image as in Figure \[cbandimage1\], overlaid on the ESO DSS-1 optical image of the Bullet cluster, from the Oschin Schmidt Telescope on Palomar Mountain and the UK Schmidt Telescope. Contour levels start at 5$\sigma$ and increase by factors of $\sqrt{2}$, with the noise rms $\sigma$ = 20 $\mu$Jy/beam.[]{data-label="overlaidimage1"}](5-5ghz_contours_on_dss01.pdf){width="8.5cm"} Diffuse emission in the closer ‘ring’ of emission in the N region is marked as ‘C’, that in the E region as ‘D’, that in the SE region as ‘E’, that in the SW region as ‘F’, and that in the NW region as ‘G’. These regions have been clearly marked in Figure \[cbandimage1\]. It is evident from Figure \[overlaidimage1\] that these ‘rings’ of diffuse emission do not follow the distribution of galaxies, and are morphologically different. Figure \[overlaidimage1\] clearly shows no correlation between diffuse emission, which is overlaid as blue contours, on the optical ESO DSS-1 image. ---------- ------------ ------------------------------ ----------------- -------------------- -------------------- ----------------- -------------------- -------------------- Source $S_{\rm 2100~MHz}$ Label h:m:s $^\circ$:$\arcmin$:$\arcsec$ $S_{\rm 4769~MHz}$ $S_{\rm 5223~MHz}$ $S_{5751~MHz}$ $S_{\rm 6193~MHz}$ $S_{\rm 9000~MHz}$ Region C 06:58:42.3 $-$55:58:37.5 7.89 $\pm$ 0.10 4.66 $\pm$ 0.10 3.80 $\pm$ 0.15 3.23 $\pm$ 0.15 2.88 $\pm$ 0.20 1.20 $\pm$ 0.15 Region D 06:58:37.6 $-$55:57:24.0 5.92 $\pm$ 0.10 2.17 $\pm$ 0.10 1.50 $\pm$ 0.10 1.15 $\pm$ 0.15 1.00 $\pm$ 0.10 0.19 $\pm$ 0.05 Region F 06:58:58.1 $-$55:55:35.1 6.27 $\pm$ 0.04 2.14 $\pm$ 0.10 1.65 $\pm$ 0.15 1.30 $\pm$ 0.15 $-$ 0.48 $\pm$ 0.05 Region G 06:58:57.8 $-$56:00:20.1 1.14 $\pm$ 0.10 0.93 $\pm$ 0.05 0.81 $\pm$ 0.15 0.54 $\pm$ 0.20 0.47 $\pm$ 0.13 0.27 $\pm$ 0.08 ---------- ------------ ------------------------------ ----------------- -------------------- -------------------- ----------------- -------------------- -------------------- Observed multi-layered ring-like radio relic formation is very unusual and reported for the first time in this work. As we can see that the bullet event is currently on-going in the cluster, the shocks arising from this event definitely cannot reach a very large distance of few Mpc from the centre (outer radio rings Fig. \[cbandimage1\]). So, how this has formed? To understand this unique event, we need a time evolution study. Therefore, numerical simulations become inevitable. Understanding observed radio emissions from a simulated bullet cluster {#model} ====================================================================== Simulation details ------------------ Bullet like merger is one of the most complex events in the structure formation history [@Mastropietro_2008MNRAS], thus cannot be understood well from only observations or from the ideal and controlled simulations. To unveil mysteries in such a rare and extreme object, we also need 3D realistic simulations in full cosmological environment. For this purpose, we have used [ENZO]{}, an N-Body plus hydrodynamic AMR code [@Bryan_2014ApJS; @O'Shea_2004astro.ph]. A flat $\Lambda$CDM background cosmology with the parameters of the $\Lambda$CDM model, derived from WMAP [@2009ApJS..180..330K] has been used. The simulations have been initialized at redshift $z = 60$ using the @1999ApJ...511....5E transfer function, and evolved up to $z = 0$. An ideal equation of state was used for the gas, with $\gamma = 5/3$. Since, the emergence and propagation of shocks are the most important events in this study, we have thus captured the shocks very efficiently and resolved the grids adaptively where ever shock is generated and passes by using the method described in [@Vazza_2009MNRAS; @Vazza_2011MNRAS]. In order to capture the correct energy distribution of Intra Cluster Medium, radiative cooling [@Sarazin_1987ApJ] and a star formation feedback scheme have been applied [@Cen1992ApJL]. A detail description of the simulation can be found in @Paul_2017MNRAS [@Paul_2018arXiva; @John_2018arXiv]. Cosmological simulations were performed to create a sky realization of 128$^3$ Mpc$^3$ volume with 0.3 millions of particles and 64$^3$ grids at the root grid level. We have further introduced 2 child grids and inside the innermost child grid of $(32 Mpc)^3$ volume, we have implemented 4 levels of AMR based on both over-densities and the shock strength. With this set-up, we have achieved about 30 kpc spatial resolution and mass resolution of about 10$^{8} M_{\odot}$ at the highest resolved grids. For further details of the simulations and resolution studies of our numerical schemes, readers are suggested to go through [@Paul_2018arXiva; @Paul_2017MNRAS; @John_2018arXiv]. Simulated Bullet {#simbullet1} ---------------- {width="9cm"} {width="9cm"}\ {width="9cm"} {width="9cm"}\ Galaxy clusters are formed through hierarchical structure formation. From the merger tree produced for our simulation, we have found the biggest cluster with a mass more than 10$^{15}M_{\odot}$, and there are several clusters of mass $\sim$ 10$^{14}M_{\odot}$ and numerous $\sim$ 10$^{13}M_{\odot}$ objects. During the formation of the biggest cluster, it has gone through several mergers. These mergers are usually in the mass ratios of 1:10 to even 1:1. Equal ratio mergers usually yield extremely high energy, but in high ratio mergers, smaller groups are severely attracted by the bigger mass clumps and attain a velocity of high-supersonic (Mach $M>3$) level similar to what has been observed in bullet cluster by @Markevitch_2002ApJ. In such cases, smaller groups ram into the larger groups and pass the cluster core extremely fast without getting merged [@Mastropietro_2008MNRAS] and it can be compared with the bullet hitting a target. We have identified a merging event at about redshift $z=0.3$ in our simulation using the merger tree, where, a smaller group hits a big cluster. The event is shown in Figure \[xry\_dm\] using four panels showing the step by step process. Here, the small group of total mass of about $2\times 10^{14}M_{\odot}$ (left corner of Fig. \[xry\_dm\](A)) started falling into a much bigger cluster with total mass little more than $8\times 10^{14}M_{\odot}$ with a relative approaching velocity of 1800 km s$^{-1}$ as shown in Figure \[xry\_dm\](A). The system has begun interacting at the redshift z=0.315 and nearest approach is at z=0.295 (Fig. \[xry\_dm\](B)). After the first passage, the smaller group rams past like a bullet with a velocity about 2700 km s$^{-1}$ creating a bow shock in the front of the bullet (Fig. \[hist\] and Fig. \[xry\_dm\](C)). The shock Mach number is around $M=3.5$, indicating a shock velocity of about 5000 km s$^{-1}$. In any normally merging clusters, it is impossible to see even a shock of Mach number $M=2$ inside the hot cluster core. We have also studied the velocity structure of the simulated cluster and plotted as a histogram in Figure \[hist\]. We found two distinctly separated peaks with the main cluster having a median speed of about 1155 km s$^{-1}$ and the bullet is having a median of about 2670 km s$^{-1}$. The bullet travelled a Mpc within 0.42 Gyr i.e. with an average speed of about 2400 km s$^{-1}$. Further, to confirm the other reported parameters of the observed bullet cluster, we studied the DM and baryon distribution in our simulated bullet. As reported, possibly bullet is the first example of a cluster where baryons get separated out from dark matter [@Markevitch_2002ApJ]. Figure \[xry\_dm\](a)$-$(d), show the evolution of baryons and dark matter in our simulated cluster. It shows that initially the peak of dark matter (see the contours) and the X-ray emitting gas were coinciding (Fig. \[xry\_dm\](A)). But, after the smaller clump passes the bigger one, the non-interacting dark matter went far ahead, while due to thermal energy loss and viscous drag, X-ray emitting baryons lagged behind, just like the observed bullet cluster (Fig. \[xry\_dm\](C)). Baryonic matter gets severely attracted back by the DM core ahead. Finally, baryons catch the dark matter core once again after almost a Gyr (see (Fig. \[xry\_dm\](D)). This is the first report of clear separation of DM and baryon in a full set up simulation with $\Lambda$CDM cosmology. Such event has been observed in (128 Mpc)$^3$ simulation, indicating a possibility of finding more such events if larger volume is simulated or if an extensive sky search is done. ![Mach number of the shocks are plotted in colour with black density contours for an area of (4 Mpc)$^2$.[]{data-label="shock-mach"}](Mach_with_density_contours_166output_4_Mpc_167.pdf){width="9cm"} ![Histogram of the velocity distribution in the simulated bullet cluster.[]{data-label="hist"}](Frequency_plot_vrl_disp.pdf){width="8.5cm"} Shocks and turbulence in the modelled ‘bullet’ cluster {#shock-turb} ------------------------------------------------------ ![[**Panel 1:**]{} Mach number of the shocks are plotted in colour with black density contours for an area of (10 Mpc)$^2$. [**Panel 2:**]{} Vorticity magnitude ($\lvert{\omega}\lvert$) of plotted for the same area.[]{data-label="Shock-vort-fig"}](Mach_with_density_contours_10Mpc_167.pdf "fig:"){width="9cm"} ![[**Panel 1:**]{} Mach number of the shocks are plotted in colour with black density contours for an area of (10 Mpc)$^2$. [**Panel 2:**]{} Vorticity magnitude ($\lvert{\omega}\lvert$) of plotted for the same area.[]{data-label="Shock-vort-fig"}](vorticity_magnitude_z_167.pdf "fig:"){width="9cm"} We have produced a video for the time evolution of thermal shocks for a period of few Gyrs (see the supporting supplementary material). It shows the evolution of the thermal properties much before and after the bullet event. While, the bullet has produced a strong shock ($M=3.5$) in the medium (see Fig. \[shock-mach\]), shocks from earlier mergers (before the bullet event has taken place) produced multiple layers of shocks that are seen to move with a supersonic speed (see the video of temperature shock), creating spherical shock waves. These shocks are the efficient in situ particle accelerator (through DSA) and can become visible in radio waves. Since, these merging shocks last for more than few Gyrs [@Paul_2011ApJ], with an average speed of 1000 km/s they are observed to reach to the periphery of the clusters and meet the accretion shock. Further, a snapshot of the simulation plotted for the Mach number at redshift z=0.27 (see Fig. \[Shock-vort-fig\], Panel 1), clearly shows the multiple supersonic shocks around the cluster and the bow shock that was seen ahead of the bullet in Figure \[shock-mach\] is placed at a very short distance from the cluster centre. Mach number of the shocks outside the virial radius (shown as the black circle), is beyond 5 and reached more than 10 for the merger shocks those meet the accretion shocks, indicating a possible show up in radio. We have also plotted the vorticity ($\vec{\omega}=\vec{\nabla} \times \vec{v}$) in the cluster after the bullet has passed the centre (see Fig. \[Shock-vort-fig\], Panel 2). Vorticity being a proxy to the solenoidal turbulence in the medium [@Vazza_2017MNRAS; @Miniati_2015Natur], it is very important for the cluster radio halo emissions through turbulent re-acceleration. We have found that the bullet has induced a high vorticity of 10$^{-15}$ s$^{-1}$ which corresponds to a turbulent velocity of more than 2000 km s$^{-1}$, in the vicinity of its movement. Rest of the cluster is at a lower level of turbulence with vorticity magnitude about an order lower. We can also observe that the vorticity is also higher in the region behind the shocks that emerged out of some previous merger indicating a better efficiency of turbulent re-acceleration and magnetic field amplification through turbulent dynamo. Radio emission computation {#radioemcompute} -------------------------- Radio emission takes place in a magnetised medium if relativistic charged particles, usually the electrons are available. The ICM is known to host magnetic field of $\mu$G order [@2004IJMPD..13.1549G] that can be achieved through turbulent dynamo model applied on primordial seed magnetic field [@Subramanian_2006MNRAS]. But, it needs a high degree of turbulence in the ICM. A saturation of magnetisation can be obtained when medium becomes fully turbulent, usually Kolmogorov type $E(k) \propto k^{-5/3}$ (where $k$ is wave number). A simple equipartition condition can be considered between magnetic energy density $\frac{B^2}{8\pi}$ and the kinetic energy density $\rho \epsilon_{turb}$ in this condition [@1998MNRAS.294..718S]. A detailed description of the method can be found in @Paul_2018arXiva. In Figure \[Shock-vort-fig\], Panel 2, we saw that the ‘bullet’ event has generated a considerable amount of turbulence in the medium. Implementing above model, we have found a few times of $\mu$G magnetic field in our simulated ‘bullet’ system. Further, to obtain the radio emissions due to the bullet event, we need an abundance of relativistic electrons that will gyrate in the said magnetic field to emit through synchrotron process [@Longair_2011]. The known best particle acceleration mechanism active in large scale structures that can elevate thermal electrons to relativistic energies are the diffusive shock acceleration (DSA) [@Drury_1983RPPh] and the turbulent re-acceleration [@Brunetti_2001MNRAS; @2011MNRAS.412..817B]. The synchrotron power spectrum in this medium will be determined by the electron energy spectrum from either DSA or the TRA. As we have noticed that both the shocks and the turbulence are available at a very high level in our simulated bullet (see section \[shock-turb\]), they could easily supply the required high energy particles to produce radio emission in the system. Thermally distributed ICM particles after shocks acceleration (DSA) get converted to a power-law energy distribution $N(E) dE \propto E^{-\delta} dE$ [@1983RPPh...46..973D]. Where, $\delta$ is the spectral index of electron energy with value 2 and more steeper. With turbulent re-acceleration (TRA), the electron energy power-law takes up the form $$\left(\frac{dn_e}{dE_e} \right) = \frac{3P_A\,c}{4 S(E_{\rm max})^{1/2}}\,E_e^{-\delta}$$ where, $P_A$ is the part of the total turbulent power going into the Alfven waves and E$_{max}$ is the maximum available electron energy. A simple assumption would be to consider a fully developed Kolmogorov type turbulence that makes the $\delta$ to be $\frac{5}{2}$ [@2016JCAP...10..004F]. Here, we did not include synchrotron ageing model, which may lead to a slightly different spectrum of radio emission. Synchrotron Radio emission is then calculated using the standard synchrotron emission formula $$\begin{aligned} \frac{d^{2}P(\nu_{obs})}{dVd\nu} &=& \frac{\sqrt{3}e^3B}{8m_e c^2}\,\int_{E_{\rm min} }^{E_{\rm max}}dE_e\,F\left(\frac{\nu_{obs}}{\nu_c}\right)\,\left(\frac{dn_e}{dE_e} \right)_{\rm inj}\end{aligned}$$ where $F(x)=x\int_x^\infty K_{5/3}(x')dx'$ is the Synchrotron function, $K_{5/3}$ is the modified Bessel function, and $\nu_c$ is the critical frequency of synchrotron emission, $$\nu_c = {3}\gamma^2{eB}/{(4\pi m_e)} = 1.6\, ( {B}/{1 \mu \rm G})({E_e}/{10 \rm GeV})^2 ~ \rm GHz$$ $\frac{dn_e}{dE_e}$ is the injected electron energy spectrum determined by either be DSA [@Hong_2015ApJ] or the TRA mechanism [@2016JCAP...10..004F],. For a detail account of the computation of radio emissions for our study, please see @Paul_2018arXiva. Simulated radio map from the modelled ‘bullet’ {#sim-res} ---------------------------------------------- Implementing above models for computing magnetic field and electron energy spectrum, radio emission has been computed on snapshots taken from our hydrodynamic simulations. For computing radio emission from DSA electrons, shocked cells are identified. Computation is done on the grid parameters of these cells and a proper weight has been used to nullify the effect of complicated resolution pattern of an AMR simulation. The computing formulae has been implemented on each grids and different output parameters have been visualized as slice or projection plots of few Mpc scales as required. {width="8.8cm"} {width="8.8cm"}\ {width="8.8cm"} {width="8.8cm"}\ Radio emission from this merging system clearly shows that the bullet is associated with a radio shock front (see Fig. \[shock-mach\]). The shock fronts are mostly having flux of 10s of $\mu$Jy in 5.5 GHz and almost half an order below in magnitude at 9 GHz. When we have plotted the same cluster with (10 Mpc)$^2$ area in Figure \[Shock-vort-fig\], Panel 1 and Figure \[rad-map\] Panel 3–4, it shows more outer shocks in radio waves. These shocks are far away from the cluster and are perpendicular to the merging axis. The order of flux is very low i.e. about few 10s nano Jy. With a beam convolution of $57\arcsec \times 57\arcsec$ at 5.5 GHz and $30\arcsec\times 30\arcsec$ at 9 GHz, we observe total flux density to be varying in the range of sub mJy to a few mJy for different regions as observed with ATCA (Table \[diffsrc\]). Morphologically, they produce multiple ‘ring’ like shock structures similar to the Figure \[cbandimage1\] at 5.5 GHz. As seen in the Figure \[Shock-vort-fig\], Panel 1, these are associated with merger shocks from earlier mergers. From the distance of these shocks from the centre of the cluster a simple estimation can be done about the look-back time for the actual merging event. Further, these mergers had taken place along another axis than the bullet, more than a few Gyr back. From the video and the radio image (in Fig. \[rad-map\], Panel 3–4) it is quite evident that the outer most relic has originated from the interaction of merger and accretion shocks that has Mach number $M>10$ and internal 2–3 layers of shocks are the multiple merger shocks with Mach number $M\eqsim2$ (see Fig. \[Shock-vort-fig\], Panel 1). Discussion and conclusion {#last} ========================= We have presented the discovery of peripheral diffuse radio emissions surrounding the Bullet cluster, in multiple ring-like structures at 5.5 and 9 GHz, using the ATCA. There are several (marked as ‘outer ring’) regions of these peripheral emissions that are greater than two virial radii away from the centre of the merging system. Origins of observed peripheral diffuse emission in a system like the bullet cluster, as reported in this paper, is not very straightforward to deduce, especially when these peripheral emissions are further away from the cluster virial radius. To explain such a unique and complicated morphology, here, for the first time, we present a simulation of a bullet like event in a full cosmological environment with N-Body plus Hydrodynamics as well as thermal physics. The structures observed in the bullet cluster have been nicely reproduced in our simulations (see Section \[radioemcompute\]). Various simulated ‘bullet’ parameters such as the bullet velocity, shock strength, DM-baryon distribution as well as the radio flux and extent of peripheral diffuse emission is strikingly similar to the observed values. Comparing bullet event ---------------------- Our model shows, a bullet is a violently merging slingshot that ramps pass the main cluster core with an average velocity of 2700 km s$^{-1}$ creating a bow shock of Mach number $M=3.5$ which corresponds to a shock velocity of about 5000 km s$^{-1}$. We have also found a clear separation of baryonic matter of bullet from the DM, similar to as reported in [@Markevitch_2002ApJ]. So, we report that the bullet event is a viable process in $\Lambda$CDM cosmology i.e. a very high velocity of 2700 km s$^{-1}$ and the temporal separation of DM and baryons both are the possible scenarios. Since, it has been observed in merely a (128 Mpc)$^3$ simulated volume, we expect that this may not be an extremely rare event or posses any challenge to $\Lambda$CDM cosmology as of now as doubted by @Lee_2010ApJ [@Kraljic_2015JCAP]. Both the authors have concluded from their DM only simulations. @Lee_2010ApJ has also mentioned that a velocity of above 1800 km s$^{-1}$ is impossible in $\Lambda$CDM, but our velocity histogram in Figure \[hist\] clearly shows a bullet with average velocity of 2700 km s$^{-1}$. It also shows that the bullet has travelled a Mpc within a time period of 0.42 Gyr (redshift $z=0.315-z=0.270$) with an average velocity of 2400 km s$^{-1}$. So, we attribute this discrepancy of findings to their DM only simulations which completely lacks the information about the transient behaviour of baryons inside a DM environment. Understanding ‘rings’ of peripheral emission -------------------------------------------- This bullet event creates the first layer of radio ring due to emergence of high Mach bow shocks as observed in Figure \[rad-map\], Panel 1 and 2 in 5.5 GHz and 9 Ghz, the same feature as it is observed in the Bullet cluster. Observed relics are close enough that one of them – Relic ‘A’ – even has a ‘radio bridge’ connecting it to the radio halo. The observed diffuse emission at about 2 Mpc away from the cluster merger centre (see Fig. \[cbandimage1\], Relic C, D, F and G) possibly be a radio relic from an earlier merger shock as shown in the simulated image in Panel 3, Figure \[rad-map\]. @2015MNRAS.449.1486S have pointed out the evidence for a second shock, but Relic ‘B’ has formed possibly at the same time as the bullet as, the shocks emerged from the oscillation of bigger cluster that was disturbed by the bullet itself. This is the reason that it is still significantly closer to the cluster centre. Shock waves due to large-scale structure accretion on to galaxy clusters have possibly been observed by @2006Sci...314..791B in the cluster/merger Abell 3376, where they also report ring-like structures surrounding the cluster, which are $\sim$ 1–2 Mpc away from the cluster centre. A two-layer shock with a possible combination of merger and accretion has also been reported by [@Bagchi_2011ApJ] in PLCK G287.0+32.9. We report another ‘ring’ of emissions (Fig. \[cbandimage1\], outer ’ring’ ) at the outermost region of our map. There is no report of a third shock in the Bullet cluster preceding to our work. This farthest one is almost two virial radius away, at the accretion zone. @2001ApJ...563..660F show that diffuse radio emission from accretion shocks can be a factor of a few order less than merger shocks. Mach numbers of accretion shocks can be significantly greater than those of merger shocks, since the gas in cluster outskirts has not been heated by shocks before[@2000ApJ...542..608M; @2006MNRAS.367..113P; @2008MNRAS.391.1511H]. At the same time, the surface brightness of any peripheral diffuse emission would fade away with time (see, e.g. Fig. 2 in @2011JApA...32..509H). The observed outer ring is also very fainter, indicating a possibility that these ‘rings’ of diffuse emission are caused by the accretion shocks in the cluster outskirts. But, the density of available high energy charged particles become extremely low in the peripheral region, so for a detectable amount of emission, it will need another power engine. So, a third model can be invoked as the interaction of merger shocks with the accretion shock that can re-energise the medium near to the accretion shock. A more clearer picture can be observed from our simulated bullet like cluster in Section \[radioemcompute\]. We found 2-3 layers of shocks in our simulations. The farthest one is almost two virial radius away and the intermediates are almost at the virial radius. Looking at the time evolution in the supplementary video, it can be clearly seen that the merger shock from a previous merger meets the peripheral accretion shock at the same instance when bullet structure has been formed. This interaction seen to increase the temperature substantially and can significantly re-energise the ambient medium as well as shock compresses the magnetic field and produce radio emissions as described briefly in Section \[intro\] and observed in our simulations (see section \[sim-res\]). Acknowledgements {#acknowledgements .unnumbered} ================ The Australia Telescope Compact Array is part of the Australia Telescope which is funded by the Commonwealth of Australia for operation as a National Facility managed by CSIRO. This research has made use of the services of the ESO Science Archive Facility, specifically, the ESO Online DSS Archive. SP would like to thank DST INSPIRE Faculty Scheme (IFA-12/PH-44) for supporting this research. SM is grateful to Mark Wieringa, Robin Wark, Maxim Voronkov and the research staff at ATCA/ATNF for guidance and help with understanding the CABB system. We thank Mark Wieringa for useful discussions. Observations and analysis were made possible by a generous grant for Astronomy by IIT Indore. SC acknowledges support by the South African Research Chairs Initiative of the Department of Science and Technology and National Research Foundation of South Africa (Grant no. 77948). SP, RSJ and PG are thankful to the Inter-University Centre for Astronomy and Astrophysics (IUCAA) for providing the HPC facility. Computations described in this work were performed using the publicly-available `Enzo` code (http://enzo-project.org) and data analysis is done with the yt-tools (http://yt-project.org/) \[lastpage\] [^1]: E-mail: surajit@physics.unipune.ac.in

[**Exponential Riesz bases of subspaces and divided differences** ]{} [^1] .3cm [S.A. Avdonin [^2], S.A.Ivanov [^3] ]{} > Abstract > > Linear combinations of exponentials $e^{i{\lambda}_kt}$ in the case where the distance between some points ${\lambda}_k$ tends to zero are studied. D. Ullrich [@Ullrich] has proved the basis property of the divided differences of exponentials in the case when $\left\{\lambda_k\right\}=\bigcup \Lambda^{(n)}$ and the groups $\Lambda^{(n)}$ consist of equal number of points all of them are close enough to $n,\, n\in > {\bf Z}.$ We have generalized this result for groups with arbitrary number of close points and obtained a full description of Riesz bases of exponential divided differences. Introduction ============ [\[n0\]]{} Families of ‘nonharmonic’ exponentials $\left\{e^{i\lambda_kt}\right\}$ appear in various fields of mathematics such as the theory of nonselfadjoint operators (Sz.-Nagy–Foias model), the Regge problem for resonance scattering, the theory of linear initial boundary value problems for partial differential equations, control theory for distributed parameter systems, and signal processing. One of the central problems arising in all of these applications is the question of the Riesz basis property of an exponential family. In the space $L^2(0,T)$ this problem was considered for the first time in the classical work of R. Paley and N. Wiener [@PW], and since then has motivated a great deal of work by many mathematicians; a number of references are given in [@KNP], [@Young] and [@AI95]. The problem was ultimately given a complete solution [@Pavlov79], [@KNP], [@Minkin] on the basis of an approach suggested by B. Pavlov. The main result in this direction can be formulated as follows [@Pavlov79]. [\[Pavlov\]]{} Let ${\Lambda}:= \{\lambda_k | k\in {{\mbox{\sf Z\kern-.45em Z}}}\}$ be a countable set of the complex plane. The family $\left\{e^{i\lambda_kt}\right\}$ forms a Riesz basis in $L^2(0,T)$ if and only if the following conditions are satisfied: \(i) ${\Lambda}$ lies in a strip parallel to the real axis, $$\sup_{k\in {{\mbox{\sf Z\kern-.45em Z}}}}|\Im {\lambda}_k| < \infty\,,$$ and is uniformly discrete (or separated), i.e. [[$${\label{discr}} \delta({\Lambda}):=\inf_{k\neq n}|{\lambda}_k - {\lambda}_n |>0\,;$$]{}]{} \(ii) there exists an entire function $F$ of exponential type with indicator diagram of width $T$ and zero set ${\Lambda}$ [(the generating function of the family $\left\{e^{i\lambda_kt}\right\}$ on the interval $(0,T)$)]{} such that, for some real $h$, the function $\left|F(x+ih)\right|^2$ satisfies the Helson–Szegö condition: functions $ u, v \in L^{\infty}({{\mbox{\rm I\kern-.21em R}}}), {\Vert v \Vert}_{L^{\infty}({{{\mbox{\sBlackboard R}}}})} < \pi/2 $ may be found such that [[$${\label{HS}} |F(x+ih)|^2=\exp\{u(x)+{\tilde}v(x)\}$$]{}]{} Here the map $v\mapsto \tilde{v}$ denotes the Hilbert transform for bounded functions: $${\tilde}v(x)=\frac 1\pi \int_{-\infty}^\infty v(t) \left \{ \frac 1{x-t}-\frac{t}{t^2+1} \right \} dt.$$ Note that (i) is equivalent to Riesz basis property of [${\mathcal E}$]{}in its span in $L^2(0,{\infty})$ and (ii) is a criterion that the orthoprojector $P_T$ from this span into $L^2(0,T)$ is an isomorphism (bounded and boundedly invertible operator). It is well known that the Helson–Szegö condition is equivalent to the Muckenhoupt condition $(A_2)$: $$\sup_{I\in {\cal J}}\Bigl\{ \frac{1}{\left|I\right|} \int_I\left|F(x+ih)\right|^2\,dx\,\,\frac{1}{\left|I\right|} \int_I\left|F(x+ih)\right|^{-2}\,dx\Bigr\}\,\, < \,\,\infty,$$ where ${\cal J}$ is the set of all intervals of the real axis. The notion of the generating function mentioned above plays a central role in the modern theory of nonharmonic Fourier series [@KNP; @AI95]. This notion plays also an important role in the theory of exponential bases in Sobolev spaces (see [@AI00; @IK; @LyubSeip00]). It is possible to write the explicit expression for this function $$F(z)=\lim_{R\to\infty}\prod_{|\lambda_k|\le R}(1-\frac{z_k}{\lambda_k})$$ (we replace the term $(1-\lambda_k^{-1}z)$ by $z$ if $\lambda_k=0$). The theory of nonharmonic Fourier series was successfully applied to control problems for distributed parameter systems and formed the base of the powerful method of moments ([@Butkovsky; @R78; @AI95]). Recent investigations into new classes of distributed systems such as hybrid systems, structurally damped systems have raised a number of new difficult problems in the theory of exponential families (see, e.g. [@HZ; @MZ; @JTZ]). One of them is connected with the properties of the family ${\ensuremath{{\mathcal E}}\xspace}=\left\{e^{i\lambda_kt}\right\}$ in the case when the set ${\Lambda}$ does not satisfy the separation condition (\[discr\]), and therefore ${\ensuremath{{\mathcal E}}\xspace}$ does not form a Riesz basis in its span in ${\ensuremath{L^2(0,T)}\xspace}$ for any $T>0$. Properties of such families in [$L^2(0,{\infty})$]{}have been studied for the first time in the paper of V. Vasyunin [@Vasyunin78] (see also [@Nik Lec. IX]). In the case when ${\Lambda}$ is a finite union of separated sets, a natural way to represent ${\Lambda}$ as a set of groups ${\Lambda}^{(p)}$ of close points was suggested. The subspaces spanned on the corresponding exponentials form a [Riesz basis]{}. This means that there exists an isomorphism mapping these subspaces into orthogonal ones. This fact together with Pavlov’s result on the orthoprojector $P_T$ (see Remark above) gives a criterion of the [Riesz basis]{}property of subspaces of exponentials in [$L^2(0,T)$]{}: the generating function have to satisfy the Helson–Szegö condition (\[HS\]). Note that for the particular case when the generating function is a sine type function (see definition in [@Levin61], [@KNP], [@AI95]), theorem of such a kind was proved by Levin [@Levin61]. Thus, Vasyunin’s result and Pavlov’s geometrical approach give us description of exponential Riesz bases of subspaces. If we do have a [Riesz basis]{}of subspaces, clearly, we can choose an orthonormalized basis in each subspace and obtain a [Riesz basis]{}of [*elements*]{}. However, this way is not convenient in applications when we need more explicit formulae. It is important to obtain description of Riesz bases of elements which are ‘simple and natural’ linear combinations of exponentials. The first result in this direction was obtained by D. Ullrich [@Ullrich] who considered sets ${\Lambda}$ of the form ${\Lambda}=\bigcup_{n\in {{{\mbox{\sBlackboard Z}}}}}{\Lambda}^{(n)}$, where subsets ${\Lambda}^{(n)}$ consist of equal number (say, N) real points ${\lambda}_1^{(n)},\ldots,{\lambda}_N^{(n)}$ close to $n$, i.e., $| {\lambda}_j^{(n)}-n| < {\varepsilon}$ for all $j$ and $n$. He proved that for sufficiently small ${\varepsilon}>0$ (no estimate of ${\varepsilon}$ was given) the family of particular linear combinations of exponentials $e^{i\lambda_kt}$ — the so–called [*divided differences*]{} constructed by subsets ${\Lambda}^{(n)}$ (see Definition \[gdd\_def\] in subsection \[n13\]) — forms a Riesz basis in $L^2(0,2\pi N)$. Such functions arise in numerical analysis [@Shilov], and the divided difference of $e^{i\mu t}$, $e^{i{\lambda}t}$ of the first order is $(e^{i\mu t}-e^{i{\lambda}t})/(\mu-{\lambda})$. In a sense, the Ullrich result may be considered as a perturbation theorem for the basis family $\l\{e^{int}, te^{int}, \dots, t^{N-1}e^{int}\r\}, n\in {\bf Z}$. The conditions of this theorem are rather restrictive and it can not be applied to some problems arising in control theory (see, e.g. [@BKL; @CZ; @JTZ; @LZ; @MZ]). In the present paper we generalize Ullrich’s result in several directions: the set ${\Lambda}$ is allowed to be complex, subsets ${\Lambda}^{(n)}$ are allowed to contain an arbitrary number of points, which are not necessarily ‘very’ close to each other (and, moreover, to some integer). Actually, we give a full description of Riesz bases of exponential divided differences and generalized divided differences (the last ones appear in the case of multiple points ${\lambda}_n$). To be more specific, we take a [*sequence*]{} ${\Lambda}$ which is ‘a union’ of a finite number of separated sets. Following Vasyunin we decompose ${\Lambda}$ into groups ${\Lambda}^{(p)}$, then choose for each group the family of the generalized divided differences (GDD) and prove that these functions form a [Riesz basis]{}in [$L^2(0,T)$]{}if the generating function of the exponential family satisfies the Helson–Szegö condition (\[HS\]). To prove that we show that GDD for points ${\lambda}_1, \dots, {\lambda}_N$ lying in a fixed ball form ‘a uniform basis’, i.e. the basis constants do not depend on the positions of ${\lambda}_j$ in the ball. Along with that, GDD depend on parameters analytically. Thus, this family is a natural basis for the situation when exponentials $e^{i{\lambda}t}, {\lambda}\in {\Lambda}$ do not form even uniformly minimal family. For the particular case ${\Lambda}=\{n\alpha\} \cup \{n\beta\}, n \in {\bf Z},$ appearing in a problem of simultaneous control, this scheme was realized in [@AT]. In the case when ${\Lambda}$ is not a finite union of separated sets, we present a negative result: for some ordering of ${\Lambda}$, GDDs do not form a uniformly minimal family. After this paper has been written, a result on Riesz bases of exponential DD in their span in $L^2(0,T)$ for large enough $T$ has been announced in [@MR1753294]. There, though ${\Lambda}$ is contained in ${{\mbox{\rm I\kern-.21em R}}}.$ For more general results in this direction see [@AM1; @AM2]. In a series of papers [@Vasyunin83], [@Vasyunin84], [@BNO], [@Hartmann96], [@Hartmann99]), the free interpolation problem has been studied and a description of traces of bounded analytic functions on a finite union of Carleson sets has been obtained in terms of divided differences. In view of well known connections between interpolation and basis properties, these results may be partially ([@Hartmann96], [@Hartmann99]) rewritten in terms of geometrical properties of exponential DD in [$L^2(0,{\infty})$]{}. Main Results ============ [\[n1\]]{} Let ${\Lambda}=\{{\lambda}_n\}$ be a sequence in $\CX$ ordered in such a way that $\Re {\lambda}_n$ form a nondecreasing sequence. We connect with ${\Lambda}$ the exponential family $${\ensuremath{{\mathcal E}}\xspace}({\Lambda})=\{ e^{i{\lambda_n}t}, te^{{\lambda_n}t}, \dots, t^{m_{{\lambda_n}}-1}e^{i{\lambda_n}t} \},$$ where $m_{{\lambda_n}}$ is the multiplicity of ${\lambda_n}\in {\Lambda}$. For the sake of simplicity, we confine ourselves to the case $\sup |\Im {\lambda}_n| < {\infty}$. The multiplication [operator]{}$f(t)\mapsto e^{-at}f(t)$ is an isomorphism in [$L^2(0,T)$]{}for any $T$ and maps exponential functions $e^{i{\lambda}_n t}$ to $e^{i({\lambda}_n + ia)t}$. Since we are interesting in the Riesz basis property of linear combinations of functions $t^r e^{i{\lambda}_n t}$ in [$L^2(0,T)$]{}, we can suppose without loss of generality that ${\Lambda}$ lies in a strip $S:=\{z|\,0<{\alpha}\le \Im {\lambda}\le {\beta}<{\infty}\}$ in the upper half plane. The sequence ${\Lambda}$ is called [*uniformly discrete*]{} or [*separated*]{} if condition (\[discr\]) is fulfilled. Note that in this case all points ${\lambda}$ are simple and we do not need differentiate between a sequence and a set. We say that ${\Lambda}$ [*relatively uniformly discrete*]{} if ${\Lambda}$ can be decomposed into a finite number of uniformly discrete subsequences. Sometimes we shall simply say that such a ${\Lambda}$ is a finite union of uniformly discrete sets, however we always consider a point $\lambda_n$ to be assigned a multiplicity. Splitting of the spectrum and subspaces of exponentials ------------------------------------------------------- [\[n12\]]{} Here we introduce notations needed to formulate the main result. For any ${\lambda}\in\CX$, denote by $ D_{\lambda}(r)$ a disk with center ${\lambda}$ and radius $r$. Let $G^{(p)}(r)$, $p=1,2,\dots,$ be the connected components of the union $\cup_ {{\lambda}\in {\Lambda}} D_{\lambda}(r)$. Write ${\Lambda}^{(p)}(r)$ for the subsequences of ${\Lambda}$ lying in $G^{(p)}$, ${\Lambda}^{(p)}(r):= \{{\lambda_n}|\, {\lambda_n}\in G^{(p)}(r)\} $, and ${\mathcal L}^{(p)}(r)$ for subspaces spanned by corresponding exponentials $\{t^n e^{i{\lambda}t} \}$, ${\lambda}\in {\Lambda}^{(p)}(r), n=0,\dots, m_{{\lambda}}-1$. [\[card\]]{} Let ${\Lambda}$ be a union of $N$ [uniformly discrete set]{}s ${\Lambda}_j$, $${\delta}_j:={\delta}({\Lambda}_j):=\inf_{{\lambda}\neq \mu;\;{\lambda},\mu\in{\Lambda}_j }|{\lambda}- \mu|, \ {\delta}:=\min_j {\delta}_j.$$ Then for $$r<r_0:=\frac{{\delta}}{2N}$$ the number ${\mathcal N^{(p)} }(r)$ of elements of ${\Lambda}^{(p)}$ is at most $N$. [[Proof:]{}]{}Let points $\mu_k$, $k=1,\dots,N+1$, belong to the same ${\Lambda}^{(p)}$. Then the distance between any two of these points is less than $2rN$ and so less than ${\delta}$. From the other hand, there at least two among $N+1$ points which belong to the same ${\Lambda}_j$ and, therefore, the distance between them is not less than ${\delta}$. This contradiction proves the lemma. $\quad$ ------------------------------------------------------------------------ \ We call by [*[[${\mathcal L}$-basis]{}]{}*]{} a family in a Hilbert space which forms a [Riesz basis]{}in the closure of its linear span. The following statement is a small modification of the theorem of Vasyunin [@Vasyunin78]. [\[Vasyunin78\]]{} Let ${\Lambda}$ be a [relatively uniformly discrete sequence]{}. Then for any $r>0$ the family of subspaces ${\mathcal L}^{(p)}(r)$ forms an [[${\mathcal L}$-basis]{}]{}in [$L^2(0,{\infty})$]{}. This statement is proved in [@Vasyunin78] and [@Nik Lect. IX] for $r=r_0/16$ and for disks in so called hyperbolic metrics, however one can easily check that the proof with obvious modifications remains valid for all $r$. In applications we often meet the case of real ${\Lambda}'s$. Then a relatively discrete set can be characterized using another parameter than in Lemma \[card\]. The following statement can be easily proved similar to Lemma \[card\]. [\[delta\]]{} A real sequence ${\Lambda}$ is a union of $N$ [uniformly discrete set]{}s ${\Lambda}_j $ if and only if $\inf_n ({\lambda}_{n+N}- {\lambda}_n):= {\tilde}{\delta}>0$ for all $n$. Along with that, $\min_j {\delta}({\Lambda}_j)\le {\tilde}{\delta}$ Divided differences ------------------- [\[n13\]]{} Let $\mu_k$, $k=1,\dots,m$, be arbitrary complex numbers, not necessarily distinct. [\[gdd\_def\]]{} Generalized divided difference (GDD) of order zero of $e^{i\mu t}$ is $[\mu_1](t):=e^{i\mu_1 t}$. GDD of the order $n-1$, $n\le m$, of $e^{i\mu t}$ is $$[\mu_1,\dots,\mu_n]:= \cases{{\displaystyle}\frac{[\mu_1,\dots,\mu_{n-1}]-[\mu_2,\dots,\mu_n] }{\mu_1-\mu_n}, &$\mu_1\ne\mu_n$,\cr \frac{\partial}{\partial \mu}[\mu,\mu_2,\dots,\mu_{n-1}]\Big |_{\mu= \mu_1}, &$\mu_1=\mu_n$. }$$ If all $\mu_k$ are distinct one can easily derive the explicit formulae for GDD: [[$${\label{dd}} [\mu_1,\dots,\mu_n]=\sum_{k=1}^n \frac{e^{i\mu_k t}}{\prod_{j \neq k} (\mu_k - \mu_j)}.$$]{}]{} For any points $\{\mu_k\}$ we can write [@Shilov p. 228] $$\begin{aligned} {\label{dd_int}} \nonumber [\mu_1,\dots,\mu_n]= \int_0^1d\tau_1 \int_0^{\tau_1}d\tau_2 \dots \int_0^{\tau_{n-2}}d\tau_{n-1} (it)^{n-1}\\ \exp\Big( it \l[ \mu_1+\tau_1(\mu_2-\mu_1) +\dots + \tau_{n-1}(\mu_n-\mu_{n-1})\r]\Big).\end{aligned}$$ [\[gdd\_1\]]{} The following statements are true. \(i) Functions ${\varphi}_1:=[\mu_1],\dots,{\varphi}_n:=[\mu_1,\dots,\mu_n]$, depend on parameters $\mu_j$ continuously and symmetrically. If points $\mu_1,\dots,\mu_n$ are in a convex domain $\Omega\subset\CX,$ then [[$${\label{iii}} |{\varphi}_j(t)|\le c_n e^{{\gamma}t}, {\gamma}:=-\inf_{z\in\Omega}\Im z.$$]{}]{} \(ii) Functions ${\varphi}_1,\dots,{\varphi}_n$, are linearly independent. \(iii) If points $\mu_1,\dots,\mu_n$ are distinct, then the family of GDD’s forms a basis in the span of exponentials $e^{i\mu_1t},\dots,e^{i\mu_n t}$. \(iv) Translation of the set $\mu_1,\dots,\mu_n$ leads to multiplying of GDD’s by an exponential: $$[\mu_1+{\lambda},\dots,\mu_n+{\lambda}]=e^{i{\lambda}t}[\mu_1,\dots,\mu_n].$$ \(v) For any ${\varepsilon}>0$ and any $N\in {\mbox{I\kern-.21em N}}$, there exists ${\delta}$ such that the estimates $$||[\mu_1,\dots,\mu_j](t)-e^{i\mu t} t^{j-1}/(j-1)!||_{{\ensuremath{L^2(0,{\infty})}\xspace}} < {\varepsilon},\ j=1,\dots,N,$$ are valid for any $\mu$ in the strip $S$ and all points $ \mu_1, \ldots , \mu_N$ belonging to the disk $D_\mu({\delta})$ of radius ${\delta}$ with the center at $\mu$. Bases of elements ----------------- [\[n14\]]{} Let ${\Lambda}^{(p)}(r)=\left\{{\lambda}_{j,p}\right\}, j=1,\ldots,{\mathcal N^{(p)} }(r)$ be subsequences described in subsection \[n12\] Denote by $\{{\ensuremath{{\mathcal E}}\xspace}^{(p)}({\Lambda},r) \}$ the family of GDD corresponding to the points ${\Lambda}^{(p)}(r)$: $${\ensuremath{{\mathcal E}}\xspace}^{(p)}({\Lambda},r)=\{ [{\lambda}_{1,p}],[{\lambda}_{1,p},{\lambda}_{2,p}],\dots, [{\lambda}_{1,p}, \dots, {\lambda}_{{\mathcal N^{(p)} },p}] \}.$$ Note that ${\ensuremath{{\mathcal E}}\xspace}^{(p)}({\Lambda},r)$ depends on enumeration of ${\Lambda}^{(p)}(r)$, although every GDD depends symmetrically on its parameters, see the assertion (i) of the last theorem. [\[main\_21\]]{} Let ${\Lambda}$ be a relatively uniformly discrete sequence and $r<r_0$. Then \(i) the family $\{{\ensuremath{{\mathcal E}}\xspace}^{(p)}({\Lambda},r)\}$ forms a [Riesz basis]{}in $L^2(0,T)$ [if and only if ]{} there exists an entire function $F$ of exponential type with indicator diagram of width $T$ and zeros at points ${\lambda_n}$ multiplicity $m_{{\lambda_n}}$ [(the generating function of the family ${\ensuremath{{\mathcal E}}\xspace}({\Lambda})$ on the interval $(0,T)$)]{} such that, for some real $h$, the function $\left|F(x+ih)\right|^2$ satisfies the Helson–Szegö condition (\[HS\]); \(ii) for any finite sequence $\{a_{p,j}\}$ the [[inequality]{}]{}$$\|\sum_{p,j}a_{p,j}e^{i{\lambda}_{j,p} t}\|^2_{L^2(0,T)} \geq C \sum_{p,j}|a_{p,j}|^2{\delta}_p^{2({\mathcal N^{(p)} }(r) -1)}$$ is valid with a constant $C$ independent of $\{a_{p,j}\}$, where $${\delta}_p:=\min\{|{\lambda}_{j,p} - {\lambda}_{k,p} |\, \Big |\,k\ne j \}.$$ Suppose now that ${\Lambda}$ is not a relatively uniformly discrete sequence. Then, $$\sup_p{\mathcal N^{(p)} }(r)={\infty}\ \ \mbox{ for any} \ \ r>0.$$ It is possible also that there is an infinite set ${\Lambda}^{(p)}$. We show that in this case the family of GDDs is not uniformly minimal even in ${\ensuremath{L^2(0,{\infty})}\xspace}$ at least, for some enumeration of points. [\[nonrel\]]{} For any $r>0$, there exists numbering of points in the subsequences ${\Lambda}^{(p)}$ such that family $\{{\ensuremath{{\mathcal E}}\xspace}^{(p)}({\Lambda},r)\}$ is not uniformly minimal in $L^2(0,{\infty})$. Moreover, for any ${\varepsilon}$, there exists a pair ${\varphi}_m,{\varphi}_{m+1}$ of GDD, corresponding to some set ${\Lambda}^{(p)}(r)$ such that $${\mathrm{angle}\,}_{{\ensuremath{L^2(0,{\infty})}\xspace}}({\varphi}_m,{\varphi}_{m+1})<{\varepsilon}.$$ Application to an observation problem ------------------------------------- [\[appl\]]{} Before starting the proofs of main results, we present an application of Theorem \[main\_21\] to observability of a coupled 1d system studied in [@KomornikL00]: $$\begin{cases} { \frac{{\partial}^2}{{\partial}t^2}u_1- \frac{{\partial}^2}{{\partial}x^2}u_1 +Au_1+Bu_2=0 & in $(0,\pi)\times{{\mbox{\rm I\kern-.21em R}}}$,\cr \frac{{\partial}^2}{{\partial}t^2}u_2+\frac{{\partial}^4}{{\partial}x^4}u_2 +Cu_1+Du_2=0 & in $(0,\pi)\times{{\mbox{\rm I\kern-.21em R}}}$,\cr u_1=u_2=\frac{{\partial}^2}{{\partial}x^2}u_2=0 & for $ x=0,\pi$,\cr u_1=y_0\in H^1_0(0,\pi),\ \frac{{\partial}}{{\partial}t}u_1=y_1\in L^2(0,\pi), & for $ t=0,$ \cr u_2=\frac{{\partial}}{{\partial}t}u_2=0 & for $ t=0$ \cr }\end{cases}$$ ($A$, $B$, $C$, $D$ are constants). We introduce the initial energy $E_0$ of the system, $E_0:=\|y_0\|^2_{H^1} +\|y_1\|^2_{L^2}$. In the paper of C. Baiocchi, V. Komornik, and P. Loreti [@KomornikL00] the partial observability, i.e. inequality [[$${\label{obs}} \l\Vert\frac{{\partial}}{{\partial}x} u_1(0,t)\r\Vert_{L^2(0,T)}^2 \geq c E_0$$]{}]{} with a constant $c$ independent of $y_0$ and $y_1$, has been proved for almost all 4–tuples $(A,B,C,D)$ and for $T>4\pi$. (It means that we can recover the initial state via the observation $\l\Vert\frac{{\partial}}{{\partial}x} u_1(0,t)\r\Vert_{L^2(0,T)}^2$ during the time $T$ and the operator: [*observation $\to$ initial state* ]{} is bounded. The authors conjectured the system is probably partially observed for $T>2\pi$. Here we demonstrate this fact using the basis property of exponential DD. [\[p\_obs\]]{} For almost all 4–tuples $(A,B,C,D)$ and for $T>2\pi$ the estimate (\[obs\]) is valid. To prove this proposition we use the representation and properties of the solution given in the paper [@KomornikL00]. To apply the Fourier method we introduce the eigenfrequencies ${\omega}_k$, $\nu_k$, $k\in {\mbox{I\kern-.21em N}}$ of the system, where $\nu_k^2$ and ${\omega}_k^2$ are the eigenvalues of the matrix $$\l({ \begin{array}{cc} k^2+A & B \cr C & k^4 +D \end{array} } \r) .$$ It is easy to see that the following asymptotic relations are valid: [[$${\label{asymp}} \nu_k=k+A/2k +O(k^{-3}), \ {\omega}_k=k^2+D/2k^2 +O(k^{-6}).$$]{}]{} We suppose that all ${\omega}_k$ and $\nu_k$ are distinct (this is true for almost all 4-tuples). Then the first component of the solution of the system can be written in the form [[$${\label{wave}} u_1(x,t)=\sum_{k\in {{\bf K}}} \l[{\alpha}_ke^{i{\omega}_kt} +{\beta}_ke^{i\nu_kt}\r] \sin kx,$$]{}]{} where $ {{\bf K}}:={{\mbox{\sf Z\kern-.45em Z}}}\backslash \{0\}, \ {\omega}_{-k}:=-{\omega}_k, \ \nu_{-k}:=-\nu_k, $ The coefficients ${\alpha}_k$, ${\beta}_k$ entered the last sum can be expressed via the initial data, and the authors of [@KomornikL00] show that under zero initial condition for $u_2$, we have [[$${\label{KL}} |{\alpha}_k|^2 + |{\alpha}_{-k}|^2\prec k^{-8}(|{\beta}_k|^2 + |{\beta}_{-k}|^2)$$]{}]{} and [[$${\label{energy}} E_0\asymp \sum_{k\in {{\bf K}}} k^2 |{\beta}_k|^2.$$]{}]{} Relations (\[KL\]) and (\[energy\]) mean, correspondingly, one–sided and two–sided inequalities with constants which do not depend on sequences $\{{\alpha}_k\}$ and $\{{\beta}_k\}$. We now have to study the exponential family $${\ensuremath{{\mathcal E}}\xspace}=\{e^{i{\lambda}t}\}_{{\lambda}\in {\Lambda}},\ {\Lambda}= M \cup {\Omega},\ M=\{\nu_k\}, \ {\Omega}=\{{\omega}_k\}, \ k\in {{\bf K}}.$$ For $r<1$ and for large enough $|{\lambda}|$ the family ${\Lambda}^{(p)}(r)$ consist of one point $\nu_k$ if k is not a full square or of two points $\nu_{{\mathrm{sign }\,}k\,k^2}$ and ${\omega}_k$. Denote by ${\ensuremath{{\mathcal E}}\xspace}^{(p)}({\Lambda},r)$ the related family of exponential DD. Suppose for a moment that there exists an entire function $F$ of exponential type with indicator diagram of width $T>2\pi$ vanishing at ${\Lambda}$ ($F$ may also have another zeros) such that $|F(x+ih)|^2$ satisfies the Helson–Szegö condition (\[HS\]) for some real $h$. Then, by Theorem \[main\_21\], the family ${\ensuremath{{\mathcal E}}\xspace}^{(p)}({\Lambda},r)$ forms an [[${\mathcal L}$-basis]{}]{}in [$L^2(0,T)$]{}. Expanding exponentials in DD of the zero and first order, $${\alpha}e^{i{\lambda}t}+{\beta}e^{i\mu t}=({\alpha}+{\beta})e^{i{\lambda}t}-{\beta}({\lambda}-\mu)[{\lambda},\mu],$$ for finite sequences $\{p_k\}$, $\{q_k\}$ we obtain the estimate [[[$${\label{2pi}} {{\begin{array} {rl} {\displaystyle}\l\Vert\sum_{k\in {{\bf K}}} \l[ p_ke^{i{\omega}_kt} +q_ke^{i\nu_kt}\r] \r\Vert^2_{L^2(0,T)} \asymp {\displaystyle}\sum_{k\in {{\bf K}}, \sqrt{|k|}\notin {\bf N} } |q_k|^2+\\ \sum_{k\in {{\bf K}}}\l[|p_k + q_{{\mathrm{sign}\,} k\,k^2}|^2 + |{\omega}_k-\nu_{{\mathrm{sign}\,} k\,k^2}|^2 |q_{{\mathrm{sign}\,} k\,k^2}|^2\r]. \end{array}}}$$]{}]{}]{} In the right hand side of this relation the first sum corresponds to one dimensional subspaces ${\mathcal L}^{(p)}(r)$ (this sum is taken for all $k$ which are not full squares). The second sum corresponds to two dimensional subspaces. Together with (\[wave\]), it follows $$\begin{aligned} \l\Vert \frac{{\partial}}{{\partial}x}u_1(0,t)\r\Vert^2_{{\ensuremath{L^2(0,T)}\xspace}}=\l\Vert \sum_{k\in {{\bf K}}}k \l[{\alpha}_ke^{i{\omega}_kt} +{\beta}_ke^{i\nu_kt}\r]\r\Vert_{{\ensuremath{L^2(0,T)}\xspace}}^2 \\ \succ \sum_{k\in {{\bf K}}, \sqrt{|k|}\notin {\bf N} } k^2 |{\beta}_k|^2 + \sum_{k}\l[ |{\omega}_k-\nu_{{\mathrm{sign}\,} k\,k^2}|^2|k^2{\beta}_{{\mathrm{sign}\,} k\,k^2}|^2+ \l|k{\alpha}_k+k^2{\beta}_{{\mathrm{sign}\,} k\,k^2}\r|^2 \r]\,.\end{aligned}$$ For $R$ large enough using (\[KL\]) we have $$\sum_{|k|>R} \l[k{\alpha}_k+k^2{\beta}_{{\mathrm{sign}\,} k\,k^2}\r] \succ \sum_{|k|>R} k^2 |{\beta}_{{\mathrm{sign}\,} k\,k^2}|^2.$$ Thus, we obtain $$\l\Vert \frac{{\partial}}{{\partial}x}u_1(0,t)\r\Vert^2_{{\ensuremath{L^2(0,T)}\xspace}}\succ \sum_{k\in {{\bf K}}} k^2 |{\beta}_k|^2 \asymp E_0.$$ In order to complete the proof of Proposition \[p\_obs\], it remains to construct the function $F$. We do that using the following proposition which is interesting in its own right. [\[mean\]]{} Let ${\Lambda}=\{{\lambda}_n\}$ be a zero set of a sine type function (see definition in [@AI95 p. 61]), with the indicator diagram of with $2\pi$, $\{{\delta}_n\}$ a bounded sequence of complex numbers, and $F$ an entire function of the Cartwright class ([@AI95 p. 60]) with the zero set $\{{\lambda}_n+{\delta}_n\}$. If $$\lim_{N\to {\infty}}\sup_n\frac1N \Big|\Re\l({\delta}_{n+1} + {\delta}_{n+2}+\dots + {\delta}_{n+N}\r)\Big|=:d<\frac14,$$ then $F$ has the indicator diagram of with $2\pi$, and for any $d_1>d$, $h$, functions $ u, v \in L^{\infty}({{\mbox{\rm I\kern-.21em R}}}), {\Vert v \Vert}_{L^{\infty}({{{\mbox{\sBlackboard R}}}})} < 2\pi d_1 $ may be found such that $$|F(x+ih)|^2=\exp\{u(x)+{\tilde}v(x)\}$$ for any real $h$ such that $|h|>\sup |\Im ({\lambda_n}+ {\delta}_n)|$. For the proof of this assertion, one argues using [@Avd74 Lemma 1], and in a similar way to the proof of [@Avd74 Lemma 2] that there exists a sine type function with zeros $\mu_n$ such that, for any $d_1>d,$ $$d_1\Re(\mu_{n-1}-\mu_n)\leq \Re ({\lambda_n}+{\delta}_n - \mu_n)\leq d_1 \Re(\mu_{n+1}-\mu_n)\,.$$ Then Proposition \[mean\] follows directly from [@AJ Lemmas 1, 2]. We begin to construct $F$ putting $$F_1(z):=z \prod_{n\in{\bf N} }(1-\frac{z^2}{\nu_n^2}).$$ In view of the asymptotics (\[asymp\]) of $\nu_k$ and Proposition \[mean\], $F_1(z)$ may be written in the form [[$${\label{HS1}} |F_1(x+ih)|^2=\exp\{u(x)+{\tilde}v(x)\}$$]{}]{} with $ u, v \in L^{\infty}({{\mbox{\rm I\kern-.21em R}}}), {\Vert v \Vert}_{L^{\infty}({{{\mbox{\sBlackboard R}}}})} < {\varepsilon}$ for any positive ${\varepsilon}$. Further, for any fixed ${\delta}>0$, we take the function $\sin \pi {\delta}z/2$ and for every $k \in {{\bf K}}$ find the zero $2n_k/ {\delta}$ of this function which is the nearest to ${\omega}_k$. We set $$F_2(z):=z \prod_{n\in{\bf N} }(1-\frac{z^2}{\mu_n^2})\,,$$ where $$\mu_n=\cases{{\displaystyle}2n/ {\delta}&\mbox{for } $n\notin \{n_k\}$ ,\cr {\omega}_k & \mbox{for } $n= n_k$. }$$ Using again Proposition \[mean\] we conclude that for any ${\varepsilon}>0$ function $|F_2(x+ih)|^2$ has the representation similar to (\[HS1\]) and the width of the indicator diagram of $F_2$ is $2{\delta}$. Taking $F=F_1F_2$ we obtain the function with indicator diagram of width $2(\pi +{\delta})$ vanishing on ${\Lambda}$ and satisfying (\[HS\]). Proposition \[p\_obs\] is proved. Proofs ====== [\[n2\]]{} Proof of Theorem \[gdd\_1\] ----------------------------- [\[n21\]]{} \(i) Continuity on parameters $\{ \mu_k\}$ and estimate (\[iii\]) follows immediately from the representation (\[dd\_int\]). From (\[dd\]) symmetry is clear if all $\{ \mu_k\}$ are different. Then, by continuity, we obtain symmetry for any points in the sense that if $\sigma$ is a permutation of $\{1,2,\dots,n\}$, then $[\mu_1,\dots, \mu_n]= [\mu_{\sigma(1)},\dots, \mu_{\sigma(n)}]$. Now we describe the structure of GDD. Let $z_1,z_2,\dots,z_n$ be complex numbers, not necessarily different. Let us represent this set as the union of $q$ different points, $\nu_1, \dots,\nu_q$, with multiplicities $m_1, \dots,m_q$; ($m_1+m_2+\dots+m_q=n$). [\[structure\]]{} The GDD $[z_1, \dots,z_n]$ of order $n-1$ is a linear combination of functions $$t^me^{i\nu_kt},\ k=1,\dots,q,\ m=0,1,\dots,m_k-1$$ and the coefficients of the leading terms $t^{m_k-1}e^{i\nu_kt}$ are not equal to zero. [Proof]{}  of the lemma. The statement is clear for $n=1$. Let us suppose that it is true for GDD of the order $n-2$. If $z_1\ne z_n$ then, by definition, $[z_1, \dots,z_n]={\displaystyle}\frac{[z_1,\dots,z_{n-1}]-[z_2,\dots,z_n] }{z_1-z_n}$ and is a linear combination of the GDD’s of order $n-2$. Multiplicity of the point $z_n$ in the set $\{z_2,\dots,z_n\}$ is more than in the set $\{z_1,\dots,z_{n-1}\}$. Let $k$ be such a number that $z_n=\nu_k$. Then the leading term $t^{m_k-1}e^{i\nu_kt}$ appears in $[z_2,\dots,z_n]$ by the inductive conjecture and does not appear in $[z_1,\dots, z_{n-1}]$. Let $z_1= z_n$. Then $[z_1,\dots,z_{n-1}]$ contains the leading term $t^{m_1-2}e^{i\nu_1t}$. After differentiation in $z_1$ we get the leading term $t^{m_1-1}e^{i\nu_1 t}$ with nonzero coefficient. Using symmetry of GDD relative to points $z_1,...,z_n$, we complete the proof of the lemma. $\quad$ ------------------------------------------------------------------------ \ We are able now, with knowledge of the structure of GDD, to continue the proof of Theorem \[gdd\_1\]. \(ii) This assertion is the consequence of the ‘triangle’ structure of GDD: if we add $n$-th point, then a GDD of the order $n-1$, in comparison with a GDD of the order $n-2$, contains either a new exponential or a term of the form $t^me^{i\nu_pt}$ with the same frequency $\nu_p$ and larger exponent $m$. \(iii) If all points $\mu_1,\dots,\mu_n$ are different, then the family of GDD’s contains all exponentials $e^{i\mu_1t},\dots,e^{i\mu_n t}$. \(iv) Immediately follows from the definition. \(v) As well known, divided differences approximate the derivatives of the corresponding order. We use the following estimate [@Ullrich]. For any ${\varepsilon}>0$, $N\in {\mbox{I\kern-.21em N}}$, there exists ${\delta}$ such that for any set $\{ z_j \}_{j=1}^N$ belonging to the disk $D_0({\delta})$ of a radius ${\delta}$ with center at the origin, the estimates [[$${\label{Ulr}} \Big|[z_1,\dots,z_j](t)- t^{j-1}/(j-1)!\Big| < {\varepsilon},\ j=1,\dots,N,$$]{}]{} are valid for $t \in [-\pi N, \pi N]$. To proceed with (v) we choose $T$ large enough that $$\begin{aligned} {\label{v_1}} { \Big\|[\mu_1,\dots,\mu_j]\Big\|_{L^2(T,{\infty})}< {\varepsilon}/3,\ \Big\|e^{i\mu t} t^{j-1}/(j-1)!\Big\|_{L^2(T,{\infty})}< {\varepsilon}/3,\ } \nonumber \\ j=1,\dots,N,\end{aligned}$$ for all $\mu,\mu_1,\ldots ,\mu_j$ lying in the strip $S$. Set ${\varepsilon}_1={\varepsilon}/3\sqrt{T}$, and let ${\delta}$ be small enough in order to (\[Ulr\]) be fulfilled for such ${\varepsilon}_1$, $ t \in [0, T]$ and any set $\{ z_j \}_{j=1}^{N}\in D_0({\delta})$. Set $z_j:= \mu_j-\mu$. In view of (iv), we obtain $$\Big|e^{-i\mu t}[\mu_1,\dots,\mu_j](t)- t^{j-1}/(j-1)!\Big| < {\varepsilon}_1,\ j=1,\dots,N,\ t\le T,$$ that implies $$\Big|[\mu_1,\dots,\mu_j](t)- e^{i\mu t}t^{j-1}/(j-1)!\Big| < {\varepsilon}/3\sqrt{T},\ j=1,\dots,N,$$ for $t\le T$. Therefore, we have $$\Big\|[\mu_1,\dots,\mu_j](t)- e^{i\mu t}t^{j-1}/(j-1)!\Big\|_{{\ensuremath{L^2(0,T)}\xspace}} < {\varepsilon}/3,\ j=1,\dots,N.$$ Taking into account (\[v\_1\]) we obtain the statement (v). $\quad$ ------------------------------------------------------------------------ \ Proof of the Theorem \[main\_21\]. ------------------------------------ [\[n22\]]{} (i) From Lemma \[Vasyunin78\] it follows that the family $\{{\mathcal L}^{(p)}\}:=\{{\mathcal L}^{(p)}(r)\}$ forms a [Riesz basis]{}in the closure of its span (i.e. an [[${\mathcal L}$-basis]{}]{}) in $L^2(0,{\infty})$. Introduce the projector $P_T$ from this closure [*into*]{} $L^2(0,T)$. Clearly, $\{{\mathcal L}^{(p)}\}$ forms a Riesz basis in $L^2(0,T)$ [if and only if ]{}$P_T$ is an isomorphism [*onto*]{} ${\ensuremath{L^2(0,T)}\xspace}$. This takes a place [if and only if ]{}the generating function of the family ${\ensuremath{{\mathcal E}}\xspace}({\Lambda})$ satisfies the Helson-Szegö condition (\[HS\]) (see [@KNP]; [@AI95], Theorems II.3.14, II.3.17). The [[${\mathcal L}$-basis]{}]{}property of the subspaces ${\mathcal L}^{(p)}$ means that for any finite number of functions $\psi_p\in {\mathcal L}^{(p)}$ we have the estimates: [[$${\label{6.1}} \|\sum_p a_p \psi_p\|_{L^2(0,{\infty})}^2\asymp \sum_p|a_p|^2 ||\psi_p ||_{L^2(0,{\infty})}^2.$$]{}]{} This relation means that there exist two positive constants $c$ and $C$ which do not depend on a sequence $\{a_p\}$ such that $$c\|\sum_p a_p \psi_p\|_{L^2(0,{\infty})}^2\leq \sum_p|a_p|^2 ||\psi_p ||_{L^2(0,{\infty})}^2 \leq C\|\sum_p a_p \psi_p\|_{L^2(0,{\infty})}^2.$$ In each subspace ${\mathcal L}^{(p)}$ we choose the family of GDD’s corresponding to ${\lambda}_{j,p}\in {\Lambda}^{(p)}$: ${\varphi}_j^{(p)}:=[{\lambda}_{1,p},{\lambda}_{2,p},\dots,{\lambda}_{j,p}]$. $j=1,\dots, {\mathcal N^{(p)} }$. In view of Theorem \[gdd\_1\], this family forms a basis in ${\mathcal L}{^{(p)}}$. Let us expand $\psi_p$ in the basis $\{{\varphi}_j{^{(p)}}\}_{j=1}^{j={\mathcal N^{(p)} }}$. Then statement (i) of the theorem is equivalent to the estimates [[$${\label{6.2}} \|\sum_{p,j} a_{p,j}{\varphi}_j^{(p)}\|_{L^2(0,{\infty})}^2\asymp \sum_{p,j}|a_{p,j}|^2.$$]{}]{} Taking into account (\[6.1\]), we see that (\[6.2\]) is equivalent to [[$${\label{6.3}} \|\sum_j a_j{\varphi}_j^{(p)}\|_{L^2(0,{\infty})}^2\asymp \sum_j|a_j|^2, \mbox {uniformly in }p.$$]{}]{} We introduce the ${\mathcal N^{(p)} }\times{\mathcal N^{(p)} }$ Gram matrices ${\Gamma}{^{(p)}}$ corresponding to the families ${\ensuremath{{\mathcal E}}\xspace}^{(p)}$, $${\Gamma}{^{(p)}}:=\l\{({\varphi}_k^{(p)},{\varphi}_j^{(p)})_{L^2(0,{\infty})}\r\}_{k,j}.$$ For an ${\mathcal N^{(p)} }$-dimensional vector $a$ we have $$\|\sum_j a_j{\varphi}_j^{(p)}\|_{L^2(0,{\infty})}^2= \langle {\Gamma}{^{(p)}}a,a\rangle,$$ where $\langle \cdot,\cdot\rangle$ is the scalar product in ${{\mbox{\rm I\kern-.21em R}}}^{{\mathcal N^{(p)} }}$. In terms of the Gram matrices, (\[6.3\]) is true [if and only if ]{}  matrices ${\Gamma}{^{(p)}}$ and their inverses are bounded uniformly in $p$. [\[gram\]]{} Let different complex points $\mu_1,\mu_2,\dots,\mu_n$ lie in a disk $D_\mu(R)\subset \CX_+$ of radius $R$ with the center $\mu$, $\mu\in S$; ${\varphi}_1,{\varphi}_2,\dots,{\varphi}_n$ are corresponding GDD’s, and ${\Gamma}$ is the Gram matrix of this family in $L^2(0,{\infty})$. Then the norms $\langle\langle{\Gamma}\rangle\rangle$ and $\langle\langle{\Gamma}^{(-1)}\rangle\rangle$ are estimated from above by constants depending only on $R$ and $n$. [[Proof:]{}]{}By Theorem \[gdd\_1\], functions ${\varphi}_1(t),{\varphi}_2(t),\dots,{\varphi}_n(t)$ are estimated above by constants depending only on $R$, $n$. Therefore the entries of ${\Gamma}$ are estimated by ${const}\,(R,n)$ and, then, the estimate of ${\Gamma}$ from above is obtained. We shall prove the estimate for the inverse matrices by contradiction. Let us fix a disk and assume that for arbitrary ${\varepsilon}>0$ there exist different points $\mu_1^{({\varepsilon})},\mu_2^{({\varepsilon})},\dots,\mu_n^{({\varepsilon})}$ lying in the disk, and normalized $n$-dimensional vectors $a^{({\varepsilon})}$ such that for the corresponding Gram matrix ${\Gamma}^{({\varepsilon})}$ we have [[$${\label{le_eps}} \langle{\Gamma}^{({\varepsilon})}a^{({\varepsilon})},a^{({\varepsilon})}\rangle\le{\varepsilon}.$$]{}]{} Using compactness arguments we can choose a sequence ${\varepsilon}_n\to 0$ such that $$a^{({\varepsilon}_n)}\to a^0, \ \mu_j^{({\varepsilon}_n)}\to \mu_j^0.$$ as $n\to {\infty}$. Then the Gram matrix tends to the Gram matrix ${\Gamma}^0$ for the limit family and from (\[le\_eps\]) we see that $$\langle{\Gamma}^0a^{0},a^0\rangle=0.$$ Then $ \sum a^0_j{\varphi}^0_j=0 $ and the limit family of GDD’s is linearly dependent that contradicts to Theorem \[gdd\_1\]. Thus, we have proved that the norm of ${\Gamma}^{(-1)}$ is estimated from above by constants depending only on $R$, $n$, and $\mu$. In view of Theorem \[gdd\_1\](iv) the constant depends actually not on $\mu$, but on $\Im \mu$. Indeed, translation of the disk on a real number, $\mu \mapsto \mu+ x$, does not change the Gram matrix. Since ${\alpha}\leq \Im \mu \leq \beta,$ the norm of ${\Gamma}^{(-1)}$ is estimated uniformly in $\mu\in S$. $\quad$ ------------------------------------------------------------------------ \ From this lemma it follows that all Gram matrices ${\Gamma}_p$, ${\Gamma}_p^{(-1)}$ are bounded uniformly in $p$. (It was supposed in the lemma that all points $\mu_1,\mu_2,\dots,\mu_n$ are distinct. For the case of multiple points we use the continuity of DD on parameters.) The assertion (i) of the Theorem \[main\_21\] is proved. \(ii) We need to estimate the Gram matrices ${\Gamma}$ for the exponential family $\{e^{i\mu_1t},\dots,e^{i\mu_n t}\}$ in $L^2(0,T)$ from below, where $\{\mu_j\}$ is a fixed set ${\Lambda}{^{(p)}}$ and $n={\mathcal N^{(p)} }$. Since the projector $P_T$ is an isomorphism (see (i)), we may do it for exponentials on the positive semiaxis, i.e., in $L^2(0,{\infty})$. Denote by $e_j(t)$ the normalized exponentials $$e_j(t):=\frac{1}{\sqrt{2\Im \mu_j}}e^{i \mu_j t}.$$ For the Gram matrices ${\Gamma}_0$, corresponding to the normalized exponentials, we have $${\Gamma}={\mathrm{diag}\,}[\sqrt{2\Im \mu_j}] \,{\Gamma}_0\, {\mathrm{diag}\,}[\sqrt{2\Im \mu_j}],$$ and we can estimate ${\Gamma}_0$ instead ${\Gamma}$, since $|\Im\mu_j|\asymp 1$. It can be easily shown that the inverse matrix is the Gram matrix for the biorthogonal family $e'_j(t)$ and $$({\Gamma}_0)^{(-1)}_{jj}=\|e'_j\|^2 = { \mathop{\Pi}\limits_{k\ne j}^{} }{\Big |\frac{\mu_k-\bar\mu_j}{\mu_k-\mu_j}\Big |}^2$$ (see [@Nik; @AI95]). Elementary calculations give $$({\Gamma}_0)^{(-1)}_{jj}=\|e'_j(t) \|^2 \prec {\delta}_p^{-2(n-1)}$$ Then $$|({\Gamma}_0)^{(-1)}_{jk}|=(e_j',e_k') \le \|e'_j \|\|e'_k \| \prec {\delta}_p^{-2(n-1)}$$ and so, $$\langle \langle {\Gamma}_0^{(-1)} \rangle\rangle \prec {\delta}_p^{-2(n-1)}.$$ The theorem is proved. $\quad$ ------------------------------------------------------------------------ \ Proof of Theorem \[nonrel\]. ------------------------------ As well known, the family $\{t^n\}$, $n=0,1,\dots,$ of powers is not minimal in ${\ensuremath{L^2(0,T)}\xspace}$ for any $T$. Moreover, direct calculations for ${\varphi}^0_j:=e^{i\mu t} t^{j-1}/(j-1)!$ give $${\mathrm{angle}\,}_{{\ensuremath{L^2(0,{\infty})}\xspace}}({\varphi}^0_m,{\varphi}^0_{m+1})\to 0,\ \ m\to{\infty},$$ uniformly in $\mu\in S$. Since ${\Lambda}$ is not a [relatively uniformly discrete sequence]{}, for any ${\delta}>0$, $m\in {\mbox{I\kern-.21em N}}$ we are able to find a disk $D_\mu({\delta})$ in the strip $S$, which contains $m$ points of ${\Lambda}$. In view of Theorem \[gdd\_1\](v), GDD corresponding to these points are ${\varepsilon}$–close to ‘unperturbed’ functions ${\varphi}^0_j$. This proves the assertion of Theorem \[nonrel\]. [99]{} S.A. Avdonin. On [R]{}iesz bases of exponentials in ${L}^2$. , (13):5–12, 1974. (Russian); English transl. in Vestnik Leningrad Univ. Math., v. 7 (1979), 203-211. S.A. Avdonin and S.A. Ivanov. . Cambridge University Press., N. Y., 1995. S.A. Avdonin and S.A. Ivanov. evin–[G]{}olovin theorem for the [S]{}obolev spaces. , 68(2):[163–172]{}, 2000. (Russian); English transl. in Math. Notes. S.A. Avdonin and I. Joó, Riesz bases of exponentials and sine type functions. [*Acta Math. Hung.*]{}, [**51**]{} (1988), 3–14. S. Avdonin and W. Moran. Simultaneous control problems for systems of elastic strings and beams. [*Systems & Control Letters*]{}, accepted. S. Avdonin and W. Moran, Ingham type inequalities and Riesz bases of divided differences. [*J. Appl. Math. Comp. Sci*]{}, accepted. S. Avdonin and M. Tucsnak. Simultaneous controllability in short time of two elastic strings. , 2000, accepted. C. Baiocchi, V. Komornik, and P. Loreti. ngham type theorem and applications to control theory. , (8)2(1):33–63, 1999. C. Baiocchi, V. Komornik, Paola Loreti. Généralisation d’un théorème de [B]{}eurling et application à la théorie du contrôle. , 330(4):281–286, 2000. Bruna J., Nicolau A., Øyma K. Note on interpolation in the hardy spaces of the unit disk. , 4:1197–1204, 1996. A.G. Butkovsky. . Nauka, Moscow, 1965. (Russian); English transl. 1969. C. Castro and E. Zuazua. Boundary controllability of a hybrid system consisting in two flexible beams connected by a point mass. , 36:1576–1595, 1998. Hartmann A. Une approche de l’interpolation libre généralisée par la théorie des opérateurs et caractérisation des traces $H^p_{\Lambda}$. , 35(2):281–316, 1996. Hartmann A. Traces of certain classes of holomorphic functions on finite unions of carleson sequences. , 41:104–114, 1999. S. Hansen and E. Zuazua. Exact controllability and stabilization of a vibrating string with an interior point mass. , 33:1357–1391, 1995. S. Ivanov and N. Kalton. Interpolation of subspaces and applications to exponential bases. In preparation. S. Jaffard, M. Tucsnak, and E. Zuazua. Singular internal stabilization of the wave equation. , 145:184–215, 1998. S. V. Khrushchev, N. K. Nikol’skii, and B. S. Pavlov. , volume 864 of [*Lecture Notes in Math.“Complex Analysis and Spectral Theory”*]{}. Springer–Verlag, Berlin/Heidelberg, 1981. pp. 214–335. V. Komornik and P. Loreti. . , 86(1–2):218–237, 2000. B.Ya. Levin. On bases of exponential functions in ${L}_2$. , 27(Ser. 4):39–48, 1961. (Russian). Yu. Lyubarskii and K. Seip. Weighted [P]{}aley–[W]{}iener spaces. Submitted. A. Lopes and E. Zuazua. Null controllability of the $1-d$ heat equation as limit of the controllability of dissipative wave equations. , 327(8):753–758, 1998. A.M. Minkin. Reflection of exponents, and unconditional bases of exponentials. , 3(5):110–135, 1992. (Russian); English transl. in St. Petersburg Math. J., v. 3, no. 5 (1992), 1043–1068. S. Micu and E. Zuazua. Boundary controllability of a linear hybrid system arising in the control of noise. , 35:1614–1637, 1997. N. K. Nikol’skiĭ. . Springer, Berlin, 1986. B.S. Pavlov. Basicity of an exponential systems and [M]{}uckenhoupt’s condition. , 247(1):37–40, 1979. (Russian); English transl. in Soviet Math. Dokl., v. 20 (1979), 655–659. R.E.A.C. Paley and N. Wiener. , volume XIX. AMS Coll. Publ., New York, 1934. D. L. Russell. Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions. , 20(4):639–739, 1978. G. E. Shilov. . Moscow, 1965. (Russian). D. Ullrich. Divided differences and system of nonharmonic [F]{}ourier series. , 80(1):47–57, 1980. V.I. Vasyunin. Unconditionally convergent spectral decomposition and interpolation problem. , 130:5–49, 1978. In Russian, English transl. in Proc. Steklov Inst. Math., 1979, N.4,(130). V.I. Vasyunin. Traces of bounded analytic functions on finite unions of carleson sets. , 126:31–34, 1983. In Russian, English transl. in J. Soviet Math., 27:2448–2450, 1984. V.I. Vasyunin. Characterization of finite union of carleson sets in terms of solvability of interpolation problems. , 135:31–35, 1984. In Russian, English transl. in J. Soviet Math., 31:2660–2662, 1985. R.M. Young. . Academic Press, New York, 1980. [^1]: This work was partially supported by the Australian Research Council (grant \# A00000723) and by the Russian Basic Research Foundation (grant \# 99-01-00744). [^2]: Department of Applied Mathematics and Control, St.Petersburg State University, Bibliotechnaya sq. 2, 198904 St.Petersburg, Russia and Department Mathematics and Statistics, The Flinders University of South Australia, GPO Box 2100, Adelaide SA 5001, Australia; email: avdonin@ist.flinders.edu.au [^3]: Russian Center of Laser Physics, St.Petersburg State University, Ul’yanovskaya 1, 198904 St.Petersburg, Russia; e-mail: sergei.ivanov@pobox.spbu.ru

--- abstract: 'A new scheme to produce very low emittance muon beams using a positron beam of about 45 GeV interacting on electrons on target is presented. One of the innovative topics to be investigated is the behaviour of the positron beam stored in a low emittance ring with a thin target, that is directly inserted in the ring chamber to produce muons. Muons can be immediately collected at the exit of the target and transported to two $\mu^+$ and $\mu^-$ accumulator rings and then accelerated and injected in muon collider rings. We focus in this paper on the simulation of the e$^+$ beam interacting with the target, the effect of the target on the 6-D phase space and the optimization of the e$^+$ ring design to maximize the energy acceptance. We will investigate the performance of this scheme, ring plus target system, comparing different multi-turn simulations. The source is considered for use in a multi-TeV collider in ref. [@DAgnolo:2268140].' author: - 'M. Boscolo' - 'M. Antonelli' - 'O.R. Blanco-García' - 'S. Guiducci' - 'S. Liuzzo' - 'P. Raimondi' - 'F. Collamati' title: Low emittance muon accelerator studies with production from positrons on target --- \[sec:level1\]Introduction ========================== Muon beams are customarily obtained via $K/\pi$ decays produced in proton interaction on target. A complete design study using this scheme, including the muon cooling system has been performed by the Muon Accelerator Program [@map; @map_ipac14]. In this paper we will investigate the possibility to produce low emittance muon beams from a novel approach, using electron-positron collisions at a centre-of-mass energy just above the $\mu^{+}\mu^{-}$ production threshold with minimal muon energy spread, corresponding to the direct annihilation of approximately 45 GeV positrons and atomic electrons in a thin target, O(0.01$\sim$ radiation lengths). Concept studies on this subject are reported in Refs. [@NIM; @IPAC17]. A feasibility study of a muon collider based on muon electro-production has been studied in Ref. [@Barletta:1993rq]. The most important key properties of the muons produced by the positrons on target are: - the low and tunable muon momentum in the centre of mass frame; - large boost, being about $\gamma\sim$200. These characteristics result in the following advantages: - the final state muons are highly collimated and have very small emittance, overcoming the need to cool them; - the muons have an average laboratory lifetime of about 500 $\mu$s at production. - the muons are produced with an average energy of 22 GeV easing the acceleration scheme. The use of a low emittance positron beam on target allows the production of low emittance muon beam and, unlike previous designs, muon cooling would not be necessary.\ The very small emittance could enable high luminosity with smaller muon fluxes, reducing both the machine background in the experiments and more importantly the activation risks due to neutrino interactions. ![Dose equivalent due to neutrino radiation at a collider depth of 100 m for a muon rate of $3\cdot 10^{13}$ s$^{-1}$ as taken from [@rolandi_silari]. Contributions from the straight section and from the arc are shown in red and blue, respectively.[]{data-label="radiolimit"}](fig01_new.pdf){width="230pt"} Figure \[radiolimit\] shows the dose equivalent due to neutrino radiation as taken from [@rolandi_silari]. It shows an increase of the dose equivalent with the muon beam energy posing a severe limit on the centre of mass energy reach of the muon collider. It has been obtained for a muon rate of $3 \cdot 10^{13}$s$^{-1}$ and a collider depth of 100 m. Such a muon rate is problematic for muon collider operations above about 6 TeV centre of mass energy [@map]. Ref. [@map] chose a collider depth of 500 m for a 6 TeV collider scenario. Figure \[radiolimit\] and Ref. [@rolandi_silari] provides a first crude approximation of the dose effect, and a more detailed calculation with consideration of mitigation strategies is needed for precise limit. To enable higher centre of mass energies the muon rate has to be reduced, thus competitive luminosity performance must be obtained by reducing the beam emittance. The current muon collider design studied by MAP [@map] foresees a normalized emittance of $25~\rm \mu m$ with a luminosity of about 10$^{35}$ cm$^{-2}$ s$^{-1}$ at 6 TeV. The muons per bunch are $2 \cdot 10^{12}$ and the muon production rate is $3 \cdot 10^{13}$s$^{-1}$. The aim of the study is to obtain comparable luminosity performances with lower fluxes and smaller emittances allowing operation at a higher centre of mass energy. One possibility is to accommodate the accelerator complex in the CERN area in existing tunnels. A preliminary proposal for a 14 TeV CERN muon collider has been described in ref [@DAgnolo:2268140]. Muon Production --------------- The cross section for continuum muon pair production $e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}$ has a maximum value of about 1 $\mu$b at $\sqrt{s} \sim$ 0.230 GeV. In our proposal these values of $\sqrt{s}$ can be obtained from fixed target interactions with a positron beam energy of $$E(e^{+})~\approx \frac{s}{2~m_{e}} \approx 45 ~\hbox{GeV}$$ where $m_{e}$ is the electron mass, with a boost of $$\gamma \approx \frac{E(e^{+})}{\sqrt{s}} \approx \frac{\sqrt{s}}{2~m_{e}} \approx 220.$$ The maximum scattering angle of the outgoing muons $\theta_{\mu}^{max}$ depends on $\sqrt{s}$. In the approximation of $\beta_{\mu}=1$, $$% \begin{aligned} \theta_{\mu}^{max}= \frac{4~m_{e}}{s} \sqrt{\frac{s}{4}-m_{\mu}^{2} } % \end{aligned}$$ where $\beta_{\mu}$ is the muon velocity. Muons produced with very small momentum in the rest frame are well contained in a cone of about $5\cdot 10^{-4}$ rad for $\sqrt{s}$=0.212 GeV, the cone size increases to $\sim 1.2\cdot 10^{-3} $ rad at $\sqrt{s}$=0.220 GeV. The difference between the maximum and the minimum energy of the muons produced at the positron target ($\Delta E_{\mu}$) also depends on $\sqrt{s}$, and with the $\beta_{\mu}=1$ approximation we get: $$\Delta E_{\mu} =\frac{\sqrt{s}}{2m_{e}} \sqrt{\frac{s}{4}-m_{\mu}^{2}}$$ The $RMS$ energy distribution of the muons increases with $\sqrt{s}$, from about 1 GeV at $\sqrt{s}$=0.212 GeV to  3 GeV at $\sqrt{s}$=0.220 GeV. Target options -------------- The number of $\mu^{+}\mu^{-}$ pairs produced per positron bunch on target is: $$\label{eq:nmu} n(\mu^{+} \mu^{-}) = n^{+}~\rho^{-}~L~\sigma(\mu^{+} \mu^{-})$$ where $n^{+}$ is the number of positrons in the bunch, $\rho^-$ is the electron density in the medium, $L$ is the thickness of the target, and $\sigma(\mu^{+} \mu^{-})$ is the muon pairs production cross section. The dominant process in $e^+$ $e^-$ interactions at these energies is collinear radiative Bhabha scattering, with a cross section of about $\sigma_{rb} \approx \rm 150~mb$ for a positron energy loss larger than 1%. This sets the value of the positron beam interaction length for a given pure electron target density value. So, it is convenient to use targets with thickness of at most one interaction length, corresponding to $L = 1/(\sigma_{rb}~\rho^-)$: $$(\rho^-~L)_{max} = 1/\sigma_{rb}\approx 10^{25} \hbox{cm}^{-2}$$ The ratio of the muon pair production cross section to the radiative bhabha cross section determines the maximum value of the [*muon conversion efficiency*]{} $\rm{eff}(\mu^+\mu^-)$ that can be obtained with a pure electrons target. In the following we will refer to $\rm{eff}(\mu^+\mu^-)$ defined as the ratio of the number of produced $\mu^+\mu^-$ pairs to the number of the incoming positrons. One can easily see that the upper limit of $\rm{eff}(\mu^+\mu^-)$ is of the order of $10^{-5}$, so that: $$n(\mu^+ \mu^-)_{max}\approx n^+~10^{-5}.$$ Electromagnetic interactions with nuclei are dominant in conventional targets. In addition, limits are present for the target thickness not to increase the muon beam emittance $\epsilon_{\mu}$. Assuming a uniform distribution in the transverse $x-x'$ plane the emittance contribution due to the target thickness is proportional to the target length and to the maximum scattering angle of the outgoing muons: $$\epsilon_{\mu}\propto{L~(\theta_{\mu}^{max})^{2}}$$ The number of $\mu^+\mu^-$ pairs produced per crossing has the form given by equation \[eq:nmu\], with $$\rho^-=N_A/A~\rho~Z$$ where $Z$ is the atomic number, $A$ the mass number, $N_A$ Avogadro’s constant, and $\rho$ the material density. In addition, the multiple scattering contributes to the emittance increase according to: $$\theta_{\rm{r.m.s}} \sim \frac{13.6~\rm{MeV} }{\beta cp}\sqrt{x/X_0}[1+0.038~\ln(x/X_0)] \label{MSscat}$$ where $p$ and $\beta c$ are the positron velocity and momentum, respectively, $x/X_0$ is the target thickness in radiation lengths. Similarly: $$x_{\rm{r.m.s.}}\sim \theta_{\rm{r.m.s}} ~0.5~L~\sqrt{3}.$$ The bremsstrahlung process governs the positron beam degradation in this case and it scales with the radiation length. Roughly, the cross section per atom increases by a factor (Z+1) with respect to the case of a pure electron target. On one side to minimize the emittance there is the need of a small length $L$ (thin target), on the other side compact materials have typically small radiation lengths causing an increase of the emittance due to multiple scattering and a fast positron beam degradation due to bremsstrahlung. Positron interactions on different targets have been studied with [`GEANT4`]{} [@geant]. The results show that the optimal target has to be thin and not too heavy [@NIM]. Carbon (C), Beryllium (Be) and Liquid Lithium (LLi) have the best performances. A 3 mm Beryllium target provides a muon conversion efficiency $\rm eff(\mu^+\mu^-)$ of about $0.7\cdot 10^{-7}$. The muon transverse phase space at the target exit, as produced by a 45 GeV positron beam, is shown in Figure \[f2\], assuming negligible emittance of the positron beam. ![Transverse phase space distribution of muons at the target exit (GEANT4). The positron beam impinging on the 3 mm Be target has negligible emittance.[]{data-label="f2"}](fig02){width="230pt"} A transverse emittance as small as $0.19\cdot 10^{-9}$ m-rad is observed for the outgoing $\sim$22 GeV muons; it represents the ultimate value of emittance that can be obtained with such a scheme.\ The very low muon production efficiency, due to the low value of the production cross section, makes convenient a scheme where positrons are recirculated after the interaction on a thin target. A 3 mm Beryllium target is considered in our studies. A preliminary layout of Low EMittance Muon Accelerator (LEMMA) is shown in Figure \[scheme\]. This scheme foresees muons produced by the interaction of positrons circulating in a storage ring on target at $\sim$ 22 GeV, then muons are accumulated in isochronous rings with a circumference of $\sim$60 m with 13 T dipoles. In addition to muons, high intensity and high energy photons are produced from the interaction of the positron beam with the target. The possibility of using these photons for an embedded positron source is extremely appealing and will be briefly described in Section \[posiso\]. However, a solution to transform the temporal structure of produced positrons to be used for the ring injection has not been found yet. ![Schematic layout for low emittance muon beam production: positron source with adiabatic matching device (AMD), $e^{+}$ ring plus target (T) for $\mu^{+}$-$\mu^{-}$ production, two $\mu^{+}$-$\mu^{-}$ accumulator rings (AR), and fast acceleration section to be followed by the muon collider.[]{data-label="scheme"}](fig03_new.png){width="230pt"} This innovative scheme has many key topics to be investigated: - a low emittance high energy acceptance 45 GeV positron ring, - O(100 kW) class target, - high momentum acceptance muon accumulator rings, - high rate positron source. A first design of a 45 GeV positron ring with low emittance and high momentum acceptance will be described in the following. The effects on beam parameters due to the target insertion will be analysed from the point of view of the positron beam lifetime and its degradation. In particular, an attempt to the control the positron emittance growth will be described. Our goal is to preserve as much as possible the ultimate value of normalized emittance, $\epsilon_\mu$=40 nm, obtained for a 3 mm Be target and to obtain a positron beam lifetime of about 250 turns. Additional emittance dilution effects due to the muon accumulation are the subject of future studies. Positron storage ring {#optics} ===================== A 45 GeV low emittance positron storage ring has been designed and the effect of the target on the beam properties has been studied. The main processes affecting the beam sizes in the target are bremsstrahlung and multiple Coulomb scattering. The effects of these two contributions have been studied separately, finding that the best location for the target corresponds to a low-$\beta$, dispersion-free region. The ring is composed of 32 lattice cells of 197 m each for a total length of 6 km with the parameters shown in Table \[tab:ringpara\]. The cell shown in Figure \[cell\] is based on the Hybrid Multi-Bend Achromat (HMBA) [@doi:10.1080/08940886.2014.970931] to minimize emittance, while keeping large momentum and dynamic acceptance. The maximum dipole field is 0.26 T, the filling factor is 77%. The maximum quadrupole, sextupole and octupole gradients are 110 T/m, 340 T/m$^2$ and 5900 T/m$^3$ respectively. The horizontal and vertical phase advance, in $2\pi$ units, between sextupoles at the peak of the horizontal dispersion is 1.5 and 0.5 respectively. Sixty-four RF cavities have been considered, each RF cavity is 5.4 m long and composed by 9 RF cells of about 7 MV/m accelerating gradient. Transverse unnormalized emittance is 5.73 $\times ~10^{-9}$ and longitudinal emittance 3 $\mu m$; then, as the positron beam passes through the target, emittance is increased as discussed later, see Figure \[horbeam\]. Longitudinal emittance is increased due to a large elongation effect due to the bremsstrahlung effect in the passage of the target. [\*[3]{}[l]{}]{} Parameter & Units &\ Energy & GeV & 45\ Circumference (32 ARCs, no IR) & m & 6300.960\ Geometrical emittance x, y & m & 5.73 $\times ~10^{-9}$\ Bunch length & mm & 3\ Beam current & mA & 240\ RF frequency & MHz & 500\ RF voltage & GV & 1.15\ Harmonic number & \# & 10508\ Number of bunches & \# & 100\ N. particles/bunch &\# & 3.15 $\times 10^{11}$\ Synchrotron tune & & 0.068\ Transverse damping time & turns & 175\ Longitudinal damping time & turns & 87.5\ Energy loss/turn & GeV & 0.511\ Momentum compaction & & 1.1 $\times 10^{-4}$\ RF acceptance & % & $\pm$ 7.2\ Energy spread & dE/E & 1 $\times 10^{-3}$\ SR power & MW & 120\ ![Optical functions of one cell of the 45 GeV e$^+$ ring. $\beta_{x}$ and $\beta_{y}$ are shown in blue and dotted red, respectively; horizontal dispersion function $\eta_{x}$ is plotted in yellow with values on the right y axis.[]{data-label="cell"}](fig04.pdf){width="230pt"} The effect of the target on the positron beam has been studied for different locations in the cell for different materials and thicknesses. This simulation has been divided in two parts: particle tracking and positron interaction with target. Particle tracking in the ring is performed with Accelerator Toolbox (AT) [@AT] and MAD-X PTC [@MADX], while positron interaction with the target is studied with either GEANT4 or FLUKA [@FLUKA]. The cycle starts by the generation of a particle distribution from the equilibrium emittances of the ring. These particles are tracked through the ring for one turn and the particle distribution is modified according to simulations to account for the passage through the target. Radiation damping is included in the simulation. Single turn tracking and target interaction are then repeated for a given number of turns. Coulomb scattering in the target changes the beam divergence and the beam size. The multiple Coulomb scattering contribution is normally distributed and uncorrelated with the beam, therefore it only depends on the target. It can be estimated as in eq. \[MSscat\]. As the beam passes through several times, the contribution from multiple scattering (MS) increases as $$\sigma_{x',y'}(MS) = \sqrt{N} ~\theta_{\rm{r.m.s}} \label{eq:10}$$ where $N$ is the turn number. Therefore, the beam divergence per turn is given by $$\sigma_{x',y'}(N) = \sqrt{\sigma^{2}_{x',y'}(MS)+\sigma^{2}_{x',y'}(0)} \label{eq:11}$$ where $\sigma_{x',y'}(0)$ is the unperturbed beam divergence in the two transverse planes. Similar effects have been studies for ionization cooling [@ionization_cooling] and for ion stripping [@strip]. While the contribution to the divergence from multiple scattering is completely determined by the target, the contribution to the beam size is expected to be proportional to the $\beta$-function at the target location. At a waist the beam width after $N$ machine turns is given by $$\sigma_{x,y}(N) = \beta_{x,y}~\sigma_{x',y'}(N).$$ This explains how a strong reduction of the beam size growth is obtained by placing the target in a low-$\beta$ location, such as the interaction point in a collider ring. To visualize this effect, Figure \[multiscat\] shows the beam size and divergence for a positron beam interacting with a 3 mm thick Beryllium target at each turn. In the case of $\beta$ value at the target (waist) equal to 1.5 m the simulated increase in the beam divergence is $\theta_{\rm{r.m.s}}\sim$25 $\mu$rad at each single pass. Lower $\beta$ functions evidently reduce this effect. Minimization of the emittance growth is obtained by placing the target in a beam waist and setting the angular contribution from the multiple scattering similar to or smaller than the beam divergence [@HWANG2014153]. This matching reduces beam filamentation [@Mohl:1005037]. Bremsstrahlung in the target causes the particles to lose energy, degrading the beam emittance and reducing its lifetime. For a 3 mm thick target of Beryllium we obtain, from tracking with MAD-X PTC, a life time between 37 and 40 turns, equivalent to 0.8 ms, in good agreement with AT results. Figure \[lifetime\] shows the number of particles per turn for several target thicknesses from which it is concluded that the beam lifetime and target thickness are inversely proportional, as expected. ![Number of e$^+$ versus machine turns for various Be target thicknesses (MAD-X PTC). For a 3 mm Be target (light blue line) lifetime is about 40 turns.[]{data-label="lifetime"}](fig06_new.pdf){width="8cm"} Beam lifetime slightly increases due to radiation damping. For a 3 mm Be target the effect is negligible because the ring longitudinal damping time is approximately twice the 40 turns we have shown, and the transverse damping time is almost four times larger. With a Be target of 500 $\mu$m the increase in beam lifetime amounts to 8% of the 140 turns obtained from simulations with MAD-X, is as shown in Figure \[damping\]. ![Beam lifetime (top) and horizontal beam size (bottom) with and without the damping effect in the tracking simulation including a 0.5 mm thick Be target.[]{data-label="damping"}](fig07_new.pdf){width="8.cm"} Lifetime is limited by the energy loss, thus it is important to maximise the ring momentum acceptance. Figure \[moma\] shows that the cell without errors reaches 8% of momentum acceptance, obtained from particle tracking along the ring with three different tracking codes: AT, MAD-X and MAD-X PTC. Good agreement is found between the codes. One of the ring cells has been modified to obtain a low-$\beta$ interaction region to place the target at its center similarly to the interaction point of a collider. Figure \[IRcell\] shows the optics and layout of the current target insertion region and 25 m of the cell. At the target location, $s=0$ m, the optical functions are $\beta_{x}=\beta_{y}=0.5$ m and $\eta_{x}=0$ mm. ![Target insertion region (target is at s=0 m) and 25 m of the cell. $\beta_{x}$ and $\beta_{y}$ functions are shown in blue and red, respectively; horizontal dispersion function $\eta_{x}$ is plotted in yellow with values on the right y axis.[]{data-label="IRcell"}](fig09_new.jpeg){width="8.5cm"} In addition, linear and non-linear terms related to the momentum deviation, $\delta p$, should be minimized because they contribute to the emittance growth. Figure \[radiative\] shows the positron beam size at the target increasing as a function of the machine turns for different values of dispersion function in the target insertion region, and with $\beta$ functions unchanged. When the dispersion at the target is cancelled, the beam size increase due to target interactions is damped. Figure \[horbeam\] shows the results of beam emittance degradation with the contributions from multiple scattering and bremsstrahlung separated. Values have been calculated using the 95 % core of the particle distribution. The current design status amounts only to a two-fold horizontal emittance increase and a four-fold in vertical by the end of the lifetime. The Bremsstrahlung energy loss in the target gives a longitudinal emittance growth. This increase of emittance and corresponding bunch lengthening can be reduced by decreasing the ring momentum compaction and increasing the RF voltage. The longitudinal emittance growth is given completely by the energy loss per turn making the beam longer. Thin low Z targets ([*i.e.*]{} thickness of the order of 0.01 Xo) based on Li, Be and C could be used for low emittance muon production. For equivalent electron density in the target, lighter materials will provide smaller beam perturbations at the cost of larger intrinsic muon beam emittances. Figure \[lifetimemat\] shows the number of survived positrons as a function of machine turns for different material targets. A 10 mm Lithium target might provide sizeably larger lifetime at the cost of a factor three increase in the intrinsic muon beam emittance. To maximize the brillance of the muon beam the positron beam spot at the target has to be minimized and the positron beam intensity maximized. Target consideration {#target} ==================== ![Number of e$^+$ vs turns number for different target materials. Target thickness has been chosen to obtain a constant muon yield. 3 mm Be, shown in this plot in dark violet, is the case studied in our beam dynamics simulations.[]{data-label="lifetimemat"}](fig12_new2.jpg){width="250pt"} Both temperature rise and thermal shock are related to the beam size on target. For a given material the lower limit on the beam size is obtained when there is no pile-up of bunches on the same target position. For this reason both the target and the positron beam have to be movable. A fast beam bump can be done after the extraction of one muon bunch from the muon accumulator ring every one muon laboratory lifetime, corresponding to approximately 2500 positron bunches. Fast moving targets can be obtained with rotating disks for solid targets or high velocity jets for liquids. A beam current of about 200 mA will provide about 100 kW of power that has to be removed to keep the target temperature under control. Be and C composites/structures are in use and under study for low Z target and collimators in accelerators for high energy physics also because of the stringent vacuum requirements in such complexes that are not easy to fulfil with Li targets. Recently developed C based materials with excellent thermo-mechanical properties are under study for the LHC upgrade collimators [@ipacT]. A 7.5 $\mu$s long beam pulse made of 288 bunches with 1.2$\times$10$^{11}$ protons per bunch, which is the full LHC injection batch extracted from SPS, has been used to test both C-based [@ipacT] and Be-based [@targetry] targets with maximum temperatures reaching 1000$^\circ$ C. Good results have been obtained with a beam spot of $0.3\times0.3$ mm$^{2}$. Liquid Lithium, LLi, is under study for divertors in tokamaks and in use for neutron production. In particular, LLi jets in the order of 1 cm thickness are used as targets for MeV proton beams to produce high flux neutron beams. Power deposited on target of about 0.5 MW and of $>$ 1 MW have been obtained or planned with LLi jet velocities of 5 m/s or 30 m/s respectively [@LLi1]. In addition to evaporation, beam spot sizes smaller than 1 mm$^2$ would reach the bubbling regime for LLi. One possibility to overcome this problem is to produce a fast LLi jet in a thin pipe. Positron source requirements {#posiso} ============================ [\*[7]{}[c]{}]{} & SLC & CLIC & ILC & LHeC & LHeC ERL& LEMMA goal\ E \[GeV\] & 1.19 & 2.86 & 4 & 140 & 60 &45\ $\gamma\epsilon_{x}$ \[$\mu$m\] & 30 & 0.66 & 10 & 100 & 50 &0.04\ $\gamma\epsilon_{y}$ \[$\mu$m\] & 2 & 0.02 & 0.04 & 100 & 50 & 0.04\ $e^{+}$\[$10^{14}$s$^{-1}$\] & 0.06 & 1.1 & 3.9 & 18 & 440 & 100\ The scheme described in the previous section relies on the possibility of enhancing the muon production by recirculating the positron beam, allowing multiple beam target interactions. However, every time the primary beam interacts in the target, positrons will lose part of their energy, and a certain fraction of them will exceed the energy acceptance of the machine, being eventually lost. An high intensity positron source is needed to replace these losses. The present record positron production rate has been reached at the SLAC linac SLC. A summary of the parameters of the positron sources for the future facilities is reported in Table  \[tab:Esources\]. The ILC positron source has been designed to provide $3.9\times 10^{14} e^+/$s. Two order of magnitudes more intense sources are foreseen for LHeC. LHeC is an electron machine although a positron option has been conceived. The required intensity is strongly related to the beam lifetime that is determined by the ring momentum acceptance and the target material, see sections \[optics\] and \[target\]. The beam lifetime with the present optics (energy acceptance about 6%) is in the range of 40-50 turns corresponding to about 2-3% of positrons lost per turn. By increasing the ring circumference we aim at increasing the energy acceptance.\ With 50 turns lifetime, the scenario requires $3 \times 10^{16}$ 45 GeV $e^+$/s . This corresponds to a beam power of 216 MW. A lifetime of 100 turns can be obtained with an H pellet target and energy acceptance of about 10%. In the interaction of the primary positron beam with the Beryllium target, Bremsstrahlung photons are produced with a strong boost along the primary beam direction. It may then possible to exploit this photon flux for an embedded positron source. A thick high Z target placed downstream of the muon target can be used for electron positron pair production. Experimental tests of such a scheme based on the adiabatic matching device collection scheme have been performed at KEK [@Chaikovska:2017amy]. However we do not yet have a system that is able to transform the temporal structure of the produced positrons to one that is compatible with the requirement of a standard positron injection chain. To evaluate the performance of such a scheme, a full Monte Carlo simulation has been performed, using GEANT4 for the part related to the positron production. A simulation of the target system has been performed with GEANT4. The geometry set-up is shown in Figure \[fig:scheme\]. A 45 GeV monochromatic positron beam is sent towards a 3 mm Beryllium target. After the target, a dipole magnet of the positron ring bends away charged particles, while the produced photons proceed straight, interacting with the Tungsten target, where $\rm e^+ e^-$ pairs are created. {width="10cm"} It has been found that in the interaction of $100$ positrons within the Beryllium target, about $11$ photons and $5$ electrons are created on average. As expected, photons are produced with a Bremsstrahlung energy spectrum and strongly boosted along the beam axis. ![[]{data-label="fig:produced"}](fig14a.pdf "fig:"){height="5.cm"} ![[]{data-label="fig:produced"}](fig14b_new.pdf "fig:"){height="5.cm"} Only photons with high boost will reach the thick Tungsten target, with an average overall yield of about 10.5 photons per 100 primary positrons, with a very small ($<0.1$ per mill) contamination of residual $\rm e^+/e^-$. In the interaction of these 10.5 photons in a $\rm 5~X_0$ Tungsten target, about $65$ positrons are produced, together with 77 electrons and 1200 photons. Figure \[fig:produced\] shows the positron energy spectra and the positron transverse phase space distribution. In conclusion, the suggested scheme produces about 60 secondary positrons from the interaction of 100 primary positrons, of which less than $\rm \sim 3$ are lost due to the interaction in the beryllium target. It would thus be enough to have a collection efficiency for these secondary positrons of about $5 \%$ in order to be able to recover the loss in the primary beam. A full simulation in GEANT4 and ASTRA [@astra] will be performed in future study, following the adiabatic matching scheme proposed in [@Chehab:2000xc]. Conclusion and Perspectives =========================== We have presented a novel scheme for the production of muons starting from a positron beam on target, discussing some of the critical aspects and key parameters of this idea and giving a consistent set of possible parameters to show its feasibility. This scheme has several advantages, the most important one is that it can provide low emittance muons without adding cooling. This innovative scheme has many key topics to be investigated: a low emittance 45 GeV positron ring, O(100 kW) class target, high rate positron source, and a 22 GeV muon accumulator ring. We presented the preliminary study of a 45 GeV positron ring with a thin Beryllium target insertion. The ring has an high momentum acceptance allowing a lifetime of about 40 turns for a 3 mm Be target. Beam emittance growth due to the interaction with target has been observed. A dedicated cell has been designed to show that the emittance growth can be contained with proper optics parameters at the target location. We have shown that this effect can be reduced by lowering the value of the $\beta$-function at the target location in addition to the minimization of the linear and high order dispersion terms. However stringent constraints on the beam size at the target are imposed by thermo-mechanical stresses, posing a lower limit on the emittance that can be obtained. Although an increase in emittance with respect to that shown in Figure \[f2\] seems unavoidable, it might be compensated with an increase of the positron beam energy up to 50 GeV where a factor of two in the muon production rate can be obtained, at the cost of more than a factor four larger emittance with almost no loss in muon collider luminosity performance. Nevertheless, one has to cope with a much larger muon energy spread, up to 18%. Further improvements in nonlinear effects corrections to increase the energy acceptance of the positron ring can be studied. We need also to reduce the momentum compaction to be as close to zero as possible in order to decrease the positron -and thus also the muon- bunch length. Beam instabilities driven by this small momentum compaction need to be studied, as well.\ The possibility of increasing the ring circumference has to be fully explored since it would allow the reduction of the synchrotron radiation power loss. Keeping the same number of positrons per bunch and the same bunch distance one would get the same muon production rate for the same positron source requirements. In addition, by increasing the ring circumference, the ring parameters could be improved, reducing transverse emittance and momentum compaction and increasing dynamical aperture and momentum acceptance. Progress need to include all other topics like target material, muon accumulation issues, positron source and injection. Acknowledgments =============== The authors thank L. Keller for useful discussions and suggestions. The authors also thank I. Chaikovska and R. Chehab for discussions on the positron source and AMD scheme, S. Gilardoni and M. Calviani for suggestions about the target issues and H. Burkhardt for discussions about GEANT4. [99]{} . http://map.fnal.gov. Mark Palmer. . In [*[Proceedings, 5th International Particle Accelerator Conference (IPAC 2014): Dresden, Germany, June 15-20, 2014]{}*]{}, page TUPME012, 2014. M. Antonelli, M. Boscolo, R. Di Nardo, and P. Raimondi. . , A807:101–107, 2016. Manuela Boscolo et al. . In [*[Proceedings, 8th International Particle Accelerator Conference (IPAC 2017): Copenhagen, Denmark, May 14-19, 2017]{}*]{}, page WEOBA3, 2017. William A. Barletta and Andrew M. Sessler. . , A350:36–44, 1994. . B. Autin, A. Blondel and J. R. Ellis, Editor: Frank Zimmermann, . . http://geant4.cern.ch. J.C. Biasci, J.F. Bouteille, N. Carmignani, J. Chavanne, D. Coulon, Y. Dabin, F. Ewald, L. Farvacque, L. Goirand, M. Hahn, J. Jacob, G. LeBec, S. Liuzzo, B. Nash, H. Pedroso-Marques, T. Perron, E. Plouviez, P. Raimondi, J.L. Revol, K. Scheidt, and V. Serrière. A low-emittance lattice for the esrf. , 27(6):8–12, 2014. Boaz Nash, N Carmignani, L Farvacque, S M Liuzzo, T Perron, P Raimondi, R Versteegen, and S White. New functionality for beam dynamics in accelerator toolbox (at). In [*[Proceedings, 6th International Particle Accelerator Conference (IPAC 2015): Richmond, VA, USA, 2015]{}*]{},IPAC15, 05 2015. . http://mad.web.cern.ch/mad. . http://www.fluka.org/fluka.php. D. Neuffer, Principles and Applications of Muon Cooling. Part. Accel.  [**14**]{}, 75 (1983). doi:10.2172/1156195 N. Munemoto [*et al.*]{}, Direct injection of fully stripped carbon ions into a fast-cycling induction synchrotron and their capture by the barrier bucket. Phys. Rev. Accel. Beams [**20**]{} (2017) no.8, 080101. D Möhl. . , 2006. Ji-Gwang Hwang, Eun-San Kim, Hye-Jin Kim, and Dong-O Jeon. Minimization of the emittance growth of multi-charge particle beams in the charge stripping section of raon. , 767:153 – 158, 2014. Fausto Maciariello et al. . In [*[Proceedings, 7th International Particle Accelerator Conference (IPAC 2016): Busan, Korea, May 8-13, 2016]{}*]{}, page TUPMB052, 2016. K. Ammigan et al. . In [*[Proceedings, 2nd North American Particle Accelerator Conference (NAPAC2016): Chicago, Illinois, USA, October 9-14, 2016]{}*]{}, page MOPOB14, 2017. E. Wakai, T. Kanemura, H. Kondo, Y. Hirakawa, Y. Ito, H. Serizawa, Y. Kawahito, T. Higashi, A. Suzuki, S. Fukada, K. Furuya, K. Esaki, J. Yagi, Y. Tsuji, T. Ito, S. Niitsuma, S. Yoshihashi-Suzuki, K. Watanabe, T. Furukawa, F. Groeschel, G. Micciche, S. Manorri, P. Favuzza, F.S. Nitti, R. Heidinger, T. Terai, H. Horiike, M. Sugimoto, S. Ohira, and J. Knaster. Engineering validation for lithium target facility of the ifmif under ifmif/eveda project. , 9:278 – 285, 2016. I. Chaikovska et al. . , B402:58–62, 2017. V. Yakimenko et al. . , C1205201:3108–3110, 2012. K. Flottmann, [ASTRA, A Space Charge Tracking Algorithm]{}. http://www.desy.de/ mpyflo/ R. Chehab. . , A451:362–366, 2000.

--- abstract: 'The possibility of an alternative way to formulate the Hawking radiation in a static Schwarzschild spacetime has been explored. To calculate the Hawking radiation, there can be two possible choices of the spacetime wedge pairs in the Krucal-Szekeres coordinates. One is the wedge pair consists of exterior spacetime of a black hole and the exterior spacetime of a white hole, and the other is that of exterior and interior spacetimes of one black hole. The radiation from the former is the Hawking’s original one. Though the the latter has been often regarded as the same phenomena as the former, the result here suggests it is not; its radiation has a temperature twice as high as the Hawking temperature.' address: 'Fukui Prefectural University, 4-1-1 Eiheiji, Fukui 910-0095, Japan' author: - 'Tadas K. Nakamura' date: '2007/10/20' title: Factor Two Discrepancy of Hawking Radiation Temperature --- 04.62.+v, 04.70.Dy Introduction ============= Right after the Hawking’s original paper [@hawking], attempts have been made to derive the Hawking radiation from a vacuum in a static Schwarzschild spacetime [@hartle; @unruh; @israel; @fulling; @kay]. The choice of the correct vacuum is crucial in such attempts. It is well known that the definition of a vacuum is not unique in curved spacetimes, and there is no general prescription to define a natural vacuum. Therefore we have to determine somehow which vacuum is actually realized, depending on each problem. The difficulty to find such a vacuum in a Schwarzschild spacetime comes from its bifurcating Killing horizons. The horizons divide the extended Schwarzschild spacetime into four spacetime wedges, and we have several choices of wedges or wedge pairs to define the vacuum. Unruh [@unruh] compared two distinct definitions of vacua, called $\eta$ definition and $\xi$ definition in his paper. The vacuum in $\xi$ definition is often referred as the Hartle-Hawking vacuum, which is calculated from analytic functions across the exterior spacetime of a black hole and the exterior spacetime of a white hole. On the other hand, it is also possible to define a vacuum state on the exterior wedge of a black hole alone, which is the $\eta$ definition; the vacuum with this definition is often called Boulware vacuum. Unruh [@unruh] considered the Haretle-Hawking vacuum is preferable because Boulware vacuum has singularity on physical quantities such as energy. This point has been examined extensively by several authors [@fulling; @kay], and it was shown that under some basic assumptions, the Hartle-Hawking vacuum is the only mathematically reasonable one in a wide class of spacetimes with Killing horizons [@kay2]. The legitimacy of the Hartle-Hawking vacuum shown in the above mentioned papers is, however, based on the particle number (or a conserved quantity along the Killing field, in general) defined on a Cauchy surface in the Kruscal-Szekeres coorinates. The vacua are calculated with analytic functions across the two exterior spacetime wedges; one is of a black hole and the other is of a white hole. Therefore, what have shown is that the Hartle-Hawking vacuum is the natural vacuum among the vacua across these two exterior spacetime wedges. In the present paper we explore another possibility of vacuum that spans across the interior and exterior spacetime wedges of a black hole; we will call this R-F vacuum hereafter. This is the vacuum defined with the particle number on a surface with $t=\textrm{constant}$. There have been several papers [@unruh2; @Fredenhagen; @jacobson] that regard the R-F vacuum as the source of the Hawking radiation, however, it seems its difference from the Hartle-Hawking vacuum was not well recognized; most of papers consider the radiation from the R-F vacuum is the same phenomena as that from the Hartle-Hawking vacuum. However, our calculation shows the temperature of the radiation from the R-F vacuum is twice as large as the one with the Hartle-Hawking vacuum, which means the R-F vacuum is not identical to the Hartle-Hawking vacuum. This temperature discrepancy of factor two was first reported by Akhmedov et al. [@akhmedov]. In relatively recent years, an approach based on the tunneling effect has been extensively investigated ([@parikh; @akhmedov] and references therein). Akhmedov et al. [@akhmedov] have carefully examined the integration contour in the tunneling calculation, and concluded the resulting radiation temperature is twice as large as the Hawking’s original value. The view point with tunneling effect may not physically well founded, however, it can be reinterpreted in the context of canonical quantization in the R-F wedge pair when back reaction of the particles to the metric is neglected. Then what calculated by Akhmedov et al. [@akhmedov] (or other papers with tunneling picture) coincides with the radiation from the R-F vacuum in the present paper. Therefore, the factor two discrepancy reported by Akhmedov et al. [@akhmedov] can be explained by the difference of vacua (R-F or Hartle-Hawking) from which the radiation comes from. We first examine the case of Unruh effect in the next section. Though the R-F vacuum has little physical significance in the Minkowski spacetime, its radiation is mathematically simpler and can be a good example for this type of vacua in other spacetimes. Its results are readily applied to the Schwarzschild spacetime since the structure near the bifurcating horizons are the same in both spacetimes. The application to the Schwarzschild spacetime is then sketched in the following section. A brief discussion on the validity of our approach is provided in the last section of this paper. Radiation from Minkowski Vacua ============================== Canonical Quantization with Horizons ------------------------------------ Suppose two dimensional coordinate system $(\eta,\xi)$ with the following metric:$$ds^{2}=A(\xi)\, d\eta^{2}-B(\xi)^{-1}d\xi^{2}.\label{eq:metric}$$ The wave equation of a massless scalar particle may be written as$$\frac{1}{A}\frac{\partial^{2}}{\partial\eta^{2}}\phi-\sqrt{\frac{B}{A}}\,\frac{\partial}{\partial\xi}\sqrt{AB}\,\frac{\partial}{\partial\xi}\phi=0.\label{eq:wave}$$ Now let us assume there is one and only one point where $A(\xi)B(\xi)=0$. We choose $\xi$ coordinate such that $\xi=0$ at that point; we call $\xi>0$ region “positive side” and $\xi<0$ region “negative side”. Separating the variables, we write eigenfunctions with respect to $\xi$ on the positive/negative side as $$u_{k}^{P}=\left\{ \begin{array}{ll} u_{k}^{P} & \;\;(\xi\ge0)\\ 0 & \;\;(\xi<0)\end{array}\right.,\;\;\; u_{k}^{N}=\left\{ \begin{array}{ll} 0 & \;\;(\xi\ge0)\\ u_{k}^{N} & \;\;(\xi<0)\end{array}\right.\;.$$ We assume the inner product $\left\langle \cdots\right\rangle $ is properly defined from the metric, and $u_{k}^{P,N}$ are normalized as $$\left\langle u_{k}^{P},u_{k'}^{P}\right\rangle =\left\langle u_{k}^{N},u_{k'}^{N}\right\rangle =\delta_{kk'}\,.$$ We can construct a solution $e^{-ik\eta}U_{k}(\xi)$ that satisfies Eq (\[eq:wave\]) over $-\infty<\xi<\infty$ across $\xi=0$ in the following:$$U_{k}=\left\{ \begin{array}{ll} \theta_{k}^{P}\, u_{k}^{P} & \;\;(\xi\ge0)\\ \theta_{k}^{N}\, u_{k}^{N} & \;\;(\xi<0)\end{array}\right..\label{eq:uk}$$ The wave equation Eq (\[eq:wave\]) is satisfied when we adjust the coefficients $\theta_{k}^{P,N}$ so that $U_{k}(\xi)$ becomes analytic across $\xi=0$; at the same time $\theta_{k}^{P,N}$ should satisfy the normalization condition $\left\langle U_{k},U_{k}\right\rangle =1$. Then we can expand a wave solution $\phi$ as$$\phi(\eta,\xi)=\sum_{k}a(\eta)\, U_{k}(\xi)$$ Decomposing $a(\eta)$ into the positive and negative frequency modes and calculating Boglubov coefficients from $\theta_{k}^{P,N}$, we can obtain the spectrum of Hawking/Unruh radiation, R-L (Right-Left) vacuum ----------------------- This vacuum corresponds to the Hartle-Hawking vacuum in a extended Schwarzschild spacetime; we call this type of vacuum R-L vacuum in this paper. Obviously this vacuum is the usual vacuum realized in a flat spacetime. Let us define the Rindler coordinate system as $$t=\xi\sinh a\eta,\;\; x=\xi\cosh a\eta\,,\label{eq:rindler}$$ where $t$ and $x$ are ordinary time and space coordinates in the Minkowski spacetime. The range of the space coordinate $\xi$ spans $-\infty<\xi<\infty$ so that the above equation covers both R and L regions (referred as Wedge R and Wedge L hereafter) illustrated in Figure 1.  The inner product is defined as$$\left\langle \phi,\phi'\right\rangle =\int_{-\infty}^{\infty}\xi^{-1}\,\phi\,\phi'^{*}\, d\xi\,.\label{eq:prod}$$ and the eigenfunctions on Wedges R and L become$$u_{k}^{R}=\exp\left(\frac{ik}{a}\,\ln\xi\right),\;\; u_{k}^{L}=\exp\left(-\frac{ik}{a}\,\ln|\xi|\right)\,.$$ It should be noted that the norm of the above eigenfunctions calculated from Eq (\[eq:prod\]) diverges as usually we encounter in this type of calculations. In the following we assume an appropriate prescription, such as wave packet treatment, has been applied implicitly to avoid this difficulty. We wish to obtain the solution across R and L Wedges in the form of $U_{k}$ in Eq (\[eq:uk\]). To this end, we must choose $\theta_{k}^{P,N}$ such that Eq (\[eq:wave\]) holds across the point of $\xi=0$. If we choose arbitrary $\theta_{k}^{P,N}$ then discontinuity occurs at $\xi=0$, and the right hand side of Eq (\[eq:wave\]) will have a $\delta$-function shaped “source term” at $\xi=0$. To avoid this we have to adjust $\theta_{k}^{P,N}$so that $U_{k}$ becomes analytic across $\xi=0$. Using the analytic continuation of the logarithmic function, $\ln(-\xi)=i\pi+\ln\xi$, such coefficients $\theta_{k}^{P,N}$ can be calculated in the following:$$\theta_{k}^{P}=\left\{ \begin{array}{ll} {\displaystyle \frac{1}{\sqrt{1-e^{-\pi k/a}}}} & \;\;(k\ge0)\\ {\displaystyle \frac{e^{\pi k}}{\sqrt{1-e^{\pi k/a}}}} & \;\;(k<0)\end{array}\right.,\quad\theta_{k}^{N}=\left\{ \begin{array}{ll} {\displaystyle \frac{e^{-\pi k/a}}{\sqrt{1-e^{-\pi k/a}}}} & \;\;(k\ge0)\\ {\displaystyle \frac{1}{\sqrt{1-e^{\pi k/a}}}} & \;\;(k<0)\end{array}\right.\;.\label{eq:coeffs}$$ The Unruh radiation spectrum becomes$$P(k)\propto\frac{1}{2}(\theta_{k}^{P\,2}+\theta_{k}^{N\,2}-1)=\frac{1}{\exp(2\pi k/a)-1}\label{eq:unruh}$$ for $k>0$ (see, eg., Birrel and Davies [@birrell], p105 for details of calculation). R-F (Right-Future) vacuum ------------------------- The above calculation has been done with the solution that spans over Wedges of R and L in Figure 1. We perform the same calculations with the solution over Wedges R and F in this subsection. This does not have physical significance for a flat spacetime, however, the similar calculation becomes important in the Schwarzschild spacetime as we will see in the next section. We examine the R-F case with a flat spacetime because it has essentially the same but mathematically simpler spacetime structure. We define the coordinates $(\eta',\xi'$) in Wedge F by$$t=\xi'\cosh a\eta',\;\; x=\xi'\sinh a\eta'\,,\label{eq:rindler2}$$ where $\eta'$ and $\xi'$ are real numbers, The eigenfunction with respect to $\xi'$ in Wedge F becomes$$u_{k}^{F}=\exp\left(\frac{ik}{a}\,\ln\xi'\right)\,.$$ Let us recall that the key point in the previous calculation is in the process to construct the solution $U_{k}$ that satisfies Eq (\[eq:wave\]) across the singular point of $\xi=0$. There we utilized the analyticity of $U_{k}$ as a function of $\xi$ across the both wedges. The reason why this works is that the coordinate $\xi$ is itself analytic across the wedges, in other words, we can express any point in both wedges with the same single expression of Eq (\[eq:rindler\]). We wish to take the same approach here, that is, to express points in Wedge F with Eq (\[eq:rindler2\]). This can be done by complexifing $\eta$ and $\xi$ as$$\eta=\frac{i\pi}{2}-\eta',\quad\xi=i\xi'\,,$$ then the complex numbers $(\eta,\xi)$ can express any points in Wedges F and R with Eq (\[eq:rindler\]). Having done that, $u_{k}^{F}$ may be expressed using logarithmic analytic continuation ($\ln i\xi'=i\pi/2+\ln\xi'$) as$$u_{k}^{F}=\exp\left(\frac{ik}{a}\,\ln\xi\right)=e^{\pi k/2a}\exp\left(\frac{ik}{a}\,\ln\xi'\right)\,.$$ Then the same calculation as in the R-L case gives the radiation spectrum as$$P(k)\propto\frac{1}{\exp(\pi k/a)-1}\,,$$ which has the temperature twice as large as in the R-L case. Schwarzschild coordinates ========================= Now let us move on to the Schwarzschild coordinate system whose metric is given by$$ds^{2}=-\left(1-\frac{2M}{r}\right)dt^{2}+\left(1-\frac{2M}{r}\right)^{-1}dr^{2}+r^{2}d\Omega^{2}\,.$$ where symbols have conventional meaning. The Rindler coordinates $(\eta,\xi)$ correspond to $(t,r-2M)$ in the above Schwarzschild coordinates. Wedges R and F in Figure 1 correspond to the interior and exterior spacetimes of a black hole respectively, and Wedges L and P correspond to the “white hole” in the extended Schwarzschild coordinates. Since the spacetime structure near Killing horizons are the same in the Rindler and Schwarzschild coordinates (see, eg., [@wald], p128), the arguments in the previous subsections are valid for the Schwarzschild coordinates with $a=1/4M$. We briefly sketch in the following the procedure of analytic continuation that leads us to this result. The the solutions to the wave equation Eq (\[eq:wave\]) have the form of $$u_{k}^{R,L}\propto\exp ik[\xi+2M+2M\ln(\xi/2M)]\label{eq:uk_sch}$$ where $\xi=r-2M$. The coordinates $(t,\xi)$ naturally covers Wedge R with $\xi>0$ and Wedge L with $\xi<0$. Using $\ln(-\xi/2M)=\ln(\xi/2M)+i\pi$, we obtain the amplitude discontinuity as $\exp(-\pi Mk)$. This means the radiation temperature is twice as large as the Hawking’s prediction. Continuation from Wedge R to Wedge L is not that straightforward. We notice in the above calculation the discontinuity comes from the analytic continuation of the logarithmic function, therefore we have to find an appropriate way to analytically extend the logarithmic function from Wedge R to Wedge L. To this end, we express the Schwarzschild coordinates in Wedge L as $(t',\xi')$ and examine the analytic relation of $\ln\xi$ in Wedge R and $\ln\xi'$ in Wedge L. The following Kruscal Szekeres coordinates $(U,V)$ are analytical across Wedges R and L, $$\begin{aligned} UV & = & -2M\xi\exp\left(\frac{\xi}{2M}+1\right)\nonumber \\ |U/V| & = & \exp\left(\frac{t}{2M}\right)\,,\end{aligned}$$ therefore, expressing $\xi$ by $U$ and $V$ gives the analyticity across the Wedges. Near $\xi=0$ we can approximate $$V-U\simeq\sqrt{2M\xi}\,.$$ at $t=0$. In Wedge R this is equivalent to$$\ln(V-U)=\frac{1}{2}\ln2M\xi\,,$$ since $V-U>0$; its analytic continuation to the region of $V-U<0$ (Wedge L) is$$\ln(V-U)=\pi i+\ln|V-U|\,.$$ Therefore the analytic continuation of the logarithmic function from Wedge R to Wedge L may be written as$$\ln\xi'=2\pi i+\ln\xi\,.$$ The above expression inserted into Eq (\[eq:uk\_sch\]) results in the amplitude jump of $\exp(-2\pi Mk)$. The rest of the calculation is the same as in the Rindler case, and we obtain the temperature predicted in the Hawking’s original paper [@hawking]. Concluding Remarks ================== The present paper has explored the possibility of an alternative vacuum to obtain Hawking radiation; the difference is in the choice of the wedge pair to define the vacuum. Most of past papers with a static Schwarzschild spacetime [@unruh; @hartle; @israel; @fulling; @kay; @kay2] were based on the vacuum across the exterior spacetime of a black hole and the exterior space of a white hole, which is referred as R-L vacuum in the present paper. On the other hand, several attempts have made to calculate the radiation with the vacuum across the interior and exterior spacetimes of a black hole (R-F vacuum) [@unruh2; @Fredenhagen; @jacobson; @parikh; @akhmedov]. To the author’s knowledge, the difference of these two vacua is not well recognized so far, and the radiation form R-L and R-F vacua are regarded as the same phenomena. The result of the present paper suggests these two are distinct; the temperature of the radiation from the R-F vacuum is twice as large as that of R-L vacuum. At the present we do not know which (R-F or R-L) vacuum should be realized around a black hole. The R-L vacuum requires the Kruscal extension, or a “white hole”, which is usually considered unphysical. On the other hand, the quantum construction for R-F vacuum may be questionable. In general, the procedure of quantization in a curved spacetime is based on a Cauchy surface with timelike normal vectors. However, if we choose the surface of $t=\textnormal{constant}$ in the Schwarzschild coordinates for a R-F vacuum, it is not a Cauchy surface because its normal vectors become spacelike in Wedge F. There may be two approaches to avoid this problem of R-F vacuum. One is to extend the method of quantization to be able to utilize surfaces with spacelike normal vectors instead of Cauchy surfaces. Recently several papers have been published proposing a quantization method in which an arbitrary closed surface can play the role of the Cauchy surface [@oeckl; @conrady; @doplicher]. This method may be applicable to the quantization for R-F vacuum. The other way is to stay in the conventional quantization with a Cauchy surface, but define the particle number on the surface with $t=\textnormal{constant}$. Then the surface to define the particle number can have spacelike normal vector within the conventional framework. Now the author of the present paper is working on this direction and the result will be reported in a forthcoming paper. Before closing this paper, let us briefly take a look at the original derivation by Hawking [@hawking]. The R-F vacuum means the ground state with respect to the total “energy” in the interior and exterior spacetimes. (Here “energy” means the conserved quantity that agrees with the usual energy in a flat spacetime at the region far away from the black hole.) Hawking [@hawking] examined the process of a star collapse, assuming a vacuum state long before the star collapse remains unchanged long after the black hole formation. With this assumption, what realized at the later time is the Hartle-Hawking vacuum. It was the ground state at the initial time, but it is not after the black hole formation; the ground sate at a later time is the R-F vacuum. The author of the present paper feels the state is likely to settle down to the ground state somehow long after the black hole formation, however, further investigation will be required to verify this point. The scope of the present paper is just to point out the possibility of R-F vacuum and cannot tell which vacuum is actually realized around black holes at the present. Acknowledgment: The author would like to thank M. Maeno for suggestions and discussions. [10]{} Hawking, S. W., Comm. Math. Phys. **43**, 199 (1975). Hartle, J. B. and S. W. Hawking, Phys. Rev. **D189**, 13 (1976). Israel, W., Phys. Lett. **A57**, 107 (1976). Unruh, W.G., Phys. Rev. **D14**, 870 (1976). Fulling, S.A., J. Phys. **A10**, 917 (1977). Kay, S.B., Commun. Math. Phys. **100**, 57 (1985). Kay, S.B and M. Wald, Phys. Reports **207** 49 (1991). Unruh, W. G., Phys. Rev. D15 **367** (1977). Fredenhagen, K., and R. Haag, Commun. Math. Phys. **127** 273 (1990). Jacobson, T., gr-qc/0308048. Parikh, M. K. and F. Wilczek, Phys. Rev. Lett. **85**, 5042 (2000), hep-th/9907001. Akhmedov E. T., V. Akhmedova, and D. Singleton, Phys. Lett. **B642**, 124 (2006), hep-th/0608098; Akhmedov E. T., V. Akhmedova, P. Terry, and D. Singleton, Int. J. Mod. Phys **A22**, 1705 (2007), hep-th/0605137. Birrell, N. D., and P. C. W. Davis, *Quantum fields in curved space*, Cambridge Univ. Prs. (1982). Wald, R. M., *Quantum field theory in curved spacetime and black hole thermodynamics,* Univ. Chicago Prs, (1994). Oeckl, R. Phys. Lett. **B575** 318 (2003). Conrady, F., C. Rovelli, Int. J. Mod. Phys. **A19** 4037 (2004), hep-th/0310246. Doplicher, L., Phys. Rev. **D70** 64037 (2004), gr-qc/0505006.

--- abstract: 'We study the Anderson localization of atomic gases exposed to three-dimensional optical speckles by analyzing the statistics of the energy-level spacings. This method allows us to consider realistic models of the speckle patterns, taking into account the strongly anisotropic correlations which are realized in concrete experimental configurations. We first compute the mobility edge $E_c$ of a speckle pattern created using a single laser beam. We find that $E_c$ drifts when we vary the anisotropy of the speckle grains, going from higher values when the speckles are squeezed along the beam propagation axis to lower values when they are elongated. We also consider the case where two speckle patterns are superimposed, forming interference fringes, and we find that $E_c$ is increased compared to the case of idealized isotropic disorder. We discuss the important implications of our findings for cold-atom experiments.' author: - 'E. Fratini and S. Pilati$^{1}$' title: 'Anderson localization of matter waves in quantum-chaos theory' --- Anderson localization is the complete suppression of wave diffusion due to destructive interferences induced by sufficiently strong disorder [@wiersma]. It was first discussed by Anderson in 1958 [@anderson] and has been observed (only much later) in various physical systems, including light waves [@maret1; @maret2; @roghini; @segev], sound waves [@tiggelen], and microwaves [@genack], and also in experiments performed with ultracold gases, first implementing an effective Anderson model [@garreau], and then observing the localization of matter waves in one dimension [@aspect; @inguscio1] and in three dimensions [@aspect2; @demarco1]. Recently, transverse Anderson localization has been realized in randomized optical fibers [@karbasi1], paving the way to potential applications in biological and medical imaging [@karbasi2].\ The key quantity which characterizes Anderson localization in three-dimensional quantum systems is the mobility edge $E_c$, which is the energy threshold that separates the localized states (with energy $E < E_c$) from the delocalized ones (with energy $E > E_c$) [@ramakrishnan]. Many accurate theoretical predictions for the value of $E_c$ exist, but most of them regard simplified toy models defined on a discrete lattice [@kramer; @romer]. These lattice models do not describe the spatial correlations, and their possible anisotropy, of the disorder present in the physical systems where Anderson localization has been observed. In fact, these features are expected to have a profound impact on the Anderson transition. For example, it is known that due to finite spatial correlations an effective mobility edge exists also in low-dimensional systems [@krokhin; @palencia2007; @palencia1; @lugan; @kenneth], while for uncorrelated disorder all states would be localized [@ramakrishnan]. According to recent results [@hilke], in continuous-space systems localization does not occur if the disorder correlation length vanishes, even for strong disorder. It is also known that the structure of the spatial correlations changes drastically the localization length and the transport properties [@piraud1; @piraud3].\ The experiments performed with ultracold atoms are emerging as the ideal experimental setup to study Anderson localization [@inguscio2; @lewenstein; @shapiro]. Different from other condensed-matter systems, atomic gases are not affected by absorption effects and permit us to suppress the interactions. Furthermore, by shining coherent light through diffusive surfaces, experimentalists are able to create three-dimensional disordered profiles (typically referred to as optical speckle patterns) with tunable intensity and to manipulate the structure of their spatial correlations.\ ![(color online) (a) Intensity profile of a speckle pattern measured on a plane orthogonal to the beam propagation axis $z$. (b) Elongated speckle pattern with anisotropy $\sigma_z/\sigma=6$, measured along a plane containing the beam axis $z$. (c) Profile resulting from two orthogonally crossed speckle patterns (see text) measured on a plane containing the second principal axis. (d) Representation of the two speckle-patterns configuration, indicating the propagation directions of the first beam ($z$) and the second beam ($y$). The red arrows indicate the 1$^{\textrm st}$ principal axis ($z'$) and the 2$^{\textrm nd}$ p. a. ($y'$). The $x$-axis enters the sheet plane. []{data-label="fig1"}](fig1.eps){width="1.0\columnwidth"} In this Rapid Communication, we investigate the Anderson localization of noninteracting atomic gases moving in three-dimensional optical speckles. We determine the single-particle energy spectrum using large-scale diagonalization algorithms. Then, by performing a statistical analysis of the spacings between consecutive energy levels, we locate the mobility edge. The study of the level-spacing statistics lies at the heart of random-matrix and quantum-chaos theories. It has permitted us to interpret the complex spectra of large nuclei, atoms, and molecular systems [@mehta; @haake]. More recently, it has been employed in the analysis of the Google Matrix [@shepelyansky1; @shepelyansky2]. Quantum-chaos theory provides a universal basis-independent criterion for the localization transition. One has to identify two kinds of level-spacing distributions, namely, the Wigner-Dyson distribution characteristic of ergodic chaotic systems, and the Poisson distribution characteristic of localized quantum systems. This method has allowed researchers to locate the localization transition in noninteracting three-dimensional lattice models (both isotropic and anisotropic) [@shore; @hofstetter; @schreiber; @schweitzer], and, more recently, also in interacting one-dimensional spin systems [@huse; @cuevas; @alet; @scardicchio]. In the present study this criterion is used to investigate the Anderson localization of matter waves, setting the basis for future investigations of many-body localization in interacting three-dimensional Fermi gases.\ First, we consider the experimental configuration with a single speckle pattern created by shining a laser through a diffusive plate. In this case the spatial correlations of the disorder are intrinsically strongly anisotropic, with cylindrical symmetry around the beam propagation axis. We find that, when the speckles are elongated along the axis, which is the typical experimental situation, the mobility edge is only moderately reduced compared to the idealized models of disorder with spherically symmetric correlations. This unexpected result indicates that the experimental setup with a single speckle pattern is quite suitable to investigate Anderson localization, despite the strong disorder anisotropy. We also consider the case where two orthogonal speckle patterns are coherently superimposed. This setup, which was originally implemented to avoid the large axial correlation length of the single-pattern configuration, generates an intricate correlation structure, with rapid oscillations of the external field due to interference fringes (see Fig. \[fig1\]) [@semeghini]. In this case we find that the mobility edge is higher than for isotropic disorder, and is similarly to the case of a single speckle pattern with axially squeezed speckle grains. This means that the two-pattern configuration provides experimentalists a handle to shift upwards the position of the mobility edge.\ The first step in the determination of $E_c$ is to compute the spectrum of the single-particle Hamiltonian $\hat{H} = -\frac{\hbar^2}{2m}\Delta + V({\bf r})$, where $\hbar$ is the reduced Planck’s constant, $m$ is the atom’s mass, and $V({\bf r})$ is the disordered potential experienced by the atoms exposed to optical speckle patterns. We consider a large box with periodic boundary conditions, which has a cubic shape (of size $L$) and parallelepiped shape, for isotropic and anisotropic speckles, respectively. We tackle this challenging computational task by representing $\hat{H}$ in momentum space, truncating the Fourier expansion at a large wave vector, carefully analyzing that the basis-truncation error is smaller than the final statistical uncertainty. To compute the eigenvalues we employ advanced numerical libraries for high-performance computers with shared-memory architectures [@plasma]. For more details on the Hamiltonian representation and on the numerical diagonalization procedure, see the Supplemental Material [@SM].\ If the speckle field is blue detuned with respect to the atomic transitions, it generates a repulsive potential with an exponential probability distribution of the local intensity, which reads $P_{\textrm{bd}}(V) = \exp\left(-V/V_0\right)/V_0$, if the intensity is $V>0$ and $P(V)=0$ otherwise. Thus, the potential has the lower bound $V({\bf r})=0$, while it is unbounded from above. The disorder strength is determined by the energy scale $V_0$, which is equal to the spatial average of the potential, $V_0=\left<V({\bf r})\right>$ and also to its standard deviation, so that $V_0^2=\left<V({\bf r})^2\right>-\left<V({\bf r})\right>^2$. For sufficiently large systems the disorder is self-averaging, and the spatial average coincides with the average over disorder realizations. Another fundamental property which characterizes the speckle pattern is the two-point spatial correlation function $\Gamma({\bf r}) = \left< V({\bf r}'+{\bf r}) V({\bf r}')\right>/V_0^2-1$. After averaging over the position of the first point ${\bf r'}$, it depends on the relative (vector) distance ${\bf r}$.\ ![(color online) Two-point spatial correlation functions of the disorder. The continuous curves represent the analytical formulas [@goodman] and the symbols represent the correlation measured on the speckle patterns generated numerically. The inset shows the correlation function of a single speckle-pattern along the beam axis $z$ for elongated speckle grains ($\sigma_z/\sigma=3$) and squeezed speckle grains ($\sigma_z/\sigma=1/3$), radial correlation, and isotropic correlation of idealized spherically-symmetric speckle patterns. The main panel shows the correlation of crossed speckle patterns along the first ($z'$) and the second principal axis ($y'$) and along the orthogonal axis $x$.[]{data-label="fig2"}](fig2.eps){width="1.0\columnwidth"} In order to make a direct comparison with a previous theoretical study based on transfer-matrix theory [@orso], we first consider an idealized isotropic model of the speckle pattern with a spherically symmetric correlation function that reads $\Gamma^{\textrm{iso}}(r) = \left[\sin(r/\sigma)/(r/\sigma)\right]^2$ (see inset in Fig. \[fig2\]). The parameter $\sigma$ fixes the length scale of the spatial correlations and therefore the typical grain size [@SM]. An efficient numerical algorithm to generate isotropic speckle patterns is described in details in Refs. [@huntley; @modugnomichele]. ![(color online). The main panel shows the ensemble-averaged adjacent-gap ratio $\left<r\right>$ as a function of the energy $E/E_\sigma$ for an isotropic speckle pattern of intensity $V_0=E_\sigma$, where $E_\sigma$ is the correlation energy. The horizontal green line is the result for the Wigner-Dyson distribution $\left<r\right>_{\textrm{WD}}$, and the dashed black line the one for the Poisson distribution $\left<r\right>_{\textrm{P}}$. The inset gives comparison between different system sizes. The vertical orange line indicates the position of the mobility edge $E_c$ (the hatched rectangle represents the error-bar). The gray bar represents the value of $E_c$ predicted in Ref. [@orso] using transfer-matrix theory. []{data-label="fig3"}](fig3.eps){width="1.0\columnwidth"} We determine the energy spectrum of a large number of realizations of the speckle pattern [@SM]. In the high-energy regime, the energy levels $E_{n}$ (listed in ascending order) fluctuate, avoiding each other, signaling the level repulsion typical of delocalized chaotic systems. The distribution of the level spacings $\delta_n=E_{n+1}-E_{n}$ should correspond to the statistics of random-matrix theory (in particular, to the Gaussian orthogonal ensemble), namely, the Wigner-Dyson distribution. Instead, in the low-energy regime the energy levels easily approach each other like independent random variables. This is a consequence of the localized character of the corresponding wave functions. In this regime the level spacings follow a Poisson distribution. In order to identify the two statistical distributions and determine the energy threshold $E_c$ which separates them, we compute the ratio of consecutive level spacings: $r = \min \left\{ \delta_n, \delta_{n-1} \right\}/\max \left\{ \delta_n, \delta_{n-1} \right\}$. The average over disorder realizations is known to be $\left< r \right>_{WD} \simeq 0.5307$ for the Wigner-Dyson distribution, and $\left< r \right>_{P} \simeq 0.38629$ for the Poisson distribution [@roux]. This statistical parameter was first introduced in Ref. [@huse] in the context of many-body localization. In Fig. \[fig3\] we show the data corresponding to the disorder strength $V_0 = E_\sigma$, where $E_\sigma = \hbar^2/m\sigma^2$ is the correlation energy. We find that the ensemble average $\left< r \right>$ changes rapidly from $\left< r \right>_{P}$ to $\left<r\right>_{WD}$ as the energy increases. While in an infinite system one would have a sudden transition between the two statistics (with a third distribution exactly at $E_c$ [@kravtsov1]), in a finite system we have a rapid but continuous crossover. For energies $E<E_C$, the data drift towards $\left< r \right>_{P}$ as the system size $L$ increases since the localized wave functions are independent only for $L\rightarrow \infty$, while they drift in the opposite direction for $E>E_c$. The crossing of the curves corresponding to different system sizes indicates the critical energy (see inset of figure \[fig3\]). To pinpoint $E_c$ we fit the data close to the transition with the scaling Ansatz $\left< r \right> = g\left[ \left(E-E_c\right)L^{1/\nu}\right]$ [@50years], where $\nu$ is the critical exponent of the correlation length and $g[x]$ is the scaling function (universal up to a rescaling of the argument) which we Taylor expand up to second order. For the case of Fig. \[fig3\], from the best-fit analysis we obtain $E_c = 0.576(10)E_\sigma$, in quantitative agreement with the result of transfer-matrix theory from Ref. [@orso]: $E_c = 0.570(7)E_\sigma$. For the critical exponent we obtain $\nu = 1.6(2)$, which is consistent with the prediction for the Anderson model: $\nu = 1.571(8)$ [@slevin]. It is worth mentioning that in the energy regime $E\sim V_0$ classical particles would be completely delocalized since the energy threshold $\epsilon_p$ for classical percolation in three-dimensional speckle patterns is extremely small, namely, $\epsilon_p\sim 10^{-4} V_0$ [@pilati1]. We consider also a red-detuned speckle field. Its distribution of intensities $P_{\textrm{rd}}(V)$ is the opposite of what corresponds to blue-detuned speckles, that is, $P_{\textrm{rd}}(V) = P_{\textrm{bd}}(-V)$. The corresponding average value is $\left<V(\bf{r})\right>=-V_0$. At the disorder strength $V_0=E_\sigma$, we obtain the mobility edge $E_c = -0.81(4)E_\sigma$, which (marginally) agrees with the result of transfer-matrix theory: $E_c = -0.863(6)E_\sigma$ [@orso]. It is worth noticing that for blue-detuned speckles the mobility edge is well below the average intensity of the potential, while for red-detuned speckles it is instead above it. This strong asymmetry was already found in Ref. [@orso] using transfer-matrix theory, but it was not captured by previous approximate calculations based on the self-consistent theory of localization. This means that predicting the position of the mobility edge requires quantitatively accurate methods.\ We now turn the discussion to concrete experimental configurations. We first consider the setup where a single laser beam with wavelength $\lambda$, propagating along the positive $z$ axis (see Fig. \[fig1\]), is transmitted through a diffusive plate and then focused onto the atomic cloud using a lens with focal length $f$. We assume the lens to be uniformly lit over a circular aperture of diameter $D$. The simplified numerical procedure employed before for isotropic models [@huntley; @modugnomichele] does not apply in this case. The complex amplitude of the speckle field at the position ${\bf r}=(x,y,z)$ measured from the focal point can be computed using the Fresnel diffraction integral [@goodman]: $$\begin{gathered} \label{fresnel} A_1\left( {\bf r} \right) = \frac{1}{i\lambda f} \exp \left[i 2\pi \left( z + f \right)/\lambda \right] \times \\ \int \int a_1\left( \alpha, \beta \right) \exp \left[ i\pi \frac{\left( x- \alpha\right)^2 +\left( y- \beta\right)^2 }{\lambda \left( z+f\right)} \right] \textrm{d}\alpha \textrm{d}\beta,\end{gathered}$$ where $a_1(\alpha,\beta)$ is the complex field amplitude at the point ${\bf l} \equiv (\alpha,\beta)$ just behind the focusing lens. The potential intensity is $V_1({\bf r}) = \left|A_1\left( {\bf r} \right)\right|^2$. Equation \[fresnel\] was derived assuming paraxial approximation. Consistently, we will consider only small (positive) displacements from the focal point: $x,y,z \ll f,D$. A convenient procedure to evaluate the Fresnel integral is to simulate the effect of a large number $N$ of scattering centers randomly placed on the aperture [@goodman; @dainty]. On the lens plane, one has: $a_1\left(\alpha, \beta \right) = \sum_{n=1}^{N} o_n \exp(i\phi_n) \delta^{(2)} \left( {\bf l} -{\bf l}_n \right)$, where ${\bf l}_n \equiv (\alpha_n,\beta_n)$ is the position of the $n$th scatterer, $o_n$ is the modulus of the corresponding scattered wave, and $\phi_n$ is its phase, which has to be sampled from a uniform random distribution in the interval from $-\pi$ to $\pi$. To simulate the effect of a uniform illumination, which is the case considered in this Rapid Communication, the moduli $o_n$ have to be identically and independently distributed random variables, while the random positions of the scattering centers must fill the aperture circle uniformly [@SM]. Substituting the expression for $a_1\left(\alpha, \beta \right) $ in eq. \[fresnel\], one obtains the complex field $A_1\left( {\bf r} \right) $ as the sum of wavelets propagating from the scattering center to the observation point. The field $A_1\left( {\bf r} \right)$ then has to be normalized to have the desired average intensity $V_0$. We verified that for $N\gg100$ the resulting potential has the statistical properties of fully developed speckle patterns [@goodman]. The intensities have the exponential probability distribution $P_{\textrm{bd}}(V)$ defined above. The spatial correlation function is anisotropic, with cylindrical symmetry around the propagation axis $z$. If one takes two points aligned in the radial direction, the correlation function reads $\Gamma^{\textrm{rad}}(r) = \left[ 2J_1\left(r/\sigma\right) /\left(r/\sigma\right) \right]^2$ [@goodman], where the correlation length is fixed by the parameters of the optical apparatus: $\sigma=\lambda f / \left(D\pi\right)$. Instead, in the axial direction the correlation function is [@goodman]: $\Gamma^{\textrm{z}}(r) = \left[\sin(r/\sigma_z)/(r/\sigma_z)\right]^2$, where the axial correlation length is $\sigma_z = 8\lambda (f/D)^2/\pi$. In current experimental implementations, the optical parameters are typically such that $\sigma_z > \sigma$, meaning that the speckle grains are elongated along the beam propagation axis. For example, in the experiment of Ref. [@demarco1] the anisotropy parameter was $\sigma_z/\sigma \approx 6$, while in the later experiment [@demarco2] it was varied in the range $1\lesssim \sigma_z/\sigma \lesssim 10$ by adjusting the aperture of the focusing length [@noteanisotropy].\ ![(color online) Mobility edge $E_c/E_{\sigma}$ as a function of the anisotropy parameter $\sigma_z/\sigma$. $\sigma_z$ and $\sigma$ are the axial and radial correlation lengths, respectively. The horizontal purple line indicates the disorder intensity $V_0 = E_\sigma$. The solid black curve is a guide to the eye (the dashed parts are an extrapolation). []{data-label="fig4"}](fig4.eps){width="1.0\columnwidth"} In order to investigate the effects due to the correlation anisotropy, we compute $E_c$ for varying values of the axial correlation length $\sigma_z$, considering both squeezed speckle grains ($\sigma_z/\sigma < 1$) and elongated speckle grains ($\sigma_z/\sigma > 1$). In our computations the box shape is adapted to the disorder anisotropy, see [@SM]. The disorder intensity is set at $V_0 = E_\sigma = \hbar^2/m\sigma^2$, defined using the (fixed) radial correlation length $\sigma$. We find that $E_c$ monotonously decreases as we increase the anisotropy parameter $\sigma_z/\sigma $. In the quasi-isotropic case $\sigma_z/\sigma = 1$, the result agrees with the idealized isotropic model considered above, while it is approximately $50\%$ larger for $\sigma_z/\sigma = 1/9$, and $15\%$ lower for $\sigma_z/\sigma = 6$ (see Fig. \[fig4\]). It is worth noticing that this dependence of $E_c$ on the anisotropy parameter is not trivially related to the scaling of the average correlation energy $E_{{\tilde \sigma}} = \hbar^2 / m {\tilde \sigma}^2$, defined from the geometric mean of the correlation lengths in the three spatial directions: ${\tilde \sigma} =\left( \sigma\sigma\sigma_z\right)^{1/3}$. This suggests that the geometric mean ${\tilde \sigma}$ is not the unique relevant length-scale, and that the structure of the spatial correlations plays a central role. We emphasize that in this Rapid Communication we are considering the speckle pattern created by a uniform aperture function. With different kinds of illumination (e.g, the Gaussian illumination [@demarco1; @aspect2]), $E_c$ might be somewhat different.\ While the reduction of $E_c$ due to a large axial correlation length could be observed using currently available experimental setups, the increase of $E_c$ is not easily accessible since the optical apparatuses do not permit us to create squeezed speckle grains. However, we can show that a similar increase in $E_c$ is induced when two orthogonal speckle patterns are superimposed. Explicitly, we numerically construct the potential due to the sum of two speckle patterns generated by laser beams with the same wavelength $\lambda$. The first pattern propagates along $z$ and the second along $y$ (see Fig. \[fig1\]), and they interfere coherently, as is the case when the two laser beams have the same linear polarization. The total complex amplitude is then [@kirchner; @okamoto]: $A_{\textrm{tot}} \left( {\bf r} \right) = A_{\textrm{1}} \left( {\bf r} \right)+ A_{\textrm{2}}\left( {\bf r} \right).$ The complex amplitude of the second speckle pattern $A_2 \left( {\bf r} \right)$ can be computed using eq. \[fresnel\] as described above, just exchanging the roles of the coordinates $y$ and $z$ in the right-hand side. For simplicity, we consider two patterns created with equal circular apertures, lit with the same (uniform) intensity, and focused using identical lenses. Thus the corresponding potentials $\left|A_1({\bf r})\right|^2$ and $\left|A_2({\bf r})\right|^2$ have the same radial correlations lengths, which we set at $\sigma \cong 0.75\lambda$. Their axial correlation lengths are extremely large, so that their variations along the respective propagation axes are irrelevant. This configuration is inspired by the experimental setup of Ref. [@semeghini]. The potential $V({\bf r}) = \left|A({\bf r})\right|^2$ corresponding to the coherent sum of two blue-detuned fields has the same exponential intensity distribution $P_{\textrm{bd}}(V)$ as a single (blue-detuned) speckle pattern [@goodman]. The structure of the spatial correlations of this total potential is instead much more intricate [@semeghini]. To describe it, it is convenient to consider the principal axes $y'$ and $z'$, obtained with a $45^\circ$ rotation of the $y$ and $z$ axes around the $x$ axis (see Fig. \[fig1\]). The correlation between two points aligned in parallel with the first principal axis $z'$ is $\Gamma^{z'}(r)=\Gamma^{\textrm{rad}}(r/\sqrt{2})$, meaning that the correlation length is $\sigma_{\textrm{p}} = \sqrt{2}\sigma$ [@semeghini]. Moving in parallel with the second principal axis $y'$, the potential is seen to oscillate rapidly due to the interference fringes and the corresponding correlation function is: $\Gamma^{y'} (r) = \left[ 2J_1\left( r/ \sigma_{\textrm{p}} \right) / \left( r / \sigma_{\textrm{p}} \right) \cos \left( \sqrt{2} \pi r/ \lambda \right) \right]^2$. The correlation function along the transverse axis $x$ is instead the same as for a single speckle pattern: $\Gamma^{x}(r)=\Gamma^{\textrm{rad}}(r)$. For our choice of parameters, the correlation function along the second principal axis $\Gamma^{y'} (r)$ touches zero four times before the first zero of the corresponding function along the transverse axis $\Gamma^{x} (r)$, indicating the strong anisotropy of the disorder correlations. We consider again a potential with average intensity $V_0 = E_\sigma$. The corresponding mobility edge is found to be $E_c = 0.67(1)E_\sigma$, significantly higher than for the idealized isotropic disorder. This result is comparable with the one for a single speckle-pattern with squeezed axial correlation length $\sigma_z/\sigma\simeq 1/3$ . We argue that this increase of $E_c$ is induced by the rapid variations of the potential due to the interference fringes, which effectively reduce the spatial correlation length along the second principal axis. Experimentalists can easily modify the width of the interference fringes, either by changing the angle between the two beams or by using lasers with different wavelengths. Observing the increase of $E_c$ is thus within experimental reach.\ We now turn the discussion to the comparison with the available experimental data. $E_c$ was first measured in Ref. [@demarco1] in the single-pattern configuration. The speckle grains were elongated, corresponding approximately to the anisotropy parameter $\sigma_z/\sigma \approx 6$ [@noteanisotropy]. In the regime of disorder strengths $V_0 \approx E_\sigma$, the results were in the range $1.5V_0 \lesssim E_c \lesssim 2V_0$. These findings do not agree with our results for strongly elongated speckle grains: $E_c \approx 0.5V_0$. Most likely, the reason of this discrepancy traces back to the procedure used to extract the values of $E_c$ from the measurement of the fraction of atoms that remain Anderson localized. In this derivation, the spectral function was approximated using the disorder-free value [@piraud2; @piraudthesis]. This approximation is not reliable at the disorder strength necessary to observe Anderson localization. Notice also that in the experiment of Ref. [@demarco1] a Gaussian pupil function was employed. More recently, the mobility edge was measured in the configuration with two crossed speckle patterns created with approximately uniform apertures [@semeghini] . For disorder intensities comparable to the correlation energy $V_0 \approx E_\sigma$, the mobility edge was found in the regime $E_c \approx V_0$. This result is significantly larger than the predictions for idealized isotropic models of the disorder and, in this sense, is consistent with our findings. However, it also overestimates our prediction for $V_0 = E_\sigma$. This discrepancy is probably due to the fact that in the experiment the two interfering speckle patterns are not equivalent because they were created using slightly different apertures and lenses with different focal lengths. Also, the width of the experimental interference fringes is slightly smaller compared that in to our model. Furthermore, an exact modeling of the experiment of Ref. [@semeghini] would require us to go beyond the paraxial approximation. All of these details of the experimental setup, once fully characterized, could be easily implemented in our formalism to compute $E_c$.\ In conclusion, we have studied the Anderson localization of matter waves exposed to optical speckles in the framework of quantum-chaos theory. We have shown that the structure of the spatial correlation of the disorder determines the position of the mobility edge, and we have described the effects induced by the correlation anisotropy in concrete experimental configurations, thus paving the way to a quantitative comparison between theory and experiment. This study sets the basis for future investigations of the effects due to interactions on the transport and on the coherence properties of disordered atomic gases [@pilati2] and on the role played by the fractality of the critical wave functions close the mobility edge [@kravtsov2].\ The authors acknowledge fruitful discussions with G. Modugno, A. Scardicchio and G. Semeghini. G. Orso is acknowledged for helpful discussions and for providing the data from Ref. [@orso]. We thank M. Atambo and I. Girotto for their help in the use of the parallel linear algebra software in our numerical calculation. [99]{} A. Lagendijk, B. A. van Tiggelen, and D. S. Wiersma, Phys. Today [**62**]{}, 24-29 (2009). P. W. Anderson, Phys. Rev. [**109**]{}, 1492 (1958). D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, Nature (London) [**390**]{}, 671 (1997). F. Scheffold, R. Lenke, R. Tweer, and G. Maret, Nature (London) [**398**]{}, 206-270 (1999). M. Störzer, P. Gross, C. M. Aegerter, and G. Maret, Phys. Rev. Lett. [**96**]{}, 063904 (2006). T. Schwartz, G. Bartal, S. Fishman, and B. Segev, Nature (London) [**446**]{}, 52 (2007). H. Hu, A. Strybulevych, J. H. Page, S. E. Skipetrov, and B. A. van Tiggelen, Nat. Phys. [**4**]{}, 945 (2008). A. A. Chabanov, M. Stoytchev, and A. Z. Genack, Nature (London) [**404**]{}, 850 (2000). J. Chabe, G. Lemarie, B. Gremaud, D. Delande, P. Szriftgiser, and J. C. Garreau, Phys. Rev. Lett. [**101**]{}, 255702 (2008). J. Billy, V. Josse, Z. Zuo, A. Bernard, B. Hambrecht, P. Lugan, D. ClŽment, L. Sanchez-Palencia, P. Bouyer and A. Aspect, Nature (London) [**453**]{}, 891 (2008). G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Forti, M Zaccanti, G. Modugno, M. Modugno and M. Inguscio, Nature (London) [**453**]{}, 895 (2008). F. Jendrzejewski, A. Bernard, K. MŸller, P. Cheinet, V. Josse, M. Piraud, L. PezzŽ, L. Sanchez-Palencia, A. Aspect, and P. Bouyer, Nat. Phys. [**8**]{}, 398-403 (2012). S. S. Kondov, W. R. McGehee, J. J. Zirbel, and B. De-Marco, Science [**334**]{}, 66 (2011). S. Karbasi, C. R. Mirr, P. G. Yarandi, R. J. Frazier, K. W. Koch, and A. Mafi, Optics Letters [**37**]{}, 2304 (2012). S. Karbasi, R. J. Frazier, K. W. Koch, T. Hawkins, J. Ballato, and A. Mafi, Nature Comm. [**5**]{}, 3362 (2014). E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Phys. Rev. Lett. [**42**]{}, 673-676 (1979). A. MacKinnon, B. Kramer, Phys. rev. Lett. [**47**]{}, 1546 (1981). A. Rodriguez, L. J. Vasquez, K. Slevin, and R. A. Römer, Phys. Rev. Lett. [**105**]{}, 046403 (2010). F. M. Izrailev and A. A. Krokhin, Phys. Rev. Lett. [**82**]{}, 4062-4065, (1999). L. Sanchez-Palencia, D. Clément, P. Lugan, P. Bouyer, G. V. Shlyapnikov, and A. Aspect, Phys. Rev. Lett. [**98**]{}, 210401 (2007). L. Sanchez-Palencia, D. Clément, P. Lugan, P. Bouyer, and A. Aspect, New J. Phys. [**10**]{}, 045019 (2008). P. Lugan, A. Aspect, L. Sanchez-Palencia, D. Delande, B. Grémaud, C. A. Müller,C. Miniatura, Phys. Rev. A [**80**]{}, 023605 (2009). E. Gurevich and O. Kenneth, Phys. Rev. A [**79**]{}, 063617 (2009). H. Eleuch and M. Hilke, arXiv:1501.05433 (2015). M. Piraud, A. Aspect, and L. Sanchez-Palencia, Phys. Rev. A [**85**]{}, 063611 (2012). M. Piraud, and and L. Sanchez-Palencia, Eur. Phys. J. Special Topics [**217**]{}, 91-102 (2013). A. Aspect, M. Inguscio, Physics Today [**62**]{}, 30 (2009) L. Sanchez-Palencia and M. Lewenstein, Nat. Phys. [**6**]{}, 87-95 (2010). B. Shapiro, J. Phys. A: Math. Theor. [**45**]{}, 143001 (2012). M. L. Mehta, Random Matrices (Elsevier-Academic Press, Amsterdam, 2004). F. Haake, Quantum Signatures of Chaos (Springer, Berlin, 2010). K. M. Frahm and D. L. Shepelyansky, Eur. Phys. J. B [**87**]{}, 93 (2014) O. V. Zhirov and D. L. Shepelyansky, arXiv:1502.00584 (2015). B. I. Shklovskii, B. Shapiro, B. R. Sears, P. Lambrianides, and H. B. Shore, Phys. Rev. B [**47**]{}, 11487 (1993). E. Hofstetter, M. Schreiber, Phys. Rev. B [**49**]{}, 14726 (1994). F. Milde, R. A. Römer, and M. Schreiber, Phys. Rev. B [**61**]{}, 6028 (2000). L. Schweitzer, H. Potempa, J. Phys.: Condens. Matter [**10**]{}, L431-L435 (1998). V. Oganesyan, D.A. Huse, Phys. Rev. B [**75**]{}, 155111 (2007). E. Cuevas, M. Feigel’man, L. Ioffe, and M. Mezard, Nat. Commun. [**3**]{}, 1128 (2012). D. J. Luitz, N. Laflorencie, F. Alet, Phys. Rev. B [**91**]{}, 081103(R) (2015). C. R. Laumann, A. Pal, A. Scardicchio, Phys. Rev. Lett. [**113**]{}, 200405 (2014). G. Semeghini, M. Landini, P. Castilho, S. Roy, G. Spagnolli, A. Trenkwalder, M. Fattori, M. Inguscio, and G. Modugno, arXiv:1404.3528v1 (2014). The Parallel Linear Algebra for Scalable Multi-core Architec- tures (PLASMA) project, http://icl.cs.utk.edu/plasma/ See Supplemental Material for more details on the box size and shape, on the boundary conditions, and on the Hamiltonian matrix representation. J. Goodman, *Speckle Phenomena in Optics: Theory and Applications*, (Roberts & Company Publishers, Dover, Englewood, Colorado, USA, 2007). D. Delande and G. Orso, Phys. Rev. Lett [**113**]{}, 060601 (2014). J. M. Huntley, Appl. Opt. [**28**]{}, 4316 (1989). M. Modugno, Phys. Rev. A [**73**]{}, 013606 (2006). Y. Y. Atas, E. Bogomolny, O. Giraud, and G. Roux, Phys. Rev. Lett. [**110**]{}, 084101 (2013). V. E. Kravtsov, I. V. Lerner, B. L. Altshuler and A. G. Aronov, Phys. Rev. Lett. [**72**]{}, 888 (1994). S. Chakravarty, in *50 Years of Anderson Localization*, ed. E. Abrahams (World Scientific Publishing, Singapore, 2010 ) K. Slevin, T. Ohtsuki, New J. Phys. [**16**]{}, 015012 (2014). S. Pilati, S. Giorgini, M. Modugno, and N. Prokof’ev, New J. Phys. [**12**]{}, 073003 (2010). J. C. Dainty, in *Progress in Optics XIV*, edited by E. Wolf (North-Holland, Amsterdam, 1977), pp 1-46. W. R. McGehee, S. S. Kondov, W. Xu, J. J. Zirbel, and B. DeMarco, Phys. Rev. Lett. [**111**]{}, 145303 (2013). The experiments of Ref. [@demarco1; @demarco2] implemented a Gaussian illumination, so the identification with the anisotropy parameter used here is only semi-quantitative. L. Leushacke, M. Kirchner, J. Opt. Soc. Am. A [**7**]{}, 827 (1990). T. Okamoto, S. Fujita, J. Opt. Soc. Am. A [**25**]{}, 3030 (2008). M. Piraud, L. Pezzé, and L. Sanchez-Palencia, Europhys. Lett. [**99**]{}, 50003 (2012). M . Piraud, Ph. D Thesis, Université Paris-Sud, (2012). S. Pilati, S. Giorgini, N. Prokof’ev, Phys. Rev. Lett. [**102**]{}, 150402 (2009). M. V. Feigel’Man, L. B. Ioffe, V. E. Kravtsov, E. Cuevas, Ann. Phys. [**325**]{}, 1390-1478 (2010). **Supplemental Material for\ Anderson localization of matter waves in quantum-chaos theory** In the following supplemental material, we provide additional technical details about the numerical procedure we employ to determine the energy spectrum and the level-spacing statistics of isotropic and anisotropic speckle patterns.\ The real-space Hamiltonian of a quantum particle moving in a speckle pattern is given by: $\hat{H} = -\frac{\hbar^2}{2m}\Delta + V({\bf r})$, where $\hbar$ is the reduced Planck’s constant, $m$ the particle’s mass, and $V({\bf r})$ is the external potential at the position ${\bf r}$ corresponding to the intensity of the optical speckle field. We consider a box with periodic boundary conditions and linear dimensions $L_x$, $L_y$ and $L_z$, in the three directions $\iota = x, y, z$.\ It is convenient to represent the Hamiltonian operator in momentum space as a large finite matrix: $ H_{\mathbf{k},\mathbf{k'}}= T_{\mathbf{k},\mathbf{k'}}+V_{\mathbf{k},\mathbf{k'}}$; $ T_{\mathbf{k},\mathbf{k'}}$ represents the kinetic energy operator, and $V_{\mathbf{k},\mathbf{k'}}$ the potential energy operator. The wavevectors form a discrete three-dimensional grid: $\mathbf{k}=(k_x,k_y,k_z)$, with the three components $k_\iota=\frac{2\pi}{L_\iota} j_\iota$, where $j_\iota=-N_\iota/2,...,N_\iota/2-1$, and the (even) number $N_\iota$ determines the size of the grid in the $\iota$ direction and, hence, the corresponding maximum wavevector. Therefore, when expanded in the square matrix format, the size of the Hamiltonian matrix $H_{\mathbf{k},\mathbf{k'}}$ is $N_{\mathrm{tot}}\times N_{\mathrm{tot}}$, where $N_{\mathrm{tot}}= N_x N_y N_z$.\ In this basis the kinetic energy operator is diagonal: $T_{\mathbf{k},\mathbf{k'}}=-\frac{\hbar^2 k^2}{2m} \delta_{{k_x},{k_x'}}\delta_{{k_y},{k_y'}}\delta_{{k_z},{k_z'}}$, where $\delta_{{k_\iota},{k_\iota'}}$ is the Kronecker delta. The element $V_{\mathbf{k},\mathbf{k'}}$ of the potential energy matrix can be computed as: $V_{\mathbf{k},\mathbf{k'}}=\tilde{v}_{\mathbf{k'}-\mathbf{k}}$, where $\tilde{v}_{\mathbf{k}}$ is the discrete Fourier transform of the speckle pattern $V({\bf r})$: $$\begin{gathered} \label{fourier} \tilde{v}_{ k_{x} k_{y} k_{z} } = N_{\mathrm{tot}}^{-1} \sum_{r_x} \, \sum_{r_y} \, \sum_{r_z} \\ v_{r_xr_yr_z} \, \exp \left[- i \left( k_{x} r_{x}+ k_{y} r_{y} + k_{z} r_{z} \right)\right] ;\end{gathered}$$ here, $v_{{r_xr_yr_z}} = V({\mathbf r=(r_x,r_y,r_z)})$ is the value of the external potential on the $N_{\mathrm{tot}}$ nodes of a regular lattice defined by $r_\iota=L_\iota n_\iota/N_\iota$, with $n_\iota=0,..,N_\iota-1$. Wavevectors differences are computed exploiting periodicity in wavevector space.\ ![Two-point spatial correlation functions of isotropic speckle patterns generated by scattering centers placed on a continuous spherical shell and on a discrete grid. The continuous blue curve represents the analytical formula $\Gamma^{\mathrm{iso}}({\bf r})$. The size of the cubic box is $L = 16\pi \sigma$. []{data-label="figS1"}](figS1.eps){width="1.0\columnwidth"} The numerical algorithm we employ to generate the speckle patterns is described in the main text. It allows us to create both isotropic and anisotropic speckles. In the former case, the scattering centers (see main text) have to be placed on a fictitious spherical shell of diameter $D$, with uniform random distribution. Also, equation (1) of the main text has to be trivially modified, arriving at: $$\begin{gathered} \label{fresnel} A_1\left( {\bf r} \right) = \frac{1}{i\lambda f} \exp \left[i 2\pi f /\lambda \right] \times \\ \sum_n o_n \exp \left( i \phi_n \right) \exp \left[ i\pi \frac{\left| {\bf r}- {\bf l}_n \right|^2 }{\lambda f} \right] ,\end{gathered}$$ where ${\bf l}_n = (\alpha_n,\beta_n,\gamma_n)$ is the position of the $n$-th scattering center on the (fictitious) spherical aperture. One obtains a speckle pattern with the isotropic correlation function $\Gamma^{\mathrm{iso}}({\bf r})$ (see definition in the main text). It is worth mentioning that in order to generate isotropic speckle patterns one could instead employ the conventional numerical recipe described in Refs. [@huntleyS; @modugnomicheleS].\ For isotropic speckles, we employ cubic boxes with $L= L_x=L_y=L_z$ (with periodic boundary conditions). The numerical recipe described here (and in the main text) creates a potential which does not necessarily satisfy periodic boundary conditions. In principle, this could introduce distortions in the Fourier transform. However, a speckle pattern compatible with periodic boundary conditions is obtained if the position of the scattering centers is discretized according to: $\alpha_n \rightarrow \mathrm{nint} \left( \alpha_n / \delta \right) \delta$, where $\mathrm{nint}\left( \cdot \right)$ is the function that returns the whole integer closest to the argument, $\delta = \lambda f/L$ is the discretization step, and the analogous operation is applied to $\beta_n$ and $\gamma_n$. In figure \[figS1\], we compare the correlation functions numerically measured in speckle patterns generated with continuous and with discrete scatterers, against the analytical formula $\Gamma^{\mathrm{iso}}({\bf r})$. The perfect agreement indicates that the discretization procedure does not alter the statistical properties of the speckles.\ ![ Analyses of the level-spacing statistics for isotropic speckle patterns generated by scattering centers placed on a continuous spherical shell and on a discrete grid. $\left<r\right>$ is the ensemble-averaged adjacent-gap ratio, which is plotted as a function of the energy $E/E_\sigma$, where $E_\sigma$ is the correlation energy. The horizontal green line is the result for the Wigner-Dyson distribution $\left<r\right>_{\textrm{WD}}$, the dashed black line the one for the Poisson distribution $\left<r\right>_{\textrm{P}}$. []{data-label="figS2"}](figS2.eps){width="1.0\columnwidth"} In figure \[figS2\], we compare the analyses of the energy-levels statistics of two speckle patterns (with average intensity $V_0 = E_\sigma$) generated with continuous and with discrete scatterers. The excellent agreement cross-validates the two procedures, meaning that they are both suitable for our purposes. This is to be expected, since the linear size of the cubic box is $L = 20\pi \sigma$ (this is a typical size we use), much larger than the correlation length of the disorder, so that border effects play a minor role. Notice that the parameter $\sigma$ characterizes the lengths scale of the disorder spatial correlation, and hence, the typical speckle size. More quantitatively, the full width at half maximum $\ell_c$, defined by the condition $\Gamma^{\mathrm{iso}}(\ell_c/2) = \Gamma^{\mathrm{iso}}(0)/2$, is $\ell_c \cong 0.89 \pi \sigma$. The discretization procedure can be easily adapted to have periodicity in anisotropic speckle patterns, provided the focal length $f$ is much larger than the box size.\ ![ Adjacent-gap ratio $\left<r\right>$ as a function of the energy $E/E_\sigma$, for an isotropic speckle pattern with intensity $V_0 = E_\sigma$. The different datasets correspond to different number of wavevectors $N$, thus to different levels of accuracy. []{data-label="figS3"}](figS3.eps){width="1.0\columnwidth"} ![Adjacent-gap ratio $\left<r\right>$ as a function of the energy $E/E_\sigma$, for an anisotropic speckle pattern in the single-beam configuration. The anisotropy parameter is $\sigma_z/\sigma = 2/9$, while the disorder intensity is $V_0 = E_\sigma$. The box sizes are $L_x=L_y=7.5\pi\sigma$ and $L_z = L_x/3$, meaning that the box shape has been only partially adapted to the disorder anisotropy. The different datasets correspond to different number of wavevectors $N_z$ in the axial direction $z$, while in the radial directions the wave-vector number is fixed at $N_x=N_y=12$. []{data-label="figS4"}](figS4.eps){width="1.0\columnwidth"} A crucial step to guarantee the accuracy of our result is to test that the number of wavevectors (and, correspondingly, the maximum wavevector) is sufficient to have an accurate representation of the speckle pattern and of the orbitals. In figure \[figS3\], we compare the level-spacing statistics for an isotropic speckle pattern obtained with different numbers of wavevectors. For isotropic speckles, it is convenient to set $N_x=N_y=N_z=N$. It is evident that we obtain convergence already for moderately large $N\approx 18$. Consequently, we can afford to perform ensemble averages over a large number of disorder realizations in a suitable computational time. For example, the data corresponding to $N=18$ in figure \[figS3\] have been obtained by averaging approximately $3\times 10^4$ disorder realizations, requiring $72$ hours on a 20-cores CPU, using the PLASMA library for linear algebra computations [@plasma]. The maximum wavevector number we consider is $N=40$, allowing us to solve approximately $13$ disorder realizations in 24 hours. Notice that, when we increase the system size $L$, we proportionally increase $N$, so that the maximum wavevector in the Hamiltonian matrix remains fixed and, therefore, we maintain the same level of accuracy.\ In the case of anisotropic speckle patterns, it is convenient to (partially) adapt the shape of the box to the disorder anisotropy, so that the system size can be set to be much larger than the disorder correlation length in each direction, without exceedingly increasing the matrix size. Also, the numbers of wavevectors in the three spatial directions $N_x$, $N_y$, and $N_z$ have to be adjusted according to the corresponding linear system sizes, $L_x$, $L_y$, and $L_z$, respectively, and also according to the correlation length in the corresponding direction. The shorter the correlation length, the larger $N_\iota$ is required. ![ Adjacent-gap ratio $\left<r\right>$ as a function of the energy $E/E_\sigma$, for the same anisotropic speckle pattern as in figure \[figS4\]. Here, the different datasets correspond to different number of wavevectors $N_x=N_y$ in the radial directions $x$ and $y$, while in the axial direction $z$ the wave-vector number is fixed at $N_z = 14$. []{data-label="figS5"}](figS5.eps){width="1.0\columnwidth"} In the single laser-beam configuration, the disorder correlations have axial symmetry around the beam propagation axis $z$; therefore we set $L_x=L_y$ and $N_x=N_y$. As an illustrative example, we consider here an anisotropic pattern with strongly squeezed grains corresponding to $\sigma_z/\sigma = 2/9$. The parameter $\sigma_z$ characterizes the correlation length in the axial direction; the full width at half maximum of the axial corresponding correlation function $\Gamma^{z}(z)$ (see definition in the main text) is $\ell_c \cong 0.89 \pi \sigma_z$. For the radial correlation function $\Gamma^{\mathrm{rad}}(r)$, one has $\ell_c \cong 1.029 \pi \sigma$. Notice that even in the case $\sigma = \sigma_z$, the disorder is not perfectly isotropic. In our anisotropic example, we employ a box with an anisotropy which is similar to that of the disorder: $L_z = L_x/3$. It is important to stress that the analysis of the effect due to the finite wavevectors number has to be performed both for the axial direction $z$ and the radial directions $x$ and $y$, separately. The former effect is analyzed in figure \[figS4\], the latter in figure \[figS5\]. Again, we observe convergence with moderately large grid sizes.\ We emphasize that the analysis of the level-spacing statistic can be used to determine the mobility edge $E_c$ both in the case of isotropic and anisotropic models; see, e.g., Ref. [@schreiberS]. In particular, it was found in Ref. [@schweitzerS] that the position of the mobility edge does not depend on the shape of the box, and that the level-spacing statistics converges in the thermodynamic limit to the Wigner-Dyson and to the Poisson distributions, in the delocalized and in the localized regimes, respectively. Instead, the level-spacing statistics exactly at the critical point $E = E_c$ (which is system-size independent) was found to depend on the box shape. This does not invalidate our finite-size scaling procedure, but likely implies that the amplitude of the parameter $\left< r \right >$ at $E_c$ for different speckle anisotropies is not universal. Results analogous to those of Ref. [@schweitzerS] were obtained in Refs. [@braunS; @schweitzerS], where the effect due to the choice of boundary conditions (e.g., periodic boundary conditions vs. hard-wall or Dirichlet boundary conditions) was analyzed. Again, it was found that the position of mobility edge is not affected by the choice of boundary conditions, while the level-spacing distribution at the critical point is. ![ Main panel: ensemble-averaged adjacent-gap ratio $\left<r\right>$ as a function of the energy $E/E_\sigma$ for an anisotropic speckle pattern of intensity $V_0=E_\sigma$ and anisotropy parameter $\sigma_z/\sigma=2/9$. The different datasets correspond to different box sizes $L_x=L_y$, with $L_z/L_x = 1/3$. Inset: comparison between different system sizes. The vertical orange line indicates the position of the mobility edge $E_c$ (the rectangle with pattern represents the error-bar). The continuous curves represent the scaling Ansatz $g[x]$ (defined in the main text) expanded to second order.[]{data-label="figS6"}](figS6.eps){width="1.0\columnwidth"} Figure \[figS6\] shows the analysis of the level-spacing statistics for our example of anisotropic speckle, and the finite-size scaling analysis (see inset) employed to locate the mobility edge. For completeness, we have also analyzed the potential effect on $E_c$ due to the box shape by repeating calculations with different $L/L_z$ ratios. Consistently with the findings of Refs. [@schweitzerS; @braunS; @schweitzerS], we also find that there is no systematic bias by changing the box shape; however, larger errorbars are obtained if the shape is not the optimal one, since one has to employ larger matrix sizes.\ In the configuration with two superimposed speckle patterns, the disorder has the intricate anisotropic spatial correlation structure described in the main text; see figure (1) \[panel (c)\] and figure (2) in the main text. In this case, we use a cubic box with $L_x=L_y=L_z$, but we rotate the speckle pattern such that second principal axis $y'$ (see panel (d) of figure (1) in the main text) is aligned with one of the sides of the box, e.g., along the $z$ direction. Then, we increase the wavevector number corresponding to this spatial direction, since the disorder has rapid oscillations and a shorter effective correlation length due to the interference fringes. For the two-pattern configuration described in the main text, we find it is necessary to use a wavevector number $N_z \approx 3N_x$.\ Finally, it is worth comparing the efficiency of our procedure to determine $E_c$, which is based on the analysis of the level-spacing statistics within quantum-chaos theory, with the one of the transfer-matrix theory employed in Ref. [@orsoS] in the case of isotropic speckle patterns. In the main text we show the quantitative comparison between the two theories both for blue-detuned and red-detuned isotropic speckle fields. The results of transfer-matrix theory have somewhat smaller errorbars, in particular for red-detuned speckles, since in our formalism they require a larger wavevector number, and hence, allow us to consider fewer disorder realizations. However, we have shown that quantum-chaos theory provides us with effective and flexible tools to address disorder patterns with intricate correlation structures and to adapt the shape of the sample to the disorder anisotropy. More importantly, our exact diagonalization study allows us to disclose interesting properties of the energy spectrum of optical speckles, thus creating a strong link between random matrix theory and ultracold atomic gases. [99]{} J. M. Huntley, Appl. Opt. [**28**]{}, 4316 (1989). M. Modugno, Phys. Rev. A [**73**]{}, 013606 (2006). http://icl.cs.utk.edu/plasma/ F. Milde, R. A. Römer, and M. Schreiber, Phys. Rev. B [**61**]{}, 6028 (2000). L. Schweitzer, H. Potempa, J. Phys.: Condens. Matter [**10**]{}, L431-L435 (1998). D. Braun, G. Montambaux, and M. Pascaud, Phys. Rev. Lett. [**81**]{}, 1062 (1998). L. Schweitzer, H. Potempa, Phys. A [**266**]{}, 486-491 (1999). D. Delande and G. Orso, Phys. Rev. Lett [**113**]{}, 060601 (2014).

--- abstract: 'Constraint programming can definitely be seen as a model-driven paradigm. The users write programs for modeling problems. These programs are mapped to executable models to calculate the solutions. This paper focuses on efficient model management (definition and transformation). From this point of view, we propose to revisit the design of constraint-programming systems. A model-driven architecture is introduced to map solving-independent constraint models to solving-dependent decision models. Several important questions are examined, such as the need for a visual high-level modeling language, and the quality of metamodeling techniques to implement the transformations. A main result is the platform that efficiently implements the chain from modeling to solving constraint problems.' title: 'Model-Driven Constraint Programming' --- \[Constraint and logic languages\] \[User interfaces\] \[Classes and objects, Constraints\] Languages Constraint Modeling Languages, Constraint Programming, Metamodeling, Model Transformation Introduction {#sec:intro} ============ In constraint programming (CP), programmers define a model of a problem using *constraints* over *variables*. The variables may take values from domains, typically boolean, integer, or rational values. The solutions to be found are tuples of values of the variables satisfying the constraints. The search process is performed by powerful solving techniques, for instance backtracking-like procedures and consistency algorithms to explore and reduce the space of potential solutions. In the past, CP has been shown to be efficient for solving hard combinatorial problems. CP systems evolved from the early days of constraint logic programming (CLP). In a CLP system, the constraint language is embedded in a logic language, and the solving procedure combines the SLD-resolution with calls to constraint solvers [@JaffarPOPL1987]. The logic language can be replaced with any computer programming language (e.g. C++ in ILOG Solver [@PugetSCIS1994] or Java in Gecode/J [@Gecode]) and even term rewriting [@FruehwirthJLP1998]. It turns out that the programming task may be hard, especially for non experts of CP or computer programming. In this approach, modeling concerns are not enough to write programs, and it is often mandatory to deal with the encoding aspects of the host language or to tune the solving strategy. In response to this problem, almost pure modeling languages have been built, such as OPL [@VanHentenryckBook1999] and Zinc [@RafehPADL07]. The design of the last generation of CP systems has been governed by the idea of separating modeling and solving capabilities (e.g. Essence [@FrischIJCAI2007] and MiniZinc [@Nethercote2007]). The system architecture has three layers, including the modeling language, the solvers, and a middle tool composed by a set of solver-translators implementing the mappings. In particular, this approach gives important benefits: The full expressiveness of CP is supported by a unique high-level modeling language, which is expected to be simple enough for non experts. The user is able to process one model with different solvers, a crucial feature for easy and fast problem experimentation. The platform is open to plug new solvers. Our work follows this solver-independent idea, but under a Model-Driven Development (MDD) approach [@OMG_MDA], which is well-known in the software engineering sphere. General requirements have been defined for MDD architectures in order to define concise models, to enable interoperability between tools, and to easily program mappings between models. The classical MDD infrastructure uses as base element the notion of a metamodel, which allows one to clearly define the concepts appearing in a model. In this paper, the MDD approach is applied to a CP system. The goal is to implement the chain from modeling to solving constraints. Our approach is to transform user solving-independent models defined through a visual modeling language to solver (executable) models using a metamodeling strategy. CP concepts like domains, variables, constraints, and relations between them are defined in a metamodel, and thus the transformation rules are able to map these concepts from a source language to a target one. It results in a flexible and extensible architecture, robust enough to support changes at the mapping tool level. Moreover, we believe that the study of metamodels for CP is of interest. These ideas have been implemented in the platform [@s-COMMA]. The front tool allows users to graphically define constraint models. It is made on top of a general object-oriented constraint language [@SotoICTAI07]. Many solvers have been plugged in the platform such as [ECL$^{i}$PS$^{e}$]{} [@Wallace97eclipse], Gecode/J [@Gecode], GNU Prolog [@DiazSAC00] and Realpaver [@GranvilliersACM2006]. Upgrades are supported at the mapping tool, new solver-translators can be added by means of the AMMA platform [@Kurtev2006]. The language for stating constraints in is clearly not the novel part of the platform, in fact it includes typical and state-of-the-art modeling constructs and features. Novelty arises from the introduction of a solver-independent visual language – which we believe is intuitive and simple enough for non experts –, and the use of a MDD approach involving metamodeling techniques to implement the mappings. The outline of this paper is as follows. The MDD architecture proposed is introduced in Section \[sec:MDD-approach\]. The modeling language and the associated graphical interface are presented in Section \[sec:modeling-tool\]. The mapping tool and the metamodeling techniques used to develop solver-translators are explained in Section \[sec:mapping-tool\]. Some experimental results are then discussed in Section \[sec:bench\]. The related work and conclusion follows. A MDD approach for CP {#sec:MDD-approach} ===================== Model-Driven Engineering (MDE) aims to consider models as first class entities. A model is defined according to the semantics of a model of models, also called a **metamodel**. A metamodel describes the concepts appearing in a model, but also the links between these concepts, such as: inheritance, composition or simple association. Figure \[fig:mda\] depicts a general Model-Driven Architecture (MDA) for model transformation. Level M1 holds the model. Level M2 describes the semantic of the level M1 and thus identifies concepts handled by this model through a metamodel. Level M3 is the specification of level M2 and is self-defined. Transformation rules are defined to translate models from a source model to a target one, the semantic of these rules is also defined by a metamodel. A major strength of using this metamodeling approach is that models are concisely represented by metamodels. This allows one to define transformation rules that only operate on the concepts of metamodels (at the M2 level of the MDA approach), not on the concrete syntax of a language. Syntax concerns are defined independently (we illustrate this in Section \[sec:mapping-tool\]). This separation is a great advantage for a clearly definition of transformation rules and grammar descriptions, which are the base of our mapping tool. Let us now illustrate how this approach is implemented in our platform. Figure \[fig:mdaSCOMMA\] shows the MDD architecture, which is composed by two main parts, a modeling tool and a mapping tool. The is our modeling tool, and it allow users to state constraint models using visual artifacts. An exactly textual representation of this language is also provided (for who does not want to use visual artifacts). Both languages are solver-independent and are designed conform to the same metamodel (see Section \[sec:modeling-tool\]). The output of the is an intermediate language which is still solver-independent but, in terms of abstraction is closer to the solver level. The goal is to simplify the development of solver-translators. is also designed conform to a metamodel (see Section \[sec:flatscomma\]). The mapping tool is composed by a set of solver-translators. Solver-translators are designed to match the metamodel concepts of to the concepts of the solver metamodel (see Section \[sec:mapping-tool\]). This process is defined conform to the general MDA for model transformation. The is written in Java (about 30000 lines) and translators are developed using the AMMA platform. The whole system allows to perform the complete process from visual models to solver models. The system involves several metamodels: an metamodel, a and solver metamodels. The metamodel has been built just for defining the concepts of the textual and visual language, it is not used to map to . For this task we already have an efficient translator. Our key aim of using metamodeling techniques is to provide an easier way to develop new solver-translators, compared to the task of writing translators by hand. In the following two sections we present the main parts of this architecture: The modeling and the mapping tool, respectively. Modeling Tool {#sec:modeling-tool} ============= We have built our modeling tool on top of the language. The language is defined through its metamodel and it has been designed to represent the concepts of constraint problems, also called constraint satisfaction problems (CSPs). In this metamodel, the CSP concepts such as variables and domains have been merged with object-oriented concepts in order to state CSPs using an object-oriented style. The result is an object-oriented visual language for modeling CSPs. These decisions are supported by the following benefits: - A problem is generally composed of several parts which may represent objects. They are naturally specified through classes. Thus, we obtain a more modular model, instead of forcing modelers to state the entire problem in a single block of code. - We gain similar benefits – constraint and variable encapsulation, composition, inheritance, reuse – to those gained by writing software in a object-oriented programming language. - Visual artifacts are more intuitive to use and give a clearer view of the complete structure of the problem. Figure \[fig:scommametamodel\] illustrates the main concepts of the metamodel using UML class diagram notation. The role of each one of these concepts is explained in the following paragraphs. s-COMMA models -------------- The metamodel defines the concepts appearing in models. Thus, conform to this metamodel an model must be composed by two main parts, the model and data. The model describes the structure of the problem and the data contain the constant values used by the model. In our front tool this problem’s structure is represented by class artifacts and the data concept is represented by the data artifact[^1] (see Figure \[fig:classartifact\]). ### Class artifacts Class artifacts have by default three compartments, the upper compartment for the class name, the middle compartment for attributes and the bottom one for constraint zones. By clicking on the class artifact its specification can be opened to define its class name, their attributes and constraint zones. Relationships can be used to define inheritance (a subclass inherits all attributes and constraint zones of its superclass) or composition between classes. ### Data artifacts Data artifacts have two compartments, one for the file name and another for both the constants and variable-assignments. Constants, also called data variables can be defined with a real, integer or enumeration type. Arrays of one dimension and arrays of two dimensions of constants are allowed. Variable-assignment corresponds to the assignment of a value to a variable of an object. Variable-assignments can also be performed if objects are inside an array (see an example in Section \[sec:stablemarriage\]). ![Class artifact used in s-COMMA GUI\[fig:classartifact\]](images/classartifact.eps){width="0.8\linewidth"} ### Attributes Attributes may represent decision variables, sets, objects or arrays. Decision variables can be defined by an integer, real or boolean type. Sets can be composed of integers or enumeration values. Objects are instances of classes which must be typed with their class name. Arrays of one and two dimensions are allowed, they can contain decision variables, sets or objects. Decision variables, sets and arrays can be constrained to a determined domain. ### Constraint Zones Constraint zones are used to group constraints encapsulating them inside a class. A constraint zone is stated with a name and it can contain the following elements: - *Constraints:* Typical operations and relations are provided to post constraints. For example, comparison relations ([``]{} [``]{}), arithmetic operations ([``]{} [``]{}), logical relations ([``]{} [``]{}), and set operations ([``]{}). - *Statements:* Forall and conditional statements are supported. The forall (e.g. [``]{}) is stated by declaring a loop-variable ([``]{}) and the set of values to be traversed ([``]{}). A loop-variable is a local variable and it is valid just inside the loop where it was declared. Conditionals are stated by means of if-else expressions. For instance, [``]{} where [``]{} is the condition, which can includes decision variables; and [``]{} and [``]{} are the alternatives, which may be statements or constraints. - *Objective:* objective functions are allowed and they can be stated by tagging the expression involved with the selected option (e.g. [``]{}). - *Global Constraints*: a basic set of global constraints (e.g. alldifferent, cumulatives) is supported. Additional constraints can be integrated to this basic set by means of extension mechanisms (for details refer to [@SotoICTAI07]). The stable marriage problem {#sec:stablemarriage} --------------------------- Let us now illustrate some of these concepts in the by means of the stable marriage problem. ![The stable marriage problem on the s-COMMA GUI\[fig:GUI\]](images/GUI.eps){width="1\linewidth"} Consider a group of $n$ women and a group of $n$ men who must marry. Each women has a preference ranking for her possible husband, and each men has a preference ranking for his possible wife. The problem is to find a matching between the groups such that the marriages are stable i.e., there are no pair of people of opposite sex that like each other better than their respective spouses. Figure \[fig:GUI\] shows a snapshot of the where the stable marriage problem is represented by a class diagram. This diagram is composed by three classes, one class to represent men, one to represent women, and a main class to describe the stable marriages. Once the user states a visual artifact, the corresponding textual version is automatically generated on the right-panel of the tool. For readability we illustrate the textual version of the problem in Fig. \[fig:stablemarriage\] //Model file 1. import StableMarriage.dat; 2. 3. class StableMarriage { 4. 5. Man man[menList]; 6. Woman woman[womenList]; 7. 8. constraint matchHusbandWife { 9. forall(m in menList) 10. woman[man[m].wife].husband = m; 11. 12. forall(w in womenList) 13. man[woman[w].husband].wife = w; 14. } 15. 16. constraint forbidUnstableCouples { 17. forall(m in menList){ 18. forall(w in womenList){ 19. man[m].rank[w] < man[m].rank[man[m].wife] -> 20. woman[w].rank[woman[w].husband] < woman[w].rank[m]; 21. 22. woman[w].rank[m] < woman[w].rank[woman[w].husband] -> 23. man[m].rank[man[m].wife] < man[m].rank[w]; 24. } 25. } 26. } 27.} 28. 29. class Man { 30. int rank[womenList]; 31. womenList wife; 32. } 33. 34. class Woman { 35. int rank[menList]; 36. menList husband; 37. } //Data file 1. enum menList := {Richard,James,John,Hugh,Greg}; 2. enum womenList := {Helen,Tracy,Linda,Sally,Wanda}; 3. Man StableMarriage.man := [Richard: {[Helen:5 ,Tracy:1, Linda:2, Sally:4, Wanda:3],_}, James : {[Helen:4 ,Tracy:1, Linda:3, Sally:2, Wanda:5],_}, John : {[Helen:5 ,Tracy:3, Linda:2, Sally:4, Wanda:1],_}, Hugh : {[Helen:1 ,Tracy:5, Linda:4, Sally:3, Wanda:2],_}, Greg : {[Helen:4 ,Tracy:3, Linda:2, Sally:1, Wanda:5],_}]; 4. Woman StableMarriage.woman := [Helen: {[Richard:1, James:2, John:4, Hugh:3, Greg:5],_}, Tracy: {[Richard:3, James:5, John:1, Hugh:2, Greg:4],_}, Linda: {[Richard:5, James:4, John:2, Hugh:1, Greg:3],_}, Sally: {[Richard:1, James:3, John:5, Hugh:4, Greg:2],_}, Wanda: {[Richard:4, James:2, John:3, Hugh:5, Greg:1],_}]; The class representing men (at line 29 in the model file) is composed by one array containing integer values which represents the preferences of a man, the array is indexed by the enumeration type [``]{} (at line 2 in the data file), thereby the 1st index of the array is [``]{}, the 2nd is [``]{}, the third is [``]{} and so on. Then, an attribute called [``]{} is defined (line 31), which represents the spouse of an object man. This variable has [``]{} as a type which means that its domain is given by the enumeration [``]{}. The definition of the class [``]{} is analogous. The class [``]{} has a more complex declaration. We first define two arrays, one called [``]{} which contains objects of the class [``]{} and other which contains objects of the class [``]{}. Each one represents the group of men and the group of women, respectively. The composition relationship between classes can be seen on the class diagram. At line 8 a constraint zone called [``]{} is stated. In this constraint zone, two [``]{} loops including a constraint are posted to ensure that the pairs man-wife match with the pairs woman-husband. The [``]{} constraint zone contains two loops including two logical formulas to ensure that marriages are stable. The data file is called by means of an import statement (at line 1). This file contains two enumeration types, [``]{} and [``]{}, which have been used in the model as a type, for indexing arrays, and as the set of values that loop-variables must traverse. [``]{} is a variable-assignment for the array called [``]{} defined at line 5 in the model file. This variable-assignment is composed by five objects (enclosed by [``]{}), one for each men of the group. Each of these objects has two elements, the first element[^2] is an array (enclosed by [``]{}). This array sets the preferences of a men, assigning the values to the array [``]{} of a [``]{} object (e.g. Richard prefers Tracy 1st, Linda 2nd, Wanda 3rd, etc). The second element is an underscore symbol (’\_’). This symbol is used to omit assignments, so the variable [``]{} remains as a decision variable of the problem i.e., a variable for which the solver must search a solution. Flat s-COMMA models {#sec:flatscomma} ------------------- Before explaining how models are mapped to their equivalent solver models, let us introduce the intermediate language. has been designed to simplify the transformation process from models to solver models. In much of the constructs supported by are transformed to simpler ones, in order to be closer to the form required by classical solver languages. is also defined by a metamodel. Figure \[fig:flatscommametamodel\] illustrates the main elements of the metamodel, where many concepts have been removed. Now, the metamodel is mainly a definition of a problem composed by variables (decision variables) and constraints. In order to transform to , several steps are involved, which are explained in the following. - Enumeration substitution: In general solvers do not support non-numeric types. So, enumerations are replaced by integer values. However, enumeration values are stored to show the results in the correct format. - Data substitution: Data variables stated in the model file are replaced by their corresponding values i.e., the value defined in the data file. - Loop unrolling: Loops are not widely supported by solvers, hence we generate an unrolled version of the forall loop. - Flattening composition: The hierarchy generated by composition is flattened. This process is done by expanding each object declared in the main class adding its attributes and constraints in the file. The name of each attribute has a prefix corresponding to the concatenation of the names of objects of origin in order to avoid name redundancy. - Conditional removal: Conditional statements are transformed to logical formulas. For instance, [``]{} is replaced by $(a\Rightarrow b) \wedge (a \vee c)$. - Logic formulas transformation: Some logic operators are not supported by solvers. For example, logical equivalence ($a \Leftrightarrow b$) and reverse implication ($a \Leftarrow b$). We transform logical equivalence expressing it in terms of logical implication ($(a \Rightarrow b) \wedge (b \Rightarrow a)$). Reverse implication is simply inverted ($b \Rightarrow a$). <!-- --> 1. variables: 2. 3. womenList man_wife[5] in [1,5]; 4. menList woman_husband[5] in [1,5]; 5. 6. constraints: 7. 8. woman_husband[man_wife[1]]=1; 9. woman_husband[man_wife[2]]=2; 10. woman_husband[man_wife[3]]=3; 11. ... 12. 13. man_wife[woman_husband[1]]=1; 14. man_wife[woman_husband[2]]=2; 15. man_wife[woman_husband[3]]=3; 16. ... 17. 18. 5<man_1_rank[man_wife[1]] -> 19. woman_1_rank[woman_husband[1]]<1; 20. 1<woman_1_rank[woman_husband[1]] -> 21. man_1_rank[man_wife[1]]<5; 22. 23. 1<man_1_rank[man_wife[1]] -> 24. woman_2_rank[woman_husband[2]]<3; 25. 3<woman_2_rank[woman_husband[2]] -> 26. man_1_rank[man_wife[1]]<1; 27. ... 28. 29. enum-types: 30. 31. menList := {Richard,James,John,Hugh,Greg}; 32. womenList := {Helen,Tracy,Linda,Sally,Wanda}; Figure \[fig:flatstablemarriage\] depicts the model of the stable marriage problem. The file is composed of two main parts, variables and constraints. Variables at lines 3-4 are generated by the flattening composition process. The array [``]{} composed by objects of type [``]{} is decomposed and transformed to a single array of decision variables. The array [``]{} contains the decision variables [``]{} of the original array [``]{}; and the array [``]{} contains the decision variables [``]{} of the original array [``]{}. The arrays rank of both objects [``]{} and [``]{} are not considered as decision variables since they have been filled with constants (at lines 3-4 of the data file in Figure \[fig:stablemarriage\]). The size of the array [``]{} is 5, this value is given by the enumeration substitution step which sets the size of the array with the size of the enumeration [``]{} ([``]{}). The domain [``]{} is also given by this step which states as domain an integer range corresponding to the number of elements of the enumeration used as a type ([``]{}) by the attribute [``]{}. The type of both arrays is maintained to give the solutions in the enumeration format. These values are stored in the block [``]{}. Lines 8-15 come from the loop unrolling phase of the forall statements of the [``]{} constraint zone. Likewise, lines 18-26 are generated by the loops of [``]{}. In these constraints, the data substitution step has replaced several constants with their corresponding integer values. Mapping Tool {#sec:mapping-tool} ============ In this section we explain the mechanisms provided by the MDD approach to develop our solver-translators. These translators are designed to perform the mapping from to solver models. We use the AMMA platform as our base tool to build them. The AMMA platform allows one to develop this task by means of two languages: KM3 [@Jouault2006KM3] and ATL [@Jouault2006ATL]. KM3 is used to define metamodels, and ATL is used to describe the transformation rules and also to generate the target file. KM3 {#sec:KM3} --- The Kernel Meta Meta Model (KM3) is a language to define metamodels. KM3 has been designed to support most metamodeling standards and it is based on the simple notion of classes to define each one of the concepts of a metamodel. These concepts will then be used by the transformation rules and to generate the target file. Figure \[fig:flatscommaKM3\] illustrates an extract of the metamodel written in KM3. 1. class Problem { 2. attribute name : String; 3. reference variables[1-*] container : Variable; 4. reference constraints[0-*] container : Constraint; 5. reference enumTypes[0-*] container : EnumType; 6. } 7. 8. class Variable { 9. attribute name : String; 10. attribute type : String; 11. reference array [0-1] container : Array; 12. reference domain container : Domain; 13. } 14. 15. class Array { 16. attribute row : Integer; 17. attribute col[0-1] : Integer; 18. } The KM3 metamodel states that the concept [``]{} is composed of one attribute and three references. The attribute [``]{} at line 2 represents the name of the model and it is declared with the basic type [``]{}. Line 3 simply states that the class [``]{} is composed by a set of objects of the class [``]{}. The reserved word [``]{} is used to declare links with instances of other classes and the statement [``]{} defines the multiplicity of the relationship. If the multiplicity statement is omitted the relationship is defined as [``]{}. Lines 4-5 are similar and define that the class [``]{} is also composed by [``]{} and [``]{} (values stored by the enumeration substitution step). Remaining classes are defined in the same way. ATL {#sec:ATL} --- The Atlas Transformation Language (ATL) allow us to define transformation rules according to one or several metamodels. The rules clearly state how concepts from source metamodels are matched to concepts of the target ones. Figure \[fig:rulesATL\] shows some of the ATL rules used to transform the concepts of the metamodel to the concepts of the Gecode/J metamodel. The metamodel of Gecode/J is not presented here since it is very close to the metamodel. Indeed, most CP solver languages are used to express quite the same concepts and is designed to be as close as possible from the solving level. This is a great asset because transformation rules become simple: we mainly need one to one transformations. 1. module FlatsComma2GecodeJ; 2. create OUT : GecodeJ from IN : FlatsComma; 3. 4. rule Problem2Problem { 5. from 6. s : FlatsComma!Problem ( 7. ) 8. to 9. t : GecodeJ!Problem( 10. name <- s.name, 11. variables <- s.variables, 12. constraints <- s.constraints, 13. enumTypes <- s.enumTypes 14. ) 15. } 16. 17. rule Variable2Variable { 18. from 19. s : FlatsComma!Variable ( 20. not s.isArrayVariable 21. ) 22. to 23. t : GecodeJ!Variable ( 24. name <- s.name, 25. type <- s.type, 26. domain <- s.domain 27 ) 28. } 29. 30. helper context FlatsComma!Variable def: 31. isArrayVariable : Boolean= 32. not self.array.oclIsUndefined(); The first line of this file specifies the name of the transformation. A [``]{} is used to define and regroup a set of rules and helpers. Rules define the mappings, and helpers allow to define factorized ATL code that can be called from different points of the ATL file (they can be viewed as the ATL equivalent to Java methods). Line 2 states the target ([``]{}) and source metamodels ([``]{}). The first rule presented is called [``]{} and defines the matching between the concepts [``]{} expressed in and Gecode/J. The source elements are stated with the reserved word [``]{} and the target ones with the reserved word [``]{}. These elements are declared like variables with a name ([``]{}) and a type corresponding to a class in a metamodel ([``]{}). In the target part of the rule the name attribute of the problem is assigned to the Gecode/J name ([``]{}), this is just an string assignment. However, the following two statement are assignments between concepts that are defined as [``]{} in the metamodel. So, they need a specific rule to carry out the transformation. For instance, the KM3 metamodel defines that the reference [``]{} is composed by a set of [``]{} elements. Thus, the statement ([``]{}) calls implicitly the rule [``]{}, which defines the match between each element of objects [``]{}. It can be highlighted that the ATL engine requires a unique name for each rule and a unique matching case: [``]{} and [``]{} blocks. When several rules can be applied a guard (the boolean test in line 20) over the from statement must remove choice ambiguities. The [``]{} rule matches three elements. The first two statements are simple string assignments and the last one is a reference assignment. Let us remark that a second rule to process array variables has been defined (but not presented here) which includes an additional statement for the array element. These two rules are distinguished according to complementary guards over the source block using the helper [``]{}. Guards act as filter on the source variable instances to process. The presented helper [``]{} applies on variable instances in models and returns true when the instance contains an array element. ATL inherits from OCL [@OMG_OCL] syntax and semantics; and most OCL functions and types are available within ATL. Although the rules used here are not complex, ATL is able to perform more difficult rules. For instance, the most difficult rule we defined, was the transformation rule from matrix containing sets, which must be unrolled in the [ECL$^{i}$PS$^{e}$]{} models (since set matrix are not supported). This unroll process is carried out by defining a single set in [ECL$^{i}$PS$^{e}$]{} for each cell in the matrix. The name of each single variable is composed by the name of the matrix, and the corresponding row and column index. 1. rule Problem2Problem { 2. from 3. s : FlatsComma!Problem ( 4. s.hasSetMatrix 5. ) 6. to 7. t : ECLiPSe!Problem ( 8. name <- s.name, 9. constraints <- s.constraints, 10. enumTypes <- s.enumTypes 11. ) 12. do { 13. t.variables <- s.variables->collect(e| 14. if e.isSetMatrix() then 15. thisModule.getMatrixCells(e)->collect(f| 16. thisModule.SetMatrixVariable2Variable(f.var,f.i,f.j) 17. ) 18. else 19. e 20. endif 21. )->flatten(); 22. } 23. } 24. 25. rule SetMatrixVariable2Variable(var : FlatsComma!Variable, 26. i : Integer, j : Integer) { 27. to 28. t : ECLiPSe!Variable( 29. name <- var.name + i.toString() + '_' + j.toString(), 30. type <- var.type, 31. domain <- var.domain, 32. ) 33. do { 34. t; 35. } 36. } Figure \[fig:setmat\] shows the rule [``]{} defined for [ECL$^{i}$PS$^{e}$]{}, this rule has a condition (line 4) to check whether set matrix are defined in the model. If the condition is true, [``]{}, [``]{} and [``]{} are matched normally, but [``]{} has a special procedure to decompose the set matrix. This procedure begins at line 12 with a [``]{} block. In this block, the [``]{} loop iterates over the variables. Then, each of these variables ([``]{}) is checked to determine whether it has been defined as a set matrix (line 14). If this occurs, the helper [``]{} calculates the set of tuples corresponding to all the cells of the matrix ([``]{} is used to call explicitly helpers or rules). Each tuple is composed of the variable ([``]{}), a row index ([``]{}) and a column index ([``]{}). Then, the rule [``]{} is applied to each tuple in order to generate the [ECL$^{i}$PS$^{e}$]{} variables. This rule does not contain a source block since the source elements are the input parameters. The rule sets to the attribute [``]{}, the concatenation of the name of the matrix with the respective row ([``]{}) and column ([``]{}). Attributes [``]{} and [``]{} are also matched. Finally, [``]{} is an OCL inherited method used to match the generated set of variables with [``]{}. ATL is also used to generate the solver target file. This is possible by defining a new ATL file (called generically ATL2Text) where we can embed the concepts of the metamodel in the syntax of the target file. This is done by means of a querying facility that enables to specify requests onto models. 1. query GecodeJ2Text = GecodeJ!Problem.allInstances()-> 2. asSequence()->first().toString2(). 3. writeTo('./GecodeJ/Samples/'+ thisModule.getFileName() + 4. '.java'); 5. 6. helper context GecodeJ!Problem def: toString2() : String= 7. 'package comma.solverFiles.gecodej;\n' + 8. 'import static org.gecode.Gecode.*;\n' + 9. 'import static org.gecode.GecodeEnumConstants.*;\n' + 10. ... 11. 12. self.variables->collect(e | e.toString2()) 13. ->iterate(e; acc:String = '' | acc +' '+e) + 14. ... 15. '}\n\n' 16. ; 17. 18. helper context GecodeJ!Variable def: toString2() : 19. String= 20. if self.array.oclIsUndefined() then 21. 'IntVar ' + self.name + ' = new IntVar(this,\"' + 22. self.name + '\",' + self.domain.toString2() +');\n' + 23. ' vars.add('+ self.name +');\n' 24. else if self.array.col.oclIsUndefined() then 25. 'VarArray<IntVar> ' + self.name + ' = initialize(\"' + 26. self.name + '\",' + self.array.toString2() + 27. ',' + self.domain.toString2()+');\n' + 28. ' vars.addAll(' + self.name + ');\n' 29. else 30. 'VarMatrix<IntVar> ' + self.name + ' = initialize(\"' + 31. self.name + '\",' + self.array.toString2() + 32. ',' + self.domain.toString2()+');\n' + 33. ' vars.addAll(' + self.name + ');\n' 34. endif endif 35. ; Figure \[fig:ATLtoText\] shows a fragment of the GecodeJ2Text definition to generate the Gecode/J file. Lines 1-4 states the query on the [``]{} concept and defines the target file. Queries are able to call helpers, which allow us to build the string to be written in the target solver file. This query calls the helper [``]{} defined for the concept [``]{}. This helper is stated at line 6 and it creates first the string corresponding to the headers (package and import statements) of a Gecode/J model. Then, at lines 12-13 the string corresponding to the variables declarations is created. This is done by iterating the collection of variables and calling the corresponding [``]{} helper for the [``]{} instances. This helper is declared at line 18, it defines three possible variable declarations, single variable ([``]{}), a one dimension array ([``]{}), and a two dimension array ([``]{}). The alternatives are chosen by means of an if-else statement. The condition [``]{} checks whether the concept array is undefined. If this occurs, the variable corresponds to a single variable. The string representing this declaration uses [``]{} which refers to the name of the variable, [``]{} calls a helper to get the string representing the domain of the variable. The next alternative tests if the attribute [``]{} of the [``]{} is undefined, in this case the variable is a one dimension array, otherwise it is a two dimension array. The call [``]{} is used in the two last alternatives, it returns the string corresponding to the size of arrays. Figure \[fig:stablegecodej\] depicts an extract of the Gecode/J file generated for the stable marriage problem. Lines 1-3 states the headers. Line 6 declares the array called [``]{}. which is initialized with size [``]{} and domain [``]{}. At line 8 the array is added to a global array called [``]{} for performing the labeling process. Lines 14-19 illustrate some constraints, which are stated by means of the [``]{} method. 1. package comma.solverFiles.gecodej; 2. import static org.gecode.Gecode.*; 3. import static org.gecode.GecodeEnumConstants.*; 4. ... 5. 6. VarArray<IntVar> man_wife = 7. initialize("man_wife",5,1,5); 8. vars.addAll(man_wife); 9. 10. VarArray<IntVar> woman_husband = 11. initialize("woman_husband",5,1,5); 12. vars.addAll(woman_husband); 13. 14. post(this, new Expr().p(get(this,woman_husband, 15. get(man_wife,1))),IRT_EQ, new Expr().p(1)); 16. post(this, new Expr().p(get(this,woman_husband, 17. get(man_wife,2))),IRT_EQ, new Expr().p(2)); 18. post(this, new Expr().p(get(this,woman_husband, 19. get(man_wife,3))),IRT_EQ, new Expr().p(3)); 20. ... TCS {#sec:TCS} --- TCS [@Jouault2006TCS] (Textual Concrete Syntax) is another language provided by the AMMA platform. TCS is not mandatory to add a new translator but it is involved in the process since it is the language used to parse the file. TCS is able to perform this task by bridging the metamodel with the grammar. 1. template Problem 2. : "variables" ":" variables 3. "constraints" ":" constraints 4. "enum-types" ":" enumTypes 5. ; 6. 7. template Variable 8. : type name (isDefined(array) ? array) "in" domain ";" 9. ; 10. 11. template Array 12. : "[" row (isDefined(col) ? "," col ) "]" 13. ; Figure \[fig:TCS\] shows an extract of the TCS file for . Each class of the metamodel has a dedicated template declared with the same name. Within templates, words between double quotes are tokens in the grammar (e.g. [``]{}, [``]{}). Words without double quotes are used to introduce the corresponding list of concepts. For instance [``]{} is defined as a reference to objects [``]{} in the class [``]{} of the metamodel. Thus, [``]{} is used to call their associate template i.e., the [``]{} template. This template defines the syntactic structure of a variable declaration. It has a conditional structure ([``]{}), which means that the template [``]{} is only called if the variable is defined as an array. Transformation process {#sec:trans-process} ---------------------- TCS and KM3 work together and their compilation generates a Java package (which includes lexers, parsers and code generators) for (FsC), which is then used by the ATL files to generate the target model. Figure \[fig:AMMA\] depicts the complete transformation process. The file is the output of the , this file is taken by the Java package which generates a XMI [^3] (XML Metadata Interchange) for , this file includes an organized representation of models in terms of their concepts in order to facilitate the task of transformation rules. Over this file ATL rules act and generate a XMI file for Gecode/J. Finally this file is taken by the Gecode/J2Text which builds the solver file. ![The AMMA model-driven process on the example of Flat s-COMMA (FsC) to Gecode/J.\[fig:AMMA\]](images/mm-amma.eps){width="1\linewidth"} The complete process involves TCS, KM3 and ATL. But, the integration of a new translator just requires KM3 and ATL (the mapping tool only needs one TCS file). As we mention in Section \[sec:ATL\], solver metamodels are almost equivalents, and ATL rules are mainly one to one mappings. As a consequence, the development of KM3 and ATL rules for new solver-translators should not be a hard task. So, we could say that the concrete work for plugging a new solver is reduced to the definition of the ATL2Text file. Currently, There are two versions of our mapping tool, one with AMMA translators and one with translators written by hand (in Java), which we got from a preliminary development phase of the system. Comparing both approaches, let us make the following concluding remarks. - The development of hand-written translators is in general a hard task. Their creation, modification and reuse require to have a deep insight in the code and in the architecture of the platform, even more if they have a specific and/or complex design. For instance, the developer may be forced to directly use lexers and parsers, or a given library which provides specific methods to generates the target files. - The development of AMMA translators does not require advanced language implementation skills. We show that the use of KM3 and ATL is not really a hard task. Moreover, AMMA is supported by a set of tools [@EclipseM2M] which provide a great framework to create and manipulate KM3, ATL and TCS models, and also for project handling. An independent definition of syntax concerns (ATL2Text) from metamodel concepts (KM3) is another advantage which gives us a more organized view that facilitates the creation and reuse of translators. - The development of hand-written translators requires more code lines. In our implementation, the source files of Java translators are approximately 60% bigger than the AMMA translators source files (ATL+KM3). Direct code generation {#sec:simpler-process} ---------------------- There is another approach to develop translators using the AMMA platform. For instance, if we want to use just the features that are supported by the solver, we can omit the transformation rules and we can apply the ATL2Text directly on the source metamodel. Figure \[fig:simpler\] shows this direct code generation process. ![Direct code generation.\[fig:simpler\]](images/mm-amma-simpler.eps){width="0.8\linewidth"} Although this approach is simpler, it is less flexible since we lose the possibility of using interesting rules transformations such as the set matrix decomposition explained in Section \[sec:ATL\]. Experiments {#sec:bench} =========== We have carried out a set of tests in order to first compare the performance of AMMA translators (using transformation rules) with translators written by hand, and second, to show that the automatic generation of solver files does not lead to a loss of performance in terms of solving time. Tests have been performed on a 3GHz Pentium 4 with 1GB RAM running Ubuntu 6.06, and benchmarks used are the following [@s-COMMA]: - Send: The cryptoarithmetic puzzle Send + More = Money. - Stable: The stable marriage problem presented. - Queens: The N-Queens problem (n=10 and n=18). - Packing: Packing 8 squares into a square of area 25. - Production: A production-optimization problem. - Ineq20: 20 Linear Inequalities. - Engine: The assembly of a car engine subject to design constraints. - Sudoku: The Sudoku logic-based number placement puzzle. - Golfers: To schedule a golf tournament. ------------ ------- ------- ------- ------- ------- sC to Benchmark FsC Java AMMA Java AMMA Send 0.237 0.052 0.688 0.048 0.644 Stable 0.514 0.137 1.371 0.143 1.386 10-Queens 0.409 0.106 1.301 0.115 1.202 18-Queens 0.659 1.122 3.194 0.272 2.889 Packing 0.333 0.172 1.224 0.133 1.246 Production 0.288 0.071 0.887 0.066 0.783 20 Ineq. 0.343 0.072 0.895 0.072 0.891 Engine 0.285 0.071 0.815 0.071 0.844 Sudoku 3.503 1.290 4.924 0.386 4.196 Golfers 0.380 0.098 1.166 0.111 1.136 ------------ ------- ------- ------- ------- ------- : Translation times (seconds)\[table:translators\] Table \[table:translators\] shows preliminary results comparing AMMA translators with translators written by hand (in Java). Column 3 and 4 give the translation times using Java and AMMA translators, from (FsC) to Gecode/J and from to [ECL$^{i}$PS$^{e}$]{}, respectively. Translation times from (sC) to are given for reference in column 2 (This process involves syntactic and semantic checking, and the transformations explained in Section \[sec:flatscomma\]). The results show that AMMA translators are slower than Java translators, this is unsurprising since Java translators have been designed specifically for . They take as input a definition and generate the solver file directly. The transformation process used by AMMA translators is not direct, it performs intermediate phases (XMI to XMI). Moreover, the AMMA tools are under continued development and many optimizations can be done especially on the parsing process of the source file (more than 60% of the time is consumed by this process). Although our primary scope is not focused on performance, we expect to improve this using the next AMMA version. However, despite of this speed difference, we believe translation times using AMMA are acceptable and this loss of performance is a reasonable price to pay for using a generic approach. In Table \[table:modelsize\] we compare the solver files generated by AMMA translators [^4] with native solver files version written by hand. The data is given in terms of *solving time(seconds)/model size(tokens)*. Results show that generated solver files are in general bigger than solver versions written by hand. This is explained by the loop unrolling and flattening composition processes presented in Section \[sec:flatscomma\]. However, this increase in terms of code size does not cause a negative impact on the solving time. In general, generated solver versions are very competitive with hand-written versions. Table \[table:modelsize\] also shows that Gecode/J files are bigger than [ECL$^{i}$PS$^{e}$]{} files, this is because the Java syntax is more verbose than the [ECL$^{i}$PS$^{e}$]{} syntax. ------------ --------- --------- -------- -------- Benchmark hand AMMA hand AMMA Send 0.002/ 0.002/ 0.01/ 0.01/ 590 615 231 329 Stable 0.005/ 0.005/ 0.01/ 0.01/ 1898 8496 1028 4659 10-Queens 0.003/ 0.003/ 0.01/ 0.01/ 460 9159 193 1958 18-Queens 0.008/ 0.008/ 0.02/ 0.02/ 460 30219 193 6402 Packing 0.009/ 0.009/ 0.49/ 0.51/ 663 12037 355 3212 Production 0.026/ 0.028/ 0.014/ 0.014/ 548 1537 342 703 20 Ineq 13.886/ 14.652/ 10.34/ 10.26/ 1576 1964 720 751 Engine 0.012/ 0.012/ 0.01/ 0.01/ 1710 1818 920 1148 Sudoku 0.007/ 0.007/ 0.21/ 0.23/ 1551/ 33192/ 797/ 11147/ Golfers 0.005/ 0.005/ 0.21/ 0.23/ 618/ 4098/ 980/ 1147/ ------------ --------- --------- -------- -------- : Solving times(seconds) and model sizes (number of tokens)\[table:modelsize\] Related Work {#sec:related} ============ is as related to solver-independent languages as object-oriented languages. In the next paragraphs we compare our approach to languages belonging to these groups. Solver-Independent Constraint Modeling -------------------------------------- Solver-independence in constraint modeling languages is a recent trend. Just a few languages have been developed under this principle. One example is MiniZinc, which is mainly a subset of constructs provided by Zinc, its syntax is closely related to OPL and its solver-independent platform allows to translate models into Gecode and [ECL$^{i}$PS$^{e}$]{} solver code. This model transformation is performed by a rule-based system called Cadmium [@BrandPADL2008] which can be regarded as an extension of Term-Rewriting (TR) [@Baader1998] and Constraint Handling Rules (CHR) [@FruehwirthJLP1998]. This process also involves an intermediate model called FlatZinc, which plays a similar role than , to facilitate the translation. The implementation of our approach is quite different to Cadmium. While Cadmium is supported by CHR and TR, our approach is based on standard model transformation techniques, which we believe give us some advantages. For instance, ATL and KM3 are strongly supported by the model engineering community. A considerable amount of documentation and several examples are available at the Eclipse IDE site [@EclipseM2M]. Tools such as Eclipse plug-ins are also available for developing and debugging applications. It is not less important to mention that ATL is considered as a standard solution for model transformation in Eclipse. On the technical side, the Cadmium system is strongly tied to MiniZinc. This is a great advantage since the rules operate directly on Zinc expression, so transformation rules are often compact. However, this integration forces to merge the metamodel concepts of MiniZinc with the MiniZinc syntax. This property makes Cadmium programs more compact but less modular than our approach, where the syntax is defined independently from the metamodel (as we have presented in Section \[sec:mapping-tool\]). Essence is another solver-independent language. Its syntax is addressed to users with a background in discrete mathematics, this style makes Essence a specification language rather than a modeling language. The Essence execution platform allows to map specifications into [ECL$^{i}$PS$^{e}$]{} and Minion solver [@GentECAI96]. A model transformation system called Conjure has been developed, but the integration of solver translators is not its scope. Conjure takes as input an Essence specification and transform it to an intermediate OPL-like language called Essence’. Translators from Essence’ to solver code are written by hand. From a language standpoint, is as expressive as MiniZinc and Essence, in fact these approaches provide similar constructs and modeling features. However, a main feature of that strongly differences it from aforementioned languages is the object-oriented framework provided and the possibility of modeling problems using a visual language. Object-Oriented Constraint Modeling and Visual Environments ----------------------------------------------------------- The capability of defining constraints in an object-oriented modeling language is the base of the object-oriented constraint modeling paradigm. The first attempt in performing this combination was on the development of ThingLab [@BorningTOPLAS1981]. This approach was designed for interactive graphical simulation. Objects were used to represent graphical elements and constraints defined the composition rules of these objects. COB [@JayaramanPADL2002] is another object-oriented language, but its framework is not purely based on this paradigm. In fact, the language is a combination of objects, first order formulas and CLP (Constraint Logic Programming) predicates. A GUI tool is also provided for modeling problems using CUML, a UML-like language. The focus of this language was the engineering design. Modelica [@FritzsonECCOP1998] is another object-oriented approach for modeling problems from the engineering field, but it is mostly oriented towards simulation. Gianna [@PaltrinieriCP1995] is a precursor visual environment for modeling CSP. But its modeling style is not object-oriented and the level of abstraction provided is lower than in UML-like languages. In this tool, CSPs are stated as constraint graphs where nodes represent the variables and the edges represent the constraints. Although these approaches do not have a system to plug-in new solvers and were developed for a specific application domain, we believe it is important to mention them. It is important to clarify too, that object-oriented capabilities are also provided by languages such as CoJava [@BrodskyCP2006]; and in libraries such as Gecode or ILOG SOLVER. The main difference here is that the host language provided is a programming language but not a high-level modeling language. As we have explained in Section \[sec:intro\], advanced programming skills may be required to deal with these tools. Conclusions and Future Work {#sec:conclusion} =========================== In this work we have presented , an extensible MDD platform for modeling CSPs. The whole system is composed by two main parts: A modeling tool and a mapping tool, which provide to the users the following three important facilities: - A visual modeling language that combines the declarative aspects of constraint programming with the useful features of object-oriented languages. The user can state modular models in an intuitive way, where the compositional structure of the problem can be easily maintained through the use of objects under constraints. - Models are stated independently from solver languages. Users are able to design just one model and to target different solvers. This clearly facilitates experimentation and benchmarking. - A model transformation system supported by the AMMA platform which follows the standards of the software engineering field. The system allows users to plug-in new solvers without writing translators by hand. Currently, we do not use as our source model, because its metamodel is quite large and defining generic mappings to different solver metamodels will be a serious challenge. However we believe that this task will lead to an interesting future work, for instance to perform reverse engineering (e.g. Gecode/J to or [ECL$^{i}$PS$^{e}$]{} to ). The use of AMMA for model optimization will be useful too, for instance to eliminate redundant or useless constraints. The definition of selective mappings is also an interesting task, for instance to decide, depending on the solver used, whether loops must be unrolled or the composition must be flattened. We are grateful to the support of this research from the “Pontificia Universidad Católica de Valpara[í]{}so" under the grant “Beca de Estudios Básica", and to Frédéric Jouault for his support on the implementation of the AMMA translators. ANTLR Reference Manual, 2007. http://www.antlr.org. F. Baader and T. Nipkow. Term rewriting and all that, Cambridge Univ. Press, 1998. A. Borning. The Programming Languages Aspects of ThingLab, a Constraint-Oriented Simulation Laboratory. *ACM Transactions on Programming Languages and Systems (ACM TOPLAS),* 3(4), pages 353–387, 1981. A. Brodsky and H. Nash. CoJava: Optimization Modeling by Nondeterministic Simulation. In *Proceedings of the 12th International Conference on Principles and Practice of Constraint Programming (CP 2006)* 3(4). LNCS, vol. 4204, pages 91–106, 2006. S. Brand, G. J. Duck, J. Puchinger and P. J. Stuckey. Flexible, Rule-Based Constraint Model Linearisation. In *Proceedings of the 10th Symposium on Practical Aspects of Declarative Languages (PADL 2008)*. LNCS, vol. 4902, pages 68–83, 2008. D. Diaz and P. Codognet. The GNU Prolog System and its Implementation. In *Proceedings of the 2000 ACM Symposium on Applied Computing (SAC 2000)*, pages 728–732, 2000. Eclipse Model-to-model transformation, 2008.\ http://www.eclipse.org/m2m/. A. M. Frisch, C. Jefferson, B. Martínez Hernández and I. Miguel. The Rules of Constraint Modelling. In *Proceedings of the 19th International Joint Conference on Artificial Intelligence (IJCAI 2005)*, pages 109–116, 2005. A. M. Frisch, M. Grum, C. Jefferson, B. Martínez Hernández and I. Miguel. The Design of ESSENCE: A Constraint Language for Specifying Combinatorial Problems. In *Proceedings of the 20th International Joint Conference on Artificial Intelligence (IJCAI 2007)*, pages 80–87, 2007. P. Fritzson and V. Engelson. Modelica – A Unified Object-Oriented Language for System Modeling and Simulation. In *Proceedings of the 12th European Conference on Object-Oriented Programming (ECOOP 1998)*. LNCS, vol. 1445, pages 67–90, 1998. T. W. Fr[ü]{}hwirth. Theory and Practice of Constraint Handling Rules. *Journal of Logic Programming* 37(1-3), pages 95–138, 1998. Gecode System, 2006. http://www.gecode.org. I. P. Gent, C. Jefferson and I. Miguel. Minion: A Fast Scalable Constraint Solver. In *Proceedings of the 17th European Conference on Artificial Intelligence (ECAI 2006)*, pages 98–102, 2006. L. Granvilliers and F. Benhamou. Algorithm 852: RealPaver: an interval solver using constraint satisfaction techniques. *ACM Transactions on Mathematical Software (ACM TOMS),* 32(1), pages 138–156, 2006. J. Jaffar and J.-L. Lassez. Constraint Logic Programming. In *Proceedings of the 14th Annual ACM Symposium on Principles of Programming Languages (POPL 1987)*, pages 111–119, 1987. B. Jayaraman and P. Tambay. Modeling Engineering Structures with Constrained Objects. In *Proceedings of the 4th Symposium on Practical Aspects of Declarative Languages (PADL 2002)*. LNCS, vol. 2257, pages 28–46, 2002. F. Jouault and I. Kurtev. Transforming Models with ATL. In *Proceedings of Satellite Events at the 8th International Conference on Model Driven Engineering Languages and Systems (MoDELS Satellite Events 2005)*. LNCS, vol. 3844, pages 128–138, 2005. F. Jouault and J. B[é]{}zivin. KM3: A DSL for Metamodel Specification. In *Proceedings of the 8th IFIP WG 6.1 International Conferenceon Formal Methods for Open Object-Based Distributed Systems (FMOODS 2006)*. LNCS, vol. 4037, pages 171–185, 2006. F. Jouault, J. B[é]{}zivin and I. Kurtev KM3: A DSL for Metamodel Specification. In *Proceedings of the 5th International Conference on Generative Programming and Component Engineering (GPCE 2006)*, pages 249–254, 2006. I. Kurtev, J. B[é]{}zivin, F. Jouault and P. Valduriez. Model-based DSL frameworks. In *Proceedings of Companion to the 21th Annual ACM SIGPLAN Conference on Object-Oriented Programming, Systems, Languages, and Applications (OOPSLA Companion 2006)*, pages 602–616, 2006. N. Nethercote, P. J. Stuckey, R. Becket, S. Brand, G. J. Duck and G. Tack. MiniZinc: Towards a Standard CP Modelling Language. In *Proceedings of the 13th International Conference on Principles and Practice of Constraint Programming (CP 2007)* 3(4). LNCS, vol. 4741, pages 529–543, 2007. OMG - Object Constraint Language (OCL) 2.0, 2006.\ http://www.omg.org/cgi-bin/doc?formal/2006-05-01 OMG - The Unified Modeling Language (UML) 2.1.1 Infrastructure Specification, 2007. http://www.omg.org/spec/UML/2.1.2/. OMG - Model Driven Architecture (MDA) Guide V1.0.1, 2003 http://www.omg.org/cgi-bin/doc?omg/03-06-01. M. Paltrinieri. A Visual Constraint-Programming Environment. In *Proceedings of the 1st International Conference on Principles and Practice of Constraint Programming (CP 1995)* 3(4). LNCS, vol. 976, pages 499–514, 1995. J.-F. Puget. A C++ implementation of CLP. In *Proceedings of the Second Singapore International Conference on Intelligent Systems (SCIS 1994)*, 1994. R. Rafeh, M. J. García de la Banda, K. Marriott and M. Wallace. From Zinc to Design Model. In *Proceedings of the 9th Symposium on Practical Aspects of Declarative Languages (PADL 2007)*. LNCS, vol. 4354, pages 215–229, 2007. s-COMMA System, 2008. http://www.inf.ucv.cl/ rsoto/s-comma. R. Soto and L. Granvilliers. The Design of COMMA: An Extensible Framework for Mapping Constrained Objects to Native Solver Models. In *Proceedings of the 19th IEEE International Conference on Tools with Artificial Intelligence (ICTAI 2007)*, pages 243–250, 2007. P. Van Hentenryck. The OPL Language. MIT Press, 1999. M. Wallace, S. Novello and J. Schimpf. Technical report, IC-Parc, Imperial College, London, 1997. [^1]: Artifacts used on the have been adapted from the class artifact provided by the UML Infrastructure Library Basic Package. This adaptation is completely allowed by the UML Infrastructure Specification [@OMG_UML]. [^2]: Let us note that we use standard modeling variable-assignments, that is, assignments are performed respecting the order of the class’ attributes: the first element of the variable-assignment is matched with the first attribute of the class, the second element of the variable-assignment with the second attribute of the class and so on. [^3]: XMI is the standard used for exchanging metadata in MDD architectures. [^4]: In the comparison, we do not consider solver files generated by Java translators. They do not have relevant differences compared to solver files generated by AMMA translators.

--- abstract: 'We show that for certain classes of actions of ${{\mathbb Z}}^d,\,\,d\ge 2$, by automorphisms of the torus any measurable conjugacy has to be affine, hence measurable conjugacy implies algebraic conjugacy; similarly any measurable factor is algebraic, and algebraic and affine centralizers provide invariants of measurable conjugacy. Using the algebraic machinery of dual modules and information about class numbers of algebraic number fields we consruct various examples of ${{\mathbb Z}}^d$-actions by Bernoulli automorphisms whose measurable orbit structure is rigid, including actions which are weakly isomorphic but not isomorphic. We show that the structure of the centralizer for these actions may or may not serve as a distinguishing measure–theoretic invariant.' address: - 'A. Katok: Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA' - 'S. Katok: Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA' - 'K. Schmidt: Mathematics Institute, University of Vienna, Strudlhofgasse 4, A-1090 Vienna, Austria, Erwin Schrödinger Institute for Mathematical Physics, Boltzmanngasse 9, A-1090 Vienna, Austria' author: - Anatole Katok - Svetlana Katok - Klaus Schmidt title: 'Rigidity of measurable structure for ${{\mathbb Z}}^d$–actions by automorphisms of a torus' --- [^1] Introduction; description of results ==================================== In the course of the last decade various rigidity properties have been found for two different classes of actions by higher–rank abelian groups: on the one hand, certain Anosov and partially hyperbolic actions of ${{\mathbb Z}}^d$ and ${{\mathbb R}}^d,\,\, d\ge 2$, on compact manifolds ([@KS1; @KS2; @KS4]) and, on the other, actions of ${{\mathbb Z}}^d,\,\, d\ge 2$, by automorphisms of compact abelian groups (cf. e.g. [@KSch; @KiS]). Among these rigidity phenomena is a relative scarcity of invariant measures which stands in contrast with the classical case $d=1$ ([@KS3]). In this paper we make the first step in investigating a different albeit related phenomenon: rigidity of the measurable orbit structure with respect to the natural smooth invariant measure. In the classical case of actions by ${{\mathbb Z}}$ or ${{\mathbb R}}$ there are certain natural classes of measure–preserving transformations which possess such rigidity: ergodic translations on compact abelian groups give a rather trivial example, while horocycle flows and other homogeneous unipotent systems present a much more interesting one [@R1; @R2; @R3]. In contrast to such situations, individual elements of the higher–rank actions mentioned above are Bernoulli automorphisms. The measurable orbit structure of a Bernoulli map can be viewed as very “soft”. Recall that the only metric invariant of Bernoulli automorphisms is entropy ([@O]); in particular, weak isomorphism is equivalent to isomorphism for Bernoulli maps since it implies equality of entropies. Furthermore, description of centralizers, factors, joinings and other invariant objects associated with a Bernoulli map is impossible in reasonable terms since each of these objects is huge and does not possess any discernible structure. In this paper we demonstrate that some very natural actions of ${{\mathbb Z}}^d,\,\,d\ge 2$, by Bernoulli automorphisms display a remarkable rigidity of their measurable orbit structure. In particular, isomorphisms between such actions, centralizers, and factor maps are very restricted, and a lot of algebraic information is encoded in the measurable structure of such actions (see Section \[s:rigidity\]). All these properties occur for broad subclasses of both main classes of actions of higher–rank abelian groups mentioned above: Anosov and partially hyperbolic actions on compact manifolds, and actions by automorphisms of compact abelian groups. However, at present we are unable to present sufficiently definitive general results due to various difficulties of both conceptual and technical nature. Trying to present the most general available results would lead to cumbersome notations and inelegant formulations. To avoid that we chose to restrict our present analysis to a smaller class which in fact represents the intersection of the two, namely the actions of ${{\mathbb Z}}^d,\,\,d\ge 2$, by automorphisms of the torus. Thus we study the measurable structure of such actions with respect to Lebesgue (Haar) measure from the point of view of ergodic theory. Our main purpose is to demonstrate several striking phenomena by means of applying to specific examples general rigidity results which are presented in Section \[s:rigidity\] and are based on rigidity of invariant measures developed in [@KS3] (see [@KaK] for further results along these lines including rigidity of joinings). Hence we do not strive for the greatest possible generality even within the class of actions by automorphisms of a torus. The basic algebraic setup for irreducible actions by automorphisms of a torus is presented in Section \[s:irred\]. Then we adapt further necessary algebraic preliminaries to the special but in a sense most representative case of Cartan actions, i.e. to ${{\mathbb Z}}^{n-1}$–actions by hyperbolic automorphisms of the $n$–dimensional torus (see Section \[s:ca\]). The role of entropy for a smooth action of a higher–rank abelian group $G$ on a finite-dimensional manifold is played by the [*entropy function*]{} on $G$ whose values are entropies of individual elements of the action (see Section \[ss:high-rank\] for more details) which is naturally invariant of isomorphism and also of weak isomorphism and is equivariant with respect to a time change. In Section \[s:examples\] we produce several kinds of specific examples of actions by ergodic (and hence Bernoulli) automorphisms of tori with the same entropy function. These examples provide concrete instances when general criteria developed in Section \[s:rigidity\] can be applied. Our examples include: - actions which are not weakly isomorphic (Section \[ss:not weakly\]), - actions which are weakly isomorphic but not isomorphic, such that one action is a maximal action by Bernoulli automorphisms and the other is not (Section \[ss:cent\]), - weakly isomorphic, but nonisomorphic, maximal actions (Section \[ss:max-Cart\]). Once rigidity of conjugacies is established, examples of type (i) appear in a rather simple–minded fashion: one simply constructs actions with the same entropy data which are not isomorphic over ${{\mathbb Q}}$. This is not surprising since entropy contains only partial information about eigenvalues. Thus one can produce actions with different eigenvalue structure but identical entropy data. Examples of weakly isomorphic but nonisomorphic actions are more sophisticated. We find them among Cartan actions (see Section \[s:ca\]). The centralizer of a Cartan action in the group of automorphisms of the torus is (isomorphic to) a finite extension of the acting group, and in some cases Cartan actions isomorphic over ${{\mathbb Q}}$ may be distinguished by looking at the index of the group in its centralizer (type (ii); see Examples 2a and 2b). The underlying cause for this phenomenon is the existence of algebraic number fields $K={{\mathbb Q}}(\lambda)$, where $\lambda$ is a unit, such that the ring of integers $\mathcal O_K\ne{{\mathbb Z}}[\lambda]$. In general finding even simplest possible examples for $n=3$ involves the use of data from algebraic number theory and rather involved calculations. For examples of type (ii) one may use some special tricks which allow to find some of these and to show nonisomorphism without a serious use of symbolic manipulations on a computer. A Cartan action $\alpha$ of ${{\mathbb Z}}^{n-1}$ on ${{\mathbb T}}^n$ is called maximal if its centralizer in the group of automorphisms of the torus is equal to $\alpha({{\mathbb Z}}^{n-1})\times\{\pm{{\rm Id}}\}$. A maximal Cartan action turns out to me maximal in the above sense: it cannot be extended to any action of a bigger abelian group by Bernoulli automorphisms. Examples of maximal Cartan actions isomorphic over ${{\mathbb Q}}$ but not isomorphic (type (iii)) are the most remarkable. Conjugacy over ${{\mathbb Q}}$ guarantees that the actions by automorphisms of the torus ${{\mathbb T}}^n$ arising from their centralizers are weakly isomorphic with finite fibres. The mechanism providing obstructions for algebraic isomorphism in this case involves the connection between the class number of an algebraic number field and $GL(n,{{\mathbb Z}})$–conjugacy classes of matrices in $SL(n,{{\mathbb Z}})$ which have the same characteristic polynomial (see Example 3). In finding these examples the use of computational number–theoretic algorithms (which in our case were implemented via the Pari-GP package) has been essential. One of our central conclusions is that for a broad class of actions of ${{\mathbb Z}}^d,\,\,d\ge 2$, (see condition $(\mathcal R)$ in Section \[ss:high-rank\]) the conjugacy class of the centralizer of the action in the group of [*affine*]{} automorphisms of the torus is an invariant of measurable conjugacy. Let $Z_\mathit{meas}({{\alpha}})$ be the centralizer of the action ${{\alpha}}$ in the group of measurable automorphisms. As it turns out in all our examples but Example 3b, the conjugacy class of the pair $(Z_\mathit{meas}({{\alpha}}),{{\alpha}})$ is a distinguishing invariant of the measurable isomorphism. Thus, in particular, Example 3b shows that there are weakly isomorphic, but nonisomorphic actions for which the affine and hence the measurable centralizers are isomorphic as abstract groups. We would like to acknowledge a contribution of J.-P. Thouvenot to the early development of ideas which led to this paper. He made an important observation that rigidity of invariant measures can be used to prove rigidity of isomorphisms via a joining construction (see Section \[ss:conj\]). Preliminaries ============= Basic ergodic theory -------------------- Any invertible (over ${{\mathbb Q}}$) integral $n\times n$ matrix $A\in M(n,{{\mathbb Z}})\cap GL(n,{{\mathbb Q}})$ determines an endomorphism of the torus ${{\mathbb T}}^n={{\mathbb R}}^n/{{\mathbb Z}}^n$ which we denote by $F_A$. Conversely, any endomorphism of ${{\mathbb T}}^n$ is given by a matrix from $A\in M(n,{{\mathbb Z}})\cap GL(n,{{\mathbb Q}})$. If, in addition, $\det A=\pm 1$, i.e. if $A$ is invertible over ${{\mathbb Z}}$, then $F_A$ is an automorphism of ${{\mathbb T}}^n$ (the group of all such $A$ is denoted by $GL(n,{{\mathbb Z}})$). The map $F_A$ preserves Lebesgue (Haar) measure $\mu$; it is ergodic with respect to $\mu$ if and only if there are no roots of unity among the eigenvalues of $A$, as was first pointed out by Halmos ([@H]). Furthermore, in this case there are eigenvalues of absolute value greater than one and $(F_A,{{\lambda}})$ is an exact endomorphism. If $F_A$ is an automorphism it is in fact Bernoulli ([@Kat]). For simplicity we will call such a map $F_A$ an [*ergodic toral endomorphism*]{} (respectively, [*automorphism*]{}, if $A$ is invertible). If all eigenvalues of $A$ have absolute values different from one we will call the endomorphism (automorphism) $F_A$ [*hyperbolic*]{}. When it does not lead to a confusion we will not distinguish between a matrix $A$ and corresponding toral endomorphism $F_A$. Let ${{\lambda}}_1,\dots , {{\lambda}}_n$ be the eigenvalues of the matrix $A$, listed with their multiplicities. The entropy $h_{\mu}(F_A)$ of $F_A$ with respect to Lebesgue measure is equal to $$\sum_{\{i:|{{\lambda}}_i|>1\}}\log|{{\lambda}}_i|.$$ In particular, entropy is determined by the conjugacy class of the matrix $A$ over ${{\mathbb Q}}$ (or over ${{\mathbb C}}$). Hence [*all ergodic toral automorphisms which are conjugate over ${{\mathbb Q}}$ are measurably conjugate with respect to Lebesgue measure.*]{} Classification, up to a conjugacy over ${{\mathbb Z}}$, of matrices in ${{SL(n,{\mathbb Z})}}$, which are irreducible and conjugate over ${{\mathbb Q}}$ is closely related to the notion of class number of an algebraic number field. A detailed discussion relevant to our purposes appears in Section \[ss:LM\]. Here we only mention the simplest case $n=2$ which is not directly related to rigidity. In this case trace determines conjugacy class over ${{\mathbb Q}}$ and, in particular, entropy. However if the class number of the corresponding number field is greater than one there are matrices with the given trace which are not conjugate over ${{\mathbb Z}}$. This algebraic distinctiveness is not reflected in the measurable structure: in fact, in the case of equal entropies the classical Adler–Weiss construction of the Markov partition in [@AW] yields metric isomorphisms which are more concrete and specific than in the general Ornstein isomorphism theory and yet not algebraic. Higher rank actions {#ss:high-rank} ------------------- Let ${{\alpha}}$ be an action by commuting toral automorphisms given by integral matrices $A_1,\dots,A_d$. It defines an embedding $\rho_{{\alpha}}:{{\mathbb Z}}^d\to GL(n,{{\mathbb Z}})$ by $$\rho_{{\alpha}}^{{\mathbf n}}=A_1^{n_1}\dots A_d^{n_d},$$ where ${{\mathbf n}}=(n_1,\dots,n_d)\in {{\mathbb Z}}^d$, and we have $${{\alpha}}^{{\mathbf n}}=F_{\rho_{{{\alpha}}}^{{\mathbf n}}}.$$ Similarly, we write $\rho_{{{\alpha}}}:{{\mathbb Z}}^d_+\to M(n,{{\mathbb Z}})\cap GL(n,{{\mathbb Q}})$ for an action by endomorphisms. Conversely, any embedding $\rho:{{\mathbb Z}}^d\to GL(n,{{\mathbb Z}})$ (respectively, $\rho:{{\mathbb Z}}^d_+\to M(n,{{\mathbb Z}})\cap GL(n,{{\mathbb Q}}))$ defines an action by automorphisms (respectively, endomorphisms) of $\mathbb{T}^n$ denoted by ${{\alpha}}_{\rho}$. Sometimes we will not explicitly distinguish between an action and the corresponding embedding, e.g. we may talk about “the centralizer of an action in $GL(n,{{\mathbb Z}})$” etc. Let $\alpha$ and $\alpha'$ be two actions of ${{\mathbb Z}}^d$ (${{\mathbb Z}}^d_+$) by automorphisms (endomorphisms) of ${{\mathbb T}}^n$ and ${{\mathbb T}}^{n'}$, respectively. The actions $\alpha$ and $\alpha'$ are [*measurably*]{} (or [*metrically*]{}, or [*measure–theoretically*]{}) [*isomorphic*]{} (or [*conjugate*]{}) if there exists a Lebesgue measure–preserving bijection $\varphi:{{\mathbb T}}^n\to{{\mathbb T}}^{n'}$ such that $\varphi\circ\alpha=\alpha'\circ\varphi$. The actions $\alpha$ and $\alpha'$ are [*measurably isomorphic up to a time change*]{} if there exist a measure–preserving bijection $\varphi:{{\mathbb T}}^n\to{{\mathbb T}}^{n'}$ and a $C\in GL(d,{{\mathbb Z}})$ such that $\varphi\circ\alpha\circ C=\alpha'\circ\varphi$. The action $\alpha'$ is a [*measurable factor*]{} of $\alpha$ if there exists a Lebesgue measure–preserving transformation $\varphi:{{\mathbb T}}^n\to{{\mathbb T}}^{n'}$ such that $\varphi\circ\alpha=\alpha'\circ\varphi$. If, in particular, $\varphi$ is almost everywhere finite–to–one, then $\alpha'$ is called a [*finite factor*]{} or a [*factor with finite fibres*]{} of $\alpha$. Actions $\alpha$ and $\alpha'$ are [*weakly measurably isomorphic*]{} if each is a measurable factor of the other. A [*joining*]{} between $\alpha$ and $\alpha'$ is a measure $\mu$ on ${{\mathbb T}}^n\times{{\mathbb T}}^{n'}={{\mathbb T}}^{n+n'}$ invariant under the Cartesian product action $\alpha\times\alpha'$ such that its projections into ${{\mathbb T}}^n$ and ${{\mathbb T}}^{n'}$ are Lebesgue measures. As will be explained in Section \[s:rigidity\], conjugacies and factors produce special kinds of joinings. These measure–theoretic notions have natural algebraic counterparts. The actions $\alpha$ and $\alpha'$ are [*algebraically isomorphic*]{} (or [*conjugate*]{}) if $n=n'$ and if there exists a group automorphism $\varphi:{{\mathbb T}}^n\to{{\mathbb T}}^{n}$ such that $\varphi\circ\alpha=\alpha'\circ\varphi$. The actions $\alpha$ and $\alpha'$ are [*algebraically isomorphic up to a time change*]{} if there exists an automorphism $\varphi:{{\mathbb T}}^n\to{{\mathbb T}}^{n}$ and $C\in GL(d,{{\mathbb Z}})$ such that $\varphi\circ\alpha\circ C=\alpha'\circ\varphi$. The action $\alpha'$ is an [*algebraic factor*]{} of $\alpha$ if there exists a surjective homomorphism $\varphi:{{\mathbb T}}^n\to{{\mathbb T}}^{n'}$ such that $\varphi\circ\alpha=\alpha'\circ\varphi$. The actions $\alpha$ and $\alpha'$ are [*weakly algebraically isomorphic*]{} if each is an algebraic factor of the other. In this case $n=n'$ and each factor map has finite fibres. Finally, we call a map $\varphi: {{\mathbb T}}^n\to {{\mathbb T}}^{n'}$ [*affine*]{} if there is a surjective continuous group homomorphism $\psi: {{\mathbb T}}^n\to {{\mathbb T}}^{n'}$ and $x'\in{{\mathbb T}}^{n'}$ s.t. $\varphi(x)=\psi(x)+x'$ for every $x\in {{\mathbb T}}^n$. As already mentioned, we intend to show that under certain condition for $d\ge 2$, measure theoretic properties imply their algebraic counterparts. We will say that an algebraic factor ${{\alpha}}'$ of ${{\alpha}}$ is a [*rank–one factor*]{} if ${{\alpha}}'$ is an algebraic factor of ${{\alpha}}$ and ${{\alpha}}'({{\mathbb Z}}^d_+)$ contains a cyclic sub–semigroup of finite index. The most general situation when certain rigidity phenomena appear is the following : $(\mathcal R')$: [*The action ${{\alpha}}$ does not possess nontrivial rank–one algebraic factors.*]{} In the case of actions by automorphisms the condition $(\mathcal R')$ is equivalent to the following condition $(\mathcal R)$ (cf. [@S]): $(\mathcal R)$: [*The action ${{\alpha}}$ contains a group, isomorphic to ${{\mathbb Z}}^2$, which consists of ergodic automorphisms.* ]{} By Proposition 6.6 in [@Sch2], Condition $(\mathcal{R})$ is equivalent to saying that [*the restriction of $\alpha $ to a subgroup isomorphic to ${{\mathbb Z}}^2$ is mixing.*]{} [*A Lyapunov exponent*]{} for an action ${{\alpha}}$ of ${{\mathbb Z}}^d$ is a function $\chi: {{\mathbb Z}}^d\to {{\mathbb R}}$ which associates to each ${\bf n}\in{{\mathbb Z}}^d$ the logarithm of the absolute value of the eigenvalue for $\rho_{{\alpha}}^{{\mathbf n}}$ corresponding to a fixed eigenvector. Any Lyapunov exponent is a linear function; hence it extends uniquely to ${{\mathbb R}}^d$. The [*multiplicity*]{} of an exponent is defined as the sum of multiplicities of eigenvalues corresponding to this exponent. Let $\chi_i,\,\,i=1,\dots, k$, be the different Lyapunov exponents and let $m_i$ be the multiplicity of $\chi_i$. Then the entropy formula for a single toral endomorphism implies that $$h_{{{\alpha}}}({{\mathbf n}})=h_\mu(\rho_{{\alpha}}^{{\mathbf n}})=\sum_{\{i: \chi_i({\bf n})> 0\}}m_i\chi_i(\bf n).$$ The function $h_{{{\alpha}}}: {{\mathbb Z}}^d\to {{\mathbb R}}$ is called [*the entropy function*]{} of the action ${{\alpha}}$. It naturally extends to a symmetric, convex piecewise linear function of ${{\mathbb R}}^d$. Any cone in ${{\mathbb R}}^d$ where all Lyapunov exponents have constant sign is called a [*Weyl chamber*]{}. The entropy function is linear in any Weyl chamber. The entropy function is a prime invariant of measurable isomorphism; since entropy does not increase for factors the entropy function is also invariant of a weak measurable isomorphism. Furthermore it changes equivariantly with respect to automorphisms of ${{\mathbb Z}}^d$. it is interesting to point out that the convex piecewise linear structure of the entropy function persists in much greater generality, namely for smooth actions on differentiable manifolds with a Borel invariant measure with compact support. Finite algebraic factors and invariant lattices ----------------------------------------------- Every algebraic action has many algebraic factors with finite fibres. These factors are in one–to–one correspondence with lattices ${{\Gamma}}\subset{{\mathbb R}}^n$ which contain the standard lattice ${{\Gamma}}_0={{\mathbb Z}}^n$, and which satisfy that $\rho_{{{\alpha}}}({{\Gamma}})\subset{{\Gamma}}$. The factor–action associated with a particular lattice ${{\Gamma}}\supset {{\Gamma}}_0$ is denoted by ${{\alpha}}_{{{\Gamma}}}$. Let us point out that in the case of actions by automorphisms such factors are also invertible: if ${{\Gamma}}\supset{{\Gamma}}_0$ and $\rho_{{{\alpha}}}({{\Gamma}})\subset{{\Gamma}}$, then $\rho_{{{\alpha}}}({{\Gamma}})={{\Gamma}}$. Let ${{\Gamma}}\supset{{\Gamma}}_0$ be a lattice. Take any basis in ${{\Gamma}}$ and let $S\in GL(n,{{\mathbb Q}})$ be the matrix which maps the standard basis in ${{\Gamma}}_0$ to this basis. Then obviously the factor–action ${{\alpha}}_{{{\Gamma}}}$ is equal to the action ${{\alpha}}_{S\rho_{{{\alpha}}}S^{-1}}$. In particular, $\rho_{{{\alpha}}}$ and $\rho_{{{\alpha}}_{{{\Gamma}}}}$ are conjugate over ${{\mathbb Q}}$, although not necessarily over ${{\mathbb Z}}$. Notice that conjugacy over ${{\mathbb Q}}$ is equivalent to conjugacy over ${{\mathbb R}}$ or over ${{\mathbb C}}$. For any positive integer $q$, the lattice $\frac 1{q}{{\Gamma}}_0$ is invariant under any automorphism in $GL(n,{{\mathbb Z}})$ and gives rise to a factor which is conjugate to the initial action: one can set $S=\frac 1{q}{{\rm Id}}$ and obtains that $\rho_{{{\alpha}}}=\rho_{{{\alpha}}_{\frac 1{q}{{\Gamma}}_0}}$. On the other hand one can find, for any lattice ${{\Gamma}}\supset{{\Gamma}}_0$, a positive integer $q$ such that $\frac 1{q}{{\Gamma}}_0\supset{{\Gamma}}$ (take $q$ the least common multiple of denominators of coordinates for a basis of ${{\Gamma}}$). Thus ${{\alpha}}_{\frac 1{q}{{\Gamma}}_0}$ appears as a factor of ${{\alpha}}_{{{\Gamma}}}$. Summarizing, we have the following properties of finite factors. \[prop-finfactors\]Let ${{\alpha}}$ and ${{\alpha}}'$ be ${{\mathbb Z}}^d$–actions by automorphism of the torus ${{\mathbb T}}^n$. The following are equivalent. 1. $\rho_{{{\alpha}}}$ and $\rho_{{{\alpha}}'}$ are conjugate over ${{\mathbb Q}}$; 2. there exists an action ${{\alpha}}''$ such that both ${{\alpha}}$ and ${{\alpha}}'$ are isomorphic to finite algebraic factors of ${{\alpha}}''$; 3. ${{\alpha}}$ and ${{\alpha}}'$ are weakly algebraically isomorphic, i.e. each of them is isomorphic to a finite algebraic factor of the other. Obviously, weak algebraic isomorphism implies weak measurable isomorphism. For ${{\mathbb Z}}$–actions by Bernoulli automorphisms, weak isomorphism implies isomorphism since it preserves entropy, the only isomorphism invariant for Bernoulli maps. In Section \[s:rigidity\] we will show that, for actions by toral automorphisms satisfying Condition $(\mathcal R)$, measurable isomorphism implies algebraic isomorphism. Hence, existence of such actions which are conjugate over ${{\mathbb Q}}$ but not over ${{\mathbb Z}}$ provides examples of actions by Bernoulli maps which are weakly isomorphic but not isomorphic. Dual modules ------------ For any action ${{\alpha}}$ of ${{\mathbb Z}}^d$ by automorphisms of a compact abelian group $X$ we denote by $\hat{{\alpha}}$ the dual action on the discrete group $\hat X$ of characters of $X$. For an element $\chi\in \hat X$ we denote $\hat X_{{{\alpha}}_,\chi}$ the subgroup of $\hat X$ generated by the orbit $\hat {{\alpha}}\chi$. The action ${{\alpha}}$ is called [*cyclic*]{} if $\hat X_{{{\alpha}}_,\chi}=\hat X$ for some $\chi\in\hat X$. Cyclicity is obviously an invariant of algebraic conjugacy of actions up to a time change. More generally, the dual group $\hat X$ has the structure of a module over the ring ${{\mathbb Z}}[u_1^{\pm1},\dots,u_d^{\pm1}]$ of Laurent polynomials in $d$ commuting variables. Action by the generators of $\hat {{\alpha}}$ corresponds to multiplications by independent variables. This module is called [*the dual module*]{} of the action ${{\alpha}}$ (cf. [@Sch1; @Sch2]). Cyclicity of the action corresponds to the condition that this module has a single generator. The structure of the dual module up to isomorphism is an invariant of algebraic conjugacy of the action up to a time change. In the case of the torus $X={{\mathbb T}}^n$ which concerns us in this paper one can slightly modify the construction of the dual module to make it more geometric. A ${{\mathbb Z}}^d$-action ${{\alpha}}$ by automorphisms of the torus ${{\mathbb R}}^n/{{\mathbb Z}}^n$ naturally extends to an action on ${{\mathbb R}}^n$ (this extension coincides with the embedding $\rho_{{{\alpha}}}$ if matrices are identified with linear transformations). This action preserves the lattice ${{\mathbb Z}}^n$ and furnishes $\mathbb{Z}^n$ with the structure of a module over the ring ${{\mathbb Z}}[u_1^{\pm1},\dots,u_d^{\pm1}]$. This module is — in an obvious sense — a *transpose* of the dual module defined above. In particular, the condition of cyclicity of the action does not depend on which of these two definitions of dual module one adopts. Algebraic and affine centralizers --------------------------------- Let ${{\alpha}}$ be an action of ${{\mathbb Z}}^d$ by toral automorphisms, and let $\rho_{{\alpha}}({{\mathbb Z}}^d)=\{\rho_{{\alpha}}^{{\mathbf n}}: n\in{{\mathbb Z}}^d\}$. The *centralizer* of ${{\alpha}}$ in the group of automorphisms of ${{\mathbb T}}^n$ is denoted by $Z({{\alpha}})$ and is not distinguished from the centralizer of $\rho_{{\alpha}}({{\mathbb Z}}^d)$ in ${{GL(n,{\mathbb Z})}}$. Similarly, the centralizer of ${{\alpha}}$ in the semigroup of all endomorphisms of ${{\mathbb T}}^n$ (identified with the centralizer of $\rho_{{\alpha}}({{\mathbb Z}}^d)$ in the semigroup $M(n,{{\mathbb Z}})\cap{{GL(n,{\mathbb Q})}}$) is denoted by $C({{\alpha}})$. The centralizer of ${{\alpha}}$ in the group of affine automorphisms of ${{\mathbb T}}^n$ will be denoted by $Z_\mathit{Aff}({{\alpha}})$. The centralizer of ${{\alpha}}$ in the semigroup of surjective affine maps of ${{\mathbb T}}^n$ will be denoted by $C_\mathit{Aff}({{\alpha}})$. Irreducible actions {#s:irred} =================== Definition ---------- The action ${{\alpha}}$ on $\mathbb{T}^n$ is called [*irreducible*]{} if any nontrivial algebraic factor of ${{\alpha}}$ has finite fibres. The following characterization of irreducible actions is useful (cf. [@B]). \[Berend\] The following conditions are equivalent: 1. ${{\alpha}}$ is irreducible; 2. $\rho_{{{\alpha}}}$ contains a matrix with characteristic polynomial irreducible over ${{\mathbb Q}}$; 3. $\rho_{{{\alpha}}}$ does not have a nontrivial invariant rational subspace or, equivalently, any ${{\alpha}}$–invariant closed subgroup of ${{\mathbb T}}^n$ is finite. \[cor:irred\] Any irreducible action $\alpha$ of ${{\mathbb Z}}^d_+,\,\,d\ge 2$, satisfies condition . A rank one algebraic factor has to have fibres of positive dimension. Hence the pre–image of the origin under the factor map is a union of finitely many rational tori of positive dimension and by Proposition \[Berend\] $\alpha$ cannot be irreducible. Uniqueness of cyclic actions ---------------------------- Cyclicity uniquely determines an irreducible action up to algebraic conjugacy within a class of weakly algebraically conjugate actions. \[prop:cyclic\]If ${{\alpha}}$ is an irreducible cyclic action of ${{\mathbb Z}}^d,\,\,d\ge 1$, on ${{\mathbb T}}^n$ and ${{\alpha}}'$ is another cyclic action such that $\rho_{{\alpha}}$ and $\rho_{{{\alpha}}'}$ are conjugate over ${{\mathbb Q}}$, then ${{\alpha}}$ and ${{\alpha}}'$ are algebraically isomorphic. For the proof of Proposition \[prop:cyclic\] we need an elementary lemma. Let $\rho:{{\mathbb Z}}^d\to {{GL(n,{\mathbb Z})}}$ be an irreducible embedding. The centralizer of $\rho$ in ${{GL(n,{\mathbb Q})}}$ acts transitively on ${{\mathbb Z}}^n\setminus\{0\}$. By diagonalizing $\rho$ over ${{\mathbb C}}$ and taking the real form of it, one immediately sees that the centralizer of $\rho$ in ${{GL(n,{\mathbb R})}}$ acts transitively on vectors with nonzero projections on all eigenspaces and thus has a single open and dense orbit. Since the centralizer over ${{\mathbb R}}$ is the closure of the centralizer over ${{\mathbb Q}}$, the ${{\mathbb Q}}$-linear span of the orbit of any integer or rational vector under the centralizer is an invariant rational subspace. Hence any integer point other than the origin belongs to the single open dense orbit of the centralizer of $\rho $ in ${{GL(n,{\mathbb R})}}$. This implies the statement of the lemma. Choose $C\in M(n,{{\mathbb Z}})$ such that $C\rho_{{{\alpha}}'}C^{-1}=\rho_{{\alpha}}$. Let $\mathbf k,\mathbf l\in {{\mathbb Z}}^n$ be cyclic vectors for $\rho_{{\alpha}}|_{{{\mathbb Z}}^n}$ and $\rho_{{{\alpha}}'}|_{{{\mathbb Z}}^n}$, respectively. Now consider the integer vector $C(\mathbf l)$ and find $D\in {{GL(n,{\mathbb Q})}}$ commuting with $\rho_{{\alpha}}$ such that $DC(\mathbf l)=\mathbf k$. We have $DC\rho_{{{\alpha}}'}C^{-1}D^{-1}=\rho_{{\alpha}}$. The conjugacy $DC$ maps bijectively the ${{\mathbb Z}}$–span of the $\rho_{{{\alpha}}'}$–orbit of $\mathbf l$ to ${{\mathbb Z}}$–span of the $\rho_{{\alpha}}$–orbit of $\mathbf k$. By cyclicity both spans coincide with ${{\mathbb Z}}^n$, and hence $DC\in {{GL(n,{\mathbb Z})}}$. Centralizers of integer matrices and algebraic number fields {#ss:irr-integers} ------------------------------------------------------------ There is an intimate connection between irreducible actions on ${{\mathbb T}}^n$ and groups of units in number fields of degree $n$. Since this connection (in the particular case where the action is Cartan and hence the number field is totally real) plays a central role in the construction of our principal examples (type (ii) and (iii) of the Introduction), we will describe it here in detail even though most of this material is fairly routine from the point of view of algebraic number theory. Let $A\in{{GL(n,{\mathbb Z})}}$ be a matrix with an irreducible characteristic polynomial $f$ and hence distinct eigenvalues. The centralizer of $A$ in $M(n,{{\mathbb Q}})$ can be identified with the ring of all polynomials in $A$ with rational coefficients modulo the principal ideal generated by the polynomial $f(A)$, and hence with the field $K={{\mathbb Q}}({{\lambda}})$, where ${{\lambda}}$ is an eigenvalue of $A$, by the map $$\label{eq:gamma} {{\gamma}}: p(A)\mapsto p({{\lambda}})$$ with $p\in{{\mathbb Q}}[x]$. Notice that if $B=p(A)$ is an integer matrix then ${{\gamma}}(B)$ is an algebraic integer, and if $B\in GL(n,{{\mathbb Z}})$ then ${{\gamma}}(B)$ is an algebraic unit (converse is not necessarily true). \[l:inj\] The map ${{\gamma}}$ in is injective. If ${{\gamma}}(p(A))=1$ for $p(A)\ne {{\rm Id}}$, then $p(A)$ has $1$ as an eigenvalue, and hence has a rational subspace consisting of all invariant vectors. This subspace must be invariant under $A$ which contradicts its irreducibility. Denote by $\mathcal O_K$ the ring of integers in $K$, by $\mathcal U_K$ the group of units in $\mathcal O_K$, by $C(A)$ the centralizer of $A$ in $M(n,{{\mathbb Z}})$ and by $Z(A)$ the centralizer of $A$ in the group $GL(n,{{\mathbb Z}})$. \[l:rings-units\] ${{\gamma}}(C(A))$ is a ring in $K$ such that ${{\mathbb Z}}[{{\lambda}}]\subset{{\gamma}}(C(A))\subset\mathcal O_K$, and ${{\gamma}}(Z(A))=\mathcal U_K\cap{{\gamma}}(C(A))$. ${{\gamma}}(C(A))$ is a ring because $C(A)$ is a ring. As we pointed out above images of integer matrices are algebraic integers and images of matrices with determinant $\pm 1$ are algebraic units. Hence ${{\gamma}}(C(A))\subset\mathcal O_K$. Finally, for every polynomial $p$ with integer coefficients, $p(A)$ is an integer matrix, hence ${{\mathbb Z}}[{{\lambda}}]\subset{{\gamma}}(C(A))$. Notice that ${{\mathbb Z}}({{\lambda}})$ is a finite index subring of $\mathcal O_K$; hence ${{\gamma}}(C(A))$ has the same property. The groups of units in two different rings, say $\mathcal O_1\subset\mathcal O_2$, may coincide. Examples can be found in the table of totally real cubic fields in [@C]. \[Dirichlet\] $Z(A)$ is isomorphic to ${{\mathbb Z}}^{r_1+r_2-1}\times F$ where $r_1$ is the number the real embeddings, $r_2$ is the number of pairs of complex conjugate embeddings of the field $K$ into ${{\mathbb C}}$, and $F$ is a finite cyclic group. By lemma \[l:rings-units\], $Z(A)$ is isomorphic to the group of units in the order $\mathcal O$, the statement follows from the Dirichlet Unit Theorem ([@BS], Ch.2, §3). Now consider an irreducible action ${{\alpha}}$ of ${{\mathbb Z}}^d$ on ${{\mathbb T}}^n$. Denote $\rho_{{{\alpha}}}({{\mathbb Z}}^d)$ by $\Gamma$, and let ${{\lambda}}$ be an eigenvalue of a matrix $A\in\Gamma$ with an irreducible characteristic polynomial. The centralizers of $\Gamma$ in $M(n,{{\mathbb Z}})$ and $GL(n,{{\mathbb Z}})$ coincide with $C(A)$ and $Z(A)$ correspondingly. The field $K={{\mathbb Q}}({{\lambda}})$ has degree $n$ and we can consider the map ${{\gamma}}$ as above. By Lemma \[l:rings-units\] ${{\gamma}}(\Gamma)\subset \mathcal U_K$. For the purposes of purely algebraic considerations in this and the next section it is convenient to consider actions of integer $n\times n$ matrices on ${{\mathbb Q}}^n$ rather than on ${{\mathbb R}}^n$ and correspondingly to think of ${{\alpha}}$ as an action by automorphisms of the rational torus ${{\mathbb T}}^n_{{{\mathbb Q}}}={{\mathbb Q}}^n/{{\mathbb Z}}^n$. Let $v=(v_1,\dots,v_n)$ be an eigenvector of $A$ with eigenvalue ${{\lambda}}$ whose coordinates belong to $K$. Consider the “projection” $\pi:{{\mathbb Q}}^n\to K$ defined by $\pi(r_1,\dots\,r_n)=\sum_{i=1}^n r_iv_i$. It is a bijection ([@W], Prop. 8) which conjugates the action of the group $\Gamma$ with the action on $K$ given by multiplication by corresponding eigenvalues $\prod_{i=1}^d{{\lambda}}_i^{k_i},\,\,k_1,\dots,k_d\in{{\mathbb Z}}$. Here $A_1,\dots,A_d\in \Gamma$ are the images of the generators of the action ${{\alpha}}$, and $A_iv={{\lambda}}_i v,\,\,i=1,\dots, d$. The lattice $\pi{{\mathbb Z}}^n\subset K$ is a module over the ring ${{\mathbb Z}}[{{\lambda}}_1,\dots,{{\lambda}}_d]$. Conversely, any such data, consisting of an algebraic number field $K={{\mathbb Q}}({{\lambda}})$ of degree $n$, a $d$-tuple $\bar{{\lambda}}=({{\lambda}}_1,\dots,{{\lambda}}_d)$ of multiplicatively independent units in $K$, and a lattice $\mathcal L\subset K$ which is a module over ${{\mathbb Z}}[{{\lambda}}_1,\dots,{{\lambda}}_d]$, determine an ${{\mathbb Z}}^d$-action ${{\alpha}}_{\bar{{\lambda}},\mathcal L}$ by automorphisms of $\mathbb{T}^n$ up to algebraic conjugacy (corresponding to a choice of a basis in the lattice $\mathcal L$). This action is generated by multiplications by ${{\lambda}}_1,\dots,{{\lambda}}_d$ (which preserve $\mathcal L$ by assumption). The action ${{\alpha}}_{\bar{{\lambda}},\mathcal L}$ diagonalizes over ${{\mathbb C}}$ as follows. Let $\phi_1={{\rm id}},\phi_2,\dots,\phi_n$ be different embeddings of $K$ into ${{\mathbb C}}$. The multiplications by ${{\lambda}}_i,\,\,i=1,\dots,d$, are simultaneously conjugate over ${{\mathbb C}}$ to the respective matrices $$\left(\begin{smallmatrix}{{\lambda}}_i & 0 & \dots & 0 \\ 0 & \phi_2({{\lambda}}_i) & \dots & 0 \\ \dots & \dots & \dots & \dots\\ 0 & 0 & \dots & \phi_n({{\lambda}}_i) \end{smallmatrix}\right),\quad i=1,\dots,d.$$ We will assume that the action is irreducible which in many interesting cases can be easily checked. Thus, all actions ${{\alpha}}_{\bar{{\lambda}},\mathcal L}$ with fixed $\bar{{\lambda}}$ are weakly algebraically isomorphic since the corresponding embeddings are conjugate over ${{\mathbb Q}}$ (Proposition \[prop-finfactors\]). Actions produced with different sets of units in the same field, say $\bar{{{\lambda}}}$ and $\bar{\mu}=(\mu_1,\dots,\mu_d)$, are weakly algebraically isomorphic if and only if there is an element $g$ of the Galois group of $K$ such that $\mu_i=g{{\lambda}}_i,\,\, i=1,\dots,d$. By Proposition \[prop:cyclic\] there is a unique cyclic action (up to algebraic isomorphism) within any class of weakly algebraically isomorphic actions: it corresponds to setting $\mathcal L={{\mathbb Z}}[{{\lambda}}_1,\dots,{{\lambda}}_d]$; we will denote this action by ${{\alpha}}_{\bar{{\lambda}}}^{\min}$. Cyclicity of the action ${{\alpha}}_{\bar{{\lambda}}}^{\min}$ is obvious since the whole lattice is obtained from its single element $1$ by the action of the ring ${{\mathbb Z}}[\lambda _1^{\pm1},\dots,\lambda _d^{\pm1}]$. Let us summarize this discussion. \[prop-class\] Any irreducible action ${{\alpha}}$ of ${{\mathbb Z}}^d$ by automorphisms of ${{\mathbb T}}^n$ is algebraically conjugate to an action of the form ${{\alpha}}_{\bar{{\lambda}},\mathcal L}$. It is weakly algebraically conjugate to the cyclic action ${{\alpha}}_{\bar{{\lambda}}}^{\min}$. The field $K={{\mathbb Q}}[{{\lambda}}_1,\dots,{{\lambda}}_d]$ has degree $n$, and the vector of units $\bar{{\lambda}}=({{\lambda}}_1,\dots,{{\lambda}}_d)$ is defined up to action by an element of the Galois group of $K:\mathbb{Q}$. Apart from the cyclic model ${{\alpha}}_{\bar{{\lambda}}}^{\min}$ there is another canonical choice of the lattice $\mathcal L$, namely the ring of integers $\mathcal O_K$. We will denote the action ${{\alpha}}_{\bar{{\lambda}},\mathcal O_K}$ by ${{\alpha}}_{\bar{{\lambda}}}^{\max}$. More generally, one can choose as the lattice $\mathcal L$ any subring $\mathcal O$ such that ${{\mathbb Z}}[{{\lambda}}_1,\dots,{{\lambda}}_d]\subset\mathcal O\subset\mathcal O_K$. \[prop:min-max\] Assume that $\mathcal O \supsetneq {{\mathbb Z}}[{{\lambda}}_1,\dots,{{\lambda}}_d]$. Then the action ${{\alpha}}_{\bar{{{\lambda}}},\mathcal O}$ is not algebraically isomorphic up to a time change to ${{\alpha}}_{\bar{{\lambda}}}^{\min}$. In particular, if $\mathcal O_K\neq{{\mathbb Z}}[{{\lambda}}_1,\dots,{{\lambda}}_d]$, then the actions ${{\alpha}}_{\bar{{\lambda}}}^{\max}$ and ${{\alpha}}_{\bar{{\lambda}}}^{\min}$ are not algebraically isomorphic up to a time change. Let us denote the centralizers in $M(n,{{\mathbb Z}})$ of the actions ${{\alpha}}_{\bar{{{\lambda}}},\mathcal O}$ and ${{\alpha}}_{\bar{{\lambda}}}^{\min}$ by $C_1$ and $C_2$, respectively. The centralizer $C_1$ contains multiplications by all elements of $\mathcal O$. For, if one takes any basis in $\mathcal O$, the multiplication by an element $\mu\in\mathcal O$ takes elements of the basis into elements of $\mathcal O$, which are linear combinations with integral coefficients of the basis elements; hence the multiplication is given by an integer matrix. On the other hand any element of each centralizer is a multiplication by an integer in $K$ (Lemma \[l:rings-units\]). Now assume that the multiplication by $\mu\in\mathcal O_K$ belongs to $C_2$. This means that this multiplication preserves ${{\mathbb Z}}[{{\lambda}}_1,\dots,{{\lambda}}_d]$; in particular, $\mu=\mu\cdot 1\in {{\mathbb Z}}[{{\lambda}}_1,\dots,{{\lambda}}_d]$. Thus $C_2$ consists of multiplication by elements of ${{\mathbb Z}}[{{\lambda}}_1,\dots,{{\lambda}}_d]$. An algebraic isomorphism up to a time change has to preserve both the module of polynomials with integer coefficients in the generators of the action and the centralizer of the action in $M(n,{{\mathbb Z}})$, which is impossible. The central question which appears in connection with our examples is the classification of weakly algebraically isomorphic Cartan actions up to algebraic isomorphism. Proposition \[prop:min-max\] is useful in distinguishing weakly algebraically isomorphic actions when $\mathcal O_K\neq{{\mathbb Z}}[{{\lambda}}_1,\dots,{{\lambda}}_d]$. Cyclicity also can serve as a distinguishing invariant. \[cor-cyclic\]The action ${{\alpha}}_{\bar{{{\lambda}}},\mathcal O}$ is cyclic if and only if $\mathcal O ={{\mathbb Z}}[{{\lambda}}_1,\dots,\linebreak[0]{{\lambda}}_d]$. The action ${{\alpha}}_{\bar{{\lambda}}}^{\min}$ corresponding to the ring ${{\mathbb Z}}[{{\lambda}}_1,\dots,{{\lambda}}_d]$ is cyclic by definition since the ring coincides with the orbit of $1$. By Proposition \[prop:cyclic\], if ${{\alpha}}_{\bar{{{\lambda}}},\mathcal O}$ were cyclic, it would be algebraically conjugate to ${{\alpha}}_{\bar{{\lambda}}}^{\min}$, which, by Proposition \[prop:min-max\], implies that $\mathcal O ={{\mathbb Z}}[{{\lambda}}_1,\dots,{{\lambda}}_d]$. The property common to all actions of the ${{\alpha}}_{\bar{{{\lambda}}},\mathcal O}$ is transitivity of the action of the centralizer $C({{\alpha}}_{\bar{{{\lambda}}},\mathcal O})$ on the lattice. Similarly to cyclicity this property is obviously an invariant of algebraic conjugacy up to a time change. \[prop-trans\] Any irreducible action ${{\alpha}}$ of ${{\mathbb Z}}^d$ by automorphisms of ${{\mathbb T}}^n$ whose centralizer $C({{\alpha}})$ in $M(n,{{\mathbb Z}})$ acts transitively on ${{\mathbb Z}}^n$ is algebraically isomorphic to an action ${{\alpha}}_{\bar{{{\lambda}}},\mathcal O}$, where $\mathcal O\subset\mathcal O_K$ is a ring which contains ${{\mathbb Z}}[{{\lambda}}_1,\dots,{{\lambda}}_{d}]$. By Proposition \[prop-class\] any irreducible action ${{\alpha}}$ of ${{\mathbb Z}}^d$ by automorphisms of ${{\mathbb T}}^n$ is algebraically conjugate to an action of the form ${{\alpha}}_{\bar{{\lambda}},\mathcal L}$ for a lattice $\mathcal L\subset K$. Let $C$ be the centralizer of ${{\alpha}}_{\bar{{\lambda}},\mathcal L}$ in the semigroup of linear endomorphisms of $\mathcal{L}$. We fix an element $\beta\in\mathcal L$ with $C(\alpha )\beta =\mathcal{L}$ and consider conjugation of the action ${{\alpha}}_{\bar{{\lambda}},\mathcal L}$ by multiplication by $\beta^{-1}$; this is simply ${{\alpha}}_{\bar{{\lambda}},\beta^{-1}\mathcal L}$. The centralizer of ${{\alpha}}_{\bar{{\lambda}},\beta^{-1}\mathcal L}$ acts on the element $1\in\beta^{-1}\mathcal L$ transitively. By Lemma \[l:rings-units\] the centralizer consists of all multiplications by elements of a certain subring $\mathcal O\subset \mathcal O_K$ which contains ${{\mathbb Z}}[{{\lambda}}_1,\dots,{{\lambda}}_{d}]$. Thus $1\in\beta^{-1}\mathcal L=\mathcal O$. Structure of algebraic and affine centralizers for irreducible actions ---------------------------------------------------------------------- By Lemma \[l:rings-units\], the centralizer $C({{\alpha}})$, as an additive group, is isomorphic to ${{\mathbb Z}}^n$ and has an additional ring structure. In the terminology of Proposition \[Dirichlet\], the centralizer $Z({{\alpha}})$ for an irreducible action ${{\alpha}}$ by toral automorphisms is isomorphic to ${{\mathbb Z}}^{r_1+r_2-1}\times F$. An irreducible action ${{\alpha}}$ has [*maximal rank*]{} if $d=r_1+r_2-1$. In this case $Z({{\alpha}})$ is a finite extension of ${{\alpha}}$. Notice that any affine map commuting with an action ${{\alpha}}$ by toral automorphisms preserves the set ${{\rm Fix}}({{\alpha}})$ of fixed points of the action. This set is always a subgroup of the torus and hence, for an irreducible action, always finite. The translation by any element of ${{\rm Fix}}({{\alpha}})$ commutes with ${{\alpha}}$ and thus belongs to $Z_\mathit{Aff}({{\alpha}})$. Furthermore, the affine centralizers $Z_\mathit{Aff}({{\alpha}})$ and $C_\mathit{Aff}({{\alpha}})$ are generated by these translations and, respectively, $Z({{\alpha}})$ and $C({{\alpha}})$. Most of the material of this section extends to general irreducible actions of ${{\mathbb Z}}^d$ by automorphisms of compact connected abelian groups; a group possessing such an action must be a torus or a solenoid ([@Sch2; @Sch3]). In the solenoid case, which includes natural extensions of ${{\mathbb Z}}^d$–actions by toral endomorphisms, the algebraic numbers ${{\lambda}}_1,\dots ,{{\lambda}}_d$ which appear in the constructions are not in general integers. As we mentioned in the introduction we restrict our algebraic setting here since we are able to exhibit some of the most interesting and striking new phenomena using Cartan actions and certain actions directly derived from them. However, other interesting examples appear for actions on the torus connected with not totally real algebraic number fields, actions on solenoids, and actions on zero-dimensional abelian groups (cf. e.g. [@KiS; @Sch1; @Sch2; @Sch3]). One can also extend the setup of this section to certain classes of reducible actions. Since some of these satisfy condition $(\mathcal R)$ basic rigidity results still hold and a number of further interesting examples can be constructed. Cartan actions {#s:ca} ============== Structure of Cartan actions --------------------------- Of particular interest for our study are abelian groups of ergodic automorphisms of ${{\mathbb T}}^n$ of maximal possible rank $n-1$ (in agreement with the real rank of the Lie group ${{SL(n,{\mathbb R})}}$). An action of ${{\mathbb Z}}^{n-1}$ on ${{\mathbb T}}^n$ for $n\ge 3$ by ergodic automorphisms is called a [*Cartan action*]{}. \[cartan-str\] Let ${{\alpha}}$ be a Cartan action on ${{\mathbb T}}^n$. 1. Any element of $\rho_{{\alpha}}$ other than identity has real eigenvalues and is hyperbolic and thus Bernoulli. 2. ${{\alpha}}$ is irreducible. 3. The centralizer of $Z({{\alpha}})$ is a finite extension of $\rho_{{\alpha}}({{\mathbb Z}}^{n-1})$. First, let us point out that it is sufficient to prove the proposition for irreducible actions. For, if ${{\alpha}}$ is not irreducible, it has a nontrivial irreducible algebraic factor of dimension, say, $m\le n-1$. Since every factor of an ergodic automorphism is ergodic, we thus obtain an action of ${{\mathbb Z}}^{n-1}$ in ${{\mathbb T}}^m$ by ergodic automorphisms. By considering a restriction of this action to a subgroup of rank $m-1$ which contains an irreducible matrix, we obtain a Cartan action on ${{\mathbb T}}^m$. By Statement 3. for irreducible actions, the centralizer of this Cartan action is a finite extension of ${{\mathbb Z}}^{m-1}$, and thus cannot contain ${{\mathbb Z}}^{n-1}$, a contradiction. Now assuming that ${{\alpha}}$ is irreducible, take a matrix $A\in \rho_{{\alpha}}({{\mathbb Z}}^{n-1})$ with irreducible characteristic polynomial $f$. Such a matrix exists by Proposition \[Berend\]. It has distinct eigenvalues, say ${{\lambda}}={{\lambda}}_1,\dots,{{\lambda}}_n$. Consider the correspondence ${{\gamma}}$ defined in . By Lemma \[l:rings-units\] for every $B\in\rho_{{\alpha}}({{\mathbb Z}}^{n-1})$ we have ${{\gamma}}(B)\in\mathcal U_K$, hence the group of units $\mathcal U_K$ in $K$ contains a subgroup isomorphic to ${{\mathbb Z}}^{n-1}$. By the Dirichlet Unit Theorem the rank of the group of units in $K$ is equal to ${r_1+r_2-1}$, where $r_1$ is the number of real embeddings and $r_2$ is the number of pairs of complex conjugate embeddings of $K$ into ${{\mathbb C}}$. Since $r_1+2r_2=n$ we deduce that $r_2=0$, so the field $K$ is totally real, that is all eigenvalues of $A$, and hence of any matrix in $\rho_{{\alpha}}({{\mathbb Z}}^{n-1})$, are real. The same argument gives Statement 3, since any element of the centralizer of $\rho_{{\alpha}}({{\mathbb Z}}^{n-1})$ in ${{GL(n,{\mathbb Z})}}$ corresponds to a unit in $K$. Hyperbolicity of matrices in $\rho_{{\alpha}}({{\mathbb Z}}^{n-1})$ is proved in the same way as Lemma \[l:inj\]. \[hyp\] Let $A$ be a hyperbolic matrix in ${{SL(n,{\mathbb Z})}}$ with irreducible characteristic polynomial and distinct real eigenvalues. Then every element of the centralizer $Z(A)$ other than $\{\pm 1\}$ is hyperbolic. Assume that $B\in Z(A)$ is not hyperbolic. As $B$ is simultaneously diagonalizable with $A$ and has real eigenvalues, it has an eigenvalue $+1$ or $-1$. The corresponding eigenspace is rational and $A$–invariant. Since $A$ is irreducible, this eigenspace has to coincide with the whole space and hence $B=\pm 1$. Cartan actions are exactly the maximal rank irreducible actions corresponding to totally real number fields. The centralizer $Z({{\alpha}})$ for a Cartan action ${{\alpha}}$ is isomorphic to ${{\mathbb Z}}^{n-1}\times\{\pm 1\}$. We will call a Cartan action ${{\alpha}}$ [*maximal*]{} if ${{\alpha}}$ is an index two subgroup in $Z({{\alpha}})$. Let us point out that $Z_\mathit{Aff}({{\alpha}})$ is isomorphic $Z({{\alpha}})\times{{\rm Fix}}({{\alpha}})$. Thus, the factor of $Z_\mathit{Aff}({{\alpha}})$ by the subgroup of finite order elements is always isomorphic to ${{\mathbb Z}}^{n-1}$. If ${{\alpha}}$ is maximal, this factor is identified with ${{\alpha}}$ itself. In the next Section we will show (Corollary \[cor-centr\]) that for a Cartan action ${{\alpha}}$ on ${{\mathbb T}}^n,\,\,n\ge 3$ the isomorphism type of the pair $(Z_\mathit{Aff}({{\alpha}}),{{\alpha}})$ is an invariant of the measurable isomorphism. Thus, in particular, for a maximal Cartan action the order of the group ${{\rm Fix}}({{\alpha}})$ is a measurable invariant. An important geometric distinction between Cartan actions and general irreducible actions by hyperbolic automorphisms is the absence of multiple Lyapunov exponents. This greatly simplifies proofs of various rigidity properties both in the differentiable and measurable context. Algebraically nonisomorphic maximal Cartan actions {#ss:LM} --------------------------------------------------- In Section \[ss:irr-integers\] we described a particular class of irreducible actions ${{\alpha}}_{\bar{{\lambda}},\mathcal O}$ which is characterized by the transitivity of the action of the centralizer $C({{\alpha}}_{\bar{{\lambda}},\mathcal O})$ on the lattice (Proposition \[prop-trans\]). In the case $\mathcal O_K={{\mathbb Z}}[{{\lambda}}]$ there is only one such action, namely the cyclic one (Corollary \[cor-cyclic\]). Now we will analyze this special case for totally real fields in detail and show how information about the class number of the field helps to construct algebraically nonisomorphic maximal Cartan actions. This will in particular provide examples of Cartan actions not isomorphic up to a time change to any action of the form ${{\alpha}}_{\bar{{\lambda}},\mathcal O}$. It is well–known that for $n=2$ there are natural bijections between conjugacy classes of hyperbolic elements in $SL(2,{{\mathbb Z}})$ of a given trace, ideal classes in the corresponding real quadratic field, and congruence classes of primitive integral indefinite quadratic forms of the corresponding discriminant. This has been used by Sarnak [@Sa] in his proof of the Prime Geodesic Theorem for surfaces of constant negative curvature (see also [@K]). It follows from an old Theorem of Latimer and MacDuffee (see [@LM], [@T], and a more modern account in [@W]), that the first bijection persists for $n>2$. Let $A$ a hyperbolic matrix $A\in {{SL(n,{\mathbb Z})}}$ with irreducible characteristic polynomial $f$, and hence distinct real eigenvalues, $K={{\mathbb Q}}({{\lambda}})$, where ${{\lambda}}$ is an eigenvalue of $A$, and $\mathcal O_K={{\mathbb Z}}[{{\lambda}}]$. To each matrix $A'$ with the same eigenvalues, we assign the eigenvector $v=(v_1,\dots,v_n)$ with eigenvalue ${{\lambda}}$: $A'v={{\lambda}}v$ with all its entries in $\mathcal O_K$, which can be always done, and to this eigenvector, an ideal in $\mathcal O_K$ with the ${{\mathbb Z}}$–basis $v_1,\dots,v_n$. The described map is a bijection between the $GL(n,{{\mathbb Z}})$–conjugacy classes of matrices in ${{SL(n,{\mathbb Z})}}$ which have the same characteristic polynomial $f$ and the set of ideal classes in $\mathcal O_K$. Moreover, it allows us to reach conclusions about centralizers as well. \[mytheor\] Let $A\in {{SL(n,{\mathbb Z})}}$ be a hyperbolic matrix with irreducible characteristic polynomial $f$ and distinct real eigenvalues, $K={{\mathbb Q}}({{\lambda}})$ where ${{\lambda}}$ is an eigenvalue of $A$, and $\mathcal O_K={{\mathbb Z}}[{{\lambda}}]$. Suppose the number of eigenvalues among ${{\lambda}}_1,\dots,{{\lambda}}_n$ that belong to $K$ is equal to $r$. If the class number $h(K)>r$, then there exists a matrix $A'\in{{SL(n,{\mathbb Z})}}$ having the same eigenvalues as $A$ whose centralizer $Z(A')$ is not conjugate in $GL(n,{{\mathbb Z}})$ to $Z(A)$. Furthermore, the number of matrices in ${{SL(n,{\mathbb Z})}}$ having the same eigenvalues as $A$ with pairwise nonconjugate in $GL(n,{{\mathbb Z}})$ centralizers is at least $[\frac{h(K)}{r}] +1$, where $[x]$ is the largest integer $<x$. Suppose the matrix $A$ corresponds to the ideal class $I_1$ with the ${{\mathbb Z}}$–basis $v^{(1)}$. Then $$A v^{(1)}={{\lambda}}v^{(1)}.$$ Since $h(K)>1$, there exists a matrix $A_2$ having the same eigenvalues which corresponds to a different ideal class $I_2$ with the basis $v^{(2)}$, and we have $$A_2 v^{(2)}={{\lambda}}v^{(2)}.$$ The eigenvectors $v^{(1)}$ and $v^{(2)}$ are chosen with all their entries in $\mathcal O_K$. Now assume that $Z(A_2)$ is conjugate to $Z(A)$. Then $Z(A_2)$ contains a matrix $B_2$ conjugate to $A$. Since $B_2$ commutes with $A_2$ we have $B_2 v^{(2)}=\mu_2 v^{(2)}$, and since $B_2$ is conjugate to $A$, $\mu_2$ is one of the roots of $f$. Moreover, since $B_2\in {{SL(n,{\mathbb Z})}}$ and all entries of $v^{(2)}$ are in $K$, $\mu_2\in K$. Thus $\mu_2$ is one of $r$ roots of $f$ which belongs to $K$. From $B_2=S^{-1} A S$ ($S\in GL(n,{{\mathbb Z}})$) we deduce that $\mu_2 (Sv^{(2)})=A(Sv^{(2)})$. Since $I_1$ and $I_2$ belong to different ideal classes, $Sv^{(2)}\ne kv^{(1)}$ for any $k$ in the quotient field of $\mathcal O_K$, and since ${{\lambda}}$ is a simple eigenvalue for $A$, we deduce that $\mu_2\ne{{\lambda}}$, and thus $\mu_2$ can take one of the $r-1$ remaining values. Now assume that $A_3$ corresponds to the third ideal class, i.e $$A_3 v^{(3)}={{\lambda}}v^{(3)},$$ and $B_3$ commutes with $A_3$ and is conjugate to $A$, and hence to $B_2$. Then $B_3 v^{(3)}=\mu_3 v^{(3)}$ where $\mu_3$ is a root of $f$ belonging to the field $K$. By the previous considerations, $\mu_3\ne{{\lambda}}$ and $\mu_3\ne\mu_2$. An induction argument shows that if the class number of $K$ is greater than $r$, there exists a matrix $A'$ such that no matrix in $Z(A')$ is conjugate to $A$, i.e. $Z(A')$ and $Z(A)$ are not conjugate in $GL(n,{{\mathbb Z}})$. Since $A'$ has the same characteristic polynomial as $A$, continuing the same process, we can find not more than $r$ matrices representing different ideal classes having centralizers conjugate to $Z(A')$, and the required estimate follows. Measure–theoretic rigidity of conjugacies, centralizers, and factors {#s:rigidity} ==================================================================== Conjugacies {#ss:conj} ----------- Suppose ${{\alpha}}$ and ${{\alpha}}'$ are measurable actions of the same group $G$ by measure–preserving transformations of the spaces $(X,\mu)$ and $(Y,\nu)$, respectively. If $H:(X,\mu)\to(Y,\nu)$ is a metric isomorphism (conjugacy) between the actions then the lift of the measure $\mu$ onto the $\mbox{graph} \,H\subset X\times Y$ coincides with the lift of $\nu$ to $\mbox{graph}\, H^{-1}$. The resulting measure $\eta$ is a very special case of a [*joining*]{} of ${{\alpha}}$ and ${{\alpha}}'$: it is invariant under the diagonal (product) action ${{\alpha}}\times{{\alpha}}'$ and its projections to $X$ and $Y$ coincide with $\mu$ and $\nu$, respectively. Obviously the projections establish metric isomorphism of the action ${{\alpha}}\times{{\alpha}}'$ on $(X\times Y, \eta)$ with ${{\alpha}}$ on $(X,\mu)$ and ${{\alpha}}'$ on $(Y,\nu)$ correspondingly. Similarly, if an automorphism $ H:(X,\mu)\to(X,\mu)$ commutes with the action ${{\alpha}}$, the lift of $\mu$ to $\mbox{graph} \,H\subset X\times X$ is a self-joining of ${{\alpha}}$, i.e. it is ${{\alpha}}\times{{\alpha}}$–invariant and both of its projections coincide with $\mu$. Thus an information about invariant measures of the products of different actions as well as the product of an action with itself may give an information about isomorphisms and centralizers. The use of this joining construction in order to deduce rigidity of isomorphisms and centralizers from properties of invariant measures of the product was first suggested in this context to the authors by J.-P. Thouvenot. In both cases the ergodic properties of the joining would be known because of the isomorphism with the original actions. Very similar considerations apply to the actions of semi–groups by noninvertible measure–preserving transformations. We will use the following corollary of the results of [@KS3]. \[corr-K\] Let ${{\alpha}}$ be an action of ${{{\mathbb Z}}^2}$ by ergodic toral automorphisms and let $\mu$ be a weakly mixing ${{\alpha}}$–invariant measure such that for some $\mathbf{m}\in {{\mathbb Z}}^2$, ${{\alpha}}^\mathbf{m}$ is a $K$-automorphism. Then $\mu$ is a translate of Haar measure on an ${{\alpha}}$–invariant rational subtorus. We refer to Corollary 5.2’ from ([@KS3], “Corrections...”). According to this corollary the measure $\mu$ is an extension of a zero entropy measure for an algebraic factor of smaller dimension with Haar conditional measures in the fiber. But since ${{\alpha}}$ contains a $K$-automorphism it does not have non–trivial zero entropy factors. Hence the factor in question is the action on a single point and $\mu$ itself is a Haar measure on a rational subtorus. Conclusion of Theorem \[corr-K\] obviously holds for any action of ${{\mathbb Z} ^d},\,\,d\ge 2$ which contains a subgroup ${{\mathbb Z}}^2$ satisfying assumptions of Theorem \[corr-K\]. Thus we can deduce the following result which is central for our constructions. \[thm-isom\] Let ${{\alpha}}$ and ${{\alpha}}'$ be two actions of ${{\mathbb Z} ^d}$ by automorphisms of ${{\mathbb T}}^n$ and ${{\mathbb T}}^{n'}$ correspondingly and assume that ${{\alpha}}$ satisfies condition $(\mathcal R)$. Suppose that $H:{{\mathbb T}}^n\to {{\mathbb T}}^{n'}$ is a measure–preserving isomorphism between $({{\alpha}}, {{\lambda}})$ and $({{\alpha}}', {{\lambda}})$, where ${{\lambda}}$ is Haar measure. Then $n=n'$ and $H$ coincides with an affine automorphism on the torus ${{\mathbb T}}^n$, and hence ${{\alpha}}$ and ${{\alpha}}'$ are algebraically isomorphic. First of all, condition $(\mathcal R)$ is invariant under metric isomorphism, hence ${{\alpha}}'$ also satisfies this condition. But ergodicity with respect to Haar measure can also be expressed in terms of the eigenvalues; hence ${{\alpha}}\times{{\alpha}}'$ also satisfies ($\mathcal R$). Now consider the joining measure $\eta$ on $\mbox{graph} \,H\subset {{\mathbb T}}^{n+n'}$. The conditions of Theorem \[corr-K\] are satisfied for the invariant measure $\eta$ of the action ${{\alpha}}\times{{\alpha}}'$. Thus $\eta$ is a translate of Haar measure on a rational ${{\alpha}}\times{{\alpha}}'$–invariant subtorus ${{\mathbb T}}'\subset {{\mathbb T}}^{n+n'}={{\mathbb T}}^n\times {{\mathbb T}}^{n'}$. On the other hand we know that projections of ${{\mathbb T}}'$ to both ${{\mathbb T}}^n$ and ${{\mathbb T}}^{n'}$ preserve Haar measure and are one–to–one. The partitions of ${{\mathbb T}}'$ into pre–images of points for each of the projections are measurable partitions and Haar measures on elements are conditional measures. This implies that both projections are onto, both partitions are partitions into points, and hence $n=n'$ and ${{\mathbb T}}'=\mbox{graph}\, I$, where $I: {{\mathbb T}}^n\to {{\mathbb T}}^n$ is an affine automorphism which has to coincide $(\textup{mod}\;0)$ with the measure–preserving isomorphism $H$. Since a time change is in a sense a trivial modification of an action we are primarily interested in distinguishing actions up to a time change. The corresponding rigidity criterion follows immediately from Theorem \[thm-isom\]. \[cor:time-change\] Let ${{\alpha}}$ and ${{\alpha}}'$ be two actions of ${{\mathbb Z} ^d}$ by automorphisms of ${{\mathbb T}}^n$ and ${{\mathbb T}}^{n'}$, respectively, and assume that ${{\alpha}}$ satisfies condition $(\mathcal R)$. If ${{\alpha}}$ and ${{\alpha}}'$ are measurably isomorphic up to a time change then they are algebraically isomorphic up to a time change. Centralizers ------------ Applying Theorem \[thm-isom\] to the case ${{\alpha}}={{\alpha}}'$ we immediately obtain rigidity of the centralizers. \[cor-centr\] Let ${{\alpha}}$ be an action of ${{\mathbb Z} ^d}$ by automorphisms of ${{\mathbb T}}^n$ satisfying condition $(\mathcal R)$. Any invertible Lebesgue measure–preserving transformation commuting with ${{\alpha}}$ coincides with an affine automorphism of ${{\mathbb T}}^n$. Any affine transformation commuting with ${{\alpha}}$ preserves the finite set of fixed points of the action. Hence the centralizer of ${{\alpha}}$ in affine automorphisms has a finite index subgroups which consist of automorphisms and which corresponds to the centralizer of $\rho_{{{\alpha}}}({{\mathbb Z}}^d)$ in $GL(n,{{\mathbb Z}})$. Thus, in contrast with the case of a single automorphism, the centralizer of such an action ${{\alpha}}$ is not more than countable, and can be identified with a finite extension of a certain subgroup of $GL(n,{{\mathbb Z}})$. As an immediate consequence we obtain the following result. For any $d$ and $k$, $2\le d\le k$, there exists a ${{\mathbb Z} ^d}$–action by hyperbolic toral automorphisms such that its centralizer in the group of Lebesgue measure–preserving transformations is isomorphic to $\{\pm 1\}\times{{\mathbb Z}}^k$. Consider a hyperbolic matrix $A\in SL(k+1,{{\mathbb Z}})$ with irreducible characteristic polynomial and real eigenvalues such that the origin is the only fixed point of $F_A$. Consider a subgroup of $Z(A)$ isomorphic to ${{\mathbb Z} ^d}$ and containing $A$ as one of its generators. This subgroup determines an embedding $\rho:{{\mathbb Z}}^d\to SL(k+1,{{\mathbb Z}})$. Since $d\ge 2$ and by Proposition \[hyp\], all matrices in $\rho({{\mathbb Z} ^d})$ are hyperbolic and hence ergodic, condition $(\mathcal R)$ is satisfied. Hence by Corollary \[cor-centr\], the measure–theoretic centralizer of the action ${{\alpha}}_{\rho}$ coincides with its algebraic centralizer, which, in turn, and obviously, coincides with centralizer of the single automorphism $F_A$ isomorphic to $\{\pm 1\}\times{{\mathbb Z}}^k$. Factors, noninvertible centralizers and weak isomorphism -------------------------------------------------------- A small modification of the proof of Theorem \[thm-isom\] produces a result about rigidity of factors. \[thm-factors\] Let ${{\alpha}}$ and ${{\alpha}}'$ be two actions of ${{\mathbb Z} ^d}$ by automorphisms of ${{\mathbb T}}^n$ and ${{\mathbb T}}^{n'}$ respectively, and assume that ${{\alpha}}$ satisfies condition $(\mathcal R)$. Suppose that $H:{{\mathbb T}}^n\to {{\mathbb T}}^{n'}$ is a Lebesgue measure–preserving transformation such that $H\circ{{\alpha}}={{\alpha}}'\circ H$. Then ${{\alpha}}'$ also satisfies $(\mathcal R)$ and $H$ coincides with an epimorphism $h:{{\mathbb T}}^n\to {{\mathbb T}}^{n'}$ followed by translation. In particular, ${{\alpha}}'$ is an algebraic factor of ${{\alpha}}$. Since ${{\alpha}}'$ is a measurable factor of ${{\alpha}}$, every element which is ergodic for ${{\alpha}}$ is also ergodic for ${{\alpha}}'$. Hence ${{\alpha}}'$ also satisfies condition $(\mathcal R)$. As before consider the product action ${{\alpha}}\times{{\alpha}}'$ which now by the same argument also satisfies $(\mathcal R)$. Take the ${{\alpha}}\times{{\alpha}}'$ invariant measure $\eta=(\mbox{Id}\times H) _*{{\lambda}}$ on $\mbox{graph} \,H$. This measure provides a joining of ${{\alpha}}$ and ${{\alpha}}'$. Since $({{\alpha}}\times{{\alpha}}', (\mbox{Id}\times H)_*{{\lambda}})$ is isomorphic to $({{\alpha}},{{\lambda}})$ the conditions of Corollary \[corr-K\] are satisfied and $\eta$ is a translate of Haar measure on an invariant rational subtorus ${{\mathbb T}}'$. Since ${{\mathbb T}}'$ projects to the first coordinate one-to-one we deduce that $H$ is an algebraic epimorphism followed by a translation. Similarly to the previous section the application of Theorem \[thm-factors\] to the case ${{\alpha}}={{\alpha}}'$ gives a description of the centralizer of ${{\alpha}}$ in the group of all measure–preserving transformations. \[cor-centrNI\] Let ${{\alpha}}$ be an action of ${{\mathbb Z} ^d}$ by automorphisms of ${{\mathbb T}}^n$ satisfying condition $(\mathcal R)$. Any Lebesgue measure–preserving transformation commuting with ${{\alpha}}$ coincides with an affine map on ${{\mathbb T}}^n$. Now we can obtain the following strengthening of Proposition \[prop-finfactors\] for actions satisfying condition $(\mathcal R)$ which is one of the central conclusions of this paper. \[thm:weak-iso\] Let ${{\alpha}}$ be an action of ${{\mathbb Z}}^d$ by automorphisms of ${{\mathbb T}}^n$ satisfying condition $(\mathcal R)$ and ${{\alpha}}'$ another ${{\mathbb Z}}^d$-action by toral automorphisms. Then $({{\alpha}},\,{{\lambda}})$ is weakly isomorphic to $({{\alpha}}',\,{{\lambda}}')$ if and only if $\rho_{{{\alpha}}}$ and $\rho_{{{\alpha}}'}$ are isomorphic over ${{\mathbb Q}}$, i.e. if ${{\alpha}}$ and ${{\alpha}}'$ are finite algebraic factors of each other. By Theorem \[thm-factors\], ${{\alpha}}$ and ${{\alpha}}'$ are algebraic factors of each other. This implies that ${{\alpha}}'$ acts on the torus of the same dimension $n$ and hence both algebraic factor–maps have finite fibres. Now the statement follows from Proposition \[prop-finfactors\]. Distinguishing weakly isomorphic actions ---------------------------------------- Similarly we can translate criteria for algebraic conjugacy of weakly algebraically conjugate actions to the measurable setting. \[thm:cyclic\]If ${{\alpha}}$ is an irreducible cyclic action of ${{\mathbb Z}}^d,\,\,d\ge 2$, on ${{\mathbb T}}^n$ and ${{\alpha}}'$ is a non–cyclic ${{\mathbb Z}}^d$-action by toral automorphisms. Then ${{\alpha}}$ and ${{\alpha}}'$ are not measurably isomorphic up to a time change. Since action ${{\alpha}}$ satisfies condition $(\mathcal R)$ (Corollary \[cor:irred\]) we can apply Theorem \[thm:weak-iso\] and conclude that we only need to consider the case when $\rho_{{{\alpha}}}$ and $\rho_{{{\alpha}}'}$ are isomorphic over ${{\mathbb Q}}$ up to a time change. But then, by Proposition \[prop:cyclic\], ${{\alpha}}$ and ${{\alpha}}'$ are not algebraically isomorphic up to a time change and hence, by Corollary \[cor:time-change\], they are not measurably isomorphic up to a time change. Combining Proposition \[prop:min-max\] and Corollary \[cor:time-change\] we immediately obtain rigidity for the minimal irreducible models. \[cor:min-max\] Assume that $\mathcal O\supsetneq {{\mathbb Z}}[{{\lambda}}_1,\dots,{{\lambda}}_d]$. Then the action ${{\alpha}}_{\bar{{\lambda}},\mathcal O}$ is not measurably isomorphic up to a time change to ${{\alpha}}_{\bar{{\lambda}}}^{\min}$. In particular, if $\mathcal O_K\supsetneq{{\mathbb Z}}[{{\lambda}}_1,\dots,{{\lambda}}_d]$, then the actions ${{\alpha}}_{\bar{{\lambda}}}^{\max}$ and ${{\alpha}}_{\bar{{\lambda}}}^{\min}$ are not measurably isomorphic up to a time change. Examples {#s:examples} ======== Now we proceed to produce examples of actions for which the entropy data coincide but which are not algebraically isomorphic, and hence by Theorem \[thm-isom\] not measure–theoretically isomorphic. Weakly nonisomorphic actions {#ss:not weakly} ---------------------------- In this section we consider actions which are not algebraically isomorphic over ${{\mathbb Q}}$ (or, equivalently, over ${{\mathbb R}}$) and hence by Theorem \[thm:weak-iso\] are not even weakly isomorphic. The easiest way is as follows. [**Example 1a.**]{} Start with any action ${{\alpha}}$ of ${{\mathbb Z} ^d},\,d\ge 2$, by ergodic automorphisms of ${{\mathbb T}}^n$. We may double the entropies of all its elements in two different ways: by considering the Cartesian square ${{\alpha}}\times{{\alpha}}$ acting on ${{\mathbb T}}^{2n}$, and by taking second powers of all elements: ${{\alpha}}_2^{{\mathbf n}}={{\alpha}}^{2{{\mathbf n}}}$ for all ${{\mathbf n}}\in{{\mathbb Z}}^d$. Obviously ${{\alpha}}\times{{\alpha}}$ is not algebraically isomorphic to ${{\alpha}}_2$, since, for example, they act on tori of different dimension. Hence by Theorem \[thm-isom\] $({{\alpha}}\times{{\alpha}}, {{\lambda}})$ is not metrically isomorphic to $({{\alpha}}_2, {{\lambda}})$ either. Now we assume that ${{\alpha}}$ contains an automorphism $F_A$ where $A$ is hyperbolic with an irreducible characteristic polynomial and distinct positive real eigenvalues. In this case it is easy to find an invariant distinguishing the two actions, namely, the algebraic type of the centralizer of the action in the group of measure–preserving transformations. By Corollary \[cor-centr\], the centralizer of ${{\alpha}}$ in the group of measure–preserving transformations coincides with the centralizer in the group of affine maps, which is a finite extension of the centralizer in the group of automorphisms. By the Dirichlet Unit Theorem, the centralizer of $Z({{\alpha}}_2)$ in the group of automorphisms of the torus is isomorphic to $\{\pm 1\}\times{{\mathbb Z}}^{n-1}$, whereas the centralizer of ${{\alpha}}\times{{\alpha}}$ contains the ${{\mathbb Z}}^{2(n-1)}$–action by product transformations ${{\alpha}}^{\mathbf{n}_1}\times{{\alpha}}^{\mathbf{n}_2},\,\,\mathbf{n}_1, \mathbf{n}_2 \in{{\mathbb Z}}^{n-1}$. In fact, the centralizer of ${{\alpha}}\times{{\alpha}}$ can be calculated explicitly: Let ${{\lambda}}$ be an eigenvalue of $A$. Then $K={{\mathbb Q}}({{\lambda}})$ is a totally real algebraic field. If its ring of integers $\mathcal O_K$ is equal to ${{\mathbb Z}}[{{\lambda}}]$ then the centralizer of ${{\alpha}}\times{{\alpha}}$ in $GL(2n,{{\mathbb Z}})$ is isomorphic to the group $GL(2,\mathcal O_K)$, i.e. the group of $2\times 2$ matrices with entries in $\mathcal O_K$ whose determinant is a unit in $\mathcal O_K$. First we notice that a matrix in block form $B=\left( \begin{smallmatrix} X & Y\\Z & T\end{smallmatrix}\right)$ with $X,Y,Z,T\in M(n,{{\mathbb Z}})$ commutes with $\left( \begin{smallmatrix} A & 0\\0 & A\end{smallmatrix}\right)$ if an only if $X,Y,Z,T$ commute with $A$ and can thus be identified with elements of $\mathcal O_K$. In this case $B$ can be identified with a matrix in $M(2,\mathcal O_K)$. Since $\det\left( \begin{smallmatrix} X & Y\\Z & T\end{smallmatrix}\right)=\det(XT-YZ)=\pm 1$ (cf. [@Ga]), the norm of the determinant of the $2\times 2$ matrix corresponding to $B$ is equal $\pm 1$. Hence this determinant is a unit in $\mathcal O_K$, and we obtain the desired isomorphism. It is not difficult to modify Example 1a to obtain weakly nonisomorphic actions with the same entropy on the torus of the same dimension. [**Example 1b.**]{} For a natural number $k$ define the action ${{\alpha}}_k$ similarly to ${{\alpha}}_2$: ${{\alpha}}_k^{{{\mathbf n}}}={{\alpha}}^{k{{\mathbf n}}}$ for all ${{\mathbf n}}\in{{\mathbb Z}}^d$. The actions ${{\alpha}}_3\times {{\alpha}}$ and ${{\alpha}}_2\times{{\alpha}}_2$ act on ${{\mathbb T}}^{2n}$, have the same entropies for all elements and are not isomorphic. As before, we can see that centralizers of these two actions are not isomorphic. In particular, the centralizer of ${{\alpha}}_3\times {{\alpha}}$ is abelian since it has simple eigenvalues, while the centralizer of ${{\alpha}}_2\times{{\alpha}}_2$ is not. Cartan actions distinguished by cyclicity or maximality {#ss:cent} ------------------------------------------------------- We give two examples which illustrate the method of Section \[ss:irr-integers\]. They provide weakly algebraically isomorphic Cartan actions of ${{\mathbb Z}}^2$ on ${{\mathbb T}}^3$ which are not algebraically isomorphic even up to a time change (i.e. a linear change of coordinates in ${{\mathbb Z}}^2$) by Proposition \[prop:min-max\]. These examples utilize the existence of number fields $K={{\mathbb Q}}({{\lambda}})$ and units $\bar{{\lambda}}=({{\lambda}}_1,{{\lambda}}_2)$ in them for which $\mathcal O_K\ne{{\mathbb Z}}[{{\lambda}}_1,{{\lambda}}_2]$. In each example one action has a form ${{\alpha}}_{\bar{{\lambda}}}^{\min}$ and the other ${{\alpha}}_{\bar{{\lambda}}}^{\max}$. Hence by Corollary \[cor:min-max\] they are not measurably isomorphic up to a time change In other words, in each example one action, namely, ${{\alpha}}_{\bar{{\lambda}}}^{\min}$, is a cyclic Cartan action, and the other is not. We will aslo show that in these examples the conjugacy type of the pair $(Z({{\alpha}}),{{\alpha}})$ distinguishes weakly isomorphic actions. Let us point out that a noncylic action for example ${{\alpha}}_{\bar{{\lambda}}}^{\max}$ may be maximal, for example when fundamental units lie in a proper subring of $\mathcal O_K$. However in our examples centralizers for the cyclic actions will be dirrefern and thus will serve as a distuinguishing invariant. The information about cubic fields is either taken from [@C] or obtained with the help of the computer package Pari-GP. Some calculations were made by Arsen Elkin during the REU program at Penn State in summer of 1999. We construct two ${{\mathbb Z}}^2$–actions, ${{\alpha}}$, generated by commuting matrices $A$ and $B$, and ${{\alpha}}'$, generated by commuting matrices $A'$ and $B'$ in $GL(3,{{\mathbb Z}})$. These actions are weakly algebraically isomorphic by Proposition \[prop-class\] since they are produced with the same set of units on two different orders, ${{\mathbb Z}}[{{\lambda}}]$ and $\mathcal O_K$, but not algebraically isomorphic by Proposition \[prop:min-max\]. In these examples the action ${{\alpha}}$ is cyclic by Corollary \[cor-cyclic\] and will be shown to be a maximal Cartan action. Thus $Z({{\alpha}})={{\alpha}}\times\{\pm{{\rm Id}}\}$. The action ${{\alpha}}'$ is not maximal, specifically, $Z({{\alpha}}')/\{\pm{{\rm Id}}\}$ is a nontrivial finite extension of ${{\alpha}}'$. [**Example 2a**]{}. Let $K$ be a totally real cubic field given by the irreducible polynomial $f(x)=x^3 + 3 x^2 - 6 x + 1$, i.e. $K={{\mathbb Q}}({{\lambda}})$ where ${{\lambda}}$ is one of its roots. The discriminant of $K$ is equal to $81$, hence its Galois group is cyclic, and $[\mathcal O_K:{{\mathbb Z}}[{{\lambda}}]]=3$. The algebraic integers ${{\lambda}}_1={{\lambda}}$ and ${{\lambda}}_2=2-4{{\lambda}}-{{\lambda}}^2$ are units with $f({{\lambda}}_1)=f({{\lambda}}_2)=0$. The minimal order in $K$ containing ${{\lambda}}_1$ and ${{\lambda}}_2$ is ${{\mathbb Z}}[{{\lambda}}_1,{{\lambda}}_2]={{\mathbb Z}}[{{\lambda}}]$, and the maximal order is $\mathcal O_K$. A basis in fundamental units is $\epsilon=\frac{{{\lambda}}^2+5{{\lambda}}+1}{3}$ and $\epsilon-1$, hence $\mathcal U_K$ is not contained in ${{\mathbb Z}}[{{\lambda}}]$. With respect to the basis $\{1,{{\lambda}},{{\lambda}}^2\}$ in ${{\mathbb Z}}[{{\lambda}}]$, multiplications by ${{\lambda}}_1$ and ${{\lambda}}_2$ are given by the matrices $$A=\left( \begin{smallmatrix} \hphantom{-}0&\hphantom{-}1&\hphantom{-}0 \\ \hphantom{-}0&\hphantom{-}0&\hphantom{-}1 \\ -1&\hphantom{-}6&-3 \end{smallmatrix} \right),\qquad B=\left( \begin{smallmatrix} 2&-4&-1 \\ 1&-4&-1 \\ 1&-5&-1 \end{smallmatrix} \right),$$ respectively (if acting from the right on row–vectors). A direct calculation shows that this action is maximal. With respect to the basis $\{-\frac 2{3}+\frac 5{3}{{\lambda}}+\frac 1{3}{{\lambda}}^2, -\frac 1{3}+\frac 7{3}{{\lambda}}+\frac 2{3}{{\lambda}}^2\}$ in $\mathcal O_K$, multiplications by ${{\lambda}}_1$ and ${{\lambda}}_2$ are given by the matrices $$A'=\left( \begin{smallmatrix} \hphantom{-}1&\hphantom{-}2&-1 \\ -1&-2&\hphantom{-}2 \\ \hphantom{-}2&\hphantom{-}5&-2 \end{smallmatrix} \right),\qquad B'=\left( \begin{smallmatrix} \hphantom{-}1&-1&-1 \\ -1&-2&-1 \\ -1&-4&-2 \end{smallmatrix} \right).$$ We have $A'=VAV^{-1}$, $B'=VBV^{-1}$ for $ V=\left( \begin{smallmatrix} 2&-2&-1 \\ 0&-3&\hphantom{-}0 \\ 1&-4&-2 \end{smallmatrix} \right)$. Since $A$ is a companion matrix of $f$, ${{\alpha}}=\langle A,B\rangle$ has a cyclic element in ${{\mathbb Z}}^3$. If $A'$ also had a cyclic element $\mathbf m=(m_1,m_2,m_3)\in{{\mathbb Z}}^3$, then the vectors $$\begin{smallmatrix} \mathbf m=(m_1,m_2,m_3),\enspace \mathbf mA'=(m_1-m_2+2m_3,2m_1-2m_2+5m_3,-m_1+2m_2-2m_3)\\ \mathbf m(A')^2=(-3m_1+5m_2-7m_3,-7m_1+12m_2-16m_3,5m_1-7m_2+12m_3), \end{smallmatrix}$$ would have to generate ${{\mathbb Z}}^3$ or, equivalently $$\begin{gathered} \det \left( \begin{smallmatrix} m_1&m_2&m_3 \\ m_1 - m_2 + 2 m_3&2 m_1 - 2 m_2 + 5 m_3&-m_1+2m_2-2m_3 \\ - 3 m_1 + 5 m_2 - 7 m_3&- 7 m_1 + 12 m_2 - 16 m_3&5 m_1 - 7 m_2 + 12 m_3 \end{smallmatrix} \right) \\ \begin{aligned} = 3 m_1^3 &+ 18 m_1^2m_3 - 9 m_1 m_2^2 - 9 m_1 m_2 m_3 \\ &+ 27 m_1 m_3^2 + 3 m_2 ^3 - 9 m_2 m_3^2+ 3 m_3^3=1. \end{aligned} \end{gathered}$$ This contradiction shows that $A'$ has no cyclic vector, and since $B'=2-4A'-{A'}^2$ , the action ${{\alpha}}'$ is not cyclic. In this example both actions ${{\alpha}}$ and ${{\alpha}}'$ have a single fixed point $(0,0,0)$, hence their linear and affine centralizers coincide, and by Corollary \[cor:time-change\] ${{\alpha}}$ and ${{\alpha}}'$ are not measurably isomorphic up to a time change. The action ${{\alpha}}'$ is not maximal beacuse $Z({{\alpha}}')$ contains fundamental units. [**Example 2b**]{}. Let us consider a totally real cubic field $K$ given by the irreducible polynomial $f(x)=x^3 - 7x^2 + 11x - 1$. Thus $K={{\mathbb Q}}({{\lambda}})$ where ${{\lambda}}$ is one of its roots. In this field the ring of integers $\mathcal O_K$ has basis $\{1,{{\lambda}},\frac 1{2}{{\lambda}}^2+\frac1{2}\}$ and hence $[\mathcal O_K:{{\mathbb Z}}[{{\lambda}}]]=2$. The fundamental units in $\mathcal O_K$ are $\{\frac 1{2}{{\lambda}}^2-2{{\lambda}}+\frac 1{2}, {{\lambda}}-2\}$. We choose the units ${{\lambda}}= {{\lambda}}_1=(\frac 1{2}{{\lambda}}^2-2{{\lambda}}+\frac 1{2})^2$ and ${{\lambda}}_2={{\lambda}}-2$ which are contained in both orders, $\mathcal O_K$ and ${{\mathbb Z}}[{{\lambda}}]$. In ${{\mathbb Z}}[{{\lambda}}]$ we consider the basis $\{1,{{\lambda}},{{\lambda}}^2\}$ relative to which the multiplication by ${{\lambda}}_1$ is represented by the companion matrix $A=\left( \begin{smallmatrix} \hphantom{-}0 & \hphantom{-}1 & \hphantom{-}0\\ \hphantom{-}0 & \hphantom{-}0 & \hphantom{-}1\\ \hphantom{-}1 & -11 & \hphantom{-}7\end{smallmatrix}\right)$ and multiplication by ${{\lambda}}_2$ is represented by the matrix $B=\left( \begin{smallmatrix} -2 & \hphantom{-}1 & \hphantom{-}0\\ \hphantom{-}0 & -2 & \hphantom{-}1\\\hphantom{-}1 & -11 & \hphantom{-}5\end{smallmatrix}\right)$. For $\mathcal O_K$ with the basis $\{1,{{\lambda}},\frac 1{2}{{\lambda}}^2+\frac1{2}\}$ multiplications by ${{\lambda}}_1$ and ${{\lambda}}_2$ are represented by the matrices $A'=\left( \begin{smallmatrix} \hphantom{-}0 & \hphantom{-}1 & \hphantom{-}0\\ -1 & \hphantom{-}0 & \hphantom{-}2\\ -3 & -5 & \hphantom{-}7\end{smallmatrix}\right)$ and $B'=\left( \begin{smallmatrix} -2 & \hphantom{-}1 & \hphantom{-}0\\ -1 & -2 & \hphantom{-}2\\ -3 & -5 & \hphantom{-}5\end{smallmatrix}\right)$. It can be seen directly that ${{\alpha}}$ and ${{\alpha}}'$ are not algebraically conjugate up to a time change since $A'$ is a square of a matrix from $SL(3,{{\mathbb Z}})$: $A'= {\left(\begin{smallmatrix}\hphantom{-}0 & -2 & \hphantom{-}1\\ -1 & -5 & \hphantom{-}3\\ -2 & -9 & \hphantom{-}6\end{smallmatrix}\right)}^2$, while $A$ is not a square of a matrix in $GL(3,{{\mathbb Z}})$, which is checked by reducing modulo $2$. In this case it is also easily seen that the action ${{\alpha}}'$ is not cyclic since the corresponding determinant is divisible by 2. The action ${{\alpha}}$ has $2$ fixed points on ${{\mathbb T}}^3$: $(0,0,0)$ and $(\frac 1{2},\frac 1{2},\frac 1{2})$, while the action ${{\alpha}}'$ has $4$ fixed points: $(0,0,0)$, $(\frac 1{2},\frac 1{2},\frac 1{2})$, $(\frac 1{2},\frac 1{2},0)$, and $(0,0,\frac 1{2})$. Hence the affine centralizer of ${{\alpha}}$ is $Z({{\alpha}})\times{{\mathbb Z}}/2{{\mathbb Z}}$, and the affine centralizer of ${{\alpha}}'$ is $Z({{\alpha}}')\times({{\mathbb Z}}/2{{\mathbb Z}}\times{{\mathbb Z}}/2{{\mathbb Z}})$. By Lemma \[hyp\], the group of elements of finite order in $Z_\mathit{Aff}({{\alpha}})$ is ${{\mathbb Z}}/2{{\mathbb Z}}\times {{\mathbb Z}}/2{{\mathbb Z}}$ and in $Z_\mathit{Aff}({{\alpha}}')$ it is ${{\mathbb Z}}/2{{\mathbb Z}}\times {{\mathbb Z}}/2{{\mathbb Z}}\times {{\mathbb Z}}/2{{\mathbb Z}}$. The indices of each action in its affine centralizer are $[Z_\mathit{Aff}({{\alpha}}):{{\alpha}}]=4$ and $[Z_\mathit{Aff}({{\alpha}}'):{{\alpha}}']=16$. This gives two alternative arguments that the actions are not measurably isomorphic up to a time change. Nonisomorphic maximal Cartan actions {#ss:max-Cart} ------------------------------------ We find examples of weakly algebraically isomorphic maximal Cartan actions which are not algebraically isomorphic up to time change. For such an action ${{\alpha}}$ the structure of the pair $(Z({{\alpha}}),{{\alpha}})$ is always the same: $Z({{\alpha}})$ is isomorphic as a group to ${{\alpha}}\times\{\pm{{\rm Id}}\}$. The algebraic tool which allows to distinguish the actions is Theorem \[mytheor\]. [**Example 3a**]{}. An example for $n=3$ can be obtained from a totally real cubic field with class number $2$ and the Galois group $S_3$. The smallest discriminant for such a field is $1957$ ([@C], Table B4), and it can be represented as $K={{\mathbb Q}}({{\lambda}})$ where ${{\lambda}}$ is a unit in $K$ with minimal polynomial $f(x)=x^3-2x^2-8x-1$. In this field the ring of integers $\mathcal O_K={{\mathbb Z}}[{{\lambda}}]$ and the fundamental units are ${{\lambda}}_1={{\lambda}}$ and ${{\lambda}}_2={{\lambda}}+2$. Two actions are constructed with this set of units (fundamental, hence multiplicatively independent) on two different lattices, $\mathcal O_K$ with the basis $\{1,{{\lambda}},{{\lambda}}^2\}$, representing the principal ideal class, and $\mathcal L$ with the basis $\{2, 1+{{\lambda}}, 1+{{\lambda}}^2\}$ representing to the second ideal class. Notice that the units ${{\lambda}}_1$ and ${{\lambda}}_2$ do not belong to $\mathcal L$, but $\mathcal L$ is a ${{\mathbb Z}}[{{\lambda}}]$-module. The first action ${{\alpha}}$ is generated by the matrices $A=\left( \begin{smallmatrix} 0 & 1& 0\\ 0 & 0 & 1\\ 1 & 8 & 2 \end{smallmatrix}\right)$ and $B=\left( \begin{smallmatrix} 2 & 1 & 0\\ 0 & 2 & 1\\ 1 & 8 & 4 \end{smallmatrix}\right)$ which represent multiplication by ${{\lambda}}_1$ and ${{\lambda}}_2$, respectively, on $\mathcal O_K$. The second action ${{\alpha}}'$ is generated by matrices $A'=\left( \begin{smallmatrix} -1 & \hphantom{-}2 & \hphantom{-}0\\ -1 & \hphantom{-}1 & \hphantom{-}1\\ -5 & \hphantom{-}9 & \hphantom{-}2\end{smallmatrix}\right)$ and $B'=\left( \begin{smallmatrix} \hphantom{-}1 & \hphantom{-}2 & \hphantom{-}0\\ -1 & \hphantom{-}3 & \hphantom{-}1\\ -5 & \hphantom{-}9 & \hphantom{-}5\end{smallmatrix}\right)$ which represent multiplication by ${{\lambda}}_1$ and ${{\lambda}}_2$, respectively, on $\mathcal L$ in the given basis. By Proposition \[prop-class\] these actions are weakly algebraically isomorphic. By Theorem \[mytheor\] they are not algebraically isomorphic. Since the Galois group is $S_3$ there are no nontrivial time changes which produce conjugacy over ${{\mathbb Q}}$. Therefore, but Theorem \[thm-isom\] the actions are not measurably isomorphic. It is interesting to point out that for actions ${{\alpha}}$ and ${{\alpha}}'$ the affine centralizers $Z_\mathit{Aff}({{\alpha}})$ and $Z_\mathit{Aff}({{\alpha}}')$ are not isomorphic as abstract groups. The action ${{\alpha}}$ has $2$ fixed points on ${{\mathbb T}}^3$: $(0,0,0)$ and $(\frac 1{2},\frac 1{2},\frac 1{2})$, while the action ${{\alpha}}'$ has a single fixed point $(0,0,0)$. Hence $Z_\mathit{Aff}({{\alpha}})$ is isomorphic to $Z({{\alpha}})\times{{\mathbb Z}}/2{{\mathbb Z}}$, $Z_\mathit{Aff}({{\alpha}}')$ is isomorphic to $Z({{\alpha}}')$. As abstract groups, $Z_\mathit{Aff}({{\alpha}})\approx {{\mathbb Z}}^2\times {{\mathbb Z}}/2{{\mathbb Z}}\times {{\mathbb Z}}/2{{\mathbb Z}}$ and $Z_\mathit{Aff}({{\alpha}}')\approx {{\mathbb Z}}^2\times {{\mathbb Z}}/2{{\mathbb Z}}$. Hence by Corollary \[cor-centr\] the measurable centralizers of ${{\alpha}}$ and ${{\alpha}}'$ are not conjugate in the group of measure–preserving transformation providing a distinguishing invariant of measurable isomorphism. [**Example 3b**]{}. This example is obtained from a totally real cubic field with class number $3$, Galois group $S_3$, and discriminant $2597$. It can be represented as $K={{\mathbb Q}}({{\lambda}})$ where ${{\lambda}}$ is a unit in $K$ with minimal polynomial $f(x)=x^3-2x^2-8x+1$. In this field the ring of integers $\mathcal O_K={{\mathbb Z}}[{{\lambda}}]$ and the fundamental units are ${{\lambda}}_1={{\lambda}}$ and ${{\lambda}}_2={{\lambda}}+2$. Three actions are constructed with this set of units on three different lattices, $\mathcal O_K$ with the basis $\{1,{{\lambda}},{{\lambda}}^2\}$, representing the principal ideal class, $\mathcal L$ with the basis $\{2, 1+{{\lambda}}, 1+{{\lambda}}^2\}$ representing the second ideal class, and $\mathcal L^2$ with the basis $\{4,3+{{\lambda}},3+{{\lambda}}^2\}$ representing the third ideal class. Multiplications by ${{\lambda}}_1$ and ${{\lambda}}_2$ generate the following three weakly algebraically isomorphic actions which are not algebraically isomorphic by Theorem \[mytheor\] even up to a time change, and therefore not measurably isomorphic: $$A=\left( \begin{smallmatrix} \hphantom{-}0 & \hphantom{-}1& \hphantom{-}0\\ \hphantom{-}0 & \hphantom{-}0 & \hphantom{-}1\\ -1 & \hphantom{-}8 & \hphantom{-}2 \end{smallmatrix}\right) \quad\text{and} \quad B=\left( \begin{smallmatrix} \hphantom{-}2 & \hphantom{-}1 & \hphantom{-}0\\ \hphantom{-}0 & \hphantom{-}2 & \hphantom{-}1\\ -1 & \hphantom{-}8 & \hphantom{-}4 \end{smallmatrix}\right);$$ $$A'=\left( \begin{smallmatrix} -1 & \hphantom{-}2 & \hphantom{-}0\\-1 & \hphantom{-}1 & \hphantom{-}1\\ -6 & \hphantom{-}9 & \hphantom{-}2 \end{smallmatrix}\right) \quad\text{and} \quad B'=\left( \begin{smallmatrix} \hphantom{-}1 & \hphantom{-}2 & \hphantom{-}0\\-1 & \hphantom{-}3 & \hphantom{-}1\\ -6 & \hphantom{-}9 & \hphantom{-}4 \end{smallmatrix}\right);$$ $$A''=\left( \begin{smallmatrix}-3 & \hphantom{-}4 & \hphantom{-}0\\ -3 & \hphantom{-}3 & \hphantom{-}1\\ -10 & \hphantom{-}11 & \hphantom{-}2 \end{smallmatrix}\right) \quad\text{and} \quad B''=\left( \begin{smallmatrix} -1 & \hphantom{-}4 & \hphantom{-}0\\ -3 & \hphantom{-}5 & \hphantom{-}1\\ -10 & \hphantom{-}11 & \hphantom{-}4 \end{smallmatrix}\right).$$ Each action has $2$ fixed point in ${{\mathbb T}}^3$, $(0,0,0)$ and $(\frac 1{2},\frac 1{2},\frac 1{2})$. Hence all affine centralizers are isomorphic as abstract groups to ${{\mathbb Z}}^2\times {{\mathbb Z}}/2{{\mathbb Z}}\times {{\mathbb Z}}/2{{\mathbb Z}}$. [**Example 3c**]{} Finally we give an example of two nonisomorphic maximal Cartan actions which come from the vector of fundamental units $\bar{{\lambda}}=({{\lambda}}_1,{{\lambda}}_2)$ in a totally real cubic field $K$ such that ${{\mathbb Z}}({{\lambda}}_1,{{\lambda}}_2)\neq \mathcal O_K$. Thus the whole group of units does not generate the ring $\mathcal O_K$. Both actions ${{\alpha}}_{\bar{{\lambda}}}^{\min}$ and ${{\alpha}}_{\bar{{\lambda}}}^{\max}$ of the group ${{\mathbb Z}}^2$ are maximal Cartan actions by Lemma \[l:rings-units\]. However by Corollary \[cor-cyclic\] the former is cyclic and the latter is not and hence they are not measurably isomorphic up to a time change by Corollary \[cor:min-max\]. For a specific example we pick the totally real cubic field $K={{\mathbb Q}}({{\alpha}})$ with class number $1$ discriminant $1304$ given by the polynomial $x^3-x^2-11x-1$. For this filed we have $[\mathcal O_K:{{\mathbb Z}}({{\alpha}})]=2$. Generators in $\mathcal O_K$ can be taken to be $\{1,{{\alpha}},\beta=\frac{{{\alpha}}^2+1}2\}$. Fundamental units are ${{\lambda}}_1=-{{\alpha}},\;{{\lambda}}_2=-5+14{{\alpha}}+10\beta=14{{\alpha}}+5{{\alpha}}^2\in{{\mathbb Z}}[{{\alpha}}]$. Thus the whole group of units lies in ${{\mathbb Z}}[{{\lambda}}]$. To construct the generators for two non–isomorphic action ${{\alpha}}_{\bar{{\lambda}}}^{\min}$ and ${{\alpha}}_{\bar{{\lambda}}}^{\max}$ we write multiplications by ${{\lambda}}_1$ and ${{\lambda}}_2$ in bases $\{1,{{\alpha}},{{\alpha}}^2\}$ and $\{1,{{\alpha}},\beta\}$, correspondingly. The resulting matrices are: $$A=\left(\begin{smallmatrix}\hphantom{-}0 & -1 & \hphantom{-}0\\ \hphantom{-}1 & \hphantom{-}0 & -1 \\ \hphantom{-}1 & \hphantom{-}11 & \hphantom{-}1 \end{smallmatrix}\right)\quad B=\left(\begin{smallmatrix} 0 & 14 & 5\\ 5 & 55 & 19\\ 19 & 214 & 74 \end{smallmatrix}\right),$$ $$A'=\left(\begin{smallmatrix}\hphantom{-}0 & -1 & \hphantom{-}0\\ \hphantom{-}1 & \hphantom{-}0 & -2 \\ \hphantom{-}0 & -6 & -1 \end{smallmatrix}\right)\quad B=\left(\begin{smallmatrix} -5 & \hphantom{-}14 & \hphantom{-}10\\ -14 & \hphantom{-}55 & \hphantom{-}38\\ -30 & \hphantom{-}114 & \hphantom{-}79 \end{smallmatrix}\right).$$ The first action has only one fixed point, the origin; the second has four fixed points $(0,0,0)$, $(\frac 1{2},\frac 1{2},\frac 1{2})$, $(\frac 1{2},\frac 1{2},0)$, and $(0,0,\frac 1{2})$. Thus we have an example of two maximal Cartan actions of ${{\mathbb Z}}^2$ which have nonisomorphic affine and hence measurable centralizers. [aa]{} R.L. Adler and B. Weiss *Entropy, a complete metric invariant for automorphisms of the torus*, Proc. Nat. Acad. Sci. **57** (1967), 1573–1576. D. Berend, *Multi-invariant sets on tori*, Trans. Amer. Math. Soc. **280** (1983), 509–532. Z.I. Borevich and I.R. Shafarevich, *Number Theory*, Academic Press, New York, 1966. H. Cohen, *A course in computational algebraic number theory* Springer, Berlin–Heidelberg–New York, 1996. F.R. Gantmacher, *Theory of matrices*, vol. 1, Chelsea, New York, 1959; vol. 2, Chelsea, New York, 1960. P.R. Halmos, *On automorphisms of compact groups*, Bull. Amer. Math. Soc. **49** (1943), 619–624. B. Kalinin and A. Katok, *Rigidity of invariant measures, factors and joinings for actions of higher–rank abelian groups*, Preprint (2000). A. Katok and K.Schmidt, *The cohomology of expansive $Z^d$ actions by automorphisms of compact abelian groups* Pacific Jour. Math. **170** (1995), 105–142. A. Katok and R.J. Spatzier, *First cohomology of Anosov actions of higher rank abelian groups and applications to rigidity* Publ. Math. IHES **79** (1994), 131–156. A. Katok and R.J. Spatzier, *Subelliptic estimates of polynomial differential operators and applications to rigidity of abelian actions* Math. Res. Lett. **1** (1994), 193–202. A. Katok and R.J. Spatzier, *Invariant measures for higher rank hyperbolic abelian actions*, Ergod. Th. & Dynam. Sys. **16** (1996), 751–778; *Corrections*: Ergod. Th. & Dynam. Sys. **18** (1998), 503–507. A. Katok and R.J. Spatzier, *Differential rigidity of Anosov actions of higher rank abelian groups and algebraic lattice actions*, Proc. Steklov Inst. Math. **216** (1997), 287–314. S. Katok, *Coding of closed geodesics after Gauss and Morse*, Geom. Dedicata **63** (1996), 123–145. I. Katznelson, *Ergodic automorphisms of $T^n$ are Bernoulli shifts*, Israel J. Math. **10** (1971), 186–195. B. Kitchens and K. Schmidt, *Automorphisms of compact groups*, Ergod. Th. & Dynam. Sys. **9** (1989), 691–735. B. Kitchens and K. Schmidt, *Isomorphism rigidity of simple algebraic $\mathbf{Z}^d$–actions*, preprint (1999). Latimer and C.C. MacDuffee, *A correspondence between classes of ideals and classes of matrices*, Ann. Math. **74** (1933), 313–316. D. Lind, K. Schmidt and T. Ward, *Mahler measure and entropy for commuting automorphisms of compact groups*, Invent. Math. **101** (1990), 593–629. D. Ornstein, *Bernoulli shifts with the same entropy are isomorphic*, Adv. Math. **4** (1970), 337–352. M. Ratner, *Rigidity of horocycle flows*, Ann. Math. **115**, (1982), 597–614. M. Ratner, *Factors of horocycle flows*, Ergod. Th. & Dynam. Sys. **2** (1982), 465–489. M. Ratner, *Horocycle flows, joinings and rigidity of products* Ann. Math. **118** (1983), 277-313. P. Sarnak, *Prime geodesic theorem*, Ph.D. thesis, Stanford, 1980. K. Schmidt, *Automorphisms of compact abelian groups and affine varieties*, Proc. London Math. Soc. **61** (1990), 480–496. K. Schmidt, *Dynamical systems of algebraic origin*, Birkhäuser Verlag, Basel-Berlin-Boston, 1995. K. Schmidt, *Measurable rigidity of algebraic $\mathbf{Z}^d$-actions*, Preprint (2000). A. Starkov, *First cohomology group, mixing and minimal sets of commutative group of algebraic action on torus*, preprint. O. Taussky, *Introduction into connections between algebraic number theory and integral matrices*, Appendix to: H. Cohn, A classical invitation to algebraic numbers and class field, Springer, New–York, 1978. D.I. Wallace, *Conjugacy classes of hyperbolic matrices in $\textsl{SL}(n,{{\mathbb Z}}$) and ideal classes in an order*. Trans. Amer. Math. Soc. **283** (1984), 177–184. [^1]: The research of the first author was partially supported by NSF grant DMS-9704776. The first two authors are grateful to the Erwin Schrödinger Institute, Vienna, and the third author to the Center for Dynamical Systems at Penn State University, for hospitality and support while some of this work was done.

--- abstract: 'We consider wavepackets composed of two modulated carrier Bloch waves with opposite group velocities in the one dimensional periodic Nonlinear Schrödinger/Gross-Pitaevskii equation. These can be approximated by first order coupled mode equations (CMEs) for the two slowly varying envelopes. Under a suitably selected periodic perturbation of the periodic structure the CMEs possess a spectral gap of the corresponding spatial operator and allow families of exponentially localized solitary waves parametrized by velocity. This leads to a family of approximate solitary waves in the periodic nonlinear Schrödinger equation. Besides a formal derivation of the CMEs a rigorous justification of the approximation and an error estimate in the supremum norm are provided. Several numerical tests corroborate the analysis.' address: - 'T. Dohnal Department of Mathematics, Technical University DortmundD-44221 Dortmund, Germany' - 'L. Helfmeier Department of Mathematics, Technical University DortmundD-44221 Dortmund, Germany' author: - Tomáš Dohnal - Lisa Helfmeier bibliography: - 'biblio\_CME1D.bib' title: Justification of the Coupled Mode Asymptotics for Localized Wavepackets in the Periodic Nonlinear Schrödinger Equation --- Introduction ============ Propagation of localized wavepackets in nonlinear media with a periodic structure is a classical problem in the field of nonlinear dispersive equations. This paper contributes to the mathematical analysis of asymptotic approximations of small wavepackets in periodic structures of finite (rather than infinitesimal) contrast. Typical examples of coherent wavepackets in nonlinear periodic media are optical pulses in nonlinear photonic crystals [@SE03] or matter waves in Bose-Einstein condensates with superimposed optical lattices [@LOKS03]. In optics such pulses can be applied as bit carriers in future devices for optical logic and computation, which are likely to heavily exploit photonic crystals [@Brod_98; @Pelusi_etal_08]. Small spatially broad wavepackets in nonlinear problems with periodic coefficients can be effectively studied using a slowly varying envelope approximation. The scaling of the envelope variables is, however, not unique and can lead to different effective amplitude equations with qualitatively different solutions. On the one hand there is the nonlinear Schrödinger scaling, in which the envelope multiplies a single selected Bloch wave and depends on the moving frame variable $x-c_g t$ and slowly on time to describe a slow temporal modulation, see e.g. [@BSTU06]. For semilinear equations of second order in space with a nonlinearity that is quadratic or cubic near zero (like, e.g., a nonlinear wave equation with the nonlinearity $u^2$ or $u^3$ or the periodic nonlinear Schrödinger equation (PNLS) with the nonlinearity $|u|^2u$) the ansatz \[E:NLS\_ans\] u(x,t)\~A((x-c\_g t),\^2 t)p(x,k\_0)\^[(k\_0x-\_0 t)]{}, 0&lt;1 leads to the cubic nonlinear Schrödinger equation (NLS) with constant coefficients for the envelope $A(X,T)$. The function $p(x,k_0)^{\ri (k_0x-\omega_0 t)}$, with $(k_0,\omega_0)$ being a point in the graph of the dispersion relation, is the selected Bloch wave and $c_g$ is its group velocity. The localized bound state solutions of the NLS predict via approximate solitary wave solutions $u$ of the original problem. At a given frequency $\omega_0$ the solitary wave has a velocity that is asymptotically close to the group velocity $c_g$ of the selected Bloch wave. In one spatial dimension, where the dispersion relation is monotonous and symmetric there is only one group velocity (up to the sign) for a given frequency. On the other hand one can consider a wavepacket composed of several carrier Bloch waves. Such coupling was studied, for instance, in [@GMS08] for the cubic PNLS in $d\in \N$ dimensions, where the Bloch waves are allowed to have different frequencies but need to form a so called closed mode system. When restricted to $d=1$ and a single frequency, the ansatz has the form \[E:CME\_ans\] u(x,t)\~\^[1/2]{} e\^[-\_0 t]{}( A\_+(x,t)p\_+(x)e\^[k\_0x]{}+A\_-(x,t)p\_-(x)e\^[-k\_0x]{}), 0&lt;1, where $p_\pm(x)e^{\ri (\pm k_0x-\omega_0 t)}$ are the two Bloch waves. Clearly, the two envelopes $A_\pm$ are not prescribed to be functions of any moving frame variable. The scaling in leads to a system of first order Dirac type equations - so called *coupled mode equations* (CMEs). In order for the ansatz to predict approximate solitary waves of the original system, e.g. the PNLS, the envelope pair $(A_+,A_-)$ has to be a solitary wave solution of the CMEs with $A_+$ and $A_-$ propagating at the same velocity. The CMEs in [@GMS08] do not possess a spectral gap of the corresponding linear spatial operator such that (exponentially) localized solitary waves of the CMEs are not expected. Indeed, the linear part of the CMEs in [@GMS08] is $$\begin{aligned} &\ri(\pa_T A_++c_g \pa_XA_+)=0\\ &\ri(\pa_T A_--c_g \pa_XA_-)=0 \end{aligned}$$ and the spectrum of $\ri c_g \bspm \pa_X & 0 \\ 0 & -\pa_X\espm$ is the whole $\R$. Note that $X:=\eps x$ and $T:=\eps t$. Our aim is to find a setting which leads to CMEs with solitary waves and to rigorously justify the asymptotic approximation. If a family of CME-solitary waves parametrized by velocity exists, then ansatz predicts a family of approximate solitary waves with frequency close to $\omega_0$ but with an $O(1)$- range of velocities, i.e. not only velocities asymptotically close to the group velocity of the Bloch wave $p_+$ or $p_-$. We restrict to the one dimensional periodic PNLS and show that a large class of $\eps$-small periodic perturbations of the underlying periodic structure leads to CMEs with the linear part $$\begin{aligned} &\ri(\pa_T A_++c_g \pa_XA_+)+\kappa A_-=0\\ &\ri(\pa_T A_--c_g \pa_XA_-)+\kappa A_+=0 \end{aligned}$$ with $\kappa>0$, where, clearly, the operator $\bspm \ri c_g \pa_X & \kappa \\ \kappa & -\ri c_g \pa_X\espm$ has the spectral gap $(-\kappa,\kappa)$. In certain cases the CMEs are identical to those for small contrast periodic structures [@AW89; @GWH01; @SU01; @Pelinov_2011], which possess explicit solitary wave families, see [@AW89] and Section \[S:num\] here. Hence, we consider \[E:PNLS\] \_t u + \_x\^2 u - (V(x) + W(x))u -(x) |u|\^2 u =0, x,t , where the real functions $V,W$, and $\sigma$ satisfy $V(x+2\pi)=V(x), \sigma(x+2\pi)=\sigma(x)$ and $W(x+2\pi/k_W)=W(x)$ for all $x\in \R$ with $k_W>0$ and where $W$ is of mean zero, i.e. \[E:W\] W(x)=\_[n{0}]{}a\_n e\^[n k\_W x]{}, a\_[-n]{}= n. We assume (see assumptions (H1)-(H4)) that $V$ and $\sigma$ are continuous, $W\in C^2(\R)$ and that if $k_W\notin \Q$, then $W$ has only finitely many nonzero Fourier components, i.e. there is an $M\in \N$ such that $a_n=0$ for all $|n|>M$. Note that there is no loss of generality in assuming $a_0=0$ since solutions for a nonzero $a_0\in \R$ can be obtained by a simple phase rotation factor $e^{-\ri \eps a_0 t}$. We point out that since $V$ is independent of $\eps$, the periodic structure has finite contrast, unlike in [@AW89; @GWH01; @SU01; @Pelinov_2011], where infinitesimal contrast is considered. Our approximate wavepacket ansatz is \[E:uapp\] (x,t) := \^[1/2]{}e\^[-\_0 t]{}(A\_+(x,t)p\_+(x)e\^[k\_+ x]{} + A\_-(x,t)p\_-(x)e\^[k\_- x]{}), where $p_\pm$ are $2\pi$-periodic Bloch eigenfunctions at the “band structure coordinates” $(k_\pm,\omega_0)$ respectively, see Sec. \[S:Bloch\] for details. This problem was previously studied also in [@D14], where a specific form of $V$ was considered, namely a finite band potential. This choice guarantees the presence of points $(k_0,\omega_0)$ in the band structure, where at $\eps=0$ eigenvalue curves cross transversally and the perturbation $\eps W$ generates a small spectral gap of the operator $-\pa_x^2+V+\eps W$. Here we show that this special choice of $V$ is not necessary for CMEs with a spectral gap. In addition we provide a rigorous justification of the CME-approximation by proving an estimate of the supremum norm of the error $u-\uapp$ on time intervals of length $O(\eps^{-1})$. In [@GMS08], where the $d-$dimensional problem with $W\equiv 0$ is considered, such a justification is performed in $H^s(\R^d)$ using mainly the semigroup theory, the unitary nature of the Gross-Pitaevskii group and the Gronwall inequality. The proof is performed in the $(x,t)$-variables. On the other hand the approach in [@BSTU06] (for a nonlinear wave equation and the ansatz ) is based on reformulating the problem into one for the Bloch coefficients. First the Bloch transform is applied to the equation as well as the ansatz. The transform is an isomorphism between $H^s(\R)$ and the $L^2(\B,H^s(0,2\pi))$-space over the Brillouin zone $\B$. Subsequently one expands the solution and the ansatz in the Bloch eigenfunctions. This leads to an infinite dimensional ODE system for the expansion coefficients parametrized by the wavenumber $k$. Due to the concentration of the ansatz near $k=k_0$ the problem can be approximately decomposed into one on a neighborhood of $k_0$ and one on neighborhoods of the $k-$points generated by the nonlinearity applied to the ansatz. The problem near $k_0$ leads to the amplitude equation, i.e. the NLS, and the problem near the generated $k-$points can be solved explicitly. The error estimate is provided also here using a Gronwall argument. We present a relatively detailed application of the Bloch transform approach to and , where besides the different scaling compared to in [@BSTU06] a major change is that the action of the potential $\eps W$ on the ansatz generates new $k-$concentration points, which need to be accounted for. Unlike [@BSTU06] we are forced to work in $L^1(\B,H^s(0,2\pi))$ as too many powers of $\eps$ are lost in $L^2$. This was observed also in [@SU01]. No isomorphism is available for the $L^1$ space and we cannot carry over estimates in the original $u(x,t)$-variable to estimates for the Bloch coefficients. Nevertheless, we take advantage of the fact that the inverse Bloch transform applied to $L^1(\B,H^s(0,2\pi))$ with $s>1/2$ produces a continuous function, see Sec. \[S:fn\_spaces\]. Hence an $L^1$ estimate of the error in the Bloch variables translates into a supremum norm estimate in the $u(x,t)$-variable. The Structure of the Paper {#S:struct} -------------------------- In Section \[S:Bloch\] the concept of Bloch waves and of the band structure is reviewed and the choice of carrier Bloch waves $p_\pm(x)e^{\ri(k_\pm x-\omega_0t)}$ for the ansatz is explained. Two cases of the choice of the wavenumbers $k_+$ and $k_-$ are distinguished, namely case (a) with simple Bloch eigenvalues at $(k_+,\omega_0)$ and $(k_-,\omega_0)$ with $k_-=-k_+$, and case (b) with a double eigenvalue at $(k_+,\omega_0)=(k_-,\omega_0)$. Section \[S:formal\] presents the formal derivation of the effective coupled mode equations and explains why it makes sense to distinguish the cases of $k_\pm$ rational and $k_\pm$ irrational. Namely, the rational case can always be reduced to case (b) with a double eigenvalue at $(0,\omega_0)$. In Section \[S:justif\] we formulate and prove the main approximation result. After defining the function spaces and the Bloch transformation in Section \[S:fn\_spaces\], we first present the proof for the case of rational $k_\pm$ in Section \[S:rational\_pf\]. In Section \[S:irrational\_pf\] we treat the case of irrational $k_\pm$, which works in an analogous way but several mainly notational changes are necessary. In Section \[S:discuss\] we discuss some extensions of the analytical results. Finally, Section \[S:num\] provides numerical examples and convergence tests confirming the analysis. Linear Bloch waves; choice of the carrier waves {#S:Bloch} =============================================== For a $P$-periodic ($P>0$) potential $V$, i.e. $V(x+P)=V(x)$ for all $x\in \R$ and the corresponding Brillouin zone $\B_{P}:=(-\pi/P,\pi/P]$, we consider first the Bloch eigenvalue problem \[E:Bloch\_ev\_prob\] (x,k)p:=-(\_x + k)\^2p +V(x)p = p, x(0,P) with $p(x+P)=p(x)$. There is a countable set of eigenvalues $\omega_n(k), n\in \N=\{1,2,\dots\}$ for each $k\in \B_{P}$, which we order in the natural way $\omega_n(k)\leq \omega_{n+1}(k)$. The graph $(k,\omega_n(k))_{n\in \N}$ over $k\in \B_P$ is called the band structure. As functions of $k$ the eigenvalues $\omega_n(k)$ are $2\pi/P-$periodic and analytic away from points of higher multiplicity, which can occur only at $k\in \{0,\pi/P\}$. Moreover, on $(0,\pi/P)$ the eigenvalues are strictly monotone, i.e. $\omega_n'(k)\neq 0$ for all $n\in \N, k\in(0,\pi/P)$ and the band structure is symmetric: $\omega_n(-k)=\omega_n(k)$ for all $k\in \R$. Due to the $2\pi/P$-periodicity we have also $\omega_n(\pi/P-k)=\omega_n(\pi/P+k)$ for all $k\in \R$. In addition, the multiplicity is at most two as the eigenvalue problem is an ordinary differential equation of second order. The $L^2$-spectrum of the operator $-\pa_x^2+V$ is $\text{spec}(-\pa_x^2+V)=\cup_{k\in \B_P} \text{spec} (\cL(\cdot,k))=\cup_{n\in \N}\omega_n(\B_{P})$. For a review of problems with periodic coefficients see [@Eastham] or [@RS4]. The $P$-periodic eigenfunction corresponding to $\omega_n(k)$ is denoted by $p_n(x,k)$ and called a Bloch eigenfunction. After normalization the eigenfunctions satisfy $$\langle p_n(\cdot,k),p_m(\cdot,k)\rangle_{P}:=\langle p_n(\cdot,k),p_m(\cdot,k)\rangle_{L^2(0,P)}=\delta_{n,m} \ \text{for all} \ k\in \B_{P}.$$ For each fixed $k\in \B_{P}$ the set $(p_n(\cdot,k))_{n\in \N}$ is complete in $L^2(0,P)$. As functions of $k$ the eigenfunctions satisfy the periodicity \[E:Bloch\_k\_per\] p\_n(x,k+2/P)=p\_n(x,k)e\^[-(2/P) x]{}. Due to the equivalence of complex conjugation of and replacing $k$ by $-k$, we have $p_n(\cdot,-k)=\overline{p_n(\cdot,k)}$ for simple eigenvalues $\omega_n(k)$. Hence, certainly, $$p_n(\cdot,-k)=\overline{p_n(\cdot,k)} \qquad \text{for all } k\in(0,\pi/P).$$ For our $2\pi$-periodic $V$ in we assume that if the selected eigenvalue $\omega_0$ of at $k=k_0\in \{0,1/2\}$ is double, i.e. $\omega_{n_0}(k_0)=\omega_{n_0+1}(k_0)$ for some $n_0\in \N$, then one can define $C^2-$smooth eigenvalue curves $\tilde{\omega}_{\pm}(k)$, see assumption (H1), as $$\label{E:om_til} \tilde{\omega}_{+}(k)=\begin{cases}\omega_{n_0}(k), &k< k_0 \\ \omega_{n_0+1}(k), &k> k_0 \end{cases} \quad \text{ and } \quad \tilde{\omega}_{-}(k)=\begin{cases}\omega_{n_0+1}(k), &k< k_0 \\ \omega_{n_0}(k), &k> k_0 \end{cases},$$ where $1-$periodicity of the eigenvalues is used if $k_0=1/2$. Also, we assume that the corresponding eigenfunction families $$\label{E:p_til} \tilde{p}_{+}(\cdot, k)=\begin{cases}p_{n_0}(\cdot,k), &k< k_0 \\ p_{n_0+1}(\cdot,k), &k> k_0 \end{cases} \quad \text{ and } \quad \tilde{p}_{-}(\cdot,k)=\begin{cases}p_{n_0+1}(\cdot,k), &k< k_0 \\ p_{n_0}(\cdot,k), &k> k_0 \end{cases}$$ are Lipschitz continuous in $k$ in the $H^2(0,2\pi)$-norm, i.e. the maps $\phi_\pm:\overline{\B}\to H^2(0,2\pi), k\mapsto \tilde{p}_\pm(\cdot,k)$ are Lipschitz continuous on $\overline{\B}$, see assumption (H1). Due to the above even symmetries of the eigenvalues each such point $k_0\in \{0,1/2\}$ of double multiplicity then satisfies $\tilde{\omega}'_+(k_0)=-\tilde{\omega}'_{-}(k_0)>0.$ The conjugation and the periodicity symmetries in $k$ then imply $$\tilde{p}_{+}(x,0)=\overline{\tilde{p}_{-}(x,0)} \quad \text{and} \quad \tilde{p}_{+}(x,1/2)=\overline{\tilde{p}_{-}(x,1/2)}e^{-\ri x}.$$ In we first choose $\omega_0\in \text{spec}(-\pa_x^2+V)$ such that there are two linearly independent eigenfunctions of at $\omega=\omega_0$ and we denote by $k_+,k_-$ the corresponding values in the level set (within $\B_{2\pi}$) of $\omega_0$. Due to the band structure symmetry $\omega_n(-k)=\omega_n(k)$ and monotonicity $\omega_n'(k)\neq 0$ for all $k\in(0,1/2)$, we get that only the following two cases are possible 1. *simple eigenvalues at $k_+,k_-=-k_+$:* $k_+ \in (0,1/2)$, $\omega_0=\omega_{n_0}(k_+)=\omega_{n_0}(k_-)$ for some $n_0\in \N$, 2. *double eigenvalue at $k_+=k_-$:* $k_+=k_- \in \{0,1/2\}$, $\omega_0=\omega_{n_0}(k_+)=\omega_{n_0+1}(k_+)$ for some $n_0\in \N$. In both cases we denote by $\tilde{\omega}_{+}(k), \tilde{\omega}_{-}(k)$ the eigenvalue curves with $C^2$-smoothness at $k=k_+$ and $k=k_-$ respectively with $\tilde{\omega}_{+}(k_+)=\tilde{\omega}_{-}(k_-)=\omega_0$. The corresponding eigenfunction families are denoted (in both cases) by $\tilde{p}_\pm(\cdot,k)$. The group velocity at $k=k_\pm$ is given by \[E:cg\] c\_g:=’\_[+]{}(k\_+)=-’\_[-]{}(k\_-)=\_[kk\_+]{}’\_[n\_0]{}(k)=-\_[kk\_-]{}’\_[n\_0]{}(k). To simplify the notation, we also define the Bloch eigenfunctions at $k=k_\pm$ of the families $\tilde{p}_\pm(\cdot,k)$ by $p_\pm$, i.e. $$p_+:=\tilde{p}_+(\cdot,k_+), \quad p_-\coloneqq\tilde{p}_-(\cdot,k_-).$$ In summary, we have for the two above cases 1. $\tilde{\omega}_+\equiv \tilde{\omega}_-\equiv \omega_{n_0}$; $\tilde{p}_+\equiv \tilde{p}_- \equiv p_{n_0}$, $p_-=\overline{p_+}$, 2. $\tilde{\omega}_\pm$ given by ; $\tilde{p}_\pm$ given by , $p_-=\overline{p_+}e^{-2\ri k_+ \cdot}$, $\langle p_+,p_-\rangle_{2\pi}=0, c_g>0$. For later use we note that differentiating the eigenvalue problem in $k$, we obtain also the formula \[E:cg\_int\] c\_g=2k\_+p\_+ -\_x p\_+, p\_+\_[2]{} = -2k\_-p\_- -\_x p\_-, p\_-\_[2]{}, where the second equality can be obtained using the above symmetries between $p_+$ and $p_-$. Formal Asymptotics {#S:formal} ================== Substituting the formal ansatz $\uapp + \eps^{3/2}u_1(x,\eps x, \eps t)e^{-\ri \omega_0 t}$ in and collecting terms on the left hand side with the same power of $\eps$, we get at $O(\eps^{1/2})$ $$e^{-\ri \omega_0 t}\sum_\pm A_\pm(X,T)(\omega_0 p_\pm +(\pa_x+\ri k_\pm)^2 p_\pm -V(x)p_\pm)e^{\ri k_\pm x}, \quad X:=\eps x, T:= \eps t$$ where the expression in the parentheses vanishes for each $\pm$ due to the choice of $p_\pm$ and $\omega_0$. For $O(\eps^{3/2})$ let us first consider the term $W\uapp$. In order to identify terms of the form of a $2\pi$-periodic function times $e^{\ri k_\pm x}$, we note that we can write \[E:W\_split\] W(x)=W\^[(1)]{}(x)+W\_\^[(2)]{}(x)e\^[-2k\_x]{} +W\^[(3)]{}\_(x), where $$\begin{aligned} & W^{(1)}(x)=\sum_{\stackrel{n\in \Z\setminus\{0\}}{nk_W\in \Z}}a_n e^{\ri nk_Wx}, \quad W^{(2)}_\pm(x)=\sum_{\stackrel{n\in \Z\setminus\{0\}}{nk_W\notin \Z, nk_W+2k_\pm\in \Z}}a_ne^{\ri (nk_W+2k_\pm)x},\\ & W^{(3)}_\pm(x)=\sum_{\stackrel{n\in \Z\setminus\{0\}}{nk_W\notin \Z, nk_W+2k_\pm\notin \Z}}a_ne^{\ri nk_W x}.\end{aligned}$$ Clearly, $W^{(1)}$ and $W^{(2)}_\pm$ are $2\pi$-periodic while $W^{(3)}_\pm$ are not $2\pi$-periodic. Equation defines two ways of splitting $W(x)$. The splitting of $W$ in is motivated by the relation $$W(x)e^{\ri k_\pm x} = W^{(1)}(x)e^{\ri k_\pm x}+W^{(2)}_\pm(x)e^{\ri k_\mp x}+W^{(3)}_\pm(x)e^{\ri k_\pm x}.$$ Hence, in the case $k_+=-k_-, k_+\in(0,1/2)$ the part $W_\pm^{(2)}(x)e^{-2\ri k_\pm x}$ guarantees coupling of the two carrier waves in because its multiplication with $p_\pm(x)e^{\ri k_\pm x}$ produces a periodic function times $e^{\ri k_\mp x}$. Therefore, in below we use the splitting with $W^{(2)}_+$ and $W^{(3)}_+$ for the multiplication of $W$ with $e^{\ri k_+ x}$ and the splitting with $W^{(2)}_-$ and $W^{(3)}_-$ for the multiplication of $W$ with $e^{\ri k_- x}$. In the case $k_+=k_-\in \{0,1/2\}$, clearly, $W^{(2)}_\pm\equiv 0$ and the coupling is provided by $W^{(1)}$. The coupling can be seen explicitly in the $\kappa$ coefficient in . For the above two cases we get 1. $k_+=-k_-\in(0,1/2) \ \Rightarrow \ W^{(2)}_-\equiv \overline{W^{(2)}_+}$ 2. $k_+=k_-\in \{0,1/2\} \ \Rightarrow \ W^{(2)}_+\equiv W^{(2)}_-\equiv 0$. Because $a_{-n}=\overline{a_n},$ we also have that $W^{(1)}$ is real. The Bloch functions $p_\pm$ have a free complex phase, which we fix by requiring $p_-=\overline{p_+}$ and $$\label{E:kap_real} \begin{aligned} &\text{(a)} \quad \text{Im}(\langle W^{(2)}_+p_+,p_-\rangle_{2\pi})\stackrel{!}{=}0 \text{ if } k_+=-k_-\in(0,1/2),\\ &\text{(b)} \quad \text{Im}(\langle W^{(1)}p_+,p_-\rangle_{2\pi})\stackrel{!}{=}0 \text{ if } k_+=k_-\in \{0,1/2\}. \end{aligned}$$ This choice makes the coefficient $\kappa$ in the effective equations for the envelopes $A_\pm$ real. Hence at $O(\eps^{3/2})$ the left hand side of is (after multiplication by $e^{\ri \omega_0 t}$) \[E:eps3/2\] &(\_0+\_x\^2 -V(x))u\_1(x,X,T)\ +&e\^[k\_+ x ]{}\ +&e\^[k\_- x ]{}\ -& (x) (p\_+\^2A\_+\^2e\^[(2k\_+-k\_-) x ]{}+p\_-\^2A\_-\^2e\^[(2k\_–k\_+) x]{})\ -& W\^[(3)]{}\_+(x)p\_+A\_+e\^[k\_+ x]{} - W\^[(3)]{}\_-(x)p\_-A\_-e\^[k\_- x]{}, where $X=\eps x, T=\eps t$ and except for the last line all factors multiplying $e^{\ri k_\pm x}$ and $e^{\ri(2k_\mp-k_\pm) x}$ are $2\pi$-periodic in $x$. Note that when $2k_+-k_- \in \{k_+,k_-\}+\Z$, then $e^{\ri(2k_+-k_-) x}$ can also be written as a $2\pi$-periodic function times $e^{\ri k_+ x}$ or $e^{\ri k_- x}$. Similarly for $e^{\ri(2k_--k_+) x}$. Within our allowed setting, i.e. within cases (a) and (b), this occurs if and only if $k_+=-k_-=1/4$ or $k_+=k_-\in\{0,1/2\}$: $$\begin{aligned} k_+=-k_-=1/4 \ \Rightarrow \ &2k_+-k_-=k_-+1, \text{ i.e. } e^{\ri(2k_+-k_-) x}= e^{\ri x}e^{\ri k_-x} \quad \text{and} \\ &2k_--k_+=k_+-1, \text{ i.e. } e^{\ri(2k_--k_+) x}= e^{-\ri x}e^{\ri k_+x},\\ k_+=k_-\in\{0,1/2\} \ \Rightarrow \ &2k_+-k_-=2k_--k_+=k_+=k_-.\end{aligned}$$ In order to set the $O(\eps^{3/2})$ terms proportional to a $2\pi$-periodic function times $e^{\ri k_+ x}$ or $e^{\ri k_- x}$ to zero, we search for $u_1$ in form $$u_1(x,X,T)=U_{1,+}(X,T)s_+(x)e^{\ri k_+ x}+U_{1,-}(X,T)s_-(x)e^{\ri k_- x}$$ with $2\pi-$periodic functions $s_\pm$, such that these terms vanish. Formally, such $u_1$ exists if the Fredholm alternative holds, i.e. if the inhomogeneous terms (independent of $u_1$) in having the form of a $2\pi$-periodic function times $e^{\ri k_\pm x}$ are $L^2(0,2\pi)$-orthogonal to $p_\pm(x)e^{\ri k_\pm x}$ respectively. In the case (b), when $k_+=k_-$, it means, of course, that all inhomogeneous terms having the form of a $2\pi$-periodic function times $e^{\ri k_+ x}=e^{\ri k_- x}$ must be orthogonal to both $p_+(x)e^{\ri k_+ x}$ and $p_-(x)e^{\ri k_- x}$. In this case we have $\langle p_+,p_-\rangle_{2\pi} =0$ and $\langle \pa_x p_+,p_-\rangle_{2\pi} =0$. The latter identity follows for $k_+=0$ from $p_-=\overline{p_+}$ because $\langle \pa_x p_+,p_-\rangle_{2\pi} =\int_0^{2\pi}p_+(x)\pa_x p_+(x)dx=\tfrac{1}{2}\int_0^{2\pi}\pa_x (p_+(x))^2dx=0$ and for $k_+=1/2$ from $p_-=\overline{p_+}e^{-\ri x}$ because $\langle \pa_x p_+,p_-\rangle_{2\pi} =\tfrac{1}{2}\int_0^{2\pi}\pa_x (p_+(x))^2e^{\ri x}dx = -\tfrac{\ri}{2}\int_0^{2\pi}p_+^2(x)e^{\ri x}dx$ and $\int_0^{2\pi}p_+^2(x)e^{\ri x}dx=\langle p_+,p_-\rangle_{2\pi} =0$. Using these identities, the orthogonality conditions become the *coupled mode equations* (CMEs) \[E:CME\] (\_T +c\_g \_X)A\_+ + A\_- + \_s A\_+ +(|A\_+|\^2+2|A\_-|\^2)A\_+ &\ + (|A\_-|\^2+2|A\_+|\^2)A\_- + A\_+\^2 +A\_-\^2 &= 0,\ (\_T -c\_g \_X)A\_- + A\_+ + \_s A\_- +(|A\_-|\^2+2|A\_+|\^2)A\_- &\ + (|A\_+|\^2+2|A\_-|\^2)A\_+ + A\_-\^2 + A\_+\^2 &= 0,\ where $$\label{E:CME_coeffs} \begin{aligned} c_g & =2\langle k_+p_+ -\ri \pa_xp_+, p_+\rangle_{2\pi}= - 2\langle k_-p_- -\ri \pa_xp_-, p_-\rangle_{2\pi}\in \R,\\ \kappa_s & = - \langle W^{(1)}p_+,p_+\rangle_{2\pi} =- \langle W^{(1)}p_-,p_-\rangle_{2\pi} \in \R, \\ \kappa & = \begin{cases} -\langle W^{(2)}_-p_-,p_+\rangle_{2\pi}=-\langle W^{(2)}_+p_+,p_-\rangle_{2\pi} \in \R \quad &\text{if } k_+=-k_-\in(0,1/2),\\ - \langle W^{(1)}p_-,p_+\rangle_{2\pi} =- \langle W^{(1)}p_+,p_-\rangle_{2\pi}\in \R \quad &\text{if } k_+=k_-\in\{0,1/2\}, \end{cases}\\ \alpha & = -\langle\sigma p_+^2,p_+^2\rangle_{2\pi}=-\langle\sigma p_-^2,p_-^2\rangle_{2\pi} = -\langle \sigma|p_-|^2p_+,p_+\rangle_{2\pi}=-\langle \sigma|p_+|^2p_-,p_-\rangle_{2\pi} \in \R,\\ \beta & = \begin{cases} 0 \quad &\text{if } k_+=-k_-\in(0,1/2),\\ -\langle \sigma|p_\pm|^2p_-,p_+\rangle_{2\pi} = - \overline{\langle \sigma |p_\pm|^2p_+,p_-\rangle_{2\pi}}\quad &\text{if } k_+=k_-\in\{0,1/2\}, \end{cases}\\ \gamma & = \begin{cases} 0 \quad &\text{if } k_+=-k_-\in(0,1/4)\cup(1/4,1/2),\\ -\langle \sigma p_-^2\overline{p_+}e^{-\ri \cdot},p_+\rangle_{2\pi} = -\overline{\langle \sigma p_+^2\overline{p_-}e^{\ri \cdot},p_-\rangle_{2\pi}} \quad &\text{if } k_+=-k_-=1/4,\\ -\langle \sigma p_-^2\overline{p_+},p_+\rangle_{2\pi} = -\overline{\langle \sigma p_+^2\overline{p_-},p_-\rangle_{2\pi}} \quad &\text{if } k_+=k_-\in\{0,1/2\}.\\ \end{cases} \end{aligned}$$ The realness of $\kappa_s$ follows from the realness of $W^{(1)}$. Note that without any loss of generality we can set $\kappa_s=0$ because solutions $(A_+,A_-)$ of with $\kappa_s\neq 0$ can then be constructed from solutions $(A^{(0)}_+,A^{(0)}_-)$ with $\kappa_s=0$ via the multiplication by a simple phase factor, namely $(A_+,A_-)=(A^{(0)}_+,A^{(0)}_-)e^{\ri \kappa_s T}$. The identities in $\kappa$ and its realness follow from and from $W^{(2)}_-=\overline{W^{(2)}_+}$ for $k_+=-k_-\in(0,1/2)$ and from the realness of $W^{(1)}$ for $k_+=k_-\in \{0,1/2\}$ . \[R:gap\] As mentioned in the introduction, the linear part of system has a spectral gap if $\kappa\neq 0$. Indeed, the spectrum of the self-adjoint operator $\bspm \ri c_g \pa_x+\kappa_s & \kappa \\ \kappa & -\ri c_g \pa_x+\kappa_s\espm$ has the gap $(\kappa_s-|\kappa|,\kappa_s+|\kappa|)$. Hence, exponentially localized solutions $(A_+,A_-)$ can be expected. This is based on the heuristic argument that in spectral gaps the linear solution modes are exponentials and in the tails of the nonlinear solution, where the cubic nonlinearity is negligible, the linear dynamics govern. To the best of our knowledge, a rigorous proof of the existence of exponentially localized solitary waves of is not in the literature. However, for $\beta=\gamma=0$ explicit families of exponentially localized solitary waves parametrized by velocity exist, see . The definition of $\kappa$ in produces a necessary and sufficient condition for a spectral gap. In case (a), i.e. if $k_+=-k_-\in(0,1/2)$, this condition is $$\langle W^{(2)}_+p_+,p_-\rangle_{2\pi} \neq 0$$ and in case (b), i.e. if $k_+=k_-\in \{0,1/2\}$, it is $$\langle W^{(1)}p_+,p_-\rangle_{2\pi}\neq 0.$$ A necessary condition is $W^{(2)}_\pm \neq 0$ in case (a) and $W^{(1)}\neq 0$ in case (b). Based on the definition of $W^{(2)}_\pm$ and $W^{(1)}$ in , we obtain in case (a) the necessary condition \[E:kW\_cond\_a\] k\_W 2k\_+ + n {m{0}: a\_m0} and in case (b) the necessary condition \[E:kW\_cond\_b\] k\_Wn {m{0}: a\_m0}. For a further discussion we note that these are clearly possible only if all $k_\pm,k_W$ are rational or all are irrational. The simplest choice which satisfies conditions and is $k_W =2k_+$ with $a_1=a_{-1}=\tfrac{a}{2}\in \R\setminus\{0\}, a_n =0$ for all $n\in \Z\setminus\{1,-1\}$, i.e. $W(x)=a\cos(2k_+x)$. The case of rational $k_+,k_-$ {#S:rational} ------------------------------ We consider the case of rational $k_\pm$ separately because a common period of the Bloch waves and the potential $V$ (and $\sigma$) can be chosen in this case and the points $k_\pm$ get mapped to zero in the Brillouin zone corresponding to this new period. Hence, for $k_\pm \in \Q$ the problem is effectively transformed to the above case (b) of a double Bloch eigenvalue at $k=0$. For $k_+\in \Q$ the functions $V$ and $e^{\ri k_+ \cdot}$ have a common period $[0,Q]$ with $Q=N2\pi$ for some $N\in \N$. Clearly, as either $k_-=-k_+$ or $k_-=k_+$, also $e^{\ri k_- \cdot}$ is $Q-$periodic. We use $Q$ as the working periodicity of the problem. The corresponding Brillouin zone is $$\B_Q:=(-\tfrac{1}{2N},\tfrac{1}{2N}].$$ The Bloch eigenvalue problem is \[E:Bloch\_prob\_Q\] (x,k)q = q, x(0,Q) for $k\in\B_Q$. The band structure on $\B_Q$ is generated from that on $\B_{2\pi}$ using the $\tfrac{1}{N}$-periodicity in the variable $k$ of the eigenvalues. The labeling of the eigenvalues changes when mapped from $\B_{2\pi}$ to $\B_Q$. We denote the band structure on $\B_Q$ by $(k, (\vartheta_n(k))_{n\in \N})$ with $k\in \B_Q$. Fig. \[F:reduce\_zone\] shows an example of a band structure on $\B_Q$ for a given band structure on $\B_{2\pi}$ and for $N=3,k_+=1/3$. Since $N k_\pm \in \Z$, we get that $$k_\pm = 0 \mod \tfrac{1}{N}.$$ The eigenvalue at $(k,\vartheta)=(0,\omega_0)$ is thus double. Because of assumption (H1) and the eigenvalues $\vartheta_n(k)$ can be relabeled to produce the transversal crossing of two $C^2$ eigenvalue curves at $k=0$. Our labeling of the eigenvalues is thus determined as follows. Let us denote the eigenvalues of at $k=0$ by $(\lambda_n)_{n\in \N}$ ordered by size and suppose $\lambda_{n_*}=\lambda_{n_*+1}=\omega_0$. We label the eigenvalue curves $(\vartheta(k))_n$ according to size for all $n<n_*$ and all $n>n_*+1$ and for $n\in\{n_*,n_*+1\}$ we label the curves such that they are smooth at $k=0$, i.e. \[E:n\_star\] &\_n(k) \_[n+1]{}(k) k\_Q, n {1,2,…,n\_\*-1,n\_\*+1,n\_\*+2,…}\ &\_[n\_\*]{}(k)&lt; \_[n\_\*+1]{}(k) k&lt;0,  \_[n\_\*]{}(k)&gt; \_[n\_\*+1]{}(k) k&gt;0. By assumption (H1) is $\vartheta_{n_*}, \vartheta_{n_*+1} \in C^2(\text{int}(\B_Q))$. Note that $$c_g= \vartheta'_{n_*}(0)=-\vartheta'_{n_*+1}(0).$$ The eigenfunctions corresponding to $\vartheta_n(k)$ are denoted by $q_n(x,k)$ and normalized via $$\langle q_n(\cdot,k), q_m(\cdot,k)\rangle_{Q}=\delta_{n,m}.$$ The $Q-$periodic eigenfunctions at $(k,\vartheta)=(0,\omega_0)$ with group velocities $\pm c_g$ are $$q_+(x):=q_{n_*}(\cdot,0) \quad \text{and} \quad q_-(x):=q_{n_*+1}(\cdot,0).$$ These are related to $p_\pm$ via $$q_\pm(x)=\frac{1}{\sqrt{N}}p_\pm(x) e^{\ri k_\pm x}, \ \text{and satisfy} \ q_-=\overline{q_+}.$$ Note that the orthogonality $\langle q_+,q_-\rangle_Q=0$ can be checked directly. It is obvious in case (b) where $k_+=k_-\in \{0,1/2\}$ and $\langle q_+,q_-\rangle_Q=\tfrac{1}{N}\langle p_+,p_-\rangle_Q=\langle p_+,p_-\rangle_{2\pi}=0$ follows from the orthogonality of eigenfunctions at each $k$. For case (a), where $k_+=-k_-\in (0,1/2)$ we argue using a Fourier series expansion. Namely, writing $p_+^2(x)=\sum_{n\in \N}b_ne^{\ri n x}$, we have $$\langle q_+,q_-\rangle_Q=\frac{1}{N}\int_0^Q p_+^2(x)e^{2\ri k_+ x} dx=\sum_{n\in \N}b_n\int_0^{2N\pi}e^{\ri(n+2k_+) x}dx =0$$ because $n+2k_+\neq 0$ and $n+2k_+\in \tfrac{1}{N}\Z$ for all $n\in \Z$. Because in the rational case our Bloch eigenfunctions $q_+,q_-$ are both at $k=0$, the splitting of $W$ using the parts $W^{(2)}_\pm$ as in in is not suitable. It is more convenient to rewrite Because in the rational case our Bloch eigenfunctions $q_+,q_-$ are both at $k=0$, the splitting of $W$ using the parts $W^{(2)}_\pm$ as in in is not suitable. It is more convenient to rewrite $$W(x)=W^{(1)}_Q(x)+W_Q^{(R)}(x), \quad W^{(1)}_Q(x):=\sum_{n\in \Z_1}a_n e^{\ri nk_Wx}, \ W^{(R)}_Q(x):=\sum_{n\in \Z_R}a_n e^{\ri nk_Wx},$$ where $$\Z_1:= \{n\in \Z\setminus\{0\}: a_n\neq 0, nk_W\in \tfrac{1}{N}\Z\}, \quad \Z_R:= \{n\in \Z\setminus\{0\}: a_n\neq 0, nk_W\notin \tfrac{1}{N}\Z\}.$$ The formal ansatz we will use in the rational case is \[E:uapp\_rat\] (x,t) = \^[1/2]{}e\^[-\_0 t]{}(A\_+(x,t)q\_+(x) + A\_-(x,t)q\_-(x)). The effective model is again the coupled mode system but the nonlinear coefficients have to be modified due to the different normalization of $p_\pm$ and $q_\pm$, i.e. we have \[E:CME\_rat\] (\_T +c\_g \_X)A\_+ + A\_- + \_s A\_+ +\_Q (|A\_+|\^2+2|A\_-|\^2)A\_+ &\ + \_Q (|A\_-|\^2+2|A\_+|\^2)A\_- + A\_+\^2 +\_Q A\_-\^2 &= 0,\ (\_T -c\_g \_X)A\_- + A\_+ + \_s A\_- +\_Q (|A\_-|\^2+2|A\_+|\^2)A\_- &\ + (|A\_+|\^2+2|A\_-|\^2)A\_+ + \_Q A\_-\^2 + A\_+\^2 &= 0,\ where $$\begin{aligned} c_g & =2\ri \langle \pa_xq_+, q_+\rangle_Q= -2\ri \langle \pa_xq_-, q_-\rangle_Q\in \R,\\ \kappa_s & = - \langle W_Q^{(1)}q_+,q_+\rangle_Q =- \langle W_Q^{(1)}q_-,q_-\rangle_Q \in \R, \\ \kappa & = -\langle W_Q^{(1)}q_-,q_+\rangle_Q=-\langle W_Q^{(1)}q_+,q_-\rangle_Q \in \R,\\ \alpha_Q & = -\langle\sigma q_+^2,q_+^2\rangle_{Q}=-\langle\sigma q_-^2,q_-^2\rangle_{Q} = -\langle \sigma|q_-|^2q_+,q_+\rangle_Q=-\langle \sigma|q_+|^2q_-,q_-\rangle_Q \in \R,\\ \beta_Q & = -\langle \sigma|q_\pm|^2q_-,q_+\rangle_Q = -\overline{\langle \sigma |q_\pm|^2q_+,q_-\rangle_Q},\\ \gamma_Q & = -\langle \sigma q_-^2\overline{q_+},q_+\rangle_Q = -\overline{\langle \sigma q_+^2\overline{q_-},q_-\rangle_Q}.\end{aligned}$$ Note that the linear coefficients $c_g,\kappa_s$ and $\kappa$ are indeed identical with those defined in using $p_\pm$ but for the nonlinear coefficients we have $$(\alpha_Q,\beta_Q,\gamma_Q)=\tfrac{1}{N}(\alpha,\beta,\gamma).$$ These identities between the linear and nonlinear coefficients can be checked by expanding the periodic parts of the integrands in a Fourier series. E.g. to show $\langle W^{(1)}_Q q_+,q_+ \rangle_Q=\langle W^{(1)} p_+,p_+ \rangle_{2\pi}$, we expand $|q_+(x)|^2=\tfrac{1}{N}|p_+(x)|^2=\sum_{m\in \Z}b_me^{\ri m x}$ and get $$\begin{aligned} \langle W^{(1)}_Q q_+,q_+ \rangle_Q&=\sum_{m\in \Z, n\in\Z_1}a_nb_m\int_0^{2N\pi} e^{\ri (m+nk_W)x}dx=\sum_{\stackrel{m\in \Z}{n\in \{j\in\Z_1:jk_W\in \Z\}}}a_nb_m\int_0^{2N\pi} e^{\ri (m+nk_W)x}dx\\ &=\frac{1}{N}\langle W^{(1)}p_+,p_+\rangle_{Q}=\langle W^{(1)} p_+,p_+ \rangle_{2\pi}\end{aligned}$$ because if $m\in \Z$, then $\int_0^{2N\pi} e^{\ri (m+nk_W)x}dx=0$ for any $nk_W\in \tfrac{1}{N}\Z \setminus \Z$. Similar arguments yield the other identities. Rigorous Justification of CMEs as an Effective Model {#S:justif} ==================================================== It is clear that the discussion and the derivation of the coupled mode equations (CME) in Section \[S:formal\] is only formal as terms which are not of the form of a $2\pi$-periodic function times $e^{\ri k_\pm x}$ in the residual as well as higher derivative terms with respect to $X$ have been ignored. To make the discussion rigorous, we will modify (and extend) the asymptotic ansatz in order to make the residual of $O(\eps^{5/2})$ in a suitable norm. After estimating the residual, we use the Gronwall lemma to show the smallness of the asymptotic error on large time scales. We make the following basic assumptions - In the case of a double eigenvalue at $(k,\omega)=(k_+,\omega_0)=(k_-,\omega_0)$ the functions $k\mapsto \tilde{\omega}_{\pm}(k)$ in have two continuous derivatives at $k=k_+=k_-$ and the mappings $\phi_\pm:\overline{\B}\to H^2(0,2\pi), k\mapsto \tilde{p}_\pm(\cdot,k)$ with $\tilde{p}_\pm$ defined in are Lipschitz continuous on $\overline{\B}$, - $W\in C^2(\R,\R), W(x+\tfrac{2\pi}{k_W})=W(x)$ for all $x\in \R$ is given by and if $k_W\notin \Q$, then there is $M\in \N$ such that $a_n=0$ for all $|n|>M$, - $V\in C(\R,\R), V(x+2\pi)=V(x)$ for all $x\in \R$. - $\sigma\in C(\R,\R), \sigma(x+2\pi)=\sigma(x)$ for all $x\in \R$. Note that in (H1) the Lipschitz continuity over the whole $\overline{\B}$ requires that if $k_+=k_-=0$ and the eigenvalue at $(k,\omega)=(1/2,\tilde{\omega}_+(1/2))$ or at $(1/2,\tilde{\omega}_-(1/2))$ is double, then the eigenvalue curves need to be smoothly extendable also across $k=1/2$. In the case of simple eigenvalues at $k_+, k_-$ the $C^2$ smoothness of $\tilde{\omega}_\pm$ and the Lipschitz continuity of the Bloch eigenfunctions in (H1) always hold, see [@Kato-1966]. \[T:main\] Assume (H1)-(H4) and let $(A_+,A_-)$ be a solution of , with $\Ahat_\pm \in C^1([0,T_0],$$L^1_{s_A}(\R)\cap L^2(\R))$ for some $T_0>0$ and some $s_A \geq 2$. There exist $c>0$ and $\eps_0>0$ such that if $u(x,0)=u_\text{app}(x,0)$ given by , then the solution $u$ of satisfies $u(x,t)\to 0$ as $|x|\to \infty$ and $$\|u(\cdot,t)-u_\text{app}(\cdot,t)\|_{C^0_b(\R)}\leq c\eps^{3/2} \quad \text{for all } \eps\in(0,\eps_0), t\in [0,\eps^{-1}T_0].$$ The functions $\Ahat_\pm$ are the Fourier transformations of $A_\pm$, where we define $$\widehat{f}(k)=\frac{1}{2\pi}\int_{-\infty}^\infty f(x)e^{-\ri kx}dx$$ with the inverse transformation $f(x)=\int_{-\infty}^\infty \widehat{f}(k)e^{\ri kx}dk$. The space $L^1_{r}, r>0$ in the theorem is defined as $$L^1_r(\R):=\left\{f\in L^1(\R): \|f\|_{L^1_r(\Omega)}:=\int_\Omega (1+|x|)^r|f(x)|dx <\infty\right\}.$$ To the best of our knowledge, existence of solutions $(A_+,A_-)$ of , with $\Ahat_\pm \in C^1([0,T_0],$$L^1_{s_A}(\R)\cap L^2(\R))$ is not covered in the existing literature. However, in the case $\beta=\gamma=0$ there are explicit smooth solutions (satisfying $\Ahat_\pm \in C^1([0,T_0],L^1_{s_A}(\R)\cap L^2(\R))$), see . Note that the case $\beta=\gamma=0$ is generic as it corresponds to $k_+=-k_-\in (0,1/4)\cup(1/4,1/2)$, see . Note also that in general (for all values of the coefficients) local existence is guaranteed for in $H^1(\R)$, i.e. for any $(A_+,A_-)(\cdot,0)\in H^1(\R)$ there is a $T_0>0$, such that has a $C^1((0,T_0),H^1(\R))$-solution. This follows from Theorem 1 in [@Reed]. Moreover, by Theorem 2 in [@Reed] the existence is either global (i.e. $T_0=\infty$ can be chosen) or $\|\Ahat_+(T)\|_{H^1(\R)}+\|\Ahat_-(T)\|_{H^1(\R)}\to \infty$ as $T\to T_0$. We are particularly interested in localized traveling waves $u$ of nearly constant shape. Such solutions are guaranteed if we find exact solitary waves of . For $\beta=\gamma=0$ system is exactly the classical coupled mode system for optical Bragg fibers with a small contrast, see e.g. [@AW89; @GWH01]. If $\kappa=\langle W^{(2)}_+p_+,p_-\rangle_{2\pi}\neq 0$, this system has an explicit family of traveling solitary waves, see [@AW89; @GWH01]. As explained above, in the generic case $k_+=-k_-\in (0,1/4)\cup(1/4,1/2)$ we always have $\beta=\gamma=0$, and hence approximate localized traveling waves $u$ of are certainly guaranteed if $k_+=-k_-\in (0,1/4)\cup(1/4,1/2)$ and $\langle W^{(2)}_+p_+,p_-\rangle_{2\pi}\neq 0$. For $\beta,\gamma\neq 0$ traveling wave solutions may be constructed by a homotopy continuation, see [@D14] for a numerical implementation. As explained in Remark \[R:gap\], a necessary condition for exponentially localized waves of the CMEs is that either all $k_\pm, k_W$ be rational or all be irrational. Theorem \[T:main\], however, holds for any $k_W\in \R$ and any $k_+=-k_-\in (0,1/2)$ or $k_+=k_-\in \{0,1/2\}$. The proof of the case of rational $k_\pm$ is technically somewhat simpler as there is a common period of $V(x)$ and $e^{\ri k_\pm x}$ and on the Brillouin zone corresponding to this common periodicity cell the wavenumbers $k_+,k_-$ both correspond (are periodic images of) the point $k=0$ as explained in Sec. \[S:rational\]. We treat the case of rational $k_\pm$ in Sec. \[S:rational\_pf\] and irrational $k_\pm$ in Sec. \[S:irrational\_pf\]. Bloch Transformation, Function Spaces {#S:fn_spaces} ------------------------------------- For a given $P>0$ the Bloch transformation is the operator $$\begin{aligned} &\cT:H^s(\R,\C) \to L^2(\B_P,H^s((0,P),\C)), u \mapsto \util :=\cT u,\\ & \util(x,k)=(\cT u)(x,k)=\sum_{j\in \tfrac{2\pi}{P}\Z}e^{\ri j x}\uhat(k+j). \end{aligned}$$ By construction of $\util$ we have \[E:util\_per\] (x+P,k)=(x,k)  (x,k+2/P)=e\^[-(2/P) x]{}(x,k) x,k . In the following we write simply $H^s(\Omega)$ for $H^s(\Omega,\C)$. The Bloch transform $\cT:H^s(\R) \to L^2(\B_P,H^s(0,P))$ is an isomorphism for $s\geq 0$, see [@RS4], and the inverse is given by $$u(x)=(\cT^{-1}\util)(x)=\int_{\B_P}e^{\ri kx}\util(x,k)dk.$$ It is easy to see that the product of two general $H^s(\R)$ functions is mapped to a convolution by $\cT$ and that $\cT$ commutes with the multiplication by a $P-$periodic function, i.e. for all $u,v \in H^s(\R)$ $$\begin{aligned} &\cT(uv)(x,k)=(\util *_{\B_P} \vtil)(x,k):=\int_{\B_P}\util(x,k-l)\vtil(x,l)dl,\\ &\cT(Vu)(x,k)=V(x)\util(x,k) \quad \text{for all } V\in C(\R) \text{ such that } V(x+P)=V(x) \ \text{for all }x\in \R, \end{aligned}$$ where in the convolution the $k-$periodicity in needs to be used when $k-l\notin \B_P$. Our analysis does not make use of the isomorphism as we work in $L^1(\B_P,H^s(0,P))$ rather than $L^2(\B_P,H^s(0,P))$. The reason is that in $L^2$ too many powers of $\eps$ are lost such that the resulting asymptotic error is not $o(1)$ on the desired time interval $[0,O(\eps^{-1})]$, see also [@SU01] for the same issue. The norm in $L^1(\B_P,H^s(0,P))$ is $$\|\util\|_{L^1(\B_P,H^s(0,P))} = \int_{\B_P} \|\util(\cdot,k)\|_{H^s(0,P)}dk.$$ Unfortunately, the isomorphism property of $\cT$ is lost when $L^2$ is replaced by $L^1$. On the other hand, for $s>1/2$ the supremum norm of $u:=\cT^{-1}\util$ can be controlled by $\|\util\|_{L^1(\B_P,H^s(0,P))}$. This means that the supremum norm of the error will be controlled if we estimate the Bloch transform of the error in $L^1(\B_P,H^s(0,P))$. Moreover, for $\util \in L^1(\B_P,H^s(0,P))$ with $s>1/2$ the function $u(x)$ decays as $|x|\to \infty$ as the next lemma shows. \[L:sup\_control\] Let $s>1/2$. There is $c>0$ such that for all $\util \in L^1(\B_P,H^s(0,P))$ which satisfy , we have for the function $u:=\cT^{-1} \util$ $$|u(x)|\leq c\|\util\|_{L^1(\B_P,H^s(0,P))} \quad \text{and} \ u(x)\to 0 \text{ as } |x| \to \infty.$$ $$\begin{aligned} \|u\|_{C^0_b(\R)}\leq \int_{\B_P}\|\util(\cdot,k)\|_{C^0_b(0,P)}dk \leq c \int_{\B_P}\|\util(\cdot,k)\|_{H^s(0,P)}dk=c\|\util\|_{L^1(\B_P,H^s(0,P))} \end{aligned}$$ due to Sobolev’s embedding. The proof of the decay follows the same lines as the proof of Riemann-Lebesgue’s lemma. One approximates $\util$ by $v\in C^\infty(\B_P,H^s(0,P))$ with $v(x,k+2\pi/P)=e^{-\ri x 2\pi/P }v(x,k)$ and $v(x+P,k)=v(x,k)$ for all $x$ and $k$, and uses integration by parts. It is also easy to establish the following algebra property of $L^1(\B_P,H^s(0,P))$, which is needed for the treatment of the nonlinearity. \[L:algeb\_L1Hs\] Let $\util,\vtil\in L^1(\B_P,H^s(0,P))$ with $s>1/2$. Then $$\|\util \ast_{\B_P}\vtil\|_{L^1(\B_P,H^s(0,P))} \leq c \|\util\|_{L^1(\B_P,H^s(0,P))}\|\vtil\|_{L^1(\B_P,H^s(0,P))}.$$ $$\begin{aligned} \|\util \ast_{\B_P}\vtil\|_{L^1(\B_P,H^s(0,P))} & \leq c\int_{\B_P}\int_{\B_P}\|\util(\cdot,k-l)\|_{H^s(0,P)}\|\vtil(\cdot,l)\|_{H^s(0,P)} dldk \\ &\leq c \|\util\|_{L^1(\B_P,H^s(0,P))}\|\vtil\|_{L^1(\B_P,H^s(0,P))}, \end{aligned}$$ where the first inequality follows by the algebra property of $H^s$ in one dimension, i.e. $\|fg\|_{H^s(\Omega)}\leq c\|f\|_{H^s(\Omega)}\|g\|_{H^s(\Omega)}$ for $\Omega\subset \R$ and $s>1/2$. The second step follows by Young’s inequality for convolutions. For each $k\in \B_P$ the eigenfunctions $(p_n(\cdot,k))_{n\in \N}$ of $L(k)$ in are complete in $L^2(0,P)$. Hence, for each $k\in \B_P$ fixed and $s\geq 0$ a function $\util(\cdot,k)\in H^s(0,P)$ can be expanded in $(p_n(\cdot,k))_{n\in \N}$. We denote the expansion operator by $D(k)$, i.e. $$D(k):\util(\cdot,k)\mapsto \Uvec(k):=(\langle\util(\cdot,k),p_n(\cdot,k)\rangle_P)_{n\in \N}.$$ As shown in Lemma 3.3 of [@BSTU06], $D(k)$ is an isomorphism between $H^s(0,P)$ and $$l_s^2=\{\vec{v}\in l^2(\C):\|\vec{v}\|^2_{l_s^2}=\sum_{n\in \N}n^{2s}|v_n|^2 <\infty\}$$ for all $s\geq 0$ with $$\|D(k)\|,\|D^{-1}(k)\|\leq c<\infty \quad \text{for all } k\in \B_P.$$ For the expansion coefficients $\Uvec$ we define the space $$\cX(s):=L^1(\B_P,l^2_s) \quad \text{with the norm } \|\Uvec\|_{\cX(s)}=\int_{\B_P} \|\Uvec(k)\|_{l^2_s} dk.$$ Our working space for $\Uvec$ will be $\cX(s)$ with $s>1/2$. As a simple consequence of the properties of $D(k)$ we have \[L:D\_isom\] The operator $\cD:L^1(\B_P,H^s(0,P))\to \cX(s), \util \mapsto \Uvec$ is an isomorphism for any $s\geq 0$. There are $c_1,c_2>0$ such that for any $\Uvec(k)=D(k) \util(\cdot,k)$ we have $\|\util(\cdot,k)\|_{H^s(0,P)}\leq c_1 \|\Uvec(k)\|_{l_s^2}$ and $\|\Uvec(k)\|_{l_s^2} \leq c_2\|\util(\cdot,k)\|_{H^s(0,P)}$ for all $k\in \B_P$. Hence $$\|\util\|_{L^1(\B_P,H^s(0,P))} \leq c_1\int_{\B_P}\|\Uvec(k)\|_{l_s^2}dk = c_1\|\Uvec\|_{\cX(s)}$$ and $$\|\Uvec\|_{\cX(s)}\leq c_2\int_{\B_P}\|\util(\cdot,k)\|_{H^s(0,P)}dk = c_2\|\util\|_{L^1(\B_P,H^s(0,P))}.$$ Note that the $k-$periodicity in implies that $\Uvec=\cD \util$ satisfies \[E:U\_per\] (k+)=(k) k. The above function spaces are used in our analysis in the following way. We define an extended (compared to $\uapp$) ansatz for the approximate solution in the $\Uvec$-variables and show that it lies in $\cX(s)$ for $s<3/2$. The residual of this ansatz in equation is then estimated in $\|\cdot\|_{\cX(s)}$, where some terms are transformed by $\cD^{-1}$ to $L^1(\B_P,H^s(0,P))$ and estimated in $\|\cdot\|_{L^1(\B_P,H^s(0,P))}$. The approximation error is then estimated in the $\Uvec$-variables (i.e. in $\|\cdot\|_{\cX(s)}$), in which the equation becomes an infinite ODE system, by Gronwall’s inequality. The supremum of the error in the physical $u$ variables then satisfies the same estimate due to Lemma \[L:sup\_control\] because $s>1/2$. Proof of Theorem \[T:main\] for Rational $k_\pm$ {#S:rational_pf} ------------------------------------------------ Let us recall that in the rational case we work with the period $P=Q=2N\pi$, see Sec. \[S:rational\]. Also $k_\pm \in \tfrac{1}{N}\Z$, such that in $\B_Q$ the points $k_\pm$ are identified with $0$. We will prove Theorem \[T:main\] for $u_\text{app}$ in and $(A_+,A_-)$ a solution of . This is, however, equivalent to the theorem since $(A_+,A_-)_{\eqref{E:CME}} = \tfrac{1}{\sqrt{N}}(A_+,A_-)_{\eqref{E:CME_rat}}$ and $p_\pm(x)e^{\ri k_\pm x}=\sqrt{N}q_\pm(x)$ such that $u_\text{app}$ in and in are identical. The rigorous justification of the asymptotic model will be carried out in the Bloch variables $\Uvec=\Uvec(k,t)$. Instead of the approximate ansatz $\Uvec_{\text{app}}:=\cD\cT(\uapp)$ we use its modification $\Uvecext$ which is supported in $k$ only near the wavenumbers of the corresponding carrier waves, i.e. near $k=0$, and includes correction terms supported near the new $k-$points generated by $W^{(R)}_Q\uapp$. Note that the nonlinearity does not generate new $k-$neighborhoods when $k_\pm =0 \mod \tfrac{1}{N}$. This is a standard approach, where the concentration points of the residual for $u_\text{app}$ are identified and the modified ansatz is chosen to be supported only in neighborhoods of these points. The problem can then be easily decomposed according to the disjoint intervals in the support on $\B_Q$. To motivate the choice of $\Uvecext$, we study the $\cT$-transform of the residual for $\uapp$ with a compact support of $\hat{A}_\pm(\cdot,T)$. Assuming that[^1] $\text{supp}(\hat{A}_\pm(\cdot,T))\subset [-\eps^{-1/2},\eps^{-1/2}]$, we get $$\begin{aligned} \cT(\uapp)(x,k,t)=&\eps^{-1/2}\sum_{\pm}q_\pm(x)\sum_{\eta\in \tfrac{1}{N}\Z}\hat{A}_\pm\left(\tfrac{k+\eta}{\eps},\eps t\right)e^{\ri(\eta x- \omega_0 t)}\\ =&\eps^{-1/2}\sum_{\pm}q_\pm(x)\hat{A}_\pm\left(\tfrac{k}{\eps},\eps t\right)e^{-\ri \omega_0 t}, \quad k \in \B_Q \end{aligned}$$ because $\eps^{-1}(k+\eta) \in \text{supp}(\hat{A}_\pm(\cdot,T))$ for some $k\in \B_Q$ and $\eta\in \tfrac{1}{N}\Z$ implies $\eta=0$. Hence, if $\text{supp}(\hat{A}_\pm(\cdot,T))\subset [-\eps^{-1/2},\eps^{-1/2}]$, then for $k\in\B_Q$ $$\begin{aligned} &\cT(\text{PNLS}(\uapp))(x,k,t)= \eps^{1/2}e^{-\ri \omega_0 t}\sum_{\pm}\left[\ri \pa_T\hat{A}_\pm(K,T)q_\pm(x)+2\ri K\hat{A}_\pm(K,T)q_\pm'(x)\phantom{\sum_{m\in \Z_R,\eta \in \cS_m}}\right.\\ &\left.-W_Q^{(1)}(x)q_\pm(x)\hat{A}_\pm(K,T)-\sum_{m\in \Z_R,\eta \in \cS_m} a_m\hat{A}_\pm\left(\frac{k-mk_W+\eta}{\eps},T\right)e^{\ri \eta x}q_\pm(x)\right]\\ & -\eps^{1/2} e^{-\ri \omega_0 t}\sigma(x) \sum_{s_1,s_2,s_3\in \{+,-\}}q_{s_1}(x)\overline{q_{s_2}}(x)q_{s_3}(x)\left(\hat{A}_{s_1}\ast \hat{\overline{A}}_{s_2}\ast \hat{A}_{s_3}\right)(K,T)\\ & - \eps^{3/2}e^{-\ri \omega_0 t}\sum_{\pm}p_\pm(x)K^2\hat{A}_\pm(K,T), \end{aligned}$$ where $$K:=\eps^{-1}k, \qquad T = \eps t, \qquad \cS_m:=\{\eta\in \tfrac{1}{N} \Z: mk_W-\eta\in \overline{\B_Q}=[-\tfrac{1}{2N},\tfrac{1}{2N}]\},$$ and the convolution $\ast$ is in the $K$-variable, $(\hat{f}\ast\hat{g})(K)=\int_\R\hat{f}(K-s)\hat{g}(s) ds$. Note that the set $\cS_m$ has at most two elements, namely $\cS_m\subset \{0,\tfrac{1}{N}\}$ or $\cS_m\subset \{0,-\tfrac{1}{N}\}$. Because of the support of $\hat{A}_\pm$ the above residual is supported in $k$ in intervals of radius at most $3\eps^{1/2}$ centered at $k=0$, $k\in k_W \Z_R$ and at all $\tfrac{1}{N}\Z$-shifts of these points, cf. the periodicity of the Bloch transform in $k$. Due to assumption (H2) in the case of irrational $k_W$ the number of distinct support centers from $k_W \Z_R + \tfrac{1}{N}\Z$ which lie in $\overline{\B_Q}$ is finite. For $k_W\in \Q$ is $\{k_W \Z_R + \tfrac{1}{N}\Z\}\cap \B_Q$ finite even if infinitely many coefficients $a_n$ are nonzero. Note that we need $\overline{\B_Q}$ rather than only $\B_Q$ because a support interval centered at $k=-\tfrac{1}{2N}$ intersects $\B_Q$ for any $\eps>0$. We denote this *finite set* of support centers by \[E:KR\] \_R:=(k\_W \_R + )={k: k=mk\_W+ m\_R, \_m} and label its elements by $\kappa_j$: $$\cK_R=\{\kappa_1,\dots, \kappa_J\} \text{ for some } J\in \N.$$ Our modified (extended) ansatz in the $\Uvec$-variables is thus for $k\in \B_Q$ \[E:Uext\] U\_[n\_\*]{}\^(k,t) &:= \^[-1/2]{}\_[n\_\*]{}(K,T)e\^[-\_0 t]{}+\^[1/2]{}\_[j=1]{}\^J \_[n\_\*,j]{}(,T)e\^[-\_0 t]{}\ U\_[n\_\*+1]{}\^(k,t) &:= \^[-1/2]{}\_[n\_\*+1]{}(K,T)e\^[-\_0 t]{}+\^[1/2]{}\_[j=1]{}\^J \_[n\_\*+1,j]{}(,T)e\^[-\_0 t]{}\ U\_[n]{}\^(k,t) &:= \^[1/2]{}\_[n]{}(K,T)e\^[-\_0 t]{}+\^[1/2]{}\_[j=1]{}\^J \_[n,j]{}(,T)e\^[-\_0 t]{}, n {n\_\*,n\_\*+1}\ with $$\begin{aligned} &\supp(\Atil_q(\cdot,T)) \cap \eps^{-1}\B_Q \subset [-\eps^{-1/2},\eps^{-1/2}], \quad q\in \{n_*,n_*+1\},\\ &\supp(\Atil_n(\cdot,T)) \cap \eps^{-1}\B_Q \subset [-3\eps^{-1/2},3\eps^{-1/2}], \quad \supp(\Atil_{n,m}(\cdot,T))\cap \eps^{-1}\B_Q \subset [-\eps^{-1/2},\eps^{-1/2}] \end{aligned}$$ for all $n\in \N,m\in \Z$. As expected from the formal asymptotics and as shown in detail below, the residual for $\Uvecext$ is small if $\Atil_{n_*}$ and $\Atil_{n_*+1}$ are selected as cut-offs of $\widehat{A}_+$ and $\widehat{A}_-$ respectively. To comply with , we define $\Uvecext$ with $\tfrac{1}{N}$-periodicity in $k$, i.e. $$\Uvecext\left(k+\tfrac{1}{N},t\right)=\Uvecext(k,t) \ \text{for all} \ k\in \R, t\in \R.$$ Below it will be useful to write \[E:Uext\_decomp\] = \^[0,]{}+\^[1,]{}, where \[E:Uext\_decomp2\] \^[0,]{}:= \^[-1/2]{}(\_[n\_\*]{}(K,T)e\_[n\_\*]{}+\_[n\_\*+1]{}(K,T)e\_[n\_\*+1]{})e\^[-\_0 t]{}  \^[1,]{}:=-\^[0,]{}. Here $e_n$ is the standard $n$-th Euclidean unit vector in $\R^\N$. In order to determine the residual for $\Uvecext$, let us first formulate equation in the $\Uvec$-variables. Applying first $\cT$ to , we get $$\left(\ri \pa_t-\cL(x,k)-\eps W^{(1)}_Q(x)\right)\util(x,k,t)-\eps \sum_{m\in \Z_R}a_m \util(x,k-mk_W,t)-\sigma(x) (\util\ast_{\B_Q}\tilde{\overline{u}}\ast_{\B_Q}\util)(x,k,t)=0.$$ Expanding $\util(x,k,t) = \sum_{n\in \N}U_n(k,t)q_n(x,k)$ leads to the (infinitely dimensional) ODE-system \[E:PNLS\_U\] (\_t -(k)-M\^[(1)]{}(k))(k,t) -\_[m\_R]{}M\^[(R,m)]{}(k) (k-mk\_W,t)+(,,)(k,t)=0 parametrized by $k$, where $$\begin{aligned} &\Theta_{j,j}(k):=\vartheta_j(k) \ \forall j\in \N, \ \Theta_{i,j}:=0 \ \forall i,j\in \N, i\neq j,\notag \\ &M^{(1)}_{i,j}(k):=\langle W^{(1)}_Qq_j(\cdot,k), q_i(\cdot,k)\rangle_{Q}, \quad M^{(R,m)}_{i,j}(k):=a_m\langle q_j(\cdot,k-mk_W), q_i(\cdot,k)\rangle_{Q},\notag\\ &F_j(\Uvec,\Uvec,\Uvec)(k,t) :=-\langle \sigma(\cdot) (\util\ast_{\B_Q}\tilde{\overline{u}}\ast_{\B_Q}\util)(\cdot,k,t),q_j(\cdot,k)\rangle_Q, \ \util(x,k,t)=\sum_{n\in \N}U_n(k,t)q_n(x,k).\label{E:Fj}\end{aligned}$$ Note that $\tilde{\overline{u}}(x,k,t)=\sum_{n\in \N}\overline{U_n}(-k,t)q_n(x,k)$. Substituting $\Uvec^{\text{ext}}$ in , we get the residual $$\begin{aligned} \vec{\Res}(k,t):=&\left(\ri \pa_t -\Theta(k)-\eps M^{(1)}(k)\right)\Uvecext(k,t) -\eps \sum_{m\in \Z_R}M^{(R,m)}(k) \Uvecext(k-mk_W,t)\\ &+\vec{F}(\Uvecext,\Uvecext,\Uvecext)(k,t) \end{aligned}$$ where in $\util^\text{ext}(x,k,t) := \sum_{n\in \N}U_n^\text{ext}(k,t)q_n(x,k)$ in $\vec{F}$. For $k\in \B_Q$ we have $$\begin{aligned} &\Res_{n_*}(k,t)= \eps^{1/2}\left[\left(\ri \pa_T -\eps^{-1}(\vartheta_{n_*}(k)-\omega_0)-M^{(1)}_{n_*,n_*}(k)\right)\Atil_{n_*}(K,T) \phantom{\sum_{m\in \Z_R,\eta\in \cS_m}}\right.\\ & + \eps^{-1/2}F_{n_*}(\UvecOext,\UvecOext,\UvecOext)(k,t)e^{\ri \omega_0 t} - M^{(1)}_{n_*,n_*+1}(k)\Atil_{n_*+1}(K,T) \\ &- \left(\vartheta_{n_*}(k)-\omega_0\right)\sum_{j=1}^J \Atil_{n_*,j}\left(\frac{k-\kappa_j}{\eps},T\right)\\ &-\sum_{m\in \Z_R,\eta\in \cS_m}M^{(R,m)}_{n_*,n_*}(k)\Atil_{n_*}\left(\frac{k-mk_W+\eta}{\eps},T\right)\\ &\left.-\sum_{m\in \Z_R,\eta\in \cS_m}M^{(R,m)}_{n_*,n_*+1}(k)\Atil_{n_*+1}\left(\frac{k-mk_W+\eta}{\eps},T\right)\right]e^{-\ri \omega_0 t} + \text{h.o.t.}, \end{aligned}$$ $$\begin{aligned} &\Res_{n_*+1}(k,t)= \eps^{1/2}\left[\left(\ri \pa_T -\eps^{-1}(\vartheta_{n_*+1}(k)-\omega_0)-M^{(1)}_{n_*+1,n_*+1}(k)\right)\Atil_{n_*+1}(K,T) \phantom{\sum_{m\in \Z_R,\eta\in \cS_m}}\right.\\ & + \eps^{-1/2}F_{n_*+1}(\UvecOext,\UvecOext,\UvecOext)(k,t)e^{\ri \omega_0 t} - M^{(1)}_{n_*+1,n_*}(k)\Atil_{n_*}(K,T) \\ &- \left(\vartheta_{n_*+1}(k)-\omega_0\right)\sum_{j=1}^J\Atil_{n_*+1,j}\left(\frac{k-\kappa_j}{\eps},T\right)\\ &-\sum_{m\in \Z_R,\eta\in \cS_m}M^{(R,m)}_{n_*+1,n_*}(k)\Atil_{n_*}\left(\frac{k-mk_W+\eta}{\eps},T\right)\\ &\left.-\sum_{m\in \Z_R,\eta\in \cS_m}M^{(R,m)}_{n_*+1,n_*+1}(k)\Atil_{n_*+1}\left(\frac{k-mk_W+\eta}{\eps},T\right)\right]e^{-\ri \omega_0 t} + \text{h.o.t.}, \end{aligned}$$ and for $n\in \N\setminus\{n_*,n_*+1\}$ $$\begin{aligned} &\Res_{n}(k,t)= \eps^{1/2}\left[(\omega_0-\vartheta_{n}(k))\Atil_{n}(K,T) -(M_{n,n_*}^{(1)}(k)\Atil_{n_*}(K,T) +M_{n,n_*+1}^{(1)}(k)\Atil_{n_*+1}(K,T))\phantom{\sum_{m\in \Z_R,\eta\in \cS_m}}\right.\\ &\left.+\eps^{-1/2}F_{n}(\UvecOext,\UvecOext,\UvecOext)(k,t)e^{\ri \omega_0 t} - \left(\vartheta_{n}(k)-\omega_0\right)\sum_{j=1}^J\Atil_{n,j}\left(\frac{k-\kappa_j}{\eps},T\right)\right.\\ &-\sum_{m\in \Z_R,\eta\in \cS_m}M^{(R,m)}_{n,n_*}(k)\Atil_{n_*}\left(\frac{k-mk_W+\eta}{\eps},T\right)\\ &\left.-\sum_{m\in \Z_R,\eta\in \cS_m}M^{(R,m)}_{n,n_*+1}(k)\Atil_{n_*+1}\left(\frac{k-mk_W+\eta}{\eps},T\right)\right]e^{-\ri \omega_0 t} + \text{h.o.t.}. \end{aligned}$$ Note that $$F_{n}(\UvecOext,\UvecOext,\UvecOext)=\eps^{-3/2}\sum_{\alpha,\beta,\gamma\in\{n_*,n_*+1\}}F_{n}(\Atil_{\alpha}e_\alpha,\Atil_{\beta}e_\beta,\Atil_{\gamma}e_\gamma)e^{-\ri \omega_0 t}$$ and $$\begin{aligned} F_{n}&(\Atil_{\alpha}e_\alpha,\Atil_{\beta}e_\beta,\Atil_{\gamma}e_\gamma)(k,t) \\ & = \int_{-2\eps^{1/2}}^{2\eps^{1/2}}\int_{-\eps^{1/2}}^{\eps^{1/2}} b_{\alpha\beta\gamma}^{(n)}(k,k-s,s-l,l)\Atil_\alpha\left(\tfrac{k-s}{\eps},T\right)\tilde{\overline{A}}_\beta\left(\tfrac{s-l}{\eps},T\right)\Atil_\gamma\left(\tfrac{l}{\eps},T\right) dl ds\\ &=\eps^2\int_{-2\eps^{-1/2}}^{2\eps^{-1/2}}\int_{-\eps^{-1/2}}^{\eps^{-1/2}} b_{\alpha\beta\gamma}^{(n)}(\eps K,\eps(K-s),\eps(s-l),\eps l)\Atil_\alpha(K-s,T)\tilde{\overline{A}}_\beta(s-l,T)\Atil_\gamma(l,T) dl ds \end{aligned}$$ with $$b_{\alpha\beta\gamma}^{(n)}(k,r,s,l):=\langle \sigma(\cdot) q_\alpha(\cdot,r)\overline{q_\beta}(\cdot,-s)q_\gamma(\cdot,l),q_n(\cdot,k)\rangle_Q.$$ Hence $\eps^{-1/2}F_{n}(\UvecOext,\UvecOext,\UvecOext)(k,t)$ is $O(1)$. Also the terms $\eps^{-1}(\vartheta_{j}(k)-\omega_0)\Atil_{j}$ for $j=n_*,n_*+1$ are $O(1)$ as can be seen by the Taylor expansion $\vartheta_{j}(k)\sim\omega_0 \pm kc_g=\omega_0\pm\eps K c_g \ (\eps \to 0, k\in \supp \Atil_j(\cdot,T)\cap \eps^{-1}\B_Q)$ for $j=n_*,n_*+1$ respectively. The “h.o.t.” in the residual stands for terms of higher order in $\eps$ and consists of the following terms \[E:hot\] &\_T \^[1,]{}, M\^[(1)]{}\^[1,]{}, \_[m\_R]{}M\^[(R,m)]{}\^[1,]{}(-mk\_W,t),\ & (,\^[1,]{},)+2(,,\^[1,]{}), and nonlinear terms quadratic or cubic in $\vec{U}^{1,\text{ext}}$. Note that if we approximate in the first two lines of $\Res_{n_*}$ the function $\vartheta_{n_*}(k)$ by $\vartheta_{n_*}(0)+k c_g$, the function $M^{(1)}_{n_*,j}(k)$ by $M^{(1)}_{n_*,j}(0)$ for $j=n_*,n_*+1$ and in $F_{n_*}$ approximate $q_j(\cdot,k)$ by $q_j(\cdot,0)$, such that $b_{\alpha\beta\gamma}^{(n_*)}(\eps K,\eps(K-s),\eps(s-l),\eps l)$ is replaced by $b_{\alpha\beta\gamma}^{(n_*)}(0,0,0,0)$, we recover the left hand side of the first equation in the CMEs . Similarly for the first two lines in $\Res_{n_*+1}$ and the second equation in . This folows from the identities \[E:coef\_ident\] &M\^[(1)]{}\_[n\_\*,n\_\*+1]{}(0)=M\^[(1)]{}\_[n\_\*+1,n\_\*]{}(0)=-, M\^[(1)]{}\_[n\_\*,n\_\*]{}(0)=M\^[(1)]{}\_[n\_\*+1,n\_\*+1]{}(0)=-\_s,\ &b\^[(n\_\*)]{}\_[n\_\*,n\_\*,n\_\*]{}(0,0,0)=b\^[(n\_\*+1)]{}\_[n\_\*+1,n\_\*+1,n\_\*+1]{}(0,0,0)=b\^[(n\_\*)]{}\_[n\_\*,n\_\*+1,n\_\*+1]{}(0,0,0)=b\^[(n\_\*+1)]{}\_[n\_\*,n\_\*,n\_\*+1]{}(0,0,0)=-\_Q\ &b\^[(n\_\*)]{}\_[n\_\*,n\_\*,n\_\*+1]{}(0,0,0)=b\^[(n\_\*)]{}\_[n\_\*+1,n\_\*+1,n\_\*+1]{}(0,0,0)=b\^[(n\_\*+1)]{}\_[n\_\*,n\_\*,n\_\*]{}(0,0,0)=b\^[(n\_\*+1)]{}\_[n\_\*+1,n\_\*+1,n\_\*]{}(0,0,0)=-\_Q\ &b\^[(n\_\*)]{}\_[n\_\*+1,n\_\*,n\_\*+1]{}(0,0,0)=b\^[(n\_\*+1)]{}\_[n\_\*,n\_\*+1,n\_\*]{}(0,0,0)=-\_Q. Therefore, the first two lines in $\Res_{n_*}$, $\Res_{n_*+1}$ will be close to zero (precisely $O(\eps^{5/2})$ in $\|\cdot\|_{L^1(\B_Q)}$ as shown below) after setting \[E:Atil\_def\] \_[n\_\*]{}(K,T):= \_[\[-\^[-1/2]{},\^[-1/2]{}\]]{}(K)\_+(K,T), \_[n\_\*+1]{}(K,T):= \_[\[-\^[-1/2]{},\^[-1/2]{}\]]{}(K)\_-(K,T). The remaining three lines in the formal $O(\eps^{1/2})$ part of $\Res_{n_*}$ and $\Res_{n_*+1}$ can then be made exactly zero by setting \[E:Atilcor\_def\] \_[q,j]{}(K,T):=(&\_0-\_q(\_j+K))\^[-1]{}\_(M\^[(R,m)]{}\_[q,n\_\*]{}(\_j+K)\_[n\_\*]{}(K,T).\ &.+M\^[(R,m)]{}\_[q,n\_\*+1]{}(\_j+K)\_[n\_\*+1]{}(K,T)), q{n\_\*,n\_\*+1}, j{1,…,J}, where the $1/N$-periodicity of $\vartheta_q$ and $M_{q,n}^{(R,m)}$ was used. Note that for $\eps>0$ small enough we have $$\min_{q\in\{n_*,n_*+1\}, j\in \{1,\dots,J\}, |K|\leq \eps^{-1/2}}\left|\vartheta_q(\kappa_j+\eps K)-\omega_0\right|>c>0$$ because $\omega_0=\vartheta_{n_*}(0)=\vartheta_{n_*+1}(0)$, the functions $\vartheta_{q}(k)$ are monotonous on $(-\tfrac{1}{2N},0)$ and $(0,\tfrac{1}{2N}]$ and $\tfrac{1}{N}$-periodic and because ${\rm dist}(\cK_R,\tfrac{1}{N}\Z)> \delta >0$. The last property follows from the finiteness of $\cK_R$. Finally, the formal $O(\eps^{1/2})$ part of $\Res_{n}, n\notin\{n_*,n_*+1\}$ vanishes if we set for $n\in \N\setminus\{n_*,n_*+1\}$ \[E:Atiln\_def\] \_[n]{}(K,T):=&(\_0-\_n(K))\^[-1]{}(M\^[(1)]{}\_[n,n\_\*]{}(K)\_[n\_\*]{}(K,T)+M\^[(1)]{}\_[n,n\_\*+1]{}(K)\_[n\_\*+1]{}(K,T) .\ &-\_[-2\^[-1/2]{}]{}\^[2\^[-1/2]{}]{}\_[-\^[-1/2]{}]{}\^[\^[-1/2]{}]{} \_[,,{n\_\*,n\_\*+1}]{}b\_\^[(n)]{}(K,(K-s),(s-l),l)\ &. \_(K-s,T)\_(s-l,T)\_(l,T) dl ds ) and \[E:Atilncor\_def\] \_[n,j]{}(K,T):=(&\_0-\_n(\_j+K))\^[-1]{}\_(M\^[(R,m)]{}\_[n,n\_\*]{}(\_j+K)\_[n\_\*]{}(K,T).\ &.+M\^[(R,m)]{}\_[n,n\_\*+1]{}(\_j+K)\_[n\_\*+1]{}(K,T) ), j{1,…,J}. The factors $(\omega_0-\vartheta_n(\eps K))^{-1}$ and $(\omega_0-\vartheta_n(\kappa_j+\eps K))^{-1}$ are again bounded because $$\min_{n\in\N\setminus\{n_*,n_*+1\}, k\in \R}\left|\omega_0-\vartheta_n(k)\right|>c>0.$$ This follows from the transversal crossing of $\vartheta_{n_*}$ and $\vartheta_{n_*+1}$ at $(0,\omega_0)$ and because eigenvalue functions do not overlap in one dimension. In the remainder of the argument, first in Lemma \[L:res\_est\] we estimate $\|\vec{\Res}(\cdot,t)\|_{\cX(s)}$ under the conditions , , , , and , and then use a Gronwall argument and an estimate of $\|\Uvecext(\cdot,t)-\Uvec^\text{app}(\cdot,t)\|_{\cX(s)}$ in Lemma \[L:diff\_app\_ext\] to conclude the proof of Theorem \[T:main\]. Below we will need some asymptotics of $\left|\vartheta_n(k)-\omega_0\right|$ for $n\to \infty$. In particular, see p. 55 in [@Hoerm_85], there are constants $c_1,c_2>0$ such that \[E:band\_as\] c\_1n\^2 \_n(k) c\_2n\^2  k\_Q, n. The ansatz components $\UvecOext$ and $\vec{U}^{1,\text{ext}}$ can be estimated as follows. \[L:ext\_est\] Assume (H2)-(H4) and let $(A_+,A_-)(X,T)$ be a solution of such that $\widehat{A}_\pm\in C([0,T_0],L^1(\R))$ and assume , , and . Then there is $\eps_0>0$ and $$c=c\left(\max_{T\in[0,T_0]}\|\hat{A}_+(\cdot,T)\|_{L^1(\R)},\max_{T\in[0,T_0]}\|\hat{A}_-(\cdot,T)\|_{L^1(\R)}\right)>0$$ such that for all $s\in (0,3/2), \eps\in(0,\eps_0)$ and all $t\in [0,\eps^{-1}T_0]$ $$\|\UvecOext\|_{\cX(s)}\leq c\eps^{1/2}, \qquad \|\vec{U}^{1,\text{ext}}\|_{\cX(s)}\leq c\eps^{3/2}.$$ From we get $$\|\UvecOext(\cdot,t)\|_{\cX(s)} \leq c \eps^{1/2}\sum_{\pm}\|\hat{A}_\pm(\cdot, \eps t)\|_{L^1(\R)}.$$ Next, note that due to the $C^2$ smoothness of $W$ there is a constant $c>0$ such that \[E:MRm\_est\] |M\^[(R,m)]{}\_[n,j]{}(k)|   j{n\_\*,n\_\*+1}, n k\_Q. Also, one obviously has \[E:Mb\_est\] |M\^[(1)]{}\_[n,j]{}(k)|c, |b\_[,,]{}\^[(n)]{}(k,l,s,r)|c for all $k,l,s,r\in \B_Q$, all $j,\alpha,\beta,\gamma\in \{n_*,n_*+1\}$ and all $n\in \N$ Finally, we use , , ,,, and to estimate $$\begin{aligned} &\|\vec{U}^{1,\text{ext}}\|_{\cX(s)}= \left\|\left(\sum_{n\in \N}n^{2s}|U_n^{1,\text{ext}}(\cdot,t)|^2\right)^{1/2}\right\|_{L^1(\B_Q)}\\ &\leq c\eps^{1/2} \left(\sum_{n\in \N}n^{2s-4}\right)^{1/2}\left(\left(\left\|\Atil_{n_*}(\eps^{-1}\cdot,T)\right\|_{L^1(\B_Q)}+\left\|\Atil_{n_*+1}(\eps^{-1}\cdot,T)\right\|_{L^1(\B_Q)}\right)\left(1+\sum_{m\in \Z_R}\frac{c}{m^2}\right)\right.\\ &\left.+\sum_{\alpha,\beta,\gamma\in \{n_*,n_*+1\}}\left\|\left(|\Atil_\alpha|*_{\B_Q}|\Atil_\beta|*_{\B_Q}|\Atil_\gamma|\right)(\eps^{-1}\cdot,T)\right\|_{L^1(\B_Q)}\right)\\ &\leq c\eps^{3/2} \left(\sum_{\pm}\|\Ahat_\pm(\cdot,T)\|_{L^1(\R)}+\sum_{\xi,\zeta,\theta\in \{+,-\}}\|\Ahat_\xi(\cdot,T)\|_{L^1(\R)}\|\Ahat_\zeta(\cdot,T)\|_{L^1(\R)}\|\Ahat_\theta(\cdot,T)\|_{L^1(\R)}\right), \end{aligned}$$ where the factor $cn^{-4}$ in the sum over $n\in \N$ comes from $(\omega_0 -\vartheta_n(\eps K))^{-2}$ using . The assumption $s<3/2$ implies the summability of the series in $n$. The bound on the convolution follows from Young’s inequality for convolutions. The estimates $\|\UvecOext(\cdot,t)\|_{\cX(s)}\leq c\eps^{1/2}, \|\vec{U}^{1,\text{ext}}(\cdot,t)\|_{\cX(s)}\leq c\eps^{3/2}$ for some $c>0$ and all $t\in [0,\eps^{-1}T_0]$ and $s\in (0,3/2)$ now follow from $\widehat{A}_\pm\in C([0,T_0],L^1(\R))$. \[L:res\_est\] Assume (H1)-(H4). Let $s\in(1/2,3/2)$ and let $(A_+,A_-)(X,T)$ be a solution of such that $\hat{A}_\pm\in C^1([0,T_0],L^1_{s_A}(\R))$ for some $T_0>0$ and some $s_A\geq 2$. Choosing $\Uvecext$ according to , , , , and , there exists $\eps_0>0$ and $$\begin{aligned} C_\text{Res}=&C_\text{Res}\left(\max_{\pm,T\in[0,T_0]}\|\hat{A}_\pm(\cdot,T)\|_{L^1_{s_A}(\R)},\max_{\pm,T\in[0,T_0]}\|\pa_T\hat{A}_\pm(\cdot,T)\|_{L^1_{s_A}(\R)}\right)>0 \end{aligned}$$ such that for all $\eps \in (0,\eps_0)$ and all $t\in [0,\eps^{-1}T_0]$ the residual $\vec{\Res}$ of $\Uvecext$ in satisfies $$\|\vec{\Res}(\cdot,t)\|_{\cX(s)} \leq C_\text{Res} \eps^{5/2}.$$ First we show that for $n=n_*,n_*+1$ the formal $O(\eps^{1/2})$ part in $\Res_{n}$ satisfies $\|\Res_{n}(\cdot,t)\|_{L^1(\B_Q)}\leq c \eps^{5/2}$ for all $t\in [0,\eps^{-1}T_0]$. Choosing $\Atil_{n}$ and $\Atil_{n,m}$ as in , , , and with $(A_+,A_-)$ being a solution of , we get $$\|\Res_{n_*}(\cdot,t)\|_{L^1(\B_Q)}\leq \eps^{-1/2}I_1(T)+\eps^{1/2}\sum_{j=2}^4I_j(T) + \text{h.o.t.},$$ where $$\begin{aligned} I_1(T)&:=\int_{-\eps^{1/2}}^{\eps^{1/2}}|(\vartheta_{n_*}(k)-\omega_0-c_g\tfrac{k}{\eps})\Atil_{n_*}\left(\tfrac{k}{\eps},T\right)|dk, \\ I_2(T)&:=\int_{-\eps^{1/2}}^{\eps^{1/2}}|(M^{(1)}_{n_*,n_*}(k)-M^{(1)}_{n_*,n_*}(0))\Atil_{n_*}(\tfrac{k}{\eps},T)|dk, \\ I_3(T)&:=\int_{-\eps^{1/2}}^{\eps^{1/2}}|(M^{(1)}_{n_*,n_*+1}(k)-M^{(1)}_{n_*,n_*+1}(0))\Atil_{n_*+1}\left(\tfrac{k}{\eps},T\right)|dk, \\ I_4(T)&:=\hspace{-0.4cm}\sum_{\alpha,\beta,\gamma\in\{n_*,n_*+1\}}\hspace{-0.1cm}J_{\alpha\beta\gamma}(T), \quad J_{\alpha\beta\gamma}(T):=\int_{-3\eps^{1/2}}^{3\eps^{1/2}}\left| \int_{-2\eps^{-1/2}}^{2\eps^{-1/2}}\int_{-\eps^{-1/2}}^{\eps^{-1/2}} b_{\alpha\beta\gamma}^{(n_*)}(k,k-\eps s,\eps(s-l),\eps l) \times \right.\\ &\left. \Atil_\alpha\left(\tfrac{k}{\eps}-s,T\right)\tilde{\overline{A}}_\beta(s-l,T)\Atil_\gamma(l,T) dl ds - b_{\alpha\beta\gamma}^{(n_*)}(0,0,0,0) (\Ahat_{\xi_\alpha}\ast \widehat{\overline{A}}_{\xi_\beta}\ast \Ahat_{\xi_\gamma})\left(\tfrac{k}{\eps},T\right) \right|dk, \end{aligned}$$ where $$\xi_\delta =\text{ ``+'' if }\delta=n_*\text{ and }\xi_\delta =\text{ ``-'' if }\delta=n_*+1.$$ For $I_1$ we use the $C^2$-smoothness assumption on $\tilde{\omega}_\pm(k)$ in (H1) and have $$\begin{aligned} \eps^{-1/2}I_1(T)&=\eps^{1/2}\int_{-\eps^{-1/2}}^{\eps^{-1/2}}|(\vartheta_{n_*}(\eps K)-\omega_0-c_g K)\Atil_{n_*}(K,T)|dK \notag\\ &\leq c \eps^{5/2}\int_{-\eps^{-1/2}}^{\eps^{-1/2}} K^2 |\Ahat_+(K,T)|dK\leq c\eps^{5/2}\|\Ahat_+(\cdot,T)\|_{L^1_2(\R)}.\label{E:I1_est}\end{aligned}$$ For $I_2$ we use the Lipschitz property in (H1). Hence there is $L>0$ such that $$\begin{aligned} &|M^{(1)}_{n_*,n_*}(k)-M^{(1)}_{n_*,n_*}(0)| \\ &\leq |\langle W_Q^{(1)}(q_{n_*}(\cdot, k)-q_{n_*}(\cdot, 0)), q_{n_*}(\cdot, k)\rangle_Q | + |\langle W_Q^{(1)}q_{n_*}(\cdot, 0), (q_{n_*}(\cdot, k)-q_{n_*}(\cdot, 0))\rangle_Q|\\ & \leq L|k| \|W^{(1)}_Q\|_{L^\infty}\left(\|q_{n_*}(\cdot,k)\|_{L^2((0,Q))}+\|q_{n_*}(\cdot,0)\|_{L^2((0,Q))}\right)= 2L|k| \|W^{(1)}_Q\|_{L^\infty} \end{aligned}$$ for all $|k|\leq \eps^{1/2}$. Hence $$\eps^{1/2}I_2(T)\leq 2L\|W^{(1)}_Q\|_{L^\infty}\eps^{5/2} \int_{-\eps^{-1/2}}^{\eps^{-1/2}}|K||\Ahat_+(K,T)|dK \leq c \eps^{5/2} \|\Ahat_+(\cdot,T)\|_{L^1_1(\R)}.$$ Similarly $$\eps^{1/2}I_3(T)\leq c \eps^{5/2} \|\Ahat_-(\cdot,T)\|_{L^1_1(\R)}.$$ For $I_4$ we first write $$\begin{aligned} &J_{\alpha\beta\gamma}\leq \eps\int_{-3\eps^{-1/2}}^{3\eps^{-1/2}}\int_{-2\eps^{-1/2}}^{2\eps^{-1/2}}\int_{-\eps^{-1/2}}^{\eps^{-1/2}} \left| b_{\alpha\beta\gamma}^{(n_*)}(\eps K,\eps(K- s),\eps(s-l),\eps l) - b_{\alpha\beta\gamma}^{(n_*)}(0,0,0,0)\right| \times \\ &\left| \Atil_\alpha(K-s)\tilde{\overline{A}}_\beta(s-l)\Atil_\gamma(l)\right| dl ds dK \\ &+ \eps |b_{\alpha\beta\gamma}^{(n_*)}(0,0,0,0)| \|\Atil_\alpha \ast \tilde{\overline{A}}_\beta\ast \Atil_\gamma-\Ahat_{\xi_\alpha}\ast \widehat{\overline{A}}_{\xi_\beta}\ast \Ahat_{\xi_\gamma}\|_{L^1(-3\eps^{-1/2},3\eps^{-1/2})}, \end{aligned}$$ where we have left out the $T$-dependence of $\Atil_\alpha$ and $\Ahat_\alpha$ for brevity. By the triangle inequality $$\begin{aligned} &\left| b_{\alpha\beta\gamma}^{(n_*)}(k,\lambda,\mu,\nu) - b_{\alpha\beta\gamma}^{(n_*)}(0,0,0,0)\right|\leq \left| b_{\alpha\beta\gamma}^{(n_*)}(k,\lambda,\mu,\nu) -b_{\alpha\beta\gamma}^{(n_*)}(0,\lambda,\mu,\nu)\right| \\ &+\left|b_{\alpha\beta\gamma}^{(n_*)}(0,\lambda,\mu,\nu) - b_{\alpha\beta\gamma}^{(n_*)}(0,0,\mu,\nu)\right|+\dots+\left|b_{\alpha\beta\gamma}^{(n_*)}(0,0,0,\nu)-b_{\alpha\beta\gamma}^{(n_*)}(0,0,0,0)\right|. \end{aligned}$$ The differences on the right hand side can be estimated using the Lipschitz continuity in $k$ of Bloch waves, the Cauchy-Schwarz inequality and the algebra property of $H^1(\R)$. For instance, for the first difference we have $$\begin{aligned} &\left| b_{\alpha\beta\gamma}^{(n_*)}(\eps K,\lambda,\mu,\nu) -b_{\alpha\beta\gamma}^{(n_*)}(0,\lambda,\mu,\nu)\right| = \left|\langle q_\alpha(\cdot,\lambda) \overline{q}_\beta(\cdot,\mu)q_\gamma(\cdot,\nu), q_{n_*}(\cdot,\eps K)-q_{n_*}(\cdot,0)\rangle_Q\right|\\ &\leq \eps L|K|\|q_{\alpha}(\cdot,\lambda)\overline{q}_{\beta}(\cdot,\mu)q_{\gamma}(\cdot,\nu)\|_{L^2(0,Q)}\\ &\leq \eps L|K| \|q_{\alpha}(\cdot,\lambda)\|_{H^1(0,Q)} \|q_{\beta}(\cdot,\mu)\|_{H^1(0,Q)}\|q_{\gamma}(\cdot,\nu)\|_{H^1(0,Q)}. \end{aligned}$$ Further, to estimate $\|\Atil_\alpha \ast \tilde{\overline{A}}_\beta\ast \Atil_\gamma-\Ahat_{\xi_\alpha}\ast \widehat{\overline{A}}_{\xi_\beta}\ast \Ahat_{\xi_\gamma}\|_{L^1(-3\eps^{-1/2},3\eps^{-1/2})}$, we set $$\Atil_\alpha(K):= \Ahat_{\xi_\alpha}(K)+e_{\xi_\alpha}(K), \text{ where } e_\pm(K):=(\chi_{[-\eps^{-1/2},\eps^{-1/2}]}(K)-1)\Ahat_\pm(K),$$ and similarly for $\Atil_\beta$ and $\Atil_\gamma$. Since $e_\pm$ satisfy \[E:epm\_est\] e\_\_[L\^1()]{}\_[|K|&gt;\^[-1/2]{}]{}|(\_[\[-\^[-1/2]{},\^[-1/2]{}\]]{}(K)-1)(1+|K|)\^[-s\_A]{}|\_\_[L\^1\_[s\_A]{}()]{}c \^[s\_A/2]{}\_\_[L\^1\_[s\_A]{}()]{}, and because $\|\Atil_\alpha \ast \tilde{\overline{A}}_\beta\ast \Atil_\gamma-\Ahat_{\xi_\alpha}\ast \widehat{\overline{A}}_{\xi_\beta}\ast \Ahat_{\xi_\gamma}\|_{L^1}\leq \|e_{\xi_\alpha} \ast \widehat{\overline{A}}_{\xi_\beta}\ast \Ahat_{\xi_\gamma}\|_{L^1} +\|\Ahat_{\xi_\alpha}\ast \overline{e_{\xi_\beta}}(-\cdot)\ast \Ahat_{\xi_\gamma}\|_{L^1}+\|\Ahat_{\xi_\alpha}\ast \widehat{\overline{A}}_{\xi_\beta}\ast e_{\xi_\gamma}\|_{L^1} + $ terms quadratic or cubic in $e_\pm$, we get, using Young’s inequality for convolutions, $$\|\Atil_\alpha \ast \tilde{\overline{A}}_\beta\ast \Atil_\gamma-\Ahat_{\xi_\alpha}\ast \widehat{\overline{A}}_{\xi_\beta}\ast \Ahat_{\xi_\gamma}\|_{L^1(-3\eps^{-1/2},3\eps^{-1/2})}\leq c\eps^{s_A/2}\|\Ahat_{\xi_\alpha}\|_{L^1_{s_A}(\R)}\|\Ahat_{\xi_\beta}\|_{L^1_{s_A}(\R)}\|\Ahat_{\xi_\gamma}\|_{L^1_{s_A}(\R)}.$$ Hence, we arrive at $$\eps^{1/2}I_4(T)\leq c\eps^{5/2}\sum_{\xi,\zeta,\theta\in \{+,-\}}\|\Ahat_{\xi}(\cdot,T)\|_{L^1_{s_A}(\R)}\|\Ahat_{\zeta}(\cdot,T)\|_{L^1_{s_A}(\R)}\|\Ahat_{\theta}(\cdot,T)\|_{L^1_{s_A}(\R)}$$ because $s_A \geq 2$. Similarly, one can show that all the formal $O(\eps^{1/2})$ terms in $\Res_{n_*+1}$ are $O(\eps^{5/2})$ in the $L^1(\B_Q)$-norm. Because in $\Res_n, n \in \N\setminus\{n_*,n_{*+1}\}$ the whole formal $O(\eps^{1/2})$-part vanishes by the choice of $\Atil_j$ and $\Atil_{j,m}$, it remains to discuss the h.o.t. terms in . Firstly, for $s\in (0,3/2)$ $$\|\eps \pa_T \vec{U}^{1,\text{ext}}\|_{\cX(s)} \leq c\eps^{5/2},$$ where $c=c(\max_{T\in [0,T_0]}\|\pa_T \Ahat_\pm(\cdot,T)\|_{L^1(\R)},\max_{T\in [0,T_0]}\|\Ahat_\pm(\cdot,T)\|_{L^1(\R)})$ similarly to the proof of Lemma \[L:ext\_est\]. Next, because $M^{(1)}\vec{U}^{1,\text{ext}}=\cD(W_Q^{(1)}\util^{1,\text{ext}})$ with $\util^{1,\text{ext}}:=\cD^{-1}\vec{U}^{1,\text{ext}}$, the isomorphic property of $\cD$ (Lemma \[L:D\_isom\]) yields \[E:M1\_U1ext\] M\^[(1)]{} \^[1,]{}\_[(s)]{}cW\_Q\^[(1)]{} \^[1,]{}\_[L\^1(\_Q,H\^s(0,Q))]{}c\^[1,]{}\_[L\^1(\_Q,H\^s(0,Q))]{}c\^[1,]{}\_[(s)]{}, where $c=c(\|\tfrac{d^{\lceil s \rceil}}{dx^{\lceil s \rceil}}W_Q^{(1)}\|_{L^2(\B_Q)})>0$. Hence, by Lemma \[L:ext\_est\], $$\|\eps M^{(1)}(\cdot) \vec{U}^{1,\text{ext}}(\cdot,t)\|_{\cX(s)}\leq c\eps^{5/2} \quad \text{for all }t\in [0,T_0\eps^{-1}].$$ Similarly \[E:MR\_U1ext\] \_[m\_R]{}M\^[(R,m)]{}() \^[1,]{}(-mk\_W,t)\_[(s)]{}&c \_[m\_R]{}|a\_m|\^[1,]{}(,-mk\_W,t)\_[L\^1(\_Q,H\^s(0,Q))]{}\ &c \_[m\_R]{} \^[1,]{}(-mk\_W,t)\_[(s)]{}m\^[-2]{}\ &c\^[5/2]{} t where we have used the decay $|a_m|\leq cm^{-2}$ for $W\in C^2$, the identity $\|\vec{U}^{1,\text{ext}}(\cdot-mk_W,t)\|_{\cX(s)}=\|\vec{U}^{1,\text{ext}}(\cdot,t)\|_{\cX(s)}$ due to the $1/N$-periodicity of $\Uvec^{1,\text{ext}}$ in $k$, and Lemmas \[L:D\_isom\] and \[L:ext\_est\]. Finally, $$\begin{aligned} \|\vec{F}(\UvecOext,\vec{U}^{1,\text{ext}},\UvecOext)\|_{\cX(s)}&\leq c|\sigma| \|\util^{0,\text{ext}}*_{\B_Q}\overline{\util}^{1,\text{ext}}*_{\B_Q}\util^{0,\text{ext}}\|_{L^1(\B_Q,H^s(0,Q))} \\ & \leq c \|\util^{1,\text{ext}}\|_{L^1(\B_Q,H^s(0,Q))}\|\util^{0,\text{ext}}\|^2_{L^1(\B_Q,H^s(0,Q))}\\ &\leq c \|\vec{U}^{1,\text{ext}}\|_{\cX(s)} \|\UvecOext\|_{\cX(s)}^2 \leq c \eps^{5/2} \quad \text{for all }t\in [0,T_0\eps^{-1}] \end{aligned}$$ using the algebra property in Lemma \[L:algeb\_L1Hs\] for $s>1/2$ and, again, Lemma \[L:ext\_est\]. Other nonlinear terms are treated analogously. We conclude that there is $C_{\text{Res}}>0$ and $\eps_0>0$ such that $\|\vec{\Res}(\cdot,t)\|_{\cX(s)}\leq C_{\text{Res}}\eps^{5/2}$ for all $\eps\in(0,\eps_0)$ and all $t\in [0,T_0\eps^{-1}]$. Finally, we complete the proof of Theorem \[T:main\]. Writing the exact solution of as a sum of the extended ansatz $\Uvecext $ and the error $\Evec$, we have $\Uvec = \Uvecext + \Evec$. In this produces \[E:err\_eq\] \_t = -(k) +(,) with $$\begin{aligned} \vec{G}(\Uvecext,\Evec)=&\vec{\Res}+\vec{F}(\Uvec,\Uvec,\Uvec) -\vec{F}(\Uvecext,\Uvecext,\Uvecext)\\ &-\varepsilon \left(M^{(1)}(k) \Evec + \sum \limits_{m \in \mathbb{Z}_R} M^{(R,m)}(k) \Evec(k-mk_W,t)\right). \end{aligned}$$ Due to the cubic structure of $\vec{F}$, Lemma \[L:ext\_est\], the algebra property of $L^1(\B_Q,H^s(0,Q))$ and the isomorphism $\cD$ we have for $s>1/2$ the existence of $c>0$ such that \[E:F\_est\] (,,) -(,,)\_[(s)]{} c(\^2\_[(s)]{}\_[(s)]{}+\_[(s)]{}\^2\_[(s)]{}+\^3\_[(s)]{}). Similarly to and we get \[E:M\_est\] ( M\^[(1)]{}() (,t) + \_[m \_R]{} M\^[(R,m)]{}() (-mk\_W,t) )\_[(s)]{} c (, t)\_[(s)]{}. Next, because $\Uvecext=\UvecOext+\vec{U}^{1,\text{ext}}$, Lemma \[L:ext\_est\] implies \[E:Uext\_est\] (,t)\_[(s)]{}c\^[1/2]{} t. Combining , , and Lemma \[L:res\_est\] provides the estimate $$\|\vec{G}(\Uvecext,\Evec)\|_{\cX(s)} \leq c_1\eps \|\Evec\|_{\cX(s)}+c_2\eps^{1/2} \|\Evec\|_{\cX(s)}^2+c_3\|\Evec\|_{\cX(s)}^3+C_{\text{Res}}\eps^{5/2}$$ on the time interval $[0,\eps^{-1}T_0]$ with some $c_1,c_2,c_3>0$ independent of $\eps$ and $t$. The operator $-\ri\Theta(k)$ generates a strongly continuous unitary group $S(t)=e^{-\ri \Theta t}: \cX(s) \rightarrow \cX(s)$ and equation reads $$\Evec(t) = \Evec(0)+\int \limits_0^t S(t-\tau)\vec{G}(\Uvecext,\Evec)(\tau) ~d\tau.$$ With the above estimates we get $$\begin{aligned} \| \Evec(t) \|_{\cX(s)} \leq \| \Evec(0) \|_{\cX(s)}+\int \limits^t_0 c_1 \varepsilon \| \Evec(\tau) \|_{\cX(s)} + c_2 \varepsilon^{1/2} \| \Evec(\tau) \|^2_{\cX(s)} +c_3 \| \Evec(\tau) \|^3_{\cX(s)} + C_{\text{Res}} \varepsilon^{5/2} ~d\tau. \end{aligned}$$ Because $\Uvec(0)=\Uvec^\text{app}(0)$, we get $\Evec(0)=\Uvec^\text{app}(0)-\Uvecext(0)$ and Lemma \[L:diff\_app\_ext\] provides $\|\Evec(0)\|_{\cX(s)}\leq C_0\eps^{3/2}$ for all $s\in (0,3/2)$ and $\eps\in (0,\eps_0)$ with some $\eps_0>0$. Given an $M>C_0$ there exists $T>0$ such that $\|\Evec(t)\|_{\cX(s)}\leq M \eps^{3/2}$ for all $t\in [0,T]$. Next, we use Gronwall’s lemma and a bootstrapping argument to choose $\eps_0>0$ and $M>0$ such that if $\eps\in (0,\eps_0)$, then $\|\Evec(t)\|_{\cX(s)}\leq M \eps^{3/2}$ for all $t\in [0,\eps^{-1}T_0]$. If $\|\Evec(t)\|_{\cX(s)} \leq M\eps^{3/2}$, then $$\begin{aligned} \| \Evec(t) \|_{\cX(s)} & \leq ~C_0 \eps^{3/2}+\int \limits^t_0 c_1 \varepsilon \| \Evec(\tau) \|_{\cX(s)} ~d\tau + t\left( c_2 \varepsilon^{7/2} M^2 +c_3 \varepsilon^{9/2} M^3 + C_{\text{Res}} \varepsilon^{5/2} \right)\\ & \leq ~ \varepsilon^{3/2} \left[ C_0 + t \eps\left( c_2 \varepsilon M^2 + c_3 \varepsilon^{2} M^3 + C_{\text{Res}} \right) \right] e^{c_1 \varepsilon t}, \end{aligned}$$ where the second inequality follows from Gronwall’s lemma. In order to achieve the desired estimate on $t\in [0,\eps^{-1}T_0]$, we redefine $$M:=C_0+T_0(C_\text{Res}+1)e^{c_1T_0}$$ and choose $\eps_0$ so small that $ c_2 \varepsilon_0 M^2 + c_3 \varepsilon_0^{2} M^3 \leq 1$. Then, clearly, $$\sup \limits_{t \in [0,\eps^{-1}T_0]} \| \Evec(t) \|_{\cX(s)} \leq M \varepsilon^{3/2}.$$ Using Lemma \[L:diff\_app\_ext\] and the triangle inequality produces $\sup_{t\in[0,\eps^{-1}T_0]}\|\Uvec(\cdot,t)-\Uvecapp(\cdot,t)\|_{\cX(s)} \leq c \eps^{3/2}$. The estimate in the $C^0_b$-norm and the decay for $|x|\to \infty$ follow from Lemmas \[L:sup\_control\] and \[L:D\_isom\] since $s>1/2$. This completes the proof of Theorem \[T:main\] for the case of $k_\pm \in \Q$. $\square$ \[L:diff\_app\_ext\] Assume (H1) and let $\Ahat_\pm(\cdot,T) \in L^1_{s_A}(\R)\cap L^2(\R)$ with $s_A\geq 2$ for all $T\in [0,T_0]$. Then there exist $c=c(\max_{T\in [0,T_0]}\|\Ahat_+(\cdot,T)\|_{L^1_{s_A}(\R)},\max_{T\in [0,T_0]}\|\Ahat_-(\cdot,T)\|_{L^1_{s_A}(\R)})$ and $\eps_0>0$ such that for $u_\text{app}$ and $\Uvecext$ given by and respectively we have $$\|\Uvecext(\cdot,t)-\Uvec^\text{app}(\cdot,t)\|_{\cX(s)}\leq c \eps^{3/2}$$ for all $s\in (0,3/2), \eps \in (0,\eps_0)$ and all $t\in [0,\eps^{-1}T_0]$. Because $\Ahat_\pm(\cdot,T) \in L^2(\R)$, it is also $A_\pm(\cdot,T)\in L^2(\R)$ and hence $u^\text{app}(\cdot,t)\in L^2(\R)$ and one can apply $\cD\cT$ to $u^\text{app}$ producing $\Uvec^\text{app}$ with $$U^\text{app}_n(k,t)=\eps^{-1/2}e^{-\ri \omega_0 t} \sum_\pm \Ahat_\pm\left(\tfrac{k}{\eps},\eps t\right)\pi_n^\pm(k), \quad \text{where } \pi_n^\pm(k):=\langle q_\pm(\cdot), q_n(\cdot,k)\rangle_Q.$$ Similarly to the decomposition of $\Uvecext$ in and we write $$\Uvecapp=\UvecOapp+\vec{U}^{1,\text{app}},$$ where $$\UvecOapp(k,t):=\eps^{-1/2}e^{-\ri \omega_0 t}\left(\Ahat_{+}(K,T)\pi_{n_*}^+(k)e_{n_*}+\Ahat_{-}(K,T)\pi_{n_*+1}^-(k)e_{n_*+1}\right), \ K=\eps^{-1}k.$$ Since $\|\vec{U}^{1,\text{ext}}(\cdot,t)\|_{\cX(s)} \leq c\eps^{3/2}$ for all $t\in [0,\eps^{-1}T_0]$ (see Lemma \[L:ext\_est\]), it remains to show that $\|\vec{U}^{1,\text{app}}(\cdot,t)\|_{\cX(s)} \leq c\eps^{3/2}$ and $\|(\UvecOext-\UvecOapp)(\cdot,t)\|_{\cX(s)}\leq c\eps^{3/2}$ for all $t\in [0,\eps^{-1}T_0]$. For $\|\vec{U}^{1,\text{app}}(\cdot,t)\|_{\cX(s)}$ note first that $$\begin{aligned} &U^{1,\text{app}}_{n_*}(k,t)=\eps^{-1/2}e^{-\ri \omega_0 t} \Ahat_-(K,T)\pi^-_{n_*}(k), \ U^{1,\text{app}}_{n_*+1}(k,t)=\eps^{-1/2}e^{-\ri \omega_0 t} \Ahat_+(K,T)\pi^+_{n_*+1}(k),\\ &U^{1,\text{app}}_{n}(k,t)=\eps^{-1/2}e^{-\ri \omega_0 t} \sum_\pm \Ahat_\pm(K,T)\pi^\pm_{n}(k) \quad \text{for } n \notin \{n_*,n_*+1\}. \end{aligned}$$ Using the Lipschitz continuity in (H1), there is some $L>0$, such that, for instance, $|\pi^-_{n_*}(k)|\leq |\langle q_-(\cdot),q_{n_*}(\cdot,0)\rangle_Q|+L |k|=L|k|$ because $q_-(x)=q_{n_*+1}(x,0)$. Hence $$\|U^{1,\text{app}}_{n_*}(\cdot,t)\|_{L^1(\B_Q)}\leq L\eps^{3/2} \int_\R |\Ahat_-(K,T)||K| dK \leq c \eps^{3/2} \|\Ahat_-(\cdot,T)\|_{L^1_1(\R)}.$$ Similarly, $\|U^{1,\text{app}}_{n_*+1}(\cdot,t)\|_{L^1(\B_Q)}\leq c\eps^{3/2} \|\Ahat_+(\cdot,T)\|_{L^1_1(\R)}$. Because $$\begin{aligned} \|\vec{U}^{1,\text{app}}(\cdot,t)\|_{\cX(s)} \leq & c\left(\sum_{n\in \{n_*,n_*+1\}}\|U^{1,\text{app}}_{n}(\cdot,t)\|_{L^1(\B_Q)} \phantom{\left\|\left(\sum_{n\in \N\setminus\{n_*,n_*+1\}}n^{2s}|U^{1,\text{app}}_n(\cdot,t)|^2\right)^{1/2}\right\|_{L^1(\B_Q)}}\right.\\ & \left. + \left\|\left(\sum_{n\in \N\setminus\{n_*,n_*+1\}}n^{2s}|U^{1,\text{app}}_n(\cdot,t)|^2\right)^{1/2}\right\|_{L^1(\B_Q)}\right), \end{aligned}$$ it remains to consider $|U^{1,\text{app}}_n|$ for $n\in \N\setminus\{n_*,n_*+1\}$. For $\pi^+_n(k)$ we have $$\begin{aligned} \pi^+_n(k)&=\langle q_{n_*}(\cdot,k),q_n(\cdot,k)\rangle_Q+\langle q_{n_*}(\cdot,0)-q_{n_*}(\cdot,k),q_n(\cdot,k)\rangle_Q =\langle q_{n_*}(\cdot,0)-q_{n_*}(\cdot,k),q_n(\cdot,k)\rangle_Q \notag\\ &=\frac{1}{\vartheta_n(k)}\langle \cL(\cdot,k)(q_{n_*}(\cdot,0)-q_{n_*}(\cdot,k)),q_n(\cdot,k)\rangle_Q.\label{E:pi+_L}\end{aligned}$$ The Lipschitz continuity in (H1) now provides $$\|\cL(\cdot,k)(q_{n_*}(\cdot,0)-q_{n_*}(\cdot,k))\|_{L^2(0,Q)}\leq \|q_{n_*}(\cdot,0)-q_{n_*}(\cdot,k)\|_{H^2(0,Q)}\leq L|k|$$ such that (using ) $$|\pi^+_n(k)|\leq \frac{c}{n^2}|k| \quad \text{for all } n\in \N \ \text{and } k\in \B_Q$$ and similarly for $|\pi^-_n|$. As a result $\left\|\left(\sum_{n\in \N\setminus\{n_*,n_*+1\}}n^{2s}|U^{1,\text{app}}_n(\cdot,t)|^2\right)^{1/2}\right\|_{L^1(\B_Q)} $ can be estimated by $$c\eps^{1/2}\left(\sum_{n\in \N}n^{2s-4}\right)^{1/2}\sum_\pm\int_{\B_Q}|\Ahat_\pm\left(\tfrac{k}{\eps},T\right)||\tfrac{k}{\eps}|dk \leq c\eps^{3/2}\sum_\pm \|\Ahat_\pm(\cdot,T)\|_{L^1_1(\R)},$$ where the last inequality uses $s<3/2$. In summary $$\|\vec{U}^{1,\text{app}}(\cdot,t)\|_{\cX(s)} \leq c \eps^{3/2} \sum_\pm \|\Ahat_\pm(\cdot,T)\|_{L^1_1(\R)}.$$ For $\|(\UvecOext-\UvecOapp)(\cdot,t)\|_{\cX(s)}$ we have $$\begin{aligned} \|(U^{0,\text{ext}}_{n_*}-& U^{0,\text{app}}_{n_*})(\cdot,t)\|_{L^1(\B_Q)} \\ &\leq \eps^{-1/2}\left(\int_{-\eps^{1/2}}^{\eps^{1/2}}|\Ahat_+\left(\tfrac{k}{\eps},\eps t\right)||1-\pi_{n_*}^+(k)|dk+\int_{\B_Q\setminus (-\eps^{1/2},\eps^{1/2})}|\Ahat_+\left(\tfrac{k}{\eps},\eps t\right)||\pi_{n_*}^+(k)|dk\right). \end{aligned}$$ Once again, by the Lipschitz continuity it is $|1-\pi_{n_*}^+(k)|=|\langle q_+(\cdot),q_{n_*}(\cdot,0)-q_{n_*}(\cdot,k)\rangle_Q|\leq L|k|$. Clearly, also $|\pi_{n_*}^+(k)|\leq 1$ for all $k\in \B_Q$. Hence $$\begin{aligned} \|(U^{0,\text{ext}}_{n_*}-U^{0,\text{app}}_{n_*})(\cdot,t)\|_{L^1(\B_Q)}&\leq c\eps^{3/2}\|\Ahat_+(\cdot,\eps t)\|_{L^1_1(\R)} + c\eps^{1/2}\|\Ahat_+(\cdot,\eps t)\|_{L^1(\R\setminus (-\eps^{-1/2}, \eps^{-1/2}))}\\ &\leq c\eps^{3/2}\|\Ahat_+(\cdot,\eps t)\|_{L^1_1(\R)} + c\eps^{1/2+s_A/2}\|\Ahat_+(\cdot,\eps t)\|_{L^1_{s_A}(\R)}, \end{aligned}$$ where in the second step we have used $\|\Ahat_+(\cdot,\eps t)\|_{L^1(\R\setminus (-\eps^{-1/2}, \eps^{-1/2}))}\leq c\eps^{s_A/2}\|\Ahat_+(\cdot,\eps t)\|_{L^1_{s_A}(\R)}$, see . Similarly, one gets $$\|(U^{0,\text{ext}}_{n_*+1}-U^{0,\text{app}}_{n_*+1})(\cdot,t)\|_{L^1(\B_Q)}\leq c\eps^{3/2}\|\Ahat_-(\cdot,\eps t)\|_{L^1_1(\R)} + c\eps^{1/2+s_A/2}\|\Ahat_-(\cdot,\eps t)\|_{L^1_{s_A}(\R)}.$$ For $s_A\geq 2$ is $1/2+s_A/2\geq 3/2$ and the lemma is proved. Proof of Theorem \[T:main\] for Irrational $k_\pm$ {#S:irrational_pf} -------------------------------------------------- The method of proof in the case of irrational $k_\pm$ is the same as in the rational case. The main difference is in the choice of the extended ansatz $\Uvec^\text{ext}(k,t)$. Therefore, we concentrate on explaining the choice of $\Uvec^\text{ext}(k,t)$ and describe where the proof differs from that in Section \[S:rational\_pf\]. When $k_\pm \notin \Q$, then clearly case (a) in Section \[S:Bloch\] applies, i.e. we have simple Bloch eigenvalues at $k=k_+=k_0$ and $k=k_-=-k_0$ for some $k_0\in (0,1/2)\setminus \Q$. The Bloch eigenfunctions are $$p_\pm(x)=p_{n_0}(x,\pm k_0).$$ Because $k_0\notin \Q$, there is no common period of the Bloch waves $p_\pm(x)e^{\pm \ri k_0 x}$ and of $V$. Hence, we use the period $P=2\pi$ corresponding to $V$ as the working period with the corresponding Brillouin zone $\B_{2\pi}=(-1/2,1/2]$. Note that the effective coupled mode equations are now with $\beta=\gamma=0$. Similarly to the beginning of Section \[S:rational\_pf\], in order to motivate the choice of the extended ansatz for the approximate solution $\Uvec^\text{ext}$, we study first the residual of the formal approximate ansatz $u_\text{app}$ with $\text{supp}(\hat{A}_\pm(\cdot,T))\subset [-\eps^{-1/2},\eps^{-1/2}]$. We have $$\begin{aligned} \cT(\uapp)(x,k,t)=&\eps^{-1/2}\sum_{\pm}p_\pm(x)\sum_{\eta\in \Z}\hat{A}_\pm\left(\tfrac{k\mp k_0+\eta}{\eps},\eps t\right)e^{\ri(\eta x- \omega_0 t)}\\ =&\eps^{-1/2}\sum_{\pm}p_\pm(x)\hat{A}_\pm\left(\tfrac{k\mp k_0}{\eps},\eps t\right)e^{-\ri \omega_0 t}, \quad k \in \B_{2\pi} \end{aligned}$$ because $\eps^{-1}(k\mp k_0+\eta) \in \text{supp}(\hat{A}_\pm(\cdot,T))$ for some $k\in \B_Q$ and $\eta\in \Z$ and with $k_0 \in (0,1/2)$ is possible only if $\eta=0$. Hence, if $\text{supp}(\hat{A}_\pm(\cdot,T))\subset [-\eps^{-1/2},\eps^{-1/2}]$, then for $k\in\B_{2\pi}$ $$\begin{aligned} &\cT(\text{PNLS}(\uapp))(x,k,t)= \eps^{1/2}\sum_{\pm}\left[\ri \pa_T\hat{A}_\pm\left(\tfrac{k\mp k_0}{\eps},T\right)p_\pm(x)+2(\ri k_0p_\pm +p_\pm')\ri \tfrac{k\mp k_0}{\eps}\hat{A}_\pm\left(\tfrac{k\mp k_0}{\eps},T\right)\phantom{\sum_{m\in \Z_3^\pm,\eta \in \cS_m^\pm}}\right.\\ &\left.-W^{(1)}(x)p_\pm(x)\hat{A}_\pm\left(\tfrac{k\mp k_0}{\eps},T\right)-W^{(2)}_\pm p_\pm(x)\hat{A}_\pm\left(\tfrac{k\pm k_0}{\eps},T\right)\right.\\ &\left.- p_\pm(x)\sum_{m\in \Z_3^\pm,\eta \in \cS_m^\pm} a_m\hat{A}_\pm\left(\frac{k\mp k_0-mk_W+\eta}{\eps},T\right)e^{\ri \eta x}\right]e^{-\ri \omega_0 t}\\ & -\eps^{1/2} e^{-\ri \omega_0 t}\sigma(x) \sum_{\stackrel{\xi,\zeta,\theta\in \{+,-\}}{\eta\in \cS_{\xi,\zeta,\theta}}}p_{\xi}(x)\overline{p_{\zeta}}(x)p_{\theta}(x)\left(\hat{A}_{\xi}\ast \hat{\overline{A}}_{\zeta}\ast \hat{A}_{\theta}\right)\left(\tfrac{k-(\xi k_0-\zeta k_0+\theta k_0) +\eta}{\eps},T\right)e^{\ri \eta x}\\ & - \eps^{3/2}e^{-\ri \omega_0 t}\sum_{\pm}p_\pm(x)\left(\tfrac{k\mp k_0}{\eps}\right)^2\hat{A}_\pm\left(\tfrac{k\mp k_0}{\eps},T\right), \end{aligned}$$ where $$\begin{aligned} &T = \eps t, \quad \Z_3^\pm := \{m\in \Z\setminus\{0\}: a_m\neq 0, mk_W\notin \Z, mk_W\pm 2k_0 \notin \Z\},\\ & \cS_m^\pm:=\{\eta\in\Z: \pm k_0 +mk_W-\eta\in \overline{\B_{2\pi}}=[-\tfrac{1}{2},\tfrac{1}{2}]\}, \\ &\cS_{\xi,\zeta,\theta}:=\{\eta\in \Z: \xi k_0-\zeta k_0+\theta k_0-\eta\in \overline{\B_{2\pi}}\} \ \text{for} \ \xi,\zeta,\theta\in\{+,-\}. \end{aligned}$$ It is $\cS_{\xi,\zeta,\theta}\subset \{0,1\}$ or $\cS_{\xi,\zeta,\theta}\subset \{0,-1\}$. In the $k-$variable the support of $\cT(\text{PNLS}(\uapp))$ within $\B_{2\pi}$ consists of intervals (with radius at most $3\eps^{1/2}$) centered at $k_0,-k_0, k_0+mk_W$ with $m\in \Z_3^{+}$, $-k_0+mk_W$ with $m\in \Z_3^{-}$, and at $3k_0$ and $-3k_0$ as well as at integer shifts of these points. Note that within $\B_{2\pi}$ there are only finitely many support intervals because $\pm k_0+mk_W+\Z, m\in \Z_3^\pm$ generates only finitely many distinct points in $\B_{2\pi}$ due to assumption (H2). Similarly to we define these sets of points by $$\cK_R^\pm:=(\pm k_0+k_W+\Z_3^\pm + \Z)\cap \overline{\B_{2\pi}}=\{k\in \overline{\B_{2\pi}}: k=\pm k_0+mk_W+\eta \text{ for some } m\in \Z_3^\pm, \eta\in \cS_m^\pm\}$$ and their elements by $$\cK_R^\pm=\{\kappa^\pm_1,\dots, \kappa^\pm_{J^\pm}\} \text{ with some } J^\pm\in \N.$$ The choice of the splitting in $W$ using $W^{(2)}_+$ and $W^{(3)}_+$ or $W^{(2)}_-$ and $W^{(3)}_-$ is motivated at the beginning of Sec. \[S:formal\]. The choice is made in order to isolate the parts of $W$ responsible for the coupling of the two modes. Analogously to we are lead to the following extended ansatz for $k\in \B_{2\pi}$ \[E:Uext\_rat\] &U\_[n\_0]{}\^(k,t) := e\^[-\_0 t]{}\_(\^[-1/2]{}\_[n\_0]{}\^(,T) +\^[1/2]{}\_[j=1]{}\^[J\^]{} \^\_[n\_0,j]{}(,T) )\ &+\^[1/2]{}e\^[-\_0 t]{} (\_[\_[+,-,+]{}]{}\^+\_[n\_0,NL]{}(,T)+\_[\_[-,+,-]{}]{}\^-\_[n\_0,NL]{}(,T)),\ &U\_[n]{}\^(k,t) := \^[1/2]{}e\^[-\_0 t]{}\_(\_[n]{}\^(,T) +\_[j=1]{}\^[J\^]{} \^\_[n,j]{}(,T) )\ &+\^[1/2]{}e\^[-\_0 t]{}(\_[\_[+,-,+]{}]{}\^+\_[n,NL]{}(,T)+\_[\_[-,+,-]{}]{}\^-\_[n,NL]{}(,T)) for $n\in \N\setminus\{n_0\}$, where $$\begin{aligned} &\supp (\Atil_{n_0}^\pm(\cdot, T)) \cap \eps^{-1}\B_{2\pi}, \quad \supp (\Atil_{n,j}^\pm(\cdot, T)) \cap \eps^{-1}\B_{2\pi} \subset [-\eps^{-1/2},\eps^{-1/2}],\\ &\supp (\Atil_{m}^\pm(\cdot, T)) \cap \eps^{-1}\B_{2\pi}, \quad \supp (\Atil_{n,NL}^\pm(\cdot,T)) \cap \eps^{-1}\B_{2\pi} \subset [-3\eps^{-1/2},3\eps^{-1/2}] \end{aligned}$$ for all $n\in \N, m\in \N\setminus\{n_0\}$ and $j\in \{1,\dots,J^\pm\}$ and where $$\Uvecext\left(k+1,t\right)=\Uvecext(k,t) \ \text{for all} \ k\in \R, t \in \R.$$ Similarly to , we decompose $\Uvecext = \vec{U}^{0,\text{ext}}+\vec{U}^{1,\text{ext}}$, where \[E:Uext\_decomp2\_rat\] &\^[0,]{}=\_+\^[0,]{}+\_-\^[0,]{},\ &\^[0,]{}\_+:= \^[-1/2]{}e\_[n\_0]{}\^+\_[n\_0]{}(,T)e\^[-\_0 t]{}, \^[0,]{}\_-:=\^[-1/2]{}e\_[n\_0]{}\^-\_[n\_0]{}(,T)e\^[-\_0 t]{},\ &\^[1,]{}:=-\^[0,]{}. In analogy to we get \[E:PNLS\_U\_rat\] (\_t -(k)-M’\^[(1)]{}(k))(k,t) -\_[mk\_W]{}M’\^[(R,m)]{}(k) (k-mk\_W,t)+(,,)(k,t)=0, where $$\begin{aligned} &\Omega_{j,j}(k):=\omega_j(k), \ \Omega_{i,j}:=0 \text{ if } i\neq j,\\ &M'^{(1)}_{i,j}(k):=\langle W^{(1)}(\cdot)p_j(\cdot,k), p_i(\cdot,k)\rangle_{2\pi}, \quad M'^{(R,m)}_{i,j}(k):=a_m\langle p_j(\cdot,k-mk_W), p_i(\cdot,k)\rangle_{2\pi},\\ &F'_j(\Uvec,\Uvec,\Uvec)(k,t) :=-\langle\sigma(\cdot) (\util\ast_{\B_{2\pi}}\tilde{\overline{u}}\ast_{\B_{2\pi}}\util)(\cdot,k,t),p_j(\cdot,k)\rangle_{2\pi}, \ \util(x,k,t)=\sum_{n\in \N}U_n(k,t)p_n(x,k). \end{aligned}$$ When studying the residual near $k=k_0$ or $k=-k_0$, we exploit both ways of splitting $W(x)$ in . Namely, we have the following two equivalent reformulations of \[E:PNLS\_U\_split1\] &(\_t -(k)-M’\^[(1)]{}(k))(k,t) -M\^[(2\_+)]{}(k)(k+2k\_0,t)\ &-\_[m \_3\^+]{}M’\^[(R,m)]{}(k) (k-mk\_W,t)+(,,)(k,t)=0 and \[E:PNLS\_U\_split2\] &(\_t -(k)-M’\^[(1)]{}(k))(k,t) -M\^[(2\_-)]{}(k)(k-2k\_0,t)\ &-\_[m \_3\^-]{}M’\^[(R,m)]{}(k) (k-mk\_W,t)+(,,)(k,t)=0, where $$\begin{aligned} &M^{(2_\pm)}_{i,j}(k):=\langle W^{(2)}_\pm p_j(\cdot,k\pm 2k_0), p_i(\cdot,k)\rangle_{2\pi}. \end{aligned}$$ As explained in Sec. \[S:formal\], the splitting of $W$ with $W^{2_-}$ will be used near $k=k_0$ because it extracts the part of $W$ which shifts $\Uvec^\text{ext}(k,t)$ in $k$ by $2k_0$ to the right and thus produces the linear $\Atil_{n_0}^-$-term in the residual near $k=k_0$. Similarly, the splitting with $W^{2_+}$ will be used near $k=-k_0$. The residual of $\Uvec^\text{ext}$ on $k\in (\pm k_0-3\eps^{1/2},\pm k_0+3\eps^{1/2})$ is given by $$\begin{aligned} &\Res_{n_0}(k,t)=\\ &\eps^{1/2}\left[\left(\ri \pa_T -\eps^{-1}(\omega_{n_0}(k)-\omega_0)-M'^{(1)}_{n_0,n_0}(k)\right)\Atil^\pm_{n_0}\left(\tfrac{k\mp k_0}{\eps},T\right) - M^{(2_\mp)}_{n_0,n_0}(k)\Atil^\mp_{n_0}\left(\tfrac{k\mp k_0}{\eps},T\right) \right]e^{-\ri \omega_0 t}\\ &+ F'_{n_0}(\UvecOext_\pm,\UvecOext_\pm,\UvecOext_\pm)(k,t)+2F'_{n_0}(\UvecOext_\mp,\UvecOext_\mp,\UvecOext_\pm)(k,t) + \text{h.o.t.} \end{aligned}$$ respectively and for $n\in \N\setminus \{n_0\}$ by $$\begin{aligned} &\Res_{n}(k,t)= \\ &\eps^{1/2}\left[(\omega_0-\omega_{n}(k))\Atil^\pm_{n}\left(\tfrac{k\mp k_0}{\eps},T\right) -M'^{(1)}_{n,n_0}(k)\Atil^\pm_{n_0}\left(\tfrac{k\mp k_0}{\eps},T\right) - M^{(2_\mp)}_{n,n_0}(k)\Atil^\mp_{n_0}\left(\tfrac{k\mp k_0}{\eps},T\right) \right]e^{-\ri \omega_0 t}\\ &+F'_{n}(\UvecOext_\pm,\UvecOext_\pm,\UvecOext_\pm)(k,t)+2F'_{n}(\UvecOext_\mp,\UvecOext_\mp,\UvecOext_\pm)(k,t) + \text{h.o.t.}. \end{aligned}$$ Note that no $M^{R',m}(k)$-terms (with $m\in \Z_3^\pm$) appear because these are not supported near $\pm k_0$. For $k\in (\kappa_j^\pm-\eps^{1/2},\kappa_j^\pm+\eps^{1/2})\cap\B_{2\pi}$ with $j\in \{1,\dots,J^\pm\}$ respectively we use the residual as given by the left hand side of and get $$\begin{aligned} &\Res_{n}(k,t)= \eps^{1/2}\left[(\omega_0-\omega_{n}(k))\Atil^\pm_{n,j}\left(\tfrac{k-\kappa_j^\pm}{\eps},T\right) - \sum_{\stackrel{m\in \Z_3^\pm}{\pm k_0+mk_W \in \kappa_j^\pm+\Z}}M'^{(R,m)}_{n,n_0}(k)\Atil^\pm_{n_0}\left(\tfrac{k-\kappa_j^\pm}{\eps},T\right) \right.\\ &\left. - \sum_{\stackrel{m\in \Z_3^\mp}{\mp k_0+mk_W \in \kappa_j^\pm+\Z}}M'^{(R,m)}_{n,n_0}(k)\Atil^\mp_{n_0}\left(\tfrac{k-\kappa_j^\pm}{\eps},T\right) \right]e^{-\ri \omega_0 t} + \text{h.o.t.}, \quad n \in \N. \end{aligned}$$ When $\kappa_j^+\in \pm 3k_0+\Z$ for some $\kappa_j^+\in \cK_R^+$ or $\kappa_j^-\in \pm 3k_0+\Z$ for some $\kappa_j^-\in \cK_R^-$, then also nonlinear terms $F'_n$ appear in this part of the residual. We treat, however, the neighborhoods of $\pm 3k_0$ separately below. Hence, all terms in the residual are accounted for. Finally, we consider the residual for $k\in (\pm 3k_0-\eta-3\eps^{1/2},\pm 3k_0-\eta+3\eps^{1/2})\cap\B_{2\pi}$ with $\eta\in \cS_{\pm,\mp,\pm}$ respectively. Here $$\begin{aligned} \Res_{n}(k,t)= &\eps^{1/2}e^{-\ri \omega_0 t}(\omega_0-\omega_{n}(k))\Atil^\pm_{n,NL}\left(\tfrac{k\mp 3k_0+\eta}{\eps},T\right) +F'_n(\UvecOext_\pm,\UvecOext_\mp,\UvecOext_\pm)(k,t)\\ & + \text{h.o.t.}, \quad n \in \N. \end{aligned}$$ In all other neighborhoods of its support the residual is of higher order in $\eps$, i.e. falls into the “h.o.t.” part. Similarly to the “h.o.t.” part consists of the following terms \[E:hot\_rat\] &\_T \^[1,]{}, M’\^[(1)]{}\^[1,]{}, \_[mk\_W]{}M’\^[(R,m)]{}\^[1,]{}(-mk\_W,t),\ & ’(,\^[1,]{},)+2’(,,\^[1,]{}), and nonlinear terms quadratic or cubic in $\vec{U}^{1,\text{ext}}$. In analogy to ,, , and we make the residual small by choosing \[E:Atil\_rat\_def\] \^\_[n\_0]{}(K,T):=\_[\[-\^[-1/2]{},\^[-1/2]{}\]]{}(K)\_(K,T), where $(A_+,A_-)(X,T)$ is a solution of , \[E:Atil\_rat\_def2\] &\^\_[n]{}(K,T)\ &:=(\_0-\_n(k\_0+K))\^[-1]{} for all $n \in \N \setminus \{n_0\}$, \[E:Atilcor\_rat\_def\] \_[n,j]{}\^(K,T):=&(\_0-\_n(\_j\^+K))\^[-1]{}, and \[E:Atilcor\_ratNL\_def\] \_[n,NL]{}\^(K,T):=(\_0-\_n(3k\_0+K))\^[-1]{}\^[-1/2]{}e\^[\_0 t]{}F’\_n(\_,\_,\_)(3k\_0+K, t),  n. The estimate of the residual and the Gronwall argument are completely analogous to the rational case and the proofs are omitted. Lemmas \[L:ext\_est\] and \[L:res\_est\] hold again with , , , , and replaced by , , , , and . Also Lemma \[L:diff\_app\_ext\] holds in the irrational case - with and replaced by and respectively. But some notational changes are needed in the proof. We list them next. For $u^\text{app}$ with $k_\pm=\pm k_0, k_0\in(0,1/2)$ and $p_\pm(x)=p_{n_0}(x,\pm k_0)$ we have $$U^\text{app}_n(k,t)=\eps^{-1/2}e^{-\ri \omega_0 t} \sum_\pm \Ahat_\pm\left(\tfrac{k\mp k_0}{\eps},\eps t\right)\pi_n^\pm(k), \quad \text{where } \pi_n^\pm(k):=\langle p_\pm(\cdot), p_n(\cdot,k)\rangle_{2\pi}.$$ We decompose $$\Uvecapp=\UvecOapp+\vec{U}^{1,\text{app}},$$ where $$\UvecOapp(k,t):=e_{n_0}U^\text{app}_{n_0}(k,t).$$ Again, we need to show that $\|\vec{U}^{1,\text{app}}(\cdot,t)\|_{\cX(s)} \leq c\eps^{3/2}$ and $\|(\UvecOext-\UvecOapp)(\cdot,t)\|_{\cX(s)}\leq c\eps^{3/2}$ for all $t\in [0,\eps^{-1}T_0]$. We have for any $n\neq n_0$ $$\begin{aligned} \pi^\pm_n(k)&=\langle p_{n_0}(\cdot,k),p_n(\cdot,k)\rangle_{2\pi}+\langle p_{n_0}(\cdot,\pm k_0)-p_{n_0}(\cdot,k),p_n(\cdot,k)\rangle_{2\pi} \notag\\ &=\frac{1}{\omega_n(k)}\langle \cL(\cdot,k)(p_{n_0}(\cdot,\pm k_0)-p_{n_0}(\cdot,k)),p_n(\cdot,k)\rangle_{2\pi}\label{E:pi+-}\end{aligned}$$ and by the $H^2$-Lipschitz continuity (in $k$) of the Bloch waves and using (which holds also for $Q=2\pi$ with $\vartheta_n$ replaced by $\omega_n$) $$|\pi^\pm_n(k)|\leq \frac{c}{n^2}|k\mp k_0| \quad \text{for all } n\in \N \ \text{and } k\in \B_{2\pi}.$$ As a result (for $s<3/2$) $$\begin{aligned} &\|\vec{U}^{1,\text{app}}(\cdot,t)\|_{\cX(s)} = \left\|\left(\sum_{n\in \N\setminus\{n_0\}}n^{2s}|U^{1,\text{app}}_n(\cdot,t)|^2\right)^{1/2}\right\|_{L^1(\B_{2\pi})} \\ &\leq c\eps^{1/2}\left(\sum_{n\in \N}n^{2s-4}\right)^{1/2}\sum_\pm\int_{\B_{2\pi}}|\Ahat_\pm\left(\tfrac{k\mp k_0}{\eps},\eps t\right)||\tfrac{k\mp k_0}{\eps}|dk \leq c\eps^{3/2}\sum_\pm \|\Ahat_\pm(\cdot,\eps t)\|_{L^1_1(\R)}. \end{aligned}$$ For $\|(\UvecOext-\UvecOapp)(\cdot,t)\|_{\cX(s)}$ we have $$\begin{aligned} \|(U^{0,\text{ext}}_{n_0}-& U^{0,\text{app}}_{n_0})(\cdot,t)\|_{L^1(\B_{2\pi})} \\ &\leq \eps^{-1/2}\sum_\pm\left(\int_{\pm k_0-\eps^{1/2}}^{\pm k_0+\eps^{1/2}}|\Ahat_\pm\left(\tfrac{k\mp k_0}{\eps},\eps t\right)||1-\pi_{n_0}^\pm(k)|dk\right.\\ & \left. +\int_{\B_{2\pi}\setminus (\pm k_0-\eps^{1/2},\pm k_0+\eps^{1/2})}|\Ahat_\pm\left(\tfrac{k\mp k_0}{\eps},\eps t\right)||\pi_{n_0}^\pm(k)|dk\right). \end{aligned}$$ By the Lipschitz continuity it is $|1-\pi_{n_0}^\pm(k)|\leq L|k\mp k_0|$ and like in $$\|\Ahat_\pm(\cdot,\eps t)\|_{L^1(\R\setminus (-\eps^{-1/2}, \eps^{-1/2}))}\leq c\eps^{s_A/2}\|\Ahat_\pm(\cdot,\eps t)\|_{L^1_{s_A}(\R)}.$$ Hence $$\begin{aligned} \|(\UvecOext-\UvecOapp)(\cdot,t)\|_{\cX(s)}&\leq c\eps^{3/2}\sum_\pm\|\Ahat_\pm(\cdot,\eps t)\|_{L^1_1(\R)} + c\eps^{1/2}\sum_\pm\|\Ahat_\pm(\cdot,\eps t)\|_{L^1(\R\setminus (-\eps^{-1/2}, \eps^{-1/2}))}\\ &\leq c\eps^{3/2}\sum_\pm\|\Ahat_\pm(\cdot,\eps t)\|_{L^1_1(\R)} + c\eps^{1/2+s_A/2}\sum_\pm\|\Ahat_\pm(\cdot,\eps t)\|_{L^1_{s_A}(\R)}. \end{aligned}$$ Discussion {#S:discuss} ========== In fact, we have proved slightly more than the supremum norm estimate of the error. Namely, our proof estimates $\|u(\cdot,t)-u_\text{app}(\cdot,t)\|_{H^s(\R)}$ for all $s\in (1/2,3/2)$. In order to provide an estimate in $H^s$ for $s\geq 3/2$, the $l^2_s$-summability of $\Uvec^{1,\text{ext}}$ and $\Uvec^{1,\text{app}}$ has to be ensured. For this one would need in the rational case a sufficient decay of $M^{(1)}_{n,j}, M^{(R,m)}_{n,j}$ and $b^{(n)}_{\alpha,\beta,\gamma}$ as $n\to \infty$ instead of the simple estimates and . This can be achieved by replacing $q_n(x,k)$ in the inner products by $(\vartheta_n(k))^{-r}\cL^r(x,k)q_n(x,k)$ with $r\in \N$ sufficiently large and moving the self-adjoint $\cL^r$ to the other argument of the inner product. This would require sufficient smoothness of $q_{n_*}$ and $q_{n_*+1}$ in $x$. Similarly, $\cL$ would be applied more times in to get a faster decay of $|\pi_n^+|$ (and similarly for $|\pi_n^-|$). This would require replacing $H^2$ by some smaller space $H^s, s >2$ in the Lipschitz assumption (H1). Analogous requirements would be needed in the irrational case on the respective matrices and vectors. Alternatively, the residual and the error can be estimated in $L^1(\B_Q,H^s(0,Q))$ instead of $\cX(s)$ in order to avoid the $l^2_s$-summability issue [@SU_book]. Because our result is sufficient to provide the physically relevant $C_b^0$ as well as $H^1$ bounds, we do not pursue the straightforward improvement to $s\geq 3/2$ here. Note also that the analysis can be easily carried over to other nonlinear equations with periodic coefficients. For instance, for the wave equation with periodic coefficients and the cubic nonlinearity $u^3$ as studied in [@BSTU06] the approach is completely analogous except for first rewriting the equation as a system of two first order equations. To obtain a real solution $u$ of the wave equation the ansatz is extended by adding the complex conjugate to . Nevertheless, this does not change the analysis as no other $k$-points are generated by $u^3$ applied to such ansatz compared to $|u|^2u$ applied to . Numerical Examples {#S:num} ================== We present numerical examples for both cases (a) and (b) from Section \[S:Bloch\]: in Section \[S:num\_a\] for case (a) with simple Bloch eigenvalues at $k=k_+$ and $k=k_-=-k_+, k_+\in(0,1/2)$ and in Section \[S:num\_b\] for case (b) with a double eigenvalue at $k=k_+=k_-\in \{0,\pi/P\}$. We solve by the Strang splitting method of second order in time, see e.g. [@WH86]. The equation is split into the part $\ri\pa_t u=-\pa_x^2u$, which is solved with a spectral accuracy in Fourier space, and into the ODE part $\ri\pa_t u =(V(x)+\eps W(x))u+\sigma(x)|u|^2u$, which is solved exactly: $u(x,t)=e^{\ri(V(x)+\eps W(x)+\sigma(x)|u_0(x)|^2)t}u_0(x)$ (for initial data $u(x,0)=u_0(x)$). We discretize with $dx=0.05$ and $dt=0.02$ and solve up to $t=2\eps^{-1} $ with the initial data $u(x,0)=u_\text{app}(x,0)$ for a range of values of $\eps$ in order to study the error convergence. The choice of a solution $(A_+,A_-)$ of the CMEs is specified in each case below. Case (a): simple eigenvalues at $k=k_\pm=\pm k_0, k_0\in (0,1/2)$ {#S:num_a} ----------------------------------------------------------------- We choose here $V(x)=2(\cos(x)+1),\sigma \equiv -1$ such that $P=2\pi$. The band structure is plotted in Fig. \[F:band\_Bloch\_cos\] (a). The carrier Bloch waves are given by the choice $k_0=0.2, n_0=2$ resulting in $\omega_0 \approx 2.645$. The Bloch function $p_+(x)=p_2(x,0.2)$ is plotted in Fig. \[F:band\_Bloch\_cos\] (b) and (c). Because $k_0\in(0,1/4)\cup(1/4,1/2)$, the resulting CMEs have $\beta=\gamma=0$ and hence reduce to the classical CMEs for envelopes of pulses in the nonlinear wave equation with an infinitesimal contrast periodicity [@AW89; @GWH01]. There is the following two-parameter family of explicit solitary waves [@AW89; @GWH01; @D14] $$\label{E:1D_gap_sol} \begin{split} A_+(X,T)&= \nu a e^{\ri\eta}\sqrt{\frac{|\kappa|}{2|\alpha|}}\sin(\delta)\Delta^{- 1}e^{\ri\nu\zeta} \text{sech}(\theta - \ri\nu\delta/2),\\ A_-(X,T)&= -a e^{\ri\eta}\sqrt{\frac{|\kappa|}{2|\alpha|}}\sin(\delta)\Delta e^{\ri\nu\zeta} \text{sech}(\theta +\ri\nu\delta/2), \end{split}$$ where $$\begin{aligned} \nu &=\text{sign}(\kappa \alpha), \quad a = \sqrt{\frac{2(1-v^2)}{3-v^2}}, \quad \Delta = \left(\frac{1-v}{1+v}\right)^{1/4}, \quad e^{\ri\eta} = \left(-\frac{e^{2\theta}+e^{-\ri\nu\delta}}{e^{2\theta}+e^{\ri\nu\delta}} \right)^{\frac{2v}{3-v^2}},\\ \theta & = \mu \kappa \sin(\delta)\left(\frac{X}{c_g}-vT\right), \quad \zeta = \mu \kappa \cos(\delta)\left(\frac{v}{c_g}X-T\right), \quad \mu = (1-v^2)^{-1/2}\end{aligned}$$ with the velocity $v\in (-1,1)$ and “detuning” $\delta \in [0,\pi]$. Next, we select two examples of the perturbations $\eps W$ of the periodic structure. ### {#S:W_cos_2k0x} For this $W$ the splitting in is given by $W^{(1)}\equiv 0, W^{(2)}_\pm \equiv \tfrac{1}{2}, W^{(3)}_\pm(x)=\tfrac{1}{2}e^{\pm 2\ri k_0 x}$. The resulting coefficients of the CMEs are \[E:CME\_coeffs\_ex\] c\_g -0.3341, 0.3826, \_s=0, 0.2509, ==0. We choose the solution with $\delta =\pi/2$ and $v=0.5$. The modulus of $A_+$ and $A_-$ as well as of the approximation $u_\text{app}$ for $\eps =0.01$ are plotted in Fig. \[F:GS\_profile\]. In Fig. \[F:eps\_conv\_Vcos\] (a) we show that the error convergence in Theorem \[T:main\] is confirmed by studying the error at $t=2\eps^{-1}$. The observed convergence rate is $\eps^{1.67}$. In order to demonstrate the approximation quality on a very large time interval Fig. \[F:eps\_conv\_Vcos\] (b) shows the numerical solution $|u(x,t)|$ for $\eps=0.01$ and $u(x,0)=u_\text{app}(x,0)$ at $t=5000=50\eps^{-1}$. The solitary wave shape is still well preserved. ### {#S:W_3terms} For $k_0=0.2$ the splitting in is $W^{(1)}(x)=\tfrac{1}{3}\cos(2x), W^{(2)}_\pm(x)=\tfrac{1}{2}, W^{(3)}_\pm(x)=\tfrac{1}{2}\cos(\tfrac{4}{5}x)$. The CME coefficients are given by except for $\kappa_s\approx 0.1324$, such that we can choose as $(A_+,A_-)$ the solutions of Sec. \[S:W\_cos\_2k0x\] multiplied by $e^{\ri \kappa_s T}$. Also here the convergence rate is confirmed, see Fig. \[F:eps\_conv\_Vcos\_W\_3terms\], where the observed convergence rate is $\eps^{1.61}$. Case (b): double eigenvalue at $k=k_+=k_-\in \{0,1/2\}$ {#S:num_b} ------------------------------------------------------- This case was previously studied numerically in [@D14]. We choose here the example 6.2 of [@D14], where $V$ is the finite band potential $V(x)=\text{sn}^2(x;1/2)$ and $\sigma \equiv -1$ such that $P\approx 3.7081$. The band structure from Fig. 2(a) in [@D14] is reproduced in Fig. \[F:band\_str\_FB\]. We choose the point $k_\pm=0, \omega_0\approx 3.428$. With $W(x)=\cos(\tfrac{4\pi}{P}x)$ we get a $P$-periodic perturbation of the potential $V$ such that setting $k_W=0$, we have in $W^{(1)}=W$ after adjusting the definition of $W^{(1)},W^{(2)}_\pm,W^{(3)}_\pm$ to the $P-$periodic case (i.e. replacing $nk_W\in \Z$ by $nk_W\in\frac{2\pi}{P}\Z$). The CME coefficients are given in Sec. 6.2 of [@D14]. Although $\beta$ and $\gamma$ are very small: $\beta\approx 6.5*10^{-4}, \gamma\approx 7*10^{-6}$, it is shown that the error convergence is suboptimal when $\beta$ and $\gamma$ are set to zero, see Sec. 6.1 in [@D14]. Solutions of CMEs for $\beta\neq 0$ or $\gamma\neq 0$ can be found by a numerical parameter continuation technique starting from the explicit CME solutions at $\beta=\gamma=0$, see [@D14]. The error convergence for the velocity $v=0.5$ and detuning $\delta =\pi/2$ is plotted in Fig. \[F:err\_conv\_FB\]. The observed rate $\eps^{1.39}$ is close to the predicted $\eps^{1.5}$. Note that in [@D14] the convergence of the error was studied in the $L^2$-norm and the rate $\eps^{0.91}$, i.e. approximately $1/2$-smaller than in $\|\cdot\|_{C^0_b}$, was observed. Although our result does not guarantee the convergence in $L^2$, this is expected because heuristically, the error should be of the form $\eps^{3/2}B(\eps x,\eps t)f(x,t)$ for some bounded function $f$ and an $L^2$-function $B$. Due to the scaling of the $L^2$-norm, we then get an $O(\eps^1)$ estimate of the $L^2$-error. Acknowledgments {#acknowledgments .unnumbered} =============== This research is supported by the *German Research Foundation*, DFG grant No. DO1467/3-1. The authors thank Guido Schneider for fruitful discussions. [^1]: The support $[-\eps^{-1/2},\eps^{-1/2}]$ has been chosen a-posteriori. If one assumes $\text{supp}(\hat{A}_\pm(\cdot,T))\subset [-\eps^{r-1},\eps^{r-1}]$ for some $r\in (0,1)$, it turns out in the proof of Lemma \[L:res\_est\] that $r\leq 1/2$ is optimal.

--- abstract: 'We construct density dependent Class $III$ charge symmetry violating (CSV) potential due to mixing of $\rho$-$\omega$ meson with off-shell corrections. Here in addition to the usual vacuum contribution, the matter induced mixing of $\rho$-$\omega$ is also included. It is observed that the contribution of density dependent CSV potential is comparable to that of the vacuum contribution.' address: 'Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Kolkata-700 064, INDIA' author: - Subhrajyoti Biswas - Pradip Roy - 'Abhee K. Dutt-Mazumder' title: Matter induced charge symmetry violating NN potential --- introduction ============ The exploration of symmetries and their breaking have always been an active and interesting area of research in nuclear physics. One of the well known examples, that can be cited here, is the nuclear $\beta$ decay which violates parity that led to the discovery of the weak interaction. Our present concern, however, is the strong interaction where, in particular, we focus attention to the charge symmetry violation (CSV) in nucleon-nucleon ($NN$) interaction. Charge symmetry implies invariance of the $NN$ interaction under rotation in isospin space, which in nature, is violated. The CSV, at the fundamental level is caused by the finite mass difference between up $(u)$ and down $(d)$ quarks [@Nolen69; @Henley69; @Henley79; @Miller90; @Machleidt89; @Miller95]. As a consequence, at the hadronic level, charge symmetry (CS) is violated due to non-degenerate mass of hadrons of the same isospin multiplet. The general goal of the research in this area is to find small but observable effects of CSV which might provide significant insight into the strong interaction dynamics. There are several experimental data which indicate CSV in $NN$ interaction. For instance, the difference between $pp$ and $nn$ scattering lengths at $^1$S$_0$ state is non-zero [@Miller90; @Howell98; @Gonzalez99]. Other convincing evidence of CSV comes from the binding energy difference of mirror nuclei which is known as Okamoto-Nolen-Schifer (ONS) anomaly [@Nolen73; @Okamoto64; @Garcia92]. The modern manifestation of CSV includes difference of neutron-proton form factors, hadronic correction to $g-2$ [@Miller06], the observation of the decay of $\Psi^{\prime} (3686) \ra (J/\Psi) \p^0 $ etc [@Miller06]. In nuclear physics, one constructs CSV potential to see its consequences on various observables. The construction of CSV potential involves evaluation of the $NN$ scattering diagrams with intermediate states that include mixing of various isospin states like $\rho$-$\omega$ or $\pi$-$\eta$ mesons. The former is found to be most dominant [@Coon87; @Henley79; @McNamee75; @Coon77; @Blunden87; @Sidney87] which we consider here. Most of the calculations performed initially to construct CSV potential considered the on-shell [@Blunden87] or constant $\rho$-$\omega$ mixing amplitude [@Sidney87], which are claimed to be successful in explaining various CSV observables [@Sidney87; @Machleidt01]. This success has been called into question [@Cohen95; @Piekarewicz92] on the ground of the use of on-shell mixing amplitude for the construction of CSV potential. First in [@Piekarewicz92] and then in [@Goldman92; @Krein93; @Connell94; @Coon97; @Hatsuda94], it is shown that the $\rho$-$\omega$ mixing has strong momentum dependence which even changes its sign as one moves away from the $\rho$ (or $\omega$) pole to the space-like region which is relevant for the construction of the CSV potential. Therefore inclusion of off-shell corrections are necessary for the calculation of CSV potential. We here deal with such mixing amplitude induced by the $N$-$N$ loop incorporating off-shell corrections. In vacuum, the charge symmetry is broken explicitly due to the non-degenerate nucleon masses. In matter, there can be another source of symmetry breaking if the ground state contains unequal number of neutrons ($n$) and protons ($p$) giving rise to ground state induced mixing of various charged states like $\rho$-$\omega$ meson even in the limit $M_n=M_p$. This additional source of symmetry breaking for the construction of CSV potential has, to the best of our knowledge, not been considered before. The possibility of such matter induced mixing was first studied in [@Abhee97] and was subsequently studied in [@Broniowski98; @Abhee01; @Kampfer04; @Roy08]. For the case of $\pi$-$\eta$ meson also such asymmetry driven mixing is studied in [@Biswas06]. But none of these deal with the construction of two-body potential and the calculations are mostly confined to the time-like region where the main motivation is to investigate the role of such matter induced mixing on the dilepton spectrum observed in heavy ion collisions, pion form factor, meson dispersion relations etc. [@Broniowski98; @Roy08]. In Ref.[@Saito03], attempt has been made to calculate the density dependent CSV potential where only the effect of the scalar mean field on the nucleon mass is considered excluding the possibility of matter driven mixing. All existing matter induced mixing calculations, however, suggest that, at least in the $\rho$-$\omega$ sector, the inclusion of such a matter induced mixing amplitude into the two body $NN$ interaction potential can significantly change the results both qualitatively and quantitatively. It is also to be noted that such mixing amplitudes, in asymmetric nuclear matter (ANM), have non-zero contribution even if the quark or nucleon masses are taken to be equal [@Abhee97; @Broniowski98; @Abhee01; @Kampfer04; @Roy08]. We consider both of these mechanisms to construct the CSV potential. Physically, in dense system, intermediate mesons might be absorbed and re-emitted from the Fermi spheres. In symmetric nuclear matter (SNM) the emission and absorption involving different isospin states like $\rho$ and $\omega$ cancel when the contributions of both the proton and neutron Fermi spheres are added provided the nucleon masses are taken to be equal. In ANM, on the other hand, the unbalanced contributions coming from the scattering of neutron and proton Fermi spheres, lead to the mixing which depends both on the density $(\rho_B)$ and the asymmetry parameter $[\a = (\rho_n - \rho_p)/\rho_B] $. Inclusion of this process is depicted by the second diagram in Fig.\[fig00\] represented by $V^{NN}_{med}$ which is non-zero even in symmetric nuclear matter if explicit mass differences of nucleons are retained. In the first diagram, $V^{NN}_{vac}$ involves NN loop denoted by the circle. The other important element which we include here is the contribution coming from the external legs. This is another source of explicit symmetry violation which significantly modify the CSV potential in vacuum as has been shown only recently by the present authors [@Biswas08]. This paper is organized as follows. In Sec.II we present the formalism where the three momentum dependent $\rho^0$-$\omega$ mixing amplitude is calculated to construct the CSV potential in matter. The numerical results are discussed in Sec.III. Finally, we summarize in Sec.IV. formalism ========= We start with the following effective Lagrangians to describe $\o NN$ and $\rho NN$ interactions: \_[øNN]{} & = & [g\_ø]{}|\_\^\_ø, \[lag:omega\]\ \_[NN]{} & = & [g\_]{} | \^\_, \[lag:rho\] where $C_{\rho} = f_{\rho}/{\rm g_{\rho}}$ is the ratio of vector to tensor coupling, $M$ is the average nucleon mass and $\vec{\t}$ is the isospin operator. $\Psi$ and $\Phi$ represent the nucleon and meson fields, respectively, and ${\rm g}$’s stand for the meson-nucleon coupling constants. The tensor coupling of $\omega$ is not included in the present calculation as it is negligible compared to the vector coupling. The matrix element, which is required for the construction of CSV $NN$ potential is obtained from the relevant Feynman diagram [@Biswas08]: \^[NN]{}\_(q) &=& \[|[u]{}\_N(p\_3)\_\^(q) u\_N(p\_1)\]  \_\^(q) \^\_(q\^2)\ && \_\^(q)  \[|[u]{}\_N(p\_4)\_\^(-q) u\_N(p\_2)\]. \[ma0\] In the limit $q_0 \ra 0$, Eq.(\[ma0\]) gives the momentum space CSV $NN$ potential, $V^{NN}_{CSV}({\bf q})$. Here $\G^\m_\o(q) = {\rm g_\o} \g^\m $, $\G^\n_\rho (q) = {\rm g_\rho} \lt[\g^\n - \f{C_\rho}{2M}i\s^{\n\l}q_\l\rt]$ denote the vertex factors, $u_N$ is the Dirac spinor and $\Dt^i_{\m\n}(q)$, $(i = \rho, \omega)$ is the meson propagator. $p_j$ and $q$ are the four momenta of nucleon and meson, respectively. In the present calculation, $\rho$-$\omega$ mixing amplitude ([*i.e.*]{} polarization tensor) $\Pi^{\m\n}_{\rho\omega}(q^2)$ is generated by the difference between proton and neutron loop contributions: \^\_(q\^2) = \^[(p)]{}\_(q\^2) -\^[(n)]{}\_(q\^2). \[se\] Explicitly, the polarization tensor is given by &&i\^[(N)]{}\_(q\^2) = ,\ &&\[pol:total0\] where $k=(k_0,{\bf k})$ denotes the four momentum of the nucleon in the loops ([*see*]{} Fig.\[fig00\]) and $G_N$ is the in-medium nucleon propagator consisting of free $(G^F_N)$ and density dependent $(G^D_N)$ parts [@Serot86], \[nucl:prop\] G\^F\_N(k)&=&, \[nucl:prop\_vac\]\ G\^D\_N(k)&=&(k/+M\_N)(k\_0-E\_N)(k\_N-|[**k**]{}|) \[nucl:prop\_med\] The subscript $N$ stands for nucleon index ([*i.e.*]{} $N=p$ or $n$), $k_N$ denotes the Fermi momentum of nucleon, nucleon energy $E_N=\sqrt{M^2_N+k^2_N}$ and nucleon mass is denoted by $M_N$. $\theta(k_N-|{\bf k}|)$ is the Fermi distribution function at zero temperature. The origin of $G^D_N(k)$, in addition to the free propagator resides in the fact that here one deals with vacuum containing real particles which when acted upon the annihilation operator does not vanish ([*see* ]{} [**Appendix A**]{} for details). The appearance of the delta function in Eq.(\[nucl:prop\_med\]) indicates the nucleons are on-shell while $\theta(k_N-|{\bf k}|)$ ensures that propagating nucleons have momentum less than $k_N$. Likewise, the polarization tensor of Eq.(\[pol:total0\]) also contains a vacuum $[\Pi^{\m\n (N)}_{vac}(q^2)]$ and a density dependent $[\Pi^{\m\n (N)}_{med}(q^2)]$ parts as shown in Fig.\[fig00\]. It is to be noted that the density dependent part given by the combination of $G^F_NG^D_N+G^D_NG^F_N$ corresponds to scattering that we have discussed already, whereas the term proportional to $G^D_NG^D_N$ vanishes for low energy excitation [@Chin77]. The vacuum part, [*viz.*]{} $[\Pi^{\m\n (N)}_{vac}(q^2)]$ on the other hand involves $G_N^FG_N^F$ which gives rise to usual CSV part of the potential due to the splitting of the neutron and proton mass. It might be worthwhile to mention here that Eq.(\[nucl:prop\_med\]) can induce charge symmetry breaking in asymmetric nuclear matter due to the appearance of the Fermi distribution function in the propagator itself which can distinguish between neutron and proton, even if their mass are taken to be degenerate. Evidently, this is an exclusive medium driven effect where, as mentioned already in the introduction, the charge symmetry is broken by the ground state. The total charge symmetry breaking would involve both the contributions where, it is clear that even for $\alpha=0$, the medium dependent term can contribute if the non-degenerate nucleon masses are considered. Note that the polarization tensor $\Pi^{\m\n}_{\rho\omega}(q^2)$ can be expressed as the sum of longitudinal component $[\Pi^L_{\rho\omega}(q^2)]$ and transverse component $[\Pi^T_{\rho\omega}(q^2)]$ which will be useful to simplify the matrix element given in Eq.(\[ma0\]). \^\_(q\^2) = \^L\_(q\^2) A\^ + \^T\_(q\^2) B\^, where $A^{\m\n}$ and $B^{\m\n}$ are the longitudinal and transverse projection operators [@Kapusta]. We define $\Pi^L_{\rw}=-\Pi^{00}_{\rw} + \Pi^{33}_{\rw}$ and $\Pi^T_{\rw}=\Pi^{11}_{\rw}=\Pi^{22}_{\rw}$. We, in the present calculation, use the average of longitudinal and transverse components of the polarization tensor instead of $\Pi^L_{\rw}$ and $\Pi^T_{\rw}$. The average mixing amplitude is denoted by |(q\^2) &=&\ &=& |\_[vac]{}(q\^2) + |\_[med]{}(q\^2). \[pol:av0\] In the last line of Eq.(\[pol:av0\]), $\bar{\Pi}_{vac}(q^2)$ and $\bar{\Pi}_{med}(q^2)$ denote the average mixing amplitudes of vacuum and density dependent parts, respectively. To obtain $\bar{\Pi}_{vac}(q^2)$ and $\bar{\Pi}_{med}(q^2)$ one would calculate the total polarization tensor given in Eq.(\[pol:total0\]). After evaluating the trace of Eq.(\[pol:total0\]), we find the following vacuum and density dependent parts of the polarization tensor. \^[(N)]{}\_[vac]{}(q\^2) &=& Q\^, \[pol:vac0\] and \^[{vv(N)}]{}\_[med]{}(q\^2) &=& 16 (k\_N-|[**k**]{}|)\ && , \[pol:dens\_vv0\]\ \^[{tv(N)}]{}\_[med]{}(q\^2) &=& 4 C\_ (k\_N-|[**k**]{}|)\ && , \[pol:dens\_tv0\] where $Q^{\m\n} = (-g^{\m\n} + q^\m q^\n/q^2)$ and $K^{\m\n} = \lt(k^\m -(q\cdot k)\f{q^\m}{q^2}\rt)\lt(k^\n -(q\cdot k)\f{q^\n}{q^2}\rt)$. It is to be mentioned that both $\Pi^{\m\n}_{vac}(q^2)$ and $\Pi^{\m\n}_{med}(q^2)$ obey the current conservation as $q_\m Q^{\m\n} = q_\n Q^{\m\n} = 0$ and $q_\m K^{\m\n} = q_\n K^{\m\n} = 0$. The superscripts $vv$ and $tv$ in Eqs.(\[pol:vac0\]) -(\[pol:dens\_tv0\]) indicate the vector-vector and tensor-vector interactions, respectively. The dimensional counting shows that vacuum part of the polarization tensor \[both $\Pi^{vv(N)}_{vac}(q^2)$ and $\Pi^{tv(N)}_{vac}(q^2)$\] is ultraviolet divergent and dimensional regularization [@Hooft73; @Peskin95; @Cheng06] is used to isolate the divergent parts. Since the mixing amplitude is generated by the difference between the proton and neutron loop contributions, the divergent parts cancel out yielding the vacuum amplitude finite. \^\_[vac]{}(q\^2) &=& \^[(p)]{}\_[vac]{}(q\^2)-\^[(n)]{}\_[vac]{}(q\^2)\ &=&q\^2Q\^\^1\_0 dx\ && ( ). \[pol:vac1\] Eq.(\[pol:vac1\]) shows the four-momentum dependent vacuum polarization tensor. From the above equation one can calculate longitudinal $(\P^L_{vac})$ and transverse $(\P^T_{vac})$ components of the vacuum mixing amplitude and in the limit $q_0 \ra 0$, $\Pi^L_{vac}({\bf q}^2)= \Pi^T_{vac}({\bf q}^2)$. Therefore, the average vacuum mixing amplitude is |\_[vac]{}([**q**]{}\^2) &=&\ &=& - (2+3C\_)()[**q**]{}\^2\ & & - \^2. \[pol:vac3\] Eq.(\[pol:vac3\]) represents the three momentum dependent vacuum mixing amplitude. This mixing amplitude vanishes for $M_n = M_p$ and then no CSV potential in vacuum will exist. To calculate density dependent mixing amplitude from Eq.(\[pol:dens\_vv0\]) and (\[pol:dens\_tv0\]) we consider $E_N \approx M_N $. In the limit $q_0 \ra 0$, one finds following expressions: \^[00(N)]{}\_[med]{}([**q**]{}\^2) &=&- \[pol:dens\_00\]\ &&\ \^[11(N)]{}\_[med]{}([**q**]{}\^2) &=& \[pol:dens\_11\] Note that the terms within the first curly braces of both Eq.(\[pol:dens\_00\]) and Eq.(\[pol:dens\_11\]) arise from the vector-vector interaction while the terms within the second curly braces arise from tensor-vector interaction of the density dependent polarization tensor. The $33$ component of the density dependent polarization tensor vanishes [*i.e.*]{} $\Pi^{33(N)}_{med}({\bf q}^2) = 0$. Now |\_[med]{}([**q**]{}\^2) = , where \^L\_[med]{}([**q**]{}\^2) &=& - ,\ \^T\_[med]{}([**q**]{}\^2) &=& - . With the suitable expansion of Eqs.(\[pol:dens\_00\]) and (\[pol:dens\_11\]) in terms of $|{\bf q}|/k_{p(n)}$ and keeping $\mathcal{O}({\bf q}^2/k^2_{p(n)})$ terms we get |\_[med]{}([**q**]{}\^2) \^- \^\^2 , \[pol:dens0\] where \^ &=& . \[dprm\]\ \^ &=&,\ && \[aprm\] Clearly, $\bar{\Pi}_{med}({\bf q}^2)$ is also three momentum dependent and if $M_n=M_p$ it vanishes in SNM but is non-vanishing in ANM. In the present calculation nucleon masses are considered non-degenerate and the asymmetry parameter $\a \neq 0$. To construct CSV potential we take non-relativistic (NR) limit of the Dirac spinors in which case we obtain u\_N() (1--) ( [c]{} 1\ \ ), \[sp\] where ${\bf \sigma}$ is the spin of nucleon. ${\bf P}$ denotes the average three momentum of the interacting nucleon pair and ${\bf q}$ stands for the three momentum of the meson. The explicit expression of full CSV potential in momentum space can be obtained from Eq.(13) of ref.[@Biswas08] by replacing the mixing amplitude $\Pi_{\rho\o}({\bf q})$ with $\bar{\Pi}({\bf q}^2)$. V\^[NN]{}\_[CSV]{}([**q**]{})&=&-\ &&. \[pot:mom\_full\] Eq.(\[pot:mom\_full\]) presents the full CSV $NN$ potential in momentum space in matter. Here $T^\pm_3 = \t_3(1)\pm\t_3(2)$ and ${\bf S}=\f{1}{2}({\bf\s}_1+{\bf\s}_2)$ is the total spin of the interacting nucleon pair. We define $M=(M_n+M_p)/2$, $\Dt M=(M_n-M_p)/2$ and $\Dt M(1,2)=-\Dt M(2,1) = \Dt M $. It is be mentioned that the spin dependent parts of the potential appear because of the contribution of the external nucleon legs shown in Fig.\[fig00\]. On the other hand, $3{\bf P}^2/2M^2_N$ and $-{\bf q}^2/8M^2_N$ arise due to expansion of the relativistic energy $E_N$ of the spinors. In matter, $V^{NN}_{CSV}({\bf q})$, consists of two parts, one contains the vacuum mixing amplitude and other contains the density dependent mixing amplitude. The former is denoted by $V^{NN}_{vac}({\bf q})$ and later by $V^{NN}_{med}({\bf q})$. Thus, $V^{NN}_{CSV}({\bf q})= V^{NN}_{vac}({\bf q}) + V^{NN}_{med}({\bf q})$. From Eq.(\[pot:mom\_full\]) we extract the following term which, in coordinate space gives rise to the $\delta$-function potential. V\^[NN]{}\_[CSV]{} &=& (+\^)\ && . \[pot:contact\] To avoid the appearance of $\delta$-function potential in coordinate space, one should introduce form factors \[$F_i({\bf q}^2)$\], for which the meson-nucleon coupling constants become momentum dependent [*i.e.*]{} ${\rm g}_i({\bf q}^2) = {\rm g}_i F_i({\bf q}^2)$. Here we use the following form factor. F\_i([**q**]{}\^2) = ( ), \[form\] where $\L_i$ is the cut-off parameter governing the range of the suppression and $m_i$ denotes the mass of exchanged meson. The full CSV potential presented in Eq.(\[pot:mom\_full\]) contains both Class $III$ and Class $IV$ potentials, and both break the charge symmetry in $NN$ interactions. The terms within the first and the second curly braces represent Class $(III)$ and Class $IV$ potentials, respectively. Class $(III)$ potential differentiates between $nn$ and $pp$ systems while Class $(IV)$ $NN$ potential exists in the $np$ system only. We, in this paper, restrict ourselves to Class $(III)$ potential only. The coordinate space potential can be easily obtained by Fourier transformation of $V^{NN}_{CSV}({\bf q})$ [*i.e.*]{} V\^[NN]{}\_[CSV]{}([**r**]{})= V\^[NN]{}\_[CSV]{}([**q**]{})  e\^[-i[**qr**]{}]{} We drop the term $3{\bf P}^2/2M^2_N$ from Eq.(\[pot:mom\_full\]) while taking the Fourier transform as it is not important in the present context. However, this term becomes important to fit $^1S_0$ and $^3P_2$ phase shifts simultaneously. Now the CSV potential in coordinate space without $\delta V^{NN}_{CSV}$ reduces to V\^[NN]{}\_[vac]{}([**r**]{}) &=& - T\^+\_3 , \[pot:vac\_cord\] and V\^[NN]{}\_[med]{}([**r**]{})&=& - T\^+\_3 , \[pot:dens\_cord\] where $x_i = m_i r$. The explicit expressions of $V_{vv}(x_i)$ and $V_{tv}(x_i)$ are given in Ref.[@Biswas08]. In Eqs.(\[pot:vac\_cord\]) and (\[pot:dens\_cord\]), the $M^{-2}_N$ independent terms represent central parts without contributions of external nucleon legs. The potentials presented in Eqs.(\[pot:vac\_cord\]) and (\[pot:dens\_cord\]) do not include the form factors so that these potentials diverge near the core. The problem of divergence near the core can be removed by incorporating form factors as discussed before. With the inclusion of form factors, $V^{NN}_{vac}({\bf r})$ and $V^{NN}_{med}({\bf r})$ take the following form: V\^[NN]{}\_[vac]{}([**r**]{}) &=& - T\^+\_3 , \[pot:vac\_ff\] and V\^[NN]{}\_[med]{}([**r**]{}) &=&-T\^+\_3 . \[pot:dens\_ff\] In Eqs.(\[pot:vac\_ff\]) and (\[pot:dens\_ff\]), $X_i = \L_i r$ and &=& (), \[lambda\]\ a\_i = (),&& b\_i = (), \[ab\]\ [where]{}  i(j) = , ø&&(i j). Note that Eqs.(\[pot:vac\_ff\]) and (\[pot:dens\_ff\]) contain the contribution of $\delta V^{NN}_{CSV}$ and the problem of divergence near the core is removed. results ======= Using Eqs.(\[pot:vac\_cord\]) and (\[pot:dens\_cord\]) we show the difference of CSV potentials between $nn$ and $pp$ systems [*i.e.*]{} $\Dt V=V^{nn}_{CSV}-V^{pp}_{CSV}$ in Fig.\[fig01\] for $^1$S$_0$ state at nuclear matter density $\rho_B = 0.148$ fm$^{-3}$ with asymmetry parameter $\a = 1/3$. The dashed and dotted curves show $\Dt V_{vac}$ and $\Dt V_{med}$, respectively. The total contribution ([*i.e.*]{} $\Dt V_{vac} + \Dt V_{med}$) is shown by the solid curve. It is observed that the density dependent CSV potential can not be neglected while estimating CSV observables such as binding energy difference of mirror nuclei.\ Fig.\[fig02\] displays $\Dt V$ for $^1$S$_0$ state including the Fourier transform of $\dt V^{NN}_{CSV}$ and form factors. $\Dt V_{vac}$ and $\Dt V_{med}$ are represented by dashed and dotted curves. Solid curve shows the sum of these two contributions. Note that incorporation of form factors remove the problem of divergence near the core. It is observed that the CSV $NN$ potential changes its sign due to inclusion of the Fourier transform of $\dt V^{NN}_{CSV}$. summary and discussion ====================== In this work we have constructed the CSV $NN$ potential in dense matter using asymmetry driven momentum dependent $\rho^0$-$\omega$ mixing amplitude within the framework of one-boson exchange model. Furthermore, the correction to the central part of the CSV potential due to external nucleon legs are also considered. The closed-form analytic expressions both for vacuum and density dependent CSV $NN$ potentials in coordinate space are presented. We have shown that the vacuum mixing amplitude and the density dependent mixing amplitude are of similar order of magnitude and both contribute with the same sign to the CSV potential. The contribution of density dependent CSV potential is not negligible in comparison to the vacuum CSV potential. The position space Fermion propagator in vacuum is given by the vacuum expectation value of the time ordered product of Fermion fields. i\_N(x-x\^)&=& 0|\[(x)|()\]|0. \[app01\] In medium, the vacuum $|0\ket$ is replaced by the ground state $|\Psi_0\ket$ which contains positive-energy particles with same Fermi momentum $k_N$ and no antiparticles. Thus, i\_N(x-x\^)&=& \_0|(x)|()|\_0(t-)\ &-& \_0||()(x)|\_0(-t). \[app02\] Note that the time-ordered product in Eq.(\[app02\]) involves negative sign for Fermions. The Fermion field contains both the positive- and negative-energy solutions. The modal expansion for the Fermion fields are, (x)&=&\_s (a\_[ks]{}u\_[ks]{}e\^[-ikx]{}+b\^\_[ks]{}v\_[ks]{}e\^[ikx]{}),\ && \[app03\]\ |(x)&=&\_s (a\^\_[ks]{}|[u]{}\_[ks]{}e\^[ikx]{}+b\_[ks]{}|[v]{}\_[ks]{}e\^[-ikx]{}).\ && \[app04\] Here $a^\dagger_{ks}$ and $a_{ks}$ are the creation and annihilation operators for particles and likewise $b^\dagger_{ks}$ and $b_{ks}$ are the creation and annihilation operators for antiparticles. The only nonvanishing anticommutation relations are {a\_[ks]{},a\^\_} = {b\_[ks]{},b\^\_} = \_[s]{}\^3(-). \[app05\] Since $|\Psi_0\ket$ contains only positive-energy particles, we have the following relations: . [ccl]{} b\_[ks]{}|\_0& = 0 & [for all]{} \ &&\ a\_[ks]{}|\_0& = 0 & [for]{} ||&gt;k\_N\ &&\ a\^\_[ks]{}|\_0& = 0 & [for]{} ||&lt;k\_N\ &&\ a\_[ks]{}a\^\_[ks]{}|\_0&= n([**k**]{})|\_0&\ . \[app06\] $n({\bf k})$ is either $0$ or $1$ and this can be accomplished with the step function $\th(k_N-|{\bf k}|)$. Using Eqs.(\[app03\])-(\[app06\]) one obtains &&\_0|(x)|()|\_0=\ &&\_[s]{} \_0|a\_[ks]{}a\^\_|\_0u\_[ks]{}|[u]{}\_ e\^[-i(kx - )]{}\ &=&(k/+M\_N) e\^[-ik(x-)]{}, \[app07\] and &&\_0||()(x)|\_0=\ &&\_[s]{}\ &=& .\[app08\] Now, (t-)e\^[-ik(x-)]{}&=&i. \[app09\]\ (-t)e\^[-ik(x-)]{} &=&-i, \[app10\]\ (-t)e\^[ik(x-)]{}&=&i . \[app11\] From Eqs.(\[app08\]), (\[app10\]) and (\[app11\]) we have && \_0||()(x)|\_0(t-)\ &=&-ie\^[-ik(x-)]{}(k/+M\_N)\ &+&i(k/-M\_N). \[app12\] Now changing $k\ra -k$ in the last integral of Eq.(\[app12\]) and substituting Eqs.(\[app07\]), (\[app09\]) and (\[app12\]) in Eq.(\[app02\]) we get, &&i\_N(x-)= i e\^[-ik(x-)]{}(k/+M\_N)   \ &&. \[app13\] In Eq.(\[app13\]) $E_k$ has been replaced by $E_N$. The first term of Eq.(\[app13\]) represents particle propagation above the Fermi sea and the second term indicates the propagation of holes inside the Fermi sea. The last term shows the propagation of holes in the infinite Dirac sea. Now, -&=&, \[app14\]\ -&=&2i(k\_0-E\_N).\[app15\] From Eqs.(\[app13\])-(\[app15\]), i\_N(x-)&=& ie\^[-ik(x-)]{} G\_N(k) \[app16\] where $G_N(k)$ is the in-medium Fermion propagator in momentum space. G\_N(k)&=&\ &+&(k/+M\_N)(k\_0-E\_N)(k\_N-||)\ &=& G\_N\^F(k)+G\_N\^D(k). \[app17\] [99]{} [Nolen69]{} J. A. Nolen and J. P. Schiffer, Annu. Rev. Nucl. Sci. [**19**]{}, 471 (1969). [Henley69]{} E. M. Henley, in [*Isospin in Nuclear Physics*]{}, edited by D. H. Wilkinson (North-Holland, Amsterdam, 1969), p. 17. [Henley79]{} E. M. Henley and G. A. Miller, in [*Mesons in Nuclei*]{}, edited by M. Rho and D. H. Wilkinson (North-Holland, Amsterdam, 1979), p. 405. [Miller90]{} G. A. Miller, M. K. Nefkens and I. Slaus, Phys. Rep. [**194**]{}, 1 (1990). [Machleidt89]{} R. Machleidt, Adv. Nucl. Phys. [**19**]{}, 189 (1989). [Miller95]{} G. A. Miller and W. H. T. van Oers, in [*Symmetries and Fundamental Interactions in Nuclei*]{}, edited by W. C. Haxton and E. M. Henley (World Scientific, Singapore, 1995), p. 127. [Howell98]{} C. R. Howell [*et. at.*]{}, Phys. Lett. [**B 444**]{}, 252 (1998). [Gonzalez99]{} D. E. Gonzalez Trotter [*et. al.*]{}, Phys. Rev. Lett. [**83**]{}, 3788 (1999). [Nolen73]{} J. A. Nolen and J. P. Schiffer, Annu. Rev. Nucl. Sci. [**19**]{}, 471 (1969). [Okamoto64]{} K. Okamoto, Phys. Lett. [**11**]{}, 150 (1964). [Garcia92]{} L.N Epele, H. Fanchlottl, C. A. Garcia Canal and G. A. Gonzfilez Sprlnberg, Phys. Lett. [**B 277**]{}, 33 (1992). [Miller06]{} G. A. Miller, A. K. Opper and E. J. Stephenson, [*nucl-ex / 0602021*]{} [Coon87]{} S. A. Coon and M. D. Scadron, Phys. Rev. [**C 26**]{}, 562, (1982). [McNamee75]{} Peter C. McNamee, Michael D. Scadron and Sydney A. Coon, Nucl. Phys. [**A 249**]{}, 483 (1975). [Coon77]{} S. A. Coon and M. D. Scadron, Nucl. Phys. [**A 287**]{}, 381 (1977). [Blunden87]{} P. G. Blunden and M. J. Iqbal, Phys. Lett. [**B 198**]{}, 14 (1987). [Sidney87]{} S. A. Coon and R. C. Barrett, Phys. Rev. [**C 36**]{}, 2189 (1987). [Machleidt01]{} R. Machleidt and H. Müther, Phys. Rev. [**C 63**]{}, 034005 (2001). [Cohen95]{} T. D. Cohen and G. A. Miller, Phys. Rev. [**C 52**]{}, 3428 (1995). [Piekarewicz92]{} J. Piekarewicz and A. G. Williams, Phys. Rev. [**C 47**]{}, R2462 (1993). [Goldman92]{} T. Goldman, J. A. Henderson and A. W. Thomas, Few-Body Syst. [**12**]{}, 123 (1992). [Krein93]{} G. Krein, A. W. Thomas and A. G. Williams, Phys. Lett. [**B 317**]{}, 293 (1993). [Connell94]{} H. B. O’Connell, B. C. Pearce, A. W. Thomas and A. G. Williams, Phys. Lett. [**B 336**]{}, 1 (1994). [Coon97]{} S. A. Coon, B. H. J. McKellar and A. A. Rawlinson, In Intersections Between Particle and Nuclear Physics, edited by T. W. Donelly, AIP Conf. Proc. No. 412 (AIP, Woodbury, NY, 1997), p368. [Hatsuda94]{} T. Hatsuda, E. M. Henley, Th. Meissner and G. Krein, Phys. Rev. [**C 49**]{}, 452 (1994). [Abhee97]{} A. K. Dutt-Mazumder, B. Dutta-Roy and A. Kundu, Phys. Lett. [**B 399**]{}, 196 (1997). [Broniowski98]{} W. Broniowski and W. Florkowski, Phys. Lett. [**B 440**]{}, 7 (1998). [Abhee01]{} A. K. Dutt-Mazumder, R. Hofmann and M. Pospelov, Phys. Rev. [**C 63**]{}, 015204 (2001). [Kampfer04]{} S. Zschocke and B. Kampfer, Phys. Rev. [**C 70**]{}, 035207 (2004). [Roy08]{} P. Roy, A. K. Dutt-Mazumder, S. Sarkar and J. Alam, J. Phys. [**G 35**]{}, 065106 (2008). [Biswas06]{} S. Biswas and A. K. Dutt-Mazumder, Phys. Rev. [**C 74**]{}, 065205 (2006). [Saito03]{} Y. Mori and K. Saito, Phys. Lett. [**B 552**]{}, 21 (2003). [Biswas08]{} S. Biswas, P. Roy and A. K. Dutt-Mazumder, Phys. Rev. [**C 78**]{}, 045207 (2008). [Serot86]{} B. D. Serot and J. D. Walecka, Adv. Nucl. Phys. [**16**]{}, 1 (1986). [Chin77]{} S. A. Chin, Ann. Phys. [**108**]{}, 301 (1977). [Kapusta]{} J. I. Kapusta, [*Finite-Temperature Field Theory*]{}, (Cambridge University Press, New York, 1989). [Hooft73]{} G. ’tHooft and M. T. Veltmann, Nucl. Phys. [**B 44**]{}, 189 (1973). [Peskin95]{} M. E. Peskin and D. V. Schroeder, [*An Introduction to Quantum Field Theory*]{}, (Addision-Wesley Publishing Company, 1995). [Cheng06]{} T. P. Cheng and L. F. Li, [*Gauge theory of elementary particles*]{}, (Clarendon Press, Oxford, New York 2006).

--- abstract: | Let $Y_1, \ldots, Y_n$ be $n$ indeterminates. For $I=(i_1, \ldots,i_p)$, $i_s\in \{1, 2$, $\ldots, n\}$, $s= 1, 2, \ldots, p$, let $Y_I$ be the monomial $Y_{i_1} Y_{i_2} \cdots Y_{i_p}$. Denote by $|I|=p$. Let $\can[Y_1, Y_2, \ldots, Y_p]$ be the ring of non-commutative series $\sum a_I Y_I$, $a_I\in \C$, such that $\sum |a_I| R^{|I|}<\infty$ for all $R>0$. On $\can[Y_1, Y_2, \ldots, Y_n]$ we have a canonical involution extending by linearity $(a_I Y_I)^* = \overline a Y_{I^{\rm op}}$, $a_I\in \C$, $I\in \cI_n$, $I=\{i_1, i_2, \ldots,i_p\}$, $I^{\rm op}= \{i_p, i_{p-1}. \ldots, i_1\}$. By $\can^{\rm sym}[Y_1, Y_2, \ldots,Y_n]$ we denote the real subspace of $\can [Y_1, Y_2, \ldots, Y_n]$ of series that are auto-adjoint. We say that two series $p, q$ are cyclic equivalent if $p-q$ is a sum (possible infinite) of scalar multiples of monomials of the type $Y_I-Y_{\tilde I}$, where $\widetilde I$ is a cyclic permutation of $I$. We call a series $q$ in $\can [Y_1, \ldots, Y_p]$ a sum of squares if $q$ is a weak limit of sums $\sum_s b^*_s b_s$, where $b_s\in \can [Y_1, \ldots, Y_p]$. We prove that if a series $p(Y_1, \ldots,Y_n)$ in $\can^{\rm sym}[Y_1, \ldots, Y_n]$ has the property that $\tau (p(X_1, \ldots, X_n))\geq 0$ for every $M$ type II$_1$ von Neumann algebra with faithful trace $\tau$ and for all selfadjoint $X_1, X_2, \ldots, X_n$ in $M$, then $p$ is equivalent to a sum of squares in $\can[Y_1, \ldots, Y_n]$. As a corollary, it follows that the Connes embedding conjecture is equivalent to a statement on the structure of matrix trace inequalities: if $p(Y_1, \ldots, Y_n)$ in $\can^{\rm sym} [Y_1, \ldots, Y_n]$ is such that $\tr p(X_1, \ldots,X_n)\geq 0$, for all selfadjont matrices $X_1, \ldots, X_n$, of any size, then $p$ should be equivalent to a sum of squares in$\can [Y_1, \ldots,Y_n]$. address: | Department of Mathematics\ University of Roma “Tor Vergata”\ Via della Ricerca Scientifica, 00133 Roma, Italy\ on leave from University of Iowa\ Iowa City, IA 52242, USA author: - Florin Rădulescu title: | A non-commutative,\ analytic version of Hilbert’s $17$-th problem\ in type II$_1$ von Neumann algebras[^1] --- Introduction ============ The Connes embedding conjecture ([@Co]) states that: [*Every type [II]{}$_1$ factor (equivalently any type [II]{}$_1$ von Neumann algebra) can be embedded into the factor $R^\omega$*]{} (see [@Co], p. 105). Equivalent forms of this conjecture have been extensively studied by Kirchberg ([@Ki]), and subsequently in [@Pi], [@Ra1], [@Ra2], [@Br], [@Oz]. In this paper we prove that the Connes embedding conjecture is equivalent to a statement on the structure of the trace inequalities on matrices. To prove this, we deduce an analogue of the Hilbert $17$-th problem in the context of type II$_1$ von Neumann algebras. More precisely, let $Y_1, \ldots,Y_n$ be $n$ indeterminates. Let $\cI_n$ be the index set of all monomials in the variables $Y_1, Y_2, \ldots, Y_n$ $$\cI _n=\{(i_1, \ldots, i_p)\mid p \in \N, \, i_1, \ldots, i_p\in \{1, 2, \ldots, n\}\}$$ (we assume that $\emptyset \in \cI_n$ and that $\emptyset$ corresponds to the monomial which is identically one). For $I=(i_1, \ldots i_p)$ let $|\cI|=p$ and let $Y_I$ denote the monomial $Y_{i_1} Y_{i_2} \cdots Y_{i_p}$. For such an $I$ define $I^{\rm op} = (i_p, \ldots, i_1)$ and define an adjoint operation on $\C[Y_1, Y_2, \ldots, Y_n]$ by putting $Y^*_I = Y_{I^{\rm op}}$. We let $\can [Y_1, \ldots, Y_n]$ be the ring of all series $$V= \Big\{\sum_{I\in I_n} a_I Y_I\mid a_I\in \C, \, \Big \| \sum a_I Y_I \Big\|_R = \sum |a_I| R^{|I|}<\infty, \, \forall \, R>0\Big\}.$$ It turns out (Section 2) that $V$ is a Fr' echet space, and hence that $V$ has a natural weak topology $\sigma (V, V^*)$. We will say that an element $q$ in $V$ is a sum of squares if $q$ is in the weak closure of the cone of sum of squares $$\sum_s p_s^* p_s, \quad p_s \in \can [Y_1, Y_2,\ldots, Y_n].$$ By $\can^{\rm sym} [Y_1, Y_2, \ldots, Y_n]$ we denote the real subspace of all analytic series that are auto-adjoint. We say that two series $p, q$ in $\can^{\rm sym}[Y_1, Y_2, \ldots, Y_n]$ are [*cyclic equivalent*]{} if $p-q$ is a weak limit of sums of scalar multiples of monomials of the form $Y_I- Y_{\tilde I}$, where $I\in \cI_n$, and $\widetilde I$ is a cyclic permutation of $I$. Our analogue of the Hilbert’s $17$-th problem is the following Let $p \in \can ^{\rm sym}[Y_1, \ldots, Y_n]$ such that, whenever $M$ is a separable type [II]{}$_1$ von Neumann algebra with faithful trace $\tau$ and $X_1, \ldots, X_n$ are selfadjoint elements in $M$, then by substituting $X_1, \ldots, X_n$ for $Y_1, \ldots, Y_n$, we obtain $\tau (p(X_1, \ldots, X_n))\geq 0$. Then $p$ is cyclic equivalent to a weak limit of a sum of squares in $\can [Y_1, \ldots, Y_n]$. As a corollary, we obtain the following statement which describes the Connes embedding conjecture strictly in terms of (finite) matrix algebras. (A similar statement has been noted in [@DH]). The Connes embedding conjecture holds if and only if whenever $p\in \can^{\rm sym}[Y_1, \ldots, Y_n]$ is such that, whenever we substitute selfadjoint matrices $X_1, X_2, \ldots, X_n$ in $M_N (\C)$ endowed with the canonical trace $\tr$, we have $\tr(p(X_1, X_2, \ldots, X_n))\geq 0$. Then $p$ should be equivalent to a weak limit of sums of squares. To prove the equivalence of the two statements we use the fact that the Connes conjecture is equivalent to show that the set of non-commutative moments of $n$ elements in a type II$_1$ factor can be approximated, in a suitable way, by moments of $n$ elements in a finite matrix algebra. Properties of the vector space $\can[Y_1, Y_2, \ldots, Y_n]$ ============================================================= We identify in this section the vector space $\can [Y_1, \ldots, Y_n]$ with the vector space $V= \Big \{ (a_I)_{I\in \cI_n}\mid \sum\limits_{I\in \cI_n} |a_I| R^I< \infty , \, R>0\Big\}$. Denote by $\|(a_I)_{I\in \cI_n} \|_R = \sum \limits_{I \in \cI_n} |a_i| R^I$ for $R>0$. Clearly, $\|\cdot\|_R$ is a norm on $V$. In the next proposition (which is probably known to specialists, but we could not provide a reference) we prove that $V$ is a Fr' echet space. For two elements $J, K \in \cI_n$, $J=(j_1, \ldots, j_n)$, $K= (k_1, \ldots, k_s)$, we denote by $K\diez J= (j_1, \ldots, j_n, k_1, \ldots, k_s)$ the concatenation of $J$ and $K$. With the norms $\| \cdot \|_R$, $V$ becomes a Fréchet space. Moreover, the operation $*$ defined for $a= (a_I)$, $b= (b_I)$ by $$(a* b)_I = \sum _{{\scriptstyle J,K \in \cI_n} \atop {\scriptstyle J\diez K=I}} a_Ja_K$$ (which in terms of $\can[Y_1, \ldots, Y_n]$ corresponds to the product of series) is continuous with respect to the norms $\|\cdot \|_R$, that is $$\|a * b\|_R\leq \|a\|_R \|b\|_R, \quad \mbox{ for all } R>0.$$ To prove that $V$ is a Fréchet vector space, consider a Cauchy sequence $b^s =(b_I^s)_{I\in \cI_n}$. Thus for all $R,\varepsilon>0$ there exist $N_{R, \varepsilon}$ in $\N$ such that for all $s, t> N_{R, \varepsilon}$ we have $\|b^s- b^t\|_R<\varepsilon$. Clearly, this implies that $(b^s_I)_{I\in \cI_n}$ converges pointwise to a sequence $(b_I)_{I\in \cI_n}$ and it remains to prove that $b$ belongs to $V$ and $b^s$ converges to $b$. To do this we may assume (by passing to a subsequence, and using a typical Cantor diagonalization process) that $$\|b^s- b^{s+1}\|_s \leq \frac 1 {2^s} \quad \mbox{for all $s$ in $\N$}.$$ But then for every $R>0$, we have that $$\begin{aligned} | b_I^s-b_I| R^{|I|} &\leq \sum_{{\scriptstyle s= [R] +1}\atop {\scriptstyle I\in \cI_n}} |b^s_I-b^{s+1}_I| R^{|I|}\\&\leq \sum_{{\scriptstyle s=[R]+1}\atop {\scriptstyle I\in \cI_n}} | b_I^s - b_I^{s+1}| s^{|I|} \leq \sum _{s= [R]+1} \frac 1 {2^s}.\end{aligned}$$ This proves that $b\in V$ and that $b^s$ converges to $b$. For the second part of the statement we have to estimate, for all $a, b\in V$ and $R>0$, the quantity $$\begin{aligned} \sum_{I\in \cI_n} \sum_{{\scriptstyle J, K\in \cI_n}\atop {\scriptstyle J\diez K=I}} |a_J| \, |b_K| R^{|I|}&= \sum_{I\in \cI_n} \sum_{{\scriptstyle J,K \in \cI_n}\atop{\scriptstyle J\diez K=I}} |a_J|\, |b_K|R^{|J|+|K|}\\&= \bigg ( \sum_{J\in \cI_n} |a_J| R^{|J|}\bigg)\bigg( \sum _{K\in \cI_n}|b_K|R^{|K|}\bigg)= \| a\|_R \|b \|_R.\end{aligned}$$ This proves that $a*b\in V$ and that $\|a*b\|_R \leq \| a\|_R \|b\|_R$. In what follows we describe the dual $V^*$ of the Fréchet space $V$. $V^*$ is identified with the space of all sequences of complex numbers $$\{(t_I)_{I\in cI_n} \mid \exists\, R>0, \, \sup\limits _{I\in \cI_n}|t_I|R^{-|I|} < \infty\}.$$ The duality between $V$ and $V^*$ is realized via the pairing $$\langle a, t\rangle = \sum_{I\in \cI_n} a_I t_I\quad \mbox{for } a\in V,\, t\in V^*,$$ which is convergent if $\sup\limits _{I\in \cI_n} |t_I|R^{-|I|}<\infty$ for some $R$. It is obvious that each sequence $(t_I)_{I\in \cI_n}$ such that $\sup\limits_{I\in \cI_n} |t_I|R^{-|I|}$ defines an element in $V^*$. Conversely, if $\varphi$ is a continuous linear functional on $V$, then there exists a semi-norm $\|\cdot\|_R$ and a positive constant $C>0$, such that $$|\varphi ((a_I)_{I\in \cI_n})| \leq C \sum_{I \in \cI_n} |a_I|R^{|I|}.$$ But then by the usual duality between $\ell^1$ and $\ell^\infty$ and since $\{ (a_I)R^{|I|})_{I\in \cI_n}\mid (a_I)_{I \in \cI_n} \in V\}$ is dense in $\ell^1(\cI_n)$, it follows that there exists $(t_I)_{I\in \cI}$ such that $\sup \limits_{I\in \cI_n} (t_I) R^{-|I|}<\infty$ and such that $\varphi ((a_I)_{I\in \cI_n}) = \sum\limits_{I\in \cI_n} a_It_I$. By $V_{\rm sym}$ we consider the real subspace of $V$ consisting of all $\{(a_I)_{I\in \cI_n}\in V\mid a_{I^{\rm op}}= \overline {a_I} , \, \forall I\in \cI_n\}$, and by $V^*_{\rm sym}$ we consider the space of all real functionals on $V_{\rm sym}$. If $\sigma$ is a permutation of $\{1,2, \ldots, p\}$ (which we denote by $\sigma \in S_p$) and $I\in \cI_n$, $I=(i_1, i_2, \ldots, i_p)$ then by $\sigma(I)$ we denote $(i_{\sigma(1)}, i_{\sigma(2)}, \ldots, i_{\sigma(p)})$. By $S_{p,{\rm cyc}}$ we denote the cyclic permutations as $S_p$. We omit the index $p$ when it is obvious. Let $W = \{(t_I)_{I\in \cI_n} \mid t_I = t_{\sigma(I)},\, \forall\, \sigma\in S_{\rm cyc} and \,\exists \, R>0$ such that $\sup |t_I|R^{-|I|} < \infty, \, t_I = \overline{t_{I^{\rm op}}}, \, \forall\, I \in \cI_n\}$. $W$ is a closed subspace of $V^*_{\rm sym}$. Moreover, if $\varphi _s=(t_I^s)_{I\in \cI_n} \in W$ converges to $\varphi \in W$ in the $\sigma (V^*, V)$ topology, then there exists $R>0$ such that $$\sup_{s, I\in \cI_n}|t_I^s| R^{-|I|} <\infty.$$ $W$ is a closed subspace of $V^*$ follows immediately from the fact that $W$ is the fixed point subspace for the actions of the finite groups $S_{p, {\rm cyc}}$, $p\geq 1$, on the components of the indices. The same is true for the third condition in the definition of $W$. The second statement is a standard consequence of the Banach-Steinhaus theorem. Indeed, if $\varphi_s \to \varphi$ in the $\sigma(V^*, V)$ topology, then $(\varphi _s)_{s\in \N}$ forms an equicontinuous family and hence there exist a semi-norm $\|\cdot\|_R$ on $V$ and a constant $C>0$ such that $$|\varphi_s((a_I)_{I\in \cI_n})| \leq C\| (a_I)_{I\in \cI_n}\|_R.$$ By the preceding section this means exactly the condition in the statement. We note the following straightforward consequence of the Bipolar theorem ([@Ru], [@Simai-Reed]): Let $K_m\subseteq K_2\subseteq W$ be closed convex cones, (which will correspond to the sets $K_{\rm mat}$ and $K_{{\rm II}_1}$ in the next section). Let $L_2=K^0_2$, $L_m=K^0_m$ be the corresponding polar sets (the polars are with respect $V_{\rm sym}$) in $V_{\rm sym}$. (In particular $W^\perp \subseteq L_2\subseteq L_m$.) By the annihilator $W^\prep$ of the space $W$ we mean the (relative) annihilator with respect to$V_{\rm sym}$. Likewise we do for the polar sets. Let $L_p$ be a $\sigma (V_{\rm sym}, V^*_{\rm sym})$ closed conex subset of $L_2$. Then to prove that $L_p+W^\perp=L_2$ is equivalent to prove that $L^0_p\cap W=K_2$ and hence it is sufficient to verify that $L^0_p \cap W\subseteq K_2$. Indeed, in this situation $L_p+W^\perp \subseteq L_2$ and since $L^0\cap W= (L_p+W^\perp)^0$ it follows immediately from the Bipolar theorem that $K_2=L_2^0\subseteq (L_p + W^\perp)^0 =L_p\cap W$. Moreover, if the last two closed convex sets are equal, i.e. if $K_2=L_p^0 \cap W$, then by the Bipolar theorem we get $L_2=K_2^0= L_p ^0 \cap W= (L_p+W^\prep )^{00}= L_p+W^\perp.$ The set of non-commutative moments for variables\ in a type II$_1$ von Neumann algebra ================================================= We consider the following subsets of the (real) vector space $W$ introduced in the preceding section: Let $$\begin{aligned} K_{{\rm II}_1} = \sigma(V^*,V)\mbox{-{\rm closure}}& \{(\tau(x_I))_{I\in \cI_n} \mid M \mbox{ type ${\rm II}_1$ separable von Neumann algebra, }\\ & \mbox{ with faithful trace } \tau, \; x_i = x_i^* \in M, \, i = 1,2,\ldots,n\}.\end{aligned}$$ Here by $x_\phi$ we mean the unit of $M$. We also consider the following set. Let $\tr$ be the normalized trace $\frac{1}{N}\trr$ on $M_N(\C)$. Then $$K_{\rm mat} = \sigma(V^*,V)\mbox{-{\rm closure}}\{(\tr(x_I))_{I\in \cI_n} \mid N \in \N, \, x_i = x_i^* \in M_N(\C)\}.$$ We clearly have that $K_{\rm mat} \subseteq K_{{\rm II}_1} \subseteq W$ and as proven in [@Ra1], $K_{\rm mat}$, $K_{{\rm II}_1}$ are convex $(\sigma(V^*,V)\mbox{-{\rm closed}}$ or $\sigma(V_{\rm sym}^*,V_{\rm sym})$ closed sets. Let $\tilde K_{{\rm II}_1}$, $\tilde K_{\rm mat}$ be the (convex) cones generated by the convex sets $K_{{\rm II}_1}$ and $ K_{\rm mat}$ We consider also the following subset of $V_{\rm sym} \subseteq V$ $$L_p = \overline{\rm co}^{\sigma(V,V^*)} \{a^* * a \mid a \in V\},$$ where $*$ is the operation introduced in Proposition 2.1. In the identification of $V$ with $\C_{\rm an}[Y_1,Y_2,\ldots,Y_n]$, $L_p$ corresponds to the series that are limits of sums of squares. Moreover, $W^\perp \cap V_{\rm sym}$ corresponds to the $\sigma(V,V^*)\mbox{-{\rm closure}}$ of the span of the selfadjoint part of monomials of the form $Y_I - Y_{\sigma(I)}$, $\sigma \in S_p$. Hence $L_p + (W^\perp \cap V_{\rm sym}$ corresponds to the series in $\C_{\rm an}^{\rm sym} [Y_1,\ldots,Y_n]$ that are equivalent to a weak $(\sigma(V,V^*))$ limit of sum of squares: $$\sum\limits_{s\in S} b_{s^*}^*b_s,\quad b_s \in \C_{\rm an}[Y_1,Y_2,\ldots,Y_n].$$ Moreover, $L_p \subseteq K_{{\rm II}_1}^0 \subseteq K_{\rm mat}^0$. Indeed the series corresponding to $a^* * a$ is $$\begin{aligned} \sum_{I\in \cI_n} (a^* * a)_I Y_I & = \sum_I \sum_{J\sharp K} a_J^* a_K Y_{J\sharp K} = \sum_{I\in \cI_n} \sum_{J\sharp K=I} \overline{a_{J^{\rm op}}} a_K Y_J Y_K \\ & = \sum_{J,K\in \cI_n} \overline{a}_{J} a_K Y_{J^{\rm op}} Y_K = \bigg[ \sum_{J\in \cI_n} (\overline{a}_J Y_J^*)\bigg]^* \bigg( \sum_{K\in \cI_n} a_K Y_K\bigg).\end{aligned}$$ Moreover, when pairing a series $\sum a_I Y_I \in \C_{\rm an}[Y_1,\ldots, Y_n]$ with $t_I = \tau(x_I)$, $I \in \cI_n$, $x_1,\ldots,x_n \in M$, $x_i = x_i^*$, the result is $\sum a_I \tau(x_I) = \tau\Big(\sum a_I x_I\Big)$. Hence an element in $L$ paired with an element in $K_{{\rm II}_1}$ is the trace of a sum squares and hence is positive. Thus, $L_p \subseteq K_{{\rm II}_1}^0$ and hence $L_p + (W^\perp \cap V_{\rm sym}) \subseteq K_{{\rm II}_1}^0$ so $L_p^0 \cap W \supseteq K_{{\rm II}_1}$. With these observations we can prove Theorem 1.1. By the above observations and by Lemma 2.4 this amounts to prove that $$L_p + (W^\perp \cap V_{\rm sym}) = K_{{\rm II}_1}^0$$ and by the previous remarks we only have to prove that $$L_p^0 \cap W \subseteq \tilde K_{{\rm II}_1}.$$ Thus we want to show that if an element $\theta=(\theta_I)_{I \in \cI}$ with $\theta_{\emptyset}=1$ belongs to $L_p^0$ then there exists a type II$_1$ von Neumann algebra $M$ with faithful trace $\tau$, and selfadjoint elements $x_1,\ldots,x_n$ in $M$ such that $$\theta_I = \tau(x_I) \quad \mbox{ for } I \in \cI_n.$$ To construct the Hilbert space on which $M$ acts consider the vector space $$\cH_0 = \C_{\rm an}[Z_1,\ldots,Z_n]$$ where $Z_1,Z_2,\ldots,Z_n$ are indetermined variables. We consider the following scalar product on $\cH_0$: $$\langle Z_I,Z_J \rangle = \theta_{J^{\rm op}\sharp I}$$ or more general for $(a_I)_{I \in \cI_n}$, $(b_I)_{I \in \cI_n}\in V$ $$\bigg\langle \sum_{I \in \cI_n} a_I Z_I, \sum_{J \in \cI_n} b_J Z_J \bigg \rangle = \langle b^* * a, \theta \rangle.$$ (Recall that $\langle \;,\;\rangle$ is the pairing between $V$ and $V^*$ introduced in Section 2). Note that the scalar product is positive since $$\bigg\langle \sum_{I \in \cI_n} a_I Z_I, \sum_{I \in \cI_n} a_I Z_I \bigg \rangle = \langle a^* * a, \theta \rangle,$$ which is positive since $t\in L_p^0 \cap W\subseteq L_p^0$. We let $\cH$ to be the Hilbert space completion of $\cH_0$ after we mod out by the elements $\xi$ with $\langle \xi,\xi\rangle =0$. Note that in this Hilbert space completion, we have obviously that $\sum\limits_{I \in \cI_n} a_I Z_I$ is the Hilbert space limit after $p \to \infty$ of $\sum\limits_{I \in \cI_n\atop |I|\leq p} a_I Z_I$. We consider the following unitary operators $U_t^i$, $i=1,2,\ldots,n$, $t\in \R$ acting on $\cH_0$ isometrically and hence on $\cH$. For $i \in \{1,2,\ldots,n\}$ and $q \in \N$ we will denote by $i^q$ the element $(i,i,\ldots,i)$ of length $q$ and belonging to $\cI_n$. The following formula defines $U_t^i$ $$U_t^i Z_I = \sum_{s=0}^\infty \frac{(it)^s}{s!} Z_{i^s \sharp I} \quad \mbox{ for } I \in \cI_n.$$ To check that $U_t^i$ is corectly defined, and extendable to an unitary in $\cH$ it is sufficient to check that $$\langle U_t^i Z_I, U_t^i Z_J \rangle = \langle Z_I, Z_J \rangle \eqno(1)$$ for all $I,J \in \cI_n$. But we have $$\begin{aligned} \langle U_t^i Z_I, U_t^i Z_J \rangle & = \sum_{p,q}\frac{(it)^p \overline{(it)^q}}{p!q!} \langle Z_{i^p\sharp I}, Z_{i^q \sharp J} \rangle = \sum_{p,q}\frac{(it)^p \overline{(it)^q}}{p!q!} \theta_{J^{\rm op} \sharp i^{p+q}\sharp I} \\ & = \sum_{p,q}\frac{i^{p-q}t^{p+q}}{p!q!} \theta_{J^{\rm op} \sharp i^{p+q}\sharp I}.\end{aligned}$$ Note that all the changes in summation are allowed and that all the series are absolutely convergent since $(t_I)_{I\in \cI} \in W$ and hence there exists $R>0$ such that $|t_I| \leq R^{|I|}$ (we also use the analiticy of the series $\sum\limits_{p\geq 0} \frac{(it)^p}{p!}$. For a fixed $k$, the coefficient of $t^k$ in the above sum is $$t^k(\theta_{J^{\rm op} \sharp i^k \sharp I}) \sum_{p+q=k}\frac{i^{p-q}}{p!q!}$$ and this vanishes unless $k=0$ because of the corresponding property of the exponential function: $e^{it} e^{-it}=1$. This proves (1) and hence that $U_t^i$ can be extended to an isometry on $\cH_0$ and hence on $\cH$. Next we check that for all $j=1,2,\ldots,n$, $$U_t^j U_s^j = U_{t+s}^j \quad \mbox{ for all } t,s \geq 0$$ for $I \in \cI_n$. But $$U_t^j U_s^j Z_I = \sum_{p,q=0}^\infty\frac{(it)^p(is)^q}{p!q!} Z_{j^p\sharp j^q \sharp I} = \sum_{k=0}^\infty \sum_{p+q=k} \frac{(it)^p(is)^q}{p!q!} Z_{j^k\sharp I}$$ and this is then equal to $U_{t+s}^j Z_I$ because of the corresponding property for the exponential function: $e^{is} e^{it}=e^{i(s+t)}$. From this it then follows that $U_t^j$ is a one parameter group of unitaries for all $j=1,2,\ldots,n$. We let $M$ be the von Neumann algebra of $B(\cH)$ generated by all $U_t^j$, $j=1,2,\ldots,n$, $t\in \R$. We will verify that $Z_\emptyset$ is a cyclic trace vector for $M$ (and hence it will be also separating, [@Sz]). To verify that $Z_\emptyset$ is a trace vector it sufficient to check that for all $p,q$ and all $i_1,i_2,\ldots,$ $i_p, j_1,j_2,\ldots,j_q \in \{1,2,\ldots,n\}$ we have that for all $t_1,\ldots,t_p, s_1,\ldots,s_q \in \R$ $$\langle U_{t_1}^{i_1} \cdots U_{t_p}^{i_p} \cdots U_{s_1}^{j_1} \cdots U_{s_q}^{j_q} Z_\Phi, Z_\Phi \rangle = \langle U_{s_1}^{j_1} \cdots U_{s_q}^{j_q} U_{t_1}^{i_1} \cdots U_{t_p}^{i_p} Z_\Phi, Z_\Phi \rangle.\eqno(2)$$ It is immediate that the right side of (2) is equal to $$\sum_{\alpha_1,\ldots,\alpha_p =1 \atop \beta_1,\ldots,\beta_q =1}^\infty \frac{(it_1)^{\alpha_1}\cdots (it_p)^{\alpha_p} (is_1)^{\beta_1}\cdots (is_q)^{\beta_q}} {\alpha_1!\cdots\alpha_p! \beta_1! \cdots \beta_q!} \theta_{i_1^{\alpha_1}\cdots i_p^{\alpha_p} j_1^{\beta_1}\cdots j_q^{\beta_q}}.$$ Because of the cyclic symmetry of $(\theta_I)_{I\in \cI_n}$ is then equal to the lefthand side of (2). The sums involved are absolutely convergent since $|t_I| \leq R^{|I|}$, $I\in \cI_n$ for some$R>0$ and because of the fact that the scalar function involved are entire analytic functions. We now prove that $Z_\Phi$ is a cyclic vector. Denote by $iy_j$ the selfadjoint generator for the group $U_t^j$, $j = 1,2,\ldots,n$. We claim that for all $I\in \cI_n$, the vector $Z_I$ belongs to the domain of $iy_j$ and that $y_j Z_I = Z_{j \sharp I}$, $j = 1,2,\ldots,n$, $I\in \cI_n$. Indeed this follows by evaluating the limit $$\lim_{t \to 0} \frac{1}{i} \frac{1}{t}(U_t^j Z_I - Z_I) - Z_{j \sharp I}.$$ But this is equal to $$t\bigg( \sum_{q=2}^\infty \frac{(it)^q}{q!} Z_{j^n \sharp I}\bigg)$$ and this converges to zero when $t\to 0$ because of the summability condition for $(t_I)_{I\in \cI_n}$. Indeed the square of norm of this term is $$t^2 \sum_{p,q=2}^\infty \frac{(it)^p \overline{(it)^q}}{p!q!} \theta_{I^* \circ j^{n+m} \circ I}.$$ From this we deduce that $Z_\Phi$ belongs to domain of $y_{j_1}\cdots y_{j_q}$ and $y_{j_1}\cdots y_{j_q}Z_\Phi = Z_{(j_1\cdots j_q)}$. This implies in particular that $Z_\Phi$ is cyclic for $M$ and also that $$\langle y_I Z_\Phi, Z_\Phi \rangle = \theta_I, \quad I\in \cI_n.$$ By [@Sz] it follows that $Z_\Phi$ is also separating for $M$ and hence that $M$ is a type II$_1$ von Neumann algebra and that the vector form $\langle \,\cdot\, Z_\Phi,Z_\Phi \rangle$ is a trace on $M$. By definition the elements $y_j$, $j = 1,2,\ldots,n$ are selfadjoint and affiliated with $M$. Moreover since $\tau(m)=\langle m Z_\Phi,Z_\Phi \rangle$ is a faithfull finite trace on $M$ and since there exist $R>0$ such that $$\tau(y_j^{2k}) = \theta_{j^{2k}} \leq R^{2k} \quad \mbox { for all } k,$$ it follows that $y_j$ are bounded elements in $M$ such that $\tau(y_I) = \theta_I$ for all $I\in \cI_n$. This proves that $(\theta_I)_{I\in \cI_n}$ belongs to $K_{{\rm II}_1}$ and hence that $L_p + (W^\perp \cap V_{\rm sym}) \subseteq \tilde K_{{\rm II}_1}$. This completes the proof of the theorem. We are now ready to prove Corollary 1.2. With our previous notations this amounts to show that the statement $L_p + (W^\perp \cap V_{\rm sym}) = K_{\rm mat}^0$ is equivalent to the Connes embedding conjecture. Since we already know that $L_p + (W^\perp \cap V_{\rm sym}) \subseteq \tilde K_{{\rm II}_1}$, this amounts to prove that the statement that $K_{\rm mat} = K_{{\rm II}_1}$ is equivalent to Connes conjecture. The only non-trivial part is that the statement $K_{\rm mat} = K_{{\rm II}_1}$ implies the Connes conjecture. So assume that for a given type II$_1$ von Neumann algebra $M$ and for given $x_1,\ldots,x_n$ in $M$ there exists $(X_i^k)_{i=1}^n$ in $M_{N_k}(\C)$ such that $(\tr(X_I^k))_{I\in \cI_n}$ converges in the $\sigma(W,V)$ topology to $(\tau(x_I))_{I\in \cI_n}$. By Lemma 2.3, we know that there exists $R>0$ such that for all $k \in \N$ $$|\tr(X_I^k)|\leq R^{|I|}$$ and hence that $$\tr((X_j^k)^{2p}) \leq R^{2p} \quad \mbox{ for all } j = 1,2,\ldots,n,\, p,k\in \N.$$ But this implies that $\|X_j^k\|\leq R$ for all $k\in \N$, $j = 1,2,\ldots,n$. But then the map $x_j \to (X_j^k)_{k\in \N}$ extends to an algebra morphism from $M$ into $R^\omega$ which preserves the trace. Acknowledgements {#acknowledgements .unnumbered} ================ We are indebted to Professors R. Curto, F. Goodman, C. Frohman, and C. Procesi for several disscutions during the elaboration of this paper. The author is grateful to the EPSRC and to the Deparment of Mathematics of the Cardiff University, where part of this work was done. [aa]{} , Connes’ embedding problem and Lance’s WEP, [*Int. Math. Res. Not.*]{} 2004, no. 10, 501–510. , Classification of injective factors. Cases $II\sb{1},$ $II\sb{\infty },$ $III\sb{\lambda },$ $\lambda \not=1$, [*Ann. of Math. (2)*]{} [**104**]{}(1976), no. 1, 73–115. , A noncommutative moment problem, [*Proc. Amer. Math. Soc.*]{} [**129**]{} (2001), no.  6, 1785–1791 , Discrete groups with Kazhdan’s property ${\rm T}$ and factorization property are residually finite, [*Math. Ann.*]{} [**299**]{}(1994), no. 3, 551–563. , Dusa Central sequences and the hyperfinite factor, [*Proc. London Math. Soc. (3)*]{} [**21**]{}(1970), 443–461. , About the QWEP conjecture, preprint, math. OA/0306067. , A simple proof of a theorem of Kirchberg and related results on $C\sp *$-norms, [*J. Operator Theory*]{} [**35**]{}(1996), no. 2, 317–335. , The invariant theory of $n\times n$ matrices, [*Advances in Math.*]{} [**19**]{}(1976), no. 3, 306–381 , Convex sets associated with von Neumann algebras and Connes’ approximate embedding problem, [*Math. Res. Lett.*]{} [**6**]{}(1999), no. 2, 229–236. , The von Neumann algebra of the non-residually finite Baumslag group $\langle a,b \mid a b^3 a^-1 = b^2 \rangle$ embeds into $R^\omega$ preprint, math. OA/0004172. , [*Functional Analysis*]{}, Second edition, International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991. , Methods of modern mathematical physics. II. Fourier analysis, self-adjointness, Academic Press, New York-London, 1975. , Lectures on von Neumann algebras, Revision of the 1975 original, translated from the Romanian by Silviu Teleman, Editura Academiei, Bucharest; Abacus Press, Tunbridge Wells, 1979. [^1]: Research partially supported by NSF GRANT no. DMS 0200741

--- abstract: 'We have studied electron scattering by out-of-plane (flexural) phonon modes in doped suspended graphene and its effect on charge transport. In the free-standing case (absence of strain) the flexural branch show a quadratic dispersion relation, which becomes linear at long wavelength when the sample is under tension due to the rotation symmetry breaking. In the non-strained case, scattering by flexural phonons is the main limitation to electron mobility. This picture changes drastically when strains above $\bar{u}=10^{-4} n(10^{12}\,\text{cm}^{-2})$ are considered. Here we study in particular the case of back gate induced strain, and apply our theoretical findings to recent experiments in suspended graphene.' address: - 'Instituto de Ciencia de Materiales de Madrid, CSIC, Cantoblanco, E-28049 Madrid, Spain' - 'Centro de Fìsica do Porto, Rua do Campo Alegre 687, P-4169-007 Porto, Portugal' - 'Radboud University Nijmegen, Institute for Molecules and Materials, NL-6525 AJ Nijmegen, The Netherlands' author: - Héctor Ochoa - 'Eduardo V. Castro' - 'M. I. Katsnelson' - 'F. Guinea' title: Scattering by flexural phonons in suspended graphene under back gate induced strain --- graphene ,phonons ,strain ,resistivity 63.22.Rc ,72.10.Di ,72.80.Vp Introduction ============ Graphene is a novel two dimensional material whose low-temperature conductivity is comparable to that of conventional metals [@Morozov_etal], despite much lower carrier concentrations. Interactions with the underlying substrate seem to be the main limitation to electron mobility, and recent experiments on suspended samples show a clear enhancement of mobility (more than one order of magnitude) at low temperatures [@Du_etal; @Bolotin_etal_1; @Bolotin_etal_2]. In suspended graphene carbon atoms can oscillate in the out-of-plane direction leading to a new class of low-energy phonons, the flexural branch [@Landau_book; @Nelson_book]. In the free standing case, these modes show a quadratic dispersion relation, so there is a high number of these low-energy phonons and the graphene sheet can be easily deformed in the out-of-plane direction. For this reason it can be expected that flexural phonons are the intrinsic strongly T-dependent scattering mechanism which ultimately limits mobility at room temperature [@Katsnelson_Geim]. However, since the scattering process always involves two flexural phonons, a membrane characteristic feature, its effect could be reduced, specially at low temperatures [@Mariani_VonOppen]. In the present manuscript we analyse theoretically the contribution of flexural modes to the resistivity in suspended graphene samples. Our results suggest, indeed, that flexural phonons are the main source of resistivity in this kind of samples. We also show how this intrinsic limitation is reduced by the effect of strain. A quantitative treatment of back gate induced strain where graphene is considered as an elastic membrane with clamped edges is given. The model ========= In order to describe long-wavelength acoustic phonons graphene can be seen as a two dimensional membrane whose elastic properties are described by the free energy [@Landau_book; @Nelson_book] $$\mathcal{F}=\frac{1}{2}\kappa\int dxdy(\nabla^{2}h)^{2}+\frac{1}{2}\int dxdy(\lambda u_{ii}^{2}+2\mu u_{ij}^{2}).\label{eq:fe}$$where $\kappa$ is the bending rigidity, $\lambda$ and $\mu$ are Lamé coefficients, $h$ is the displacement in the out of plane direction, and $u_{ij} = 1/2 \left[ \partial_i u_j + \partial_j u_i + ( \partial_i h ) ( \partial_j h ) \right]$ is the strain tensor. Typical parameters for graphene [@Zakharchenko_etal] are $\kappa \approx 1\,$eV, and $\mu \approx 3 \lambda \approx 9\,$eV$\,$Å$^{-2}$. The mass density is $\rho = 7.6 \times 10^{-7}\,$Kg/m$^2$. The longitudinal and transverse in-plane phonons show the usual linear dispersion relation with sound velocities $v_L = \sqrt{\frac{\lambda+2\mu}{\rho}} \approx 2.1 \times 10^4$m/s and $v_T = \sqrt{\frac{\mu}{\rho}} \approx 1.4 \times 10^4$m/s. Flexural phonons have the dispersion $$\omega_{\bf q}^F=\alpha \left|{\bf q}\right|^2 \label{eq:flexural}$$ with $\alpha = \sqrt{\frac{\kappa}{\rho}} \approx 4.6 \times 10^{-7}$m$^2$/s. The quadratic dispersion relation is strictly valid in the absence of strain. At finite strain the dispersion relation of flexural phonons becomes linear at long-wavelength due to rotation symmetry breaking. Let us assume a slowly varying strain field $u_{ij} ( \mathbf{r} )$. The dispersion in Eq.  is changed to: $$\omega_{\bf q}^F ( \mathbf{r} )=| \mathbf{q} | \sqrt{\frac{\kappa}{\rho} | \mathbf{q} |^2 + \frac{\lambda}{\rho} u_{ii} ( \mathbf r ) + \frac{2 \mu}{\rho} u_{ij} ( \mathbf{r} ) \frac{q_i q_j}{| \mathbf{q} |^2}} \label{eq:flexuralstr}$$ In order to keep an analytical treatment we assume uniaxial strain ($u_{xx} \equiv \bar{u}$, and the rest of strain components zero), and drop the anisotropy in Eq.  by considering the effective dispersion relation $$\omega_{\bf q}^F=q\sqrt{\alpha^2q^2+\bar{u}v_L^2}. \label{eq:flexuralstrapp}$$ Long-wavelength phonons couple to electrons in the effective Dirac-like Hamiltonian [@review] through a scalar potential (diagonal in sublattice indices) called the deformation potential, which is associated to the lattice volume change and hence it can be written in terms of the trace of the strain tensor [@Suzuura_Ando; @Manes] $$V ( {\bf r} )=g_0 \left[ u_{xx} ( \mathbf{r} ) + u_{yy} ( \mathbf{r} ) \right]$$ where $g_0\approx 20 - 30\,$eV [@Suzuura_Ando]. Phonons couple also to electrons through a vector potential associated to changes in bond length between carbon atoms, and whose components are related with the strain tensor as [@Manes; @Vozmediano_etal] $${\bf A} ( \mathbf{r} )=\frac{\beta}{a} \left\{ \frac{1}{2} \left[ u_{xx} ( \mathbf{r} ) - u_{yy} ( \mathbf{r} ) \right] , u_{xy} ( \mathbf{r} ) \right\}$$ where $a \approx 1.4\,$Åis the distance between nearest carbon atoms, $\beta =\partial \log ( t ) / \partial \log ( a ) \approx 2-3$ [@Heeger_etal], and $t \approx 3\,$eV is the hopping between electrons in nearest carbon $\pi$ orbitals. Quantizing the displacements fields in terms of the usual bosonic $a_{\vec{\bf q}}^{i=L,T,F}$ operators for phonons of momentum $\mathbf{q}$ we arrive at the interaction Hamiltonian. The term which couples electrons and flexural phonons reads $$\begin{aligned} {\cal H}_{e-ph}^F &= \sum_{\mathbf{k},\mathbf{k}'} \sum_{\mathbf{q},\mathbf{q}'} \left( a_{\mathbf{q}}^F + {a_{- \mathbf{q}}^F}^\dag \right) \left( a_{\mathbf{q}'}^F + {a_{- \mathbf{q}'}^F}^\dag \right) \delta_{\mathbf{k}',\mathbf{k} - \mathbf{q} - \mathbf{q}'} \nonumber \\ &\times \left[\sum_{c=a,b} V^F_{1,\mathbf{q},\mathbf{q}'} c^\dag_{\mathbf{k}} c_{\mathbf{k}'} + \left( V^F_{2,\mathbf{q},\mathbf{q}'} a^\dag_{\mathbf{k}} b_{\mathbf{k}'} + h.c. \right)\right], \label{eq:coupling}\end{aligned}$$ where operators $a_{\mathbf{k}}^\dag$ and $b_{\mathbf{k}}^\dag$ create electrons in Bloch waves with momentum $\vec{\bf k}$ in the $A$ and $B$ sublattices respectively. The matrix elements are $$\begin{aligned} V_{1,\mathbf{q},\mathbf{q}'}^{F} &= -\frac{g_0}{2\varepsilon(\mathbf{q}+\mathbf{q}')}qq'\cos(\phi-\phi')\frac{\hbar}{2\mathcal{V}\rho\sqrt{\omega_{\mathbf{q}}^{F}\omega_{\mathbf{q}'}^{F}}},\nonumber \\ V_{2,\mathbf{q},\mathbf{q}'}^{F} &= -v_{F}\frac{\hbar\beta}{a}\frac{1}{4}qq'e^{i(\phi-\phi')}\frac{\hbar}{2\mathcal{V}\rho\sqrt{\omega_{\mathbf{q}}^{F}\omega_{\mathbf{q}'}^{F}}} \label{eq:potential}\end{aligned}$$ where $\phi_{\bf q} = \arctan \left( q_y / q_x \right)$ and $\mathcal{V}$ is the volume of the system. The effect of screening has been taken into account in the matrix elements of deformation potential through a Thomas-Fermi -like dielectric function $\varepsilon\left(\mathbf q\right)=1+\frac{e^2\mathcal{D}\left(E_F\right)}{2\epsilon_0q}$, where $\mathcal{D}\left(E_F\right)$ is the density of states at Fermi energy. Note that $g = g_0 / \varepsilon(k_F) \approx 3\,$eV in agreement with recent *ab initio* results [@Choi_etal]. Resistivity in the absence of strain {#sec:notstrain} ===================================== From the linearized Boltzmann equation we can calculate the resistivity as $\varrho=\frac{2}{e^{2}v_F^{2}\mathcal{D}(E_{F})}\frac{1}{\tau(k_{F})}$, where $v_F \approx 10^6\,$m/s is the Fermi velocity. Our aim is to compute the inverse of the scattering time of quasiparticles, given by $\tau_{\mathbf{k}}^{-1} = \sum_{\mathbf{k}'} (1-\cos\theta_{\mathbf{k},\mathbf{k}'}) {\cal W}_{\mathbf{k},\mathbf{k}'}$, where ${\cal W}_{\mathbf{k},\mathbf{k}'}$ is the scattering probability per unit time, which can be calculated through the Fermi’s golden rule. For scattering processes mediated by two flexural phonons, within the quasi-elastic approximation, we obtain $$\begin{aligned} {\cal W}_{\mathbf{k},\mathbf{k}'} &= \frac{4 \pi}{\hbar} \sum_{i=1,2} \sum_{\mathbf{q},\mathbf{q}'} \left| V^F_{i,\mathbf{q},\mathbf{q}'} \right|^2 f^{(i)}_{\mathbf{k},\mathbf{k}'} \times \nonumber \\ & \times n_{\mathbf{q}} ( n_{\mathbf{q}' } +1 ) \delta_{\mathbf{k}',\mathbf{k} - \mathbf{q} - \mathbf{q}'} \delta \left(E_{\mathbf{k}} - E_{\mathbf{k}'} \right) \label{rate}\end{aligned}$$ where $f^{(1)}_{\bf k,\bf k'} = 1 + \cos\theta_{\mathbf{k},\mathbf{k}'}$ and $f^{(2)}_{\mathbf{k},\mathbf{k}'} = 1$, $n_{\mathbf q}$ is the Bose distribution, and $E_{\mathbf{k}} = v_F \hbar k$ is the quasi-particle dispersion for the Dirac-like Hamiltonian [@review]. In order to obtain analytical expressions for the scattering rates it is useful to introduce the Bloch-Grüneisen temperature $T_{BG}$. If we take into account that the relevant phonons which contribute to the resistivity are those of momenta $q\gtrsim 2 k_F$ then we have $k_B T_{BG}=\hbar \omega_{2k_F}$. For in-plane longitudinal (transverse) phonons $T_{BG}=57 \sqrt n \,\,\text{K}$ ($T_{BG}=38 \sqrt n \,\,\text{K}$), where $n$ is expressed in $10^{12}$ cm$^{-2}$. For flexural phonons in the absence of strain $T_{BG}=0.1n \,\,\text{K}$. From the last expression it is obvious that for carrier densities of interest the experimentally relevant regime is $T\gg T_{BG}$, so let us concentrate on this limit. In the case of scattering by in-plane phonons at $T\gg T_{BG}$ the scattering rate is given by [@us] $$\frac{1}{\tau_I} \approx \left[ \frac{g^2}{2v_L^2} + \frac{\hbar^2 v_F^2 \beta^2}{4 a^2}\left(\frac{1}{v_L^2}+\frac{1}{v_T^2}\right) \right] \frac{E_F}{2 \rho \hbar^3 v_F^2} k_B T, \label{eq:tau_inplane}$$ where now $g \approx 3\,$eV is the screened deformation potential constant. At $T\ll T_{BG}$ the scattering rate behaves as $\tau^{-1}\sim T^4$, where only the gauge potential contribution is taken into account since the deformation potential is negligible in this regime due to screening effects ($\tau^{-1}\sim T^{6}$ [@Hwang_DasSarma]). In the case of flexural phonons in the non-strained case (in practice $\bar{u}\ll 10^{-4}n$ with $n$ in $10^{12}\,\text{cm}^{-2}$), the scattering rate at $T\gg T_{BG}$ reads [@us] $$\begin{aligned} \frac{1}{\tau_{F}} &\approx \left( \frac{g^2}{2} + \frac{\hbar^2 v_F^2 \beta^2}{4 a^2} \right) \frac{( k_B T )^2}{64 \pi \hbar \kappa^2 E_F} \ln \left( \frac{k_B T}{\hbar \omega_c} \right) + \nonumber \\ &+ \left( \frac{g^2}{4} + \frac{\hbar^2 v_F^2 \beta^2}{4 a^2} \right) \frac{k_B T E_F}{32 \pi v_F^2 \kappa \sqrt{\rho \kappa}} \ln \left( \frac{k_B T}{\hbar \omega_c} \right) \label{eq:tau_flexural}\end{aligned}$$ where we have taken into account two contributions, one coming from the absorption or emission of two thermal phonons, and other involving one non-thermal phonon. The first one dominates over the second at $T\gg T_{BG}$. It is necessary to introduce an infrared cutoff frequency $\omega_c$, where for small but finite strain $\bar{u}\ll 10^{-4}n(10^{12}\,\text{cm}^{-2})$ is just the frequency below which the flexural phonon dispersion becomes linear. From Eq.  we deduce a resistivity which behaves as $\varrho\sim T$, with no dependence on $n$, whereas from Eq.  we have (neglecting the logarithmic correction) $\varrho\sim T^2/n$, as it was deduced for classical ripples in [@Katsnelson_Geim]. As it can be seen in Fig. \[comp\], the resistivity due to scattering by flexural phonons dominates over the in-plane contribution. However, this picture changes considerably if one considers strain above $10^{-4}n(10^{12}\,\text{cm}^{-2})$, as is discussed in the next section. \[comp\] {width="0.7\columnwidth"} Resistivity at finite strains {#sec:strain} ============================= Scattering rate {#subsec:sr} --------------- The Bloch-Grüneisen temperature for flexural phonons at finite strains $\bar{u} \gtrsim 10^{-4}n(10^{12}\,\text{cm}^{-2})$ is $T_{BG} = 28 \sqrt{\bar u n}\,\, \text{K}$. In the relevant high-temperature regime, $T\gg T_{BG}$ the scattering rate can be written as [@us] $$\begin{aligned} \frac{1}{\tau_{F}^{str}} &\approx \left( \frac{g^2}{4} + \frac{\hbar^2 v_F^2 \beta^2}{4 a^2} \right) \frac{E_F ( k_B T )^4}{16\pi \rho^2 \hbar^5 v_F^2 v_L^6 \bar{u}^3} \times \nonumber\\ &\times \left[\mathcal{R}_2\left(\frac{\alpha k_BT}{\hbar v_L^2 \bar{u}}\right) + \mathcal{R}_1\left(\frac{\alpha k_BT}{\hbar v_L^2 \bar{u}} \right) \right] \label{eq:tau_flexural_2}\end{aligned}$$ where $\mathcal{R}_{n}(\gamma)=\int_{0}^{\infty}dx\,\frac{x^{3}}{(\gamma^{2}x^{2}+1)[\exp(\sqrt{\gamma^{2}x^{4}+x^{2}})-1]^{n}}$. The two terms in Eq.  come from the same processes as in Eq.  described above. It is possible to obtain asymptotic analytical expressions for Eq. . For instance, in the limit $T\ll T^* = \frac{\hbar v_L^2\bar{u}}{\alpha k_B}\approx 7\times 10^3 \bar{u}\,\,\text{K}$ the scattering rate behaves as $\tau^{-1}\sim \frac{T^4}{\bar{u}^3}$, whereas in the opposite limit it behaves as $\tau^{-1}\sim \frac{T^2}{\bar{u}}$. The temperature $T^*$ characterises the energy scale at which the flexural phonon dispersion under strain Eq.  cross over from linear to quadratic. It is pertinent to compute the crossover temperature $T^{**}$ above which scattering by flexural phonons dominates when strain is induced. This can be inferred by comparing Eq.  with Eq.  and imposing $\tau_{I}/\tau_F \approx 1$. The numerical solution give for the corresponding crossover $T^{**} \approx 10^6\bar{u}\,\, {\rm K}$. Since $T^{**} \gg T^*$ we can use the respective asymptotic expression for Eq. , $\tau^{-1}\sim \frac{T^2}{\bar{u}}$ to obtain $T^{**} \approx 32 \pi \kappa \bar u /k_B \approx 10^6\bar{u}\,\, {\rm K}$. A remarkable conclusion may then be drawn: scattering due to flexural phonons can be completely suppressed by applying strain as low as $\bar u \gtrsim 0.1\% $. Back gate induced strain ------------------------ {width="0.7\columnwidth"} \[fig:strain\] In order to compute the strain induced by the back gate we consider the simplest case of a suspended membrane with clamped edges. A side view of the system is given in the inset of Fig. \[fig:strain\]. The static height profile is obtained by minimising the free energy, Eq.  in the presence of the load $P=e^{2}n^{2}/(2\varepsilon_{0})$ due to the back gate induced electric field. The built up strain is related with the applied load as [@Landau_book; @Fogler], $$\bar{u}=\frac{PL^{2}}{8h_{0}(\lambda+2\mu)} \approx5\times10^{-5}\frac{[n(10^{12}\mbox{cm}^{-2}) L(\mu\mbox{m})]^{2}}{h_{0}(\mu\mbox{m})}, \label{eq:uP}$$ where $L$ is the length of the trench over which graphene is clamped and $h_{0}$ is the maximum deflection (see the inset of Fig. \[fig:strain\]). We assume the length of the suspended graphene region in the undeformed case to be $L + \Delta L$, where the $\Delta L$ can be either positive or negative. Under the approximation of nearly parabolic deformation (which can be shown to be the relevant case here [@Fogler]) the maximum deflection $h_{0}$ is given by the positive root of the cubic equation $$\left(h_{0}^{2}-\frac{3}{8}L\Delta L\right)h_{0} = \frac{3PL^{4}}{64(\lambda+2\mu)}, \label{eq:root}$$ with trench/suspended-region length mismatch $\Delta L$ such that $\Delta L\ll L$. If $\Delta L=0$ then Eq.  can be easily solved and we obtain for strain $$\bar{u}=\frac{1}{2\sqrt[3]{3}}\left(\frac{PL}{\lambda+2\mu}\right)^{2/3} \approx2\times10^{-3}(n^{2}L)^{2/3}, \label{eq:strainNoSlack}$$ with $n$ in $10^{12}\,\mbox{cm}^{-2}$ and $L$ in $\mu\mbox{m}$. In Fig. \[fig:strain\] the back gate induced strain is plotted as a function of the respective carrier density. For a typical density $n\sim 10^{11}\, \text{cm}^{-2}$ and $\Delta L =0$ we see that a back gate induces strain $\bar u \sim 10^{-4}$. This would imply a crossover from in-plane dominated resistivity $\varrho \sim T$ to $\varrho \sim T^2 /\bar u$ due to flexural phonons at $T^{**} \sim 100\,\,\text{K}$, well within experimental reach. In the next section we will argue that the experimental data in Ref. [@Bolotin_etal_2] can be understood within this framework. Note, however, that gated samples can also fall in the category of non-strained system if $\Delta L > 0$. This is clearly seen in Fig. \[fig:strain\] for $\Delta L$ as small as $\Delta L / L \approx 0.3\%$. Resistivity estimates: comparison with experiment ------------------------------------------------- ![Left: Temperature dependent resistivity from Ref. [@Bolotin_etal_2] at different gate voltages; the inset shows the same in log log scale. Right: Result of Eq. \[eq:rhofit\]; the inset shows the back gate induced strain as given by Eq. .](fig3.eps){width="0.9\columnwidth"} \[fig:expfit\] Bolotin et al. [@Bolotin_etal_2] have recently measured the temperature dependent resistivity in doped suspended graphene. The experimental results are shown in the left panel of Fig. \[fig:expfit\] in linear scale, and the inset shows the same in log log scale. In Ref. [@Bolotin_etal_2] the resistivity was interpreted as linearly dependent on temperature for $T\gtrsim 50\,\,\text{K}$. In the left inset of Fig. \[fig:expfit\], however, it becomes apparent that the behaviour is closer to the $T^2$ dependence in the high temperature regime (notice the slopes of $T^4$ and $T$ indicated in full lines and that of $T^2$ indicated as dashed lines). Within the present framework the obvious candidates to explain the quadratic temperature dependence are flexural phonons. Since the measured resistivity is too small to be due to scattering by non-strained flexural phonons we are left with the case of flexural phonons under strain, where the strain can be naturally assigned to the back gate. In the right panel of Fig. \[fig:expfit\] we show the theoretical $T-$dependence of the resistivity taking into account scattering by in-plane phonons and flexural phonons with finite strain, $$\varrho = \frac{2}{e^{2}v_F^{2}\mathcal{D}(E_{F})} \left(\frac{1}{\tau_I} + \frac{1}{\tau_F^{str}}\right), \label{eq:rhofit}$$ where $1/\tau_I$ is given by Eq.  and $1/\tau_F^{str}$ by Eq. . We calculated the back gate induced strain via Eq. , and related the density and gate voltage as in a parallel plate capacitor model, $n\simeq C_g(V_g-V_{NP})/e$ [@Bolotin_etal_1; @Bolotin_etal_2] ($C_g = 60\,\, \text{aF}/\mu \text{m}^2$ and $V_{NP} \approx -0.4\,\,\text{V}$). The obtained strain is shown in the right inset of Fig. \[fig:expfit\] versus applied gate voltage. It is seen that the system is well in the region where Eq. \[eq:tau\_flexural\_2\] is valid. The agreement between left and right panels in Fig. \[fig:expfit\] for realistic parameter values [@params] is an indication that we are indeed observing the consequences of scattering by flexural phonons at finite, though very small strains. Full quantitative agreement is not aimed, however, since our two side clamped membrane is a very crude approximation to the real device [@Bolotin_etal_1; @Bolotin_etal_2]. Conclusions =========== Our theoretical results suggest that scattering by flexural phonons constitute the main limitation to electron mobility in doped suspended graphene. This picture changes drastically when the sample is strained. In that case, strains with not too large values, as those induced by the back gate, can suppress significantly this source of scattering. This result opens the door to the possibility of modify locally the resistivity of a suspended graphene by strain modulation. [00]{} S.V. Morozov, et al., Phys. Rev. Lett. 100 (2008) 016602. X. Du, et al., Nature Nanotech. 3 (2008) 491. K.I. Bolotin, et al., Solid State Commun. 146 (2008) 351. K.I. Bolotin, et al., Phys. Rev. Lett. 101 (2008) 096802. L. D. Landau and E. M. Lifschitz, *Theory of Elasticity* (Pergamon Press, Oxford, 1959). D. Nelson, in *Statistical Mechanics of Membranes and Surfaces*, edited by D. Nelson, T. Piran, and S. Weinberg (World Scientific, Singapore, 1989). M.I. Katsnelson and A.K. Geim, Phil. Trans. R. Soc. A 366 (2008) 195. E. Mariani and F. Von Oppen, Phys. Rev. Lett. 100 (2008) 076801; 100 (2008) 249901(E). K.V. Zakharchenko, M.I. Katsnelson, and A. Fasolino, Phys. Rev. Lett. 102 (2009) 046808. A.H. Castro Neto, et al., Rev. Mod. Phys. 81 (2009) 109. H. Suzuura and T. Ando, Phys. Rev. B 65 (2002) 235412. J.L. Mañes, Phys. Rev. B 76 (2007) 045430. S.-M. Choi, S.-H. Jhi, and Y.-W. Son, Phys. Rev. B 81 (2010) 081407. M.A.H. Vozmediano, M.I. Katsnelson, and F. Guinea, Phys. Rep., in press, doi:10.1016/j.physrep.2010.07.003. A.J. Heeger, et al., Rev. Mod. Phys. 60 (1988) 781. Derivation details will be given elsewhere. E.H. Hwang and S. Das Sarma, Phys. Rev. B 77 (2008) 115449. M.M. Fogler, F. Guinea, M.I. Katsnelson, Phys. Rev. Lett. 101 (2008) 226804. We used $g = 3\, \text{eV}$, $\beta = 3$, $\Delta L = 0$, and $L = 0.3\,\mu\text{m}$. The latter parameter is interpreted as an effective length mimicking the difference between our two side clamped membrane and the real four point clamped device.

--- abstract: 'The S-expansion method is a generalization of the Inönü-Wigner (IW) contraction that allows to study new non-trivial relations between different Lie algebras. Basically, this method combines a Lie algebra $\mathcal{G}$ with a finite abelian semigroup $S$ in such a way that a new S-expanded algebra $\mathcal{G}_{S}$ can be defined. When the semigroup has a zero-element and/or a specific decomposition, which is said to be resonant with the subspace structure of the original algebra, then it is possible to extract smaller algebras from $\mathcal{G}_{S}$ which have interesting properties. Here we give a brief description of the S-expansion, its applications and the main motivations that lead us to elaborate a Java library, which automatizes this method and allows us to represent and to classify all possible S-expansions of a given Lie algebra.' address: - '$^1$ Departamento de Física, Universidad de Concepción, Casilla 160-C, Concepción, Chile' - '$^2$ Grupo de Matemática Aplicada, Departamento de Ciencias Básicas, Univerdidad del Bío-Bío, Campus Fernando May, Casilla 447, Chillán, Chile' - '$^3$ APC, Universite Paris Diderot, 10, rue Alice Domon et Leonie Duquet, 75205 Paris Cedex 13, France' - | $^4$ Instituto de Física Corpuscular (IFIC), Edificio Institutos de Investigación,\ c/ Catedrático José Beltrán, 2, E-46980 Paterna, España author: - | Carlos Inostroza$^{1}$, Igor Kondrashuk$^{2},$ Nelson Merino$^3$\ and Felip Nadal$^4$ title: 'On a Java library to perform S-expansions of Lie algebras' --- Introduction ============ As is well-known, the theory of Lie groups and algebras plays an essential role in Physics: it represents the mathematical tool allowing to describe the continues symmetries of a physical system and, via the Noether theorem, this is directly connected with the corresponding conservation laws of the system. Since the fifties, different mechanisms allowing to establish non-trivial relations between Lie algebras appeared and were very useful to understand interrelations between different physical theories. The original idea was introduced in Ref. [@Segal], where it was pointed out that if two physical theories are related by means of a limit process (like Newtonian mechanics and special relativity which are related by the limit where the speed of light $c$ goes to infinite), then the corresponding symmetry groups (the Galilean and Poincaré groups, in the example) under which those theories are invariant should also be related by means of a similar limit process. This process was formally introduced in Ref. [@IW; @IW2] and it is known nowdays as Inönü-Wigner (IW) contraction. Many generalized contractions were introduced in the literature during the last decades. In particular, the Weimar-Woods (WW) contraction [@WW2; @WW; @WW3] is one of the most general realization of this method. Starting with an algebra which have a certain subspace structure, the WW contraction performs a suitable rescaling with a real parameter on the generators of each subspace. Then, a special limit for that parameter leads to the contracted algebra, which has the same dimension than the original one, but very different properties[^1]. Another peculiar generalization of the contraction, known as *expansion* method, was parallelly introduced in the context of string theory [@hs] and supergravity [@aipv1]. This procedure not only is able to reproduce the WW contractions when the dimension is preserved in the process, but also may lead to expanded algebras whose dimension is higher than the original one. The main difference with the contraction method is that, using the dual description of a Lie algebra in terms of Maurer-Cartan (MC) forms, the rescaling is performed on some coordinates of the Lie group manifold and not on the generators of the Lie algebra. Here we focus on an even more general procedure called *S-expansion*[^2] [@irs], which combines the structure constants of the original algebra with the inner multiplication law of an abelian finite semigroup of order $n$ to define a new *S-expanded algebra* $\mathcal{G}_{S}$. When the semigroup have a zero-element and/or a specific decomposition, which is said to be resonant with the subspace structure of the original algebra, then it is possible to extract smaller algebras, called *resonant subalgebras* and *reduced algebras*. In particular, the previous expansion method [@hs; @aipv1] can be reproduced as a $0_{S}$-reduction of the resonant subalgebra for an expansion with a special family of semigroups denoted by $S_{E}^{\left( N\right)}$. During the last decade, many applications using the S-expansion has been performed [@Izaurieta:2006aj; @Izaurieta:2009hz; @Quinzacara:2012zz; @Concha:2013uhq; @Concha:2014zsa; @Quinzacara:2013uua; @Crisostomo:2014hia; @Crisostomo:2016how], mainly in the context of modified theories of gravity. At the begining, they considered only the family of semigroups $S_{E}^{(N)}$, which lead to the definition of the so called $\mathfrak{B}_{N}$ algebras. The use of other abelian semigroups to perform S-expansions of Lie algebras was first considered in Refs. [@Caroca:2011qs], where it was pointed out that possible new applications could be performed if the following question is analyzed: *given two Lie algebras, is it possible to find a suitable semigroup that relates them by means of an S-expansion?* Of course, the answer depends on the specific algebras that we want to connect and involves to consider all possible finite abelian semigroups in each order[^3] (for a brief review about that classification see the presentation [@PresCI], by C. Inostroza). In the same line, a study of the general properties $S$-expansion method (in the context of the classification of Lie algebras) was made in Ref. [@Andrianopoli:2013ooa] and revealed that some properties of the semigroup allow to determine if a given property of the original algebra (like semisimplicity, compactness, etc) will be preserved or not under the expansion process. These results were shown to be useful as criteria to answer if two given algebras can be S-related. In particular, semigroups preserving semisimplicity were identified. The results given in Refs. [@Caroca:2011qs] and [@Andrianopoli:2013ooa] (and in particular the use of semigroups preserving semisimplicity) were used in Ref. [@Diaz:2012zza] to show that the semisimple version of the so called Maxwell algebra (introduced in [@Soroka:2006aj]) can be obtained as an expansion of the AdS algebra. Later, this result was generalized in Refs. [@Salgado:2014qqa; @Concha:2016hbt] to new families of semigroups generating algebras denoted by $\mathfrak{C}_{N}$ and $\mathfrak{D}_{N}$ which have been useful to construct new (super)gravity models [@Salgado:2014jka; @Fierro:2014lka; @Concha:2014vka; @Concha:2014xfa; @Concha:2014tca; @Gonzalez:2014tta; @Concha:2015tla; @Concha:2015woa; @Concha:2016kdz; @Gonzalez:2016xwo; @Durka:2016eun; @Concha:2016tms; @Ipinza:2016con; @Concha:2016zdb; @Penafiel:2016ufo; @Artebani:2016gwh; @Ipinza:2016bfc]. These several new applications show the importance of considering also semigroups outside of the $S_{E}^{(N)}$ family. Motivated by this, we have constructed a Java library [@webJava; @Inostroza:2017ezc] that automatizes the S-expansion procedure and that is aimed to create a general picture about all possible S-relations between Lie algebras. Brief review of the S-expansion method ====================================== Consider a Lie algebra $\mathcal{G}$ with generators $\{X_{i}\}$ and Lie product $\left[ X_{i},X_{j}\right] =C_{ij}^{k}X_{k}$ where $C_{ij}^{k}$ are the structure constants. Consider also a finite abelian semigroup $S=\{\lambda_{\alpha},\alpha=1,\ldots,n\}$. An informal way (but useful for our purposes) of expressing the semigroup multiplication law is by means of quantities called *selectors*[^4], denoted by $K_{\alpha\beta}^{\kappa}$ and defined by the relation $\lambda_{\alpha}\cdot\lambda_{\beta} =\lambda_{\gamma\left( \alpha,\beta\right) }=K_{\alpha\beta}^{\rho}\lambda_{\rho}$, where $K_{\alpha\beta}^{\rho}=1$ if $\rho=\gamma\left( \alpha,\beta\right) $ and $K_{\alpha\beta}^{\rho}=0$ if $\rho\neq\gamma\left( \alpha,\beta\right) \,$. Then, the S-expansion consists of the following steps. **Step I: Constructing S-expanded algebra.** As shown in [@irs], the set $\mathcal{G}_{S}$ $\mathcal{=}$ $S\otimes\mathcal{G}$ (with $\otimes$ being the Kronecker product between the matrix representation of $S$ and $\mathcal{G}$) is also Lie algebra, which is called *expanded algebra*, if the basis elements are defined as $X_{\left( i,\alpha\right) }\equiv\lambda_{\alpha}\otimes X_{i}$ and Lie product by $\left[ X_{\left( i,\alpha\right) },X_{\left( j,\beta\right) }\right] \equiv\lambda_{\alpha}\cdot\lambda_{\beta}\otimes\left[ X_{i},X_{j}\right] =C_{\left( i,\alpha\right) \left( j,\beta\right) }^{\left( k,\gamma \right) }X_{\left( k,\gamma\right) }\label{z3}$. The structure constants of the expanded algebra $\mathcal{G}_{S}$ are fully determined by the selectors and the structure constants of the original Lie algebra $\mathcal{G}$, i.e., $C_{\left( i,\alpha\right) \left( j,\beta\right) }^{\left( k,\gamma\right) }=K_{\alpha\beta}^{\gamma}% C_{ij}^{k}\,$. **Step II: Extraction of the resonant subalgebra.** Consider the case where the original algebra has the subspace decomposition $\mathcal{G}=V_{0}\oplus V_{1}$ with the following structure $$\left[ V_{0},V_{0}\right] \subset V_{0}\,,\ \ \ \left[ V_{0},V_{1}\right] \subset V_{1}\,,\ \ \ \left[ V_{1},V_{1}\right] \subset V_{0}\label{r1}%$$ Suppose also that a given semigroup has a decomposition in subsets $S=S_{0}\cup S_{1}$ satisfying $$S_{0}\cdot S_{0}\in S_{0}\,,\ \ \ S_{0}\cdot S_{1}\in S_{1}\,,\ \ \ S_{1}\cdot S_{1}\in S_{0}\label{r2}%$$ which is called *resonant condition*, because of the similarity with the subspace structure (\[r1\]) of the algebra[^5]. Then, as shown in [@irs], the set $\mathcal{G}_{S,R}=\left( S_{0}\otimes V_{0}\right) \oplus\left(S_{1}\otimes V_{1}\right)$ is a subalgebra of the expanded algebra $\mathcal{G}_{S}$ $\mathcal{=}% $ $S\otimes\mathcal{G}$. **Step III: Extraction of the** $0_{S}$-**reduced algebra.** If semigroup contains an element $0_{S}$ satisfying $\lambda_{\alpha}% \cdot0_{S}=0_{S}$ for any element $\lambda_{\alpha}\in S$, then this element is called a zero element. In that case, the sector $0_{S}\otimes\mathcal{G}$ can be removed from the expanded algebra in such a way that what is left is also a Lie algebra, called the $0_{S}$*-reduced algebra*. Remarkably, the reduced algebra is not necessarily a subalgebra of the expanded algebra. Need of automizing the procedure ================================ First we notice that the steps II and III are independent, but can also be applied simultaneously. This means that, depending on the semigroup that is going to be used in the expansion, the following algebras can be obtained: 1. the expanded algebra $\mathcal{G}_{S} = S \otimes \mathcal{G}$, 2. the resonant subalgebra $\mathcal{G}_{S,R}=\left( S_{0}\otimes V_{0}\right) \oplus\left(S_{1}\otimes V_{1}\right)$, if the semigroup have at least one resonant decomposition, 3. the $0_{S}$-reduced algebra if the semigroup have a zero element $0_{S}$, 4. the $0_{S}$-reduction of the resonant subalgebra, if the semigroup has simultaneously a zero element and at least one resonant decomposition. In order to study all possible S-expansions (i-iv) of a given algebra, one should consider the full set of abelian semigroups. As it will be reviewed in the presentation by C. Inostroza [@PresCI], the problem of enumerating the all non-isomorphic finite semigroups of a certain order is a non-trivial problem because the number of semigroups increases very quickly with the order of the semigroup. Thus, this task can be performed only up to a certain order and, in particular, we have used the program *gen.f* of Ref. [@Hildebrant] to generate the files *sem.2*, *sem.3*, *sem.4*, *sem.5* and *sem.6* which contain all the non isomorphic semigroups up to order 6. Those files can be used as input data for many of the programs that compose our library [@webJava; @Inostroza:2017ezc]. For example, in the order 3 there are 18 non-isomorphic semigroups denoted by $S_{(3)}^{a} $ with $a=1,...18$, from which only 12 of them are abelian[^6]. Thus, the different type of expansions that can be done with them are represented in the figure \[fig:fig0\] where, for reasons of space, we use only the label ‘$a$’ to name the different semigroups in the horizontal axis, while in the vertical axis we represent the different kinds of expansions that can be performed with them. With a *gray number* it has also been identified the expansions that preserve the semisimplicity. ![Representing different kind of expansions with semigroups of order $3$.[]{data-label="fig:fig0"}](figure1.eps) To identify all possible resonant decompositions of all non-isomorphic semigroups in higher orders is not a trivial task. For this reason, we have constructed a Java library [@webJava; @Inostroza:2017ezc] which allows us to study all possible S-expansions of the type (i-iv) using the mentioned lists *sem.n*. On the maximal order of semigroups which the Java Library can perform ===================================================================== In some cases it is useful to have the full lists of non-isomorphic semigroups up to certain order. However, there is an intrinsic computational limitation to construct these lists for order 8 and higher. Indeed, the number of non-isomorphic semigroups in those orders has been evaluated only by using indirect techniques. For lower orders, instead, the construction of these tables is possible. For example, in Ref. [@Hildebrant] a fortran program gen.f was proposed, which, according to what is claimed in this reference, is able to generate these lists up to order 7. However, after running that fortran program we were able to obtain those lists only up to order 6. These lists are used as inputs for our library [@webJava] although its methods are not restricted to the order 6. Indeed, the methods of our library also allow us to work with semigroup of higher orders. The only issue is that we do not have the full list of non-isomorphic tables for those higher orders. For example, our methods are able to read a multiplication table of order 20 and show in few seconds if that table is associative, commutative, to identify if there is a zero element and also allow to check if a given decomposition is resonant. The only task that involves more computations and time is the method [*findAllResonances*]{} that finds out all possible resonant decompositions of a given semigroup. In that case, one can obtain the answer in few minutes for semigroups of order 8, during about one hour for semigroups of order 9, and in our case we were unable to obtain an answer over 3 days for semigroups of order 10. All the computations above were made in usual laptop computers. Conclusions =========== We have presented a brief description of the S-expansion method, its applications and the motivations to automatize this procedure. We have made this with a set of methods organized in the form of Java Library [@webJava; @Inostroza:2017ezc] which is able not only to represent and perform basic operations with semigroups and Lie algebras but also to perform operations that are intrinsic of the S-expansion method, such as the characterization of resonances and the corresponding representation of the resonant subalgebra and the reduced algebras. The main features of this library will be described in the related talk by C. Inostroza [@PresCI]. References {#references .unnumbered} ========== [99]{} Segal I E 1951 A class of operator algebras which are determined by groups [*Duke Math. J.*]{} [**18**]{} 221-65 Inönü E and Wigner E P 1953 On the contraction of groups and their representations [*Proc. Nat. Acad. Sci. USA*]{} [**39**]{} 510-24 Inönü E 1964 Contractions of Lie groups and their representations [*Group theoretical concepts in elementary particle physics*]{}, ed Gürsey F, (New York: Gordon and Breach) pp 391-402 Weimar-Woods E 1991 Contractions of Lie algebras: generalized Inonu-Wigner contractions versus graded contractions [*J. Math. Phys.*]{} [**32**]{} 2028 Weimar-Woods E 1995 Contractions of Lie algebras: generalized Inonu-Wigner contractions versus graded contractions [*J. Math. Phys.*]{} [**36**]{} 4519-48 Weimar-Woods E 2000 Contractions, generalized Inönü and Wigner contractions and deformations of finite-dimensional Lie algebras [*Rev. Math. Phys.*]{} [**12**]{} 1505-29 Hatsuda M and Sakaguchi M 2003 Wess-Zumino term for the AdS superstring and generalized Inonu-Wigner contraction [*Prog. Theor. Phys.*]{} [**109**]{} 853 ([*Preprint*]{} hep-th/0106114 ) de Azcarraga J A, Izquierdo J M, Picon M and Varela O 2003 Generating Lie and gauge free differential (super)algebras by expanding Maurer-Cartan forms and Chern-Simons supergravity [*Nucl. Phys.*]{} B [**662**]{} 185 ([*Preprint*]{} hep-th/0212347) Izaurieta F, Rodriguez E and Salgado P 2006 Expanding Lie (super)algebras through Abelian semigroups [*J. Math. Phys.*]{} [**47**]{} 123512 ([*Preprint*]{} hep-th/0606215) Izaurieta F, Rodriguez E, Perez A and Salgado P 2009 Dual Formulation of the Lie Algebra S-expansion Procedure [*J. Math. Phys.*]{} [**50**]{} 073511 ([*Preprint*]{} arXiv:0903.4712 \[hep-th\]) Caroca R, Merino N and Salgado P 2009 S-Expansion of Higher-Order Lie Algebras [*J. Math. Phys.*]{} [**50**]{} 013503 ([*Preprint*]{} arXiv:1004.5213 \[math-ph\]) Caroca R, Merino N, Perez A and Salgado P 2009 Generating Higher-Order Lie Algebras by Expanding Maurer Cartan Forms [*J. Math. Phys.*]{} [**50**]{} 123527 ([*Preprint*]{} arXiv:1004.5503 \[hep-th\]) Caroca R, Merino N, Salgado P and Valdivia O 2011 Generating infinite-dimensional algebras from loop algebras by expanding Maurer-Cartan forms [*J. Math. Phys.*]{} [**52**]{} 043519 Izaurieta F, Rodriguez E and Salgado P 2008 Eleven-dimensional gauge theory for the M algebra as an Abelian semigroup expansion of osp(32|1) [*Eur. Phys. J.*]{} C [**54**]{} 675 ([*Preprint*]{} hep-th/0606225) Izaurieta F, Rodriguez E, Minning P, Salgado P and Perez A 2009 Standard General Relativity from Chern-Simons Gravity [*Phys. Lett.*]{} B [**678**]{} 213 ([*Preprint*]{} arXiv:0905.2187 \[hep-th\] ) Quinzacara C A C and Salgado P 2012 Black hole for the Einstein-Chern-Simons gravity [*Phys. Rev.*]{} D [**85**]{} 124026 ([*Preprint*]{} arXiv:1401.1797 \[gr-qc\] ) Concha P K, Peñafiel D M, Rodríguez E K and Salgado P 2013 Even-dimensional General Relativity from Born-Infeld gravity [*Phys. Lett.*]{} B [**725**]{} 419 ([*Preprint*]{} arXiv:1309.0062 \[hep-th\]) Concha P K, Peñafiel D M, Rodríguez E K and Salgado P 2015 Generalized Poincaré algebras and Lovelock-Cartan gravity theory [*Phys. Lett.*]{} B [**742**]{} 310 ([*Preprint*]{} arXiv:1405.7078 \[hep-th\]) Quinzacara C A C and Salgado P 2013 Stellar equilibrium in Einstein-Chern-Simons gravity [*Eur. Phys. J.*]{} C [**73**]{} 2479 Crisóstomo J, Gómez F, Salgado P, Quinzacara C, Cataldo M and del Campo S 2014 Accelerated FRW Solutions in Chern-Simons Gravity [*Eur. Phys. J.*]{} C [**74**]{} 3087 ([*Preprint*]{} arXiv:1401.2128 \[gr-qc\]) Crisóstomo J, Gómez F, Quinzacara C and Salgado P 2016 Static solutions in Einstein-Chern-Simons gravity [*J. Cosmol. Astropart. Phys.*]{} JCAP 1606 (2016) no.06, 049 ([*Preprint*]{} arXiv:1601.06592 \[gr-qc\]) Caroca R, Kondrashuk I, Merino N and Nadal F 2013 Bianchi spaces and their three-dimensional isometries as S-expansions of two-dimensional isometries [*J. Phys.*]{} A [**46**]{} 225201 ([*Preprint*]{} arXiv:1104.3541 \[math-ph\]) Bianchi L 1898 Sugli spazi a tre dimensioni che ammettono un gruppo continuo di movimenti [*Memorie  di Matematica e di Fisica della Societa Italiana delle Scienze*]{} Serie Terza [**Tomo XI**]{} 267-352 Inostroza C, Kondrashuk I, Merino N and Nadal F 2017 On the algorithm to find S-related Lie algebras [*Merged talk by C Inostroza at ACAT 2017*]{} Andrianopoli L, Merino N, Nadal F and Trigiante M 2013 General properties of the expansion methods of Lie algebras [*J. Phys.*]{} A [**46**]{} 365204 ([*Preprint*]{} arXiv:1308.4832 \[gr-qc\]) Díaz J D, Fierro O, Izaurieta F, Merino N, Rodríguez E, Salgado P and Valdivia O 2012 A generalized action for (2 + 1)-dimensional Chern-Simons gravity [*J. Phys.*]{} A [**45**]{} 255207 ([*Preprint*]{} arXiv:1311.2215 \[gr-qc\]) Soroka D V and Soroka V A 2009 Semi-simple extension of the (super)Poincare algebra [*Adv. High Energy Phys.*]{} 234147 ([*Preprint*]{} hep-th/0605251) Salgado P and Salgado S 2014 $\mathfrak{so}(D-1,1)\otimes\mathfrak{so}(D-1,2)$ algebras and gravity [*Phys. Lett.*]{} B [**728**]{} 5 Concha P K, Durka R, Merino N and Rodríguez E K 2016 New family of Maxwell like algebras [*Phys. Lett.*]{} B [**759**]{} 507 ([*Preprint*]{} arXiv:1601.06443 \[hep-th\]) Salgado P, Szabo R J and Valdivia O 2014 Topological gravity and transgression holography [*Phys. Rev.*]{} D [**89**]{} 084077 ([*Preprint*]{} arXiv:1401.3653 \[hep-th\]) Fierro O, Izaurieta F, Salgado P and Valdivia O 2014 (2+1)-dimensional supergravity invariant under the AdS-Lorentz superalgebra ([*Preprint*]{} arXiv:1401.3697 \[hep-th\]) Concha P K, Peñafiel D M, Rodríguez E K and P. Salgado P 2014 Chern-Simons and Born-Infeld gravity theories and Maxwell algebras type [*Eur. Phys. J.*]{} C [**74**]{} 2741 ([*Preprint*]{} arXiv:1402.0023 \[hep-th\]) Concha P K and Rodríguez E K 2014 Maxwell Superalgebras and Abelian Semigroup Expansion [*Nucl. Phys.*]{} B [**886**]{} 1128 ([*Preprint*]{} arXiv:1405.1334 \[hep-th\]) Concha P K and Rodríguez E K 2014 N = 1 Supergravity and Maxwell superalgebras [*J. High Energy Phys.*]{} JHEP09(2014)090 ([*Preprint*]{} arXiv:1407.4635 \[hep-th\]) González N, Salgado P, Rubio G and Salgado S 2014 Einstein-Hilbert action with cosmological term from Chern-Simons gravity [*J. Geom. Phys.*]{} [**86**]{} 339 Concha P K, Rodríguez E K and Salgado P 2015 Generalized supersymmetric cosmological term in $N=1$ Supergravity [*J. High Energy Phys.*]{} JHEP08(2015)009 ([*Preprint*]{} arXiv:1504.01898 \[hep-th\]) Concha P K, Fierro O, Rodríguez E K and Salgado P 2015 Chern-Simons supergravity in $D=3$ and Maxwell superalgebra [*Phys. Lett.*]{} B [**750**]{} 117 ([*Preprint*]{} arXiv:1507.02335 \[hep-th\]) Concha P K, Durka R, Inostroza C, Merino N and Rodríguez E K 2016 Pure Lovelock gravity and Chern-Simons theory [*Phys. Rev.*]{} D [**94**]{} 024055 ([*Preprint*]{} arXiv:1603.09424 \[hep-th\] ) González N, Rubio G, Salgado P and Salgado S 2016 Generalized Galilean algebras and Newtonian gravity [*Phys. Lett.*]{} B [**755**]{} 433 ([*Preprint*]{} arXiv:1604.06313 \[hep-th\]) Durka R 2017 Resonant algebras and gravity [*J. Phys.*]{} A [**50**]{} 145202 ([*Preprint*]{} arXiv:1605.00059 \[hep-th\]) Concha P K, Merino N and Rodríguez E K 2017 Lovelock gravities from Born-Infeld gravity theory [*Phys. Lett.*]{} B [**765**]{} 395-401 ([*Preprint*]{} arXiv:1606.07083 \[hep-th\]) Ipinza M C, Concha P K, Ravera L and Rodríguez E K 2016 On the Supersymmetric Extension of Gauss-Bonnet like Gravity [*J. High Energy Phys.*]{} JHEP09(2016)007 ([*Preprint*]{} arXiv:1607.00373 \[hep-th\]) Concha P K, Fierro O and Rodríguez E K 2016 Inönü-Wigner Contraction and $D=2+1$ Supergravity [*Eur. Phys. J.*]{} C [**77**]{} 48 ([*Preprint*]{} arXiv:1611.05018 \[hep-th\]) Peñafiel D M and Ravera L 2017 Generalized Inönü-Wigner Contraction as $S$-Expansion with Infinite Semigroup and Ideal Subtraction [*J. Math. Phys.*]{} [**58**]{} 081701 ([*Preprint*]{} arXiv:1611.05812 \[hep-th\]) Artebani M, Caroca R, Ipinza M C, Peñafiel D M and Salgado P 2016 Geometrical aspects of the Lie algebra S-expansion procedure [*J. Math. Phys.*]{} [**57**]{} 023516 ([*Preprint*]{} arXiv:1602.04525 \[math-ph\]) Ipinza M C, Lingua F, Peñafiel D M and Ravera L 2016 An Analytic Method for $S$-Expansion involving Resonance and Reduction [*Fortsch. Phys.*]{} [**64**]{} 854-80 ([*Preprint*]{} arXiv:1609.05042 \[hep-th\]) https://github.com/SemigroupExp/Sexpansion/releases/tag/v1.0.0 Inostroza C, Kondrashuk I, Merino N and Nadal F 2017 A Java library to perform S-expansions of Lie algebras [*Preprint*]{} arXiv:1703.04036 \[cs.MS\] Hildebrant J 2001 Handbook of Finite Semigroup Programs [*LSU Mathematics Electronic Preprint Series*]{} [*Preprint*]{} 2001-24 [^1]: For example, if the original algebra is semisimple, then the contracted algebra is usually non-semisimple. [^2]: The dual formulation of the S-expansion in terms of MC forms was performed in [@irs2], while in Refs. [@Caroca:2010ax; @Caroca:2010kr; @Caroca:2011zz] the method was also extended to other mathematical structures, like the case of higher order Lie algebras and infinite dimensional loop algebras. [^3]: An example, it was considered the 2 and 3-dimensional that acts transitively on 2 and 3-dimensional spaces, i.e., the two 2-dimensional algebras $[X_{1},X_{2}]=0$, $[X_{1},X_{2}]=X_{1}$ and the ten 3-dimensional algebras classified by Bianchi [@bian]. It was shown, that only four Bianchi algebras can be obtained as S-expansions of the the 2-dimensional algebras if one uses some specific semigroups of order 4, which do not belong in general to the $S_{E}^{(N)}$ family. Thus, this result cannot be obtained neither by a contraction nor by the expansion method [@aipv1]. The procedure used to construct by hand the multiplication table of those semigroups also made clear that, if a given problem involves the use of semigroups of higher order, then the use of computer programs is needed. [^4]: In particular, the selectors provide a matrix representation for the semigroup $\lambda_{\alpha}\rightarrow\left( \lambda_{\alpha}\right) _{\ \beta}^{\rho}=K_{\alpha\beta}^{\rho}$, which is used in our library to represent a given semigroup. [^5]: As it can be seen in [@irs], one can deal with algebras and semigroups having decompositions which are more general, but in the first version of our library we only consider the case given by (\[r1\]) and (\[r2\]). [^6]: The non-isomorphic abelian semigroups of order 3 are: $$ \[c\][l|lll]{}$S_{\left( 3\right) }^{1}$ & $\lambda_{1}$ & $\lambda_{2}$ & $\lambda_{3}% $\ $\lambda_{1}$ & $\lambda_{1}$ & $\lambda_{1}$ & $\lambda_{1}$\ $\lambda_{2}$ & $\lambda_{1}$ & $\lambda_{1}$ & $\lambda_{1}$\ $\lambda_{3}$ & $\lambda_{1}$ & $\lambda_{1}$ & $\lambda_{1}$    \[c\][l|lll]{}$S_{\left( 3\right) }^{2}$ & $\lambda_{1}$ & $\lambda_{2}$ & $\lambda_{3}% $\ $\lambda_{1}$ & $\lambda_{1}$ & $\lambda_{1}$ & $\lambda_{1}$\ $\lambda_{2}$ & $\lambda_{1}$ & $\lambda_{1}$ & $\lambda_{1}$\ $\lambda_{3}$ & $\lambda_{1}$ & $\lambda_{1}$ & $\lambda_{2}$    \[c\][l|lll]{}$S_{\left( 3\right) }^{3}$ & $\lambda_{1}$ & $\lambda_{2}$ & $\lambda_{3}% $\ $\lambda_{1}$ & $\lambda_{1}$ & $\lambda_{1}$ & $\lambda_{1}$\ $\lambda_{2}$ & $\lambda_{1}$ & $\lambda_{1}$ & $\lambda_{1}$\ $\lambda_{3}$ & $\lambda_{1}$ & $\lambda_{1}$ & $\lambda_{3}$    \[c\][l|lll]{}$S_{\left( 3\right) }^{6}$ & $\lambda_{1}$ & $\lambda_{2}$ & $\lambda_{3}% $\ $\lambda_{1}$ & $\lambda_{1}$ & $\lambda_{1}$ & $\lambda_{1}$\ $\lambda_{2}$ & $\lambda_{1}$ & $\lambda_{1}$ & $\lambda_{2}$\ $\lambda_{3}$ & $\lambda_{1}$ & $\lambda_{2}$ & $\lambda_{3}$   $$$$ \[c\][l|lll]{}$S_{\left( 3\right) }^{7}$ & $\lambda_{1}$ & $\lambda_{2}$ & $\lambda_{3}% $\ $\lambda_{1}$ & $\lambda_{1}$ & $\lambda_{1}$ & $\lambda_{1}$\ $\lambda_{2}$ & $\lambda_{1}$ & $\lambda_{2}$ & $\lambda_{1}$\ $\lambda_{3}$ & $\lambda_{1}$ & $\lambda_{1}$ & $\lambda_{3}$   \[c\][l|lll]{}$S_{\left( 3\right) }^{9}$ & $\lambda_{1}$ & $\lambda_{2}$ & $\lambda_{3}% $\ $\lambda_{1}$ & $\lambda_{1}$ & $\lambda_{1}$ & $\lambda_{1}$\ $\lambda_{2}$ & $\lambda_{1}$ & $\lambda_{2}$ & $\lambda_{2}$\ $\lambda_{3}$ & $\lambda_{1}$ & $\lambda_{2}$ & $\lambda_{2}$   \[c\][l|lll]{}$S_{\left( 3\right) }^{10}$ & $\lambda_{1}$ & $\lambda_{2}$ & $\lambda_{3}% $\ $\lambda_{1}$ & $\lambda_{1}$ & $\lambda_{1}$ & $\lambda_{1}$\ $\lambda_{2}$ & $\lambda_{1}$ & $\lambda_{2}$ & $\lambda_{2}$\ $\lambda_{3}$ & $\lambda_{1}$ & $\lambda_{2}$ & $\lambda_{3}$   \[c\][l|lll]{}$S_{\left( 3\right) }^{12}$ & $\lambda_{1}$ & $\lambda_{2}$ & $\lambda_{3}% $\ $\lambda_{1}$ & $\lambda_{1}$ & $\lambda_{1}$ & $\lambda_{1}$\ $\lambda_{2}$ & $\lambda_{1}$ & $\lambda_{2}$ & $\lambda_{3}$\ $\lambda_{3}$ & $\lambda_{1}$ & $\lambda_{3}$ & $\lambda_{2}$  $$$$ \[c\][l|lll]{}$S_{\left( 3\right) }^{15}$ & $\lambda_{1}$ & $\lambda_{2}$ & $\lambda_{3}% $\ $\lambda_{1}$ & $\lambda_{1}$ & $\lambda_{1}$ & $\lambda_{3}$\ $\lambda_{2}$ & $\lambda_{1}$ & $\lambda_{1}$ & $\lambda_{3}$\ $\lambda_{3}$ & $\lambda_{3}$ & $\lambda_{3}$ & $\lambda_{1}$   \[c\][l|lll]{}$S_{\left( 3\right) }^{16}$ & $\lambda_{1}$ & $\lambda_{2}$ & $\lambda_{3}% $\ $\lambda_{1}$ & $\lambda_{1}$ & $\lambda_{1}$ & $\lambda_{3}$\ $\lambda_{2}$ & $\lambda_{1}$ & $\lambda_{2}$ & $\lambda_{3}$\ $\lambda_{3}$ & $\lambda_{3}$ & $\lambda_{3}$ & $\lambda_{1}$   \[c\][l|lll]{}$S_{\left( 3\right) }^{17}$ & $\lambda_{1}$ & $\lambda_{2}$ & $\lambda_{3}% $\ $\lambda_{1}$ & $\lambda_{1}$ & $\lambda_{2}$ & $\lambda_{2}$\ $\lambda_{2}$ & $\lambda_{2}$ & $\lambda_{1}$ & $\lambda_{1}$\ $\lambda_{3}$ & $\lambda_{2}$ & $\lambda_{1}$ & $\lambda_{1}$   \[c\][l|lll]{}$S_{\left( 3\right) }^{18}$ & $\lambda_{1}$ & $\lambda_{2}$ & $\lambda_{3}% $\ $\lambda_{1}$ & $\lambda_{1}$ & $\lambda_{2}$ & $\lambda_{3}$\ $\lambda_{2}$ & $\lambda_{2}$ & $\lambda_{3}$ & $\lambda_{1}$\ $\lambda_{3}$ & $\lambda_{3}$ & $\lambda_{1}$ & $\lambda_{2}$  $$

--- abstract: 'We investigate the inhomogeneous chiral dynamics of the O(4) linear sigma model in 1+1 dimensions using the time dependent variational approach in the space spanned by the squeezed states. We compare two cases, with and without the Gaussian approximation for the Green’s functions. We show that mode-mode correlation plays a decisive role in the out-of-equilibrium quantum dynamics of domain formation and squeezing of states.' author: - 'N. Ikezi$^{1,2}$' - 'M. Asakawa$^2$' - 'Y. Tsue$^3$' date: Received 10 October 2003 title: | Nonequilibrium chiral dynamics by the time dependent variational approach\ with squeezed states --- The possibility of the formation of the disoriented chiral condensate (DCC) in high energy heavy ion collisions has been extensively studied with various methods. In classical approximation [@ref:classical01; @ref:classical02], it has been shown that the amplification of long wavelength modes of the pion fields takes place when the system starts with the nonequilibrium initial condition, quench initial condition [@ref:classical01; @ref:classical02]. In addition to the amplification, spatial correlation of the fields has been also shown to grow. Although the classical approximation is expected to work well in incorporating nonequilibrium aspects of the system when pion density is large, it is still desirable to include quantum effects. In fact, investigations in this direction have been also carried out extensively with the Hartree approximation, the large $N$ approximation, and so on [@boyanovski01; @cooper01]. In most of the previous studies which include quantum effects, however, it has been assumed that the system is spatially homogeneous. Problems such as insufficient thermalization at late times and impossibility to describe domain structures have been recognized. It has not been conclusive whether there is a chance for the correlations to grow through nonequilibrium time evolution. There are at least two ways for possible improvement. One is to include higher order quantum corrections and the other is to accommodate spatial inhomogeneity. We will pursue the latter in this paper. Recently, the dynamics of spatially inhomogeneous system has been studied by several groups quantum mechanically [@berges; @cooper02; @smit; @bettencourt] and it has been shown that the thermalization of the quantum fields can occur. In these works, the Gaussian approximation, in which the Green’s functions are assumed to be diagonal in momentum space, has been adopted because of computational reasons. Physically, it corresponds to ignoring correlations between modes with different momenta, and under the approximation different modes can interact only through the mean fields. However, it is possible that the direct coupling of modes through the off-diagonal correlations is important for the time evolution of the system when the system does not possess translational invariance. To see if such an effect is substantial, we study the dynamics of chiral phase transition in spatially inhomogeneous systems with off-diagonal components of the Green’s function in momentum space fully taken into account. In this paper, we take the O(4) linear sigma model as a low energy effective theory of QCD and apply the method of the time dependent variational approach (TDVA) with squeezed states. This method was originally developed by Jackiw and Kerman as an approximation in the functional Schrödinger approach [@JK] and later it was shown to be equivalent to TDVA with squeezed states by Tsue and Fujiwara [@TF]. In this approach, the trial state is a squeezed state $$\begin{aligned} | \Phi (t) \rangle & = & \prod_{a}| \Phi_{a} (t) \rangle, \nonumber \\ | \Phi_{a} (t) \rangle & = & \exp \bigl[S_a(t)\bigr] \cdot N_a (t) \cdot \exp \bigl[T_a (t)\bigr] |0 \rangle, \nonumber \\ S_a (t) & = & i \int d \vec{x} \bigl[ C_{a}(\vec{x},t) {\phi_a}(\vec{x}) - D_{a}(\vec{x},t){\pi_a}(\vec{x}) \bigr], \nonumber \\ T_a (t) & = & \int d \vec{x} d \vec{y} \phi_{a}(\vec{x}) \Bigl[ - \frac{1}{4} \bigl[ G_{a} ^{-1}(\vec{x},\vec{y},t) \nonumber \\ & & - G_{a}^{(0)-1}(\vec{x},\vec{y}) \bigr] + i \Pi_{a}(\vec{x},\vec{y},t) \Bigr] \phi_{a} (\vec{y}) \label{squeezedstate},\end{aligned}$$ where $a$ runs from 0 to 3. $a=0$ is for the sigma field and $a = 1-3$ are for the pion fields. $|0 \rangle$ is the reference vacuum and $G_{a}^{(0)}(\vec{x},\vec{y}) = \langle 0|\phi_a(\vec{x})\phi_a(\vec{y})|0 \rangle $. $\phi_a(\vec{x})$ and $\pi_a(\vec{x})$ are the field operator and conjugate field operator for the field $a$, respectively. $C_{a}(\vec{x},t)$, $D_{a}(\vec{x},t)$, $G_{a}(\vec{x},\vec{y},t)$, and $\Pi_{a}(\vec{x},\vec{y},t)$ are the mean field variables for $\phi_{a}$ field at $\vec{x}$ and $t$, the canonical conjugate variable for the mean field, the quantum correlation for $\vec{x} \neq \vec{y}$ (fluctuation around the mean field for $\vec{x} = \vec{y}$), and the canonical conjugate variable for $G_{a}(\vec{x},\vec{y},t)$, respectively, and all of them are real functions. $N_a (t)$ is a normalization constant. $T(t)$ is an operator that describes the squeezing, and if $T_a (t)$ is set to 0, $| \Phi_{a} (t) \rangle $ is reduced to a coherent state with the expectation values of $\phi(\vec{x},t)$ and $\pi(\vec{x},t)$ given by $C_{a}(\vec{x},t)$ and $D_{a}(\vec{x},t)$, respectively. The Hamiltonian $H$ of the O(4) linear sigma model is given by $$\begin{aligned} H & = &\int d \vec{x} \sum_{a=0}^3 \biggl \{ \frac{1}{2}\pi_a(\vec{x})^2 + \frac{1}{2}\vec{\nabla}\phi_a(\vec{x})\cdot \vec{\nabla}\phi_a(\vec{x}) \nonumber \\ & & + \lambda \bigl[ \phi_a(\vec{x})^2 - v^2 \bigr]^2 - h\phi_0(\vec{x}) \biggr \}. \label{hamiltonian:o4}\end{aligned}$$ As shown in Eq. (\[squeezedstate\]), the trial state is specified by $C_{a}(\vec{x},t)$, $D_{a}(\vec{x},t)$, $G_{a}(\vec{x},\vec{y},t)$, and $\Pi_{a}(\vec{x},\vec{y},t)$. Their time evolution is determined through the time dependent variational principle: $$\delta \int dt \langle \Phi (t) | i \frac{\partial}{\partial t} - H | \Phi (t) \rangle = 0. \label{TDVP}$$ In this approach, correlation between different modes in momentum space arises through the scattering of quanta caused by the nonlinear coupling term in the model Hamiltonian even if there is initially no such correlation. This can be seen best from the following equations of motion in momentum space, $$\begin{aligned} \ddot{C}_{a}(\vec{k},t) & = & - \vec{k}^{2} - \mathcal{M}_{a}^{(1)}(\vec{k},t) ,\nonumber \\ \dot{G}_{a}(\vec{k},\vec{k}',t) & = & 2 \langle \vec{k} | \left[ G_{a}(t) \Pi_{a}(t) + \Pi_{a}(t) G_{a}(t) \right] | \vec{k}' \rangle , \nonumber \\ \dot{\Pi}_{a}(\vec{k},\vec{k}',t)& = & \mbox{$\frac{1}{8}$} \langle \vec{k} | G_{a}^{-1}(t) G_{a}^{-1}(t) | \vec{k}' \rangle - 2 \langle \vec{k} | \Pi_{a}(t) \Pi_{a}(t) | \vec{k}' \rangle \nonumber \\ & &- \vec{k}^{2} \delta ^{3} ( \vec{k} - \vec{k}' ) - \mbox{$\frac{1}{2}$} \mathcal{M}_{a}^{(2)} (\vec{k}-\vec{k}',t), \nonumber \\ \mathcal{M}_{a}^{(1)}(\vec{k},t) & = & \Bigl[- m^{2} + 4 \lambda C_{a}^{2}(\vec{k},t) + 12 \lambda G_{a}(\vec{k},\vec{k},t) \nonumber \\ & & + 4 \lambda \sum_{b (\neq a)} \bigl( C_{b}^{2}(\vec{k},t) + G_{b}(\vec{k},\vec{k},t) \bigr) \Bigr] C_{a}(\vec{k},t) \nonumber \\ & &- h\delta_{a0} , \nonumber \\ \mathcal{M}_{a}^{(2)}(\vec{k},t) & = & - m^{2} + 12 \lambda C_{a}^{2}(\vec{k},t) + 12 \lambda G_{a}(\vec{k},\vec{k},t) \nonumber \\ & & + 4 \lambda \sum_{b (\neq a)} \left( C_{b}^{2}(\vec{k},t) + G_{b}(\vec{k},\vec{k},t) \right), \label{eom}\end{aligned}$$ where $m^2 = 4 \lambda v^2$, and $C_a(\vec{k},t)$, $G_a(\vec{k},\vec{k}',t)$, and $\Pi_a(\vec{k},\vec{k}',t)$ are the mean fields for the $\phi_a$ field with momentum $\vec{k}$, the correlation between modes with momenta $\vec{k}$ and $\vec{k}'$ for $\vec{k} \neq \vec{k}'$ (the quantum fluctuation around the mean field for $\vec{k} = \vec{k}'$), the canonical conjugate variable for $G_a(\vec{k},\vec{k}',t)$, respectively. In Eq. (\[eom\]), we have used the notation, $$\langle \vec{k} | H(t) I(t) | \vec{k}' \rangle = \int \frac{d \vec{k}''}{(2\pi)^{3}} H(\vec{k},\vec{k}'',t) I(\vec{k}'',\vec{k}',t).$$ In the Gaussian approximation, $G_{a}(\vec{k},\vec{k}',t)$ and $\Pi_{a}(\vec{k},\vec{k}',t)$ are set to zero for $\vec{k}\neq \vec{k}'$ and correlations between different modes in momentum space are ignored. However, $\mathcal{M}^{(2)}(\vec{k}-\vec{k}',t)$ in Eq. (\[eom\]), which originates from the four-point interaction terms in the O(4) linear sigma model, couples modes with different momenta and correlations between them develop even if initially there exists no correlation among them. In numerical calculation, we have assumed the one-dimensional spatial dependence for the mean fields and the Green’s functions for computational simplicity. In addition, we have imposed the periodic boundary condition for the mean fields and the Green’s functions. We have carried out calculation on a lattice with the lattice spacing $d=1.0$ fm and the total length $L=64$ fm, which leads to the momentum cutoff $\Lambda = 1071$ MeV. The parameters $\lambda$, $v$, and $h_0$ are determined so that they give the pion mass $M_{\pi} = 138$ MeV, the sigma meson mass $M_{\sigma}=500 $ MeV, and the pion decay constant $f_{\pi} = 93 $ MeV in the vacuum following the prescription given in Ref. [@TKI], and we have obtained $\lambda = 3.44$, $v = 110$ MeV, and $h_0 = (103 ~\rm{MeV})^{3}$ . There are several scenarios for the DCC formation in high energy heavy ion collisions. Here we adopt the quench scenario. In this scenario, the chiral order parameters remain around the top of the Mexican hat potential after the rapid change of the effective potential from the chirally symmetric phase to the chirally broken phase. In order to take this situation into account, we have used the following initial condition; at each lattice point, the mean field variable for the chiral fields and their conjugate variables $C_a(\vec{x},0)$ and $D_a(\vec{x},0)$ are randomly distributed according to the Gaussian form with the following parameters [@AM], $$\begin{aligned} \langle C_a(\vec{x},0) \rangle &=& 0, \nonumber \\ \langle C_a(\vec{x}, 0)^2 \rangle - \langle C_a(\vec{x}, 0) \rangle ^2 &=& \delta^2 , \nonumber \\ \langle D_a(\vec{x}, 0) \rangle &=& 0 , \nonumber \\ \langle D_a(\vec{x}, 0)^2 \rangle - \langle D_a(\vec{x},0) \rangle ^2 &=& \frac{D}{d^2} \delta ^2 , \label{initial:mf4}\end{aligned}$$ where $D = 1$ is the spatial dimension and $\delta$ is the Gaussian width. We shall use $\delta = 0.19 v$ in the following calculations. In relating the Gaussian widths of $C_a(\vec{x}, 0)$ and $D_a(\vec{x}, 0)$, we have taken advantage of the virial theorem [@AM]. As for the initial conditions for the quantum fluctuation and correlation, we have assumed that their values are those realized in the case where each state in momentum space were in a coherent state with a degenerate mass $m_{0}$ for the sigma meson and pions, namely, $$\begin{aligned} & & G_a(\vec{x}, \vec{y},0) = \int_{0}^{\Lambda} \frac{d \vec{k}}{(2 \pi)^{3}} \frac{1}{2 \omega_k} e ^{i \vec{k}\cdot(\vec{x}- \vec{y})} , \nonumber \\ & & \Pi_a(\vec{x}, \vec{y}, 0) = 0 , \label{coh_fluc}\end{aligned}$$ where $\omega _{k} = \sqrt{m_{0}^2 + \vec{k}^2}$. We adopt $m_{0} = 200$ MeV. As shown above, the Green’s functions $G_{a}(\vec{x},\vec{y},0)$ and $\Pi_a(\vec{x},\vec{y},0)$ are initially diagonal in momentum space. The off-diagonal components appear in the course of the time evolution of the system due to the direct mode-mode coupling induced by $\mathcal{M}^{(2)}$ in Eq. (\[eom\]). We have carried out two sets of numerical calculations. In one case, we have taken into account all components of the two-point Green’s functions (case I), while in the other case only the diagonal components of the two-point Green’s functions (case II) were included in the calculation as in most of the preceding works. In Fig. \[fig:meanfields01\], we show the time evolution of the mean fields for the sigma and the third component of the pion fields ($\phi_0$ and $\phi_3$, respectively) obtained with the calculation including all components of the Green’s functions (case I). As a whole, the expectation value of the sigma field approaches a constant. On the other hand, that of the pion field oscillates around zero and shows a domain structure with long correlation length. This is the formation of DCC domains. It is observed that the domain structure continues to grow till as late as $t \sim 40$ fm. In Fig. \[fig:meanfields02\], we show the time evolution of the same mean fields obtained with the Gaussian approximation, i.e., without the off-diagonal components of the Green’s functions (case II). At the beginning, the behavior of the mean fields is similar to that in case I. However, after a few femtometers, there appears a clear difference between the two cases. In case II, short range fluctuation is dominant and no long length correlation is observed. No qualitative change in the behavior of the mean fields takes place in case II after a few femtometers. This tells us that the mode-mode correlation plays a decisive role in the formation of DCC domains. To examine the growth of spatial correlation of the pion fields more quantitatively, we define the following spatial correlation function $C(r, t)$, $$C(r, t) = \frac{\int {\vec{\phi} }(\vec{x}) \!\cdot\! {\vec{\phi}}(\vec{y}) \delta (|\vec{x} - \vec{y}| - r) d\vec{x} d\vec{y} } {\int | {\vec\phi}(\vec{x}) | | {\vec\phi}(\vec{y}) | \delta (|\vec{x} - \vec{y}| - r) d\vec{x} d\vec{y} },$$ where ${\vec{\phi} }(\vec{x})\!\cdot\! {\vec{\phi}}(\vec{y}) = \sum_{i=1}^{3} {\phi_i}(\vec{x}) {\phi_i}(\vec{y})$ and $|{\vec\phi}(\vec{x}) | = \sqrt{ \sum_{i=1}^{3} {\phi_i^2}(\vec{x}) }$. In Figs. \[fig:correlation1\](a) and \[fig:correlation1\](b), we show this spatial correlation function in cases I and II, respectively. The correlation functions are calculated by taking average over 10 events. Substantial generation of the correlation takes place after the typical time scale of the initial rolling down of the chiral fields, say, a few femtometers in case I, while the growth of the spatial correlation ends in case II by $t\sim5$ fm. The domain formation of DCC shown in Fig. \[fig:meanfields01\](b) and this growth of spatial correlation shown in Fig. \[fig:correlation1\](a) beyond the rolling down time scale may be related to the parametric resonance [@mrowczynski95; @hiro-oka00]. We are currently investigating this possibility. ![(Color online) The spatial correlations for the pion fields in case I (a) and case II (b).[]{data-label="fig:correlation1"}](cr.eps){width="65mm"} Next we show the time evolution of the quantum fluctuation, which is represented by the same-point Green’s function $G_{a}(\vec{x}, \vec{x}, t)$. We define the spatially averaged fluctuation function $\langle F_a(t)\rangle _{{\rm space}}$ at time $t$ as $$\langle F_{a}(t) \rangle_{{\rm space}} = \frac{1}{V} \int G_{a}(\vec{x} , \vec{x}, t) d{\vec{x}} ,$$ where $V$ is the volume of the system. In Fig. \[fig:quantum\_fluctuation\], we compare the time evolution of the spatially averaged quantum fluctuation of the the third component of the pion field, $\langle F_3 (t) \rangle_{{\rm space}}$ in the two cases. We observe remarkable increase of quantum fluctuation in case I, while only small amplification is seen in case II. This increase also lasts until about $t = 40$ fm. The comparison between case I and case II tells us that the off-diagonal correlations take an important role also for the enhancement of the quantum fluctuation. We have in fact found the growth of off-diagonal components of the Green’s functions in momentum space. Note that this phenomenon cannot be described unless the off-diagonal correlation is introduced. This has an important meaning also for the identical particle correlation in high energy heavy ion collisions. Usually it is assumed that identical particles with different momenta are emitted independently. However, if there is quantum correlation between two different modes, this assumption becomes invalid, and it will be necessary to reformulate the theory for the identical particle correlation. In summary, we have studied the inhomogeneous chiral dynamics of the O(4) linear sigma model in 1+1 dimensions using TDVA with squeezed states. We have compared two cases. One is a general case in which both the mean fields and the Green’s functions are inhomogeneous, and the other is a case with the Gaussian approximation, where translational invariance is imposed on the Green’s functions. We have shown for the first time that the large correlated domains can be realized in the quench scenario with quantum mechanical treatment. More specifically, we have shown that the large amplification of quantum fluctuation and large domain structure emerge when all components of the Green’s functions are retained, while only small quantum fluctuation and small and noisy domain structure are seen in the case with the Gaussian approximation. The dynamics in $1+2$ and $1+3$ dimensional cases is of great interest and indispensable for the understanding of the DCC formation in ultrarelativistic heavy ion collisions. We expect more enhanced domain growth in $1+2$ and $1+3$ dimensional cases because of the existence of more mode-mode correlations in such cases. We plan to confirm it by actual numerical calculation. N.I. thanks the nuclear theory group at Kyoto University for encouragement. M.A. is partially supported by the Grants-in-Aid of the Japanese Ministry of Education, Science and Culture, Grants Nos. 14540255. Y.T. is partially supported by the Grants-in-Aid of the Japanese Ministry of Education, Science and Culture, Grants No. 13740159 and 15740156. Numerical calculation was carried out at Yukawa Institute for Theoretical Physics at Kyoto University and Tohoku-Gakuin University. We thank T. Otofuji for making it possible to use the computing facility at Tohoku-Gakuin University. \[sec:TeXbooks\] K. Rajagopal and F.  Wilczek, Nucl. Phys. [**B404**]{}, 577 (1993). M. Asakawa, Z. Huang, and X.-N. Wang, Phys. Rev. Lett. [**74**]{}, 3126 (1995). D. Boyanovsky, H. J. de Vega, and R. Holman, Phys. Rev. D [**49**]{}, 2769 (1994); D. Boyanovsky, H. J. de Vega, R. Holman, D. S. Lee, and A. Singh, *ibid.* [**51**]{}, 4419 (1995); D. Boyanovsky, H. J. de Vega, R. Holman, and J. Salgado, *ibid.* [**59**]{}, 125009 (1999). F. Cooper, S. Habib, Y. Kluger, E. Mottola, J. P. Paz, and P. R. Anderson, Phys. Rev. D [**50**]{}, 2848 (1994); F. Cooper, Y. Kluger, E. Mottola, and J. P. Paz, *ibid.* [**51**]{}, 2377 (1995). J. Berges, Nucl. Phys. [**A699**]{}, 847 (2002); G. Aarts, D. Ahrensmeier, R. Baier, J. Berges, and J. Serreau, Phys. Rev. D [**66**]{}, 045008 (2002). F. Cooper, J. F. Dawson, and B. Mihaila, Phys. Rev. D [**67**]{}, 056003 (2003). M. Salle, J. Smit, and J. C. Vink, Phys. Rev. D [**64**]{}, 025016 (2001); M. Salle, J. Smit, and J. C. Vink Nucl. Phys. [**B625**]{}, 495 (2002); G. Aarts and J. Smit, Phys. Rev. D [**61**]{}, 025002 (2001). L. M. A. Bettencourt, K. Pao, and J. G. Sanderson, Phys. Rev. D [**65**]{}, 025015 (2002). R. Jackiw and A. Kerman, Phys. Lett. [**A71**]{}, 158 (1979). Y. Tsue and Y. Fujiwara, Prog. Theor. Phys. [**86**]{}, 443 (1991); *ibid.* [**86**]{}, 469 (1991). Y. Tsue, A. Koike, and N. Ikezi, Prog. Theor. Phys. [**106**]{}, 469 (2001) 807. M. Asakawa, H. Minakata, and B.Müller, Phys. Rev. D [**58**]{}, 094011 (1998). S. Mrówczyński and B. Müller, Phys. Lett. B [**363**]{}, 1 (1995). H. Hiro-Oka and H. Minakata, Phys. Rev. C [**61**]{}, 044903 (2000).

--- abstract: 'Excess work is a non-diverging part of the work during transition between nonequilibrium steady states (NESSs). It is a central quantity in the steady state thermodynamics (SST), which is a candidate for nonequilibrium thermodynamics theory. We derive an expression of excess work during quasistatic transitions between NESSs by using the macroscopic linear response relation of NESS. This expression is a line integral of a vector potential in the space of control parameters. We show a relationship between the vector potential and the response function of NESS, and thus obtain a relationship between the SST and a macroscopic quantity. We also connect the macroscopic formulation to microscopic physics through a microscopic expression of the nonequilibrium response function, which gives a result consistent with the previous studies.' author: - | Tatsuro Yuge\ Department of Physics, Osaka University, 1-1, Toyonaka, Osaka, 560-0043, Japan title: An expression of excess work during transition between nonequilibrium steady states --- Introduction ============ The second law of thermodynamics gives a fundamental limit to thermodynamic operations on systems in equilibrium states. One of its formulations provides the lower bound for the work performed on a system during a thermodynamics operation that induces a transition between equilibrium states; the lower bound is given by the change in the free energy and is achieved for quasistatic operations. To establish analogous thermodynamic theory for nonequilibrium steady state (NESS) is one of the challenging problems in physics. In recent attempts [@Esposito_etal07; @KNST; @EspositoBroeck; @DeffnerLutz2010; @DeffnerLutz2012; @Takara_etal; @SaitoTasaki; @SagawaHayakawa; @Nakagawa; @VerleyLacoste; @Boksenbojm_etal; @Mandal; @MaesNetocny; @Bertini_etal; @Yuge_etal], particularly investigated are relations to be satisfied by the work and entropy production (or heat) for transitions between NESSs. One of the candidates for nonequilibrium thermodynamics is the steady state thermodynamics (SST) proposed in [@OonoPaniconi]. A central idea of the SST is to use excess work (and excess heat), which is defined as follows. In nonequilibrium states, work is continuously supplied to the system, so that the total work $W_{\rm tot}$ during the transition between NESSs diverges. Therefore, for the construction of a meaningful thermodynamic theory for the transition, it is necessary to take a finite part out of $W_{\rm tot}$. To this end, the excess work $W_{\rm ex}$ is defined by subtracting from $W_{\rm tot}$ the integral of the steady work flow in the instantaneous NESS at each point of the operation [@OonoPaniconi; @Landauer]. Studies along this idea have been developed in Refs. [@KNST; @SagawaHayakawa; @SaitoTasaki; @MaesNetocny; @Bertini_etal; @Yuge_etal; @HatanoSasa; @SasaTasaki]. We here concentrate our attention on quasistatic transitions. In the regime near to equilibrium, the work version of the results in Refs. [@KNST; @SaitoTasaki] states that the excess work $W_{\rm ex}$ for quasistatic transitions is given by the change in a certain scalar potential. In the regime far from equilibrium, by contrast, the work version of Refs. [@SagawaHayakawa; @Yuge_etal] states that in general it is not equal to the change in any scalar function but equal to a geometrical quantity; i.e., it is given by a line integral of a vector potential $\bm{A}^W$ in the operation parameter space. This suggests that $\bm{A}^W$ plays an important role in SST. Since these studies are based on the microscopic dynamics, the relationship with the microscopic state has been developed. In contrast, the relationship with macroscopic quantities has been less clear although there exist some studies on this issue [@Boksenbojm_etal; @Mandal; @Maes]. Particularly important is to clarify how the transport coefficients and response functions are treated in the framework of SST, because the transport coefficients are main quantities to characterize NESS and are accessible in experiments. In this paper, as a first step toward this issue, we derive an expression of $W_{\rm ex}$ for quasistatic transitions in terms of the linear response function of NESS. We show that $W_{\rm ex}$ is equal to the line integral of a vector potential $\bm{A}^W$ and clarify the relation between $\bm{A}^W$ and the response function. Since the derivation relies only on the macroscopic phenomenological equation (linear response relation), the result is universally valid (independent of microscopic detail). We also show that in the regime near to equilibrium $W_{\rm ex}$ is given by the change of a scalar potential thanks to the reciprocal relation. Furthermore, we connect the macroscopic theory for $\bm{A}^W$ to microscopic physics by using a microscopic expression of the response function, which is called response-correlation relation (RCR) [@ShimizuYuge; @Yuge]. We obtain a microscopic expression of $\bm{A}^W$ that is consistent with the work version of the results in Refs. [@SagawaHayakawa; @Yuge_etal]. Setup ===== We consider a system S that is in contact with multiple (say $n$) reservoirs. A schematic diagram of the setup is shown in Fig. \[fig:schematic\]. The $i$th reservoir is in the equilibrium state characterized by the chemical potential $\mu_i$. We denote the set of the chemical potentials by $\mu$, i.e., $\mu=\{ \mu_i \}_{i=1}^n$. The temperatures of all the reservoirs are set to the same value. We denote the particle current between S and the $i$th reservoir by $I_i$, where we take the sign of $I_i$ positive when it flows from the reservoir to S. More precisely, $I_i$ at time $t$ is defined as $I_i(t) = -dN_i(t)/dt$, where $N_i$ is the particle number in the $i$th reservoir. We assume that the reservoirs are sufficiently large so that they are not affected by the change in $N_i$ and remain in the equilibrium states on the time scale of interest. We also assume that for a fixed $\mu$ a stable NESS is realized in S uniquely and independently of initial states after a relaxation time. We note that in the NESS $\sum_i \langle I_i \rangle_\mu^{\rm ss} = 0$ holds due to the particle number conservation in the total system (S plus reservoirs), where $\langle I_i \rangle_\mu^{\rm ss}$ is the expectation value of the current $I_i$ in the NESS characterized by $\mu$. ![ A schematic diagram of the setup. The system S is connected to $n$ reservoirs. The $i$th reservoir is characterized by the chemical potential $\mu_i$. The particle current $I_i$ flows from the $i$th reservoir into the system. []{data-label="fig:schematic"}](fig1.eps "fig:"){width="0.6\linewidth"} \[fig:illustration1\] Examples of such a setup are seen in field-effect semiconductor devices. A typical one is the modulation-doped field-effect transistor (MODFET) [@Sze; @Davies]. In this example, the system S is realized as the two-dimensional electron system, and the reservoirs are the electrodes (source, drain, and gate). Transition between NESSs ------------------------ Suppose that at the initial time $t_0$ the system S is in a NESS characterized by $\mu$. The difference $\mu_i - \mu_j$ may be so large that the initial NESS is far from equilibrium. At $t_0+0$, we change the chemical potentials from $\mu$ to $\mu' = \{ \mu_i + \delta\mu_i \}_{i=1}^n$ with small constants $\{ \delta\mu_i \}_{i=1}^n$. Then the state of the system S varies in time for $t>t_0$, and after a sufficiently long time it settles to a new NESS characterized by $\mu'$. In this paper we investigate the work $W$ done on S during the transition between the NESSs. For this purpose, we here consider the expectation value of the current $I_i$ from the macroscopic viewpoint of the linear response relation. To the linear order in $\delta\mu$, we can express the expectation value $\langle I_i \rangle_{\mu'}^t$ at $t>t_0$ as $$\begin{aligned} \langle I_i \rangle_{\mu'}^t = \langle I_i \rangle_\mu^{\rm ss} + \sum_{j=1}^n \Psi_{ij}(t-t_0) \delta\mu_j + O(\delta\mu^2), \label{linearResponse}\end{aligned}$$ where $\Psi_{ij}(\tau) \equiv \int_0^\tau d\tau' \Phi_{ij}(\tau')$, and $\Phi_{ij}(\tau)$ is the linear response function of the NESS [@Marconi_etal; @ChetriteGupta; @Baiesi_etal; @BaiesiMaes], which satisfies the causality relation $\Phi_{ij}(\tau<0)=0$. We note that the linear response relation (\[linearResponse\]) is a relation around a NESS (not equilibrium state) and is valid for a NESS even far from equilibrium if the NESS is stable to perturbations. Also we can express the expectation value of the current $I_i$ in the final NESS (characterized by $\mu'$) by the long-time limit of Eq. (\[linearResponse\]): $$\begin{aligned} \langle I_i \rangle_{\mu'}^{\rm ss} = \langle I_i \rangle_\mu^{\rm ss} + \sum_j \tilde{\Phi}_{ij} \delta\mu_j + O(\delta\mu^2), \label{steadyCurrentAfter}\end{aligned}$$ where $\tilde{\Phi}_{ij} \equiv \lim_{\tau\to\infty}\Psi_{ij}(\tau)$ is the transport coefficient (differential conductivity) of the initial NESS (characterized by $\mu$). We again note that the limit exists if the NESS is stable. Excess work =========== General case ------------ In nonequilibrium states, work $W$ is continuously supplied to the system S from the reservoirs, accompanied by the particle current to S. We can use the equilibrium thermodynamics to estimate the work done by the reservoirs since they are in the equilibrium states; When the particle number in the $i$th reservoir increases by $\Delta N_i$, the work done by the $i$th reservoir is given by $W_i = - \mu'_i \Delta N_i$ Therefore, the unit-time work by the $i$th reservoir is $J_i^W = \mu'_i I_i$. From Eq. (\[linearResponse\]), we obtain the average work flow $\langle J^W \rangle_{\mu'}^t = \sum_i \langle J_i^W \rangle_{\mu'}^t$ at time $t>t_0$ as $$\begin{aligned} \langle J^W \rangle_{\mu'}^t &= \sum_i \langle I_i \rangle_{\mu'}^t \bigl( \mu_i + \delta\mu_i \bigr) \nonumber\\ &\simeq \sum_i \langle I_i \rangle_\mu^{\rm ss} \bigl( \mu_i + \delta\mu_i \bigr) + \sum_{ij} \mu_i \Psi_{ij}(t-t_0) \delta\mu_j, %+ O(\delta\mu^2). \label{workFlow_t}\end{aligned}$$ where we neglected the $O(\delta\mu^2)$ terms in the second line. The total work $W_{\rm tot}$ during the transition between the NESSs is given by the time-integral of Eq. (\[workFlow\_t\]). However, $W_{\rm tot}$ is a diverging quantity because the system S remains to be supplied with the work from the reservoirs after it reaches the final NESS. To extract a finite quantity intrinsic to the transition, we employ the idea of the SST [@OonoPaniconi]; we subtract from $W_{\rm tot}$ the contribution of the steady work flow $\langle J^W \rangle_{\mu'}^{\rm ss}$ in the final NESS: $$\begin{aligned} W_{\rm ex} \equiv \int_{t_0}^\infty dt \Bigl( \langle J^W \rangle_{\mu'}^t - \langle J^W \rangle_{\mu'}^{\rm ss} \Bigr). \label{excessWork}\end{aligned}$$ We refer to this quantity as the excess work. We note that $W_{\rm ex}$ is related to the excess heat $Q_{\rm ex}$ as $W_{\rm ex} + Q_{\rm ex} = \Delta U$, where $\Delta U$ is the change in the energy of S between the NESSs, due to the energy conservation in the transition between the NESSs and the energy balance in the steady states. Our definition (\[excessWork\]) of $W_{\rm ex}$ is consistent with the definition of $Q_{\rm ex}$ in Refs. [@KNST; @SaitoTasaki; @SagawaHayakawa; @Yuge_etal]. We also note that the steady flow $\langle J^W \rangle_{\mu'}^{\rm ss}$ is equal to the long-time limit of Eq. (\[workFlow\_t\]): $$\begin{aligned} \langle J^W \rangle_{\mu'}^{\rm ss} &= \sum_i \langle I_i \rangle_\mu^{\rm ss} \bigl( \mu_i + \delta\mu_i \bigr) + \sum_{ij} \mu_i \tilde{\Phi}_{ij} \delta\mu_j. %+ O(\delta\mu^2). \label{workFlow_ss}\end{aligned}$$ By substituting Eqs. (\[workFlow\_t\]) and (\[workFlow\_ss\]) into Eq. (\[excessWork\]), we obtain $$\begin{aligned} W_{\rm ex} = \sum_j A_j^W \delta\mu_j, \label{geometricalExcessWork}\end{aligned}$$ where the $j$th component $A_j^W$ of the vector potential $\bm{A}^W$ is given by $$\begin{aligned} A_j^W = \sum_i \mu_i \int_{t_0}^\infty dt \Bigl[ \Psi_{ij}(t-t_0) - \tilde{\Phi}_{ij} \Bigr]. \label{vectorPotentialResponse}\end{aligned}$$ Equation (\[geometricalExcessWork\]) indicates that the excess work during quasistatic transitions between NESSs is not given by the difference of some scalar function $F$ but given by the geometrical quantity unless $A_j^W$ is equal to the $\mu_j$-derivative of $F$ for all $j$. This is consistent with the results in [@SagawaHayakawa; @Yuge_etal]. Equation (\[vectorPotentialResponse\]) relates the nonequilibrium linear response function $\Phi$ with the vector potential $\bm{A}^W$ in the expression (\[geometricalExcessWork\]). Therefore $\bm{A}^W$ can be experimentally determined in principle, because $\Phi$ is measurable. The sufficient condition for $A_j^W = \partial_j F$ for all $j$ is that $$\begin{aligned} \partial_i A_j^W = \partial_j A_i^W \label{MaxwellRelation}\end{aligned}$$ holds for all $i,j$, where $\partial_j$ is the abbreviation of $\partial/\partial \mu_j$. Weakly nonequilibrium case -------------------------- In the regime near to equilibrium (linear response regime), we can use the response function $\Phi^{\rm eq}$ of the equilibrium state in Eq. (\[vectorPotentialResponse\]). Then Eq. (\[MaxwellRelation\]) is valid because $\Phi^{\rm eq}$ is independent of $\mu_i$ and the reciprocal relation $\Phi^{\rm eq}_{ij} = \Phi^{\rm eq}_{ji}$ holds. Therefore the extension of the Clausius equality is possible in this regime, which is consistent with the results in Refs. [@KNST; @SaitoTasaki]. Connection to microscopic physics {#sec:micro} ================================= Up to here our formulation is closed on the macroscopic level. Now we connect it to microscopic physics. In this paper we assume that the microscopic dynamics of the system S is governed by the quantum master equation (QME) [@BreuerPetruccione]: $$\begin{aligned} \frac{\partial \hat{\rho}}{\partial t} = \mathcal{K}\hat{\rho}. \label{QME}\end{aligned}$$ Here, $\hat{\rho}$ is the density matrix of S, and the generator $\mathcal{K}$ is written as $\mathcal{K} = [\hat{H}, ~]/i\hbar + \sum_j \mathcal{L}_j$, where $\hat{H}$ is the Hamiltonian of S and $\mathcal{L}_j$ is the dissipator induced by the interaction with the $j$th reservoir. As in the previous sections, we assume that there exists a unique steady state in the QME. The steady state density matrix $\hat{\rho}_{\rm ss}$ satisfies $\mathcal{K}\hat{\rho}_{\rm ss}=0$. First, we consider the connection to microscopic physics through the response function $\Phi$ of NESS. For this purpose we employ the response-correlation relation (RCR) [@ShimizuYuge; @Yuge], which is a microscopic expression of $\Phi$. In the framework of the QME and for the response of the current $I_i$ from the $i$th reservoir into S, the RCR reads $$\begin{aligned} \Phi_{ij}(\tau) = {\rm Tr} \bigl[ (\partial_j \hat{I}_i) \hat{\rho}_{\rm ss} \bigr] \delta(\tau) + {\rm Tr} \bigl[ \hat{I}_i e^{\mathcal{K}\tau} (\partial_j \mathcal{K}) \hat{\rho}_{\rm ss} \bigr], \label{RCR}\end{aligned}$$ where ${\rm Tr}$ is the trace over S and $\hat{I}_i \equiv \mathcal{L}_i^\dag \hat{N}$ with $\hat{N}$ being the particle number operator in S. See \[appendix:RCR\] for the derivation of Eq. (\[RCR\]). Note that $\hat{I}_i$ can be regarded as the particle current operator from the $i$th reservoir into S because it satisfies the continuity equation: $(\partial/\partial t) {\rm Tr} [ \hat{N} \hat{\rho}(t)] = \sum_i {\rm Tr} [\hat{I}_i \hat{\rho}(t)]$. Substituting Eq. (\[RCR\]) into Eq. (\[vectorPotentialResponse\]), we obtain $$\begin{aligned} A_j^W = \sum_i \mu_i \int_{t_0}^\infty dt \biggl\{ & \int_{t_0}^t dt' {\rm Tr} \bigl[ \hat{I}_i e^{\mathcal{K}(t-t')} (\partial_j \mathcal{K}) \hat{\rho}_{\rm ss} \bigr] + {\rm Tr} \bigl[ \hat{I}_i \mathcal{R} (\partial_j \mathcal{K}) \hat{\rho}_{\rm ss} \bigr] \biggr\}. \label{vectorPotentialRCR}\end{aligned}$$ Note that the contribution from the first term in Eq. (\[RCR\]) vanishes. Here we defined $$\begin{aligned} \mathcal{R} \equiv -\lim_{T\to\infty}\int_{t_0}^T dt' e^{\mathcal{K}(T-t')} \mathcal{Q}_0, \label{R}\end{aligned}$$ and $\mathcal{Q}_0 = 1- \mathcal{P}_0$, where the projection superoperator $\mathcal{P}_0$ is defined such that $\mathcal{P}_0 \hat{X}= \hat{\rho}_{\rm ss} {\rm Tr}\hat{X}$ holds for any linear operator $\hat{X}$. See \[appendix:inverse\] for the fact that $\mathcal{R}$ is a well-defined superopertor. To rewrite Eq. (\[vectorPotentialRCR\]) further, we note the following relation: $$\begin{aligned} \frac{d}{dt'}{\rm Tr}\bigl[ \hat{I}_i e^{\mathcal{K}(t-t')} \mathcal{R} (\partial_j \mathcal{K}) \hat{\rho}_{\rm ss} \bigr] &= -{\rm Tr}\bigl[ \hat{I}_i e^{\mathcal{K}(t-t')} \mathcal{K} \mathcal{R} (\partial_j \mathcal{K}) \hat{\rho}_{\rm ss} \bigr] \nonumber\\ &= -{\rm Tr}\bigl[ \hat{I}_i e^{\mathcal{K}(t-t')} \mathcal{Q}_0 (\partial_j \mathcal{K}) \hat{\rho}_{\rm ss} \bigr] \nonumber\\ &= -{\rm Tr}\bigl[ \hat{I}_i e^{\mathcal{K}(t-t')} (\partial_j \mathcal{K}) \hat{\rho}_{\rm ss} \bigr] + {\rm Tr}\bigl[ \hat{I}_i \hat{\rho}_{\rm ss} \bigr] {\rm Tr}\bigl[(\partial_j \mathcal{K}) \hat{\rho}_{\rm ss} \bigr] \nonumber\\ &= -{\rm Tr}\bigl[ \hat{I}_i e^{\mathcal{K}(t-t')} (\partial_j \mathcal{K}) \hat{\rho}_{\rm ss} \bigr]. \label{integrand_vec}\end{aligned}$$ In the third line we used $$\begin{aligned} \mathcal{R}\mathcal{K} = \mathcal{K}\mathcal{R} = \mathcal{Q}_0. \label{RK}\end{aligned}$$ See \[appendix:inverse\] for the derivation of Eq. (\[RK\]). In the last line of Eq. (\[integrand\_vec\]) we used ${\rm Tr}\bigl[(\partial_j \mathcal{K}) \hat{\rho}_{\rm ss} \bigr] = \partial_j {\rm Tr}\bigl[\mathcal{K} \hat{\rho}_{\rm ss} \bigr] - {\rm Tr}\bigl[\mathcal{K} \partial_j \hat{\rho}_{\rm ss} \bigr] = 0$. This follows from $\mathcal{K} \hat{\rho}_{\rm ss}=0$ and ${\rm Tr}\bigl[\mathcal{K} \hat{X}]=0$ for any $\hat{X}$ (trace-preserving property of the QME). Integrating Eq. (\[integrand\_vec\]), we can rewrite the first term on the right hand side of Eq. (\[vectorPotentialRCR\]) as $$\begin{aligned} \int_{t_0}^t dt' {\rm Tr} \bigl[ \hat{I}_i e^{\mathcal{K}(t-t')} (\partial_j \mathcal{K}) \hat{\rho}_{\rm ss} \bigr] = {\rm Tr} \bigl[ \hat{I}_i e^{\mathcal{K}(t-t_0)} \mathcal{R} (\partial_j \mathcal{K}) \hat{\rho}_{\rm ss} \bigr] -{\rm Tr} \bigl[ \hat{I}_i \mathcal{R} (\partial_j \mathcal{K}) \hat{\rho}_{\rm ss} \bigr]. \nonumber\end{aligned}$$ The second term on the right hand side of this equation cancels out the second term on the right hand side of Eq. (\[vectorPotentialRCR\]). We thus rewrite Eq. (\[vectorPotentialRCR\]) as $$\begin{aligned} A_j^W = - \sum_i \mu_i {\rm Tr} \left[ \hat{I}_i \mathcal{R}^2 (\partial_j \mathcal{K}) \hat{\rho}_{\rm ss} \right], \label{vectorPotentialRR}\end{aligned}$$ where we used $\int_{t_0}^\infty dt e^{\mathcal{K}(t-t_0)}\mathcal{Q}_0 = -\mathcal{R}$. This is a microscopic expression of the vector potential $\bm{A}^W$. Next, we investigate the consistency of Eq. (\[vectorPotentialRR\]) with the results in Refs. [@SagawaHayakawa; @Yuge_etal]. In a manner almost the same as those in Refs. [@SagawaHayakawa; @Yuge_etal], we can derive another microscopic expression of $\bm{A}^W$ without relying on Eq. (\[vectorPotentialResponse\]): $$\begin{aligned} A_j^W = -{\rm Tr} \left( \hat{\ell}^{\prime\dag}_0 \partial_j \hat{\rho}_{\rm ss} \right), \label{vectorPotentialQME}\end{aligned}$$ where $\hat{\ell}^\prime_0 \equiv \partial \hat{\ell}^\chi_0/\partial (i\chi) |_{\chi=0}$. Here, $\chi$ is the counting field in the full counting statistics (FCS) of the work $W$ from the reservoirs, and $\hat{\ell}^\chi_0$ is the left eigenvector of $\mathcal{K}^\chi$ corresponding to the eigenvalue $\lambda^\chi_0$ that has the maximum real part. $\mathcal{K}^\chi=[\hat{H},~]/i\hbar + \sum_j \mathcal{L}_j^\chi$ is the $\chi$-modified generator, which is introduced for the FCS in the framework of the QME [@EspositoHarbolaMukamel_RMP]. See \[appendix:vectorPotentialQME\] for the details and derivation. We note that $\hat{\ell}^{\chi=0}_0=\hat{1}$ (identity operator), $\lambda^{\chi=0}_0 = 0$, and $\partial \lambda^\chi_0 / \partial (i\chi) |_{\chi=0} = \langle J^W \rangle^{\rm ss}_\mu$. In the following we rewrite Eq. (\[vectorPotentialQME\]) to show its equivalence to Eq. (\[vectorPotentialRR\]). We first rewrite $\hat{\ell}^{\prime\dag}_0$ in Eq. (\[vectorPotentialQME\]). By differentiating the left eigenvalue equation $(\mathcal{K}^\chi)^\dag \hat{\ell}^\chi_0 = (\lambda^\chi_0)^* \hat{\ell}^\chi_0$ with respect to $i\chi$ and setting $\chi=0$, we obtain $$\begin{aligned} \mathcal{K}^\dag \hat{\ell}^\prime_0 = - (\mathcal{K}')^\dag \hat{1} - \langle J^W \rangle^{\rm ss}_\mu \hat{1}. \label{KDag_ellPrime}\end{aligned}$$ Here $\mathcal{K}' = \partial \mathcal{K}^\chi / \partial (i\chi) |_{\chi=0}$ and the adjoint $\mathcal{O}^\dag$ of a superoperator $\mathcal{O}$ is defined by ${\rm Tr} [(\mathcal{O}^\dag \hat{X}_1)^\dag \hat{X}_2] = {\rm Tr} [ \hat{X}_1^\dag \mathcal{O} \hat{X}_2]$ for any pair $(\hat{X}_1, \hat{X}_2)$ of linear operators. By operating on the both sides of Eq. (\[KDag\_ellPrime\]) with $\mathcal{R}^\dag$, we obtain $$\begin{aligned} \hat{\ell}^\prime_0 = - \mathcal{R}^\dag (\mathcal{K}')^\dag \hat{1} + c \hat{1}, \label{ellPrime}\end{aligned}$$ where $c = - \langle J^W \rangle^{\rm ss}_\mu + {\rm Tr} [\hat{\rho}_{\rm ss} \hat{\ell}^\prime_0]$ and we used Eq. (\[RK\]). Substituting Eq. (\[ellPrime\]) into Eq. (\[vectorPotentialQME\]) and using ${\rm Tr} [\partial_j \hat{\rho}_{\rm ss}] = \partial_j {\rm Tr} [\hat{\rho}_{\rm ss}] = 0$, we have $A_j^W = {\rm Tr} \bigl[ \bigl( (\mathcal{K}')^\dag \hat{1} \bigr)^\dag \mathcal{R} \partial_j \hat{\rho}_{\rm ss} \bigr]$. Furthermore we can show $(\mathcal{K}')^\dag \hat{1} = \sum_i \mu_i \mathcal{L}_i^\dag \hat{N} = \sum_i \mu_i \hat{I}_i$. With this equation we have $$\begin{aligned} A_j^W = \sum_i \mu_i {\rm Tr} [ \hat{I}_i \mathcal{R} \partial_j \hat{\rho}_{\rm ss} ]. \label{vectorPotentialQME_middle}\end{aligned}$$ We next rewrite $\partial_j \hat{\rho}_{\rm ss}$. By differentiating the steady-state equation $\mathcal{K} \hat{\rho}_{\rm ss} = 0$ with respect to $\mu_j$ and operating on it with $\mathcal{R}$, we have $\partial_j \hat{\rho}_{\rm ss} = - \mathcal{R} (\partial_j \mathcal{K}) \hat{\rho}_{\rm ss}$. Substituting this equation into Eq. (\[vectorPotentialQME\_middle\]) we obtain $$\begin{aligned} A_j^W = - \sum_i \mu_i {\rm Tr} \left[ \hat{I}_i \mathcal{R}^2 (\partial_j \mathcal{K}) \hat{\rho}_{\rm ss} \right]. \nonumber\end{aligned}$$ This is the same as Eq. (\[vectorPotentialRR\]). We thus show that Eq. (\[vectorPotentialRR\]) \[and therefore Eq. (\[vectorPotentialResponse\])\] is consistent with the results in Refs. [@SagawaHayakawa; @Yuge_etal]. Concluding remarks ================== We have derived an expression of the excess work for quasistatic transitions between NESSs in particle transport systems on the basis of the linear response relation. We have related the vector potential $\bm{A}^W$ in the expression with the response function. We note that it is possible to extend our formulation to situations where other control parameters for transition between NESS are varied. In particular, we can obtain a similar result in heat conducting systems, where the temperatures of heat reservoirs are changed. We finally make remarks. First, the relationship between the excess work and the response function suggests that the response functions can be calculated in the framework of the SST. We expect that this expression becomes a first step for understanding of how transport phenomena are treated in the SST. Second, as is mentioned below Eq. (\[excessWork\]), our definition of the excess work is consistent with the definition of the excess heat in Refs. [@KNST; @SaitoTasaki; @SagawaHayakawa; @Yuge_etal]. However the definition of the excess work and heat is not unique; e.g., there are Hatano-Sasa type [@HatanoSasa; @DeffnerLutz2012; @Mandal] and Maes-Netočný type [@MaesNetocny] approaches. Recently, Ref. [@Mandal] gave evidence that the Hatano-Sasa approach is appropriate for the definition. Since the Hatano-Sasa approach relies on microscopic information (e.g., steady-state distribution and transition rate), the connection to macroscopic quantities is not clear. It is therefore important to investigate the definition from the viewpoint of response function as a future work. Third, in recent years the nonequilibrium response function is a hot topic in the statistical physics [@Marconi_etal; @ChetriteGupta; @Baiesi_etal; @BaiesiMaes]. One of the points in recent works is decomposition of the response function [@Baiesi_etal; @BaiesiMaes]. We expect that the application of these results to the expression of the excess work would lead to a further decomposition of the work that is appropriate for the construction of the SST. Acknowledgments {#acknowledgments .unnumbered} =============== The author thanks K. Akiba and M. Yamaguchi for helpful discussions. This work was supported by a JSPS Research Fellowship for Young Scientists (No. 24-1112), KAKENHI (No. 26287087), and ImPACT Program of Council for Science, Technology and Innovation (Cabinet Office, Government of Japan). Linear response function of NESS in quantum master equation approach {#appendix:RCR} ==================================================================== Here we derive Eq. (\[RCR\]), the RCR in QME. We consider the QME (\[QME\]) that depends on multiple parameters $\bm{\alpha}=(\alpha_1,\alpha_2,\alpha_3,...)$ like chemical potentials; i.e., we assume that the generator of the QME depends on these parameters: $\mathcal{K}=\mathcal{K}(\bm{\alpha})$. Suppose that at time $t \le t_0$ the system S is in the NESS with $\bm{\alpha}=\bm{\alpha}^0$; i.e., $\hat{\rho}(t) = \hat{\rho}_{\rm ss}^0$ for $t \le t_0$, where $\hat{\rho}_{\rm ss}^0$ satisfies $\mathcal{K}^0 \hat{\rho}_{\rm ss}^0=0$ with $\mathcal{K}^0 \equiv \mathcal{K}(\bm{\alpha}^0)$. For $t>t_0$, we weakly modulate the parameters in time: $\alpha_l(t) = \alpha_l^0 + f_l(t)$ ($l=1,2,3,...$), where $\max_t |f_l(t)|$ is much smaller than a typical value of $\alpha_l$. Then we can expand the generator $\mathcal{K}$ around $\bm{\alpha}^0$ in terms of $f$: $$\begin{aligned} \mathcal{K}\bigl(\bm{\alpha}(t)\bigr) \simeq \mathcal{K}^0 + \sum_l f_l(t) \partial_l \mathcal{K}^0, \label{expandK}\end{aligned}$$ where $\partial_l \mathcal{K}^0 \equiv \partial \mathcal{K}(\bm{\alpha})/\partial \alpha_l |_{\bm{\alpha}=\bm{\alpha}^0}$. To solve the QME (\[QME\]) with the weakly time-dependent $\bm{\alpha}$ and the initial condition $\hat{\rho}(t_0) = \hat{\rho}_{\rm ss}^0$, we transform the QME into an “interaction picture”. That is, we introduce $\breve{\rho}(t) = e^{-\mathcal{K}_0(t-t_0)} \hat{\rho}(t)$. Then, from the QME (\[QME\]), we have the equation of motion for $\breve{\rho}$ as $$\begin{aligned} \frac{\partial \breve{\rho}(t)}{\partial t} = \sum_l f_l(t) e^{-\mathcal{K}_0(t-t_0)} (\partial_l \mathcal{K}^0) e^{\mathcal{K}_0(t-t_0)} \breve{\rho}(t), \end{aligned}$$ where we used Eq. (\[expandK\]). By integrating this equation form $t_0$ to $t$ with $\breve{\rho}(t_0) = \hat{\rho}_{\rm ss}^0$, we obtain $$\begin{aligned} \breve{\rho}(t) &= \hat{\rho}_{\rm ss}^0 + \sum_l \int_{t_0}^t dt' f_l(t') e^{-\mathcal{K}_0(t'-t_0)} (\partial_l \mathcal{K}^0) e^{\mathcal{K}_0(t'-t_0)} \breve{\rho}(t') \nonumber\\ &\simeq \hat{\rho}_{\rm ss}^0 + \sum_l \int_{t_0}^t dt' f_l(t') e^{-\mathcal{K}_0(t'-t_0)} (\partial_l \mathcal{K}^0) \hat{\rho}_{\rm ss}^0. \end{aligned}$$ In going from the first line to the second, we approximately replaced $\breve{\rho}(t')$ in the integral with $\hat{\rho}_{\rm ss}^0$. This approximation corresponds to the first-order time-dependent perturbation theory in quantum mechanics. Going back to the Schrödinger picture, we have $$\begin{aligned} \hat{\rho}(t) &= \hat{\rho}_{\rm ss}^0 + \sum_l \int_{t_0}^t dt' f_l(t') e^{\mathcal{K}_0(t-t')} (\partial_l \mathcal{K}^0) \hat{\rho}_{\rm ss}^0. \end{aligned}$$ We thus obtain the time dependence of the expectation value of a quantity $\hat{X}$ that is independent of $\bm{\alpha}$: $$\begin{aligned} \langle X \rangle^t_{\bm{\alpha}(t)} &= {\rm Tr} \big[ \hat{X} \hat{\rho}(t) \bigr] \nonumber\\ &= \langle X \rangle^{\rm ss}_{\bm{\alpha}_0} + \sum_l \int_{t_0}^t dt' {\rm Tr} \Big[ \hat{X} e^{\mathcal{K}^0(t-t')} (\partial_l \mathcal{K}^0) \hat{\rho}_{\rm ss}^0 \Bigr] f_l(t') \label{linearResponseQME1} \\ &= \langle X \rangle^{\rm ss}_{\bm{\alpha}_0} + \sum_l \int_{t_0}^t dt' {\rm Tr} \Big[ \Bigl\{ (\partial_l \mathcal{K}^0)^\dag e^{\mathcal{K}^{0\dag}(t-t')} \hat{X} \Bigr\} \hat{\rho}_{\rm ss}^0 \Bigr] f_l(t'). \label{linearResponseQME2}\end{aligned}$$ Equations (\[linearResponseQME1\]) and (\[linearResponseQME2\]) give the RCR in the QME. We note that Eq. (\[linearResponseQME2\]) reduces to the Kubo formula if $\hat{\rho}_{\rm ss}^0$ is an equilibrium state (i.e., when we consider the response of an equilibrium state) [@ShimizuYuge]. Now we consider the current $\hat{I}_i = \mathcal{L}_i^\dag \hat{N}$ from the $i$th reservoir into the system S. We note that $\hat{I}_i$ is dependent on $\bm{\alpha}$ because so is $\mathcal{L}_i$. Therefore we have the average current at time $t$ as $$\begin{aligned} &\langle \hat{I}_i \rangle^t_{\bm{\alpha}(t)} - \langle \hat{I}_i \rangle^{\rm ss}_{\bm{\alpha}_0} \nonumber\\ &= {\rm Tr} \big[ \hat{I}_i\bigl(\bm{\alpha}(t)\bigr) \hat{\rho}(t) \bigr] - {\rm Tr} \big[ \hat{I}_i\bigl(\bm{\alpha}_0\bigr) \hat{\rho}^0_{\rm ss} \bigr] \nonumber\\ &= \sum_l {\rm Tr} \big[ (\partial_l \hat{I}_i^0) \hat{\rho}^0_{\rm ss} \bigr] f_l(t) + {\rm Tr} \big[ \hat{I}_i^0 \hat{\rho}(t) \bigr] - {\rm Tr} \big[ \hat{I}_i^0 \hat{\rho}^0_{\rm ss} \bigr] \nonumber\\ &= \sum_l {\rm Tr} \big[ (\partial_l \hat{I}_i^0) \hat{\rho}^0_{\rm ss} \bigr] f_l(t) + \sum_l \int_{t_0}^t dt' {\rm Tr} \Big[ \hat{I}_i^0 e^{\mathcal{K}^0(t-t')} (\partial_l \mathcal{K}^0) \hat{\rho}_{\rm ss}^0 \Bigr] f_l(t'), \end{aligned}$$ where $\hat{I}_i^0 \equiv \hat{I}_i(\bm{\alpha}_0)$ and $\partial_l \hat{I}_i^0 \equiv \partial \hat{I}_i(\bm{\alpha})/\partial \alpha_l |_{\bm{\alpha}=\bm{\alpha}_0}$. We used Eq. (\[linearResponseQME1\]) in the third line. Finally, by performing the functional differentiation with respect to $f_j(t')$, we obtain $$\begin{aligned} \Phi_{ij}(t-t') &= \frac{\delta \langle \hat{I}_i \rangle^t_{\bm{\alpha}(t)}}{\delta f_j(t')} \nonumber\\ &= {\rm Tr} \bigl[ (\partial_j \hat{I}_i^0) \hat{\rho}_{\rm ss}^0 \bigr] \delta(t-t') + {\rm Tr} \bigl[ \hat{I}_i^0 e^{\mathcal{K}(t-t')} (\partial_j \mathcal{K}) \hat{\rho}_{\rm ss}^0 \bigr]. \nonumber\end{aligned}$$ This is equivalent to Eq. (\[RCR\]). Inverse-like superoperator in quantum master equation approach {#appendix:inverse} ============================================================== First we show that $\mathcal{R}$ in Eq. (\[R\]) is well defined. To this end, we denote the eigenvalue and corresponding left and right eigenvectors of $\mathcal{K}$ as $\lambda_m$, $\hat{\ell}_m$, and $\hat{r}_m$. We assign the steady state of $\mathcal{K}$ to the index $m=0$; i.e., $\lambda_0 = 0$, $\hat{\ell}_0=\hat{1}$, and $\hat{r}_0 = \hat{\rho}_{\rm ss}$. By the assumption of the unique existence of the stable steady state, ${\rm Re}\lambda_m < 0$ for $m \neq 0$. Then, for any linear operator $\hat{X}$, we obtain the following equation: $$\begin{aligned} \mathcal{R} \hat{X} &= - \int_0^\infty dt e^{\mathcal{K}t} \mathcal{Q}_0 \hat{X} \nonumber\\ &= - \int_0^\infty dt e^{\mathcal{K}t} \sum_{m\neq 0} {\rm Tr}[\hat{\ell}_m^\dag \hat{X}] \hat{r}_m \nonumber\\ &= - \int_0^\infty dt \sum_{m\neq 0} e^{\lambda_m t} {\rm Tr}[\hat{\ell}_m^\dag \hat{X}] \hat{r}_m \nonumber\\ &= \sum_{m\neq 0} \frac{{\rm Tr}[\hat{\ell}_m^\dag \hat{X}]}{\lambda_m} \hat{r}_m.\end{aligned}$$ Since this is not diverging, $\mathcal{R}$ is well defined. We here show that Eq. (\[RK\]) holds for the generator $\mathcal{K}$ of the QME. We first note that $\mathcal{R} \mathcal{K} = \mathcal{K} \mathcal{R}$ follows from $\mathcal{Q}_0 \mathcal{K} = \mathcal{K} \mathcal{Q}_0= \mathcal{K}$, which we can derive from the fact that $$\begin{aligned} \mathcal{P}_0 \mathcal{K} \hat{X} &= \hat{\rho}_{\rm ss} {\rm Tr} [ \mathcal{K} \hat{X} ] = 0, \label{PKX} \\ \mathcal{K} \mathcal{P}_0 \hat{X} &= \mathcal{K} \hat{\rho}_{\rm ss} {\rm Tr} \hat{X} = 0, \label{KPX}\end{aligned}$$ hold for any linear operator $\hat{X}$. Equation (\[PKX\]) follows from ${\rm Tr} [ \mathcal{K} \hat{X} ] = 0$ (trace-preserving property of QME), and Eq. (\[KPX\]) from $\mathcal{K} \hat{\rho}_{\rm ss}=0$ (steady-state equation). Then we can show Eq. (\[RK\]) as follows: $$\begin{aligned} \mathcal{R} \mathcal{K} = \mathcal{K} \mathcal{R} &= \lim_{T\to\infty} \int_{t_0}^T dt' \frac{d}{dt'}e^{\mathcal{K}(T-t')} \mathcal{Q}_0 \nonumber\\ &= \Bigl( 1 - \lim_{T\to\infty} e^{\mathcal{K}(T-t_0)} \Bigr) \mathcal{Q}_0 \nonumber\\ &= ( 1 - \mathcal{P}_0 ) \mathcal{Q}_0. \nonumber\\ &= \mathcal{Q}_0. \label{RK_a}\end{aligned}$$ Here the third line follows from the convergence theorem of the Markov process, which we can derive from the fact that for any linear operator $\hat{X}$ the following equation holds: $$\begin{aligned} \lim_{T\to\infty} e^{\mathcal{K}(T-t_0)} \hat{X} &= \lim_{T\to\infty} e^{\mathcal{K}(T-t_0)} \sum_m {\rm Tr}[\hat{\ell}_m^\dag \hat{X}] \hat{r}_m \nonumber\\ &= \sum_m {\rm Tr}[\hat{\ell}_m^\dag \hat{X}] \hat{r}_m \lim_{T\to\infty} e^{\lambda_m(T-t_0)} \nonumber\\ &= \hat{\rho}_{\rm ss} {\rm Tr}\hat{X}. \end{aligned}$$ This gives the third line in Eq. (\[RK\_a\]). We note that Eq. (\[RK\_a\]) leads to $\mathcal{K} \mathcal{R} \mathcal{K} = \mathcal{K}$. This implies that $\mathcal{R}$ satisfies one of the conditions for the Moore-Penrose pseudoinverse of $\mathcal{K}$. Derivation of Eq. (\[vectorPotentialQME\]) {#appendix:vectorPotentialQME} ========================================== For completeness we here derive Eq. (\[vectorPotentialQME\]), the work version of the results in Refs. [@SagawaHayakawa; @Yuge_etal]. First we note that we can measure the work $W$ during varying the chemical potentials $\mu=\{\mu_i\}_i$ with a time interval $\tau$ as follows. At the initial time $t=t_0$, we perform a projection measurement of reservoir particle numbers $\{\hat{N}_i\}_i$ to obtain measurement outcomes $\{N_i(t_0)\}_i$. For $t>t_0$, we vary $\mu$, where the system evolves with interacting with the reservoirs. At $t=t_0+\tau$, we again perform a measurement of $\hat{N}_i$ to obtain outcomes $\{N_i(t_0+\tau)\}_i$. The difference of the outcomes gives the work $W=\sum_i [\mu_i(t_0+\tau) N_i(t_0+\tau) - \mu_i(t_0) N_i(t_0)]$. Repeating the measurements, we obtain a probability distribution $p_\tau(W)$. The average work is given by $\langle W \rangle_\tau = \int dW p_\tau(W) W$, and the average work flow in a NESS is given by $J_W = \lim_{\tau\to\infty} \langle W \rangle_\tau / \tau$ with $\mu$ being fixed. In the following, we calculate the average work by $\langle W \rangle_\tau = \partial G_\tau(\chi) / \partial(i\chi)|_{\chi=0}$, where $G_\tau(\chi) \equiv \ln \int dW p_\tau(W) e^{i\chi W}$ is the cumulant generating function and $\chi$ is the counting field. By using the full counting statistics [@EspositoHarbolaMukamel_RMP], we can calculate $G_\tau(\chi)$ by $$\begin{aligned} G_\tau(\chi) = \ln {\rm Tr}_{\rm S} \hat{\rho}^{\chi}(\tau). \end{aligned}$$ Here $\hat{\rho}^\chi$ is the solution of the generalized quantum master equation (GQME): $$\begin{aligned} \frac{\partial \hat{\rho}^{\chi}(t)}{\partial t} = \mathcal{K}^\chi\bigl(\mu(t)\bigr) \hat{\rho}^{\chi}(t), \label{GQME}\end{aligned}$$ where the generalized generator is given by $\mathcal{K}^\chi = [ \hat{H}, ~] /i\hbar + \sum_j \mathcal{L}_j^\chi$, with the generalized dissipator $\mathcal{L}_j^\chi \hat{\rho} \equiv - (1/\hbar^2) \int_0^\infty dt' {\rm Tr}_j \bigl[ \hat{H}_{{\rm S}j} , [ \breve{H}_{{\rm S}j}(-t') , \hat{\rho} \otimes \hat{\rho}_j (\mu_j ) ]_\chi \bigr]_\chi$. ${\rm Tr}_j$ is the trace over the $j$th reservoir, $\hat{H}_{{\rm S}j}$ is the interaction Hamiltonian between the system and the $j$th reservoir, $\breve{H}_{{\rm S}j}$ is its interaction picture, $\hat{\rho}_i (\mu_j )$ is the thermal equilibrium state of the $j$th reservoir with the chemical potential $\mu_j$, $[\hat{O}_1 , \hat{O}_2]_\chi \equiv \hat{O}_1^\chi \hat{O}_2 - \hat{O}_2 \hat{O}_1^{-\chi}$, and $\hat{O}^\chi \equiv e^{-i\chi \sum_j \mu_j \hat{N}_j/2} \hat{O} e^{i\chi \sum_j \mu_j \hat{N}_j/2}$. Note that, if we set $\chi=0$, the GQME (\[GQME\]) reduces to the original QME (\[QME\]), and $\mathcal{K}^\chi$, $\hat{\ell}^\chi_0$, and $\hat{r}^\chi_0$ also reduce to $\mathcal{K}$, $\hat{1}$, and $\hat{\rho}_{\rm ss}$, respectively. For fixed $\mu$, we can define the left and right eigenvectors of $\mathcal{K}^\chi(\mu)$ corresponding to the eigenvalue $\lambda^\chi_m(\mu)$, which are respectively denoted by $\hat{\ell}^\chi_m(\mu)$ and $\hat{r}^\chi_m(\mu)$. They are normalized as ${\rm Tr} (\hat{\ell}^{\chi\dag}_m \hat{r}^\chi_n) = \delta_{mn}$. We assign the label for the eigenvalue with the maximum real part to $m=0$. It is known that $\lim_{\tau\to\infty} G_\tau(\chi)/\tau =\lambda^\chi_0$ holds [@EspositoHarbolaMukamel_RMP]. Therefore, the average work flow $J_W$ in the NESS can be calculated by $$\begin{aligned} J_W(\mu) = \frac{\partial \lambda^\chi_0(\mu)}{\partial (i\chi)}\bigg|_{\chi=0}. \label{workFlux}\end{aligned}$$ We now derive Eq. (\[vectorPotentialQME\]). We first note that the excess work can be written as $W_{\rm ex} = \partial G_{\rm ex}(\chi) / \partial(i\chi)|_{\chi=0}$, where $G_{\rm ex}(\chi) \equiv G_\tau(\chi) - \Lambda^\chi_0(\tau)$ and $\Lambda^\chi_m(t) \equiv \int_{t_0}^{t_0+t} dt' \lambda^\chi_m \bigl(\mu(t')\bigr)$. This is because $\langle W \rangle_\tau = \partial G_\tau(\chi) / \partial(i\chi)|_{\chi=0}$ and Eq. (\[workFlux\]). To calculate $G_{\rm ex}(\chi)$, we solve the GQME (\[GQME\]). For this purpose we expand $\hat{\rho}^\chi(t)$ as $$\begin{aligned} \hat{\rho}^\chi(t) = \sum_m c_m(t) e^{\Lambda^\chi_m(t)} \hat{r}^\chi_m \bigl(\mu(t)\bigr). \label{expansion_rho}\end{aligned}$$ Substituting this expansion into Eq. (\[GQME\]) and taking the Hilbert-Schmidt inner product with $\hat{\ell}^\chi_0 \bigl(\mu(t)\bigr)$, we obtain $$\begin{aligned} \frac{d{c}_0(t)}{dt} = - \sum_m c_m(t) e^{\Lambda^\chi_m(t)-\Lambda^\chi_0(t)} {\rm Tr}_{\rm S} \left[ \hat{\ell}^{\chi\dag}_0 \bigl(\mu(t)\bigr) \dot{\hat{r}}^\chi_m \bigl(\mu(t)\bigr) \right]. \nonumber\end{aligned}$$ If the time scale of varying $\mu$ is sufficiently slower than the relaxation time of the system, we can approximate the sum on the RHS by the contribution only from the term with $m=0$ (adiabatic approximation). By solving the approximate equation we obtain $$\begin{aligned} c_0 (t_0+\tau) = c_0(t_0) \exp\left\{ - \int_{C} {\rm Tr}_{\rm S} \left[ \hat{\ell}^{\chi\dag}_0(\mu) d \hat{r}^\chi_0(\mu) \right] \right\}, \label{c_0}\end{aligned}$$ where $C$ is a path connecting $\mu(t_0)$ and $\mu(t_0+\tau)$ in the parameter space and $d \hat{r}^\chi_0(\mu) \equiv \sum_j \bigl(\partial \hat{r}^\chi_0(\mu) / \partial \mu_j \bigr) d \mu_j$. If $\hat{\rho}^\chi(t_0) = \hat{\rho}_{\rm ss} \bigl(\mu(t_0)\bigr)$, then $c_0(t_0) = {\rm Tr} \Bigl[ \hat{\ell}^{\chi\dag}_0\bigl(\mu(t_0)\bigr) \hat{\rho}_{\rm ss}\bigl(\mu(t_0)\bigr) \Bigr]$. At long time, only the $m=0$ term remains in Eq. (\[expansion\_rho\]) since $\Lambda^\chi_0(t)$ has the maximum real part. Therefore we obtain $$\begin{aligned} \hat{\rho}^\chi(t_0+\tau) &\simeq c_0 (t_0+\tau) e^{\Lambda^\chi_0(\tau)} \hat{r}^\chi_0 \bigl(\mu(t_0+\tau)\bigr).\end{aligned}$$ Substituting Eq. (\[c\_0\]) into this equation we obtain an expression for $G_{\rm ex}(\chi) = \ln {\rm Tr}_{\rm S} \hat{\rho}^\chi(t_0+\tau) - \Lambda^\chi_0(\tau)$ as $$\begin{aligned} G_{\rm ex}(\chi) = & - \int_C {\rm Tr}_{\rm S} \Bigl[ \hat{\ell}_0^{\chi\dag}(\mu) d \hat{r}_0^\chi(\mu) \Bigr] + \ln {\rm Tr}_{\rm S} \Bigl[ \hat{\ell}_0^{\chi\dag} \bigl(\mu(t_0)\bigr) \hat{\rho}_{\rm ss} \bigl(\mu(t_0)\bigr) \Bigr] \nonumber\\ &+ \ln {\rm Tr}_{\rm S} \hat{r}_0^{\chi} \bigl(\mu(t_0+\tau)\bigr) . \label{CGF_ex}\end{aligned}$$ Finally, by differentiating Eq. (\[CGF\_ex\]) with respect to $i\chi$ and setting $\chi=0$, we obtain an expression for the excess work: $$\begin{aligned} W_{\rm ex} &= - \int_C {\rm Tr}_{\rm S}\left[ \hat{\ell}_0^{\prime\dag}(\mu) d\hat{\rho}_{\rm ss}(\mu) \right]. \label{excess_derived}\end{aligned}$$ We thus obtain Eq. (\[vectorPotentialQME\]). [99]{} Esposito M, Harbola U and Mukamel S 2007 Phys. Rev. E [**76**]{} 031132 Esposito M and Van den Broeck C 2010 Phys. Rev. Lett. [**104**]{} 090601 Deffner S and Lutz E 2010 Phys. Rev. Lett. [**105**]{} 170402 Takara K, Hasegawa H H and Driebe D J 2010 Phys. Lett. A, [**375**]{} 88 Deffner S and Lutz E 2012 arXiv:1201.3888 Nakagawa N 2012 Phys. Rev. E [**85**]{} 051115 Verley G and Lacoste D 2012 Phys. Rev. E [**86**]{} 051127 Boksenbojm E, Maes C, Netočný K and Pešek J 2011 Europhys. Lett. [**96**]{} 40001 Mandal D 2013 Phys. Rev. E [**88**]{} 062135 Komatsu T S, Nakagawa N, Sasa S and Tasaki H 2008 Phys. Rev. Lett. [**100**]{} 230602 Saito K and Tasaki H 2011 J. Stat. Phys. [**145**]{} 1275 Sagawa T and Hayakawa H 2011 Phys. Rev. E [**84**]{} 051110 Maes C and Netočný K 2014 J. Stat. Phys. [**154**]{} 188 Bertini L, Gabrielli D, Jona-Lasinio G and Landim C 2013 Phys. Rev. Lett. [**110**]{} 020601 Yuge T, Sagawa T, Sugita A, and Hayakawa H 2013 J. Stat. Phys. [**153**]{} 412 Oono Y and Paniconi M 1998 Prog. Theor. Phys. Suppl. [**130**]{} 29 Landauer R 1978 Phys. Rev. A [**18**]{} 255 Hatano T and Sasa S 2001 Phys. Rev. Lett. [**86**]{} 3463 Sasa S and Tasaki H 2006 J. Stat. Phys. [**125**]{} 125 Maes C 2014 J. Stat. Phys. [**154**]{} 705 Shimizu A and Yuge T 2010 J. Phys. Soc. Jpn. [**79**]{} 013002 Yuge T 2010 Phys. Rev. E [**82**]{} 051130 Sze S M 2002 [*Semiconductor devices: physics and technology*]{}, 2nd ed. (Wiley, New York) Davies J H 1998 [*The Physics of Low-dimensional Semiconductors: An Introduction*]{} (Cambridge University Press, New York) Marconi U M B, Puglisi A, Rondoni L, and Vulpiani A 2008 Phys. Rep. [**461**]{} 111 Chetrite R and Gupta S 2011 J. Stat. Phys. [**143**]{} 543 Baiesi M, Maes C and Wynants B 2009 Phys. Rev. Lett. [**103**]{} 010602 Baiesi M and Maes C 2013 New J. Phys. [**15**]{} 013004 Breuer H P and Petruccione F 2002 [*The Theory of Open Quantum Systems*]{} (Oxford University Press, London) Esposito M, Harbola U and Mukamel S 2009 Rev. Mod. Phys. [**81**]{} 1665

epsf [SLAC–PUB–8553\ August 2000\ ]{} [**Light Quark Fragmentation in Polarized $Z^{0}$ Decays at SLD [^1]**]{} M. Kalelkar\ Rutgers University, Piscataway, NJ 08854\ Representing The SLD Collaboration$^{**}$\ Stanford Linear Accelerator Center, Stanford University, Stanford, CA 94309\ [Abstract ]{} [ We report results on two physics topics from the SLD experiment at the SLAC Linear Collider, using our full sample of 550,000 events of the type $e^{+}e^{-} \rightarrow Z^{0} \rightarrow q\bar{q}$. The electron beam was polarized, enabling the quark and antiquark hemispheres to be tagged in each event. One physics topic is the first study of rapidities signed such that positive rapidity is along the quark rather than antiquark direction. Distributions of ordered differences in signed rapidity between pairs of particles are analyzed, providing the first direct observation of baryon number ordering along the $q\bar{q}$ axis. The other topic is the first direct measurement of $A_{s}$, the parity-violating coupling of the $Z^{0}$ to strange quarks, by measuring the left-right forward-backward production asymmetry in polar angle of the tagged $s$ quark. We obtain $A_{s}$ = $0.895\pm 0.066(stat.)\pm 0.062(syst.)$, which is consistent with the Standard Model and is currently the most precise measurement of this quantity. ]{} [*Presented at the International Euroconference on Quantum Chromodynamics (QCD 00), 6-13 July 2000, Montpellier, France*]{} Introduction ============ In this paper we report new results on two physics topics from the SLD experiment at the SLAC Linear Collider, using our full data sample of 550,000 hadronic decays of $Z^{0}$ bosons produced in $e^{+}e^{-}$ annihilations. A unique feature of the experiment was a highly longitudinally polarized electron beam, with an average magnitude of polarization of 73%. One physics topic is a study of signed rapidities. The rapidity of a particle is typically defined with an arbitrary sign. If a sign could be given to each measured rapidity such that, for example, positive (negative) rapidity corresponds to the initial quark (antiquark) direction, then one could measure the extent to which a leading particle has higher rapidity than its associated antiparticle, and the extent to which low-momentum particles in jets remember the initial quark/antiquark direction. The other physics topic in this paper is the first direct measurement of the strange quark asymmetry parameter $A_{s}$. Measurements of the fermion production asymmetries in the process $e^{+}e^{-} \rightarrow Z^{0} \rightarrow f\bar{f}$ provide information on the extent of parity violation in the coupling of the $Z^{0}$ bosons to fermions of type $f$. The differential production cross section can be expressed in terms of $x = \cos\theta$, where $\theta$ is the polar angle of the final state fermion $f$ with respect to the electron beam direction: $${d\sigma\over{dx}} \propto (1-A_{e}P_{e})(1+x^{2}) + 2A_{f}(A_{e}-P_{e})x$$ where $P_{e}$ is the longitudinal polarization of the electron beam, the positron beam is assumed unpolarized, and the asymmetry parameters $A_{f} = 2v_{f}a_{f}/(v_{f}^{2} + a_{f}^{2})$ are defined in terms of the vector and axial-vector couplings of the $Z^{0}$ to fermion $f$. The Standard Model predictions for the values of the asymmetry parameters, assuming $\sin^{2}\theta_{w} = 0.23$, are $A_{e} = A_{\mu} = A_{\tau} = 0.16$, $A_{u} = A_{c} = A_{t} = 0.67$, and $A_{d} = A_{s} = A_{b} = 0.94$. For a given final state $f\bar{f}$, if one measures the polar angle distributions in equal luminosity samples taken with negative and positive beam polarization, then one can derive the left-right forward-backward asymmetry: $$\tilde{A}^{f}_{FB} = {3\over{4}}\mid P_{e}\mid A_{f}$$ which is insensitive to the initial state coupling. A number of previous measurements have been made of the leptonic asymmetries and the heavy-flavor asymmetries, but very few measurements exist for the light quark flavors, due to the difficulty of tagging specific light flavors. We present a direct measurement of the strange quark asymmetry parameter $A_{s}$, in which $Z^{0} \rightarrow s\bar{s}$ events were tagged by the absence of $B$ or $D$ hadrons and the presence in each hemisphere of a high-momentum $K^{\pm}$ or $K^{0}_{S}$. Particle Identification ======================= A description of the SLD detector, trigger, track and hadronic event selection, and Monte Carlo simulation is given in Ref. [@impact]. The identification of $\pi^{\pm}$, K$^{\pm}$, p, and $\bar{\rm p}$ was achieved by reconstructing emission angles of individual Cherenkov photons radiated by charged particles passing through liquid and gas radiator systems of the SLD Cherenkov Ring Imaging Detector (CRID) [@crid]. In each momentum bin, identified $\pi$, K, and p were counted, and these were unfolded using the inverse of an identification efficiency matrix [@pavel], and corrected for track reconstruction efficiency. The elements of the identification efficiency matrix were mostly measured from data, using selected $K^{0}_{S}$, $\tau$, and $\Lambda$ decays. A detailed Monte Carlo simulation was used to derive the unmeasured elements in terms of these measured ones. $K^{0}_{S}\rightarrow\pi^{+}\pi^{-}$ and $\Lambda^{0} (\bar{\Lambda}^{0}) \rightarrow p\pi^{-} (\bar{p}\pi^{+})$ decays were reconstructed as described in Ref. [@bfp; @staengle] by examining appropriate invariant mass distributions. Signed Rapidities ================= We next tagged the quark (vs. antiquark) direction in each hadronic event by using the electron beam polarization for that event, exploiting the large forward-backward quark production asymmetry in $Z^{0}$ decays. If the beam was left(right)-handed, then the thrust axis was signed such that $\cos\theta_{T}$ was positive (negative). Events with $|\cos\theta_{T}| < 0.15$ were removed, as the production asymmetry is small in this region. The probability to tag the quark direction correctly in these events was 73%, assuming Standard Model couplings at tree-level. For each identified particle the rapidity $y =$ 0.5 ln$((E+p_{\|})/(E-p_{\|}))$ was calculated using the measured momentum and its projection $p_{\|}$ along the thrust axis, and the appropriate hadron mass. The rapidity with respect to the signed thrust axis is naturally signed such that positive rapidity corresponds to the hemisphere in the tagged direction of the initial quark, and negative rapidity corresponds to the tagged antiquark hemisphere. The signed rapidity distributions for identified $K^{+}$ and $K^{-}$ are shown in Fig. \[fig1\]. =15.0cm There is a clear difference between the two, with more $K^{-}$ than $K^{+}$ in the quark hemisphere, as expected due to leading $K^{-}$ produced in $s$-quark jets [@bfp; @leading]. The difference between the two distributions is also shown in the figure and is compared with the prediction of the JETSET [@jetset] simulation, which is consistent with the data. For pairs of identified particles, one can define an ordered rapidity difference. For particle-antiparticle pairs, we define $\Delta y^{+-} = y_{+} - y_{-}$ as the difference between the signed rapidities of the positively charged particle and the negatively charged particle. In Fig. \[fig2\] we show the distribution of $\Delta y^{+-}$ for $\pi^{+}\pi^{-}$, $K^{+}K^{-}$ and $p\bar{p}$ pairs. =15.0cm Asymmetries in these distributions are indications of ordering along the event axis, and the differences between the positive and negative sides of these distributions are also shown. The predictions of the simulation are also shown and are consistent with the data at high $|\Delta y^{+-}|$. The negative difference at high $|\Delta y^{+-}|$ for the $K^{+}K^{-}$ pairs can be attributed to the fact that leading kaons are produced predominantly in $s\bar{s}$ events. For $\pi^{+}\pi^{-}$ pairs we observe a large positive difference at high $|\Delta y^{+-}|$ rather than the expected small negative difference, and we have confirmed that this effect is due entirely to $c\bar{c}$ events. Our sample of $uds$-tagged events does show the expected small negative difference. The positive difference in the lowest $|\Delta y^{+-}|$ bins for the $p\bar{p}$ pairs indicates that the baryon in an associated baryon-antibaryon pair follows the quark direction more closely than the antibaryon. This could be due to leading baryon production and/or to baryon numbering ordering along the entire fragmentation chain. We find a significant effect in all of our momentum bins, and the bulk of the observed difference occurs at low momentum. We therefore conclude that both of these effects contribute; this is the first direct observation of baryon number ordering along the entire chain. The prediction of the simulation is low by a factor of two at low $|\Delta y^{+-}|$. Strange Quark Asymmetry ======================= For the measurement of the strange quark asymmetry parameter $A_{s}$, the first step was to select $s\bar{s}$ events and tag the $s$ and $\bar{s}$ jets. We used the SLD Vertex Detector [@vxd] to measure each track’s impact parameter $d$ in the plane perpendicular to the beam direction. We then removed $c\bar{c}$ and $b\bar{b}$ events by requiring no more than one well-measured track with $d$ larger than 2.5 times its uncertainty. Each remaining event was divided into two hemispheres by the plane perpendicular to the thrust axis. In each hemisphere we searched for the strange particle with the highest momentum. If it was a charged kaon, we required it to have $p>9$ GeV/c, while if it was a $K^{0}_{S}$ or $\Lambda^{0}/\bar{\Lambda}^{0}$ it was required to have $p>5$ GeV/c. An event was tagged as $s\bar{s}$ if one hemisphere contained a $K^{\pm}$ selected as just described, and the other contained either an oppositely charged $K^{\pm}$ or a $K^{0}_{S}$. The thrust axis, signed so as to point into the hemisphere containing (opposite) the $K^{-}(K^{+})$, was used as an estimate of the initial $s$-quark direction. Fig. \[fig3\] shows the distributions of the measured $s$-quark polar angle $\theta_{s}$ for the $K^{+}K^{-}$ and $K^{\pm}K^{0}_{S}$ modes. =15.0cm In each case, production asymmetries of opposite sign and different magnitude for left- and right-polarized $e^{-}$ beams are visible. $A_{s}$ was extracted from these distributions by a simultaneous maximum likelihood fit, the result of which is shown as a histogram in the figure. The fit quality was good, with a $\chi^{2}$ of 42 for 48 bins. Also shown in the figure are our estimates [@staengle] of the non-$s\bar{s}$ backgrounds, which were mostly determined from data. Our final result is $A_{s} = 0.895 \pm 0.066(stat.) \pm 0.062(syst.)$. This result is consistent with the Standard Model expectation of 0.935 for $A_{s}$, and with less precise previous measurements [@delphi; @opal]. Conclusions =========== We have presented results from a sample of 550,000 $e^{+}e^{-} \rightarrow Z^{0} \rightarrow q\bar{q}$ events produced with a longitudinally polarized electron beam. Polarization enables us to give a sign to rapidities so that positive rapidity corresponds to the quark (rather than antiquark) direction. The distribution of the difference between the signed rapidities of $K^{+}$ and $K^{-}$ shows a large asymmetry at large values of the absolute rapidity difference, a direct indication that the long-range correlated $KK$ pairs are dominated by $s\bar{s}$ events. There is a large asymmetry at small rapidity difference for $p\bar{p}$ pairs, a clear indication of ordering of baryons along the event axis. We have also performed a measurement of $A_{s}$, the parity-violating coupling of the $Z^{0}$ to strange quarks, obtained directly from the left-right forward-backward production asymmetry in polar angle of the tagged $s$ quark. Our result is $A_{s}$ = $0.895\pm 0.066(stat.)\pm 0.062(syst.)$, which is consistent with the Standard Model expectation, and with less precise previous measurements. Questions and Answers ===================== Question (from M. Boutemeur, Munich): How do the $\pi^{\pm}$, $K^{\pm}$, and $p/\bar{p}$ rates compare in quark and gluon jets? Answer: My graduate student Hyejoo Kang is working on this very topic for her Ph.D. thesis. We expect to have results ready in time for the Moriond Conference in early 2001. [99]{} SLD Collab.: K. Abe [*et al.*]{}, Phys. Rev. D [**53**]{} (1996) 1023. K. Abe [*et al.*]{}, Nucl. Inst. Meth. [**A343**]{} (1994) 74. T.J. Pavel, Ph.D. Thesis, Stanford University, 1996, SLAC-R-491 (1996). SLD Collab.: K. Abe [*et al.*]{}, Phys. Rev. D [**59**]{} (1999) 52001. H. Staengle, Ph.D. Thesis, Colorado State University (1999), SLAC-R-549. SLD Collab.: K. Abe [*et al.*]{}, Phys. Rev. Lett. [**78**]{} (1997) 3442. T. Sjöstrand, Comp. Phys. Comm. [**82**]{} (1994) 74. C.J.S. Damerell [*et al.*]{}, Nucl. Inst. Meth. [**A288**]{} (1990) 236. P. Abreu [*et al.*]{}, Eur. Phys. J. [**C14**]{} (2000) 613. K. Ackerstaff [*et al.*]{}, Z. Phys. [**C76**]{} (1997) 387. $^{**}$List of Authors {#list-of-authors .unnumbered} ====================== =.75 *=.75 Aomori University, Aomori, 030 Japan, University of Bristol, Bristol, United Kingdom, Brunel University, Uxbridge, Middlesex, UB8 3PH United Kingdom, Boston University, Boston, Massachusetts 02215, University of Colorado, Boulder, Colorado 80309, Colorado State University, Ft. Collins, Colorado 80523, INFN Sezione di Ferrara and Universita di Ferrara, I-44100 Ferrara, Italy, INFN Laboratori Nazionali di Frascati, I-00044 Frascati, Italy, Johns Hopkins University, Baltimore, Maryland 21218-2686, Lawrence Berkeley Laboratory, University of California, Berkeley, California 94720, University of Massachusetts, Amherst, Massachusetts 01003, University of Mississippi, University, Mississippi 38677, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, Institute of Nuclear Physics, Moscow State University, 119899 Moscow, Russia, Nagoya University, Chikusa-ku, Nagoya, 464 Japan, University of Oregon, Eugene, Oregon 97403, Oxford University, Oxford, OX1 3RH, United Kingdom, INFN Sezione di Perugia and Universita di Perugia, I-06100 Perugia, Italy, Rutherford Appleton Laboratory, Chilton, Didcot, Oxon OX11 0QX United Kingdom, Rutgers University, Piscataway, New Jersey 08855, Stanford Linear Accelerator Center, Stanford University, Stanford, California 94309, Soongsil University, Seoul, Korea 156-743, University of Tennessee, Knoxville, Tennessee 37996, Tohoku University, Sendai, 980 Japan, University of California at Santa Barbara, Santa Barbara, California 93106, University of California at Santa Cruz, Santa Cruz, California 95064, Vanderbilt University, Nashville,Tennessee 37235, University of Washington, Seattle, Washington 98105, University of Wisconsin, Madison,Wisconsin 53706, Yale University, New Haven, Connecticut 06511.* [^1]: Work supported in part by Department of Energy contract DE-AC03-76SF00515.

--- abstract: | Adaptive wave model for financial option pricing is proposed, as a high-complexity alternative to the standard Black–Scholes model. The new option-pricing model, representing a controlled Brownian motion, includes two wave-type approaches: nonlinear and quantum, both based on (adaptive form of) the Schrödinger equation. The nonlinear approach comes in two flavors: (i) for the case of constant volatility, it is defined by a single adaptive nonlinear Schrödinger (NLS) equation, while for the case of stochastic volatility, it is defined by an adaptive Manakov system of two coupled NLS equations. The linear quantum approach is defined in terms of de Broglie’s plane waves and free-particle Schrödinger equation. In this approach, financial variables have quantum-mechanical interpretation and satisfy the Heisenberg-type uncertainty relations. Both models are capable of successful fitting of the Black–Scholes data, as well as defining Greeks.\ **Keywords:** Black–Scholes option pricing, adaptive nonlinear Schrödinger equation,\ adaptive Manakov system, quantum-mechanical option pricing, market-heat potential\ **PACS:** 89.65.Gh, 05.45.Yv, 03.65.Ge author: - | Vladimir G. Ivancevic\ [Defence Science & Technology Organisation, Australia]{} title: | Adaptive Wave Models for Option Pricing Evolution:\ Nonlinear and Quantum Schrödinger Approaches --- Introduction ============ Recall that the celebrated Black–Scholes partial differential equation (PDE) describes the time–evolution of the market value of a *stock option* [@BS; @Merton]. Formally, for a function $u=u(t,s)$ defined on the domain $0\leq s<\infty ,~0\leq t\leq T$ and describing the market value of a stock option with the stock (asset) price $s$, the *Black–Scholes PDE* can be written (using the physicist notation: $\partial _{z}u=\partial u/\partial z$) as a diffusion–type equation: $$\partial _{t}u=-\frac{1}{2}(\sigma s)^{2}\,\partial _{ss}u-rs\,\partial _{s}u+ru, \label{BS}$$ where $\sigma >0$ is the standard deviation, or *volatility* of $s$, $r$ is the short–term prevailing continuously–compounded risk–free interest rate, and $T>0$ is the time to maturity of the stock option. In this formulation it is assumed that the *underlying* (typically the stock) follows a *geometric Brownian motion* with ‘drift’ $\mu $ and volatility $\sigma $, given by the stochastic differential equation (SDE) [@Osborne] $$ds(t)=\mu s(t)dt+\sigma s(t)dW(t), \label{gbm}$$ where $W$ is the standard Wiener process. The Black-Scholes PDE (\[BS\]) is usually derived from SDEs describing the geometric Brownian motion ([gbm]{}), with the stock-price solution given by: $$s(t)=s(0)\,\mathrm{e}^{(\mu -\frac{1}{2}\sigma ^{2})t+\sigma W(t)}.$$ In mathematical finance, derivation is usually performed using Itô lemma [@Ito] (assuming that the underlying asset obeys the Itô SDE), while in physics it is performed using Stratonovich interpretation [@Perello; @Gardiner] (assuming that the underlying asset obeys the Stratonovich SDE [@Stratonovich]). The Black-Sholes PDE (\[BS\]) can be applied to a number of one-dimensional models of interpretations of prices given to $u$, e.g., puts or calls, and to $s$, e.g., stocks or futures, dividends, etc. The most important examples are European call and put options, defined by: $$\begin{aligned} &&u_{\mathrm{Call}}(s,t)=s\,\mathcal{N}(\mathrm{d_{1}})\,\mathrm{e}% ^{-T\delta }-k\,\mathcal{N}(\mathrm{d_{2}})\,\mathrm{e}^{-rT}, \label{call} \\ &&u_{\mathrm{Put}}(s,t)=k\,\mathcal{N}(-\mathrm{d_{2}})\,\mathrm{e}^{-rT}-s\,% \mathcal{N}(-\mathrm{d_{1}})\,\mathrm{e}^{-T\delta }, \label{put} \\ &&\,\mathcal{N}(\lambda )=\frac{1}{2}\left( 1+\mathrm{erf}\left( \frac{% \lambda }{\sqrt{2}}\right) \right) , \notag \\ &&\mathrm{d_{1}}=\frac{\ln \left( \frac{s}{k}\right) +T\left( r-\delta +% \frac{\sigma ^{2}}{2}\right) }{\sigma \sqrt{T}},\qquad \mathrm{d_{2}}=\frac{% \ln \left( \frac{s}{k}\right) +T\left( r-\delta -\frac{\sigma ^{2}}{2}% \right) }{\sigma \sqrt{T}}, \notag\end{aligned}$$where erf$(\lambda )$ is the (real-valued) error function, $k$ denotes the strike price and $\delta $ represents the dividend yield. In addition, for each of the call and put options, there are five Greeks (see, e.g. [@Kelly; @IvCogComp]), or sensitivities, which are partial derivatives of the option-price with respect to stock price (Delta), interest rate (Rho), volatility (Vega), elapsed time since entering into the option (Theta), and the second partial derivative of the option-price with respect to the stock price (Gamma). ![Fitting the Black–Scholes call option with $\protect\beta (w)$-adaptive PDF of the shock-wave NLS-solution (\[tanh1\]).[]{data-label="fitCall"}](fitCall){width="10cm"} ![Fitting the Black–Scholes put option with $\protect\beta (w)$–adaptive PDF of the shock-wave NLS $\protect\psi _{2}(s,t)$ solution (\[tanh1\]). Notice the kink near $s=100$.[]{data-label="fitPut"}](fitPut){width="10cm"} Using the standard *Kolmogorov probability* approach, instead of the market value of an option given by the Black–Scholes equation (\[BS\]), we could consider the corresponding probability density function (PDF) given by the backward Fokker–Planck equation (see [@Gardiner; @Voit]). Alternatively, we can obtain the same PDF (for the market value of a stock option), using the *quantum–probability* formalism [ComplexDyn,QuLeap]{}, as a solution to a time–dependent linear or nonlinear *Schrödinger equation* for the evolution of the complex–valued wave $\psi -$function for which the absolute square, $|\psi |^{2},$ is the PDF. The adaptive nonlinear Schrödinger (NLS) equation was recently used in [@IvCogComp] as an approach to option price modelling, as briefly reviewed in this section. The new model, philosophically founded on adaptive markets hypothesis [@Lo1; @Lo2] and Elliott wave market theory [@Elliott1; @Elliott2], as well as my own recent work on quantum congition [@LSF; @QnnBk], describes adaptively controlled Brownian market behavior. This nonlinear approach to option price modelling is reviewed in the next section. Its important limiting case with low interest-rate reduces to the linear Schrödinger equation. This linear approach to option price modelling is elaborated in the subsequent section. Nonlinear adaptive wave model for general option pricing ======================================================== Adaptive NLS model ------------------ The adaptive, wave–form, nonlinear and stochastic option–pricing model with stock price $s,$ volatility $\sigma $ and interest rate $r$ is formally defined as a complex-valued, focusing (1+1)–NLS equation, defining the time-dependent *option–price wave function* $\psi =\psi (s,t)$, whose absolute square $|\psi (s,t)|^{2}$ represents the probability density function (PDF) for the option price in terms of the stock price and time. In natural quantum units, this NLS equation reads: $$\mathrm{i}\partial _{t}\psi =-\frac{1}{2}\sigma \partial _{ss}\psi -\beta |\psi |^{2}\psi ,\qquad (\mathrm{i}=\sqrt{-1}),\qquad \label{nlsGen}$$ where $\beta=\beta (r,w) $ denotes the adaptive market-heat potential (see [@KleinertBk]), so the term $V(\psi )=-\beta |\psi |^{2}$ represents the $% \psi -$dependent potential field. In the simplest nonadaptive scenario $\beta $ is equal to the interest rate $r$, while in the adaptive case it depends on the set of adjustable synaptic weights $\{w^i_j\}$ as: $$\beta (r,w)=r\sum_{i=1}^{n}w_{1}^{i}\,\text{erf}\left( \frac{w_{2}^{i}s}{% w_{3}^{i}}\right) . \label{betaW}$$ Physically, the NLS equation (\[nlsGen\]) describes a nonlinear wave (e.g. in Bose-Einstein condensates) defined by the complex-valued wave function $\psi (s,t)$ of real space and time parameters. In the present context, the space-like variable $s$ denotes the stock (asset) price. The NLS equation (\[nlsGen\]) has been exactly solved using the power series expansion method [@LiuEtAl01; @LiuFan05] of [Jacobi elliptic functions]{} [@AbrSte]. Consider the $\psi -$function describing a single plane wave, with the wave number $k$ and circular frequency $% \omega $: $$\psi (s,t)=\phi (\xi )\,\mathrm{e}^{\mathrm{i}(ks-\omega t)},\qquad \text{% with \ }\xi =s-\sigma kt\text{ \ and \ }\phi (\xi )\in \Bbb{R}. \label{subGen}$$ Its substitution into the NLS equation (\[nlsGen\]) gives the nonlinear oscillator ODE: $$\phi ^{\prime \prime }(\xi )+[\omega -\frac{1}{2}\sigma k^{2}]\,\phi (\xi )+\beta \phi ^{3}(\xi )=0. \label{15e}$$ We can seek a solution $\phi (\xi )$ for (\[15e\]) as a linear function [@LiuFan05] $$\phi (\xi )=a_{0}+a_{1}\mathrm{sn}(\xi ),$$ where $\mathrm{sn}(s)=\mathrm{sn}(s,m)$ are Jacobi elliptic sine functions with *elliptic modulus* $m\in \lbrack 0,1]$, such that $\mathrm{sn}% (s,0)=\sin (s)\ $and $\mathrm{sn}(s,1)=\mathrm{\tanh }(s)$. The solution of (\[15e\]) was calculated in [@IvCogComp] to be $$\begin{aligned} \phi (\xi ) &=&\pm m\sqrt{\frac{-\sigma }{\beta }}\,\mathrm{sn}(\xi ),\qquad ~\text{for~~}m\in \lbrack 0,1];~~\text{and} \\ \phi (\xi ) &=&\pm \sqrt{\frac{-\sigma }{\beta }}\,\mathrm{\tanh }(\xi ),\qquad \text{for~~}m=1.\end{aligned}$$ This gives the exact periodic solution of (\[nlsGen\]) as [@IvCogComp] $$\begin{aligned} \psi _{1}(s,t) &=&\pm m\sqrt{\frac{-\sigma }{\beta (w)}}\,\mathrm{sn}% (s-\sigma kt)\,\mathrm{e}^{\mathrm{i}[ks-\frac{1}{2}\sigma t(1+m^{2}+k^{2})]},\qquad ~~\text{for~~}m\in \lbrack 0,1); \label{sn1} \\ \psi _{2}(s,t) &=&\pm \sqrt{\frac{-\sigma }{\beta (w)}}\,\mathrm{\tanh }% (s-\sigma kt)\,\mathrm{e}^{\mathrm{i}[ks-\frac{1}{2}\sigma t(2+k^{2})]},\qquad \qquad \text{for~~}m=1, \label{tanh1}\end{aligned}$$ where (\[sn1\]) defines the general solution, while (\[tanh1\]) defines the *envelope shock-wave*[^1] (or, ‘dark soliton’) solution of the NLS equation (\[nlsGen\]). Alternatively, if we seek a solution $\phi (\xi )$ as a linear function of Jacobi elliptic cosine functions, such that $\mathrm{cn}(s,0)=\cos (s)$ and $% \mathrm{cn}(s,1)=\mathrm{sech}(s)$,[^2] $$\phi (\xi )=a_{0}+a_{1}\mathrm{cn}(\xi ),$$ then we get [@IvCogComp] $$\begin{aligned} \psi _{3}(s,t) &=&\pm m\sqrt{\frac{\sigma }{\beta (w)}}\,\mathrm{cn}% (s-\sigma kt)\,\mathrm{e}^{\mathrm{i}[ks-\frac{1}{2}\sigma t(1-2m^{2}+k^{2})]},\qquad \text{for~~}m\in \lbrack 0,1); \label{cn1} \\ \psi _{4}(s,t) &=&\pm \sqrt{\frac{\sigma }{\beta (w)}}\,\mathrm{sech}% (s-\sigma kt)\,\mathrm{e}^{\mathrm{i}[ks-\frac{1}{2}\sigma t(k^{2}-1)]},\qquad \qquad \text{for~~}m=1, \label{sech1}\end{aligned}$$ where (\[cn1\]) defines the general solution, while (\[sech1\]) defines the *envelope solitary-wave* (or, ‘bright soliton’) solution of the NLS equation (\[nlsGen\]). In all four solution expressions (\[sn1\]), (\[tanh1\]), (\[cn1\]) and (\[sech1\]), the adaptive potential $\beta (w)$ is yet to be calculated using either unsupervised Hebbian learning, or supervised Levenberg–Marquardt algorithm (see, e.g. [@NeuFuz; @CompMind]). In this way, the NLS equation (\[nlsGen\]) becomes the *quantum neural network* (see [@QnnBk]). Any kind of numerical analysis can be easily performed using above closed-form solutions $\psi _{i}(s,t)~~(i=1,...,4)$ as initial conditions. The adaptive NLS–PDFs of the shock-wave type (\[tanh1\]) has been used in [@IvCogComp] to fit the Black–Scholes call and put options (see Figures \[fitCall\] and \[fitPut\]). Specifically, the adaptive heat potential (\[betaW\]) was combined with the spatial part of (\[tanh1\]) $$\phi (s)=\left| \sqrt{\frac{\sigma }{\beta }}\tanh (s-kt\sigma )\right| {}^{2}, \label{kink2}$$ while parameter estimates where obtained using 100 iterations of the Levenberg–Marquardt algorithm. ![Smoothing out the kink in the put option fit, by combining the shock-wave solution with the soliton solution, as defined by (\[kinksech\]).[]{data-label="tanSechPut"}](tanSechPut){width="10cm"} As can be seen from Figure (\[fitPut\]) there is a kink near $s=100$. This kink, which is a natural characteristic of the spatial shock-wave (\[kink2\]), can be smoothed out (Figure \[tanSechPut\]) by taking the sum of the spatial parts of the shock-wave solution (\[tanh1\]) and the soliton solution (\[sech1\]) as: $$\phi (s)=\left\vert \sqrt{\frac{\sigma }{\beta }}\left[d_{1}\tanh (s-kt\sigma )+d_{2}\,\text{sech}(s-kt\sigma )\right] \right\vert {}^{2}. \label{kinksech}$$ The adaptive NLS–based Greeks (Delta, Rho, Vega, Theta and Gamma) have been defined in [@IvCogComp], as partial derivatives of the shock-wave solution (\[tanh1\]). Adaptive Manakov system ----------------------- Next, for the purpose of including a *controlled stochastic volatility*[^3] into the adaptive–NLS model (\[nlsGen\]), the full bidirectional quantum neural computation model [@QnnBk] for option-price forecasting has been formulated in [@IvCogComp] as a self-organized system of two coupled self-focusing NLS equations: one defining the *option–price wave function* $\psi =\psi (s,t)$ and the other defining the *volatility wave function* $\sigma =\sigma (s,t)$: $$\begin{aligned} \text{Volatility NLS :}\quad \mathrm{i}\partial _{t}\sigma &=&-\frac{1}{2}% \partial _{ss}\mathcal{\sigma }-\beta (r,w)\left( |\mathcal{\sigma }% |^{2}+|\psi |^{2}\right) \mathcal{\sigma }, \label{stochVol} \\ \text{Option price NLS :}\quad \mathrm{i}\partial _{t}\psi &=&-\frac{1}{2}% \partial _{ss}\psi -\beta (r,w)\left( |\mathcal{\sigma }|^{2}+|\psi |^{2}\right) \psi . \label{stochPrice}\end{aligned}$$ In this coupled model, the $\sigma $–NLS (\[stochVol\]) governs the $% (s,t)-$evolution of stochastic volatility, which plays the role of a nonlinear coefficient in (\[stochPrice\]); the $\psi $–NLS (\[stochPrice\]) defines the $(s,t)-$evolution of option price, which plays the role of a nonlinear coefficient in (\[stochVol\]). The purpose of this coupling is to generate a *leverage effect*, i.e. stock volatility is (negatively) correlated to stock returns[^4] (see, e.g. [@Roman]). This bidirectional associative memory effectively performs quantum neural computation [@QnnBk], by giving a spatio-temporal and quantum generalization of Kosko’s BAM family of neural networks [@Kosko1; @Kosko2]. In addition, the shock-wave and solitary-wave nature of the coupled NLS equations may describe brain-like effects frequently occurring in financial markets: volatility/price propagation, reflection and collision of shock and solitary waves (see [@Han]). The coupled NLS-system (\[stochVol\])–(\[stochPrice\]), without an embedded $w-$learning (i.e., for constant $\beta =r$ – the interest rate), actually defines the well-known *Manakov system*,[^5] proven by S. Manakov in 1973 [@manak74] to be completely integrable, by the existence of infinite number of involutive integrals of motion. It admits ‘bright’ and ‘dark’ soliton solutions. The simplest solution of (\[stochVol\])–(\[stochPrice\]), the so-called *Manakov bright 2–soliton*, has the form resembling that of the sech-solution (\[sech1\]) (see [@Benney; @Zakharov; @Hasegawa; @Radhakrishnan; @Agrawal; @Yang; @Elgin]), and is formally defined by: $$\mathbf{\psi }_{\mathrm{sol}}(s,t)=2b\,\mathbf{c\,}\mathrm{sech}(2b(s+4at))\,% \mathrm{e}^{-2\mathrm{i}(2a^{2}t+as-2b^{2}t)}, \label{ManSol}$$ where $\mathbf{\psi }_{\mathrm{sol}}(s,t)=\left( \begin{array}{c} \sigma (s,t) \\ \psi (s,t) \end{array} \right) $, $\mathbf{c}=(c_{1},c_{2})^{T}$ is a unit vector such that $% |c_{1}|^{2}+|c_{2}|^{2}=1$. Real-valued parameters $a$ and $b$ are some simple functions of $(\sigma ,\beta ,k)$, which can be determined by the Levenberg–Marquardt algorithm. I have argued in [@IvCogComp] that in some short-time financial situations, the adaptation effect on $\beta$ can be neglected, so our option-pricing model (\[stochVol\])–(\[stochPrice\]) can be reduced to the Manakov 2–soliton model (\[ManSol\]), as depicted and explained in Figure \[SolitonCollision\]. ![Hypothetical market scenario including sample PDFs for volatility $|% \mathcal{\protect\sigma }|^{2}$ and $|\protect\psi |^{2}$ of the Manakov 2–soliton (\[ManSol\]). On the left, we observe the $(s,t)-$evolution of stochastic volatility: we have a collision of two volatility component-solitons, $S_{1}(s,t)$ and $S_{2}(s,t)$, which join together into the resulting soliton $S_{2}(s,t)$, annihilating the $S_{1}(s,t)$ component in the process. On the right, we observe the $(s,t)-$evolution of option price: we have a collision of two option component-solitons, $S_{1}(s,t)$ and $S_{2}(s,t)$, which pass through each other without much change, except at the collision point. Due to symmetry of the Manakov system, volatility and option price can exchange their roles.[]{data-label="SolitonCollision"}](SolitonCollision){width="12cm"} Quantum wave model for low interest-rate option pricing ======================================================= In the case of a low interest-rate $r\ll 1$, we have $\beta (r)\ll 1$, so $% V(\psi )\rightarrow 0,$ and therefore equation (\[nlsGen\]) can be approximated by a quantum-like *option wave packet.* It is defined by a continuous superposition of *de Broglie’s plane waves*, ‘physically’ associated with a free quantum particle of unit mass. This linear wave packet, given by the time-dependent complex-valued wave function $\psi =\psi (s,t)$, is a solution of the *linear Schrödinger equation* with zero potential energy, Hamiltonian operator $\hat{H}$ and volatility $\sigma $ playing the role similar to the Planck constant. This equation can be written as: $$\mathrm{i}\sigma \partial _{t}\psi =\hat{H}\psi ,\qquad \text{where}\qquad \hat{H}=-\frac{\sigma ^{2}}{2}\partial _{ss}. \label{sch1}$$ Thus, we consider the $\psi -$function describing a single de Broglie’s plane wave, with the wave number $k$, linear momentum $p=\sigma k,$ wavelength $\lambda _{k}=2\pi /k,$ angular frequency $\omega _{k}=\sigma k^{2}/2,$ and oscillation period $T_{k}=2\pi /\omega _{k}=4\pi /\sigma k^{2}$. It is defined by (compare with [@Griffiths; @Thaller; @QuLeap]) $$\psi _{k}(s,t)=A\mathrm{e}^{\mathrm{i}(ks-\omega _{k}t)}=A\mathrm{e}^{% \mathrm{i}(ks-{\frac{\sigma k^{2}}{2}}t)}=A\cos (ks-{\frac{\sigma k^{2}}{2}}% t)+A\mathrm{i}\sin (ks-{\frac{\sigma k^{2}}{2}}t), \label{Broglie}$$ where $A$ is the amplitude of the wave, the angle $(ks-\omega _{k}t)=(ks-{% \frac{\sigma k^{2}}{2}}t)$ represents the phase of the wave $\psi _{k}$ with the *phase velocity:* $v_{k}=\omega _{k}/k=\sigma k/2.$ The space-time wave function $\psi (s,t)$ that satisfies the linear Schrödinger equation (\[sch1\]) can be decomposed (using Fourier’s separation of variables) into the spatial part $\phi (s)\,$ and the temporal part $\mathrm{e}^{-\mathrm{i}\omega t}\ $as: $$\psi (s,t)=\phi (s)\,\mathrm{e}^{-\mathrm{i}\omega t}=\phi (s)\,\mathrm{e}^{-% \frac{\mathrm{i}}{\sigma }Et}.$$ The spatial part, representing *stationary* (or,* amplitude*)* wave function*, $\phi (s)=A\mathrm{e}^{\mathrm{i}ks},$ satisfies the *linear harmonic oscillator,* which can be formulated in several equivalent forms: $$\phi ^{\prime\prime}+k^{2}\phi =0,\qquad \phi ^{\prime\prime}+\left( \frac{p% }{\sigma }\right) ^{2}\phi =0,\qquad \phi ^{\prime\prime}+\left( \frac{% \omega _{k}}{v_{k}}\right) ^{2}\phi =0,\qquad \phi ^{\prime\prime}+\frac{% 2E_{k}}{\sigma ^{2}}\phi =0. \label{stac}$$ Planck’s *energy quantum* of the option wave $\psi _{k}$ is given by: $ E_{k}=\sigma \omega _{k}=\frac{1}{2}(\sigma k)^{2}. $ From the plane-wave expressions (\[Broglie\]) we have: $\psi _{k}(s,t)=A% \mathrm{e}^{\frac{\mathrm{i}}{\sigma }(ps-E_{k}t)}-$ for the wave going to the ‘right’ and $\psi _{k}(s,t)=A\mathrm{e}^{-\frac{\mathrm{i}}{\sigma }% (ps+E_{k}t)}-$ for the wave going to the ‘left’. The general solution to (\[sch1\]) is formulated as a linear combination of de Broglie’s option waves (\[Broglie\]), comprising the option wave-packet: $$\psi (s,t)=\sum_{i=0}^{n}c_{i}\psi _{k_{i}}(s,t),\qquad (\text{with}\ n\in \Bbb{N}). \label{w-pack}$$ Its absolute square, $|\psi (s,t)|^{2},$ represents the probability density function at a time $t.$ The *group velocity* of an option wave-packet is given by: $\ v_{g}=d\omega _{k}/dk.$ It is related to the phase velocity $v_{k}$ of a plane wave as: $% v_{g}=v_{k}-\lambda _{k}dv_{k}/d\lambda _{k}.$ Closely related is the *center* of the option wave-packet (the point of maximum amplitude), given by: $% s=td\omega _{k}/dk.$ The following quantum-motivated assertions can be stated: 1. Volatility $\sigma $ has dimension of *financial action*, or *energy* $\times $* time*. 2. The total energy $E$ of an option wave-packet is (in the case of similar plane waves) given by Planck’s superposition of the energies $E_{k}$ of $n$ individual waves: $E=n\sigma \omega _{k}=\frac{n}{2}(\sigma k)^{2},$ where $L=n\sigma $ denotes the *angular momentum* of the option wave-packet, representing the shift between its growth and decay, and *vice versa.* 3. The average energy $\left\langle E\right\rangle $ of an option wave-packet is given by Boltzmann’s partition function: $$\left\langle E\right\rangle =\frac{\sum_{n=0}^{\infty }nE_{k}\mathrm{e}^{-% \frac{nE_{k}}{bT}}}{\sum_{n=0}^{\infty }\mathrm{e}^{-\frac{nE_{k}}{bT}}}=% \frac{E_{k}}{\mathrm{e}^{\frac{E_{k}}{bT}}-1},$$ where $b$ is the Boltzmann-like kinetic constant and $T$ is the market temperature. 4. The energy form of the Schrödinger equation (\[sch1\]) reads: $% E\psi =\mathrm{i}\sigma \partial _{t}\psi $. 5. The eigenvalue equation for the Hamiltonian operator $\hat{H}$ is the *stationary Schrödinger equation:* $\ $$$\hat{H}\phi (s)=E\phi (s),\qquad \text{or}\qquad E\phi (s)=-\frac{\sigma ^{2}% }{2}\partial _{ss}\phi (s),$$ which is just another form of the harmonic oscillator (\[stac\]). It has oscillatory solutions of the form: $$\phi _{E}(s)=c_{1}\mathrm{e}^{\frac{\mathrm{i}}{\sigma }\sqrt{2E_{k}}% \,s}+c_{2}\mathrm{e}^{-\frac{\mathrm{i}}{\sigma }\sqrt{2E_{k}}\,s}\,,$$ called *energy eigen-states* with energies $E_{k}$ and denoted by: $\hat{H}% \phi _{E}(s)=E_{k}\phi _{E}(s).$ ![Fitting the Black–Scholes put option with the quantum PDF given by the absolute square of (\[w-pack\]) with $n=7$.[]{data-label="PutQuant"}](PutQuant){width="10cm"} ![Fitting the Black–Scholes call option with the quantum PDF given by the absolute square of (\[w-pack\]) with $n=3$. Note that fit is good in the realistic stock region: $s\in [75,140]$.[]{data-label="QuantumCall"}](QuantumCall){width="10cm"} The Black–Scholes put and call options have been fitted with the quantum PDFs (see Figures \[PutQuant\] and \[QuantumCall\]) given by the absolute square of (\[w-pack\]) with $n=7$ and $n=3$, respectively. Using supervised Levenberg–Marquardt algorithm and *Mathematica* 7, the following coefficients were obtained for the Black–Scholes put option: $\sigma^* = -0.0031891,~t^*= -0.0031891,~k_1= 2.62771,~k_2= 2.62777,~k_3= 2.65402,$ $k_4= 2.61118,~k_5= 2.64104,~k_6= 2.54737,~k_7= 2.62778,~c_1= 1.26632,~c_2= 1.26517,$ $c_3= 2.74379,~c_4= 1.35495,~c_5= 1.59586,~c_6= 0.263832,~c_7= 1.26779,$ with  $\sigma_{BS}=-94.0705\sigma^*,~t_{BS}=-31.3568t^*.$ Using the same algorithm, the following coefficients were obtained for the Black–Scholes call option: $\sigma^* = -11.9245,~t^*= -11.9245,~k_1= 0.851858,~k_2= 0.832409,$ $k_3= 0.872061,~c_1= 2.9004,~c_2= 2.72592,~c_3= 2.93291,$ with $\sigma_{BS}-0.0251583 \sigma^*,~t=-0.00838609 t^*.$ Now, given some initial option wave function, $\psi (s,0)=\psi _{0}(s),$ a solution to the initial-value problem for the linear Schrödinger equation (\[sch1\]) is, in terms of the pair of Fourier transforms $(\mathcal{F},% \mathcal{F}^{-1}),$ given by (see [@Thaller]) $$\psi (s,t)=\mathcal{F}^{-1}\left[ \mathrm{e}^{-\mathrm{i}\omega t}\mathcal{F}% (\psi _{0})\right] =\mathcal{F}^{-1}\left[ \mathrm{e}^{-\mathrm{i}{\frac{% \sigma k^{2}}{2}}t}\mathcal{F}(\psi _{0})\right] . \label{Fouri}$$ For example (see [@Thaller]), suppose we have an initial option wave-function at time $t=0$ given by the complex-valued Gaussian function: $$\psi (s,0)=\mathrm{e}^{-as^{2}/2}\mathrm{e}^{\mathrm{i}\sigma ks},$$ where $a$ is the width of the Gaussian, while $p$ is the average momentum of the wave. Its Fourier transform, $\hat{\psi}_{0}(k)=\mathcal{F}[\psi (s,0)],$ is given by $$\hat{\psi}_{0}(k)=\frac{\mathrm{e}^{-\frac{(k-p)^{2}}{2a}}}{\sqrt{a}}.$$ The solution at time $t$ of the initial value problem is given by $$\psi (s,t)=\frac{1}{\sqrt{2\pi a}}\int_{-\infty }^{+\infty }\mathrm{e}^{% \mathrm{i}(ks-{\frac{\sigma k^{2}}{2}}t)}\,\mathrm{e}^{-\frac{a(k-p)^{2}}{2a}% }\,dk,$$ which, after some algebra becomes $$\psi (s,t)=\frac{\mathrm{\exp }(-\frac{as^{2}-2\mathrm{i}sp+\mathrm{i}p^{2}t% }{2(1+\mathrm{i}at)})}{\sqrt{1+\mathrm{i}at}},\qquad (\text{with \ }p=\sigma k).$$ As a simpler example,[^6] if we have an initial option wave-function given by the real-valued Gaussian function, $$\psi (s,0)=\frac{\mathrm{e}^{-s^{2}/2}}{\sqrt[4]{\pi }},$$ the solution of (\[sch1\]) is given by the complex-valued $\psi -$function, $$\psi (s,t)=\frac{\mathrm{\exp }(-\frac{s^{2}}{2(1+\mathrm{i}t)})}{\sqrt[4]{% \pi }\sqrt{1+\mathrm{i}t}}.$$ From (\[Fouri\]) it follows that a stationary option wave-packet is given by: $$\phi (s)=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{+\infty }\mathrm{e}^{\frac{% \mathrm{i}}{\sigma }ks}\,\hat{\psi}(k)\,dk,\qquad \text{where}\qquad \hat{% \psi}(k)=\mathcal{F}[\phi (s)].$$ As $|\phi (s)|^{2}$ is the stationary stock PDF, we can calculate the *expectation values* of the stock and the wave number of the whole option wave-packet, consisting of $n$ measured plane waves, as: $$\left\langle s\right\rangle =\int_{-\infty }^{+\infty }s|\phi (s)|^{2}ds\qquad \text{and}\qquad \left\langle k\right\rangle =\int_{-\infty }^{+\infty }k|\hat{\psi}(k)|^{2}dk. \label{means}$$ The recordings of $n$ individual option plane waves (\[Broglie\]) will be scattered around the mean values (\[means\]). The width of the distribution of the recorded $s-$ and $k-$values are uncertainties $\Delta s$ and $\Delta k,$ respectively. They satisfy the Heisenberg-type uncertainty relation: $$\Delta s\,\Delta k\geq \frac{n}{2},$$ which imply the similar relation for the total option energy and time: $$\Delta E\,\Delta t\geq \frac{n}{2}.$$ Finally, Greeks for both put and call options are defined as the following partial derivatives of the option $\psi-$function PDF:\ ${\text{Delta}}=\partial _{s}|\psi (s,t)|^{2}=\newline {2\mathrm{i}\sum_{j=1}^{n}c_{j}k_{j}\,\mathrm{e}^{k_{j}\left( \mathrm{i}s-% \text{${\frac{1}{2}}$}\mathrm{i}\sigma k_{j}t\right) }\text{Abs}\left[ \sum_{j=1}^{n}c_{j}\,\mathrm{e}^{k_{j}\left( \mathrm{i}s-\text{${\frac{1}{2}} $}\mathrm{i}\sigma k_{j}t\right) }\right] \text{Abs}^{\prime }\left[ \sum_{j=1}^{n}c_{j}\,\mathrm{e}^{k_{j}\left( \mathrm{i}s-\text{${\frac{1}{2}} $}\mathrm{i}\sigma k_{j}t\right) }\right] }$\ ${\text{Vega}}=\partial _{{\text{$\sigma $}}}|\psi (s,t)|^{2}=% \newline {-\text{it}\sum_{j=1}^{n}c_{j}k_{j}{}^{2}\,\mathrm{e}^{k_{j}\left( \mathrm{i}% s-\text{${\frac{1}{2}}$}\mathrm{i}\sigma k_{j}t\right) }\text{Abs}\left[ \sum_{j=1}^{n}c_{j}\,\mathrm{e}^{k_{j}\left( \mathrm{i}s-\text{${\frac{1}{2}} $}\mathrm{i}\sigma k_{j}t\right) }\right] \text{Abs}^{\prime }\left[ \sum_{j=1}^{n}c_{j}\,\mathrm{e}^{k_{j}\left( \mathrm{i}s-\text{${\frac{1}{2}} $}\mathrm{i}\sigma k_{j}t\right) }\right] }$\ ${\text{Theta}}=\partial _{{\text{$t$}}}|\psi (s,t)|^{2}=\newline {-\text{\textrm{i}$\sigma $}\sum_{j=1}^{n}c_{j}k_{j}{}^{2}\,\mathrm{e}% ^{k_{j}\left( \mathrm{i}s-\text{${\frac{1}{2}}$}\mathrm{i}\sigma k_{j}t\right) }\text{Abs}\left[ \sum_{j=1}^{n}c_{j}\,\mathrm{e}^{k_{j}\left( \mathrm{i}s-\text{${\frac{1}{2}}$}\mathrm{i}\sigma k_{j}t\right) }\right] \text{Abs}^{\prime }\left[ \sum_{j=1}^{n}c_{j}\,\mathrm{e}^{k_{j}\left( \mathrm{i}s-\text{${\frac{1}{2}}$}\mathrm{i}\sigma k_{j}t\right) }\right] }$\ ${\text{Gamma}}=\partial _{ss}|\psi (s,t)|^{2}=\newline {-2\sum_{j=1}^{n}c_{j}k_{j}{}^{2}\,\mathrm{e}^{k_{j}\left( \mathrm{i}s-\text{% ${\frac{1}{2}}$}\mathrm{i}\sigma k_{j}t\right) }\text{Abs}\left[ \sum_{j=1}^{n}c_{j}\,\mathrm{e}^{k_{j}\left( \mathrm{i}s-\text{${\frac{1}{2}} $}\mathrm{i}\sigma k_{j}t\right) }\right] \text{Abs}^{\prime}\left[ \sum_{j=1}^{n}c_{j}\,\mathrm{e}^{k_{j}\left( \mathrm{i}s-\text{${\frac{1}{2}} $}\mathrm{i}\sigma k_{j}t\right) }\right] -}\newline {\left( \sum_{j=1}^{n}c_{j}k_{j}\,\mathrm{e}^{k_{j}\left( \mathrm{i}s-\text{$% {\frac{1}{2}}$}\mathrm{i}\sigma k_{j}t\right) }\right) {}^{2}\text{Abs}% ^{\prime }\left[ \sum_{j=1}^{n}c_{j}\,\mathrm{e}^{k_{j}\left( \mathrm{i}s-% \text{${\frac{1}{2}}$}\mathrm{i}\sigma k_{j}t\right) }\right] {}^{2}-}% \newline {\left( \sum_{j=1}^{n}c_{j}k_{j}\,\mathrm{e}^{k_{j}\left( \mathrm{i}s-\text{$% {\frac{1}{2}}$}\mathrm{i}\sigma k_{j}t\right) }\right) {}^{2}\text{Abs}\left[ \sum_{j=1}^{n}c_{j}\,\mathrm{e}^{k_{j}\left( \mathrm{i}s-\text{${\frac{1}{2}} $}\mathrm{i}\sigma k_{j}t\right) }\right] \text{Abs}^{\prime \prime }\left[ \sum_{j=1}^{n}c_{j}\,\mathrm{e}^{k_{j}\left( \mathrm{i}s-\text{${\frac{1}{2}} $}\mathrm{i}\sigma k_{j}t\right) }\right] },$\ where $\text{Abs}$ denotes the absolute value, while $\text{Abs}^{\prime}$ and $\text{Abs}^{\prime \prime }$ denote its first and second derivatives. Conclusion ========== I have proposed an adaptive–wave alternative to the standard Black-Scholes option pricing model. The new model, philosophically founded on adaptive markets hypothesis [@Lo1; @Lo2] and Elliott wave market theory [@Elliott1; @Elliott2], describes adaptively controlled Brownian market behavior. Two approaches have been proposed: (i) a nonlinear one based on the adaptive NLS (solved by means of Jacobi elliptic functions) and the adaptive Manakov system (of two coupled NLS equations); (ii) a linear quantum-mechanical one based on the free-particle Schrödinger equation and de Broglie’s plane waves. For the purpose of fitting the Black-Scholes data, the Levenberg-Marquardt algorithm was used. The presented adaptive and quantum wave models are spatio-temporal dynamical systems of much higher complexity [@ComNonlin] then the Black-Scholes model. This makes the new wave models harder to analyze, but at the same time, their immense variety is potentially much closer to the real financial market complexity, especially at the time of financial crisis. [99]{} F. Black, M. Scholes, The Pricing of Options and Corporate Liabilities, J. Pol. Econ. **81**, 637-659, (1973) R.C. Merton, Bell J. Econ. and Management Sci. **4**, 141-183, (1973) M.F.M. Osborne, Operations Research **7**, 145-173, (1959) K. Itô, Mem. Am. Math. Soc. **4**, 1-51, (1951) J. Perello, J. M. Porra, M. Montero, J. Masoliver, Physica A **278**, **1-2**, 260-274, (2000) C.W. Gardiner, Handbook of Stochastic Methods, Springer, Berlin, (1983) J. Voit, The Statistical Mechanics of Financial Markets. Springer, (2005) R.L. Stratonovich, SIAM J. Control **4**, 362-371, (1966) M. Kelly, Black-Scholes Option Model & European Option Greeks. The Wolfram Demonstrations Project, http://demonstrations.wolfram.com/EuropeanOptionGreeks, (2009) V. Ivancevic, Cogn. Comput. (in press) arXiv.q-fin.PR:0911.1834 V. Ivancevic, T. Ivancevic, Complex Dynamics: Advanced System Dynamics in Complex Variables. Springer, Dordrecht, (2007) V. Ivancevic, T. Ivancevic, Quantum Leap: From Dirac and Feynman, Across the Universe, to Human Body and Mind. World Scientific, Singapore, (2008) A.W. Lo, J. Portf. Manag. **30**, 15-29, (2004) A.W. Lo, J. Inves. Consult. **7**, 21-44, (2005) A.J. Frost, R.R. Prechter, Jr., Elliott Wave Principle: Key to Market Behavior. Wiley, New York, (1978); (10th Edition) Elliott Wave International, (2009) P. Steven, Applying Elliott Wave Theory Profitably. Wiley, New York, (2003) V. Ivancevic, E. Aidman, Physica A **382**, 616–630, (2007) V. Ivancevic, T. Ivancevic, Quantum Neural Computation, Springer, (2009) H. Kleinert, H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets (3rd ed), World Scientific, Singapore, (2002) S. Liu, Z. Fu, S. Liu, Q. Zhao, Phys. Let. A **289**, 69–74, (2001) G-T. Liu, T-Y. Fan, Phys. Let. A **345**, 161–166, (2005) M. Abramowitz, I.A. Stegun, (Eds): Jacobian Elliptic Functions and Theta Functions. Chapter 16 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (9th ed). Dover, New York, 567-581, (1972) V. Ivancevic, T. Ivancevic, Neuro-Fuzzy Associative Machinery for Comprehensive Brain and Cognition Modelling. Springer, Berlin, (2007) V. Ivancevic, T. Ivancevic, Computational Mind: A Complex Dynamics Perspective. Springer, Berlin, (2007) V. Ivancevic, T. Ivancevic, Complex Nonlinearity: Chaos, Phase Transitions, Topology Change and Path Integrals, Springer, (2008) F. Black, 1976 Meet. Ame. Stat. Assoc. Bus. Econ. Stat. 177—181, (1976) H.E. Roman, M. Porto, C. Dose, EPL **84**, 28001, (5pp), (2008) B. Kosko, IEEE Trans. Sys. Man Cyb. **18**, 49–60, (1988) B. Kosko, Neural Networks, Fuzzy Systems, A Dynamical Systems Approach to Machine Intelligence. Prentice–Hall, New York, (1992) S.-H. Hanm, I.G. Koh, Phys. Rev. E **60**, 7608–7611, (1999) S.V. Manakov, (in Russian) Zh. Eksp. Teor. Fiz. **65** 505?516, (1973); (transleted into English) Sov. Phys. JETP **38**, 248–253, (1974) M. Haelterman, A.P. Sheppard, Phys. Rev. E **49**, 3376?3381, (1994) J. Yang, Physica D **108**, 92?112, (1997) D.J. Benney, A.C. Newell, J. Math. Phys. **46**, 133-139, (1967) V.E. Zakharov, S.V. Manakov, S.P. Novikov, L.P. Pitaevskii, Soliton theory: inverse scattering method. Nauka, Moscow, (1980) A. Hasegawa, Y. Kodama, Solitons in Optical Communications. Clarendon, Oxford, (1995) R. Radhakrishnan, M. Lakshmanan, J. Hietarinta, Phys. Rev. E. **56**, 2213, (1997) G. Agrawal, Nonlinear fiber optics (3rd ed.). Academic Press, San Diego, (2001). J. Yang, Phys. Rev. E **64**, 026607, (2001) J. Elgin, V. Enolski, A. Its, Physica D **225** (22), 127-152, (2007) D.J. Griffiths, Introduction to Quantum Mechanics (2nd ed.), Pearson Educ. Int., (2005) B. Thaller, Visual Quantum Mechanics, Springer, New York, (2000) [^1]: A shock wave is a type of fast-propagating nonlinear disturbance that carries energy and can propagate through a medium (or, field). It is characterized by an abrupt, nearly discontinuous change in the characteristics of the medium. The energy of a shock wave dissipates relatively quickly with distance and its entropy increases. On the other hand, a soliton is a self-reinforcing nonlinear solitary wave packet that maintains its shape while it travels at constant speed. It is caused by a cancelation of nonlinear and dispersive effects in the medium (or, field). [^2]: A closely related solution of an anharmonic oscillator ODE: $$\phi ^{\prime \prime }(s)+\phi (s)+\phi ^{3}(s)=0$$ is given by $$\phi (s)=\sqrt{\frac{2m}{1-2m}}\,\text{cn}\left( \sqrt{1+\frac{2m}{1-2m}}% ~s,\,m\right) .$$ [^3]: Controlled stochastic volatility here represents volatility evolving in a stochastic manner but within the controlled boundaries. [^4]: The hypothesis that financial leverage can explain the leverage effect was first discussed by F. Black [@Bl76]. [^5]: Manakov system has been used to describe the interaction between wave packets in dispersive conservative media, and also the interaction between orthogonally polarized components in nonlinear optical fibres (see, e.g. [@Kerr; @Yang1] and references therein). [^6]: An example of a more general Gaussian wave-packet solution of (\[sch1\]) is given by: $$\psi (s,t)=\sqrt{\frac{\sqrt{a/\pi }}{1+\mathrm{i}at}}\,\exp \left( \frac{-% \frac{1}{2}a(s-{s_{0}})^{2}-\frac{\mathrm{i}}{2}p_{0}^{2}t+\mathrm{i}p_{0}(s-% {s_{0}})}{1+\mathrm{i}at}\right) ,$$ where $s_{0},p_{0}$ are initial stock-price and average momentum, while $a$ is the width of the Gaussian. At time $t=0$ the ‘particle’ is at rest around $s=0$, its average momentum $p_{0}=0$. The wave function spreads with time while its maximum decreases and stays put at the origin. At time $-t$ the wave packet is the complex-conjugate of the wave-packet at time $t$.

--- abstract: 'We study genuine tripartite entanglement shared among the spins of three localized fermions in the non-interacting Fermi gas at zero temperature. Firstly, we prove analytically with the aid of entanglement witnesses that in a particular configuration the three fermions are genuinely tripartite entangled. Then various three-fermion configurations are investigated in order to quantify and calculate numerically the amount of genuine tripartite entanglement present in the system. Further we give a lower and an upper limit to the maximum diameter of the three-fermion configuration below which genuine tripartite entanglement exists and find that this distance is comparable with the maximum separation between two entangled fermions. The upper and lower limit turn to be very close to each other indicating that the applied witness operator is well suited to reveal genuine tripartite entanglement in the collection of non-interacting fermions.' author: - 'T. Vértesi' title: 'Genuine tripartite entanglement in the non-interacting Fermi gas' --- Introduction {#sec:intro} ============ Entanglement is in the heart of quantum mechanics and of great importance in quantum information theory. Two entangled particles already offer a valuable resource to perform several practical tasks such as quantum teleportation, quantum cryptography or quantum computation [@NC]. However, the multipartite setting due to the much richer structure suggests many new possibilities and phenomena over the bipartite case. Indeed, multipartite states may contradict local realistic models in a qualitatively different and stronger way [@GHSZ]. Moreover, this feature allows to implement novel quantum information processing tasks such as quantum computation based on cluster states [@RB], entanglement enhanced measurements [@GLM], quantum communication without a common reference frame [@BRS] and open-destination teleportation [@Zhao04]. Though the characterization of multipartite entanglement is studied in great depth [@Acin; @PV], still rarely investigated in solid state systems (see [@GTB] and references therein), however, it definitely plays an essential role in quantum phase transitions [@ON02; @Oli06] and might well be a key ingredient to unresolved problems in physics such as high temperature superconductivity [@Ved04]. In this article we investigate genuine multipartite entanglement shared among the spins of three fermions in the Fermi gas of non-interacting particles at zero temperature (degenerate Fermi gas) following the work of Refs. [@Ved03; @LBV]. We apply an entanglement witness developed in Ref. [@GTB] (and also appeared in Ref. [@TG]) in order to reveal genuine tripartite quantum correlations in the collection of fermions and using the approach in Ref. [@Bra05] we also characterize quantitatively the amount of it. The article is organized as follows: In Sec. \[sec:analysis\] according to Refs. [@Ved03; @LBV] we present the three-spin reduced density matrix of the degenerate Fermi gas, and show by analytical means (extending the related results of Ref. [@LBV]) that both $GHZ$-type and $W$-type witnesses are unable to detect genuine tripartite entanglement (GTE) among the spins of three localized fermions. On the other, we demonstrate that for specific configurations of the three fermions the GTE witness of Ref. [@GTB] is capable to signal GTE both for the two- and three-dimensional (2D and 3D) degenerate Fermi gases. In Sec. \[sec:er\] on the basis of this witness a formula is constructed to the lower bound of the generalized robustness ($E_R$) of genuine tripartite entanglement. With the aid of this formula in Sec. \[sec:num\] we quantify numerically genuine tripartite quantum correlations for various arrangements of the three particles. In Sec. \[sec:gte\] we determine lower and upper bounds to the GTE distance (i.e., to the largest diameter of the three-fermion configuration below which GTE is still present in the system) both in the 2D and 3D degenerate Fermi gases. The paper concludes in Sec. \[sec:disc\] with a brief summary of the results obtained and discusses possible schemes to extract GTE from the system. Analysis of the density matrix for three fermions {#sec:analysis} ================================================= Three-spin reduced density matrix {#subsec:3spin} --------------------------------- Consider a system of $N$ non-interacting fermions in a box with volume $V$. At zero temperature the ground state of the system is $|\phi_0\rangle = \Pi_k^{k_F} \hat{c}_{k,\sigma}^\dagger|vac\rangle$, where $k_F = (3\pi^2 N/V)^{1/3}$ is the Fermi momentum and $|vac\rangle$ denotes the vacuum state. From the ground state of the system $|\phi_0\rangle$ one obtains the three-spin reduced density matrix (up to normalization) between the three fermions localized at positions $\mathbf{r}$, $\mathbf{r}'$, and $\mathbf{r}''$, $$\begin{aligned} &\rho_3(s,s',s'';t,t',t'') = \nonumber \\ &\langle\phi_0| \hat{\psi}_{t}^\dagger(\mathbf{r}) \hat{\psi}_{t'}^\dagger(\mathbf{r}') \hat{\psi}_{t''}^\dagger(\mathbf{r}'') \hat{\psi}_{s''}(\mathbf{r}'') \hat{\psi}_{s'}(\mathbf{r}') \hat{\psi}_{s}(\mathbf{r}) |\phi_0\rangle \;, \label{rho3gen}\end{aligned}$$ where $\hat{\psi}_{s}(\mathbf{r})$, $\hat{\psi}_{s}^\dagger(\mathbf{r})$ are field annihilation/creation operators for a particle with spin $s$ located at position $\mathbf{r}$ satisfying $\{ \hat{\psi}_{s}(\mathbf{r}),\hat{\psi}_{s}^\dagger(\mathbf{r})\}=\delta_{s,s'}\delta(\mathbf r-\mathbf r')$. The justification, that the above matrix elements indeed describe three-qubit quantum states is discussed in the Appendix of Ref [@Cav05]. Following Refs. [@Ved03; @LBV] the explicit formula for the three-spin reduced density matrix $\rho_3$ is given by $$\begin{aligned} \rho_3=(1-p)\frac{\mathbb I_8}{8} &+p_{12}|\Psi_{12}^-\rangle \langle \Psi_{12}^-|\otimes\frac{\mathbb I_4}{2} + p_{13}|\Psi_{13}^-\rangle \langle \Psi_{13}^-|\otimes\frac{\mathbb I_4}{2} \nonumber \\ & +p_{23}|\Psi_{23}^-\rangle \langle \Psi_{23}^-|\otimes\frac{\mathbb I_4}{2} \;, \label{rho3}\end{aligned}$$ where $p=p_{12}+p_{13}+p_{23}$ and $\mathbb I_n$ denotes the $n\times n$ identity matrix. Further, $\Psi_{ij}^-=(|\uparrow\downarrow\rangle-|\downarrow\uparrow\rangle)/\sqrt 2$ is the singlet state of the pair $ij$ in the orthonormal basis $\{|\uparrow\rangle,|\downarrow\rangle\}$. The value $p_{ij}$ depends only on the relative distance between the three fermions and can be written explicitly for the fermion pair $ij$ as [@LBV] $$p_{ij}=\frac{-f_{ij}^2+f_{ij}f_{ik}f_{jk}}{-2+f_{ij}^2+f_{ik}^2+f_{jk}^2-f_{ij}f_{ik}f_{jk}}\;, \label{pij}$$ where the analytic form of $f_{ij}$ depends on the spatial dimension of the system, that is we may write $$\begin{aligned} f^{\mathrm{2D}}_{ij} &= 2J_1(k_F r_{ij})/k_F r_{ij} \nonumber \\ f^{\mathrm{3D}}_{ij} &= 3j_1(k_F r_{ij})/k_F r_{ij} \label{fij}\end{aligned}$$ in the case of the two- and three-dimensional Fermi gases [@OK]. In the above formulae $j_1$ and $J_1$ denote the spherical and the first order Bessel function of the first kind, respectively. Actually, owing to the collective $SU(2)$ rotational symmetry of the model Hamiltonian of non-interacting fermions, many matrix elements of $\rho_3$ in (\[rho3\]) are forced to be zero. Explicitly, the states which are invariant under collective $SU(2)$ rotation of the three qubits are the three-qubit Werner states [@EW], and they can be given in the form [@EW] $$\rho=\sum_{k=+,0,1,2,3}{\frac{r_k}{4}R_k}\;, \label{rho}$$ where $R_k$ are certain linear combinations of permutation operators and $r_k(\rho)=\mathrm{Tr}(\rho R_k)$. Using the definitions for $R_k$ from Ref. [@EW] and the explicit form of the state $\rho_3$ from (\[rho3\]), we are able to calculate the parameters $r_k$ for the three-spin reduced density matrix $\rho_3$, which read as follows $$\begin{aligned} r_+ & =\frac{1-p}{2} \nonumber \\ r_0 & =\frac{1+p}{2} \nonumber \\ r_1 & =\frac{p_{12}+p_{13}-2p_{23}}{2} \nonumber \\ r_2 & =\frac{3}{2\sqrt 3}(p_{13}-p_{12}) \nonumber \\ r_3 & =0 \;. \label{rk}\end{aligned}$$ Possible range of parameters $p_{ij}$ {#subsec:param} ------------------------------------- According to the Lemma 2 of Ref. [@EW] $\rho$ in (\[rho\]) is a density matrix only if $r_+,r_0\geq 0$. These inequalities imply for the state $\rho_3$ by the virtue of (\[rk\]) that $p$ lies in the interval $$-1\leq p\leq +1\;. \label{p}$$ Let us observe in (\[fij\]) that $|f_{ij}|\leq 1$ both for the 2D and 3D Fermi gases. This fact together with the bound to $p$ in (\[p\]) and also the definition $p=p_{12}+p_{13}+p_{23}$, after some algebraic manipulations (which are not detailed here), lead to the bounds $$-1\leq p_{ij}\leq 1 \label{pijbound}$$ for the three different fermion pairs $ij=12,13,23$. Thus, the parameters $p_{ij}$ appearing in state $\rho_3$ are limited by the values $\pm1$. Let us introduce the class of biseparable three-qubit states $B$, i.e., the states $$\rho = \sum_i{p_i |\psi_i\rangle\langle \psi_i|}\;, \label{bisep}$$ which can be expressed as a convex sum of projectors onto product and bipartite entangled vectors [@Acin]. In definition (\[bisep\]) the pure states $|\psi_i\rangle$ are separable on the Hilbert-space of three qubits $1,2,3$ with respect to one of the three bipartitions $1|23$, $12|3$ or $13|2$ and $p_i\geq 0$ adding up to $1$. We say that a general three-qubit state is genuine tripartite entangled when it is not in the class of biseparable states $B$, that is they cannot be constructed by mixing pure states containing bipartite entanglement at most. Clearly, if $p_{ij}$ in (\[rho3\]) was positive for each of the three different pairs (whose sum $p$ is upper bounded by $+1$ according to (\[p\])), then $\rho_3$ should define a biseparable state. Note, that the bound to $p_{ij}$’s in (\[pijbound\]) may not be tight, therefore it is not evident whether $p_{ij}$ can take up negative values at all. However, by arranging the three fermions in a particular geometry, we demonstrate that $p_{ij}$ may take the value $-1/3$ as well: Taking the first and second derivatives of the functions $f_{ij}(k_F r)$ (defined for both the 2D and 3D Fermi gases under equations (\[fij\])) with respect to $x=k_F r$, we observe that in the limit $x\rightarrow 0$ they behave as $$\lim_{x\rightarrow 0}f_{ij}(x)=1,\;\; \lim_{x\rightarrow 0}f'_{ij}(x)=0,\;\; \lim_{x\rightarrow 0}f''_{ij}(x)\neq0 \;. \label{limits}$$ Now let us place the three particles on a line so that particle $2$ would lie just at the midpoint between particle $1$ and particle $3$, and let the relative distance between these outer particles tends to zero. Actually, in the limit $x\rightarrow 0$, $p_{ij}$ in (\[pij\]) can be given explicitly by applying l’Hospital’s rule twice and by taking account the limiting values (\[limits\]). As a result we obtain $$p_{12}=p_{23}=2/3,\hspace{0.5cm}p_{13}=-1/3 \label{plimit}$$ both for the 2D and 3D Fermi gases. In the next two subsections we propose witness operators in order to reveal GTE in the degenerate Fermi systems. An observable which is well suited for signaling genuine tripartite quantum correlations in Heisenberg spin lattices [@GTB] turns out to detect GTE in the degenerate Fermi gas as well. Generalized $GHZ$ and $W$ witnesess {#subsec:ghzw} ----------------------------------- For deciding whether the state $\rho_3$ with the explicit parameters $p_{12}=p_{23}=2/3$, $p_{13}=-1/3$ in (\[plimit\]) is genuine tripartite entangled, we will use entanglement witnesses. A witness of genuine tripartite entanglement is an observable $\Pi$ with a positive mean value on all biseparable states so a negative expectation value $\mathrm{Tr}(\rho \Pi)$ guarantees that the state $\rho$ carries genuine tripartite entanglement [@Terhal]. Thus a witness operator which separates genuine tripartite entangled states from the biseparable set $B$ (defined by equation (\[bisep\])) can be given in the form [@Bou04] $$W_\psi = \Lambda \mathbb I_8 - |\psi\rangle\langle\psi|\;, \label{wit}$$ where $$\Lambda = \max_{|\phi\rangle \in B} |\langle \phi|\psi\rangle|^2\;.$$ A simple method has been found in Ref. [@Bou04] to determine $\Lambda$ for any pure genuine tripartite entangled state $|\psi\rangle$. In particular, let $|\psi\rangle$ be the $GHZ$-like and the $W$-like state [@CC05], which are respectively $$\begin{aligned} |GHZ(\alpha)\rangle &= \frac{|\mathbf n_1,\mathbf n_2, \mathbf n_3\rangle+ e^{i\alpha}|-\mathbf n_1,-\mathbf n_2,-\mathbf n_3\rangle}{\sqrt 2} \nonumber \\ |W(\beta,\gamma)\rangle &= \frac{|\mathbf n_1,\mathbf n_2,-\mathbf n_3\rangle +e^{i\beta}|\mathbf n_1,-\mathbf n_2,\mathbf n_3\rangle+e^{i\gamma} |-\mathbf n_1,\mathbf n_2,\mathbf n_3\rangle}{\sqrt 3} \;, \label{GHZW}\end{aligned}$$ where $\{\mathbf n_i,-\mathbf n_i\}$ denotes an arbitrary local orthonormal basis in the Hilbert space of qubit $i$. Note, that for the original $GHZ$ state the phase $\alpha = 0$ and for the original $W$ state the phases $\beta=\gamma=0$, and the corresponding parameters $\Lambda$ appearing in the witness operator (\[wit\]) are $1/2$ and $2/3$, respectively [@Acin]. The states (\[GHZW\]), however, can be transferred to the original ones by local unitary operations, which leave the parameters $\Lambda$ unchanged, that is we have the witness operators $$\begin{aligned} W_{GHZ(\alpha)} &= \frac{1}{2} \mathbb I_8 -|GHZ(\alpha)\rangle\langle GHZ(\alpha)| \nonumber \\ W_{W(\beta,\gamma)} &= \frac{2}{3} \mathbb I_8 -|W(\beta,\gamma)\rangle\langle W(\beta\gamma)| \label{witGHZW}\end{aligned}$$ for the $GHZ$- and $W$-like states, respectively. Lunkes et al. [@LBV] applying these witness operators for the special case $\alpha=\beta=\gamma=0$ and $|\mathbf n_i\rangle=|\uparrow\rangle$, $i=1,2,3$ established that neither $\mathrm{ Tr}(\rho_3 W_{GHZ})$ nor $\mathrm{Tr}(\rho_3 W_W)$ can become negative in the permitted range of parameters $p_{ij}$, $ij=12,13,23$, hence no GTE could be revealed in the three-spin reduced density matrix $\rho_3$ by the application of these witnesses. We confirm and extend this result by generalizing the $GHZ$ and $W$ witnesses to the form (\[witGHZW\]) with the corresponding values $\alpha,\beta,\gamma \in [0,2\pi]$ and with the arbitrary local bases $|\mathbf n_i\rangle = \cos (\theta_i/2)|\uparrow\rangle + e^{i\phi_i}\sin(\theta_i/2)|\downarrow\rangle,\; i=1,2,3$. Then the trace of $\rho_3 \Pi$, where $\Pi$ denotes either $W_{GHZ(\alpha)}$ or $W_{W(\beta,\gamma)}$ and $\rho_3$ is the state (\[rho3\]), is a linear combination of the trigonometric functions cosine/sine with arguments $\theta_i,\phi_i,\;i=1,2,3$. Owing to convexity arguments $\mathrm{Tr}(\rho_3 \Pi)$ can be extremal only if $p_{ij}\in\{+1,-1\},\; ij=12,13,23$ in the permitted range (\[pijbound\]) and $\theta_i,\phi_i \in \{0, \pi/2, \pi, 3\pi/2, 2\pi\}, \; i=1,2,3$. Moreover, we may fix three parameters, e.g., $\phi_1=\phi_2=0$ and $\theta_1=0$, owing to the invariance of the state $\rho_3$ under the collective $SU(2)$ rotation. Considering all the possible combinations of the remaining parameters $p_{ij}$, $ij=12,13,23$ and $\theta_2,\theta_3,\phi_3$ from the set above, we found that $\mathrm{Tr}(\rho_3 \Pi)\geq 0$ is always true for the witness operators $\Pi$ in (\[witGHZW\]). Therefore, genuine tripartite entanglement could not be witnessed by the general $GHZ$-type and $W$-type witnesses (\[witGHZW\]), as well. Naturally, we may ask whether there exist other witnesses over the $GHZ/W$-types in (\[witGHZW\]) which are better suited for detecting GTE in the state $\rho_3$. Indeed, in Ref. [@CC05] it has been shown that there are several genuine triparite entangled states which are not witnessed by the operators (\[witGHZW\]). Note, that nonlinear entanglement witnesses [@GL06] may show improvement with respect to linear witnesses in the multipartite case as well. However, if we stick to linear functionals we are even able to reveal GTE in the state $\rho_3$, as it will be discussed in the next subsection. The witness operator of Gühne et al. {#subsec:witness} ------------------------------------ Gühne et al. [@GTB] showed that the internal energy is a good indicator of genuine tripartite entanglement in macroscopic spin systems. The idea was to write the internal energy in terms of the mean value of the observables $W_{ijk}= \vec{\sigma}^i\cdot\vec{\sigma}^j + \vec{\sigma}^j \cdot \vec{\sigma}^k$, where $\vec{\sigma}^i=(\sigma_x^i,\sigma_y^i,\sigma_z^i)$ is the vector of Pauli spin operators associated with the qubit $i$, and the absolute value of $\langle W_{ijk} \rangle$ has been shown to be a witness itself, capable to detect GTE. Namely, it has been proven [@GTB] that if the inequality $$|\langle W_{ijk} \rangle|> 1+\sqrt 5\simeq 3.236 \label{Wijk}$$ holds, the qubits $i,j,k$ are genuinely tripartite entangled. Let us use this inequality (\[Wijk\]) in order to reveal GTE in the degenerate Fermi gas among the spins of the three fermions $i,j,k$. Plugging the state $\rho_3$ in (\[rho3\]) into the expectation $\langle W_{ijk}\rangle =\mathrm{Tr}(\rho_3 W_{ijk})$ one has $\langle W_{ijk} \rangle=3(p_{ij}+p_{jk})$, which by substitution back into (\[Wijk\]) gives the condition $$|\langle W_{ijk} \rangle|:= 3|p_{ij}+p_{jk}|> 1+\sqrt 5 \label{Wijk3}$$ for the existence of genuine tripartite entanglement in the degenerate Fermi gas. Next we discuss from the viewpoint of witnessed GTE by the mean of this condition two different three-fermion configurations: \(a) Consider the case investigated before in Section \[subsec:param\], that three particles lie evenly spaced on a line close to each other. Choosing $ijk=123$ and recalling $p_{12}=p_{23}=2/3$ from (\[plimit\]), by the virtue of (\[Wijk3\]), $|\langle W_{123} \rangle|=4 > 1+\sqrt 5$, hence the three-fermion state $\rho_3$ corresponding to this arrangement of particles is genuine tripartite entangled. \(b) In this case the particles are separated from each other by equal distances, i.e., the particles are put on the vertices of an equilateral triangle. Owing to three-fold symmetry of this configuration the state $\rho_3$ contains an equal mixture of maximally entangled states $|\Psi^-\rangle$, that is, all $p_{ij},\;ij=12,13,23$ in (\[rho3\]) must have the same value. Further, considering the constraint $|p|=|p_{12}+p_{13}+p_{23}|\leq 1$ in (\[p\]) and also owing to the left-hand side of (\[Wijk3\]) we have $|\langle W_{123}\rangle|=|\langle W_{231} \rangle|=|\langle W_{132}\rangle|=2|p|\leq 2$ implying that in this case all possible $|\langle W_{ijk}\rangle|$ (with different permutations of $ijk$) are smaller than the bound $1+\sqrt 5$. Consequently, no GTE can be revealed by the witness (\[Wijk\]) of Gühne et al., no matter how far the fermions are separated from each other. It is reasonable to think that there is indeed no GTE associated with this highly symmetrical configuration, as it has been argued in Ref. [@LBV] by attributing it to the Pauli principle. In the next section we construct from the observable $W_{ijk}$ a witness operator $\tilde{W}_{ijk}$, which has a maximum eigenvalue smaller than unity ($\tilde{W}_{ijk}\leq \mathbb I_8$) and with the aid of it a lower bound is given for the amount of GTE in the state $\rho_3$ quantified by an entanglement monotone, the generalized robustness $E_R$. Deriving a lower bound to the generalized robustness $E_R$ {#sec:er} ========================================================== Up to this point the observable $W_{ijk}$ was applied for witnessing genuine tripartite entanglement. On the other, they are also good for quantifying it [@Bra05] (see [@RGFGC] as an application to a magnetic material). The maximum eigenvalue of the operator $-W_{ijk}$ is 4, thus considering (\[Wijk\]) we may construct the following witness operator, $$\tilde{ W}_{ijk} = \frac{(1+\sqrt 5)\mathbb I_8 - W_{ijk}}{5+\sqrt 5}\;,\label{tildeWijk}$$ whose negative mean value $\mathrm{Tr}(\rho \tilde{W}_{ijk})$ guarantees that the three-qubit state $\rho$ is genuine tripartite entangled. Further the witness operator is normalized so that $\tilde{W}_{ijk}\leq \mathbb I_8$. It is apparent that provided the mean value $\langle W_{ijk} \rangle \geq 0$ for state $\rho_3$ (i.e., $p_{ij}+p_{jk} \geq 0$ according to calculations in Sec. \[subsec:witness\]), the witness operator (\[tildeWijk\]) is just as powerful to detect GTE associated with state $\rho_3$ as $|\langle W_{ijk}\rangle|$ in the inequality (\[Wijk\]). The generalized robustness $E_R$ as a GTE measure quantifies how robust the genuine tripartite entangled state $\rho$ is under the influence of noise, and also has a geometrical meaning measuring the distance of $\rho$ from the biseparable set $B$ [@CBC]. According to Ref. [@Bra05] $E_R$ can be expressed in a Lagrange dual representation $$E_R(\rho) = \max\{0,-\min_{\Pi\in M}\mathrm{Tr}(\rho\Pi)\}\;, \label{witent}$$ where the set $M$ is given by the restriction $\Pi\leq \mathbb {I}_8$ for the GTE witnesses $\Pi$. Since $\tilde{W}_{ijk}$ with any permutation of $ijk$ defines a valid GTE witness with maximum eigenvalue smaller than unity, we are able to develop the lower bound $$E_{R,\min}(\rho) = \max\{0,-\min_{ijk \in \{123,231,132 \}} \mathrm{Tr}(\rho \tilde{W}_{ijk})\} \label{boundER}$$ to the generalized robustness (\[witent\]) of an arbitrary state $\rho$ on qubits $123$, as it is discussed in Refs. [@CC06; @EBA]. This lower bound by applying (\[tildeWijk\]) for the particular state $\rho_3$ in (\[rho3\]) reads as $$E_{R,\min}(\rho_3) = \max_{ijk\in\{123,231,132\}}\{0,\frac{3(p_{ij}+p_{jk})-1-\sqrt 5}{5+\sqrt 5} \}\;. \label{boundER3}$$ In the next section this formula will be applied to give explicitly a lower bound to the generalized robustness $E_R$ of GTE for various configurations of three fermions, associated with the reduced state $\rho_3$ of the degenerate Fermi gas. Numerical calculations to the lower bound of $E_R$ {#sec:num} ================================================== Fermion moving on a straight line {#subsec:line} --------------------------------- Now we concentrate on two different kinds of arrangements of the three fermions in the 3D degenerate Fermi gas, which configurations were also investigated in Ref. [@LBV] from the viewpoint of bipartite entanglement shared between two arbitrary groups of three fermions. \(a) In the first instance a collinear arrangement is considered, namely we put three fermions on a straight line numbering them in the order $1,2$, and $3$. The distance between particles $1$ and $3$ is $r$, and the intermediate particle $2$ is by a distance of $x$ away from particle $1$ (shown by the geometrical picture of Fig. \[fig-line\].(a)). In Fig. \[fig-line\].(a), the lower bound to GTE quantified by $E_R$ is plotted in the $3D$ Fermi gas according to the formula (\[boundER3\]) in the function of $x/r$ for different values $k_Fr$ of the external fermions. The calculations can be in general performed only numerically, however for the limiting value $k_Fr\rightarrow 0$ one obtains $$\max_{ijk\in 123, 231, 132}{\{p_{ij}+p_{jk}\}} = p_{12}+p_{23} = \frac{1}{1-x/r+(x/r)^2}\;.\label{p123x}$$ Substitution of this expression into (\[boundER3\]) gives analytically the curve corresponding to $k_Fr\rightarrow 0$. Note that formula (\[p123x\]) holds true independently of the dimensionality of the Fermi gas (i.e., both for the 2D and 3D cases). The curves produced in Fig. \[fig-line\].(a) exhibit two essential features: For any given value of $k_Fr$, the maximum of $E_{R,\min}$ is achieved by the symmetrical configuration (i.e., particle $2$ is located at the midpoint of the line connecting particles $1$ and $3$), still presenting GTE in the system by the dimensionless distance $k_Fr=2.59$. On the other, when fermion $2$ starting from this midpoint is moved toward fermion $1$ in the case $k_Fr\rightarrow 0$, the curve falls off to zero by the value $x/r =1/2 (1 - \sqrt{3(\sqrt 5 -2)}) \simeq 0.08 $. That is, if fermions $1$ and $3$ are a distance $r$ away from each other and fermion $2$ becomes closer than $x \simeq 0.08r$ to fermion $1$ (or to fermion $3$ in the symmetrically equivalent situation) formula (\[boundER3\]) does not indicate GTE among the three fermions. This result fits to the monogamy property of entanglement [@CKW], as in the case $x/r\rightarrow 0$ fermion $2$ becomes maximally entangled with fermion $1$, excluding the existence of any higher order entanglement in the system. \(b) Now let the three particles lie on the vertices of an isosceles triangle fermions $1$ and $3$ forming its base with length $r$, and fermion $2$ is positioned by a distance of $y$ from the midpoint of the base as it is illustrated in the geometrical part of Fig. \[fig-line\].(b). The curves $E_{R,\min}$ in Fig. \[fig-line\].(b) are plotted against $y/r$ for different values of $k_Fr$. As it can be observed, all the curves $E_{R,\min}$ plotted are monotonically decreasing in the function of the ratio $y/r$ for any given $k_Fr$. Similarly to case (a) the curve corresponding to $k_Fr\rightarrow 0$ can be treated analytically, and one obtains vanishing GTE beyond the value $y/r =1/2 (\sqrt{3(\sqrt 5 -2)}) \simeq 0.42$ (both in the 2D and 3D Fermi gases). This supports the result of case (b) in Sec. \[subsec:witness\] that three fermions located at the vertices of an equilateral triangle (where $y/r = \sqrt 3/2 \simeq 0.866$) are not genuine tripartite entangled independent of the separation distance $r$. It is also apparent from Fig. \[fig-line\].(b) that for a fixed ratio $y/r$, $E_{R,\min}$ is maximal when $k_Fr\rightarrow 0$, for in this case the antisymmetrization effect between the three particular fermions and the rest of the fermions (which reduces the amount of quantum correlations shared among the three fermions) becomes negligible. 2.8in Fermion moving in a plane {#subsec:plane} ------------------------- We now turn to the situation (pictured in the geometrical part of Fig. \[fig-plane\]) when fermion $2$ is allowed to move in the two-dimensional plane given by polar coordinates ($\theta,q$) with an origin at the midpoint of the line of length $r$ connecting fermion $1$ and fermion $3$. Let us restrict fermion $2$ to be positioned within the circle with radius $r/2$. Recalling the definition of the GTE distance from Sec. \[sec:intro\], in this case the GTE distance is equal to the maximum separation length $r$ between the two external fermions, below which the three fermions are genuine tripartite entangled. In what follows, we inquire the shape of region within particle $2$ could be located so that fermions $1,2$, and $3$ by a fixed relative distance $r$ would be genuine tripartite entangled. Further, owing to the symmetry of the configuration it suffices to study the interval $\theta \in [0,\pi/2]$, i.e., particle $2$ is restricted to be situated on a quarter-disk of radius $r/2$. The polar plot of Fig. \[fig-plane\] represents curves which distinguish regions with (left side) and without (right side) witnessed GTE in the system, by the following values of the dimensionless distances $k_Fr\rightarrow 0$ and $k_Fr=1,2,2.5,2.59$. The witnessed GTE (i.e., to decide whether $E_{R,\min}(\rho_3)>0$) corresponding to these curves for the various values of $k_Fr$ was calculated according to the formula (\[boundER3\]). The case $k_Fr\rightarrow 0$ can be treated analytically yielding the result (as it can be read off from Fig. \[fig-plane\]) that by $k_Fr\rightarrow 0$ GTE is present in the system provided fermion $2$ is located inside a disk with radius $q\simeq 0.42r$ (generalizing the results $x\simeq 0.08r$ and $y\simeq 0.42r$ in the limit $k_Fr\rightarrow 0$ obtained in Sec. \[subsec:line\]). Now one can see from the shape of the polar curves that for greater $k_Fr$ the corresponding disk associated with GTE squeezes toward the axis of particles $1$ and $3$, eventually contracting on the origin $q=0$. It is noted that the same behavior would have been observed for the case of 2D Fermi gas, as well. This implies that if in this particular case we were searching for the GTE distance (i.e., the maximum separation $r$, below which GTE exists) then we should focus on the case where particle $2$ is located at the midpoint between particles $1$ and $3$. This task will be performed in the sequel both for the 2D and 3D degenerate Fermi gases. 3.2in An upper and a lower bound to the GTE distance in the 2D and 3D degenerate Fermi gases {#sec:gte} ====================================================================================== Lower bound {#subsec:lower} ----------- In the present section the three-fermion configuration is investigated, where the particles $1,2$ and $3$ are positioned evenly spaced on a straight line, with a relative distance $r$ between the external particles $1$ and $3$. We develop both for the 2D and 3D Fermi gases an upper and a lower bound to the relative distance $r$, beyond which GTE disappears. Let us first calculate numerically and plot the lower bound to the generalized robustness of GTE defined by (\[boundER3\]) in the function of $k_Fr$. In Fig. \[fig-distance\] the respective curves are plotted both for the two- and three-dimensional degenerate Fermi gases, and numerics shows that $E_{R,\min}(\rho_3)$ vanishes beyond, i.e., the lower bound to the GTE distance is $$\begin{aligned} r_{\min}^{\mathrm{2D}} &=2.3588/k_F\;, \nonumber \\ r_{\min}^{\mathrm{3D}} &=2.5964/k_F\;, \label{rmin}\end{aligned}$$ respectively. Let us compare these values (\[rmin\]) with the bipartite entanglement distance (i.e., the maximal separation distance between two entangled fermions), which are slightly smaller, and are given explicitly by the values [@LBV] $1.6163/k_F$ and $1.8148/k_F$ for the 2D and 3D Fermi gases, respectively. For the 3D Fermi gas the $p_{ij}$ parameters of the state $\rho_3$ corresponding to $r_{\min}^{\mathrm{3D}}$ are $$p_{12}=p_{23}=0.539345 ,\;\; p_{13}=-0.160702\;. \label{p123min}$$ We can also observe in Fig. \[fig-distance\], as one may expect, that the curves are monotonically decreasing, such as in the bipartite case for the entanglement measure negativity [@LBV]. 2.8in Upper bound {#subsec:upper} ----------- We continue with studying the particular three-fermion configuration discussed in the previous subsection in order to establish an upper bound to the GTE distance, beside the lower bound already obtained. Exploiting the mirror symmetry of the configuration, for any given distance $r$ between the positions of particle $1$ and particle $3$ we have $p_{12}=p_{23}$, thus in this case parameters $r_k$ defined by (\[rk\]) in Section \[subsec:3spin\] can be expressed through $p_{12}$ and $p_{13}$ alone, and we obtain the relation $$r_2=\sqrt 3 r_1\;. \label{rsym}$$ On the other, by applying Theorem 7 of Eggeling and Werner [@EW] it is asserted that a three-qubit state is biseparable with respect to the partition $1|23$ if the following inequalities are satisfied: $$\begin{aligned} -1< r_1-2r_+ < 0\;, \nonumber \\ 3r_2^2 +3r_3^2+(1-3r_+)^2 \leq (r_1-2r_+)^2\;. \label{EWineq}\end{aligned}$$ Now let us calculate the boundary of the area in the plane $(r_1,r_2)$ described by these inequalities (\[EWineq\]) by $r_+=0.041$ and $r_3=0$, which parameters correspond to the symmetrical configuration of separation $r_{\min}^{\mathrm{3D}}$ between fermion $1$ and fermion $3$ with parameters (\[p123min\]). The solution of the inequality (\[EWineq\]) by $r_+=0.041$, $r_3=0$ corresponds to the leftmost yellow shaded semi-disk in Fig. \[fig-polygon\], representing states in the section $r_+=0.041$ and $r_3=0$ that are separable with respect to the partition $1|23$. The other two disks (representing biseparable states with respect to partitions $12|3$, $13|2$) can be obtained through $\pm 2\pi/3$ rotations around the origin of the plane $(r_1,r_2)$ [@TA] owing to the permutation symmetry of the three subsystems. The boundary of the convex hull of these semi-disks are indicated by solid blue line segments in Fig. \[fig-polygon\]. All the three-qubit states in the section $r_+=0.041$ and $r_3=0$ which lie inside this polygon are in the class of biseparable states $B$. However, since this area corrresponds to a section, biseparable states in this section may exist outside the polygon as well. Next let us determine the explicit position of the point corresponding to the mirror-symmetrical configuration with separation $r_{\min}^{\mathrm{3D}}$ between the two external fermions. In this particular symmetrical configuration according to (\[rsym\]) the ratio $r_2/r_1=\sqrt 3$ and we also have $r_1=(p_{13}-p_{12})/2$. The above ratio has been displayed in Fig. \[fig-polygon\] by a dashed line and the point on this line with coordinate $r_1 = -0.35$ (obtained by plugging the values (\[p123min\]) into the above formula for $r_1$) corresponding to the separation $r_{\min}^{\mathrm{3D}}$ with $E_{R,\min}=0$ has been designated by the red cross marker. Numerics shows, that this point lies outside the solid blue polygon, as it ought to owing to $E_R\geq 0$ associated with the point. By inspection, on the other, this point is very close to the border of the polygon. Indeed, explicit numerical calculations yield that the distance $r$ between the two outer fermions corresponding to the border of the polygon is $r_{\max}^{\mathrm{3D}}=2.5988/k_F$. This value was obtained by tuning the value of $r_+$ from $0.041$ up to $\simeq 0.0415$ by the mean of increasing the separation $r$ starting from the value $r_{\min}^{\mathrm{3D}}$ so that by $r_{\max}^{\mathrm{3D}}$ the point represented by the the cross marker would lie just on the edge of the polygon. However, this distance $r_{\max}^{\mathrm{3D}}=2.5988/k_F$ is just an upper bound to the GTE distance for the 3D Fermi gas. On the other, similar evaluations give the value $r_{\max}^{\mathrm{2D}}=2.3599/k_F$ for the 2D Fermi gas. Comparing these values with the ones in (\[rmin\]) corresponding to the lower bound, it shows that the upper and lower bounds to the GTE distance are indeed very close to each other both for the 2D and 3D Fermi gases. Hence, this implies that the witness of Gühne et al. [@GTB] in our particular problem ought to be close to an optimal one. 3.0in Discussion {#sec:disc} ========== Previous works (e.g., [@Ved03; @LBV]) explored that bipartite entanglement exists within the order of the Fermi wavelength $1/k_F$ at zero temperature in the non-interacting Fermi gas and may even persist for nonzero temperatures (e.g., [@LBVa; @OK]). Since the system consists of non-interacting fermions, entanglement is purely due to particle statistics and not to any physical interaction between the particles. In the present work we found the result as an extension of the formerly studied bipartite case that particle statistics is capable to generate genuine tripartite entanglement (GTE) as well. Furthermore, it has been found that the diameter of the three-fermion configuration wherein GTE is present (a lower bound to the maximum diameter is given explicitly by (\[rmin\])) is comparable with the maximum relative distance between two entangled fermions, both in the 2D and 3D Fermi gases. Looking at higher order entanglement as a useful resource, the presence of GTE in Fermi systems would be promising to allow for performing new quantum information processing tasks, exemplified by the $GHZ$ paradox [@GHSZ]. However, in order to do so, the amount of entanglement stored by three fermions should be somehow extracted from the system. In the present article, though, we did not consider this problem some explicit schemes has been put forward recently in the bipartite setting [@DDW; @Cav06; @CV], some of which might be extended to the tripartite setting as well. Namely, in Ref. [@DDW] it has been shown that bipartite entanglement can exist between non-interacting fermions on a lattice and can extend over multiple lattice sites even if the entanglement is quantified by the most restrictive measure, the entanglement of particles [@WV]. Further, considering that in the continuum limit the entanglement of particles corresponds to the entanglement in the spin reduced density matrix [@DDW], by continuity arguments genuine tripartite entanglement, quantified by the measure entanglement of particles, should exist in the lattice system as well. Thus, in the near future optical lattice implementations may offer a simulation technics to observe the phenomenon of genuine tripartite entanglement among non-interacting fermions in a lattice. On the other, in the continuum limit the extraction of genuine tripartite entangled particles seems to be a more difficult problem: As it has been shown [@Ved03] the entanglement distance between two fermions is inversely proportional to the Fermi momentum $k_F$, and $k_F^3$ in turn is proportional to the density of particles. In the case of conduction electrons in a usual metal the density is very large indicating an entanglement distance of the order of a few angstroms. This failure might be avoided by using 2D electron gas formed in GaAs heterostructure, where the entanglement distance is in the order of hundred angstroms [@OK] or using stored ultra-cold neutrons in a carefully devised experiment [@CV]. Also, note the intriguing proposal, exploiting decoherence effects to extract bipartite entanglement created merely by particle statistics from semiconductor quantum wells [@Cav06]. Although the GTE distance in (\[rmin\]) is comparable (even greater) than the bipartite entanglement distance both for the 2D and 3D Fermi gases, technically these proposals appear to be very demanding when applied to the three-party setting. Finally, we would like to mention interesting future directions as a continuation of the present work. One could for example apply the same methods as in the present article for determining GTE distance in Fermi gases trapped in a harmonic trap [@Yi] or considering GTE not only in spin, but in other internal degrees of freedom as well [@CWZ]. Also the possible existence of genuine multipartite entanglement beyond the three-party scenario remains to be explored. This work was supported by the Grant Öveges of the National Office for Research and Technology. [99]{} M.A. Nielsen and I.L. Chuang, [*Quantum Computation and Quantum Information*]{}, (Cambridge University Press, 2000). D.M. Greenberger, M.A. Horne, A. Shimony, A. Zeilinger, Am. J. Phys. [**58**]{} 1131 (1990). R. Raussendorf and H.J. Briegel, Phys. Rev. Lett, [**86**]{} 5188 (2001). V. Giovannetti, S. Lloyd, L. Maccone, Science [**306**]{} 1330 (2004). S.D. Bartlett, T. Rudolph, and R.W. Spekkens, Phys. Rev. Lett. [**91**]{} 027901 (2003). Z. Zhao, Y.-A. Chen, A.-N. Zhang, T. Yang, H.J. Briegel, and J.-W. Pan, Nature [**430**]{} 54 (2004). A. Acín, D. Bruß, M. Lewenstein, and A. Sanpera, Phys. Rev. Lett. [**87**]{} 040401 (2001). M.B. Plenio, S. Virmani, Quant. Inf. Comp. [**7**]{} 1 (2007). O. Gühne, G. Tóth, and H.J. Briegel, New. J. Phys. [**7**]{} 229 (2005) T.J. Osborne and M.A. Nielsen, Phys. Rev. A [**66**]{} 032110 (2002). T.R. de Oliveira, G. Rigolin, M.C. de Oliveira, E. Miranda, Phys. Rev. Lett. [**97**]{} 170401 (2006). V. Vedral, New J. Phys. [**6**]{} 102 (2004). V. Vedral, Cent. Eur. J. Phys. [**1**]{} 289 (2003) C. Lunkes, Č. Brukner, and V. Vedral, Phys. Rev. Lett. [**95**]{} 030503 (2005). G. Tóth and O. Gühne, Appl. Phys. B [**82**]{} 237 (2006). F.G.S.L. Brandão, Phys. Rev. A [**72**]{} 022310 (2005). T.G. Rappoport, L. Ghivelder, J.C. Fernandes, R.B. Guimarães, M.A. Continentino, ArXiv: quant-ph/0608403. D. Cavalcanti, M. França Santos, M.O. Terra Cunha, C. Lunkes, V. Vedral, Phys. Rev. A. [**72**]{} 062307 (2005). S. Oh and J. Kim, Phys. Rev. A. [**69**]{} 054305 (2004). T. Eggeling and R.F. Werner, Phys. Rev. A. [**63**]{} 042111 (2001). B.M. Terhal, Phys. Lett. A [**271**]{} 319 (2000). M. Bourennane, M. Eibl, C. Kurtsiefer, S. Gaertner, H. Weinfurter, O. Gühne, P. Hyllus, D. Bruß, M. Lewenstein, and A. Sanpera, Phys. Rev. Lett. [**92**]{} 087902 (2004). D. Cavalcanti and M.O. Terra Cunha, ArXiv: quant-ph/0506035. O. Gühne, N. Lütkenhaus, ArXiv: quant-ph/0612108. D. Cavalcanti, F.G.S.L. Brandão, and M.O. Terra Cunha, ArXiv: quant-ph/0510068. D. Cavalcanti and M.O. Terra Cunha, Appl. Phys. Lett. [**89**]{} 084102 (2006). J. Eisert, F.G.S.L. Brandão, and K.M.R. Audenart, ArXiv: quant-ph/0607167. V. Coffman, J. Kundu, and W.K. Wootters, Phys. Rev. A [**61**]{} 052306 (2000). G. Tóth and A. Acín, Phys. Rev. A [**74**]{} 030306(R) (2006). C. Lunkes, Č. Brukner, and V. Vedral, Phys. Rev. A [**71**]{} 034309 (2005). M.R. Dowling, A.C. Doherty, H.M. Wiseman, Phys. Rev. A. [**73**]{} 052323 (2006). D. Cavalcanti, L.M. Moreira, F. Matinaga, M.O. Terra Cunha, M. França Santos, ArXiv: quant-ph/0608141. M.O. Terra Cunha and V. Vedral, ArXiv: quant-ph/0607224. H.M. Wiseman and J.A. Vaccaro, Phys. Rev. Lett. 91 097902 (2003). X.X. Yi, Eur. J. Phys. D [**39**]{} 465 (2006). D.-M. Chen, W.-H. Wang, and L.-J. Zou, ArXiv: cond-mat/0605378.

--- abstract: 'The fidelity susceptibility measures sensitivity of eigenstates to a change of an external parameter. It has been fruitfully used to pin down quantum phase transitions when applied to ground states (with extensions to thermal states). Here we propose to use the fidelity susceptibility as a useful dimensionless measure for complex quantum systems. We find analytically the fidelity susceptibility distributions for Gaussian orthogonal and unitary universality classes for arbitrary system size. The results are verified by a comparison with numerical data.' author: - Piotr Sierant - Artur Maksymov - Marek Kuś - Jakub Zakrzewski bibliography: - 'ref2018.bib' title: Fidelity susceptibility in Gaussian Random Ensembles --- The discovery of many body localization (MBL) phenomenon resulting in non-ergodicity of the dynamics in many body systems [@Basko06] restored also the interest in purely ergodic phenomena modeled by Gaussian random ensembles (GRE) [@Mehtabook] and in possible measures to characterize them. The gap ratio between adjacent level spacings [@Oganesyan07] was introduced precisely for that purpose as it does not involve the so called unfolding [@Haake] necessary for meaningful studies of level spacing distributions and yet often leading to spurious results [@Gomez02]. Still, the level spacing distribution belongs to the most popular statistical measures used for single particle quantum chaos studies [@Bohigas84; @Bohigas93; @Stockmann99; @Guhr98] and also in the transition to MBL [@Serbyn16; @Bertrand16; @Sierant18b; @SierantPRB]. A particular place among different measures was taken by those characterizing level dynamics for a Hamiltonian $H(\lambda)$ dependent on some parameter $\lambda$. In Pechukas-Yukawa formulation [@Pechukas83; @Yukawa85] energy levels are positions of a fictitious gas particles, derivatives with respect to the fictitious time $\lambda$ are velocities (level slopes), the second derivatives describe curvatures of the levels (accelerations). Simons and Altschuler [@Simons93] put forward a proposition that the variance of velocities distribution is an important parameter characterizing universality of level dynamics. This led to predictions for distributions of avoided crossings [@Zakrzewski93c] and, importantly, curvature distributions postulated first on the basis of numerical data for GRE [@Zakrzewski93] and then derived analytically *via* supersymmetric method by von Oppen [@vonOppen94; @vonOppen95] (for alternative techniques see [@Fyodorov95; @Fyodorov11]). Curvature distributions were recently addressed in MBL studies [@Filippone16; @Monthus17]. Apart from quantum chaos studies in the eighties and nineties of the last millennium, another “level dynamics” tool has been introduced in the quantum information area, i.e. the fidelity [@Uhlmann76]. It compares two close (possibly mixed) quantum states. If these states are dependent on a parameter $\lambda$ it is customary to introduce a fidelity susceptibility $\chi$. For sufficiently small $\lambda$, in a finite system, one has $${\cal F}(\hat \rho (0) , \hat \rho(\lambda))= 1 - \frac{1}{2} \chi \lambda^{2} + O(\lambda^{3}). \label{eq:rozwiniecie}$$ Fidelity susceptibility is directly related to the quantum Fisher information (QFI), $G$, being directly proportional to the Bures distance between density matrices at slightly differing values of $\lambda$ [@Hubner93; @Invernizzi08], with $G(\lambda)=4 \chi $. Fidelity susceptibility emerged as a useful tool to study quantum phase transitions as at the transition point the ground state changes rapidly leading to the enhancement of $\chi$ [@Zanardi2006; @You-Li-Gu; @Paris; @Invernizzi08; @Salvatori2014; @Bina2016; @Kraus16; @sanpera_2016]. All of these studies were restricted to ground state properties while MBL considers the bulk of excited states (for a discussion of thermal states see [@Zanardi07; @Quan09; @Sirker10; @Rams18]). In the context of MBL we are aware of a single study which considered the mean fidelity susceptibility across the MBL transition [@Hu16]. In particular, nobody addressed the issue of fidelity susceptibility behavior for GRE. The aim of this letter is to fill this gap and to provide analytic results for the fidelity susceptibility distributions for the most important physically, orthogonal and unitary ensembles. This provides novel characteristics of GRE as well as a starting point for the study of fidelity susceptibility in the transition to and within the MBL domain [@Maksymov18]. Consider $H=H_0+\lambda H_1$ with $H_0, H_1$ corresponding to the orthogonal (unitary) class of GRE i.e., Gaussian Orthogonal Ensemble (GOE) corresponding to level repulsion parameter $\beta=1$ or Unitary Ensemble (GUE) with $\beta=2$. For such a Hamiltonian one may easily prove that fidelity susceptibility of $n$-th eigenstate of $H_0$ is given by $$\label{fidelity} \chi_n=\sum_{m\ne n}\frac{|H_{1,nm}|^2}{(E_n-E_m)^2},$$ with $E_n$ being the $n$-th eigenvalue of $H_0$. We aim at calculating the probability distribution of the fidelity susceptibility $$\label{probability} P(\chi,E)=\frac{1}{N\rho(E)}\left\langle \sum_{n=1}^N\delta(\chi-\chi_n)\delta(E-E_n) \right\rangle$$ at the energy $E$. The averaging is over two, independent GRE ($\beta=1,2$) $$\label{pH1} P(H_a)\sim\exp\left( -\frac{\beta}{4J^2}\operatorname{Tr}{H_a^2}\right), \quad H_a=[H_{a,nm}]$$ with $a=0,1$. Using Fourier representation for $\delta(\chi-\chi_n)$, the average over $H_1$ reduces to calculation of Gaussian integrals. Since the formula (\[fidelity\]) involves only the eigenvalues of $H_0$, the averaging over $H_0$ can be expressed as an average over the well-known joint probability density of eigenvalues [@Haake] for a suitable GRE. At the center of the spectrum ($E=0$), after straightforward algebraic manipulations (see [@Suppl] for details) we get $$\label{s6} P(\chi)\sim \int_{-\infty}^{\infty}d\omega e^{-i\omega\chi}\left\langle \left[\frac{\det \bar{H}^2}{\det\left(\bar{H}^2-\frac{2i\omega J^2}{\beta} \right)^{\frac{1}{2}}} \right]^\beta \right\rangle_{N-1},$$ where the averaging is now over $(N-1)\times (N-1)$ matrix $\bar{H}$ from an appropriate Gaussian ensemble. This derivation parallels the similar steps used in the derivation of curvature distributions [@vonOppen94; @vonOppen95; @Fyodorov95]. ![ \[fig: smallGOE\] [Fidelity susceptibility $P^O_N(\chi)$ distribution for GOE matrices of small size $N$. Numerical data denoted by markers. Solid lines correspond to with $\mathcal{I}^{O,2}_{N}$ given by . ]{}](fid_suscGOEsmall.pdf){width="1\linewidth"} To perform the average in we employ technique developed in [@Fyodorov95] and express the denominator as a Gaussian integral over a vector $\textbf {z} \in \mathbb{R}^{N-1}$ for $\beta=1$ or $\textbf{z}\in \mathbb{C}^{N-1}$ for $\beta=2$. Employing the invariance of GRE with respect to an adequate class (orthogonal or unitary) of transformations allows us to choose $\textbf{z}=r[1,0,\ldots,0]^T$, hence we arrive at $$P(\chi) \sim \int_{0}^{\infty} dr r^{s} \delta \left(\chi - 2J^2 r^2/\beta\right) \left \langle \mathrm{det}\bar H^{2\beta} \mathrm{e}^{-r^2 X} \right \rangle_{N-1}, \label{s6a}$$ where $X=\sum_{j=1}^{N-1} |\bar H_{1j}|^2$ depends on the first row of $\bar H$ only, and $s = \beta(N-1)-1$. After calculating the ensuing Gaussian integrals over $\bar H_{1j}$ we can reduce the averaging to one over $(N-2)\times(N-2)$ block of $\bar H$, $V_{ij}=\bar H_{i+1,j+1}$ for $1 \leq i,j \leq N-2$), using the expression $\det \bar H=\det V (\bar H_{11}- \sum_{j,k=2}^{N-1} \bar H_{1j} V^{-1}_{jk} \bar H_{1k}^*) $ for a determinant of a block matrix. Integrating over $r$ we find (details described in [@Suppl]) that the desired fidelity susceptibility distribution $P^O_N(\chi)$ for GOE reads $$\begin{aligned} \nonumber P^O_N(\chi) = \frac{C^{O}_N}{\sqrt{\chi}} \left(\frac{\chi}{1+\chi} \right)^{\frac{N-2}{2}}\left( \frac{1}{1+2\chi}\right)^{\frac{1}{2}}\\ \left[ \frac{1}{1+2\chi} + \frac{1}{2} \left(\frac{1}{1+\chi}\right)^{2} \mathcal{I}^{O,2}_{N-2}\right], \label{eq: 5mt}\end{aligned}$$ where $C^{O}_N$ is a normalization constant and $$\mathcal{I}^{O,2}_{N}= \langle \mathrm{det}V^2\left(2 \mathrm{Tr}V^{-2}+\left(\mathrm{Tr}V^{-1}\right)^2 \right)\rangle_{N}/ \langle \mathrm{det}V^2\rangle_N. \label{5c}$$ ![ \[fig: largeGOE\] Fidelity susceptibility $P^O_N(\chi)$ distribution for GOE matrices of different sizes as indicated in the Figure. Panels a$)$ and b$)$ correspond to lin-lin and log-log scales allowing for a detailed test of accuracy both for the bulk and for the tails of the distribution. Solid lines correspond to with $\mathcal{I}^{O,2}_{N}$ given by . ](fid_susc_lin-lin.pdf "fig:"){width="0.52\linewidth"}![ \[fig: largeGOE\] Fidelity susceptibility $P^O_N(\chi)$ distribution for GOE matrices of different sizes as indicated in the Figure. Panels a$)$ and b$)$ correspond to lin-lin and log-log scales allowing for a detailed test of accuracy both for the bulk and for the tails of the distribution. Solid lines correspond to with $\mathcal{I}^{O,2}_{N}$ given by . ](fid_susc_log-log.pdf "fig:"){width="0.5\linewidth"} ![ \[fig: 2\] [ Distribution of rescaled fidelity susceptibility $P^O(x)$ for GOE, numerical data denoted by markers, solid lines – formula . ]{}](fid_susc_lin-lin_scaled.pdf "fig:"){width="0.5\linewidth"}![ \[fig: 2\] [ Distribution of rescaled fidelity susceptibility $P^O(x)$ for GOE, numerical data denoted by markers, solid lines – formula . ]{}](fid_susc_log-log_scaled_N200.pdf "fig:"){width="0.51\linewidth"} The form of is suited for a random matrix theory calculation of $ \mathcal{I}^{O,2}_{N}$. However, to obtain $\mathcal{I}^{O,2}_{N}$ it suffices to note that our calculation implies that $$\left \langle \mathrm{det}\bar H^{2} \mathrm{e}^{-r^2 X} \right \rangle_{N-1} {\Big|_{r=0}}= \left\langle \det \bar V^2 \right \rangle_{N-2} \mathcal{I}^{O,2}_{N-2}-2, \label{eq: Io2n}$$ showing that $\mathcal{I}^{O,2}_{N}$ is actually determined by the second moments of determinants of matrices of appropriate sizes from GOE. Moments as well as the full probability distribution of determinant of GOE matrices were obtained in [@Delannay00] for arbitrary $N$. Using the expression for the second moment in we get $$\mathcal{I}^{O,2}_{N}= \begin{cases} 2p\frac{p+1}{ p+3/4}, \quad \quad \quad N=2p, \\ (2 p+3/2),\quad \quad N=2p+1. \end{cases} \label{c9}$$ The formula is exact for arbitrary $N\geq 2$, for smaller $N$ one gets $\mathcal{I}^{O,2}_{1}=\frac{3}{2}$ and $\mathcal{I}^{O,2}_{0}=0$. Inserting appropriate values of $\mathcal{I}^{O,2}_{N}$ into we obtain an exact formula for the fidelity susceptibility distribution $P^O_N(\chi)$ for GOE matrix of arbitrary size $N$. Comparison of the resulting distribution $P^O_N(\chi)$ with numerically generated fidelity susceptibility distributions for small matrix sizes $N\leq20$ is shown in Fig. \[fig: smallGOE\]. However, it is the large $N$ regime which is interesting from the point of view of the potential applications. For $N \gg 1$ the $\mathcal{I}^{O,2}$ scales linearly $\mathcal{I}^{O,2}_{N}=NJ^2$ with the matrix size $N$. This, together with the form of $P^O_N(\chi)$ implies that $P^O_{\alpha N}(\alpha \chi) \approx P^O_{N}(\chi)$. Indeed, the distribution $P(\chi)$ shifts linearly with $N$ as visible in Fig. \[fig: largeGOE\]. This scaling is a direct consequence of $J^2$ scaling of the squared matrix element in and the GRE $\sqrt{N/J^2}$ scaling of the density of states in the center of the Wigner semicircle distribution [@Bohigas93] implying $J^2/N$ scaling of the squared spacings in . The linear in $N$ scaling of $\chi$ suggests to introduce scaled fidelity susceptibility, $x={\chi}/{N}$. Inserting it into and taking $N\rightarrow \infty$ limit one obtains $$P^O(x)=\frac{1}{6} \frac{1}{x^2}\left(1+\frac{1}{x}\right) \exp\left(-\frac{1}{2x}\right), \label{eq: 7mt}$$ which is the final, simple, analytic result for a large size GOE matrix. It performs remarkably well also for modest size matrices e.g. $N=200$ – compare Fig. \[fig: 2\]. For smaller matrices – for instance for $N = 20$, the rescaled distribution $P(x)$ has a correct large $x$ tail and a nonzero slope at $x=0$ as compared to nonanalytic behavior of $P^O(x)$ at $x=0$ in . Observe also that the mean scaled fidelity susceptibility does not exist as the corresponding integral diverges logarithmically showing the importance of the heavy tail of the distribution. Starting from for GUE ($\beta=2$), after a few technical steps (described in detail in [@Suppl]) we obtain the following expression for fidelity susceptibility distribution: $$P^U_N(\chi)= C^{U}_N \left(\frac{\chi}{1+\chi}\right)^{N-2} \left( \frac{1}{1+2\chi}\right)^{\frac{1}{2}} \left[ \frac{3}{4} \left( \frac{1}{1+2\chi}\right)^{2} + \frac{3}{2} \frac{1}{1+2\chi} \left( \frac{1}{1+\chi} \right)^2 \mathcal{I}^{U,2}_{N-2} + \frac{1}{4} \left( \frac{1}{1+\chi} \right)^4 \mathcal{I}^{U,4}_{N-2} \right], \label{eq: 10}$$ where $C^{U}_N$ is a normalization constant. The fidelity susceptibility distribution $P_U(\chi)$ for GUE depends on two quantities $\mathcal{I}^{U,2}_{N-2}$ and $\mathcal{I}^{U,4}_{N-2}$ which makes the calculation slightly more complicated. The formula is exact for arbitrary $N$. However, the argument with ratio of second moments of determinants which allowed us to obtain the exact expression for $\mathcal{I}^{O,2}_{N}$ suffices only to identify the leading contribution in $N$ for the GUE case for $N \gg 1$ $$\mathcal{I}^{U,4}_{N-2} = N^2 \label{iu4a}$$ where expressions for the second moment of determinant of GUE matrix [@Mehta98; @Cicuta00] were used (for details see [@Suppl]). To complete the calculation of the fidelity susceptibility distribution $P_U(\chi)$ for GUE for large $N$ we need to determine $\mathcal{I}^{U,2}_{N}$. The first step is to show that $$\mathcal{I}^{U,2}_N = J^2 \frac{\left \langle \mathrm{det}H^4 \left( \mathrm{Tr}(H^{-2} + (\mathrm{Tr}H)^{-2}\right) \right \rangle_{N}} {\left \langle \mathrm{det}H^{4} \right \rangle_{N}}, \label{eq: I42a}$$ ![ \[fig: 3a\] Fidelity susceptibility distribution $P^U_N(\chi)$ for GUE, numerically generated data denoted by markers, solid lines – formula with $\mathcal{I}^{U,2}_N $ and $\mathcal{I}^{U,2}_N $ given by and respectively. ](GUEfid_susc_lin_lin.pdf "fig:"){width="0.5\linewidth"}![ \[fig: 3a\] Fidelity susceptibility distribution $P^U_N(\chi)$ for GUE, numerically generated data denoted by markers, solid lines – formula with $\mathcal{I}^{U,2}_N $ and $\mathcal{I}^{U,2}_N $ given by and respectively. ](GUEfid_susc_log_log.pdf "fig:"){width="0.5\linewidth"} ![ \[fig: 3\] [ Distribution of rescaled fidelity susceptibility $P_U(x)$ for GUE, numerical data denoted by markers, solid lines – formula . ]{}](GUEfid_susc_lin_lin_scaled.pdf "fig:"){width="0.5\linewidth"}![ \[fig: 3\] [ Distribution of rescaled fidelity susceptibility $P_U(x)$ for GUE, numerical data denoted by markers, solid lines – formula . ]{}](GUEfid_susc_log_log_scaled.pdf "fig:"){width="0.5\linewidth"} where $H$ is $N \times N$ GUE matrix. Introducing the following generating function $$Z_N(j_1,j_2) = \left \langle \det H^2 \det(H-j_1) \det(H-j_2) \right \rangle_{N}, \label{eq: 7.1mt}$$ we immediately verify that $$\mathcal{I}^{U,2}_N = \frac{ J^2}{ Z_N(0,0) } \left ( 2 \frac{\partial^2}{\partial j_1 \partial j_2} Z_N(0,0) -\frac{\partial^2}{\partial j_1^2 } Z_N(0,0) \right ). \label{eq: 7.5mt}$$ The generating function $Z_N(j_1,j_2)$ is actually a correlation function of a characteristic polynomial of the $H$ matrix. It was shown in [@Brezin00; @Fyodorov03] that such quantities can be calculated exactly for arbitrary matrix sizes and number of determinants as determinants of appropriate orthogonal polynomials. A kernel structure of those expressions has been identified in [@Strahov03] leading to formulas most convenient in our calculation of $Z_N(j_1,j_2)$. The generating function $Z(j_1,j_2)$ is given by $$\begin{aligned} \nonumber Z_N(j_1, j_2) & = & \frac{C_{N,2}}{(j_1-j_2)} \lim_{\mu_2 \rightarrow 0}\frac{\partial}{\partial \mu_2} \\ & \det &\left[ \begin{array}{cc} W_{N+2}(j_1,0) & W_{N+2}(j_2, 0) \\ W_{N+2}(j_1,\mu_2) & W_{N+2}(j_2,\mu_2) \\ \end{array} \right], \label{eq: 8.3}\end{aligned}$$ with the kernel $W_{N+2}(\lambda, \mu)$ defined as $$W_{N+2}(\lambda, \mu) = \frac{ H_{N+2}(\lambda) H_{N+2}(\mu) - H_{N+2}(\mu) H_{N+1}(\lambda)}{\lambda-\mu}. \label{eq: 8.2}$$ The Hermite polynomials $H_N(\lambda)$ are orthogonal with respect to the measure $\mathrm{e}^{-\frac{1}{2J^2} x^2} \mathrm{d}x$ and normalized in such a way that the coefficient in front of $\lambda^N$ is equal to unity. In principle, we could calculate the $2\times2$ determinant in and the ensuing derivatives in order to obtain the generating function $Z_N(j_1,j_2)$ for arbitrary $N$. However, we are predominantly interested in the case of $N\gg1$, therefore we use the following asymptotic form for Hermite polynomials $$H_N(x)\approx\frac{1}{\sqrt{\pi}} \Gamma \left(\frac{N+1}{2}\right) \mathrm{e}^{\frac{x^2}{4 J^2}} \cos \left(\frac{\pi N}{2}-\sqrt{\frac{N}{J^2}} x\right). \label{eq: 8.4}$$ Using the above expression in we obtained a closed formula for the generating function $Z_N(j_1,j_2)$ (see [@Suppl] for details). Calculating the derivatives according to and taking the limits $j_1 \rightarrow 0$ and $j_2 \rightarrow 0$ we obtain $$\mathcal{I}^{U,2}_N = \frac{1}{3}N. \label{eq: 8.9mt}$$ The distribution together with expressions , for $\mathcal{I}^{U,4}_N$ and $\mathcal{I}^{U,2}_N$ is the desired fidelity susceptibility distribution for GUE. As shown in Fig. \[fig: 3a\] the expression is confirmed by numerical data for different system sizes $N \gg 1$. Moreover, similarly to the GOE case, it shifts linearly with increasing $N$. Therefore, considering again the scaled fidelity susceptibility $ x=\chi/N$ we arrive at the large $N$ limit of the simple form $$P_U(x)=\frac{1}{3\sqrt{\pi}}\frac{1}{x^{5/2}}\left(\frac{3}{4}+\frac{1}{x} +\frac{1}{x^2}\right) \exp\left(-\frac{1}{x}\right) \label{eq: 11mt}$$ which works well for GUE data as shown in Fig. \[fig: 3\]. To conclude, we have derived closed formulae for fidelity susceptibility distributions corresponding to level dynamics for both the orthogonal and the unitary class of Gaussian random ensembles. Particularly simple analytic expressions are found in the large $N$ limit. The fidelity susceptibility distributions obtained for quantally chaotic systems may be compared with the results found for GOE (GUE) in order to characterize the degree to which a given system is faithful to random matrix predictions. The obtained distributions also open a way to address level dynamics in the transition between delocalized – ergodic and many-body localized regimes. The work in this direction is already in progress [@Maksymov18]. We thank Dominique Delande for careful reading of this manuscript. P. S. and J. Z. acknowledge support by PL-Grid Infrastructure and EU project the EU H2020-FETPROACT-2014 Project QUIC No.641122. This research has been supported by National Science Centre (Poland) under projects 2015/19/B/ST2/01028 (P.S. and A.M.), 2018/28/T/ST2/00401 (doctoral scholarship – P.S.), 2017/25/Z/ST2/03029 (J.Z.), and 2017/27/B/ST2/0295 (M.K.). Supplementary material to\ “Fidelity susceptibility in Gaussian Random Ensembles” ====================================================== Derivation of formulas and --------------------------- To obtain the equation we use Fourier representation for $\delta(\chi-\chi_n)$ rewriting as $$\label{s1} P(\chi,E)=\frac{1}{2\pi N\rho(E)}\sum_{n=1}^N\int_{-\infty}^{\infty}d\omega e^{-i\omega\chi}\left\langle\delta(E-E_n)\exp\left(i\omega\sum_{m\ne n} \frac{|H_{1,nm}|^2}{(E_n-E_m)^2} \right) \right\rangle.$$ The averaging over $H_1$ with the probability density (\[pH1\]) reduces to a Gaussian integral and gives, $$\label{s2} P(\chi,E)=\frac{1}{2\pi N\rho(E)}\sum_{n=1}^N \int_{-\infty}^{\infty}d\omega e^{-i\omega\chi}\left\langle\delta(E-E_n) \prod_{m\ne n}\left(1-\frac{2i\omega J^2}{\beta(E_n-E_m)^2} \right)^{-\frac{\beta}{2}} \right\rangle.$$ The remaining averaging over the distribution $P(H_0)$ reduces to average over eigenvalues $E_1,\ldots, E_N$ of $H_0$ $$\begin{aligned} \label{s3} P(\chi,E)\sim\sum_{n=1}^N\int_{-\infty}^{\infty} d\omega e^{-i\omega\chi} \int \prod_{j=1}^N dE_j \delta(E-E_n) \prod_{k<l} \left| E_k-E_l\right|^\beta \mathrm{e}^{-\frac{\beta}{4J^2}\sum_k E_k^2} \prod_{m\ne n}\left(1-\frac{2i\omega J^2}{\beta(E_n-E_m)^2} \right)^{-\frac{\beta}{2}}.\end{aligned}$$ Now we can perform the integral over $E_n$. There are $N$ such integrals due to the summation from $n=1$ to $n=N$ at the beginning of the formula. So let’s take $E_n=E_1$. Due to the delta function we can substitute $E_1=E$ and rewrite the averaging over the eigenvalues as $$\begin{aligned} \label{s4} &&\int dE_1\cdots dE_N\delta(E-E_1)\prod_{k<l}\left| E_k-E_l\right|^\beta\exp\left(-\frac{\beta}{4J^2}\sum_k E_k^2\right)\prod_{m\ne n}\left(1-\frac{2i\omega J^2}{\beta(E_n-E_m)^2} \right)^{-\frac{\beta}{2}} = \nonumber \\ &=& e^{-\frac{\beta E^2}{4J^2}}\int dE_2\cdots dE_N\prod_{m=2}\left| E-E_m\right|^\beta\prod_{m=2}\left(1-\frac{2i\omega J^2}{\beta(E-E_m)^2} \right)^{-\frac{\beta}{2}}\prod_{2\le k< l}\left| E_k-E_l\right|^\beta\exp\left(-\frac{\beta}{4J^2}\sum_{k=2}E_k^2\right)= \nonumber \\ &=&e^{-\frac{\beta E^2}{4J^2}}\left\langle\prod_{m=2}\left(1-\frac{2i\omega J^2}{\beta(E-E_m)^2} \right)^{-\frac{\beta}{2}}\left| E-E_m\right|^\beta \right\rangle, \end{aligned}$$ where the averaging goes over the joint probability of the remaining eigenvalues $E_2,\ldots,E_n$. At the center of the spectrum $E=0$ the averaged quantity reads $$\label{s5} \prod_{m=2}\left[ \frac{\left|E_m\right|}{\left(1-\frac{2i\omega J^2}{\beta E_m^2} \right)^{\frac{1}{2}}}\right]^\beta= \left[\frac{\det \bar{H}^2}{\det\left(\bar{H}^2-\frac{2i\omega J^2}{\beta}\right)^{\frac{1}{2}}} \right]^\beta$$ Plugging into , we finally arrive at . The denominator in can be expressed in the form of a Gaussian integral $$\label{s7} \det\left(\bar{H}^2-\frac{2i\omega J^2}{\beta}\right)^{-\frac{\beta}{2}}\sim\int d\mathbf{z}\, \exp\left[-\mathbf{z}^\dagger\left(\bar{H}^2-\frac{2i\omega J^2}{\beta}\right)\mathbf{ z} \right]=\int d\mathbf{z}\, \exp\left(-\mathbf{z}^\dagger\bar{H}^2\mathbf{z}\right)e^{\frac{2i\omega J^2|\mathbf{ z}|^2}{\beta}},$$ where $\mathbf{ z}$ is a $N-1$-dimensional vector, real for $\beta=1$ and complex for $\beta=2$. Due to the invariance of the ensembles with respect to appropriate ($O(N-1)$ or $U(N-1)$) rotations the average does not depend on the direction of $\mathbf{z}$, but only on its norm $|\mathbf{z}|^2$ $$\label{s9} P(\chi)\sim\int_{-\infty}^{\infty}d\omega e^{-i\omega\chi}\left\langle\det \bar{H}^2\int d\mathbf{z}\, \exp\left(-|\mathbf{z}|^2 X\right)e^{\frac{2i\omega J^2|\mathbf{ z}|^2}{\beta}}\right\rangle,$$ where $X$ is some quadratic form in the elements of $\bar{H}$ specified below. In the spherical coordinates $d\mathbf{z}\sim drr^{\beta(N-1)\beta-1}$ (where $r:=|\mathbf{z}|$), integrating over $\omega$ results in $\delta\left(\chi-2J^2r^2/\beta\right)$ and thus we arrive at . Fidelity susceptibility distribution for GOE -------------------------------------------- For GOE ($\beta=1$), choosing $\mathbf{z} = r[1,0,0..]^T$ we rewrite the average in as $$\left \langle \mathrm{det}\bar H^{2} \mathrm{e}^{-r^2 X} \right \rangle_{N-1} = \int d\bar H_{11}\mathrm{e}^{-A \bar H_{11}^2}\prod_{j=2}^{N-1} d \bar H_{1j}\mathrm{e}^{-B \sum_{j=2}^{N-1} \bar H_{1j}^2} \mathrm{det}\bar H^{2} D^{N-2}V, \label{eq: 2}$$ with $A = \frac{1}{4J^2} +r^2$, $B=\frac{1}{2J^2} +r^2$, $X=\sum_{j=1}^{N-1} | \bar H_{1j}|^2$ and $$\label{s13} \bar{H}=\left[ \begin{array}{cc} H_{11} & H_{1j} \\ H_{1k} & V \end{array} \right].$$ The block $V$ is itself a $(N-2)\times(N-2)$ GOE matrix (with the GOE density $\mathrm{D}^{N-2}V=\prod_{k<j} dV_{kj}\exp(-\frac{1}{4J^2}\operatorname{Tr}V^2)$). Using the general formula for the determinant of a block matrix $$\det\left[\begin{array}{cc} \mathbf{A} & \mathbf{B} \\ \mathbf{C} & \mathbf{D} \end{array} \right]=\det\left(\mathbf{A}-\mathbf{B}\mathbf{D}^{-1}\mathbf{C}\right)\det\mathbf{D}$$ we get (since the upper diagonal block is in fact one-dimensional, $\mathbf{A}=H_{11}$), $$\label{s15} \det\bar{H}=\det V\left(\bar{H}_{11}-\sum_{j,k=2}^{N-1}\bar{H}_{1j}V^{-1}_{jk}\bar{H}_{1k}\right).$$ Thus, becomes $$\left \langle \mathrm{det}\bar H^{2} \mathrm{e}^{-r^2 X} \right \rangle_{N-1}= \int d\bar H_{11} \mathrm{e}^{-A \bar H_{11}^2} \prod_{j=2}^{N-1} d \bar H_{1j} \mathrm{e}^{-B \sum_{j=2}^{N-1} | \bar H_{1j}|^2} \left \langle \mathrm{det}V^{2} \left( \bar H_{11}- \sum_{j,k=2}^{N-1} \bar H_{1j} V^{-1}_{jk} \bar H_{1k} \right)^{2} \right \rangle_{N-2}, \label{eq: 3}$$ where the average is now taken over the matrix $V$. Changing variables $\bar H_{1j} = (\frac{\pi}{B})^{\frac{1}{2}}y_j$ (only terms with even powers of $\bar H_{11} $ survive the integration over $\bar H_{11} $) $$\left \langle \mathrm{det}\bar H^{2} \mathrm{e}^{-r^2 X} \right \rangle_{\bar H}= \frac{\langle \mathrm{det}V^2\rangle_V}{A^{1/2}} \left(\frac{\pi}{B}\right)^{\frac{N-2}{2}} \int \prod_{j=2}^{N-1} d y_j \mathrm{e}^{ -\pi \sum_{j=2}^{N-1} y_j^2} \left( \frac{1}{2A}+ \left(\frac{\pi}{B}\right)^{2} \frac{\left \langle \left ( \sum_{j,k=2}^{N-1} y_j V^{-1}_{jk} y_k \right)^2 \right \rangle_{N-2}}{\langle \mathrm{det}V^2\rangle_{N-2}} \right). \label{eq: 4}$$ Denote $$\mathcal{I}^{O,2}_{N-2} = 4\pi^2 J^2\int \prod_{j=2}^{N-1} d y_j \mathrm{e}^{ -\pi \sum_{j=2}^{N-1} y_j^2} \frac{\left \langle \left ( \sum_{j,k=2}^{N-1} y_j V^{-1}_{jk} y_k \right)^2 \right \rangle_{N-2}}{\langle \mathrm{det}V^2\rangle_{N-2}} \label{eq: 4a}.$$ Changing the order of integration and averaging in , integration over $y_j$ can be done in the following way $$\begin{aligned} \nonumber 4\pi^2 \int \prod_{j=2}^{N-1} d y_j \mathrm{e}^{ -\pi \sum_{j=2}^{N-1} y_j^2} \left ( \sum_{j,k=2}^{N-1} y_j V^{-1}_{jk} y_k \right)^2 = 4\pi^2\int \prod_{j=2}^{N-1} d \xi_j \mathrm{e}^{ -\pi \sum_{j=2}^{N-1} \xi_j^2} \sum_{j,k=2}^{N-1} \xi^2_j \xi^2_k E^{-1}_{j} E^{-1}_{j} =\\ = 3 \sum_{j=2}^{N-1} E^{-2}_{j} + \sum_{j,k=2, j\neq k}^{N-1} E^{-1}_{j} E^{-1}_{j} = 2 \operatorname{Tr}V^{-2}+\left( \operatorname{Tr}V^{-1}\right)^2 \label{eq: 4b},\end{aligned}$$ where a change of variables $z_j = O \xi_j$ such that $O^{T}V^{-1}O=\mathrm{diag}\left( E^{-1}_{2},\ldots, E^{-1}_{N-1}\right) $ was performed. Thus $$\mathcal{I}^{O,2}_{N-2} =J^2\frac{\langle \mathrm{det}V^2\left(2 \mathrm{Tr}V^{-2}+\left(\mathrm{Tr}V^{-1}\right)^2 \right)\rangle_{N-2}} {\langle \mathrm{det}V^2\rangle_{N-2}}, \label{eq: 4b}$$ which is precisely the form of . The averages in contain functions of eigenvalues of $V$ – therefore this formula is suited for averaging over joint probability distribution of eigenvalues for GOE. However, we can proceed in an easier way. Plugging in definitions of $A$ and $B$, becomes $$\frac{ \left \langle \mathrm{det}\bar H^{2} \mathrm{e}^{-r^2 X} \right \rangle_{\bar H}}{\langle \mathrm{det}V^2\rangle_V } = \left( \frac{1}{4 J^2 \pi } \right)^{\frac{1}{2} } \left( \frac{1}{2 J^2 \pi} \right)^{\frac{N-2}{2}} \left( \frac{4J^2\pi }{1+4J^2r^2}\right)^{\frac{1}{2}} \left(\frac{2J^2 \pi}{1+2J^2r^2}\right)^{\frac{N-2}{2}} \left( \frac{2J^2 }{1+4J^2r^2}+ \left(\frac{1}{1+2J^2r^2}\right)^{2} J^2 \mathcal{I}^{O,2}_N \right), \label{eq: 3.2}$$ where all of the normalization constants are kept. Putting $r=0$ in this formula we arrive at which allows for straightforward (and *exact*) calculation of $\mathcal{I}^{O,2}_N$. Moreover, using in , remembering that $\delta\left( \chi - 2J^2 r^2\right) \propto \left(\frac{1}{\chi}\right)^{\frac{1}{2}} \delta \left( r- \left(\frac{\chi}{2J^2}\right)^{\frac{1}{2}} \right)$ we obtain the fidelity susceptibility distribution for GOE . We finally note that the form of $P^O(x)$ is such that distribution of $t=\frac{1}{x}$ is many aspects simpler: $$P(t)=\frac{1}{6}\left(1+t\right) \exp\left(-\frac{t}{2}\right), \label{eq: 8}$$ which suggests that further inquires of properties of fidelity susceptibility outside the realm of GRE could be done for $t=\frac{N}{\chi}$ variable. Calculation and results for GUE ------------------------------- Writing for GUE - $\beta =2$, one gets choosing $\mathbf{z} = r[1,0,0..]^T$ $$\left \langle \mathrm{det}\bar H^{4} \mathrm{e}^{-r^2 X} \right \rangle_{\bar H} = \int d\bar H_{11}\mathrm{e}^{-A \bar H_{11}^2}\prod_{j=2}^{N-1} d \bar H^R_{1j} d \bar H^I_{1j} \mathrm{e}^{-B \sum_{j=2}^{N-1} | \bar H_{1j}|^2} \mathrm{det}\bar H^{4} D^{N-2}V, \label{eq: 9p}$$ with $A = \frac{1}{2J^2} +r^2$ and $B=\frac{1}{J^2} +r^2$. Changing variables: $\bar H_{1j} =\bar H^R_{1j}+i\bar H^I_{1j}= (\frac{\pi}{B})^{\frac{1}{2}}(x_j+ \mathrm{i}y_j) =(\frac{\pi}{B})^{\frac{1}{2}}z_j$ and using the formula for determinant of block matrix one gets $$\left \langle \mathrm{det}\bar H^{4} \mathrm{e}^{-r^2 X} \right \rangle_{\bar H} / \left \langle \mathrm{det}V^{4} \right \rangle_{N-2}= \left(\frac{\pi}{A}\right)^{\frac{1}{2}} \left(\frac{\pi}{B}\right)^{N-2} \int \prod_{j=2}^{N-1} d x_j d y_j \mathrm{e}^{ -\pi \sum_{j=2}^{N-1}|z_j|^2} \times \nonumber$$ $$\left( \frac{3}{4A^2} + 6\frac{1}{2 A} \left( \frac{\pi}{B} \right)^2 \frac{\left \langle \mathrm{det}V^4 \left ( \sum_{j,k=2}^{N-1} z_j V^{-1}_{jk} z^*_k \right)^2 \right \rangle_{N-2} } {\left \langle \mathrm{det}V^{4} \right \rangle_{N-2}} +\left( \frac{\pi}{B} \right)^4 \frac{ \left \langle \mathrm{det}V^4 \left ( \sum_{j,k=2}^{N-1} z_j V^{-1}_{jk} z^*_k \right)^4 \right \rangle_{N-2} }{\left \langle \mathrm{det}V^{4} \right \rangle_{N-2} } \right). \label{eq: 9}$$ Denote $$\mathcal{I}^{U,2}_{N-2} = J^2\pi^2 \int \prod_{j=2}^{N-1} d x_j d y_j \mathrm{e}^{ -\pi \sum_{j=2}^{N-1}|z_j|^2} \frac{\left \langle \mathrm{det}V^4 \left ( \sum_{j,k=2}^{N-1} z_j V^{-1}_{jk} z^*_k \right)^2 \right \rangle_{N-2} } {\left \langle \mathrm{det}V^{4} \right \rangle_{N-2}} \label{eq: I42}$$ and $$\mathcal{I}^{U,4}_{N-2} = J^4\pi^4 \int \prod_{j=2}^{N-1} d x_j d y_j \mathrm{e}^{ -\pi \sum_{j=2}^{N-1}|z_j|^2} \frac{ \left \langle \left( \mathrm{det}V^4 ( \sum_{j,k=2}^{N-1} z_j V^{-1}_{jk} z^*_k \right)^4 \right \rangle_{N-2} }{\left \langle \mathrm{det}V^{4} \right \rangle_{N-2} }. \label{eq: I44}$$ Expressing $A$ and $B$ in terms of $J^2$ and $r^2$ results in $$\left \langle \mathrm{det}\bar H^{4} \mathrm{e}^{-r^2 X} \right \rangle_{N-1} / \left \langle \mathrm{det}V^{4} \right \rangle_{N-1}=\left( \frac{1}{2 J^2 \pi } \right)^{\frac{1}{2} } \left( \frac{1}{ J^2 \pi} \right)^{N-2} \times \nonumber$$ $$\left(\frac{2J^2\pi}{1+2J^2r^2}\right)^{\frac{1}{2}} \left(\frac{J^2\pi}{1+J^2r^2}\right)^{N-2} \left[ \frac{3}{4} \left( \frac{2J^2}{1+2J^2r^2}\right)^{2} + 3 \frac{2J^2}{1+2J^2r^2} \left( \frac{J^2}{1+J^2r^2} \right)^2 \mathcal{I}^{U,2}_N +\left( \frac{J^2}{1+J^2r^2} \right)^4 \mathcal{I}^{U,4}_N \right]. \label{eq: 99}$$ First of all, this equation used in implies the form of the fidelity susceptibility distribution for GUE . Moreover, taking $r=0$ in and using expression for the second moment of determinant of GUE matrix from [@Mehta98; @Cicuta00] implies that $$\left \langle \mathrm{det}\bar H^{4} \right \rangle_{N-1} / \left \langle \mathrm{det}V^{4} \right \rangle_{N-1} = J^4 N^2, \label{eq: 99c}$$ which, in the $N\gg1$ limit is equivalent to . To complete the derivation of fidelity susceptibility we need to address the task of calculating $\mathcal{I}^{U,2}_{N}$ to which we turn now. Let us start by expressing $\mathcal{I}^{U,2}_N$ in terms of invariants ($H$ is now $N\times N$ GUE matrix), $$\mathcal{I}^{U,2}_N =\frac{ J^2 \pi^2}{\left \langle \mathrm{det}H^{4} \right \rangle_{N}} \left \langle \mathrm{det}H^4 \int \prod_{j=1}^{N} d x_j d y_j \mathrm{e}^{ -\pi \sum_{j=1}^{N}|z_j|^2} \left ( \sum_{j,k=1}^{N} z_j H^{-1}_{jk} z^*_k \right)^2 \right \rangle_{N} \equiv \frac{ J^2}{\left \langle \mathrm{det}H^{4} \right \rangle_{N}} \left \langle \mathrm{det}H^4 I^{U,2}_N \right \rangle_{N}. \label{eq: 5.1s}$$ Substituting $z_i = U \xi_i$ with $U$ such that $UH^{-1}U^{\dag} = \mathrm{diag} \left( E^{-1}_1,..., E^{-1}_N\right)$ and then putting $\xi_i=r_i \mathrm{e}^{\mathrm{i}\phi_i}$ one gets $$I^{U,2}_N = \pi^2 \int \prod_{j=1}^{N} d r_j d \phi_j r_j \mathrm{e}^{ -\pi \sum_{j=1}^{N}r_j^2} \sum_{j,l}r_j^2 r_l^2 E_j^{-1} E_l^{-1}. \label{eq: 5.2}$$ One can integrate over the phases $\phi_j$, resulting in a factor $(2\pi)^N$ which cancels out with $1/(2\pi)^N$ arising in substitution $t_i=\pi r_i^2$ so that the integral becomes $$I^{U,2}_N = \int \prod_{j} d t_j \mathrm{e}^{ - \sum_{j}t_j} \sum_{j}t_j t_l E_j^{-1} E_l^{-1}= m_2\sum_{j} E_j^{-2} + m_1^2 \sum_{j\neq l}E_j^{-1} E_l^{-1}, \label{eq: 5.3}$$ where $m_2$ and $m_1$ are the second and the first moments of $\mathrm{e}^{-t}$ distribution. Using in , remembering that $m_2=2$ and $m_1=1$ one obtains the following expression $$\mathcal{I}^{U,2}_N = \frac{ J^2}{\left \langle \mathrm{det}H^{4} \right \rangle_{N}} \left \langle \mathrm{det}H^4 \left(\mathrm{Tr}H^{-2} + ( \mathrm{Tr}{H^{-1}} )^2 \right) \right \rangle_{N}, \label{eq: 5.1}$$ demonstrating validity of . The generating function ----------------------- Consider the generating function $$Z_N(j_1,j_2) = \left \langle \det H^2 \det(H-j_1) \det(H-j_2) \right \rangle_{N}. \label{eq: 7.1}$$ Using the equality $$\frac{\partial}{\partial j} \det(H-j) = \frac{\partial}{\partial j} \prod_{k=1}^N (E_k - j)= -\sum_l \frac{\prod_{k=1}^N (E_k - j)}{E_l-j} = -\det(H-j) \mathrm{Tr}(H-j)^{-1} \label{eq: 7.2}$$ we verify that indeed holds. Moreover, as a side product one gets $$\frac{\left \langle \det H \right \rangle_{N+1}}{\left \langle \det H \right \rangle_{N}} =\frac{Z_{N+1}(0,0)}{Z_N(0,0)} \stackrel{\lim_{N\rightarrow\infty}}{=}J^4\mathcal{I}^{U,4}_N \label{eq: 7.6}$$ which can be used as a validation of our calculation by comparison of the result with . Calculation of generating function ---------------------------------- Formulas best suited for our task of finding $Z(j_1,j_2)$ are worked out in [@Strahov03]: $$\left \langle \prod_{j=1}^K \det(H - \lambda_j)\det(H - \mu_j) \right \rangle_N= \frac{C_{N,K}}{\Delta(\lambda_1,...,\lambda_K) \Delta(\mu_1,...,\mu_K)} \det\left[W_{N+K}(\lambda_i, \mu_j) \right]_{i,j=1,...,K}, \label{eq: 8.1}$$ where $\Delta(\lambda_1,...,\lambda_K)$ is Vandermonde determinant and the kernel $W_{N+K}$ reads $$W_{N+K}(\lambda, \mu) = \frac{1}{\lambda-\mu} \left[ \Pi_{N+K}(\lambda) \Pi_{N+K-1}(\mu) - \Pi_{N+K}(\mu) \Pi_{N+K-1}(\lambda) \right] \label{eq: 8.2c}$$ where $\Pi_{M}(\lambda)$ are monic polynomials orthogonal with respect to a measure $\mathrm{e}^{-V(x)} \mathrm{d}x$ and $C_{N,K}$ is a constant. For the GUE case $V(x)=\frac{1}{2J^2}x^2$. Using the equations , , – we obtain the following closed analytical expression for the generating function $Z(j_1,j_2)$ $$Z_N(j_1, j_2) = 4 C_{N,2} N^{2 N+3} \left(J^2\right)^{2 N+3} \exp \left(\frac{j_1^2+j_2^2}{4 J^2}-2 N\right)\frac{1}{j_1^2 \left(j_1-j_2\right) j_2^2}\times \nonumber$$ $$\left(j_1 j_2 \sqrt{\frac{N}{J^2}} \sin \left(j_1 \sqrt{\frac{N}{J^2}}\right) \cos \left(j_2 \sqrt{\frac{N}{J^2}}\right)-\sin \left(j_2 \sqrt{\frac{N}{J^2}}\right) \left(\left(j_1-j_2\right) \sin \left(j_1 \sqrt{\frac{N}{J^2}}\right)+j_1 j_2 \sqrt{\frac{N}{J^2}} \cos \left(j_1 \sqrt{\frac{N}{J^2}}\right)\right)\right). \label{eq: 8.5}$$ It is interesting to note that von Oppen, during his calculation of distribution of curvatures for GUE [@vonOppen94] calculated $$\left \langle \det H^3(\det H - j_2)\right \rangle_{N} \sim \frac{\sin\left( \sqrt{\frac{N}{J^2}} j_2 \right) -\sqrt{\frac{N}{J^2}}j_2 \cos\left( \sqrt{\frac{N}{J^2}} E_2\right)}{(\sqrt{\frac{N}{J^2}}E_2)^3} \label{q2d}$$ using technique of suppersymmetric integrals (for a pedagogical introduction of this technique see [@Haake]). The formula for $Z(j_1,j_2)$ derived by us is an extension of the above expression– one can show that in the limit $\lim_{j_1 \rightarrow 0}Z_N(j_1,j_2)$ one recovers the von Oppen’s formula . Calculating the limit: $\lim_{j_1 \rightarrow0}\lim_{j_2 \rightarrow0} Z_N(j_1,j_2)$ one obtains for large $N$: $$\mathcal{I}^{U,4}_N=\frac{1}{J^4} \frac{Z_{N+1}(0,0)}{Z_N(0,0)} =N^2, \label{q6}$$ where we have also used $C_{N+1,2}/C_{N,2} \stackrel{N\rightarrow \infty}{\rightarrow} 1$. Moreover, $$\frac{\partial^2}{\partial j_1 \partial j_2} Z_N(0,0) = \frac{1}{15}\frac{N}{J^2} \label{eq: 8.7}$$ and $$\frac{\partial^2}{\partial j_2^2 } Z_N(0,0) = -\frac{1}{5}\frac{N}{J^2} \label{eq: 8.8}$$ which via implies that $$\mathcal{I}^{U,2}_N = \frac{1}{3}N. \label{eq: 8.9}$$

--- abstract: 'The period-luminosity (P-L) relation for Cepheid variables is important in modern astrophysics. In this work, we present the multi-band P-L relations derived from the Large Magellanic Cloud (LMC) and Small Magellanic Cloud (SMC) Cepheids, based on the latest release of OGLE-III catalogs. In addition to the $VI$ band mean magnitudes adopted from OGLE-III catalogs, we also cross-matched the LMC and SMC Cepheids to the 2MASS point source catalogs and publicly available [*Spitzer*]{} catalogs from SAGE program. Mean magnitudes for these Cepheids were corrected for extinction using available extinction maps. When comparing the P-L slopes, we found that the P-L slopes in these two galaxies are consistent with each others within $\sim2.5\sigma$ level.' address: '$^1$Graduate Institute of Astronomy, National Central University, Jhongli City, 32001, Taiwan' author: - 'Chow-Choong Ngeow$^1$' title: 'Period-Luminosity Relations For Magellanic Clouds Cepheids Based on OGLE-III Data: A Comparison' --- Introduction ============ The period-luminosity (P-L, also known as Leavitt Law) relation for Cepheid variables is important in modern astrophysics, as it is the first rung in the distance scale ladder. The P-L relations can also be used to constrain the theoretical PL relations based on stellar pulsation and evolution models. One important issue in the application of P-L relation in distance scale work is its universality – is the slope of P-L relation independent of metallicity? In this work, the multi-band P-L relations were derived for the Large Magellanic Cloud (LMC) and Small Magellanic Cloud (SMC) Cepheids, based on the latest release of OGLE-III (the third phase of Optical Gravitational Lensing Experiment) catalogs. Large numbers ($\sim 10^3$) of Cepheids in the LMC and SMC permit the determination of accurate P-L slopes, hence testing the universality of the P-L relation in low metallicity galaxies. Data and Methods ================ Periods and intensity mean magnitudes in $VI$ bands were available from OGLE-III catalogs for $\sim1800$ LMC Cepheids [@sos08] and $\sim2600$ SMC Cepheids [@sos10]. All of these Cepheids are fundamental mode Cepheids. SMC Cepheids with $\log(P)<0.4$, however, were removed from the sample, as they followed different P-L relation [@bau99]. These Cepheids were cross-matched to 2MASS point source catalog. The random-phase $JHK$ photometry were converted to mean magnitudes using the prescription given in [@sos05]. The mid-infrared photometry were available from [*Spitzer*]{} archival data – SAGE [@mei06] and SAGE-SMC. Zaritsky’s extinction maps for LMC [@zar04] and SMC [@zar02] were used for extinction corrections. In addition, the extinction-free P-L relation – the Wesenheit Function in the form of $W=I-1.55(V-I)$, was also derived. Outliers presented in the P-L plane were removed using an iterative sigma clipping algorithm [@nge09]. Additional period cuts need to be applied in certain bands, as the faint end (hence shorter period) of these P-L relations may be affected by incompleteness bias. Results and Conclusion ====================== [llll]{} Band & LMC P-L Slopes & SMC P-L Slopes & Slope Difference\ $V$ & $-2.769\pm0.023$ & $-2.672\pm0.036$ & $ 0.097\pm0.043$\ $I$ & $-2.961\pm0.015$ & $-2.926\pm0.028$ & $ 0.035\pm0.032$\ $J$ & $-3.115\pm0.014$ & $-3.062\pm0.024$ & $ 0.053\pm0.028$\ $H$ & $-3.206\pm0.013$ & $-3.171\pm0.023$ & $ 0.035\pm0.026$\ $K$ & $-3.194\pm0.015$ & $-3.231\pm0.039$ & $ 0.037\pm0.038$\ $3.6\mu\mathrm{m}$ & $-3.253\pm0.010$ & $-3.226\pm0.019$ & $0.027\pm0.021$\ $4.5\mu\mathrm{m}$ & $-3.214\pm0.010$ & $-3.184\pm0.020$ & $0.030\pm0.022$\ $5.8\mu\mathrm{m}$ & $-3.182\pm0.020$ & $-3.235\pm0.042$ & $0.053\pm0.047$\ $8.0\mu\mathrm{m}$ & $-3.197\pm0.036$ & $-3.281\pm0.062$ & $0.084\pm0.072$\ $W$ & $-3.313\pm0.008$ & $-3.319\pm0.018$ & $ 0.006\pm0.020$\ Comparison of the multi-band P-L slopes for these two metal poor galaxies are presented in Table \[tab\]. Note that the LMC P-L slopes have been published in [@nge09]. As can be seen from this Table, the P-L slopes are within $\sim2.5\sigma$ in all bands between the LMC and SMC Cepheids. The $W$ band P-L slopes are almost identical, suggesting the extinction-free $W$ band P-L relation is a good choice for distance scale application. CCN thanks the funding from National Science Council (of Taiwan) under the contract NSC 98-2112-M-008-013-MY3. References {#references .unnumbered} ========== [8]{} Bauer, F., et al. 1999, , 348, 175 Meixner, M., et al. 2006, , 132, 2268 Ngeow, C., et al. 2009, , 693, 691 Soszynski, I., et al. 2005, , 117, 823 Soszynski, I., et al. 2008, Acta Astron., 58, 163 Soszynski, I., et al. 2010, Acta Astron., 60, 17 Zaritsky, D., et al. 2002, , 123, 855 Zaritsky, D., et al. 2004, , 128, 1606

--- abstract: 'We report the first sub-arcsecond VLA imaging of 6 GHz continuum, methanol maser, and excited-state hydroxyl maser emission toward the massive protostellar cluster NGC6334I following the recent 2015 outburst in (sub)millimeter continuum toward MM1, the strongest (sub)millimeter source in the protocluster. In addition to detections toward the previously known 6.7 GHz Class II methanol maser sites in the hot core MM2 and the UCHII region MM3 (NGC6334F), we find new maser features toward several components of MM1, along with weaker features $\sim1''''$ north, west, and southwest of MM1, and toward the non-thermal radio continuum source CM2. None of these areas have heretofore exhibited Class II methanol maser emission in three decades of observations. The strongest MM1 masers trace a dust cavity, while no masers are seen toward the strongest dust sources MM1A, 1B and 1D. The locations of the masers are consistent with a combination of increased radiative pumping due to elevated dust grain temperature following the outburst, the presence of infrared photon propagation cavities, and the presence of high methanol column densities as indicated by ALMA images of thermal transitions. The non-thermal [radio emission]{} source CM2 ($2''''$ north of MM1) also exhibits new maser emission from the excited 6.035 and 6.030 GHz OH lines. Using the Zeeman effect, we measure a line-of-sight magnetic field of +0.5 to +3.7 mG toward CM2. In agreement with previous studies, we also detect numerous methanol and excited OH maser spots toward the UCHII region MM3, with predominantly negative line-of-sight magnetic field strengths of $-2$ to $-5$ mG and an intriguing south-north field reversal.' author: - 'T. R. Hunter, C. L. Brogan, G. C. MacLeod, C. J. Cyganowski, J. O. Chibueze, R. Friesen, T. Hirota, D. P. Smits, C. J. Chandler, R. Indebetouw' title: | The extraordinary outburst in the massive protostellar system NGC6334I-MM1:\ emergence of strong 6.7 GHz methanol masers --- Introduction ============ [lcc]{} Project code & 16B-402 & 10C-186\ Observation date(s) & 2016 Oct 29, 2016 Nov 19 & 2011 May 23\ Mean epoch & 2016.9 & 2011.4\ Configuration & A & BnA\ Time on source (min) & 58, 58 & 92\ Number of antennas & 26, 27 & 27\ FWHM primary beam ($\arcmin$) & 7 & 2\ Baseband center frequencies (GHz) & 5, 7 & 24.059, 25.406\ Polarization products & dual circular & dual circular\ Gain calibrator & J1717-3342 & J1717-3342\ Bandpass calibrator & J1924-2914 & J1924-2914\ Flux calibrator & 3C286 & 3C286\ Spectral windows & 32 & 16\ Digitizer resolution & 3-bit & 8-bit\ Channel spacing (narrow, wide kHz) & 1.953, 1000 & 31.25\ Total bandwidth (GHz) & 3.9 & 0.128\ Proj. baseline range (k$\lambda$) & 4.6 - 980 & 12 - 1030\ Robust clean parameter & $-1.0$ & +1.0\ Cont. Resolution ($\arcsec\times\arcsec$ (P.A.$\arcdeg$)) & $0.63\times 0.14$ ($-2.9$) & ...\ Cont. RMS noise ([mJy beam$^{-1}$]{}) & 0.022 & ...\ Line Resolution ($\arcsec\times\arcsec$ (P.A.$\arcdeg$)) & $0.76\times 0.21$($-3.2$) \[6.7 GHz\] & $0.35\times 0.26$($+3.3$) \[24.51 GHz\]\ Line channel width ([km s$^{-1}$]{}) & 0.15 & 2.0\ RMS noise per channel ([mJy beam$^{-1}$]{}) & 2.2 \[6.7 GHz\] & 0.5 Episodic accretion in protostars is increasingly recognized as being an essential phenomenon in star formation [@Kenyon90; @Evans09]. The total luminosity of a protostar scales with the instantaneous accretion rate, so variations in that rate will lead to observable brightness changes [e.g., @Offner11]. The classical manifestations of these changes [ in low-mass protostars]{} are the FU Orionis stars [FUors, @Herbig77], which exhibit optical flares of 5 or more magnitudes followed by a slow decay, and the EX Lupi stars [EXors, @Herbig89], which exhibit smaller, 2-3 magnitude flares but in many cases have been seen to repeat. In recent years, near-infrared surveys such as the Vista Variables in the Via Lactea [VVV, @Minniti10] and the United Kingdom Infrared Deep Sky Survey [UKIDSS, @Lucas08] have identified hundreds of high-amplitude variables, most of which have been shown to be protostars in earlier evolutionary states [Class I and II, @Contreras17; @Lucas17]. Outbursts in even younger and more deeply-embedded low-mass protostars have recently been detected via large increases in their mid-infrared or submillimeter emission, including a Class 0 source [HOPS-383, @Safron15] from the Herschel Orion Protostar Survey [HOPS, e.g., @Stutz13 and references therein] and a Class I source [EC53, @Yoo17] from the James Clerk Maxwell Telescope (JCMT) Transient Survey [@Mairs17]. [ In more massive protostars, indirect evidence for episodic accretion has been inferred from outflow features seen toward high mass young stellar objects (HMYSOs). Examples include interferometric millimeter CO images of massive bipolar outflows that show high velocity bullets [e.g. @Qiu09], and near-infrared CO spectra of HMYSOs that show multiple blue-shifted absorption features analogous to the absorption features seen in FUOrs [e.g., @Ellerbroek11; @Thi10; @Mitchell91].]{} Recently, it has become [ more directly]{} clear that massive protostars also exhibit outbursts as evidenced by the 4000 [$L_\odot$]{} erratic variable V723 Carinae [@Tapia15], the infrared flare from the 20 [$M_\odot$]{} protostar powering S255IR-NIRS3 [@Caratti16], and the ongoing (sub)millimeter flare of the deeply-embedded source [[[NGC6334]{}]{}I]{}-MM1 [@Hunter17]. The large increases in bolometric luminosity observed in these events provide evidence for accretion outbursts similar to those predicted by hydrodynamic simulations of massive star formation [@Meyer17]. Identifying additional phenomena associated with these events will help to explore the mechanism of the outbursts, particularly if spectral line tracers can be identified, as they can potentially trace gas motions at high angular resolution. Among the strongest molecular lines emitted from regions of massive star formation are a number of maser transitions [@Reid81; @Elitzur92]. Ever since its discovery, maser emission [ from protostars]{} has exhibited eruptive phenomena, such as the three past outbursts observed in the 22 GHz water masers in Orion KL [@Abraham81; @Omodaka99; @Hirota14], multiple events in W49N [@Honma04; @Liljestrom00], and most recently the outburst in G25.65+1.05 [@Volvach17a; @Volvach17b; @Lekht17]. The repeating nature of the Orion maser outbursts [@Tolmachev11], as well as the periodic features seen toward many [ HMYSOs]{} in the 6.7 GHz methanol maser line [e.g., @Goedhart04], strongly suggest that variations in the underlying protostar could be responsible for the changes in maser emission. [ Such a link was suggested by @Fujisawa12 to explain the methanol maser flare in the HMYSO G33.64-0.21.]{} While @Kumar16 noted a similarity between the infrared light curves of several low-mass protostars in the VVV survey and the periodic methanol maser light curves of [ HMYSOs]{}, these phenomena have never been observed in the same object. However, the recent methanol maser flare in S255IR [@Fujisawa15] associated with the infrared outburst in S255IR-NIRS3 has provided the first direct link between protostellar accretion outbursts and maser flares [@Moscadelli17]. Beginning in early 2015, a strong flaring of many species of masers toward the millimeter protocluster [[[NGC6334]{}]{}I]{} [@Brogan16; @Hunter06] was discovered via a regular program of single-dish maser monitoring at the Hartebeesthoek Radio Observatory [@MacLeod17]. Of the four major millimeter sources in this deeply-embedded protocluster, only the UCHII region MM3 [also known as NGC6334F, @Rodriguez82] is detected [ in the near- and mid-]{}infrared [ [e.g., @Willis13; @Walsh99; @Tapia96], despite observations at wavelengths]{} as long as 18 [$\mu$m]{} [@DeBuizer02; @Persi98]. [ While this protocluster is associated with a high-velocity, 0.5 pc-scale NE/SW bipolar outflow [@Zhang14; @Qiu11; @Beuther08; @Leurini06; @Ridge01; @McCutcheon00], it remains uncertain which protostar is the driving source; a component within MM1 seems the most likely, especially considering the presence of water masers there [@Brogan16]. ]{} The fact that the velocity range of the flaring masers encompassed the LSR velocity of the hot molecular core thermal gas [$-$7.2 [km s$^{-1}$]{}, @McGuire17; @Zernickel12; @Beuther07] provided strong evidence for an association [ between the flaring masers and]{} the (sub)millimeter continuum outburst in the MM1 hot core [@Hunter17]. However, the angular resolution of the maser observations was insufficient to be certain. In this paper, we present follow-up observations of [[[NGC6334]{}]{}I]{} with the National Science Foundation’s Karl G. Jansky Very Large Array (VLA) in 5 cm continuum, the 6.7 GHz [CH$_3$OH]{} transition, and multiple transitions of excited-state OH. The relatively nearby distance to the source [1.3$\pm$0.1 kpc, inferred from the weighted mean of the two maser parallax measurements of the neighboring source I(N), @Chibueze14; @Reid14] allows us to achieve a linear resolution of $\approx500$ au. We confirm that MM1 now exhibits strong 6.7 GHz [CH$_3$OH]{} maser emission. This is the first such detection of a Class II [CH$_3$OH]{} maser from this member of the protocluster in three decades of previous interferometric observations. We also find a new source of excited OH maser emission toward the non-thermal radio source CM2 to the north of MM1. The locations and relative strengths of the new maser spots in relation to images of thermally excited methanol offer a unique test of maser pumping theory and provide important clues to the origin of maser emission and its response to protostellar outbursts. Observations ============ VLA --- The VLA observing parameters in the two frequency bands (C and K) are presented in Table \[obs\]. In addition to the coarse resolution spectral windows used to image the continuum emission in C-band, we observed several maser transitions with high spectral resolution 2 MHz-wide correlator windows using a correlator recirculation factor of 8. The 6.66852 GHz (hereafter 6.7 GHz) transition of [CH$_3$OH]{} 5(1)-6(0) A$^+$ ([$E_{Lower}\widehat{=}$]{}49 K) was observed with a channel spacing of 1.953 kHz (0.0878 [km s$^{-1}$]{}) over a span of 90 [km s$^{-1}$]{} centered on the local standard of rest (LSR) velocity of $-7$ [km s$^{-1}$]{}. Three lines of excited-state OH maser emission were also observed: $J$=1/2-1/2, $F$=0-1 at 4.66024 GHz ([$E_{Lower}\widehat{=}$]{}182 K) observed with 0.0628 [km s$^{-1}$]{} channels over a span of 64 [km s$^{-1}$]{}, and $J$=5/2-5/2, $F$=2-2 at 6.03075 GHz and $F$=3-3 at 6.03509 GHz (both having [$E_{Lower}\widehat{=}$]{}120 K) observed with 0.0970 [km s$^{-1}$]{} channels over a span of 99 [km s$^{-1}$]{}. The 6 GHz lines were observed using the 8-bit digitizers. The data were calibrated using the VLA pipeline[^1] with some additional flagging applied (the pipeline [ task that attempts to flag]{} radio frequency interference was not applied [ to the science target in order to avoid flagging the maser features]{}). The pipeline applies Hanning smoothing to the data, which reduces ringing from strong spectral features. The bright 6.7 GHz maser emission was used to iteratively self-calibrate the 6.7 GHz line emission and the resulting solutions were applied to the other line and continuum data. After excising line emission from the broad band continuum spectral windows, multi-term multi-scale clean was used with scales of 0, 5, and 15 times the image pixel size to produce the continuum image. The bright UCHII region (MM3 = NGC 6334F) limits the dynamic range of the continuum image causing the rms noise in the image to vary significantly as a function of position. After subtracting the continuum in the uv-plane, spectral cubes of the maser emission were made with a channel spacing (and effective spectral resolution) of 0.15 [km s$^{-1}$]{}. Stokes I cubes were made for the relatively unpolarized 6.7 GHz transition, while right and left circular polarization (RCP and LCP) cubes were made for the strongly polarized excited OH transitions. The typical noise in a signal-free channel is 2.2 [mJy beam$^{-1}$]{} for the [CH$_3$OH]{} Stokes I cube, and 3.1 [mJy beam$^{-1}$]{} for the RCP and LCP 6 cm excited OH cubes. For all maser lines, although recirculation was used, the velocity extent of the emission is much smaller than the [total bandwidth]{} of the spectral window employed, thus none of the maser channels are affected by any spectral artifacts at high dynamic range due to phase serialization in the correlator [@Sault13]. The epoch 2011.6 K-band (1.3 cm) observations reported here employed narrow spectral windows (with limited total bandwidth) to target a range of spectral lines at high spectral resolution, including the H64$\alpha$ radio recombination line (RRL) at 24.50990 GHz that we discuss in this work. These data were calibrated in CASA using manually generated scripts prior to the development of the pipeline. The H64$\alpha$ cube was created with a velocity resolution of 2.0 [km s$^{-1}$]{}. We estimate that the absolute position uncertainties for all of the the VLA data reported here are smaller than 50 mas. ALMA ---- We also present spectral line data from ALMA project 2015.A.00022.T, for which the observing parameters are described in more detail in @Hunter17. The thermal transition of methanol 11(2)-10(3) $A^-$ at 279.35191 GHz ([$E_{Lower}\widehat{=}$]{}177 K) was observed with a channel spacing of 976.5625 kHz (1.05 [km s$^{-1}$]{}) and a velocity resolution of 1.21 [km s$^{-1}$]{}. The data cube was cleaned with a robust parameter of 0 and has a beamsize of $0\farcs29 \times 0\farcs22$ at position angle $-84^\circ$. The rms noise achieved is 1.7 [mJy beam$^{-1}$]{} in 1.1 [km s$^{-1}$]{} wide channels. We constructed a moment image of the peak intensity across the 22 channels spanning the entire line. We estimate that the absolute position uncertainties for the ALMA data presented here are smaller than 50 mas. Results ======= {width="1.0\linewidth"} [ccccccc]{} 1 & UCHII-Met4 & -18.80 & 17:20:53.3729 & -35:47:01.474 & 0.0119 (0.0015)\ 2 & UCHII-Met4 & -18.65 & 17:20:53.3721 & -35:47:01.382 & 0.00893 (0.00148)\ 3 & UCHII-Met4 & -17.75 & 17:20:53.3703 & -35:47:01.456 & 0.0114 (0.0015)\ 4 & UCHII-Met4 & -17.60 & 17:20:53.3715 & -35:47:01.356 & 0.0119 (0.0015)\ 5 & UCHII-Met4 & -17.45 & 17:20:53.3705 & -35:47:01.393 & 0.00988 (0.00147)\ 6 & UCHII-Met4 & -17.30 & 17:20:53.3696 & -35:47:01.361 & 0.0110 (0.0015)\ 7 & UCHII-Met4 & -17.15 & 17:20:53.3663 & -35:47:01.416 & 0.0109 (0.0015)\ 8 & UCHII-Met4 & -17.00 & 17:20:53.3710 & -35:47:01.394 & 0.0111 (0.0015)\ 9 & UCHII-Met4 & -16.85 & 17:20:53.3685 & -35:47:01.503 & 0.0150 (0.0014)\ 10 & UCHII-Met4 & -16.70 & 17:20:53.3684 & -35:47:01.496 & 0.0171 (0.0015)\ {width="0.49\linewidth"} {width="0.49\linewidth"} {width="0.49\linewidth"} {width="0.49\linewidth"} Radio continuum --------------- Contours from the new 5 cm continuum image are shown in Fig. \[methanolimage\] alongside those from the epoch 2011.4 image [@Brogan16]. The morphology [ and intensity]{} of the emission are consistent given the difference in synthesized beams, and the lower sensitivity and image fidelity of the earlier data. In particular, the compact non-thermal source CM2 reported by @Brogan16 has persisted to the current epoch, as has the north-south jet-like feature centered on the grouping of millimeter sources designated MM1. The fitted position of the continuum emission from the point source CM1 located outside the field of view of Fig. \[methanolimage\] to the west of the central protocluster [@Brogan16] agrees between the 2011.4 and 2016.9 epochs to within $0\farcs03$; thus, there is no evidence for any significant instrumental shift in astrometry between the two epochs. We also note that the proper motion of the parent cloud [measured by the Very Long Baseline Array, @Wu14] accumulated over the 5.5 years between the two epochs corresponds to only (+1.7, $-$12) milliarcsec, which is undetectable at the current resolution. Further details about the centimeter wavelength continuum emission will be presented in a future multi-wavelength paper [ (Brogan et al. 2018, in preparation)]{}. [ccccccc]{} CM2-Met1 & 17:20:53.381 & -35:46:55.68 & 2.00 & -8.00, -7.55, -7.40 & 5.03 & 0.849\ CM2-Met2 & 17:20:53.413 & -35:46:55.84 & 2.72 & -6.80, -6.65, -5.30 & 4.98 & 0.840\ CM2 Total & ... & ... & 4.72 & -8.00, -7.55, -5.30 & ... & ...\ MM1-Met1 & 17:20:53.417 & -35:46:56.98 & 741 & -8.60, -7.25, -1.55 & 545 & 92.0\ MM1-Met2 & 17:20:53.327 & -35:46:58.01 & 70.8 & -11.00, -9.35, -2.75 & 55.4 & 9.35\ MM1-Met3 & 17:20:53.499 & -35:46:57.90 & 0.0925 & -4.55, -3.50, -3.20 & 0.111 & 0.0188\ MM1-MetC & 17:20:53.439 & -35:46:58.30 & 0.924 & -8.30, -6.20, -5.60 & 1.51 & 0.255\ MM1 Total & ... & ... & 813 & -11.00, -7.25, -1.55 & ... & ...\ MM2-Met1 & 17:20:53.174 & -35:46:59.17 & 644 & -15.95, -11.15, -7.70 & 583 & 98.4\ MM2-Met2 & 17:20:53.139 & -35:46:58.39 & 3.55 & -8.30, -7.85, -2.30 & 5.59 & 0.943\ MM2 Total & ... & ... & 648 & -15.95, -11.15, -2.30 & ... & ...\ UCHII-Met1 & 17:20:53.430 & -35:47:03.46 & 87.7 & -9.35, -6.65, -5.60 & 93.0 & 15.7\ UCHII-Met2 & 17:20:53.410 & -35:47:02.56 & 13.7 & -9.35, -8.00, -6.35 & 10.9 & 1.83\ UCHII-Met3 & 17:20:53.405 & -35:47:02.04 & 0.203 & -8.15, -8.00, -8.00 & 0.707 & 0.119\ UCHII-Met4 & 17:20:53.370 & -35:47:01.52 & 1760 & -18.80, -10.40, -8.15 & 1820 & 307\ UCHII-Met5 & 17:20:53.430 & -35:47:01.49 & 61.2 & -9.50, -8.60, -6.80 & 88.9 & 15.0\ UCHII-Met6 & 17:20:53.389 & -35:47:00.91 & 5.80 & -10.10, -10.10, -8.00 & 16.9 & 2.86\ UCHII-Met7 & 17:20:53.451 & -35:47:00.53 & 23.2 & -10.25, -10.10, -9.05 & 36.3 & 6.13\ UCHII Total & ... & ... & 1950 & -18.80, -10.40, -5.60 & ... & ...\ Grand Total & ... & ... & 3420 & -18.80, -10.40, -1.55 & ... & ...\ 6.7 GHz methanol masers ----------------------- Because the degree of circular polarization of the 6.7 GHz maser emission is very low, we analyzed the Stokes I cube. [Strong emission appears in many directions within the central protocluster. The spectrum toward any given pixel typically extends over several [km s$^{-1}$]{} with components of rather different strength and width appearing at different velocities, and thus is not amenable to Gaussian spectral profile fitting. Instead,]{} we fit each channel of the cube that had significant emission ($> 6\sigma$, measured independently for each channel) with an appropriate number of [two-dimensional]{} Gaussian sources. Because the spots of maser emission are likely smaller than our synthesized beam (Table \[obs\]), we fixed the fitted shape parameters to the synthesized beam. We used the pixel position of each emission peak as the initial guess for the centroid position of each fit. In channels with complicated emission, we also set the intensity of each peak as an initial guess. After manual inspection of all the fits, a few of the weakest features that formally met the sensitivity criterion could be attributed to imaging artifacts in dynamic-range limited channels and were subsequently discarded. The fitted [ per-channel]{} positions and [ per-channel]{} flux densities of the masers are given in Table \[methanoltable\], and their positions and relative intensities are shown in Figure \[methanolimage\][ b]{}. For comparison, we also show previously published maser spots from two representative studies in Figure \[methanolimage\]a. As this figure demonstrates, the general location and strength of the masers toward MM2 and MM3 are consistent with past observations [@Walsh98; @Green15]. To summarize the salient maser properties in a compact format we have identified several distinct maser “associations” toward each of the continuum sources with detected 6.7 GHz maser emission: MM1, MM2, CM2, and the UCHII region MM3. The assignment of each fitted spot to an association is provided in a column in Table \[methanoltable\], and the properties and general locations of the associations are summarized in Table \[methanolassoc\] and Figure \[methanolzoom\]a, b, c, and d, respectively. Lower limits to the peak 6.7 GHz maser brightness temperatures ($T_B$) for the associations are also given in Table \[methanolassoc\] derived using the peak intensity and the synthesized beam size (Table \[obs\]). Even given our modest resolution, lower limits as high as $T_B = 8.5\times 10^5$, $1.1\times 10^8$, $1.5\times 10^8$, and $3.1\times 10^8$ K are found for the CM2, MM1, MM2, and MM3-UCHII regions, respectively. A number of studies using very long baseline interferometry (VLBI) have explored the intrinsic size of 6.7 GHz masers, with several finding that significant flux (up to 50%) can be missed by very high angular resolution observations, suggesting a core-halo type of morphology with a very compact core of a few au and extended halos of a few hundred au (i.e., of order the minor axis of our synthesized beam, $0\farcs2$) [see for example @Minier02; @HarveySmith06]. Therefore, we are unlikely to be missing significant flux, but it is possible that the maser spot cores have significantly higher $T_B$ than our current lower limits. ![Integrated Stokes I spectra from the epoch 2016.9 6.7 GHz methanol maser cube integrated over all the maser features (black spectrum) and over an elliptical region surrounding only the maser features associated with MM1 (red spectrum). \[methanol\_spectra\]](fig3.eps){width="1.0\linewidth"} ### MM1 and CM2 region In epoch 2016.9, many 6.7 GHz [CH$_3$OH]{} maser features have appeared within the dust continuum source MM1 (including the associations MM1-Met1 and MM1-MetC) and along its western periphery (MM1-Met2), eastern periphery (MM1-Met3), and northern peripheries (CM2-Met1, and CM2-Met2). A zoom to the area around MM1 and CM2 is shown in Figure \[methanolzoom\]a. These are the first detections of Class II methanol maser emission toward this region within the NGC6334I protocluster from nearly 30 years of interferometric observations, all of which were sufficiently sensitive to detect masers of this strength. Previous epochs at 6.7 GHz include January 1992 [@Norris93], May 1992 [@Ellingsen96; @Ellingsen96etal], July 1994 [@Walsh98], July 1995 [@Caswell97], September 1999 [@Dodson12], March 2005 [@Krishnan13], May 2011 [@Brogan16], and August 2011 [@Green15]. In all of these epochs, detections of this maser transition in this field originate from the UCHII region (NGC6334F) and MM2. In addition, the original VLBI imaging of the Class II 12.2 GHz methanol maser [ transition]{} did not detect any features toward MM1 [@Norris88], [ nor did the May 1992 ATCA observations [@Ellingsen96etal]]{}. The brightest 6.7 GHz features in MM1 (peak = 545 Jy at $-7.25$ [km s$^{-1}$]{}) reside in the valley of dust continuum emission between the 1 mm dust continuum sources MM1F and MM1G (in the MM1-Met1 association). This area lies about 1000 au north of the hypercompact HII region MM1B, which is the proposed central driving source of the millimeter outburst [@Hunter17]. The velocity of the brightest feature coincides with the LSR velocity of the thermal molecular gas [-7.3 [km s$^{-1}$]{}, @Zernickel12]. The integrated spectrum of MM1 (all associations shown in Fig. \[methanolzoom\]a) compared to the total field (Figure \[methanol\_spectra\]) further demonstrates that the dominant emission from this region arises from $-6$ to $-8$ [km s$^{-1}$]{}. Within the MM1-Met1 association (see Fig. \[methanolzoom\]b), a line of strong spots form the western side of a “V”-shaped structure, with a weaker line of redshifted features south of MM1F forming the eastern side. Additional strong features appear on the dust continuum source MM1F while weaker features near the LSR velocity lie around MM1G and extend northward from MM1F (including the association CM2-Met2). The northernmost spots (in association CM2-Met1) are coincident with the non-thermal radio source CM2. Another set of near-LSR features (MM1-MetC) coincide with the millimeter source MM1C, however the three brightest and most central millimeter sources (MM1A, B, and D) are notably lacking in any maser emission. A large number of moderate strength, primarily blueshifted masers (MM1-Met2) lie in cavities of the dust emission located west and southwest of MM1A. Finally, a number of weak, moderately redshifted features (MM1-Met3) lie eastward in an depression in the dust emission that originates between MM1C and MM1E. ### MM2 region A zoom to the area around MM2 is shown in Figure \[methanolzoom\]c. The strongest masers (MM2-Met1 in Tables \[methanoltable\] and \[methanolassoc\]) are situated between the 1 mm source MM2A and the 7 mm source MM2B and are predominantly blueshifted by 5-10 [km s$^{-1}$]{} from the LSR velocity of the thermal molecular gas. A subgroup of masers near the LSR velocity are centered just west of the MM2B. Weaker masers lie $\sim1''$ to the northwest (MM2-Met2 in Tables \[methanoltable\] and \[methanolassoc\]) including two weak redshifted spots. ### MM3 region (UCHII region) A zoom to the area around MM3 is shown in Figure \[methanolzoom\]d. The arrangement of 6.7 GHz maser spots toward the UCHII region is strikingly filamentary, and seems to closely trace the shape of the 5 cm continuum emission, especially toward the southern and middle regions. [ These masers may arise in a layer of compressed gas just outside the UCHII region.]{} Most of these spots lie near the LSR velocity, becoming somewhat blueshifted in the northern half. The strongest spots (Met4) are significantly blueshifted (by up to 12 [km s$^{-1}$]{}) and cluster around a small knot in the continuum emission located about $1''$ northwest of the 5 cm UCHII region peak. {width="0.96\linewidth"} [ccccccc]{} 1 & UCHII-OH4 & -11.00 & 17:20:53.3718 & -35:47:01.573 & 0.0241 (0.0025)\ 2 & UCHII-OH4 & -10.85 & 17:20:53.3722 & -35:47:01.573 & 0.0903 (0.0022)\ 3 & UCHII-OH6 & -10.85 & 17:20:53.3975 & -35:47:00.737 & 0.0294 (0.0022)\ 4 & UCHII-OH4 & -10.70 & 17:20:53.3712 & -35:47:01.531 & 0.326 (0.002)\ 5 & UCHII-OH6 & -10.70 & 17:20:53.3967 & -35:47:00.768 & 0.203 (0.002)\ 6 & UCHII-OH4 & -10.55 & 17:20:53.3709 & -35:47:01.525 & 2.84 (0.01)\ 7 & UCHII-OH6 & -10.55 & 17:20:53.3972 & -35:47:00.775 & 0.348 (0.004)\ 8 & UCHII-OH4 & -10.40 & 17:20:53.3709 & -35:47:01.525 & 13.3 (0.1)\ 9 & UCHII-OH6 & -10.40 & 17:20:53.3979 & -35:47:00.780 & 0.253 (0.007)\ 10 & UCHII-OH4 & -10.25 & 17:20:53.3709 & -35:47:01.524 & 16.6 (0.1)\ [ccccccc]{} 1 & UCHII-OH4 & -11.30 & 17:20:53.3687 & -35:47:01.536 & 0.0305 (0.0021)\ 2 & UCHII-OH4 & -11.15 & 17:20:53.3704 & -35:47:01.533 & 0.113 (0.002)\ 3 & UCHII-OH4 & -11.00 & 17:20:53.3709 & -35:47:01.527 & 0.973 (0.003)\ 4 & UCHII-OH4 & -10.85 & 17:20:53.3709 & -35:47:01.525 & 6.56 (0.01)\ 5 & UCHII-OH4 & -10.70 & 17:20:53.3709 & -35:47:01.524 & 13.0 (0.1)\ 6 & UCHII-OH4 & -10.55 & 17:20:53.3709 & -35:47:01.523 & 6.78 (0.01)\ 7 & UCHII-OH4 & -10.40 & 17:20:53.3708 & -35:47:01.527 & 0.639 (0.003)\ 8 & UCHII-OH6 & -10.40 & 17:20:53.3964 & -35:47:00.796 & 0.128 (0.003)\ 9 & UCHII-OH4 & -10.25 & 17:20:53.3702 & -35:47:01.547 & 0.0517 (0.0027)\ 10 & UCHII-OH6 & -10.25 & 17:20:53.3967 & -35:47:00.776 & 0.332 (0.003)\ [ccccccc]{} 1 & UCHII-OH4 & -12.05 & 17:20:53.3719 & -35:47:01.507 & 0.0231 (0.0022)\ 2 & UCHII-OH4 & -11.90 & 17:20:53.3709 & -35:47:01.555 & 0.0281 (0.0024)\ 3 & UCHII-OH4 & -11.75 & 17:20:53.3717 & -35:47:01.551 & 0.0376 (0.0024)\ 4 & UCHII-OH4 & -11.60 & 17:20:53.3718 & -35:47:01.558 & 0.0672 (0.0021)\ 5 & UCHII-OH4 & -11.45 & 17:20:53.3723 & -35:47:01.560 & 0.0972 (0.0025)\ 6 & UCHII-OH4 & -11.15 & 17:20:53.3638 & -35:47:01.863 & 0.895 (0.009)\ 7 & UCHII-OH4 & -11.15 & 17:20:53.3712 & -35:47:01.544 & 5.35 (0.01)\ 8 & UCHII-OH4 & -11.00 & 17:20:53.3631 & -35:47:01.858 & 1.06 (0.01)\ 9 & UCHII-OH4 & -11.00 & 17:20:53.3712 & -35:47:01.543 & 30.0 (0.1)\ 10 & UCHII-OH4 & -10.85 & 17:20:53.3712 & -35:47:01.542 & 36.9 (0.1)\ [ccccccc]{} 1 & UCHII-OH4 & -12.05 & 17:20:53.3731 & -35:47:01.504 & 0.0391 (0.0023)\ 2 & UCHII-OH4 & -11.90 & 17:20:53.3723 & -35:47:01.565 & 0.0624 (0.0025)\ 3 & UCHII-OH4 & -11.75 & 17:20:53.3718 & -35:47:01.597 & 0.0718 (0.0025)\ 4 & UCHII-OH4 & -11.45 & 17:20:53.3644 & -35:47:01.856 & 0.617 (0.004)\ 5 & UCHII-OH4 & -11.45 & 17:20:53.3713 & -35:47:01.539 & 2.03 (0.01)\ 6 & UCHII-OH4 & -11.30 & 17:20:53.3643 & -35:47:01.870 & 1.78 (0.01)\ 7 & UCHII-OH4 & -11.30 & 17:20:53.3712 & -35:47:01.541 & 17.5 (0.1)\ 8 & UCHII-OH4 & -11.15 & 17:20:53.3712 & -35:47:01.540 & 31.2 (0.1)\ 9 & UCHII-OH4 & -11.15 & 17:20:53.3662 & -35:47:01.899 & 0.773 (0.018)\ 10 & UCHII-OH4 & -11.00 & 17:20:53.3671 & -35:47:01.929 & 0.727 (0.007)\ Excited-state OH 6.030 and 6.035 GHz transitions ------------------------------------------------ [[[NGC6334]{}]{}I]{} is one of the strongest known sources of the 5 cm excited-state OH masers [@Caswell95; @Zuckerman72; @Gardner70]. [Although the excited OH spectra are generally simpler than those of the 6.7 GHz methanol, in order to match the methanol maser fitting procedure, we also fit the excited OH maser properties using the channel by channel approach.]{} Because the emission in these OH transitions is strongly circularly polarized, we fit the image cubes of RCP and LCP emission independently. The fitted positions and flux densities of the two transitions (6.030 and 6.035 GHz) are given in Tables \[oh6030LLtable\], \[oh6030RRtable\], \[oh6035LLtable\] and \[oh6035RRtable\]. The maser spots are shown in Figure \[ohmasers\]. [ Note we have not attempted to correct the flux densities for beam squint, but this effect will be small ($\sim 1.2\%$) toward the inner parts of the $7\arcmin$ primary beam where the masers are located (beam squint does not affect the Zeeman frequency splitting). ]{} Since all of the excited OH masers are found in the same general areas as the 6.7 GHz methanol detections, the labels for the OH associations have the same numerical index for a given region, for example UCHII-Met1 has an analogous excited OH maser association UCHII-OH1. The spatial distribution of the two OH transitions is similar, with 6.035 GHz being significantly stronger in most locations and showing additional associations (OH5 and OH7) toward the northern portion of the UCHII region (Figure \[ohcompare\]). The positions of the seven associations UCHII-OH1 through UCHII-OH7 are closely matched to those of the 6.7 GHz maser associations UCHII-Met1 through UCHII-Met7. {width="0.96\linewidth"} Integrated spectra of the cubes are shown in Figure \[oh\_spectra\]. These are similar to previous single-dish spectra toward this region [@Caswell03; @Avison16], with the exception of the strong new association toward CM2, which now dominates the emission near the LSR velocity, peaking at $-7.7$ [km s$^{-1}$]{}. Both lines were also previously observed interferometrically [@Knowles73; @Caswell97; @Green15], with the highest resolution images produced with the Long Baseline Array (LBA) in January 2001 [@Caswell11], which showed the ridge of spots along the UCHII region, but no feature toward CM2 has ever been reported. We thus conclude that the CM2 feature is associated with the millimeter outburst. It is notable that the flux ratio of the 6.035/6.030 lines toward CM2 is abnormally low ($1.2-1.8$) compared to the rest of the spots, which have a median ratio of 23, similar to the historical ratio of 25 noted by @Knowles76. At the current resolution, a lower limit to the peak brightness temperature of the 6.030 and 6.035 GHz masers is $3\times 10^6$ K and $5\times 10^7$ K, respectively (UCHII-OH4-1b, see § \[Zeemansplitting\]). From LBA observations toward NGC6334I, @Caswell11 found the excited OH spot sizes to be smaller than 20 mas for the UCHII region masers, and intensities within a factor of 2 of those reported here. From much higher angular resolution European VLBI Network (EVN) observations of the same excited OH transitions in W3(OH), @Fish07 find typical maser spot sizes of order 5 mas or smaller, [ which would translate to 7.5 mas or smaller at the nearer distance of NGC6334I]{}. Thus, the true OH brightness temperatures are likely to be orders of magnitude higher than reported here. {width="0.49\linewidth"} {width="0.49\linewidth"} ### Zeeman splitting {#Zeemansplitting} [lccccccccc]{} & & & & & OH 6.035 GHz\ UCHII-OH1-1 & 17:20:53.422 & -35:47:03.20 & 0.67 (0.03) & 0.11 & -8.762 (0.008) & 0.63 (0.03) & 0.11 & -8.520 (0.010) & -4.32 (0.24)\ UCHII-OH1-2 & 17:20:53.430 & -35:47:03.41 & 0.13 (0.02) & 0.022 & -7.533 (0.040) & 0.16 (0.01) & 0.027 & -7.391 (0.027) & -2.54 (0.86)\ UCHII-OH1-3 & 17:20:53.421 & -35:47:03.37 & 0.16 (0.02) & 0.027 & -8.061 (0.038) & 0.21 (0.02) & 0.036 & -7.888 (0.033) & -3.08 (0.90)\ UCHII-OH1-4 & 17:20:53.426 & -35:47:03.32 & 0.59 (0.02) & 0.10 & -6.923 (0.007) & 0.55 (0.02) & 0.094 & -6.644 (0.008) & -4.98 (0.19)\ UCHII-OH2-1a & 17:20:53.399 & -35:47:02.76 & 0.49 (0.03) & 0.083 & -9.441 (0.010) & 0.55 (0.04) & 0.094 & -9.308 (0.010) & -2.38 (0.25)\ UCHII-OH2-1b & & & 0.52 (0.02) & 0.089 & -8.752 (0.013) & 0.45 (0.03) & 0.077 & -8.531 (0.018) & -3.94 (0.40)\ UCHII-OH2-1c & & & 0.75 (0.03) & 0.13 & -7.692 (0.007) & 0.75 (0.04) & 0.13 & -7.386 (0.008) & -5.46 (0.18)\ UCHII-OH2-2a & 17:20:53.414 & -35:47:02.66 & 2.58 (0.04) & 0.44 & -8.734 (0.004) & 2.20 (0.02) & 0.37 & -8.522 (0.003) & -3.78 (0.08)\ UCHII-OH2-2b & & & 1.52 (0.07) & 0.26 & -8.078 (0.005) & 1.41 (0.03) & 0.24 & -7.831 (0.003) & -4.40 (0.11)\ UCHII-OH2-2c & & & 4.92 (0.05) & 0.84 & -7.671 (0.002) & 4.78 (0.03) & 0.81 & -7.375 (0.001) & -5.30 (0.04)\ UCHII-OH3-1 & 17:20:53.417 & -35:47:01.92 & 3.78 (0.07) & 0.64 & -8.713 (0.003) & 3.98 (0.05) & 0.68 & -8.523 (0.002) & -3.40 (0.06)\ UCHII-OH4-1a & 17:20:53.371 & -35:47:01.53 & 30.76 (0.95) & 5.24 & -11.156 (0.005) & 39.11 (1.09) & 6.7 & -10.899 (0.004) & -4.60 (0.12)\ UCHII-OH4-1b & & & 236.89 (0.96) & 40 & -10.539 (0.001) & 298.65 (1.08) & 51 & -10.266 (0.001) & -4.89 (0.02)\ UCHII-OH4-2a & 17:20:53.365 & -35:47:01.93 & 11.07 (0.33) & 1.9 & -11.158 (0.006) & 13.61 (0.38) & 2.3 & -10.901 (0.005) & -4.58 (0.13)\ UCHII-OH4-2b & & & 78.61 (0.36) & 13 & -10.542 (0.001) & 98.51 (0.39) & 17 & -10.268 (0.001) & -4.90 (0.02)\ UCHII-OH4-3 & 17:20:53.352 & -35:47:02.24 & 1.52 (0.03) & 0.26 & -10.490 (0.006) & 2.54 (0.03) & 0.43 & -10.246 (0.003) & -4.36 (0.13)\ UCHII-OH5-1 & 17:20:53.434 & -35:47:01.46 & 2.29 (0.13) & 0.39 & -8.913 (0.008) & 3.46 (0.04) & 0.59 & -8.645 (0.004) & -4.79 (0.16)\ UCHII-OH5-2 & 17:20:53.420 & -35:47:01.39 & 0.90 (0.03) & 0.15 & -8.059 (0.008) & 0.55 (0.02) & 0.094 & -7.924 (0.010) & -2.41 (0.23)\ UCHII-OH6-1 & 17:20:53.397 & -35:47:00.81 & 12.51 (0.17) & 2.1 & -9.993 (0.002) & 11.25 (0.14) & 1.9 & -10.217 (0.002) & +3.99 (0.06)\ UCHII-OH6-3 & 17:20:53.404 & -35:47:00.68 & 1.19 (0.09) & 0.20 & -9.212 (0.010) & 0.93 (0.09) & 0.16 & -9.311 (0.012) & +1.77 (0.29)\ UCHII-OH7-1 & 17:20:53.459 & -35:47:00.48 & 0.53 (0.01) & 0.090 & -10.200 (0.005) & 0.45 (0.01) & 0.077 & -10.463 (0.004) & +4.70 (0.11)\ UCHII-OH7-2 & 17:20:53.451 & -35:47:00.62 & 3.74 (0.05) & 0.64 & -9.507 (0.002) & 3.65 (0.04) & 0.62 & -9.768 (0.002) & +4.66 (0.05)\ UCHII-OH7-3 & 17:20:53.494 & -35:47:00.31 & 0.08 (0.01) & 0.014 & -8.281 (0.013) & 0.09 (0.01) & 0.015 & -8.071 (0.013) & -3.76 (0.33)\ UCHII-OH7-4 & 17:20:53.480 & -35:47:00.52 & 0.10 (0.01) & 0.017 & -7.227 (0.013) & 0.09 (0.01) & 0.015 & -7.416 (0.012) & +3.38 (0.31)\ UCHII-OH7-5 & 17:20:53.435 & -35:47:00.36 & 0.98 (0.01) & 0.17 & -5.117 (0.001) & 0.93 (0.01) & 0.16 & -5.175 (0.001) & +1.03 (0.03)\ CM2-OH1-1 & 17:20:53.392 & -35:46:55.61 & 4.94 (0.09) & 0.84 & -7.705 (0.005) & 4.63 (0.07) & 0.79 & -7.731 (0.004) & +0.47 (0.12)\ CM2-OH1-2 & 17:20:53.378 & -35:46:55.62 & 10.54 (0.20) & 1.8 & -7.722 (0.005) & 8.33 (0.17) & 1.4 & -7.758 (0.006) & +0.65 (0.14)\ CM2-OH1-3 & 17:20:53.395 & -35:46:55.63 & 0.76 (0.06) & 0.13 & -6.641 (0.014) & 0.71 (0.05) & 0.12 & -6.849 (0.011) & +3.71 (0.31)\ & & & & & OH 6.030 GHz\ UCHII-OH1-4 & 17:20:53.426 & -35:47:03.32 & 0.08 (0.01) & 0.13 & -6.994 (0.012) & 0.09 (0.01) & 0.015 & -6.556 (0.010) & -5.54 (0.20)\ UCHII-OH2-1b & 17:20:53.413 & -35:47:02.65 & 0.57 (0.01) & 0.097 & -8.780 (0.004) & 0.66 (0.01) & 0.11 & -8.407 (0.004) & -4.71 (0.07)\ UCHII-OH2-2a & 17:20:53.415 & -35:47:02.66 & 0.58 (0.01) & 0.099 & -8.774 (0.004) & 0.68 (0.01) & 0.12 & -8.406 (0.003) & -4.66 (0.07)\ UCHII-OH2-2b & & & 0.54 (0.01) & 0.092 & -8.181 (0.004) & 0.54 (0.01) & 0.092 & -7.811 (0.003) & -4.68 (0.06)\ UCHII-OH2-2c & & & 1.35 (0.01) & 0.23 & -7.750 (0.002) & 1.33 (0.01) & 0.23 & -7.327 (0.001) & -5.35 (0.03)\ UCHII-OH3-1 & 17:20:53.417 & -35:47:01.93 & 0.58 (0.02) & 0.099 & -8.726 (0.004) & 0.65 (0.01) & 0.11 & -8.432 (0.003) & -3.72 (0.06)\ UCHII-OH4-1b & 17:20:53.371 & -35:47:01.52 & 12.93 (0.02) & 2.2 & -10.699 (0.000) & 17.92 (0.04) & 3.0 & -10.304 (0.000) & -5.00 (0.01)\ UCHII-OH6-1 & 17:20:53.397 & -35:47:00.79 & 0.34 (0.00) & 0.058 & -10.153 (0.002) & 0.34 (0.00) & 0.058 & -10.491 (0.003) & +4.28 (0.05)\ CM2-OH1-1 & 17:20:53.376 & -35:46:55.64 & 2.98 (0.02) & 0.51 & -7.692 (0.001) & 2.67 (0.03) & 0.45 & -7.763 (0.003) & +0.90 (0.04)\ CM2-OH1-2 & 17:20:53.391 & -35:46:55.60 & 8.84 (0.07) & 1.50 & -7.723 (0.001) & 6.27 (0.11) & 1.1 & -7.799 (0.004) & +0.96 (0.05) The identification of Zeeman pairs can be challenging with an angular resolution of $0\farcs79\times 0\farcs25$ ($\sim 520$ au), i.e. significantly larger than the likely physical size of a single maser “spot”. We have nevertheless used the following procedure to kinematically and spatially separate (within the limitations of the data) the individual masing regions. For both transitions, we first identified maser “spots” of contiguous velocity components with fitted positions that were within 1/3 of the minor axis of the synthesized beam, independently for RCP and LCP. These independent RCP and LCP spot positions were then compared with each other and deemed to be a Zeeman pair if their intensity weighted centroid positions were within 1/3 of the minor axis of the synthesized beam. Then the RCP and LCP profiles for each Zeeman pair were independently fitted using Gaussian profiles. In a few cases, spatially coincident Zeeman pairs consisted of more than one velocity component and these were further separated into independent “spots”, though the spectral Gaussian fits per polarization were derived simultaneously. The derived parameters, including the fitted flux density and line center velocity ($V_{center}$) for the Zeeman pairs for both transitions are given in Table \[blos\]. The spots have been named according to their association, and then an additional number in increasing velocity order; spots with multiple velocity components (at the current resolution) are further appended with a letter of the alphabet in increasing velocity order. The line-of-sight magnetic field strength reported in Table \[blos\] was derived using $B_{los}=$($V_{center}(RCP) - V_{center}(LCP)$)$/Z$ where $Z = 0.056$ [km s$^{-1}$]{}mG$^{-1}$ for the 6.035 GHz transition and $0.079$ [km s$^{-1}$]{}mG$^{-1}$ for the 6.030 GHz transition [@Yen69]. @Caswell11 also report $B_{los}$ for these excited OH transitions from observations using three antennas of the LBA. Although these data have superior angular resolution ($0\farcs05\times 0\farcs02$), the absolute position accuracy was estimated to be as poor as $\pm0\farcs2$. In Figure \[methoh\] we have over-plotted the @Caswell11 Zeeman pairs using $\diamond$ symbols, after applying a -01 shift in right ascension in order to align them with the (more accurate) VLA Zeeman pairs from Table \[blos\]. From this figure, it is clear that toward the UCHII region many @Caswell11 Zeeman pairs are still present for both transitions, and with similar velocities. Comparison of the derived $B_{los}$ between the present work and @Caswell11 also yields excellent overall agreement. For example, for spots that are in common between the two epochs at 6.035 GHz in the associations UCHII-OH2, UCHII-OH3, UCHII-OH4, UCHII-OH5, differences in $B_{los}$ are less than $10\%$. Only toward UCHII-OH6 are there notable differences in $B_{los}$ but this region also showed the largest variation in $B_{los}$ in the @Caswell11 data. For the less prevalent 6.030 GHz transition, the agreement in the derived $B_{los}$ (for matching spots) with @Caswell11 is also within $10\%$. Regarding our VLA data, while most 6.030 GHz spots have derived $B_{los}$ that agree with the 6.035 GHz measurements, at a few locations the magnitude of the 6.030 GHz-derived field is notably larger (UCHII-OH2-1b, UCHII-OH2-2a, and especially CM2-OH1). This discrepancy may be a result of spectral/spatial blending for the stronger and more confused 6.035 GHz spots. Additionally, we confirm the finding of @Caswell11 of a reversal in the sign of $B_{los}$ between spots located toward the Southern portions (negative field: UCHII-OH1 to UCHII-OH5) and Northern portions (positive field: UCHII-OH6 and UCHII-OH7) of the [H[ii]{}]{} region. Interestingly, the most northerly of the spots detected in UCHII-OH7 reveals a second reversal back to negative value [this spot was not detected by @Caswell11]. Toward CM2-OH1, the $B_{los}$ is also positive, but somewhat weaker than for the majority of spots toward the [H[ii]{}]{} region (as discussed previously, this is the first time excited OH masers have been resolved toward the CM2 region of the protocluster). Excited-state OH 4.660 GHz transition ------------------------------------- We did not detect the 4.660 GHz OH transition to a $3\sigma$ limit of 10 [mJy beam$^{-1}$]{}. This transition is rarely seen in surveys [see @Cohen95 for an earlier single-dish upper limit of 0.17 Jy]. However, it was detected at a peak flux density of 2.2 Jy in the field of [[[NGC6334]{}]{}I]{} by the HartRAO single dish monitoring during the initial weeks of the flare, making it only the fifth object to have ever exhibited this transition [see @MacLeod17 and references therein]. The emission velocity of the transient 4.660 GHz maser was consistent with the flaring [CH$_3$OH]{} and 6 GHz OH masers, suggesting that they were co-located. But because no high-resolution observations exist, we lack definitive proof of this association. ![Similar to Fig. \[methanolzoom\]d except the RCP OH 6.035 GHz maser spots (o symbols) are overplotted in orange for comparison with the positions of the methanol masers ($+$ symbols). \[methoh\]](fig7.eps){width="0.99\linewidth"} Discussion {#disc} ========== Implications from maser pumping schemes --------------------------------------- The rapid onset of methanol masers on and around MM1 raises the question as to what conditions changed to support the maser inversion. The 6.7 GHz methanol transition is radiatively pumped by mid-infrared photons [ [@Sobolev97]]{}. The inversion requires [$T_{dust}$]{} above  120 K and can occur across a wide range of gas volume densities (up to about $n=10^{ 9}$ [cm$^{-3}$]{}) and kinetic temperatures (at least 25 to 250 K), as long as the [CH$_3$OH]{} abundance is greater than 10$^{-7}$ [@Cragg05]. Prior to the millimeter continuum outburst, when the dust and gas temperatures were presumably in better equilibrium, the gas temperature provides an estimate for the dust temperature. In the line survey of [[[NGC6334]{}]{}I]{} by @Zernickel12, the [ complex organic molecules with the most compact emission]{} exhibited model excitation temperatures of 100 K ([CH$_3$OH]{}) to 150 K (CH$_3$CN). Furthermore, the abundance of [CH$_3$OH]{} was modeled as $4.7 \times 10^{-6}$. Following the millimeter continuum outburst, the dust temperature of MM1 inferred directly from the 1.3 mm continuum brightness temperature exceeds 250 K toward the central components, and was above 150 K over a several square arcsecond region [@Brogan16]. Thus, in [[[NGC6334]{}]{}I]{}-MM1, the rapid heating of the dust grains by $\gtrapprox 100$ K by the recent accretion outburst can plausibly explain the appearance of 6.7 GHz masers in the gas in the vicinity of the powering source of the outburst, MM1B. Furthermore, the lack of masers toward the central dust peaks (MM1A, 1B and 1D) suggests that the density is too high toward these objects to support the inversion, because the dust temperature is surely high enough. ![The colorscale shows the peak intensity of the thermal methanol transition 11(2)-10(3) with an [$E_{Lower}\widehat{=}$]{}177 K. The same magenta contours, maser spot locations, and velocity color mapping from Fig. \[methanolimage\]b are also shown. Blue letters denote the 1.3 mm continuum peaks in MM1 defined by @Brogan16. \[thermal\]](fig8.eps){width="0.99\linewidth"} When compared to the ALMA 1 mm dust distribution (Fig. \[methanolzoom\]a), the appearance of 6.7 GHz masers to the north, west, and southwest of MM1 may seem perplexing, as the dust emission is relatively weak at all of these locations. In fact, over 80% of the masers reside outside of the 40% level of the continuum, and over 50% reside outside of the 5% level of the continuum. However, strong thermal molecular line emission is significantly more widespread than the bright dust emission. In Figure \[thermal\] we show the peak line intensity for a representative line from the ALMA 1 mm data (from whence the dust image also came). The transition shown is CH$_3$OH 11(2)-10(3) with an [$E_{Lower}\widehat{=}$]{}177 K (of order the expected gas kinetic temperature). From comparison of the distribution of 6.7 GHz masers versus the distribution of thermal CH$_3$OH (Fig. \[thermal\]), it is clear that the masers do lie in regions of high molecular column density. The critical density ($n_{crit}$) of the CH$_3$OH 11(2)-10(3) transition can be computed from the ratio of the Einstein A coefficient ($3.46 \times 10^5$ s$^{-1}$) and the collisional cross section ($6\times10^{-13}$ cm$^{2}$) where the values are taken from @Rabli10 via the LAMDA database [@Schoier10], yielding $n_{crit} = 6\times10^7$ cm$^{-3}$. [ While the density at which strong emission can begin to arise from millimeter transitions [the effective excitation density, @Shirley15] will be somewhat below $n_{crit}$, both values are]{} within the required range of $n$ for pumping the maser [ [@Cragg05]]{}. There is a notable lack of maser emission in the bright thermal gas between MM1 and MM3, suggesting that the gas density is too high here or the dust temperature is too low. {width="0.49\linewidth"} {width="0.49\linewidth"} {width="0.49\linewidth"} {width="0.49\linewidth"} As demonstrated in Figure \[methoh\], the 6.7 GHz and excited OH masers are coincident to within $<0\farcs1$ at a number of locations toward the MM3-UCHII region [see also @Caswell97], though it is notable that higher angular resolution studies generally find they are not exactly coincident at milliarcsecond scales, e.g. W3(OH) [@Etoka05]. Figure \[profiles\] shows example spectra comparing the velocities of the ionized gas (as traced by the H64$\alpha$ RRL), thermal gas (as traced by the CH$_3$OH 11(2,10)-10(3,7) transition), and 6.7 GHz and RCP for the 6.035 GHz maser gas toward several of the maser associations coincident with the MM3-UCHII region. Toward all of the maser associations (including those not shown), the velocities of both species of maser emission and of thermal [CH$_3$OH]{} emission are in excellent agreement, suggesting that these gas components are also at similar locations along the line-of-sight. Toward the maser associations UCHII-Met/OH2, UCHII-Met/OH5, and UCHII-Met/OH7, the neutral gas is notably blueshifted with respect to the ionized gas. Assuming a typical expanding UCHII region scenario, this suggests that the neutral gas emission originates on the front side of MM3-UCHII. This effect is most dramatic toward the northern UCHII-Met/OH7 association where the neutral gas is blueshifted by $\gtrsim 10$ [km s$^{-1}$]{} compared to the ionized gas, such that there is little overlap. It is notable that it is the ionized gas velocity that has changed compared to the other regions, rather than the maser velocities. The significant reddening of the ionized gas velocity toward the northern part of the MM3-UCHII region was also observed by @dePree95 from VLA observations of the H76$\alpha$ line. The brightest 6.7 GHz and OH 6.035 GHz masers toward the UCHII region are found along its extreme western boundary (UCHII-Met/OH4), coincident with a $\sim 0\farcs25$ diameter blister-like bulge in the 5 cm continuum; unfortunately the H64$\alpha$ line was not detected at this location, so the relationship between the masers and ionized gas cannot be determined. The RRL was not detected toward the CM2 region [ (nor toward MM1)]{}, though as for the UCHII region, the positions and velocities of the [ CM2]{} 6.7 GHz and OH 6.035 GHz masers are similar, and their peak velocities match the thermal methanol that is present there. Excited state OH masers are efficiently pumped for a lower range of kinetic (gas) temperatures than the 6.7 GHz [CH$_3$OH]{} transition: 25 - 70 K [see for example @Cragg02]. Therefore, the excellent kinematic agreement between the two maser species suggests that the molecular gas toward the MM3-UCHII region is cooler than toward MM1 (where excited OH masers are not detected). The maser models of @Cragg02 [@Cragg05] also suggest that the excited 6.030 and 6.035 GHz OH transitions are only effectively pumped toward the upper end of the density range predicted for 6.7 GHz [CH$_3$OH]{} masers ($10^{6.5}$ to $10^{8.3}$ [cm$^{-3}$]{} compared to $10^{4}$ to $10^{8.3}$ [cm$^{-3}$]{}). Interestingly, the @Cragg02 models also suggest that the ratio of 6.035/6.030 GHz OH maser brightness temperature is minimized, and indeed becomes nearly equal, for the highest densities, which may explain the anomalous ratio observed toward CM2 if the gas density is higher there. Future multi-transition analysis of the thermal molecular gas throughout the protocluster will help to verify that the observed physical conditions meet the predictions of maser pump models. [ The range of the magnetic field strengths we measure in excited OH (0.5-5 mG) is very similar to that found in other high-mass star forming regions observed interferometrically in these transitions [e.g., W51 Main, W3 (OH), and ON1, @Etoka12; @Fish07; @Green07]. These values are about five times higher than those measured in single-dish CN observations of massive star forming regions [@Falgarone08], however those observations sample gas at larger scales and hence lower average density (1-20 $\times 10^5$ [cm$^{-3}$]{}). Because magnetic field strength is expected to scale with the square root of density [@Crutcher99], the excited OH masers apparently arise from gas at densities that are $\sim$25 times higher, i.e. 0.2-5 $\times 10^7$ [cm$^{-3}$]{}. Radiative transfer modeling of these transitions predict that OH densities of $\sim$10 [cm$^{-3}$]{} are required to pump these masers, which are consistent with H$_2$ densities of $5 \times 10^7$ [cm$^{-3}$]{} using the nominal OH abundance of $2 \times 10^{-7}$ [@Etoka12]. Thus, the density implied by the measured excited OH field strengths is in good agreement with the value required for the thermal methanol emission and the pumping models for excited OH maser emission.]{} Spatial and kinematic structures -------------------------------- The sudden appearance of strong 6.7 GHz methanol masers on and around MM1 demonstrates an important new characteristic of the Class II maser phenomenon, which has presented an ongoing enigma to observers. While Class I methanol masers [@Leurini16] trace ambient-velocity shocked gas at the interfaces of outflow structures and can be located far from the driving source [@Cyganowski09; @Rodriguez17], Class II methanol masers are always found in close proximity to other tracers of massive protostars such as submillimeter continuum [@Urquhart13], mid-infrared continuum [@Bartkiewicz14], or hot core molecular gas [@Chibueze17]. An early imaging survey with the ATCA at $\sim1''$ resolution found linear distributions of fitted spots across (up to) a few synthesized beams, which were interpreted as a indicator of either edge-on protoplanetary disks or collimated jets [@Norris93]. Interestingly, two of the 10 so-called “linear” sources were the two concentrations of emission in NGC6334F (the previously-known masers associated with MM2 and MM3). Although only half of the cases exhibited a corresponding velocity gradient, the edge-on disk interpretation became preferred in the literature [@Norris98]. However, the disk hypothesis was subsequently tested by searching for outflow lobes perpendicular to the linear structures by imaging the shock-tracing near-infrared H$_2$ line in 28 such maser sources [@DeBuizer03]. Only two sources showed outflowing gas in the perpendicular direction, while in contrast 80% of the cases (with radiatively excited H$_2$) showed emission parallel to the maser axis, instead suggesting an association with the outflow. Further evidence for an association between 6.7 GHz masers and outflows has come from other studies of individual sources [e.g. NGC7538 IRS1, @DeBuizer05]. Nevertheless, the disk interpretation continues to be invoked to explain recent observations of individual massive protostars, such as Cepheus A HW2 [@Sanna17] and G353.273+0.641 [@Motogi17]; however in these cases, the proposed disks have an inclined or face-on geometry. In one case, proper motion studies suggest masers reside in both the disk and outflow [IRAS 20126+4104, @Moscadelli11]. Recently, a VLBI imaging survey of 31 methanol masers identified a new class of structure: a ring-like distribution of spots in 29% of the sample [@Bartkiewicz09], with the most striking example being G23.657-0.127 [@Bartkiewicz05]. With typical radii of $\lesssim0\farcs1$, these rings are smaller than the scales probed by the VLA or ATCA. A follow-up study in the near-IR and mid-IR again found no evidence to support the idea that these rings trace protostellar disks [@DeBuizer12]. Proper motion studies show that the ring of masers in G23.657-0.127 is expanding [@Szymczak17] and while the interpretation is still unclear, it may be the result of a disk wind [@Bartkiewicz17]. [ Before interpreting the location of the flaring masers, we summarize the current prior knowledge of star formation in MM1. The original ALMA observations resolved MM1 into multiple dust continuum components, which we modeled as seven two-dimensional Gaussian sources at 1.3 mm [A-F, @Brogan16]. Since the molecular hot core emission encompasses all these components, each one should be considered as a candidate protostar. However, the water maser emission from MM1 (VLA epoch 2011.7) was associated with only the two brightest components, B and D. The small upper limit to the size of B at 7 mm ($<$018 $\sim230$ au) led us to model its spectral energy distribution (SED) as a dense hypercompact HII (HCHII) region, while the SED and larger size of D were more consistent with emission from a jet. A single component fit to the elongated 5 cm emission feature (epoch 2011.4) peaks near D, but extends down to B and up to an area between F and G. We interpreted this emission as a jet, but its origin and whether it is symmetric with respect to its driving source (D) or more asymmetric (B) were unclear. Besides B and D, the only other source detected at 7 mm is F, whose SED is consistent with dust emission alone, suggesting an earlier stage protostar. The other 5 cm source, CM2, was enigmatic as its emission is non-thermal and it exhibits the brightest water maser emission yet no compact dust emission.]{} The distribution of the new flaring masers in MM1 exhibits a variety of shapes. There are a few linear arrangements of spots, perhaps the most striking of which is the “V”-shaped structure of masers between MM1F and 1G (Fig. \[methanolzoom\]b). Because this structure lies at the northern end of the 5 cm continuum emission, [ and along an axis parallel to this emission that intersects MM1B]{}, it may indicate an association between the masers and a jet originating from [ the HCHII region]{} MM1B. [ This axis also passes close to the non-thermal radio source CM2, suggesting that it traces a shock against the ambient medium, as proposed for the non-thermal component in the HH 80-81 jet [@RodriguezKamenetzky]. How this jet relates to the large scale NE/SW bipolar outflow remains uncertain from these data, but we note a similarity to the G5.89-0.39 protocluster, which exhibits a young compact outflow and an older larger scale outflow at very different position angles [@Hunter08; @Puga06].]{} While it is [ also]{} unclear if the maser spots near MM1F are associated with an [ early stage]{} protostar at that location, the thread of spots that continues northward (CM2-Met2) curving toward the non-thermal continuum source CM2 (Fig. \[methanolzoom\]a) supports an association with the jet. The spots north of MM1G partially form a similar structure west of the jet, which suggests that both sets of masers follow the walls of an outflow cavity. Proper motion studies with VLBI would be helpful to test this hypothesis [ (Chibueze et al., in preparation).]{} Shorter linear collections of maser spots also appear in MM1-MetC and MM1-Met3. The dust emission from MM1C indicates there may be a protostar associated with MM1-MetC, however the velocity range of these spots is rather small (2.7 [km s$^{-1}$]{}) and lacks a systematic gradient. MM1-Met3 does show an east/west velocity gradient, but the velocity range is even smaller (1.3 [km s$^{-1}$]{}) and there is no evidence for a protostar as a point source in the dust emission. Linear features with such narrow velocity ranges are similar to those seen in LBA images of five other HMYSOs by @Dodson04, who conclude that they trace planar shocks propagating perpendicular to the line of sight rather than disks. Similar to MM1-Met3, the copious maser spots of MM1-Met2 have no direct association with a protostar, since there are no millimeter point sources in that area to very sensitive levels [@Brogan16]. Instead, these maser spots look more like they arise from a collection of filaments, similar to those seen in W3OH where masers trace filaments extending up to 3100 au [@HarveySmith06]. It would appear that the presence of maser filaments can occur over a broad range in evolutionary state of massive protostars since the masers in MM3-UCHII (Met1 through Met7) appear to delineate a filament nearly 5000 au in length (Fig. \[methoh\]). In the other HMYSO hosting methanol masers in this protocluster, MM2, we see a similar dichotomy as in MM1, with the spots of MM2-Met1 appearing close to the protostar(s) traced by the continuum sources MM2A and 2B, while those of MM2-Met2 appear further afield ($>1000$ au). So the primary conclusion we draw from this outburst is that although some 6.7 GHz masers do reside close to massive protostars (within 500 au), significant amounts of maser emission can occur in the gas further away as long as the radiative pumping from nearby protostars is sufficient in those locations. The presence of outflow cavities driven by a central protostar likely creates propagation paths for infrared photons that increase the range of the required radiative pumping to these larger distances. Indeed, the brightest masers can lie along these paths, as evidenced by MM1-Met1, which lies $\approx$1000 au from the outburst source MM1B. This result is strikingly similar to the other recent flaring maser source S255 NIRS3, in which the brightest maser emission also lies 500-1000 au from the flaring protostar [@Moscadelli17]. Origin of the 1999 methanol maser flare --------------------------------------- @Goedhart04 presented single-dish monitoring data of the 6.7 GHz line from 28 February 1999 to 27 March 2003 and noted a large flare in the velocity component at $-5.88$ [km s$^{-1}$]{}. Re-analysis of these data [@MacLeod17] indicates that the flare peaked on 19 November 1999 (epoch 1999.88) at a velocity of $-5.99$ [km s$^{-1}$]{} with a peak flux density 230 times higher than the pre-burst value of $1.6\pm0.4$ Jy. While there are no published interferometric maps from that epoch, we can examine the location of the emission at this velocity in our recent epoch 2016.9 VLA data. In the $-5.90$ [km s$^{-1}$]{} channel, we find a total flux density of 6.41 Jy, only 10% of which originates from any of the UCHII associations (only UCHII-Met1). The majority (5.41 Jy) originates from MM1, and the rest (0.34 Jy) from CM2-Met2. Since the rest of the emission (0.70 Jy, from UCHII-Met1 and MM2-Met2) is consistent with the quiescent single-dish value found after the 1999 flare ($0.55\pm0.20$ Jy), it is quite plausible that this flare could have also originated entirely from MM1 and CM2. This result supports the hypothesis of @MacLeod17 that the 1999 and 2015 events arise from the same physical location and are due to a common repeating mechanism such as the decaying orbit of a binary system embedded in dense gas [@Stahler10]. Continued long-term monitoring of the maser emission is essential to test this scenario. It is particularly important to measure how long the current maser flare lasts compared to the more limited 1999 event. If this flare persists for many years, it increases the likelihood that a significant population of the masers found in surveys arise from protostars that have recently undergone a large accretion outburst. Conclusions =========== The recent extraordinary outburst in the massive protostellar cluster NGC6334I provides an unprecedented window into the central environment of a massive star-forming region. We have obtained high resolution ($\sim$500 au) VLA images of multiple maser transitions which reveal the first appearance of Class II methanol maser emission toward the MM1 protostellar system in over 30 years of past observations. The 6.7 GHz masers are distributed toward particular parts of MM1 (including MM1C, F and G), on its northern and western peripheries, and toward the non-thermal radio source CM2. The masers are strikingly absent from the brightest millimeter components (MM1A, B and D). We find that current models of maser pumping can explain the general location of the masers in the context of the recent luminous outburst from MM1B. The spatio-kinematic structures traced by the 6.7 GHz masers are varied but support the idea that while some masers do reside close to massive protostars (within 500 au) traced by compact dust emission, significant maser action can also occur in more extended areas associated with strong thermal gas emission. Also, the presence of jets and outflow cavities driven by a central protostar increases the range of infrared pumping photons, allowing strong masers to appear further away ($>1000$ au). Our simultaneous observations of two of the 6 GHz excited state OH transitions reveal no emission toward MM1, but we do detect a strong new maser at $-7.7$ [km s$^{-1}$]{} arising from the non-thermal radio source CM2. Not previously detected by the LBA, it was apparently excited by the recent millimeter outburst. By analyzing the RCP and LCP data cubes, we identify several Zeeman pairs to measure the line of sight magnetic field in the UCHII region and CM2. We confirm the field strengths and the reversal in field direction across the UCHII region identified by @Caswell11. The magnetic field toward CM2 is significantly weaker than toward the majority of positions across the UCHII region, and the flux ratio between the 6.035 and 6.030 GHz lines is anomalous, with the lines being comparable in strength toward CM2. OH maser pumping models suggest that the gas density is significantly higher toward CM2, consistent with compression of gas by the propagation of the jet from MM1B. Future high resolution observations of the [CH$_3$OH]{} and OH masers will be essential to obtain proper motions of the new features in order to determine which ones may be tracing structures associated with an outflow versus those that may be bound to a protostar. Comparison with thermal gas outflow tracers imaged by ALMA will then be possible. Continued single dish monitoring is also critical to measure the lifetime of the current maser flare and to search for future events in order to test the hypothesis that a periodic phenomenon such as an eccentric protostellar binary orbit may be responsible for repeating accretion outbursts. [ We thank the anonymous referee for a thorough review which has improved the manuscript.]{} The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under agreement by the Associated Universities, Inc. This paper makes use of the following ALMA data: ADS/JAO.ALMA\#2015.A.00022.T. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada) and NSC and ASIAA (Taiwan) and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ. This research made use of NASA’s Astrophysics Data System Bibliographic Services, the SIMBAD database operated at CDS, Strasbourg, France, [ Astropy, a community-developed core Python package for Astronomy [@astropy], and APLpy, an open-source plotting package for Python hosted at http://aplpy.github.com. T. Hirota is supported by the MEXT/JSPS KAKENHI grant No. 17K05398.]{} C.J. Cyganowski acknowledges support from the STFC (grant number ST/M001296/1). Facilities: , . Abraham, Z., Cohen, N. L., Opher, R., Raffaelli, J. C., & Zisk, S. H. 1981, , 100, L10 Avison, A., Quinn, L. J., Fuller, G. A., et al. 2016, , 461, 136 Bartkiewicz, A., et al. 2017, IAU Symp. 336 Bartkiewicz, A., Szymczak, M., & van Langevelde, H. J. 2014, , 564, A110 Bartkiewicz, A., Szymczak, M., van Langevelde, H. J., Richards, A. M. S., & Pihlstr[ö]{}m, Y. M. 2009, , 502, 155 Bartkiewicz, A., Szymczak, M., & van Langevelde, H. J. 2005, , 442, L61 Beuther, H., Walsh, A. J., Thorwirth, S., et al. 2007, , 466, 989 Brogan, C. L., Hunter, T. R., Cyganowski, C. J., et al. 2016, , 832, 187 Caratti o Garatti, A., Stecklum, B., Garcia Lopez, R., et al. 2017, Nature Physics, 13, 276 Caswell, J. L., & Vaile, R. A. 1995, , 273, 328 Caswell, J. L. 1997, , 289, 203 Caswell, J. L., Kramer, B. H., & Reynolds, J. E. 2011, , 414, 1914 Caswell, J. L. 2003, , 341, 551 Chibueze, J. O., Csengeri, T., Tatematsu, K., et al. 2017, , 836, 59 Chibueze, J. O., Omodaka, T., Handa, T., et al. 2014, , 784, 114 Cohen, R. J., Masheder, M. R. W., & Caswell, J. L. 1995, , 274, 808 Contreras Pe[ñ]{}a, C., Lucas, P. W., Minniti, D., et al. 2017, , 465, 3011 Cragg, D. M., Sobolev, A. M., & Godfrey, P. D. 2005, , 360, 533 Cragg, D. M., Sobolev, A. M., & Godfrey, P. D. 2002, , 331, 521 Cyganowski, C. J., Brogan, C. L., Hunter, T. R., & Churchwell, E. 2009, , 702, 1615 De Buizer, J. M., Bartkiewicz, A., & Szymczak, M. 2012, , 754, 149 De Buizer, J. M., & Minier, V. 2005, , 628, L151 De Buizer, J. M. 2003, , 341, 277 De Buizer, J. M., Radomski, J. T., Pi[ñ]{}a, R. K., & Telesco, C. M. 2002, , 580, 305 de Pree, C. G., Rodriguez, L. F., Dickel, H. R., & Goss, W. M. 1995, , 447, 220 Dodson, R., & Moriarty, C. D. 2012, , 421, 2395 Dodson, R., Ojha, R., & Ellingsen, S. P. 2004, , 351, 779 Elitzur, M. 1992, , 30, 75 Ellingsen, S. 1996, Ph.D. Thesis, University of Tasmania Ellingsen, S. P., Norris, R. P., Diamond, P. J., et al. 1996, Proc. of Third Asia-Pacific Telescope Workshop, arXiv:astro-ph/9604024 Etoka, S., Cohen, R. J., & Gray, M. D. 2005, , 360, 1162 Evans, N. J., II, Dunham, M. M., J[ø]{}rgensen, J. K., et al. 2009, , 181, 321-350 Fish, V. L., & Sjouwerman, L. O. 2007, , 668, 331 Fujisawa, K., Yonekura, Y., Sugiyama, K., et al. 2015, The Astronomer’s Telegram, 8286 Fujisawa, K., Sugiyama, K., Aoki, N., et al. 2012, , 64, 17 Gardner, F. F., Ribes, J. C., & Goss, W. M. 1970, , 7, 51 Goedhart, S., Gaylard, M. J., & van der Walt, D. J. 2004, , 355, 553 Gray, M. D., Baudry, A., Richards, A. M. S., et al. 2016, , 456, 374 Green, J. A., Caswell, J. L., & McClure-Griffiths, N. M. 2015, , 451, 74 Harvey-Smith, L., & Cohen, R. J. 2006, , 371, 1550 Herbig, G. H. 1977, , 217, 693 Herbig, G. H. 1989, European Southern Observatory Conference and Workshop Proceedings, 33, 233 Hirota, T., Tsuboi, M., Kurono, Y., et al. 2014, , 66, 106 Honma, M., Yoon, K. C., Bushimata, T., et al. 2004, , 56, L15 Hunter, T. R., Brogan, C. L., MacLeod, G., et al. 2017, , 837, L29 , T. R., [Brogan]{}, C. L., [Megeath]{}, et al., 2006, , 649, 888 Kenyon, S. J., Hartmann, L. W., Strom, K. M., & Strom, S. E. 1990, , 99, 869 Knowles, S. H., Caswell, J. L., & Goss, W. M. 1976, , 175, 537 Knowles, S. H., Johnson, K. J., Moran, J. M., & Ball, J. A. 1973, , 180, L117 Krishnan, V., Ellingsen, S. P., Voronkov, M. A., & Breen, S. L. 2013, , 433, 3346 Kumar, M. S. N., Contreras Pe[ñ]{}a, C., Lucas, P. W., & Thompson, M. A. 2016, , 833, 24 Leurini, S., Menten, K. M., & Walmsley, C. M. 2016, , 592, A31 Liljestr[ö]{}m, T., & Gwinn, C. R. 2000, , 534, 781 Lucas, P. W., Smith, L. C., Contreras Pena, C., et al. 2017, , 472, 2990 Lucas, P. W., Hoare, M. G., Longmore, A., et al. 2008, , 391, 136 Macleod, G. et al., [ 2018]{}, submitted to MNRAS Mairs, S., Lane, J., Johnstone, D., et al. 2017, , 843, 55 McGuire, B. A., et al. 2017, , 851, L46 Meyer, D. M.-A., Vorobyov, E. I., Kuiper, R., & Kley, W. 2017, , 464, L90 Minier, V., Booth, R. S., & Conway, J. E. 2002, , 383, 614 Minniti, D., Lucas, P. W., Emerson, J. P., et al. 2010, New Astronomy, 15, 433 Moscadelli, L., Sanna, A., Goddi, C., et al. 2017, , 600, L8 Moscadelli, L., Cesaroni, R., Rioja, M. J., Dodson, R., & Reid, M. J. 2011, , 526, A66 Motogi, K., Hirota, T., Sorai, K., et al. 2017, , 849, 23 Norris, R. P., Caswell, J. L., Wellington, K. J., McCutcheon, W. H., & Reynolds, J. E. 1988, , 335, 149 Norris, R. P., Whiteoak, J. B., Caswell, J. L., Wieringa, M. H., & Gough, R. G. 1993, , 412, 222 Norris, R. P., Byleveld, S. E., Diamond, P. J., et al. 1998, , 508, 275 Offner, S. S. R., & McKee, C. F. 2011, , 736, 53 Omodaka, T., Maeda, T., Miyoshi, M., et al. 1999, , 51, 333 Persi, P., Tapia, M., Felli, M., Lagage, P. O., & Ferrari-Toniolo, M. 1998, , 336, 1024 Rabli, D., & Flower, D. R. 2010, , 406, 95 Reid, M. J., Menten, K. M., Brunthaler, A., et al. 2014, , 783, 130 Reid, M. J., & Moran, J. M. 1981, , 19, 231 Rodr[í]{}guez-Garza, C. B., Kurtz, S. E., G[ó]{}mez-Ruiz, A. I., et al. 2017, , 233, 4 Rodriguez, L. F., Canto, J., & Moran, J. M. 1982, , 255, 103 Safron, E. J., Fischer, W. J., Megeath, S. T., et al. 2015, , 800, L5 Sanna, A., Moscadelli, L., Surcis, G., et al. 2017, , 603, A94 Sault, R.J. & Sowinski, K. 2013, EVLA Memo 169 Sch[ö]{}ier, F., van der Tak, F., van Dishoeck, E., & Black, J. 2010, Astrophysics Source Code Library, ascl:1010.077 Shirley, Y. L. 2015, , 127, 299 Sobolev, A. M., Cragg, D. M., & Godfrey, P. D. 1997, , 324, 211 Stahler, S. W. 2010, , 402, 1758 Stutz, A. M., Tobin, J. J., Stanke, T., et al. 2013, , 767, 36 Szymczak, M. et al., 2017, MNRAS, submitted Tapia, M., Roth, M., & Persi, P. 2015, , 446, 4088 Tolmachev, A. 2011, The Astronomer’s Telegram, 3177 Urquhart, J. S., Moore, T. J. T., Schuller, F., et al. 2013, , 431, 1752 Volvach, A. E., Volvach, L. N., MacLeod, G., et al. 2017b, The Astronomer’s Telegram, 10853 Walsh, A. J., Burton, M. G., Hyland, A. R., & Robinson, G. 1998, , 301, 640 Wu, Y. W., Sato, M., Reid, M. J., et al. 2014, , 566, A17 Yen, J. L., Zuckerman, B., Palmer, P., & Penfield, H. 1969, , 156, L27 Yoo, H., Lee, J.-E., Mairs, S., et al. 2017, , 849, 69 Zernickel, A., Schilke, P., Schmiedeke, A., et al. 2012, , 546, A87 Zuckerman, B., Yen, J. L., Gottlieb, C. A., & Palmer, P. 1972, , 177, 59 [^1]: See <https://science.nrao.edu/facilities/vla/data-processing/pipeline/scripted-pipeline> for more information.

--- author: - | ***H. S. Abdel-Aziz*** and ***M. Khalifa Saad***[^1]\ [*Dept. of Math., Faculty of Science, Sohag Univ., 82524 Sohag, Egypt*]{} title: '[**Smarandache curves of some special curves in the Galilean 3-space**]{}' --- **Abstract.** In the present paper, we consider a position vector of an arbitrary curve in the three-dimensional Galilean space $G_{3}$. Furthermore, we give some conditions on the curvatures of this arbitrary curve to study special curves and their Smarandache curves. Finally, in the light of this study, some related examples of these curves are provided and plotted. **Keywords.** Galilean space, Smarandache curves, Frenet frame. **MSC(2010):** 51B20, 53A35. Introduction ============ Discovering Galilean space-time is probably one of the major achievements of non relativistic physics. Nowadays Galilean space is becoming increasingly popular as evidenced from the connection of the fundamental concepts such as velocity, momentum, kinetic energy, etc. and principles as indicated in [@IY]. In recent years, researchers have begun to investigate curves and surfaces in the Galilean space and thereafter pseudo-Galilean space.\ In classical curve theory, the geometry of a curve in three-dimensions is essentially characterized by two scalar functions, curvature $\kappa$ and torsion $\tau$ as well as its Frenet vectors. A regular curve in Euclidean space whose position vector is composed by Frenet frame vectors on another regular curve is called a Smarandache curve. Smarandache curves have been investigated by some differential geometers [@AT; @TY]. M. Turgut and S. Yilmaz defined a special case of such curves and call it Smarandache $TB_{2}$ curves in the space $E_{1}^{4}$ [@TY]. They studied special Smarandache curves which are defined by the tangent and second binormal vector fields. Additionally, they computed formulas of this kind curves. In [@AT], the author introduced some special Smarandache curves in the Euclidean space. He studied Frenet-Serret invariants of a special case.\ In the field of computer aided design and computer graphics, helices can be used for the tool path description, the simulation of kinematic motion or the design of highways, etc. [@XY]. The main feature of general helix or slope line is that the tangent makes a constant angle with a fixed direction in every point which is called the axis of the general helix. A classical result stated by Lancret in $1802$ and first proved by de Saint Venant in 1845 says that: A necessary and sufficient condition that a curve be a general helix is that the ratio ($\kappa / \tau$) is constant along the curve, where $\kappa$ and $\tau$ denote the curvature and the torsion, respectively. Also, the helix is also known as circular helix or W-curve which is a special case of the general helix [@DS].\ Salkowski (resp. Anti-Salkowski) curves in Euclidean space are generally known as family of curves with constant curvature (resp. torsion) but nonconstant torsion (resp. curvature) with an explicit parametrization.They were defined in an earlier paper [@ES].\ In this paper, we compute Smarandache curves for a position vector of an arbitrary curve and some of its special curves. Besides, according to Frenet frame [$\textbf{T}$, $\textbf{N}$, $\textbf{B}$]{} of the considered curves in the Galilean space $G_{3}$, the meant Smarandache curves $\textbf{TN}$, $\textbf{TB}$ and $\textbf{TNB}$ are obtained. We hope these results will be helpful to mathematicians who are specialized on mathematical modeling. Preliminaries ============= Let us recall the basic facts about the three-dimensional Galilean geometry $G_{3}$. The geometry of the Galilean space has been firstly explained in [@OR]. The curves and some special surfaces in $G_{3}$ are considered in [@DM]. The Galilean geometry is a real Cayley-Klein geometry with projective signature $(0,0,+,+)$ according to [@EM]. The absolute of the Galilean geometry is an ordered triple $({w,f,I}$ $)$ where $w$ is the ideal (absolute) plane $(x_{0}=0)$, $f$ is a line in $w$ $(x_{0}=x_{1}=0)$ and $I$ is elliptic $((0:0:x_{2}:x_{3})\longrightarrow (0:0:x_{3}:-x_{2}))$ involution of the points of $f$ . In the Galilean space there are just two types of vectors, non-isotropic $\mathbf{x}(x,y,z)$ (for which holds $x\neq 0 $). Otherwise, it is called isotropic. We do not distinguish classes of vectors among isotropic vectors in $G_{3}$. A plane of the form $x=const$. in the Galilean space is called Euclidean, since its induced geometry is Euclidean. Otherwise it is called isotropic plane. In affine coordinates, the Galilean inner product between two vectors $P=(p_{1},p_{2},p_{3})$ and $Q=(q_{1},q_{2},q_{3})$ is defined by [@PK]: $$\langle P,Q\rangle _{G_{3}}=\left\{ \begin{array}{c} p_{1}q_{1}\ \ \ \ \ \ \ \ \ \ \ \ \text{if}\ p_{1}\neq 0\vee q_{1}\neq 0, \\ p_{2}q_{2}+p_{3}q_{3}\ \ \text{if\ }p_{1}=0\wedge q_{1}=0.\end{array}\right.$$And the cross product in the sense of Galilean space is given by: $$\left( P\times Q\right) _{G_{3}}=\left\{ \begin{array}{c} \left\vert \begin{array}{ccc} 0 & e_{2} & e_{3} \\ p_{1} & p_{2} & p_{3} \\ q_{1} & q_{2} & q_{3}\end{array}\right\vert \ ;\ \ \ \ \ \text{if}\ p_{1}\neq 0\vee q_{1}\neq 0, \\ \\ \left\vert \begin{array}{ccc} e_{1} & e_{2} & e_{3} \\ p_{1} & p_{2} & p_{3} \\ q_{1} & q_{2} & q_{3}\end{array}\right\vert \ ;\ \ \ \ \ \text{if\ }p_{1}=0\wedge q_{1}=0. \end{array} \right.$$ A curve $\eta (t)=(x(t),y(t),z(t))$ is admissible in $G_{3}$ if it has no inflection points $(\dot{\eta}(t)\times \ddot{\eta}(t)\neq 0)$ and no isotropic tangents $(\dot{x}(t)\neq 0)$. An admissible curve in $G_{3}$ is an analogue of a regular curve in Euclidean space. For an admissible curve $\eta $ $:I$  $\rightarrow $ $G_{3},$ $I\subset R $ parameterized by the arc length $s$ with differential form $dt=ds$, given by $$\eta (s)=(s,y(s),z(s)).$$ The curvature $\kappa (s)$ and torsion $\tau (s)$ of $\eta $ are defined by $$\begin{aligned} \kappa (s) &=&\left\Vert \eta ^{^{\prime \prime }}(s)\right\Vert =\sqrt{y^{^{\prime \prime }}(s)^{2}+z^{^{\prime \prime }}(s)^{2}}, \notag \\ \tau (s) &=&\frac{det(\eta^{\prime}(s),\eta^{\prime \prime}(s),\eta^{\prime \prime \prime}(s))}{\kappa ^{2}(s) }.\end{aligned}$$ Note that an admissible curve has non-zero curvature. The associated trihedron is given by $$\begin{aligned} \mathbf{T}(s) &=&\eta ^{^{\prime }}(s)=(1,y^{^{\prime }}(s),z^{^{\prime }}(s)), \notag \\ \mathbf{N}(s) &=&\frac{\eta ^{^{\prime \prime }}(s)}{\kappa (s)}=\frac{(0,y^{^{\prime \prime }}(s),z^{^{\prime \prime }}(s))}{\kappa (s)}, \notag \\ \mathbf{B}(s) &=&\frac{(0,-z^{^{\prime \prime }}(s),y^{^{\prime \prime }}(s))}{\kappa (s)}.\end{aligned}$$ For derivatives of the tangent $\mathbf{T}$, normal $\mathbf{N}$ and binormal $\mathbf{B}$ vector field, the following Frenet formulas in the Galilean space hold [@OR] $$\left[ \begin{array}{c} \mathbf{T} \\ \mathbf{N} \\ \mathbf{B}\end{array}\right] ^{{\large {\prime }}}=\left[ \begin{array}{ccc} 0 & \kappa & 0 \\ 0 & 0 & \tau \\ 0 & -\tau & 0\end{array}\right] \left[ \begin{array}{c} \mathbf{T} \\ \mathbf{N} \\ \mathbf{B}\end{array}\right] .$$From $(2.5)$ and $(2.6)$, we derive an important relation $$\eta ^{\prime \prime \prime }(s)=\kappa ^{\prime }(s)\mathbf{N}(s)+\kappa (s)\tau (s)\mathbf{B}(s).$$ In [@TY] authors introduced: A regular curve in Minkowski space-time, whose position vector is composed by Frenet frame vectors on another regular curve, is called a Smarandache curve. In the light of the above definition, we adapt it to admissible curves in the Galilean space as follows: let $\eta =\eta (s)$ be an admissible curve in $G_{3}$ and $\{\mathbf{T}, \mathbf{N},\mathbf{B}\}$ be its moving Frenet frame. Smarandache $\mathbf{TN} ,\mathbf{TB}$ and $\mathbf{TNB}$ curves are respectively, defined by\ $$\begin{aligned} \eta _{\mathbf{TN}} &=&\frac{\mathbf{T}+\mathbf{N}}{\left\Vert \mathbf{T}+\mathbf{N}\right\Vert },\ \ \ \notag \\ \eta _{\mathbf{TB}} &=&\frac{\mathbf{T}+\mathbf{B}}{\left\Vert \mathbf{T}+\mathbf{B}\right\Vert }, \notag \\ \eta _{\mathbf{TNB}} &=&\frac{\mathbf{T}+\mathbf{N+B}}{\left\Vert \mathbf{T}+\mathbf{N+B}\right\Vert }.\end{aligned}$$ Smarandache curves of an arbitrary curve in $G_{3}$ =================================================== In this section, we consider the position vector of an arbitrary curve with curvature $\kappa (s)$ and torsion $\tau (s)$ in the Galilean space $G_{3}$ which introduced by [@AT] as follows $$\textbf{r}(s)=\left( s,\int \left( \int \kappa (s)\cos \left( \int \tau (s)ds\right) ds\right) ds, \int \left( \int \kappa (s)\sin \left( \int \tau (s)ds\right) ds\right) ds \right).$$ The derivatives of this curve are respectively, given by $$\textbf{r}^{\prime }(s)=\left( 1,\int \kappa (s)\,\cos \left( \int \tau (s)\,ds\right) ds, \int \kappa (s)\,\sin \left( \int \tau (s)\,ds\right) \,ds \right),$$ $$\textbf{r}^{\prime \prime }(s)=\left( 0,\kappa (s)\cos \left( \int \tau (s)\,ds\right) ,\kappa (s)\,\sin \left( \int \tau (s)\,ds\right) \right),$$ $$\textbf{r}^{\prime \prime \prime }(s)=\left( \begin{array}{c} 0,\kappa ^{\prime }(s)\cos \left( \int \tau (s)\,ds\right) -\kappa (s)\tau (s)\sin \left( \int \tau (s)\,ds\right) , \\ \kappa ^{\prime }(s)\sin \left( \int \tau (s)\,ds\right) +\kappa (s)\tau (s)\cos \left( \int \tau (s)\,ds\right)\end{array} \right).$$ The frame vector fields of $\textbf{r}$ are as follows\ $$\textbf{T}_{\textbf{r}}=\left( 1,\int \kappa (s)\cos \left( \int \tau (s)\,ds\right) ds, \int \kappa (s)\sin \left( \int \tau (s)\,ds\right) \,ds \right),$$ $$\textbf{N}_{\textbf{r}}=\left( 0,\cos \left( \int \tau (s)\,ds\right) ,\sin \left( \int \tau (s)\,ds\right) \right),$$ $$\textbf{B}_{\textbf{r}}=\left( 0,-\sin \left( \int \tau (s)\,ds\right) ,\cos \left( \int \tau (s)\,ds\right) \right).$$ By Definition (2.2), the $\textbf{TN}$, $\textbf{TB}$ and $\textbf{TNB}$ Smarandache curves of $r$ are respectively, written as $$\textbf{r}_{\textbf{TN}}=\left( \begin{array}{c} 1,\cos \left( \int \tau (s)\,ds\right) +\int \kappa (s)\cos \left( \int \tau (s)\,ds\right) \,ds, \\ \int \kappa (s)\sin \left( \int \tau (s)\,ds\right) \,ds+\sin \left( \int \tau (s)\,ds\right)\end{array}\right),$$ $$\textbf{r}_{\textbf{TB}}=\left( \begin{array}{c} 1,\int \kappa (s)\cos \left( \int \tau (s)\,ds\right) \,ds-\sin \left( \int \tau (s)\,ds\right) , \\ \cos \left( \int \tau (s)\,ds\right) +\int \kappa (s)\sin \left( \int \tau (s)\,ds\right) ds\end{array}\right),$$ $$\textbf{r}_{\textbf{TNB}}=\left( \begin{array}{c} 1,\cos \left( \int \tau (s)\,ds\right) +\int \kappa (s)\cos \left( \int \tau (s)\,ds\right) \,ds \\ -\sin \left( \int \tau (s)\,ds\right) ,\cos \left( \int \tau (s)\,ds\right) + \\ \int \kappa (s)\sin \left( \int \tau (s)\,ds\right) ds+\sin \left( \int \tau (s)\,ds\right) \end{array}\right).$$ Smarandache curves of some special curves in $G_{3}$ ==================================================== Smarandache curves of a general helix ------------------------------------- Let $\alpha (s)$ be a general helix in $G_{3}$ with ($\tau /\kappa =m=const.$)which can be written as $$\alpha (s)=\left( \begin{array}{c} s,\frac{1}{m}\int \sin \left( m\int \kappa (s)\,ds\right) \,ds, \\ \frac{-1}{m}\int \cos \left( m\int \kappa (s)\,ds\right) \,ds \end{array} \right).$$ Then $\alpha ^{\prime }, \alpha ^{\prime \prime }, \alpha ^{\prime \prime \prime }$ for this curve are respectively, expressed as $$\alpha ^{\prime }(s)=\left( 1,\frac{1}{m}\sin \left( m\int \kappa (s)\,ds\right) ,\frac{-1}{m}\cos \left( m\int \kappa (s)\,ds\right) \right),$$ $$\alpha ^{\prime \prime }(s)=\left( 0,\kappa (s)\cos \left( m\int \kappa (s)\,ds\right) ,\kappa (s)\sin \left( m\int \kappa (s)\,ds\right) \right),$$ $$\alpha ^{\prime \prime \prime }(s)=\left( \begin{array}{c} 0,\kappa ^{\prime }(s)\cos \left( m\int \kappa (s)\,ds\right) - \\ m~\kappa ^{2}(s)\sin \left( m\int \kappa (s)\,ds\right) , \\ \kappa ^{\prime }(s)\sin \left( m\int \kappa (s)\,ds\right) + \\ m~\kappa ^{2}(s)\cos \left( m\int \kappa (s)\,ds\right) \end{array} \right).$$ The moving Frenet vectors of $\alpha (s)$ are given by $$\textbf{T}_{\alpha}=\left( 1,\frac{1}{m}\sin \left( m\int \kappa (s)\,ds\right) ,\frac{-1}{m}\cos \left( m\int \kappa (s)\,ds\right) \right),$$ $$\textbf{N}_{\alpha}=\left( 0,\cos \left( m\int \kappa (s)\,ds\right) ,\sin \left( m\int \kappa (s)\,ds\right) \right),$$ $$\textbf{B}_{\alpha}=\left( 0,-\sin \left( m\int \kappa (s)\,ds\right) ,\cos \left( m\int \kappa (s)\,ds\right) \right).$$ From which, Smarandache curves are obtained $$\alpha_{\textbf{TN}}=\left( \begin{array}{c} 1,\cos \left( m\int \kappa (s)\,ds\right) +\frac{1}{m}\sin \left( m\int \kappa (s)\,ds\right), \\ \frac{-1}{m}\cos \left( m\int \kappa (s)\,ds\right) +\sin \left( m\int \kappa (s)\,ds\right) \end{array} \right),$$ $$\alpha_{\textbf{TB}}=\left( 1,-\left( \frac{m-1}{m}\right) \sin \left( m\int \kappa (s)\,ds\right) , \left( \frac{m-1}{m}\right) \cos \left( m\int \kappa (s)\,ds\right) \right),$$ $$\alpha_{\textbf{TNB}}=\left( \begin{array}{c} 1,\cos \left( m\int \kappa (s)\,ds\right) -\left( \frac{m-1}{m}\right) \sin \left( m\int \kappa (s)\,ds\right), \\ \left( \frac{m-1}{m}\right) \cos \left( m\int \kappa (s)\,ds\right) +\sin \left( m\int \kappa (s)\,ds\right) \end{array} \right).$$ Smarandache curves of a circular helix -------------------------------------- Let $\beta (s)$ be a circular helix in $G_{3}$ with ($\tau=a=const., \kappa =b=const.$) which can be written as $$\beta (s)=\left( s,a\int \left( \int \cos (bs)\,ds\right) \,ds,a\int \left( \int \sin (bs)\,ds\right) \,ds\right).$$ For this curve, we have $$\beta ^{\prime }(s)=\left( 1,\frac{a}{b}\sin (bs),-\frac{a}{b}\cos (bs)\right),$$ $$\beta ^{\prime \prime }(s)=\left( 0,a\cos (bs),a\sin (bs)\right),$$ $$\beta ^{\prime \prime \prime }(s)=\left( 0,-ab\sin (bs),ab\cos (bs)\right).$$ Making necessary calculations from above, we have $$\textbf{T}_{\beta}=\left( 1,\frac{a}{b}\sin (bs),-\frac{a}{b}\cos (bs)\right),$$ $$\textbf{N}_{\beta}=\left( 0,\cos (bs),\sin (bs)\right),$$ $$\textbf{B}_{\beta}=\left( 0,-\sin (bs),\cos (bs)\right).$$ Considering the last Frenet vectors, the $\textbf{TN}$, $\textbf{TB}$ and $\textbf{TNB}$ Smarandache curves of $\beta$ are respectively, as follows $$\beta_{\textbf{TN}}=\left( 1,\cos (bs)+\frac{a}{b}\sin (bs),-\frac{a}{b}\cos (bs)+\sin (bs)\right),$$ $$\beta_{\textbf{TB}}=\left( 1,\left( \frac{a-b}{b}\right) \sin (bs),\left( \frac{b-a}{b} \right) \cos (bs)\right),$$ $$\beta_{\textbf{TNB}}=\left( \begin{array}{c} 1,\cos (bs)+\left( \frac{a-b}{b}\right) \sin (bs), \\ \left( \frac{b-a}{b}\right) \cos (bs)+\sin (bs) \end{array} \right).$$ Smarandache curves of a Salkowski curve --------------------------------------- Let $\gamma (s)$ be a Salkowski curve in $G_{3}$ with ($\tau=\tau (s), \kappa =a=const.$) which can be written as $$\gamma (s)=\left( \begin{array}{c} s,a\int \left( \int \cos \left( \int \tau (s)\,ds\right) ds\,\right) ds, \\ a\int \left( \int \sin \left( \int \tau (s)\,ds\right) ds\,\right) ds\end{array} \right).$$ If we differentiate this equation three times, one can obtain $$\gamma ^{\prime }(s)=\left( 1,a\int \cos \left( \int \tau (s)\,ds\right) ds\,,a\int \sin \left( \int \tau (s)\,ds\right) ds\right),$$ $$\gamma ^{\prime \prime }(s)=\left( 0,a\cos \left( \int \tau (s)\,ds\right) ,a\sin \left( \int \tau (s)\,ds\right) \right),$$ $$\gamma ^{\prime \prime \prime }(s)=\left( \begin{array}{c} 0,-a~\tau (s)\sin \left( \int \tau (s)\,ds\right) , \\ a~\tau (s)\cos \left( \int \tau (s)\,ds\right) \end{array} \right).$$ In addition to that, the tangent, principal normal and binormal vectors of $\gamma$ are in the following forms $$\textbf{T}_{\gamma}=\left( 1,a\int \cos \left( \int \tau (s)\,ds\right) ds,a\int \sin \left( \int \tau (s)\,ds\right) ds\right),$$ $$\textbf{N}_{\gamma}=\left( 0,\cos \left( \int \tau (s)\,ds\right) ,\sin \left( \int \tau (s)\,ds\right) \right),$$ $$\textbf{B}_{\gamma}=\left( 0,-\sin \left( \int \tau (s)\,ds\right) ,\cos \left( \int \tau (s)\,ds\right) \right).$$ Furthermore, Smarandache curves for $\gamma$ are $$\gamma_{\textbf{TN}}=\left( \begin{array}{c} 1,\cos \left( \int \tau (s)\,ds\right) +a\int \cos \left( \int \tau (s)\,ds\right) ds, \\ a\int \sin \left( \int \tau (s)\,ds\right) ds+\sin \left( \int \tau (s)\,ds\right) \end{array} \right),$$ $$\gamma_{\textbf{TB}}=\left( \begin{array}{c} 1,a\int \cos \left( \int \tau (s)\,ds\right) ds-\sin \left( \int \tau (s)\,ds\right) , \\ \cos \left( \int \tau (s)\,ds\right) +a\int \sin \left( \int \tau (s)\,ds\right) ds \end{array} \right),$$ $$\gamma_{\textbf{TNB}}=\left( \begin{array}{c} 1,\cos \left( \int \tau (s)\,ds\right) +a\int \cos \left( \int \tau (s)\,ds\right) \,ds \\ -\sin \left( \int \tau (s)\,ds\right) ,\cos \left( \int \tau (s)\,ds\right) + \\ a\int \sin \left( \int \tau (s)\,ds\right) ds+\sin \left( \int \tau (s)\,ds\right) \end{array} \right).$$ Smarandache curves of Anti-Salkowski curve ------------------------------------------ Let $\delta (s)$ be Anti-Salkowski curve in $G_{3}$ with ($\kappa=\kappa (s), \tau =a=const.$) which can be written as $$\delta (s)=\left( \begin{array}{c} s,\int \left( \int \kappa (s)\cos (as)ds\right) \,ds, \\ \int \left( \int \kappa (s)\sin (as)ds\right) \,ds \end{array} \right).$$ It gives us the following derivatives $$\delta ^{\prime }(s)=\left( 1,\int \kappa (s)\cos (as)ds,\int \kappa (s)\sin (as)ds\right),$$ $$\delta ^{\prime \prime }(s)=\left( 0,\kappa (s)\cos (as),\kappa (s)\sin (as)\right),$$ $$\delta ^{\prime \prime \prime }(s)=\left( \begin{array}{c} 0,\kappa ^{\prime }(s)\cos (as)-a~\kappa (s)\sin (as), \\ \kappa ^{\prime }(s)\sin (as)+a~\kappa (s)\cos (as)\end{array} \right).$$ Further, we obtain the following Frenet vectors $\textbf{T}$, $\textbf{N}$, $\textbf{B}$ in the form $$\textbf{T}_{\delta}=\left( 1,\int \kappa (s)\cos (as)ds,\int \kappa (s)\sin (as)ds\right),$$ $$\textbf{N}_{\delta}=\left( 0,\cos (as),\sin (as)\right),$$ $$\textbf{B}_{\delta}=\left( 0,-\sin (as),\cos (as)\right).$$ Thus the above computations of Frenet vectors are give Smarandache curves by $$\delta_{\textbf{TN}}=\left( 1,\cos (as)+\int \kappa (s)\cos (as)\,ds, \int \kappa (s)\sin (as)\,ds+\sin (as)\, \right)$$ $$\delta_{\textbf{TB}}=\left( 1,\int \kappa (s)\cos (as)\,\,ds-\sin (as), \cos (as)+\int \kappa (s)\sin (as)ds \right)$$ $$\delta_{\textbf{TNB}}=\left( \begin{array}{c} 1,\cos (as)+\int \kappa (s)\cos (as)\,ds-\sin (as), \\ \cos (as)+\int \kappa (s)\sin (as)ds+\sin (as)\end{array} \right)$$ Examples ======== Let $\alpha :I\longrightarrow G_{3}$ be an admissible curve and $\kappa \neq 0$ of class $C^{2},\tau \neq 0$ of calss $C^{1}$ its curvature and torsion, respectively written as $$\alpha (s)=\left( s,\frac{s}{10}\left( -2\cos (2\ln s)+\sin (2\ln s)\right) , -\frac{s}{10}\left( \cos (2\ln s)+2\sin (2\ln s)\right) \right)$$ By differentiation, we get $$\alpha ^{\prime }(s)=\left( 1,\cos (\ln s)+\sin (\ln s),-\frac{1}{2}\cos (2\ln s)\right),$$ $$\alpha ^{\prime \prime }(s)=\left( 0,\frac{\cos (2\ln s)}{s},\frac{\sin (2\ln s)}{s}\right),$$ $$\alpha ^{\prime \prime \prime }(s)=\left( 0,-\frac{\cos (2\ln s)+2\sin (2\ln s)}{s^{2}}, \frac{2\cos (2\ln s)-\sin (2\ln s)}{s^{2}} \right).$$ Using $(2.5)$ to obtain $$\textbf{T}_{\alpha}=\left( 1,\cos (\ln s)\sin (\ln s),-\frac{1}{2}\cos (2\ln s)\right),$$ $$\textbf{N}_{\alpha}=\left( 0,\cos (2\ln s),\sin (2\ln s)\right),$$ $$\textbf{B}_{\alpha}=\left( 0,-\sin (2\ln s),\cos (2\ln s)\right).$$ The natural equations of this curve are given by $$\kappa_{\alpha}=\frac{1}{s}, \tau_{\alpha}=\frac{2}{s}.$$ Thus, the Smarandache curves of $\alpha$ are respectively, given by $$\alpha_{\textbf{TN}}=\left( 1,\cos (2\ln s)+\cos (\ln s)\sin (\ln s), -\frac{1}{2}\cos (2\ln s)+\sin (2\ln s) \right),$$ $$\alpha_{\textbf{TB}}=\left( 1,-\cos (\ln s)\sin (\ln s),\frac{1}{2}\cos (2\ln s)\right),$$ $$\alpha_{\textbf{TNB}}=\left( 1,\cos (2\ln s)-\cos (\ln s)\sin (\ln s), \frac{1}{2}\cos (2\ln s)+\sin (2\ln s) \right).$$ The curve $\alpha$ and their Smarandache curves are shown in Figures 1,2.  \[fig:gh\]  \[fig:tngh\]  \[fig:tbgh\]  \[fig:tnbgh\] For an admissible curve $\delta (s)$ in $G_{3}$ parameterized by $$\delta (s)=\left( s,\frac{e^{-s}}{25}(-3\cos (2s)-4\sin (2s)), \frac{e^{-s}}{25}(4\cos (2s)-3\sin (2s)) \right),$$ we use the derivatives of $\delta ; \delta ^{\prime }, \delta ^{\prime \prime }, \delta ^{\prime \prime \prime }$ to get the associated trihedron of $\delta$ as follows $$\textbf{T}_{\delta}=\left\{ 1,-\frac{e^{-s}}{5}(\cos (2s)-2\sin (2s)), -\frac{e^{-s}}{5}(2\cos (2s)+\sin (2s)) \right\},$$ $$\textbf{N}_{\delta}=\left( 0,\cos (2s),\sin (2s)\right),$$ $$\textbf{B}_{\delta}=\left( 0,-\sin (2s),\cos (2s)\right).$$ Curvature $\kappa(s)$ and torsion $\tau(s)$ are obtained as follows $$\kappa_{\delta}=e^{-s}, \tau_{\delta}=2.$$ According to the above calculations, Smarandache curves of $\delta $ are $$\delta_{\textbf{TN}}=\left( \begin{array}{c} 1,\cos (2s)-\frac{1}{5}e^{-s}(\cos (2s)-2\sin (2s)), \\ \sin (2s)-\frac{1}{5}e^{-s}(2\cos (2s)+\sin (2s)) \end{array} \right),$$ $$\delta_{\textbf{TB}}=\left( \begin{array}{c} 1,-\frac{e^{-s}}{5}\left( \cos (2s)+\left( -2+5e^{s}\right) \sin (2s)\right) , \\ \cos (2s)-\frac{e^{-s}}{5}(2\cos (2s)+\sin (2s))\end{array} \right),$$ $$\delta_{\textbf{TNB}}=\left( \begin{array}{c} 1,\cos (2s)-\frac{e^{-s}}{5}(\cos (2s)-2\sin (2s))-\sin (2s), \\ \cos (2s)+\sin (2s)-\frac{e^{-s}}{5}(2\cos (2s)+\sin (2s)) \end{array} \right).$$  \[fig:as\]  \[fig:tnas\]  \[fig:tbas\]  \[fig:tnbas\] Conclusion ========== In the three-dimensional Galilean space, Smarandache curves of an arbitrary curve and some special curves such as helix, circular helix, Salkowski and Ant-Salkowski curves have been studied. To confirm our main results, two examples (helix and Anti-Salkowski curves) have been given and illustrated. [99]{} , *Special Smarandache curves in the Euclidean space*, [International Journal of Mathematical Combinatorics]{}, **2** (2010), 30-36. , *Position vectors of curves in the Galilean space $G_{3}$*, [Matematicki Vesnik]{}, **64(3)** (2012), 200-210. , *Minding’s isometries of ruled surfaces in Galilean and pseudo-Galilean space*, [J. Geom.]{}, **77** (2003), 35-47. , *The equiform differential geometry of curves in the Galilean space $G_{3}$*, [Glasnik Matematicki]{}, **22(42)** (1987), 449-457. , *The projective interpretation of the eight 3-dimensional homogeneous geometries*, [Beiträge Algebra Geom.]{}, **38(2)** (1997), 261-288. , *Zur transformation von raumkurven*, [ Math. Ann.]{}, **66** (1909), 517-557. , *A simple non-Euclidean geometry and its physical basis*, [Springer-Verlag]{}, [New York]{}, 1979. , *Smarandache curves in Minkowski space-time*, [International Journal of Mathematical Combinatorics]{} , **3** (2008), 51-55. , *Differential geometry of curves and surfaces*, [Prentice Hall]{}, [Englewood Cliffs, NJ]{}, 1976. , *Die Geometrie des Galileischen Raumes*, [Habilitationsschrift]{}, [Institut für Math. und Angew. Geometrie, Leoben]{}, 1984. , *Lectures in Classical Differential Geometry*, [Addison,-Wesley]{}, [Reading, MA]{}, 1961. , *High accuracy approximation of helices by quintic curve*, [Comput. Aided Geomet. Design]{}, **20** (2003), 303-317. , *Zur transformation von raumkurven*, [Mathematische Annalen]{}, **66(4)** (1909), 517-557. [^1]:  E-mail address: mohamed\_khalifa77@science.sohag.edu.eg

--- abstract: 'This paper presents a detailed analysis of two-armed spiral structure in a sample of galaxies from the Spitzer Infrared Nearby Galaxies Survey (SINGS), with particular focus on the relationships between the properties of the spiral pattern in the stellar disc and the global structure and environment of the parent galaxies. Following Paper I we have used a combination of Spitzer Space Telescope mid-infrared imaging and visible multi-colour imaging to isolate the spiral pattern in the underlying stellar discs, and we examine the systematic behaviours of the observed amplitudes and shapes (pitch angles) of these spirals. In general, spiral morphology is found to correlate only weakly at best with morphological parameters such as stellar mass, gas fraction, disc/bulge ratio, and v$_{flat}$. In contrast to weak correlations with galaxy structure a strong link is found between the strength of the spiral arms and tidal forcing from nearby companion galaxies. This appears to support the longstanding suggestion that either a tidal interaction or strong bar is a necessary condition for driving grand-design spiral structure. The pitch angles of the stellar arms are only loosely correlated with the pitch angles of the corresponding arms traced in gas and young stars. We find that the strength of the shock in the gas and the contrast in the star formation rate are strongly correlated with the stellar spiral amplitude.' author: - | S. Kendall$^1$ , C. Clarke$^1$[^1] ,R. C. Kennicutt$^1$\ $^1$ Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA date: 'submitted, accepted' title: | Spiral structure in nearby galaxies\ II. comparative analysis and conclusions --- \[firstpage\] galaxies:individual–galaxies:spiral–galaxies:structure–infrared:galaxies Introduction. {#intro} ============= This paper is the second of two detailing an observational study of spiral structure for galaxies in the *Spitzer* Infrared Nearby Galaxies Survey (SINGS - Kennicutt et al. (2003). This work also builds directly on an earlier study of M81 presented in Kendall et al. (2008). In the first paper (Kendall et al. 2011, hereafter Paper I), the methods and results from individual galaxies were presented together with a comparison between those galaxies in which grand design spiral structure was detected and those that lacked such structure. In this paper we focus on the subset of $13$ galaxies (henceforth the ‘detailed sample’) in which it has been possible to characterise the properties of the spiral structure in detail and present an analysis of any trends within this sample. Observational results have been crucial to the advances in the study of spiral structure, and the launch of the *Spitzer* Space Telescope provided a new resource in its high quality infrared data. This paper aims to build on previous studies in the optical and infrared, and provides a new dataset against which numerical simulations of spiral structure can be tested. There have been numerous observational studies of spiral structure, some key results are as follows: Schweizer (1976) analysed six galaxies using optical data to produce profiles of surface brightness against azimuth. From these data Schweizer concluded that there must be an old disc component in the spiral arms of each of the galaxies studied and that the spiral arms reach amplitudes as large as 30 per cent relative to the disc. Other studies have confirmed this result; the measured ratio of arm to disc mass varies with galaxy and the exact method and wavelength used, but as a rule the contrast between arm and inter-arm regions in the optical is in the range 20 to 100 per cent (e.g Schweizer (1976); Elmegreen & Elmegreen (1984); Grosbol et al. (2004)), with more recent studies suggesting that an upper limit of 40 to 50 per cent is appropriate in the majority of cases (note that contamination from star forming regions can cause this quantity to be over-estimated particularly in the early studies which used the *I* band rather than near infrared (NIR) data). The amplitude of spiral arms is potentially important since it might be expected to directly affect the global star formation rate (SFR) in the galaxy: larger contrasts between the stellar arm-interarm regions are expected to cause the gas to shock more strongly, become more dense, and thus increase the star formation (bearing in mind that according to the Kennicutt-Schmidt law (Kennicutt 1998), star formation varies as $\Sigma^{1.4}$.) This association between star formation and spiral structure would appear to be borne out in cases like M81 and M51, where the vast majority of star formation lies on well defined spiral arms. Some studies have attempted to correlate the rates of star formation with the strength of the stellar spiral arms: for example, Cepa & Beckman (1990) found that star formation is triggered preferentially by spiral arms (with a ‘non-linear dependency’). Seigar & James (1998) found a link between SFR measured from far-infrared luminosity and *K* band arm strength, and found that the SFR increases with arm strength. Similarly, Seigar & James (2002) reported that the spiral arm strength (in *K*) correlates with H$\alpha$ (a SFR tracer) suggesting that stronger potential variations (and associated shocks) lead to a larger SFR (up to a limiting threshold, above which the SFR is constant). Nevertheless, the picture does not always appear to be this simple: studies of flocculent galaxies showing underlying m=2 spiral structure in the mass density as traced by near infrared emission (Elmegreen et al. 1999, Thornley 1996) clearly show that star formation is not always preferentially located on two well defined spiral arms. More recently Elmegreen et al (2011) analysed NIR imaging for a sample of nearby galaxies from the Spitzer S$^4$G survey (Sheth et al. 2010) and found that the distinctions between grand design and flocculent arm structures seen in the visible extend to the NIR structure as well. Another property of spiral galaxies that has been extensively studied is the pitch angle (i.e. the angle between spiral features and the tangential direction): Kennicutt & Hodge (1982) measured pitch angles and compared them against theoretical predictions from both QSSS (quasi-stationary spiral structure - Lindblad 1964; Lin & Shu 1964,1966) and SSPSF (stochastic self-propagating star formation - Mueller & Arnett 1976; Gerola & Seiden 1978) theories and found qualitative trends that were in agreement with both theories but poor detailed agreement in both cases. Similarly, Kennicutt (1981) compared pitch angle against RSA type (largely dependent on disc resolution), and found a good correlation on average, but with a large degree of scatter. A correlation between pitch angle, *i*, and bulge-to-disc ratio was also found, where a larger pitch angle (more open spiral pattern) is associated with a smaller bulge fraction; again there is a large scatter in the pitch angle at given bulge to disc ratio. In addition the maximum rotational velocity was found to correlate strongly with pitch angle, with galaxies with large rotational velocities having more tightly wound spirals. However, there are indications that the appearance of spiral arms in the NIR and optical do not always correlate (an extreme case being the optically flocculent galaxies with NIR arms already noted above). Some studies using NIR data (e.g. Seigar & James (1998) have not found the same trends as Kennicutt. Block & Wainscoat (1991) found evidence that the spiral arms are less tightly wound in the NIR than the optical. Likewise, Block et al. (1994) compare the NIR and optical morphology for a sample of galaxies and, for a number of spirals, found significant differences between the pitch angles of the arms in the optical and NIR. They found that galaxies generally appear to be of an earlier Hubble type when viewed in the NIR (*K’* band) than in the optical, the traditional wavelength for such classifications. On the other hand, more recent studies have found a generally good agreement between the pitch angles measured in the NIR and B band (Seigar et al. 2006, Davis et al. 2012); in the few cases where the two values are significantly different, there is a mild preference for the NIR pitch angles to be larger. Turning now to correlations with galactic parameters, Seigar & James (1998) found no evidence of a correlation between NIR pitch angle and bulge fraction or Hubble type. Seigar et al. (2005) however reported a convincing connection between pitch angle in the NIR and the morphology of the galactic rotation curve, with open arms correlating with rising rotation curves, while pitch angles decline for flat and falling rotation curves. The goal of this paper is to use the new measurements of the properties of the underlying [*stellar*]{} spiral arms from Paper I to re-examine the degrees to which the strengths and shapes of these arms are driven by the structural and environmental properties of their host galaxies. This is done mainly by examining correlations between the amplitude and pitch angles of the stellar arms with measures of galaxy morphology (independent of spiral structure) and mass, and with a measure of tidal perturbation from nearby companion galaxies. The same data are also used here to compare the shapes of the stellar arms with those traced by young gaseous and stellar components, and to constrain the degree to which the response of the gas discs and the local star formation rates are influenced by the amplitude of the spiral perturbation in the stellar discs. The data and methods used in this work are described in detail in Paper I. A brief summary is as follows; 3.6 and 4.5$\mu$m NIR data from the *Spitzer* Infrared Array Camera (IRAC) are used in conjunction with complementary optical data in order to trace the stellar mass. In addition, IRAC 8$\mu$m data are used to trace the gas response, with particular emphasis on shocks. The data were analysed using azimuthal profiles (defined from fitting isophotal ellipses to the axisymmetric components of the galaxies) or radial profiles in the case that difficulties were encountered using azimuthal profiles. Azimuthal profile data were further examined by calculating the Fourier components in order to examine the amplitude and phase of individual modes, particularly the m=2 component. This Fourier analysis was not possible for radial profile data; instead the total amplitude and phase were analysed. In this paper we consider only the $13$ galaxies that constituted the so-called ‘detailed sample’ discussed in Paper I. We list various properties of these galaxies (which we go on to correlate with the properties of their spiral structure) in Table 1. The methods developed to isolate, identify, and parametrise the massive stellar arms are not well suited to strongly barred galaxies (which can distort arm shapes very significantly from a logarithmic shape), so this paper focuses mainly on normal and weakly barred galaxies; the role of bars in spiral structure is another important question awaiting further study (see Elmegreen et al. 2011). Galaxy HT EC $v_{flat}$ C log ($M_*$) $A/\omega $ gf SSFR ---------- ----- ---- ------------ ------ ------------- ------------- ------- ------------------------- NGC 0628 Sc 9 217 2.95 10.1 0.39 0.32 $ 6.4 \times 10^{-11} $ NGC 1566 Sbc 12 196 3.57 10.7 0.55 0.18 - NGC 2403 Scd 4 134 3.16 9.7 0.36 0.39 $7.6 \times 10^{-11}$ NGC 2841 Sb 3 302 3.27 10.8 0.48 0.17 $ 1.2 \times 10^{-11}$ NGC 3031 Sab 12 229 3.91 10.3 0.58 0.04 $2 \times 10^{-11}$ NGC 3184 Scd 9 210 2.40 10.3 0.21 0.22 $4.5 \times 10^{-11}$ NGC 3198 Sc 9 150 2.54 10.1 0.21 0.51 $7.4 \times 10^{-11}$ NGC 3938 Sc 9 196 2.95 10.1 0.41 0.39 $9.5 \times 10^{-11}$ NGC 4321 Sbc 12 222 3.03 10.9 0.36 0.21 $6.9 \times 10^{-11}$ NGC 4579 Sb 9 288 3.92 10.9 0.41 0.06 $1.1 \times 10^{-11}$ NGC 5194 Sc 12 219 3.0 10.6 0.55 0.125 $7.8 \times 10^{-11}$ NGC 6946 Sd 9 186 2.83 10.5 0.27 0.25 $ 1.0 \times 10^{-10}$ NGC 7793 Sb 2 115 2.48 9.5 0.22 0.29 $7.4 \times 10^{-11}$ The relationship between galactic properties and the nature of spiral structure. {#ch5_influence} ================================================================================ Previous studies in the optical and infrared have found conflicting trends relating the properties of spiral features with a range of galactic parameters such as Hubble type, galaxy mass and rotation curve morphology. This section sets out the findings from this work and discusses them in the context of previous results. Amplitude and radial extent of spiral structure {#ch5_strength} ----------------------------------------------- Figure 1 presents data on the relative amplitude of the m=2 component at $3.6 \mu$m as a function of radius. The axisymmetric components used to normalise the m=2 components contain contributions from the disc and bulge; the halo contribution is not included. In M81, the only galaxy where the halo contribution has been considered (Kendall et al. 2008), adding the halo to the axisymmetric components makes very little difference to the relative amplitude, although this is not necessarily true in all cases. Data is presented for the $8$ galaxies in the ‘detailed’ sample for which it was possible to analyse azimuthal profile data and thus deduce the amplitude of various Fourier modes. ![Relative amplitude of the m=2 component of the 3.6$\mu$m (stellar mass) data for all the ‘grand design’ spirals which could be analysed with Fourier components. Note this does not include data obtained from radial profiles. The data are shown for the radial range over which a logarithmic spiral could be traced.[]{data-label="m_2_plot"}](kckf1r.eps){width="83mm"} The galaxies with much less power in m=2 are normally those in which there is no single dominant Fourier component and the power in modes m=1, 3 and 4 is similar to that in m=2 (these tend to be the optically flocculent galaxies like NGC 7793). The galaxy with a smaller radial range than many but a higher relative amplitude is NGC 1566. . \[m\_2\_plot\] In what follows we represent the ‘average’ amplitude of spiral structure in each waveband and for each galaxy as a straight average of the values obtained over the range of radii for which a logarithmic spiral pattern is detectable. These averages are listed for each galaxy in Table 2. Most galaxies thus have amplitudes in three bands (optical, IRAC1 and IRAC2) which are denoted by cross, dot and star symbols in each case. The data for each galaxy are linked by vertical lines. The larger symbols and dashed vertical lines correspond to the galaxies whose structure has been determined through radial profile analysis. The largest crosses denote the optical data for the three galaxies (NGC 3031,NGC 4321 and NGC 5194) which are apparently undergoing the strongest tidal interactions (i.e. with $P > 10^{-3}$ (equation 1); see Table 4 and Figure 17). In almost all cases the main uncertainty is associated with the range of values at different radii (see Figure 1) and the issue of how datapoints at small and large radii should be weighted. The difference in mean amplitude between different wavebands is mainly due to the fact that in some galaxies the radial range over which a logarithmic spiral can be traced is different in different bands. The difference between amplitudes in different wavebands is therefore a measure of the uncertainty associated with radial averaging within the radial range studied (which is generally less than $R_{25}$; see Figure 1). Note that in the case of galaxies that overlap with the sample of Elmegreen et al. (2011), the amplitude of variation at given radius is in good agreement between the two papers (once account is taken of the difference in the definition of arm contrast); the average values quoted in the Elmegreen et al. study are however generally higher due to the fact that the spiral structure is generally analysed out to larger radius. As noted above, in the bulk of our plots the vertical lines link average values in different wavebands; in order to avoid a confusing plethora of vertical lines we show the errors associated with radial averaging (as fine vertical lines) only in a single plot (Figure 4). The measurement errors for [*individual*]{} datapoints at given radius are generally much smaller than the uncertainties associated with radial averaging, being much less than 0.1 magnitudes in most of the azimuthal profile galaxies: see individual profiles and associated errors in Paper I. Galaxy $\Delta_V$ $\Delta_{3.6}$ $\Delta_{4.5}$ $\Delta \phi_v$ $\Delta \phi_{3.6}$ $\Delta \phi_{4.5}$ ---------- ------------ ---------------- ---------------- ----------------- --------------------- --------------------- NGC 0628 0.131 0.198 0.187 6.25 8.33 8.22 NGC 1566 0.247 0.284 0.287 3.92 4.50 4.57 NGC 2403 - 0.119 0.130 3.89 3.89 3.83 NGC 2841 - 0.069 0.076 2.85 2.85 3.60 NGC 3031 0.260 0.224 0.207 1.79 1.89 2.19 NGC 3184 0.225 0.282 0.304 6.57 5.55 5.49 NGC 3198 0.131 0.213 0.200 2.98 5.69 5.42 NGC 3938 0.091 0.093 0.086 5.01 4.49 4.37 NGC 4321 0.284 0.327 0.351 4.59 4.46 4.19 NGC 4579 0.177 0.188 0.166 3.06 3.50 3.76 NGC 5194 0.299 0.416 0.425 4.89 5.96 5.28 NGC 6946 0.166 0.214 0.222 1.97 1.56 1.55 NGC 7793 0.085 0.071 0.077 4.78 4.93 4.97 : The parameters of spiral structure in the visible, $3.6 \mu$m (IRAC1) and $4.5 \mu$m (IRAC2) bands: $\Delta$ is the mean spiral amplitude (see text) and $\Delta \phi$ is the total angle subtended by the arms in radians ### Dependence on Hubble type and Elmegreen type. We first plot the amplitudes of spiral structure as a function of Hubble type and Elmegreen type in Figures 2 and 3. There is no obvious correlation with Hubble type unless one excludes the three galaxies of type Sb and earlier. The correlation between spiral amplitude and Elmegreen class is much more evident: this is unsurprising since Elmegreen class is based purely on spiral morphology (albeit in the optical) with patchy, wispy spirals corresponding to arm class 1-4 and classes 5-12 corresponding to galaxies with increasingly prominent two armed grand design spiral structure. The fact that this classification scheme tracks the NIR amplitudes is a confirmation that the over-all level of spiral structure from galaxy to galaxy is reasonably well correlated in the optical and the NIR. We also note that the three ‘strongly interacting galaxies’ are associated with high Elmegreen class (i.e. well defined grand design structure). These results are broadly consistent with those from an independent study by Elmegreen et al. (2011), based on Spitzer S$^4$G observations. In particular both studies show the absence of any correlation between arm amplitude and Hubble type when all spiral arm types are considered together, though Elmegreen et al. (2011) do see evidence for a possible correlation between arm-interarm contrast and Hubble type when flocculent galaxies are considered separately. Our observation of a correlation between arm amplitude and Elmegreen spiral type is also consistent with a trend observed in the Elmegreen et al. (2011) sample. ![Average m=2 relative amplitude plotted against Hubble type. The smaller symbols correspond to the $8$ galaxies analysed with the azimuthal profile method, while the larger symbols denote the $5$ galaxies analysed with radial profiles. Each galaxy has three data points shown, with separate amplitude estimates from 3.6$\mu$m, 4.5$\mu$m and optical colour corrected data being denoted by dots, stars and crosses respectively; datapoints for a given galaxy are linked with vertical lines (dashed in the case of the radial profile galaxies). Note that for two galaxies ( NGC 2403 and NGC 2841) no optical data are presented since these two galaxies were too flocculent in the optical to be able to extract reliable spiral parameters (see Paper I). The largest crosses denote optical data for the three galaxies with $P > 10^{-3}$ (equation (1)) that are apparently undergoing tidal interactions (see Figure 17 and Table 4). []{data-label="hub_av2"}](kckf2r.ps){width="83mm"} {width="83mm"} {width="83mm"} {width="83mm"} {width="83mm"} ![Dimensionaless shear (see text) plotted against average m=2 amplitude. Solid body rotation, flat rotation curves and falling rotation curves correspond to $A/\omega$ values of $0,0.5$ and $>0.5$ respectively. See Figure 2 for explanation of symbols.[]{data-label="conC_av2"}](kckf7r.ps){width="83mm"} {width="83mm"} {width="83mm"} ### Dependence on galactic stellar mass and rotational velocity Figures 4 and 5 plot the m=2 amplitude against the galactic rotation velocity evaluated in the flat region of the rotation curve, $v_{flat}$, and the stellar mass $M_*$, with both these quantities being derived, where possible, from Leroy et al. (2008). Leroy et al. calculate $v_{flat}$ by fitting the rotation curve with a function of the form $v_{rot}(r) = v_{flat} (1 - exp (-r/r_{flat})$ where $v_{flat}$ and $r_{flat}$ are free parameters. For galaxies not included in the Leroy et al. sample, the same function is fit to the kinematic data from the sources given in Paper I, with errors estimated at the $10$ per cent level. The stellar masses are calculated by integrating the galactic models produced by GALFIT (Peng et al. 2002) and applying the prescription to convert to stellar mass given in Leroy et al. (2008). In both plots there is a clear trend for m=2 average amplitude to increase with stellar rotation velocity and stellar mass, particularly if one excludes the three galaxies of Hubble type Sb or earlier. It should not be surprising that the trend is roughly the same in both plots since the Tully -Fisher relation (Tully & Fisher 1977) implies that $v_{flat}$ is an approximate tracer of stellar mass. The same approximate trend was found by Elmegreen & Elmegreen (1987) who demonstrated that spiral arm amplitude is an increasing function of galaxy ‘size’ (proxied by the product of $R_{25}$ and $v_{flat}$). This trend is consistent with the decline in spiral amplitudes towards later Hubble types seen in Figure 2 since the general relationship between Hubble type and luminosity implies that later type galaxies are generally less massive. A qualitative correlation between the strength of spiral structure and parent galaxy luminosity has been recognised for decades (van den Bergh 1960a, b), and forms the basis of the van den Bergh luminosity classification of galaxies. The trends seen in Figures 4 and 5 can be regarded as a quantitative manifestation of this correlation. Figure 4 is particularly suggestive in that $11/13$ of the galaxies appear to broadly follow a trend of increasing amplitude of spiral features with increasing rotation velocity; the low amplitude of spiral features for the two objects with the largest rotational velocities (the Sb galaxies NGC 4579 and NGC 2841) are notably discrepant. We however caution against the drawing of any but the most tentative conclusions from this small sample. ### Dependence on concentration and galactic shear Another factor which may influence the amplitude of spiral structure is the degree of central concentration of galaxies inasmuch as this affects the morphology of the galactic rotation curve. Kormendy & Norman (1979) argued that spiral waves are damped at the Inner Lindblad Resonance (ILR) and thus predicted that spiral structure (other than that driven by companions or bars) should be restricted to regions of galaxies with nearly solid body rotation (for which there is no ILR). Their optical data showed some support for this hypothesis although this may be partly driven by the fact that low shear conditions favour more open spirals (which are more readily identified observationally). On the other hand this association between spiral structure and solid body rotation was [*not*]{} confirmed by the infrared study of Seigar et al. (2003) who found that spiral structure extends well beyond the region of solid body rotation. On a theoretical level, Sellwood & Carlberg (2014) have argued against the significance of ILRs for wave damping and indeed report a diminished amplitude of spiral structure in regions of low shear (Sellwood & Carlberg 1984), ascribing this result to the relative ineffectiveness of swing amplification in conditions of low shear (Toomre 1981). Here we investigate possible dependences on the rotation curve in two ways. We plot the amplitude of spiral structure against galaxy concentration parameter (C) in Figure 6. We use the IRAC 3.6$\mu$m concentration index (C) data from Bendo et al. (2007), which is defined as the ratio of the radius containing 80 per cent of the light to that containing 20 per cent of the light, i.e. $\frac{r_{80}}{r_{20}}$. Higher values of C imply more concentrated emission and hence presumably also mass; the dynamical influence of a strong central mass concentration is to promote differential rotation (angular velocity declining with radius) and thus - according to the Kormendy & Norman argument - might be expected to be associated with lower amplitude spiral structure. Figure 6 however shows no evidence of an obvious (anti-)correlation between spiral amplitude and concentration index. Figure 7 investigates the dependence of the strength of spiral structure on rotation curve morphology more directly, by evaluating for each galaxy the [*dimensionless shear*]{} defined by Seigar et al. (2005) as: $\frac{A}{\omega} = \frac{1}{2}(1 - \frac{R}{V}\frac{dV}{dR})$ where A is Oort’s constant, $\omega$ is the angular velocity and $V$ the local rotational velocity. In each case this quantity is evaluated for the region of the galaxy over which a spiral structure is detected. Since $A/\omega$ is equal to $-0.5 \times$ the power law index for the dependence of angular velocity on radius, it follows that solid body rotation corresponds to dimensionless shear of zero, a flat rotation curve to $0.5$ and a falling rotation curve to larger values of $\frac{A}{\omega}$. The Kormendy & Norman hypothesis would preferentially associate prominent spiral structure with low values of the shear but Figure 7 shows no evidence for this. The amplitude of the m=2 component is only one measure of the strength of the spiral structure; another is the extent (radial or azimuthal) of the spiral arms. Seigar & James (1998) found ‘a hint of a deficit’ of galaxies with a large bulge to disc ratio and extended spiral arms, quantifying the latter in terms of the angle subtended by the arms. Figure 8 shows the angle subtended by the spiral arms plotted against concentration parameter for this sample (with the fine vertical liens representing the errorbars on the angle subtended), and provides some support for the Seigar & James result. However, if the angle subtended by the arms is plotted against dimensionless shear as shown in Figure 9, the trend is much less convincing, suggesting that disc shear is in fact not an important contributory factor to spiral arm extent. The mild (anti-) correlation between spiral arm extent and concentration parameter would then need an alternative explanation. In summary then, we find no relationship between spiral arm amplitude or angle subtended by the arms with dimensionless shear and no relationship between spiral arm amplitude and concentration parameter. There is arguably a weak anti-correlation between the angle subtended by the arms and concentration parameter but this does not obviously relate to the rotation curve morphology. ![Gas fraction, (M$_{HI + H_{2}}$)/(M$_{HI + H_{2}}$+ M$_{*}$), against average m=2 relative amplitude. The arrow indicates that the gas fraction data for NGC 1566 give a lower limit (see Figure 2 for explanation of symbols ).[]{data-label="gasfrac_av2"}](kckf10r.ps){width="83mm"} ### Dependence on gas fraction and specific star formation rate The gas fraction is plotted against the relative amplitude of the m=2 Fourier component in Figure 10. Where galaxies are not found in Leroy et al. (2008) the HI and H$_2$ masses are taken from Kennicutt et al. (2003). For galaxies that are common to both datasets there is a scatter of up to 0.5 dex (but generally more like 0.2 dex) in gas masses. H$_2$ masses are not given for NGC 1566 and NGC 3031: in these cases the total gas mass plotted is in fact only the HI mass. NGC 3031 has very little detected CO, and so this should not be significant, but for NGC 1566 the difference may be larger: we indicate the fact that the gas fraction is a lower limit in this object via the arrow in Figure 10. Figure 10 suggests a trend of decreasing amplitude of spiral arms with gas fraction (with again the three earliest type galaxies having discrepantly low spiral arm amplitude for their gas fraction values). This trend is in apparent contradiction to the notion that spiral structure is maintained by dynamical cooling provided by the continued production of stars from a dynamically cold gas reservoir. We also note that the three strongly interacting galaxies each lie at the top of the range of spiral amplitude at given gas fraction. In Figure 11 we plot the m=2 amplitude against specific star formation (normalised by the stellar mass). SFR data come from Leroy et al. (2008) except in the cases of NGC 3031 and NGC 1566: in NGC 3031 the SFR data derive from Perez-Gonzalez et al. (2006) and references therein. No SFR data are available for NGC 1566. We note that in principle a positive correlation between spiral arm amplitude and specific star formation rate could be driven by incomplete removal of young stellar emission from the NIR images. However Figure 11 does not suggest any correlations between these quantities (in contrast to the mild positive correlation seen by Seigar & James 1998). This apparent lack of evidence for spiral structure enhancing the global SFR is discussed further in Section 4.2. ![Average m=2 amplitude plotted against specific star formation rate (yr$^{-1}$). See Figure 2 for explanation of symbols. []{data-label="spsfr_av2"}](kckf11r.ps){width="83mm"} Pitch angles {#ch5_pitch} ------------ The pitch angles are calculated by fitting a straight line to the data in the $\phi$ vs ln(R) plots presented in Paper I. The data from each galaxy and wavelength are presented in Table 3. Galaxy method ------------ ---------------------- ---------------------- ---------------------- --- NGC 0628 16.2$_{16.2}^{16.3}$ 16.4$_{16.3}^{16.5}$ 17.3$_{17.3}^{17.3}$ A NGC 0628 15.5$_{14.6}^{16.5}$ 15.7$_{14.8}^{16.7}$ 15.3$_{14.2}^{16.5}$ R NGC 1566 19.7$_{19.6}^{19.7}$ 19.4$_{19.3}^{19.4}$ 22.3$_{22.2}^{22.4}$ A NGC 2403 19.6$_{19.3}^{19.8}$ 19.9$_{19.6}^{20.1}$ - A NGC 2841 6.7$_{ 6.3}^{ 7.2}$ 6.5$_{ 6.1}^{ 7.0}$ - R NGC 2841\* 9.4$_{ 8.5}^{10.4}$ 8.1$_{ 7.5}^{ 8.8}$ - R NGC 3031 23.6$_{23.3}^{23.9}$ 20.7$_{20.7}^{20.7}$ 22.6$_{22.6}^{22.7}$ A NGC 3184 19.5$_{18.9}^{20.0}$ 19.9$_{18.5}^{21.6}$ 18.2$_{17.3}^{19.2}$ R NGC 3198 15.8$_{15.7}^{15.8}$ 15.5$_{15.4}^{15.7}$ 18.2$_{18.2}^{18.2}$ A NGC 3938 15.0$_{14.9}^{15.2}$ 15.4$_{15.2}^{15.5}$ 17.8$_{17.8}^{17.8}$ A NGC 4321 21.3$_{19.4}^{23.7}$ 22.2$_{20.1}^{24.8}$ 16.3$_{14.8}^{18.0}$ R NGC 4579 20.3$_{17.9}^{23.3}$ 16.6$_{14.9}^{18.7}$ 17.9$_{15.2}^{21.6}$ R NGC 5194 13.7$_{12.8}^{14.7}$ 13.6$_{12.6}^{14.8}$ 13.6$_{13.5}^{13.8}$ R NGC 6946 29.3$_{28.8}^{29.8}$ 29.5$_{28.8}^{30.2}$ 24.0$_{23.5}^{24.6}$ A NGC 7793 15.7$_{15.5}^{15.9}$ 15.6$_{15.4}^{15.8}$ 16.2$_{16.1}^{16.3}$ A : Measured pitch angles at $3.6 \mu$m, $4.5 \mu$m and optical (V band). Method; this column lists the methods used to analyse the galaxy; A= azimuthal profiles, R=radial profiles. It is worth noting that the upper and lower bounds quoted in the Table result from accounting for statistical errors; the systematic errors (discussed in the text) may be larger, especially for highly inclined galaxies. \*NGC 2841; the first entry gives the pitch angle calculated for R$>$0.45R$_{25}$, the second entry for R$<$0.45R$_{25}$ (the data used in the figures are those for R$\ge$0.45R$_{25}$). Possibly the first point that is worth commenting on is that, as was seen in Paper I, all these galaxies have phase-radius relationships that are close to logarithmic in nature ($\phi \propto ln(R)$) over at least part of the disc of the galaxy. Although logarithmic spiral arms are predicted by QSSS theories (and subsequent related theories such as global spiral modes) this does not automatically lead to the conclusion that spiral arms must obey these relationships; that they do is notable. It is unlikely that this is merely a selection bias since the radial ranges for detectable spiral structure quoted throughout this work are not [*defined*]{} as regions of logarithmic spiral structure, even though this turns out to be the case in practice. There is one galaxy, NGC 2841 (pitch angle $\sim$7$^o$), for which the pitch angle should be treated with an extra degree of caution, because as described in Paper I the high inclination and complex structure make the extraction of a single pitch angle problematical (see caption to Table 3). The errors quoted for all galaxies in Table 3 are the errors associated with accurately determining the phase of the m=2 component in the Fourier fits or phase determined from radial profiles; these errors do not take into account the uncertainties in the galaxies’ ellipticity or position angle. It is difficult to quantify the associated uncertainties in the pitch angle due to inaccuracies in galaxy orientation, although Block et al. (1999) found that by varying the inclination and position angle used to fit a highly inclined galaxy (i$\ge$60$^o$) the pitch angle could change by as much as 10 per cent. Nevertheless, it is striking that the pitch angles in the two infrared bands and optical (V) band are in general so similar to each other, with the differences being similar to the errors quoted in Table 3. It is worth considering whether there is a systematic variation in pitch angle measured from the azimuthal profile and radial profiles. If anything, the radial method should produce pitch angles that are systematically smaller, due to measuring more of the contamination from young populations which are expected to be more tightly wound. The only galaxy for which a direct comparison is available is NGC 0628: the small difference ($1 \deg$) is within the errors. ### Comparison with previous (H$\alpha$ and K band ) determinations We now compare the pitch angles found in this work to results from optical (H $\alpha$) data (Figure 12) for the set of $9$ galaxies which overlap our sample and that of Kennicutt (1981). It can be seen that the late type spirals agree well, but that there is a systematic difference for the early types, Sab and Sb. The two galaxies that are most discrepant (NGC 3031 (Sab) and NGC 4579 (Sb)) are notably those that were shown in Paper I to exhibit the most convincing [*offsets*]{} between gas shocks and stellar density maxima. The sense of the offset (gas shock precedes stellar density maximum inward of corotation) combined with the fact that one would expect star formation to follow the shock after a fixed time (and therefore at an angular offset that decreases with radius for a differentially rotating galaxy) are both such as to explain smaller pitch angles in the H$\alpha$ emission (assuming this traces recent star formation). Our sample size does not allow us to say whether this effect is a general feature of early type galaxies. We can also compare some of our optical pitch angle determinations with those reported in the literature (see Ma 2001 and Davis et al. 2012: $7$ members of our sample have B band pitch angle determinations also listed in either or both of these studies). As reported in the latter paper (Table 3) the agreement between our results and these other determinations is generally good (to within a few degrees or better in most cases). Turning now to previous pitch angle determinations in the NIR we note that there is no overlap in sources between the galaxies listed in Table 3 and the galaxies previously studied in the K band by Seigar & James (1998) and Seigar et al. (2005). These two previous K band studies report rather different pitch angle distributions: Seigar & James 1998 record low values (generally in the $5-10 \deg$ range) while the sample of Seigar et al. is broadly distributed in pitch angle, including a number of objects with $i > 30 \deg$. In that respect the latter distribution is similar to that found in the B and I band by Davis et al. (2012). Our own results follow a much narrower distribution ( mainly lying in the range $15-20 \deg$ in the optical as well as NIR bands). This result is most likely a consequence of our small sample size. ![Pitch angles plotted against data from Kennicutt (1981) obtained using H$\alpha$ data. The pitch angles of late type spirals are consistent (to within the errors). Early type spirals have systematically larger pitch angles measured in the NIR.[]{data-label="pitch_compare"}](kckf12r.ps){width="60mm"} ![Pitch angles plotted against Hubble type. See Figure 2 for an explana tion of the symbols[]{data-label="hub_pitch"}](kckf13r.ps){width="83mm"} ![Pitch angles plotted against mass concentration index C. See Figure 2 for an explanation of the symbols.[]{data-label="conC_pitch"}](kckf14r.ps){width="83mm"} ![Pitch angles plotted against the dimensionless shear parameter. See Figure 2 for explanation of symbols.[]{data-label="hub_pitch2"}](kckf15r.ps){width="83mm"} {width="83mm"} ### Dependence on Hubble type Figure 13 plots the pitch angle of the m=2 components from galaxies in the detailed sample against Hubble type. Roberts et al. (1975) proposed a relationship between Hubble type and pitch angle under the assumption that the latter is fundamentally a measure of bulge to disc ratio and using the dispersion relation of Lin & Shu (1964) to quantify the expectation that more loosely wound patterns are to be found in systems that are more disc dominated. The appearance of Figure 13 is strongly driven by the two galaxies with unusually large and small pitch angles in our sample (NGC 6941 and NGC 2841) and the fact that the former is of later Hubble type creates an impression of a correlation. However, the small range of measured pitch angles for the rest of the sample undermines the case for a correlation. Note that previous NIR results ( Seigar & James 1998) also found no clear evidence of such a correlation (and again for the reason that the bulk of objects in the sample were concentrated over a narrow range of pitch angles). Moreover the $H \alpha$ study of Kennicutt (1981) found only a weak relationship between Hubble type and pitch angle ( which is surprising given that the Hubble classification system implicitly considers the pitch angle of the spiral arms). ### Dependence on rotation curve morphology and concentration. Figure 14 shows likewise that there is no obvious relationship between pitch angles and central concentration , again contradicting the expectation that disc dominated systems should be associated with more open spiral patterns. This is consistent with the NIR results of Seigar & James (1998) and also with Kennicutt (1981) who found only a weak correlation with bulge to disc ratio. In Figure 15 we plot pitch angles against the dimensionless shear parameter of Seigar et al. (2005). These data show some weak evidence for more open structures in the case of galaxies with strongly rising rotation curves (although this impression is strongly driven by the two galaxies with outlying pitch angle values, NGC 2841 and NGC 6946). The relationship between pitch angles and galactic shear is much less evident than in the study of Seigar et al. (2005) and in particular we do not find the large pitch angles ($i > 30$ deg.) at low shear ($< 0.4$) that were reported in this work. An association between shear and pitch angle (in the sense found by Seigar et al. ) is expected theoretically (Lin & Shu 1964, Bertin & Lin 1996, Fuchs 2000, Baba et al. 2013) and has recently been explored numerically by Grand et al. (2013) and Michikoshi & Kokubo (2014): these simulations show much larger pitch angles in the case of systems with rising rotation curves than we find here. Finally, we plot in Figure 16 the dependence of pitch angle upon the galactic rotation velocity in the flat portion of the rotation curve (see Section 2.1.2). The appearance of an anti-correlation is strongly driven by one galaxy (NGC 2841) which combines a small pitch angle with a large rotational velocity, though as noted above the pitch angle in this case is particularly uncertain due to its large inclination and complex structure (Paper I). We thus find no evidence for the strong anti-correlation between pitch angle and $v_{flat}$ that was found in the optical by Kennicutt & Hodge (1982). ### Summary The results presented in this section show no strong correlations between pitch angles and other galaxy parameters. The presence or absence of apparent correlations in Figures 13 -16 need to be interpreted in the context of the fact that $11/13$ of our galaxies have pitch angles within a rather narrow range, with only two galaxies having values that are either $< 10 \deg$ or considerably greater than $20 \deg$ (as such, our sample is almost all consistent with the predictions of swing amplification theory which predicts maximum pitch angles in the range $15-20 \deg$; Toomre 1981, Oh et al. 2008). However, as discussed in Section 2.2.1 above, we believe that this narrow range reflects the small sample size rather than differences in analysis method or the different range of wavebands employed compared with previous studies. Nevertheless the strongly bunched distribution means that the placing of the two most discrepant galaxies in each plot is a strong driver of whether there is an apparent correlation or not. With this caveat, we find no evidence to support the association between large pitch angles and rising rotation curves found by Seigar et al. (2005) in the near infrared, nor between large pitch angles low rotation velocities found by Kennicutt & Hodge (1982) in the optical. We have also drawn attention to the suggestion that early type galaxies may be more tightly wound in H$\alpha$ than in the near infrared. Finally, it is evident that the three strongly interacting galaxies do not occupy a distinctive region in Figures 13-16: clearly spiral structure that is driven by galaxy interactions cannot be distinguished from that in isolated galaxies on the grounds of near infrared pitch angle. \[hub\_pitch2\] The relationship between galactic environment and the nature of spiral structure ================================================================================ In the preceding plots we differentiated the three galaxies which we judge to be undergoing the strongest tidal interactions. Here we present the analysis of galactic environment that led us to this conclusion and examine how the strength of tidal interaction affects the amplitude of spiral structure. We used the NASA/IPAC Extragalactic Database to search for companions to the $13$ galaxies considered in this paper. The companions identified are detailed in Table 4 and all lie within within $\pm$400kms$^{-1}$ of the target galaxy. The mass ratio was calculated using the relative *B* band magnitudes and assumed a constant mass -to-light ratio. Byrd & Howard (1992) proposed the tidal parameter $P$ as a measure of the strength of galactic interactions: $$P = \frac{M_{c}/M_{g}}{(r/R)^{3}} \label{ch5_eq1}$$ where $M_{g}$ is the galaxy mass, $M_{c}$ the companion mass, $R$ the galaxy radius, and $r$ is the distance of closest approach. Byrd & Howard suggested a minimum value for P of 0.01 and 0.03 for pro- and retro-grade interactions respectively to induce a spiral response to the centre of the galaxy. In Paper I we analysed the incidence of ‘strong interactions’ (corresponding to galaxies with $P$ values exceeding a given threshold) in the sample of galaxies that did and did not exhibit grand design spiral structure. We found that this incidence was only different (and higher in grand design spirals) if a threshold value of $P = 0.01$ was adopted, thus confirming the estimate of Byrd & Howard regarding the value of $P$ required for significant tidal perturbation of the galaxy. Whereas in Paper I we assessed companions down to a limiting magnitude of $B=15$ and such that they lay within 10 scalelengths ( $\frac{R_{proj}}{R_{25}}$ $<$ 10.0) of the main galaxy, this criterion did not correspond to a fixed $P$ value (the $6$ galaxies in our sample that had companions conforming to this criterion are marked with a $+$ symbol in Table 4). In the present paper our focus is to discover whether ([*within the sample of grand design spirals*]{}) there is a significant correlation between the tidal parameter $P$ and the amplitude of spiral structure. Accordingly we examine all the companions in the correct velocity range lying within $20$ scale lengths and evaluate the maximum $P$ value that is contributed by any of the companions. The mass ratios, normalised distances and $P$ values are listed in Table 4 and $P$ is plotted against spiral amplitude in Figure 17. We see that most of the sample have $P$ values of around $10^{-5}$ and indeed we found in Paper I that such values are also typical of the sample in which grand design spiral structure was not detected. We also note that a difficulty in relating the observed $P$ value to the importance of tidal interactions: while the projected distance may be a substantial under-estimate of the three-dimensional separation, spiral structure is likely to reflect the strength of the interaction at a previous pericentre passage and thus the projected separation is in this sense an over-estimate of the relevant value. Galaxy Companion $\frac{R_{proj}}{R_{25}}$ Relative mass P ------------ -------------- --------------------------- --------------- ----------------------- NGC 0628 + UGC 01176 9.6 0.02 $2.2 \times 10^{-5}$ NGC 1566 + NGC 1581 9.7 0.05 $5.5 \times 10^{-5}$ NGC 2403 NGC 2366 20. 0.1 $1.3 \times 10^{-5}$ NGC 2841 UGC 4932 6.1 0.01 $4.4 \times 10^{-5}$ NGC 3031 + M82 2.7 0.28 $1.4 \times 10^{-2}$ NGC 3184 KHG 1013+414 7.4 0.01 $2.4 \times 10^{-5}$ NGC 3198 \* \* \* $5.5 \times 10^{-5}$ NGC 3938 - - - - NGC 4321 + NGC 4323 1.4 0.01 $3.4 \times 10^{-3}$ NGC 4579 + NGC 4564 9.8 0.25 $2.6 \times 10^{-4}$ NGC 5194 + NGC 5195 0.8 0.25 $4.9 \times 10^{-1}$ NGC 6946 - - - - NGC 7793 - - - - : Values of $P$ together with list of companions, mass ratios and normalised separations taken from the literature for all galaxies with a companion contributing $ P > 10^{-5}$ (the $6$ galaxies marked with a $+$ sign are those that were deemed to be interacting according to the ‘inclusive’ definition of Paper I: see text for details). \* In the case of NGC 3198 there are two galaxies (SDSS J 101848.78+452137.1 and VV 834 NEDOR) that contribute nearly equal $P$ values and the quoted value is the sum of these. ![P (see equation (1) plotted against m=2 relative amplitude. P gives a measure of the tidal interaction between two galaxies, and correlates strongly with the strength of the spiral. The galaxies with the largest P values ( NGC 5194, NGC 3031 and NGC 4321 in descending order) are marked with the large crosses in the other Figures.[]{data-label="P_av2"}](kckf17r.ps){width="83mm"} Despite these caveats, Figure 17 demonstrates that the amplitude of spiral responses does indeed correlate positively with the tidal $P$ parameter (in fact Figure 17 shows a more convincing correlation than any of the other plots we present here). The fact that $3/13$ galaxies have $P < 10^{-5}$ and yet display a significant spiral pattern implies that tidal interaction [*is clearly not a pre-requisite*]{} for inducing spiral structure (and indeed we have seen in Paper I that at least half the sample of galaxies without detected grand design structure have $P$ values of $10^{-5}$ or above). On the other hand, Figure 17 presents convincing evidence that where there is strong interaction the [*amplitude of spiral structure is enhanced*]{}. We have seen that the pitch angles of such ‘induced’ spirals are indistinguishable from spiral patterns in apparently isolated galaxies. We have also seen that where we have tentatively identified trends in the amplitude of spiral features (i.e. the positive correlations with $v_{flat}$ and stellar mass shown in Figures 4 and 5 and the mild anti-correlation with gas content shown in Figure 10) then the strongly interacting galaxies (i.e. those with $P > 10^{-3}$) roughly follow these trends but at enhanced amplitude. Once again, we however caution against over-interpretation given our limited sample size. \[hubb\_m3m2\] \[hub\_pitch2\] \[rotgrad\_amp2\] The response to stellar spiral structure: gaseous shocks and associated star formation. {#ch5_responses} ======================================================================================= The first half of this paper studied the effects of galaxy type, morphology and environment on the stellar spiral structure. However, as well as being influenced by the galaxy, a global spiral is expected to influence the galaxy in which it is located. The following section considers the effect of the spiral on the structure of the gas and the distribution of star formation in galaxies. Gas response {#ch5_gas} ------------ ### Azimuthal offsets in the location of spiral shocks Here we briefly recapitulate the results of Paper I with regard to the location of shocks in the gaseous component (as traced by $8 \mu$m emission) with respect to the spiral in the stellar mass density (as traced at $3.6 \mu$m). Of the $13$ galaxies in the detailed sample considered here, $4$ galaxies (NGC 2403, NGC 2841, NGC 6946, NGC 7793) exhibited a flocculent response at $8 \mu$m so it is not possible to talk meaningfully about the azimuthal alignment between the spirals manifest in the gas and the stars. Amongst the remaining $9$ galaxies there is a general tendency for the $8 \mu$m spiral to be located on the trailing (concave) side of the stellar ($3.6 \mu$m) spiral, i.e. (at radii inwards of corotation) being [*upstream*]{} of the stellar spiral. Moreover the $8 \mu$m spiral is generally somewhat more tightly wound. This is most clearly demonstrated in the case of NGC 3031 (M81) where the azimuthal offset increased systematically with radius and whose behaviour was the subject of detailed analysis in Kendall et al. 2008. Four other galaxies (NGC 3184, NGC 3198, NGC 3938 and NGC 4579) also demonstrated evidence of a systematic increase of offset angle with radius. In the remaining cases, the scatter in azimuthal offset is sufficiently large to prevent the identification of an obvious radial trend. The possible significance of an upstream azimuthal offset between the shock in the gas and the maximum stellar density is discussed in Kendall et al. 2008. A scenario that predicts such an offset (with magnitude increasing with radius) is the case of a rigidly rotating spiral mode whose lifetime is sufficiently long to permit the gas to achieve a steady state flow in the imposed potential. In this case the solutions of Roberts (1969) and Shu et al. (1972) predict that the gas shocks upstream of the gas inwards of corotation, with the magnitude of this offset increasing with radius, a result that has been verified through hydrodynamic simulations (Gittins & Clarke 2004). Kendall et al. (2008) used these results to infer the corotation radius in NGC 3031 and obtained a value that is in reasonable accordance with other estimates in the literature. On the other hand, Clarke & Gittins (2006) and Dobbs & Bonnell (2008) undertook hydrodynamic simulations of gas in the case that the stellar potential derived from a high resolution re-simulation of the N-body calculations reported in Sellwood & Carlberg (1984). Notably in these simulations with a ‘live’ galactic potential, the gas shocks tended to trace the regions of instantaneous maximum stellar density and exhibited no systematic azimuthal offset with respect to the stellar spiral. A clear difference between this simulation and the idealised case involving a single long lived spiral mode is that, in the N-body calculations, spiral features come and go on a timescale of a few orbits. Clarke & Gittins related the gas morphology in this case to the fact that the gas did not have time to adjust to a steady state configuration on the timescale on which individual spiral features established and dissolved. In the case of spiral structure induced by a strong galactic encounter (as in the modeling of M51 - NGC 5194 - by Dobbs et al. 2010) no azimuthal offset was found between the gas and stellar spirals. How do these theoretical results relate to the range of results that we find in our sample? We can divide our sample (as detailed above) into $4$ galaxies with flocculent $8 \mu$m structure, $5$ with evidence for a systematic upstream offset with magnitude broadly increasing with radius and the remaining $4$ as having no simple offset pattern (we denote these groups F(locculent), R(egular) and C(omplex): see Paper I for the phase diagrams on which this classification is based. We find no systematic differences between the galactic properties in these three groups apart from the fact that all the galaxies in the F(locculent) category are isolated ($P < 10^{-5}$; see Table 2); the R(egular) and C(omplex) galaxies occupy a wide range of $P$ values, although notably the galaxy which is undergoing by far the strongest interaction (NGC 5194) is in the C(omplex) category, in line with the simulations of this encounter by Dobbs et al. (2010). We conclude from this that many of the isolated galaxies exhibit the kind of gas response seen in simulations based on live galactic potentials and that the most strongly interacting galaxy also agrees with simulations inasmuch as it exhibits no clear offset; the R(egular) group however exhibit a trend suggestive of a more long lived spiral pattern. We re-emphasise that this group is heterogeneous in terms of its galactic properties. To date there has been no detailed quantitative study of how tightly the offset data can be used to constrain the lifetime and nature of spiral structure in simulations. ### The strength of the shock as traced by $8 \mu$m emission In addition to the position of the shock, it is instructive to look at the strength of the gas shocks as a function of the strength of the stellar spiral, since a larger spiral amplitude is expected to trigger a larger shock. We compared these properties for all galaxies in the sample, not just those for which the offset work was carried out. In contrast to the stellar mass maps, the gas response cannot be normalised by an axisymmetric component, so instead we simply divide the 8$\mu$m azimuthal profile into ‘on’ and ‘off’ arm regions; ‘on’ arm is defined as being angles $\pm\frac{\pi}{4}$ of the m=2 peaks (‘off’ arm is the other $\pi$ degrees, in two arcs). The response is calculated by taking the ratio of the highest peaks in the ‘on’ arm region to the average ‘off’ arm flux. Note that we use azimuthal profile data for analysis of the $8 \mu$m emission for all galaxies in the sample, including those whose amplitude at $3.6 \mu$m is obtained through radial profile analysis. ![The $8 \mu$m gas shock strength indicator (defined as the average of the four highest peaks in the 8$\mu$m profile in the ‘on’ arm region, normalised by the average value of the 8$\mu$m profile in the ‘off’ arm region) plotted against average m=2 amplitude at $3.6 \mu$m. See Figure 2 for an explanation of the symbols.[]{data-label="shockresp_m2"}](kckf18r.ps){width="83mm"} ![The $24 \mu$m excess ratio (see text for definition). See Figure 2 for an explanation of the symbols.[]{data-label="shockresp_m2"}](kckf19r.ps){width="83mm"} Figure 18 shows the trends in the gas response as a function of m=2 amplitude and show that, as expected, the gas response will tend to be stronger for a larger stellar wave amplitude. (Note that we expect such a correlation to be weakened somewhat by the fact that $8 \mu$m PAH features may be depleted by the strong UV emission associated with the enhanced star formation in shocked gas.) Star formation. {#ch5_SFR} --------------- In this section we apply the same method as in Section 4.1.2 to define ‘on’ and ‘off’ arm regions, using the *Spitzer* MIPS (Rieke et al. 2004) 24$\mu$m wavelength as a tracer of star formation (Calzetti et al. 2007). $24 \mu$m emission is dominated by reprocessed radiation from young massive stars as demonstrated by its excellent spatial correlation with other diagnostics associated with such stars (see Relano & Kennicutt 2009 for such a demonstration in the case of H$\alpha$ emission in M33). Helou et al. (2004) argued that the stellar continuum contributes at most a few per cent at $24 \mu$m In Figure 19 we plot the ratio of the [*excess*]{} emission at 24$\mu$m in the arm compared with the inter-arm region (where the excess is defined as the mean emission levels in that region minus the minimum emission level in that radial ring: note that in the case of the $8 \mu$m data this minimum level is essentially zero in many regions and so in contrast Figure 18 presents the ratio of unsubtracted values.) Figure 19 is qualitatively similar to Figure 18 and shows a clear dependence of the star formation response (as measured at $24 \mu$m) and the amplitude of variations in the stellar potential. The levels of contrast can be used to interpret Figure 11, noting that the dispersion in specific star formation rates and in the $24 \mu$m excess ratios are comparable. This suggests that strong spiral structure can contribute significantly to the dispersion in specific star formation rates in our sample. Nevertheless, it cannot be the only driver since there are some notable counter-examples (for example, NGC 3031 with its strong $24 \mu$m excess ratio and low specific star formation rate. Conclusions {#discuss} =========== By far the strongest correlation that we have identified (Figure 17) is between the [*amplitude*]{} of spiral structure and the strength of tidal interaction as measured by the $P$ parameter (equation (1)). We conclude - in line with our conclusions in Paper I - that a close tidal encounter appears to be a sufficient but not necessary condition for prominent spiral features. NGC 3198 is a good example of a galaxy that is unbarred and isolated and yet shows the ‘grand design structure’ that permits its inclusion in the detailed sample studied in this paper. Nevertheless, the amplitude of its spiral features is admittedly lower than the strongly interacting galaxies in our sample and moreover the $m=2$ mode is less dominant than in these galaxies. We have indicated the galaxies with the three largest values of $P$ (the strongly interacting galaxies) by the large crosses in all the other correlation plots. These galaxies are not atypical for the sample with regard to their pitch angles and the radial extent of spiral structure (as measured by the azimuthal angle subtended by the arms in Figures 8 and 9) and so do not strongly influence these plots. They are however all rather massive and gas deficient galaxies of type Sbc or earlier and we must therefore be mindful that apparent correlations may be driven by this association. This may be particularly relevant to the interpretation of Figures 4, 5 and 10. Our sample covers a rather narrow range of pitch angles and it is thus unsurprising that we have not identified any correlations between pitch angles and other galactic parameters. Figure 12 may throw some light on previous comparisons between the pitch angles of spiral structure in the optical and NIR: for example in the most recent and sophisticated analysis of this issue by Davis et al. (2012) there is generally good agreement between pitch angles in the B and I bands. Discrepant galaxies have a tendency to have larger pitch angles at longer wavelengths. Figure 12 (which compares NIR pitch angles with pitch angles measured in H$\alpha$, an optical star formation indicator) suggests that in our sample a discrepancy of this sign is evident only for earlier Hubble types. This would suggest - if borne out in larger samples - that an association between early Hubble types and tightly wound arms may only be manifest in the optical (Clarke et al. 2010). In the most discrepant galaxies (NGC 3031 and NGC 4579) this difference in pitch angles can be explained by our previous demonstration (Kendall et al. 2008, 2011) that in these galaxies the spiral pattern traced at $8 \mu$m in shocked gas is significantly displaced upstream with respect to the NIR (stellar) arms. In a differentially rotating galaxy it would then be expected that the spiral traced by star formation indicators should be more tightly wound than the stellar arms. The origin of azimuthal offsets is discussed in our previous papers; it is currently unclear whether there is a more general association between such offsets and galaxies of early Hubble type. We also draw attention to the other strong correlation that we have found, i.e. the demonstration in Figures 18 and 19 of a clear dependence of the amplitudes of spiral structure in star formation ($24 \mu$m) and shock ($8 \mu$m) tracers on the amplitude of potential variations as traced in the NIR. The amplitude of spiral structure at $24 \mu$m demonstrates that the star formation within the arms cannot on its own account for all the variation in specific star formation rate exhibited by the galaxies in our sample and there are indeed objects with strong arm contrast yet low specific star formation rates. However the range of arm amplitudes at $24 \mu$m suggest that variation of strength of spiral features plays a significant contributory role in setting the specific star formation rates in disc galaxies. Acknowledgments. ================ Many thanks to Bob Carswell for much useful advice regarding the use of IDL, in particular the CURVEFIT program. We are indebted to Hans-Walter Rix for the idea of using colour-correction to make mass maps from optical data. Many thanks also go to Jim Pringle, Clare Dobbs, Stephanie Bush, Giuseppe Bertin and Jerry Sellwood for useful discussions over the course of this work and to Phil James and Marc Seigar for valuable observational input. We also thank the referee for constructive comments. This work makes use of IRAF. IRAF is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation. Baba, J., Saitoh, T., Wada, K., 2013. ApJ 763,46 Bendo, G., Calzetti, D., Englebracht, C. et al., 2007. MNRAS 380, 1313 Bertin, G., Lin, C., 1996. in Spiral structure in galaxies a density wave theory, Publisher: Cambridge, MA MIT Press. Block D. L., Bertin G., Stockton A., Grosbol P., Moorwood A. F. M., Peletier R. F., 1994, A&A, 288, 365 Block D. L., Bertin G., Stockton A., Grosbol P., Moorwood A. F. M., Peletier R. F., 1994, A&A, 288, 365. Block D. L., Puerari I., Frogel J. A., Eskridge P. B., Stockton A., Fuchs B., 1999, Ap&SS, 269, 5 Block D. L., Wainscoat R. J., 1991, Nature, 353, 48 Byrd G. G., Howard S., 1992, AJ, 103, 1089 Calzetti, D., Kennicutt, R., Engelbracht, C. et al. , 2007. Ap J 666, 870 Cepa J., Beckman J. E., 1990, ApJ, 349, 497 Clarke, C., Gittins, D., 2006. MNRAS 371,530 Clarke, C., Kendall, S., Kennicutt, R., 2010 in AIP Conf. Proceedings 1242,129 Davis, B., Berrier, J., Shields, D. et al., 2012. ApJ S 199,33 Dobbs C. L., Bonnell I. A., 2008, MNRAS, 385, 1893 Dobbs, C.L., Theis, C., Pringle, J., Bate, M., 2010. MNRAS 403, 625 Elmegreen D. M., Chromey F. R., Bissell B. A., Corrado K., 1999, AJ,118,2618 Elmegreen D. M., Elmegreen B. G., 1984, ApJS, 54, 127 - 1987, ApJ, 314, 3 Elmegreen, D.M. et al 2011, ApJ, 737, A32 Fuchs, B., 2000. In Dynamics of Galaxies from the Early Universe to the Present, 15th IAP meeting, eds. Combes, F., Mamon, G., Charmandaris, V., p.53 Gerola H., Seiden P. E., 1978, ApJ, 223, 129 Gittins D. M., Clarke C. J., 2004, MNRAS, 349, 909 Grand, M., Kawata, D., Cropper, M., 2013. A & A 553, A77 Grosbol P., Patsis P. A., Pompei E., 2004, A&A, 423, 849 Helou, G., Roussel, H., Appleton, P. et al., 2004. ApJS 154,253 Kendall S., Kennicutt R. C., Clarke C., Thornley M. D., 2008, MNRAS, 387, 1007 Kendall, S., Kennicutt, R., Clarke, C., 2011. MNRAS 414,538 (Paper I) Kennicutt, Jr R. C., Hodge P., 1982, ApJ, 253, 101 - 1981, AJ, 86, 1847 Kennicutt Jr. R. C, 1998, ApJ, 498, 541 Kennicutt Jr. R. C., Armus L., Bendo G., et al. 2003. PASP 115,928 Kormendy J., Norman C. A., 1979, ApJ, 233, 539 Leroy, A., Walter, F., Brinks, E., et al. 2008. AJ 136,2782 Lin C. C., Shu F. H., 1964, ApJ, 140, 646 - 1966, Proceedings of the National Academy of Science, 55,229 Lindblad B., 1964, Astrophysica Norvegica, 9, 103 Ma, J., 2001. ChJAA, 1, 395 Michikoshi, S., Kokubo, E., 2014. ApJ 787,174. Mueller M. W., Arnett W. D., 1976, ApJ, 210, 670 Oh S. H., Kim W.-T., Lee H. M., Kim J., 2008, ApJ, 683, 94 Perez-Gonzalez P. G., Kennicutt Jr. R. C., Gordon K. D., et al., 2006. ApJ 648,987 Relano, M., Kennicutt, R. Jr., 2009. ApJ 699,1125 Rieke G. H., Young E. T., Engelbracht C. W., et al. 2004. ApJ S 154,25 Roberts, W. W., 1969, ApJ, 158, 123 Roberts Jr. W. W., Roberts M. S., Shu F. H., 1975, ApJ, 196, 381 Schweizer, F., 1976. ApJS, 31, 313 Seigar M. S., Block D. L., Puerari I., Chorney N. E., James P. A.,2005, MNRAS, 359.1065 Seigar, M., Bullock, J., Barth, A., Ho, L., 2006. ApJ 645,1012 Seigar M. S., Chorney N. E., James P. A., 2003, MNRAS, 342, 1 Seigar M. S., James P. A., 1998, MNRAS, 299, 685 - , 2002, MNRAS, 337, 1113 Sellwood, J., Carlberg, R., 1984. ApJ 282,61 -, 2014. ApJ 785,137 Sheth, K. et al. 2010, PASP, 122, 1397 Shu, F., Millone, V., Gebel, W., Yuan, C., Goldsmith, D., Roberts, W., 1972. ApJ 173,557 Thornley M. D., 1996, ApJ, 469, L45 Toomre A., 1981, in Structure and Evolution of Normal Galaxies, Fall S. M., Lynden-Bell D., eds., pp. 111-136 Tully R. B., Fisher J. R., 1977, A&A, 54, 661 van den Bergh, S., 1960a, ApJ 131,215 van den Bergh, S., 1960b. ApJ 131,558 \[lastpage\] [^1]: E-mail:cclarke@ast.cam.ac.uk

--- abstract: | We describe the second telescope of the Wise Observatory, a 0.46-m Centurion 18 (C18) installed in 2005, which enhances significantly the observing possibilities. The telescope operates from a small dome and is equipped with a large-format CCD camera. In the last two years this telescope was intensively used in a variety of monitoring projects. The operation of the C18 is now automatic, requiring only start-up at the beginning of a night and close-down at dawn. The observations are mostly performed remotely from the Tel Aviv campus or even from the observer’s home. The entire facility was erected for a component cost of about 70k\$ and a labor investment of a total of one man-year. We describe three types of projects undertaken with this new facility: the measurement of asteroid light variability with the purpose of determining physical parameters and binarity, the following-up of transiting extrasolar planets, and the study of AGN variability. The successful implementation of the C18 demonstrates the viability of small telescopes in an age of huge light-collectors, provided the operation of such facilities is very efficient. author: - Noah Brosch - David Polishook - Avi Shporer - Shai Kaspi - Assaf Berwald - Ilan Manulis title: The Centurion 18 telescope of the Wise Observatory --- Introduction ============ The Wise Observatory (WO, see http://wise-obs.tau.ac.il) began operating in 1971 as a Tel-Aviv University (TAU) research laboratory in observational optical astronomy. It is located on a high plateau in the central part of the Negev desert (longitude 34$^\circ$45’ 48“ E, latitude 30$^\circ$35’45” N, altitude 875 m, time zone is -2 hours relative to Universal Time). The site is about 5 km west of the town of Mitzpe Ramon, 200 km south of Tel-Aviv and 86 km south of Beer Sheva. An image of the Wise Observatory is shown in Figure \[fig:WiseObs\]. [ ]{} The WO was originally equipped with a 40-inch telescope (T40). The Boller and Chivens telescope is a wide-field Ritchey-Chrétien reflector mounted on a rigid, off-axis equatorial mount. The optics are a Mount Wilson/Palomar Observatories design, consisting of a 40-inch diameter clear aperture f/4 primary mirror, a 20.1-inch diameter f/7 Ritchey-Chrétien secondary mirror, and a quartz corrector lens located 4 inches below the surface of the primary mirror, providing a flat focal field of up to 2.5 degrees in diameter with a plate scale of 30 arcsec mm$^{-1}$. An f/13.5 Cassegrain secondary mirror is also available, but is hardly used nowadays. This telescope was originally a twin of the Las Campanas 1m Swope telescope, described by Bowen & Vaughan (1973), though the two instruments diverged somewhat during the years due to modifications and upgrades. The telescope is controlled by a control system located in the telescope room. In its 36 years of existence the observatory has kept abreast of developments in the fields of detectors, data acquisition, and data analysis. In many instances the observations can now be performed remotely from Tel Aviv, freeing the observer from the necessity to travel to the observing site. The efficiency of modern detectors implies that almost every photon collected by the telescope can be used for scientific analysis. The on-going modernization process allowed landmark studies to be performed and generations of students to be educated in the intricacies of astronomy and astrophysics. Some of these students are now staff members of the Physics and Astronomy Department at TAU or at other academic institutions in Israel or overseas. The one-meter telescope is over-subscribed, with applications for observing time exceeding by $\sim$50% the number of available nights. This demonstrates the vitality of the observatory as a research and academic education facility, even though on a world scale the size of the telescope shrank from being a medium-sized one in the early-1970s to being a “small” telescope nowadays. The Wise Observatory operates from a unique location, in a time zone between India and Greece and in a latitude range from the Caucasus to South Africa, where no other modern observatories exist, and at a desert site with a large fraction of clear nights. Thus, even though its 1m telescope is considered small, it is continuously producing invaluable data for the study of time-variable phenomena, from meteors and extrasolar planet searches to monitoring GRBs and microlensing events, to finding distant supernovae and “weighing” black holes in AGNs. A cursory literature search shows that the Wise Observatory has one of the greatest scientific impacts among 1m-class telescopes. One research aspect that developed into a major WO activity branch is of time-series studies of astronomical phenomena. A project to monitor photometrically and spectroscopically Active Galactic Nuclei (AGNs) is still running, following about 30 years of data collection. Other major projects include searches for supernovae or for extrasolar planets (using transits or lensing events), observations of novae and cataclysmic variables, studies of star-forming galaxies in a variety of environments, and studies of Near Earth Objects (NEOs) and other asteroids. These studies, mainly part of PhD projects, are observation-intensive and require guaranteed telescope access for a large number of nights and for a number of years. The oversubscription of the available nights on the T40, the need to follow-up possible discoveries by small telescopes with long observing runs, and a desire to provide a fallback capability in case of major technical problems with the T40, required therefore the expansion of the WO observational capabilities. Small automatic, or even robotic, instruments provide nowadays significant observing capability in many astronomy areas. Combining these small telescopes into a larger network can increase their scientific impact many-fold. This is currently done with the GCN (Barthelmy et al. 1998). Examples of such instruments are REM (Zerbi 2001), WASP and super-WASP (e.g., Pollacco 2006), HAT (Bakos 2004), and ROTSE-III (e.g., Akerlof 2003; Yost 2006), and very recently KELT (Pepper 2007). Among the larger automatic instruments we mention the Liverpool Telescope (e.g., Steele 2004) and its clones. Such automatic instruments enable the exploration of the last poorly studied field of astronomy: the temporal domain. However, a first step before developing a network of automatic telescopes is demonstrating feasibility at a reasonable cost for a single, first instrument. This paper describes such an experiment at the Wise Observatory in Israel. The telescope {#txt:Telescope} ============= To enhance the existing facilities with special emphasis on time-series astronomy, we decided in 2002 to base an additional observing facility on as many off-the-shelf hardware and software components as possible, to speed the development and bring the new facility on-line as soon as possible. We decided to acquire a Centurion 18 (C18) telescope manufactured by AstroWorks, USA. The telescope was delivered towards the end of 2003 and operated for a year in a temporary enclosure. From 2005 the telescope operates in its permanent dome, which is described below. The C18 has a prime-focus design with an 18-inch (0.46-m) hyperbolic primary mirror figured to provide an f/2.8 focus. The light-weighted mirror reflects the incoming light to the focal plane through a doublet corrector lens. The telescope is designed to image on detectors as wide as regular (35-mm) camera film though with significant edge-of-field vignetting, but we now use a much smaller detector, and the images are very reasonable indeed (FWHM=$\sim$2".9 at the edge of the CCD, only 13% worse than at field center, with most image size attributable to local seeing). [ ]{} The focal plane is maintained at the proper distance from the primary mirror by a carbon-reinforced epoxy plastic (CREP) truss tube structure. The support tubes allow the routing of various wires and tubes (see below) through the structure, providing a neat construction and much less vignetting than otherwise. The CREP structure has a very low expansion coefficient; this implies that the distance between the primary and the focal plane hardly changes with temperature. The stiff structure and the relatively low loading of the focal plane imply that the telescope hardly flexes with elevation angle. The optical assembly uses a fork mount, with the right ascension (RA) and declination (DEC) aluminum disk drives being of pressure-roller types. The steel rollers are rotated by stepping motors and both axes are equipped with optical sensors, thus the motions of the telescope can be controlled by computer. The fork mount and the truss structure limit the telescope pointing to north of declination -33$^{\circ}$. The focal plane assembly permits fine focusing using a computer-controlled focuser connected to the doublet corrector and equipped with a stepper motor. The C18 was originally supplied with a primary mirror metal cover that had to be removed manually. Since then, an electrically-operated, remotely-commanded mirror cover was installed. The CCD {#txt:CCD} ======= The telescope was equipped from the outset with a Santa Barbara Instrument Group (SBIG) ST-10 XME USB CCD camera that was custom-fitted to our specific telescope by AstroWorks and was delivered together with the telescope. This thermoelectrically-cooled chip has 2184$\times$1472 pixels each 6.8 $\mu$m wide, which convert to 1.1 arcsec at the f/2.8 focus of the telescope. The chip offers, therefore, a 40’.5$\times$27’.3 field of view. A second, smaller CCD, mounted next to the science CCD, allows guiding on a nearby star using exactly the same optical assembly. The CCD is used in “white light” with no filter.The readout noise is 10 electrons per pixel and the gain is 1.37 electrons per count. Each FITS image is 6.4 MB and the read out time is $\sim$15 sec using the [*MaximDL*]{} package. The CCD is mounted behind the doublet corrector lens and the focusing is achieved by moving the lens. In practice, even though the beam from the telescope is strongly converging, the focus is fairly stable throughout the night despite ambient temperature excursions of $\sim10^{\circ}$C or more. In any case, refocusing is very easy using the automatic focuser. After a few months of test operations, we decided to add water cooling to the ST-10. This was achieved by feeding the inlet water port of the SBIG CCD with an antifreeze solution from a one-liter reservoir that was continuously circulated by an aquarium pump. The addition of water cooling reduced the CCD temperature by an additional 10$^{\circ}$C, lowering the dark counts to below 0.5 electrons pixel$^{-1}$ sec$^{-1}$. We emphasize that the water circulation operates continuously, 24 hours per day, irrespective of whether the telescope is used for observations or not. Since its implementation, we changed the cooling fluid only once when it seemed to develop some kind of plaque. The CCD is used in “white light” without filters to allow the highest possible sensitivity. The chip response reaches 87% quantum efficiency near 630 nm, implying an overall response similar to a “wide-R” band. [ ]{} The effective area of the telescope and CCD is shown in Figure \[fig:C18\_response\]. This was calculated using the generic CCD response from the SBIG web site for the Kodak enhanced KAF-3200ME chip, assuming 7% areal obscuration of the primary mirror by the secondary mirror baffle and CCD, 80% mirror reflectivity, and 5% attenuation at each surface by the doublet lens and CCD window. The dome {#txt:Dome} ======== Given the small size of the telescope and the high degree of automation desired, we chose a small dome that would not allow routine operation with a human inside, but would allow unrestricted access to the sky for the C18. From among the off-the-shelf domes we chose a Prodome 10-foot dome from Technical Innovations USA[^1], equipped with two wall rings to provide sufficient height for the C18 in all directions. The fiberglass dome was equipped by Technical Innovations with an electrical shutter and with the necessary sensors to operate the [*Digital Dome Works (DDW)*]{} software bundled with the dome; this allows control of all the dome functions from a computer equipped with a four-port Multi-I/O card, adding four serial ports to the two already available on the operating computer. In addition, we equipped the dome with a weather station mounted on a nearby mast, a video camera with a commandable light source that allows remote viewing of the telescope and of part of the dome to derive indications about its position, and a “Robo reboot” device for [*DDW*]{}. The latter was installed to allow the remote initialization of the dome functions in case of a power failure. Later, we added a Boltwood Cloud Sensor[^2] that measures the amount of cloud cover by comparing the temperature of the sky to that of the ambient ground. The sky and ground temperatures are determined by measuring the amount of radiation in the 8 to 14 micron infrared band. A large difference indicates clear skies, whereas a small difference indicates dense, low-level clouds. This allows the sensor to continuously monitor the clarity of the skies, and to trigger appropriate alerts on the control computer. The device also includes a moisture sensor that directly detects rain drops. The dome and the operating computer are connected to the mains power supply through a “smart” UPS that will shut down the observatory in case of an extended power outage. The dome and telescope are mounted on a circular reinforced concrete slab with a 3.5-m diameter and 0.5-m thickness. The concrete was poured directly on the bedrock, which forms the surface ground layer at the WO site. The dome and the concrete slab are shown in Figure \[fig:C18\_dome\]. [ ]{} The operating software {#txt:Software} ====================== We decided that all the software would be tailored into a suite of operating programs that would conform to the [*ASCOM*]{} standards[^3]. This is in contrast with other similar, but significantly more expensive, small automatic observatories (e.g., Akerloff 2003), which chose various flavors of LINUX. Our choice saved the cost in money and time of developing specialized software by using off-the-shelf products. It also facilitated a standard interface to a range of astronomy equipment including the dome, the C18 mount, the focuser and the camera, all operating in a Microsoft Windows environment. The [*ACP*]{} (Astronomer’s Control Program) software is a product of DC-3 Dreams[^4]. [*ACP*]{} controls the telescope motion and pointing, and can change automatically between different sky fields according to a nightly observing plan. [*ACP*]{} also solves astrometricaly the images collected by the CCD, and improves automatically the pointing of the telescopes using these solutions. The program communicates with and controls the operating software of the dome, interfacing with [*DDW*]{}, enabling one to open and close the dome shutter, and commanding the dome to follow the telescope or to go to a “home” position. In addition, the [*ACP*]{} software is a gateway to the [*MaximDL*]{}[^5] program that operates the CCD. Different types of exposures, guiding and cooling of the CCD can be commanded manually using [*MaximDL*]{}, but in most cases we use the [*ACP*]{} code envelope (the [*AcquireImages*]{} script) to operate the entire system. The focusing is also done automatically using the freeware [*FocusMax*]{}[^6], a software package that operates the Robo-focus and searches automatically for the best FWHM of a selected star. To enable power shut-down by remote users we connected the telescope, dome, CCD, focuser and the telescope’s cover to a relay box that is connected to the six serial ports on the computer. Using a self-written code ([*C18 control*]{}) the remote user can enable or disable the electrical power supply to the different components of the system and open or close the telescope cover. This guarantees the safety of the equipment during daytime and enables the astronomer to fully operate the system from a remote location using any VNC viewer software. Figure \[fig:C18\_SW\] exhibits the different programs making up the software environment, their connections and their hierarchy. {width="8.5cm"} Cost ==== The affordability of an automatic telescope is an important consideration for many observatories. We found that it was not possible to collect sufficient funds to cover the high cost of an off-the-shelf automatic telescope with similar capabilities to those of the installation described here. The C18 telescope, CCD, and dome, including all the electronic add-ons and the software, added up to slightly more than 70k\$. To this one should add significant in-house contributions; the dome was received in segments that were erected and bolted on the concrete slab by WO staff Ezra Mash’al and Sammy Ben-Guigui together with NB; the installation, tuning-up and interfacing of the various software components were done mainly by IM, DP, and AB; the polar alignment of the C18 was done (twice) by AB who also developed the electrically-operated primary mirror cover. All these and more would add up to about a person-year of work by very experienced personnel. Performance =========== The automated operation mode of the C18 makes it an easy telescope to use, demanding from the astronomer only to watch the weather conditions when these are likely to change. Overall, the C18’s performance and output are satisfying. The telescope slewing time is about 5$^{\circ}$ sec$^{-1}$ on both axes and the settling-down time is 5-8 sec. The dome rotates at a rate of 3.33 degrees per second in each direction, while its response time in following the telescope motion is 2-3 sec. Opening the dome shutter requires 55 sec and closing it 70 sec, with the dome parked at its “home” location. Opening or closing the electrical telescope cover requires 20 sec. The auto-guiding system, which uses a smaller CCD in the same camera head, maintains round stellar images even for the longest exposures that are limited by the sky background, although the telescope is not perfectly aligned to the North. Without the guider, one can expect round star images only for exposure times of 90 sec or shorter. In some cases, telescope shake is experienced due to wind blows. Since the wind at the WO usually blows from the North-West, and usually slows down a few hours into the night, selecting targets away from this direction for the first half of the night decreases the number of smeared images to a minimum. The pointing errors of the C18 are of order 10 seconds of time in RA and 30 to 60 arcsec in DEC. However, astrometrically solving the images using the [*PinPoint*]{}[^7] engine, which can be operated automatically by the [*ACP*]{} software immediately following the image readout, re-points the telescope to a more accurate position for the following images of the same field. Since the CCD cooling is thermoelectric with water-assistance, with the cooling water at ambient temperature, the CCD temperature dependents very much on the weather. The usual values run between -15$^{\circ}$C in the summer nights with ambient temperatures of +30$^{\circ}$C to -30$^{\circ}$C in the winter (-5$^{\circ}$C ambient; see Figure \[fig:CCDtemp\]). All images are acquired with the CCD cooling-power at less then 100% capacity, assuring a steady chip temperature throughout the observation. The image read-out using MaximDL takes 15 sec. In regular observations an additional 10 sec interval is required for the astrometric solution of the image and another 5 sec to write the image on the local computer. A five-sec delay is required by the CCD before it continues to the next image for guider activation. Since the performance of the astrometric solution and the activation of the auto-guiding operation are user-selectable, the off-target time between sequential images is 15–30 sec, yielding a duty fraction of 83–92% on-sky time for typical 180 sec exposures. The CCD bias values are 990$\pm$5 counts, a value which is cooling-dependent. The bias values changed from an original $\sim$110 to the present value after one year of use and a rebuild of the software following a disk crash. The CCD flat field (FF) shows a 1.2% standard deviation from the image median. The FF gradient is steeper at the southern corners of the images than at the northern corners, as seen in Figure \[fig:CCD\_FF\]. This probably reflects some additional vignetting in the telescope on top of the prime focus baffle, possibly caused by the asymmetric blockage by the SBIG CCD, and by the pick-off prism edge for the TC-237H tracking CCD, mounted next to the science CCD, which is used for guiding. Twilight flats give the best results, provided the CCD cooling is started at least 30 min before obtaining the flat field images. {width="8.5cm"} On a dark night with photometric conditions the sky background is 10–15 counts per pixel per second. Comparing with the R-band magnitude of standard stars (Landolt 1992) the sky as measured on the C18 images in the “wide-R” band is about 20.4 mag arcsec$^{-1}$ and stars of 19.5 magnitudes are detectable with S/N$\simeq$25. This fits well previous measurements of the sky brightness at the Wise Observatory ($\mu_R\simeq$21.2 mag arcsec$^{-1}$ in 1989; Brosch 1992), accounting for the slight site deterioration in the last 15 years due to ambient light sources and for the wider spectral bandpass. Lessons learned =============== Dome electrical contacts ------------------------ The dome shutter opening mechanism receives its 12V DC electrical supply from two spring contacts that touch metal pads on the dome inner circumference. These spring contacts are made of slightly elastic copper strips twisted into rings and are fixed to the dome. To maintain electrical contact, the copper strips brush against a scouring pad before reaching the fixed contacts on the dome rim to scrape off accumulated dirt and oxide. During the two years of operating the C18 we have had the strips break a number of times. This interrupts the electrical contacts to the dome shutter and may prove dangerous when bad weather arrives in case nobody is present at the WO, since it prevents the remote shutting of the dome. The remedy is replacing the copper strips at regular intervals, even though they may appear intact. Dome lift-offs -------------- We experienced two instance of the dome lifting off its track on the stationary part of the dome enclosure. In these instances the dome slips off its track and has to be reseated manually. Since the fiberglass structure is very light, this operation can be done by a single person. We suspect that the dome lift-off events were caused by operator error. The dome rides on a track that is segmented and passes over a short door that must be latched shut when rotating the dome. If this is not done, the door might open slightly causing the dome to derail. Sticky dome shutter ------------------- We experienced a number of instances when the dome indicated that the shutter was closed, but inspection with the video camera showed that the shutter was still open. We have not yet tracked down the roots of this problem but it may be linked to a limitation of the electrical power to the shutter motor, perhaps connected with the operation of a low-voltage DC motor in conditions of high humidity. Note that this type of fault requires rapid human intervention to prevent damage to the telescope and electronics in changing atmospheric conditions. Operation in a dusty and hot desert environment ----------------------------------------------- Upon ordering of the PD-10 dome we worried about the possibility of dust entering the dome while closed, specifically in case of a dust storm. These storms are rare but can deposit significant amounts of dust on optics and mechanisms in a short time. This is why we specified that the dome be equipped with thick brushes that would prevent most of the dust from entering the (closed) dome. The two-year operational experience showed that the anti-dust brushes do indeed protect the interior of the dome and that the dust deposit on surfaces within the PD-10 dome is similar to that in the T40 dome. However, after two years of operation, the brush hairs are no longer straight but show curling and the dust prevention is no longer optimal. The brushes probably require replacing. The daytime external temperature at the WO can reach +40$^{\circ}$C and the inside of the dome can indeed become very hot. To prevent this, we plan to air condition the dome during daytime. This will also cool the huge concrete slab on which the telescope and dome are mounted, providing a cold reservoir for night-time operation. Daytime cooling will hopefully improve the night-time seeing by removing part of the “dome seeing” of the C18. Anti-collision switches ----------------------- In remote or automatic operation there exists a possibility that the telescope may become stuck in tracking mode, bypassing the software horizon limits. In this case, it is possible that the C18 may drive itself into the concrete floor or into the telescope fork causing instrumental damage. To prevent this, contact switches were installed on three corners at the top of the truss structure. A similar microswitch is activated by the truss structure if it approaches the base of the telescope fork (see Figure \[fig:C18\_in\_dome\]). These switches cut off instantly the power supply through the UPS, in case any of the switches are activated. Since such an event is primarily an operator error, we arranged that the reset of the mechanism can only be done manually, by someone physically pressing the reset button from within the dome. Scientific results ================== The C18 is used by different researchers at the Tel-Aviv University for various goals. Some examples, with representative outputs, are given below. We describe studies of physical parameters of asteroids, the investigation of extrasolar planets, and the monitoring of variable AGNs. NEOs and other asteroids {#NEOs} ------------------------ Understanding the potential danger of asteroid collisions on Earth has encouraged extensive research in recent years. Knowledge of physical properties of asteroids such as size, density and structure is critical for any future mitigation plan. At WO we focus on photometry, which allows the derivation of valuable data on asteroid properties: periodicity in light curves of asteroids coincides with their spins; shape is determined by examining the light curve amplitude; axis orientation is derived by studying changes in the light curve amplitude; special features in the light curves, such as eclipses, may suggest binarity and can shed light on the object structure and density. The C18 is mainly used for differential photometry of known asteroids as a primary target, but also enables the detection of new asteroids. The large field of the CCD allows the asteroids to cross one CCD field or less per night, even for the fast-moving Near-Earth Objects (NEOs) that can traverse at angular velocities of 0.1” sec$^{-1}$ or slower. This ensures that the same comparison stars are used while calibrating the differential photometry. Exposure times between 30 to 180 sec are fixed for every night, depending on the object’s expected magnitude and angular velocity, the nightly seeing, and the sky background. The lower limit for the signal to noise ratio of acceptable images is $\sim$10, thus asteroids that move too fast ($\geq$0.1”/sec) or are too faint ($\geq$18.5 mag) are de-selected for observation. Objects brighter than 13 mag are also avoided, to prevent CCD saturation. The images are reduced using bias, dark and normalized flat field images. Time is fixed at mid-exposure for each image. The IRAF [*phot*]{} function is used for the photometric measurements. Apertures of four pixel ($\sim$4.5 arcsec) radius are usually chosen. The mean sky value is measured in an annulus with an inner radius of 10 pixels and a width of 10 pixels around the asteroid. The photometric values are calibrated to a differential magnitude level using 100-500 local comparison stars that are also measured on every image of a specific field. For each image a magnitude shift is calculated, compared to a good reference image. Stars whose magnitude shift is off by more than 0.02 mag from the mean shift value of the image are removed at a second calibration iteration. This primarily solves the question of transient opacity changes, and results in a photometric error of $\sim$0.01 mag. Most of the observed asteroids are followed-up on different nights; this changes the background star field. Some asteroids are also observed at different phase angles and their brightness can change dramatically from one session to another. To allow comparisons and light curve folding to determine the asteroid spin, the instrumental differential photometric values are calibrated to standard R-band magnitudes using $\sim$20 stars from the Landolt equatorial standards (Landolt 1992). These are observed at air masses between 1.1 to 2.5, while simultaneously observing the asteroid fields that include the same local comparison stars used for the relative calibration. Such observations are done only on photometric nights. The extinction coefficients and the zero point are obtained using the Landolt standards after measuring them as described above. From these, the absolute magnitudes of the local comparison stars of each field are derived, followed by a calculation of the magnitude shift between the daily weighted-mean magnitudes and the catalog magnitudes of the comparison stars. This magnitude shift is added to the photometric results of the relevant field and asteroid. The procedure introduces an additional photometric error of 0.02-0.03 mag. Since the images are obtained in white light, they are calibrated by the Landolt standards assuming the measurements are in the Cousins R system. In addition, the asteroid magnitudes are corrected for light travel time and are reduced to a Solar System absolute magnitude scale at a 1 a.u. geocentric and heliocentric distance, to yield H(1,$\alpha^0$) values (Bowell et al. 1989). {width="8.5cm"} To retrieve the light curve period and amplitude, the data analysis includes folding all the calibrated magnitudes to one rotation phase, at zero phase angle, using two basic techniques: a Fourier decomposition to determine the variability period(s) (Harris & Lupishko 1989) and the H-G system for calibrating the phase angle influence on the magnitude (Bowel et al. 1989). The best match of the model light curve to the observations is chosen by least squares. An example is shown on Figure \[fig:LC\_fit\] were a simple model was fitted to the observed data points of the asteroid (106836) 2000 YG$_8$. Figure \[fig:LC\_folded\] displays the folded light curve from which the rotation period P is deduced (here P=20.2$\pm$0.2 hours). {width="8.5cm"} While the main advantage of the wide field of view of the C18 is the ability to observe even a fast-moving NEO in the same field during one night (Polishook & Brosch 2007), the instrument allows also the simultaneous monitoring of the light variations from several asteroids in the same field of view. Looking at Main Belt asteroids, many objects can be seen sharing the same field (our recent record is 11 objects in one field; see Figure \[fig:11asteroids\]). An exposure time of $\sim$180 seconds is needed to detect this amount of asteroids while avoiding the smearing of their images due to their angular motion. [ ]{} In addition to the rapid increase of our asteroid light curve database, due to this efficient observing method, new asteroids are discovered on the same images used for light curve derivation, proving that 18-inch wide-field telescopes can contribute to the detection of unknown asteroids even in the age of Pan-STARRS and other big, automated NEO-survey observatories. We emphasize that the discovery of new asteroids is [**not**]{} a goal of our research programs, but is a side-benefit. Our records indicate that by observing in the direction of the main belt we can detect a new asteroid in every 2-3 fields. These objects are reported to the MPC, but are not routinely followed up. The [*ACP*]{} software can move the telescope automatically between different fields, increasing the number of measured asteroids. With the telescope switching back and forth between two fields, the photometric cost is the increase of the statistical error by $\sqrt{2}$, since the exposure time is reduced by half for each field, but the number of measured light curves obtained every night is very high. Extrasolar planets ------------------ A *transiting* extrasolar planet crosses the parent star’s line-of-sight once every orbital revolution. During this crossing, referred to as a transit, which usually lasts a few hours, the planet blocks part of the light coming from the stellar disk, inducing a $\sim$1% decrease in the star’s observed intensity. By combining the photometric measurement of the transit light curve with a spectroscopic measurement of the planet’s orbit, both the planet radius $r_p$ and mass $m_p$ can be derived. Such an intrinsic planetary characterization can only be done for transiting planets. By comparing the measured $r_p$ and $m_p$ with planetary models (e.g., Guillot et al. 2005, Fortney et al.2007) the planet’s structure and composition can be inferred. This, in turn, can be used to test predictions of planetary formation and evolution theories (Pollack et al. 1996, Boss 1997). In addition, transiting planets allow the study of their atmospheres (Charbonneau et al. 2007 and references therein), the measurement of the alignment, or lack of it, between the stellar spin and planetary orbital angular momentum (e.g., Winn et al.2005) and the search for a second planet in the system (Agol et al. 2005, Holman & Murray 2005). For the reasons detailed above it is clear that transiting planets are an important tool for extending our understanding of the planet phenomenon, hence the importance of searching for them. The combination of the light collecting area, large FOV (0.31 deg$^2$) and short read-out time, makes the C18 a useful tool for obtaining high-quality transit light curves for relatively bright stars. The large FOV is especially important, since it allows one to observe many comparison stars, similar in brightness and color to the target, which is crucial for the accurate calibration of the target’s brightness. Currently the C18 is used for obtaining transit light curves for several projects, two of which are: - *Photometric follow-up of transiting planet candidates*: Over the last few years small-aperture wide-field ground-based telescopes have been used to detect $\sim 15$ transiting planets[^8]. This is about half of all known such planets and the discovery rate of these instruments is increasing. Once a transit-like light curve is identified by these small telescopes it is listed as a transiting planet *candidate*, to be followed-up photometrically and spectroscopically, discriminating between true planets and false positives (e.g., O’Donovan et al. 2007), and measuring the system’s parameters. Photometric follow-up observations to obtain a high-quality transit light curve are carried out to verify the detection and measure several system parameters, including the planet radius and mid-transit time. Spectroscopic follow-up is used for measuring the companion’s orbit from which its mass is inferred. The C18 is used for photometric following-up of candidates identified in data obtained by the WHAT telescope (Shporer et al. 2007), in collaboration with the HATNet telescopes (Bakos et al. 2006). - *Photometric follow-up of planets discovered spectroscopically*: Most of the $\sim 270$ extrasolar planets known today were discovered spectroscopically, i.e., through the radial velocity (RV) modulation of the star induced by the planet, as the two bodies orbit the joint center of mass. The probability that such a planet will show transits is $\frac{r_s + r_p}{d_{tr}}$, where $r_s$ and $r_p$ are the stellar and planetary radii, respectively, and $d_{tr}$ is the planet-star distance at the predicted transit time. This probability is about 1:10 for planets with close-in orbits, with orbital periods of several days. Interestingly, the first extrasolar planet for which transits were observed, HD209458 (Charbonneau et al. 2000, Henry et al. 2000) was the tenth close-in planet discovered spectroscopically (Mazeh et al. 2000). The C18 is taking part in follow-up observations of planets detected spectroscopically, especially those newly discovered, in order to check whether they show a transit signal. The remote operation makes it possible to carry out observations on very short notice. Observations made with the C18 were part of the discovery of the transiting nature of Gls 436 b (Gillon et al. 2007), a Neptune-mass planet orbiting an M dwarf (Butler et al. 2004, Maness et al. 2007). During an observation of a transit, the PSF is monitored and the exposure time is adjusted from time to time, keeping the target count level from changing significantly. A guide star is usually used. After bias, dark and flat-field corrections, images are processed with the IRAF/phot task, using a few trial aperture radii. The target light curve is calibrated using a few dozen low-RMS stars similar in brightness to the target. As a final step, the out-of-transit measurements are fitted to several parameters, such as airmass, HJD and PSF FWHM, and all measurements are divided by the fit. An example of a transit light curve obtained with the C18 is shown in Figure \[fig:trlc\]; this is the light curve of a HATNet candidate (Internal ID HTR176-003). The top panel shows the actual light curve and the bottom panel shows the light curve binned in 5 minute bins. The residual RMS of the unbinned measurements is 0.22 %, or 2.4 milli-magnitude, and for the binned light curve it is 0.09 %, or 1.0 milli-magnitude. Figure \[fig:rmsmean\] presents the RMS vs. mean $V$ magnitude for all the stars identified in this field. The X-axis is in instrumental magnitude, which is close to the real magnitude. The RMS of the brightest stars reaches below the 3 milli-magnitude level (see also Winn 2007). ![Light curve of a HATNet transiting planet candidate obtained by the C18. Both panels show relative flux vs. time. The top panel presents the actual measurements (blue points) overplotted by a fitted model (solid red line). Residuals from the fitted model are also plotted (black points), shifted to a zero point of 0.965. In the bottom panel, blue points and error bars represent a 5 min mean and RMS. The same model is overplotted (solid red line) and residuals (in black) are also shifted to a zero point of 0.965. Each of the 5 min bins includes 5.3 measurements on average. The RMS of the residuals of the actual measurements is 0.22 %, or 2.4 milli-magnitude, and for the binned light curve, 0.09 %, or 1.0 milli-magnitude. This light curve includes 400 individual measurements taken during 6.2 hours, with an exposure time of $\sim 25$ sec for each frame.[]{data-label="fig:trlc"}](lc.eps){width="8.5cm"} ![RMS vs. mean $V$ magnitude for all stars identified in the field of the transiting planet candidate shown in Fig. \[fig:trlc\]. Y-axis is in log scale and X-axis is in instrumental magnitude, although this is close to the real magnitude. Observations were done on a single night with a typical exposure time of 25 seconds. For the brightest stars, the RMS reaches below the 3 milli-magnitude level.[]{data-label="fig:rmsmean"}](rmsmean.eps){width="8.5cm"} AGN monitoring -------------- The C18 fits the requirements for monitoring Active Galactic Nuclei (AGNs). Such objects are known for their continuum variability and, though the origin of this variability is still unclear, it is possible to use it to study, using various techniques, the physical conditions in the AGNs and their properties. One such technique, which enables the study of the emission-line gas and the measuring of the central supermassive black hole mass in AGNs, is the “reverberation mapping” (e.g., Peterson et al. 1993). In such studies, the time lag between the variations in the continuum flux and the emission-line fluxes is used to estimate the Broad Line Region (BLR) size, and to map its geometry. Combining such time lags with the BLR velocity (measured from the width of the emission-lines) allows the determination of the black hole mass in an AGN (e.g., Kaspi et al. 2000, Peterson et al. 2004). The WO took a leading role in such studies, carrying out about half of all reverberation mapping studies ever done. An important empirical size-mass-luminosity relation, spanning a broad luminosity range, has been derived for AGNs based on 36 AGNs with reverberation mapping data. This is now widely used to determine the black hole mass from single-epoch spectra for large samples of AGNs and distant quasars, allowing the study of black hole growth and its effect on galaxy evolution. However, more reverberation mapping studies are needed to expand the luminosity range and to re-define the current relation with better statistics (i.e., adding more objects to the 36 already studied). One crucial element in reverberation mapping studies is the continuous monitoring of the continuum flux variations. For low-luminosity AGNs, which vary on timescales of hours to days, it is important that the monitoring period of several days will be densely covered at a sampling rate of several minutes. To achieve such coverage the close collaboration of several observatories around the world is essential. One such study, carried out with the C18, is the monitoring of the low-mass candidate AGN SDSS J143450.62+033842.5 (Greene & Ho 2004; $z=0.0286$, $m_r=15$). During a one-week period in April 2007 this AGN was monitored with a typical exposure time of 240 sec. The images were reduced in the same way as described in Section \[NEOs\]. The typical uncertainty of the measurements is about 0.1 magnitude which is sufficient to detect the variability of this AGN. Figure \[AGN-LC\] shows the light curve, derived using differential photometry and the “daostat” program described in Netzer et al. (1996, section 2.3). ![Light curve of the low-luminosity AGN SDSS J143450.62+033842.5. The full-resolution light curve of the entire monitoring run is shown in the top panel and a zoomed-in view of the third monitoring night is shown in the bottom panel. Flux variations on timescales of hours to days are clearly detected.[]{data-label="AGN-LC"}](plot5.eps){width="8.5cm"} Combining this light curve with additional data from the Crimean Astrophysical Observatory in Ukraine, the MAGNUM-2m telescope in Hawaii, and the 2m Faulkes Telescope in Australia, produced a fully-sampled continuum light curve for SDSS J143450.62+033842.5 covering several days. Together with an emission-line light curve obtained at the Magellan telescope, the central mass of the black hole in the center of this AGN will be determined. The C18 based at the WO played a crucial part in this monitoring campaign and proved that it can be most useful in producing AGN light curves, which are used in state-of-the-art studies of supermassive black holes. Conclusions {#txt:Conclusions} =========== We described the construction of a secondary observing facility at the Wise Observatory, consisting of a 0.46m Prime Focus f/2.8 telescope equipped with a good quality commercial CCD camera. The C18 telescope and camera are sited in a small fiberglass dome, with a suite of programs using ASCOM interface capability orchestrating the automatic operation. The combination offers a cost-effective way of achieving a limited goal: the derivation of high-quality time-domain sampling of various astronomical sources. The automatic operation, despite the few snags discovered following two years of operation, proves to be very user-friendly since it allows the collection of many observations without requiring the presence of the observer at the telescope or even at the Wise Observatory. This was achieved with a very modest financial investment and may be an example for other astronomical observatories to follow. We presented results from three front-line scientific projects performed with the new facility, which emphasize the value of an automated and efficient, relatively wide-field, small telescope. Acknowledgments {#acknowledgments .unnumbered} =============== The C18 telescope and most of its equipment were acquired with a grant from the Israel Space Agency (ISA) to operate a Near-Earth Asteroid Knowledge Center at Tel Aviv University. DP acknowledges an Ilan Ramon doctoral scholarship from ISA to study asteroids. We are grateful to the Wise Observatory Technical Manager Mr. Ezra Mash’al, and the Site Manager Mr. Sammy Ben-Guigui, for their dedicated contribution in erecting the C18 facility and repairing the many small faults discovered during the first two years of operation. [99]{} Agol, E., Steffen, J., Sari, R., & Clarkson, W. 2005, MNRAS, 359, 567 Akerlof, C. W., et al.2003, , 115, 132 Bakos, G., Noyes, R. W., Kov[á]{}cs, G., Stanek, K. Z., Sasselov, D. D., & Domsa, I.2004, PASP, 116, 266 Bakos, G., Noyes, R. W., Latham, D. W., Cs[á]{}k, B., G[á]{}lfi, G., & P[á]{}l, A. 2006, Tenth Anniversary of 51 Peg-b: Status of and prospects for hot Jupiter studies, 184 Barthelmy, S.D. et al. 1998, in [*Gamma-Ray Bursts: 4$^{th}$ Huntsville Symposium*]{}, C.A. Meegan, R.D. Preece, T.M. Koshut (eds.), AIP Conf. Proc. [**428**]{}, 99 Boss, A. P. 1997, Science, 276, 1836 Bowell, E., Hapke, B., Domingue, D., Lumme, K., Peltoniemi, J. and Harris, A. W., 1989. [*Application of photometric models to asteroids.*]{} In Asteroids II (R. P. Binzel, T. Gehrels and M. S. Matthews, Eds.), pp. 524-556. Univ. of Arizona Press, Tucson Bowen, I. and Vaughan, A. H., Jr., 1973, Applied Optics [**12**]{}, 1430. Brosch, N. 1992, Q.J.R.astr.Soc. [**33**]{}, 27 Butler, R. P., Vogt, S. S., Marcy, G. W., Fischer, D. A., Wright, J. T., Henry, G. W., Laughlin, G., & Lissauer, J. J. 2004, ApJ, 617, 580 Charbonneau, D., Brown, T. M., Latham, D. W., & Mayor, M. 2000, ApJ, 529, L45 Charbonneau, D., Winn, J. N., Everett, M. E., Latham, D. W., Holman, M. J., Esquerdo, G. A., & O’Donovan, F. T. 2007, ApJ, 658, 1322 Gillon, M., et al. 2007, A&A, 472, L13 Greene, J. E., & Ho, L. C. 2004, ApJ, 610, 722 Guillot, T. 2005, Annual Review of Earth and Planetary Sciences, 33, 493 Fortney, J. J., Marley, M. S., & Barnes, J. W. 2007, ApJ, 659, 1661 Harris, A. W. and Lupishko, D. F., 1989. [*Photometric light curve observations and reduction techniques.*]{} In Asteroids II (R. P. Binzel, T. Gehrels and M. S. Matthews, Eds.), pp. 39-53. Univ. of Arizona Press, Tucson. Henry, G. W., Marcy, G. W., Butler, R. P., & Vogt, S. S. 2000, ApJ, 529, L41 Holman, M. J., & Murray, N. W. 2005, Science, 307, 1288 Kaspi, S., Smith, P. S., Netzer, H., Maoz, D., Jannuzi, B. T., & Giveon, U. 2000, ApJ, 533, 631 Landolt, A., 1992. AJ [**104**]{}, no. 1, 340 Maness, H. L., Marcy, G. W., Ford, E. B., Hauschildt, P. H., Shreve, A. T., Basri, G. B., Butler, R. P., & Vogt, S. S. 2007, PASP, 119, 90 Mazeh, T., et al.2000, ApJ, 532, L55 Netzer, H., et al.1996, MNRAS, 279, 429 O’Donovan, F. T., et al. 2007, ApJ, 662, 658 Pepper, J., et al.2007, PASP, 119, 923 Peterson, B. M. 1993, PASP, 105, 247 Peterson, B. M., et al. 2004, ApJ, 613, 682 Polishook, D. & Brosch N., Icarus, in press (astro-ph/0607128) Pollacco D. L., et al., 2006, PASP, 118, 1407. Pollack, J. B., Hubickyj, O., Bodenheimer, P., Lissauer, J. J., Podolak, M., & Greenzweig, Y. 1996, Icarus, 124, 62 Shporer, A., Mazeh, T., Moran, A., Bakos, G., & Kovacs, G. 2007, Transiting Extrapolar Planets Workshop, 366, 99 Steele I. A., et al., 2004, SPIE, 5489, 679. Yost S. A., et al., 2006, AN, 327, 803. Zerbi R. M., et al., 2001, AN, 322, 275. Winn, J. N., et al.  2005, ApJ, 631, 1215 Winn, J. N., Holman, M. J., & Roussanova, A. 2007, ApJ, 657, 1098 [^1]: http://www.homedome.com/ [^2]: http://www.cyanogen.com/products/cloud\_main.htm [^3]: AStronomy Common Object Model, http://ascom-standards.org/ [^4]: http://acp.dc3.com/ [^5]: http://www.cyanogen.com/products/maxim\_main.htm [^6]: http://users.bsdwebsolutions.com/ larryweber/ [^7]: http://pinpoint.dc3.com/ [^8]: For an updated list of known transiting planets see: http://obswww.unige.ch/$\sim$pont/TRANSITS.htm

--- abstract: 'We propose a model of sub-diffusion in which an external force is acting on a particle at all times not only at the moment of jump. The implication of this assumption is the dependence of the random trapping time on the force with the dramatic change of particles behavior compared to the standard continuous time random walk model. Constant force leads to the transition from non-ergodic sub-diffusion to seemingly ergodic diffusive behavior. However, we show it remains anomalous in a sense that the diffusion coefficient depends on the force and the anomalous exponent. For the quadratic potential we find that the anomalous exponent defines not only the speed of convergence but also the stationary distribution which is different from standard Boltzmann equilibrium.' author: - 'Sergei Fedotov, Nickolay Korabel' title: 'Sub-diffusion in External Potential: Anomalous hiding behind Normal' --- Recently it has become clear that anomalous diffusion measured by a non-linear growth of the ensemble averaged mean squared displacement $% \left\langle x^{2}\right\rangle \sim t^{\mu}$ with the anomalous exponent $% \mu \neq 1$ is as widespread and important as normal diffusion with $\mu =1$ [@Kla08]. Sub-diffusion with $\mu <1$ was observed in many physical and biological systems such as porous media [@Dra99], glass-forming systems [@Weeks02], motion of single viruses in the cell [@Sei01], cell membranes [@Sax97; @Rit05], and inside living cells [Wei03,Gol06,Tolic04]{}. Many examples of sub-diffusive processes in biological systems can be found in recent reviews [@Fra13; @Bar12]. Nowadays new tools are available including super-resolution light optical microscopy techniques to deal with biological in vivo data which allows to monitor a large number of trajectories at the single-molecule level and at nanometer resolution [@Ser08; @Jaq08; @Nor14]. Using these techniques it is possible to discriminate between anomalous ergodic processes where the ensemble and time averages coincide and non-ergodic processes where ensemble and time averages have different behavior [@He08; @Lub08; @Mer13]. Two important observations have been made about anomalous transport in living cells: (1) anomalous transport is usually a transient phenomenon before transition to normal diffusion or saturation due to confined space [Bro09,Neu08,Sax01]{} (2) ergodic and non-ergodic processes may coexist as it was observed in plasma membrane [@Wei11]. Several models are proposed to describe ergodic and non-ergodic anomalous processes such as non-ergodic continuous time random walk (CTRW) with power-law tail waiting times, ergodic anomalous process generated by fractal structures, fractional Brownian-Langevin motion characterized by long correlations and time dependent diffusion coefficient [@Kla08; @Sok12; @Bre13]. The standard CTRW model for sub-diffusion of a particle in an external field $F(x)$ randomly moving along discrete one-dimensional lattice can be described by the generalized master equation for the probability density $p(x,t)$ to find the particle at position $x$ at time $t$ $$\frac{\partial p}{\partial t}=-i(x,t)+w^{+}(x-a)i(x-a,t)+w^{-}(x+a)i(x+a,t), \label{MM}$$where $a$ is the lattice spacing and $i(x,t)$ is the total escape rate from $% x$ $$i(x,t)=\frac{1}{\Gamma(1-\mu)\tau _{0}^{\mu }}\mathcal{D}_{t}^{1-\mu }p(x,t). \label{ieq}$$Here $\tau _{0}$ is a constant timescale and $\mathcal{D}_{t}^{1-\mu }$ is the Riemann-Liouville fractional derivative defined by $$\mathcal{D}_{t}^{1-\mu }p(x,t)=\frac{1}{\Gamma (\mu )}\frac{\partial }{% \partial t}\int_{0}^{t}\frac{p(x,\tau )}{(t-\tau )^{1-\mu }}d\tau .$$The probabilities of jumping to the right $w^{+}(x)$ and to the left $w^{-}(x)$ are $$w^{+}(x)=\frac{1}{2}+\beta a F(x),\quad w^{-}(x)=\frac{1}{2}-\beta a F(x). \label{ww}$$Series expansion of Eq. (\[MM\]) together with Eq. (\[ieq\]) and Eq. (\[ww\]) leads to the fractional Fokker-Planck equation (FFPE) [@Met99; @Met00] $$\frac{\partial p}{\partial t}=D_{\mu }\left[ \frac{\partial ^{2}}{\partial x^{2}}-\beta \frac{\partial }{\partial x}F(x)\right] \mathcal{D}_{t}^{1-\mu }p, \label{eq:FFPE}$$where the generalized diffusion $D_{\mu }=a^{2}/(2\;\Gamma (1-\mu )\tau _{0}^{\mu }).$ The stationary solution of Eq. (\[eq:FFPE\]) is the Boltzmann distribution. There exist a huge literature on this equation [Met99,Met00]{} and its generalization for time dependent forces [Hei07,Mag08,Henry10,Eule09,Sok06,Shu08,Shkilev12]{}. One of the main assumptions in this literature, which is not always clearly stated is that, as long as a random walker is trapped at a particular point $% x$, the external force $F(x)$ does not influence the particle. It is clear from Eq. (\[ieq\]) that the escape rate $i(x,t)$ does not depend on the external force $F(x).$ The force only acts at the moment of escape inducing a bias. The question is how to take into account the dependence of the escape rate on $% F(x)?$ To the author’s knowledge this is still an open question. One of the main aims of this Letter is to propose a model which deals with this problem. We find that the dependence of escape rate on force drastically changes the form of the master equation (\[MM\]) and FFPE (\[eq:FFPE\]). We observe transient anomalous diffusion and transition from non-ergodic to normal ergodic behavior. However, we show that this seemingly normal process could be still anomalous masked by normal behavior. Our findings suggest that a closer inspection of experimental results could be necessary in order to discriminate between normal and anomalous processes. *Model.—* We consider a random particle moving on a one dimensional lattice under assumption that an external force acts on a particle at all times not only at the moment of jump as in Eq. (\[MM\]). The implication of this assumption is the dependence of the random trapping time on the external force (not just jumping probabilities as in (\[ww\])). Some discussion of situation when the external force influence the rates and jumps can be found in [@Hei07]. The main physical idea behind our model is that there exists two independent mechanism of escaping from the point $x$ with two different random residence times. The first mechanism is due to external force with the escape rate proportional to $F(x).$ The second one is the sub-diffusive mechanism involving the rate inversely proportional to the residence time. The latter generates the power law waiting time distribution with the infinite first moment. Regarding the first mechanism, we define the jump process from the point $x$ as follows. We assume that the rate of jump to the right $\mathbb{T}_{x}^{+}$ from $x$ to $x+a$ is $\nu aF(x)$ when $F(x)\geq 0$ and the rate of jump to the left $\mathbb{T}_{x}^{-}$ from $x$ to $x-a$  is $-\nu aF(x)$ when $F(x)\leq 0.$ For this jump model the random waiting time $T_{F}$ at the point $x$ is defined by the exponential survival probability $\Psi _{F}(x,\tau )$ involving the external force $F(x)$ $$\Psi _{F}(x,\tau )=\Pr \left\{ T_{F}>\tau \right\} =\exp \left( -\nu a|F(x)|\tau \right).$$where $\nu $ is the intensity of jumps due to force field. For example, one can think of the escape rate $\mathbb{T}_{x}^{+}$ that is defined in terms of the potential field $U(x)$ that is $\mathbb{T}_{x}^{+}=-\nu \left[ U(x+a)-U\left( x\right) \right] >0,$ there $F(x)=-U^{\prime }(x)+o(a^{2})$ for $U^{\prime }(x)\leq 0.$ The second mechanism involves the sub-diffusive random walk with the escape rate $\lambda (x,\tau )$ from the point $x$, which is inversely proportional to the residence time $\tau.$ In this case the random waiting time $T_{\lambda }$ at the point $x$ is defined by the survival probability $$\Psi _{\lambda }(x,\tau )=\Pr \left\{ T_{\lambda }>\tau \right\} =\exp \left( -\int_{0}^{\tau }\lambda (x,s)ds\right) . \label{FFFF}$$The question now is how to implement the jumping process due to external force into the sub-diffusive random walk scheme? When the random walker makes a jump to the point $x$, it spends some random time (residence time) before making another jump to $x+a$ or $x-a$. Let us denote this residence time $T_{x}$. The key point of our model is that we define this residence time as the minimum of two: $T_{\lambda }$ and $T_{F}$ $$T_{x}=\min \left( T_{\lambda },T_{F}\right) . \label{min}$$For the anomalous sub-diffusive case this model could lead to the drastic change in the form of the fractional master equation. The main reason for this is that the external force $F(x)$ plays the role of tempering factor preventing the random walker to be anomalously trapped at point $x$. Because of the independence of two mechanisms, in our model the rate of jump $\mathbb{T}_{x}^{+}$ to the right from $x$ to $x+a$ and the rate of jump $% \mathbb{T}_{x}^{-}$ to the left from $x$ to $x-a$ can be written as the sum$$\mathbb{T}_{x}^{+}=% \begin{cases} \omega ^{+}(x)\lambda (x,\tau )+\nu aF(x), & F(x)\geq 0, \\ \omega ^{-}(x)\lambda (x,\tau ),\quad & F(x)<0% \end{cases} \label{T+}$$and$$\mathbb{T}_{x}^{-}=% \begin{cases} \omega ^{+}(x)\lambda (x,\tau ), & F(x)\geq 0, \\ \omega ^{-}(x)\lambda (x,\tau )-\nu aF(x),\quad & F(x)<0.% \end{cases} \label{T-}$$Although it is straightforward to consider general $\omega ^{+}(x)$, $\omega ^{-}(x)$, for simplicity in what follows we consider $\omega ^{+}(x)=\omega ^{-}(x)=1/2$. In our model the asymmetry of random walk occur only from the force dependent rate. Let us explain the main idea of Eqs. (\[T+\]) and (\[T-\]). The external force $F(x)\geq 0$ increases the sub-diffusive rate of jumps to the right $\lambda (x,\tau )/2$ and does not change the sub-diffusive rate of jumps to the left. The essential property of Eqs. (\[T+\]) and (\[T-\]) is that the rate $\lambda (x,\tau )$ depends on the residence time variable $\tau $. This dependence makes any model involving the probability density $p(x,t)$ non-Markovian. For the Markov case with $% F(x)=0,$ $\lambda ^{-1}(x)$ has a meaning of the mean residence time at the point $x$. When the parameter $\nu =0$ and the rates are $\mathbb{T}% _{x}^{+}=w^{+}(x)\lambda (x,\tau ),$ $\mathbb{T}_{x}^{-}=w^{-}(x)\lambda (x,\tau )$, we obtain the standard fractional Fokker-Planck equation ([eq:FFPE]{}). Notice that Eq. (\[min\]) is consistent with the expression for the effective escape rate $\mathbb{T}_{x}^{+}+\mathbb{T}_{x}^{-}$ as a sum of two rates $\lambda (x,\tau )+\nu a|F(x)|$. Similar situation has been considered in [@Fed13]. After incorporation of the force dependent escape rates we can obtain generalized master equation (see Supplementary Materials for the derivation). By expanding the RHS of the master equation to the second order in jump size $a$, we get a fractional diffusion equation $$\frac{\partial p}{\partial t}=\frac{\partial ^{2}}{\partial x^{2}}\left[ D_{\mu }e^{-\nu a|F(x)|t}\mathcal{D}_{t}^{1-\mu }\left[ p(x,t)e^{\nu a|F(x)|t}\right] \right] - \label{main_eq}$$$$-a^{2} \nu \frac{\partial }{\partial x}\left[ F(x)p(x,t)\right] .$$ This equation is fundamentally different from the classical FFPE ([eq:FFPE]{}) because it involves the external force in both terms on the right hand side. One can see that the force $F(x)$ not only determines the advection term as in Eq. (\[eq:FFPE\]), but also plays the role of tempering parameter through the factor $e^{\nu a|F(x)|t}$. Similar factor occurs in sub-diffusive equation with the death or evanescent process [@Abad10; @Fal13]. However, here we consider the system with constant total number of particle. The stationary solution $p_{st}(x)$ of Eq. (\[main\_eq\]) obeys the standard equation$$-a^{2}\nu F(x)p_{st}(x)+\frac{d}{dx}\left[ D_{F}(x)p_{st}(x)\right] =0. \label{stat_eq}$$(see a supplement material for details). Interesting property of this equation is that the effective diffusion constant $D_{F}(x)$ depends on the external force and anomalous exponent $$D_{F}(x)=D_{\mu }\left( \nu a|F(x)|\right) ^{1-\mu }. \label{D_F}$$This fact implies that the Boltzmann distribution is no longer stationary solution of (\[stat\_eq\]). For the quadratic potential $U(x)=\kappa x^{2}/2 $ with $F(x)=kx,$ we find that for large $x$ the stationary density $% p_{st}(x)$ has the form $$p_{st}(x)\sim \exp (-A|x|^{1+\mu }), \label{stat}$$where $A>0$ is a constant. One can see that the form of stationary density is determined by the anomalous exponent $\mu $. In this case the particles spread further compared to the Boltzmann case. The reason is the dependence of the effective diffusion constant $D_{F}(x)$ on force $F(x).$ Note that for the sub-diffusive fractional Fokker-Planck equation (\[eq:FFPE\]) the anomalous exponent only determines the slow power law relaxation rate, while the stationary density converges to Boltzmann equilibrium which does not depend on $\mu .$ *Numerical simulations.—* We consider two particular cases:  (1) constant force $F$ corresponding to the linear potential and (2) the quadratic potential $U(x)=\kappa x^{2}/2$ both in the infinite domain. We concentrate on the behavior of the density function $p(x,t)$, the mean $% \left\langle x(t)\right\rangle $ and the variance $\sigma (t)=\left\langle x^{2}\right\rangle -\left\langle x\right\rangle ^{2}$ calculated using an ensemble of trajectories from the initial distribution $p(x,0)=\delta (x)$. We also calculate the time averaged variance of a single trajectory of length $T$, $\sigma _{T}(\Delta ,T)=\delta ^{2}(\Delta ,T)-(\delta (\Delta ,T))^{2}$, where $\delta ^{n}(\Delta ,T)=\int_{0}^{T-\Delta }(x(t+\Delta )-x(t))^{n}dt/(T-\Delta )$, $n=1,2$. This quantity become a standard tool to assess ergodic properties of a system been equivalent to its ensemble averaged counterpart only for ergodic case. When the external force $F$ is constant, we observe the transition from sub-diffusion at short times to seemingly normal diffusion at long times. The density function changes from the distinct sub-diffusive shape for short times to the Gaussian shape propagator at longer times (see the inset of figure \[FIG1\]). The average position of the ensemble behaves as $\left<x(t)\right>=F t$. The ensemble averaged variance $\sigma(t)$ grows as a power law for short times, $\sigma(t) \sim t^{\eta}$, and transition to a normal diffusive linear growth $\sigma (t)\sim 2D_{F}t$ for longer times. However, in this case the diffusion coefficient $D_{F}$ depends on the force $F$ and anomalous exponent $\mu $. We conclude that although the variance $% \sigma (t)$ is linearly proportional to time, this dependence reveals the anomalous nature of the process even in the limit $t\rightarrow \infty $. Numerical calculations confirms the analytical result for the diffusion coefficient Eq. (\[D\_F\]) (see figure \[FIG1\]). Second observation is that the power law behavior at short times involves the exponent $\eta (F)>\mu $ which depends on force $F$. For $\mu =0.3$ they are estimated to be $\eta \approx 0.39$ for $F=0.0001$, $\eta \approx 0.47$ for $F=0.001$ and $\eta \approx 0.6$ for $F=0.01$. This can be interpreted as an enhancement of sub-diffusion coursed by the constant force. Such enhancement should be taken into account in the analysis of biological experiments where sub-diffusion usually appears as transient before the transition to the normal diffusion [@Fra13]. For the large value of $F$ the exponent $\eta $ tends to one while in the small force limit $\eta \rightarrow \mu $. The time averaged variance calculated for constant force grows linearly $\sigma _{T}(\Delta ,T)\sim \Delta $ and shows minor scatter between single trajectories (figure \[FIG2\]). After averaging over different trajectories, it grows with the coefficient $2 D_F$ which is equal to the ensemble average value. This shows that the non-ergodic sub-diffusive system becomes an ergodic one. Now we consider the quadratic potential $U(x)=\kappa x^{2}/2.$ The system becomes again non-ergodic despite the tempering affect of the force. To confirm this we calculate the time averaged variance (inset of figure [FIG2]{}). As expected it shows large fluctuations among different trajectories typical for non-ergodic systems. Note that even with this typical behavior, it can be easily distinguished in experiments since in our case the mean of the time averaged variance converges to a constant, while for standard CTRW in a bounded region it grows as a power of the anomalous exponent, $\left\langle \sigma_{T}(\Delta )\right\rangle \sim \Delta ^{1-\mu }$. Regarding the shape of the stationary density, numerical simulations are in good agreement with analytical results Eq. (\[stat\]) (see figure [FIG3]{}). *Summary.—* In this Letter we have presented a model of anomalous sub-diffusive transport in which the force acts on the particle at all times not only at the moment of jump. This leads to the dependence of jumping rate on the force with the dramatic change of particles behavior compared to the standard CTRW model. We have derived a new type of fractional diffusion equation which is fundamentally different from the classical fractional Fokker-Planck equation. In our model the force $F(x)$ not only appears in the drift term as in Eq. (\[eq:FFPE\]), but also determines the structure of the diffusion term controlling the spread of particles. The constant external force leads to the natural tempering of the broad waiting time distribution and, as a result, to the transition to a seemingly normal diffusion (linear growth of the mean squared displacement) and equivalence of the time and ensemble averages. However, this may lead to a wrong conclusion in analyzes of experimental results on transient sub-diffusion [@Fra13] that the process is normal for large times. We have found that contrary to normal diffusion process in the external force field, the diffusion coefficient depends on the force and anomalous exponent. This fact implies that the Boltzmann distribution is no longer stationary solution. External perturbations and noise fluctuations are not separable which reflects the non-Markovian nature of the process even for large times. Our results would be possible to test in experiments, for example, by considering a bead which is moving sub-diffusively in an actin network. The motion of such a beat can be described by a random walk type of dynamics [@Wong04]. Force-measurements could be realized by using optical trap and tweezers which are the nano-tools capable of performing such measurements on individual molecules and organelles within the living cell [@Nor14]. When the force is constant the dependence of the measures diffusion coefficient on the strength of the force would reveal the predicted power law behavior $F^{1-\mu }$. For quadratic potential it could be possible to retrieve the form of the stationary profile (\[stat\]) with the slow decay compared to Boltzmann distribution for large $x$. Acknowledgements {#acknowledgements .unnumbered} ================ SF and NK acknowledge the support of the EPSRC Grant EP/J019526/1. [10]{} *Anomalous Transport: Foundations and Applications*, edited by R. Klages, G. Radons, and I.M. Sokolov (Wiley-VCH, Weinheim, 2007). G. Drazer and D.H. Zanette, Phys. Rev. E **60**, 5858 (1999). E.R. Weeks and D.A. Weitz, Chem. Phys. **284**, 361 (2002). G. Seisenberger *et al.*, Science **294**, 1929 (2001). M.J. Saxton and K. Jacobson, Ann. Rev. Biophys. Biomol. Struct. **26**, 373 (1997). K. Ritchie *et al.*, Biophys. J. **88**, 2266 (2005). M. Weiss, H. Hashimoto, and T. Nilsson, Biophys. J. **84**, 4043 (2003). I. Golding and E.C. Cox, Phys. Rev. Lett. **96**, 098102 (2006). I.M. Tolic-Norrelykke *et al.*, Phys. Rev. Lett. **93**, 078102 (2004). F. Höfling and T. Franosch, Rep. Prog. Phys. **76**, 046602 (2013). E. Barkai, Y. Garini, and R. Metzler, Phys. Today **65** 29 (2012). A. Sergé *et al.*, Nat. Methods **5**, 687 (2008). K. Jaqaman *et al.*, Nat. Methods **5**, 695 (2008). K. Norregaard *et al.*, Phys. Chem. Chem. Phys. **16**, 12614 (2014). Y. He *et al.*, Phys. Rev. Lett.**101**, 058101 (2008). A. Lubelski, I.M. Sokolov, and J. Klafter, Phys. Rev. Lett.**100**, 250602 (2008). Y. Meroz, I.M. Sokolov and J. Klafter, Phys. Rev. Lett. **110**, 090601 (2013). I. Bronstein *et al.*, Phys. Rev. Lett. **103**, 018102 (2009). T. Neusius *et al.*, Phys. Rev. Lett. 100, 188103 (2008). M. Saxton, Biophys. J. **81**, 2226 (2001). A.V. Weigel *et al.*, Proc. Natl. Acad. Sci. USA **108**, 6438 (2011). I.M. Sokolov, Soft Matter [**8**]{}, 9043 (2012). P.C. Bressloff and J.M. Newby, Rev. Mod. Phys. [**85**]{}, 135 (2013). R. Metzler, E. Barkai, and J. Klafter, Phys. Rev. Lett.**82**, 3563 (1999). R. Metzler and J. Klafter, Phys. Reports **339**, 1 (2000). E. Heinsalu *et al.*, Phys. Rev. Lett. **99**, 120602 (2007). M. Magdziarz, A. Weron, and J. Klafter, Phys. Rev. Lett.**101**, 210601 (2008). B.I. Henry, T.A.M. Langlands, and P. Straka, Phys. Rev. Lett. **105**, 170602 (2010). S. Eule and R. Friedrich, Euro. Phys. Lett. **86**, 30008 (2009). I.M. Sokolov and J. Klafter, Phys. Rev. Lett. **97**, 140602 (2006). A.I. Shushin, Phys. Rev. E **78**, 051121 (2008). V.P. Shkilev, Journal of Experimental and Theoretical Physics, [**114**]{}, 830 (2012). S. Fedotov, A.O. Ivanov and A.Y. Zubarev, Math. Model. Nat. Phenom. [**8**]{}, 28 (2013). E. Abad, S.B. Yuste, and K. Lindenberg, Phys. Rev. E [**81**]{}, 031115 (2010). S. Fedotov and S. Falconer, Phys. Rev. E [**87**]{}, 052139 (2013). T. Neusius, I.M. Sokolov and J.C. Smith, Phys. Rev. E **80**, 011109 (2009). I.Y. Wong *et al.*, Phys. Rev. Lett. **92**, 178101 (2004). D.R. Cox and H.D. Miller, [*The Theory of Stochastic Processes*]{} (Methuen, London, 1965). A.V. Chechkin, R. Gorenflo, and I.M. Sokolov, J. Phys. A: Math. Gen. [**38**]{}, L679 (2005). S. Fedotov, Phys. Rev. E [**88**]{}, 032104 (2013). SUPPLEMENTARY MATERIALS {#supplementary-materials .unnumbered} ======================= To derive Eq. (\[main\_eq\]) we use the structured probability density function $\xi (x,t,\tau )$ with the residence time $\tau $ as auxiliary variable. This density gives the probability that the particle position $% X(t) $ at time $t$ is at the point $x$ and its random residence time $T_{x}$ at point $x$ is in the interval $(\tau ,\tau +d\tau ).$ The density $\xi (x,t,\tau )$ obeys the balance equation $$\frac{\partial \xi }{\partial t}+\frac{\partial \xi }{\partial \tau }% =-\left( \mathbb{T}_{x}^{+}(x,\tau )+\mathbb{T}_{x}^{-}(x,\tau )\right) \xi . \label{basic}$$We consider only the case when the residence time of random walker at $t=0$ is equal to zero, so the initial condition is $$\xi (x,0,\tau )=p_{0}(x)\delta (\tau ), \label{initial}$$where $p_{0}(x)$ is the initial density. The boundary condition in terms of residence time variable ($\tau =0)$ can be written as [@Cox] $$\begin{aligned} \xi (x,t,0) &=&\int_{0}^{t}\mathbb{T}_{x}^{+}(x-a,\tau )\xi (x-a,t,\tau )d\tau + \notag \\ &&\int_{0}^{t}\mathbb{T}_{x}^{-}(x+a,\tau )\xi (x+a,t,\tau )d\tau . \label{arr1}\end{aligned}$$We solve (\[basic\]) by the method of characteristics for $\tau <t$ $$\xi (x,t,\tau )=j\left( x,t-\tau \right) \Psi_{\lambda} (x,\tau )e^{-\Phi(x) \tau },\quad \tau <t, \label{je}$$where $$\Phi(x) = \nu a |F(x)|.$$The solution Eq. (\[je\]) is written in terms of the integral arrival rate $% j(x,t)=\xi (x,t,0)$ and in terms of the survival function Eq. (\[FFFF\]) $$\Psi_{\lambda} (x,\tau ) = e^{-\int_{0}^{\tau } \lambda (x,s) ds}.$$Our purpose now is to derive the master equation for the probability density$$p(x,t)=\int_{0}^{t^{+}}\xi (x,t,\tau )d\tau . \label{denG}$$Let us introduce the integral escape rate to the right $i^{+}(x,t)$ and the integral escape rate to the left $i^{-}(x,t)$ as$$i^{\pm }(x,t)=\int_{0}^{t^{+}} \omega^{\pm}(x) \lambda(x,\tau )\xi (x,t,\tau )d\tau . \label{i1}$$Note that the integration with respect to the residence time $\tau $ in ([denG]{}) and (\[i1\]) involves the upper limit $\tau =t,$ where we have a singularity due to the initial condition (\[initial\]). Then the boundary conditions (\[arr1\]) can be written in a simple form: $$\begin{aligned} j(x,t) &=&i^{+}(x-a,t)+i^{-}(x+a,t) \notag \\ &&+% \begin{cases} \nu aF(x-a)p(x-a,t), & F\geq 0, \\ -\nu aF(x+a)p(x+a,t),\quad & F<0.% \end{cases} \label{jj}\end{aligned}$$It follows from (\[initial\]), (\[je\]) and (\[i1\]) that $$\begin{aligned} i^{\pm }(x,t) &=&\int_{0}^{t}\psi ^{\pm }(x,\tau )j(x,t-\tau )e^{-\Phi(x) \tau }d\tau \notag \\ &&+\psi ^{\pm }(x,t)p_{0}(x)e^{-\Phi(x) t}, \label{i55}\end{aligned}$$where $\psi ^{+}(x,\tau )=\omega^{+}(x) \lambda(x,\tau ) \Psi_{\lambda} (x,\tau )$ and $\psi^{-}(x,\tau )=\omega^{-}(x) \lambda(x,\tau )\Psi_{\lambda} (x,\tau ).$ Substitution of ([initial]{}) and (\[je\]) to (\[denG\]), gives $$\begin{aligned} p(x,t) &=&\int_{0}^{t}\Psi_{\lambda} (x,\tau )j(x,t-\tau )e^{-\Phi(x) \tau }d\tau \notag \\ &&+\Psi_{\lambda} (x,t)p_{0}(x)e^{-\Phi(x) t}. \label{p11}\end{aligned}$$The balance equation for probability density $p(x,t)$ can be written as$$\frac{\partial p}{\partial t}=-i^{+}(x,t)-i^{-}(x,t)+j(x,t)-\Phi(x) p(x,t). \label{balan}$$Let us find a closed equation for $p(x,t)$ by expressing integral rates $% i^{\pm }(x,t)$ and $j(x,t)$ in terms of the density $p(x,t).$ We apply the Laplace transform $\hat{f}(s)=\int_{0}^{\infty }f (\tau )e^{-s\tau }d\tau $ to (\[i55\]), and (\[p11\]), and obtain $$\hat{i}^{\pm }(x,s)=\frac{\hat{\psi}^{\pm }(x,s+\Phi(x))}{\hat{\Psi}% (x,s+\Phi(x))}\hat{p}(x,s), \label{new55}$$which after the inversion of the Laplace transform and using the shift theorem gives $$i^{\pm }(x,t)=\int_{0}^{t}K^{\pm }(x,t-\tau )e^{-\Phi(x) (t-\tau)} p(x,\tau) d\tau.$$The memory kernels $K^{+}(x,t)$ and $K^{-}(x,t)$ are defined by Laplace transforms $$\hat{K}^{\pm }\left( x,s\right) =\hat{\psi}^{\pm }(x,s)/\hat{\Psi}_{\lambda}\left( x,s\right). \label{new5}$$Now we consider the sub-diffusive case where $\lambda (\tau)$ is inversely proportional to the residence time $\tau:$ $$\lambda (\tau )= \mu/(\tau _{0}+\tau),\qquad 0<\mu <1. \label{s33}$$For simplicity we consider $$\omega^{-}=\omega^{+}=1\slash2.$$ It is straightforward to generalize to non-homogeneous systems by considering space dependent $\lambda(x)$ and space dependent anomalous exponent $\mu(x)$, this case we consider elsewhere [@Che05; @Fed13]. From Eqs. (\[FFFF\]) and ([s33]{}) it follows that the survival function has a power-law dependence $$\Psi_{\lambda} (\tau)= \tau _{0}^{\mu} \left( \tau _{0}+\tau \right)^{-\mu}.$$ The waiting time density functions $\psi^{\pm} (\tau )$ are $$\psi^{+}(\tau)=\psi^{-}(\tau)= \mu \tau _{0}^{\mu} (\tau _{0}+\tau )^{-1-\mu}\slash2. \label{Pareto}$$Using the Tauberian theorem their Laplace transforms are $\hat{\psi}% ^{\pm}\left( s\right) \simeq (1- g s^{\mu})/2$ as $s\rightarrow 0$, where $g = \Gamma (1-\mu )\tau _{0}^{\mu}$. From (\[new5\]) we obtain the Laplace transforms $$\hat{K}^{+}(s) = \hat{K}^{-} (s) \simeq s^{1-\mu}\slash (2 g),\qquad s\rightarrow 0.$$Therefore, the integral escape rates to the right $i^{+}$ and to the left $% i^{-}$ in the sub-diffusive case are $$i^{+}(x,t)=i^{-}(x,t) = e^{-\Phi(x) t}\mathcal{D}_{t}^{1-\mu}\left[ p(x,t)e^{\Phi(x) t}\right] \slash (2 g). \label{gen_i}$$By introducing the total integral escape rate $$i(x,t)=i^{+}(x,t)+i^{-}(x,t),$$and expanding the right-hand side of Eq. (\[balan\]) to second order in jump size $a$ we obtain the following fractional equation $$\frac{\partial p}{\partial t} = - a^{2}\nu \frac{\partial }{\partial x}\left[ F(x)p(x,t)\right] + \frac{a^{2}}{2}\frac{\partial ^{2}i}{\partial x^{2}}, \label{sup_main}$$which using Eq. (\[gen\_i\]) leads to the main equation of the paper Eq. (\[main\_eq\]). Now we derive the equation for the stationary solution Eq. (\[stat\_eq\]). Writing the escape rate $i(x,t)$ in Laplace form $$\hat{i}(x,s) = \frac{(s+\Phi(x))^{1-\mu}}{g} \hat{p}(x,s)$$and taking the limit $s \rightarrow 0$ corresponding to $t \rightarrow \infty$, we obtain the stationary escape rate $$i_{st}(x) = \frac{\Phi(x)^{1-\mu}}{g} p_{st}(x). \label{i_st}$$where the stationary density is defined in a standard way $p_{st}(x) = \lim_{s \rightarrow 0} s \hat{p}(x,s)$. Taking the time derivative in Eq. (\[sup\_main\]) to zero and substituting Eq. (\[i\_st\]) we obtain the stationary advection-diffusion equation $$-a^{2}\nu \frac{d}{dx}\left[ F(x) p_{st}(x) \right] + \frac{d^2}{dx^2}\left[ D_{F}(x)p_{st}(x)\right] =0. \label{stat_eq_2}$$Integrating Eq. (\[stat\_eq\_2\]) and taking into account that the flux of the particles is zero we obtain Eq. (\[stat\_eq\]).

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--- author: - 'Lahcen Assoud, René Messina, Hartmut Löwen' title: 'Stable crystalline lattices in two-dimensional binary mixtures of dipolar particles' --- While the freezing transition and the corresponding crystal lattice in one-component systems is well-understood by now [@review; @Dietrich_PRE_2001], binary mixtures of two different particle species exhibits a much richer possibility of different solid phases For example, while a one-component hard sphere system freezes into the close-packed face-centered-cubic lattice [@Pronk], binary hard sphere mixtures exhibit a huge variety of close-packed structures depending on their diameter ratio. These structures include $AB_{n}$ superlattices, where $A$ are the large and $B$ the small spheres, with $n=1,2,5,6,13$. These structures were found in theoretical calculations [@Xu], computer simulations [@Eldridge1; @Eldridge2] and in real-space experiments on sterically stabilized colloidal suspensions [@Bartlett1; @Bartlett2]. Much less is known for soft repulsive interparticle interactions; most recent studies on crystallization include attractions and consider Lennard-Jones mixtures [@Ref27_SFB_Antrag_D1; @Harrowell] or oppositely charged colloidal particles [@Dijkstra1; @Dijkstra2; @Leunissen]. In this letter we explore the phase diagram of a binary mixture interacting via a soft repulsive pair potential proportional to the inverse cube of the particle separation. Using lattice sums, we obtain the zero-temperature phase diagram as a function of composition and asymmetry, i.e. the ratio of the corresponding prefactors in the particle-particle interaction. Our motivation to do so is threefold: i\) First there is an urgent need to understand the effect of softness in general and in particular in two spatial dimensions. The case of hard interactions in two spatial dimensions, namely binary hard disks, has been obtained by Likos and Henley [@RefDA12] for a large range of diameter ratios. A complex phase behavior is encountered and it is unknown how the phase behavior is affected and controlled by soft interactions. ii\) The model of dipolar particles considered in this letter is realized in quite different fields of physics. Dipolar [*colloidal particles*]{} can be realized by imposing a magnetic field [@Maret_1]. In particular, our model is realized by micron-sized superparamagnetic colloidal particles which are confined to a planar water-air interface and exposed to an external magnetic field parallel to the surface normal [@Maret_1; @Maret_2; @Maret_3; @Koppl_PRL_2007; @Magnot_PCCP_2004]. The magnetic field induces a magnetic dipole moment on the particles whose magnitude is governed by the magnetic susceptibility. Hence their interaction potential scales like that between two parallel dipoles with the inverse cube of the particle distance. Binary mixtures of colloidal particles with different susceptibilities have been studied for colloidal dynamics [@Naegele], fluid clustering [@Hoffmann_PRL_2006; @Hoffmann2], and the glass transition [@Koenig]. A complementary way to obtain dipolar colloidal particles is a fast alternating electric field which generates effective dipole moments in the colloidal particles [@Blaaderen]. This set-up has been applied for two-dimensional binary mixtures in Ref. [@Kusner]. In [*granular matter*]{}, the model has been realized by mixing millimeter-sized steel and brass spheres [@Hay] which are placed on a horizontal plate and exposed to a vertical magnetic field such that the repulsive dipole-dipole interaction is the leading term. Stable triangular $AB_2$ crystalline lattice were found [@Hay]. Layers of [*dusty plasmas*]{} involve particles whose interactions can be dominated by that of dipoles [@Resendes; @Shukla; @Kalman]. Other situations where two-dimensional mixtures of parallel dipoles are relevant concern [*amphiphiles*]{} (confined to a monomolecular film at an air-water interface [@Seul]), [*binary monolayers*]{} [@Israelachvili; @Keller; @Clarke], [*ferrofluid*]{} monolayers [@Elias] exposed to a perpendicular magnetic field, or thin films of [*molecular*]{} mixtures (e.g. of boron nitride and hydrocarbon molecules) with a large permanent dipole moment [@Wiechert]. Hence, in principle, our results can be directly compared to various experiments of (classical) dipolar particles with quite different size in quite different set-ups. iii\) It is important to understand the different crystalline sub-structures in detail, since a control of the colloidal composite lattices may lead to new optical band-gap materials (so-called photonic crystals) [@Pine] to molecular-sieves [@Kecht] and to micro- and nano-filters with desired porosity [@Goedel]. Nano-sieves and filters can be constructed on a colloidal monolayer confined at interfaces [@Goedel]. Their porosity is directly coupled to their crystalline structure. For these applications, it is mandatory to understand the different stable lattice types which occur in binary mixtures. As a result, we find a variety of different stable composite lattices. They include $A_mB_n$ structures with, for instance, $n=1,2,4,6$ for $m=1$. Their elementary cells consist of (equilateral) triangular, square, rectangular and rhombic lattices of the $A$ particles. These are highly decorated by a basis involving either $B$ particles alone or both $B$ and $A$ particles. The topology of the resulting phase diagram differs qualitatively from that of hard disk mixtures [@RefDA12]. For small (dipolar) asymmetries, for instance, we find intermediate $AB_2$ and $A_2B$ structures besides the pure triangular $A$ and $B$ lattices which are absent for hard disks. Our calculations admit more candidate phases than considered in earlier investigations [@RefDA20] where two-dimensional quasicrystals were shown to be metastable. We further comment that we expect that colloidal glasses in binary mixtures of magnetic colloids [@Koenig] are metastable as well but need an enormous time to phase separate into their stable crystalline counterparts. The model systems used in our study are binary mixtures of dipolar particles made up of two species denoted as $A$ and $B$. Each component $A$ and $B$ is characterized by its dipole moment ${\mathbf m}_A$ and ${\mathbf m}_B$, respectively. The particles are confined to a two-dimensional plane and the dipole moments are fixed in the direction perpendicular to the plane. Thereby the dipole-dipole interaction is repulsive. Introducing the ratio $m=m_B/m_A$ of dipole strengths $m_A$ and $m_B$, the pair interaction potentials between two $A$ dipoles, a $A$- and $B$-dipole, and two $B$-dipoles at distance $r$ are $$\begin{aligned} \label{eq_dipol} V_{AA}(r) \lefteqn{=V_0\varphi(r), \quad V_{AB}(r)=V_0m\varphi(r),}\nonumber\\ & & V_{BB}(r)=V_0m^2\varphi(r),\end{aligned}$$ respectively. The dimensionless function $\varphi(r)$ is equal $\ell^3/r^3$, where $\ell$ stands for a unit length. The amplitude $V_0$ sets the energy scale. Our task is to find the stable crystalline structures adopted by the system at zero temperature. We consider a parallelogram as a primitive cell which contains $n_A$ $A$-particles and $n_B$ $B$-particles. This cell can be described geometrically by the two lattice vectors ${\mathbf a}=a(1,0)$ and ${\mathbf b}=a\gamma(\cos{\theta},\sin{\theta})$, where $\theta$ is the angle between ${\mathbf a}$ and ${\mathbf b}$ and $\gamma$ is the aspect ratio ($\gamma=|{\mathbf b}|/|{\mathbf a}|$). The position of a particle $i$ (of species $A$) and that of a particle $j$ (of species $B$) in the parallelogram is specified by the vectors ${\mathbf r}_{\rm i}^A=(x_i^{A},y_i^{A})$ and ${\mathbf r}_{\rm j}^B=(x_j^{B},y_j^{B})$, respectively. The total internal energy (per primitive cell) $U$ has the form $$\begin{aligned} \label{eq_energy} \lefteqn{U=\frac{1}{2}\sum_{J=A,B} \sum_{i,j=1}^{n_J}\sideset{}{'}\sum_{\mathbf{R}}V_{JJ} \left( \left| \mathbf{r}^J_i-\mathbf{r}^J_j+\mathbf{R} \right| \right)} \nonumber\\ & & + \sum_{i=1}^{n_A}\sum_{j=1}^{n_B}\sum_{{\mathbf R}} V_{AB}(\left| \mathbf{r}^A_i-\mathbf{r}^B_j+\mathbf{R} \right|),\end{aligned}$$ where ${\mathbf R}=k{\mathbf a}+l{\mathbf b}$ with $k$ and $l$ being integers. The sums over ${\mathbf R}$ in Eq. \[eq\_energy\] run over all lattice cells where the prime indicates that for ${\mathbf R=0}$ the terms with $i=j$ are to be omitted. In order to handle efficiently the long-range nature of the dipole-dipole interaction, we employed a Lekner-summation [@Lekner; @Lekner_dip_2d]. We choose to work at prescribed pressure $p$ and zero temperature ($T=0$). Hence, the corresponding thermodynamic potential is the Gibbs free energy $G$. Additionally, we consider interacting dipoles at composition $X:=n_B/(n_A+n_B)$, so that the (intensive) Gibbs free energy $g$ per particle reads: $g=g(p,m,X)=G/(n_A+n_B)$. At $T=0$, $g$ is related to the internal energy per particle $u=U/(n_A+n_B)$ through $ g=u+p/\rho $, where the pressure $p$ is given by $p=\rho^2(\partial u/\partial\rho)$, and $\rho=(n_A+n_B)/|{\mathbf a} \times {\mathbf b}|$ is the total particle density. The Gibbs free energy per particle $g$ has been minimized with respect to $\gamma$, $\theta$ and the position of particles of species $A$ and $B$ within the primitive cell. To reduce the complexity of the energy landscape, we have limited the number of variables and considered the following candidates for our binary mixtures: $A_4B$, $A_3B$, $A_2B$, $A_4B_2$, $A_3B_2$, $AB$, $A_2B_2$, $A_2B_3$, $AB_2$, $A_2B_4$, $AB_3$, $AB_4$ and $AB_6$. For the $AB_6$ case we considered a triangular lattice formed by the $A$ particles. Phase Bravais lattice \[basis\] ----------------------- ----------------------------------------- [**T**]{}($A$) Triangular for $A$ \[one $A$ particle\] [**T**]{}($B$) Triangular for $B$ \[one $B$ particle\] [**S**]{}($AB$) Square for $A$ and $B$ together \[one $A$ and one $B$ particles\] [**S**]{}$(A)B_n$ Square for $A$ \[one $A$ and $n$ $B$ particles\] [**Re**]{}$(A)A_mB_n$ Rectangular for $A$ \[$(m+1)$ $A$ and $n$ $B$ particles\] [**Rh**]{}$(A)A_mB_n$ Rhombic for $A$ \[$(m+1)$ $A$ and $n$ $B$ particles\] [**P**]{}$(A)AB_4$ Parallelogram for $A$ \[two $A$ and four $B$ particles\] [**T**]{}$(AB_2)$ Triangular for $A$ and $B$ together \[one $A$ and two $B$ particles\] [**T**]{}$(A_2B)$ Triangular for $A$ and $B$ together \[two $A$ and one $B$ particles\] [**T**]{}$(A)B_n$ Triangular for $A$ \[one $A$ and $n$ $B$ particles\] \[tab.struct\] : The stable phases with their Bravais lattice and their basis. The final phase diagram in the $(m,X)$-plane has been obtained by using the common tangent construction. The dipole-strength ratio $m$ can vary between zero and unity. A low value of $m$ (i.e., close to zero) corresponds to a large dipole-strength asymmetry, whereas a high one (i.e., close to unity) indicates a weak dipole-strength asymmetry. Our calculations show that all the mixtures, except $AB_3$ and $A_4B_2$, are stable. Their corresponding crystalline lattices are depicted in figure \[fig.struct\] and the nomenclature is explained in Table \[tab.struct\]. For the one component case \[$X=0$ (pure $A$) and $X=1$ (pure $B$), see figure \[fig.PD\]\], we found an equilateral triangular lattice [**T**]{}($A$) and [**T**]{}($B$), respectively, as expected (see figure \[fig.struct\]). The most relevant and striking findings certainly concern the phase behavior at weak dipole-strength asymmetry ($ 0.5 \lesssim m < 1$), see figure \[fig.PD\]. Thereby, the only stable mixture $AB_2$ over such a large range of $m$ corresponds to the (“globally” triangular) phase ${\bf T}(AB_2)$ (see figure \[fig.struct\] and figure \[fig.PD\]). This is in strong contrast to what occurs with hard disk potentials [@RefDA12], where $no$ mixture sets in at low size asymmetry. At sufficiently low dipole-strength asymmetry ($m>0.88$), see figure \[fig.PD\], the mixture $A_2B$, that also corresponds to a globally triangular crystalline structure \[namely ${\bf T}(A_2B)$, see figure \[fig.struct\]\], is equally stable. The stability in the limit $m\to1$ of those globally triangular structures are fully consistent with the fact that one-component dipolar systems are triangular. In the regime of strong dipole-strength asymmetry ($ 0.06 < m \lesssim 0.5$), see figure \[fig.PD\], the stability of the composition $X=1/2$, corresponding to the mixtures $AB$ and $A_2B_2$, is dominant and the phase diagram gets richer involving all the different structures \[except ${\bf T}(AB_2)$\] shown in figure \[fig.struct\]. More specifically, for the composition $X=1/2$, we have two phases ${\bf S} (AB)$ and ${\bf Rh}(A)AB_2$. The transition between these two phases is continuous as marked by a symbol $\#$ in figure \[fig.PD\]. For $X=2/3$, many stable phases emerge as depicted in figure \[fig.PD\]. In the $B$-rich region at large asymmetry, the stability will involve many different structures which are probably not considered here. Therefore we leave this region open, see the gray box in figure \[fig.PD\]. Below $X=2/3$, at large asymmetry ($m \lesssim 0.2$), the true phase diagram will also involve a very dense spectrum of stable compositions, as suggested by the already many stable compositions (see figure \[fig.PD\]), which are not among the candidate structures considered here. This feature is very similar to the behavior reported in hard disk mixtures [@RefDA12], where a [*continuous*]{} spectrum of stable mixtures is found for $X \leq 2/3$ at high size asymmetry. In the limit $m \to 0$, a triangular lattice for the $A$ particles will be stable with an increasingly complex substructure of $B$ particles. In conclusion, the ground-state phase diagram of a monolayer of two-dimensional dipolar particles shows a variety of different stable solid lattices. The topology of the phase diagram is different from that of hard disks. Whereas short-ranged interactions lead to a phase separation into pure $A$ and $B$ crystals at low asymmetries, there are two intermediate $A_2B$ and $AB_2$ mixtures for softer interactions. This explains the experimental findings of Hay and coworkers [@Hay] who found an $AB_2$ crystal structure in millimeter-sized steel and brass spheres [@Hay] which does not occur to be stable for hard particles. A further more quantitative experimental confirmation of our theoretical predictions are conceivable either in suspension of magnetic colloids or for binary charged colloidal suspensions [@Palberg] confined between two parallel glass plates [@Palberg2] or for any other situation where two-dimensional dipolar particles are involved. We finish with a couple of remarks: First, based on the present studies it would be interesting to study the behavior of tilted dipoles where anisotropies and attraction play a significant role [@Froltsov]. Our data may also serve as a benchmark to perform further studies on melting of the composite crystals and crystal nucleation out of the melt in two spatial dimensions. The extension to one-component bilayers [@goldoni_prb; @messina_prl] made up of dipolar particles would certainly be relevant. It would also be interesting to apply the method of evolutionary algorithms [@Kahl] to the present problem in order to increase the basket of candidate phases. We thank Christos Likos, Hans-Joachim Schöpe and Thomas Palberg for helpful discussions. This work has been supported by the DFG within the SFB-TR6, Project Section D1. [10]{} . North-Holland, Amsterdam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ; . . . . . . . . . .

--- bibliography: - 'thesis.bib' title: '[**Thermodynamics of QCD-inspired theories**]{}' ---

--- abstract: 'Local structure in the colossal thermal expansion material Ag$_3$\[Co(CN)$_6$\] is studied here using a combination of neutron total scattering and reverse Monte Carlo (RMC) analysis. We show that the large thermal variations in cell dimensions occur with minimal distortion of the \[Co(CN)$_6$\] coordination polyhedra, but involve significant flexing of the Co–CN–Ag–NC–Co linkages. We find real-space evidence in our RMC configurations for the importance of low-energy rigid unit modes (RUMs), particularly at temperatures below 150K. Using a reciprocal-space analysis we present the phonon density of states at 300K and show that the lowest-frequency region is dominated by RUMs and related modes. We also show that thermal variation in the energies of Ag$\ldots$Ag interactions is evident in both the Ag partial pair distribution function and in the Ag partial phonon density of states. These findings are discussed in relation to the thermodynamic properties of the material.' address: - '$^1$ Department of Earth Sciences, University of Cambridge, Downing Street, Cambridge CB2 3EQ, U. K.' - '$^2$ ISIS Facility, Rutherford Appleton Laboratory, Harwell Science and Innovation Campus, Didcot, Oxfordshire OX11 0QX, U.K.' - '$^3$ Department of Physics, Oxford University, Clarendon Laboratory, Parks Road, Oxford OX1 3PU, U.K.' - '$^4$ Department of Chemistry, University of Durham, University Science Laboratories, South Road, Durham DH1 3LE, U.K.' author: - 'Michael J Conterio$^1$, Andrew L Goodwin$^{1}$, Matthew G Tucker$^2$, David A Keen$^{2,3}$, Martin T Dove$^1$, Lars Peters$^4$ and John S O Evans$^4$' title: 'Local structure in Ag$_3$\[Co(CN)$_6$\]: Colossal thermal expansion, rigid unit modes and argentophilic interactions' --- Introduction ============ Flexibility plays a key role in determining the physical properties of framework materials. In many oxide-containing frameworks, for example, the underconstraint of linear cation–oxygen–cation (M–O–M) linkages means that the M–O–M angle is especially sensitive to changes in temperature, pressure and composition. A direct consequence of this underconstraint is the rich series of polyhedral-tilting transitions known to pervade the broad and technologically-important families of perovskite materials [@Mitchell_2002] and silicate frameworks [@Hammonds_1996; @Dove_1997]. A similar example is the well-known negative thermal expansion (NTE) material ZrW$_2$O$_8$, where the same local flexibility of M–O–M linkages is implicated in a number of unusual physical properties: NTE itself [@Mary_1996; @Pryde_1996; @Tucker_2005], elastic constant softening under pressure [@Pantea_2006], and pressure-induced amorphisation [@Keen_2007]. It is becoming increasingly evident that the two-atom metal–cyanide–metal (M–CN–M) linkages found in many transition metal cyanides [@Sharpe_1976] can produce a similar, and often more extreme, array of unusual physical behaviour. Because the cyanide molecule is able to translate as an independent unit, not only is the linkage especially flexible but much of the correlation between displacement patterns around neighbouring M centres is lost [@Goodwin_2006]. What this means is that properties such as NTE appear to be inherently more common amongst cyanides than amongst oxide-containing frameworks, and also the magnitude of these effects can be much larger [@Hibble_2002; @Margadonna_2004; @Goodwin_2005; @Goodwin_2005b; @Pretsch_2006]. For example, isotropic NTE behaviour in the Zn$_x$Cd$_{1-x}$(CN)$_2$ family is more than twice as strong, and in the case of single-network Cd(CN)$_2$ more than three times as strong, as that in ZrW$_2$O$_8$ [@Goodwin_2005; @Phillips_2008]. The concept of increasing framework flexibility—and hence the likelihood and extent of unusual physical behaviour—by employing linear structural motifs of increasing length has led us recently to discover colossal positive and negative thermal expansion in Ag$_3$\[Co(CN)$_6$\] [@Goodwin_2008]. The covalent framework structure of this material, discussed in more detail below, is assembled from Co–CN–Ag–NC–Co linkages, so that the transition metal centres are now connected by linear chains containing five atoms. The framework lattice produced is so flexible that it can adopt a wide variety of geometries with essentially no difference in lattice enthalpy. This serves to amplify the thermodynamic role of low-energy bonding interactions, in this case the dispersion-like “argentophilic” interactions between neighbouring Ag$\ldots$Ag pairs. The net effect is that the lattice expands in some directions at a rate typical of those observed in van-der-Waals solids (*e.g.* Xe [@Sears_1962]) and this expansion is coupled via flexing of the framework lattice to an equally strong NTE effect along a perpendicular direction: the coefficients of thermal expansion, $\alpha$, were found to be roughly linear and equal to $\pm130$MK$^{-1}$ (1MK$^{-1}=10^{-6}$K$^{-1}$) over much of the temperature range 16–500K. It is an interesting and, as yet, unresolved issue as to how these large changes in framework geometry are accommodated at the atomic scale. The separations between some neighbouring pairs of atoms must be affected just as strongly by temperature as the overall lattice dimensions, and the important question is how these changes can be correlated throughout the crystal lattice with such a low energy penalty. This is essentially a *local structure* problem, and as such one needs to employ experimental techniques capable of probing local correlations in order to address these issues. In previous studies of flexible materials, NTE phases and other disordered systems, we have seen that neutron total scattering can be an invaluable tool when studying local structure and dynamics [@Tucker_2005; @Keen_2007; @Tucker_2000; @Tucker_2001; @Goodwin_2007]. Indeed the technique is ideally suited to thermal expansion studies because it probes at once both average and local structure, and so can be used to produce atomic-level descriptions of the crystal structure that are entirely consistent with the known changes in average structure. In this paper, we use a combination of neutron total scattering and reverse Monte Carlo (RMC) analysis [@Dove_2002; @Tucker_2007] to study local structure in Ag$_3$\[Co(CN)$_6$\] over the temperature range 10–300K. Our interest is in understanding how the local arrangements of atoms in the material and their correlated behaviour vary as a function of temperature, with particular emphasis on: (i) thermal changes in the geometry of the Co–CN–Ag–NC–Co linkages, (ii) the extent to which transverse vibrational motion of these linkages helps moderate expansion along the $\langle101\rangle$ crystal axes, (iii) the rigidity (or otherwise) of \[CoC$_6$\] coordination octahedra, and hence whether so-called rigid unit modes (RUMs [@Hammonds_1996]) play an important role in the lattice dynamics, and (iv) evidence for the anharmonicity of Ag$\ldots$Ag interactions. Our paper begins with a review of the crystal structure of Ag$_3$\[Co(CN)$_6$\] and its thermal expansion behaviour. We also include a brief discussion regarding bonding in the material, based largely on a parallel density functional theory (DFT) investigation by one of our groups [@Calleja_2008]. Section \[methods\] then describes the various experimental and computational methods used in our study, including our methods of overcoming the severe anisotropic peak broadening effects observed in low-temperature diffraction patterns. Our experimental results and discussion are given in section \[results\], where we report the bond length and bond angle distributions extracted from our total scattering data, together with the results of both real- and reciprocal-space analyses of correlated behaviour in our RMC configurations. The implications of these experimental and computational results are discussed with particular reference to our previous study of colossal thermal expansion behaviour in Ag$_3$\[Co(CN)$_6$\] [@Goodwin_2008]. Structure and Bonding in Ag$_3$\[Co(CN)$_6$\]: A Review {#structurereview} ======================================================= Crystal structure ----------------- The first crystallographic study of Ag$_3$\[Co(CN)$_6$\] was reported in 1967 by Ludi and co-workers [@Ludi_1967], but the validity of the structural model proposed in this paper was questioned in a subsequent re-evaluation by Pauling and Pauling [@Pauling_1968]. The Paulings’ alternative solution, which was confirmed shortly afterwards [@Ludi_1968] and appears consistent with modern x-ray and neutron diffraction data [@Goodwin_2008], describes a three-dimensional framework assembled from Co–CN–Ag–NC–Co linkages. Each Co$^{3+}$ centre is connected to six other Co atoms via these linkages, forming an extended covalent framework with a cubic network topology analogous to that of the Prussian Blues. The “cube edges” of the framework—namely the Co–CN–Ag–NC–Co linkages—are so long ($\simeq10$Å) that two additional, identical framework lattices can be accommodated within the open cavities of the first, producing the triply-interpenetrating motif illustrated in Fig. \[fig1\](a). ![\[fig1\]Representations of the crystal structure of Ag$_3$\[Co(CN)$_6$\]. (a) Extended Co–CN–Ag–NC–Co linkages connect to form a three-dimensional framework whose topology is described by three interpenetrating cubic ($\alpha$-Po) nets. (b) The trigonal unit cell, with the cyanide linkages oriented along the $\langle101\rangle$ directions. (c) Elongation of the cubic nets along the trigonal axis produces a layered structure in which triangular arrays of hexacyanocobaltate anions (filled polyhedra) alternate with Ag-containing Kagome nets.](fig1.jpg) Threefold interpenetration of this type distinguishes one of the cube body diagonals from the remaining three, removing all but one of the original threefold axes of the cubic network. Consequently the highest compatible crystal symmetry is not a cubic space group, but rather the experimentally-observed trigonal group $P\bar31m$, with the $\bar3$ axis oriented parallel to the unique cube body diagonal. Consequently there is no symmetry constraint to prevent distortion of the cubic networks along the trigonal axis, and indeed at room temperature one observes a Co$\ldots$Co$\ldots$Co “cube” angle of 74.98$^\circ$ [@Goodwin_2008]. Inspection of the crystallographic unit cell determined from our recent neutron diffraction study \[Fig. \[fig1\](b)\] suggests that this deviation from pseudo-cubic symmetry is accommodated by flexing of the Co–CN–Ag linkages (which now run parallel to the $\langle101\rangle$ axes of the trigonal lattice) rather than any significant distortion of the \[CoC$_6$\] coordination geometries: the C–Co–C angles deviate by less than 3$^\circ$ from the expected octahedral values [@Goodwin_2008]. By distorting in this way, the framework assumes a layered structure in which alternate sheets of Ag$^+$ and \[Co(CN)$_6$\]$^{3-}$ ions are stacked parallel to the unique axis of the trigonal cell \[Fig. \[fig1\](c)\]. The silver atoms within a given layer lie at the vertices of a Kagome lattice, and the octahedral hexacyanocobaltate ions form a triangular lattice with vertices above and below the hexagonal Kagome “holes”. This same layered motif is seen in the isostructural rare-earth salts La\[Ag$_x$Au$_{1-x}$(CN)$_2$\]$_3$ [@Colis_2005; @Colis_2005b; @Larochelle_2006], Eu\[Ag$_x$Au$_{1-x}$(CN)$_2$\]$_3$ [@Colis_2005c] and L\[M$^\prime$(CN)$_2$\]$_3$ (L=Tb, Gd, Y; M$^\prime$ = Ag, Au) [@RawashdehOmary_2000], and again in the chiral $P312$ transition-metal derivatives KCo\[Au(CN)$_2$\]$_3$ [@Abrahams_1980], RbCd\[Ag(CN)$_2$\]$_3$ [@Hoskins_1994], KFe\[Au(CN)$_2$\]$_3$ [@Dong_2003] and KMn\[Ag(CN)$_2$\]$_3$ [@Geiser_2003]. In this second group of compounds, the negative charge of the transition-metal-cyanide framework is counterbalanced by the inclusion of alkali earth cations within the framework cavities (the inclusion being responsible for the lower symmetry). Amongst all these structurally-related materials, Ag$_3$\[Co(CN)$_6$\] is unique in terms of the orientation of the cyanide ions: the $d^{10}$ centres are coordinated by the N atoms of the cyanide groups. That we have a Co–CN–Ag arrangement, as opposed to Co–NC–Ag, is very unusual for an Ag-containing cyanide compound, which would normally be expected to contain the C-bound dicyanoargentate \[Ag(CN)$_2$\]$^-$ species [@Sharpe_1976; @Geiser_2003]. The different arrangement seen here may be forced by the coordinatively-inert \[Co(CN)$_6$\]$^{3-}$ salt from which the material is prepared [@Goodwin_2008]. Perhaps a synthetic route involving Co$^{3+}$ and \[Ag(CN)$_2$\]$^-$ ions would yield the “expected” Co\[Ag(CN)$_2$\]$_3$ isomer; however, DFT calculations suggest that the Co–CN–Ag bonding arrangement is in fact thermodynamically favoured [@Calleja_2008]. A second point of interest, and one that is discussed in more detail below, is the relatively short nearest-neighbour Ag$\ldots$Ag distances: the separation at room temperature is approximately 3.5Å, which is only marginally greater than the van der Waals limit (3.4Å [@Bondi_1964]), despite the coulombic repulsion. This close approach between $d^{10}$ centres is indicative of $d^{10}\ldots d^{10}$ “metallophilic” interactions [@Schmidbaur_2001; @Pyykko_2004]. Our previous neutron scattering analysis [@Goodwin_2008] showed that anharmonicity of these interactions is implicated in the unusual thermal expansion behaviour of the material. Similarly, our separate DFT study [@Calleja_2008] showed the interactions also help “soften” the material by flattening the potential energy surface, facilitating the large changes in cell dimensions observed experimentally. The crystal symmetry imposes a number of constraints on the framework geometry that restrict how it can respond to changes in temerapture. These can be summarised as follows: 1. [The Co atoms (which are all equivalent) have $\bar3m$ point symmetry, so that each \[CoC$_6$\] coordination octahedron may distort along the trigonal axis, but all Co–C bonds are equivalent and there are only three internal C–Co–C angles: $(90+\delta)^\circ$, $(90-\delta)^\circ$ and $180^\circ$.]{} 2. [The Ag atoms (which also are equivalent) are coplanar; with a single nearest-neighbour Ag distance and all nearest-neighbour Ag–Ag–Ag angles equal to $60^\circ$, $120^\circ$ or $180^\circ$.]{} 3. [The Ag site has $2/m$ point symmetry; consequently, all atoms within a given Co–CN–Ag–NC–Co linkage are coplanar, the N–Ag–N angle is equal to 180$^\circ$ and all cyanide groups are equivalent.]{} Thermal expansion behaviour --------------------------- Our primary interest in Ag$_3$\[Co(CN)$_6$\] has been its highly unusual thermal expansion behaviour. When heated, the crystal lattice responds by expanding remarkably quickly along the $\mathbf a$ and $\mathbf b$ crystal axes, while contracting equally strongly along the $\mathbf c$ (trigonal) axis. In absolute terms, x-ray diffraction data show the unit cell parameters to vary between extremes of $a=6.7572(8), c=7.3731(13)$Å at 16K and $a=7.1934(9), c=6.9284(15)$Å at 496K [@Goodwin_2008]. Over much of this temperature range, the coefficients of thermal expansion are found to be essentially constant and equal to $\alpha_a=+132$MK$^{-1}$ and $\alpha_c=-130$MK$^{-1}$; these values correspond to a volume coefficient of thermal expansion of $\alpha_V=+134$MK$^{-1}$ and an equivalent isotropic expansion of $\alpha_\ell=+45$MK$^{-1}$. For the vast majority of useable materials, uniaxial thermal expansivities lie in the range $0<\alpha<+20$MK$^{-1}$ [@Barron_1999; @Krishnan_1979], and so the behaviour exhibited by Ag$_3$\[Co(CN)$_6$\] is really rather exceptional. Indeed the positive thermal expansion (PTE) effect is similar in magnitude to that of the most weakly bound solids (*e.g.* Xe [@Sears_1962]), while the NTE effect along $\mathbf c$ is unprecedented for a crystalline material. The unusual behaviour of Ag$_3$\[Co(CN)$_6$\] appears to be intimately related to the flexibility of its framework lattice. In particular, the coupling between expansion along $\mathbf a$ and contraction along $\mathbf c$ means the overall behaviour can be interpreted as a variation of the Co$\ldots$Co$\ldots$Co “cube” angle \[Fig. \[fig1\](a)\]. Because any such variation does not significantly affect the strongest bonding interactions—namely those along the Co–CN–Ag–NC–Co linkages—this process can occur with minimal cost in lattice enthalpy. What this means is that the absolute value of the Co$\ldots$Co$\ldots$Co angle at any given temperature (and hence the absolute unit cell dimensions) is essentially determined only by the weak Ag$\ldots$Ag interactions that occur *between* interpenetrated networks (see below). Bonding ------- As for all bridging transition-metal cyanides, one expects some degree of covalency in the Co$^{3+}\ldots$(CN)$^-$ and Ag$^+\ldots$(NC)$^-$ interactions. This expectation is consistent with the results of DFT calculations [@Calleja_2008], which give Mulliken bond orders (overlap populations) of 0.25 (Co–C), 1.80 (C–N) and 0.34 (Ag–N) [@Segall_1996; @Segall_1996b]. The difference in Co–C and Ag–N bond orders reflects the low electron density near the Co centre: the Co atom has a relatively high nominal charge, and at the same time the cyanide ligand is polarised such that the C atom is more electron-poor than the N atom. The same DFT calculations give Mulliken charges of $Q_{\rm Co}=+1.16$, $Q_{\rm Ag}=+0.65$, $Q_{\rm C}=-0.01$ and $Q_{\rm N}=-0.51$, which suggest there exists a substantial degree of charge transfer from the cyanide ligands to the cobalt atoms. The covalency reflected in these DFT results is responsible for strengthening the bonding along the cube “edges” of the framework—the Co–CN–Ag–NC–Co linkages—and hence the coupling between thermal expansion behaviour along the $\mathbf a$ and $\mathbf c$ axes. It will also mean that deformation of the transition metal coordination geometries will carry a higher energy penalty than would otherwise be the case. In terms of the lattice dynamics, this would see phonon modes that preserve these coordination geometries (namely the RUMs) emerge as some of the lowest-energy lattice vibrations. Our recent structural investigation suggested that the only significant bonding interactions *between* interpenetrated frameworks are dispersion-like forces between neighbouring Ag atoms [@Goodwin_2008]. Rather than involving shared electrons, these “argentophilic” bonds arise from multipolar interactions between excited electronic states of the $d^{10}$ centres. As such, the interactions are not explicitly taken into account at the DFT level; indeed DFT gives the anticipated Ag$\ldots$Ag bond order of 0.0 (reflecting the absence of any covalency), together with a separation closer to that expected in the absence of any attractive terms [@Calleja_2008]. In fact the large difference between DFT and experimental cell parameters is actually strong evidence for the existence and dispersion-like nature of these interactions [@Calleja_2008]. It is difficult to estimate the strength of these Ag$\ldots$Ag interactions based on the DFT results alone; a comprehensive *ab initio* study of \[Cl–Ag–PH$_3$\]$_2$ dimers suggests a bond energy of approximately 15kJmol$^{-1}$ [@OGrady_2004], but larger values can also be found elsewhere in the literature [@Schmidbaur_2001; @Pyykko_2004]. In either case, what is clear is that the energies involved are at least an order of magnitude lower than typical values for covalent or ionic bonds, and this results in a heightened sensitivity to the effects of *e.g.* temperature and pressure. Even without any explicit consideration of the dispersion-like Ag$\ldots$Ag interactions, DFT calculations show that the framework can adopt a wide variety of different geometries with essentially equivalent lattice enthalpies [@Calleja_2008]. For each geometry, the covalent Co–C, C–N and Ag–N interactions are unchanged (and the Ag$\ldots$Ag overlap populations remain equal to zero), but the Ag atoms show considerable tolerance to a range of Ag$\ldots$Ag distances. To some extent this will reflect the relatively high polarisability of the Ag$^+$ cation [@Iwadate_1982], which translates to a “fluffiness” of its valence electron cloud. This “fluffiness” may well contribute to the superionic behaviour of some Ag$^+$ salts [@Hull_2004], where cation polarisability can play a key role in diffusion mechanisms [@Castiglione_1999]. The relevance here is that this polarisability will facilitate flexibility within the Ag$_3$\[Co(CN)$_6$\] framework by allowing relatively large displacements of the Ag atoms as the material is heated. Methods ======= Neutron total scattering ------------------------ The neutron time-of-flight diffractometer GEM [@Williams_1998; @Day_2004; @Hannon_2005] at the ISIS pulsed spallation source was used to collect total scattering patterns from a polycrystalline Ag$_3$\[Co(CN)$_6$\] sample, prepared as in [@Goodwin_2008]. Data were collected over a large range of scattering vectors of magnitudes $0.3\leq Q\leq50$Å$^{-1}$, giving a real-space resolution of order $\Delta r\simeq3.791/Q_{\rm max}\simeq0.08$Å. For the experiment, approximately 3g of sample was placed within a cylindrical thin-walled vanadium can of 8.3mm diameter and 5.8cm height, which was in turn mounted within a top-loading CCR. Data suitable for subsequent RMC analysis were collected at temperatures of 10, 50, 150 and 300K. Following their collection, the total scattering data were corrected using standard methods, taking into account the effects of background scattering, absorption, multiple scattering within the sample, beam intensity variations, and the Placzek inelasticity correction [@Dove_2002]. These corrected data were then converted to experimental $F(Q)$ and $T(r)$ functions [@Dove_2002; @Keen_2001], which are related to the radial distribution function $G(r)$ by the equations $$\begin{aligned} F(Q)&=&\rho_0\int_0^\infty4\pi r^2G(r)\frac{\sin Qr}{Qr}{\rm d}r,\\ T(r)&=&4\pi r\rho_0\left[G(r)+\left(\textstyle\sum_mc_mb_m\right)^2\right],\end{aligned}$$ where $\rho_0$ is the number density, $c_m$ the concentration of each species $m$ and $b_m$ the corresponding neutron scattering lengths. The Bragg profiles for each data set were extracted from the scattering data collected by the detector banks centred on scattering angles $2\theta=63.62$, $91.37$ and $154.46^\circ$. Average structure refinement ---------------------------- The experimental Bragg diffraction profiles were fitted with the [gsas]{} Rietveld refinement program [@Larson_2000] using the published structural model [@Pauling_1968]. As we found in our previous crystallographic study [@Goodwin_2008], the room temperature diffraction pattern could be fitted well using a standard peak shape function, while data collected at lower temperatures required special treatment \[Fig. \[fig2\]\]. ![\[fig2\]Bragg profile refinements of Ag$_3$\[Co(CN)$_6$\] neutron diffraction data collected at 10K: (top) using a single-phase [gsas]{} refinement with variable time-of-flight peak shape parameters, and (bottom) using our multiple-phase [gsas]{} refinement and time-of-flight peak shape parameters obtained from data measured at room temperature. Experimental data are given as points, the fitted profile is shown as a solid line, and the difference (fit$-$data) is shown beneath each curve.](fig2.jpg) In order to refine those data collected at lower temperatures, and their increased variation in peak widths, we considered that the polycrystalline sample contained a distribution of lattice parameters [@Stephens_1999]. Accordingly, we employed a multi-phase model in our structural refinements in which the Ag$_3$\[Co(CN)$_6$\] powder sample is treated as a mixture of phases with different unit cell parameters but with common relative atomic positions and atomic displacement parameters (at a given temperature). There is an implicit assumption here that the same set of atomic coordinates can be used for a distribution of cell parameters at a given temperature. While this will translate to some additional uncertainty in *e.g.* bond lengths, we found the concession to be necessary in order to obtain fits to data that might be considered at all reasonable. For a given temperature, the contribution of each phase (calculated using the room temperature peak-shape parameters) is added to produce a composite diffraction profile that is capable of assuming anisotropic peak shapes. The [gsas]{} refinement process then involved variation not only of the atomic positions and displacement parameters, but also of the relative populations of each phase. As such, we avoided refining a single set of lattice parameters, obtaining instead a distribution profile across a range of unit cell dimensions. For each data set, we refined the populations of eight separate phases, choosing lattice parameters for each phase according to the following algorithm, developed in order to model the observed anisotropy most satisfactorily. First, a structure refinement was performed using a standard t.o.f. peak shape function, giving a “compromise” set of lattice parameters $a_{\rm n},c_{\rm n}$ that resemble a weighted mean across the entire distribution. In each case, these values reflected a more moderate thermal expansion behaviour than those values $a_{\rm x},c_{\rm x}$ obtained in the [topas]{} [@Coelho_2000] refinements of x-ray data published elsewhere [@Goodwin_2008] (*i.e.* $a_{\rm x}<a_{\rm n}$ and $c_{\rm x}>c_{\rm n}$). We note that this is to be expected since the x-ray values correspond to the single-crystal limit; *i.e.*, the values expected in the absence of strain effects. The differences $\delta_a=a_{\rm x}-a_{\rm n}$ and $\delta_c=c_{\rm x}-c_{\rm n}$ between these two lattice parameters are then used to produce a set of eight lattice parameters for the [gsas]{} multi-phase refinement: $\{(a_{\rm n}+\epsilon\delta_a,c_{\rm n}+\epsilon\delta_c); \epsilon=\frac{4}{3},\frac{2}{3},0,-\frac{2}{3},-\frac{4}{3},-\frac{6}{3},-\frac{8}{3},-\frac{10}{3}\}$. This range of $\epsilon$ values was judged the minimum necessary to model the peak asymmetry. The values of $a_{\rm x}$, $a_{\rm n}$, $c_{\rm x}$ and $c_{\rm n}$ and the resultant $\delta_a$, $\delta_c$ are listed in Table \[table\_gsasparams\]; the refined populations for different values of $\epsilon$ are given in Table \[table\_epsilons\]. Because of the relatively small values for $\delta_a$ and $\delta_c$ at 300K, we used only a single-phase refinement (with lattice parameters $a_{\rm n}$ and $c_{\rm n}$) for this data set. $T$/K $a_{\rm n}$/Å $a_{\rm x}$/Å $\delta_a$/Å $c_{\rm n}$/Å $c_{\rm x}$/Å $\delta_c$/Å ------- --------------- --------------- -------------- --------------- --------------- --------------- 10 6.8256(5) 6.7537(8) 0.0719(13) 7.3294(11) 7.3806(13) $-$0.0512(24) 50 6.8485(5) 6.7805(7) 0.0680(12) 7.3052(10) 7.3522(12) $-$0.0470(22) 150 6.9239(3) 6.8879(5) 0.0360(8) 7.2307(6) 7.2587(9) $-$0.0281(15) 300 7.03066(9) 7.0255(5) 0.0052(6) 7.11748(17) 7.1251(11) $-$0.0076(13) : \[table\_gsasparams\] Lattice parameter values $a,c$ refined from neutron (subscript “n”) and x-ray (subscript “x”) [@Goodwin_2008] diffraction patterns, together with their differences $\delta$. The neutron-derived values are “compromise” values, obtained using a standard time-of-flight peak shape that does not account for the distribution in lattice parameters observed at low temperatures \[*cf.* the top panel in Fig. \[fig2\] and see text for further details\]. [cccc]{} $\epsilon$&\ \ &10&50&150\ $\frac{4}{3}$&0.1281(21)&0.1360(17)&0.2610(21)\ $\frac{2}{3}$&0.2785(23)&0.2386(10)&0.1628(16)\ $0$&0.2147(24)&0.2209(17)&0.2105(23)\ $-\frac{2}{3}$&0.1207(24)&0.1555(12)&0.0836(18)\ $-\frac{4}{3}$&0.0956(26)&0.0931(20)&0.106(3)\ $-\frac{6}{3}$&0.0845(26)&0.0739(20)&0.047(3)\ $-\frac{8}{3}$&0.0332(26)&0.0312(20)&*0*\ $-\frac{10}{3}$&0.0447(21)&0.0509(17)&0.1288(20)\ Reverse Monte Carlo refinement ------------------------------ The reverse Monte Carlo refinement method as applied to crystalline materials, together with its implementation in the program [rmcp]{}rofile have been described in detail elsewhere [@Dove_2002; @Tucker_2007]. The basic refinement objective is to produce atomistic configurations that account simultaneously for the experimental $F(Q)$, $T(r)$ and Bragg profile $I(t)$ functions. This is achieved by accepting or rejecting random atomic moves subject to the metropolis Monte Carlo algorithm, where in this case the Monte Carlo “energy” function is determined by the quality of the fits to data. In light of the anisotropic peak broadening effects at low temperatures, we used a modified version of the [rmcp]{}rofile code that was capable of taking into account the lattice parameter distributions described above. The experimental data were compared against an appropriately weighted sum of individual $F(Q)$, $T(r)$ and $I(t)$ functions, each corresponding to a different set of lattice parameters but calculated from a common atomistic configuration. This approach was deemed necessary because anisotropic variation of the lattice parameters affects the powder-averaged pair distribution function in a non-trivial manner. Our starting configurations for the RMC process were based not on the crystallographic unit cell shown in Fig. \[fig1\](b), but on a supercell with orthogonal axes given by the transformation $$\left[\begin{array}{l}\mathbf a\\ \mathbf b\\ \mathbf c\end{array}\right]_{\rm RMC}=\left[\begin{array}{ccc}6&0&0\\ -4&8&0\\ 0&0&6\end{array}\right]\times\left[\begin{array}{l}\mathbf a\\ \mathbf b\\ \mathbf c\end{array}\right]_{P\bar31m}.$$ The use of orthogonal axes facilitates the preparation of RMC configurations that are approximately the same length in each direction, maximising the pair distribution cut-off value $r_{\rm max}$ for a given number of atoms. Our configurations contained 4608 atoms and extended approximately 42Å in each direction. Prior to RMC refinement, small random initial displacements were applied to each atom in the configuration, and a set of data-based “distance window” constraints were set in place in order to maintain an appropriate framework connectivity throughout the refinement process [@Tucker_2007; @Goodwin_2005c]. The values used for these constraints are given in Table \[table1\], where they are shown not to interfere with the relevant bond-length distributions. We chose not to include “soft” bond-length or bond-angle restraints in the refinement, in order to avoid the incorporation of any dynamical bias [@Goodwin_2005c]. Atom pair $d_{\rm min}$/Å $d_{\rm max}$/Å $\bar d$/Å $\sigma$/Å ----------- ----------------- ----------------- ------------ ------------ Co–C 1.76 2.22 1.928 0.085 C–C 2.44 3.00 2.718 0.135 C–N 1.00 1.29 1.161 0.055 Ag–N 1.76 2.22 2.084 0.070 Ag–Ag 3.00 4.10 3.519 0.156 : \[table1\] “Distance window” parameters $d_{\rm min},d_{\rm max}$ used in our Ag$_3$\[Co(CN)$_6$\] RMC refinements and the corresponding mean pair separations $\bar d$ and their standard deviations $\sigma$ at 300K (where the distributions are broadest). The refinement process was allowed to continue until no further improvements in the fits to data were observed. The real-space fits obtained for each of the four temperature points are illustrated in Fig. \[fig3\], where we have converted the experimental and modelled $T(r)$ functions to the related functions $$D(r)=T(r)-4\pi r\rho_0\left(\textstyle\sum_mc_mb_m\right)^2=4\pi r\rho_0G(r)$$ for ease of representation [@Dove_2002; @Keen_2001]. The $F(Q)$ and Bragg profile $I(t)$ functions were modelled similarly successfully, and the quality of these fits was just as consistent across the four data sets. A final check of the integrity of the RMC models is to determine whether any anomalous regions of “damage” or unphysical atomic displacements have evolved during the refinement process, as discussed in more detail elsewhere with respect to RMC refinements of SrTiO$_3$ [@Goodwin_2005c]. Here, visual inspection of our RMC configurations confirms the absence of any such regions; a representative section of a 300K RMC configuration is illustrated in Fig. \[fig4\]. Furthermore, to ensure reproducibility in our results, all positional, bond length and bond angle distributions presented in our analysis were calculated as averages over 10 independent RMC refinements. ![\[fig3\]Experimental $D(r)$ data (points) and RMC fits (solid lines) obtained using [rmcp]{}rofile as described in the text. Data for successive temperature points have been shifted vertically by three units in each case.](fig3.jpg) ![\[fig4\]A $(001)$ slice of a 300K RMC configuration, viewed along a direction parallel (top) and perpendicular (bottom) to the trigonal axis $\mathbf c$.](fig4.jpg){width="10cm"} Phonon analysis --------------- The [rmcp]{}rofile implementation of the reverse Monte Carlo (RMC) structural refinement method [@Tucker_2007], modified as described above, was used to generate ensembles of atomistic configurations consistent with the neutron $T(r)$, $F(Q)$ and $I(t)$ data. Each ensemble (corresponding to a single temperature point) contained approximately 1000 independent configurations. The atomic positions in these ensembles were analysed via the reciprocal-space approach given in [@Goodwin_2005c; @Goodwin_2004] to yield a set of phonon frequencies and mode eigenvectors for each point in reciprocal space allowed by the supercell geometry. Essentially what this process does is to translate the magnitude of correlated atomic displacements into the energy of lattice vibrations: larger displacements will correspond to lower phonon energies, and smaller displacements to higher energies. There is a difficulty in comparing the phonon frequencies obtained for different temperature points in that RMC configurations will contain a level of baseline “noise” that arises because, amongst other effects, instrument resolution is not explicitly taken into account. This has the effect of making the displacements at low temperatures appear to be much greater than one would expect, while those at higher temperatures (where the “noise” becomes less significant with respect to the “true” correlated motion) become increasingly accurate. Consequently, one observes a systematic shift in phonon frequencies for variable-temperature studies that can make quantitative analysis difficult [@Goodwin_2005c]. Here we have scaled the phonon frequencies at temperatures below 300K such that they give the same mean phonon frequency as observed at 300K itself (where we expect this value to be most robust). This means that changes in the total phonon energy are not recoverable from the analysis, but we can still see how the energies of partial densities of states, *i.e.* individual components of the phonon spectrum, are affected by temperature *relative to one another*. Results and Discussion {#results} ====================== Average structure ----------------- By analysing our neutron scattering data using both [gsas]{} and [rmcp]{}rofile approaches, we obtain two subtly different, but complementary, descriptions of the average structure (*i.e.* the distribution of atomic positions). On the one hand, [gsas]{} refinements give accurate measurements of the lattice parameters and the average positions of atoms in the unit cell. On the other hand, [rmcp]{}rofile gives a three-dimensional positional distribution function for each atom: it models thermal displacements as a distribution of partial atom occupancies rather than as ellipsoidal Gaussian functions. The average atomic coordinates refined using [gsas]{} are given in Table \[avgcoords\]. Perhaps the most interesting result from these refinements is the variation in average geometry of the Co–CN–Ag–NC–Co linkage as a function of temperature, illustrated in Fig. \[fig5\]. What is apparent from this figure is that the lattice parameter variations occur with minimal effect on the Co–C and C–N bonding geometries. Instead, the C–N–Ag angle increases with increasing temperature, such that the linkage approaches a more linear average coordination geometry. We note that the same qualitative behaviour was observed in our parallel DFT study [@Calleja_2008]. Because of these geometric changes in the Co–CN–Ag–NC–Co linkage, the actual thermal expansion behaviour of the Co$\ldots$Co vector is relatively complicated. The move towards a more linear geometry at higher temperatures will produce a PTE effect by pushing the Co atoms apart as the linkage straightens. However, we would also expect transverse vibrational motion of the C, N and Ag atoms to draw the Co ends closer together at the same time [@Goodwin_2006]. What happens in practice—illustrated most clearly in our previous x-ray diffraction results [@Goodwin_2008]—is that the expansion behaviour is very much weaker than the colossal effects along the crystal axis, and there is also a subtle alternation between low-level PTE and NTE behaviour, which is entirely compatible with the existence of two competing effects. $T$/K $x_{\rm C}$ $z_{\rm C}$ $x_{\rm N}$ $z_{\rm N}$ ---------------- ------- ------------- ------------- ------------- ------------- [gsas]{} 10 0.21994(26) 0.15280(23) 0.34167(20) 0.26553(15) 50 0.21968(26) 0.15301(23) 0.34033(20) 0.26589(15) 150 0.21376(25) 0.15742(27) 0.33778(20) 0.26653(17) 300 0.21160(21) 0.16080(19) 0.33377(13) 0.26737(17) [rmcp]{}rofile 10 0.22195(12) 0.15587(13) 0.33855(17) 0.26443(12) 50 0.22256(13) 0.15559(14) 0.33784(18) 0.26475(13) 150 0.22083(8) 0.15683(13) 0.33498(15) 0.26517(13) 300 0.21993(16) 0.15712(15) 0.33089(24) 0.26644(18) : \[avgcoords\]Relative average atomic coordinates from multiple-phase [gsas]{} and [rmcp]{}rofile refinements of Ag$_3$\[Co(CN)$_6$\]: Co at $(0,0,0)$, Ag at $(\frac{1}{2},0,\frac{1}{2})$, C at $(x_{\rm C},0,z_{\rm C})$ and N at $(x_{\rm N},0,z_{\rm N})$. ![\[fig5\]Variation in the average geometry of the Co–CN–Ag–NC–Co linkage in Ag$_3$\[Co(CN)$_6$\] over the temperature range 10–300K as determined by [gsas]{} refinements of Bragg profile data.](fig5.jpg) The atomic distributions calculated by collapsing representative RMC configurations onto a single unit cell are shown in Fig. \[fig6\]. These illustrate the absence of any split-site disorder, with thermal displacements centred on the average crystallographic positions. As expected, these displacements are rather anisotropic in the case of C and N atoms (and, to a lesser extent, for the Ag atoms too). The distributions at low temperatures are perhaps a little broader than expected; the origin of this broadening is understood [@broadening_note] and is seen in other RMC studies (*e.g.* [@Goodwin_2005c]). The average atomic coordinates obtained using [rmcp]{}rofile are given in Table \[avgcoords\], where they are compared with the values obtained from [gsas]{} refinements. The RMC values show a smoother variation with respect to temperature than the [gsas]{} results. However the general changes in structure, such as flexing of the Co–CN–Ag–NC–Co linkages as illustrated in Fig. \[fig5\], remain consistent between the two approaches. ![\[fig6\]Atomic distributions obtained by collapsing RMC configurations onto a single unit cell, viewed down the crystallographic $\mathbf c$ axis.](fig6.jpg) Local structure: bond length and bond angle distributions --------------------------------------------------------- The “bond lengths” calculated from average structure determinations such as Rietveld refinement really only represent the separations between average atomic positions, denoted by the nomenclature $\langle$A$\rangle$–$\langle$B$\rangle$. RMC refinement of total scattering data allows us to measure the true average bond lengths $\langle$A–B$\rangle$ because these correspond directly to peak positions in the experimental $T(r)$ function. In this respect, the Co–C, C–N and Ag–N bond lengths are of particular interest because transverse vibrational motion of the C and N atoms means that the true distances cannot easily be determined from average structure analysis. This is especially difficult here because the spread in lattice parameters at low temperatures gives rise to a corresponding additional uncertainty in the bond lengths; indeed the scatter observed in our [gsas]{} analysis is too large to consider the absolute values meaningful. On the other hand, the true average bond lengths $\langle$Co–C$\rangle$, $\langle$C–N$\rangle$ and $\langle$Ag–N$\rangle$ determined from our RMC models behave smoothly with respect to temperature \[Fig. \[fig7\]\]. What is clear is that all bond lengths increase linearly across the temperature range studied. The rate of increase of the Co–C separation approximately double that of the Ag–N separation, which in turn increases more strongly than the C–N separation. ![\[fig7\]Thermal variation in the average $\langle$C–N$\rangle$, $\langle$Co–C$\rangle$ and $\langle$Ag–N$\rangle$ bond lengths as given by the corresponding RMC pair distribution functions.](fig7.jpg) The magnitude of the PTE effect observed for the Co–C and Ag–N bonds is perhaps surprising: the corresponding coefficients of thermal expansion are approximately equal to +45 and +25MK$^{-1}$, respectively. By way of a comparison, the coefficient of thermal expansion for the Zn–C/N bond in Zn(CN)$_2$ was found to be equal to +10.2(10)MK$^{-1}$ in an x-ray total scattering study [@Chapman_2005]. The accuracy of our values will be affected by the overlap between the Co–C and Ag–N distribution functions; we note also that the magnitude of the changes is small with respect to the real-space resolution given by $2\pi/Q_{\rm max}$. A direct fit of two appropriately-weighted Gaussian distributions to the observed $T(r)$ data gives a similar value for expansion of the Ag–N bond (which is more strongly weighted in the neutron data), but a reduced expansivity for the Co–C bond of *ca* +10MK$^{-1}$. The changes in the Ag$\ldots$Ag distribution function are fundamentally different to the behaviour along the Co–CN–Ag linkage: the essentially-Gaussian distribution moves to higher $r$ values with increasing temperature as quickly as its breadth increases \[Fig. \[fig8\]\]. By comparison, the variations in bond lengths seen in Fig. \[fig7\] are all small with respect to the breadths of the corresponding partial pair distribution functions. It is important to keep in mind that the distributions at low temperature are artificially broadened as discussed above; consequently the “true” 10K Ag$\ldots$Ag distribution function would be much sharper than that shown in Fig. \[fig8\]. However the precision with which we can determine the mid-point of a distribution function is much smaller than the width of the distribution, and so the mean values are still well-constrained by the $T(r)$ data. In the harmonic limit, one expects these distribution functions to remain centred on the same value of $r$, and to broaden symmetrically with increasing temperature; here the behaviour reflects the anharmonicity of nearest-neighbour Ag$\ldots$Ag bonding interactions. ![\[fig8\]Ag$\ldots$Ag (left) and Co$\ldots$Ag (right) nearest-neighbour pair distribution functions extracted from our RMC configurations. In each case, the broader distributions correspond to higher temperatures. The values determined from [gsas]{} refinement of the average structure are given as open circles.](fig8.jpg) Conversely, the Co$\ldots$Ag distribution function shows that the separation between Co and Ag atoms remains essentially constant with respect to temperature, with the only noticeable thermal effect being an increased broadening at higher temperatures. Indeed the variation in mean values is found to be less than $0.005$Å, which is similar to the changes in C–N bond lengths but substantially smaller than the increase in Co–C and Ag–N distances along the Co–CN–Ag linkages. This discrepancy can only be understood by allowing for a strong NTE effect of transverse vibrational motion of the C and N atoms. Various bond angle distributions are shown in Fig. \[fig9\], where they are given in a form that takes into account the fact that the number of angles around any particular value $\theta$ is proportional to $\sin\theta$. Considering first the intra-octahedral C–Co–C angles, we find that these behave much as might be anticipated in that the distributions, centred around 90$^\circ$ and 180$^\circ$, broaden very slowly with increasing temperature. There are two crystallographically-distinct C–Co–C angles near 90$^\circ$ (one corresponding to pairs of C atoms on the same side of the $z=0$ plane, and one to pairs of C atoms on opposite sides), and a slight difference in the breadths of the corresponding distributions may be responsible for the asymmetric peak shape observed. The fact that the experimental distribution functions do not change significantly across our different RMC refinements strongly indicates a relatively constrained \[CoC$_6$\] coordination geometry that is not easily affected by temperature. ![\[fig9\]RMC bond angle distributions: intra-octahedral C–Co–C angles and Co–C–N, C–N–Ag and N–Ag–N angles. Successive data sets are shifted vertically by multiples of 6 units. In all cases, the broader distributions correspond to higher temperatures.](fig9.jpg) We find much greater variation in bond angles along the Co–CN–Ag–NC–Co linkages. There are three symmetry-independent angles to consider—namely, Co–C–N, C–N–Ag and N–Ag–N—and the corresponding distributions are also shown in Fig. \[fig9\]. Again, the lowest-temperature distributions will be artificially broadened, and keeping this in mind we see that all three angles reflect a significant flexibility along the entire linkage. The strongest changes are observed for the C–N–Ag angles. This is entirely consistent with the conclusions drawn from average structure analysis, where we expected the large changes in framework geometry to be accommodated primarily via hingeing of the linear linkages at the N atom \[Fig. \[fig5\]\]. As anticipated for a linear coordination geometry, the distribution in N–Ag–N angles shows a much stronger thermal variation than the intra-polyhedral \[CoC$_6$\] angles. That the C–Co–C angle distributions are essentially temperature-independent despite the large C atom displacements seen in the average structure analysis is a clear indication that the atomic displacements in our RMC configurations are strongly correlated. However, it is difficult to obtain a quantitative handle on these correlations from a consideration of pair and triplet distribution functions alone. Instead we turn to geometric algebra, which permits a more sophisticated real-space analysis. Geometric algebra analysis: Rigid unit modes -------------------------------------------- By comparing the orientations of \[CoC$_6$\] octahedra within an ensemble of RMC configurations, it is possible to quantify two aspects of their dynamical behaviour: first, we obtain a measure of the magnitude of octahedral tilts for each octahedron; second, we can determine the extent to which each octahedron distorts via stretching of the Co–C bonds and/or bending of the C–Co–C angles. We carry out these calculations using a method based on geometric algebra [@Wells_2002], as implemented in the program [gasp]{} [@Wells_2004]. For each octahedron, the program first computes an associated “rotor”, an algebraic quantity whose components represent, to first order, the degree of rotation about the three cartesian axes. These rotor components are then used by the program to determine the magnitudes of octahedral translations, rotations and distortions (due both to bending and to stretching of bonds). By analysing how these values are affected by temperature, we can determine whether the dominant lattice vibrations involve octahedral deformation modes, or alternatively whether the Co coordination geometries are reasonably well preserved. The individual rotor components also indicate whether any particular rotation directions are preferred. The rotational, translational and distortive (C–Co–C bending and Co–C stretching) components of the octahedral motion calculated from our RMC configurations are illustrated in Fig. \[fig10\]. In this figure, the various components are compared with the values calculated from reference configurations, which share the same set of atomic displacements given by RMC but where these have now been distributed randomly throughout the configuration. Both RMC and “reference” configurations share the same average structure, but higher-order correlations will be constrained in the former via refinement against the experimental $T(r)$ function. This comparison then helps illustrate the way in which $T(r)$ data provide information about local correlations. In this specific case, it is apparent that the extent to which \[CoC$_6$\] octahedra are distorted by bond stretching and bond bending displacements is substantially smaller in our RMC configurations than is required by the average structure information. Instead, the atomic displacements appear to be correlated more strongly in the form of octahedral rotations, which occur with a significantly greater probability than the average structure demands. This is strong evidence of the RUM-type vibrations discussed above. It is straightforward to show that one expects the same degree of polyhedral translation in both cases (since this depends essentially on the root-mean-squared Co displacement, which is preserved), and that this is indeed the case is a good internal check on our analysis. ![\[fig10\]\[CoC$_6$\] octahedral displacement components calculated using [gasp]{} for our RMC configurations (filled circles, solid lines) and for “reference” configurations (open squares, dashed lines): (a) rotations, (b) translations, (c) bond-bending deformations, and (d) bond-stretching deformations.](fig10.jpg) As explained above our RMC configurations are known to contain a certain degree of random “noise”, responsible for adding a baseline level of atomic motion that is particularly noticeable at low temperatures. Together with the baseline contribution due to zero point motion, these effects explain why our geometric algebra analysis shows non-zero levels of polyhedral motion and deformation even at low temperatures. By subtracting the projected 0K values, we arrive at the baseline-corrected plot shown in Fig. \[fig11\](a). This representation of our [gasp]{} analysis shows that on heating Ag$_3$\[Co(CN)$_6$\], one populates vibrational modes that involve translations and rotations of \[CoC$_6$\] in preference to C–Co–C bending modes and Co–C stretching modes. This is all very intuitive, and certainly consistent with the expectations of a RUM analysis [@Goodwin_2006].  Thermal variation in \[CoC$_6$\] octahedral displacement patterns, after correction for the 0K RMC background: translations (open circles), rotations (filled circles), bond-bending deformations (crosses) and bond-stretching deformations (filled squares). The right-hand panels show the distributions of \[CoC$_6$\] rotation angles around (b) axes perpendicular to $\mathbf c$ and (c) an axis parallel to $\mathbf c$. In panels (b) and (c), the broader distributions correspond to higher temperatures.](fig11.jpg) As discussed above, it is possible to determine the extent to which \[CoC$_6$\] octahedra rotate in different directions. By virtue of the $\bar3m$ site symmetry at the Co position, there are two key axes to consider: these correspond to rotations around $\mathbf a$ and others around $\mathbf c$. Panels (b) and (c) of Fig. \[fig11\] give plots of the distribution of rotation angles around both of these axes. This figure shows that the distribution around $\mathbf c$ is broader at each temperature point than that around $\mathbf a$, and hence the root-mean-squared deviation is larger, but that these axial rotations may have saturated by 150K. We note that with only one temperature point above 150K it is impossible for us to determine whether this saturation is indeed a real effect. Phonon analysis --------------- With 16 atoms in the primitive unit cell, the lattice dynamics of Ag$_3$\[Co(CN)$_6$\] is characterised by a set of 48 phonon modes. As such, this is a substantially more complex system than either MgO or SrTiO$_3$ (6 and 15 modes, respectively), for which a phonons-from-total-scattering (PFTS) approach has been applied previously [@Goodwin_2005c; @Goodwin_2004]. The overall range in phonon frequencies is similar in all cases, and so an increase in the number of phonon modes corresponds to a finer separation between mode frequencies across the phonon spectrum. In terms of our phonon calculations this makes meaningful analysis of the three-dimensional RMC phonon dispersion curves particularly difficult, precisely because the confidence in mode frequency values approaches the separation between successive modes. Consequently, our PFTS analysis focusses on the phonon density of states, which helps condense the quantity of vibrational information into a more manageable form. Considering first the overall structure of the phonon density of states, we obtain the most accurate picture from the ensemble of RMC configurations that corresponds to a temperature of 300K, where the effects of anisotropic peak broadening are negligible. The overall density of states reveals three main features: (i) a sharp peak at approximately 3THz; (ii) a broader distribution of modes, peaking at approximately 10THz and with an upper limit of *ca* 25THz; and (iii) a very high frequency component, composed of a peak at 35THz together with a broader distribution centred around the same value \[Fig. \[fig12\](a)\]. Random “noise” in the configuration has the effect of adding an exponential decay function to the PFTS density of states, and this is why the form of the d.o.s. function appears perhaps slightly unfamiliar. What we are really interested in is the deviation from this exponential background, which reflects the energies of correlated atomic displacements.  Powder-averaged phonon density of states calculated from an ensemble of 1000 separate 300K RMC configurations. The three energy regimes referred to in the text are labelled. (b) Scaled partial phonon densities of states, shifted vertically by multiples of 25 units. Each scaled partial density of states (thick line) is superposed on the overall density of states (thin line) for comparison. For the RUM curve, the component due to rotations around $\mathbf c$, which oscillates more strongly than the overall RMC curve, is also shown.](fig12.jpg) In order to determine the types of atomic displacements characterised by the different frequency regimes of this density of states, we calculated a series of partial phonon densities of states for different types of displacement. We calculated these partial densities of states by weighting each mode by the projection of its eigenvector onto a relevant set of displacement patterns. So, for example, a mode whose eigenvector could be represented as a linear combination of the RUMs would project strongly onto the set of RUM eigenvectors, and so its contribution to the RUM partial density of states would be large. The corresponding functions are shown in Fig. \[fig12\](b) for Co atoms, Ag atoms, CN groups and also for RUM-type rotations of \[CoC$_6$\] octahedra (essentially a subset of the CN group displacements). Naturally each of these types of displacements contribute to modes at many different wave-vectors, but some overall features do emerge. What we are comparing at each stage is the likelihood of observing particular displacement patterns at a given energy value relative to the overall density of phonon modes at that energy. Our analysis shows that the peak at lowest frequencies projects most strongly onto the RUM displacements, and especially those corresponding to rotations around $\mathbf c$. This reflects some of the results of the [gasp]{} analysis above: the population of RUMs at low temperatures, and also the broader distributions for rotations around $\mathbf c$ than around $\mathbf a$. The mid-frequency region contains components of all types of displacements, but the contribution of RUMs falls to zero most quickly on moving to higher energies. The highest-frequency regime corresponds exclusively to stretching modes of the CN groups. The energy calculated for these modes comes directly from the width of the CN peak at 1.15Å in the experimental $T(r)$ [@CN_peak_note]. This peak width is especially sensitive to the effects of truncation ripples in the Fourier transform from $F(Q)$, and so we expect the corresponding phonon frequencies to be substantially lower than experimental values. Here we obtain a mean value of approximately 35THz, which underestimates the literature values by around 10THz [@Sharpe_1976]. This difference corresponds to a broadening of the CN peak of less than 30%, which is not at all unreasonable. One of the key results in our previous thermal expansion study was the decrease in Ag displacement frequencies with increasing temperatures, which provided a dynamical mechanism for overcoming the small changes in lattice enthalpies associated with thermal expansion of the lattice [@Goodwin_2008]. Here we can observe this softening directly by considering the Ag partial phonon density of states calculated for each temperature point in our RMC analysis; the corresponding plots are given in Fig. \[fig13\]. Again the form of these partial density of states contains an exponential background contribution due to “noise” in the RMC configurations. Not withstanding this effect, there is certainly an overall decrease in energy scale (the weighted mean frequencies corresponding precisely to those given in [@Goodwin_2008]). However, the lowest-frequency modes actually increase in energy with increasing temperature, despite the overall decrease in energy for Ag displacements. This indicates that there are modes with negative Gr[ü]{}neisen parameters that involve Ag translations (remembering that the material as a whole shows PTE), and also that these modes occur at low energies. A natural candidate for these phonon modes is the collection of transverse vibrations of Co–CN–Ag–NC–Co linkages, which may also couple with RUMs of the \[CoC$_6$\] octahedra. ![\[fig13\]Energy-corrected Ag partial phonon densities of states calculated from our RMC configurations. Successive curves have been raised by 2.5 units in each case to aid visibility. Dashed lines are guides to the eye that connect equivalent modes in each set of data.](fig13.jpg) It is also interesting to note that the magnitude of the shifts in phonon frequencies can be moderately large: one can estimate from Fig. \[fig13\] that values of some mode Gr[ü]{}neisen parameters $\gamma=-\partial(\ln \omega)/\partial(\ln V)$ are of the order of at least $\gamma=\pm3$. Consequently the magnitude of the overall thermal expansion behaviour ($\gamma$ of the order of 0.5, subject to the difficulty in obtaining a measure of the true compressibility [@Calleja_2008]) reflects a difference between competing thermal expansion effects of quite different types of lattice vibrations. This result contrasts sharply the behaviour of other NTE systems such as ZrW$_2$O$_8$ and Zn(CN)$_2$, where the overall behaviour is dominated by contributions from sets of phonon modes with similar Gr[ü]{}neisen parameters [@Ravindran_2000; @Zwanziger_2007]. This means that the extent of thermal expansion in Ag$_3$\[Co(CN)$_6$\] may be very sensitive to changes in composition, because subtle variations in the magnitudes of individual Gr[ü]{}neisen parameters will have a much greater relative impact on their differences than on their sum. We conclude this section by commenting that the similarity in structure of the Ag partial phonon density of states at each temperature point is encouraging in terms of the robustness of the PFTS approach in this case. Uncorrelated atomic motion produces a density of states that is an exponential decay function, with none of the reproducible structure seen in Fig. \[fig13\]. The level of detail of the phonon spectrum could certainly be improved if we were to extend our analysis to larger RMC configurations and/or larger configurational ensembles (while suffering the associated cost in additional computation time). However, the reproducibility we see at the level of analysis given here lends us some confidence that our results in terms of the key dynamical features and their general temperature dependencies are sufficiently robust for the purposes of this investigation. Conclusions =========== This study has shown that the thermal expansion behaviour of Ag$_3$\[Co(CN)$_6$\] actually represents a somewhat complex synthesis of various different local effects. For example, it was always suspected that covalent bonding interactions would constrain thermal expansion along the Co–CN–Ag–NC–Co linkages, but we see here that actually the expansion of individual bonds is quite large, and that this PTE effect is counteracted by the effects of transverse vibrational motion (*cf*. the local structure behaviour in Zn(CN)$_2$ [@Chapman_2005]). Similarly, the decrease in energy at higher temperatures seen for the Ag partial phonon density of states occurs as the (perhaps fortuitous) balance between vibrational modes with moderately-large negative *and* positive Gr[ü]{}neisen parameters. Our analysis has also illustrated the complementary roles played by what one might term “structural” and “dynamical” flexibility of the Co–CN–Ag–NC–Co linkages. By “structural flexibility” what we mean is that the average geometry of these linkages is able to vary in order to preserve the \[CoC$_6$\] coordination geometries as the lattice parameters change with temperature. This was seen in particular in terms of thermal variation of the Ag–N–C bond angles. Alternatively, by “dynamical flexibility” we mean the ability to support low-energy transverse vibrational modes, which help constrain thermal expansion of the Co$\ldots$Co vector. We showed in particular the importance of RUMs, where transverse vibrational motion is correlated around individual metal centres in the form of polyhedral rotations and translations. Our results strongly suggest that similarly atypical behaviour may be found in many other extended framework materials with flexible structures. Indeed in our parallel DFT study it emerged that the Au-containing analogue Ag$_3$\[Co(CN)$_6$\] is a very likely a colossal thermal expansion material [@Calleja_2008]. The same principles may yet prove relevant to other quite different materials, such as the increasingly diverse and widely-studied family of metal-organic frameworks (MOFs) [@Roswell_2004; @Dubbeldam_2007]. References {#references .unnumbered} ========== [10]{} Mitchell R H 2002 [*Perovskites Modern and Ancient*]{} (Ontario: Almaz Press) Hammonds K D, Dove M T, Giddy A P, Heine V and Winkler B 1996 [*Am. Min.*]{} [**81**]{}, 1057 Dove M T 1997 [*Am. Min.*]{} [**82**]{}, 213 Mary T A, Evans J S O, Vogt T and Sleight A W 1996 [*Science*]{} [**272**]{}, 90 Pryde A K A, Hammonds K D, Dove M T, Heine V, Gale J D and Warren M C 1996 [*J. Phys.: Condens. Matt.*]{} [**8**]{}, 10973 Tucker M G, Goodwin A L, Dove M T, Keen D A, Wells S A and Evans J S O 2005 [*Phys. Rev. Lett.*]{} [**95**]{}, 255501 Pantea C A, Migliori A, Littlewood P B, Zhao Y, Ledbetter H, Lashley J C, Kimura T, Van Duijn J, Kowach G R 2006 [*Phys. Rev. B*]{} [**73**]{}, 214118 Keen D A, Goodwin A L, Tucker M G, Dove M T, Evans J S O, Crichton W A, Brunelli M 2007 [*Phys. Rev. Lett.*]{} [**98**]{}, 225501 Sharpe A G 1976 [*The Chemistry of Cyano Complexes of the Transition Metals*]{} (London: Academic Press) Goodwin A L 2006 [*Phys. Rev. B*]{} [**74**]{}, 134302 Hibble S J, Cheyne S M, Hannon A C and Eversfield S G 2002 [*Inorg. Chem.*]{} [**41**]{}, 1042 Margadonna S, Prassides K and Fitch A N 2004 [*J. Am. Chem. Soc.*]{} [**126**]{}, 15390 Goodwin A L and Kepert C J 2005 [*Phys. Rev. B*]{} [**71**]{}, 140301(R) Goodwin A L, Chapman K W and Kepert C J 2005 [*J. Am. Chem. Soc.*]{} [**127**]{}, 15630 Pretsch T, Chapman K W, Halder G J and Kepert C J 2006 [*Chem. Commun.*]{} 1857 Phillips A E, Goodwin A L, Halder G J, Southon P D and Kepert C J 2008 [*Angew. Chem. Int. Ed.*]{} [**47**]{}, 1396 Goodwin A L, Calleja M, Conterio M J, Dove M T, Evans J S O, Keen D A, Peters L and Tucker M G 2008 [*Science*]{} [**319**]{}, 794 Sears D R and Klug H P 1962 [*J. Chem. Phys.*]{} [**37**]{}, 3002 Tucker M G, Dove M T and Keen D A 2000 [*J. Phys.: Condens. Matt.*]{} [**12**]{}, L723 Tucker M G, Squires M D, Dove M T and Keen D A 2001 [*J. Phys.: Condens. Matt.*]{} [**13**]{}, 403 Goodwin A L, Withers R L and Nguyen H-B 2007 [*J. Phys.: Condens. Matt.*]{} [**19**]{}, 335216 Dove M T, Tucker M G and Keen D A 2002 [*Eur. J. Mineral.*]{} [**14**]{}, 331 Tucker M G, Keen D A, Dove M T, Goodwin A L and Hui Q 2007 [*J. Phys.: Condens. Matt.*]{} [**19**]{}, 335218 Calleja M, Goodwin A L and Dove M T 2008 [*J. Phys.: Condens. Matt.*]{} (submitted) Ludi A, G[ü]{}del H U and Dvo[ř]{}[á]{}k V 1967 [*Helv. Chim. Acta*]{} [**50**]{}, 2035 Pauling L and Pauling P 1968 [*Proc. Natl. Acad. Sci., U.S.A.*]{} [**60**]{}, 362 Ludi A and G[ü]{}del H U 1968 [*Helv. Chim. Acta*]{} [**51**]{}, 1762 Colis J C F, Larochelle C, Staples R, Herbst-Irmer R and Patterson H, 2005 [*J. Chem. Soc., Dalton Trans.*]{}, 675 Colis J C F, Larochelle C, Fern[á]{}ndez E J, L[ó]{}pez-de-Luzuriaga J M, Monge M, Laguna A, Tripp C and Patterson H 2005 [*J. Phys. Chem. B*]{} [**109**]{}, 4317 Larochelle C L and Patterson H H 2006 [*Chem. Phys. Lett.*]{} [**429**]{}, 440 Colis J C F, Staples R, Tripp C, Labrecque D and Patterson H 2005 [*J. Phys. Chem. B*]{} [**109**]{}, 102 Rawashdeh-Omary M A, Larochelle C L and Patterson H H 2000 [*Inorg. Chem.*]{} [**39**]{}, 4527 Abrahams S C, Bernstein J L, Liminga R and Eisenmann E T 1980 [*J. Chem. Phys.*]{} [**73**]{}, 4585 Hoskins B F, Robson R and Scarlett V Y 1994 [*J. Chem. Soc., Chem. Commun.*]{} 2025 Dong W, Zhu L-N, Sun Y-Z, Liang M, Liu Z-Q, Liao D-Z, Jiang Z-H, Yan S-P and Cheng P 2003 [*Chem. Commun.*]{} 2544 Geiser U and Schlueter J A 2003 [*Acta Crystllagr.*]{} [**C59**]{}, i21 Bondi A 1964 [*J. Chem. Phys.*]{} [**41**]{}, 3199 O’Grady E and Kaltsoyannis N 2004 [*Phys. Chem. Chem. Phys.*]{} [**6**]{}, 680 Schmidbaur H 2001 [*Nature*]{} [**413**]{}, 31 Pyykk[ö]{} 2004 [*Angew. Chem. Int. Ed.*]{} [**43**]{}, 4412 Barron T H K and White G K 1999 [*Heat Capacity and Thermal Expansion at Low Temperatures*]{} (New York: Kluwer Academic / Plenum Press) Krishnan R S, Srinivasan R and Devanarayanan S 1979 [*Thermal Expansion of Crystals*]{} (Oxford: Pergamon Press) Segall M D, Pickard C J, Shah R and Payne M C 1996 [*Mol. Phys.*]{} [**89**]{}, 571 Segall M D, Shah R, Pickard C J and Payne M C 1996 [*Phys. Rev. B*]{} [**54**]{}, 16317 Iwadate Y, Kawamura K, Murawami K, Igarashi K and Mochinaga J 1982 [*J. Chem. Phys.*]{} [**77**]{}, 6177 Hull S 2004 [*Rep. Prog. Phys.*]{} [**67**]{}, 1233 Castiglione M J, Wilson M and Madden P A 1999 [*J. Phys.: Condens. Matt.*]{} [**11**]{}, 9009 Williams W G, Ibberson R I, Day P and Enderby J E 1998 [*Physica B*]{} [**241–243**]{}, 234 Day P, Enderby J E, Williams W G, Chapon L C, Hannon A C, Radaelli P G and Soper A K 2004 [*Neutron News*]{} [**15**]{}, 19 Hannon A C 2005 [*Nucl. Instrum. Methods Phys. Res. A*]{} [**551**]{}, 88 Keen D A 2001 [*J. Appl. Crystallgr.*]{} [**34**]{}, 172 Larson A C and von Dreele R B 2000 [*General Structure Analysis System (GSAS)*]{}, Los Alamos National Laboratory Report No. LAUR 86-748 (unpublished) Stephens P 1999 [*J. Appl. Cryst.*]{} [**32**]{}, 281 Coelho A A 2000 [*TOPAS v2.0: General profile and structure analysis software for powder diffraction data*]{}, Karslruhe Goodwin A L, Tucker M G, Cope E R, Dove M T and Keen D A 2005 [*Phys. Rev. B*]{} [**72**]{}, 214304 There are two effects responsible for this broadening. The first is that Fourier truncation in $Q$-space broadens the low-$r$ peaks in the $T(r)$ data; the second is that, by not taking into account the effects of instrument resolution on different detector banks (a limitation we are working towards overcoming), there is a similar dampening of the $T(r)$ function that will serve to broaden the RMC distributions. Because these broadening effects are independent of temperature, the *relative* difference between “ideal” and experimental peak widths is greatest at low temperatures, where the peaks in the $T(r)$ function should be much sharper than is observed. Accordingly, the low-temperature partial pair distributions are fitted most satisfactorily during the RMC refinement process by increasing atomic motion to what may be a slightly unphysical level. At later stages in our analysis, we will see that this gives rise to broadening of the pair and triplet correlation functions at low temperatures, a higher-than-expected degree of polyhedral motion and deformation, and also a decrease in the calculated phonon frequencies. These effects are really only noticeable because we are attempting to extract very detailed and subtle information from our total scattering data. Goodwin A L, Tucker M G, Dove M T and Keen D A 2004 [*Phys. Rev. Lett.*]{} [**93**]{}, 075502 Champan K W, Chupas P J and Kepert C J 2005 [*J. Am. Chem. Soc.*]{} [**127**]{}, 15630 Wells S A, Dove M T and Tucker M G 2002 [*J. Phys.: Condens. Matter*]{} [**14**]{}, 4567 Wells S A, Dove M T and Tucker M G 2004 [*J. Appl. Crystallogr.*]{} [**37**]{}, 536 Moreover, because the energy of the CN stretch is so much higher than thermal energy at all of the temperatures considered here, actually the phonon mode frequency calculated from the CN peak width is independent of temperature. We do not observe any significant change at all in the CN peak widths between 10 and 300K, and indeed the CN stretching frequencies consistently emerged in the region 35–40THz for each of our temperature points. Ravindran T R, Arora A K, Marty T A 2000 [*Phys. Rev. Lett.*]{} [**84**]{}, 3879 Zwanziger J W 2007 [*Phys. Rev. B*]{} [**76**]{}, 052102 Roswell J and Yaghi O M 2004 [*Micro. Meso. Mater.*]{} [**73**]{}, 3 Dubbeldam D, Walton K S, Ellis D E and Snurr R Q 2007 [*Angew. Chem. Int. Ed.*]{} [**46**]{}, 4496

--- abstract: 'Joint multi-messenger observations with gravitational waves and electromagnetic data offer new insights into the astrophysical studies of compact objects. The third Advanced LIGO and Advanced Virgo observing run began on April 1, 2019; during the eleven months of observation, there have been 14 compact binary systems candidates for which at least one component is potentially a neutron star. Although intensive follow-up campaigns involving tens of ground and space-based observatories searched for counterparts, no electromagnetic counterpart has been detected. Following on a previous study of the first six months of the campaign, we present in this paper the next five months of the campaign from October 2019 to March 2020. We highlight two neutron star - black hole candidates (S191205ah, S200105ae), two binary neutron star candidates (S191213g and S200213t) and a binary merger with a possible neutron star and a “MassGap” component, S200115j. Assuming that the gravitational-wave candidates are of astrophysical origin and their location was covered by optical telescopes, we derive possible constraints on the matter ejected during the events based on the non-detection of counterparts. We find that the follow-up observations during the second half of the third observing run did not meet the necessary sensitivity to constrain the source properties of the potential gravitational-wave candidate. Consequently, we suggest that different strategies have to be used to allow a better usage of the available telescope time. We examine different choices for follow-up surveys to optimize sky localization coverage vs. observational depth to understand the likelihood of counterpart detection.' author: - 'Michael W. Coughlin' - Tim Dietrich - Sarah Antier - Mouza Almualla - Shreya Anand - Mattia Bulla - Francois Foucart - Nidhal Guessoum - Kenta Hotokezaka - Vishwesh Kumar - Geert Raaijmakers - Samaya Nissanke bibliography: - 'references.bib' title: 'Implications of the search for optical counterparts during the second part of the Advanced LIGO’s and Advanced Virgo’s third observing run: lessons learned for future follow-up observations' --- Introduction ============ The observational campaigns of Advanced LIGO [@aLIGO] and Advanced Virgo [@adVirgo] revealed the existence of a diverse population of compact binary systems. Thanks to the continuous upgrades of the detectors from the first observing run (O1) over the second observing run (O2) up to the recent third observational campaign (O3), the gain in sensitivity leads to an increasing number of compact binary mergers candidates: 16 alerts of gravitational-wave (GW) candidates were sent to the astronomical community during O1 and O2, covering a total of 398 days [@AbEA2019], compared to 80 alerts for O3a and O3b, covering a total of 330 days. Some of the candidates found during the online searches were retracted after further analysis, e.g., only 10 out of the 16 alerts were confirmed as candidates during the O1 and O2 runs [@AbEA2019; @LIGOScientific:2018mvr]. Additional compact binary systems were found during the systematic offline analysis performed with re-calibrated data, e.g., @LIGOScientific:2018mvr, resulting in 11 confirmed GW events. During O3a and O3b, 24 of 80 alerts have already been retracted due to data quality issues, e.g. @2019GCN.26413....1L [@2020GCN.26665....1L]. GW detections improve our understanding of binary populations in the nearby Universe (distances less than $\sim$2 Gpc), and cover a large range of masses; these cover from $\sim$1–2.3 solar masses, e.g. @La12 [@OzFr16; @MaMe2017; @ReMo2017; @AbEA2019_GW190425], for binary neutron stars (BNSs) to $\sim$100 solar masses for the most massive black hole remnants. They may also potentially constrain black hole spins [@2019ApJ...882L..24A]. For mergers including NSs, electromagnetic (EM) observations provide a complementary view, providing precise localizations of the event, required for redshift measurements which are important for cosmological constraints [@Sch1986]; these observations may last for years at wavelengths outside the optical spectrum; for instance, X-ray photons were detected almost 1000 days post-merger in the case of GW170817 [@troja2020thousand]. The success of joint GW and EM observations to explore the compact binaries systems has been demonstrated by the success of GW170817, AT2017gfo, and GRB170817A, e.g., @AbEA2017b [@GBM:2017lvd; @ArHo2017; @2017Sci...358.1556C; @LiGo2017; @MoNa2017; @SaFe2017; @SoHo2017; @TaLe2013; @TrPi2017; @VaSa2017]. GRB170817A, a short $\gamma$-ray burst (sGRB) [@1989Natur.340..126E; @Paczynski1991; @1992ApJ...395L..83N; @MocHer1993; @LeeRR2007; @Nakar2007], and AT2017gfo, the associated kilonova [@SaPiNa1998; @LiPa1998; @MeMa2010; @RoKa2011; @KaMe2017], were the EM counterparts of GW170817. Overall, this multi-messenger event has been of interest for many reasons: to place constraints on the supranuclear equation of state describing the NS interior (e.g. @Abbott:2018wiz [@RaPe2018; @RaDa2018; @BaJu2017; @MaMe2017; @ReMo2017; @CoDi2018; @CoDi2018b; @Capano:2019eae; @DiCo2020]), to determine the expansion rate of the Universe [@AbEA2017g; @Hotokezaka:2018dfi; @CoDi2019; @DhBu2019; @DiCo2020], to provide tests for alternative theories of gravity [@EzMa17; @BaBe17; @CrVe17; @AbEA2019b], to set bounds on the speed of GWs [@GBM:2017lvd], and to prove BNS mergers to be a production side for heavy elements, e.g., @2017Natur.551...67P [@WaHa19]. Numerical-relativity studies reveal that not all binary neutron star (BNS) and black hole- neutron star (BHNS) collisions will eject a sufficient amount of material to create bright EM signals, e.g., @Bauswein:2013jpa [@HoKi13; @DiUj2017; @AbEA2017f; @Koppel:2019pys; @Agathos:2019sah; @Kawaguchi:2016ana; @Foucart:2018rjc; @KrFo2020]. For example, there will be no bright EM signal if a black hole (BH) forms directly after merger of an almost equal-mass BNS, since the amount of ejected material and the mass of the potential debris disk are expected to be very small. Whether a merger remnant undergoes a prompt collapse depends mostly on its total mass but also seems to be sub-dominantly affected by the mass-ratio [@Kiuchi:2019lls; @Beea20]. EM bright signatures originating from BHNS systems depend on whether the NS gets tidally disrupted by the BH and thus ejects a large amount of material and forms a massive accretion disk. If the neutron star falls into the BH without disruption, EM signatures will not be produced. This outcome is mostly determined by the mass ratio of the binary, the spin of the black hole, and the compactness of the NS, with disruption being favored for low-mass, rapidly rotating BH and large NS radii [@Etienne:2008re; @Pannarale:2010vs; @Foucart:2012nc; @Kyutoku:2015gda; @Kawaguchi:2016ana; @Foucart:2018rjc]. In addition, beamed ejecta from the GRB can be weakened by the jet break [@2006ApJ...653..468B; @2018ApJ...866L..16M] and may not escape from the “cocoon”, which would change the luminosity evolution of the afterglow. The observability and detectability of the EM signature depends on a variety of factors. First, and most practically, the event must be observable by telescopes, e.g., not too close to the Sun or majorly overlapping with the Galactic plane; 20% of the O3 alerts were not observable by any of three major sites of astronomy; e.g. Palomar, the Cerro Tololo Inter-American Observatory, and Mauna Kea [@GRANDMA2020]. Secondly, the identification of counterparts depends on the duty cycle of instruments and the possibility to observe the skymap shortly after merger. For example, $\gamma$-ray observatories such as the [*Fermi*]{} Gamma-Ray Burst Monitor [@Connaughton12] can cover up to 70% of the full sky, but due to their altitude and pointing restrictions, their field of view can be occluded by the Earth or when the satellite is passing through the South Atlantic Anomaly [@2019GCN.26342....1M; @2020GCN.27361....1L]. The ability for telescopes to observe depends on the time of day of the event. For example, between 18hr – 15hr UTC (although this level of coverage is available only portions of the year, and even then, it is twilight at the edges), both the Northern and Southern sky can, in theory, be covered thanks to observatories in South Africa, the Canary islands, Chile and North America; at other times, such as when night passes over the Pacific ocean or the Middle East, the dearth of observatories greatly reduces the chances of a ground detection. Third, counterpart searches are also affected by the viewing angle of the event with respect to the line of sight towards Earth. While the beamed jet of the burst can be viewed within a narrow cone, the kilonova signature is likely visible from all viewing angles; however, its color and luminosity evolution is likely to be viewing angle dependent [@RoKa2011; @Bul2019; @DaKa20; @KaSh20; @KoWo20]. Finally, as the distance of the event changes, the number of instruments sensitive enough to perform an effective search changes. For example, compared to GW170817, detected at 40 Mpc, the O3 BNS candidates reported so far (with a BNS source probability of $>50\%$) have median estimated distances $\sim150-250$ Mpc. ![Timeline of O3 alerts with highest probability as being BNS, BHNS or MassGap, with highlights of some of the exceptional candidates released. The candidates, if astrophysical, on the top half of the plot are most likely BNSs (or a NS-MassGap candidate in the case of S200115j [@gcn26759]), while the candidates on the bottom half are most likely BHNSs. We highlight GW190425 [@AbEA2019_GW190425], GW190814 [@AbEA2020], S200105ae [@gcn26640; @gcn26688], and S200115j [@gcn26759]. We note that the initial estimate of p(remnant) for S200105ae was 12% [@gcn26640], but is now $<$1% [@gcn26688]. It also has a significance likely greatly underestimated due to it being a single-instrument event, and a chirp-like structure in the spectrograms as mentioned in the public reports [@gcn26640; @gcn26657].\ $^*$We note that GW190814 contains either the highest mass neutron star or lowest mass black hole known [@AbEA2020].[]{data-label="fig:timeline"}](O3_timeline.pdf){width="3.5in"} Despite those observational difficulties, the O3a and O3b observational campaigns were popular for searches of EM counterparts associated with the GW candidates (see Figure \[fig:timeline\] for a timeline for candidates with at least one NS component expected). They mobilized $\sim$100 groups covering multiple messengers, including neutrinos, cosmic rays, and the EM spectrum; about half of the participating groups are in the optical. In total, GW follow-up represented $\sim$50% of the GCN service traffic (Gamma-ray Coordinates Network) with 1,558 circulars. The first half of the third observation run (O3a) brought ten compact binary merger candidates that were expected to have low-mass components, including GW190425 [@SiEA2019a; @SiEA2019b], S190426c [@ChEA2019a; @ChEA2019b], S190510g [@gcn24489], GW190814 [@AbEA2020], S190901ap [@gcn25606], S190910h [@gcn25707], S190910d [@gcn25695], S190923y [@gcn25814], and S190930t [@gcn25876]. The follow-up campaigns of these candidates have been extensive, with a myriad of instruments and teams scanning the sky localizations.[^1] The follow-up of O3a yielded a number of interesting searches. For example, GW190425 [@AbEA2019_GW190425] brought stringent limits on potential counterparts from a number of teams, including GROWTH [@CoAh2019b] and MMT/SOAR [@HoCo2019]. GW190814, as a potential, well-localized BHNS candidate, also had extensive follow-up from a number of teams, including GROWTH [@Andreoni2020], ENGRAVE [@AcAm2020], GRANDMA [@GRANDMAO3A] and Magellan [@GoHo2019]. S190521g brought the first strong candidate counterpart to a BBH merger [@GrFo2020]. The second half of the third observation run (O3b) has brought 23 new publicly announced compact binary merger candidates for which observational facilities performed follow-up searches, including two new BNS candidates, S191213g [@gcn26402] and S200213t [@gcn27042] and two new BHNS candidates, S191205ah [@gcn26350] and S200105ae [@gcn26640; @gcn26688]. S200115j is special for having one NS component and one component object likely falling in the “MassGap” regime, indicating it is between 3–5 M$_\odot$[@gcn26759]. After the second half of this intensive campaign, no significant counterpart (either GRB or kilonovae) was found. While this might be caused by the fact that the GW triggers have not been accompanied by bright EM counterparts, a likely reason for this lack of success in finding optical counterparts is the limited coordination of global EM follow-up surveys and the limited depth of the individual observations. In this article, we build on our summary of the O3a observations [@CoDi2019b] to explore constraints on potential counterparts based on the wide field-of-view telescope observations during O3b, and provide analyses summarizing how we may improve existing strategies with respect to the fourth observational run of advanced LIGO and advanced Virgo (O4). In Sec. \[sec:EM\_follow\_up\_campaigns\], we review the optical follow-up campaigns for these sources. In Sec. \[sec:limits\], we summarize parameter constraints that are possible to achieve based on these follow-ups assuming that the candidate location was covered during the observations. In Sec. \[sec:strategies\], we use the results of these analyses and others to inform future observational strategies trying to determine the optimal balance between coverage and exposure time. Finally, in Sec. \[sec:summary\], we summarize our findings. EM follow-up campaigns {#sec:EM_follow_up_campaigns} ====================== Name p(BNS) p(BHNS) p(terr.) p(HasRemn.) ------------------------------------------------------------------- --------- ---------- ---------- ------------- [S191205ah](https://gracedb.ligo.org/superevents/S191205ah/view/) $0\%$ $93\%$ $7\%$ $< 1\%$ [S191213g](https://gracedb.ligo.org/superevents/S191213g/view/) $77\%$ $ < 1\%$ $23\%$ $> 99\%$ [S200105ae](https://gracedb.ligo.org/superevents/S200105ae/view/) $0\%$ $3\%$ $97\%$ $< 1\%$ [S200115j](https://gracedb.ligo.org/superevents/S200115j/view/) $< 1\%$ $< 1\%$ $< 1\%$ $> 99\%$ [S200213t](https://gracedb.ligo.org/superevents/S200213t/view/) $63\%$ $< 1\%$ $37\%$ $> 99\%$ : Current overview of non-retracted GW triggers with large probabilities of being BNS or BHNS systems. The individual columns refer to: The name of the event, an estimate using the most up-to-date classification for the event to be a BNS \[p(BNS)\], a BHNS \[p(BHNS)\], or terrestrial noise \[p(terrestrial)\] [@KaCa2019], and an indicator to estimate the probability of producing an EM signature assuming the candidate is of astrophysical origin \[p(HasRemnant)\] [@ChSh2019], whose definition is in the [LIGO-Virgo alert userguide](https://emfollow.docs.ligo.org/userguide/content.html). Note that S200115j can also be classified as “MassGap,” completing the possible classifications. During O3b, a change in the template bank used led to a simplified version of the classification scheme where all of the astrophysical probabilities but one became 0, whereas during O3a, accounting for the mass uncertainty, more than one non-zero astrophysical class probability was generally obtained. \[tab:allevents\] **S200115j** **S200213t**\ {width="3.5in"} {width="3.5in"}\ We summarize the EM follow-up observations of the various teams that performed synoptic coverage of the sky localization area and circulated their findings in publicly available circulars during the second half of Advanced LIGO and Advanced Virgo’s third observing run. The LIGO-Scientific and Virgo collaborations used the same near-time alert system during O3b as during O3a, releasing alerts within 2–6 minutes in general (with an important exception, S200105ae, discussed below). For a summary of the second observing run, please see [@AbEA2019], and for the first six months of the third observing run, see [@CoDi2019b] and references therein. In addition to the classifications for the event in categories BNS, BHNS, “MassGap,” or terrestrial noise [@KaCa2019] and an indicator to estimate the probability of producing an EM signature assuming the candidate is of astrophysical origin, p(HasRemnant) [@ChSh2019], skymaps using BAYESTAR [@SiPr2014] are also released. At later times, updated LALInference [@VeRa2015] skymaps are also sent to the community. In addition to the summaries below, we provide Table \[tab:allevents\], displaying source properties based on publicly available information in GCNs and Table \[tab:Tableobs\], displaying the results of follow-up efforts for the relevant candidates. All numbers listed regarding coverage of the localizations refer explicitly to the 90% credible region. We treat S200115j as a BHNS candidate despite its official classification as a “Mass-Gap” event; it has p(HasRemnant) value close to 1, indicating the presence of a NS, but with a companion mass between 3 and 5 solar masses. In addition, we compare the limiting magnitudes and probabilities covered for S200115j and S200213t in Figure \[fig:efficiency\_2\], highlighted as example BHNS and BNS candidates with deep limits from a number of teams. As a point of reference, we include the apparent magnitude of an object with an absolute magnitude of $-16$ with distances ($\pm$1$\sigma$ error bars) consistent with the respective events. As a more physical visualization of the coordinated efforts that go into the follow-up process, we provide Figure \[fig:S200213t\_tiles\]; this representation displays the tiles observed by various telescopes for the BNS merger candidate S200213t, along with a plot of the integrated probability and sky area that was covered over time by each of the telescopes. The black line is the combination of observations made by the telescopes indicated in the caption. These plots are also reminiscent of public, online visualization tools such as GWSky[^2], the Transient Name Server (TNS)[^3], and the Gravitational Wave Treasure Map [@2020arXiv200100588W]. **S200115j** **S200213t**\ **Within 12 hours of merger**\ {width="3.5in"} {width="3.5in"}\ **Within 24 hours of merger**\ {width="3.5in"} {width="3.5in"}\ **Within 48 hours of merger**\ {width="3.5in"} {width="3.5in"} **S191205ah** ------------- LIGO/Virgo S191205ah was identified by the LIGO Hanford Observatory (H1), LIGO Livingston Observatory (L1), and Virgo Observatory (V1) at 2019-12-05 21:52:08 UTC [@gcn26350] with a false alarm rate of one in two years. It has been so far categorized as a BHNS signal $(93\%)$ with a small probability of being terrestrial $(7\%)$. The distance is relatively far at $385 \pm 164$Mpc, and the event localization is coarse, covering nearly $6400$ square degrees. No update of the sky localisation and alert properties have been released by the LVC. 23 groups participated in the follow-up of the event including 3 neutrinos observatories (including IceCube and ANTARES; @gcn26352 [@gcn26349]), two VHE $\gamma$-ray observatories, eight $\gamma$-ray instruments, two X-ray telescopes and ten optical groups (see the [list of GCNs for S191205ah](https://gcn.gsfc.nasa.gov/other/S191205ah.gcn3)). No candidates were found for the neutrinos, high-energy and $\gamma$-ray searches. Five of the optical groups have been engaged for the search of EM counterparts: GRANDMA network, MASTER network, SAGUARO, SVOM-GWAC, and the Zwicky Transient Facility (see Table \[tab:Tableobs\]). The MASTER-network led the way in covering a significant fraction of the localization area, observing $\approx56$% down to 19 in a clear filter and within 144 h [@gcn26353]. Seven transient candidates were reported by the Zwicky Transient Facility [@gcn26416], as well as four transient candidates reported by Gaia [@gcn26397], and one candidate from the SAGUARO Collaboration [@gcn26360], although none displayed particularly interesting characteristics encouraging further follow-up; all of the candidates for which spectra were obtained were ultimately ruled out as unrelated to S191205ah [@gcn26405; @gcn26421; @gcn26422]. **S191213g** ------------ LIGO/Virgo S191213g was identified by H1, L1, and V1 at 2019-12-13 04:34:08 UTC [@gcn26402]. It has been so far categorized as a BNS signal $(77\%)$ with a moderate probability of being terrestrial $(23\%)$, as well as a note that scattered light glitches in the LIGO detectors may have affected the estimated significance and sky position of the event. As expected for BNS candidates, the distance is more nearby (initially $195 \pm 59$Mpc, later updated to be $201 \pm 81$Mpc with the LALInference map @gcn26417). The updated map covered $\sim 4500$ square degrees. Since the updated skymap was released $\sim$1 day after trigger time, much of the observations made in the first night used the initial BAYESTAR map. While it was the first BNS alert during the second half of the O3 campaign, the response to this alert was relatively tepid, likely due to the scattered light contamination. However, 53 report circulars have been distributed for this event due to the presence of an interesting transient found by the Pan-STARRS Collaboration PS19hgw/AT2019wxt, finally classified as supernovae IIb due to the photometry evolution and spectroscopy characterization [@gcn26485; @gcn26508; @GRANDMA2020]. In total, three neutrinos, one VHE, eight $\gamma$-rays, two X-rays, 19 optical and one radio groups participated to the S191213g campaign (see the [list of GCNs for S191213g](https://gcn.gsfc.nasa.gov/other/S191213g.gcn3)). No significant neutrino, VHE and $\gamma$-ray GW counterpart was found in the archival analysis. A moderate fraction of the localization area was covered using a tiling approach (GRANDMA, Master-Network, ZTF) (see Table \[tab:Tableobs\]). The MASTER-network covered $\approx41$% within 144 h down to 19 in clear [@gcn26400], and the Zwicky Transient Facility covered $\approx28$% down to 20.4 in $g$- and $r$-band [@gcn26424; @gcn26437; @Kasliwal2020]. The search yielded 19 candidates of interest from ZTF, as well as the transient counterpart AT2019wxt from the Pan-STARRS Collaboration [@gcn26485]. It was shown that all ZTF candidates were in fact unrelated with the GW candidate S191213g [@gcn26426; @gcn26429; @gcn26432]. In addition to searches by wide field of view telescopes, there was also galaxy-targeted follow-up performed by the J-GEM Collaboration, observing 57 galaxies [@gcn26477], and the GRANDMA citizen science program, observing 16 galaxies [@gcn26558] within the localization of S191213g. **S200105ae** ------------- LIGO/Virgo S200105ae was identified by L1 (with V1 also observing) at 2020-01-05 16:24:26 UTC as a subthreshold event with a false alarm rate of 24 per year; if it is astrophysical, it is most consistent with being an BHNS. However, its significance is likely underestimated due to it being a single-instrument event. This candidate was most interesting due to the presence of chirp-like structure in the spectrograms [@gcn26640; @gcn26657]. The first public notice was delivered 27.2 h after the GW trigger impacting significantly the follow-up campaign of the event. In addition, the most updated localization was very coarse, spanning $\sim7400$ square degrees with a distance of $283 \pm 74$Mpc [@gcn26688]. S200105ae follow-up activity was comparable to S191205ah’s: 25 circular reports were associated to the S200105ae in the GCN service with the search of counterpart engaged by two neutrinos, one VHE, seven $\gamma$-ray, one X-ray and five optical groups (see the [list of GCNs for S200105ae](https://gcn.gsfc.nasa.gov/other/S200105ae.gcn3)). No significant neutrino, VHE and $\gamma$-ray GW counterpart was found in the archival analysis. Various groups participated to the search of optical counterpart with ground-based observatories: GRANDMA, Master-Network, and the Zwicky Transient Facility (see Table \[tab:Tableobs\]). The alert space system for Gaia was also activated [@gcn26686]. The MASTER-network covered $\approx43$% down to 19.5 in clear and within 144 h [@gcn26646]. The telescope network was already observing at the time of the trigger and because its routine observations were compatible with the sky localization of S200105ae, the delay was limited to 3hr. GRANDMA-TCA telescope was triggered as soon as the notice comes out, and the full GRANDMA network totalized 12.5 % of the full LALInference skymap down to 17 mag in clear and within 60 h [@GRANDMA2020]. The Zwicky Transient Facility covered $\approx52$% of the LALInference skymap down to 20.2 in both $g$- and $r$-bands [@gcn26673; @Kasliwal2020] and with a delay of 10 h. There were 23 candidate transients reported by ZTF, as well as one candidate from the Gaia Alerts team [@gcn26686] out of which ZTF20aaervoa and ZTF20aaertpj were both quite interesting due to their red colors ($g-r$= 0.66 and 0.35 respectively), and absolute magnitudes ($-16.4$ and $-15.9$ respectively) [@gcn26673]. ZTF20aaervoa was soon classified as a SN IIp $\sim3$ days after maximum, and ZTF20aaertpj as a SN Ib close to maximum [@gcn26702; @gcn26703]. **S200115j** ------------ LIGO/Virgo S200115j, a MassGap signal $(99\%)$ with a very high probability $(99\%)$ of containing a NS as well, was identified by H1, L1, and V1 at 2020-01-15 04:23:09.742 UTC [@gcn26759]. As discussed before, it can be considered as a BHNS candidate. Due to its discovery by multiple detectors, the sky location is well-constrained; the most updated map spans $\sim765$ square degrees, with most of the probability shifting towards the southern lobe in comparison to the initial localization, and has a distance of $340 \pm 79$ Mpc. With a very high p$_{\mathrm{remnant}} > 99$% [@gcn26807] and good localization, many space and ground instruments/telescopes followed up this signal: 33 circular reports were associated to the event in the GCN service with the search of counterpart engaged by two neutrinos, three VHE, five $\gamma$-ray, two X-ray and eight optical groups (see the [list of GCNs for S200115j](https://gcn.gsfc.nasa.gov/other/S200115j.gcn3)). INTEGRAL was not active during the time of the event [@gcn26766] and so was unable to report any prompt short GRB emission. No significant neutrino, VHE and $\gamma$-ray GW counterpart was found in the archival analysis. Swift satellite was also pointed toward the best localization region for finding X-ray and UVOT counterpart. Some candidates were reported: one of them was detected in the optical by Swift/UVOT and the Zwicky Transient Facility, but was concluded to likely be due to AGN activity [@gcn26808; @gcn26863]. Various groups participated to the search of optical counterpart with ground observatories: GOTO, GRANDMA, Master-Network, Pan-Starrs, SVOM-GWAC and the Zwicky Transient Facility (see Table \[tab:Tableobs\]). GOTO [@gcn26794] covered $\approx50$% down to 19.5 in $G$-band, starting almost immediately the observations, while the SVOM-GWAC team covered $\approx40$% of the LALInference sky localization down to 16 in R-band using the SVOM-GWAC only 16h after the trigger time [@gcn26786]. In addition, a list of 20 possible host galaxies for the trigger was produced by convolving the GW localization with the 2MPZ galaxy catalog [@gcn26763; @BiJa2014]; 12 of these galaxies were observed by GRAWITA [@gcn26823] in the r-sdss filter. **S200213t** ------------ S200213t was identified by H1, L1, and V1 at 2020-02-13 at 04:10:40 UTC [@gcn27042]. It has been categorized as a BNS signal $(63\%)$ with a moderate probability of being terrestrial $(37\%)$. The LALInference localization spanned $\sim2326$ square degrees, with a distance of $201 \pm 80$ Mpc [@gcn27096]. A total of 51 circular reports were associated to this event including two neutrinos, two VHE, eight $\gamma$-rays, two X-ray, and eleven optical groups (see the [list of GCNs for S200213t](https://gcn.gsfc.nasa.gov/other/S200213t.gcn3)). Fermi and Swift were both transiting the South Atlantic Anomaly at the time of event, and so were unable to observe and report any GRBs coincident with S200213t [@gcn27056; @gcn27058]. No significant counterpart candidate was found during archival analysis: IceCube detected muon neutrino events, but it was shown that they have not originated from the GW source [@gcn27043]. With a very high p$_{\mathrm{remnant}} > 99$% and probable BNS classification, many telescopes followed-up this signal: DDOTI/OAN, GOTO, GRANDMA, MASTER and ZTF. DDTOI/OAN covered $\approx40$% of the LALInference skymap starting less than 1h after the trigger time down to 19.2 in $w$-band [@gcn27061], GRANDMA covered 32% of the LALInference area within $\approx$ 26h down to 18 mag in clear (TCA) and down to 21 mag in R-band (OAJ). GOTO covered $\approx54$% of bayestar skymap down to 18.4 in $G$-band [@gcn27069]. 15 candidate transients were reported by ZTF [@gcn27051; @gcn27065; @gcn27068], as well as one by the MASTER-network [@gcn27077]. All were ultimately ruled out as possible counterparts to S200213t through either spectroscopy or due to pre-discovery detections [@gcn27060; @gcn27063; @gcn27074; @gcn27075; @gcn26839; @gcn27085]. Galaxy targeted observations were conducted by several observatories: examples include KAIT, which observed 108 galaxies [@gcn27064], Nanshan-0.6m, which observed a total of 120 galaxies [@gcn27070], in addition to many other teams [@gcn27066; @gcn27067]. Kilonova Modeling and possible ejecta mass limits {#sec:limits} ================================================= Following [@CoDi2019b], we will compare the upper limits described in Section \[sec:EM\_follow\_up\_campaigns\] to different kilonova models. We seek to measure “representative constraints,” limited by the lack of field and time-dependent limits. To do so, we approximate the upper limits in a given passband as one-sided Gaussian distributions. We take the sky-averaged distance in the GW localizations to determine the transformation from apparent to absolute magnitudes. To include the uncertainty in distance, we sample from a Gaussian distribution consistent with this uncertainty and add it to the model lightcurves. In this analysis, we employ three kilonova models based on [@KaMe2017], [@Bul2019], and [@HoNa2019], in order to compare any potential systematic effects. These models use similar heating rates [@MeMa2010; @KoRo2012], while using different treatments of the radiative transfer. **Model I** **Model II** **Model III**\ **S191213g**\ {width="3.0in" height="2.2in"} {width="3.0in" height="2.2in"} {width="3.0in" height="2.2in"}\ **S200213t**\ {width="3.0in" height="2.2in"} {width="3.0in" height="2.2in"} {width="3.0in" height="2.2in"} \[fig:violin\_constraints\_BNS\] **Model I** **Model II** **Model III**\ **S191205ah**\ {width="3.0in" height="2.2in"} {width="3.0in" height="2.2in"} {width="3.0in" height="2.2in"}\ **S200105ae**\ {width="3.0in" height="2.2in"} {width="3.0in" height="2.2in"} {width="3.0in" height="2.2in"}\ **S20015j**\ {width="3.0in" height="2.2in"} {width="3.0in" height="2.2in"} {width="3.0in" height="2.2in"}\ \[fig:violin\_constraints\_NSBH\] We will show limits as a function of one parameter for each model chosen to maximize its impact on the predicted kilonova brightness and color, marginalizing out the other parameters when performing the sampling. For the models based on [@KaMe2017] and [@Bul2019], as grid-based models, we interpolate these models by creating a surrogate model using a singular value decomposition (SVD) and Gaussian Process Regression (GPR) based interpolation [@DoFa2017] that allows us to create lightcurves for arbitrary ejecta properties within the parameter space of the model  [@CoDi2018b; @CoDi2018]. We refer the reader to [@CoDi2019b] for more details about the models, but we will also briefly describe them in the following for completeness. Model I [@KaMe2017] depends on the ejecta mass $M_{\rm ej}$, the mass fraction of lanthanides $X_{\rm lan}$, and the ejecta velocity $v_{\rm ej}$. We allow the sampling to vary within $-3 \leq \log_{10} (M_{\rm ej}/M_\odot) \leq 0$ and $ 0 \leq v_{\rm ej} \leq 0.3$$c$, while restricting the lanthanide fraction to $X_{\rm lan}$ = \[ $10^{-9}$, $10^{-5}$, $10^{-4}$, $10^{-3}$, $10^{-2}$, $10^{-1} ]$. Model II [@Bul2019] assumes an axi-symmetric geometry with two ejecta components, one component representing the dynamical ejecta and one the post-merger wind ejecta. Model II depends on four parameters: the dynamical ejecta mass $M_{\rm ej,dyn}$, the post-merger wind ejecta mass $M_{\rm ej,pm}$, the half-opening angle of the lanthanide-rich dynamical-ejecta component $\phi$ and the inclination angle $\theta_{\rm obs}$ (with $\cos\theta_{\rm obs}=0$ and $\cos\theta_{\rm obs}=1$ corresponding to a system viewed edge-on and face-on, respectively). We refer the reader to [@DiCo2020] for a more detailed discussion of the ejecta geometry. In this study, we fix the dynamical ejecta mass to the best-fit value from [@DiCo2020], $M_{\rm ej,dyn}=0.005\,M_\odot$, and allow the sampling to vary within $-3 \leq \log_{10} (M_{\rm ej,pm}/M_\odot) \leq 0$ and $0^\circ\leq \phi\leq90^\circ$, while restricting the inclination angle to $\theta_{\rm obs} = [0^\circ, 30^\circ, 60^\circ, 90^\circ]$. To facilitate comparison with the other models, we will provide constraints on the total ejecta mass $M_{\rm ej}=M_{\rm ej,dyn}+M_{\rm ej,pm}$ for Model II. We note that the model adopted here is more tailored to BNS than BHNS mergers given the relatively low dynamical ejecta mass, $M_{\rm ej,dyn}=0.005\,M_\odot$. However, for a given $M_{\rm ej,pm}$, the larger values of $M_{\rm ej,dyn}$ predicted in BHNS are expected to produce longer lasting kilonovae more easily detectable. Therefore, the ejecta mass upper limits derived below for BHNS systems should be considered conservative. Model III [@HoNa2019] depends on the ejecta mass $M_{\rm ej}$, the dividing velocity between the inner and outer component $v_{\rm ej}$, the lower and upper limit of the velocity distribution $v_{\text{min}}$ and $v_{\text{max}}$, and the opacity of the 2-components, $\kappa_{\rm low}$ and $\kappa_{\rm high}$. We allow the sampling to vary $-3 \leq \log_{10} (M_{\rm ej}/M_\odot) \leq 0$, $ 0 \leq v_{\rm ej} \leq 0.3$$c$, $0.1 \leq v_{\text{min}}/v_{\text{ej}} \leq 1.0$ and $1.0 \leq v_{\text{max}}/v_{\text{ej}} \leq 2.0$. We restrict $\kappa_{\rm low}$ and $\kappa_{\rm high}$ to a set of representative values in the analysis, i.e. 0.15 and 1.5, 0.2 and 2.0, 0.3 and 3.0, 0.4 and 4.0, 0.5 and 5.0, and 1.0 and 10 cm$^2$/g. Figure \[fig:violin\_constraints\_BNS\] shows the ejecta mass constraints for BNS events, S191213g and S200213t, while Figure \[fig:violin\_constraints\_NSBH\] shows them for NSBH events, S191205ah, S200105ae, and S200115j. We mark each $90\%$ confidence with a horizontal dashed line. As a brief reminder, given that the entire localization region is not covered for these limits, and the limits implicitly assume that the region containing the counterpart was imaged, these should be interpreted as optimistic scenarios. It is also simplified to assume that the light curve can not exceed the stated limit at any point in time. Similar to what was found during the analysis of O3a [@CoDi2019b], the constraints are not particularly strong, predominantly due to the large distances for many of the candidate events. Given the focus of these systems on the bluer optical bands, the constraints for the bluer kilonova models (low $X_{\rm lan}$, low $\theta_{\rm obs}$ and low $\kappa_{\rm low}/\kappa_{\rm high}$) tend to be stronger. **S191205ah:** The left column of Figure \[fig:violin\_constraints\_NSBH\] shows the ejecta mass constraints for S191205ah based on observations from ZTF (left, @gcn26416) and SAGUARO (right, @gcn26360). For all models we basically recover our prior, i.e., no constraint on the ejecta mass can be given. **S191213g:** The middle column of Figure \[fig:violin\_constraints\_BNS\] shows the ejecta mass constraints for S191213g based on the observations from ZTF [@gcn26424; @gcn26437] and the MASTER-Network [@gcn26400]. Interestingly, Model II allows us for small values of $\theta_{\rm obs}$ (brighter kilonovae) to constrain ejecta masses above $\sim 0.3$ $M_\odot$, however for larger angles, no constraint can be made. For Model III we obtain even tighter ejecta mass limits between $0.2$ $M_\odot$ and $0.3$ $M_\odot$, where generally for potentially lower opacity ejecta we obtain better constraints. While $0.2$ $M_\odot$ rules out systems producing very large ejecta masses, e.g., highly unequal mass systems, AT2017gfo was triggered by only about a quarter of the ejecta mass and our best bound for GW190425 [@CoDi2019b] was a factor of a few smaller. Thus, we are overall unable to extract information that help us to constrain the properties of the GW trigger S191213g. **S200105ae:** The right column of Figure \[fig:violin\_constraints\_NSBH\] shows the ejecta mass constraints for S200105ae based on observations from ZTF [@gcn26673] and the MASTER-network [@gcn26646]. As for S191205ah our analysis recovers basically the prior and no additional information can be extracted. **S200115j:** The left column of Figure \[fig:violin\_constraints\_NSBH\] shows the ejecta mass constraints for S200115j based on observations from ZTF [@gcn26767] and GOTO [@gcn26794]. Model II allows us for small values of $\theta_{\rm obs}$ (brighter kilonovae) to constrain ejecta masses above $\sim 0.1$ $M_\odot$, however for larger angle, no constraint can be made; similar constraints (ejecta masses below $0.15$ $M_\odot$) are also obtained with Model III. As for S191213g, the obtained bounds are not strong enough to reveal interesting properties about the source properties. **S200213t:** The right column of Figure \[fig:violin\_constraints\_BNS\] shows the ejecta mass constraints for S200213t based on observations from ZTF [@gcn27051] and GOTO [@gcn27069]. As for S191205ah and S191213g, our analysis recovers basically the prior and no additional information can be extracted for Model I and Model II.[^4] Model III allows us to rule out large ejecta masses $> 0.15$ $M_\odot$ for low opacities. **Summary:** In conclusion, we find that for the follow-up surveys to the important triggers of O3b, the derived constraints on the ejecta mass are too weak to extract any information about the sources as it was possible for GW190425 [@CoDi2019b]. This is likely due to a number of different circumstances: a reduction number of observations from O3a to O3b, e.g., three GW events out of five were happening around 4 h UTC, leading to an important delay of observations for all facilities located in Asia and Europe. Furthermore, the distance to most of the events was quite far (around 200 Mpc) and there was the possibility that in many cases a non-astrophysical origin caused the GW alert. Also, the weather was particularly problematic for a number of the promising events (see above). Unfortunately, we also observed that some groups were less rigorous in their report compared to O3a and did not report all observations publicly, which clearly hinders the analysis outlined above. Overall, some of the observational strategies were not optimal and motivates a more detailed discussion in Section \[sec:strategies\]. While these analyses do not evaluate the joint constraints possible based on multiple systems, under the assumption that different telescopes observed the same portion of the sky in different bands (or at different times), it makes sense that improved constraints on physical parameters are possible. To demonstrate this, we show the ejecta mass constraints for GW190425 based on observations from ZTF (left, [@gcn24191]) and PS1 (right, [@gcn24210]) and the combination of the two. While the constraints for the low lanthanide fractions are stronger than available for the “red kilonovae” for all examples, the combination of $g$- and $r$-band observations from ZTF and $i$-band from PS1 yield stronger constraints across the board. ![Probability density for the total ejecta mass for GW190425 based on the [@KaMe2017] model using the ZTF (left, [@gcn24191]), PS1 (right, [@gcn24210]), and joint ZTF and PS1 limits.[]{data-label="fig:violin_constraints_GW190425"}](Ka2017_GW190425_ZTF_PS1.pdf){width="3.6in"} Using the kilonova models to inform observational strategies {#sec:strategies} ============================================================ **OAJ PS1**\ {width="3.5in" height="2.0in"} {width="3.5in" height="2.0in"}\ **ZTF ZTF+PS1+OAJ**\ {width="3.5in" height="2.0in"} {width="3.5in" height="2.0in"}\ Given the relatively poor limits on the ejecta masses, we are interested in understanding how optimized scheduling strategies can aid in obtaining higher detection efficiencies of kilonova counterparts. Similar but slightly stronger constraints were obtained during the analysis of the first six months of O3 [@CoDi2019b], where we advocated for longer observations at the cost of a smaller sky coverage. For our investigation, we use the codebase `gwemopt`[^5] (Gravitational-Wave ElectroMagnetic OPTimization) [@CoTo2018], which has been developed to schedule Target of Opportunity (ToO) telescope observations after the detection of possible multi-messenger signals, including neutrinos, gravitational waves, and $\gamma$-ray bursts. There are three main aspects to this scheduling: tiling, time allocation, and scheduling of the requested observations. Multi-telescope, network-level observations [@CoAn2019] and improvements for scheduling in the case of multi-lobed maps [@AlCo2020] are the most recent developments in these areas. We note that `gwemopt` naturally accounts for slew and read out times based on telescope-specific configuration parameters, which are important to account for inefficiencies in either long slews or when requesting short exposure times. We now perform a study employing these latest scheduling improvements to explore realistic schedules, analyzing them with respect to exposure time in order to determine the time-scales required to make kilonova detections. We will use four different types of lightcurve models to explore this effect. The first is based on a “top hat” model, where a specific absolute magnitude is taken as constant over the course of the observations; in this paper, we take an absolute magnitude (in all bands) of $-16$, which is roughly the peak magnitude of AT2017gfo [@ArHo2017]. The second is similar: a base absolute magnitude of $-16$ is taken at the start of observation, but the magnitude decays linearly over time at a decay rate of 0.5mag/day. These agnostic models depend only on the intrinsic luminosity and luminosity evolution of the source. The third and fourth model types are derived from our Model II [@Bul2019]. We use two different values of the post-merger wind ejecta component to explore the dependence on the amount of ejecta, one with dynamical ejecta $M_{\rm ej,dyn}=0.005\,M_\odot$ and post-merger wind ejecta $M_{\rm ej,pm}=0.01\,M_\odot$ and the other with $M_{\rm ej,dyn}=0.005\,M_\odot$ and $M_{\rm ej,pm}=0.05\,M_\odot$, similar to that found for AT2017gfo [@DiCo2020]. As mentioned in Section \[sec:limits\], dynamical ejecta masses of $M_{\rm ej,dyn}=0.005\,M_\odot$ are more typical for BNS than BHNS mergers, and therefore we restrict our analysis to a BNS event (see below). Figure \[fig:efficiency\] shows the efficiency of transient discovery for these models as a function of exposure time for a BNS event occurring at a distance similar to that of S200213t, 224 $\pm$ 90Mpc. We inject kilonovae according to the 3D probability distribution in the final LALInference localization of S200213t and generate a set of tilings for each telescope (with fixed exposure times) through scheduling algorithms. Here, the detection efficiency corresponds to the total number of detected kilonovae divided by the total number of simulated kilonovae, which is a proxy for the probability that the telescope covered the correct sky location during observations to a depth sufficient to detect the transient. We show the total integrated probability that the event was part of the covered sky area as a black line, and the probabilities for all four different lightcurve models as colored lines.[^6] For our study, we use OAJ (top left), PS1 (top right), and ZTF (bottom left), and a network consisting of all three telescopes. As expected, there is a trade-off between exposure time and the ability to effectively cover a large sky area. Both of these contribute to the overall detection efficiency, given that the depths required for discovery are quite significant. In order to rule out moving objects (e.g., asteroids) during the transient-filtering process, it is important to have at least 30 min gaps between multi-epoch observations; opting for longer exposure times can render this close to impossible, and hinder achieving coverage of the 90% credible region during the first 24 hours, especially for larger localizations. There are also observational difficulties, as field star-based guiding is not available on all telescopes, so some systems are not able to exceed exposure durations of a few minutes without sacrificing image quality. Therefore, we are interested in pinpointing the approximate peaks in efficiency so as to find a balance between the depth and coverage attained, and ultimately increase the possibility of a kilonova detection. It is important to note that the comparably “close” distance of S200213t (listed in Section \[sec:EM\_follow\_up\_campaigns\]) must be taken into account in this analysis, as farther events will likely favor relatively longer exposure times to achieve the depth required. In addition to exposure time, visibility constraints also contribute to the maximum probability coverage observable from a given site. Only taking into consideration the single-telescope observations shown in Figure \[fig:efficiency\], we find that as expected, the peak differs considerably depending on the telescope, by virtue of its configuration. The results with PS1, for example, are illustrative of its lower field of view in combination with its higher limiting magnitude of 21.5 (assuming optimal conditions), leading to both a quick decline in coverage for longer exposure times, and sufficient depth achieved at shorter exposure times. As a result, the efficiency peaks at a much earlier range of $\sim$30-100s for this event. In the case of OAJ, the similar field of view to PS1 but relatively lower limiting magnitude supports opting for exposure times of $\sim$160 - 300s —in which one expects to reach $\sim$20.8 - $\sim$21.5mag— to not lose out on coverage to the point of jeopardizing the detection efficiency for this skymap. ZTF’s 47-square-degree field of view, however, allows for longer exposure times to be explored while maintaining an increase in efficiency. Generally, ZTF ToO follow-ups have used $\sim$120 - 300s exposures [@CoAh2019b], expected to reach $\sim$21.5 $-$ $\sim$22.4mag, but going for even longer exposure times appears beneficial to optimizing counterpart detection for ZTF. The bottom right panel, which shows the joint analysis, aptly re-emphasizes the potential benefit of multi-telescope coordination through the gain in detection efficiency due to the ability to more effectively cover a large sky area; additionally, since achieving significant coverage is no longer an issue, pushing for longer exposure times will only positively affect the chances of detecting a transient counterpart. As grounds for comparison, we also performed identical simulations for BNS event GW190425 [@SiEA2019b] (with a sizable updated localization of $\sim$7500 square degrees) in order to investigate the effects of the skymap’s size on the peak efficiencies and the corresponding exposure times. The results are compared using a single telescope configuration (ZTF) vs. a multi-telescope configuration (ZTF, PS1, and OAJ) for different lightcurve models. For the Tophat model with a decay rate of 0.5 mag/day, the detection efficiency peaked at $\sim$70s for ZTF, with both the integrated probability and detection efficiency at 27%. However, under identical conditions, the telescope network configuration peaked at a detection efficiency and integrated probability of 34% at $\sim$40s. Using the synthetic lightcurve adopted from Model II, with dynamical ejecta of $M_{\rm ej,dyn}=0.005\,M_\odot$ and post-merger wind ejecta of $M_{\rm ej,pm}=0.01\,M_\odot$, ZTF attained a peak efficiency of 25% at $\sim$170s. On the other hand, the telescope network resulted in a higher detection efficiency of 29% at $\sim$100s due to the increased coverage. It is clear that regardless of the model adopted, there is some benefit in utilizing telescope networks to optimize the search for counterparts, especially in the case of such large localizations; however, truly maximizing this benefit requires the ability to optimize exposure times on a field-by-field (or at least, telescope-by-telescope) basis. This also requires that the telescopes coordinate their observations, or in other words, optimize their joint observation schedules above and beyond optimization of individual observation schedules.\ {width="3.5in" height="2.0in"} {width="3.5in" height="2.0in"}\ Finally, we want to show the impact of observation conditions on the peak detection efficiencies and the corresponding exposure times in Fig. \[fig:efficiency\_bad\_good\]. We uses two baselines for ZTF magnitude limits, with one corresponding to 19.5, the median $-1\sigma$ and the other to 20.5, the median $+1\sigma$. Our analysis shows that for good conditions (left panel), the performance for ToOs is reasonable, although especially optimal towards the upper end of the 120 - 300s range. For relatively poor conditions (right panel), longer exposure times are required, which is now possible due to the significant work that has gone into improving ZTF references to adequate depths for these deeper observations. One more point of consideration is the distance information for the event; a kilonova with twice the luminosity distance will produce four times less flux, and this will affect the depth required to possibly detect the transient. This aspect of the analysis does not overshadow the importance of prioritizing longer exposure times (in particular under bad observational conditions). We note that the quoted limits for S200213t are $\sim$20.7mag in 120s from ZTF [@gcn27051]; this corresponds to $\sim$19.2 expected for 30s exposures, and therefore sub-optimal conditions. Summary {#sec:summary} ======= In this paper, we have presented a summary of the searches for EM counterparts during the second half of the third observing run of Advanced LIGO and Advanced Virgo; we focus on the gravitational-wave event candidates which are likely to be the coalescence of compact binaries with at least one neutron star component. We used three different, independent kilonova models [@KaMe2017; @Bul2019; @HoNa2019] to explore potential ejecta mass limits based on the non-detection of kilonova counterparts of the five potential GW events S191205ah, S191213, S200105ae, S200115j, and S200213t by comparing apparent magnitude limits from optical survey systems to the gravitational-wave distances. While the models differ in their radiative transfer treatment, our results show that the publicly-available observations do not provide any strong constraints on the quantity of mass ejected during the possible events, assuming the source was covered by those observations. The most constraining measurement is obtained for S200115j thanks to the observations of ZTF and GOTO; the model of @Bul2019 excludes an ejecta of more than 0.1$M_\odot$ for some viewing angles. In general, the reduced number of observations between O3a and O3b, the delay of observations, the shallower depth of observations, and large distances of the candidates, which yield faint kilonovae, explain the minimal constraints for the compact binary candidates. However, it shows the benefit of a systematic diagnostic about quantity of ejecta thanks to the observations, as was done in the analysis of O3a [@CoDi2019b]. Although the strategy of follow-up employed by the various teams and their instrument capabilities did not evolve significantly in the eleven months of O3, it is clear that a global coordination of the observations would yield expected gains in efficiency, both in terms of coverage and sensitivity. Given the uninformative constraints, we explored the depths that would be required to improve the detection efficiencies at the cost of coverage of the sky location areas for both single telescopes and network level observations. We find that exposure times of $\sim$3-10min would be useful for ZTF to maximize its sensitivity for the events discussed here, depending on the model and atmospheric conditions, which is a factor of 6-20$\times$ longer than survey observations, and up to a factor of 2$\times$ longer than for current ToO observations; the result is similar for OAJ. For PS1, on the other hand, its larger aperture leads to the conclusion that its natural survey exposure time is about right for events in the BNS distance range. Our results also highlight the advantages of telescope networks in increasing coverage of the localization and thereby allowing for longer exposure times to be used, thus leading to a corresponding increase in detection efficiencies. It is also important to connect our results to conclusions drawn in other works: @CaBu2020 showed that detections of a AT2017gfo-like light curve at 200 Mpc requires observations down to limiting magnitudes of 23mag for lanthanide-rich viewing angles and 22mag for lanthanide-free viewing angles. The authors point out that because the optical lightcurves of kilonovae become red in a matter of few days, observing in red filters, such as inclusion of $i$-band observations, results in almost double the detections as compared to observations in $g$- and $r$-band only. They propose that observations of rapid decay in blue bands, followed by longer observations in redder bands is therefore an ideal strategy for searching for kilonovae. This strategy can be combined with the exposure time measurements here to create more optimized schedules. @Kasliwal2020 also demonstrate that under the assumption that the GW events are astrophysical, strong constraints on kilonova luminosity functions are possible by taking multiple events and considering them together, even when the probabilities and depths covered on individual events are not always strong. This motivates future work where ejecta mass constraints can be made on a population basis by considering the joint constraints over all events. Building in field-dependent exposure times will be critical for improving the searches for counterparts. While our estimates are clearly model dependent (e.g., by assuming an absolute magnitude, a decay rate for candidate counterparts, and a particular kilonova model), it is clear that deeper observations are required, especially with the future upgrades of the GW detectors, to improve detection efficiencies when the localization area and telescope configuration allow for it. Telescope upgrades alone do not guarantee success, as detecting more marginal events at further distances will not necessarily yield better covered skymaps. Smaller localizations from highly significant, nearby events are key, perhaps with the inclusion of more information to differentiate those most likely to contain counterpart, such as the chirp mass [@MaMe2019], to support the follow-up. Acknowledgements {#acknowledgements .unnumbered} ================ SA is supported by the CNES Postdoctoral Fellowship at Laboratoire Astroparticle et Cosmologie. MB acknowledges support from the G.R.E.A.T research environment funded by the Swedish National Science Foundation. MWC acknowledges support from the National Science Foundation with grant number PHY-2010970. FF gratefully acknowledges support from NASA through grant 80NSSC18K0565, from the NSF through grant PHY-1806278, and from the DOE through CAREER grant DE-SC0020435. SGA acknowledges support from the GROWTH (Global Relay of Observatories Watching Transients Happen) project funded by the National Science Foundation under PIRE Grant No 1545949. G.R. and S.N. are grateful for support from the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO) through the VIDI and Projectruimte grants (PI Nissanke). The lightcurve fitting / upper limits code used here is available at: <https://github.com/mcoughlin/gwemlightcurves>. We also thank Kerry Paterson, Samuel Dilon Wyatt and Owen McBrien for giving explanations of their observations. Data Availability {#data-availability .unnumbered} ================= The data underlying this article are derived from public code found here: <https://github.com/mcoughlin/gwemlightcurves>. The simulations resulting will be shared on reasonable request to the corresponding author. [cccccccc]{} \[tab:Tableobs\]\ GRANDMA-TCA & Clear & $18$ & $18.9$ & $50$ & bayestar ini & $3$ & @GRANDMA2020 \ GRANDMA-TCH & Clear & $18$ & $2.9$ & $54$ & bayestar ini & $1$ & @GRANDMA2020 \ MASTER-network & Clear & $ \approx 19$ & $ \approx 0.1$ & $144$ & bayestar ini & $\approx56$ & @gcn26353 \ SAGUARO & G-band & $21.3$ & 4.4 & $0.5$ & bayestar ini & $9$ & @gcn26360 [@2020arXiv200100588W], this work \ SVOM-GWAC & R-band & $16$ & $\approx0.1$ & $23$ & bayestar ini & $28$ & @gcn26386, this work \ Zwicky Transient Facility & g/r-band & $17.9$ & $10.7$ & $167$ & bayestar ini & $6$ & @gcn26416 [@Kasliwal2020] \ & & & & & & &\ \ GRANDMA-TCA & Clear & $18$ & $47.6$ & $73$ & LALInference & $1$ & @GRANDMA2020 \ MASTER-network & Clear & $\approx 18.5$ & $0.4$ & $144$ & LALInference & $41$ & @gcn26400 \ Zwicky Transient Facility & g/r-band & $20.4$ & $0.01$ & $\approx27.8$ & LALInference & $28$ & @gcn26424 [@Kasliwal2020] \ & & & & & & &\ \ GRANDMA-TCA & Clear & $18$ & $27.5$ & $50.4$ & LALInference & $3$ & @GRANDMA2020 \ GRANDMA-TCH & Clear & $18$ & $59.0$ & $53.8$ & LALInference & $3$ & @GRANDMA2020 \ GRANDMA-TRE & Clear & $17$ & $48.0$ & $26.5$ & LALInference & $10$ & @GRANDMA2020 \ MASTER-network & Clear & $\approx 19.5$ & $\approx3.2$ & $144$ & LALInference & $43$ & @gcn26646 \ Zwicky Transient Facility & g/r-band & $20.2$ & $9.96$ & $34.6$ & LALInference & 52 & @gcn26673 [@2020arXiv200100588W; @Kasliwal2020], \ & & & & & & &\ \ GOTO & g-band & $19.5$ & $0.2$ & $26.4$ & bayestar ini & $52$ & @gcn26794 \ GRANDMA-FRAM-A & R-band & $18$ & $20.8$ & $1.7$ & LALInference & $2$ & @GRANDMA2020 \ GRANDMA-TCA & Clear & $18$ & $12.9$ & $72.2$ & LALInference & $4$ & @GRANDMA2020 \ GRANDMA-TCH & Clear & $18$ & $0.3$ & $148.3$ & LALInference & $7$ & @GRANDMA2020 \ GRANDMA-TRE & Clear & $17$ & $11.7$ & $28.7$ & LALInference & $7$ & @GRANDMA2020 \ MASTER-network & P/Clear & $ \approx 15/19$ & $\approx0.1$ & $144$ & LALInference & $62$ & @gcn26755 \ Pan-STARRS & w-band & $21$ & $\approx72$ & $\approx 24$ & $-$ & $-$ & @gcn26839 \ SVOM-GWAC & R-band & $\approx16$ & $7.1$ & $10.6$ & LALInference & $41$ & @gcn26786, this work \ Swift-UVOT & u-band & $19.6$ & $2.0$ & $80.1$ & LALInference & $3$ & @gcn26798 [@2020arXiv200100588W], this work \ Zwicky Transient Facility & g/r-band & $20.8$ & $0.24$ & $> 1$ & LALInference & $22$ & @gcn26767 [@Kasliwal2020] \ & & & & & & &\ \ DDOTI/OAN & w-filter & $\approx 19$ & $0.75$ & $2.17$ & LALInference & $\approx 41$ & @gcn27061, this work \ GOTO & G-band & $18.4$ & $\approx0$ & $26.5$ & bayestar ini & $54$ & @gcn27069 \ GRANDMA-FRAM-C & R-band & $17$ & $15.3$ & $1.5$ & LALInference & $4$ & @GRANDMA2020 \ GRANDMA-OAJ & r-band & $21$ & $15$ & $1.5$ & LALInference & $18$ & @GRANDMA2020 \ GRANDMA-TCA & Clear & $18$ & $0.4$ & $43.6$ & LALInference & $30$ & @GRANDMA2020 \ MASTER-network & Clear & $\approx 18.5$ & $0.1$ & $144$ & LALInference & $87$ & @gcn27041 \ Zwicky Transient Facility & g/r-band & $21.2$ & $0.4$ & $<25.7$ & LALInference & $72$ & @gcn27051 [@Kasliwal2020] \ & & & & & & &\ [^1]: Amongst the wide field-of-view telescopes, ATLAS [@gcn24197; @gcn24517; @gcn25922; @gcn25375], ASAS-SN [@gcn24309], CNEOST [@gcn24285; @gcn24465; @gcn24286], Dabancheng/HMT [@gcn24476], DESGW-DECam [@gcn25336], DDOTI/OAN [@gcn24310; @gcn25352; @gcn25737], GOTO [@gcn24224; @gcn25654; @gcn24291; @gcn25337; @GoCu2020], GRANDMA [@GRANDMAO3A; @GRANDMA2020], GRAWITA-VST [@gcn24484; @gcn25371], GROWTH-DECAM [@AnGo2019; @GoAn2019], GROWTH-Gattini-IR [@gcn24187; @gcn24284; @gcn25358], GROWTH-INDIA [@gcn24258], HSC [@gcn24450], J-GEM [@gcn24299], KMTNet [@gcn24466; @gcn25342], MASTER-network [@gcn24167; @gcn24436; @gcn25609; @gcn25712; @gcn25712; @gcn24236; @gcn25322; @gcn25694; @gcn25812], MeerLICHT [@gcn25340], Pan-STARRS [@gcn24210; @gcn24517; @gcn24517], SAGUARO [@2019arXiv190606345L], SVOM-GWAC [@gcn25648], Swope [@gcn25350], Xinglong-Schmidt [@gcn24190; @gcn24475], and the Zwicky Transient Facility [@gcn24191; @gcn25616; @gcn25722; @gcn25899; @gcn24283; @gcn25343; @gcn25706; @CoAh2019b] participated. [^2]: https://github.com/ggreco77/GWsky [^3]: <https://wis-tns.weizmann.ac.il> [^4]: While the 90% indicates that the prior is recovered, the shape of the posterior distributions suggest that the parameter space is somewhat constrained, disfavoring the high ejecta masses somewhat, but not enough to affect the limits. [^5]: <https://github.com/mcoughlin/gwemopt> [^6]: For intuition purposes: a tourist observing the full night sky at Mauna Kea in Hawaii would have reached 70% for the integrated probability, but a detection efficiency of 0% (since the typical depth reached by the human eye is about 7 mag), whereas a $\sim$one arcminute field observed by Keck, a 10m-class telescope on the mountain near to them, would have reached the necessary sensitivity but covered close to 0% of the integrated probability.

--- abstract: 'Subspace-based signal processing has a rich history in the literature. Traditional focus in this direction has been on problems involving a few subspaces. But a number of problems in different application areas have emerged in recent years that involve significantly larger number of subspaces relative to the ambient dimension. It becomes imperative in such settings to first identify a smaller set of *active subspaces* that contribute to the observations before further information processing tasks can be carried out. We term this problem of identification of a small set of active subspaces among a huge collection of subspaces from a single (noisy) observation in the ambient space as *subspace unmixing*. In this paper, we formally pose the subspace unmixing problem, discuss its connections with problems in wireless communications, hyperspectral imaging, high-dimensional statistics and compressed sensing, and propose and analyze a low-complexity algorithm, termed *marginal subspace detection* (MSD), for subspace unmixing. The MSD algorithm turns the subspace unmixing problem into a multiple hypothesis testing (MHT) problem and our analysis helps control the family-wise error rate of this MHT problem at any level $\alpha \in [0,1]$. Some other highlights of our analysis of the MSD algorithm include: ($i$) it is applicable to an arbitrary collection of subspaces on the Grassmann manifold; ($ii$) it relies on properties of the collection of subspaces that are computable in polynomial time; and ($iii$) it allows for linear scaling of the number of active subspaces as a function of the ambient dimension. Finally, we also present numerical results in the paper to better understand the performance of the MSD algorithm.' author: - 'Waheed U. Bajwa and Dustin G. Mixon[^1][^2]' title: 'A Multiple Hypothesis Testing Approach to Low-Complexity Subspace Unmixing' --- Average mixing coherence; family-wise error rate; Grassmann manifold; interference subspaces; local 2-subspace coherence; multiple hypothesis testing; subspace detection; subspace unmixing Introduction ============ Subspace models, in which it is assumed that signals of interest lie on or near a low-dimensional subspace of a higher-dimensional Hilbert space $\cH$, have a rich history in signal processing, machine learning, and statistics. While much of the classical literature in detection, estimation, classification, dimensionality reduction, etc., is based on the subspace model, many of these results deal with a small number of subspaces, say, $\cX_{N}:= \{\cS_1,\cS_2,\dots,\cS_{N}\}$ with each $\cS_i$ a subspace of $\cH$, relative to the dimension of the Hilbert space: $\text{dim}(\cH) := {D}\geq N$. Consider, for instance, the classical subspace detection problem studied in [@Scharf.Friedlander.ITSP1994]. In this problem, one deals with two subspaces—the *signal* subspace and the *interference* subspace—and a solution to the detection problem involves a low-complexity generalized likelihood ratio test [@Scharf.Friedlander.ITSP1994]. However, proliferation of cheap sensors and low-cost semiconductor devices in the modern world means we often find ourselves dealing with a significantly larger number of subspaces relative to the extrinsic dimension, i.e., ${D}\ll {N}$. But many of the classical subspace-based results do not generalize in such “${D}$ smaller than ${N}$” settings either because of the breakdown of the stated assumptions or because of the prohibitive complexity of the resulting solutions. In fact, without additional constraints, information processing in such settings might well be a daunting, if not impossible, task. One constraint that often comes to our rescue in this regard in many applications is the “principle of parsimony”: *while the total number of subspaces might be large, only a small number of them, say, ${n}\propto {D}$, tend to be “active” at any given instance*. Mathematically, the ${D}$-dimensional observations $y \in \cH$ in this parsimonious setting can be expressed as $y \in \cF\big(\cX_{\cA}\big) + \text{noise}$, where $\cX_{\cA}:= \big\{\cS_i \in \cX_{N}: \cS_i \text{ is active}\big\}$ denotes the set of active subspaces with $n := \#\big\{i : \cS_i \text{ is active}\big\} \ll {D}\ll {N}$ and $\cF(\cdot)$ denotes the rule that relates the set of active subspaces $\cX_{\cA}$ to the (noisy) observations. It is easy to convince oneself in this case that the classical subspace-based computational machinery for information processing becomes available to us as soon as we have access to $\cX_{\cA}$. One of the fundamental challenges for information processing in the “${D}$ smaller than ${N}$” setting could then be described as the recovery of the set of active subspaces, $\cX_{\cA}\subset \cX_{N}$, from the ${D}$-dimensional observations $y \in \cF\big(\cX_{\cA}\big) + \text{noise}$. We term this problem of the recovery of $\cX_{\cA}$ from noisy observations as *subspace unmixing*. In this paper, we study a special case of subspace unmixing that corresponds to the *subspace sum model*, i.e., $\cF(\cX_{\cA}) := \sum_{i \in {\cA}} \cS_i$, where ${\cA}:= \{i : \cS_i \text{ is active}\}$ denotes the indices of active subspaces. Before describing our main contributions in relation to subspace unmixing from $y \in \sum_{i \in {\cA}} \cS_i + \text{noise}$, we discuss some of its applications in different areas. ### Multiuser Detection in Wireless Networks Consider a wireless network comprising a large number of users in which some of the users simultaneously transmit data to a base station. It is imperative for the base station in this case to identify the users that are communicating with it at any given instance, which is termed as the problem of multiuser detection. This problem of multiuser detection in wireless networks can also be posed as a subspace unmixing problem. In this context, users in the network communicate with the base station using $D$-dimensional codewords in $\cH$, each individual user is assigned a codebook that spans a low-dimensional subspace $\cS_i$ of $\cH$, the total number of users in the network is $N$, the number of active users at any given instance is $n \ll N$, and the base station receives $y \in \sum_{i \in {\cA}} \cS_i + \text{noise}$ due to the superposition property of the wireless medium, where ${\cA}$ denotes the indices of the users actively communicating with the base station. ### Spectral Unmixing in Hyperspectral Remote Sensing Hyperspectral remote sensing has a number of civilian and defense applications, which typically involve identifying remote objects from their spectral signatures. Because of the low spatial resolution of hyperspectral imaging systems in most of these applications, individual hyperspectral pixels tend to comprise multiple objects (e.g., soil and vegetation). Spectral unmixing is the problem of decomposition of a “mixed” hyperspectral pixel into its constituent objects. In order to pose this spectral unmixing problem into the subspace unmixing problem studied in this paper, we need two assumptions that are often invoked in the literature. First, the spectral variability of each object in different scenes can be captured through a low-dimensional subspace. Second, the mixture of spectra of different objects into a hyperspectral pixel can be described by a linear model. The spectral unmixing problem under these assumptions is the subspace unmixing problem, with $y \in \cH$ denoting the $D$-dimensional hyperspectral pixel of an imaging system with $D$ spectral bands, $\{\cS_i \subset \cH\}_{i=1}^{N}$ denoting the low-dimensional subspaces of $\cH$ associated with the spectra of individual objects, $N$ denoting the total number of objects of interest, and $y \in \sum_{i \in {\cA}} \cS_i + \text{noise}$ with $n := |\cA| \ll N$ since only a small number of objects are expected to contribute to a single hyperspectral pixel. ### Group Model Selection in High-Dimensional Statistics Model selection in statistical data analysis is the problem of learning the relationship between the samples of a dependent or response variable (e.g., the malignancy of a tumor, the health of a network) and the samples of independent or predictor variables (e.g., the expression data of genes, the traffic data in the network). There exist many applications in statistical model selection where the implication of a single predictor in the response variable implies presence of other related predictors in the true model. In such situations, the problem of model selection is often reformulated in a “group” setting. This problem of group model selection in high-dimensional settings, where the number of predictors tends to be much larger than the number of samples, can also be posed as the subspace unmixing problem. In this context, $y \in \cH$ denotes the $D$-dimensional response variable with $D$ representing the total number of samples, $N$ denotes the total number of groups of predictors that comprise the design matrix, $\{\cS_i \subset \cH\}_{i=1}^{N}$ denotes the low-dimensional subspaces of $\cH$ spanned by each of the groups of predictors, and $y \in \sum_{i \in {\cA}} \cS_i + \text{noise}$ with ${\cA}$ denoting the indices of the groups of predictors that truly affect the response variable. ### Sparsity Pattern Recovery in Block-Sparse Compressed Sensing Compressed sensing is an alternative sampling paradigm for signals that have sparse representations in some orthonormal bases. In recent years, the canonical compressed sensing theory has been extended to the case of signals that have block-sparse representations in some orthonormal bases. Sparsity pattern recovery in block-sparse compressed sensing is the problem of identifying the nonzero “block coefficients” of the measured signal. The problem of sparsity pattern recovery in block-sparse compressed sensing, however, can also be posed as the subspace unmixing problem. In this context, $y \in \cH$ denotes the $D$-dimensional measurement vector with $D$ being the total number of measurements, $N$ denotes the total number of blocks of coefficients, $\{\cS_i \subset \cH\}_{i=1}^{N}$ denotes the low-dimensional subspaces of $\cH$ spanned by the “blocks of columns” of the composite matrix $\Phi\Psi$ with $\Phi$ being the measurement matrix and $\Psi$ being the sparsifying basis, and $y \in \sum_{i \in {\cA}} \cS_i + \text{noise}$ with ${\cA}$ denoting the indices of the nonzero blocks of coefficients of the signal in $\Psi$. Relationship to Prior Work and Our Contributions ------------------------------------------------ Since the subspace unmixing problem has connections to a number of application areas, it is invariably related to prior works in some of those areas. In the context of multiuser detection, the work that is most closely related to ours is [@Applebaum.etal.PC2011]. However, the setup of [@Applebaum.etal.PC2011] can be considered a restrictive version of the general subspace unmixing problem posed in here. Roughly speaking, the setup in [@Applebaum.etal.PC2011] can be described as a *randomly modulated* subspace sum model, $y \in \sum_{i \in {\cA}} \varepsilon_i \cS_i + \text{noise}$ with $\{\varepsilon_i\}_{i=1}^N$ being independent and identically distributed isotropic random variables. In addition, the results of [@Applebaum.etal.PC2011] rely on parameters that cannot be easily translated into properties of the subspaces alone. Finally, [@Applebaum.etal.PC2011] relies on a convex optimization procedure for multiuser detection that has superlinear (in $D$ and $N$) computational complexity. In the context of group model selection and block-sparse compressed sensing, our work can be considered related to [@Yuan.Lin.JRSSSB2006; @Bach.JMLR2008; @Nardi.Rinaldo.EJS2008; @Huang.Zhang.AS2010; @Eldar.etal.ITSP2010; @Ben-Haim.Eldar.IJSTSP2011; @Elhamifar.Vidal.ITSP2012; @Cotter.etal.ITSP2005; @Tropp.etal.SP2006; @Tropp.SP2006; @Gribonval.etal.JFAA2008; @Eldar.Rauhut.ITIT2010; @Obozinski.etal.AS2011; @Davies.Eldar.ITIT2012]. None of these works, however, help us understand the problem of subspace unmixing in its most general form. Some of these works, when translated into the general subspace unmixing problem, either consider only random subspaces [@Bach.JMLR2008; @Nardi.Rinaldo.EJS2008; @Huang.Zhang.AS2010] or study subspaces generated through a Kronecker operation [@Davies.Eldar.ITIT2012; @Obozinski.etal.AS2011; @Eldar.Rauhut.ITIT2010; @Tropp.etal.SP2006; @Tropp.SP2006; @Cotter.etal.ITSP2005; @Gribonval.etal.JFAA2008]. Further, some of these works either focus on the randomly modulated subspace sum model [@Eldar.Rauhut.ITIT2010; @Gribonval.etal.JFAA2008] or generate results that suggest that, fixing the dimensions of subspaces, the total number of active subspaces can at best scale as $O\left(\sqrt{D}\right)$ [@Huang.Zhang.AS2010; @Eldar.etal.ITSP2010; @Ben-Haim.Eldar.IJSTSP2011; @Elhamifar.Vidal.ITSP2012]—the so-called “square-root bottleneck.” Finally, many of these works either focus on computational approaches that have superlinear complexity [@Obozinski.etal.AS2011; @Yuan.Lin.JRSSSB2006; @Bach.JMLR2008; @Huang.Zhang.AS2010; @Elhamifar.Vidal.ITSP2012; @Nardi.Rinaldo.EJS2008; @Tropp.SP2006] or suggest that low-complexity approaches suffer from the “dynamic range of active subspaces” [@Gribonval.etal.JFAA2008; @Ben-Haim.Eldar.IJSTSP2011]. In contrast to these and other related earlier works, the main contributions of this paper are as follows. First, it formally puts forth the problem of subspace unmixing that provides a mathematically unified view of many problems studied in other application areas. Second, it presents a low-complexity solution to the problem of subspace unmixing that has linear complexity in $D$, $N$, and the dimensions of the individual subspaces. Third, it presents a comprehensive analysis of the proposed solution, termed *marginal subspace detection* (MSD), that makes no assumptions about the structures of the individual subspaces. In particular, the resulting analysis relies on geometric measures of the subspaces that can be computed in polynomial time. Finally, the analysis neither suffers from the square-root bottleneck nor gets affected by the dynamic range of the active subspaces. We conclude by pointing out that a preliminary version of this work appeared in [@Bajwa.Conf2012]. However, that work was focused primarily on group model selection, it did not account for noise in the observations, and the ensuing analysis lacked details in terms of the metrics of multiple hypothesis testing. Notation and Organization ------------------------- The following notational convention is used throughout the rest of this paper. We use the standard notation $:=$ to denote definitions of terms. The notation $|\cdot|$ is used for both the cardinality of a set and the absolute value of a real number. Similarly, $\|\cdot\|_2$ is used for both the $\ell_2$-norm of a vector and the operator 2-norm of a matrix. The notation $\setminus$ denotes the set difference operation. Finally, we make use of the following “*Big–O*” notation for scaling relations: $f(n) = O(g(n))$ if $\exists c_o > 0, n_o : \forall n \geq n_o, f(n) \leq c_o g(n)$, $f(n) = \Omega(g(n))$ (alternatively, $f(n) \succeq g(n)$) if $g(n) = O(f(n))$, and $f(n) = \Theta(g(n))$ if $g(n) = O(f(n))$ and $f(n) = O(g(n))$. The rest of this paper is organized as follows. In Sec. \[sec:prob\_form\], we rigorously formulate the problem of subspace unmixing and define the relevant metrics used to measure the performance of subspace unmixing algorithms. In Sec. \[sec:MSD\], we describe our proposed algorithm for subspace unmixing and provide its analysis in terms of the requisite performance metrics. In Sec. \[sec:geometry\], we expand on our analysis of the proposed subspace unmixing algorithm and provide results that help understand its significance. In Sec. \[sec:num\_res\], we present some numerical results to support our analysis and we finally conclude in Sec. \[sec:conc\]. Problem Formulation {#sec:prob_form} =================== Consider a ${D}$-dimensional real Hilbert space $\cH$ and the Grassmann manifold $\fG({d}, \cH)$ that denotes the collection of all ${d}$-dimensional subspaces of $\cH$.[^3] Since finite-dimensional Hilbert spaces are isomorphic to Euclidean spaces, we will assume without loss of generality in the following that $\cH = \R^{D}$ and hence $\fG({d}, \cH) = \fG({d}, {D})$. Next, consider a collection of ${N}\gg {D}/{d}\gg 1$ subspaces given by $\cX_{N}= \big\{\cS_i \in \fG({d},{D}), i=1,\dots,{N}\big\}$ such that $\cS_1,\dots,\cS_{N}$ are pairwise disjoint: $\cS_i \cap \cS_j = \{0\}~\forall i,j=1,\dots,{N}, i \not= j$. Heuristically, this means each of the subspaces in $\cX_{N}$ is low-dimensional and the subspaces collectively “fill” the ambient space $\R^D$. The fundamental assumptions in the problem of subspace unmixing in this paper are that only a small number ${n}< {D}/{d}\ll {N}$ of the subspaces are active at any given instance and the observations $y \in \R^{D}$ correspond to a noisy version of an $x \in \R^{D}$ that lies in the sum of the active subspaces. Mathematically, we can formalize these assumptions by defining ${\cA}= \{i : \cS_i \in \cX_{N}\text{ is active}\}$, writing $x \in \sum_{i \in {\cA}} \cS_i$, and stating that the observations $y = x + \eta$, where $\eta \in \R^{D}$ denotes noise in the observations. For the sake of this exposition, we assume $\eta$ to be either bounded energy, deterministic error, i.e., $\|\eta\|_2 < \epsilon_\eta$, or independent and identically distributed (i.i.d.) Gaussian noise with variance $\sigma^2$, i.e., $\eta \sim \cN(0, \sigma^2 I)$. The final detail we need in order to complete formulation of the problem of subspace unmixing is a mathematical model for generation of the “noiseless signal” $x \in \sum_{i \in {\cA}} \cS_i$. In this paper, we work with the following generative model: - ***Mixing Bases:*** Each subspace $\cS_i$ in the collection $\cX_{N}$ is associated with an orthonormal basis $\Phi_i \in \R^{{D}\times {d}}$, i.e., $\tspan(\Phi_i) = \cS_i$ and $\Phi_i^\tT \Phi_i = I$. - ***Activity Pattern:*** The set of indices of the active subspaces ${\cA}$ is a random ${n}$-subset of $\{1,\dots,{N}\}$ with $\Pr({\cA}= \{i_1,i_2,\dots,i_{n}\}) = 1/\binom{{N}}{{n}}$. - ***Signal Generation:*** There is a deterministic but unknown collection of “mixing coefficients” $\{\theta_j \in \R^{d}, j=1,\dots,{n}\}$ such that the noiseless signal $x$ is given by $x = \sum_{j=1}^{n}\Phi_{i_j} \theta_j$, where $\cA = \{i_1,i_2,\dots,i_{n}\}$. Given this generative model, the goal of subspace unmixing in this paper is to identify the set of active subspaces $\cX_{\cA}= \big\{\cS_i \in \cX_{N}: i \in {\cA}\big\}$ using knowledge of the collection of subspaces $\cX_{N}$ and the noisy observations $y \in \R^{D}$. In particular, our focus is on unmixing solutions with linear (in $d$, $N$, and $D$) computational complexity. A few remarks are in order now regarding the stated assumptions. First, the assumption of pairwise disjointness of the subspaces is much weaker than the assumption of linear independence of the subspaces, which is typically invoked in the literature on subspace-based information processing [@Scharf.Friedlander.ITSP1994; @Manolakis.etal.ITGRS2001].[^4] In particular, while pairwise disjointness implies pairwise linear independence, it does not preclude the possibility of an element in one subspace being representable through a linear combination of elements in two or more subspaces. Second, while the generative model makes use of the mixing bases, these mixing bases might not be known to the unmixing algorithms in some applications; we will briefly discuss this further in the sequel. Third, the rather mild assumption on the randomness of the activity pattern can be interpreted as the lack of a priori information concerning the activity pattern of subspaces. Finally, those familiar with detection under the classical linear model [@Kay.Book1998 Sec. 7.7] will recognize the assumption $x = \sum_{j=1}^{n}\Phi_{i_j} \theta_j$ as a simple generalization of that setup for the problem of subspace unmixing. Performance Metrics {#ssec:perf_metrics} ------------------- In this paper, we address the problem of subspace unmixing by transforming it into a multiple hypothesis testing problem (cf. Sec. \[sec:MSD\]). While several measures of error have been used over the years in multiple hypothesis testing problems, the two most widely accepted ones in the literature remain the *family-wise error rate* () and the *false discovery rate* () [@Farcomeni.SMiMR2008]. Mathematically, if we use ${\widehat{{\cA}}}\subset \{1,\dots,{N}\}$ to denote an estimate of the indices of active subspaces returned by an unmixing algorithm then controlling the  at level $\alpha$ in our setting means $\FWER := \Pr({\widehat{{\cA}}}\not\subset {\cA}) \leq \alpha$. In words, $\FWER \leq \alpha$ guarantees that the probability of declaring even one inactive subspace as active (i.e., a single *false positive*) is controlled at level $\alpha$. On the other hand, controlling the  in our setting controls the *expected proportion* of inactive subspaces that are incorrectly declared as active by an unmixing algorithm [@Benjamini.Hochberg.JRSSSB1995]. While the  control is less stringent than the  control [@Benjamini.Hochberg.JRSSSB1995], our goal in this paper is control of the . This is because control of the  in the case of dependent test statistics, which will be the case in our setting (cf. Sec. \[sec:MSD\]), is a challenging research problem [@Benjamini.etal.B2006]. Finally, once we control the  at some level $\alpha$, our goal is to have as large a fraction of active subspaces identified as active by the unmixing algorithm as possible. The results reported in the paper in this context will be given in terms of the *non-discovery proportion* (), defined as $\NDP := \frac{|{\cA}\setminus {\widehat{{\cA}}}|}{|{\cA}|}$. Preliminaries {#ssec:prelim} ------------- In this section, we introduce some definitions that will be used throughout the rest of this paper to characterize the performance of our proposed approach to subspace unmixing. It is not too difficult to convince oneself that the “hardness” of subspace unmixing problem should be a function of the “similarity” of the underlying subspaces: *the more similar the subspaces in $\cX_{N}$, the more difficult it should be to tell them apart*. In order to capture this intuition, we work with the similarity measure of *subspace coherence* in this paper, defined as: $$\begin{aligned} \gamma(\cS_i, \cS_j) := \max_{w \in \cS_i, z \in \cS_j} \frac{|\langle w,z \rangle|}{\|w\|_2\|z\|_2},\end{aligned}$$ where $(\cS_i,\cS_j)$ denote two $d$-dimensional subspaces in $\R^D$. Note that $\gamma : \fG({d}, D) \times \fG({d}, D) \rightarrow [0,1]$ simply measures cosine of the smallest principal angle between two subspaces and has appeared in earlier literature [@Drmac.SJMAA2000; @Elhamifar.Vidal.ITSP2012]. In particular, given (any arbitrary) orthonormal bases $U_i$ and $U_j$ of $\cS_i$ and $\cS_j$, respectively, it follows that $\gamma(\cS_i, \cS_j) := \|U_i^\tT U_j\|_2$. Since we are interested in unmixing *any* active collection of subspaces, we will be stating our main results in terms of the *local $2$-subspace coherence* of individual subspaces, defined in the following. Given a collection of subspaces $\cX_{N}= \big\{\cS_i \in \fG({d},{D}), i=1,\dots,N\big\}$, the local $2$-subspace coherence of subspace $\cS_i$ is defined as $\gamma_{2,i} := \max_{j \not=i ,k \not= i: j \not= k} \big[\gamma(\cS_i, \cS_j) + \gamma(\cS_i, \cS_k)\big]$. In words, $\gamma_{2,i}$ measures closeness of $\cS_i$ to the worst pair of subspaces in the collection $\cX^{-i}_{N}:= \cX_{N}\setminus \{\cS_i\}$. It also follows from the definition of subspace coherence that $\gamma_{2,i} \in [0,2]$, with $\gamma_{2,i} = 0$ if and only if every subspace in $\cX^{-i}_{N}$ is orthogonal to $\cS_i$, while $\gamma_{2,i} = 2$ if and only if two subspaces in $\cX^{-i}_{N}$ are the same as $\cS_i$. Because of our assumption of pairwise disjointness, however, we have that $\gamma_{2,i}$ is strictly less than $2$ in this paper. We conclude our discussion of the local $2$-subspace coherence by noting that it is trivially computable in polynomial time. The next definition we need to characterize the performance of subspace unmixing is *active subspace energy*. Given the set of indices of active subspaces $\cA = \{i_1,i_2,\dots,i_{n}\}$ and the noiseless signal $x = \sum_{j=1}^{n}\Phi_{i_j} \theta_j$, the energy of the $i_j$-th active subspace is defined as $\cE_{i_j} := \|\Phi_{i_j} \theta_j\|_2^2$. Inactive subspaces of course contribute no energy to the observations, i.e., $\cE_i = 0~\forall i \not\in \cA$. But it is important for us to specify the energy of active subspaces for subspace unmixing. Indeed, active subspaces that contribute too little energy to the final observations to the extent that they get buried in noise cannot be identified using any computational method. Finally, note that $\cE_{i_j} \equiv \|\theta_j\|_2^2$ due to the orthonormal nature of the mixing bases. Finally, the low-complexity algorithm proposed in this paper requires two additional definitions. The first one of these is termed *average mixing coherence* of individual subspaces, which measures the “niceness” of the mixing bases in relation to each of the subspaces in the collection $\cX_{N}$. Given a collection of subspaces $\cX_{N}= \big\{\cS_i \in \fG({d},{D}), i=1,\dots,N\big\}$ and the associated mixing bases $\cB_{N}:= \big\{\Phi_i: \tspan(\Phi_i) = \cS_i, \Phi_i^\tT \Phi_i = I, i=1,\dots,N\big\}$, the average mixing coherence of subspace $\cS_i$ is defined as $\rho_i := \frac{1}{N-1}\left\|\sum_{j\not=i} \Phi_i^\tT \Phi_j\right\|_2$. Since we are introducing average mixing coherence for the first time in the literature,[^5] it is worth understanding its behavior. First, unlike (local $2$-)subspace coherence, it is not invariant to the choice of the mixing bases. Second, note that $\rho_i \in [0,1]$. To see this, observe that $\rho_i = 0$ if the subspaces in $\cX_{N}$ are orthogonal to each other. Further, we have from triangle inequality and the definition of subspace coherence that $\rho_i \leq \sum_{j\not=i} \gamma(\cS_i,\cS_j)/(N-1) \leq 1$. Clearly, the *average subspace coherence* of the subspace $\cS_i$, defined as $\overline{\gamma}_i := \sum_{j\not=i} \gamma(\cS_i,\cS_j)/(N-1)$, is a trivial upper bound on $\rho_i$ and we will return to this point in the sequel. We once again conclude by noting that both the average mixing coherence, $\rho_i$, and the average subspace coherence, $\overline{\gamma}_i$, are trivially computable in polynomial time. The final definition we need to characterize subspace unmixing is that of *cumulative active subspace energy*. Given the set of indices of active subspaces $\cA$, the cumulative active subspace energy is defined as $\cE_\cA := \sum_{i \in \cA} \cE_i$. In words, cumulative active subspace energy can be considered a measure of “signal energy” and together with the noise energy/variance, it characterizes signal-to-noise ratio for the subspace unmixing problem. Marginal Subspace Detection for Subspace Unmixing {#sec:MSD} ================================================= In this section, we present our low-complexity approach to subspace unmixing and characterize its performance in terms of the parameters introduced in Sec. \[ssec:prelim\]. Recall that the observations $y \in \R^D$ are given by $y = x + \eta$ with $x \in \sum_{i \in {\cA}} \cS_i$. Assuming the cardinality of the set of indices of active subspaces, $n = |\cA|$, is known, one can pose the subspace unmixing problem as an $M$-ary hypothesis testing problem with $M = \binom{N}{n}$. In this formulation, we have that the $k$-th hypothesis, $\cH_k,\,k=1,\dots,M$, corresponds to one of the $M$ possible choices for the set $\cA$. While an optimal theoretical strategy in this setting will be to derive the $M$-ary maximum likelihood decision rule, this will lead to superlinear computational complexity since one will have to evaluate $M = \binom{N}{n} \succeq \left(\frac{N}{n}\right)^n$ test statistics, one for each of the $M$ hypotheses, in this formulation. Instead, since we are interested in low-complexity approaches in this paper, we approach the problem of subspace unmixing as $N$ individual binary hypothesis testing problems. An immediate benefit of this approach, which transforms the problem of subspace unmixing into a multiple hypothesis testing problem, is the computational complexity: *we need only evaluate $N$ test statistics in this setting*. The challenges in this setting of course are specifying the decision rules for each of the $N$ binary hypotheses and understanding the performance of the corresponding low-complexity approach in terms of the  and the . We address these challenges in the following by describing and analyzing this multiple hypothesis testing approach. Marginal Subspace Detection --------------------------- In order to solve the problem of subspace unmixing, we propose to work with $N$ binary hypothesis tests on the observations $y = x + \eta$, as defined below. $$\begin{aligned} \cH^k_0 \ &: \ x = \sum_{j=1}^{n}\Phi_{i_j} \theta_j \quad \text{s.t.} \quad k \not\in \cA = \{i_1,i_2,\dots,i_{n}\} , \quad k=1,\dots,N,\\ \cH^k_1 \ &: \ x = \sum_{j=1}^{n}\Phi_{i_j} \theta_j \quad \text{s.t.} \quad k \in \cA = \{i_1,i_2,\dots,i_{n}\} , \quad k=1,\dots,N.\end{aligned}$$ In words, the null hypothesis $\cH^k_0$ being true signifies that subspace $\cS_k$ is not active, while the alternative hypothesis $\cH^k_1$ being true signifies that $\cS_k$ is active. Note that if we fix a $k \in \{1,\dots,N\}$ then deciding between $\cH^k_0$ and $\cH^k_1$ is equivalent to detecting a subspace $\cS_k$ in the presence of an interference signal and additive noise. While this setup is reminiscent of the subspace detection problem studied in [@Scharf.Friedlander.ITSP1994], the fundamental differences between the binary hypothesis test(s) in our problem and that in [@Scharf.Friedlander.ITSP1994] are that: ($i$) the interference signal in [@Scharf.Friedlander.ITSP1994] is assumed to come from a *single*, known subspace, while the interference signal in our problem setup is a function of the underlying activity pattern of the subspaces; and ($ii$) our problem setup involves multiple hypothesis tests (with dependent test statistics), which therefore requires control of the . Nonetheless, since *matched subspace detectors* are known to be (quasi-)optimal in subspace detection problems [@Scharf.Friedlander.ITSP1994], we put forth the test statistics for our $N$ binary hypothesis tests that are based on matched subspace detectors. **Input:** Collection $\cX_{N}= \big\{\cS_i \in \fG({d},{D}), i=1,\dots,N\big\}$, observations $y \in \R^D$, and thresholds $\big\{\tau_i > 0\big\}_{i=1}^{N}$\ **Output:** An estimate $\whcA \subset \{1,\dots,N\}$ of the set of indices of active subspaces $U_k \leftarrow \text{An orthonormal basis of the subspace } \cS_k, \ k=1,\dots,N$ $T_k(y) \leftarrow \|U_k^\tT y\|_2^2, \ k=1,\dots,N$ $\whcA \leftarrow \big\{k \in \{1,\dots,N\}: T_k(y) > \tau_k\big\}$ Specifically, in order to decide between $\cH^k_0$ and $\cH^k_1$ for any given $k$, we compute the test statistic $T_k(y) := \|U_k^T y\|_2^2$, where $U_k$ denotes any orthonormal basis of the subspace $\cS_k$. Notice that $T_k(y)$ is invariant to the choice of the basis $U_k$ and therefore it can be computed without knowledge of the set of mixing bases $\cB_{N}$. In order to relate this test statistic to the classical subspace detection literature, note that $T_k(y) = \|U_k U_k^T y\|_2^2 = \|\cP_{\cS_k} y\|_2^2$. That is, the test statistic is equivalent to projecting the observations onto the subspace $\cS_k$ and computing the energy of the projected observations, which is the same operation that arises in the classical subspace detection literature [@Scharf.Friedlander.ITSP1994; @Schaum.Conf2001]. The final decision between $\cH^k_0$ and $\cH^k_1$ then involves comparing the test statistic against a threshold $\tau_k$: $$\begin{aligned} T_k(y) \underset{\cH^k_0}{\overset{\cH^k_1}{\gtrless}} \tau_k, \quad k=1,\dots,N.\end{aligned}$$ Once we obtain these marginal decisions, we can use them to obtain an estimate of the set of indices of the active subspaces by setting $\whcA = \{k : \cH^k_1 \text{ is accepted}\}$. We term this entire procedure, outlined in Algorithm \[alg:MSD\], as *marginal subspace detection* (MSD) because of its reliance on detecting the presence of subspaces in the active set using marginal test statistics. The challenge then is understanding the behavior of the test statistics for each subspace under the two hypotheses and specifying values of the thresholds $\{\tau_k\}$ that lead to acceptable  and  figures. Further, a key aspect of any analysis of MSD involves understanding the number of active subspaces that can be tolerated by it as a function of the subspace collection $\cX_{N}$, the ambient dimension $D$, the subspace dimension $d$, etc. In order to address these questions, one would ideally like to understand the distributions of the test statistics for each of the ${N}$ subspaces under the two different hypotheses. However, specifying these distributions under the generative model of Sec. \[sec:prob\_form\] *and* ensuring that ($i$) the final results can be interpreted in terms of the geometry of the underlying subspaces, and ($ii$) the number of active subspaces can be allowed to be almost linear in $\frac{D}{d}$ appears to be an intractable analytical problem. We therefore instead focus on characterizing the (left and right) tail probabilities of the test statistics for each subspace under the two hypotheses. Tail Probabilities of the Test Statistics ----------------------------------------- In this section, we evaluate $\Pr\left(T_k(y) \geq \tau \big| \cH^k_0\right)$ and $\Pr\left(T_k(y) \leq \tau \big| \cH^k_1\right)$. To this end, we assume an arbitrary (but fixed) $k \in \{1,\dots,N\}$ in the following and first derive the right-tail probability under the null hypothesis, i.e., $y = \sum_{j=1}^{n}\Phi_{i_j} \theta_j + \eta$ and $k \not\in \cA = \{i_1,i_2,\dots,i_{n}\}$. In order to facilitate the forthcoming analysis, we note that since $T_k(y)$ is invariant to the choice of $U_k$, we have $T_k(y) = \big\|\sum_{j=1}^{n}U_k^\tT\Phi_{i_j} \theta_j + U_k^\tT\eta\big\|^2_2 \equiv \big\|\sum_{j=1}^{n}\Phi_k^\tT\Phi_{i_j} \theta_j + \Phi_k^\tT\eta\big\|^2_2$. We now state the result that characterizes the right-tail probability of $T_k(y)$ under the null hypothesis, $\cH_0^k$. \[lemma:right\_tail\_null\_hypo\] Under the null hypothesis $\cH_0^k$ for any fixed $k \in \{1,\dots,N\}$, the test statistic has the following right-tail probability: 1. In the case of bounded deterministic error $\eta$ and the assumption $\tau > (\epsilon_\eta + \rho_k \sqrt{n \cE_\cA})^2$, we have $$\begin{aligned} \Pr\left(T_k(y) \geq \tau \big| \cH^k_0\right) \leq e^2 \exp\left(-\frac{c_0(N-n)^2\big(\sqrt{\tau} - \epsilon_\eta - \rho_k \sqrt{n \cE_\cA}\big)^2}{N^2 \gamma^2_{2,k} \cE_\cA}\right).\end{aligned}$$ 2. In the case of i.i.d. Gaussian noise $\eta$, define $\epsilon := \sigma\sqrt{d + 2\delta + 2\sqrt{d\delta}}$ for any $\delta > 0$. Then, under the assumption $\tau > (\epsilon + \rho_k \sqrt{n \cE_\cA})^2$, we have $$\begin{aligned} \Pr\left(T_k(y) \geq \tau \big| \cH^k_0\right) \leq e^2 \exp\left(-\frac{c_0(N-n)^2\big(\sqrt{\tau} - \epsilon - \rho_k \sqrt{n \cE_\cA}\big)^2}{N^2 \gamma^2_{2,k} \cE_\cA}\right) + \exp(-\delta).\end{aligned}$$ Here, the parameter $c_0 := \frac{e^{-1}}{256}$ is an absolute positive constant. We begin by defining $\wtT_k(y) := \sqrt{T_k(y)}$ and noting $\wtT_k(y) \leq \big\|\sum_{j=1}^{n}\Phi_k^\tT\Phi_{i_j} \theta_j\big\|_2 + \big\|\Phi_k^\tT\eta\big\|_2$. In order to characterize the right-tail probability of $T_k(y)$ under $\cH_0^k$, it suffices to characterize the right-tail probabilities of $Z_1^k := \big\|\sum_{j=1}^{n}\Phi_k^\tT\Phi_{i_j} \theta_j\big\|_2$ and $Z_2^k := \big\|\Phi_k^\tT\eta\big\|_2$ under $\cH_0^k$. This is rather straightforward in the case of $Z_2^k$. In the case of deterministic error $\eta$, we have $Z_2^k \geq \epsilon_\eta$ with zero probability. In the case of $\eta$ being distributed as $\cN(0, \sigma^2 I)$, we have that $\eta_k := \Phi_k^\tT\eta \in \R^{d}\sim \cN(0, \sigma^2 I)$. In that case, the right-tail probability of $Z_2^k$ can be obtained by relying on a concentration of measure result in [@Laurent.Massart.AS2000 Sec. 4, Lem. 1] for the sum of squares of i.i.d. Gaussian random variables. Specifically, it follows from [@Laurent.Massart.AS2000] that $\forall \delta_2 > 0$, $$\begin{aligned} \label{eqn:laurent_massart_bound} \Pr\left(Z_2^k \geq \sigma\sqrt{d + 2\delta_2 + 2\sqrt{d\delta_2}}\right) \leq \exp(-\delta_2).\end{aligned}$$ We now focus on the right-tail probability of $Z_1^k$, conditioned on the null hypothesis. Recall that $\cA$ is a random ${n}$-subset of $\{1,2,\dots,{N}\}$ with $\Pr({\cA}= \{i_1,i_2,\dots,i_{n}\}) = 1/\binom{{N}}{{n}}$. Therefore, defining $\bar{\Pi} := \left(\pi_1,\dots,\pi_{N}\right)$ to be a random permutation of $\{1,\dots,{N}\}$ and using $\Pi := \left(\pi_1,\dots,\pi_{n}\right)$ to denote the first $n$-elements of $\bar{\Pi}$, the following equality holds in distribution: $$\begin{aligned} \label{eqn:eq_in_distrib_1} \Big\|\sum_{j=1}^{n}\Phi_k^\tT\Phi_{i_j} \theta_j \Big\|_2 \ : \ k \not\in \cA \ \overset{dist}{=} \ \Big\|\sum_{j=1}^{n}\Phi_k^\tT\Phi_{\pi_j} \theta_j\Big\|_2 \ : \ k \not\in \Pi.\end{aligned}$$ We now define a probability event $E^k_0 := \big\{\Pi = \left(\pi_1,\dots,\pi_{n}\right) : k \not\in \Pi\big\}$ and notice from that $$\begin{aligned} \label{eqn:prob_bd_H0} \Pr(Z_1^k \geq \delta_1 \big| \cH_0^k) = \Pr\Bigg(\Big\|\sum_{j=1}^{n}\Phi_k^\tT\Phi_{\pi_j} \theta_j\Big\|_2 \geq \delta_1 \big| E^k_0\Bigg).\end{aligned}$$ The rest of this proof relies heavily on a Banach-space-valued Azuma’s inequality (Proposition \[prop:azumaineq\]) stated in Appendix \[app:azuma\]. In order to make use of Proposition \[prop:azumaineq\], we construct an $\R^{d}$-valued Doob’s martingale $\left(M_0,M_1,\dots,M_{n}\right)$ on $\sum_{j=1}^{n}\Phi_k^\tT\Phi_{\pi_j} \theta_j$ as follows: $$\begin{aligned} M_0 &:= \sum_{j=1}^{n}\Phi_k^\tT \E\big[\Phi_{\pi_j}\big|E_0^k\big] \theta_j, \quad \text{and}\\ M_{\ell} &:= \sum_{j=1}^{n}\Phi_k^\tT \E\big[\Phi_{\pi_j}\big|\pi_1^\ell, E_0^k\big] \theta_j, \ \ell=1,\dots,{n},\end{aligned}$$ where $\pi_1^\ell := (\pi_1,\dots,\pi_\ell)$ denotes the first $\ell$ elements of $\Pi$. The next step involves showing that the constructed martingale has bounded $\ell_2$ differences. In order for this, we define $$\begin{aligned} M_\ell(u) := \sum_{j=1}^{n}\Phi_k^\tT \E\big[\Phi_{\pi_j}\big|\pi_1^{\ell-1}, \pi_\ell = u, E_0^k\big] \theta_j\end{aligned}$$ for $u \in \{1,\dots,{N}\} \setminus \{k\}$ and $\ell=1,\dots,{n}$. It can then be established using techniques very similar to the ones used in the *method of bounded differences* for scalar-valued martingales that [@McDiarmid.SiC1989; @Motwani.Raghavan.Book1995] $$\begin{aligned} \label{eqn:martingale_bdd_1} \|M_\ell - M_{\ell-1}\|_2 \leq \sup_{u,v} \|M_\ell(u) - M_\ell(v)\|_2.\end{aligned}$$ In order to upper bound $\|M_\ell(u) - M_\ell(v)\|_2$, we define a ${D}\times {d}$ matrix $\tPhi_{\ell,j}^{u,v}$ as $$\begin{aligned} \tPhi_{\ell,j}^{u,v} := \E\big[\Phi_{\pi_j}\big|\pi_1^{\ell-1}, \pi_\ell = u, E_0^k\big] - \E\big[\Phi_{\pi_j}\big|\pi_1^{\ell-1}, \pi_\ell = v, E_0^k\big], \quad \ell=1,\dots,{n},\end{aligned}$$ and note that $\tPhi_{\ell,j}^{u,v} = 0$ for $j < \ell$ and $\tPhi_{\ell,j}^{u,v} = \Phi_u - \Phi_v$ for $j = \ell$. In addition, notice that the random variable $\pi_j$ conditioned on $\big\{\pi_1^{\ell-1}, \pi_\ell = u, E_0^k\big\}$ has a uniform distribution over $\{1,\dots,{N}\} \setminus \{\pi_1^{\ell-1}, u, k\}$, while $\pi_j$ conditioned on $\big\{\pi_1^{\ell-1}, \pi_\ell = v, E_0^k\big\}$ has a uniform distribution over $\{1,\dots,{N}\} \setminus \{\pi_1^{\ell-1}, v, k\}$. Therefore, we get $\forall j > \ell$, $$\begin{aligned} \tPhi_{\ell,j}^{u,v} = \frac{1}{N-\ell-1}\left(\Phi_u - \Phi_v\right).\end{aligned}$$ It now follows from the preceding discussion that $$\begin{aligned} \nonumber \|M_\ell(u) - M_\ell(v)\|_2 = \big\|\sum_{j=1}^{n}\Phi_k^\tT \tPhi_{\ell,j}^{u,v} \theta_j\big\|_2 &\stackrel{(a)}{\leq} \sum_{j=1}^{n}\big\|\Phi_k^\tT \tPhi_{\ell,j}^{u,v}\big\|_2 \|\theta_j\|_2\\ \nonumber &\leq \big\|\Phi_k^\tT\left(\Phi_u - \Phi_v\right)\big\|_2 \|\theta_\ell\|_2 + \frac{\sum_{j > \ell} \big\|\Phi_k^\tT\left(\Phi_u - \Phi_v\right)\big\|_2 \|\theta_j\|_2}{N-\ell-1}\\ \label{eqn:martingale_bdd_2} &\leq \big(\gamma(\cS_k, \cS_u) + \gamma(\cS_k, \cS_v)\big)\left(\|\theta_\ell\|_2 + \frac{\sum_{j > \ell} \|\theta_j\|_2}{N-\ell-1}\right),\end{aligned}$$ where $(a)$ is due to the triangle inequality and submultiplicativity of the operator norm. It then follows from , and definition of the local $2$-subspace coherence that $$\begin{aligned} \label{eqn:martingale_bdd_3} \|M_\ell - M_{\ell-1}\|_2 \leq \underbrace{\gamma_{2,k}\left(\|\theta_\ell\|_2 + \frac{\sum_{j > \ell} \|\theta_j\|_2}{N-\ell-1}\right)}_{b_\ell}.\end{aligned}$$ The final bound we need in order to utilize Proposition \[prop:azumaineq\] is that on $\|M_0\|_2$. To this end, note that $\pi_j$ conditioned on $E_0^k$ has a uniform distribution over $\{1,\dots,{N}\} \setminus \{k\}$. It therefore follows that $$\begin{aligned} \label{eqn:bound_M0_H0} \|M_0\|_2 = \Big\|\sum_{j=1}^{n}\Phi_k^\tT \big(\sum_{\substack{q=1\\q\not=k}}^{{N}}\frac{\Phi_q}{{N}- 1}\big) \theta_j\Big\|_2 \stackrel{(b)}{\leq} \frac{1}{N-1}\Big\|\sum_{\substack{q=1\\q\not=k}}^{{N}} \Phi_k^\tT \Phi_q\Big\|_2 \Big\|\sum_{j=1}^{n}\theta_j\Big\|_2 \stackrel{(c)}{\leq} \rho_k \sqrt{n \cE_\cA}.\end{aligned}$$ Here, $(b)$ is again due to submultiplicativity of the operator norm, while $(c)$ is due to definitions of the average mixing coherence and the cumulative active subspace energy as well as the triangle inequality and the Cauchy–Schwarz inequality. Next, we make use of [@Donahue.etal.CA1997 Lemma B.1] to note that $\zeta_{\cB}(\tau)$ defined in Proposition \[prop:azumaineq\] satisfies $\zeta_{\cB}(\tau) \leq \tau^2/2$ for $(\cB, \|\cdot\|) \equiv \big(L_2(\R^d), \|\cdot\|_2\big)$. Consequently, under the assumption $\delta_1 > \rho_k \sqrt{n \cE_\cA}$, it can be seen from our construction of the Doob martingale $\left(M_0,M_1,\dots,M_{n}\right)$ that $$\begin{aligned} \nonumber \Pr\Bigg(\Big\|\sum_{j=1}^{n}\Phi_k^\tT\Phi_{\pi_j} \theta_j\Big\|_2 \geq \delta_1 \big| E^k_0\Bigg) &= \Pr\big(\|M_n\|_2 \geq \delta_1 \big| E^k_0\big) = \Pr\big(\|M_n\|_2 - \|M_0\|_2 \geq \delta_1 - \|M_0\|_2\big| E^k_0\big)\\ \nonumber &\stackrel{(d)}{\leq} \Pr\left(\|M_n - M_0\|_2 \geq \delta_1 - \rho_k \sqrt{n \cE_\cA} \,\big| E^k_0\right)\\ \label{eqn:azuma_1} &\stackrel{(e)}{\leq} e^2 \exp\left(-\frac{c_0\big(\delta_1 - \rho_k \sqrt{n \cE_\cA}\big)^2}{\sum_{\ell=1}^{{n}} b_\ell^2}\right),\end{aligned}$$ where $(d)$ is mainly due to the bound on $\|M_0\|_2$ in , while $(e)$ follows from the Banach-space-valued Azuma inequality in Appendix \[app:azuma\]. In addition, we can establish using , the inequality $\sum_{j > \ell} \|\theta_j\|_2 \leq \sqrt{n \cE_\cA}$, and some tedious algebraic manipulations that $$\begin{aligned} \label{eqn:azuma_bds_1} \sum_{\ell=1}^{{n}} b_\ell^2 = \gamma^2_{2,k} \sum_{\ell=1}^{{n}} \left(\|\theta_\ell\|_2 + \frac{\sum_{j > \ell} \|\theta_j\|_2}{N-\ell-1}\right)^2 \leq \gamma^2_{2,k} \cE_\cA \left(\frac{{N}}{{N}-{n}}\right)^2.\end{aligned}$$ Combining , and , we therefore obtain $\Pr(Z_1^k \geq \delta_1 \big| \cH_0^k) \leq e^2 \exp\left(-\frac{c_0(N-n)^2\big(\delta_1 - \rho_k \sqrt{n \cE_\cA}\big)^2}{N^2 \gamma^2_{2,k} \cE_\cA}\right)$. We now complete the proof by noting that $$\begin{aligned} \nonumber \Pr\left(T_k(y) \geq \tau \big| \cH^k_0\right) &= \Pr\left(\wtT_k(y) \geq \sqrt{\tau} \big| \cH^k_0\right) \leq \Pr\left(Z_1^k + Z_2^k \geq \sqrt{\tau} \big| \cH^k_0\right)\\ \nonumber &\leq \Pr\left(Z_1^k + Z_2^k \geq \sqrt{\tau} \big| \cH^k_0, Z_2^k < \epsilon_2\right) + \Pr\left(Z_2^k \geq \epsilon_2 \big| \cH^k_0\right)\\ &\leq \Pr\left(Z_1^k \geq \sqrt{\tau} - \epsilon_2 \big| \cH^k_0\right) + \Pr\left(Z_2^k \geq \epsilon_2\right).\end{aligned}$$ The two statements in the lemma now follow from the (probabilistic) bounds on $Z_2^k$ established at the start of the proof and the probabilistic bound on $Z_1^k$ obtained in the preceding paragraph. Our next goal is evaluation of $\Pr\left(T_k(y) \leq \tau \big| \cH^k_1\right)$. In this regard, we once again fix an arbitrary $k \in \{1,\dots,N\}$ and derive the left-tail probability under the alternative hypothesis, $\cH^k_1$, i.e., $y = \sum_{j=1}^{n}\Phi_{i_j} \theta_j + \eta$ such that the index $k \in \cA = \{i_1,i_2,\dots,i_{n}\}$. \[lemma:left\_tail\_alt\_hypo\] Under the alternative hypothesis $\cH_1^k$ for any fixed $k \in \{1,\dots,N\}$, the test statistic has the following left-tail probability: 1. In the case of bounded deterministic error $\eta$ and under the assumptions $\cE_k > (\epsilon_\eta + \rho_k \sqrt{{n}(\cE_\cA - \cE_k)})^2$ and $\tau < (\sqrt{\cE_k} - \epsilon_\eta - \rho_k \sqrt{{n}(\cE_\cA - \cE_k)})^2$, we have $$\begin{aligned} \label{eqn:lemma:left_tail_alt_hypo:1} \Pr\left(T_k(y) \leq \tau \big| \cH^k_1\right) \leq e^2 \exp\left(-\frac{c_0({N}-{n})^2\big(\sqrt{\cE_k} - \sqrt{\tau} - \epsilon_\eta - \rho_k \sqrt{{n}(\cE_\cA - \cE_k)}\big)^2}{(2{N}- {n})^2 \gamma^2_{2,k} (\cE_\cA - \cE_k)}\right).\end{aligned}$$ 2. In the case of i.i.d. Gaussian noise $\eta$, define $\epsilon := \sigma\sqrt{d + 2\delta + 2\sqrt{d\delta}}$ for any $\delta > 0$. Then, under the assumptions $\cE_k > (\epsilon + \rho_k \sqrt{{n}(\cE_\cA - \cE_k)})^2$ and $\tau < (\sqrt{\cE_k} - \epsilon - \rho_k \sqrt{{n}(\cE_\cA - \cE_k)})^2$, we have $$\begin{aligned} \label{eqn:lemma:left_tail_alt_hypo:2} \Pr\left(T_k(y) \leq \tau \big| \cH^k_1\right) \leq e^2 \exp\left(-\frac{c_0({N}-{n})^2\big(\sqrt{\cE_k} - \sqrt{\tau} - \epsilon - \rho_k \sqrt{{n}(\cE_\cA - \cE_k)}\big)^2}{(2{N}- {n})^2 \gamma^2_{2,k} (\cE_\cA - \cE_k)}\right) + \exp(-\delta).\end{aligned}$$ Here, the parameter $c_0 := \frac{e^{-1}}{256}$ is an absolute positive constant. We once again define $\wtT_k(y) := \sqrt{T_k(y)}$ and note that $\wtT_k(y) \geq \big\|\sum_{j=1}^{n}\Phi_k^\tT\Phi_{i_j} \theta_j\big\|_2 - \big\|\Phi_k^\tT\eta\big\|_2$. Therefore, characterization of the left-tail probability of $Z_1^k := \big\|\sum_{j=1}^{n}\Phi_k^\tT\Phi_{i_j} \theta_j\big\|_2$ and the right-tail probability of $Z_2^k := \big\|\Phi_k^\tT\eta\big\|_2$ under $\cH_1^k$ helps us specify the left-tail probability of $T_k(y)$ under $\cH_1^k$. Since the right-tail probability of $Z_2^k$ for both deterministic and stochastic errors has already been specified in the proof of Lemma \[lemma:right\_tail\_null\_hypo\], we need only focus on the left-tail probability of $Z_1^k$ under $\cH_1^k$ in here. In order to characterize $\Pr(Z_1^k \leq \delta_1 \big| \cH_1^k)$, we once again define $\bar{\Pi} := \left(\pi_1,\dots,\pi_{N}\right)$ to be a random permutation of $\{1,\dots,{N}\}$ and use $\Pi := \left(\pi_1,\dots,\pi_{n}\right)$ to denote the first $n$-elements of $\bar{\Pi}$. We then have the following equality in distribution: $$\begin{aligned} \label{eqn:left_tail_eq_in_distrib_1} \Big\|\sum_{j=1}^{n}\Phi_k^\tT\Phi_{i_j} \theta_j \Big\|_2 \ : \ k \in \cA \ \overset{dist}{=} \ \Big\|\sum_{j=1}^{n}\Phi_k^\tT\Phi_{\pi_j} \theta_j\Big\|_2 \ : \ k \in \Pi.\end{aligned}$$ We now define a probability event $E^k_1 := \big\{\Pi = \left(\pi_1,\dots,\pi_{n}\right) : k \in \Pi\big\}$ and notice from that $$\begin{aligned} \label{eqn:prob_bd_H1} \Pr(Z_1^k \leq \delta_1 \big| \cH_1^k) = \Pr\Bigg(\Big\|\sum_{j=1}^{n}\Phi_k^\tT\Phi_{\pi_j} \theta_j\Big\|_2 \leq \delta_1 \big| E^k_1\Bigg).\end{aligned}$$ Next, we fix an arbitrary $i \in \{1,\dots,{n}\}$ and define another probability event $E_2^i := \{\pi_i = k\}$. It then follows that $$\begin{aligned} \nonumber \Pr\Bigg(\Big\|\sum_{j=1}^{n}\Phi_k^\tT\Phi_{\pi_j} \theta_j\Big\|_2 \leq \delta_1 \big| E^k_1\Bigg) &= \sum_{i=1}^{{n}} \Pr\Bigg(\Big\|\sum_{j=1}^{n}\Phi_k^\tT\Phi_{\pi_j} \theta_j\Big\|_2 \leq \delta_1 \big| E^k_1, E_2^i\Bigg) \Pr(E_2^i \big| E^k_1)\\ \nonumber &= \sum_{i=1}^{{n}} \Pr\Bigg(\Big\|\theta_i + \sum_{\substack{j=1\\j\not=i}}^{n}\Phi_k^\tT\Phi_{\pi_j} \theta_j\Big\|_2 \leq \delta_1 \big| E^k_1, E_2^i\Bigg) \Pr(E_2^i \big| E^k_1)\\ \label{eqn:prob_bd_H1_2} &\stackrel{(a)}{\leq} \sum_{i=1}^{{n}} \Pr\Bigg(\Big\|\sum_{\substack{j=1\\j\not=i}}^{n}\Phi_k^\tT\Phi_{\pi_j} \theta_j\Big\|_2 \geq \sqrt{\cE_k} - \delta_1 \big| E_2^i\Bigg) \Pr(E_2^i \big| E^k_1),\end{aligned}$$ where $(a)$ follows for the facts that ($i$) $\|\theta_i + \sum_{j\not=i} \Phi_k^\tT\Phi_{\pi_j} \theta_j\|_2 \geq \|\theta_i\|_2 - \|\sum_{j\not=i} \Phi_k^\tT\Phi_{\pi_j} \theta_j\|_2$, ($ii$) $\|\theta_i\|_2$ conditioned on $E_2^i$ is $\sqrt{\cE_k}$, and ($iii$) $E_2^i \subset E^k_1$. It can be seen from and that our main challenge now becomes specifying the right-tail probability of $\|\sum_{j\not=i} \Phi_k^\tT\Phi_{\pi_j} \theta_j\|_2$ conditioned on $E_2^i$. To this end, we once again rely on Proposition \[prop:azumaineq\] in Appendix \[app:azuma\]. Specifically, we construct an $\R^{d}$-valued Doob martingale $(M_0,M_1,\dots,M_{n-1})$ on $\sum_{j\not=i} \Phi_k^\tT\Phi_{\pi_j} \theta_j$ as follows. We first define $\Pi^{-i} := \left(\pi_1,\dots,\pi_{i-1},\pi_{i+1},\dots,\pi_{n}\right)$ and then define $$\begin{aligned} M_0 &:= \sum_{\substack{j=1\\j\not=i}}^{n}\Phi_k^\tT \E\big[\Phi_{\pi_j}\big|E_2^i\big] \theta_j, \quad \text{and}\\ M_{\ell} &:= \sum_{\substack{j=1\\j\not=i}}^{n}\Phi_k^\tT \E\big[\Phi_{\pi_j}\big|\pi_1^{-i,\ell}, E_2^i\big] \theta_j, \ \ell=1,\dots,{n}-1,\end{aligned}$$ where $\pi_1^{-i,\ell}$ denotes the first $\ell$ elements of $\Pi^{-i}$. The next step in the proof involves showing $\|M_\ell - M_{\ell-1}\|_2$ is bounded for all $\ell \in \{1,\dots,{n}-1\}$. To do this, we use $\pi_\ell^{-i}$ to denote the $\ell$-th element of $\Pi^{-i}$ and define $$\begin{aligned} M_{\ell}(u) := \sum_{\substack{j=1\\j\not=i}}^{n}\Phi_k^\tT \E\big[\Phi_{\pi_j}\big|\pi_1^{-i,\ell-1}, \pi_\ell^{-i} = u, E_2^i\big] \theta_j\end{aligned}$$ for $u \in \{1,\dots,{N}\} \setminus \{k\}$ and $\ell=1,\dots,{n}-1$. It then follows from the argument in Lemma \[lemma:right\_tail\_null\_hypo\] that $\|M_\ell - M_{\ell-1}\|_2 \leq \sup_{u,v} \|M_\ell(u) - M_\ell(v)\|_2$. We now define a ${D}\times {d}$ matrix $\tPhi_{\ell,j}^{u,v}$ as $$\begin{aligned} \tPhi_{\ell,j}^{u,v} := \E\big[\Phi_{\pi_j}\big|\pi_1^{-i,\ell-1}, \pi_\ell^{-i} = u, E_2^i\big] - \E\big[\Phi_{\pi_j}\big|\pi_1^{-i,\ell-1}, \pi_\ell^{-i} = v, E_2^i\big], \quad \ell=1,\dots,{n}.\end{aligned}$$ It is then easy to see that $\forall j > \ell + 1, j \not= i$, the random variable $\pi_j$ conditioned on the events $\{\pi_1^{-i,\ell-1}, \pi_\ell^{-i} = u, E_2^i\}$ and $\{\pi_1^{-i,\ell-1}, \pi_\ell^{-i} = v, E_2^i\}$ has a uniform distribution over the sets $\{1,\dots,{N}\} \setminus \{\pi_1^{-i,\ell-1}, u, k\}$ and $\{1,\dots,{N}\} \setminus \{\pi_1^{-i,\ell-1}, v, k\}$, respectively. It therefore follows $\forall j > \ell + 1, j \not= i$ that $\tPhi_{\ell,j}^{u,v} = \frac{1}{{N}- \ell - 1}(\Phi_u - \Phi_v)$. In order to evaluate $\tPhi_{\ell,j}^{u,v}$ for $j \leq \ell+1, j \not= i$, we need to consider three cases for the index $\ell$. In the first case of $\ell \geq i$, it can be seen that $\tPhi_{\ell,j}^{u,v} = 0~\forall j \leq \ell$ and $\tPhi_{\ell,j}^{u,v} = \Phi_u - \Phi_v$ for $j = \ell + 1$. In the second case of $\ell = i - 1$, it can similarly be seen that $\tPhi_{\ell,j}^{u,v} = 0~\forall j < \ell$ and $j = \ell + 1$, while $\tPhi_{\ell,j}^{u,v} = \Phi_u - \Phi_v$ for $j = \ell$. In the final case of $\ell < i - 1$, it can be further argued that $\tPhi_{\ell,j}^{u,v} = 0~\forall j < \ell$, $\tPhi_{\ell,j}^{u,v} = \Phi_u - \Phi_v$ for $j = \ell$, and $\tPhi_{\ell,j}^{u,v} = \frac{1}{{N}- \ell - 1}(\Phi_u - \Phi_v)$ for $j = \ell + 1$. Combining all these facts together, we have the following upper bound: $$\begin{aligned} \nonumber \|M_\ell(u) - M_\ell(v)\|_2 &= \big\|\sum_{\substack{j=1\\j\not=i}}^{n}\Phi_k^\tT \tPhi_{\ell,j}^{u,v} \theta_j\big\|_2 \stackrel{(b)}{\leq} \sum_{\substack{j\geq\ell\\j\not=i}} \big\|\Phi_k^\tT \tPhi_{\ell,j}^{u,v}\big\|_2 \|\theta_j\|_2\\ \nonumber &\stackrel{(c)}{\leq} \big\|\Phi_k^\tT\left(\Phi_u - \Phi_v\right)\big\|_2 \Bigg(\|\theta_\ell\|_2 1_{\{\ell \not= i\}} + \|\theta_{\ell+1}\|_2 1_{\{\ell \not= i-1\}} + \sum_{\substack{j>\ell+1\\j\not=i}} \frac{\|\theta_j\|_2}{{N}- \ell - 1}\Bigg)\\ \label{eqn:martingale_bdd_H1_1} &\leq \big(\gamma(\cS_k, \cS_u) + \gamma(\cS_k, \cS_v)\big) \Bigg(\|\theta_\ell\|_2 1_{\{\ell \not= i\}} + \|\theta_{\ell+1}\|_2 1_{\{\ell \not= i-1\}} + \sum_{\substack{j>\ell+1\\j\not=i}} \frac{\|\theta_j\|_2}{{N}- \ell - 1}\Bigg).\end{aligned}$$ Here, $(b)$ and $(c)$ follow from the preceding facts that $\tPhi_{\ell,j}^{u,v} = 0~\forall j < \ell$ and $\big\|\Phi_k^\tT \tPhi_{\ell,j}^{u,v}\big\|_2 \leq \big\|\Phi_k^\tT\left(\Phi_u - \Phi_v\right)\big\|_2$ for $j = \ell$ and $j = \ell + 1$. Consequently, it follows from and definition of the local $2$-subspace coherence that $$\begin{aligned} \label{eqn:martingale_bdd_H1} \|M_\ell - M_{\ell-1}\|_2 \leq \underbrace{\gamma_{2,k}\Bigg(\|\theta_\ell\|_2 1_{\{\ell \not= i\}} + \|\theta_{\ell+1}\|_2 1_{\{\ell \not= i-1\}} + \sum_{\substack{j>\ell+1\\j\not=i}} \frac{\|\theta_j\|_2}{{N}- \ell - 1}\Bigg)}_{b_\ell}.\end{aligned}$$ The next step needed to utilize Proposition \[prop:azumaineq\] involves an upper bound on $\|M_0\|_2$, which is given as follows: $$\begin{aligned} \label{eqn:bound_M0_H1} \|M_0\|_2 = \Big\|\sum_{j \not= i} \Phi_k^\tT \big(\sum_{\substack{q=1\\q\not=k}}^{{N}}\frac{\Phi_q}{{N}- 1}\big) \theta_j\Big\|_2 \leq \frac{1}{N-1}\Big\|\sum_{\substack{q=1\\q\not=k}}^{{N}} \Phi_k^\tT \Phi_q\Big\|_2 \Big\|\sum_{j\not= i}\theta_j\Big\|_2 \stackrel{(d)}{\leq} \rho_k \sqrt{({n}-1)(\cE_\cA - \cE_k)}.\end{aligned}$$ Here, $(d)$ primarily follows from the fact that, conditioned on $E_2^i$, $\sum_{j\not= i} \|\theta_j\|_2^2 = \cE_\cA - \cE_k$ Our construction of the Doob martingale, Proposition \[prop:azumaineq\] in Appendix \[app:azuma\], [@Donahue.etal.CA1997 Lemma B.1] and the assumption $\sqrt{\cE_k} - \delta_1 > \rho_k \sqrt{{n}(\cE_\cA - \cE_k)}$ now provides us the following upper bound: $$\begin{aligned} \nonumber \Pr\Bigg(\Big\|\sum_{\substack{j=1\\j\not=i}}^{n}\Phi_k^\tT\Phi_{\pi_j} \theta_j\Big\|_2 \geq \sqrt{\cE_k} - \delta_1 \big| E_2^i\Bigg) &= \Pr\big(\|M_{n-1}\|_2 \geq \sqrt{\cE_k} - \delta_1 \big| E_2^i\big)\\ \nonumber &= \Pr\big(\|M_{n-1}\|_2 - \|M_0\|_2 \geq \sqrt{\cE_k} - \delta_1 - \|M_0\|_2\big| E_2^i\big)\\ \nonumber &\stackrel{(e)}{\leq} \Pr\left(\|M_{n-1} - M_0\|_2 \geq \sqrt{\cE_k} - \delta_1 - \rho_k \sqrt{{n}(\cE_\cA - \cE_k)} \,\big| E^k_0\right)\\ \label{eqn:azuma_2} &\leq e^2 \exp\left(-\frac{c_0\big(\sqrt{\cE_k} - \delta_1 - \rho_k \sqrt{{n}(\cE_\cA - \cE_k)}\big)^2}{\sum_{\ell=1}^{{n}-1} b_\ell^2}\right),\end{aligned}$$ where $(e)$ is primarily due to the bound on $\|M_0\|_2$ in . Further, it can be shown using , the inequality $\sum_{\ell=1}^{{n}-1}\|\theta_\ell\|_2 1_{\{\ell \not= i\}}\cdot\|\theta_{\ell+1}\|_2 1_{\{\ell \not= i-1\}} \leq (\cE_\cA - \cE_k)$, and some tedious manipulations that the following holds: $$\begin{aligned} \label{eqn:azuma_bds_H1} \sum_{\ell=1}^{{n}-1} b_\ell^2 \leq \gamma^2_{2,k}(\cE_\cA - \cE_k)\left(\frac{2{N}- {n}}{{N}-{n}}\right)^2.\end{aligned}$$ Combining , , and , we obtain $\Pr(Z_1^k \leq \delta_1 \big| \cH_1^k) \leq e^2 \exp\left(-\frac{c_0({N}-{n})^2\big(\sqrt{\cE_k} - \delta_1 - \rho_k \sqrt{{n}(\cE_\cA - \cE_k)}\big)^2}{(2{N}- {n})^2 \gamma^2_{2,k} (\cE_\cA - \cE_k)}\right)$. The proof of the lemma can now be completed by noting that $$\begin{aligned} \nonumber \Pr\left(T_k(y) \leq \tau \big| \cH^k_1\right) &= \Pr\left(\wtT_k(y) \leq \sqrt{\tau} \big| \cH^k_1\right) \leq \Pr\left(Z_1^k - Z_2^k \leq \sqrt{\tau} \big| \cH^k_1\right)\\ \nonumber &\leq \Pr\left(Z_1^k - Z_2^k \leq \sqrt{\tau} \big| \cH^k_1, Z_2^k < \epsilon_2\right) + \Pr\left(Z_2^k \geq \epsilon_2 \big| \cH^k_1\right)\\ &\leq \Pr\left(Z_1^k \leq \sqrt{\tau} + \epsilon_2 \big| \cH^k_1\right) + \Pr\left(Z_2^k \geq \epsilon_2\right).\end{aligned}$$ The two statements in the lemma now follow from the (probabilistic) bounds on $Z_2^k$ established at the start of the proof of Lemma \[lemma:right\_tail\_null\_hypo\] and the probabilistic bound on $Z_1^k$ obtained in the preceding paragraph. Performance of Marginal Subspace Detection ------------------------------------------ In this section, we will leverage Lemma \[lemma:right\_tail\_null\_hypo\] to control the  of MSD at a prescribed level $\alpha$. In addition, we will make use of Lemma \[lemma:left\_tail\_alt\_hypo\] to understand the  performance of MSD when its  is controlled at level $\alpha$. Before proceeding with these goals, however, it is instructive to provide an intuitive interpretation of Lemmas \[lemma:right\_tail\_null\_hypo\] and \[lemma:left\_tail\_alt\_hypo\] for individual subspaces (i.e., in the absence of a formal correction for multiple hypothesis testing [@Farcomeni.SMiMR2008; @Benjamini.Hochberg.JRSSSB1995]). We provide such an interpretation in the following for the case of bounded deterministic error $\eta$, with the understanding that extensions of our arguments to the case of i.i.d. Gaussian noise $\eta$ are straightforward. Lemma \[lemma:right\_tail\_null\_hypo\] characterizes the probability of *individually* rejecting the null hypothesis $\cH_0^k$ when it is true (i.e., declaring the subspace $\cS_k$ to be active when it is inactive). Suppose for the sake of argument that $\cH_0^k$ is true and $\cS_k$ is orthogonal to every subspace in $\cX_{N}\setminus \{\cS_k\}$, in which case the $k$-th test statistic reduces to $T_k(y) \equiv \|\eta\|_2^2$. It is then easy to see in this hypothetical setting that the decision threshold $\tau_k$ must be above the *noise floor*, $\tau_k > \epsilon_\eta^2$, to ensure one does not reject $\cH_0^k$ when it is true. Lemma \[lemma:right\_tail\_null\_hypo\] effectively generalizes this straightforward observation to the case when the $\cS_k$ cannot be orthogonal to every subspace in $\cX_{N}\setminus \{\cS_k\}$. First, the lemma states in this case that an *effective noise floor*, defined as $\epsilon^2_{\text{eff}} := (\epsilon_\eta + \rho_k \sqrt{n \cE_\cA})^2$, appears in the problem and the decision threshold must now be above this effective noise floor, $\tau_k > \epsilon_{\text{eff}}^2$, to ensure one does not reject $\cH_0^k$ when it is true. It can be seen from the definition of the effective noise floor that $\epsilon_{\text{eff}}$ has an intuitive additive form, with the first term $\epsilon_\eta$ being due to the additive error $\eta$ and the second term $\rho_k \sqrt{n \cE_\cA}$ being due to the mixing with non-orthogonal subspaces. In particular, $\epsilon_{\text{eff}} \searrow \epsilon_\eta$ as the average mixing coherence $\rho_k \searrow 0$ (recall that $\rho_k \equiv 0$ for the case of $\cS_k$ being orthogonal to the subspaces in $\cX_{N}\setminus \{\cS_k\}$). Second, once a threshold above the effective noise floor is chosen, the lemma states that the probability of rejecting the true $\cH_0^k$ decreases exponentially as the gap between the threshold and the effective noise floor increases and/or the local $2$-subspace coherence $\gamma_{2,k}$ of $\cS_k$ decreases. In particular, the probability of rejecting the true $\cH_0^k$ in this case has the intuitively pleasing characteristic that it approaches zero exponentially fast as $\gamma_{2,k} \searrow 0$ (recall that $\gamma_{2,k} \equiv 0$ for the case of $\cS_k$ being orthogonal to the subspaces in $\cX_{N}\setminus \{\cS_k\}$). We now shift our focus to Lemma \[lemma:left\_tail\_alt\_hypo\], which specifies the probability of individually rejecting the alternative hypothesis $\cH_1^k$ when it is true (i.e., declaring the subspace $\cS_k$ to be inactive when it is indeed active). It is once again instructive to first understand the hypothetical scenario of $\cS_k$ being orthogonal to every subspace in $\cX_{N}\setminus \{\cS_k\}$. In this case, the $k$-th test statistic under $\cH_1^k$ being true reduces to $T_k(y) \equiv \|x_{\cS_k} + U_k^\tT \eta\|_2^2$, where $x_{\cS_k}$ denotes the component of the noiseless signal $x$ that is contributed by the subspace $\cS_k$. Notice in this hypothetical setting that the rotated additive error $U_k^\tT \eta$ can in principle be antipodally aligned with the signal component $x_{\cS_k}$, thereby reducing the value of $T_k(y)$. It is therefore easy to argue in this idealistic setup that ensuring one does accept $\cH_1^k$ when it is true requires: ($i$) the energy of the subspace $\cS_k$ to be above the *noise floor*, $\cE_k > \epsilon_\eta^2$, so that the test statistic remains strictly positive; and ($ii$) the decision threshold $\tau_k$ to be *below* the *subspace-to-noise gap*, $\tau_k < (\sqrt{\cE_k} - \epsilon_\eta)^2$, so that the antipodal alignment of $U_k^\tT \eta$ with $x_{\cS_k}$ does not result in a false negative. We now return to the statement of Lemma \[lemma:left\_tail\_alt\_hypo\] and note that it also effectively generalizes these straightforward observations to the case when the $\cS_k$ cannot be orthogonal to every subspace in $\cX_{N}\setminus \{\cS_k\}$. First, similar to the case of Lemma \[lemma:right\_tail\_null\_hypo\], this lemma states in this case that an *effective noise floor*, defined as $\epsilon^2_{\text{eff}} := (\epsilon_\eta + \rho_k \sqrt{{n}(\cE_\cA - \cE_k)})^2$, appears in the problem and the energy of the subspace $\cS_k$ must now be above this effective noise floor, $\cE_k > \epsilon^2_{\text{eff}}$, to ensure that the test statistic remains strictly positive. In addition, we once again have an intuitive additive form of $\epsilon_{\text{eff}}$, with its first term being due to the additive error $\eta$, its second term being due to the mixing with non-orthogonal subspaces, and $\epsilon_{\text{eff}} \searrow \epsilon_\eta$ as the average mixing coherence $\rho_k \searrow 0$. Second, the lemma states that the decision threshold must now be below the *subspace-to-effective-noise gap*, $\tau_k < (\sqrt{\cE_k} - \epsilon_{\text{eff}})^2$. Third, once a threshold below the subspace-to-effective-noise gap is chosen, the lemma states that the probability of rejecting the true $\cH_1^k$ decreases exponentially as the gap between $(\sqrt{\cE_k} - \epsilon_{\text{eff}})^2$ and the threshold increases and/or the local $2$-subspace coherence $\gamma_{2,k}$ of $\cS_k$ decreases. In particular, Lemma \[lemma:left\_tail\_alt\_hypo\] once again has the intuitively pleasing characteristic that the probability of rejecting the true $\cH_1^k$ approaches zero exponentially fast as $\gamma_{2,k} \searrow 0$. Roughly speaking, it can be seen from the preceding discussion that increasing the values of the decision thresholds $\{\tau_k\}$ in MSD should decrease the . Such a decrease in the  of course will come at the expense of an increase in the . We will specify this relationship between the $\tau_k$’s and the  in the following. But we first characterize one possible choice of the $\tau_k$’s that helps control the  of MSD at a predetermined level $\alpha$. The following theorem makes use of Lemma \[lemma:right\_tail\_null\_hypo\] and the Bonferroni correction for multiple hypothesis testing [@Farcomeni.SMiMR2008]. \[thm:FWER\_MSD\] The family-wise error rate of the marginal subspace detection (Algorithm \[alg:MSD\]) can be controlled at any level $\alpha \in [0,1]$ by selecting the decision thresholds $\{\tau_k\}_{k=1}^{{N}}$ as follows: 1. In the case of bounded deterministic error $\eta$, select $$\tau_k = \left(\epsilon_\eta + \rho_k \sqrt{n \cE_\cA} + \frac{\gamma_{2,k}N}{N-n}\sqrt{c_0^{-1}\cE_\cA\log\big(\tfrac{e^2 N}{\alpha}\big)}\right)^2, \quad k=1,\dots,{N}.$$ 2. In the case of i.i.d. Gaussian noise $\eta$, select $$\tau_k = \left(\sigma\sqrt{d + 2\log\big(\tfrac{2N}{\alpha}\big) + 2\sqrt{d\log\big(\tfrac{2N}{\alpha}\big)}} + \rho_k \sqrt{n \cE_\cA} + \frac{\gamma_{2,k}N}{N-n}\sqrt{c_0^{-1}\cE_\cA\log\big(\tfrac{e^2 2N}{\alpha}\big)}\right)^2, \quad k=1,\dots,{N}.$$ The Bonferroni correction for multiple hypothesis testing dictates that the  of the MSD is guaranteed to be controlled at a level $\alpha \in [0,1]$ as long as the probability of false positive of each *individual* hypothesis is controlled at level $\frac{\alpha}{N}$ [@Farcomeni.SMiMR2008], i.e., $\Pr\left(T_k(y) \geq \tau_k \big| \cH^k_0\right) \leq \frac{\alpha}{N}$. The statement for the bounded deterministic error $\eta$ can now be shown to hold by plugging the prescribed decision thresholds into Lemma \[lemma:right\_tail\_null\_hypo\]. Similarly, the statement for the i.i.d. Gaussian noise $\eta$ can be shown to hold by plugging $\delta := \log\big(\frac{2N}{\alpha}\big)$ and the prescribed decision thresholds into Lemma \[lemma:right\_tail\_null\_hypo\]. A few remarks are in order now regarding Theorem \[thm:FWER\_MSD\]. We once again limit our discussion to the case of bounded deterministic error, since its extension to the case of i.i.d. Gaussian noise is straightforward. In the case of deterministic error $\eta$, Theorem \[thm:FWER\_MSD\] requires the decision thresholds to be of the form $\tau_k = (\epsilon_\eta + \epsilon_{m,1} + \epsilon_{m,2})^2$, where $\epsilon_\eta$ captures the effects of the additive error, $\epsilon_{m,1}$ is due to the mixing with non-orthogonal subspaces, and $\epsilon_{m,2}$ captures the effects of both the mixing with non-orthogonal subspaces and the  $\alpha$.[^6] Other factors that affect the chosen thresholds include the total number of subspaces, the number of active subspaces, and the cumulative active subspace energy. But perhaps the most interesting aspect of Theorem \[thm:FWER\_MSD\] is the fact that as the mixing subspaces become “closer” to being orthogonal, the chosen thresholds start approaching the noise floor $\epsilon_\eta^2$: $\tau_k \searrow \epsilon_\eta^2$ as $\rho_k, \gamma_{2,k} \searrow 0$. While Theorem \[thm:FWER\_MSD\] helps control the  of MSD, it does not shed light on the corresponding  figure for MSD. In order to completely characterize the performance of MSD, therefore, we also need the following theorem. \[thm:NDP\_MSD\] Suppose the family-wise error rate of the marginal subspace detection (Algorithm \[alg:MSD\]) is controlled at level $\alpha \in [0,1]$ by selecting the decision thresholds $\{\tau_k\}_{k=1}^{{N}}$ specified in Theorem \[thm:FWER\_MSD\]. Then the estimate of the indices of active subspaces returned by MSD satisfies $\whcA \supset \cA_*$ with probability exceeding $1 - \varepsilon$, where: 1. In the case of bounded deterministic error $\eta$, we have $\varepsilon := N^{-1} + \alpha$ and $$\cA_* := \left\{i \in \cA : \cE_i > \left(2\epsilon_\eta + \rho_i \sqrt{n \cE_{1,i}} + \frac{\gamma_{2,i}N}{N-n}\sqrt{c_0^{-1}\cE_{2,i}}\right)^2\right\}$$ with parameters $\cE_{1,i} := \Big(\sqrt{\cE_\cA} + \sqrt{\cE_\cA - \cE_i}\Big)^2$ and $\cE_{2,i} := \Big(\sqrt{\cE_\cA\log(\tfrac{e^2 N}{\alpha})} + (2-\frac{n}{N})\sqrt{2(\cE_\cA - \cE_i)\log(eN)}\Big)^2$. 2. In the case of i.i.d. Gaussian noise $\eta$, we have $\varepsilon := N^{-1} + \frac{3}{2}\alpha$ and $$\cA_* := \left\{i \in \cA : \cE_i > \left(2\epsilon + \rho_i \sqrt{n \cE_{1,i}} + \frac{\gamma_{2,i}N}{N-n}\sqrt{c_0^{-1}\cE_{2,i}}\right)^2\right\}$$ with the three parameters $\epsilon := \sigma\sqrt{d + 2\log\big(\tfrac{2N}{\alpha}\big) + 2\sqrt{d\log\big(\tfrac{2N}{\alpha}\big)}}$, $\cE_{1,i} := \Big(\sqrt{\cE_\cA} + \sqrt{\cE_\cA - \cE_i}\Big)^2$ and $\cE_{2,i} := \Big(\sqrt{\cE_\cA\log(\tfrac{e^2 2 N}{\alpha})} + (2-\frac{n}{N})\sqrt{2(\cE_\cA - \cE_i)\log(eN)}\Big)^2$. In order to prove the statement for the bounded deterministic error $\eta$, pick an arbitrary $i \in \cA_*$ and notice that the assumptions within Lemma \[lemma:left\_tail\_alt\_hypo\] for the subspace $\cS_i \in \cX_{N}$ are satisfied by virtue of the definition of $\cA_*$ and the choice of the decision thresholds in Theorem \[thm:FWER\_MSD\]. It therefore follows from in Lemma \[lemma:left\_tail\_alt\_hypo\] that $i \not\in \whcA$ with probability at most $N^{-2}$. We can therefore conclude by a simple union bound argument that $\cA_* \not\subset \whcA$ with probability at most $N^{-1}$. The statement now follows from a final union bound over the events $\cA_* \not\subset \whcA$ and ${\widehat{{\cA}}}\not\subset {\cA}$, where the second event is needed since we are *simultaneously* controlling the  at level $\alpha$. Likewise, the statement for the i.i.d. Gaussian noise $\eta$ can be shown to hold by first plugging $\delta := \log\big(\frac{2N}{\alpha}\big)$ into in Lemma \[lemma:left\_tail\_alt\_hypo\] and then making use of similar union bound arguments. An astute reader will notice that we are being loose in our union bounds for the case of i.i.d. Gaussian noise. Indeed, we are double counting the event that the sum of squares of $d$ i.i.d. Gaussian random variables exceeds $\epsilon^2$, once during Lemma \[lemma:right\_tail\_null\_hypo\] (which is used for  calculations) and once during Lemma \[lemma:left\_tail\_alt\_hypo\] (which is used for this theorem). In fact, it can be shown through a better bookkeeping of probability events that $\varepsilon = N^{-1} + \alpha$ for i.i.d. Gaussian noise also. Nonetheless, we prefer the stated theorem because of the simplicity of its proof. It can be seen from Theorem \[thm:NDP\_MSD\] that if one controls the  of the MSD using Theorem \[thm:FWER\_MSD\] then its  figure satisfies $\NDP \leq \frac{|\cA \setminus \cA_*|}{n}$ with probability exceeding $1 - N^{-1} - \Theta(\alpha)$. Since $\cA_* \subset \cA$, it then follows that the  figure is the smallest when the cardinality of $\cA_*$ is the largest. It is therefore instructive to understand the nature of $\cA_*$, which is the set of indices of active subspaces that are guaranteed to be identified as active by the MSD algorithm. Theorem \[thm:NDP\_MSD\] tells us that *any* active subspace whose energy is not “too small” is a member of $\cA_*$. Specifically, in the case of bounded deterministic error, the threshold that determines whether the energy of an active subspace is large or small for the purposes of identification by MSD takes the form $(2\epsilon_\eta + \tilde{\epsilon}_{m,1} + \tilde{\epsilon}_{m,2})^2$. Here, similar to the case of Theorem \[thm:FWER\_MSD\], we observe that $\tilde{\epsilon}_{m,1}$ and $\tilde{\epsilon}_{m,2}$ are *pseudo-noise terms* that appear *only* due to the mixing with non-orthogonal subspaces and that depend upon additional factors such as the total number of subspaces, the number of active subspaces, the cumulative active subspace energy, and the .[^7] In particular, we once again have the intuitive result that $\tilde{\epsilon}_{m,1}, \tilde{\epsilon}_{m,2} \searrow 0$ as $\rho_i, \gamma_{2,i} \searrow 0$, implying that any active subspace whose energy is on the order of the noise floor will be declared as active by the MSD algorithm in this setting. Since this is the best that any subspace unmixing algorithm can be expected to accomplish, one can argue that the MSD algorithm performs near-optimal subspace unmixing for the case when the average mixing coherences and the local $2$-subspace coherences of individual subspaces in the collection $\cX_{N}$ are significantly small. Finally, note that this intuitive understanding of MSD can be easily extended to the case of i.i.d. Gaussian noise, with the major difference being that $\epsilon_\eta$ in that case gets replaced by $\epsilon = \sigma\sqrt{d + 2\log\big(\tfrac{2N}{\alpha}\big) + 2\sqrt{d\log\big(\tfrac{2N}{\alpha}\big)}}$. Marginal Subspace Detection and Subspace Coherence Conditions {#sec:geometry} ============================================================= We have established in Sec. \[sec:MSD\] that the  of MSD can be controlled at any level $\alpha \in [0,1]$ through appropriate selection of the decision thresholds (cf. Theorem \[thm:FWER\_MSD\]). Further, we have shown that the selected thresholds enable the MSD algorithm to identify all active subspaces whose energies exceed *effective* noise floors characterized by additive error/noise, average mixing coherences, local $2$-subspace coherences, etc. (cf. Theorem \[thm:NDP\_MSD\]). Most importantly, these effective noise floors approach the “true” noise floor as the average mixing coherences and the local $2$-subspace coherences of individual subspaces approach zero, suggesting near-optimal nature of MSD for such collections of mixing subspaces in the “${D}$ smaller than ${N}$” setting. But we have presented no mathematical evidence to suggest the average mixing coherences and the local $2$-subspace coherences of individual subspaces can indeed be small enough for the effective noise floors of Theorem \[thm:NDP\_MSD\] to be on the order of $\big(\text{true noise floor} + o(1)\big)$. Our goal in this section, therefore, is to provide evidence to this effect by arguing for the existence of collection of subspaces whose average mixing coherences and local $2$-subspace coherences approach zero at significantly fast rates. Recall from the statement of Theorem \[thm:NDP\_MSD\] and the subsequent discussion that the effective noise floor for the $i$-th subspace involves additive pseudo-noise terms of the form $$\begin{aligned} \label{eqn:effective_noise_term} \epsilon^i_f := \rho_i \sqrt{n \cE_{1,i}} + \frac{\gamma_{2,i}N}{N-n}\sqrt{c_0^{-1}\cE_{2,i}},\end{aligned}$$ where $\sqrt{\cE_{1,i}} = \Theta\Big(\sqrt{\cE_\cA}\Big)$ and $\sqrt{\cE_{2,i}} = \Theta\Big(\sqrt{\cE_\cA \log(N/\alpha)}\Big)$. Since we are assuming that the number of active subspaces $n = O(N)$, it follows that $\epsilon^i_f = \Theta\Big(\rho_i \sqrt{n \cE_\cA}\Big) + \Theta\Big(\gamma_{2,i} \sqrt{\cE_\cA \log(N/\alpha)}\Big)$. In order to ensure $\epsilon^i_f = o(1)$, therefore, we need the following two conditions to hold: $$\begin{aligned} \label{eqn:mixing_coh_cond} \rho_i &= O\left(\frac{1}{\sqrt{n \cE_\cA}}\right), \quad \text{and}\\ \label{eqn:local_2_coh_cond} \gamma_{2,i} &= O\left(\frac{1}{\sqrt{\cE_\cA \log(N/\alpha)}}\right).\end{aligned}$$ Together, we term the conditions and as *subspace coherence conditions*. Both these conditions are effectively statements about the geometry of the mixing subspaces and the corresponding mixing bases. In order to understand the implications of these two conditions, we parameterize the cumulative active subspace energy as $\cE_\cA = \Theta(n^\delta)$ for $\delta \in [0,1]$. Here, $\delta = 0$ corresponds to one extreme of the cumulative active subspace energy staying constant as the number of active subspaces increases, while $\delta = 1$ corresponds to other extreme of the cumulative active subspace energy increasing linearly with the number of active subspaces. We now turn our attention to the extreme of $\delta = 1$, in which case the subspace coherence conditions reduce to $\rho_i = O(n^{-1})$ and $\gamma_{2,i} = O(n^{-1/2}\log^{-1/2}(N/\alpha))$. We are interested in this setting in understanding whether there indeed exist subspaces and mixing bases that satisfy these conditions. We have the following theorem in this regard, which also sheds light on the maximum number of active subspaces that can be tolerated by the MSD algorithm. \[thm:lin\_scaling\] Suppose the number of active subspaces satisfies $n \leq \min\left\{\sqrt{N} - 1,\frac{c_1^2 D(N - 1)}{(Nd - D)\log(N/\alpha)}\right\}$ for some constant $c_1 \in (0,1)$. Then there exist collections of subspaces $\cX_{N}= \big\{\cS_i \in \fG({d},{D}), i=1,\dots,{N}\big\}$ and corresponding mixing bases $\cB_{N}= \big\{\Phi_i: \tspan(\Phi_i) = \cS_i, \Phi_i^\tT \Phi_i = I, i=1,\dots,N\big\}$ such that $\rho_i \leq n^{-1}$ and $\gamma_{2,i} \leq c_2 n^{-1/2}\log^{-1/2}(N/\alpha)$ for $i=1,\dots,{N}$, where $c_2 \geq \max\{2c_1,1\}$ is a positive numerical constant. The proof of this theorem follows from a combination of results reported in [@Calderbank.etal.unpublished2014]. To begin, note from the definition of local $2$-subspace coherence that $\frac{\gamma_{2,i}}{2} \leq \mu(\cX_{N}) := \max_{i\not=j} \gamma(\cS_i, \cS_j)$. We now argue there exist $\cX_{N}$’s such that $\mu(\cX_{N}) = 0.5 c_2 n^{-1/2}\log^{-1/2}(N/\alpha)$, which in turn implies $\gamma_{2,i} \leq c_2 n^{-1/2}\log^{-1/2}(N/\alpha)$ for such collections of subspaces. The quantity $\mu(\cX_{N})$, termed *worst-case subspace coherence*, has been investigated extensively in the literature [@Lemmens.Seidel.IMP1973; @Calderbank.etal.unpublished2014]. The first thing we need to be careful about is the fact from [@Lemmens.Seidel.IMP1973 Th. 3.6][@Calderbank.etal.unpublished2014 Th. 1] that $\mu(\cX_{N}) \geq \sqrt{\frac{Nd - D}{D(N-1)}}$, which is ensured by the conditions $n \leq \frac{c_1^2 D(N - 1)}{(Nd - D)\log(N/\alpha)}$ and $c_2 \geq 2c_1$. The existence of such collections of subspaces now follows from [@Calderbank.etal.unpublished2014], which establishes that the worst-case subspace coherences of many collections of subspaces (including subspaces drawn uniformly at random from $\fG({d},{D})$) come very close to meeting the lower bound $\sqrt{\frac{Nd - D}{D(N-1)}}$. In order to complete the proof, we next need to establish that if a collection of subspaces has $\mu(\cX_{N}) = 0.5 c_2 n^{-1/2}\log^{-1/2}(N/\alpha)$ then there exists *at least* one corresponding mixing bases for that collection such that $\rho_i \leq n^{-1}$. In this regard, note that $\rho_i \leq \nu(\cB_{N}) := \max_i \rho_i$. The quantity $\nu(\cB_{N})$, termed *average group/block coherence*, was introduced in [@Bajwa.Mixon.Conf2012] and investigated further in [@Calderbank.etal.unpublished2014]. In particular, it follows from [@Calderbank.etal.unpublished2014 Lemma 7] that every collection of subspaces $\cX_{N}$ has at least one mixing bases with $\nu(\cB_{N}) \leq \frac{\sqrt{N}+1}{N-1}$, which can in turn be upper bounded by $n^{-1}$ for $n \leq \sqrt{N} - 1$. Recall that our problem formulation calls for ${n}< {D}/{d}\ll {N}$. Theorem \[thm:lin\_scaling\] helps quantify these inequalities for the case of linear scaling of cumulative active subspace energy. Specifically, note that $\frac{D(N - 1)}{(Nd - D)\log(N/\alpha)} = O\left(\frac{D}{d\log(N/\alpha)}\right)$ for large $N$. We therefore have that Theorem \[thm:lin\_scaling\] allows the number of active subspaces to scale linearly with the extrinsic dimension $D$ modulo a logarithmic factor. Stated differently, Theorem \[thm:lin\_scaling\] establishes that the total number of *active dimensions*, $nd$, can be proportional to the extrinsic dimension $D$, while the total number of subspaces in the collection, ${N}$, affect the number of active dimensions only through a logarithmic factor. Combining Theorem \[thm:lin\_scaling\] with the earlier discussion, therefore, one can conclude that the MSD algorithm does not suffer from the “square-root bottleneck” of $nd = O(\sqrt{D})$ despite the fact that its performance is being characterized in terms of polynomial-time computable measures. Finally, we note that the constraint $n = O(\sqrt{N})$ in Theorem \[thm:lin\_scaling\] appears due to our use of [@Calderbank.etal.unpublished2014 Lemma 7], which not only guarantees existence of appropriate mixing bases but also provides a polynomial-time algorithm for obtaining those mixing bases. If one were interested in merely proving existence of mixing bases then this condition can be relaxed to $n = O(N)$ by making use of [@Calderbank.etal.unpublished2014 Th. 8] instead in the proof. Since Theorem \[thm:lin\_scaling\] guarantees existence of subspaces and mixing bases that satisfy the subspace coherence conditions for $\delta = 1$, it also guarantees the same for any other sublinear scaling $(0 \leq \delta < 1)$ of cumulative active subspace energy. Indeed, as $\delta \searrow 0$, the subspace coherence conditions (cf.  and ) only become more relaxed. In fact, it turns out that the order-wise performance of the MSD algorithm no longer remains a function of the mixing bases for certain collections of subspaces when cumulative active subspace energy reaches the other extreme of $\delta = 0$. This assertion follows from the following theorem and the fact that $\delta = 0$ reduces the subspace coherence conditions to $\rho_i = O(n^{-1/2})$ and $\gamma_{2,i} = O(\log^{-1/2}(N/\alpha))$. \[thm:constant\_energy\] Suppose the number of active subspaces satisfies $n \leq \frac{c_3 D(N - 1)}{Nd - D}$ for some constant $c_3 \in (0,1)$ and the total number of subspaces in the collection $\cX_{N}$ satisfies $N \leq \alpha \exp(n/4)$. In such cases, there exist collections of subspaces that satisfy $\mu(\cX_{N}) := \max_{i\not=j} \gamma(\cS_i, \cS_j) \leq n^{-1/2}$. Further, all such collections satisfy $\rho_i \leq n^{-1/2}$ and $\gamma_{2,i} \leq \log^{-1/2}(N/\alpha)$ for $i=1,\dots,{N}$. The proof of this theorem also mainly follows from [@Calderbank.etal.unpublished2014], which establishes that there exist many collections of subspaces for which $\mu(\cX_{N}) = \sqrt{\frac{Nd - D}{c_3 D(N-1)}}$ for appropriate constants $c_3 \in (0,1)$. Under the condition $n \leq \frac{c_3 D(N - 1)}{Nd - D}$, therefore, it follows that $\mu(\cX_{N}) \leq n^{-1/2}$. Since $\gamma_{2,i} \leq 2 \mu(\cX_{N})$, we in turn obtain $\gamma_{2,i} \leq \log^{-1/2}(N/\alpha)$ under the condition $N \leq \alpha \exp(n/4)$. Finally, we have from the definition of the average mixing coherence that $\rho_i \leq \mu(\cX_{N})$, which in turn implies $\rho_i \leq n^{-1/2}$ and this completes the proof of the theorem. Once again, notice that Theorem \[thm:constant\_energy\] allows linear scaling of the number of active dimensions as a function of the extrinsic dimension. In words, Theorem \[thm:constant\_energy\] tells us that MSD performs well for unmixing of collections of subspaces that are *approximately equi-isoclinic* [@Lemmens.Seidel.IMP1973], defined as ones with same principal angles between any two subspaces, regardless of the underlying mixing bases as long as the cumulative active subspace energy does not scale with the number of active subspaces. Numerical Results {#sec:num_res} ================= -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- [(a)]{} [(b)]{} ![Plots of local 2-subspace coherences and average mixing coherences for different values of $d$, $D$, and $N$. (a) and (b) correspond to local 2-subspace coherences for $N=1500$ and $N=2000$, respectively. (c) and (d) correspond to average mixing coherences for $N=1500$ and $N=2000$, respectively. The error bars in the plots depict the range of the two coherences for the different subspaces.[]{data-label="fig:coh_errorbars"}](local2sscoh_N1500.eps "fig:"){width="3in"} ![Plots of local 2-subspace coherences and average mixing coherences for different values of $d$, $D$, and $N$. (a) and (b) correspond to local 2-subspace coherences for $N=1500$ and $N=2000$, respectively. (c) and (d) correspond to average mixing coherences for $N=1500$ and $N=2000$, respectively. The error bars in the plots depict the range of the two coherences for the different subspaces.[]{data-label="fig:coh_errorbars"}](local2sscoh_N2000.eps "fig:"){width="3in"} [(c)]{} [(d)]{} ![Plots of local 2-subspace coherences and average mixing coherences for different values of $d$, $D$, and $N$. (a) and (b) correspond to local 2-subspace coherences for $N=1500$ and $N=2000$, respectively. (c) and (d) correspond to average mixing coherences for $N=1500$ and $N=2000$, respectively. The error bars in the plots depict the range of the two coherences for the different subspaces.[]{data-label="fig:coh_errorbars"}](avemixingcoh_N1500.eps "fig:"){width="3in"} ![Plots of local 2-subspace coherences and average mixing coherences for different values of $d$, $D$, and $N$. (a) and (b) correspond to local 2-subspace coherences for $N=1500$ and $N=2000$, respectively. (c) and (d) correspond to average mixing coherences for $N=1500$ and $N=2000$, respectively. The error bars in the plots depict the range of the two coherences for the different subspaces.[]{data-label="fig:coh_errorbars"}](avemixingcoh_N2000.eps "fig:"){width="3in"} -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ -- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- [(a) $D=600$]{} [(b) $D=1400$]{} ![Histograms of local 2-subspace coherences and average mixing coherences for different values of $d$. (a) and (c) correspond to local 2-subspace coherences and average mixing coherences, respectively, for $N=2000$ and $D=600$. (b) and (d) correspond to local 2-subspace coherences and average mixing coherences, respectively, for $N=2000$ and $D=1400$.[]{data-label="fig:coh_hists"}](local2sscohHist_N2000D600.eps "fig:"){width="3in"} ![Histograms of local 2-subspace coherences and average mixing coherences for different values of $d$. (a) and (c) correspond to local 2-subspace coherences and average mixing coherences, respectively, for $N=2000$ and $D=600$. (b) and (d) correspond to local 2-subspace coherences and average mixing coherences, respectively, for $N=2000$ and $D=1400$.[]{data-label="fig:coh_hists"}](local2sscohHist_N2000D1400.eps "fig:"){width="3in"} [(c) $D=600$]{} [(d) $D=1400$]{} ![Histograms of local 2-subspace coherences and average mixing coherences for different values of $d$. (a) and (c) correspond to local 2-subspace coherences and average mixing coherences, respectively, for $N=2000$ and $D=600$. (b) and (d) correspond to local 2-subspace coherences and average mixing coherences, respectively, for $N=2000$ and $D=1400$.[]{data-label="fig:coh_hists"}](avemixingcohHist_N2000D600.eps "fig:"){width="3in"} ![Histograms of local 2-subspace coherences and average mixing coherences for different values of $d$. (a) and (c) correspond to local 2-subspace coherences and average mixing coherences, respectively, for $N=2000$ and $D=600$. (b) and (d) correspond to local 2-subspace coherences and average mixing coherences, respectively, for $N=2000$ and $D=1400$.[]{data-label="fig:coh_hists"}](avemixingcohHist_N2000D1400.eps "fig:"){width="3in"} ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ -- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- [(a)]{} [(b)]{} ![Plots of , , and  as a function of the number of active subspaces, $n$, for $D=600$ (solid) and $D=1400$ (dashed).[]{data-label="fig:MSD_perf"}](MSDperf_N2000d3.eps "fig:"){width="3in"} ![Plots of , , and  as a function of the number of active subspaces, $n$, for $D=600$ (solid) and $D=1400$ (dashed).[]{data-label="fig:MSD_perf"}](MSDperf_N2000d15.eps "fig:"){width="3in"} --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- In this section, we report results of numerical experiments that further shed light on the relationships between the local 2-subspace coherences, the average mixing coherences, and the MSD algorithm for the problem of subspace unmixing. The subspaces used in all these experiments are independently drawn at random from $\fG({d}, {D})$ according to the natural uniform measure induced by the Haar measure on the *Stiefel manifold* $\mathbb{S}({d},{D})$, which is defined as $\mathbb{S}({d},{D}) := \{U \in \R^{{D}\times {d}}: U^\tT U = I\}$. Computationally, we accomplish this by resorting to the numerical algorithm proposed in [@Mezzadri.NotA2007] for random drawing of elements from $\mathbb{S}({d},{D})$ according to the Haar measure. In doing so, we not only generate subspaces $\cX_{N}= \{\cS_i\}_{i=1}^{N}$ from $\fG({d}, {D})$, but we also generate the associated mixing bases $\cB_{N}= \{\Phi_i\}_{i=1}^{N}$ from $\mathbb{S}({d},{D})$. Mathematically, given a subspace $\cS_i \in \fG({d}, {D})$ and its equivalence class in the Stiefel manifold $[\cS_i] \subset \mathbb{S}({d},{D})$, its associated mixing basis $\Phi_i \in \mathbb{S}({d},{D})$ is effectively drawn at random from $[\cS_i]$ according to the Haar measure on $[\cS_i]$. It is important to note here that once we generate the $\cS_i$’s and the $\Phi_i$’s, they remain fixed throughout our experiments. In other words, our results are not averaged over different realizations of the subspaces and the mixing bases; rather, they correspond to a *fixed* set of subspaces and mixing bases. Our first set of experiments evaluates the local 2-subspace coherences of the $\cS_i$’s and the average mixing coherences of the corresponding $\Phi_i$’s for different values of $d$, $D$, and $N$. The results of these experiments are reported in Figs. \[fig:coh\_errorbars\] and \[fig:coh\_hists\]. Specifically, Fig. \[fig:coh\_errorbars\](a) and Fig. \[fig:coh\_errorbars\](b) plot $\sum_{i=1}^{N} \gamma_{2,i}/N$ as well as the range of the $\gamma_{2,i}$’s using error bars for $N = 1500$ and $N = 2000$, respectively. Similarly, Fig. \[fig:coh\_errorbars\](c) and Fig. \[fig:coh\_errorbars\](d) plot $\sum_{i=1}^{N} \rho_i/N$ as well as the range of the $\rho_i$’s using error bars for $N = 1500$ and $N = 2000$, respectively. It can be seen from these figures that both the local 2-subspace coherence and the average mixing coherence decrease with an increase in $D$, while they increase with an increase in $d$. In addition, it appears from these figures that the $\gamma_{2,i}$’s and the $\rho_i$’s start concentrating around their average values for large values of $D$. Careful examination of Fig. \[fig:coh\_errorbars\], however, suggests a contrasting behavior of the two coherences for increasing $N$. While increasing $N$ from 1500 to 2000 seems to increase the $\gamma_{2,i}$’s slightly, this increase seems to have an opposite effect on the $\rho_i$’s. We attribute this behavior of the average mixing coherence to the “random walk nature” of its definition, although a comprehensive understanding of this phenomenon is beyond the scope of this paper. One of the most important things to notice from Fig. \[fig:coh\_errorbars\] is that the average mixing coherences tend to be about two orders of magnitude smaller than the local 2-subspace coherences, which is indeed desired according to the discussion in Sec. \[sec:geometry\]. Finally, since the error bars in Fig. \[fig:coh\_errorbars\] do not necessarily give an insight into the distribution of the $\gamma_{2,i}$’s and the $\rho_i$’s, we also plot histograms of the two coherences in Fig. \[fig:coh\_hists\] for $N=2000$ corresponding to $D=600$ (Figs. \[fig:coh\_hists\](a) and \[fig:coh\_hists\](c)) and $D=1400$ (Figs. \[fig:coh\_hists\](b) and \[fig:coh\_hists\](d)). Our second set of experiments evaluates the performance of the MSD algorithm for subspace unmixing. We run these experiments for *fixed* subspaces and mixing bases for the following four sets of choices for the $(d,D,N)$ parameters: $(3, 600, 2000), (3, 1400, 2000), (15, 600, 2000)$, and $(15, 1400, 2000)$. The results reported for these experiments are averaged over 5000 realizations of subspace activity patterns, mixing coefficients, and additive Gaussian noise. In all these experiments, we use $\sigma = 0.01$ and $\cE_\cA = n$, divided equally among all active subspaces, which means that all active subspaces lie above the additive noise floor. In terms of the selection of thresholds for Algorithm \[alg:MSD\], we rely on Theorem \[thm:FWER\_MSD\] with a small caveat. Since our analysis uses a number of bounds, it invariably results in conservative thresholds. In order to remedy this, we use the thresholds $\tilde{\tau}_k := c_1^2 \tau_k$ with $\tau_k$ as in Theorem \[thm:FWER\_MSD\] *but* using $c_0 = 1$ and $c_1 \in (0,1)$. We learn this new parameter $c_1$ using cross validation and set $c_1 = 0.136$ and $c_1 = 0.107$ for $d=3$ and $d=15$, respectively. Finally, we set the final thresholds to control the  in all these experiments at level $\alpha = 0.1$. The results of these experiments for our choices of the parameters are reported in Fig. \[fig:MSD\_perf\](a) and Fig. \[fig:MSD\_perf\](b) for $d=3$ and $d=15$, respectively. We not only plot the  and the  in these figures, but we also plot another metric of *false-discovery proportion* (), defined as $\FDP := \frac{|{\widehat{{\cA}}}\setminus {\cA}|}{|{\widehat{{\cA}}}|}$, as a measure of the . Indeed, the expectation of the  is the  [@Farcomeni.SMiMR2008]. It is instructive to compare the  plots for $D=600$ and $D=1400$ in these figures. We can see from Fig. \[fig:coh\_hists\] that the $\gamma_{2,i}$’s and the $\rho_i$’s are smaller for $D=1400$, which means that the thresholds $\tilde{\tau}_k$’s are also smaller for $D=1400$ (cf. Theorem \[thm:FWER\_MSD\]). But Fig. \[fig:MSD\_perf\] shows that the  for $D=1400$ mostly remains below $D=600$, which suggests that Theorem \[thm:FWER\_MSD\] is indeed capturing the correct relationship between the  of MSD and the properties of the underlying subspaces. In addition, the  plots in these figures for $D=600$ and $D=1400$ also help validate Theorem \[thm:NDP\_MSD\]. Specifically, Theorem \[thm:NDP\_MSD\] suggests that the  of MSD should remain small for larger values of $n$ as long as the $\gamma_{2,i}$’s and the $\rho_i$’s remain small. Stated differently, since the $\gamma_{2,i}$’s and the $\rho_i$’s are smaller for $D=1400$ than for $D=600$ (cf. Fig \[fig:coh\_hists\]), Theorem \[thm:NDP\_MSD\] translates into a smaller  figure for larger values of $n$ for $D=1400$. It can be seen from the  plots in Fig. \[fig:MSD\_perf\] that this is indeed the case. Conclusion {#sec:conc} ========== In this paper, we motivated and posed the problem of subspace unmixing as well as discussed its connections with problems in wireless communications, hyperspectral imaging, high-dimensional statistics and compressed sensing. We proposed and analyzed a low-complexity algorithm, termed *marginal subspace detection* (MSD), that solves the subspace unmixing problem under the subspace sum model by turning it into a multiple hypothesis testing problem. We showed that the MSD algorithm can be used to control the family-wise error rate at any level $\alpha \in [0,1]$ for an arbitrary collection of subspaces on the Grassmann manifold. We also established that the MSD algorithm allows for linear scaling of the number of active subspaces as a function of the ambient dimension. Numerical results presented in the paper further validated the usefulness of the MSD algorithm and the accompanying analysis. Future work in this direction includes design and analysis of algorithms that perform better than the MSD algorithm as well as study of the subspace unmixing problem under mixing models other than the subspace sum model. Banach-Space-Valued Azuma’s Inequality {#app:azuma} ====================================== In this appendix, we state a Banach-space-valued concentration inequality from [@Naor.CPaC2012] that is central to some of the proofs in this paper. \[prop:azumaineq\] Fix $s > 0$ and assume that a Banach space $(\cB, \|\cdot\|)$ satisfies $$\begin{aligned} \zeta_{\cB}(\tau) := \sup_{\substack{u,v\in\cB\\\|u\|=\|v\|=1}} \left\{\frac{\|u + \tau v\| + \|u - \tau v\|}{2} - 1\right\} \leq s\tau^2\end{aligned}$$ for all $\tau > 0$. Let $\{M_k\}_{k=0}^{\infty}$ be a $\cB$-valued martingale satisfying the pointwise bound $\|M_k - M_{k-1}\| \leq b_k$ for all $k \in \N$, where $\{b_k\}_{k=1}^{\infty}$ is a sequence of positive numbers. Then for every $\delta > 0$ and $k \in \N$, we have $$\begin{aligned} \Pr\left(\|M_k - M_0\| \geq \delta\right) \leq e^{\max\{s,2\}} \exp\bigg(-\frac{c_0\delta^2}{\sum_{\ell=1}^{k} b_\ell^2}\bigg),\end{aligned}$$ where $c_0 := \frac{e^{-1}}{256}$ is an absolute constant. Theorem 1.5 in [@Naor.CPaC2012] does not explicitly specify $c_0$ and also states the constant in front of $\exp(\cdot)$ to be $e^{s+2}$. Proposition \[prop:azumaineq\] stated in its current form, however, can be obtained from the proof of Theorem 1.5 in [@Naor.CPaC2012]. [10]{} \[1\][\#1]{} url@samestyle \[2\][\#2]{} \[2\][[l@\#1=l@\#1\#2]{}]{} W. U. Bajwa and D. Mixon, “Group model selection using marginal correlations: [T]{}he good, the bad and the ugly,” in *Proc. $50$th Annu. Allerton Conf. Communication, Control, and Computing*, Monticello, IL, Oct. 2012, pp. 494–501. L. L. Scharf and B. Friedlander, “Matched subspace detectors,” *[IEEE]{} Trans. Signal Processing*, vol. 42, no. 8, pp. 2146–2157, Aug. 1994. L. Applebaum, W. U. Bajwa, M. F. Duarte, and R. Calderbank, “Asynchronous code-division random access using convex optimization,” *Phy. Commun.*, vol. 5, no. 2, pp. 129–147, Jun. 2012. M. Yuan and Y. Lin, “Model selection and estimation in regression with grouped variables,” *J. Roy. Statist. Soc. Ser. B*, vol. 68, no. 1, pp. 49–67, 2006. F. Bach, “Consistency of the group lasso and multiple kernel learning,” *J. Machine Learning Res.*, vol. 9, no. 6, pp. 1179–1225, Jun. 2008. Y. Nardi and A. Rinaldo, “On the asymptotic properties of the group lasso estimator for linear models,” *Electron. J. Stat.*, vol. 2, pp. 605–633, 2008. J. Huang and T. Zhang, “The benefit of group sparsity,” *Ann. Statist.*, vol. 38, no. 4, pp. 1978–2004, Aug. 2010. Y. C. Eldar, P. Kuppinger, and H. Bölcksei, “Block-sparse signals: [U]{}ncertainty relations and efficient recovery,” *[IEEE]{} Trans. Signal Processing*, vol. 58, no. 6, pp. 3042–3054, Jun. 2010. Z. Ben-Haim and Y. C. Eldar, “Near-oracle performance of greedy block-sparse estimation techniques from noisy measurements,” *[IEEE]{} J. Select. Topics Signal Processing*, vol. 5, no. 5, pp. 1032–1047, Sep. 2011. E. Elhamifar and R. Vidal, “Block-sparse recovery via convex optimization,” *[IEEE]{} Trans. Signal Processing*, vol. 60, no. 8, pp. 4094–4107, Aug. 2012. S. Cotter, B. Rao, K. Engan, and K. Kreutz-Delgado, “Sparse solutions to linear inverse problems with multiple measurement vectors,” *[IEEE]{} Trans. Signal Processing*, vol. 53, no. 7, pp. 2477–2488, Jul. 2005. J. Tropp, A. Gilbert, and M. Strauss, “Algorithms for simultaneous sparse approximation. [P]{}art [I]{}: [G]{}reedy pursuit,” *Signal Processing*, vol. 86, no. 3, pp. 572–588, Apr. 2006. J. Tropp, “Algorithms for simultaneous sparse approximation. [P]{}art [II]{}: [C]{}onvex relaxation,” *Signal Processing*, vol. 86, no. 3, pp. 589–602, Apr. 2006. R. Gribonval, H. Rauhut, K. Schnass, and P. Vandergheynst, “Atoms of all channels, unite! [A]{}verage case analysis of multi-channel sparse recovery using greedy algorithms,” *J. Fourier Anal. Appl.*, vol. 14, no. 5-6, pp. 655–687, Dec. 2008. Y. C. Eldar and H. Rauhut, “Average case analysis of multichannel sparse recovery using convex relaxation,” *[IEEE]{} Trans. Inform. Theory*, vol. 56, no. 1, pp. 505–519, Jan. 2010. G. Obozinski, M. Wainwright, and M. Jordan, “Support union recovery in high-dimensional multivariate regression,” *Ann. Statist.*, vol. 39, no. 1, pp. 1–47, Jan. 2011. M. Davies and Y. Eldar, “Rank awareness in joint sparse recovery,” *[IEEE]{} Trans. Inform. Theory*, vol. 58, no. 2, pp. 1135–1146, Feb. 2012. W. U. Bajwa, “Geometry of random [T]{}oeplitz-block sensing matrices: Bounds and implications for sparse signal processing,” in *Proc. SPIE Defense, Security, and Sensing Conf. Compressive Sensing*, Baltimore, MD, Apr. 2012, pp. 1–7. D. Manolakis, C. Siracusa, and G. Shaw, “Hyperspectral subpixel target detection using the linear mixing model,” *[IEEE]{} Trans. Geoscience Remote Sens.*, vol. 39, no. 7, pp. 1392–1409, Jul. 2001. S. M. Kay, *Fundamentals of Statistical Signal Processing: Detection Theory*.1em plus 0.5em minus 0.4emUpper Saddle River, NJ: Prentice Hall, 1998. A. Farcomeni, “A review of modern multiple hypothesis testing, with particular attention to the false discovery proportion,” *Statistical Methods in Medical Research*, vol. 17, no. 4, pp. 347–388, Aug. 2008. Y. Benjamini and Y. Hochberg, “Controlling the false discovery rate: [A]{} practical and powerful approach to multiple testing,” *J. Roy. Statist. Soc. Ser. B*, vol. 57, no. 1, pp. 289–300, 1995. Y. Benjamini, A. M. Krieger, and D. Yekutieli, “Adaptive linear step-up procedures that control the false discovery rate,” *Biometrika*, vol. 93, no. 3, pp. 491–507, 2006. Z. Drmač, “On principal angles between subspaces of [E]{}uclidean space,” *SIAM J. Matrix Analy. App.*, vol. 22, no. 1, pp. 173–194, 2000. R. Calderbank, A. Thompson, and Y. Xie, “On block coherence of frames,” 2014, in review. \[Online\]. Available: <arXiv:1307.7544v4> A. P. Schaum, “Spectral subspace matched filtering,” in *Proc. SPIE 4381, Algorithms for Multispectral, Hyperspectral, and Ultraspectral Imagery VII*, Orlando, FL, Apr. 2001, pp. 1–17. B. Laurent and P. Massart, “Adaptive estimation of a quadratic functional by model selection,” *Ann. Statist.*, vol. 28, no. 5, pp. 1302–1338, Oct. 2000. C. McDiarmid, “On the method of bounded differences,” in *Surveys in Combinatorics*, J. Siemons, Ed.1em plus 0.5em minus 0.4em Cambridge University Press, 1989, pp. 148–188. R. Motwani and P. Raghavan, *Randomized Algorithms*.1em plus 0.5em minus 0.4emNew York, NY: Cambridge University Press, 1995. M. Donahue, C. Darken, L. Gurvits, and E. Sontag, “Rates of convex approximation in non-[H]{}ilbert spaces,” in *Constructive Approximation*.1em plus 0.5em minus 0.4emNew York, NY: Springer, Jun. 1997, vol. 13, no. 2, pp. 187–220. P. W. H. Lemmens and J. J. Seidel, “Equi-isoclinic subspaces of [E]{}uclidean spaces,” *Indagationes Mathematicae (Proceedings)*, vol. 76, no. 2, pp. 98–107, 1973. F. Mezzadri, “How to generate random matrices from the classical compact groups,” *Notices of the AMS*, vol. 54, no. 5, pp. 592–604, May 2007. A. Naor, “On the [B]{}anach-space-valued [A]{}zuma inequality and small set isoperimetry of [A]{}lon–[R]{}oichman graphs,” *Combinatorics, Probability and Computing*, vol. 21, no. 04, pp. 623–634, Jul. 2012. [^1]: Preliminary versions of some of the results reported in this paper were presented at the $50$th Annual Allerton Conference on Communication, Control, and Computing, Monticello, IL, Oct. 1–5, 2012 [@Bajwa.Mixon.Conf2012]. WUB is with the Department of Electrical and Computer Engineering, Rutgers, The State University of New Jersey, Piscataway, NJ 08854 (Email: [waheed.bajwa@rutgers.edu]{}). DGM is with the Department of Mathematics and Statistics, Air Force Institute of Technology, Dayton, OH 45433 (Email: [dustin.mixon@afit.edu]{}). [^2]: The research of WUB is supported in part by the National Science Foundation under grant CCF-1218942 and by the Army Research Office under grant W911NF-14-1-0295. The research of DGM is supported in part by the National Science Foundation under grant DMS-1321779. [^3]: Note that all results presented in this paper can be extended in a straightforward manner to the case of a complex Hilbert space. [^4]: The other commonly invoked assumption of orthogonal subspaces is of course impossible in the ${D}/{d}\ll {N}$ setting. [^5]: We refer the reader to our preliminary work [@Bajwa.Mixon.Conf2012] and a recent work [@Calderbank.etal.unpublished2014] for a related concept of *average group/block coherence*. [^6]: In here, we are suppressing the dependence of $\epsilon_{m,1}$ and $\epsilon_{m,2}$ on the subspace index $k$ for ease of notation. [^7]: We are once again suppressing the dependence of $\tilde{\epsilon}_{m,1}$ and $\tilde{\epsilon}_{m,2}$ on the subspace index for ease of notation.

--- abstract: 'We describe a measurement of the angular power spectrum of anisotropies in the Cosmic Microwave Background (CMB) from $0.3^{\circ}$ to $\sim 10^{\circ}$ from the North American test flight of the [[Boomerang]{} ]{}experiment. [[Boomerang]{} ]{}is a balloon-borne telescope with a bolometric receiver designed to map CMB anisotropies on a Long Duration Balloon flight. During a 6-hour test flight of a prototype system in 1997, we mapped $> 200$ square degrees at high galactic latitudes in two bands centered at 90 and 150 GHz with a resolution of 26 and 16.6 arcmin FWHM respectively. Analysis of the maps gives a power spectrum with a peak at angular scales of $\sim 1$ degree with an amplitude $\sim 70 \mu$K$_{CMB}$.' author: - 'P.D. Mauskopf, P.A.R. Ade, P. de Bernardis, J.J. Bock, J. Borrill, A. Boscaleri, B.P. Crill, G. DeGasperis, G. De Troia, P. Farese, P. G. Ferreira, K. Ganga, M. Giacometti, S. Hanany, V.V. Hristov, A. Iacoangeli, A. H. Jaffe, A.E. Lange, A. T. Lee, S. Masi, A. Melchiorri, F. Melchiorri, L. Miglio, T. Montroy, C.B. Netterfield, E. Pascale, F. Piacentini, P. L. Richards, G. Romeo, J.E. Ruhl, E. Scannapieco, F. Scaramuzzi, R. Stompor and N. Vittorio' title: | Measurement of a Peak in the Cosmic Microwave Background\ Power Spectrum from the North American test flight of [[Boomerang]{}]{} --- Introduction ============ Measurements of Cosmic Microwave Background (CMB) anisotropies have the potential to reveal many of the fundamental properties of the universe (e.g. see [@review] and references therein). Since the measurement of the large scale anisotropies by COBE-DMR ([@cobe2], [@bennett], etc.), many ground-based and balloon-borne experiments have continued to develop the technology necessary to produce accurate measurements of CMB structure on smaller angular scales. Recent results from these experiments have shown evidence for the existence of a peak in the power spectrum of CMB fluctuations at a multipole, $\ell \sim 200$ ([@cbn], [@QMAP], [@MSAMI], [@TOCO97], [@pythonv], [@MAT]). In this paper, we present results from the test flight of [[Boomerang]{} ]{}that constrain the position and amplitude of this peak. In fact, the results from this data set alone can be used to constrain cosmological models and provide evidence for a flat universe, $\Omega = \Omega_M + \Omega_{\Lambda} \simeq 1$ (see the companion paper, Melchiorri, et al. 1999). [[Boomerang]{} ]{}is a millimeter-wave telescope and receiver system designed for a Long Duration Balloon (LDB) flight from Antarctica of up to two weeks duration. The data presented here are from a prerequisite test flight of the payload in August, 1997, which lasted 6 hours at float altitude. A prototype of the LDB focal plane was used to test the experimental strategy of making a map and measuring the angular power spectrum of CMB anisotropy with a slowly-scanned telescope and total-power bolometric receiver. Instrumentation =============== [[Boomerang]{} ]{}employs bolometric detectors cooled to 0.3 K, coupled to an off-axis telescope and operated as total power radiometers. Electronic modulation and synchronous demodulation of the bolometer signals provide low frequency stability of the detector system. The optical signal is modulated by slowly scanning the telescope ($\sim 1$ deg/s) in azimuth. A description of the instrument can be found in [@b99tech]. The [[Boomerang]{} ]{}telescope consists of three off-axis, aluminum mirrors: an ambient temperature 1.3 m primary mirror and two smaller ($15$ cm) 2 K reimaging mirrors. The telescope is shielded from ground radiation by extensive low emissivity baffles. Incoming radiation is reflected by the primary mirror into the entrance window of the cryostat and reimaged by the 2 K optics onto the 0.3 K focal plane. The tertiary mirror is positioned at an image of the primary mirror and forms a throughput-limiting Lyot stop illuminating the central 85 cm of the primary mirror. The inside of the 2 K optics box has a thick coating of absorbing material ([@bockblack]) to intercept stray light. A calibration lamp mounted in the Lyot stop of the optical system is pulsed for $\sim$ 0.5 s every 15 minutes to produce a stable, high signal-to-noise ratio (${\lower.5ex\hbox{$\; \buildrel > \over \sim \;$}}1000$) calibration transfer. For the test flight, the focal plane contained silicon nitride micromesh bolometers ([@bolo1], [@bolo2]) fed by multi-mode feed horns with channels at 90 and 150 GHz. Fourier transform spectroscopy of each channel determined the average band centers and effective band widths for sources with a spectrum of CMB anisotropy (Table 1). A long duration $^3He$ cryostat ([@cryo1], [@cryo2]) maintained the focal plane at $0.285 \pm 0.005$ K during the flight. The bolometers were biased with an AC current and the signals were demodulated, amplified and passed through a 4-pole Butterworth low-pass filter with a cut-off frequency of $20$ Hz for the 150 GHz channels and $10$ Hz for the 90 GHz channels. The bolometer DC levels were removed with a single-pole, high-pass filter with a cut-on frequency of $16$ mHz and re-amplified before digital sampling at 62.5 Hz with 16 bit resolution. The attitude control system included a low-noise two-axis flux-gate magnetometer, a CCD star camera and three orthogonal (azimuth, pitch, roll) rate gyroscopes. The azimuth gyroscope drove the azimuth feedback loop, which actuated two torque motors, one of which drove a flywheel while the other transferred angular momentum to the balloon flight line. The gyroscope was referenced every scan to the magnetometer azimuth to remove long time scale drifts. Pendulations of the payload were reduced by a oil-filled damper mounted near the main bearing and sensed by the pitch and roll gyroscopes outside of the control loop. The CCD frame had $480 \times 512$ pixels, with a resolution of 0.81 arcmin/pixel (azimuth) and 0.65 arcmin/pixel (elevation). The limiting magnitude of stars used for pointing reconstruction was $m_V \sim 5$. The coordinates of the two brightest stars in each CCD frame were identified by on-board software and recorded at 5 Hz, for post-flight attitude reconstruction. The primary mirror and the receiver were mounted on an inner frame which could be rotated to position the elevation boresight of the telescope between $35^{\circ} - 55^{\circ}$. During observations, the inner frame was held fixed while the entire gondola slowly scanned in azimuth. The scan speed was set to the maximum allowed by the beam size and bolometer time constants. The window functions for the 90 and 150 GHz channels in this flight were rolled off at $\ell {\lower.5ex\hbox{$\; \buildrel > \over \sim \;$}}300$ and $500$, respectively, by the 26 and 16.5 arcmin beams and were limited to $\ell {\lower.5ex\hbox{$\; \buildrel > \over \sim \;$}}20$ by the system 1/f noise. Observations ============ The instrument was launched at 00:25 GMT on August 30, 1997 from the National Scientific Balloon Facility in Palestine, Texas. Observations started at 3:50 GMT and continued until the flight was terminated at 9:50 GMT. Sunset was at 01:00 GMT and the moon rose at 10:42 GMT. The average altitude at float was 38.5 km. We observed in three modes: $\sim 4.5$ hours of CMB scans, 40 minutes of calibration scans on Jupiter, and four ten-minute periods of full-sky rotations of the gondola. The CMB observations consisted of smoothed triangle wave scans in azimuth with a peak-to-peak amplitude of 40 degrees centered on due South at an elevation angle of $45$ degrees. The scan rate was $\sim 2.1$ deg/s in azimuth ($1.4$ deg/s on the sky) in the linear portion of the scan (63$\%$ of the period). These scans, combined with the earth’s rotation, covered a wide strip of sky, from $-73^{\circ}< {\rm RA} < 23^{\circ}$ and $-20^{\circ} < {\rm DEC} < -16^{\circ}$. Data reduction ============== We obtain a first order reconstruction of the attitude from the elevation encoder and magnetometer data. Using this, the stars recorded by the CCD camera are identified. The data from the gyroscopes are used to interpolate between star identifications. The offset between the microwave telescope and the star camera is measured from the observation of Jupiter. The attitude solution obtained in this way is accurate to $\sim$ 1 arcmin rms. The results presented in this paper are primarily from the highest sensitivity 150 GHz channel. Data from a 90 GHz channel are presented as a systematic check. The data at float consist of $\sim$ 1.4 million samples for each bolometer. These data are searched for large ($>5 \sigma$) deviations such as cosmic ray events and radio frequency interference and flagged accordingly. Flags are also set for events in the auxiliary data such as calibration lamp signals and elevation changes, as well as for the different scan modes. Unflagged data, used for the CMB analysis, are 54$\%$ of the total at 150 GHz and 60$\%$ of the total at 90 GHz. We deconvolve the transfer functions of the readout electronics and the bolometer thermal response from the data, and apply a flat phase numeric filter to reduce high frequency noise and slow drifts. The time constants of the bolometers given in Table 1 are measured in flight from the response to cosmic rays and to fast (18 deg/s) scans over Jupiter. [cccccc]{} $\nu_0$ & $\Delta \nu$ & FWHM & $\Omega$ & $\tau_b$ & NET$_{\rm CMB}$\ (GHz) & (GHz) & ($'$) & $10^{-5}$ sr & (ms) & ($\mu$K$/\sqrt{\rm Hz}$)\ 96 & 33 & 26 & $6.47 \pm 0.27$ & $71 \pm 8$ & $400$\ 153 & 42 & 16.5 & $2.63 \pm 0.10$ & $83 \pm 12$ & $250$\ The primary calibration is obtained from scans of the telescope over Jupiter, which were performed from 3:32 U.T. to 3:53 U.T. (1.2 deg/s azimuth scans). Jupiter was also re-observed later during regular CMB scans (from 5:58 to 6:18 U.T.) and during full sky rotations. Jupiter was rising during the first observation (from an elevation of 36.9$^{\circ}$ to 38.2$^{\circ}$) and setting during the second one (from an elevation of 40.1$^{\circ}$ to 39.0$^{\circ}$). The deconvolved and filtered data from these scans are triangle-interpolated on a regular grid centered on the optical position of Jupiter to make a beam map. The beams are symmetric with minor and major axes equivalent within $5$%. Solid angles given in Table 1 are computed by integration of the interpolated data. The beam map is used to derive the window function, $W_{\ell}$, and an overall normalization. We use a brightness temperature of $T_{\rm eff} = 173$ K for both the 90 and 150 GHz bands ([@Ulich], [@Griffin], [@Cole]) and assume an uncertainty of $5\%$ ([@MSAM]). We also assign a $5\%$ error in the conversion from brightness temperature to CMB temperature due to statistical noise in the measurements of the band pass. Errors in the determination of the receiver transfer function are largest at the highest temporal frequencies where signals are attenuated by the bolometer time constants. These errors affect the calibration of the window function and the CMB power spectrum measurements at the largest multipoles. These should cancel because the beam maps of Jupiter are made from scans at the same scan speed as the CMB scans. In simulations, a 15$\%$ error in the determination of the bolometer time constant produces a maximum error $<1.5\%$ in the normalization and ${\lower.5ex\hbox{$\; \buildrel < \over \sim \;$}}6 \%$ in the value of $W_{\ell}$ at $\ell < 300$ near scan turnarounds where the scan speed was only 1 deg/s. The final precision of the calibration including errors in the temperature of Jupiter, $T_{\rm eff}$, the transfer function used for deconvolution, the measurement of the filter pass bands and the beam solid angle is 8.1$\%$ and 8.5$\%$ at 150 GHz and 90 GHz, respectively. Signals from the internal calibration lamp are used to correct for a slow linear drift in detector responsivity during the flight, mainly due to variation (0.29-0.28 K) in the base temperature of the fridge. The change in response to the calibration lamp from the beginning to the end of the flight is $+9\%, -1\%$ at 90 GHz, and $+8\%, -1.5\%$ at 150 GHz. Map Making and Power Spectrum Estimates ======================================= Current and future CMB missions require new methods of analysis able to incorporate the effects of correlated instrument noise and new implementations capable of processing large data sets (Bond et al. 1999). The analysis of the [[Boomerang]{} ]{}test flight data provides a test of these methods on a moderate size ($\sim 25,000$ pixels) data set. The calibrated time stream data are processed to produce a pixelized map, and from this a measurement of the angular power spectrum, using the MADCAP software package of Borrill (1999a, 1999b) (see http://cfpa.berkeley.edu/$\sim$borrill/cmb/madcap.html) on the Cray T3E-900 at NERSC and the Cray T3E-1200 at CINECA. Noise correlation functions are estimated from the time stream under the assumption that in this domain the signal is small compared to the noise. For the 150 GHz channel, the maximum likelihood map with 23,561, 1/3-beam sized ($6.9'$) pixels is calculated from the noise correlation function, the bolometer signal (excluding flagged data and all data within 2 degrees of Jupiter) and the pointing information using the procedures described in Wright (1996), Tegmark (1997) and Ferreira & Jaffe (1999). The maximum likelihood power spectrum is estimated from the map using a Newton-Raphson iterative maximization of the $C_\ell$ likelihood function following Bond, Jaffe & Knox (1998) (BJK98) and Tegmark (1998). The power spectrum analysis requires approximately 12 hours on sixty-four T3E-900 processors. [ ]{} [F[IG]{}. 1.— Power spectrum of the [[Boomerang]{} ]{}150 GHz map with 6 arcminute pixelization. The solid curve is a marginally closed model with ($\Omega_b,\Omega_M,\Omega_{\Lambda},n_S,h) = (0.05, 0.26, 0.75, 0.95, 0.7)$. The dotted curve is standard CDM with $(0.05, 0.95, 0.0, 1.0, 0.65)$. Dashed curves are open and closed models with fixed $\Omega_{\rm total} = 0.66$ and $1.55$, respectively (see the companion paper, [@parameters]). \[fig:powspec\]]{} Due to finite sky coverage, individual $C_\ell$ values are not independent, and we estimate the power in eight bins over $25<\ell<1125$, binning more finely at low $\ell$ than high. In each bin we calculate the maximum likelihood amplitude of a flat power spectrum in that bin, i.e. $\ell(\ell+1)C_{\ell} = $constant (Table 2). Given our insensitivity to low $\ell$ signal, and in order to ameliorate the effects of low-frequency noise (which translates into low-$\ell$ structure for this scanning strategy), we marginalize over power at $\ell<25$ and diagonalize the bin correlation matrix using a variant of techniques discussed in BJK98. Bin to bin signal correlations are small but non-negligible, with each top-hat bin anticorrelated with its nearest neighbors at approximately the 10% level. We calculate likelihood functions for each bin using the offset log-normal distribution model of BJK98. We quote errors for the power spectrum from $\pm 1 \sigma$ errors assuming simple Gaussian likelihood functions and from 68% confidence intervals for the full likelihood functions. Plots of the likelihood functions are presented in the companion paper (Melchiorri, et al. 1999) and full spectral data including information on the shape of the likelihood function (Bond, Jaffe & Knox 1998b) and appropriate window and filter functions (Knox 1999) will be available at the [[Boomerang]{} ]{}web site (http://boom.physics.ucsb.edu or http://oberon.roma1.infn.it/boom). In Figure 1, we show the calculated power spectrum for the 150 GHz channel 6 arcminute pixel map. Systematics checks and Foregrounds ================================== The scan strategy and analysis allow for a variety of checks for systematic effects in the data. For all systematic tests we produce small ($\sim 4000$ pixel) maps with $16^{\prime}$ pixels which can be generated and analyzed quickly ($\sim$ 30 minutes on four T3E-900 processors). Monte-Carlo simulations show the effects of the 16$'$ pixelization are negligible for values of $\ell < 400$. Compared to the level of statistical noise, we find that the derived power spectrum from the 150 GHz channel is insensitive to (i) the size of the excluded region around Jupiter, (ii) the choice of spectral shape within the multipole bins, and (iii) the method of marginalizing over low $\ell$ signals. We also analyze maps made from combinations of data that we expect to produce null power spectra in the absence of spurious sources of noise power: (i) data from a dark detector and (ii) the difference between left and right-going scans at 150 GHz. In each case (Table 3), we find a power spectrum consistent with zero. [lcccc]{} $[\ell_{\rm min}, \ell_{\rm max}]$ & 150 GHz & 90 GHz & Dark & 150 (L-R)/2\ & ($\mu$K$^2/100$) & ($\mu$K$^2/100$) & ($\mu$K$^2/100$) & ($\mu$K$^2/100$)\ $\left[25,75\right]$ & $10\pm7$ & $16\pm12$ & $-1\pm1$ & $4\pm7$\ $\left[76,125\right]$ & $23\pm9$ & $25\pm13$ & $1\pm1$ & $-9\pm4$\ $\left[126,175\right]$ & $46\pm13$ & $50\pm20$ & $1\pm2$ & $-7\pm9$\ $\left[176,225\right]$ & $50\pm14$ & $53\pm25$ & $6\pm4$ & $12\pm17$\ $\left[226,275\right]$ & $29\pm13$ & $21\pm29$ & $-1\pm5$ & $-22\pm18$\ $\left[276,325\right]$ & $23\pm15$ & $5\pm40$ & $2\pm7$ & $-30\pm26$\ $\left[326,475\right]$ & $8\pm12$ & $3\pm48$ & $7\pm5$ & $-78\pm21$\ $\left[476,1125\right]$ & $2\pm27$ & $2\pm253$ & $9\pm12$ & $-169\pm52$\ $\left[25,475\right]$ & $31\pm5$ & $25\pm7$ & $-1\pm1$ & $-4\pm2$\ The sky strip observed is extended in Galactic latitude from $b \sim 15^\circ $ to $b \sim 80^\circ $. From IRAS/DIRBE map extrapolation (Schlegel et al. 1998) dust emission is not expected to produce significant contamination in this region of the sky. We find an amplitude for a flat power spectrum from $25<\ell<400$ of $\ell (\ell+1)C_{\ell}/2\pi = (3580\pm706) \mu$K$^2$ and $(2982\pm702) \mu$K$^2$ from the half of the map near the galactic plane ($15^{\circ}<b<45^{\circ}$) and the higher-latitude half ($45^{\circ}<b<80^{\circ}$), respectively. Because of sky rotation, this also corresponds to separately analyzing the data from the first half and second half of the flight. Jupiter is more than 1000 times brighter than degree-scale anisotropy in the CMB and is in the middle of the lowest latitude half of the map. We test for sidelobe contamination by varying the size of the circular avoidance zone around Jupiter from $\phi = 0.5-3$ degrees for the CMB maps, and find no significant change in the power spectrum as long as we remove data within $\sim 1$ degree. Discussion and Conclusions ========================== Data obtained in $\sim 4.5$ hours of CMB scans during the [[Boomerang]{} ]{}test flight have been analyzed to produce maps of the millimeter-wave sky at 90 and 150 GHz, with 26 and 16 arcmin FWHM resolution, respectively. The instrument was calibrated using observations of Jupiter. The experiment employed a new scan strategy and detector technology designed to give maximum coverage of angular scales. The data analysis implements new techniques for making maximum likelihood maps from low signal-to-noise time stream data over large numbers of pixels. The power spectrum $C_\ell$ obtained from the maps extends from $\ell \sim 50$ to $\ell \sim 800$ and shows a peak at $\ell \sim 200$. Since the [[Boomerang]{}/NA]{}test flight in August, 1997, we have obtained $> 200$ hours of data from the LDB flight of [[Boomerang]{} ]{}([[Boomerang]{} ]{}/LDB), carried out in Antarctica at the end of 1998. The focal plane for the LDB flight contained 2, 6, and 3 detectors at 90, 150, and 240 GHz, each with better sensitivity to CMB, faster time constants, and lower 1/f noise than the best channel in the [[Boomerang]{}/NA]{}flight, as well as 3 detectors at 400 GHz to provide a monitor of interstellar dust and atmospheric emission. The results from this flight will be reported elsewhere. We thank the staff of the NASA NSBF for their excellent support of the test flight. We are also grateful for the help and support of a large group of students and scientists (a list of contributors to the [[Boomerang]{} ]{}experiment can be found at http://astro.caltech.edu/$\sim$bpc/boom98.html). The [[Boomerang]{} ]{}program has been supported by Programma Nazionale Ricerche in Antartide, Agenzia Spaziale Italiana and University of Rome La Sapienza in Italy; by NASA grant numbers NAG5-4081, NAG5-4455, NAG5-6552, NAG53941, by the NSF Science & Technology Center for Particle Astrophysics grant number SA1477-22311NM under AST-9120005, by NSF grant 9872979, and by the NSF Office of Polar Programs grant number OPP-9729121 in the USA; and by PPARC in UK. This research also used resources of the National Energy Research Scientific Computing Center, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC03-76SF00098. Additional computational support for the data analysis has been provided by CINECA/Bologna. Bennett, C. L. et al. 1996, , 464, 1 Bock, J. J., et al. 1998, ESA proceedings Bock, J. J. 1994, PhD. Thesis, Berkeley, CA Bond, J. R., Jaffe A. H., & Knox L. 1998, Phys. Rev. D, 57, 2117 Bond, J. R., Jaffe, A. H., & Knox, L. 1998, , in press, astro-ph/9808264 Bond D., Crittenden, Jaffe A., Knox L., 1999, Computing in Science and Engineering, 1, 21, astro-ph/9811148 Borrill, J. 1999b, [*The Challenge of Data Analysis for Future CMB Observations*]{}, in “Proceedings of the 3K Cosmology: EC-TMR Conference”, Rome, Italy edited by L. Maiani, F. Melchiorri, and N.Vittorio AIP Conference Proceedings, astro-ph/9903204 Borrill, J. 1999b, [*MADCAP: The Microwave Anisotropy Dataset Computation Analysis Package*]{}, in “Proceedings of the Fifth European SGI/Cray MPP Workshop” Borrill, J. 1999a, Phys Rev D, 59, 7302, astro-ph/9712121 Coble, K., et al. 1999, , submitted, astro-ph/9902125 de Oliveira-Costa A. et al., 1998, , 509, L77 Ferreira, P. G. & Jaffe A. H. 1999, MNRAS, submitted astro-ph/9909250 Goldin, A. B., et al. 1997, , 488, 161 Griffin, M. 1986, Icarus, 65, 244 Hildebrand R. 1985, Icarus, 64, 64 Knox, 1999, Phys. Rev. D, submitted, astro-ph/9902046 Kamionkowski, M. & Kosowsky, A., 1999, To appear in [*Annu. Rev. Nucl. Part. Sci.*]{}, astro-ph/9904108 Masi, S. et al. 1998, Cryogenics, 38, 319 Masi, S. et al. 1999, [*BOOMERanG: a scanning telescope for 10 arcminutes resolution CMB maps*]{}, in “Proceedings of the 3K Cosmology : EC-TMR Conference”, Rome, Italy edited by L. Maiani, F. Melchiorri, and N.Vittorio AIP Conference Proceedings 237 Masi S. et al. 1999, Cryogenics, 39, 217 Mauskopf P., et al., Applied Optics, 36, 765, 1997 Melchiorri A., et al. 1999, , submitted, astro-ph9911445 Miller A.D. et al. 1999, , submitted, astro-ph/9906421 Netterfield, C. B., et. al. 1997, , 474, 47 Ostriker, J.P. & Steinhardt P. J. 1995, Nature 377, 600 Schlegel D.J., Douglas P.F., Davis M. 1998, , 500, 525 Torbet, E. et al. 1999, , 521, 79 Turner, M.S. & White M. 1997, Phys. Rev. D., 56, R4439 Ulich B.L. 1981, AJ, 86, 1619 Wilson G.W. et al., preprint 1999, astro-ph/9902047 Tegmark M. 1998, [*CMB and LSS Power Spectrum Analysis*]{}, in “Astrophysics and Algorithms: a DIMACS Workshop on Massive Astronomical Data Sets”, 17 Tegmark M. 1997, , 480, 87 Wright, E. L., Smooth, G. F., Bennett, C. L., Lubin, P. M. 1994, , 436, 443 Wright, E. 1996, astro-ph/9612006

--- abstract: | Let $p\neq2$ be a prime. We show a technique based on local class field theory and on the expansions of certain resultants which allows to recover very easily Lbekkouri’s characterization of Eisenstein polynomials generating cyclic wild extensions of degree $p^2$ over ${\mathbb{Q}}_p$, and to extend it to the case of the base field $K$ being an unramified extension of ${\mathbb{Q}}_p$. Furthermore, when a polynomial satisfies only some of the stated conditions, we show that the first unsatisfied condition gives information about the Galois group of the normal closure. This permits to give a complete classification of Eisenstein polynomials of degree $p^2$ whose splitting field is a $p$-extension, providing a full description of the Galois group and its higher ramification subgroups. We then apply the same methods to give a characterization of Eisenstein polynomials of degree $p^3$ generating a cyclic extension. In the last section we deduce a combinatorial interpretation of the monomial symmetric function evaluated in the roots of the unity which appear in certain expansions. address: 'Scuola Normale Superiore di Pisa - Piazza dei Cavalieri, 7 - 56126 Pisa' author: - Maurizio Monge bibliography: - 'biblio.bib' title: 'A characterization of Eisenstein polynomials generating cyclic extensions of degree $p^2$ and $p^3$ over an unramified ${\mathfrak{p}}$-adic field' --- Introduction ============ In this paper we explore the techniques which can be used to deduce necessary and sufficient conditions for a polynomial to have a certain Galois group as group of the splitting field over a ${\mathfrak{p}}$-adic field. Lbekkouri gave in [@Lbekkouri2009a] congruence conditions for Eisenstein polynomials of degree $p^2$ with coefficients in the rational ${\mathfrak{p}}$-adic field ${\mathbb{Q}}_p$, which are satisfied if and only if the generated extension is Galois. Since the multiplicative group $U_{1,{\mathbb{Q}}_p}$ of $1$-units of ${\mathbb{Q}}_p$ has rank $1$ as ${\mathbb{Z}}_p$-module and in particular ${\nicefrac{U_{1,{\mathbb{Q}}_p}({\mathbb{Q}}_p^\times)^p}{({\mathbb{Q}}_p^\times)^p}}\cong{\nicefrac{{\mathbb{Z}}}{p{\mathbb{Z}}}}$, we have by local class field theory that every Galois totally ramified extension of degree $p^2$ over ${\mathbb{Q}}_p$ is cyclic, and consequently over ${\mathbb{Q}}_p$ the problem is reduced to finding conditions for Eisenstein polynomials of degree $p^2$ to generate a cyclic extension. If the base field $K$ is a proper extension of ${\mathbb{Q}}_p$ this is no longer true, so the restriction of considering polynomials which generate cyclic extensions has to be added explicitly. If $K$ is ramified over ${\mathbb{Q}}_p$ even the characterization of the possible upper ramification jumps is a non-trivial problem (see [@Maus1971; @Miki1981]) and the problem seems to be very difficult for a number of other reasons, so we will only consider fields $K$ which are finite unramified extensions over ${\mathbb{Q}}_p$, with residue degree $f=f(K/{\mathbb{Q}}_p)=[K:{\mathbb{Q}}_p]$. In this setting the problem is still tractable without being a trivial generalization of the case over ${\mathbb{Q}}_p$, and we will show a technique which allows to handle very easily the case of degree $p^2$. During the proof the Artin-Hasse exponential function comes into play, and we use it to clarify the connection between the image of the norm map and the coefficients of the Eisenstein polynomial. While some condition are necessary to force the splitting field to be a $p$-extension, the remaining conditions can be tested in order, and the first which fails gives information on the Galois group of the splitting field. Taking into account another family of polynomial which can never provide a cyclic extension of degree $p^2$, we give a full classification of the polynomials of degree $p^2$ whose normal closure is a $p$-extension, providing a complete description of the Galois group of the normal closure with its ramification filtration, see [@caputo2007classification] for an abstract classification of all such extensions when the base field is ${\mathbb{Q}}_p$. We then show how the same methods apply to characterize Eisenstein polynomials of degree $p^3$ generating a cyclic extension. This case a substantially more complicated is obtained, but the strategy used in degree $p^2$ can still be applied in a relatively straightforward way. We plan to show in a forthcoming paper the how the techniques used here can be extended to give a characterization of polynomials generating all remaining groups of order $p^3$, including the non-abelian ones. In the last section we give a combinatorial interpretation of certain sums of roots of the unity which appear during the proof, it is actually much more than needed but it has some interest on its own. Acknowledgements ---------------- We feel indebted with Luca Caputo and François Laubie for some fruitful discussions and for the motivation to try to recover and generalize Lbekkouri’s result. We would also like to thank Philippe Cassou-Noguès, Ilaria Del Corso, Roberto Dvornicich and Boas Erez for various discussions on this topic, and the Institut de Mathématiques de Bordeaux for hospitality while conceiving this work. Preliminaries ============= Let $K$ be a ${\mathfrak{p}}$-adic field, we denote by ${\mathcal{O}}_K$ the ring of integers, by ${\mathfrak{p}}_K$ the maximal ideal, by $U_{i,K}$ the $i$-th units group, by ${\kappa}_K$ the residue field ${\nicefrac{{\mathcal{O}}_K}{{\mathfrak{p}}_K}}$, and for $x\in{}{\mathcal{O}}_K$ let $\bar{x}$ be its image in ${\kappa}_K$. We will denote with $[K^\times]_K$ the group of $p$-th power classes ${\nicefrac{K^\times}{(K^\times)^p}}$. For integers $a,b$, it will be convenient to denote by ${\llbracket a,b\rrbracket}$ the set of integers $a\leq{}i\leq{}b$ such that $(i,p)=1$. We start computing modulo which power of $p$ we must consider the coefficients of an Eisenstein polynomial (this computation is very well known and we repeat it only to keep the paper self-contained, see [@Krasner1962] for more details): let $f(X)=\sum_{i=0}^nf_{n-i}X^i$ and $g(X)=\sum_{i=0}^n g_{n-i}X^i$ be Eisenstein polynomials of degree $n$ say, $\rho$ a root of $g$, $\pi=\pi_1,\pi_2,\dots$ the roots of $f$ with $\pi$ the most near to $\rho$, and put $L=K(\pi)$. Let $v$ be the biggest lower ramification jump and ${\mathscr{D}}_f=f'(\pi)$ be the different, if $$\left|(f_{n-i}-g_{n-i})\pi^i\right| < \left|\pi^{v+1}{\mathscr{D}}_f\right|$$ then being $$f(\rho) = f(\rho)-g(\rho) = \sum_{i=0}^n \left(f_{n-i}-g_{n-i}\right)\rho^i$$ we obtain $|f(\rho)|<|\pi^v{\mathscr{D}}_f|$. We have $$\left|(\rho-\pi)\cdot \prod_{i=2}^n(\pi-\pi_i)\right| \leq \left|\prod_{i=1}^n(\rho-\pi_i)\right| < \left|\pi^{v+1}{\mathscr{D}}_f\right|,$$ in fact $|\pi-\pi_i|\leq|\rho-\pi_i|$ for $i\geq2$ or we would contradict the choice of $\pi=\pi_1$. Consequently $|\rho-\pi|<|\pi^{v+1}|$ which is equal to the minimum of the $|\pi-\pi_i|$, and hence $K(\rho)\subseteq{}K(\pi)$ by Krasner’s lemma, and $K(\rho)=K(\pi)$ having the same degree. Let now $K$ be unramified over ${\mathbb{Q}}_p$, then $U_{1,K}^{p^i}=U_{i+1,K}$, and consequently by local class field theory the upper ramification jumps of a cyclic $p$-extension are $1,2,3,\dots$, and the lower ramification jumps are $1,p+1,p^2+p+1,\dots$. For an extension of degree $p^k$ with lower ramification jumps $t_0\leq{}t_1\leq\dots\leq{}t_{k-1}$ we can compute $v_L({\mathscr{D}}_{L/K})$ as $\sum_{i=1}^k (p^i-p^{i-1})t_{k-i}$, which for a cyclic $L/K$ of degree $p^2$ or $p^3$ is $3p^2-p-2$ (resp. $4p^3-p^2-p-2$), while $v_L(\pi^{v+1}{\mathscr{D}}_{L/K})$ is respectively $3p^2=v_L(p^3)$ and $4p^3=v_L(p^4)$. Hence we obtain the condition on the precision of the coefficients, which we state in a proposition for convenience: \[prop\_coeff\] Let $L/K$ be a totally ramified cyclic extension of degree $n=p^2$ (resp. $n=p^3$) determined by the Eisenstein polynomial $f(X)=\sum_{i=0}^nf_{n-i}X^n$. Then the lower ramification jumps are $1,p+1$ (resp. $1,p+1,p^2+p+1$), $v_L({\mathscr{D}}_{L/K})$ is equal to $3p^2-p-2$ (resp. is $4p^3-p^2-p-2$), and the extension is uniquely determined by the classes of $f_n\imod{p^4}$ and $f_i\imod{p^3}$ for $0\leq{}i<n$ (resp. by the classes of $f_n\imod{p^5}$ and $f_i\imod{p^4}$ for $0\leq{}i<n$, for $n=p^3$). Additive polynomials -------------------- We will need a few facts about additive polynomials, and in particular some formulas to express in terms of the coefficients the condition that and additive polynomial has range contained in the range of some other additive polynomial. We resume what we need in the following \[prop\_add\] Let $A(Y)=a_pY^p+a_1Y$ be an additive polynomial in ${\kappa}_K[Y]$ such that $A'(0)\neq0$ and all the roots of $A(Y)$ are in ${\kappa}_K$, and let $B(Y)=b_pY^p+b_1Y$, $C(Y)=c_{p^2}Y^{p^2}+c_pY^p+c_1Y$ and $D(Y)=d_{p^3}Y^{p^3}+d_{p^2}Y^{p^2}+d_pY^p+d_1Y$ be any three other additive polynomials in ${\kappa}_K[Y]$. Then - $B({\kappa}_K)\subseteq A({\kappa}_K)$ if and only if $b_p=a_p({\nicefrac{b_1}{a_1}})^p$, and in this case $B(Y)$ is equal to $A({\nicefrac{b_1}{a_1}}Y)$, - $C({\kappa}_K)\subseteq A({\kappa}_K)$ if and only if $c_p=a_p({\nicefrac{c_1}{a_1}})^p+a_1({\nicefrac{c_{p^2}}{a_p}})^{{\nicefrac{1}{p}}}$, and in this case $C(Y)$ can be written as $A(\beta{}Y^p+{\nicefrac{c_1}{a_1}}Y)$ with $\beta=({\nicefrac{c_{p^2}}{a_p}})^{{\nicefrac{1}{p}}}$ or equivalently $\beta={\nicefrac{c_p}{a_1}}-{\nicefrac{a_p}{a_1}}({\nicefrac{c_1}{a_1}})^p$. - $D({\kappa}_K)\subseteq A({\kappa}_K)$ if and only if ${\nicefrac{a_1}{a_p}}({\nicefrac{d_{p^3}}{a_p}})^{{\nicefrac{1}{p}}}+({\nicefrac{d_{p}}{a_1}})^p={\nicefrac{d_{p^2}}{a_p}}+ ({\nicefrac{a_p}{a_1}})^p({\nicefrac{d_1}{a_1}})^{p^2}$. Note that being ${\kappa}_K$ finite and hence perfect the map $x\mapsto{}x^p$ is an automorphism, and we just denote by $x\mapsto{}x^{{\nicefrac{1}{p}}}$ the inverse automorphism. Since $A'(0)\neq0$ and all the roots of $A(Y)$ are in ${\kappa}_K$ we have from the theory of additive polynomials (see [@fesenko2002local Chap. 5, §2, Corollary 2.4]) that if $B({\kappa}_K)\subseteq{}A({\kappa}_K)$ then $B(Y)=A(G(Y))$ for some other additive polynomial $G(Y)$ which should be linear considering the degrees, $G(Y)=\alpha{}Y$ say. Consequently it has to be $B(Y)=a_p\alpha^p{}Y^p+a_1\alpha{}Y$, and comparing the coefficients we obtain that $\alpha^p=({\nicefrac{b_1}{a_1}})^p$ and should also be equal to ${\nicefrac{b_p}{a_p}}$. Similarly if $C({\kappa}_K)\subseteq{}A({\kappa}_K)$ we should have $$C(Y)=A(\beta{}Y^p+\alpha{}Y)=a_p\beta^p{}Y^{p^2}+(a_p\alpha^p+a_1\beta)Y^p+a_1\alpha{}Y,$$ and we deduce $\alpha={\nicefrac{c_1}{a_1}}$, $\beta^p={\nicefrac{c_{p^2}}{a_p}}$, and we obtain the condition substituting $\alpha,\beta$ in $c_p=a_p\alpha^p+a_1\beta$. If $D({\kappa}_K)\subseteq{}A({\kappa}_K)$ then $D(Y)$ should be $A(\gamma{}Y^{p^2}+\beta{}Y^p+\alpha{}Y)$ and hence $$a_p\gamma^p{}Y^{p^3}+(a_p\beta^p+a_1\gamma){}Y^{p^2}+(a_p\alpha^p+a_1\beta)Y^p+a_1\alpha{}Y,$$ $\alpha={\nicefrac{d_1}{a_1}}$, $\gamma=({\nicefrac{d_{p^3}}{a_p}})^{{\nicefrac{1}{p}}}$, and $\beta^p$ can be written in two different ways as $${\nicefrac{d_{p^2}}{a_p}}-{\nicefrac{a_1}{a_p}}({\nicefrac{d_{p^3}}{a_p}})^{{\nicefrac{1}{p}}} = ({\nicefrac{d_{p}}{a_1}} - {\nicefrac{a_p}{a_1}}({\nicefrac{d_1}{a_1}})^p)^p.$$ The condition is clearly also sufficient. The following proposition will also be useful, it gives a criterion to verify if the splitting field of an additive polynomial of degree $p^2$ is a $p$-extension (that is, either trivial of cyclic of degree $p$) which is slightly easier to test than the condition itself: \[add\_gal\] Let $A(Y)=Y^{p^2}+aY^p+bY$ be an additive polynomial in ${\kappa}_K[Y]$, than the splitting field is a $p$-extension over ${\kappa}_K$ precisely when $A(Y)$ has a root in ${\kappa}_K^\times$, and $b\in({\kappa}_K^\times)^{p-1}$. If the Galois group is a $p$-group then its orbits have cardinality divisible by $p$ and the action on the roots of $A(Y)$ should clearly have some fixed point other than $0$, $\eta\in{\kappa}_K^\times$ say. If $\beta=\eta^{p-1}$ then the roots of $Y^p-\beta{}Y$ are roots of $A(Y)$, so by [@fesenko2002local Chap. 5, §2, Prop. 2.5] $A(Y)$ is $B(Y^p-\beta{}Y)$ for some additive polynomial $B(Y)$ which has to be monic too, $B(Y)=Y^p-\alpha{}Y$ say. The roots of $B(Y)$ have to be in ${\kappa}_K$ or it, and hence $A(Y)$, would generate an extension of order prime with $p$, and consequently $\alpha$ has to be in $({\kappa}_K^\times)^{p-1}$, and $b=\alpha\beta$ as well. On the other hand if a root $\eta$ is in ${\kappa}_K$ we can write $A(Y)=B(Y^p-\beta{}Y)$ for $\beta=\eta^{p-1}$, and replacing $Y$ by $\eta{}Z$ we can consider $B(\eta^p(Z^p-Z))$, and the extension is obtained as an Artin-Schreier extension over the extension determined by $B(Y)$. Consequently we only need the extension determined by $B(Y)$ to be trivial, which is the case when $b\in({\kappa}_K^\times)^{p-1}$. Sum of roots of the unity ------------------------- Let $\zeta_\ell$ be a primitive $\ell$-th root of the unity for some $\ell$, we define for each tuple $\lambda=(\lambda_1,\lambda_2,\dots,\lambda_r)$ of $r$ integers the sum $$\Sigma_\lambda(\ell) = \sum_{\iota=(\iota_1,\dots,\iota_r)} \zeta_\ell^{\iota_1\lambda_1+\iota_2\lambda_2+\dots+\iota_r\lambda_r},$$ where the sum ranges over all the $r$-tuples $\iota=(\iota_1,\dots,\iota_r)$ such that $0\leq\iota_i\leq\ell-1$ for each $i$, and the $\iota_i$ are all distinct. We deduce some property of the sums $\Sigma_\lambda(\ell)$ to help expanding the expressions that will appear. For each $\lambda=(\lambda_1,\lambda_2,\dots)$ and integer $k$ put $k\lambda$ for the partition $(k\lambda_1,k\lambda_2,\dots)$. For integers $\ell,k,m$ let’s define the functions $$\delta^{[m]}_{\ell,k} = \begin{cases} \ell & \text{ if }\ell\geq{}m\text{ and }\ell\mid{}k, \\ 0 & \text{in any other case,} \end{cases}$$ and put $\delta_{\ell,k}=\delta^{[1]}_{\ell,k}$ for short. Then we have \[prop\_sums\] Assume $(\ell,p)=1$, $\ell>1$. For any partition $\lambda$ we have $\Sigma_{p\lambda}(\ell)=\Sigma_{\lambda}(\ell)$. For $k\geq1$ we have $\Sigma_{(k)}(\ell)=\delta_{\ell,k}$, and $\Sigma_{(k,1)}(\ell)=\delta^{[2]}_{\ell,k+1}$, and if $(k,p)=1$ we also have $\Sigma_{(k,p)}(\ell)=\delta^{[2]}_{\ell,k+p}$ and $\Sigma_{(k,p^2)}(\ell)=\delta^{[2]}{\ell,k+p^2}$. We also have $\Sigma_{(1,1,1)}(\ell)=\delta^{[3]}_{\ell,3}$, $\Sigma_{(p,1,1)}(\ell)=\delta^{[3]}_{\ell,p+2}$ and $\Sigma_{(p,p,1)}(\ell)=\delta^{[3]}_{\ell,2p+1}$. The proof can be obtained via an easy computation, but we omit it being also an immediate consequence of Lemma \[combinat\] proved in the last section. Polynomials of degree $p^2$ generating a cyclic extension ========================================================= Since the different $f'(\pi)$ has valuation $3p^2-p-2$ it must come from a term $f_{p+1}X^{p^2-p-1}$ with $v_p(f_{p+1})=2$, and we must have $v_p(f_i)\geq2$ for all $(i,p)=1$, and $v_p(f_i)\geq3$ if furthermore $i>p+1$. Since the first jump is at $1$, the $p$-th coefficient of the ramification polynomial $f(X+\pi)$ needs to have valuation exactly equal to $(p^2-p)\cdot2=2p^2-2p$, and observe that a monomial $f_{p^2-i}(X+\pi)^i$ contributes at most one term $\binom{i}{p}f_{p^2-i}\pi^{i-p}X^p$ in $X^p$. The valuations of these terms have different remainders modulo the degree $p^2$, and consequently the minimal valuation of the $\binom{i}{p}f_{p^2-i}\pi^{i-p}$ has to be $2p^2-2p$, is achieved for $i=p^2-p$ and we must have $v_p(f_p)=1$, while $v_p(f_{pk})\geq2$ for all $2\leq{}k\leq{}p-1$. So we have to respect the following We must have - $v_p(f_{p})=1$, and $v_p(f_{pi})\geq2$ for $i\in{\llbracket 2,p-1\rrbracket}$, - $v_p(f_{i})\geq2$ for $i\in{\llbracket 1,p-1\rrbracket}$, $v(f_{p+1})=2$ and $v_p(f_{i})\geq3$ for $i\in{\llbracket p+2,p^2-1\rrbracket}$. In other words, turning to $0$ all the $f_i$ divisible by $p^3$ for $i>1$, that we are allowed to do by Prop. \[prop\_coeff\], $f(X)$ can be written as $$\label{f1} f(X) = X^{p^2} + \underbrace{f_{p}X^{p^2-p} +\phantom{\bigg|}\! f_{p^2}}_{\substack{{\rotatebox{270}{$\in$}}\\p{\mathcal{O}}[X]}} + \underbrace{\sum_{j\in{\llbracket 2,p-1\rrbracket}} f_{pj}X^{p^2-pj} + \sum_{k\in{\llbracket 1,p+1\rrbracket}} f_{k} X^{p^2-k}}_{\substack{{\rotatebox{270}{$\in$}}\\p^2{\mathcal{O}}[X]}}.$$ Suppose now $L$ to be an arbitrary extension determined by a root $\pi$ of the polynomial $f(X)$, by local class field theory it is a totally ramified abelian extension precisely when $N_{L/K}(L^\times)\cap{}U_{1,K}=N_{L/K}(U_{1,L})$ has index $p^2$ in $U_{1,K}$ and the corresponding quotient is cyclic. Being $U_{i+1,K}=U_{1,K}^{p^{i}}$ for $i\geq1$, to have a cyclic extension $N_{L/K}(U_{1,L})U_{2,K}$ shall have index $p$ in $U_{1,K}$, and $N_{L/K}(U_{1,L})\cap{}U_{2,K}$ index $p$ in $U_{2,K}$. Let’s recall that for each $i\geq0$ we have a natural map $\times{}p:{\nicefrac{{\mathfrak{p}}^i_K}{{\mathfrak{p}}^{i+1}_K}}\rightarrow{\nicefrac{{\mathfrak{p}}^{i+1}_K}{{\mathfrak{p}}^{i+2}_K}}$ induced by multiplication by $p$, and for $i\geq1$ being $(1+\theta{}p^i)^p\equiv1+\theta{}p^{i+1}+{\mathcal{O}}(p^{i+2})$ we have a natural map ${\mathop{\uparrow{}p}}:{\nicefrac{U_{i,K}}{U_{i+1,K}}}\rightarrow{\nicefrac{U_{i+1,K}}{U_{i+2,K}}}$ induced by taking $p$-th powers and a commutative diagram $$\label{p_maps} \begin{array}{c} \xymatrix@!0@=25pt{ {\nicefrac{{\mathfrak{p}}^i_K}{{\mathfrak{p}}^{i+1}_K}} \ar@{->}[rrr]^{\times p} \ar@{->}[dd]_{\mu_i} & & & {\nicefrac{{\mathfrak{p}}^{i+1}_K}{{\mathfrak{p}}^{i+2}_K}}\ar@{->}[dd]_{\mu_{i+1}} \\ \\ {\nicefrac{U_{i,K}}{U_{i+1,K}}} \ar@{->}[rrr]^{{\mathop{\uparrow{}p}}} & & & {\nicefrac{U_{i+1,K}}{U_{i+2,K}}} }\end{array}$$ where $\mu_i$ is induced by $x\mapsto1+x$. Since $N_{L/K}(U_{1,L})\cap{}U_{2,K}$ will certainly contain $N_{L/K}(U_{1,L})^pU_{3,K}$ which has index $p$ in $U_{2,K}$ and consequently has to be equal, for $L/K$ to be Galois cyclic we need $$\label{normc} N_{L/K}(U_{1,L})\subseteq{}1+pV,\qquad N_{L/K}(U_{1,L})\cap{}U_{2,K}\subseteq{}1+p^2V$$ for some ${\mathbb{F}}_p$-subspace $V$ of ${\nicefrac{{\mathcal{O}}_K}{{\mathfrak{p}}_K}}$ of codimension $1$. Note that $V$ is uniquely determined by $N_{L/K}(U_{1,L})U_{2,K}$ as a subgroup of $U_{1,L}$, we commit the abuse of denoting also as $V$ its preimage in ${\mathcal{O}}_L$, and the meaning of the expressions $1+p^iV$ should be clear. If $i\geq1$ then $N_{L/K}(U_{i+1,L})\subseteq{}U_{\phi_{L/K}(i)+1,K}$ (see [@fesenko2002local Chap. 3, §3.3 and §3.4]), and in our case we have $N_{L/K}(U_{2,L})\subseteq{}U_{2,K}$ and $N_{L/K}(U_{p+2,L})\subseteq{}U_{3,K}$. Consequently we can prove that $L/K$ is Galois by showing that the norms of elements whose images generate ${\nicefrac{U_{1,L}}{U_{2,L}}}$ are contained in $1+pV$ for some $V$, and that for any $x$ obtained as combination of a set of elements whose images generate ${\nicefrac{U_{1,L}}{U_{p+2,L}}}$ and such that $N_{L/K}(x)\in{}U_{2,K}$ we actually have $N_{L/K}(x)\in{}1+p^2V$. We will take as generators the elements of the form $(1-\theta\pi^\ell)$ for $\ell\in{\llbracket 1,p+1\rrbracket}$, plus those of the forms $(1-\theta\pi)^p$ for $\theta$ in the set of multiplicative representative. Those of the form $(1-\theta\pi)^p$ can be discarded considering that we are already requesting $N_{L/K}(1-\theta\pi)\in1+pV$, so their norm will certainly be in $1+p^2V$. The norm of an element of the form $1-\theta\pi^\ell$ can be expressed as $$N_{L/K}(1-\theta\pi^\ell) = \prod_{\pi_i \mid f(\pi_i)=0}(1-\theta\pi_i^\ell) = {\mathop{\mathrm{Res}}\nolimits}_X(1-\theta{}X^\ell, f(X)),$$ where $\pi_i$ are the roots of $f(X)$ and we denote by ${\mathop{\mathrm{Res}}\nolimits}_X$ the resultant in $X$. For a polynomial $a(X)$ of degree $d$ let’s denote by $\tilde{a}(X)$ the conjugate polynomial $X^da(X^{-1})$, then for each pair of polynomials $a(X), b(X)$ we have $Res_X(a(X),b(X))=Res_X(\tilde{b}(X),\tilde{a}(X))$. Consequently $N_{L/K}(1-\theta\pi^\ell)$ can also we written as $${\mathop{\mathrm{Res}}\nolimits}_X(\tilde{f}(X),X^\ell-\theta) = \prod_{i=0}^{\ell-1} \tilde{f}(\zeta_{\ell}^i\theta^{{\nicefrac{1}{\ell}}})$$ for some primitive $\ell$-th root of the unity. In the expansion of the second term only integral powers of $\theta$ appear being invariant under the substitution $\theta^{{\nicefrac{1}{p}}}\rightarrow\zeta_{\ell}\theta^{{\nicefrac{1}{\ell}}}$. In the same way while the terms in the right hand side live in $K(\zeta_\ell)$ the result always lives in $K$, and the above expansion should be rather considered as a combinatorial expedient. Put $T=\theta^{{\nicefrac{1}{\ell}}}$ and consider it as an indeterminate, from the expression for $f(X)$ in the we have that $N_{L/K}(1-\theta\pi^\ell)$ is $$\prod_{i=0}^{\ell-1} \bigg(1 + \underbrace{f_{p}\zeta_\ell^{ip}T^p +\phantom{\bigg|}\! f_{p^2}\zeta_\ell^{ip^2}T^{p^2} }_{\substack{{\rotatebox{270}{$\in$}}\\{\mathfrak{p}}_K}} + \underbrace{\sum_{j\in{\llbracket 2,p-1\rrbracket}} f_{pj}\zeta_\ell^{ipj}T^{pj} + \sum_{k\in{\llbracket 1,p+1\rrbracket}} f_{k} \zeta_\ell^{ik}T^{k}}_{\substack{{\rotatebox{270}{$\in$}}\\{\mathfrak{p}}_K^2}} \bigg).$$ For each tuple $\lambda=(\lambda_1,\lambda_2,\dots,\lambda_r)$ of $r$ integers put $f_{\lambda}T^{|\lambda|}$ for the term $\prod_{i=1}^rf_{\lambda_i}T^{\lambda_i}$ of the expansion, its coefficient is computed over all the ways we can partitioning the $1,\zeta_\ell,\zeta_\ell^2,\dots$ in sets of cardinality $m_k(\lambda)$, the number of parts $\lambda_i$ equal to $k$, for all $k$. Computing the ratio to the ordered choices of $r$ distinct elements, which is the collection over which we are iterating while computing $\Sigma_\lambda(\ell)$, we have that the coefficient of $f_\lambda{}T^{|\lambda|}$ in the expansion is $\frac{1}{\prod_{k\geq1}m_k(\lambda)!}\cdot\Sigma_\lambda(\ell)$. In particular we have that discarding the terms with valuation $\geq3$ and subtracting $1$ the above product can be expanded modulo $p^3$ as $$\begin{aligned} {\mathfrak{p}}_K\ni &\left[\qquad \begin{gathered} \Sigma_{(p)}(\ell)\cdot{}f_{p}T^p + \Sigma_{(p^2)}(\ell)\cdot{}f_{p^2}T^{p^2} \end{gathered} \right. \\ {\mathfrak{p}}_K^2\ni &\left[ \begin{gathered} +\frac{1}{2}\Sigma_{(p,p)}(\ell)\cdot{}f_{p}^2T^{2p} +\Sigma_{(p^2,p)}(\ell)\cdot{}f_{p}f_{p^2}T^{p^2+p} +\frac{1}{2}\Sigma_{(p^2,p^2)}(\ell)\cdot{}f_{p^2}^2T^{2p^2} \\ +\sum_{j\in{\llbracket 2,p-1\rrbracket}} \left( \Sigma_{(pj)}(\ell)\cdot f_{pj}T^{pj} \right) + \sum_{k\in{\llbracket 1,p+1\rrbracket}} \left( \Sigma_{(k)}(\ell)\cdot f_{k} T^{k}\right), \end{gathered}\right.\end{aligned}$$ which applying Prop. \[prop\_sums\] can be rewritten as $$\label{expa1} \begin{aligned} {\mathfrak{p}}_K\ni &\left[\qquad \begin{gathered} \qquad\delta_{\ell,1}f_{p}T^p + \delta_{\ell,1}f_{p^2}T^{p^2} \end{gathered} \right. \\ {\mathfrak{p}}_K^2\ni &\left[ \begin{gathered} -\frac{1}{2}\delta^{[2]}_{\ell,2}f_{p}^2T^{2p} -\delta^{[2]}_{\ell,p+1}f_{p}f_{p^2}T^{p^2+p} -\frac{1}{2}\delta^{[2]}_{\ell,2}f_{p^2}^2T^{2p^2} \\ +\sum_{j\in{\llbracket 2,p-1\rrbracket}} \delta_{\ell,j}f_{pj}T^{pj} + \sum_{k\in{\llbracket 1,p+1\rrbracket}} \delta_{\ell,k} f_{k} T^{k}. \end{gathered}\right. \end{aligned}$$ The expansion for $\ell=1$ and modulo $p^2$ tells us that the norms in $U_{1,K}$ are of the form $1+f_pT^{p}+f_{p^2}T^{p^2}+{\mathcal{O}}(p^2)$ for some $T$, consequently put $F_p=\overline{{\nicefrac{f_{p}}{p}}}$, $F_{p^2}=\overline{{\nicefrac{f_{p^2}}{p}}}$ and consider the additive polynomial $$\label{cond1} A(Y)= F_{p^2}Y^p+F_pY$$ over the residue field. It defines a linear function, if $V$ is the range $A({\kappa}_K)$ then $N_{L/K}(U_{1,L})U_{2,K}$ is contained in $1+pV$, and $V$ has codimension $1$ precisely when the map defined by $A$ has a kernel of dimension $1$, that is when $-{\nicefrac{F_p}{F_{p^2}}}$ is a $(p-1)$-th power (if $K={\mathbb{Q}}_p$ we shall have $V=0$ and the condition is $F_{p^2}=-F_p$). We must have $-{\nicefrac{F_p}{F_{p^2}}}\in{}{\kappa}_K^{p-1}$. Now for $\ell\geq2$ the first part of the expansion is $0$, while the remaining part shall be contained in $p^2V$ for each $\ell$ and each specialization of $T^\ell=\theta$. Note that we only consider the $\ell$ prime with $p$, and all the $\delta^{[m]}_{\ell,i}$ appearing for some $m,i$ have been reduced with no loss of generality to have $(i,p)=1$. Consequently for $\ell\geq2$ the expansion can be written as a polynomial $C_\ell(T^\ell)=\ell\sum_{(k,p)=1}c_{k\ell}(T^{k\ell})$, where for each $\ell$ prime with $p$ we denote by $c_\ell(T^\ell)$ the polynomial of $T^\ell$ obtained by the evaluating equation , but changing the definition of $\delta^{[m]}_{a,b}$ to be $1$ if $a=b$, and $0$ if $a\neq{}b$. Fix $\ell$, then $c_\ell(T^\ell)$ can be obtained via a sort of Möbius inversion $$\begin{aligned} \sum_{(k,p)=1} \mu(k) \frac{C_{k\ell}(T^{k\ell})}{k\ell} &= \sum_{(k,p)=1}\left( \mu(k) \cdot \sum_{(j,p)=1} c_{jk\ell}(T^{jk\ell}) \right) \\ &= \sum_{(i,p)=1} \left(c_{i\ell}(T^{i\ell})\cdot \sum_{k\mid{}i}\mu(k) \right)\end{aligned}$$ by change of variable $i=jk$, obtaining $c_\ell(T^\ell)$ by the properties of the Möbius function $\mu$. In view of the isomorphism ${\nicefrac{{\mathfrak{p}}^2_K}{{\mathfrak{p}}^3_K}}\rightarrow{\nicefrac{U_{2,K}}{U_{3,K}}}$ induced by $x\mapsto{}1+x$ and specializing the argument $T^{k\ell}$ of $C_{k\ell}(T^{k\ell})$ to $\theta'=\frac{1}{\ell}\theta^k$ we have that $$1+\frac{1}{k\ell}C_{k\ell}(\theta^k)\equiv N_{L/K}\left(1-\frac{1}{\ell}\theta^k\pi^{k\ell}\right)^{{\nicefrac{1}{k}}} \imod{p^3},$$ for each $\ell,k$ prime with $p$ and each $\theta=T^\ell$. Consequently $1+c_\ell(\theta)$ is congruent modulo $p^3$ to the norm of $$\prod_{(k,p)=1}\left(1-\frac{1}{\ell}\theta^k\pi^{k\ell}\right)^{{\nicefrac{\mu(k)}{k}}} = E(\theta\pi^\ell),$$ where $E(x)$ is the Artin-Hasse exponential function (in its original form, according to [@fesenko2002local Chap. 3, §9.1]). Note that we can equivalently require that all the $N_{L/K}(E(\theta\pi^\ell))$ are in $1+p^2V$, for $\ell\in{\llbracket 2,p+1\rrbracket}$ and residue representative $\theta\in{}K$. Put $A_\ell(Y)=\overline{{\nicefrac{c_\ell(Y)}{p^2}}}$, we obtain depending on $\ell$ the additive polynomials $$\begin{gathered} \qquad\qquad -F_pF_{p^2}Y^p + G_{p+1}Y \qquad\qquad\qquad\qquad \ell=p+1, \\ \qquad\qquad\qquad G_{p\ell}Y^p + G_\ell Y \qquad \qquad\qquad\qquad \ell\in{\llbracket 3, p-1\rrbracket},\\ -\frac{1}{2}F_{p^2}^2Y^{p^2}+\left(G_{2p}-\frac{1}{2}F_p^2\right)Y^p+G_{2}Y \qquad \ell=2,\end{gathered}$$ where for convenience we have put $G_i=\overline{{\nicefrac{f_i}{p^2}}}$ for each $i\neq{}p,p^2$. Hence we have obtained the For each $\ell\in{\llbracket 2,p+1\rrbracket}$ we shall have $A_\ell({\kappa}_K)\subseteq{}A({\kappa}_K)$. We are left to ensure that norms are in $1+p^2V$ for $\ell=1$ with $T=\theta$ for some $\theta$ such that $\theta^{p^2-p}\equiv-{\nicefrac{f_p}{f_{p^2}}}\imod{p}$. Consider again the , considering the definition of the $c_k(T)$ we have that $\sum_{(k,p)=1}c_{k}(T^{k})$ differs from $C_1(T)=N_{L/K}(1-T\pi)-1$ by the extra term $$- \frac{1}{2}f_{p}^2T^{2p} - f_{p}f_{p^2}T^{p^2+p} - \frac{1}{2}f_{p^2}^2T^{2p^2} = -\frac{1}{2}\left(f_{p}T^{p} + f_{p^2}T^{p^2}\right)^2,$$ which is however even contained in ${\mathfrak{p}}^4$ for $T=\theta$. Since we already required the polynomials $c_k(Y^{k})$ to take values in $p^2V$ identically for $k\geq2$, our requirement becomes that $$c_1(\theta) = f_{p^2}\theta^{p^2} + f_{p}\theta^{p} + f_{1}\theta$$ shall be contained in $p^2V$ too. Hence we have the Let $\theta$ be such that $\theta^{p^2-p}\equiv-{\nicefrac{f_p}{f_{p^2}}}\imod{p}$, then we must have $\overline{{\nicefrac{c_1(\theta)}{p^2}}}\in{}V$. Collecting all the above conditions and applying Prop. \[prop\_add\] to obtain conditions on the coefficients we have the following theorem: \[theo1\] The Eisenstein polynomial $f(X)=X^{p^2}+f_1X^{p^2-1}+\dots+f_{p^2-1}X+f_{p^2}$ determines a Galois extension of degree $p^2$ over $K$ if and only if 1. \[tcr1\] $v_p(f_{p})=1$, and $v_p(f_{pi})\geq2$ for $i\in{\llbracket 2,p-1\rrbracket}$, 2. \[tcr2\] $v_p(f_{i})\geq2$ for $i\in{\llbracket 1,p-1\rrbracket}$, $v(f_{p+1})=2$ and $v_p(f_{i})\geq3$ for $i\in{\llbracket p+2,p^2-1\rrbracket}$, putting $F_p=\overline{{\nicefrac{f_p}{p}}}$, $F_{p^2}=\overline{{\nicefrac{f_{p^2}}{p}}}$, and $G_i=\overline{{\nicefrac{f_i}{p^2}}}$ for all $i\neq{}p,p^2$ we have 1. \[tcgal\] $-{\nicefrac{F_p}{F_{p^2}}}\in{}{\kappa}_K^{p-1}$, 2. \[tcpp1\] $G_{p+1}^p=-F_p^{p+1}$, 3. \[tcell\] $G_{p\ell}=F_{p^2}\left({\nicefrac{G_\ell}{F_p}}\right)^p$, for all $\ell\in{\llbracket 3,p-1\rrbracket}$, 4. \[tc2\] $G_{2p}=F_{p^2}\left({\nicefrac{G_2}{F_p}}\right)^p+\frac{1}{2}F_p\left(F_p-F_{p^2}^{{\nicefrac{1}{p}}}\right)$, if $\theta$ is such that $\bar\theta^{p(p-1)}=-{\nicefrac{F_p}{F_{p^2}}}$ we have that (for any choice of $\theta$) 1. \[tc1\] $F_{p^2}X^p+F_pX-\overline{\frac{1}{p^2}\left(f_{p^2}\theta^{p^2}+f_{p}\theta^{p}\right)}-G_1\bar\theta$ has a root in ${\kappa}_K$. Polynomials of degree $p^2$ whose Galois group is a $p$-group ============================================================= In the proof of Theorem \[theo1\] we obtained a list of requirements on an Eisenstein equation of degree $p^2$ that guarantee that the generated extension is Galois cyclic over $K$. Let’s keep the hypotheses on the ramification numbers (and consequently conditions \[tcr1\] and \[tcr2\] of the Theorem), it is a natural question to describe the Galois group of the normal closure when some of these hypotheses are not satisfied. Put $L=K(\pi)$, we will also keep the condition \[tcgal\], which we can see immediately to be satisfied if and only if $L$ contains a Galois extension of degree $p$ of $K$, which is a necessary condition for the normal closure of $L/K$ to be a $p$-group. Note that this hypothesis is always satisfied for $f(X)$ if $K$ is replaced with a suitable unramified extension. We can notice from the proof that the first unsatisfied requirement among the conditions \[tcpp1\], \[tcell\] with $\ell$ as big as possible, \[tc2\] and \[tc1\] in Theorem \[theo1\] gives us information about the biggest possible $\ell$ such that $N_{L/K}(U_{\ell,L})\cap{}U_{2,K}$ is not contained in $1+p^2V$, with $V$ defined as in the proof. We expect this fact to allow us to deduce information about the Galois group of the normal closure. Let $F$ be the Galois extension of degree $p$ of $K$ contained in $L$, it is unique or ${\mathop{\mathrm{Gal}}\nolimits}(L/K)$ would be elementary abelian, which is not possible considering the ramification breaks. Then $F/K$ has ramification number $1$ and $L/F$ ramification number $p+1$. Before continuing we prove a proposition that will also be of use later, for which we only assume $L/F$ to have ramification break $>1$ and $F/K$ to be Galois with break at $1$, with group generated by $\sigma$ say. Similarly we have that the only $f_i$ with $v_p(f_i)=1$ are $f_p$ and $f_{p^2}$. For some $\theta\in{}K$ we can write $$\begin{aligned} \pi_F^{(\sigma-1)} &= 1-\theta^p\pi_F+{\mathcal{O}}(\pi_F^2)\\ &= N_{L/F}(1-\theta\pi)+{\mathcal{O}}(\pi_F^2) \end{aligned}$$ in view of [@fesenko2002local Chap. 3, §1, Prop. 1.5] and having $L/F$ ramification break $>1$. Since $\pi_F^{(\sigma-1)}$ is killed by $N_{F/K}$ and $N_{F/K}(U_{2,F})\subseteq{}U_{2,K}$ we should have $N_{L/K}(1-\theta\pi)\in{}U_{2,K}$ and hence $\bar\theta^{p(p-1)}=-{\nicefrac{F_p}{F_{p^2}}}$, as we have expanding $\tilde{f}(\theta)=N_{L/F}(1-\theta\pi)$ like in the proof of Theorem \[theo1\]. We obtain inductively that \[prop\_shift\] For each $1\leq\ell<p$ we have $$\pi_F^{(\sigma-1)^\ell}=1-k\theta^{p\ell}\pi_F^\ell+{\mathcal{O}}(\pi_F^{\ell+1}),$$ for some integer $k$ prime with $p$, where $\bar\theta^{p(p-1)}=-{\nicefrac{F_p}{F_{p^2}}}$. We return to our main problem, so let $L/F$ have ramification break at $p+1$. We require $L/F$ to be Galois: by local class field theory this is the case precisely when the map ${\nicefrac{U_{p+1,L}}{U_{p+2,L}}}\rightarrow{\nicefrac{U_{p+1,F}}{U_{p+2,F}}}$ induced by $N_{L/F}$ is not surjective. Since the map ${\nicefrac{U_{p+1,F}}{U_{p+2,F}}}\rightarrow{\nicefrac{U_{2,K}}{U_{3,K}}}$ induced by $N_{F/K}$ is an isomorphism by [@fesenko2002local Chap. 3, §1, Prop. 1.5], we are reduced to study the image of ${\nicefrac{U_{p+1,L}}{U_{p+2,L}}}$ in ${\nicefrac{U_{2,K}}{U_{3,K}}}$. Considering the norms of the usual $1+\theta\pi^{p+1}$, we have from the proof of Theorem \[theo1\] that this map is described by the additive polynomial $A_{p+1}(Y)$, and is non-surjective precisely when ${\nicefrac{G_{p+1}}{F_pF_{p^2}}}$ is in ${\kappa}_K^{p-1}$. Consequently we will always assume the We require ${\nicefrac{G_{p+1}}{F_pF_{p^2}}}\in{\kappa}_K^{p-1}$. which is necessary and sufficient for the Galois closure of $L/K$ to be a $p$-group, and again is always satisfied if we replace $K$ by a suitable unramified extension. For an ${\mathbb{F}}_p[G]$-module $M$ we respectively denote by ${\mathop{\mathrm{soc}}\nolimits}^iM$ and ${\mathop{\mathrm{rad}}\nolimits}^iM$ the $i$-th socle and radical of $M$. If $\sigma$ is a generator of $G$, the radical of ${\mathbb{F}}_p[G]$ is generated by $\sigma-1$, and we have $${\mathop{\mathrm{rad}}\nolimits}^i M = M^{(\sigma-1)^i},\qquad {\mathop{\mathrm{soc}}\nolimits}^i M = \left\{ x : x^{(\sigma-1)^i} = 0 \right\}.$$ Let $G={\mathop{\mathrm{Gal}}\nolimits}(F/K)$ and $\tilde{L}$ be the Galois closure of $L$ over $K$, we want to compute the length of ${\mathop{\mathrm{Gal}}\nolimits}(\tilde{L}/F)$ as a ${\mathbb{F}}_p[G]$-module, which we will also show to determine completely ${\mathop{\mathrm{Gal}}\nolimits}(\tilde{L}/K)$ in the present case. If $F^{(p)}$ is the maximal abelian elementary $p$-extension of $F$, this amounts to computing the smallest $m$ such that $rad^m{\mathop{\mathrm{Gal}}\nolimits}(F^{(p)}/F)$ is contained in ${\mathop{\mathrm{Gal}}\nolimits}(F^{(p)}/L)$. For $0\leq{}i\leq{}p$ let’s consider the submodules $S_i={\mathop{\mathrm{soc}}\nolimits}^{p-i}P_F$ of $P_F=[F^\times]_F$ (which is canonically identified with ${\mathop{\mathrm{Gal}}\nolimits}(F^{(p)}/F)$ via local class field theory), and let $K_i$ be the class field corresponding to $S_i$ over $F$. For $0\leq{}i<p$ we have $[U_{i+1,F}]\subseteq{}S_i$ and thus the highest upper ramification break of ${\mathop{\mathrm{Gal}}\nolimits}(K_i/F)$ is $i$ for $i<p$, and in particular being $p+1$ the unique ramification break of $L/F$ we have that $K_i\nsupseteq{}L$ for $i<p$. Note also that $K_1$ is the maximal elementary abelian $p$-extension of $K$. Let $K'$ be the field corresponding to ${\mathop{\mathrm{rad}}\nolimits}^1P_F$, it is the maximal $p$-elementary abelian extension of $F$ which is abelian over $K$, and it corresponds to $N_{F/K}(F^\times)^p$ via the class field theory of $K$. Considering the structure of $P_F\cong{\mathbb{F}}_p[G]^{\oplus{}f}\oplus{\mathbb{F}}_p$ as a Galois module we have that $$rad^iP_F = {\mathop{\mathrm{soc}}\nolimits}^{p-i}P_F \cap {\mathop{\mathrm{rad}}\nolimits}^1P_F = S_i\cap {\mathop{\mathrm{rad}}\nolimits}^1P_F,$$ for each $i$, and $rad^iP_F$ corresponds to $K'K_i$ via class field theory, so we are looking for the smallest $m$ such that $L\subset{}K'K_m$. Since $L$ and $K'$ are never contained in $K_i$ for $i<p$ and $K'$ has degree $p$ over $K_1$, this inclusion holds if and only if $L$ and $K'$ generate the same extension over $K_m$ (and $\tilde{L}$ will too, being $K'K_m$ Galois over $K$). This is the case if and only if $K'\subset{}LK_m$, and this condition is consequently equivalent to the ${\mathbb{F}}_p[G]$-module ${\mathop{\mathrm{Gal}}\nolimits}(\tilde{L}/F)$ having length $\leq{}m$ We can now show that if $K'\subset{}LK_m$ for some $m<p$, then ${\mathop{\mathrm{Gal}}\nolimits}(\tilde{L}/K)$ cannot be the semidirect product: indeed ${\mathop{\mathrm{Gal}}\nolimits}(K_{m+1}/K)$ lives in the exact sequence $$1 \rightarrow {\nicefrac{P_F}{S_{m+1}}} \rightarrow {\mathop{\mathrm{Gal}}\nolimits}(K_{m+1}/K) \rightarrow G \rightarrow 1,$$ and all $p$-th powers in ${\mathop{\mathrm{Gal}}\nolimits}(K_{m+1}/K)$ are clearly $G$-invariant elements of ${\nicefrac{P_F}{S_{m+1}}}$, and hence contained in ${\nicefrac{S_m}{S_{m+1}}}$, and this shows that the quotient ${\mathop{\mathrm{Gal}}\nolimits}(K_m/K)$ has exponent $p$ since we quotiented out all $p$-th powers. On the other hand ${\mathop{\mathrm{Gal}}\nolimits}(K'/K)$ has exponent $p^2$ so if $K'\subset{}LK_m$ then also ${\mathop{\mathrm{Gal}}\nolimits}(\tilde{L}K_m/K)$ does, and ${\mathop{\mathrm{Gal}}\nolimits}(\tilde{L}/K)$ should also have exponent $p^2$ or the $p$-th power of any element of the absolute Galois group would act trivially on $\tilde{L}$, $K_m$, and consequently on $\tilde{L}K_m$, which is impossible. Note that if ${\mathop{\mathrm{Gal}}\nolimits}(\tilde{L}/F)$ has maximal length $m=p$ there is only one possible isomorphism class of possible $p$-groups, which is the wreath product of two cyclic groups of order $p$, see [@waterhouse1994normal; @minac2005galois] for more information about these groups. The above observation can be viewed as the fact that, for $m<p$, $K_m$ is the compositum of all the extensions of degree $p$ whose normal closure has group over $F$ of length $\leq{}m$ as ${\mathbb{F}}_p[G]$-module, and whose group over $K$ is the semidirect product extension (and hence has exponent $p$). The extensions whose group of the normal closure over $K$ is not the semidirect product are obtained via a sort of twist with $K'$, which is non-trivial when $L\nsubseteq{}K_m$. Now $K'$ is not contained in $LK_m$ precisely when there exist an element in ${\mathop{\mathrm{Gal}}\nolimits}(K^{\mathrm{ab}}/K)$ fixing $LK_m$ but not $K'$, any such element can be lifted to ${\mathop{\mathrm{Gal}}\nolimits}(L^{\mathrm{ab}}/L)$. Since the image of the Artin map $\Psi_L:L^\times\rightarrow{\mathop{\mathrm{Gal}}\nolimits}(L^{\mathrm{ab}}/L)$ is dense in ${\mathop{\mathrm{Gal}}\nolimits}(L^{\mathrm{ab}}/L)$ we can take such element of the form $\Psi_L(\alpha)$ for some $\alpha\in{}L^\times$. Having to fix $K_1$ we will have $N_{L/K}(\alpha)\in{}(K^\times)^p$ by the functoriality of the reciprocity map (see [@fesenko2002local Chap. 4, Theorem 4.2]), $[N_{L/F}(\alpha)]_F\in{}S_m$ because $K_m$ is fixed, and $N_{L/K}(\alpha)\notin{}N_{F/K}(F^\times)^p$ because the action is non-trivial on $K'$, and on the other hand the existence of such an element ensures that $K'\notin{}LK_m$. If $L$ and $K$ are as above, we have proved the \[clos1\] Let $1\leq{}m\leq{}p$ be the smallest possible integer such that for all $\alpha\in{}L^\times$ such that $N_{L/K}(\alpha)\in(K^\times)^p$ and $[N_{L/F}(\alpha)]_F\in{}S_m$ we also have $N_{L/K}(\alpha)\in{}N_{F/K}(F^\times)^p$. Then ${\mathop{\mathrm{Gal}}\nolimits}(\tilde{L}/K)$ is the unique $p$-group which has exponent $p^2$ and is an extension of $G={\mathop{\mathrm{Gal}}\nolimits}(F/K)$ by an indecomposable ${\mathbb{F}}_p[G]$-module of length $m$. We now determine the $(p-m)$-th socle $S_m$ of $P_F$ for each $0\leq{}m\leq{}p$, and deduce the ramification jumps of the normal closure. Consider the images $V_i=[U_{i,F}]_F$ of the $U_{i,F}$ in $P_F$ for $i\geq{}1$, and put $V_0=P_F$ for convenience. If $G$ is generated by $\sigma$ say, the radical of ${\mathbb{F}}_p[G]$ is generated by $(\sigma-1)$ and we have $V_i^{\sigma-1}\subseteq{}V_{i+1}$. Since $V_p=V_{p+1}$ and $V_{p+2}=1$ we have that $V_p$ is killed by $\sigma-1$, $V_{p-1}$ by $(\sigma-1)^2$ and so on, so that $V_{k+1}\subseteq{}{\mathop{\mathrm{soc}}\nolimits}^{p-k}P_F=S_{k}$ for $0\leq{}k<p$, while clearly $S_p=0$. Furthermore if $\pi_F$ is a uniformizing element of $F$ we have $\pi_F^{(\sigma-1)^k}\in{}V_k\setminus{}V_{k+1}$ and $\pi_F^{(\sigma-1)^{k}}\in{}S_k$ for $0\leq{}k<p$, so comparing the dimensions we have that $$S_k = \langle\pi_F^{(\sigma-1)^k}\rangle + V_{k+1}.$$ If $m$ is like in the proposition and $\geq2$, take in $L^\times$ an element $\alpha$ contradicting the proposition for $m-1$ and such that $t=v_L(1-N_{L/F}(\alpha))$ is as big as possible. Then $\psi_{LK_{m-1}/F}(t)$ is the ramification break of $K'LK_{m-1}/LK_{m-1}$, which is also equal to that of $LK'K_{m-1}/K'K_{m-1}$ considering that $K'K_{m-1}/K_{m-1}$ and $LK_{m-1}/K_{m-1}$ have the same ramification break equal to $\psi_{K_{m-1}/F}(p+1)$, and the total set of breaks has to be preserved. By the definition of $S_{m-1}$ and $S_m$ we have that $t$ can be either $m-1$ or $m$, unless $m=p$ where $t$ is either $p-1$ or $p+1$. By local class field theory $K'K_{m-1}/F$ corresponds to the subgroup $A=rad^{m-1}P_F$ of $P_F$, and $LK'K_{m-1}/F$ to another subgroup $B$ with index $p$ in $A$, and $t$ is the biggest $t$ such that some $x\in{}V_t\cap{}A$ has non-trivial image in $A/B$. Passing to the groups $A'$ and $B'$ of the elements sent by $\sigma-1$ into $A$ and $B$ respectively, $A'=soc^{p-m+2}P_F$ corresponds to $K_{m-2}$, and $B'$ to $L'K_{m-2}$ where $L'$ is the subfield of $\tilde{L}$ corresponding to ${\mathop{\mathrm{soc}}\nolimits}^1{\mathop{\mathrm{Gal}}\nolimits}(\tilde{L}/F)$ as ${\mathbb{F}}_p[G]$-module. The upper ramification break of the new relative extension is $\psi_{K_{m-2}/F}(s)$ where $s$ is the biggest so that some $y\in{}V_s\cap{}A'$ is nontrivial in $A'/B'$. Being $A=rad^{m-1}P_F$ each $x\in{}A\setminus{}B$ is of the form $x=y^{\sigma-1}$ for some $y\in{}A'\setminus{}B'$, so $s=t-1$ unless $t=p+1$ which becomes $s=p-1$. Since ${\mathop{\mathrm{Gal}}\nolimits}(L'/F)$ has length $m-1$ and the field $L''$ corresponding to ${\mathop{\mathrm{soc}}\nolimits}^1{\mathop{\mathrm{Gal}}\nolimits}(L'/L)$ is contained in $K_{m-2}$, and $V_{m-2}\supseteq{}A'\supseteq{}V_{m-1}$, we have that $s$ is also the ramification number of $L'/L''$. Repeating this observation for $m-1$ steps we have that the ramification breaks over $F$ are either $1,2,\dots,m-1,p+1$, either $0,1,\dots,m-2,p+1$ depending on whether an element $\alpha\in{}S_{m-1}$ contradicting the proposition can be found in $V_m$ or not, where for convenience a “ramification break” of $0$ indicates an unramified extension. We proved the \[clos2\] Let $1\leq{}m\leq{}p$ be like in the Prop. \[clos1\], if we can find an $\alpha$ such that $N_{L/K}(\alpha)\in{}(K^\times)^p\setminus{}N_{F/K}(F^\times)^p$ such that $[N_{L/F}(\alpha)]_F\in{}V_m$, then the normal closure $\tilde{L}/F$ is totally ramified with breaks $1,2,\dots,m-1,p+1$. If not, then $\tilde{L}/F$ is formed by an unramified extension of degree $p$ and an extension with breaks $1,2,\dots,m-2,p+1$. We will look for the biggest $1\leq{}m\leq{}p-1$ such that we can find an $\alpha$ contradicting the requests of the Prop. \[clos1\]. For all $\ell=p-1,\dots,2,1$ in descending order, if we cannot find a suitable $\alpha$ with $[N_{L/F}(\alpha)]_F\in{}V_{\ell+1}$, we inductively test $S_{\ell}\supset{}V_{\ell+1}$ (and ${\mathop{\mathrm{Gal}}\nolimits}(\tilde{L}/F)$ has length $\ell+1$ and there is an unramified part), and then $V_{\ell}\supseteq{}S_\ell$ (and in this case ${\mathop{\mathrm{Gal}}\nolimits}(\tilde{L}/F)$ has length $\ell$ and the extension is totally ramified). Verifying that we cannot find $\alpha$ with $[N_{L/F}(\alpha)]_F\in{}V_{\ell+1}$ is easy, and is the condition of the theorem connected to $A_{p+1}(Y)$ for $\ell=p-1$, or to $A_{\ell+1}$ if $\ell<p-1$. We then allow $[N_{L/F}(\alpha)]_F$ to be in $S_\ell=\langle\pi_F^{(\sigma-1)^{\ell}}\rangle+V_{\ell+1}$: by Prop. \[prop\_shift\] for $\bar\theta^{p(p-1)}=-{\nicefrac{F_p}{F_{p^2}}}$ and for some $k$ prime with $p$ we have $$\begin{aligned} \pi_F^{(\sigma-1)^\ell} &= 1-k\theta^{p\ell}\pi_F^\ell+{\mathcal{O}}(\pi_F^{\ell+1})\\ &= N_{L/F}(1-k\theta^\ell\pi^\ell)+{\mathcal{O}}(\pi_F^{\ell+1}),\end{aligned}$$ in view of [@fesenko2002local Chap. 3, §1, Prop. 1.5] and being $\ell$ smaller than the ramification number $p+1$. In particular the image of $N_{L/F}(1-\theta^\ell\pi^\ell)$ generates ${\nicefrac{S_\ell}{V_{\ell+1}}}$, and testing the condition for $S_\ell$ is equivalent to verifying that $A_{\ell}(\bar\theta^\ell)\in{}V$. Note that $A_2(\bar\theta^2)$ has the simplified form $G_{2p}\bar\theta^{2p}+G_{2}\bar\theta^2$, and testing if $F_{p^2}X^p+F_pX=A_{\ell}(\bar\theta^\ell)$ has solution in ${\kappa}_K$ is equivalent to checking, after replacing $X$ by $\bar\theta^\ell{}X$ and dividing by $\bar\theta^\ell$, if there are solutions to $$F_{p^2}(-{\nicefrac{F_p}{F_{p^2}}})^{{\nicefrac{\ell}{p}}}X^p + F_pX - G_{p\ell}(-{\nicefrac{F_p}{F_{p^2}}})^{{\nicefrac{\ell}{p}}} - G_\ell = 0.$$ Note that for $\ell=1$ we just test if $\overline{{\nicefrac{c_1(\theta)}{p^2}}}$ is in $V$, like in the last condition of Theorem \[theo1\]. We have the \[theo2\] Assume $f(X)$ to satisfy conditions \[tcr1\], \[tcr2\], \[tcgal\] of Theorem \[theo1\], and keeping the notation assume additionally that 1. ${\nicefrac{G_{p+1}}{F_pF_{p^2}}}\in{\kappa}_K^{p-1}$. Let $L$ be the extension determined by $f(X)$, $\tilde{L}$ the normal closure over $K$, and $F$ the unique subextension of degree $p$ contained in $L$. Then ${\mathop{\mathrm{Gal}}\nolimits}(\tilde{L}/K)$ is an extension of $G={\mathop{\mathrm{Gal}}\nolimits}(F/K)$ by the indecomposable ${\mathbb{F}}_p[G]$-module $M={\mathop{\mathrm{Gal}}\nolimits}(\tilde{L}/F)$, ${\mathop{\mathrm{Gal}}\nolimits}(\tilde{L}/K)$ has exponent $p^2$ and is a non-split extension unless $M$ has length $p$. Furthermore 1. if $G_{p+1}^p\neq-F_p^{p+1}$ then $M$ has length $p$ and $L/F$ is totally ramified with upper ramification breaks $1,2,\dots,p-1,p+1$; assuming equality in the previous condition, 1. if $G_{p\ell}\neq{}F_{p^2}\left({\nicefrac{G_\ell}{F_p}}\right)^p$ for some $\ell\in{\llbracket 3,p-1\rrbracket}$ that we take as big as possible, or $G_{2p}\neq{}F_{p^2}\left({\nicefrac{G_2}{F_p}}\right)^p+\frac{1}{2}F_p\left(F_p-F_{p^2}^{{\nicefrac{1}{p}}}\right)$ and we put $\ell=2$, let $$U(X) = F_{p^2}(-{\nicefrac{F_p}{F_{p^2}}})^{{\nicefrac{\ell}{p}}}X^p + F_pX - G_{p\ell}(-{\nicefrac{F_p}{F_{p^2}}})^{{\nicefrac{\ell}{p}}} - G_\ell.$$ We have that - if $U(X)$ has no root in ${\kappa}_K$, then $M$ has length $\ell+1$ and $\tilde{L}/F$ is formed by an unramified extension followed by a totally ramified extension with upper ramification breaks $1,2,\dots,\ell-1,p+1$, - if $U(X)$ has some root in ${\kappa}_K$, then $M$ has length $\ell$ and $\tilde{L}/F$ is a totally ramified extension with upper ramification breaks $1,2,\dots,\ell-1,p+1$, assuming equality in the previous conditions, and for $\bar\theta^{p(p-1)}=-{\nicefrac{F_p}{F_{p^2}}}$, 1. if $F_{p^2}X^p+F_pX-\overline{\frac{1}{p^2}\left(f_{p^2}\theta^{p^2}+f_{p}\theta^{p}\right)}-G_1\bar\theta$ has no root in ${\kappa}_K$, then $M$ has length $2$ and $\tilde{L}/F$ is formed by an unramified extension followed by a totally ramified extension with upper ramification break $p+1$. All conditions pass precisely when all requirements of Theorem \[theo1\] are satisfied, and in this case $L/F$ is Galois cyclic. It turns out that we just worked out the hard case of the classification of all polynomials of degree $p^2$ whose Galois group is a $p$-group. We keep the notation of the previous part of this section. We have classified in Theorem \[theo2\] all polynomials such that $L/F$ has ramification break at $p+1$ and the normal closure is a $p$-group, and it turned out that the condition on the ramification number is sufficient to guarantee that the Galois group of the normal closure has exponent $p^2$. Conversely if the ramification number is $\leq{}p-1$ then either $L\subset{}K_m$ for some $m<p$ and ${\mathop{\mathrm{Gal}}\nolimits}(\tilde{L}/F)$ has length $\leq{}m$ and ${\mathop{\mathrm{Gal}}\nolimits}(\tilde{L}/K)$ is the splitting extension of $G$, either ${\mathop{\mathrm{Gal}}\nolimits}(\tilde{L}/F)$ has length $p$ and there is only one possibility for ${\mathop{\mathrm{Gal}}\nolimits}(\tilde{L}/K)$ which is both a split extension and has exponent $p^2$, and is a wreath product of two cyclic groups of order $p$. Again as above, assume $m$ to be the smallest integer such that $[N_{L/F}(L^\times)]_F$ contains $V_{m+1}$. The ramification number of $L/F$ is $m$, and the length of ${\mathop{\mathrm{Gal}}\nolimits}(\tilde{L}/K)$ as $G$-module can be $m$ when the norms also contain $S_m$, or $m+1$ if this is not the case. Since $S_\ell=\langle\pi_F^{(\sigma-1)^{\ell}}\rangle+V_{\ell+1}$ to resolve this ambiguity we should test whether $[\pi_F^{(\sigma-1)^{\ell}}]_F\in{}[N_{L/F}(L^\times)]_F$. Since $\pi_F^{(\sigma-1)^{\ell}}\in{}U_{\ell,F}$ and $N_{L/F}(L^\times)\supset{}U_{\ell+1,F}$ we can just test if $$N_{L/F}(1+\theta\pi^{\ell}) = \pi_F^{(\sigma-1)^{\ell}} + {\mathcal{O}}(\pi^{\ell+1})$$ for some unit $\theta\in{}K$. Factorizing in $L$ the ramification polynomial $f(X+\pi)$ over the Newton polygon we have that $f(X+\pi)=Xg(X)h(X)$, where $g(X)$ has degree $p-1$ with roots of valuation $\ell+1$ and $h(X)$ degree $p^2-p$ and roots with valuation $2$. We can take $g(X)$ to be monic and with roots $\tau^i(\pi)-\pi$, where $\tau$ is an automorphism of order $p$ of the normal closure of $L$ over $F$ and $1\leq{}i<p$, and $L/F$ is Galois if and only if $g(X)=X^{p-1}+\dots+g_1X+g_0$ splits in linear factors in $L$. If we can write $\tau(\pi)-\pi=\eta\pi^{\ell+1}+{\mathcal{O}}(\pi^{\ell+2})$ with $\eta\in{}K$, then $\tau(\pi)$ can be approximated in $L$ better than by any other conjugate, and consequently $L/F$ is Galois by Krasner lemma. On the other hand if $L/K$ is Galois we certainly have such an expression for some $\eta$. Since $$\begin{aligned} g_0&=\prod_{i=1}^{p-1}(\tau^i\pi-\pi)\\ &\equiv\prod_{i=1}^{p-1}i\eta\pi^{\ell+1}\equiv -\eta^{p-1}\pi^{(p-1)(\ell+1)} \end{aligned}$$ we have that $L/F$ is Galois if and only if $-g_0$ is a $(p-1)$-th power. The monomial in $X^p$ of $f(X+\pi)$ is $$\binom{p^2-p}{p}f_p\pi^{p^2-2p}X^p=h_0X^p$$ where $h_0$ is the constant term of $h(X)$, while the monomial in $X$ is $$Xf'(\pi) = (p^2-r)f_r\pi^{p^2-r-1}X = g_0h_0X$$ where $r$ should be $p^2-(p-1)\ell+p$ and $v_p(f_{p^2-r})=2$, considering that $v_L(f'(\pi))$ is $(p^2-p)\cdot2+(p-1)\cdot(\ell+1)$. Since $\binom{p^2-p}{p}\equiv-1\imod{p}$ by the definition of $r$ we have taking the ratio of the coefficients of the monomials above that $$\begin{aligned} \frac{g_0}{\pi^{(p-1)(\ell+1)}} &= \frac{-rf_r\pi^{(p-1)\ell-p-1}}{-f_p\pi^{p^2-2p}}\cdot\pi^{-(p-1)(\ell+1)} + {\mathcal{O}}(\pi)\\ &= {\nicefrac{rf_r}{f_p}} \cdot\pi^{-p^2} + {\mathcal{O}}(\pi) = -{\nicefrac{rf_r}{f_pf_{p^2}}} + {\mathcal{O}}(\pi),\end{aligned}$$ being $\pi^{p^2}=-f_0+{\mathcal{O}}(\pi)$. Since $r\equiv\ell\imod{p}$ we obtained that $\bar\eta^{p-1}$ is equal to $\overline{{\nicefrac{\ell{}f_r}{f_pf_{p^2}}}}$, and it is contained in ${\kappa}_K^{p-1}$ if and only if $g_0$ is a $p-1$-th power. Put again $F_p=\overline{{\nicefrac{f_p}{p}}}$, $F_{p^2}=\overline{{\nicefrac{F_{p^2}}{p}}}$ and $G_i=\overline{{\nicefrac{f_i}{p^2}}}$ for $i\neq{}p,p^2$. $L/F$ is Galois if and only if ${\nicefrac{\ell{}G_r}{F_pF_{p^2}}}$ is in ${\kappa}_K^{p-1}$, where $r$ is equal to $p^2-(p-1)\ell+p$. Let’s recall that from [@fesenko2002local Chap. 3, §1, Prop. 1.5] we have that $$N_{L/F}(1+\theta\pi^\ell) = 1+(\theta^p-\eta^{p-1}\theta)\pi_F^\ell+{\mathcal{O}}(\pi_F^{\ell+1}),$$ while $$\pi_F^{(\sigma-1)^\ell}=1-k\rho^\ell\pi_F^{\ell}+{\mathcal{O}}(\pi_F^{\ell+1})$$ for $\bar\rho^{p-1}=-{\nicefrac{F_p}{F_{p^2}}}$ and some integer $k$ prime with $p$, by Prop. \[prop\_shift\]. From what observed at the beginning, we obtain that the length of ${\mathop{\mathrm{Gal}}\nolimits}(\tilde{L}/K)$ is $\ell$ when $X^p-{\nicefrac{\ell{}G_r}{F_pF_{p^2}}}X=\bar\rho^\ell$ has solution in ${\kappa}_K$, and $\ell+1$ if this is not the case. Replacing $X$ by $\bar\rho^\ell{}X$ and dividing by $\bar\rho^\ell$ this is equivalent to testing if $$(-{\nicefrac{F_p}{F_{p^2}}})^\ell{}X^p - {\nicefrac{\ell{}G_r}{F_pF_{p^2}}}X = 1$$ has solution in ${\kappa}_K$. Consequently we obtain \[theo3\] Let $2\leq\ell\leq{}p-1$ an let $r=p^2-(p-1)\ell+p$, and assume that $f(X)$ is such that 1. $v_p(f_{p})=1$, and $v_p(f_{pi})\geq2$ for $i\in{\llbracket 2,p-1\rrbracket}$, 2. $v_p(f_{i})\geq2$ for $i\in{\llbracket 1,r-1\rrbracket}$, $v(f_r)=2$ and $v_p(f_{i})\geq3$ for $i\in{\llbracket r+1,p^2-1\rrbracket}$, putting $F_p=\overline{{\nicefrac{f_p}{p}}}$, $F_{p^2}=\overline{{\nicefrac{f_{p^2}}{p}}}$, and $G_i=\overline{{\nicefrac{f_i}{p^2}}}$ for all $i\neq{}p,p^2$ we have 1. $-{\nicefrac{F_p}{F_{p^2}}}=\bar\rho^{p-1}$ for some $\bar\rho\in{\kappa}_K^\times$, 2. ${\nicefrac{\ell{}G_r}{F_pF_{p^2}}}=\bar\eta^{p-1}$ for some $\bar\eta\in{\kappa}_K^\times$. Let $L$ be the extension determined by $f(X)$, $\tilde{L}$ the normal closure over $K$, and $F$ the unique subextension of degree $p$ contained in $L$. Then ${\mathop{\mathrm{Gal}}\nolimits}(\tilde{L}/K)$ is a split extension of $G={\mathop{\mathrm{Gal}}\nolimits}(F/K)$ by the indecomposable ${\mathbb{F}}_p[G]$-module $M={\mathop{\mathrm{Gal}}\nolimits}(\tilde{L}/F)$ and furthermore defining $$U(X) = (-{\nicefrac{F_p}{F_{p^2}}})^\ell{}X^p - {\nicefrac{\ell{}G_r}{F_pF_{p^2}}}X - 1$$ we have that if - $U(X)$ has no root in ${\kappa}_K$, then $M$ has length $\ell+1$ and $\tilde{L}/K$ is formed by an unramified extension followed by a totally ramified with upper ramification breaks $1,2,\dots,\ell$, - $U(X)$ has some root in ${\kappa}_K$, then $M$ has length $\ell$ and $\tilde{L}/K$ is totally ramified with upper ramification breaks $1,2,\dots,\ell$. What is left is the easy case for $\ell=1$, which is considered separately. In this case $L/K$ has $1$ as unique ramification jump, $v_p(f_1)=1$ while $v_p(f_i)\geq2$ for $i\in{\llbracket 2,p^2-1\rrbracket}$, and consequently put $F_i=\overline{{\nicefrac{f_i}{p}}}$ for $i=1,p,p^2$. The map ${\nicefrac{U_{1,L}}{U_{2,L}}}\rightarrow{\nicefrac{U_{1,K}}{U_{2,K}}}$ induced by $N_{L/K}$ is described by the additive polynomial $A(Y)=F_{p^2}Y^{p^2}+F_pY^p+F_1Y$, and $L/K$ is Galois precisely when $N_{L/K}(U_{1,L})=1+pW$ for a subspace $W$ of codimension $2$ in ${\kappa}_K$, that is when $A(Y)$ splits completely in ${\kappa}_K$. On the other hand the normal closure $\tilde{L}/K$ is a $p$-extension if and only if $L$ becomes abelian elementary over the unique unramified extension of degree $p$ of $K$, or equivalently if $A(Y)$ splits completely over the unique extension of degree $p$ of ${\kappa}_K$. \[theo4\] Assume that $f(X)$ is such that 1. $v_p(f_{p})\leq{}1$, and $v_p(f_{pi})\geq2$ for $i\in{\llbracket 2,p-1\rrbracket}$, 2. $v_p(f_1)=1$, and $v(f_i)\geq2$ for $i\in{\llbracket 2,p^2-1\rrbracket}$, and putting $F_i=\overline{{\nicefrac{f_i}{p}}}$ for $i=1,p,p^2$ 1. the polynomial $F_{p^2}Y^{p^2}+F_pY^p+F_1Y$ has a root in ${\kappa}_K^\times$, and ${\nicefrac{F_1}{F_{p^2}}}\in({\kappa}_K^\times)^{p-1}$. Let $L$ be the extension determined by $f(X)$, $\tilde{L}$ the normal closure over $K$, and $F$ the unique subextension of degree $p$ contained in $L$. Then - if $F_{p^2}Y^{p^2}+F_pY^p+F_1Y$ does not split in ${\kappa}_K$ then $M$ has length $2$, and $\tilde{L}/F$ is formed by an unramified extension followed by a totally ramified extension with upper ramification break $1$, and ${\mathop{\mathrm{Gal}}\nolimits}(\tilde{L}/K)$ is a split extension of ${\mathop{\mathrm{Gal}}\nolimits}(F/K)$ by ${\mathop{\mathrm{Gal}}\nolimits}(\tilde{L}/F)$, - if $F_{p^2}Y^{p^2}+F_pY^p+F_1Y$ has all roots in ${\kappa}_K$ then $M$ has length $1$ and $L/K$ is an abelian elementary $p$-extension. Theorems \[theo2\], \[theo3\] and \[theo4\] cover all possible ramification jumps of the extension $L/F$, so they completely describe the Galois groups of polynomials of degree $p^2$ whose splitting field is a $p$-extension. Polynomials of degree $p^3$ generating a cyclic extension ========================================================= We proceed with the same strategy used for the polynomials of degree $p^2$, starting from the conditions on the valuations of the coefficients. Let $f(X)=X^{p^3}+\dots+f_{p^3-1}X+f_{p^3}$, since the different has now valuation $4p^3-p^2-p-2$ it will be determined by the monomial $f_{p^2+p+1}X^{p^3-p^2-p-1}$, $v_p(f_{p^2+p+1})=3$, $v_p(f_i)\geq3$ if $(i,p)=1$ and $v_p(f_i)\geq4$ if furthermore $i>p^2+p+1$. Let $\pi$ be a root, the coefficients of the term of degree $p$ of the ramification polynomial $f(X+\pi)$ will have valuation $(p^3-p^2)\cdot2+(p^2-p)\cdot{}(p+1)=3p^3-p^2-2p$ and has to come from a monomial $f_{p^3-i}(X+\pi)^i$ contributing the term $\binom{i}{p}f_{p^3-i}X^p\pi^{i-p}$, and we deduce the $i$ has to be $i=p^3-p^2-p$, that $v_p(f_{p^2+p})=2$, that $v_p(f_{pi})\geq2$ for $(i,p)=1$ and $v_p(f_{pi})\geq3$ if furthermore $i\geq{}p+2$. Similarly considering the coefficient of the term of degree $p^2$ of the ramification polygon which shall have valuation $\geq2p^3-2p^2$ we obtain that $v_p(f_{p^2})=1$ and $v_p(f_{p^2i})\geq2$ for all indices such that $(i,p)=1$. We must have 1. $v_p(f_{p^2})=1$ and $v_p(f_{p^2i})\geq{}2$ for $i\in{\llbracket 2,p-1\rrbracket}$, 2. $v_p(f_{pi})\geq{}2$ for all $i\in{\llbracket 1,p-1\rrbracket}$, $v_p(f_{p^2+p})=2$, and $v_p(f_{pi})\geq{}3$ for all $i\in{\llbracket p+1,p^2-1\rrbracket}$, 3. $v_p(f_{i})\geq{}3$ for all $i\in{\llbracket 1,p^2+p-1\rrbracket}$, $v_p(f_{p^2+p+1})=3$ and $v_p(f_{i})\geq{}4$ for all $i\in{\llbracket p^2+p+2,p^3-1\rrbracket}$. Again working like in degree $p^2$, we shall require $N_{L/K}(U_{1,L})^{p^{i-1}}\cap{}U_{i+1,L}$ to be contained in $1+p^iV$ for $1\leq{}i\leq3$ and some ${\mathbb{F}}_p$-vector space $V$, and after determining $V$ we will have to verify the condition on the combinations of the norms of elements of the form $1+\theta\pi^{\ell}$ for a unit $\theta$, and $1\leq\ell\leq{}p^2+p+1$ and $(\ell,p)=1$. Let’s expand again $\prod_{i=0}^\ell{}\tilde{f}(\zeta_\ell^i{}T)$ modulo $p^4$, taking into account the valuations of the $f_i$ and evaluating directly the $\Sigma_\lambda(\ell)$ via Prop. \[prop\_sums\] it can be written with the terms in increasing valuation as $$\begin{aligned} {\mathfrak{p}}_K\!\ni\! &\left[\qquad\qquad\qquad\qquad\qquad \begin{gathered} + \delta_{\ell,1}f_{p^2}T^{p^2} + \delta_{\ell,1}f_{p^3}T^{p^3} \end{gathered} \right.\\ {\mathfrak{p}}_K^2\!\ni\! &\left[\qquad\qquad \begin{gathered} + \frac{1}{2}\delta_{\ell,2}^{[2]}f_{p^2}^2T^{2p^2} + \delta_{\ell,p+1}^{[2]}f_{p^2}f_{p^3}T^{p^3+p^2}+ \frac{1}{2}\delta_{\ell,2}^{[2]}f_{p^3}^2T^{2p^3} \\ + \sum_{j\in{\llbracket 2,p-1\rrbracket}} \delta_{\ell,j}f_{p^2j}T^{p^2j} + \sum_{k\in{\llbracket 1,p+1\rrbracket}} \delta_{\ell,k}f_{pk}T^{pk} \end{gathered} \right.\\ {\mathfrak{p}}_K^3\!\ni\! &\left[ \begin{gathered} + \frac{1}{3}\delta_{\ell,3}^{[3]}f_{p^2}^3T^{3p^2} + \delta_{\ell,p+2}^{[3]}f_{p^3}f_{p^2}^2T^{p^3+2p^2} + \delta_{\ell,2p+1}^{[3]}f_{p^3}^2f_{p^2}T^{2p^3+p^2} + \frac{1}{3}\delta_{\ell,3}^{[3]}f_{p^3}^3T^{3p^3}\\ + \!\!\!\!\sum_{j\in{\llbracket 2,p-2\rrbracket}}\! \!\delta_{\ell,j+1}^{[2]} f_{p^2}f_{p^2j}T^{p^2+p^2j} + \delta_{\ell,1}^{[2]} f_{p^2}f_{p^3-p^2}T^{p^3} + \!\!\!\!\sum_{k\in{\llbracket 1,p+1\rrbracket}}\!\! \delta_{\ell,k+p}^{[2]} f_{p^2}f_{pk}T^{p^2+pk}\\ + \sum_{j\in{\llbracket 2,p-1\rrbracket}} \delta_{\ell,j+p}^{[2]} f_{p^3}f_{p^2j}T^{p^3+p^2j} + \sum_{k\in{\llbracket 1,p+1\rrbracket}} \delta_{\ell,k+p^2}^{[2]} f_{p^3}f_{pk}T^{p^3+pk}\\ + \sum_{j\in{\llbracket p+2,p^2-1\rrbracket}} \delta_{\ell,j} f_{pj}T^{pj} + \sum_{k\in{\llbracket 1,p^2+p+1\rrbracket}} \delta_{\ell,k} f_{k}T^{k} \end{gathered} \right.\end{aligned}$$ While this expansion looks scary we can start noticing that since raising to a $p$-th power induces an automorphism on the set of multiplicative representatives we have considering the expansion modulo $p^3$ that the conditions stated in Theorem \[theo1\] must be satisfied with $f_{pi}$in place of $f_i$. Consequently put $F_{i}=\overline{{\nicefrac{f_{i}}{p}}}$ for $i=p^2,p^3$, $G_i=\overline{{\nicefrac{g_i}{p^2}}}$ for $i\in{}p{\llbracket 1,p+1\rrbracket}$ or $i\in{}p^2{\llbracket 2,p-1\rrbracket}$, let $A(Y)=F_{p^3}Y^p+F_{p^2}Y$ and put $V=A({\kappa}_K)$. Such conditions are satisfied if and only if $V$ has codimension $1$ in ${\kappa}_K$ and the norms contained in $U_{1,K}$ or $U_{2,K}$ are respectively in $1+pV$ and $1+p^2V$. Similarly to the case in degree $p^2$, for $\ell\geq2$ this sum can be written as $D_\ell(\theta)=\sum_{\ell\mid{}k}d_k(\theta^k)$ where the $d_\ell(T^\ell)$ are the polynomial obtained if every $\delta^{[m]}_{\ell,i}$ is interpreted as a Kronecker’s delta and $1+d_\ell(\theta)\equiv{}N(E(\theta\pi^\ell))\imod{p^4}$. For $\ell=1$ there are exceptions because $\delta^{[m]}_{\ell,i}=0$ for $\ell<m$. We require the norms in $U_{3,K}$ to be in $1+p^3V$, and let’s concentrate first on the case of $\ell\in{\llbracket p+2,p^2+p+1\rrbracket}$ so that the norms $N_{L/K}(1+\theta\pi^\ell)$ already live in $U_{3,K}$, and the first few terms of the expansion disappear. For such indices $\ell$, $d_\ell(\theta)$ shall be in $p^3V$ for each representative $\theta$, and dividing by $p^3$ we can consider the additive polynomials $A_\ell(Y)=\overline{{\nicefrac{d_\ell(Y)}{p^3}}}$ which depending on $\ell$ are $$\begin{gathered} \qquad \qquad\qquad -G_{p(\ell-p^2)}F_{p^3}Y^p + H_{\ell}Y \qquad\qquad\qquad \qquad \ell\in{\llbracket p^2+1,p^2+p+1\rrbracket},\\ \qquad\qquad\qquad\qquad H_{p\ell}Y^p + H_{\ell}Y \qquad\qquad\qquad\qquad\qquad \ell\in{\llbracket 2p+2,p^2-1\rrbracket},\\ \qquad F_{p^3}^2F_{p^2}Y^{p^2}+(H_{p\ell}-F_{p^2}G_{p(p+1)})Y^p + H_{\ell}Y \quad\qquad \ell=2p+1,\\ \qquad -F_{p^3}F_{p^2(\ell-p)}Y^{p^2}+(H_{p\ell}-F_{p^2}G_{p(\ell-p)})Y^p + H_{\ell}Y \quad\qquad \ell\in{\llbracket p+3,2p-1\rrbracket},\\ (F_{p^3}F_{p^2}^2-F_{p^3}F_{2p^2})Y^{p^2}+(H_{p\ell}-F_{p^2}G_{2p})Y^p + H_{\ell}Y \qquad \ell=p+2,\end{gathered}$$ where we have put $H_k=\overline{{\nicefrac{f_k}{p^3}}}$ for $k\in{\llbracket 1,p^2+p+1\rrbracket}$ and $k\in{}p{\llbracket p+2,p^2-1\rrbracket}$. For each $\ell\in{\llbracket p+2,p^2+p+1\rrbracket}$ we shall have $A_\ell({\kappa}_K)\subseteq{}A({\kappa}_K)$. For $\ell\leq{}p+1$ the question is a bit more complicate because in general the norms of $1-\theta\pi^\ell$ will not be contained in $U_{3,K}$, but a proper combination of norms of elements of this form may be, and we should require it to be in $1+p^3V$. However for $\eta$ varying the elements $N_{L/K}(1-\eta\pi)^p$ have norms which cover all classes modulo $U_3$, and consequently each $N_{L/K}(1-\theta\pi^\ell)$ can be reduced into $U_{3,K}$ by multiplication by an suitable $N_{L/K}(1-\eta\pi)^p$ for some $\eta$, and we should verify that all such reduction are actually in $1+p^3V$. The condition for more complicated combinations will certainly also be ensured. Since the map ${\nicefrac{{\mathfrak{p}}_K^2}{{\mathfrak{p}}_K^4}}\rightarrow{\nicefrac{U_{2,K}}{U_{4,K}}}$ induced by $x\mapsto1+x$ is still an isomorphism we have that a proper adjustment of $1+\theta\pi^\ell$ (e.g. via the Artin-Hasse exponential) has norm of the form $1+c_\ell(\theta)+d_\ell(\theta)$, where $c_\ell(\theta)$ is obtained like in the case $p^2$ by with $f_i$ replaced by $f_{pi}$ and $T$ by $T^p$, plus some modification for $\ell=2,3$. In other words depending on $2\leq\ell\leq{}p+1$ we have that the remaining term which we call $g_\ell(Y)$ is $$\label{condb2} \begin{array}{cc} \left\{-f_{p^3}f_{p^2}Y^{p^2}+f_{p^2+p}Y^p\right\} -f_{p^2}f_pY^p + f_{p+1}Y & [p+1], \\ \left\{f_{p^2\ell}Y^{p^2}+ f_{p\ell}Y^p\right\} - f_{p^2}f_{p^2(\ell-1)}Y^{p^2} + f_{\ell}Y & [4,p-1], \\ \left\{f_{3p^2}Y^{p^2}+f_{3p}Y^p\right\} + \frac{1}{3}f_{p^3}^3Y^{p^3} + \frac{1}{3}f_{p^2}^3Y^{p^2} -f_{p^2}f_{2p^2}Y^{p^2} + f_{3}Y & [3], \\ \left\{-\frac{1}{2}f_{p^3}^2Y^{p^3}+\left(f_{2p^2} -\frac{1}{2}f_{p^2}^2\right)Y^{p^2}+f_{2p}Y^p\right\} + f_{2}Y & [2], \\ \end{array}$$ where under braces are the terms which are not identically in ${\mathfrak{p}}_{K}^3$. On the other hand $$N(1+\eta\pi) = 1+f_{p^3}\eta^{p^3} + f_{p^2}\eta^{p^2} + f_{p}\eta^{p} \mod p^2V$$ and consequently $$N(1+\eta\pi)^p = 1+\left\{pf_{p^3}\eta^{p^3} + pf_{p^2}\eta^{p^2}\right\} + pf_{p}\eta^{p} \mod p^3V,$$ with again under braces are the terms which are not identically in ${\mathfrak{p}}_{K}^3$. Consequently let’s consider the polynomial $$h(Z) = \{pf_{p^3}Z^{p^2} + pf_{p^2}Z^{p}\} + pf_{p}Z,$$ we are looking for values of $Z=\phi_\ell(Y)$, that will be the lifting of some additive polynomials in $Y$, such that $g_\ell(Y)-h(\phi_\ell(Y))\in{\mathfrak{p}}^3_K$, to impose the condition that it shall be in $p^3V$ as well. The connected additive polynomials $\overline{{\nicefrac{g_\ell(Y)}{p^2}}}$, which we denote by $B_\ell(Y^p)$ replacing $Y^p$ by $Y$, are forced to have image contained $V$ which is the image of $\overline{{\nicefrac{h(Y)}{p^2}}}=A(Y^{p})$, and the condition is that $B_\ell(Y)=A(D_\ell(Y))$ for some other additive polynomial $D_\ell(Y)$ whose coefficients can be deduced easily. In particular, being $A(Y)=F_{p^3}Y^p+F_{p^2}Y$ and $B_\ell(Y)$ the polynomials $$ \begin{array}{cc} -F_{p^2}F_{p^3}Y^p + G_{p^2+p}Y & [p+1], \\ G_{p^2\ell}Y^p + G_{p\ell} Y & [3,\ p-1],\\ -\frac{1}{2}F_{p^3}^2Y^{p^2}+\left(G_{2p^2}-\frac{1}{2}F_{p^2}^2\right)Y^p+G_{2p}Y & [2], \end{array}$$ in view of Prop. \[prop\_add\] we can take as $D_\ell(Y)$ respectively the polynomials $$ \begin{array}{cl} {\nicefrac{G_{p\ell}}{F_{p^2}}}Y & [3,\ p+1], \\ -\frac{1}{2}F_{p^3}^{{\nicefrac{1}{p}}}Y^{p^2}+{\nicefrac{G_{2p}}{F_{p^2}}}Y & [2]. \end{array}$$ Now, $B_\ell(Y^p)=A((D_\ell^{{\nicefrac{1}{p}}}(Y))^p)$ where $D_\ell^{{\nicefrac{1}{p}}}(Y)$ is $D_\ell(Y)$ with the map $x\mapsto{}x^{{\nicefrac{1}{p}}}$ applied to the coefficients. Given the definitions of $A(Y)$ and $B_\ell(Y)$ in terms of the $h(Y)$ and $g_\ell(Y)$, we have that we can take as $\phi_\ell(Y)$ any lifting of $D_\ell^{{\nicefrac{1}{p}}}(Y)$ to ${\mathcal{O}}_K[Y]$. For $3\leq\ell\leq{}p+1$ let’s take a $\rho\in{\mathcal{O}}_K$ such that $\bar\rho^p={\nicefrac{G_{p\ell}}{F_{p^2}}}=\overline{{\nicefrac{f_{p\ell}}{pf_{p^2}}}}$, then $D_\ell(Y)=\bar\rho{}^pY$ and we can take $\phi_\ell(Y)=\rho{}Y$, and the polynomials $\overline{\frac{1}{p^3}(g_\ell(Y)-h(\phi_\ell(Y)))}$ should take values in $V$. Considering that $$h(\phi_\ell(Y)) = \left\{pf_{p^3}\rho^{p^2}Y^{p^2} + pf_{p^2}\rho^{p}Y^{p}\right\} + pf_{p}\rho{}Y,$$ depending on $\ell$ they are $$\begin{gathered} \overline{\left(-{\nicefrac{f_{p^3}f_{p^2}}{p^3}}-{\nicefrac{f_{p^3}\rho^{p^2}}{p^2}}\right)}Y^{p^2} \\ +\left[\overline{\left({\nicefrac{f_{p^2+p}}{p^3}}-{\nicefrac{f_{p^2}\rho^p}{p^2}}\right)}-F_{p^2}G_p\right]Y^p + H_{p+1}Y \end{gathered}$$ for $\ell=p+1$, $$\begin{gathered} \left[\overline{\left(-{\nicefrac{f_{p^2\ell}}{p^3}}-{\nicefrac{f_{p^3}\rho^{p^2}}{p^2}}\right)}-F_{p^2}G_{p^2(\ell-1)}\right]Y^{p^2}\\ +\overline{\left({\nicefrac{f_{p\ell}}{p^3}}-{\nicefrac{f_{p^2}\rho^p}{p^2}}\right)}Y^p + H_\ell{} Y, \end{gathered}$$ for $4\leq\ell=p-1$, and $$\begin{gathered} \frac{1}{3}F_{p^3}^3Y^{p^3} + \left[\overline{\left({\nicefrac{f_{3p^2}}{p^3}}-{\nicefrac{f_{p^3}\rho^{p^2}}{p^2}}\right)} + \frac{1}{3}F_{p^2}^3 -F_{p^2}G_{2p^2} \right]Y^{p^2} \\ + \overline{\left({\nicefrac{f_{3p}}{p^3}}-{\nicefrac{f_{p^2}\rho^p}{p^2}}\right)}Y^p + H_{3}Y \end{gathered}$$ for $\ell=3$. For $\ell=2$ let’s take $\rho,\tau\in{\mathcal{O}}_K$ such that $\bar\rho^p={\nicefrac{G_{p\ell}}{F_{p^2}}}=\overline{{\nicefrac{f_{p\ell}}{pf_{p^2}}}}$ and $\bar\tau^{p^2}=-\frac{1}{2}{F_{p^3}}=-\frac{1}{2}\overline{{\nicefrac{f_{p^3}}{p^3}}}$. Then $D_\ell(Y)=\bar\tau^pY^p+\bar\rho^pY$ so that we can take $\phi_2(\ell)=\tau{}Y^p+\rho{}Y$, and we have $$\begin{gathered} h(\phi_2(Y)) = \left\{pf_{p^3}(\tau{}Y^p+\rho{}Y)^{p^2} + pf_{p^2}(\tau{}Y^p+\rho{}Y)^{p}\right\} + pf_{p}(\tau{}Y^p+\rho{}Y)\\ = pf_{p^3}\tau^{p^3}Y^{p^3}+pf_{p^3}\rho^{p^2}Y^{p^2} + pf_{p^3} \sum_{i=1}^{p-1} \binom{p^2}{ip} \tau^{ip}\rho^{(p-i)p}Y^{ip^2+(p-1)p} + {\mathcal{O}}(p^4) \\ + pf_{p^2}\tau^{p^2}Y^{p^2}+pf_{p^2}\rho^{p}Y^{p} + pf_{p^2} \sum_{i=1}^{p-1} \binom{p}{i} \tau^{i}\rho^{(p-i)}Y^{ip+(p-1)} \\ + pf_{p}\tau{}Y^p+pf_{p}\rho{}Y.\end{gathered}$$ Considering that $\frac{1}{p}\binom{p}{i} \equiv \frac{1}{p}\binom{p^2}{ip}\imod{p}$ and the terms in the sums can be paired in elements that are $pf_{p^3}\binom{p^2}{ip}Z^p+pf_{p^2}\binom{p}{i}Z$ for $Z=\tau^{i}\rho^{(p-i)}Y^{ip+(p-1)}$ and hence in $p^3V$ for each $Z$, we have that up to some element in $p^3V$ we can write $h(\phi_2(Y))$ as $$pf_{p^3}\tau^{p^3} Y^{p^3} +(pf_{p^3}\rho^{p^2} + pf_{p^2}\tau^{p^2}) Y^{p^2} +(pf_{p^2}\rho^{p} + pf_{p}\tau{}) Y^p + pf_{p}\rho{}Y.$$ Consequently up to some element of $V$ the polynomial $\overline{\frac{1}{p^3}(g_\ell(Y)-h(\phi_2(Y)))}$ is the $$\begin{gathered} \overline{\left(-\frac{1}{2}{\nicefrac{f_{p^3}^2}{p^3}}-{\nicefrac{f_{p^3}\tau^{p^3}}{p^2}}\right)}Y^{p^3} +\overline{\left({\nicefrac{f_{2p^2}}{p^3}}-\frac{1}{2}{\nicefrac{f_{p^2}^2}{p^3}} -{\nicefrac{f_{p^3}\rho^{p^2}}{p^2}}-{\nicefrac{f_{p^2}\tau^{p^2}}{p^2}}\right)}Y^{p^2}\\ +\left(\overline{\left({\nicefrac{f_{2p}}{p^3}}-{\nicefrac{f_{p^2}\rho^{p}}{p^2}}\right)}-G_p\tau\right)Y^p + \left(H_{2}-G_p\rho \right)Y, \end{gathered}$$ which is required to take values in $V$. One last effort is required: for $\ell=1$ in the case that $1-\theta\pi$ has norm in $U_{2,K}$ (and hence in $1+p^2V$), that is when $\theta$ is such that $A(\bar\theta^{p^2})=0$, we should also have that taking $\eta$ such that $(1-\theta\pi)(1-\eta\pi)^{-p}$ has norm in $U_{3,K}$, than that norm is required to be actually in $1+p^3V$. Let $\theta=T$ be as required, the terms which disappear because $\ell=1$ are $$\frac{1}{2}f_{p^2}^2T^{2p^2} + f_{p^3}f_{p^2}T^{p^3+p^2}+\frac{1}{2}f_{p^3}^2T^{2p^3} =\frac{1}{2}\left(f_{p^2}T^{p^2}+f_{p^3}T^{p^3}\right)^2,$$ then $$\frac{1}{3}f_{p^2}^3T^{3p^2} + f_{p^3}f_{p^2}^2T^{p^3+2p^2}+ f_{p^3}^2f_{p^2}T^{2p^3+p^2} +\frac{1}{3}f_{p^3}^3T^{3p^3} =\frac{1}{3}\left(f_{p^2}T^{p^2}+f_{p^3}T^{p^3}\right)^3,$$ and the sums can be decomposed as sums of $(f_{p^2}T^{p^2}+f_{p^3}T^{p^3})f_{p^2j}T^{p^2j}$ and of $(f_{p^2}T^{p^2}+f_{p^3}T^{p^3})f_{pk}T^{pk}$, and in particular all such terms are in ${\mathfrak{p}}_K^4$ considering the hypotheses on $T$. Consequently such terms can be assumed to be present, and removing the extra terms we already studied (or considering the norm of $E(\theta\pi)$) the remaining terms are $$w(T) = f_{p^3}T^{p^3} + f_{p^2}T^{p^2}-f_{p^2}f_{p^3-p^2}T^{p^2}+f_pT^p+f_1{}T.$$ Assume $\overline{\frac{1}{p^2}\left(f_{p^3}\theta^{p^3}+f_{p^3}\theta^{p^2}+f_p\theta^p\right)}$ can be written as $F_{p^3}\bar\alpha^{p^2}+F_{p^2}\bar\alpha^p$ for some $\bar\alpha$, then taking any lift $\alpha$ of $\bar\alpha$ we can consider $w(\theta)-h(\alpha)$ which comes from a norm of the required type, and should be in $p^3V$. At last, we can state the \[theox\] The Eisenstein polynomial $f(X)=X^{p^3}+f_1X^{p^3-1}+\dots+f_{p^3-1}X+f_{p^3}$ determines a Galois extension of degree $p^3$ over $K$ if and only if 1. $v_p(f_{p^2})=1$ and $v_p(f_{p^2i})\geq{}2$ for $i\in{\llbracket 2,p-1\rrbracket}$, 2. $v_p(f_{pi})\geq{}2$ for all $i\in{\llbracket 1,p-1\rrbracket}$, $v_p(f_{p^2+p})=2$, and $v_p(f_{pi})\geq{}3$ for all $i\in{\llbracket p+1,p^2-1\rrbracket}$, 3. $v_p(f_{i})\geq{}3$ for all $i\in{\llbracket 1,p^2+p-1\rrbracket}$, $v_p(f_{p^2+p+1})=3$ and $v_p(f_{i})\geq{}4$ for all $i\in{\llbracket p^2+p+2,p^3-1\rrbracket}$, putting $F_{p^2}=\overline{{\nicefrac{f_{p^2}}{p}}}$, $F_{p^3}=\overline{{\nicefrac{f_{p^3}}{p}}}$, and $\overline{G_i={\nicefrac{f_i}{p^2}}}$ for all $i$ in $p^2{\llbracket 2,p-1\rrbracket}$ or in $p{\llbracket 1,p+1\rrbracket}$ we have 1. $-{\nicefrac{F_{p^2}}{F_{p^3}}}\in{}{\kappa}_K^{p-1}$, 2. $G_{p(p+1)}^p=-F_{p^2}^{p+1}$, 3. $G_{p^2\ell}=F_{p^3}\left({\nicefrac{G_{\ell{}p}}{F_{p^2}}}\right)^p$ for $\ell\in{\llbracket 3,p-1\rrbracket}$, 4. $G_{2p^2}=F_{p^3}\left({\nicefrac{G_{2p}}{F_{p^2}}}\right)^p+\frac{1}{2}F_{p^2}\left(F_{p^2}-F_{p^3}^{{\nicefrac{1}{p}}}\right)$, if $\rho$ is such that $\bar\rho^{p(p-1)}=-{\nicefrac{F_{p^2}}{F_{p^3}}}$ we have (independently of $\rho$) 1. $\overline{\frac{1}{p^2}\left(f_{p^3}\rho^{p^2}+f_{p^2}\rho^{p}+f_p\rho\right)}=F_{p^3}\alpha^p+F_{p^2}\alpha$ for some $\alpha\in{}{\kappa}_K$, putting $H_i=\overline{{\nicefrac{f_i}{p^3}}}$ for $i$ in ${\llbracket 1,p^2+p+1\rrbracket}$ or in $p{\llbracket p+2,p^2-1\rrbracket}$ we have 1. $-G_{p(\ell-p^2)}F_{p^3}=F_{p^3}({\nicefrac{H_{\ell}}{F_{p^2}}})^p$ for $\ell\in{\llbracket p^2+1,p^2+p+1\rrbracket}$, 2. $H_{p\ell}=F_{p^3}({\nicefrac{H_{\ell}}{F_{p^2}}})^p$ for $\ell\in{\llbracket 2p+2,p^2-1\rrbracket}$, 3. $H_{p(2p+1)}-F_{p^2}G_{p(p+1)} =F_{p^3}({\nicefrac{H_{2p+1}}{F_{p^2}}})^p+F_{p^2}(F_{p^3}F_{p^2})^{{\nicefrac{1}{p}}}$, 4. $H_{p\ell}-F_{p^2}G_{p(\ell-p)} = F_{p^3}({\nicefrac{ H_{\ell} }{F_{p^2}}})^p - F_{p^2}(F_{p^2(\ell-p)})^{{\nicefrac{1}{p}}}$ for $\ell\in{\llbracket p+3,2p-1\rrbracket}$, 5. $H_{p(p+2)}-F_{p^2}G_{2p} = F_{p^3}({\nicefrac{ H_{p+2} }{F_{p^2}}})^p + F_{p^2}(F_{p^2}^2-F_{2p^2})^{{\nicefrac{1}{p}}}$, for each $\ell\in{\llbracket 3,p+1\rrbracket}$, let $\rho_\ell$ be such that $\bar\rho_\ell^p={\nicefrac{G_{p\ell}}{F_{p^2}}}$. Then 1. putting $$Q_{p+1} = \overline{\left({\nicefrac{f_{p^2+p}}{p^3}}-{\nicefrac{f_{p^2}\rho_{p+1}^p}{p^2}}\right)}-F_{p^2}G_p, \qquad R_{p+1} = \overline{\left(-{\nicefrac{f_{p^3}f_{p^2}}{p^3}}-{\nicefrac{f_{p^3}\rho_{p+1}^{p^2}}{p^2}}\right)},$$ we have $Q_{p+1} = F_{p^3}({\nicefrac{ H_{p+1} }{F_{p^2}}})^p + F_{p^2}({\nicefrac{ R_{p+1} }{F_{p^3}}})^{{\nicefrac{1}{p}}}$, 2. for each $4\leq\ell\leq{}p-1$ putting $$Q_\ell = \overline{\left({\nicefrac{f_{p\ell}}{p^3}}-{\nicefrac{f_{p^2}\rho_\ell^p}{p^2}}\right)}-F_{p^2}G_p, \qquad R_\ell = \overline{\left(-{\nicefrac{f_{p^2\ell}}{p^3}}-{\nicefrac{f_{p^3}\rho_\ell^{p^2}}{p^2}}\right)}-F_{p^2}G_{p^2(\ell-1)},$$ we have $Q_\ell = F_{p^3}({\nicefrac{ H_{\ell} }{F_{p^2}}})^p + F_{p^2}({\nicefrac{ R_\ell }{F_{p^3}}})^{{\nicefrac{1}{p}}}$, 3. putting $$Q_3 = \overline{\left({\nicefrac{f_{3p}}{p^3}}-{\nicefrac{f_{p^2}\rho_3^p}{p^2}}\right)}, \qquad R_3 = \overline{\left({\nicefrac{f_{3p^2}}{p^3}}-{\nicefrac{f_{p^3}\rho_3^{p^2}}{p^2}}\right)} + \frac{1}{3}F_{p^2}^3 -F_{p^2}G_{2p^2}$$ we have $\frac{1}{3}F_{p^2}(F_{p^3}^2)^{{\nicefrac{1}{p}}}+F_{p^3}({\nicefrac{Q_3}{F_{p^2}}})^p=R_3+ F_{p^3}({\nicefrac{F_{p^3}}{F_{p^2}}})^p({\nicefrac{H_3}{F_{p^2}}})^{p^2}$, let $\rho_2,\tau_2\in{\mathcal{O}}_K$ such that $\bar\rho_2^p={\nicefrac{G_{p\ell}}{F_{p^2}}}$ and $\bar\tau_2^{p^2}=-\frac{1}{2}{F_{p^3}}$. Then 1. putting $$\begin{gathered} P_2 = H_{2}-G_p\bar\rho,\qquad Q_2 = \overline{\left({\nicefrac{f_{2p}}{p^3}}-{\nicefrac{f_{p^2}\rho^{p}}{p^2}}\right)}-G_p\bar\tau,\\ R_2 = \overline{\left({\nicefrac{f_{2p^2}}{p^3}}-\frac{1}{2}{\nicefrac{f_{p^2}^2}{p^3}} -{\nicefrac{f_{p^3}\rho^{p^2}}{p^2}}-{\nicefrac{f_{p^2}\tau^{p^2}}{p^2}}\right)},\qquad S_2 = \overline{\left(-\frac{1}{2}{\nicefrac{f_{p^3}^2}{p^3}}-{\nicefrac{f_{p^3}\tau^{p^3}}{p^2}}\right)} \end{gathered}$$ we have $F_{p^2}({\nicefrac{S_2}{F_{p^3}}})^{{\nicefrac{1}{p}}}+F_{p^3}({\nicefrac{Q_2}{F_{p^2}}})^p=R_2+ F_{p^3}({\nicefrac{F_{p^3}}{F_{p^2}}})^p({\nicefrac{P_2}{F_{p^2}}})^{p^2}$, if $\rho,\xi$ are such that $\bar\rho^{p^2(p-1)}=-{\nicefrac{F_{p^2}}{F_{p^3}}}$ and $$\overline{\frac{1}{p^2}\left(f_{p^3}\rho^{p^3}+f_{p^2}\rho^{p^2}+f_p\rho^p\right)} =F_{p^3}\bar\xi^{p^2}+F_{p^2}\bar\xi^p,$$ 1. we have that $$\overline{\frac{1}{p^3}\left(f_{p^3}(\rho^{p^3}-\xi^{p^2})+f_{p^2}(\rho^{p^2}-\xi^{p})+f_p(\rho^p-\xi) -f_{p^2}f_{p^3-p^2}\rho^{p^2}+f_1\rho\right)}$$ is also of the form $F_{p^3}\bar\omega^p+F_{p^2}\bar\omega$ for some $\bar\omega\in{\kappa}_K$. <!-- --> 1. $b_p = F_{p^3}({\nicefrac{ b_1 }{F_{p^2}}})^p$, 2. $c_p = F_{p^3}({\nicefrac{ c_1 }{F_{p^2}}})^p + F_{p^2}({\nicefrac{ c_{p^2} }{F_{p^3}}})^{{\nicefrac{1}{p}}}$, 3. $F_{p^2}({\nicefrac{d_{p^3}}{F_{p^3}}})^{{\nicefrac{1}{p}}}+F_{p^3}({\nicefrac{d_{p}}{F_{p^2}}})^p=d_{p^2}+ F_{p^3}({\nicefrac{F_{p^3}}{F_{p^2}}})^p({\nicefrac{d_1}{F_{p^2}}})^{p^2}$. Sums of roots of unity ====================== We finally prove the lemma about the $\Sigma_\lambda(\ell)$, it is actually much more than needed but nevertheless is has a nice statement which could still be useful in similar circumstances: \[combinat\] Let $\lambda=(\lambda_1,\lambda_2,\dots,\lambda_r)$ be a partition, then $$\Sigma_\lambda(\ell) = \sum_{\lambda=\sqcup_{j\in{}J}\lambda^{(j)}} \ell^{\#J} \cdot \prod_{j\in J} (-1)^{\#\lambda^{(j)}-1} (\#\lambda^{(j)}-1)!$$ where the sum is over all the partitions $\lambda=\bigsqcup_{j\in{}J}\lambda^{(j)}$ (as set) such that for each $j\in{}J$ the sum $|\lambda^{(j)}|$ of the elements in $\lambda^{(j)}$ is multiple of $\ell$ and $\#\lambda^{(j)}$ is the cardinality of the subset $\lambda^{(j)}$. Let $A_{(i,j)}$ be the sets (indexed by the pairs $(i,j)$) of indices $(\iota_1,\dots,\iota_r)$ such that $\iota_i=\iota_j$, let $A_0$ be the set of all possible indices, and for $A\subseteq{}A_0$ denote by $\Sigma(A)$ the sum over all the indices in $A$. By inclusion-exclusion we have that $$\begin{aligned} \Sigma_\lambda(\ell) &= \Sigma(A_0)-\Sigma(\cup_{(i,j)} A_{(i,j)}) \\ &= \Sigma(A_0)-\sum_{(i,j)}\Sigma(A_{(i,j)}) + \sum_{(i,j)\neq (i',j')}\Sigma(A_{(i,j)}\cap A_{(i',j')}) - \dots\end{aligned}$$ Now let $A$ be the intersection of all the sets $A_{(i_k,j_k)}$ for a collection of pairs $P=\{(i_1,j_1),\dots,(i_s,j_s)\}_{s\in{}S}$, if we consider the graph with $R=\{1,\dots,r\}$ as vertices and the $(i_k,j_k)$ as edges we have that if we split $R$ in connected components $R=\sqcup_{t\in{T}}R_t$ then the allowed indices $\iota$ are those constant on each $R_t$, and calling $\iota_t$ the value taken on $R_t$ the sum $\Sigma(A)$ becomes $$\Sigma(A) = \prod_{t\in T} \sum_{\iota_t=0}^{\ell-1}\zeta_\ell^{\iota_t(\sum_{r\in{}R_t}\lambda_r)},$$ and this sum is $\ell^{\#T}$ when all the $\sum_{r\in{}R_t}\lambda_r$ are multiple of $\ell$ and $0$ if not. Note that $\Sigma(A)$ appears with sign equal to $(-1)^{\#S}$ in the inclusion-exclusion, so for each partition of $R$ in sets $R_t$ such that the sum of $\lambda_r$ for $r\in{}R_t$ is multiple of $\ell$ we have that to consider the all graphs with set of vertices $R$ and such that each $R_t$ is a connected component, and count the number of graphs with an even number of edges minus those with a odd number of edges. Now the total difference is the product of the differences over all the connected components, so we have $$\Sigma_\lambda(\ell) = \sum_{\lambda=\sqcup_{j\in{}J}\lambda^{(j)}} \ell^{\#J} \cdot \prod K_{\#\lambda^{(j)}}$$ where for each $i$ we denote by $K_i$ the difference of the number of connected graphs on $i$ vertices having an even and odd number of edges. The difference of the number of connected graphs $K_i$ on $i$ vertices with an even or odd number of vertices can be computed fixing an edge, and considering the graphs obtained adding or removing that edge. Those such that with or without it are connected come in pairs with an even and odd number of edges, the other graphs are obtained connecting two other connected graphs on $j$ and $i-j$ vertices. In particular choosing $j-1$ vertices to make one component with the first vertex of our distinguished edges we obtain $$K_{i+2} = - \sum_{j}^{i} \binom{i}{j} K_{i-j+1}K_{j+1}$$ for $i\geq0$, and $K_1=1$. Calling $G(X)$ the exponential generating function $\sum_{i=0}^\infty\frac{K_{i+1}}{i!}X^i$ we obtain that $$\frac{d}{dX}G(X) = -G(X)^2$$ with the additional condition that $K_1=1$, and this equation is clearly satisfied by ${\nicefrac{1}{1+X}}$ which can be the only solution. Consequently $K_{i+1}=(-1)^i\cdot{}i!$ and the lemma is proved.

--- abstract: 'Merging the [*Chandra*]{} and [*XMM–Newton*]{} deep surveys with the previously identified [*ROSAT*]{} surveys a unique sample of almost 1000 AGN–1 covering five orders of magnitude in 0.5–2 keV flux limit and six orders of magnitude in survey solid angle with $\sim95\%$ completeness has been constructed. The luminosity–redshift diagram is almost homogeneously filled. AGN–1 are by far the largest contributors to the soft X–ray selected samples. Their evolution is responsible for the break in the total 0.5–2 keV source counts. The soft X–ray AGN–1 luminosity function shows a clear change of shape as a function of redshift, confirming earlier reports of luminosity–dependent density evolution for optical quasars and X–ray AGN. The space density evolution with redshift changes significantly for different luminosity classes, showing a strong positive evolution, i.e. a density increase at low redshifts up to a certain redshift and then a flattening. The redshift, at which the evolution peaks, changes considerably with X–ray luminosity, from $z\approx$0.5–0.7 for luminosities $\log L_x$=42–43 erg s$^{-1}$ to $z\approx2$ for $\log L_x$=45–46 erg s$^{-1}$. The amount of density evolution from redshift zero to the maximum space density also depends strongly on X–ray luminosity, more than a factor of 100 at high luminosities, but less than a factor of 10 for low X–ray luminosities. For the first time, a significant decline of the space density of X–ray selected AGN towards high redshift has been detected in the range $\log L_x$=42–45 erg s$^{-1}$, while at higher luminosities the survey volume at high–redshift is still too small to obtain meaningful densities. A comparison between X–ray and optical properties shows now significant evolution of the X–ray to optical spectral index for AGN–1. The constraints from the AGN luminosity function and evolution in comparison with the mass function of massive dark remnants in local galaxies indicates, that the average supermassive black hole has built up its mass through efficient accretion ($\epsilon\sim10\%$) and is likely rapidly spinning.' author: - Günther Hasinger title: 'When Supermassive Black Holes were growing: Clues from Deep X–ray Surveys' --- Introduction ============ In recent years the bulk of the extragalactic X–ray background in the 0.1-10 keV band has been resolved into discrete sources with the deepest [*ROSAT*]{}, [*Chandra*]{} and [*XMM–Newton*]{} observations [@has98; @mus00; @gia01; @gia02; @has01; @ale03; @wor04; @bau04]. Optical identification programmes with Keck [@schm98; @leh01; @bar01a; @bar03] and VLT [@szo04; @fio03] find predominantly unobscured AGN–1 at X–ray fluxes $S_X>10^{-14}$ erg cm$^{-2}$ s$^{-1}$, and a mixture of unobscured and obscured AGN–2 at fluxes $10^{-14}>S_X>10^{-15.5}$ erg cm$^{-2}$ s$^{-1}$ with ever fainter and redder optical counterparts, while at even lower X–ray fluxes a new population of star forming galaxies emerges [@hor00; @ros02; @ale02; @hor03; @nor04; @bau04]. At optical magnitudes R$>$24 these surveys suffer from large spectroscopic incompleteness, but deep optical/NIR photometry can improve the identification completeness significantly, even for the faintest optical counterparts [@zhe04; @mai04]. A recent review article [@bra05] summarizes the current status of X–ray deep surveys. The AGN/QSO luminosity function and its evolution with cosmic time are key observational quantities for understanding the origin of and accretion history onto supermassive black holes, which are now believed to occupy the centers of most galaxies. X–ray surveys are practically the most efficient means of finding active galactic nuclei (AGNs) over a wide range of luminosity and redshift. Enormous efforts have been made by several groups to follow up X–ray sources with major optical telescopes around the globe, so that now we have fairly complete samples of X–ray selected AGNs. The most complete and sensitive sample was compiled recently by Hasinger, Miyaji and Schmidt [@has05], concentrating on unabsorbed (type–1) AGN selected in the soft (0.5–2 keV) X–ray band, where due to the previous [*ROSAT*]{} work [@paper1; @paper2] complete samples exist, with sensitivity limits varying over five orders of magnitude in flux, and survey solid angles ranging from the whole high galactic latitude sky to the deepest pencil-beam fields. These samples allowed to construct luminosity functions over cosmological timescales, with an unprecedented accuracy and parameter space. ------------------ -------------- ------------------- --------------- --------------------------- ------------------------ -- -- Survey$^{\rm a}$ Solid Angle $S_{\rm X14,lim}$ $N_{\rm tot}$ $N_{\rm AGN-1}$$^{\rm b}$ $N_{\rm unid}^{\rm c}$ \[deg$^2$\] \[cgs\] RBS 20391 $\approx 250$ 901 203 0 SA–N 684.0–36.0 47.4–13.0 380 134 5 NEPS 80.7–1.78 21.9–4.0 262 101 9 RIXOS 19.5–15.0 10.2–3.0 340 194 14 RMS 0.74–0.32 1.0–0.5 124 84 7 RDS/XMM 0.126–0.087 0.38–0.13 81 48 8 CDF–S 0.087–0.023 0.022–0.0053 293 113 1 CDF–N 0.048–0.0064 0.030–0.0046 195 67 21 Total 2566 944 57 ------------------ -------------- ------------------- --------------- --------------------------- ------------------------ -- -- : The soft X–ray sample \[tab:samp\] $^{\rm a}$ Abbreviations – RBS: The [*ROSAT*]{} Bright Survey [@schw00]; SA–N: [*ROSAT*]{} Selected Areas North [@app98]; NEPS: [*ROSAT*]{} North Ecliptic Pole Survey [@gio03]; RIXOS: [*ROSAT*]{} International X–ray Optical Survey [@mas00], RMS: [*ROSAT*]{} Medium Deep Survey, consisting of deep PSPC pointings at the North Ecliptic Pole [@bow96], the UK Deep Survey [@mch98], the Marano field [@zam99] and the outer parts of the Lockman Hole [@schm98; @leh00]; RDS/XMM: [*ROSAT*]{} Deep Survey in the central part of the Lockman Hole, observed with [*XMM–Newton*]{} [@leh01; @mai02; @fad02]; CDF–S: The [*Chandra*]{} Deep Field South [@szo04; @zhe04; @mai04]; CDF–N: The [*Chandra*]{} Deep Field North [@bar01a; @bar03]. $^{\rm b}$ Excluding AGNs with $z<0.015$. $^{\rm c}$ Objects without redshifts, but hardness ratios consistent with type–1 AGN. The X–ray selected AGN–1 sample =============================== For the derivation of the X–ray luminosity function and cosmological evolution of AGN well–defined flux–limited samples of active galactic nuclei have been chosen, with flux limits and survey solid angles ranging over five and six orders of magnitude, respectively (see Table 1). To be able to utilize the massive amount of optical identification work performed previously on a large number of shallow to deep [*ROSAT*]{} surveys, the analysis was restricted to samples selected in the 0.5–2 keV band. In addition to the [*ROSAT*]{} surveys already used in [@paper1; @paper2], data from the recently published [*ROSAT*]{} North Ecliptic Pole Survey (NEPS) [@gio03; @mul04], from an [*XMM–Newton*]{} observation of the Lockman Hole [@mai02] as well as the [*Chandra*]{} Deep Fields South (CDF–S) [@szo04; @zhe04; @mai04] and North (CDF–N) [@bar01a; @bar03] were included. In order to avoid systematic uncertainties introduced by the varying and a priori unknown AGN absorption column densities only unabsorbed (type–1) AGN, classified by optical and/or X–ray methods were selected. We are using here a definition of type–1 AGN, which is largely based on the presence of broad Balmer emission lines and small Balmer decrement in the optical spectrum of the source (optical type–1 AGN, e.g. the ID classes a, b, and partly c in [@schm98], which largely overlaps the class of X–ray type–1 AGN defined by their X–ray luminosity and unabsorbed X–ray spectrum [@szo04]. However, as Szokoly et al show, at low X–ray luminosities and intermediate redshifts the optical AGN classification often breaks down because of the dilution of the AGN excess light by the stars in the host galaxy (see e.g. [@mor02]), so that only an X–ray classification scheme can be utilized. Schmidt et al. [@schm98] have already introduced the X–ray luminosity in their classification. For the deep XMM–Newton and [*Chandra*]{} surveys in addition the X–ray hardness ratio was used to discriminate between X–ray type–1 and type–2 AGN, following [@szo04]. Most of the extragalactic sources found in both the deep and wider surveys with Chandra and XMM-Newton are AGN of some type. Starburst and normal galaxies make increasing fractional contributions at the faintest flux levels, but even in the they represent of all sources (and create $\sim 5$% of the XRB). The observed AGN sky density in the deepest surveys is $\approx 7200$ deg$^{-2}$, about an order of magnitude higher than that found at any other wavelength [@bau04]. This exceptional effectiveness at finding AGN arises because selection (1) has reduced absorption bias and minimal dilution by host-galaxy starlight, and (2) allows concentration of intensive optical spectroscopic follow-up upon high-probability AGN with faint optical counterparts (i.e., it is possible to probe further down the luminosity function). ![(a) Cumulative number counts N($>$S) for the total sample (upper blue thin line), the AGN–1 subsample (lower black thick line), the AGN–2 subsample (red dotted line) and the galaxy subsample (green dashed line). (b) Differential number counts of the total sample of X–ray sources (open squares) and the AGN–1 subsample (filled squares). The dot-dashed lines refer to broken powerlaw fits to the differential source counts (see text). The dashed red line shows the prediction for type–1 AGN (from [@has05]). []{data-label="fig:ns"}](hasinger_F1a.eps "fig:"){width="6cm"} ![(a) Cumulative number counts N($>$S) for the total sample (upper blue thin line), the AGN–1 subsample (lower black thick line), the AGN–2 subsample (red dotted line) and the galaxy subsample (green dashed line). (b) Differential number counts of the total sample of X–ray sources (open squares) and the AGN–1 subsample (filled squares). The dot-dashed lines refer to broken powerlaw fits to the differential source counts (see text). The dashed red line shows the prediction for type–1 AGN (from [@has05]). []{data-label="fig:ns"}](hasinger_F1b.eps "fig:"){width="6cm"} Number Counts and Resolved Background Fraction ============================================== Based on deep surveys with Chandra and XMM-Newton, the log(N)–log(S) relation has now been determined down to fluxes of $2.4\times10^{-17}$ erg cm$^{-2}$ s$^{-1}$, $2.1\times10^{-16}$ erg cm$^{-2}$ s$^{-1}$, and $1.2\times10^{-15}$ erg cm$^{-2}$ s$^{-1}$ in the 0.5–2, 2-10 and 5-10 keV band, respectively [@bra01b; @has01; @ros02; @mor03; @bau04]. Figure \[fig:ns\]a shows the normalized cumulative source counts $N(>S_{\rm X14}) S_{\rm X14}^{1.5}$. The total differential source counts, normalized to a Euclidean behaviour (dN/d$S_{\rm X14} \times S_{\rm X14}^{2.5}$) is shown with open symbols in Figure \[fig:ns\]b. Euclidean source counts would correspond to horizontal lines in these graphs. For the total source counts, the well-known broken powerlaw behaviour is confirmed with high precision. A broken power law fitted to the differential source counts yields power law indices of $\alpha_b=2.34\pm0.01$ and $\alpha_f=1.55\pm0.04$ for the bright and faint end, respectively, a break flux of $S_{\rm X14}= 0.65\pm0.10$ and a normalisation of dN/d$S_{\rm X14} = 103.5 \pm 5.3$ deg$^{-2}$ at $S_{\rm X14} = 1.0$ with a reduced $\chi^2$=1.51. We see that the total source counts at bright fluxes, as determined by the [*ROSAT*]{} All-Sky Survey data, are significantly flatter than Euclidean, consistent with the discussion in [@has93]. Moretti et al. [@mor03], on the other hand, have derived a significantly steeper bright flux slope ($\alpha_b \approx 2.8$) from [*ROSAT*]{} HRI pointed observations. This discrepancy can probably be attributed to the selection bias against bright sources, when using pointed observations where the target area has to be excised. The ROSAT HRI Ultradeep Survey had already resolved 70-80% of the extragalactic 0.5–2 keV XRB into discrete sources, the major uncertainty being in the absolute flux level of the XRB. The deep Chandra and XMM-Newton surveys have now increased the resolved fraction to 85-100% [@mor03; @wor04]. Above 2 keV the situation is complicated on one hand by the fact, that the HEAO-1 background spectrum [@mar80], used as a reference over many years, has a $\sim 30\%$ lower normalization than several earlier and later background measurements (see e.g. [@mor03]). Recent determinations of the background spectrum with XMM-Newton [@mol04] and RXTE [@rev03] strengthen the consensus for a 30% higher normalization, indicating that the resolved fractions above 2 keV have to be scaled down correspondingly. On the other hand, the 2-10 keV band has a large sensitivity gradient across the band. A more detailed investigation, dividing the recent 770 ksec XMM-Newton observation of the Lockman Hole into finer energy bins, comes to the conclusion, that the resolved fraction decreases substantially with energy, from over 90% below 2 keV to less than 50% above 5 keV [@wor04]. Type–1 AGN are the most abundant population of soft X–ray sources. For the determination of the AGN–1 number counts we include those unidentified sources, which have hardness ratios consistent with AGN–1 (a contribution of $\sim 6\%$, see Table \[tab:samp\]). Figure \[fig:ns\] shows, that the break in the total source counts at intermediate fluxes is produced by type–1 AGN, which are the dominant population there. Both at bright fluxes and at the faintest fluxes, type–1 AGN contribute about 30% of the X–ray source population. At bright fluxes, they have to share with clusters, stars and BL-Lac objects, at faint fluxes they compete with type–2 AGN and normal galaxies (see Fig. \[fig:ns\]a and [@bau04]). A broken power law fitted to the differential AGN–1 source counts yields power law indices of $\alpha_b=2.55\pm0.02$ and $\alpha_f=1.15\pm0.05$ for the bright and faint end, respectively, a break flux of $S_{\rm X14}= 0.53\pm0.05$, consistent with that of the total source counts within errors, and a normalisation of of dN/d$S_{\rm X14} = 83.2 \pm 5.5$ deg$^{-2}$ at $S_{\rm X14} = 1.0$ with a reduced $\chi^2$=1.26. The AGN–1 differential source counts, normalized to a Euclidean behaviour (dN/d$S_{\rm X14} \times S_{\rm X14}^{2.5}$) is shown with filled symbols in Figure \[fig:ns\]. The Soft X–ray Luminosity Function and Space Density Evolution {#sec:exlf} ============================================================== {width="12cm"} Hasinger, Miyaji and Schmidt [@has05] have employed two different methods to derive the AGN–1 X–ray luminosity function and its evolution. The first method uses a variant of the $1 \over V_a$ method, which was developed in [@paper1]. The binned luminosity function in a given redshift bin $z_i$ is derived by dividing the observed number $N_{obs}(L_x,z_i)$ by the volume appropriate to the redshift range and the survey X–ray flux limits and solid angles. To evaluate the bias in this value caused by a gradient of the luminosity function across the bin, each of the luminosity functions is fitted by an analytical function. This function is then used to predict $N_{mdl}(L_x,z_i)$. Correcting the luminosity function by the ratio $N_{obs}/N_{mdl}$ takes care of the bias to first order. The second method uses unbinned data. Individual $V_{\rm max}$ of the RBS sources are used to evaluate the zero-redshift luminosity function. This is free of the bias described above: using this luminosity function to derive the number of expected RBS sources matches the observed numbers precisely. In the subsequent derivation of the evolution, i.e., the space density as a function of redshift, binning in luminosity and redshift is introduced to allow evaluation of the results. Bias at this stage is avoided by iterating the parameters of an analytical representation of the space density function. Together with the zero-redshift luminsity function this is used to predict $N_{mod}(L_x,z_i)$ for the surveys. The observed densities in the bins are derived by multiplying the space density value by the ratio $N_{obs}(L_x,z_i)/N_{mod}(L_x,z_i)$. At this stage, none of the densities are derived by dividing a number by a volume. The other difference between the two methods is in the treatment of missing redshifts for optically faint objects. In the binned method, all AGN without redshift with $R > 24.0$ were assigned the central redshift of each redshift bin to derive an upper boundary to the luminosity function. In the unbinned method, the optical magnitudes of the RBS sources were used to derive the optical redshift limit corresponding to $R = 24.0$. The $V_{\rm max}$ values for surveys (such as CDF–N) spectroscopically incomplete beyond $R = 24.0$ were based on the smaller of the X–ray and optical redshift limits. ![Comparison between the space densities derived with two different methods. The blue datapoints with error bars refer to the binned treatment using the N$_{\rm obs}$/N$_{\rm mdl}$ method, the dashed horizontal lines corresponding to the maximum contribution of unidentified sources. The thin and thick red lines and dots refer to the unbinned method (from [@has05]).[]{data-label="fig:comp_ma_ta"}](hasinger_F3.eps){width="12cm"} Figure \[fig:exlf\] shows the luminosity function derived this way in different redshift shells. A change of shape of the luminosity function with redshift is clearly seen and can thus rule out simple density or luminosity evolution models. In a second step, instead of binning into redshift shells, the sample has been cut into different luminosity classes and the evolution of the space density with redshift was computed. Figure \[fig:comp\_ma\_ta\] shows a direct comparison between the binned and unbinned determinations of the space density, which agree very well within statistical errors. The fundamental result is, that the space density of lower–luminosity AGN–1 peaks at significantly lower redshift than that of the higher–luminosity (QSO–type) AGN. Also, the amount of evolution from redshift zero to the peak is much less for lower–luminosity AGN. The result is consistent with previous determinations based on less sensitive and/or complete data, but for the first time our analysis shows a high-redshift decline for all luminosities $L_X < 10^{45}$ erg s$^{-1}$ (at higher luminosities the statistics is still inconclusive). Albeit the different approaches and the still existing uncertainties, it is very reassuring that the general properties and absolute values of the space density are very similar in the two different derivations in. A luminosity-dependent density evolution (LDDE) model has been fit to the data. Even though the sample is limited to soft X–ray-selected type–1 AGN, the parameter values of the overall LDDE model are surprisingly close to those obtained by Ueda et al. 2003 for the intrinsic (de-absorbed) luminosity function of hard X–ray selected obscured and unobscured AGN, except for the normalization, where Ueda et al. reported a value about five times higher. These new results paint a dramatically different evolutionary picture for low–luminosity AGN compared to the high–luminosity QSOs. While the rare, high–luminosity objects can form and feed very efficiently rather early in the Universe, with their space density declining more than two orders of magnitude at redshifts below z=2, the bulk of the AGN has to wait much longer to grow with a decline of space density by less than a factor of 10 below a redshift of one. The late evolution of the low–luminosity Seyfert population is very similar to that which is required to fit the Mid–infrared source counts and background [@fra02] and also the bulk of the star formation in the Universe [@mad98], while the rapid evolution of powerful QSOs traces more the merging history of spheroid formation [@fra99]. This kind of anti–hierarchical Black Hole growth scenario is not predicted in most of the semi–analytic models based on Cold Dark Matter structure formation models (e.g. [@kau00; @wyi03]). This could indicate two modes of accretion and black hole growth with radically different accretion efficiency (see e.g. [@dus02]). A self–consistent model of the black hole growth which can simultaneously explain the anti–hierarchical X–ray space density evolution and the local black hole mass function derived from the $M_{BH}-\sigma$ relation assuming two radically different modes of accretion has recently been presented in [@mer04]. Optical versus X–ray selection of AGN–1 ======================================= ![Comparison of the space density of luminous QSOs between optically selected and soft X–ray selected samples (from [@has05]). The X–ray number densities are plotted for the luminosity class ${\rm log\,}L_{\rm x}=44-45$, both for the binned and unbinned analysis with the same symbols as in Fig. \[fig:comp\_ma\_ta\]. The dashed lines represent the one sigma range for $M_{b_{r,j}}<-26.0$ from [@cro04], multiplied by a factor of 16 to match the X–ray space density at z=2. The triangles at $z>2.7$ with 1$\sigma$ errors are from [@ssg95] (SSG95) and [@fan01] after a cosmology conversion (see text) and a scaling by a factor of 40 to match with the soft X–ray density at $z\sim 2.7$. As discussed in this paper, both the rise and the decline of the space density, behavior changes with $L_x$ and therefore that the comparison can only be illustrative.[]{data-label="fig:zevcomp"}](hasinger_F4.eps){width="12cm"} The space density of soft X–ray selected QSOs from the Hasinger et al. sample is compared to the one of optically-selected QSOs at the most luminous end in Fig. \[fig:zevcomp\]. The $z<2$ number density curve for optically selected QSOs ($M_{b_{\rm J}}<-26.0$) is from the combination of the 2dF and 6dF QSO redshift surveys [@cro04]. The $z>2.7$ number densities from [@ssg95] and [@fan01] have been originally given for $H_0$=50 km s$^{-1}$ Mpc$^{-1}, \Omega_m=1, \Omega_\Lambda=0$. Their data points have been converted to $H_0$=70 km s$^{-1}$ Mpc$^{-1}, \Omega_m=0.3, \Omega_\Lambda=0.7$ and the $M_{\rm B}$ threshold has been re-calculated with an assumed spectral index of $\alpha_o=-0.79$ ($f_\nu\propto\nu^{\alpha_o}$), following e.g. [@vig03]. The plotted curve from [@ssg95; @fan01] is for $M_B<-26.47$ under these assumptions. A small correction of densities due to the cosmology conversion causing redshift-dependent luminosity thresholds has also been made, assuming $d\Phi/dlogL_B\propto L_B^{-1.6}$ [@fan01]. The space density for the soft X–ray QSOs for the luminosity class $44<logL_x<45$ has been plotted both for the binned and unbinned determination. The Croom et al. [@cro04] space density had to be scaled up by a factor of 16 in order to match the X–ray density at $z\sim 2$. The Schmidt, Schneider & Gunn / Fan et al. data points have been scaled by a factor of 40 to match the soft X–ray data at $z=2.7$ in the plot. There is relatively little difference in the density functions between the X–ray and optical QSO samples, although we have to keep in mind, that both the rise and the decline of the space density is varying with X–ray luminosity, so that this comparison can only be illustrative until larger samples of high–redshift X–ray selected QSOs will be available. ![(a): comparison of X–ray fluxes and AB$_{2500}$ UV magnitudes for the sample of $\sim 1000$ X–ray selected type-1 AGN from [@has05] (blue points) with the ROSAT-observed optically selected SDSS QSOs [@vig03]. Filled red squares give the standard SDSS QSO, while open red squares give the specifically selected high-redshift SDSS sample (see [@vig03]). (b): Monochromatic 2 keV X–ray versus 2500 $\AA$  UV luminosity for the same samples. The blue (dark) solid line shows a linear relation between the two luminosities, while the yellow (light) solid line gives the non-linear relation $L_X \sim L_{UV}^{0.75}$ from the literature [@vig03]. []{data-label="fig:x_uv"}](hasinger_F5.eps){width="12cm"} As a next step we directly study the X–ray and optical fluxes and luminosities of our sample objects and compare this with the optically selected QSO sample of Vignali et al. [@vig03] based on SDSS-selected AGN serendipitously observed in ROSAT PSPC pointings. Because of the inhomogeneous nature and different systematics of the different surveys entering our sample, the optical/UV magnitudes of our objects have unfortunately much larger random and systematic errors and are based on fewer colours than the excellent SDSS photometry. In our preliminary analysis we therefore calculated the AB$_{2500}$ magnitudes simply extrapolating or interpolating the observed magnitudes in the optical filters closest to the redshifted 2500 $\AA$ band, assuming an optical continuum with a power law index of -0.7, i.e. not utilizing the more complicated QSO spectral templates including emission lines which have been used in [@vig03]. A spectroscopic correction for the host galaxy contamination, as done for the SDSS sample, was also not possible for our sample, however, for a flux and redshift-selected sample of 94 RBS Seyferts [@salvato] we have morphological determinations of the nuclear versus host magnitudes (see below). In all other aspects of the analysis we follow the Vignali et al. treatment. Figure \[fig:x\_uv\]a shows 0.5–2 keV X–ray fluxes versus AB$_{2500}$ magnitudes for our sample objects (blue stars) in comparison with the Vignali et al. SDSS sample (X–ray detections are shown as filled red squares, upper limits as down–pointing triangles). It is obvious, that our multi-cone survey sample covers a much wider range in X–ray and optical fluxes than the magnitude-limited SDSS sample. Unlike the SDSS sample, our sample shows a very clear correlation between X–ray and optical fluxes, but also a wider scatter in this correlation. Figure \[fig:x\_uv\]b shows the monochromatic X–ray versus UV luminosity for the same data. Now the X–ray and optically selected samples cover a similar parameter range at the high luminosity end, but the X–ray selected data reach significantly lower X–ray and UV luminosities than the optically selected sample. Again, there is a larger scatter in the X–ray selected sample. The figure also shows two analytic relations between X–ray and UV luminosity: a linear relation L$_X \propto $L$_{UV}$ and the non–linear behaviour L$_X \propto $L$_{UV}^{0.75}$ found in the literature (e.g. [@vig03]). While the Vignali et al. optically selected sample clearly prefers the non-linear dependence (see also [@bra04]), this is not true for the X-ray selected sample, which is consistent with a linear relation, apart from the behaviour at low luminosities, where significant contamination from the host galaxy is expected. ![X–ray to optical spectral index $\alpha_{ox}$ as a function of redshift for different luminosity classes for the Hasinger et al. sample of $\sim1000$ soft X–ray selected type-1 AGN. Blue (dark) stars give the values derived using the total integrated optical light, while the green triangles give the values using only the nuclear component from 93 RBS AGN, derived by Salvato [@salvato] from detailed imaging decomposition. The solid black squares with error bars give the median and variance of the X-ray selected sample. Open red squares with error bars show the average $\alpha_{ox}$ values for the Vignali et al. optically selected QSOs.[]{data-label="fig:z_aox"}](hasinger_F6.eps){width="12cm"} To check on any evolution of the optical to X–ray spectral index with redshift we calculated $\alpha_{ox}$ values, following [@vig03] for all our sample objects. In order to see possible luminosity–dependent evolution effectssimilar to those observed in the space density evolution, we divided our sample into the same luminosity classes as in Section \[sec:exlf\]. Figure \[fig:z\_aox\] shows the $\alpha_{ox}$ values determined for objects in the luminosity class 43–44 and 44–45, respectively, as a function of redshift. Apart from a few wiggles, which are likely due to the omission of the emission lines in the optical AGN continuum, there is no significant evolution with redshift. The optically selected sample, on the other hand, shows a significant trend with redshift and average values inconsistent with the X-ray selected sample for most of the redshift range. The diagram also shows, that this discrepancy is likely not caused by the missing host galaxy contamination correction in our analysis. From the relatively small number of nearby (z$<$0.1) of RBS sources, where a morphological fitting procedure has been used to subtract the host emission from the total magnitude [@salvato] we can estimate the host dilution effect, which is clearly larger at lower X–ray luminosities and makes the discrepancy even larger. The immediate conclusion is, that the average optical to X-ray sample properties are dependent on systematic sample selection effects. Constraints on the Growth of SMBH ================================== The AGN luminosity function can be used to determine the masses of remnant black holes in galactic centers, using Sołtan’s continuity equation argument [@sol82] and assuming a mass-to-energy conversion efficiency $\epsilon$. For a non-rotating Schwarzschild BH, $\epsilon$ is expected to be 0.054, while for a maximally rotating Kerr BH, $\epsilon$ can be as high as 0.37 [@tho74]. The AGN demography predicted, that most normal galaxies contain supermassive black holes (BH) in their centers, which is now widely accepted (e.g. [@kor01] and references therein). Recent determinations of the accreted mass from the optical QSO luminosity function are around $2 \epsilon_{0.1}^{-1} \cdot 10^5 M_\odot Mpc^{-3}$ [@cho92; @yut02]. Estimates from the X–ray background spectrum, including obscured accretion power obtain even larger values: 6-9 [@fab99] or 8-17 [@elv02] in the above units, and values derived from the infrared band [@hae01] or multiwavelength observations [@bar01b] are similarly high (8-9). Probably the most reliable recent determination comes from an integration of the X–ray luminosity function. Using the Ueda et al. [@ued03] hard X–ray luminosity function including a correction for Compton–thick AGN normalized to the X–ray background, as well as an updated bolometric correction ignoring the IR dust emission, Marconi et al [@mar04] derived $\rho_{accr} \sim 3.5 \epsilon_{0.1}^{-1} \cdot 10^5 M_\odot Mpc^{-3}$. The BH masses measured in local galaxies are tightly correlated to the galactic velocity dispersion [@fer00; @geb00], and less tightly to the mass and luminosity of the host galaxy bulge (however, see [@mar03]). Using these correlations and galaxy luminosity (or velocity) functions, the total remnant black hole mass density in galactic bulges can be estimated. Scaled to the same assumption for the Hubble constant (H$_0$=70 km s$^{-1}$ Mpc$^{-1}$), recent papers arrive at different values, mainly depending on assumptions about the intrinsic scatter in the BH–galaxy correlations: $\rho_{BH}$ = ($2.4\pm0.8$), ($2.9\pm0.5$) and $(4.6^{+1.9}_{-1.4} h_{70}^2 \cdot 10^5 M_\odot Mpc^{-3}$, respectively [@all02; @yut02; @mar04]. The local dark remnant mass function is fully consistent with the above accreted mass function, if black holes accrete with an average energy conversion efficiency of $\epsilon=0.1$ [@mar04], which is the classically assumed value and lies between the Schwarzschild and the extreme Kerr solution. However, taking also into account the widespread evidence for a significant kinetic AGN luminosity in the form of jets and winds, it is predicted, that the average supermassive black hole should be rapidly spinning fast (see also [@elv02; @yut02]). Recently, using XMM-Newton, a strong relativistic Fe K$\alpha$ line has been discovered in the average rest–frame spectra of AGN–1 and AGN–2 [@stre05], which can be best fit by a rotating Kerr solution consistent with this conjecture. [*Acknowledgements*]{} I thank my colleagues in the [*Chandra Deep Field South*]{} and [*XMM–Newton Lockman Hole*]{} projects, as well as Maarten Schmidt, Takamitsu Miyaji and Niel Brandt for the excellent cooperation on studies of the X–ray background. [8.]{} Alexander D.M., Aussel, H., Bauer F.E., et al., ApJ, 568, L85 (2002) Alexander D.M., Bauer F.E., Brandt W.N., et al., AJ, 126, 539 (2003) Aller, M. C., Richstone, D., AJ 124, 3035 (2002) Appenzeller, I., Thiering, I., Zickgraf, F.-J., et al., A&A (Suppl.) 364, 443 (1998) Barger A. J., Cowie L. L., Mushotzky R. F., Richards, E. A., AJ 121, 662 (2001a) Barger A J, Cowie L L, Bautz M W, Brandt W N, Garmire G P, et al., AJ, [**122**]{}, 2177 (2001b) Barger A.J., Cowie L.L., Capak P., et al., AJ, 126, 632 (2003) Bauer F.E., Alexander D.M., Brandt W.N., et al., AJ in press, astro-ph/0408001 (2004) Bower R. G., Hasinger, G., Castander, F. J. et al., MNRAS 281, 59 (1996) Brandt W. N., Alexander, D. M., Hornschemeier, A. E., Garmire, G. P., Schneider, D. P., et al., AJ 122, 2810 (2001) Brandt, W.N., Vignali C., Lehmer B.D., et al., this volume (2005) Brandt, W. N., Hasinger, G., ARAA in press (2005) Chokshi, A., Turner, E. L., MNRAS 259, 421 (1992) Croom, S. M., Smith, R. J., Boyle, B. J., Shanks, T., Miller, L., et al., MNRAS 349, 1397 (2004) De Luca, A., Molendi, S., A&A 419, 837 Duschl, W.J., Strittmatter, P.A., in Active Galactic Nuclei: from Central Engine to Host Galaxy Abstract Book, meeting held in Meudon, France, July 23-27, 2002, Eds.: S. Collin, F. Combes and I. Shlosman. To be published in ASP Conference Series, p. 76 (2002) Elvis, M., Risaliti, G., & Zamorani, G., ApJ 565, L75 (2002) Fabian, A. C., Iwasawa, K., MNRAS 303, 34 (1999) Fadda, D., Flores, H., Hasinger, G., et al., A&A 383, 838 (2002) Fan, X., Strauss, M. A., Schneider, D. P., Gunn, J. E., Lupton, R. H., et al., AJ 121, 54 (2001) Ferrarese, L., Merritt, D., ApJ 539, L9 (2000) Fiore F., Brusa M., Cocchia F., et al., A&A 409, 79 (2003) Franceschini, A., Hasinger, G., Miyaji, T., Malquori, D., MNRAS 310, L15 (1999) Franceschini A., Braito V. & Fadda D., MNRAS, 335, L51 (2002) Gebhardt, K., Bender, R., Bower, G., Dressler, A., Faber S. M., et al., ApJ 539, L13 (2000) Giacconi R., Rosati P., Tozzi P., et al., ApJ, 551, 624 (2001) Giacconi, R., Zirm, A., Wang, J.X., et al., ApJS, 139, 369 (2002) Gilli, R., Cimatti, A., Daddi, E., et al., ApJ 592, 721 (2003) Gioia, I., Henry, J.P., Mullis, C.R., et al., ApJS 149, 29 (2003) Haehnelt, M. G., Kauffmann, G., in [*Black Holes in Binaries and Galactic Nuclei*]{}, ed. L. Kaper, E. P. J. van den Heuvel, & P. A. Woudt, p.364 (Berlin: Springer) (2001) Hasinger G., Burg R., Giacconi R., et al., A&A 275, 1 (1993) Hasinger G., Burg R., Giacconi R., et al., A&A 329, 482 (1998) Hasinger G., Altieri B., Arnaud M., et al., A&A 365, 45 (2001) Hasinger, G., Miyaji, T., Schmidt, M., A&A submitted (2005) Hornschemeier A.E., Brandt W.N., Garmire G.P., et al., ApJ 541, 49 (2000) Hornschemeier A.E., Bauer F.E., Alexander D.M., et al., AJ 126, 575 (2003) Kauffmann, G. & Haehnelt, M., MNRAS 311, 576 (2000) Kormendy, J., Gebhardt, K., [*AIP conference proceedings*]{} 586, 363 (2001) Lehmann I., Hasinger G., Schmidt M., et al., A&A 354, 35 (2000) Lehmann I., Hasinger G., Schmidt M., et al., A&A 371, 833 (2001) Madau, P., Pozzetti, L., Dickinson, M., ApJ 498, 106 (1998) Mainieri V., Bergeron J., Rosati P., et al., A&A 393, 425 (2002) Mainieri V., Rosati P., Tozzi, P., et al., A&A submitted (2004) Marconi, A., Hunt, L. K. ApJ 589, L21 (2003) Marconi, A., Risaliti, G., Gilli, R., Hunt, L. K., Maiolino, R., Salvati, M., MNRAS 351, 169 (2004) Marshall, F. E., Boldt, E. A., Holt, S. S., et al., ApJ 235, 4 (1980) Mason K. O., Carrera, F. J., Hasinger, G., et al., MNRAS 311, 456 (2000) McHardy I.M., Jones, L. R., Merrifield, M. R., et al., MNRAS, 295, 641 (1998) Merloni, A., MNRAS 353, 1035 (2004) Miyaji, T., Hasinger G., Schmidt M., A&A 353, 25 (2000) Miyaji, T., Hasinger, G., Schmidt, M., A&A 369, 49 (2001) Moran, E. C., Filippenko, A. V. & Chornock, R., ApJ 579, L71 (2002) Moretti, A., Campana, S., Lazzati, D., et al., ApJ 588, 696 (2003) Mullis, C. R., Henry, J. P., Gioia, I. M., et al., ApJ submitted (astro-ph/0408304) (2004) Mushotzky, R. F., Cowie, L. L., Barger, A. J., Arnaud, K. A., Nature 404, 459 (2000) Norman, C., Ptak, A., Hornschemeier, A., et al., ApJ 607, 721 (2004) Revnivtsev, M., Gilfanov, M., Sunyaev, R., et al., A&A 411, 329 (2003) Rosati P., Tozzi P., Giacconi R., et al., ApJ 566, 667 (2002) Salvato, M., Dissertation, Potsdam University (2002) Schmidt, M., Schneider, D. P., Gunn, J. E., AJ 110, 68 (1995) Schmidt, M., Hasinger, G., Gunn, J. E., et al., A&A 329, 495 (1998) Schwope A., Hasinger, G., Lehmann, I., et al., AN, 321, 1 (2000) Sołtan, A., MNRAS 200, 115 (1982) Streblyanska, A., Hasinger, G., Finoguenov, A. et al., A&A in press (2005) Szokoly G., Bergeron J., Hasinger G., et al., ApJS in press (astro-ph/0312324) (2004) Thorne, K. S., ApJ 191, 507 (1974) Tozzi, P., Rosati, P., Nonino, M., et al., ApJ 562, 42 (2001) Ueda, Y., Akiyama, M., Ohta, K., Miyaji, T., ApJ 598, 886 (2003) Vignali C., Brandt W.N., Schneider D.P., AJ 125, 433 (2003) Worsley, M., Fabian, A.C., Mateos, S., et al., MNRAS 352, L28 (2004) Wyithe, J.S.B., Loeb, A., ApJ 595, 614 (2003) Yang, Y., Mushotzky, R. F., Barger, A. J., et al., ApJ 585, L85 (2003) Yu, Q., Tremaine, S., MNRAS 335, 965 (2002) Zamorani, G., et al. A&A, 346, 731 (1999) Zheng W., Mikles, V. J., Mainieri, V., et al., ApJ in press, astro-ph/0406482 (2004)

amstex =1200 **GEORG CANTOR AND HIS HERITAGE1** **Yu. I. Manin** [*God is no geometer, rather an unpredictable poet.*]{} [*(Geometers can be unpredictable poets, so there could be room for compromise.)*]{} **Introduction** Georg Cantor’s grand meta-narrative, Set Theory, created by him almost singlehandedly in the span of about fifteen years, resembles a piece of high art more than a scientific theory. Using a slightly modernized language, basic results of set theory can be stated in a few lines. Consider the category of all sets with arbitrary maps as morphisms. Isomorphism classes of sets are called [*cardinals*]{}. Cardinals are well–ordered by the sub–object relation, and the cardinal of the set of all subsets of $U$ is strictly larger than that of $U$ (this is of course proved by the famous diagonal argument). This motivates introduction of another category, that of well ordered sets and monotone maps as morphisms. Isomorphism classes of these are called [*ordinals*]{}. They are well–ordered as well. The Continuum Hypothesis is a guess about the order structure of the initial segment of cardinals. Thus, exquisite minimalism of expressive means is used by Cantor to achieve a sublime goal: understanding infinity, or rather infinity of infinities. A built–in self–referentiality and the forceful extension of the domain of mathematical intuition (principles for building up new sets) add to this impression of combined artistic violence and self–restraint. Cantor himself would have furiously opposed this view. For him, the discovery of the hierarchy of infinities was a revelation of God–inspired Truth. But mathematics of the XXth century reacted to Cantor’s oeuvre in many ways that can be better understood in the general background of various currents of contemporary science, philosophical thought, and art. Somewhat provocatively, one can render one of Cantor’s principal insights as follows: *$2^x$ is considerably larger than $x$.* Here $x$ can be understood as an integer, an arbitrary ordinal, or a set; in the latter case $2^x$ denotes the set of all subsets of $x$. Deep mathematics starts when we try to make this statement more precise and to see [*how much larger*]{} $2^x$ is. If $x$ is the first infinite ordinal, this is the Continuum Problem. I will argue that properly stated for finite $x$, this question becomes closely related to a universal $NP$ problem. I will then discuss assorted topics related to the role of set theory in contemporary mathematics and the reception of Cantor’s ideas. **Axiom of Choice and $P/NP$–problem** *or* **finite as poor man’s transfinite2** In 1900, at his talk at the second ICM in Paris, Hilbert put the Continuum Hypothesis at the head of his list of 23 outstanding mathematical problems. This was one of the highlights of Cantor’s scientific life. Cantor invested much effort consolidating the German and international mathematical community into a coherent body capable of counterbalancing a group of influential professors opposing set theory. The opposition to his theory of infinity, however, continued and was very disturbing to him, because the validity of Cantor’s new mathematics was questioned. In 1904, at the next International Congress, König presented a talk in which he purported to show that continuum could not be well–ordered, and therefore the Continuum Hypothesis was meaningless. Dauben writes (\[D\], p. 283): “The dramatic events of König’s paper read during the Third International Congress for Mathematicians in Heidelberg greatly upset him \[Cantor\]. He was there with his two daughters, Else and Anna–Marie, and was outraged at the humiliation he felt he had been made to suffer.” It turned out that König’s paper contained a mistake; Zermelo soon afterwards produced a proof that any set could be well ordered using his then brand new Axiom of Choice. This axiom essentially postulates that, starting with a set $U$, one can form a new set, whose elements are pairs $(V,v)$ where $V$ runs over all non–empty subsets of $U$, and $v$ is an element of $V$. A hundred years later, the mathematical community did not come up with a compendium of new problems for the coming century similar to the Hilbert’s list. Perhaps, the general vision of mathematics changed — already in Hilbert’s list a considerable number of items could be better described as research programs rather than well–defined problems, and this seems to be a more realistic way of perceiving our work in progress. Still, a few clearly stated important questions remain unanswered, and recently seven such questions were singled out and endowed with a price tag. Below I will invoke one of these questions, the $P/NP$ problem and look at it as a finitary travesty of Zermelo’s Axiom of Choice. Let $U_m=\bold{Z}_2^m$ be the set of $m$–bit sequences. A convenient way to encode its subsets is via Boolean polynomials. Using the standard – and more general – language of commutative algebra, we can identify each subset of $U_m$ with the $0$–level of a unique function $f\in B_m$ where we define the algebra of Boolean polynomials as $$B_m:=\bold{Z}_2[x_1,\dots ,x_m]/(x_1^2+x_1, \dots ,x_m^2+x_m).$$ Hence Zermelo’s problem – [*choose an element in each non–empty subset of $U$*]{} – translates into: [*for each Boolean polynomial, find a point at which the polynomial equals 0, or prove that the polynomial is identically 1*]{}. Moreover, we want to solve this problem in time bounded by a polynomial of the bit size of the code of $f$. This leads to a universal (“maximally difficult”) $NP$–problem if one writes Boolean polynomials in the following version of disjunctive normal form. Code of such a form is a family $$u =\{m; (S_1,T_1),\dots ,(S_N, T_N)\}, \ m\in \bold{N}; \, S_i,T_i\subset \{1,\dots ,m\}.$$ The bit size of $u$ is $mN$, and the respective Boolean polynomial is $$f_u:=1+ \prod_{i=1}^N\left(1+\prod_{k\in S_i}(1+x_k)\prod_{j\in T_i}x_j\right)$$ This encoding provides for a fast check of the inclusion relation for the elements of the respective level set. The price is that the uniqueness of the representation of $f$ is lost, and moreover, the identity $f_u=f_v (?)$ becomes a computationally hard problem. In particular, even the following weakening of the finite Zermelo problem becomes $NP$–complete and hence currently intractable: [*check whether a Boolean polynomial given in disjunctive normal form is non–constant.*]{} Zermelo’s Axiom of Choice aroused a lively international discussion published in the first issue of Mathematische Annalen of 1905. A considerable part of it was focused on the psychology of mathematical imagination and on the reliability of its fruits. Baffling questions of the type: “How can we be sure that during the course of a proof we keep thinking about the same set?” kept emerging. If we imagine that at least a part of computations that our brains perform can be adequately modeled by finite automata, then quantitative estimates of the required resources as provided by the theory of polynomial time computability might eventually be of use in neuroscience and by implication in psychology. A recent paper in “Science” thus summarizes some experimental results throwing light on the nature of mental representation of mathematical objects and physiological roots of divergences between, say, intuitionists and formalists: “\[...\] our results provide grounds for reconciling the divergent introspection of mathematicians by showing that even within the small domain of elementary arithmetic, multiple mental representations are used for different tasks. Exact arithmetic puts emphasis on language–specific representations and relies on a left inferior frontal circuit also used for generating associations between words. Symbolic arithmetic is a cultural invention specific to humans, and its development depended on the progressive improvement of number notation systems. \[...\] Approximate arithmetic, in contrast, shows no dependence on language and relies primarily on a quantity representation implemented in visuo–spatial networks of the left and right parietal lobes.” (\[DeSPST\], p. 973). In the next section, I will discuss an approach to the Continuum Conjecture which is clearly inspired by the domination of the visuo–spatial networks, and conjecturally better understood in terms of probabilistic models than logic or Boolean automata. [*Appendix: some definitions.*]{} For completeness, I will remind the reader of the basic definitions related to the $P/NP$ problem. Start with an infinite constructive world $U$ in the sense of \[Man\], e.g. natural numbers $\bold{N}$. A subset $E\subset U$ [*belongs to the class $P$*]{} if it is decidable and the values of its characteristic function $\chi_E$ are computable in polynomial time on all arguments $x\in E.$ Furthermore, $E\subset U$ [*belongs to the class $NP$*]{} if it is a polynomially truncated projection of some $E'\subset U\times U$ belonging to $P$: for some polynomial $G$, $$u\in E\ \Leftrightarrow\ \exists\,(u,v)\in E'\ \roman{with}\ |v|\le G(|u|)$$ where $|v|$ is the bit size of $v$. In particular, $P\subset NP$. Intuitively, $E\in NP$ means that for each $u\in E$ [*there exists a polynomially bounded proof*]{} of this inclusion (namely, the calculation of $\chi_{E'} (u,v)$ for an appropriate $v$). However, to find such a proof (i. e. $v$) via the naive search among all $v$ with $|v|\le G(|u|)$ can take exponential time. The set $E\subset U$ is called [*NP–complete*]{} if, for any other set $D\subset V, D\in NP,$ there exists a polynomial time computable function $f:\,V\to U$ such that $D=f^{-1}(E)$, that is, $\chi_D(v)=\chi_E(f(v)).$ The encoding of Boolean polynomials used above is explained and motivated by the proof of the $NP$–completeness: see e. g. \[GaJ\], sec. 2.6. **The Continuum Hypothesis and random variables** Mumford in \[Mum\], p. 208, recalls an argument of Ch. Freiling (\[F\]) purporting to show that Continuum Hypothesis is “obviously” false by considering the following situation: “Two dart players independently throw darts at a dartboard. If the continuum hypothesis is true, the points $P$ on the surface of a dartboard can be well ordered so that for every $P$, the set of $Q$ such that $Q<P$, call it $S_Q$, is countable. Let players 1 and 2 hit the dart board at points $P_1$ and $P_2$. Either $P_1<P_2$ or $P_2<P_1$. Assume the first holds. Then $P_1$ belongs to a countable subset $S_{P_2}$ of the points on on the dartboard. As the two throws were independent, we may treat throw 2 as taking place first, then throw 1. After throw 2, this countable set $S_{P_2}$ has been fixed. But every countable set is measurable and has measure 0. The same argument shows that the probability of $P_2$ landing on $S_{P_1}$ is 0. Thus almost surely neither happened and this contradicts the assumption that the dartboard is the first uncountable cardinal! \[...\] I believe \[...\] his ‘proof’ shows that if we make random variables one of the basic elements of mathematics, it follows that the C.H. is false and we will get rid of one of the meaningless conundrums of set theory.”3 Freiling’s work was actually preceded by that of Scott and Solovay, who recast in terms of “logically random sets” P. Cohen’s forcing method for proving the consistency of the negated CH with Zermelo–Frenkel axioms. Their work has shown that one can indeed put random variables in the list of basic notions and use them in a highly non–trivial way. P. Cohen himself ended his book by suggesting that the view that CH is “obviously false” may become universally accepted. However, whereas the Scott–Solovay reasoning proves a precise theorem about the formal language of the Set Theory, Freiling’s argument appeals directly to our physical intuition, and is best classified as a [*thought experiment.*]{} It is similar in nature to some classical thought experiments in physics, deducing e.g. various dynamic consequences from the impossibility of [*perpetuum mobile*]{}. The idea of thought experiment, as opposed to that of logical deduction, can be generally considered as a right–brain equivalent of the left brain elementary logical operations. A similar role is played by good metaphors. When we are comparing the respective capabilities of two brains, we are struck by what I called elsewhere ‘the inborn weakness of metaphors’: they resist becoming building blocks of a system. One can only more or less artfully put one metaphor upon another, and the building will stand or crumble upon its own weight independently of its truth or otherwise. Physics disciplines thought experiments as poetry disciplines metaphors, but only logic has an inner discipline. Successful thought experiments produce mathematical truths which, after being accepted, solidify into axioms, and the latter start working on the treadmill of logical deductions. **Foundations and Physics** I will start this section with a brief discussion of the impact of Set Theory on the foundations of mathematics. I will understand “foundations” neither as the para–philosophical preoccupation with nature, accessibility, and reliability of mathematical truth, nor as a set of normative prescriptions like those advocated by finitists or formalists. I will use this word in a loose sense as a general term for the historically variable conglomerate of rules and principles used to organize the already existing and always being created anew body of mathematical knowledge of the relevant epoch. At times, it becomes codified in the form of an authoritative mathematical text as exemplified by Euclid’s Elements. In another epoch, it is better expressed by nervous self–questioning about the meaning of infinitesimals or the precise relationship between real numbers and points of the Euclidean line, or else, the nature of algorithms. In all cases, foundations in this wide sense is something which is of relevance to a working mathematician, which refers to some basic principles of his/her trade, but which does not constitute the essence of his/her work. In the XXth century, all of the main foundational trends referred to Cantor’s language and intuition of sets. The well developed project of Bourbaki gave a polished form to the notion that [*any*]{} mathematical object $\Cal{X}$ (group, topological space, integral, formal language ...) could be thought of as a set $X$ with an additional structure $x$. This notion had emerged in many specialized research projects, from Hilbert’s [*Grundlagen der Geometrie*]{} to Kolmogorov’s identification of probability theory with measure theory. The additional structure $x$ in $\Cal{X}=(X,x)$ is an element of another set $Y$ belonging to the [*échelle*]{} constructed from $X$ by standard operations and satisfying conditions (axioms) which are also formulated entirely in terms of set theory. Moreover, the nature of elements of $X$ is inessential: a bijection $X\to X'$ mapping $x$ to $x'$ produces an isomorphic object $\Cal{X}'=(X',x')$. This idea played a powerful unifying and clarifying role in mathematics and led to spectacular developments far outside the Bourbaki group. Insofar as it is accepted in thousands of research papers, one can simply say that the language of mathematics is the language of set theory. Since the latter is so easily formalized, this allowed logicists to defend the position that their normative principles should be applied to all of mathematics and to overstate the role of “paradoxes of infinity” and Gödel’s incompleteness results. However, this fact also made possible such self–reflexive acts as inclusion of metamathematics into mathematics, in the form of model theory. Model theory studies special algebraic structures – formal languages – considered in turn as mathematical objects (structured sets with composition laws, marked elements etc.), and their interpretations in sets. Baffling discoveries such as Gödel’s incompleteness of arithmetics lose some of their mystery once one comes to understand their content as a statement that a certain algebraic structure simply is not finitely generated with respect to the allowed composition laws. When at the next stage of this historical development, sets gave way to categories, this was at first only a shift of stress upon morphisms (in particular, isomorphisms) of structures, rather than on structures themselves. And, after all, a (small) category could itself be considered as a set with structure. However, primarily thanks to the work of Grothendieck and his school on the foundations of algebraic geometry, categories moved to the foreground. Here is an incomplete list of changes in our understanding of mathematical objects brought about by the language of categories. Let us recall that generally objects of a category $C$ are not sets themselves; their nature is not specified; only morphisms $Hom_C(X,Y)$ are sets. A. An object $X$ of the category $C$ can be identified with the functor it represents: $Y\mapsto Hom_C(Y,X)$. Thus, if $C$ is small, initially structureless $X$ becomes a structured set. This external, “sociological” characterization of a mathematical object defining it through its interaction with all objects of the same category rather than in terms of its intrinsic structure, proved to be extremely useful in all problems involving e. g. moduli spaces in algebraic geometry. B. Since two mathematical objects, if they are isomorphic, have exactly the same properties, it does not matter how many pairwise isomorphic objects are contained in a given category $C$. Informally, if $C$ and $D$ have “the same” classes of isomorphic objects and morphisms between their representatives, they should be considered as equivalent. For example, the category of “all” finite sets is equivalent to any category of finite sets in which there is exactly one set of each cardinality 0,1,2,3, $\dots$ . This “openness” of a category considered up to equivalence is an essential trait, for example, in the abstract computability theory. Church’s thesis can be best understood as a postulate that there is an open category of “constructive worlds” — finite or countable structured sets and computable morphisms between them — such that any infinite object in it is isomorphic to the world of natural numbers, and morphisms correspond to recursive functions (cf. \[Man\] for more details). There are many more interesting infinite constructive worlds determined by widely diverging internal structures: words in a given alphabet, finite graphs, Turing machines, etc. However, they are all isomorphic to $\bold{N}$ due to the existence of computable numerations. C. The previous remark also places limits on the naive view that categories “are” special structured sets. In fact, if it is natural to identify categories related by an equivalence (not necessarily bijective on objects) rather than isomorphism, then this view becomes utterly misleading. More precisely, what happens is the slow emergence of the following hierarchical picture. Categories themselves form objects of a larger category $Cat$ morphisms in which are functors, or “natural constructions” like a (co)homology theory of topological spaces. However, functors do not form simply a set or a class: they also form objects of a category. Axiomatizing this situation we get a notion of [*2–category*]{} whose prototype is $Cat$. Treating 2–categories in the same way, we get 3–categories etc. The following view of mathematical objects is encoded in this hierarchy: there is no equality of mathematical objects, only equivalences. And since an equivalence is also a mathematical object, there is no equality between them, only the next order equivalence etc., [*ad infinitim*]{}. This vision, due initially to Grothendieck, extends the boundaries of classical mathematics, especially algebraic geometry, and exactly in those developments where it interacts with modern theoretical physics. With the advent of categories, the mathematical community was cured of its fear of classes (as opposed to sets) and generally “very large” collections of objects. In the same vein, it turned out that there are meaningful ways of thinking about “all” objects of a given kind, and to use self–reference creatively instead of banning it completely. This is a development of the old distinction between sets and classes, admitting that at each stage we get a structure similar to but not identical with the ones we studied at the previous stage. In my view, Cantor’s prophetic vision was enriched and not shattered by these new developments. What made it recede to the background, together with preoccupations with paradoxes of infinity and intuitionistic neuroses, was a renewed interaction with physics and the transfiguration of formal logic into computer science. The birth of quantum physics radically changed our notions about relationships between reality, its theoretical descriptions, and our perceptions. It made clear that Cantor’s famous definition of sets (\[C\]) represented only a distilled classical mental view of the material world as consisting of pairwise distinct things residing in space: [*“Unter einer ‘Menge’ verstehen wir jede Zusammenfassung $M$ von bestimmten wohlunterschiedenen Objekten $m$ unserer Anschauung oder unseres Denkens (welche die ‘Elemente’ von $M$ genannt werden) zu einem Ganzen.”*]{} “By a ‘set’ we mean any collection $M$ into a whole of definite, distinct objects $m$ (called the ‘elements’ of $M$) of our perception or our thought.” Once this view was shown to be only an approximation to the incomparably more sophisticated quantum description, sets lost their direct roots in reality. In fact, the structured sets of modern mathematics used most effectively in modern physics are not sets of things, but rather sets of [*possibilities*]{}. For example, the phase space of a classical mechanical system consists of pairs [*(coordinate, momentum)*]{} describing all possible states of the system, whereas after quantization it is replaced by the space of complex probability amplitudes: the Hilbert space of $L_2$–functions of the coordinates or something along these lines. The amplitudes are all possible quantum superpositions of all possible classical states. It is a far cry from a set of things. Moreover, requirements of quantum physics considerably heightened the degree of tolerance of mathematicians to the imprecise but highly stimulating theoretical discourse conducted by physicists. This led, in particular, to the emergence of Feynman’s path integral as one of the most active areas of research in topology and algebraic geometry, even though the mathematical status of path integral is in no better shape than that of Riemann integral before Kepler’s “Stereometry of Wine Barrels”. Computer science added a much needed touch of practical relevance to the essentially hygienic prescriptions of formal logic. The introduction of the notion of “success with high probability” into the study of algorithmic solvability helped to further demolish mental barriers which fenced off foundations of mathematics from mathematics itself. [*Appendix: Cantor and physics.*]{} It would be interesting to study Cantor’s natural philosophy in more detail. According to \[D\], he directly referred to possible physical applications of his theory several times. For example, he proved that that if one deletes from a domain in $\bold{R}^n$ any dense countable subset (e.g. all algebraic points), then any two points of the complement can be connected by a continuous curve. His interpretation: continuous motion is possible even in discontinuous spaces, so “our” space might be discontinuous itself, because the idea of its continuity is based upon observations of continuous motion. Thus a revised mechanics should be considered. At a meeting of GDNA in Freiburg, 1883, Cantor said: “One of the most important problems of set theory \[...\] consists of the challenge to determine the various valences or powers of sets present in all of Nature in so far as we can know them” (\[D\], p. 291). Seemingly, Cantor wanted atoms (monads) to be actual points, extensionless, and in infinite number in Nature. “Corporeal monads” (massive particles? Yu.M.) should exist in countable quantity. “Aetherial monads” (massless quanta? Yu.M.) should have had cardinality aleph one. **Coda: Mathematics and postmodern condition** Already during Cantor’s life time, the reception of his ideas was more like that of new trends in the art, such as impressionism or atonality, than that of new scientific theories. It was highly emotionally charged and ranged from total dismissal (Kronecker’s “corrupter of youth”) to highest praise (Hilbert’s defense of “Cantor’s Paradise”). (Notice however the commonly overlooked nuances of both statements which subtly undermine their ardor: Kronecker implicitly likens Cantor to Socrates, whereas Hilbert with faint mockery hints at Cantor’s conviction that Set Theory is inspired by God.) If one accepts the view that Bourbaki’s vast construction was the direct descendant of Cantor’s work, it comes as no surprise that it shared the same fate: see \[Mas\]. Especially vehement was reaction against “new maths”: an attempt to reform the mathematical education by stressing precise definitions, logic, and set theoretic language rather than mathematical facts, pictures, examples and surprises. One is tempted to consider this reaction in the light of Lyotard’s (\[L\]) famous definition of the postmodern condition as “incredulity toward meta–narratives” and Tasić’s remark that mathematical truth belongs to “the most stubborn meta–narratives of Western culture” (\[T\], p. 176). In this stubbornness lies our hope. **References** \[B\] H. Bloom. [*The Western Canon.*]{} Riverhead Books, New York, 1994. \[C\] G. Cantor. [*Beiträge zur Begründung der transfiniten Mengenlehre.*]{} Math. Ann. 46 (1895), 481– 512; 49 (1897), 207 –246. \[D\] J. W. Dauben. [*Georg Cantor: his mathematics and philosophy of the infinite.*]{} Princeton UP, Princeton NJ, 1990. \[DeSPST\] S. Dehaene, E. Spelke, P. Pinet, R. Stanescu, S. Tsivkin. [*Sources of mathematical thinking: behavioral and brain–imaging evidence.*]{} Science, 7 May 1999, vol. 284, 970–974. \[F\] C. Freiling. [*Axioms of symmetry: throwing darts at the real line.*]{} J. Symb. Logic, 51 (1986), 190–200. \[GaJ\] M. Garey, D. Johnson. [*Computers and intractability: a guide to the theory of $NP$–completeness.*]{} W. H. Freeman and Co., San–Francisco, 1979. \[L\] J.-F. Lyotard. [*The postmodern condition: a report on knowledge.*]{} University of Minneapolis Press, Minneapolis, 1984. \[Man\] Yu. Manin. [*Classical computing, quantum computing, and Shor’s factoring algorithm.*]{} Séminaire Bourbaki, no. 862 (June 1999), Astérisque, vol 266, 2000, 375–404. \[Mas\] M. Mashaal. [*Bourbaki.*]{} Pour la Science, No 2, 2000. \[Mum\] D. Mumford. [*The dawning of the age of stochasticity.*]{} In: Mathematics: Frontiers and Perspectives 2000, AMS, 1999, 197–218. \[PI\] W. Purkert, H. J. Ilgauds. [*Georg Cantor, 1845 – 1918.*]{} Birkhäuser Verlag, Basel – Boston – Stuttgart, 1987. \[T\] V. Tasić. [*Mathematics and the roots of postmodern thought.*]{} Oxford UP, 2001. **Appendix: Chronology of Cantor’s life and mathematics** *(following \[PI\] and \[D\])* March 3, 1845: Born in St Petersburg, Russia. 1856: Family moves to Wiesbaden, Germany. 1862 – 1867: Cantor studies in Zürich, Berlin, Göttingen and again Berlin. 1867 – 1869: First publications in number theory, quadratic forms. 1869: Habilitation in the Halle University. 1870 – 1872: Works on convergence of trigonometric series. 1872 – 1879: Existence of different magnitudes of infinity, bijections $\bold{R}\to \bold{R}^n$, studies of relations between continuity and dimension. November 29, 1873: Cantor asked in a letter to Dedekind whether there might exist a bijection between $\bold{N}$ and $\bold{R}$ (\[D\], p.49). Shortly after Christmas he found his diagonal procedure (\[D\], p. 51 etc.). 1874: First publication on set theory. 1879 – 1884: Publication of the series [*Über unendliche lineare Punktmannichfaltigkeiten*]{}. 1883: [*Grundlagen einer allgemeinen Mannigfaltigkeitslehre. Ein mathematisch–philosophischer Versuch in der Lehre des Unendlichen.*]{} May 1884: First nervous breakdown, after a successful and enjoyable trip to Paris: depression lasting till Fall: \[D\]. p. 282. 1884–85: Contact with Catholic theologians, encouragement from them, but isolation in Halle. \[D\], p. 146: “... early in 1885, Mittag–Leffler seemed to close the last door on Cantor’s hopes for understanding and encouragement among mathematicians.” September 18, 1890: Foundation of the German Mathematical Society; Cantor becomes its first President. 1891: Kronecker died. 1895 – 1895: [*“Beiträge zur Begründung der transfiniten Mengenlehre”*]{} (Cantor’s last major mathematical publication). 1897: The first ICM. Set theory is very visible. 1897: “Burali–Forti \[...\] was the first mathematician to make public the paradoxes of transfinite set theory” (\[D\]). He argued that [*all*]{} ordinals, if any pair of them is comparable, would form an Ordinal which is greater than itself. He concluded that not all ordinals are comparable. Cantor instead believed that all ordinals do not form an ordinal, just as all sets do not form a set. 1899: Hospitalizations in Halle Nervenklinik before and after the death of son Rudolph. 1902–1903, winter term: Hospitalization. Oct. 1907 – June 1908: Hospitalization. Sept. 1911 – June 1912: Hospitalization. 1915: Celebration of the 70th anniversary of Cantor’s birth, on a national level because of WW I. May 1917 – Jan. 6, 1918: Hospitalization; Cantor dies at Halle Klinik. *Max–Planck–Institut für Mathematik, Bonn, Germany* and Northwestern University, Evanston, USA. [*e-mail:*]{} maninmpim-bonn.mpg.de

--- abstract: 'Many studies have shown that either the nearby astrophysical source or dark matter (DM) annihilation/decay is required to explain the origin of high energy cosmic ray (CR) $e^\pm$, which are measured by many experiments, such as PAMELA and AMS-02. Recently, the Dark Matter Particle Explorer (DAMPE) collaboration has reported its first result of the total CR $e^\pm$ spectrum from $25 \,\GeV$ to $4.6 \,\TeV$ with high precision. In this work, we study the DM annihilation and pulsar interpretations of the DAMPE high energy $e^\pm$ spectrum. In the DM scenario, the leptonic annihilation channels to $\tau^+\tau^-$, $4\mu$, $4\tau$, and mixed charged lepton final states can well fit the DAMPE result, while the $\mu^+\mu^-$ channel has been excluded. In addition, we find that the mixed charged leptons channel would lead to a sharp drop at $\sim$ TeV. However, these DM explanations are almost excluded by the observations of gamma-ray and CMB, unless some complicated DM models are introduced. In the pulsar scenario, we analyze 21 nearby known pulsars and assume that one of them is the primary source of high energy CR $e^\pm$. Considering the constraint from the Fermi-LAT observation of the $e^\pm$ anisotropy, we find that two pulsars are possible to explain the DAMPE data. Our results show that it is difficult to distinguish between the DM annihilation and single pulsar explanations of high energy $e^\pm$ with the current DAMPE result.' author: - 'Bing-Bing Wang' - 'Xiao-Jun Bi' - 'Su-Jie Lin' - 'Peng-fei Yin' bibliography: - 'paper.bib' title: 'Explanations of the DAMPE high energy electron/positron spectrum in the dark matter annihilation and pulsar scenarios' --- Introduction {#section_introduction} ============ After the Cosmic Ray (CR) electron/positron excess above $10\GeV$ was confirmed by PAMELA and AMS-02 with high precision [@adriani2009anomalous; @aguilar2014electron], many studies on its origin have been proposed in the literature. Two kinds of interpretations, including dark matter (DM) annihilation/decay in the Galactic halo [@bergstrom2008new; @barger2009pamela; @cirelli2009model; @yin2009pamela; @zhang2009discriminating; @ArkaniHamed:2008qn; @Pospelov:2008jd] and nearby astrophysical sources [@yuksel2009tev; @hooper2009pulsars; @profumo2012dissecting; @malyshev2009pulsars; @blasi2009origin; @hu2009e+], are widely studied. Although the measurements of AMS-02 are unprecedentedly precise, those results are not yet sufficient to distinguish the two explanations [@lin2015quantitative]. The DArk Matter Particle Explorer (DAMPE) satellite launched on Dec.17, 2015 is a multipurpose detector, which consists of a Plastic Scintillator strip Detector (PSD), a Silicon-Tungsten tracker-converter (STK), a BGO imaging calorimeter, and a Neutron Detector (NUD). Comparing to the AMS-02 experiment, DAMPE has a better energy resolution and could measure CR electrons and positrons at higher energies up to $10\,\TeV$. Recently, the DAMPE collaboration reported its first result of the total CR $e^\pm$ spectrum in the energy range from 25 $\GeV$ to 4.6 $\TeV$ [@Ambrosi:2017wek]. Many studies have been performed to explain the tentative features in this spectrum  [@Yuan:2017ysv; @Fang:2017tvj; @Fan:2017sor; @Duan:2017pkq; @Athron:2017drj; @Cao:2017ydw; @Liu:2017rgs; @Huang:2017egk; @Niu:2017hqe; @Gao:2017pym; @Yang:2017cjm; @Cao:2017sju; @Ghorbani:2017cey; @Nomura:2017ohi; @Gu:2017lir; @Zhu:2017tvk; @Li:2017tmd; @Chen:2017tva; @Jin:2017qcv; @Duan:2017qwj; @Zu:2017dzm; @Ding:2017jdr; @Gu:2017bdw; @Chao:2017yjg; @Tang:2017lfb; @Gu:2017gle; @Liu:2017obm; @Ge:2017tkd]. In this work, we explain the DAMPE $e^\pm$ result in the DM annihilation and pulsar scenarios, and perform a fit to the DAMPE $e^\pm$ spectrum and the AMS-02 positron fraction. The tentative feature at $\sim 1.4$ TeV in the DAMPE spectrum is not considered in this analysis. Several leptonic DM annihilation channels to $\mu^+\mu^-$, $\tau^+\tau^-$, $4\mu$, $4\tau$, and mixed charged lepton final states $e^+e^-+\mu^+\mu^-+\tau^+\tau^-$ (denoted by $e\mu\tau$ for simplicity) are considered. Although all these channels except for the $\mu^+\mu^-$ channel could provide a good fit to the DAMPE data, they would be excluded by the observations of gamma-ray and CMB in ordinary DM models. For the pulsar scenario, we investigate 21 nearby pulsars in the ATNF catalog [@manchester2005australia][^1] and find out some candidates that can be the single source of the high energy $e^\pm$. Furthermore, since the nearby pulsars may lead to significant anisotropy in the $e^\pm$ flux, we also adopt the anisotropy measurement of Fermi-LAT to explore the pulsar origin of high energy $e^\pm$, and find that some pulsars accounting for the DAMPE data could evade the current anisotropy constraint. Since the best-fit spectra resulted from some pulsars are very similar to those induced by DM annihilation, our results show that it is difficult to distinguish between these two explanations of high energy $e^\pm$ with the current DAMPE result. This paper is organized as follows. In Section \[section\_electron\_(positron)\_propagation\_in\_galaxy\], we introduce the CR propagation model adopted in our analysis. In Section \[section\_cr\_injection\_sources\_\], we outline the injection spectra of $e^{\pm}$ for the background, DM annihilation, and single nearby pulsar. In Section \[section\_can\_local\_known\_single\_pulsar\_confuse\_dampe\_to\_confirm\_dm\_signal\], we use the DAMPE data to investigate the DM annihilation and the single pulsar explanations for high energy $e^\pm$. Finally, the conclusion is given in Section \[section\_conclusion\]. CR $e^\pm$ propagation in the Galaxy {#section_electron_(positron)_propagation_in_galaxy} ==================================== Galactic supernova remnants (SNRs) are generally believed to be the main source of primary CR particles with energies below $\sim 10^{17}\, \ev$. After leaving the source, CR particles travel along the trajectories which are tangled by the Galactic magnetic field, and thus diffusively propagate. Furthermore, they would also suffer from the so-called re-acceleration effect by scattering with the moving magnetic turbulence and gaining energy through the second order Fermi acceleration. CRs propagate within a magnetic cylindrical diffusion halo with a characteristic radius of $20\, \kpc$ and a half height $z_h \sim \mathcal{O}(1)$ kpc. At the boundary of the propagation halo, CRs would freely escape. During the journey to the earth, CRs lose their energies by a variety of effects; the primary nucleons fragment through inelastic collisions with the interstellar medium (ISM) and create secondary CR particles. ------------------------------ ----------------------------- -------------------- $D_0$ $10^{28} {\cm}^2 \sec^{-1}$ $4.16 \pm 0.57$ $\delta$ $0.500 \pm 0.012$ $z_h$ $\kpc$ $5.02 \pm 0.86$ $v_A$ $\km \, \sec^{-1}$ $18.4 \pm 2.0$ $R_0$ $\GV$ 4 $\eta$ $-1.28 \pm 0.22$ $\mathrm{log(A_{\rho})}$[^2] $-8.334 \pm 0.002$ $\nu_1$ $2.04 \pm 0.03$ $\nu_2$ $2.33 \pm 0.01$ $\mathrm{log(R_{br}^p)}$[^3] $4.03 \pm 0.03$ ------------------------------ ----------------------------- -------------------- : The mean values and $1\sigma$ uncertainties of the propagation and proton injection parameters for the DR2 propogation model. []{data-label="tab:pro"} Involving the diffusion, re-acceleration, momentum loss and fragmentation effects, the transport equation can be described as [@ginzburg1964origin; @ginzburg1990astrophysics] $$\label{eq:transp} \frac{\partial \Psi(\vec{r},p,t)}{\partial t} = Q(\vec r,p) + \nabla\cdot(D_{xx}\nabla\Psi) + \frac{\partial} {\partial p}p^2 D_{pp} \frac{\partial}{\partial p} \frac{1}{p^2}\Psi -\frac{\partial}{p} \dot{p} \Psi -\frac{\Psi}{\tau_f} ,$$ where $\Psi(\vec{r},p,t)$ is the CR density per unit momentum interval at $\vec r$, $Q(\vec r,p)$ is the source term including primary and spallation contributions, $D_{xx}$ is the spatial diffusion coefficient, $D_{pp}$ is the diffusion coefficient in momentum space, $\tau_f$ is the time scale for the loss by fragmentation. We use the public code GALPROP [@strong1999galprop; @strong2009galprop; @strong2015recent][^4] to numerically solve this equation. The spatial diffusion coefficient is described by $$D_{xx} = {\beta}^{\eta} D_0 {(\frac{R}{R_0})}^{\delta} ,$$ where $\beta = {\upsilon}/{c}$ is velocity of particle in unit of the speed of light, $D_0$ is a normalization constant, $R = {pc}/{ze} $ is the rigidity, $R_0$ is the reference rigidity and $\eta$ describes the velocity dependence of the diffusion coefficient. The re-acceleration effect can be described by the diffusion in the momentum space. The momentum diffusion coefficient $D_{pp}$ and spatial diffusion coefficient $D_{xx}$ are related by [@seo1994stochastic] $$\label{eq:re} D_{pp} = \frac{1}{D_{xx}} \cdot \frac{4p^2 {V_A}^2 }{3 \delta(4-\delta)(4-{\delta}^2)\omega},$$ where $\omega$ denotes the level of the interstellar turbulence. Absorbing $\omega$ to $V_A$ and referring $V_A$ characterizes the re-acceleration strength. In Ref. [@Yuan:2017ozr] we have systematically studied the typical propagation models and nuclei injection spectra using the latest Boron-to-Carbon ratio $\mathrm{B/C}$ data from AMS-02 and the proton fluxes from PAMELA and AMS-02. We find that the DR2 model including the re-acceleration and velocity depending diffusion effects gives the best fit to all the data. The posterior mean values and $68\%$ confidence interval of the model parameters are given in table \[tab:pro\]. The local interstellar flux is given by $\Phi = \Psi(r_{\odot})c/4\pi$. Before the local interstellar (LIS) CRs arrive at the earth, they suffer from the solar modulation effect within the heliosphere. We employ the force field approximation, which is described by a solar modulation potential $\phi$, to deal with this effect. CR injection sources {#section_cr_injection_sources_} ===================== The observed CR $e^\pm$ consist of three components: the primary electrons produced by SNRs; the secondary electrons and positrons from primary nuclei spallation processes in the ISM; $e^\pm$ pairs generated from exotic sources such as the DM or pulsar. The sum of the first two components is treated as the background. In this Section, we outline the injection CR $e^{\pm}$ spectra for the backgrounds, DM annihilations, and single nearby pulsar. The $e^{\pm}$ background spectrum --------------------------------- Ordinary CR sources are expected to be located around the Galactic disk, following the SNR radial distribution given by [@trotta2011constraints] $$\label{eq:frz} f(r,z) = (r/r_{\odot})^{1.25} \exp(-3.56 \cdot \frac{r-r_{\odot}}{r_{\odot}}) \exp(-\frac{|z|}{z_s}) ,$$ where $r_{\odot} = 8.3\, \kpc$ is distance between the sun and the Galactic center, $z_s = 0.2 \, \kpc$ is the characteristic height of the Galactic disk. These sources are able to accelerate the high energy CR electrons through the first order Fermi shock acceleration, which would result in a power law spectrum. The previous studies have found that a three-piece broken is enough to describe the injection spectrum of electrons below $\sim\TeV$[@Moskalenko:1997gh; @lin2015quantitative]. The break at a few $\GeV$ is used to fit the low energy data, while the hardening around hundreds of $\GeV$ is introduced to account for the effect of possible nearby sources [@Bernard:2012pia] or non-linear particle acceleration [@Ptuskin:2012qu]. Above $\TeV$, the contribution to observed CR electrons would be dominated by several nearby SNRs due to the serious energy loss effect. The electron spectra from these SNRs may depend on their properties. The detailed discussions can be found in Ref. [@fang2016perspective; @Fang:2017tvj]. In this work, we simply introduce an exponential cutoff $\sim\TeV$ to describe the behaviour of high energy injection spectra from the nearby SNRs. Thus, the injection spectrum of the primary electron component follows the form $$\label{eq:prminj} q(R) \propto \left\{ \begin{array}{c} (R/R_{br})^{-\gamma_0}\exp(E/E_{bgc}) ,\quad R<=R_{br0} \\ (R/R_{br})^{-\gamma_1}\exp(E/E_{bgc}) ,\quad R<=R_{br} \\ (R/R_{br})^{-\gamma_2}\exp(E/E_{bgc}) ,\quad R>R_{br} \end{array} \right. .$$ The source function $Q(\vec r ,p)$ in Eq. \[eq:transp\] for SNRs is given by $Q(\vec r,p) = f(r,z)\cdot q(R)$. The secondary $e^\pm$ are produced by the spallation of primary particles (mainly protons and helium nuclei) in the ISM. The steady-state production rate of the secondary $e^\pm$ at the position $\vec r$ is $$\label{eq:secinj} Q_{sec}(\vec r,E) = 4\pi \sum_{ij} \int \mathrm{d}E^{'} \Phi_i(E^{'},\vec r) \frac{\mathrm{d}\sigma_{ij}(E^{'},E)}{\mathrm{d}E} n_j(\vec r),$$ where $\mathrm{d}\sigma_{ij}(E',E)/\mathrm{d}E$ is the differential cross section for $e^\pm$ with the kinetic energy of $E$ from the interaction between the CR particle $i$ with the energy of $E'$ and ISM target $j$, and $n_j$ is the number density of the ISM target $j$. $e^{\pm}$ from DM annihilations ------------------------------- The Galaxy is embedded in a huge DM halo. If DM particles have some interactions with standard model particles, DM annihilations could produce CRs as an exotic source. The source term for DM annihilations is given by $$\label{eq:dmsource} Q_{DM}(\vec r,E) = \frac{1}{2} {\frac{ \rho_{DM}(\vec r)}{m_{DM}}}^2 \langle {\sigma \upsilon}\rangle \sum_k{ B_k \frac{dN^k_{e^\pm}}{dE}} ,$$ where $\mathrm{d}N^k_{e\pm}/\mathrm{d}E$ denotes the $e^{\pm}$ energy spectrum from a single annihilation with final states $k$, $B_k$ is the corresponding branching fraction, $\rho$ is the DM density, and $\langle {\sigma \upsilon}\rangle$ is thermal averaged velocity-weighted annihilation cross section. In our analysis, we adopt the Navarro-Frenk-White (NFW) density profile [@navarro1997universal] $$\label{eq:profile} \rho_{DM}(\vec r) = \frac{\rho_s}{(|\vec r|/r_s){(1+ |\vec r|/r_s)}^2},$$ where $r_s = 20 \,\kpc$, and the local DM density is normalized to $\rho_{\odot}= 0.4 \,\GeV\cm^{-3}$ consistent with the dynamical constraints [@nesti2013dark; @sofue2012grand; @weber2010determination]. The initial energy spectra of DM annihilations are taken from PPPC 4 DM ID [@cirelli2011pppc] which includes the electroweak corrections. We also use GALPROP to simulate the propagation of such emissions from DM annihilation. $e^{\pm}$ from the nearby pulsar {#sec:pssoc} -------------------------------- The pulsar is a rotating neutron star surrounded by the strong magnetic field. It can produce $e^{\pm}$ pairs through the electromagnetic cascade and accelerate them by costing the spin-down energy. These high energy $e^{\pm}$ pairs are injected into the pulsar wind nebula (PWN) and finally escape to the ISM. A burst-like spectrum of the electron and positron is adopted to describe the pulsar injection, and is usually assumed to be a power law with an exponential cutoff $$\label{eq:pulsarinj} Q_{psr}(\vec r,E,t) = Q_0 E^{-\alpha} \exp{(-\frac{E}{E_c})} \delta(\vec r) \delta(t),$$ where $Q_0$ is the normalization factor, $\alpha$ is the spectrum index, and $E_c$ is the cutoff energy. The total energy output is related to the spin-down energy $W_0$ by assuming that a fraction $f$ of $W_0$ would be transferred to $e^{\pm}$ pairs, so that $$\int_{0.1\GeV}^{\infty} \mathrm{d}E E Q_0 E^{-\alpha} \exp{(-\frac{E}{E_c})} = \frac{f}{2} W_0.$$ $W_0$ can be derived from $\tau_0 \dot{E}(1+t/{\tau_0})^2$, where $t$ is the pulsar age and the typical pulsar decay time is taken to be $t_0 = 10^{4}\, \kyr$ here [@aharonian1995high; @profumo2012dissecting]. Due to the high energy loss rate, the energetic $e^{\pm}$ could only propagates over a small distance of $\mathcal{O}(1)$ kpc [@yin2013pulsar; @feng2016pulsar]. For the local sources, we adopt an analytic solution of the propagation equation. The re-acceleration effect is neglected for particles with energies above $10 \,\GeV$ [@delahaye2009galactic]. The Green function solution without boundary condition is given by [@delahaye2010galactic; @fang2016perspective] $$\label{eq:locflux} \Psi(r_{\odot},E) = \frac{1}{(\pi \lambda^2(E,E_s))^{3/2}}\cdot \exp{(-\frac{r^2}{\lambda^2(E,E_s)})} \cdot Q_{psr}(E_s),$$ where $r$ is the distance to the pulsar, $E_s$ is the initial $e^{\pm}$ energy from the source, and $\lambda$ is the diffusion length defined as $$\lambda^2 \equiv 4 \int_E^{E_s} \mathrm{d}E^{'} {D(E')}/{b(E^{'})}.$$ The $E_s$ could be derived from the propagation time $t$ and the final energy $E$ by the relation of $$\label{eq:Es} \int_{E_s}^E - \frac{\mathrm{d} E}{b(E)} = \int_{-t}^0\mathrm{d}t,$$ where $b(E)\equiv -\mathrm{d}E/\mathrm{d}t$ is the energy loss rate of $e^{\pm}$. We include the energy loss induced by the synchrotron radiation in the Galactic magnetic field and the inverse Compton scattering with the ambient photon field. The local interstellar radiation field (ISRF) is taken from M1 model in Ref. [@delahaye2010galactic] and the magnetic field is assumed to be $4 \,\mu G$. A detailed discussion on the relativistic energy loss rate is given in appendix \[sec:lossrate\]. As no simple analytic solution is available, we use GNU Scientific Library (GSL) [^5] to numerically solve Eq.\[eq:Es\]. Explanations of the DAMPE $e^\pm$ spectrum {#section_can_local_known_single_pulsar_confuse_dampe_to_confirm_dm_signal} ========================================== The DAMPE collaboration has reported the $e^\pm$ spectrum with high resolution from $25 \GeV$ to $4.6 \TeV$ [@Ambrosi:2017wek]. In this section, we fit the DAMPE data in the DM annihilation and single pulsar scenarios. The AMS-02 positron fraction  [@accardo2014high] is also considered in the fit. Since we focus on the high energy $e^\pm$, the positron fraction data below 10 $\GeV$ are not adopted. There are 81 data points involved in the analysis. The DM scenario {#subsection_dm_scenarios} --------------- DM particles in the propagation halo may decay or annihilate to standard model particles and contribute to the finally observed CR leptons [@Bergstrom:2000pn; @Bertone:2004pz; @Bergstrom:2009ib]. In this work, we discuss the DM annihilation channels to $\mu^+\mu^-$, $\tau^+\tau^-$, $e\mu\tau$, $4\mu$, and $4\tau$. In the $e\mu\tau$ channel, we set the branch ratios of $e^+e^-$, $\mu^+\mu^-$ and $\tau^+\tau^-$ final states to be free parameters. With the propagation parameters and proton injections given in Table. \[tab:pro\], we vary the injection of primary electrons, DM parameters and solar modulation potential to obtain the best-fit though the Monte Carlo Markov Chain (MCMC) method. Note that we only consider the energy region above $10\GeV$, thus the low energy break around several $\GeV$ in the background electron spectrum is neglected. In addition, a rescale factor $c_{e^\pm}$ is introduced to indicate the uncertainty of the hadronic collisions. The degree of freedom (d.o.f.) in this fitting is thus 72.[^6] $\mu^{+}\mu^{-}$ $\tau^{+}\tau^{-}$ $e\mu\tau$ $4\mu$ $4\tau$ -------------------------------------------------------------- ------------------ -------------------- ----------------- ----------------- ----------------- $\gamma_1$ $3.05 \pm 0.02$ $2.93 \pm 0.02$ $2.93 \pm 0.03$ $3.06 \pm 0.01$ $2.90 \pm 0.02$ $\gamma_2$ $2.57 \pm 0.02$ $2.50 \pm 0.02$ $2.53 \pm 0.02$ $2.53 \pm 0.02$ $2.50 \pm 0.02$ $\log(R_{br}^e/\mathrm{MV})$ $4.69\pm0.03$ $4.70\pm0.04$ $4.73\pm0.03$ $4.71 \pm 0.03$ $4.68 \pm 0.04$ $c_{e^{\pm}}$ $3.66 \pm 0.03$ $2.82 \pm 0.09$ $3.07 \pm 0.19$ $3.67 \pm 0.04$ $2.65 \pm 0.03$ $E_{bgc}/\TeV$ $4.42 \pm 0.96$ $6.2 \pm 2.0$ $10.9\pm2.2$ $4.17 \pm 1.09$ $5.58 \pm 0.71$ $\phi/\mathrm{GV}$ $1.48\pm0.02$ $0.77 \pm 0.10$ $1.16 \pm 0.17$ $1.30 \pm 0.08$ $0.67\pm 0.07$ $m_{DM}/\mathrm{GeV}$ $1891\pm71$ $3210\pm316$ $1560\pm178$ $3243 \pm 290$ $5366 \pm 338$ $\langle\sigma\nu\rangle/(10^{-23}{\mathrm{cm}}^3\sec^{-1})$ $1.37\pm0.09$ $5.24\pm0.82$ $1.46\pm0.31$ $2.25\pm0.37$ $7.96\pm1.01$ $\chi^2$ 113.30 68.71 59.69 89.87 69.56 : Posterior mean and $68\%$ credible uncertainties of the model parameters and $\chi^2$ value in the DM scenarios, with d.o.f. of 72.[]{data-label="tab:dmfit"} We list the best-fit results in Table. \[tab:dmfit\] and show the spectra in Fig. \[fig:dm\_best\]. With a large $\chi^2$ valued 113.3, the $\mu^+\mu^-$ channel is excluded with more than $3\sigma$ confidence, while all the other channels provide reasonable fits. This is because that the $\mu^+\mu^-$ channel induces a harder $e^\pm$ spectrum than all the other channels, and tends to produce too many positrons at high energies when explaining the positron fraction below $100\GeV$. Note that the fit for annihilation channels with hard DM contributions is sensitive to the secondary positrons. In Ref. [@Yuan:2017ysv], Yuan et al. performed a similar analysis but with a smaller diffusion coefficient power index $\delta=1/3$. Since this $\delta$ leads to a harder secondary positron spectrum, the $\mu^+\mu^-$ channel work well in that analysis. ![The spectra of the best-fit restuffs in the DM scenario. Top panels show the positron fractions in comparison with the AMS-02 data [@accardo2014high], while the bottom panels show the total $e^\pm$ spectra in comparison with the DAMPE data [@Ambrosi:2017wek]. The dashed, dotted-dashed and solid lines represent the backgrounds, DM contributions and total results, respectively.[]{data-label="fig:dm_best"}](dm1_posif.eps "fig:"){width="45.00000%"} ![The spectra of the best-fit restuffs in the DM scenario. Top panels show the positron fractions in comparison with the AMS-02 data [@accardo2014high], while the bottom panels show the total $e^\pm$ spectra in comparison with the DAMPE data [@Ambrosi:2017wek]. The dashed, dotted-dashed and solid lines represent the backgrounds, DM contributions and total results, respectively.[]{data-label="fig:dm_best"}](dm2_posif.eps "fig:"){width="45.00000%"}\ ![The spectra of the best-fit restuffs in the DM scenario. Top panels show the positron fractions in comparison with the AMS-02 data [@accardo2014high], while the bottom panels show the total $e^\pm$ spectra in comparison with the DAMPE data [@Ambrosi:2017wek]. The dashed, dotted-dashed and solid lines represent the backgrounds, DM contributions and total results, respectively.[]{data-label="fig:dm_best"}](dm1_totep.eps "fig:"){width="45.00000%"} ![The spectra of the best-fit restuffs in the DM scenario. Top panels show the positron fractions in comparison with the AMS-02 data [@accardo2014high], while the bottom panels show the total $e^\pm$ spectra in comparison with the DAMPE data [@Ambrosi:2017wek]. The dashed, dotted-dashed and solid lines represent the backgrounds, DM contributions and total results, respectively.[]{data-label="fig:dm_best"}](dm2_totep.eps "fig:"){width="45.00000%"} In the fit for the $e\mu\tau$ channel, since the sharp shapes of the injection spectra from the $\mu^\pm$ and $e^\pm$ final states are not favoured here, the $\tau^\pm$ final states are dominant with a branch ratio of 0.755, while he branch ratios of $e^\pm$ and $\mu^\pm$ are 0.094 and 0.151 respectively . A recent work also analysed this channel but found that the branch ratio of $\tau^\pm$ is suppressed [@Niu:2017hqe]. This different conclusion may be attributed by that their background secondary positron spectrum is much harder than ours. It is interesting to note that the contribution from the $e^\pm$ final states would indicate a distinct drop in the spectrum at the DM mass, as shown in the bottom left panel of Fig. \[fig:dm\_best\]. It is possible to check such spectral feature in the results of DAMPE or HERD in the future. In Fig. \[fig:dm\_contour\], we show the $68\%$ and $95\%$ confidence regions for the DM mass and thermally averaged annihilation cross section. Note that the DM implication for the CR $e^\pm$ excess has been strongly constrained by many other observations, such as the cosmic microwave background (CMB) [@komatsu2009five; @ade2016planck; @slatyer2016indirect], dwarf galaxy gamma-ray  [@collaboration2015searching], and diffuse gamma-ray observations [@ackermann2012constraints]. We also show the corresponding constraints in Fig. \[fig:dm\_contour\]. To calculate the constraints from dwarf galaxies, we use PPPC 4 DM ID [@Cirelli:2010xx] to produce the DM gamma-ray spectra and adopt the likelihood results from the combined analysis given in  [@Fermi-LAT:2016uux]. The constraints from the CMB observations are taken from Ref. [@Yuan:2017ysv], where the Planck 2015 results with an energy deposition efficiency $f_\mathrm{eff}$ ranged from 0.15 to 0.7 [@Ade:2015xua] are adopted. The best-fit regions for many channels shown in Fig. \[fig:dm\_contour\] seem to be excluded. However, the tensions between these observations can be reconciled in the velocity-dependent annihilation scenario due to the fact that DM particles contributing to these observations have different typical relative velocities [@Bai:2017fav; @xiang2017dark]. ![The $68\%$ and $95\%$ confidence regions of the DM mass and thermally averaged annihilation cross section for different annihilation channels. The corresponding limits derived from the Fermi-LAR gamma-ray observations of dwarf galaxies [@Fermi-LAT:2016uux] are also shown. The dark green shade region represents the constraint from the CMB observations, with an energy deposition efficiency $f_\mathrm{eff}$ ranged from 0.15 to 0.7 [@Ade:2015xua].[]{data-label="fig:dm_contour"}](contour.eps){width="70.00000%"} On the other hand, the constraints from the diffuse gamma-ray observation [@ackermann2012constraints] cannot be easily avoided in the velocity-dependent annihilation scenario. However, the astrophysical uncertainties of this analysis arising from the Galactic CR model are not negligible. The solid constraints given by the Fermi-LAT collaboration without modeling of the astrophysical background cannot exclude the DM implication accounting for the CR $e^\pm$ excess. When the contributions of the pion decay and inverse Compton scattering from Galactic CRs are considered, the constraints given by the Fermi-LAT collaboration become stringent and strongly disfavor the $\tau^+\tau^-$ channel. However, these constraints depend on the CR distribution and prorogation models. In order to reduce the related uncertainties, more precise data and further studies on the CR model will be needed. The single pulsar scenario -------------------------- The nearby pulsars are possible sources of high energy CR positrons. We consider the cases in which a nearby mature pulsar is the primary source of high energy $e^\pm$. The 21 known pulsars within $1\, \kpc$ with characteristic ages in the range of $10^3-10^6$ years are considered. Their properties, such as the distance, age and spin-down luminosity, are taken from the ATNF catalog [@manchester2005australia], which includes the most exhaustive and updated list of known pulsars. In this section, we fit the DAMPE total $e^\pm$ spectrum and the AMS-02 positron fraction using one of the 21 selected pulsars. Comparing with the DM scenario with two free parameters, there are three free parameters ($f$, $\alpha$, $E_c$) in the pulsar scenario. Thus the d.o.f. is 71 in the fit. Then, we drop all the candidates which are not acceptable at 95.4% C.L., and list the best-fit result of the five left pulsars in Table. \[tab:psfit\]. Note that there are uncertainties in the estimation for the spin-down energies of pulsars. Therefore, a transfer fraction $f$ larger than 1 within a tolerance range may be allowed. Except for J0954-5430 with a very large transfer fraction is 7.58, we accept the other four pulsars listed in Table. \[tab:psfit\]. For these four acceptable pulsars, we show the best-fit spectra in Fig. \[fg:psfit\_best\]. We find that B1001-47, which is older than the others, would lead to a drop around $1.2\TeV$ at the $e^\pm$ spectrum, due to the energy loss effect. From Fig. \[fig:dm\_best\] and Fig. \[fg:psfit\_best\], it can be seen that the best-fit spectra resulted from some pulsars are very similar to those induced by DM annihilation. It is still difficult to distinguish between these two explanations of high energy $e^\pm$ with the current DAMPE result. ![The same as Fig. \[fig:dm\_best\], but for the single pulsar scenario.[]{data-label="fg:psfit_best"}](ps_posif.eps "fig:"){width="45.00000%"} ![The same as Fig. \[fig:dm\_best\], but for the single pulsar scenario.[]{data-label="fg:psfit_best"}](ps_totep.eps "fig:"){width="45.00000%"}\ In order to take account the uncertainty from the injection spectrum of the pulsar, the 68% and 95% confidence regions for the power index $\alpha$ and cutoff energy $E_c$ are shown in Fig. \[fig:ps\_contour\]. The favoured values of injection parameters depend on the age and distance of the pulsar. For the far or young pulsars, such as J1732-3131 and J0940-5428, the injected low energy positrons are difficult to reach the Earth, therefore a large $\alpha$ is needed to result in enough positrons at low energies. In addition, the contributions from such pulsars would suffer a serious energy loss effect, thus they require a high energy cutoff $E_c$ to ensure enough high energy positrons. ![Similar to Fig. \[fig:dm\_contour\], but for the $\alpha$ and $E_c$ in single pulsar scenario.[]{data-label="fig:ps_contour"}](galp_ps.eps){width="80.00000%"} The local young astrophysical sources may induce the observable dipole anisotropy in the CR arrival direction [@shen1971anisotropy; @kobayashi2004most; @di2011implications; @manconi2017dipole]. Therefore, the four pulsars considered above would be constrained by the currently anisotropy observations. Both Fermi and AMS-02 have not detected a significant anisotropy, and set upper limits on the CR electron-positron, positron and electron dipole anisotropies [@ackermann2010searches; @abdollahi2017search; @aguilar2013alpat; @la2016search]. For the AMS-02 experiment [@la2016search], the $95\%$ C.L limit on the integrated positron dipole anisotropy based on the first five years data is about $0.02$ above $16 \,\GeV$. Recently, the Fermi collaboration has released the latest result of the dipole $e^\pm$ anisotropy, using seven years data reconstructed by Pass 8 in the energy region from $42 \,\GeV$ to $2 \,\TeV$ [@abdollahi2017search]. The $95\%$ C.L upper limit ranges from $3\times 10^{-3}$ to $3\times 10^{-2}$. For the single nearby source dominating the CR flux, the dipole anisotropy is given by [@shen1971anisotropy] $$\Delta = \frac{3D}{c}|\frac{\nabla\Phi}{\Phi}| .$$ By using the flux in Eq. \[eq:locflux\], we explicitly express the dipole anisotropy as $$\Delta (E) = \frac{3D(E)}{c} \frac{2d_s}{\lambda^2(E,E_s)} .$$ \[fg:ps\_aniso\] ![The $e^\pm$ anisotropies for the pulsars. The 95% C.L. upper limits given by Fermi-LAT using the log-likelihood ratio and Bayesian methods [@abdollahi2017search] are also shown for comparison.](ps_aniso.eps "fig:"){width="70.00000%"} We estimate the positron anisotropy for the four pulsars, and find all of them would result in an integrated anisotropy obviously smaller than 0.02 at $16\GeV$, which evade the limit of AMS-02. In addition, we show the expected $e^\pm$ anisotropy from each pulsar comparing with the measurement of Fermi-LAT in Fig. \[fg:ps\_aniso\]. Note that the dipole anisotropy is proportional to the distance-to-age ratio ${r}/{t}$ of the pulsar. We find that the anisotropy induced by J0940-5428 with a ${r}/{t}\sim 9\times 10^{-3} $ kpc/kyr has a tension with the Fermi upper limit. For J1732-3131 with a larger age of $111\,\kyr$ and a smaller ${r}/{t}\sim 5.8\times 10^{-3}$ kpc/kyr, there is a slight tension as the Fermi-LAT Bayesian limit excludes this source while the LLR limit does not. The other pulsars with a very small ${r}/{t}\sim 2.5 \times 10^{-3}$ kpc/kyr provides a very small anisotropy, which is far from the Fermi-LAT sensitivity. As a conclusion, B0656+14 and B1001-47 survive from all the limits considered. Conclusion {#section_conclusion} ========== In this paper, we study the DM annihilation and pulsar interpretations of high energy CR $e^\pm$ observed by DAMPE. We investigate the $e^\pm$ contributions from several DM annihilation channels and known nearby pulsars in the ATNF catalog, and find some allowed realizations. For the DM scenario, we investigate the $\mu^+\mu^-$, $\tau^+\tau^-$, $e\mu\tau$, $4\mu$, and $4\tau$ annihilation channels. We find that all these channels except for the $\mu^+\mu^-$ channel can explain the DAMPE data. In the $e\mu\tau$ mixing channel, the $\tau^\pm$ final states are dominant. However, the contribution from the $e^\pm$ final states in this channel would lead to a distinct drop in the spectrum. Such spectral feature may be detected in the future measurements with larger statistics, such as DAMPE and HERD. The constraints from the diffuse $\gamma$ ray, the $\gamma$ ray of dwarf galaxy and the CMB observations are discussed. We find that many channels have been excluded and some complicated DM models are necessary to reconcile the tension between different observations. For the pulsar scenario, we find five single pulsars that are acceptable at 94.5% C.L. . Among these pulsars, J0954-5430 requires a too large transfer efficiency which is unacceptable. We also investigate the $e^\pm$ anisotropy from the single pulsar, and find that J1732-3131 and J0940-5428 are not favored by the Fermi-LAT observations. As a conclusion, B0656+14 and B1001-47 are possible to explain the current observations of DAMPE and AMS-02. Our results show that it is difficult to distinguish between the DM annihilation and single pulsar explanations of high energy $e^\pm$ with the current DAMPE result. In the future, the combination of the $e^\pm$ spectra and anisotropy measurements with large statistics like HERD may be useful to further discriminate the origins of high energy CR $e^\pm$. Acknowledgment {#acknowledgment .unnumbered} ============== This work is supported by the National Key Program for Research and Development (No. 2016YFA0400200), by the 973 Program of China under Grant No. 2013CB837000, and by the National Natural Science Foundation of China under Grants No. 11475189, 11475191, Appendix ======== Energy loss and cutoff {#sec:lossrate} ---------------------- By using Eq. \[eq:Es\] we can derive the $e^\pm$ energy $E_s$ in the source. The energy loss of $e^\pm$ above few $\GeV$ is mainly caused by the synchrotron radiation and inverse Compton scattering processes. The synchrotron radiation energy loss is given by $$\label{eq:synloss} -{(\frac{dE}{dt})}_{syn} = \frac{4}{3} \sigma_T c U_B {\gamma_e}^2 ,$$ where $\sigma_T$ is the Thomson scattering cross section ($\sigma_T=6.65\times 10^{-25}\cm^2$), $c$ is the speed of light, $U_B$ is the magnetic field energy density, and $\gamma_e = E/{m_e c^2}$ is the Lorentz factor. The energy loss caused by the inverse Compton scattering of $e^\pm$ with energies $E<<{(m_e c^2)}^2/{k_b T}$ in the Thomson regime can be expressed by Eq. \[eq:synloss\]) with replacing $U_B$ to the radiation field energy density ($U_{rad}$). The energy density distribution of ISRF is described in section \[sec:pssoc\]. In the Thomson approximation the energy loss rate can be given by $-\frac{dE}{dt} = b_0 E^2$, where $b_0 = \frac{4}{3 m_e c^2} \sigma_T c (U_B+U_{rad})$ is a constant. This number is often taken to be $\mathrm{O}(10^{-16}) \, \GeV^{-1}/{\mathrm{s}}$ [@blasi2009origin; @feng2016pulsar]. Then we can derive the maximum energy of $e^\pm$ arriving at the solar system as $$\int_{E_s}^E \frac{dE}{-b_0 E^2} = \int_{-t}^0 dt \quad \rightarrow \quad E = \frac{1}{b_0 t + 1/E_s} < \frac{1}{b_0 t} .$$ This indicates that the maximum energies of observed $e^\pm$ are determined by $1/{b_0 t}$ due to the energy loss and are almost independent of their initial energies from the source. However, under the extreme Klein-Nishina limit with $E>>{(m_e c^2)}^2/{k_b T}$, the energy loss rate is $$-{(\frac{dE}{dt})}_{KN} = \frac{\sigma_T}{16} \frac{{(m_e c k_b T)}^2}{\hbar^3} {\mathrm{ln}\frac{4\gamma_e k_b T}{m_e c^2} - 1.9805}.$$ In this case the energy loss rate only increases logarithmically with $E$, while it increases with $E^2$ under the Thomson limit. Thus the efficiency of inverse Compton scattering would be strongly reduced; this effect is referred to as “KLein-Nishina cutoff”[@schlickeiser2013cosmic] and has been discussed in Ref. [@kobayashi2004most; @schlickeiser2010klein; @delahaye2010galactic; @khangulyan2014simple]. In this work we adopt the parametrization expression in [@delahaye2010galactic] to accurately calculate the energy loss rate caused by the inverse Compton scattering. For instance, we illustrate the Klein-Nishina effect for Geminga, which is a pulsar with a distance of $0.25 \, \kpc$, an age of $342 \, \kyr$, and a spin-down luminosity of $3.2 \times 10^{34} \, \erg/\sec$. The energy transfer efficiency $f$ is taken as $30\%$ and the injection spectrum energy cutoff is assumed to be $10 \, \TeV$. As can be seen in figure \[fg:cutoff\], since Thomson approximation results in a higher energy loss rate, the spectrum cutoff is sharper than that derived from the Klein-Nishina effect especially for a hard injection spectrum. ![Comparison between different treatments for the inverse Compton energy loss of the Geminga electron spectrum. The solid and dashed lines represented results for the relativistic energy loss rate and the Thomson approximation energy, respectively. $\alpha$ is the index of the injection spectrum.[]{data-label="fg:cutoff"}](cutoff.eps){width="100.00000%"} [^1]: <http://www.atnf.csiro.au/people/pulsar/psrcat/> [^2]: Propagated flux normalization at $100\, \GeV$ in unit of $\cm^{-2}\sec^{-1}\sr^{-1} {\MeV}^{-1}$ [^3]: Break rigidity of proton injection spectrum in unit of MV [^4]: <http:galprop.stanford.edu/> [^5]: <https://www.gnu.org/software/gsl/> [^6]: There is also a normalize factor of the injection spectrum $A_e$, which is not quite relevant but is still taken as a free parameter.

--- author: - 'John D. Hefele' - Francesco Bortolussi - Simon Portegies Zwart bibliography: - 'bibliography.bib' date: 'Received 29 May 2019 / Accepted 9 December 2019' title: 'Identifying Earth-impacting asteroids using an artificial neural network\' --- Introduction {#SEC:Introduction} ============ In 1990 the US Congress requested for NASA to establish two workshops to focus on the identification of potentially hazardous small bodies and on methods of altering their orbits to prevent impact [@milani2002]. The workshops led to the establishment of the *Sentry earth impact monitoring* system [@Sentry]. If a hazardous asteroid is identified early enough prior to impact, it would be possible to mitigate the impact by means of an appropriate space mission to alter the asteroid’s orbit through a gravitational tugboat [@10.2307/26060526] or by obliterating it with a nuclear warhead [@BARBEE201837]. Both mitigation strategies require many years of preparation, which makes the early detection of hazardous objects vital for allowing ample time to prepare such missions. The Sentry system adopts a Monte Carlo approach in which millions of virtual objects are launched with orbital parameters that are statistically sampled from within the error ellipse of the observed asteroids. The impact probability is subsequently determined based on the fraction of virtual asteroids that reach Earth within some predetermined striking distance [@milani2002]. In this approach, the orbits of many asteroids are integrated numerically and the final parameter space is considered to represent the probability-density distribution of the respective objects. The calculation of this probability density distribution relies on the algorithm and implementation used to integrate the orbits of the asteroids. The time scale over which such integrations remain reliable depends on the degree to which the asteroid’s orbit is chaotic, that is, it depends on the value of the largest positive Lyapunov exponent. Additionally, the reliability of such integrations depends on the ability of the integrator to obtain a solution, such that the integration complies to the concept of nagh Hoch[^1] [@PORTEGIESZWART2018160]. Both of these concepts are not guaranteed with regard to the adopted numerical schemes and the results reach questionable proportions as soon as the asteroid experiences a close encounter with any object other than the Earth. In the latter case, the phase space of possible solutions grows exponentially due to the chaotic nature of the equations of motion. Establishing the chaotic nature of an asteroid is limited by the accuracy of its orbital determination. This is generally realized by observing any particular asteroid a number of times. These observations result in a data arc, the fraction of the orbit over which the object has been observed. The adopted Monte-Carlo method used in the Sentry system is expected to be reliable for at most a few dozen years [@HorizonsManual] for asteroids whose observed data arc is shorter than a month, which comprises 12.9% of all smallbodies [@dastcom5]. Considering the high degree of chaotic motion (small Lyapunov time scale) in asteroids and the consequential exponential divergence of its orbit, one might wonder if it is worth the effort to perform extensive computer simulations to track the orbital trajectories of a large number of particles so long as the veracity of the orbital integration cannot be guaranteed. For the most chaotic asteroids, the impact probability depends acutely on the statistics of the adopted method and a more coarse grained approach to identify potentially hazardous objects may suffice. This approach would free up computer time to provide a more reliable impact probability for the most promising candidate impostors. We explore the population of asteroids and, in particular, the potentially dangerous ones by means of automatic machine recognition through a combination of numerical integrations and a trained neural network similar to the architectures described in @ref1 and @ref2, which were used for classifying hazardous taxonomy and solar sail transfer time estimation respectively. It is a statistical approach in which we determine the prospect for impact of the known population of asteroids gathered from the *dastcom5* off-line database [@dastcom5]. Our analysis is mediated by an artificial neural-network dubbed HOI[^2] for Hazardous Object Identifier, which was trained on a population of known impactors (KI) and a random sample from the observed database using the *TensorFlow* framework [@TensorFlow]. The KIs are machine-generated from an integrated population of asteroids that start their orbit on a random position of Earth’s surface and are launched radially away with the varying speeds. These objects are subsequently integrated backward in time together with the planets in the Solar System for up to 20,000 years. To train HOI, these computer generated KIs are then mixed with a subset of observed asteroids, which we assume to be known non-impacting objects. The trained network is then used on another random selection of observed asteroids in order to identify potential impactors (PIs). All the objects that were not identified by the model as PIs, which were not initially labeled as KIs, are referred to as unidentified objects (UOs). We begin by describing HOI’s architecture in Section\[SEC:arch\], followed by a discussion of the generation of the small-body datasets in Section\[SEC:Data\]. The results are examined in Section\[SEC:Results\] and conclusions are drawn in Section\[SEC:Conclusions\]. All the code used to train the neural network, generate data, and evaluate the results are publicly available on GitHub[^3]. Hazardous Object Identifier (HOI) {#SEC:arch} ================================= In general, neural networks are particularly well-suited for recognizing complex patterns hidden in multidimensional datasets. In our particular case, we strive to identify observed objects that have topologically similar trajectories to the trajectories of the population of KIs. Because we are no longer reliant on calculations that attempt to estimate the asteroids position at a particular point in time, the network is more resilient to perturbations of the initial conditions, that is, chaotic motion. The problem at hand is a discrete binary classification task, where the two mutually exclusive classes for the observed objects are either potential impactors (PIs) or unidentified objects (UOs). For the purpose of our experiments, the UOs are what we would consider “benign objects”, meaning objects that are identified as having a negligible chance of colliding with the Earth. To quantify the network’s accuracy, the standard cross-entropy cost function is used. This is defined as: $$\label{cost_function} H(y,\hat{y})=-\sum_i^{N} y_i \text{ln}(\hat{y}_i)+(1-y_i)\text{ln}(1-\hat{y}_i).$$ Here $y$ is the actual value, or label, $\hat{y}$ is the predicted value, and $N$ is the total amount of predictions. This cost function has the convenient property that its derivative with respect to some input weight, $w$, scales linearly with the difference between the label and predicted value [@Nielson]: $$\frac{\partial C}{\partial w}=\frac{1}{N}\sum^{N}_i x(\hat{y}_i-y_i)$$ Here $x$ is the input value by which $w$ is multiplied. To minimize (\[cost\_function\]), the *Adam Optimizer* is used, which expands upon naïve stochastic gradient descent by adapting its learning rate based on both the average of the first and second moments of the gradients [@AdamOptimizer]. Empirically, it is observed that this optimizer reduces the cost function to the lowest value with the fewest number of iterations relative to the other algorithms available in TensorFlow. Each object fed into the HOI is represented by a five-element vector where each vector is the Keplerian elements of the asteroid around the sun including the semi-major axis (*a*), eccentricity (*e*), inclination (*i*), the mean speed (*N*), and the specific angular momentum (*H*). These five orbital elements fully characterize the shape of an asteroid trajectory around the sun, but not its orientation as the longitude of the ascending node $\Omega$ and argument of periapsis $\omega$ are omitted. ![\[FIG:HOI\_Design\] HOI network architecture. The input layer is comprised of five nodes, which is followed by two hidden layers of seven and three nodes, and an output layer of a single node.](1_network_hoi.png){width="83mm"} A diagram showing the HOI architecture is presented in Fig. \[FIG:HOI\_Design\]. The input layer is a vector of $five$ neurons that matches the dimensionality of the input, which is followed by two hidden layers that are composed of seven and three neurons, respectively, from the input layer. The output layer is composed of a single neuron whose values are restrained between 0 and 1 by virtue of the sigmoid function. Here, objects with a rating of 0.5 or above are classified as PI while those below the threshold are classified as UO. This neural network architecture was arrived at by a combination of empirical experimentation and the incorporation of domain knowledge. We wanted to provide the network with enough degrees of freedom to properly generalize the orbital elemental profiles of KI but to avoid giving it so many degrees of freedom that the network would overfit to the training datasets. The described architecture results in 69 free parameters: 59 weights and ten biases [^4]. To optimize these parameters, the network is trained on five randomly selected sub-sets of 100,000 observed and KI objects over 20 epochs, which took less than five minutes on a CPU-type laptop without a GPU. The training was halted when the relative loss decrease per epoch was less than $1\%$ to prevent overfitting. At the end of the training process, the network’s performance was validated with a subset of 20,000 KI and 20,000 observed objects that had been held out of the training process. Furthermore, all potentially hazardous objects (PHOs)[^5] were held out of the training process and used exclusively for testing purposes. Fig. \[FIG:loss\] shows how the training and validation loss decreased per training epoch, while the fraction of PHO hazardous objects identified simultaneously increased. ![\[FIG:loss\] Normalized training and validation losses plotted against the training epoch number, along with the fraction of PHOs identified by the network.](cost_curve_val.png){width="83mm"} We gave the observed objects and KIs labels of 0.1 and 0.9, respectively. Here, higher numbers correspond with a larger probability of colliding with Earth. The label of 0.9 was chosen for the KIs to represent calculations of the KI trajectories which are not converged solutions [@2014ApJ...785L...3P] and to show that several perturbing effects in the Solar System were neglected during the simulations, implying that all of the KIs will, in fact, not collide with Earth when their respective velocities are negated. To arrive at the label of 0.1 for the observed objects, we assumed that any individual observed object is very likely to be benign by the following logic: first, all of the PHOs which have considerably larger probability to collide with the Earth compared with the rest of the observed population are not used in HOI training. As a result, their labeling does not degrade the network’s ultimate performance. Second, impacts from large objects are rare [@Chapman] as the impact frequency of an asteroid collision decreases with the cube of an asteroid’s diameter. Earth collisions with 5 kilometer asteroids occur approximately every 20 million years, while those with a 100 meter asteroids occur every 500 years [@Bostrom]. Because 98.4% of the observed objects used for our experiments are greater than 100 meters in diameter[^6], we can use the following formula to estimate an upper-bound of the number of expected Earth impacts from asteroids in our sample within the next 20,000 years: $$\label{num_collisions} N_{collisions}=\int^{\infty}_{100}\frac{4\times10^7}{D^3}=2000,$$ Where $D$ is the diameter of an asteroid. Given that over 700,000 objects were used in HOI training, the number of 2000 mislabeled objects implies that 0.3% of the observed labels are inaccurate. As discussed further in the following sections, although our sample contains only a small fraction of misclassified non-impactors, they still may effect the ability of HOI to accurately identify an impactor. Data generation and acquisition {#SEC:Data} =============================== Observed objects ---------------- We extracted $736,496$ minor bodies from NASA’s *dastcom5* database [@dastcom5]. A percentage of 95.5% of the extracted objects are main-belt asteroids, 3.2% are asteroids that are not in the main belt (such as Apollo or Trojan asteroids), 0.7% are comets, 0.2% are Kuiper-belt objects, and the remaining 0.4% is composed of a plethora of miscellaneous objects, such as planetary satellites and centaurs [@SolarSystemObjects]. These proportions, however, are not representative of the actual small-body populations because there is considerable observational bias towards the closer main-belt asteroids in comparison with more distant objects [@KBO_Population]. Generating a database of known impactors ---------------------------------------- We generated an ensemble of 330,000 KIs according to Algorithm \[generate-ki\] to act as examples of hazardous objects. Here virtual objects are launched from future positions of Earth’s surface and then integrated backward in time to the present era. The idea is that the virtual objects’ trajectories would be similar to that of an asteroid observed in the present that would strike the Earth or come very close to it at some point in the future. [^7] The future launch dates, defined by the orientation of the Solar System, are evenly distributed between 300 and 20,000 years in the future, which correspond to $T_0$ and $T_1$ values of 2318 and 22018, respectively. The launching velocities are selected to bracket the Earth’s and Solar System’s escape speeds of 11.2 and 42.5km/s, respectively. We deliberately did not attempt to mimic the observed asteroid impact velocities to allow the neural network to learn from the full range of parameters, rather than just based on a hand-selected subsample. Results {#SEC:Results} ======= Identifying Earth-impacting asteroids ------------------------------------- The training of the network led to the positive identification of 95.25% of the KIs that were not part of the training and 90.99% of the PHOs as PIs. Additionally, 1.94% of the observed objects that were not classified as PHOs were identified as PIs. The high fraction of correctly identified KIs indicates that HOI positively recognizes most objects that are constructed to strike Earth. This result is not unexpected because HOI was specifically tuned to identify artificial KI objects. A more meaningful metric of performance is the percentage of PHOs identified. Although 9.01% PHOs were not classified as potential impactors, HOI is approximately 47 (90.99/1.94) times more likely to select a PHO over some other observed object. To further evaluate the effectiveness of HOI, we performed simulations to compare the closest Earth approaches of PIs and UOs. To run these simulations, we began by loading the positions and velocities of the asteroids and other Solar System objects corresponding to January 1, 2018. We then integrated all of the bodies forward in time for a thousand years while saving the closest approach that the asteroids made relative to Earth. The trajectories of all the 14,680 observed PIs and an equal number of randomly selected UO asteroids were computed. The distributions of the closest Earth approaches achieved during these simulations are plotted in Fig. \[FIG:Closeness\_Histogram\]. ![\[FIG:Closeness\_Histogram\] Closest approaches to Earth achieved in the next 1000 years for all the observed PIs and an equal number of randomly selected UOs. 108 PIs and 884 UOs are not plotted because their closest approaches exceeded the x-axis limits of 2 au. Every object that reach Earth within 0.01au and 99.9% of objects within 0.05au are identified by HOI as PIs. ](3_Closest_Approach_Together.pdf){width="83mm"} To investigate why HOI only identified approximately nine-tenths of PHOs as PIs, the thousand-year integrations described above were additionally performed for all PHOs. We present in Fig. \[FIG:Closeness\_PHOs\] the distributions of these closest approaches. The distributions of identified PHOs and unidentified PHOs are similar, therefore the fraction of PHOs identified as PIs could be used as a measure of the network’s performance. Additionally, all objects that did not approach Earth within at least 0.5au could be considered misclassified PIs. This cut-off is not arbitrary but based, rather, on the minimum distance achieved by approximately 99.7%, or $3\sigma$, of PHOs. In the case of HOI, 12.2% of the PIs are outside of this threshold and are therefore considered misclassified. The root of this misclassification likely stems from the approximations made in the labeling schemes described in Section \[SEC:arch\]. ![\[FIG:Closeness\_PHOs\] Closest approach distances to Earth reached for PHOs in the coming 1000 years.](closest_approach_PHO.png){width="85mm"} A total of $13,258$ asteroids identified by HOI as KIs are not listed by NASA as PHOs. In our thousand-year integrations, $4472$ of these objects approached within 0.05au of Earth while $2015$ approached within 0.02au. In Table.\[TAB:Short\_List\] we present a short list of 11 notable asteroids with absolute magnitudes of less than 22, data arcs of less than 31 days, and closest approaches less than 0.02au. -------------------------------------------------------------------------- -- -- -- -- **[Designation]{} & **[CA]{} & **[$t_{\rm CA}$]{} & **[H]{} & **[arc]{}\ & \[au\] & \[Year\] & \[mag\] & \[day\]\ 2005 RV24 & 0.020 & Feb. 2374 & 20.60 & 28\ 2008 UV99 & 0.013 & April 2332 & 20.03 & 1\ 2011 BU10 & 0.006 & April 2920 & 21.30 & 18\ 2011 HH1 & 0.012 & July 2923 & 21.7 & 13\ 2011 WC44 & 0.018 & Feb. 2679 & 20.5 & 31\ 2013 AG76 & 0.013 & Dec. 2638 & 20.3 & 24\ 2014 GL35 & 0.018 & July 2556 & 20.6 & 23\ 2014 TW57 & 0.017 & Sept. 2165 & 20.1 & 24\ 2014 WD365 & 0.017 & Sept. 2735 & 19.7 & 5\ 2017 DQ36 & 0.013 & Dec. 2131 & 19.3 & 29\ 2017 JE3 & 0.016 & July 2741 & 21.9 & 23\ ********** -------------------------------------------------------------------------- -- -- -- -- : Potential impactor shortlist: relatively large minor bodies with a short data arcs that were identified as PIs by HOI but are not considered PHOs. Along with their closest approaches (CA) in au, the month and year that their closest approaches occurred ($t_{\rm CA}$), their absolute magnitudes (H), and their data arc lengths in days (arc) are tabulated. \[TAB:Short\_List\] The absolute magnitude threshold of 22 was chosen so that only asteroids that have the potential of causing regional devastation unprecedented in human history would make the shortlist. Assuming a geometric albedo between 0.05 and 0.25 and a spherical shape, objects with an absolute magnitude of 22 are estimated to have diameters between from 100m to 236m. For perspective, Tunguska object which flattened 2,000 square kilometers of forest in Siberia was estimated to have a diameter of between 50-80m [@Tunguska]. The month long data-arc limit is selected because the Monte-Carlo method adopted by NASA is particularly ill-suited for calculating the impact probabilities of such uncertain orbits. As a consequence, these objects are the most likely to be overlooked as PHOs. Comparing various populations of object --------------------------------------- The characteristics of the simulated KIs and the observed objects are compared to better understand how HOI differentiates between the two populations. In Fig. \[FIG:Object\_Trajectories\] we present 100 trajectories of observed objects and KIs. {width="170mm"} There are profound differences between the orbital elements of the two distinct populations of objects. Our artificial population of objects launched from Earth tend to have highly eccentric and inclined orbits, whereas the observed objects tend to have circular orbits confined near the ecliptic plane. For the observed objects, the orbital plane is essentially empty within approximately 2au of the Sun, while for the KIs this is the most densely occupied space. This object distribution should be expected considering that all the KIs were generated $1\pm0.017$au away from the Sun along the Earth’s orbit and that the integration times were not sufficiently long enough to allow considerable outward migration of the objects. The *a* versus *e* ratio is an important factor in an object’s identification, as illustrated in Fig. \[FIG:AvsE\]. A curve is drawn to highlight an apparent “classification boundary”, which is above 95.2% of PI and below 90.3% of unidentified observed objects. Although the boundary is an indicator of an object’s potential classification, it is not definite, which is understandable considering that HOI takes five orbital elements as input for each object instead of just the *a* and *e* orbital elements. Conclusions {#SEC:Conclusions} =========== We designed, constructed, and trained a fairly simple neural network aimed at classifying asteroids with the potential to impact the Earth over the coming $20,000$ years. Our method takes the observed orbital elements as input and provides a classifier for the expectation value for the object’s striking Earth. The network was able pick out 95.25% of the KIs when mixed into a set of observed asteroids which are not expected to strike Earth. When applied to the entire population of observed asteroids, the network was able to identify approximately nine-tenths of the asteroids identified by NASA as PIs and along with virtually every other observed asteroid that approached within 0.05au of Earth. We generated a short list of network identified PIs which NASA does not label as PHOs, mainly because the observed orbital elements are so uncertain that NASA’s Monte Carlo approach to determine their Earth-striking probability fails. The network classifies an object as a PI or UO within $0.25$ milliseconds, which is negligible compared to the time required for the Monte-Carlo method employed by NASA. {width="170mm"} Follow-up calculations over a time-span of 1000 years revealed that 12.2% of the PIs identified by the network did not come within 0.5au of Earth. This may imply that thee asteroids pose no direct threat on the time scale considered. Integrating their orbits for a longer time-frame, however, this is impractical because of the large uncertainty in their orbital elements and the relatively small Lyapunov timescale for these objects. We look forward to improving the network’s classification accuracy. The network, as we show in Fig.\[FIG:HOI\_Design\], is the result of a great deal of experimentation in network depth, width, and (sub)selection input parameters. It is possible that the structure preserving mimetic architectures motivated by the underlying Keplerian topology of the orbits could allow us to achieve a higher quality of prediction accuracy but this still requires a considerable degree of further experimentation. Another improvement could be carried out by considering a stricter labeling scheme in which some probability statistics for impacting the Earth could be taken into account. We thank the Microsoft Cooperation for access to the Azure cloud on which many of the calculations presented here are performed. John D. Hefele thanks Sander van den Hoven for his mentoring during his internship at Microsoft Amsterdam. This work was supported by the Netherlands Research School for Astronomy (NOVA), NWO (grant \# 621.016.701 \[LGM-II\]). [^1]: Nagh Hoch is a concept stating that an ensemble of random initial realizations in a wide range of parameters gives statistically the same result as the converged solutions of the same ensemble of realizations. [^2]: This also means “Hello” in the Dutch language. [^3]: <https://github.com/mrteetoe/HOI> [^4]: Following the architecture described, the number of free parameters can be calculated as follows: the input is fed through layers which are comprised of 7, 3, and 1 neuron(s). This results in 5$\times$7+7$\times$3+3$\times$1 weights and 7+3 biases, as only the hidden layers have bias parameters. [^5]: All objects with a minimum orbit intersection distance of 0.05 AU or less and an absolute magnitude (H) of 22.0 or less are considered PHOs [@NasaPHA]. [^6]: This assumes an albedo of 0.15 for all small bodies. [^7]: An object, for example, that is launched from the Solar System at the year 2318, and is then integrated backwards in time 300 years, would create an example of a present day asteroid that would strike the Earth in 300 years after the velocity vectors are negated to account for the time reversal. As explained in Section \[SEC:arch\], the asteroids are not guaranteed to collide with Earth due to the finite precision of the integrations.

--- author: - 'Iryna Boiarska,' - 'Kyrylo Bondarenko,' - 'Alexey Boyarsky,' - 'Volodymyr Gorkavenko,' - 'Maksym Ovchynnikov,' - Anastasia Sokolenko bibliography: - 'ship.bib' title: 'Phenomenology of GeV-scale scalar portal' --- Introduction ============ We review and revise the phenomenology of the scalar portal – a gauge singlet scalar particle $S$ that couples to the Higgs boson and can play a role of a mediator between the Standard model and a dark sector (see e.g. [@Bird:2004ts; @Pospelov:2007mp; @Krnjaic:2015mbs]) or be involved in the cosmological inflation [@Shaposhnikov:2006xi; @Bezrukov:2009yw; @Bezrukov:2013fca]. We focus here on the mass range $\lesssim 10\text{ GeV}$ (see however section \[sec:quartic\_coupling\] for a discussion of larger masses). The interaction of the $S$ particle with the Standard model particles is similar to the interaction of a light Higgs boson but is suppressed by a small mixing angle $\theta$. Namely, the Lagrangian of the scalar portal is $$\mathcal{L} = \mathcal{L}_{SM} + \frac{1}{2} \partial_{\mu} S \partial^{\mu} S + (\alpha_1 S + \alpha_{2} S^2) (H^{\dagger} H) -\frac{m_{S}^{2}}{2}S^2. \label{eq:L1}$$ After the electroweak symmetry breaking the Higgs doublet gains a non-zero vacuum expectation value $v$. As a result, the $SHH$ interaction  provides a mass mixing between $S$ and the Higgs boson $h$. Transforming the Higgs field into the mass basis, $h \to h + \theta S$, one arrives at the following interaction of $S$ with the SM fermions and gauge bosons: $$\begin{aligned} & \mathcal{L}_{SM}^{S} = -\theta\frac{m_f}{v} S \bar{f}f + 2\theta\frac{ m_W^2}{v} S W^+ W^- + \theta\frac{m_Z^2}{v} S Z^2 + \alpha\left(\frac{1}{4v}S^2 h^2 + \frac{1}{2}S^2 h\right), \label{g01}\end{aligned}$$ where $\alpha \equiv 2\alpha_{2} v$. These interactions also mediate effective couplings of the scalar to photons, gluons, and flavor changing quark operators, see Fig. \[fig:Seffective\]. Additionally, the effective proton-scalar interaction that originates from the interaction of scalars with quarks and gluons (see Fig. \[fig:Sproton\]) will also be relevant for our analysis. The effective Lagrangian for these interactions is discussed in Appendix \[sec:effective-interactions\].  \ ![An example of a diagram for the effective interaction of a proton with a scalar, see Appendix \[sec:effective-interactions-nucleons\] for details.[]{data-label="fig:Sproton"}](scalar-nucleon-nucleon-vertex.pdf){height="42.00000%"} Searches for light scalars have been previously performed by CHARM, LHCb and Belle [@Schmidt-Hoberg:2013hba; @Clarke:2013aya], CMS [@Sirunyan:2018owy] and ATLAS [@Aad:2015txa; @Aaboud:2018sfi] experiments. Significant progress in searching for light scalars can be achieved by the proposed and planned intensity-frontier experiments SHiP [@Alekhin:2015byh; @Anelli:2015pba; @SHiP:2018xqw], CODEX-b [@Gligorov:2017nwh], MATHUSLA [@Curtin:2018mvb; @Chou:2016lxi; @Curtin:2017izq; @Evans:2017lvd; @Helo:2018qej; @Lubatti:2019vkf], FASER [@Feng:2017uoz; @Feng:2017vli; @Kling:2018wct], SeaQuest [@Berlin:2018pwi], NA62 [@Mermod:2017ceo; @CortinaGil:2017mqf; @Drewes:2018gkc], DUNE [@Adams:2013qkq]. The phenomenology of light GeV-like scalars has been studied in [@Bezrukov:2009yw; @Clarke:2013aya; @Bezrukov:2018yvd; @Batell:2009jf; @Winkler:2018qyg; @Monin:2018lee; @Bird:2004ts], and in [@Voloshin:1985tc; @Raby:1988qf; @Truong:1989my; @Donoghue:1990xh; @Willey:1982ti; @Willey:1986mj; @Grzadkowski:1983yp; @Leutwyler:1989xj; @Haber:1987ua; @Chivukula:1988gp] in the context of a light Higgs boson. However, in the literature, there are still conflicting results, both for the scalar production and decay. In this work, we reanalyze the phenomenology of light scalars and present the results in the form directly suitable for experimental sensitivity estimates. Scalar production {#sec:production} ================= Mixing with the Higgs boson --------------------------- In this section, we will discuss the scalar production channels that are defined by the mixing between a scalar and the Higgs boson. In proton-proton or proton-nucleus collisions, a scalar particle: (a) can be emitted by the proton, (b) produced from photon-photon, gluon-gluon or quark-antiquark fusion in proton-proton or proton-nucleus interactions or (c) produced in the decay of the secondary particles, see Fig. \[fig:Sproduction\]. Let us compare these three types of the scalar production mechanisms depending on the collision energy and the scalar mass. In the following we will present the results for three referent proton-proton center-of-mass energies: $\sqrt{s_{pp}} \approx 16\text{ GeV}$ (corresponding to the beam energy of the DUNE experiment), $\sqrt{s_{pp}} \approx 28\text{ GeV}$ (SHiP) and $\sqrt{s_{pp}} = 13\text{ TeV}$ (LHC). \ \ **The proton bremsstrahlung (the case (a))** is a process of a scalar emission by a proton in proton-proton interaction. For small masses of scalars, $m_S < 1$ GeV, this is a usual bremsstrahlung process described by elastic nucleon-scalar interaction with a coupling constant $\theta g_{SNN}\sim \theta m_{p}/v$, see Appendix \[sec:effective-interactions-nucleons\]. However, the probability of elastic interaction decreases with the scalar mass and we need to take into account inelastic processes. The probability for the bremsstrahlung is $$P_{\text{brem}} = \theta^{2} g_{SNN}^{2} \mathcal{P}_{\text{brem}}(m_{S},s_{pp}), \label{eq:Pbrem}$$ where $\mathcal{P}_{\text{brem}}$ is a proton bremsstrahlung probability for the case $\theta=g_{SNN}=1$ (see Appendix \[sec:bremsstrahlung\]). This quantity varies from $10^{-2}$ for DUNE and SHiP to $10^{-1}$ for the LHC, see Appendix \[sec:bremsstrahlung\]. [^1] ---------------------------- ----------------------------------- ----------------------------------- ----------------------------------- Experiment DUNE SHiP LHC $(\sqrt{s_{pp}} = 16\text{ GeV})$ $(\sqrt{s_{pp}} = 28\text{ GeV})$ $(\sqrt{s_{pp}} = 13\text{ TeV})$ $W_{GG}(m_S=2\text{ GeV})$ $1.2\cdot 10^{3}$ $1.4\cdot 10^{3}$ $6.2\cdot 10^{10}$ $W_{GG}(m_S=5\text{ GeV})$ $1.5\cdot 10^{-1}$ $7.9$ $3.9 \cdot 10^{10}$ ---------------------------- ----------------------------------- ----------------------------------- ----------------------------------- : Factors $W_{GG}$ (see Eq. ) for the DUNE, SHiP and LHC experiments.[]{data-label="tab:WDIS"} **For the case (b)** we have to distinguish photon-photon fusion that can occur for an arbitrary transferred momentum and, therefore, an arbitrary scalar mass (as electromagnetic interaction is long-range), and gluons or quark-antiquarks fusion (the so-called deep inelastic scattering processes (DIS)), which is effective only for $m_S \gtrsim 1 \text{ GeV}$. The scalar production in the DIS process can be estimated as $P_{\text{DIS}} = (\sigma_{\text{DIS},G}+\sigma_{\text{DIS},q})/\sigma_{pp}$, where $\sigma_{pp}$ is the total proton-proton cross section and $\sigma_{\text{DIS},X}$ is the cross section of scalar production in the DIS process, $$\sigma_{\text{DIS},G} \sim \frac{\theta^2 \alpha_{s}^2(m_S) m_S^2}{s_{pp} v^2} W_{GG}(m_S,s_{pp}), \quad \sigma_{\text{DIS},q} \sim \frac{\theta^2 m_q^2}{s_{pp} v^2} W_{q\bar{q}}(m_S,s_{pp}). \label{eq:DISsigma}$$ Here, $\sqrt{s_{pp}}$ is the center-of-mass energy of colliding protons and $W_{XX}$ given by Eq.  is a dimensionless combinatorial factor that, roughly, counts the number of the parton pairs in two protons that can make a scalar. The values of $W_{XX}$ factors for some scalar masses and experimental energies are given in Table \[tab:WDIS\]. In Fig. \[fig:dis-production-plot\] we show the ratio between cross sections of gluon-gluon and quark-antiquark fusion. We see that quark fusion is relevant only for low scalar masses, while for $m_S \gtrsim 2$ GeV the gluon fusion dominates for all collision energies considered. ![The ratio of cross sections of the scalar production in deep inelastic scattering via gluon and quark fusion. The dashed line corresponds to a ratio equal to unity. “LHC”, “SHiP” and “DUNE” denote correspondingly the results for the proton-proton center-of-mass energies $\sqrt{s_{pp}} = 13\text{ TeV}$, $\sqrt{s_{pp}} = 28\text{ GeV}$ and $\sqrt{s_{pp}} = 16\text{ GeV}$.[]{data-label="fig:dis-production-plot"}](dis-gluon-quark-comparison.pdf){width="60.00000%"} In the case of *the production of a scalar in photon fusion*, the most interesting process is the coherent scattering on the whole nucleus, as its cross section is enhanced by a factor $Z^2$, where $Z$ is the charge of the nucleus. The electromagnetic process $p+Z \to p+Z+S$ involves the effective $S\gamma\gamma$ vertex proportional to $\theta \alpha_{\text{EM}}$, see Appendix \[sec:effective-interactions-gauge-bosons\]. The probability of the process is $P_{\gamma\text{ fus}} = \sigma_{\gamma\text{ fus}}/\sigma_{pZ}$, where the fusion cross section $\sigma_{\gamma\text{ fus}}$ has a structure similar to that of gluon fusion : $$\sigma_{\gamma\text{ fus}} \sim 10^{-2} \frac{\theta^2 Z^{2}\alpha_{\text{EM}}^{4}m_{S}^{2}}{v^{2}s_{pZ}} W_{\text{coh}},$$ where $\sqrt{s_{pZ}}$ is the CM energy of the proton and nucleus, and $W_{\text{coh}}$ given by Eq.  is a dimensionless combinatorial factor that counts the number of pair of photons that can form a scalar. It ranges from $10^{6}$ for the DUNE energies to $10^{14}$ for the LHC energies. Let us compare the probabilities of photon fusion and proton bremsstrahlung, $$\begin{aligned} \frac{P_{\gamma\text{ fus}}}{P_{\text{brem}}} &\sim 10^{-2}\frac{Z^{2}}{s_{pZ}\sigma_{pZ}} \frac{\alpha_{\text{EM}}^{4}}{g_{SNN}^{2}} \frac{m_{S}^{2}}{v^{2}} \frac{W_{\text{coh}}}{\mathcal{P}_{\text{brem}}} \sim \\ &\sim 10^{-15}\frac{(100\text{ GeV})^{2}}{s_{pp}}\frac{Z^{2}}{A^{1.77}}\left(\frac{m_{S}}{1\text{ GeV}}\right)^{2}\frac{W_{\text{coh}}}{\mathcal{P}_{\text{brem}}} \lesssim 10^{-4}\end{aligned}$$ for all three energies considered. Here we used $s_{pZ}\approx A s_{pp}$, where $A$ is the nucleus mass number. The proton-nucleus cross section $\sigma_{pZ}$ weakly depends on energy and can be estimated as $\sigma_{pZ} \simeq 50\text{ mb}\ A^{0.77}$ [@Tanabashi:2018oca; @Carvalho:2003pza]. This ratio is smaller than $10^{-4}$ for all energies and scalar masses of interest. Next, comparing the probabilities of the production in photon fusion and in DIS, we obtain $$\frac{P_{\gamma \text{ fus}}}{P_{\text{DIS}}}\sim \frac{Z^2 \alpha_{\text{EM}}^4}{\alpha_s^2} \frac{s_{pp}}{s_{pZ}} \frac{\sigma_{pp}}{\sigma_{pZ}} \frac{W_{\text{coh}}}{W_{\text{DIS}}} \sim 10^{-8}\frac{Z^{2}}{A^{1.77}}\frac{W_{\text{coh}}}{W_{\text{DIS}}} \lesssim 10^{-4},$$ where we used that $s_{pZ}/s_{pp}\approx A$ and $W_{\text{coh}}/W_{\text{DIS}} \lesssim 1$ for all three energies considered, see Appendix \[sec:production-direct-coherent\]. The proton-proton cross section also depends on the energy weakly, and we can estimate $\sigma_{pZ}/\sigma_{pp}\sim A^{0.77}$ (see Appendix \[sec:bremsstrahlung\]). *We conclude that the scalar production in photon fusion is always sub-dominant for the considered mass range of scalar masses and beam energies.* Let us now compare gluon fusion and proton bremsstrahlung with the **production from secondary mesons (type (c))**. The latter can be roughly estimated using “inclusive production”, i.e. production from the decay of a free heavy quark, without taking into account that in reality this quark is a part of different mesons with different masses. This is only an order of magnitude estimate that breaks down for $m_S \gtrsim m_q -\Lambda_{\text{QCD}}$, so it can be used only for $D$ and $B$ mesons. We will see however that such an estimate is sufficient to conclude that for the energies of SHiP and LHC the production from mesons dominates and we need to study it in more details (see more detailed analysis below). The inclusive branching $\text{BR}_{\text{incl}}(X_{Q_{i}} \to X_{Q_{j}}S)$ can be estimated using the corresponding quark level process $Q_{i} \to Q_{j} S$. To minimize QCD uncertainty we follow [@Chivukula:1988gp; @Curtin:2018mvb] and estimate the inclusive branching ratio as $$\text{BR}_{\text{incl}}(X_{Q_{i}} \to X_{Q_{j}}S) \simeq \frac{\Gamma(Q_{i}\to Q_{j}S )}{\Gamma(Q_{i} \to Q' e\bar{\nu}_{e})}\times\text{BR}_{\text{incl}}(X\to X_{Q'}e\bar{\nu}_{e}), \label{eq:prodBRinclusiveB}$$ where $\Gamma(Q_{i}\to Q'e\bar{\nu}_{e})$ is the semileptonic decay width of a quark $Q_{i}$ calculated using the Fermi theory and $\text{BR}_{\text{incl}}(X\to X_{Q'}e\bar{\nu}_{e})$ is the experimentally measured inclusive branching ratio. As both the quark decay widths in  get the QCD corrections, their total effect in  is expected to be minimal [@Chivukula:1988gp] For $D$ and $B$ mesons decays the inclusive production probabilities are [@Chivukula:1988gp; @Evans:2017lvd] $$\begin{aligned} P_{D} &\sim 2\chi_{c\bar{c}} \times \text{BR}(c\to Su) \sim 6\cdot 10^{-11} \ \theta^{2} \chi_{c\bar{c}} \left( 1-\frac{m_{S}^{2}}{m_{c}^{2}}\right)^{2}, \label{eq:PD} \\ P_{B} &\sim 2\chi_{b\bar{b}} \times \text{BR}(b\to Ss) \sim 13 \ \theta^{2}\chi_{b\bar{b}}\left( 1-\frac{m_{S}^{2}}{m_{b}^{2}}\right)^{2}, \label{eq:PB}\end{aligned}$$ where $\chi_{q\bar{q}}$ is the production fraction of the $q\bar{q}$ pair in $pp$ collisions, see Table \[tab:meson-amounts\]. The difference in $10^{-11}$ orders of magnitude is mostly coming from $(m_b/m_t)^4\sim 10^{-7}$ and $|V_{ub}|^2/|V_{ts}|^2\sim 10^{-2}$ (see Appendix \[sec:fcnc\] for details). In fact for $D$ mesons the leptonic decay $D\to S e\nu$ with $\text{BR}(D\to S e\nu)\sim 5\cdot 10^{-9}\theta^{2}$ is more important than , see Appendix \[sec:leptonic-decays\] for details. We see that the production from $D$ mesons may be important only if the number of $B$ mesons is suppressed by $10^9$ times, which is possible only if the center-of-mass energy of $p$-$p$ collisions is close to the $b\bar{b}$ pair production threshold. Let us compare the production from $B$ mesons with the production from proton bremsstrahlung and DIS. Using Eqs. ,  and  for masses of scalar below $b$ quark kinematic threshold we get $$\frac{P_{\text{brem}}}{P_{B}} \sim \frac{g_{SNN}^2}{\text{BR}(b\to Ss)} \frac{\mathcal{P}_{\text{brem}}}{\chi_{b\bar{b}}} \sim 10^{-7} \frac{\mathcal{P}_{\text{brem}}}{\chi_{b\bar{b}}}, \label{eq:Pbrem2B}$$ $$\frac{P_{\text{DIS}}}{P_{B}} \sim 10^{-6} \frac{1}{s_{pp} \sigma_{pp}} \left(\frac{m_S}{1\text{ GeV}}\right)^2 \frac{W_{GG}(m_S,s_{pp})}{\chi_{b\bar{b}}}. \label{eq:pDIS-to-B}$$ The ratios  and   depend on the center-of-mass energy of the experiment (see Tables \[tab:WDIS\] and \[tab:meson-amounts\]). *We conclude that for the experiments with high beam energies, like SHiP or LHC, the most relevant production channel is a production of scalars from secondary mesons. For experiments with smaller energies like, e.g., DUNE the dominant channel is the direct production of scalars in proton bremsstrahlung and in DIS.* ------------------- ----------------------------------- ----------------------------------- ----------------------------------- Experiment DUNE SHiP LHC $(\sqrt{s_{pp}} = 16\text{ GeV})$ $(\sqrt{s_{pp}} = 28\text{ GeV})$ $(\sqrt{s_{pp}} = 13\text{ TeV})$ $\chi_{c\bar{c}}$ $1.0\cdot 10^{-4}$ $3.9\cdot 10^{-3}$ $2.9\cdot 10^{-2}$ $\chi_{b\bar{b}}$ $1.0\cdot 10^{-10}$ $2.7\cdot 10^{-7}$ $8.6 \cdot 10^{-3}$ ------------------- ----------------------------------- ----------------------------------- ----------------------------------- : Production fractions of the $q\bar{q}$ pair, $\chi_{q\bar{q}} = \sigma_{q\bar{q}}/\sigma_{pp}$, for the DUNE, SHiP and LHC experiments. We took the production fractions for the DUNE and SHiP experiments from [@Lourenco:2006vw; @CERN-SHiP-NOTE-2015-009]. To estimate the production fractions for the LHC, we calculated the total cross section of $B$ and $D$ production using FONLL [@Cacciari:1998it] and took the total cross section of the $pp$ collisions at the LHC energies from [@Tanabashi:2018oca].[]{data-label="tab:meson-amounts"} **Production from decays of different mesons.**\ Let us discuss the production of scalars from decays of mesons in more details. The calculation of branching ratios for two-body decays of mesons is summarized in Appendix \[sec:meson-two-body-decays\]. Above we made an estimate for the cases of $D$ and $B$ mesons that are the most efficient production channels for larger masses of $S$. Instead, for scalar masses $m_S<m_K-m_\pi$ the main production channel is the decay of kaons, $K \to S \pi$, see Table \[tab:BR\] with the relevant information about these production channels. Numerically, the branching ratio of the production from kaons is suppressed by 3 orders of magnitude in comparison to the branching ratio of the production from $B$ mesons, but for the considered energies the number of kaons is at least $10^{3}$ times larger than the number of $B$ mesons. Process $\text{BR}(m_S=0)/\theta^2$ Closing mass \[GeV\] Appendix ------------------------------------- ------------------------------------ ---------------------- ---------------------- $K^\pm\to S\pi^{\pm}$ $1.7 \cdot 10^{-3}$ $0.350$ \[app:pseudoscalar\] $K^0_L\to S\pi^0$ $7 \cdot 10^{-3}$ $0.360$ \[app:pseudoscalar\] $B^{\pm} \to S K^{\pm}_{1}(1270)$ $(9.1^{+3.6}_{-4.0})\cdot 10^{-1}$ 3.82 \[app:pseudovector\] $B^{\pm} \to S K^{*,\pm}_{0}(700)$ $7.6\cdot 10^{-1}$ 4.27 \[app:scalar\] $B^{\pm} \to S K^{*,\pm}(892)$ $(4.7^{+0.9}_{-0.8})\cdot 10^{-1}$ 4.29 \[app:vector\] $B^{\pm} \to S K^{\pm}$ $(4.3^{+1.1}_{-1.0})\cdot 10^{-1}$ 4.79 \[app:pseudoscalar\] $B^{\pm} \to S K^{*,\pm}_{2}(1430)$ $3.0\cdot 10^{-1}$ 3.85 \[app:tensor\] $B^{\pm} \to S K^{*,\pm}(1410)$ $(2.1^{+0.6}_{-1.1})\cdot 10^{-1}$ 3.57 \[app:vector\] $B^{\pm} \to S K^{*,\pm}(1680)$ $(1.3^{+0.5}_{-0.4})\cdot 10^{-1}$ 3.26 \[app:vector\] $B^{\pm} \to S K^{*,\pm}_{0}(1430)$ $8.1\cdot 10^{-2}$ 3.82 \[app:scalar\] $B^{\pm} \to S K^{\pm}_{1}(1400)$ $(1.6^{+0.6}_{-1.1})\cdot 10^{-2}$ 2.28 \[app:pseudovector\] $B^{\pm} \to S \pi^{\pm}$ $(1.3^{+0.3}_{-0.3})\cdot 10^{-2}$ 5.14 \[app:pseudoscalar\] : Properties of the main production channels of a scalar $S$ from kaons and $B$ mesons through the mixing with the Higgs boson. *First column*: decay channels; *Second column*: branching ratios of 2-body meson decays evaluated at $m_{S}=0$ using formula  and normalized by $\theta^2$. For $B$ mesons the numerical values are given for $B^{\pm}$ mesons; in the case of $B^0$ meson all the given branching ratios should be multiplied by a factor of $0.93$ that comes from the difference in total decay widths of $B^{\pm}$ and $B^0$ mesons [@Tanabashi:2018oca]. Uncertainties (where available) follow from uncertainties in meson transition form-factors; *Third column*: effective closing mass, i.e. a mass of a scalar at which the branching ratio of the channel decreases by a factor of $10$; *Fourth column*: a reference to the appendix with details about form-factors used.[]{data-label="tab:BR"} ![Branching ratios of the 2-body decays $B^{+}\to S X_{s/d}$, where $X_{q}$ is a hadron that contains a quark $q$. By the $K^{*}$ channel, we denote the sum of the branching ratios for $K^{*}(892)$, $K^{*}(1410)$, $K^{*}(1680)$ final states, by $K^{*}_{0}$ – for $K^{*}_{0}(700)$, $K_{0}^{*}(1430)$, and by $K_{1}$ – for $K_{1}(1270)$, $K_{1}(1400)$. The “Inclusive” line corresponds to the branching ratio  obtained using the free quark model.[]{data-label="fig:plot-b-production"}](scalar-production-b-mesons-mixing.pdf){width="80.00000%"} ![The probabilities of a scalar production in proton bremsstrahlung process  (solid lines), DIS process  (dotted lines) and decays of $B$ mesons (dashed lines) versus the scalar mass. “LHC”, “SHiP” and “DUNE” denote correspondingly results for the proton-proton center-of-mass energies $\sqrt{s_{pp}} = 13\text{ TeV}$, $\sqrt{s_{pp}} = 28\text{ GeV}$ and $\sqrt{s_{pp}} = 16\text{ GeV}$. The gray line corresponds to the extrapolation of the bremsstrahlung production probability assuming unit value of the proton elastic form-factor, see text for details.[]{data-label="fig:dis-vs-b-meson"}](prod-dune.pdf "fig:"){width="49.00000%"} ![The probabilities of a scalar production in proton bremsstrahlung process  (solid lines), DIS process  (dotted lines) and decays of $B$ mesons (dashed lines) versus the scalar mass. “LHC”, “SHiP” and “DUNE” denote correspondingly results for the proton-proton center-of-mass energies $\sqrt{s_{pp}} = 13\text{ TeV}$, $\sqrt{s_{pp}} = 28\text{ GeV}$ and $\sqrt{s_{pp}} = 16\text{ GeV}$. The gray line corresponds to the extrapolation of the bremsstrahlung production probability assuming unit value of the proton elastic form-factor, see text for details.[]{data-label="fig:dis-vs-b-meson"}](prod-ship.pdf "fig:"){width="49.00000%"}\ ![The probabilities of a scalar production in proton bremsstrahlung process  (solid lines), DIS process  (dotted lines) and decays of $B$ mesons (dashed lines) versus the scalar mass. “LHC”, “SHiP” and “DUNE” denote correspondingly results for the proton-proton center-of-mass energies $\sqrt{s_{pp}} = 13\text{ TeV}$, $\sqrt{s_{pp}} = 28\text{ GeV}$ and $\sqrt{s_{pp}} = 16\text{ GeV}$. The gray line corresponds to the extrapolation of the bremsstrahlung production probability assuming unit value of the proton elastic form-factor, see text for details.[]{data-label="fig:dis-vs-b-meson"}](prod-lhc.pdf "fig:"){width="49.00000%"} For scalar masses $m_K-m_\pi < m_S < m_B$ the main scalar production channel from hadrons is the production from $B$ mesons. Inclusive estimate at the quark level, that we made above (see Eq. ), contains an unknown QCD uncertainty and completely breaks down for scalar masses $m_{b} - m_{S} \simeq \Lambda_{\text{QCD}}$. Below we discuss therefore decays of different mesons containing the b quark $B \to X_{s/d}S$. We consider kaon and its resonances as the final states $X_{s}$: - Pseudoscalar meson $K$; - Scalar mesons $K_{0}^{*}(700), K_{0}^{*}(1430)$ (here assuming that $K_{0}^{*}(700)$ is a di-quark state); - Vector mesons $K^{*}(892), K^{*}(1410), K^{*}(1680)$; - Axial-vector mesons $K_{1}(1270), K_{1}(1400)$; - Tensor meson $K_{2}^{*}(1430)$. We also consider the meson $X_{d} = \pi$. Although the rate of the corresponding process $B\to \pi S$ is suppressed in comparison to the rate of $B \to X_{s}S$, it may be important since it has the largest kinematic threshold $m_{S}\lesssim m_{B}-m_{\pi}$. We calculate the branching ratios $\text{BR}(B^{+}\to X_{s/d}S)$ at $m_{S} \ll m_{K}$ using Eq.  and give the results in Table \[tab:BR\]. The main uncertainty of this approach is related to form factors describing meson transitions $X_{Q_{i}} \to X'_{Q_{j}}$, see Appendix \[sec:hadronic-form-factors\] for details. They are calculated theoretically using approaches of light cone fum rules and covariant quark model, and indirectly fixed using experimental data on rare mesons decays [@Ball:2004rg; @Ball:2004ye; @Cheng:2010yd; @Sun:2010nv; @Lu:2011jm]. The errors given in Table \[tab:BR\] result from uncertainties in the meson transition form-factors $F_{BX_{s/d}}$ (see Appendix \[sec:hadronic-form-factors\]). Since $F_{BX_{s/d}}$ are the same for $B^{+}$ and $B^{0}$ mesons, the branching ratios $\text{BR}(B^{0}\to X'^{0}S)$ differ from $\text{BR}(B^{0}\to X_{s/d}^{0}S)$ only by the factor $\Gamma_{B^{+}}/\Gamma_{B^{0}} \approx 0.93$. The values of the branching ratios for the processes $B \to KS$, $B \to K^{*}S$ are found to be in good agreement with results from the literature [@Batell:2009jf; @Winkler:2018qyg]. We conclude that the most efficient production channels of light scalars with $m_{S}\lesssim 3\text{ GeV}$ are decays $B \to K_{0}^{*}S, \ B\to K_{1}S$ and $B\to K^{*}S$; the channel $B \to KS$, considered previously in the literature, is sub-dominant. Summing over all final $K$ states, in the limit $m_{S}\ll m_{B}$ for the total branching ratio we have $$\label{eq:total-exclusive} \text{BR}(B \to SX_{s}) \equiv \sum_{X_{s}}\text{BR}(B\to SX_{s}) \approx 3.3^{+0.8}_{-0.7}\ \theta^{2}.$$ Using the estimate , for the ratio of the central value of the branching ratio  and the inclusive branching ratio at $m_{S} \ll m_{B}$ we find $$\text{BR}(B \to SX_{s})/ \text{BR}_{\text{incl}}(B \to SX_{s}) \approx 0.5. \label{eq:exclusive-to-inclusive}$$ Provided that the inclusive estimate of the branching ration has a large uncertainty, we believe that Eq.  suggests that we have taken into account all main decay channels of this type. Our results are summarized in Table \[tab:BR\] and Fig. \[fig:plot-b-production\]. We have found that the channels with $K^{*}$, $K_{0}^{*}$ and $K_{1}$ give the main contribution to the production branching ratio for small scalar masses $m_{S} \lesssim 3\text{ GeV}$, while for larger masses the main channel is decay to $K$. The comparison between the probability of the production from mesons and our estimates for bremsstrahlung and DIS for three center of mass energies are shown in Fig. \[fig:dis-vs-b-meson\]. Quartic coupling {#sec:quartic_coupling} ---------------- \ Above we discussed the production of scalars only through the mixing with the Higgs boson. One more interaction term in the Lagrangian , $$\mathcal{L}_{\text{quartic}} = \frac{\alpha}{2} S^2 h, \label{eq:quartic-coupling-1}$$ (the so-called “quartic coupling” that originates from the term $S^2 H^{\dagger}H$ in the Lagrangian ) affects the production of scalars from decays of mesons and opens an additional production channel - production from Higgs boson decays, see Fig. \[fig:Sproduction-quartic\]. **The production from the Higgs boson (case (a))** can be important for high-energy experiments like LHC. The branching ratio is $$\text{BR}(h\to S S) = \frac{\alpha^2|\bm{p}_S|}{16\pi m_h^2 \Gamma_h} \approx 2.0\cdot 10^{-2}\left(\frac{\alpha}{1\text{ GeV}}\right)^{2}\sqrt{1 - \frac{4m_{S}^{2}}{m_{h}^{2}}}, \label{eq:BR-h-to-SS}$$ where $\bm{p}_S$ is a momentum of a scalar in the rest frame of the Higgs boson and we used the SM decay width of the Higgs boson $\Gamma_h \approx 4$ MeV [@Denner:2011mq]. If the decay length of the scalar is large enough $c\gamma\tau_S \gtrsim 1$ m this decay channel manifests itself at ATLAS and CMS experiments as an invisible Higgs boson decay. The invisible Higgs decay is constrained at CMS [@Sirunyan:2018owy], the $2\sigma$ upper bound is $$\text{BR}(h \to \text{invis.}) < 0.19. \label{eq:higgs_inv}$$ This puts an upper bound $\alpha< 3.1$ GeV for the scalar masses $m_S < m_h/2$. The production probability $P_{h\to SS} = \chi_h \times \text{BR}(h\to S S)$, where $\chi_h$ is a production fraction of the Higgs bosons. Comparing with the production from $B$ mesons for a scalar mass below the $B$ threshold estimated by the inclusive production  we get $$\frac{P_{h\to SS}}{P_B} \sim 10^{-3} \frac{1}{\theta^2} \left(\frac{\alpha}{1\text{ GeV}}\right)^{2} \frac{\chi_h}{\chi_{b\bar{b}}} \sim 10^{-10} \frac{1}{\theta^2} \left(\frac{\alpha}{1\text{ GeV}}\right)^{2}, \label{eq:Pss-to-PB}$$ where we used $\chi_{h}\sim 10^{-9}$ for the LHC energy [@Heinemeyer:2013tqa] and $\chi_{b\bar{b}} \sim 10^{-2}$ (see Table \[tab:meson-amounts\]). Also, **the quartic coupling generates new channels of scalar production in meson decays (case (b))**. In addition to the process $X\to X'S$ shown in Fig. \[fig:Sproduction\] (c) the quartic coupling enables also additional processes $X\to SS$ and $X\to X'SS$ shown in Fig. \[fig:Sproduction-quartic\] (b)  [@Bird:2004ts; @Bird:2006jd; @Kim:2009qc; @He:2010nt; @Batell:2009jf; @Badin:2010uh]. First, let us make a simple comparison between the branching ratios for the scalar production from mesons in the case of mixing with the Higgs boson and quartic coupling. Comparing Feynman diagrams in Figs. \[fig:Sproduction\] (c) and \[fig:Sproduction-quartic\] (b) we see that for the case $m_S\ll m_X$ $$\frac{\text{BR}(X\to X'SS)}{\text{BR}(X\to X'S)} \sim \frac{\alpha^2 m_X^2}{\theta^2 m_h^4} \sim 10^{-9} \frac{1}{\theta^2} \left(\frac{\alpha}{1\text{ GeV}}\right)^{2} \left(\frac{m_X}{1\text{ GeV}}\right)^{2}, \label{eq:quartic-to-mixing1}$$ $$\frac{\text{BR}(X\to SS)}{\text{BR}(X\to X'S)} \sim \frac{\alpha^2 f_X^2}{\theta^2 m_h^4} \sim 10^{-9} \frac{1}{\theta^2} \left(\frac{\alpha}{1\text{ GeV}}\right)^{2} \left(\frac{f_X}{1\text{ GeV}}\right)^{2}, \label{eq:quartic-to-mixing2}$$ where $f_X$ is a meson’s decay constant (see Appendix \[sec:quartic-coupling\]). The channel $X\to X'SS$ is very similar to the channel $X\to X'S$ from Fig. \[fig:Sproduction\] (c). By the same reasons, this process is strongly suppressed for $D$-mesons and is efficient only for kaons and $B$ mesons. The decay $X\to SS$ is possible only for $K^0$, $D^0$, $B^0$ and $B^0_s$ due to conservation of charges. The production from $D^0$ mesons is strongly suppressed by the same reason as above (small Yukawa constant and CKM elements in the effective interaction, see Appendix \[sec:fcnc\]). Our results for the branching ratio of the scalar production from mesons decays in the case of the quartic coupling are presented in Table \[tab:b-production-quartic\] and in Fig. \[fig:plot-b-production-quartic\], the formulas for the branching ratios and details of calculations are given in Appendix \[sec:quartic-coupling\]. The results are shown for the value of coupling constant $\alpha = 1$ GeV which corresponds to the Higgs boson branching ratio $\text{BR}(h\to SS) \approx 0.02$ (see Eq. ). Process $\text{BR}(m_S=0)$ Closing mass \[GeV\] Appendix -------------------------------------- ------------------------------------- ---------------------- -------------------------- $K^0_L\to SS$ $4.4\cdot 10^{-13}$ $0.252$ \[sec:quartic-coupling\] $K^0_L\to SS\pi^0$ $6.6\cdot 10^{-15}$ $0.140$ \[app:pseudoscalar\] $K^\pm\to SS\pi^{\pm}$ $1.4\cdot 10^{-15}$ $0.136$ \[app:pseudoscalar\] $K^{0}_{S}\to SS$ $7.8\cdot 10^{-16}$ $0.252$ \[sec:quartic-coupling\] $K^0_S\to SS\pi^0$ $1.2\cdot 10^{-17}$ $0.140$ \[app:pseudoscalar\] $B_{s}\to SS$ $4.0\cdot 10^{-10}$ $2.670$ \[sec:quartic-coupling\] $B^{\pm} \to SS K^{\pm}$ $1.4\cdot 10^{-10}$ $1.998$ \[app:pseudoscalar\] $B^{\pm} \to SS K^{*,\pm}_{0}(700)$ $1.2\cdot 10^{-10}$ $1.621$ \[app:scalar\] $B^{\pm} \to SS K^{\pm}_{1}(1270)$ $(1.2^{+0.5}_{-0.5})\cdot 10^{-10}$ $1.478$ \[app:pseudovector\] $B^{\pm} \to SS K^{*,\pm}(892)$ $9.1\cdot 10^{-11}$ $1.701$ \[app:vector\] $B^{\pm} \to SS K^{*,\pm}_{0}(1430)$ $3.5\cdot 10^{-11}$ $1.621$ \[app:scalar\] $B^{\pm} \to SS K^{*,\pm}(1410)$ $(1.9^{+0.6}_{-0.5})\cdot 10^{-11}$ $1.358$ \[app:vector\] $B^{\pm} \to SS K^{*,\pm}_{2}(1430)$ $2.5\cdot 10^{-11}$ $1.499$ \[app:tensor\] $B_{0}\to SS$ $1.1\cdot 10^{-11}$ $2.624$ \[sec:quartic-coupling\] $B^{\pm} \to SS K^{*,\pm}(1680)$ $(9.9^{+0.4}_{-0.3})\cdot 10^{-12}$ $1.240$ \[app:vector\] $B^{\pm} \to SS \pi^{\pm}$ $(4.7^{+1.2}_{-1.1})\cdot 10^{-12}$ $2.149$ \[app:pseudoscalar\] $B^{\pm} \to SS K^{\pm}_{1}(1400)$ $(7.3^{+0.3}_{-0.3})\cdot 10^{-13}$ $1.545$ \[app:pseudovector\] : Properties of the main production channels of a scalar $S$ from kaons and $B$ mesons through quartic coupling . *First column*: decay channels; *Second column*: branching ratios of 2-body meson decays evaluated at $m_{S}=0$ and $\alpha = 1\text{ GeV}$, see Eqs. , . For $B$ mesons the numerical values are given for $B^{\pm}$ mesons; in the case of decays of $B^0$ mesons, all the given branching ratios should be multiplied by a factor $0.93$ that comes from the difference in total decay widths of $B^{\pm}$ and $B^0$ mesons [@Tanabashi:2018oca]. Uncertainties (where available) follow from uncertainties in meson transition form-factors; *Third column*: effective closing mass, i.e. a mass of a scalar at which the branching ratio of the channel decreases by a factor of $10$. *Fourth column*: a reference to appendix with details about form-factors used.[]{data-label="tab:b-production-quartic"} ![Branching ratios of decays $B^{+}\to SS X_{s/d}$ and $B \to SS$, where $X_{q}$ is a hadron that contains a quark $q$, versus the scalar mass. By the $K^{*}$ channel, we denote the sum of the branching ratios for $K^{*}(892)$, $K^{*}(1410)$, $K^{*}(1680)$ final states, by $K^{*}_{0}$ – for $K^{*}_{0}(700)$, $K_{0}^{*}(1430)$, and by $K_{1}$ – for $K_{1}(1270)$, $K_{1}(1400)$. The values of the branching ratios are given at $\alpha = 1\text{ GeV}$.[]{data-label="fig:plot-b-production-quartic"}](scalar-production-b-mesons-quartic.pdf){width="\textwidth"} Scalar decays ============= The main decay channels of the scalar are decays into photons, leptons and hadrons, see Appendix \[sec:decay\]. In the mass range $m_{S} \lesssim 2m_{\pi}$ the scalar decays into photons, electrons and muons, see Appendix \[sec:decayleptonphoton\]. For masses small enough compared to the cutoff $\Lambda_{\text{QCD}}\approx 1~$GeV, ChPT (Chiral Perturbation Theory) can be used to predict the decay width into pions [@Voloshin:1985tc]. For masses of order $m_{S} \gtrsim \Lambda_{\text{QCD}}$ a method making use of dispersion relations was employed in [@Raby:1988qf; @Truong:1989my; @Donoghue:1990xh] to compute hadronic decay rates. As it was pointed out in [@Monin:2018lee] and later in [@Bezrukov:2018yvd], the reliability of the dispersion relation method is questionable for $m_{S}\gtrsim 1\text{ GeV}$ because of lack of the experimental data on meson scattering at high energies and unknown contribution of scalar hadronic resonances, which provides significant theoretical uncertainties. To have a concrete benchmark – although in the light of the above the result should be taken with a grain of salt – we use the decay width into pions and kaons from [@Winkler:2018qyg], see Fig. \[fig:Winkler-Br\], which combines the next-to-leading order ChPT with the analysis of dispersion relations for the recent experimental data. For the ratio of the decay widths into neutral and charged mesons we have $$\Gamma_{S\to \pi^{0}\pi^{0}}/\Gamma_{S\to \pi^{+}\pi^{-}} = 1/2, \quad \Gamma_{S\to K^{0}\bar{K}^{0}}/\Gamma_{S\to K^{+}K^{-}} = 1.$$ For scalar masses above $f_0(1370)$ the channel $S\to 4\pi$ becomes important and should be taken into account [@Moussallam:1999aq]. The decay width of this channel is currently not known; its contribution can be approximated by a toy-model formula [@Winkler:2018qyg] $$\Gamma_{\text{multi-meson}} = C\theta^{2}m_{S}^{3}\beta_{2\pi}, \quad \beta_{2\pi} = \sqrt{1-(4m_{\pi})^{2}/m_{S}^{2}}, \label{eq:Gmultimeson}$$ where a dimensional constant $C$ is chosen so that the total decay width is continuous at large $m_S$ that will be described by perturbation QCD, see Fig. \[fig:decays\]. ![The ratio of the decay widths of a light scalar into pions, kaons and into muons obtained in [@Winkler:2018qyg]. We summed over all final meson states, i.e. for decays into pions we summed over $\pi^{+}\pi^{-}, \pi^{0}\pi^{0}$. A peak in the decay width corresponding to the channel $S\to \pi \pi$ around $m_{S} \simeq 1\text{ GeV}$ is caused by the narrow $f_{0}(980)$ resonance.[]{data-label="fig:Winkler-Br"}](Winkler.pdf){width="55.00000%"} For $m_S \geq \Lambda_{S}^{\text{pert}} \simeq 2-4$ GeV hadronic decays of a scalar can be described perturbatively using decays into quarks and gluons, see Appendix \[sec:decay-perturbative\]. Currently, the value of $\Lambda_{S}^{\text{pert}}$ is not known because of lack of knowledge about scalar resonances which can mix with $S$ and enhance the scalar decay width. In [@Winkler:2018qyg] the value of $\Lambda_{S}^{\text{pert}}$ is set to $2\text{ GeV}$, in [@Bezrukov:2009yw] it is $\Lambda_{S}^{\text{pert}} = 2.5\text{ GeV}$, while in [@Evans:2017lvd] its value is $\Lambda_{S}^{\text{pert}} = 2m_{D}$. This scale certainly should above the mass of the heaviest known scalar resonance $f_{0}(1710)$, so in this work we choose $\Lambda_{S}^{\text{pert}} = 2$ GeV. The summary of branching ratios of various decay channels of the scalar and the total lifetime of the scalar is shown in Fig. \[fig:decays\]. = [|c|c|c|c|c|c|]{} \ Channel & Opens at& Relevant from & Relevant to & Max BR & Reference\ & \[MeV\] & \[MeV\] & \[MeV\] & \[%\] & in text\ \ Channel & Open, MeV & Rel. from, MeV & Rel. to, MeV & Max BR, % & Formula $S\to 2\gamma$ & 0 & 0 & 1.02 & 100 & (\[eq:Sto2gamma\])\ $S\to e^+ e^-$ & 1.02 & 1.02 & 212 & $\approx 100$ & (\[eq:Gll\])\ $S\to \mu^+ \mu^-$ & 211 & 211 and 1668 & 564 and 2527 & $\approx 100$ & (\[eq:Gll\])\ $S\to \pi^+ \pi^-$ & 279 & 280 & 1163 & 65.5 & Fig. \[fig:Winkler-Br\]\ $S\to 2 \pi^0$ & 270 & 280 & 1163 & 32.8 & Fig. \[fig:Winkler-Br\]\ $S\to K^+ K^-$ & 987 & 996 & $\Lambda_{S}^{\text{pert}} = 2000$ & 36.8 & Fig. \[fig:Winkler-Br\]\ $S\to K^0 \bar{K}^0$ & 995 & 1004 & $\Lambda_{S}^{\text{pert}} = 2000$ & 36.8 & Fig. \[fig:Winkler-Br\]\ & [ 550]{} & [ 1210]{} & [ $\Lambda_{S}^{\text{pert}} = 2000$]{} & [ 52.4]{} &\ $S\to GG$ & $275$ & $\Lambda_{S}^{\text{pert}} = 2000$ & 4178 & 68.6 & (\[eq:decay-gluons\])\ $S\to s \bar{s}$ & 990 & $\Lambda_{S}^{\text{pert}} = 2000$ & 3807 & 42.5 & (\[eq:decay-quarks\])\ $S\to \tau^+\tau^-$ & 3560 & 3608 & — & 45.7 & (\[eq:Gll\])\ $S\to c \bar{c}$ & 3740 & 3797 & — & 50.6 & (\[eq:decay-quarks\])\ ![Left panel: branching ratios of decays of a scalar $S$ as a function of its mass. For decays into hadrons up to $m_{S} =2\text{ GeV}$ we used results from [@Winkler:2018qyg], while for decays into hadrons in the mass range $m_{S} > 2\text{ GeV}$ we used perturbative decays into quarks and gluons (see Sec. \[sec:decay-perturbative\]). In order to match these two regimes, we added a toy-model contribution to the total decay width that imitates multi-meson decay channels, see Eq. . Right panel: the lifetime of a scalar $S$ as a function of its mass with the mixing angle $\theta^2 = 1$. Solid blue line denotes the lifetime calculated using decays into leptons, kaons and pions from [@Winkler:2018qyg] and fictitious multi-meson channel, see Eq. , while solid red line – the lifetime obtained using decays into quarks and gluons within the framework of perturbative QCD. The filled gray regions on the plot correspond to the domain of the scalar masses for which there are significant theoretical uncertainties in hadronic decays.[]{data-label="fig:decays"}](Scalar-decay-br.pdf){width="\textwidth"} ![Left panel: branching ratios of decays of a scalar $S$ as a function of its mass. For decays into hadrons up to $m_{S} =2\text{ GeV}$ we used results from [@Winkler:2018qyg], while for decays into hadrons in the mass range $m_{S} > 2\text{ GeV}$ we used perturbative decays into quarks and gluons (see Sec. \[sec:decay-perturbative\]). In order to match these two regimes, we added a toy-model contribution to the total decay width that imitates multi-meson decay channels, see Eq. . Right panel: the lifetime of a scalar $S$ as a function of its mass with the mixing angle $\theta^2 = 1$. Solid blue line denotes the lifetime calculated using decays into leptons, kaons and pions from [@Winkler:2018qyg] and fictitious multi-meson channel, see Eq. , while solid red line – the lifetime obtained using decays into quarks and gluons within the framework of perturbative QCD. The filled gray regions on the plot correspond to the domain of the scalar masses for which there are significant theoretical uncertainties in hadronic decays.[]{data-label="fig:decays"}](tau-scalar.pdf){width="\textwidth"} Conclusion ========== In this paper, we have reviewed and revised the phenomenology of the scalar portal, a simple extension of the Standard Model with a scalar $S$ that is not charged under the SM gauge group, for masses of scalar $m_S \lesssim 10$ GeV. We considered three examples of experimental setup that correspond to DUNE (with proton-proton center of mass energy $\sqrt{s_{pp}} \approx 16$ GeV), SHiP ($\sqrt{s_{pp}} \approx 28$ GeV) and LHC based experiments ($\sqrt{s_{pp}} = 13$ TeV). Interactions of a scalar $S$ with the Standard Model can be induced by the mixing with the Higgs boson and the interaction $S h^2$ (the “quartic coupling”), see Lagrangian . The mixing with the Higgs boson is relevant for a scalar production and decay, while the quartic coupling could be important only for the scalar production. For the scalar production through the mixing with the Higgs boson, we have explicitly compared decays of secondary mesons, proton bremsstrahlung, photon-photon fusion, and deep inelastic scattering. For the energy of the SHiP experiment, the most relevant production channel is the production in decays of secondary mesons, specifically kaons and $B$ mesons. For smaller energies (corresponding in our examples to the DUNE experiment) the situation is more complicated, and direct production channels from $p-p$ collisions (proton bremsstrahlung, deep inelastic scattering) give the main contribution to the production of scalars, see Fig. \[fig:dis-vs-b-meson\]. Our results for various channels of the scalar production from mesons via mixing with the Higgs boson are summarized in Table \[tab:BR\] and in Fig. \[fig:plot-b-production\]. The results for decays $B\to KS$, $B\to K^{*}(892)S$ agree with the references [@Bezrukov:2009yw; @Clarke:2013aya; @Winkler:2018qyg; @Bird:2004ts; @Pospelov:2007mp; @Batell:2009jf], while other decay channels have not been studied in these papers. For the LHC based experiments, an important contribution to the production of scalars is given by the production in decays of Higgs bosons that may be possible due to non-zero quartic coupling. This production channel, when allowed by the energy of an experiment, allows to search for scalars that are heavier than $B$ mesons. It may also significantly increase the experimental sensitivity in the region of the lower bound of the sensitivity curve, where production through the mixing with the Higgs boson is less efficient. Also the quartic coupling gives rise to meson decay channels $X\to SS$ and $X\to X' SS$ that are important for scalar masses $m_S \lesssim m_B/2$. Our results for these channels are presented in Table \[tab:b-production-quartic\] and in Fig. \[fig:plot-b-production-quartic\]. The description of scalar decays is significantly affected by two theoretical uncertainties: (i) the decay width of a scalar into mesons like $S\to \pi \pi$ and $S \to KK$ (that may be uncertain more than by an order of magnitude for masses of a scalar around 1 GeV) and (ii) the uncertainty in the scale $\Lambda_S^{\text{pert}}$ at which perturbative QCD description can be used. As a benchmark, for decays into mesons we use results of [@Winkler:2018qyg] and choose $\Lambda_{s}^{\text{pert}} = 2\text{ GeV}$, but we stress that the correct result is not really known for such masses. The main properties of scalar decays are summarized in Table \[tab:decaychannels\] and Fig. \[fig:decays\]. Acknowledgements {#acknowledgements .unnumbered} ---------------- We thank O. Ruchayskiy, F. Bezrukov, J. Bluemlein, A. Manohar, A. Monin for fruitful discussions and J.-L. Tastet for careful reading of the manuscript. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (GA 694896). Effective interactions {#sec:effective-interactions} ====================== Photons and gluons {#sec:effective-interactions-gauge-bosons} ------------------ ![Diagrams of the interaction of the scalar $S$ with photons and gluons.[]{data-label="fig:scalar-gauge-bosons-interactions"}](scalar-photon-photon-vertex.pdf "fig:"){width="32.00000%"} ![Diagrams of the interaction of the scalar $S$ with photons and gluons.[]{data-label="fig:scalar-gauge-bosons-interactions"}](scalar-photon-photon-vertex2.pdf "fig:"){width="32.00000%"} ![Diagrams of the interaction of the scalar $S$ with photons and gluons.[]{data-label="fig:scalar-gauge-bosons-interactions"}](scalar-gluon-gluon-vertex.pdf "fig:"){width="32.00000%"} The effective lagrangian of the interaction of $S$ with photons and gluons is generated by the diagrams \[fig:scalar-gauge-bosons-interactions\]. It reads $$\label{eq:LggS} \mathcal{L} = \theta S C_{S\gamma\gamma}\frac{\alpha_{\text{EM}}}{4\pi v}F_{\gamma}F_{\mu\nu}F^{\mu\nu}+\theta SC_{SGG}\frac{\alpha_{s}}{4\pi v}F_{G}\sum_{a}G_{\mu\nu}^{a}G^{\mu\nu,a}.$$ Here the effective vertices $F_{\gamma}, F_{G}$ are [@Bezrukov:2009yw; @Spira:2016ztx] $$\label{eq:photon-gluon-loop-factor} F_{\gamma} = \sum_{l = e,\mu,\tau}F_{l}+N_{c}\sum_{q}F_{q} +F_{W}, \quad F_{G} = \sum_q F_{q},$$ where $$F_{f}(l_{f}) = -2l_{f}(1+(1-l_{f})x^{2}(l_{f})), \quad F_{W} = 2+3l_{W}(1+(2-l_{W})x^{2}(l_{W})), \quad l_{X} = 4m_{X}^{2}/m_{S}^{2},$$ and $$\label{xnotation} x(l) = \begin{cases}\arctan\left(\frac{1}{\sqrt{l-1}}\right), \quad l > 1, \\ \frac{1}{2}\left( \pi + i\ \ln\left[\frac{1+\sqrt{1-l}}{1-\sqrt{1-l}}\right]\right), \quad l<1\end{cases}$$ Their behavior in dependence on the scalar mass is shown in Fig. \[fig:quark-gluon-loop-factor\]. ![Dependence of the photon and gluon loop factors  on the scalar mass.[]{data-label="fig:quark-gluon-loop-factor"}](gluon-photon-loop-factor.pdf){width="70.00000%"} The values of the constants $C_{SGG}$ and $C_{S\gamma\gamma}$ vary in the literature. Namely, in [@Bezrukov:2009yw] they are $C_{S\gamma\gamma} = 1, C_{SGG} = 1/\sqrt{8}$. From the other side, in [@Spira:2016ztx] predicts $|C_{S\gamma\gamma}| = 1/2, |C_{SGG}| = 1/4$. Calculating the decay branching ratio of the Higgs boson into two photons, we found that the value $C_{S\gamma\gamma} = 1/2$ is consistent with experimental results for the signal strength of the process $p + p \to h +X, \ h \to \gamma\gamma$ [@Khachatryan:2016vau].[^2] The gluon loop factor in the triangle diagram \[fig:scalar-gauge-bosons-interactions\] differs from the photon loop factor by the factor $\text{tr}[t_{a}t_{a}] = \frac{1}{2}$, where $t_{a}$ is the QCD gauge group generators, and therefore $C_{Sgg} = 1/4$. Nucleons {#sec:effective-interactions-nucleons} -------- Consider the low-energy interaction Lagrangian between the nucleons $N$ and the scalar: $$\label{eq:SNN} \mathcal{L}_{S NN} = g_{S NN}\theta S\bar{N}N$$ The coupling $g_{SNN}$ is defined as $$\label{eq:SNN-coupling} g_{S NN} \equiv \frac{1}{v}\lim_{p\to p'}\langle N(p)|\sum_{q}m_{q}\bar{q}q|N(p')\rangle \equiv \frac{1}{v}\langle N|\sum_{q}m_{q}\bar{q}q|N\rangle,$$ where the shorthand notation $\langle N|..|N\rangle \equiv \lim_{p\to p'}\langle N(p)|..|N(p')\rangle$ was used. The applicability of the effective interaction  is $m_{S}^{2}\lesssim r_{N}^{-2} \simeq 1\text{ GeV}^{2}$. Above this scale the elastic $SNN$ vertex competes with the inelastic processes on partonic level and hence it is suppressed. For energy scales of order of the nucleon mass, the $u,d,s$ quarks are light, while the $c,b,t$ quarks are heavy. Therefore, the latter can contribute to the effective coupling  only through effective interactions involving the lighs quarks and gluons. The latter can be obtained using the heavy quarks expansion [@Vainshtein:1975sv; @Witten:1975bh]. Keeping only the leading $1/m_{q_{\text{heavy}}}$ term, for the effective interaction operator we obtain [@Shifman:1978zn] $$\label{eq:hqeft} \sum_{q = c,b,t}m_{q}\bar{q}q \to -n_{\text{heavy}}\cdot \frac{\alpha_{s}}{12\pi}G_{\mu\nu}^{a}G^{\mu\nu,a} + O\left(\frac{1}{m_{q_{\text{heavy}}}^{2}}\right).$$ Here $\alpha_{s}$ is the QCD interaction constant evaluated on the scale of the hadronic mass, $G_{\mu\nu}^{a}$ is the gluon strength tensor and $n_{\text{heavy}} = 3$ is the number of the heavy quarks. Therefore, in the leading order of $1/m_{q_{\text{heavy}}}$ expansion the coupling  takes the form [@Shifman:1978zn; @Cheng:1988im] $$\label{eq:SN1} g_{SNN} = \frac{\theta}{v}\langle N|\sum_{q = u,d,s}m_{q}\bar{q}q -n_{\text{heavy}}\frac{\alpha_{s}}{12\pi}G_{\mu\nu}^{a}G^{\mu\nu,a}|N\rangle.$$ The last expression we can written in terms of the nucleon mass $m_{N}$, $$m_{N} \equiv \langle N|\theta_{\mu\mu}|N \rangle,$$ where $\theta_{\mu\mu}$ is the trace of the stress-energy tensor in the QCD [@Shifman:1978zn] $$\label{eq:trace} \theta_{\mu\mu} = \sum_{q = u,d,s}m_{q}\bar{q}q +\frac{\beta_{s}}{4 \alpha_{s}}G_{\mu\nu}^{a}G^{\mu\nu,a},$$ where $\beta_{s}$ is the QCD $\beta$ function, $$\beta_{s} = -\left(9-\frac{2}{3}n_{\text{heavy}}\right)\frac{\alpha_{s}^{2}}{2\pi},$$ in the leading order by $\alpha_s$. Therefore, we get [@Shifman:1978zn; @Cheng:1988im] $$g_{S NN} =\frac{2}{9}\frac{m_{N}}{v}\left( 1 + \frac{7}{2}\sum_{q = u,d,s}\frac{m_{q}}{m_{N}}\langle N|\bar{q}q|N\rangle\right).$$ The numerical value is $g_{SNN} \approx 1.2\cdot 10^{-3}$ [@Cheng:2012qr]. In order to incorporate effects of non-zero momentum transfer $q^{2}$ in the $SNN$ vertex, we need to take into account an scalar nucleon form-factor $F_{N,S}(q^{2})$: $$g_{SNN} \to g_{SNN}F_{N,S}(q^{2}), \quad F_{N,S}(0) = 1.$$ We have not found any paper discussing the form-factor $F_{N,S}$. From general ground we expect that it incorporates a mixing with scalar resonances $f_{0}$ which causes peaks at $q^{2} = m_{f_{0}}^{2}$. For large momentum transfers $F_{N,S}$ should vanish. Flavor changing effective Lagrangian {#sec:fcnc} ------------------------------------ ![Diagrams of the production of the scalar $S$ in flavor changing quarks transitions in the unitary gauge.[]{data-label="fig:3diagrams"}](scalar_production.pdf){width="\textwidth"} A light scalar $S$ can be produced from a hadron via flavor changing quarks transitions (see diagrams in Fig. \[fig:3diagrams\]). The flavor changing amplitude was calculated using different techniques in many papers [@Willey:1982ti; @Willey:1986mj; @Grzadkowski:1983yp; @Leutwyler:1989xj; @Haber:1987ua; @Chivukula:1988gp]. The corresponding effective Lagrangian of flavor changing quark interactions with the $S$ particle is $$\label{eq:g05} \mathcal{L}_{eff}^{Sqq}= \theta\ \frac{S}{v}\sum_{i,j}\xi_{ij}m_{Q_j}\bar{Q}_{i}P_{R}Q_{j}+h.c.,$$ where $Q_i$ and $Q_j$ are both upper or lower quarks and $P_R \equiv (1+\gamma_{5})/2$ is a projector on the right chiral state. The effective coupling $\xi_{ij}$ is defined as $$\xi_{ij} = \frac{3G_{F}\sqrt{2}}{16\pi^{2}}\sum_{k}V_{ki}^{*}m_{k}^{2}V_{kj},$$ where $Q_k$ are the lower quarks if $Q_i$ and $Q_j$ are the upper and vice versa, $V_{ij}$ are the elements of the CKM matrix, and $G_F$ is the Fermi constant. One power of the quark mass in the expression for $\xi$ comes from the $h\bar{q}q$ coupling, while another one comes from the helicity flip on the quark line in the diagrams in Fig. \[fig:3diagrams\]. Because of such behavior, the quark transition generated by the Lagrangian  is more probable for lower quarks than for upper ones, since the former goes through the virtual top quark. Numerical values of some of $\xi_{ij}$ constants are given in Table \[tab:xi\]. $\xi_{ij}$ $\xi_{ds}$ $\xi_{uc}$ $\xi_{db}$ $\xi_{sb}$ ------------ --------------------- --------------------- --------------------- --------------------- Value $3.3 \cdot 10^{-6}$ $1.4 \cdot 10^{-9}$ $7.9 \cdot 10^{-5}$ $3.6 \cdot 10^{-4}$ : Numerical values of $\xi_{ij}$ constants in effective Lagrangian .[]{data-label="tab:xi"} Scalar production from mesons {#sec:production-hadronic-decays} ============================= In the scalar production from hadron decays, the main contribution comes from the lightest hadrons in each flavor, which are mesons.[^3] The list of the main hadron candidates is the following (the information is given in the format “Hadron name (quark contents, mass in MeV)”): - s-mesons $K^-(s\bar{u}, 494)$, $K^0_{S,L}(s\bar{d}, 498)$; - c-mesons $D^0(c\bar{u},1865)$, $D^+(c\bar{d},1870)$, $D_s(c\bar{s},1968)$, $J/\psi(c\bar{c},3097)$; - b-mesons $B^-(b\bar{u},5279)$, $B^0(b\bar{d},5280)$, $B_s (b\bar{s},5367)$, $B_c (b\bar{c},6276)$, $\Upsilon (b\bar{b},9460)$. The production of a scalar from mesons is possible through the flavor changing neutral current \[sec:fcnc\], so the production from $D_s$, $J/\psi$, $B_s$, $B_c$ and $\Upsilon$ mesons does not have any advantage with respect to the production from $D^0$, $D^+$, $B^-$ and $B^0$, while their amount at any experiment is significantly lower. Therefore, we will discuss below only production from later mesons. Inclusive production {#sec:inclusive-br-calculation} -------------------- The decay widths for the processes $Q_{i}\to Q_{j}S$, $Q_{i}\to Q'e\bar{\nu}_{e}$ are $$\Gamma_{Q_{i}\to Q_{j} +S} = \frac{|\mathcal{M}_{Q_{i}\to Q_{j}S}|^{2}}{8\pi m_{b}}\frac{|\bm{p}_{S}|}{m_{b}} \approx |\xi_{bs}|^{2}\frac{m_{b}^{3}\left(1-\frac{m_{S}^{2}}{m_{b}^{2}} \right)^{2}}{32 \pi v^{2}}\theta^{2},$$ $$\begin{gathered} \Gamma_{Q_{i}\to Q_{k} +e+\bar{\nu}_{e}} = \frac{1}{(2\pi)^{3}}\int \limits_{m_{Q_{k}}^{2}}^{m_{Q_{i}}^{2}}ds_{Q_{k}e} \int \limits_{s_{e\nu,\text{min}}}^{s_{e\nu,\text{max}}} ds_{e\nu} \frac{|\mathcal{M}_{Q_{i}\to Q_{k}+e+\bar{\nu}_{e}}|^{2}}{32m_{Q_{i}}^{3}} \approx \\ \approx \frac{G_{F}^{2}|V_{Q_{i}Q_{k}}|^{2}m_{Q_{i}}^{5}}{192\pi^{3}}\times f(m_{Q_{k}}/m_{Q_{i}}),\end{gathered}$$ where $\bm{p}_{S}$ is the $S$ particle momentum at the rest frame of the meson $X$, $$|\bm{p}_{S}| = \frac{\sqrt{(m_{X}^{2}-(m_{S}+m_{X'})^{2})((m_{X}^{2}-(m_{S}-m_{X'})^{2})}}{2m_{X}},$$ the integration limits are $$s_{e\nu,\text{min}} = 0, \quad s_{e\nu,\text{max}}= m_{Q_{i}}^{2}+m_{Q_{i}}^{2}-s_{Q_{k}e}-\frac{m_{Q_{i}}^{2}m_{Q_{k}}^{2}}{s_{Q_{k}e}},$$ and $$f(m_{Q_{k}}/m_{Q_{i}}) = \bigg(1-8\frac{m_{Q_{k}}^{2}}{m_{Q_{i}}^{2}}-24\frac{m_{Q_{k}}^{4}}{m_{Q_{i}}^{4}}\ln\left(\frac{m_{Q_{k}}}{m_{Q_{i}}}\right)+8\frac{m_{Q_{k}}^{6}}{m_{Q_{i}}^{6}}-\frac{m_{Q_{k}}^{8}}{m_{Q_{i}}^{8}} \bigg) \approx 1/2$$ is the phase space factor. Scalar production in two-body mesons decays {#sec:meson-two-body-decays} ------------------------------------------- Let us consider exclusive 2-body decay of a meson $$X_{Q_i}\to X'_{Q_j} +S, \label{eq:exclusive-process}$$ corresponding to the transition $Q_{i} \to Q_{j}+S$. Here and below, $X_{Q}$ denotes a meson which contains a quark $Q$. The Feynman diagram of the process is shown in Fig. \[fig:Sproduction\] (c). Using the Lagrangian , for the matrix element we have $$\mathcal{M} (X_{Q_{i}} \to S X'_{Q_{j}}) = \frac{\theta}{2}\frac{m_{Q_{i}}}{v}\times \xi_{ij}\times M_{XX'}(m_{S}^{2}), \label{eq:matrixelementhhS}$$ where $$M_{XX'}((p_{X}-p_{X'})^{2}) \equiv \langle X'(p_{X'})|\bar{Q}_{i}(1+\gamma_{5})Q_{j}|X(p_{X})\rangle \label{eq:hadronic-matrix-element-1}$$ is the matrix element of the transition $X_{Q_{i}}\to X'_{Q_{j}}$. Expressions for these matrix elements for different initial and final mesons are given in Appendix \[sec:hadronic-form-factors\]. So, we can calculate the branching fraction of the corresponding process by the formula  [@Tanabashi:2018oca] $$\text{BR}(X_{Q_{i}}\to X'_{Q_{j}}+S) = \frac{1}{\Gamma_{X}}\theta^{2}\frac{|\xi_{ij}|^{2}m_{Q_{i}}^{2}|M_{XX'}(m_{S}^{2})|^{2}}{32\pi v^{2}}\frac{|\bm{p}_{S}|}{m_{X}^{2}}, \label{eq:productionBRmesons}$$ where $\Gamma_{X}$ is the decay width of the meson $X$. We use the lifetimes of mesons from [@Tanabashi:2018oca]. For the kaons, the only possible 2-body decay is the process $$K \to \pi +S$$ There are 3 types of the kaons – $K^{\pm}, K^{0}_{L}, K^{0}_{S}$. Although the decay width for each of them is by given by the same loop factor, $\xi_{sd}$, the branching ratios differ. The first reason is that these kaons have different decay widths. The second reason is that the $K^{0}_{S}$ is approximately the $CP$-even eigenstate. Therefore the decay $K^{0}_{S}\to \pi S$ is proportional to the CKM $CP$-violating phase and is strongly suppressed [@Leutwyler:1989xj]. Further we assume that the corresponding branching ratio vanishes. See Table \[tab:BR\] for the branching ratios of $K^{0}_{L},K^{\pm}$. Scalar production in leptonic decays of mesons {#sec:leptonic-decays} ---------------------------------------------- Consider the process $X \to S e \nu$. Its branching ratio is [@Dawson:1989kr; @Cheng:1989ib] $$\text{BR} (X \rightarrow S e \nu ) = \dfrac{\sqrt{2} G_F m_X^4}{96 \pi^2 m_{\mu}^2 (1 - m_{\mu}^2/m_h^2)^2} \times \text{BR} (X \rightarrow \mu \nu) \left(\frac{7}{9}\right)^2 f\left( \frac{m_S^2}{m_X^2}\right),$$ where $f(x) = (1 - 8x + x^2) (1 - x^2) - 12 x^2 \text{ln} x$. The values of the branching ratios for different types of the mesons are shown in Table \[tab:BR-3-body\]. Meson BR$(h \rightarrow S e \nu )/ f\left( x\right) \theta^2$ ------------------------- --------------------------------------------------------- $D \to S e \nu$ $5.2 \cdot 10^{-9}$ $K \rightarrow S e \nu$ $4.1 \cdot 10^{-8}$ $B \rightarrow S e \nu$ $< 7.4 \cdot 10^{-10}$ : Branching ratios of 3-body meson decay. From experimental data we have only upper bound on the $\text{BR}(B\to\mu\nu)$, so we put upper bound on $B \rightarrow S e \nu$ decay.[]{data-label="tab:BR-3-body"} However, although for the $D$ this channel enhances the production in $\simeq \mathcal{O}(100)$ times, the production from $D$ is still sub-dominant. DIS {#sec:production-direct-dis} === The scalar production in the DIS is driven by the interaction with the quarks and gluons: $$\mathcal{L} = S\theta\sum_{q}\frac{m_{q}}{v}\bar{q}q + \theta\frac{S\alpha_{s}}{16\pi v}F_{G}(m_{S})G_{\mu\nu}^{a}G^{\mu\nu ,a}, \label{eq:dis-lagrangian}$$ where $F_{G}$ is a loop factor being of order of $|F_{G}|^{2} \simeq 10-20$ for the scalars in the mass range $m_{S} \lesssim 10\text{ GeV}$ (see Appendix \[sec:effective-interactions-gauge-bosons\]). Processes of the scalar production in DIS are quark and gluon fusions: $$q+\bar{q}\to S, \quad G + G \to S \label{eq:dis-processes}$$ ![The diagrams of the production of the scalar in deep inelastic scattering.[]{data-label="fig:dis-production"}](scalar-production-dis.pdf){width="\textwidth"} Corresponding diagrams are shown in Fig. \[fig:dis-production\] and the matrix elements are $$\mathcal{M}(GG\to S) = 4\frac{F_{G}(m_{S}) \alpha_s}{16 \pi} \frac{\theta}{v} [(k_2^{\mu} \cdot k_1^{\nu}) - g^{\mu \nu} (k_1 \cdot k_2)] \epsilon_{\mu}(k_1) \epsilon_{\nu} (k_2),$$ $$i\mathcal{M}(q\bar{q}\to S) = \overline{v}(k_2) \left( \frac{-i \theta \overline{m}_q}{v} \right) u(k_1).$$ The differential cross section is given by $$d \sigma(s_{YY}) = \frac{(2 \pi)^4}{4} \frac{\overline{|\mathcal{M}(YY\to S)|^2}}{\sqrt{(k_1 \cdot k_2)^2}} d \Phi (k_1 + k_2,p_{S}) = \frac{\pi \overline{|\mathcal{M}(YY\to S)|^2}}{ m_S^2} \delta(s_{YY} - m_S^2), \label{eq:differential-cross-section-dis}$$ where Y denotes a quark/antiquark or a gluon, $\overline{|\mathcal{M}(GG\to S)|^{2}}$ is the squared matrix element averaged over gluon or quark polarizations and $$d \Phi(k_1 + k_2,p_{S}) = \delta^4(k_1 + k_2 - p_{S}) \dfrac{d^3 \bm{p}_{S}}{(2 \pi)^3 2 E_s}.$$ The hard cross sections for the gluon and quark fusions are thus $$\label{sigmagg} \sigma_{G}(s_{GG}) = \delta(s_{GG} - m_S^2) \frac{|F_{G}(m_{S})|^2 \alpha_s^2 \theta^2 m_S^2}{128\pi v^2},$$ $$\label{hard_cross_section_quarks} \sigma_{q}(s_{qq}) = \frac{\pi}{m_S^2} \delta(s_{qq} - m_S^2) \frac{\theta^2 \overline{m}_q^2}{2v^2} m_S^2.$$ Using hard cross sections and , one can calculate the total cross section of the production in DIS as $$\sigma_{\text{DIS},Y}=g_{Y}\int \sigma_{Y}(s)f_{Y_{1}}(\sqrt{s_{Y_{1}Y_{2}}},x_{1}) f_{Y_{2}}(\sqrt{s_{Y_{1}Y_{2}}},x_{2}) dx_1 dx_2.$$ Here, $f_{Y}(Q,x)$ is the parton distribution function (pdf) of the parton $Y$ carrying the momentum fraction x at the scale $Q$. $g_{q} = 2, g_{G} = 1$; $g_{q}$ is a combinatorial factor taking into account that the quark/antiquark producing a scalar can be stored in both of colliding protons. The result is $$\sigma_{\text{DIS},q} (s) = \frac{\pi}{m_S^2} \frac{\theta^2 \overline{m}_q^2 m_S^2}{2v^2 s} \times W_{q\bar{q}}, \ \sigma_{\text{DIS},G} (s) = \theta^{2}\frac{|F_{G}(m_{S})|^{2}\alpha_{s}^{2}(m_{S})m_{S}^{2}}{128\pi sv^{2}}\times W_{GG}. \label{eq:dis-cross-sections}$$ Here, $s$ denotes the $pp$ CM energy, $\overline{m}_{q}$ is the $\overline{\text{MS}}$ quark mass at the scale $m_{S}$, and $$W_{XX}(s,m_{S}) \equiv \int\limits_{m_S^2/s}^1 \frac{dx}{x} f_{X}(m_{S},x) f_{X}\left(m_{S}, \frac{m_S^2}{sx} \right) \label{eq:partonic-weight}$$ is the partonic weight of the process. Since the partonic model breaks down at scales $Q \lesssim 1\text{ GeV}$, the description of the scalar production in DIS presented in this section is valid only for scalars with masses $m_{S}\gtrsim 1\text{ GeV}$. For numerical estimates we have used LHAPDF package [@Buckley:2014ana] with CT10NLO pdf set [@Lai:2010vv]. The main contribution to the DIS cross section comes from gluons. To see this, let us compare the gluon cross section $\sigma_{\text{DIS},G}$ with the s-quark cross section $\sigma_{\text{DIS},S}$, which is the largest quark cross section.[^4] Their ratio is $$\frac{\sigma_{\text{DIS},G}}{\sigma_{\text{DIS},s}}\approx 0.6\left(\frac{\alpha_{s}(m_{S})}{0.4}\right)^{2}\frac{|F_{G}(m_{S})|^{2}}{20}\times \left(\frac{m_{S}}{1\text{ GeV}}\right)^{2}\frac{W_{GG}}{W_{ss}}. \label{eq:dis-gluon-to-quark}$$ The product $|F_{G}|^{2} \alpha_{s}^{2}$ changes with $m_{S}$ relatively slowly, and therefore the ratio  is determined by the product $(m_{S}/1\text{ GeV})^{2}W_{GG}/W_{ss}$. It is larger than one for the masses $m_{S}\gtrsim 2 \text{ GeV}$ in broad CM energy range, see Fig. \[fig:dis-production-plot\]. Having the cross sections , we calculate the DIS probability as $$P_{\text{DIS}} = \frac{\sum_{q}\sigma_{\text{DIS},q}+\sigma_{\text{DIS},G}}{\sigma_{pp}}, \label{eq:dis-probability}$$ where for the total proton-proton cross section $\sigma_{pp}$ we used the data from [@Tanabashi:2018oca]. Scalar production in proton bremsstrahlung {#sec:bremsstrahlung} ========================================== ![A diagram of the production of a scalar in the proton bremsstrahlung process.[]{data-label="fig:bremsstrahlung"}](scalar-production-bremsstrahlung.pdf){width="60.00000%"} A scalar $S$ can be produced through the $SNN$ vertex (see Sec. \[sec:effective-interactions-nucleons\]) in proton-proton bremsstrahlung process $$p+p \to S+X, \label{eq:bremsstrahlung-process}$$ with the diagram of the process shown in Fig. \[fig:bremsstrahlung\]. Corresponding probability can be estimated using generalized Weizsacker-Williams method, allowing to express the cross section of the given process by the cross section of its sub-process [@Altarelli:1977zs; @Blumlein:2013cua; @Blumlein:2011mv; @Chen:1975sh; @Kim:1973he; @Frixione:1993yw; @Baier:1973ms]. Namely, let us denote the momentum of the incoming proton in the rest frame of the target proton by $p_{p}$, the fraction of $p_{p}$ carried by $S$ as $z$ and the transverse momentum of $S$ as $p_{T}$. Then, under conditions $$\frac{p_{T}^{2}}{4p_{p}^{2}}\ll z(1-z)^{2}, \quad \frac{m_{S}^{2}}{4p_{p}^{2}}\ll z(1-z), \quad \frac{m_{p}^{2}}{4p_{p}^{2}}\ll \frac{(1-z)^{2}}{z} \label{eq:bremsstrahlung-domain}$$ the differential production cross section of $S$ production can be written as (see Appendix \[sec:bremsstrahlung-derivation\]) $$d\sigma_{\text{brem}} \approx \sigma_{pp_{t}}(s')\times P_{p\to pS}(p_{T},z)dp_{T}^{2}dz,$$ where we denoted a target proton as $p_{t}$, $\sigma_{pp_{t}}$ is the total $p$-$p$ cross section, $s' = 2m_{p}p_{p}(1-z)+2m_{p}^{2}$ and the differential splitting probability of the proton to emit a scalar is $$P_{p\to pS}(p_{T},z) \approx |F_{pS}(m_{S}^{2})|^{2} \frac{g_{SNN}^{2}\theta^{2}}{8\pi^{2}}z\frac{m_{p}^{2}(2-z)^{2}+p_{T}^{2}}{(m_{p}^{2}z^{2}+m_{S}^{2}(1-z)+p_{T}^{2})^{2}},$$ with $g_{SNN}$ being low-energy proton-scalar coupling, and $F_{pS}$ the scalar-proton form-factor, see Appendix \[sec:effective-interactions-nucleons\]. For the total $pp$ cross section we use experimental fit $$\sigma_{pp}(s) = Z+B \ln^{2}\left(\frac{s}{s_{0}}\right)+C_{1}\left( \frac{s_{1}}{s}\right)^{\eta_{1}}-C_{2}\left( \frac{s_{1}}{s}\right)^{\eta_{2}},$$ where $Z = 35.45\text{ mb}$, $B = 0.308\text{ mb}$, $C_{1}=42.53\text{ mb}$, $C_{2} = 33.34\text{ mb}$, $\sqrt{s_{0}}=5.38\text{ GeV}$, $\sqrt{s_{1}} = 1\text{ GeV}$, $\eta_{1}=0.458$ and $\eta_{2} = 0.545$ [@Tanabashi:2018oca]. This cross section is shown in Fig. \[fig:pp-cross-section\], where we see that it is almost constant for a wide range of energies. The total cross section can be written in the form $$\sigma_{\text{brem}} = g_{SNN}^{2}\theta^{2}|F_{pS}(m_{S}^{2})|^{2}\sigma_{pp}(s)\mathcal{P}_{\text{brem}}(s,m_{S}),$$ where $$\mathcal{P}_{\text{brem}}(s,m_{S}) = \frac{1}{g_{SNN}^{2}\theta^{2}}\int dp_{T}^{2}dz P_{p\to pS}(p_{T},z)\frac{\sigma_{pp}(s')}{\sigma_{pp}(s)}.$$ ![Proton-proton total cross section as a function of the center of mass energy $\sqrt{s_{pp}}$.[]{data-label="fig:pp-cross-section"}](pp-cross-section.pdf){width="60.00000%"} The domain of the definition of $p_{T}$ and $z$ is determined by the conditions . For definiteness, we fix the domain of integration by the requirement $$\frac{m_{S}^{2}(1-z)+m_{p}^{2}z^{2}+p_{T}^{2}}{4p_{p}^{2}z(1-z)^{2}}<0.1.$$ The probability of a scalar production in proton bremsstrahlung is $$P_{\text{brem}} = \frac{\sigma_{\text{brem}}}{\sigma_{pp}(s)} \approx g_{SNN}^{2}\theta^{2}|F_{pS}(m_{S}^{2})|^{2}\mathcal{P}_{\text{brem}}(s,m_{S}) ,$$ where $s$ is the CM energy of two protons. We show its dependence on the scalar mass and the incoming beam energy in Fig. \[fig:bremsstrahlung-probability\]. ![The probability of the production of a scalar $S$ in bremsstrahlung process versus the scalar mass.[]{data-label="fig:bremsstrahlung-probability"}](scalar-production-bremsstrahlung.pdf){width="70.00000%"} Splitting probability derivation {#sec:bremsstrahlung-derivation} -------------------------------- Following the approach described in [@Altarelli:1977zs], let us consider the process  within the old-fashioned perturbation theory. The corresponding diagrams are shown in Fig. \[fig:bremsstrahlung-old-fashioned\]. ![The lowest order old-fashioned perturbation theory diagrams for the bremsstrahlung process . Vertical dotted lines denote the intermediate states.[]{data-label="fig:bremsstrahlung-old-fashioned"}](bremsstrahlung-old-fashioned.pdf) The matrix element has the form $\mathcal{V}_{pp_{t}\to SX} = \mathcal{V}_{a}+\mathcal{V}_{b}$, where $$\mathcal{V}_{a} = \frac{\mathcal{M}_{p\to p'S}\mathcal{M}_{p'p_{t}\to X}}{2E_{p'}(E_{p'}+E_{S}-E_{p})}\bigg|_{\bm{p}_{p'} = \bm{p}_{p}-\bm{p}_{S}}, \quad \mathcal{V}_{b} = \frac{\mathcal{M}_{p_{t}\to p'X}\mathcal{M}_{p'p\to S}}{2E_{p'}(E_{p}+E_{p'}-E_{S})}\bigg|_{\bm{p}_{p'} = \bm{p}_{p}-\bm{p}_{S}}. \label{eq:matrix-element-old-fashioned}$$ Here, $\mathcal{M}$ denotes Lorentz-invariant amplitude of the processes. There exists a kinematic domain at which $|\bm{M}_{b}| \ll |\bm{M}_{a}|$. Namely, let us consider an ultrarelativistic incoming $p$, and write the 4-momenta of $p$, $S$ and intermediate $p'$ as $$\begin{aligned} P_{p}^{\mu} &= \left(p_{p}+ \frac{m_{p}^{2}}{p_{p}^{2}}, \bm{0}, p_{p}\right), \\ P_{S}^{\mu} &= \left(p_{p}z +\frac{p_{T}^{2}+m_{S}^{2}}{2p_{p}z},\bm{p}_{T},zp_{p}\right), \\ P_{p'}^{\mu} &= \left((1-z)p_{p}+\frac{m_{p}^{2}+p_{T}^{2}}{2p_{p}(1-z)},-\bm{p}_{T},(1-z)p_{p} \right), \end{aligned}$$ where $p_{T}$ is a transverse momentum of $S$ and $z$ is a fraction of a parallel momentum carried by $S$. Then the energy denominators in  are $$\Delta E_{a} = E_{p'}+E_{S}-E_{p} \approx \frac{p_{T}^{2}+(1-z)m_{S}^{2}+z^{2}m_{p}^{2}}{2p_{p}z(1-z)}, \quad \Delta E_{b} = E_{p}+E_{p'}-E_{S} \approx 2p_{p}(1-z).$$ Assuming that $\Delta E_{a}\ll \Delta E_{b}$ we can neglect the matrix element $\mathcal{V}_{b}$. Once we neglect $\mathcal{V}_{b}$, it is possible to relate the differential cross section of the process  to the total $pp$ scattering cross section. Indeed, let us consider a corresponding process $pp_{t}\to X$, which is a sub-process of  obtained by removing the in $p$ line and out $S$ line, see Fig. \[fig:bremsstrahlung-process-sub-process\]. ![Diagrams the bremsstrahlung process  (left) and its sub-process $pp_{t} \to X$ describing a proton-proton collision (right).[]{data-label="fig:bremsstrahlung-process-sub-process"}](bremsstrahlung-process-sub-process.pdf){width="\textwidth"} The matrix element for this process is simply $$\mathcal{V}_{pp_{t}\to S} = \mathcal{M}_{pp_{t}\to X}. \label{eq:matrix-element-pp-inelastic}$$ Using , , for the corresponding differential cross sections we obtain $$\begin{gathered} d\sigma_{pp_{t}\to SX} = \frac{1}{4E_{p}E_{p_{t}}}\frac{|\mathcal{M}_{p\to p'S}|^{2}|\mathcal{M}_{p'p_{t}\to X}|^{2}}{(2E_{p'})^{2}(E_{p'}+E_{S}-E_{p})^{2}}\times \\ (2\pi)^{4}\delta^{(4)}\left(p_{p}+p_{p_{t}}-p_{S}-\sum_{X}p_{X}\right) \frac{d^{3}\bm{p}_{S}}{(2\pi)^{3}2E_{S}}\times \prod_{X}\frac{d^{3}\bm{p}_{X}}{(2\pi)^{3}2E_{X}}, \label{eq:sub-process-a}\end{gathered}$$ $$d\sigma_{p'p_{t}\to X} = \frac{1}{4E_{p'}E_{p_{t}}}|\mathcal{M}_{p'p_{t}\to X}|^{2}\times (2\pi)^{4}\delta^{(4)}\left(p_{p'}+p_{p_{t}}-\sum_{X}p_{X}\right) \prod_{X}\frac{d^{3}\bm{p}_{X}}{(2\pi)^{3}2E_{X}} \label{eq:sub-process-b}$$ Neglecting the difference in the energy conservation arguments in the delta-functions that are of order $\mathcal{O}(m_{p/S}^{2}/p_{p}^{2},p_{T}^{2}/p_{p}^{2})$, we can relate these two cross sections as $$d\sigma_{pp_{t}\to SX} = dP_{p\to p'S}(z,p_{T})d\sigma_{p'p_{t}\to X}(p_{T},z), \label{eq:differential-cross-section}$$ where we introduced differential splitting probability $dP_{p\to p'S}$: $$dP_{p\to p'S}(p_{T},z) \equiv 2\frac{|\mathcal{M}_{p\to p'S}|^{2}}{4E_{p}E_{p'}(E_{p'}+E_{S}-E_{p})^{2}}\frac{d^{3}\bm{p}_{S}}{(2\pi)^{3}2E_{S}}. \label{eq:splitting-probability}$$ Here a factor of $2$ is combinatorial factor taking into account that a scalar can be produced from both the legs of colliding protons. Integrating the differential cross section  over the momenta of the final states particles $X$ and summing over all possible sets $\{X\}$, we finally arrive at $$d\sigma_{pp_{t}\to SX} \approx P_{p\to p'S}(z,p_{T})dp_{T}^{2}dz\sigma_{pp}(s'),$$ where $s' \approx 2m_{p}p_{p}(1-z)+2m_{p}^{2}$ [^5] and $\sigma_{pp}(s')$ is the total proton-proton cross section. Let us now find explicit expression for the splitting probability . Using the expressions , we find $$\frac{d^{3}\bm{p}_{S}}{(2\pi)^{3}2E_{S}} \approx \frac{dp_{T}^{2}dz}{16 \pi^{2}z}, \quad |\mathcal{M}_{p\to pS}|^{2} \approx 2g_{SNN}^{2}\theta^{2}|F_{pS}(m_{S}^{2})|^{2}(m_{p}^{2}+(P_{p}\cdot P_{p'})).$$ Finally, we arrive at $$P_{p\to p'S} \approx \frac{g_{SNN}^{2}\theta^{2}|F_{pS}(m_{S}^{2})|^{2}}{8\pi^{2}}z\frac{m_{p}^{2}(2-z)^{2}+p_{T}^{2}}{(m_{p}^{2}z^{2}+m_{S}^{2}(1-z)+p_{T}^{2})^{2}}.$$ Scalar production in photon fusion {#sec:production-direct-coherent} ================================== A scalar can be produced elastically in $pp$ collisions through the $S\gamma\gamma$ vertex (see Appendix \[sec:effective-interactions-gauge-bosons\]). The production process is $$p+Z \to p + Z +S, \label{eq:photon-fusion}$$ with the corresponding diagram shown in Fig. \[fig:photon-fusion\]. ![A diagram of the production of a scalar in photon fusion.[]{data-label="fig:photon-fusion"}](scalar-production-photon-fusion.pdf){width="40.00000%"} To find the number of produced scalars in the photon fusion, we will use the equivalent photon approximation (EPA), which provides a convenient framework for studying processes involving photons emitted from fast-moving charges [@Budnev:1974de; @Martin:2014nqa; @Dobrich:2015jyk]. The basic idea of the EPA is a replacement of the charged particle $Y$ in the initial and final state, that interacts through the virtual photon carrying the virtuality $q$ and the fraction of charge’s energy $x$, by the almost real photon with a distribution $n_{Y}(x; q^2)$ that depends on the type of the charged particle, see Fig. \[fig:EPA\]. [ ![The idea of the equivalent photon approximation. If a charge with the momentum $k$, emitting the virtual photon with the virtuality $q$, is ultrarelativistic, then the cross section of the process (a) can be expressed in terms of the cross section of the process (b). The remained effect of the charge is the distribution function $n_{\text{charge}}(x,q^{2})$, where $x$ is the energy fraction carried by photon.[]{data-label="fig:EPA"}](Equivalent-photon.pdf "fig:"){width="70.00000%"} ]{} The magnitude of the momentum transfer carried by the virtual photon can be approximated as $$\label{eq:phf1} q^{2}\approx \frac{q_t^2+x^2 m_{Y}^2}{1-x},$$ where $q_t$ is the transverse component of the spatial momentum of the photon with respect to the spatial momentum of the particle $Y$, and $m_{Y}$ is the mass of $Y$. Conditions for validity of the EPA are $x\ll 1$ and $q_t\lesssim x m_{Y}$ [@Dobrich:2015jyk]. The distribution $n_{Y}(x,q_{t}^{2})$ of the emitted photons can be described by $$\label{eq:phf3} n_Y(x; q_t^2)=\frac{\alpha_{\text{EM}}}{2\pi} \frac{1 + (1 -x)^2}{x(q_t^2+x^2 m_{X}^2)}\left[ \frac{q_t^2}{q_t^2+x^2 m_Y^2}D_{Y}(q^2)+\frac{x^2}2 C_{Y}(q^2)\right],$$ where $C(q^{2}),D(y^{2})$ are appropriate form-factors. We take the proton and nucleus form-factors from [@Dobrich:2015jyk]. Within the EPA, we approximate the cross section of the process  by $$\label{eq:phf9} \sigma_{pZ\rightarrow SpZ}=\int dx_1 dx_2 d\vec q^{\,\,2}_{1t} d\vec q_{2t}^{\,\,2} \gamma_p(q_{1t}^2,x_1)\gamma_Z(q_{2t}^2,x_2)\sigma_{\gamma\gamma\to S}(s_{\gamma\gamma}).$$ Here $$\label{eq:sigmaphiphi} \sigma_{\gamma\gamma\to S}(s_{\gamma\gamma}) = \frac{\pi}{ m_S^2} \frac{|F_\gamma(m_S)|^{2} \alpha_{\text{EM}}^2 \theta^2 m_S^4}{256 \pi^2 v^2}\,\delta(s_{\gamma\gamma} - m_S^2) \equiv\frac{1}{x_{1}}\Sigma_{\gamma\gamma} \frac{\delta\left(x_2 - \frac{m_S^2}{x_1 s_{p}Z}\right)}{x_{1}s_{pY}},$$ where $s_{\gamma\gamma}=(q_1+q_2)^2 \approx 4x_1x_2E_{p}^{\text{CM}}E_{Y}^{\text{CM}} \approx x_{1}x_{2}s_{pY}$, and $$\Sigma_{\gamma\gamma} = \theta^{2}\frac{|F_{\gamma}|^{2}\alpha_{\text{EM}}^{2}m_{S}^{2}}{256 \pi v^{2}s_{pZ}}.$$ Let us discuss the boundaries of integration in Eq. . Following [@Dobrich:2015jyk], for the upper limit of $q$ we choose $q_{\text{max}} = 1\text{ GeV}$ for the maximal virtuality of a photon emitter by the proton and $q_{\text{max}} = 4.49/R_{1}$ for a photon emitted by the nucleus. Using , we get $x_{p,max}\approx 0.63$, $x_{Z,max}=0.018$. The lower bound on $q$, it is given by the kinematic threshold for the $S$ particle production. For the nucleus, there is additional constraint $q^{2} \gtrsim r_{\text{s}}^{-2}$, where $r_{s} \simeq 10\text{ keV}$ is the inverse radius of the electron shell (at larger scales the nucleus is screened by electrons). Substituting the photon fusion cross section  into , for the $pZ$ cross section we get $$\sigma_{pZ\rightarrow pZS}=Z^{2}\alpha_{\text{EM}}^{2}\Sigma_{\gamma\gamma} \times W_{\text{coh}}, \label{eq:coherent-cross-section}$$ where $$W_{\gamma\text{ fusion}} = \frac{(2\pi)^2}{Z^{2}\alpha_{\text{EM}}^{2}}\int\limits_0^{q_{1t,\text{max}}} q_{1t}\, dq_{1t} \int\limits_0^{q_{2t,\text{max}}} q_{2t}\, dq_{2t}\int\limits_{\frac{m_S^2}{x_{2,max} s_{pZ} }}^{x_{1,max}} \frac{dx_1}{x_1}\, \gamma_p(x_1,q_{1t})\gamma_{Z}\left(\frac{m_S^2}{x_{1}s_{pZ}},q_{2t}\right) \label{eq:coherent-weight}$$ is the integrated form-factor. Here we simplified the integration domain for $p_{t}$ assuming $q_{1t,\text{max}},q_{2t,\text{max}} = 1\text{ GeV}$, since the integrand is nonzero only in some region of parameters within the integration area, and therefore by increasing of integration limits we will not affect the result. The production probability is calculated using the cross section  as $$P_{\gamma\text{ fusion}} = \frac{\sigma_{pZ\to pZS}}{\sigma_{pZ}},$$ where $\sigma_{pZ} \approx 53\ A^{0.77}\text{ mb}$ is the total $pZ$ cross section, with $A$ being the mass number of the nucleus target [@Carvalho:2003pza]. The dependence of $P_{\gamma \text{ fusion}}$ on the scalar mass and collision energy is shown in Fig. \[fig:gamma-fusion-probability\] ![The production probabilities of the scalar in photon fusion process versus the scalar mass. We consider $\text{Mo}$ nucleus ($Z = 42$, $A = 96$).[]{data-label="fig:gamma-fusion-probability"}](gamma-fus-probability.pdf){width="70.00000%"} Form-factors for the flavor changing neutral current meson decays {#sec:hadronic-form-factors} ================================================================= Consider matrix elements $$M_{XX'}^{P=P'} =\langle X'(p_{X'})|\bar{Q}_{j}Q_{i}|X(p_{X})\rangle, \quad M_{XX'}^{P \neq P'}= \langle X'(p_{X'})|\bar{Q}_{j}\gamma_{5}Q_{i}|X(p_{X})\rangle \label{eq:meson-decay-scalar-matrix-element}$$ describing transitions of mesons $X(Q_{i}) \to X'(Q_{j})$ in the case of the same and opposite parities $P$, $P'$ correspondingly. These matrix elements can be related to the matrix elements $$M^{\mu}_{XX'} \equiv \langle X'(p_{X'})|\bar{Q}_{i}\gamma^{\mu}Q_{j}|X(p_{X})\rangle, \quad M^{\mu 5}_{XX'} \equiv \langle X'(p_{X'})|\bar{Q}_{i}\gamma^{\mu}\gamma_{5}Q_{j}|X(p_{X})\rangle \label{eq:meson-decay-vector-matrix-element}$$ describing the weak charged current mediating mesons transition $X \to X'$. To derive the relation, we follow [@Bobeth:2001sq] in which a relation for pseudoscalar transition $X'$ was obtained. We generalize this approach to the arbitrary final-state meson. We first notice that $$\begin{aligned} M_{XX'}^{P = P'} \equiv \frac{1}{m_{Q_{i}} - M_{Q_{j}}}\langle X'(p_{X'})|\bar{Q}_{j}\slashed{p}_{Q}Q_{i}|X(p_{X})\rangle, \\ M_{XX'}^{P \neq P'} \equiv \frac{1}{m_{Q_{i}} + m_{Q_{j}}}\langle X'(p_{X'})|\bar{Q}_{j}\gamma_{5}\slashed{p}_{Q}Q_{i}|X(p_{X})\rangle, \label{eq:meson-decay-scalar-pseudoscalar-matrix-element}\end{aligned}$$ where $p_{Q}^{\mu} \equiv p^{\mu}_{Q_{i}} - p^{\mu}_{Q_{j}}$ and we used the Dirac equation for free quarks. Using then the identity $$\bar{Q}_{j}\slashed{p}_{Q}Q_{i} \equiv \hat{P}_{\mu}\bar{Q}_{j}\gamma^{\mu}Q_{i} \equiv [\hat{P}_{\mu},\bar{Q}_{j}\gamma^{\mu}Q_{i}],$$ where $\hat{P}_{\mu} \equiv i\partial_{\mu}$ is the momentum operator, we find $$\begin{gathered} M_{XX'}^{P = P'} = \frac{1}{m_{Q_{i}} - m_{Q_{j}}}\langle X'(p_{X'})|[\hat{P}_{\mu},\ \bar{Q}_{j}\gamma^{\mu}Q_{i}]|X(p_{X})\rangle = \\ = -\frac{1}{m_{Q_{i}} - m_{Q_{j}}}(p_{X}-p_{X'})_{\mu}\langle X'(p_{X'})| \bar{Q}_{j}\gamma^{\mu}Q_{i}|X(p_{X})\rangle \equiv -\frac{1}{m_{Q_{i}} - m_{Q_{j}}}q^{\mu}M_{\mu}, \label{eq:trick}\end{gathered}$$ where $q_{\mu} \equiv p_{X'\mu} - p_{X\mu}$; for deriving the expression we have acted by $\hat{P}_{\mu}$ on the meson states $|X\rangle, |X'\rangle$. Similarly, for $P\neq P'$ we find $$M_{XX'}^{P \neq P'} = -\frac{1}{m_{Q_{i}} + m_{Q_{j}}}q^{\mu}M_{XX\mu}^{5}$$ Further we will assume that $X$ is a pseudoscalar, and therefore transitions in pseudoscalar, pseudovector mesons $X'$ are parity even, while transitions in scalar, vector and tensor mesons are parity odd. Scalar and pseudoscalar final meson state ----------------------------------------- ### Pseudoscalar {#app:pseudoscalar} In the case of the pseudoscalar meson, $X' = P$, we have [@Ebert:1997mg] $$\begin{gathered} M^{\mu}_{XP} = \langle P(p_{P})|\bar{Q}_{i}\gamma^{\mu}Q_{j}|X(p_{X})\rangle = \\ =\left[(p_{X}+p_{P})^{\mu} - \frac{m_{X}^{2} - m_{P}^{2}}{q^{2}}q^{\mu}\right]f^{XP}_{1}(q^{2})+\frac{m_{X}^{2} - m_{P}^{2}}{q^{2}}q^{\mu}f^{XP}_{0}(q^{2}),\end{gathered}$$ where $q = p_{X} - p_{P}$. Contracting it with $q_{\mu}$, we obtain $$q_{\mu}M^{\mu}_{XP} = (m_{X}^{2} - m_{P}^{2})f^{XP}_{0}(q^{2})$$ Therefore $$M_{XP} = \frac{m_{X}^{2} - m_{P}^{2}}{m_{Q_{j}} - m_{Q_{i}}}f^{XP}_{0}(q^{2})$$ We take the expression for the form-factor $f^{XP}_{0}(q^{2})$ from [@Ball:2004ye]: $$\label{eq:pseudoscalar-form-factor} f^{XP}_{0}(q^{2})=\frac{F_{0}^{XP}}{1-q^{2}/(m^{X}_{\text{fit}})^{2}}$$ The values of the parameters $m_{\text{fit}}^{X}$, $F_{0}^{XP}$ for different $X,P$ are summarized in Table \[tab:pseudoscalar-form-factor-parameters\]. $X, P$ $B^{+/0}, K^{+/0}$ $B^{+/0},\pi^{+/0}$ $K,\pi$ ---------------------------------- -------------------- --------------------- ---------- -- $m_{\text{fit}}^{X},\text{ GeV}$ $6.16$ $6.16$ $\infty$ $F_{0}^{XP}$ $0.33\pm 0.04$ $0.258\pm 0.031$ $0.96$ : Values of the parameters in the form-factor  for different $X,P$. We use [@Ball:2004ye; @Marciano:1996wy].[]{data-label="tab:pseudoscalar-form-factor-parameters"} ### Scalar {#app:scalar} For the scalar meson $X' = \tilde{S}$ we have [@Sun:2010nv] $$M^{\mu}_{X\tilde{S}} = -i\left[(p_{X}+p_{\tilde{S}})^{\mu} -q^{\mu}\right]f^{X\tilde{S}}_{+}(q^{2})$$ (here we used $f_{+}(q^{2}) = -f_{-}(q^{2})$ in Eq. (6) of [@Sun:2010nv]). Similarly to the case $h' = P$, $$M_{X\tilde{S}} = i\frac{m_{X}^{2} - m_{\tilde{S}}^{2} - q^2}{m_{Q_{j}} + m_{Q_{i}}}f^{X\tilde{S}}_{+}(q^{2}).$$ Consider the transition $B\to K_{0}^{*}S$. There is an open question whether hypothetical $K_{0}^{*}(700)$ is a state formed by two or four quarks, see, e.g. [@Daldrop:2012sr], discussions in [@Sun:2010nv; @Cheng:2013fba] and references therein. We assume that $K_{0}^{*}(700)$ is a di-quark state and $K_{0}^{*}(1430)$ is its excited state. There are no experimentally observed decays $B \to K_{0}^{*}(700)X$, and therefore there is quite large theoretical uncertainty in determination of the form-factors (see a discussion in [@Issadykov:2015iba]). We will use [@Sun:2010nv], where there are results for $B\to K_{0}^{*}(700)$ and $B\to K_{0}^{*}(1430)$, and the results for the latter are in good agreement with the experimental data for $B \to K_{0}^{*}(1430)\eta'$ decay. We fit the $q^{2}$ dependence of $f^{BK_{0}^{*}}_{+}$ from [@Sun:2010nv] by the standard pole-like function that is used in the literature discussing the $B \to K^{*}_{0}$ transitions (see, e.g., [@Cheng:2013fba]): $$f^{BK_{0}^{*}}_{+}(q^{2}) = \frac{F^{BK_{0}^{*}}_{0}}{1-a\frac{q^{2}}{m_{B}^{2}}+b\left(\frac{q^{2}}{m_{B}^{2}}\right)^{2}}, \label{eq:scalar-form-factor}$$ where $m_{B} = 5.3\text{ GeV}$ is the mass of the $B^{+}$ meson. The fit parameters are given in Table \[tab:scalar-form-factor-parameters\]. $\tilde{S}$ $F_{0}^{B\tilde{S}}$ $a$ $b$ ------------------- ---------------------- ------- -------- -- $K_{0}^{*}(700)$ $0.46$ $1.6$ $1.35$ $K_{0}^{*}(1430)$ $0.17$ $4.4$ $6.4$ : Values of the parameters in the form-factor  for $B = B^{+}$, $\tilde{S} = K_{0}^{*0}(700)$, $K_{0}^{*}(1430)$. We used [@Sun:2010nv].[]{data-label="tab:scalar-form-factor-parameters"} Vector and pseudovector final meson state ----------------------------------------- ### Vector {#app:vector} For the vector final state, $X' = V$, we have [@Ebert:1997mg; @Ball:2004rg] $$\begin{gathered} \langle V(p_{V})| \bar{Q}_{i}\gamma^{\mu}\gamma_{5}Q_{j} |X(p_{X})\rangle = (m_{X}+m_{V})\epsilon^{\mu *}(p_{V})A_{1}(q^{2}) - \\ -(\epsilon^{*}(p_{V})\cdot q)(p_{X} + p_{V})^{\mu}\frac{A_{2}(q^{2})}{m_{X}+m_{V}}-2m_{V}\frac{\epsilon^{*}(p_{V})\cdot q}{q^{2}}q^{\mu}(A_{3}(q^{2})-A_{0}(q^{2})), \label{eq:vector-form-factor-axial-part}\end{gathered}$$ $$\langle V(p_{V})| \bar{Q}_{i}\gamma^{\mu}Q_{j} |X(p_{X})\rangle= \frac{2V(q^{2})}{m_{X}+m_{V}}i\epsilon^{\mu\nu\rho\sigma}\epsilon^{*}_{\nu}(p_{V})p_{X, \rho}p_{V, \sigma}, \label{eq:vector-form-factor-vector-part}$$ where $\epsilon_{\mu}(p_{V})$ is the polarization vector of the vector meson, and $A_{i}, V$ are the form-factors. The form-factor $A_{3}$ is related to $A_{1}$ and $A_{2}$ as $$A_{3}(q^{2}) = \frac{m_{X}+m_{V}}{2m_{V}}A_{1}(q^{2})-\frac{m_{X} - m_{V}}{2m_{V}}A_{2}(q^{2}) \label{eq:a3-a0}$$ Contracting  and  with $q_{\mu}$, we obtain that the vector part of the matrix element vanishes, while for the axial-vector part we find $$M_{XV} = \langle V(p_{V})| \bar{Q}_{i}\gamma_{5}Q_{j} |X(p_{X})\rangle = -\frac{(\epsilon^{*}(p_{V}) \cdot p_{X})}{m_{Q_{i}}+m_{Q_{j}}}2m_{V}A^{XV}_{0}(q^{2}),$$ where we used the relation . Consider a scalar product $(\epsilon^{*}(p_{V}) \cdot p_{X})$ in the rest frame of the meson $X$. In this case only longitudinal polarization of $\epsilon^{*}_{\mu}(p_{V})$ contributes. Using $\epsilon_{\mu}^{L,*}(p_{V}) = \left(\frac{|\bm{p}_{V}|}{m_{V}},\frac{\bm{p}_{V}}{|\bm{p}_{V}|}\frac{E_{V}}{m_{V}}\right)$ we obtain $$M_{XV} = -\frac{2m_{X}|\bm{p}_{V}|}{m_{Q_{i}}+m_{Q_{j}}}A_{0}(q^{2})$$ For the case $B\to K^{*}(892)$, we follow [@Ball:2004rg] and parametrize the form-factor as $$A^{BK^{*}(892)}_{0}(q^{2}) = \frac{r_{1}}{1-q^{2}/m_{R}^{2}}+\frac{r_{2}}{1-q^{2}/(m_{\text{fit}}^{A_{0}})^{2}}. \label{eq:vector-form-factor-1}$$ The values of parameters are given in Table \[tab:vector-form-factor-parameters1\]. For the case $B\to V = K^{*}(1410), K^{*}(1680)$, we use an expression for the form-factors [@Lu:2011jm; @Hatanaka:2009gb]: $$A^{BV}_{0}(q^{2}) = \left( 1-\frac{2m_{V}^{2}}{m_B^2 + m_V^2 - q^2}\right)\xi_{||}(q^{2})+\frac{m_{V}}{m_{B}}\xi_{\perp}(q^{2}), \label{eq:vector-form-factor-2}$$ where $$\xi_{\perp/||}(q^{2}) = \frac{\xi_{\perp/||}(0)}{1-q^{2}/m_{B}^{2}}$$ The values of the parameters are given in Table \[tab:vector-form-factor-parameters2\]. $V$ $r_{1}$ $r_{2}$ $m_{R}$, GeV $m_{\text{fit}}, \text{ GeV}$ $A^{BV}_{0}(0)$ -------------- --------- --------- -------------- ------------------------------- --------------------------- $K^{*}(892)$ $1.364$ $-0.99$ $m_{B}^{+}$ $\sqrt{36.8}$ $0.374^{+0.033}_{-0.033}$ : Values of the parameters in the vector form-factor  from [@Ball:2004rg].[]{data-label="tab:vector-form-factor-parameters1"} $V$ $\xi_{\perp}(0)$ $\xi_{||}(0)$ $A^{BV}_{0}(0)$ --------------- ------------------------ ------------------------ ------------------------- $K^{*}(1410)$ $0.28^{+0.04}_{-0.04}$ $0.22^{+0.03}_{-0.03}$ $0.3^{+0.036}_{-0.036}$ $K^{*}(1680)$ $0.24^{+0.05}_{-0.05}$ $0.18^{+0.03}_{-0.03}$ $0.22^{+0.04}_{-0.04}$ : Values of the parameters in the vector form-factors  from [@Lu:2011jm; @Hatanaka:2009gb].[]{data-label="tab:vector-form-factor-parameters2"} ### Pseudo-vector {#app:pseudovector} For the pseudo-vector mesons, $X' = A$, the expansion of the matrix elements is similar to , , but the expressions for the vector and axial-vector matrix elements are interchanged [@Bashiry:2009wq; @Hatanaka:2008gu], $$\begin{gathered} \langle A(p_{A})| \bar{Q}_{i}\gamma^{\mu}Q_{j} |X(p_{X})\rangle = (m_{X}+m_{A})\epsilon^{\mu *}(p_{A})V_{1}(q^{2}) - \\ -(\epsilon^{*}(p_{A})\cdot q)(p_{X} + p_{A})^{\mu}\frac{V_{2}(q^{2})}{m_{X}+m_{A}}-2m_{A}\frac{\epsilon^{*}(p_{A})\cdot q}{q^{2}}q^{\mu}(V_{3}(q^{2})-V_{0}(q^{2})), \label{eq:pseudovector-form-factor-axial-part}\end{gathered}$$ $$\langle A(p_{A})| \bar{Q}_{i}\gamma^{\mu}\gamma_{5}Q_{j} |X(p_{X})\rangle= \frac{2A(q^{2})}{m_{X}+m_{A}}i\epsilon^{\mu\nu\rho\sigma}\epsilon^{*}_{\nu}(p_{A})p_{X, \rho}p_{A, \sigma}, \label{eq:pseudovector-form-factor-vector-part}$$ with the same relation between $V_i$ as for $A_i$ in the case of vector mesons . We therefore obtain $$M_{XA} = \frac{2m_{X}|\bm{p}_{A}|}{m_{Q_{j}}-m_{Q_{i}}}V^{XA}_{0}(q^{2}),$$ We will consider two lightest pseudo-vector resonances $K_{1}(1270), K_{1}(1400)$, each of which is the mixture of unphysical $K_{1A}$ and $K_{1B}$ states [@Bashiry:2009wq], $$\begin{pmatrix} |K_{1}(1270) \rangle \\ |K_{1}(1400) \rangle \end{pmatrix} = \begin{pmatrix} \sin(\theta_{K_{1}}) & \cos(\theta_{K_{1}}) \\ \cos(\theta_{K_{1}}) & -\sin(\theta_{K_{1}})\end{pmatrix} \begin{pmatrix} |K_{1A} \rangle \\ |K_{1B}\rangle \end{pmatrix},$$ The form-factors $V_{0}^{BK_{1}}$ can be related to the form-factors $V_{0}^{A/B}$ of the $K_{1A}, K_{1B}$ as $$\begin{aligned} V^{BK_{1}(1270)}_{0}(q^{2}) =\frac{1}{m_{K_{1}(1270)}}\left[\sin(\theta_{K_{1}})m_{K_{1A}}V_{0}^{A}(q^{2})+\cos(\theta_{K_{1}})m_{K_{1B}}V_{0}^{B}(q^{2})\right], \label{eq:pseudovector-form-factors1} \\ V^{BK_{1}(1400)}_{0}(q^{2}) =\frac{1}{m_{K_{1}(1400)}}\left[\cos(\theta_{K_{1}})m_{K_{1A}}V_{0}^{A}(q^{2})-\sin(\theta_{K_{1}})m_{K_{1B}}V_{0}^{B}(q^{2})\right], \label{eq:pseudovector-form-factors2}\end{aligned}$$ where $$V_{0}^{A/B}(q^{2}) = \frac{F^{A/B}_{0}}{1-a_{A/B}\frac{q^{2}}{m_{B}^{2}}+b_{A/B}\left(\frac{q^{2}}{m_{B}^{2}}\right)^{2}}. \label{eq:pseudovector-form-factors-unphys}$$ The values of all relevant parameters are given in Tables \[tab:pseudovector-form-factors-parameters1\], \[tab:pseudovector-form-factors-parameters2\]. $F_{0}^{A}$ $F_{0}^{B}$ $a_{A}$ $a_{B}$ $b_{A}$ $b_{B}$ ------------------------ ------------------------- --------- --------- --------- --------- $0.22^{+0.04}_{-0.04}$ $-0.45^{+0.12}_{-0.08}$ $2.4$ $1.34$ $1.78$ $0.69$ : Values of the parameters in the vector form-factors  from [@Bashiry:2009wq].[]{data-label="tab:pseudovector-form-factors-parameters1"} $\theta_{K_1}$ $m_{K_{1A}}$ $m_{K_{1B}}$ $V_{0}^{BK_{1}(1270)}(0)$ $V_{0}^{BK_{1}(1400)}(0)$ ----------------------------- -------------- -------------- --------------------------- --------------------------- $-34^{\circ}\pm 13^{\circ}$ 1.31 1.34 $-0.52^{+0.13}_{-0.09}$ $-0.07^{+0.033}_{-0.012}$ : Values of the parameters in the vector form-factors , from [@Bashiry:2009wq].[]{data-label="tab:pseudovector-form-factors-parameters2"} Tensor final meson state {#app:tensor} ------------------------ For the tensor meson, $X' = T$, the expansion of the matrix element is [@Li:2010ra; @Cheng:2010yd] $$\begin{gathered} \langle T(p_{T})| \bar{Q}_{i}\gamma^{\mu}\gamma_{5}Q_{j} |X(p_{X})\rangle = (m_{X}+m_{T})\epsilon^{\mu *,s}_{T}(p_{T})A_{1}(q^{2}) - \\ -(\epsilon^{*,s}_{T}(p_{T})\cdot q)(p_{X} + p_{T})^{\mu}\frac{A_{2}(q^{2})}{m_{X}+m_{T}}-2m_{T}\frac{\epsilon^{*,s}_{T}(p_{T})\cdot q}{q^{2}}q^{\mu}(A_{3}(q^{2})-A_{0}(q^{2})) \label{eq:tensor-form-factor-axial-part}\end{gathered}$$ Here, $\epsilon^{s}_{T\mu}(p_{T})$ is a vector defined by $$\epsilon_{T\mu}^{s}(p_{T}) \equiv \frac{1}{m_{X}}\epsilon_{\mu\nu}^{s}(p_{T})p_{X}^{\nu},$$ with $\epsilon_{\mu\nu}^{s}$ being the polarization tensor of $T$ satisfying $p_{\mu}\epsilon^{\mu\nu,s}(p) = 0$ and $\epsilon^{\mu\nu,s} = \epsilon^{\nu\mu,s}$, $\epsilon^{\mu, \ s}_{\ \mu} = 0$. For particular polarizations $s = \pm 2, \pm 1, 0$ we have [@Li:2010ra] $$\epsilon^{\pm 2}_{T\mu} = 0, \quad \epsilon_{T\mu}^{\pm 1} = \frac{1}{m_{h}\sqrt{2}}(\epsilon^{0}\cdot p_{X})\epsilon^{\pm 1}_{\mu}, \quad \epsilon^{0}_{T\mu} = \sqrt{\frac{2}{3}}\frac{\epsilon^{0}\cdot p_{X}}{m_{X}}\epsilon_{\mu}^{0},$$ where $$\epsilon^{\pm 1}_{\mu} = \frac{1}{\sqrt{2}}(0, \mp 1, i, 0), \quad \epsilon_{\mu}^{0} = \frac{1}{m_{T}}(|\bm{p}_{T}|,0,0,E_{T}).$$ Repeating the same procedure as in the previous section, we find that to $q_{\mu}M^{\mu,s}_{XT}$ contributes only the polarization $s = 0$, and therefore $$M_{XT} = -\frac{q_{\mu}M^{\mu, 0}_{XT}}{m_{Q_{i}}+m_{Q_{j}}}= -\frac{1}{m_{Q_{i}}+m_{Q_{j}}}\sqrt{\frac{2}{3}}\frac{m_{X}|\bm{p}_{T}|^{2}}{m_{T}}2 A^{XT}_{0}(q^{2}).$$ The parametrization of the form-factor $A^{XT}_{0}$ is [@Li:2010ra; @Cheng:2010yd] $$A^{XT}_{0}(q^{2}) = \frac{F^{XT}_{0}}{\left( 1-\frac{q^{2}}{m_{X}^{2}}\right)\left( 1-a_{T}\frac{q^{2}}{m_{X}^{2}}+b_{T}\left(\frac{q^{2}}{m_{X}^{2}}\right)^{2}\right)}$$ For the transition $B\to K_{2}^{*}(1430)$ we use the values $F^{BK_{2}^{*}}_{0} = 0.23$, $a_{T} = 1.23$, $b_{T} = 0.76$ from [@Cheng:2010yd]. Production from mesons through quartic coupling {#sec:quartic-coupling} =============================================== The quartic coupling $$\mathcal{L}_{\text{quartic}} = \frac{\alpha}{2}hS^{2} \label{eq:quartic-coupling}$$ generates new production channels from the mesons $$X_{Q_i} \to X'_{Q_j}SS, \quad X\to SS, \label{eq:quartic-coupling-processes}$$ that are described by Feynman diagrams in Fig. \[fig:Sproduction-quartic\] (b). The matrix element for decays $X_{Q_{i}}\to X'_{Q_{j}}SS$ can be written in terms of the matrix element $M_{XX'}$ of hadronic transitions given by Eq. : $$\mathcal{M}(X_{Q_{i}} \to X'_{Q_{j}}SS) \approx \frac{\alpha}{m_{h}^{2}}\frac{m_{Q_i}}{2v}\xi_{ij}M_{XX'}(q^{2}),$$ where $q^{2}$ is invariant mass of scalars pair, $M_{XX'}(q^{2})$ is the matrix element of hadronic transitions $X_{Q_{i}} \to X'_{Q_{j}}$ given by Eq. . The matrix element for a process $X_{Q_{i}Q_{j}} \to SS$ can be expressed in terms of the decay constant $f_{X}$ of the meson $X$. Namely, $f_{X}$ is defined by $$\langle 0|\bar{Q}_{i}\gamma_{\mu}\gamma_{5}Q_{j} |X(p)\rangle \equiv if_{X}p_{\mu}$$ Contracting it with $p_{\mu}$ and using the same trick as in Eq. , we obtain $$\langle 0|\bar{Q}_{i}\gamma_{5}Q_{j} |X(p)\rangle \equiv -\frac{if_{X}m_{X}^{2}}{m_{Q_{i}} - m_{Q_{j}}}$$ Therefore, the matrix element $\mathcal{M}(X_{Q_{i}Q_{j}} \to SS)$ is $$\mathcal{M}(X_{Q_{i}Q_{j}} \to SS)= \frac{m_{Q_{i}}\xi_{ij}}{2v m_{h}^{2}} \langle 0|\bar{Q}_{i}\gamma_{5}Q_{j} |X(p)\rangle \approx i\frac{\alpha f_{X}m_{X}^{2}}{2v m_{h}^{2}}\xi_{ij},$$ The values of $f_{X}$ are summarized in Table \[tab:meson-decay-constants\]. Meson $X$ $B_{0}$ $B_{s}$ $K_{0}$ --------------------- --------- --------- --------- -- -- $f_{X},\text{ GeV}$ $0.19$ $0.23$ $0.16$ : Values of meson decay constants. We use [@Chang:2018aut] and references therein.[]{data-label="tab:meson-decay-constants"} For the decay width of the process $X_{Q_{i}Q_{j}} \to SS$ we find $$\Gamma (X_{Q_{i}Q_{j}}\to SS) = \frac{m_{X}^3}{v^2} \frac{|\xi_{ij}|^2 f_{X}^2 \alpha^2}{128\pi m_h^4}\sqrt{1-\frac{4m_{S}^{2}}{m_{X}^{2}}} \label{eq:bstossbranching}$$ The decay width for the process $X_{Q_{i}} \to X'_{Q_{j}}SS$ can be calculated using the formulas from Appendices \[sec:inclusive-br-calculation\]. Namely, we have $$\Gamma_{X_{Q_{i}} \to X'_{Q_{j}}SS} = \frac{|\xi_{ij}|^{2}m_{Q_{i}}^{2}\alpha^{2}}{512\pi^{3}m_{X}^{3}v^{2}m_{h}^{4}}\int \limits_{4m_{S}^{2}}^{(m_{X}-m_{X'})^{2}} |M_{XX'}(q^{2})|^{2}\sqrt{(E_{2}^{*})^{2}-m_{S}^{2}}\sqrt{(E_{3}^{*})^{2}-m_{X'}^{2}}dq^{2}, \label{eq:br-quartic-3-body}$$ where $q^{2}$ is the squared invariant mass of two scalars, and $$E_{2}^{*} =\frac{\sqrt{q^{2}}}{2}, \quad E_{3}^{*} = \frac{m_{X}^{2}-q^{2}-m_{X'}^{2}}{2\sqrt{q^{2}}}$$ Decays of a scalar {#sec:decay} ================== Decay into leptons and photons {#sec:decayleptonphoton} ------------------------------ The decay width of the $S$ particle into leptons pair simply follows from the Lagrangian  and reads $$\Gamma (S \rightarrow l^+ l^-) = \frac{ \theta^2 y_{f}^{2} m_S}{8 \pi} \beta_{l}^{3}, \label{eq:Gll}$$ where $\beta_{l} = \left( 1 - \frac{4 m_l^2}{m_S^2} \right)^{1/2}$. The decay width into photons is $$\Gamma (S \rightarrow \gamma \gamma) = |F_{\gamma}(m_{S})|^2 \left( \frac{\alpha_{\text{EM}}}{8 \pi} \right)^2 \frac{\theta^2 m_S^3}{8 \pi v^2}, \label{eq:Sto2gamma}$$ Where $F_{\gamma}$ is given by Eq. . Decays into quarks and gluons {#sec:decay-perturbative} ----------------------------- The decay width into quarks in leading order in $\alpha_{s}$ can be obtained directly from the Lagrangian ; the QCD corrections were obtained in [@Spira:1997dg]. In order to take into account the quark hadronization, we follow [@Winkler:2018qyg; @Alekhin:2015byh; @Gunion:1989we] and use the mass of the lightest hadron $m_{M_{q}}$ containing quark $q$ instead of the quark mass $m_{q}$ in the kinematical factors. The result is $$\Gamma (S \rightarrow \bar{q} q) = N_c \frac{ \theta^2 m_S \overline{m}_q^2(m_S)}{8 \pi v^2} \left( 1 - \dfrac{4 \overline{m}_{M_{q}}^2}{m_S^2} \right)^{3/2} \big(1 + \Delta_{\text{QCD}} + \Delta_t\big), \label{eq:decay-quarks}$$ where $M_{q} = K$ for the $s$ quark and $D$ for $c$ quark, the factor $N_c = 3$ stays for the number of the QCD colors, $$\begin{aligned} \Delta_{\text{QCD}} &= 5.67 \frac{\alpha_s(m_S)}{\pi} + (35.94 - 1.36 N_f) \left(\frac{\alpha_s(m_S)}{\pi}\right)^2 \nonumber\\ &+ (164.14 - 25.77 N_f + 0.259 N_f^2) \left(\frac{\alpha_s(m_S)}{\pi}\right)^3, \\ \Delta_t &= \left(\frac{\alpha_s(m_S)}{\pi}\right)^2 \left( 1.57 - \frac{2}{3} \text{log}\frac{m_S^2}{m_t^2} + \frac{1}{9} \text{log}^2 \frac{\overline{m}_q^2(m_S)}{m_S^2}\right),\end{aligned}$$ and the running mass [@Spira:1997dg] $\overline{m}_q(m_S)$ is given by $$\overline{m}_q(m_S) = \overline{m}_q(Q) \frac{c(\alpha_s(m_S)/\pi)}{c(\alpha_s(Q)/\pi)},$$ with the coefficient $c$, which is equal to $$\begin{aligned} c(x) &= \left(\frac{9}{2} x\right)^{4/9} (1 + 0.895 x + 1.371 x^2 + 1.952 x^3), &\text{ for } &m_s < m_S < m_{c}, \\ c(x) &= \left(\frac{25}{6} x\right)^{12/25} (1 + 1.014 x + 1.389 x^2+ 1.091 x^3), &\text{ for } &m_{c} < m_S < m_{b},\\ c(x) &= \left(\frac{23}{6} x\right)^{12/23} (1 + 1.175 x + 1.501 x^2 + 0.1725 x^3), &\text{ for } &m_{b} < m_S < m_t.\end{aligned}$$ We use the MS-mass at $Q = 2 \text{ GeV}$ scale [@Sanfilippo:2015era]: $\overline{m}_c = 1.23 \text{ GeV}$ and $\overline{m}_s = 0.0924 \text{ GeV}$. For decays into gluons, using the effective couplings , summing over all gluon species (which gives a factor of 8) and including QCD corrections, we obtain [@Spira:1997dg] $$\Gamma (S \rightarrow GG) = |F_{G}(m_{S})|^2 \left( \frac{\alpha_s}{4 \pi} \right)^2 \frac{\theta^2 m_S^3}{8 \pi v^2} \left( 1 + \frac{m_t^2}{8 v^2 \pi^2} \right), \label{eq:decay-gluons}$$ Where $F_{G}$ is given by Eq. . [^1]: In this estimate we neglect possible effects of QCD scalar resonances that could significantly enhance scalar production for some scalar masses. [^2]: We used the Higgs boson decay width $\Gamma_{h,\text{SM}} = 4\text{ MeV}$. [^3]: Indeed, if $X$ is the lightest hadron in the family, it could decay *only* through weak interaction, so it has small decay width $\Gamma_X$ (in comparison to hadrons that could decay through electromagnetic or strong interactions). The probability of light scalar production from hadron is inversely proportional to hadron decay width thus the light scalar production from the lightest hadrons is the most efficient. [^4]: Indeed, the quark cross sections are proportional to the Yukawa constant squared $y_{q}^{2}$, and the large ratio $(y_{s}/y_{u,d})^{2}$ compensates smaller partonic weight $W_{ss}/W_{u,d}$. [^5]: Here we neglected the $p_{T}$ dependence in $\sigma_{pp}$.

--- abstract: 'We consider a real, massive scalar field in BTZ spacetime, a 2+1-dimensional black hole solution of the Einstein’s field equations with a negative cosmological constant. First, we analyze the space of classical solutions in a mode decomposition and we characterize the collection of all admissible boundary conditions of Robin type which can be imposed at infinity. Secondly, we investigate whether, for a given boundary condition, there exists a ground state by constructing explicitly its two-point function. We demonstrate that for a subclass of the boundary conditions it is possible to construct a ground state that locally satisfies the Hadamard property. In all other cases, we show that bound state mode solutions exist and, therefore, such construction is not possible.' author: - Francesco Bussola - Claudio Dappiaggi - 'Hugo R. C. Ferreira' - Igor Khavkine title: | Ground state for a massive scalar field in BTZ spacetime\ with Robin boundary conditions --- Introduction ============ Quantum field theory on curved backgrounds is a well-established branch of theoretical and mathematical physics which allows to study matter systems in the presence of a non vanishing gravitational field (for a recent review see Ref. [@Benini:2013fia]). In this framework it is always assumed both that no proper quantum gravitational effect has to be accounted for and that the backreaction in the Einstein’s equations is negligible. Although this entails that the geometry of the spacetime is fixed, it is not at all necessary to consider metrics which are small perturbations over a flat background. Actually quantum field theory in the presence of a strong gravitational field, [*e.g.*]{}, a black hole, is of great interest since one can unveil some novel phenomena, the most famous example being Hawking radiation [@Hawking:1974sw], which have no counterpart on Minkowski spacetime. For this reason a lot of attention has always been given to the investigation and to the formulation of quantized free field theories on black hole spacetimes. Especially under the additional assumption of spherical symmetry, many results have been obtained leading to an almost complete understanding of these matter systems both at the structural and at the physical level [@Christensen:1976vb; @Candelas:1980zt; @Howard:1984qp; @Fredenhagen:1989kr; @Anderson:1990jh; @Anderson:1993if; @Anderson:1993if; @Anderson:1994hg]. Much more complicated is the scenario when the underlying black hole solution of the Einstein’s equations is rotating, hence only axisymmetric, the most notable example being Kerr spacetime. In this case even the analysis of free quantum theories is more elusive and simple questions like the construction of a ground state in the region outside the event horizon are difficult to answer [@Kay:1988mu]. As a byproduct, the computation of renormalized physical observables has been a daunting task [@Frolov:1982pi; @Ottewill:2000qh; @Duffy:2005mz; @Ferreira:2014ina] and only very recently a promising renormalization scheme has been applied with success [@Levi:2015eea; @Levi:2016exv]. The main reason for this quandary can be ascribed to a peculiarity of such rotating geometries, namely the absence of a complete, everywhere timelike Killing field. If it existed, the latter would allow for the identification of a canonical and natural choice for the notion of positive frequency, which can be used in turn to select a distinguished two-point function for the underlying theory. This defines uniquely and unambiguously a full-fledged quantum state, dubbed the [*ground state*]{}, with notable physical and structural properties [@Sahlmann:2000fh]. Among them we recall in particular that all quantum observables have finite fluctuations and that, starting from such a state, it is possible to construct the algebra of all Wick polynomials, including relevant objects such as the stress-energy tensor [@Khavkine:2014mta]. Therefore, on account of the lack of a global timelike Killing field, our understanding of quantum field theories in presence of rotating black holes is not as advanced as one could hope. The main goal of this paper is to discuss a concrete scenario where most of the problems mentioned above can be circumvented. We refer to the so-called BTZ black-hole [@Banados:1992wn; @Banados:1992gq], a solution of the (2+1)-dimensional Einstein’s equations with a negative cosmological constant. This geometry possesses some rather peculiar features. On the one hand, it can be obtained directly from the anti-de Sitter (AdS) metric with an appropriate identification of boundaries—see [@Banados:1992gq]—, hence locally it is a region of constant curvature. On the other hand, the BTZ solution is both stationary and axisymmetric and possesses an inner and an outer horizon, as well as two canonical Killing fields, say $\partial_t$ and $\partial_\phi$, associated to these symmetries. In addition, contrary to what happens in the Kerr spacetime, although none of these vector fields is everywhere timelike, there exists a suitable linear combination which enjoys such property everywhere in the region exterior of the black hole. This feature prompts the possibility of analyzing free field theories in the BTZ background constructing an associated ground state. In this paper we will address this issue thoroughly for the case of a real, massive Klein-Gordon field obeying Robin boundary conditions at conformal infinity. In this respect, we generalize and complement the results of Ref. [@Lifschytz:1993eb], which considers a massless, conformally coupled scalar field with either Dirichlet or Neumann boundary conditions at infinity. Our analysis starts from the construction, via a mode expansion, of the space of solutions for the underlying equation of motion. This must be approached delicately, since the underlying spacetime shares both locally and asymptotically the geometry of anti-de Sitter spacetime. In particular, this entails that a BTZ black hole spacetime is not globally hyperbolic, which is tantamount to saying that an acausal spacelike surface can be at most partially Cauchy (at least some complete timelike curves will never intersect it) and that the solutions of the equations of motion for a free field theory cannot be obtained only by imposing suitable initial data on such a partial Cauchy surface. As a matter of fact, the existence of a timelike conformal boundary at infinity requires additional boundary conditions thereon. Here, we follow the same path taken in the analysis of a pure AdS spacetime, recently investigated in [@Dappiaggi:2016fwc; @Dappiaggi:2017wvj]. This was based on a careful use of the Sturm-Liouville theory for ordinary differential equations (ODEs) [@Zettl:2005; @weidmann] which complements the earlier analyses in [@Ishibashi:2003jd; @Ishibashi:2004wx; @Seggev:2003rp]. By using and extending similar methods, we will show that also in the BTZ spacetime there exists a one parameter family of admissible boundary conditions, of Robin type, depending on the value of the effective squared mass in the Klein-Gordon equation. From a physical point of view, this guarantees vanishing energy flux through conformal infinity [@Ferreira:2017]. These conditions are different from the “transparent boundary conditions” used in Ref. [@Steif:1993zv] to compute the renormalized stress-energy tensor for a massless, conformally coupled scalar field in the BTZ black hole. A thorough discussion of this feature is of paramount importance for the core goal of this work: the construction of the two-point function of a ground state. In fact, each different boundary condition identifies, for all practical purposes, a separate dynamical theory. To each of these, one can compute a distinguished two-point function associated to the ground state defined with respect to the timelike Killing field which exists in the region outside the outer horizon. In the main body of the paper, we not only construct such two-point functions explicitly, but we also investigate their physical properties. Most notably, we show that, for a large class of the Robin boundary conditions, including the Dirichlet one, only positive frequencies contribute to the mode expansion of the two-point function. Hence, for each of these admissible boundary conditions, we identify a full-fledged ground state, which moreover is locally of Hadamard form on account of some structural results of quantum field theory on curved backgrounds proven in [@Sahlmann:2000fh]. By saying *locally*, we distinguish from the *global* feature observed in [@Dappiaggi:2016fwc; @Dappiaggi:2017wvj] for a free quantum field theory in the Poincaré patch of anti-de Sitter spacetime: on account of the presence of the boundary and independently of the chosen boundary condition, the two-point function is singular not only at those pairs of points connected by a null geodesic, but also at those which can be reached after such a geodesic is reflected at the conformal boundary. The presence of these additional singularities cannot be inferred from the standard structural properties proven in [@Sahlmann:2000fh] and it requires a more involved mathematical analysis, which is outside the scope of the present work. In addition, we confirm the existence of a rather peculiar feature which was already observed in the analysis of a real, massive scalar field in the Poincaré patch of anti-de Sitter spacetime [@Dappiaggi:2016fwc]. There exists a class of Robin boundary conditions for which the mode expansion of the two-point function necessarily includes the contribution of bound state mode solutions. For these boundary conditions, one cannot claim that the constructed two-point function is that of a ground state and, more importantly, that it is of Hadamard form. The paper will be organized as follows. In Section \[sec:BTZ\_geometry\] we review the geometry of a BTZ black hole, emphasizing in particular the presence of an everywhere timelike Killing field in the exterior region of the black hole. In Section \[sec:KGeq\], we analyze the massive Klein-Gordon equation with an arbitrary coupling to scalar curvature on this background. Via a Fourier expansion, the field equation is reduced to an ODE in the radial direction, which can be solved explicitly. The solutions are classified in terms of their square integrability near the horizon and the conformal infinity, which gives us the range of the effective squared mass of the scalar field for which Robin boundary conditions have to be imposed at conformal infinity. Finally, in Section \[Sec:ground\_state\], we obtain our main result, namely the explicit construction for the two-point function of the ground state for a large class of Robin boundary conditions. Those not in this set are shown not to possess a ground state, given the presence of bound state mode solutions. In Section \[sec:conclusions\], we draw our conclusions. In Appendix \[apx:deltaexpansion\] we discuss how to handle a key technical problem in our construction of the two-point function: contrary to what happens when dealing with a scalar field in a static spacetime, the ODE obtained out of the Fourier analysis cannot be interpreted as a simple eigenvalue problem with $\omega^2$ as the spectral parameter, where $\omega$ is the frequency. In fact, the ensuing equation, having also a linear dependence in $\omega$, can be read as a so-called quadratic operator pencil. In Appendix \[apx:calculation-delta-expansion\] we present all the steps of the calculation of the two-point function for the ground state, whose results are presented in Section \[Sec:ground\_state\]. We leave some of the most mathematical details for Appendices \[apx:S1\], \[apx:S2\] and \[apx:S3\]. Throughout the paper we employ natural units in which $c = G_{\rm N} = \hbar = 1$ and a metric with signature $({-}{+}{+})$. BTZ black hole and 2+1 geometry {#sec:BTZ_geometry} =============================== ![\[fig:CPdiagrams\] Penrose diagrams of the BTZ black hole for the rotating $0<r_-<r_+$ (left) and the static $0=r_-<r_+$ (right) cases.](Figs/BTZ-penrose-diagrams) The BTZ black hole is a stationary, axisymmetric, (2+1) dimensional solution of the vacuum Einstein field equations with a negative cosmological constant $\Lambda=-1/\ell^2$ [@Banados:1992gq; @Banados:1992wn]. It is diffeomorphic as a manifold to $M\equiv\bR\times I\times\mathbb{S}^1$, where $I$ is an open interval of the real line. Its metric $g$ can be realized in several, different, albeit equivalent ways, [*e.g*]{} by a suitable identification of points in the Poincaré patch of the three-dimensional AdS spacetime [@Banados:1992gq]. The ensuing line element reads $$\label{metric} \dd s^2 = -N^2 \dd t^2 + N^{-2} \dd r^2 + r^2 \big(\dd\phi+ N^{\phi} \dd t \big)^2 \ ,$$ where $t\in\mathbb{R}$, $\phi\in(0,2\pi)$, $r\in(r_+,\infty)$, while $$\label{eq:metric_functions} N^2 = -M+\frac{r^2}{\ell^2}+\frac{J^2}{4r^2} \ , \qquad N^\phi = -\frac{J}{2r^2} \ ,$$ $M$ being interpreted as the mass of the black hole and $J$ as its angular momentum. The value of $r_+$ can be inferred, observing that, in the range $M>0$, $|J|\le M\ell$, $N$ vanishes at $$r^2_{\pm}= \frac{\ell^2}{2} \left( M \pm \sqrt{M^2-\frac{J^2}{\ell^2}} \right) \ .$$ These *loci* are coordinate singularities and, thus, as customary in rotating black hole spacetimes, the BTZ solution possesses an inner $(r=r_-)$ and an outer horizon $(r=r_+)$. The Penrose diagrams of this spacetime are shown in Fig. \[fig:CPdiagrams\]. In addition, the event horizon turns out to be a Killing horizon whose generator reads $$\label{eq:Killing_Field} \chi \doteq \partial_t +\Omega_{\mathcal{H}} \partial_\phi \ ,$$ where $\Omega_{\mathcal{H}} \doteq N^\phi(r_{+}) = \frac{r_{-}}{\ell r_{+}}$ is the angular velocity of the horizon. It is of paramount relevance for this paper that $\chi$ is a well-defined, [*global, timelike Killing vector field*]{} across the whole exterior region $(r>r_+)$ of BTZ spacetime. This is the sharpest difference in comparison to other models of rotating black hole spacetimes, [*e.g.*]{}, the Kerr solution of the Einstein’s equation with vanishing cosmological constant. In these cases one is forced to cope with the existence of a speed of light surface at which the analogue Killing field is null. In view of the distinguished role of $\chi$, it is natural to introduce the new coordinate system $(\tilde{t},r,\tilde{\phi})$, which is related to $(t,r,\phi)$ in such a way that $\partial_{\tilde{t}}=\chi$. The simplest choice consists of defining $t=\widetilde{t}$ and $\phi=\tilde{\phi}+\Omega_{\mathcal{H}}\tilde{t}$; the line element becomes $$\label{eq:new_line_element} \dd s^2 = -N^2 \dd\tilde{t}^2 + N^{-2} \dd r^2 + \left(\dd\tilde{\phi}+(N^\phi+\Omega_{\mathcal{H}})\dd\tilde{t}\right)^2 \, .$$ Observe that, while the range of $\tilde{t}$ is still $\bR$, that of $\tilde{\phi}$ is no longer simply the interval $(0,2\pi)$, rather $(-\Omega_\H \tilde{t}, 2\pi -\Omega_\H \tilde{t})$, with the end points still identified. Especially in the next section, we will be working mainly with , although, when we will be addressing the construction of a ground state, will turn out to be extremely useful. Massive scalar field in BTZ {#sec:KGeq} =========================== Klein-Gordon equation --------------------- We consider a real, massive scalar field $\Phi: M\to\bR$ satisfying the Klein-Gordon equation, $$\label{KG} P\Phi = (\Box_g - m^2-\xi R)\Phi =0\ ,$$ where $\Box_g$ and $R$ are respectively the D’Alembert wave operator and the scalar curvature built out of , $\xi\in\bR$ while $m^2$ is the mass parameter of the scalar field. Since $R=-6/\ell^2$, it is convenient to introduce the dimensionless parameter $\mu^2 \doteq m^2 \ell^2-6\xi$. In addition, we assume that $m^2$ and $\xi$ are such that the Breitenlohner-Freedman bound $\mu^2 \geqslant -1$ holds [@Breitenlohner:1982jf]. For our ultimate goal of quantizing and constructing the associated ground state(s) the first step in this direction consists of a careful study of the solutions of the Klein-Gordon equation. Since the underlying spacetime is not globally hyperbolic, these cannot be constructed only by assigning initial data, for example on a constant time-$t$ hypersurface. One needs to supplement such information with the choice of an admissible boundary condition. A priori it is not obvious how to proceed since one might wish to assign such a condition either at the horizon $r=r_+$, at infinity $r\to\infty$ or possibly at both ends. This quandary is easily solved by showing that can be reduced to a second order ODE, whose boundary conditions are much easier to analyze. To this end, we work with the coordinates $(t,r,\phi)$, so that reads $$\begin{gathered} \left[-\frac{1}{N^2}\partial^2_t+\frac{1}{r}\partial_r\left(rN^2\right)\partial_r+\left(\frac{1}{r^2}-\frac{N^\phi}{N}\right)^2\partial^2_\phi \right. \notag\\ +\left.2\frac{N^\phi}{N^2}\partial_t\partial_\phi-\frac{\mu^2}{\ell^2}\right]\Phi=0 \, . \label{eq:PDE_KG} \end{gathered}$$ Since both $\partial_t$ and $\partial_\phi$ are Killing fields of we can take a Fourier expansion of $\Phi$, $$\Phi(t,r,\phi)=\frac{1}{2\pi}\sum_{k\in\mathbb{Z}}\int_\mathbb{R} \dd\omega\; e^{-i\omega t + i k \phi} \, \Psi_{\omega k}(r) \ .$$ It is convenient to introduce a new coordinate $z\in (0,1)$, $$\label{eq:z_coordinate} z \doteq \frac{r^2-r_+^2}{r^2-r_-^2} \, ,$$ so that, starting from , $\Psi_{\omega k}(z)$ obeys $$\label{eq:Sturm_Liouville_form} L_\omega \Psi_{\omega k}(z) \doteq \frac{\dd}{\dd z}\left(z\frac{\dd\Psi_{\omega k}(z)}{\dd z}\right)+q(z)\Psi_{\omega k}(z)=0 \, ,$$ with $$\begin{gathered} q(z) = \frac{1}{4(1-z)} \left[ \frac{\ell^2(\omega\ell r_{+} - k r_{-})^2}{(r_{+}^2-r_{-}^2)^2 z} \right. \\ \left. -\frac{\ell^2(\omega\ell r_{-} - k r_{+})^2}{(r_{+}^2-r_{-}^2)^2} -\frac{\mu^2}{1-z} \right] \, , \end{gathered}$$ This is indeed the sought second order ODE, written in Sturm-Liouville form, defined on the interval $(0,1)$. We need to clarify which are the admissible boundary conditions that can be assigned at $z=0$ (the horizon) or at $z=1$ (infinity). For ordinary differential equations this problem can be solved in full generality by using Sturm-Liouville theory, see [*e.g.*]{}, [@Zettl:2005; @weidmann] or [@Dappiaggi:2016fwc] for an application to the study of a real, massive scalar field in the Poincaré patch of anti-de Sitter spacetime of arbitrary dimension. The nomenclature and the procedure that we will be using is the same employed in the last reference. For the sake of brevity, we will not recapitulate it fully here and we refer the reader to the works cited above. Solutions --------- The next step consists of identifying a basis of the vector space of solutions of . Using Froebenius method, we infer that $\Psi_{\omega k}(z)=z^\alpha(1-z)^\beta F_{\omega k}(z)$, with $$\label{eq:Frobenius_prequel} \alpha^2=-\frac{\ell^4 r_+^2 \tilde{\omega}^2}{4 (r_{+}^2 - r_{-}^2)^2} \, , \quad \beta^2+\beta-\frac{\mu^2}{4}=0 \, ,$$ where we define $\tilde{\omega} \doteq \omega-k\Omega_{\mathcal{H}}$ to be the square root of $\tilde{\omega}^2$ such that $\Im[\omega]=\Im[\tilde{\omega}] \geqslant 0$. By setting $$\label{eq:Frobenius} \alpha = -i \, \frac{\ell^2 r_+ \tilde{\omega}}{2 (r_{+}^2 - r_{-}^2)}\ , \quad \beta = \frac{1}{2}\left(1 + \sqrt{1+\mu^2}\right)$$ and plugging $\Psi_{\omega k}(z)$ in , we obtain the Gaussian hypergeometric equation, $$\label{radialeq} z(1-z)\partial^2_z F_{\omega k} + [c -(a+b+1)z]\partial_z F_{\omega k} -ab F_{\omega k} =0 ,$$ where $$\label{eq:parameters} \begin{cases} \displaystyle a= \frac{1}{2}\left(1 + \sqrt{1 + \mu^2} - i\ell \, \frac{\tilde{\omega}\ell}{r_{+}-r_{-}} + i\ell\frac{k}{r_+}\right)\ , \\ \displaystyle b= \frac{1}{2}\left(1 + \sqrt{1 + \mu^2} - i\ell \, \frac{\tilde{\omega}\ell}{r_{+} + r_{-}} - i\ell\frac{k}{r_+}\right)\ , \\ \displaystyle c= 1 - i \, \frac{\ell^2 r_+ \tilde{\omega}}{r_{+}^2 - r_{-}^2}\ . \end{cases}$$ For future convenience, we note that under the substitution $\tomega \mapsto \overline{\tomega}$, these parameters behave as $$\label{eq:parameters-cc} \begin{aligned} a &\mapsto \overline{b-c+1} , & \alpha &\mapsto -\overline{\alpha} , \\ b &\mapsto \overline{a-c+1} , & \beta &\mapsto \beta , \\ c &\mapsto \overline{2-c} . \end{aligned}$$ Generic solutions of can be written in closed form in terms of Gaussian hypergeometric functions that depend on the three parameters $a,b$ and $c$ of the equation. When choosing two linearly independent solutions, the dependence on these parameters forces us to disentangle two cases, accordingly to the values of $\mu^2$. ### General case: $\mu^2\neq (n-1)^2-1, \ n=1,2,3,\dots$ In this case, we choose as basis of solutions \[eq:z1gen\] $$\begin{aligned} \Psi_1(z) & =z^\alpha(1-z)^\beta F(a,b,a+b-c+1;1-z) \label{Psi1a} \, , \\ \Psi_2(z) & =z^\alpha(1-z)^{1-\beta}\notag \\ &\quad \times F(c-a,c-b,c-a-b+1;1-z) \label{Psi2a} \, , \end{aligned}$$ For future reference and inspired by the terminology used for Sturm-Liouville problems [@Zettl:2005], we call $\Psi_1$ the *principal solution* at $z=1$, that is, the unique solution (up to scalar multiples) such that $\lim_{z\to 1} \Psi_1(z)/\Psi(z)=0$ for every solution $\Psi$ that is not a scalar multiple of $\Psi_1$. Note that $\Psi_2$ is not defined when the third argument becomes a non positive integer [@NIST]. From , we get $\sqrt{1+\mu^2}+1\not\in\mathbb{N}\cup\{0\}$, which identifies exactly the special range of values for $\mu^2$ which has been excluded. Observe in particular that this set includes the case $\mu^2=-1$ which saturates the BF bound [@Breitenlohner:1982jf]. This is a very special case, which would require a lengthy analysis on its own. For this reason, we will not consider it further in this paper. Note that the above definitions obey $\Psi_1 \mapsto \overline{\Psi_1}$ and $\Psi_2 \mapsto \overline{\Psi_2}$ under the substitution $\tomega \mapsto \overline{\tomega}$. This can be checked using the conjugation identities , the symmetry $F(a,b,c;z) = F(b,a,c;z)$ and the second equality from (15.10.13) of [@NIST]: $$\begin{gathered} F\left(a,b, a+b-c+1;1-z\right) \\ =z^{1-c}F\left(a-c+1,b-c+1, a+b-c+1;1-z\right) . \end{gathered}$$ ### Special cases: $\mu^2 = (n-1)^2-1, \ n=2,3,\dots$ In this case, we choose the following basis of solutions for (see [@NIST §15.10.8]): \[eq:z1spec\] $$\begin{aligned} \Psi_1(z) & =z^\alpha(1-z)^\beta F(a,b,n;1-z) \label{Psi1b} \, , \\ \Psi_2(z) & =z^\alpha(1-z)^{\beta} \notag \\ &\quad \times \left[ F(a,b,n;1-z) \log(1-z) + K_n(z) \right]\label{Psi2b} \, , \end{aligned}$$ where $$\begin{aligned} K_n(z) &= -\sum_{p=1}^{n-1}\frac{(n-1)!(p-1)}{(n-p-1)!(1-a)_p(1-b)_p}(z-1)^{-p} \notag \\ &\quad +\sum_{p=0}^{\infty} \frac{(a)_p(b)_p}{(n)_p p!} f_{p,n} \, (1-z)^p \, , \label{eq:G} \end{aligned}$$ while $(a)_p = \Gamma(a+p)/\Gamma(a)$, $$f_{p,n} = \psi(a+p)+\psi(b+p) -\psi(1+p)-\psi(n+p) \, ,$$ and $\psi$ is the digamma function. Observe that, also in these special cases, $\Psi_1$ is the principal solution at $z=1$. Note that $\Psi_1 \mapsto \overline{\Psi_1}$ under the substitution $\tomega \mapsto \overline{\tomega}$, by the same argument as in the generic case. We do not need to check this property for $\Psi_2$ since, as it will be clear in next section, $\Psi_1$ is the only solution which plays a role for the admissible boundary conditions. End point classification {#sec:endpoints} ------------------------ Having specified a basis of solutions of , we can continue in our quest to identify the admissible boundary conditions at the end points 0 and 1 for . These will depend on the square integrability of the solutions near the end points, in a completely analogous way to the case of the Poincaré patch of AdS analyzed in [@Dappiaggi:2016fwc]. We start by identifying the fall-off behavior of the solutions of separately at the end points $z=0$ and at $z=1$. This allows to classify the end points in the following way: we call the end point 0 (respectively 1) *limit circle* if, for some $\tilde{\omega} \in \bC$, all solutions of are in $L^2((0,z_0);\mathcal{J}(z)\dd z)$ for some $z_0 \in (0,1)$ \[respectively $L^2((z_1,1);\mathcal{J}(z)\dd z)$ for some $z_1 \in (0,1)$\]; otherwise, we call it *limit point*. The measure $\mathcal{J}(z)\dd z$, with $$\label{eq:measure} \mathcal{J}(z) = \frac{1}{1-z}+\frac{r^2_+}{z(r^2_+-r^2_-)} \, ,$$ satisfies the relation $\dd\nu(g)=\pi_I^*(\mathcal{J}(z)\dd z) \dd \varphi$, where $\dd \nu(g) = r/N^2 \, \dd r \dd\varphi$ and $\pi_I:M\to I$ is the projection along the $z$-direction. Notice that the operator $S_\tomega\Psi(z) \doteq \frac{1}{\mathcal{J}(z)} L_\tomega\Psi(z)$, with $L_\tomega$ from , is Hermitian with respect to the measure $\mathcal{J}(z)\dd z$. A direct inspection of and as well as of and , supplemented with the asymptotic behavior of the hypergeometric function at $z=0$ and $z=1$, yields the sought result for the basis elements of the space of solutions of . For convenience we summarize the results described below in table \[tab:BCsummary\]. ### End point $z=1$ At $z=1$, since the hypergeometric function is equal to $1$ when evaluated at the origin, the behavior of and can be inferred from that of $(1-z)^\beta$ and $(1-z)^{1-\beta}$ respectively. By accounting also for the integration measure and using , it turns out that $\Psi_1$ lies in $L^2((z_1,1);\mathcal{J}(z)\dd z)$ for all values of $\mu^2>-1$ and regardless of $z_1\in(0,1)$ and of $\tilde{\omega}$. On the contrary, $\Psi_2$ lies in $L^2((z_1,1);\mathcal{J}(z)\dd z)$ if $-1<\mu^2<0$, again regardless of $z_1\in(0,1)$ and of $\tilde{\omega}$. Therefore, we say that $z=1$ is limit point if $\mu^2\geqslant 0$ while it is limit circle if $-1<\mu^2<0$. For the special cases $\mu^2=(n-1)^2-1$, $n=2,3,\ldots$, the first basis element $\Psi_1$ as in behaves exactly like . At the same time, $\Psi_2$ as in never lies in $L^2((z_1,1);\mathcal{J}(z)\dd z)$ on account of the singularities of $K_n(z)$. Hence, $z=1$ is always limit point of $\mu^2 \geqslant 0$. ### End point $z=0$ {#sec:endpoint0} In order to understand the behavior of the solutions of at $z=0$, we need to consider a different, more convenient basis, \[Psi0\] $$\begin{aligned} \Psi_3(z) &= z^\alpha(1-z)^\beta F(a,b,c;z) \, , \label{Psi10} \\ \Psi_4(z) &= z^{-\alpha}(1-z)^\beta \notag \\ &\quad \times F(a-c+1,b-c+1,2-c;z) \label{Psi20} \, . \end{aligned}$$ where $a,b,c$ are defined in . Observe that $\Psi_3$ and $\Psi_4$ form a well-defined basis of solutions for all $\mu^2 > -1$, except when $c=1$ ($\alpha = 0$), whose case is dealt with separately below. Since the hypergeometric function is equal to $1$ when evaluated at $z=0$, the leading behavior of the two solutions at the origin is regulated by $z^\alpha$ in the first case and by $z^{-\alpha}$ in the second one. It is easy to verify that $\Psi_3 \in L^2((0,z_0),\mathcal{J}(z)\dd z)$ for $\Im[\tilde{\omega}]>0$, irrespectively of $z_0\in(0,1)$, while $\Psi_4 \in L^2((0,z_0),\mathcal{J}(z)\dd z)$ if $\Im[\tilde{\omega}]<0$. For $\Im[\tilde{\omega}]=0$ none of the solutions is square integrable since a logarithmic singularity occurs. Therefore, we say that $z=0$ is limit point. If $c=1$, then $\omega=k\frac{r_-}{\ell r_+} = k \Omega_{\mathcal{H}}$ satisfies a synchronization condition with the black hole angular velocity, a case extensively studied in [@Ferreira:2017]. The solutions $\Psi_3$ and $\Psi_4$ no longer form a basis of solutions of , hence, we consider the following basis [@NIST §15.10.8]: $$\begin{gathered} (1-z)^\beta F(a,b,1;z) \, , \label{Psi01b} \\ (1-z)^\beta\left[F(a,b,1;z)\log(z)+K_1(1-z)\right] \, , \label{Psi02b} \end{gathered}$$ where $K_1$ is as in . A close inspection of these two solutions unveils that the leading behavior at $z=0$ is dominated by a constant in the first case and by $\log (z)$ in the second one. Hence, none of the solutions lies in $L^2((0,z_0),\mathcal{J}(z)\dd z)$ regardless of $z_0\in(0,1)$. This is in agreement with the previous point. Range of $\mu^2$ Range of $\tilde{\omega}$ $L^2$ at $z=0$ $L^2$ at $z=1$ --------------------- --------------------------- ---------------- ----------------------- $\Im[\tilde{\omega}] > 0$ $\Psi_3$ $\Psi_1$ and $\Psi_2$ $-1<\mu^2<0$ $\Im[\tilde{\omega}] = 0$ none $\Psi_1$ and $\Psi_2$ $\Im[\tilde{\omega}] < 0$ $\Psi_4$ $\Psi_1$ and $\Psi_2$ $\Im[\tilde{\omega}] > 0$ $\Psi_3$ $\Psi_1$ $\mu^2 \geqslant 0$ $\Im[\tilde{\omega}] = 0$ none $\Psi_1$ $\Im[\tilde{\omega}] < 0$ $\Psi_4$ $\Psi_1$ Note that for $\tomega \not\in \bR$, hence excluding the $c=1$ case, the above definitions obey $\Psi_3 \mapsto \overline{\Psi_4}$ and $\Psi_4 \mapsto \overline{\Psi_3}$ under the substitution $\tomega \mapsto \overline{\tomega}$. This can be checked using the conjugation identities  and the symmetry $F(a,b,c;z) = F(b,a,c;z)$. Robin boundary conditions ------------------------- We can address finally the question of which are the admissible boundary conditions at the two end points $z=0$ and $z=1$. Tentatively, as in the simple example of a massive scalar field in the Poincaré patch of AdS studied in [@Dappiaggi:2016fwc], we wish to impose Robin boundary conditions at $z=1$ for a range of the mass parameter $\mu^2$ of the scalar field. In fact, as pointed out in [@Ferreira:2017], imposing Robin boundary conditions is equivalent to requiring zero energy flux through the conformal boundary, a natural physical condition. To start with we focus our attention on the ODE at fixed value of $\tomega$ and $k$. Since we deal with a singular Sturm-Liouville problem, it is not possible to assign Robin boundary conditions by specifying the value of a linear combination between a solution and its derivative. This statement is supported also by the observation that at $z\to 1$ both solutions $\Psi_2(z)$ as per and per are divergent. This problem can be overcome by using Sturm-Liouville theory. While we do not wish to enter in a full explanation of the technical details, which are fully accounted for in [@Dappiaggi:2016fwc] and in [@Zettl:2005], we outline the main idea of the procedure. The rationale consists of observing that, in a so-called regular Sturm-Liouville problem, a generic Robin boundary condition can be expressed equivalently either in terms of a linear combination between a solution and its derivative or in terms of a linear combination between the Wronskians of such solution with respect to two linearly independent solutions, one of which is chosen to be the principal solution. In the case at hand, this translates to the following: we say that a solution $\Psi_{\zeta}$ of satisfies a [*Robin boundary condition*]{} at $z=1$ parametrized by $\zeta \in [0,\pi)$ if $$\label{eq:RBC} \lim_{z \to 1} \left\{ \cos(\zeta) \mathcal{W}_z[\Psi_{\zeta}, \Psi_1] + \sin(\zeta) \mathcal{W}_z[\Psi_{\zeta}, \Psi_2] \right\} = 0 \, ,$$ where $\Psi_1$ is the principal solution at $z=1$ \[ or \], $\Psi_2$ is a second linearly independent solution \[for instance, or \] and both are square integrable in a neighborhood of $z=1$. Here, $\mathcal{W}_z[u,v] \doteq u(z)v'(z) - v(z)u'(z)$ is the Wronskian computed with respect to two differentiable functions $u$ and $v$. As a consequence, the solution $\Psi_{\zeta}$ may be written as $$\label{eq:Robin_solution} \Psi_{\zeta}(z) = \cos(\zeta)\Psi_1(z)+\sin(\zeta)\Psi_2(z) \, .$$ We note that $\zeta=0$ corresponds to the standard Dirichlet boundary condition since it guarantees that $\Psi_{\zeta}$ coincides with $\Psi_1$. At the same time, if $\zeta=\frac{\pi}{2}$, we say that $\Psi_{\zeta}$ satisfies a Neumann boundary condition, coinciding with $\Psi_2$. Yet, contrary to the Dirichlet boundary condition, this is not a universal assignment as it depends on the choice of $\Psi_2$. The requirement of square-integrability of both $\Psi_1$ and $\Psi_2$ near $z=1$ implies that a Robin boundary condition can only be applied when $-1 < \mu^2 < 0$, as analyzed in the last section. For $\mu^2 \geqslant 0$, only the principal solution $\Psi_1$ is square integrable near $z=1$ and, hence, no boundary condition is required. In practice, this is as if the Dirichlet boundary condition had been chosen. A similar reasoning could be applied at $z=0$, but, as we have shown in the preceding subsection, if we focus only on square integrable solutions, only one exists, provided that $\Im[\tilde{\omega}] \neq 0$. Therefore, at $z=0$ there is no need to impose any boundary condition. Two-point Function {#Sec:ground_state} ================== In this section, we address the main question of this paper, namely the construction of a class of two-point functions, investigating whether they define a ground state for a real, massive scalar field in the BTZ black hole spacetime. We will follow the same procedure employed in [@Dappiaggi:2016fwc] in the Poincaré patch of an AdS spacetime of arbitrary dimension. As we will point out in the subsequent discussion, the main structural difference lies in the underlying metric being stationary, unless one considers the static case ($J=0$) in . Dropping for the moment the requirement of individuating a ground state, in general, by [*two-point function*]{} (or Wightman function) we refer to a bidistribution $G^+\in\mathcal{D}^\prime(M\times M)$ such that $$\label{eq:eom_G} (P\otimes\mathbb{I})G^+=(\mathbb{I}\otimes P)G^+=0 \, ,$$ and $$\label{eq:pos_G} G^+(f,f)\geq 0 \, , \quad \forall f\in C^\infty_0(M) \, .$$ In addition, the antisymmetric part of $G^+$ is constrained to coincide with the commutator distribution, in order to account for the canonical commutation relations (CCRs) of the underlying quantum field theory. In order to make this last requirement explicit, let us consider the coordinate system $(t,z,\phi)$ introduced in with $r$ replaced by $z$ as in . Working at the level of the integral kernel for $G^+$ and imposing the CCRs is tantamount to requiring that the antisymmetric part $iG(x,x^\prime)$, $x,x^\prime\in M$, where $$i G(x,x^\prime) = G^+(x,x^\prime)-G^+(x^\prime,x)$$ satisfies together with the initial conditions \[eq:initial\_conditions\_E\] $$\begin{aligned} G(x,x^\prime)|_{t=t^\prime} &= 0, \label{eq:initial_conditions_E_1} \\ -\partial_t G(x,x^\prime)|_{t=t^\prime} &=\partial_{t^\prime}G(x,x^\prime)|_{t=t^\prime}=\frac{\delta(z-z^\prime)\delta(\phi-\phi^\prime)}{\mathcal{J}(z)}, \label{eq:initial_conditions_E_2}\end{aligned}$$ with $\mathcal{J}(z)$ as in . In order to construct explicitly the two-point function we assume that $G^+$ admits a mode expansion $$\begin{aligned} G^+(x,x^\prime) &= \lim_{\epsilon \to 0^+} \sum\limits_{k\in\mathbb{Z}} \int_\bR \frac{\dd\omega}{(2\pi)^2} \, e^{-i\omega (t-t^\prime-i\epsilon)+ik(\phi-\phi^\prime)} \notag \\ &\quad \times \widehat{G}_{\omega k}(z,z^\prime) \, , \label{eq:2-pt_modes}\end{aligned}$$ where $x,x^\prime\in M$, $i\epsilon$ has been added as a regularization while the limit has to be taken in the weak sense. At this point, it is convenient to recall that, although both $\partial_t$ and $\partial_\phi$ are global Killing vector fields, a more prominent physical role is played by the globally timelike Killing vector field $\chi$ defined in . More precisely, in the construction of a ground state, the notion of positive frequencies is played by $\tilde{\omega}=\omega-k\Omega_{\mathcal{H}}$ which is subordinated to $\chi$. Hence, in order to make the role of $\tilde{\omega}$ manifest, following the discussion of Section \[sec:BTZ\_geometry\], we change from $(\omega, k)$ to $(\tilde{\omega},k)$ and from the coordinates $(t,r,\phi)$ to $(\tilde{t},r,\tilde{\phi})$, where $\tilde{\phi}=\phi-\Omega_{\mathcal{H}} t$ and $\tilde{t}=t$. Moreover, since only the positive $\tilde{\omega}$-frequencies contribute to the two-point function of the ground state, we can write $\widehat{G}_{\omega k}(z,z^\prime) \doteq \widetilde{G}_{\tilde{\omega} k}(z,z^\prime) \Theta(\tilde{\omega})$, with $\widetilde{G}_{\tilde{\omega} k}(z,z^\prime)$ defined for all $\tilde{\omega} \in \bR$. Taking into account these comments and recalling that the antisymmetric part ought to satisfy , a natural requirement consists of looking for $\widetilde{G}_{\tilde{\omega} k}(z,z^\prime)$ which is symmetric for exchange of $z$ and $z^\prime$ and such that $\widetilde{G}_{-\tilde{\omega},-k}(z,z^\prime) = -\widetilde{G}_{\tilde{\omega} k}(z,z^\prime)$. In this way, the commutator distribution reads $$\begin{aligned} i G(x,x^\prime) &= \lim_{\epsilon \to 0^+} \sum\limits_{k\in\mathbb{Z}} \int_\bR \frac{\dd\tilde{\omega}}{(2\pi)^2} e^{-i\tilde{\omega} (t-t^\prime-i|\tomega|\epsilon) + ik(\tilde{\phi}-\tilde{\phi^\prime})} \notag \\ &\quad \times \widetilde{G}_{\tilde{\omega} k}(z,z^\prime) \, , \label{eq:propagator_modes}\end{aligned}$$ where $\widetilde{G}_{\omega k}(z,z^\prime)$ is a mode bidistribution chosen in such a way that, *c.f.* Eq. , $$\label{eq:delta-resolution} \int_\bR \frac{\dd\tilde{\omega}}{2\pi} \, \tilde{\omega} \, \widetilde{G}_{\tilde{\omega} k}(z,z^\prime) = \frac{\delta(z-z^\prime)}{\mathcal{J}(z)} \, .$$ This identity, together with the Fourier series for the delta distribution along the angular coordinates, guarantees that finding $\widetilde{G}_{\tilde{\omega} k}(z,z^\prime)$ is tantamount to constructing a full-fledged two-point function $G^+$, provided that positivity as in is satisfied. In addition, entails that the mode bidistribution is such that $$(L_\tomega\otimes\mathbb{I})\widetilde{G}_{\tilde{\omega} k}(z,z^\prime)=(\mathbb{I}\otimes L_\tomega)\widetilde{G}_{\tilde{\omega} k}(z,z^\prime)=0 \, ,$$ where $L_\tomega$ is defined in . Our next goal will be to use this information to construct explicitly $\widetilde{G}_{\tomega k}(z,z^\prime)$ in terms of solutions of . Our strategy, as in [@Dappiaggi:2016fwc], will be to obtain an integral representation for the delta distribution on the RHS of , from which we can read off $\widetilde{G}_{\tilde{\omega} k}(z,z^\prime)$. However, and contrarily to the case of pure AdS analyzed in [@Dappiaggi:2016fwc], we face a technical hurdle. When dealing with the static case $J=0$, the ODE can be treated as an eigenvalue problem with spectral parameter $\tilde{\omega}^2$ and it is possible to express the delta distribution as an expansion in terms of the eigenfunctions of $L_\tomega$ (resolution of the identity). But this is not possible when dealing with the non static case $J \neq 0$, in which case the ODE has linear terms in $\tilde{\omega}$. Instead, we may treat $L_\tomega$ as a *quadratic operator pencil*, *i.e.* a differential operator with quadratic dependence on the spectral parameter $\tilde{\omega}$. In Appendix \[apx:deltaexpansion\], it is described how to obtain the expansion of the delta distribution in terms of eigenfunctions of an operator of this type. In the following, we present the results for the resolution of the identity and for the mode expansion of the two-point function for a fixed Robin boundary condition. We start from the simplest scenario, $\mu^2\geqslant 0$, for which no boundary condition needs to be imposed to the solutions of at $z=1$, and then consider the more interesting case $-1 < \mu^2 < 0$. The full details of the calculation can be consulted in Appendix \[apx:calculation-delta-expansion\]. Case $\mu^2\geqslant 0$ ----------------------- For $\mu^2 \geqslant 0$ both $z=0$ and $z=1$ in the Sturm-Liouville problem associated to are of limit point type. Using the results of Appendix \[apx:calculation-delta-expansion\] in the case $\zeta=0$, it is possible to obtain an integral representation of $\delta(z-z')$ in terms of eigenfunctions of $L_\tomega$, $$\frac{\delta(z-z^\prime)}{\mathcal{J}(z)} = \int_{\bR} \frac{\dd\tilde{\omega}}{2\pi i} \, \tilde{\omega} \left(\frac{A}{B}-\frac{\overline{A}}{\overline{B}}\right) C \, \Psi_1(z)\Psi_1(z^\prime) \, ,$$ where the constants $A$, $B$ and $C$ are defined as \[eq:A\_B\_constants\] $$\begin{aligned} A &= \frac{\Gamma(c-1)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)} \, , \\ B &=\frac{\Gamma(c-1)\Gamma(a+b-c)}{\Gamma(a)\Gamma(b)} \, , \\ C &= \frac{\ell^4}{4(r_+^2-r_-^2)\sqrt{1+\mu^2}} \, .\end{aligned}$$ Comparing with , we can read off $\widetilde{G}_{\tilde{\omega} k}(z,z^\prime)$ and write the two-point function as $$\begin{aligned} G^+(x,x^\prime) &= \lim_{\epsilon \to 0^+} \sum\limits_{k\in\mathbb{Z}} e^{ik\left(\tilde{\phi}-\tilde{\phi}^\prime\right)}\int_0^\infty \frac{\dd\tilde{\omega}}{(2\pi)^2} \, e^{-i\tilde{\omega}\left(\tilde{t}-\tilde{t}^\prime-i\epsilon\right)} \notag \\ &\quad \times \left(\frac{A}{B}-\frac{\overline{A}}{\overline{B}}\right) C \, \Psi_1(z)\Psi_1(z^\prime) \, . \label{eq:state_Dirichlet_v2}\end{aligned}$$ The mode decomposition of $G^+$ in contains only positive $\tilde{\omega}$-frequencies and, per construction, its antisymmetric part satisfies . Hence, it is legitimate to call the state associated with $G^+$ the [*ground state*]{} for a real, massive scalar field in the BTZ spacetime with $\mu^2 \geqslant 0$. An important related question consists of whether $G^+$ is locally of Hadamard form. Such property is desirable not only at a structural level but also for constructing Wick polynomials, the building blocks for dealing with interactions at a perturbative level. In Ref. [@Sahlmann:2000fh] it is proven under rather general hypotheses that a ground state, such as the one defined by in particular, is always of *local* Hadamard form, namely $G^+$ identifies a Hadamard state in every globally hyperbolic subregion of BTZ (for the definition of Hadamard state refer to Ref. [@Khavkine:2014mta]). A more difficult task is to verify if this ground state satisfy a *global* Hadamard condition such as the one proposed in [@Dappiaggi:2016fwc] and [@Dappiaggi:2017wvj] for a quantum state in anti-de Sitter spacetime. Although we conjecture that to be the case, we leave a rigorous verification for future work. Case $-1<\mu^2<0$ ----------------- For $-1<\mu^2<0$, a Robin boundary condition needs to be imposed on solutions at $z=1$ and therefore the analysis of the previous section is changed as we obtain a different two-point function for each possible Robin boundary condition. We have to consider separately two regimes, $\zeta\in[0,\zeta_*)$ and $\zeta\in[\zeta_*,\pi)$, with $$\begin{aligned} \label{eq:zetacritical} \zeta_* \doteq \arctan\left( \frac{\Gamma\left(2\beta-1\right)\left|\Gamma\left(1-\beta+i\ell\frac{k}{r_+}\right)\right|^2}{\Gamma\left(1-2\beta\right)\left|\Gamma\left(\beta+i\ell\frac{k}{r_+}\right)\right|^2} \right) \, ,\end{aligned}$$ where $\beta = \frac{1}{2}+\frac{1}{2}\sqrt{1+\mu^2}$ was defined in . Since $\mu^2 \in (-1,0)$ and thus $\beta \in (\frac{1}{2},1)$, it follows that $\zeta_* \in (\frac{\pi}{2}, \pi)$. ### Case $\zeta\in[0,\zeta_*)$ For Robin boundary conditions such that $\zeta\in[0,\zeta_*)$, it turns out that the spectrum of the operator $L_\tomega$ in is only $\tilde{\omega} \in \bR$ and does not include any isolated eigenvalue in $\bC \setminus \bR$, which would correspond to poles in the Green’s distribution associated with $L_\tomega$ (see Appendices \[apx:calculation-delta-expansion\] and \[apx:S2\] for more details). Observe that, since $\zeta_* \in (\frac{\pi}{2}, \pi)$, this scenario includes both the Dirichlet and the Neumann boundary conditions. This situation is structurally identical to the one investigated in the previous section for $\mu^2 \geqslant 0$. Using the results of Appendix \[apx:calculation-delta-expansion\] we obtain the following resolution of the identity $$\label{eq:identity_resolution} \frac{\delta(z-z^\prime)}{\cJ(z)} = \int_{\bR} \frac{\dd\tilde{\omega}}{2\pi i} \, \tilde{\omega} \, \frac{\left(A\overline{B}-\overline{A}B\right) C}{|{\cos(\zeta) B-\sin(\zeta)A}|^2} \Psi_{\zeta}(z)\Psi_{\zeta}(z') \, ,$$ where the constants $A$, $B$ and $C$ are the same as in . We can use this result in combination with and to obtain, for each $\zeta\in [0,\zeta_*)$, $$\begin{aligned} G^+_\zeta(x,x^\prime) &= \lim_{\epsilon \to 0^+} \sum_{k\in\mathbb{Z}} e^{ik\left(\tilde{\phi}-\tilde{\phi}^\prime\right)} \int_0^{\infty} \frac{\dd\tilde{\omega}}{(2\pi)^2} \, e^{-i\tilde{\omega} \left(\tilde{t}-\tilde{t}^\prime-i\epsilon\right)} \notag \\ &\quad \times \frac{\left(A\overline{B}-\overline{A}B\right) C}{|{\cos(\zeta) B-\sin(\zeta)A}|^2} \Psi_{\zeta}(z)\Psi_{\zeta}(z') \, . \label{eq:state_Robin_no_BS} \end{aligned}$$ Note that this two-point function, valid for scalar fields with $-1<\mu^2<0$, coincides with the one for scalar fields with $\mu^2 \geqslant 0$ obtained in if $\zeta=0$, that is, for Dirichlet boundary conditions. {width="0.44\linewidth"} {width="0.44\linewidth"} ### Case $\zeta\in[\zeta_*,\pi)$ For Robin boundary conditions such that $\zeta\in[\zeta_*,\pi)$, it turns out that the spectrum of the operator $L_\tomega$ in not only contains all $\tilde{\omega} \in \bR$ but it includes also two isolated eigenvalues in $\bC \setminus \bR$, complex conjugate to each other, which correspond to poles in the Green’s distribution associated with $L_\tomega$ (see Appendices \[apx:calculation-delta-expansion\] and \[apx:S2\] for more details). Denote those eigenvalues by $\tilde{\omega}_\zeta$ and $\overline{\tilde{\omega}_\zeta}$ such that $\Im[\tilde{\omega}_\zeta]>0$. They are dubbed [*bound state frequencies*]{} and their corresponding eigensolutions are called [*bound state mode solutions*]{}. The existence of bound state mode solutions was also verified in [@Dappiaggi:2016fwc] for the case of a massive scalar field in the Poincaré patch of AdS when Robin boundary conditions parametrized with $\zeta \in (\frac{\pi}{2},\pi)$ are imposed at conformal infinity. Unfortunately, an analytic expression for $\tilde{\omega}_\zeta$ cannot be found since, for $\Im[\tilde{\omega}_\zeta]>0$ and fixed $\zeta$, one needs to invert the equality $$\tan(\zeta) = \left.\frac{B}{A}\right|_{\tilde{\omega} = \tilde{\omega}_\zeta} \, ,$$ where the constants $A$ and $B$ are the same as in . This operation can only be completed numerically (except in very particular cases such as $\zeta=0$ and $\zeta=\pi/2$) and a representative example is shown in Fig. \[fig:bound-state\]. A more qualitative discussion of the behavior of the solutions $\tomega_\zeta$ as a function of $\zeta$ can be found in Appendix \[apx:S2\]. As a consequence of these bound state frequencies, the resolution of the identity acquires an extra term in comparison to , which, following Appendix \[apx:calculation-delta-expansion\], can be computed via Cauchy’s residue theorem, yielding $$\begin{aligned} \frac{\delta(z-z^\prime)}{\mathcal{J}(z)} &= \int_{\bR} \frac{\dd\tilde{\omega}}{2\pi i} \, \tilde{\omega} \, \frac{\left(A\overline{B}-\overline{A}B\right) C}{|{\cos(\zeta) B-\sin(\zeta)A}|^2} \Psi_{\zeta}(z)\Psi_{\zeta}(z') \notag \\ &\quad + \left. \Re\big[\tilde{\omega} \, C D(\tilde{\omega})\Psi_{\zeta}(z)\Psi_{\zeta}(z^\prime)\big]\right|_{\tilde{\omega} = \tilde{\omega}_\zeta} \, , \label{eq:delta-res-withBS}\end{aligned}$$ where we used the identity $\Psi_{\zeta}(z)|_{\tilde{\omega} = \overline{\tilde{\omega}_\zeta}} = \overline{\Psi_{\zeta}(z)|_{\tilde{\omega} = \tilde{\omega}_\zeta}}$. The remaining term $D(\tilde{\omega}_\zeta)$ cannot be expressed analytically, but can be defined implicitly (see Appendix \[apx:calculation-delta-expansion\]). Finally, the bound state mode solutions will also contribute to the two-point function so that its antisymmetric part still obeys and, consequently, the CCRs of the quantum field theory are satisfied. Using all the above information in combination with and , the two-point function for the putative ground state may be written, for each $\zeta\in[\zeta_*,\pi)$, $$\begin{aligned} G^+_\zeta(x,x^\prime) &= \lim_{\epsilon \to 0^+} \sum_{k\in\mathbb{Z}} e^{ik\left(\tilde{\phi}-\tilde{\phi}^\prime\right)} \int_0^{\infty} \frac{\dd\tilde{\omega}}{(2\pi)^2} \, e^{-i\tilde{\omega} \left(\tilde{t}-\tilde{t}^\prime-i\epsilon\right)} \frac{\left(A\overline{B}-\overline{A}B\right) C}{|{\cos(\zeta) B-\sin(\zeta)A}|^2} \Psi_{\zeta}(z)\Psi_{\zeta}(z') \notag \\ &\quad + i \sum_{k\in\mathbb{Z}} e^{ik\left(\tilde{\phi}-\tilde{\phi}^\prime\right)} \left(e^{-i\tilde{\omega}_{\zeta}(\tilde{t}-\tilde{t}^\prime)} + e^{-i\overline{\tilde{\omega}_{\zeta}}(\tilde{t}-\tilde{t}^\prime)}\right) \Re\big[ C D(\tilde{\omega})\Psi_{\zeta}(z)\Psi_{\zeta}(z^\prime)\big]\big|_{\tilde{\omega} = \tilde{\omega}_\zeta} \, . \label{eq:state_Robin_with_BS}\end{aligned}$$ Notice that for $\zeta=\zeta_*$ the two bound state frequencies both coincide with the real value $\tilde{\omega} =0$. In this case the integral over positive $\tilde{\omega}$-frequencies has to be interpreted as a Cauchy principal value for $\tilde{\omega}=0$, while the contribution of the bound state mode solutions is calculated using the Sokhotsky-Plemelj formula for distributions. To conclude this section we comment on the physical significance of the two-point functions obtained in and . In the first case, we are dealing with a generalization of to Robin boundary conditions. Hence, is a genuine ground state built only out of positive $\tilde{\omega}$-frequencies and, using once more the results of [@Sahlmann:2000fh], it satisfies the local Hadamard condition. On the contrary, in there is an additional contribution due to bound state frequencies $\tilde{\omega}_\zeta$, whose existence spoils the property of $G^+_\zeta$ of being a ground state. For this reason, it is not possible to conclude directly whether, in presence of bound sate frequencies, we have constructed a Hadamard, hence physically satisfactory, state. We plan to investigate this issue in future work. Conclusions {#sec:conclusions} =========== In this paper we have addressed two different, albeit related, questions. The first concerns the structural properties of a real, massive scalar field in BTZ spacetime, with an arbitrary coupling to scalar curvature. More precisely, since the underlying background is not globally hyperbolic, the equation of motion ruling the dynamics cannot be solved only assigning initial data on a partial Cauchy surface (a codimension $1$, acausal, spacelike surface that is intersected by any complete timelike curve at most once), but also a boundary condition at infinity has to be imposed. In this work, we focused our attention on those of Robin type, proving under which constraints on the parameters of the theory they can be imposed, and subsequently constructing explicitly the associated solutions of the equation of motion. In the second part of the paper, we have used this result to address whether it is possible to associate to a real, massive scalar field in BTZ spacetime a two-point function, which can be in turn read as the building block of a ground state. We have given a positive and explicit answer to this query for a large class of Robin boundary conditions. Nonetheless, we have highlighted that there exists of a range of boundary conditions that must be excluded, those for which bound state mode solutions occur. When this is not the case, the two-point function possesses some nice physical properties, the most notable one that of being of local Hadamard form. Hence, the states that we have constructed are suitable for defining an algebra of Wick polynomials which are the key ingredient to discuss interactions at a perturbative level. Besides offering one of the first examples of a ground state for a quantum field theory in the exterior region of a rotating black hole, this work prompts several future directions of investigation. On the one hand, one could prove the existence of a thermal counterpart of our ground states, hence obtaining in this framework the analogue of the Hartle-Hawking state in Schwarzschild spacetime. On the other hand, one could investigate Hawking radiation in this context and its interplay with the rotation of the black hole, by using the method of Parikh and Wilczek [@Parikh:1999mf], recently extended to the framework of algebraic quantum field theory in [@Moretti:2010qd]. A more long term and ambitious goal is the explicit construction of a regularized stress-energy tensor, to be used in the analysis of the semiclassical Einstein’s equations, extending the work of [@Binosi:1998yu]. We hope to come back to these problems in the near future. We are grateful to Valter Moretti and Nicola Pinamonti for enlightening discussions. We also thank Nicoló Drago, Felix Finster, Klaus Fredenhagen, Alan Garbarz, Carlos Herdeiro, Jorma Louko and Elizabeth Winstanley for comments and discussions. The work of F. B. was supported by a Ph.D. fellowship of the University of Pavia, that of C. D. was supported by the University of Pavia. The work of H. F. was supported by the INFN postdoctoral fellowship “Geometrical Methods in Quantum Field Theories and Applications” and by the “Progetto Giovani GNFM 2017 – Wave propagation on lorentzian manifolds with boundaries and applications to algebraic QFT” fostered by the National Group of Mathematical Physics (GNFM-INdAM). The work of I. K. was in part supported by the National Group of Mathematical Physics (GNFM-INdAM). Delta function as an expansion in eigenfunctions of a quadratic eigenvalue problem {#apx:deltaexpansion} ================================================================================== Our goal in this appendix to give a formula for the expansion of the delta distribution in terms of eigenfunctions of a differential operator with quadratic dependence on the spectral parameter like in , as it is necessary in the calculations of Section \[Sec:ground\_state\]. More precisely, we want to obtain the spectral resolution of the identity for quadratic operator pencils, specifically concentrating on the case of unbounded operators coming from Sturm-Liouville ODEs as the one above. While the spectral theory of polynomial operator pencils has been widely studied [@keldysh; @markus], it is not a topic often covered in standard references on spectral theory [@reedsimon; @kato]. Consider a family of operators defined on a Hilbert space $\H$, referred to as a *quadratic operator pencil*, $$S_\tomega = P + \tomega \cR_1 + \tomega^2 \cR_2 \, ,$$ with (S1) $\cR_1$, $\cR_2$ and $\cR_2^{-1}$ all bounded and self-adjoint, and $P$ unbounded, closed and hermitian on a dense domain $D(S_\tomega) \subset \H$, as is the case with our main example $S_\tomega = \cJ^{-1} L_\tomega$ on $\H = L^2((0,1); \cJ(z)\,\dd{z})$, where $L_\tomega$ is defined in  and $\cJ(z)$ in . Define the *resolvent* of $S_\tomega$ as $T_\tomega = S_\tomega^{-1}$, when it exists. The *resolvent set* $\rho(S_\tomega) \subset \bC$ consists of all values of $\tomega \in \bC$ such that $T_\tomega$ exists and is a bounded operator. As usual, we define the *spectrum* $\sigma(S_\tomega) = \bC \setminus \rho(S_\tomega)$. We will show that, when (S2) $\sigma(S_\tomega)$ consists only of a subset of $\bR$ together with a finite number of isolated points in $\bC \setminus \bR$ symmetric with respect to complex conjugation, the identity operator can be represented by the integral $$\begin{aligned} \label{eq:res-ident} I &= \lim_{\varsigma\to\oo} \int_{-\varsigma}^\varsigma \frac{\dd\tomega}{2\pi i} \, \lim_{\epsilon\to 0^+} \tomega (T_{\tomega-i\epsilon} - T_{\tomega+i\epsilon}) \cR_2 \notag \\ &\quad + \oint_{\mathring{C}} \frac{\dd\tomega}{2\pi i} \tomega T_\tomega \, ,\end{aligned}$$ where the contour $\mathring{C}$ illustrated in Figure \[fig:contour\] positively and simply encircles the non-real part of the spectrum, the inner $\epsilon \to 0^+$ limit is taken in the sense of distributions in $\tomega$ (boundary values of holomorphic functions define a special kind of distribution [@hoermander Ch.IX]) and the outer $\varsigma \to \oo$ limit is taken in the sense of the strong operator topology. ![Contour for the integral representation of the identity operator in .[]{data-label="fig:contour"}](Figs/cinf-contour) The key idea is to *linearize* the quadratic operator pencil to a linear operator pencil $\bS_\tomega$, while doubling the size of the Hilbert space, in a way that keeps the spectral problems of $S_\tomega$ and $\bS_\tomega$ equivalent. Since the spectral theory of linear operator pencils (essentially, generalized eigenvalue problems) is well known, we can leverage this equivalence to obtain the desired formulas for $S_\tomega$. More precisely, consider the following linear operator pencil defined on $\H^2 = \H \oplus \H$, $$\label{eq:linearpencil} \bS_\tomega = \P + \tomega \R = \begin{bmatrix} P & 0 \\ 0 & -\cR_2 \end{bmatrix} + \tomega \begin{bmatrix} \cR_1 & \cR_2 \\ \cR_2 & 0 \end{bmatrix} .$$ The linear pencil $\bS_\tomega$ is related to the quadratic one $S_\tomega$ by the basic identities $$\begin{aligned} \bS_\tomega \begin{bmatrix} I \\ \tomega \end{bmatrix} \Psi &= \begin{bmatrix} I \\ 0 \end{bmatrix} S_\tomega \Psi \, , \\ S_\tomega \begin{bmatrix} I & 0 \end{bmatrix} \begin{bmatrix} \Psi \\ \Phi \end{bmatrix} &= \begin{bmatrix} I & \tomega \end{bmatrix} \bS_\tomega \begin{bmatrix} \Psi \\ \Phi \end{bmatrix} .\end{aligned}$$ It is easy to see that, when $\cR_1$ and $\cR_2$ are bounded and self-adjoint, so is $\R$, and when in addition $P$ is closed and hermitian on $D(S_\tomega)$, so is $\P$ on $D(\bS_\tomega) = D(S_\tomega) \oplus \H$. While, there are many possible linearizations of a quadratic operator pencil, we have chosen this one to preserve these self-adjointness properties. Define the resolvent $\T_\tomega = \bS_\tomega^{-1}$, when it exists. The spectrum and resolvent set $\sigma(\bS_\tomega), \, \rho(\bS_\tomega) \subset \bC$ are defined in the usual way, essentially exactly as above. Direct calculation shows that, when both exist, the resolvents of $S_\tomega$ and $\bS_\tomega$ are related to each other by $$\begin{aligned} \notag \T_\tomega &= \begin{bmatrix} I \\ \tomega \end{bmatrix} T_\tomega \begin{bmatrix} I & \tomega \end{bmatrix} + \begin{bmatrix} 0 & 0 \\ 0 & -\cR_2^{-1} \end{bmatrix} \\ &= \begin{bmatrix} T_\tomega & \tomega T_\tomega \\ \tomega T_\tomega & \tomega^2 T_\tomega - \cR_2^{-1} \end{bmatrix} , \label{eq:quadR2linR}\end{aligned}$$ $$\begin{aligned} T_\tomega &= \begin{bmatrix} I & 0 \end{bmatrix} \T_\tomega \begin{bmatrix} I \\ 0 \end{bmatrix} = \frac{1}{\tomega} \begin{bmatrix} I & 0 \end{bmatrix} \T_\tomega \begin{bmatrix} 0 \\ I \end{bmatrix} .\end{aligned}$$ From the above formulas it is clear that when $T_\tomega$ exists and is bounded, so is $\T_\tomega$, and vice-versa. Thus $\rho(S_\tomega) = \rho(\bS_\tomega)$ and, necessarily, $\sigma(S_\tomega) = \sigma(\bS_\tomega)$, which makes precise the sense in which the spectral problems of the two operator pencils equivalent. Once we know what $\rho(S_\tomega)$ is, using the boundedness of $\R$, a variant of Theorem VI.5 of [@reedsimon] shows that $\T_\tomega$ is analytic on $\rho(S_\tomega)=\rho(\bS_\tomega)$, which implies by the explicit relationship between them that $T_\tomega$ is also analytic on $\rho(S_\tomega)$. Let $\nu \in \rho(\bS_\tomega) = \rho(S_\tomega)$ and let $C_\nu \subset \rho(\bS_\tomega) = \rho(S_\tomega)$ be a contour that simply encircles $\nu$, though in the negative direction, meaning that, upon deformation, $C_\nu$ has a chance of simply and positively encircling $\sigma(S_\tomega)$. Though, since our $\sigma(S_\tomega)$ is unbounded, the deformation of the contour will have to go through a limiting procedure. There is no need for $C_\nu$ to be connected. In fact, it is advantageous to have a connected component of $C_\nu$ contained in each connected component of $\rho(S_\tomega)$. Provided that the resolvent $\G_\tomega$ is analytic on $\rho(S_\tomega)$, the Cauchy residue formula gives $$\T_\nu \R = -\oint_{C_\nu} \frac{\dd\tomega}{2\pi i} \frac{1}{\tomega -\nu} \T_\tomega \R \, .$$ Multiplying both sides by $\R^{-1}\bS_\nu$, we get $$\label{eq:cauchy-Cnu} \mathbf{I} = \oint_{C_\nu} \frac{\dd\tomega}{2\pi i} \left( \T_\tomega \R - \frac{\mathbf{I}}{\tomega-\nu} \right) ,$$ where the contour $C_\nu$ can be deformed at will, as long as it remains within $\rho(S_\tomega) \setminus \{\nu\}$. We can deform the contour $C_\nu$ to the desired limiting form in  if we can take advantage of an abstract spectral representation for the operator $\R^{-1}\P$, that is (S3) there exists a projection operator valued measure $\E(\nu)$ on $\sigma(\bS_\tomega)$, satisfying the usual commutation and monotonicity conditions, giving the spectral representation $\R^{-1}\P = \int_{\sigma(\bS_\tomega)} \nu \, \dd \E(\nu)$. As a consequence, we also get the spectral representation $\T_\tomega \R = \int_{\sigma(\bS_\tomega)} \frac{1}{\nu+\tomega} \dd \E(\nu)$. If we let $\E_\varsigma = \E(\{\nu \in \bC \mid |\nu|<\varsigma\})$, then $\E_\varsigma \to \mathbf{I}$ strongly as $\varsigma \to \oo$ and $\bigcup_{\varsigma>0} \operatorname{ran} \E_\varsigma $ is dense in $\H^2$. Another consequence of the abstract spectral representation is that $\T_\tomega \R \E_\varsigma$ is now analytic for $|\tomega| > \varsigma$ and has the strong asymptotic expansion $\T_\tomega \R \E_\varsigma = \frac{1}{\tomega} \E_\varsigma + \mathcal{O}(\frac{1}{\tomega^{2}})$. Multiplying both sides of  by $\E_\varsigma$ we get $$\E_\varsigma = \oint_{C_\nu} \frac{\dd\tomega}{2\pi i} \left( \E_\varsigma \T_\tomega \R - \frac{\E_\varsigma}{\tomega-\nu} \right) - \oint_{C_\nu} \frac{\dd\tomega}{2\pi i} \frac{\mathbf{I}-\E_\varsigma}{\tomega-\nu} \, .$$ The second integral can be evaluated immediately and combined with the left-hand side. In the first integral, we can deform the contour $C_\nu$ to the contour $C_\varsigma \cup C^\epsilon_\varsigma \cup \mathring{C}$, as illustrated in Figure \[fig:contour\]. Because the asymptotics mentioned above, the integral over the large circle $C_\varsigma$ contributes at the order $\mathcal{O}(\frac{1}{\varsigma})$. On the other hand, the term $\frac{\E_\varsigma}{\tomega-\nu}$ is analytic over the contours $\mathring{C}$, $C^\epsilon_\varsigma$ and their interiors, so its contribution vanishes, which leaves us with $$\label{eq:cauchy-Ceps} \mathbf{I} = \oint_{C^\epsilon_\varsigma} \frac{\dd\tomega}{2\pi i} \, \T_\tomega \R \E_\varsigma + \oint_{\mathring{C}} \frac{\dd\tomega}{2\pi i} \, \T_\tomega \R \E_\varsigma + \mathcal{O}(\varsigma^{-1}) .$$ Next, before taking the limits $\epsilon \to 0^+$ and $\varsigma \to \oo$, we multiply both sides of  by an arbitrary $\mathbf{v}_{\varsigma'} \in \H^2$ such that $\mathbf{v}_{\varsigma'} = \E_{\varsigma} \mathbf{v}_{\varsigma'}$ for any $\varsigma > \varsigma'$, so that $$\begin{aligned} \mathbf{v}_{\varsigma'} &= \oint_{C^\epsilon_\varsigma} \frac{\dd\tomega}{2\pi i} \, \T_\tomega \R \E_\varsigma \mathbf{v}_{\varsigma'} + \oint_{\mathring{C}} \frac{\dd\tomega}{2\pi i} \, \T_\tomega \R \E_\varsigma \mathbf{v}_{\varsigma'} + \mathcal{O}(\varsigma^{-1}) \\ &= \int_{-\varsigma}^\varsigma \frac{\dd\tomega}{2\pi i} \, \lim_{\epsilon\to 0^+} (\T_{\tomega-i\epsilon} - \T_{\tomega+i\epsilon}) \R \E_\varsigma \mathbf{v}_{\varsigma'} \\ & \quad {} + \oint_{\mathring{C}} \frac{\dd\tomega}{2\pi i} \, \T_\tomega \R \E_\varsigma \mathbf{v}_{\varsigma'} + \mathcal{O}(\varsigma^{-1}) \\ &= \lim_{\varsigma\to \oo} \int_{-\varsigma}^\varsigma \frac{\dd\tomega}{2\pi i} \lim_{\epsilon\to 0^+} (\T_{\tomega-i\epsilon} - \T_{\tomega+i\epsilon}) \R \mathbf{v}_{\varsigma'} \\ & \quad {} + \oint_{\mathring{C}} \frac{\dd\tomega}{2\pi i} \, \T_\tomega \R \mathbf{v}_{\varsigma'} \, .\end{aligned}$$ Note that the $\epsilon \to 0^+$ limit is taken in the distributional sense with respect to $\tomega$. Finally, using a variant of the Banach-Steinhaus theorem (Theorem 2.11.4 of  [@HillePhillips]), we obtain the following strong limit $$\R^{-1} = \lim_{\varsigma \to \oo} \int_{-\varsigma}^\varsigma \frac{\dd \tomega}{2\pi i} \, \lim_{\epsilon\to 0^+} (\T_{\tomega-i\epsilon} - \T_{\tomega+i\epsilon}) + \oint_{\mathring{C}} \frac{\dd\tomega}{2\pi i} \, \T_\tomega \, ,$$ where to apply the theorem we need to recall that finite linear combinations of vectors like $\mathbf{v}_{\varsigma'}$ are dense in $\H^2$ and note that due to  the norms of the integrals $$\int_{-\varsigma}^\varsigma \frac{\dd \tomega}{2\pi i}\, \lim_{\epsilon\to 0^+} (\T_{\tomega-i\epsilon} - \T_{\tomega+i\epsilon}) + \oint_{\mathring{C}} \frac{\dd\tomega}{2\pi i} \, \T_\tomega$$ are uniformly bounded for large $\sigma$. Using the second equality in  and the formula $$\R^{-1} = \begin{bmatrix} \cR_1 & \cR_2 \\ \cR_2 & 0 \end{bmatrix}^{-1} = \begin{bmatrix} 0 & \cR_2^{-1} \\ \cR_2^{-1} & -\cR_2^{-1} \cR_1 \cR_2^{-1} \end{bmatrix}$$ finally gives us the desired identity . The argument we have just presented, for the linear operator pencil, mimicks that of [@weidmann Ch.9]. There, the existence of the spectral measure $\E(\nu)$ followed from the standard spectral theorem for self-adjoint operators on a Hilbert space, with the operator $\R^{-1} \P$ being self-adjoint with respect to the weighted inner product $[\mathbf{v},\mathbf{u}] = (\mathbf{v}, \R \mathbf{u})$, which was assumed to be positive definite. In our case, $[\mathbf{v},\mathbf{u}]$ is clearly indefinite and thus defines a Krein space $\cK = (\H^2, [-,-])$ rather than a Hilbert space. Fortunately, in the Krein space setting we can still appeal to a spectral theorem, provided that the operator $\R^{-1} \P$ is definitizable. This will indeed be the case for the specific operators defined in Appendix \[apx:calculation-delta-expansion\]. Though, since verifying the necessary hypothesis is rather technical, we relegate them to Appendix \[apx:S3\]. A more hands-on alternative to hypothesis (S3) would be a direct estimate of the form $\T_\tomega = \frac{1}{\tomega} \R^{-1} + \mathcal{O}(\frac{1}{\tomega^2})$ that is uniform over a neigborhood of $\tomega = \oo$ minus a sector of positive angle containing the real axis. Such an estimate could be obtained by a WKB analysis of the differential operators discussed in Appendix \[apx:calculation-delta-expansion\], which may be considered in future work. Explicit calculation of the delta integral representation {#apx:calculation-delta-expansion} ========================================================= In this appendix, we show in detail the procedure to compute the delta integral representation and, hence, the mode expansion of the two-point function for the case in which the mass parameter is such that $-1 < \mu^2 < 0$ and Robin boundary conditions parametrized by $\zeta \in [0,\pi)$ are imposed at $z=1$. The results for $\mu^2 \geqslant 0$ may be simply obtained by setting $\zeta = 0$. Now, let us apply the general discussion from Appendix \[apx:deltaexpansion\] to the differential operator $L_\tomega$ introduced in , which we write for convenience as $$\begin{gathered} L_\tomega\Psi(z) = \frac{\dd}{\dd z} \left( z \frac{\dd \Psi(z)}{\dd z} \right) - \bigg[ \frac{\ell^2 k^2(1-z)-r_+^2\mu^2}{4 r_+^2(1-z)} \\ - \frac{\tilde{\omega} \ell^3 k r_-}{2r_+ (r_+^2-r_-^2) (1-z)} - \frac{\tilde{\omega}^2\ell^4 \cJ(z)}{4(r_+^2-r_-^2)} \bigg] \Psi(z) , \label{eq:B1}\end{gathered}$$ with $\cJ(z)$ the same as in . We let the Hilbert space be $\H = L^2((0,1); \cJ(z)\dd{z})$ and we let the quadratic operator pencil be $$\label{eq:SLop} S_{\tilde{\omega}}\Psi(z) = \frac{1}{\cJ(z)} L_\tomega\Psi(z) .$$ This operator satisfies the hypotheses (S1), (S2) and (S3) from Appendix \[apx:deltaexpansion\]. The verification of the hypotheses is of a much more technical nature and is relegated to Appendix \[apx:S1\], Appendix \[apx:S2\] and Appendix \[apx:S3\], respectively. We want to construct a Green’s distribution $\cG_{\tomega,\zeta}$ associated to $L_\tomega$ consisting of the product of square integrable solutions of $L_\tomega \Psi = 0$ at both $z=0$ and $z=1$. For that we introduce $$u_{\tilde{\omega}}(z) = \begin{cases} \Psi_3(z) \, , & \Im[\tilde{\omega}] > 0 \, , \\ \Psi_4(z) \, , & \Im[\tilde{\omega}] < 0 \, , \end{cases} \label{eq:functionu}$$ with $\Psi_3$ and $\Psi_4$ defined in , which is uniquely chosen by the property of being $L^2$ at $z=0$, as seen in Section \[sec:endpoints\]. We also introduce $$\label{eq:functionv} \Psi_{\tomega,\zeta}(z) = \cos(\zeta)\Psi_1(z)+\sin(\zeta)\Psi_2(z) \, ,$$ with $\Psi_1$ and $\Psi_2$ defined either by  or , which is uniquely chosen by the property of being $L^2$ at $z=1$ when $-1<\mu^2<0$ and satisfying Robin boundary conditions parametrized by $\zeta\in [0,\pi)$. Note that, given the identity $\overline{L_{\tomega}} = L_{\overline{\tomega}}$, one has $u_{\overline{\tomega}} = \overline{u_{\tomega}}$ and $\Psi_{\overline{\tomega},\zeta} = \overline{\Psi_{\tomega,\zeta}}$. The Green’s distribution $\cG_{\tomega,\zeta}$ may then be written as $$\cG_{\tomega,\zeta}(z,z') = \begin{cases} \mathcal{N}^{-1}_{\tomega,\zeta}\ u_\tomega(z) \Psi_{\tomega,\zeta}(z') \ , & z\leqslant z' \ , \\ \mathcal{N}^{-1}_{\tomega,\zeta}\ u_\tomega(z') \Psi_{\tomega,\zeta}(z) \ , & z\geqslant z' \ , \end{cases} \label{eq:radial_Green_BC}$$ with $$\begin{aligned} \mathcal{N}_{\tilde{\omega},\zeta} = - z \mathcal{W}_z \left[u_{\tilde{\omega}},\Psi_{\zeta}\right] = \begin{cases} \cos(\zeta) \, \dfrac{\Gamma(c)\Gamma(a+b-c+1)}{\Gamma(a)\Gamma(b)} + \sin(\zeta) \, \dfrac{\Gamma(c)\Gamma(c-a-b+1)}{\Gamma(c-a)\Gamma(c-b)} \, , & \Im[\tilde{\omega}] > 0 \, , \\ \cos(\zeta) \, \dfrac{\Gamma(2-c)\Gamma(a+b-c+1)}{\Gamma(a-c+1)\Gamma(b-c+1)} + \sin(\zeta) \, \dfrac{\Gamma(2-c)\Gamma(c-a-b+1)}{\Gamma(1-a)\Gamma(1-b)} \, , & \Im[\tilde{\omega}] < 0 \, , \end{cases} \label{eq:Wronski_BC} \end{aligned}$$ where the parameters $a,b,c$ are as in . The normalization constant $\mathcal{N}_{\tilde{\omega},\zeta}$ was evaluated using the intermediate result $$\mathcal{W}_z[\Psi_1,\Psi_2] = \frac{a+b-c}{z} = \frac{\sqrt{1+\mu^2}}{z} \, ,$$ and the following connection formulas of hypergeometric functions (see Eqs. (15.10.17-18) of [@NIST]): $$\begin{aligned} \Psi_3(z) &= \frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}\Psi_1(z) \notag \\ &\quad +\frac{\Gamma(c)\Gamma(a+b-c)}{\Gamma(a)\Gamma(b)}\Psi_2(z) \, , \label{eq:connection3} \\ \Psi_4(z) &= \frac{\Gamma(2-c)\Gamma(c-a-b)}{\Gamma(1-a)\Gamma(1-b)}\Psi_1(z) \notag \\ &\quad +\frac{\Gamma(2-c)\Gamma(a+b-c)}{\Gamma(a-c+1)\Gamma(b-c+1)}\Psi_2(z) \, . \label{eq:connection4} \end{aligned}$$ By inspection of  and , one has that $\overline{\mathcal{N}_{\tomega,\zeta}} = \mathcal{N}_{\overline{\tomega},\zeta}$ and $\overline{\cG_{\tomega,\zeta}}(z,z') = \cG_{\overline{\tomega},\zeta}(z',z)$. Moreover, as noted in Appendix \[apx:S2\], $\mathcal{N}_\tomega$ is analytic on $\Im[\tomega] \ne 0$ and has at most two isolated zeros, the *bound state frequencies*, that are reflection symmetric about the real axis, forming a set $\BS_\zeta \subset \bC$ such that $\BS_\zeta = \BS_\zeta^+ \cup \overline{\BS_\zeta^+}$ with $\Im[\BS_\zeta^+] > 0$. We can now apply formula to write the following integral representation of the delta distribution: $$\begin{aligned} \frac{4(r_+^2 - r_-^2)}{\ell^4 \cJ(z)} \delta(z-z') & = -\int_{\bR} \frac{\dd\tilde{\omega}}{2\pi i} \, \tilde{\omega} \, \Delta \cG_{\tomega,\zeta}(z,z') \notag \\ &\quad + \oint_{\mathring{C}} \frac{\dd\tomega}{2\pi i} \, \tomega \mathcal{G}_{\tomega,\zeta}(z,z') \, ,\end{aligned}$$ where the contour $\mathring{C}$ illustrated in Figure \[fig:contour\] positively and simply encircles the bound state frequencies in $\BS_\zeta$, and $$\Delta \cG_{\tomega,\zeta}(z,z') \doteq \lim_{\epsilon \to 0^+} [\cG_{\tomega+i\epsilon,\zeta}(z,z') - \cG_{\tomega-i\epsilon,\zeta}(z,z')]$$ should be interpreted as a distribution in $\tomega$. An application of Cauchy’s residue theorem gives the integral representation $$\begin{aligned} \frac{\delta(z-z')}{\cJ(z)} \notag &= -\frac{\ell^4}{4(r_+^2-r_-^2)} \Bigg[ \int_\bR \frac{\dd\tomega}{2\pi i} \tomega \, \Delta \cG_{\tomega,\zeta}(z,z') \\ & \quad + \sum_{\tomega'\in \BS_\zeta} \Res_{\tomega=\tomega'}[ \tomega \, \cG_{\tomega,\zeta}(z,z')] \Bigg] \ , \label{eq:id-res}\end{aligned}$$ Both integrands in  can be computed rather explicitly, except for analytic expressions for the bound state frequencies (see Appendix \[apx:S2\]). Introducing $$A=\frac{\Gamma(c-1)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)} , \quad B=\frac{\Gamma(c-1)\Gamma(a+b-c)}{\Gamma(a)\Gamma(b)} ,$$ and using the connection formulas and , one may write $$u_{\tilde{\omega}}(z) = \begin{cases} (c-1) \left[A \Psi_1(z) + B \Psi_2(z) \right] \, , & \Im[\tilde{\omega}] > 0 \, , \\ (1-c) \left[\overline{A} \Psi_1(z) + \overline{B} \Psi_2(z) \right] \, , & \Im[\tilde{\omega}] < 0 \, , \end{cases}$$ and $$\begin{aligned} \mathcal{N}_{\tilde{\omega},\zeta} = \begin{cases} (1-c) \sqrt{1+\mu^2} \big[{\cos(\zeta)} B - \sin(\zeta) A \big], & \Im[\tilde{\omega}] > 0 , \\ (c-1) \sqrt{1+\mu^2} \big[{\cos(\zeta)} \overline{B} - \sin(\zeta) \overline{A} \big], & \Im[\tilde{\omega}] < 0 . \end{cases}\end{aligned}$$ Hence, for $z < z'$, $$\begin{aligned} \Delta \mathcal{G}_{\tilde{\omega}}(z,z') &= - \frac{1}{\sqrt{1+\mu^2}} \left[ \frac{A \Psi_1(z) + B \Psi_2(z)}{\cos(\zeta) B - \sin(\zeta) A} \right. \notag \\ &\quad \left. - \frac{\overline{A} \Psi_1(z) + \overline{B} \Psi_2(z)}{\cos(\zeta) \overline{B} - \sin(\zeta) \overline{A}} \right] \Psi_{\zeta}(z') \notag \\ &= \frac{\overline{A}B-A\overline{B}}{\left|{\cos(\zeta) B-\sin(\zeta) A}\right|^2} \frac{\Psi_{\zeta}(z)\Psi_{\zeta}(z')}{\sqrt{1+\mu^2}} \, ,\end{aligned}$$ and the result is also valid for $z > z'$. Now, let us consider the residues at a bound state frequency $\tomega_\zeta \in \BS^+_\zeta$. When it exists, it is an isolated root of $\mathcal{N}_{\tomega,\zeta} = 0$ and $$\Res_{\tomega=\tomega_\zeta} [\tomega \cG_{\tomega,\zeta}(z,z')] = \frac{\tomega_\zeta}{2} D(\tomega_\zeta) \Psi_{\tomega_\zeta,\zeta}(z) \Psi_{\tomega_\zeta,\zeta}(z') ,$$ where $D(\tomega_\zeta) = D_2(\tomega_\zeta)/D_1(\tomega_\zeta)$. From the Laurent series of $\mathcal{N}_{\tomega,\zeta}$ we get $$\begin{aligned} D_1(\tilde{\omega_\zeta}) &\doteq \frac{\ell^2 \sqrt{1+\mu^2}}{i(r_+^2-r_-^2)} \, \big\{{\sin(\zeta)}\, A \big[(r_++r_-)\psi(c-a) \\ &\qquad +(r_+-r_-)\psi(c-b)-2r_+\psi(c) \big] (1-c) \\ &\quad - \cos(\zeta)\,B \big[(r_++r_-)\psi(b)+(r_+-r_-)\psi(a) \\ &\qquad -2r_+\psi(c)\big] (1-c)\big\}|_{\tomega=\tomega_\zeta} \, ,\end{aligned}$$ where $\psi$ is the digamma function. Since $\mathcal{N}_{\tomega_\zeta,\zeta} = 0$, the solutions $u_{\tomega_\zeta}$ and $\Psi_{\tomega_\zeta,\zeta}$ are no longer linearly independent and their ratio (recall that $\Im[\tomega_\zeta] > 0$) is $$\begin{aligned} D_2(\tilde{\omega_\zeta}) &\doteq \frac{u_{\tomega_\zeta}(z)}{\Psi_{{\tomega_\zeta},\zeta}(z)}\\ &= \begin{cases} \sec(\zeta) (c-1) A|_{\tomega=\tomega_{\zeta}} \, , & \cos(\zeta) \ne 0 \, , \\ \csc(\zeta) (c-1) B|_{\tomega=\tomega_{\zeta}} \, , & \sin(\zeta) \ne 0 \, . \end{cases}\end{aligned}$$ Finally, the spectral resolution of the delta distribution takes the form $$\begin{gathered} \frac{\delta(z-z^\prime)}{\cJ(z)} = \frac{\ell^4}{4(r_+^2-r_-^2)} \\ \times \Bigg[ \int_{\bR} \frac{\dd\tilde{\omega}}{2\pi i} \, \tilde{\omega} \, \frac{A\overline{B}-\overline{A}B}{|{\cos(\zeta) B-\sin(\zeta) A}|^2} \frac{\Psi_\zeta(z)\Psi_\zeta(z^\prime)}{\sqrt{1+\mu^2}} \\ + \sum_{\tomega_{\zeta} \in \BS^+_{\zeta}} \Re\big[ \tilde{\omega}_{\zeta} D(\tilde{\omega}_{\zeta})\Psi_{\tomega_{\zeta},\zeta}(z)\Psi_{\tomega_{\zeta},\zeta}(z^\prime)\big] \Bigg] \, .\end{gathered}$$ We have taken advantage of the fact that bound state frequencies come in complex conjugate pairs, $\BS_{\zeta} = \BS^+_{\zeta} \cup \overline{\BS^+_{\zeta}}$, and of the identities $\overline{D(\tomega_{\zeta})} = D(\overline{\tomega_{\zeta}})$, $\overline{\Psi_{\tomega_{\zeta},\zeta}(z)} = \Psi_{\overline{\tomega_{\zeta}},\zeta}(z)$. Check of hypothesis (S1) {#apx:S1} ======================== In this appendix we show that hypothesis (S1) of Appendix \[apx:deltaexpansion\] is verified for the quadratic operator pencil $S_{\tilde{\omega}}$. First, we discuss the relation of the domain of $S_{\tilde{\omega}}$, $D(S_{\tilde{\omega}})\subset\mathcal{H} = L^2((0,1); \cJ(z)\dd{z})$, to the choice of boundary conditions for $L_\tomega$ in . By standard arguments [@weidmann Ch.3], each choice of boundary conditions will give a closed operator realization of $S_\tomega$ on a dense domain $D(S_\tomega)$. Then, if there exists at least one $\tomega \in \bC$ such that $\tomega,\overline{\tomega} \in \rho(S_\tomega)$ and the corresponding bounded resolvents satisfy $T_\tomega^* = T_{\overline{\tomega}}$, the closed operator $S_\tomega$ will be self-adjoint, in the sense that $S^*_\tomega = S_{\overline{\tomega}}$ and $D(S_\tomega^*) = D(S_\tomega)$. Hence, we need to check that (a) the Green’s distribution associated to $L_\tomega$, $\cG_\tomega$, exists for at least one $\tomega \in \bC$, that (b) $T_\tomega = \cG_\tomega \cJ$ is bounded for at least one $\tomega \in \bC$ and that (c) we can satisfy $\overline{\cG_\tomega}(z,z') = \cG_{\overline{\tomega}}(z',z)$ and hence $T^*_{\tomega} = T_{\overline{\tomega}}$. The properties (a) and (c) are explicitly checked in Appendix \[apx:calculation-delta-expansion\] for each choice of Robin boundary conditions parametrized by $\zeta$. In order to check property (b), we need to prove the boundedness of the resolvent $T_\tomega=\mathcal{G}_\tomega\mathcal{J}$. Using the same notation of Appendix \[apx:calculation-delta-expansion\], for a given $\tomega$ with $\Im[\tomega] \ne 0$, provided that $u_\tomega$ and $\Psi_{\tomega,\zeta}$ introduced in and are linearly independent, that is, $\mathcal{N}_{\tomega,\zeta}$ in does not vanish, we can get boundedness starting from the more precise asymptotic estimates: \[eq:estimates\] $$\begin{aligned} |u_\tomega(z)| &\lesssim z^{\lambda} (1-z)^{1-\beta-\epsilon} \ , \\ |\Psi_{\tomega,\zeta}(z)| &\lesssim \begin{cases} z^{-\lambda} (1-z)^{\beta} \ , & \zeta=0 \ , \\ z^{-\lambda} (1-z)^{1-\beta-\epsilon} \ , & \zeta\ne0 \ . \end{cases} \end{aligned}$$ Here, $\lambda \doteq \ell^2 r_+ \left|\Im\tomega\right| / 2(r_+^2-r_-^2)$ and the symbol $\lesssim$ denotes an inequality up to a multiplicative constant, uniform over $z\in (0,1)$ where applicable. The constant $\epsilon > 0$ helps to cover the cases with logarithmic singularities and it could be chosen to depend on other parameters. Using the same notation, we also have $$|\cJ(z)| \lesssim z^{-1} (1-z)^{-1} \, .$$ The strategy to show boundedness of $T_\tomega = \cG_\tomega \cJ$ is to apply the so-called *weighted Schur test* (Theorem 5.2 of  [@halmos]). The inequalities, where, after a factorization, we apply the Cauchy-Schwarz inequality, $$\begin{aligned} \|T_\tomega\Psi\|^2 &= \int_0^1\dd{z} \cJ(z) \left|\int_0^1 \dd{z'} \cG_\tomega(z,z') \cJ(z') \Psi(z') \right|^2 \\ &\leqslant \int_0^1\dd{z} \cJ(z) \left(\int_0^1\dd{z'} \left|\cG_\tomega(z,z')\right| \cJ(z') \cJ_1(z') \right) \left(\int_0^1\dd{z'} \left|\cG_\tomega(z,z')\right| \frac{\cJ(z')}{\cJ_1(z')} \left|\Psi(z)\right|^2 \right) \\ &\leqslant \int_0^1\dd{z'} \left(\int_0^1\dd{z} \cJ(z) \cJ_2(z) \left|\cG_\tomega(z,z')\right|\right) \frac{\cJ(z')}{\cJ_1(z')} \left|\Psi(z')\right|^2 \\ &\leqslant \int_0^1\dd{z'} \frac{\cJ_3(z')}{\cJ_1(z')} \cJ(z') \left|\Psi(z')\right|^2 \, , \end{aligned}$$ show that $\|T_\tomega \Psi\|^2 \lesssim \|\Psi\|^2$, provided we can find functions $\cJ_1(z)$, $\cJ_2(z)$, $\cJ_3(z)$ satisfying the estimates $$\begin{gathered} \int_0^1 \dd{z'} \left|\cG_\tomega(z,z')\right| \cJ(z') \cJ_1(z') \lesssim \cJ_2(z) \, , \\ \int_0^1 \dd{z} \cJ(z) \cJ_2(z) \left|\cG_\tomega(z,z')\right| \lesssim \cJ_3(z') \, , \\ \frac{\cJ_3(z')}{\cJ_1(z')} \lesssim 1 \, .\end{gathered}$$ The only free choice is actually in $\cJ_1$, since $\cJ_2$ and $\cJ_3$ (or rather their lower bounds) are then determined by the properties of $\cG_\tomega(z,z')$. Given the estimates  and formula , it is straightforward to show that the following choices work as desired: $$\begin{aligned} \zeta=0 &\colon \begin{cases} \cJ_1(z) = 1 , \\ \cJ_2(z) = (1-z)^{\min(\beta,1-2\epsilon)} , \\ \cJ_3(z) = (1-z)^{\min(\beta,2-4\epsilon)} , \end{cases} \\ \zeta\ne0 &\colon \begin{cases} \cJ_1(z) = 1 , \\ \cJ_2(z) = (1-z)^{1-\beta-\epsilon} , \\ \cJ_3(z) = (1-z)^{1-\beta-\epsilon} , \end{cases}\end{aligned}$$ where, for $\zeta\ne0$, we restrict to $\beta \in (\frac{1}{2}, 1)$ and we choose $\epsilon < 1-\beta$. Check of hypothesis (S2) {#apx:S2} ======================== In this appendix we show that the hypotesis (S2) of Appendix \[apx:deltaexpansion\] is verified, namely that the spectrum of $S_{\tomega}$ consists only of $\mathbb{R}$ together with at most two isolated points in $\mathbb{C}\setminus\mathbb{R}$, symmetric with respect to complex conjugation. The Green’s distribution $\mathcal{G}_{\tilde{\omega},\zeta}$ computed in Appendix \[apx:calculation-delta-expansion\] has a branch cut at $\Im[\tomega]=0$ and for certain values of $\zeta$ it can have poles with $\Im[\tomega] \ne 0$, which from the explicit calculations of Appendix \[apx:calculation-delta-expansion\] coincide with the zeros of the normalization coefficient $\mathcal{N}_{\tomega,\zeta}$ in . By direct inspection, we know that $\mathcal{N}_{\tomega,\zeta}$ has at most isolated zeros, that are reflection symmetric about the real axis. These bound state frequencies form a set $\BS_\zeta \subset \bC$, with $\BS_\zeta = \BS_\zeta^+ \cup \overline{\BS_\zeta^+}$ with $\Im[\BS_\zeta^+] > 0$. We conclude that $\sigma(S_\tomega) = \mathbb{R} \cup \BS_\zeta$. By general arguments from Appendix \[apx:deltaexpansion\], the resolvent $T_{\tomega}=S^{-1}_{\tomega}$ is analytic on its resolvent set $\rho(S_\tomega) = \bC \setminus \sigma(S_\tomega)$. We will now argue that either $\BS_\zeta^+ = \varnothing$ or $\BS_\zeta^+ = \{ \tomega_\zeta \}$ consists of a single point. Using the notation from Appendix \[apx:calculation-delta-expansion\], the zeros of $\mathcal{N}_{\tomega,\zeta}$ are precisely the solutions of the transcendental equation $$\label{eq:BS-zeta} \tan(\zeta) = \frac{B}{A} \doteq \Theta(\tomega)$$ in the upper half complex plane, $\Im[\tomega] > 0$ and $\zeta \in [0,\pi)$, together with their complex conjugates. $A$ and $B$ are as in . When $\zeta = \pi/2$, we interpret any $\tomega$ at which $\Theta(\tomega)$ has a pole as a solution of . When written out explicitly, the RHS of  is a ratio of products of gamma functions with $\tomega$-dependent parameters. Its main characteristics are that, for generic values of the parameters, it has only the simple zeros at $\tomega_\pm(n)$ and the simple poles at $\tomega^\pm(n)$ for $n=0,1,2,\ldots$, where $$\begin{aligned} \label{eq:zeros} \tomega_\pm(n) &= \pm \frac{k}{\ell} -k\Omega_\H - 2i(n+\beta) \frac{(r_+ \mp r_-)}{\ell^2} \ , \\ \label{eq:poles} \tomega^\pm(n) &= \pm \frac{k}{\ell} -k\Omega_\H - 2i(n+1-\beta) \frac{(r_+ \mp r_-)}{\ell^2} \ ,\end{aligned}$$ as well as the asymptotic behavior $$\begin{gathered} \label{eq:asymptTheta} \Theta(\tomega) = \frac{\Gamma(\sqrt{\mu^2+1})}{\Gamma({-\sqrt{\mu^2+1}})} \left(\frac{\ell^4 (-i\tomega)^2}{4(r_+^2-r_-^2)}\right)^{-\sqrt{\mu^2+1}} \\ \times [1 + \mathcal{O}(|\tomega|^{-1})]\end{gathered}$$ for $|\tomega| \to \oo$, which follows from the Stirling asymptotic formula. The branch of the power function must agree with the principal branch when $-i\tomega > 0$. Some of the poles or zeros may merge for special values of the parameters. The zeros and poles of $\Theta(\tomega)$ give us the explicit solutions of , respectively, for $\zeta = 0$ (Dirichlet) and $\zeta = \pi/2$ (Neumann) boundary conditions. For a general value of $\zeta$, the transcendental nature of equation  prevents us from giving explicit solutions. Although this equation could certainly be solved numerically for any value of the parameters $\mu^2$, $\ell$, $r_+$, $r_-$ and $k$ describing the BTZ black hole and the scalar field, we can make the following qualitative conclusions. Since $\zeta$ is always real, $\tomega \in \bC$ for which $\Theta(\tomega) \not\in \bR$ is never a solution of . On the other hand, when $\Theta(\tomega)$ is real, equation  is certainly satisfied for $\zeta = \arctan(\Theta(\tomega))$. Thus, for fixed $\zeta$, the solutions of  exist and lie on the lines of real phase $\arg [\Theta(\tomega)] = 0$ or $\pi$. Roughly speaking, lines of real phase stretch between the poles and zeros of $\Theta(\tomega)$, also with one such line stretching to $\oo$ through the upper half plane from the pole with the largest $\Im[\tomega]$, as can be deduced by . In the case $\mu^2 \geqslant 0$, only the $\zeta = 0$ (Dirichlet) boundary condition is allowed (see Section \[sec:endpoints\]), which corresponds to zeros of $\Theta(\tomega)$. As it can be seen from , all of the zeros are confined to ther lower half complex plane and so there are no solutions of  with $\Im[\tomega] > 0$. Therefore, in this case, there are no bound state frequencies, $\BS_\zeta^+ = \varnothing$. When $-1 < \mu^2 < 0$, all the poles and zeros lie in the lower half complex plane and closest to the real axis is the pole at $$\tomega^+(0) = \frac{k}{\ell} - k \Omega_\H -i\left(1-\sqrt{\mu^2+1}\right) \frac{(r_+-r_-)}{\ell^2} \ .$$ The solutions with $\Im[\tomega] > 0$ must lie on the single line of real phase stretching from this pole and are parametrized by $\zeta \in [\zeta_*,\pi)$. This phase line crosses the $\Im[\tomega] = 0$ line at $\tomega = 0$, where $$\Theta(0) = \frac{\Gamma\left(2\beta-1\right)\left|\Gamma\left(1-\beta+i\ell\frac{k}{r_+}\right)\right|^2}{\Gamma\left(1-2\beta\right)\left|\Gamma\left(\beta+i\ell\frac{k}{r_+}\right)\right|^2} = \tan(\zeta_*) \ .$$ Since $\beta \in (\frac{1}{2},1)$, then $\zeta_* \in (\frac{\pi}{2}, \pi)$. Qualitatively, we also see that the solution $\tomega = \tomega_\zeta$ is simple[^1] and of course isolated. Hence, in this case $\BS_\zeta = \{\tomega_\zeta , \overline{\tomega_\zeta} \}$. The real and imaginary parts of $\tomega_\zeta$ are plotted as a function of $\zeta$ in Figure \[fig:bound-state\] for a particular value of other parameters. Check of hypothesis (S3) {#apx:S3} ======================== In this appendix we show that the hypothesis (S3) of Appendix \[apx:deltaexpansion\] is verified, namely that there exists a spectral measure for the linearized pencil $\bS_\tomega$ in . Following the notation of Appendix \[apx:deltaexpansion\], the inner product space $\cK = (\H^2, [-,-])$, with bounded bilinear form $[\textbf{v}, \textbf{u}] = (\textbf{v}, \R \textbf{u})$, defines a *Krein space* [@bognar; @Langer:1982], that is, a Banach (in this case Hilbert) space with a bounded hermitian scalar product that need not be positive definite. The spectral problem of the linear operator pencil $\bS_\tomega = \P + \tomega \R$ is equivalent to the standard spectral problem $-\R^{-1} \P = \tomega \mathbf{I}$, where the operator $\A \doteq -\R^{-1} \P$ is now self-adjoint with respect to the Krein space scalar product $[-,-]$. Unfortunately, unlike the Hilbert space case, there is no spectral theorem available for an arbitrary self-adjoint operator on a Krein space. However, there are some special cases where the spectral theorem, and hence the existence of a spectral measure $\E(\nu)$ as requested by hypothesis (S3) in Appendix \[apx:deltaexpansion\], is available. One such case is when $\A$ is *definitizable*, that is, when there exists a degree $k$ polynomial $p(\tomega)$ with real coefficients such that $[\mathbf{u}, p(\A)\mathbf{u}] \geqslant 0$ for each $\mathbf{u} \in D(\A^k)$. The corresponding spectral theorem can be found in [@Langer:1982] and [@KaltenbaeckPruckner:2015]. Below, we give a brief argument verifying that the operator $\A$ discussed in Appendices \[apx:deltaexpansion\] and \[apx:calculation-delta-expansion\] is definitizable, hence fulfilling hypothesis (S3). The argument is as follows. First, suppose that there exists a definitizable closed restriction $\A_0$ of $\A$ to a smaller domain $D(\A_0) \subset D(\A)$, since $[\mathbf{u}, (-\A_0) \mathbf{u}] \geqslant 0$ for all $\mathbf{u} \in D(\A_0)$. While $\A_0$ itself may no longer be self-adjoint, the Krein space analog of the Friedrichs extension [@Curgus:1989] then gives us a self-adjoint extension $\A_1$ that is still satisfies $[\mathbf{u}, (-\A_1) \mathbf{u}]$ on its domain. Second, since $\A$ is essentially defined by an ordinary differential operator, the difference of the resolvents $$\label{eq:finite-rank} (\A_1 - \tomega \mathbf{I})^{-1} - (\A - \tomega \mathbf{I})$$ is an operator of finite rank, which is described by the so-called *Krein resolvent formula* [@AkhiezerGlazman 106]. The finiteness of the rank comes from the fact that an ordinary differential operator has a finite dimensional space of solutions. Finally, it is also known that when at least one of the Krein self-adjoint operators $\A_1$ or $\A$ is definitizable and the difference of their resolvents  has finite rank for at least one $\tomega$ common to both resolvent sets, then both operators are definitizable [@JonasLanger:1979]. Recall that we are working with $\H = L^2((0,1); \cJ(z)\, \dd{z})$ and consider $\mathbf{u} = [\begin{matrix} \Psi & \Phi \end{matrix}]^T \in D(\A_0)$ consisting of smooth functions with compact support. Unwinding all the definitions from Appendices \[apx:deltaexpansion\] and \[apx:calculation-delta-expansion\], and writing out $[\mathbf{u}, (-\A) \mathbf{u}]$ explicitly and using integration by parts, we get $$\begin{gathered} (\Psi, (-\cJ^{-1} L_{\tomega=0}) \Psi) + (\Phi, \cR_2 \Phi) = \int_0^1 \dd{z} \left[z \left|\frac{\dd \Psi(z)}{\dd z}\right|^2 \right. \\ \left. + \left(\frac{\ell^2 k^2}{4 r_+^2} + \frac{\mu^2}{4(1-z)}\right) |\Psi(z)|^2 + \frac{\ell^4 \cJ(z) |\Phi(z)|^2}{4 (r_+^2-r_-^2)} \right] \, . \label{eq:integrand}\end{gathered}$$ When $\mu^2\geqslant 0$, all the terms appearing under the integral are manifestly non-negative, meaning that so is the whole integral. When $-1 < \mu^2 < 0$, the integrand is still non-negative. This can be proven observing that, being the term proportional to $k^2$ in the integrand strictly greater than $0$, it suffices to show positivity for $-\mathcal{J}^{-1}L_{\tomega=0}$ when $k=0$. Yet, in this case, in view of and of the results of Appendix \[apx:deltaexpansion\], $-\mathcal{J}^{-1}L_{\tomega=0}$ is a self-adjoint operator with strictly positive spectrum, which is tantamount to saying that it is a positive operator. Thus, the restriction of $\A$ to $D(\A_0)$ does satisfy $[\mathbf{u}, (-\A_0) \mathbf{u}] \geqslant 0$ for all $\mathbf{u} \in D(\A_0)$. By the preceding reasoning, this finally implies that $\A$ is definitizable. [99]{} M. Benini, C. Dappiaggi and T. -P. Hack, Int. J. Mod. Phys. A [**28**]{} 1330023 (2013) \[gr-qc/1306.0527\]. S. W. Hawking,   Commun. Math. Phys.  [**43**]{}, 199 (1975)   Erratum: \[Commun. Math. Phys.  [**46**]{}, 206 (1976)\].    S. M. Christensen,   Phys. Rev. D [**14**]{}, 2490 (1976). P. Candelas,   Phys. Rev. D [**21**]{}, 2185 (1980). K. W. Howard and P. Candelas,   Phys. Rev. Lett.  [**53**]{}, 403 (1984).    K. Fredenhagen and R. Haag,   Commun. Math. Phys.  [**127**]{}, 273 (1990).    P. R. Anderson,   Phys. Rev. D [**41**]{}, 1152 (1990).    P. R. Anderson, W. A. Hiscock and D. A. Samuel,   Phys. Rev. Lett.  [**70**]{}, 1739 (1993). P. R. Anderson, W. A. Hiscock and D. A. Samuel,   Phys. Rev. D [**51**]{}, 4337 (1995).    B. S. Kay and R. M. Wald,   Phys. Rept.  [**207**]{}, 49 (1991). V. P. Frolov,   Phys. Rev. D [**26**]{}, 954 (1982). A. C. Ottewill and E. Winstanley,   Phys. Rev. D [**62**]{}, 084018 (2000)   \[gr-qc/0004022\]. G. Duffy and A. C. Ottewill,   Phys. Rev. D [**77**]{}, 024007 (2008)     \[gr-qc/0507116\].    H. R. C. Ferreira and J. Louko,   Phys. Rev. D [**91**]{}, no. 2, 024038 (2015)   \[gr-qc/1410.5983\]. A. Levi and A. Ori,   Phys. Rev. D [**91**]{}, 104028 (2015)   \[gr-qc/1503.02810\]. A. Levi, E. Eilon, A. Ori and M. van de Meent,   Phys. Rev. Lett.  [**118**]{}, no. 14, 141102 (2017)   \[gr-qc/1610.04848\]. H. Sahlmann and R. Verch, Commun. Math. Phys.  [**214**]{} (2000) 705 \[math-ph/0002021\]. I. Khavkine and V. Moretti, in Chapter 5 of R. Brunetti, C. Dappiaggi, K. Fredenhagen and J. Yngvason, ed., [*Advances in Algebraic Quantum Field Theory*]{}, Springer Verlag (2015), \[math-ph/1412.5945\]. M. Banados, C. Teitelboim and J. Zanelli, Phys. Rev. Lett.  [**69**]{} (1992) 1849 \[hep-th/9204099\]. M. Banados, M. Henneaux, C. Teitelboim and J. Zanelli, Phys. Rev. D [**48**]{} (1993) 1506 \[gr-qc/9302012\]. G. Lifschytz and M. Ortiz,   Phys. Rev. D [**49**]{}, 1929 (1994)     \[gr-qc/9310008\]. C. Dappiaggi and H. R. C. Ferreira, Phys. Rev. D [**94**]{}, no. 12, 125016 (2016) \[gr-qc/1610.01049 \]. C. Dappiaggi and H. R. C. Ferreira, (2016)   \[math-ph/1701.07215\]. A. Zettl, [*Sturm-Liouville Theory,*]{} American Mathematical Society, (2005). J. Weidmann, [*Spectral Theory of Ordinary Differential Operators*]{}, Springer (2006). A. Ishibashi and R. M. Wald, Class. Quant. Grav.  [**20**]{} (2003) 3815 \[gr-qc/0305012\]. A. Ishibashi and R. M. Wald, Class. Quant. Grav.  [**21**]{} (2004) 2981 \[hep-th/0402184\]. I. Seggev,   Class. Quant. Grav.  [**21**]{}, 2651 (2004)     \[gr-qc/0310016\]. H. R. C. Ferreira and C. A. R. Herdeiro, (2017)   \[gr-qc/1707.08133\]. A. R. Steif, Phys. Rev. D [**49**]{} (1994) 585 \[gr-qc/9308032\]. P. Breitenlohner and D. Z. Freedman, Annals Phys.  [**144**]{}, 249 (1982). F. Olver, [*NIST Handbook of Mathematical Functions,*]{} Cambridge University Press (2010). M. K. Parikh and F. Wilczek, Phys. Rev. Lett.  [**85**]{} (2000) 5042 \[hep-th/9907001\]. V. Moretti and N. Pinamonti, Commun. Math. Phys.  [**309**]{} (2012) 295 \[gr-qc/1011.2994\]. D. Binosi, V. Moretti, L. Vanzo and S. Zerbini, Phys. Rev. D [**59**]{} (1999) 104017 \[gr-qc/9809041\]. A. S. Markus, [*Introduction to the Spectral Theory of Polynomial Operator Pencils*]{}, American Mathematical Soc. (2012). M. V. Keldysh, Dokl. Akad. Nauk SSSR [**77**]{} 11 (1951). M. Reed, B. Simon, [*Methods of Modern Mathematical Physics, Volume I: Functional Analysis*]{}, Elsevier (1978). T. Kato, [*Perturbation theory for linear operators,*]{} Springer (2013). L. Hörmander, [*The Analysis of Linear Partial Differential Operators [I]{}: Distribution Theory and Fourier Analysis*]{}, Springer (1998). E. Hille, R. S. Phillips, [*Functional analysis and semi-groups*]{}, American Mathematical Soc. (2008). P. R. Halmos, V. S. Sunder, [*Bounded Integral Operators on $L^2$ Spaces*]{}, Springer (1978). H. Langer, pp.1–46 in [*Functional analysis (Dubrovnki, 1981)*]{}, A. Dold and B. Eckmann (eds.), Springer (1982). J. Bognár, [*Indefinite inner product spaces*]{}, Springer (1974). M. Kaltenbäck, R. Pruckner, Integral Equations and Operator Theory [**83**]{} 451 (2015). \[math-FA/1502.03222\] B. Ćurgus, Integral Equations and Operator Theory [**12**]{} 615 (1989). N. I. Akhiezer, I. M. Glazman, [*Theory of Linear Operators in Hilbert Space*]{}, Dover (1993). P. Jonas, H. Langer, J. Operator Theory [**2**]{} 63 (1979). [^1]: This could be rigorously established by a careful application of the *argument principle*, which we omit for brevity, to the function $f(\tomega) = \tan(\zeta) - \Theta(\tomega)$, which confirms the existence of a single simple zero $\tomega_\zeta \in \Im[\tomega]$ provided the integrals $\oint \frac{f'(\tomega)}{f(\tomega)} \frac{\dd \tomega}{2\pi i}$ stabilize to the value $1$ over a sequence of simple closed and positive contours whose interior exhausts the upper half complex plane.

--- abstract: 'Graphs change over time, and typically variations on the small multiples or animation pattern is used to convey this dynamism visually. However, both of these classical techniques have significant drawbacks, so a new approach, Storyline Visualization of Events on a Network (SVEN) is proposed. SVEN builds on storyline techniques, conveying nodes as contiguous lines over time. SVEN encodes time in a natural manner, along the horizontal axis, and optimizes the vertical placement of storylines to decrease clutter (line crossings, straightness, and bends) in the drawing. This paper demonstrates SVEN on several different flavors of real-world dynamic data, and outlines the remaining near-term future work.' author: - | Dustin L. Arendt[^1]\ Air Force Research Laboratory - | Leslie M. Blaha[^2]\ Air Force Research Laboratory bibliography: - 'refs.bib' title: 'SVEN: Informative Visual Representation of Complex Dynamic Structure [^3] ' --- Introduction ============ Graphs have become an essential mathematical tool to aid in the representation and analysis of relationships between discrete entities such as people, companies, computers, etc. Not surprisingly, visualization of graphs (i.e., graph drawing) has been an active area of research for the past several decades. However, the world is a dynamic place, so graphs that model objects and their relationships in the world often change over time. To account for this, one often chooses a particular time interval of interest and constructs a single graph capturing the relationships observed in that interval. This is the [*de facto*]{} method for social network analysis, where the graph is constructed from a series of interviews, questionnaires, or observations occurring within some time interval [@wasserman1994]. Visualization of the graph often plays a crucial role in the initial understanding of the phenomena of interest. There is no reason to require that such data must be compressed into a single graph aside from convenience. On the contrary, one would expect a [*longitudinal*]{} analysis to potentially yield more insight into dynamic phenomena than a more restrictive static analysis. A typical way to understand a dynamic graph is to induce a sequence of subgraphs $$\{G_1=(V_1,E_1), G_2=(V_2,E_2), ...\},$$ by partitioning a time interval into several smaller non-overlapping time windows. Alternatively we can consider a sequence of interaction events on a continuous timescale as a set of time-edge tuples of the form $$\{(t_1,(u_1,v_1)), (t_2,(u_2,v_2)), ...\},$$ where the $i^{th}$ interaction occurs at time $t_i$ and involves $u_i$ and $v_i$, and reads as “at time $t_i$ node $u$ interacted with node $v$.” For simplicity we assume that interactions are pairwise and of negligible duration relative to the timescale of the visualization; this type of network has been referred to as a “contact sequence” [@holme2012temporal], and an example is shown in Fig. \[fig:multigraph\]. Many relevant real-world phenomena can be described in this manner, such as a group of authors publishing a paper together, money being wired between bank accounts, one or more proteins activating or inhibiting another protein, and dialogue between characters a play or novel. ![Representation of a sequence of events on a network as a directed labeled multigraph. Labels on edges represent the time of the event.[]{data-label="fig:multigraph"}](multigraph-example.png){width=".35\textwidth"} Given that node-link visualizations of static graphs are wildly popular, it is not surprising that demand for effective dynamic graph visualization capability is increasing. Perhaps what is surprising is that, to date, no general purpose framework for dynamic graph visualization has become widely accepted in the same way that force directed placement has been for static graph visualization. Research in dynamic graph visualization began with showing the changes to the graph either as a sequence of still images (e.g., small multiples) for each $G_i$, or as an animation that interpolates between these drawings [@moody2005dynamic; @bender2006art]. While this approach to dynamic graph visualization problem inherits the challenges of optimizing the aesthetic properties of the static views of the graphs, it also introduces a new problem of how to change those views in a way that doesn’t confuse or mislead the viewer. It is hypothesized that the change between consecutive drawings should small to help preserve the user’s “mental map” [@purchase2007important; @archambault2011]. However, even if the mental map is well preserved, both small multiples and animation have significant drawbacks. The small multiples technique must decrease the size of the individual drawings in order to fit the entire collection on screen, and it can be tasking to trace nodes and their relationships as they jump from one view to the next. With animated drawings, motion can distract the user when that motion is an artifact of the layout algorithm and not reflective of an important change to the graph structure. Also, the user will be burdened by seeking back and forth within the animation to understand the important changes to the graph. While annotating an image is as trivial writing on a printed version, annotating an animation requires specialized software, and the animation should be rendered in a standard format. Furthermore, animated visualizations prevent multiple viewers from simultaneously viewing different times within the animation. The issues created by the “classical” approaches to dynamic network visualization can be partially addressed through effective interactions and algorithms, and much research has been devoted to such iterative improvements. However, the visualization community is still awaiting broad acceptance of a “workhorse visualization” for dynamic graph visualization. While not originally intended as general purpose solution for dynamic graph visualization, storyline visualization [@ogawa2010software; @kim2010tracing; @tanahashi2012design; @liu2013storyflow] holds promise for this. The contribution of this paper is the adaptation of storyline visualization techniques to dynamic network visualization–storyline visualization of events on a network (SVEN)–an effective visualization technique for both flavors of dynamic networks described above. Consistent with previous storyline visualization techniques, SVEN draws nodes as contiguous lines, and time is encoded on the horizontal dimension. Nodes that are interacting are drawn close to each other, and interactions are drawn as curved vertical lines that connect the interacting nodes at the appropriate time. SVEN contains an optimization algorithm to improve the aesthetic quality of the drawing by re-arranging and straightening the storylines, and renders the storyline using a metro map styling. What follows is a discussion of relevant literature related to dynamic graph visualization, a brief outline of the requirements and design of SVEN, and an overview SVEN’s layout algorithm, followed by several examples of layouts generated by SVEN. Integration of SVEN into a visual analytics software system and evaluation of SVEN is left as future work. Related Work ============ Dynamic graph visualization has its origin in static graph visualization, and many modern graph drawing algorithms have their basis in the original Kamada-Kawai [@kamada] and Fruchterman-Reingold [@fruchterman1991drawing] “force-directed” algorithms. These algorithms define a model of a physical system from the graph whose energy can be measured and consequently minimized to produce an aesthetically pleasing drawing, ideally. Popular, modern, freely available graph drawing packages include GraphViz [@ellson2002], Gephi [@bastian2009], and D$^3$ [@bostock2011; @dwyer2009]. Though not as mature as static graph visualization, the field of dynamic graph visualization is growing quickly. As stated previously, the majority of dynamic graph drawing approaches use small multiples [@chi1999] or animation [@moody2005dynamic; @bender2006art], but few have effective overviews that portray all the changes to the graph at once. Among techniques that do, an often repeated technique is to layer the 2-D graph drawings from each time slice into a third dimension [@brandes2003; @erten2004]. This approach shows connections between nodes during a specific time slice as well as the persistence of nodes over time slices, but the layering creates occlusion which decreases usability. CiteSpace-II analyzes and visualizes trends in academic research over time [@chen2006]. One of its visualizations overlays a node-link diagram on a timeline, with nodes representing events such as the publication of a scientific paper. Network analysts can often still gain insight when provided only the ego networks (the subgraph containing the ego and its direct neighbors) of individuals instead of the entire network. Based on this, Shi et al. developed a “1.5-dimensional” dynamic network visualization capable of showing the ego network of a particular individual of interest over time in a single picture [@shi2011dynamic]. Burch et al. adapted the classical parallel coordinates visualization for dynamic networks [@burch2011parallel]. In their system, vertices are drawn on parallel axes, the area between each axis represents a time interval, and edges connect vertices in adjacent axes. The inevitable problem of having an overwhelming number of edge crossings for larger datasets is addressed by reducing the opacity of the lines drawn. Recently, there has been increased interest in automated algorithms for “storyline visualization,” a technique where time is encoded in the horizontal dimension and characters are drawn as contiguous lines [@ogawa2010software; @kim2010tracing; @tanahashi2012design; @liu2013storyflow] inspired by an XKCD web comic showing presence of characters in scenes throughout various popular films [@xkcd]. When groups of characters interact, their storylines are closer within that time interval, relative to time intervals in which they are not interacting. However, before detailing these works it is useful to provide a more historical context, as there are several older techniques that are similar. Predecessors to Storyline Techniques ------------------------------------ To the author’s knowledge, the earliest portrayal of events on a network with a spatial encoding of time are “sequence diagrams,” developed in the 1970s, which are used to understand timing and synchronization in distributed systems [@lamport1978time]. These visualizations were originally hand drawn; nodes were represented as vertical parallel lines with time moving from the bottom to the top of the page. Communication events between processes were represented as wavy lines connecting connecting the processes sending and receiving the message at the local times the message was sent and received. Sequence diagrams are effective for understanding trivially small networks, but become difficult to scale up as more nodes are added. The ordering of nodes in the diagram can be chosen to minimize crossings, but this is the Traveling Salesman Problem, and even if an optimal solution was found, there is no guarantee that the resulting diagram would be interpretable for large or dense networks. Visibility representations, which have existed since the mid 1980s, are also visually similar to storylines [@tamassia1986visibility]. These are representations of planar graphs where nodes are drawn as horizontal lines, and edges are drawn as vertical lines connecting their endpoints. Node-edge crossings are not allowed, so non-planar graphs do not have visibility representations. Due to this constraint, it is not apparent how visibility representations can be used as a general purpose solution for the dynamic graph visualization problem. Nor is it apparent how to overload the horizontal axis in a visibility representation to also encode time. Dwyer and Eades investigated using “columns and worms” to visualize dynamic network induced by movements of fund managers over time [@dwyer2002visualising]. This is a 3-D visualization where nodes are represented as contiguous elements in the time dimension, and re-arranged in two spatial dimensions to optimize force-based graph aesthetics. A disadvantage of this technique is that perspective distortion and occlusion can make it very difficult to accurately interpret timing and ordering of events. Some researchers have simplified the dynamic graph visualization problem by increasing the level of abstraction and portraying how the network communities change over time [@rosvall2010mapping; @reda2011visualizing]. Visualizing this information is inherently simpler than visualizing the underlying dynamic network. Rosvall and Bergstrom apply a significance clustering technique to the dynamic network at fixed time windows to partition the vertices into groups. Then the flow of nodes between clusters at consecutive time windows is visualized using a technique similar to Sankey diagrams [@schmidt2008sankey] or parallel sets [@bendix2005parallel], which they call “alluvial diagrams.” However, the authors make no attempt to improve the aesthetic quality of their visualizations by re-arranging nodes to reduce clutter (crossings between time windows). Instead nodes are ordered within each time window according to cluster size, which causes the diagrams to scale poorly as a function of the number of communities and time windows. Another disadvantage of Rosvall and Bergstrom’s technique is that significance clustering is performed on time windows independently, which might introduce noise (nodes oscillating between clusters over time), adding clutter and artifacts to the visualization. Berger-Wolf and Saia introduced an optimization based approach for dynamic community detection that overcomes this pitfall [@berger2006framework], and this technique was improved in [@tantipathananandh2007framework; @tantipathananandh2009constant]. This framework was utilized to visualize dynamic communities for several datasets. Communities, which span time, are represented as stacked rectangles that span the horizontal space of the visualization. Similar to storyline visualizations, nodes are represented as horizontal lines, placed inside communities, and can only exist in one community at a time. When a node changes communities a diagonal line is used to represent this change. Edges in the network are not shown, and no attempt is made to optimize the ordering of the communities or the nodes within communities. This would likely reduce clutter and improve the aesthetic properties of the visualization. Storyline Techniques -------------------- Several recent visualizations [@kim2010tracing; @ogawa2010software; @tanahashi2012design; @liu2013storyflow] appear to have been directly inspired by a series of hand drawn XKCD web comics [@xkcd] detailing character interactions in several popular films. These visualizations all have a very similar visual appearance, and will be referred to as “Storyline Visualizations” in this paper. Kim et al. use storylines to simultaneously portray birth, death, marriage, and divorce within a genealogical diagram [@kim2010tracing]. Their layout method is effective at producing an aesthetically pleasing drawing, but makes structural assumptions about the genealogical graph, preventing this technique from being applied to the general dynamic graph drawing problem. Concurrently, Ogawa and Ma developed “software evolution storylines,” intended to be more generally applicable to dynamic graph drawing [@ogawa2010software], which was later improved by Tanahashi and Ma [@tanahashi2012design]. Due to the challenging multi-objective nature of the problem, their approach uses an evolutionary algorithm to simultaneously improve line wiggles, line crossovers, and whitespace gaps. However, evolutionary algorithms are often computationally intensive, and when alternative solutions exist, they are frequently desirable. For this reason, Liu et al. improved on previous work by developing a multi-stage hybrid optimization technique with essentially the same problem domain and aesthetic criteria [@liu2013storyflow]. Their contributions were algorithmic complexity improvements and the inclusion of hierarchical constraints on the interactions to improve the correctness of the drawings. These latter two storyline visualization approaches both assume the data can be described in terms of “interaction sessions,” which are essentially data about the time, duration, and participants in an interaction event. However, it is assumed that an individual cannot be in more than one interaction session at any given time, which continues to impose a certain structure on the data being visualized. Thus, none of the storyline visualization techniques to date can be considered solutions for general purpose dynamic graph visualization. Requirements & Design ===================== Dynamic graph visualizations based on small multiples or animation optimize aesthetics, but do not encode time spatially. Conversely, many of the dynamic graph visualization techniques (besides storyline visualizations) that do encode time spatially do not effectively optimize aesthetics. Therefore, we propose two fundamental requirements for effective dynamic graph visualization: - Use space to represent time ([**R1**]{}), and - Reduce clutter ([**R2**]{}) by minimizing the following - node and edge crossings ([**R2a**]{}) - node bends ([**R2b**]{}) - edge length ([**R2c**]{}) We make a spatial encoding of time the first and most important requirement for effective dynamic graph visualization. Encoding time spatially has one of the longest lasting precedents in the entire field of information visualization–the technique has been used for hundreds of years and many historical hand drawn examples exist [@aigner2011timebook]. The second requirement inherits directly from the established visual aesthetics from classical node-link drawing techniques, and states that effective dynamic graph visualization should contain an algorithm that rearranges visual elements to decrease line crossings, line bends, and line length. Below we describe how SVEN is designed with these requirements in mind. Javed and Elmqvist compiled a useful set of common design patterns for composite visualizations [@javed2012exploring] that we find helpful–they allow us to describe how SVEN is constructed from simple visualization patterns. In the language of their framework, SVEN consists of juxtaposed arc diagrams (one per time window) with integrated nodes. Arc diagrams are a well known graph drawing technique where nodes are ordered and drawn in one dimension, and links are drawn as arcs of varying radius depending on the distance between nodes [@wattenberg2002arc]. Juxtaposition describes the placement of two or more views side-by-side, and integration is the practice of drawing lines to connect related objects in separate views. In our case, when one node is present in two adjacent (juxtaposed) arc diagrams, they are connected by a line segment. When detailed time information about events within a time window is known, the arcs within the arc diagram are shifted to the left or right accordingly. An example of constructing a SVEN drawing by juxtaposing arc diagrams is shown in Fig. \[fig:arc-diagrams\]. For clarity, the four nodes are assigned the same four levels in each arc diagram. However, this is not always the case, which may cause the integrating lines to cross, or to be drawn diagonally (instead of horizontally). In SVEN, the integrating lines are always horizontal within a time window; crossing and diagonal lines occur only between time windows. We refer to the crossing of integration lines between time windows as “node-node crossings.” Additionally, edges (arcs in the time windows) may be crossed by nodes (lines integrating adjacent time windows), and we refer to this as “node-edge crossings.” A third type of crossing occurs between edges within each time widow are “edge-edge crossings,” but at the present time SVEN does not directly minimize edge-edge crossings (Fig. \[fig:arc-diagrams\] contains several edge-edge crossings). Fig. \[fig:cross\] shows a idealized SVEN drawing that contains one node-node and one node-edge crossing. ![(a) A single arc diagram (b) Juxtaposed arc diagrams (c) Integrated & juxtaposed arc diagrams (SVEN)[]{data-label="fig:arc-diagrams"}](arc-diagrams.png){width=".5\textwidth"} ![An simple SVEN rendering with two time windows that contains one node-node crossing (a) and one node-edge crossing (b).[]{data-label="fig:cross"}](crossings.png){width=".5\textwidth"} The ordering, juxtaposition, and integration of arc diagrams for each time window directly supports [**R1**]{}, by providing a spatial representation of time in the composite visualization. Furthermore, the use of arc diagrams (instead of node-link diagrams, for example) makes more effective use of screen space by constructing a two-dimensional drawing out of a sequence of one dimensional drawings. The ordering of each node within each time window can be adjusted to minimize node-node and node-edge crossings, supporting [**R2a**]{} and [**R2c**]{}. The spacing between nodes within each time window can be adjusted to straighten the integrating lines, supporting [**R2b**]{}. Past storyline visualization techniques [@kim2010tracing; @ogawa2010software; @tanahashi2012design; @liu2013storyflow] require that storylines move closer while the participants are interacting and then move apart afterwards. We do not make this an explicit requirement in SVEN because interaction events are visually mapped onto arcs in the diagram, and for very complex interactions it may impossible or ambiguous to encode interaction as proximity. Rather, proximity is more of a secondary visual cue in SVEN, and is minimized indirectly during the determination of node order within each time window. We also note that many two-dimensional static graph drawing algorithms make the same tradeoff via force directed energy minimization schemes. Few algorithms explicitly explicitly solve the layout based on the constraint that the nearest neighbors of a node in the graph drawing are also its neighbors in the graph, with Ref. [@shaw2009structure] being a rare exception. Finally, we note that omitting this requirement simplifies SVEN’s layout algorithm, which we describe in detail in the following section. Layout Algorithm ================ SVEN has five basic steps to transform raw data into a dynamic graph drawing: 1. [**Ingest:**]{} Raw data is transformed into an graph sequence or event sequence partitioned into time windows, and from this an aggregate graph is constructed. 2. [**Order:**]{} The ordering of nodes within each time window is chosen to minimize line crossings and length. 3. [**Align:**]{} As many nodes as possible are chosen for alignment to minimize line bends. 4. [**Place:**]{} Spacing between nodes is chosen to assign positions based on the ordering and alignment, and to further minimize edge length. 5. [**Render:**]{} The nodes and links are drawn, colored, labeled, and interpolated as appropriate. All optimization of aesthetics ([**R2**]{}) occurs during the ordering, alignment, and placement phase, which is similar to the multi-stage approach taken by Liu et al. [@liu2013storyflow]. They first order and align the storylines using discrete optimization techniques, and then determine the placement of the storylines by adjusting the space between the storylines using a continuous optimization strategy (i.e., quadratic programming). Our overall approach is similar, but the techniques we use within these stages differ significantly, and where appropriate we highlight these differences. Below we describe our five steps in detail. Ingestion --------- The data ingestion phase is responsible for taking the raw data from the user and transforming it into a sequence of events or induced graphs. This phase is application specific; interested readers can find a comprehensive discussion of dynamic networks and their applications in Ref [@holme2012temporal]. Below we describe some of the preprocessing that occurs immediately after ingestion. ### Aggregate Graph Given a sequence of graphs induced from time windows, an aggregate graph $G_{agg}=(V_{agg},E_{agg})$ is constructed. If an edge $(u,v)\in E_i$ then $((i,u),(i,v))\in E_{agg}$ with the same weight as the original edge. Additionally, if $u\in V_i \cap V_{i+1}$ then $((i,u),(i+1,u))\in E_{agg}$. The weight of this edge can be specified by the user, and larger values will make the corresponding node more likely to be straightened across time windows. An example of an aggregate graph is shown in Fig. \[fig:agg\]. ### Continuation & Discretization Optionally, a more classical view of the storyline can be accomplished via continuation of the storylines. This is done by calculating $v_{min}$ and $v_{max}$, the first and last appearances of a given node $v$. For a given time window $i$, if $v_{min} < i < v_{max}$, then $v$ is added to $V_i$ if it is not already present. The inverse of this operation is discretization, where $v$ is removed from $V_i$ if its degree in $G_i$ is 0. When necessary, these operations are performed prior to construction of the aggregate graph. ![The aggregate graph for the first three acts of [*Macbeth*]{}. Solid lines indicate edges within the original graphs, and dashed lines indicated edges between the adjacent graphs.](aggregate-graph.pdf){width=".85\textwidth"} \[fig:agg\] Ordering -------- The purpose of the ordering phase is to determine an ordering of the nodes within each time window in a manner that supports [**R2**]{}. A good ordering can greatly reduce crossings, of which there are two types: node-node and node-edge. Node-node crossings occur between time windows when two nodes exist in adjacent scenes and their relative ordering is flipped going from one scene to the next. Node-edge crossings occur within a time window and are a result of connected nodes not being placed at adjacent levels in the graph. An example of the two types of crossings is shown in Fig. \[fig:cross\]. The problem of minimizing node-edge crossings and node-node crossings exists in a tradeoff space. If each node was assigned a unique level, then there would be no node-node crossings, but this would most likely produce a drawing with a large number of node-edge crossings and poor aesthetics. Similarly, if the node edge crossings were somehow minimized for each time window independently, then we would likely see many node-node crossings. Therefore, because these two aesthetic properties are in competition with each other, it is the job of the ordering algorithm to balance the two, and provide a means for the user to determine a weighting on their relative importance. We approach this problem by performing matrix seriation [@liiv2010seriation] on the aggregate graph, which induces an ordering of the nodes within each time window. There are several approaches to the matrix seriation problem, and we consider two of the most popular: spectral seriation and dendrogram seriation. Our spectral seriation approach is equivalent to a one dimensional “spectral layout” of the weighted graph [@hagberg-2008-exploring], and is also very similar to the “Laplacian eigenmap” embedding and dimension reduction technique [@belkin2001laplacian]. Essentially, the eigenvector corresponding to the smallest nonzero eigenvalue of the graph laplacian matrix is found. The nodes are then sorted according to this eigenvector to produce the ordering. This technique results in a very good approximation of the globally optimal solution, and scalable sparse implementations are readily available. Liu et al. illustrated the usefulness of placing hierarchical constraints on the ordering of storylines in order to maintain correctness of the drawing. For this reason we consider an alternative ordering method, dendrogram seriation, which is most commonly known for its use in the “cluster heat map” [@wilkinson2009history]. Dendrogram seriation addresses the problem of ordering the leaves in a tree to place similar leaves (according to some arbitrary metric) near to each other. This ordering must be consistent with the tree structure, and only $2^{n-1}$ of the $n!$ possible permutations of the leaves meet this requirement. There are several algorithms which can solve this problem optimally in polynomial time [@bar2001fast; @brandes2007optimal]. Then there is the matter of finding the tree in the first place. The tree may be determined fully or partially from the hierarchical constraints placed on the nodes by the user. However, it may also be useful to generate the tree using a community detection algorithm, and many algorithms exist for this problem [@noack2009multi; @fortunato2010community]. Many of these algorithms are designed to find partitions of the network with high modularity–where clusters of nodes have many intra-cluster edges and few inter-cluster edges. This property is desirable to help minimize edge crossing ([**R2a**]{}) and length ([**R2c**]{}), as these nodes will be placed close to each other. Alignment --------- The purpose of the alignment stage is to improve the readability of the drawing by straightening nodes, or removing “wiggles.” The drawing has a wiggle whenever a node is placed at a different height in two adjacent time windows. In Lin et al.’s alignment phase, wiggles are minimized by solving the longest common subsequence (LCS) problem once per time window, assuming the node order in the previous and current time window are the sequences. While apparently effective for the datasets presented in the paper, we believe this is not the best approach for the problem. Fig. \[fig:lcs-counter\] demonstrates a typical case where LCS alignment performs poorly. In general the LCS technique will not scale effectively, as only one (potentially small) subsequence gets aligned, leaving the remaining nodes unaligned. ![Longest common subsequence alignment vs maximum weighted independent set (optimal). For this example, the longest common subsequence has length 1, and only one node will be aligned regardless of which subsequence is chosen. A greedy MWIS algorithm will find two nodes to be aligned, which is optimal for this example.[]{data-label="fig:lcs-counter"}](lcs-counter.png){width=".5\textwidth"} Instead, we propose solving this problem by framing the alignment problem as a maximum weighted independent set problem (MWIS). To do so we must first understand that two nodes cannot be simultaneously aligned if those two nodes cross (e.g., nodes 0 and 1 in Fig. \[fig:lcs-counter\]), because doing so would change their ordering. Therefore, we can find all pairs of nodes that cross and construct a constraint graph $G_c$. For a pair of adjacent time windows, the nodes of the constraint graph are those nodes that are present in both time windows (i.e., nodes that will be connected with integrating lines). Two nodes are connected in the constraint graph if their integrating lines cross, which can be found directly from the ordering of the nodes. A set of feasible nodes to align is a subgraph of $G_c$ that has no edges. Finding the largest such subgraph will align the most nodes, which is precisely the maximal independent set problem. If the nodes are weighted according to importance by the user (e.g., how important it is for a particular node to be aligned), the problem becomes the MWIS problem where the objective is to find a subgraph of $G_c$ that contains no edges and whose node weight sum is the largest. While being an NP-hard problem and the subject of much research, several simple greedy heuristics suffice as a fast approximations with good provable bounds [@sakai2003note]. Placement --------- In the placement phase the vertical positions of each node are determined, subject to the ordering and alignment constraints found in the previous stages of the layout algorithm. In this stage, the empty space between nodes should be effectively used to shorten edge lengths as much as possible, supporting [**R2c**]{}. Similarly, Liu et al. discuss a “compaction” phase, whose objective is to minimize wiggles and unnecessary whitespace. This is the continuous portion of their hybrid layout algorithm, and they rely on a quadratic program solver to perform the optimization. Instead, we are able to phrase our placement problem in a graphical, rather than numerical framework, allowing the optimal result to be quickly found. Given the ordering and alignment constraints we first construct a weighted directed acyclic graph (DAG). Then, we use the network simplex algorithm [@gansner1993technique] to compute an optimal level assignment for the DAG. The levels found are used directly as the heights of the nodes in the final SVEN layout. We found the naive $O(|V|^2\cdot |E|)$ version of the network simplex algorithm was straightforward to implement and ran quick enough to be effective for the datasets we examined. Next we provide some more details on how to construct the DAG described above. The first step is to determine the groups of nodes that are being aligned. This can be accomplished by constructing a graph containing an edge $((i,v),(i+1,v))$ if node $v$ was aligned between time window $i$ and $i+1$. We denote $C(i,v)$ to be the connected component class of node $v$ in time window $i$ and $s(i,j)$ as the $j^{th}$ node in time window $i$. The DAG, is constructed by adding edges between connected component classes that enforce the ordering previously found for each time window, and these edges have the form $(C(i,s(i,j)),C(i,s(i,j+1)))$. An edge $(u,v)$ in $G_i$ contributes a weight of $w(u,v)$ to all edges in the DAG of the form $(C(i,s(i,j)),C(i,s(i,j+1)))$ where $s(i,u) \leq s(i,j) < s(i,v)$. For clarity, Fig. \[fig:dag\] illustrates how to construct the DAG containing the ordering and alignment constraints. ![Illustration of how to construct a weighted DAG representing the ordering and alignment constraints. (a) The connected component classes of the aligned nodes are connected with directed edges when adjacent. The small circles represent nodes within each time window, and the larger rounded boxes represent the connected component classes resulting from the alignment. (b) Each edge in $E_i$ from time window $i$ contributes its weight to each edge in the DAG it spans.[]{data-label="fig:dag"}](dag-explanation.png){width=".5\textwidth"} Rendering --------- We render the storylines using a metro map design, wherever possible drawing straight lines with smooth bends and nameable colors. When a node appears in consecutive time windows, their storyline is interpolated between the time windows. A node’s storyline disappears when that node is not present in the following time window. The ends of the storylines are capped with a arrows if node’s storyline reappears later (or previously), otherwise they are capped with circles. There are likely to be more nodes than nameable colors, so an application specific means for assigning colors is needed. By default, we assign nameable colors to the most frequently occurring nodes in the graph, and a light gray color to the remaining nodes. Results ======= Here we briefly show several examples of layouts generated using SVEN from a variety of datasets. Fig. \[fig:ncaaf\] demonstrates the utility of SVEN’s alignment and placement stages. It shows changes in the weekly NCAA Football rankings from the USA Today Poll[^4] throughout the 2013 season. Only the top 10 teams from each week are shown. Fig. \[fig:ncaaf-noopt\] shows a direct mapping of the changing rankings using SVEN’s metro map rendering style without any optimization (i.e., there is 1 unit of space between each team). Fig. \[fig:ncaaf-best\], demonstrates the use of SVEN’s alignment and placement stages, while maintaining the ordering from the ranking. We noticed that in this dataset, generally rank increases in the top ranked teams are small whereas rank decreases are large. A rank increase for some team $A$ will generally occur when another higher ranked team $B$ loses a game, causing $B$ to drop below $A$. Therefore, in MWIS algorithm nodes are weighted to prefer aligning nodes with rank increases over time (i.e., weights provided to the MWIS algorithm are $1$ unless a team’s rank decreases, in which case the weight is $0$), which is shown in Fig. \[fig:ncaaf-truthy\]. The effect of this weighting is a representation of the data that highlights sudden decreases in a team’s rank. We note that the representation in Fig. \[fig:ncaaf-truthy\] is not a linear encoding of the ranks, which makes reading the true rank of the team directly from the diagram more difficult, but relative rankings are more readily apparent. Fig. \[fig:yeast\] is a representation of the dynamics occurring during cell division for a simplified model of budding yeast [@Li2004]. During this process, various proteins interact by activating or inhibiting other proteins, or naturally decay via degradation. This carefully choreographed sequence of events is responsible for regulating each phase of the cell division. The process can involve hundreds of different proteins, but the dynamical model developed by Li et al. [@Li2004] greatly summarizes the dynamics in terms of just a handful of key players. The result of their analysis is a discrete-time discrete-state description of the cell-cycle dynamics. Combined with the protein interaction network, we can deduce which proteins were responsible for activation and inhibition between each discrete time step, and we represent these as $(t,(u,v))$ events, to be read as “at time $t$ protein $u$ contributed to the activation or inhibition of protein $v$.” Note that more than one protein can contribute (i.e., $v$ can have an in-degree that is greater than 1). We also ensure that if protein $u$ is denoted as active by the dynamical system model at time step $i$, then node $u$ is present in $G_i$ in the visualization. Additionally, because the dynamics represent a cycle, where the final state and starting states are consistent, we ensure that this is accounted for in the ordering, and alignment stages in SVEN. The effect of this is seen in Fig. \[fig:yeast\] where Sic1 and Cdh1 are present at the same level within the first and last time window. ![Budding yeast cell cycle modeled by [@Li2004]. Edges represent activation or inhibition, which occurs between discrete time steps in the model.[]{data-label="fig:yeast"}](cell-cycle-storyline.pdf){width=".85\textwidth"} The “Newcomb Fraternity” dataset is a classical longitudinal study of social dynamics [@nordlie1958longitudinal; @newcomb1961acquaintance; @boorman1976social]. It contains fifteen $17\times 17$ subjective ranking matrices that represent seventeen individuals’ rankings of the other 16 fraternity members, where a low ranking indicates that member views the other member as a close friend. The fifteen versions of the ranking matrices represent measurement of these rankings over time. This dataset is transformed into a sequence of graphs as follows. Let $r_{ij}^t$ be the subjective ranking of individual $j$ by individual $i$ at time $t$. Then $E_t$ contains the edge $(i,j)$ if and only if $r_{ij}^t \leq \epsilon$ and $r_{ji}^t \leq \epsilon$, where $\epsilon$ is an arbitrarily chosen threshold. This results in an induced graph for each time window that represents close and reciprocated friendships (we let $\epsilon = 2$, as this is the smallest $\epsilon$ that allows for non-trivial structures). Fig. \[fig:frat\] shows the SVEN visualization of this dataset. ![The Newcomb Fraternity longitudinal study [@nordlie1958longitudinal; @newcomb1961acquaintance; @boorman1976social]. The sequence of graphs is induced by considering close reciprocated friendships.[]{data-label="fig:frat"}](newcomb.pdf){width=".85\textwidth"} Dynamic network visualizations can be useful to understand the evolution of discourse (i.e., a sequence of speakers) to understand who talks to whom, and how these relationships change over time. To process dialogue into a dynamic network, first a parser must extract the sequence of speakers $\{s_1, s_2, ...\}$ from the raw text data. Then, this sequence is transformed into a set of network events of the form $(t,(s_i,s_{i+1}))$ where $t$ is the line number and $s_i$ and $s_{i+1}$ are consecutive speakers. The event sequence is then partitioned into time windows; a good discretization of time would be to follow the act or scene breaks (in the case of a play) or chapters (in the case of a book), as these are the natural temporal breaks decided upon by the author. Fig. \[fig:plays\] shows SVEN visualizations of the dialogue in two well known Shakespeare plays[^5]. Conclusions & Future Work ========================= We have presented a preliminary algorithm for adapting storyline visualization techniques to the problem of dynamic graph visualization. Our solution is derived from two basic requirements: representing time using the horizontal axis ([**R1**]{}) and reducing clutter by rearranging the storylines along the vertical axis ([**R2**]{}). We approach this challenging problem by breaking the optimization into multiple stages (ordering, alignment, and placement) similar to previous work [@liu2013storyflow]. However, we differ significantly from previous work in how we accomplish these sub-tasks. Ordering of nodes is quickly and effectively accomplished by seriation of the aggregate graph matrix using spectral methods (alternatively, hierarchical graph clustering and dendrogram seriation can be used). Alignment of the nodes is performed by determining node crossings between adjacent time windows, and solving the maximum weighted independent set (MWIS) problem to maximize the number of straightened nodes. Then the placement of the nodes is optimized using the network simplex algorithm. We have applied SVEN to a variety of datasets. We showed how SVEN can be used to optimize the placement of rankings over time and that the weighting of nodes for the MWIS algorithm can have a beneficial effect on the readability of the diagram when properly chosen. We utilized SVEN to visualize the complex dynamics produced by a model of the budding yeast cell cycle, clarifying the sequence of activation, inhibition, and degradation that regulates the process. SVEN was used to visualize a classical longitudinal study of social acquaintances in a fraternity, highlighting close friendships and changing groups. Finally, SVEN was used to show the sequence of speakers in two well known Shakespeare plays. Much future work remains, and this work falls into three categories: implementing interactivity, performing evaluation, and addressing issues pertaining to scalability. Interactivity ------------- Interactivity is crucial to accommodate some of the shortcomings inherent to SVEN visualizations of dynamic networks. Many potentially useful interactions fall into the category of “details on demand,” where additional information can be given to the user when they request it. For example, when a node is absent from a time window, but present before and after, that node’s storyline will be broken making the line harder to trace through the visualization. A solution could be to complete the storyline for a given node at the user’s request. Another useful way to provide details is to allow the user to brush across the visualization, selecting a subset of the nodes and time simultaneously. Then an alternative visualization, such as a node link diagram or adjacency matrix, can be shown to the user for this particular subset where it may be more informative than the overview provided by SVEN. Temporal networks can have a large number of edges which can cause clutter. A solution would be to use transparency or other techniques to de-emphasize these edges, but then user may want to re-emphasize all the edges incident to a particular node over time or at a particular time upon request. Other interactions could involve supporting various analyses relevant to temporal networks, such as temporal shortest path. For example, the user could select a node at a particular time and request that the visualization shows when other nodes can be reached, the minimum number of steps this requires, and the particular optimal path(s) to accomplish this. Additional application specific details should also be presented directly in the visualization, or made available through interaction. Other standard interactions such as adding/deleting nodes/links and re-arranging the visualization would be important features of a fully functioning visual analytics tool for dynamic network visualization. Evaluation ---------- Evaluation of SVEN is an important future research activity–many unanswered questions remain. Crucially, we would like empirical evidence about what combinations of data and task allow users of SVEN to attain good performance relative to existing techniques. This can be accomplished with a controlled user study with simple tasks that have measurable performance. In this case it may be important to generate the datasets from a parametrized random model of a temporal networks, rather than using real-world datasets. To get a better understanding of how SVEN could provide insight to an analyst, the opposite is the case–datasets should be taken from the real world with a particular application and challenging question in mind. Evaluation by the “analysts” would be more subjective compared to the controlled user study, but potentially more informative. Additionally, objective evaluation of the layout algorithm against previous work, different node/link weightings, or different heuristics can be accomplished by measuring the aesthetic qualities of the resulting layouts (e.g., crossings, wiggles, line length). Scalability ----------- From a computational standpoint, SVEN is scalable to large datasets[^6]. The ordering stage’s time complexity, when using the spectral layout method, which requires computing the eigenvectors of a matrix, is $O(|V|^3)$ or better depending on sparsity. The alignment stage requires determining the set of nodes that cross between adjacent time windows. Our naive implementation simply compares all pairs of nodes, which has a time complexity of $O(|V|^2)$; a more sophisticated algorithm using bisection would have a time complexity of $O(|V|\log |V|)$ assuming the number of crossings per node is bounded by a small constant. Following this, the MWIS algorithm is run; our naive implementation of this is $O(|V|\cdot |E|)$, but can be improved by using a heap implementation of a priority queue with a increase/decrease key functionality. The placement stage requires running the network simplex algorithm; our naive implementation has a time complexity of $O(|V|^2 \cdot |E|)$, making this the most computationally intensive component of SVEN. However, even with an overall cubic worst case complexity, the layout can be computed under interactive conditions (i.e., latency $< 500$ms) for thousands of nodes and edges. We would expect the visualization to become cluttered and unusable long before the layout algorithm starts to running too slowly from the user’s perspective. Therefore, we suggest that future work should prioritize addressing the problem of scalability through effective interaction and abstraction techniques over fine tuning of the layout algorithm. Common ways of addressing scalability issues are through filtering and abstraction. For example, filtering can be used to reduce the number of nodes and edges to be shown, which in turn makes the layout easier to compute and potentially less cluttered. Filtering can be automatic by using degree of interest (DOI) techniques to assign a score to data elements, and visualizing a relevant subset of this data [@van2009search]. For networks, abstraction is frequently accomplished through clustering [@fortunato2010community; @noack2004energy; @abello2006ask; @henry2007nodetrix; @balzer2007level], where groups of nodes are aggregated together, and represented as a super node. In the visualization, more emphasis is placed on the clusters and the inter-cluster links, because clusters are assumed to be dense, and intra-cluster links are less informative. This technique can be adapted to dynamic networks by leveraging dynamic network clustering techniques. A simple approach is to compute a hierarchical clustering directly on the aggregate graph, and to use dendrogram seriation to compute the ordering of the nodes, allowing groups to be drawn directly on the visualization in a consistent manner. Acknowledgements {#acknowledgements .unnumbered} ================ The authors would like to thank Paul Havig at AFRL for his guidance and helpful comments throughout the project. This research is supported by AFOSR LRIR 12RH12COR to L.M.B., and was performed while D.L.A. held a National Research Council Research Associateship Award at the the Air Force Research Laboratory, Wright-Patterson AFB, Ohio. [^1]: D. Arendt is currently a research scientist at Pacific Northwest National Laboratory. email: dustin.arendt@pnnl.gov [^2]: email: leslie.blaha@us.af.mil [^3]: Distribution A: Approved for public release; distribution unlimited. 88ABW Cleared 10/31/2014; 88ABW-2014-5108 [^4]: source: ESPN <http://espn.go.com/college-football/rankings> [^5]: source: Project Gutenberg [www.gutenberg.org](www.gutenberg.org) [^6]: In the analysis of SVEN’s time complexity, we assume the number of time windows is a small constant, which is bounded more by the available screen space than the data.

--- abstract: 'We study the HBT interferometry of ultra-relativistic nuclear collisions using a freezeout model in which free pions emerge in the course of the last binary collisions in the hadron gas. We show that the HBT correlators of both identical and non-identical pions change with respect to the case of independent pion production. Practical consequences for the design of the event generator with the built in Bose-Einstein correlations are discussed. We argue that the scheme of inclusive measurement of the HBT correlation function does not require the symmetrization of the multi-pion transition amplitudes (wave-functions).' address: 'Department of Physics and Astronomy, Wayne State University, Detroit, MI 48202' author: - 'A. Makhlin and E. Surdutovich' date: 'September 4, 1998' title: | Preprint WSU-NP-4-98\ Sensitivity of HBT Interferometry to the Microscopic Dynamics of Freeze-out. --- Introduction {#sec:SNIn} ============ Pion interferometry is expected to provide important information about the space-time picture of ultrarelativistic nuclear collisions. It has already proved to be a sensitive tool for the detection of the collective motion of the matter created in the course of ultrarelativistic nuclear collisions [@MS; @AMS; @NA35; @NA44]. The primary goal of this paper is to show that under favorable circumstances, interferometry is capable of detecting the difference between various mechanisms of freezeout which is the last transient phase before the regime of free streaming of hadrons. The effect we rely upon is entirely due to real interactions at the freezeout stage, and its magnitude may serve as a measure of these interactions. We predict that the normalized two-particle correlator of [*non-identical*]{} pions (and even for pairs like $\pi p$, $pn$, etc.) must differ from the reference unit value at any difference $\Delta k$ of the pions momenta. The normalized correlator of identical pions does not approach unity at large $\Delta k$ and slightly exceeds value of 2 at $\Delta k=0$. (In the idealized model which is used in this paper to [*demonstrate*]{} these effects, they are small.) The second goal is to discuss the effect of the multiparticle final states on the one- and two-pion inclusive spectra. We demonstrate that the contribution of these states depends on the microscopic dynamical mechanism of freezeout. We explicitly show that the effect of the multi-particle final states is limited by the actual range of the interactions at the freezeout stage. No particles, which are causally or (and) dynamically disconnected from the two-pion inclusive probe, can contribute this effect. Our general conclusion is that the problem of the analysis of HBT data cannot even be posed without reference to an explicit dynamical model. Finally, our model may serve as a prototype of a realistic dynamical mechanism of freezeout, provided that the preceding stage of evolution is a hot gas of hadrons. Our results can be used for the simulation of truly quantum two-pion distributions with the input from the event generators based on semi-classical dynamics. The paper is organized as follows: In Sec. \[sec:SN1\], the effect is explained at the introductory level. We pose the problem of interferometry (in its most rigorous form) as a problem of a quantum transition specified by the observables and the initial data in Sec. \[sec:SN2\]. Sec. \[sec:SN3\] deals with the formulation of the pion interferometry problem and the results of calculations for the hydrodynamic theory of multiple production. A more realistic model of freezeout, which incorporates some elements of hadron kinetics, is considered in Sec. \[sec:SN4\]. The analytic answer for the model-based calculations is obtained by the end of this section. In Sec. \[sec:SN5\], we discuss the effect of multi-pion final states on the inclusive two-pion spectrum and speculate about physical phenomena that may lead to the increase of the effect. A practical issue of the event generator, with the built-in Bose-Einstein correlations, is discussed in Sec. \[sec:SN6\]. Physical motivation {#sec:SN1} =================== The simplest and most popular introductory explanation of the principles of HBT interferometry is as follows: The amplitude to emit (“prepare”) the pion at the point $x_N$ and to detect it with the momentum $k_1$ is $a_N(k_1)e^{-ik_1x_N}$.[^1] For a system of two pions prepared at points $x_N$ and $x_M$, and detected with the momenta $k_1$ and $k_2$, the transition amplitude (wave function) is a superposition of two indistinguishable amplitudes, $$\begin{aligned} {\cal A}_{NM}(k_1,k_2) = a_N(k_1)e^{-ik_1x_N}a_M(k_2)e^{-ik_2x_M} + a_N(k_2)e^{-ik_2x_N}a_M(k_1)e^{-ik_1x_M}~, \label{eq:E1.1}\end{aligned}$$ and for a system of distributed pion sources, the two-particle inclusive spectrum is $$\begin{aligned} {dN^{(2)} \over d{\bbox k}_{1} d{\bbox k}_{2} } =2~\sum_{N,M} \big\{ |a_N(k_1)a_M(k_2)|^2 +{\rm Re}~ \big[(a_N(k_1)a_M(k_2)a^*_N(k_2)a^*_M(k_1)e^{-i(k_1-k_2)(x_N-x_M)}\big]\big\}~. \label{eq:E1.2}\end{aligned}$$ This scheme carries an implicit assumption that the pions are truly independently [*created*]{} in the state of free propagation. Unfortunately, in the literature, this property is mostly taken for granted, though in the real world, it takes place only in very special occasions. For example, independence of pion production in nuclear collisions has been used to describe Bose-Einstein correlations at Bevalac energies when the excited nuclear matter is dominated by nucleons [@GKW]. In that case, the bremsstrahlung of pions, that accompanies scattering of nucleons, was suggested to provide independent pion sources. At RHIC energies (100 GeV/nucleon), we anticipate that the nuclear matter before the freezeout is totally dominated by pions and only very few nucleons are expected in the central rapidity region. There is a long-standing conjecture that the last form of the matter, before the freezeout, is a hot expanding hadronic gas. If this is indeed the case, then the main type of interaction at the freezeout stage is the binary collisions of pions. Therefore, the free propagation of any pion starts after it has experienced the “last” collision in the hadronic gas, and there is no really independent free pion production. Indeed, in this case, at least four particles are involved in the interference process, because at least two particles appear in the final state in each binary collision. If two pions (say, $\pi^+\pi^+$) are detected, then there are at least two more undetected particles emerging from two different collisions, and these particles may be identical as well. Thus, we encounter an additional (hidden) interference which affects the measured inclusive two-particle cross section.[^2] Moreover, even if the detected particles are different (e.g., $\pi^+\pi^-$), their partners still may be identical (e.g., the process $\pi^+\pi^0\to\pi^+\pi^0$ takes place at the coordinate $x_N$, and $\pi^-\pi^0\to\pi^-\pi^0$ takes place at the coordinate $x_M$). Therefore, if the last interaction is the binary collision in the pion gas, the naïve scheme (\[eq:E1.1\]) has to be modified. In this modified case, there are four interfering amplitudes, $$\begin{aligned} {\cal A}_{NM}(k_1,k_2;q_1,q_2) = a_N(k_1,q_1)a_M(k_2,q_2)e^{-i(k_1+q_1)x_N} e^{-i(k_2+q_2)x_M} \nonumber\\ +a_N(k_1,q_2)a_M(k_2,q_1)e^{-i(k_1+q_2)x_N} e^{-i(k_2+q_1)x_M}\nonumber\\ +a_N(k_2,q_1)a_M(k_1,q_2)e^{-i(k_2+q_1)x_N} e^{-i(k_1+q_2)x_M}\nonumber\\ +a_N(k_2,q_2)a_M(k_1,q_1)e^{-i(k_2+q_2)x_N} e^{-i(k_1+q_1)x_M}~. \label{eq:E1.3} \end{aligned}$$ The expression for the two-particle spectrum thus becomes more complicated, $$\begin{aligned} {dN^{(2)} \over d{\bbox k}_{1} d{\bbox k}_{2} } = \sum_{N,M}\sum_{q_1,q_2}|{\cal A}_{NM}(k_1,k_2;q_1,q_2)|^2 =4 \sum_{N,M}\sum_{q_1,q_2}\big\{~|a_N(k_1,q_1)a_M(k_2,q_2)|^2\nonumber\\ +{\rm Re}~[~a_N(k_1,q_1)a_M(k_2,q_2)a^*_N(k_2,q_1)a^*_M(k_1,q_2) e^{-i(k_1-k_2)(x_N-x_M)}\nonumber\\ +a_N(k_1,q_1)a_M(k_2,q_2)a^*_N(k_1,q_2)a^*_M(k_2,q_1) e^{-i(q_1-q_2)(x_N-x_M)}\nonumber\\ + a_N(k_1,q_1)a_M(k_2,q_2)a^*_N(k_2,q_2)a^*_M(k_1,q_1) e^{-i(k_1+q_1-k_2-q_2)(x_N-x_M)}~]~\big\}~, \label{eq:E1.4} \end{aligned}$$ and contains [*three*]{} interference terms (see Fig. \[fig:fig1\]). The first interference term corresponds to the familiar “opened” interference in the subsystem of the detected pions. It is present only if the detected pions are identical. The two other terms describe the “hidden” interference in the subsystem of undetected pions which, however, are unavoidable partners of the detected pions in the process of their creation in the final states. These terms are present even if the detected pions are different. In fact, we deal with the [*dynamically generated*]{} situation when the system of two pions is described by a density matrix and not by the wave function. When the dynamics of the system is driven by binary collisions, the two pions just cannot be found in a pure state![^3] A full scenario of ultrarelativistic heavy-ion collisions is still absent and an understanding of the quantum kinetics from microscopic quantum field theory, rather than from semi-classical approach, is only beginning to emerge. Currently, the most self-consistent scenario is based on an assumption that equilibration happens very early. Then it becomes possible to appeal to the equation of state of nuclear matter at various stages of the scenario, starting from the quark-gluon plasma (QGP) and ending up with the hadronic gas (as is done, e.g., in Ref.[@Shuryak].) If this is not the case (no hadronic gas occurs), the pions may be created in the course of hadronization and immediately freely propagate. In this situation, it is more likely that the pions are created independently. Thus, the difference between Eqs. (\[eq:E1.1\]), (\[eq:E1.2\]) and (\[eq:E1.3\]), (\[eq:E1.4\]) becomes of practical importance. Pion interferometry provides a tool which is capable of distinguishing between these two scenaria. In Sec. \[sec:SN5\], we argue that the effect of hidden interference may become even stronger if the system approaches the freezeout stage with the “soft” equation of state when the correlation length is long (as is, e.g., in the vicinity of a phase transition). Furthermore, it is easy to understand that the correlations due to the hidden interference exist even, e.g., between pions and protons, protons and neutrons, etc. All these effects, which may carry significant information about the freeze-out dynamics, require (and deserve) special study, which is beyond the scope of this paper. We insist that it is highly desirable (though not easy) either to measure all these correlation effects or to establish their absence at a high level of confidence. Both results are important since they would allow one to constrain the full self-consistent scenario of heavy-ion collisions which is absent now. These constraints may also affect theoretical predictions of the photon and dilepton yields in heavy-ion collisions. Theoretical background: Definition of observables {#sec:SN2} ================================================= A precise definition of observables is extremely important, because the HBT interferometry does not allow one to pose the mathematically unambiguous inverse problem. In general, we have to formulate a model and compare the solution with the data, relying on common sense and physical intuition. The full set of assumptions that accompany the formulation of the model is never articulated in full. Particularly, the question about the nature of states in which the particles are created is never discussed. However, this is the key issue. Interferometry is a consequence of the interference, and, in quantum mechanics, the latter cannot be even addressed without direct reference to the quantum states. Indeed, interference takes place every time when, with a given initial state, there are at least two alternative histories of evolution to a given final state. The wave function is nothing but a transition amplitude between the two states. HBT interferometry studies the two-particle wave functions (or, more precisely, the two-particle density matrix). Therefore, the most natural way to avoid any ambiguity is to pose the whole problem as a problem of a quantum transition.[^4] Let $|{\rm in}\rangle$ be one of the possible initial states of the system emitting a pion field which has three isospin components $$\begin{aligned} \mbox{\boldmath $\pi$}(x)\equiv\{\pi_i(x)\} =\{\pi_+(x),\pi_0(x),\pi_-(x)\}~, \label{eq:E2.1}\end{aligned}$$ where $\pi_-(x)=\pi_{+}^{\dag}(x)$, and $\pi_0(x)=\pi_{0}^{\dag}(x)$. At the moment of the measurement ($t_f\to +\infty$) each component of the pion field can be decomposed into the system of analyzer eigenfunctions $f_{\bbox k}(x)$, $$\begin{aligned} \pi_i(x) =\int d{\bbox k}[A_i({\bbox k})f_{\bbox k}(x)+ A^{\dag}_{i}({\bbox k})f^{\ast}_{\bbox k}(x)],~~~ f_{\bbox k}(x)=(2\pi)^{-3/2}(2k_{0})^{-1/2}e^{-i\,k\cdot x}~. \label{eq:E2.3}\end{aligned}$$ This expansion holds only after freezeout. In general, ${\bbox\pi}(x)$ is a multiplet of Heisenberg operators driven by the evolution operator $S$. The annihilation operators $A_i({\bbox k})$ of the pion field are given by $$\begin{aligned} A_{i}({\bbox k})\;=\;\int_{x^0=t_f} d^{3}x\; f^{*}_{\bbox k}(x)~i \tensor{\partial_{x}^{0}}~\pi_i(x)~,~~~~%\nonumber\\ A_{i}^{\dag}({\bbox k})\;=\;\int_{x^0=t_f} d^{3}x\;\pi_{i}^{\dag}(x)~ i\tensor{\partial_{x}^{0}}~f_{\bbox k}(x)~. \label{eq:E2.4}\end{aligned}$$ The operator $A_i({\bbox k})$ describes the effect of a detector (analyzer) far from the point of emission, so, by definition, the pion is detected on mass-shell, $k^0=({\bbox k}^2 +m^2)^{1/2}$. The inclusive amplitudes to find one pion with momentum ${\bbox k}$ and two pions with the momenta ${\bbox k}_1$ and ${\bbox k}_2$ in the final state are $$\begin{aligned} \langle X| A_i ({\bbox k}) S |{\rm in}\rangle ~~~{\rm and}~~~ \langle X| A_i ({\bbox k}_1) A_j ({\bbox k}_2)S |{\rm in}\rangle~, \label{eq:E2.5}\end{aligned}$$ respectively. Here, the states $|X\rangle$ form a complete set of all possible secondaries. Summing the squared moduli of these amplitudes over all (undetected) states $|X\rangle$, and averaging over the initial ensemble, we find the one-particle inclusive spectrum $$\begin{aligned} { dN_{i}^{(1)} \over d{\bbox k} } = {\rm Tr} {\hat \rho}_{\rm in} \sum_{X} S^\dagger A_{i}^{\dag}({\bbox k})|X\rangle\langle X|A_i({\bbox k})S ={\rm Tr} {\hat \rho}_{\rm in} S^\dagger A_{i}^{\dag} ({\bbox k}) A_i({\bbox k})S~, \label{eq:E2.6}\end{aligned}$$ where the density operator ${\hat \rho}_{\rm in}$ describes the emitting system. Of these two equations, the second one describes the algorithm of the measurement. In other words, we deal with the operator $~N_i({\bbox k}) = A_{i}^{\dag}({\bbox k}) A_i ({\bbox k})$, which gives the number of pions of the $i$’th kind detected by an analyzer tuned to momentum ${\bbox k}$. The first equation indicates that all effects of multiparticle production are accounted for in this [*basic definition*]{} of the inclusive one-particle spectrum. The multiparticle states are encoded in a sum over the complete set of the unobserved states, $$\begin{aligned} \sum_{X}|X\rangle\langle X|=1~, \label{eq:E2.6a}\end{aligned}$$ where they enter with an [*a priori*]{} assumption that they are [*identically weighted*]{}, and all together form a unit operator. In the same way, one may obtain the inclusive two-pion spectrum $$\begin{aligned} {dN^{(2)}_{ij} \over d{\bbox k}_{1} d{\bbox k}_{2} } = {\rm Tr} {\hat \rho}_{\rm in} \sum_{X} S^\dagger A_{i}^{\dag}({\bbox k}_1)A_{j}^{\dag}({\bbox k}_{2}) |X\rangle\langle X| A_{j}({\bbox k}_{2}) A_i({\bbox k}_1)S = {\rm Tr} {\hat \rho}_{\rm in}\, S^\dag A_{i}^{\dag}({\bbox k}_{1}) A_{j}^{\dag}({\bbox k}_{2}) A_{j}({\bbox k}_{2}) A_{i}({\bbox k}_{1}) S~. \label{eq:E2.7}\end{aligned}$$ Once again, this equation defines the number of pairs, $$\begin{aligned} N_{ij}({\bbox k}_1,{\bbox k}_2)= A_{i}^{\dag}({\bbox k}_{1}) A_{j}^{\dag}({\bbox k}_{2}) A_{j}({\bbox k}_{2}) A_{i}({\bbox k}_{1})= N_i({\bbox k}_1)[N_j({\bbox k}_2)-\delta_{ij} \delta({\bbox k}_1-{\bbox k}_2)]~, \label{eq:E2.8}\end{aligned}$$ as the observable, and incorporates all multiparticle effects by its derivation. This is a function of actual dynamics which is driven by the evolution operator $S$ to select the states which physically contribute to the process of the measurement, and to assign them the [*dynamically generated*]{} weights. Equations (\[eq:E2.6\]) and (\[eq:E2.7\]) are universal in a sense that the standard observables of interferometry are expressed in terms of their Heisenberg operators. In order to make them useful, we have to specify both the evolution operator $S$ and the initial data embodied in the density matrix $\rho_{\rm in}$. In other words, the theory that claims to have any predictive power must incorporate physical information about the freezeout dynamics. Two models which reflect our current vision of the heavy-ion scenario, are explored in the next two sections. Naïve freezeout. {#sec:SN3} ================ The simplest and the most naïve model of freezeout has been explained in detail in Ref. [@MS]. In order to have a reference point for the discussion of the more involved situation detailed in the next section, we review this model below with new updated emphases. In fact, this model is not dynamical. No microscopic mechanism of free pion production is specified. We just declare the pion field to become free after some time (or, in a more intelligent way, starting from some space-like surface $\Sigma_c$). This is what is done technically, but certain assumptions must be kept in mind. First, up to the freezeout surface $\Sigma_c$, the system is assumed to be a continuous medium which obeys relativistic hydrodynamic equations. The hypersurface $\Sigma_c$ corresponds to some critical temperature $T_c$ at which all interactions that maintained the local thermal equilibrium in the expanding medium are switched off. Consequently, we assume the pion distribution over the momenta (not in phase space!) at this moment to be (almost) thermal. It is incorporated into the initial data for the future free propagation. Second, the correlation length along $\Sigma_c$ is finite and much less than the total size of the system which is a dynamical consequence of the interactions before the freezeout. In order to incorporate this property into the naïve picture of the instantaneous freezeout, we must consider the whole system as a collection of boxes (fluid elements) filled by free particles which are opened when their world lines reach $\Sigma_c$. Third, even though we may wish to disregard the quantum nature of the pions in the fluid expansion phase, we have to account for it when we pose the problem of interferometry. In other words, the pions have to be produced in certain [*states*]{}. This (perhaps, most important) goal is also achieved if we mimic freezeout by the model of the opening boxes. Practically, we act as follows: The field $\pi_i(x)$ in Eqs.(\[eq:E2.4\]) has to be evolved in time starting from the initial data on the hypersurface $\Sigma_c$, i.e., $$\begin{aligned} \pi_i(x) = \int d\Sigma_{\mu}(y)\:G_{\rm ret}(x-y)~ \tensor{\partial^{\mu}_{y}}~\pi_i(y)~. \label{eq:E3.1}\end{aligned}$$ The space of states in which the density matrix acts is also defined on this surface. Substituting Eqs. (\[eq:E3.1\]) to (\[eq:E2.4\]), we find the pion Fock operators, expressed in terms of the initial fields, $$\begin{aligned} S^\dag A_i({\bbox k})\, S = \int d^{\,3}x\: f^{*}_{\bbox k}(x)~i\tensor{\partial_{x}^{\,0}}~ \int d\Sigma_{\mu}(y)\:G_{\rm ret}(x-y)~ \tensor{\partial^{\mu}_{y}}~ \pi_{i} (y)~. \label{eq:E3.2}\end{aligned}$$ This equation may be simplified using the explicit form of the free pion propagator, $$\begin{aligned} S^\dag A_i({\bbox k}) S = \theta(x^0-y^0)\int d\Sigma_{\mu}(y) f^*_{\bbox k}(y)~ i\tensor{\partial^{\mu}_{y}}~\pi_{i} (y)~. \label{eq:E3.4}\end{aligned}$$ The answer is simple because the whole problem of evolution is reduced to just the free propagation of the pion field. Now, we have to incorporate the idea of the absence of long-range order on $\Sigma_c$. This is done in three steps. First, we replace the continuous integral over $\Sigma_c$ by the sum of integrals over the “cells” labeled by a discrete index $N$, $$\begin{aligned} \int_{\Sigma_c} d\Sigma (y) \to \sum_{N}\int_{V^{\#}_N}d^3 y~, \label{eq:E3.5} \end{aligned}$$ where $V^{\#}_N$ is the three-dimensional volume of the $N$-th cell on the hypersurface $\Sigma_c$. (Hereafter, the $\#$-labeled quantities are related to the local reference frame with the time-axis normal to the hypersurface $\Sigma_c$.) The pion field within the $N$-th cell also acquires an additional label $N$. Next, we perform the second quantization of the pion field within each cell independently, $$\begin{aligned} \pi_N(y)=\sum_{\bbox p}[a_N({\bbox p})\phi_{\bbox p}(y)+a^{\dag}_{N}({\bbox p}) \phi^{\ast}_{\bbox p}(y)]~,\nonumber\\ \phi_{\bbox p}(y)= (2~V^*_N p_{0})^{-1/2}e^{-i\,p\cdot x}~,~~~~~ p_{0}^{2}={\bbox p}^2+m^2~, \label{eq:E3.6} \end{aligned}$$ where the components of vector ${\bbox p}$ take discrete values defined by the boundary conditions on the walls of each box, and the $\ast$-labeled quantities are related to the local rest-frame of a fluid element. Finally, we impose the commutation relations on the pion field, $$\begin{aligned} [a_N({\bbox p}_1),a^{\dag}_{M}({\bbox p}_2)]= \delta_{NM}\delta_{{\bbox p}_1{\bbox p}_2}~, \label{eq:E3.7} \end{aligned}$$ which is equivalent to the dynamical independence of the fields belonging to different cells. Each pion is created and propagates independently of all others. Thus, the model is completely defined. It is mathematically very simple and reflects the main features of the physical process. There are several scales in this model, the size $\ell$ of the elementary cell, the freezeout temperature, $T_c$, and the geometric parameters $L$ of the flow. (The correlation length is of the order $\ell\agt 1/T_c$ and, in general, we must require that $\ell \ll L$.) The interplay of these parameters should be explicitly accounted for in the course of calculations. The one-particle spectrum of pions is $$\begin{aligned} {dN^{(1)} \over d{\bbox k} } = \sum_{N}\sum_{\bbox p}\langle a_N({\bbox p})a^{\dag}_{N}({\bbox p})\rangle {(k^{\#}_{0}+p^{\#}_{0})^2\over 4 V^{\ast}_N p^{\ast}_{0}} \int_{V^{\#}_N}d^3y e^{-i(k-p)y} \int_{V^{\#}_N}d^3y e^{+i(k-p)y}~. \label{eq:E3.8}\end{aligned}$$ At this point, we must treat one of the integrals as a delta-function which sets the momenta ${\bbox p}^{\#}$ and ${\bbox k}^{\#}$ equal, while the second integral becomes just the volume $V^{\#}_N$ of the fluid cell. This procedure requires that $|{\bbox k}|,~|{\bbox p}|~\gg~1/\ell\sim T_c~$, i.e., the measured momenta should be sufficiently high. In the same way, we obtain the expression for the two-particle spectrum. Since by virtue of Eq. (\[eq:E3.7\]) we have $$\begin{aligned} \langle a_N({\bbox p}_1)a^{\dag}_{M}({\bbox p_2}) a_{M'}({\bbox p'}_2)a^{\dag}_{N'}({\bbox p'_1})\rangle=\hspace{5cm}\nonumber\\ = (\delta_{NN'} \delta_{MM'} \delta_{{\bbox p}_1{\bbox p'}_1}\delta_{{\bbox p}_2{\bbox p'}_2}\!\! +\delta_{NM'} \delta_{MN'} \delta_{{\bbox p}_1{\bbox p'}_2}\delta_{{\bbox p}_2{\bbox p'}_1}) \langle a_N({\bbox p}_1)a^{\dag}_{N}({\bbox p_1})\rangle \langle a_M({\bbox p}_2)a^{\dag}_{M}({\bbox p_2})\rangle ,\nonumber\end{aligned}$$ the two-pion inclusive spectrum becomes, $$\begin{aligned} {dN^{(2)}_{ij} \over d{\bbox k}_{1} d{\bbox k}_{2} } = {dN^{(1)} \over d{\bbox k}_1 } {dN^{(1)} \over d{\bbox k}_2 } + \sum_{NM}\sum_{\bbox p_1,p_2} \langle a_N({\bbox p}_1)a^{\dag}_{N}({\bbox p_1})\rangle \langle a_M({\bbox p}_2)a^{\dag}_{M}({\bbox p_2})\rangle \nonumber\\ \times {(k^{0\#}_{1}+p^{0\#}_{1}) (k^{0\#}_{1}+p^{0\#}_{2}) (k^{0\#}_{2}+p^{0\#}_{2}) (k^{0\#}_{2}+p^{0\#}_{1}) \over 4^2 V^{\ast}_N V^{\ast}_M p^{0\ast}_{1} p^{0\ast}_{2}} \cos(k_1-k_2)(x_N-x_M) \nonumber \\ \times \int_{V^{\#}_N}d^3y e^{-i(k_1-p_1)y} \int_{V^{\#}_N}d^3y e^{+i(k_1-p_2)y} \int_{V^{\#}_M}d^3y e^{-i(k_2-p_2)y} \int_{V^{\#}_M}d^3y e^{+i(k_2-p_1)y}~. \label{eq:E3.10}\end{aligned}$$ Thus, we have reproduced Eq. (\[eq:E1.2\]) which followed from an intuitive conjecture (\[eq:E1.1\]) about the interference of two indistinguishable amplitudes. These amplitudes, by their design, reflect all relevant properties of the interaction which prepare the two-pion system and a device which detects this system. This [*single device*]{} consists of two detectors (e.g., two tracks) tuned to the pions with momenta ${\bbox k_1}$ and ${\bbox k_2}$. Since, by the definition of the inclusive measurement, nothing else is measured, there are two and only two interfering amplitudes. Regardless of how large the total number, ${\cal N}$, of pions produced in a particular event is, only the two-pion transition amplitude has to be symmetrized since the remaining ${\cal N}-2$ pions are not measured. In order to avoid any misunderstanding, we remind the reader that the quantum-mechanical measurement of some observable, by definition, includes the procedure of averaging over an ensemble. A single element of this ensemble carries no quantum-mechanical information. Following Ref.[@TEV], we can argue that the inclusive measurement explores all quantum mechanical fluctuations which can dynamically develop before the moment of measurement and are consistent with the detector response. From this point of view, nothing but a causal chain of real interactions can affect the detector response, and only dynamical histories can interfere. There is a significant difference between the cases when one-, two-, or three-particle inclusive distributions are measured. Each of these measurements is unique in a sense that they are all [*mutually exclusive*]{} even if they are obtained from the same ensemble of multiparticle events. For example, if the three-pion distribution is measured (i.e., the average over an ensemble is accomplished), then the integration over the momentum of the third pion does not result in the two-pion inclusive distribution [@review]. This example reflects a qualitative difference between classical and quantum distributions. The former are defined immediately in terms of [*probabilities*]{} while the later are defined via [*transition amplitudes*]{} with the probabilities playing the secondary role. The same conclusions follow from the field-theory formalism we employ. The interference emerges as a strict consequence of the commutation relations (\[eq:E3.7\]) which incorporate a distinctive property of the hydrodynamic model to localize the freezeout point for each pion independently. We start from the quantum operator (\[eq:E2.8\]) of the measured observable and trace the two-pion signal back to its origin. Eventually, we arrive at the square of the symmetrized two-pion amplitude, once again, regardless of the total number ${\cal N}$of pions in a particular event. The states with many pions are completely accounted for in Eqs. (\[eq:E2.5\]) and (\[eq:E2.7\]) and, consequently, in (\[eq:E3.8\]) and (\[eq:E3.10\]). However, they can contribute to the one- and two-pion observables only via real interactions which are absent in our oversimplified model of a naïve freezeout. A more realistic example with the interaction is considered in the next section. In Eq.(\[eq:E3.10\]), we seemingly encounter a problem. If, e.g., the first of the integrals is considered as the delta-function which sets momenta ${\bbox p}^{\#}_{1}$ and ${\bbox k}^{\#}_{1}$ equal, then the value of the second one is not obvious. It occurs that the second integral over the volume $V^{\#}_N$ becomes $$\begin{aligned} \int_{V^{\#}_N}d^3y e^{+i(k_1-k_2)y}=V^{\#}_N. \label{eq:E3.11}\end{aligned}$$ To prove this, we need more physical information. The interferometry would have been impossible if we could have traced each pion back to the coordinate of its emission or, in other words, if we could have built an “optical image” of the source. According to the Rayleigh criterion, the latter is possible only if $ |\bbox {k_1-k_2}|~L~ \agt 1~$. Therefore, we may observe interference only provided the last inequality does not hold. Thus we have $ |\bbox {k_1-k_2}|~ \alt 1/L ~ \ll~ 1/\ell$, which proves (\[eq:E3.11\]). In the continuous limit, by introducing an auxiliary “emission function” $J(k_1,k_2)$, $$\begin{aligned} J(k_1,k_2)\;=\;\int_{\Sigma_{c}} d\Sigma_{\mu}(x)\: {k^{\mu}_{1}+ k^{\mu}_{2}\over{2}} \: n(k_{1}\cdot u(x))\: e^{-i(k_{1}-k_{2})x}~, \label{eq:E3.14}\end{aligned}$$ the expressions for the one- and two-particle inclusive spectra can be rewritten in a form which is convenient for numerical computations: $$\begin{aligned} k^{0}\,{{dN_{1}} \over {d{\vec k}} } \;=\; J(k,k)~ ~, \label{eq:E3.15}\end{aligned}$$ and $$\begin{aligned} k^{0}_{1} k^{0}_{2}\,{{dN_{2}} \over {d{\vec k}_{1} d{\vec k}_{2} } }\; =\; J(k_{1},k_{1})\, J(k_{2},k_{2})\:+\: {\rm Re}\bigg [J(k_{1},k_{2})\,J(k_{2},k_{1})\bigg ]~ ~ ~, \label{eq:E3.16}\end{aligned}$$ respectively. In Ref. [@AMS], these equations were used to study interferometry for several types of one–dimensional flow. Here, we are interested only in the case of the boost-invariant geometry which must be very close to the reality of RHIC. Indeed, any scenario initiated with a strong Lorentz contraction (up to 0.1 fm!) of the nuclei cannot possess a scale associated with the initial state. Hence, both at classical and quantum levels, the system must evolve with the preserved boost-invariance. In terms of the hydrodynamic theory of multiple production [@Landau], the absence of scale in initial data of the relativistic hydrodynamic equations immediately leads to the Bjorken self-similar solution [@Bjorken] as the only possible solution. Recent analysis of quantum fluctuations at the earliest stage of heavy-ion collisions [@TEV] indicates the same. The overlap of Lorentz-contracted nuclei converts them into a system of modes of [*expanding plasma*]{} with the global boost-invariant geometry. This is a single quantum transition and it is not compatible with the picture of gradually developing parton cascade [@Geiger]. Theoretical analysis of two-pion correlations, in the picture of “naïve freezeout,” from the phase of expanding hot pion gas was done in Refs. [@MS; @AMS]. One of the predictions of these studies, the so-called $m_t$-scaling of the pion and kaon correlators, was confirmed by the data of the NA35 and NA44 collaborations [@NA35; @NA44] obtained from Au-Au collisions at SPS energies ($\sim$10 GeV/nucleon). These data carry two important messages. First, even when the nuclei are Lorentz-contracted only up to the size of 1 fm, this is almost enough to bring about the boost invariant regime of collective flow. There is no doubts that this picture will be even more pronounced at RHIC. The second, less trivial consequence of the observed $m_t$-scaling is that [*the freezeout is sharp*]{}; if the creation of the final-state pions were extended in (local) time, the $m_t$-scaling would vanish [@MSW]. Thus, addressing the pion production at RHIC, we have every reason to rely on two facts: (i) the hadronic matter before the freezeout forms a collective system, and (ii) this system is in the state of the self-similar boost-invariant expansion. For the immediate goals of this study, an advantage of the intensive longitudinal flow is that it provides a window in the phase-space of the two pions where the effect of the hidden interference (explained in Sec. \[sec:SN1\], and computed in more detail in Sec. \[sec:SN4\]) is most visible. This window corresponds to the measurement of the transverse size of the longitudinally-expanding pipe using pairs of pions with the same rapidity. In this way, we employ a distinctive feature of the hydrodynamic-type sources to localize the emission spectrum. The required localization in rapidity is achieved by choosing pions with large transverse momenta ($p_t\agt 3T\sim 3m_\pi$). This has a very simple physical explanation. If the emitting system is kinetically equilibrated (or even sufficiently chaotized) then the mean energy per particle is limited from above. Thus, in the rest-frame of a fluid element, the particles with large transverse momenta can have, on average, only small longitudinal momenta. In other words, such particles are effectively frozen into the collective hydrodynamic motion in the laboratory frame.[^5] Measuring there longitudinal rapidity after freezeout, we are most likely to measure the longitudinal rapidity of the fluid at the emission site. The parameters of the model with the Bjorken geometry are the critical temperature, $T_{c}~ (\sim m_{\pi})$, and the space–like freeze–out hypersurface, defined by $t^{2}-z^{2} = \tau^{2} = {\rm const}$. The coordinates and the four-velocity of the fluid are parameterized as $x^\mu=(\tau\cosh\eta,{\vec r},\tau\sinh\eta)$, and $u^\mu (x)=(\cosh\eta,{\vec 0},\sinh\eta)$, respectively. The rapidity, $\eta$, of a fluid cell is restricted to $\pm Y$ in the center-of-mass frame. We assume an axially-symmetric distribution of hot matter in a pipe with area $S_{\bot} = \pi R^{2}_{\bot}$. The particles are described by their momenta, $k^{\mu}_{i} = (k^{0}_{i},{\vec k}_{i},k^{z}_{i}) \equiv (m_{i}\cosh\theta_{i}, {\vec k}_{i}, m_{i}\sinh\theta_{i})$, where ${\vec k}_{i}$ is the two-dimensional vector of the transverse momentum, $\theta_{i}$ is the particle rapidity in $z$-direction, and $m^2_i \equiv m_{\perp i}^{2} = m^{2}+ {\vec k}_{i}^{2}$ is the transverse mass. Let us introduce $2\alpha = \theta_1-\theta_2$, $2\theta = \theta_{1}+\theta_{2}$. The one-particle distribution is expected to be close to a thermal distribution of pions at temperature $T=T_c$. Since the $m_\bot$ values of interest are larger than $T_c$, we may take this distribution in a Boltzmann form, and use the saddle-point method to estimate the integrals (\[eq:E3.16\]). For the one-particle distribution these two steps yield $$\begin{aligned} {{dN}\over{d\theta_1 d{\vec k}_1}} \approx \tau S_\bot m_1 \int^{Y}_{-Y} d\eta \cosh(\theta_1-\eta) e^{-m_1\cosh(\theta_1-\eta)/T_c} \approx \tau S_{\bot}m_1 \sqrt{2\pi T_{c}\over{m_{1}}} \: e^{-m_{1}/T_c}~. \label{eq:E3.19} \end{aligned}$$ The general expression for $J(k_1,k_2)$ is $$\begin{aligned} J(k_1,k_2) = {1\over 2} \int\!\!d{\vec r}_1 f(r_1) e^{i({\vec k}_1-{\vec k}_2){\vec r}_1}\tau \int^{\eta }_{-\eta }\!\! d\eta \ \bigg [(m_1+m_2) \cosh(\theta-\eta )\,\cosh\alpha - (m_{1}-m_{2})\sinh(\eta -\theta)\sinh\alpha \bigg] \nonumber\\ \times\exp \bigg \{ -{1\over{T_c}} \bigg [(m_1+iF(m_1-m_2)) \cosh(\eta -\theta)\cosh\alpha -(m_1+iF(m_1+m_2)) \sinh(\eta -\theta)\sinh\alpha\bigg ] \bigg \}~, \label{eq:E3.20}\end{aligned}$$ where $F=\tau T_c$. Once we consider only those pions with large transverse momenta, the integral over the rapidity $\eta$ can (and should) be computed in the saddle-point approximation. This yields $$\begin{aligned} R(k_{1},k_{2}) = {dN_2\over d y_1 d{\vec p}_1 d y_2 d{\vec p}_2}/ \bigg [ {dN_1\over d y_1 d{\vec p}_1} {{dN_1}\over{d y_2 d{\vec p}_2}}\bigg ] -1 = {1\over 4} \; f(|{\vec q}_{\bot}|R_{\bot}) \; {{g(z)\,g(1/z)} \over {[h(z)\,h(1/z)]^{3/2}}} \nonumber \\ \times \exp \bigg \{ -{\mu \over {T_{c}}}\: \bigg [ h(z)\cos {H(z)\over{2}} +h({1\over{z}}) \cos {{H(1/z)}\over{2}}-z-{1\over z}\bigg ]\bigg \} \nonumber \\ \times \cos\bigg \{ {\mu \over {T_{c}}}\bigg [h(z)\sin {H(z)\over{2}} +h({1\over{z}}) \sin {{H(1/z)}\over{2}} \bigg ] + {3\over 4}\bigg [H(z)+H({1\over {z}})\bigg ] +G(z)+G({1\over{z}}) \bigg \}~, \label{eq:E3.21} \end{aligned}$$ where $\mu =(m_{1}m_{2})^{1/2}$ and $z = (m_{1}/m_{2})^{1/2} $, and we have introduced the functions $$\begin{aligned} h(z) = \bigg \{ \bigg [ z^{2}-F^{2}(z-{1\over{z}})^{2}+4F^2\sinh^{2}\alpha \bigg ]^2 +4F^2(z^2-\cosh 2\alpha)^2 \bigg \}^{1/4};\nonumber\\ g(z) = \bigg [(z^{2}+\cosh 2\alpha)^{2} +F^{2}(z^{2}-{1\over{z^{2}}})^{2}\bigg ]^{1/2};\nonumber\\ \tan H(z) = {2F(\cosh 2\alpha-z^{2}) \over {z^{2}-F^{2}(z-{1\over {z}})^{2}+4F^{2}\sinh^{2}\alpha}};~~~~~ \tan G(z) = {F(z^{2}- 1/z^{2}) \over {z^2+\cosh 2\alpha}}~. \label{eq:E3.22} \end{aligned}$$ When the transverse momenta of two pions are equal, then $m_1=m_2$, and Eq. (\[eq:E3.21\]) reproduces the result of Ref. [@AMS]. The function $f$ depends on how we parameterize the transverse distribution of the matter in the longitudinally expanding pipe. If $f(r) = \theta (R^{2}_{\bot}-r^{2})$ then $~f(|{\vec \Delta k}_{\bot}|R_{\bot})= [2 J_1(|\Delta {\vec k}|R_{\bot})/|\Delta{\vec k}|R_{\bot}]^2~$, where $J_1$ is a Bessel function. This HBT correlator possesses a natural property of any two-point function in the boost-invariant geometry; it depends on the difference of rapidities and not on the difference of the longitudinal momenta. The $m_t$-scaling of the “visible longitudinal size” is just a synonym for this property, which supports a picture of the sharp freeze-out of matter in the state dominated by the boost-invariant-like longitudinal expansion. Two remarks are in order: 1\. The shape of the correlator given by Eqs. (\[eq:E3.21\]) and (\[eq:E3.22\]) is manifestly non-Gaussian. Thus, it is plausible to get rid of the intermediate Gaussian fit and to use these equations to fit the data as the first step. 2\. A main deficiency of the correlator (\[eq:E3.21\]) is that the possible transverse expansion has not been taken into account. This is not easy to do since the initial data in the transverse plane are, as yet, poorly understood and the amount of matter involved in the collective transverse motion at the freezeout stage may vary depending on the initial conditions, equation of state, etc. We consider this issue still open. Dynamical freezeout. {#sec:SN4} ==================== Now, let us formulate a more realistic model for the freezeout, relying on the following phenomenological input. Let the system before the collision be a pion-dominated hadronic gas. Therefore, since pions are the lightest hadrons, the regime of continuous medium is supported mainly due to $\pi\pi$-scattering. In the framework of chiral perturbation theory there are two main channels contributing to this process. The first channel is $s$-wave scattering which we shall model using the sigma-model. The second channel is the $p$-wave $\pi\pi$-scattering via the $\rho$-meson. We shall use a model interaction Lagrangian of the form $$\begin{aligned} {\cal L}_{\rm int}= {g_{4\pi}\over 4}(\bbox{\pi^\dag\cdot\pi})^2+ g_\sigma \sigma (\bbox{\pi^\dag\cdot\pi})+ {f_{\rho\pi\pi}\over 2} \bbox{\rho}^\mu {\bbox\cdot}[\partial_\mu \bbox{\pi^\dag\times\pi} + \bbox{\pi^\dag\times}\partial_\mu \bbox{\pi}]+ \cdot\cdot\cdot~, \label{eq:E4.1} \end{aligned}$$ where $\sigma (x)$ is the field of the scalar $\sigma$-mesons, and ${\bbox\rho}^\mu=(\rho^{-},\rho^{0},\rho^{+})$ is the field of the $\rho$-mesons. The reader can easily recognize here that part of the Lagrangian of the sigma-model which is necessary to reproduce the tree-level (or even skeleton) amplitudes of the $s$- and $p$-wave $\pi\pi$-scattering. The evolution operator is usually defined as $$\begin{aligned} S=T \exp\{i\int {\cal L}_{\rm int}(x)d^4x\}~. \label{eq:E4.2} \end{aligned}$$ Applying the standard commutation formulae [@Bogol], $$\begin{aligned} A_{[i]}({\bbox k}) S - S A_{[i]}({\bbox k}) =\int d^4 x {\delta S \over \delta \pi^{\dag}_{i}(x)} f^{*}_{\bbox k}(x)~,~~~ S^{\dag} A^{\dag}_{[i]}({\bbox k})- A^{\dag}_{[i]}({\bbox k})S^{\dag}= \int d^4 x f_{\bbox k}(x){\delta S^{\dag} \over \delta \pi_{[i]}(x)}~, \label{eq:E4.3} \end{aligned}$$ to the Eqs. (\[eq:E2.6\]) and (\[eq:E2.7\]) we obtain $$\begin{aligned} { dN_{[a]}^{(1)} \over d{\bbox k} } = \int d^4y_1d^4y_2 f_{\bbox k}(y_1) \bigg\langle {\delta S^{\dag}\over \delta\pi_{[a]}(y_1)} {\delta S\over \delta\pi_{[a]}^{\dag}(y_2)}\bigg\rangle f^{*}_{\bbox k}(y_2)~, \label{eq:E4.4}\end{aligned}$$ for the one-particle spectrum of the pions of kind $a$ $(\pi^{+},\pi^{0},\pi^{-})$, and $$\begin{aligned} {dN^{(2)}_{[ab]} \over d{\bbox k}_{1} d{\bbox k}_{2} } = \int d^4y_1 d^4y_2 d^4y_3 d^4y_4 f_{{\bbox k}_1}(y_1)f_{{\bbox k}_2}(y_3) \bigg\langle {\delta^2 S^{\dag} \over \delta\pi_{[a]}(y_1) \delta\pi_{[b]}(y_3)} {\delta^2 S\over \delta\pi_{[a]}^{\dag}(y_2) \delta\pi_{[b]}^{\dag}(y_4)}\bigg\rangle f^{*}_{{\bbox k}_1}(y_2)f^{*}_{{\bbox k}_2}(y_4) ~, \label{eq:E4.5}\end{aligned}$$ for the spectrum for pairs of pions of kinds $a$ and $b$. According to our agreement, the angular brackets denote an average weighted by the density matrix $\rho_{\rm in}$. \[In order to avoid confusion, we place the isospin indices in square brackets.\] Since the goal of this paper is to demonstrate a physical effect, we shall limit ourselves, in what follows, with the ${\bbox \pi}^4$ interaction term. In this case, we are able to do most calculations analytically, which is necessary in order to understand the interplay of the many parameters. More involved calculations will require a realistic description of the $\pi\pi$-scattering and Monte-Carlo computation of multiple integrals. Calculation of the one-pion spectrum is relatively simple. A direct computation of the functional derivatives in Eq.(\[eq:E4.4\]) leads to $$\begin{aligned} { dN_{[a]}^{(1)} \over d\bbox k } = {g_{4\pi}^2\over 4}\int d^4x d^4y {e^{-ik(x-y)}\over 2k^0 (2\pi)^3}\sum_{i,j} \langle T^\dag(S^\dag:\pi^{\dag[i]}(x)\pi^{[i]}(x)\pi^{\dag[a]}(x):) T(:\pi^{[a]}(y)\pi^{\dag[j]}(y)\pi^{[j]}(y):S)\rangle~, \label{eq:E4.6}\end{aligned}$$ where the symbols $T$ and $T^\dag$ denote the time and anti-time orderings, respectively. Coupling the field operators according to the Wick theorem (to the lowest order of the perturbation expansion we have to put $S=1$), the average in Eq. (\[eq:E4.6\]) becomes, $$\begin{aligned} \langle ~\cdot\cdot\cdot~\rangle= i^3[G^{[aa]}_{01}(y,x) G^{[ij]}_{10}(x,y) G^{[ji]}_{01}(y,x)+ G^{[ja]}_{01}(y,x) G^{[ij]}_{10}(x,y) G^{[ai]}_{01}(y,x)]~, \label{eq:E4.7}\end{aligned}$$ where any pair of arguments $x$ and $y$ lie within the same cell of a size defined by the correlation length in the system under consideration. This is a consequence of the cellular structure of the density operator ${\hat\rho_{\rm in}}$ which determines the average $\langle\cdot\cdot\cdot\rangle$. The correlators $G_{AB}(x,y)$ are defined (using the Keldysh technique [@Keld]; see also Ref. [@TEV]) as $$\begin{aligned} G^{[ij]}_{AB}(x,y)= -i\langle T_c (\pi^{[i]}(x_A)\pi^{\dag[j]}(y_B)\rangle~. \label{eq:E4.7a}\end{aligned}$$ Taking for $a=1$ $(\pi^+)$ and using the facts that $G^{[ij]}\propto\delta^{ij}$, $~G^{[33]}_{01}(y,x)=G^{[11]}_{10}(x,y)$, and $~G^{[33]}_{10}(y,x)=G^{[11]}_{01}(x,y)~$ we arrive at $$\begin{aligned} \langle ~\cdot\cdot\cdot~\rangle= i^3 G^{[11]}_{01}(y,x) [3 G^{[11]}_{10}(x,y) G^{[11]}_{01}(y,x)+ G^{[00]}_{10}(x,y) G^{[00]}_{01}(y,x)]~. \label{eq:E4.8}\end{aligned}$$ Next, we have to insert this expression into Eq. (\[eq:E4.6\]) and pass over to the momentum representation (cell-by-cell) according to $$\begin{aligned} G^{[11]}_{01}(p)= G^{[33]}_{10}(-p)= -2\pi i[\theta(p^0)n^{(+)}(p) +\theta(-p^0)]~,\nonumber\\ G^{[11]}_{10}(p)=G^{[33]}_{01}(-p)= -2\pi i [\theta(p^0) +\theta(-p^0)n^{(-)}(p)]~, \nonumber\\ G^{[00]}_{01}(p)= -2\pi i[\theta(p^0)n^{(0)}(p) +\theta(-p^0)]~,\nonumber\\ G^{[00]}_{10}(p)= -2\pi i [\theta(p^0) +\theta(-p^0)n^{(0)}(-p)]~. \label{eq:E4.9}\end{aligned}$$ These equations reflect a very important feature of the process under investigation (already accounted for in Eqs. (\[eq:E4.4\]) and (\[eq:E4.5\]). ) Namely, the initial states in the system of colliding pions are populated with the densities $n_N^{(\pm)}(p)$ of charged pions and $n_N^{(0)}(p)$ of neutral pions. All the final states of free propagation [*are not occupied*]{}. They are virtually present with an [*a priori*]{} unit weight in the expansion (\[eq:E2.6a\]) of the unit operator. Access to the states with many pions is provided dynamically by real interactions, and these states show up in higher orders of the evolution operator expansion. Depending on the type of commutation relations, these states always appear in a properly symmetrized form, and there is no need to start with any [*ad hoc*]{} symmetrization. By definition, the wave function is the transition amplitude between the prepared initial state and detected final state. Unless the transition is explicitly measured, the wave function does not exist at all and there is no object for symmetrization. Therefore, there is no statistical correlations associated with the multiple production of pions. All correlations are dynamical (see further discussion in Sec. \[sec:SN5\]). After some manipulations, we obtain for the one-particle spectrum, $$\begin{aligned} { dN_{\pi^{+}}^{(1)} \over d\bbox k }={g_{4\pi}^2\over 4}\sum_{N} V^{(4)}_{N} \int {d^4p_1 d^4p_2 d^4q\over 2k^0(2\pi)^{8} }\delta(k+q-p_1-p_2) \delta(p_1^2-m^2) \delta(p_2^2-m^2) \delta(q^2-m^2)\nonumber\\ \times[3n^{(+)}_{N} (p_1)n^{(+)}_{N}(p_2) + 6n^{(+)}_{N}(p_1)n^{(-)}_{N}(p_2)+ 2n^{(+)}_{N} (p_1)n^{(0)}_{N}(p_2))+n^{(0)}_{N}(p_1)n^{(0)}_{N}(p_2))]~. \label{eq:E4.10}\end{aligned}$$ Here, as in the model of the “naïve freezeout” discussed in Sec. \[sec:SN3\], one of the integrals over the space-time volume $V^{(4)}_N$ of the domain (where the individual collision occurs), is treated as the delta-function (viz., momentum conservation), while the second integral becomes just the four-volume $V^{(4)}_{N}$ of the interaction domain. (More details are given in Appendix A.) In what follows, we shall limit ourselves to a simplified picture when all the statistical weights of the initial state are taken in the Boltzmann form and the chemical potential is zero. In this case, we gain an overall factor of 3+6+2+1=12 and arrive at $$\begin{aligned} { dN_{\pi^{+}}^{(1)} \over d\bbox k } = {12g_{4\pi}^2\over 4}\sum_{N} V^{(4)}_{N} \int {d^4p_1 d^4p_2 d^4q\over 2k^0(2\pi)^{8} }\delta(k+q-p_1-p_2) \delta(p_1^2-m^2) \delta(p_2^2-m^2)\delta(q^2-m^2) e^{-\beta_N u_N(p_1+p_2)}~. \label{eq:E4.11}\end{aligned}$$ Now, the integration over $l=p_1-p_2$ can be explicitly carried out (see Eq. (\[eq:A2.0\]) $$\begin{aligned} { dN_{\pi^{+}}^{(1)} \over d{\bbox k }} = {12g_{4\pi}^2\over 4}\sum_{N} V^{(4)}_{N} e^{-\beta_N (ku_N)}\int { d^4q\over 2k^0(2\pi)^{8} }{\pi\over 4} e^{-\beta_N (qu_N)}\delta(q^2-m^2)\sqrt{1-{4m^2\over (k+q)^2}}~. \label{eq:E4.12}\end{aligned}$$ Introducing an auxiliary function, $$\begin{aligned} I(k,N,x_{NM})=\int d^4q\delta_+(q^2-m^2)e^{-\beta_N(qu)}~e^{iqx_{NM}} \sqrt{1-{4m^2\over (k+q)^2}}~, \label{eq:E4.12a}\end{aligned}$$ which is computed in Appendix B, we may write explicitly, $$\begin{aligned} { dN_{\pi^{+}}^{(1)} \over d{\bbox k }} = {12\pi g_{4\pi}^2\over 16}\sum_{N} {V^{(4)}_{N}\over 2k_0(2\pi)^8} e^{-\beta_N (ku_N)}I(k,N,0) = {12g_{4\pi}^2\over 16}2\pi m^2{K_1({m / T})\over m/T} \int {d^4x\over 2k_0(2\pi)^8} e^{-ku(x)/T}\nonumber\\ = {12g_{4\pi}^2\over 16(2\pi)^8}2\pi m^2{K_1({m / T})\over m/T} \tau \Delta\tau S_\bot \sqrt{{2\pi T\over m_\bot}} e^{-m_\bot/T}~, \label{eq:E4.12b}\end{aligned}$$ where, with reference to the data discussed in Sec \[sec:SN3\], we assume that the freezeout occurs in a very small interval $\Delta\tau$ of the proper time $\tau$. Additionally, we further simplify the model by taking the temperature the same throughout the entire freezeout domain. Calculation of the two-pion spectrum is very cumbersome but follows exactly the same guideline, $$\begin{aligned} { dN_{\pi^{a} \pi^{b}}^{(2)} \over d\bbox k_1 d\bbox k_2} = {g_{4\pi}^4 \over 16}\int d^4y_1 d^4y_2 d^4y_3 d^4y_4 {e^{-ik_1(y_1-y_2)} e^{-ik_2(y_3-y_4)} \over 4 k_1^0 k_2^0 (2\pi)^6}\nonumber\\ \times\sum_{i,j,l,n} \langle T^\dag(S^\dag{\bbox :}\pi^{\dag [i]}(y_1)\pi^{[i]}(y_1) \pi^{\dag [a]}(y_1){\bbox ::} \pi^{\dag [l]}(y_3)\pi^{[l]}(y_3)\pi^{\dag [b]}(y_3){\bbox :})\nonumber\\ \times T({\bbox :}\pi^{[a]}(y_2)\pi^{\dag [j]}(y_2)~\pi^{[j]}(y_2){\bbox ::} \pi^{[b]}(y_4)\pi^{\dag [n]}(y_4)\pi^{[n]}(y_4){\bbox :}S)\rangle~, \label{eq:E4.13}\end{aligned}$$ where, as before, we must put $S=1$ in the lowest order of the perturbation expansion. If the observed pions are identical, then the result of coupling is a very long expression which is given in its full form in Appendix A, Eq. (\[eq:A1.1\]). The next step is to integrate over the elementary volumes ${\cal V}_{N}$. This is done according to Eq. (\[eq:A1.3\]). Finally, using the Boltzmann approximation for the statistical weights, we carry out an explicit integration over $l_1=s_1-s_2$ and $l_2=p_1-p_2$ (see Eq. (\[eq:A2.0\])). The result is as follows, $$\begin{aligned} { dN_{\pi^{+} \pi^{+}}^{(2)} \over d\bbox k_1 d\bbox k_2} = { dN_{\pi^{+}}^{(1)} \over d\bbox k_1 } { dN_{\pi^{+}}^{(1)} \over d\bbox k_2 } ~+~{\pi^2g_{4\pi}^4\over 16^2}\sum_{N,M}V^{(4)}_{N}V^{(4)}_{M} e^{-\beta_N (k_1 u_N)}e^{-\beta_M (k_2 u_M)}\nonumber\\ \times \int { d^4q_1 d^4q_2 \over 4k_1^0 k_2^0(2\pi)^{16} } \sqrt{1-{4m^2\over (k_1+q_1)^2}}\sqrt{1-{4m^2\over (k_2+q_2)^2}} e^{-\beta_N (q_1 u_N)}e^{-\beta_M (q_2 u_M)} \delta_+(q_1^2-m^2) \delta_+(q_2^2-m^2) \nonumber\\ \times {\rm Re}\{144~e^{-i(k_1-k_2)(x_N-x_M)} +44 e^{-i(q_1-q_2)(x_N-x_M)} +44e^{-i(q_1+k_1-q_2-k_2)(x_N-x_M)}\}~. \label{eq:E4.14}\end{aligned}$$ In this equation, we recognize the symmetrized transition amplitude for all four pions engaged in the process of the two pion production. Each term acquires a factor associated with the isospin algebra. Though all multi-pion states are included in the definitions (\[eq:E4.4\]) and (\[eq:E4.5\]) of the observables, it is neither necessary nor even possible to symmetrize them, if only one or two pions are measured. The next orders of the evolution operator expansion will bring in additional final-state particles. They will all be properly symmetrized. However, these terms will be more and more suppressed (discussion in Sec. \[sec:SN5\]). In a similar way, the original expression, (\[eq:A1.2\]), is transformed to the inclusive spectrum of two different pions, $$\begin{aligned} { dN_{\pi^{+} \pi^{-}}^{(2)} \over d\bbox k_1 d\bbox k_2} = { dN_{\pi^{+}}^{(1)} \over d\bbox k_1 } { dN_{\pi^{-}}^{(1)} \over d\bbox k_2 } ~+~{g_{4\pi}^4\over 16^2}\sum_{N,M} V^{(4)}_{N}V^{(4)}_{M} e^{-\beta_N (k_1 u_N)}e^{-\beta_M (k_2 u_M)}\nonumber\\ \times \int {d^4q_1 d^4q_2 \over 4k_1^0 k_2^0(2\pi)^{16} } \sqrt{1-{4m^2\over (k_1+q_1)^2}}\sqrt{1-{4m^2\over (k_2+q_2)^2}} e^{-\beta_N (q_1 u_N)}e^{-\beta_M (q_2 u_M)} \delta_+(q_1^2-m^2) \delta_+(q_2^2-m^2)\nonumber\\ \times {\rm Re}[ 16~ e^{-i(q_1-q_2)(x_N-x_M)}+ 26~e^{-i(q_1+k_1-q_2-k_2)(x_N-x_M)}] \label{eq:E4.15}\end{aligned}$$ Next, we have to find the net yield of the interference terms after the momenta of the unobserved pions are integrated out. As a first step, we can use an auxiliary function $I(k,x,\Delta x)$, (\[eq:E4.12a\]) and rewrite Eqs. (\[eq:E4.14\]) and (\[eq:E4.15\]), replacing the discrete sums by the integration over the continuous distribution of the collision points, $$\begin{aligned} { dN_{\pi^{+} \pi^{+}}^{(2)} \over d\bbox k_1 d\bbox k_2} = {\pi^2g_{4\pi}^4\over 16^2}\int {d^4x_1 d^4x_1\over 4k_1^0 k_2^0(2\pi)^{16} } e^{-(k_1 u(x_1))/T_c} e^{-(k_2 u(x_2))/T_c}\nonumber\\ \times {\rm Re}\{144 I(k_1,x_1,0) I(k_2,x_2,0) [1+e^{-i(k_1-k_2)(x_1-x_2)}]\nonumber\\ +44I(k_1,x_1,(x_1-x_2))I^\ast(k_2,x_2,(x_1-x_2)) [1+e^{-i(k_1-k_2)(x_1-x_2)}]\}~, \label{eq:E4.16}\end{aligned}$$ for identical pions, and $$\begin{aligned} { dN_{\pi^{+} \pi^{-}}^{(2)} \over d\bbox k_1 d\bbox k_2} = {\pi^2g_{4\pi}^4\over 16^2}\int {d^4x_1 d^4x_1\over 4k_1^0 k_2^0(2\pi)^{16} } e^{-(k_1 u(x_1))/T_c} e^{-(k_2 u(x_2))/T_c}\nonumber\\ \times {\rm Re}\{144 I(k_1,x_1,0) I(k_2,x_2,0) +I(k_1,x_1,(x_1-x_2))I^\ast(k_2,x_2,(x_1-x_2)) [16+26 e^{-i(k_1-k_2)(x_1-x_2)}]\}~, \label{eq:E4.17}\end{aligned}$$ for non-identical pions. These equations are written in the same approximation as the one-pion inclusive spectrum (\[eq:E4.12b\]). The first two terms of Eq. (\[eq:E4.16\]) have a well known form of the product of the intensities of two independent sources and an interference function $[1+cos\Delta k \Delta x]$ and correspond to the standard scheme of HBT interferometry. In these terms, the space-time integration over coordinates $x_1$ and $x_2$ is factorized and reproduces the result of the naïve model of freezeout given by Eq.(\[eq:E3.21\]). The only difference is an inessential kinematic factor and a form-factor $$\begin{aligned} I(k_1,x_1,0) I(k_2,x_2,0)= 4\pi^2 m^4\bigg[ {K_1(m/T)\over m/T}\bigg]^2~, \label{eq:E4.18}\end{aligned}$$ which eventually cancels out in the normalized correlator. In the last two terms of Eq. (\[eq:E4.16\]), we encounter an additional factor ${\cal F}(x_1,x_2)=I(k_1,x_1,\Delta x)I^\ast(k_2,x_2,\Delta x)$ which, in general, does not allow one to factorize the space-time integrations. In our special geometry of the freezeout, however, some further simplifications are possible, provided the transverse momenta $k_{1t}$ and $k_{2t}$ are large. Indeed, integrating with respect to $x_1$, we end up with a modified version of the emission function (\[eq:E3.14\]), $$\begin{aligned} J(k_1,k_2;x_2)=\int d^4 x_1 e^{-k_{1}\cdot u(x_1))/T} e^{-i(k_{1}-k_{2})x_1}{\cal F}(x_1,x_2)\nonumber\\ =\Delta\tau\int d^2{\vec r_1}f({\vec r_1}) e^{i{\vec r_1}({\vec k_1}-{\vec k_2})}\int_{-Y}^{Y}\tau d\eta_1 e^{-(m_{1}/T)\cosh(\eta_1-\theta_1)-i\tau[m_{1}\cosh(\eta_1-\theta_1)- m_{2}\cosh(\eta_1-\theta_2)]}{\cal F}(\Delta\eta,\Delta{\vec x})~, \label{eq:E4.19}\end{aligned}$$ where the form-factor ${\cal F}$ can be conveniently rewritten as $$\begin{aligned} {\cal F}(\Delta\eta,{\vec \rho})= 4\pi^2 m^4 \bigg| {K_1(m\sqrt{U^2})\over m\sqrt{U^2}}\bigg|^2~, \label{eq:E4.20}\end{aligned}$$ where $$\begin{aligned} U^2={1\over T^2} +{\vec \rho}^{~2}+ 4T\tau(T\tau+i)\sinh^2{\eta_1-\eta_2\over 2}~,\nonumber\end{aligned}$$ and we denoted ${\vec \rho}={\vec r_1} -{\vec r_2}$. The saddle point of the integration over $\eta_1$ in Eq. (\[eq:E4.19\]) is defined by the equation,[^6] $$\begin{aligned} \tanh (\eta_1-\theta)={1+iT\tau[1+(m_2/m_1)] \over 1+iT\tau[1-(m_2/m_1)]}~\tanh\alpha~, \label{eq:E4.21}\end{aligned}$$ and is not affected by the form-factor (\[eq:E4.20\]). Thus, if we select pions with the same rapidity, then $2\alpha=\theta_1-\theta_2=0$ and the saddle point occurs at the point $\eta_1=\theta=\theta_1=\theta_2$. Integrating in the same way with respect to $x_2$, we come to the conclusion that $\eta_2=\theta$. The form-factor then becomes independent of rapidities $\eta$, and its argument becomes a real function,[^7] $$\begin{aligned} {\cal F}(0,{\vec \rho})= 4\pi^2 m^4 {K_1^2(m\sqrt{(1/T^2)+{\vec \rho}^{~2}})\over m^2[(1/T^2)+{\vec \rho}^{~2}]}~. \label{eq:E4.22}\end{aligned}$$ From now on, we will deal only with pions of the same rapidity, which means that we physically take aim at two scattering processes which occur in one narrow slice of the longitudinally expanding pipe. In this particular case, the very cumbersome formulae (like (\[eq:E3.21\])) become drastically simplified. The spectrum of two identical pions reads as $$\begin{aligned} { dN_{\pi^{+} \pi^{+}}^{(2)} \over d\bbox k_1 d\bbox k_2} = { dN_{\pi^{+}}^{(1)} \over d\bbox k_1 }{ dN_{\pi^{+}}^{(1)}\over d\bbox k_2 } ~+~{\pi^2g_{4\pi}^4\over 16^2} {144 \over 4k_1^0 k_2^0(2\pi)^{16} } 4\pi^2 m^4 (\Delta\tau)^2 \tau^2{2\pi T\over \sqrt{m_1m_2}} e^{-(m_1+m_2)/T_c} \nonumber\\ \times\bigg\{{\rm Re}{\sqrt{m_1m_2}\over [m_1m_2+F(F-i)(m_1-m_2)^2]^{1/2}} \bigg[ \bigg(2\pi\int rdr f(r)J_0(|\Delta{\vec k}|r)\bigg)^2 \bigg[ {K_1(m/T)\over m/T}\bigg]^2 \nonumber\\ +{44\over 144} 2\pi \int r_1dr_1 r_2dr_2 f(r_1)f(r_2) \int_{0}^{2\pi}\!\!\!\! d\phi~ {K_1^2(m\sqrt{(1/T^2)+{\vec \rho}^{~2}})\over m^2[(1/T^2)+{\vec \rho}^{~2}]}J_0(|\Delta{\vec k}|\rho)\bigg] \nonumber\\ +{44\over 144} 2\pi \int r_1dr_1 r_2dr_2 f(r_1)f(r_2) \int_{0}^{2\pi}\!\!\!\!d\phi~ {K_1^2(m\sqrt{(1/T^2)+{\vec \rho}^{~2}})\over m^2[(1/T^2)+{\vec \rho}^{~2}]}\bigg\}~, \label{eq:E4.23}\end{aligned}$$ where $\rho^2=r_1^2+r_2^2-2r_1r_2 \cos\phi$. In the same way, we obtain the two-particle spectrum of non-identical pions, $$\begin{aligned} { dN_{\pi^{+} \pi^{-}}^{(2)} \over d\bbox k_1 d\bbox k_2} = { dN_{\pi^{+}}^{(1)} \over d\bbox k_1 }{ dN_{\pi^{-}}^{(1)}\over d\bbox k_2 } ~+~{\pi^2g_{4\pi}^4\over 16^2} {144 \over 4k_1^0 k_2^0(2\pi)^{16} } 4\pi^2 m^4 (\Delta\tau)^2 \tau^2{2\pi T\over \sqrt{m_1m_2}} e^{-(m_1+m_2)/T_c} \nonumber\\ \times 2\pi \int r_1dr_1 r_2dr_2 f(r_1)f(r_2) \int_{0}^{2\pi}\!\!\!\! d\phi~ {K_1^2(m\sqrt{(1/T^2)+{\vec \rho}^{~2}})\over m^2[(1/T^2)+{\vec \rho}^{~2}]}\nonumber\\ \times\bigg\{{16\over 144} + {26\over 144}~{\rm Re}\bigg[{\sqrt{m_1m_2}\over [m_1m_2+F(F-i)(m_1-m_2)^2]^{1/2}}\bigg]~J_0(|\Delta{\vec k}|\rho)\bigg\}~. \label{eq:E4.24}\end{aligned}$$ In order to give a quantitative estimate of the effect (yet with all reservations intact due to the imperfection of our model description of the $\pi\pi$-interaction) we shall assume that the distribution of the longitudinally expanding hot pipe in the transverse direction is homogeneous within a cylinder of radius $R_\bot$. We also neglect the transverse flow. Next, we normalize the two-pion spectrum by the product of the one-pion spectra and introduce $$\begin{aligned} C_{ab}({\bbox k_1},{\bbox k_2})={dN^{(2)}_{ab}/d{\bbox k_1} d{\bbox k_2}\over (dN^{(1)}_{a}/d{\bbox k_1}) (dN^{(1)}_{b}/d{\bbox k_2})}~. \label{eq:E4.25}\end{aligned}$$ Finally, we consider only the case of $|{\bbox k_1}|=|{\bbox k_2}|$. Then, $m_1=m_2$ and we arrive at $$\begin{aligned} C_{\pi^{+}\pi^{+}}= 1+ \bigg({2J_1(|\Delta{\vec k}|R) \over |\Delta{\vec k}|R}\bigg)^2 +{11\over 36}\int_{0}^{R}{r_1dr_1 r_2dr_2\over (\pi R)^2} \int_{0}^{2\pi}\!\!\!\!\! d\phi~ {2\pi\over K_1^2(m/T)}~ {K_1^2[(m/T)\sqrt{1+T^2{\vec \rho}^{~2}}]\over 1+T^2{\vec \rho}^{~2}} [1+J_0(|\Delta{\vec k}|\rho)]~, \label{eq:E4.26}\end{aligned}$$ $$\begin{aligned} C_{\pi^{+}\pi^{-}}= 1+ \int_{0}^{R}{r_1dr_1 r_2dr_2\over (\pi R)^2} \int_{0}^{2\pi}\!\!\!\!\! d\phi~ {2\pi\over K_1^2(m/T)}~ {K_1^2[(m/T)\sqrt{1+T^2{\vec \rho}^{~2}}]\over 1+T^2{\vec \rho}^{~2}} \bigg[{1\over 9}+{13\over 72}J_0(|\Delta{\vec k}|\rho)\bigg]~, \label{eq:E4.27}\end{aligned}$$ where we remind the reader that the longitudinal rapidities of the pions are set equal. The remaining integrations in Eqs. (\[eq:E4.26\]) and (\[eq:E4.27\]) are performed numerically and the results are plotted in Figs. \[fig:fig2\] and \[fig:fig3\]. Possible contributions of the multiparticle states. {#sec:SN5} =================================================== As we have seen in the previous section, the multi-pion final states indeed contribute to the one- and two-pion inclusive cross sections. This contribution is due to the real interactions, and therefore, we may speculate about its magnitude in different situations and at different conditions of observation. Within the unrealistic ${\bbox\pi}^4$-model we employed to make analytic solution possible, the effect is understandably small. Indeed, the additional pions emitted in the collision process lead to the form-factor (\[eq:E4.22\]), ${\cal F}({\vec\rho})$, in the interference term. Let us consider the formal limit of $T\to\infty$. Then the form-factor becomes $$\begin{aligned} {\cal F}({\vec\rho},T\to\infty)= 4\pi^2 m^4 {K_1^2(\;m\sqrt{{\vec\rho}^{~2}}\;)\over m^2{\vec\rho}^{~2}}~. \label{eq:E5.1} \end{aligned}$$ This expression is immediately recognized as the square of the pion vacuum correlator, $$\begin{aligned} G_1({\vec\rho})=-i\langle 0|\pi ({\vec\rho} )\pi^\dag(0)+ \pi^\dag(0)\pi({\vec\rho})|0\rangle~, \nonumber\end{aligned}$$ which represents the density of states in the pion vacuum. These are exactly the states which are excited as the final states of free propagation in the course of the $\pi\pi$-collision, and the characteristic scale of correlation for these vacuum fluctuations can be nothing but $m_{\pi}^{-1}$. At $T\to\infty$, the colliding pions are (formally) very hard (their thermal wave-length becomes very short) and they cannot explore any scale except this one. Thus, the “hidden interference” is active only if two collisions occur within the range of a typical fluctuation in the “trivial” pion vacuum. Otherwise, these collisions are dynamically disconnected and the interference becomes strongly suppressed. At finite temperatures, the behavior of the form-factor at the origin is less singular, but the main scale for the distance $\rho$ remains $m_\pi^{-1}$, which defines the smallness of the effect in the oversimplified model ${\bbox\pi}^4$. The parameter which regulates the magnitude of the excess of the [*normalized*]{} correlators (\[eq:E4.26\]) and (\[eq:E4.27\]) is the ratio of the effective radius of the dynamical correlations to the full transverse size $R_\bot$ of the expanding pipe. As is seen from the plot of Fig. \[fig:fig3\], at $T=m_\pi$ the excess reaches 0.08 when $R_\bot\simeq m_{\pi}^{-1}$. This understanding of the nature of the scale of correlation shows that the magnitude of the effect may be quite different when the $\pi\pi$-interaction is treated more adequately. For example, with properly fitted parameters, the $\sigma$-model provides a good description of many data which depend on $\pi\pi$-interaction. Most of low-energy interactions of elementary particles are mediated via wide resonances like $f_0$ (which is sometimes identified with $\sigma$), $a_1$, $\rho$, etc. In this picture, the $\pi\pi$-interaction is not as local as in the ${\bbox\pi}^4$-model, and more scales show up resulting in the dependence of the $\pi\pi$-cross section on the momentum transfer. The low-energy dynamics is rich and includes a chiral phase transition. In the vicinity of this phase transition the correlation radii should increase. Thus, on the one hand, we may speculate about various dynamical phenomena which may lead to an increase of the correlation radii. On the other hand, the system may develop a strong transverse hydrodynamic motion. In this case, the “measured $R_\bot$” also becomes a dynamically defined effective quantity, which is (though not as much as in the longitudinal direction) smaller than the natural transverse size. Until now, we have considered the lowest order (with respect to the interaction) contributions to the one- and two-pion inclusive distributions. The next orders can be accounted for by the expansion of the evolution operator in the Eqs. (\[eq:E4.6\]) and (\[eq:E4.13\]). Even without explicit calculations, it is clear that there will be two types of corrections. The virtual radiative corrections should be absorbed into the definitions of the dressed propagators and vertices. Within the framework of our phenomenological approach, these are of no interest or value. The real corrections are connected with the production of additional pions and involve integration over the positions of the additional points of emission (collision) and over the momenta of the additional non-observed pions. The corresponding complicated transition amplitudes will be properly symmetrized. However, the result of the integration over the unobserved momenta will again be a set of form-factors which will cut off all distances between the points of scattering which exceed the correlation length. Thus, the higher order effects will always be suppressed with respect to the lowest order. Our last remark is about the possibility of the induced emission of pions. To address this question, one has to realize that the so-called Bose-enhancement has two different (and mutually exclusive) aspects. If we deal with the measured states, then the symmetrization is solely due to the measurement. This is the essence of the HBT effect. If the states are not measured, then we indeed deal with the induced emission which is due to the factor $\sqrt{n+1}$ in the equation $~a^\dag|n\rangle = \sqrt{n+1}|n+1\rangle~$, which requires that the process of the emission of the $(n+1)$-th quantum physically occurs in the field of all $n$ [*identical*]{} quanta emitted before. This is the basic idea behind the laser. If the pions are produced with space-like separation, then stimulated emission is impossible. Bose-Einstein correlations in event generators {#sec:SN6} ============================================== Our estimates show that the effect of hidden interference is not expected to be large unless the system undergoes the freezeout starting from a soft mode with long-range correlations. This possibility is very appealing due to its possible physical richness. Careful simulations will be required in order to estimate the signal-to-noise ratio in a real detector. This job is usually done with the aid of the so-called event generators. A complexity stemming from an additional interference makes simulations more challenging. We address this issue below. Over the last decade, many computer codes which are supposed to mimic the particle yields in heavy-ion collisions have been developed. These codes rely on classical propagation of the particles in hadronic cascades. They do not describe the quantum properties of the measurement which lead to the interference of amplitudes. At best, the output of an event generator is the list of particles with one-to-one correspondence between their momenta and the coordinates of their emission. Such a list is equivalent to a complete optical image of the source. Therefore, there are no alternative histories in the free propagation. An “afterburner” has to be added to incorporate quantum effects. In order to recover the alternative histories and thus simulate the two-pion correlations in a model with independent pion production, one has to know four one-particle amplitudes, $a_N({\bbox k})$. They are necessary to recover the two-pion amplitude ${\cal A}_{NM}(k_1,k_2)$ (see Eq. (\[eq:E1.1\]) ). Hence, one has to store some information about the pion sources before the emission which allows for such a reconstruction. In the model of binary collisions, the parameters of parents (their ID, momenta, and the coordinates of the reaction volume) have to be stored for each pion from the detected pair. Otherwise, it is impossible to reconstruct all four amplitudes which interfere in the process of the two-pion measurement. Eventually, one has to average the square of the inclusive amplitude over an ensemble of all events. Then, this procedure has to be repeated for every point in the two-particle momentum space. In more sophisticated models, with the pions coming from decays of resonances, we may have even more interfering amplitudes and each case requires special consideration [@RES]. The requirement that the quantum mechanics of the last interaction, which forms the two-pion spectrum, must be recovered, follows from first principles and cannot be removed. In some particular cases, one may take a short cut and weaken the requirements of the complete description. However, every precaution should be taken that the initial state of the interfering system is described as a [*quantum state*]{}. Finally, we have to mention that in the literature, another kind of correlation function, which has been introduced [@Zaic] (see also Refs. [@Pratt; @Wiedem]) in connection with the problem of simulation of Bose-Einstein correlations in multi-pion events, is widely discussed. The algorithmic definition of this object (which is different from the inclusive differential distribution used in this paper) has several steps: (i) The events are sorted by the total number ${\cal N}$ of pions in the event. (ii) The correlation function $C_{\cal N}(k_1,...,k_{\cal N})$ is [*measured*]{}, i.e., the ${\cal N}$-pion amplitude is fully symmetrized and its squared modulus is averaged over the subset of events with this given ${\cal N}$. (iii) All but two arguments of $C_{\cal N}(k_1,...,k_{\cal N})$ are integrated out, which results in a set of two-pion correlators $C_{\cal N}(k_1,k_2)$, still dependent on ${\cal N}$. (iv) This set is averaged over ${\cal N}$, in order to obtain $C(k_1,k_2)$. We could not find a quantum observable which would correspond to this correlator. We also could not establish a direct connection between this correlator and the parameters of the emitting system. Conclusions {#sec:SN7} =========== In this paper, we show that precise measurements of two-particle correlators, both for identical and non-identical particles, may uncover important information about the freezeout dynamics. A reliable theoretical prediction for the magnitude of these correlations can be made only within the framework of an elaborated scenario of the heavy-ion collision which is currently absent. Data may be very helpful for the theoretical design of the fully self-consistent scenario. It is equally important to measure the magnitude of the effect of hidden interference or to establish its absence at a high level of confidence. From this point of view, our calculations carry an important message that there are no theorems which require the normalized two-pion correlator to approach unity for large differences of momenta. Neither should it take the value of 2 at $\Delta k=0$. The latter value is the norm (charge) of the state with two identical pions in a [*pure*]{} state only. In general, two final-state pions are described by a density matrix. For the same reason, the normalized correlator of non-identical pions should not equal 1. Moreover, the correlator should not equal 1 even for the couples like $\pi p$, $pn$, etc. Therefore, the technique of “mixing of events” should be used with extreme caution, if used at all. The idea of a universal Gaussian fit of the HBT correlators can hardly be fruitful as well. (The authors fully realize that there may be many technical difficulties in the practical implementation of this advice.) One more practical lesson of our analysis concerns the theory of multi-pion correlations. This issue cannot even be addressed without an explicit account of the dynamical mechanism responsible for the creation of multi-particle states. Our general conclusion is that HBT interferometry cannot be model-independent. [**ACKNOWLEDGMENTS**]{} We are grateful to Rene Bellwied, Scott Pratt, Claude Pruneau, Edward Shuryak and Vladimir Zelevinsky for many stimulating discussions. Conversations with the members of the STAR collaboration helped us realize many practical problems of measurements in HBT. We appreciate the help of Scott Payson for critically reading the manuscript. This work was supported by the U.S. Department of Energy under Contract No. DE–FG02–94ER40831. Appendix A. {#sec:A1 .unnumbered} =========== Here, we reproduce lengthy expressions for the two-particle inclusive spectra as they appear after performing all couplings. We do not use them in this form throughout the paper. However, they are the starting point, would we wish to design a code for the simulation of the two-pion correlations. In this code, the particle densities $n_N(p)$ will serve as the frequencies for the initial-state pions to appear within the range of the last interaction. At least some part of the integrations with respect to the momenta of the unobserved final-state pions is preferable to do analytically in order to simplify the remaining Monte-Carlo integrations. The full expression for the two-particle spectrum of identical pions as it appears after performing all couplings in Eq.(\[eq:E4.13\]), is $$\begin{aligned} { dN_{\pi^{+} \pi^{+}}^{(2)} \over d\bbox k_1 d\bbox k_2} = { dN_{\pi^{+}}^{(1)} \over d\bbox k_1 } { dN_{\pi^{+}}^{(1)} \over d\bbox k_2 } ~+~{g_{4\pi}^4\over 16}\sum_{N,M} \int {d^4s_1 d^4s_2 d^4p_1 d^4p_2 d^4q_1 d^4q_2 \over 4k_1^0 k_2^0(2\pi)^{24} }\nonumber\\ \times\delta_+(q_1^2-m^2) \delta_+(q_2^2-m^2)\delta_+(s_1^2-m^2) \delta_+(s_2^2-m^2) \delta_+(p_1^2-m^2) \delta_+(p_2^2-m^2) \nonumber\\ \times \bigg\{[3n^{(+)}_{N} (s_1)n^{(+)}_{N}(s_2) + 6n^{(+)}_{N}(s_1)n^{(-)}_{N}(s_2)+ 2n^{(+)}_{N}(s_1)n^{(0)}_{N}(s_2)+ n^{(0)}_{N}(s_1)n^{(0)}_{N}(s_2)]\nonumber\\ \times[3n^{(+)}_{M} (p_1)n^{(+)}_{M}(p_2) + 6n^{(+)}_{M}(p_1)n^{(-)}_{M}(p_2)+ 2n^{(+)}_{M}(p_1)n^{(0)}_{M}(p_2)+ n^{(0)}_{M}(p_1)n^{(0)}_{M}(p_2)]\nonumber\\ \times e^{-i(k_1-k_2)x_{NM}} \int_{{\cal V}_N}\!\! dy_1 e^{-i(k_1+q_1-s)y_1} \int_{{\cal V}_N}\!\! dy_2 e^{i(k_2+q_1-s)y_2} \int_{{\cal V}_M}\!\! dy_3 e^{-i(k_2+q_2-p)y_3} \int_{{\cal V}_M}\!\! dy_4 e^{i(k_1+q_2-p)y_4}\nonumber\\ +[26 n^{(+)}_{N} (s_1)n^{(-)}_{N}(s_2) n^{(+)}_{M} (p_1)n^{(-)}_{M}(p_2) + 5 n^{(+)}_{N} (s_1)n^{(+)}_{N}(s_2) n^{(+)}_{M}(p_1)n^{(+)}_{M}(p_2)\nonumber\\ + 5 n^{(+)}_{N} (s_1)n^{(-)}_{N}(s_2) n^{(0)}_{M}(p_1)n^{(0)}_{M}(p_2) + 5 n^{(0)}_{N} (s_1)n^{(0)}_{N}(s_2) n^{(+)}_{M}(p_1) n^{(-)}_{M}(p_2)\nonumber\\ +2n^{(0)}_{N}(s_1)n^{(+)}_{N}(s_2) n^{(0)}_{M}(p_1)n^{(+)}_{M}(p_2) + n^{(0)}_{N}(s_1)n^{(0)}_{N}(s_2) n^{(0)}_{M}(p_1)n^{(0)}_{M}(p_2)]\nonumber\\ \times\bigg[ e^{-i(q_1-q_2)x_{NM}} \int_{{\cal V}_N}\!\! dy_1 e^{-i(k_1+q_1-s)y_1} \int_{{\cal V}_N}\!\! dy_2 e^{i(k_1+q_2-s)y_2} \int_{{\cal V}_M}\!\! dy_3 e^{-i(k_2+q_2-p)y_3} \int_{{\cal V}_M}\!\! dy_4 e^{i(k_2+q_1-p)y_4}\nonumber\\ +e^{-i(q_1+k_1-q_2-k_2)x_{NM}} \int_{{\cal V}_N}\!\! dy_1 e^{-i(k_1+q_1-s)y_1} \int_{{\cal V}_N}\!\! dy_2 e^{i(k_2+q_2-s)y_2} \int_{{\cal V}_M}\!\! dy_3 e^{-i(k_2+q_2-p)y_3} \int_{{\cal V}_M}\!\! dy_4 e^{i(k_1+q_1-p)y_4}\bigg]\bigg\}~, \label{eq:A1.1}\end{aligned}$$ where $s=s_1+s_2$, $p=p_1+p_2$; $x_N$ and $x_M$ are the “central” coordinates of two cells, and $x_{NM}=x_N-x_M$. A quite long expression emerges for the correlator of two different pions as well. $$\begin{aligned} { dN_{\pi^{+} \pi^{-}}^{(2)} \over d\bbox k_1 d\bbox k_2} = { dN_{\pi^{+}}^{(1)} \over d\bbox k_1 } { dN_{\pi^{-}}^{(1)}\over d\bbox k_2 } ~+~{g_{4\pi}^4\over 6}\sum_{N,M} \int {d^4s_1 d^4s_2 d^4p_1 d^4p_2 d^4q_1 d^4q_2 \over 4k_1^0 k_2^0(2\pi)^{24} }\nonumber\\ \times\delta_+(q_1^2-m^2) \delta_+(q_2^2-m^2)\delta_+(s_1^2-m^2) \delta_+(s_2^2-m^2) \delta_+(p_1^2-m^2) \delta_+(p_2^2-m^2) \nonumber\\ \times \bigg[ e^{-i(q_1-q_2)x_{NM}} [7n^{(+)}_{N} (s_1)n^{(+)}_{N}(s_2) n^{(+)}_{M}(p_1)n^{(+)}_{M}(p_2)\nonumber\\ +6 n^{(+)}_{N} (s_1)n^{(-)}_{N}(s_2) n^{(-)}_{M}(p_1)n^{(-)}_{M}(p_2)+ n^{(+)}_{N} (s_1)n^{(-)}_{N}(s_2) n^{(0)}_{M}(p_1)n^{(0)}_{M}(p_2)+ n^{(0)}_{N} (s_1)n^{(0)}_{N}(s_2) n^{(0)}_{M}(p_1)n^{(0)}_{M}(p_2)]\nonumber\\ \times\int_{{\cal V}_N}\!\! dy_1 e^{-i(k_1+q_1-s)y_1} \int_{{\cal V}_N}\!\! dy_2 e^{i(k_1+q_2-s)y_2} \int_{{\cal V}_M}\!\! dy_3 e^{-i(k_2+q_2-p)y_3} \int_{{\cal V}_M}\!\! dy_4 e^{i(k_2+q_1-p)y_4}\nonumber\\ +e^{-i(q_1+k_1-q_2-k_2)x_{NM}} [17n^{(+)}_{N}(s_1)n^{(-)}_{N}(s_2) n^{(+)}_{M}(p_1)n^{(-)}_{M}(p_2)\nonumber\\ +4n^{(+)}_{N} (s_1)n^{(-)}_{N}(s_2) n^{(0)}_{M}(p_1)n^{(0)}_{M}(p_2) +4 n^{(0)}_{N} (s_1)n^{(0)}_{N}(s_2) n^{(+)}_{M}(p_1)n^{(-)}_{M}(p_2) +n^{(0)}_{N} (s_1)n^{(0)}_{N}(s_2) n^{(0)}_{M}(p_1)n^{(0)}_{M}(p_2)]\nonumber\\ \times\int_{{\cal V}_N}\!\! dy_1 e^{-i(k_1+q_1-s)y_1} \int_{{\cal V}_N}\!\! dy_2 e^{i(k_2+q_2-s)y_2} \int_{{\cal V}_M}\!\! dy_3 e^{-i(k_2+q_2-p)y_3} \int_{{\cal V}_M}\!\! dy_4 e^{i(k_1+q_1-p)y_4}\bigg]~. \label{eq:A1.2}\end{aligned}$$ In Sec. \[sec:SN4\], we proceed with a simplified version of these equations which emerges when the distributions $n^{(j)}_{N}(p)$ are taken in Boltzmann form with an additional assumption that the system is locally neutral. This results in a common weight function and coefficients which are just the numbers of terms in different groups in Eqs. (\[eq:A1.1\]) and (\[eq:A1.2\]). Integration over the elementary volumes in Eqs. (\[eq:A1.1\]) and (\[eq:A1.2\]) requires special discussion: There is an important difference between the volumes $V_N$ in equations of Sec. \[sec:SN3\] and the volumes ${\cal V}_N$ here. The former are related to the elementary fluid cells where the distributions $n_{N}(p)$ of particles over their momenta are established due to many collisions inside the $N$’th cell , and we do not (and even are prohibited to) localize the coordinates of individual collisions within the cell explicitly. The latter correspond to the opposite case, when we locate the space-time coordinates of individual collisions. Therefore, the volumes ${{\cal V}_N}$ correspond to the actual range of the pion interaction potential and are much smaller than the volumes $V_N$ of the fluid cells required by the hydrodynamic picture. Hence, the meaning of the distribution functions $n_{N}(p)$ changes as well; now they correspond to the probability for the pion with momentum ${\bbox p}$ to penetrate the “reaction domain” near the space-time point $x_N$. In fact, the volume ${\cal V}_N$ is defined by the cross section of the $\pi\pi$-interaction itself. This picture is consistent only if the pions themselves are considered as wave packets and not as plane waves. The volume ${\cal V}_N$ is that volume where the incoming packets are identified by the interaction and where the outgoing packets are completely formed. With this picture in mind, we can integrate, e.g., $$\begin{aligned} \int_{{\cal V}_N}\!\! dy_1 e^{-i(k_1+q_1-s)y_1} \int_{{\cal V}_N}\!\! dy_2 e^{i(k_1+q_2-s)y_2}= (2\pi)^4\delta(k_1+q_1-s)\int_{{\cal V}_N}\!\! dy_2 e^{-i(q_1-q_2)y_2} = (2\pi)^4\delta(k_1+q_1-s){\cal V}_N~. \label{eq:A1.3}\end{aligned}$$ In the first equation, we assume that the relation between the volume of integration and momenta allows one to verify the conservation of momentum in the collision. In the second equation, we took into account that the actual range for the momenta $q_1$ and $q_2$ is of the order of $2T\sim 2m_\pi$, while the length of the $\pi$-$\pi$ scattering is $\sim 1/5m_\pi$. In all subsequent calculations we can rescale ${\cal V}_N$ back to the elementary volume $V_N$ and thus restore the status of $n_N(p)$ as the distribution function. Appendix B. Miscellaneous formulae. {#sec:A2 .unnumbered} =================================== 1.  If we adopt the Boltzmann shape of the distribution function, then the product of the statistical weights $n(p_1)n(p_2)$ becomes a function of $p=p_1+p_2$ and it is possible to integrate $l=p_1-p_2$ out. We have an integral, $$\begin{aligned} \int d^4p_1 d^4p_2 f(p) \delta_+(p_1^2-m^2) \delta_+(p_2^2-m^2)= {1\over 8}\int d^4p f(p) \int d^4l \delta(p^2+l^2-4m^2)\delta(pl)\nonumber\\ =\int d^4p f(p){\pi\over 2p^0}\int_{0}^{\infty} \delta(p^2-4m^2+{\bbox l}^2)~{\bbox l}^2~d |{\bbox l}|= {\pi \over 4}\int d^4p f(p) \sqrt{1-{4m^2\over p^2}}~, \label{eq:A2.0}\end{aligned}$$ where in the second equation we integrate using a special reference frame where ${\bbox p}=0$, and restore the invariant form in the final answer. 2.  The correlation functions of Sec. \[sec:SN4\] are expressed via the emission function $I(k,N,x_{NM})$ which is computed below: $$\begin{aligned} I(k,N,x_{NM})=\int d^4q\delta_+(q^2-m^2)e^{-\beta(qu)}~e^{iqx} \sqrt{1-{4m^2\over (k+q)^2}}~, \label{eq:A2.1}\end{aligned}$$ where $\beta=\beta_N$, $u=u_N$, and $x=x_{NM}$. The integral is convenient to compute in a special reference frame $ \overcirc{\cal R}$ where the time-like vector $k$ has no spatial components, ${\overcirc{\bbox k}}=0$, ${\overcirc k}_0=m$. Using the delta-function to integrate out $q^0$, we arrive at $$\begin{aligned} I(k,N,x_{NM})=\int_{0}^{2\pi}d\phi\int_{0}^{\pi}\sin\alpha d\alpha \int_{0}^{\infty}{qdq\over 2\sqrt{q^2+m^2}}\nonumber\\ \times\exp\{-\sqrt{q^2+m^2}(\beta -i{\overcirc x}_0)+ q[\beta |{\overcirc{\bbox u}}|\cos\alpha-i |{\overcirc{\bbox x}}|(\cos\alpha\cos\psi -\sin\alpha\sin\psi\cos\phi)]\}~. \label{eq:A2.2}\end{aligned}$$ The first step is to integrate over $\phi$ which results in the Bessel function, $2\pi J_0(|{\overcirc{\bbox x}}|q\sin\alpha\sin\psi)$. The next integration follows a known formula, $$\begin{aligned} \int_{0}^{\pi}d\alpha~ \sin\alpha~ e^{ib\cos\alpha} J_0(a\sin\alpha)= 2{\sin\sqrt{a^2+b^2}\over\sqrt{a^2+b^2}}~, \label{eq:A2.3}\end{aligned}$$ which holds for arbitrary complex $a$ and $b$. In our case, $a^2+b^2=-(\beta {\overcirc{\bbox u}}-i{\overcirc{\bbox x}})^2=-{\overcirc{\bbox U}^2}$ where we have introduced the complex four-vector $U^\mu=\beta u^\mu - i x^\mu$. Finally, changing the variable of integration, $q=m\sinh v$, we arrive at $$\begin{aligned} I(k,N,x_{NM})={2\pi m\over|\overcirc{\bbox U}|} \int_{0}^{\infty}(\cosh v -1) dv~ e^{-m {\overcirc U}_0 \cosh v} \sinh(m |\overcirc{\bbox U}|\sinh v)\nonumber\\ ={2\pi m\over \sqrt{U^2}\sinh\chi} \bigg[\int_{0}^{\infty}dv~ (\cosh\chi \cosh v -1)e^{-m \sqrt{U^2} \cosh v}- \int_{0}^{\infty}dv~(\cosh v -1) e^{-m \sqrt{U^2} \cosh (v+\chi)}\bigg]~. \label{eq:A2.4}\end{aligned}$$ The Lorentz-invariant expressions for the quantities defined in the reference frame $ \overcirc{\cal R}$ are $$\begin{aligned} m{\overcirc U}_0 = (kU)=m\sqrt{U^2} \cosh\chi,~~ m|\overcirc{\bbox U}| = \sqrt{(kU)^2-m^2U^2}=m\sqrt{U^2}\sinh\chi~. \label{eq:A2.5}\end{aligned}$$ The first of the integrals in Eq. (\[eq:A2.4\]) is calculated exactly. The second one can be estimated by means of integration by parts. Since the integrand has the root of the second order at $v=0$, we have to integrate by parts three times before the first non-vanishing term shows up. The net yield of this procedure is $$\begin{aligned} -{4\pi m\over |\overcirc{\bbox U}|}\int_{0}^{\infty}dv~\sinh^2{ v \over 2} e^{-m \sqrt{U^2} \cosh (v+\chi)}= -{4\pi m^2\over m^4|\overcirc{\bbox U}|^4}e^{-m {\overcirc U}_0} +\cdot\cdot\cdot~. \label{eq:A2.6}\end{aligned}$$ Since $m {\overcirc U}_0= (kU)\equiv (ku)/T -i(kx_{NM})$, this term is strongly suppressed (both by the exponent and by at least the fourth power of the small number $m/k_t$). Finally, we have to high accuracy, $$\begin{aligned} I(k,N,x_{NM})={2\pi m\over \sqrt{U^2}}\bigg[\coth\chi K_1(m\sqrt{U^2})- {1\over \sinh\chi}K_0(m\sqrt{U^2})\bigg]~. \label{eq:A2.7}\end{aligned}$$ Here, $K_n(z)$ is the standard notation for the modified Bessel function, and we remind the reader that $U^2$ and $\chi$ (as well as ${\overcirc U}_0$ and $|\overcirc{\bbox U}|$) all are complex quantities. When $x\equiv x_{NM}=0$, we have $U^2=\beta^2$, $\cosh\chi=(ku)/m=(m_t/m)\cosh (\theta -\eta)\gg 1$, and consequently, $\sinh\chi\gg 1$, $\coth\chi\sim 1$. Therefore, $$\begin{aligned} I(k,N,0)=2\pi m^2 {K_1(\beta m)\over \beta m}~. \label{eq:A2.8}\end{aligned}$$ In the particular cases considered in Sec. \[sec:SN4\], the second term in Eq. (\[eq:A2.7\]) is always smaller than the first one by the factor $m/k_t$. An important property of the final answer (\[eq:A2.7\]) is that the quantity $$\begin{aligned} U^2=\beta^2(x_1) -(x_1-x_2)^2-2i\beta (x_1)(u(x_1)\cdot(x_1-x_2))\nonumber\end{aligned}$$ in the argument of function $K_1$, does not depend on the large parameter $k_t/m$. Therefore, the saddle point of the integration over the coordinate rapidities $\eta$ is never affected by this function. A. Makhlin, and Yu. Sinyukov, Sov. J. Nucl. Phys. [**46**]{}, 354 (1987). V. Averchenkov, A. Makhlin, and Yu. Sinyukov, Sov. J. Nucl. Phys. [**46**]{} (1987) 905; A. Makhlin, and Yu. Sinyukov, Z. Phys., [**C39**]{}, 69 (1988) The NA35 collaboration: M. Gazdzicki [*et al.*]{}, in Proceedings of Quark Matter ’95, Nucl. Phys. [**A590**]{}, 197c(1995). The NA44 collaboration: B.V. Jacak [*et al.*]{}, in Proceedings of Quark Matter ’95, Nucl. Phys. [**A590**]{}, 215c(1995). M. Gyulassy, S.K. Kauffmann, and L.W. Wilson, Phys.Rev.[**C20**]{}, 2267 (1979) Yu.L. Klimontovich, Kinetic theory of non-ideal gases and non-ideal plasmas, Oxford, New York : Pergamon Press, 1982. C.M. Hung and E.V. Shuryak Phys.Rev.[**C57**]{}, 1891 (1998) A. Makhlin, Sov. J. Nucl. Phys. [**49**]{}, 238 (1989). A. Makhlin, E. Surdutovich, Phys. Rev. [**C58**]{}, 389 (1998). L.D. Landau, Izv.Akad.Nauk Ser.Fiz. [**17**]{}, 51 (1953): J.D. Bjorken, Phys. Rev. [**D27**]{}, 140 (1998) M.I. Podgoretskii, Sov. J. Part. Nucl. [**20**]{}, 266 (1989). K. Geiger, Phys. Rev. [**D 46**]{}, 4965 and 4986 (1992). A. Makhlin, E. Surdutovich, G. Welke, in Proceedings of HIPAGS ’96, p.173, Detroit, 1996. N.N. Bogolyubov, D.V. Shirkov, Introduction to the theory of quantized fields, Interscience, NY,1959 L.V. Keldysh, Sov. Phys. JETP [**20**]{}, 1018 (1964); E.M. Lifshits, L.P. Pitaevsky, Physical kinetics, Pergamon Press, Oxford, 1981. A. Makhlin, E. Surdutovich, in preparation W.A. Zajc, Phys. Rev. [**D35**]{}, 3396 (1987). S. Pratt, Phys. Lett. [**B301**]{}, 159 (1993). U.A. Wiedemann, Phys. Rev. [**C57**]{}, 3324 (1998); R.L. Ray, Phys. Rev. [**C57**]{}, 2523 (1998) [^1]: We essentially base this on the Dirac definition of the wave function as a transition amplitude, $\psi_{k}(x)\equiv\langle k|x\rangle =\psi^{*}_{x}(k)$, which allows for two complementary interpretations: (i) the particle is [*prepared*]{} with momentum $k$ and [*detected*]{} at the point $x$, or, equivalently, (ii) the particle is [*prepared*]{} at the point $x$ and [*detected*]{} with the momentum $k$. The same interpretation remains valid for the entire hierarchy of the multi-particle wave functions, $\langle k_1,k_2|x_1,x_2\rangle$, etc. [^2]: A similar mechanism is known to provide corrections to the collision term in kinetic equations as well as the first virial corrections to the equation of state of quantum gases [@Klimontovich]. [^3]: At high pion multiplicity, we can neglect an exceptional case when two detected pions come from the same binary collision. [^4]: We closely follow (especially, in Sec. \[sec:SN4\]) Ref. [@phot], where this approach was first used for photon interferometry of the QGP. There, the photon emission was due to the processes $q{\bar q}\to\gamma g$ and $qg\to\gamma q$, and the identical gluons or quarks of the final state would interfere as well, if they had not become the constituents of the thermalized system. [^5]: This is the physical origin of the so-called $m_t$-scaling and the qualitative basis for the saddle-point calculations below. [^6]: The exponential in Eq.(\[eq:E4.19\]) is the same as in Eq.(\[eq:E3.20\]). Thus, it is exactly the saddle point (\[eq:E4.21\]), which leads to Eqs. (\[eq:E3.21\]) and (\[eq:E3.22\]). [^7]: When $\eta_1=\eta_2$, the form-factor ${\cal F}$ is a smoothly decreasing function. When $\eta_1\neq\eta_2$, it acquires an additional oscillatory pattern in the $\Delta\eta$-direction.

--- abstract: 'We investigate here how the current flows over a bilayer graphene in the presence of an external electric field perpendicularly applied (biased bilayer). Charge density polarization between layers in these systems is known to create a layer pseudospin, which can be manipulated by the electric field. Our results show that current does not necessarily flow over regions of the system with higher charge density. Charge can be predominantly concentrated over one layer, while current flows over the other layer. We find that this phenomenon occurs when the charge density becomes highly concentrated over only one of the sublattices, as the electric field breaks layer and sublattice symmetries for a Bernal-stacked bilayer. For bilayer nanoribbons, the situation is even more complex, with a competition between edge and bulk effects for the definition of the current flow. We show that, in spite of not flowing trough the layer where charge is polarized to, the current in these systems also defines a controllable layer pseudospin.' author: - 'C. J. Páez' - 'D. A. Bahamon' - 'Ana L. C. Pereira' title: 'Current flow in biased bilayer graphene: the role of sublattices' --- Introduction {#sec:Intro} ============ For electronic devices of reduced dimensions, the spatial mapping of charge current is of paramount importance. In a quantum point contact, for example, electrons flow through narrow branching channels rather than the expected uniform propagation[@TopLS00; @TopLW01]; these measurements are crucial to understand how the geometry and impurities of the device affect its performance. Graphene, due to its exceptional electronic properties, has been pointed out to have great potential to replace existing materials in traditional electronics [@CasGPN09], as well as to be used in new pseudospintronic devices[@RycTB07; @SanPM09; @AkhB07; @GunW11; @Sch10; @PesM12]. Common to these traditional and new applications, local aspects of charge flow in graphene have to be understood. Studies on zigzag graphene nanoribbons have shown, for low energies, dispersionless and sublattice polarized edge states[@CasPL08]. Notably the overlap of these no current-carrying states from opposite edges creates charge flow through the centre of the nanoribbon[@ZarN07]. This charge-current asymmetry has been ignored for bilayer graphene (BLG), which offers better options for digital electronics. A remarkable property of BLG -which potentiates its use in future graphene based electronics- is the possibility of opening and controlling a band gap with a potential difference applied between top and bottom layers (biased bilayer) [@McC06; @CasNN10; @Sch10; @DonVJ13; @OhtBA06; @NilCG08; @McCK13]. The externally applied perpendicular electric field breaks the inversion symmetry of the system [@ZhaTG09] allowing to define a layer pseudospin, at least for energies below the interlayer coupling energy [@PesM12; @AbdPM07; @SanPM09]. Therefore, many devices based on these systems have been proposed recently, which involve the ability to control this layer pseudospin (the charge density polarization between layers induced by the bias) for different bias layouts[@MiyTK10; @XiaFL10; @Vel12], such as the creation of electron highways [@QiaJN11] or pseudospin-valve devices [@RycTB07; @SanPM09; @LiZX10]. Experimentally, charge localization over different layers and different sublattices due to a bias voltage has been observed in these systems by STM images[@KimKW13], indicating the possibility of controlling layer and sublattice pseudospins in real samples. Even though all the attention that has been given to the possibilities of controlling charge densities in BLG through the bias, the charge flow has been neglected. The purpose of this paper is to analyze and quantify the main transport features of pristine biased BLG. In particular we are enticed to unveil the relation between charge density and charge flow. Although charge and current are intimately linked by the continuity equation, when the electric field localizes charge over different sublattices in different layers, it is not evident how current density is distributed. Using the lattice Green’s functions we are able to map charge density over each sublattice site in both layers, conjointly with the current flowing towards its neighbors. Our results show that current does not necessarily flow in the regions with higher charge density, and this would have a fundamental role when devising electronics. At low energies, for bulk biased BLG, we observe that charge is primarily concentrated over one layer while current flows over the other layer. We show that this is a consequence of an important concentration of charge in only one of the sublattices in the layer with more charge density. This picture is enriched in biased BLG nanoribbon with zigzag edges where additional sublattice polarized edge states [@NakFD96; @LimSF10; @LvL12; @WanSL11] compete with bulk sublattice polarization. The distribution of the current for edge states is found to also depend whether the most external atom of the edge corresponds to a coupled or uncoupled sublattice in the AB stacking. We show results as a function of energy around Fermi energy, and also as a function of nanoribbon width and bias strength ($V$), elucidating the behavior of current flow and the main role of sublattices. The effects of disorder and next-nearest neighbor hoppings are also discussed. ![(color online)(a) Schematic representation of a BLG nanoribbon, with zigzag edges and width W, between left (L) and right (R) semi-infinite contacts. There is an electrostatic potential difference of 2V between the two layers. Different sublattices, A and B, are represented in different colors (b) Detail of the sublattices A and B in top and bottom layer, indicating of the nearest hoppings: $\gamma_0$ in-plane, and $\gamma_1$ coupling the dimer sites interlayer.[]{data-label="fig:device"}](Fig1.eps){width="0.95\columnwidth"} Model {#sec:theory} ===== We consider a BLG nanorribon with zigzag edges and Bernal AB stacking, as shown in [Fig. \[fig:device\]]{}(a). The width $W=(N-1)a\frac{\sqrt{3}}{2}$ is defined by the number of sites $N$ in the transversal direction and $a=2.46$ [Å]{} is the lattice constant for graphene. The infinite BLG zigzag nanoribbon is modelled by the tight binding Hamiltonian: $$\label{eq:hamiltonian} \begin{array}{c} H=-\gamma_0\displaystyle\sum_{m,i,j}(a_{m,i}^{\dagger}b_{m,j}+H.c.) -\gamma_1\displaystyle\sum_{j}(a_{T,j}^ {\dagger} b_{B,j}+H.c.)\\ + V \displaystyle\sum_{i}(a_{T,i}^{\dagger}a_{T,i}+b_{T,i}^{\dagger}b_{T,i})-V \displaystyle\sum_{i}(a_{B,i}^{\dagger}a_{B,i}+b_{B,i}^{\dagger}b_{B,i}), \end{array}$$ Here, the first term refers to individual graphene layer (top and bottom), the second term describes the interlayer coupling, and the last two terms introduce the interlayer bias which induces an energy difference between layers parameterized by $V$. Field operators $a^{\dagger}_{m,i}$ ($a_{m,i}$), $b^{\dagger}_{m,i}$ ($b_{m,i}$) create (annihilate) one electron in sublattice A or B *i-th* site of the top ($m=T$) or bottom ($m=B$) layer. We use the intralayer nearest-neighbor hopping $\gamma_0=3.16$ eV and the interlayer coupling $\gamma_1$= 0.381 eV [@KuzCV09]. From our Hamiltonian [eq. (\[eq:hamiltonian\])]{} the Bernal AB stacking is easily recognised, as shown in [Fig. \[fig:device\]]{}(b), sublattice sites A in the top layer ($A_T$) are on top of sublattice sites B in the bottom layer ($B_B$). We refer to this coupled($A_T-B_B$) sites as dimer sites while non-coupled sites ($A_B$ or $B_T$) are non-dimer sites [@CasPL08; @McCK13]. The Introduction of next-nearest neighbor hopping in each layer and further interlayer couplings, as well as of on site disorder, is discussed in section VI. To account for the electronic transport properties, the infinite BLG zigzag nanoribbon is divided in three regions; a finite central region and two semi-infinite ribbons acting as contacts [@Datta99]. Although [eq. (\[eq:hamiltonian\])]{} describes the dynamics of electrons in the three regions and no qualitative differences can be found among them; it is mandatory to have a finite central region to calculate its Green’s function [@Datta99; @FG99; @LewM13] in order to extract conductance, local density of states (LDOS) ($\rho$), charge density ($\rho_c$) and current density ($\vec{J}$)[@BahPS11]. Despite the fact the transport properties are calculated for the central region, these can be extended to the whole BLG zigzag nanoribbon. The retarded Green’s function is calculated as $G^r=[E-H_c-\Sigma_L-\Sigma_R]^{-1}$ where $H_c$ is the Hamiltonian of the central region and $\Sigma_{L(R)}$ is the left (right) contact self-energy [@Datta99]. Charge and current are intimately related through the continuity equation, its lattice version can be written as $\frac{dc^{\dagger}_nc_n}{dt} + [\hat{J}_{nn'}-\hat{J}_{n'n}]=0$ where $\hat{J}_{nn'}=\frac{e}{i\hbar}[t_{n'n}c^{\dagger}_{n'}c_n-t_{nn'}c^{\dagger}_nc_{n'}]$ is the bond charge current operator. $\hat{J}_{nn'}$ results, exactly as one would expect, from the difference of electrons flow in opposite directions. The connection with the Green’s function arises because the quantum statistical average of the bond charge current operator of the form $\langle c^{\dagger}_{n}c_{n'} \rangle$ are related to the lesser Green’s function $G^<_{n'n}(E)$ [@HauJ08; @Datta99], in steady state the bond charge current including spin degeneracy is: $$J_{nn'}=I_0\int_{E_F-eV/2}^{E_F + eV/2} dE[t_{n'n}G_{nn'}^<(E)-t_{nn'}G_{n'n}^<(E)] \label{eq:current}$$ $I_0$=2e/h = 77.48092 $\mu$A/eV is the natural unit of bond charge current density. The lesser Green’s function in the absence of interactions can be resolved exactly as $G^<(E)=G^r(E)[\Gamma_Lf_L+\Gamma_Rf_R]G^a(E)$ where $\Gamma_{L(R)}=i(\sum_{L(R)}-\sum_{L(R)}^{\dagger})$ is the left (right) contact broadening function and $f_{L(R)}$ is the Fermi distribution of the left (right) contact. $t_{n'n}$ is the hopping parameter between sites $n'$ and $n$, in our BLG zigzag nanorribon represents $\gamma_0$ for intralayer bond current and $\gamma_1$ for interlayer bond current. In order to quantify the electron flow in a layer we defined the layer current density as: $$\label{eq:currentfield2} I_m=\sum_{k \in m}J_{k},$$ $k$ represents a site in the central region of the nanoribbon in layer $m=T,B$; $J_{k}=\sum_{n'}J_{kn'}$ is the total current at site $k$ calculated adding the bond current ([eq. (\[eq:current\])]{}) between site $k$ and its neighboring sites $n'$. Once again, since we are working on a pristine nanoribbon the current density in any slide of our device is exactly the same, because of that we associated it to a layer current density in [eq. (\[eq:currentfield2\])]{}. Complementary to current density, charge density at site $k$ can also be expressed using the lesser Green’s function as: $$\label{eq:cchargedensity} \tilde{\rho}_c(k)=\frac{e}{2\pi i}\int_{E_F-eV/2}^{E_F + eV/2} dE{G_{k,k}^<(E)}.$$ At equilibrium, all states are occupied as specified by the Fermi-Dirac distribution ($f(E)$) and the lesser Green’s function acquires the simple form $G^<(E)=if(E)A(E)$. Where $A(E)=i(G^r-G^a)$ is the spectral function, which is related to the LDOS as $\rho(r,E) = \frac{1}{2\pi}A(r,E)$ [@Datta99]. It is noteworthy that at low bias and low temperature $\rho_c \approx e^2V\rho(E_F)$, and clearly it is observed that charge density ($\rho_c$) has the same local distribution of LDOS ($\rho$). Given we are interested in how charge and current distributions are related; with no loss of generality, to keep explanations and figures as simples as possible, we will refer from now on to LDOS as charge distribution. To quantify and visualize how charge (LDOS) is distributed over one layer it is defined the charge density per layer as: $$\label{eq:cchargedensity2} \rho_m =\sum_{k \in m}\tilde{\rho}_k.$$ The Green’s function formalism has succeed in reproducing scanning probe microscopy experiments[@CreFG03; @MetB05] providing a framework to interpret the measured charge map, electron flow as well as predicting new effects. Results for Charge and current density {#sec:Charge and current density} ======================================= In [Fig. \[fig:Fig2\]]{} we show the band structure and details of the charge and the current densities for BLG under the influence of an applied voltage difference of $2V=0.14$ eV between the two layers. The right column shows the results for a BLG nanoribbon with zigzag edges and width of $N=300$ atoms, while the results at the left column are for a bulk BLG (for which the same width of $N=300$ atoms was considered with periodical boundary conditions). ![ (color online) The left column shows results for a bulk biased BLG, while the right column are for BLG nanoribbon with zigzag edges (for both cases a potential difference V=0.07 eV and a width of $N=300$ atoms is considered. (a)-(b) Band structures. (c)-(d) Zoom into the band structure’s regions marked by the dashed lines in (a) and (b). $\Delta$ is the minimum separation between the flat and the dispersive edge state bands for the zigzag ribbon. (e)-(f) Percentage of the charge density in each layer. (g)-(h) Current density on each layer. (i)-(j) Total current density on the BLG. (k)-(l) The contribution of each sublattice to the charge density.[]{data-label="fig:Fig2"}](Fig2.eps){width="0.945\columnwidth"} Band Structures --------------- The band structure for the bulk BLG in [Fig. \[fig:Fig2\]]{}(a) evidences the opening of the energy gap of approximately 2V (observe that the energy scale is normalized by the bias voltage V). The presence of the zigzag edges introduces edge states in the gap region of the band structure, as observed in [Fig. \[fig:Fig2\]]{}(b). Zooms into the band structure’s regions marked by the dashed lines in [Fig. \[fig:Fig2\]]{}(a) and (b) are shown in [Fig. \[fig:Fig2\]]{}(c) and (d), respectively. For the bulk, we see in [Fig. \[fig:Fig2\]]{}(c) the well-known “Mexican-hat" structure, due to the applied bias, mixed to other higher bands for this system size and bias (the wider the nanoribbon considered the higher is the density of bands and the band mixing in this region). For the zigzag case, one can see in more detail in [Fig. \[fig:Fig2\]]{}(d) that in addition to the usual band structure this region contains two edge states energy bands: a flat band at $E/V=1$ and a dispersive band for $E/V \leq 1 - \Delta/V$, in agreement with previous works [@CasPL08; @CasNM10; @YaoYN09; @LiZX10; @RhiK08]. The minimum separation between the edge states bands is $\Delta$=2V$\frac{\gamma_1^2}{\gamma_1^2+\gamma_0^2}$ (this expression is derived from the difference between dispersive band and flat band [@CasPL08] at $ka/2\pi=0.5$). Observe that $\Delta$ increases linearly with the external bias V and does not depend on the nanoribbon width. Charge and Current Density in each Layer ---------------------------------------- In [Fig. \[fig:Fig2\]]{}(e) and (f) we show the percentage of the total charge density of the bilayer which is accumulated in each of the layers (top or bottom) - once charge in each layer is obtained from [eq. (\[eq:cchargedensity2\])]{}, its proportion with respect to the total charge in both layers is calculated . Both for the system with zigzag edges and for the bulk there is a clear unbalance between layers, with electronic charge density being concentrated predominantly (from 75 to 100$\%$) over the top layer for the entire energy range shown. The percentage of the current density over each layer is calculated in the same way from [eq. (\[eq:currentfield2\])]{} and the results are shown in Fig. \[fig:Fig2\](g) and (h), for bulk and zigzag, respectively. Comparing charge and current densities in each layer, one can see that although the charge densities are highly concentrated over the top layer, the current densities are higher on the bottom layer for a wide energy range, both for bulk or zigzag BLG. This behavior is counterintuitive and contradicts the most basic theoretical model of charge flow. [Fig. \[fig:Fig2\]]{}(i) and (j) show the total current density ($I$ divided by $I_0$), which corresponds to the summation of the currents in the top and the bottom layers. The total current density is directly proportional to the conductance of the system. The Role of the Sublattices --------------------------- To investigate the origin of the discrepancy between the charge density and current density in each layer, we compute separately the contribution of each sublattice to the charge density, as shown in [Fig. \[fig:Fig2\]]{}(k) and (l). This gives us an important clue to understand the phenomenon: the charge is not only predominantly over one layer (the top layer), there is also a sublattice polarization in this layer. For the bulk, [Fig. \[fig:Fig2\]]{}(k), we observe that the charge on the top layer is entirely over the sublattice $B_T$, while sublattice $A_T$ shows zero contribution to the charge density in the entire energy range shown. This effect comes from the sublattice asymmetry introduced by the AB-stacking in BLG[@CasNM10]: the dimer sites ($A_T$ and $B_B$) hybridized their orbitals to form higher energy bands, being the charge density for low energy states located mostly on non-dimer sites ($B_T$ and $A_B$) [@McC06; @KosA06]. Here we show in [Fig. \[fig:Fig2\]]{}(k) how this sublattice asymmetry is preserved in the top layer after the application of the voltage difference between the layers. We see that although the charge on top layer keeps completely located over only one sublattice ( non-dimer $B_T$), the charge density over the bottom layer is mostly sublattice unpolarized, ie, equally shared between the two sublattices. This interesting characteristic of these systems, which has already been point out in previous experimental[@KimKW13], analytical[@McCK13; @CasNM10] and numerical[@RamNT11] works, can possibly explain why the current goes preferentially over the bottom layer. For the zigzag nanoribbon, the density distribution in each sublattice becomes even more interesting, [Fig. \[fig:Fig2\]]{}(l), with a clear competition between the bulk effect in the AB-stacking just described and the additional sublattice polarization that is well known to occur around the zigzag edges [@CasPL08], as we will show. For energies $E/V > 1 - \Delta/V$, we see from [Fig. \[fig:Fig2\]]{}(l) that the sublattice distribution is similar to that described to the bulk, with charge on top layer only over $B_T$ sublattice. However, for energies $E/V \leq 1 - \Delta/V$, while [Fig. \[fig:Fig2\]]{}(f) tells us that the charge is still over the top layer (more than $90\%$), [Fig. \[fig:Fig2\]]{}(l) shows that there is an inversion in the subltattice: $A_T$ sublattice is now predominant, with some oscillations. Comparing to the band structure in [Fig. \[fig:Fig2\]]{}(d), we see that in this region the dispersive edge-state band plays an important role. The energy split of size $\Delta$ corresponds in fact to the split of states localized on opposite edges of the top layer. The states from the flat band at $E/V=1$ are located on the edge of the top layer where the outermost atoms are $B_T$ atomic sites, the same sublattice that is privileged by the AB-stacking. On the other hand, as in zigzag graphene nanoribbons the outermost atoms in opposite edges belong to different sublattices, the states from the dispersive band at $E/V \leq 1 - \Delta/V$ are located on the edge where the outermost atoms of the top layer are $A_T$ sites. And here is where the competition between edge and bulk arises, leading to the oscillations between sublattices observed, with an advantage to the $A_T$ sites, i.e., the edge state localization effect being more robust than the bulk effect. Mapping the Spatial Distribution of Charge and Current over the BLG ===================================================================  In a lattice, we can imagine one electron injected from the left contact hopping from site to site until reaching the right contact. Clearly one electron is enabled to hop on its nearest neighbor if there are electronic states available there; in this regard the spatial distribution of charge and current densities are expected to be related to each other. In the discussion of the previous section we have already identified that the polarization of the charge density to only one of the sublattices plays an important role in the discrepancies observed between charge and current densities in each layer. Here, [Fig. \[fig:Fig3\]]{} helps us to observe in more detail the spatial distribution of charge and current densities over each layer (and each sublattice) of the bilayer systems. The systems considered in this calculation of [Fig. \[fig:Fig3\]]{} are exactly the same from [Fig. \[fig:Fig2\]]{}: biased BLG of 300 carbon atoms in width, with periodical boundary conditions for the bulk and zigzag edges for the nanoribbon, V=0.07eV. For the representations of the spatial distribution of the charge density, the density on each atomic site is shown here as proportional to the radius of the disk and its color stands for sublattice: red for $A_T$ and $A_B$ and blue for $B_T$ and $B_B$ (same color scheme shown in [Fig. \[fig:device\]]{}(b)). Initially, in [Fig. \[fig:Fig3\]]{}(a) and (b) we map charge and current spatial density for a bulk BLG, avoiding in this case any complication introduced by the edge states. This distribution corresponds to the energy $E/V=1.003$ - at this energy, Fig2(e) tells us that 80% of the charge is located on top layer, while Fig.2(g) shows that current density is much higher in bottom layer and nearly zero on top layer. [Fig. \[fig:Fig3\]]{}(a) shows that on the top layer charge is completely located on non-dimer $B_T$ sites, being homogeneously distributed over the layer. On the bottom layer, although it is not appreciated in [Fig. \[fig:Fig3\]]{}(a) because $\rho_B$ is approximately 10 times smaller than $\rho_T$; there is a homogeneous charge density on $A_B$ and $B_B$ sites (9% each, see [Fig. \[fig:Fig2\]]{}(k)). When current density is calculated, as shown in [Fig. \[fig:Fig3\]]{}(b), it is appreciated that current is homogeneously distributed over bottom layer while is nearly zero over top layer. This can be understood on account of charge is completely localized on non-dimer sites ($B_B$). Electrons on these sites can not jump on its nearest neighbors, causing no electron flow over top layer. On the bottom layer, in spite of $\rho_B < \rho_T$, charge is homogeneously distributed over both sublattices, allowing electron hopping among sites. For the BLG zigzag nanoribbons, bias lifts edge states degeneracy. We see from [Fig. \[fig:Fig3\]]{}(c) and (e) that the charge densities for energies corresponding to the split edge state bands, $E/V=0.965 \approx 1-\Delta/V$ and $E/V=1.003 \approx 1$, are highly localized on opposite zigzag edges of the top layer, in agreement to previous calculations [@CasNM10; @YaoYN09; @RhiK08; @LiZX10]. For the first energy one can see from [Fig. \[fig:Fig2\]]{}(l) that nearly 80% of the charge density is located on dimer sites $A_T$ while 18% of the charge is located on non-dimer sites $B_T$. Spatial mapping of the charge density reveals in [Fig. \[fig:Fig3\]]{}(c) an edge state located on only one of the edges of the top layer: the edge whose outermost atoms are from sublattice $A_T$. Once again due to the considerable difference between the densities in the two sublattices, only the edge state is appreciable. However, charge density is also homogeneously distributed on non-dimer sites ($B_T$) of the top layer. Overlapping of the exponentially decaying edge state ($A_T$) and the homogeneously distributed state ($B_T$) creates a high current density on this edge of the top layer, as depicted on Fig. \[fig:Fig3\](d). This figure also shows a high current density on the bottom layer right bellow this edge, its origin is similar to the top layer current: this edge at bottom layer terminates at a non-dimer sites $A_B$ sustaining edge states while dimer sites $B_B$ have an enhanced charge density caused by the top layer edge state; these two states overlap creating the highly charge current observed. When next to nearest neighbors are included in monolayer zigzag nanoribbon edge states acquire velocity, this however does not affect charge or current density. The effect of the sublattice symmetry breaking is also observed for $E/V=1.003 \approx 1$. At this energy, edge states localize on the other edge, the one whose outermost atoms are from $B_T$ sublattice, as shown in [Fig. \[fig:Fig3\]]{}(e). Considering that this sublattice $B_T$ corresponds to non-dimer atoms, current on bottom layer is not affected for this energy, as shown in [Fig. \[fig:Fig3\]]{}(f): current is distributed over the whole layer. Over the top layer there is no current because charge is completely localized on $B_T$ sites, this situation is reminiscent of bulk biased BLG. Current dependence on bias voltage and size =========================================== ![(color online) (a)-(b) Percentage of charge density on top and bottom layer as a function of the bias $V$ applied between the layers, calculated for $E/V=1$. (c)-(d) Total current density on top and bottom layer, as a function of bias V for $E/V=1$ . Several widths of zigzag biased BLG are shown: from $N=20$ to $N=640$. (e) Band structure for a BLG nanoribbon with zigzag edges, $V=0.07$ eV, for two different sizes: $N=80$ and $N=160$.[]{data-label="fig:Fig4"}](Fig4a.eps "fig:"){width="1\columnwidth"} ![(color online) (a)-(b) Percentage of charge density on top and bottom layer as a function of the bias $V$ applied between the layers, calculated for $E/V=1$. (c)-(d) Total current density on top and bottom layer, as a function of bias V for $E/V=1$ . Several widths of zigzag biased BLG are shown: from $N=20$ to $N=640$. (e) Band structure for a BLG nanoribbon with zigzag edges, $V=0.07$ eV, for two different sizes: $N=80$ and $N=160$.[]{data-label="fig:Fig4"}](Fig4b.eps "fig:"){width="1\columnwidth"} In this section we focus on the effects of the bias voltage strength and the width of the nanoribbons on the current and charge densities. For this purpose, we need first to choose a fixed energy. The experimental observation of a current density highly localized on one of the edges of a BLG nanoribbon, like the current shown in [Fig. \[fig:Fig3\]]{}(d), would require an extremely clean sample, as edge disorder would scatter electrons, degrading the current and destroying its spatial localization [@MucCL09; @CreR09; @MucL10]. On the other hand, setting $E/V \approx \pm1$ for a BLG nanorribon offers control of the layer pseudospin: the layer in which charge current is conducted (top or bottom), in a similar way observed for bulk systems (as observed for the current in Figs. \[fig:Fig2\] and \[fig:Fig3\]) and avoiding the edge disorder sensitivity. Therefore, we choose to investigate here how the current and the charge densities vary with system size and with bias voltage at energy $E/V =1$: results are shown in [Fig. \[fig:Fig4\]]{}. Figures \[fig:Fig4\](a) and (b) show that for wider nanoribbons, bias voltage variation and ribbon width do not modify the complete charge density polarization on the top layer. As we have seen, for this energy, charge mostly localizes on non-dimer sites of top layer reducing nearly to zero the current density over top layer. Here we show in Figs. \[fig:Fig4\](c) that this characteristic is maintained with increasing bias and system sizes. [Fig. \[fig:Fig4\]]{}(d) shows a different evolution for the current density on the bottom layer: for wider nanoribbons or larger bias voltages, the current density rises in steps. This dependence is understood from the band structure of the biased BLG nanoribbon, as seen in [Fig. \[fig:Fig4\]]{}(e) for $V=0.07$eV and widths corresponding to $N=80$ and $N=160$. It is appreciated that flat bands are fixed at $E/V=1$ and do not depend on the nanoribbon width. On the other hand, the number of dispersive bands around $E/V=1$ increases with N, adding more conducting channels. For that reason current density evolves in a plateau-like structure. The peaks observed, at the beginning of each plateau, for wider nanoribbons $N=320$ and $N=640$ are created by the “Mexican-hat” structure of bands crossing $E/V=1$. EFFECTS OF DISORDER AND NEXT-NEAREST HOPPINGS {#Disorder-nnn} ============================================= In this section we discuss how disorder and next-nearest neighbor hoppings affect the picture presented in the previous sections. Mainly, we show here that although there are important features introduced by disorder and by further hoppings in the model, the assymmetry between charge and current density distributions is still present, therefore, the effects previously discussed are robust. In [Fig. \[fig:Fig5\]]{} we show the comparison between a non-disorderd BLG (dashed lines) and a disordered system (solid lines), again analysing both for a bulk and for a zig-zag nanoribbon. Each layer of these systems here have width of N = 80 atoms and 40 atoms in the length between the contacts. To account for disorder, we introduce a Gaussian-correlated on-site disorder for each layer[@PerS08b], with site energies ramdomly sorted in a range of width $W/\gamma_0= 0.5$ and correlation length $\lambda=2a$ for bottom layer and $\lambda=5a$ for top layer ($a = 2.46 \mathring{A}$). Larger correlation length for the top layer is due to its higher distance from the substrate. For the charge density distribution, [Fig. \[fig:Fig5\]]{}(a) and (b), we see that the disorder does not alter significantly the clear concentration of the charge on the top layer for all the energy range shown (except for the bulk at energy exactly E/V=1). For the current density, one can see that, for the bulk, [Fig. \[fig:Fig5\]]{}(c), disorder not only does not alter the fact that about 80% of the current goes through bottom layer for $E/V>1$ but also disorder destroys the equilibrium of the currents that appears at this system size for $E/V<1$, producing again a predominance of the current over bottom layer and an unbalance between charge and current in the two layers. In the presence of edges, for the zigzag case shown in [Fig. \[fig:Fig5\]]{}(d), we see that although the current in the disordered system still flows predominantly through the bottom layer, the percentages are smaller than in the non-disordered. Edge states are probably the most affected by the disorder, however further investigations would be necessary to clarify the role of disorder separately on edge and bulk current states. ![(color online) Dashed lines are for non-disordered systems, while solid lines show the effects of the inclusion of on-site correlated disorder ($W/\gamma_0=0.5$) in the biased BLG (V=0.07 eV). Systems considered here have 80x40 sites in each layer. The left column shows results for a bulk, while the right column is for BLG nanoribbon with zigzag edges. (a)-(b) Percentage of the charge density in each layer. (c)-(d) Current density on each layer.[]{data-label="fig:Fig5"}](Fig5.eps){width="1\columnwidth"} In [Fig. \[fig:Fig6\]]{} we turn our attention to the effects of the inclusion of further hoppings in the tight-binding model. These results are for a non-disordered BLG system of the same size (N=300) and same bias voltage (V=0.07 eV) considered in [Fig. \[fig:Fig2\]]{}. The difference is that now we include next-nearest neighbors in each layer ($\gamma_2=0.316$eV) and also two extra interlayer coupling parameters: $\gamma_3 = 0.38$eV, the interlayer coupling between non-dimer sites $A_B$ and $B_T$, and $\gamma_4 = 0.14$eV, the interlayer coupling between dimer and non-dimer sites $A_T$ and $A_B$, or $B_T$ and $B_B$ [@McCK13; @KuzCV09]. These induce a trigonal warping and give rise to electron-hole asymmetry[@McCK13; @CasGPN09], as observed in[Fig. \[fig:Fig6\]]{}(a) and (b). For the bulk, we see once again the opening of the energy gap of approximately 2V, due to the applied bias. The band structure is not anymore symmetrical around E/V=0, there is an energy shift of the band structure to around $E/V\approx13.5$ [@NilCG08]. We can see the mixing of higher bands for this system size and bias. For the zigzag case, one can see that edge states acquire velocity[@SasMS06]. [Fig. \[fig:Fig6\]]{} (c) and (d) are zooms into the band structure’s regions marked by the dashed lines. In [Fig. \[fig:Fig6\]]{}(e) and (f) we show the percentage of the total charge density of the bilayer which is accumulated in each of the layers (top or bottom). Comparing them to [Fig. \[fig:Fig2\]]{}(e) and (f), one observe that both for the bulk system and for the one with zigzag edges, the further hoppings here do not affect at all the polarization of charge towards the top layer. The distribution of the current density in each layer is shown in [Fig. \[fig:Fig6\]]{}(g) and (h), for bulk and zigzag, respectively. One can see now that, in general, the current is not as polarized toward the bottom layer as it is for the nearest neighbor hopping seen in [Fig. \[fig:Fig2\]]{}(g) and (h). Nevertheless, the unbalance between charge and current densities is still clear, and even considering the non-vertical interlayer couplings and the next-nearest intralayer hoppings, there are clear energy regions where the current flows more throughout the bottom layer. There are also switches to energy regions where the current is shared between both layers or is predominant over top layer. Further investigations are important here to elucidate the exact mechanisms causing these switches. ![(color online) Analysis with the inclusion of next-nearest neighbor hoppings in the model for the biased BLG. The left column is for a bulk, while the right column is for BLG nanoribbon with zigzag edges (for both cases V=0.07 eV and a width of N = 300 atoms is considered). (a)-(b) Band structures. (c)-(d) Zoom into the band structure’s regions marked by the dashed lines in (a) and (b). (e)-(f) Percentage of the charge density in each layer. (g)-(h) Current density on each layer.[]{data-label="fig:Fig6"}](Fig6.eps){width="1\columnwidth"} Conclusions =========== An external electric field perpendicular to a Bernal-stacked bilayer graphene, breaks layer and sublattice symmetry; localizing charge over two sublattices in different layers. Under this charge landscape, it is not obvious how charge current flows within the sample. We have shown here that current distribution is highly affected by the polarization of charge to only one sublattice, as well as by the geometry of the system (width considered and presence or not of edges) and by the strength of the electric field. We demonstrate that current does not necessarily flow over regions of the system with higher charge density, even when next-nearest neighbor hoppings are included in our model. For some energy ranges, charge can be polarized to one layer, while the current is equally distributed over both layers. There are also considerable energy ranges for which the current flows predominantly over the layer with much lower charge density. We show that this effect can be explained by the sublattice polarization of charge in the AB-stacking biased BLG, and that it is robust against disorder. Therefore, to design applications of bilayer graphene in digital electronics, it is essential to calculate not only the charge distribution in each layer, but also the current density distribution in each layer, as it presents much richer details than the more monotonic behavior of the charge distribution. The authors are grateful to P. A. Schulz for a critical reading of the manuscript. C.J.P.acknowledges financial support from FAPESP (Brazil), A.L.C.P. acknowledges partial support from CNPq (Brazil). Part of the numerical simulations were performed at the computational facilities from CENAPAD-SP, at Campinas State University. 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--- abstract: | The BitTorrent mechanism effectively spreads file fragments by copying the rarest fragments first. We propose to apply a mathematical model for the diffusion of fragments on a P2P in order to take into account both the effects of peer distances and the changing availability of peers while time goes on. Moreover, we manage to provide a forecast on the availability of a torrent thanks to a neural network that models the behaviour of peers on the P2P system. The combination of the mathematical model and the neural network provides a solution for choosing file fragments that need to be copied first, in order to ensure their continuous availability, counteracting possible disconnections by some peers. author: - 'an unwanted space [^1] [^2]' - bibliography: - 'IEEEabrv.bib' - 'p2pbib.bib' title: - Improving files availability for BitTorrent using a diffusion model - Improving files availability for BitTorrent using a diffusion model --- [Shell : Bare Demo of IEEEtran.cls for Journals]{} [PREPRINT VERSION\ ]{} @incollection{Napoli2014Improving\ year={2014},\ isbn={978-1-4799-4249-7/14},\ doi={10.1109/WETICE.2014.65},\ booktitle={23rd International WETICE Conference},\ title={Improving files availability for BitTorrent using a diffusion model},\ publisher={IEEE},\ author={Napoli, Christian and Pappalardo, Giuseppe and Tramontana Emiliano},\ pages={191-196}\ } Published version copyright © 2014 IEEE\ UPLOADED UNDER SELF-ARCHIVING POLICIES\ NO COPYRIGHT INFRINGEMENT INTENDED\ Dependability, distributed caching, P2P, neural networks, wavelet analysis. Introduction ============ In peer to peer (P2P) systems using BitTorrent, a shared content (named torrent) becomes split into fragments and the rarest fragments are automatically chosen to be sent first to users requesting the file. Fragments availability is given by the number of peers storing a fragment at a given moment, and periodically computed by a server storing peer ids, fragments held and requested [@Cohen03]. For computing the priority of fragments to spread, at best availability has been freshly updated, however peers often leave the system hence file fragments availability quickly changes, possibly before the least available fragments have been spread [@Kaune10]. This occurs so frequently that such a fundamental BitTorrent mechanism may become ineffective, and as a result some fragments can quickly become unavailable. Moreover, when choosing fragments to spread, communication *latencies* [@repliche08; @Novelli07] among peers are not considered, therefore fragments spreading will surely occur sooner on peers nearby the one holding the fragment to spread. As a result, the furthest peers could disconnect before receiving the whole fragment. This paper proposes a model for spreading file fragments that considers both a time-dependent priority for a fragment to spread and latencies among nodes. I.e. the priority of fragments to spread gradually changes according to the passing time and fragments availability. The priority variation is regulated in such a way that the availability of fragments is maximised while time goes. Fragments to spread are selected according to the results of our proposed diffusion model, developed by analogy to a diffusion model on a porous medium. Moreover, we propose to characterise the typical availability of a torrent observed on the P2P system, by using an appropriate neural network. Then, the selection of fragments to spread aims at counteracting their decreasing availability estimated for a later time. Therefore, the proposed work aims at supporting *Quality of Service* and *dependability* of P2P systems by attributing a priority on both the fragment to spread and the destination peer. This in turn increases *availability* and *performances* [@GiuntaMPT12aug; @ccpe13; @Kaqudai], as well as consistency [@BannoMPT10b]. The rest of the paper is organised as follow. Next section provides the mathematical representation for the proposed model. Section \[diffusion\] develops the model for the diffusion of contents on a P2P system. Section \[wrnn\] introduces the neural network that predicts the user behavious. Section \[experiments\] describes some experiments based on our proposed model and neural network predictions, as well as the preliminary results. Section \[related\] compares with related work, and finally conclusions are drawn in Section \[conclusion\]. Mathematical Representation =========================== In order to put forward our analogy between BitTorrent and a physical system, some conventions must be chosen and some extrapolations are needed. We first describe a continuum system using a continuum metric, however later on we will single out a few interesting discrete points of the continuum. Due to the analogy to BitTorrent, we use a distance metric (named $\delta$), which will be assimilated to the network latency among nodes, i.e. the hosts on a network holding seeds, peers or leeches. For the nodes we use notations $n^i$ or $n^i_\alpha$: the first indicates a generic $i$-esime node on the BitTorrent network, the second indicates the $\alpha$-esime node as seen from the $i$-esime node. Of course, $n^i_\alpha$ and $n^j_\alpha$ could be different nodes if $i \neq j$. Double indexing is needed since when we use something like $\delta^{ij}$, it will be representing the distance of the $j$-esime node as measured by the $i$-esime node. Moreover, let us express $P^{ij}_k$ as the probability of diffusion of the $k$-esime file fragment from the $i$-esime node to the $j$-esime node. Finally, we distinguish between time and time steps: the first will be used for a continuum measure of temporal intervals and we will use for it the latin letter $t$, the second will indicate computational time steps (e.g. the steps of an iterative cycle) and we will use for it the greek letter $\tau$. Therefore, while $\delta^{ij}(t)$ will represent the continuous evolution during time $t$ of the network latency $\delta$, which is measured by the $i$-esime node for the distance with the $j$-esime node, the notation $\delta^{ij}(\tau)$ will represent the latency measured at the $\tau$-esime step, i.e. the time taken by a ping from the $i$-esime node to the $j$-esime node, only for a specific time step $\tau$. Finally, we will suppose that each node has the fragment $z_k$ of a file $z$ and is interested in sharing or obtaining other portions of the same file, hence we will compute the probability-like function that expresses how easily the $k$-esime shared fragment is being copied from the $i$-esime node to the $j$-esime node at a certain step $\tau$ and we will call it $P^{ij}_k(\tau)$. Eventually, we are interested in an analytical computation for the urgency to share a fragment $z_k$ from $n^i$ to $n^j$ during a time set $\tau$, and we will call it $\chi^{i,j}_k(\tau)$. In the following sections we will distinguish between an actually measured value and a value predicted by a neural network using a tilde for predicted values as in $\tilde x$. Fragments diffusion on a P2P network {#diffusion} ==================================== In our work we compare the file fragments of a shared file to the diffusion of mass through a porous means. To embrace this view, it is mandatory to develop some mathematical tools, which we will explain in the following. Spaces and metrics ------------------ Users in a P2P BitTorrent network can be represented as elements of a space where a metric could be given by the corresponding network communication latency. Therefore, for each node $n^i \in N$, set of the nodes, it is possible to define a function $$\delta:N\times N \rightarrow \mathbb{R} ~~/~~ \delta(n^i,n^j)= \delta^{ij} ~~~ \forall~ n^i,n^j \in N \label{eq:delta}$$ where $\delta^{ij}$ is the amount of time taken to bring a minimum amount of data (e.g. as for a ping) from $n^i$ to $n^j$. By using the given definition of distance, for each node $n^i$, it is also possible to obtain an ordered list $\Omega^i$ so that $$\Omega^i = \Big\{ n^i_\alpha \in N \Big\}_{\alpha=0}^{|N|} : \delta(n^i,n^i_\alpha)\leq\delta(n^i,n^i_{\alpha+1}) \label{eq:omega}$$ In such a way, the first item of the list will be $n^i_0=n^i$ and the following items will be ordered according to their network latency as measured by $n^i$. Using this complete ordering of peers, it is possible to introduce the concept of content permeability and diffusion. Let us consider the files shared by one user of a P2P system: each file consists of fragments that can be diffused. Then, the diffusion of a file fragment can be analysed in terms of Fick’s law. Fick’s law and its use for P2P ------------------------------ Fick’s second law is commonly used in physics and chemistry to describe the change of concentration per time unit of some element diffusing into another [@vazquez2006porous]. This work proposes an analogy between a P2P system and a physical system. The key idea is to model the sharing file fragments as the diffusion of a substance into a porous means along one dimension. Different places of the porous means would represent different P2P nodes, whereas distances along such a one-dimension would be proportional to the network latencies. Then, P2P entities would be accommodated into the formalism of equations  and . Using both the First and Second Fick’s laws, the diffusion of a substance into a means is given as the solution of the following vectorial differential equation $$\frac{\partial \Phi}{\partial t} = \nabla \cdot (D \nabla \Phi) \label{eq:nabla}$$ where $\Phi$ is the concentration, $t$ the time and $D$ the permeability of the means to the diffusing matter. Since this is a separable equation and we use a $1$–dimensional metric based on the distance $\delta$, and assuming $D$ as constant among the nodes, equation  can be written as a scalar differential equation $$\frac{\partial \Phi}{\partial t} = D \frac{\partial^2 \Phi}{\partial \delta^2} \label{eq:partial}$$ This partial differential equation, once imposed the initial and boundaries conditions, admits at least Green’s Function as a particular solution [@bokshtein2005thermodynamics]. Green’s Function lets us study the diffusion dynamics of a single substance and can be rewritten as solution of the equation  in the form: $$\Phi(\delta,t)=\Phi_0 \Gamma \Big( \frac{\delta}{ \sqrt{4Dt}} \Big)~~,~~~~ \Phi_0 = \frac{1}{\sqrt{4 \pi D t}} \label{eq:phi}$$ where $\Gamma$ is the complementary gaussian error function. $\Gamma$ function should then be computed by means of a Taylor expansion. However, to avoid such a computationally difficult task, we use an approximation proposed in [@Chiani03], where a pure exponential approximation for $\Gamma(x)$ has been obtained, having an error on the order of $10^{-9}$. Then, it is possible to have the following equation $$\left \{ \begin{array}{rl} \Phi(\delta,t)\approx&\Phi_0 \left[ \frac{1}{6} e^{\left(\Phi_0\delta\right)^2} + \frac{1}{2}e^{-\frac{4}{3}\left(\Phi_0\delta\right)^2} \right] \\ \\ \Phi_0 =&(4 \pi D t)^{-\frac{1}{2}} \end{array} \right . \label{eq:phisys}$$ for every node at a certain distance $\delta \in \mathbb{R}^+$ at a time $t \in \mathbb{R}^+$. From concentration to probability --------------------------------- In equation  the scaling factor $\Phi_0$ is a function of the time $t$. On the other hand, the used formalism was developed mainly to focus on the distance $\delta$ and managing $t$ merely as a parameter. The above mathematical formalism is valid as long as the distances $\delta(n^i,n^j)$ remain time-invariant. The common practice considers the distance between nodes $\delta$ as time-invariant, however the actual network latencies vary (almost) continuously, with time, and a stationary $\Omega^i$ ordered set is a very unlikely approximation for the network. In our solution, we make time-dependent the latency embedded into our model. In turn, this makes it possible to choose different fragments to be shared as time goes. For the P2P system, the equation  states that a certain file fragment $z^i_k$ in a node $n^i$ at a time $t_0$ has a probability $P^{ij}_k(t_0,t)$ to be given (or diffused) to node $n^j$, at a distance $\delta^{ij}(t_0)$ from $n^i$, within a time $t$, which is proportional to the $\Phi(\delta, t)$ so that $$P^{ij}_k(t_0,t)= p^{ij}_k \left[ \frac{1}{6} e^{\left(p^{ij}_k~\delta^{ij}\right)^2} + \frac{1}{2}e^{-\frac{4}{3}\left(p^{ij}_k~\delta^{ij}\right)^2} \right] \\ \label{eq:prob}$$ where the function $p^{ij}_k=p^{ij}_k(t_0,t)$ carries both the diffusion factors and the temporal dynamics. And since we are interested in a simple proportion, not a direct equation, we can also neglect the factor $4\pi$ and then write $p^{ij}_k$ in the normalised form $$p^{ij}_k(t,t_0)= \frac{1}{\sqrt{4\pi}} \cdot \frac{1}{\sqrt{D_k(t_0)}} \cdot \frac{1}{\sqrt{t}}$$ It is now important to have a proper redefinition of the coefficient $D$. Let us say that $T_k$ is the number of users using file fragment $z_k$ (whether asking or offering it), $S_k$ is the number of seeders for the file fragment and $\rho_k$ is the mean share ratio of the file fragment among peers (and leeches), then it is possible to consider the ’urge’ to share the resource as an osmotic pressure which, during time, varies the coefficient of permeability of the network $D$. In order to take into account the mutable state in a P2P system, $D$ should vary according to the amounts of available nodes and file fragments. We have chosen to define $D$ as $$D_k (t_0) \triangleq \frac{T_k(t_0)}{S_k(t_0)+\left[T_k(t_0)- S_k(t_0)\right]\rho_k(t_0)}$$ by a formal substitution of $D$ with $D_k$ in $\Phi_0$, we obtain the analytical form of the term $p^{ij}_k$. Discrete time evolution on each node ------------------------------------ Indeed, the physical nature of the adopted law works in the entire variable space, however for the problem at hand discrete-time simplifications are needed. Let us suppose that for a given discrete time step $\tau=0$ node $n^i$ effectively measures the network latencies of a set of nodes $\{n^j\}$, then an ordered set $\Omega^i$ as in equation  is computed. Now, every node $n^i$ computes probability $P^{ij}_k$ for each of its own file fragment $z_k$ and for every node $n^j$. This probability corresponds to a statistical *prevision* of the *possible* file fragments spreading onto other nodes. Suppose that for a while no more measures for $\delta$ have been taken, and at a later discrete time step $\tau$ file fragment $z^i_k$ be copied to the first node to be served, which is chosen according to the minimum probability of diffusion, latencies and time since last measures were taken (see following subsection and equation ). Moreover, such a file fragment is reaching other nodes if the latency for such nodes is less than time $t^i$, computed as $$t^i_k(\tau)=\sum\limits_{\alpha_k=0}^\tau \delta(n^i,n^i_\alpha) \label{eq:tik}$$ Index $k$ is used in equation  to refer to file fragment $z^i_k$. Indeed, since nodes need and offer their own file fragments, the ordered set of nodes referred by a given node depend on resource $z_k$, i.e. $\Omega^i_k= \{n^i_{\alpha_k}\}$. It is now possible to have a complete mapping of the probability of diffusion by reducing the time dependence from $(t_0,t)$ to a single variable dependence from the discrete time-step $\tau$. For each resource $z_k$ as $P^{ij}_k(\tau)$ stated that it is possible to reduce $D_k(t_0,t)$ to a one-variable function $D_k(\tau)$ by assuming that at $t_0$ we have $\tau=0$ and considering only the values of $D_k(t_0,t)$ when $t$ is the execution moment of a computational step $\tau$. Assigning priorities and corrections ------------------------------------ Once all the $P^{ij}_k(\tau)$ have been computed, and values stored to a proper data structure, it is actually simple to determine the most urgent file fragment to share, which is the resource that has the least probability to be spread, i.e. the $k$ for which $P^{ij}_k(\tau)$ is minimum. Furthermore, we consider that while time goes on an old measured $\delta$ differs from the actual value, hence the measure becomes less reliable. To take into account the staleness of $\delta$ values, we gradually consider less bound to $\delta$ the choice of a fragment, and this behaviour is provided by the negative exponential in equation . Given enough time, the choice will be based only on the number of available fragments. However, we consider that by that time a new value for $\delta$ would have been measured and incorporated again into the model choosing the fragment. Generally, for nodes having the highest latencies with respect to a given node $n^i$, more time will be needed to receive a fragment from the node $n^i$. We aim at compensating such a delay by incorporating into our model the inescapable latencies of a P2P network. Therefore, the node that will receive a fragment first will be chosen according to its distance. In order to model the fact that distant nodes, having the highest values for $\delta$, will take more time to send or receive file fragments, we have chosen a decay law. Now it is possible to obtain a complete time-variant analytical form of the spreading of file fragments $$\chi^{ij}_k(\tau) = \frac{e^{-c \tau \delta^{ij}}}{P^{ij}_k(\tau)} \label{eq:expneg}$$ where the decay constant $c$ can be chosen heuristically, without harming the said law, and tuned according to other parameters. If $k$ indicates a file fragment and $k^*$ the index of the most urgent file fragment to share, this latter is trivially found as the solution of a maximum problem so that $$k^* : \chi^{ij}_{k^*}(\tau) = \max\limits_{k} \left\{ \chi^{ij}_k(\tau) \right\} \label{eq:min}$$ Of course, all the priorities depend on the value of the bi-dimensional matrix of values of $P^{ij}_k$ (we mark that the index $i$ does not variate within the same node $n^i$). Among these values, there is no need to compute elements where $j=i$ and for those elements where the node $n^j$ is not in the queue for resource $z_k$. In both these cases it is assumed $P^{ij}_k = 1$. Moreover, after $n^i$ having completed to transfer $z_k$ to the node $n^j$, the element of indexes $(j,k)$ is set to 1. In a similar fashion, each peer is able to identify a possible resource to ask for in order to maximize the diffusion of rare ones instead of common ones. WRNN predictors and Users behavior {#wrnn} ================================== In order to make the P2P system able to properly react to peaks of requests, as well as very fast changes of fragments availability and/or share ratio, we propose an innovative solution based on *Wavelet Recurrent Neural Networks* (WRNN) to characterise the user behaviour and producing a short-term forecast. For a given torrent, the *wavelet analysis* provides compression and denoising on the observed time series of the amount of users prividing or requesting fragments; a proper *recurrent neural network* (RNN), trained with the said observed time series, provides well-timed estimations of future data. The ensamble of the said wavelet analysis and RNN is called WRNN [@capizzi12a; @napoli10b; @napoli10a] and provides forecasts for the number of users that will share a fragment. Several neural networks have been employed to find polaritons propagation and metal thickness correspondence [@Bonanno14]; to predict the behaviour of users requesting resources [@napoli13], to perform wavelet transform in a recursive lifting procedure [@capizzi12b; @Sweldens98]. ![Topology of the WRNN[]{data-label="f:nnet2"}](nnet){width=".36\textwidth"} WRNN setup ---------- For this work, the initial datasets consists of a time series representing requests for a torrent, coming from peers, or given declaration of availability for the torrent coming from both peers and seeds. Independently of the specificities of such data, let us call this series $x(\tau)$, where $\tau$ is the discrete time step of data, sampled with intervals of one hour. A biorthogonal wavelet decomposition of the time series has been computed to obtain the correct input set for the WRNN as required by the devised architecture shown in Figure \[f:nnet2\]. This decomposition can be achieved by applying the wavelet transform as a recursive couple of conjugate filters in such a way that the $i$-esime recursion $\hat{W}_i$ produces, for any time step of the series, a set of coefficients $d_i$ and residuals $a_i$, and so that $$\hat{W}_i [a_{i-1}(\tau)] = [d_i(\tau) , a_i(\tau) ] ~~~~~ \forall~ i \in [1,M]\cap\mathbb{N} \label{eq:xtau}$$ where we intend $a_0(\tau)=x(\tau)$. The input set can then be represented as an $N \times (M+1)$ matrix of $N$ time steps of a $M$ level wavelet decomposition, where the $\tau$-esime row represents the $\tau$-esime time step as the decomposition $$\mathbf{u}(\tau) = \left [ d_1(\tau), d_2(\tau) , \ldots , d_M(\tau) ,a_M(\tau) \right ] \label{eq:utau}$$ Each row of this dataset is given as input value to the $M$ input neurons of the proposed WRNN (Figure \[f:nnet2\]). The properties of this network make it possible, starting from an input at a time step $\tau_n$, to predict the effective number of requests (or offers) at a time step $\tau_{n+r}$. In this way the WRNN acts like a functional $$\hat{N}[\mathbf{u}(\tau_n)] = \tilde{x}(\tau_{n+r}) \label{eq:Nfunctional}$$ where $r$ is the number of time steps of forecast in the future and $\tilde{x}$ the predicted serie. Predicted user behaviour ------------------------ As described by equation , it is then possible to obtain a future prediction of the number of requests for a specific torrent, as well as its availability in the future. In fact, by considering both the predicted $\tilde{x}_k(\tau_{n+r})$ and the modeled $\chi_k(\tau_{n+r})$, it is possible, at a time step $\tau_n$, to take counteracting actions and improve the probability of diffusion for a rare torrent. This is achieved, in practice, by using altered values for $D_k(\tau_{n+r})$, which account for the forecast of future time steps. Such modified values are computed by our WRNN, e.g. in a computing node of a cloud. Therefore, predicted values for $T_k(\tau_{n+r})$, $S_k(\tau_{n+r})$ and $\rho_k(\tau_{n+r})$ are sent to each node acting as a peer. Each time a new torrent becomes shared on the P2P network, then a new WRNN is created and trained on a server, e.g. requested from a cloud system, to provide predictions related to the availability and peers of the novel set of shared fragments for that torrent. The predictions will be sent to the peers periodically, and allow peers to update the values of $D_k(\tau)$. The update frequency can be tuned in order to correctly match the dynamic of hosts. {width=".24\textwidth"}  {width=".24\textwidth"}  {width=".24\textwidth"}\ {width=".24\textwidth"}  {width=".24\textwidth"}  {width=".24\textwidth"} Experiments =========== As shown in Figure \[f:sim\], for the initial condition of the P2P system some file fragments for a torrent happen to be heterogeneously spread among peers (e.g. no one shares fragment n. 2, and very few nodes have fragment n. 5). We report a simulation comprising 11 peers and 5 file fragments, and an evolution in only 5 time steps. In the order, step after step, peer $n^0$ selects file fragment $z_4$ and sends this to peer $n^{10}$. Both the fragment to spread and the destination peer have been chosen according to equation . Later on, as soon as possible, peer $n^0$ selects another fragment, i.e. $z_3$ to spread. Such a fragment could be send just after the transfer of the previous fragment has been completed, or concurrently to the first transmission. The shown evolution does not consider that the file fragment could have been passed, e.g., to node $n^{10}$, and so that for the next time step the value of $\chi$ for $n^{10}$ would drop to zero. Note that the highest values of $\chi$ are an indication of the urgency of receiving a fragment. The described model and formula allow subsequent sharing activities, after the initial time steps, to be determined, in terms of which fragments should be sent. Figure \[f:sim\] shows that after the first time steps it becomes more and more urgent for node $n^0$ to obtain the missing fragments $z_2$ and $z_5$. It is possible to see that the highest priority is for fragment $z_2$ since its share ratio and the relative availability are very low with respect to fragment $z_5$. This was the expected behaviour of the developed model. For the simulation shown in Figure \[f:sim\], all fragments, except $z_2$ since it is actually unavailable, would be spread to peers in a very low number of time steps. Figure \[f:decay\] shows the decay of several computed $\chi$ for different peers requiring fragment number 3. On the long run, this law will benefit nearby nodes, while on the short term, distant nodes are given the highest priority. ![Time decay of normalised $\chi^{j}_k(\tau)$ for increasing time steps $\tau$[]{data-label="f:decay"}](chi3tau_norm){width=".34\textwidth"} Related Work {#related} ============ Several studies have analysed the behaviour of BitTorrent systems from the point of view of fairness, i.e. how to have users contribute with contents that can be uploaded by other users, levelling the amount of downloads with that of upload. Fewer works have studied the problem of unavailability of contents in P2P BitTorrent networks. In [@Qiu04], authors proposed to order peers according to their uploading bandwidth, hence when providing contents the selection of peers is performed accordingly. One of the mechanism proposed to increase files availability has been to use multi-torrent, i.e. for ensuring fairness, instead of forcing users stay longer, they provide their contribution to uploaders for fragments belonging to different files [@Guo05]. Similarly, in [@Kaune10] authors show that by using multi-torrent availability can be easily increased, and confirm that fast replication of rare fragments is essential. Furthermore, bundling, i.e. the dissemination of a number of related files together, has been proposed to increase availability [@Menasche09]. The above proposed mechanisms differ from our proposal, since we take into account several novel factors: the dynamic of data exchange between distant peers, a decay for the availability of peers, and the forecast of contents availability. Such factors have been related into a proposed model that manages to select the rarest content to be spread taking into account the future availability, and the peers that should provide and take such a content. Conclusions {#conclusion} =========== This paper proposed to improve availability of fragments on a P2P system by adopting a mathematical model and a neural network. The former describes the fragments diffusion and the urgency to share fragments, whereas the latter provides an estimation of the availability of peers, and hence fragments, at later time. By using the estimate of future availability into the diffusion model, we can select the fragments that need to be spread to counteract their disappearance due to users disconnections. Acknowledgment {#acknowledgment .unnumbered} ============== This work has been supported by project PRISMA PON04a2 A/F funded by the Italian Ministry of University and Research within PON 2007-2013 framework. [^1]: \*Email: napoli@dmi.unict.it. [^2]: 2014 IEEE International Conference on Enabling Technologies: Infrastructure for Collaborative Enterprises (WETICE)

--- abstract: 'Tensors describing boost-invariant and cylindrically symmetric expansion of a relativistic dissipative fluid are decomposed in a suitable chosen basis of projection operators. This leads to a simple set of scalar equations which determine the fluid behavior. As special examples, we discuss the case of the Israel-Stewart theory and the model of highly-anisotropic and strongly-dissipative hydrodynamics ADHYDRO. We also introduce the matching conditions between the ADHYDRO description suitable for the very early stages of heavy-ion collisions and the Israel-Stewart theory applicable for later stages when the system is close to equilibrium.' author: - Wojciech Florkowski - Radoslaw Ryblewski date: 'November 22, 2011' title: ' Projection method for boost-invariant and cylindrically symmetric dissipative hydrodynamics [^1]' --- Introduction {#sect:intro} ============ Soft-hadronic observables measured in the ultra-relativistic heavy-ion experiments may be very well described by the standard perfect-fluid hydrodynamics (for a recent review see [@Florkowski:2010zz]) or by dissipative hydrodynamics with a small viscosity to entropy ratio [@Chaudhuri:2006jd; @Dusling:2007gi; @Luzum:2008cw; @Song:2007fn; @Bozek:2009dw; @Schenke:2010rr]. These approaches assume generally that the produced system reaches a state of local thermal equilibrium within a fraction of a fermi [^2]. On the other hand, many microscopic approaches assume that the produced system is initially highly anisotropic in the momentum space, e.g., see [@Bjoraker:2000cf]. High anisotropies present at the early stages of relativistic heavy-ion collisions exclude formally the application of the perfect-fluid and dissipative hydrodynamics. This situation has triggered development of several approaches which combine the anisotropic early evolution with a later perfect-fluid [@Sinyukov:2006dw; @Gyulassy:2007zz; @Broniowski:2008qk; @Ryblewski:2010tn] or viscous [@Martinez:2009ry; @Bozek:2010aj] description. Very recently, a concise model describing consistently different stages of heavy-ion collisions has been proposed in Refs. [@Florkowski:2010cf; @Ryblewski:2010bs; @Ryblewski:2010ch; @Ryblewski:2011aq] (a highly-Anisotropic and strongly-Dissipative HYDROdynamics, ADHYDRO), see also a similar work that has been presented in Refs. [@Martinez:2010sc; @Martinez:2010sd]. In this paper, in order to analyze in more detail the connections between different effective descriptions of very early stages of heavy-ion collisions we consider a simplified case of the boost-invariant and cylindrically symmetric expansion of matter. In the first part of the paper, we introduce tensors that form a suitable basis for decompositions of various tensors characterizing dissipative fluids. Then, we use this basis to analyze the Israel-Stewart [@Israel:1979wp; @Muronga:2003ta] and ADHYDRO equations [@Florkowski:2010cf; @Ryblewski:2010bs; @Ryblewski:2010ch; @Ryblewski:2011aq]. Finally, we show how the solutions of the ADHYDRO model may be matched with the solutions of the Israel-Stewart theory. The last result may be treated as a generalization of the approach presented in [@Martinez:2009ry] where no transverse expansion of matter was considered. The formal results presented in this paper, when implemented as numerical procedures, may be used to model the behavior of matter produced at the very early stages of heavy-ion collisions. Of course, the use of boost-invariance and cylindrical symmetry implies that this description should be limited at the moment to central collisions and the central rapidity region. A generalization of our framework to more complicated geometries is a work in progress. Boost-invariant and cylindrically symmetric flow {#sect:flow} ================================================ The space-time coordinates and the four-vector describing the hydrodynamic flow are denoted in the standard way as $x^\mu = \left( t, x, y, z \right)$ and $$U^\mu = \gamma (1, v_x, v_y, v_z), \quad \gamma = (1-v^2)^{-1/2}. \label{Umu}$$ For boost-invariant and cylindrically symmetric systems, we may use the following parametrization $$\begin{aligned} U^0 &=& \cosh \theta_\perp \cosh \eta_\parallel, \nonumber \\ U^1 &=& \sinh \theta_\perp \cos \phi, \nonumber \\ U^2 &=& \sinh \theta_\perp \sin \phi, \nonumber \\ U^3 &=& \cosh \theta_\perp \sinh \eta_\parallel, \label{Umu}\end{aligned}$$ where $\theta_\perp$ is the transverse fluid rapidity defined by the formula $$v_\perp = \sqrt{v_x^2+v_y^2} = \tanh \theta_\perp, \label{thetaperp}$$ $\eta_\parallel$ is the space-time rapidity, $$\begin{aligned} \eta_\parallel = \frac{1}{2} \ln \frac{t+z}{t-z}, \label{etapar} \end{aligned}$$ and $\phi$ is the azimuthal angle $$\phi = \arctan \frac{y}{x}. \label{phi}$$ In addition to $U^\mu$ we define three other four-vectors. The first one, $Z^\mu$, defines the longitudinal direction that plays a special role due to the initial geometry of the collision, $$\begin{aligned} Z^0 &=& \sinh \eta_\parallel, \nonumber \\ Z^1 &=& 0, \nonumber \\ Z^2 &=& 0, \nonumber \\ Z^3 &=& \cosh \eta_\parallel. \label{Zmu}\end{aligned}$$ The second four-vector, $X^\mu$, defines a transverse direction to the beam, $$\begin{aligned} X^0 &=& \sinh \theta_\perp \cosh \eta_\parallel, \nonumber \\ X^1 &=& \cosh \theta_\perp \cos \phi, \nonumber \\ X^2 &=& \cosh \theta_\perp \sin \phi, \nonumber \\ X^3 &=& \sinh \theta_\perp \sinh \eta_\parallel, \label{Xmu}\end{aligned}$$ while the third four-vector, $Y^\mu$, defines the second transverse direction [^3], $$\begin{aligned} Y^0 &=& 0, \nonumber \\ Y^1 &=& -\sin \phi, \nonumber \\ Y^2 &=& \cos \phi, \nonumber \\ Y^3 &=& 0. \label{Ymu}\end{aligned}$$ The four-vector $U^\mu$ is time-like, while the four-vectors $Z^\mu, X^\mu, Y^\mu$ are space-like. In addition, they are all orthogonal to each other, $$\begin{aligned} U^2 &=& 1, \quad Z^2 = X^2 = Y^2 = -1, \nonumber \\ U \cdot Z &=& 0, \quad U \cdot X = 0, \quad U \cdot Y = 0, \nonumber \\ Z \cdot X &=& 0, \quad Z \cdot Y = 0, \quad X \cdot Y = 0. \label{norm}\end{aligned}$$ All these properties are most easily seen in the [*local rest frame*]{} of the fluid element (LRF), where we have $\theta_\perp = \eta_\parallel = \phi = 0$ and $$\begin{aligned} U &=& (1,0,0,0), \nonumber \\ Z &=& (0,0,0,1), \nonumber \\ X &=& (0,1,0,0), \nonumber \\ Y &=& (0,0,1,0). \label{LRF}\end{aligned}$$ In the formalism of dissipative hydrodynamics one uses the operator $ \Delta^{\mu \nu} = g^{\mu \nu} - U^\mu U^\nu$, that projects on the three-dimensional space orthogonal to $U^\mu$. It can be shown that  [^4] $$\Delta^{\mu \nu} = -X^\mu X^\nu - Y^\mu Y^\nu - Z^\mu Z^\nu. \label{Delta}$$ Using Eqs. (\[norm\]) we find that $Z^\mu, X^\mu$ and $Y^\mu$ are the eigenvectors of $\Delta^{\mu \nu}$, $$\Delta^{\mu \nu} X_\nu = X^\mu, \quad \Delta^{\mu \nu} Y_\nu = Y^\mu, \quad \Delta^{\mu \nu} Z_\nu = Z^\mu. \label{eigen}$$ Expansion and shear tensors {#sect:exp-shear} =========================== In this Section we follow the standard definitions of the expansion and shear tensors [@Muronga:2003ta] and show that they can be conveniently decomposed in the basis of the tensors obtained as products of the four-vectors $X^\mu$, $Y^\mu$ and $Z^\mu$. The [*expansion*]{} tensor $\theta_{\mu \nu}$ is defined by the expression $$\theta_{\mu \nu} = \Delta^\alpha_\mu \Delta^\beta_\nu \partial_{(\beta} U_{\alpha)}, \label{theta-munu}$$ where the brackets denote the symmetric part of $\partial_{\beta} U_{\alpha}$. Using Eqs. (\[Umu\]) in the definition of the expansion tensor (\[theta-munu\]) and also using Eqs. (\[Zmu\])–(\[Ymu\]), we may verify that the following decomposition holds $$\theta^{\mu \nu} = \theta_X X^\mu X^\nu + \theta_Y Y^\mu Y^\nu + \theta_Z Z^\mu Z^\nu, \label{theta-dec}$$ where $$\theta_X = X_\mu X_\nu \theta^{\mu \nu} = - \frac{\partial \theta_\perp}{\partial r} \cosh \theta_\perp - \frac{\partial \theta_\perp}{\partial \tau} \sinh \theta_\perp, \label{thetaX}$$ $$\theta_Y = Y_\mu Y_\nu \theta^{\mu \nu} = - \frac{\sinh \theta_\perp}{r}, \label{thetaY}$$ and $$\theta_Z = Z_\mu Z_\nu \theta^{\mu \nu} = - \frac{\cosh \theta_\perp}{\tau}. \label{thetaZ}$$ The contraction of the tensors $\Delta^{\mu \nu}$ and $\theta^{\mu \nu}$ gives the volume expansion parameter $$\theta = \Delta^{\mu \nu} \theta_{\mu \nu}. \label{theta1}$$ Equations (\[Delta\]) and (\[theta1\]) yield $$\theta = -\theta_X - \theta_Y - \theta_Z. \label{theta2}$$ Substituting Eqs. (\[thetaX\])–(\[thetaZ\]) in Eq. (\[theta2\]) we find that this formula is consistent with the definition $\theta = \partial_\mu U^\mu$. The [*shear*]{} tensor $\sigma_{\mu \nu}$ is defined by the formula $$\sigma_{\mu \nu} = \theta_{\mu \nu} - \frac{1}{3} \Delta_{\mu \nu} \theta. \label{sigma1}$$ With the help of the decompositions (\[Delta\]) and (\[theta-dec\]) we may write $$\sigma^{\mu \nu} = \sigma_X X^\mu X^\nu + \sigma_Y Y^\mu Y^\nu + \sigma_Z Z^\mu Z^\nu, \label{sigma-dec}$$ where $$\begin{aligned} \sigma_X &=& \frac{\theta}{3}+\theta_X = \frac{\cosh \theta_\perp}{3 \tau} + \frac{\sinh\theta_\perp}{3r} \label{sigmaX} \\ && -\frac{2}{3} \frac{\partial\theta_\perp}{\partial \tau} \sinh\theta_\perp -\frac{2}{3} \frac{\partial\theta_\perp}{\partial r} \cosh\theta_\perp , \nonumber \end{aligned}$$ $$\begin{aligned} \sigma_Y &=& \frac{\theta}{3}+\theta_Y = \frac{\cosh \theta_\perp}{3 \tau} - \frac{2 \sinh\theta_\perp}{3r} \label{sigmaY} \\ && +\frac{1}{3} \frac{\partial\theta_\perp}{\partial \tau} \sinh\theta_\perp +\frac{1}{3} \frac{\partial\theta_\perp}{\partial r} \cosh\theta_\perp , \nonumber \end{aligned}$$ and $$\begin{aligned} \sigma_Z &=& \frac{\theta}{3}+\theta_Z \label{sigmaZ} = -\frac{2\cosh \theta_\perp}{3 \tau} + \frac{\sinh\theta_\perp}{3r} \\ && +\frac{1}{3} \frac{\partial\theta_\perp}{\partial \tau} \sinh\theta_\perp +\frac{1}{3} \frac{\partial\theta_\perp}{\partial r} \cosh\theta_\perp . \nonumber \end{aligned}$$ In agreement with general requirements we find that $$\sigma_X+\sigma_Y+\sigma_Z=0. \label{sum-sigma}$$ In the case where the radial flow is absent and , which agrees with earlier findings [@Muronga:2003ta]. Energy-momentum tensor {#sect:Tmunu} ====================== The energy-momentum tensors of the systems considered below in this paper have the following structure $$\begin{aligned} T^{\mu \nu} = \varepsilon U^\mu U^\nu + P_X X^\mu X^\nu + P_Y Y^\mu Y^\nu + P_Z Z^\mu Z^\nu. \label{enmomten}\end{aligned}$$ The quantity $\varepsilon$ is the energy density, while $P_X, P_Y$ and $P_Z$ are three different pressure components. In LRF the energy-momentum tensor has the diagonal structure, $$T^{\mu \nu} = \left( \begin{array}{cccc} \varepsilon & 0 & 0 & 0 \\ 0 & P_X & 0 & 0 \\ 0 & 0 & P_Y & 0 \\ 0 & 0 & 0 & P_Z \end{array} \right). \label{Tmunuarray}$$ Since we consider boost-invariant and cylindrically symmetric systems, $\varepsilon, P_X, P_Y$ and $P_Z$ may depend only on the (longitudinal) proper time $$\tau = \sqrt{t^2 - z^2} \label{tau}$$ and radial distance $$r = \sqrt{x^2 + y^2}. \label{r}$$ The hydrodynamic equations include the energy-momentum conservation law $$\partial_\mu T^{\mu \nu} = 0. \label{enmomCL1}$$ Using the form of the energy-momentum tensor (\[enmomten\]) in (\[enmomCL1\]) and projecting the result on $U_\nu$, $Z_\nu$, $X_\nu$ and $Y_\nu$ one gets four equations $$\begin{aligned} && {\dot \varepsilon} + \varepsilon \partial_\mu U^\mu + P_Z U_\nu Z^\mu \partial_\mu Z^\nu \label{enmomCLU} \\ && \quad + P_X U_\nu X^\mu \partial_\mu X^\nu + P_Y U_\nu Y^\mu \partial_\mu Y^\nu = 0, \nonumber \\ && \varepsilon Z_\nu {\dot U}^\nu - Z^\mu \partial_\mu P_Z - P_Z \partial_\mu Z^\mu \label{enmomCLZ} \\ && \quad + P_X Z_\nu X^\mu \partial_\mu X^\nu + P_Y Z_\nu Y^\mu \partial_\mu Y^\nu = 0, \nonumber \\ && \varepsilon X_\nu {\dot U}^\nu - X^\mu \partial_\mu P_X - P_X \partial_\mu X^\mu \label{enmomCLX} \\ && \quad + P_Y X_\nu Y^\mu \partial_\mu Y^\nu + P_Z X_\nu Z^\mu \partial_\mu Z^\nu = 0, \nonumber \\ && \varepsilon Y_\nu {\dot U}^\nu - Y^\mu \partial_\mu P_Y - P_Y \partial_\mu Y^\mu \label{enmomCLY} \\ && \quad + P_X Y_\nu X^\mu \partial_\mu X^\nu + P_Z Y_\nu Z^\mu \partial_\mu Z^\nu = 0. \nonumber\end{aligned}$$ Here the dot denotes the total time derivative (the operator $U^\alpha \partial_\alpha$). With the help of the relations $$\begin{aligned} && Z^\mu \partial_\mu = \frac{\partial}{\tau \partial \eta_\parallel}, \quad \partial_\mu Z^\mu = 0, \quad {\dot Z}^\nu = 0, \nonumber \\ && Y^\mu \partial_\mu = \frac{\partial}{r \partial \phi}, \quad \partial_\mu Y^\mu = 0, \quad {\dot Y}^\nu = 0, \nonumber \\ && X^\mu \partial_\mu Z^\nu = 0, \quad X^\mu \partial_\mu Y^\nu = 0,\end{aligned}$$ one can show that Eqs. (\[enmomCLZ\]) and (\[enmomCLY\]) are automatically fulfilled. In this way, we are left with only two independent equations: $$\begin{aligned} && \left( \cosh \theta_\perp \frac{\partial}{\partial \tau} + \sinh \theta_\perp \frac{\partial}{\partial r} \right) \varepsilon \label{enmomCLU1} \\ && + \varepsilon \left[ \cosh \theta_\perp \left( \frac{1}{\tau} + \frac{\partial \theta_\perp}{\partial r} \right) + \sinh \theta_\perp \left( \frac{1}{r} + \frac{\partial \theta_\perp}{\partial \tau} \right) \right] \nonumber \\ && + P_X \left( \cosh \theta_\perp \frac{\partial \theta_\perp }{\partial r} + \sinh \theta_\perp \frac{\partial \theta_\perp }{\partial \tau} \right) \nonumber \\ && P_Y \frac{\sinh \theta_\perp}{r} +P_Z \frac{\cosh \theta_\perp}{\tau} = 0 \nonumber\end{aligned}$$ and $$\begin{aligned} && \left( \sinh \theta_\perp \frac{\partial}{\partial \tau} + \cosh \theta_\perp \frac{\partial}{\partial r} \right) P_X \label{enmomCLX1} \\ && + \varepsilon \left( \sinh \theta_\perp \frac{\partial \theta_\perp }{\partial r} + \cosh \theta_\perp \frac{\partial \theta_\perp }{\partial \tau} \right) \nonumber \\ && + P_X \left[ \sinh \theta_\perp \left( \frac{1}{\tau} + \frac{\partial \theta_\perp}{\partial r} \right) + \cosh \theta_\perp \left( \frac{1}{r} + \frac{\partial \theta_\perp}{\partial \tau} \right) \right] \nonumber \\ && - P_Y \frac{\cosh \theta_\perp}{r} - P_Z \frac{\sinh \theta_\perp}{\tau} = 0. \nonumber\end{aligned}$$ Israel-Stewart theory {#sect:df} ===================== Stress tensor {#sect:stress} ------------- In the Israel-Stewart theory , the crucial role is played by the stress tensor $\pi^{\mu \nu}$ that satisfies the following differential equation [@Israel:1979wp; @Muronga:2003ta] $$\tau_\pi \Delta^\alpha_\mu \Delta^\beta_\nu {\dot \pi}_{\alpha \beta} + \pi_{\mu \nu} = 2 \eta \sigma_{\mu \nu} + F_\eta \pi_{\mu \nu}. \label{pi-eq-1}$$ Here $\tau_\pi$ is the relaxation time, $\eta$ is the shear viscosity, and $F_\eta$ is our abbreviation for the scalar quantity $$F_\eta = -\eta T \partial_\lambda \left( \frac{\alpha_1}{T} U^\lambda \right), \label{Feta}$$ where $T$ is the temperature and $\alpha_1$ is one of the kinetic coefficients appearing in the Israel-Stewart theory. The structure of the shear tensor, Eq. (\[sigma-dec\]), suggests that we may seek the solutions of Eq. (\[pi-eq-1\]) in the form analogous to Eqs. (\[theta-dec\]) and (\[sigma-dec\]), namely $$\pi^{\mu \nu} = \pi_X X^\mu X^\nu + \pi_Y Y^\mu Y^\nu + \pi_Z Z^\mu Z^\nu. \label{pi-dec}$$ The condition $\Delta^{\mu \nu} \pi_{\mu \nu} =0$ leads to the constraint $$\pi_X + \pi_Y + \pi_Z = 0. \label{sum-pi}$$ The time derivative of $\pi^{\mu \nu}$ generates nine terms. Since $X^\mu, Y^\mu$ and $Z^\mu$ are the eigenvectors of the projection operator $\Delta^{\mu \nu}$, see Eq. (\[eigen\]), we find $$\begin{aligned} && \Delta^\alpha_\mu \Delta^\beta_\nu {\dot \pi}_{\alpha \beta} = \nonumber \\ && \,\,\, {\dot \pi}_X X_\mu X_\nu + \pi_X \Delta^\alpha_\mu {\dot X}_\alpha X_\nu + \pi_X X_\mu \Delta^\beta_\nu {\dot X}_\beta \nonumber \\ && + {\dot \pi}_Y Y_\mu Y_\nu + \pi_Y \Delta^\alpha_\mu {\dot Y}_\alpha Y_\nu + \pi_Y Y_\mu \Delta^\beta_\nu {\dot Y}_\beta \nonumber \\ && + {\dot \pi}_Z Z_\mu Z_\nu + \pi_Z \Delta^\alpha_\mu {\dot Z}_\alpha Z_\nu + \pi_Z Z_\mu \Delta^\beta_\nu {\dot Z}_\beta. \nonumber \\ \label{sigma-dec-1}\end{aligned}$$ For boost-invariant and cylindrically symmetric systems, the explicit calculations show that the terms $\Delta^\alpha_\mu {\dot X}_\alpha$, $\Delta^\alpha_\mu {\dot Y}_\alpha$, and $\Delta^\alpha_\mu {\dot Z}_\alpha$ vanish. Therefore, the decomposition (\[pi-dec\]) is indeed appropriate, and Eq. (\[pi-eq-1\]) splits into three scalar equations $$\begin{aligned} \tau_\pi {\dot \pi}_X + \pi_X &=& 2 \eta \sigma_X + F_\eta \pi_X, \nonumber \\ \tau_\pi {\dot \pi}_Y + \pi_Y &=& 2 \eta \sigma_Y + F_\eta \pi_Y, \nonumber \\ \tau_\pi {\dot \pi}_Z + \pi_Z &=& 2 \eta \sigma_Z + F_\eta \pi_Z. \label{pi-eq-2}\end{aligned}$$ Due to the constraints (\[sum-sigma\]) and (\[sum-pi\]), only two equations in (\[pi-eq-2\]) are independent. Bulk viscosity {#sect:bulk} -------------- The isotropic correction to pressure, $\Pi$, satisfies the equation $$\tau_\Pi {\dot \Pi} + \Pi = -\zeta \theta + F_\zeta \Pi, \label{Pi-eq-1}$$ where $\tau_\Pi$ is the relaxation time for $\Pi$, $\zeta$ is the bulk viscosity, and $$F_\zeta = -\frac{1}{2} \zeta T \partial_\lambda \left( \frac{\alpha_0}{T} U^\lambda \right), \label{Fzeta}$$ where $\alpha_0$ is another kinetic coefficient appearing in the Israel-Stewart theory. Energy-momentum tensor and hydrodynamic equations {#sect:enmomtenIS} ------------------------------------------------- In the Israel-Stewart theory, the energy momentum tensor has the following form $$T^{\mu\nu} = \varepsilon U^\mu U^\nu - P_{\rm eq} \Delta^{\mu \nu} + \pi^{\mu \nu} -\Pi \Delta^{\mu \nu}, \label{enmomtenIS}$$ where $P_{\rm eq} $ is the equilibrium pressure connected with the energy density by the equation of state, $P_{\rm eq}=P_{\rm eq}(\varepsilon)$, and $\Pi$ is the isotropic correction to pressure. A simple comparison of Eqs. (\[enmomtenIS\]) and (\[enmomten\]) leads to the identifications $$\begin{aligned} P_X &=& P_{\rm eq} + \Pi + \pi_X, P_Y = P_{\rm eq} + \Pi + \pi_Y, \nonumber \\ P_Z &=& P_{\rm eq} + \Pi + \pi_Z = P_{\rm eq} + \Pi - \pi_X - \pi_Y. \label{PXYZ_IS}\end{aligned}$$ Substituting Eqs. (\[PXYZ\_IS\]) into Eqs. (\[enmomCLU1\]) and (\[enmomCLX1\]) we obtain two equations for five unknown functions: $\varepsilon$, $\theta_\perp$, $\Pi$, $\pi_X$, and $\pi_Y$. The two first equations in (\[pi-eq-2\]) as well as Eq. (\[Pi-eq-1\]) should be included as the three extra equations needed to close this system. ADHYDRO model {#sect:adhydro} ============= The Israel-Stewart theory describes the system that is close to local equilibrium. Formally, this means that the corrections to pressure (the quantities $\Pi$, $\pi_X$, and $\pi_Y$) should be small compared to $P_{\rm eq}$. This condition cannot be fulfilled at the very early stages of heavy ion collisions. In the limit $\tau \to 0$ the components of the shear tensor, see Eqs. (\[sigmaX\])–(\[sigmaZ\]), diverge and induce very large changes of $\pi_X$, $\pi_Y$, and $\pi_Z$ through Eqs. (\[pi-eq-2\]). This leads to strong deviations from local equilibrium. In this situation, one tries to construct phenomenological models of the very early stages which grasp the essential features of the produced matter and may describe effectively the early dynamics. On the basis of microscopic models of heavy-ion collisions, we expect that the system formed at the very early stages of heavy-ion collisions is highly anisotropic — the two transverse pressures are equal and much larger than the longitudinal pressure. Such anisotropic systems are described most often by the anisotropic distribution functions which have the form of the squeezed or stretched Boltzmann equilibrium distributions for massless partons (this is often called the Romatschke-Strickland ansatz [@Romatschke:2003ms]) $$\begin{aligned} f &=& g \exp\left[ -\frac{1}{\lambda_\perp}\sqrt{(p \cdot U)^2 + (x-1) (p \cdot Z)^2}\right]. \label{f}\end{aligned}$$ Here $p$ is the particle’s four momentum, $g$ denotes the number of the internal degrees of freedom, $\lambda_\perp$ may be interpreted as the temperature of the transverse degrees of freedom, and $x$ is the anisotropy parameter. The energy-momentum tensor of the system described by the distribution function (\[f\]) has the form [@Florkowski:2010cf; @Ryblewski:2010bs; @Ryblewski:2010ch; @Ryblewski:2011aq] [^5] $$T^{\mu \nu} = \left( \varepsilon + P_\perp\right) U^{\mu}U^{\nu} - P_\perp \, g^{\mu\nu} - (P_\perp - P_\parallel) Z^{\mu}Z^{\nu}. \label{TmunuA1}$$ This form agrees with Eq. (\[enmomten\]) if we set $$P_X = P_Y = P_\perp, \quad P_Z = P_\parallel. \label{PXYZ_ADHYDRO}$$ In the case described by the distribution function (\[f\]), the energy density and the two pressures may be expressed as functions of the non-equilibrium entropy density $\sigma$ and the anisotropy parameter $x$ [@Ryblewski:2011aq], $$\begin{aligned} \varepsilon(\sigma,x) &=& \varepsilon_{\rm eq}(\sigma) r(x), \label{epsilon2a} \\ \nonumber \\ P_\perp(\sigma,x) &=& P_{\rm eq}(\sigma) \left[r(x) + 3 x r^\prime(x) \right], \label{PT2a} \\ \nonumber \\ P_\parallel(\sigma,x) &=& P_{\rm eq}(\sigma) \left[r(x) - 6 x r^\prime(x) \right]. \label{PL2a} \end{aligned}$$ We emphasize that $\varepsilon_{\rm eq}(\sigma)$ and $P_{\rm eq}(\sigma)$ are equilibrium expressions for the energy density and pressure [@Florkowski:2010zz] but the argument is a non-equilibrium value of the entropy density $$\varepsilon_{\rm eq}(\sigma) = 3 P_{\rm eq}(\sigma) = \frac{3g}{\pi^2} \left( \frac{\pi^2 \sigma}{4g} \right)^{4/3}.$$ The function $r(x)$ has the form $$\begin{aligned} r(x) = \frac{x^{-\frac{1}{3}}}{2} \left[ 1 + \frac{x \arctan\sqrt{x-1}}{\sqrt{x-1}}\right]. \label{rx} \\ \nonumber\end{aligned}$$ Substituting Eqs. (\[PXYZ\_ADHYDRO\])–(\[rx\]) into Eqs. (\[enmomCLU1\]) and (\[enmomCLX1\]) (which follow directly from the energy-momentum conservation law) we obtain two equations for three unknown functions: $\sigma$, $x$, and $\theta_\perp$. The third equation follows from the ansatz describing the entropy production in the system, $$\begin{aligned} \partial_\mu \sigma^{\mu} &=& \Sigma(\sigma,x). \label{engrow}\end{aligned}$$ The entropy source $\Sigma$ is taken in the form $$\Sigma(\sigma,x) = \frac{(1-\sqrt{x})^2}{\sqrt{x}} \frac{\sigma}{\tau_{\rm eq}}. \label{ansatz1}$$ The quantity ${\tau_{\rm eq}}$ is a timescale parameter. The form (\[ansatz1\]) guarantees that $\Sigma \geq 0$ and $\Sigma(\sigma,x=1)=0$. We stress that ADHYDRO accepts other reasonable definitions of the entropy source. In particular, it would be interesting in this context to use the forms motivated by the AdS/CFT correspondence [@Heller:2011ju]. Matching conditions between ADHYDRO model and Israel-Stewart theory {#sect:adhydro} =================================================================== In this Section, we show how the initial evolution of the system described by the ADHYDRO model may be matched to a later non-equilibrium evolution governed by the Israel-Stewart theory. We propose to do the matching at the transition proper time $\tau_{\rm tr}$ when the anisotropy parameter $x$ becomes close to unity in the whole space, i.e., when the condition $|x(\tau=\tau_{\rm tr},r)-1| \ll 1$ is satisfied for all values of $r$. Our earlier calculations done within the ADHYDRO framework [@Ryblewski:2011aq] show that if the initial value of $x$ is independent of $r$, the later values of $x$ depend weakly on $r$, hence, it makes sense to use the value of the transition time that is to large extent independent of $r$. Certainly, one should always check the sensitivity of the obtained results with respect to the chosen value of $\tau_{\rm tr}$. The acceptable results should exhibit weak dependence on $\tau_{\rm tr}$. We emphasize that the proposed matching procedure differs from our previous strategy where the ADHYDRO model was used to describe the whole evolution of the system; from a highly-anisotropic initial stage to hadronic freeze-out [@Florkowski:2010cf; @Ryblewski:2010bs; @Ryblewski:2010ch; @Ryblewski:2011aq]. The use of the ADHYDRO model [*alone*]{} implies a smooth switching from a highly-anisotropic phase (where $x \gg 1$ or $x \ll 1$) to the phase described by the perfect-fluid hydrodynamics (where $x \approx 1$). Moreover, as it has been shown in [@Florkowski:2010cf; @Martinez:2010sc], for purely longitudinal and boost-invariant expansion of matter, ADHYDRO agrees with the Israel-Stewart framework in the intermediate region where $|x-1| \ll 1$. On the other hand, if the transverse expansion is included, the presence of non-negligible shear viscosity triggers differences between the two components of the transverse pressure, $P_X$ and $P_Y$. This effect is not included in the ADHYDRO model. Therefore, in the approaches that include transverse expansion and noticeable effects of viscosity, the matching proposed below is in our opinion more appropriate than the use of ADHYDRO alone. However, the use of the ADHYDRO model is reasonable for the situations where the effects of viscosity at the later stages of the collisions may be neglected. Energy-momentum matching {#sect:enmom-match} ------------------------ In order to connect the solutions of the ADHYDRO model with the solutions of the Israel-Stewart theory, we demand that all thermodynamics- and hydrodynamics-like quantities are continuous across the transition boundary fixed at the transition time $\tau_{\rm tr}$, namely: i\) the energy density $\varepsilon$ is the same on both sides of the transition and identified with the equilibrium energy density in the Israel-Stewart theory, $\varepsilon = \varepsilon_{\rm eq}$, ii\) the transverse flow, quantified by the value of $\theta_\perp$, is the same at the end of the ADHYDRO stage and at the beginning of the Israel-Stewart stage, iii\) also the three components of pressure are the same, namely $$\begin{aligned} P_{\rm eq} + \Pi + \pi_X &=& P_\perp, \nonumber \\ P_{\rm eq} + \Pi + \pi_Y &=& P_\perp, \label{pressure-match} \\ P_{\rm eq} + \Pi + \pi_Z &=& P_\parallel. \nonumber\end{aligned}$$ Here the values on the right-hand-side are obtained at the end of the ADHYDRO evolution, at $\tau=\tau_{\rm tr}$, and treated as the input for the stage described by the Israel-Stewart equations for $\tau \geq \tau_{\rm tr}$. From (\[pressure-match\]) we find first that $\pi_Y = \pi_X$ and $\pi_Z = -2 \pi_X$. In the next step we find that $$\pi_X = \frac{P_\perp-P_\parallel}{3} = \varepsilon_{\rm eq}(\sigma) x r'(x) \approx \varepsilon_{\rm eq}(\sigma_{\rm eq}) \frac{4(x-1)}{45} \label{pi-X-match}$$ and $$P_{\rm eq} + \Pi = \frac{2 P_\perp + P_\parallel}{3} = \frac{\varepsilon}{3} = \frac{\varepsilon_{\rm eq}}{3}. \label{Peq+pi}$$ For massless particles considered here, $P_{\rm eq} = \varepsilon_{\rm eq}/3$ which implies $\Pi=0$. Finally, we demand that iv) the entropy density is the same before and after the transition. This is discussed in more detail in the next Section. Entropy matching {#sect:entropy-match} ---------------- It is interesting to see in more detail how the last condition is realized in practice. First, we start with the ADHYDRO formulation. If the energy density $\varepsilon$ corresponds to the equilibrium energy density, see the condition i), the corresponding equilibrium entropy density may be obtained with the help of the inverse function to the function $\varepsilon_{\rm eq}(\sigma)$. In this way, expanding the function $r(x)$ at $x=1$, we find $$\begin{aligned} \sigma_{\rm eq} &=& \varepsilon_{\rm eq}^{-1}\left[ \varepsilon_{\rm eq}(\sigma) r(x) \right] \nonumber \\ &\approx & \varepsilon_{\rm eq}^{-1}\left[ \varepsilon_{\rm eq}(\sigma) + \varepsilon_{\rm eq}(\sigma) \frac{2 (x-1)^2}{45} \right] \label{entr-match-1} \\ &\approx & \sigma + [ d\varepsilon_{\rm eq}(\sigma)/d\sigma]^{-1} \varepsilon_{\rm eq}(\sigma) \frac{2 (x-1)^2}{45}. \nonumber \end{aligned}$$ In the leading order in deviations from the equilibrium, we may replace $\sigma$ by $\sigma_{\rm eq}$ in the second term in the last line of (\[entr-match-1\]). Using the thermodynamic identity for the system of massless particles we find $$[d\varepsilon_{\rm eq}(\sigma_{\rm eq})/d\sigma_{\rm eq}]^{-1} \varepsilon_{\rm eq}(\sigma_{\rm eq}) = \frac{\varepsilon_{\rm eq}(\sigma_{\rm eq}) }{T} = \frac{3}{4} \sigma_{\rm eq}.$$ Combing the last two results we find $$\sigma = \sigma_{\rm eq} \left(1 - \frac{(x-1)^2}{30} \right). \label{sigma-adhydro}$$ Now we calculate the connection between the equilibrium and non-equilibrium entropy density using the Israel-Stewart theory. The basic relation in this context has the form [@Muronga:2003ta] $$\sigma = \sigma_{\rm eq} - \beta_0 \frac{\Pi^2}{2T} -\beta_2 \frac{ \pi_{\lambda \nu} \pi^{\lambda \nu} }{2T}.$$ In our case we have $\beta_0 = \tau_\Pi/\zeta$, $\beta_2 = \tau_\pi/(2\eta)$, and $\pi_{\lambda \nu} \pi^{\lambda \nu}=6\pi_X^2$. We also use the result $$\tau_\pi = \frac{5\eta}{T \sigma_{\rm eq}}, \label{taupiMS}$$ which has been derived in Ref. [@Martinez:2009ry]. Following [@Martinez:2009ry] we note that expansion of anisotropic distributions around the equilibrium backgrounds does not lead to the situation described by the 14 Grad’s ansatz, hence (\[taupiMS\]) differs from the standard result by a factor of $6/5$. The relations listed above allow us to write $$\sigma = \sigma_{\rm eq} \left(1 - \frac{135 \, \pi_X^2}{32 \, \varepsilon_{\rm eq}^2} \right) - \frac{\tau_\Pi \Pi^2}{2\zeta T}.$$ Using Eq. (\[pi-X-match\]) in this equation we find $$\sigma = \sigma_{\rm eq} \left(1 - \frac{(x-1)^2}{30} \right) - \frac{\tau_\Pi \Pi^2}{2\zeta T}. \label{sigma-IS}$$ Thus, we conclude that matching between ADHYDRO and the Israel-Stewart theory is continuous if the condition (\[taupiMS\]) is fulfilled and $\Pi=0$ at the transition time (in agreement with our remarks following Eq. (\[Peq+pi\])). We note that the condition (\[taupiMS\]) guarantees that the entropy production in ADHYDRO has the same form as in the Israel-Stewart theory, which has been already shown in Refs. [@Florkowski:2010cf; @Martinez:2010sc]. We also note that for $\tau \geq \tau_{\rm tr}$ , the two transverse pressure start to differ from each other, since their dynamics is governed by different components of the shear tensor. Conclusions {#sect:con} =========== In this paper we have introduced a basis of projection operators which allows for simple analysis of dissipative fluid dynamics of boost-invariant and cylindrically symmetric systems. We have used this basis to analyze the equations of the Israel-Stewart theory and the ADHYDRO model. We have shown how the very early evolution of matter produced in heavy-ion collisions may be described by the ADHYDRO equations combined with a later Israel-Stewart dynamics. [20]{} W. Florkowski, [*Phenomenology of Ultra-Relativistic Heavy-Ion Collisions*]{}, World Scientific, Singapore, 2010. A. K. Chaudhuri, Phys. Rev. C [**74**]{}, 044904 (2006). K. Dusling, D. Teaney, Phys. Rev. C [**77**]{}, 034905 (2008). M. Luzum, P. Romatschke, Phys. Rev. C [**78**]{}, 034915 (2008). H. Song, U. Heinz, Phys. Lett. B [**658**]{}, 279 (2008) P. Bozek, Phys. Rev. C [**81**]{}, 034909 (2010). B. Schenke, S. Jeon, C. Gale, Phys. Rev. Lett. [**106**]{}, 042301 (2011). J. Bjoraker, R. Venugopalan, Phys. Rev. C [**63**]{} 024609 (2001). Y. M. Sinyukov, Acta Phys. Polon. B [**37**]{} 3343 (2006). M. Gyulassy, Y. M. Sinyukov, I. Karpenko, A. V. Nazarenko, Braz. J. Phys. [ **37**]{}, 1031 (2007). W. Broniowski, W. Florkowski, M. Chojnacki, A. Kisiel, Phys. Rev. C [**80**]{}, 034902 (2009). R. Ryblewski, W. Florkowski, Phys. Rev. C [**82**]{}, 024903 (2010). M. Martinez, M. Strickland, Phys.Rev. C [**81**]{}, 024906 (2010). P. Bozek, I. Wyskiel-Piekarska, Phys. Rev. C [**83**]{}, 024910 (2011). W. Florkowski, R. Ryblewski, Phys. Rev. C [**83**]{}, 034907 (2011). R. Ryblewski, W. Florkowski, J. Phys. G [**38**]{}, 015104 (2011). R. Ryblewski, W. Florkowski, Acta Phys. Pol. B [**42**]{}, 115 (2011). R. Ryblewski, W. Florkowski, Eur. Phys. J. C [**71**]{}, 1761 (2011). M. Martinez, M.Strickland, Nucl. Phys. A [**848**]{}, 183 (2010). M. Martinez, M.Strickland, Nucl. Phys. A [**856**]{}, 68 (2011). W. Israel, J.M. Stewart, Annals Phys. [**118**]{}, 341 (1979). A. Muronga, Phys. Rev. C [**69**]{}, 034903 (2004). P. Romatschke, M. Strickland, Phys. Rev. D [**68**]{} 036004 (2003). M.  Heller, R. Janik, P. Witaszczyk, arXiv:1103.3452. [^1]: This work was supported in part by the Polish Ministry of Science and Higher Education under Grants No. N N202 263438 and No. N N202 288638. [^2]: We use the natural system of units where $\hbar=c=1$. The metric tensor $g_{\mu \nu} = \hbox{diag}(1,-1,-1,-1)$. [^3]: By analogy to the terminology used in interferometry (HBT) studies, one may say that the four-vector $X^\mu$ defines the [*out*]{} direction, while $Y^\mu$ defines the [*side*]{} direction. [^4]: One may check easily that (\[Delta\]) holds in LRF. Hence, as a tensor equation, (\[Delta\]) should hold in all reference frames. [^5]: In the original papers the four-vector $Z^\mu$ is denoted as $V^\mu$.

--- abstract: 'Spectral methods based on integral transforms may be efficiently used to solve differential equations in some special cases. This paper considers a different approach in which algorithms are proposed to calculate integral Laguerre transform by solving a one-dimensional transport equation. In contrast to the direct calculation of improper integrals of rapidly oscillating functions, these procedures make it possible to calculate the expansion coefficients of a Laguerre series expansion with better stability, higher accuracy, and less computational burden.' address: - 'Institute of Computational Mathematics and Mathematical Geophysics, 630090, Novosibirsk, Russia' - 'Novosibirsk State Technical University, 630073, Novosibirsk, Russia' author: - 'Andrew V. Terekhov' bibliography: - 'base.bib' title: 'Generating the Laguerre expansion coefficients by solving a one-dimensional transport equation.' --- Integral Laguerre transform ,Fast algorithms ,Transport equation 02.60.Dc ,02.60.Cb ,02.70.Bf ,02.70.Hm Introduction ============ The Laguerre integral transform has been used in various fields of mathematical simulation to solve acoustics and elasticity equations [@fatab2011; @Terekhov:2013; @Terekhov2015206; @Mikhailenko1999], Maxwell and heat conduction equations [@Mikhailenko2008; @Colton1984], and spectroscopy problems [@Jo2006]. The Laguerre transform has proved to be a very efficient tool in constructing a stable algorithm of wave field continuation when solving inverse problems of seismic prospecting [@Terekhov2017; @Terekhov2018] and many others. The Laguerre transform has served as a basis for the development of numerical methods of inversion of Laplace [@Weeks1966; @Abate1996; @Strain1992] and Fourier [@Weber1980] transforms. In numerically solving differential equations by applying the Laguerre transform in time and approximating space derivatives one has to solve definite well-conditioned systems of linear algebraic equations. For the latter one can use fast convergent algorithms of computational linear algebra [@Golub1989; @Samarski_Nikolaev]. In addition, in contrast to the Fourier transform, to calculate the coefficients of the Laguerre series one and the same operator, which does not depend on the number of the harmonic being calculated, is inverted several times. On the contrary, the operator obtained by the Fourier transform will depend on the frequency. This property of the Laguerre transform allows using efficient parallel preconditioning procedures to solve systems of linear algebraic equations, for instance, on the basis of a dichotomy algorithm [@Terekhov:2013; @terekhov:Dichotomy; @Terekhov2016], which was specially developed to invert one and the same matrix for different right-hand sides. Consider Laguerre functions [@abramowitz+stegun], which are defined as $$l_n(t)=e^{-t/2}L_n(t),\quad t\geq 0 \label{laguerre_function}$$ where $L_n(t)$ is the Laguerre polynomial of degree $n$, which is defined by the Rodrigues formula $$L_n(t)=\frac{ e^{t}}{n!}\frac{d^n}{dt^n}\left(t^ne^{-t}\right)=\frac{1}{n!}\left(\frac{d}{dt}-1\right)^nt^n.$$ We will use $L_2[0,\infty)$ to denote the space of square integrable functions $f:[0,\infty)\rightarrow \mathbb{C}$ $$L_2[0,\infty)=\left\{f:\int_{0}^{\infty}|f(t)|^2dt\le \infty\right\}.$$ The Laguerre functions are a complete orthonormal system in $ L_2[0,\infty)$ $$\int_{0}^{\infty}l_m(t)l_n(t)dt= \left\{\begin{array}{ll} 0,& m \neq n ,\\ 1,& m=n, \end{array}\right. \label{lag_orho}$$ This guarantees that for any function $f(t)\in L_2[0,\infty)$ there is a Laguerre expansion \[series\_lag\] [align]{} \[series\_lag.sum\] f(t)\~\_[m=0]{}\^|[a]{}\_m l\_m(t), t 0,  &gt; 0,\ |[a]{}\_m=\_[0]{}\^f(t)l\_m(t)dt , \[series\_lag.int\] where $\eta$ is a scaling parameter for the Laguerre functions to increase the convergence rate of the series (\[series\_lag.sum\]). The Laguerre function values for large $n$ are bounded from above, since an asymptotic representation [@Szegoe1975] $$\label{eq:assymptotic} l_n(t)=\frac{1}{\pi^{1/2}(nt)^{1/4}}\left(\cos(2\sqrt{nt}-\pi/4)\right)+O\left(\frac{1}{n^{3/4}}\right),\quad t\in[a,b],\quad 0< a< b < \infty,$$ is valid. However, one of the problems of numerical implementation of the transform (\[series\_lag.int\]) is that in calculating the Laguerre functions for $t>1$ the values of the Laguerre polynomials $L_n(t)$ rapidly increase with increasing $n$, which leads to an error of “overflow”. On the contrary, in calculating the multiplier $\exp({-t/2})$ there may be an error of “underflow”. For small $n$ and $t$ the Laguerre functions can be calculated by the formula $$l_n(t)=\left[e^{-t/4}L_n(t)\right]e^{-t/4}. \label{eq:little_laguerre_function}$$ Specifically, first we calculate the expression in the square brackets using a second order recurrence formula [@Rainville1971]: $$\begin{array}{l} (n+1)L_{n+1}(t)=(2n+1-t)L_n(t)-nL_{n-1}(t), \quad n\geq 1,\\\\ L_1(t)=1-t,\quad L_0(t)=e^{-t/4}. \end{array} \label{recurrence_laguerre}$$ Then, multiplying the result by the second exponential multiplier, we calculate the Laguerre function. If the calculations are made at $128$-bit real computer precision, this method excludes situations of the “overflow” and “underflow” types for $n$-values that do not exceed several thousand and $t<20$, $\eta<1200$. Since high-precision arithmetic is used, as a rule, with software (but not hardware), the use of high precision considerably decreases the efficiency of calculations. Therefore, to save the calculation time $128$-bit arithmetic should only be used to calculate the Laguerre functions, whereas the summation in approximating the integral (\[series\_lag.int\]) can be made using standard $64$-bit precision. Another problem of implementing the Laguerre transform is caused by the fact that the Laguerre functions of the $n$-th order on the interval $ 0<t<4n$ oscillate [@Temme1990], and the strongest oscillations are near zero (see Fig. \[pic:laguerre\_function\]). This brings up a problem of finding a method to integrate rapidly oscillating functions. To overcome this difficulty, an algorithm to calculate the integral (\[series\_lag.int\]) is proposed in [@Litko1989]. This algorithm is based on quadratures of high-order accuracy, which make it possible to calculate the Laguerre series expansion coefficients whose number $n$ is not greater than several hundreds. However, one should take into account that the quadratures of high orders are defined on nonuniform grids, which may not allow their use if the function to be approximated is given in the form of a time series for equal-spaced time intervals. For analytical functions, approaches based on the Laplace transform and Cauchy’s integral formula can be used [@Abate1996]: [0.47]{} ![Laguerre functions of various orders for a transformation parameter .[]{data-label="pic:laguerre_function"}](Laguerre10_corel.eps "fig:"){width="\textwidth"}   [0.47]{} ![Laguerre functions of various orders for a transformation parameter .[]{data-label="pic:laguerre_function"}](Laguerre25_corel.eps "fig:"){width="\textwidth"} [0.47]{} ![Laguerre functions of various orders for a transformation parameter .[]{data-label="pic:laguerre_function"}](Laguerre50_corel.eps "fig:"){width="\textwidth"}   [0.47]{} ![Laguerre functions of various orders for a transformation parameter .[]{data-label="pic:laguerre_function"}](Laguerre100_corel.eps "fig:"){width="\textwidth"} $$\label{lag_lap} \bar{a}_n=\frac{1}{2\pi \mathrm{I}}\int_{C_r}\left[\frac{\hat{f}((1+z)/2(1-z))}{1-z}\right]z^{-(n+1)}dz.$$ Here the expression in the square brackets is a generating Laguerre function, which is analytical in the circle $C_r$ of radius $r$ with the center at the origin of coordinates, and $\hat{f}(s)$ is the Laplace transform for the function $f(t)$. The imaginary unit is denoted by $\mathrm{I}=\sqrt{-1}$. In calculating the integral (\[lag\_lap\]), the necessary preliminary Laplace transform for a function expanded into a Laguerre series makes it difficult to use this algorithm. Another method of calculating the expansion coefficients is given by an integral of the form [@Weber1980] $$\label{lag_four} \bar{a}_n=\frac{1}{2\pi}\int_{0}^{2\pi}\left[\frac{1}{2}\left(1+\mathrm{I}\cot\frac{z}{2}\right)f\left(\frac{1}{2}\cot\frac{z}{2}\right)\right]e^{-\mathrm{i}nz}dz.$$ Note that the cotangent function has singularities at points $0$ and $\pi$. This complicates the calculation of the expansion coefficients if the function being approximated is discrete. One more method based on the Laplace and Fourier transforms was proposed in [@Weeks1966]. Thus, the above-mentioned approaches are most suited for approximating smooth analytical functions, for which the Laplace or Fourier transform is known. However, in solving applied problems the initial data may be specified in the form of time series with low smoothness, which calls for the development of additional procedures for this case. In this paper, a new method to calculate the Laguerre series coefficients is proposed. It is based on solving a one-dimensional transport equation. This is a distinguishing feature of the approach, since the integral transforms are used, as a rule, to solve differential equations. In contrast to this, the one-dimensional transport equation is solved to implement the integral Laguerre transform. With this approach to the problem, a stable and rather accurate algorithm which is less expensive than the direct calculation of the integral (\[series\_lag.int\]) is proposed. In addition, an efficient variant of the method will be considered for the approximation of functions on large intervals. Expansion algorithms ==================== Main formulas ------------- Consider the following initial boundary value problem for a one-dimensional transport equation: $$\left\{ \begin{array}{ll} \displaystyle \frac{\partial v}{\partial t }-\frac{\partial v}{\partial x }=0, \quad t>0,\quad -\infty< x < +\infty,\\\\ \displaystyle v(x,0)=f(x). \end{array}\right. \label{advection_eq}$$ On taking the Laguerre transform in time of the problem (\[advection\_eq\]), it can be written in the form [@Mikhailenko1999] $$\label{advection_laguerre} \left(\frac{\eta}{2}-\partial_x\right)\bar{v}_m=-\Phi(\bar{v}_m),$$ where $$\Phi(\bar{v}_m)=\sqrt{\eta}f+\eta\sum_{j=0}^{m-1}\bar{v}_j. \label{phi_function}$$ Taking into consideration $$\Phi(\bar{v}_{m})=\eta \bar{v}_{m-1}+\Phi(\bar{v}_{m-1}),$$ let us turn to another form of (\[advection\_laguerre\]) \[advection\_laguerre3\] [align]{} \[advection\_laguerre3.a\] (-\_x)|[v]{}\_[0]{}+f=0,\ \[advection\_laguerre3.b\] (-\_x)|[v]{}\_[m]{}=(--\_x)|[v]{}\_[m-1]{}, m=1,2,... Then, taking the Fourier transform in the variable $x$, we have \[advection\_laguerre2\] [align]{} \[advection\_laguerre2.a\] (-k)|[V]{}\_[0]{}(k)+(k)=0,\ \[advection\_laguerre2.b\] (-k)|[V]{}\_[m]{}(k)=(--k)|[V]{}\_[m-1]{}(k), m=1,2,..., where $k$ is the wavenumber. Expressing the sought-for function in explicit form, we have $$\begin{array}{c}\displaystyle \bar{V}_{m}(k)=\sqrt{\eta}\tilde{f}(k){\left({-\frac{\eta}{2}-\mathrm{I}k}\right)^m}/{\left({\frac{\eta}{2}-\mathrm{I}k}\right)^{m+1}}. \end{array} \label{lag_fourier}$$ Again, consider the problem (\[advection\_eq\]), but with periodic boundary conditions of the form , where $T$ determines the boundary of the interval of approximation of the function $f(t)$, $t \in [0,T]$. In this case the solution to equation (\[advection\_laguerre\]) has the form of summation of solutions of the form (\[lag\_fourier\]) for a discrete set of frequencies, $k_j=2 \pi j/T$, $j=0,1,...,N_x$: $$\label{main_formula} \bar{v}_{m}(p)\approx\sum_{j=0}^{N_x}\tilde{V}_{m}(k_{j})\exp\left(\mathrm{I}\frac{2\pi j p}{T}\right).$$ Subject to the solution (\[main\_formula\]) for the transport equation, the function $f(t)$, given as an initial condition, will move in the direction $x=0$. By writing the solution to the transport equation at the point $x=0$, we see that the sought-for coefficients of the expansion (\[series\_lag.int\]) for the function $f(t)$ can be calculated as $\bar{a}_m=\bar{v}_{m}(0)$. Although the expansion coefficients are calculated by formulas (\[lag\_fourier\]), (\[main\_formula\]) with $O(nN_x)$ arithmetic operations, that is, the algorithm is not fast, the above method, proposed for implementing the Laguerre transform, has some important advantages over the direct calculation of the improper integral of the rapidly oscillating function (\[series\_lag.int\]). First, from the definition of the absolute value of a complex number we have the identity $$\left|{\left({-\frac{\eta}{2}-\mathrm{I}k}\right)}/{\left({\frac{\eta}{2}-\mathrm{I}k}\right)^{}}\right|\equiv1,$$ which guarantees stability of the calculation and the absence of “overflow” or “underflow” situations for any $T$ and $n$, which is a problem in calculating the Laguerre functions by formula (\[laguerre\_function\]). Second, as it will be shown below, the calculations by formulas (\[lag\_fourier\]),(\[main\_formula\]) can be made with single $32$-bit real precision, which increases the accuracy of the calculations by using higher vectorization. On the contrary, the considerable spread in the Laguerre function values calls for $64$-bit precision calculations. Third, despite the presence of strong oscillations of the Laguerre functions at the origin of coordinates (Fig. \[pic:laguerre\_function\]), the spectral approach does not require using nonuniform grids or quadratures of high-order accuracy to retain a given accuracy on the entire approximation interval. From a practical viewpoint, it is much more convenient to specify the number of harmonics $N_x$ of the Fourier series instead of the grid size, since the boundaries of the spectrum of the function being approximated are, as a rule, either known beforehand or can be determined in an efficient way. A shortcoming of the computational model being considered is that this method of calculating the expansion coefficients of the Laguerre series adds a fictitious periodicity of the form $f(t)=f(t+bT)$, where b is any nonnegative integer. To remove the undesirable periodicity, two fundamentally different approaches will be proposed below. Energy-dependent truncation of Laguerre Series {#section:enetgy_truncation} ---------------------------------------------- Consider an approach which allows removing the fictitious periodicity in the calculation of the expansion coefficients of the Laguerre series by formula (\[main\_formula\]). Fig. \[pic:conjg222\]b shows the Laguerre series expansion coefficients for the function $f(t)$ in Fig. \[pic:conjg222\]b specified by the formula $$f(t)=\exp\left[-\frac{(2\pi f_0(t-t_0))^2}{g^2}\right]\sin(2\pi f_0(t-t_0)), \label{source}$$ where $t_0=0.5,\, g=4,\,f_0=30$. It is evident from Fig. \[pic:conjg222\]b that to exclude the undesirable periodicity it is sufficient to increase the calculation interval from $[0,T]$ to $[0,3T]$ assuming that for $t \in [T,3T]$ the function is zero. Then, once the expansion coefficients have been calculated, remove the coefficients with numbers $m>m_0 \approx 400$. In the calculations for smaller approximation intervals, for instance $[0,T]$ or $[0,2T]$, the fictitious periods of the function in the spectral domain cannot be separated, since the abrupt truncation of the series will cause oscillations in the entire approximation interval. [0.47]{} ![a) function (\[source\]) and its approximation by formula (\[main\_formula\]) for intervals of various lengths, b) Laguerre spectrum.[]{data-label="pic:conjg222"}](Test2.eps "fig:"){width="\textwidth"}   [0.47]{} ![a) function (\[source\]) and its approximation by formula (\[main\_formula\]) for intervals of various lengths, b) Laguerre spectrum.[]{data-label="pic:conjg222"}](Test2_L.eps "fig:"){width="\textwidth"} To automatically determine the number of the remaining expansion coefficients, we use Parseval’s relation $$\int_{0}^{\infty} v^2(t) dt=\sum_{m=0}^{\infty}\left(\bar{v}_{m}\right)^2, \label{pseudo_energy}$$ On the basis of this relation the maximum number of the Laguerre series coefficients $m_0$ is determined from the condition $$\operatorname*{arg\,min}_{m_0}\left|{\int_{0}^{T} v^2(t) dt-\sum_{m=0}^{m_0}\left(\bar{v}_{m}\right)^2}{}\right|. \label{pseudo_energy2}$$ Now let us formulate an algorithm of expanding the function $f(t)$ in a Laguerre series.\ \ [**Algorithm 1**]{} to approximate a function $f(t)$ on the interval $t\in\left[0,T\right]$ by a Laguerre series: 1. Calculate $\tilde{f}=FFT(f)$ on the basis of a fast algorithm of the discrete Fourier transform. 2. Calculate the expansion coefficients of the series (\[series\_lag.sum\]) by formula \[main\_formula\]) and the equality $\bar{a}_m=\bar{v}_{m}(0)$. 3. On the basis of formula (\[pseudo\_energy2\]), leave intact only the first $m_0$ coefficients of the series (\[series\_lag.sum\]). The above-considered a posteriori method of removing the fictitious periodicity is not convenient from a practical viewpoint, since the spectra of nonsmooth functions may be rather large. This may not allow separating the first period of the function being approximated from the subsequent fictitious periods in the spectral domain. Also, in approximating functions of various smoothness it is not clear how many times the approximation interval must be increased to reliably remove the fictitious periodicity. In this case too great increase in the approximation interval length may cause a considerable increase in the computational costs. To solve these problems, an alternative procedure of removing the fictitious periodicity not requiring a posteriori analysis of the Laguerre spectrum will be developed. Shift and Conjugation procedures -------------------------------- Let us develop two auxiliary procedures to modify the Laguerre series coefficients, which we call shift and conjugation. These will allow us to propose an alternative procedure of removing the fictitious periodicity, as well as a procedure of reducing the computational costs when a function is expanded in a series for large approximation intervals. Consider an analytical solution to the following initial boundary value problem $$\label{boundary3} \left\{ \begin{array}{ll} \displaystyle \frac{\partial v}{\partial t }+\frac{\partial v}{\partial x }=0,& t>0,\ x>0,\\\\ v(0,t)=f(t), & t\geq 0,\\ v(x,0)=0, & x\geq 0,\\ f(0)=0. \end{array}\right.$$ As in solving the problem (\[advection\_eq\]), we again apply the Laguerre transform in time to the transport equation, and obtain the equation $$\label{advection_laguerre222} \left(\frac{\eta}{2}+\partial_x\right)\bar{v}_m=-\Phi(\bar{v}_m).$$ For the boundary conditions (\[boundary3\]) to be satisfied, we use the Laguerre transform but not the Fourier transform to calculate the functions $\bar{v}_m(x)$ , that is, search for a solution of the form $$\bar{v}_m(x)=\sum_{j=0}^{\infty}W_{m,j}l_j(\kappa x),\quad m=0,1,2..., \label{laguerre_x_expansion}$$ where the transformation parameter $\kappa>0$. Then, on applying the Laguerre spatial transform to equation (\[boundary3\]), we have \[lagx\] [align]{} \[lagx.1\] (+)W\_[m,0]{}=(-+)W\_[m-1,0]{}+2(|[f]{}\_m- |[f]{}\_[m-1]{}),&m=0,1,...,\ \[lagx.2\] (+)W\_[m,j]{}+2(W\_[m,j]{})=(-+)W\_[m-1,j]{}+2(W\_[m-1,j]{}),&m=0,1,...; j=1,2,.., where $$\Upsilon\left(W_{m,j}\right)=\kappa \sum_{i=0}^{j-1}{W_{m,i}}=\kappa W_{m,j-1}+\Upsilon\left(W_{m,j-1}\right), \label{eq1}$$ and $W_{m,j}\equiv0,\;\bar{f}_{m}\equiv0,\quad \forall \; m<0.$\ \ Taking (\[eq1\]) into account, equation (\[lagx.2\]) takes the following form: $$\left(\eta+\kappa\right)W_{m,j}+\left(\eta-\kappa\right){W}_{m-1,j}= \left(\eta-\kappa\right)W_{m,j-1}+\left(\eta+\kappa\right)W_{m-1,j-1},\quad m=0,1,...;\,j=1,2,...$$ Taking $\kappa=\eta$, we finally obtain $$\label{lagx3} \left\{\begin{array}{ll} W_{m,0}=\kappa^{-1/2}\left(\bar{f}_m-\bar{f}_{m-1}\right),& m=0,1,..., \\ W_{m,j}=W_{m-1,j-1},& m=0,1,...;\; j=1,2,... \end{array} \right.$$ Based on (\[series\_lag.sum\]), (\[laguerre\_x\_expansion\]) and (\[lagx3\]), the final solution to problem (\[boundary3\]) in the time domain is as follows: $$\label{solution1} v(x,t)=\sum_{m=0}^{\infty} \left(\sum_{j=0}^mW_{m-j,0}l_{j}(\kappa x)\right)l_{m}(\eta t).$$ Changing the order of summation, we can also write $$\label{solution2} v(x,t)=\sum_{j=0}^\infty\left(\sum_{m=0}^{\infty}W_{m,0}l_{m+j}(\eta t)\right)l_{j}(\kappa x).$$ It follows from formulas (\[solution1\]) and (\[solution2\]) that the expressions in the brackets are the Laguerre series coefficients. Then, taking into account the relations (\[lagx3\]), we introduce two transforms with a parameter $\tau\geq 0$: [align]{} \[shift\_proc\] {|[a]{}\_m;}=\_[j=0]{}\^m(|[a]{}\_[m-j]{}-|[a]{}\_[m-j-1]{})l\_[j]{}() ,\ {|[a]{}\_j;}=\_[m=0]{}\^(|[a]{}\_m-|[a]{}\_[m-1]{})l\_[m+j]{}(),  |[a]{}\_[-1]{}0. \[reverse\_proc\] \[transform2\] One can see in Fig. \[pic:shift\_test\] for formula (\[shift\_proc\]) that the expansion coefficients $\bar{g}_m=\mathbb{S}\left\{\bar{f}_m;\tau\right\}$ correspond to a function $g(t) = f(t-\tau)$, where $f(t)\equiv 0$ for $t < 0$. One can see in Fig. \[pic:reverse\_test\] for formula (\[reverse\_proc\]) that the expansion coefficients $\bar{h}_m=\mathbb{Q}\left\{\bar{f}_m;\tau\right\}$ approximate a function $h(t)=f(\tau-t)$, where $f(t) \equiv 0$ for $t < 0$. The transform $\mathbb{S}\left\{\cdot;\tau\right\}$ will be called a shift. The transform $\mathbb{Q}\left\{\cdot;\tau\right\}$ will be called conjugation for the interval $[0,\tau]$, since the transform $\mathbb{Q}\left\{\cdot;\tau\right\}$ is an analog of complex conjugation for the coefficients of the trigonometric Fourier series. To implement the transforms (\[shift\_proc\]) and (\[reverse\_proc\]), $O(n \log n)$ operations are needed, if we use algorithms based on the fast Fourier transform [@Nussbaumer1982] to calculate the linear convolution (\[shift\_proc\]) and the correlation (\[reverse\_proc\]). [0.47]{} ![a) Function (\[source\]) and b) Laguerre spectrum for various values of parameter $\tau$ of shift operator $\mathbb{S}\{\bar{f}_m;\tau\}$.[]{data-label="pic:shift_test"}](Shift_t.eps "fig:"){width="\textwidth"}   [0.47]{} ![a) Function (\[source\]) and b) Laguerre spectrum for various values of parameter $\tau$ of shift operator $\mathbb{S}\{\bar{f}_m;\tau\}$.[]{data-label="pic:shift_test"}](Shift_l.eps "fig:"){width="\textwidth"} [0.47]{} ![ a) Function (\[source\]) and b) Laguerre spectrum for various values of parameter $\tau$ of conjugation operator $\mathbb{Q}\{\bar{f}_m;\tau\}$.[]{data-label="pic:reverse_test"}](Reverse_t.eps "fig:"){width="\textwidth"}   [0.47]{} ![ a) Function (\[source\]) and b) Laguerre spectrum for various values of parameter $\tau$ of conjugation operator $\mathbb{Q}\{\bar{f}_m;\tau\}$.[]{data-label="pic:reverse_test"}](Reverse_L.eps "fig:"){width="\textwidth"} Time-dependent truncation of Laguerre Series -------------------------------------------- To remove the fictitious periodicity of the function being approximated, a procedure was developed in Section \[section:enetgy\_truncation\]. This procedure, by analyzing the Laguerre series spectra, limits the number of expansion coefficients to separate the first period of the function being approximated from all subsequent fictitious periods. Here we propose another algorithm to remove the periodicity with less computational costs without any additional increase in the approximation interval. Consider a procedure which, for a given parameter $\tau>0$, transforms the Laguerre series coefficients to make the series for the function $f(t)$ approximate the function $r(t)=H(-t+\tau)f(t)$, where $H(t)$ is the Heaviside function. This is equivalent to nullifying the values of the series $ \forall \ t>\tau>0$. This can be achieved by successively applying two conjugation operations of the form $\mathbb{Q}^2\left\{\cdot;\tau\right\}\equiv\mathbb{Q}\left\{\mathbb{Q}\left\{\cdot;\tau\right\};\tau\right\}$ to the Laguerre series. [0.47]{} ![ Function $\mathbb{Q}^2\left\{\bar{f}_m;1/2\right\}$ before and after applying the operator and b) Laguerre spectrum.[]{data-label="pic:conjg2"}](Cut_Test.eps "fig:"){width="\textwidth"}   [0.47]{} ![ Function $\mathbb{Q}^2\left\{\bar{f}_m;1/2\right\}$ before and after applying the operator and b) Laguerre spectrum.[]{data-label="pic:conjg2"}](Cut_Test_L.eps "fig:"){width="\textwidth"} It follows from Fig. \[pic:conjg2\]a that once the operation $\mathbb{Q}^2\left\{\bar{f}_m;1/2\right\}$ is applied, the values of the series become zero, $ \forall \ t>1/2$. The local smoothness of the function $r(t)$ in the vicinity of the point $t=1/2$ decreases, which increases the spectrum width (Fig. \[pic:conjg2\]b). However, if there are no additional discontinuities of the function and its derivatives at the point $t=\tau$ (Fig. \[pic:conjg3\]a), the spectrum width does not increase. [0.47]{} ![ a) Function(\[source\]) before and after applying the operator $\mathbb{Q}^2\left\{\bar{f}_m;0.7\right\}$ and b) Laguerre spectrum.[]{data-label="pic:conjg3"}](Cut_Test_t3.eps "fig:"){width="\textwidth"}   [0.47]{} ![ a) Function(\[source\]) before and after applying the operator $\mathbb{Q}^2\left\{\bar{f}_m;0.7\right\}$ and b) Laguerre spectrum.[]{data-label="pic:conjg3"}](Cut_Test_L3.eps "fig:"){width="\textwidth"} As a result, one can preliminarily calculate the expansion coefficients by formula (\[main\_formula\]), and then apply the operation $\mathbb{Q}^2\left\{\cdot;T\right\}$ to remove the fictitious periodicity. The operation $\mathbb{Q}^2$ uses $O(n\log n)$ arithmetic operations, which is much less than the computational costs for formula (\[main\_formula\]). Therefore, the total costs of the approach being proposed will increase insignificantly. To avoid any additional discontinuities and decreases in the smoothness of the function being approximated and, hence, increases in the Laguerre spectrum width, the function being expanded in the series is locally multiplied by an exponentially attenuating multiplier on the right boundary of the approximation interval. In solving practical problems of seismic prospecting, it is often necessary to perform integral transforms for a set of independent time series, called seismic traces. In this case the procedure of removing the periodicity can be implemented in a more efficient way. For this formula (\[main\_formula\]) is rewritten in matrix form as follows: $$\label{main_matirx} \left( \begin{array}{c} \bar{a}_0 \\ \bar{a}_1 \\ ... \\ \bar{a}_{n-1} \\ \bar{a}_n \\ \end{array} \right)=\left( \everymath{\displaystyle} \begin{array}{ccccc} \displaystyle \frac{1}{\left({\mathrm{I}k_0+{\eta}/{2}}\right)} &\frac{1}{\left({\mathrm{I}k_1+{\eta}/{2}}\right)} & ... & \frac{1}{\left({\mathrm{I}k_{N_x}+{\eta}/{2}}\right)} \\\\ \displaystyle \frac{\left({\mathrm{I}k_0-{\eta}/{2}}\right)}{\left({\mathrm{I}k_0+{\eta}/{2}}\right)^{2}} &\frac{\left({\mathrm{I}k_1-{\eta}/{2}}\right)}{\left({\mathrm{I}k_1+{\eta}/{2}}\right)^{2}} & ... & \frac{\left({\mathrm{I}k_{N_x}-{\eta}/{2}}\right)}{\left({\mathrm{I}k_{N_x}+{\eta}/{2}}\right)^{2}} \\ ... & ... & ... & ... \\ \displaystyle \frac{\left({\mathrm{I}k_0-{\eta}/{2}}\right)^n}{\left({\mathrm{I}k_0+{\eta}/{2}}\right)^{n+1}} &\frac{\left({\mathrm{I}k_1-{\eta}/{2}}\right)^n}{\left({\mathrm{I}k_1+{\eta}/{2}}\right)^{n+1}} & ... & \frac{\left({\mathrm{I}k_{N_x}-{\eta}/{2}}\right)^n}{\left({\mathrm{I}k_{N_x}+{\eta}/{2}}\right)^{n+1}} \\ \end{array} \right)\left( \begin{array}{c} \tilde{f}_0 \\ \tilde{f}_1 \\ ... \\ \tilde{f}_{N_x-1} \\ \tilde{f}_{N_x} \\ \end{array} \right)=M\tilde{F}.$$ Consider the matrix $\check{M}$ whose columns are obtained from the columns of the matrix $M$ by applying the operations $\mathbb{Q}^2\left\{\cdot;T\right\}$. Instead of applying the operation $\mathbb{Q}^2\left\{\cdot;T\right\}$ to the calculated coefficients $\bar{a}_n$, one can preliminarily calculate the matrix $\check{M}$ and then calculate the expansion coefficients without the fictitious periodicity. This method is used if the number of columns of the matrix $M$ is much less than the number of functions to be approximated. Now let us formulate algorithms to approximate a function $f(t)$, $t\in\left[0,T\right]$ by a Laguerre series, where to remove the fictitious periodicity we use the operator $\mathbb{Q}^2\left\{\cdot;T\right\}$.\ \ [**Algorithm 2**]{} to approximate a function $f(t)$ on an interval $t\in\left[0,T\right]$ by a Laguerre series 1. [**Preparation stage:**]{} 1.1 Create a matrix $M$ of the form (\[main\_matirx\]). 1.2 Calculate the modified matrix $\check{M}$ by making the transform $\mathbb{Q}^2\left\{\cdot;T\right\}$ for each column of the matrix $M$. 2. [**For each of the functions $f(t)$ being approximated:**]{} 2.1 Calculate $\tilde{f}=FFT(f)$ using a fast algorithm of the discrete Fourier transform. 2.2 Calculate the Laguerre series coefficients as $\left(\bar{a}_0,\bar{a}_1,...,\bar{a}_n\right)^{T}=\hat{M}\times\left(\tilde{f}_0,\tilde{f}_1,...,\tilde{f}_{N_x}\right)^T$. If the number of functions to be approximated is smaller than the number of columns of the matrix $M$, the following algorithm, which does not calculate the matrix $\check{M}$, is more efficient:\ \ [**Algorithm 3**]{} to approximate a function $f(t)$ on an interval $t\in\left[0,T\right]$ by a Laguerre series: 1. Calculate $\tilde{f}=FFT(f)$ using a fast algorithm of the discrete Fourier transform. 2. Calculate the Laguerre series coefficients as $\left(\bar{a}_0,\bar{a}_1,...,\bar{a}_n\right)^{T}=M\times\left(\tilde{f}_0,\tilde{f}_1,...,\tilde{f}_{N_x}\right)^T$. 3. Transform $\mathbb{Q}^2\left\{\bar{a}_n;T\right\}$ to exclude the fictitious periodicity. A generalization for the expansion algorithms {#section:stable_laguerre_evaluation} --------------------------------------------- Algorithms 1, 2, and 3 can be used when the function to be approximated by a Laguerre series can be represented by a Fourier series as well. For the Laguerre series coefficients to decrease rapidly enough, the function being expanded must tend to zero exponentially in the vicinity of the right boundary of the approximation interval [@Boyd2001]. This can be achieved by locally multiplying the function by a factor of the form $\exp(-\mu t),\ \mu >0$. On the other hand, since the trigonometric interpolation is periodic, the condition $f(0) = f(T) = 0$ must be satisfied. This imposes constraints on the form of the function being approximated. For instance, if the above-considered algorithms are applied to the function shown in Fig. \[pic:conjg4\]a, there will be oscillations on both boundaries of the expansion interval (see Fig. \[pic:conjg4\]b). The loss of accuracy can be avoided if the calculations are made by the following scheme. [0.47]{} ![a) Function to be approximated, b) incorrect approximation of the initial function by a Laguerre series with artefacts shown by arrows, c) auxiliary function including an additional interval, d) correct approximation of the initial function by a Laguerre series after removing the auxiliary interval.[]{data-label="pic:conjg4"}](nonzero1.eps "fig:"){width="\textwidth"}   [0.47]{} ![a) Function to be approximated, b) incorrect approximation of the initial function by a Laguerre series with artefacts shown by arrows, c) auxiliary function including an additional interval, d) correct approximation of the initial function by a Laguerre series after removing the auxiliary interval.[]{data-label="pic:conjg4"}](nonzero2.eps "fig:"){width="\textwidth"} First the initial function is shifted to the right by $\Delta t$. Then, on the interval $t\in[0,\Delta t]$, a smooth function taking a zero value at $t=0$ is added (see Fig. \[pic:conjg4\]c where a scaled quarter-period function of $\cos^2(t)$ is specified on the interval $t\in[0,\Delta t]$. If the modified function is expanded in a Laguerre series using algorithm 2 or 3, the operation $\mathbb{Q}^2\left\{\bar{a}_m;T\right\}$ is used instead of the operation $\mathbb{Q}\left\{\mathbb{Q}\left\{\bar{a}_m;T+\Delta t\right\};T\right\}$. This will make it possible to remove both the fictitious periodicity and the auxiliary interval $t\in[0,\Delta t]$. If it is planned to use algorithm 1, for which the operation $\mathbb{Q}^2\left\{\bar{a}_m;T\right\}$ is not needed, the additional interval $t\in[0,\Delta t]$ is excluded by calculating the expansion coefficients by formula (\[main\_formula\]), but setting $\bar{a}_m=\bar{v}_m(-\Delta t)$ instead of $\bar{a}_m=\bar{v}_m(0)$. Stable calculation of Laguerre functions for any order and argument value ------------------------------------------------------------------------- Consider a problem of calculating Laguerre functions by performing the operations $\mathbb{S}\left\{\cdot;\tau\right\}$ and $\mathbb{Q}\left\{\cdot;\tau\right\}$. If the argument of the functions $l_m(\eta\tau)$ for (\[shift\_proc\]), (\[reverse\_proc\]) is too large, then (as noted in the introduction) there emerges an error of “overflow” in calculating the function $L_n(\eta\tau)$ or an error of “underflow” in calculating $\exp{(-\eta\tau/2)}$. The use of $128$-bit arithmetic does not exclude errors of these types for larger values of the argument or the order of the Laguerre function. Therefore, we consider a more universal approach. It follows from the relation $ l_m(\eta t_0)=\int_{0}^{\infty}\delta(t-t_0)l_m(\eta t)dt, $ where $\delta(t)$ is the delta function, that the coefficients $\bar{a}_k=l_m(0)=1$ of the Laguerre series (\[series\_lag.sum\]) correspond to $\delta(0)$. Then the Laguerre function can be calculated for any values of the argument using a series of shifts of the form $$\label{eq:shift_laguerre_evaluation} \{l_m(t_0)\}=\mathbb{S}\left\{...\mathbb{S}\left\{\mathbb{S}\left\{l_m(0);\tau_1\right\};\tau_2\right\}...;\tau_p\right\},\quad t_0=\sum_{i=1}^{p}\tau_i.$$ The maximum value of the shift parameter $\tau_i$ for $64$-bit arithmetic is limited by the capacity of representing the quantity $\exp(-\eta\tau_i/4)$ for real numbers. According to the IEEE standard describing a representation of real numbers with $64$-bit precision, by choosing $\eta\tau_i\leq 4\left|\ln(2.225\times10^{-308})\right|\approx 2600$ $l_n(\eta\tau_i)$ can be calculated by (\[eq:little\_laguerre\_function\]) without situations of the “underflow” or “overflow” type. [0.47]{} ![a) Function $l_m(\eta t)$ versus m for a constant value of argument $\eta t=2000\times 16$, b) difference of Laguerre function values calculated by formula (\[eq:reccurence\_shift\]) in $64$-bit arithmetic and formula (\[eq:little\_laguerre\_function\]) in $128$-bit arithmetic.[]{data-label="pic:conjg24"}](laguerre_funct.eps "fig:"){width="\textwidth"}   [0.47]{} ![a) Function $l_m(\eta t)$ versus m for a constant value of argument $\eta t=2000\times 16$, b) difference of Laguerre function values calculated by formula (\[eq:reccurence\_shift\]) in $64$-bit arithmetic and formula (\[eq:little\_laguerre\_function\]) in $128$-bit arithmetic.[]{data-label="pic:conjg24"}](Laguerre_error.eps "fig:"){width="\textwidth"} To decrease the total number of shifts and, hence, the computational costs, it makes sense to perform the shifts recurrently: $$\{l_m(2^p\eta\tau)\}=\mathbb{S}\left\{\mathbb{S}\left\{\mathbb{S}\left\{\mathbb{S}\left\{l_m(\eta\tau);\tau\right\};2\tau\right\},4\tau\right\}...;2^{p-1}\tau\right\}. \label{eq:reccurence_shift}$$ In comparison to formula(\[eq:shift\_laguerre\_evaluation\]), the number of calculations can be reduced owing to the fact that the Laguerre function values obtained at the previous step are used in formula (\[shift\_proc\]) to make the shift at the current step of implementing formula (\[eq:reccurence\_shift\]). Thus, the shift value at each step is doubled, which decreases the total number of shifts with each of them requiring $O(n\log n )$ operations. Note that for the first shift $l_m(\eta\tau)$ must always be calculated by formulas (\[eq:little\_laguerre\_function\]) and (\[recurrence\_laguerre\]). Fig. \[pic:conjg24\]a shows the result of calculation of the functions $l_m(2200\times16)$ by formula (\[eq:reccurence\_shift\]). The absolute difference of the values for formula (\[eq:reccurence\_shift\]) in $64$-bit arithmetic and formulas (\[eq:little\_laguerre\_function\]), (\[recurrence\_laguerre\]) in $128$- bit arithmetic is shown in Fig. \[pic:conjg24\]b. It is evident from this figure that both approaches give practically the same results. Thus, the above algorithm does not use high-precision arithmetic in performing stable calculations of Laguerre functions of any order for any values of the argument. Moreover, some test calculations have shown that, in comparison to $128$-bit arithmetic, the above calculation method needs several times less calculation time if, in particular, $32$-bit arithmetic is used to organize the shift procedure. Optimization for a large interval approximation ----------------------------------------------- It is well-known that, owing to the high performance and stability of the algorithm of fast Fourier transform, it has been widely used in many branches of computational mathematics, whereas no algorithm for the Laguerre transform having comparable efficiency has been developed so far. Although general methods of fast polynomial transforms were proposed long ago [@GOHBERG1994411], they are of theoretical rather than practical importance. This is because they use numerically unstable efficient procedures of multiplying matrices $V$ and $V^T$ by a vector, where $V$ is an ill-conditioned Vandermonde matrix [@Pan2016; @Gautschi2011]. For instance, fast multiplication by the matrix $V$ can be performed by using an algorithm [@Borodin1974] whose computational complexity is of the order of $O(n\log^2n)$ operations. Unfortunately, this method is unstable, since one of its stages includes a recursive use of the operation of polynomial division. Multiplication of the matrix $V^T$ by a vector can be reduced to solving systems of linear algebraic equations with a Vandermonde matrix with an operation count of the order of $O(n\log^2n)$ [@GOHBERG1994411; @Gohberg1994; @Pan1993]. This approach also cannot be recommended for practical use due to its numerical instability. The condition number for Laguerre functions is greater than that for the other classical orthogonal polynomials [@Gautschi1983]. Therefore, the problem of stability of fast algorithms for the Laguerre transform is probably one of the most difficult ones. By now, fast transforms have been developed for Chebyshev, Legendre, and Hermite polynomials [@Alpert1991; @Hale2016; @Leibon2008]. In these cases the arithmetic complexity of the algorithms is of the order of $O(n\log n)$ or $O(n \log^2 n)$ operations. Fast algorithms of changing from one orthogonal polynomial basis specified by a three-term recurrence relation to another one have also been developed [@Bostan2010]. In paper [@ONeil2010], an algorithm for fast polynomial transforms based on an approximate factorization of the matrices $V$ or $V^T$ was proposed. In some cases the authors managed to decrease the computational costs to a level of $O(n \log n)$ arithmetic operations. However, the computational complexity may vary widely for various orthogonal polynomials and expansion interval lengths. Also, the algorithm becomes efficient in comparison to the direct method of multiplying a matrix by a vector, for $n\ge n_0$, where $n_0$ is of the order of several thousand. To expand a function into a Laguerre series by formula (\[main\_formula\]), about $O(n N_x)$ arithmetic operations are needed, where n is the number of expansion terms of the Laguerre series and $N_x$ is the number of harmonics of the auxiliary Fourier series. Approximation of the function for longer intervals calls for specifying larger values of $n$ and $N_x$, which makes the Laguerre transform inefficient. To decrease the calculation time when performing the Laguerre transform, we consider an algorithm of the “divide and conquer” type [@Smith1985]. The general idea of this approach is that at the first stage the initial problem is divided into independent subproblems with much less computational costs needed for their solution. At the second stage the solution to the initial problem is assembled from the solutions to the subproblems. This approach was successfully used, for instance, in papers in which a parallel dichotomy algorithm was proposed to solve systems of linear algebraic equations with three-diagonal [@terekhov:Dichotomy], block-diagonal [@Terekhov:2013], and Toeplitz matrices [@Terekhov2016].\ \ [**Algorithm 4.**]{} to approximate a function $f(t)$ on an interval $t\in\left[0,T\right]$ by a Laguerre series: 1. Decompose the approximation interval $t \in[0,T]$ into $p=2^s$ overlapping subintervals of lengths $\Delta t_i=\beta_i-\alpha_{i}$ (Fig. \[pic:fast1\]). In this case the function must smoothly tend to zero on the subinterval boundaries in the buffer zones so that the sum of the two local functions remains equal to the value of the function being approximated. 2. The local function $f_i(t)$ specified on the subinterval with number $i$ is expanded in a Laguerre series on the auxiliary interval $[0,\Delta t_i]$ by algorithm 1, 2, or 3. 3. Shift the local functions by changing from the interval $[0,\Delta t_i]$ to the subinterval $[\alpha_i,\beta_i]$. This is done by a series of shifts of the function $f_i(t)$ using the scheme presented in Fig. \[pic:fast2\], which gives an example of four subintervals. The process of assembly consists of $\log_2p$ steps, where $p$ is the number of subintervals. Hence, two steps will be needed for the example being considered. At the first step the sequences of the Laguerre series coefficients for the local functions $f_2(t)$ and $f_4(t)$ are supplemented by zeroes to double the number of expansion coefficients. Then the thus expanded series are shifted using the procedures $\mathbb{S}\left\{\bar{a}_{n/2};\alpha_2\right\}$ and $\mathbb{S}\left\{\bar{a}_{n/2};\alpha_4-\alpha_3\right\}$. After this the corresponding coefficients of the first and second series and of the third and fourth series are added pairwise. This results in two intervals of larger lengths. At the second step this process is used for the new second series, and after it is shifted by $\mathbb{S}\left\{\bar{a}_n;\alpha_3\right\}$ the expansion coefficients of the first and second series are added. Thus, all local functions will be shifted to their initial positions with respect to the variable $t$, and the thus obtained series will approximate the initial function $f(t)$ with some accuracy.\ [**Remark.**]{} To execute one shift $\mathbb{S}\{\bar{a}_n;\tau\}$ using the fast Fourier transform, $O(n\log n)$ arithmetic operations are needed. One can see in Fig. \[pic:shift\_test\] that a shift of the function to the right increases the number of coefficients of the Laguerre series needed to approximate the shifted function with the previous accuracy. For the calculation scheme in Fig. \[pic:fast2\] every shift will double the minimum number of the Laguerre series terms. Therefore, before making a shift the sequence of coefficients of the Laguerre series must be added by zeros (zero padding). After making the shift the zero values of the added expansion coefficients will become nonzero ones. ![Decompositions of the initial approximation interval into four overlapping subintervals.[]{data-label="pic:fast1"}](split.eps){width="\textwidth"} ![Approximation construction scheme for function $f(t)$ with precalculated approximations for local functions $f_i(t)$.[]{data-label="pic:fast2"}](scheme3.eps){width="\textwidth"} For larger values of $n$ and $N_x$ the computational complexity of algorithm 4 will be of the order $O(nN_x/p +n\log_2 n\log_2 p)$ vs. $O(nN_x)$, where $p$ is the number of subintervals. The first term is the costs to approximate the local functions $f_i(t),\ t \in [0,\Delta t_i]$, and the second one is the costs to perform a series of shift operations to transform the local expansion coefficients to the expansion coefficients for the initial function $f(t)$. However, this algorithm has the following shortcoming: the number of Laguerre series coefficients to approximate the local functions $f_i(t)$ on the subintervals $[0,\Delta t_i]$ depends not only on the lengths of the subintervals, but also on the smoothness of the functions $f_i(t)$. Taking into account that in solving practical problems the function to be approximated may have low smoothness, the convergence of the series may be not high. This results in the fact that at the same accuracy the total number of expansion coefficients for the local problems for algorithm 4 will be greater than the number of expansion coefficients when using algorithm 1, 2, or 3. Thus, the division will require additional computational costs, which can be estimated in computational experiments. 3. Computational experiments ============================ To estimate the accuracy of the approximation and the efficiency of the methods being proposed, let us perform a series of computational experiments to approximate functions of various smoothness on intervals of various lengths. The numerical procedures to calculate the Laguerre coefficients will be performed with single and double precision. Algorithms 2 and 3 give the same results in calculating the Laguerre series coefficients and, therefore, no separate testing of algorithm 3 will be considered. Inversion of Laguerre transform -------------------------------- Consider the problem of calculating the inverse Laguerre transform (\[series\_lag.sum\]). In contrast to the direct transform, in the summation of the series there only remains the problem of calculating the Laguerre functions of high orders for larger argument values. This problem can be solved in several ways. If the calculations are made with $128$-bit real precision by formula (\[eq:little\_laguerre\_function\]), the Laguerre functions of high orders can be calculated for rather large values of the argument without errors of “overflow” and “underflow”. Another method is to use asymptotic expansions [@Temme1990; @Gil2017] to calculate the Laguerre polynomials $L_n(\eta t)$, whence, multiplying by $\exp{(-\eta t/2)}$, we obtain the Laguerre functions. If a function approximated by a Laguerre series can be represented by a Fourier series, one can change from the Laguerre coefficients to Fourier coefficients using the following formula: $$\label{matrix_inv} (\tilde{f}_0,\tilde{f}_1,...,\tilde{f}_{N_x})^T=M^{T}(\bar{a}_0,\bar{a}_1,...,\bar{a}_n)^T,$$ where $\tilde{M}$ is a modified matrix of the form (\[main\_matirx\]). In this case a major problem is in the emergence of discontinuities of the function on the boundaries of the approximation interval, $t\in[0,T]$. Finally, we can use the stable method of calculating the Laguerre functions by formulas (\[eq:shift\_laguerre\_evaluation\]) or (\[eq:reccurence\_shift\]) considered in Section \[section:stable\_laguerre\_evaluation\]. As noted above, although the fast Fourier transform is needed to calculate the linear convolution, this method of organizing the calculations requires less calculation time than when using $128$-bit arithmetic and formula (\[eq:little\_laguerre\_function\]). Test 1. Expansion of a smooth function -------------------------------------- As a first test, consider, on the interval $t \in [0,1]$, an approximation of a function $f(t)$ of the form (\[source\]) with parameters $f_0=30,\ g=4,\ t_0=0.5$. The discretization step of the function $ht=0.002$. The approximation error is estimated by the formula $$\displaystyle \epsilon=\sqrt{\frac{\sum_{i=1}^{s}\left(f(t_i)-\sum_{j=1}^{n}\bar{a}_j l_n(\eta t_j)\right)^2}{\sum_{i=1}^{s}f^2(t_i)}},$$ where $f(t_i)$ is the function to be expanded in a Laguerre series, which is specified on a set of values $ t_i \in [0,T], i=1,2,...s$, $t_1=0,\ t_s=T $. Fig. \[pic:test1\] shows the error versus the number of expansion coefficients of the Laguerre series for various values of the scaling parameter $\eta \in [50,1600]$. The calculations were made both with double real precision (Fig. \[pic:test1\]a,b) and single precision (Fig.\[pic:test1\]c,d). To exclude the fictitious periodicity in algorithm 1, the initial approximation interval was increased to $t \in [0,2]$ , where $f(t)\equiv 0$ for $t \in [1,2]$. One can see in Fig. \[pic:test1\]a that an error of the order $\epsilon=10^{-14}$ was obtained with algorithm 1 for parameters $\eta=1600$ and $n=380\div920$, as well as for $\eta=800$ and $n=420\div440$. As the number of expansion coefficients for $n>920$ and $\eta=1600$ and for $n>440$ and $\eta=800$ increases, the approximation accuracy abruptly decreases, due to the fictitious periodicity and an abrupt break in the values of the series coefficients. As shown in Fig. \[pic:test11\], the smaller is a given value of the parameter $\eta$, the longer is the spectrum. In this case the spectra of two periods of the function intersect at smaller values of $n$ and, starting with some number $n>n_0(\eta)$, the Laguerre series does not converge to the function being approximated. In contrast to algorithm 1, the use of algorithm 2 (Fig. \[pic:test1\]b,d) did not require any additional increase in the approximation interval. The accuracy level of algorithm 2 is the same both in double and single real arithmetic and is of the order of $\epsilon=10^{-7}$, not decreasing to $\epsilon=10^{-14}$ as for algorithm 1. This is explained by the fact that when using formula (\[reverse\_proc\]) the sequence $l_m(\eta t_0), \ m=0,1,2... $ is the expansion coefficients for the delta function $\delta(t_0)$, for which (as shown in Fig. \[pic:conjg24\]) the Laguerre spectrum is infinitely long and slowly attenuating. Therefore, the finite number of expansion terms is a source of an additional error. However, the behavior of the error for algorithm 2 is more regular, since no fictitious periodicity and no abrupt break of the spectrum are observed for this calculation method. [0.47]{} ![Approximation error for function (\[source\]) versus the number of terms of Laguerre series for various values of the transform parameter, $\eta=50,100,...,1600$, a) algorithm 1 with 64-bit precision, b) algorithm 1 with 32-bit precision, c) algorithm 2 with 64-bit precision, d) algorithm 2 with 32-bit precision.[]{data-label="pic:test1"}](Energy2T_Test1.eps "fig:"){width="\textwidth"}   [0.47]{} ![Approximation error for function (\[source\]) versus the number of terms of Laguerre series for various values of the transform parameter, $\eta=50,100,...,1600$, a) algorithm 1 with 64-bit precision, b) algorithm 1 with 32-bit precision, c) algorithm 2 with 64-bit precision, d) algorithm 2 with 32-bit precision.[]{data-label="pic:test1"}](Energy2T_Test1_real4.eps "fig:"){width="\textwidth"} [0.47]{} ![Approximation error for function (\[source\]) versus the number of terms of Laguerre series for various values of the transform parameter, $\eta=50,100,...,1600$, a) algorithm 1 with 64-bit precision, b) algorithm 1 with 32-bit precision, c) algorithm 2 with 64-bit precision, d) algorithm 2 with 32-bit precision.[]{data-label="pic:test1"}](TimeTest1.eps "fig:"){width="\textwidth"}   [0.47]{} ![Approximation error for function (\[source\]) versus the number of terms of Laguerre series for various values of the transform parameter, $\eta=50,100,...,1600$, a) algorithm 1 with 64-bit precision, b) algorithm 1 with 32-bit precision, c) algorithm 2 with 64-bit precision, d) algorithm 2 with 32-bit precision.[]{data-label="pic:test1"}](TimeTest1_real4.eps "fig:"){width="\textwidth"} [0.47]{} ![Laguerre spectrum for various values of the transform parameter, $\eta=50,100,...,1600$ for function (\[source\]) for a) algorithm 1 and b) algorithm 2.[]{data-label="pic:test11"}](Spectrum_energy.eps "fig:"){width="\textwidth"}   [0.47]{} ![Laguerre spectrum for various values of the transform parameter, $\eta=50,100,...,1600$ for function (\[source\]) for a) algorithm 1 and b) algorithm 2.[]{data-label="pic:test11"}](Spectrum_time.eps "fig:"){width="\textwidth"} [0.47]{} ![Approximation error versus the number of terms of Laguerre series for the function in Fig. \[pic:conjg4\]. Calculations were made by algorithms 1 and 2 for a) 32-bit precision, b) 64-bit precision.[]{data-label="pic:test12"}](Test12_error_r4.eps "fig:"){width="\textwidth"}   [0.47]{} ![Approximation error versus the number of terms of Laguerre series for the function in Fig. \[pic:conjg4\]. Calculations were made by algorithms 1 and 2 for a) 32-bit precision, b) 64-bit precision.[]{data-label="pic:test12"}](Test12_error.eps "fig:"){width="\textwidth"} Consider an approximation of the function in Fig. \[pic:conjg4\], for which $f(0) = 0$. For this function, a comparison of the accuracy of algorithms 1 and 2 was made, with direct calculation of the integral (\[series\_lag.int\]) by the method of rectangles. Since the method of rectangles has the first order of accuracy, an integration step $4\times10^5$ times smaller than the discretization step for algorithms 1 and 2 was taken. Such a fine step is needed to provide high accuracy in calculating the Laguerre coefficients for a method of first order accuracy. To perform numerical integration of rapidly oscillating functions with a larger step, it is necessary to use quadratures of very high order of accuracy. In paper [@Litko1989] it was proposed to use quadratures of the $256$th order of accuracy to calculate a Laguerre series of length $n=128$. However, in solving practical problems nonsmooth functions have to be approximated. This calls into question whether it is reasonable to use high-accuracy quadratures for which error estimation implies the presence of high-order derivatives of the function to be expanded in the series. By means of calculations with single real precision the calculation time can be decreased using a higher degree of vectorization of the calculations. In this case the error of algorithm 1 for $\eta=800,1600$ increases from $\epsilon=10^{-14}$ to $10^{-7}$, that is, to the level of single computer precision, and the error of algorithm 2 remains at a level of the order of $\epsilon=10^{-7}$. It is important that the stability of all algorithms proposed still holds. Note that computer precision in calculations for nonsmooth functions may not be achieved, which eliminates the need for double real precision. To demonstrate this, in the test below we consider an approximation of a time series from a set of test seismograms for a velocity model called “Sigsbee”[@Paffenholz]. Test 2. Expansion of a non-smooth function ------------------------------------------ In the first test an approximation of a smooth function on the interval $[0,1]$ was considered. Now let us test the above developed algorithms for a nonsmooth function (see Fig. \[pic:test2\])) specified on the interval $[0,12]$ with a discretization step $h_t=0.008$. This function corresponds to the first seismic trace from a test set of seismograms for the velocity model Sigsbee [@Paffenholz]. The seismograms for the SigSbee model have single real accuracy. Therefore, we consider an implementation of algorithms 1 and 2 only with single precision. [0.47]{} ![a) First trace from seismograms for the velocity model Sigsbee, b) approximation error versus the number of terms of the Laguerre series for the algorithms and transformation parameter $\eta=900,1800$ for first trace from seismograms for the velocity model Sigsbee.[]{data-label="pic:test2"}](Test2_Sigsbe.eps "fig:"){width="\textwidth"}   [0.47]{} ![a) First trace from seismograms for the velocity model Sigsbee, b) approximation error versus the number of terms of the Laguerre series for the algorithms and transformation parameter $\eta=900,1800$ for first trace from seismograms for the velocity model Sigsbee.[]{data-label="pic:test2"}](Test2_error2.eps "fig:"){width="\textwidth"} For calculations by algorithm 2 the approximation interval was not changed, whereas for algorithm 1 the approximation interval was increased by a factor of three up to $[0,36]$ by adding zero values (zero padding). One can see in Fig. \[pic:test2\] that algorithm 2 is approximately an order of magnitude more accurate than algorithm 1. Also, algorithm 2 demonstrates more regular behavior of the error, which considerably simplifies the process of finding an optimal number of coefficients of the Laguerre series. The smaller accuracy of algorithm 1 is caused, first, by the long Laguerre spectrum for the nonsmooth function, which does not make it possible to separate the spectra for different periods and exclude the influence of the fictitious periodicity. Second, the threefold increase in the approximation interval increases the number of terms of the series (\[main\_formula\]), which is a source of additional error due to the corresponding increase in the total number of operations. The accuracy of calculating the expansion coefficients by the above proposed algorithms and that of calculating the integral (\[series\_lag.int\]) by the method of rectangles with an integration step $ht=8\times 10^{-7}$ was also compared. One can see in Fig. \[pic:test2\]b that, despite the fact that the discretization step of the function is much smaller, the accuracy of calculation by the method of rectangles is much lower than for algorithms 1 and 2, whereas the calculation burden is several orders of magnitude greater. It follows from formula (\[eq:assymptotic\]) and Fig. \[pic:laguerre\_function\] that as $n$ increases, the oscillation frequency of the functions $l_n(\eta t)$ also increases. Therefore, to calculate every subsequent expansion coefficient one has to either decrease the discretization step of the integrand or increase the order of the quadrature formula. However, when solving practical problems one should take into account that limited smoothness of the functions to be approximated may not allow using the maximum order of accuracy of the quadrature formula. Also, there are additional difficulties in using high-accuracy quadratures, which are caused by the need to calculate the integrand on a nonuniform grid, whereas a discrete function to be approximated is, as a rule, specified for equidistant values of the argument. In summary, we can say that the use of single precision for algorithms 1 and 2 is justified, since the observed error level is acceptable in solving practical problems. The use of double precision may decrease the error, but only in the approximation of very smooth functions. -------------------------- ------------------------ ------------------ --------------------- --------------------- --------------------- ----------------- ---------------------- --------------------- -------------------- --------------------- ------------------- ------------------ -- (r)[2-7]{} (r)[8-13]{} p $ {\mathrm{\epsilon}}$ $\mathrm{Prec.}$ $ \mathrm{Step\ 1}$ $ \mathrm{Step\ 2}$ $ {\mathrm{Total}}$ $\mathrm{Rel.}$ $ \mathrm{\epsilon}$ $ {\mathrm{Prec.}}$ $\mathrm{Step\ 1}$ $ \mathrm{Step\ 2}$ $ \mathrm{Total}$ $ \mathrm{Rel.}$ 1 1.7E-3 1.8 29.0 - 29.0 - 2.5E-6 3.9 52.0 - 52.0 - 2 2.9E-3 0.3 15.6 0.9 16.5 1.7 1.5E-5 1.1 27.6 2.1 29.7 1.7 3 5.5E-3 5.7E-2 8.3 3.6 11.9 2.4 9.4E-5 0.2 15.1 9 24.1 2.1 8 7.8E-3 1.2E-2 5.3 7.8 13.1 2.2 1.2E-4 3.9E-2 8.7 17 25.7 2.0 16 2.2E-2 8.9E-2 3.5 15.5 19.1 1.5 2.1E-4 9.1E-3 5.7 34 39.7 1.3 -------------------------- ------------------------ ------------------ --------------------- --------------------- --------------------- ----------------- ---------------------- --------------------- -------------------- --------------------- ------------------- ------------------ -- : Estimates of calculation time and accuracy of algorithm 4. The number of auxiliary intervals [*p*]{} versus: approximation accuracy; preparatory calculation time needed to calculate local matrix $\tilde{M}$; calculation time of local approximation; calculation time of the sequence of shifts for constructing the global approximation; total calculation time of algorithm 4; ratio between calculation time for algorithm 2 and calculation time for algorithm 4.[]{data-label="table33"} -------------------------- ------------------------ ------------------ --------------------- --------------------- --------------------- ----------------- ---------------------- --------------------- -------------------- --------------------- ------------------- ------------------ -- (r)[2-7]{} (r)[8-13]{} p $ {\mathrm{\epsilon}}$ $\mathrm{Prec.}$ $ \mathrm{Step\ 1}$ $ \mathrm{Step\ 2}$ $ {\mathrm{Total}}$ $\mathrm{Rel.}$ $ \mathrm{\epsilon}$ $ {\mathrm{Prec.}}$ $\mathrm{Step\ 1}$ $ \mathrm{Step\ 2}$ $ \mathrm{Total}$ $ \mathrm{Rel.}$ 1 5.3E-6 77 877 - 877 - 3.4E-6 359 3455 - 3455 - 2 8.1E-6 19 449 25 474 1.8 4.6E-6 78.7 1754 58 1792 1.9 3 6.1E-5 4.8 231 44 275 3.1 9.7E-6 19.4 826 107 933 3.7 8 6.5E-5 1.2 124 63 187 4.6 6.6E-5 4.8 463 144 607 5.7 16 5.8E-5 2.5 79 80 159 5.5 6.8E-5 1.2 242 180 422 8.2 32 1.9E-4 4.2E-2 52 91 143 6.1 6.9E-5 2.6E-1 158 212 370 9.3 64 5.8E-4 9.6E-3 25 100 125 7.0 1.2E-4 4.2E-2 105 239 344 10 128 3.9E-3 1.7E-3 17 112 129 6.8 5.8E-4 9.6E-3 50 265 315 11 -------------------------- ------------------------ ------------------ --------------------- --------------------- --------------------- ----------------- ---------------------- --------------------- -------------------- --------------------- ------------------- ------------------ -- : Estimates of calculation time and accuracy of algorithm 4. The number of auxiliary intervals [*p*]{} versus: approximation accuracy; preparatory calculation time needed to calculate local matrix $\tilde{M}$; calculation time of local approximation; calculation time of the sequence of shifts for constructing the global approximation; total calculation time of algorithm 4; ratio between calculation time for algorithm 2 and calculation time for algorithm 4.[]{data-label="table34"} In testing of algorithm 4, Tables \[table33\] and \[table34\] present the calculation times and accuracy estimates in the approximation of all $152684$ seismic traces for the Sigsbee model. For the interval $[0,12]$ the numbers of coefficients of the series were $n=4096$ and $8192$, and for the intervals $[0,60]$ and $[0,120]$ the numbers of coefficients of the series were specified as $n=32768$ and $65536$, respectively. Initial seismic traces for the Sigsbee model were specified for $t \in [0,12]$. To obtain the time series for the intervals $ [0,60]$ and $[0,120]$, the initial trace was supplemented by four or eleven identical copies of the initial signal, respectively. One can see from the data presented that, although algorithm 4 does not belong to the class of fast algorithms, it allows a slight decrease in the calculation time, especially for large time intervals. Also note that when using algorithm 4 the time of the preparatory calculations needed to modify the matrix $M$ in implementing algorithm 2 decreases considerably, since a matrix of smaller order is required to approximate the local functions. Nevertheless, it follows from Tables \[table33\] and \[table34\] that the approximation accuracy $\epsilon$ decreases as the number of auxiliary intervals increases. This is caused, first, by the presence of auxiliary buffers, in which multiplication by an exponentially decreasing factor is made for the Laguerre spectrum of a local function not to be infinite because of the discontinuities of the function values on the boundaries of the subintervals. On each subinterval the local function is approximated by a Laguerre series with a number of coefficients of $n/p$, where $p$ is the number of subintervals. However, $n/p$ expansion coefficients may be insufficient to approximate a nonsmooth local function, which results in loss in approximation accuracy. Nevertheless, if one has to approximate a time series with an accuracy of the order of $\epsilon=10^{-3}\div10^{-5}$ (which is sufficient for practical calculations [@Terekhov2017; @Terekhov2018]), it is recommended to use algorithm 4. To multiply the matrix $\tilde{M}$ by a vector, a numerical procedure from BLAS MKL library was used. Taking into account high degree of optimization of the BLAS procedure for a specific processor model, the calculation of Laguerre coefficients is performed very fast. At the same time, optimization of the algorithm of fast Fourier transform is a more complicated problem. This decreases the degree of vectorization of the calculations at the second step of algorithm 4. As a result, the speedup of algorithm 4 also decreases. If an internal Fortran procedure, such as “matlmul”, had been used for matrix multiplication instead of that from BLAS library, the speedup coefficient of algorithm 4 would have been much larger (although the total calculation time also increases), since the computational costs of the “matmul” function are several times greater than those of the procedure from BLAS MKL library. Conclusions =========== In this paper, new algorithms to calculate the integral Laguerre transform by solving a one-dimensional transport equation have been developed. The main idea of the above proposed approach is that the calculation of improper integrals of rapidly oscillating functions is replaced by solving an initial boundary value problem for the transport equation using spectral algorithms. This approach has made it possible to successfully avoid the problems formulated in the introduction and associated with numerical implementation of the Laguerre transform. It would have been impossible to implement the above proposed computational model without the development of auxiliary procedures that allow removing the fictitious periodicity resulting from periodic boundary conditions. One of the correcting procedures is based on solving the transport equation, whereas the other one is based on a posteriori analysis of the Laguerre spectrum energy. Test calculations have shown that the first method of removing the periodicity is more reliable, accurate, and efficient, since it does not require increasing the approximation interval. Although the above algorithms do not belong to the class of fast algorithms, the number of arithmetic operations has been considerably decreased, since there is no need in calculations with small grid steps or quadrature formulas of high orders of accuracy to calculate rapidly oscillating improper integrals. Additionally, an approach has been developed to decrease the computational costs in making the Laguerre transform for large approximation intervals by solving the transport equation. The test calculations have also confirmed that all developed algorithms can be used both with single and double real precision without loss of numerical stability. This approach provides efficient use of both CPU and GPU resources. Thus, if a large set of functions is approximated by a Laguerre series (for instance, in solving problems of seismic prospecting), the above proposed algorithms allow saving the calculation time considerably. This fact makes this approach attractive from both theoretical and practical viewpoints. Acknowledgements ================ The work was supported by a grant from the Russian Ministry of Education no. MK-152.2017.5.

--- abstract: 'The dynamical classification of rational maps is a central concern of holomorphic dynamics. Much progress has been made, especially on the classification of polynomials and some approachable one-parameter families of rational maps; the goal of finding a classification of general rational maps is so far elusive. Newton maps (rational maps that arise when applying Newton’s method to a polynomial) form a most natural family to be studied from the dynamical perspective. Using Thurston’s characterization and rigidity theorem, a complete combinatorial classification of postcritically finite Newton maps is given in terms of a finite connected graph satisfying certain explicit conditions.' address: - 'Jacobs University Bremen, Campus Ring 1, Bremen 28759, Germany' - 'Jacobs University Bremen, Campus Ring 1, Bremen 28759, Germany' - 'Jacobs University Bremen, Campus Ring 1, Bremen 28759, Germany' bibliography: - 'C:/Users/Russell/Desktop/Lodge.bib' title: A classification of postcritically finite Newton maps --- [Russell Lodge]{} [Yauhen Mikulich]{} [Dierk Schleicher]{}

--- abstract: 'Recently, unsupervised pre-training is gaining increasing popularity in the realm of computational linguistics, thanks to its surprising success in advancing natural language understanding (NLU) and the potential to effectively exploit large-scale unlabelled corpus. However, regardless of the success in NLU, the power of unsupervised pre-training is only partially excavated when it comes to natural language generation (NLG). The major obstacle stems from an idiosyncratic nature of NLG: Texts are usually generated based on certain context, which may vary with the target applications. As a result, it is intractable to design a universal architecture for pre-training as in NLU scenarios. Moreover, retaining the knowledge learned from pre-training when learning on the target task is also a non-trivial problem. This review summarizes the recent efforts to enhance NLG systems with unsupervised pre-training, with a special focus on the methods to catalyse the integration of pre-trained models into downstream tasks. They are classified into architecture-based methods and strategy-based methods, based on their way of handling the above obstacle. Discussions are also provided to give further insights into the relationship between these two lines of work, some informative empirical phenomenons, as well as some possible directions where future work can be devoted to.' author: - | Yuanxin Liu^1,2^, Zheng Lin^1^\ ^1^Institute of Information Engineering, Chinese Academy of Sciences\ ^2^School of Cyber Security, University of Chinese Academy of Sciences\ bibliography: - 'neurips\_2019.bib' title: 'Unsupervised Pre-training for Natural Language Generation: A Literature Review' --- Introduction {#sec:intro} ============ Unsupervised pre-training has sparked a sensational research interest in the natural language processing (NLP) community. This technology provides a promising way to exploit linguistic information from large-scale unlabelled textual data, which can serve as an auxiliary prior knowledge to benefit a wide range of NLP applications. In the literature, language modeling (LM) is a prevalent task for pre-training, where the target words are predicted conditioned on a given context. Therefore, it is intuitive to employ the pre-trained LMs for natural language generation, as the pre-training objective naturally accords with the goal of NLG. However, revolutionary improvements are only observed in the field of NLU. The primary factor that impedes the progress of unsupervised pre-training in NLG is an idiosyncratic nature of text generation: Basically, we do not write words from scratch, but instead based on particular context, e.g., the source language sentences for translation, the dialog histories for response generation, and the visual scenes for image captioning, among others. In unsupervised pre-training, the task-specific context is not available, which leads to a discrepancy between pre-training and training in the target task. More precisely, the challenges posed by the discrepancy can be reflected in two aspects: First, the diverse context makes it intractable to design a universal representation extractor as in the case of NLU, and the pre-trained language generators may have to modify their inner structures to deal with the task-specific context. Second, the mismatch in data distribution and objective between the two training stages might result in the performance on the pre-training tasks being compromised during fine-tuning, which is dubbed as the *catastrophic forgetting* problem [@Goodfellow13]. In response to the above challenges, two lines of work are proposed by resorting to architecture-based and strategy-based solutions, respectively. Architecture-based methods either try to induce task-specific architecture during pre-training (task-specific methods), or aim at building a general pre-training architecture to fit all downstream tasks (task-agnostic methods). Strategy-based methods depart from the pre-training stage, seeking to take advantage of the pre-trained models during the process of target task learning. The approaches include fine-tuning schedules that elaborately design the control of learning rates for optimization, proxy tasks that leverage labeled data to help the pre-trained model better fit the target data distribution, and knowledge distillation approaches that ditch the paradigm of initialization with pre-trained parameters by adopting the pre-trained model as a teacher network. The remainder of this review is organized as follows: In Section \[sec:background\], we will introduce the background knowledge about unsupervised pre-training for NLU, followed by a sketch of how the pre-trained models are employed through parameter initialization for NLG in Section \[sec:direct\]. In Section \[sec:architecture\], we will describe the architecture-based methods, and the strategy-based methods are presented in Section \[sec:strategy\]. Section \[sec:discussion\] provides some in-depth discussions, and Section \[sec:conclusion\] concludes this review. Background: Unsupervised Pre-training for NLU {#sec:background} ============================================= Learning fine-grained language representations is a perennial topic in natural language understanding. In restrospect, compelling evidences suggest that good representations can be learned through unsupervised pre-training. Early work focused on word-level representations [@MikolovSCCD13; @PenningtonSM14], which encodes each word independently. For sentence-level representations, there are roughly two kinds of pre-training objectives, namely discriminative pre-training and generative pre-training. Discriminative pre-training distinguishes context sentence(s) for a given sentence from non-context sentence(s) [@DevlinCLT19; @LogeswaranL18], with the aim to capture inter-sentence relationships. Generative pre-training follows the language model paradigm: $$\max_{\theta} \sum_{t=1}^{T} \log P\left(x_{t} | C; \theta\right)$$ where $x_{t}$ is the $t^{th}$ word in the textual sequence to generate, $T$ indicates sequence length, $\theta$ stands for learnable parameters, and $C$ is the context information, which is defined by the pre-training objective. ELMo [@PetersNIGCLZ18] and GPT (short for Generative Pre-training) [@Radford18] adopt uni-directional Transformer [@VaswaniSPUJGKP17] and bi-directional LSTM [@HochreiterS97] language models, respectively. In this case, the context is defined as $x_{1:t}$ or $x_{t+1:T}$. BERT [@DevlinCLT19] is trained with a novel masked language model (MLM), which is a non-autoregressive way of generation. Specifically, MLM randomly replaces a fixed proportion of tokens in each sentence with a special \[MASK\] token or a random token, which results in a corrupted sentence $X_{\text{mask}}$, and predicts each replaced token based on the same context $X_{\text{mask}}$. To alleviate the inconsistency with target tasks caused by the introduction of \[MASK\] token, XLNet [@Yangzhilin] introduces permutation-based language model, which conducts autoregressive language modeling over all possible permutations of the original word sequence. This gives rise to a context $C=X_{\mathbf{z}_{1:t-1}}$, where $\mathbf{z}$ is a certain permutation of $[1,2, \ldots, T]$, according to the definitions in [@Yangzhilin]. [@DaiL15] and [@KirosZSZUTF15] pre-trained an encoder-decoder framework to reconstruct the input sentence and the surrounding sentence, respectively, and the encoded input sentence thereby is included in the context $C$. The sentence representations learned by LMs [^1] can be used to perform many NLU tasks by adding a simple linear classifier. Despite the objective of language modeling, the pre-trained representations and have successfuly pushed the state-of-the-art on multiple benchmarks . Unsupervised Pre-training and Parameter Initialization for NLG {#sec:direct} ============================================================== NLG systems are usually built with an encoder-decoder framework, where the encoder reads the context information and the decoder generates the target text from the encoded vectorial representations. A direct way to utilize the pre-trained models is to initialize part of the encoder (when dealing with textual context) and/or the decoder with pre-trained parameters. For the encoder, pre-training is expected to provide better sentence representations, as we discussed in Section \[sec:background\]. For the decoder, the intuition is to endow the model with some rudimentary ability for text generation. [@LiuYang] employed BERT as the encoder for abstractive text summarization, with some additional techniques to help integrate the BERT-initialized encoder with the randomly initialized decoder, which we will explicate in Section \[fine-tune\]. GPT-2 [@Radford19] inherited the left-to-right LM pre-training objective from GPT and extended the application to NLG, where the pre-trained LM directly serves as the language generator, with some special symbols to identify task-specific contexts. In the case of zero-shot task transfer, preliminary experiments showed that straightforward adaption of GPT-2 compares unfavorably with other unsupervised baselines. [@RamachandranLL17] is among the first attempts to investigate unsupervised pre-training for sequence to sequence (Seq2Seq) learning. They used pre-trained LSTM-based LMs to initialize the first layer of the encoder and the decoder, which act as representation extractors. An additional LSTM layer, which is randomly initialized, is then added on top of the pre-trained LMs to build the Seq2Seq framework. To make use of the text generation ability of LMs, the output softmax layer of the decoder LM is also retained. Some recent endeavours [@Lample19; @Rothe191] explored multiple combinations of GPT- and BERT-based models to initialize the encoder and the decoder, respectively. Although remarkable results are observed, the separately pre-trained LMs are still inconsistent with the Seq2Seq framework. Architecture-based Methods {#sec:architecture} ========================== Inducing Task-Specific Architecture in Pre-training --------------------------------------------------- Separately initializing the encoder and the decoder with LMs neglects the interaction between the two modules at the pre-training stage, which is sub-optimal. For NLG tasks that can be modeled as Seq2Seq learning, it is feasible to jointly pre-train the encoder and the decoder. Existing approaches for this sake can be categorized into three variants: Denoising autoencoders (DAEs), conditional masked language models (CMLMs) and sequence to sequence language models (Seq2Seq LMs). {width="100.00000%"} ### Denoising Autoencoder An intuitive way to conduct unsupervised Seq2Seq learning is to train an autoencoder (AE) based on encoder-decoder framework. Different from AEs, DAEs take a corrupted sentence as input and reconstruct the original sentence. The advantage is that the corrupted input will force the decoder to extract relevant information from the source side for text generation. To obtain the corrupted sentence, [@Wang] designed three noising functions: *shuffle*, *delete* and *replace* (the left plot of Figure \[fig:structure\] gives an illustration), each of which is controlled by a pre-defined probability distribution. To be more specific, each token in the raw sequence is assigned with a new index based on a gaussion distribution $N(0, \sigma)$; the delete and replace operations of a token are determined by a Bernoulli distribution $B(p)$ with Beta distribution as prior. The three functions are applied to the raw sequences in random order. ### Conditional Masked Language Model CMLM [@SongTQLL19] extends the single model MLM proposed by [@DevlinCLT19] to the encoder-decoder setting, where the masked text sequence is read by the encoder, and the decoder only reconstructs the masked tokens, in construct to the entire sequence in DAEs. As the middle plot of Figure \[fig:structure\] shows, CMLM masks consecutive tokens [^2], and the unmasked tokens in the encoder side are masked when being feed to the decoder. Following the notations in [@SongTQLL19], let us assume that the tokens with index from $u$ to $v$ are masked from the raw sentence $X$, which results in $X^{\backslash u: v}$, and $X^{u: v}$ denotes the decoder input. Then, when predicting each masked token $x_{t}$ ($u \leq t \leq v$), the context is $X^{u: v}_{<t}$ and $X^{\backslash u: v}$. The underlying motivation, as [@SongTQLL19] argued, is to force the encoder to understand the meaning of the unmasked tokens, which is achieved by encoder side masks, and encourage the decoder to refer to the source information rather than the leftward target tokens, which is achieved by decoder side masks. ### Sequence to Sequence Language Model Seq2Seq LM [@Dong] performs Seq2Seq modeling using a single Transformer model, with the concatenation of source sentence and target sentence as input. To simulate Seq2Seq learning with encoder-decoder frameworks, the attention span of each target token is constrained to the source tokens and the leftward target tokens, which is achieved by self-attention masks (see the right plot of Figure \[fig:structure\]). In this way, the ability to extract language representation and generate texts are integrated into a single model. It is worth mentioning that Seq2Seq LM does not auto-regressively generate the target sentence, but instead predicting masked tokens based on the contexts controlled by self-attention masks. In other words, Seq2Seq LM still belongs into the family of MLMs. Apart from Seq2Seq LM, [@Dong] also explored uni-directional LM and bi-directional LM structures to perform the MLM-based cloze task, and incorporated the three kinds of LMs to build the final pre-training objective. Encoder-Agnostic Architectures for Adaptation --------------------------------------------- Although the Seq2Seq-based pre-training methods exhibit strong performance, they are confined to text-to-text generation. In order to encompass more diverse contexts, some researches began to investigate encoder-agnostic pre-training architectures [@Golovanov19; @Ziegler19]. *Context Attention* and *Pseudo Self-Attention* are two typical variants presented by [@Ziegler19], which differ in the way that the task-specific context is injected (see Figure \[fig:agnostic\]). Context Attention takes the form of a standard Transformer decoder, with the layer that attends to the encoder outputs being randomly initialized. Pseudo Self-Attention considers the context vectors and the previous layer decoder outputs as an integral input, and the attended results are computed as follows: $$\operatorname{PSA}(C, Y)=\operatorname{softmax}\left(\left(Y W_{q}\right)\left[\begin{array}{c}{C W^{c}_{k}} \\ {Y W^{y}_{k}}\end{array}\right]^{\top}\right)\left[\begin{array}{c}{C W^{c}_{v}} \\ {Y W^{y}_{v}}\end{array}\right]$$ where $C \in \mathbb{R}^{|C| \times d_{c}}$ and $Y \in \mathbb{R}^{|Y| \times d_{y}}$ are the context vectors and representations of the target textual sequence, respectively. The linear transformation matrices $W^{c}_{k}, W^{c}_{v} \in \mathbb{R}^{|C| \times d_{model}}$ with respect to $C$ are added to project the context to the self-attention space, and $W_{q}, W^{y}_{k}, W^{y}_{v} \in \mathbb{R}^{|Y| \times d_{model}}$ are part of the pre-trained model. Except for the performance on target tasks, an alternative metric to gauge the quality of encoder-agnostic architectures is the degree to which the pre-trained parameters have to change, in order to inject the task-specific context. [@Ziegler19] compared the parameter changes of Context Attention and Pseudo Self-Attention in the feed forward layer, and discovered that Pseudo Self-Attention is more robust under this evaluation. {width="100.00000%"} Strategy-based Methods {#sec:strategy} ====================== Fine-tuning Schedules for Adaption {#fine-tune} ---------------------------------- When the pre-trained model is only a part of the target task system, fine-tuning requires joint learning of the components initialized in different fashion, which can make the training process unstable. The pre-trained model may also suffer from aggravated catastrophic forgetting problem as it has to coordinate with other components during fine-tuning [@EdunovBA19; @Yangjiacheng]. From the perspective of optimization, it is unreasonable to schedule the pre-trained components and the newly-introduced components with the same learning rate, considering that the former have already possessed some unique knowledge. A common assumption is that the pre-trained parameters should be updated at a slower learning rate and with smoother decay [@LiuYang; @Yangjiacheng]. The rationale behind such setting is that fine-tuning with more accurate gradient can prevent the pre-trained parameters from deviating too faraway from the original point, and the newly-introduced components need to quickly converge to the target parameter space. To this end, [@LiuYang] adopted two Adam optimizers with different learning rates for the pre-trained encoder and the randomly initialized decoder. The learning rates are scheduled as in [@VaswaniSPUJGKP17] with different warming up steps: $$\begin{array}{l}{l r_{\operatorname{Enc}}=\tilde{l} r_{\operatorname{Enc}} \cdot \min \left(\operatorname{step}^{-0.5}, { step } \cdot { warmup }_{\operatorname{Enc}}^{-1.5}\right)} \\ {l r_{\operatorname{Dec}}=\tilde{l} r_{\operatorname{Dec}} \cdot \min \left({step}^{-0.5}, { step } \cdot { warmup }_{\operatorname{Dec}}^{-1.5}\right)}\end{array}$$ where ${warmup}_{\operatorname{Enc/Dec}}$ and $\tilde{l}r_{\operatorname{Enc/Dec}}$ determine the speed of learning rate changes and the max learning rates, respectively. Proxy Tasks for Adaption ------------------------ Large-scale unlabelled data provides generic linguistic knowledge, but the target tasks have unique data distribution and objectives. An effective way to bridge this gap is to introduce proxy tasks with moderate changes to the pre-training objectives, but at the same time take the labeled data into account [@Lample19; @Anonymous]. *Translation Language Modeling* (TLM) [@Lample19] is a special generalization of MLM in the cross-lingual situation. It leverages the paralleled machine translation corpus for further training of the LMs that are pre-trained on monolingual corpora. Specifically, the source language sentence and the corresponding target language sentence are fed to the model in parallel, with random tokens from each language being masked to perform the cloze-style prediction as in MLM. Different from monolingual MLM, TLM encourages word predictions to rely on the interdependence from two languages, therefore the sentence representations learned from separate languages can be well aligned. For some particular NLG tasks, existing proxy tasks designed under the supervised setup can also work with unsupervised pre-training models. For instance, in neural text summarization, the combination of extractive and abstractive[^3] objectives can generate better summaries [@LiXLW18; @GehrmannDR18]. Inspired by this, [@LiuYang] introduced extractive summarization as a proxy task to fine-tune the pre-trained BERT, before adopting it as the abstractive summarization encoder. Compared with the original BERT features, the representations learned from extractive summarization contain more task-specific information, therefore conveying the meaning of source texts better. Knowledge Distillation for Adaption ----------------------------------- The aforementioned methods are diverse in implementation, but share the common idea of employing the pre-trained models through parameter initialization. An alternative way to exploit the pre-trained models is using the knowledge distillation technique [@HintonVD15]. Knowledge distillation is a special form of training, where a *student* network learns from the supervision signals produced by a *teacher* network. Taking BERT as an example, the pre-trained MLM contains global information, which can teach the autoregressive Seq2Seq models to “see from the future” [@Anonymous]. In practice, the probability distribution predicted by BERT is regarded as a soft label to compute the cross-entropy loss function : $$\mathcal{L}^{cross}_{kd}(\theta)=-\sum_{t=1}^{|Y|}\sum_{w \in \mathcal{V}}\left[P(y_{t}=w | Y^{masked}, X ; \phi ) \cdot \log P\left(y_{t}=w | Y_{1: t-1}, X ; \theta \right)\right]$$ where $X$, $Y$ and $Y^{masked}$ are the source sequence, the raw target sequence and the masked target sequence, respectively. $\mathcal{V}$ denotes the output vocabulary. $\theta$ indicates the parameters of the student network (Seq2Seq), which are learnable, and $\phi$ indicates the BERT parameters, which are fixed. In this way, the knowledge from unsupervised pre-training can be flexibly transferred to the target tasks, dispensing with the size and architecture limitations. The supervision can also be derived from the hidden representations [@Yangjiacheng], with a mean-squared-error (MSE) distillation loss: $$\mathcal{L}^{mse}_{kd}=-\left\|{h}^{bert}_{m}-h^{seq2seq}_{n}\right\|_{2}^{2}$$ where $m$ and $n$ are hyper-parameters denoting the layer subscripts. Compared with the probability soft labels, the representation distillation method requires the Seq2Seq model to have the same hidden size with BERT, which is a more strict constrain. Combining the knowledge distillation loss and the standard generative loss for Seq2Seq learning gives rise to the final objective to optimize: $$\mathcal{L}(\theta)=\alpha \mathcal{L}_{kd}(\theta)+(1-\alpha) \mathcal{L}_{seq2seq}(\theta)$$ where $\alpha$ is the weighting term that balances the contribution of the two kinds of loss functions. Discussions {#sec:discussion} =========== The Relationship between Architecture- and Strategy-based Methods ----------------------------------------------------------------- We have analysed two major challenges faced by the application of unsupervised pre-training to NLG (see Section \[sec:intro\]). On this basis, we introduced existing methodologies from the architecture and strategy considerations. The architecture-based methods are mainly proposed in response to the first challenge. Since the architecture of pre-trained model has a significant effect on the downstream task (when the pre-trained parameters are used for initialization), architecture designings have to plan in advance to narrow the discrepancy between pre-training and training on target tasks. This motivation has shown great effectiveness on the Seq2Seq framework [@Wang; @SongTQLL19; @Dong]. The strategy-based methods focus on the second challenge. They take a postprocessing point of view, with the aim to make the best of the pre-trained model at the target task training stage. It is noteworthy that the challenges are not independent inherently, and the two types of methods can actually work as complement to each other. For example, the fine-tuning schedules can alleviate the negative effects caused by the modification of pre-trained structures, and the catastrophic forgetting problem can also seek solution by devising a general task-agnostic architecture. Experimental Phenomenons ------------------------ Existing researches on unsupervised pre-training for NLG are conducted on various tasks for different purposes. Probing into the assorted empirical results may help us discover some interesting phenomenons: - The advantage of pre-training gradually diminishes with the increase of labeled data [@RamachandranLL17; @Wang; @SongTQLL19]. - Fixed representations yield better results than fine-tuning in some cases [@EdunovBA19]. - Overall, pre-training the Seq2Seq encoder outperforms pre-training the decoder [@EdunovBA19; @Wang; @Lample19; @Rothe191]. The first two phenomenons attest to the catastrophic forgetting theory. Thanks to the access to large-scale unlabeled corpora, unsupervised pre-training is able to excel at zero/low-shot settings, while the pre-trained models can only achieve few gains when abundant labeled data is available. This can be explained by the high quality of the dataset and the capacity of the task-specific models, which leave little space for improvement. Nonetheless, the increased supervision from labeled data can also influence the performance on pre-training tasks. By fixing the pre-trained parameters, the learned representations will not be affected by the numerous iterations of training on the target task, which makes them work better without fine-tuning. The third phenomenon is kind of counter-intuitive, as the generative pre-training objectives are more similar to the decoder’s function. There is no unanimous theory to explain why the encoder is a more important element to pre-train. But this discovery suggests that the pre-trained LMs are more robust when acting as representation extractors, while they are more sensitive the the change of context when acting as conditional language generators. Future Directions ----------------- The diversity of NLG applications poses challenges on the employment of unsupervised pre-training, yet it also raises more scientific questions for us to explore. In terms of the future development of this technology, we emphasize the importance of answering four questions: 1) How to introduce unsupervised pre-training into NLG tasks with cross-modal context? 2) How to design a generic pre-training algorithm to fit a wide range of NLG tasks? 3) How to reduce the computing resources required for large-scale pre-training? 4) What aspect of knowledge do the pre-trained models provide for better language generation? NLG tasks can be defined by the context features and mapping functions. The introduction of cross-lingual textual features [@Lample19] and task-specific Seq2Seq architectures [@SongTQLL19; @Wang; @Dong] in the pre-training stage has successfully boosted the performance on text-to-text generation. For NLG tasks concerning multiple modalities, it is conceivable that pre-training methods could also benefit from the joint consideration of cross-modal features. For example, in the vision-and-language field, the learning of cross-modal representations has proven to be highly effective [@LiuFenglin; @LuJiasen], but such representations can not yet be extracted from unpaired images and texts for image-grounded text generation, to the best of our knowledge. In NLU, it is possible to pre-train one model to obtain language representations once and for all. As for NLG, a task-agnostic pre-training algorithm should transcend the purpose of representation learning, and consider the general ability for language generation. The notion of “encoder-agnostic adaption” [@Ziegler19] makes a preliminary step towards this direction, but still remains far from approaching the equivalent performance as its NLU counterparts [@PetersNIGCLZ18; @DevlinCLT19; @Radford18; @Yangzhilin]. Due to the colossal scale of the pre-training corpora, including a large number of parameters is essential to achieve favorable performance. As a result, the model size usually costs at least 8 GPU cards [@Dong; @SongTQLL19; @Lample19] in the pre-training for NLG systems, and it also hinders real-world applications. To reduce the memory consumption problem, existing work resorted to knowledge distillation to transfer the knowledge from a large teacher network to a small student network [@Sunsiqi; @Jiaoxiaoqi], or parameter reduction techniques to prune the model size in a more direct way [@LanZhenzhong]. However, the research context is limited to the NLU scenarios, and same endeavours are necessary to NLG applications. Another important branch of researches on unsupervised pre-training in NLP try to explain what kind of knowledge can be learned from pre-training. Related work has been done on the basis of both language understanding [@JawaharSS19; @Petroni] and generation [@Abigail]. Specially, [@Abigail] analysed the characters of texts generated from a pre-trained GPT-2 by evaluating them over a wide spectrum of metrics. We argue that deeper understanding the way in which unsupervised pre-training contributes to better text generation, and the intrinsic mechanisms of the pre-trained models are also crucial to future work. Conclusion {#sec:conclusion} ========== Unsupervised pre-training has defined the state-of-the-arts on a variety NLP tasks. However, in the field of NLG, the diversity of context information is still impeding the the application of unsupervised pre-training. The major challenges exist in designing model architectures to cater for the assorted context, and retaining the general knowledge learned from pre-training. In this review, we survey the recent unsupervised methods to utilize large-scale corpora for NLG purposes, with a highlight on those aiming at facilitating the integration of pre-trained models with downstream tasks. We propose to classify them into architecture- and strategy-based methods, followed with detailed introductions and discussions of their pros and cons. Based on the comparison of these methods and analyses of some informative experimental results from previous publications, we summarize some scientific questions that has not yet been well understood, and suggest attention being paid to these questions by future work. [^1]: In the rest of this review, we refer to unsupervised pre-training as the generative LMs, unless otherwise specified. [^2]: CMLM can also be implemented in other manners [@Anonymous; @Marjan], where discrete tokens in the textual sequence are masked. [^3]: Extractive text summarization selects sub-sequences from the source texts, while abstractive text summarization treats the task as a Seq2Seq learning problem.

--- abstract: 'The electronic structure of the ferromagnetic semiconductor EuO is investigated by means of spin- and angle-resolved photoemission spectroscopy and density functional theory (GGA+$U$). Our spin-resolved data reveals that, while the macroscopic magnetization of the sample vanishes at the Curie temperature, the exchange splitting of the O 2$p$ band persists up to $T_{C}$. Thus, we provide evidence for short-range magnetic order being present at the Curie temperature.' author: - Tristan Heider - Timm Gerber - Markus Eschbach - Ewa Mlynczak - 'Patrick L[ö]{}mker' - Pika Gospodaric - Mathias Gehlmann - 'Moritz Pl[ö]{}tzing' - 'Okan K[ö]{}ksal' - Rossitza Pentcheva - Lukasz Plucinski - 'Claus M. Schneider' - 'Martina M[ü]{}ller' title: | Does Exchange Splitting persist above $T_C$?\ A spin-resolved photoemission study of EuO --- Introduction ============ The monoxide of europium (EuO) belongs to the rare class of ferromagnetic semiconductors which makes it an attractive material for fundamental research in the field of spintronics [@Moodera2007; @Mueller2009; @Schmehl2007]. Only very recently, direct integration with silicon [@Averyanov2015; @Caspers2016] and with other relevant semiconductors has been achieved [@Swartz2010; @Swartz2012; @Gerber2016]. For applications such as spin-filter tunneling, the exchange spitting of the unoccupied Eu 5$d$6$s$ conduction band is of central importance. It has been reported that the exchange splitting of this band decreases with temperature and vanishes at $T_{C} = \SI{70}{K}$. This correlation explains well the $R(T)$ behavior of EuO tunnel junctions [@Moodera2007; @Mueller2009] as well as EuO’s metal-insulator transition (MIT) [@Steeneken2002b; @Mairoser2010]. Also, the red-shift of the EuO optical absorption edge [@Freiser1969] is in accordance with vanishing exchange splitting of the Eu-derived 5$d$6$s$ conduction band. Moreover, it was recently claimed that the exchange splitting of the O 2$p$ valence band in EuO also vanishes at the Curie temperature [@Miyazaki2009]. EuO is considered a textbook example for the Heisenberg model of localized magnetism [@Kasuya1970; @Mauger1986]. Within this context, the vanishing of the exchange splitting in the Eu 5$d$6$s$ conduction band and the O 2$p$ valence band which has been described above [@Moodera2007; @Mueller2009; @Steeneken2002b; @Mairoser2010; @Freiser1969; @Miyazaki2009] is rather surprising. Namely, for other metallic 4$f$ Heisenberg ferromagnets, such as Gd [@Maiti2002; @Andres2015], and Tb [@Teichmann2015], it has been reported that the exchange splitting is only weakly affected by temperature and may exist around or even above the Curie temperature. The observation of a persisting exchange splitting can be described, for instance, in terms of a band-mirroring model (also referred to as fluctuating band theory [@Capellmann1982; @Prange1979] or spin-mixing [@Maiti2002; @Andres2015]), in which thermally excited collective spin fluctuations reduce the net magnetization. In this scenario, short-range magnetic order may persist above the transition temperature resulting in differently oriented microscopic areas, so-called ’spin blocks’ which fluctuate in space and time [@Stoehr2006]. With the fluctuation frequency being orders of magnitude smaller than the typical time-scale of the experiment, and a spatial average over several spin blocks being typically measured, this short-range order is difficult to observe experimentally. However, it manifests itself in an exchange splitting which is less affected by temperature and which may exist around or even above the Curie temperature, although, the net magnetization vanishes. Band-mirroring is not exclusive to Heisenberg ferromagnets. Also for the elemental 3$d$ ferromagnets it has been reported that the exchange splitting may persist above the Curie temperature. Examples include Co [@Schneider1991; @Eich2016], Fe [@Pickel2010], and Ni [@Aebi1996]. The latter is a particularly interesting example, because Ni shows, both, temperature independent *and* collapsing exchange splittings. Thus, depending on the position of the particular band in $k$-space, the persistence of local moments and a Stoner-type behavior is observed at the same time. Spin-resolved photoemission has proven to be a powerful tool for resolving the electronic structure of magnetic systems [@Schneider1991; @Eich2016; @Maiti2002; @Andres2015]. Yet, we are only aware of a single report on spin-resolved photoemission on EuO: in an angle-integrated study it was reported that the spectral polarization is temperature dependent and vanishes at $T_C$ [@Lee2007]. Moreover, the same study reports that the splitting of the O 2$p$ majority and minority bands is inverted with the minority band being located at lower binding energy, and the majority band at higher binding energy, while one would expect exactly the opposite[@Ingle2008]. {width="\textwidth"} In this letter, we present a complete picture of the EuO valence band evolution through the paramagnetic-ferromagnetic phase transition. We report on spin- and angle-resolved photoemission spectroscopy (spinARPES) measurements, which reveal the relation between the macroscopic magnetic properties and the local electronic structure of a prototypical 4$f$ Heisenberg ferromagnet. Experimental Details ==================== EuO thin films were grown in an MBE system with a residual gas pressure $p < 2 \times 10^{-10}$ mbar on a Cu(001) single crystal substrate. The films are epitaxial with EuO(001)$||$Cu(001). The films’ thickness is $d=25$nm resulting in a bulk like electronic structure [@Prinz2016]. Stoichiometry of the EuO films was achieved by using the distillation method [@Ulbricht2008; @Sutarto2009; @Caspers2013] and was confirmed by in-situ XPS. High-resolution spin- and angle-resolved photoemission spectroscopy (spinARPES) measurements were carried out in another UHV chamber ($p < 1 \times 10^{-10}$mbar). Sample transfer was done using a transportable UHV ’shuttle’. The spectra were taken using non-monochromatized He I$\alpha$ resonance radiation with h$\nu=21.22$eV. The energy and angular resolution were 10 meV and $\le$0.4 $^{\circ}$, respectively. Spin-polarization of the photoelectrons was analyzed at k$_{\parallel}$ = 0 with a FERRUM spin detector [@Escher2011]. The films were magnetized in-plane along the EuO(100) direction. This is a hard magnetic axis of bulk EuO [@Pieper1995], but parallel to one of the quantization axes of the spin detector. All measurements were performed in remanence. First-principles DFT-based calculations were performed using the full-potential linearized augmented plane wave method as implemented in the WIEN2k code [@Blaha2001]. For the exchange-correlation functional we used the generalized gradient approximation (GGA) [@Perdew1996]. In order to correctly describe the relative positions of bands as measured in photoemission experiments, we have included static local electronic correlations to the GGA potential in the GGA+$U$ method [@Larson2006] with $U = 5.7$eV (Eu 4$f$) and $U = 3.4$eV (Eu 5$d$). Results ======= We start by analyzing the results of our spin-resolved photoemission measurements. Fig.\[ferrum\](a) shows the spin-resolved spectra for the sample in the ferromagnetic state at $T= \SI{40}{K}$. We observe two bands located at binding energies of 2eV and 5eV which can be assigned to the Eu 4$f$ and O 2$p$ bands, respectively. The O 2$p$ band shows a substructure which is qualitatively well reproduced by our DFT calculation as shown in Fig.\[dft\]. If one compares the majority (spin-up) and minority (spin-down) spectra, the O 2$p$ bands appear to be slightly shifted against each other. The Eu 4$f$ band shows a spin-polarization of 50% proving that the sample is ferromagnetic and remanently magnetized. In contrast to our theoretical calculation which shows 100% spin-polarization of the Eu 4$f$ band at $T= \SI{0}{K}$ (see Fig.\[dft\] and below), the polarization deduced from our experimental measurement is reduced due to the fact that the sample was measured in a remanent state. As the sample had been magnetized along a hard magnetic axis, domain formation is expected [@Muehlbauer2008], which reduces the projection of the spin-polarization onto the quantization axis of the spin detector. In addition to that the finite temperature ($T= \SI{40}{K}$) gives rise to spin waves which further suppress the polarization. Figs.\[ferrum\](a-d) show how the the spin-resolved spectra evolve with increasing temperature. Figs.\[ferrum\](e-h) compile the corresponding spin polarization. It is evident that the spin-polarization converges to 0 upon approaching $T_C= \SI {70}{K}$. As expected for a Heisenberg ferromagnet, the sample’s magnetization and, thus, the spin-polarization vanishes at the Curie temperature. ![Spin-resolved density of states as calculated by DFT for EuO in the ferromagnetic ground state ($T=\SI {0}{K}$). Upper/lower panel corresponds to majority/minority density of states, respectively.[]{data-label="dft"}](figure_dft5.pdf){width="0.82\columnwidth"} Fig.\[dft\] shows the calculated spin-resolved density of states as obtained by DFT+$U$ for the ferromagnetic ground state of EuO at $T=\SI{0}{K}$. The Eu 4$f$ band is 100% spin polarized as its counterpart is located ca. 9eV above the Fermi level. The O 2$p$ band is exchange split by $ \sim$0.25eV. Further, this band shows a sub-structure which can be attributed to the bonding ($\sim$4eV) and anti-bonding ($ \sim$3eV) O 2$p$ orbitals. All of these findings are in agreement with previous theoretical studies [@Ingle2008]. ![Simplified spin-resolved density of states for (a) the regular EuO domains magnetized along one direction (relative intensity $r=0.66$ ), (b) the mirrored domains magnetized along the opposite direction (relative intensity $m=0.33$ ), (c) superimposed density of states as seen by a spatially averaging photoemission experiment. Blue/red corresponds to majority/minority density of states, respectively.[]{data-label="mirroring"}](figure_mirroring_only3.pdf){width="0.82\columnwidth"} In order to describe the spin-resolved density of states at $T_C$ we assume the band-mirroring scenario: Due to the short-range magnetic order which is present around and above the transition temperature [@Maiti2002; @Andres2015; @Capellmann1982; @Prange1979] the sample decomposes into small spin-blocks, each of which has a magnetic moment of constant magnitude but fluctuating direction. The net magnetization of the sample vanishes because it sums over all spin blocks with their randomly oriented magnetic moments. However, the density of states in each of these microscopic areas consists of a 100% polarized Eu 4$f$ band and an exchange-split O 2$p$ band. In the photoemission experiment we therefore access a snapshot of the superposition of the fundamental spin-resolved density of states of the individual spin blocks. This model can also be used to extract the fundamental density of states of domains with opposite magnetization directions from spatially averaged spin-polarized photoemission spectra at temperatures below $T_C$ (compare Fig.\[ferrum\] (a-d)). In the temperature range $\SI {0}{K}<T_{\mathrm{exp}}<T_C$, the sample is in a ferromagnetic state but the magnetization is reduced due to magnetic domain formation and thermal excitations. Averaging over magnetic domains below $T_C$ would have a similar effect as averaging over spin blocks at (or above) $T_C$. To clarify how the superposition of fundamental densities of states is defined, Fig.\[mirroring\] illustrates the situation exemplarily: A sample may contain 66% domains magnetized along one direction (regular) and 33% domains magnetized along the opposite direction (mirrored), the resulting net sample magnetization is $M=0.33$. Fig.\[mirroring\] (a) and (b) show a simplified version of the fundamental density of states for the regular and the mirrored domain, respectively. Fig.\[mirroring\] (c) shows the superimposed density of states (as seen by a spatially averaging photoemission experiment) which exhibits a complex shape due to band mixing. This shape can be described analytically by the following set of equations: $$%Channel_{maj} = a \cdot Spectrum_{maj} + b \cdot Spectrum_{min} SP_{maj} = R_{maj} + M_{maj} = r \cdot GS_{maj} + m \cdot GS_{min}$$ $$%Channel_{min} = b \cdot Spectrum_{maj} + a \cdot Spectrum_{min} SP_{min} = R_{min} + M_{min} = r \cdot GS_{min} + m \cdot GS_{maj}$$ where $GS_{maj, min}$ are the spin-resolved DOS of the EuO ground state, $r$ and $m$ are the relative abundances of regular and mirrored domains (note that $r+m\equiv1$), $R_{maj, min}$ and $M_{maj, min}$ are the spinDOS of these domains, and $SP_{maj, min}$ are the spin-resolved DOS resulting from the superposition of $R$ and $M$. In order to derive the spectra of the regular domains, we can solve the system of linear equations (1) and (2) yielding: $$R_{maj} = (SP_{maj}-\frac{m}{r} SP_{min})\cdot(1-\frac{m^2}{r^2})^{-1} \label{eq3}$$ $$R_{min} = (SP_{min}-\frac{m}{r} SP_{maj})\cdot(1-\frac{m^2}{r^2})^{-1} \label{eq4}$$ For the case of domains with a magnetization vector M not parallel to the quantization axis Q of the spin-detector, the above reasoning still applies when the projections of M onto Q are taken into account. As we are able to determine $r$ and $m$ from the intensity of the Eu 4$f$ peak in each of the spin spectra shown in Figs.\[ferrum\](a-d), Eqs.\[eq3\] and \[eq4\] allow us to to retrieve the spin-polarized spectra of the regular domains *only*. We want to emphasize that it is not straightforward to apply this formalism to every other magnetic sample as the values for $r$ and $m$ are typically not known. Here, these values can be derived due to the fact that the Eu 4$f$ band is 100% polarized in the ground state. The resulting spectra of the regular domains are shown in the bottom panel of Fig.\[ferrum\]. We observe that the shape of the spin-up and spin-down bands is in very good agreement with the results our DFT calculations (compare Fig.\[dft\]). The O 2$p$ bands are shifted against each other by ca. 0.3eV. Thus, this representation of our data is able to reveal the exchange splitting of the oxygen valence band. The spectra of the regular domains show almost no temperature dependence. Most importantly, the exchange splitting of the O 2$p$ bands is unaffected. We note that the signal to noise ratio of the corrected spectra is reduced with increasing temperature. This is due to the fact that the difference between the spin polarized photoemission spectra (spin-up and spin-down) decreases with decreasing average sample magnetization, a fact which does not affect any of our conclusions. In order to quantify the exchange splitting of the O 2$p$ bands, a fitting procedure was applied to the regular domain spectra shown in Figs.\[ferrum\](i-k) (see Fig.S1 of the Supporting Information). We found that two fit components were necessary to reproduce the shape of the oxygen bands. Peak parameters were restrained to be the same for all spectra except for the positions of the peaks. The relative shift of the peaks in each pair of spin-resolved spectra gives us the exchange splitting \[shown as filled squares in Fig.\[fit\_results\](b)\], which we find to be temperature independent.The signal-to-noise ratio decreases from Fig.\[ferrum\](i) to (l) so that we chose not to evaluate the spectra in Fig.\[ferrum\](l).  (b) O 2$p$ exchange-splitting as obtained by fitting the spectra of the regular domains (black squares, see Fig.S1) and by fitting the O 2$p$ bands according to the band-mirroring model (open squares, see Fig.S2). The light grey lines are guides to the eye.[]{data-label="fit_results"}](FitResults_Tristan_ZoomIn.pdf){width="1.1\columnwidth"} The fitting yielded a total of four peaks (two in each spin-spectrum). We applied a fitting procedure based on these four peaks to the spatially integrated data (Fig.\[ferrum\](a-d), see Fig.S2 of the Supporting Information). As the peak positions were not restricted during this fitting procedure, another value for the exchange splitting of the O 2$p$ bands is obtained \[open squares in Fig.\[fit\_results\](b)\]. This value shows qualitative agreement with the exchange splitting obtained by fitting the O 2$p$ band in the regular domain spectra (filled squares). In addition, the relative intensity of the peaks in two spin channels of the Eu 4$f$ band provides a value for its spin-polarization \[black diamonds in Fig.\[fit\_results\](a)\]. These values show the reduction of spin polarization with increasing temperature and, thus, the vanishing of magnetization at the Curie temperature ). To summarize our spinARPES measurements: While the spin-polarization and, thus, net magnetization of the sample vanishes, we find clear evidence for the persistence of exchange splitting in the O 2$p$ bands *at the Curie temperature*. This observation is clear evidence for the existence of short range magnetic order likely due to spin-blocks. Discussion ========== We want to review our results in the context of existing literature. The only available study on spin-resolved photoemission on EuO reported on an inverted splitting of the O 2$p$ majority and minority bands [@Lee2007]. Here, we have shown the correct splitting, which is the minority band being located at higher binding energy, and the majority band at lower binding energy. This finding is in line with our DFT results as well as with other DFT studies [@Ingle2008]. How does our result of persisting exchange splitting in the O 2$p$ band relate to previous results? As mentioned in the introduction, a vanishing of the exchange splitting of the O 2$p$ at the Curie temperature was reported [@Miyazaki2009]. We note that the exchange splitting is roughly a factor of five smaller than the O 2$p$ bandwidth which hampers the observation of the splitting in any spin-integrated measurement. Therefore, the conclusions in ref.12 are based on the second derivative of energy distribution curves, a procedure which might overestimate certain features, while other features can disappear due to the lack of spin resolution. From the shape of the O 2$p$ valence band as shown in Fig.\[dft\] it is evident that the application of a 2nd derivative would not yield a meaningful result. As a last point, it was also reported that the exchange splitting of the Eu 5$d$6$s$ conduction band decreases with temperature and vanishes at $T_{C} = \SI{70}{K}$, and that this behavior is actually crucial to describe a number of phenomena i.e. the $R(T)$ behavior of EuO tunnel junctions [@Moodera2007; @Mueller2009], the metal-insulator transition (MIT) [@Steeneken2002b; @Mairoser2010], and the red-shift of the EuO optical absorption edge [@Freiser1969]. Thus, the Eu 5$d$6$s$ shows a completely different behavior than what we have observed for the O 2$p$ bands. We might only speculate on the origin of this contrasting behavior, but it is conceivable that the vanishing splitting of the Eu 5$d$ conduction band is related to the delocalized nature of this particular band, which is in stark contrast to the localized nature of the O 2$p$ band. In the light of our findings, it would be very interesting to revisit the exchange splitting in the Eu 5$d$6$s$ conduction band by methods that are able to probe unoccupied states directly, such as spin-resolved two-photon photoemission [@Pickel2010] or spin-resolved inverse-photoemission [@Stolwijk2013]. Summary ======= We were able to measure in situ spin-resolved photoemission spectra on epitaxial EuO thin films and investigated the evolution of the spin polarized density of states with increasing temperature. Using the band mirroring model enabled us to decompose the measured spin spectra into components corresponding to areas with magnetization directions parallel and antiparallel to the quantization axis of our spin detector. A closer look at the density of states in the parallel magnetized areas (regular domains) reveals that the exchange splitting of approximately 0.3eV in the O 2$p$ band of EuO persists *up to* $T_C$. This observation is clear evidence for the existance of short-range magnetic order being present *at* the Curie temperature and even above. Supporting Information ====================== In order to quantify the exchange splitting of the O 2$p$ bands, a fitting procedure was applied to the regular domain spectra shown in Figs.\[ferrum\](i-k) (see Fig.S1). We found that two fit components were necessary to reproduce the shape of the oxygen bands. Peak parameters were restrained to be the same for all spectra except for the positions of the peaks. The relative shift of the peaks in each pair of spin-resolved spectra gives us the exchange splitting \[shown as filled squares in Fig.\[fit\_results\](b)\], which we find to be temperature independent. For the measurement at the Curie temperature shown in Fig.\[ferrum\](d) the polarization of the sample is almost zero, so that $r \approx m$ and application of Eqs.\[eq3\] and \[eq4\] yields a very noisy result \[Fig.\[ferrum\](l)\] which we chose not to evaluate. The fitting yielded a total of four peaks (two in each spin-spectrum). We applied a fitting procedure based on these four peaks to the spatially integrated data (Fig.\[ferrum\](a-d), see Fig.S2). As the peak positions were not restricted during this fitting procedure, another value for the exchange splitting of the O 2$p$ bands is obtained \[open squares in Fig.\[fit\_results\](b)\]. This value shows qualitative agreement with the exchange splitting obtained by fitting the O 2$p$ band in the regular domain spectra (filled squares). In order to remove any doubts, we also tested a Stoner-model by trying to fit only two peaks (instead of four) to the data. Although we did not put any restraints on the exchange splitting parameter in order to account for its possible decrease, this approach is not able to reproduce the data. {width="76.00000%"} {width="104.00000%"} ![Band mixing model for the density of states as calculated by DFT. Mixing ratios are 90/10, 70/30, and 50/50 (from top to bottom). Regular domains are shown in light blue, mirrored domains in light red, and the superpositions are shown as thick blue and red lines, respectively.[]{data-label="mirroring_dft"}](band_mixing_DFT.pdf){width="0.82\columnwidth"}

--- abstract: 'We propose channel matrices by using unfolding matrices from their reduced density matrices. These channel matrices can be a criterion for a channel whether the channel can teleport or not any qubit state. We consider a special case, teleportation of the arbitrary two-qubit state by using the four-qubit channel. The four-qubit channel can only teleport if the rank of the related channel matrix is four.' author: - | Bayu D. Hatmoko, Agus Purwanto, Bintoro Subagyo and Rafika Rahmawati\    \ Department of Physics, Institut Teknologi Sepuluh Nopember\ Kampus ITS Sukolilo, Surabaya 60111,\ Indonesia title: 'The Criteria for Quantum Teleportation of an Arbitrary Two-Qubit State Information Based on The Channel Matrices' --- Introduction ============ Quantum teleportation is a transmission of the information by using previous shared entangled state and the classical channel between sender and receiver. Quantum teleportation protocol was proposed theoretically by Bennett et al., in 1993 [@b.1]. One qubit quantum state is transmitted from Alice to Bob via Einstein-Podolsky-Rosen (EPR)-states as quantum channels. Subsequently, Bouwmeester successfully to demonstrated the experimental of teleportation scheme by using photons as quantum information and channels [@b.2]. Then, both theoretical [@b.3; @b.4; @b.5; @b.6; @b.7; @b.8] and experimental [@b.9; @b.10] research takes much attention in this field. Initially, the one-qubit state was teleported via the channel of two-qubits entangled state. Further development was carried out by Karlsson [@b.11] transmitting one-qubit state via the three-qubit channel, i.e GHZ-state. Moreover, Karlsson proposed the third person. In the process, Alice sent one-qubit state to Cliff via Bob as an intermediary. The advantage of the scheme is Cliff as the receiver only apply unitary operations based on Bob’s measurement results. Different from Karlsson, Joo [@b.3] investigated two quantum teleportation schemes by utilizing the W-state as a quantum channel. As a result, Joo shows the success of teleporting an unknown qubit via the W-state depends on the type of measurement performed by Alice. However, the teleportation scheme proposed in the reference [@b.3; @b.11] only consider for sending the one-qubit state. Next, two-qubit quantum teleportation via four-qubit channels was proposed by Rigolin et al. [@b.12]. The arbitrary two-qubit states were successfully teleported by using sixteen orthogonal general states constructed from Bell’s states as quantum channels. On the other hands, Zha and Song [@b.13] expanded the four-qubit quantum channel not only considering the Bell-pair states but also non-bell-pair states as quantum channels. The quantum teleportation can occur if only if Bob’s transformation matrix can be determined through the “transformation operator”. Furthermore, Zha and Ren [@b.14] analyzed the relationship between “transformation operator” and an invariant under Stochastic Local Operation and Classical Communication (SLOCC). The other approach shows the allowed criteria for quantum teleportation depend on the channel matrices, the measurement matrices, and the collapsed matrices [@b.15]. The entanglement is the heart of the quantum teleportation and quantum information in general. For two-qubit state, we just have a simple case that is entangled or separable state. However, higher-qubit cases are more complicated. Dur et al. [@b.16] proposed the entanglement class constructed by local unitary operations and classical communication. In general, the determination of entanglement and separability of multipartite state cannot be represented as a direct product $\left| \psi \right\rangle = \left| \psi_1 \right\rangle \otimes |\left| \psi_2 \right\rangle \otimes \left| \psi_3 \right\rangle \otimes \cdots$. The separability of the state can be obtained by exploring the rank of the corresponding reduced density matrix. Purwanto et al. [@b.17] revealed if the rank of each single reduced density matrix of a multipartite state is one, then the state is completely separable. Contrary, if the rank of each single reduced density matrix is not equal to one, then the type of entangled state shall be determined by investigating various higher dimensional reduced density matrices. In order to teleport an arbitrary two-qubit state via the four-qubit channel, the rank and the entanglement of the channel matrix have to be considered [@b.13; @b.14; @b.15]. In this paper, the reduced density matrix [@b.17] are analyzed further to obtain the rank of the four-qubit entangled channel. The density matrix can be expressed as the multiplication of various channel matrices and their Hermitian conjugate. In fact, the channel matrices are nothing but the unfolding matrix [@b.18]. The organization of the paper is as follows. Section 2 describes how the double reduced density matrices can be re-expressed as the multiplication of the channel matrix and its Hermite conjugate. Section 3 demonstrates the implementation of the channel matrix analysis on the teleportation of an arbitrary two-qubit quantum state. The last section serves the conclusions of our work. The unfolding channel matrices for four single reduced density matrices present in the appendix. Channel matrices ================ The entanglement of multipartite has been studied by employing the rank of the density matrix [@b.17]. If the state of n-partite is completely decomposed, then the rank of every single reduced density matrix is one. Otherwise, if it is not equal to one, the state of n-partite is entangled. However, the entanglement is not necessarily perfect but possibly to be the combination of few entangled sub-states. In this article, we focus on the state of four-qubit as follows: $$\left| \phi\right>=\sum_{ijkl=0}^1 c_{ijkl} \left| ijkl\right>, \label{eq.1}$$ with the complex coefficient, $C_{ijkl}$ is satisfied by $|c_{0000} |^2+|c_{0001} |^2+\cdots+|c_{1111} |^2=1$. The density matrix is given by $$\rho =\left| \phi\right> \left< \phi\right|=\sum_{ijkl}^1\sum_{pqrs}^1 c_{ijkl} c_{pqrs}^{*} \left| ijkl\right> \left< pqrs\right| \label{eq.2}$$ which is a $16\times 16$ matrix with rank one. If the state of Eq. (\[eq.1\]) is completely not entangled, the rank of four single reduced density matrices is one. The rank of all higher reduced density matrices is also equal to one. Since this state cannot teleport, we do not discuss this state further. We are interested in the rank analysis of the entangled state of Eq. (\[eq.1\]). In such a situation, the rank of four single reduced density matrices is two. Although not specifically discussed, we show the single reduced density matrix in the Appendix. Based on the work of Purwanto et al. [@b.17], we investigate the rank of three double reduced density matrix, i.e., (AB), (AC), and (AD) sub-states. If the rank each of them are not equal to one, the state of Eq. (\[eq.1\]) is completely entangled. On the other hand, if the rank of one of the doubled reduced density matrices is one, then the state of Eq. (\[eq.1\]) contain the combination of two entangled states, AB-CD, AC-BD, or AD-BC respectively depend on which rank of the double reduced density matrix is equal to one. The AB doubled sub-states is defined as: $$\begin{aligned} \rho_{AB}&=\sum_{mn}^1 \left(I \otimes I \otimes \left< mn \right|\right) \rho \left(I \otimes I \otimes \left| mn \right> \right) \nonumber \\ &=\sum_{ijpq}^1\sum_{mn}^1 c_{ijmn} c_{pqmn}^* \left| ij\right> \left< pq \right|, \label{eq.3}\end{aligned}$$ which is the $4\times4$ matrix with the index of row $(ij)$ and column $(pq)$. If the rank of this matrix is one, the density matrix rank of its pair (CD) is also one and the state of Eq. (\[eq.1\]) is combined entangled AB-CD sub-states. However, if the rank is not equal to one, we cannot make any conclusion about the entanglement of state Eq. (\[eq.1\]) yet. Further investigation of reduced density matrix Eq. (\[eq.3\]) results $$\rho_{AB}=C_{AB} C_{AB}^\dagger, \label{eq.4}$$ where $C_{AB}$ is defined as a channel matrix (AB), $$C_{AB} = \left(\begin{array}{cccc} c_{0000}&c_{0001}&c_{0010}&c_{0011}\\ c_{0100}&c_{0101}&c_{0110}&c_{0111}\\ c_{1000}&c_{1001}&c_{1010}&c_{1011}\\ c_{1100}&c_{1101}&c_{1110}&c_{1111} \end{array}\right) \label{5}$$ From Eq. (\[eq.4\]), it is clear that the rank of the density matrix is equal to the rank of the channel matrix. Using this channel matrix, the rank of the density matrix becomes easier to be determined by just considering the channel matrix element. Two others double reduced density matrices are: $$\rho_{AC}=C_{AC} C_{AC}^\dagger$$ And $$\rho_{AD}=C_{AD} C_{AD}^\dagger$$ with AC and AD channel matrices as follows: $$C_{AC} = \left(\begin{array}{cccc} c_{0000}&c_{0001}&c_{0100}&c_{0101}\\ c_{0010}&c_{0011}&c_{0110}&c_{0111}\\ c_{1000}&c_{1001}&c_{1100}&c_{1101}\\ c_{1010}&c_{1011}&c_{1110}&c_{1111} \end{array}\right). \label{8}$$ $$C_{AD} = \left(\begin{array}{cccc} c_{0000}&c_{0010}&c_{0100}&c_{0110}\\ c_{0001}&c_{0011}&c_{0101}&c_{0111}\\ c_{1000}&c_{1010}&c_{1100}&c_{1110}\\ c_{1001}&c_{1011}&c_{1101}&c_{1111} \end{array}\right). \label{9}$$ Each of these three channel matrices has sixteen elements in the form of the component of $c_{ijkl}$ from the four-qubit state of Eq. (\[eq.1\]). The channel matrices in Eq. (\[5\]), (\[8\]) and (\[9\]) is arranged as usual matrix $C_{IJ}$ in such a way with row $I$-th and column $J$-th are (AB) and (CD), (AC) and (BD) as well as (AD) and (BC) respectively for channel matrices $C_{AB}$, $C_{AC}$ and $C_{AD}$.\ We consider some of the following examples. First, the GHZ-like state $$\left|\phi\right> =\frac{1}{\sqrt{2}} \left(\left| 0000\right> + \left| 1111\right>\right), \label{10}$$ then each of three-channel matrices has rank two and form $$C_{AB}=C_{AC}=C_{AD} = \frac{1}{\sqrt{2}} \left(\begin{array}{cccc} 1&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&1 \end{array}\right) \label{11}$$ It means that the state of Eq. (\[10\]) is completely entangled since none of the double reduced density matrices has rank one. Moreover, it can be verified easily using the unfolding matrix as given in the Appendix that the rank of all single reduced density matrices is equal to two.\ The second example is W-like state as given by $$\left| \phi \right> =\frac{1}{2} \left(\left| 0001\right> + \left| 0010 \right> + \left| 0100\right> + \left| 1000\right>\right), \label{12}$$ The sub-states corresponding to Eq. (\[12\]) then take the form $$C_{AB}=C_{AC}=C_{AD} = \frac{1}{2} \left(\begin{array}{cccc} 0&1&1&0\\ 1&0&0&0\\ 1&0&0&0\\ 0&0&0&0 \end{array}\right) \label{13}$$ One can see that the three-channel matrices have the same form and rank two. Then, the state Eq. (\[12\]) is completely entangled. Following the same argument for the state Eq. (\[10\]), the rank of all of its single reduced density matrices is two.\ The third example is the state $$\left| \phi \right> =\frac{1}{2} \left(\left| 0000 \right> +\left| 0011 \right> + \left| 1100\right> + \left| 1111\right> \right). \label{14}$$ Subsequently, the corresponding sub-states are $$C_{AB}=\frac{1}{2} \left(\begin{array}{cccc} 1&0&0&1\\ 0&0&0&0\\ 0&0&0&0\\ 1&0&0&1 \end{array}\right) \label{15}$$ $$C_{AC}=C_{AD} = \frac{1}{2} \left(\begin{array}{cccc} 1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1 \end{array}\right) \label{16}$$ The rank of (AB) channel matrix is equal to one while the rank of (AC) and (AD) channel matrix rank is equal to four. It means that the entangled state Eq. (\[14\]) is the combination of two entangled states, those are AB-CD, $$\left| \phi \right> =\frac{1}{\sqrt{2}} \left(\left| 00\right> +\left| 11 \right> \right) \frac{1}{\sqrt{2}} \left(\left| 00\right> +\left| 11 \right> \right). \label{17}$$ The fourth example is the state $$\left| \phi \right> = \frac{1}{2} \left( \left| 0000 \right> + \left| 0011 \right> + \left| 1100 \right> - \left| 1111 \right> \right). \label{18}$$ By comparing the forms of Eq. (\[14\]) and Eq. (18), as well as channel matrix in Eq. (\[15\]) and Eq. (\[16\]), it can be seen that the rank of (AB) channel matrix of the state (\[18\]) is two while the rank of (AC) and (AD) channel matrices are four. It means, different from the state (\[14\]), the state (\[18\]) is completely entangled and known as a cluster [@b.19; @b.20].\ The fifth example is the state $$\left| \phi \right> = \frac{1}{2} \left(\left| 0000 \right> + \left| 0101 \right> + \left| 1010 \right> + \left| 1111 \right> \right). \label{19}$$ Subsequently, the corresponding sub-states take the form $$C_{AB}=\frac{1}{2} \left(\begin{array}{cccc} 1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1 \end{array}\right) \label{20}$$ $$C_{AC}=\frac{1}{2} \left(\begin{array}{cccc} 1&0&0&1\\ 0&0&0&0\\ 0&0&0&0\\ 1&0&0&1 \end{array}\right) \label{21}$$ $$C_{AD}=\frac{1}{2} \left(\begin{array}{cccc} 1&0&0&0\\ 0&0&1&0\\ 0&1&0&0\\ 0&0&0&1 \end{array}\right) \label{22}$$ From these three channel matrices, one can obtain that the state of four-qubit (\[19\]) is AC-BD combined entangled state.\ The last example is a more elaborate state varied from the state of Eq. (\[19\]) as $$\begin{aligned} \left| \phi \right> &=\frac{1}{4} \left(\left| 0000\right> + \left| 0001 \right> +\left| 0010 \right> +\left| 0011 \right> \right. \nonumber\\ &+ \left| 0100\right> - \left| 0101 \right> +\left| 0110 \right> -\left| 0111\right> \nonumber\\ & + \left| 1000 \right> + \left| 1001 \right> - \left| 1010 \right> -\left| 1011\right> \nonumber\\ &\left.+\left| 1100 \right> - \left| 1101 \right> -\left| 1110\right> +\left| 1111\right> \right). \label{23}\end{aligned}$$ Subsequently, the corresponding sub-states yield $$C_{AB}=\frac{1}{4} \left(\begin{array}{rrrr} 1&1&1&1\\ 1&-1&1&-1\\ 1&1&-1&-1\\ 1&-1&-1&1 \end{array}\right) \label{24}$$ And $$C_{AC}=\frac{1}{2} \left(\begin{array}{rrrr} 1&1&1&-1\\ 1&1&1&-1\\ 1&1&1&-1\\ -1&-1&-1&1 \end{array}\right) \label{25}$$ $$C_{AD}=\frac{1}{2} \left(\begin{array}{rrrr} 1&1&1&1\\ 1&1&-1&-1\\ 1&-1&1&-1\\ 1&-1&-1&1 \end{array}\right) \label{26}$$ Further evaluation of the three-channel matrices gives the same rank four of (AB) and (AD), while the (AC) channel matrix of rank one. This implies that the state of Eq. (\[23\]) have the $AC-BD$ pattern of entanglement. Teleportation of Arbitrary Two-Qubit State ========================================== In this Section, we apply the rank of channel matrix analysis in the previous section on the teleportation of an arbitrary two-qubit state. We consider the state belongs to Alice is given by $$\left| \chi_A \right>_{12}=\left(x_o \left| 00 \right> +x_1 \left| 01 \right> +x_2 \left| 10 \right> +x_3 \left| 11 \right> \right)_{12}. \label{27}$$ As a channel, we adopt the state of four-qubit of Eq. (\[eq.1\]) with first two-qubit (34) belong to Alice and the other two-qubit (56) belong to Bob, $$\left| \phi \right>_{3456}=\sum_{ijkl}^1 c_{ijkl} \left| ijkl \right>_{3456}. \label{28}$$ There are two other possible qubit combinations belong to Alice, (35) and (36). The combination determines the suitable channel matrices. The main requirement for channel to be able to teleport is that it should be entangled and the entanglement should be between Alice’s qubit and Bob’s qubit, not between Alice’s qubit or Bob’s qubit itself. Furthermore, Alice join her state to the channel, get $$\left| \psi\right>_{123456}= \left| \chi_A \right>_{12} \left| \phi\right>_{3456}. \label{29}$$ Since the last (56) qubits belong to Bob, Alice’s measurement is applied to the first four qubits (1234) with the projection operator, $$\left| \mu\right>_{1324}=\sum_{ijkl=0}^1 m_{ijkl} \left| ijkl \right>_{1324}, \label{30}$$ with $\sum_{ijkl=0}^1 |m_{ijkl} |^2=1 $. The index (1324) means that the first two qubits of projection operator are projected on the first qubit (first information qubit) and the third qubit (Alice first canal qubit). In other hand, the next two qubits are projected on the second qubit (second information qubit) and the fourth qubit (Alice forth channel qubit). For instance, Alice qubit is (35) then the projection operator become $\left| \mu\right>_{1325}$\ Following, Alice perform the measurement of Eq. (\[29\]) by Eq. (\[30\]) yield, $${}_{1324}\left< \mu | \psi\right>_{123456}\equiv \left| \chi \right>_{56}= \sigma_B^{-1} \left| \chi_B \right>_{56} \label{31}$$ The ket $\left| \chi\right>_{56}$ and $\left| \chi_B \right>_{56}$ is Alice’s measurement result and the Bob’s received information respectively. The $\sigma_B^{-1}$ is Bob’s transformation. This equation shows that the teleportation occurs if there is an invertible classical transformation matrix of Bob, $\sigma_B$. For the measurement based on the Bell and non-Bell state \[13\], the invertible $\sigma_B$ can be obtained when the channel matrix has rank four. On the other words, teleportation can be held if the rank of the channel matrix is four. Now, we consider the qubit structure of the channel Eq. (\[27\]). The qubits (34) belong to Alice and the qubits (56) belong to Bob. In such a configuration, the channel matrix (AB) will determine whether the channel can teleport or not. The teleportation of arbitrary two-qubits information via four-qubits channel can only occur if rank $C_{AB}$ is equal to four. The $C_{AC}$ channel matrix is used to replace $C_{AB}$ when the (35) channel qubit belongs to Alice and the other two channel qubit belong to Bob. The $C_{AD}$ is used in the remaining configuration. For the measurement based on the state of Bell, $$\left| \mu \right>_{1324}^{ij}=\left(\left| \beta^i \right> \left| \beta^j \right> \right)_{1324} \label{32}$$ with $$\left| \beta^{1,2} \right>=\frac{1}{\sqrt{2}} \left(\left| 00 \right> \pm \left| 11 \right> \right), \label{33}$$ $$\left| \beta^{3,4} \right>=\frac{1}{\sqrt{2}} \left(\left| 01 \right> \pm \left| 10 \right> \right). \label{44}$$ We apply these measurements to the channels state mentioned in section 2. First, we consider the channel of Eq. (\[10\]) and Eq. (\[12\]). In this case, the rank of all the channel matrices $C_{AB}$, $C_{AC}$ and $C_{AD}$ are not equal to four but two. Then, both of the channels cannot teleport the state of Eq. (\[27\]) for all measurement. In other word, no invertible $\sigma_B$ satisfy Eq. (\[31\]). Second, the channel of Eq. (\[14\]) has rank one of $C_{AB}$. However, the rank of $C_{AC}$ and $C_{AD}$ are equal to four. Since Alice’s channel qubit is the first and the second qubit then the appropriate channel matrix is $C_{AB}$. The rank of $C_{AB}$ is not equal to four then the channel can not teleport the state of Eq. (\[27\]) even though the rank of $C_{AC}$ and $C_{AD}$ are equal to four. Third, the channel of Eq. (\[19\]) can teleport since the rank of $C_{AB}$ is equal to four. In Principe, we can obtain invertible $\sigma_B$ for all measurement. For example, the measurement $\left| \mu \right>_{1324}^{13}$, using Eq. (\[31\]) yield the rank four of the channel matrix $C_{AB}$ with $$\sigma_B = 4 \left( \begin{array}{cccc} 0&1&0&0\\ 1&0&0&0\\ 0&0&0&1\\ 0&0&1&0\\ \end{array}\right) \label{35}$$ Finally, similar to the previous case, the rank of $C_{AB}$ from the state of Eq. (\[23\]) is also four. For projection operator $\left| \mu \right>_{1324}^{11}$, we obtain the invertible $\sigma_B$, $$\sigma_B = 2 \left( \begin{array}{rrrr} 1&1&1&1\\ 1&-1&1&-1\\ 1&1&-1&-1\\ 1&-1&-1&1\\ \end{array}\right) \label{36}$$ For the measurement based on the non-Bell state [@b.13]: $$\left| \phi\right> =\frac{1}{2} \left( \left| 0000 \right> + \left| 0101 \right> + \left| 1011 \right> + \left| 1110 \right> \right), \label{37}$$ then, we apply the above measurement of Eq. (\[37\]) to the channel state in Eq. (\[19\]) to obtain $$\sigma_B = 4 \left( \begin{array}{cccc} 1&0&0&0\\ 0&1&0&0\\ 0&0&0&1\\ 0&0&1&0\\ \end{array}\right)$$ Furthermore, applying Eq. (\[37\]) to channel state in Eq. (\[23\]), we get: $$\sigma_B = 2 \left( \begin{array}{rrrr} 1&1&1&1\\ 1&-1&1&-1\\ 1&-1&-1&1\\ 1&1&-1&-1\\ \end{array}\right)$$ For the measurement based on non-Bell state in Eq. (\[37\]), and channel state in Eq. (\[19\]) and (\[23\]), it is clear that the teleportation is success. Conclusion and Outlook ====================== In this work, we investigate the suitable channel to teleport using the rank of the channel matrix using Bell states as a measurement. The single reduced density matrix of the four-qubit channel state can be expressed in the $2\times8$ unfolding matrices multiplication. The elements of the matrices are in the form of channel state coefficient. In particular, employing the unfolding matrix to the channel state, the rank of single reduced density matrix can be determined easily by regarding the component of channel matrices without any calculation. Moreover, the double reduced density matrices can be expressed by multiplication of $4\times 4$ channel unfolding like matrices. The quantum teleportation of an arbitrary two-qubit state is available if and only if the appropriate $4\times4$ channel unfolding like matrix has rank four. The method proposed in this work is easier than previous work by Zha [@b.13; @b.14; @b.15], and we suggest it can be applied in more qubit states, for instance, in [@b.6] and [@b.7]. Acknowledgement {#acknowledgement .unnumbered} =============== This work is supported partially by the Ministry of Research, Technology and Higher Education, Indonesia and BDH thanks to Dr. Lila Yuwana for a fruitful discussion. Appendix {#appendix .unnumbered} ======== A single reduced density matrix of Eq. (\[eq.2\]) is defined by $$\begin{aligned} \rho_A =\sum_{mnt}^1\left(I\otimes \left< mnt\right> \right) \rho \left(I \otimes \left| mnt \right>\right) \nonumber\\ =\sum_{ip}^1 \sum_{mnt}^1 c_{imnt} c_{pmnt}^{*} \left| i \right> \left< p \right| \label{41}\end{aligned}$$ In a more explicit form, $\rho_A$ is given by $$\begin{aligned} \rho_A &= \left(\begin{array}{cc} \sum_{mnt}^1 c_{0mnt}c_{0mnt}^{*}&\sum_{mnt}^1 c_{0mnt}c_{1mnt}^{*}\\ \sum_{mnt}^1 c_{1mnt}c_{0mnt}^{*}&\sum_{mnt}^1 c_{1mnt}c_{1mnt}^{*}\\ \end{array}\right) \nonumber\\ &= C_A C_A^\dagger\end{aligned}$$ With $$\label{A3} C_A = \left( \begin{array}{cccccccc} c_{0000}&c_{0001}&c_{0010}&c_{0011}&c_{0100}&c_{0101}&c_{0110}&c_{0111}\\ c_{1000}&c_{1001}&c_{1010}&c_{1011}&c_{1100}&c_{1101}&c_{1110}&c_{1111} \end{array}\right)\\$$ In a similar manner, the other three single reduced density matrices result $$\begin{aligned} \rho_B = C_B C_B^\dagger\\ \rho_C = C_C C_C^\dagger\\ \rho_D = C_D C_D^\dagger\end{aligned}$$ with $$\label{A7} C_B = \left( \begin{array}{cccccccc} c_{0000}&c_{1000}&c_{0001}&c_{1001}&c_{0010}&c_{1010}&c_{0011}&c_{1011}\\ c_{0100}&c_{1100}&c_{0101}&c_{1101}&c_{0110}&c_{1110}&c_{0111}&c_{1111} \end{array}\right)\\$$ $$\label{A8} C_C = \left( \begin{array}{cccccccc} c_{0000}&c_{0100}&c_{1000}&c_{1100}&c_{0001}&c_{0101}&c_{1001}&c_{1101}\\ c_{0010}&c_{0110}&c_{1010}&c_{1110}&c_{0011}&c_{0111}&c_{1011}&c_{1111} \end{array}\right)$$ $$\label{A9} C_D = \left( \begin{array}{cccccccc} c_{0000}&c_{0010}&c_{0100}&c_{0110}&c_{1000}&c_{1010}&c_{1100}&c_{1110}\\ c_{0001}&c_{0011}&c_{0101}&c_{0111}&c_{1001}&c_{1011}&c_{1101}&c_{1111} \end{array}\right)$$ These four matrices $C_A$, $C_B$, $C_C$ and $C_D$ are nothing but unfolding matrices [@b.18]. For example, applying to the channel state of Eq. (\[10\]), we obtain $$\begin{aligned} C_A = C_B = C_C = C_D &\nonumber\\ &= \frac{1}{\sqrt{2}} \left( \begin{array}{cccccccc} 1&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&1 \end{array} \right) \label{50}\end{aligned}$$ with rank two. For channel state of Eq. (\[8\]), we have $$\begin{aligned} C_A = C_B = C_C = C_D&\nonumber\\ &= \frac{1}{2} \left( \begin{array}{cccccccc} 0&1&1&0&1&0&0&0\\ 1&0&0&0&0&0&0&0 \end{array} \right)\label{51}\end{aligned}$$ with rank two. We can know the rank of the channel matrices of single reduced density matrix only by easily seeing the component in the matrix of Eq. (\[50\]-\[51\]). In the state of Eq. (\[14\]), we have $$\begin{aligned} C_A = C_C = \frac{1}{2} \left( \begin{array}{cccccccc} 1&0&0&1&0&0&0&0\\ 0&0&0&0&1&0&0&1 \end{array} \right) \label{52}\end{aligned}$$ $$\begin{aligned} C_B = C_D = \frac{1}{2} \left( \begin{array}{cccccccc} 1&0&0&0&0&0&1&0\\ 0&1&0&0&1&0&0&1 \end{array} \right) \label{53}\end{aligned}$$ Similarly, by seeing at a glance the Eq. (\[52\]-\[53\]), we know the rank of the channel matrices are equal to two. Below is the example of non-entangled state, $$\left| \phi \right> =\frac{1}{2} \left(\left| 0001 \right> + \left| 0011 \right> + \left| 0101 \right> + \left| 0111 \right> \right) \label{54}$$ Then $$C_A = \frac{1}{2} \left( \begin{array}{cccccccc} 0&1&0&1&0&1&0&1\\ 0&0&0&0&0&0&0&0 \end{array}\right)$$ $$C_B = \frac{1}{2} \left( \begin{array}{cccccccc} 0&0&1&0&0&0&1&0\\ 0&0&1&0&0&0&1&0 \end{array}\right)$$ $$C_C = \frac{1}{2} \left( \begin{array}{cccccccc} 0&0&0&0&1&1&0&0\\ 0&0&0&0&1&1&0&0 \end{array}\right)$$ $$C_D = \frac{1}{2} \left( \begin{array}{cccccccc} 0&0&0&0&0&0&0&0\\ 1&1&1&1&0&0&0&0 \end{array}\right)$$ with each of channel matrices rank is equal to one. The state of Eq. (\[54\]) is decomposed into $$\left| \phi\right> = \left| 0 \right> \frac{1}{\sqrt{2}} \left(\left| 0 \right> + \left| 1 \right> \right) \frac{1}{\sqrt{2}} \left(\left| 0 \right> + \left| 1 \right> \right) \left| 1 \right>.$$ [99]{} Bennett, C. H., Brassard, G., Crépeau, C., Jozsa, R., Peres, A. & Wootters, W. K., [*Phys. Rev. Lett.*]{}, 70(13) 1895. Bouwmeester, D., Pan, J. W., Mattle, K., Eibl, M., Weinfurter, H., & Zeilinger, A., [*Nature.*]{}, 390 (6660) 1997 575-579. Joo, J., Park, Y. J., Oh, S., & Kim, J., [*New Journal of Physics.*]{}, 5(1) 2003 136. Liu, J. M., Feng, X. L., & Oh, C. H., [*EPL (Europhysics Letters).*]{}, 87(3) 2009 30006. Zhan, Y. B. [*EPL (Europhysics Letters).*]{}, 98(4) 2012 40005. Zha, X. W., Zou, Z. C., Qi, J. X., & Song, H. Y. [*International Journal of Theoretical Physics.*]{}, 52(6) 2013 1740-1744. Zhou, R. G., Qian, C., & Ian, H., [*IEEE Access*]{}, 7 2019 42445-42449. Zhou, R. G., Qian, C., & Ian, H., [*International Journal of Theoretical Physics.*]{}, 58(1) 2019 150-156. Kim, Y. H., Kulik, S. P., & Shih, Y., [*Physical Review Letters*]{}, 86(7) 2001 1370. Furusawa, A., Sørensen, J. L., Braunstein, S. L., Fuchs, C. A., Kimble, H. J., & Polzik, E. S., [*Science.*]{}, 282 (5389) 1998 706-709. Karlsson, A., & Bourennane, M. [*Physical Review A.*]{}, 58(6) 1998 4394. Rigolin, G. [*Physical Review A.*]{}, 71(3) 2005 032303. Zha, X. W., & Song, H. Y. [*Physics Letters A.*]{}, 369(5-6) 2007 377-379. Zha, X. W., & Ren, K. F [*Physical Review A.*]{}, 77(1) 2008 014306. Zha, X. W., Qi, J. X., & Song, H. Y. arXiv:[1402.1204]{}\[quant-ph\] Dür, W., Vidal, G., & Cirac, J. I.[*Physical Review A.*]{}, 62(6) 2000 062314. Purwanto, A., Sukamto, H., & Yuwana, L. [*International Journal of Theoretical Physics*]{} 57(8) 2018 2426-2436. De Lathauwer, L., De Moor, B., & Vandewalle, J. [*SIAM journal on Matrix Analysis and Applications*]{}21(4) 2000 1253-1278. Raussendorf, R., & Briegel, H. J. [*Physical Review Letters.*]{}, 86(22) 2001 5188. Tang, L., Chen, Z., & Chen, Z. B. arXiv:0812.4915 \[quant-ph\].

--- author: - 'Yuwei Fan[^1],   Julian Koellermeier[^2],   Jun Li[^3],   Ruo Li[^4],   Manuel Torrilhon[^5]  ' bibliography: - 'article.bib' title: Model Reduction of Kinetic Equations by Operator Projection --- [^1]: School of Mathematical Sciences, Peking University, Beijing, China, email: [ywfan@pku.edu.cn]{}. [^2]: Center for Computational Engineering Science, RWTH Aachen University, Aachen, Germany, email: [koellermeier@mathcces.rwth-aachen.de]{} [^3]: School of Mathematical Sciences, Peking University, Beijing, China, email: [lijun609@pku.edu.cn]{}. [^4]: CAPT, LMAM & School of Mathematical Sciences, Peking University, Beijing, China, email: [rli@math.pku.edu.cn]{}. [^5]: Center for Computational Engineering Science, RWTH Aachen University, Aachen, Germany, email: [mt@mathcces.rwth-aachen.de]{}

--- abstract: 'Solitons are studied in a model of a fiber Bragg grating (BG) whose local reflectivity is subjected to periodic modulation. The superlattice opens an infinite number of new bandgaps in the model’s spectrum. Averaging and numerical continuation methods show that each gap gives rise to gap solitons (GSs), including asymmetric and double-humped ones, which are not present without the superlattice.Computation of stability eigenvalues and direct simulation reveal the existence of *completely stable* families of fundamental GSs filling the new gaps – also at negative frequencies, where the ordinary GSs are unstable. Moving stable GSs with positive and negative effective mass are found too.' author: - Kazuyuki Yagasaki - 'Ilya M. Merhasin' - 'Boris A. Malomed' - Thomas Wagenknecht - 'Alan R. Champneys' title: Gap solitons in Bragg gratings with a harmonic superlattice --- Bragg gratings (BGs) are distributed reflecting structures produced by periodic variation of the refractive index of an optical fiber or waveguide. Devices based on fiber gratings, such as dispersion compensators, sensors and filters, are widely used in optical systems [@Kashyap]. Gap solitons (GSs) in fiber gratings are supported (in the temporal domain) by the balance between BG-induced linear dispersion, which includes a bandgap in the system’s spectrum, and the Kerr nonlinearity of the fiber material. Analytical solutions for BG solitons in the standard model of the fiber grating, based on coupled-mode equations for the right- and left-traveling waves, are well known [@AW-CJ:89]. Solitons with positive frequency $\omega $ are stable, while those with $\omega <0$ have an instability to perturbations that is characterized by a complex growth rate [@BPZ:98]. Solitons in fiber gratings have been created experimentally, with spatial and temporal widths of the order of a few mm and$~50$ ps, respectively [@experiment]. Spatial GSs were observed in photorefractive media with an induced photonic lattice [@Segev], and in waveguide arrays [@Silberberg]. (Indeed, the new families of GSs reported in the present work may be realized in the spatial domain too, in addition to their straightforward implementation as temporal solitons in fiber gratings). Besides their relevance to optical media, GSs were also predicted [@BEC] and created [@Oberthaler] in a Bose-Einstein condensate (BEC) trapped in a periodic potential. An issue of particular technological importance is the development of methods for the control of BG solitons – in particular, using *apodization* [@experiment; @express; @WMak], i.e.  gradual variation of the grating’s reflectivity along the fiber. In an appropriately apodized BG, one can slow down solitons and, eventually, bring them to a halt [@WMak]. Experimentally, it has been demonstrated that apodization helps to couple solitons into the grating [@experiment]. Moreover, technologies are available that allow one to fabricate fiber gratings with *periodic* apodization, thus creating an effective *superlattice* built on top of the BG [@Russell]. An asymptotic analysis of light propagation in such a *superstructure grating* was developed in Ref. [@SSG]. It was shown that the superstructure gives rise to extra gaps in the system’s spectrum (“Rowland ghost gaps"). Solitons in the gaps were sought by assuming that the soliton is a slowly varying envelope of the superlattice’s Bloch function, which applies to the description of GSs near bandgap edges. Related problems were considered in Refs. [@Louis], which treat BEC models with a doubly periodic optical lattice that opens up an additional narrow “mini-gap", in which stable solitons may be found. The subject of this Letter is the investigation of GSs in harmonic (sinusoidal) superlattices created on top of the ordinary BG in fibers with Kerr nonlinearity. The analysis of the bandgap structure in this model constitutes, by itself, an interesting extension of the classical spectral theory for the Mathieu equation [@Mathieu]. We will demonstrate that various families of GSs exist in the superlattice. Most importantly, in each newly opened gap we find a family of fundamental symmetric GSs, which fill the entire gap and are *completely stable*. Remarkably, they are stable not only at positive frequencies $\omega $, but also at $\omega <0$, where the ordinary GSs are unstable [@BPZ:98]. The superlattice may be implemented via the creation of beatings in the optical interference pattern that burns the grating into the fiber’s cladding. A model corresponding to this physical situation may be derived using standard coupled-mode equations for the amplitudes $u$ and $v$ of the right- and left-traveling electromagnetic waves. Inclusion of both Kerr nonlinearity and Bragg reflection terms leads to the following dimensionless equations: $$\begin{array}{rcl} iu_{t}+iu_{x}+\left[ 1-{\varepsilon}\cos \left( kx\right) \right] v+\mu \cos \left( kx+\delta \right) ~u+\left( |v|^{2}+|u|^{2}/2\right) u & = & 0, \\ iv_{t}-iv_{x}+\left[ 1-{\varepsilon}\cos \left( kx\right) \right] u+\mu \cos \left( kx+\delta \right) ~v+\left( |u|^{2}+|v|^{2}/2\right) v & = & 0.\end{array} \label{e:uv}$$ Here ${\varepsilon}$ is the amplitude of the periodic modulation of the BG strength, while the nonlinearity coefficient and average reflectivity are normalized, as usual, to be $1$. Accordingly, $k$ measures the ratio of the modulation period $L$ to the BG reflection length $l$ (usually, $L$ $\lesssim 1$ cm [@Russell; @SSG] and $l\sim 1$ mm, while the underlying BG period is $\sim 1$ $\mu $m, whereas the total length of the grating may be up to $1$ m). We have included additional terms in Eqs. (\[e:uv\]) with amplitude $\mu $ and phase shift $\delta $ to describe another possible control mechanism for optical pulses in BGs [@Kashyap; @Sterke], namely a periodic *chirp* of the BG, i.e. a local variation of the grating’s period. In this work we focus on the reflectivity superlattice, setting $\mu =0$; the general situation including the periodic chirp will be considered elsewhere. Thus ${\varepsilon}$ and $k$ are the parameters of the model, which we assume to be positive without loss of generality. According to the above physical considerations, typically ${\varepsilon}$ is small, but $k$ is not. Note that we should only expect a quiescent GS to be stable in this kind of nonuniform BG if the position of the soliton’s center is located at a local minimum of the reflectivity [@WMak], i.e. at $x=2\pi n/k$ for some integer $n$. Hence we shall restrict our numerical search for GSs to just such a case. Lastly, note that Eqs. (\[e:uv\]) conserve two dynamical invariants, the Hamiltonian and norm (usually called *energy* in fiber optics), $E=\int_{-\infty }^{+\infty} \left( |u(x)|^{2}+|v(x)|^{2}\right) dx$. We shall seek stationary GS in the form $\left\{ u(x,t),v(x,t)\right\} =\exp (-i\omega t)\left\{ U(x),V(x)\right\}$, substitution of which into Eqs. (\[e:uv\]) yields $$\begin{array}{rcl} \omega U+iU^{\prime }+\left[ 1-{\varepsilon}\cos (kx)\right] V+\left( |V|^{2}+|U|^{2}/2\right) U & = & 0, \\ \omega V-iV^{\prime }+\left[ 1-{\varepsilon}\cos (kx)\right] U+\left( |U|^{2}+|V|^{2}/2\right) V & = & 0,\end{array} \label{e:uvans}$$ where prime stands for $d/dx$. First, it is necessary to find bandgaps in the system’s spectrum. In the unperturbed problem (${\varepsilon}=0)$, the dispersion relation for linear waves with a propagation constant $q$ is $\omega ^{2}=q^{2}+1$, thus producing the well-known bandgap, $-1<\omega <1$. The spectrum for ${\varepsilon}>0$ should be found from the linearization of Eqs. (\[e:uvans\]). As in the Mathieu equation, gaps in these equations emerge due to the parametric resonance caused by the cosine modulation. Straightforward considerations show that new bandgaps open up at points $\omega =\omega _{m}\equiv \mathrm{sgn}(m)\sqrt{1+(mk)^{2}/4}$, with $m=\pm 1,\pm 2,\ldots $ (the extra modulation terms $\sim \mu $ in Eqs. (\[e:uv\] ) open gaps at the same positions). We will designate these gaps as $\mathbf{\ m}^{\pm }$. Using perturbation theory, one can find their boundaries for small ${\varepsilon}$ and demonstrate that their widths scale as ${\varepsilon}^{|m|} $. In particular, gaps $\mathbf{1}^{\pm}$ are found in the region $|\chi |\leq 1$, $\chi \equiv 2\left( \omega -\omega _{\pm 1}\right) /{\varepsilon}$, to leading-order in ${\varepsilon}$. Equations (\[e:uvans\]) allow the invariant reduction $V=-U^{\ast }$, which reduces them to a single equation, $$\omega U+iU^{\prime }-\left[ 1-{\varepsilon}\cos (kx)\right] U^{\ast }+(3/2)|U|^{2}U=0 \label{e:unl}$$ (the reduction with $V=U^{\ast }$ leads to an equation with the opposite sign in front of $U^{\ast }$, which can be cast back in the form of Eq. (\[e:unl\]) by the substitution $U\equiv i\tilde{U} $; neither reduction is valid for moving solitons, see below). The linearization of Eq. (\[e:unl\]) is sufficient for the analysis of the spectrum. Bandgap regions in the $(\omega ,{\varepsilon})$-parameter plane were computed using the software package AUTO with driver HomMap [@Auto]. We detected points at which the linearization of the Poincaré map [@GuHo:83] around the origin ($U=0$) has eigenvalues $\pm 1$, and then continued such points to identify the bandgap boundaries. The results are displayed in Fig. \[f:lingap\], which shows the first five gaps. In this and subsequent examples, we set $k=1$, as this value adequately represents the generic situation and is physically meaningful for the application to fiber gratings. In the central gap, which we designate $\mathbf{0}$, standard calculations using Melnikov’s method [@GuHo:83] reveal that the GS solutions, which are known in an exact  form for ${\varepsilon}=0$ [@AW-CJ:89], extend to ${\varepsilon}>0$. They were continued numerically up to ${\varepsilon}=1$, and were found to *completely fill* the central gap. That is, a GS exists for each $\{\omega,{\varepsilon}\}$-value belonging to the gap. We have also computed eigenvalues that determine stability of these solutions against small perturbations, and found that only GSs with $\omega >0$ are stable. That is, the border between stable and unstable GSs in the central gap for ${\varepsilon}>0 $ is found to remain, up to numerical accuracy, at $\omega =0$ as it is for ${\varepsilon}=0$ [@BPZ:98]. Inside gaps $\mathbf{1^{\pm }}$, one can use the method of averaging [@GuHo:83] to demonstrate the existence of GSs for small ${\varepsilon}$. To this end, we represent a solution to Eq. (\[e:unl\]), $U(x)\equiv a(x)+ib(x)$, as $$\left( \begin{array}{c} a \\ b\end{array} \right) =\sqrt{{\varepsilon}}\exp \left( \left[ \begin{array}{cc} 0 & -(\omega _{\pm 1}+1) \\ \omega _{\pm 1}-1 & 0\end{array} \right] x\right) \left( \begin{array}{c} \Xi /k \\ \Theta /\left( 2\left( \omega _{\pm 1}+1\right) \right)\end{array} \right) . \label{e:trans}$$ With constant amplitudes $\Xi $ and $\Theta $, we recover a solution to the linearized equation (\[e:unl\]) with ${\varepsilon}=0$ and $\omega =\omega _{\pm }$. The averaging method supposes that $\Xi $ and $\Theta $ are functions of a slow coordinate, $z\equiv x/\left[ 2k(\omega _{\pm 1}+1) \right] $, which leads to equations $$\frac{d}{dz}\left( \begin{array}{c} \Xi \\ \Theta\end{array} \right) =\left( \begin{array}{c} \alpha (1-\chi )\Theta -\beta \left( \Xi ^{2}+\Theta ^{2}\right) \Theta \\ \alpha (1+\chi )\Xi +\beta \left( \Xi ^{2}+\Theta ^{2}\right) \Xi\end{array} \right) , \label{e:av}$$ where $\alpha \equiv {\varepsilon}\left[ k^{2}+4(\omega _{\pm 1}+1)^{2}\right]$, and $\beta \equiv 3{\varepsilon}\lbrack 3k^{4}+8(\omega _{\pm 1}+1)^{2}k^{2}+48(\omega _{\pm 1}+1)^{4}]/[8k(\omega _{\pm 1}+1)]^{2}$. These equations conserve their Hamiltonian, $H=\alpha \left[ (1+\chi )\Xi ^{2}-(1-\chi )\Theta ^{2}\right] +(\beta /2)\left( \Xi ^{2}+\Theta ^{2}\right) ^{2}$. As gaps $\mathbf{1}^{\pm }$ exist for $|\chi |<1$, the coefficients $1\mp \chi $ in Eqs. (\[e:av\]) and in $H$ are positive. Therefore, the origin $(\Xi ,\Theta )=(0,0)$ is a saddle in Eqs. (\[e:av\]), and a pair of homoclinic orbits to this saddle can be found in the exact form $$\begin{aligned} (\tilde{\Xi}_{\pm }(t),\tilde{\Theta}_{\pm }(t))=& \left( \pm 2\sqrt{\frac{ \alpha }{3\beta }}\sin \theta \sin \frac{\theta }{2}\,\frac{\sinh (\alpha t\sin \theta )}{\cosh (2\alpha t\sin \theta )+\cos \theta },\right. \\ & \qquad \left. \mp 2\sqrt{\frac{\alpha }{3\beta }}\sin \theta \cos \frac{ \theta }{2}\,\frac{\cosh (\alpha t\sin \theta )}{\cosh (2\alpha t\sin \theta )+\cos \theta }\right) ,\end{aligned}$$ with $\chi \equiv \cos \theta $. On application of the transformation (\[e:trans\]), these orbits correspond to solitons in the full system (\[e:unl\]). In particular, there are solutions with $a(x)=\mathrm{Re}\,U(x)$ odd and $b(x)= \mathrm{Im}\,U(x)$ even. Thus, GSs exist in the entire gaps $\mathbf{1^{\pm } }$ for sufficiently small ${\varepsilon}$. A similar analysis can be performed for higher bandgaps, $\mathbf{2^{\pm }}$ , $\mathbf{3^{\pm }}$, etc., using a higher-order averaging method [@Y-YK:99]. The respective analytical computations show that each averaged system again generates solitons. They have either even real and odd imaginary parts or vice versa, depending on whether the gap’s number $m$ is odd or even. We employed the AUTO driver HomMap [@Auto] to continue numerical soliton solutions of the full system (\[e:unl\]), varying $\omega $ and ${\varepsilon}$ . In so doing, a multitude of symmetric and *asymmetric* families of GS solutions, including higher-order ones (bound states) were found in each new gap. Here, we display numerical results for fundamental solitons, as their bound states are likely to be unstable. It was found that, as predicted above, each gap $\mathbf{1^{\pm }}$, $\mathbf{2^{\pm }}$, etc. is completely filled with a single family of symmetric GSs. The families in gaps $\mathbf{1}^{\pm }$ are represented by solitons displayed in Figs. \[f:avdisc\](a$^{\pm }$) and (b$^{\pm }$), for small and larger ${\varepsilon}$, respectively, while Figs. \[f:avdisc\](c$^{\pm }$) displays the fundamental GSs in gaps $\mathbf{2}^{\pm }$. Note that stable fundamental solitons may feature a *double-humped structure* in terms of $|U(x)|$ at relatively large ${\varepsilon}$, which is the case for the soliton in Fig. \[f:avdisc\](b$^{+}$ ), see Fig. \[fig:stable\](b) below (however, the $|U(x)|$ profile of the GS displayed, for the same ${\varepsilon}$ but opposite $\omega $, in panel \[f:avdisc\](b$^{-}$), remains single-humped, see Fig. \[fig:stable\](a)). The homoclinic orbits of averaged system (\[e:av\]), also shown in a) and c), provide a good match to the envelopes of the numerical solutions. \ a) $\omega =\mp 1.118$, ${\varepsilon}=0.045$, $k=1$ (points $\times$ in Fig.  \[f:lingap\])\ \ b) $\omega =\mp 1.118$, ${\varepsilon}=0.4$, $k=1$ (points $\triangledown$ in Fig. \[f:lingap\])\ \ c) $\omega=\mp 1.42$, ${\varepsilon}=0.4$, $k=1$ (points $\times \hspace*{-1.9ex} + $ in Fig. \[f:lingap\]) Recall that the ordinary GSs in gap $\mathbf{0}$ with ${\varepsilon}=0$ cannot be asymmetric, and they do not form bound states either [@Sterke]. To explain where the asymmetric and higher-order GSs for ${\varepsilon}>0$ come from, we note that the new symmetric fundamental GSs, found above in gaps $\mathbf{1}^{\pm }$, correspond to transversal intersections of stable and unstable manifolds in the Poincaré map associated with Eq. (\[e:unl\] ). Such intersections naturally occur in pairs, one representing a symmetric soliton and the other one its asymmetric counterpart. Moreover, the transversality of the intersection implies the presence of a Smale horseshoe [@GuHo:83], which, in turn, ensures the existence of infinitely many higher-order GSs, that may be realized as bound states of symmetric or asymmetric fundamental solitons. Stability of the GSs was tested by direct simulations of Eqs. (\[e:uv\]), and verified through computation of the eigenvalues of the equations linearized around a GS, which govern the growth rate of small perturbations. Using this approach, it was found that the *entire* families of the symmetric fundamental solitons that fill the gaps $\mathbf{1^{+}}$ and $\mathbf{1^{-}}$ are stable. The significance of this result is that we can have *stable GSs *with $\omega <0$, something, that is not possible in the ordinary BG model without the periodic modulation terms. This finding is illustrated in Fig. \[fig:stable\](a), taking as an example the soliton with $\omega=-1.118$ depicted in Fig. \[f:avdisc\](b$^{-}$). Regarding asymmetric solitons in gaps $\mathbf{m^{\pm }}$ with $m\geq 1$, it was found that some of them are stable and some are not, the unstable ones being completely destroyed by growing perturbations (not shown here). An important issue for practical application is the possibility of finding moving GSs. (In experiments on ordinary fiber gratings, only moving solitons have so far been observed [@experiment]). In the presence of the superlattice, soliton mobility is a nontrivial problem, because the GS has to move in a periodically nonuniform medium. (Recall, though, that solitons belonging to gaps $\mathbf{m}^{\pm }$ with $m\geq 1$ do not exist at all without the superlattice). We have used numerical simulation to test for mobility of the stable GSs we have so-far found. Specifically, a quiescent stable soliton was multiplied by $\exp (ipx)$, with $p>0$, which implies a sudden application of a “shove" to the soliton, giving it momentum $P=i\int_{-\infty }^{+\infty }\left( uu_{x}^{\ast }+vv_{x}^{\ast }\right) dx+\mathrm{c.c.}\equiv 2pE$, where $E$ is the soliton’s energy defined above. It was observed that GSs belonging to central gap $\mathbf{0}$ are readily set in stable motion by the shove. This accords with the situation for ${\varepsilon}=0$, where exact solutions for moving GSs exist with any velocity $c $ from $-1<c<+1$ [@AW-CJ:89]. In our simulations, we found that no soliton in any gap could be made to move with velocity exceeding $1$ (a very strong shove does not make the GS “superluminal", but simply destroys it). For the stable GS belonging to gap $\mathbf{1}^{+}$, it was found that the application of the shove caused the soliton to split into a quiescent part and a moving part; see Fig. \[fig:stable\](b). A remarkable observation from these results is that the moving soliton (which retains $\simeq 2/3$ of the initial energy) has a negative average velocity, $c\approx -0.9$, hence its effective mass $M\sim P/c$ is *negative* too. (Simulations of the moving GSs belonging to gap $\mathbf{0}$ have positive mass, as is the case for ${\varepsilon}=0$ [@WMak]). We remark that the mass of a GS may be negative too in one- and two-dimensional models of BEC in an ordinary optical lattice, and that this gives rise to nontrivial effects such as stable confinement of solitons in an *anti-trapping* external potential [@HS]. Finally, the shove applied to a stable GS belonging to gap $\mathbf{1}^{-}$ was found to split into three solitons, one remaining quiescent while the other two move off in opposite directions (not shown here). Detailed results for moving solitons will be reported elsewhere. \ (a) \ (b) In conclusion, we have investigated a model for a Bragg grating optical fibre with a superlattice written on top. First, we identified a system of new bandgaps in the fiber’s spectrum, using an extension of the bandgap theory for the Mathieu equation. Combining averaging methods and numerical continuation, we have found that each new gap is completely filled with fundamental symmetric solitons, which at some parameter values may have a double-humped shape. In addition, new types of gap solitons (GSs) were found that do not exist without the superlattice, such as asymmetric GSs and bound states of fundamental GSs. An important finding is that the entire families of fundamental GSs in the new bandgaps are stable, including negative frequencies, where ordinary GSs are unstable. Stable moving solitons were found too, including ones with a negative mass. Finally, it is pertinent to point out that creation of the newly predicted GSs ought to be perfectly feasible using presently available experimental techniques. In particular, in a weak superlattice, with say, ${\varepsilon}\simeq 0.1$, an estimate shows that the solitons in the new bandgaps $\mathbf{1}^{\pm}$ can be observed if the fiber grating, on top of which the superlattice is to be imposed, is of length $\simeq 10$ cm. K.Y. acknowledges support from the Japan Society for the Promotion of Science for his stay at the University of Bristol. I.M.M. and B.A.M. appreciate support from the Israel Science Foundation, through the Center-of-Excellence grant No. 8006/03, and from EPSRC, UK. They thank the Department of Engineering Mathematics at the University of Bristol for its hospitality. T.W. acknowledges EPSRC support. [99]{} . . ; . ; . , , 1627 (1996); . , ; , . . ; , . , . , . ; , . . . ; ; , , in press (article PLA15176). 3rd. ed. . , , available at `http://cmvl.cs.concordia.ca/auto/`; . . ; . .

--- abstract: 'An experimental setup is being developed to extract Ba ions from a high-pressure Xe gas environment. It aims to transport Ba ions from 10bar Xe to vacuum conditions. The setup utilizes a converging-diverging nozzle in combination with a radio-frequency (RF) funnel to move Ba ions into vacuum through the pressure drop of several orders of magnitude. This technique is intended to be used in a future multi-ton detector investigating double-beta decay in $^{136}$Xe. Efficient extraction and detection of Ba ions, the decay product of Xe, would allow for a background-free measurement of the $^{136}$Xe double-beta decay.' address: - 'Dept. of Physics, Stanford University, Stanford, CA, USA' - 'Institute for Theoretical and Experimental Physics, Moscow, Russia' - 'Facility for Antiproton and Ion Research in Europe (FAIR), Darmstadt, Germany ' - 'Dept. of Physics, Carleton University, Ottawa, ON, Canada' - 'TRIUMF, Vancouver, BC, Canada' author: - 'T. Brunner' - 'D. Fudenberg' - 'A. Sabourov' - 'V.L. Varentsov' - 'G. Gratta' - 'D. Sinclair' - for the EXO collaboration bibliography: - 'bibliography.bib' title: 'Ba-ion extraction from a high pressure Xe gas for double-beta decay studies with EXO' --- RF confinement,Double-beta decay,RF funnel,EXO experiment Introduction {#intro} ============ Several double-beta ($\beta\beta$) decay experiments are currently trying to determine the nature of neutrinos. An unambiguous observation of the lepton-number violating zero-neutrino $\beta\beta$ decay would require neutrinos to be Majorana particles. Furthermore, if the main contribution to this process is the exchange of light Majorana neutrinos one can extract the effective Majorana neutrino mass from the half life of the decay. However, half lives greater than $10^{25}$years pose great experimental challenges. In order to reach the sensitivities needed to observe such decays requires a significant increase in detector mass and observation time relative to existing experiments, while minimizing naturally occurring radioactive backgrounds. Conventional detectors are limited by natural backgrounds thus the reachable sensitivity to the effective Majorana neutrino mass scales as $\left(N\ t\right)^{-1/4}$ with $N$ being the number of mother nuclei and $t$ being the observation time. In the case of a background-free experiment this situation is improved and the sensitivity scales with $\left(N\ t\right)^{-1/2}$ [@Dan00]. Of all the isotopes under consideration for $\beta\beta$-decay studies $^{136}$Xe is the only one that allows one to extract and identify the daughter of the decay immediately after the decay occurred [@Moe91]: $$^{136}Xe \rightarrow\ ^{136}Ba^{++}+2e^-+\left(2\ \textnormal{or}\ 0\right)\ \bar{\nu_e}.$$ A positive identification of the decay product $^{136}$Ba$^{++}$ allows the differentiation between $\beta\beta$ decays and natural background and hence a background-free measurement. EXO-200, one of the experiments investigating the $\beta\beta$ decay of $^{136}$Xe, first determined the half life of $2\nu\beta\beta$ [@Ack11] and later published a limit for the half life of $0\nu\beta\beta$ in $^{136}$Xe [@Aug12]. In parallel to the operation of EXO-200 the collaboration is developing techniques to tag, i.e., extract from the volume and identify, Ba ions in both liquid and gaseous Xe (gXe). This work presents the concept and status of Ba-tagging endeavors in gXe, focusing on the extraction of Ba ions from a high-pressure Xe environment. ![Calculated Xe-gas velocity flow field of the converging-diverging nozzle in combination with the electrode stack.[]{data-label="fig:Simulation"}](Simulation){width=".45\textwidth"} The general concept for a multi-ton gXe detector consists of a time-projection chamber (TPC) operated at 10bar enriched $^{136}$Xe. This records the energy and the position of events occurring in the active detector volume by conventional TPC techniques. If an event occurs within an energy window around $Q_{\beta\beta}(^{136}\textnormal{Xe})$, the electric field inside the TPC is modified to move ions from the decay volume to an exit port where they are flushed out of the TPC in a continuous flow of Xe gas. This Xe gas, carrying the extracted ions, is injected into a RF funnel through a converging-diverging nozzle. In the RF funnel, Xe gas is removed through spaces in the funnel’s electrode stack while Ba ions are confined by the applied RF field. Following the RF funnel Ba ions are transported to a downstream chamber filled with triethylamine (TEA). There, Ba$^{++}$ is converted to Ba$^{+}$ [@Sin11]. After the charge exchange process, the Ba ions are captured in a linear Paul trap [@Fla07] where they are identified by means of laser spectroscopy [@Gre07]. Ba$^+$ extraction from Xe gas - a prototype =========================================== At Stanford a setup has been developed to operate a 10bar natural Xe-gas jet with the ability to recycle Xe gas. The setup consists of the 10bar Xe chamber (referred to as chamber A) that is connected to a second vacuum chamber (referred to as chamber B) through a nozzle. The second vacuum vessel has a diameter of about 603 mm and a cryo pump mounted in it. During gas-jet operation, Xe freezes to the cryo pump. At gas flow rates of 0.5g/s a base pressure in the mbar range is present in the chamber B. At this flow rate 1.5kg of Xe allows the operation of the jet at 10bar stagnation pressure in chamber A for about 30 min. Afterwards, the Xe-gas storage bottles are submerged in LN$_2$ while the cryo pump is warmed up. The Xe gas condenses in the storage gas bottles for reuse. A SAES purifier installed along the Xe supply line cleans contaminants from the Xe gas each time the gas jet is operated. Concept of an RF funnel ----------------------- An ion beam extraction system for gas cells based on the RF funnel has been suggested in 2001 [@Var01]. The operation of this system is described in details elsewhere (see e.g. Refs. [@Var02; @Var04]). The RF-only ion funnel device, which consists of a stack of 301 electrodes, is placed on the converging-diverging nozzle axis in immediate vicinity of the nozzle exit plane. Ba ions are injected into the funnel via the supersonic Xe-gas jet. An RF voltage is applied with alternating phases delivered to neighboring electrodes. The funnel here works as a very good filter because the most part of the Xe-gas is evacuated through the gaps between funnel electrodes by pumping while the repelling force, which is generated by the RF field near the inner surface of the funnel electrodes and directed axially inwards, confines the Ba ions inside the funnel. At the same time the Xe-gas flow inside the funnel is strong enough to rapidly and very efficiently transport the Ba ions through the funnel into high vacuum conditions, eliminating the need for an additional electro-static drag potential. The geometry has been optimized in Monte-Carlo simulations for the operation in Stanford’s Xe-gas recirculation setup described above. A simulated gas flow map is presented in Fig.\[fig:Simulation\]. At an RF frequency of $f=2.6$MHz and an amplitude of V$_{\textnormal{PP}}$= 50V the simulated transport efficiency is 72.0(7)%. Higher frequencies and amplitudes show simulated efficiencies of more than 95%. {width=".75\textwidth"} The RF funnel ------------- Based on the RF-funnel simulations a mechanical setup has been designed and built that mounts inside chamber B. A section view of the model is presented in Fig.\[fig:whole-concept\]. The RF funnel is placed at the center of chamber B above the cryo pump. Xe gas is injected through the nozzle that is mounted on the high pressure chamber A. Downstream of the funnel an ion guide transports the ions through a differential pumping section before they are detected with an ion detector. This is an intermediate step to test the performance of the funnel. The converging-diverging nozzle was machined in a special CF2.75$''$ flange through electrode-spark discharge machining[^1]. The half-angles of subsonic and supersonic part are $45^{\circ}$ and $26.6^{\circ}$, respectively. The lengths of subsonic and supersonic part are 0.5mm and 15.5mm, respectively. The throat diameter is 0.3mm and the exit diameter is 16.0mm. The 301 funnel electrodes were photo-etched[^2] in a 0.1016 ($\pm$0.0025)mm thick stainless steel sheet[^3] with manufacturing tolerances better than 0.025mm [@Eng12]. The outer diameter of each ring electrode is 28mm. Each electrode has an aperture etched at the center. The aperture diameter decreases from 16.0mm to 1mm in steps of 0.05mm per electrode. Three mounting latches with holes are equally spaced on the electrode. Each electrode has its number etched into it to simplify the assembly process. A picture of electrode \# 295 is shown in Fig.\[fig:electrode\]. The electrodes were held in place in the metal sheet by three small ’legs’. For the installation these legs were cut close to the electrode. The electrodes were stacked alternating between two sets of three mounting rods. Both sets are electrically insulated. Within one stack the electrodes are spaced by 0.6096(+0.0051/-0.0102)mm thick stainless steel spacers[^4] that were photo-etched. This results in a distance of 0.25mm between the faces of two neighboring electrodes in the stack. The RF funnel stack was fixed to the nozzle, which was then mounted on the downstream vacuum chamber by three 3/8$''$ threaded rods. The funnel was mounted in such a way that the exit aperture of the funnel is concentric with the entrance hole of the downstream vacuum chamber. The distance between electrode \#301 and downstream vacuum chamber is 0.2388mm. A picture of high-pressure chamber, nozzle and funnel mounted on the downstream chamber is presented in Fig.\[fig:FunnelPic\]. The capacitance of this assembly outside of chamber B is 6.055nF, resulting in a resonance frequency close to 2.6MHz. At this frequency amplitudes up to 80V$_{PP}$ were successfully applied to the funnel. In a separate test, N$_2$ gas of up to 12bar was applied in chamber A, resulting in a gas jet on the funnel. During testing no electrical short between the stacks was observed due to electrode movement. Outlook ======= An RF-funnel setup to extract Ba$^{+(+)}$ from a 10bar Xe gas jet is currently being designed and constructed. The RF-funnel is fully assembled and installed inside chamber B. The initial RF tests will be repeated with and without Xe gas jet operation. A Gd driven Ba-ion source [@Mon10] is currently being fabricated. This source will be placed right at the nozzle as shown in Fig.\[fig:whole-concept\]. For the downstream ion guide a sextupole-ion guide is under development. The geometry of this guide is being optimized in SimIon simulations [@Dah00]. First ion extractions are planned for summer 2013. Acknowledgments =============== The Ba-tagging in Xe gas project is funded by NSF. We thank K. Merkle, J. Kirk, and R. Conley for their help in machining parts. We also thank M. Brodeur, T. Dickel, J. Dilling, M. Good, R. Ringle, P. Schury, S. Schwarz and K. Skarpaas for fruitful discussions and their support of this project. ![Photo of the RF funnel installed in its final configuration. The high-pressure Xe will be injected from the right side, pass through the nozzle, then the RF funnel, and proceed into the vacuum chamber on the left, where the downstream ion guide will be installed.[]{data-label="fig:FunnelPic"}](FunnelPic){width=".45\textwidth"} [^1]: EDN Labs Ltd., www.edmlabsltd.com [^2]: Newcut Inc. New York [^3]: 305mm x 305mm alloy 316, ESPI Metals [^4]: 203mm x 305mm alloy 316, ESPI Metals

--- abstract: | The model parameters of convolutional neural networks (CNNs) are determined by backpropagation (BP). In this work, we propose an interpretable feedforward (FF) design without any BP as a reference. The FF design adopts a data-centric approach. It derives network parameters of the current layer based on data statistics from the output of the previous layer in a one-pass manner. To construct convolutional layers, we develop a new signal transform, called the Saab ([**S**]{}ubspace [**a**]{}pproximation with [**a**]{}djusted [**b**]{}ias) transform. It is a variant of the principal component analysis (PCA) with an added bias vector to annihilate activation’s nonlinearity. Multiple Saab transforms in cascade yield multiple convolutional layers. As to fully-connected (FC) layers, we construct them using a cascade of multi-stage linear least squared regressors (LSRs). The classification and robustness (against adversarial attacks) performances of BP- and FF-designed CNNs applied to the MNIST and the CIFAR-10 datasets are compared. Finally, we comment on the relationship between BP and FF designs. Interpretable machine learning, convolutional neural networks, principal component analysis, linear least-squared regression, cross entropy, dimension reduction. address: 'University of Southern California, Los Angeles, CA 90089-2564, USA' author: - 'C.-C. Jay Kuo, Min Zhang, Siyang Li, Jiali Duan and Yueru Chen' bibliography: - 'CNN2.bib' title: Interpretable Convolutional Neural Networks via Feedforward Design --- Introduction {#sec:introduction} ============ Convolutional neural networks (CNNs) have received a lot of attention in recent years due to their outstanding performance in numerous applications. We have also witnessed the rapid development of advanced CNN models and architectures such as the generative adversarial networks (GANs) [@goodfellow2014generative], the ResNets [@He_2016_CVPR] and the DenseNet [@huang2017densely]. A great majority of current CNN literature are application-oriented, yet efforts are made to build theoretical foundation of CNNs. Cybenko [@cybenko1989approximation] and Hornik [*et al.*]{} [@hornik1989multilayer] proved that the multi-layer perceptron (MLP) is a universal approximator in late 80s. Recent studies on CNNs include: visualization of filter responses at various layers [@simonyan2013deep; @zeiler2014visualizing; @zhou2014object], scattering networks [@mallat2012group; @bruna2013invariant; @wiatowski2015mathematical], tensor analysis [@cohen2015expressive], generative modeling [@dai2014generative], relevance propagation [@bach2015pixel], Taylor decomposition [@montavon2015explaining], multi-layer convolutional sparse modeling [@sulam2017multi] and over-parameterized shallow neural network optimization [@soltanolkotabi2018theoretical]. More recently, CNN’s interpretability has been examined by a few researchers from various angles. Examples include interpretable knowledge representations [@zhang2017interpretable], critical nodes and data routing paths identification [@wang2018interpret], the role of nonlinear activation [@kuo2016understanding], convolutional filters as signal transforms [@kuo2017cnn; @kuo2018data], etc. Despite the above-mentioned efforts, it is still challenging to provide an end-to-end analysis to the working principle of deep CNNs. Given a CNN architecture, the determination of network parameters can be formulated as a non-convex optimization problem and solved by backpropagation (BP). Yet, since non-convex optimization of deep networks is mathematically intractable [@soltanolkotabi2018theoretical], a new methodology is adopted in this work to tackle the interpretability problem. That is, we propose an interpretable feedforward (FF) design without any BP and use it as a reference. The FF design adopts a data-centric approach. It derives network parameters of the current layer based on data statistics from the output of the previous layer in a one-pass manner. This complementary methodology not only offers valuable insights into CNN architectures but also enriches CNN research from the angles of linear algebra, statistics and signal processing. To appreciate this work, a linear algebra viewpoint on machine learning is essential. An image, its class label and intermediate representations are all viewed as high-dimensional vectors residing in certain vector spaces. For example, consider the task of classifying a set of RGB color images of spatial resolution $32 \times 32$ into $10$ classes. The input vectors are of dimension $32 \times 32 \times 3=3,072$. Each desired output is the unit dimensional vector in a 10-dimensional (10D) space. To map samples from the input space to the output space, we conduct a sequence of vector space transformations. Each layer provides one transformation. A [*dimension*]{} of a vector space can have one of three meanings depending on the context: a [*representation*]{} unit, a [*feature*]{} or a class [*label*]{}. The term “dimension" provides a unifying framework for the three different concepts in traditional pattern recognition. In our interpretation, each CNN layer corresponds to a vector space transformation. To take computer vision applications as an example, CNNs provide a link from the input image/video space to the output decision space. The output can be an object class (e.g., object classification), a pixel class (e.g., semantic segmentation) or a pixel value (e.g. depth estimation, single image super-resolution, etc.) The training data provide a sample distribution in the input space. We use the input data distribution to determine a proper transformation to the output space. The transformation is built upon two well-known ideas: 1) dimension reduction through subspace approximations and/or projections, and 2) training sample clustering and remapping. The former is used in convolutional layers construction (e.g., filter weights selection and max pooling) while the latter is adopted to build FC layers. They are elaborated a little more below. The convolutional layers offer a sequence of spatial-spectral filtering operations. Spatial resolutions become coarser along this process gradually. To compensate for the loss of spatial resolution, we convert spatial representations to spectral representations by projecting pixels in a window onto a set of pre-selected spatial patterns obtained by the principal component analysis (PCA). The transformation enhances discriminability of some dimensions since a dimension with a larger receptive field has a better chance to “see" more. We develop a new transform, called the Saab ([**S**]{}ubspace [**a**]{}pproximation with [**a**]{}djusted [**b**]{}ias) transform, in which a bias vector is added to annihilate nonlinearity of the activation function. The Saab transform is a variant of PCA, and it contributes to dimension reduction. The FC layers provide a sequence of operations that involve “sample clustering and high-to-low dimensional mapping". Each dimension of the output space corresponds to a ground-truth label of a class. To accommodate intra-class variability, we create sub-classes of finer granularity and assign pseudo labels to sub-classes. Consider a three-level hierarchy – the feature space, the sub-class space and the class space. We construct the first FC layer from the feature space to the sub-class space using a linear least-squared regressor (LSR) guided by pseudo-labels. For the second FC layer , we treat pseudo-labels as features and conduct another LSR guided by true labels from the sub-class space to the class space. A sequence of FC layers actually corresponds to a multi-layer perceptron (MLP). To the best of our knowledge, this is the first time to construct an MLP in an FF manner using multi-stage cascaded LSRs. The FF design not only reduces dimensions of intermediate spaces but also increases discriminability of some dimensions gradually. Through transformations across multiple layers, it eventually reaches the output space with strong discriminability. LeNet-like networks are chosen to illustrate the FF design methodology for their simplicity. Examples of LeNet-like networks include the LeNet-5 [@LeNet1998] and the AlexNet [@NIPS2012_AlexNet]. They are often applied to object classification problems such as recognizing handwritten digits in the MNIST dataset and 1000 object classes in the ImageNet dataset. The classification and robustness (against adversarial attacks) performances of BP- and FF-designed CNNs on the MNIST and the CIFAR-10 datasets are reported and compared. It is important to find a connection between the BP and the FF designs. To shed light on their relationship, we measure cross-entropy values at dimensions of intermediate vector spaces (or layers). The rest of this paper is organized as follows. The related background is reviewed in Sec. \[sec:interpretation\]. The FF design of convolutional layers is described in Sec. \[sec:Saab\]. The FF design of FC layers is presented in Sec. \[sec:l3sr\]. Experimental results for the MNIST and the CIFAR-10 datasets are given in Sec. \[sec:experiments\]. Follow-up discussion is made in Sec. \[sec:discussion\]. Finally, concluding remarks are drawn in Sec. \[sec:conclusion\]. Background {#sec:interpretation} ========== Computational neuron -------------------- The computational neuron serves as the basic building element of CNNs. As shown in Fig. \[fig:neuron\], it consists of two stages: 1) affine computation and 2) nonlinear activation. The input is an $N$-dimensional random vector ${\bf x}=(x_0, x_1, ... , x_{N-1})^T$. The $k$th neuron has $N$ filter weights that can be expressed in vector form as ${\bf a}_k=(a_{k,0}, a_{k,1}, \cdots , a_{k,N-1})^T$, and one bias term $b_k$. The affine computation is $$\label{eq:affine} y_k = \sum_{n=0}^{N-1} a_{k,n} x_n + b_k = {\bf a}_k^T {\bf x} + b_k, \quad k=0, 1, \cdots, K-1,$$ where ${\bf a}_k$ is the filter weight vector associated with the $k$th neuron. With the ReLU nonlinear activation function, the output after ReLU can be written as $$\label{eq:na} z_k= \phi (y_k) = \max (0, y_k).$$ In the following discussion, we assume that input ${\bf x}$ is a zero-mean random vector and the set of filter weight vectors is normalized to to be with unit length ([*i.e.*]{} $||{\bf a}_k||=1$). The filter weight vectors are adjustable in the training yet they are given and fixed in the testing. To differentiate the two situations, we call them anchor vectors in the testing case. Efforts have been made to explain the roles played by anchor vector ${\bf a}_k$ and nonlinear activation $\phi(\cdot)$ in [@kuo2016understanding; @kuo2017cnn; @kuo2018data]. ![Illustration of a computational neuron: (a) the input-output connection, and (b) the block diagram inside one neuron.[]{data-label="fig:neuron"}](fig-neuron-1.pdf "fig:"){width="8cm"}\ (a)\ ![Illustration of a computational neuron: (a) the input-output connection, and (b) the block diagram inside one neuron.[]{data-label="fig:neuron"}](fig-neuron-2.pdf "fig:"){width="12cm"}\ (b) Linear space spanned by anchor vectors -------------------------------------- It is easier to explain the role of anchor vectors by setting the bias term to zero. This constraint will be removed in Sec. \[sec:Saab\]. If $b_k=0$, Eq. (\[eq:affine\]) reduces to $y_k = {\bf a}_k^T {\bf x}$. Suppose that there are $K$ anchor vectors with $k=0, 1, \cdots, K-1$ in a convolutional layer. One can examine the role of each anchor vector individually and all anchor vectors jointly. They lead to two different interpretations. 1. A set of parallel correlators [@kuo2016understanding; @kuo2017cnn]\ If the correlation between ${\bf a}_k$ and ${\bf x}$ is weak (or strong), response $y_k$ will have a small (or large) magnitude. Thus, each anchor vector can be viewed as a correlator or a matched filter. We can use a set of correlators to extract pre-selected patterns from input ${\bf x}$ by thresholding $|y_k|$. These correlators can be determined by BP. In a FF design, one can apply the $k$-means algorithm to input samples to obtain $K$ clusters and set their centroids to ${\bf a}_k$. 2. A set of unit vectors that spans a linear subspace [@kuo2018data]\ By considering the following set of equations jointly: $$\label{eq:projection} y_k = {\bf a}_k^T {\bf x}, \quad k=0, 1, \cdots K-1,$$ the output vector, ${\bf y}=(y_0, y_1, \cdots , y_{K-1})^T$, is the projection of the input vector ${\bf x}$ onto a subspace spanned by ${\bf a}_k$, $k=0, 1, \cdots, K-1$. One can get an approximate to ${\bf x}$ in the space spanned by anchor vectors from the projected output ${\bf y}$. With this interpretation, we can use the principal component analysis (PCA) to find a subspace and determine the anchor vector set accordingly. The above two interpretations allow us to avoid the BP training procedure for filter weights in convolutional layers. Instead, we can derive anchor vectors from the statistics of input data. That is, we can compute the covariance matrix of input vectors ${\bf x}$ and use the eigenvectors as the desired anchor vectors ${\bf a}_k$. This is a data-centric approach. It is different from the traditional BP approach that is built upon the optimization of a cost function defined at the system output. Role of nonlinear activation ---------------------------- The need of nonlinear activation was first explained in [@kuo2016understanding]. The main result is summarized below. Consider two inputs ${\bf x}_1={\bf x}$ and ${\bf x}_2=-{\bf x}$ with the simplifying assumption: $$\label{eq:simplifying} {\bf a}_k^T {\bf x} \approx 1, \mbox{ and } {\bf a}_{k'}^T {\bf x} \approx 0 \mbox{ if } k' \ne k.$$ The two inputs, ${\bf x}_1$ and ${\bf x}_2$, are negatively correlated. For example, if ${\bf x}_1$ is a pattern composed by three vertical stripes with the middle one in black and two side ones in white, then ${\bf x}_2$ is also a three-vertical-stripe pattern with the middle one in white and two side ones in black. They are different patterns, yet one can be confused for the other if there is no ReLU. To show this, we first compute their corresponding outputs $$\begin{aligned} \label{eq:outputs} {\bf y}_1^T & = & ({\bf a}_1^T, \cdots, {\bf a}_k^T, \cdots, {\bf a}_K^T){\bf x}_1 \approx (0, \cdots, 0, 1, 0, \cdots, 0), \\ {\bf y}_2^T & = & ({\bf a}_1^T, \cdots, {\bf a}_k^T, \cdots, {\bf a}_K^T){\bf x}_2 \approx (0, \cdots, 0, -1, 0, \cdots, 0),\end{aligned}$$ where $1$ and $-1$ appear in the $k$th element as shown above. Vectors ${\bf y}_1$ and ${\bf y}_2$ will serve as inputs to the next stage. The outgoing links from the $k$th node can take positive or negative weights. If there is no ReLU, a node at the second layer cannot differentiate whether the input is ${\bf x}_1$ or ${\bf x}_2$ since the following two situations yield the same output: - a positive correlation (${\bf y}_1$) followed by a positive outgoing link from node $k$; and - a negative correlation (${\bf y}_2$) followed by a negative outgoing link from node $k$. Similarly, the following two situations will also yield the same output: - a positive correlation (${\bf y}_1$) followed by a negative outgoing link from node $k$; and - a negative correlation (${\bf y}_2$) followed by a positive outgoing link from node $k$. This is called the sign confusion problem. The ReLU operator plays the role of a rectifier that eliminates case (b). In the example, input ${\bf x}_2$ will be blocked by the system. A trained CNN has its preference on images over their foreground/background reversed ones. A classification example conducted on the original and reversed MNIST datasets was given in [@kuo2016understanding] to illustrate this point. To resolve the sign confusion problem, a PCA variant called the Saak transform was proposed in [@kuo2018data]. The Saak transform augments transform kernel ${\bf a}_k$ with its negative $-{\bf a}_k$, leading to $2K$ transform kernels in total. Any input vector, ${\bf x}$, will have a positive/negative correlation pair with kernel pair $({\bf a}_k,-{\bf a}_k)$. When a correlation is followed by ReLU, one of the two will go through while the other will be blocked. In other words, the Saak transform splits positive/negative correlations, ${\bf y}_k={\bf a}_k^T {\bf x}$, into positive/negative two channels. To resolve the sign confusion problem, it pays the price of doubled spectral dimensions. In the following section, we will introduce another PCA variant called the Saab (Subspace approximation with adjusted bias) transform. The Saab transform can address the sign confusion problem and avoid the spectral dimension doubling problem at the same time. ![The LeNet-5 architecture [@LeNet1998], where the convolutional layers are enclosed by a blue parallelogram.[]{data-label="fig:LeNet-5-Conv"}](fig-LeNet-5-Conv.pdf){width="12cm"} Feedforward design of convolutional layers {#sec:Saab} ========================================== In this section, we study the construction of convolutional layers in the LeNet-5 as shown in the enclosed parallelogram in Fig. \[fig:LeNet-5-Conv\]. We first examine the spatial-spectral filtering in Sec. \[subsec:filtering\]. The Saab transform and its bias selection is studied in Sec. \[subsec:bias\]. Then, we discuss the max pooling operation in Sec. \[subsec:pooling\]. Finally, we comment on the effect of cascaded convolutional layers in Sec. \[subsec:compound\]. Spatial-spectral filtering {#subsec:filtering} -------------------------- There is a clear distinction between representations and features in traditional image processing. Image representations are obtained by transforms such as the Fourier transform, the discrete Cosine transform and the wavelet transform, etc. Image transforms are invertible. Image features are extracted by detectors such as those for edges, textures and salient points. The distinction between image representations and image features is blurred in CNNs as mentioned in Sec. \[sec:introduction\]. An input image goes through a sequence of vector space transformations. Each dimension of intermediate spaces can be viewed as either a representation or a feature. The cascade of spatial-spectral filtering and spatial pooling at a convolutional layer provides an effective way in extracting discriminant dimensions. Without loss of generality, we use the LeNet-5 applied to the MNIST dataset as an illustrative example. It is not a good idea to conduct pixel-wise image comparison to recognize handwritten digits since there exists a wide range of varieties in the spatial domain for the same digit. Besides, the pixel-wise comparison operation is sensitive to small translation and rotation. For example, if we shift the same handwritten digit image “1" horizontally by several pixels, the original and shifted images will have poor match in the spatial domain. A better way is to consider the neighborhood of a pixel, and find a spectral representation for the neighborhood. For example, we use a patch of size $5 \times 5$ centered at a target pixel as its neighborhood. The stroke inside a patch is a 2D pattern. We can use PCA to find a set of dominant stroke patterns (called anchor vectors) to form a vector space and represent any neighborhood pattern as a linear combination of anchor vectors. In the multi-stage design, the convolutional layers provide a sequence of spatial-spectral transformations that convert an input image to its joint spatial-spectral representations layer by layer. Spatial resolutions become lower gradually. To compensate the spatial resolution loss, we trade spatial representations for spectral representations by projecting local spatial-spectral cuboids onto PCA-based kernels. The main purpose is to enhance discriminability of some dimensions. Generally, a spatial-spectral component with a larger receptive field has a better chance to be discriminant since it can “see" more. Another advantage of the PCA-based subspace approximation is that it does not demand image labels. One related question is whether to conduct the transform in overlapping or non-overlapping windows. Signal transforms are often conducted in non-overlapping windows for computational and storage efficiency. For example, the block discrete Cosine transform (DCT) is adopted in image/video compression. For the same reason, the Saak transform in [@kuo2018data] is conducted on non-overlapping windows.However, the LeNet-5 uses overlapping windows. The MNIST dataset contains gray-scale images of dimension $32 \times 32$. At the first convolutional layer, the LeNet-5 has 6 filters of size $5\times 5$ with stride equal to one. The output image cuboid has a dimension of $28 \times 28 \times 6$ by taking the boundary effect into account. If we ignore the boundary effect and apply $K$ spatial-spectral filtering operations at every pixel, the output data dimension is enlarged by a factor of $K$ with respect to the input. This redundant representation seems to be expensive in terms of higher computational and storage resources. However, it has one advantage. That is, it provides a [*richer*]{} feature set for selection. Redundancy is controlled by the stride parameter in CNN architecture specification. Saab transform and bias selection {#subsec:bias} --------------------------------- We repeat the affine transform in Eq. (\[eq:affine\]) below: $$\label{eq:affine_r} y_k = \sum_{n=0}^{N-1} a_{k,n} x_n + b_k = {\bf a}_k^T {\bf x} + b_k, \quad k=0, 1, \cdots, K-1,$$ The Saab transform is nothing but a specific way in selecting anchor vector ${\bf a}_k$ and bias term $b_k$. They are elaborated in this subsection. [**Anchor vectors selection.**]{} By following the treatment in [@kuo2016understanding; @kuo2017cnn; @kuo2018data], we first set $b_k=0$ and divide anchor vectors into two categories: - DC anchor vector ${\bf a}_0 = \frac{1}{\sqrt{N}} (1, \cdots, 1)^T$. - AC anchor vectors ${\bf a}_k$, $k=1, \cdots K-1$. The terms “DC" and “AC" are borrowed from circuit theory, and they denote the “direct current" and the “alternating current", respectively. Based on the two categories of anchor vectors, we decompose the input vector space, $\mathcal{S}=R^N$, into the direct sum of two subspaces: $$\label{eq:space_decomposition} \mathcal{S}=\mathcal{S}_{DC} \oplus \mathcal{S}_{AC},$$ where $\mathcal{S}_{DC}$ is the subspace spanned by the DC anchor and and $\mathcal{S}_AC$ is the subspace spanned by the AC anchors. They are called the DC and AC subspaces accordingly. For any vector ${\bf x} \in R^N$, we can project ${\bf x}$ to ${\bf a}_0$ to get its DC component. That is, we have $$\label{eq:mean} {\bf x}_{DC}={\bf x}^T {\bf a}_0 = \frac{1}{\sqrt{N}} \sum_{n=0}^{N-1} x_n.$$ Subspace $\mathcal{S}_{AC}$ is the orthogonal complement to $\mathcal{S}_{DC}$ in $\mathcal{S}$. We can express the AC component of ${\bf x}$ as $$\label{eq:AC} {\bf x}_{AC}={\bf x} - {\bf x}_{DC}.$$ Clearly, we have ${\bf x}_{DC} \in \mathcal{S}_{DC}$ and ${\bf x}_{AC} \in \mathcal{S}_{AC}$. We conduct the PCA on all possible ${\bf x}_{AC}$ and, then, choose the first $(K-1)$ principal components as AC anchor vectors ${\bf a}_k$, $k=1, \cdots K-1$. [**Bias selection.**]{} Each bias term, $b_k$, in Eq. (\[eq:affine\_r\]), provides one extra degree of freedom per neuron for the end-to-end penalty minimization in the BP design. Since it plays no role in linear subspace approximation, it was ignored in [@kuo2016understanding; @kuo2017cnn; @kuo2018data]. Here, the bias term is leveraged to overcome the sign confusion problem in the Saab transform. We impose two constraints on the bias terms. - Positive response constraint\ We choose the $k$th bias, $b_k$, to guarantee the $k$th response a non-negative value. Mathematically, we have $$\label{eq:affine2} y_k=\sum_{n=0}^{N-1}a_{k,n}x_n+b_k={\bf a}_k^T {\bf x}+b_k \ge 0,$$ for all input ${\bf x}$. - Constant bias constraint\ We demand that all bias terms are equal; namely, $$\label{eq:b_constant} b_0=b_1= \cdots = b_{K-1} \equiv d \sqrt{K}.$$ Due to Constraint (B1), we have $$\label{eq:na2} z_k= \phi (y_k) = \max (0, y_k) = y_k.$$ That is, we have the same output with or without the ReLU operation. This simplifies our CNN analysis greatly since we remove nonlinearity introduced by the activation function. Under Constraint (B2), we only need to determine a single bias value $d$ (rather than $K$ different bias values). It makes the Saab transform design easier. Yet, there is a deeper meaning in (B2). Let ${\bf x}=(x_0, \cdots, x_{N-1})$ lie in the AC subspace of the input space, $R^N$. We can re-write Eq. (\[eq:affine\_r\]) in vector form as $$\label{eq:vector} {\bf y} = {\bf x}^T {\bf a}_0 + \sum_{k=1}^{K-1} {\bf x}^T {\bf a}_k + d \sqrt{K} {\bf l},$$ where ${\bf l}=\frac{1}{\sqrt{K}}(1, \cdots, 1)^T$ is the unit constant-element vector in the output space $R^K$. The first term in Eq. (\[eq:vector\]) is zero due to the assumption. The second and third terms lie in the AC and DC subspaces of the output space, $R^K$, respectively. In other words, the introduction of the constant-element bias vector has no impact on the AC subspace but the DC subspace when multiple affine transforms are in cascade. It is essential to impose (B2) so that the multi-stage Saab transforms in cascade are mathematically tractable. That is, the constant-element bias vector ${\bf b}=d {\bf 1}$ always lies in the DC subspace, $\mathcal{S}_{DC}$. It is completely decoupled from $\mathcal{S}_{AC}$ and its PCA. We conduct PCA on the AC subspace, $\mathcal{S}_{AC}$, layer by layer to obtain AC anchor vectors at each layer. The addition of a constant-element bias vector to the transformed output vector is nothing but shift each output response by a constant amount. We derive a lower bound on this amount in the appendix. The bias selection rule can be stated below. - [**Bias selection rule**]{}\ All bias terms should be equal (i.e., $b_0=b_1= \cdots = b_{K-1}$), and they should meet the following constraint: $$\label{eq:d3} b_k \ge \max_{\bf x} || {\bf x} ||, \quad k=0, \cdots, K-1.$$ where ${\bf x} \in R^N$ is an input vector and $K$ is the dimension of the output space $R^K$. Spatial pooling {#subsec:pooling} --------------- [**Rationale of maximum pooling.**]{} Spatial pooling helps reduce computational and storage resources. However, this cannot explain a common observation: “Why maximum pooling outperforms the average pooling?" Here, we interpret the pooling as another filtering operation that preserves significant patterns and filters out insignificant ones. Fig. \[fig:pooling\] shows four spatial locations, denoted by A, B, C, D, in a representative $2 \times 2$ non-overlapping block. Suppose that $K$ filters are used to generate responses at each location. Then, we have a joint spatial-spectral response vector of dimension $4 \times K$. Pooling is used to reduce the spatial dimension of this response vector from $4$ to $1$ so that the new response vector is of dimension $1 \times K$. It is performed at each spectral component independently. Furthermore, all response values are non-negative due to the ReLU function. ![Illustration of the spatial pooling, where A, B, C, D denote four spatial locations within a $2 \times 2$ block where the maximum pooling operation is conducted at each spectral component independently.[]{data-label="fig:pooling"}](fig-pooling.pdf){width="6cm"} Instead of viewing convolution and pooling operations as two individual ones, we examine them as a whole. The new input is the union of the four neighborhoods, which is a patch of size $36=6 \times 6$. The compound operations, consisting of convolution and pooling, is to project the enlarged neighborhood patch of dimension $36$ to a subspace of dimension $K$ spanned by $K$ anchor vectors (or filters). The convolution-plus-pooling operations is equivalent to: 1. Projecting a patch of size $36=6 \times 6$ to any smaller one of size $5 \times 5$ centered at locations A-D with zero padding in uncovered areas; 2. Projecting the patch of size $5 \times 5$ to all anchor vectors. Each spectral component corresponds to a visual pattern. For a given pattern, we have four projected values obtained at locations A, B, C, D. By maximum pooling, we choose the maximum value among the four. That is, we search the target visual pattern in a slightly larger window (i.e. of size $6 \times 6$) and use the maximum response value to indicate the best match within this larger window. On the other hand, infrequent visual patterns that are less relevant to the target task will be suppressed by the compound operation of spatial-spectral filtering (i.e. the Saab transform) and pooling. The convolutional kernels of the Saab transform are derived from the PCA. The projection of these patterns on anchor vectors tend to generate small response values and they will be removed after pooling. The same principle can be generalized to pooling in deeper layers. A spatial location in a deep layer corresponds to a receptive field in the input source image. The deeper the layer the larger the receptive field. A spectral component at a spatial location can be interpreted as a projection to a dominant visual pattern inside its receptive field. A $2 \times 2$ block in a deep layer correspond to the union of four overlapping receptive fields, each of which is associated with one spatial location. To summarize, the cascade of spatial-spectral filtering and maximum pooling can capture “visually similar but spatially displaced" patterns. ![Left: Display of nine top-ranked images that have the strongest responses with respect to a certain convolutional filter (called the cat face filter) in the 5th convolutional layer of the AlexNet. Right: Their corresponding filter responses.[]{data-label="fig:cat-face"}](fig-cat-face.pdf){width="12cm"} Multi-layer compound filtering {#subsec:compound} ------------------------------ The cascade of multiple convolutional layers can generate a rich set of image patterns of various scales as object signatures. We call them compound filters. One can obtain interesting compound filters through the BP design. To give an example, we show the nine top-ranked images that have the strongest responses with respect to a certain convolutional filter (called the cat face filter) in the 5th convolutional layer of the AlexNet [@NIPS2012_AlexNet] and their corresponding filter responses in Fig. \[fig:cat-face\]. We see cat face contours clearly in the right subfigure. They are of size around $100 \times 100$. The compound filtering effect is difficult to implement using a single convolutional layer (or a single-scale dictionary). In the FF design, a target pattern is typically represented as a linear combination of responses of a set of orthogonal PCA filters. These responses are signatures of the corresponding receptive field in the input image. There is no need to add the bias term in the last convolutional layer since a different design methodology is adopted for the construction of FC layers. The block-diagram of the FF design of the first two convolutional layers of the LeNet-5 is shown in Fig. \[fig:ff-block-diagram\]. ![Summary of the FF design of the first two convolutional layers of the LeNet-5.[]{data-label="fig:ff-block-diagram"}](fig-conv-layer.pdf){width="12cm"} Feedforward design of FC layers {#sec:l3sr} =============================== The network architecture the LeNet-5 is shown in Fig. \[fig:LeNet-FC\], where FC layers enclosed by a blue parallelogram. The input to the first FC layer consists of data cuboids of dimension $5\times5\times 16$ indicated by S4 in the figure. The output layer contains 10 output nodes, which can be expressed as a 10-dimensional vector. There are two hidden layers between the input and the output of dimensions 120 and 84, respectively. We show how to construct FC layers using a sequence of label-guided linear least-squared regressors. ![The LeNet-5 architecture [@LeNet1998], where the FC layers are enclosed by a blue parallelogram.[]{data-label="fig:LeNet-FC"}](fig-LeNet-5-MLP.pdf){width="12cm"} Least-squared regressor (LSR) {#subsec:l2sr} ----------------------------- In the FF design, each FC layer is treated as a linear least-squared regressor. The output of each FC layer is a one-hot vector whose elements are all set to zero except for one element whose value is set to one. The one-hot vector is typically adopted in the output layer of CNNs. Here, we generalize the concept and use it at the output of each FC layer. There is one challenge in this generalization. That is, there is no label associated with an input vector when the output is one of the hidden layers. To address it, we conduct the k-means clustering on the input, and group them into $Q$ clusters where $Q$ is the number of output nodes. Then, each input has two labels – the original class label and the new cluster label. We combine the class and the cluster labels to create $K$ pseudo-labels for each input. Then, we can set up a linear least-squared regression problem using the one-hot vector defined by $K$ pseudo-labels at this layer. Then, we can conduct the label-guided linear least-squared regression in multiple stages (or multi-layers). To derive a linear least-squared regressor, we set up a set of linear equations that relates the input and the output variables. Suppose that ${\bf x}=(x_1, x_2, \cdots, x_n)^T \in R^n$ and ${\bf y}=(y_1, y_2, \cdots, y_c)^T \in R^c$ are input and output vectors. That is, we have $$\label{eq:l3sr} \left[ \begin{array}{ccccc} a_{11} & a_{12} & \cdots & a_{1n} & w_1 \\ a_{21} & a_{22} & \cdots & a_{2n} & w_2 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ a_{c1} & a_{c2} & \cdots & a_{cn} & w_c \end{array} \right] \left[\begin{array}{c} x_{1} \\ x_{2} \\ \vdots \\ x_{n} \\ 1 \end{array} \right] = \left[\begin{array}{c} y_{1} \\ y_{2} \\ \vdots \\ y_{c} \\ \end{array} \right],$$ where $w_1$, $w_2$, $\cdots$, $w_c$ are scalars to account for $c$ bias terms. After nonlinear activation, each FC layer is a rectified linear least-squared regressor. There are three FC layers in cascade in the LeNet-5. - First FC layer (or Stage I): $n_1=375$ and $m_1=120$. - Second FC layer (or Stage II): $n_2=120$ and $m_2=84$. - Third FC layer (or Stage III): $n_3=84$ and $m_3=10$. The input data to the first FC layer is a data cuboid of dimension $5 \times 5 \times 15=375$ since the DC responses are removed (or dropped out). The output data of the third FC layer is the following ten one-hot vector of dimension $10$: $$(1, 0, \cdots, 0)^T, (0, 1, 0, \cdots, 0)^T, \cdots, (0, \cdots, 0, 1, 0)^T, (0, \cdots, 0, 1)^T.$$ Each one-hot vector denotes the label of a hand-written digit. For example, we can use the above ten one-hot vector to denote digits “0", “1", $\cdots$, “9", respectively. By removing the two middle hidden layers of dimensions 120 and 84 and conducting the linear least-squared regression on the input feature vector of dimension 375 with 10 one-hot vectors as the desired output directly, this is nothing but traditional one-stage least-squared regression. However, its performance is not very good. The advantage of introducing hidden layers and the way to generate more pseudo-labels will be discussed below. ![Illustration of intra-class variabilities [@kuo2017cnn].[]{data-label="fig:block-diagram"}](fig-snake.pdf){width="12cm"} Pseudo-labels generation {#subsec:pseudo} ------------------------ To build least-squared regressors in Stages I and II, we need to define labels for input. Here, we consider the combination of two factors: the original label (denoted by $0, 1, \cdots, 9$) and the auxiliary label (denoted by i, ii, $\cdots$, x, xi, xii, etc.) of an input data sample. Each input training sample has its own original label. To create its auxiliary label, we conduct the k-means clustering algorithm on training samples of the same class. For example, we can divide samples of the same digit into 12 clusters to generate 12 pseudo classes. The centroid of a class provides a representative sample for that class. The reason to generate pseudo classes is to capture the diversity of a single class with more representative samples, say, from one to twelve in the LeNet-5. As a result, we can set up 120 linear equations to map samples in the input feature space to 120 one-hot vectors, which form the output feature space. A one-hot vector is a unit vector in an orthogonal feature space. Ideally, an LSR is constructed to map all samples in a pseudo-class to the unit vector of the target dimension and block the mapping of these samples to other unit vectors. This is difficult to achieve in practice since its performance is highly correlated with the feature distribution in the input space. If two pseudo classes have strong overlaps in the feature space (say, some 7’s and 9’s are visually similar), we expect a significant projection of samples in one pseudo class to the one-hot vector of another pseudo-class, vice versa. This is called “leakage" (with respect to the original pseudo class) or “interference" (with respect to other pseudo classes). The output of the last convolutional layer has a physical explanation in the FF design. It is a spatial-spectral transform of an input image. It provides a feature vector for the classifier in later layers. Once it is fed to the FC layer, a mathematical model is used to align the input and output feature spaces by mapping samples from a pseudo class to one of orthogonal unit vectors that span the output feature space. The purpose of feature space alignment is to lower the cross-entropy value and produce more discriminant features. Thus, the cascade of LSRs conducted in multiple FC layers is nothing but a sequential feature space alignment procedure. In contrast, BP-designed CNNs do not have such a clear cut in roles played by the convolutional and the FC layers. Experimental Results {#sec:experiments} ==================== We will show experimental results conducted on two popular datasets: the MNIST dataset[^1] and the CIFAR-10 dataset [^2]. Our implementation codes are available in the GitHub website. [**Network architectures.**]{} The LeNet-5 architecture targets at gray-scale images only. Since the CIFAR-10 dataset is a color image dataset, we need to modify the network architecture slightly. The parameters of the original and the modified LeNet-5 are compared in Table \[table:mLeNet-5\]. Note that the modified LeNet-5 keeps the architecture of two convolutional layers and three FC layers, which include the last output layer. The modification is needed since the input images are color images in the CIFAR-10 dataset. We use more filters at all layers, which are chosen heuristically. Architecture Original LeNet-5 Modified LeNet-5 ---------------------------- ----------------------- ------------------------ 1st Conv Layer Kernel Size $5 \times 5 \times 1$ $5 \times 5 \times 3$ 1st Conv Layer Kernel No. $6$ $32$ 2nd Conv Layer Kernel Size $5 \times 5 \times 6$ $5 \times 5 \times 32$ 2nd Conv Layer Kernel No. $16$ $64$ 1st FC Layer Filter No. $120$ $200$ 2nd FC Layer Filter No. $84$ $100$ Output Node No. $10$ $10$ : Comparison of the original and the modified LeNet-5 architectures.[]{data-label="table:mLeNet-5"} Datasets MNIST CIFAR-10 ---------- ------- ---------- FF 97.2% 62% Hybrid 98.4% 64% BP 99.9% 68% : Comparison of testing accuracies of BP and FF designs for the MNIST and the CIFAR-10 datasets.[]{data-label="table:classification"} [0.4]{} ![Adversarial attacks to MNIST and CIFAR-10 datasets using BP-designed model parameters (from left to right): original input images, images attacked by the FGS, BIM and DeepFool.[]{data-label="fig:attack"}](fig-attack-m.pdf "fig:"){width="\linewidth"} [0.4]{} ![Adversarial attacks to MNIST and CIFAR-10 datasets using BP-designed model parameters (from left to right): original input images, images attacked by the FGS, BIM and DeepFool.[]{data-label="fig:attack"}](fig-attack-c.pdf "fig:"){width="\linewidth"} [0.4]{} ![Adversarial attacks to MNIST and CIFAR-10 datasets using FF-designed model parameters (from left to right): original input images, images attacked by the FGS, BIM and DeepFool.[]{data-label="fig:attack-2"}](fig-attack-m2.pdf "fig:"){width="\linewidth"} [0.4]{} ![Adversarial attacks to MNIST and CIFAR-10 datasets using FF-designed model parameters (from left to right): original input images, images attacked by the FGS, BIM and DeepFool.[]{data-label="fig:attack-2"}](fig-attack-c2.pdf "fig:"){width="\linewidth"} [**Classification performance.**]{} The testing accuracies of the BP and the FF designs for the MNIST and the CIFAR-10 datasets are compared in Table \[table:classification\]. Besides, we introduce a hybrid design that adoptes the FF design in the first two convolutional layers to extract features. Then, it uses the multi-layer perceptron (MLP) the last three FC layers with a built-in BP mechanism. By comparing the FF and the Hybrid designs, we observe the performance degradation due to a poorer decision subnet without BP optimization. We see a drop of 1.2% and 2% in classification accuracies for MNIST and CIFAR-10, respectively. Next, we can see the performance degradation primarily due to a poorer feature extraction subnet without BP optimization. We see a drop of 1.5% and 4% for MNIST and CIFAR-10, respectively. Performance degradation due to the FF design is well expected. The performance gaps between the FF and the BP designs for MNIST and CIFAR-10 are, respectively, 2.7% and 6%. Attacks CNN Design MNIST CIFAR-10 ---------- ------------ ------- ---------- FGS BP 6 % 15 % FGS FF 56 % 21 % BIM BP 1 % 12 % BIM FF 46 % 31 % Deepfool BP 2 % 15 % Deepfool FF 96 % 59 % : Comparison of testing accuracies of BP and FF designs against FGS, BIM and Deepfool three adversarial attacks targeting at the BP design.[]{data-label="table:robustness"} Attacks CNN Design MNIST CIFAR-10 ---------- ------------ ------- ---------- FGS BP 34 % 11 % FGS FF 4 % 6 % BIM BP 57 % 14% BIM FF 1 % 12 % Deepfool BP 97 % 68 % Deepfool FF 2 % 16 % : Comparison of testing accuracies of BP and FF designs against FGS, BIM and Deepfool three adversarial attacks targeting at the FF design.[]{data-label="table:robustness-2"} [**Robustness against adversarial attacks.**]{} Adversarial attacks have been extensively examined since the pioneering study in [@szegedy2013intriguing]. We examine three attacks: the fast gradient sign (FGS) method [@goodfellow2014explaining], the basic iterative method (BIM) [@kurakin2016adversarial] and the Deepfool method [@moosavi2016deepfool]. We generate adversarial attacks targeting at the BP and the FF designs individually. Exemplary attacked images targeting at BP and FF designs are shown in Fig. \[fig:attack\] and Fig. \[fig:attack-2\], respectively. As shown in Figs. \[fig:attack\] and \[fig:attack-2\], the Deepfool provides attacks of least visual distortion. The quality of the FGS- and BIM-attacked images is poor. Since one can develop algorithms to filter out poor quality images, we should be more concerned with the Deepfool attack. The classification accuracies of BP and FF designs against attacked images are compared in Tables \[table:robustness\] and \[table:robustness-2\]. The classification performance of BP and FF designs degrades significantly against their individual attacks. This example indicates that, once CNN model parameters are known, one can design powerful adversarial attacks to fool the recognition system yet high-quality adversarial attacks have little effect on the HVS. The fact that CNNs are vulnerable to adversarial attacks of good visual quality has little to do model parameters selection methodologies. It is rooted in the end-to-end interconnection architecture of CNNs. To mitigate catastrophic performance degradation caused by adversarial attacks, one idea is to consider multiple models as explained below. [**Robustness enhancement via ensemble methods.**]{} One can generate different FF-designed CNN models using multiple initializations for the k-means clustering in the multi-stage LSR models. For example, we construct the following three FF-designed CNNs that share the same convolutional layer but different FC layers. - FF-1: Adopt the k-means++ initialization [@arthur2007k] in the k-means clustering; - FF-2: Adopt the random initialization in the k-means clustering; - FF-3: Cluster five digits into 13 clusters and the other five digits into 11 clusters in the design of the first AC layer (with 120 output dimensions) and adopt the k-means++ initialization. MNIST FF-1 FF-2 FF-3 ------------------------- ------ ------ ------ Deepfool attacking FF-1 2% 79% 81% Deepfool attacking FF-2 78% 2% 81% Deepfool attacking FF-3 79% 80% 2% : Comparison of MNIST testing accuracies of three FF designs, against the Deepfool adversarial attacks targeting at FF-1 (the first row), FF-2 (the second row) and FF-3 (the third row), respectively.[]{data-label="table:robustness-3"} CIFAR-10 FF-1 FF-2 FF-3 ------------------------- ------ ------ ------ Deepfool attacking FF-1 16% 47% 48% Deepfool attacking FF-2 49% 17% 49% Deepfool attacking FF-3 48% 49% 15% : Comparison of CIFAR-10 testing accuracies of three FF designs, against the Deepfool adversarial attacks targeting at FF-1 (the first row), FF-2 (the second row) and FF-3 (the third row), respectively.[]{data-label="table:robustness-4"} If the model parameters of the three FF designs are known, attackers can design adversarial attacks by targeting at each of them individually. The test accuracies of FF-1, FF-2 and FF-3 against the Deepfool attack for the MNIST and the CIFAR-10 datasets are given in Tables \[table:robustness-3\] and \[table:robustness-4\], respectively. We see that an adversarial attack is primarily effective against its target design. To enhance robustness, we may adopt an ensemble method by fusing their classification results (e.g., majority voting, bagging, etc.). This is a rich topic that goes beyond the scope of this work. We will report our further investigation in a separate paper. Although the ensemble method applies to both BP and FF designs, the cost of building an ensemble system of multiple FF networks is significantly lower than that of multiple BB networks. Discussion {#sec:discussion} ========== Some follow-up discussion comments are provided in this section. First, comparisons between FF and BP designs are made in Sec. \[subsec:comparison\]. Then, further insights into the BP and FF designs are given in Sec. \[subsec:relationship\]. General comparison {#subsec:comparison} ------------------ Design BP FF --------------------- ----------------------------- ------------------------------- Principle System optimization centric Data statistics centric Math. Tools Non-convex optimization Linear algebra and statistics Interpretability Difficult Easy Modularity No Yes Robustness Low Low Ensemble Learning Higher Complexity Lower Complexity Training Complexity Higher Lower Architecture End-to-end network More flexible Generalizability Lower Higher Performance State-of-the-art To be further explored : Property comparison of BP and FF designs.[]{data-label="table:comparison"} The properties of FF and BP designs are compared in various aspects in Table \[table:comparison\]. They are elaborated below. [**Principle.**]{} The BP design is centered on three factors: a set of training and testing data, a selected network architecture and a selected cost function at the output end for optimization. Once all three are decided, the BP optimization procedure is straightforward. The datasets are determined by applications. Most research contributions come from novel network architectures and new cost functions that achieve improved target performance. In contrast, the FF design exploits data statistics (i.e. the covariance matrix) to determine a sequence of spatial-spectral transformations in convolutional layers with two main purposes – discriminant dimension generation and dimension reduction. The PCA-based convolution and maximum spatial pooling can be well explained accordingly. No data labels are needed in the FF design of convolutional layers. After that, the FC layers provide sequential LSR operations to enable a multi-stage decision process. Data labels are needed in the FF design of FC layers. They are used to form clusters of data of the same class to build the LSR models. Clustering is related to the sample distribution in a high-dimensional space. It is proper to examine clustering and LSR from a statistical viewpoint. [**Mathematical Tools.**]{} The BP design relies on the stochastic gradient descent technique in optimizing a pre-defined cost function. When a CNN architecture is deeper, the vanishing gradient problem tends to occur. Several advanced network architectures such as the ResNets [@He_2016_CVPR] were proposed to address this issue. The mathematical tool used in this work is basically linear algebra. We expect statistics to play a significant role when our focus moves to training data collection and labeling for a certain task. [**Interpretability.**]{} As stated in Sec. \[sec:introduction\], interpretability of CNNs based on the BP design (or CNN/BP in short) was examined by researchers, e.g., [@zhang2017interpretable; @wang2018interpret]. Although these studies do shed light on some properties of CNN/BP, a full explanation of CNN/BP is very challenging. CNNs based on the FF design (or CNN/FF in short) are mathematically transparent. Since the filter weights selection strategy in CNN/FF is different from that in CNN/BP, our understanding of CNN/FF is not entirely transferable to that of CNN/BP. However, we can still explain the benefit of the multi-layer CNN architecture. Furthermore, the connection between FF and BP designs will be built by studying cross-entropy values of intermediate layers in Sec. \[subsec:relationship\]. [**Modularity.**]{} The BP design relies on the input data and the output labels. The model parameters at each layer are influenced by the information at both ends. Intuitively speaking, the input data space has stronger influence on parameters of shallower layers while the output decision space has has stronger impact on parameters of deeper layers. Shallower and deeper layers capture low-level image features and semantic image information, respectively. The whole network design is end-to-end tightly coupled. The FF design decouples the whole network into two modules explicitly. The subnet formed by convolutional layers is the data representation (or feature extraction) module that has nothing to do with decision labels. The subnet formed by FC layers is the decision module that builds a connection between the extracted feature space and the decision label space. This decoupling strategy is in alignment with the traditional pattern recognition paradigm that decomposes a recognition system into the “feature extraction" module and the ‘classification/regression" module. [**Robustness.**]{} We showed that both BP and FF designs are vulnerable to adversarial attacks in Sec. \[sec:experiments\] when the network model is fixed and known to attackers. We provided examples to illustrate that adversarial attacks lead to catastrophic performance degradation for its target network but not others. This suggests the adoption of an ensemble method by fusing results obtained by multiple FF networks. The cost of generating multiple network models by changing the initialization schemes for the k-means clustering in the FC layers is very low. This is an interesting topic worth further investigation. [**Training Complexity.**]{} The BP is an iterative optimization process. The network processes all training samples once in an epoch. Typically, it demands tens or hundreds of epochs for the network to converge. The FF design is significantly faster than the BP design based on our own experience. We do not report the complexity number here since our FF design is not optimized and the number could be misleading. Instead, we would like to say that it is possible to reduce complexity in the FF design using statistics. That is, both PCA and LSR can be conducted on a small set of training data (rather than all training data). Take the filter design of the first convolutional layer as an example. The input samples of the CIFAR-10 dataset are patches of size $5 \times 5 \times 3 =75$. We need to derive a covariance matrix of dimension $75 \times 75$ to conduct the PCA. It is observed that the covariance matrix converges quickly with hundreds of training images, which is much less than 50,000 training images in the CIFAR-10 dataset. One reason is that each training image provides $28 \times 28=784$ training patches. Furthermore, there exists an underlying correlation structure between training images and patches in the dataset. Both high dimensional covariance matrix estimation [@fan2008high] and regression analysis with selected samples [@cameron2013regression] are topics in statistics. Powerful tools developed therein can be leveraged to lower the complexity of the FF design. [**Architecture.**]{} The BP design has a constraint on the network architecture; i.e. to enable end-to-end BP optimization. Although we apply the FF design to CNNs here, the FF design is generally applicable without any architectural constraint. For example, we can replace the FC layers with the random forest (RF) and the support vector machine (SVM) classifiers. [**Generalizability.**]{} There is a tight coupling between the data space and the decision space in the BP design. As a result, even with the same input data space, we need to design different networks for different tasks. For example, if we want to conduct object segmentation, classification and tracking as well as scene recognition and depth estimation simultaneously for a set of video clips, we need to build multiple CNNs based on the BP design (i.e., one network for one task) since the cost function for each task is different. In contrast, all tasks can share the same convolutional layers in the FF design since they only depend on the input data space. After that, we can design different FC layers for different tasks. Furthermore, we can fuse the information obtained by various techniques conveniently in the FF design. For example, features obtained by traditional image processing techniques such as edge and salient points (e.g. SIFT features) can be easily integrated with FF-based CNN features without building larger networks that combine smaller networks to serve different purposes. [**Performance.**]{} The BP design offers state-of-the-art performances for many datasets in a wide range of application domains. There might be a perception that the performance of the BP design would be difficult to beat since it adopts an end-to-end optimization process. However, this would hold under one assumption. That is, the choice of the network architecture and the cost function is extremely relevant to the desired performance metric (e.g., the mean Average Precision in the object detection problem). If this is not the case, the BP design will not guarantee the optimal performance. The FF design is still in its infancy and more work remains to be done in terms of target performance. One direction for furthermore performance boosting is to exploit ensemble learning. This is particularly suitable for the FF design since it is easier to adopt multiple classifiers after the feature extraction stage accomplished by multiple convolutional layers. Further Insights {#subsec:relationship} ---------------- [**Degree of Freedom and Overfitting.**]{} The number of CNN parameters is often larger than that of training data. This could lead to overfitting. The random dropout scheme [@JMLR:v15:srivastava14a] provides an effective way to mitigate overfitting. We should point out that the number of CNN model parameters is not the same as its degree of freedom in the FF design. This is because that the FF design is a sequential process. Filter weights are determined layer by layer sequentially using PCA or LSR. Since the output from the previous layer serves as the input to the current layer, filter weights of deeper layers are dependent on those of shallower layers. Furthermore, PCA filters are determined by the covariance matrix of the input of the current layer. They are correlated. The LeNet-5 has two convolutional layers. The inputs of the first and the second convolutional layers have a dimension of $5 \times 5=25$ and $5 \times 5 \times 6=150$, respectively. Thus, the covariance matrices are of dimension $25 \times 25$ and $150 \times 150$, respectively. Once these two covariance matrices are estimated, we can find the corresponding PCA filters accordingly. As to FC layers, the dimension of the linear LSR matrix is roughly equal to the product of the input vector dimension and the output vector dimension. For example, for the first FC layer of the LeNet-5, the number of parameters of its LSR model is equal to $$\label{eq:B} (5 \times 5 \times 15 +1) \times 120=45,120,$$ where the DC channel is removed. It is less than 60,000 training images in the MNIST dataset. The parameter numbers of the second FC layer and the output layer are equal to $121 \times 84=10,164$ and $85 \times 10=850$, respectively. [**Discriminability of dimensions.**]{} To understand the differences between the BP and the FF designs, it is valuable to study the discriminant power of dimensions of various layers. The cross-entropy function provides a tool to measure the discriminability of a random variable by considering the probability distributions of two or multiple object classes in this random variable. The lower the cross entropy, the higher its discriminant power. It is often adopted as a cost function at the output of a classification system. The BP algorithm is used to lower the cross entropy of the system so as to boost its classification performance. We use the cross-entropy value to measure the discriminant power of dimensions of an intermediate space. By definition, the cross-entropy function can be written as $$\label{eq:cross-entropy} L = \sum_{i=1}^{N_i} L_i,$$ where $i$ is the sample index, $N_i$ is the total number of data samples, and $L_i$ is the loss of the $i$th sample in form of $$\label{eq:cross-entropy-i} L_i = - \sum_{c=1}^{N_c} y_{i,c} \log p_{i,c},$$ where $c$ is the class index, $N_c$ is the total number of object classes, $p_{i,c}$ is the probability for the $i$th sample in class $c$ and $y_{i,c}$ is a binary indicator ($0$ or $1$). We have $y_{i,c}=1$ if it is a correct classification. Otherwise, $y_{i,c}=0$. In practice, we only need to sum up all correct classifications in computing $L_i$. To compute the cross-entropy value in Eq. (\[eq:cross-entropy\]), we adopt the k-means clustering algorithm, partition samples in the target dimension into $Q$ intervals, and use the majority voting rule to predict its label in each interval. The majority voting rule is adopted due to its simplicity. Since we have the ground-truth of all training samples, we know whether they are correctly classified. Consequently, the correct classification probability, $p_{i,c}$, can be computed based on the classification results in all intervals. ![Comparison of rank-ordered cross-entropy values of BP and FF designs (from left to right): at the output of the second convolutional layer, the first and the second FC layers and the output layer. This illustrative example is obtained by applying the LeNet-5 to the MNIST dataset.[]{data-label="fig:centropy_comparison"}](fig-centropy.pdf){width="\linewidth"} Without loss of generality, we use the LeNet-5 applied to the MNIST dataset as an example and plot rank-ordered cross-entropy values of four vector spaces in Fig. \[fig:centropy\_comparison\]. They are: the output of the second convolutional layer (of dimension $5 \times 5 \times 15 = 375$), the first and the second FC layers and the output layer. The dimensions of the last three are 120, 84 and 10, respectively. We omit the plot of cross-entropy values of the first layers since each dimension corresponds to a local $5 \times 5$ receptive field and their discriminant power is quite weak. We see from Fig. \[fig:centropy\_comparison\] that the cross-entropy values of the FF design are higher than those of the BP design at the output of the 2nd convolutional layers as shown in the leftmost subfigure. This is because the FF design does not use any label information in the convolutional layer. Thus, the discriminability of dimensions (or features) of the FF design is poorer. The cross-entropy values of the FF design in the middle two subfigures are much flatter than those of the BP design. Its discriminant power is more evenly distributed among different dimensions of the first and the second FC layers. The distributions of cross-entropy values of the BP design are steeper with a larger dynamic range. The BP design has some favored dimensions, which are expected to play a significant role in inference in the BP-designed network. The cross-entropy values continue to go lower from the shallow to the deep layers, and they become the lowest at the output layer. We also see that the difference between the first and the second FC layers is not significant. Actually, the removal of the second FC layer has only a small impact to the final classification performance. [**Signal representation and processing.**]{} We may use an analogy to explain the cross-entropy plots of the BP algorithm. For a given terrain specified by a random initialization seed, the BP procedure creates a narrow low-cross-entropy path that connects the input and the output spaces through iterations. Such a path is critical to the performance in the inference stage. Representation BP FF -------------------- ------------------------ ------------------------------ Analogy Matched Filtering Projection onto Linear Space Search Iterative optimization PCA and LSR Sparsity Yes No Redundancy Yes No Orthogonality No Yes Signal Correlation Stronger Weaker Data Flow Narrow-band Broad-band : Comparison of BP and FF designs from the signal representation and processing viewpoint.[]{data-label="table:sp"} We compare the BP and the FF designs from the signal representation and processing viewpoint in Table \[table:sp\]. The BP design uses an iterative optimization procedure to find a set of matched filters for objects based on the frequency of visual patterns and labels. When cat faces appear very frequently in images labeled by the cat category, the system will gradually able to extract cat face contours to represent the cat class effectively. The FF design does not leverage labels in finding object representations but conducts the PCA. As a result, it uses values projected onto the subspace formed by principal components for signal representation. The representation units are orthogonal to each other in each layer. The representation framework is neither sparse nor redundant. Being similar to the sparse representation, the BP design has a built-in dictionary. As compared to the traditional sparse representation, it has a much richer set of atoms by leveraging the cascade of multi-layer filters. The representation framework is sparse and redundant. Signal correlation is stronger in the BP design and weaker in the FF design. Finally, the representation can be viewed as narrow-band and broad-band signals in the BP and FF designs, respectively. ![Hierarchical image segmentation based on the HEVC video coding tool.[]{data-label="fig:HEVC"}](fig-HEVC.pdf){width="0.45\linewidth"} [**Heterogeneous spatial-spectral filtering.**]{} Although CNNs have multiple filters at a convolutional layer, they apply to all spatial locations. It is designed to capture the same pattern in different locations. However, it is not effective for large images with heterogeneous regions. As shown in Fig. \[fig:HEVC\], an image can be segmented into homogeneous regions with the HEVC coding tool [@sullivan2012overview]. Codebooks of PCA filters of different sizes can be developed in the FF design. PCA filters of the first convolutional layers describe the pixel combination rules while those of the second, third and higher convolutional layers characterize the combination rules of input spatial-spectral cuboids. The segmentation-guided FF-CNN design is natural and interesting. Conclusion and Future Work {#sec:conclusion} ========================== An interpretable CNN design based on the FF methodology was proposed in this work. It offers a complementary approach in CNN filter weights selection. We conducted extensive comparison between these two design methodologies. The new FF design sheds light on the traditional BP design. As future extensions, it is worthwhile to develop an ensemble method to improve classification performance and tackle with adversarial attacks as discussed in Sec. \[sec:experiments\]. It will be interesting to provide an interpretable design for advanced CNN architectures such as ResNet, DenseNet and GANs. Furthermore, it is beneficial to introduce more statistical tools. Today’s CNN research has focused much on better and better performances for benchmark datasets with more and more complicated network architectures with respect to target applications. Yet, it is equally important to examine the setup of a CNN solution from the data viewpoint. For a given application or task, what data to collect? What data to label? How many are sufficient? Statistics is expected to play a key role in answering these questions. The FF design methodology is still in its infancy. Our work aims at basic CNN research. It is of exploratory nature. By providing a new research and development topic with preliminary experimental results, we hope that it will inspire more follow-up work along this direction. Appendix: Bias Selection {#subsec:saab .unnumbered} ======================== We use ${\bf x}_{AC} \in R^N$ and ${\bf y}_{AC} \in R^K$ to denote input and output flattened AC random vectors defined on 3D cuboids with two spatial dimensions and one spectral dimension. Anchor vectors, ${\bf a}_k \in R^N$, $1 \le k \le K-1$, are the unit-length vectors obtained by the principal component analysis of ${\bf x}_{AC}$ and used in the Saab transform from ${\bf x}_{AC}$ to ${\bf y}_{AC}$. We would like to add a sufficiently large bias to guarantee that all response elements are non-negative. The correlation output between ${\bf x}_{AC}$ and anchor vector ${\bf a}_k$ can be written as $$\label{eq:corr1} y_{k} = {\bf a}^T_k {\bf x}_{AC}, \quad k=1, \cdots, K-1.$$ Clearly, ${\bf y}_{AC}=(0, y_1, \cdots, y_{K-1})^T$ is in the AC subspace of $R^K$ since ${\bf x}$ is in the AC subspace of $R^N$. Let ${\bf 1}= K^{-1/2} (1, \cdots , 1)^T$ be the constant-element unit vector in $R^K$. We add a constant displacement vector ${\bf d} = d {\bf 1}$ of length $d$ to ${\bf y}_{AC}=(0, y_1, \cdots, y_{K-1})^T + {\bf d}$ and get a shifted output vector ${\bf y}_d$ whose elements are $$\label{eq:corr2} y_{d,k} = {\bf a}^T_k {\bf x} + d \sqrt{K}, \quad k=0, 1, \cdots, K-1.$$ To ensure that $y_{d,k}$ is non-negative, we demand $$\label{eq:d1} d \sqrt{K} \ge - {\bf a}^T_k {\bf x}, \quad k=0, 1, \cdots, K-1.$$ The displacement length, $d$, can be easily bound by using the following inequality: $$\label{eq:ineq1} - ||{\bf a}^T_k|| ||{\bf x}|| \le ||{\bf a}^T_k {\bf x}|| \le ||{\bf a}^T_k|| ||{\bf x}||.$$ Since $||{\bf a}^T_k||=1$, we have $$\label{eq:ineq2} - \max || {\bf x} || \le \max_{\bf x} || {\bf a}^T_k {\bf x} || \le \max || {\bf x} ||.$$ By combining Eqs. (\[eq:d1\]) and (\[eq:ineq2\]), we obtain $$\label{eq:d2} d \ge \frac{1}{\sqrt{K}} \max_{\bf x} || {\bf x} ||.$$ Finally, we have the lower bound on $b_k$ as $$\label{eq:ab2} b_k = d {\sqrt{K}} \ge \max_{\bf x} || {\bf x} ||, \quad k=0, 1, \cdots, K-1.$$ This result is repeated in Eq. (\[eq:d3\]). Without loss of generality, we choose $$\label{eq:ab3} b_k = \max_{\bf x} || {\bf x} || + \delta, \quad \delta > 0, \quad k=0, 1, \cdots, K-1,$$ where $\delta$ is a small positive number in our experiment. Acknowledgment {#acknowledgment .unnumbered} ============== This material is partially based on research sponsored by DARPA and Air Force Research Laboratory (AFRL) under agreement number FA8750-16-2-0173. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of DARPA and Air Force Research Laboratory (AFRL) or the U.S. Government. The authors would also like to give thanks to Dr. Pascal Frossard and Dr. Mauro Barni for their valuable comments to the draft of this work. [^1]: http://yann.lecun.com/exdb/mnist/ [^2]: https://www.cs.toronto.edu/ kriz/cifar.html

--- author: - 'Behnam Pourhassan,' - 'Sanjib Dey,' - 'Sumeet Chougule,' - Mir Faizal title: Quantum Corrections to a Finite Temperature BIon --- Introduction ============ In string theory, it is possible to analyze certain physical objects in a region of space-time in terms of very different objects. Thus, it is possible to analyze a system of many coincident strings in terms of D-brane geometry, and this is done in the BIon solution [@Callan_Maldacena; @Gibbons]. So, this BIon solution can describe an F-string coming out of the D3-brane or a D3-brane parallel to an anti-D3-brane, such that they are connected by a wormhole with F-string charge. This configuration is called as the brane-antibrane-wormhole configuration. It is also possible to use such a solution to analyze D-branes probing a thermal background [@BIon01; @bion]. This can be done using the blackfold approach [@q1; @q2; @q5; @q4]. In this method, a large number of coincident D-branes form a brane probe. Furthermore, as this probe is in thermal equilibrium with the background, this method has been used to heat up a BIon. This was done by putting it in a hot background. It is also possible to analyze the thermodynamics of this finite temperature BIon solution [@BIon01; @bion]. In this paper, we will analyze the effects of thermal fluctuations on the thermodynamics of this system.\ The entropy-area law of black holes thermodynamics [@Hawking; @Bekenstein], is expected to get modified near Planck scale due to quantum fluctuations [@Rama]. These quantum fluctuations in the geometry of any black object are expected to produce thermal fluctuations in its associated thermodynamics. It is interesting to note that the thermal fluctuations produce a logarithmic correction term to the thermodynamics of black objects [@1; @2; @3; @3a]. The consequences of such logarithmic correction have been studied for a charged AdS black hole [@12], charged hairy black hole [@12a], a black saturn [@13], a Hayward black hole [@18] and a small singly spinning Kerr-AdS black hole [@19], and a dyonic charged AdS black hole [@4]. In non-perturbative quantum general relativity, the density of microstates was associated with the conformal blocks has been used to obtain logarithmic corrections to the entropy [@Ashtekar]. It has also been demonstrated that the Cardy formula can produce logarithmic correction terms for all black objects whose microscopic degrees of freedom are characterized by a conformal field theory [@Govindarajan]. The logarithmic correction has also been studied from the black hole in the presence of matter fields [@Mann_Solodukhin] and dilatonic black holes [@Jing_Yan]. Leading order quantum corrections to the semi-classical black hole entropy have been obtained [@5], and applied to Gödel black hole [@6; @koka2]. The logarithmic corrections were also used to study different aspects of regular black holes satisfying the weak energy condition [@10], three-dimensional black holes with soft hairy boundary conditions [@new2], and certain aspects of Kerr/CFT correspondence [@11]. The logarithmic corrected entropy also corrects some hydrodynamical quantities, and so the the field theory dual to such corrected solutions has also been studied [@23; @24; @27; @28; @29].\ The logarithmic corrections to the entropy of a various black hole have also been obtained using the Euclidean Quantum Gravity [@32; @8; @9]. In this approach, Euclidean Quantum Gravity [@hawk] is used to obtain the partition function for the black hole, which is then used to obtain the logarithmic corrections to the thermodynamics of that black hole. As the logarithmic corrections occur almost universally in the thermodynamics of black objects, in this paper we will analyze the consequences of such corrections for a thermal BIon. We compute the quantum correction to the black hole entropy, internal energy, specific heat using the Euclidean Quantum Gravity [@hawk]. We find that the logarithmic correction affects the critical points, and the corrections significantly change the stability of this system. Euclidean Quantum Gravity ========================= Now, we start with the Euclidean Quantum Gravity that is obtained by performing a Wick rotation on the temporal coordinates in the usual path integral. Thus, we obtain gravitational partition function in Euclidean Quantum Gravity [@hawk], $$\label{PF} Z = \int [\mathcal{D}]e^{-\mathcal{I}_\text{E}}= \int_0^\infty \rho(E)e^{-\beta E} \text{d}E,$$ where $\mathcal{I}_\text{E}$ is the Euclidean action for the BIon solution [@BIon01; @bion], and $\beta\propto1/T$. The density of states $\rho (E)$ is easily obtained from (\[PF\]) by performing an inverse Laplace transform, so that one obtains $$\label{CompInt} \rho(E) =\frac{1}{2 \pi i} \int^{a+ i\infty}_{a -i\infty}e^{S(\beta)}\text{d}\beta.$$ Here, $S(\beta)$ is the entropy and its exact form is given in terms of the partition function and the total energy as $S(\beta)=\beta E+\ln Z$, where $S_{0}=S(\beta_{0})$. The complex integral (\[CompInt\]) can be evaluated using the steepest decent method around the saddle point $\beta_0$ so that $[\partial S(\beta)/\partial\beta]_{\beta=\beta_0}$ vanishes and the equilibrium relation $E=-[\partial\ln Z(\beta)/\partial\beta]_{\beta=\beta_0}$ is satisfied. Therefore, the equilibrium temperature is given by $T_{0}=1/\beta_0$, and we can expand the entropy $S(\beta)$ around the equilibrium point $\beta_0$ as follows $$\label{a1} S(\beta)=S_0+\frac{1}{2}(\beta-\beta_0)^2 \left[\frac{\partial^2 S(\beta)}{\partial \beta^2 }\right]_{\beta=\beta_0}+\cdots.$$ Here, the first term $S_0=S(\beta_0)$ denotes the entropy at the equilibrium, the second term represents the first order correction over it. If we restrict ourselves to this first order and replace (\[a1\]) into (\[CompInt\]), we obtain $$\rho(E) = \frac{e^{S_{0}}}{\sqrt{2\pi}} \left\{\left[\frac{\partial^2 S(\beta)}{\partial \beta^2 }\right]_{\beta = \beta_0}\right\}^{- \frac{1}{2}},$$ for $[\partial^2 S(\beta)/\partial\beta^2]_{\beta=\beta_0}>0$, where we choose $a=\beta_0$ and $\beta-\beta_0=ix$ with $x$ being a real variable. Thus, the expression of the microcanonical entropy $\mathcal{S}$ turns out to be [@1; @2] $$\label{MicorEntropy} \mathcal{S}=\ln\rho(E)=S_0-\frac{1}{2}\ln\left\{\left[\frac{\partial^2 S(\beta)}{\partial \beta^2 }\right]_{\beta = \beta_0}\right\}.$$ Note that the entropy $S(\beta)$ given in (\[a1\]) is different from $\mathcal{S}$ as given by (\[MicorEntropy\]), the former $S(\beta)$ being the entropy at any temperature, whereas the latter one, $\mathcal{S}$ is the corrected microcanonical entropy at equilibrium, which is computed by incorporating small fluctuations around thermal equilibrium. However, the result obtained in (\[MicorEntropy\]) is completely model independent and, it can be applied to any canonical thermodynamical system including a BIon solution. Thus, the first order correction is solely governed by the term $[\partial^2 S(\beta)/\partial\beta^2]_{\beta=\beta_0}$. This can be simplified to a generic form of the entropy correction given by $\ln (CT^2)$ [@1; @2].\ Furthermore, it can be demonstrated that for any black object whose degrees of freedom can be counted using a CFT, we can write $ \ln [\partial^2 S(\beta)/\partial\beta^2]_{\beta=\beta_0} = \gamma \ln (S_0 T^2) $ [@2; @3; @3a], where $\gamma$ is a parameter added by hand [@12] to track correction term, hence it is equal one for the corrected entropy, while zero for the original entropy. Now, as this holds for any black object whose degrees of freedom can be analyzed using a CFT [@2; @3; @3a], and it has been argued that degrees of freedom of a BIon can also be analyzed using using a CFT [@cft1y; @cft2y], we can use a similar equation for BIon. So, we propose that the quantum correction to the entropy of a BIon can be expressed as $$\label{Corrected-Entropy} \mathcal{S}=S_{0}-\frac{\gamma}{2}\ln{(S_0T^{2}) \mathcal{Y}} \sim S_{0}-\frac{\gamma}{2}\ln{(S_0T^{2}) - \frac{\gamma}{2} \mathcal{Y}},$$ where $S_0$ is the original entropy of the BIon solution [@BIon01; @bion], and $\mathcal{Y}$ is, in general, a function of other quantities such as the dependence on the D3-brane and string charges. Thus, a full analysis of this system should incorporate such a quantities, but as a toy model, we will analyze the effect of temperature and entropy on the thermodynamics of such a system, and neglect the effect of $\mathcal{Y}$. This can possible be justified by fixing certain quantities in the system, and analyzing it as a toy model. However, the last term of (\[Corrected-Entropy\]) behaves as higher order corrected of entropy [@hnew] which may be ignored.\ It may be noted that such logarithmic corrections terms are universal, and occur in almost all approaches to quantum gravity. However, the coefficient of such logarithmic correction term is model dependent. As the expression used in this paper involves a free parameter $\gamma$, it will hold even using different approaches. As any other approaches to this problem can only change the value of this coefficient $\gamma$, which is not fixed in this paper. Thus, the validity of the (\[Corrected-Entropy\]) can be argued on general grounds, and the main aim of the paper is to analyze the effects of such a logarithmic corrections on the thermodynamics of a BIon solution.\ So, to obtain quantum corrections to the entropy of a BIon solution, we need to use the original entropy $S_0$ of the BIon solution [@BIon01; @bion]. Now a BIonic system is a configuration in a flat space of the D-brane which is parallel to anti-D-brane and they are connected by a wormhole which has a F-string charge. Geometrically, it is composed of $\mathcal{N}$ coincident D-branes which are infinitely extended and has $\mathcal{K}$ units of F-string charge, ending in a throat with minimal radius $\sigma_0$ and at temperature $T$. To construct a wormhole solution from this, all we have to do, is to attach a mirror solution at the end of the throat.\ It is well known that the blackfold action can be used to describe the D-brane for probing the zero temperature background. However, it was shown that one can also use DBI action for probing the thermal backgrounds [@bion], where it is ensured that the brane is not affected by the thermal background, but the degrees of freedom living on the brane are ‘warmed up’ due to the temperature of thermal background. Thus, the thermal background acts as a heat bath to the D-brane probe, and due to this, the probe stays in thermal equilibrium with the thermal background, which is a ten dimensional hot flat space. This is constructed in the blackfold approach, which is a general description for the black holes in the regime where they can be approximated to black brane curved along the sub-manifold of the space-time background. The thermal generalization of BIon solution has also been carried out and the thermodynamic quantities for this configuration are given by [@BIon01; @bion] $$\begin{aligned} M &=& \frac{4T^{2}_{D3}}{\pi T^{4}}\int_{\sigma_{0}}^{\infty}d\sigma \frac{\sigma^{2}(4cosh^{2}\alpha +1)F(\sigma)}{\sqrt{F^{2}(\sigma)-F^{2}(\sigma_{0})}\cosh^{4}\alpha}, \\ S_{0} &=& \frac{4T^{2}_{D3}}{\pi T^{5}}\int_{\sigma_{0}}^{\infty}d\sigma \frac{4\sigma^{2}F(\sigma)}{\sqrt{F^{2}(\sigma)-F^{2}(\sigma_{0})}\cosh^{4}\alpha }, \label{a} \\ \mathcal{F} &=& \frac{4T^{2}_{D3}}{\pi T^{4}}\int_{\sigma_{0}}^{\infty}d\sigma \sqrt{1+z'^{2}(\sigma)} F(\sigma),\end{aligned}$$ which are the total mass, entropy and free energy, respectively. Here, $T_{D3}$ is the D3-brane tension, $z$ is a transverse coordinate to the branes and $F(\sigma)=\sigma^{2}(4\cosh^{2}\alpha-3)/\cosh^{4}\alpha$, with $\sigma$ being the world volume coordinate and $\sigma_{0}$ being the minimal sphere radius of the throat or wormhole. Here $\alpha$ is a function of the temperature. The chemical potentials for the D3-brane and F-string are as follows $$\begin{aligned} \mu_{D3} &=& 8\pi T_{D3}\int_{\sigma_{0}}^{\infty}d\sigma \frac{\sigma^{2} \tanh \alpha \cos \zeta F(\sigma)}{\sqrt{F^{2}(\sigma)-F^{2}(\sigma_{0})}} \\ \mu_{F1} &=& 2 T_{F1}\int_{\sigma_{0}}^{\infty}d\sigma \frac{\tanh \alpha \cos \zeta F(\sigma)}{\sqrt{F^{2}(\sigma)-F^{2}(\sigma_{0})}}\end{aligned}$$ It should be noted that these relations satisfy the first law of thermodynamics $dM=TdS_{0}+\mu_{D3}d\mathcal{N} +\mu_{F1}d\mathcal{K}$ as well as the Smarr relation, $4(M-\mu_{D3} \mathcal{N}-\mu_{F1} \mathcal{K})-5TS_{0}=0$. One can also calculate internal energy and the specific heat of the BIon solution as $$\begin{aligned} {1} U_{0} &=\frac{4T^{2}_{D3}}{\pi T^{4}}\int_{\sigma_{0}}^{\infty}d\sigma F(\sigma)\left[\sqrt{1+z'^{2}(\sigma)}+\frac{4\sigma^{2}}{\sqrt{F^{2}(\sigma)-F^{2}(\sigma_{0})}\cosh^{4}\alpha}\right], \\ C_{0} &= T\bigg(\frac{dS_{0}}{dT}\bigg)=-\frac{20T^{2}_{D3}}{\pi T^{5}}\int_{\sigma_{0}}^{\infty}d\sigma \frac{4\sigma^{2}F(\sigma)}{\sqrt{F^{2}(\sigma)-F^{2}(\sigma_{0})}\cosh^{4}\alpha},\end{aligned}$$ which indicates that the system has a negative specific heat. Corrected Thermodynamics for the BIon {#sec4} ===================================== Let us now look for the thermal corrections to the above equations by considering logarithmic correction to the entropy $S$ given by the equation (\[Corrected-Entropy\]). The entropy (\[a\]) of $\mathcal{N}$ coincident D-branes with a throat solution gets corrected as $$\label{Corrected-Entropy-222} S=\frac{4T^{2}_{D3}}{\pi T^{5}}\int_{\sigma_{0}}^{\infty}d\sigma \frac{4\sigma^{2}F(\sigma)}{\cosh^{4}\alpha\sqrt{F^{2}(\sigma)-F^{2}(\sigma_{0})}}-\frac{\gamma}{2} \ln \left[\frac{4T^{2}_{D3}}{\pi T^{3}}\int_{\sigma_{0}}^{\infty}d\sigma \frac{4\sigma^{2}F(\sigma)}{\cosh^{4}\alpha\sqrt{F^{2}(\sigma)-F^{2}(\sigma_{0})}}\right].$$ In order to analyze the expression for the corrected entropy we can assume $\cosh^{2}\alpha(\sigma_{0})=\frac{3}{4}$, which means that $F(\sigma_{0})=0$ leading to a relatively easy solution. It is indeed a possible solution of the equation of motion at $\sigma_{0}$ [@BIon01]. However, we would like to work on the regime where the branch connected to the extremal BIon, which was also utilized in [@BIon01]. In this formulation one obtains $ \begin{array}{cccc} \includegraphics[width=54 mm]{1-1.eps}\includegraphics[width=54 mm]{1-2.eps}\includegraphics[width=54 mm]{1-3.eps} \end{array}$ $ \begin{array}{cccc} \includegraphics[width=75 mm]{2-1.eps}\includegraphics[width=75 mm]{2-2.eps} \end{array}$ $$\cosh^{2}\alpha=\frac{3}{2}\frac{\cos{\frac{\delta}{3}}+\sqrt{3}\cos{\frac{\delta}{3}}}{\cos\delta},$$ where $\cos\delta={\bar{T}}^{4}\sqrt{1+\mathcal{K}^2/\sigma^{4}}$, with ${\bar{T}}^{4}=9\pi^{2}T^4\mathcal{N}/(4\sqrt{3}T_{D3})$. We further assume an infinitesimal $\delta$ (corresponding to $\sigma^{2}>\mathcal{K}$ at low temperature $0\leq{\bar{T}}<1$), so that $\cos\delta\approx1$ and $\sin\delta\approx\delta\approx{\bar{T}}^{4}[1+1/(2\sigma^{4})]$. Here we have considered $\mathcal{K}=1$, as it can always be absorbed in $\sigma$ by means of a re-scaling. Now the remaining parameter is $T_{D3}$, which we set $T_{D3}=1$ and plot the corrected entropy (\[Corrected-Entropy-222\]) by varying the $\bar{T}$, $\sigma_{0}$ and $\gamma$ as depicted in Figs.\[fig1\],\[fig2\] and \[fig3\]. In Fig.\[fig1\], we draw the corrected entropy in terms of $\bar{T}$ for different values of $\sigma_{0}$. We should note that there is a minimal radius $\sigma_{min}={\bar{T}}^{2}(1-{\bar{T}}^{8})^{-\frac{1}{4}}$ corresponding to each plots, with $\sigma_{0}\geq \sigma_{min}$. In the panel (a) of Fig.\[fig1\] we have considered $\sigma_{0}=0.2$, such that ${\bar{T}}\geq0.45$ and, in such a situation, we can see that the corrected entropy is larger than the uncorrected one. It means that the logarithmic corrections has increased the value of the entropy. Similar thing happens in the panel (b) also, where $\sigma_{0}=1$ and, thus, ${\bar{T}}\geq0.9$. However, the behavior becomes exactly converse when we consider $\sigma_{0}=2$ and ${\bar{T}}\geq0.99$, as shown in Fig.\[fig1\](c), where we see that corrected entropy is smaller than the uncorrected entropy. Therefore, it suggests that the behavior of the entropy after the logarithmic correction depends on both of the parameters, namely, the temperature and $\sigma_{0}$. In Fig.\[fig2\], we plot the corrected entropy as a function of $\sigma_{0}$ for ${\bar{T}}\geq0.5$, i.e. $\sigma_{min}=0.25$. In this case, we see that there exists a critical $\sigma_{c}$ for each of the plots, and when the value of $\sigma_{0}$ is less than the value of $\sigma_{c}$, the corrected entropy is larger than the uncorrected one. Whereas, when $\sigma_{0}>\sigma_{c}$, the corrected entropy is less. However, the critical points depend on the temperature, for instance, in the case when ${\bar{T}}=0.5$, i.e. in Fig.\[fig2\](a), we notice that $\sigma_{c}\approx0.7$, while in Fig.\[fig2\](b), i.e. for ${\bar{T}}\approx1$, $\sigma_{c}\approx1.6$. We should note that the region compatible with our assumption is $\sigma_{0}>1$. Fig.\[fig3\] demonstrates the behavior of the entropy with respect to $\gamma$, which is the coefficient that determines the amount of correction that is imposed into the system. By choosing ${\bar{T}}\geq1$, we can see that the entropy is a decreasing function of $\gamma$ for $\sigma_{0}=2$, while it is an increasing function of $\gamma$ for $\sigma_{0}=1$. It means that for the small throat (smaller than $\sigma_{c}$), the effect of the thermal fluctuation is to increase the entropy which may yield more stability to the system with maximum value of the entropy. On the other hand, a bigger throat may render the instability to the system. $ \begin{array}{cccc} \includegraphics[scale=0.4]{3.eps} \end{array}$ The logarithmic correction also modifies the internal energy and the specific heat. Let us analyze the effects for the modification of the internal energy first, which we compute as follows $ \begin{array}{cccc} \includegraphics[width=75 mm]{4-1.eps}\includegraphics[width=75 mm]{4-2.eps} \end{array}$ $ \begin{array}{cccc} \includegraphics[width=75 mm]{5-1.eps}\includegraphics[width=75 mm]{5-2.eps} \end{array}$ $ \begin{array}{cccc} \includegraphics[scale=0.4]{6.eps} \end{array}$ $$\begin{aligned} U&=\frac{4T^{2}_{D3}}{\pi T^{4}}\int_{\sigma_{0}}^{\infty}d\sigma F(\sigma)\left[\sqrt{1+z'^{2}(\sigma)}+\frac{4\sigma^{2}}{\sqrt{F^{2}(\sigma)-F^{2}(\sigma_{0})}\cosh^{4}\alpha}\right] \nonumber \\ &~~~~-\frac{\gamma T}{2} \ln \left[\frac{4T^{2}_{D3}}{\pi T^{3}}\int_{\sigma_{0}}^{\infty}d\sigma \frac{4\sigma^{2}F(\sigma)}{\sqrt{F^{2}(\sigma)-F^{2}(\sigma_{0})}\cosh^{4}\alpha}\right].\end{aligned}$$ We can perform a graphical analysis similar to the entropy to give similar results. However, we focus only on the behavior of the internal energy with the variation of the parameter $\gamma$. Fig. \[fig4\] shows that, although the variation of the internal energy is smaller with the logarithmic corrections, however, the slope of the increasing or decreasing functions depends on the value of the temperature and the radius of the throat, as expected. For larger radius, the entropy is decreased due to the thermal fluctuations, while for the smaller radius the entropy is increased. Let us now see the effects on the specific heat. The exact expression of the corrected specific heat is given by $$\begin{aligned} \label{heat} C=T\frac{d}{dT}\left(S-\frac{\gamma}{2}\ln [ST^{2}]\right),\end{aligned}$$ which has been analyzed in Fig.\[fig5\]. Here, we show the effect of the logarithmic correction on the specific heat inside the allowed region $1\leq\sigma_{0}$ for Fig.\[fig5\](a), and for the whole range in Fig.\[fig5\](b) in order to explore the general approximate behavior. In the case of $\gamma=0$, we find that the specific heat is entirely negative, however, in the presence of the thermal fluctuations there is some region where it is positive. It means that in the presence of the logarithmic correction, there is a special radius $\sigma_{s}$ for which the specific heat is negative, $\sigma_{0}>\sigma_{s}$, while it is positive for $\sigma_{0}<\sigma_{s}$. As always, the value of the $\sigma_{s}$ depends on the temperature, for instance, in Fig.\[fig5\](a) it is obvious that when we choose ${\bar{T}}=0.9$, we obtain $\sigma_{s}\approx1.03$. On the other hand, in Fig.\[fig5\](b), when we increase $\sigma_{0}$, the difference between the corrected and uncorrected case slowly vanishes. It indicates that the thermal fluctuation becomes relevant for the smaller radius. Also, we can see from the Fig.\[fig5\](b) an asymptotic behavior which may be interprets as a phase transition as found in [@BIon01], which we show that it is due to the thermal fluctuations. Finally, we demonstrate the variation of the specific heat with the parameter $\gamma$ in Fig.\[fig6\]. It is obvious that the effect of the logarithmic correction is to increase the specific heat and, for $\gamma>0.65$ (approximately) the specific heat is completely positive, while for $\gamma=0$ it is completely negative. Conclusion ========== Quantum fluctuation is an important phenomenon while dealing with objects of very small length scales (close to the Planck length). It can be neglected while the object is large enough compared to the Planck scale, however, for small objects the quantum fluctuation become important. We analyze the quantum corrections for the BIonic systems. We also identify the critical points by including the correction terms over the standard thermodynamical quantities. Moreover, by analyzing the stability conditions, we show how relevant these fluctuation are in the given context. As it turns out that the quantum fluctuation highly affects the critical points and, thus, the stability of the system. The stability is increased under certain relevant conditions, for instance, when the throat is smaller, the inclusion of fluctuation effects increases the stability. Our analysis explicitly shows how the quantum fluctuation terms dominate with the decrease of the radius. As for instance, Fig.\[fig5\](b) tells that while we increase the radius $\sigma_0$, the correction term slowly vanishes and the result merges with that of the uncorrected one. Apart from the stability analysis, we have computed the corrections to the internal energy and specific heat due to the quantum fluctuation. We have also demonstrated the change of the behavior of the corrected system with the temperature. It is possible to construct a BIon solution in M-theory using a system of M2-branes and M5-branes [@m2]. It would be interesting to analyze the system at a finite temperature. Then the thermodynamics of this system can be studied. It would be possible to study the quantum fluctuations to the geometry of a BIon in M-theory, and these could produce thermal fluctuations in the thermodynamics of this system. It would be interesting to analyze the critical points for such a system, and study the effects of these fluctuations on the stability of this system. It may also be noted that the thermodynamics of AdS black hole been studied in M-theory [@iib1; @iib2]. It is possible to analyze the quantum corrections to these black holes, and this can also be done in Euclidean Quantum Gravity. It would also be interesting to generalize the work of this paper to such AdS black holes in M-theory.\ \ **** S.D. acknowledges the support of research grant (DST/INSPIRE/04/2016/ 001391) from the Department of Science and Technology, Govt. of India. S.C. is funded by Conicyt grant 21181211. The Centro de Estudios Científicos (CECs) is funded by the Chilean Government through the Centers of Excellence Base Financing Program of Conicyt. B. Pourhassan would like to thanks Iran Science Elites Federation. [99]{} J. Callan, Curtis G. and J. M. Maldacena, Nucl. Phys. B 513 198 (1998) G. W. Gibbons, Nucl. Phys. B 514, 603 (1998) G. Grignani, T. Harmark, A. Marini, N. A. Obers and M. Orselli, Nucl.Phys.B [851]{}, 462 (2011) G. Grignani, T. Harmark, A. Marini, N. A. Obers, and M. Orselli, JHEP [1106]{}, 058 (2011) R. Emparan, T. Harmark, V. Niarchos, N. A. Obers and M. J. Rodriguez, JHEP 10, 110 (2007) R. Emparan, T. Harmark, V. Niarchos and N. A. Obers, Phys. Rev. Lett. 102, 191301 (2009) R. Emparan, T. Harmark, V. Niarchos and N. A. Obers, JHEP 03, 063 (2010) R. Emparan, T. Harmark, V. Niarchos and N. A. Obers, JHEP 04, 046 (2010) S.W.Hawking, Nature [248]{} 30 (1974) J.D.Bekenstein, Phys.Rev.D [23]{}, 287 (1981) S.K.Rama, Phys.Lett.B [457]{}, 268 (1999) S.S.More, Class. Quant. Grav. [22]{}, 4129 (2005) S.Das, P.Majumdar and R.K.Bhaduri, Class. Quant. Grav. [19]{}, 2355 (2002) S. Carlip, Class. Quant. Grav. [17]{}, 4175 (2000) B. Pourhassan, M. Faizal, Z. Zaz and A. Bhat, Phys.Lett.B [773]{}, 325 (2017) E. Poppitz and M. Unsal, JHEP [ 0907]{}, 060 (2009) M. M. Anber and E. Poppitz, JHEP [ 1106]{}, 136 (2011) B.Pourhassan and M.Faizal, Europhys.Lett. [111]{}, 40006 (2015) J.Sadeghi, B.Pourhassan and F.Rahimi, Can.J.Phys. [92]{}, 1638 (2014) M.Faizal and B.Pourhassan, Phys.Lett.B [751]{}, 487 (2015) B.Pourhassan, M.Faizal and U.Debnath, Eur.Phys.J.C [76]{}, 145 (2016) B.Pourhassan and M.Faizal, Nucl.Phys.B [913]{}, 834 (2016) J.Sadeghi, B.Pourhassan and M.Rostami, Phys.Rev.D [94]{}, 064006 (2016) A.Ashtekar, [*Lectures on non-perturbative canonical gravity*]{}, World Scientific: Singapore (1991) T.R.Govindarajan, R.K.Kaul and V.Suneeta, Class.Quantum Gravity [18]{}, 2877 (2001) R.B.Mann and S.N.Solodukhin, Nucl.Phys.B [523]{}, 293 (1998) J.Jing and M.L.Yan, Phys.Rev.D [63]{}, 24003 (2001) R.K.Kaul and P.Majumdar, Phys.Rev.Lett. [84]{}, 5255 (2000) A. Pourdarvish, J. Sadeghi, H. Farahani and B. Pourhassan, Int.J.Theor.Phys. [52]{}, 3560 (2013) B.Pourhassan, K.Kokabi and Z.Sabery, Ann.Phys. [399]{}, 181 (2018). N. Morales-Durán, A. F. Vargas, P. Hoyos-Restrepo and P. Bargueno, Eur.Phys.J.C [76]{}, 559 (2016) D. Grumiller, A. Perez, D. Tempo and R. Troncoso, JHEP [08]{}, 107 (2017) A. Pathak, A. P. Porfyriadis, A. Strominger and O. Varela, JHEP [04]{}, 090 (2017) G.Policastro, D.T.Son and A.O.Starinets, JHEP [09]{}, 043 (2002) P.Kovtun, D.T.Son and A.O.Starinets, JHEP [10]{}, 064 (2003) A.Buchel and J.T.Liu, Phys.Rev.Lett. [93]{}, 090602 (2004) J.Sadeghi, B.Pourhassan and S.Heshmatian, Adv.High Energy Phys. [2013]{}, 759804 (2013) B.Pourhassan, M.Faizal, Eur.Phys.J.C [77]{}, 96 (2017) J. R. Mureika, J. W. Moffat and M. Faizal, Phys. Lett. B [757]{}, 528 (2016) A.Sen, Gen.Rel.Grav. [44]{}, 1947 (2012) A.Sen, JHEP [04]{}, 156 (2013) G.W.Gibbons and S.W.Hawking, Phys.Rev.D [15]{}, 2752 (1977) B. Pourhassan, S. Upadhyay, H. Farahani, Int. J. Mod. Phys. A \[arXiv:1701.08650\] C. Krishnan, Nucl. Phys. Proc. Suppl.  [ 192]{}, 132 (2009) S. Katmadas and A. Tomasiello, JHEP [ 1512]{}, 111 (2015) A. Belhaj, M. Chabab, H. El Moumni, K. Masmar and M. B. Sedra, Eur. Phys. J. C [76]{}, no. 2, 73 (2016)

--- abstract: 'In $\delta$ Scuti star models the photometric amplitudes and phases exhibit a strong dependence on convection, which enters through the complex parameter $f$, that describes the bolometric flux variation. We present a new method of extracting $\ell$ and $f$ from multi-colour data and apply it to several $\delta$ Scuti stars. The inferred values of $f$ are sufficiently accurate to yield an useful constraint on models of stellar convection.' author: - | J. Daszy[ń]{}ska-Daszkiewicz$^{1,2}$, W.A. Dziembowski$^{3,4}$,\ A.A. Pamyatnykh$^{3,5,6}$ title: 'On application of multi-colour photometry of $\delta$ Scuti stars' --- \#1[[*\#1*]{}]{} \#1[[*\#1*]{}]{} = \#1 1.25in .125in .25in Introduction ============ Until now the main application of multi-colour photometry of $\delta$ Scuti stars has been the determination of the spherical harmonic degree, $\ell$, of the observed modes (e.g. Balona & Evers 1999, Garrido 2000). The amplitudes and phases from measurements in various passbands do indeed contain a signature of the $\ell$ value, but there is more information to be extracted from these quantities. In principle we may calculate their theoretical counterparts. For this we need appropriate stellar models and a linear nonadiabatic code for calculating model oscillations. From such calculations we derive the complex ratio of the local flux variation to the radial displacement at the photosphere, $f$. In the case of $\delta$ Scuti stars, however, there is a large uncertainty arising from lack of adequate theory of stellar convection. The uncertainity is reflected in a strong sensitivity of $f$ to the mixing-length prameter, $\alpha$, which translates to calculated mode positions in the amplitude ratio vs. phase difference diagrams. This strong sensitivity to the treatment of convection is not necessarily bad news, if we can determine simultaneously $\ell$ and $f$. For this aim we need data from at least three passbands. If we succeed, then the $f$-value yields a useful constraint on convection. In addition, if the identified mode is radial, the multi-passband data may be used to refine stellar parameters. Finally, the use of radial velocity measurements in our method improves significantly determination of $\ell$ and $f$. Method for inferring $f$ parameter from observations ==================================================== The presented method is based on $\chi^2$ minimization assuming trial values of $\ell$. We regard the $\ell$ degree and the associated complex $f$ value as the solution if it corresponds to $\chi^2$ minimum, which is much deeper than at other values of $\ell$. We write the photometric complex amplitude in the form of the linear observational equation for a number of passbands, $\lambda$, $${\cal D}_{\ell}^{\lambda} ({\tilde\varepsilon} f) +{\cal E}_{\ell}^{\lambda} {\tilde\varepsilon} = A^{\lambda},\eqno(1)$$ where $${\tilde\varepsilon}\equiv \varepsilon Y^m_{\ell}(i,0),~~~~ {\cal D}_{\ell}^{\lambda}f\equiv b_{\ell}^{\lambda} D_{1,\ell}^{\lambda},~~~~ {\cal E}_{\ell}^{\lambda}\equiv b_{\ell}^{\lambda}(D_{2,\ell}+D_{3,\ell}^{\lambda}).$$ On the right-hand side we have measured amplitudes, $A^\lambda$, expressed in the complex form. The quantities to be determined are $({\tilde\varepsilon} f)$ and ${\tilde\varepsilon}$. If we have data on spectral line variations, the set of equations (1) may be supplemented with an expression for the first moment, ${\cal M}_1^{\lambda}$, $${\rm i}\omega R \left( u_{\ell}^{\lambda} + \frac{GM v_{\ell}^{\lambda} }{R^3\omega^2} \right) \tilde\varepsilon ={\cal M}_1^{\lambda}.\eqno(2)$$ For more details and generalization of the method to the case of modes coupled by rotation see Daszyńska-Daszkiewicz et al. (2003). Applications ============ Here we rely on Kurucz (1998) models of stellar atmospheres and Claret (2000) computations of limb-darkening coefficients. For application of the method we used $uvby$ Strömgren photometry in all cases. [**$\beta$ Cas**]{}: $\beta$ Cas is a near and bright $\delta$ Scuti star, therefore we have rather precise parameters for it. Its estimated value of mass is $1.95 M_{\odot}$. In the left panel of Fig.1 we show $\chi^2$ as a function of $\ell$ obtained for three models. The minimum at $\ell=1$ is the deepest one, particularly at lower $T_{\rm eff}$. Because the star is a relatively rapid rotator ($v_{\rm rot} >70$ km/s), we have considered the possibility that the mode is an $\ell =0$ and 2 combination resulting from the rotatational coupling. We found, however, that at no inclination is the $\chi2$ value as low as at a single $\ell=1$. Thus, the latter identification is most likely. In the right panel we show a comparison of the $f$ values inferred from observations for $\beta$ Cas with the theoretical values calculated with the five indicated values of the MLT parameter, $\alpha$. The observed values of $f_R$ are closer to those calculated with $\alpha=0$, however values of $f_I$ require rather higher values of $\alpha$.\ [**20 CVn**]{}: 20 CVn is a $\delta$ Scuti variable regarded to be monoperiodic with a metal abundance of \[m/H\]=0.5 $(Z\approx0.06)$. For models on the edges of the error box at $Z=0.06$, the minimum of the $\chi^2$ is clearly at $\ell=0$. Having such a mode identification we can refine stellar parameters by fitting them to reproduce the observed period. In the HR diagram, shown in the left panel of Fig. 2, we plot the error box and the lines of the constant period for radial orders $n=3,4,5$ at $Z=0.06$. Only models along these lines are allowed. We cas see that we have two possibilities: $n=3$ or $n=4$. The models yielding the lowest $\chi^2$ are those marked with numbers 1 and 3, both equally probable. In the case of $Z=0.04$ only $n=4$ is allowed, but the $\chi^2$ is much higher. Thus from our method we have also some constraints on metallicity. In the right panel we compare the empirical and theoretical values of the $f$ parameter for models corresponding to the deepest $\chi^2$ minima. The positions are qualitatively similar to those in $\beta$ Cas, but considerably higher as a consequence of higher radial order. [**1 Mon**]{}: 1 Mon has three almost equally distant frequencies. Here we present results for the dominant mode. In the left panel of Fig. 5 we show $\chi^2$ as a function of $\ell$ obtained from $uvby$ photometry, for the models on the edges of error box. In the right panel we have the same, but after including data on radial velocity changes. Discrimination is now much better. Especially the higher values of $\ell$ and some sets of stellar parameters are clearly excluded. The empirical values of the $f$ parameter are now very close to those calculated with $\alpha=0.0$. We obtained $f_{\rm obs}=(-4.9\pm 0.7,~11.3\pm0.8)$ and $f_{\rm calc}=(-2.4,~11.2)$. Conclusions =========== We have argued that multi-colour photometry of $\delta$ Scuti stars may give us useful information not only about the excited modes but also about stars themselves. We believe that the most interesting is the prospect of probing the efficiency of convective transport in outer layers. We also showed that inferrence on mode degree and stellar properties is much improved by supplementing photometric data with radial velocity measurements. The KBN grant No.5 P03D 012 20 is acknowledged. Balona, L.A., Evers, E.A. 1999, , 302, 349 Claret, A. 2000, , 363, 1081 Daszyńska-Daszkiewicz, J., Dziembowski, W.A., Pamyatnykh, A.A. 2003, , 407, 999 Garrido, R., 2000, ASP Conf. Ser. 210, Delta Scuti and Related Stars, eds. M. Breger& M.H. Montgomery, (San Francisco ASP), 67 Kurucz, R.L. 1998, http://cfaku5.harvard.edu

--- address: ' $^{1}$Visiting Astronomer, Canada–France–Hawaii Telescope, which is operated by the National Research Council of Canada, le Centre National de Recherche Scientifique, and the University of Hawaii. $^{2}$Department of Astronomy, University of Toronto, Toronto ON, M5S 3H8 Canada. $^{3}$Center for Astrophysics & Space Astronomy, University of Colorado, CO 80309, USA. $^{4}$Dominion Astrophysical Observatory, Herzberg Institute of Astrophysics, National Research Council of Canada, 5071 West Saanich Road, Victoria, BC, V8X 4M6, Canada. $^{5}$Institute of Astronomy, Madingley Road, Cambridge CB3 OHA, UK. $^{6}$Department of Physics & Astronomy, University of Victoria, Victoria, BC, V8W 3P6, Canada. $^{7}$Department of Physics & Astronomy, University of California, Irvine, CA 92717, USA. $^{8}$[*on leave at:*]{} Carnegie Observatories, 813 Santa Barbara Street, Pasadena, CA 91101, USA. ' author: - 'R. G. Carlberg, H. K. C. Yee, E. Ellingson, S. L. Morris, H. Lin, M. Sawicki, D. Patton, G. Wirth, R. Abraham, P. Gravel, C. J. Pritchet, T. Smecker-Hane D. Schade, F. D. A. Hartwick J. E. Hesser, J. B. Hutchings, & J. B. Oke' title: 'The $\Omega_M-\Omega_\Lambda$ Constraint from CNOC Clusters' --- 1 200 \#1[to 0pt[\#1]{}]{} \#1[$^{#1}$]{} Introduction ============ In the Friedmann-Robertson-Walker solution for the structure of the universe the geometry and future of the expansion uniquely depend on the mean mass density, $\rho_0$, and a possible cosmological constant. It is a statement of arithmetic[@oort] that at redshift zero $\Omega_M \equiv\rho_0/\rho_c= M/L \times j/\rho_c$, where $M/L$ is the average mass-to-light ratio of the universe and $\rho_c/j$ is the closure mass-to-light ratio, with $j$ being the luminosity density of the universe. Estimates of the value of $\Omega_M$ have a long history with a substantial range of cited results[@bld]. Both the “Dicke coincidence” and inflationary cosmology would suggest that $\Omega_M=1$. The main thrust of our survey is to clearly discriminate between $\Omega_M=1$ and the classical, possibly biased, indicators that $\Omega_M\simeq 0.2$. Rich galaxy clusters are the largest collapsed regions in the universe and are ideal to make an estimate of the cluster $M/L$ which can be corrected to the value which should apply to the field as a whole. To use clusters to estimate self-consistently the global $\Omega_M$ we must, as a minimum, perform four operations. Survey Design ============= The Canadian Network for Observational Cosmology (CNOC) designed observations to make a conclusive measurement of $\Omega_M$ using clusters[@yec]. The clusters are selected from the X-ray surveys, primarily the Einstein Medium Sensitivity Survey[@emss1; @emss2; @gl], which has a well defined flux-volume relation. The spectroscopic sample, roughly one in two on the average, is drawn from a photometric sample which goes nearly 2 magnitudes deeper, thereby allowing an accurate measurement of the selection function. The sample contains 16 clusters spread from redshift 0.18 to 0.55, meaning that evolutionary effects are readily visible, and any mistakes in differential corrections should be more readily detectable. For each cluster, galaxies are sampled all the way from cluster cores to the distant field. This allows testing the accuracy of the virial mass estimator and the understanding of the differential evolution process. We introduce some improvements to the classical estimates of the velocity dispersion and virial radius estimators, which have somewhat better statistical properties. A critical element is to assess the errors in these measurements. The random errors are relatively straightforward and are evaluated using either the statistical jackknife or bootstrap methods[@et], which follow the entire complex chain of analysis from catalogue to result. The data are designed to correct from the $M/L$ values of clusters to the field $M/L$. Results ======= We find[@profile] that $\Omega_M=0.19\pm0.06$ (in a $\Omega_\Lambda=0$ cosmology, which is the formal $1\sigma$ error. In deriving this result we apply a variety of corrections and tests of the assumptions. These results rule out $\Omega_M=1$ in any component with a velocity dispersion less than about 1000 . $\Omega_\Lambda$ dependence =========================== The luminosity density, $j$, contains the cosmological volume element which has a very strong cosmology dependence. The cosmological dependence of the $\Omega_e(z)$ can be illustrated by expanding the cosmological terms to first order in the redshift, $z$, $$\Omega_e(z) \simeq \Omega_M [1 + {3\over 4}(\Omega_M^i -\Omega_M+2\Omega_\Lambda)z], \label{eq:ao}$$ where $\Omega_M$ and $\Omega_\Lambda$ are the true values and $\Omega_M^i$ with $\Lambda=0$ is the cosmological model assumed for the sake of the calculation[@lambda]. If there is a non-zero $\Lambda$ then $\Omega_e(z)$ will vary with redshift. The available data are the CNOC1 cluster $M/L$ values and the 3000 galaxies of the preliminary CNOC2[@cnoc2_pre] field sample for $j$. To provide a well defined $\Omega_e(z)$ we limit both the field and cluster galaxy luminosities at $M_r^{k,e}\le -19.5$ mag, which provides a volume limited sample over the entire redshift range. A crucial advantage is of using high luminosity galaxies alone is that they are known to have a low average star formation rate and evolve slowly with redshift, hence their differential corrections are small, and reasonably well determined[@profile]. The results are displayed in Figure 1. The fairly narrow redshift range available does not provide a very good limit on $\Omega_\Lambda$, although values $\Omega_\Lambda>1.5$ are ruled out. The power of this error ellipse is to use it in conjunction with other data, such as the SNIa results which provide complementary constraints on the $\Omega_M-\Omega_\Lambda$ pair. 8truecm [Figure 1: The CNOC1 cluster $M/L$ values combined with the CNOC2 measurements of $j$ for $M_r^{k,e}\le -19.5$ mag galaxies, gives an $\Omega_e(z)$ which leads to the plotted $\chi^2$ (68% and 90% confidence) contours.]{} The limit on the $\Omega_M-\Omega_\Lambda$ pair in Figure 1 has been corrected for known systematic errors which are redshift independent scale errors in luminosity and mass. The high luminosity galaxies in both the cluster and field populations are evolving at a statistically identical rate with redshift, which is close passive evolution. If the cluster galaxies are becoming more like the field with redshift ([*i. e.*]{} the Butcher-Oemler effect, which is partially shared with the field), so that they need less brightening to be corrected to the field, then that would raise the estimated $\Omega_\Lambda$, although the correction is so small that the correction would be $\Delta\Omega_\Lambda\simeq0.3$ over this redshift interval. The results are completely insensitive to galaxy merging that produces no new stars. The data indicate that there is no excess star formation in cluster galaxies over the observed redshift range, with galaxies fading as they join the cluster[@profile; @balogh]. The fact that evolution of the high luminosity field galaxies is very slow and consistent with pure luminosity evolution[@profile] (Lin, in preparation) gives us confidence that the results are reasonably well understood. It will be very useful to have data that extends to both higher and lower redshift, which would allow a measurement of $\Omega_\Lambda$ and better constraints on any potential systematic errors. References {#references .unnumbered} ========== [99]{} Abraham, R. G., Smecker-Hane, T. A., Hutchings, J. B., Carlberg, R. G., Yee, H. K. C., Ellingson, E., Morris, S. L., Oke, J. B., & Rigler, M. 1996, , 471, 694 Bahcall, N. A., Lubin, L. M., & Dorman, V. 1995, , 447, L81 Balogh, M. L., Morris, S. L., Yee, H. K. C., Carlberg, R. G., & Ellingson, E. 1997, ApJ 488, L75 Carlberg, R. G. 1997, , submitted (astroph/9708054) Carlberg, R. G., Yee, H. K. C., Ellingson, E., Abraham, R., Gravel, P., Morris, S. L., & Pritchet, C. J. 1996, , 462, 32 Carlberg, R. G., Yee, H. K. C., & Ellingson, E. 1997, , 478, 462 Carlberg, R. G., Morris, S. L., Yee, H. K. C., & Ellingson, E. 1997b, , 479, L19 Carlberg, R. G., Yee, H. K. C., Ellingson, E., Morris, S. L., Abraham, R., Gravel, P., Pritchet, C. J., Smecker-Hane, T., Hartwick, F. D. A., Hesser, J. E., Hutchings, J. B., & Oke, J. B. 1997, , 485, L13 Carlberg, R. G., Yee, H. K. C., Ellingson, E., Morris, S. L., Abraham, R., Gravel, P., Pritchet, C. J., Smecker-Hane, T., Hartwick, F. D. A., Hesser, J. E., Hutchings, J. B., & Oke, J. B. 1997, , submitted Efron, B. & Tibshirani, R. 1986, [*Statistical Science*]{}, 1, 54 Gioia, I. M. & Luppino, G. A. 1994, , 94, 583 Gioia, I. M., Maccacaro, T., Schild, R. E., Wolter, Stocke, J. T., Morris, S. L., & Henry, J. P. 1990, , 72, 567 Henry, J. P., Gioia, I. M., Maccacaro, T., Morris, S. L., Stocke, J. T., & Wolter, A. 1992, , 386, 408 Lin, H., Yee., H. K. C., Carlberg, R. G., & Ellingson, E. 1997, , 475, 494 Navarro, J. F., Frenk, C. S. & White, S. D. M. 1997, , 490, 493 Oort, J. H. 1958, in [*La Structure et L’Évolution de L’Univers*]{}, Onzième Conseil de Physique, ed. R. Stoops (Solvay: Bruxelles) p. 163 Press, W. H. & Schechter, P. 1974, , 187, 425 Schade, D., Carlberg, R. G., Yee, H. K. C., López-Cruz, O. & Ellingson, E. 1996a, , 464, L63 Schade, D., Carlberg, R. G., Yee, H. K. C., López-Cruz, O. & Ellingson, E. 1996b, , 465, L103 Yee, H. K. C., Ellingson, E. & Carlberg, R. G. 1996, , 102, 269 Yee, H. K. C., Sawicki, M. J., Carlberg, R. G., Morris, S. L., Patton, D. R., Wirth, G. D., Shepherd, C. W., Ellingson, E., Schade, D., & Marzke, R. 1997 in [*Redshift Surveys in the 21st Century*]{}, JD11 at the 23rd IAU General Assembly (astro-ph/971036)

--- abstract: 'We study the spin and thermal conductivity of spin-1/2 ladders at finite temperature. This is relevant for experiments with quantum magnets. Using a state-of-the-art density matrix renormalization group algorithm, we compute the current autocorrelation functions on the real-time axis and then carry out a Fourier integral to extract the frequency dependence of the corresponding conductivities. The finite-time error is analyzed carefully. We first investigate the limiting case of spin-1/2 XXZ chains, for which our analysis suggests non-zero dc-conductivities in all interacting cases irrespective of the presence or absence of spin Drude weights. For ladders, we observe that all models studied are normal conductors with no ballistic contribution. Nonetheless, only the high-temperature spin conductivity of XX ladders has a simple diffusive, Drude-like form, while Heisenberg ladders exhibit a more complicated low-frequency behavior. We compute the dc spin conductivity down to temperatures of the order of $T\sim 0.5J$, where $J$ is the exchange coupling along the legs of the ladder. We further extract mean-free paths and discuss our results in relation to thermal conductivity measurements on quantum magnets.' author: - 'C. Karrasch' - 'D. M. Kennes' - 'F. Heidrich-Meisner' bibliography: - 'references.bib' title: Spin and thermal conductivity of quantum spin chains and ladders --- Introduction ============ Low-dimensional quantum magnetism is a field in which an extraordinary degree of quantitative agreement between experimental results and theory has been achieved due to the availability of both high-quality samples and powerful theoretical tools such as bosonization [@giamarchi], Bethe ansatz [@kluemper-book], series expansion methods [@knetter00; @trebst00], or the density matrix renormalization group [@white92; @schollwoeck05]. This includes the thermodynamics [@johnston00a; @johnston00], inelastic neutron scattering data [@notbohm07; @lake13], as well as various other spectroscopic methods [@windt01]. While there are also exciting experimental results for spin diffusion probed via NMR [@thurber01; @branzoli11] or $\mu$sr [@maeter; @pratt06] as well as for the thermal conductivity [@sologubenko07; @hess07], the calculation of finite-temperature linear-response transport coefficients poses a formidable problem for theorists (see Refs.  for a review), which is further complicated by the need to account for phonons and impurities (see Refs.  for work in this direction). Very recently, significant progress has been made in the computation of linear response transport properties of the seemingly simplest one-dimensional model, the integrable spin-1/2 XXZ chain with an exchange anisotropy $\Delta$. Its Hamiltonian reads: $$\label{eq:hxxz} H = J \sum_{n=1}^{L-1} \left[ S^x_{n}S^x_{n+1} + S^y_{n}S^y_{n+1} + \Delta S^z_{n}S^z_{n+1}\right]\,,$$ where $S^{x,y,z}_{n}$ is a spin-1/2 operator acting on site $n$. The spin and thermal conductivities generally take the form $$\label{eq:sigma}\begin{split} \mbox{Re}\,\sigma(\omega) &= 2\pi D_{\textnormal}{s\phantom{h}}\delta(\omega) + \sigma_{\textnormal}{reg}(\omega)\,,\\ \mbox{Re}\,\kappa(\omega) &= 2\pi D_{\textnormal}{th}\delta(\omega) + \kappa_{\textnormal}{reg}(\omega)\,, \end{split}$$ where $D_{\textnormal}{s,th}$ denote the Drude weights, and $\sigma_{\textnormal}{reg}$ and $\kappa_{\textnormal}{reg}$ are the regular parts. The exact conservation of the energy current [@zotos97] of the XXZ chain renders the zero-frequency thermal conductivity strictly divergent at all temperatures, i.e., $D_{\textnormal}{th}>0$, $\kappa_{\textnormal}{reg}=0$. The thermal Drude weight has been calculated exactly [@kluemper02; @sakai03]. For spin transport, the following picture emerges: while there is a regular contribution $\sigma_{\textnormal}{reg}>0$ for all $|\Delta|>0$ [@naef98], the Drude weight $D_{\textnormal}{s}$ is non-zero for $|\Delta|<1$ but vanishes for $|\Delta|>1$. Initially, these results were largely based on numerical simulations [@zotos96; @narozhny98; @hm03; @heidarian07; @herbrych11; @karrasch12; @karrasch13] as well as analytical approaches using the Bethe ansatz [@zotos99; @peres99; @benz05]. Recently, a rigorous proof of finite spin Drude weights for $|\Delta|<1$ has been obtained [@prosen11; @prosen13] by relating $D_{\textnormal}{s}>0$ to the existence of a novel family of quasi-local conservation laws via the Mazur inequality [@zotos97]. For the experimentally most relevant case of the spin-1/2 Heisenberg chain ($\Delta=1$), it is still debated whether or not a ballistic contribution exists at finite temperatures (see Refs.  for recent work). The same questions of diffusive versus ballistic transport can be addressed in non-equilibrium setups [@langer09; @jesenko11; @langer11; @karrasch14] or for open quantum systems [@prosen09; @znidaric11; @znidaric13a; @mendoza-arenas13]. A recent quantum gas experiment, in which the ferromagnetic Heisenberg chain was realized with a two-component Bose gas, studied the decay of a spin spiral, and the results were interpreted in terms of diffusion [@hild14]. Other non-equilibrium experiments with quantum gases have investigated the mass transport of fermions [@schneider12] and bosons [@ronzheimer13] in optical lattices. Another very interesting question pertains to the functional form of the regular part $ \sigma_{\rm reg }(\omega)$. A field-theoretical study [@sirker09; @sirker11], which incorporates the leading irrelevant umklapp term, suggests that $ \sigma_{\rm reg }(\omega)$ has a simple diffusive form at low temperatures $T\ll J$. This is consistent with early results [@giamarchi91] for the generic behavior of a Luttinger liquid in the presence of umklapp scattering as well as with Quantum Monte Carlo simulations for $\Delta=1 $ [@grossjohann10]. At higher temperatures, a suppression of weight at low frequencies according to $ \sigma_{\rm reg }(\omega) \propto \omega^2$ has been suggested [@herbrych12]. Most studies of $|\Delta|>1$ are interpreted in terms of diffusive spin dynamics, i.e., finite dc-conductivities [@prosen09; @steinigeweg09; @znidaric11; @steinigeweg12; @karrasch14; @karrasch14a]; however, indications of an anomalous low-frequency response were reported in Ref. . The theory by Sachdev and Damle provides a semi-classical interpretation for the emergence of diffusive dynamics in gapped spin models and predictions for the low-temperature dependence of the diffusion constant [@sachdev97; @damle98; @damle05]. Many transport experiments on quantum magnets probe materials which are described by quasi-one dimensional models more complicated than the integrable XXZ chain. Most notably, very large thermal conductivities due to spin excitations have been observed in spin-ladder compounds [@sologubenko00; @hess01; @hess07], which more recently have also been investigated using real-time techniques [@otter09; @montagnese13; @hohensee14]. Most theoretical studies of non-integrable models suggest the absence of ballistic contributions [@zotos96; @rosch00; @hm03; @hm04; @zotos04; @karadamoglou04; @jung06] (possible exceptions have been proposed in Refs. ). Numerical results for the expansion of local spin and energy excitations in real space are consistent with diffusive dynamics [@langer09; @karrasch14]. A qualitatively similar picture has emerged from studies of transport in open quantum systems [@prosen09; @mendoza-arenas14]. Despite the relevance for experiments, however, transport properties of generic non-integrable systems are still not fully understood quantitatively. Two important and largely open problems in the realm of spin ladders are (a) the question of whether they exhibit standard diffusive dynamics or a more complicated low-frequency behavior, and (b) a quantitative calculation of their dc spin and thermal conductivities. It turns out that a Drude-like $\sigma_{\rm reg}(\omega)$ rarely exists in quasi one-dimensional spin Hamiltonians with short-range interactions (see, e.g., Refs. ). A notable example in which standard diffusion is realized in the high-temperature regime is the XX spin ladder [@steinigeweg14a], which is equivalent to hard-core bosons and thus relevant for recent experiments on mass transport of strongly interacting bosons in optical lattices in one and two dimensions [@ronzheimer13; @vidmar13]. The main goal of our work is to compute the frequency dependence of the spin and thermal conductivity of spin ladders as well as of the spin-1/2 XXZ chain. We use a finite-temperature, real-time version of the density matrix renormalization group method (DMRG) [@karrasch12; @barthel13; @karrasch13a; @kennes14] based on the purification trick [@verstraete04]. This method allows one to calculate both thermodynamics [@feiguin05] but also the time dependence of current autocorrelation functions. We calculate the conductivities from Kubo formulae. For the accessible time scales, our results are free of finite-size effects [@karrasch13] and thus effectively describe systems in the thermodynamic limit. Exploiting several recent methodological advances and using an optimized and parallelized implementation allows us to access larger time scales than in earlier applications of the method [@karrasch12; @karrasch13; @karrasch14]. Our data agree well with exact diagonalization approaches [@zotos04] for the thermal conductivity of spin ladders and the spin transport in XX ladders [@steinigeweg14a]. The latter results have been obtained from a pure state propagation method based on the dynamical typicality approach, which has recently been applied to the calculation off transport coefficients [@steinigeweg14; @steinigeweg14a; @steinigeweg14b]. Our key results are as follows. For the spin-1/2 XXZ chain with $ 0<\Delta < 1$, we provide evidence that $\sigma_{\rm reg }(\omega)$ remains finite in the dc limit, but its low-frequency behavior is not of a simple Lorentzian form. For $\Delta=0.5$, we observe a suppression of weight for $\omega\ll J$ in the high-temperature regime. In the case of spin ladders, $\sigma_{\rm reg}(\omega)$ also generically exhibits a complicated low-frequency dependence, and a simple Drude-like form is recovered only in the XX case $\Delta=0$ in agreement with the results of Ref. . We extract the dc spin conductivity for temperatures $T \geq 0.5J$ and discuss how it depends on the exchange anisotropy $\Delta$. We translate the high-$T$ spin and thermal conductivities of the Heisenberg ladder into mean-free paths by fitting to a simple phenomenological expression often used in the interpretation of experimental data [@hess01]. It turns out that the values of the mean-free paths depend on which type of transport is considered. The structure of this exposition is as follows. We introduce the model and definitions in Sec. \[sec:def\]. Section \[sec:num\] provides details on our numerical method. Our results are summarized in Sec. \[sec:results\], where we discuss the real-time dependence of current correlations and the methods to convert them into frequency-dependent conductivities, which we then study for spin chains and ladders. Our conclusions are presented in Sec. \[sec:sum\]. Model and definitions {#sec:def} ===================== The prime interest of this work is in two-leg spin ladders governed by the Hamiltonian $H=\sum_{n=1}^{L-1} h_n$ and local terms $$\label{eq:hlad}\begin{split} h_n = J & \sum_{\lambda=1,2} \left[ S^x_{n,\lambda}S^x_{n+1,\lambda} + S^y_{n,\lambda}S^y_{n+1,\lambda} + \Delta S^z_{n,\lambda}S^z_{n+1,\lambda}\right] \\ + \frac{J_\perp}{2} & \sum_{m=n,n+1} \left[S^x_{m,1}S^x_{m,2} + S^y_{m,1}S^y_{m,2} + \Delta S^z_{m,1}S^z_{m,2}\right]\,, \end{split}$$ where $S^{x,y,z}_{n,\lambda}$ are spin-$1/2$ operators acting on the rung $\lambda=1,2$. The model is non-integrable and gapped for all $J_\perp>0$. At $J_\perp=0$, one recovers two identical, decoupled XXZ chains, which (at zero magnetization) are gapless for $|\Delta|\leq1$ and gapped otherwise [@giamarchi]. Both the Drude weights and the regular parts of the spin (s) and thermal (th) conductivities defined in Eq. (\[eq:sigma\]) can be obtained from the corresponding current correlation function $C_{\textnormal}{s,th}(t)$. Their long-term asymptote is related to $D_{\textnormal}{s,th}$ via $$\begin{split}\label{eq:dw} D_{\textnormal}{s,th} & = \lim_{t\to\infty}\lim_{L\to\infty} \frac {C_{\rm s,th}(t) }{2T^{\alpha_{\textnormal}{s,th}}},~ C_{\rm s,th}(t) = \frac{\mbox{Re} \, \langle I_{\textnormal}{s,th}(t) I_{\textnormal}{s,th}\rangle}{L}, \end{split}$$ where $\alpha_{\textnormal}{s}=1$ and $\alpha_{\textnormal}{th}=2$. The regular part of the conductivity is determined by $$\label{eq:sigma1}\begin{split} &{\textnormal}{Re}\,\left\{{{\sigma_{\textnormal}{reg}(\omega)}\atop{\kappa_{\textnormal}{reg}(\omega)}}\right\} =\, \frac{1-e^{-\omega/T}}{\omega T^{\alpha_{\textnormal}{s,th}-1} }\times \\ &~~~~~{\textnormal}{Re} \int_0^{\infty}dte^{i\omega t} \lim_{L\to\infty}\left[ C_{\textnormal}{s,th}(t) - 2T^{\alpha_{\textnormal}{s,th}}D_{\textnormal}{s,th}\right]\,. \end{split}$$ Only finite times can be reached in the DMRG calculation of $C_{\textnormal}{s,th}(t)$, which leads to a ‘finite-time’ error of $\sigma_{\textnormal}{reg}(\omega)$ that can be assessed following Ref. . We will elaborate on this below. The current operators $I_{\textnormal}{s,th} = \sum_n j_{({\textnormal}{s,th}),n}$ are defined via the respective continuity equations [@zotos97]. The local spin-current operators of the XXZ chain take the well-known form $j_{\textnormal{s},n} = i J S^x_{n} S^y_{n+1} +{\textnormal}{h.c.}$. For the spin ladder, one finds $$\begin{aligned} j_{\textnormal{s},n} &=& i J \sum_\lambda \big(S^x_{n,\lambda} S^y_{n+1,\lambda} - S^y_{n,\lambda}S^x_{n+1,\lambda} \big)\,, \\ j_{\textnormal{th},n} &=& i [h_{n}, h_{n+1}]\,.\end{aligned}$$ Note that our definition for the local energy density $h_n$ preserves all spatial symmetries of the ladder, and our energy-current operator $I_{\textnormal}{th}$ is the same as the one used in Ref. . The full expression for $I_{\textnormal{th}}$ is lengthy and not given here. Numerical method {#sec:num} ================ We compute the spin- and energy-current correlation function $$\begin{aligned} \label{eq:time} \langle I_{\textnormal}{s,th}(t) I\rangle &\sim& {\textnormal}{Tr}\big[ e^{-H/T}e^{iHt}I_{\textnormal}{s,th}e^{-iHt}I_{\textnormal}{s,th}\big]\end{aligned}$$ using the time-dependent [@vidal04; @white04; @daley04; @schmitteckert04; @vidal07] density matrix renormalization group [@white92; @schollwoeck05] in a matrix-product state [@fannes91; @ostlund91; @verstraete06; @verstraete08; @schollwoeck11] implementation. Finite temperatures [@verstraete04; @white09; @barthel09; @zwolak04; @sirker05; @white09; @barthel13] are incorporated via purification of the thermal density matrix. Purification is a concept from quantum information theory in which the physical system is embedded into an environment. The wave-function of the full system is then a pure state and the mixed state describing the system is obtained by tracing out the degrees of freedom of the environment. When using this approach in DMRG, one typically simply chooses a copy of the system degrees of freedom to be the environment. Details of purification based finite-$T$ DMRG methods can be found in Refs. [@verstraete04; @feiguin05; @schollwoeck11; @barthel13; @karrasch14]. Our actual implementation follows Ref. [@karrasch14]. The real- and imaginary time evolution operators in Eq. (\[eq:time\]) are factorized by a fourth-order Trotter-Suzuki decomposition with a step size of $dt=0.05,\ldots, 0.2$. We keep the discarded weight during each individual ‘bond update’ below a threshold value $\epsilon$. This leads to an exponential increase of the bond dimension $\chi$ during the real-time evolution. In order to access time scales as large as possible, we employ the finite-temperature disentangler introduced in Ref. , which uses the fact that purification is not unique to slow down the growth of $\chi$. Moreover, we ‘exploit time translation invariance’ [@barthel13], rewrite $\langle I_{\textnormal}{s,th}(t)I_{\textnormal}{s,th}(0)\rangle=\langle I_{\textnormal}{s,th}(t/2)I_{\textnormal}{s,th}(-t/2)\rangle$, and carry out two independent calculations for $I_{\textnormal}{s,th}(t/2)$ as well as $I_{\textnormal}{s,th}(-t/2)$. Our calculations are performed using a system size of $L=100$ for spin ladders and $L=200$ for the XXZ chain, respectively. By comparing to other values of $L$ we have ensured that $L$ is large enough for the results to be effectively in the thermodynamic limit [@karrasch13]. Results {#sec:results} ======= Current autocorrelation functions {#sec:current} --------------------------------- Figure \[Fig1\] shows typical results for the decay of spin-current autocorrelations of the XXZ chain as a function of time. Some of these data have previously been shown in Refs.  and are here included for comparison. For $\Delta<1$, we clearly observe the saturation of $C_{\rm s}(t)$ at a non-zero value at long times that at $T=\infty$ agrees well with an improved lower bound [@prosen13] for $\lim_{T\to\infty} TD_{\textnormal}{s}(T)$ and Zotos’ Bethe-ansatz calculation [@zotos99]. At $\Delta=0.5$, the values for $D_{\textnormal}{s}$ obtained in Ref.  coincide with our tDMRG data also for the finite temperatures $T< \infty$ considered here (see the inset to Fig. \[Fig1\]; compare Ref. ). In the case of $\Delta>1$, the current correlators appear to decay to zero, consistent with predictions of a vanishing finite-temperature Drude weight in this regime [@zotos96; @peres99; @hm03; @karrasch14]. At the isotropic point $\Delta=1$, $C_{\rm s}(t)$ does not saturate to a constant on the time scale reached in the simulations [@karrasch13], and no conclusion on the presence or absence of a ballistic contribution is possible. We next turn to the case of spin ladders. Exemplary DMRG data for $C_{\rm s}$ and $C_{\rm th}$ at three different temperatures $T\in\{\infty,J,0.5J\}$ are shown in Fig. \[Fig2\]. The thermal current autocorrelation function is strictly time-independent in the chain limit $J_\perp=0$ [@zotos97] (data not shown in the figure) but decays to zero for any $J_\perp>0$, which is consistent with earlier studies that suggested the absence of ballistic contributions in spin-ladder systems [@hm03; @zotos04]. For the isotropic ladder $\Delta=J_\perp/J=1$ at high temperatures, this decay takes place on a fairly short time scale $t J\lesssim 8$ \[see Fig. \[Fig2\](a)\]. In Figs. \[Fig2\](b,c), we compare the behavior of $C_{\rm s}(t)$ on chains to isotropic ladders ($J_\perp=J$) for two different exchange anisotropies $\Delta=0.5$ and $\Delta=1$. In both cases, $C_{\rm s}(t)$ decays much faster if $J_\perp>0$, and our data suggest the absence of ballistic contributions to spin transport in agreement with Refs. . Moreover, oscillations in $C_{\rm s}(t)$ emerge in the case of ladders. They become very pronounced at lower temperatures and are related to the existence of a spin gap for $J_\perp>0$. Extraction of conductivities {#sec:extract} ---------------------------- ![(Color online) Real-time spin current correlation functions of the XXZ chain \[see Eq. (\[eq:hxxz\])\] at infinite temperature $T=\infty$ (main panel) and fixed exchange anisotropy $\Delta=0.5$ (inset). The model is integrable, and the spin Drude weight $D_{\textnormal}{s}$ is finite for $|\Delta|<1$ [@herbrych11; @prosen11; @karrasch13; @prosen13]. The horizontal lines show the lower bounds for $D_{\textnormal}{s}$ established in Ref.  as well as the Bethe-ansatz result from Ref. . []{data-label="Fig1"}](fig1.eps){width="0.90\columnwidth"} We compute the spin and heat conductivities from the corresponding real-time current correlation functions via Eq. (\[eq:sigma1\]). However, only finite times $t<t_{\rm max}$ can be reached in the DMRG calculation of $C_{\textnormal}{s,th}(t)$, which gives rise to a ‘finite-time error’ in $\sigma_{\textnormal}{reg}(\omega)$ and $\kappa_{\textnormal}{reg}(\omega)$. We assess this error as follows. Our data suggest (in agreement with the results of Refs. ) that for any $J_\perp>0$ the spin and thermal Drude weights vanish; the current correlators decay to zero for $t\to\infty$. We first compute the frequency integral in Eq. (\[eq:sigma1\]) using only the finite-time data. Thereafter, we extrapolate $C_{\textnormal}{s,th}(t)$ to $t=\infty$ using linear prediction [@barthel09] and re-compute the frequency integral. Linear prediction attempts to obtain data for correlation functions of interest at times $t>t_{\rm max}$ as a linear combination of the available data for a discrete set of times points $t_n<t_{\rm max}$ (see [@barthel09] for details). We perform the linear prediction for a variety of different fitting parameters (such as the fitting interval) and then define the error bar as twice the largest deviation to the conductivity computed without any extrapolation at all. For the XXZ chain with $|\Delta|>1$, the Drude weight also vanishes, and the finite-time error of $\sigma_{\textnormal}{reg}(\omega)$ can be assessed analogously to ladders. The same holds at $|\Delta|<1$ and $T=\infty$ where a lower bound for $D_{\textnormal}{s}$ is known analytically from Prosen’s work (see the discussion in Sec. \[sec:current\]). For the values of $\Delta$ considered here, this bound agrees with the Drude weight computed using other methods [@zotos99; @herbrych11; @karrasch13]; hence, we assume that it is exhaustive, which allows us to subtract $D_{\textnormal}{s}$ in Eq. (\[eq:sigma\]). At $|\Delta|<1$ and $T<\infty$, the Drude weight needs to be extracted from the numerical data [@karrasch12; @karrasch13], which is an additional source of error, or it has to be taken from other methods such as the Bethe-ansatz calculation of Ref. . We estimate the corresponding uncertainty of the conductivity as follows. For $\Delta=0.5$ and $T=\infty$, $C_{\textnormal}{s}(t)$ oscillates around the value for $TD_{\textnormal}{s}$ known from the improved lower bound from [@prosen13] (see Fig. \[Fig1\]); for finite temperatures $T\in\{3.3J,J,J/2\}$, $C_{\textnormal}{s}(t)$ oscillates around the Bethe-ansatz result $D_{\textnormal}{s}^{\textnormal}{BA}$ of Ref. . An upper as well as a lower bound $D_{\textnormal}{s}^{\textnormal}{u,l}$ can be determined from the magnitude of the oscillations. For each $D_{\textnormal}{s}^{\textnormal}{BA}$, $D_{\textnormal}{s}^{\textnormal}{u}$, and $D_{\textnormal}{s}^{\textnormal}{l}$, we carry out the procedure used at $T=\infty$ and define the uncertainty in $\sigma_{\textnormal}{reg}(\omega)$ as either twice the difference between the curves computed with and without extrapolation or twice the maximum difference between the curves at the different $D_{\textnormal}{s}^{\textnormal}{BA,u,l}$, whatever is larger. ![(Color online) Current correlation functions of two-leg spin ladders governed by the Hamiltonian of Eq. (\[eq:hlad\]). $\Delta$ and $J_\perp$ denote the exchange anisotropy and the rung coupling, respectively. (a) Energy current autocorrelation function of isotropic ladders $J_\perp/J=1, \Delta=1$ for various $T$. (b,c) Spin current autocorrelation functions at $\Delta=1$ and $\Delta=0.5$, respectively. For $J_\perp=0$, one recovers two identical, decoupled XXZ chains. []{data-label="Fig2"}](fig2.eps){width="0.90\columnwidth"} For other exchange anisotropies, the accessible time scales are either too short to fully resolve the oscillations around $D_{\rm s}$, or $C_{\textnormal}{s}(t)$ decays monotonicly for large times. The latter seems to be true, in particular, close to the isotropic point $\Delta=1$. For $\Delta=0.901$ (see Fig. \[Fig1\]) and at $T=\infty$, the value of $C_{\textnormal}{s}(t)$ at the largest time reached is approximately $10$ percent larger than the improved bound from [@prosen13]. For finite but not too small $T/J$, we assume that $C_{\textnormal}{s}(t)/(2T)=rD_{\textnormal}{s}$ at the maximal time reached, where we typically choose $r\sim1.2$. Given this estimate for $D_{\textnormal}{s}$, we assess the error analogously to the infinite-temperature case. Note that the larger $r$, the larger the error bars. We stress that this way of estimating the error is less controlled than in those cases for which the value of the Drude weight is known. Exemplary error bars are shown in Figs. \[Fig3\]-\[Fig6\]. The data for $\sigma_{\textnormal}{reg}(\omega)$ displayed in the figures are the ones obtained using linear prediction; the conductivities for the XXZ chain at $\Delta=0.5$ and $T<\infty$ shown in Fig. \[Fig3\](c) were calculated using the Bethe-ansatz value of Ref.  for the Drude weight. Note that the numerical error of the bare DMRG data for $C_{\textnormal}{s,th}(t)$ is negligible compared to the finite-time error. The finite-time data used for the above procedure is the DMRG data up to the maximum time $t_{\rm max}$ reached in the simulation. In the case of the XXZ chain, this time is fairly large compared to $1/J$, and it is thus instructive to re-calculate $\sigma_{\textnormal}{reg}(\omega)$ using only the data for half of the maximum time (both with and without extrapolation). Results are shown in the insets to Fig. \[Fig3\]; they illustrate that linear prediction provides a fairly reliable way to estimate the error. As an additional test for the accuracy of the conductivities, one can verify the optical sum rule. In the spin case it reads $$\label{eq:sumrule} \int_0^\infty d\omega\, \mbox{Re}\,\sigma(\omega) = \frac{\pi\langle -\hat T \rangle }{2L}\,,$$ where $\hat T$ is the kinetic energy, i.e., all terms in Eq. (\[eq:hlad\]) computed at $\Delta=0$. We show exemplary data for the validity of the sum rule in the insets of Figs. \[Fig3\](c) and \[Fig5\](b), which illustrate that Eq. (\[eq:sumrule\]) holds with great accuracy. ![(Color online) Regular part of the spin conductivity of the XXZ chain at (a) infinite temperature and various values of $\Delta$, and (b,c) at a fixed value of $\Delta<1$ and various $T$. The ‘finite-time’ error can be estimated following the procedure outlined in Sec. \[sec:extract\]. The insets to (a) and (b) show the conductivity obtained from the finite-time data $t<t_m$ without extrapolation (‘no LP’) as well as from using linear prediction to extrapolate to $t\to\infty$ (‘LP’). The inset to (c) illustrates that the optical sum rule Eq.  is fulfilled accurately. []{data-label="Fig3"}](fig3.eps){width="0.90\columnwidth"} Spin conductivity of the spin-1/2 XXZ chain ------------------------------------------- Figure \[Fig3\] gives an overview over the behavior of $\sigma_{\textnormal}{reg}(\omega)$ for the spin-1/2 XXZ chain for various values of the exchange anisotropy $\Delta$ (results for $\Delta>1$ have previously been shown in Ref. ). At infinite temperature \[see Fig. \[Fig3\](a)\] and for all $\Delta>0$ considered, we find a finite dc conductivity $\sigma_{\rm dc} =\lim_{\omega\to 0} \sigma_{\rm reg}(\omega)$ within the error bars of our extrapolation method. This is at odds with the predictions of Ref. , where $\sigma_{\rm reg}(\omega)\propto \omega^2$ was suggested in the low-frequency limit. For the special value of $\Delta=0.5$, however, there clearly is a suppression of weight around $\omega=0$ accompanied by a pronounced maximum at $\omega \approx 0.25J$. For other values of $\Delta<1$, $\sigma_{\rm reg}(\omega)$ seems to exhibit a global maximum at $\omega=0$ as well as additional lower maxima at higher frequencies that shift to larger values of $\omega$ as $\Delta$ increases. The spin conductivity at $\Delta=2$ has also been analyzed in Ref. , and large anomalous, $L$-dependent fluctuations in Re$\,\sigma(\omega)$ have been observed at low frequencies. Those are not present in our data. Returning to the regime of $\Delta<1$, we cannot rule out that the disagreement between our result for the low-frequency behavior of the conductivity and the prediction of Ref.  is attributed to finite-time effects. However, there is no obvious indication for this in our data: At $T=\infty$, the Drude weight is known from [@prosen13], and $\lim_{T\to \infty}[T\sigma_{\textnormal}{dc}]$ is simply given by the integral of $C_{\textnormal}{s}(t)$ with $2TD_{\textnormal}{s}$ subtracted. The real-time data are shown in Fig. \[Fig1\]; the errors due to the finite system size and the finite discarded weight are negligible. As illustrated in the insets to Fig. \[Fig3\](a,b), our extrapolation scheme using linear prediction provides a stable and meaningful way to establish finite-time error bars. As an additional test, it is instructive to assume that the improved lower bound – which at $T=\infty$ and for the exchange anisotropies considered here coincides with the Bethe-ansatz result of Ref.  – is not fully saturated. At $\Delta=0.5$, an *upper* bound $D_{\textnormal}{s}^{\textnormal}{u}$ to the Drude weight can be estimated from the magnitude of the oscillations of $C_{\textnormal}{s}(t)$. Using $D_{\textnormal}{s}^{\textnormal}{u}$ instead of the lower bound from [@prosen13] decreases $\sigma_{\textnormal}{dc}$ by 15% but does not yield $\sigma_{\textnormal}{dc}=0$. ![(Color online) Spin conductivity of two-leg ladders at infinite temperature $T=\infty$ for (a) fixed rung coupling $J_\perp=J$ and (b) fixed anisotropy $\Delta=1$. At $\Delta=0$ and $J_\perp\lesssim J$, $\sigma_{\textnormal}{reg}(\omega)$ is of the simple Lorentzian form (see the inset). []{data-label="Fig4"}](fig4.eps){width="0.90\columnwidth"} To summarize, a vanishing dc conductivity suggested by Ref.  could only be caused by oscillations at large times around the asymptote $2TD_{\textnormal}{s}$, which would need to cancel out the large positive contribution from times $tJ\lesssim40$. Put differently, if $\sigma_{\textnormal}{reg}(\omega)\sim\omega^2$ holds, it only holds for very small frequencies $\omega\ll J$. This is further corroborated by the fact that the optical sum rule of Eq. (\[eq:sumrule\]) is fulfilled accurately \[see the inset to Fig. \[Fig3\](c)\]. Even though $\sigma_{\rm dc}>0$ is supported by our tDMRG calculation in combination with the results for $D_{\rm s}(T)$ from Refs. , the emergence of very narrow peaks in the data for $\Delta=0.309, 0.901$ at low frequencies should be taken with some caution. For these parameters, the time scale $tJ\lesssim t_{\rm max}=40/J$ reached in the simulation is too short to resolve potential oscillations around the long-time asymptote. A very conservative estimate of the accessible frequencies is $\omega_{\rm min}= 2\pi/t_{\rm max} \sim 0.15 J$. It is possible that redistributions of weight below $\omega_{\rm min}$ would occur if longer times were available. The temperature dependence of $\sigma_{\rm reg}(\omega)$ is shown in Figs. \[Fig3\](b,c) for two different exchange anisotropies. At $\Delta=0.901$, the global maximum is always at $\omega=0$, $\sigma_{\textnormal}{dc}$ increases with decreasing temperature, and the $\omega$-dependence seems to become smoother the smaller $T$ is. For $\Delta=0.5$, the suppression of weight at low frequencies survives down to temperatures of $T\gtrsim0.5J$ (at $T=0.5J$, the error bars become too large to draw any conclusions). ![(Color online) Spin conductivity of two-leg ladders with fixed $J_\perp=J$ but various $T$ and anisotropies ranging from $\Delta=0$ (XX ladder) to $\Delta=1$ (isotropic ladder). Note that the curve at $\Delta=1$, $T=0.29J$ (at $\Delta=0$, $T=0.5J$) is plotted only for frequencies $\omega\geq J$ ($\omega\geq 0.2J$). The insets show the DC conductivity, the optical sum rule, and the $\Delta$-dependence of the spin gap (calculated for $L=128$ at $J>0$), respectively.[]{data-label="Fig5"}](fig5.eps){width="0.90\columnwidth"} To guide our ensuing discussion of ladders, we summarize the $\Delta$-dependence of $\sigma(\omega)$ in the chain limit $J_\perp=0$. At $\Delta=0$, Re$\,\sigma(\omega)= 2\pi D_s(T) \delta(\omega)$, and the perturbation $J_\perp>0$ thus breaks both the integrability of the model and the conservation of the spin current. For $\Delta>0$, the spin current is no longer conserved even for $J_\perp=0$, which gives rise to a non-zero regular contribution $\sigma_{\rm reg}(\omega)$ to the conductivity. According to recent studies [@prosen11; @prosen13; @karrasch13; @herbrych11], the Drude weight is finite for any $0\leq|\Delta|<1$, but no final conclusion on $D_{\textnormal}{s}(T)$ at $\Delta=1$ has been reached yet. At $T=\infty$, the relative contribution of $\sigma_{\rm reg} (\omega)$ to the total spectral weight increases monotonicly from zero at $\Delta=0$ to a value of the order of 90% close to $\Delta=1$ [@karrasch13]. For $\Delta>1$, the commonly expected picture is that $D_{\textnormal}{s}(T>0)=0$; hence, all weight is concentrated in the regular part. Based on these qualitative differences of $\sigma(\omega)$ that depend on $\Delta$ and the interplay of the ballistic contribution with finite-frequency weight at small $\omega$, we expect significant changes in the spin conductivity of ladders as a function of $\Delta$. Spin conductivity of ladders ---------------------------- We now turn to the spin conductivity $\sigma(\omega)$ of two-leg ladders and contrast our results to the limiting case of isolated chains ($J_\perp=0$), where the behavior of $\sigma(\omega)$ crucially depends on $\Delta$. We first discuss the infinite-temperature case; data for $J_{\perp}=J$ are presented in Fig. \[Fig4\](a). At $\Delta=0$, $\sigma_{\rm reg}(\omega)$ has a simple Lorentzian shape \[see the inset to Fig. \[Fig4\](a)\]: $$\label{eq:lorentz} \mbox{Re}\, \sigma(\omega) = \frac{\pi\sigma_{\rm dc}/\tau^2}{\omega^2 + (1/\tau)^2}\,.$$ This follows directly from the results of Ref. , where the spin-autocorrelation function of the XX two-leg ladder was studied numerically and analytically as a function of $J_{\perp}/J$. It turned out that $C_{\textnormal}{s}(t)$ decays exponentially at small values of $J_\perp\lesssim J$ and with a Gaussian for larger values of $J_\perp$. The results of Ref.  in conjunction with our data altogether identify the XX spin-1/2 ladder as a textbook realization of a diffusive conductor with a [*single*]{} relaxation time $\tau\propto (J/J_\perp)^2$. Systems with $\Delta=0$ are rarely found in real materials, but the XX model on a ladder can easily be realized with hard-core bosons in optical lattices (see, e.g., Ref.  and the discussion in Refs. ). For the special case of $\Delta=1$, we show exemplary data for $J_{\perp} \not= J$ in Fig. \[Fig4\](b). Even at $T=\infty$, the conductivity does not have a simple functional form but features side maxima at finite frequencies that shift to larger $\omega$ as $J_{\perp}/J$ increases. In Figs. \[Fig5\](a)-(c), we illustrate how $\sigma_{\rm reg}(\omega)$ of isotropic ladders $J_{\perp}=J$ evolves as the temperature decreases from $T=\infty$ down to $T=0.29J$. It turns out that it is easier to reach low temperatures for larger values of $\Delta$. In the case of $\Delta=0$ \[see Fig. \[Fig5\](a)\], we observe a Drude-like conductivity down to temperatures of $T\sim 3J$. At lower temperatures, however, Re$\sigma(\omega)$ deviates from a simple Lorentzian (see Ref.  for similar observations for a chain with a staggered field). This is a consequence of the existence of a spin gap $\Delta_{\rm spin}$ in the two-leg ladder which at low temperatures manifests itself by a suppression of weight below the optical $2\Delta_{\rm spin}$ (see, e.g., the case of dimerized chains studied in Ref. ) and a sharp increase of $\sigma_{\rm reg}(\omega)$ at $\omega \sim 2\Delta_{\rm spin}$. As a consequence, the dc conductivity is expected to diverge with $T^{-\alpha}$, $\alpha>0$ as $T$ is lowered [@sachdev97; @damle98; @damle05; @karrasch14a]. Next, we investigate how the Drude-like conductivity observed for $\Delta=0$ evolves as $\Delta$ increases. We find that (i) the current autocorrelations at $\Delta=0.5$ and $\Delta=1$ do not follow a simple exponential or Gaussian decay even at infinite temperature, and hence (ii) the low-frequency conductivity is not well-described by a simple Lorentzian. Pragmatically, we associate the (zero-frequency) current relaxation time $\tau$ with the inverse of the half-width-half-maximum of the zero-frequency peak in Re$\,\sigma(\omega)$ for $\Delta \gg 0$. The presence of these two scales, the optical gap $2\Delta_{\rm spin}$ and the inverse high-temperature relaxation time $1/\tau$, which controls the low-frequency behavior, is more visible in the data for $\Delta=0.5$ and $\Delta=1$ \[shown in Figs. \[Fig5\](b) and (c)\] even at the highest temperatures $T=3.3J$. Clearly, there are two maxima in Re$\,\sigma(\omega)$, one at $\omega=0$ and one at $\omega \gtrsim 2\Delta_{\rm spin}$ (in fact, at very low $T$, Re$\,\sigma(\omega)$ has a an edge at the optical gap). The reason is the dependence of the optical gap on the exchange anisotropy $\Delta$. The spin gap in a two-leg ladder as a function of $\Delta$ is, in the limit of $J=0$, given by $$\Delta_{\rm spin} = \frac{J_\perp} {2} (1+\Delta)\,.$$ This monotonic dependence of $\Delta_{\rm spin}$ on $\Delta$ survives at finite values of $J_\perp\sim J$. This is shown in the inset of Fig. \[Fig5\](c), which has been obtained from $\Delta_{\rm spin} = E_0(S^z=1)-E_0(S^z=0)$ using standard DMRG [@white92; @schollwoeck05], where $E_0(S^z)$ is the ground state in the subspace with total magnetization $S^z$ for $L=128$. ![(Color online) Thermal conductivity of two-leg ladders at fixed $\Delta=1$ and $T=\infty$ but various $J_\perp$. We compare our data with the exact diagonalization result of Ref. . []{data-label="Fig6"}](fig6.eps){width="0.90\columnwidth"} Our data are compatible with a leading temperature dependence of the form $\sigma_{\rm dc}(T) \propto 1/T$. Moreover, $\sigma_{\rm dc}$ is a monotonicly decreasing function of $\Delta$ in the high-temperature regime. The latter can be understood by the nature of the single-particle spin-1 excitations of the two-leg ladder that originate from the local triplet excitations of the $J/J_\perp\to 0$ limit. Finite values of $J$ render these triplets dispersive and give rise to interactions between the quasi-particles. A nonzero value of $\Delta$ introduces additional scattering terms, and it is thus intuitive to expect smaller quasi-particle life times and hence also smaller dc-conductivities. Thermal conductivity of Heisenberg ladders ------------------------------------------ For experiments with quantum magnets, the thermal conductivity is the most easily accessible transport coefficient, which has been investigated in a large number of experiments on ladders [@sologubenko00; @hess01], chains [@solo00a; @sologubenko07a; @hess07a; @hlubek10], and two-dimensional antiferromagnets [@hess03; @sales02] (see Refs.  for a review). These experiments have clearly established that magnetic excitations can dominantly contribute to the thermal conductivity of these insulating materials at elevated temperatures, exceeding the phononic contribution (see, e.g., [@hlubek10]). The contribution of magnetic excitations to the full thermal conductivity in these low-dimensional systems manifests itself via a prominent anisotropy of the thermal conductivity measured along different crystal axes [@hess07; @sologubenko07]. Open and timely questions include a comprehensive and quantitative theoretical explanation for the magnitude of the thermal conductivity, a theory of relevant scattering channels beyond pure spin systems (see, e.g., [@shimshoni03; @chernyshev05; @rozhkov05a; @boulat07; @gangadharaiah10; @bartsch13; @rezania13]), a full understanding of the spin-phonon coupling including spin-drag effects [@boulat07; @bartsch13; @gangadharaiah10], and the understanding of a series of experiments studying the effect of doping with nonmagnetic or magnetic impurities and disorder onto the thermal conductivity (see, e.g., [@hess04; @hess06]). Here we solely focus on pure spin Hamiltonians. Given that most of the materials realize spin Hamiltonians that are more complicated than just spin chains with nearest-neighbor interactions only, one needs to resort to numerical methods to get a quantitative picture. The real-time energy current autocorrelations for the Heisenberg ladder ($\Delta=1$) are shown in Fig. \[Fig2\](a). At $T=\infty$, $C_{\rm th}(t)$ decays fast, and $\kappa(\omega)$ can be obtained down to sufficiently low frequencies. For lower temperatures, however, the accessible time scales are at present too short to reach the dc-limit in a reliable way. We therefore focus on $T=\infty$. Our results for the thermal conductivity are shown in Fig. \[Fig6\] for $J_\perp/J=0.5,1,2$. They are in reasonable agreement with the exact diagonalization data of Ref.  that were obtained using a micro-canonical Lanczos method for $L=14$ sites. Note that our data for $\kappa_{\rm dc}$ is typically larger than the ED results. The behavior at low frequencies is anomalous – it does not follow a Drude-like Lorentzian shape (this was already pointed out in Ref. ). The actual form of the low-frequency dependence of $\kappa_{\rm reg}(\omega)$ (discussed in Ref. [@zotos04]) cannot be clarified using the existing data. The knowledge of the infinite temperature dc-conductivity $\kappa_{\rm dc}^{\infty}$ still gives access to a wide temperature regime since the leading term is $\kappa_{\rm dc}(T) = \kappa_{\rm dc}^{\infty}/T^2$, and we can therefore address the question of mean-free paths. Mean-free paths for the Heisenberg spin ladder ---------------------------------------------- In the analysis of experimental data for $\kappa$, one often uses a kinetic equation to extract magnetic mean-free paths [@hess01; @hess07]. An analogous equation can also be employed for $\sigma$, and we obtain the following set of kinetic equations: $$\begin{aligned} \kappa &=& \frac{1}{L}\sum_k v_k \frac{d(\epsilon_k n_k)}{dT} l_{\kappa,k} \label{eq:lkappa}\\ \sigma &=& \frac{1}{L}\sum_k v_k \left(-\frac{dn_k}{d\epsilon_k}\right) l_{\sigma,k}\,, \label{eq:lsigma}\end{aligned}$$ where $\epsilon_k$ is the dispersion of the threefold degenerate triplet excitations, $v_k =\partial_k \epsilon_k$, and $n_k$ is a distribution function which accounts for the hard-core boson nature of the triplets [@hess01]: $$n_k = \frac{3}{\mbox{exp}(\beta \epsilon_k)+3}\,.$$ The actual form of the dispersion is not important since $v_k$ drops out in one dimension when the integration over $k$ is replaced by an integral over energy $\epsilon$. The mean-free paths $l_{\kappa,k}$ and $l_{\sigma,k}$ are taken to be independent of quasi-momentum, $l_{\kappa(\sigma),k}=l_{\kappa(\sigma), \rm mag}$. In order to analyze the total thermal conductivity measured experimentally, one assumes $\kappa_{\rm total } = \kappa_{\rm ph} + \kappa_{\rm mag}$, where $\kappa_{\rm ph}$ and $\kappa_{\rm mag}$ represent the phononic and magnetic contribution, respectively. Such a separation is an approximation and should be understood as an operational means to extract mean-free paths – in general, spin-drag effects can lead to additional contributions to $\kappa_{\rm total }$ [@chernyshev05; @boulat07; @bartsch13]. In the high-temperature limit, one needs to keep only the leading terms in a $1/T$ expansion of Eqs.  and . The mean-free paths can then be extracted from $\kappa_{\rm dc} = \kappa_{\rm dc}^\infty/T^2$ and $\sigma_{\rm dc} = \sigma_{\rm dc}^\infty/T$ via $$\begin{aligned} \kappa_{\rm dc}^{\infty} = \frac{1}{16\pi} (\epsilon_{\rm max}^3 -\epsilon_{\rm min }^3) l_{\kappa,\rm mag }\,, \\ \sigma_{\rm dc}^{\infty} = \frac{3}{4 \pi} (\epsilon_{\rm max} -\epsilon_{\rm min}) l_{\sigma,\rm mag }\,,\end{aligned}$$ where $\epsilon_{\rm max}$ and $\epsilon_{\rm min}$ are the band minimum and band maximum of the single-triplet dispersion, respectively. For an isotropic ladder system such as the one realized in La$_5$Ca$_{9}$Cu$_{24}$O$_{41}$ (the actual Hamiltonian is more complicated, though [@notbohm07]), $\epsilon_{\rm min } =\Delta_{\rm spin}\approx J/2$ and $\epsilon_{\rm max} \approx 2J$ are reasonable estimates [@knetter01] for $J_\perp = J$. We can thus approximate $\epsilon_{\rm max} -\epsilon_{\rm min}= 3J/2$ and $\epsilon_{\rm max}^3 -\epsilon_{\rm min }^3 \approx 8 J^3$, which then leads to $$\begin{aligned} \kappa_{\rm dc}^{\infty} = \frac{J^3}{2\pi} l_{\kappa,\rm mag }\,, \\ \sigma_{\rm dc}^{\infty} = \frac{9 J}{8 \pi} l_{\sigma,\rm mag }\,.\end{aligned}$$ For the isotropic ladder $J_\perp/J=1, \Delta=1$, we have $\kappa_{\textnormal}{dc}^\infty\approx0.66J^3$ and $\sigma_{\textnormal}{dc}^\infty\approx0.39J^2$ and thus $l_{\kappa,{\textnormal}{mag}}\approx 4.2 $ and $l_{\sigma,{\textnormal}{mag}}\approx 1.1$. Hence, $ l_{\kappa,{\textnormal}{mag}} > l_{\sigma,{\textnormal}{mag}}$ such that the (averaged) mean-free paths differ from each other. In this framework the mean-free paths in the high-temperature regime are $T$-independent, which seems reasonable since at large $T\gg J,J_{\perp}$ (i.e., $T$ larger than the band-width of triplets) all states are populated equally. In other words, the qualitative difference with phonons, the most typical bosonic quasi-particle that contributes to the thermal conductivity in solids, is that the number of triplet excitations saturates at large $T$ due to their hard-core nature, reflecting the fact that the spin system has a spectrum that is bounded from above. Our results demonstrate that the extraction of mean-free paths as commonly employed in the analysis of the experimental data, while providing very useful intuition, cannot easily be related to single-excitation mean-free paths, due to the different results obtained for $\kappa$ and $\sigma$ and the gap in the excitation spectrum (see also the discussion in Ref. ). We stress that the observation of different mean-free paths for different transport channels is not unusual. Even in metals (more generally, Fermi-liquids) momentum and energy can relax differently via inelastically scattering processes [@ashcroft]. Moreover, more dramatic deviations from the Wiedemann-Franz law are well-known for non-Fermi-liquids (see, e.g., [@hill]), in Luttinger liquids [@kane96; @wakeham11] and mesoscopic systems [@vavilov; @kubala08]. Summary {#sec:sum} ======= In this work, we studied the spin and thermal conductivity of spin chains and ladders using finite-temperature, real-time density matrix renormalization group techniques. We first computed the spin conductivity of the spin-1/2 XXZ chain as a function of the exchange anisotropy $\Delta>0$. Our data suggest finite dc-conductivities for all $\Delta>0$, yet a suppression of weight at low frequencies for special values such as $\Delta=0.5$. While the main drawback of the numerical method is that only finite times can be reached in the simulations, the comparison of various schemes to extract the frequency dependence supports our conclusion. Our results for two-leg spin ladders are consistent with the absence of ballistic contributions in agreement with Refs. . At high-temperatures, the XX ladder – which is equivalent to a system of hard-core bosons – exhibits a simple, Drude-like spin conductivity [@steinigeweg14a]. This property is lost as either the temperature is lowered or the exchange anisotropy is increased. At low temperatures, the spin conductivity features a two-peak structure with a maximum at $\omega=0$ and a large weight for frequencies above the optical spin gap. We further computed the dc spin conductivity; it decreases as the exchange anisotropy increases from $\Delta =0$ towards $\Delta=1$ and is a monotonicly increasing function of temperature. The thermal conductivity was obtained in the infinite-temperature limit, and our data agree reasonably well with earlier exact diagonalization results [@zotos04]. We extracted estimates for mean-free paths via kinetic equations that are used in the analysis of experimental data [@hess01]. The (momentum-averaged) mean-free paths $l_{\textnormal}{mag}$ obtained from $\kappa$ are larger than the ones calculated from $\sigma$. Thus, $l_{\textnormal}{mag}$ depends on the type of transport considered, and it is therefore not obvious that values for $l_{\textnormal}{mag}$ can directly be interpreted as a mean-free path of single-particle excitations. Future time- and real-space experiments could provide additional insight into the connection between single-excitations and the mean-free paths observed in transport measurements. [*Acknowledgment.*]{} We thank W. Brenig, P. Prelovšek, T. Prosen, R. Steinigeweg, and X. Zotos for very useful discussions. We are further indepted to X. Zotos for sending us exact diagonalization data from Ref.  and Bethe ansatz results for $D_{\rm s}(T)$ computed with the methods of Ref.  for comparison. We acknowledge support by the Nanostructured Thermoelectrics program of LBNL (C.K.) as well as by the DFG through the Research Training Group 1995 (D.M.K) and through FOR 912 via grant HE-5242/2-2 (F.H.-M.).

--- abstract: 'In this paper, we aim to understand and explain the decisions of deep neural networks by studying the behavior of predicted attributes when adversarial examples are introduced. We study the changes in attributes for clean as well as adversarial images in both standard and adversarially robust networks. We propose a metric to quantify the robustness of an adversarially robust network against adversarial attacks. In a standard network, attributes predicted for adversarial images are consistent with the wrong class, while attributes predicted for the clean images are consistent with the true class. In an adversarially robust network, the attributes predicted for adversarial images classified correctly are consistent with the true class. Finally, we show that the ability to robustify a network varies for different datasets. For the fine grained dataset, it is higher as compared to the coarse grained dataset. Additionally, the ability to robustify a network increases with the increase in adversarial noise.' author: - Sadaf Gulshad - Zeynep Akata - Jan Hendrik Metzen - Arnold Smeulders bibliography: - 'egbib.bib' title: Understanding Misclassifications by Attributes --- Introduction ============ Understanding neural networks is crucial in applications like autonomous vehicles, health care, robotics, for validating and debugging, as well as for building the trust of users [@kim2018textual; @uzunova2019interpretable]. This paper strives to understand and explain the decisions of deep neural networks by studying the behavior of predicted attributes when adversarial examples are introduced. We argue that even if no adversaries are being inserted in real world applications, adversarial examples can be exploited for understanding neural networks in their failure modes. Most of the state of the art approaches for interpreting neural networks work by focusing on features to produce saliency maps by considering class specific gradient information [@selvaraju2017grad; @simonyan2013deep; @sundararajan2017axiomatic], or by finding the part of the image which influences classification the most and removing it by adding perturbations [@zeiler2014visualizing; @fong2017interpretable]. These approaches reveal the part in the image where there is support to the classification and visualize the performance of known good examples. This tells a little about the boundaries of a class where dubious examples reside. However, humans motivate their decisions through semantically meaningful observations. For example, this type of bird has a blue head and red belly so, this must be a painted bunting. Hence, we study changes in the predicted attribute values of samples under mild modification of the image through adversarial perturbations. We believe this alternative dimension of study can provide a better understanding of how misclassification in a deep network can best be communicated to humans. Note that, we consider adversarial examples that are generated to fool only the classifier and not the interpretation (attributes) mechanism. Interpreting deep neural network decisions for adversarial examples helps in understanding their internal functioning [@tao2018attacks; @du2018towards]. Therefore, we explore ![ Our study with interpretable attribute prediction-grounding framework shows that, for a clean image predicted attributes “red belly” and “blue head” are coherent with the ground truth class (painted bunting), and for an adversarial image “white belly” and “white head” are coherent with the wrong class (herring gull) .[]{data-label="fig:Motivation"}](WACV_Motivation.pdf){width="\linewidth"} *How do the attribute values change under an adversarial attack on the standard classification network?* However while, describing misclassifications due to adversarial examples with attributes helps in understanding neural networks, assessing whether the attribute values still retain their discriminative power after making the network robust to adversarial noise is equally important. Hence, we also ask *How do the attribute values change under an adversarial attack on a robust classification network?* To answer these questions, we design experiments to investigate which attribute values change when an image is misclassified with increasing adversarial perturbations, and further when the classifier is made robust against an adversarial attack. Through these experiments we intend to demonstrate what attributes are important to distinguish between the right and the wrong class. For instance, as shown in Figure \[fig:Motivation\], “blue head” and ”red belly” associated with the class “painted bunting” are predicted correctly for the clean image. On the other hand, due to predicting attributes incorrectly as “white belly” and “white head”, the adversarial image gets classified into “herring gull” incorrectly. After analysing the changes in attributes with a standard and with a robust network we propose a metric to quantify the robustness of the network against adversarial attacks. Therefore, we ask *Can we quantify the robustness of an adversarially robust network?* In order to answer the third question, we design a robustness quantification metric for both standard as well as attribute based classifiers. To the best of our knowledge we are the first to exploit adversarial examples with attributes to perform a systematic investigation on neural networks, both *quantitatively* and *qualitatively*, for not only *standard*, but also for *adversarially robust* networks. We explain the decisions of deep computer vision systems by identifying what attributes change when an image is perturbed in order for a classification system to produce a specific output. Our results on three benchmark attribute datasets with varying size and granularity elucidate why adversarial images get misclassified, and why the same images are correctly classified with the adversarially robust framework. Finally we introduce a new metric to quantify the robustness of a network for both general as well as attribute based classifiers. Related Work ============ In this section, we discuss related work on interpretability and adversarial examples. [[**Interpretability.**]{}]{} Explaining the output of a decision maker is motivated by the need to build user trust before deploying them into the real world environment. Previous work is broadly grouped into two: 1) *rationalization*, that is, justifying the network’s behavior and 2) *introspective explanation*, that is, showing the causal relationship between input and the specific output [@du2018techniques]. Text-based class discriminative explanations [@hendricks2016generating; @park2016attentive], text-based interpretation with semantic information [@dong2017improving] and counter factual visual explanations [@goyal2019counterfactual] fall in the first category. On the other hand activation maximization [@simonyan2013deep; @zintgraf2017visualizing], learning the perturbation mask [@fong2017interpretable], learning a model locally around its prediction and finding important features by propagating activation differences [@ribeiro2016should; @shrikumar2017learning] fall in the second group. The first group has the benefit of being human understandable, but it lacks the causal relationship between input and output. The second group incorporates internal behavior of the network, but lacks human understandability. In this work, we incorporate human understandable justifications through attributes and causal relationship between input and output through adversarial attacks. [[**Interpretability of Adversarial Examples.**]{}]{} After analyzing neuronal activations of the networks for adversarial examples in [@dong2017towards] it was concluded that the networks learn recurrent discriminative parts of objects instead of semantic meaning. In [@jiang2018recent], the authors proposed a datapath visualization module consisting of the layer level, feature level, and the neuronal level visualizations of the network for clean as well as adversarial images. In [@zhang2019interpreting], the authors investigated adversarially trained convolutional neural networks by constructing images with different textural transformations while preserving the shape information to verify the shape bias in adversarially trained networks compared with standard networks. Finally, in [@tsipras2018robustness], the authors showed that the saliency maps from adversarially trained networks align well with human perception. These approaches use saliency maps for interpreting the adversarial examples, but saliency maps [@selvaraju2017grad] are often weak in justifying classification decisions, especially for fine-grained adversarial images. For instance, in Figure \[fig:saliency\] the saliency map of a clean image classified into the ground truth class, “red winged blackbird”, and the saliency map of a misclassified adversarial image, look quite similar. Instead, we propose to predict and ground attributes for both clean and adversarial images to provide visual as well as attribute-based interpretations. In fact, our predicted attributes for clean and adversarial images look quite different. By grounding the predicted attributes one can infer that the “orange wing” is important for “red winged blackbird” while the “red head” is important for “red faced cormorant”. Indeed, when the attribute value for orange wing decreases and red head increases the image gets misclassified. [[**Adversarial Examples.**]{}]{} Small carefully crafted perturbations, called *adversarial perturbations*, when added to the inputs of deep neural networks, result in *adversarial examples*. These adversarial examples can easily drive the classifiers to the wrong classification [@szegedy2013intriguing]. Such attacks involve iterative fast gradient sign method (IFGSM) [@kurakin2016adversarial], Jacobian-based saliency map attacks [@papernot2016limitations], one pixel attacks [@su2019one], Carlini and Wagner attacks [@carlini2017towards] and universal attacks [@moosavi2016deepfool]. We select IFGSM for our experiments, but our method can also be used with other types of adversarial attacks. Adversarial examples can also be used for understanding neural networks. [@anonymous2020evaluations] aims at utilizing adversarial examples for understanding deep neural networks by extracting the features that provide the support for classification into the target class. The most salient features in the images provide the way to interpret the decision of a classifier, but they lack human understandability. Additionally, finding the most salient features is computationally rather expensive. The crucial point, however, is that if humans explain classification by attributes, they are also natural candidates to study misclassification and robustness. Hence, in this work in order to understand neural networks we utilize adversarial examples with attributes which explain the misclassification due to adversarial attacks. Method ====== {width="\linewidth"} \[fig:saliency\] In this section, in order to explain what attributes change when an adversarial attack is performed on the classification mechanism of the network, we detail a two-step framework. First, we perturb the images using adversarial attack methods and robustify the classifiers via adversarial training. Second, we predict the class specific attributes and visually ground them on the image to provide an intuitive justification of why an image is classified as a certain class. Finally, we introduce our metric for quantifying the robustness of an adversarially robust network against adversarial attacks. Adversarial Attacks and Robustness ---------------------------------- Given a clean $n\text{-th}$ input $x_n$ and its respective ground truth class $y_n$ predicted by a model $f(x_n)$, an adversarial attack model generates an image $\hat{x}_n$ for which the predicted class is $y$, where $y \neq y_n$. In the following, we detail an adversarial attack method for fooling a general classifier and an adversarial training technique that robustifies it. [[**Adversarial Attacks.**]{}]{} The iterative fast gradient sign method (IFGSM) [@kurakin2016adversarial] is leveraged to fool only the classifier network. IFGSM solves the following equation to produce adversarial examples: $$\begin{aligned} & \hat{x}^0 =x_n \nonumber \\ & \hat{x}_n^{i+1}=\text{Clip}_{\epsilon}\{\hat{x}_n^{i}+\alpha\text{Sign}(\bigtriangledown{\hat{x}_n^i}\mathcal{L}(\hat{x}_n^i,y_{n}))\} \end{aligned}$$ where $\bigtriangledown{\hat{x}_n^i}\mathcal{L}$ represents the gradient of the cost function w.r.t. perturbed image $\hat{x}_n^i$ at step $i$. $\alpha$ determines the step size which is taken in the direction of sign of the gradient and finally, the result is clipped by epsilon $\text{Clip}_{\epsilon}$. [[**Adversarial Robustness.**]{}]{} We use *adversarial training* as a defense against adversarial attacks which minimizes the following objective [@43405]: $$\begin{aligned} \mathcal{L}_{adv}(x_n,y_n) & = \alpha \mathcal{L}(x_n,y_n) + (1-\alpha)\mathcal{L}(\hat{x}_n,y) \end{aligned}$$ where, $\mathcal{L}(x_n,y_n)$ is the classification loss for clean images, $\mathcal{L}(\hat{x}_n,y)$ is the loss for adversarial images and $\alpha$ regulates the loss to be minimized. The model finds the worst case perturbations and fine tunes the network parameters to reduce the loss on perturbed inputs. Hence, this results in a robust network $f^r(\hat{x})$, using which improves the classification accuracy on the adversarial images. {width="\linewidth"} \[fig:ADV\_SJE\] Attribute Prediction and Grounding ---------------------------------- Our attribute prediction and grounding model uses attributes to define a joint embedding space that the images are mapped to. [[**Attribute prediction.**]{}]{} The model is shown in Figure \[fig:ADV\_SJE\]. During training our model maps clean training images close to their respective class attributes, e.g. “painted bunting” with attributes “red belly, blue head”, whereas adversarial images get mapped close to a wrong class, e.g. “herring gull” with attributes “white belly, white head”. We employ structured joint embeddings (SJE) [@akata2015evaluation] to predict attributes in an image. Given the input image features $\theta(x_n) \in \mathcal{X}$ and output class attributes $\phi(y_n) \in \mathcal{Y}$ from the sample set $\mathcal{S}=\{(\theta(x_n),\phi(y_n),n=1...N \}$ SJE learns a mapping $\mathbb{f}:\mathcal{X} \to \mathcal{Y}$ by minimizing the empirical risk of the form $\frac{1}{N}\sum_{n=1}^N \Delta(y_n,\mathbb{f}(x_n))$ where $\Delta: \mathcal{Y} \times \mathcal{Y} \to \mathbb{R} $ estimates the cost of predicting $\mathbb{f}(x_n)$ when the ground truth label is $y_n$. A compatibility function $F:\mathcal{X}\times\mathcal{Y}\to \mathbb{R}$ is defined between input $\mathcal{X}$and output $\mathcal{Y}$ space: $$F(x_n,y_n;W)=\theta(x_n)^TW\phi(y_n)$$ Pairwise ranking loss $\mathbb{L}(x_n,y_n,y)$ is used to learn the parameters $(W)$: $$\Delta(y_n,y)+\theta(x_n)^TW\phi(y_n)-\theta(x_n)^TW\phi(y)$$ Attributes are predicted for both clean and adversarial images by: $$\vspace{-3mm} \bold{A}_{n,y_n}=\theta(x_n)W \, , \bold{\hat{A}}_{n,y}=\theta(\hat{x}_n)W$$ The image is assigned to the label of the nearest output class attributes $\phi(y_n)$. [[**Attribute grounding.**]{}]{} In our final step, we ground the predicted attributes on to the input images using a pre-trained Faster RCNN network and visualize them as in [@anne2018grounding]. The pre-trained Faster RCNN model $\mathcal{F}(x_n)$ predicts the bounding boxes denoted by $b^j$. For each object bounding box it predicts the class $\mathbb{Y}^j$ as well as the attribute $\mathbb{A}^j$  [@anderson2018bottom]. $$\vspace{-1.5mm} b^j,\mathbb{A}^j,\mathbb{Y}^j=\mathcal{F}(x_n)$$ where, $j$ is the bounding box index. The most discriminative attributes predicted by SJE are selected based on the criteria that they change the most when the image is perturbed with noise. For clean images we use: $$q=\underset{i}{\mathrm{argmax}}(\bold{A}_{n,y_n}^i-\phi(y^i)) \label{eq:att_sel1} \vspace{-1mm}$$ and for adversarial images we use: $$p=\underset{i}{\mathrm{argmax}}(\bold{\hat{A}}_{n,y}^i-\phi(y_n^i)). \label{eq:att_sel2} \vspace{-1mm}$$ where $i$ is the attribute index, $q$ and $p$ are the indexes of the most discriminative attributes predicted by SJE and $\phi(y^i)$, $\phi(y_n^i)$ are wrong class and ground truth class attributes respectively. Then we search for selected attributes $\bold{A}_{x_n,y_n}^q, \bold{A}_{\hat{x}_n,y}^p$ in attributes predicted by Faster RCNN for each bounding box $\mathbb{A}_{x_n}^j, \mathbb{A}_{\hat{x}_n}^j$, and when the attributes predicted by SJE and Faster RCNN are found, that is $\bold{A}_{x_n,y_n}^q = \mathbb{A}_{x_n}^j$, $\bold{A}_{\hat{x}_n,y}^p = \mathbb{A}_{\hat{x}_n}^j$ we ground them on their respective clean and adversarial images. Note that the adversarial images being used here are generated to fool only the general classifier *and not the attribute predictor nor the Faster RCNN*. Robustness Quantification ------------------------- To describe the ability of a network for robustification, independent of its performance on a standard classifier we introduce a metric called *robust ratio*. We calculate the loss of accuracy $L_R$ on a robust classifier, by comparing a standard classifier $f(x_n)$ on clean images with the robust classifier $f^r(\hat{x}_n)$ on the adversarially perturbed images as given below: $$L_R=f(x_n)-f^r(\hat{x}_n)$$ And then we calculate the loss of accuracy $L_S$ on a standard classifier, by comparing its accuracy on the clean and adversarially perturbed images: $$L_S=f(x_n)-f(\hat{x}_n)$$ The ability to robustify is then defined as: $$R=\frac{L_R}{L_S}$$ $R$ is the robust ratio. It indicates the fraction of the classification accuracy of the standard classifier recovered by the robust classifier when adding noise. Experiments =========== {width="0.324\linewidth"} {width="0.324\linewidth"} \[fig:acc\_plots\] In this section, we perform experiments on three different datasets and analyse the change in attributes for clean as well as adversarial images. We additionally analyse results for our proposed robustness quantification metric on both general and attribute based classifiers. [[**Datasets.**]{}]{} We experiment on three datasets, Animals with Attributes 2 (AwA) [@lampert2009learning], Large attribute (LAD) [@zhao2018large] and Caltech UCSD Birds (CUB) [@wah2011caltech]. AwA contains 37322 images (22206 train / 5599 val / 9517 test) with 50 classes and 85 attributes per class. LAD has 78017 images (40957 train / 13653 val / 23407 test) with 230 classes and 359 attributes per class. CUB consists of 11,788 images (5395 train / 599 val / 5794 test) belonging to 200 fine-grained categories of birds with 312 attributes per class. All the three datasets contain real valued class attributes representing the presence of a certain attribute in a class. Visual Genome Dataset [@krishna2017visual] is used to train the Faster-RCNN model which extracts the bounding boxes using 1600 object and 400 attribute annotations. Each bounding box is associated with an attribute followed by the object, e.g. a brown bird. [[**Image Features and Adversarial Examples.**]{}]{} We extract image features and generate adversarial images using the fine-tuned Resnet-152. Adversarial attacks are performed using IFGSM method with epsilon $\epsilon$ values $0.01$, $0.06$ and $0.12$. The $\l_\infty $ norm is used as a similarity measure between clean input and the generated adversarial example. [[**Adversarial Training.**]{}]{} As for adversarial training, we repeatedly computed the adversarial examples while training the fine-tuned Resnet-152 to minimize the loss on these examples. We generated adversarial examples using the projected gradient descent method. This is a multi-step variant of FGSM with epsilon $\epsilon$ values $0.01$, $0.06$ and $0.12$ respectively for adversarial training as in [@madry2017towards]. Note that we are not attacking the attribute based network directly but we are attacking the general classifier and extracting features from it for training the attribute based classifier. Similarly, the adversarial training is also performed on the general classifier and the features extracted from this model are used for training the attribute based classifier. [[**Attribute Prediction and Grounding.**]{}]{} At test time the image features are projected onto the attribute space. The image is assigned with the label of the nearest ground truth attribute vector. The predicted attributes are grounded by using Faster-RCNN pre-trained on Visual Genome Dataset since we do not have ground truth part bounding boxes for any of attribute datasets. Results ======= We investigate the change in attributes quantitatively (i) by performing classification based on attributes and (ii) by computing distances between attributes in embedding space. We additionally investigate changes qualitatively by grounding the attributes on images for both standard and adversarially robust networks. At first, we compare the general classifier $f(x_n)$ and the attribute based classifier $\mathbb{f}(x_n)$ in terms of the classification accuracy on clean images. Since the attribute based model is a more explainable classifier, it predicts attributes, compared to general classifier, which predicts the class label directly. Therefore, we first verify whether the attribute based classifier performs equally well as the general classifier. We find that, the attribute based and general classifier accuracies are comparable for AWA (general: 93.53, attribute based: 93.83). The attribute based classifier accuracy is slightly higher for LAD (general: 80.00, attribute based: 82.77), and slightly lower for CUB (general: 81.00, attribute based: 76.90) dataset. ![ **Attribute distance plots for standard learning frameworks.** Standard learning framework plots are shown for the clean and the adversarial image attributes.[]{data-label="fig:standardattr"}](standard_quantitative.pdf){width="\linewidth"} {width="\linewidth"} To qualitatively analyse the predicted attributes, we ground them on clean and adversarial images. We select our images among the ones that are correctly classified when clean and incorrectly classified when adversarially perturbed. Further we select the most discriminative attributes based on equation \[eq:att\_sel1\] and \[eq:att\_sel2\]. We evaluate $50$ attributes that change their value the most for the CUB, $50$ attributes for the AWA, and $100$ attributes for the LAD dataset. Adversarial Attacks on Standard Network --------------------------------------- ### Quantitative Analysis [[**By Performing Classification based on Attributes.**]{}]{} With adversarial attacks, the accuracy of both the general and attribute based classifiers drops with the increase in perturbations see Figure \[fig:acc\_plots\] (blue curves). The drop in accuracy of the general classifier for the fine grained CUB dataset is higher as compared to the coarse AWA dataset which confirms our hypothesis. For example, at $\epsilon=0.01$ for the CUB dataset the general classifier’s accuracy drops from $81\%$ to $31\%$ ($\approx 50\%$ drop), while for the AWA dataset it drops from $93.53\%$ to $70.54\%$ ($\approx 20\%$ drop). However, the drop in accuracy with the attribute based classifier is almost equal for both, $\approx 20\%$ . We propose one of the reasons behind the smaller drop of accuracy for the CUB dataset with the attribute based classifier compared to the general classifier is that for fine grained datasets there are many common attributes among classes. Therefore, in order to misclassify an image a significant number of attributes need to be changed. For a coarse grained dataset, changing a few attributes is sufficient for misclassification. Another reason is that there are $9\%$ more attributes per class in the CUB dataset as compared to the AWA dataset. For the coarse dataset the attribute based classifier shows comparable performance with the general classifier. While for the fine grained dataset the attribute based classifier shows better performance than the general classifier so a large change in attributes is required to cause misclassification with attributes. Overall, the drop in the accuracy with the adversarial attacks demonstrates that, with adversarial perturbations, the attribute values change towards those that belong to the new class and cause the misclassification. [[**By Computing Distances in Embedding Space.**]{}]{} In order to perform analysis on attributes in embedding space, we consider the images which are correctly classified without perturbations and misclassified with perturbations. Further, we select the top $20\%$ of the most discriminative attributes using equation \[eq:att\_sel1\] and \[eq:att\_sel2\]. Our aim is to analyse the change in attributes in embedding space. ![ **Attribute distance plots for robust learning frameworks.** Robust learning framework plots are shown only for the adversarial image attributes but for adversarial images misclassified with the standard features and correctly classified with the robust features.[]{data-label="fig:robustattr"}](robust_quantitative.pdf){width="\linewidth"} We contrast the Euclidean distance between predicted attributes of clean and adversarial samples: $$d_1 = d\{\bold{A}_{n,y_n},\bold{\hat{A}}_{n,y}\} =\parallel \bold{A}_{n,y_n}-\bold{\hat{A}}_{n,y} \parallel_2 \label{eq:d1_1}$$ with the Euclidean distance between the ground truth attribute vector of the correct and wrong classes: $$d_2 = d\{\phi(y_n),\phi(y)\}=\parallel\phi(y_n)-\phi(y)) \parallel_2 \label{eq:d2_1}$$ and show the results in Figure \[fig:standardattr\]. Where, $\bold{A}_{n,y_n}$ denotes the predicted attributes for the clean images classified correctly, and $\bold{\hat{A}}_{n,y}$ denotes the predicted attributes for the adversarial images misclassified with a standard network. The correct ground truth class attribute is referred to as $\phi(y_n)$ and wrong class attributes are $\phi(y)$. {width="\linewidth"} We observe that for the AWA dataset the distances between the predicted attributes for adversarial and clean images $d_1$ are smaller than the distances between the ground truth attributes of the respective classes $d_2$. The closeness in predicted attributes for clean and adversarial images as compared to their ground truths shows that attributes change towards the wrong class but not completely. This is due to the fact that for coarse classes, only a small change in attribute values is sufficient to change the class. The fine-grained CUB dataset behaves differently. The overlap between $d_1$ and $d_2$ distributions demonstrates that attributes of images belonging to fine-grained classes change significantly as compared to images from coarse categories. Although the fine grained classes are closer to each other, due to the existence of many common attributes among fine grained classes, attributes need to change significantly to cause misclassification. Hence, for the coarse dataset, the attributes change minimally, while for the fine grained dataset they change significantly. ### Qualitative Analysis We observe in Figure \[fig:Qualitative-1\] that the most discriminative attributes for the clean images are coherent with the ground truth class and are localized accurately; however, for adversarial images they are coherent with the wrong class. Those attributes which are common among both clean and adversarial classes are localized correctly on the adversarial images; however, the attributes which are not related to the ground truth class, the ones that are related to the wrong class can not get grounded as there is no visual evidence that supports the presence of these attributes. For example “brown wing, long wing, long tail” attributes are common in both classes; hence, they are present both in the clean image and the adversarial image. On the other hand, “has a brown color” and “a multicolored breast” are related to the wrong class and are not present in the adversarial image. Hence, they can not be grounded. Similarly, in the second example none of the attributes are grounded. This is because attributes changed completely towards the wrong class and the evidence for those attributes is not present in the image. This indicates that attributes for the clean images correspond to the ground truth class and for adversarial images correspond to the wrong class. Additionally, only those attributes common among both the wrong and the ground truth classes get grounded on adversarial images. Similarly, our results on the LAD and AWA datasets in the second row of Figure \[fig:Qualitative-1\] show that the grounded attributes on clean images confirm the classification into the ground truth class while the attributes grounded on adversarial images are common among clean and adversarial images. For instance, in the first example of AWA, the “is black” attribute is common in both classes so it is grounded on both images, but “has claws” is an important attribute for the adversarial class. As it is not present in the ground truth class, it is not grounded. {width="0.35\linewidth"} {width="0.35\linewidth"} Compared to misclassifications caused by adversarial perturbations on CUB, images do not necessarily get misclassified into the most similar class for the AWA and LAD datasets as they are coarse grained datasets. Therefore, there is less overlap of attributes between ground truth and adversarial classes, which is in accordance with our quantitative results. Furthermore, the attributes for both datasets are not highly structured, as different objects can be distinguished from each other with only a small number of attributes. Adversarial Attacks on Robust Network ------------------------------------- ### Quantitative Analysis [[**By Performing Classification based on Attributes.**]{}]{} Our evaluation on the standard and adversarially robust networks shows that the classification accuracy improves for the adversarial images when adversarial training is used to robustify the network: \[fig:acc\_plots\] (purple curves). For example, in Figure  \[fig:acc\_plots\] for AWA the accuracy of the general classifier improved from $ 70.54\%$ to $92.15\%$ ($\approx 21\%$ improvement) for adversarial attack with $\epsilon=0.01$. As expected for the fine grained CUB dataset the improvement is $\approx 31\%$ higher than the AWA dataset. However, for the attribute based classifier, the improvement in accuracy for AWA ($\approx 18.06\%$) is almost double that of the CUB dataset ($\approx 7\%$). We propose this is because the AWA dataset is coarse, so in order to classify an adversarial image correctly to its ground truth class, a small change in attributes is sufficient. Conversely the fine grained CUB dataset requires a large change in attribute values to correctly classify an adversarial image into its ground truth class. Additionally, CUB contains $9\%$ more per class attributes. For a coarse AWA dataset the attributes change back to the correct class and represent the correct class accurately. While for the fine grained CUB dataset, a large change in attribute values is required to correctly classify images. This shows that with a robust network, the change in attribute values for adversarial images indicate to the ground truth class, resulting in better performance. Overall, we observe by analysing attribute based classifier accuracy that with the adversarial attacks the change in attribute values indicates in which wrong class it is assigned and with the robust network the change in attribute values indicates towards the ground truth class. [[**By Computing Distances in Embedding Space**]{}]{} We compare the distances between the predicted attributes of only adversarial images that are classified correctly with the help of an adversarially robust network $\bold{\hat{A}}^{{r}}_{n,y_n}$ and classified incorrectly with a standard network $\bold{\hat{A}}_{n,y}$: $$\label{eq:d1_3} d_1 = d\{\bold{\hat{A}}^{{r}}_{n,y_n},\bold{\hat{A}}_{n,y}\}=\parallel \bold{\hat{A}}^{{r}}_{n,y_n}-\bold{\hat{A}}_{n,y} \parallel_2 {\vspace{-2.5mm}}$$ with the distances between the ground truth target class attributes $\phi(y_n)$ and ground truth wrong class attributes $\phi(y)$: $$\label{eq:d2_3} d_2 = d\{\phi(y_n),\phi(y)\}=\parallel\phi(y_n)-\phi(y)) \parallel_2$$ The results are shown in Figure \[fig:robustattr\]. By comparing Figure \[fig:robustattr\] with Figure \[fig:standardattr\] we observe a similar behavior. The plots in Figure \[fig:standardattr\] are plotted between clean and adversarial image attributes. While plots in Figure \[fig:robustattr\] are plotted between only adversarial images but classified correctly with an adversarially robust network and misclassified with a standard network. This shows that the adversarial images classified correctly with a robust network behave like clean images, i.e. a robust network predicts attributes for the adversarial images which are closer to their ground truth class. ### Qualitative Analysis Finally, our analysis with correctly classified images by the adversarially robust network shows that adversarial images with the robust network behave like clean images also visually. In the Figure \[fig:Qualitative-2\], we observe that the attributes of an adversarial image with a standard network are closer to the adversarial class attributes. However, the grounded attributes of adversarial image with a robust network are closer to its ground truth class. For instance, the first example contains a “blue head” and a “black wing” whereas one of the most discriminating properties of the ground truth class “blue head” is not relevant to the adversarial class. Hence this attribute is not predicted as the most relevant by our model, and thus our attribute grounder did not ground it. This shows that the attributes for adversarial images classified correctly with the robust network are in accordance with the ground truth class and hence get grounded on the adversarial images. Analysis for Robustness Quantification -------------------------------------- The results for our proposed robustness quantification metric are shown in Figure \[fig:Robustifiability\]. We observe that the ability to robustify a network against adversarial attacks varies for different datasets. The network with fine grained CUB dataset is easy to robustify as compared to coarse AWA and LAD datasets. For the general classifier as expected the ability to robustify the network increases with the increase in noise. For the attribute based classifier the ability to robustify the network is high with the small noise but it drops as the noise increases (at $\epsilon=0.06$) and then again increases at high noise value (at $\epsilon=0.12$). Conclusion ========== In this work we conducted a systematic study on understanding the neural networks by exploiting adversarial examples with attributes. We showed that if a noisy sample gets misclassified then its most discriminative attribute values indicate to which wrong class it is assigned. On the other hand, if a noisy sample is correctly classified with the robust network then the most discriminative attribute values indicate towards the ground truth class. Finally, we proposed a metric for quantifying the robustness of a network and showed that the ability to robustify a network varies for different datasets. Overall the ability to robustify a network increases with the increase in adversarial perturbations.

--- abstract: 'Motivated by this, we present *aggregation, mode decomposition and projection* (AMP) a feature extraction technique particularly suited to intermittent time-series data which contain time-frequency patterns. For our method all individual time-series within a set are combined to form a non-intermittent *aggregate*. This is decomposed into a set of components which represent the intrinsic time-frequency signals within the data set. . Using synthetically generated data we show that a clustering approach which uses the features derived from AMP significantly outperforms traditional clustering methods. Our technique is further exemplified on a real world data set where AMP can be used to discover groupings of individuals which correspond to real world sub-populations.' author: - | Duncan S. Barrack\ \ \ \ James Goudling\ \ \ \ Keith Hopcraft\ \ \ \ - | Simon Preston\ \ \ \ Gavin Smith\ \ \ \ bibliography: - 'barrack.bib' date: 03 June 2015 title: ' a new time-frequency feature extraction method for intermittent time-series data' --- MiLeTS workshop in conjunction with KDD’ 15 August 10-13 2015, Sydney, Australia Introduction ============ Extracting numerical features from time-series data is desirable for a number of reasons including revealing human interpretable characteristics of the data [@xing2011extracting], data compression [@fu2011review] as well as clustering and classification [@garrett2003comparison; @liao2005clustering; @wang2006characteristic]. It is often useful to divide the different feature extraction approaches into frequency domain and time domain based methods. Frequency domain extraction techniques include the discrete Fourier transform [@vlachos2005periodicity; @vlachos2006structural] and wavelet transform [@misiti2007clustering]. Examples of time domain techniques are model based approaches [@macdonald1997hidden] and more recently shapelets [@mueen2012clustering; @ye2009time]. Of particular interest to this paper are intermittent time-series data, such as that derived from human behavioural (inter-) actions, e.g. communications and retail transaction logs. Data of this type contains oscillatory time-frequency patterns corresponding to human behavioural patterns such as the 24 hour circadian rhythm, or 7 day working week/weekend. It is also characterised by short periods of high activity followed by long periods of inactivity (intermittence) [@barabasi2005origin; @jo2012circadian; @karsai2012universal; @vazquez2006modeling]. Such characteristics mean that intermittent time-series feature sharp transitions in the dependent variable. When frequency based feature extraction techniques underpinned by the Fourier or wavelet transforms are applied, the transforms produce *ringing* artefacts (a well known example in Fourier analysis is the Gibbs phenomena [@gibbs1899fourier]) which results in spurious signals being produced in the spectra. These rogue signals make it extremely difficult to determine what the genuine frequency patterns in the data are. Furthermore, such signals are extremely damaging to clustering and classification techniques which use frequency or time-frequency features as inputs. The paper is structured as follows. In section \[sec:prob statement\] we demonstrate the issue of using traditional feature extraction techniques on intermittent data with a focus on the use of derived features for clustering. After discussing related work in Section \[sec:related\_work\] we introduce our ameliorative strategy in the form of Aggregation, mode decomposition and projection (AMP) in Section \[sec:method\]. We show in Section \[sec:eval\] that when features derived from AMP are used for clustering synthetically generated intermittent time-series data, results are significantly better than those which use traditional time-series clustering techniques. In this section, we also demonstrate that AMP gives promising results when applied a real communications data set. We conclude with a discussion in Section \[sec:conclusions\]. Background {#sec:prob statement} ========== Although the concept of intermittence has received some examination across various fields [@scott2013encyclopedia; @kicsi2009neural] no accepted definition for the term currently exists. Consequently, in this work, we introduce our own expression which can be used to quantify intermittence in time-series. Before this is formally defined, to explain our rationale behind it we refer the reader to plots of three time different time-series in Figure \[fig:time\_series\_IM\_egs\], These observations leads us to construct a intermittence measure based on the total proportion of the time domain that a time-series takes its most frequent value. In particular, if we regard a discrete time-series as a vector $\mathbf{x}=(x_{1},x_{2},\ldots,x_{n})$ of real valued elements sampled at equal spaces in time as a realisation of a random process, then we define a measure of its intermittence by $$\begin{aligned} \phi (\mathbf{x}) = P(x_{i}=M(\mathbf{x})), \label{eq:interittence}\end{aligned}$$ where $M(\mathbf{x})$ is the mode, or most common value, in vector $\mathbf{x}$ and $P(x_{i}=M(\mathbf{x}))$ is the empirical probability, or relative frequency, that a randomly selected element $x_{i}$ of $\mathbf{x}$ has this value. As a time-series becomes increasing intermittent $\phi$ will tend to 1. With this definition, time-series (a) in Figure \[fig:time\_series\_IM\_egs\] has a value for $\phi$ of 0.0001 reflecting the fact it is not intermittent. The intermittence measure of time-series (c) (0.7675) is higher than time-series (b) (0.4860) reflecting our observation that (c) is more intermittent than (b). ![Three example time-series illustrating the distinctions between a non-intermittent times series (a), partially intermittent time-series (b) and an extremely intermittent time-series (c).[]{data-label="fig:time_series_IM_egs"}](time_series_IM_egs.pdf){width="50.00000%"} To illustrate the negative impact that intermittence has on the pertinence of features extracted using traditional techniques, first consider two distinct sets of non-intermittent time-series. We investigate what affect increasing intermittence has on clustering results which use features obtained via the Fourier and wavelet transforms as well as clustering approaches which use the Euclidean and dynamic time warping (DTW) distance between individual time series. The first set of time-series data is composed of 100 realisations of an [*almost periodically-driven*]{} stochastic process [@bezandry2011almost] (see Section \[sec:syn\_generation\] for full details of this procedure), with period ranging linearly from 2 at the beginning of the simulation to 4 at the end (time-series from this set are depicted diagrammatically in blue). The second set also contains 100 time-series generated from an identical process, except for a period which ranges linearly from 8 to 16 (depicted diagrammatically in red). Two examples from each set are illustrated in Figure \[fig:intermittency\_examples\]a. Each of the time-series is plotted using the values for the first two dimensions obtained from classical multi-dimensional scaling (MDS) [@cox2010multidimensional] of every coefficient value of each term in their direct discrete Fourier and wavelet decompositions (see Section \[sec:synth\_eval\] for full details of this procedure) in Figures \[fig:intermittency\_examples\]b and \[fig:intermittency\_examples\]c respectively. Additionally MDS results are shown where the Euclidean distance and DTW distance are used as the similarity measure between time series (Figures \[fig:intermittency\_examples\]d and \[fig:intermittency\_examples\]e respectively). Clearly, in this instance, a clustering approach based on any of these techniques is sufficient to discriminate the time-series from the two groups. Next we consider what effect time-series with a greater value of $\phi$ (and hence higher intermittence) has upon clustering. These have the same time-frequency patterns as the corresponding time-series in Figure \[fig:intermittency\_examples\]a but are more intermittent. Examples are presented in Figure \[fig:intermittency\_examples\]f and illustrate the sharp transitions and long periods for which the time-series take a constant value (some examples marked in the figure) that begin to occur in the data. Although it is still possible to discriminate the time-series in the MDS plots (Figures \[fig:intermittency\_examples\]g-j), the sharp transitions in the data introduce ringing artefacts in the frequency based decompositions which results in less well separated clusters (compare Figure \[fig:intermittency\_examples\]g with \[fig:intermittency\_examples\]b and Figure \[fig:intermittency\_examples\]h with \[fig:intermittency\_examples\]c). Furthermore, the large periods of constant values act to degrade the discriminative power of Euclidean and DTW based methods (compare Figure \[fig:intermittency\_examples\]i with \[fig:intermittency\_examples\]d and Figure \[fig:intermittency\_examples\]j with \[fig:intermittency\_examples\]e). By the time we have increased intermittency further still to generate sets of 100 highly intermittent time-series (Figure \[fig:intermittency\_examples\]k) the negative impact of intermittency on clustering is severe and neither frequency domain based, Euclidean or DTW based methods can be used to separate the data (see Figure \[fig:intermittency\_examples\]l-o). time-series MDS scatter plots\ ![The impact of intermittency on cluster analysis. Plot (a) shows two non-intermittent time-series from a set of 200 which were generated via an almost periodically driven stochastic process, with periods ranging linearly from 2 to 4 for one half the set (blue) and 8 to 16 for the other half (red). Each time-series in the set of 200 is plotted using the values for the first two dimensions obtained from classical MDS of every coefficient value of the terms in their direct discrete Fourier (plot (b)) and wavelet decomposition (plot (c)) as well as of the Euclidean (plot (d)) and DTW (plot (e)) distance matrices. Plot (f) shows time-series with increased intermittency but with the same time-frequency pattern as in (a). The corresponding MDS plots are shown in plots (g-j). Finally, plot (k) shows highly intermittent time-series with associated MDS scatter plots in (l-o) illustrating the collapse in efficacy of cluster analysis. The value for the intermittence $\phi$ (equation (\[eq:interittence\])) for the time-series are given in the figure insets.[]{data-label="fig:intermittency_examples"}](arrow_no_int.pdf "fig:"){width="2.50000%"} ![The impact of intermittency on cluster analysis. Plot (a) shows two non-intermittent time-series from a set of 200 which were generated via an almost periodically driven stochastic process, with periods ranging linearly from 2 to 4 for one half the set (blue) and 8 to 16 for the other half (red). Each time-series in the set of 200 is plotted using the values for the first two dimensions obtained from classical MDS of every coefficient value of the terms in their direct discrete Fourier (plot (b)) and wavelet decomposition (plot (c)) as well as of the Euclidean (plot (d)) and DTW (plot (e)) distance matrices. Plot (f) shows time-series with increased intermittency but with the same time-frequency pattern as in (a). The corresponding MDS plots are shown in plots (g-j). Finally, plot (k) shows highly intermittent time-series with associated MDS scatter plots in (l-o) illustrating the collapse in efficacy of cluster analysis. The value for the intermittence $\phi$ (equation (\[eq:interittence\])) for the time-series are given in the figure insets.[]{data-label="fig:intermittency_examples"}](time_series1.pdf "fig:"){width="22.00000%"} ![The impact of intermittency on cluster analysis. Plot (a) shows two non-intermittent time-series from a set of 200 which were generated via an almost periodically driven stochastic process, with periods ranging linearly from 2 to 4 for one half the set (blue) and 8 to 16 for the other half (red). Each time-series in the set of 200 is plotted using the values for the first two dimensions obtained from classical MDS of every coefficient value of the terms in their direct discrete Fourier (plot (b)) and wavelet decomposition (plot (c)) as well as of the Euclidean (plot (d)) and DTW (plot (e)) distance matrices. Plot (f) shows time-series with increased intermittency but with the same time-frequency pattern as in (a). The corresponding MDS plots are shown in plots (g-j). Finally, plot (k) shows highly intermittent time-series with associated MDS scatter plots in (l-o) illustrating the collapse in efficacy of cluster analysis. The value for the intermittence $\phi$ (equation (\[eq:interittence\])) for the time-series are given in the figure insets.[]{data-label="fig:intermittency_examples"}](eg1_scatter.pdf "fig:"){width="22.00000%"} ![The impact of intermittency on cluster analysis. Plot (a) shows two non-intermittent time-series from a set of 200 which were generated via an almost periodically driven stochastic process, with periods ranging linearly from 2 to 4 for one half the set (blue) and 8 to 16 for the other half (red). Each time-series in the set of 200 is plotted using the values for the first two dimensions obtained from classical MDS of every coefficient value of the terms in their direct discrete Fourier (plot (b)) and wavelet decomposition (plot (c)) as well as of the Euclidean (plot (d)) and DTW (plot (e)) distance matrices. Plot (f) shows time-series with increased intermittency but with the same time-frequency pattern as in (a). The corresponding MDS plots are shown in plots (g-j). Finally, plot (k) shows highly intermittent time-series with associated MDS scatter plots in (l-o) illustrating the collapse in efficacy of cluster analysis. The value for the intermittence $\phi$ (equation (\[eq:interittence\])) for the time-series are given in the figure insets.[]{data-label="fig:intermittency_examples"}](arrow_some_int.pdf "fig:"){width="2.50000%"} ![The impact of intermittency on cluster analysis. Plot (a) shows two non-intermittent time-series from a set of 200 which were generated via an almost periodically driven stochastic process, with periods ranging linearly from 2 to 4 for one half the set (blue) and 8 to 16 for the other half (red). Each time-series in the set of 200 is plotted using the values for the first two dimensions obtained from classical MDS of every coefficient value of the terms in their direct discrete Fourier (plot (b)) and wavelet decomposition (plot (c)) as well as of the Euclidean (plot (d)) and DTW (plot (e)) distance matrices. Plot (f) shows time-series with increased intermittency but with the same time-frequency pattern as in (a). The corresponding MDS plots are shown in plots (g-j). Finally, plot (k) shows highly intermittent time-series with associated MDS scatter plots in (l-o) illustrating the collapse in efficacy of cluster analysis. The value for the intermittence $\phi$ (equation (\[eq:interittence\])) for the time-series are given in the figure insets.[]{data-label="fig:intermittency_examples"}](time_series2.pdf "fig:"){width="22.00000%"} ![The impact of intermittency on cluster analysis. Plot (a) shows two non-intermittent time-series from a set of 200 which were generated via an almost periodically driven stochastic process, with periods ranging linearly from 2 to 4 for one half the set (blue) and 8 to 16 for the other half (red). Each time-series in the set of 200 is plotted using the values for the first two dimensions obtained from classical MDS of every coefficient value of the terms in their direct discrete Fourier (plot (b)) and wavelet decomposition (plot (c)) as well as of the Euclidean (plot (d)) and DTW (plot (e)) distance matrices. Plot (f) shows time-series with increased intermittency but with the same time-frequency pattern as in (a). The corresponding MDS plots are shown in plots (g-j). Finally, plot (k) shows highly intermittent time-series with associated MDS scatter plots in (l-o) illustrating the collapse in efficacy of cluster analysis. The value for the intermittence $\phi$ (equation (\[eq:interittence\])) for the time-series are given in the figure insets.[]{data-label="fig:intermittency_examples"}](eg2_scatter.pdf "fig:"){width="22.00000%"} ![The impact of intermittency on cluster analysis. Plot (a) shows two non-intermittent time-series from a set of 200 which were generated via an almost periodically driven stochastic process, with periods ranging linearly from 2 to 4 for one half the set (blue) and 8 to 16 for the other half (red). Each time-series in the set of 200 is plotted using the values for the first two dimensions obtained from classical MDS of every coefficient value of the terms in their direct discrete Fourier (plot (b)) and wavelet decomposition (plot (c)) as well as of the Euclidean (plot (d)) and DTW (plot (e)) distance matrices. Plot (f) shows time-series with increased intermittency but with the same time-frequency pattern as in (a). The corresponding MDS plots are shown in plots (g-j). Finally, plot (k) shows highly intermittent time-series with associated MDS scatter plots in (l-o) illustrating the collapse in efficacy of cluster analysis. The value for the intermittence $\phi$ (equation (\[eq:interittence\])) for the time-series are given in the figure insets.[]{data-label="fig:intermittency_examples"}](arrow_high_int.pdf "fig:"){width="2.50000%"} ![The impact of intermittency on cluster analysis. Plot (a) shows two non-intermittent time-series from a set of 200 which were generated via an almost periodically driven stochastic process, with periods ranging linearly from 2 to 4 for one half the set (blue) and 8 to 16 for the other half (red). Each time-series in the set of 200 is plotted using the values for the first two dimensions obtained from classical MDS of every coefficient value of the terms in their direct discrete Fourier (plot (b)) and wavelet decomposition (plot (c)) as well as of the Euclidean (plot (d)) and DTW (plot (e)) distance matrices. Plot (f) shows time-series with increased intermittency but with the same time-frequency pattern as in (a). The corresponding MDS plots are shown in plots (g-j). Finally, plot (k) shows highly intermittent time-series with associated MDS scatter plots in (l-o) illustrating the collapse in efficacy of cluster analysis. The value for the intermittence $\phi$ (equation (\[eq:interittence\])) for the time-series are given in the figure insets.[]{data-label="fig:intermittency_examples"}](time_series3.pdf "fig:"){width="22.00000%"} ![The impact of intermittency on cluster analysis. Plot (a) shows two non-intermittent time-series from a set of 200 which were generated via an almost periodically driven stochastic process, with periods ranging linearly from 2 to 4 for one half the set (blue) and 8 to 16 for the other half (red). Each time-series in the set of 200 is plotted using the values for the first two dimensions obtained from classical MDS of every coefficient value of the terms in their direct discrete Fourier (plot (b)) and wavelet decomposition (plot (c)) as well as of the Euclidean (plot (d)) and DTW (plot (e)) distance matrices. Plot (f) shows time-series with increased intermittency but with the same time-frequency pattern as in (a). The corresponding MDS plots are shown in plots (g-j). Finally, plot (k) shows highly intermittent time-series with associated MDS scatter plots in (l-o) illustrating the collapse in efficacy of cluster analysis. The value for the intermittence $\phi$ (equation (\[eq:interittence\])) for the time-series are given in the figure insets.[]{data-label="fig:intermittency_examples"}](eg3_scatter.pdf "fig:"){width="22.00000%"} Related work {#sec:related_work} ============ Numerous techniques for *non-intermittent* time-series feature extraction, both time and frequency domain based, have been proposed. The most prevalent use of these within the machine learning community is to obtain numerical features for use as inputs for clustering and classification algorithms. The most simple time domain feature extraction techniques involve extracting summary statistics such as the mean, variance, as well as other higher order moments of the time-series data. Such features have been used for time-series classification [@wang2006characteristic]. Other more complex time-domain features such as the Lyapunov exponent [@wolf1985determining] have also been used for machine learning [@schreiber1997classification]. Recently, shapelets which represent local features in the data, have been used for classification [@ye2009time] and unsupervised learning [@mueen2012clustering] with promising results. Model based approaches, where time-series data is fitted to a statistical model are also common. For example the linear predictive coding cepstrum coefficients obtained from fitting data to the autoregressive integrated moving average model have been used for clustering [@kalpakis2001distance]. Although not strictly based on a feature extraction technique, an effective approach with regards to time series learning is to use the raw, un-transformed time series data itself. It has been known for some time that using the Euclidean distance as a similarity measure between time-series data can lead to extremely good clustering results [@keogh2003need]. Elastic measures including DTW and edit distance, where the temporal alignment of data points isn’t respected, are also popular. An empirical study conducted on the data contained within the UCR time-series data mining archive [@keogh2002ucr] where the performance of numerous static and elastic measures on classification was investigated suggested that DTW distance is the best measure [@ding2008querying]. Frequency domain based approaches are most commonly underpinned by the discrete Fourier or wavelet transformation of the data. For example Vlachos *et al* used periodic features obtained partly via the direct Fourier decomposition for clustering of MSN query log and electrocardiography time-series data [@vlachos2005periodicity]. Features derived from wavelet representations have also been used to cluster synthetic and electrical signals [@misiti2007clustering]. These works show that time domain and frequency domain based features and distance measures can be extremely effective inputs to classification and clustering algorithms when time-series are non-intermittent. However, we not aware of any work which investigates how these approaches stand up against [*intermittent*]{} data or present feature extraction approaches designed specifically for data of this type. Aggregation, Mode Decomposition and Projection (AMP) {#sec:method} ==================================================== As a way of dealing with the issues discussed in Section \[sec:prob statement\], in this section, we outline our time-frequency feature extraction method. Firstly, all intermittent time-series are pooled into a non-intermittent aggregate. A set of vectors corresponding to pertinent time-frequency pattens is then learnt from the aggregate. By projecting the individual time-series data onto this set, we obtain a set of fitted coefficient values. These act as a feature vector indicating the degree to which each time-frequency pattern of the aggregate is expressed in each individual time-series. The values of the features are suitable for further analysis, e.g. to cluster or classify intermittent time-series data. Such an approach makes three main assumptions (1) a non-intermittent time-series can be obtained from the aggregation of a set of intermittent time-series; (2) the decomposition of the aggregate contains components which correspond only to the underlying time-frequency patterns of the data and not to spurious signals; and (3) the time-frequency patterns of the intermittent data (which due to intermittency are difficult to identify directly) are represented in the aggregate (which because of it non-intermittent nature are far easier to identify directly). The stages of the AMP method are described below.\ **Aggregate time-series generation.** Given a set of $m$ discrete time-series $\mathbf{X} = \{\mathbf{x_{1}}, \mathbf{x_{2}}, \ldots , \mathbf{x_{m}}\}$, where each time-series $\mathbf{x_{i}}=(x_{i1}, x_{i2}, \cdots x_{in})$ is represented by a vector of length $n$ with real valued elements, an aggregate is constructed as follows $$\mathbf{a}=\sum\limits_{i=1}^{m} \mathbf{x_{i}}. \label{eq:aggregate_time}$$ Under the assumption that each time-series $i$ is a realisation of a stochastic process with a positive probability that it will take a value other than the modal value, as $m \to \infty$, equation (\[eq:aggregate\_time\]) will yield an *non-intermittent* aggregate. There is evidence to support the notion that much data can be regarded as realisations of stochastic processes with a positive probability that an event will take place at any time. For example the times at which emails are sent has been modelled with a cascading non-homogeneous Poisson process with a positive rate function [@malmgren2008poissonian]. This model led to results with characteristics which were consistent with the characteristics of empirical data. **Time-frequency feature learning.** Using a signal decomposition technique (e.g. Fourier or wavelet decomposition) $\mathbf{a}$ is decomposed into $l$ components $$\mathbf{a}=\sum\limits_{j=1}^l \mathbf{b_{j}}, \label{eq:decompose_aggregate}$$ where each vector $\mathbf{b_{j}}=(b_{j1}, b_{j2}, \cdots, b_{jn})$ corresponds to a different time-frequency component. To ensure that (\[eq:decompose\_aggregate\]) does not include a constant term corresponding to the mean of the signal and only includes components corresponding to time-frequency patterns, $\mathbf{a}$ is mean centred (also know as ‘average centring’) prior to its decomposition. Series (\[eq:decompose\_aggregate\]) is ordered in descending order of the total energy of each component signal, i.e. $\sum\limits_{k=1}^{n} \mid b_{1k} \mid ^{2} \geq \sum\limits_{k=1}^{n} \mid b_{2k} \mid ^{2} \geq \cdots \geq \sum\limits_{k=1}^{n} \mid b_{lk} \mid ^{2}$. We discard time-frequency components that are not, or only minimally, expressed in the aggregate (these correspond to signals with the lowest energies) as such terms are often an artifact of noise in the data or the decomposition process itself. This is achieved by selecting the first $p$ terms of (\[eq:decompose\_aggregate\]) (where $p \leq l$)\ such that $\left(\sum\limits_{j=1}^p \sum\limits_{k=1}^{n} \mid b_{jk} \mid ^{2} \right) \Big{/} \sum\limits_{k=1}^{n} \mid a_{k} \mid ^{2} \geq E_{t}$,\ where $E_{t} \in [0,1]$ represents a selected threshold. This procedure ensures that only the components which correspond to the most salient time-frequency patterns of the aggregate are selected. Otherwise, the inclusion of terms corresponding to low energy time-frequency patterns in the subsequent projection step of AMP will result in the fitting of intermittent time-series to these unimportant patterns. In this work we set $E_{t}=0.9$, as we find such a value is sufficient to omit low energy signals. Next, each retained component of (\[eq:decompose\_aggregate\]) is normalised (i.e. $\hat{\mathbf{b_{j}}}=\mathbf{b_{j}}/\mid \mathbf{b_{j} \mid}$). This step is key as it ensures that, during the next step of our method, where each individual time-series is projected onto a set of basis vectors made up of the retained components, each basis vector will have equal weight. This ensures that basis vectors corresponding to components with extreme amplitudes will not skew results in the projection step. **Basis vector projection.** The final step in our method is to obtain a set of numerical features for each time-series $\mathbf{x_{i}}$ which indicate how much each time-frequency feature learnt from the aggregate is present in them. This is achieved by projecting each $\mathbf{x_{i}}$ on to the set of normalised basis vectors. In particular we seek the linear combination of basis vectors which is closest in the least-squares sense to the original observation, i.e. we minimise $$\mid \mid \mathbf{x}{_{i}}^{\text{T}}-\mathbf{\widehat{B}}\boldsymbol{c}{ _{i}}^{\text{T}} \mid \mid \label{eq:matrix}$$ where the $n \times p$ matrix $\mathbf{\widehat{B}} = (\hat{\mathbf{b}}_1^{\text{T}}, \hat{\mathbf{b}}_2^{\text{T}}, \ldots, \hat{\mathbf{b}}_p^{\text{T}})$ is comprised of normalised basis vectors learnt from the aggregate. $\mathbf{c_{i}}=(c_{i1}, c_{i2}, \ldots , c_{ip})$ is a vector of fitted coefficients which form the feature vector. The value of element $c_{ij}$ indicates the degree to which the time-frequency signal corresponding to normalised basis vector $j$ is expressed in time-series $i$. Fitting all $m$ time-series to the set basis of vectors, as described above, yields the set of features {$\boldsymbol{c}_{1}$, $\boldsymbol{c}_{2}$, $\boldsymbol{c}_{3}$,…,$\boldsymbol{c}_{m}$}. This feature set therefore represents the extent to which an individual time-series expresses the time-frequency patterns present within the overall population. Clustering on this set will result in the grouping together of time-series with similar time-frequency patterns and the clustering into different groups of those which exhibit different time-frequency patterns.\ **Choice of decomposition method for the aggregate time-series.** We consider four methods for the decomposition of the aggregate $\mathbf{a}$, which were selected based on the high prevalence in which they appear in the signal processing literature. These are described below. [**Discrete Fourier decomposition.**]{} Using the discrete Fourier transform (DFT) [@oppenheim1989discrete] the aggregate is decomposed into a Fourier series. We set the number of Fourier components $l=1022$. This ensures that the Fourier series approximates the aggregate extremely well for all data considered in this paper whilst, at the same time, being relatively computationally inexpensive to obtain. We refer to the variant of AMP which uses Fourier decomposition for the aggregate as discrete Fourier transform AMP (DFT-AMP). [**Discrete wavelet decomposition.**]{} This decomposition procedure takes a wavelet function and decomposes a time-series in terms of a set of scaled (stretched and compressed) and translated versions of this function [@sheng2000wavelet]. Because of it’s prevalence of use within the scientific literature we use the Haar wavelet [@mallat1999wavelet] for the mother wavelet. For consistency with DFT-AMP we ensure that the discrete wavelet transform produces 1022 components. This approach as discrete wavelet transform AMP (DWT-AMP). [**Discrete wavelet packet decomposition.**]{} The wavelet packet transform [@oppenheim1995wavelets] is a generalisation of the wavelet transform which provides a more flexible data adaptive decomposition of a signal. It can be used to produce a sparser representation and consequently it is preferred to the wavelet transform when signal compression is the goal. Unlike the DWT there is no fixed relationship between the number of basis functions at each scale. The set of wavelet packet basis functions is selected according to the minimisation of a cost function. We again use the Haar wavelet and select the optimal basis set using the Shannon entropy criteria for the cost function [@coifman1992entropy]. The variant of AMP which uses DWPT is referred to as discrete wavelet packet transform AMP (DWPT-AMP). [**Empirical Mode Decomposition.**]{} In contrast to Fourier and wavelet decomposition, empirical mode decomposition (EMD) [@huang1998empirical] makes no *a priori* assumptions about the composition of the time-series signal and as such is completely non-parametric. The method proceeds by calculating the envelope of the signal via spline interpolation of its maxima and minima. The mean of this envelope corresponds to the intrinsic mode of the signal with the highest frequency and it is designated the first [*intrinsic mode function*]{} (IMF). The first IMF is then removed from the signal and lower frequency IMFs are found by iteratively applying the mean envelope calculation step of the method. The number of IMFs produced is not fixed and depends on the number of intrinsic modes of the data. This variant of AMP is referred to as empirical mode decomposition AMP (EMD-AMP).\ Empirical Evaluation {#sec:eval} ==================== , showing that it outperforms traditional frequency domain and time domain based clustering techniques. We also show that across all variants, EMD-AMP is the most effective in partitioning data into groups with similar time-frequency patterns. With this demonstrated, EMD-AMP is then applied to a real world data set made up of the phone call logs of Massachusetts Institute of Technology (MIT) faculty and students [@eagle2009inferring]. The population of MIT individuals is clustered according to the IMFs they most express with scatter plots revealing two distinct groupings that correspond to different departments in which the staff and students work. Synthetic data {#sec:synth_eval} -------------- compared to using traditional time-frequency feature extraction methods and time domain based clustering approaches. The traditional methods considered for comparison are: [**Fourier power clustering (Four. pow.).**]{} Here each time-series $\mathbf{x_{i}}$ is decomposed into a Fourier series using the DFT. The power of the components at each frequency is used as a feature for clustering. For consistency with the DFT-AMP approach, 1022 Fourier components are used.\ [**Wavelet coefficient clustering (wav. coef.).**]{} Each time-series is decomposed via the DWT using the Haar wavelet and the coefficient values for each wavelet term are used as features for clustering. Again, each time-series is decomposed into 1022 basis vectors. [**Euclidean distance clustering (Euc.).**]{} Because of its simplicity and the fact such an approach can give excellent results [@keogh2003need], we consider a clustering approach based on the Euclidean distances between time-series. [**Dynamic time warping distance clustering (DTW)**]{} Four. pow., wav. coef., Euc. and DTW were used to obtain the results in Figure \[fig:intermittency\_examples\] in the introduction. ### Data set generation {#sec:syn_generation} Such a model allows us to control the the intermittency as well as the stationarity of the data, which is particularly useful given real world human activity data is often non-stationary [@malmgren2009universality; @stehle2010dynamical; @zhao2011social]. We datasets, the first in which all data is stationary ([**Syn1**]{}), the second in which non-stationary data is considered ([**Syn2**]{}), and the final set where noise is added to non-stationary data ([**Syn3**]{}). In order to investigate the impact of intermittency on the results we also vary the amount of intermittency the data exhibits. To generate data sets of intermittent time-series with known time-frequency features (and hence known cluster memberships) we, in the first instance, generate temporal point process data [@daley2007introduction] with a prescribed generating function which controls the time-frequency patterns in the data. Synthetic time-series are then created by mapping the point process data to a continuous function by convolving the data with a kernel [@silverman1986density] as follows $$x_{i}(t)=\frac{1}{\theta_{i}} \sum\limits_{k=1}^{n_{i}} K \left(\frac{t-t_{ik}}{h} \right), \label{eq:kd_ind}$$ where, $t$ is time, $t_{ik}$ is the k$^{\mbox{th}}$ point process event attributed to time-series $i$, $\theta_{i}$ the total number of events generated and $K$ is the standard normal density function with bandwidth $h$. Function (\[eq:kd\_ind\]) is then sampled at $n$ equally spaced points in time to obtain the discrete time-series $\mathbf{x_{i}}$. The $t_{ik}$’s are generated using a non-homogeneous Poisson process [@ross2006introduction]. By utilising the rate function for the non-homogeneous Poisson process we can prescribe different time-frequency patterns in the synthetic time-series data. $$\begin{aligned} \text{\textbf{group 1:}}\: \nonumber \lambda_{1}(t)&=\varphi \left( \gamma \text{sin}^{2}(\frac{\pi t}{T_{1}(t)}) + (1-\gamma) \text{sin}^{2}(\frac{\pi t}{T_{2}(t)}) \right ) \: \\ \text{if} \: i\le m/2, \label{eq:population1} \\ \text{\textbf{group 2:}}\: \nonumber \lambda_{2}(t)&=\varphi \left( (1-\gamma) \text{sin}^{2}(\frac{\pi t}{T_{1}(t)}) + \gamma \text{sin}^{2}(\frac{\pi t}{T_{2}(t)}) \right) \: \\ \text{if} \: i> m/2, \label{eq:population2}\end{aligned}$$ $$\begin{aligned} T_{1}(t)&=T_{1}'+\alpha_{1} t, \: \: \: T_{2}(t)&=T_{2}'+\alpha_{2} t, \label{eq:period_func} \label{eq:periods}\end{aligned}$$ where $T_{1}'$ and $T_{2}'$ are constants. The coefficients $\alpha_{1}$ and $\alpha_{2}$ act to allow non-stationary scenarios to be considered where the period of oscillation of the rate functions (\[eq:population1\]) and (\[eq:population2\]) change with time. \[sec:dgen\] [**Syn1:**]{} : Stationary, No Noise\ In this dataset, $\alpha_{1}$ and $\alpha_{2}$ from equation (\[eq:periods\]) are both set to 0 to ensure that the period of the rate functions of the non-homogeneous Poisson processes is fixed. $T'_{1}$ and $T'_{2}$ are set to 2 and 8 respectively. [**Syn2:**]{} : Non-Stationary, No Noise\ As for [**Syn1**]{}, except $\alpha_{1}= 0.0078$ and $\alpha_{2}= 0.0314$ which ensures that the period of the rate functions are an increasing linear function of time. In particular, the period of rate function $\lambda_1(t)$ (*resp.* $\lambda_2(t)$) from equations (\[eq:population1\]) and (\[eq:population2\]) ranges from 2 (4) at $t=0$ to 4 (8) at $t=255$. This means that the time-series are characterised by time-frequency patterns with period which increases with time. [**Syn3:**]{} : Non-Stationary, Noisy\ As for [**Syn2**]{}, except noise is incorporated in one tenth of the time-series within the set. In particular, $m/10$ time-series were selected at random. Of the temporal events from which these time-series were formed, 50 are selected at random and an additional 41 events (equally distributed over a period of 0.02 time units) are introduced starting from the selected time point. This gives a total of 2050 additional events per time-series selected. These manifests themselves as ‘spikes’ in the time-series where the value of the dependent variable rises and falls extremely quickly. ### Results {#sec:syn_results} synthetic experiments are shown in Figure \[fig:beta\_varphi\]. The performance of the methods presented in this paper are measured firstly by the mean silhouette score [@rousseeuw1987silhouettes] for all data points in a set against the true clustering in two dimensions (obtained via classical MDS where applicable). A score of 1 indicates maximal distance between the two true clusters (i.e. between the data points of groups 1 and 2), with 0 corresponding to maximal mixing between the clusters. The performance is also measured via the Rand index [@rand1971objective] between the true clustering and that obtained from the application of $k$-means ($k=2$) [@macqueen1967some] to the full set of features outputted by each method. Here, 1 corresponds to perfect agreement between the $k$-means results and the true clustering. For each data set the effects of varying the mixing parameter $\gamma$ and amplitude parameter $\varphi$ (equations (\[eq:population1\]) and (\[eq:population2\])) are also considered. **Syn. 1** **Syn. 2** **Syn. 3**\ Performance against ‘mixing parameter’ $\gamma$\ ![Plots (a)-(f) show the mean silhouette index score for the two dimensional representations of the feature values obtained using all methods detailed in Sections \[sec:method\] and \[sec:eval\]. Scores are plotted as a function of the mixing parameter $\gamma$ (a-c) and amplitude parameter $\varphi$ (d-f). Plots (g)-(l) show the Rand index scores obtained by comparing the results of $k$-means clustering applied to the full set of coefficient scores for each method to the true clustering. The average intermittence score $\bar{\phi}$ (equation (\[eq:interittence\])) for all time-series in a set is also shown in each plot (value given in the right axes). For each value of $\gamma$ and $\varphi$ considered, ten simulations were run and an average taken for the mean silhouette score, Rand index and mean intermittence score. Parameter values for all simulations are $m=4000$, $h=0.05$ and where given in Section \[sec:syn\_generation\] and in the plots. All simulations were run for 256 time units.[]{data-label="fig:beta_varphi"}](sil_index_stationary_beta.pdf "fig:"){width="14.00000%"} ![Plots (a)-(f) show the mean silhouette index score for the two dimensional representations of the feature values obtained using all methods detailed in Sections \[sec:method\] and \[sec:eval\]. Scores are plotted as a function of the mixing parameter $\gamma$ (a-c) and amplitude parameter $\varphi$ (d-f). Plots (g)-(l) show the Rand index scores obtained by comparing the results of $k$-means clustering applied to the full set of coefficient scores for each method to the true clustering. The average intermittence score $\bar{\phi}$ (equation (\[eq:interittence\])) for all time-series in a set is also shown in each plot (value given in the right axes). For each value of $\gamma$ and $\varphi$ considered, ten simulations were run and an average taken for the mean silhouette score, Rand index and mean intermittence score. Parameter values for all simulations are $m=4000$, $h=0.05$ and where given in Section \[sec:syn\_generation\] and in the plots. All simulations were run for 256 time units.[]{data-label="fig:beta_varphi"}](sil_index_nonstationary_beta.pdf "fig:"){width="14.00000%"} ![Plots (a)-(f) show the mean silhouette index score for the two dimensional representations of the feature values obtained using all methods detailed in Sections \[sec:method\] and \[sec:eval\]. Scores are plotted as a function of the mixing parameter $\gamma$ (a-c) and amplitude parameter $\varphi$ (d-f). Plots (g)-(l) show the Rand index scores obtained by comparing the results of $k$-means clustering applied to the full set of coefficient scores for each method to the true clustering. The average intermittence score $\bar{\phi}$ (equation (\[eq:interittence\])) for all time-series in a set is also shown in each plot (value given in the right axes). For each value of $\gamma$ and $\varphi$ considered, ten simulations were run and an average taken for the mean silhouette score, Rand index and mean intermittence score. Parameter values for all simulations are $m=4000$, $h=0.05$ and where given in Section \[sec:syn\_generation\] and in the plots. All simulations were run for 256 time units.[]{data-label="fig:beta_varphi"}](sil_index_nonstationary_noise_beta.pdf "fig:"){width="14.00000%"}\ \ Performance against parameter $\varphi$\ ![Plots (a)-(f) show the mean silhouette index score for the two dimensional representations of the feature values obtained using all methods detailed in Sections \[sec:method\] and \[sec:eval\]. Scores are plotted as a function of the mixing parameter $\gamma$ (a-c) and amplitude parameter $\varphi$ (d-f). Plots (g)-(l) show the Rand index scores obtained by comparing the results of $k$-means clustering applied to the full set of coefficient scores for each method to the true clustering. The average intermittence score $\bar{\phi}$ (equation (\[eq:interittence\])) for all time-series in a set is also shown in each plot (value given in the right axes). For each value of $\gamma$ and $\varphi$ considered, ten simulations were run and an average taken for the mean silhouette score, Rand index and mean intermittence score. Parameter values for all simulations are $m=4000$, $h=0.05$ and where given in Section \[sec:syn\_generation\] and in the plots. All simulations were run for 256 time units.[]{data-label="fig:beta_varphi"}](sil_index_stationary_varphi.pdf "fig:"){width="14.00000%"} ![Plots (a)-(f) show the mean silhouette index score for the two dimensional representations of the feature values obtained using all methods detailed in Sections \[sec:method\] and \[sec:eval\]. Scores are plotted as a function of the mixing parameter $\gamma$ (a-c) and amplitude parameter $\varphi$ (d-f). Plots (g)-(l) show the Rand index scores obtained by comparing the results of $k$-means clustering applied to the full set of coefficient scores for each method to the true clustering. The average intermittence score $\bar{\phi}$ (equation (\[eq:interittence\])) for all time-series in a set is also shown in each plot (value given in the right axes). For each value of $\gamma$ and $\varphi$ considered, ten simulations were run and an average taken for the mean silhouette score, Rand index and mean intermittence score. Parameter values for all simulations are $m=4000$, $h=0.05$ and where given in Section \[sec:syn\_generation\] and in the plots. All simulations were run for 256 time units.[]{data-label="fig:beta_varphi"}](sil_index_nonstationary_varphi.pdf "fig:"){width="14.00000%"} ![Plots (a)-(f) show the mean silhouette index score for the two dimensional representations of the feature values obtained using all methods detailed in Sections \[sec:method\] and \[sec:eval\]. Scores are plotted as a function of the mixing parameter $\gamma$ (a-c) and amplitude parameter $\varphi$ (d-f). Plots (g)-(l) show the Rand index scores obtained by comparing the results of $k$-means clustering applied to the full set of coefficient scores for each method to the true clustering. The average intermittence score $\bar{\phi}$ (equation (\[eq:interittence\])) for all time-series in a set is also shown in each plot (value given in the right axes). For each value of $\gamma$ and $\varphi$ considered, ten simulations were run and an average taken for the mean silhouette score, Rand index and mean intermittence score. Parameter values for all simulations are $m=4000$, $h=0.05$ and where given in Section \[sec:syn\_generation\] and in the plots. All simulations were run for 256 time units.[]{data-label="fig:beta_varphi"}](sil_index_nonstationary_noise_varphi.pdf "fig:"){width="14.00000%"}\ \ Performance against ‘mixing’ parameter $\gamma$\ ![Plots (a)-(f) show the mean silhouette index score for the two dimensional representations of the feature values obtained using all methods detailed in Sections \[sec:method\] and \[sec:eval\]. Scores are plotted as a function of the mixing parameter $\gamma$ (a-c) and amplitude parameter $\varphi$ (d-f). Plots (g)-(l) show the Rand index scores obtained by comparing the results of $k$-means clustering applied to the full set of coefficient scores for each method to the true clustering. The average intermittence score $\bar{\phi}$ (equation (\[eq:interittence\])) for all time-series in a set is also shown in each plot (value given in the right axes). For each value of $\gamma$ and $\varphi$ considered, ten simulations were run and an average taken for the mean silhouette score, Rand index and mean intermittence score. Parameter values for all simulations are $m=4000$, $h=0.05$ and where given in Section \[sec:syn\_generation\] and in the plots. All simulations were run for 256 time units.[]{data-label="fig:beta_varphi"}](rand_index_stationary_beta.pdf "fig:"){width="14.00000%"} ![Plots (a)-(f) show the mean silhouette index score for the two dimensional representations of the feature values obtained using all methods detailed in Sections \[sec:method\] and \[sec:eval\]. Scores are plotted as a function of the mixing parameter $\gamma$ (a-c) and amplitude parameter $\varphi$ (d-f). Plots (g)-(l) show the Rand index scores obtained by comparing the results of $k$-means clustering applied to the full set of coefficient scores for each method to the true clustering. The average intermittence score $\bar{\phi}$ (equation (\[eq:interittence\])) for all time-series in a set is also shown in each plot (value given in the right axes). For each value of $\gamma$ and $\varphi$ considered, ten simulations were run and an average taken for the mean silhouette score, Rand index and mean intermittence score. Parameter values for all simulations are $m=4000$, $h=0.05$ and where given in Section \[sec:syn\_generation\] and in the plots. All simulations were run for 256 time units.[]{data-label="fig:beta_varphi"}](rand_index_nonstationary_beta.pdf "fig:"){width="14.00000%"} ![Plots (a)-(f) show the mean silhouette index score for the two dimensional representations of the feature values obtained using all methods detailed in Sections \[sec:method\] and \[sec:eval\]. Scores are plotted as a function of the mixing parameter $\gamma$ (a-c) and amplitude parameter $\varphi$ (d-f). Plots (g)-(l) show the Rand index scores obtained by comparing the results of $k$-means clustering applied to the full set of coefficient scores for each method to the true clustering. The average intermittence score $\bar{\phi}$ (equation (\[eq:interittence\])) for all time-series in a set is also shown in each plot (value given in the right axes). For each value of $\gamma$ and $\varphi$ considered, ten simulations were run and an average taken for the mean silhouette score, Rand index and mean intermittence score. Parameter values for all simulations are $m=4000$, $h=0.05$ and where given in Section \[sec:syn\_generation\] and in the plots. All simulations were run for 256 time units.[]{data-label="fig:beta_varphi"}](rand_index_nonstationary_noise_beta.pdf "fig:"){width="14.00000%"}\ \ Performance against parameter $\varphi$\ ![Plots (a)-(f) show the mean silhouette index score for the two dimensional representations of the feature values obtained using all methods detailed in Sections \[sec:method\] and \[sec:eval\]. Scores are plotted as a function of the mixing parameter $\gamma$ (a-c) and amplitude parameter $\varphi$ (d-f). Plots (g)-(l) show the Rand index scores obtained by comparing the results of $k$-means clustering applied to the full set of coefficient scores for each method to the true clustering. The average intermittence score $\bar{\phi}$ (equation (\[eq:interittence\])) for all time-series in a set is also shown in each plot (value given in the right axes). For each value of $\gamma$ and $\varphi$ considered, ten simulations were run and an average taken for the mean silhouette score, Rand index and mean intermittence score. Parameter values for all simulations are $m=4000$, $h=0.05$ and where given in Section \[sec:syn\_generation\] and in the plots. All simulations were run for 256 time units.[]{data-label="fig:beta_varphi"}](rand_index_stationary_varphi.pdf "fig:"){width="14.00000%"} ![Plots (a)-(f) show the mean silhouette index score for the two dimensional representations of the feature values obtained using all methods detailed in Sections \[sec:method\] and \[sec:eval\]. Scores are plotted as a function of the mixing parameter $\gamma$ (a-c) and amplitude parameter $\varphi$ (d-f). Plots (g)-(l) show the Rand index scores obtained by comparing the results of $k$-means clustering applied to the full set of coefficient scores for each method to the true clustering. The average intermittence score $\bar{\phi}$ (equation (\[eq:interittence\])) for all time-series in a set is also shown in each plot (value given in the right axes). For each value of $\gamma$ and $\varphi$ considered, ten simulations were run and an average taken for the mean silhouette score, Rand index and mean intermittence score. Parameter values for all simulations are $m=4000$, $h=0.05$ and where given in Section \[sec:syn\_generation\] and in the plots. All simulations were run for 256 time units.[]{data-label="fig:beta_varphi"}](rand_index_nonstationary_varphi.pdf "fig:"){width="14.00000%"} ![Plots (a)-(f) show the mean silhouette index score for the two dimensional representations of the feature values obtained using all methods detailed in Sections \[sec:method\] and \[sec:eval\]. Scores are plotted as a function of the mixing parameter $\gamma$ (a-c) and amplitude parameter $\varphi$ (d-f). Plots (g)-(l) show the Rand index scores obtained by comparing the results of $k$-means clustering applied to the full set of coefficient scores for each method to the true clustering. The average intermittence score $\bar{\phi}$ (equation (\[eq:interittence\])) for all time-series in a set is also shown in each plot (value given in the right axes). For each value of $\gamma$ and $\varphi$ considered, ten simulations were run and an average taken for the mean silhouette score, Rand index and mean intermittence score. Parameter values for all simulations are $m=4000$, $h=0.05$ and where given in Section \[sec:syn\_generation\] and in the plots. All simulations were run for 256 time units.[]{data-label="fig:beta_varphi"}](rand_index_nonstationary_noise_varphi.pdf "fig:"){width="14.00000%"}\ \ ![Plots (a)-(f) show the mean silhouette index score for the two dimensional representations of the feature values obtained using all methods detailed in Sections \[sec:method\] and \[sec:eval\]. Scores are plotted as a function of the mixing parameter $\gamma$ (a-c) and amplitude parameter $\varphi$ (d-f). Plots (g)-(l) show the Rand index scores obtained by comparing the results of $k$-means clustering applied to the full set of coefficient scores for each method to the true clustering. The average intermittence score $\bar{\phi}$ (equation (\[eq:interittence\])) for all time-series in a set is also shown in each plot (value given in the right axes). For each value of $\gamma$ and $\varphi$ considered, ten simulations were run and an average taken for the mean silhouette score, Rand index and mean intermittence score. Parameter values for all simulations are $m=4000$, $h=0.05$ and where given in Section \[sec:syn\_generation\] and in the plots. All simulations were run for 256 time units.[]{data-label="fig:beta_varphi"}](legend_all_sim_results.pdf "fig:"){width="20.00000%"} The results illustrate that all AMP variants consistently outperform state of the art techniques (wav. coef., Four. pow, Euc. and DTW) in every plot except for the DWT-AMP variant, the performance of which is comparable to wav. coef. and Four. pow. Of all variants, EMD-AMP is the best performer. This is particularly noticeable when the data sets are both non-stationary and contain noise (see plots (c,f,i,l). All frequency domain based methods outperform the time domain based methods (Euc. and DTW). . As expected, the performance of every method decreases as the mixing parameter $\gamma$ increases and amplitude parameter $\varphi$ (which is inversely related to intermittence) decreases. in producing accurate time-frequency features for intermittent data and why, of all the variants, EMD-AMP produces the most accurate features, we consider a specific instance of **Syn3**, the non-stationary and noisy data set. We use the same parameter values for synthetic data generation as those used in Figure \[fig:beta\_varphi\]l . The period of oscillation of group 1 time-series ranges linearly from 2 at $t=0$ to 4 at $t=255$ and group 2 data from 4 to 8. A sample of individual time-series from each group, together with the aggregate and spectrograms obtained from its decomposition are shown in Figure \[fig:nonstationary\_noise\_agg\_signal\_plus\_analysis\]. Despite the intermittent nature of the individual time-series, the aggregate is clearly non-intermittent (Figure \[fig:nonstationary\_noise\_agg\_signal\_plus\_analysis\]b). Furthermore, the noise present in some time-series has been suppressed by the aggregation process. The non-intermittence of the aggregate time-series is one of the strengths of the AMP approach as it permits decompositions which do not contain spurious signals corresponding to ringing artefacts. In particular wavelet and EMD decompositions reveal only the two time-frequency patterns (one with period ranging from 2 to 4 and the other with period from 4 to 8) which are present within the data set (Figures \[fig:nonstationary\_noise\_agg\_signal\_plus\_analysis\]d and \[fig:nonstationary\_noise\_agg\_signal\_plus\_analysis\]e respectively). Because the aggregated signal is non-stationary with time varying frequency components, the Fourier spectrum picks up the range in frequencies of the underlying time-frequency patterns . Scatter plots of the time-frequency features obtained using every AMP method considered in this work are shown in Figure \[fig:nonstationary\_noise\_scatter\]. These have been symbolised based on whether the time-series are members of group 1 (blue symbols) or group 2 (red symbols). EMD-AMP (Figure \[fig:nonstationary\_noise\_scatter\]d) clearly clusters the two groups according to the time-frequency patterns they most express. So do DFT-AMP (a) and DWPT-AMP (c), but not to the same extent. DWT-AMP (b) fails to cluster the data correctly in this instance. The success of EMD-AMP is related to the fact that its basis vectors permit a more parsimonious model of the data’s underlying time-frequency patterns. Indeed for all cases considered in this section, applying EMD to the aggregate produces just two IMFs - each corresponding to one of the two intrinsic time-frequency patterns of the data. This is still true even when such patterns are non-stationary with time varying frequencies. In contrast, Fourier basis vectors (with their fixed frequencies) are incapable of succinctly modelling the intrinsic time-frequency patterns that exist in data with a frequency that varies over time. Similarly, the rigidity of the DWT means it produces wavelet vectors which individually only model an underlying time-frequency pattern for a small proportion of time *and* for a small proportion of its frequency band. The basis vectors produced via the DWPT can only individually model either (i) a proportion of the frequency of an underlying pattern over the whole time domain, (ii) all of the underlying patterns frequency band but only for a short period of time, or (iii) neither. This means that no single basis vector obtained via the DFT, DWT or DWPT may individually capture the complete time-frequency patterns that underpin non-stationary data. Despite these weaknesses, these basis vectors are sufficiently similar to underlying time-frequency patterns within the data for DFT-AMP, DWT-AMP and DWPT-AMP to still yield reasonable results. It is also notable that for stationary data (see the results in Figure \[fig:beta\_varphi\] (a,d,g,j)) the performances of EMD-AMP, DFT-AMP and DWPT-AMP are almost identical. In this instance, this is due to each method decomposing the aggregate into two almost identical components: one corresponding to the intrinsic oscillation with fixed period 2 and the other to the oscillation with fixed period 4. Individual time-series examples\ ![The top plots shows four individual time-series, two from group 1 (blue lines) and two from group 2 (red lines) illustrating the intermittent nature of the data. The first of these was generated from data containing noise which manifests itself as ‘spikes’ (some examples marked) in the time-series. The plot also shows the aggregate (equation (\[eq:aggregate\_time\])) obtained by combining all 4000 intermittent time-series in the set. The non-intermittent aggregate permits wavelet and empirical model decompositions which reveal the two underlying time-frequency patterns (indicated by blue and red broken lines in the spectrum) of the data set. Note, the edge effects in the EMD plot are artefacts resulting from the discrete Hilbert transform of the IMFs. The Fourier spectra is also shown and this picks up the range of the frequencies of the two time-frequency patterns (indicated by blue and red broken lines).[]{data-label="fig:nonstationary_noise_agg_signal_plus_analysis"}](time_series_egs.pdf "fig:"){width="53.00000%"}\ ![The top plots shows four individual time-series, two from group 1 (blue lines) and two from group 2 (red lines) illustrating the intermittent nature of the data. The first of these was generated from data containing noise which manifests itself as ‘spikes’ (some examples marked) in the time-series. The plot also shows the aggregate (equation (\[eq:aggregate\_time\])) obtained by combining all 4000 intermittent time-series in the set. The non-intermittent aggregate permits wavelet and empirical model decompositions which reveal the two underlying time-frequency patterns (indicated by blue and red broken lines in the spectrum) of the data set. Note, the edge effects in the EMD plot are artefacts resulting from the discrete Hilbert transform of the IMFs. The Fourier spectra is also shown and this picks up the range of the frequencies of the two time-frequency patterns (indicated by blue and red broken lines).[]{data-label="fig:nonstationary_noise_agg_signal_plus_analysis"}](agg_sig_nonstationary_noise.pdf "fig:"){width="23.00000%"} ![The top plots shows four individual time-series, two from group 1 (blue lines) and two from group 2 (red lines) illustrating the intermittent nature of the data. The first of these was generated from data containing noise which manifests itself as ‘spikes’ (some examples marked) in the time-series. The plot also shows the aggregate (equation (\[eq:aggregate\_time\])) obtained by combining all 4000 intermittent time-series in the set. The non-intermittent aggregate permits wavelet and empirical model decompositions which reveal the two underlying time-frequency patterns (indicated by blue and red broken lines in the spectrum) of the data set. Note, the edge effects in the EMD plot are artefacts resulting from the discrete Hilbert transform of the IMFs. The Fourier spectra is also shown and this picks up the range of the frequencies of the two time-frequency patterns (indicated by blue and red broken lines).[]{data-label="fig:nonstationary_noise_agg_signal_plus_analysis"}](four_anal_nonstationary_noise.pdf "fig:"){width="23.00000%"} ![The top plots shows four individual time-series, two from group 1 (blue lines) and two from group 2 (red lines) illustrating the intermittent nature of the data. The first of these was generated from data containing noise which manifests itself as ‘spikes’ (some examples marked) in the time-series. The plot also shows the aggregate (equation (\[eq:aggregate\_time\])) obtained by combining all 4000 intermittent time-series in the set. The non-intermittent aggregate permits wavelet and empirical model decompositions which reveal the two underlying time-frequency patterns (indicated by blue and red broken lines in the spectrum) of the data set. Note, the edge effects in the EMD plot are artefacts resulting from the discrete Hilbert transform of the IMFs. The Fourier spectra is also shown and this picks up the range of the frequencies of the two time-frequency patterns (indicated by blue and red broken lines).[]{data-label="fig:nonstationary_noise_agg_signal_plus_analysis"}](wav_anal_nonstationary_noise.pdf "fig:"){width="23.00000%"} ![The top plots shows four individual time-series, two from group 1 (blue lines) and two from group 2 (red lines) illustrating the intermittent nature of the data. The first of these was generated from data containing noise which manifests itself as ‘spikes’ (some examples marked) in the time-series. The plot also shows the aggregate (equation (\[eq:aggregate\_time\])) obtained by combining all 4000 intermittent time-series in the set. The non-intermittent aggregate permits wavelet and empirical model decompositions which reveal the two underlying time-frequency patterns (indicated by blue and red broken lines in the spectrum) of the data set. Note, the edge effects in the EMD plot are artefacts resulting from the discrete Hilbert transform of the IMFs. The Fourier spectra is also shown and this picks up the range of the frequencies of the two time-frequency patterns (indicated by blue and red broken lines).[]{data-label="fig:nonstationary_noise_agg_signal_plus_analysis"}](emd_nonstationary_noise.pdf "fig:"){width="23.00000%"} ![Scatter plots indicating that EMD-AMP (d) and, to a lesser extent, DFT-AMP (a) and DWPT-AMP (c) can be used to cluster data according to the time-frequency pattens most expressed. Results obtained by plotting the coefficient values outputted by each method (after classical MDS to two dimensions where appropriate) for 200 randomly selected time-series (100 from each group 1 (blue symbols) and 100 from group 2 (red symbols)). The average value for intermittency over all time-series in the set $\bar{\phi}$ is 0.8697. Parameter values for synthetic data generation as for Figure \[fig:beta\_varphi\]l except $\varphi=1.5$.[]{data-label="fig:nonstationary_noise_scatter"}](change_freq_noise.pdf "fig:"){width="45.00000%"}\ ![Scatter plots indicating that EMD-AMP (d) and, to a lesser extent, DFT-AMP (a) and DWPT-AMP (c) can be used to cluster data according to the time-frequency pattens most expressed. Results obtained by plotting the coefficient values outputted by each method (after classical MDS to two dimensions where appropriate) for 200 randomly selected time-series (100 from each group 1 (blue symbols) and 100 from group 2 (red symbols)). The average value for intermittency over all time-series in the set $\bar{\phi}$ is 0.8697. Parameter values for synthetic data generation as for Figure \[fig:beta\_varphi\]l except $\varphi=1.5$.[]{data-label="fig:nonstationary_noise_scatter"}](legend_change_freq_noise.pdf "fig:"){width="10.00000%"} Matlab code used to produce the synthetic data and obtain the results in this section is available at https://github.com/duncan-barrack/AMP. MIT reality mining data set {#sec:real_world} --------------------------- In order to provide evidence that AMP can be used to achieve meaningful results when applied to real world data we consider the MIT Reality Mining dataset. This set comprises event data pertaining to the times and dates at which MIT staff and students made a total of 54 440 mobile phone calls over a period from mid 2004 until early 2005. The average intermittency measure across the population is high ($\bar{\phi}=0.544$) and thus the data set is an excellent candidate for the AMP method. While no ground truth exists for this data, we utilise additional co-variates within the data set as a qualitative proxy for a ground truth for a useful segmentation. In particular, we use participants’ affiliation (Media lab or Sloan business school, see [@eagle2009inferring] for details) as the proxy. Because of its excellent performance in the previous section, we choose to use the EMD-AMP variant. ### Results {#sec:MIT_results} The aggregate and normalised IMFs of the MIT data are shown in Figure \[fig:mit\_sig\_and\_decomp\]. Interestingly, many of the IMFs have a physical interpretation. The first IMF has a period of almost exactly a day. This is most likely generated by the natural 24-hour circadian rhythm which will cause individuals to make a large proportion of their phone calls during the day and early evening. IMF 3 has a period of one week and most likely corresponds to the propensity of study participants to make more phone calls during the working week than at weekends. IMF 6 peaks in September/October before falling again in December/January. It is likely that this function corresponds to the changes in activity between the Fall term (September to December/January) and the holiday periods (over the summer and after Christmas) at MIT. The extracted feature values for the 65 individuals who make the most phone calls are plotted in Figure \[fig:mit\_coef\_dist\], together with two clearly intermittent time-series of two randomly chosen individuals. The scatter plots have been symbolised based on whether the individuals were members of the reality mining group or the Sloan business school at MIT. Interestingly, from the three dimensional representation (Figure \[fig:mit\_coef\_dist\]b) of the feature values, with the exception of a handful of individuals, the individuals from the two groups are separated from each other. Recall that the higher the feature value, the more the corresponding IMF is expressed in that individual’s communications activity. From Figure \[fig:mit\_coef\_dist\]a it can be seen that Sloan business school affiliates (red symbols) have, on average, larger coefficient values corresponding to IMFs 4-6 than the Media lab affiliates (black symbols). From this we can infer that the frequency patterns corresponding to IMFs 4-6 are expressed more strongly in the communication patterns of members of the Sloan business school. ![Aggregate signal and decomposition obtained via EMD revealing six intrinsic time-frequency patters of the MIT communications data. IMF 1 has a period of 1 day and corresponds to the circadian cycle, while IMF 3 has a period of a week and corresponds to the seven day working week/weekend cycle.[]{data-label="fig:mit_sig_and_decomp"}](mit_aggregate.pdf "fig:"){width="55.00000%"} ![Aggregate signal and decomposition obtained via EMD revealing six intrinsic time-frequency patters of the MIT communications data. IMF 1 has a period of 1 day and corresponds to the circadian cycle, while IMF 3 has a period of a week and corresponds to the seven day working week/weekend cycle.[]{data-label="fig:mit_sig_and_decomp"}](mit_imfs.pdf "fig:"){width="55.00000%"} ![Scatter plots indicating that Media lab and Sloan business school affiliates can be clustered according the the time-frequency patterns they most express. a) shows fitted coefficient values corresponding to each IMF. b) is a three dimensional representation of this data obtained via classical MDS. The bottom plots show the time-series of two study participants which are both clearly intermittent (intermittency measure values $\phi$ are given in the plots).[]{data-label="fig:mit_coef_dist"}](mit_coeff_dist.pdf "fig:"){width="23.00000%"} ![Scatter plots indicating that Media lab and Sloan business school affiliates can be clustered according the the time-frequency patterns they most express. a) shows fitted coefficient values corresponding to each IMF. b) is a three dimensional representation of this data obtained via classical MDS. The bottom plots show the time-series of two study participants which are both clearly intermittent (intermittency measure values $\phi$ are given in the plots).[]{data-label="fig:mit_coef_dist"}](mit_mds.pdf "fig:"){width="23.00000%"} ![Scatter plots indicating that Media lab and Sloan business school affiliates can be clustered according the the time-frequency patterns they most express. a) shows fitted coefficient values corresponding to each IMF. b) is a three dimensional representation of this data obtained via classical MDS. The bottom plots show the time-series of two study participants which are both clearly intermittent (intermittency measure values $\phi$ are given in the plots).[]{data-label="fig:mit_coef_dist"}](mit_ind_sig1.pdf "fig:"){width="23.00000%"} ![Scatter plots indicating that Media lab and Sloan business school affiliates can be clustered according the the time-frequency patterns they most express. a) shows fitted coefficient values corresponding to each IMF. b) is a three dimensional representation of this data obtained via classical MDS. The bottom plots show the time-series of two study participants which are both clearly intermittent (intermittency measure values $\phi$ are given in the plots).[]{data-label="fig:mit_coef_dist"}](mit_ind_sig2.pdf "fig:"){width="23.00000%"} Conclusions and further work {#sec:conclusions} ============================ In this work we have addressed the problem of extracting pertinent features from intermittent time-series data containing . We have introduced a new approach entitled aggregation, mode decomposition and projection (AMP). The efficacy of AMP has been demonstrated by applying it to extensive synthetic data as well as to a real world communications data set with intermittent characteristics. . We note that even though our intermittence measure (equation (\[eq:aggregate\_time\])) may not capture the degree of intermittence of all types of intermittent data, AMP will still be effective in these situations. In terms of further work, it would be interesting to compare the performance of AMP clustering to techniques not considered in this paper such as a clustering approach underpinned by the Lomb-Scargle Periodogram [@lomb1976least] which was developed specifically for irregularly sampled data. Another time-domain based clustering method which we did not consider is the recently proposed shapelet based clustering approach [@mueen2012clustering] where ‘local patterns’ in the data are exploited. However, as the values of intermittent times-series fluctuate only briefly from the the modal value, data rarely displays meaningful local patterns and our intuition is that shapelet based clustering is not appropriate in this instance. A comparison with model based approaches such those underpinned by switching Kalman filters [@murphy1998switching] would also be interesting. has been on using AMP derived features for clustering. However, AMP features are also suited to the classification task. There is huge scope to apply an AMP based clustering and/or classification approach to many other types of intermittent time-series data and this would provide an interesting avenue for future work. For example, retail transaction data is characterised by the sporadic activity of customers who make a small number of purchases over a long period of time. The timings of these purchases will be dictated by time-frequency patterns corresponding to human behavioural patterns such as the 24 hour circadian rhythm, or 7 day working week/weekend. Another example is the number of trips made by an individual on public transport. These are unlikely to total more than a handful per day, but the timings of these trips is likely to follow time-frequency patterns corresponding to the traveller’s commuting behaviour or leisure plans. Acknowledgements ================ This work was funded by EPSRC grant EP/G065802/1 - Horizon: Digital Economy Hub at the University of Nottingham and EPSRC grant EP/L021080/1 - Neo-demographics: Opening Developing World Markets by Using Personal Data and Collaboration.

[ **Chiral extrapolation of the leading hadronic contribution to the muon anomalous magnetic moment** ]{} Maarten Golterman,$^a$ Kim Maltman,$^{b,c}$ Santiago Peris$^d$\ [*$^a$Department of Physics and Astronomy\ San Francisco State University, San Francisco, CA 94132, USA\ $^b$Department of Mathematics and Statistics\ York University, Toronto, ON Canada M3J 1P3\ $^c$CSSM, University of Adelaide, Adelaide, SA 5005 Australia\ $^d$Department of Physics and IFAE-BIST, Universitat Autònoma de Barcelona\ E-08193 Bellaterra, Barcelona, Spain*]{}\ [ABSTRACT]{}\ > A lattice computation of the leading-order hadronic contribution to the muon anomalous magnetic moment can potentially help reduce the error on the Standard Model prediction for this quantity, if sufficient control of all systematic errors affecting such a computation can be achieved. One of these systematic errors is that associated with the extrapolation to the physical pion mass from values on the lattice larger than the physical pion mass. We investigate this extrapolation assuming lattice pion masses in the range of 200 to 400 MeV with the help of two-loop chiral perturbation theory, and find that such an extrapolation is unlikely to lead to control of this systematic error at the 1% level. This remains true even if various tricks to improve the reliability of the chiral extrapolation employed in the literature are taken into account. In addition, while chiral perturbation theory also predicts the dependence on the pion mass of the leading-order hadronic contribution to the muon anomalous magnetic moment as the chiral limit is approached, this prediction turns out to be of no practical use, because the physical pion mass is larger than the muon mass that sets the scale for the onset of this behavior. \[introduction\] Introduction ============================= Recently, there has been an increasing interest in a high-precision lattice computation of the leading-order hadronic vacuum polarization (HVP) contribution to the muon anomalous magnetic moment, $a_\mu^{\rm HLO}$. We refer to Ref. [@Wittiglat16] for a recent review, and to Refs. [@AB2007; @Fengetal; @UKQCD; @DJJW2012; @FHHJPR; @HPQCDstrange; @BE; @HPQCDdisconn; @RBCdisconn; @HPQCD; @RBCstrange; @BMW16] for efforts in this direction. The aim is to use the methods of lattice QCD to arrive at a value for $a_\mu^{\rm HLO}$ with a total error of one-half to one percent or less. Such a result would help solidify, and eventually reduce, the total error on the Standard-Model value of the total muon anomalous magnetic moment $a_\m$, which is currently dominated by the error on the HVP contribution. This desired accuracy requires both a high-statistics computation of the HVP, in particular at low momenta (or, equivalently, at large distance), as well as a theoretically clean understanding of the behavior of the HVP as a function of the Euclidean squared-momentum $Q^2$ [@Pade; @taumodel; @strategy], in order to help in reducing systematic errors. In addition, it is important to gain a thorough understanding of various other systematic errors afflicting the computation, such as those caused by a finite volume [@BM; @Mainz; @ABGPFV; @Lehnerlat16], scale setting uncertainties, and the use of lattice ensembles with light quark masses larger than their physical values. Isospin breaking, electromagnetic effects, the presence of dynamical charm and the contribution of quark-disconnected diagrams also all enter at the percent level, and thus also have to be understood quantitatively with sufficient precision. In this article, we consider the extrapolation of $a_\mu^{\rm HLO}$ from heavier than physical pion masses to the physical point, with the help of chiral perturbation theory (ChPT). While lattice computations are now being carried out on ensembles with light quark masses chosen such that the pion mass is approximately physical, a number of computations obtain the physical result via extrapolation from heavier pion masses, while others incorporate results from heavier pion masses in the fits used to convert near-physical-point to actual-physical-point results. The use of such heavier-mass ensembles has a potential advantage since increasing pion mass typically corresponds to decreasing statistical errors on the corresponding lattice data. It is thus important to investigate the reliability of extrapolations of $a_\mu^{\rm HLO}$ from such heavier masses, say, $m_\pi\approx 200$ MeV or above, to the physical pion mass. The leading hadronic contribution is given in terms of the hadronic vacuum polarization, and can be written as [@ER; @TB2003] \[amu\] $$\begin{aligned} a_\m^{\rm HLO}&=&-4\a^2\int_0^\infty \frac{dQ^2}{Q^2}\,w(Q^2)\, \P_{\rm sub}(Q^2)\ ,\label{amua}\\ w(Q^2)&=&\frac{m_\m^2Q^4Z^3(Q^2)(1-Q^2Z(Q^2))}{1+m_\m^2Q^2Z^2(Q^2)}\ , \label{amub}\\ Z(Q^2)&=&\frac{\sqrt{Q^4+4m_\m^2Q^2}-Q^2}{2m_\m^2Q^2}\ ,\label{amuc}\\ \P_{\rm sub}(Q^2)&=&\P(Q^2)-\P(0)\ ,\label{amud}\end{aligned}$$ where $\P(Q^2)$, defined by $$\label{vacpol} \P_{\m\n}(Q)=(Q^2\d_{\m\n}-Q_\m Q_\n)\P(Q^2)\ ,$$ is the vacuum polarization of the electromagnetic (EM) current, $\a$ is the fine-structure constant, and $m_\m$ is the muon mass. If we wish to use ChPT, we are restricted to considering only the low-$Q^2$ part of this integral, because the ChPT representation of $\P_{\rm sub}(Q^2)$ is only valid at sufficiently low values of $Q^2$ (as will be discussed in more detail in Sec. \[comparison\]). In view of this fact, we will define a truncated $a_\m^{\rm HLO}(Q^2_{max})$: $$\label{amutrunc} a_\m^{\rm HLO}(Q^2_{max})=-4\a^2\int_0^{Q^2_{max}} \frac{dQ^2}{Q^2}\,w(Q^2)\, \P_{\rm sub}(Q^2)\ ,$$ and work with $Q^2_{max}$ small enough to allow for the use of ChPT. In order to check over which $Q^2$ range we can use ChPT, we need data to compare with. Here, we will compare to the subtracted vacuum polarization obtained using the non-strange $I=1$ hadronic vector spectral function measured in $\t$ decays by the ALEPH collaboration [@ALEPH13]. Of course, in addition to the $I=1$ part $\P^{33}(Q^2)$, the vacuum polarization also contains an $I=0$ component $\P^{88}(Q^2)$ (and, away from the isospin limit, a mixed isovector-isoscalar component as well). In the isospin limit,[^1] $$\label{EMVP} \P_{\rm EM}(Q^2)=\half\,\P^{{33}}(Q^2)+\frac{1}{6}\,\P^{{88}}(Q^2)\ ,$$ where $\P^{33}$ and $\P^{88}$ are defined from the octet vector currents $$\begin{aligned} \label{currents} V_\m^{3}&=&\frac{1}{\sqrt{2}}\left(V_\m^{uu}-V_\m^{dd}\right)\ ,\\ V_\m^{8}&=&\frac{1}{\sqrt{6}}\left(V_\m^{uu}+V_\m^{dd}-2V_\m^{ss}\right)\ ,\nonumber\end{aligned}$$ with the EM current given by $$\label{emcurrent} V^{\rm EM}_\m=\frac{1}{\sqrt{2}}\left(V^{3}_\m+ \frac{1}{\sqrt{3}}\,V^{8}_\m\right)\ .$$ Here $V_\m^{uu}=\overline{u}\g_\m u$, $V_\m^{dd}=\overline{d}\g_\m d$ and $V_\m^{ss}=\overline{s}\g_\m s$. The quantity we will thus primarily consider in this article is[^2] $$\label{amutilde} \ta_\m(Q^2_{max})=-4\a^2\int_0^{Q^2_{max}} \frac{dQ^2}{Q^2}\,w(Q^2)\, \P^{{33}}_{\rm sub}(Q^2)\ ,$$ where we will choose $Q^2_{max}=0.1$ GeV$^2$ ( Sec. \[comparison\] below). In Sec. \[EM\] we will consider also the inclusion of the $I=0$ contribution. It is worth elaborating on why we believe the quantity $\ta_\m(Q^2_{max}=0.1~\mbox{GeV}^2)$ will be useful for studying the extrapolation to the physical pion mass, in spite of the fact that it constitutes only part of $a_\m^{\rm HLO}$. First, the $I=1$ threshold is $s=4m_\p^2$, while that for $I=0$ is $s=9m_\p^2$.[^3] This suggests that the $I=1$ part of $a_\m^{\rm HLO}$ should dominate the chiral behavior. Second, from the dispersive representation, it is clear that the relative contributions to $\P_{\rm sub}(Q^2)$ from the region near the two-pion threshold are larger at low $Q^2$ than they are at high $Q^2$. Contributions to $a_\m^{\rm HLO}$ from the low-$Q^2$ part of the integral in Eq. (\[amua\]) are thus expected to be relatively more sensitive to variations in the pion mass than are those from the rest of the integral. The part of the integral below $Q^2_{max}=0.1$ GeV$^2$, moreover, yields about 80% of $a_\m^{\rm HLO}$. We thus expect a study of the chiral behavior of $\ta_\m(Q^2_{max}=0.1~\mbox{GeV}^2)$ to provide important insights into the extrapolation to the physical pion mass. This leaves out the contribution from the integral above $0.1$ GeV$^2$, which can be accurately computed directly from the lattice data using a simple trapezoidal rule evaluation [@taumodel; @strategy]. Its pion mass dependence is thus not only expected to be milder, for the reasons given above, but also to be amenable to a direct study using lattice data. In light of these comments, it seems to us highly unlikely that adding the significantly smaller ($\sim 20\%$) $Q^2>0.1$ GeV$^2$ contributions, with their weaker pion mass dependence, could produce a complete integral with a significantly reduced sensitivity to the pion mass. This paper is organized as follows. In Sec. \[chpt\] we collect the needed expressions for the HVP to next-to-next-to-leading order (NNLO) in ChPT, and derive a formula for the dependence of $a_\m^{\rm HLO}$ on the pion mass in the limit $m_\pi\to 0$. In Sec. \[ALEPH\] we compare the $I=1$ ChPT expression with the physical $\P^{{33}}_{\rm sub}(Q^2)$, constructed from the ALEPH data, and argue that $\ta_\m(Q^2_{max}=0.1~\mbox{GeV}^2)$ can be reproduced to an accuracy of about 1% in ChPT. Section \[extrapolation\] contains the study of the extrapolation of $\ta_\m(Q^2_{max}=0.1~\mbox{GeV}^2)$ computed at pion masses typical for the lattice, also considering various tricks that have been considered in the literature to modify $a_\m^{\rm HLO}$ at larger pion mass in such a way as to weaken the pion-mass-dependence of the result and thus improve the reliability of the thus-modified chiral extrapolation. We end this section with a discussion of the inclusion of the $I=0$ part. We present our conclusions in Sec. \[conclusion\], and relegate some technical details to an appendix. \[chpt\] The vacuum polarization in chiral perturbation theory ============================================================== In this section we collect the NNLO expressions for $\P^{{33}}(Q^2)$ and $\P^{{88}}(Q^2)$ as a function of Euclidean $Q^2$, summarizing the results of Refs. [@GK95; @ABT]. Using the conventions of Ref. [@ABT], one has $$\begin{aligned} \label{Pi1} \P^{{33}}_{\rm sub}(Q^2)&=&-8\hB(Q^2,m_\p^2)-4\hB(Q^2,m_K^2)\\ &&+\frac{16}{f_\p^2}\,L_9^r \,Q^2\left(2B(Q^2,m_\p^2)+B(Q^2,m_K^2) \right)\nonumber\\ &&-\frac{4}{f_\p^2}\,Q^2\left(2B(Q^2,m_\p^2)+B(Q^2,m_K^2) \right)^2\nonumber\\ &&+8C_{93}^rQ^2+C^r(Q^2)^2\ , \nonumber\end{aligned}$$ and $$\begin{aligned} \label{Pi0} \P^{{88}}_{\rm sub}(Q^2)&=&-12\hB(Q^2,m_K^2)\\ &&+\frac{48}{f_\p^2}\,L_9^r\,Q^2B(Q^2,m_K^2) -\frac{36}{f_\p^2}\,Q^2\left(B(Q^2,m_K^2)\right)^2 \nonumber\\ &&+8C_{93}^rQ^2+C^r(Q^2)^2\ , \nonumber\end{aligned}$$ where $B(Q^2,m^2)=B(0,m^2)+\hB(Q^2,m^2)$ is the subtracted standard equal-mass, two-propagator, one-loop integral, with $$\begin{aligned} \label{B} B(0,m^2)&=&\frac{1}{192\p^2}\left(1+\log{\frac{m^2}{\m^2}}\right)\ ,\\ \hB(Q^2,m^2)&=&\frac{1}{96\p^2} \left(\left(\frac{4 m^2}{Q^2}+1\right)^{3/2} \mbox{coth}^{-1} \sqrt{1+\frac{4m^2}{Q^2}}- \frac{4m^2}{Q^2}-{\frac{4}{3}}\right)\ ,\nonumber\end{aligned}$$ and the low-energy constants (LECs) $L_9^r$ and $C_{93}^r$ are renormalized at the scale $\m$, in the “$\overline{MS}+1$” scheme employed in Ref. [@ABT]. Note that these are the only two LECs appearing in the subtracted versions of the $I=1$ and $I=0$ non-strange vacuum polarizations to NNLO. As in Ref. [@strategy], we have added an analytic NNNLO term, $C^r(Q^2)^2$, to $\P^{33}_{\rm sub}(Q^2)$ and $\P^{88}_{\rm sub}(Q^2)$ in order to improve, in the $I=1$ case, the agreement with $\P^{{33}}_{\rm sub}(Q^2)$ constructed from the ALEPH data, as we will see in Sec. \[ALEPH\] below. Such a contribution, which would first appear at NNNLO in the chiral expansion, and be produced by six-derivative terms in the NNNLO Lagrangian, will necessarily appear with the same coefficient in both the $I=0$ and $I=1$ polarizations at NNNLO.[^4] Following Ref. [@strategy], we will refer to the expressions with $C^r=0$ as “NNLO,” and with the $C^r$ term included as “NN$^\prime$LO.” From these expressions, it is clear that the chiral behavior of $a_\m^{\rm HLO}$, which we expect to be primarily governed by the pion, rather than kaon, contributions, will be dominated by the $I=1$ component. In fact, in the limit that $m_\p\to 0$ with $m_\m$ fixed, we find $$\begin{aligned} \label{amufinal} a_\m^{I=1}&\equiv&\ta_\m(Q^2_{max}=\infty)=\\ &&\frac{\a^2}{12\p^2}\left(-\log{\frac{m_\p^2}{m_\m^2}} -\frac{31}{6}+3\p^2\sqrt{\frac{m_\p^2}{m_\m^2}} +O\left(\frac{m_\p^2}{m_\m^2}\log^2{\frac{m_\p^2}{m_\m^2}}\right)\right)\ . \nonumber\end{aligned}$$ A derivation of this result is given in App. \[appendix\]. We note that the scale for the chiral extrapolation is set by the muon mass, and thus Eq. (\[amufinal\]) applies to the region $m_\p\ll m_\m$. Therefore, this result is unlikely to be of much practical value. Indeed, our tests in Sec. \[extrapolation\] below will confirm this expectation. \[ALEPH\] Comparison with ALEPH data for hadronic $\t$ decays ============================================================= In this section, we will construct $\P^{{33}}_{\rm sub}(Q^2)$ from the non-strange, vector spectral function $\r^{{33}}_V$ measured by ALEPH in hadronic $\t$ decays [@ALEPH13], using the once-subtracted dispersion relation $$\label{dispPiV} \P^{{33}}_{\rm sub}(Q^2)=-Q^2\int_{4m_\p^2}^\infty ds\, \frac{\r^{{33}}_V(s)}{s(s+Q^2)}\ .$$ Since the spectral function is only measured for $s\le m_\t^2$, this is not entirely trivial, and we will describe our construction in more detail in Sec. \[data\]. We then compare the data with ChPT in Sec. \[comparison\]. 0.8cm \[data\] $\P^{{33}}_{\rm sub}(Q^2)$ from ALEPH data --------------------------------------------------- In order to construct $\r^{{33}}_V(s)$ for $s>m_\t^2$, we follow the same procedure as in the $V-A$ case considered in Refs. [@L10; @VmAALEPH]. For a given $s_{min}\le m_\t^2$, we switch from the data representation of $\r^{{33}}_V(s)$ to a theoretical representation given by the sum of the QCD perturbation theory (PT) expression $\r^{{33}}_{V,{\rm PT}}(s)$ and a “duality-violating” (DV) part $\r^{{33}}_{V,{\rm DV}}(s)$ that represents the oscillations around perturbation theory from resonances, and which we model as $$\label{DV} \r^{{33}}_{V,{\rm DV}}(s)=e^{-\d_V-\g_V s}\sin{(\a_V+\b_V s)}\ .$$ The perturbative expression is known to order $\a_s^4$ [@PT], where $\a_s=\a_s(m_\t^2)$ is the strong coupling. Fits to the ALEPH data determining the parameters $\a_s$, $\a_V$, $\b_V$, $\g_V$ and $\d_V$ have been extensively studied in Ref. [@alphas14], with the goal of a high-precision determination of $\a_s$ from hadronic $\t$ decays. Here we will use the values obtained from the FOPT $s_{min}=1.55$ GeV$^2$ fit of Table 1 of Ref. [@alphas14], $$\begin{aligned} \label{param} \a_s(m_\t^2)&=&0.295(10) \ ,\\ \a_V&=&-2.43(94) \ ,\nonumber\\ \b_V&=&4.32(48)\ \mbox{GeV}^{-2} \ ,\nonumber\\ \g_V&=&0.62(29)\ \mbox{GeV}^{-2} \ ,\nonumber\\ \d_V&=&3.50(50) \ .\nonumber\end{aligned}$$ The match between the data and theory representations of the spectral function in the window $s_{min}\le s\le m_\t^2$ is excellent, and there is no discernible effect on $\P^{{33}}_{\rm sub}(Q^2)$ for the values of $Q^2$ smaller than $0.2$ GeV$^2$ of interest in the comparison to ChPT below if we use a different switch point from data to theory inside this interval, switch to a CIPT instead an FOPT fit, or if we use the parameter values of one of the other optimal fits in Ref. [@alphas14].[^5] Results for $\P^{{33}}_{\rm sub}(Q^2)$ in the region below $Q^2=0.2$ GeV$^2$, at intervals of $0.01$ GeV$^2$, are shown in Fig. \[PiVdata\]. The errors shown are fully correlated, taking into account, in particular, correlations between the parameters of Eq. (\[param\]) and the data. {width="14cm"} > [*$\P^{{33}}_{\rm sub}(Q^2)$ as a function of $Q^2$. Black points: data, constructed as explained in Sec. \[data\]; red (lower) curve: [NNLO]{} ChPT representation with $C^r=0$; blue (upper) curve: [NN$^\prime$LO]{} ChPT representation with $C^r$ as determined from the data. For the ChPT representations, see Sec. \[comparison\].*]{} 0.8cm \[comparison\] Comparison with ChPT ----------------------------------- In order to compare the data for $\P^{{33}}_{\rm sub}(Q^2)$ with ChPT, we need values for $L_9^r$, $C^r_{93}$ and $C^r$. Although, in principle, they can all be obtained from a fit to the ALEPH data, in practice $L_9^r$ and $C^r_{93}$ turn out to be strongly anti-correlated, making it difficult to determine these two LECs separately from these data. We thus, instead, use an external value for $L_9^r$ taken from the NNLO analysis of Ref. [@BTL9]: $$\label{L9} L^r_9(\m=0.77~\mbox{GeV})=0.00593(43)\ .$$ With this value, a fit to the slope and curvature at $Q^2=0$ of $\P^{{33}}_{\rm sub}(Q^2)$ is straightforward, and we find $$\begin{aligned} \label{Cs} C^r_{93}(\m=0.77~\mbox{GeV})&=&-0.0154(4)\ \mbox{GeV}^{-2}\ ,\\ C^r(\m=0.77~\mbox{GeV})&=&0.29(3)\ \mbox{GeV}^{-4}\ .\nonumber\end{aligned}$$ The determination of $C^r_{93}$ is new, and will be discussed in more detail in a forthcoming publication [@C93]. Here, we will use the central values in a comparison between the data and ChPT, in order to see with what accuracy $\ta_\m(Q^2_{max})$ can be represented in ChPT, as a function of $Q^2_{max}$. The two curves in Fig. \[PiVdata\] show ChPT representations of $\P^{{33}}_{\rm sub}(Q^2)$. The blue solid curve corresponds to NN$^\prime$LO ChPT, employing the values (\[L9\]) and (\[Cs\]), the red dashed curve to NNLO ChPT, obtained by dropping the $C^r(Q^2)^2$ contribution from the fitted NN$^\prime$LO result. It is clear that allowing for the analytic NNNLO term in ChPT helps improve the agreement with the data, even though this falls short of a full NNNLO comparison. We may now compare values of $\ta_\m(Q^2_{max})$ computed from the data and from ChPT, as a function of $Q^2_{max}$. For $Q^2_{max}=0.1$ GeV$^2$ we find $$\label{01} \ta_\m(0.1\ \mbox{GeV}^2)=\left\{ \begin{array}{ll} 9.81\times 10^{-8} & \qquad\mbox{data\ ,} \\ 9.73\times 10^{-8} & \qquad\mbox{NN$^\prime$LO ChPT}\ ,\\ 10.23\times 10^{-8} & \qquad\mbox{NNLO ChPT}\ . \end{array}\right.$$ We do not show errors, because we are only interested in the ChPT values for $\ta_\m(Q^2_{max})$ as a model to study the pion mass dependence. However, Eq. (\[01\]) shows that NN$^\prime$LO ChPT reproduces the data value for $\ta_\m(0.1\ \mbox{GeV}^2)$ to about 1%, and that the addition of the $C^r$ term to Eq. (\[Pi1\]) improves this agreement from about 4%. With the value of $\ta_\m=\ta_\m(\infty)=11.95\times 10^{-8}$ computed from the data we also see that the $Q^2_{\max}=0.1$ GeV$^2$ value amounts to 82% of the full integral. For $Q^2_{max}=0.2$ GeV$^2$ we find, similarly, $$\label{02} \ta_\m(0.2\ \mbox{GeV}^2)=\left\{ \begin{array}{ll} 10.96\times 10^{-8} & \qquad\mbox{data\ ,} \\ 10.77\times 10^{-8} & \qquad\mbox{NN$^\prime$LO ChPT}\ ,\\ 11.61\times 10^{-8} & \qquad\mbox{NNLO ChPT}\ . \end{array}\right.$$ For $Q^2_{\max}=0.2$ GeV$^2$ the presence of $C^r$ improves the agreement between the ChPT and data values from about 6% to about 2%, and the truncated integral provides 92% of the full result. It is remarkable that ChPT does such a good job for $\ta_\m(Q^2_{max})$ for these values of $Q^2_{max}$, and that such low values of $Q^2_{max}$ already represent such a large fraction of the integral (\[amutilde\]) for $Q^2_{max}=\infty$. The reason is that the integrand of Eq. (\[amutilde\]) is strongly peaked at $Q^2\approx m_\m^2/4=0.0028$ GeV$^2$. Below, we will use values of $\ta_\m(Q^2_{max})$ computed with $Q^2_{\max}=0.1$ GeV$^2$ for our study of the pion mass dependence. \[extrapolation\] Chiral extrapolation of $\ta_\m(Q^2_{max})$ ============================================================= The pion mass dependence of $\ta_\m(Q^2_{max})$, as it would be computed on the lattice, has a number of different sources. Restricting ourselves to $Q^2_{max}=0.1$ GeV$^2$, we can use ChPT to trace these sources. In addition to the explicit dependence on $m_\p$ in Eq. (\[Pi1\]), $m_K$ and $f_\p$ also depend on the pion mass.[^6] Although $C_{93}^r$ and $C^r$ represent LECs of the effective chiral Lagrangian and hence are mass-independent, the data includes contributions of all chiral orders. Thus, when we perform fits using the truncated NNLO and NN$^\prime$LO forms, the resulting LEC values, in general, will become effective ones, in principle incorporating mass-dependent contributions from terms higher order in ChPT than those shown in Eq. (\[Pi1\]). These will, in general, differ from the true mass-independent LECs $C_{93}^r$ and $C^r$ due to residual higher-order mass-dependent effects. These same effects would also cause the values obtained from analogous fits to lattice data for ensembles with unphysical pion mass to differ from the true, mass-independent values. We will denote the general mass-dependent effective results by $C_{93,{\rm eff}}^{r}$ and $C^{r}_{\rm eff}$, and model their mass dependence by assuming the fitted values in Eq. (\[Cs\]) are dominated by the contributions of the $\r$ resonance [@strategy; @ABT]. With this assumption, $$\begin{aligned} \label{CVMD} C_{93,{\rm eff}}(\m=0.77~\mbox{GeV})&=&-\frac{f_\r^2}{4m_\r^2}\ ,\\ C^{r}_{\rm eff}(\m=0.77~\mbox{GeV})&=&\frac{2f_\r^2}{m_\r^4}\ ,\nonumber\end{aligned}$$ with $m_\rho$ and $f_\rho$ in general dependent on the pion mass. We will suppress the explicit $m_\pi$ dependence of $C_{93,{\rm eff}}^r$ and $C^{r}_{\rm eff}$ except where a danger of confusion exists. For physical light quark mass, with $f_\r\approx 0.2$ and $m_\r=0.775$ GeV, we find $C_{93,{\rm eff}}(m_\pi^2)\approx-0.017$ GeV$^{-2}$, and $C^{r}_{\rm eff}(m_\pi^2)\approx 0.22$ GeV$^{-4}$. These values are in quite reasonable agreement with Eq. (\[Cs\]). On the lattice, one finds that $m_\r$ is considerably more sensitive to the pion mass than is $f_\r$ [@Fengetal]. We thus model the pion mass dependence of $C^r_{93,{\rm eff}}$ and $C^r_{\rm eff}$ by assuming the effective $\m=0.77$ GeV values are given by $$\begin{aligned} \label{Ceff} C^r_{93,{\rm eff}}(m_{\pi ,latt}^2)&=& C^r_{93,{\rm eff}}(m_\pi^2)\,\frac{m_\r^2}{m_{\r,{\rm latt}}^2}\ ,\\ C^r_{\rm eff}(m_{\pi ,latt}^2)&=& C^r_{{\rm eff}}(m_\pi^2)\,\frac{m_\r^4}{m_{\r,{\rm latt}}^4}\ .\nonumber\end{aligned}$$ where $m_{\r,{\rm latt}}$ is the $\r$ mass computed on the lattice. This strategy allows us to generate a number of fake lattice data for $\ta_\m(Q^2_{max})$ using ChPT. For each $m_\pi$ in the range of interest, the corresponding $m_\rho$ is needed to compute $C_{93,{\rm eff}}^r$ and $C_{\rm eff}^r$ via Eqs. (\[Ceff\]). This information is available, over the range of $m_\pi$ we wish to study, for the HISQ ensembles of the MILC collaboration [@MILC], and we thus use the following set of values for $m_\p$, $f_\p$, $m_K$ and $m_\r$, corresponding to those ensembles: $$\label{MILC} \begin{array}{|c|c|c|c|} \hline m_\p\ (\mbox{MeV}) & f_\p\ (\mbox{MeV}) & m_K\ (\mbox{MeV}) & m_\r\ (\mbox{MeV}) \\ \hline 223 & 98 & 514 & 826 \\ 262 & 101 & 523 & 836 \\ 313 & 104 & 537 & 859^* \\ 382 & 109 & 558 & 894 \\ 440 & 114 & 581 & 929\\ \hline \end{array} \vspace{2ex}$$ The statistical errors on these numbers are always smaller than 1%, except for the $\r$ mass marked with an asterisk. In fact, the (unpublished) MILC value for this $\r$ mass is $834(30)$ MeV. Since we are interested in constructing a model, we corrected this value by linear interpolation in $m_\p^2$ between the two neighboring values, obtaining the value $859$ MeV, which is consistent within errors with the MILC value. With this correction, $f_\p$, $m_K$ and $m_\r$ are all approximately linear in $m_\p^2$. 0.8cm \[ETMC\] The ETMC trick ----------------------- Before starting the numerical study of our ChPT-based model, we outline a trick aimed at modifying $a_\m^{\rm HLO}$ results at heavier pion masses in such a way as to weaken the resulting pion mass dependence, and thus improve the reliability of the extrapolation to the physical pion mass. The trick, first introduced in Ref. [@Fengetal], is best explained using an example. Consider the following very simple vector-meson dominance (VMD) model for $\P^{{33}}(Q^2)$ [@MHA]: $$\label{VMD} \P^{{33}}_{\rm VMD,sub}(Q^2)=-\frac{2f_\r^2 Q^2}{Q^2+m_\r^2}-\frac{1}{4\p^2}\log{\left(1+\frac{Q^2}{8\p^2 f_\r^2 m_\r^2}\right)}\ .$$ The logarithm is chosen such that it reproduces the parton-model logarithm while at the same time generating no $1/Q^2$ term for large $Q^2$. The (simplest version of the) ETMC trick consists of inserting a correction factor $m_{\r,{\rm latt}}^2/m_\r^2$ in front of $Q^2$ in the subtracted HVP before carrying out the integral over $Q^2$ in Eq. (\[amu\]). If we assume that $f_\r$ does not depend on $m_\p$ but $m_\r$ does, it is easily seen that the resulting ETMC-modified version of $\P^{{33}}_{\rm VMD,sub}(Q^2)$, $\P^{{33}}_{\rm VMD,sub} \left( {\frac{m_{\r,{\rm latt}}^2}{m_\r^2}} Q^2\right)$, is completely independent of $m_\pi$. With the VMD form known to provide a reasonable first approximation to $\P^{{33}}_{\rm sub}(Q^2)$, the application of the ETMC trick to actual lattice results is thus expected to produce a modified version of $a_\m^{\rm HLO}$ displaying considerably reduced $m_\pi$ dependence. In Ref. [@Fengetal] a further change of variable was performed to shift the modification factor out of the argument of the HVP and into that of the weight function, the result being a replacement of the argument $Q^2$ in $w(Q^2)$ by $\left( m_\r^2/m_{\r,{\rm latt}}^2\right) Q^2$. We do not perform this last change of variable since, in our study, we cut off the integral at $Q^2=Q^2_{max}$,  Eqs. (\[amutrunc\]) and (\[amutilde\]). In Ref. [@HPQCD] a variant of this trick was used as follows. First, the HVP was modified to remove what was expected to be the strongest pion mass dependence by subtracting from the lattice version of $\P^{{33}}_{\rm sub}(Q^2)$ the NLO pion loop contribution (effectively, from our perspective, the first term of Eq. (\[Pi1\])), evaluated at the lattice pion mass, $m_{\p,{\rm latt}}$. The ETMC rescaling, $Q^2\to (m_{\r,{\rm latt}}^2/m_\r^2)Q^2$, was then applied to the resulting differences and the extrapolation to physical pion mass performed on these results. Finally, the physical mass version of the NLO pion loop contribution (again, effectively the first term of Eq. (\[Pi1\]), now evaluated at physical $m_\p$) was added back to arrive at the final result for $a_\m^{\rm HLO}$.[^7] This sequence of procedures is equivalent, in our language, to employing the modified HVP $$\label{pioncorr} \P_{\rm sub,corr}^{33}(Q^2)=\P_{\rm sub}^{33} \left(\frac{m_{\r,{\rm latt}}^2}{m_\r^2}\,Q^2\right)+8\left( \hB\left(\frac{m_{\r,{\rm latt}}^2}{m_\r^2}\,Q^2,m_{\p,{\rm latt}}^2\right) -\hB(Q^2,m_\p^2)\right)\ .$$ We will refer to this version of the ETMC trick as the HPQCD trick. \[I=1\] The $I=1$ case ---------------------- We have generated three “data” sets based on the results for $\ta_\m\equiv\ta_\m(Q^2_{max}=0.1$ GeV$^2)$, at the five values of $m_\p$ given in Eq. (\[MILC\]), using Eq. (\[Pi1\]) with the effective LECs (\[Ceff\]). One set consists simply of the five unmodified results for $\ta_\m$, the other two of the ETMC- and HPQCD-modified version thereof, all obtained using Eq. (\[Pi1\]) with the effective LECs (\[Ceff\]). We will refer to these three data sets as unimproved, ETMC-improved, and HPQCD-improved in what follows. To avoid a proliferation of notation, and since it should cause no confusion to do so, the ETMC- and HPQCD-modified versions of $\ta_\m$ will also be denoted by $\ta_\m$ in what follows. We performed three fits on each of these three data sets, using the following three functional forms for the dependence on $m_\p$: \[fits\] $$\begin{aligned} \ta_\m^{\rm quad}&=&Am_{\p,{\rm latt}}^4+Bm_{\p,{\rm latt}}^2+C\hspace{4cm}(\mbox{quadratic})\ ,\label{fitsa}\\ \ta_\m^{\rm log}&=&A\log{(m_{\p,{\rm latt}}^2/m_\p^2)}+Bm_{\p,{\rm latt}}^2+C \hspace{2.23cm}(\mbox{log})\ ,\label{fitsb}\\ \ta_\m^{\rm inv}&=&\frac{A}{m_{\p,{\rm latt}}^2}+Bm_{\p,{\rm latt}}^2+C \hspace{4.19cm}(\mbox{inverse})\ .\label{fitsc}\end{aligned}$$ The “log” fit is inspired by Eq. (\[amufinal\]). The “inverse” fit is essentially that used by HPQCD in Ref. [@HPQCD]. Explicily, without scaling violations, and assuming a physical strange quark mass, the HPQCD fit function takes the form $$\label{HPQCD} a_\m^{\rm HLO}\left(1+c_\ell\,\frac{\d m_\ell}{\L}+{\tilde c}_\ell\,\frac{\d m_\ell}{m_\ell} \right)\ ,$$ with $\d m_\ell=m_\ell-m_\ell^{\rm phys}$ and $m_\ell$ the average of the up and down quark masses on the lattice.[^8] Assuming a linear relation between $m_\ell$ and $m_\p^2$, this form can be straightforwardly rewritten in the form Eq. (\[fitsc\]). {width="7.4cm"} {width="7.4cm"}\ {width="7.4cm"} > [*The unmodified and ETMC- and HPQCD-improved versions of $\ta_\m$ as a function of $m_\p^2$. In each plot, the upper (magenta) data points are HPQCD-improved, the middle (red) data points ETMC-improved and the lower (blue) data points unimproved. The (black) point in the upper left corner of each plot is the “physical” point, $\ta_\m=9.73\times 10^{-8}$ ( Eq. (\[01\])). Fits are “quadratic” (upper left panel), “log” (upper right panel) and “inverse” (lower panel). For further explanation, see text.*]{} All three fits on all three data sets are shown in Fig. \[fitfig\]. Clearly, as expected, the ETMC trick, and even more so the HPQCD trick, improve (, reduce) the pion mass dependence of the resulting modified $\ta_\m$: the values at larger pion masses are closer to the correct “physical” value shown as the black point in the upper left corner of all panels. All fits look good,[^9] and the log fits to the ETMC- or HPQCD-improved data approach the correct value. However, the coefficient of the logarithm in Eq. (\[fitsb\]) falls in the range $-1.5\times 10^{-8}$ to $-0.8\times 10^{-8}$, more than an order of magnitude smaller than the value $-4.5\times 10^{-7}$ predicted by Eq. (\[amufinal\]). In this respect, we note that the expansion (\[amufinal\]) has a chance of being reliable for $m_\p<m_\m$; the log fits, however, are carried out for lattice pion masses which are larger than the physical pion mass, which, in turn, is larger than $m_\m$. We also note that the form (\[fitsc\]) is more singular than predicted by Eq. (\[amufinal\]). We conclude that all three fits are at best phenomenological, with none of the fit forms in Eq. (\[fits\]) theoretically preferred. We also note that replacing the linear term in $m_\p^2$ in Eq. (\[fitsb\]) by a term linear in $m_\p$, as suggested by Eq. (\[amufinal\]), does not improve the mismatch between the theoretical and fitted values of the coefficient of the logarithm. All this suggests that the systematic error from the extrapolation to the physical pion mass is hard to control, at least when the lowest lattice pion mass is around $200$ MeV (or, when the statistical error on a value closer to the physical pion mass is too large to sufficiently constrain the extrapolation). unimproved data ETMC-improved data HPQCD-improved data ----------- ----------------- -------------------- --------------------- quadratic 8.26 8.91 9.38 log 8.96 9.55 9.77 inverse 9.93 10.46 10.33 In Table \[tab1\] we show the values for $\ta_\m$ at the physical pion mass obtained from the three types of fit to the three data sets. The log-fit value to the HPQCD-improved data is particularly good, missing the correct value by only $0.4\%$. However, as we have seen, the log fit is not theoretically preferred, and without knowledge of the correct value, the only way to obtain an estimate for the systematic error associated with the extrapolation in the real world would be by comparing the results obtained using different fit forms. Discarding the inverse fit as too singular, one may take the (significantly smaller) variation between the quadratic and log fits as a measure of the systematic error. This spread is equal to 8%, 7% and 4%, respectively, for the unimproved, ETMC-improved and HPQCD-improved data sets. Therefore, even though the ETMC and HPQCD tricks do improve the estimated accuracy, they are not sufficiently reliable to reach the desired level of sub-1% accuracy. We have also carried out the same fits omitting the highest pion mass (of 440 MeV,  Eq. (\[MILC\])), and find this makes very little difference. The extrapolated values reported in Table \[tab1\] do not change by more than about $0.5$ to 1%, and there is essentially no change in the systematic uncertainty estimated using the variation with the fit-form choice as we did above. \[EM\] The electromagnetic case ------------------------------- We may repeat the analysis of Sec. \[I=1\] for the electromagnetic case, , using $\P_{\rm EM}(Q^2)$ instead of $\P^{{33}}(Q^2)$. The only difference is that in this case we do not have a “data” value as in Eqs. (\[01\]) and (\[02\]), and we have to rely on ChPT alone. {width="7.4cm"} {width="7.4cm"}\ {width="7.4cm"} > [*The unmodified and ETMC- and HPQCD-improved versions of $a^{\rm EM}_\m$ as a function of $m_\p^2$. In each plot, the upper (magenta) data points are HPQCD-improved, the middle (red) data points ETMC-improved and the lower (blue) data points unimproved. The (black) point in the upper left corner of each plot is the “physical” point, $a^{\rm EM}_\m=6.00\times 10^{-8}$. Fits are “quadratic” (upper left panel), “log” (upper right panel) and “inverse” (lower panel). For further explanation, see text.*]{} Defining the shorthand $a^{\rm EM}_\m=a^{\rm EM}_\m(Q^2_{max}=0.1$ GeV$^2)$, and using the same notation for the ETMC- and HPQCD-modified versions thereof, we show the quadratic, log and inverse fits for this case in Fig. \[fitfigEM\]. We see again that the use of the ETMC and HPQCD tricks significantly reduces the pion mass dependence of the resulting modified data, with the HPQCD improvement being especially effective in this regard. The predicted value of the coefficient of the logarithm in the EM analogue of Eq. (\[amufinal\]) is now half the value shown in that equation ( Eq. (\[EMVP\])), equal to $-2.2\times 10^{-7}$, while the fitted coefficients for the log fits range between $-0.8\times 10^{-8}$ and $-0.4\times 10^{-8}$. The relative difference is of the same order as in the $I=1$ case, and the log fit should thus, as before, be considered purely phenomenological in nature. unimproved data ETMC-improved data HPQCD-improved data ----------- ----------------- -------------------- --------------------- quadratic 5.19 5.59 5.82 log 5.55 5.91 6.02 inverse 6.04 6.37 6.30 In Table \[tab2\] we show the values for $a^{\rm EM}_\m$ at the physical pion mass obtained from the three types of fit to the three data sets. The log-fit value to the HPQCD-improved data is particularly good, missing the correct value by only $0.3\%$. Discarding again the inverse fit as too singular, and taking the variation between the quadratic and log fits as a measure of the systematic error, the spread is 6%, 5% and 3%, respectively, for the unimproved, ETMC-improved and HPQCD-improved data sets. Therefore, even though the ETMC and HPQCD tricks do improve the estimated accuracy, this improvement is not sufficient to reach the desired target of sub-1% accuracy. Again, removing the highest pion mass points from the fits makes no signficant difference in these conclusions. We conclude that, in the electromagnetic case, the situation is slightly better than in the $I=1$ case, no doubt because of the larger relative weight of contributions which are less sensitive to the pion mass (such as the two-kaon contribution). While the ETMC and HPQCD tricks do again improve the estimated accuracy, these improvements remain insufficient to reliably reach the desired sub-1% level. 0.8cm \[conclusion\] Conclusion ========================= In this paper, we used a ChPT-inspired model to investigate the extrapolation of the leading-order hadronic contribution to the muon anomalous magnetic moment, $a_\m^{\rm HLO}$, from lattice pion masses of order 200 to 400 MeV to the physical pion mass. We found that such pion masses are too large to allow for a reliable extrapolation, if the aim is an extrapolation error of less than 1%. This is true even if various tricks to improve the extrapolation are employed, such as those proposed in Ref. [@Fengetal] and Ref. [@HPQCD]. In order to perform our study, we had to make certain assumptions. First, we assumed that useful insight into the pion mass dependence could be obtained by focussing on the contribution to $a_\m^{\rm HLO}$ from $Q^2$ up to $Q^2_{max}=0.1$ GeV$^2$. This restriction is necessary if we want to take advantage of information on the mass dependence from ChPT, since it is only in this range that ChPT provides a reasonable representation of the HVP. We believe this is not a severe restriction, since that part of the integral yields over 80% of $a_\m^{\rm HLO}$, and it is clear that it is the low-$Q^2$ part of the HVP which is most sensitive to the pion mass. Changing $Q^2_{max}$ to $0.2$ GeV$^2$ makes no qualitative difference to our conclusions. Second, we assumed Eq. (\[Ceff\]) for the dependence of the effective LECs $C^r_{93,{\rm eff}}$ and $C^r_{\rm eff}$ on the pion mass. While this is a phenomenological assumption, we note that this assumption is in accordance with the ideas underlying the ETMC and HPQCD tricks, so that those tricks should work particularly well if indeed this assumption would be correct in the real world. There are two reasons that the modified extrapolations nevertheless do not work well enough to achieve the desired sub-1% accuracy. One is the fact that in addition to the physics of the $\r$, the two-pion intermediate state contributing to the non-analytic terms in Eq. (\[Pi1\]), not just at one loop, but also beyond one loop, plays a significant role as well. This is especially so because of the structure of the weight function $w(Q^2)$ in Eq. (\[amu\]). The second reason is that, although ChPT provides a simple functional form for the chiral extrapolation of $\tilde{a}_\m^{\rm HLO}$ for pion masses much smaller than the muon mass ( Eq. (\[amufinal\])), this is not useful in practice, so that one needs to rely on phenomenological fit forms, such as those of Eq. (\[fits\]). In order to eliminate the systematic error from the chiral extrapolation, which we showed to be very difficult to estimate reliably, one needs to compute $a_\m^{\rm HLO}$ at, or close to, the physical pion mass. This potentially increases systematic errors due to finite-volume effects, but it appears these may be more easily brought under theoretical control [@ABGPFV; @Lehnerlat16; @BRFV] than the systematic uncertainties associated with a long extrapolation to the physical pion mass. Contrary to the experience with simpler quantities such as, , meson masses and decay constants, even an extrapolation from approximately 200 MeV pions turns out to be a long extrapolation. It would be interesting to consider the case in which extrapolation from larger than physical pion masses is combined with direct computation at or very near the physical pion mass in order to reduce the total error on the final result. This case falls outside the scope of the study presented here, because in this case the trade-off between extrapolation and computation at the physical point is expected to depend on the statistical errors associated with the ensembles used for each pion mass. However, our results imply that also in this case a careful study should be made of the extrapolation. The methodology developed in this paper can be easily adapted to different pion masses and extended to take into account lattice statistics, and thus should prove very useful for such a study. [**Acknowledgments**]{} We would like to thank Christopher Aubin, Tom Blum and Cheng Tu for discussions, and Doug Toussaint for providing us with unpublished hadronic quantities obtained by the MILC collaboration. This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of High Energy Physics, under Award Number DE-FG03-92ER40711 (MG). KM is supported by a grant from the Natural Sciences and Engineering Research Council of Canada. SP is supported by CICYTFEDER-FPA2014-55613-P, 2014-SGR-1450 and the CERCA Program/Generalitat de Catalunya. \[appendix\] Chiral behavior of $a_\m^{\rm HVP}$ ================================================ In this appendix, we derive the dependence of $a_\m^{I=1}$ on $m_\p$, for $m_\p\to 0$ (in particular, $m_\p\ll m_\m$), using the lowest order pion-loop expression for the $I=1$ HVP, which we will denote by $\P^{33,{\rm NLO}}_{\rm sub}(Q^2)$. Writing this as a dispersive integral, $$\begin{aligned} \label{disp} \P^{33,{\rm NLO}}_{\rm sub}(Q^2)&=& -Q^2\int_{4m_\p^2}^\infty\frac{dt}{t}\,\frac{\r^{33}(t)}{t+Q^2}\ ,\\ \r^{33}\left(\frac{4m_\p^2}{t}\right)&=&\frac{\a}{6\p}\left(1-\frac{4m_\p^2}{t}\right)^{3/2}\ ,\nonumber\end{aligned}$$ and using Eq. (\[amu\]), the integral for $a_\m^{I=1}$ can be written as [@EduardodeR] \[amudisp\] $$\begin{aligned} a_\m^{I=1}&=&\frac{\a}{\pi}\int_0^\infty\frac{d\o}{\o}\,f(\o) \int_1^\infty\frac{d\t}{\t}\,\frac{\r^{33}(\t)}{1+\frac{\z\t}{\o}}\ ,\label{amudispa}\\ f(\o)&=&w(m_\m^2\o)=\sqrt{\frac{\o}{4+\o}}\left(\frac{\sqrt{4+\o}-\sqrt{\o}}{\sqrt{4+\o}+\sqrt{\o}}\right)\ , \label{amudispb}\\ \t&=&\frac{t}{4m_\p^2}\ ,\label{amudispc}\\ \z&=&\frac{4m_\p^2}{m_\m^2}\ .\label{amudispd}\end{aligned}$$ Employing the Mellin–Barnes representation [@Greynat] $$\label{MB} \frac{1}{1+\frac{\z\t}{\o}}=\frac{1}{2\p i}\int_C ds\left(\frac{\z\t}{\o}\right)^{-s} \G(s)\G(1-s)\ ,$$ with $C$ a line parallel to the imaginary axis with $\Re(s)$ inside the fundamental strip $0<\Re(s)<1$, we find an expression for $a_\m^{I=1}$ after performing the integrals over $\t$ and $\o$, \[MBamu\] $$\begin{aligned} a_\m^{I=1}&=&\frac{\a^2}{6\p}\,\frac{1}{2\p i}\int_Cds\,\z^{-s}M(s)\ ,\label{MBamua}\\ M(s)&=&3\cdot 4^{s-1}s(s-1)\,\frac{\G^2(s)\G(1-s)\G\left(\half+s\right)\G(-2-s)}{\G\left(\frac{5}{2}+s\right)}\ . \label{MBamub}\end{aligned}$$ The singular expansion consisting of the sum over all singular terms from a Laurent expansion around each of the singularities of $M(s)$ equals $$\label{singexp} M(s)\asymp\frac{1}{2s^2}+\frac{\log{2}-\frac{31}{12}}{s}+\frac{3\p^2}{4}\frac{1}{s+\half} +O\left(\frac{1}{(s+1)^3}\right)\ .$$ Using that $$\label{simple} \frac{1}{2\p i}\oint ds\,\frac{\z^{-s}}{(s+a)^{k+1}}=\frac{(-1)^k}{k!}\,\z^a\log^k{\z}\ ,$$ and closing the contour in Eq. (\[MBamua\]) to the left, we find $$\label{amuzeta} a_\m^{I=1}= \frac{\a^2}{12\p^2}\left(-\log{\z} +2\log{2}-\frac{31}{6}+\frac{3\p^2}{2}\sqrt{\z} +O\left(\z\log^2{\z}\right)\right)\ .$$ Substituting the expression given in Eq. (\[amudispd\]) for $\z$ yields Eq. (\[amufinal\]). [99]{} H. Wittig, https://conference.ippp.dur.ac.uk/event/470/session/1/contribution/31 . C. Aubin and T. Blum, Phys. Rev. D [**75**]{}, 114502 (2007) \[hep-lat/0608011\]. X. Feng, K. Jansen, M. Petschlies and D. B. Renner, Phys. Rev. Lett.  [**107**]{}, 081802 (2011) \[arXiv:1103.4818 \[hep-lat\]\]. P. Boyle, L. Del Debbio, E. Kerrane and J. Zanotti, Phys. Rev. D [**85**]{}, 074504 (2012) \[arXiv:1107.1497 \[hep-lat\]\]. M. Della Morte, B. Jäger, A. Jüttner and H. Wittig, JHEP [**1203**]{}, 055 (2012) \[arXiv:1112.2894 \[hep-lat\]\]. X. Feng, S. Hashimoto, G. Hotzel, K. Jansen, M. Petschlies and D. B. Renner, Phys. Rev. D [**88**]{}, 034505 (2013) \[arXiv:1305.5878 \[hep-lat\]\]. B. Chakraborty [*et al.*]{} \[HPQCD Collaboration\], Phys. Rev. D [**89**]{}, no. 11, 114501 (2014) \[arXiv:1403.1778 \[hep-lat\]\]. G. Bali and G. Endrödi, Phys. Rev. D [**92**]{}, no. 5, 054506 (2015) \[arXiv:1506.08638 \[hep-lat\]\]. B. Chakraborty, C. T. H. Davies, J. Koponen, G. P. Lepage, M. J. Peardon and S. M. Ryan, Phys. Rev. D [**93**]{}, no. 7, 074509 (2016) \[arXiv:1512.03270 \[hep-lat\]\]. T. Blum [*et al.*]{}, Phys. Rev. Lett.  [**116**]{}, no. 23, 232002 (2016) \[arXiv:1512.09054 \[hep-lat\]\]. B. Chakraborty, C. T. H. Davies, P. G. de Oliviera, J. Koponen and G. P. Lepage, arXiv:1601.03071 \[hep-lat\]. T. Blum [*et al.*]{} \[RBC/UKQCD Collaboration\], JHEP [**1604**]{}, 063 (2016) \[arXiv:1602.01767 \[hep-lat\]\]. S. Borsanyi [*et al.*]{}, arXiv:1612.02364 \[hep-lat\]. C. Aubin, T. Blum, M. Golterman and S. Peris, Phys. Rev. D [**86**]{}, 054509 (2012) \[arXiv:1205.3695 \[hep-lat\]\]. M. Golterman, K. Maltman and S. Peris, Phys. Rev. D [**88**]{}, no. 11, 114508 (2013) \[arXiv:1309.2153 \[hep-lat\]\]. M. Golterman, K. Maltman and S. Peris, Phys. Rev. D [**90**]{}, no. 7, 074508 (2014) \[arXiv:1405.2389 \[hep-lat\]\]. D. Bernecker and H. B. Meyer, Eur. Phys. J. A [**47**]{}, 148 (2011) \[arXiv:1107.4388 \[hep-lat\]\]. A. Francis, B. Jaeger, H. B. Meyer and H. Wittig, Phys. Rev. D [**88**]{}, 054502 (2013) \[arXiv:1306.2532 \[hep-lat\]\]. C. Aubin, T. Blum, P. Chau, M. Golterman, S. Peris and C. Tu, Phys. Rev. D [**93**]{}, no. 5, 054508 (2016) \[arXiv:1512.07555 \[hep-lat\]\]. C. Lehner, https://conference.ippp.dur.ac.uk/event/470/session/9/contribution/18 . B. E. Lautrup, A. Peterman and E. de Rafael, Nuovo Cim. A [**1**]{}, 238 (1971). T. Blum, Phys. Rev. Lett.  [**91**]{}, 052001 (2003) \[hep-lat/0212018\]. M. Davier, A. Hoecker, B. Malaescu, C. Z. Yuan and Z. Zhang, Eur. Phys. J. C [**74**]{}, 2803 (2014) \[arXiv:1312.1501 \[hep-ex\]\]. E. Golowich and J. Kambor, Nucl. Phys. B [**447**]{}, 373 (1995) \[hep-ph/9501318\]. G. Amoros, J. Bijnens and P. Talavera, Nucl. Phys. B [**568**]{}, 319 (2000) \[hep-ph/9907264\]. D. Boito, M. Golterman, M. Jamin, K. Maltman and S. Peris, Phys. Rev. D [**87**]{}, no. 9, 094008 (2013) \[arXiv:1212.4471 \[hep-ph\]\]. D. Boito, A. Francis, M. Golterman, R. Hudspith, R. Lewis, K. Maltman and S. Peris, Phys. Rev. D [**92**]{}, no. 11, 114501 (2015) \[arXiv:1503.03450 \[hep-ph\]\]. P. A. Baikov, K. G. Chetyrkin and J. H. Kühn, Phys. Rev. Lett.  [**101**]{} (2008) 012002 \[arXiv:0801.1821 \[hep-ph\]\]. D. Boito, M. Golterman, K. Maltman, J. Osborne and S. Peris, Phys. Rev. D [**91**]{}, 034003 (2015) \[arXiv:1410.3528 \[hep-ph\]\]. J. Bijnens and P. Talavera, JHEP [**0203**]{}, 046 (2002) \[hep-ph/0203049\]. M. Golterman, K. Maltman and S. Peris, work in progress. A. Bazavov [*et al.*]{} \[MILC Collaboration\], arXiv:1503.02769 \[hep-lat\]; private correspondence with Doug Toussaint, for MILC. S. Peris, M. Perrottet and E. de Rafael, JHEP [**9805**]{}, 011 (1998) \[hep-ph/9805442\]; M. Golterman, S. Peris, B. Phily and E. de Rafael, JHEP [**0201**]{}, 024 (2002) \[hep-ph/0112042\]. J. Bijnens and J. Relefors, arXiv:1611.06068 \[hep-lat\]. E. de Rafael, Phys. Lett. B [**736**]{}, 522 (2014) \[arXiv:1406.4671 \[hep-lat\]\]. S. Friot, D. Greynat and E. de Rafael, Phys. Lett. B [**628**]{}, 73 (2005) \[hep-ph/0505038\]. [^1]: In Ref. [@ABT], $\P^{33}$ and $\P^{88}$ are denoted as $\P^{(1)}_{V\p}$ and $\P^{(1)}_{V\eta}$, respectively. [^2]: This quantity, at varying values of $Q^2_{max}$, was considered before in Refs. [@taumodel; @strategy]. [^3]: In fact, to NNLO in ChPT, the threshold is $s=4m_K^2$. [^4]: Singling out the $C^r(Q^2)^2$ term from amongst the full set of NNNLO contributions introduces a phenomenological element to our extended parametrization. As noted in Ref. [@strategy], the fact that one finds $C_{93}^r$ to be dominated by the contribution of the $\rho$ resonance leads naturally to the expectation that the next term in the expansion of the $\rho$ contribution at low $Q^2$, which has precisely the form $C^r (Q^2)^2$, should begin to become numerically important already for $Q^2$ as low as $\sim 0.1\ {\rm GeV}^2$. [^5]: This stability is not surprising since (i) for small $Q^2$, the weight in the dispersive representation (\[dispPiV\]) falls of as $1/s^2$ for larger $s$, and (ii) the DV and perturbative contributions to $\r^{{33}}_V(s)$ are small relative to the leading parton model contribution in the higher-$s$ region where the PT+DV representation is used. [^6]: We will assume lattice computations with the strange quark fixed at its physical mass, and with isospin symmetry, in which only the light quark mass (, the average of the up and down quark masses) is varied. [^7]: In Ref. [@HPQCD] the ETMC rescaling was actually done at the level of the moments used to construct Padé approximants for $\P_{\rm sub}(Q^2)$. The two procedures are equivalent if the Padé approximants converge. [^8]: Ref. [@HPQCD] assumed exact isospin symmetry in their computation of $a_\m^{\rm HVP}$. [^9]: Of course, in this study there are no statistical errors, and we can only judge this by eye.

--- abstract: 'A new system model reflecting the clustered structure of distributed storage is suggested to investigate interplay between storage overhead and repair bandwidth as storage node failures occur. Large data centers with multiple racks/disks or local networks of storage devices (e.g. sensor network) are good applications of the suggested clustered model. In realistic scenarios involving clustered storage structures, repairing storage nodes using intact nodes residing in other clusters is more bandwidth-consuming than restoring nodes based on information from intra-cluster nodes. Therefore, it is important to differentiate between intra-cluster repair bandwidth and cross-cluster repair bandwidth in modeling distributed storage. Capacity of the suggested model is obtained as a function of fundamental resources of distributed storage systems, namely, node storage capacity, intra-cluster repair bandwidth and cross-cluster repair bandwidth. The capacity is shown to be asymptotically equivalent to a monotonic decreasing function of number of clusters, as the number of storage nodes increases without bound. Based on the capacity expression, feasible sets of required resources which enable reliable storage are obtained in a closed-form solution. Specifically, it is shown that the cross-cluster traffic can be minimized to zero (i.e., intra-cluster local repair becomes possible) by allowing extra resources on storage capacity and intra-cluster repair bandwidth, according to the law specified in the closed-form. The network coding schemes with zero cross-cluster traffic are defined as *intra-cluster repairable codes*, which are shown to be a class of the previously developed *locally repairable codes*.' author: - 'Jy-yong Sohn,  Beongjun Choi,  Sung Whan Yoon,  and Jaekyun Moon, [^1]' bibliography: - 'IEEEabrv.bib' - 'IEEE\_T\_IT2017.bib' title: Capacity of Clustered Distributed Storage --- [Shell : Bare Demo of IEEEtran.cls for IEEE Journals]{} Capacity, Distributed storage, Network coding Introduction ============ enterprises, including Google, Facebook, Amazon and Microsoft, use cloud storage systems in order to support massive amounts of data storage requests from clients. In the emerging Internet-of-Thing (IoT) era, the number of devices which generate data and connect to the network increases exponentially, so that efficient management of data center becomes a formidable challenge. However, since cloud storage systems are composed of inexpensive commodity disks, failure events occur frequently, degrading the system reliability [@ghemawat2003google]. In order to ensure reliability of cloud storage, distributed storage systems (DSSs) with erasure coding have been considered to improve tolerance against storage node failures [@bhagwan2004total; @dabek2004designing; @rhea2003pond; @shvachko2010hadoop; @huang2012erasure; @muralidhar2014f4]. In such systems, the original file is encoded and distributed into multiple storage nodes. When a node fails, a newcomer node regenerates the failed node by contacting a number of survived nodes. This causes traffic burden across the network, taking up significant repair bandwidth. Earlier distributed storage systems utilized the 3-replication code: the original file was replicated three times, and the replicas were stored in three distinct nodes. The 3-replication coded systems require the minimum repair bandwidth, but incur high storage overhead. Reed-Solomon (RS) codes are also used (*e.g.* HDFS-RAID in Facebook [@borthakur2010hdfs]), which allow minimum storage overhead; however, RS-coded systems suffer from high repair bandwidth. The pioneering work of [@dimakis2010network] on distributed storage systems focused on the relationship between two required resources, the storage capacity $\alpha$ of each node and the repair bandwidth $\gamma$, when the system aims to reliably store a file $\mathcal{M}$ under node failure events. The optimal $(\alpha, \gamma)$ pairs are shown to have a fundamental trade-off relationship, to satisfy the maximum-distance-separable (MDS) property (i.e., any $k$ out of $n$ storage nodes can be accessed to recover the original file) of the system. Moreover, the authors of [@dimakis2010network] obtained capacity $\mathcal{C}$, the maximum amount of reliably storable data, as a function of $\alpha$ and $\gamma$. The authors related the failure-repair process of a DSS with the multi-casting problem in network information theory, and exploited the fact that a cut-set bound is achievable by network coding [@ahlswede2000network]. Since the theoretical results of [@dimakis2010network], explicit network coding schemes [@rashmi2009explicit; @rashmi2011optimal; @shah2012interference] which achieve the optimal $(\alpha, \gamma)$ pairs have also been suggested. These results are based on the assumption of homogeneous systems, i.e., each node has the same storage capacity and repair bandwidth. However, in real data centers, storage nodes are dispersed into multiple clusters (in the form of disks or racks) [@ford2010availability; @huang2012erasure; @muralidhar2014f4], allowing high reliability against both node and rack failure events. In this clustered system, repairing a failed node gives rise to both intra-cluster and cross-cluster repair traffic. While the current data centers have abundant intra-rack communication bandwidth, cross-rack communication is typically limited. According to [@rashmi2013solution], nearly a $180$TB of cross-rack repair bandwidth is required everyday in the Facebook warehouse, limiting cross-rack communication for foreground map-reduce jobs. Moreover, surveys [@ahmad2014shufflewatcher; @benson2010network; @vahdat2010scale] on network traffic within data centers show that cross-rack communication is *oversubscribed*; the available cross-rack communication bandwidth is typically $5-20$ times lower than the intra-rack bandwidth in practical systems. Thus, a new system model which reflects the imbalance between intra- and cross-cluster repair bandwidths is required. Main Contributions ------------------ This paper suggests a new system model for *clustered DSS* to reflect the clustered nature of real distributed storage systems wherein an imbalance exists between intra- and cross-cluster repair burdens. This model can be applied to not only large data centers, but also local networks of storage devices such as the sensor networks or home clouds which are expected to be prevalent in the IoT era. This model is also more general in the sense that when the intra- and cross-cluster repair bandwidths are , the resulting structure reduces to the original DSS model of [@dimakis2010network]. The main contributions of this paper can be seen as twofold: one is the derivation of a closed-form expression for capacity, and the other is the analysis on feasible sets of system resources which enable reliable storage. ### Closed-form Expression for Capacity storage capacity $\mathcal{C}$ of the clustered DSS is obtained as a function of node storage capacity $\alpha$, intra-cluster repair bandwidth $\beta_I$ and cross-cluster repair bandwidth $\beta_c$. The existence of the cluster structure manifested as the imbalance between intra/cross-cluster traffics makes the capacity analysis challenging; Dimakis’ proof in [@dimakis2010network] cannot be directly extended to handle the problem at hand. We show that symmetric repair (obtaining the same amount of information from each helper node) is optimal in the sense of maximizing capacity given the storage node size and total repair bandwidth, as also shown in [@ernvall2013capacity] for the case of varying repair bandwidth across the nodes. However, we stress that in most practical scenarios, the need is greater for reducing cross-cluster communication burden, and we show that this is possible by trading with reduced overall storage capacity and/or increasing intra-repair bandwidth. Based on the derived capacity expression, we analyzed how the storage capacity $\mathcal{C}$ changes as a function of $L$, the number of clusters. It is shown that the capacity is asymptotically equivalent to $\underline{C}$, some monotonic decreasing function of $L$. ### Analysis on Feasible $(\alpha, \beta_I, \beta_c)$ Points Given the need for reliably storing file $\mathcal{M}$, the set of required resource pairs, node storage capacity $\alpha$, intra-cluster repair bandwidth $\beta_I$ and cross-cluster repair bandwidth $\beta_c$, which enables $\mathcal{C}(\alpha, \beta_I, \beta_c) \geq \mathcal{M}$ is obtained in a closed-form solution. In the analysis, we introduce $\epsilon = \beta_c/\beta_I$, a useful parameter We here stress that the special case of $\epsilon=0$ corresponds to the scenario where repair is done only locally via intra-cluster communication, i.e., when a node fails the repair process requires intra-cluster traffic only without any cross-cluster traffic. Thus, the analysis on the $\epsilon=0$ case provides a guidance on the network coding for data centers for the scenarios where the available cross-cluster (cross-rack) bandwidth is very scarce. Similar to the non-clustered case of [@dimakis2010network], the required node storage capacity and the required repair bandwidth show a trade-off relationship. In the trade-off curve, two extremal points - the minimum-bandwidth-regenerating (MBR) point and the minimum-storage-regenerating (MSR) point - have been further analyzed for various $\epsilon$ values. Moreover, from the analysis on the trade-off curve, it is shown that the minimum storage overhead $\alpha = \mathcal{M}/k$ is achievable if and only if $\epsilon \geq \frac{1}{n-k}$. This implies that in order to reliably store file $\mathcal{M}$ with minimum storage $\alpha = \mathcal{M}/k$, sufficiently large cross-cluster repair bandwidth satisfying $\epsilon \geq \frac{1}{n-k}$ is required. Finally, for the scenarios with the abundant intra-cluster repair bandwidth, the minimum required cross-cluster repair bandwidth $\beta_c$ to reliably store file $\mathcal{M}$ is obtained as a function of node storage capacity $\alpha$. Related Works ------------- Several researchers analyzed practical distributed storage systems with a goal in mind to reflect the non-homogeneous nature of storage nodes [@ernvall2013capacity; @yu2015tradeoff; @akhlaghi2010fundamental; @shah2010flexible; @gaston2013realistic; @prakash2016generalization; @prakash2017storage; @choi2017secure; @hu2017optimal; @calis2016architecture; @ye2017explicit; @tamo2016optimal]. A heterogeneous model was considered in [@ernvall2013capacity; @yu2015tradeoff] where the storage capacity and the repair bandwidth for newcomer nodes are generally non-uniform. Upper/lower capacity bounds for the heterogeneous DSS are obtained in [@ernvall2013capacity]. An asymmetric repair process is considered in [@akhlaghi2010fundamental], coining the terms, cheap and expensive nodes, based on the amount of data that can be transfered to any newcomer. The authors of [@shah2010flexible] considered a flexible distributed storage system where the amount of information from helper nodes may be non-uniform, as long as the total repair bandwidth is bounded from above. The view points taken in these works are different from ours in that we adopt a notion of cluster and introduce imbalance between intra- and cross-cluster repair burdens. Compared to the conference version [@sohn2016capacity] of the current work, this paper provides the formal proofs for the capacity expression, and obtains the feasible $(\alpha, \gamma)$ region for $0 \leq \epsilon \leq 1$ setting[^2] (only $\epsilon=0$ is considered in [@sohn2016capacity]). The present paper also shows the behavior of capacity as a function of $L$, the number of clusters, and provides the sufficient and necessary conditions on $\epsilon = \beta_c/\beta_I$, to achieve the minimum storage overhead $\alpha = \mathcal{M}/k$. Finally, the asymptotic behaviors of the MBR/MSR points are investigated in this paper, and the connection between what we call the intra-cluster repairable codes and the existing locally repairable codes [@papailiopoulos2014locally] is revealed. Organization ------------ This paper is organized as follows. Section \[Section:Background\] reviews preliminary materials about distributed storage systems and the information flow graph, an efficient tool for analyzing DSS. Section \[Section:Capacity\_of\_Clustered\_DSS\] proposes a new system model for the clustered DSS, and derives a closed-form expression for the storage capacity of the clustered DSS. The behavior of the capacity curves is also analyzed in this section. Based on the capacity expression, Section \[Section:analysis on feasible points\] provides results on the feasible resource pairs which enable reliable storage of a given file. Further research topics on clustered DSS are discussed in Section \[Section:Future\_Works\], and Section \[Section:Conclusion\] draws conclusions. Background {#Section:Background} ========== Distributed Storage System -------------------------- Distributed storage systems can maintain reliability by means of erasure coding [@dimakis2011survey]. The original data file is spread into $n$ potentially unreliable nodes, each with storage size $\alpha$. When a node fails, it is regenerated by contacting $d < n$ helper nodes and obtaining a particular amount of data, $\beta$, from each helper node. The amount of communication burden imposed by one failure event is called the repair bandwidth, denoted as $\gamma = d \beta$. When the client requests a retrieval of the original file, assuming all failed nodes have been repaired, access to any $k<n$ out of $n$ nodes must guarantee a file recovery. The ability to recover the original data using any $k<n$ out of $n$ nodes is called the maximal-distance-separable (MDS) property. Distributed storage systems can be used in many applications such as large data centers, peer-to-peer storage systems and wireless sensor networks [@dimakis2010network]. Information Flow Graph {#Section:Info_flow_graph} ---------------------- Information flow graph is a useful tool to analyze the amount of information flow from source to data collector in a DSS, as utilized in [@dimakis2010network]. It is a directed graph consisting of three types of nodes: data source $\mathrm{S}$, data collector $\mathrm{DC}$, and storage nodes $x^i$ as shown in Fig. \[Fig:information\_flow\_graph\]. Storage node $x^{i}$ can be viewed as consisting of input-node $x_{in}^i$ and output-node $x_{out}^i$, which are responsible for the incoming and outgoing edges, respectively. $x_{in}^i$ and $x_{out}^i$ are connected by a directed edge with capacity identical to the storage size $\alpha$ of node $x^i$. Data from source $S$ is stored into $n$ nodes. This process is represented by $n$ edges going from $\mathrm{S}$ to $\{x^i\}_{i=1}^n$, where each edge capacity is set to infinity. A failure/repair process in a DSS can be described as follows. When a node $x^{j}$ fails, a new node $x^{n+1}$ joins the graph by connecting edges from $d$ survived nodes, where each edge has capacity $\beta$. After all repairs are done, data collector $\mathrm{DC}$ chooses arbitrary $k$ nodes to retrieve data, as illustrated by the edges connected from $k$ survived nodes with infinite edge capacity. Fig. \[Fig:information\_flow\_graph\] gives an example of information flow graph representing a distributed storage system with $n=4, k=3, d=3$. ![Information flow graph ($n=4, k=3, d=3$)[]{data-label="Fig:information_flow_graph"}](information_flow_graph.pdf){height="35mm"} ![Clustered distributed storage system ($n=15,L=3$)[]{data-label="Fig:Layered_DSS"}](2D_structure.pdf){height="25mm"} Notation used in the paper {#Subsection: notation} -------------------------- This paper requires many notations related to graphs, because it deals with information flow graphs. Here we provide the definition of each notation used in the paper. For the given system parameters, we denote $\mathcal{G}$ as the set of all possible information flow graphs. A graph $G\in \mathcal{G}$ is denoted as $G=(V,E)$ where $V$ is the set of vertices and $E$ is the set of edges in the graph. For a given graph $G \in \mathcal{G}$, we call a set $c \subset E$ of edges as *cut-set* [@bang2008digraphs] if it satisfies the following: every directed path from $\mathrm{S}$ to $\mathrm{DC}$ includes at least one edge in $c$. An arbitrary cut-set $c$ is usually denoted as $c=(U, {\mkern 1.5mu\overline{\mkern-1.5muU\mkern-1.5mu}\mkern 1.5mu})$ where $U \subset V$ and ${\mkern 1.5mu\overline{\mkern-1.5muU\mkern-1.5mu}\mkern 1.5mu} = V\setminus U$ (the complement of $U$) satisfy the following: the set of edges from $U$ to ${\mkern 1.5mu\overline{\mkern-1.5muU\mkern-1.5mu}\mkern 1.5mu}$ is the given cut-set $c$. The set of all cut-sets available in $G$ is denoted as $C(G)$. For a graph $G\in \mathcal{G}$ and a cut-set $c \in C(G)$, we denote the sum of edge capacities for edges in $c$ as $w(G,c)$, which is called the *cut-value* of $c$. A vector is denoted as $\textbf{v}$ using the bold notation. For a vector $\textbf{v}$, the transpose of the vector is denoted as $\textbf{v}^T$. A set is denoted as $X = \{x_1, x_2, \cdots, x_k\}$, while a sequence $x_1, x_2, \cdots, x_N$ is denoted as $(x_n)_{n=1}^{N}$, or simply $(x_n)$. For given sequences $(a_n)$ and $(b_n)$, we use the term “$a_n$ is asymptotically equivalent to $b_n$" [@erdélyi1956asymptotic] if and only if $$\lim\limits_{n \rightarrow \infty} \frac{a_n}{b_n} = 1.$$ We utilize a useful notation: $$\mathds{1}_{i=j} = \begin{cases} 1, & \text{ if } i=j \\ 0, & \text{ otherwise.} \end{cases}$$ For a positive integer $n$, we use $[n]$ as a simplified notation for the set $\{1,2,\cdots, n\}$. For a non-positive integer $n$, we define $[n] = \emptyset$. Each storage node is represented as either $x^t = (x_{in}^t, x_{out}^t)$ as defined in Section \[Section:Info\_flow\_graph\], or $N(i,j)$ as defined in (\[Eqn:set\_of\_nodes\]). Finally, important parameters used in this paper are summarized in Table \[Table:Params\]. \[Table:Params\] $n$ number of storage nodes -------------------------------- ------------------------------------------------------------ $k$ number of DC-contacting nodes $L$ number of clusters $n_I = n/L$ number of nodes in a cluster $d_I = n_I - 1$ number of intra-cluster helper nodes $d_c = n- n_I$ number of cross-cluster helper nodes $\mathbb{F}_q$ $\alpha$ storage capacity of each node $\beta_I $ intra-cluster repair bandwidth (per node) $\beta_c $ cross-cluster repair bandwidth (per node) $\gamma_I = d_I\beta_I$ intra-cluster repair bandwidth $\gamma_c = d_c\beta_c$ cross-cluster repair bandwidth $\gamma = \gamma_I + \gamma_c$ repair bandwidth $\epsilon = \beta_c/\beta_I$ ratio of $\beta_c$ to $\beta_I$ ($0 \leq \epsilon \leq 1$) $\xi = \gamma_c/\gamma$ ratio of $\gamma_c$ to $\gamma$ ($ 0 \leq \xi < 1$) $R=k/n$ ratio of $k$ to $n$ ($0 < R \leq 1$) : Parameters used in this paper Capacity of Clustered DSS {#Section:Capacity_of_Clustered_DSS} ========================= Clustered Distributed Storage System {#Section:Clustered_DSS} ------------------------------------ A distributed storage system with multiple clusters is shown in Fig. \[Fig:Layered\_DSS\]. Data from source $\mathrm{S}$ is stored at $n$ nodes which are grouped into $L$ clusters. The number of nodes in each cluster is fixed and denoted as $n_I = n/L$. The storage size of each node is denoted as $\alpha$. When a node fails, a newcomer node is regenerated by contacting $d_I$ helper nodes within the same cluster, and $d_c$ helper nodes from other clusters. The amount of data a newcomer node receives within the same cluster is $\gamma_I = d_I \beta_I$ (each node equally contributes to $\beta_I$), and that from other clusters is $\gamma_c = d_c \beta_c$ (each node equally contributes to $\beta_c$). Fig. \[Fig:Repair Process in clustered DSS\] illustrates an example of information flow graph representing the repair process in a clustered DSS. ![Repair process in clustered DSS ($n=4, L=2, d_I =1, d_c = 2$) []{data-label="Fig:Repair Process in clustered DSS"}](repair_process_information_graph_ver3.pdf){height="30mm"} Assumptions for the System {#Section:assumptions} -------------------------- We assume that $d_c$ and $d_I$ have the maximum possible values ($d_c = n-n_I, d_I = n_I - 1$), since this is the capacity-maximizing choice, as formally stated in the following proposition. The proof of the proposition is in Appendix \[Section:max\_helper\_node\_assumption\]. \[Prop:max\_helper\_nodes\] Consider a clustered distributed storage system with given $\gamma$ and $\gamma_c$. Then, setting both $d_I$ and $d_c$ to their maximum values maximizes storage capacity. Note that the authors of [@dimakis2010network] already showed that in the non-clustered scenario with given repair bandwidth $\gamma$, maximizing the number of helper nodes $d$ is the capacity-maximizing choice. Here, we are saying that a similar property also holds for clustered scenario considered in the present paper. Under the setting of the maximum number of helper nodes, the overall repair bandwidth for a failure event is denoted as $$\label{Eqn:gamma} \gamma = \gamma_I + \gamma_c = (n_I-1)\beta_I + (n-n_I)\beta_c$$ Data collector $\mathrm{DC}$ contacts any $k$ out of $n$ nodes in the clustered DSS. Given that the typical intra-cluster communication bandwidth is larger than the cross-cluster bandwidth in real systems, we assume $$\beta_I \geq \beta_c$$ throughout the present paper; Moreover, motivated by the security issue, we assume that a file cannot be retrieved entirely by contacting any single cluster having $n_I$ nodes. Thus, the number $k$ of nodes contacted by the data collector satisfies $$\begin{aligned} \label{Eqn:k_constraint} k &> n_I.\end{aligned}$$ We also assume $$\label{Eqn:L_constraint} L \geq 2$$ which holds for most real DSSs. Usually, all storage nodes cannot be squeezed in a single cluster, i.e., $L=1$ rarely happens in practical systems, to prevent losing everything when the cluster is destroyed. Note that many storage systems [@huang2012erasure; @muralidhar2014f4; @rashmi2013solution] including those of Facebook uses $L=n$, i.e., every storage node reside in different racks (clusters), . Finally, according to [@rashmi2013solution], nearly $98 \%$ of data recoveries in real systems deal with single node recovery. In other words, the portion of simultaneous multiple nodes failure events is small. Therefore, the present paper focuses single node failure events. The closed-form solution for Capacity ------------------------------------- Consider a clustered DSS with fixed $n, k, L$ values. In this model, we want to find the set of *feasible* parameters ($\alpha, \beta_I, \beta_c$) which enables storing data of size $\mathcal{M}$. In order to find the feasible set, min-cut analysis on the information flow graph is required, similar to [@dimakis2010network]. Depending on the failure-repair process and $k$ nodes contacted by $\mathrm{DC}$, various information flow graphs can be obtained. Let $\mathcal{G}$ be the set of all possible flow graphs. Consider a graph $G^* \in \mathcal{G}$ with minimum min-cut, the construction of which is specified in Appendix \[Section:Proof of Thm 1\]. Based on the max-flow min-cut theorem in [@ahlswede2000network], the maximum information flow from source to data collector for arbitrary $G \in \mathcal{G}$ is greater than or equal to $$\label{Eqn:capacity expression} \mathcal{C}(\alpha, \beta_I, \beta_c) \coloneqq \text{min-cut of }G^*,$$ which is called the *capacity* of the system. In order to send data $\mathcal{M}$ from the source to the data collector, $\mathcal{C} \geq \mathcal{M}$ should be satisfied. Moreover, if $\mathcal{C} \geq \mathcal{M}$ is satisfied, there exists a linear network coding scheme [@ahlswede2000network] to store a file with size $\mathcal{M}$. Therefore, the set of $(\alpha, \beta_I, \beta_c)$ points which satisfies $\mathcal{C} \geq \mathcal{M}$ is *feasible* in the sense of reliably storing the original file of size $\mathcal{M}$. Now, we state our main result in the form of a theorem which offers a closed-form solution for the capacity $\mathcal{C}(\alpha, \beta_I, \beta_c)$ of the clustered DSS. Note that setting $\beta_I = \beta_c$ reduces to capacity of the non-clustered DSS obtained in [@dimakis2010network]. \[Thm:Capacity of clustered DSS\] The capacity of the clustered distributed storage system with parameters $(n,k,L,\alpha, \beta_I, \beta_c)$ is $$\label{Eqn:Capacity of clustered DSS_rev} \mathcal{C}(\alpha, \beta_I, \beta_c) = \sum_{i=1}^{n_I} \sum_{j=1}^{g_i} \min \{\alpha, \rho_i\beta_I + (n-\rho_i - j - \sum_{m=1}^{i-1}g_m) \beta_c \},$$ where $$\begin{aligned} \rho_i &= n_I - i, \label{Eqn:rho_i}\\ g_m &= \begin{cases} \floor{\frac{k}{n_I}} + 1, & m \leq (k {\ \mathrm{mod}\ n_I}) \\ \floor{\frac{k}{n_I}}, & otherwise.\label{Eqn:g_m} \end{cases} \end{aligned}$$ The proof is in Appendix \[Section:Proof of Thm 1\]. Note that the parameters used in the statement of Theorem \[Thm:Capacity of clustered DSS\] have the following property, the proof of which is in Appendix \[Section:proof\_of\_omega\_bound\_gamma\]. \[Prop:omega\_i\_bounded\_by\_gamma\] For every $(i,j)$ with $i \in [n_I], j \in [g_i]$, we have $$\label{Eqn:omega_is_bounded_by_gamma} \rho_i \beta_I + (n-\rho_i - j - \sum_{m=1}^{i-1}g_m)\beta_c \leq \gamma.$$ Moreover, $$\label{Eqn:sum of g is k} \sum_{m=1}^{n_I} g_m = k$$ holds. Relationship between $\mathcal{C}$ and $\epsilon = \beta_c/\beta_I$ {#Section:C_versus_kappa} ------------------------------------------------------------------- In this subsection, we analyze the capacity of a clustered DSS as a function of an important parameter $$\label{Eqn:epsilon} \epsilon \coloneqq \beta_c/\beta_I,$$ the cross-cluster repair burden per intra-cluster repair burden. In Fig. \[Fig:capacity\_versus\_kappa\_plot\], capacity is plotted as a function of $\epsilon$. From (\[Eqn:gamma\]), the total repair bandwidth can be expressed as $$\begin{aligned} \label{Eqn:capacity versus kappa} \gamma &= \gamma_I + \gamma_c = (n_I-1)\beta_I + (n-n_I)\beta_c \nonumber\\ &= \Big( n_I - 1 + (n-n_I)\epsilon \Big) \beta_I.\end{aligned}$$ Using this expression, the capacity is expressed as $$\label{Eqn:capacity_kappa} \mathcal{C}(\epsilon) = \sum_{i=1}^{n_I} \sum_{j=1}^{g_i} \min \Big\{\alpha, \frac{(n-\rho_i - j - \sum_{m=1}^{i-1}g_m)\epsilon + \rho_i }{(n-n_I)\epsilon + n_I - 1} \gamma \Big\}.$$ For fair comparison on various $\epsilon$ values, capacity is calculated for a fixed ($n, k, L, \alpha, \gamma$) set. The capacity is an increasing function of $\epsilon$ as shown in Fig. \[Fig:capacity\_versus\_kappa\_plot\]. This implies that for given resources $\alpha$ and $\gamma$, allowing a larger $\beta_c$ (until it reaches $\beta_I$) is always beneficial, in terms of storing a larger file. For example, under the setting in Fig. \[Fig:capacity\_versus\_kappa\_plot\], allowing $\beta_c = \beta_I$ (i.e., $\epsilon = 1$) can store $\mathcal{M}=48$, while setting $\beta_c = 0$ (i.e., $\epsilon = 0$) cannot achieve the same level of storage. This result is consistent with the previous work on asymmetric repair in [@ernvall2013capacity], which proved that the symmetric repair maximizes capacity. Therefore, when the total communication amount $\gamma$ is fixed, a loss of storage capacity is the cost we need to pay in order to reduce the communication burden $\beta_c$ across different clusters. ![Capacity as a function of $\epsilon$, under the setting of $n=100, k=85, L=10, \alpha = 1, \gamma = 1$[]{data-label="Fig:capacity_versus_kappa_plot"}](capacity_versus_kappa_plot_4.pdf){width="90mm"} Relationship between $\mathcal{C}$ and $L$ {#Section:C_versus_L} ------------------------------------------ In this subsection, we analyze the capacity of a clustered DSS as a function of $L$, the number of clusters. For fair comparison, ($n, k, \alpha, \gamma$) values are fixed for calculating capacity. In Fig. \[Fig:capacity\_versus\_L\_plot\], capacity curves for two scenarios are plotted over a range of $L$ values. First, the solid line corresponds to the scenario when the system has abundant cross-rack bandwidth resources $\gamma_c$. In this ideal scenario which does not suffer from the over-subscription problem, the system can store $\mathcal{M}=80$ irrespective of the dispersion of nodes. However, consider a practical situation where . The dashed line in Fig. \[Fig:capacity\_versus\_L\_plot\] corresponds to this scenario where the system has not enough cross-rack bandwidth resources. In this practical scenario, reducing $L$ (i.e., gathering the storage nodes into a smaller number of clusters) increases capacity. However, note that sufficient dispersion of data into a fair number of clusters is typically desired, in order to guarantee the reliability of storage in rack-failure events. Finding the optimal number $L^*$ of clusters in this trade-off relationship remains as an important topic for future research. ![Capacity as a function of $L$, under the setting of $n=100, k=80,\alpha = 1, \gamma = 10$[]{data-label="Fig:capacity_versus_L_plot"}](capacity_versus_L_ver2.pdf){width="85mm"} ![An example of DSS with dispersion ratio $\sigma = 5/3$, when the parameters are set to $L=3, n_I = 5, n=n_IL = 15$[]{data-label="Fig:sigma"}](sigma_example_ver3.pdf){width="65mm"} In Fig. \[Fig:capacity\_versus\_L\_plot\], the capacity is a monotonic decreasing function of $L$ when the system suffers from an over-subscription problem. However, in general $(n,k,\alpha,\gamma)$ parameter settings, capacity is not always a monotonic decreasing function of $L$. Theorem \[Thm:cap\_dec\_ftn\_L\] illustrates the behavior of capacity as $L$ varies, focusing on the special case of $\gamma = \alpha.$ Before formally stating the next main result, we need to define $$\label{Eqn:sigma} \sigma \coloneqq \frac{L}{n_I} = \frac{L^2}{n},$$ the *dispersion factor* of a clustered storage system, as illustrated in Fig. \[Fig:sigma\]. In the two-dimensional representation of a clustered distributed storage system, $L$ represents the number of rows (clusters), while $n_I = n/L$ represents the number of columns (nodes in each cluster). The dispersion factor $\sigma$ is the ratio of the number of rows to the number of columns. If $L$ increases for a fixed $n_I$, then $\sigma$ grows and the nodes become more dispersed into multiple clusters. Now we state our second main result, which is about the behavior of $\mathcal{C}$ versus $L$. \[Thm:cap\_dec\_ftn\_L\] In the asymptotic regime of large $n$, capacity $\mathcal{C}(\alpha, \beta_I, \beta_c)$ is asymptotically equivalent to $$\underline{C} = \frac{k}{2} \left(\gamma + \frac{n-k}{n(1-1/L)} \gamma_c \right), \label{Eqn:cap_lower}$$ a monotonically decreasing function of $L$. This can also be stated as $$\mathcal{C} \sim \underline{C}\label{Eqn:Cap_asymp_equiv}$$ as $n \rightarrow \infty$ for a fixed $\sigma$. $$\begin{aligned} n_I&= \frac{n}{L} = \frac{n}{\sqrt{n\sigma}} = \Theta(\sqrt{n}), \nonumber\\ k&= nR = \Theta(n), \quad \alpha= \gamma = \text{constant}, \nonumber\\ \beta_I &= \frac{\gamma_I}{n_I-1} = \Theta(\frac{1}{\sqrt{n}}), \quad \beta_c = \frac{\gamma_c}{n-n_I} = \Theta(\frac{1}{n})\end{aligned}$$ The proof of Theorem \[Thm:cap\_dec\_ftn\_L\] is based on the following two lemmas. \[Lemma:cap\_upper\_lower\] , capacity $ \mathcal{C}(\alpha,\beta_I,\beta_c)$ is upper/lower bounded as $$\label{Eqn:cap_upper_lower_bound} \underline{C} \leq \mathcal{C}(\alpha,\beta_I,\beta_c) \leq \underline{C} + \delta$$ where $$\begin{aligned} \delta &= n_I^2(\beta_I-\beta_c)/8 \label{Eqn:cap_delta_val} \end{aligned}$$ and $\underline{C}$ is defined in (\[Eqn:cap\_lower\]). \[Lemma:Scale\] , $\underline{C}$ and $\delta$ defined in (\[Eqn:cap\_lower\]), (\[Eqn:cap\_delta\_val\]) satisfy $$\begin{aligned} \underline{C} &= \Theta(n), \nonumber\\ \delta &= O(n_I) \nonumber\end{aligned}$$ The proofs of these lemmas are in Appendix \[Section:Proofs\_of\_Lemmas\]. Here we provide the proof of Theorem \[Thm:cap\_dec\_ftn\_L\] by using Lemmas \[Lemma:cap\_upper\_lower\] and \[Lemma:Scale\]. From Lemma \[Lemma:Scale\], $$\label{Eqn:delta_over_C} \delta / \underline{C} = O(1/L) = O(\sqrt{1/n\sigma}) \rightarrow 0$$ as $n\rightarrow \infty$ for a fixed $\sigma$. Moreover, dividing (\[Eqn:cap\_upper\_lower\_bound\]) by $\underline{C}$ results in $$\label{Eqn:cap_bound_revisited} 1 \leq \mathcal{C}/\underline{C} \leq 1 + \delta/\underline{C}.$$ Putting (\[Eqn:delta\_over\_C\]) into (\[Eqn:cap\_bound\_revisited\]) completes the proof. Discussion on Feasible ($\alpha, \beta_I, \beta_c$) {#Section:analysis on feasible points} =================================================== In the previous section, we obtained the capacity of the clustered DSS. This section analyzes the feasible ($\alpha, \beta_I, \beta_c$) points which satisfy $\mathcal{C}(\alpha, \beta_I, \beta_c) \geq \mathcal{M}$ for a given file size $\mathcal{M}$. Using $$\gamma = \gamma_I + \gamma_c = (n_I-1)\beta_I + (n-n_I)\beta_c$$ in (\[Eqn:gamma\]), the behavior of the feasible set of $(\alpha, \gamma)$ points can be observed. Essentially, the feasible points demonstrate a trade-off relationship. Two extreme points  minimum storage regenerating (MSR) point and minimum bandwidth regenerating (MBR) point  of the trade-off has been analyzed. Moreover, a special family of network codes which satisfy $\epsilon=0$, which we call the *intra-cluster repairable codes*, is compared with the *locally repairable codes* considered in [@papailiopoulos2014locally]. Finally, the set of feasible $(\alpha, \beta_c)$ points is discussed, when the system allows maximum $\beta_I$. Set of Feasible $(\alpha, \gamma)$ Points {#Section:nonzero_gammac} ----------------------------------------- We provide a closed-form solution for the set of feasible $(\alpha, \gamma)$ points which enable reliable storage of data $\mathcal{M}$. Based on the range of $\epsilon$ defined in (\[Eqn:epsilon\]), the set of feasible points show different behaviors as stated in Corollary \[Corollary:Feasible Points\_large\_epsilon\]. \[Corollary:Feasible Points\_large\_epsilon\] Consider a clustered DSS for storing data $\mathcal{M}$, when $\epsilon = \beta_c/\beta_I$ satisfies $0 \leq \epsilon \leq 1$. For any $\gamma \geq \gamma^*(\alpha)$, the data $\mathcal{M}$ can be reliably stored, i.e., $\mathcal{C} \geq \mathcal{M}$, while it is impossible to reliably store data $\mathcal{M}$ when $\gamma < \gamma^*(\alpha)$. The threshold function $\gamma^*(\alpha)$ can be obtained as: 1. $\frac{1}{n-k} \leq \epsilon \leq 1$ $$\label{Eqn:Feasible Points Result} \gamma^*(\alpha) = \begin{cases} \infty, & \ \alpha \in (0,\frac{M}{k}) \\ \frac{M-t\alpha}{s_t}, & \ \alpha \in [\frac{M}{t+1+s_{t+1}y_{t+1}} ,\frac{M}{t+s_{t}y_{t}} ) ,\\ & \ \ \ \ \ \ \ \ \ \ (t = k-1, k-2, \cdots, 1)\\ \frac{M}{s_0 }, & \ \alpha \in [\frac{M}{s_0}, \infty) \end{cases}$$ 2. $0 \leq \epsilon < \frac{1}{n-k}$) $$\label{Eqn:Feasible Points Result_intermediate_epsilon} \gamma^*(\alpha) = \begin{cases} \infty, & \ \alpha \in (0,\frac{M}{\tau + \sum_{i=\tau+1}^{k}z_i}) \\ \frac{\mathcal{M} - \tau \alpha}{s_{\tau}}, & \ \alpha \in [\frac{M}{\tau + \sum_{i=\tau+1}^{k}z_i},\frac{\mathcal{M}}{\tau + s_{\tau}y_{\tau}}) \\ \frac{M-t\alpha}{s_t}, & \ \alpha \in [\frac{M}{t+1+s_{t+1}y_{t+1}} ,\frac{M}{t+s_{t}y_{t}} ) ,\\ & \ \ \ \ \ \ \ \ \ \ (t = \tau-1, \tau-2, \cdots, 1)\\ \frac{M}{s_0 }, & \ \alpha \in [\frac{M}{s_0}, \infty) \end{cases}$$ where $$\begin{aligned} \tau & = \max \{ t \in \{0,1,\cdots, k-1\} : z_t \geq 1 \}\label{Eqn:tau}\\ s_t &= \begin{cases} \frac{\sum_{i=t+1}^{k} z_i}{(n_I-1) + \epsilon (n-n_I)}, & t =0, 1, \cdots, k-1 \label{Eqn:s_t}\\ 0, & t=k \end{cases}\\ y_t &= \frac{(n_I-1)+\epsilon (n-n_I)}{z_t}\label{Eqn:y_t},\\ z_t &= \begin{cases} (n-n_I-t+h_t)\epsilon + (n_I-h_t), & t \in [k] \\ \infty, & t=0 \label{Eqn:z_t} \end{cases}\\ h_t &= \min \{s \in [n_I]: \sum_{l=1}^s g_l \geq t \}, \label{Eqn:h_t} \end{aligned}$$ and $\{g_l\}_{l=1}^{n_I}$ is defined in (\[Eqn:g\_m\]). ![Optimal tradeoff between node storage size $\alpha$ and total repair bandwidth $\gamma$, under the setting of $n=15, k=8, L=3, \mathcal{M} = 8$[]{data-label="Fig:kappa_various_latex"}](various_epsilon_ext_rev.pdf){height="55mm"} ![Optimal tradeoff between node storage size $\alpha$ and cross-rack repair bandwidth $\gamma_c$, under the setting of $n=15, k=8, L=3, \mathcal{M} = 8$[]{data-label="Fig:gamma_c_various_latex"}](gamma_c_various_epsilon.pdf){height="50mm"} An example for the trade-off results of Corollary \[Corollary:Feasible Points\_large\_epsilon\] is illustrated in Fig. \[Fig:kappa\_various\_latex\], for various $0 \leq \epsilon \leq 1$ values. Here, the $\epsilon = 1$ (i.e., $\beta_I = \beta_c$) case corresponds to the symmetric repair in the non-clustered scenario [@dimakis2010network]. The plot for $\epsilon = 0$ (or $\beta_c = \gamma_c = 0$) shows that the cross-cluster repair bandwidth can be reduced to zero with extra resources ($\alpha$ or $\gamma_I$), where the amount of required resources are specified in Corollary \[Corollary:Feasible Points\_large\_epsilon\]. From Fig. \[Fig:kappa\_various\_latex\], we can confirm that as $\epsilon$ decreases, extra resources ($\gamma$ or $\alpha$) are required to reliably store the given data $\mathcal{M}$. Moreover, Corollary \[Corollary:Feasible Points\_large\_epsilon\] suggests a mathematically interesting result, stated in the following Theorem, the proof of which is in Appendix \[Section:proof\_of\_thm\_condition\_for\_min\_storage\]. \[Thm:condition\_for\_min\_storage\] A clustered DSS can reliably store file $\mathcal{M}$ with the minimum storage overhead $\alpha = \mathcal{M}/k$ if and only if $$\epsilon \geq \frac{1}{n-k}.$$ Note that $\alpha = \mathcal{M}/k$ is the minimum storage overhead which can satisfy the MDS property, as stated in [@dimakis2010network]. The implication of Theorem \[Thm:condition\_for\_min\_storage\] is shown in Fig. \[Fig:kappa\_various\_latex\]. Under the $\mathcal{M} = k = 8$ setting, data $\mathcal{M}$ can be reliably stored with minimum storage overhead $\alpha=\mathcal{M}/k=1$ for $\frac{1}{n-k} \leq \epsilon \leq 1$, while it is impossible to achieve minimum storage overhead $\alpha=\mathcal{M}/k=1$ for $0 \leq \epsilon < \frac{1}{n-k}$. Finally, since reducing the cross-cluster repair burden $\gamma_c$ is regarded as a much harder problem compared to reducing the intra-cluster repair burden $\gamma_I$, we also plotted feasible $(\alpha, \gamma_c)$ pairs for various $\epsilon$ values in Fig. \[Fig:gamma\_c\_various\_latex\]. The plot for $\epsilon = 0$ obviously has zero $\gamma_c$, while $\epsilon > 0$ cases show a trade-off relationship. As $\epsilon$ increases, the minimum $\gamma_c$ value increases gradually. Minimum-Bandwidth-Regenerating (MBR) point and Minimum-Storage-Regenerating (MSR) point {#Section:MSR_MBR} --------------------------------------------------------------------------------------- ![Set of Feasible $(\alpha, \gamma)$ Points[]{data-label="Fig:MBR_MSR_points"}](MBR_MSR_plot_latest.pdf){height="50mm"} According to Corollary \[Corollary:Feasible Points\_large\_epsilon\], the set of feasible $(\alpha, \gamma)$ points shows a trade-off curve as in Fig. \[Fig:MBR\_MSR\_points\], for arbitrary $n,k,L,\epsilon$ settings. Here we focus on two extremal points: the minimum-bandwidth-regenerating (MBR) point and the minimum-storage-regenerating (MSR) point. As originally defined in [@dimakis2010network], we call the point on the trade-off with minimum bandwidth $\gamma$ as MBR. Similarly, we call the point with minimum storage $\alpha$ as MSR[^3]. Let $(\alpha_{msr}^{(\epsilon)}, \gamma_{msr}^{(\epsilon)})$ be the $(\alpha, \gamma)$ pair of the MSR point for given $\epsilon$. Similarly, define $(\alpha_{mbr}^{(\epsilon)}, \gamma_{mbr}^{(\epsilon)})$ as the parameter pair for MBR points. According to Corollary \[Corollary:Feasible Points\_large\_epsilon\], the explicit $(\alpha, \gamma)$ expression for the MSR and MBR points are as in the following Corollary, the proof of which is given in Appendix \[Section:Proof of Corollary\_msr\_mbr\_points\]. \[Coro:msr\_mbr\_points\] For a given $0 \leq \epsilon \leq 1$, we have $$\begin{aligned} &(\alpha_{msr}^{(\epsilon)}, \gamma_{msr}^{(\epsilon)}) \nonumber\\ &= \begin{cases} (\frac{\mathcal{M}}{\tau + \sum_{i=\tau+1}^{k}z_i}, \frac{\mathcal{M}}{\tau + \sum_{i=\tau+1}^{k}z_i} \frac{\sum_{i=\tau+1}^{k}z_i}{s_{\tau}} ), & 0 \leq \epsilon < \frac{1}{n-k} \\ (\frac{\mathcal{M}}{k}, \frac{\mathcal{M}}{k}\frac{1}{s_{k-1}}), & \frac{1}{n-k} \leq \epsilon \leq 1 \end{cases} \label{Eqn:MSR_point}\end{aligned}$$ $$\begin{aligned} (\alpha_{mbr}^{(\epsilon)}, \gamma_{mbr}^{(\epsilon)}) &= (\frac{\mathcal{M}}{s_0}, \frac{\mathcal{M}}{s_0}). \label{Eqn:MBR_point}\end{aligned}$$ Now we compare the MSR and MBR points for two extreme cases of $\epsilon = 0$ and $\epsilon = 1$. Using $R\coloneqq k/n$ and the dispersion ratio $\sigma$ defined in (\[Eqn:sigma\]), the asymptotic behaviors of MBR and MSR points are illustrated in the following theorem, the proof of which is in Appendix \[Proof\_of\_Thm\_MSR\_MBR\]. \[Thm:MSR\_MBR\] Consider the MSR point $(\alpha_{msr}^{(\epsilon)}, \gamma_{msr}^{(\epsilon)})$ and the MBR point $(\alpha_{mbr}^{(\epsilon)}, \gamma_{mbr}^{(\epsilon)})$ for $\epsilon = 0, 1$. The minimum node storage for $\epsilon=0$ is asymptotically equivalent to the minimum node storage for $\epsilon=1$, i.e., $$\label{Eqn:msr_property} \alpha_{msr}^{(0)} \sim \alpha_{msr}^{(1)} = \mathcal{M}/k$$ as $n \rightarrow \infty$ for arbitrary fixed $\sigma$ and $R=k/n$. Moreover, the MBR point for $\epsilon=0$ approaches the MBR point for $\epsilon=1$, i.e., $$\label{Eqn:mbr_property} (\alpha_{mbr}^{(0)}, \gamma_{mbr}^{(0)}) \rightarrow (\alpha_{mbr}^{(1)}, \gamma_{mbr}^{(1)}),$$ as $R=k/n \rightarrow 1$. The ratio between $\gamma_{mbr}^{(0)}$ and $\gamma_{mbr}^{(1)}$ is expressed as $$\label{Eqn:mbr_ratio} \frac{\gamma_{mbr}^{(0)}}{\gamma_{mbr}^{(1)}} \leq 2- \frac{k-1}{n-1}$$ ![Optimal trade-off curves for $\epsilon = 0, 1$[]{data-label="Fig:tradeoff_min"}](tradeoff_min_latest.pdf){height="50mm"} $$\begin{aligned} \alpha_{msr}^{(1)}&= \text{constant}, \quad k= nR = \Theta(n), \quad \mathcal{M}= \Theta(n)\end{aligned}$$ According to Theorem \[Thm:MSR\_MBR\], the minimum storage for $\epsilon=0$ can achieve $\mathcal{M}/k$ as $n \rightarrow \infty$ with fixed $R=k/n$. This result coincides with the result of Theorem \[Thm:condition\_for\_min\_storage\]. According to Theorem \[Thm:condition\_for\_min\_storage\], the sufficient and necessary condition for achieving the minimum storage of $\alpha=\mathcal{M}/k$ is $$\epsilon \geq \frac{1}{n-k} = \frac{1}{n(1-R)}.$$ As $n$ increases with a fixed $R$, the lower bound on $\epsilon$ reduces, so that in the asymptotic regime, $\epsilon=0$ can achieve $\alpha = \mathcal{M}/k$. Moreover, Theorem \[Thm:MSR\_MBR\] states that the MBR point for $\epsilon=0$ approaches the MBR point for $\epsilon=1$ as $R=k/n$ goes to 1. Fig. \[Fig:MBR\_coding\_schemes\] provides two MBR coding schemes with $(n,k,L) = (6,5,2)$, which has different $\epsilon$ values; one coding scheme in Fig. \[Fig:mbr\_epsilon\_1\] satisfies $\epsilon = 1$, while the other in Fig. \[Fig:mbr\_epsilon\_0\] satisfies $\epsilon=0$. The RSKR coding scheme [@rashmi2009explicit] is applied to the six nodes in Fig. \[Fig:mbr\_epsilon\_1\]. Each node (illustrated as a rectangular box) contains five $c_i$ symbols, where each symbol $c_i$ consists of two sub-symbols, $c_i^{(1)}$ and $c_i^{(2)}$. Note that any symbol $c_i$ is shared by exactly two nodes in Fig. \[Fig:mbr\_epsilon\_1\], which is due to the property of RSKR coding. This system can reliably store fifteen symbols $\{c_i\}_{i=1}^{15}$, or $\mathcal{M} = 30$ sub-symbols $\{c_i^{(1)}, c_i^{(2)}\}_{i=1}^{15}$, since it satisfies two properties  the exact repair property and the data recovery property  as illustrated below. First, when a node fails, five other nodes transmit five symbols (one distinct symbol by each node), which exactly regenerates the failed node. Second, we can retrieve data, $\mathcal{M} = 30$ sub-symbols $\{c_i^{(1)}, c_i^{(2)}\}_{i=1}^{15}$, by contacting any $k=5$ nodes. In Fig. \[Fig:mbr\_epsilon\_0\], each node contains two $e_i$ symbols, where each symbol $e_i$ consists of five sub-symbols, $\{e_i^{(j)}\}_{j=1}^{5}$. Note that in Fig. \[Fig:mbr\_epsilon\_0\], any symbol $e_i$ is shared by exactly two nodes which reside in the same cluster. This is because we applied RSKR coding at each cluster in the system of Fig. \[Fig:mbr\_epsilon\_0\]. This system can reliably store six symbols $\{e_i\}_{i=1}^{6}$, or $\mathcal{M} = 30$ sub-symbols $\bigcup_{i \in [6], j\in [5]} \{e_i^{(j)}\}$, since it satisfies the exact repair property and the data recovery property. Note that both DSSs in Fig. \[Fig:MBR\_coding\_schemes\] reliably store $\mathcal{M} = 30$ sub-symbols, by using the node capacity of $\alpha = 10$ sub-symbols and the repair bandwidth of $\gamma = 10$ sub-symbols. However, the former system requires $\gamma_c = 6$ cross-cluster repair bandwidth for each node failure event, while the latter system requires $\gamma_c = 0$ cross-cluster repair bandwidth. For example, if the leftmost node of the $1^{st}$ cluster fails in Fig. \[Fig:mbr\_epsilon\_1\], then four sub-symbols $\{c_1^{(i)}, c_2^{(i)}\}_{i=1}^2$ are transmitted within that cluster, while six sub-symbols $\{c_3^{(i)}, c_4^{(i)}, c_5^{(i)}\}_{i=1}^2$ are transmitted from the $2^{nd}$ cluster. In the case of $\epsilon=0$ in Fig. \[Fig:mbr\_epsilon\_0\], ten sub-symbols $\{e_1^{(i)}, e_2^{(i)}\}_{i=1}^5$ are transmitted within the $1^{st}$ cluster and no sub-symbols are transmitted across the clusters, when the leftmost node of the $1^{st}$ cluster fails. Thus, transition from the former system (Fig. \[Fig:mbr\_epsilon\_1\]) to the latter system (Fig. \[Fig:mbr\_epsilon\_0\]) reduces the cross-cluster repair bandwidth to zero, while maintaining the storage capacity $\mathcal{M}$ and the required resource pair $(\alpha, \gamma)$. Likewise, we can reduce the cross-cluster repair bandwidth to *zero* while maintaining the storage capacity, in the case of $R=k/n \rightarrow 1$. Note that $\gamma_{mbr}^{(\epsilon)} = \alpha_{mbr}^{(\epsilon)}$ for $0 \leq \epsilon \leq 1$, from (\[Eqn:MBR\_point\]). Thus, the result of (\[Eqn:mbr\_ratio\]) in Theorem \[Thm:MSR\_MBR\] can be expressed as Fig. \[Fig:MBR\_relationship\] at the asymptotic regime of large $n,k$. According to Fig. \[Fig:MBR\_relationship\], intra-cluster only repair ($\epsilon = 0$ or $\beta_c = 0$) is possible by using additional resources ($\alpha$ and $\gamma$) in the $1-R$ portion, compared to the symmetric repair ($\epsilon =1$) case. ![Relationship between MBR point of $\epsilon = 0$ and that of $\epsilon = 1$[]{data-label="Fig:MBR_relationship"}](mbr_point_relationship_ver3.pdf){height="50mm"} Intra-cluster Repairable Codes versus Locally Repairable Codes [@papailiopoulos2014locally] {#Section:IRC_versus_LRC} ------------------------------------------------------------------------------------------- Here, we define a family of network coding schemes for clustered DSS, which we call the *intra-cluster repairable codes*. In Corollary \[Corollary:Feasible Points\_large\_epsilon\], we considered DSSs with $\epsilon=0$, which can repair any failed node by using intra-cluster communication only. The optimal $(\alpha, \gamma)$ trade-off curve which satisfies $\epsilon=0$ is illustrated as the solid line with cross markers in Fig. \[Fig:kappa\_various\_latex\]. Each point on the curve is achievable (i.e., there exists a network coding scheme), according to the result of [@ahlswede2000network]. We call the network coding schemes for the points on the curve of $\epsilon=0$ the *intra-cluster repairable codes*, since these coding schemes can repair any failed node by using intra-cluster communication only. The relationship between the intra-cluster repairable codes and the locally repairable codes (LRC) of [@papailiopoulos2014locally] are investigated in Theorem \[Thm:IRC\_versus\_LRC\], the proof of which is given in Appendix \[Section:Proof\_of\_IRC\_versus\_LRC\]. \[Thm:IRC\_versus\_LRC\] The intra-cluster repairable codes with storage overhead $\alpha$ are the $(n,l_0,m_0,\mathcal{M},\alpha)$-LRC codes of [@papailiopoulos2014locally] where $$\begin{aligned} l_0 &= n_I-1, \\ m_0 &= n-k+1. \end{aligned}$$ It is confirmed that Theorem 1 of [@papailiopoulos2014locally], $$\label{Eqn:locality_ineq} m_0 \leq n - \ceil[\bigg]{\frac{\mathcal{M}}{\alpha}} - \ceil[\bigg]{\frac{\mathcal{M}}{l_0\alpha}} + 2,$$ holds for every intra-cluster repairable code with storage overhead $\alpha$. Moreover, the equality of (\[Eqn:locality\_ineq\]) holds if $$\begin{aligned} \alpha = \alpha_{msr}^{(0)}, (k {\ \mathrm{mod}\ n_I}) \neq 0. \nonumber \end{aligned}$$ Required $\beta_c$ for a given $\alpha$ --------------------------------------- Here we focus on the following question: when the available intra-cluster repair bandwidth is abundant, how much cross-cluster repair bandwidth is required to reliably store file $\mathcal{M}$? We consider scenarios when the intra-cluster repair bandwidth (per node) has its maximum value, i.e., $\beta_I = \alpha$. Under this setting, Theorem \[Thm:beta\_c\_alpha\_trade\] specifies the minimum required $\beta_c$ which satisfies $\mathcal{C}(\alpha, \beta_I, \beta_c) \geq \mathcal{M}$. The proof of Theorem \[Thm:beta\_c\_alpha\_trade\] is given in Appendix \[Section:Proof\_of\_Thm\_beta\_c\_alpha\_trade\]. \[Thm:beta\_c\_alpha\_trade\] Suppose the intra-cluster repair bandwidth is set at the maximum value, i.e., $\beta_I = \alpha$. For a given node capacity $\alpha$, the clustered DSS can reliably store data $\mathcal{M}$ if and only if $\beta_c \geq \beta_c^*$ where $$\begin{aligned} \beta_c^* &= \begin{cases} \frac{\mathcal{M} - (k-1)\alpha}{n-k}, & \text{ if } \alpha \in [\frac{\mathcal{M}}{k}, \frac{\mathcal{M}}{f_{k-1}} ) \\ \frac{\mathcal{M} - m\alpha}{\sum_{i=m+1}^{k} (n-i)}, & \text{ if } \alpha \in [\frac{\mathcal{M}}{f_{m+1}}, \frac{\mathcal{M}}{f_m} ) \\ & \quad \quad (m=k-2, k-3, \cdots, s+1) \\ \frac{\mathcal{M} - k_0 \alpha}{\sum_{i=k_0+1}^{k} (n-i)}, & \text{ if } \alpha \in [\frac{\mathcal{M}}{f_{k_0+1}}, \frac{\mathcal{M}}{k_0} ) \\ 0, & \text{ if } \alpha \in [ \frac{\mathcal{M}}{k_0}, \infty ), \end{cases} \label{Eqn:beta_c_star}\\ f_m &= m + \frac{\sum_{i=m+1}^{k} (n-i)}{n-m}, \label{Eqn:f_m}\\ k_0 &= k-\floor{\frac{k}{n_I}} \label{Eqn:k_0}. \end{aligned}$$ ![Optimal tradeoff between cross-cluster repair bandwidth and node capacity, when $n=100, k=85, L=10, \mathcal{M} = 85$ and $\beta_I = \alpha$[]{data-label="Fig:beta_c_alpha"}](beta_c_alpha_latest.pdf){height="50mm"} Fig. \[Fig:beta\_c\_alpha\] provides an example of the optimal trade-off relationship between $\beta_c$ and $\alpha$, explained in Theorem \[Thm:beta\_c\_alpha\_trade\]. For $\alpha \geq \mathcal{M}/k_0 = 1.104$, the cross-cluster burden $\beta_c$ can be reduced to zero. However, as $\alpha$ decreases from $\mathcal{M}/k_0$, the system requires a larger $\beta_c$ value. For example, if $\alpha = \beta_I= 1.05$ in Fig. \[Fig:beta\_c\_alpha\], $\beta_c \geq 0.03$ is required to satisfy $\mathcal{C}(\alpha, \beta_I, \beta_c) \geq \mathcal{M} = 85$. Thus, for each node failure event, a cross-cluster repair bandwidth of $\gamma_c = (n-n_I)\beta_c \geq 2.7$ is required. Theorem \[Thm:beta\_c\_alpha\_trade\] provides an explicit equation for the cross-cluster repair bandwidth we need to pay, in order to reduce node capacity $\alpha$. Further Comments & Future Works {#Section:Future_Works} =============================== Explicit coding schemes for clustered DSS ----------------------------------------- According to the part I proof of Theorem \[Thm:Capacity of clustered DSS\], there exists an information flow graph $G^*$ which has the min-cut value of $\mathcal{C}$, the capacity of clustered DSS. Thus, according to [@ahlswede2000network], there exists a linear network coding scheme which achieves capacity $\mathcal{C}$. Although the existence of a coding scheme is verified, explicit network coding schemes which achieve capacity need to be specified for implementing practical systems. Optimal number of clusters -------------------------- According to Theorem \[Thm:cap\_dec\_ftn\_L\], capacity $\mathcal{C}$ is asymptotically a monotonically decreasing function of $L$, the number of clusters. Thus, reducing the number of clusters (i.e., gathering storage nodes into a smaller number of clusters) increases storage capacity. However, as mentioned in Section \[Section:C\_versus\_L\], we typically want to have a sufficiently large $L$, to tolerate the failure of a cluster. Then, the remaining question is in finding optimal $L^*$ which not only allows sufficiently large storage capacity, but also a tolerance to cluster failures. We regard this problem as a future research topic, the solution to which will provide a guidance on the strategy for distributing storage nodes into multiple clusters. Extension to general $d_I, d_c$ settings ---------------------------------------- The present paper assumed a maximum helper node setting, $d_I = n_I-1$ and $d_c = n-n_I$, since it maximizes the capacity as stated in Proposition \[Prop:max\_helper\_nodes\]. However, waiting for all helper nodes gives rise to a latency issue. If we reduce the number of helper nodes $d_I$ and $d_c$, low latency repair would be possible, while the achievable storage capacity decreases. Thus, we consider obtaining the capacity expression for general $d_I, d_c$ settings, and discover the trade-off between capacity and latency for various $d_I, d_c$ values. Scenarios of aggregating the helper data within each cluster ------------------------------------------------------------ Conclusion {#Section:Conclusion} ========== This paper considered a practical distributed storage system where storage nodes are dispersed into several clusters. Noticing that the traffic burdens of intra- and cross-cluster communications are typically different, a new system model for clustered distributed storage systems is suggested. Based on the cut-set bound analysis of information flow graph, the storage capacity $\mathcal{C}(\alpha, \beta_I, \beta_c)$ of the suggested model is obtained in a closed-form, as a function of three main resources: node storage capacity $\alpha$, intra-cluster repair bandwidth $\beta_I$ and cross-cluster repair bandwidth $\beta_c$. It is shown that the asymmetric repair ($\beta_I > \beta_c$) degrades capacity, which is the cost for lifting the cross-cluster repair burden. Moreover, in the asymptotic regime of a large number of storage nodes, capacity is shown to be asymptotically equivalent to a monotonic decreasing function of $L$, the number of clusters. Thus, reducing $L$ (i.e., gathering nodes into less clusters) is beneficial for increasing capacity, although we would typically need to guarantee sufficiently large $L$ to tolerate rack failure events. Using the capacity expression, we obtained the feasible set of ($\alpha, \beta_I, \beta_c$) triplet which satisfies $\mathcal{C}(\alpha, \beta_I, \beta_c) \geq \mathcal{M}$, i.e., it is possible to reliably store file $\mathcal{M}$ by using the resource value set ($\alpha, \beta_I, \beta_c$). The closed-form solution on the feasible set shows a different behavior depending on $\epsilon = \beta_c/\beta_I$, the ratio of cross- to intra-cluster repair bandwidth. It is shown that the minimum storage of $\alpha = \mathcal{M}/k$ is achievable if and only if $\epsilon \geq \frac{1}{n-k}$. Moreover, in the special case of $\epsilon = 0$, we can construct a reliable storage system without using cross-cluster repair bandwidth. A family of network codes which enable $\epsilon = 0$, called the *intra-cluster repairable codes*, has been shown to be a class of the *locally repairable codes* defined in [@papailiopoulos2014locally]. Proof of Theorem \[Thm:Capacity of clustered DSS\] {#Section:Proof of Thm 1} ================================================== Here, we prove Theorem \[Thm:Capacity of clustered DSS\]. First, denote the right-hand-side (RHS) of (\[Eqn:Capacity of clustered DSS\_rev\]) as $$\label{Eqn:Capacity_value} T \coloneqq \sum_{i=1}^{n_I} \sum_{j=1}^{g_i} \min \{\alpha, \rho_i\beta_I + (n-\rho_i - j - \sum_{m=1}^{i-1}g_m) \beta_c \}.$$ For other notations used in this proof, refer to subsection \[Subsection: notation\]. The proof proceeds in two parts. *Part I*. Show an information flow graph $G^* \in \mathcal{G}$ and a cut-set $c^* \in C(G^*)$ such that $w(G^*, c^*) = T$: Consider the information flow graph $G^*$ illustrated in Fig. \[Fig:partI\_proof\_Thm1\], which is obtained by the following procedure. First, data from source node $S$ is distributed into $n$ nodes labeled from $x^1$ to $x^n$. As mentioned in Section \[Section:Info\_flow\_graph\], the storage node $x^i = (x^i_{in}, x^i_{out})$ consists of an input-node $x_{in}^i$ and an output-node $x_{out}^i$. Second, storage node $x^t$ fails and is regenerated at the newcomer node $x^{n+t}$ for $t \in [k]$. The newcomer node $x^{n+t}$ connects to $n-1$ survived nodes $\{x^{m}\}_{m=t+1}^{n+t-1}$ to regenerate $x^t$. Third, data collector node $DC$ contacts $\{x^{n+t}\}_{t=1}^{k}$ to retrieve data. This whole process is illustrated in the information flow graph $G^*$. ![2-dim. structure representation[]{data-label="Fig:partI_proof_Thm1_2"}](proof_thm1_part1_2_ver3.pdf){height="45mm"} To specify $G^*$, here we determine the 2-dimensional location of the $k$ newcomer nodes $\{x^{n+t}\}_{t=1}^{k}$. First, consider the 2-dimensional structure representation of clustered distributed storage, illustrated in Fig. \[Fig:partI\_proof\_Thm1\_2\]. In this figure, each row represents each cluster, and each node is represented as a 2-dimensional $(i,j)$ point for $i \in [L]$ and $j \in [n_I]$. The symbol $N(i,j)$ denotes the node at $(i,j)$ point. Here we define the set of $n$ nodes, $$\label{Eqn:set_of_nodes} \mathcal{N} \coloneqq \{N(i,j) : i \in [L], j \in [n_I] \}.$$ For $t \in [k] $, consider selecting the newcomer node $x^{n+t}$ as $$\label{Eqn:mapping_btw_t_and_ij} x^{n+t} = N(i_t,j_t)$$ where $$\begin{aligned} i_t &= \min\{\nu \in [n_I] : \sum_{m=1}^{\nu} g_m \geq t \},\label{Eqn:i_from_t}\\ j_t &= t - \sum_{m=1}^{i_t-1} g_m \label{Eqn:j_from_t},\end{aligned}$$ and $g_m$ used in the method is defined in (\[Eqn:g\_m\]). The location of $k$ newcomer nodes selected by this method are illustrated in Fig. \[Fig:partI\_proof\_Thm1\_3\]. Moreover, for the $n=12, L=3, k=9$ case, the newcomer nodes $\{x^{n+t}\}_{t=1}^k$ are depicted in Fig. \[Fig:partI\_proof\_Thm1\_4\]. In these figures, the node with number $t$ inside represents the newcomer node labeled as $x^{n+t}$. ![The location of $k$ newcomer nodes: $x^{n+1}, \cdots, x^{n+k}$[]{data-label="Fig:partI_proof_Thm1_3"}](proof_thm1_part1_3_ver5.pdf){width="85mm"} ![The location of $k$ newcomer nodes for $n=12, L=3, k=9$ case[]{data-label="Fig:partI_proof_Thm1_4"}](proof_thm1_part1_4.pdf){height="20mm"} For the given graph $G^*$, now we consider a cut-set $c^* \in C(G^*)$ defined as below. The cut-set $c^*=(U,{\mkern 1.5mu\overline{\mkern-1.5muU\mkern-1.5mu}\mkern 1.5mu})$ can be defined by specifying $U$ and ${\mkern 1.5mu\overline{\mkern-1.5muU\mkern-1.5mu}\mkern 1.5mu}$ (complement of $U$), which partition the set of vertices in $G^*$. First, let $x^{i}_{in}, x^{i}_{out} \in U$ for $i \in [n]$ and $x^{n+i}_{out} \in {\mkern 1.5mu\overline{\mkern-1.5muU\mkern-1.5mu}\mkern 1.5mu}$ for $ i \in [k]$. Moreover, let $S \in U$ and $DC \in {\mkern 1.5mu\overline{\mkern-1.5muU\mkern-1.5mu}\mkern 1.5mu}$. See Fig. \[Fig:cut\_analysis\]. Let $U_0 = \bigcup_{i=1}^n \{x^i_{out}\} $. For $t \in [k]$, let $\omega_t^*$ be the sum of capacities of edges from $U_0$ to $x^{n+t}_{in}$. If $\alpha \leq \omega_t^*$, then we include $x_{in}^{n+t}$ in $U$. Otherwise, we include $x^{n+t}_{in}$ in ${\mkern 1.5mu\overline{\mkern-1.5muU\mkern-1.5mu}\mkern 1.5mu}$. Then, the cut-set $c^*$ has the cut-value of $$\label{Eqn:optimal_omega_t} w(G^*,c^*) = \sum_{t=1}^{k} \min \{\alpha, \omega_t^*\}.$$ All that remains is to show that (\[Eqn:optimal\_omega\_t\]) is equal to the expression in (\[Eqn:Capacity\_value\]). In other words, we will obtain the expression for $\omega_t^*$. Recall that in the generation process of $G^*$, any newcomer node $x^{n+t}$ connects to $n-1$ helper nodes $\{x^{m}\}_{m=t+1}^{n+t-1}$ to regenerate $x^t$. Among the $n-1$ helper nodes, the $n_I - 1$ nodes reside in the same cluster with $x^t$, while the $n-n_I$ nodes are in other clusters. From our system setting in Section \[Section:Clustered\_DSS\], the helper nodes in the same cluster as the failed node help by $\beta_I$, while the helper nodes in other clusters help by $\beta_c$. Therefore, the total repair bandwidth to regenerate any failed node is $$\label{Eqn:gamma_recall} \gamma = (n_I-1)\beta_I + (n-n_I)\beta_c$$ as in (\[Eqn:gamma\]). The newcomer node $x^{n+1}_{in}$ connects to $\{x_{out}^{m}\}_{m=2}^n$, all of which are included in $U_0$. Therefore, $\omega_1^* = \gamma = (n_I - 1)\beta_I + (n-n_I)\beta_c$ holds. Next, $x^{n+2}_{in}$ connects to $n-2$ nodes $\{x_{out}^{m}\}_{m=3}^{n}$ from $U_0$ and one node $x_{out}^{n+1}$ from ${\mkern 1.5mu\overline{\mkern-1.5muU\mkern-1.5mu}\mkern 1.5mu}$. Define variable $\beta_{lm}$ as the repair bandwidth from $x_{out}^{n+l}$ to $x_{in}^{n+m}$. Then, $\omega_2^* = \gamma - \beta_{12}$. From (\[Eqn:mapping\_btw\_t\_and\_ij\]), we have $x^{n+1} = N(1,1)$ and $x^{n+2} = N(1,2)$. Therefore, $x^{n+1}$ and $x^{n+2}$ are in different clusters, which result in $\beta_{12} = \beta_c$. Therefore, $\omega_2^* = \gamma - \beta_{12} = \gamma - \beta_c.$ ![The cut-set $c^* = (U, {\mkern 1.5mu\overline{\mkern-1.5muU\mkern-1.5mu}\mkern 1.5mu})$ for the graph $G^*$[]{data-label="Fig:cut_analysis"}](cut_analysis_ver2.pdf){width="90mm"} In general, $x^{n+t}_{in}$ connects to $n-t$ nodes $\{x_{out}^{m}\}_{m = t+1}^{n}$ from $U_0$, and $t-1$ nodes $\{x_{out}^{n+m}\}_{m=1}^{t-1}$ from ${\mkern 1.5mu\overline{\mkern-1.5muU\mkern-1.5mu}\mkern 1.5mu}$. Thus, $\omega_t^*$ for $t \in [k]$ can be expressed as $\omega_t^* = \gamma - \sum_{l=1}^{t-1} \beta_{lt}$ where $$\beta_{lt} = \begin{cases} \beta_I, & \text{ if } x^{n+l} \text{ and } x^{n+t} \text{ are in the same cluster} \\ \beta_c, & \text{ otherwise.} \end{cases}$$ Recall Fig. \[Fig:partI\_proof\_Thm1\_3\]. For arbitrary newcomer node $x^{n+t} = N(i_t, j_t)$, the set $\{x^{n+m}\}_{m=1}^{t - 1}$ contains $i_t - 1$ nodes which reside in the same cluster with $x^{n+t}$, and $t - i_t$ nodes in other clusters. Therefore, $\omega_t^*$ can be expressed as $$\omega_t^* = \gamma - (i_t-1)\beta_I - (t-i_t)\beta_c$$ where $i_t$ is defined in (\[Eqn:i\_from\_t\]). Combined with (\[Eqn:gamma\_recall\]) and (\[Eqn:j\_from\_t\]), we get $$\begin{aligned} \omega_t^* &= (n_I - i_t)\beta_I + (n-n_I - t + i_t)\beta_c \nonumber\\ &= (n_I - i_t)\beta_I + (n-n_I - j_t - \sum_{m=1}^{i_t-1} g_m + i_t)\beta_c.\end{aligned}$$ Then, (\[Eqn:optimal\_omega\_t\]) can be expressed as $$\begin{aligned} \label{Eqn:optimal_mincut} & w(G^*, c^*) = \nonumber\\ & \sum_{i=1}^{n_I} \sum_{t \in T_i} \min \{\alpha, (n_I-i)\beta_I + (n-n_I-j_t - \sum_{m=1}^{i-1}g_m + i) \beta_c\}\end{aligned}$$ where $T_i = \{t \in [k]: i_t = i\}$. From the definition of $i_t$ in (\[Eqn:i\_from\_t\]), we have $$T_i = \{\sum_{m=1}^{i-1} g_m + 1, \sum_{m=1}^{i-1} g_m + 2, \cdots, \sum_{m=1}^{i} g_m \}.$$ Thus, $j_t = 1,2,\cdots, g_i$ for $t \in T_i$. Therefore, (\[Eqn:optimal\_mincut\]) can be expressed as $$\begin{aligned} & w(G^*, c^*) = \\ & \sum_{i=1}^{n_I} \sum_{j=1}^{g_i} \min \{\alpha, (n_I-i)\beta_I + (n-n_I-j - \sum_{m=1}^{i-1}g_m + i) \beta_c\}, \end{aligned}$$ which is identical to $T$ in (\[Eqn:Capacity\_value\]), where $\rho_i$ used in this equation is defined in (\[Eqn:rho\_i\]). Therefore, the specified information flow graph $G^*$ and the specified cut-set $c^*$ satisfy $\omega(G^*, c^*) = \sum_{t=1}^k \min\{\alpha, \omega_t^*\} = T$. *Part II*. Show that for every information flow graph $G \in \mathcal{G}$ and for every cut-set $c \in C(G)$, the cut-value $w(G,c)$ is greater than or equal to $T$ in (\[Eqn:Capacity\_value\]). In other words, $\forall G \in \mathcal{G}, \forall c \in C(G)$, we have $w(G,c) \geq T$. The proof is divided into 2 sub-parts: Part II-1 and Part II-2. *Part II-1*. Show that $\forall G \in \mathcal{G}, \forall c \in C(G),$ we have $w(G,c) \geq B(G,c)$ where $B(G,c)$ is in (\[Eqn:Lower\_bound\]): Consider an arbitrary information flow graph $G \in \mathcal{G}$ and an arbitrary cut-set $c \in C(G)$ of the graph $G$. Denote the cut-set as $c=(U, {\mkern 1.5mu\overline{\mkern-1.5muU\mkern-1.5mu}\mkern 1.5mu})$. Consider an output node $x_{out}^{i}$ connected to $DC$. If $x_{out}^{i} \in U$, then the cut-value $w(G,c)$ is infinity, which is a trivial case for proving $w(G,c) \geq B(G,c)$. Therefore, the $k$ output nodes connected to $DC$ are assumed to be in ${\mkern 1.5mu\overline{\mkern-1.5muU\mkern-1.5mu}\mkern 1.5mu}$. In other words, at least $k$ output nodes exist in ${\mkern 1.5mu\overline{\mkern-1.5muU\mkern-1.5mu}\mkern 1.5mu}$. Note that every directed acyclic graph can be topologically sorted [@bang2008digraphs], where vertex $u$ is followed by vertex $v$ if there exists a directed edge from $u$ to $v$. Consider labeling the topologically first $k$ output nodes in ${\mkern 1.5mu\overline{\mkern-1.5muU\mkern-1.5mu}\mkern 1.5mu}$ as $v_{out}^{1}, \cdots,v_{out}^{k}$. Similar to the notation for a storage node $x^{i} = (x^{i}_{in}, x^{i}_{out})$ in Section \[Section:Info\_flow\_graph\], we denote the storage node which contains $v^{i}_{out}$ as $v^{i} = (v^{i}_{in}, v^{i}_{out})$. $$\begin{aligned} \label{Eqn:V_k} \mathcal{V}_k = \{(v^1, v^2, \cdots, v^k) : & \quad v^t \in \mathcal{N} \text{ for } t \in [k] \nonumber\\ & \quad v^{t_1} \neq v^{t_2} \text{ for } t_1 \neq t_2 \}.\end{aligned}$$ We also define $u_i$, the sum of capacities of edges from $U$ to $v_{in}^i$. See Fig. \[Fig:proof\_thm1\_part2\_1\]. ![Arbitrary information flow graph $G \in \mathcal{G}$ and arbitrary cut-set $c \in C(G)$[]{data-label="Fig:proof_thm1_part2_1"}](proof_thm1_part2_1_ver3.pdf){height="30mm"} If $v_{in}^1 \in U$, then the cut-set $c =(U, {\mkern 1.5mu\overline{\mkern-1.5muU\mkern-1.5mu}\mkern 1.5mu})$ should include the edge from $v_{in}^1$ to $v_{out}^1$, which has the edge capacity $\alpha$. Otherwise (i.e., $v_{in}^1 \in {\mkern 1.5mu\overline{\mkern-1.5muU\mkern-1.5mu}\mkern 1.5mu}$), the cut-set $c =(U, {\mkern 1.5mu\overline{\mkern-1.5muU\mkern-1.5mu}\mkern 1.5mu})$ should include the edges from $U$ to $v_{in}^1$. If $v_{in}^1$ node is directly connected to the source node $S$, the cut-value $w(G,c)$ is infinity (trivial case for proving $w(G,c) \geq B(G,c)$). Therefore, $v_{in}^1$ node is assumed to be a newcomer node helped by $n-1$ helper nodes. Note that all helper nodes of $v_{in}^1$ are in $U$, since $v_{out}^1$ is the topologically first output node in ${\mkern 1.5mu\overline{\mkern-1.5muU\mkern-1.5mu}\mkern 1.5mu}$. Thus, the cut-set $c$ should include the edges from $U$ to $v_{in}^1$, where the sum of capacities of these edges are $$u_1 = \gamma = (n_I - 1)\beta_I + (n-n_I)\beta_c.$$ If $v_{in}^2 \in U$, then the cut-set $c =(U, {\mkern 1.5mu\overline{\mkern-1.5muU\mkern-1.5mu}\mkern 1.5mu})$ should include the edge from $v_{in}^2$ to $v_{out}^2$, which has the edge capacity $\alpha$. Otherwise (i.e., $v_{in}^2 \in {\mkern 1.5mu\overline{\mkern-1.5muU\mkern-1.5mu}\mkern 1.5mu}$), the cut-set $c =(U, {\mkern 1.5mu\overline{\mkern-1.5muU\mkern-1.5mu}\mkern 1.5mu})$ should include the edges from $U$ to $v_{in}^2$. As we discussed in the case of $v_{in}^1$, we can assume $v_{in}^2$ is a newcomer node helped by $n-1$ helper nodes. Since $v_{in}^2$ is the topologically second node in ${\mkern 1.5mu\overline{\mkern-1.5muU\mkern-1.5mu}\mkern 1.5mu}$, it may have one helper node $v_{out}^1 \in {\mkern 1.5mu\overline{\mkern-1.5muU\mkern-1.5mu}\mkern 1.5mu}$; at least $n-2$ helper nodes in $U$ help to generate $v_{in}^2$. Note that the total amount of data coming into $v_{in}^2$ is $\gamma = (n_I - 1)\beta_I + (n-n_I)\beta_c$, while the amount of information coming from $v_{out}^1$ to $v_{in}^2$, denoted $\beta_{12}$, is as follows: if $v^1$ and $v^2$ are in the same cluster, $\beta_{12} = \beta_I$, otherwise $\beta_{12} = \beta_c$. Recall that the cut-set should include the edges from $U$ to $v_{in}^2$. The sum of capacities of these edges are $$u_2 \geq \gamma - \beta_{12},$$ while the equality holds if and only if $v_{out}^1$ helps $v_{in}^2$. In a similar way, for $i \in [k]$, $u_i$ can be bounded as $$\label{Eqn:u_t_lower_bound} u_i \geq \omega_i$$ where $$\begin{aligned} \omega_i &= \gamma - \displaystyle\sum_{j=1}^{i-1} \beta_{ji},\label{Eqn:omega_i}\\ \beta_{ji} &= \begin{cases} \beta_I, & v^j \text{ and } v^i \text{ are in the same cluster} \\ \beta_c, & \text{otherwise.} \\ \end{cases}\label{Eqn:beta_ji_value}\end{aligned}$$ The equality in (\[Eqn:u\_t\_lower\_bound\]) holds if and only if $v_{out}^{j}$ helps $v_{in}^i$ for $j \in [i-1]$. Thus, $v_{out}^i$ contributes at least $\min \{\alpha, \omega_i\}$ to the cut value, for $i \in [k]$. In summary, for arbitrary graph $G \in \mathcal{G}$, an arbitrary cut-set $c$ has cut-value $w(G,c)$ of at least $\sum_{i=1}^k \min \{\alpha, \omega_i\}$: $$\label{Eqn:omega_bound} w(G,c) \geq \sum_{i=1}^k \min \{\alpha, \omega_i\}, \quad \quad \forall G \in \mathcal{G}, \forall c \in C(G).$$ Note that $\{\omega_i\}$ depends on the relative position of $\{v_{out}^i\}_{i=1}^k$, which is determined when an arbitrary information flow graph $G \in \mathcal{G}$ and arbitrary cut-set $c \in C(G)$ are specified. This relationship is illustrated in Fig. \[Fig:Dependency\_graph\]. Therefore, we define $$\label{Eqn:Lower_bound} B(G,c) = \sum_{i=1}^k \min \{\alpha, \omega_i\}$$ for arbitrary $G \in \mathcal{G}$ and arbitrary $c \in C(G)$. Combining (\[Eqn:omega\_bound\]) and (\[Eqn:Lower\_bound\]) completes the proof *part II-1*. ![Dependency graph for variables in the proof of Part II.[]{data-label="Fig:Dependency_graph"}](dependency_graph_ver2.pdf){width="85mm"} *Part II-2*. $\displaystyle\min_{G \in \mathcal{G}} \ \min_{c \in C(G)} B(G,c) = R$: Assume that $\alpha$ and $k$ are fixed. See Fig. \[Fig:Dependency\_graph\]. Note that for a given graph $G \in \mathcal{G}$ and a cut-set $c = (U, {\mkern 1.5mu\overline{\mkern-1.5muU\mkern-1.5mu}\mkern 1.5mu})$ with $ c \in C(G)$, the sequence of topologically first $k$ output nodes $(v_{out}^i)_{i=1}^k$ in ${\mkern 1.5mu\overline{\mkern-1.5muU\mkern-1.5mu}\mkern 1.5mu}$ is determined. Moreover, for a given sequence $(v_{out}^i)_{i=1}^k$, we have a fixed $(\omega_i)_{i=1}^k$, which determines $B(G,c)$ in (\[Eqn:Lower\_bound\]). Thus, $\displaystyle\min_{G \in \mathcal{G}} \ \min_{c \in C(G)} B(G,c)$ can be obtained by finding the optimal $(v_{out}^i)_{i=1}^k$ sequence which minimizes $B(G,c)$. It is identical to finding the optimal $(v^i)_{i=1}^k$, the sequence of $k$ different nodes out of $n$ existing nodes in the system. Therefore, based on (\[Eqn:Lower\_bound\]) and (\[Eqn:omega\_i\]), we have $$\label{minmin_equiv_1} \displaystyle\min_{G \in \mathcal{G}} \ \min_{c \in C(G)} B(G,c) = \displaystyle\min_{(v^i)_{i=1}^k \in \mathcal{V}_k} \left(\sum_{i=1}^k \min \{\alpha, \gamma - \sum_{j=1}^{i-1}\beta_{ji}\}\right)$$ where $$\label{Eqn:beta_ji_orig} \beta_{ji} = \begin{cases} \beta_I, & v^j \text{ and } v^i \text{ are in the same cluster} \\ \beta_c, & \text{otherwise.} \\ \end{cases}$$ holds as defined in (\[Eqn:beta\_ji\_value\]), and $\mathcal{V}_k$ is defined in (\[Eqn:V\_k\]). In order to obtain the solution for RHS of (\[minmin\_equiv\_1\]), all we need to do is to find the optimal sequence $(v^i)_{i=1}^k$ of $k$ different nodes, which can be divided into two sub-problems: $i)$ finding the optimal way of selecting $k$ nodes $\{v^i\}_{i=1}^k$ out of $n$ nodes, and $ii)$ finding the optimal order of selected $k$ nodes. Note that there are $n \choose k$ selection methods and $k!$ ordering methods. Each selection method can be assigned to a *selection vector* $\bm{s}$ defined in Definition \[Def:Selection vector\], and each ordering method can be assigned to an *ordering vector* $\bm{\pi}$ defined in Definition \[Def:Ordering vector\]. First, we define a selection vector $\bm{s}$ for a given $\{v^i\}_{i=1}^k$. \[Def:Selection vector\] Assume arbitrary $k$ nodes are selected as $\{v^{i}\}_{i=1}^k$. Label each cluster by the number of selected nodes in a descending order. In other words, the $1^{st}$ cluster contains a maximum number of selected nodes, and the $L^{th}$ cluster contains a minimum number of selected nodes. Under this setting, define the selection vector $\bm{s} = [s_1, s_2, \cdots, s_L]$ where $s_i$ is the number of selected nodes in the $i^{th}$ cluster. ![Obtaining the selection vector for given $k$ output nodes $\{v_{out}^{i}\}_{i=1}^k$ ($n=15, k=8, L=3$)[]{data-label="Fig:Selection Vector"}](selection_vector.pdf){height="25mm"} Fig. \[Fig:Selection Vector\] shows an example of selection vector $\bm{s}$ corresponding to the selected $k$ nodes $\{v^{i}\}_{i=1}^k$. From the definition of the selection vector, the set of possible selection vectors can be specified as follows. $$\begin{aligned} \mathcal{S} = \big\{\bm{s} &=[s_1, \cdots, s_L] : 0 \leq s_i \leq n_I, s_{i+1} \leq s_{i}, \sum_{i=1}^{L} s_i = k \big\}. \end{aligned}$$ Note that even though ${n\choose k}$ different selections exist, the $\{\omega_i\}$ values in (\[Eqn:omega\_i\]) are only determined by the corresponding selection vector $\bm{s}$. This is because $\{\omega_i\}$ depends only on the relative positions of $\{v^{i}\}_{i=1}^k$, whether they are in the same cluster or in different clusters. Therefore, comparing the $\{\omega_i\}$ values of all $\vert \mathcal{S} \vert$ possible selection vectors $\bm{s}$ is enough; it is not necessary to compare the $\{\omega_i\}$ values of ${n\choose k}$ selection methods. Now, we define the ordering vector $\bm{\pi}$ for a given selection vector $\bm{s}$. \[Def:Ordering vector\] Let the locations of $k$ nodes $\{v^{i}\}_{i=1}^k$ be fixed with a corresponding selection vector $\bm{s} = [s_1, \cdots, s_L]$. Then, for arbitrary ordering of the selected $k$ nodes, define the ordering vector $\bm{\pi} = [\pi_1, \cdots, \pi_k]$ where $\pi_i$ is the index of the cluster which contains $v^{i}$. For a given $\bm{s}$, the ordering vector $\bm{\pi}$ corresponding to an arbitrary ordering of $k$ nodes is illustrated in Fig. \[Fig:Ordering Vector\]. In this figure (and the following figures in this paper), the number $i$ written inside each node means that the node is $v^{i}$. From the definition, an ordering vector $\bm{\pi}$ has $s_l$ components with value $l$, for all $l\in [L]$. The set of possible ordering vectors can be specified as $$\label{Eqn:set of ordering vectors} \Pi(\bm{s}) = \big\{ \bm{\pi} = [\pi_1, \cdots, \pi_k] : \sum_{i=1}^{k} \mathds{1}_{\pi_i = l} = s_l, \ \forall l \in [L] \big\}$$ Note that for given $k$ selected nodes, there exists $k! $ different ordering methods. However, the $\{\omega_i\}$ values in (\[Eqn:omega\_i\]) are only determined by the corresponding ordering vector $\bm{\pi} \in \Pi(\bm{s})$, by similar reasoning for compressing ${n\choose k}$ selection methods to $\vert \mathcal{S} \vert$ selection vectors. Therefore, comparing the $\{\omega_i\}$ values of all possible ordering vectors $\bm{\pi}$ is enough; it is not necessary to compare the $\{\omega_i\}$ values of all $k!$ ordering methods. ![Obtaining the ordering vector for given an arbitrary order of $k$ output nodes ($n=15, k=8, L=3$)[]{data-label="Fig:Ordering Vector"}](ordering_vector.pdf){height="25mm"} Thus, finding the optimal sequence $(v^i)_{i=1}^k$ is identical to specifying the optimal ($\bm{s, \pi}$) pair, which is obtained as follows. Recall that from the definition of $\bm{\pi} = [\pi_1, \pi_2, \cdots, \pi_k]$ in Definition \[Def:Ordering vector\], $\pi_i = \pi_j$ holds if and only if $v^i$ and $v^j$ are in the same cluster. Therefore, $\beta_{ji}$ in (\[Eqn:beta\_ji\_orig\]) can be expressed by using $\bm{\pi}$ notation: $$\label{Eqn:beta_ji} \beta_{ji} (\bm{\pi})= \begin{cases} \beta_I & \text{ if } \pi_i = \pi_j\\ \beta_c & \text{ otherwise} \end{cases}$$ Thus, the $\sum_{j=1}^{i-1}\beta_{ji}$ term in (\[minmin\_equiv\_1\]) can be expressed as $$\begin{aligned} \label{Eqn:sum_beta_ji} \sum_{j=1}^{i-1}\beta_{ji}(\bm{\pi}) &= (\sum_{j=1}^{i-1} \mathds{1}_{\pi_j = \pi_i}) \beta_I + (i - 1 - \sum_{j=1}^{i-1} \mathds{1}_{\pi_j = \pi_i}) \beta_c. \end{aligned}$$ Combining (\[Eqn:gamma\]), (\[minmin\_equiv\_1\]) and (\[Eqn:sum\_beta\_ji\]), we have $$\displaystyle\min_{G \in \mathcal{G}} \ \min_{c \in C(G)} B(G,c) = \displaystyle\min_{\bm{s} \in S} \ \min_{\bm{\pi} \in \Pi(\bm{s})} L (\bm{s},\bm{\pi})$$ where $$\begin{aligned} L (\bm{s},\bm{\pi}) &= \sum_{i=1}^k \min\{\alpha, \omega_i(\bm{\pi})\}, \label{Eqn:lower_bound}\\ \omega_i(\bm{\pi}) &= \gamma - \sum_{j=1}^{i-1} \beta_{ji} (\bm{\pi})= a_i(\bm{\pi}) \beta_I + (n-i-a_i(\bm{\pi})) \beta_c, \label{Eqn:weight vector}\\ a_i(\bm{\pi}) &= n_I - 1 - \sum_{j=1}^{i-1} \mathds{1}_{\pi_j = \pi_i}. \label{Eqn:a_i}\end{aligned}$$ Therefore, the rest of the proof in *part II-2* shows that $$\label{Eqn:Capacity_expression_minmin} \displaystyle\min_{\bm{s} \in S} \ \min_{\bm{\pi} \in \Pi(\bm{s})} L (\bm{s},\bm{\pi}) = R$$ holds. We begin by stating a property of $\omega_i(\bm{\pi})$ seen in (\[Eqn:weight vector\]). \[Prop:weight vector\] Consider a fixed selection vector $\bm{s}$. We claim that $\sum_{i=1}^k \omega_i(\bm{\pi})$ is constant irrespective of the ordering vector $\bm{\pi} \in \Pi(\bm{s})$. Let $\bm{s}= [s_1, \cdots, s_L]$. For an arbitrary ordering vector $\bm{\pi} \in \Pi(\bm{s})$, let $b_i(\bm{\pi})= n-i-a_i(\bm{\pi})$ where $a_i(\bm{\pi})$ is as given in (\[Eqn:a\_i\]). For simplicity, we denote $a_i(\bm{\pi})$, $b_i(\bm{\pi})$ and $\omega_i(\bm{\pi})$ as $a_i$, $b_i$ and $\omega_i$, respectively. Then, $$\label{Eqn:proof of proposition 2} \sum_{i=1}^{k}(a_i + b_i) = \sum_{i=1}^{k}(n-i) = \textit{constant (const.)}$$ for fixed $n,k$. Note that $$\label{Eqn:sum_a_i} \sum_{i=1}^{k}a_i = k(n_I - 1) - \sum_{i=1}^{k} \sum_{j=1}^{i-1} \mathds{1}_{\pi_j = \pi_i}$$ from (\[Eqn:a\_i\]). Also, from the definition of $\Pi(\bm{s})$ in (\[Eqn:set of ordering vectors\]), an arbitrary ordering vector $\bm{\pi} \in \Pi(\bm{s})$ has $s_l$ components with value $l$, for all $l\in [L]$. If we define $$\label{Eqn:I_l} I_l (\bm{\pi}) = \{ i \in [k] : \pi_i = l \},$$ then $|I_l(\bm{\pi})| = s_l$ holds for $l \in [L]$. Then, $$\sum_{i\in I_l(\bm{\pi})} \sum_{j=1}^{i-1} \mathds{1}_{\pi_j = \pi_i} = 0 + 1 + \cdots + (s_l-1) = \sum_{t=0}^{s_l - 1} t.$$ Therefore, $$\sum_{i=1}^{k} \sum_{j=1}^{i-1} \mathds{1}_{\pi_j = \pi_i} = \sum_{l=1}^{L} \sum_{i\in I_l(\bm{\pi})} \sum_{j=1}^{i-1} \mathds{1}_{\pi_j = \pi_i} = \sum_{l=1}^{L} \sum_{t=0}^{s_l-1} t = \textit{const.}$$ for fixed $L, \bm{s}$. Combining with (\[Eqn:sum\_a\_i\]), $$\label{Eqn:sum_a_i_const} \sum_{i=1}^{k}a_i = k(n_I - 1) - \sum_{l=1}^{L} \sum_{t=0}^{s_l-1} t = \textit{const.}$$ for fixed $n,k,L,\bm{s}$. From (\[Eqn:proof of proposition 2\]) and (\[Eqn:sum\_a\_i\_const\]) we get $$\sum_{i=1}^k b_i = \textit{const.}$$ Therefore, from (\[Eqn:weight vector\]), $$\sum_{i=1}^k \omega_i = (\sum_{i=1}^k a_i) \beta_I + (\sum_{i=1}^k b_i) \beta_c = \textit{const.}$$ for every ordering vector $\bm{\pi} \in \Pi(\bm{s})$, if $n,k,L,\beta_I, \beta_c$ and $\bm{s}$ are fixed. Now, we define a special ordering vector $\bm{\pi}_v$ called *vertical ordering vector*, which is shown to be the optimal ordering vector $\bm{\pi}$ which minimizes $L (\bm{s},\bm{\pi}) $ for an arbitrary selection vector $\bm{s}$. \[Def:vertical ordering vector\] For a given selection vector $\bm{s} \in \mathcal{S}$, the corresponding vertical ordering vector $\bm{\pi}_v (\bm{s})$, or simply denoted as $\bm{\pi}_v$, is defined as the output of Algorithm \[Alg:vertical ordering\]. The vertical ordering vector $\bm{\pi}_v$ is illustrated in Fig. \[Fig:vertical ordering\], for a given selection vector $\bm{s}$ as an example. For $\bm{s}=[4,3,1]$, Algorithm 1 produces the corresponding vertical ordering vector $\bm{\pi}_v = [1,2,3,1,2,1,2,1]$. Note that the order of $k=8$ output nodes is illustrated in Fig. \[Fig:vertical ordering\], as the numbers inside each node. Although the vertical ordering vector $\bm{\pi}_v$ depends on the selection vector $\bm{s}$, we use simplified notation $\bm{\pi}_v$ instead of $\bm{\pi}_v(\bm{s})$. From Fig. \[Fig:vertical ordering\], obtaining $\bm{\pi}_v$ using Algorithm \[Alg:vertical ordering\] can be analyzed as follows. Moving from the leftmost column to the rightmost column, the algorithm selects one node per cluster. After selecting all $k$ nodes, $\pi_i$ stores the index of the cluster which contains the $i^{th}$ selected node. Now, the following Lemma shows that the vertical ordering vector $\bm{\pi}_v$ is optimal in the sense of minimizing $L (\bm{s},\bm{\pi}) $ for an arbitrary selection vector $\bm{s}$. **Input:** $\bm{s} = [s_1, \cdots, s_L]$ **Output:** $\bm{\pi}_v = [\pi_1, \cdots, \pi_k]$ Initialization: $l \leftarrow 1$ $l \leftarrow 1$ $\pi_i \leftarrow l$ (Store the index of cluster) $s_{\pi_i} \leftarrow s_{\pi_i} - 1$ (Update the remaining node info.) $l \leftarrow (l {\ \mathrm{mod}\ L}) + 1$   (Go to the next cluster) ![The vertical ordering vector $\bm{\pi}_v $ for the given selection vector $\bm{s}=[4,3,1]$ (for $n=15, k=8, L=3$ case) []{data-label="Fig:vertical ordering"}](vertical_ordering.pdf){height="25mm"} \[Lemma:optimal ordering vector\] Let $\bm{s} \in \mathcal{S}$ be an arbitrary selection vector. Then, a vertical ordering vector $\bm{\pi}_v$ minimizes $ L (\bm{s},\bm{\pi}) $. In other words, $ L (\bm{s},\bm{\pi}_v) \leq L (\bm{s},\bm{\pi}) $ holds for arbitrary $\bm{\pi} \in \Pi(\bm{s})$. In the case of $\beta_c = 0$, we show that $L (\bm{s},\bm{\pi}) $ is constant for every $\bm{\pi} \in \Pi(\bm{s})$. From (\[Eqn:lower\_bound\]), $$\label{Eqn:lower_bound_zero_betac} L(\bm{s}, \bm{\pi}) = \sum_{i=1}^k \min \{\alpha, a_i(\bm{\pi}) \beta_I\}$$ holds for $\beta_c = 0$. Using $I_l(\bm{\pi})$ in (\[Eqn:I\_l\]), note that $[k]$ can be partitioned into $L$ disjoint subsets as $[k] = \bigcup_{l=1}^L I_l(\bm{\pi}).$ Therefore, (\[Eqn:lower\_bound\_zero\_betac\]) can be written as $$L(\bm{s}, \bm{\pi}) = \sum_{l=1}^L \sum_{i \in I_l(\bm{\pi})} \min \{\alpha, a_i(\bm{\pi}) \beta_I\}.$$ Recall that $\bm{\pi} \in \Pi(\bm{s})$ contains $s_l$ components with value $l$ for $l \in [L]$. Thus, from (\[Eqn:a\_i\]), $$\bigcup_{i\in I_l (\bm{\pi})} \{a_i(\bm{\pi})\} = \{n_I - 1, n_I-2, \cdots, n_I - s_l\}$$ for $l \in [L]$. Therefore, $\sum_{i \in I_l(\bm{\pi})} \min \{\alpha, a_i(\bm{\pi}) \beta_I\}$ is constant $\forall \bm{\pi} \in \Pi(\bm{s})$ for arbitrary $l \in [L]$. In conclusion, $L (\bm{s},\bm{\pi})$ in (\[Eqn:lower\_bound\_zero\_betac\]) is constant irrespective of $\bm{\pi} \in \Pi(\bm{s})$ for $\beta_c = 0$. The rest of the proof deals with the $\beta_c \neq 0$ case. For a given arbitrary $\bm{s} \in \mathcal{S}$, define two subsets of $\Pi(\bm{s})$ as $$\begin{aligned} &\Pi_r = \{\bm{\pi^*} \in \Pi(\bm{s}) :\ S_t(\bm{\pi}) \leq S_t (\bm{\pi^*}) , \forall t \in [k], \ \forall \bm{\pi} \in \Pi(\bm{s}) \}\label{Eqn:running_sum_maximizer}\\ &\Pi_m = \{\bm{\pi^*} \in \Pi(\bm{s}) : L (\bm{s},\bm{\pi}) \geq L (\bm{s},\bm{\pi}^*), \forall \bm{\pi} \in \Pi(\bm{s}) \} \label{Eqn:min_cut_minimizer} \end{aligned}$$ where $S_t(\bm{\pi}) = \sum_{i=1}^t w_i (\bm{\pi})$ is the running sum of $w_i(\bm{\pi})$. Here, we call $\Pi_r$ the *running sum maximizer* and $\Pi_m$ the *min-cut minimizer*. Now the proof proceeds in two steps. The first step proves that the *running sum maximizer* minimizes min-cut, i.e., $\Pi_r \subseteq \Pi_m$. The second step proves that the vertical ordering vector is a running sum maximizer, i.e., $\bm{\pi}_v \in \Pi_r$. **Step 1.** Prove $\Pi_r \subseteq \Pi_m$: Define two index sets for a given ordering vector $\bm{\pi}$: $$\begin{aligned} \label{Eqn:Omega_L,s} \Omega_L (\bm{\pi}) &= \{ i \in [k] : w_i (\bm{\pi}) \geq \alpha \} \nonumber\\ \Omega_s (\bm{\pi}) &= \{ i \in [k] : w_i (\bm{\pi}) < \alpha \} \end{aligned}$$ Now define a set of ordering vectors as $$\label{Eqn:partitionable} \Pi_p = \{ \bm{\pi} \in \Pi(\bm{s}) : i \leq j \ \forall i \in \Omega_L (\bm{\pi}), \ \forall j \in \Omega_s (\bm{\pi}) \}.$$ The rest of the proof is divided into 2 sub-steps. **Step 1-1.** Prove $\Pi_m \subseteq \Pi_p$ and $ \Pi_r \subseteq \Pi_p$ by transposition: Consider arbitrary $\bm{\pi}=[\pi_1, \cdots, \pi_k] \in \Pi_p^c$. Use a short-hand notation $\omega_i$ to represent $\omega_i(\bm{\pi})$ for $i \in [k]$. From (\[Eqn:partitionable\]), there exists $i > j$ such that $i \in \Omega_L(\bm{\pi})$ and $j \in \Omega_s(\bm{\pi})$. Therefore, there exists $t \in [k-1]$ such that $t+1 \in \Omega_L(\bm{\pi})$ and $t \in \Omega_s(\bm{\pi})$ hold. By (\[Eqn:Omega\_L,s\]), $$\label{Eqn:omega_relative} \omega_{t+1} \geq \alpha > \omega_{t}$$ for some $t \in [k-1]$. Note that from (\[Eqn:weight vector\]), $\pi_t = \pi_{t+1}$ implies $\omega_{t+1} = \omega_t - \beta_I < \omega_t$. Therefore, $$\label{Eqn:pi_relative} \pi_t \neq \pi_{t+1}$$ should hold to satisfy (\[Eqn:omega\_relative\]). Define an ordering vector $\bm{\pi'} = [\pi'_1, \cdots, \pi'_k] $ as $$\begin{aligned} \label{Eqn:pi_and_pi_prime} \begin{cases} \pi'_i = \pi_i & i \neq t, t+1 \\ \pi'_t = \pi_{t+1} \\ \pi'_{t+1} = \pi_{t}. \end{cases} \end{aligned}$$ Use a short-hand notation $\omega'_i$ to represent $\omega_i(\bm{\pi}')$ for $i \in [k]$. Note that $\{\omega'_i\}$ satisfies $$\begin{aligned} \label{Eqn:omega_and_omega_prime} \begin{cases} \omega'_i = \omega_i & i \neq t, t+1 \\ \omega'_{t} = \omega_{t+1} + \beta_c \\ \omega'_{t+1} = \omega_{t} - \beta_c \end{cases} \end{aligned}$$ for the following reason. First, use simplified notations $a_i$ and $a'_i$ to mean $a_i(\bm{\pi})$ and $a_i(\bm{\pi}')$, respectively. Then, using (\[Eqn:a\_i\]), (\[Eqn:pi\_relative\]) and (\[Eqn:pi\_and\_pi\_prime\]), we have $$\begin{aligned} \label{Eqn:a_t_a_t+1} a_{t}' &= n_I - 1 - \sum_{j=1}^{t-1} \mathds{1}_{\pi_j' = \pi_{t}'} = n_I - 1 - \sum_{j=1}^{t-1} \mathds{1}_{\pi_j = \pi_{t+1}}\nonumber\\ &= n_I - 1 - \sum_{j=1}^{t} \mathds{1}_{\pi_j = \pi_{t+1}} = a_{t+1}. \end{aligned}$$ Similarly, $$\begin{aligned} \label{Eqn:a_t+1_a_t} a_{t+1}' &= n_I - 1 - \sum_{j=1}^{t} \mathds{1}_{\pi_j' = \pi_{t+1}'} = n_I - 1 - \sum_{j=1}^{t-1} \mathds{1}_{\pi_j' = \pi_{t+1}'}\nonumber\\ &= n_I - 1 - \sum_{j=1}^{t-1} \mathds{1}_{\pi_j = \pi_{t}} = a_{t}. \end{aligned}$$ Therefore, from (\[Eqn:omega\_i\]), (\[Eqn:a\_t\_a\_t+1\]) and (\[Eqn:a\_t+1\_a\_t\]), we have $$\begin{aligned} \omega'_t &= a'_t \beta_I + (n-t-a'_t)\beta_c \\ & = a_{t+1}\beta_i + ( n-t-a_{t+1} ) \beta_c = \omega_{t+1} + \beta_c. \end{aligned}$$ Similarly, $\omega'_{t+1} = \omega_{t} - \beta_c$ holds. This proves (\[Eqn:omega\_and\_omega\_prime\]). Thus, from (\[Eqn:omega\_relative\]) and (\[Eqn:omega\_and\_omega\_prime\]), $$L(\bm{s}, \bm{\pi}) = \sum_{i=1}^k \min\{\alpha, \omega_i\} = \sum_{i \notin \{t,t+1\} } \min\{\alpha, \omega_i\} + \omega_t + \alpha,$$ $$\begin{aligned} L(\bm{s}, \bm{\pi}') &= \sum_{i=1}^k \min\{\alpha, \omega_i'\} \nonumber\\ &= \sum_{i \notin \{t,t+1\} } \min\{\alpha, \omega_i'\} + \alpha + (\omega_t - \beta_c) \nonumber\\ &= \sum_{i \notin \{t,t+1\} } \min\{\alpha, \omega_i\} + \alpha + (\omega_t - \beta_c). \end{aligned}$$ Therefore, $L(\bm{s}, \bm{\pi}) > L(\bm{s}, \bm{\pi}')$ holds for $\beta_c > 0$. In other words, if $\pi \in \Pi_p^c$, then $\pi \in \Pi_m^c$. This proves that $\Pi_p^c \subseteq \Pi_m^c$ holds for $\beta_c \neq 0$. Similarly, $\Pi_p^c \subseteq \Pi_r^c$ can be proved as follows. For the pre-defined ordering vectors $\bm{\pi}$ and $\bm{\pi'}$, we have $S_t(\bm{\pi}) = \sum_{i=1}^{t-1}\omega_i + \omega_t$ and $S_t(\bm{\pi'}) = \sum_{i=1}^{t-1}\omega_i + \omega_{t+1} + \beta_c$. Using (\[Eqn:omega\_relative\]), we have $S_t(\bm{\pi}) < S_t(\bm{\pi'})$, so that $\bm{\pi}$ cannot be a running-sum maximizer. Therefore, $\Pi_p^c \subseteq \Pi_r^c$ holds. **Step 1-2.** Prove that $L(\bm{s, \pi^*}) \leq L(\bm{s, \pi})$, $\forall \bm{\pi}^* \in \Pi_r, \forall \bm{\pi} \in \Pi_p \cap \Pi_r^c$ : Consider arbitrary $\bm{\pi}^* \in \Pi_r$ and $\bm{\pi} \in \Pi_p \cap \Pi_r^c$. For $i \in [k]$, let $\omega_i^*$ and $\omega_i$ be short-hand notations for $\omega_i(\bm{\pi}^*)$ and $\omega_i(\bm{\pi})$, respectively. Note that from Proposition \[Prop:weight vector\], $$\label{Eqn:omega_invariant} \sum_{i=1}^{k} \omega_i = \sum_{i=1}^{k} \omega_i^*.$$ Let $$\begin{aligned} \label{Eqn:t_and_t^*} t & = \max \{i \in [k] : w_i \geq \alpha \}, \nonumber\\ t^* & = \max \{i \in [k] : w_i^* \geq \alpha \} \end{aligned}$$ Then, from (\[Eqn:running\_sum\_maximizer\]), $$\label{Eqn:omega_running_sum_compare} \sum_{i=1}^{t} \omega_i \leq \sum_{i=1}^{t} \omega_i^*.$$ Combining with (\[Eqn:omega\_invariant\]), we obtain $$\label{Eqn:omega_compare_result} \sum_{i=t+1}^k (\omega_i - \omega_i^*) \geq 0.$$ Note that from the result of Step 1-1, both $\bm{\pi}$ and $\bm{\pi}^*$ are in $\Pi_p$. Therefore, $\omega_i \geq \alpha$ for $i \in [t]$. Similarly, $\omega_i^* \geq \alpha$ for $i \in [t^*]$. Therefore, (\[Eqn:lower\_bound\]) can be expressed as $$\begin{aligned} L(\bm{s}, \bm{\pi}) &= \sum_{i=1}^k \min\{\alpha, \omega_i\} = \sum_{i=1}^t \alpha + \sum_{i=t+1}^k \omega_i, \label{Eqn:lower_bound_pi}\\ L(\bm{s}, \bm{\pi}^*) &= \sum_{i=1}^k \min\{\alpha, \omega_i^*\} = \sum_{i=1}^{t^*} \alpha + \sum_{i=t^*+1}^k \omega_i^*. \label{Eqn:lower_bound_pi^*} \end{aligned}$$ If $t = t^*$, then we have $$L(\bm{s, \pi}) - L(\bm{s, \pi^*}) = \sum_{i=t+1}^{k} (\omega_i - \omega_i^*) \geq 0$$ from (\[Eqn:omega\_compare\_result\]). If $t > t^*$, we get $$L(\bm{s, \pi}) - L(\bm{s, \pi^*}) = \sum_{i=t^* + 1}^t (\alpha - \omega_i^*) + \sum_{i=t + 1}^k (\omega_i - \omega_i^*).$$ From (\[Eqn:t\_and\_t\^\*\]), $\omega_i^* < \alpha$ holds for $i > t^*$. Therefore, $$L(\bm{s, \pi}) - L(\bm{s, \pi^*}) > \sum_{i=t + 1}^k (\omega_i - \omega_i^*) \geq 0$$ from (\[Eqn:omega\_compare\_result\]). In the case of $t < t^*$, define $\Delta = \sum_{i=1}^t (\omega_i - \alpha)$ and $\Delta^* = \sum_{i=1}^t (\omega_i^* - \alpha)$. From (\[Eqn:lower\_bound\_pi\]), $$L(\bm{s, \pi}) = \sum_{i=1}^t \alpha + \sum_{i=t+1}^k \omega_i = ( \sum_{i=1}^k \omega_i ) - \Delta.$$ Similarly, from (\[Eqn:lower\_bound\_pi\^\*\]), $$\begin{aligned} L(\bm{s, \pi^*}) &= \sum_{i=1}^k \omega_i^* - \sum_{i=1}^{t^*} (\omega_i^* - \alpha)\\ &= \sum_{i=1}^k \omega_i^* - \Delta^* - \sum_{i=t+1}^{t^*} (\omega_i^* - \alpha) \leq \sum_{i=1}^k \omega_i^* - \Delta^* \end{aligned}$$ where the last inequality is from (\[Eqn:t\_and\_t\^\*\]). Combined with (\[Eqn:omega\_invariant\]) and (\[Eqn:omega\_running\_sum\_compare\]), we obtain $$L(\bm{s, \pi}) - L(\bm{s, \pi^*}) \geq \Delta^* - \Delta \geq 0.$$ In summary, $L(\bm{s, \pi^*}) \leq L(\bm{s, \pi})$ irrespective of the $t,t^*$ values, which completes the proof for Step 1-2. From the results of Step 1-1 and Step 1-2, the relationship between the sets can be depicted as in Fig. \[Fig:set\_relationship\]. ![Relationship between sets[]{data-label="Fig:set_relationship"}](set_relationship_ver2.pdf){height="30mm"} Consider $\bm{\pi}_0 \in \Pi_r$ and $\bm{\pi}^* \in \Pi_m \cap \Pi_r^c$. Then, $L(\bm{s}, \bm{\pi}_0) \leq L(\bm{s}, \bm{\pi}^*)$ from the result of Step 1-2. Based on the definition of $\Pi_m$ in (\[Eqn:min\_cut\_minimizer\]), we can write $ L(\bm{s}, \bm{\pi_0}) \leq L(\bm{s}, \bm{\pi})$ for every $\bm{\pi} \in \Pi(\bm{s}).$ In other words, $ \pi_0 \in \Pi_m$ holds for arbitrary $\pi_0 \in \Pi_r$. Therefore, $\Pi_r \subseteq \Pi_m$ holds. **Step 2.** Prove $\bm{\pi}_v \in \Pi_r$: ![Set of $s_1$ lines where $(i,\omega_i)$ points can position[]{data-label="Fig:omega_I_line"}](omega_I_line.pdf){height="40mm"} For a given selection vector $\bm{s}=[s_1, \cdots, s_L]$, consider an arbitrary ordering vector $\bm{\pi} = [\pi_1, \cdots, \pi_k] \in \Pi(\bm{s})$. The corresponding $\omega_i(\bm{\pi})$ defined in (\[Eqn:weight vector\]) is written as $$\label{Eqn:omega_i_notation_j} w_i = (n_I - j) \beta_I + (n-i-n_I + j)\beta_c$$ where $j = \sum_{t=1}^{i} \mathds{1}_{\pi_t = \pi_i} $. Consider a set of lines $\{l_j\}_{j=1}^{n_I}$, where line $l_j$ represents an equation: $w_i = (n_I - j) \beta_I + (n-i-n_I + j)\beta_c$. Since we assume $\beta_I \geq \beta_c$, these lines can be illustrated as in Fig. \[Fig:omega\_I\_line\]. For a given $\bm{\pi}$, consider marking a $(i,\omega_i)$ point for $i \in [k]$. Note that the $(i, \omega_i)$ point is on line $l_j$ if and only if $$\label{Eqn:l_j_condition} j = \sum_{t=1}^{i} \mathds{1}_{\pi_t = \pi_i}$$ where the summation term in (\[Eqn:l\_j\_condition\]) represents the number of occurrence of $\pi_i$ value in $\{\pi_t\}_{t=1}^{i}$. For the example in Fig. \[Fig:vertical ordering\], when $\bm{s} = [4, 3, 1]$ and $\bm{\pi} = [1, 2, 3, 1, 2, 1, 2, 1] \in \Pi(\bm{s})$, line $l_3$ contains the point $(i,\omega_i)=(6,\omega_6)$ since $3 = \sum_{t=1}^{6} \mathds{1}_{\pi_t = \pi_6}$. Recall that $ I_l (\bm{\pi}) = \{ i \in [k] : \pi_i = l \}$, as defined in (\[Eqn:I\_l\]), where $|I_l (\bm{\pi})| = s_l$ holds $\forall l \in [L]$. For $j \in [n_I]$, consider $l \in [L]$ with $s_l \geq j$. Let $i_0$ be the $j^{th}$ smallest element in $I_l (\bm{\pi})$. Then, $ j = \sum_{t=1}^{i_0} \mathds{1}_{\pi_t = l}$ and $\pi_{i_0} = l$ hold. Thus, the $(i_0, \omega_{i_0})$ point is on line $l_j$. Similarly, we can find $$\label{Eqn:point vector components} p_j = \sum_{l=1}^{L} \mathds{1}_{s_l \geq j}$$ points on line $l_j$, irrespective of the ordering vector $\bm{\pi} \in \Pi(\bm{s})$. Note that $$\label{Eqn:point vector property1} \sum_{j=1}^{n_I} p_j = \sum_{l=1}^{L} \sum_{j=1}^{n_I} \mathds{1}_{s_l \geq j} = \sum_{l=1}^{L} s_l = k,$$ which confirms that Fig. \[Fig:omega\_I\_line\] contains $k$ points. Moreover, $$\label{Eqn:point vector property2} \forall j \in [n_I-1], \ p_j \geq p_{j+1}$$ holds from the definition in (\[Eqn:point vector components\]). ![Optimal packing of $k$ points[]{data-label="Fig:omega_I_line_packing"}](omega_I_line_packing.pdf){height="42mm"} In order to maximize the running sum $S_t(\bm{\pi}) = \sum_{i=1}^{t} w_i(\bm{\pi})$ for every $t$, the optimal ordering vector packs $p_1$ points in the leftmost area ($i=1, \cdots, p_1$), pack $p_2$ points in the leftmost remaining area ($i=p_1 + 1, \cdots, p_1 + p_2$), and so on. This packing method corresponds to Fig. \[Fig:omega\_I\_line\_packing\]. Note that from the definition of $p_j$ in (\[Eqn:point vector components\]) and Fig. \[Fig:vertical ordering\], vertical ordering $\bm{\pi}_v$ in Definition \[Def:vertical ordering vector\] first chooses $p_1$ points on line $l_1$, then chooses $p_2$ points on line $l_2$, and so on. Thus, $\bm{\pi}_v$ achieves optimal packing in Fig. \[Fig:omega\_I\_line\_packing\], which maximizes the running sum $S_t(\bm{\pi})$. Therefore, vertical ordering is a running sum maximizer, i.e., $\bm{\pi}_v \in \Pi_r$. Combining Steps 1 and 2, we conclude that $\bm{\pi}_v$ minimizes $L (\bm{s},\bm{\pi})$ among $\bm{\pi} \in \Pi(\bm{s})$ for arbitrary $\bm{s} \in S$. Now, we define a special selection vector called the *horizontal selection vector*, which is shown to be the optimal selection vector which minimizes $L (\bm{s},\bm{\pi}_v)$. \[Def:horizontal selection vector\] The horizontal selection vector $\bm{s}_h = [s_1, \cdots, s_L] \in \mathcal{S}$ is defined as: $$s_i = \begin{cases} n_I, & i \leq \floor{\frac{k}{n_I}} \nonumber\\ (k {\ \mathrm{mod}\ n_I}), & i = \floor{\frac{k}{n_I}} + 1 \nonumber\\ 0 & i > \floor{\frac{k}{n_I}} + 1. \end{cases}$$ ![The optimal selection vector $\bm{s}_h$ and the optimal ordering vector $\bm{\pi}_v $ (for $n=15, k=8, L=3$ case)[]{data-label="Fig:horizontal selection"}](horizontal_selection.pdf){height="25mm"} The graphical illustration of the horizontal selection vector is on the left side of Fig. \[Fig:horizontal selection\], in the case of $n=15, k=8, L=3$. The following Lemma states that the horizontal selection vector minimizes $L (\bm{s},\bm{\pi}_v)$. \[Lemma:optimal selection vector\] Consider applying the vertical ordering vector $\bm{\pi}_v$. Then, the horizontal selection vector $\bm{s}_h$ minimizes the lower bound $L (\bm{s},\bm{\pi})$ on the min-cut. In other words, $L(\bm{s}_h, \bm{\pi}_v) \leq L(\bm{s}, \bm{\pi}_v ) \ \forall \bm{s}\in \mathcal{S}$. From the proof of Lemma \[Lemma:optimal ordering vector\], the optimal ordering vector turns out to be the vertical ordering vector, where the corresponding $\omega_i(\bm{\pi}_v)$ sequence is illustrated in Fig. \[Fig:omega\_I\_line\_packing\]. Depending on the selection vector $\bm{s} = [s_1, \cdots, s_L]$, the number $p_j$ of points on each line $l_j$ changes. Consider an arbitrary selection vector $\bm{s}$. Define a point vector $\bm{p(s)}=[p_1,\cdots, p_{n_I}]$ where $p_j$ is the number of points on $l_j$, as defined in (\[Eqn:point vector components\]). Similarly, define $\bm{p(s_h)}=[p_1^*,\cdots, p_{n_I}^*]$. Using Definition \[Def:horizontal selection vector\] and (\[Eqn:point vector components\]), we have $$\label{Eqn:point vector of horizontal selection vector} p_j^* = \begin{cases} \floor{\frac{k}{n_I}} + 1, & \ j \leq (k {\ \mathrm{mod}\ n_I})\\ \floor{\frac{k}{n_I}} & \ otherwise \end{cases}$$ Now, we prove $$\label{Eqn:Lemma_2_proof_1} \forall t \in [n_I], \ \sum_{j=1}^t p_j^* \leq \sum_{j=1}^t p_j.$$ The proof is divided into two steps: *base case* and *inductive step*. **Base Case:** We wish to prove that $p_1^* \leq p_1$. Suppose $p_1^* > p_1$ (i.e., $p_1 \leq p_1^* - 1$). Then, $$\label{Eqn:base_case_eqn1} \sum_{l=1}^{n_I}p_l \leq \sum_{l=1}^{n_I}p_1 \leq \sum_{l=1}^{n_I} (p_1^* - 1)$$ where the first inequality is from (\[Eqn:point vector property2\]). Note that if $(k {\ \mathrm{mod}\ n_I}) = 0$, then $$\label{Eqn:base_case_eqn2} \sum_{l=1}^{n_I} p_l^* = \sum_{l=1}^{n_I} p_1^* > \sum_{l=1}^{n_I} (p_1^* - 1).$$ Otherwise, $$\begin{aligned} \sum_{l=1}^{n_I} p_l^* &= \sum_{l=1}^{(k {\ \mathrm{mod}\ n_I})} p_l^* + \sum_{l=(k {\ \mathrm{mod}\ n_I}) + 1}^{n_I} p_l^* \nonumber\\ &= \sum_{l=1}^{(k {\ \mathrm{mod}\ n_I})} (\floor{\frac{k}{n_I}} + 1) + \sum_{l=(k {\ \mathrm{mod}\ n_I}) + 1}^{n_I} \floor{\frac{k}{n_I}} \nonumber\\ &> \sum_{l=1}^{n_I} \floor{\frac{k}{n_I}} = \sum_{l=1}^{n_I} (p_1^* - 1).\label{Eqn:base_case_eqn3} \end{aligned}$$ Therefore, combining (\[Eqn:base\_case\_eqn1\]), (\[Eqn:base\_case\_eqn2\]) and (\[Eqn:base\_case\_eqn3\]) results in $\sum_{l=1}^{n_I}p_l < \sum_{l=1}^{n_I} p_l^* = k$, which contradicts (\[Eqn:point vector property1\]). Therefore, $p_1^* \leq p_1$ holds. **Inductive Step:** Assume that $\sum_{l=1}^{l_0} p_l^* \leq \sum_{l=1}^{l_0} p_l$ for arbitrary $l_o \in [n_I-1]$. Now we prove that $\sum_{l=1}^{l_0+1} p_l^* \leq \sum_{l=1}^{l_0+1} p_l$ holds. Suppose not. Then, $$\label{Eqn:point_vector_inequality} p_{l_0 + 1}^* - \Theta - 1 \geq p_{l_0 + 1}$$ holds where $$\label{Eqn:inductive_step_eqn2} \Theta = \sum_{l=1}^{l_0} (p_l - p_l^*).$$ Using (\[Eqn:point vector property2\]) and (\[Eqn:point\_vector\_inequality\]), we have $$\begin{aligned} \sum_{l=l_0 + 1}^{n_I}p_l &\leq \sum_{l=l_0 + 1}^{n_I}p_{l_0 + 1} \leq \sum_{l=l_0 + 1}^{n_I} (p_{l_0 + 1}^* - 1 - \Theta )\nonumber\\ &\leq \sum_{l=l_0 + 1}^{n_I} (p_{l_0 + 1}^* - 1) - \Theta \label{Eqn:inductive_step_eqn1} \end{aligned}$$ where equality holds for the last inequality iff $l_0 = n_I - 1$. Using analysis similar to (\[Eqn:base\_case\_eqn2\]) and (\[Eqn:base\_case\_eqn3\]) for the *base case*, we can find that $ \sum_{l=l_0 + 1}^{n_I} (p_{l_0 + 1}^* - 1) < \sum_{l=l_0 + 1}^{n_I} p_l^*.$ Combining with (\[Eqn:inductive\_step\_eqn1\]), we get $$\label{Eqn:inductive_step_eqn3} \sum_{l=l_0+1}^{n_I} p_l < \sum_{l=l_0 + 1}^{n_I} p_l^* - \Theta.$$ Equations (\[Eqn:inductive\_step\_eqn2\]) and (\[Eqn:inductive\_step\_eqn3\]) imply $ \sum_{l=1}^{n_I}p_l < \sum_{l=1}^{n_I} p_l^* = k,$ which contradicts (\[Eqn:point vector property1\]). Therefore, (\[Eqn:Lemma\_2\_proof\_1\]) holds. Now define $$\begin{aligned} f_i &= \min \{s \in [n_I]: \sum_{l=1}^s p_l \geq i \} \label{Eqn:t_i}\\ h_i &= \min \{s \in [n_I]: \sum_{l=1}^s p_l^* \geq i \} \label{Eqn:t_i^*} \end{aligned}$$ for $i\in[k]$. Then, $$\label{Eqn:Lemma_2_proof_2} \forall i \in [k], \ h_i \geq f_i$$ holds directly from (\[Eqn:Lemma\_2\_proof\_1\]). Note that since $p_i^*$ in (\[Eqn:point vector of horizontal selection vector\]) is identical to $g_i$ in (\[Eqn:g\_m\]), $h_i$ can be written as $$h_i = \min \{s \in [n_I]: \sum_{l=1}^s g_l \geq i \} \label{Eqn:t_i^*_reform}$$ Consider $\bm{\pi}_v(\bm{s})$, the vertical ordering vector for a given selection vector $\bm{s}$. Recall that as in Fig. \[Fig:omega\_I\_line\_packing\], vertical ordering packs the leftmost $p_1$ points on line $l_1$, packs the next $p_2$ points on line $l_2$, and so on. Using (\[Eqn:omega\_i\_notation\_j\]), we can write $$\begin{aligned} \omega_i = \begin{cases} (n_I-1)\beta_I + (n-i-n_I+1)\beta_c, \quad & \text{if } i \in [p_1] \\ (n_I-2)\beta_I + (n-i-n_I+2)\beta_c, \quad & \text{if } i - p_1 \in [p_2] \\ \quad \quad \vdots \\ 0 \cdot \beta_I + (n-i)\beta_c, & \text{if } i - \sum_{t=1}^{n_I-1}p_t \\ & \quad \quad \in [p_{n_I}] \end{cases} \end{aligned}$$ Therefore, using (\[Eqn:t\_i\]), we further write $$\label{Eqn:Lemma_2_proof_3} \omega_i(\bm{\pi}_v(\bm{s})) = (n_I - f_i) \beta_I + (n-i-n_I + f_i) \beta_c.$$ Similarly, $$\label{Eqn:Lemma_2_proof_4} \omega_i(\bm{\pi}_v(\bm{s}_h)) = (n_I - h_i) \beta_I + (n-i-n_I + h_i) \beta_c.$$ Combining (\[Eqn:Lemma\_2\_proof\_2\]), (\[Eqn:Lemma\_2\_proof\_3\]) and (\[Eqn:Lemma\_2\_proof\_4\]), we have $$\omega_i(\bm{\pi}_v(\bm{s}_h)) \leq \omega_i(\bm{\pi}_v(\bm{s})) \ \forall i=1,\cdots,k, \forall \bm{s} \in \mathcal{S},$$ since we assume $\beta_I \geq \beta_c$. Therefore, combining with (\[Eqn:lower\_bound\]), we conclude that $L(\bm{s}_h, \bm{\pi}_v) \leq L(\bm{s}, \bm{\pi}_v)$ for arbitrary $\bm{s} \in \mathcal{S}$, which completes the proof of Lemma \[Lemma:optimal selection vector\]. From Lemmas \[Lemma:optimal ordering vector\] and \[Lemma:optimal selection vector\], we have $$\forall \bm{s} \in \mathcal{S}, \forall \bm{\pi} \in \Pi(\bm{s}), \ L(\bm{s}_h, \bm{\pi}_v) \leq L(\bm{s}, \bm{\pi}).$$ All that remains is to compute $L(\bm{s}_h, \bm{\pi}_v)$ and check that it is identical to (\[Eqn:Capacity\_value\]). From (\[Eqn:lower\_bound\]), $L(\bm{s}_h, \bm{\pi}_v)$ can be written as $$\label{Eqn:lower_bound_optimal} L(\bm{s}_h, \bm{\pi}_v) = \sum_{i=1}^k \min \{\alpha, \omega_i(\bm{\pi}_v(\bm{s}_h) )\}$$ where $\omega_i(\bm{\pi}_v(\bm{s}_h) )$ is defined in (\[Eqn:Lemma\_2\_proof\_4\]). From $h_i$ in (\[Eqn:t\_i\^\*\_reform\]), we have $$\label{Eqn:t_i^*_cases} h_i = \begin{cases} 1, & i \in [g_1] \\ 2, & i - g_1 \in [g_2] \\ & \vdots \\ n_I, & i - \sum_{t=1}^{n_I-1}g_t \in [g_{n_I}] \end{cases}$$ If we define $$I_m^* = \{i \in [k] : h_i = m \},$$ then $L(\bm{s}_h, \bm{\pi}_v)$ in (\[Eqn:lower\_bound\_optimal\]) can be expressed as $$\begin{aligned} \label{Eqn:lower_bound_intermediate} & L(\bm{s}_h, \bm{\pi}_v) = \sum_{i=1}^k \min\{\alpha, (n_I-h_i)\beta_I + (n-n_I-i+h_i)\beta_c\} \nonumber\\ &= \sum_{m=1}^{n_I} \sum_{i \in I_m^*} \min \{\alpha, (n_I-m)\beta_I + (n-n_I-i+m)\beta_c\}\nonumber\\ &= \sum_{m=1}^{n_I} \sum_{i \in I_m^*} \min \{\alpha, \rho_m\beta_I + (n-\rho_m-i)\beta_c\}\end{aligned}$$ where $\rho_m$ is defined in (\[Eqn:rho\_i\]). Using (\[Eqn:t\_i\^\*\_cases\]), we have $$I_m^* = \{\sum_{t=1}^{m-1}g_t + 1, \cdots, \sum_{t=1}^{m-1}g_t + g_m\}.$$ for $m \in [n_I]$. Therefore, $i \in I_m^*$ can be represented as $$i = \sum_{t=1}^{m-1} g_t + l$$ for $l \in [g_m]$. Using this notation, (\[Eqn:lower\_bound\_intermediate\]) can be written as $$\begin{aligned} \label{Eqn:capacity_final} L(\bm{s}_h, \bm{\pi}_v) = \sum_{m=1}^{n_I} \sum_{l=1}^{g_m} \min \{\alpha, \rho_m\beta_I + (n-\rho_m - s_m^{(l)})\beta_c\}\end{aligned}$$ where $s_m^{(l)} = \sum_{t=1}^{m-1}g_t + l$. This expression reduces to (\[Eqn:Capacity\_value\]). This completes the proof of Part II-2. Therefore, the storage capacity of clustered DSS is as stated in Theorem \[Thm:Capacity of clustered DSS\]. Proof of Theorem \[Thm:condition\_for\_min\_storage\] {#Section:proof_of_thm_condition_for_min_storage} ===================================================== We begin with introducing properties of the parameters $z_t$ and $h_t$ defined in (\[Eqn:z\_t\]) and (\[Eqn:h\_t\]). $$\begin{aligned} h_k &= n_I, \label{Eqn:h_k} \\ z_k &= (n-k)\epsilon. \label{Eqn:z_k} \end{aligned}$$ Since we consider $k > n_I$ case as stated in (\[Eqn:k\_constraint\]), we have $$g_i \geq 1 \quad \forall i \in [n_I]$$ for $\{g_i\}_{i=1}^{n_I}$ defined in (\[Eqn:g\_m\]). Combining with (\[Eqn:sum of g is k\]) and (\[Eqn:h\_t\]), we can conclude that $h_k = n_I$. Finally, $z_k = (n-k)\epsilon$ is from (\[Eqn:z\_t\]) and (\[Eqn:h\_k\]). First, consider the $\epsilon \geq \frac{1}{n-k}$ case. From (\[Eqn:Feasible Points Result\]), data $\mathcal{M}$ can be reliably stored with node storage $\alpha = \mathcal{M}/k$ if the repair bandwidth satisfies $\gamma \geq \gamma^*$, where $$\begin{aligned} \gamma^* &= \frac{\mathcal{M} - (k-1)\mathcal{M}/k}{s_{k-1}} = \frac{\mathcal{M}}{k} \frac{1}{s_{k-1}}\nonumber\\ &= \frac{\mathcal{M}}{k} \frac{(n-k)\epsilon}{(n_I-1) + (n-n_I)\epsilon}\end{aligned}$$ where the last equality is from (\[Eqn:s\_t\]) and (\[Eqn:z\_k\]). Thus, $\alpha = \mathcal{M}/k$ is achievable with finite $\gamma$, when $\epsilon \geq \frac{1}{n-k}$. Second, we prove that it is impossible to achieve $\alpha = \mathcal{M}/k$ for $0 \leq \epsilon < \frac{1}{n-k}$, in order to reliably store file $\mathcal{M}$. Recall that the minimum storage for $0 \leq \epsilon < \frac{1}{n-k}$ is $$\label{Eqn:alpha_MSR} \alpha = \frac{M}{\tau + \sum_{i=\tau+1}^{k}z_i}$$ from . From (\[Eqn:tau\]) and (\[Eqn:z\_k\]), we have $z_i < 1$ for $i=\tau+1, \tau+2, \cdots, k$. Therefore, $$\tau + \sum_{i=\tau+1}^{k}z_i < \tau + (k-(\tau+1)+1) = k$$ holds, which result in $$\frac{M}{\tau + \sum_{i=\tau+1}^{k}z_i} > \frac{\mathcal{M}}{k}.$$ Thus, the $0 \leq \epsilon \leq \frac{1}{n-k}$ case has the minimum storage $\alpha$ greater than $\mathcal{M}/k$, which completes the proof of Theorem \[Thm:condition\_for\_min\_storage\]. Proof of Theorem \[Thm:MSR\_MBR\] {#Proof_of_Thm_MSR_MBR} ================================= First, we prove (\[Eqn:msr\_property\]). To begin with, we obtain the expression of $\alpha_{msr}^{(\epsilon)}$, for $\epsilon = 0, 1$. From (\[Eqn:MSR\_point\]), we obtain $$\begin{aligned} \alpha_{msr}^{(0)} &= \frac{\mathcal{M}}{\tau + \sum_{i=\tau+1}^{k}z_i}, \label{Eqn:msr_epsilon_0_alpha} \\ \alpha_{msr}^{(1)} &= \frac{\mathcal{M}}{k} \nonumber.\end{aligned}$$ We further simplify the expression for $\alpha_{msr}^{(0)}$ as follows. Recall $$\label{Eqn:z_t_epsilon0} z_t = n_I - h_t$$ for $t \in [k]$ from , when $\epsilon=0$ holds. Note that we have $$\label{Eqn:z_t_epsilon0_cases} z_t = \begin{cases} 0, & t \geq k- \floor{\frac{k}{n_I}} + 1 \\ 1, & t = k- \floor{\frac{k}{n_I}} \end{cases}$$ from the following reason. First, from and , $z_t = 0$ holds for $$\begin{aligned} t &\geq \sum_{l=1}^{n_I-1} g_l + 1 = \sum_{l=1}^{n_I} g_l - g_{n_I} + 1 = k - \floor{\frac{k}{n_I}} + 1\end{aligned}$$ where the last equality is from and . Similarly, we can prove that $z_t = 1$ holds for $t = k - \floor{\frac{k}{n_I}}$. From and , we obtain $$\label{Eqn:tau_epsilon0} \tau = k - \floor{\frac{k}{n_I}}$$ when $\epsilon=0$. Combining , and , we have $$\alpha_{msr}^{(0)} = \frac{\mathcal{M}}{k - \floor{\frac{k}{n_I}}}.$$ Then, using $R=k/n$ and $\sigma = L^2/n$, $$\begin{aligned} \frac{\alpha_{msr}^{(0)}}{\alpha_{msr}^{(1)}} &= \frac{k}{k-\floor{\frac{k}{n_I}}} = \frac{nR}{nR - \floor{RL}} \nonumber\\ &= \frac{nR}{nR - \floor{R\sqrt{n\sigma}}} = \frac{R}{R - \floor{R\sqrt{n\sigma}}/n}.\end{aligned}$$ Thus, for arbitrary fixed $R$ and $\sigma$, $$\lim\limits_{n \rightarrow \infty} \frac{\alpha_{msr}^{(0)}}{\alpha_{msr}^{(1)}} = 1.$$ Therefore, $\alpha_{msr}^{(0)}$ is asymptotically equivalent to $\alpha_{msr}^{(1)}$. Second, we prove (\[Eqn:mbr\_property\]). Note that from (\[Eqn:MBR\_point\]), $\alpha_{mbr}^{(\epsilon)} = \gamma_{mbr}^{(\epsilon)}$ holds for arbitrary $0 \leq \epsilon \leq 1$. Therefore, all we need to prove is $$\label{Eqn:gamma_asymptote} \gamma_{mbr}^{(0)} \rightarrow \gamma_{mbr}^{(1)}.$$ To begin with, we obtain the expression for $\gamma_{mbr}^{(\epsilon)}$, when $\epsilon = 0, 1$. For $\epsilon = 1$, $z_t$ in (\[Eqn:z\_t\]) is $$\label{Eqn:z_t_kappa_1} z_t = n-t$$ for $t \in [k]$. Moreover, from (\[Eqn:s\_t\]), $$\label{Eqn:s_0} s_0 = \frac{\sum_{i=1}^{k} z_i}{n-1}$$ for $\epsilon = 1$. Therefore, from (\[Eqn:MBR\_point\]), (\[Eqn:z\_t\_kappa\_1\]) and (\[Eqn:s\_0\]), $$\begin{aligned} \gamma_{mbr}^{(1)}=\frac{\mathcal{M}}{s_0} &= \frac{(n-1)\mathcal{M}}{\sum_{i=1}^{k} (n-i)} = \frac{\mathcal{M}}{k} \frac{2(n-1)}{2n-k-1}. \label{Eqn:gamma_mbr_1}\end{aligned}$$ Now we focus on the case of $\epsilon=0$. First, let $q$ and $r$ be $$\begin{aligned} q &\coloneqq \floor{\frac{k}{n_I}}, \label{Eqn:quotient}\\ r &\coloneqq (k {\ \mathrm{mod}\ n_I}) , \label{Eqn:remainder}\end{aligned}$$ which represent the quotient and remainder of $k/n_I$. Note that $$\label{Eqn:quotient_remainder_relation} qn_I + r = k.$$ Then, we have $$\begin{aligned} \label{Eqn:sum of weighted g} \sum_{t=1}^{n_I} t g_t &= \sum_{t=1}^{r} (q+1) t + \sum_{t=r+1}^{n_I} qt = q \sum_{t=1}^{n_I} t + \sum_{t=1}^{r} t \nonumber\\ &= q \frac{n_I (n_I+1)}{2} + \frac{r (r+1)}{2} = \frac{1}{2}(qn_I^2 + r^2 + k)\end{aligned}$$ where the last equality is from . From and , we have $$\begin{aligned} \label{Eqn:sum_of_z} \sum_{i=1}^k z_i &= \sum_{t=1}^{n_I} (n_I-t) g_t = n_I k - \frac{1}{2} (qn_I^2 + r^2 + k) \nonumber\\ &= (n_I-1) k - \frac{1}{2} (qn_I^2 + r^2 - k)\end{aligned}$$ where the second last equality is from and . Moreover, using , we have $$\begin{aligned} \label{Eqn:bound} qn_I^2 + r^2 - k &= qn_I^2 + r^2 - qn_I - r \nonumber\\ &= (qn_I + r) (n_I-1) - r(n_I-r) \nonumber\\ &\leq (qn_I+r)(n_I-1) = k(n_I-1)\end{aligned}$$ where the equality holds if and only if $r=0$. Furthermore, $$\label{Eqn:s_0_0} s_0 = \frac{\sum_{i=1}^{k} z_i}{n_I-1}$$ for $\epsilon = 0$ from (\[Eqn:s\_t\]). Combining , , and result in $$\begin{aligned} \gamma_{mbr}^{(0)}=\frac{\mathcal{M}}{s_0} &\leq \frac{2\mathcal{M}}{k}. \label{Eqn:gamma_mbr_0}\end{aligned}$$ From (\[Eqn:gamma\_mbr\_1\]) and (\[Eqn:gamma\_mbr\_0\]), we have $$\begin{aligned} \label{Eqn:gamma_ratio} \gamma_{mbr}^{(0)} - \gamma_{mbr}^{(1)} & \leq \frac{\mathcal{M}}{k} \left( 2 - \frac{2(n-1)}{2n-k-1} \right) = \frac{\mathcal{M}}{k} \frac{2(n-k)}{2n-k-1} \nonumber\\ &= \frac{\mathcal{M}}{nR} \frac{2n(1-R)}{n(2-R)-1}\end{aligned}$$ where $R=k/n$. Thus, for arbitrary $n$, $$\gamma_{mbr}^{(0)} \rightarrow \gamma_{mbr}^{(1)}$$ as $R \rightarrow 1$. This completes the proof of (\[Eqn:mbr\_property\]). Finally, (\[Eqn:gamma\_mbr\_1\]) and (\[Eqn:gamma\_mbr\_0\]) provides $$\begin{aligned} \frac{\gamma_{mbr}^{(0)}}{\gamma_{mbr}^{(1)}} &\leq \frac{2n-k-1}{n-1} = 2- \frac{k-1}{n-1}, \end{aligned}$$ which completes the proof of (\[Eqn:mbr\_ratio\]). Proof of Theorem \[Thm:IRC\_versus\_LRC\] {#Section:Proof_of_IRC_versus_LRC} ========================================= Moreover, recall that the *repair locality* of a code is defined as the number of nodes to be contacted in the node repair process [@papailiopoulos2014locally]. Since each cluster contains $n_I$ nodes, every node in a DSS with $\epsilon = 0$ has the repair locality of $$\label{Eqn:locality_val} l_0 = n_I - 1.$$ Moreover, note that for any code with minimum distance $m$, the original file $\mathcal{M}$ can be retrieved by contacting $n-m+1$ coded symbols [@moon2005error]. Since the present paper considers DSSs such that contacting any $k$ nodes can retrieve the original file $\mathcal{M}$, we have the minimum distance of $$\label{Eqn:min_dist_val} m_0 = n-k+1.$$ Thus, the intra-cluster repairable code defined in Section \[Section:IRC\_versus\_LRC\] is a $(n,l_0,m_0,\mathcal{M},\alpha)$-LRC. Now we show that (\[Eqn:locality\_ineq\]) holds. Note that from Fig. \[Fig:tradeoff\_min\], we obtain $\alpha \geq \alpha_{msr}^{(0)}$ for $\epsilon=0$, where $$\label{Eqn:alpha_val} \alpha_{msr}^{(0)} = \frac{\mathcal{M}}{k-\floor{\frac{k}{n_I}}}$$ holds according to (\[Eqn:msr\_epsilon\_0\_alpha\]). Thus, (\[Eqn:locality\_ineq\]) is proven by showing $$\label{Eqn:local_repair_code_ineq} m_0 \leq n - \ceil[\bigg]{\frac{\mathcal{M}}{\alpha_{msr}^{(0)}}} - \ceil[\bigg]{\frac{\mathcal{M}}{l_0\alpha_{msr}^{(0)}}} + 2.$$ By plugging (\[Eqn:locality\_val\]), (\[Eqn:min\_dist\_val\]) and (\[Eqn:alpha\_val\]) into (\[Eqn:local\_repair\_code\_ineq\]), we have $$\begin{aligned} \label{Eqn:locality_confirm} n-k+1 & \leq n-\ceil[\bigg]{k-\floor[\bigg]{\frac{k}{n_I}}} - \ceil[\bigg]{\frac{k-\floor{\frac{k}{n_I}}}{n_I-1}} + 2 \nonumber\\ & = n - k + \floor[\bigg]{\frac{k}{n_I}} - \ceil[\bigg]{\frac{k-\floor{\frac{k}{n_I}}}{n_I-1}} + 2.\end{aligned}$$ Therefore, all we need to prove is $$\label{Eqn:LRC_WTP} 0 \leq \floor[\bigg]{\frac{k}{n_I}} - \ceil[\bigg]{\frac{k-\floor{\frac{k}{n_I}}}{n_I-1}} + 1,$$ which is proved as follows. Using $q$ and $r$ defined in and , the right-hand-side (RHS) of (\[Eqn:LRC\_WTP\]) is $$\begin{aligned} RHS &= q - \ceil[\bigg]{\frac{qn_I+r-q}{n_I-1}} + 1 \nonumber\\ &= \begin{cases} q - (q+1) + 1 = 0, & \quad r \neq 0\\ q - q + 1 = 1, & \quad r = 0\\ \end{cases}\end{aligned}$$ Thus, (\[Eqn:LRC\_WTP\]) holds, where the equality condition is $r \neq 0$, or equivalently $(k {\ \mathrm{mod}\ n_I}) \neq 0$. Therefore, (\[Eqn:locality\_ineq\]) holds, where the equality condition is $\alpha = \alpha_{msr}^{(0)}$ and $(k {\ \mathrm{mod}\ n_I}) \neq 0$. This completes the proof of Theorem \[Thm:IRC\_versus\_LRC\]. Proof of Theorem \[Thm:beta\_c\_alpha\_trade\] {#Section:Proof_of_Thm_beta_c_alpha_trade} ============================================== According to (\[Eqn:lower\_bound\]) and (\[Eqn:capacity\_final\]) in Appendix B, the capacity can be expressed as $$\label{Eqn:cap_final} \mathcal{C} = \sum_{i=1}^k \min \{\alpha, \omega_i(\bm{\pi}_v (\bm{s}_h)) \}.$$ Using (\[Eqn:Lemma\_2\_proof\_4\]) and (\[Eqn:t\_i\^\*\_cases\]), $\omega_i(\bm{\pi}_v (\bm{s}_h))$ in (\[Eqn:cap\_final\]), or simply $\omega_i$ has the following property: $$\label{Eqn:omega_i_relation} \omega_{i+1} = \begin{cases} \omega_i - \beta_I, & i \in I_{G} \\ \omega_i - \beta_c, & i \in [k] \setminus I_{G} \end{cases}$$ where $I_{G} = \{g_m\}_{m=1}^{n_I-1}.$ Note that $g_{n_I} = \floor{\frac{k}{n_I}}$ from (\[Eqn:g\_m\]). Therefore, $k_0$ in (\[Eqn:k\_0\]) can be expressed as $$\label{Eqn:k_0_revisited} k_0 = k - \floor{\frac{k}{n_I}} = k - g_{n_I} \leq k - 1$$ where the last inequality is from (\[Eqn:k\_constraint\]). Combining (\[Eqn:h\_t\]) and (\[Eqn:k\_0\_revisited\]) result in $$\label{Eqn:h_k_0} h_{k_0} = n_I - 1.$$ From (\[Eqn:Lemma\_2\_proof\_4\]), (\[Eqn:k\_0\_revisited\]) and (\[Eqn:h\_k\_0\]), we have $$\begin{aligned} \label{Eqn:omega_k_0} \omega_{k_0} &= (n_I - h_{k_0}) \beta_I + (n-k_0- n_I + h_{k_0}) \beta_c \nonumber\\ &= \beta_I + (n-k_0 -1) \beta_c \geq \beta_I + (n-k)\beta_c \geq \beta_I =\alpha.\end{aligned}$$ where the last equality holds due to the assumption of $\beta_I = \alpha$ in the setting of Theorem \[Thm:beta\_c\_alpha\_trade\]. Since $(\omega_i)_{i=1}^k$ is a decreasing sequence from (\[Eqn:omega\_i\_relation\]), the result of (\[Eqn:omega\_k\_0\]) implies that $$\omega_i \geq \alpha, \quad \quad 1 \leq i \leq k_0.$$ Thus, the capacity expression in (\[Eqn:cap\_final\]) can be expressed as $$\begin{aligned} \label{Eqn:cap_final_cases} \mathcal{C} &= \begin{cases} k_0 \alpha + \sum_{i=k_0+1}^{k} \omega_i, & \omega_{k_0+1} \leq \alpha \\ m \alpha + \sum_{i=m+1}^{k} \omega_i, & \omega_{m+1} \leq \alpha < \omega_{m} \\ &\ \ \ (k_0 + 1 \leq m \leq k-1)\\ k\alpha, & \alpha < \omega_k \end{cases}\end{aligned}$$ Note that from (\[Eqn:Lemma\_2\_proof\_4\]) and (\[Eqn:t\_i\^\*\_cases\]), we have $\omega_i = (n-i)\beta_c$ for $i = k_0 + 1, k_0 + 2, \cdots, k$. Therefore, $\mathcal{C}$ in (\[Eqn:cap\_final\_cases\]) is $$\begin{aligned} \label{Eqn:cap_final_cases_revisited} \mathcal{C} &= \begin{cases} k_0 \alpha + \sum_{i=k_0+1}^{k} (n-i)\beta_c, & 0 \leq \beta_c \leq \frac{\alpha}{n-k_0-1}\\ m \alpha + \sum_{i=m+1}^{k} (n-i)\beta_c, & \frac{\alpha}{n-m} < \beta_c \leq \frac{\alpha}{n-m-1} \\ &\ \ \ (k_0 + 1 \leq m \leq k-1)\\ k\alpha, & \frac{\alpha}{n-k} < \beta_c, \end{cases}\end{aligned}$$ which is illustrated as a piecewise linear function of $\beta_c$ in Fig. \[Fig:beta\_c\_cap\]. Based on (\[Eqn:cap\_final\_cases\_revisited\]), the sequence $(T_m)_{m=k_0}^{k}$ in this figure has the following expression: $$\label{Eqn:T_m} T_m = \begin{cases} k_0 \alpha , & m = k_0 \\ (m + \frac{\sum_{i=m+1}^{k} (n-i)}{n-m}) \alpha, & k_0+1 \leq m \leq k-1 \\ k \alpha, & m = k \end{cases}$$ ![Capacity as a function of $\beta_c$[]{data-label="Fig:beta_c_cap"}](beta_c_cap_curve.pdf){width="85mm"} From Fig. \[Fig:beta\_c\_cap\], we can conclude that $\mathcal{C} \geq \mathcal{M}$ holds if and only if $\beta_c \geq \beta_c^*$ where $$\label{Eqn:beta_c_star_prev} \beta_c^* = \begin{cases} 0, & \mathcal{M} \in [0, T_{k_0}] \\ \frac{\mathcal{M} - m \alpha}{\sum_{i=m+1}^{k} (n-i)}, & \mathcal{M} \in (T_{m}, T_{m+1}] \\ & \quad (m = k_0, k_0 + 1, \cdots, k-1) \\ \infty, & \mathcal{M} \in (T_k, \infty). \end{cases}$$ Using $T_m$ in (\[Eqn:T\_m\]) and $f_m$ in (\[Eqn:f\_m\]), (\[Eqn:beta\_c\_star\_prev\]) reduces to (\[Eqn:beta\_c\_star\]), which completes the proof. Proofs of Corollaries ===================== Proof of Corollary \[Corollary:Feasible Points\_large\_epsilon\] {#Section: Proof of Corollary:Feasible Points} ---------------------------------------------------------------- From the proof of Theorem \[Thm:Capacity of clustered DSS\], the capacity expression is equal to (\[Eqn:lower\_bound\_intermediate\]), which is $$\mathcal{C} = \sum_{i=1}^k \min \{\alpha, (n_I-h_i)\beta_I + (n-n_I-i+h_i)\beta_c\}.$$ where $h_i$ is defined in (\[Eqn:t\_i\^\*\]). Using $\epsilon = \beta_c/\beta_I$ and (\[Eqn:gamma\]), this can be rewritten as $$\label{Eqn:capacity_kappa_general} \mathcal{C} = \sum_{i=1}^k \min \{\alpha, \frac{(n-n_I-i+h_i) \epsilon + (n_I - h_i)}{(n-n_I)\epsilon + (n_I-1)} \gamma \}.$$ Using $\{z_t\}$ and $\{y_t\}$ defined in (\[Eqn:z\_t\]) and (\[Eqn:y\_t\]), the capacity expression reduces to $$\label{Eqn:capacity_simple} \mathcal{C} (\alpha, \gamma)= \sum_{t=1}^{k} \min \{\alpha, \frac{\gamma}{y_t} \},$$ which is a continuous function of $\gamma$. \[Rmk:z\_dec\_y\_inc\] $\{z_t\}$ in (\[Eqn:z\_t\]) is a decreasing sequence of $t$. Moreover, $\{y_t\}$ in (\[Eqn:y\_t\]) is an increasing sequence. Note that from (\[Eqn:t\_i\^\*\_reform\]), $$h_{t+1} = \begin{cases} h_t + 1, & t \in T \\ h_t, & t \in [k-1] \setminus T \end{cases}$$ where $T = \{g_1, g_1+g_2, \cdots, \sum_{m=1}^{n_I-1} g_m\}. $ Therefore, $\{z_t\}$ in (\[Eqn:z\_t\]) is a decreasing function of $t$, which implies that $\{y_t\}$ is an increasing sequence. Moreover, note that $\beta_I \leq \alpha$ holds from the definition of $\beta_I$ and $\alpha$ in Table \[Table:Params\]. Thus, combined with $\epsilon = \beta_I/\beta_c$, it is shown that $\gamma$ in (\[Eqn:gamma\]) is lower-bounded as $$\begin{aligned} \gamma &= (n_I-1)\beta_I + (n-n_I)\beta_c \nonumber\\ &= \{n_I - 1 + (n-n_I)\epsilon\} \beta_I \leq \{n_I - 1 + (n-n_I)\epsilon\} \alpha.\end{aligned}$$ Here, we define $${\mkern 1.5mu\overline{\mkern-1.5mu\gamma\mkern-1.5mu}\mkern 1.5mu} = \{n_I - 1 + (n-n_I)\epsilon\} \alpha.$$ Then, the valid region of $\gamma$ is expressed as $\gamma \leq {\mkern 1.5mu\overline{\mkern-1.5mu\gamma\mkern-1.5mu}\mkern 1.5mu}, $ as illustrated in Figs. \[Fig:corollary\_proof\_1\] and \[Fig:corollary\_proof\_2\]. The rest of the proof depends on the range of $\epsilon$ values; we first consider the $\frac{1}{n-k} \leq \epsilon \leq 1$ case, and then consider the $0 \leq \epsilon < \frac{1}{n-k}$ case. ### If $ \frac{1}{n-k} \leq \epsilon \leq 1$ Using (\[Eqn:z\_k\]), $z_k = (n-k)\epsilon \geq 1$ holds. Combining with (\[Eqn:y\_t\]), we have $y_k \leq n_I-1 + \epsilon (n-n_I),$ or equivalently, $y_k \alpha \leq {\mkern 1.5mu\overline{\mkern-1.5mu\gamma\mkern-1.5mu}\mkern 1.5mu}.$ If $y_t \alpha < \gamma \leq y_{t+1}\alpha$ for some $t \in [k-1]$, then (\[Eqn:capacity\_simple\]) can be expressed as $$\begin{aligned} \mathcal{C} (\alpha, \gamma)&= t\alpha + \sum_{m=t+1}^k \frac{\gamma}{y_m} \\ &= t\alpha + \frac{\gamma \ (\sum_{m=t+1}^k z_m)}{(n_I-1) + \epsilon (n-n_I)} = t\alpha + s_t \gamma\end{aligned}$$ where $\{s_t\}$ is defined in (\[Eqn:s\_t\]). If $0 \leq \gamma \leq y_1\alpha$, then $$\begin{aligned} \mathcal{C}(\alpha,\gamma) &= \sum_{m=1}^k \frac{\gamma}{y_m} = \frac{\gamma \ (\sum_{m=1}^k z_m)}{(n_I-1) + \epsilon (n-n_I)} = s_0 \gamma.\end{aligned}$$ If $y_k \alpha < \gamma \leq {\mkern 1.5mu\overline{\mkern-1.5mu\gamma\mkern-1.5mu}\mkern 1.5mu}$, then $\mathcal{C}(\alpha, \gamma) = \sum_{m=1}^k \alpha = k\alpha. $ In summary, capacity is $$\begin{aligned} \label{Eqn:capacity_cases} \mathcal{C}(\alpha, \gamma) &= \begin{cases} s_0 \gamma, & 0 \leq \gamma \leq y_1\alpha \\ t\alpha + s_t \gamma, & y_t \alpha < \gamma \leq y_{t+1}\alpha \\ & \ \ \ \ (t = 1, 2, \cdots, k-1)\\ k\alpha, & y_k \alpha < \gamma \leq {\mkern 1.5mu\overline{\mkern-1.5mu\gamma\mkern-1.5mu}\mkern 1.5mu} \end{cases}\end{aligned}$$ which is illustrated in Fig. \[Fig:corollary\_proof\_1\]. Since $\{z_t\}$ is a decreasing sequence from Remark \[Rmk:z\_dec\_y\_inc\], we have $ z_t \geq z_k = (n-k)\epsilon > 0$ for $t \in [k]$. Thus, $\{s_t\}_{t=1}^k$ defined in (\[Eqn:s\_t\]) is a monotonically decreasing, non-negative sequence. This implies that the curve in Fig. \[Fig:corollary\_proof\_1\] is a monotonic increasing function of $\gamma$. ![Capacity as a function of $\gamma$, for $\frac{1}{n-k} \leq \epsilon \leq 1$[]{data-label="Fig:corollary_proof_1"}](corollary_proof_1_ver2.pdf){width="70mm"} From Fig. \[Fig:corollary\_proof\_1\], it is shown that $\mathcal{C}(\alpha, \gamma) \geq \mathcal{M}$ holds if and only if $\gamma \geq \gamma^*(\alpha)$. From (\[Eqn:capacity\_cases\]), the threshold value $\gamma^*(\alpha)$ can be expressed as $$\label{Eqn:gamma_th} \gamma^*(\alpha) = \begin{cases} \frac{\mathcal{M}}{s_0}, & \mathcal{M} \in [0, E_1]\\ \frac{\mathcal{M}- t \alpha}{s_t}, & \mathcal{M} \in (E_{t}, E_{t+1}]\\ & \ \ \ \ (t = 1, 2, \cdots, k-1)\\ \infty, & \mathcal{M} \in (E_k, \infty). \end{cases}$$ where $$\label{Eqn:E_t} E_t = \mathcal{C}(\alpha, y_t \alpha) = (t-1+s_{t-1}y_t) \alpha$$ for $t \in [k]$. The threshold value $\gamma^*(\alpha)$ in (\[Eqn:gamma\_th\]) can be expressed as (\[Eqn:Feasible Points Result\]), which completes the proof. ### Otherwise (if $0 \leq \epsilon < \frac{1}{n-k}$) Using (\[Eqn:z\_k\]), $$\label{Eqn:z_k_small_epsilon} z_k = (n-k)\epsilon < 1$$ holds. Since $\{z_t\}$ is a decreasing sequence from Remark \[Rmk:z\_dec\_y\_inc\], there exists $\tau \in \{0, 1, \cdots, k-1\}$ such that $z_{\tau+1} < 1 \leq z_{\tau}$ holds, or equivalently, $y_{\tau}\alpha \leq {\mkern 1.5mu\overline{\mkern-1.5mu\gamma\mkern-1.5mu}\mkern 1.5mu} < y_{\tau+1}\alpha.$ Using the analysis similar to the $\frac{1}{n-k} \leq \epsilon \leq 1$ case, we obtain $$\begin{aligned} \label{Eqn:capacity_cases_2} \mathcal{C}(\alpha, \gamma) &= \begin{cases} s_0 \gamma, & 0 \leq \gamma \leq y_1\alpha \\ t\alpha + s_t \gamma, & y_t \alpha < \gamma \leq y_{t+1}\alpha \\ & \ \ \ \ (t = 1, 2, \cdots, \tau-1)\\ \tau\alpha + s_{\tau} \gamma, & y_{\tau} \alpha < \gamma \leq {\mkern 1.5mu\overline{\mkern-1.5mu\gamma\mkern-1.5mu}\mkern 1.5mu} \end{cases}\end{aligned}$$ which is illustrated in Fig. \[Fig:corollary\_proof\_2\]. ![Capacity as a function of $\gamma$, for $0 \leq \epsilon < \frac{1}{n-k} $ case[]{data-label="Fig:corollary_proof_2"}](corollary_proof_2.pdf){width="70mm"} From Fig. \[Fig:corollary\_proof\_2\], it is shown that $\mathcal{C}(\alpha, \gamma) \geq \mathcal{M}$ holds if and only if $\gamma \geq \gamma^*(\alpha)$. From (\[Eqn:capacity\_cases\_2\]), the threshold value $\gamma^*(\alpha)$ can be expressed as $$\label{Eqn:gamma_th_2} \gamma^*(\alpha) = \begin{cases} \frac{\mathcal{M}}{s_0}, & \mathcal{M} \in [0, E_1]\\ \frac{\mathcal{M}- t \alpha}{s_t}, & \mathcal{M} \in (E_{t}, E_{t+1}]\\ & \ \ \ \ (t = 1, 2, \cdots, \tau-1)\\ \frac{\mathcal{M}- \tau \alpha}{s_{\tau}}, & \mathcal{M} \in (E_{\tau}, {\mkern 1.5mu\overline{\mkern-1.5muE\mkern-1.5mu}\mkern 1.5mu}]\\ \infty, & \mathcal{M} \in ({\mkern 1.5mu\overline{\mkern-1.5muE\mkern-1.5mu}\mkern 1.5mu}, \infty). \end{cases}$$ where $\{E_t\}$ is defined in (\[Eqn:E\_t\]), and $${\mkern 1.5mu\overline{\mkern-1.5muE\mkern-1.5mu}\mkern 1.5mu} = \mathcal{C}(\alpha, {\mkern 1.5mu\overline{\mkern-1.5mu\gamma\mkern-1.5mu}\mkern 1.5mu}) = \tau \alpha + s_{\tau} \alpha \{n_I-1 + (n-n_I)\epsilon\}.$$ The threshold value $\gamma^*(\alpha)$ in (\[Eqn:gamma\_th\_2\]) can be expressed as (\[Eqn:Feasible Points Result\_intermediate\_epsilon\]), which completes the proof. Proof of Corollary \[Coro:msr\_mbr\_points\] {#Section:Proof of Corollary_msr_mbr_points} -------------------------------------------- First, we focus on the MSR point illustrated in Fig. \[Fig:MBR\_MSR\_points\]. From (\[Eqn:Feasible Points Result\]), the MSR point for $\frac{1}{n-k} \leq \epsilon \leq 1$ is $$\begin{aligned} (\alpha_{msr}^{(\epsilon)}, \gamma_{msr}^{(\epsilon)}) &= (\frac{\mathcal{M}}{k+s_ky_k}, \frac{\mathcal{M} - (k-1)\alpha_{msr}^{(\epsilon)}}{s_{k-1}}) \nonumber\\ &= (\frac{\mathcal{M}}{k}, \frac{\mathcal{M}}{k} \frac{1}{s_{k-1}}) \label{Eqn:proof_of_corollary_MSR} \end{aligned}$$ where the last equality is from $s_k = 0$ in (\[Eqn:s\_t\]). Moreover, from (\[Eqn:Feasible Points Result\_intermediate\_epsilon\]), the MSR point for $0 \leq \epsilon < \frac{1}{n-k}$ is $$\begin{aligned} (\alpha_{msr}^{(\epsilon)}, \gamma_{msr}^{(\epsilon)}) &= (\frac{M}{\tau + \sum_{i=\tau+1}^{k}z_i}, \frac{\mathcal{M} - \tau\alpha_{msr}^{(\epsilon)}}{s_{\tau}}) \nonumber\\ &= (\frac{M}{\tau + \sum_{i=\tau+1}^{k}z_i}, \frac{M}{\tau + \sum_{i=\tau+1}^{k}z_i} \frac{\sum_{i=\tau+1}^{k}z_i}{s_{\tau}}). \label{Eqn:proof_of_corollary_MSR_2} \end{aligned}$$ Equations (\[Eqn:proof\_of\_corollary\_MSR\]) and (\[Eqn:proof\_of\_corollary\_MSR\_2\]) proves (\[Eqn:MSR\_point\]). The expression (\[Eqn:MBR\_point\]) for the MBR point is directly obtained from Corollary \[Corollary:Feasible Points\_large\_epsilon\] and Fig. \[Fig:MBR\_MSR\_points\]. Proofs of Propositions {#Section:proofs_of_remarks} ====================== Proof of Proposition \[Prop:max\_helper\_nodes\] {#Section:max_helper_node_assumption} ------------------------------------------------ As in (\[Eqn:Capacity\_expression\_minmin\]), capacity $\mathcal{C}$ for maximum $d_I, d_c$ setting ($d_I = n_I-1, d_c = n-n_I$) is expressed as $$\mathcal{C} = \displaystyle\min_{\bm{s} \in S, \bm{\pi} \in \Pi(\bm{s})} L (\bm{s},\bm{\pi})$$ where $$\begin{aligned} L (\bm{s},\bm{\pi}) &= \sum_{i=1}^k \min\{\alpha, \omega_i(\bm{\pi})\}, \nonumber\\ \omega_i(\bm{\pi}) &= \gamma - e_i(\bm{\pi}) \beta_I - (i - 1 - e_i(\bm{\pi})) \beta_c, \label{Eqn:omega_i_max}\\ e_i(\bm{\pi}) &= \sum_{j=1}^{i-1} \mathds{1}_{\pi_j = \pi_i},\nonumber\end{aligned}$$ as in (\[Eqn:lower\_bound\]), (\[Eqn:weight vector\]) and (\[Eqn:sum\_beta\_ji\]). Consider a general $d_I, d_c$ setting, where each newcomer node is helped by $d_I$ nodes in the same cluster, receiving $\beta_I$ information from each node, and $d_c$ nodes in other clusters, receiving $\beta_c$ information from each node. Under this setting, the coefficient of $\beta_I$ in (\[Eqn:omega\_i\_max\]) cannot exceed $d_I$. Similarly, the coefficient of $\beta_c$ in (\[Eqn:omega\_i\_max\]) cannot exceed $d_c$. Thus, the capacity for general $d_I, d_c$ is expressed as $$\label{Eqn:capacity_general_helper_nodes} \mathcal{C}(d_I, d_c) = \displaystyle\min_{\bm{s} \in S, \bm{\pi} \in \Pi(\bm{s})} \ L (d_I, d_c, \bm{s},\bm{\pi})$$ where $$\begin{aligned} L (d_I, d_c,\bm{s},\bm{\pi}) &= \sum_{i=1}^{k} \min \{\alpha, \omega_i(d_I, d_c, \bm{s}, \bm{\pi})\}, \nonumber\\ \omega_i(d_I, d_c, \bm{s}, \bm{\pi}) &= \gamma - \min\{d_I, e_i(\bm{\pi})\}\beta_I \nonumber\\ & \quad \quad - \min\{d_c, i-1-e_i(\bm{\pi})\} \beta_c,\label{Eqn:omega_i_general}\\ e_i(\bm{\pi}) &= \sum_{j=1}^{i-1} \mathds{1}_{\pi_j = \pi_i}.\nonumber\end{aligned}$$ Consider arbitrary fixed $\bm{s}, \bm{\pi}$ and $d_c$. Since $\gamma$ and $\gamma_c=d_c\beta_c$ are fixed in the basic setting of Proposition \[Prop:max\_helper\_nodes\], only $d_I$ and $\beta_I$ are variables in (\[Eqn:omega\_i\_general\]), while other parameters are constants. Then, (\[Eqn:omega\_i\_general\]) can be expressed as $$\begin{aligned} \omega_i(d_I, d_c, \bm{s}, \bm{\pi}) &= C_1 - \min\{d_I, e_i(\bm{\pi})\}\beta_I \nonumber\\ & = C_1 - \frac{\min\{d_I, e_i(\bm{\pi})\}}{d_I}C_2 \label{Eqn:omega_general_constants}\end{aligned}$$ where $C_1 = \gamma - \min\{d_c, i-1-e_i(\bm{\pi})\} \beta_c $ and $C_2 = \gamma_I = \gamma - \gamma_c$ are constants. Note that $$\frac{\min\{d_I, e_i(\bm{\pi})\}}{d_I} = \begin{cases} 1, & \text{ if } d_I \leq e_i(\bm{\pi}), \\ \frac{e_i(\bm{\pi})}{d_I}, & \text{ otherwise } \end{cases}$$ is a non-increasing function of $d_I$. Thus, $\omega_i(d_I, d_c, \bm{s}, \bm{\pi})$ in (\[Eqn:omega\_general\_constants\]) is a non-decreasing function of $d_I$ for arbitrary fixed $\bm{s}, \bm{\pi}, d_c$ and $i \in [k]$. Since the maximum $d_I$ value is $n_I-1$, we have $$\begin{aligned} L(d_I,d_c,\bm{s},\bm{\pi}) &= \sum_{i=1}^{k} \min \{\alpha, \omega_i(d_I, d_c, \bm{s}, \bm{\pi})\} \nonumber\\ &\leq \sum_{i=1}^{k} \min \{\alpha, \omega_i(n_I-1, d_c, \bm{s}, \bm{\pi})\} \nonumber\\ &= L(n_I-1,d_c,\bm{s},\bm{\pi})\end{aligned}$$ for $d_I \in [n_I-1] $. In other words, for all $\bm{s}, \bm{\pi}, d_c$, we have $$\underset{d_I \in [n_I-1]}{\arg \max}\ L(d_I,d_c,\bm{s},\bm{\pi}) = n_I-1. $$ Similarly, for all $\bm{s}, \bm{\pi}, d_I$, $$\underset{d_c \in [n-n_I]}{\arg \max}\ L(d_I,d_c,\bm{s},\bm{\pi}) = n-n_I $$ holds. Therefore, for all $\bm{s},\bm{\pi}$, $$\label{Eqn:lower_bound_argmax} \underset{[d_I, d_c]}{\arg \max}\ L(d_I,d_c,\bm{s},\bm{\pi}) = [n_I-1, n-n_I]. $$ Let $$\label{Eqn:optimal_s_pi} [\bm{s}^*, \bm{\pi}^*] = \underset{\bm{s} \in S, \bm{\pi} \in \Pi(\bm{s})}{\arg \min}\ \ L (n_I-1, n-n_I, \bm{s},\bm{\pi}). $$ Then, from (\[Eqn:capacity\_general\_helper\_nodes\]), (\[Eqn:lower\_bound\_argmax\]) and (\[Eqn:optimal\_s\_pi\]), $$\begin{aligned} \mathcal{C}(d_I, d_c) &= \displaystyle\min_{\bm{s} \in S, \bm{\pi} \in \Pi(\bm{s})} \ L (d_I, d_c, \bm{s},\bm{\pi}) \nonumber\\ & \leq L (d_I, d_c, \bm{s}^*,\bm{\pi}^*) \leq L (n_I-1, n-n_I, \bm{s}^*,\bm{\pi}^*) \nonumber\\ &= \displaystyle\min_{\bm{s} \in S, \bm{\pi} \in \Pi(\bm{s})} \ L (n_I-1, n-n_I, \bm{s},\bm{\pi}) \nonumber\\ & = \mathcal{C}(n_I-1, n-n_I)\end{aligned}$$ for all $d_I \in [n_I-1] $ and $d_c \in [n-n_I] $. Therefore, choosing $d_I = n_I -1$ and $d_c = n-n_I$ maximizes storage capacity when the available resources, $\gamma$ and $\gamma_c$, are given. Proof of Proposition \[Prop:omega\_i\_bounded\_by\_gamma\] {#Section:proof_of_omega_bound_gamma} ---------------------------------------------------------- First, we prove (\[Eqn:omega\_is\_bounded\_by\_gamma\]). Recall $\rho_i$ and $g_m$ defined in (\[Eqn:rho\_i\]) and (\[Eqn:g\_m\]). Consider the *support set* $S$, which is defined as $$S = \{i \in [n_I] : g_i \geq 1 \}.\label{Eqn:Support_Set}$$ Then, we have $$\begin{aligned} \rho_i = n_I - i &\leq n_I - 1, \label{rho_i_inequality} \\ \sum_{m=1}^{i-1}g_m &\geq i-1 \label{Eqn:t_ij_inequality}\end{aligned}$$ for every $i \in S$. Therefore, by combining (\[Eqn:t\_ij\_inequality\]) and (\[Eqn:rho\_i\]), $$\begin{aligned} \label{phi_ij_inequality} n-\rho_i-j-\sum_{m=1}^{i-1}g_m & \leq n- (i-1) - j - \rho_i \nonumber\\ &= n-n_I - (j-1) \leq n-n_I\end{aligned}$$ holds for every $i \in S, j \in [g_i] $. Combining (\[Eqn:gamma\]), (\[rho\_i\_inequality\]) and (\[phi\_ij\_inequality\]) results in $$\label{Eqn:omega_i_bounded_by_gamma} \rho_i \beta_I + (n-\rho_i-j-\sum_{m=1}^{i-1}g_m)\beta_c \leq (n_I-1)\beta_I + (n-n_I)\beta_c = \gamma$$ for arbitrary $i \in S, j \in [g_i] $. Since $ [g_i] = \emptyset$ holds for $i \in [n_I] \setminus S$, we conclude that (\[Eqn:omega\_i\_bounded\_by\_gamma\]) holds for $(i,j)$ with $i \in [n_I]$, $j \in [g_i]$. Second, we prove (\[Eqn:sum of g is k\]). Using $q$ and $r$ in (\[Eqn:quotient\]) and (\[Eqn:remainder\]), $$\label{Eqn:g_m_simple} g_i = \begin{cases} q+1, & i \leq r \\ q, & \text{otherwise} \end{cases}$$ Therefore, $\sum_{i=1}^{n_I}g_i = (q+1)r + q(n_I-r) = r + qn_I = k,$ where the last equality is from (\[Eqn:quotient\_remainder\_relation\]). Proofs of Lemmas {#Section:Proofs_of_Lemmas} ================ Proof of Lemma \[Lemma:cap\_upper\_lower\] {#Section:proof_of_prop_bound} ------------------------------------------ Using (\[Eqn:gamma\]), $\underline{C}$ in (\[Eqn:cap\_lower\]) can be expressed as $$\begin{aligned} \underline{C} &= \frac{k}{2} \left( (n_I-1)\beta_I + (n-n_I)(1+\frac{n-k}{n(1-1/L)}) \beta_c \right) \nonumber\\ &= \frac{k}{2} \left( (n_I-1)\beta_I + (n-n_I)(1+\frac{n-k}{n-n_I}) \beta_c \right) \nonumber\\ &= \frac{k}{2} \left\{ (n_I-1)\beta_I + (2n-n_I-k) \beta_c \right\}. \label{Eqn:cap_lower_revisited}\end{aligned}$$ According to (\[Eqn:lower\_bound\]) and (\[Eqn:capacity\_final\]) in Appendix B, the capacity can be expressed as $$\label{Eqn:cap} \mathcal{C} = L(\bm{s}_h, \bm{\pi}_v) = \sum_{i=1}^{k} \min \{\alpha, \omega_i (\bm{\pi}_{v} (\bm{s}_h))\}.$$ From (\[Eqn:weight vector\]), we have $$\omega_i (\bm{\pi}_{v} (\bm{s}_h)) \leq \gamma$$ for $i \in [k]$. Therefore, when $\alpha = \gamma$, the capacity expression in (\[Eqn:cap\]) reduces to $$\label{Eqn:cap_bw_limited} \mathcal{C} = \sum_{i=1}^k \omega_i (\bm{\pi}_{v} (\bm{s}_h)).$$ Recall (\[Eqn:Lemma\_2\_proof\_4\]): $$\label{Eqn:opt_omega_i_revisited} \omega_i(\bm{\pi}_v(\bm{s}_h)) = (n_I - h_i) \beta_I + (n-i-n_I + h_i) \beta_c.$$ From the expression of $(h_i)_{i=1}^k$ in (\[Eqn:t\_i\^\*\_cases\]), we have $$\label{Eqn:A_0_part} \sum_{i=1}^k (n_I - h_i) = \sum_{s=1}^{n_I} g_s (n_I-s)$$ since $\sum_{m=1}^k g_m = k$ from (\[Eqn:sum of g is k\]), Using (\[Eqn:opt\_omega\_i\_revisited\]) and (\[Eqn:A\_0\_part\]), the capacity expression in (\[Eqn:cap\_bw\_limited\]) is expressed as $$\begin{aligned} \label{Eqn:cap_bw_limited_factored} \mathcal{C} &= \left( \sum_{i=1}^k (n_I-h_i) \right) \beta_I + \left( \sum_{i=1}^k (n-i - (n_I-h_i)) \right) \beta_c \nonumber\\ &= A_0 \beta_I + B_0 \beta_c\end{aligned}$$ where $A_0 = \sum_{s=1}^{n_I} g_s (n_I-s)$ and $$\begin{aligned} B_0 &= \sum_{i=1}^k (n-i) - A_0. \label{Eqn:B_0}\end{aligned}$$ Similarly, $\underline{C}$ in (\[Eqn:cap\_lower\_revisited\]) can be expressed as $$\label{Eqn:cap_underbar_factored} \underline{C} = A_0'\beta_I + B_0'\beta_c$$ where $A_0' = (n_I-1)\frac{k}{2}$ and $$\begin{aligned} B_0' &= (2n-n_I-k)\frac{k}{2} = \sum_{i=1}^{k}(n-i)-A_0'. \label{Eqn:B_0_prime}\end{aligned}$$ First, we show that $\mathcal{C} \geq \underline{C}$ holds. From (\[Eqn:cap\_bw\_limited\_factored\]), (\[Eqn:B\_0\]), (\[Eqn:cap\_underbar\_factored\]) and (\[Eqn:B\_0\_prime\]), $$\begin{aligned} \label{Eqn:c-c'} \mathcal{C} &- \underline{C} = (A_0 - A_0')\beta_I + (B_0 - B_0')\beta_c \nonumber\\ &= (A_0 - A_0')\beta_I - (A_0 - A_0')\beta_c = (A_0-A_0') (\beta_I - \beta_c).\end{aligned}$$ Since we consider the $\beta_I \geq \beta_c $ case, all we need to prove is $$A_0-A_0' \geq 0.$$ Using $(g_i)_{i=1}^k$ expression in (\[Eqn:g\_m\_simple\]), $A_0$ can be rewritten as $$\begin{aligned} A_0 & = \sum_{s=1}^{n_I}g_s(n_I-s)= q \sum_{s=1}^{n_I} (n_I-s) + \sum_{s=1}^{r} (n_I-s) \nonumber\\ &= q \frac{n_I(n_I-1)}{2} +\sum_{s=1}^{r} (n_I-s) \nonumber \\ &= q \frac{n_I(n_I-1)}{2} + \frac{r((n_I-1)+(n_I-r))}{2} \nonumber \\ &= \left(qn_I + r\right) \frac{n_I-1}{2} + \frac{r(n_I-r) }{2} \nonumber\\ & = \frac{k(n_I-1)}{2} + \frac{r(n_I-r) }{2} = A_0' + \frac{r(n_I-r) }{2}.\end{aligned}$$ Since $0 \leq r < n_I$, we have $$\label{A-A'} A_0-A_0' = \frac{r(n_I-r)}{2} \geq 0.$$ Therefore, $\mathcal{C} \geq \underline{C} $ holds. Second, we prove $\mathcal{C} \leq \underline{C} + n_I^2(\beta_I-\beta_c)/8$. Note that $A_0-A_0'$ in (\[A-A’\]) is maximized when $r = \lfloor n_I/2 \rfloor$ holds. Thus, $A_0-A_0' \leq \frac{\lfloor n_I/2 \rfloor (n_I - \lfloor n_I/2 \rfloor)}{2} \leq n_I^2/8.$ Combining with (\[Eqn:c-c’\]), $$\begin{aligned} \mathcal{C} - \underline{C} &= (A_0-A_0')(\beta_I-\beta_c) \leq n_I^2(\beta_I-\beta_c)/8.\end{aligned}$$ Proof of Lemma \[Lemma:Scale\] {#Section:proof_of_prop_scale} ------------------------------ The expression for $\underbar{C}$ in (\[Eqn:cap\_lower\]) can be expressed as $$\begin{aligned} \underline{C} &= \frac{k}{2} ( \gamma + \frac{n-k}{n(1-\frac{1}{L}) }\gamma_c ) = \frac{nR}{2} ( \gamma + \frac{1-R}{1-\frac{1}{L}}\gamma_c ) \nonumber\\ &= \gamma \frac{nR}{2} ( 1 + \frac{1-R}{(1-\frac{1}{L}) }\xi) \label{Eqn:C_underbar_monotone} \end{aligned}$$ Note that (\[Eqn:C\_underbar\_monotone\]) is a monotonic decreasing function of $L$. Moreover, we consider the $L \geq 2$ case, as mentioned in (\[Eqn:L\_constraint\]). Thus, $\underline{C}$ is upper/lower bounded by expressions for $L=2$ and $L=\infty$, respectively: $$\gamma \frac{nR}{2} ( 1 + (1-R)\xi) < \underline{C} \leq \gamma \frac{nR}{2} ( 1 + 2(1-R)\xi).$$ Therefore, $\underline{C} = \Theta(n)$ holds. Moreover, the expression for $\delta$ in (\[Eqn:cap\_delta\_val\]) is $$\begin{aligned} \nonumber \delta & = \frac{n_I^2(\beta_I-\beta_c)}{8} = \frac{n_I^2}{8}(\frac{\gamma_I}{n_I-1} - \frac{\gamma_c}{n-n_I}) \nonumber \\ &= \frac{n_I^2}{8}(\frac{\gamma (1-\xi)}{n_I-1} - \frac{ \gamma\xi}{n-n_I}) \nonumber\\ &= \frac{n_I^2 \gamma}{8} \frac{(1-\xi)(n-n_I) - \xi(n_I-1)}{(n_I-1) (n-n_I)}. \label{Eqn:delta}\end{aligned}$$ Putting $n = n_IL$ into (\[Eqn:delta\]), we get $$\begin{aligned} \delta &= \frac{n_I \gamma}{8} \frac{(1 - \xi)n_I(L-1) - \xi(n_I-1)}{(n_I-1)(L-1)} = O(n_I). \nonumber\end{aligned}$$ [Jy-yong Sohn]{} (S’15) received the B.S. and M.S. degrees in electrical engineering from the Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea, in 2014 and 2016, respectively. He is currently pursuing the Ph.D. degree in KAIST. His research interests include coding for distributed storage and distributed computing, massive MIMO effects on wireless multi cellular system and 5G Communications. He is a recipient of the IEEE international conference on communications (ICC) best paper award in 2017. [Beongjun Choi]{} (S’17) received the B.S. and M.S. degrees in mathematics and electrical engineering from the Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea, in 2014 and 2017. He is currently pursuing the electrical engineering Ph.D degree in KAIST. His research interests include error-correcting codes, distributed storage system and information theory. He is a co-recipient of the IEEE international conference on communications (ICC) best paper award in 2017. [Sung Whan Yoon]{} (M’17) received the M.S. and Ph.D. degrees in electrical engineering from the Korea Advanced Institute of Science and Technology (KAIST), Daejeon, South Korea, in 2013 and 2017 respectively. He is currently a postdoctoral researcher in KAIST from 2017. His research interests are in the area of coding theory, distributed system and artificial intelligence, with focusing on polar codes, distributed storage system and meta-learning algorithm of neural network. Especially for the area of artificial intelligence, his primary interests include information theoretic analysis and algorithmic development of meta-learning. He was a co-recipient of the IEEE International Conference on Communications best Paper Award in 2017. [Jaekyun Moon]{} (F’05) received the Ph.D degree in electrical and computer engineering at Carnegie Mellon University, Pittsburgh, Pa, USA. He is currently a Professor of electrical engineering at KAIST. From 1990 through early 2009, he was with the faculty of the Department of Electrical and Computer Engineering at the University of Minnesota, Twin Cities. He consulted as Chief Scientist for DSPG, Inc. from 2004 to 2007. He also worked as Chief Technology Officer at Link-A-Media Devices Corporation. His research interests are in the area of channel characterization, signal processing and coding for data storage and digital communication. Prof. Moon received the McKnight Land-Grant Professorship from the University of Minnesota. He received the IBM Faculty Development Awards as well as the IBM Partnership Awards. He was awarded the National Storage Industry Consortium (NSIC) Technical Achievement Award for the invention of the maximum transition run (MTR) code, a widely used error-control/modulation code in commercial storage systems. He served as Program Chair for the 1997 IEEE Magnetic Recording Conference. He is also Past Chair of the Signal Processing for Storage Technical Committee of the IEEE Communications Society. He served as a guest editor for the 2001 IEEE JSAC issue on Signal Processing for High Density Recording. He also served as an Editor for IEEE TRANSACTIONS ON MAGNETICS in the area of signal processing and coding for 2001-2006. He is an IEEE Fellow. [^1]: The authors are with the School of Electrical Engineering, Korea Advanced Institute of Science and Technology, Daejeon, 34141, Republic of Korea (e-mail: {jysohn1108, bbzang10, shyoon8}@kaist.ac.kr, jmoon@kaist.edu). A part of this paper was presented [@sohn2016capacity] at the IEEE Conference on Communications (ICC), Paris, France, May 21-25, 2017. This work is in part supported by the National Research Foundation of Korea under Grant No. 2016R1A2B4011298, and in part supported by the ICT R&D program of MSIP/IITP \[2016-0-00563, Research on Adaptive Machine Learning Technology Development for Intelligent Autonomous Digital Companion\]. [^2]: [^3]: .

--- abstract: 'Supercontinuum generation in integrated photonic waveguides is a versatile source of broadband light, and the generated spectrum is largely determined by the phase-matching conditions. Here we show that quasi-phase-matching via periodic modulations of the waveguide structure provides a useful mechanism to control the evolution of ultrafast pulses during supercontinuum generation. We experimentally demonstrate quasi-phase-matched supercontinuum to the TE~20~ and TE~00~ waveguide modes, which enhances the intensity of the SCG in specific spectral regions by as much as 20 dB. We utilize higher-order quasi-phase-matching (up to the 16^th^ order) to enhance the intensity in numerous locations across the spectrum. Quasi-phase-matching adds a unique dimension to the design-space for SCG waveguides, allowing the spectrum to be engineered for specific applications.' author: - 'Daniel D. Hickstein' - 'Grace C. Kerber' - 'David R. Carlson' - Lin Chang - Daron Westly - Kartik Srinivasan - Abijith Kowligy - 'John E. Bowers' - 'Scott A. Diddams' - 'Scott B. Papp' bibliography: - 'Zotero.bib' title: 'Quasi-phase-matched supercontinuum-generation in photonic waveguides' --- Supercontinuum generation (SCG) is a $\chi^{(3)}$ nonlinear process where laser pulses of relatively narrow bandwidth can be converted into a continuum with large spectral span [@alfano_supercontinuum_2016; @dudley_supercontinuum_2006; @agrawal_nonlinear_2007]. SCG has numerous applications, including self-referencing frequency combs [@jones_carrier-envelope_2000; @holzwarth_optical_2000; @diddams_direct_2000], microscopy [@betz_excitation_2005], spectroscopy [@coddington_dual-comb_2016], and tomography [@moon_ultra-high-speed_2006]. SCG is traditionally accomplished using bulk crystals or nonlinear fiber, but recently, “photonic waveguides” (on-chip waveguides produced using nanofabrication techniques) have proven themselves as a versatile platform for SCG, offering small size, high nonlinearity, and increased control over the generated spectrum [@klenner_gigahertz_2016; @porcel_two-octave_2017; @mayer_frequency_2015; @epping_chip_2015; @kuyken_octave-spanning_2015-1; @boggio_dispersion_2014; @carlson_photonic-chip_2017; @carlson_self-referenced_2017; @oh_coherent_2017; @hickstein_ultrabroadband_2017]. The spectral shape and efficiency of SCG is determined by the input pulse parameters, the nonlinearity of the material, and the refractive index of the waveguide, which determines the phase-matching conditions. ![\[fig:overview\] a,b) Quasi-phase-matching (QPM) of supercontinuum generation in on-chip photonic waveguides can be achieved via waveguide-width modulation (a) or cladding-modulation (b). c) When the effective index of the soliton in the $\mathrm{TE_{00}}$ mode (“soliton index”) intersects the effective index of a waveguide mode (black curves), phase matching to dispersive waves (DWs) is achieved (green squares) and a spectrum with several peaks (d) is generated. The periodic modulation of the waveguide can enable numerous QPM orders, both positive and negative, which can allow QPM-DW generation to the fundamental mode and to higher-order modes (circles), producing a spectrum with many peaks (e). Note: the index curvature is exaggerated to better show phase-matching.](waveguides.png "fig:"){width="\linewidth"} ![\[fig:overview\] a,b) Quasi-phase-matching (QPM) of supercontinuum generation in on-chip photonic waveguides can be achieved via waveguide-width modulation (a) or cladding-modulation (b). c) When the effective index of the soliton in the $\mathrm{TE_{00}}$ mode (“soliton index”) intersects the effective index of a waveguide mode (black curves), phase matching to dispersive waves (DWs) is achieved (green squares) and a spectrum with several peaks (d) is generated. The periodic modulation of the waveguide can enable numerous QPM orders, both positive and negative, which can allow QPM-DW generation to the fundamental mode and to higher-order modes (circles), producing a spectrum with many peaks (e). Note: the index curvature is exaggerated to better show phase-matching.](concept.png "fig:"){width="\linewidth"} Specifically, when phase-matching between a soliton and quasi-continuous-wave (CW) light is achieved, strong enhancements of the intensity of the supercontinuum spectrum can occur in certain spectral regions. These spectral peaks are often referred to as a dispersive waves (DWs) [@dudley_supercontinuum_2006; @driben_resonant_2015; @mclenaghan_few-cycle_2014; @akhmediev_cherenkov_1995], and they are often crucial for providing sufficient spectral brightness for many applications. The soliton-DW phase-matching condition is typically satisfied by selecting a material with a favorable refractive index profile and engineering the dimensions of the waveguide to provide DWs at the desired wavelengths [@boggio_dispersion_2014; @carlson_photonic-chip_2017]. However, there are limitations to the refractive index profile that can be achieved by adjusting only the waveguide cross-section. Quasi-phase-matching (QPM) takes a different approach, utilizing periodic modulations of the material nonlinearity to achieve an end-result similar to true phase-matching [@boyd_nonlinear_2008; @franken_optical_1963; @armstrong_interactions_1962; @fejer_quasi-phase-matched_1992]. QPM is routinely employed to achieve high conversion efficiency for nonlinear processes such as second harmonic generation and different frequency generation, and QPM can also be used to satisfy the phase-matching conditions for DW generation [@kudlinski_parametric_2015; @luo_resonant_2015; @conforti_multiple_2016; @wright_ultrabroadband_2015; @droques_dynamics_2013]. Here we show that periodic modulations of the effective mode area can enable QPM of DW generation in photonic silicon nitride ($\mathrm{Si_3N_4}$) waveguides, enhancing the intensity of the supercontinuum in specific spectral regions determined by the modulation period. Experimentally, we utilize a sinusoidal modulation of the waveguide width to enable first-order QPM to the $\mathrm{TE_{20}}$ mode. Additionally, we demonstrate that periodic $\mathrm{SiO_2}$ under-cladding provides numerous orders of QPM to both the $\mathrm{TE_{20}}$ and $\mathrm{TE_{00}}$ modes. This quasi-phase-matched dispersive-wave (QPM-DW) scheme provides a fundamentally different approach to phase-matching in SCG, allowing light to be generated outside the normal wavelength range, and providing separate control over the group-velocity dispersion (GVD) of the waveguide (which influences soliton propagation) and DW phase-matching, capabilities that are desirable for many applications of SCG. In the regime of anomalous GVD, the nonlinearity of the material can balance GVD and allow pulses to propagate while remaining temporally short. Such solitons can propagate indefinitely, unless perturbed [@agrawal_nonlinear_2007; @dudley_supercontinuum_2006]. However, in the presence of higher-order dispersion, some wavelengths of quasi-CW light may propagate at the same phase velocity as the soliton. Light at these wavelengths can “leak out” of the soliton, in the form of DW radiation [@dudley_supercontinuum_2006], which is also referred to as “resonant radiation” [@driben_resonant_2015; @mclenaghan_few-cycle_2014] or “optical Cherenkov radiation” [@akhmediev_cherenkov_1995]. In the SCG process, higher-order solitons undergo soliton fission and can convert significant amounts of energy into DW radiation [@dudley_supercontinuum_2006]. The phase-matching condition for DW generation (in the absence of QPM) is simply [@akhmediev_cherenkov_1995] $$\label{eq:dw} n(\lambda_\mathrm{s}) + (\lambda - \lambda_\mathrm{s}) \frac{dn}{d\lambda}(\lambda_\mathrm{s}) + \gamma \, p \, \lambda = n(\lambda),$$ where $\lambda$ is wavelength, $\lambda_\mathrm{s}$ is the center wavelength of the soliton, $n$ is the effective index of the waveguide, $\frac{dn}{d\lambda}(\lambda_\mathrm{s})$ is the slope of the $n$-versus-$\lambda$ curve evaluated at $\lambda_\mathrm{s}$, $\gamma$ is the effective nonlinearity of the waveguide, and $p$ is the peak power. The left side of Eq. \[eq:dw\] represents the effective index of the soliton while the right side represents the effective index of the DW. This equation has a simple graphical interpretation; because all wavelengths in the soliton travel with the same group velocity, the effective index of the soliton is simply a straight line (Fig. \[fig:overview\]c). Where this line crosses the refractive index curve for any waveguide mode (black lines in Fig. \[fig:overview\]c), DW generation is phase matched. Periodic modulations of the waveguide change the effective area of the mode, modulating both $\gamma$ and the GVD, enabling QPM-DW generation (see Supplemental Materials, section IV). The contribution of a modulation (with period $\Lambda$) to the wavevector phase-mismatch will be $k_\mathrm{QPM} = 2\pi q/\Lambda$, where $q$ is the QPM order and can be any positive or negative integer. Thus, the phase-matching condition for dispersive wave generation, in the presence of QPM is $$\label{eq:qpm} \smash{\underbrace{n(\lambda_\mathrm{s}) + (\lambda-\lambda_\mathrm{s}) \frac{dn}{d\lambda} (\lambda_\mathrm{s}) + \gamma \, p\lambda_\mathrm{s}}_{\text{Soliton index}} = \underbrace{n(\lambda)}_{\text{DW index}} + \underbrace{\frac{2\pi q \lambda}{\Lambda}.}_{\text{QPM}}}$$\ This phase-matching condition has a similar graphical interpretation; an additional line is drawn for each QPM order, with curve-crossings indicating QPM of DWs (Fig. \[fig:overview\]c). In addition to satisfying the QPM condition (Eq. \[eq:qpm\]), three additional requirements must be met for efficient DW generation. (1) The QPM order $q$ must be a strong Fourier component of the periodic modulation. (2) There must be overlap between the spatial mode of the soliton and the mode of the DW. (3) The DW must be located in a spectral region where the soliton has significant intensity, a requirement that applies regardless of the phase-matching method. ![\[fig:wiggle\] a) The spectrum of supercontinuum generation from a width-modulated waveguide as a function of input power. The arrow indicates the quasi-phase-matched dispersive wave (QPM-DW) to the $\mathrm{TE_{20}}$ mode. b) The effective refractive index of various modes of the waveguide as a function of wavelength. When the index of the soliton including a first order grating effect from the 6.2- width-modulation (red line) crosses the $\mathrm{TE_{20}}$ mode, a QPM-DW is generated. c) The calculated spectral location of the $\mathrm{TE_{20}}$ QPM-DW as a function of the width-modulation period is in agreement with experimental results. The slight difference in slope may arise from irregularities in the dimensions of the waveguides.](wiggle.png){width="\linewidth"} Experimentally, we explore two different approaches for QPM-DW generation in $\mathrm{Si_3N_4}$ waveguides: width-modulated waveguides and cladding-modulated waveguides. The width-modulated $\mathrm{Si_3N_4}$ waveguides (Fig. \[fig:overview\]a) are fully $\mathrm{SiO_2}$-clad and have a thickness of 750 nm, a maximum width of 1500 nm, and an overall length of 15 mm. Over a 6-mm central region, the width is modulated sinusoidally from 1250 to 1500 nm. Multiple waveguides are fabricated on the same silicon chip, and each waveguide has a different width modulation period, which ranges from 5.5 to . Each cladding-modulated waveguide (Fig. \[fig:overview\]b) consists of a 700-nm-thick Si$_3$N$_4$ waveguide that is completely air-clad, except for underlying SiO$_2$ support structures. The support structures are placed every along the waveguide, and each one contacts the Si$_3$N$_4$ waveguide for approximately . For the cladding-modulated waveguides, the modulation period is kept constant, but several waveguide widths are tested, ranging from 3000 to 4000 nm. We generate supercontinuum by coupling $\sim$80 fs pulses of 1560-nm light from a compact 100 MHz Er-fiber frequency comb [@sinclair_compact_2015]. The power is adjusted using a computer-controlled rotation-mount containing a half-waveplate, which is placed before a polarizer. The polarization is set to horizontal (i.e., parallel to the Si-wafer surface and along the long dimension of the rectangular Si$_3$N$_4$ waveguide), which excites the lowest order quasi-transverse-electric ($\mathrm{TE_{00}}$) mode of the waveguide. We record the spectrum at many increments of the input power using an automated system [@hickstein_gracefulosa_2017] that interfaces with both the rotation mount and the optical spectrum analyzers. The waveguide modes (and their effective indices) are calculated using a vector finite-difference modesolver [@fallahkhair_vector_2008; @bolla_empy_2017], using published refractive indices for $\mathrm{Si_3N_4}$ [@luke_broadband_2015] and $\mathrm{SiO_2}$ [@malitson_interspecimen_1965]. Further experimental details are found in the SM. For the width-modulated waveguides, a narrow peak appears in the spectrum in the 630-nm region (Fig. \[fig:wiggle\]a), and the location of this peak changes with the width-modulation period (Fig. \[fig:wiggle\]c). By calculating the refractive index of the higher order modes of the waveguide, and including the QPM effect from the periodic width modulation (Fig. \[fig:wiggle\]b and Eq. \[eq:qpm\]), we find the QPM-DW generation to the $\mathrm{TE_{20}}$ mode is a likely mechanism for the appearance of this peak (Fig. \[fig:wiggle\]c). The preference for QPM-DW generation to the $\mathrm{TE_{20}}$ mode is a result of the modal overlap [@agrawal_nonlinear_2007; @lin_nonlinear_2007] (Fig. \[fig:modes\]) between the $\mathrm{TE_{20}}$ mode at the DW wavelength ($\sim$630 nm) and the $\mathrm{TE_{00}}$ mode at the soliton wavelength (1560 nm). In general, modes that are symmetric in both the vertical and horizontal (such as the $\mathrm{TE_{20}}$ mode) will have much higher overlap to the fundamental mode than antisymmetric modes (see SM, section III), and are consequently the most commonly used for model phase-matching schemes [@guo_second-harmonic_2016]. ![\[fig:modes\] Electric field profiles for the waveguide modes for a fully $\mathrm{SiO_2}$-clad $\mathrm{Si_3N_4}$ waveguide. a) The $\mathrm{TE_{00}}$ mode at 1560 nm, which is the expected mode of the soliton. b-i) The electric field for various higher order modes at 630 nm, which is the approximate wavelength for the QPM-DW observed for the width-modulated waveguides. The result of the overlap integral of each mode with the $\mathrm{TE_{00}}$ mode at 1560 nm is listed. Only the $\mathrm{TE_{00}}$ and $\mathrm{TE_{20}}$ modes have overlap integrals that are not vanishingly small. Note: for TM modes, $E_y$ is shown, while $E_x$ is shown for TE modes.](modes.png){width="\linewidth"} ![\[fig:suspended\] a) Supercontinuum generation from a 4000-nm-width cladding-modulated waveguide, showing many QPM-DW peaks resulting from QPM orders 6-to-15 to the $\mathrm{TE_{20}}$ mode. The locations of the QPM-DWs predicted by theory (dashed lines) agree with the experiment. DWs corresponding to phase-matching to the soliton and to –1 order QPM to the $\mathrm{TE_{00}}$ mode are indicated with arrows. b) The effective index of the $\mathrm{TE_{00}}$ and $\mathrm{TE_{20}}$ modes, compared with the effective index of the soliton for various QPM orders (colorful lines). The dots indicate the location of the QPM-DW in the $\mathrm{TE_{20}}$ mode. c) Calculations indicate that the locations of the QPM-DWs change as a function of waveguide width, in agreement with experiment. Note: the sharp peaks in the 530-nm region of (a) are a result of third-harmonic generation to higher-order spatial modes [@carlson_self-referenced_2017].](suspended.png){width="\linewidth"} For the cladding-modulated waveguides, many QPM-DW peaks are seen in the supercontinuum spectrum (Fig \[fig:suspended\]a). In some cases, the enhancement in the spectral intensity is as high as 20 dB. Similarly to the width-modulated waveguides, an analysis of the refractive index profile indicates that the $\mathrm{TE_{20}}$ mode is responsible for the QPM-DW generation (Fig. \[fig:suspended\]). Interestingly, peaks are observed corresponding to both odd- and even-order QPM, and effects up to the 16$^\mathrm{th}$ QPM order are detectable. This situation differs from the preference for low, odd-ordered QPM effects in typical QPM materials (such as periodically poled lithium niobate, PPLN [@fejer_quasi-phase-matched_1992]), which usually employ a 50-percent duty-cycle modulation. In contrast, the cladding-modulated $\mathrm{Si_3N_4}$ waveguides have short regions of oxide cladding, followed by long regions of fully air-clad $\mathrm{Si_3N_4}$. This high-duty-cycle square wave is composed of both even- and odd-order harmonics, and consequently provides both even- and odd-order QPM. The simulated QPM-DW positions are in good agreement with the experiment for all grating orders and waveguide widths. Importantly, we see that QPM can still produce strong DWs with QPM orders of 8 or more, indicating that QPM for strongly phase-mismatched processes could be achieved with higher-order QPM instead of short modulation periods, potentially avoiding fabrication difficulties and scattering loss. We also observe QPM-DW generation to the $\mathrm{TE_{00}}$ mode (Fig. \[fig:suspended\]a), which is reproduced by numerical simulations using the nonlinear schrödinger equation [@heidt_efficient_2009; @hult_fourth-order_2007; @ycas_pynlo_2016] (See Supplementary Material, Fig. S2). This is the first demonstration of QPM to produce DWs in on-chip waveguides, but it is interesting to note that the QPM-DWs have been observed in a variety of situations. Indeed, the “Kelly sidebands” [@kelly_characteristic_1992] seen in laser cavities and sidebands seen during soliton propagation in long-distance fiber links [@matera_sideband_1993] are both examples of QPM-DWs. QPM has also been demonstrated for both modulation-instability as well as soliton-DW phase-matching using width-oscillating fibers [@kudlinski_parametric_2015; @conforti_multiple_2016; @droques_dynamics_2013] and for fiber-Bragg gratings [@westbrook_supercontinuum_2004; @kim_improved_2006; @zhao_observation_2009; @yeom_tunable_2007; @westbrook_perturbative_2006]. QPM has also been seen in Kerr frequency comb generation [@huang_quasi-phase-matched_2017]. On-chip waveguides provide a powerful new platform for QPM-DW generation, offering straightforward dispersion engineering, access to a range of modulation periods, scalable fabrication, and the ability to access well-defined higher-order modes. In this first demonstration, the spectral brightness of the QPM-DWs was limited by several factors. First, most of the QPM-DWs were generated in the $\mathrm{TE_{20}}$ mode, which doesn’t have optimal overlap with the $\mathrm{TE_{00}}$ mode. Indeed, in the case where a $-1$-order QPM-DW is generated in the $\mathrm{TE_{00}}$ mode (Fig. \[fig:suspended\]a), the intensity of the light is higher. Second, for the cladding-modulated waveguides, we rely on a QPM structure with a high duty cycle, which effectively spreads the available QPM efficiency over many QPM orders, sacrificing efficiency in one particular order. Third, our waveguides only made modest changes to the effective mode area, and stronger QPM could be likely achieved with a deeper width modulation or stronger change in the cladding index. Finally, the QPM-DWs are often produced far from the pump wavelength, in a spectral region where the soliton is dim. In future designs, optimized strategies for QPM-DW generation could utilize somewhat longer modulation periods, allowing QPM to the $\mathrm{TE_{00}}$ mode, thereby maximizing mode-overlap and allowing the QPM-DWs to be located closer to the soliton central wavelength. Additionally, the waveguide modulation could be designed such that the efficiency of $\pm1$ order QPM is optimized. All of these parameters can be modeled using software that calculates the modes of the waveguide. This fact, combined with the massive scalability of lithographic processing, should allow for rapid progress in designing optimized photonic waveguides for SCG. Currently, designers of waveguide-SCG sources work in a limited parameter space: selecting materials and selecting the dimensions of the waveguide cross section. QPM opens a new dimension in the design-space for photonic waveguides, one that is largely orthogonal to the other design dimensions. This orthogonality exists both in real-space, since the QPM-modulations exist in the light-propagation direction, but also in the waveguide-design-space, as it provides a simple vertical shift of the phase-matching conditions with no bending of the index curve (Fig. \[fig:overview\]c). Consequently, it allows the spectral location of DWs to be modified with minimal effect on the GVD at the pump wavelength, which enables the soliton propagation conditions to be controlled separately from the DW phase-matching conditions. For example, using QPM, DWs could be produced even for purely anomalous GVD, greatly relaxing the requirements for material dispersion and waveguide cross section. Importantly, since the GVD at the pump is known to affect the noise properties of the SCG process [@dudley_supercontinuum_2006], the ability to manipulate the locations of the DWs separately from the GVD could enable SCG sources that are simultaneously broadband and low-noise. In addition, since similar phase-matching conditions apply to SCG with picosecond pulses or continuous-wave lasers [@dudley_supercontinuum_2006], QPM of the SCG process is likely not restricted to the regime of femtosecond pulses. In summary, here we demonstrated that quasi-phase-matching is a powerful tool for controlling the supercontinuum generation process in on-chip photonic waveguides. We experimentally verified that a periodic modulation of either the waveguide width or cladding can allow $\mathrm{Si_3N_4}$ waveguides to produce dispersive wave light at tunable spectral locations. By allowing dispersive waves to be quasi-phase-matched without significantly modifying the dispersion at the pump wavelength, this approach provides independent control over soliton compression and the spectral location of dispersive waves. Thus, quasi-phase-matching provides a new dimension in the design-space for on-chip waveguides and allows supercontinuum sources to be tailored for the specific needs of each application. We thank Jordan Stone, Nate Newbury, Michael Lombardi, and Chris Oates for providing helpful feedback on this manuscript. We thank Alexandre Kudlinski for his insightful comments on a draft of this manuscript. This work is supported by AFOSR under award number FA9550-16-1-0016, DARPA (DODOS and ACES programs), NIST, and NRC. This work is a contribution of the U.S. government and is not subject to copyright in the U.S.A. **Supplemental Materials:\ Quasi-phase-matched supercontinuum-generation in photonic waveguides** Waveguide fabrication ===================== The width-modulated $\mathrm{Si_3N_4}$ waveguides were fabricated by Ligentec, using the “Photonic Damascene” process [@pfeiffer_photonic_2016]. They are fully $\mathrm{SiO_2}$-clad, and have a thickness of 750 nm, and a maximum width of 1500 nm. The end-sections are tapered to 150 nm at the input and exit facets in order to expand the mode and allow for improved coupling efficiency, which is typically –2 dB per facet. The overall length of the waveguides is 15 mm, and, over a 6-mm central region, the width is modulated sinusoidally from 1250 to 1500 nm. The modulation period ranges from 5.5 to . The “suspended” cladding-modulated waveguides were fabricated at NIST Gaithersburg. First, 700 nm of Si$_3$N$_4$ was deposited onto thermally oxidized silicon wafers using low pressure chemical vapor deposition (LPCVD), and the waveguides were patterned with electron-beam lithography and reactive ion etching. Plasma enhanced chemical vapor deposition (PECVD) was then used to selectively deposit a SiO$_2$ cladding on the input and output coupling regions. The sample was then coated with photoresist and patterned to define the suspended regions. After development, the sample was soaked for 30 minutes in a 6:1 solution of buffered oxide etch (BOE) to release the waveguides. The waveguides are 12 mm in total length, and the first and last 0.5 mm are clad with SiO$_2$ in order to provide a symmetric mode profile that improves coupling efficiency. The SiO$_2$ supports are on a pitch of and are in contact with the $\mathrm{Si_3N_4}$ waveguide for approximately . Experiment ========== The light is coupled into each waveguide using an aspheric lens (numerical aperture of 0.6) designed for 1550 nm. The light is collected by butt-coupling an $\mathrm{InF_3}$ multimode fiber (numerical aperture of 0.26) at the exit facet of the chip. The waveguide output is then recorded using two optical spectrum analyzers (OSAs); a grating-based OSA (Ando 6315E) is used for the spectrum across the visible and near-infrared regions, while a Fourier-transform OSA (Thorlabs OSA205) extends the coverage to 5600 nm. The angle of the half-waveplate is controlled using a Thorlabs K10CR1 rotation stage. For easy visualization of the spectral enhancement due to QPM, Fig. \[fig:lineouts\] shows representative spectra from both the width-modulated and cladding-modulated waveguides. These spectra are single rows of Fig. 2a and 4a in the main text and in each case, the QPM-enabled spectral peaks can have heights of 20 dB or more relative to the supercontinuum. ![\[fig:lineouts\] Experimental supercontinuum spectra showing quasi-phase-matched dispersive-waves (QPM-DWs) due to waveguide modulation. a) Spectrum of the supercontinuum light produced from the width-modulated waveguide shown in Fig. 2a (6.2- width-modulation period) with 65.0 mW incident power. Enhancement of the spectral flux to the $\mathrm{TE_{20}}$ mode can be seen as a sharp peak near 630 nm. b) Spectrum of the supercontinuum light produced by the cladding-modulated waveguide shown in Fig. 4a (width of 4000-nm) with 50.5 mW of incident power. Numerous QPM orders can be seen with peak heights of $>$20 dB in some cases. The dashed red lines show the theoretically predicted position of each $\mathrm{TE_{20}}$ QPM order. ](lineouts.png){width="\linewidth"} Waveguide modes =============== We label the quasi-transverse-electric (TE, horizontal polarization) and quasi-transverse-magnetic (TM, vertical polarizaion) waveguides modes using subscripts that indicate the number of nodes in the $x$- and $y$-directions respectively (Fig. 3). For example, the $\mathrm{TE_{20}}$ mode (Fig. 3e) has two nodes in the $x$-direction and zero nodes in the $y$-direction. Experimentally, we observe quasi-phase-matched dispersive wave (QPM-DW) generation from the $\mathrm{TE_{00}}$ mode to the $\mathrm{TE_{20}}$, which seems somewhat counter-intuitive, since in a waveguide, all modes *at the same wavelength* are orthogonal. However, it is possible for one mode at a certain wavelength to have a nonzero overlap with other modes at a different wavelength. For a simple rectangular waveguide with symmetric cladding (Fig. 3), overlap to the $\mathrm{TE_{00}}$ mode can only occur for symmetric modes, i.e., modes that have even numbers of $x$ and $y$ nodes (where zero is an even number). For example, if we consider the antisymmetric $\mathrm{TE_{10}}$ mode (Fig. 3c), we find that the mode overlap integral will be strictly zero, since any overlap on the right side will be precisely canceled by the antisymmetric left side. In contrast, the $\mathrm{TE_{20}}$ mode (Fig. 3e) can have nonzero overlap with the $\mathrm{TE_{00}}$ mode. Thus, to generate light into higher order modes using a lowest-order-mode pump, symmetric modes such as the $\mathrm{TE_{20}}$ mode offer the best overlap [@guo_second-harmonic_2016]. QPM mechanism ============= The width- and cladding-modulated waveguides provide QPM for the DW phase-matching condition. However, the experimental results do not specify if the effect is due to the modulation of the intensity of the light (and therefore modulation of the effective nonlinearity of the waveguide, $\gamma$) or if it is due to the modulation of the waveguide group-velocity dispersion (GVD), which changes the effective phase-mismatch for soliton-DW phase-matching along the waveguide length. Depending on the specific situation, it is conceivable that either effect causes the experimentally observed QPM. Thus, we employ numerical solutions to the Nonlinear Schrödinger equation (NLSE) [@heidt_efficient_2009; @hult_fourth-order_2007; @ycas_pynlo_2016] to investigate which effect plays the most important role for our waveguides. We run NLSE simulations in three cases: 1. Including the modulation of both the GVD and $\gamma$, 2. Including the modulation of $\gamma$ only, 3. Including the modulation of the GVD only. Our simulations only consider the fundamental mode of the waveguide and do not take into account effects due to higher order modes, which prevents them from modeling the QPM-DWs in the $\mathrm{TE_{20}}$ mode. Fortunately, in the case of the cladding-modulated waveguides, QPM-DWs to the fundamental ($\mathrm{TE_{00}}$) mode are experimentally observed on the short-wavelength side of the main DW (Fig. 4a). These features are reproduced by the NLSE (Fig. \[fig:NLSE\]a,c) *only* when the modulation of the GVD is taken into account. When the simulations include the modulation of $\gamma$ alone, no obvious QPM effects are seen, suggesting that for the cladding-modulated waveguides, phase-matching is enabled by GVD modulation. {width="\linewidth"} For the width modulated waveguides, the only experimentally observed QPM-DWs are those in the $\mathrm{TE_{20}}$ mode, and thus we do not attempt to precisely model the experiment, since the simulations do not take into account higher order modes. Instead, we model a waveguide with a similar sinusoidal width modulation (1250 to 1500 nm), but with a modulation period of , which enables a QPM-DW in the $\mathrm{TE_{00}}$ mode. Again, the NLSE indicates that the modulation of the GVD is the dominant effect (Fig. \[fig:NLSE\]b,d), in agreement with previous studies of QPM-DW generation [@kudlinski_parametric_2015; @conforti_multiple_2016; @droques_dynamics_2013]. We expect that the GVD will also provide QPM for the QPM-DWs in the $\mathrm{TE_{20}}$ mode. This method of QPM via GVD modulation can be understood by considering that the soliton and the DW are phase-mismatched, and that the GVD determines the degree of mismatch. As a result, the DW light experiences regions of constructive and destructive interference along the length of the waveguide. By modulating the GVD, the regions of constructive interference can be made slightly longer than the regions of destructive interference. The net result is that, while the intensity of the DW still oscillates to some degree, there is a constructive build-up of light intensity [@droques_dynamics_2013]. The oscillation and build-up of the QPM-DW light as a function of propagation length can be seen in Figs. \[fig:NLSE\]c and \[fig:NLSE\]d in a region near the point of soliton fission (approximately 3 mm in Fig. \[fig:NLSE\]b and 2 mm in Fig. \[fig:NLSE\]d). Disclaimer ========== Certain commercial equipment, instruments, or materials are identified here in order to specify the experimental procedure adequately. Such identification is not intended to imply recommendation or endorsement by the National Institute of Standards and Technology, nor is it intended to imply that the materials or equipment identified are necessarily the best available for the purpose. This work is a contribution of the United States government and is not subject to copyright in the United States of America.

--- author: - 'Andrei Linde,' - 'Dong-Gang Wang,' - 'Yvette Welling,' - Yusuke Yamada - and Ana Achúcarro bibliography: - 'bibfile.bib' --- Introduction ============ In this paper we will continue our investigation of multifield $\alpha$-attractors following the earlier work [@Achucarro:2017ing]. Cosmological $\alpha$-attractors form a broad recently discovered class of inflationary models [@Kallosh:2013hoa; @Ferrara:2013rsa; @Kallosh:2013yoa; @Cecotti:2014ipa; @Galante:2014ifa; @Kallosh:2016gqp; @Kallosh:2015zsa; @Ferrara:2016fwe; @Kallosh:2017ced; @Kallosh:2017wnt]. These models give nearly model-independent cosmological predictions $n_s\approx 1-{2\over N}$, $r\approx {12\alpha \over N^2}$, providing a very good match to the recent observational data [@Ade:2015lrj; @Ade:2015ava]. These models are most naturally formulated in terms of the theory of scalar fields with hyperbolic geometry  [@Kallosh:2015zsa; @Ferrara:2016fwe; @Kallosh:2017ced]. For example, in the simplest supergravity embedding of these models, the potential depends on the complex scalar $Z= |Z| \, e^{i\theta} $, where $Z$ belongs to the Poincaré disk with $|Z| < 1$, and the kinetic terms are 3  . \[geometry\] In many versions of these models, the field $\theta$ is heavy and stabilized at $\theta = 0$, so that the inflationary trajectory corresponds to the evolution of the single real field $Z = \bar Z$. This field is not canonically normalized, but one can easily express everything in terms of the canonically normalized field $\varphi$, where \[ZZ\] Z =  . Then the potential of the inflaton field $V(Z,\bar Z)$ along the inflaton direction can be represented as \[TT\] V = V()  . Since $\tanh{{\varphi}\over \sqrt{6\alpha}}\to \pm 1$ in the limit ${\varphi}\to \pm \infty$ (corresponding to $Z\to \pm 1$), inflationary predictions are mainly determined by the behavior of $V(Z,\bar Z)$ at the boundary of the moduli space $|Z| \to 1$. This explains the stability of these predictions with respect to modifications of the original potential $V(Z,\bar Z)$ everywhere outside a small vicinity of the points $Z = \pm 1$. However, there is another class of models, where the field $\theta$ is not stabilized and may remain light during the cosmological evolution. A particularly interesting case is $\alpha=1/3$, which has a fundamental origin from maximal ${\mathcal{N}}=4$ superconformal symmetry and from maximal ${\mathcal{N}}=8$ supergravity [@Kallosh:2015zsa]. For $\alpha=1/3$, a class of supergravity embeddings are known to possess an unbroken or slightly broken $U(1)$ symmetry, which makes both ${\varphi}$ and $\theta$ light [@Kallosh:2015zsa; @Achucarro:2017ing]. In that case, the inflationary evolution may involve both fields, which would require taking into account the multi-field effects, as discussed in [@Gordon:2000hv; @GrootNibbelink:2000vx; @GrootNibbelink:2001qt; @Bartolo:2001rt; @Lalak:2007vi; @Achucarro:2010jv; @Achucarro:2010da; @Peterson:2010np]. However, in contrast with the naive expectation, the cosmological predictions of the simplest class of such models are very stable not only with respect to modifications of the potential of the field $|Z|$, but also with respect to strong modifications of the potential of the field $\theta$ [@Achucarro:2017ing]. The predictions coincide with those of the single-field $\alpha$-attractors: $n_s\approx 1-{2\over N}$, $r\approx {12\alpha \over N^2}$. Thus, this class of models exhibits double-attractor behavior, providing universal cosmological predictions, which are stable with respect to strong modifications of the potential $V(Z,\bar Z)$. Other closely related investigations can be found in [@Kallosh:2013daa; @Christodoulidis]. This result is valid for all models with hyperbolic geometry [(\[geometry\])]{} and an unbroken or slightly broken $U(1)$ symmetry, for any $\alpha \lesssim O(1)$ [@Achucarro:2017ing]. In supergravity such models were known only for $\alpha = 1/3$, but recently such models were constructed for all $\alpha < 1$ [@Yamada:2018nsk], so now we have a broad class of supergravity models with the double attractor behavior found in [@Achucarro:2017ing]. In this paper, we will further generalize this construction. In the models considered in [@Achucarro:2017ing], we had only one stage of inflation, which ended at ${\varphi}= 0$. Instead of that, we will consider Mexican hat potentials $V({\varphi}, \theta)$ with a pseudo-Goldstone direction along the minimum with respect to the field ${\varphi}$, see Figs. \[fig:Higgs\], \[fig:Higgs2\]. In such models the first stage of $\alpha$-attractor inflation driven by the field ${\varphi}$ can be followed by a subsequent stage of inflation driven by the field $\theta$, as in the natural inflation scenario [@Freese:1990rb]. This leads to a novel realization of the natural inflation scenario in supergravity. Natural inflation is already in tension with the latest CMB data, except possibly for super-Planckian values of the axion decay constant, which are difficult (if not impossible) to achieve in string theory [@Banks:2003sx; @Conlon:2012tz]. A number of extensions of natural inflation have tried to address this problem in different ways [@Kim:2004rp; @Dimopoulos:2005ac; @Czerny:2014qqa; @Li:2015mwa; @Achucarro:2015rfa; @Achucarro:2015caa; @Conlon:2012tz; @Cicoli:2014sva; @Bachlechner:2014gfa; @Bachlechner:2015qja; @Long:2016jvd; @Dias:2018pgj]. Here, the hyperbolic field metric results in an effective axion decay constant that is very different from the naive one. Our construction is embedded in $N=1$ supergravity, but whether it admits an embedding in string theory remains an open question here as much as for other extensions of natural inflation. We will not discuss this question in our paper, trying to solve one problem at a time. Indeed, theoretical problems with natural inflation appear already at the level of supergravity. Historically, one could always postulate the periodic potential required for natural inflation [@Freese:1990rb]. But it was very difficult to stabilize the inflaton direction in supergravity-based versions of natural inflation. This problem was solved only few years ago [@Kallosh:2014vja] (see also [@Kallosh:2007ig; @Kallosh:2007cc]). However, in all natural inflation models in supergravity constructed in [@Kallosh:2014vja], the inflaton field was different from the angular direction $\theta$ in the Mexican hat potential as originally proposed back in [@Freese:1990rb]. This problem, which remained unsolved since 1990, will be solved in our paper. However, as we will discuss in the end of the paper, embedding of natural inflation in the theory of $\alpha$-attractors does not remove the tension between natural inflation and the latest observational data. Moreover, adding a stage of natural inflation [*after*]{} the stage of inflation driven by $\alpha$-attractors tends to decrease the predicted value of $n_{s}$. From this perspective, inflationary models where the last 60 e-foldings of inflation are driven by $\alpha$-attractors, without an additional stage of natural inflation, seem to provide a better fit to the existing observational data [@Ade:2015lrj; @Ade:2015ava]. {#sT} There are two different ways to obtain $U(1)$ symmetric potentials for $\alpha$ attractors in supergravity. The simplest one is to consider models with $\alpha = 1/3$ with the  potential and superpotential \[k1\] K=- (1-Z|Z -S|S)  , W= S f(Z)  , where $S$ is a nilpotent field and $f(Z)$ is a real holomorphic function. In these models, the potential $V(Z,\bar Z) = |f(Z)|^{2}$. For the simplest functions $f(Z) = Z^{n}$, we find $\theta$-independent potentials $V(Z,\bar Z) = |f(Z)|^{2} = f^{2}(\tanh{{\varphi}\over \sqrt{6\alpha}})$. However, for more complicated functions, such as $1-Z^{2}$, which we may need to reproduce the natural inflation potential, the $U(1)$ symmetry of the potential is strongly broken. Therefore it is essential for us to use the novel method developed in [@Achucarro:2017ing] for $\alpha = 1/3$ and generalized in for all $\alpha < 1$ [@Yamada:2018nsk]. For a detailed discussion of this construction we refer the readers to the original papers where this formalism was developed [@Achucarro:2017ing; @Yamada:2018nsk], and also to the closely related works [@McDonough:2016der; @Kallosh:2017wnt]. Here is a short summary of the results required for our work. Following [@Yamada:2018nsk], we will consider supergravity with the  potential and superpotential K=-3(1-Z|Z)+S+|S+G\_[S|S]{}S|S  , W=W\_0  .\[KW\] The function $G_{S\bar S}$ is given by G\_[S|S]{}=. In this setting, the inflaton potential is given by $V(Z,\bar Z)$, where $ V(Z,\bar Z)$ is an arbitrary real function of $Z$ and $\bar{Z}$. This formulation is valid for any $\alpha < 1$. For the special case $\alpha = 1/3$, it coincides with the formulation given in [@Achucarro:2017ing]. A detailed explanation of notations and basic principles of this general approach to inflation in supergravity can be found in [@Kallosh:2017wnt]. In our paper, we will focus on constructing specific potentials $ V(Z,\bar Z)$ suitable for implementation of natural inflation in this context. The kinetic term, which plays important role in this scenario , is given by $$\label{kin} 3\alpha\frac{\partial Z \partial \bar{Z}}{(1-Z\bar{Z})^2} = \frac{1}{2}\left(\partial {\varphi}\right)^2 + \frac{3\alpha}{4} \sinh^2 \left(\sqrt{\frac{2}{3\alpha}}{\varphi}\right)\left(\partial \theta\right)^2 \ .$$ Importantly, for all functions $V$ depending only on the product $Z\bar Z$, the potential is $U(1)$ invariant, i.e. it does not depend on the angular variable $\theta$. That is exactly what we need as a first step towards the theory of natural inflation: a potential with a flat Goldstone direction. To give a particular example, we will begin with the Higgs-type potential \[h\] V(Z, |Z) =V\_0 (1- c\^[-2]{} Z|Z)\^[2]{} with $|Z|< 1$. In terms of ${\varphi}$ and $\theta$, the potential is given by \[toymodelpotential\] V(, )=V\_0 (1-c\^[-2]{}\^[2]{} )\^[2]{} . One may call it “hyperhiggs” potential. It has a $\theta$-independent minimum at \[minimum\] = c  , which is shown as a red circle in Fig. \[fig:Higgs\]. ![The shape of the hyperbolic generalization of the Higgs potential $V_0 \left(1-c^{-2}\tanh^{2} \frac{{\varphi}}{\sqrt{6\alpha}} \right)^{2}$ for $c = 0.8$ and $\alpha = 1/3$.[]{data-label="fig:Higgs"}](AlphaHiggs.jpg){width="50.00000%"} Natural inflation in disk variables: T-models with a Mexican hat potential {#sec:toy} ========================================================================== As a next step, we will introduce a small term breaking the $U(1)$ symmetry. Here again one can use several different methods. For example, one can add a term proportional to $ Z$ to the superpotential in . One can show that this leads to the natural inflation potential. Instead of that, we will use a simpler phenomenological method, which we already used in [@Achucarro:2017ing]: We will add to the hyperhiggs potential [(\[h\])]{} a small term containing a sum $Z^n+\bar{Z}^n$, which breaks the $U(1)$ symmetry. More specifically, one may consider the following potential on the unit disk as a toy model for natural inflation V(Z, |Z) =V\_0 , with $c < 1$ and $0<A\ll 1$. In terms of ${\varphi}$ and $\theta$, the potential is given by V(, )=V\_0  . \[toymodelpotential2\]The term breaking the rotational symmetry was chosen so that for $A>0$ it is positive and monotonically grows with ${\varphi}$ everywhere except $\theta = {(2k+1)\pi\over n}$, where $k, n$ are integers, and $n > 1$. The shape of this potential is shown in Fig. \[fig:Higgs2\] for $A \ll1$ and $n = 1, 2$ and $8$. ![The natural inflation potential [(\[toymodelpotential2\])]{} with one, two, and eight minima (shown by red) along the pseudo-Goldstone (axion) direction.[]{data-label="fig:Higgs2"}](1min.jpg "fig:"){width="32.00000%"} ![The natural inflation potential [(\[toymodelpotential2\])]{} with one, two, and eight minima (shown by red) along the pseudo-Goldstone (axion) direction.[]{data-label="fig:Higgs2"}](2min.jpg "fig:"){width="32.00000%"} ![The natural inflation potential [(\[toymodelpotential2\])]{} with one, two, and eight minima (shown by red) along the pseudo-Goldstone (axion) direction.[]{data-label="fig:Higgs2"}](8min.jpg "fig:"){width="30.00000%"} If $A$ is sufficiently small, and $1-c\ll 1$, the potential has a flat direction at $\tanh \frac{{\varphi}}{\sqrt{6\alpha}} = c$, with a periodic potential \[natpot\] V() = 4V\_[0]{}A c\^[[n+2]{}]{}\^[2]{} [n2]{}  . Naively, one could expect that the circumference of the circle with any particular value of the field ${\varphi}$ is given by $L=2\pi {\varphi}$. However, because of the hyperbolic geometry reflected in the kinetic term [(\[kin\])]{}, the circumference of the red circle shown in Fig. \[fig:Higgs\] in terms of the canonically normalized angular field is greater by the factor $ \sqrt{\frac{3\alpha}{2}} \sinh \left(\sqrt{\frac{2}{3\alpha}}{\varphi}\right)$. Using equation [(\[minimum\])]{}, one finds that the circumference of the red circle is given by $$\label{kin2} L =2\pi { \sqrt {6\alpha}\, c \over 1-c^{2}} \ .$$ Thus for $1-c\ll 1$ one finds the potential where the length of the Goldstone direction is extremely large. This is the key feature of similar potentials to be considered in this paper, which allows to implement natural inflation scenario in this context. Usually one represents the length of the circle as $L = 2\pi f_{a}$, where $f_{a}$ is the axion decay constant. However, the possibility to implement the natural inflation scenario depends not on the full length of the circle, but on the distance between the two nearby minima, $$\label{kin2n} L_{n} = {2\pi\over n} { \sqrt {6\alpha}\, c \over 1-c^{2}} \ .$$ Therefore in this paper we will use an alternative definition of the axion decay constant, such that $L_{n} = 2\pi f_{a}$, where $$f_{a} = \frac{\sqrt{6\alpha}\, c}{n(1-c^2)} \ . \label{beta}$$ Natural inflation requires that the absolute value of the mass squared of the field $\theta$ at its maximum at $\theta = 0$ should be smaller than its height. This implies that natural inflation may occur only for $f_{a} > 1/2$. To give a particular example, this condition can be met for $\alpha = 1/3$, $0.5 \lesssim c < 1$. Natural inflation in half-plane variables: E-models {#sec:E} =================================================== Following [@Yamada:2018nsk], we will use the Kähler potential and superpotential[^1] K=-3(T+|T)+S + |S + G\_[S|S]{}S|S  , W=W\_0  . \[KE\] Here G\_[S|S]{}=  ,\[GE\] and $T$ is the half-plane variable, $T = e^{-\sqrt{2\over 3 \alpha}{\varphi}} + i\theta$, where ${\varphi}$ is a canonically normalized inflaton field. The kinetic term is given by $$\label{kinT2} 3\alpha\frac{\partial T \partial \bar{T}}{(T+\bar{T})^2} = \frac{1}{2}\left(\partial {\varphi}\right)^2 + \frac{3\alpha}{4} e^{2\sqrt{\frac{2}{3\alpha}}{\varphi}} \left(\partial \theta\right)^2 \ .$$ The kinetic coefficient of $\theta$ plays an important role as in the T-model case. ![The shape of the E-model $\alpha$-attractor with a flat axion valley at ${\varphi}= c$, for $c=10$ and $\alpha = 1/3$.[]{data-label="fig:E1"}](flat.pdf){width="54.00000%"} As a starting point, we will consider a potential \[Eflat\] V=V\_[0]{}(1 - [T + |T2]{} e\^[c]{})\^[2]{} = V\_[0]{}(1- e\^[-(-c)]{})\^[2]{}  . This is an E-model $\alpha$-attractor potential with respect to the field ${\varphi}$, which has a flat axion direction at ${\varphi}= c$, see Fig. \[fig:E1\]. ![The shape of the E-model $\alpha$-attractor with a modulated axion valley at ${\varphi}=c$ and a natural inflation potential for the field $\theta$, for $c=10$ and $\alpha = 1/3$. []{data-label="fig:E2"}](nat.pdf){width="55.00000%"} To describe natural inflation, one may add to [(\[Eflat\])]{} a term $4V_{0}A \cosh^{2} {n(T- \bar T)\over 4} $. The resulting potential shown in see Fig. \[fig:E2\] is given by \[Eax\] V=V\_[0]{}(1- e\^[-(-c)]{})\^[2]{} +4 V\_[0 ]{}A \^[2]{}[n2]{}  . At large ${\varphi}$, the last term in this expression coincides with the last term in . This potential is periodic in $\theta$, with the period ${2\pi\over n}$, as in the T-models considered earlier. The physical distance between the two nearby minima of the periodic potential is given by L\_[n]{} = [2n]{} e\^[c]{}  . Thus for $c \gg \sqrt{3\alpha \over 2}$, the physical distance between the minima becomes exponentially large, and one can have natural inflation driven by the field $\theta$. We define the decay constant as $$f_a={1\over n}\sqrt{\frac{3\alpha}{2}}e^{\sqrt{\frac{2}{3\alpha}}c},$$ and at ${\varphi}=c$, the canonically normalized axion $\hat\theta$ has the potential $$\label{natpot2} V= 4 A\,\cos^{2}\left(\frac{\hat\theta}{2f_a}\right).$$ Equations of motions and the slow-roll regime {#EOM} ============================================== Inflation in this model consists of two stages. First, the field ${\varphi}$ rolls down from its large values, whereas the field $\theta$ remains nearly constant, in accordance with [@Achucarro:2017ing]. Then when the field comes close to the minimum, the field $\theta$ begins to evolve, and a stage of natural inflation may occur. The details of the process depend on initial conditions and on the parameters of the model. We performed a full analytical and numerical analysis for the models discussed in the present paper. Fortunately, the main conclusions can be understood using the simple arguments following [@Achucarro:2017ing], which we present in this section. Here we begin with the action of the scalar field $Z$ in T-models $$\int d^4x\sqrt{-g}\left(-\frac12\partial^\mu{\varphi}\partial_\mu{\varphi}-\frac{3\alpha}4\sinh^2 \left(\sqrt{\frac{2}{3\alpha}}{\varphi}\right)\partial_\mu\theta\partial^\mu\theta-V({\varphi},\theta)\right)~,$$ where we have used the parametrization $Z=e^{i\theta}\tanh\frac{{\varphi}}{\sqrt{6\alpha}}$. The equations of motion for the homogeneous fields ${\varphi}$ and $\theta$ in this theory look as follows: \[s2a\] + 3H- ([2]{}) \^[2]{} = - V\_ , \[s1\] +3H+ 2 [ ]{} = - [V\_\^[2]{} ]{}  , where we have used the flat FLRW metric $g_{\mu\nu}=\text{diag}(-1,a^2(t),a^2(t),a^2(t))$, $H\equiv \dot a/a$ and the dot denotes a time derivative. We will study the evolution of the fields during inflation at asymptotically large ${\varphi}$, where the terms $ \frac{2V_\theta}{3\alpha\sinh^2 \left(\sqrt{\frac{2}{3\alpha}}{\varphi}\right)}$ and $V_{{\varphi}}$ in the right hand sides of these equations are exponentially small. An investigation of the combined evolution of the fields ${\varphi}$ and $\theta$ in this case shows that if the evolution begins at sufficiently large values of the field ${\varphi}$ in the region with $V >0$, then after a short period of relaxation, the fields approach a slow roll regime where the kinetic energy of both fields is much smaller than $V$. Under this assumption, one can neglect the first and the third terms in equation (\[s1\]) and the first term in equation (\[s2a\]). Taking into account that $\tanh \sqrt{\frac{2}{3\alpha}}{\varphi}\approx 1$ and $\sinh \left(2\sqrt{\frac{2}{3\alpha}}{\varphi}\right) \sim 2 \sinh^{2} \left(\sqrt{\frac{2}{3\alpha}}{\varphi}\right)\sim {1\over 2}\, e^{2\sqrt {2\over {3\alpha}}\varphi}$ at ${\varphi}\gg \sqrt{\alpha}$, one can represent equations for the fields $\theta$ and ${\varphi}$ in the slow roll approximation as follows: \[ts1\] 3H= e\^[2]{} \^[2]{} - V\_ , \[o1\] 3H= -[83]{} e\^[-2]{}  V\_  , where $H^{2} = V({\varphi},\theta)/3$. Using the last of these two equations, the first one can be simplified: \[simp\] 3H= [829]{}[ [V\^[2]{}\_]{} V]{} e\^[-2]{} - V\_ . Note that in the $\alpha$-attractor scenario the potential $V$ is a function of $\tanh {\frac{{\varphi}}{\sqrt{6\alpha}}}$, which behaves as $1-2 e^{-\sqrt {2\over 3\alpha}{\varphi}}$ at large ${\varphi}$. As a result, its derivative $V_{{\varphi}}$ is suppressed by the factor $e^{-\sqrt {2\over 3\alpha}{\varphi}}$. Consider for example the theory [(\[toymodelpotential2\])]{}. In this model, at ${\varphi}\gg \sqrt { \alpha}$, one has $V_{\theta} \sim V_{0}$, and $V_{{\varphi}} \sim V_{0}e^{-\sqrt {2\over 3\alpha}{\varphi}}$, up to some factors depending on $A$, $c$, and $\theta$. Therefore the second term in [(\[simp\])]{} is suppressed by an extra factor $e^{-\sqrt {2\over 3\alpha}{\varphi}}$ as compared to $V_{{\varphi}}$, so it can be neglected, and one can write the last equation at large ${\varphi}$ as follows: \[o2\] 3H= - V\_ . Comparing [(\[o1\])]{} and [(\[o2\])]{} one finds that during the early stage of inflation the value of $\dot\theta$ is suppressed by the exponentially small factor $e^{-\sqrt {2\over 3\alpha}{\varphi}}$ with respect to $\dot{\varphi}$. In other words, in the regime ${\varphi}\gg \sqrt {3\alpha\over 2}$, the field evolution is almost straight in the radial direction, [*i.e.*]{} it can be “rolling on the ridge", despite of the angular dependence of the potential [@Achucarro:2017ing]. An analogous result is valid for inflation in multi-field E-models. The action of the scalar fields is $$\int d^4x\sqrt{-g}\left(-\frac12\partial^\mu{\varphi}\partial_\mu{\varphi}-\frac{3\alpha}{4}e^{2\sqrt{\frac{2}{3\alpha}}{\varphi}}\partial_\mu\theta\partial^\mu\theta-V({\varphi},\theta)\right) \ .$$ The equations of motion for the homogeneous fields ${\varphi}$ and $\theta$ are $$\ddot{\varphi}+3H\dot{\varphi}-\sqrt{\frac{3\alpha}{2}}e^{2\sqrt{\frac{2}{3\alpha}}{\varphi}}\, \dot\theta^{2}+V_{\varphi}=0 \ ,$$ $$\ddot\theta+3H\dot\theta+2\sqrt{\frac2{3\alpha}}\dot{\varphi}\dot\theta+\frac{2}{3\alpha}e^{-2\sqrt{\frac{2}{3\alpha}}{\varphi}}V_\theta=0 \ .$$ At large ${\varphi}\gg \sqrt{\alpha}$ in the slow-roll approximation one has $$\label{Es1} 3H\dot{\varphi}-\sqrt{\frac{3\alpha}{2}}e^{2\sqrt{\frac{2}{3\alpha}}{\varphi}} \dot\theta^{2}+V_{\varphi}=0 \ ,$$ $$\label{Es2} 3H\dot\theta+\frac{2}{3\alpha}e^{-2\sqrt{\frac{2}{3\alpha}}{\varphi}}V_\theta=0 \ .$$ These two equations coincide with equations [(\[o1\])]{}, [(\[simp\])]{} up to factors $\mathcal{O}(1)$. Just as in the T-models, equation [(\[Es1\])]{} can be approximated by [(\[o2\])]{}, and the value of $\dot \theta$ typically is suppressed by the factor of $e^{-\sqrt {2\over 3\alpha}{\varphi}}$ with respect to $\dot{\varphi}$. Therefore the field evolution in E-models with $\theta$-dependence also shows the “rolling on the ridge” behavior at ${\varphi}\gg \sqrt{\alpha}$. This fact is responsible for the main result derived in [@Achucarro:2017ing] for T-models: general inflationary predictions of the new class of models coincide with general predictions of single field $\alpha$-attractors for large number of e-foldings $N$: n\_s= 1-[2N]{} , r= [12N\^2]{}  . The analysis performed above shows that the same conclusion should be valid for E-models as well. For a better understanding of this result, one should take into account that in the simplest $\alpha$-attractor models the exponential suppression factor described above is given by e\^[-]{} \~[38 N]{}  , where $N$ is the number of remaining e-foldings of inflation at the time when the inflaton field is given by ${\varphi}$. For example, for $\alpha = 1/3$ and $N \sim 55$, this suppression factor is about $2\times 10^{{-3}}$, which explains why the field $\theta$ practically does not move at the early stages of inflation. However, one can always construct models where some of the assumptions of our analysis are not satisfied. This may happen, in particular, in the models where inflation in the $\alpha$-attractor regime driven by the field ${\varphi}$ is followed by another stage of inflation, such as the natural inflation scenario driven by the angular component of the inflaton field. We will discuss this possibility now. Phenomenology {#sec:pheno} ============= In this section we describe the phenomenology of our toy models for natural inflation. The phenomenology of natural inflation in the E-model is very similar to that of the $U(1)$ symmetric T-model, and our results are phrased in such a way that they apply to both. Let us first study the simplest $U(1)$ symmetric models with $A= 0$. In that case, after a long stage of inflation, the value of $\dot \theta$ vanishes, and inflation is driven by the field ${\varphi}$ [^2]. After the end of inflation, the field ${\varphi}$ falls to the minimum of its potential (the Goldstone valley with respect to the field $\theta$), and oscillates there, which leads to reheating. In that case, the observational predictions of inflation coincide with the prediction of the single field $\alpha$-attractors, despite the existence of the massless particles $\theta$ produced by reheating. Clearly, for sufficiently small values of $A$ these conclusions will remain valid, unless the stage of $\alpha$-attractor inflation driven by the field ${\varphi}$ is followed by another stage of inflation involving the field $\theta$. For example, after the stage of oscillations, the field ${\varphi}$ may settle near the minimum of its potential, but the field $\theta$ may be away from its minimum. Then the field $\theta$ will start moving, driving a stage of natural inflation in the potential [(\[natpot\])]{}, [(\[natpot2\])]{}. As we already mentioned, natural inflation is possible only for $f_{a} > 1/2$. Thus for sufficiently small $A$ and $f_{a} < 1/2$ we have only one stage of inflation. It is driven by the field $\phi$ and leads to the universal predictions $$n_s = 1-\frac{2}{N}, \quad r = \frac{12\alpha}{N^2} . \label{universalpredictions}$$ Meanwhile for $f_{a} \gg 1$, the $\alpha$-attractor regime can be followed by the stage of natural inflation. In that case, the results depend on the duration of this stage. If this stage is longer than $N \sim 55$ e-foldings, all observational predictions will be determined by the stage of natural inflation, and the first stage driven by the field ${\varphi}$ will be forgotten. As we will show in the Appendix, this regime occurs for $f_{a} \gg \sqrt{N}/\pi \sim 3$. In that case, if we start with a typical initial value of $\theta$ in the range $|\theta| \sim \pi$, the stage of natural inflation will last more than $N \sim 55$ e-foldings. Thus we have found a rather non-trivial realization of natural inflation in the new context provided by the theory of $\alpha$-attractors. The latest observational data do not offer much support to natural inflation as compared to the simplest $\alpha$-attractor models, but more data are needed to reach a definite conclusion [@Ade:2015lrj; @Ade:2015ava]. Finally, there is an intermediate regime $1/2\lesssim f_{a} \lesssim \sqrt{N}/\pi \sim 3$, where the duration of natural inflation typically is smaller than $N \sim 55$ e-foldings. In this regime, the large scale CMB anisotropy will be described by the universal $\alpha$-attractor results \[universalpredictions\]. However, in these results instead of the full number $N \sim 55$ one should use $N_{\alpha}=N_{total}-N_{{natural}}$, where $N_{total} \sim 55$, and $N_{{natural}}$ is the total number of e-foldings during the stage of natural inflation. Unless $N_{\alpha} \gg N_{{natural}}$, the corresponding predictions will lead to unacceptably small values of $n_{s}$. From this perspective, more complicated models studied in this paper do not offer any advantages as compared to the simplest $\alpha$-attractor models. Nevertheless, we believe that it is important that natural inflation is indeed possible in a broad class of models based on supergravity, including the T-models with the Mexican hat potentials. Moreover, as we will explain in the next section, the new methods of inflationary model construction described above may be helpful for development of a more general class of inflationary models. Multifield evolution for large $A$ ================================== In the analysis of phenomenology, we studies only the small $A$ case. In the situations where the amplitude $A$ of the modulations of the potential is large, the inflationary evolution can be quite complicated. One of the examples is shown in Fig. \[fig:E2u\]. We considered the theory [(\[Eax\])]{} with $\alpha = 1/3$, $A = 1$ and $c = -3$, and plotted the height of the potential in units of $V_{0}$. Note that the minimum of the potential is at ${\varphi}= -3$. ![A model with two different $\alpha$-attractor stages separated by an intermediate inflationary regime driven by a combined motion of the fields ${\varphi}$ and $\theta$.[]{data-label="fig:E2u"}](d1.pdf){width="55.00000%"} Initially the field ${\varphi}$ was rolling along the ridge of the potential, as shown either by a red line, or a purple line. The field $\theta$ was not evolving because of the factor $e^{-\sqrt {2\over 3\alpha}{\varphi}}$ in its equations of motion suppressing its evolution. However, when the field ${\varphi}$ decreases, the factor $e^{-\sqrt {2\over 3\alpha}{\varphi}}$ decreases, and the field $\theta$ rapidly falls down. Because of the interaction between the derivatives of ${\varphi}$ and $\theta$ in the equations of motion of these fields, the field ${\varphi}$ moves back a bit. This unusual effect is similar to the behavior found in a closely related context in [@Kallosh:2014qta; @Christodoulidis]. When the field $\theta$ reaches the minimum of its potential at $\theta = \pi$, a new stage of inflation driven by the field ${\varphi}$ begins. If this stage is long enough (as is the case for the model described here), the observational data will be determined by this last stage of $\alpha$-attractor inflation. On the other hand, if we take the same potential with $c = +3$, the first stage of $\alpha$-attractor inflation will continue for a long time, until the field ${\varphi}$ falls down to its minimum at ${\varphi}= c$, and then the new stage of inflation begins, which will be driven by the field $\theta$ with the natural inflation potential, see Fig. \[fig:E2v\]. ![$\alpha$-attractor inflation along each of the ridges of the potential, shown either by a red line, or a purple line, follows by natural inflation driven by the field $\theta$.[]{data-label="fig:E2v"}](d2.pdf){width="55.00000%"} Models with a finite plateau {#singul} ============================ Until now we studies the standard $\alpha$-attractor models with nonsingular potentials in terms of the original geometric variables. However, one of the way to describe the E-models such as the Starobinsky model is to start with a T-model and assume that the original potential is singular in one of the two directions [@Kallosh:2015zsa]. Let us see what may happen in the context of the $U(1)$-symmetric models discussed in this paper. As an example, consider the T-model scenario discussed in Section \[sT\], with \[sing\] V(Z, |Z) =V\_0  Z|Z (1+ [A1- Z|Z]{}). If the parameter $A$ is very small, the last term is important only in the vicinity of the singularity. The potential in terms of the inflaton field ${\varphi}$ looks as follows: V(, )=V\_0 ( \^[2]{} + A\^[2]{} ) . \[toymodelpotentialhyper\] This function looks strikingly similar to some of the potentials studied in the previous sections, and yet it is very much different, see Fig. \[fig:E2w\]. ![A model with an $\alpha$-attractor plateau potential bounded by an exponentially steep wall, which emerges because of the singularity of the potential [(\[sing\])]{} at $|Z| = 1$. []{data-label="fig:E2w"}](TmodelHyperinflation.pdf){width="55.00000%"} At the first glance, this potential looks like a good candidate for the role of a potential supporting hyperinflation [@Brown:2017osf]. A preliminary investigation of this issue indicates that it does not satisfy the required conditions formulated in [@Mizuno:2017idt], though one might achieve this goal by modifying the nature of the singularity of the potential. However, the structure of the potential [(\[toymodelpotentialhyper\])]{} suggests another interesting possibility. At large ${\varphi}$ this potential has asymptotic behavior V(, )= [V\_0 A4]{} e\^ . \[toymodelpotentialhyper2\] Cosmological evolution in such potentials is well known [@Liddle:1988tb]: \[power\] a(t) = a\_[0]{} t\^[3]{}  . For $\alpha = 1/3$ we have a regime with $a \sim t$, exactly at the boundary between the accelerated and decelerated expansion of the universe. In that case, the energy of the homogeneous component of the scalar field decreases as $a^{{-2}}$, i.e. much more slowly than the energy of dust $\sim a^{{-3}}$ and of the relativistic gas $\sim a^{{-4}}$. This makes the solution of the problem of initial conditions proposed in [@Carrasco:2015rva; @East:2015ggf] much simpler: If initially the energy of the homogeneous field was comparable to other types of energy, then in an expanding universe it gradually starts to dominate. Meanwhile for $\alpha > 1/3$ the power-law solution [(\[power\])]{} describes inflation. It may begin already at the Planck density, which solves the problem of initial conditions in this class of models along the lines of [@Linde:1985ub]. For $\alpha = 1/3$, this potential provides the power-law expansion with $a(t) \sim t$, exactly at the boundary between the accelerated and decelerated expansion of the universe. In this regime, the energy density of the universe decreases as $1/a^{2}$, similar to what happens in the open universe scenario. This regime is very helpful for solving the problem of initial conditions for the subsequent stage of inflation supported by the plateau potential [@Carrasco:2015rva; @East:2015ggf; @Linde:2017pwt]. Meanwhile for any value $\alpha > 1/3$ one can have inflation starting at the Planck density, which immediately solves the problem of initial conditions for inflation in this scenario [@Linde:2017pwt]. Similar conclusions are valid not only for T-models, but also for E-models with singular potentials, or for the single-field $\alpha$-attractors with singular potentials [@Linde:2017pwt]. In particular, if one adds to the E-model potential [(\[Eflat\])]{} a term $\sim(T+\bar T)^{{-1}}$, which is singular at $T \to 0$, the potential will also acquire an exponentially growing correction $\sim e^{{\sqrt{2\over 3\alpha}}{\varphi}}$, just as in the T-model discussed above. Note that the possibility to have inflation at asymptotically large values of the fields in such models does depend on $\alpha$ and on the behavior of the potential near the singularity. For example for the T-models with the singularity $(1- Z\bar Z)^{n}$, the inflationary regime at the asymptotically large ${\varphi}$ is possible for $\alpha > n/3$. The scenario outlined above may have non-trivial observational implications. The amplitude of scalar perturbations $A_{s}$ in the slow-roll approximation is given by A\_[s]{} = [V\^[3]{}12\^[2]{}V’\^[2]{}]{}  . In the theory without the singular term (i.e. with $A = 0$), the amplitude grows monotonically at large ${\varphi}$ since $V$ approaches a plateau and $V'$ decreases. Meanwhile in the theory with the potential [(\[toymodelpotentialhyper\])]{}, the value of $V'$ exponentially decreases, and therefore $A_{s}({\varphi})$ reaches a maximum and then falls down exponentially at large $\phi$. This effect suppress the large wavelength amplitude of inflationary perturbations. If the transition between the two inflationary regimes is sufficiently sharp (which depends on $\alpha$ and the nature of the singularity of the potential) and corresponds to $N \sim 55$, this effect may help to suppress the low-$\ell$ CMB anisotropy [@Linde:1998iw; @Contaldi:2003zv]. Other two-field attractors ========================== Finally, we should mention a more general class of models which can be constructed by our methods: Instead of the term $4A \cosh^{2} {n(T- \bar T)\over 4} $, one can add to [(\[Eflat\])]{} an arbitrary function $f\bigl[{1\over 2}((T+\bar T),-{i\over 2}(T-\bar T)\bigr]$, which leads to the potential \[Eaxa\] V=V\_[0]{}(1- e\^[-(-c)]{})\^[2]{} +f(, ) . In particular, for $f =- {m^{2}\over 8} (T-\bar T)^{2}$ one has a combination of the E-model $\alpha$-attractor potential and a quadratic potential, \[Eaxbb\] V=V\_[0]{}(1- e\^[-(-c)]{})\^[2]{} +[m\^[2]{}2]{}\^[2]{}  , and for $f =- {m^{2}\over 8} (T-\bar T)^{2} + 4V_{0}A \cosh^{2} {n(T- \bar T)\over 4}$ one has a “monodromy” potential [@Silverstein:2008sg; @McAllister:2008hb] shown in Fig. \[fig:E2pp\]. \[Eaxcc\] V=V\_[0]{}(1- e\^[-(-c)]{})\^[2]{} +4 V\_[0 ]{}A \^[2]{}[n2]{} +[m\^[2]{}2]{}\^[2]{}  . ![An example of the $\alpha$-attractor monodromy potential [(\[Eaxcc\])]{}.[]{data-label="fig:E2pp"}](mono.pdf){width="55.00000%"} Thus one can find a large class of consistent inflationary models without making a requirement that the axion field $\theta$ must be strongly stabilized during inflation. Conclusions =========== In this paper we have shown how to realize $U(1)$ symmetry breaking and shift symmetric potentials in the framework of $\alpha$-attractors. This might lead to a novel realization of natural inflation in supergravity, in a similar way as originally proposed in [@Freese:1990rb] through non-perturbative instanton corrections. To illustrate the phenomenology of these models of natural inflation, we consider a concrete example of a Higgs-type potential with a circular minimum at radius $|Z| = c$, together with sinusoidal corrections of amplitude $A$ and $n$ minima. If we take a simple and natural value of $0<c<1$, not too close to 1, we recover the results we found in [@Achucarro:2017ing]. We find universal predictions fairly insensitive to modifications of the potential. This regime could be interesting for applications such as suppression of isocurvature perturbations in the axion dark mater [@Linde:1991km; @Ema:2016ops; @Yamada:2018nsk; @Ema:2018abj; @LindeYamada]. If we fine-tune $1-c^2 \ll 1$, the circumference of the valley and the axion decay constant $f_{a}$ become very large because of the hyperbolic geometry. This opens up the possibility of inflation that proceeds in two stages. For example, inflation may start in the $\alpha$-attractors regime. Then, after the fields have reached the valley of the Higgs potential, the stage of natural inflation may begin. For naturally small values of $A$ we find the following generic behavior. - If $f_a \gtrsim \sqrt{N}$, starting with a random initial angle we typically find the predictions of natural inflation with a large effective decay constant $$\label{predictionsnaturalinflation} n_s = 1-\frac{2}{N}\ , \qquad r = \frac{8}{N} \ .$$ Here $N$ is the number of efolds before the end of inflation where the observable scales cross the horizon. - If $\frac{1}{2} \lesssim f_a \lesssim \sqrt{N}$ or if we happen to start close to one of the minima, we will find the $\alpha$-attractor predictions with a reduced number of efolds $$n_s \approx 1-\frac{2}{N-N_{II}}\ , \qquad r \approx \frac{12\alpha}{(N-N_{II})^2}\ ,$$ where $N_{II}$ is determined by the initial angle. These predictions are shifted left upwards in the $(n_s, r)$ plane with respect to the predictions of $\alpha$-attractor. - In the simplest case $f_a \lesssim \frac{1}{2}$ natural inflation does not occur, and the predictions coincide with the standard $\alpha$-attractor predictions $$\label{alp} n_s \approx 1-\frac{2}{N}\ , \qquad r \approx \frac{12\alpha}{N^2}\ .$$ Our results extend to any potential with a flat minimum well in the $\alpha$-attractor regime, and any type of small angular correction breaking the shift symmetry. In particular, we find exactly the same predictions for natural inflation in the E-models. The cosmological evolution in the models with very large $A$ sometimes involves a more complicated behavior. However, in all cases where the last 50-60 e-foldings of inflation occur in the $\alpha$-attractor regime, the cosmological predictions are quite stable with respect to modifications of the inflaton potential and coincide with the universal single-field $\alpha$-attractor predictions [(\[alp\])]{}, in agreement with the general conclusions of [@Achucarro:2017ing]. The new class of models discussed in this paper allow various generalizations. In particular, in Section \[singul\] we described a new class of models, where the potentials contain small terms which become singular at the boundary of the moduli space. Such models allow to describe $\alpha$-attractors with potentials having a plateau of a finite size. We found that in certain cases the exponentially rising potentials at asymptotically large values of the field ${\varphi}$ can support inflation all the way up to the Planck density, which provides a simple solution to the problem of initial conditions for inflation in such models [@Linde:2017pwt]. [**[Acknowledgements:]{}**]{} We thank Renata Kallosh, Diederik Roest and Alexander Westphal for stimulating discussions and collaborations on related work. The work of AL and YY is supported by SITP and by the US National Science Foundation grant PHY-1720397. DGW and YW are supported by a de Sitter Fellowship of the Netherlands Organization for Scientific Research (NWO). The work of AA is partially supported by the Netherlands’ Organization for Fundamental Research in Matter (FOM), by the Basque Government (IT-979-16) and by the Spanish Ministry MINECO (FPA2015-64041-C2-1P). {#section} = Parameter ranges {#sec:parameterranges} ---------------- We shortly discuss the range of values the model parameters can take to which our generic results in Section \[subsec:singlefield\] apply. The parameter $A$ determines the size of the correction to the potential [(\[toymodelpotential2\])]{} and [(\[Eax\])]{}. For our generic results to apply, we only need to ensure that we stay away from the multi-field regime in the valley of the potential. In case of the T-model, this leads to the constraint[^3] $$A \ll \frac{6}{n^2 N}, \label{improvedconstraintA}$$ with $N\sim 60$ the number of efolds between horizon crossing and the end of inflation. Similarly, for the E-model a sufficient condition is $$\alpha A\ll 1. \label{EimprovedconstraintA}$$ In Section \[subsec:singlefield\] we assume these conditions are obeyed and study the generic behavior of this system. Phenomenology in the effectively single field regime {#subsec:singlefield} ---------------------------------------------------- We assume that the condition is obeyed. We generically first have a stage of inflation in the two field $\alpha$-attractor regime until $\varphi$ reaches the valley of the symmetry breaking potential. Using the results of [@Achucarro:2017ing], we know the system behaves effectively like the single field alpha attractor. We outlined the derivation of this result in Section \[EOM\]. Consequently we enter a second stage of natural inflation in the bottom of the Higgs potential. The value of the effective axion decay constant $f_{a}$ determines the phenomenology of these family of models. To see this, the phenomenologically distinct regimes we are interested in are: 1. The natural inflation potential allows for slow roll inflation to happen: $\eta_\theta \ll 1$ at the top of the hill of the angular potential at the minimum of the Higgs potential. $$\eta_\theta \sim -\frac{V_{\theta\theta}/V}{\frac{3\alpha}{2}\sinh^2\sqrt{\frac{2}{3\alpha}}\varphi} \ll 1 \ .$$ Natural inflation is possible for $ f_{a}\gg 1/2$. 2. The stretching of the angular potential is large enough to allow for a quadratic expansion about the origin of the natural inflation potential in the observable range of scales: $$N/ f_\ast^2 \lesssim 1, \quad \text{or} \quad f_{a} \gtrsim \sqrt{N},$$ with $N \sim 60$ the number of efolds before the end of inflation where the observable modes cross the horizon. 3. The probability to have $N > 60$ e-folds of natural inflation starting with a random initial angle is close to 1 (assuming the quadratic expansion around the minima holds true): $$\theta_\text{can}(N) = \sqrt{4 N} \quad \longrightarrow \quad \theta(N) = \left.\frac{\sqrt{4 N}}{\sqrt{\frac{3\alpha}{2}}\sinh\sqrt{\frac{2}{3\alpha}}\varphi}\right|_{ f_{a}=f_\ast}.$$ This leads to $$P \equiv 1 - \frac{\theta(N) }{2\pi c / n}= 1-\frac{\sqrt{N}}{ f_{a}\ \pi},$$ with $2\pi c / n$ the distance between the minima. Therefore in order to have $P$ close to 1 we need the second contribution to be negligible $f_{a} \gg \frac{\sqrt{N}}{\pi}.$ This is automatically satisfied, since we already assumed the quadratic approximation. Therefore, for small enough $A$, the behavior of our system is very simple and splits in three regimes determined by the value of $ f_{a}$. It can be summarized as follows. - If there is no axion potential or if $ f_{a} \lesssim \frac{1}{2}$ we recover the universal predictions of $\alpha$ attractors [(\[universalpredictions\])]{}, as shown in our previous paper $$n_s = 1-\frac{2}{N}, \quad r = \frac{12\alpha}{N^2} . \label{universalpredictions2}$$ - If $ f_{a} \gtrsim \sqrt{N}$, we would generically expect to have sufficient efolds in the valley of the potential where natural inflation takes place. We get the same predictions as natural inflation with an large effective axion decay constant, which coincide with those of chaotic inflation $$n_s = 1-\frac{2}{N}\ , \qquad r = \frac{8}{N} \ . \label{naturalinflationpredictions}$$ - If $ f_{a}$ takes values in between, we most likely have not sufficient number of efolds in the regime of natural inflation. Therefore the predictions for the large-scale perturbations are the same as for the $\alpha$-attractor predictions, but with a reduced number of e-folds. If we denote $N_{II}$ as the number of e-folds in the natural inflation potential, we find $$n_s \approx 1-\frac{2}{N-N_{II}}, \quad r \approx \frac{12\alpha}{(N-N_{II})^2}. \label{alphaattractorpredictions}$$ This means that the predictions move left upwards in the $(n_s, r)$-plane with respect to the $\alpha$-attractor predictions. The initial value of the axion $\theta_\ast$ determines the duration $N_{II}$ of the second stage of inflation. The precise dependence can be computed by using the slow roll approximation of natural inflation: $N_{II}(\theta_\ast) = -f^2_{a} \log\left(\frac{1-\cos(n\theta_\ast)}{2}\right).$ Some comments are in order. When the field reaches the valley of the Higgs potential, we may end inflation for a short while. This is more likely to happen if $A$ is really small, because its value determines the Hubble friction, which may or may not prevent inflation to end. We have checked numerically using methods of [@Dias:2015rca] that it does not affect the initial value of $\theta$ when natural inflation starts. Moreover, the field will oscillate strongly for a short amount of time. For fine-tuned initial conditions this might happen exactly during the observable regime and it will leave features in the spectra of perturbations [@Zelnikov:1991nv; @Polarski:1992dq]. In this case the predictions become model dependent, and we cannot make definite statements. Appendix B: Other ways to generate natural inflation potentials =============================================================== By using the methods described in this paper as well as in [@Achucarro:2017ing; @Yamada:2018nsk] one can construct a very broad class of potentials, including the potentials of natural inflation with a weakly broken $U(1)$ symmetry, as well as the models where this symmetry is completely absent. Indeed, any potential $V(Z \bar Z)$ or $V(T+\bar T)$ is $U(1)$-symmetric, but our approach is equally valid for general potentials $V(Z,\bar Z)$ or $V(T,\bar T)$ without any $U(1)$ symmetry. Therefore one could construct the potentials of natural inflation with weakly broken $U(1)$ symmetry without modifying any parts of our construction but the potential $V$. Other methods can also be used to reach a similar goal. For example, one may start with a $U(1)$-symmetric potential $V(Z \bar Z)$ or $V(T+\bar T)$, and then add a small symmetry breaking term to the superpotential $W$. In particular, one can keep the $U(1)$ invariant function $V = V_0 \left(1- c^{-2}Z\bar Z\right)^{2} +C$ of the T-model, and generate the natural inflation potential by adding a small correction term $A\, Z$ to the superpotential $W = W_{0}$. Here $C$ is an arbitrary constant that can be added to the potential without affecting its $U(1)$ symmetry. Similarly, one may consider the $U(1)$ invariant potential $V = V_{0}\Big(1 - {T + \bar T\over 2}\, e^{\sqrt{2\over 3 \alpha}c}\Big)^{2} +C$ of the E-model, and generate the natural inflation potential by adding a small term $A\, e^{{-T}}$ to the superpotential. We found that this method works well for $\alpha \leq 1/3$. However, if one considers $\alpha > 1/3$, the potential at large ${\varphi}$ becomes unbounded from below. Thus one may either use the methods developed in the main part of our paper, or try to improve these models using singular potentials discussed in Section \[singul\] and in [@Yamada:2018nsk], or consider the models with $\alpha \leq 1/3$. Note that the models with $\alpha = 1/3$ studied in [@Achucarro:2017ing] have a very interesting interpretation in terms of M-theory and extended supergravity and provide specific targets for the search of B-modes, as emphasized in [@Ferrara:2016fwe; @Kallosh:2017ced]. [^1]: Although these are different fields and parameters, in this section we abuse notation and use the variables $S,\ \varphi, \ \theta, \ A$ and $c$ again. This turns out to be convenient for describing the phenomenology for both models in Section \[sec:pheno\]. [^2]: It is straightforward to verify that this regime does not suffer from geometric destabilization, see eq. (B.8) in [@Achucarro:2017ing]. [^3]: \[footnote\]In the slow roll regime we can roughly estimate the size of the effective mass $\mu$ of the radial perturbations as $\frac{\mu^2}{H^2} \approx 3\frac{V_{\varphi\varphi}}{V}$. For simplicity we neglect the corrections from the curvature of the field space, which scale like $\epsilon/\alpha$, and the turn rate, which is smaller than $H$. Using $\rho \approx c\sim 1$ this becomes $\frac{\mu^2}{H^2} \approx \frac{6}{A f_a^2 \cos^2(n \theta/2)}$ for the T-model [(\[toymodelpotential2\])]{}. As soon as this ratio becomes order one we have to be careful about the multi-field effects and perform a numerical analysis. To find the constraint on $A$ we first solve for the evolution of $\theta(N)$ in the gradient flow approximation to find $\cos^2(n \theta/2) = 1 - e^{-n^2 N/f_a^2}$. In the regime $f_a^2 \gtrsim n^2 N$ the condition $\mu^2/H^2\gg 1$ translates to $A \ll \frac{6}{n^2 N}$. In the regime $f_a^2 \lesssim n^2 N$ we instead find $A \ll \frac{6}{f_a^2}$. Together they lead to the result quoted in . For the E-model [(\[Eax\])]{} we can estimate $\frac{\mu^2}{H^2} \approx \left|3\frac{V_{{\varphi}{\varphi}}}{V}\right|_{{\varphi}=c} \approx \frac{1}{\alpha A\cos^2(n\theta/2)}.$ Hence, we need to have $\alpha A\cos^2(n\theta/2) \ll 1$. Therefore is a sufficient condition.

--- abstract: | The aim of this Note is to prove by a perturbation method the existence of solutions of the coupled Einstein-Dirac-Maxwell equations for a static, spherically symmetric system of two fermions in a singlet spinor state and with the electromagnetic coupling constant $\left(\frac{e}{m}\right)^2<1$. We show that the nondegenerate solution of Choquard’s equation generates a branch of solutions of the Einstein-Dirac-Maxwell equations. [**Une méthode de perturbation pour les solutions localisées des équations d’Einstein-Dirac-Maxwell.**]{} <span style="font-variant:small-caps;">Résumé.</span> Le but de cette Note est de démontrer par une méthode de perturbation l’existence de solutions des équations d’Einstein-Dirac-Maxwell pour un système statique, à symétrie sphérique de deux fermions dans un état de singulet et avec une constante de couplage électromagnétique $\left(\frac{e}{m}\right)^2<1$. On montre que la solution non dégénérée de l’équation de Choquard génère une branche de solutions des équations d’Einstein-Dirac-Maxwell. address: - 'Ceremade (UMR CNRS 7534), Université Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, 75775 Paris Cedex 16, France' - 'Dipartimento di Matematica, Università di Milano, Via Saldini 50, 20133 Milan, Italy' author: - Simona Rota Nodari date: 'December 18, 2009' title: 'Perturbation Method for Particle-like Solutions of the Einstein-Dirac-Maxwell Equations' --- Version française abrégée {#version-française-abrégée .unnumbered} ========================= Dans un papier récent [@rotanodari], par une méthode de perturbation, on a montré de manière rigoureuse l’existence de solutions des équations d’Einstein-Dirac pour un système statique, à symétrie sphérique de deux fermions dans un état de singulet. Dans cette Note, on généralise ce résultat aux équations d’Einstein-Dirac-Maxwell et on montre, dans le cas particulier d’un couplage électromagnétique faible, l’existence des solutions obtenues numériquement par F. Finster, J. Smoller et ST. Yau dans [@finsmoyaumax]. Plus précisément, en utilisant l’idée introduite par Ounaies pour une classe d’équations de Dirac non linéaires (voir [@ounaies]) et adaptée dans [@rotanodari] aux équations d’Einstein-Dirac, on obtient le théorème suivant. \[th:solutionfr\] Soient $e,m,\omega$ tels que $e^2-m^2<0$, $0<\omega< m$ et supposons $m-\omega$ assez petit; alors il existe une solution non triviale de (\[eq:einsteindiracmaxwelleq1\]-\[eq:einsteindiracmaxwelleq5\]). Dans cette Note, on décrit la méthode utilisée pour démontrer ce théorème. Premièrement, par un changement d’échelle, on transforme les équations d’Einstein-Dirac-Maxwell (\[eq:einsteindiracmaxwelleq2.1\]-\[eq:einsteindiracmaxwelleq2.4\]) en un système perturbé qui s’écrit sous la forme (\[eq:systemepsilon\]). On choisit $\varepsilon=m-\omega$ comme paramètre de perturbation. Deuxièment, on remarque que, pour $\varepsilon=0$ et $\left(\frac{e}{m}\right)^2<1$, ce système est équivalent au système (\[eq:systemepsilonzero2\]) où l’équation pour la variable $\varphi$ est l’équation de Choquard. Il est bien connu que l’équation de Choquard a une solution radiale positive. De plus, dans l’espace des fonctions radiales, cette solution est non dégénérée, dans le sens où le noyau de la linéarisation de l’équation contient seulement la fonction identiquement nulle. On appelle $\phi_0$ la solution du système (\[eq:systemepsilonzero2\]). Ensuite, on observe que le système perturbé s’écrit sous la forme $D(\varepsilon,\varphi,\chi,\tau,\zeta)=0$ avec $D$ un opérateur non linéaire de classe $\mathcal{C}^1$, pour un bon choix d’espaces fonctionnels. On prouve que cet opérateur satisfait les hypothèses du théorème des fonctions implicites. En particulier, on montre que la linéarisation de l’opérateur $D$ par rapport à $(\varphi,\chi,\tau,\zeta)$ en $(0,\phi_0)$, $D_{\varphi,\chi,\tau,\zeta}(0,\phi_0)$, est une injection, grâce à la non-dégénérescence de la solution de l’équation de Choquard, et s’écrit comme somme d’un isomorphisme et d’un opérateur compact; donc $D_{\varphi,\chi,\tau,\zeta}(0,\phi_0)$ est un isomorphisme. En appliquant le théorème des fonctions implicites, on déduit que, pour $\varepsilon$ assez petit et $e^2-m^2<0$, le système (\[eq:systemepsilon\]) a une solution. En conclusion, pour $e^2-m^2<0$, $0<\omega< m$ et $m-\omega$ assez petit, les équations d’Einstein-Dirac-Maxwell possèdent une solution non triviale. Introduction ============ In a recent paper [@rotanodari], using a perturbation method, we proved rigorously the existence of solutions of the coupled Einstein-Dirac equations for a static, spherically symmetric system of two fermions in a singlet spinor state. In this Note, we extend our result to the Einstein-Dirac-Maxwell equations and we prove, in the particular case of a weak electromagnetic coupling, the existence of the solutions obtained numerically by F. Finster, J. Smoller and ST. Yau in [@finsmoyaumax]. The general Einstein-Dirac-Maxwell equations for a system of $n$ Dirac particles take the form $$\begin{array}{lll} (G-m)\psi_a=0, & R^i_j-\frac{1}{2}R\delta^i_j=-8\pi T^i_j, &\nabla_kF^{jk}=4\pi e\sum\limits_{a=1}^n\overline{\psi_a}G^j\psi_a \end{array}$$ where $G^j$ are the Dirac matrices, $G$ denote the Dirac operator, $\psi_a$ are the wave functions of fermions of mass $m$ and charge $e$, $F_{jk}$ is the electromagnetic field tensor and, finally, $T^i_j$ is the sum of the energy-momentum tensor of the Dirac particle and the Maxwell stress-energy tensor. In [@finsmoyaumax], the metric, in polar coordinates $(t,r,\vartheta,\varphi)$, is given by $$ds^2=\frac{1}{T^2}\,dt^2-\frac{1}{A}\,dr^2-r^2\,d\vartheta^2-r^2\sin^2\vartheta\,d\varphi^2$$ with $A=A(r)$, $T=T(r)$ positive functions; moreover, using the ansatz from [@finsmoyau], Finster, Smoller and Yau describe the Dirac spinors with two real radial functions $\Phi_1(r)$, $\Phi_2(r)$ and they assume that the electromagnetic potential has the form $\mathcal{A}=(-V,0)$, with $V$ the Coulomb potential. In this case the Einstein-Dirac-Maxwell equations can be written as $$\begin{aligned} \label{eq:einsteindiracmaxwelleq1} \sqrt{A}\Phi_1'&=&\frac{1}{r}\Phi_1-((\omega-eV) T+m)\Phi_2\\ \label{eq:einsteindiracmaxwelleq2} \sqrt{A}\Phi_2'&=&((\omega-eV)T-m)\Phi_1-\frac{1}{r}\Phi_2\\ \label{eq:einsteindiracmaxwelleq3} rA'&=&1-A-16\pi(\omega-eV) T^2\left(\Phi_1^2+\Phi_2^2\right)-r^2AT^2\left(V'\right)^2\\ \label{eq:einsteindiracmaxwelleq4} 2rA\frac{T'}{T}&=&A-1-16\pi(\omega-eV) T^2\left(\Phi_1^2+\Phi_2^2\right)+32\pi\frac{1}{r} T\Phi_1\Phi_2\nonumber\\ &&+16\pi m T\left(\Phi_1^2-\Phi_2^2\right)+r^2AT^2\left(V'\right)^2\\ \label{eq:einsteindiracmaxwelleq5} r^2AV''&=&-8\pi e \left(\Phi_1^2+\Phi_2^2\right)- \left(2rA+r^2A\frac{T'}{T}+\frac{r^2}{2}A'\right)V'\end{aligned}$$ with the normalization condition $\int_0^{\infty}{|\Phi|^2\frac{T}{\sqrt A}\,dr}=\frac{1}{4\pi}$.\ In order that the metric be asymptotically Minkowskian and the solutions have finite (ADM) mass, Finster, Smoller and Yau assume $$\lim\limits_{r\rightarrow \infty}T(r)=1$$ and $$\lim\limits_{r\rightarrow \infty}\frac{r}{2}(1-A(r))<\infty.$$ Finally, they also require that the electromagnetic potential vanishes at infinity. In this Note, using the idea introduced by Ounaies for a class of nonlinear Dirac equations (see [@ounaies]) and adapted in [@rotanodari] to the Einstein-Dirac equations, we obtain the following result. \[th:solution\] Given $e,m,\omega$ such that $e^2-m^2<0$, $0<\omega< m$ and $m-\omega$ is sufficiently small, there exists a non trivial solution of (\[eq:einsteindiracmaxwelleq1\]-\[eq:einsteindiracmaxwelleq5\]). Perturbation method for the Einstein-Dirac-Maxwell equations {#section:EinsteinDiracMaxwell} ============================================================ First of all, we observe that, writing $T(r)=1+t(r)$ and integrating the equation (\[eq:einsteindiracmaxwelleq5\]), the Einstein-Dirac-Maxwell equations become $$\begin{aligned} \label{eq:einsteindiracmaxwelleq2.1} \sqrt{A}\Phi_1'&=&\frac{1}{r}\Phi_1-((\omega-eV)(1+t)+m)\Phi_2\\ \label{eq:einsteindiracmaxwelleq2.2} \sqrt{A}\Phi_2'&=&((\omega-eV)(1+t)-m)\Phi_1-\frac{1}{r}\Phi_2\\ \label{eq:einsteindiracmaxwelleq2.3} 2rAt'&=&(A-1)(1+t)-16\pi(\omega-eV) (1+t)^3\left(\Phi_1^2+\Phi_2^2\right)\nonumber\\ &&+32\pi\frac{1}{r} (1+t)^2\Phi_1\Phi_2+16\pi m (1+t)^2\left(\Phi_1^2-\Phi_2^2\right)\nonumber\\ &&+r^2A(1+t)^3\left(V'\right)^2\\ \label{eq:einsteindiracmaxwelleq2.4} \sqrt A (1+t) V'&=&-\frac{8\pi e}{r^2} \int_0^r \left(\Phi_1^2+\Phi_2^2\right) \frac{(1+t)}{\sqrt A}\,ds.\end{aligned}$$ where $A(r)=1+a(r)$ and $$\begin{aligned} \label{eq:defa} a(r)&=&-\frac{1}{r}\exp{(-F(r))}\int_0^{r}\left[16\pi(\omega-eV) (1+t)^2\left(\Phi_1^2+\Phi_2^2\right)\right.\nonumber\\ &&\left.+s^2(1+t)^2\left(V'\right)^2\right]\exp{(F(s))}\,ds\end{aligned}$$ with $F(r)=\int_0^r s(1+t)^2\left(V'\right)^2\,ds$. After that, we introduce the new variable $(\varphi,\chi,\tau,\zeta)$ such that $$\begin{array}{llll} \Phi_1(r)=\varepsilon^{1/2}\varphi(\varepsilon^{1/2} r), &\Phi_2(r)=\varepsilon\chi(\varepsilon^{1/2} r), &t(r)=\varepsilon\tau(\varepsilon^{1/2}r), &V(r)=\varepsilon \zeta(\varepsilon^{1/2}r) \end{array}$$ where $\Phi_1,\Phi_2, t,V$ satisfy (\[eq:einsteindiracmaxwelleq2.1\]-\[eq:einsteindiracmaxwelleq2.4\]) and $\varepsilon=m-\omega$. Using the explicit expression of $a(r)$, given in (\[eq:defa\]), we write $$a(\Phi_1,\Phi_2,t,V)=\varepsilon \alpha(\varepsilon,\varphi,\chi,\tau,\zeta)$$ with $\alpha(0,\varphi,\chi,\tau,\zeta)=-\frac{16\pi m}{r}\int_0^{r} \varphi^2\,ds$ . It is now clear that if $\Phi_1,\Phi_2,t,V$ satisfy (\[eq:einsteindiracmaxwelleq2.1\]-\[eq:einsteindiracmaxwelleq2.4\]), then $\varphi,\chi,\tau,\zeta$ satisfy the system $$\label{eq:systemepsilon} \left\{ \begin{array}{l} (1+\varepsilon\alpha(\varepsilon,\varphi,\chi,\tau,\zeta))^{1/2}\frac{d}{dr}\varphi-\frac{1}{r}\varphi+2m\chi+K_1\left(\varepsilon,\varphi,\chi,\tau,\zeta \right)=0\\[5pt] (1+\varepsilon\alpha(\varepsilon,\varphi,\chi,\tau,\zeta))^{1/2} \frac{d}{dr}\chi+\frac{1}{r}\chi+\varphi-m\varphi\tau+e\varphi\zeta+K_2\left(\varepsilon,\varphi,\chi,\tau,\zeta \right)=0\\[5pt] (1+\varepsilon\alpha(\varepsilon,\varphi,\chi,\tau,\zeta))\frac{d}{dr}\tau-\frac{\alpha(\varepsilon,\varphi,\chi,\tau,\zeta)}{2r}+K_3\left(\varepsilon,\varphi,\chi,\tau,\zeta \right)=0\\[5pt] (1+\varepsilon\alpha(\varepsilon,\varphi,\chi,\tau,\zeta))^{1/2}(1+\varepsilon\tau)\frac{d}{dr}\zeta+\frac{8\pi e}{r^2}\int_0^r\varphi^2\,ds+K_4\left(\varepsilon,\varphi,\chi,\tau,\zeta \right)=0 \end{array} \right.$$ where $K_1\left(0,\varphi,\chi,\tau,\zeta \right) = K_2\left(0,\varphi,\chi,\tau,\zeta \right) =K_3\left(0,\varphi,\chi,\tau,\zeta \right) =K_4\left(0,\varphi,\chi,\tau,\zeta \right)=0 $. Then, for $\varepsilon=0$, (\[eq:systemepsilon\]) becomes $$\label{eq:systemepsilonzero2} \left\{ \begin{array}{l} -\frac{d^2}{dr^2}\varphi+2m\varphi+16\pi (e^2-m^2)m\left(\int_{0}^{\infty}\frac{\varphi^2}{\max(r,s)}\,ds\right)\varphi=0\\[5pt] \chi(r)=\frac{1}{2m}\left(\frac{1}{r}\varphi-\frac{d}{dr}\varphi\right)\\[5pt] \tau(r)=8\pi m \int_{0}^{\infty}\frac{\varphi^2}{\max(r,s)}\,ds\\[5pt] \zeta(r)=8\pi e \int_{0}^{\infty}\frac{\varphi^2}{\max(r,s)}\,ds\\[5pt] \end{array} \right.$$ We remark that if $e^2-m^2<0$ the first equation of the system (\[eq:systemepsilonzero2\]) is the Choquard equation $$\label{eq:choquard} \begin{array}{cc} -\triangle u+2m u-4(m^2-e^2)m\left(\int_{\mathbb{R}^3}\frac{\left|u(y)\right|^2}{|x-y|}\,dy\right)u=0&\mbox{in}\ H^1\left(\mathbb{R}^3\right) \end{array}$$ with $u(x)=\frac{\varphi(|x|)}{|x|}$. It is well known that Choquard’s equation (\[eq:choquard\]) has a unique radial, positive solution $u_0$ with $\int|u_0|^2=N$ for some $N>0$ given. Furthermore, $u_0$ is infinitely differentiable, goes to zero at infinity and is a radial nondegenerate solution; by this we mean that the linearization of (\[eq:choquard\]) around $u_0$ has a trivial nullspace in $L^2_{r}(\mathbb{R}^3)$ (see [@liebcho], [@Lionscho], [@Lenzmann] for more details).\ Let $\phi_0=(\varphi_0,\chi_0,\tau_0,\zeta_0)$ be the ground state solution of (\[eq:systemepsilonzero2\]). The main idea is that the solutions of (\[eq:systemepsilon\]) are the zeros of a $\mathcal{C}^1$ operator $D:\mathbb{R}\times X_{\varphi}\times X_{\chi}\times X_{\tau}\times X_{\zeta}\rightarrow Y_{\varphi}\times Y_{\chi}\times Y_{\tau}\times Y_{\zeta}$. So, to obtain a solution of (\[eq:systemepsilon\]) from $\phi_0$, we define the operators $$\begin{aligned} L_1(\varepsilon,\varphi,\chi,\tau,\zeta)&=&(1+\varepsilon\alpha(\varepsilon,\varphi,\chi,\tau,\zeta))^{1/2}\frac{1}{r}\frac{d}{dr}\varphi-\frac{\varphi}{r^2}+2m\frac{\chi}{r}\\&&+\frac{1}{r}K_1\left(\varepsilon,\varphi,\chi,\tau,\zeta \right)\\ L_2(\varepsilon,\varphi,\chi,\tau,\zeta)&=&(1+\varepsilon\alpha(\varepsilon,\varphi,\chi,\tau,\zeta))^{1/2}\frac{1}{r} \frac{d}{dr}\chi+\frac{\chi}{r^2}+\frac{\varphi}{r}-m\frac{\varphi}{r}\tau+e\frac{\varphi}{r}\zeta\\&&+\frac{1}{r}K_2\left(\varepsilon,\varphi,\chi,\tau,\zeta \right)\\ L_3(\varepsilon,\varphi,\chi,\tau,\zeta)&=&(1+\varepsilon\alpha(\varepsilon,\varphi,\chi,\tau,\zeta))\frac{d}{dr}\tau-\frac{\alpha(\varepsilon,\varphi,\chi,\tau,\zeta)}{2r}+K_3\left(\varepsilon,\varphi,\chi,\tau,\zeta \right)\\ L_4(\varepsilon,\varphi,\chi,\tau,\zeta)&=&(1+\varepsilon\alpha(\varepsilon,\varphi,\chi,\tau,\zeta))^{1/2}(1+\varepsilon\tau)\frac{d}{dr}\zeta+\frac{8\pi e}{r^2}\int_0^r\varphi^2\,ds\\&&+K_4\left(\varepsilon,\varphi,\chi,\tau,\zeta \right)\end{aligned}$$ and $$D(\varepsilon,\varphi,\chi,\tau,\zeta)=\left(L_1(\varepsilon,\varphi,\chi,\tau,\zeta), L_2(\varepsilon,\varphi,\chi,\tau,\zeta),L_3(\varepsilon,\varphi,\chi,\tau,\zeta),L_4(\varepsilon,\varphi,\chi,\tau,\zeta)\right),$$ with $X_{\varphi}$, $X_{\chi}$, $X_{\tau}$, $Y_{\varphi}, Y_{\chi}, Y_{\tau}$ defined as in [@rotanodari] and $$\begin{aligned} X_\zeta&=&\left\{\zeta:(0,\infty)\rightarrow\mathbb{R}\left|\lim_{r\rightarrow\infty}\zeta(r)=0, \frac{d}{dr}\zeta\in L^1((0,\infty),dr)\cap L^2((0,\infty),rdr )\right.\right\}\\ Y_\zeta&=&L^1((0,\infty),dr)\cap L^2((0,\infty),rdr )\end{aligned}$$ with their natural norms. Next, we linearize the operator $D$ on $(\varphi,\chi,\tau,\zeta)$ around $(0,\phi_0)$: $$\begin{aligned} D_{\varphi,\chi,\tau,\zeta}(0,\phi_0)(h,k,l,z)&=&\left( \begin{array}{c} \frac{1}{r}\frac{d}{dr}h-\frac{h}{r^2}+2m\frac{k}{r}\\[5pt] \frac{1}{r}\frac{d}{dr}k+\frac{k}{r^2}+\frac{h}{r}-m\frac{\varphi_0}{r} l+e\frac{\varphi_0}{r} z\\[5pt] \frac{d}{dr}l\\[5pt] \frac{d}{dr}z \end{array} \right)\\ &&+ \left( \begin{array}{c} 0\\[5pt] -m \frac{h}{r}\tau_0+e \frac{h}{r}\zeta_0\\[5pt] \frac{16\pi m}{r^2}\int_0^r\varphi_0h\,ds\\[5pt] \frac{16\pi e}{r^2}\int_0^r\varphi_0h\,ds\\ \end{array} \right).\end{aligned}$$ We observe that, thanks to the nondegeneracy of the solution of Choquard’s equation, $D_{\varphi,\chi,\tau,\zeta}(0,\phi_0)$ is a one-to-one operator. Moreover, it can be written as a sum of an isomorphism and a compact operator. It is thus an isomorphism. Finally, the application of the implicit function theorem yields the following result, which is equivalent to Theorem \[th:solution\]. \[th:principalth1\] Suppose $e^2-m^2<0$ and let $\phi_0$ be the ground state solution of (\[eq:systemepsilonzero2\]), then there exists $\delta>0$ and a function $\eta\in\mathcal{C}((0,\delta),X_\varphi\times X_\chi\times X_\tau\times X_\zeta)$ such that $\eta(0)=\phi_0$ and $D(\varepsilon,\eta(\varepsilon))=0$ for $0\leq \varepsilon <\delta$. Acknowledgements {#acknowledgements .unnumbered} ================ The author would like to thank professor Eric Séré and professor Bernhard Ruf for helpful discussions. [00]{} H. Ounaies, *Perturbation method for a class of non linear Dirac equations.* Differential Integral Equations 13 (2000), no. 4-6, 707–720. E. Lenzmann, *Uniqueness of ground states for pseudo-relativistic Hartree equations.* Preprint (2008). E. H. Lieb, *Existence and Uniqueness of the Minimizing Solution of Choquard’s Nonlinear Equation.* Studies in Applied Mathematics 57 (1977), 93–105. P.L. Lions, *The Choquard equation and related questions.* Nonlinear Analysis. Theory, Methods and Applications 4 (1980), no. 6, 1063–1073. S. Rota Nodari, *Perturbation Method for Particle-like Solutions of the Einstein-Dirac Equations.* Ann. Henri Poincaré (2009), DOI 10.1007/s00023-009-0015-x. F. Finster, J. Smoller, S.T. Yau, *Particlelike solutions of the Einstein-Dirac equations.* Physical Review. D. Particles and Fields. Third Series 59 (1999). F. Finster, J. Smoller, S.T. Yau, *Particle-like solutions of the Einstein-Dirac-Maxwell equations.* Phys. Lett. A. 259 (1999), no. 6, 431–436.

--- abstract: 'Multiple initial state parton interactions in p(d)+A collisions are calculated in a Glauber-Eikonal formalism, which incorporates the competing pattern of low-$p_T$ suppression due geometrical shadowing, and a moderate-$p_T$ Cronin enhancement of hadron spectra. Dynamical shadowing effects, which are not included in the computation, may be extracted by comparing experimental data to the baseline provided by the Glauber-Eikonal model. Data for $\pi^0$ production at midrapidity show absence of dynamical shadowing in the RHIC energy range, $\sqrt{s}\sim 20-200$ GeV. Recent preliminary data at forward rapidity are addressed, and their interpretation discussed.' address: | Columbia University, Department of Physics\ 538 West 120th Street, New York, NY 10027, USA author: - Alberto Accardi and Miklos Gyulassy title: 'Cronin effect and geometrical shadowing in d+Au collisions: pQCD vs. CGC' --- In proton ($p$), or deuteron ($d$), reactions involving heavy nuclei ($A$$\sim$$200$) at $\sqrt{s}<40$ AGeV, the moderate transverse momentum ($p_T$$\sim$2-6 GeV) spectra are enhanced relative to linear extrapolation from $p+p$ reactions. This “Cronin effect” [@Cronin] is generally attributed to multiple scattering of projectile partons propagating through the target nucleus [@Accardi02]. In this talk, we discuss multiple parton scattering in the Glauber-Eikonal (GE) approach [@AG03b; @GEmodels], in which sequential multiple partonic collisions are computed in pQCD, and unitarity is naturally preserved. The low-$p_T$ spectra in $p+A$ collisions are suppressed by unitarity. At moderate $p_T$, the accumulation of transverse momentum leads to an enhancement of transverse spectra. At high $p_T$ the binary scaled $p+p$ spectrum is recovered: no high-$p_T$ shadowing is predicted in this approach.\ [**Hadron production in p+p collisions.**]{} The first step to understand $p+A$ collisions is to understand $p+p$ collisions. The pQCD formula for the single inclusive hadron transverse spectrum is: $$\fl \frac{d\sigma}{dp_T^2 dy}^{\hspace{-0.3cm}pp'\rightarrow hX} \hspace{-0.7cm} = \sum_{i=q,g}\Bigg\{ {\langle xf_{i/p} \rangle}_{y_i,p_T} \, \frac{d\sigma^{\,ip'}}{dy_i d^2p_T} { \bigg| _{y_i=y} } + {\langle xf_{i/p'} \rangle}_{y_i,p_T} \, \frac{d\sigma^{\,ip}}{dy_i d^2p_T}{ \bigg| _{y_i=-y} } \Bigg\} \otimes D_{i\rightarrow h}(z,Q_h^2) \ . \label{ppcoll_pt}$$ Here we considered only elastic parton-parton subprocesses, which contribute to more than 98% of the cross section at midrapidity. In [Eq. (\[ppcoll\_pt\])]{}, $$\begin{aligned} \fl {\langle xf_{i/p} \rangle}_{y_i,p_T} &=& {\displaystyle}{K \over \pi} \sum_j \frac{1}{1+\delta_{ij}} \int dy_2 x_1f_{i/p}(x_1,Q_p^2) \frac{d\hat\sigma}{d\hat t}^{ij} \hspace{-0.2cm} (\hat s,\hat t,\hat u) \, x_2f_{j/p'}(x_2,Q_p^2) \Bigg/ \frac{d\sigma^{ip'}}{d^2p_T dy_i} \label{avflux} \\ \fl \frac{d\sigma^{ip'}}{d^2p_T dy_i} &=& {K \over \pi} \sum_j \frac{1}{1+\delta_{ij}} \int dy_2 \frac{d\hat\sigma}{d\hat t}^{ij} \hspace{-0.2cm} (\hat s,\hat t,\hat u) \, x_2f_{j/p'}(x_2,Q_p^2) \label{iNxsec}\end{aligned}$$ are interpreted, respectively, as the average flux of incoming partons of flavour $i$ from the hadron $p$, and the cross section for the parton-hadron scattering. The rapidities of the $i$ and $j$ partons in the final state are labelled by $y_i$ and $y_2$. Infrared regularization is performed by adding a small mass to the gluon propagator and defining the Mandelstam variables $\hat t (\hat u) = - m_T^2 (1+e^{\mp y_i \pm y_2})$, with $m_{T}=\sqrt{p_T^2+p_0^2}$. For more details, see [Ref. [@AG03b]]{}. Finally, inclusive hadron production is computed as a convolution of Eq. (\[ppcoll\_pt\]) with a fragmentation function $D_{i\rightarrow h}(z,Q_h^2)$. In Eqs. (\[ppcoll\_pt\])-(\[iNxsec\]), we have two free parameters, $p_0$ and the K-factor $K$, and a somewhat arbitrary choice of the factorization and fragmentation scales, $Q_p=Q_h=m_T/2$. After making this choice, we fit $p_0$ and $K$ to hadron production data in $pp$ collisions at the energy and rapidtiy of interest. Equation (\[ppcoll\_pt\]) is very satisfactory for $q_T\gtrsim 5$ GeV, but overpredicts the curvature of the hadron spectrum in the $q_T$=1-5 GeV range. This can be corrected for by considering an intrinsic transverse momentum, $k_T$, for the colliding partons [@Owens87]. We found that a fixed ${\langle k_T^2 \rangle}=0.52$ GeV leads to a dramatic improvement in the computation, which now agrees with data at the $\pm$40% level [@AG03b]. Finally, we obtain, at midrapidity $\eta=0$, $p_0=0.7\pm0.1$ GeV and $K=1.07\pm 0.02$ at Fermilab, and $p_0=1.0\pm0.1$ GeV and $K=0.99\pm0.03$ at RHIC.\ [**From p+p to p+A collisions.**]{} Having fixed all parameters in p+p collisions, and defined the parton-nucleon cross section (\[iNxsec\]), the GE expression for a parton-nucleus scattering is [@AT01b]: $$\begin{aligned} \fl \frac{d\sigma^{\,iA}}{d^2p_Tdyd^2b} = \sum_{n=1}^{\infty} \frac{1}{n!} \int & d^2b \, d^2k_1 \cdots d^2k_n \, \delta\big(\sum _{i=1,n} {\vec k}_i - {\vec p_T}\big) \nonumber \\ \fl & \times \frac{d\sigma^{\,iN}}{d^2k_1} T_A(b) \times \dots \times \frac{d\sigma^{\,iN}}{d^2k_n} T_A(b) \, e^{\, - \sigma^{\,iN}(p_0) T_A(b)} \, \ , \label{dWdp}\end{aligned}$$ where $T_A(b)$ is the target nucleus thickness function at impact parameter $b$. The exponential factor in [Eq. (\[dWdp\])]{} represents the probability that the parton suffered no semihard scatterings after the $n$-th one, and explicitly implements unitarity at the nuclear level. Assuming that the partons from $A$ suffer only one scattering on $p$ or $d$, we may generalize [Eq. (\[ppcoll\_pt\])]{} as follows, without introducing further parameters: $$\begin{aligned} \fl \frac{d\sigma}{d^2p_T dy d^2b}^{\hspace{-0.5cm}pA\rightarrow iX} \hspace{-0.5cm}=\Bigg\{ {\langle xf_{i/p} \rangle}_{y_i,p_T} \, \frac{d\sigma^{\,iA}}{d^2p_T dy_i d^2b} { \bigg| _{y_i=y} } \hspace*{-.6cm} + \hspace{.1cm} T_A(b) \sum_{b}\, {\langle xf_{i/A} \rangle}_{y_i,p_T} \, \frac{d\sigma^{\,ip}}{d^2p_T dy_i} { \bigg| _{y_i=-y} } \Bigg\} \otimes D_{i\rightarrow h} \nonumber\end{aligned}$$ Unitarity introduces a suppression of parton yields compared to the binary scaled p+p case. This is best seen in integrated parton yields: $d\sigma^{iA}/{dyd^2b} \approx 1 - e^{\, - \sigma^{\,iN}(p_0) T_A(b)}$. At low opacity $\chi = \sigma^{\,iN}(p_0) T_A(b) \ll 1$, i.e.,when the number of scatterings per parton is small, the binary scaling is recovered. However, at large opacity, $\chi \gtrsim 1$, the parton yield is suppressed: $d\sigma^{iA}/{dyd^2b} \ll 1 < \sigma^{\,iN}(p_0) T_A(b)$. This suppression is what we call “geometrical shadowing”, since it is driven purely by the geometry of the collision through the thickness function $T_A$. As the integrated yield is dominated by small momentum partons, geometrical shadowing is dominant at low $p_T$. Beside the geometrical quark and gluon shadowing, which is automatically included in GE models, at low enough $x$ one expects genuine dynamical shadowing due to non-linear gluon interactions as described in Colour Glass Condensate (CGC) models [@CGC]. However, it is difficult to disentangle these two sources of suppression of $p_T$ spectra, and to understand where dynamical effects begin to play a role beside the ubiquitous geometrical effects. The GE model computation outlined above can be used as a baseline to extract the magnitude of dynamical effects by comparison with experimental data.\ -.2cm . -1.2cm [**Cronin effect at Fermilab and RHIC.**]{} The Cronin effect may be quantified by taking the ratio of hadron $p_T$ spectrum in p(d)+A collision, and dividing it by the binary scaled p+p spectrum: $$R_{pA} \simeq {\frac{d\sigma}{dq_T^2 dy d^2b} ^{\hspace{-0.5cm}dAu\rightarrow h X} \hspace{-0.8cm} (b=\hat b)} \Bigg/ T_{A}(\hat b) {\frac{d\sigma}{dq_T^2 dy} ^{\hspace{-0.3cm}pp\rightarrow h X} \hspace{-0.8cm} } \ , \label{CroninRatio}$$ where $\hat b$ is the average impact parameter in the centrality bin in which experimental data are collected (see [@AG03b]). The GE model reproduces quite well both Fermilab and PHENIX data at $\eta=0$ (Fig.1, left and center). It also describe the increase of the Cronin effect with increasing centrality (Fig.1, right). If dynamical shadowing as predicted in CGC models was operating in this rapidity and energy range, the central/peripheral ratio should be smaller than the GE result: the more central the collision, the higher the parton density in the nucleus, the larger the non-linear effects. Therefore, we conclude that there is no dynamical shadowing nor Colour Glass Condensate at RHIC midrapidity. To address the preliminary BRAHMS data at forward rapidity $\eta\approx3.2$ [@BRAHMSfwd], we would first need to fit $p_0$ and $K$ in p+p collisions at the same pseudo-rapidity. Unfortunately the available data $p_T$-range $0.5 \lesssim p_T \lesssim 3.5$ Ge is not large enough $p_T$ for the fit to be done. Therefore, we use the parameters extracted at $\eta=0$. The resulting Cronin ratio, shown by the solid line in Figure 2, overestimates the data at such low-$p_T$. This may be corrected in part by considering elastic energy loss [@CT04]. [l]{}[6.3cm]{} \[fig:fwdrap\] -.4cm\ [ Cronin ratio at $\eta=3.2$.]{} -.4cm Furthermore, the opacity $\chi_0=0.95$ might be underestimated, due to the use of the mid-rapidity parameters: to check this we tripled the opacity, but the resulting dashed line still overestimates the data. Does the discrepancy between our calculations and the BRAHMS data prove that a CGC has been observed? It is too early to tell. An important observation [@BRAHMSfwd] is that the yield of positive charged hadrons exceeds the yield of negative charged hadrons at moderate pt by just the factor expected in HIJING due to dominance of valence quark fragmentation effects: gluon fragmentation is subdominant mechanism in this kinematic range. In addition, current simplified CGC scenrarios predict asymptotic $1/p_T^4$ absolute behavior which overestimates by an order of magnitude the observed absolute cross sections. In our pQCD approach the absolute cross sections in both p+p and d+Au collisions are much closer to the data, though the modest discrepancies shown in Figure 2 remain. Gluon shadowing appears to be needed in this $x\sim 10^{-3}-10^{-2}$ regime, though more data will be required to quantify the effect. The optimal region to study the onset of dynamical shadowing is $3 \lesssim p_T \lesssim 6$ GeV, where the GE model expects the peak of the Cronin ratio to be. If in this region the final data at forward rapidity $0 \lesssim \eta \lesssim 4$ explored by the four RHIC collaboration will reveal a consistent pattern of suppression compared to the GE computation (as preliminary data seem to suggest), and nonperturbative fragmentation effects will not be able to explain it, then the case for the CGC will be made more solid. The case would be even stronger if at the same time a progressive disappearance of back-to-back jets in favour of an increased dominance of monojet production was observed.  \ [99]{} J. W. Cronin , Phys. Rev. D [**11**]{} (1975) 3105; A. Accardi, in CERN Yellow Report on Hard Probes in Heavy Ion Collisions at the LHC \[arXiv:hep-ph/0212148\]. A. Accardi and M. Gyulassy, arXiv:nucl-th/0308029. A. Krzywicki, J. Engels, B. Petersson and U. Sukhatme, Phys. Lett. B [**85**]{} (1979) 407. M. Lev and B. Petersson, Z. Phys. [**C21**]{} (1983) 155; A. Accardi and D. Treleani, Phys. Rev. D [**64**]{} (2001) 116004; M. Gyulassy, P. Levai and I. Vitev, Phys. Rev. D [**66**]{} (2002) 014005. D. Antreasyan , Phys. Rev. D [**19**]{}, 764 (1979); P. B. Straub , Phys. Rev. Lett.  [**68**]{} (1992) 452. S. S. Adler [*et al.*]{} \[PHENIX\], Phys. Rev. Lett.  [**91**]{} (2003) 241803. J. F. Owens, Rev. Mod. Phys.  [**59**]{} (1987) 465. Yu. K. Kovchegov, [*these proceedings*]{}; D. Kharzeev, [*these proceedings*]{}. S. S. Adler , arXiv:nucl-ex/0306021; C. Klein-Bösing \[PHENIX\], [*these proceedings*]{}. R. Debbe \[BRAHMS\], [*these proceedings*]{} and talk at APS Division of Nuclear physics meeting, Tucson AZ, USA, Oct.30-Nov.1, 2004. E. Cattaruzza and D. Treleani, arXiv:hep-ph/0401067.

--- abstract: 'Production distributed systems are challenging to formally verify, in particular when they are based on distributed protocols that are not rigorously described or fully understood. In this paper, we derive models and properties for two core distributed protocols used in eventually consistent production key-value stores such as Riak and Cassandra. We propose a novel modeling called certified program models, where complete distributed systems are captured as programs written in traditional systems languages such as concurrent C. Specifically, we model the read-repair and hinted-handoff recovery protocols as concurrent C programs, test them for conformance with real systems, and then verify that they guarantee eventual consistency, modeling precisely the specification as well as the failure assumptions under which the results hold.' author: - Edgar Pek - Pranav Garg - Muntasir Raihan Rahman - | \ Karl Palmskog - Indranil Gupta - 'P. Madhusudan' bibliography: - 'refs.bib' title: | Inferring Formal Properties of\ Production Key-Value Stores ---

--- address: | $^1$University of Karlsruhe, D-76128 Karlsruhe, Germany\ $^2$Physics Department, Brookhaven National Laboratory, Upton, NY 11973\ $^3$University of Hamburg, D-22761 Hamburg, Germany author: - 'K.G. Chetyrkin$^1$, R. Harlander$^{1,2}$, T. Seidensticker$^1$, M. Steinhauser$^3$' title: - | Second Order QCD Corrections to the Top Decay Rate$^\dagger$ **TTP99-41\ **BNL-HET-99/29\ **October 1999\ **hep-ph/9910339******** - 'Second Order QCD Corrections to the Top Decay Rate$^\dag$' --- Introduction ============ The top quark is currently the heaviest known elementary particle. Thus top physics is a very promising field with regard to physics beyond the Standard Model (SM). For example, its large Higgs coupling makes one hope to learn something about the Higgs spectrum in particular, or mass generating mechanisms in general. Furthermore, its large mass is an important premise for decays into non-standard particles. Another exceptional property of the top quark is its large decay width ($\Gamma_{\rm Born} \approx 1.56$ GeV) as predicted by the SM. Instead of undergoing the process of hadronization, the top quark is hardly affected by the non-perturbative regime of QCD and decays almost exclusively into a bottom quark and a $W$ boson by weak interaction. Known results ============= The ${{\cal O}(\alpha_s)}$ corrections [@JK:twb] to the decay rate $\Gamma(t\to bW)$ amount to $-9\%$ of the Born result. This is comparable to the expected experimental accuracy at an NLC which is around $10\%$. Thus one should make sure that the QCD corrections are reliable, in the sense that the series in $\alpha_s$ converges sufficiently fast. This is the main motivation for investigating the $\alpha_s^2$ corrections to this process. The $-9\%$ from above may be split into a contribution for $M_W=0$ ($-11\%$) and the effects induced by a non-vanishing $W$ mass ($+2\%$). The electroweak corrections [@twb:ew1l] at one-loop level are about $2\%$. For the ${{\cal O}(\alpha_s^2)}$ corrections there exists a result for $M_W=0$ [@CzaMel:twb]. It was obtained by performing an expansion in the limit $$(M_t^2 - M_b^2)/M_t^2 \ll 1$$ and taking into account a sufficiently large number of terms in the expansion. The result reads $$\Gamma_t = \Gamma_{\rm Born}(1-0.09-0.02)\,,$$ where the three numbers correspond to the Born, ${{\cal O}(\alpha_s)}$, and ${{\cal O}(\alpha_s^2)}$ terms, respectively. The aim of the calculation of [@CHSS:twb] was, on the one hand, to perform an independent check of the ${{\cal O}(\alpha_s^2)}$ terms at $M_W=0$, and, on the other hand, to take into account effects induced by a non-vanishing $W$ mass at this order. Method {#sec::method} ====== While in [@CzaMel:twb] vertex diagrams for $t\to bW$ were computed, in [@CHSS:twb] the top quark decay rate was obtained via the optical theorem which relates it to the imaginary part of the top quark propagator $\Sigma = q\hspace{-1.15ex}/ \Sigma_{\rm V} + M_t \Sigma_{\rm S}$: $$\Gamma_t \propto {{\rm Im}}(\Sigma_{\rm V} + \Sigma_{\rm S})\big|_{q^2=M_t^2}\,.$$ This means that one should calculate $\Sigma$ at the point $q^2=M_t^2$ up to ${{\cal O}({G_{\rm F}}\alpha_s^2)}$. An example for a diagram that contributes to this order is shown in Fig. \[fig::dia\]. The $b$ quark mass can safely be set to zero, and for the moment also $M_W=0$ will be assumed. The analytic evaluation of diagrams like the one shown in Fig. \[fig::dia\] is currently neither available for general $q^2$, nor for the special case of interest, $q^2=M_t^2$. The only limiting cases that are accessible are $q=0$ or $M_t=0$. However, asymptotic expansions provide an efficient tool to obtain approximate results also away from these extreme choices. Their application yields series in $q^2/M_t^2$ (or $M_t^2/q^2$), with the individual coefficients containing the non-analytic structures in terms of logarithms of $\mu^2/q^2$ and $\mu^2/M_t^2$ ($\mu$ is the renormalization scale). Employing the analyticity properties of the approximated function, one obtains the region of convergence for the corresponding series. Within this region, the full result can be approximated with arbitrary accuracy by including sufficiently many terms in the expansion. Examples demonstrating the quality of such approximations can be found in [@CHKS:m12; @HSS:zbb]. =4.cm Applying this strategy to top quark decay, in a first step one should compute as many terms as possible in the expansion around $q^2/M_t^2$. Note that one cannot approach the point $q^2=M_t^2$ from the opposite side (i.e. $q^2/M_t^2 > 1$), because this implies top quarks in the final state which is kinematically forbidden. The second step is to extrapolate the result from the small-$q^2$ region to the point $q^2=M_t^2$. At ${{\cal O}(\alpha_s)}$ this could be done by explicitly resumming the full series in $q^2/M_t^2$ [@CzaMel:proc]. At ${{\cal O}(\alpha_s^2)}$, however, a different strategy was pursued in [@CHSS:twb] by performing a Padé approximation in the variables $z=q^2/M_t^2$ and $\omega = (1-\sqrt{1-z})/(1+\sqrt{1-z})$. Effects of a non-vanishing $W$ mass can be taken into account by applying asymptotic expansions w.r.t. the relation $M_t^2\gg q^2\gg M_W^2$. In this way one obtains a nested series in $M_W^2/M_t^2$ and $q^2/M_t^2$. The above procedure can then be applied to each coefficient of $M_W^2/M_t^2$ separately. One of the questions one is faced with when following this approach is gauge (in)dependence. The off-shell fermion propagator is not a gauge invariant quantity, and only in the limit $q^2=M_t^2$ the QCD gauge parameter $\xi$ formally drops out. Due to the fact that one works with a limited number of terms here, gauge parameter dependence does not vanish exactly but is only expected to decrease gradually as soon as a sufficiently large number of expansion terms is included. Nevertheless, the claim is that by a reasonable choice of the gauge parameter the predictive power of the result is preserved. At ${{\cal O}(\alpha_s^2)}$ the calculation cannot be performed for general $\xi$ due to the enormous increase in required computer resources. Thus one cannot decide upon the choice of $\xi$ [*a posteriori*]{}, but has to set it to some definite value from the very beginning. On the other hand, it is expected that the behavior of the $\alpha_s^2$ terms w.r.t. the choice of $\xi$ is similar to the ${{\cal O}(\alpha_s)}$ result. Therefore, in [@CHSS:twb] the gauge parameter dependence was studied in some detail at ${{\cal O}(\alpha_s)}$. This study was mainly based on the stability of the Padé results $[m/n]$ upon variation of $m$ and $n$ for different values of $\xi$. Another way to find a “reasonable” choice for $\xi$ is to study the $q^2$-dependence of the Padé results close to the physically relevant point $q^2=M_t^2$. The extrapolation to $q^2=M_t^2$ is expected to work best if the variation of the approximating function close to this point is smoothest. For some values of $\xi$, the $q^2$ dependence of the one-loop result near $q^2/M_t^2=1$ is shown in Fig. \[fig::xi\]. The results of [@CHSS:twb] were obtained by setting $\xi=0$. The validity of this choice is further justified by the perfect agreement of the approximation to the exact result at ${{\cal O}(\alpha_s)}$ (see Fig. \[fig::xi\] and the results in the following section). =6.cm Concerning the technical realization of the calculation it heavily relies on automatic Feynman diagram evaluation with the help of algebraic programs [@HS:review]. For details we refer to [@CHSS:twb]. Results ======= The result for the top decay rate to ${{\cal O}(\alpha_s^2)}$ will be written in the following way ($y\equiv M_W^2/M_t^2$, $y_0 = (80.4/175)^2$): $$\Gamma_t = \Gamma_0\left(\delta^{(0)}(y) + {\alpha_s\over\pi}\delta^{(1)}(y) + \left({\alpha_s\over\pi}\right)^2 \delta^{(2)}(y) \right)\,, \label{eq::gammat}$$ with $\delta^{(0)}(y_0) = 0.885\ldots$. Following the method described in Section \[sec::method\], one obtains $\delta^{(1)}(y_0) = -2.20(3)$ which is to be compared with the exact number, reading $-2.220\ldots$. This demonstrates the validity and the accuracy of the underlying approach. At ${{\cal O}(\alpha_s^2)}$, the result of [@CHSS:twb] reads $$\begin{aligned} \lefteqn{\delta^{(2)}(y) = -16.7(8) + 5.4(4)\, y + y^2\,(11.4(5.0)} \nonumber\\&&\mbox{}+ 7.3(1)\,\ln y) + {{\cal O}(y^3)} \stackrel{y=y_0}{=} -15.6(1.1).\,\,\,\,\,\,\label{eq::top2l}\end{aligned}$$ The first number corresponds to the case of vanishing $W$ mass. It agrees perfectly with the result of [@CzaMel:twb] which is $-16.7(5)$. The uncertainty quoted in [@CHSS:twb] is slightly larger than the one of [@CzaMel:twb] but is still negligible as compared to the expected experimental accuracy at future colliders. The series in $M_W^2/M_t^2$ is converging very quickly, and the total effect of the $M_W$-suppressed terms is small. The errors in (\[eq::top2l\]) are added linearly, yielding a bigger uncertainty for the sum than for the $M_W=0$ result. Adding the errors quadratically instead, the uncertainty again would be $0.8$. Estimate for ============= ---------------- =5.5cm \[1em\] =5.5cm ---------------- The integration of Eq. (\[eq::gammat\]) over $y$ from $0$ to $1$ is directly related to the decay rate for $b\to ul\bar \nu$ at ${{\cal O}(\alpha_s^2)}$ (and with obvious modifications to the two-loop QED corrections to $\Gamma(\mu\to e\nu\bar \nu))$. However, since the $y$ dependence of $\delta^{(2)}$ is only known up to $y^2$, it is more promising to pull out $\delta^{(0)}(y)$ and to consider the Taylor expansion of $\delta^{(i)}/\delta^{(0)}$ which is expected to vary only slightly in the relevant $y$-range [@CzaMel:twb]. On the other hand, if more terms in $y$ are included, the Taylor series of $\delta^{(i)}/\delta^{(0)}$ becomes ill-behaved, and one should directly expand $\delta^{(i)}$. These observations are illustrated for $i=1$ in Fig. \[fig::a0a1\]. In [@CHSS:twb] both methods were applied, and the results for $\Gamma(b\to ul\bar \nu)$ and $\Gamma(\mu\to e\nu\bar \nu)$ agree with the ones of $\cite{RitStu:bmudecay}$ to about $10\%$. This constitutes a stringent check on both calculations. Also a direct application of the method described in Section \[sec::method\] to the (four-loop!) diagrams corresponding to $b\to ul\bar\nu$ and $\mu\to e\nu\bar \nu$ was performed in [@KueSeiSte:mu], and full agreement with $\cite{RitStu:bmudecay}$ was found. \#1\#2\#3[[*Act. Phys. Pol. *]{}[**B \#1**]{} (\#2) \#3]{} \#1\#2\#3[[*Act. Phys. Austr. *]{}[**\#1**]{} (\#2) \#3]{} \#1\#2\#3[[*Comm. Math. Phys. *]{}[**\#1**]{} (\#2) \#3]{} \#1\#2\#3[[*Comp. Phys. Commun. *]{}[**\#1**]{} (\#2) \#3]{} \#1\#2\#3[[*Eur. Phys. J. *]{}[**C \#1**]{} (\#2) \#3]{} \#1\#2\#3[[*Fortschr. Phys. *]{}[**\#1**]{} (\#2) \#3]{} \#1\#2\#3[[*Int. J. Mod. Phys. *]{}[**C \#1**]{} (\#2) \#3]{} \#1\#2\#3[[*Int. J. Mod. Phys. *]{}[**A \#1**]{} (\#2) \#3]{} \#1\#2\#3[[*J. Comp. Phys. *]{}[**\#1**]{} (\#2) \#3]{} \#1\#2\#3[[*JETP Lett. *]{}[**\#1**]{} (\#2) \#3]{} \#1\#2\#3[[*Mod. Phys. Lett. *]{}[**A \#1**]{} (\#2) \#3]{} \#1\#2\#3[[*Nucl. Inst. Meth. *]{}[**A \#1**]{} (\#2) \#3]{} \#1\#2\#3[[*Nucl. Phys. *]{}[**B \#1**]{} (\#2) \#3]{} \#1\#2\#3[[*Nuovo Cim. *]{}[**\#1A**]{} (\#2) \#3]{} \#1\#2\#3[[*Phys. Lett. *]{}[**B \#1**]{} (\#2) \#3]{} \#1\#2\#3[[*Phys. Reports* ]{}[**\#1**]{} (\#2) \#3]{} \#1\#2\#3[[*Phys. Rev. *]{}[**D \#1**]{} (\#2) \#3]{} \#1\#2\#3[[*Phys. Rev. *]{}[**\#1**]{} (\#2) \#3]{} \#1\#2\#3[[*Phys. Rev. Lett. *]{}[**\#1**]{} (\#2) \#3]{} \#1\#2\#3[[*Phys. Reports* ]{}[**\#1**]{} (\#2) \#3]{} \#1\#2\#3[[*Prog. Theor. Phys. *]{}[**\#1**]{} (\#2) \#3]{} \#1\#2\#3[[*Prog. Part. Nucl. Phys. *]{}[**\#1**]{} (\#2) \#3]{} \#1\#2\#3[[*Sov. J. Nucl. Phys. *]{}[**\#1**]{} (\#2) \#3]{} \#1\#2\#3[[*Teor. Mat. Fiz. *]{}[**\#1**]{} (\#2) \#3]{} \#1\#2\#3[[*Yad. Fiz. *]{}[**\#1**]{} (\#2) \#3]{} \#1\#2\#3[[*Z. Phys. *]{}[**C \#1**]{} (\#2) \#3]{} \#1\#2\#3[[ibid. ]{}[**\#1**]{} (\#2) \#3]{} [99]{} M. Jeżabek and J.H. Kühn, . A. Denner and T. Sack, ;\ G. Eilam, R.R. Mendel, R. Migneron, and A. Soni, . A. Czarnecki and K. Melnikov, . K.G. Chetyrkin, R. Harlander, T. Seidensticker, and M. Steinhauser, Report Nos. BUTP-99/10, TTP99-25 (Bern, Karlsruhe, 1999), hep-ph/9906273; [*Phys. Rev. *]{}[**D**]{} (in print). K.G. Chetyrkin, R. Harlander, J.H. Kühn and M. Steinhauser, . R. Harlander, T. Seidensticker, and M. Steinhauser, . A. Czarnecki and K. Melnikov, talk given at the Lake Louise Winter Institute: [*Quantum Chromodynamics*]{}, Lake Louise, Alberta, Canada, 15-21 Feb 1998. Report Nos. BNL-HET-98-22, TTP98-23 (BNL, Karlsruhe, 1998), hep-ph/9806258. R. Harlander and M. Steinhauser, . T. van Ritbergen and R. Stuart, ; T. van Ritbergen, . T. Seidensticker and M. Steinhauser, Report Nos. BUTP99-17, TTP99-38 (Bern, Karlsruhe, 1999), hep-ph/9909436, [*Phys. Lett.*]{} [**B**]{} (in print).

--- abstract: 'Using the 870-$\mu$m APEX Telescope Large Area Survey of the Galaxy (ATLASGAL), we have identified 577 submillimetre continuum sources with masers from the methanol multibeam (MMB) survey in the region $280\degr < \ell < 20\degr$; $|\,b\,| < 1.5\degr$. 94percent of methanol masers in the region are associated with sub-millimetre dust emission. We estimate masses for $\sim$450 maser-associated sources and find that methanol masers are preferentially associated with massive clumps. These clumps are centrally condensed, with envelope structures that appear to be scale-free, the mean maser position being offset from the peak column density by $0 \pm 4$. Assuming a Kroupa initial mass function and a star-formation efficiency of $\sim$30percent, we find that over two thirds of the clumps are likely to form clusters with masses $>$20[M$_\odot$]{}. Furthermore, almost all clumps satisfy the empirical mass-size criterion for massive star formation. Bolometric luminosities taken from the literature for $\sim$100 clumps range between $\sim$100 and 10$^6$[L$_\odot$]{}. This confirms the link between methanol masers and massive young stars for 90percent of our sample. The Galactic distribution of sources suggests that the star-formation efficiency is significantly reduced in the Galactic-centre region, compared to the rest of the survey area, where it is broadly constant, and shows a significant drop in the massive star-formation rate density in the outer Galaxy. We find no enhancement in source counts towards the southern Scutum-Centaurus arm tangent at $\ell \sim 315\degr$, which suggests that this arm is not actively forming stars.' author: - | J.S.Urquhart$^{1}$[^1], T.J.T.Moore$^{2}$, F.Schuller$^{3}$, F.Wyrowski$^{1}$, K.M.Menten$^{1}$, M.A.Thompson$^{4}$, T.Csengeri$^{1}$, C.M.Walmsley$^{5,6}$, L.Bronfman$^{7}$, C.König$^{1}$\ $^{1}$ Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, Bonn, Germany\ $^{2}$Astrophysics Research Institute, Liverpool John Moores University, Twelve Quays House, Egerton Wharf, Birkenhead, CH411LD, UK\ $^{3}$European Southern Observatory, Alonso de Cordova 3107, Vitacura, Santiago, Chile\ $^{4}$Science and Technology Research Institute, University of Hertfordshire, College Lane, Hatfield, AL10 9AB, UK\ $^{5}$Osservatorio Astrofisico di Arcetri, Largo E. Fermi, 5, 50125 Firenze, Italy\ $^{6}$Dublin Institute for Advanced Studies, Burlington Road 10, Dublin 4, Ireland\ $^{7}$Departamento de Astronomía, Universidad de Chile, Casilla 36-D, Santiago, Chile bibliography: - 'mmb.bib' date: 'Accepted ??. Received ??; in original form ??' title: 'ATLASGAL — Environments of 6.7GHz methanol masers' --- \[firstpage\] Stars: formation – Stars: early-type – Galaxy: structure – ISM: molecules – ISM: submillimetre. Introduction ============ Massive stars ($>$ 8[M$_\odot$]{} and $>$10$^3$[L$_\odot$]{}) play a hugely important role in many astrophysical processes from the formation of the first solid material in the early Universe (@dunne2003); to their substantial influence upon the evolution of their host galaxies and future generations of star formation (@kennicutt2005). Given the profound impact massive stars have, not only on their local environment, but also on a Galactic scale, it is crucial to understand the environmental conditions and processes involved in their birth and the earliest stages of their evolution. However, massive stars form in clusters and are generally located at greater distances than regions of low-mass star formation and therefore understanding how these objects form is observationally much more challenging (see @zinnecker2007 for a review). Moreover, massive stars are rare and they evolve much more quickly than low-mass stars, reaching the main sequence while still deeply embedded in their natal environment. As a consequence of their rarity and relatively short evolution large spatial volumes need to be searched in order to identify a sufficient number of sources in each evolutionary stage. Only then can we begin to understand the processes involved in the formation and earliest stages of massive star formation. There have been a number of studies over the last two decades or so that have been used to identify samples of embedded young massive stars utilizing the presence of methanol masers (e.g., @walsh1997), IRAS, MSX or GLIMPSE infrared colours (e.g., @molinari1996 and @bronfman1996, @lumsden2002 and @robitaille2008, respectively) and compact radio emission (e.g., @wood1989). Although these methods have had some success, they tend to focus on a particular evolutionary type and in the case of the infrared colour selected samples are biased away from complex regions as a result of confusion in the images due to the limited angular resolution of the surveys. This is particularly acute for IRAS selected samples. Some of these surveys suffer from significant biases. For example, methanol masers require a strong mid-infrared source for the creation of high enough methanol abundances and to pump the maser transitions, while a hot ionising star must be present to ionize an [UCH [ii]{}]{} region. Surveys such as these preclude the possibility of identifying the very earliest pre-stellar clumps that would need to be included in any complete evolutionary sequence for massive star formation. All of the earliest stages of massive star formation take place within massive clumps of dust and gas, which can be traced by their thermal dust continuum emission. Dust emission is generally optically thin at (sub)millimetre wavelengths and is therefore an excellent tracer of column density and total clump mass. As well as including all of the embedded stages, dust emission observations are also sensitive to the colder pre-stellar phases and so provide a means to study the whole evolutionary sequence of the massive star formation. Until recently there were no systematic and unbiased surveys of dust emission. Most studies to date that have been undertaken consisted of targeted observations of IRAS or maser selected samples (e.g., @sridharan2002 [@faundez2004; @hill2005; @thompson2006]), and so suffer from the same problems mentioned in the previous paragraph, or have concentrated on a single, often exceptionally rich region (e.g., @motte2007), which may not be representative. However, there are now two large unbiased (sub)millimetre surveys available: the APEX Telescope Large Area Survey of the Galaxy (ATLASGAL; @schuller2009) at 870[$\mu$m]{} and the Bolocam Galactic Plane Survey (BGPS; @aguirre2011) at 1.1mm. These surveys have identified many thousands of sources across the Galaxy with which to compile the large samples of massive clumps required to build up a comprehensive understanding of massive star formation, and test the predictions of the main two competing theoretical models (i.e., competitive accretion (@bonnell1997 [@bonnell2001]) and monolithic collapse (@mckee2003)). The complete ATLASGAL catalogue consists of approximately 12,000 compact sources (see @contreras2013 for details) distributed across the inner Galaxy. This is the first of a series of papers that will investigate the dust properties and Galactic distribution of massive star formation. Here we use the association of methanol masers, that are considered to be an excellent tracer of the early stages of massive star formation (@minier2003), to identify a large sample of massive clumps. The structure of this paper is as follows: in Sect.2 we provide a brief summary of the two surveys used to select the sample of massive star forming clumps. In Sect.3 we describe the matching procedure and discuss sample statistics, while in Sect.4 we derive the physical properties of the clumps and their associated masers. In Sect.5 we evaluate the potential of the clumps to form massive stars and investigate their Galactic distribution with reference to the large scale structural features of the Milky Way. We present a summary of the results and highlight our main findings in Sect.6. Survey descriptions =================== ATLASGAL Survey --------------- The APEX Telescope Large Area Survey of the Galaxy (ATLASGAL; @schuller2009) is the first systematic survey of the inner Galactic plane in the submillimeter wavelength range. The survey was carried out with the Large APEX Bolometer Camera (LABOCA; @siringo2009), an array of 295 bolometers observing at 870$\mu$m (345 GHz). The 12-metre diameter telescope affords an angular resolution of $19\rlap{.}''2$ FWHM. The initial survey region covered a Galactic longitude region of $300\degr < \ell < 60\degr$ and $|b| < 1.5\degr$, but this was extended to include $280\degr < \ell < 300\degr$, however, the latitude range was shifted to $-2\degr < b < 1\degr$ to take account of the Galactic warp in this region of the plane and is not as sensitive as for the inner Galaxy ($\sim$60 and 100mJy beam$^{-1}$ for the $300\degr < \ell < 60\degr$ and $280\degr < \ell < 300\degr$ regions, respectively). @contreras2013 produced a compact source catalogue for the central part of the survey region (i.e., 330 $ <\ell <$ 21) using the source extraction algorithm [`SExtractor`]{} [@bertin1996]. Signal-to-noise maps that had been filtered to remove the large scale variations due to extended diffuse emission were used by [`SExtractor`]{} to detect sources above a threshold of 3$\sigma$, where $\sigma$ corresponds to the background noise in the maps. Source parameters were determined for each detection from the dust emission maps. This catalogue consists of 6,639 sources and is 99percent complete at $\sim$6$\sigma$, which corresponds to a flux sensitivity of 0.3-0.4Jybeam$^{-1}$. We have used the same source extraction algorithm and method described by @contreras2013 to produce a catalogue for the currently unpublished 280 $ <\ell <$ 330 and 21 $ <\ell <$ 60 regions of the survey. When the sources identified in these regions are combined with those identified by @contreras2013 we obtain a final compact source catalogue of some 12,000 sources (full catalogue will be presented in Csengeri et al. 2013 in prep.). The telescope has an [r.m.s.]{} pointing accuracy of $\sim$2, which we adopted as the positional accuracy for the catalogue. This catalogue provides a complete census of dense dust clumps located in the inner Galaxy and includes all potential massive star forming clumps with masses greater than 1,000[M$_\odot$]{} out to 20kpc. MMB Survey {#sect:mmb_description} ---------- Methanol masers are well-known indicators of the early phases of high-mass star formation, in particular sources showing emission in the strong 6.7GHz Class-II maser transition [@menten1991]. The Methanol Multibeam (MMB) Survey mapped the Galactic plane for this maser transition using a 7-beam receiver on the Parkes telescope with a sensitivity of 0.17Jybeam$^{-1}$ and a half-power beamwidth of 3.2 (@green2009). All of these initial maser detections were followed up at high-resolution ($\sim$2) with the Australia Telescope Compact Array (ATCA) to obtain sub-arcsec positional accuracy (0.4 [r.m.s.]{}; @caswell2010b). To date the MMB is complete between $186\degr < \ell\ < 20\degr$ and $|b| < 2\degr$ (@caswell2010b [@green2010; @caswell2011; @green2012]) and has reported the positions of 707 methanol maser sites; these sites consist of groups of maser spots that can be spread up to 1 in size but are likely to be associated with a single object. The MMB catalogue gives the velocity of the peak component and the flux density as measured from both the high sensitivity ATCA follow-up observations and those measured from the initial lower sensitivity Parkes observations. These observations were taken over different epochs up to two years apart and consequently the measured values can be affected by variability of the maser. We have used the velocity and peak flux densities measured from the ATCA data for all MMB sources except for MMBG321.704+01.168 as it was not detected in the ATCA observations; for this source we use the values recorded from the Parkes observations. ATLASGAL-MMB associations {#sect:atlas-mmb} ========================= Matching statistics {#sect:matching_stats} ------------------- Of the 707 methanol masers currently identified by the MMB survey 671 are located in the Galactic longitude and latitude range surveyed at 870$\mu$m by the APEX telescope as part of the ATLASGAL project. This represents $\sim$95percent of the entire published MMB catalogue. As a first step to identify potential matches between the methanol masers and the ATLASGAL sources we used a matching radius of 120, which is the maximum radius of sources found in the ATLASGAL catalogue [@contreras2013]. We used the peak 870$\mu$m flux as the ATLASGAL source position for these matches. In cases where a methanol maser was found to be located within this search radius of two or more ATLASGAL sources the nearest submillimetre source was selected as the most likely association. This simple radius search identified 637 potential ATLASGAL-MMB associations from the possible 671 methanol masers. To verify that these associations are genuine, we extracted $3\arcmin\times3\arcmin$ regions from the ATLASGAL emission maps and inspected these by eye to confirm the emission region and position of the methanol maser are coincident. We failed to find an ATLASGAL source at the MMB position towards nine MMB sources and therefore removed these from the associated sample; these masers form part of the sample of unassociated masers discussed in Sect.\[sect:unmatched\_mmb\]. For the potential associations located in the part of the Galaxy surveyed by GLIMPSE (i.e., $\ell > 295$ and $|b| < 1$; @benjamin2003_ori) we extracted $3\arcmin\times3\arcmin$ mid-infrared datasets from the project archive and produced false colour images of their environments. We present a sample of these images in Fig.\[fig:irac\_images\_associated\_atlasgal\] overplotted with contours of the submillimetre emission. In the left panels of Fig.\[fig:irac\_images\_associated\_atlasgal\] we present some examples of the rejected matches. The final sample consists of 628 methanol masers that are positionally coincident with 577 ATLASGAL sources (see middle and right panels of Fig.\[fig:irac\_images\_associated\_atlasgal\] for examples of these associations), with two or more methanol masers found toward 44 clumps. This sample includes $\sim$94percent of the MMB masers located in the surveys’ overlap region, however, this represents only $\sim$7percent of the ATLASGAL sources in the same region. For two of the ATLASGAL sources associated with multiple methanol masers we find the maser velocities disagree by more that would be expected if the masers were associated with the same molecular complex (i.e., $|\Delta v| > 10$[kms$^{-1}$]{}). These are: AGAL355.538$-$00.104 which is associated with MMB355.538$-$00.105 (3.8kms$^{-1}$) and MMB355.545$-$00.103 ($-$28.2kms$^{-1}$); and AGAL313.766$-$00.862 which is associated with MMB313.767$-$00.863 ($-$56.3kms$^{-1}$) and MMB313.774$-$00.863 ($-$44.8kms$^{-1}$). For each of these we have adopted the velocity of the maser with the smallest angular offset from the submillimetre peak. In the left panel of Fig.\[fig:mmb\_atlas\_offset\] we present a plot showing the offsets in Galactic longitude and latitude between the peak positions of the ATLASGAL and methanol maser emission. The positional correlation between the two tracers is excellent with the mean (indicated by the blue cross) centred at zero ($\Delta \ell = - 0.3\arcsec\pm0.4$ and $\Delta b=+0.18\arcsec\pm0.37$). In the right panel of Fig.\[fig:mmb\_atlas\_offset\] we present a plot showing the MMB angular surface density as a function of angular separation between the maser position and the peak of the 870$\mu$m emission of the associated ATLASGAL source. This plot reveals a strong correlation between the position of the methanol maser and the peak submillimetre continuum emission. The distribution peaks at separations less than 2 and falls off rapidly as the angular separation increases to $\sim$12 after which the distribution flattens off to an almost constant background level close to zero. We find that $\sim$87percent of all ATLASGAL-MMB associations have an angular separation $<$12 (which corresponds to 3$\sigma$, where $\sigma$ is the standard deviation of the offsets weighted by the surface density). The high concentration and small offsets between the masers and the peak dust emission reveals that the methanol masers are embedded in the brightest emission parts of these submillimetre clumps. This would suggest that the protostars giving rise to these methanol masers are preferentially found towards the centre of their host clumps. This is in broad agreement with the predictions of the competitive accretion model (@bonnell1997 [@bonnell2001]) where the deeper gravitational potential at the centre of the clump is able to significantly increase the gas density by funneling material from the whole cloud toward the centre. However, high angular resolution interferometric observations (e.g., with ALMA) are required to conclusively prove this hypothesis by measuring the density distribution for the clumps on small spatial scales. The larger angular offsets are found for approximately 80 ATLASGAL-MMB associations, which could indicate the presence of clumpy substructure that has not been properly identified by SExtractor. Alternatively, these matches with large offsets could be the result of a chance alignment of a nearby clump with an MMB source that is associated with more distant dust clump that falls below the ATLASGAL sensitivity limit. In the right panels of Fig.\[fig:irac\_images\_associated\_atlasgal\] we present mid-infrared images towards two sources where the methanol maser is offset from the peak position of the dust emission by more than 12. Inspection of the dust emission (shown by the contours) does reveal the presence of weak localised peaks close to that of the methanol maser position, which would suggest that these two particular sources do possess substructure that has not been identified independently by the ATLASGAL source extraction method. These two sources are fairly typical of the matches with offsets greater than 12 and therefore we would conclude that this is the most likely explanation, however, higher sensitivity observations are required to confirm this, and to properly characterise the dust properties of these methanol maser sites. In Fig.\[fig:flux\_density\_mmb\_atlasgal\] we present a plot comparing the methanol flux density and the 870$\mu$m peak flux density of the ATLASGAL-MMB associations. There is no apparent correlation between these parameters from a visual inspection of this plot, however, the correlation coefficient is 0.19 with a significance value of 5$\times$10$^{-6}$ and so there is a weak correlation. Methanol masers are found to be associated with a range of evolutionary stages of massive star formation (i.e., from the hot molecular core (HMC) through to the ultra-compact (UC) HII region stage). Therefore the low level of correlation could simply be a reflection of the spread in evolutionary stages covered by the methanol masers. It is also important to bear in mind that these methanol masers are likely to be associated with the circumstellar envelope/disk of a single embedded source, and given the resolution of the ATLASGAL survey it is almost certain that the measured submillimetre flux is related to the mass of the whole clump, which is likely to go on to form an entire cluster. It is therefore not surprising that we find the flux densities of the masers and the dust clumps to be only weakly correlated. To investigate whether there is a correlation between the submillimetre and methanol maser fluxes and the mid-infrared properties of the source we have cross-correlated the matched sources with the @gallaway2013 catalogue. Using this catalogue we separate our matched sample into three groups, those associated with mid-infrared emission, infrared dark sources and those located outside the region covered by the GLIMPSE Legacy project, upon which the work of @gallaway2013 is based. The distribution of these three groups are shown in Fig.\[fig:flux\_density\_mmb\_atlasgal\] as red, purple and black symbols, respectively. Comparing the flux distributions of the infrared bright and dark samples with a Kolmogorov-Smirnov (KS) test we do not find them to be significantly different. ATLASGAL 870$\mu$m flux distribution {#sect:flux_dist} ------------------------------------ In Fig.\[fig:flux\_density\] we present plots of the 870$\mu$m peak and integrated flux distribution (upper and lower panels, respectively) of the ATLASGAL catalogue (grey histogram) and ATLASGAL-MMB associations (blue histogram). It is clear from these plots that the methanol masers are preferentially associated with the brighter ATLASGAL sources in the sense that the probability of an association with a maser approaches 100percent for brighter clumps. This is particularly evident in the peak flux distribution, which reveals that only a relatively small number of submillimetre sources brighter than $\sim$7Jybeam$^{-1}$ are not associated with a methanol maser. It is also clear from these plots that there is a stronger correlation between the brightest peak flux density ATLASGAL sources and the presence of a methanol maser than between the integrated flux and the presence of a maser. The integrated flux is a property of the whole clump/cloud, whereas the peak flux is more likely to be associated with the highest column density and/or warmest regions of the clump where star formation is taking place. Furthermore, some of the ATLASGAL sources with the highest integrated fluxes are large extended sources, that can have relatively low peak flux densities. In Fig.\[fig:peak\_flux\_density\_association\_ratio\] we present a plot showing the ratio of ATLASGAL sources found to be associated with a methanol maser as a function of peak 870$\mu$m flux density. Although the errors in the ratios for the higher flux bins are relatively large, due to the smaller numbers of sources they contain, there is still clearly a strong correlation between bright submillimetre emission and the presence of a methanol maser. Given that the association rate of ATLASGAL sources with methanol masers increases rapidly with peak flux density ($\sim$100percent for sources above 20Jybeam$^{-1}$) an argument can be made for the maser emission being effectively isotropic. Although the radiation beamed from individual maser spots is highly directional, there are many very high resolution studies that have revealed significant numbers of maser spots to be associated with a single source (e.g., @goddi2011). These individual maser spots are distributed around the central embedded protostar and are therefore spatially distinct. Unassociated MMB Sources {#sect:unmatched_mmb} ------------------------ [43]{} MMB masers could not be matched with an ATLASGAL source. These masers have integrated flux densities between 0.43 and 15.65Jy, with a mean and median value of 2.2 and 1.5Jy, respectively. In Fig.\[fig:mmb\_flux\_distribution\] we present a histogram showing the distribution of maser fluxes for the whole MMB sample (grey filled histogram) and of the unassociated masers (red histogram). The flux densities of the unmatched masers are significantly above the MMB survey’s sensitivity of 0.17Jybeam$^{-1}$, and although they are principally found towards the lower flux end of the distribution, they are not the weakest masers detected. For comparison the mean and median fluxes for the whole maser sample are 47.8 and 5.1Jy, respectively. Using a KS test to compare the distribution of the dustless MMB masers with that of the whole MMB catalogue we are able to reject the null hypothesis that these are drawn from the same population with greater than three sigma confidence. It is widely accepted that methanol masers are almost exclusively associated with high-mass star forming regions (e.g., @minier2003 [@pandian2010]). If this is the case then we might expect all of these unmatched sources to be located at the far side of the Galaxy where their dust emission falls below the ATLASGAL detection sensitivity, however, it is possible that these are associated with more evolved stars (e.g., @walsh2003). We will investigate the nature of these unassociated methanol masers in more detail in Sect.5.4. Physical properties =================== In the previous section we identified two subsamples based on the possible combinations of ATLASGAL and MMB associations: 1) methanol masers associated with the thermal continuum emission from cold dust and likely tracing star formation; 2) apparently dustless methanol masers which could be a combination of masers associated with more distant dust clumps not currently detected. These two subsamples consist of approximately 94percent and 6percent of the MMB sources in the overlap region, respectively. In this section we will concentrate on the ATLASGAL-MMB associations to determine the physical properties of their environments; these are given for each clump in Table\[tbl:derived\_clump\_para\] while in Table\[tbl:derived\_para\] we summarise the global properties. [ll.........]{} & & & & & &&& & &\ & & & & & &&& & &\ & & & & & &&& & &\ AGAL281.709$-$01.104 & MMB281.710$-$01.104 & 2.3 & 1.6 & 3.44 & 4.2 & 8.7 & 0.50 & 22.97 & 2.94 & 1.88\ AGAL284.352$-$00.417 & MMB284.352$-$00.419 & 6.5 & 1.5 & 10.44 & 5.2 & 8.8 & 1.05 & 22.84 & 3.49 & 2.93\ AGAL284.694$-$00.359 & MMB284.694$-$00.361 & 6.2 & 1.0 & 2.00 & 6.3 & 9.2 & & 22.41 & 2.51 & 3.22\ AGAL285.339$-$00.001 & MMB285.337$-$00.002 & 8.3 & 1.7 & 5.29 & 5.1 & 8.7 & 0.53 & 22.63 & 2.96 & 3.56\ AGAL287.372+00.646 & MMB287.371+00.644 & 6.8 & 1.1 & 2.59 & 5.2 & 8.6 & 0.19 & 22.67 & 2.72 & 4.56\ AGAL291.272$-$00.714 & MMB291.270$-$00.719 & 19.6 & 1.4 & 11.25 & 1.0 & 8.2 & 0.36 & 23.88 & 3.13 & 2.00\ AGAL291.272$-$00.714 & MMB291.274$-$00.709 & 20.5 & 1.4 & 11.25 & 1.0 & 8.2 & 0.36 & 23.88 & 3.13 & 2.94\ AGAL291.579$-$00.432 & MMB291.579$-$00.431 & 4.3 & 1.4 & 5.05 & 8.1 & 9.4 & 2.02 & 23.59 & 4.31 & 2.92\ AGAL291.579$-$00.432 & MMB291.582$-$00.435 & 14.8 & 1.4 & 5.05 & 7.7 & 9.1 & 1.90 & 23.59 & 4.26 & 3.31\ AGAL291.636$-$00.541 & MMB291.642$-$00.546 & 28.0 & 1.5 & 14.72 & 7.8 & 9.2 & 2.94 & 23.33 & 4.49 & 2.37\ AGAL291.879$-$00.809 & MMB291.879$-$00.810 & 2.7 & 1.2 & 2.89 & 9.9 & 10.4 & & 22.61 & 3.27 & 3.22\ AGAL292.074$-$01.129 & MMB292.074$-$01.131 & 8.1 & 2.1 & 3.18 & 3.2 & 7.9 & & 22.34 & 2.05 & 2.04\ AGAL293.828$-$00.746 & MMB293.827$-$00.746 & 1.4 & 1.1 & 2.48 & 10.7 & 10.7 & 0.94 & 23.10 & 3.75 & 3.56\ AGAL293.941$-$00.874 & MMB293.942$-$00.874 & 3.7 & 1.2 & 3.10 & 11.2 & 10.9 & 0.86 & 22.81 & 3.59 & 3.83\ AGAL294.336$-$01.705 & MMB294.337$-$01.706 & 5.6 & 1.1 & 1.72 & 1.0 & 8.1 & & 22.59 & 1.03 & 0.11\ AGAL294.511$-$01.622 & MMB294.511$-$01.621 & 3.3 & 1.1 & 3.60 & 1.0 & 8.1 & 0.14 & 23.11 & 1.89 & 1.98\ AGAL294.976$-$01.734 & MMB294.977$-$01.734 & 3.1 & 1.3 & 6.73 & 0.2 & 8.4 & 0.05 & 23.20 & 1.02 & -1.10\ AGAL294.989$-$01.719 & MMB294.990$-$01.719 & 2.2 & 1.1 & 4.87 & 1.1 & 8.1 & 0.17 & 23.10 & 2.03 & 2.19\ AGAL296.893$-$01.306 & MMB296.893$-$01.305 & 2.4 & 1.3 & 0.98 & 10.0 & 9.8 & & 22.52 & 2.71 & 3.18\ AGAL297.391$-$00.634 & MMB297.406$-$00.622$^\dagger$ & 68.2 & 1.6 & 10.82 & 10.7 & 10.1 & 0.50 & 22.38 & 3.66 & 3.31\ AGAL298.182$-$00.786 & MMB298.177$-$00.795$^\dagger$ & 37.7 & 1.0 & 3.60 & 10.4 & 9.9 & 1.49 & 23.15 & 3.94 & 3.56\ AGAL298.224$-$00.339 & MMB298.213$-$00.343$^\dagger$ & 41.4 & 1.3 & 9.89 & 11.4 & 10.5 & 3.89 & 23.31 & 4.61 & 3.33\ AGAL298.263+00.739 & MMB298.262+00.739$^\dagger$ & 2.5 & 1.3 & 3.23 & 4.0 & 7.5 & 0.37 & 22.92 & 2.83 & 3.47\ AGAL298.631$-$00.362 & MMB298.632$-$00.362$^\dagger$ & 6.0 & 1.6 & 1.31 & 11.9 & 10.8 & & 22.31 & 2.77 & 3.38\ AGAL298.724$-$00.086 & MMB298.723$-$00.086$^\dagger$ & 5.0 & 1.1 & 1.84 & 10.6 & 9.9 & 0.65 & 22.94 & 3.45 & 3.20\ AGAL299.012+00.127 & MMB299.013+00.128$^\dagger$ & 3.5 & 1.9 & 5.29 & 10.2 & 9.6 & 1.66 & 22.72 & 3.66 & 4.03\ AGAL300.504$-$00.176 & MMB300.504$-$00.176$^\dagger$ & 1.4 & 1.3 & 4.83 & 9.6 & 9.0 & 2.16 & 23.11 & 3.95 & 3.67\ AGAL300.969+01.146 & MMB300.969+01.148$^\star$ & 7.0 & 1.3 & 6.49 & 4.3 & 7.3 & 1.25 & 23.41 & 3.69 & 3.05\ AGAL301.136$-$00.226 & MMB301.136$-$00.226$^\dagger$ & 2.3 & 1.3 & 2.59 & 4.3 & 7.3 & 0.78 & 23.90 & 3.77 & 2.59\ AGAL302.032$-$00.061 & MMB302.032$-$00.061$^\dagger$ & 1.2 & 1.2 & 3.94 & 4.5 & 7.2 & 0.70 & 23.00 & 3.10 & 3.43\ \ \ $^{\rm{a}}$ Sources with a superscript have been searched for mid-infrared emission by @gallaway2013: $\dagger$ and $\ddagger$ indicate infrared bright and infrared dark sources, respectively, and $\star$ identifies the sources they were unable to classify.\ Notes: Only a small portion of the data is provided here, the full table is available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.125.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/MNRAS/. [lc.......]{} & & & & & & &\ Radius (pc) & 375& 1.27& 0.05 & 1.01 & 0.97 & 0.01 & 5.69\ Aspect Ratio & 577& 1.51& 0.02 & 0.40 & 1.40 & 1.01 & 3.39\ $Y$-factor & 577& 5.63& 0.16 & 3.89 & 4.55 & 0.98 & 24.66\ Log\[Clump Mass ([M$_\odot$]{})\] & 442& 3.27& 0.04 & 0.77 & 3.36 & -2.00 & 5.43\ Log\[Column Density (cm$^{-2}$)\]& 577& 22.86& 0.04 & 1.06 & 22.88 & 4.16 & 24.74\ Log\[$L_{\rm{MMB}}$ (Jykpc$^2$)\] & 442& 3.48 & 0.05 & 0.99 & 3.47 & -1.30 & 6.34\ \ \ Distances {#sect:distance} --------- Using the velocity of the peak maser component and a Galactic rotation model (e.g., @brand1993 [@reid2009]) it is possible to estimate a particular source’s kinematic distance. However, for sources located within the solar circle (i.e., $<$ 8.5kpc of the Galactic centre) there is a two fold degeneracy as the source velocity corresponds to two distances equally spaced on either side of the tangent position. In a follow-up paper to the MMB survey @green2011b used archival HI data taken from the Southern and VLA Galactic Plane Surveys (SGPS (@mcclure2005) and VGPS (@stil2006), respectively) to resolve the distance ambiguities to a large number of methanol masers. @green2011b examined HI spectra for 734 methanol masers, of which 204 are located at Galactic longitudes between $\ell=20\degr$-60 and therefore are not included in the current MMB catalogue and are not considered here. However, the @green2011b study does include 525 of the 671 MMB sources located in the overlapping ATLASGAL and MMB region. Breaking this down further we find that this includes 506 of the 628 MMB sources associated with an ATLASGAL source as discussed in Sect.\[sect:atlas-mmb\], and 31 of the [43]{} methanol masers not associated with submillimetre emission mentioned in Sect.\[sect:unmatched\_mmb\]. @green2011b used the Galactic rotation model of @reid2009 to determine the kinematic distances to their sample of methanol masers. However, this model’s assumption of a flat rotation curve with a high rotational velocity leads to very noticeable differences in the fourth quadrant of the Galaxy between the model-derived tangent velocities and the empirically derived values determined by @mcclure2007 from the HI termination velocities (see their Fig.8). This leads to the near and far distance being located farther from the tangent position than would otherwise be expected, and produces a large hole in the Galactic distribution around the tangent positions (see Fig.4 of @green2011b). In order to avoid this we have used the Galactic rotation curve of @brand1993 as its model tangent velocities are a much closer match to the HI termination velocities. The use of a different rotation model in most cases results in only a slight change in the estimated kinematic distances from those given by @green2011b. In addition to the minor difference in kinematic distances imposed by the change of rotation model ($\sim$1kpc) we also: 1) place any source with a velocity within 10[kms$^{-1}$]{} of the tangent point at the tangent distance, and 2) place any sources within the solar circle at the near distance if a far-distance allocation would lead to a height above the mid-plane larger than 4 times the scale height of young massive stars (i.e., $\sim$30pc; @reed2000 [@urquhart2011]). @green2011b provides distance solutions for 385 of the ATLASGAL-MMB associations and 16 of the unassociated MMB sources. We have adopted their distances for 378 of the ATLASGAL-MMB associations and find the values determined from the difference rotation curves, after applying the two criteria mentioned in the previous paragraph, agree within 1kpc in every case. We only disagree with the distance allocations given by @green2011b for seven sources. Of these, we have associated four sources with the G305 complex (AGAL305.361+00.186, AGAL305.362+00.151, AGAL305.799$-$00.244 and AGAL305.887+00.016), three located at the tangent position (AGAL309.384$-$00.134, AGAL311.627+00.266 and AGAL336.916$-$00.022). Only one source (AGAL351.774$-$00.537) is placed at the near distance of 0.4kpc by the @reid2009 model but is placed outside the solar circle by the @brand1993 model at a distance of 17.4kpc. There are 83 ATLASGAL-MMB associations that have not been assigned a distance by @green2011b. We have allocated distances to 64 of these. Fourteen have been placed at the tangent position, fourteen have been associated with a well-known complex (i.e., G305 and W31) and 19 have been placed at the near distance. Fifteen of the sources placed at the near distance are because a far-distance allocation would place than more then 120pc from the Galactic mid-plane. Finally, the velocities of seventeen sources places them outside the solar circle, and thus, these sources do not suffer from the kinematic distance ambiguity problem. In total we have distances to 442 ATLASGAL-MMB associated sources and their distribution is shown in Fig.\[fig:atlas\_mmb\_distance\_hist\] (grey filled histogram). The distribution is shown to be bimodal, with peaks at 3-4kpc and 11-12kpc. The low number of sources between these two peaks is due to an almost total lack of any MMB sources within 3kpc of the Galactic centre. In addition to the distances we have determined for the ATLASGAL-MMB associations, we have obtained distances for 33 of the 43 MMB masers not associated with thermal dust emission. Nineteen of these were drawn from @green2011b, eleven are found to be located in the outer Galaxy using the @brand1993 model, two others are located at the tangent position and one is placed at the near distance as a far distance would place more than 120pc from the Galactic mid-plane. It is clear from the distribution of this sample of sources (green histogram shown in Fig.\[fig:atlas\_mmb\_distance\_hist\]) that the majority have distances larger than 9kpc. In the previous section (i.e., Sect.3.3) we suggested that one possible explanation for the non-detection of submillimetre dust emission from these sources could be that they are located at larger distances; this distribution offers some support for that hypothesis. The median value for the distance for these unassociated MMB sources is $\sim$13kpc compared with a median value of $\sim$5kpc for the ATLASGAL-MMB associations. A KS test on the two samples shows their distance distributions are significantly different and we are able to reject the null hypothesis that they are drawn from the same population. Sizes and morphology {#sect:size} -------------------- Using the kinematic distances discussed in the previous section and the effective angular radii derived for the clumps by [`SExtractor`]{} we are able to estimate their physical sizes (see @rosolowsky2010 for definition of effective radius). The distribution of sizes is presented in the upper panel of Fig.\[fig:physical\_size\] and ranges from 0.1 to several pc, with a peak at $\sim$1pc. The dashed vertical lines shown on this plot at radii of 0.15 and 1.25pc indicate the boundary between cores and clumps, and clumps and clouds, respectively (as adopted by @dunham2011 from Table1 of @bergin2007). However, we note that the distribution is continuous with no features at these sizes so the definitions are probably somewhat arbitrary. As also reported by [@tackenberg2012], from a study of ATLASGAL candidate starless clumps we find no correlation between angular size and distance, which results in an approximately linear correlation between physical sizes and distance. It is important to bear in mind that at distances of a kpc or so we are primarily resolving structures on the size scale of cores, at intermediate distances, clumps, and at greater distances, entire cloud structures. This will have an effect on some of the derived parameters such as column and volume densities. However, although the sample covers a large range of sizes, the majority falls into the size range suggested for clumps, which are more likely to be in the process of forming clusters rather than a single massive star. For simplicity we refer to our sample as clumps, with the caveat that it includes a large range of physical sizes. In the lower panel of Fig.\[fig:physical\_size\] we plot the source radius as a function of heliocentric distance. The colours of the symbols used in this plot give an indication of the aspect ratio of each source, the values of which can be read off from the colour bar to the right of the plot. This plot clearly illustrates that at larger distances we are probing larger physical structures, however, the aspect ratio of this sample of objects does not appear to have a significant distance dependence. This may suggest that even at larger distance the molecular clouds associated with MMB sources are still single structures. In the upper panel of Fig.\[fig:morphology\] we present the distribution of the aspect ratios of ATLASGAL-MMB sources (blue histogram), which is shown against that of the whole population of compact ATLASGAL sources (grey filled histogram). It is clear from this plot that the ATLASGAL-MMB sources have a significantly smaller aspect ratio than the general population, which would suggest they are more spherical in structure. The mean (median) values for the ATLASGAL-MMB and the whole population are 1.51$\pm$0.02 (1.40) and 1.69$\pm$0.01 (1.55), respectively. A similar median value of 1.3 was found by @thompson2006 from a programme of targeted SCUBA submillimetre observations of clumps associated with [UCH [ii]{}]{} regions. Another way to compare the morphology of the ATLASGAL-MMB associations with the general population of ATLASGAL sources is to look at the ratio of their integrated to peak fluxes; this is referred to as the $Y$-factor. This has been used to investigate the general extent of submillimetre clumps associated with high-mass protostellar cores (HMPOs) and [UCH [ii]{}]{} regions by @williams2004 and @thompson2006, respectively. In the lower panel of Fig.\[fig:morphology\] we plot the $Y$-factor for the whole ATLASGAL compact source population (filled grey histogram) and the ATLASGAL-MMB associated sources (blue histogram). The $Y$-factors for both distributions peak at similar values, between 2-4, however, the ATLASGAL-MMB associated sources have significantly lower overall $Y$-factors (with a median value of 4.8 compared to 6.7 for the whole ATLASGAL compact source population). A KS test is able to reject the null hypothesis that these are drawn from the same population with greater than three sigma confidence. Peaks in the $Y$-factor around 3 are also seen in the HMPO and [UCH [ii]{}]{} region samples of @williams2004 and @thompson2006 and appear to have a similar distribution to our sample sample. The large range of heliocentric distances over which their HMPO sample is distributed led @williams2004 to suggest that the envelope structures may be scale-free. The ATLASGAL clumps are similarly located over a range of heliocentric distances and thus the ATLASGAL beam traces structure on scales from a few times 0.1 pc to a several pc. As we observe similar Y-factors for a range of spatial resolutions this suggests that the radial density distribution follows a similar power law over a range of spatial scales. Hence the structure of the clump *envelopes* (at least on pc scales) are likely to follow a scale-free power law. The fact that all three samples (ATLASGAL-MMB, HMPO and [UCH [ii]{}]{} regions) are likely to cover the whole range of embedded star formation and broadly show the same $Y$-factor properties would seem to support this. Similarly to @williams2004 and @thompson2006, we find that a significant amount of the mass associated with the ATLASGAL-MMB sources is found in the outer regions of the clumps, from which we conclude that this situation does not change significantly over the evolution of the embedded stars. In summary we have determined that overall the ATLASGAL-MMB associated sources are roughly spherical, centrally condensed clumps that appear to have a scale-free envelope with a methanol maser coincident with the peak of the submillimetre emission. Isothermal clump masses ----------------------- In calculating masses and column densities we have assumed that all of the measured flux arises from warm dust, however, free-free emission from embedded ionised gas and/or molecular line emission could make a significant contribution for broadband bolometers such as LABOCA. @schuller2009 considered these two forms of contaminating emission and concluded that even in the most extreme case of the giant HII region associated with W43, and the CO (3-2) lines associated with extreme outflows, hot cores and photon-dominated regions, the contribution from free-free emission and molecular lines is of the order 20 and 15percent, respectively, and in the majority of cases will be almost negligible (see also @drabek2012). Assuming the dust is generally optically thin and can be characterised by a single temperature we are able to estimate the isothermal dust masses for the ATLASGAL-MMB associated sources. Following @hildebrand1983, the total mass in a clump is directly proportional to the total flux density integrated over the source: $$M \, = \, \frac{D^2 \, S_\nu \, R}{B_\nu(T_D) \, \kappa_\nu},$$ where $S_\nu$ is the integrated 870$\mu$m flux, $D$ is the heliocentric distance to the source, $R$ is the gas-to-dust mass ratio (assumed to be 100), $B_\nu$ is the Planck function for a dust temperature $T_D$, and $\kappa_\nu$ is the dust absorption coefficient taken as 1.85cm$^2$g$^{-1}$ (this value was derived by @schuller2009 by interpolating to 870$\mu$m from Table1, Col.5 of @ossenkopf1994). As reliable dust temperatures are not available for the majority of our sample, we make the simplifying assumption that all of the clumps have approximately the same temperature and set this to be 20K. Single-dish ammonia studies have derived kinetic gas temperatures for a large of number massive star formation regions that cover the full range of evolutionary stages. These include methanol masers (@pandian2012); 1.1mm thermal dust sources identified by the Bolocam Galactic Plane Survey (BGPS; @dunham2011b); 870$\mu$m ATLASGAL sources (@wienen2012) and the Red MSX Source Survey (RMS; @urquhart2011b).[^2] For the methanol masers mean and median kinetic temperatures of 26K and 23.4K were reported by @pandian2012, and for the massive young stellar objects (MYSOs) and [UCH [ii]{}]{} regions have mean and median kinetic temperatures of 22.1 and 21.4K @urquhart2011b, while @wienen2012 report kinetic temperatures of $\sim$24K for a subsample of ATLASGAL sources associated with methanol masers. @dunham2011b report lower kinetic gas temperatures 15.6$\pm$5.0K, however, their sample includes a larger fraction of starless clumps and so the lower mean temperature is expected. A kinetic gas temperature of $\sim$25K would seem to characterise the clumps that show evidence of star formation, however, this temperature is likely to be an upper limit to the clump-averaged kinetic temperature as these observations were pointed at the peak emission of the clumps and the kinetic temperature is likely to be significantly lower towards the edges of the clumps (@zinchenko1997). @dunham2011b estimate that using the peak kinetic temperature for the whole clump may underestimate the isothermal mass by up to a factor of two. Therefore we have chosen to use a value of 20K, consistent with a number of similar studies (cf. @motte2007 [@hill2005]). Given that the true clump-averaged kinetic temperatures of the sample are likely to range between 15 and 25K the resulting uncertainties in the derived isothermal clump masses of individual sources are $\pm$43percent (allowing for an uncertainty in temperature of $\pm$5K which is is added in quadrature with the 15percent flux measurement uncertainty). However, this is unlikely to have a significant impact on the overall mass distribution or the statistical analysis of the masses. We note that 10percent of the ATLASGAL-MMB associations are also associated with embedded UCHII regions. However, @urquhart2011b found that the presence of an [UCH [ii]{}]{} region only results in an increase in clump-averaged kinetic temperatures of a few Kelvin. In the left panel of Fig.\[fig:clump\_mass\] we present a plot of the isothermal dust mass distribution, while in the right panel we show the mass distribution as a function of heliocentric distance. It is clear from the right panel of this figure that we are sensitive to all ATLASGAL-MMB associated clumps with masses above 1,000[M$_\odot$]{} across the Galaxy, and our statistics should be complete above this level. (This completeness limit is indicated on the left panel of Fig.\[fig:clump\_mass\] by the vertical blue line.) In this regard it is interesting to note that the mass distribution peaks at several thousand solar masses (see left panel of Fig.\[fig:clump\_mass\]), which is significantly above the completeness limit, and so the drop off in the source counts between the completeness limit and the peak is likely to be real. This is an important point as it confirms that the methanol masers are preferentially associated with very massive clumps. According to @lada2003 and @motte2003 the radius and mass required to form stellar clusters is of the order 0.5-1pc and 100-1,000[M$_\odot$]{}, respectively. Given the sizes and masses of the ATLASGAL-MMB associated clumps it is highly likely that the majority are in the process of forming clusters. Assuming a star formation efficiency (SFE) of 30percent and an initial mass function (IMF; @kroupa2001), @tackenberg2012 estimate that a clump mass of $\sim$1,000[M$_\odot$]{} is required to have the potential to form at least one 20[M$_\odot$]{} star, while a clump of $\sim$3,000[M$_\odot$]{} is required to form at least one star more massive than 40[M$_\odot$]{}. It is consistent with the assumption that methanol masers are associated mainly with high-mass star formation to find that the majority of ATLASGAL-MMB associations ($\sim$72percent) have masses larger than $\sim$1,000[M$_\odot$]{}, and thus, satisfy the mass requirement for massive star formation. We also note that approximately a third of the ATLASGAL-MMB associations ($\sim$28percent) have masses below what is thought to be required to form at least one massive star. However, all of these sources also tend to be more compact objects (in most cases $\le$0.3pc) and may have a higher star formation efficiency and be forming smaller stellar systems where the stellar IMF does not apply (e.g., @motte2007). ### Peak column densities We estimate column densities from the peak flux density of the clumps using the following equation: $$N_{H_2} \, = \, \frac{S_\nu \, R}{B_\nu(T_D) \, \Omega \, \kappa_\nu \, \mu \, m_H},$$ where $\Omega$ is the beam solid angle, $\mu$ is the mean molecular weight of the interstellar medium, which we assume to be equal to 2.8, and $m_H$ is the mass of an hydrogen atom, while $\kappa_\nu$ and $R$ are as previously defined. We again assume a dust temperature of 20K. The derived column densities are in the range $\sim$10$^{22-24}$cm$^{-2}$, peaking at 10$^{23}$cm$^{-2}$. This corresponds to a surface density of few times 0.1gcm$^{-2}$, which is a factor of a few lower than the value of 1gcm$^{-2}$ predicted to be the lower limit for massive star formation (i.e., @mckee2003 [@krumholz2008]). However, we should not read too much into this as the column densities of the more distant sources can be affected by beam dilution. This effect is nicely illustrated in the right panel of Fig.\[fig:clump\_mass\] where we use colours to show the column densities as a function of distance (see colour bar for values). There is clearly a dependence of column density on distance as, given the resolution of the survey, we are sampling larger scale physical structures as the distance increases, which preferentially reduces the column densities of more distant sources. It is likely that these more distant sources would fragment into smaller and denser core-like structures at higher resolution (e.g., @motte2007). Therefore one should exercise caution when drawing conclusions from beam-averaged quantities such as the column and volume densities (cf. @tackenberg2012). ### Clump mass function In Fig.\[fig:mass\_radius\_dndm\_histogram\] we present the differential mass distribution for the ATLASGAL-MMB associated clumps. On this plot the dashed vertical blue line indicates the completeness limit and the solid red line shows the results of a linear least-squares fit to the bins above the peak in the mass distribution shown in the left panel of Fig.\[fig:clump\_mass\] ($\sim$3,000[M$_\odot$]{}). This line provides a reasonable fit to all of the bins above the peak mass. It does not fit the two bins just above the completeness limit, which may suggest that a second power law is required to account for these mass bins. The derived exponent ($\alpha$ where ${\rm{d}}N/{\rm{d}}M \propto M^{\alpha}$) of the fit to the high-mass tail for the ATLASGAL-MMB associated sources is $-2.0\pm0.1$, which is similar to values ($-2.0$ to $-2.3$) derived by @williams2004, @reid2005, @beltran2006 from studies of more evolved stages and @tackenberg2012 who targeted a sample of starless clumps identified from ATLASGAL data. This similarity between the clump mass function over the different stages of massive star formation would suggest that the CMF does not change significantly as the embedded star formation evolves. Methanol maser luminosity ------------------------- In the left panel of Fig.\[fig:mmb\_luminosity\] we present the 6.7GHz luminosity distribution of the ATLASGAL-MMB associated masers for which we have derived a distance. These “luminosities” have been calculated assuming the emission is isotropic using: $$L_{\rm{MMB}} \, = \, 4\pi D^2 S_\nu$$ where $D$ is the heliocentric distance in kpc and $S_\nu$ is the methanol maser peak flux density in Jy. $L_{\rm{MMB}}$ thus has units of Jykpc$^2$. The distribution shown in the left panal of Fig.\[fig:mmb\_luminosity\] peaks at $\sim$3,000Jykpc$^2$, but is skewed to higher luminosities with a mean value of $3\times10^4$Jykpc$^2$. In the right panel of Fig.\[fig:mmb\_luminosity\] we show the methanol maser luminosities as a function of heliocentric distance. We can see from this plot that the MMB survey is effectively complete across the Galaxy ($\sim$20kpc) to all methanol masers with luminosities greater than $\sim$1,000Jykpc$^2$. This completeness limit is shown by the dashed vertical line plotted on the luminosity distribution shown in the left panel of this figure. ### Luminosity-volume density correlation {#lum_vol_correlation} There has been a number of recent methanol and water-maser and dust-clump studies that have reported a trend towards lower mean volume densities with increasing maser luminosity, which has been interpreted as the result of the evolution of the embedded star formation (i.e., @breen2011_water [@breen2011_methanol; @breen2012]). To test this trend for our sample of methanol masers, we plot the maser luminosity as a function of clump-averaged volume density in Fig.\[fig:maser\_lum\_volume\_density\] for the whole sample of 374 ATLASGAL-MMB associations for which we have sufficient data to derive the maser luminosity and volume density.[^3] We more or less replicate the correlation coefficient and least-squares fit gradient reported by @breen2011_methanol using the whole sample. Correlation coefficients are $-$0.43 and $-$0.46 and gradients $-0.78\pm0.09$ and $-0.85\pm0.16$ for the fit presented in Fig.\[fig:maser\_lum\_volume\_density\] and Fig.2 of @breen2011_methanol, respectively. However, from a casual inspection of the mean volume densities (see colour bar of the right panel of Fig.\[fig:mmb\_luminosity\]) it is clear that there is a significant distance dependency. It would appear that the poorer sensitivity to lower luminosity masers, and the increase in the spatial volume being sampled at larger distances, results in a decrease in the mean volume-densities of clumps and an increase in maser luminosities. To test this further, we have performed a partial Spearman correlation test of the mean volume-densities and maser luminosities to remove their mutual dependence on the distance (e.g., @yates1986), of the form $r_{\rm AB,C}$, where $$r_{\rm AB,C} \; = \; \frac {r_{\rm AB} - r_{\rm AC} r_{\rm BC}} {[(1-r^2_{\rm AC})(1-r^2_{\rm BC})]^{1/2}},$$ where A, B and C are the maser luminosity, mean particle density and distance respectively, and $r_{\rm AB}$, $r_{\rm AC}$ and $r_{\rm BC}$ are the Spearman rank correlation coefficients for each pair of parameters. The significance of the partial rank correlation coefficients is estimated using assuming it is distributed as Student’s t statistic (see @collins1998 for more details). This returned a partial correlation coefficient value of $-$0.06 and a $p$-value $\sim$0.6 and we are therefore unable to exclude the null hypothesis that the sample is drawn from a population where rho=0. We would conclude that there is no intrinsic correlation between maser luminosity and clump-averaged volume density. We additionally performed tests on distance selected subsamples and obtained the same result. The scales traced by the ATLASGAL observations are much larger than those of either an HMC or an UCHII region, which by definition are $< 0.1$pc. We do not see evidence for radical changes in the density distribution of the envelope on pc scales with evolutionary state (the distribution of Y values being similar). Hence, the density distribution of the envelope does not evolve appreciably over the relevant timescale. For the clump-averaged volume density, the volume density of the envelope dominates over the much smaller core, and so we would not expect volume densities derived from single dish observations to change appreciably either. Our conclusion that there is no correlation between clump density and methanol maser luminosity is supported by a recent study by @cyganowski2013 who also failed to find a correlation between water maser luminosities and clump densities towards a sample of extended green objects (EGOs; @cyganowski2008). This is not surprising, since for the masers to work, the density must be within a narrow range, which has little to do with the density sampled in the ATLASGAL beam. Since the H$_2$ number densities form a central part of the evolutionary arguments put forward by Breen et al. the results presented here cast significant doubt on some of their conclusions. Maser luminosity-clump mass: completeness {#sect:completeness} ----------------------------------------- In Fig.\[fig:maser\_mass\_plot\] we present a plot of the maser luminosity as a function of clump mass for the 442 ATLASGAL-MMB associations for which we have a distance. The upper right region outlined by the grey dashed line of this figure indicates the region of parameter space that we are complete across the Galaxy to both methanol maser luminosity and clump mass; this region contains 280 sources. Applying a partial correlation function to the clump mass and maser luminosity to remove the dependence of these two parameters on distance (@collins1998) we obtain a partial correlation coefficient value of 0.37 with a $p$-value $\ll$ 0.01, suggesting that there is a weak correlation between the maser luminosity and clump mass. We fit these data with a linear least squared function the result of which is shown in Fig.\[fig:maser\_mass\_plot\] as a blue dashed line. The parameters of the fit are: Log($L_{\rm{MMB}})=(0.857\pm0.046)\,M_{\rm{clump}}+(0.692\pm0.155)$. Within the errors the fit to the data has a gradient close to one and so the relationship between mass and maser luminosity is relatively linear. As discussed in Sect.4.3 the most massive clumps are likely to be forming more massive stars (assuming a Kroupa IMF and a SFE of 30percent) and therefore the weak correlation between the clump mass and maser luminosity may be related to this. This suggests that higher (isotropic) maser luminosity is related to higher stellar luminosity in some way, perhaps via the pumping mechanism or maybe the larger clump/core just provides a longer maser amplification column. Discussion ========== There have been a number of studies that have tried to firmly establish a connection between the presence of a methanol maser and ongoing massive star formation and these have been relatively successful. Most have searched for methanol masers towards low luminosity protostellar sources and, when no masers were detected, have concluded that methanol masers are exclusively associated with high-mass protostars (e.g., @minier2003 [@bourke2005]). However, most of these surveys have focused on small samples, with poorly defined selection criteria, and often use IRAS fluxes to determine the luminosity of the embedded source and so the luminosities may have been overestimated. So, although these studies have set a lower limit to the luminosity of the associated protostar, it is unclear whether the results obtained are applicable to the whole methanol maser population. In this subsection we will draw on the results presented in the previous sections to test this hypothesis. The ATLASGAL-MMB sample presented in this paper includes 94percent of the MMB sources in the overlap region of the two surveys, and 90percent of the entire MMB published catalogue. Moreover, since the number of methanol masers in the whole Galaxy is not expected to exceed a few thousand, our sample is likely to incorporate a large fraction of the whole Galactic population. Therefore any statistical results drawn from this sample will be a fair reflection of the properties of the general population. Empirical mass-size relationship for massive star formation {#mass-size-relationship} ----------------------------------------------------------- ### Criterion for massive star formation In two papers @kauffmann2010a [@kauffmann2010b] investigated the mass-radius relationship of nearby ($<$500pc) molecular cloud complexes (i.e., Ophiuchus, Perseus, Taurus and the Pipe Nebula) and found that clouds that were devoid of any high-mass star formation generally obeyed the following empirical relationship: $$m(r) \le 580\,{\rm{M}}_\odot\,(R_{\rm{eff}}/{\rm{pc}})^{1.33}$$ where $R_{\rm{eff}}$ is the effective radius as defined by @rosolowsky2010.[^4] Comparing this mass-size relation with samples of known high-mass star forming regions such as those of @beuther2002, @hill2005, @motte2007 and @Mueller2002, @kauffmann2010b found that all of these regions occupied the opposing side of the parameter space (i.e., where $m(r) \ge 580$[M$_\odot$]{} $(R_{\rm{eff}}/{\rm{pc}})^{1.33}$. This led them to suggest that the relation may approximate a requirement for massive star formation, where only more massive clumps have the potential to form massive stars. However, @kauffmann2010b states that larger samples of massive star forming clouds are required to strengthen this hypothesis. In Fig.\[fig:mass\_radius\_distribution\], we present the mass-size relationship for the ATLASGAL-MMB associated sources. This sample consists of 375 clumps that have distance estimates and are spatially resolved in the APEX beam. Of these, we find that 363 have masses larger than the limiting mass for their size, as determined by @kauffmann2010b for massive star formation. This corresponds to $\sim$97percent of the sample and, although this in itself does not confirm that the embedded source is a high-mass protostar, it does at least suggest that these clumps have the potential to form one. This also supports our earlier statement (in Sect.4.3) that the less massive clumps (i.e., less than 1,000[M$_\odot$]{}) also have the potential to form massive stars, as long as they are relatively compact. Only 6 ATLASGAL-MMB associated sources are in the part of the parameter space that was found to be devoid of massive stars. All of these sources have been placed at the near distance and if the wrong distance has been assigned then this would explain their location in the mass-radius plot. Checking the confidence flag given by @green2011b we find that three of these sources have been given a flag of $b$, indicating their distance assignments are less reliable, and may explain why these sources fail to satisfy @kauffmann2010b mass-radius requirement for massive star formation. Turning our attention back to Fig.\[fig:mass\_radius\_distribution\] we see that the ATLASGAL-MMB data form a fairly continuous distribution over almost four orders of magnitude in mass and two orders of magnitude in radius. The dashed blue line overlaid on this plot shows the result of a linear least-squares fit to the data. This fit provides a good description of the mass-size relationship for these objects (Log($M_{\rm{clump}}) = 3.4\pm0.013 + (1.67\pm0.036)\times {\rm{Log}}(R_{\rm{eff}}$)). The upper and lower red diagonal lines indicate constant surface densities, $\Sigma({\rm{gas}})$, of 1gcm$^{-2}$ and 0.05gcm$^{-2}$, respectively. These two lines provide fairly reliable empirical upper and lower bounds for the clump surface densities required for massive star formation. Furthermore, the lower bound of 0.05gcm$^{-2}$ provides a better constraint than Kauffmann et al. for the high-mass end of the distribution (i.e., $R_{\rm{eff}} > $ $\sim$0.5pc or $M_{\rm{clump}} > $ $\sim$500[M$_\odot$]{}). The thick pink line shows the threshold derived by @lada2010 and @heiderman2010 (116 and 129[M$_\odot$]{}pc$^{-2}$, respectively; hereafter LH threshold) for “efficient” star formation. Above this threshold the observed star formation in nearby molecular clouds (d $\le 500$pc) is linearly proportional to cloud mass and negligible below. This line corresponds to an extinction threshold of A$_K\simeq0.9$mag or visual extinction, A$_V$, of $\simeq8$mag. As discussed in the previous paragraph, the lower mass-radius envelope of the ATLASGAL-MMB associations is well modeled by $\Sigma({\rm{gas}})=0.05$gcm$^{-2}$, which is approximately twice the LH threshold. However, we note that the star formation associated with the nearby molecular clouds used to determine the LH threshold is likely to be predominantly low-mass, while the threshold determined from the ATLASGAL-MMB associations may be a requirement for efficient formation of intermediate- and high-mass stars. This result suggests that a volume density threshold may apply (e.g., @parmentier2011), however, with the data presented here it is not possible to determine the density distribution or the volume density at the scales involved with star formation. Given that this sample includes a large range of physical sizes (cores, clumps and clouds) and evolutionary stages (HMC, HMPO and [UCH [ii]{}]{} regions) it is somewhat surprising to find such a strong correlation. However, as mentioned in Sections 4.2 and 4.3, the clumps appear to have a scale-free envelope structure that is not significantly changed as the embedded YSOs evolve towards the main sequence. ### Precursors to young massive clusters [l...c]{} & & & &\ & & & &\ AGAL010.472+00.027 & 11.0 & 3.0 & 4.77 & 1\ AGAL328.236$-$00.547 & 11.4 & 4.9 & 4.99& 1\ AGAL329.029$-$00.206 & 11.7 & 3.2 & 4.68& 1\ AGAL345.504+00.347 & 10.8 & 5.7 & 4.93& 1\ AGAL350.111+00.089 & 11.4 & 2.1 & 4.55& 1\ AGAL351.774$-$00.537 & 17.4 & 4.8 & 5.43& 1\ AGAL352.622$-$01.077 & 19.4 & 3.3 & 4.79& 1\ G000.253+00.016 & 8.4 & 2.8 & 5.10 & 2\ G010.472+00.026 & 10.8 & 2.1 & 4.58 & 3\ G043.169+00.009 & 11.4 & 2.2&5.08 & 3\ G049.489$-$00.370$^a$& 5.4 & 1.6 & 4.68 & 3\ G049.489$-$00.386$^a$& 5.4 & 1.6 & 4.72&3\ \ \ References: (1) this work, (2) @longmore2012a, (3) @ginsburg2012\ Notes: $^a$These two sources are considered as a single MPC in the discussion presented by @ginsburg2012. The green shaded area in the upper left part of Fig.\[fig:mass\_radius\_distribution\] indicates the region of the mass-radius parameter space in which young massive proto-cluster (MPC) candidates are thought to be found (see @bressert2012 for details). MPCs are massive clumps with sufficient mass that, assuming a fairly typical SFE (i.e., $\sim$30percent), have the potential to be the progenitors of future young massive clusters (YMCs) that have masses of $>$10$^4$[M$_\odot$]{} such as the Arches and Quintuplet clusters (@portegies_zwart2010). We consider both starless and star-forming clumps to be MPCs as anything with a gas envelope is a possible precursor. In an analysis of the BGPS data, @ginsburg2012 identified only three MPC candidates in a longitude range of $\ell = 6$-90 from a sample of $\sim$6,000 dust clumps. Using their detection statistics, @ginsburg2012 go on to estimate the number of MPCs in the Galaxy to be $\leq 10\pm6$, however, they state that a similar study of the southern Galactic plane is needed. Given that the lifetime of the starless massive clumps is relatively short ($\sim$0.5Myr; @tackenberg2012 [@ginsburg2012]) it is probably safe to assume that the ATLASGAL-MMB sample probably includes all of the likely MPC candidates found in the $20\degr > \ell > 280\degr$ of the Galaxy. Applying the @bressert2012 criteria to our ATLASGAL-MMB sample, we have identified 7 young massive proto-cluster candidates, one of which (i.e., AGAL010.472+00.027) is located in the overlapping regions analysed by @ginsburg2012 who also identified this source as an MPC candidate. The ATLASGAL clump names and their derived parameters are tabulated in Table\[tbl:MPC\_derived\_para\] along with the MPC candidates identified by @ginsburg2012 and @longmore2012a. We note that the detection rate of young massive proto-cluster candidates identified from the ATLASGAL-MMB sample is twice that identified from the BGPS (@ginsburg2012). Both studies covered a similar area of the Galactic disk (approximately 30percent each between galactocentric radii of 1-15kpc), however, ATLASGAL covers a much broader range of Galactic latitude ($|b|<1.5\degr$ and $|b|<0.5\degr$ for ATLASGAL and the BGPS, respectively) and we note that three of the MPC candidates identified here have latitudes greater than 0.5. The difference in MPC candidate detection rates between the two surveys is likely a result of the difference in latitude coverage. It is therefore also likely that @ginsburg2012 have underestimated the number of MPC candidates in the first quadrant by a factor of two. Combining the number of MPCs identified and the volume coverage we estimate the total number of MPC candidates in the Galaxy is $\leq 20\pm6$. The number of MPC candidates is a factor of two larger than the number of YMCs known in the Galaxy, but similar to the number of embedded clusters in the Galaxy estimated by @longmore2012a (i.e., $\sim$25). However, this is almost an order of magnitude greater than the number of MPCs expected assuming a YMC lifetime of 10Myr and a formation time of $\sim$1Myr. This would suggest that either only a small number of the MPC candidates will evolve into YMCs, which would require a SFE lower than the assumed 30percent, or there are many more YMCs that have as yet not been identified. Galactic distribution --------------------- \ \ In the previous subsection we have shown that the clumps identified from their association with methanol masers are likely to be involved with the formation of the next generation of massive stars. Massive star formation has been found to be almost exclusively associated with the spiral arms of nearby analogues of the Milky Way (@kennicutt2005) and therefore the Galactic distribution of this sample of ATLASGAL-MMB associated clumps may provide some insight into where in the Galaxy massive star formation is taking place and its relation to the spiral arms. Figs.\[fig:gal\_lat\] and \[fig:gal\_long\] we show the distribution in Galactic latitude and longitude of ATLASGAL sources and ATLASGAL-MMB associations in our sample, in the range $20^{\circ} \ge \ell \ge 280^{\circ}$. We find no significant difference between the ATLASGAL-MMB sample’s latitude distributions and the overall ATLASGAL source distribution; which has been previously commented on by @beuther2012 and @contreras2013 and so will not be discussed further here. The distribution of source counts at 2-degree resolution is shown in the lower panel of Figs.\[fig:gal\_long\] and the fraction of ATLASGAL sources with MMB associations in the upper panel of this figure. While the total number of sources is dependent on both the local source density and the line-of-sight distribution, the ratio is independent of these and shows the incidence rate of methanol masers in dense clumps. The distribution of ATLASGAL sources as a function of Galactic longitude in Fig.\[fig:gal\_long\] (lower panel) reflects Galactic structure, with a strong, narrow peak towards the Galactic Centre and a broader maximum centred around $\ell\,\sim\,330$, which corresponds to a group of sources in the Scutum-Centaurus arm and possibly along the Norma arm tangent, which is close to the same direction (see @beuther2012 and @contreras2013 for further discussion). Since methanol masers are associated with the early stages of massive star formation (@minier2003), maser source counts can be taken as a measure of the massive star-formation rate (SFR). If the total maser counts trace the massive SFR on this spatial scale, then this fraction traces an analogue of the star-formation efficiency (SFE), i.e. the rate at which dense, submillimetre-traced clumps are producing massive YSOs, within the timescale appropriate to the methanol-maser stage of evolution ($\sim$2.5-$4.5\times 10^4$yr; @walt2005). The MMB fraction dips significantly at longitudes $|\,\ell\,|\,<$4, and perhaps within 10, suggesting that the SFE is significantly reduced near the Galactic centre. Outside this narrow zone, there is little evidence of any significant changes in SFE on these scales, associated with other features of Galactic structure within the survey area. In particular, the ratio is more or less constant across both the peak in source counts seen at $\ell \simeq 330^{\circ}$ mentioned above, and the clear drop in counts at larger longitudes. Similar results were reported by @beuther2012 from a comparison of GLIMPSE source counts to the ATLASGAL longitude distribution. @moore2012 report increases in the large-scale SFE associated with some spiral-arm structures, while @eden2012 found no significant variations associated with major features of Galactic structure in the fraction of molecular cloud mass in the form of dense clumps. The lower SFE towards the Galactic centre inferred from the ratio of ATLASGAL-MMB associated clumps is supported by a recent study presented by @longmore2012b. These authors used the integrated emission of the inversion transition of ammonia (1,1) from HOPS (@walsh2011 [@purcell2012]) and the 70-500[$\mu$m]{} data from the Hi-GAL survey (@molinari2010a) to trace the Galactic distribution of dense gas and compared this to star formation tracers such as the water, methanol masers and [UCH [ii]{}]{} regions identified by HOPS, MMB and the Green Bank Telescope HII region Discovery Survey (HRDS; @bania2010), respectively, to derive the star formation rate per unit gas mass. The overall distribution of the integrated ammonia emission is very similar to that of the ATLASGAL source distribution showing a very strong peak towards the Galactic centre (cf. Fig.3 @purcell2012). Although @longmore2012b found the ammonia and 70-500[$\mu$m]{} dust emission to be highly concentrated towards the Galactic centre they found the distribution of masers and [H [ii]{}]{} regions to be relatively uniform across the Galaxy. Measuring the SFR, @longmore2012b found it to be an order of magnitude lower than what would be expected given the region’s surface and volume densities. We are unable to independently estimate the absolute SFR from the MMB and ATLASGAL source counts. However, we can state that we find the fraction of clumps associated with the formation of massive stars is a factor of 3-4 lower in the Galactic centre compared with that found for the rest of the Galaxy. The lower SFE seen towards the Galactic centre is likely a result of the extreme environmental conditions found in this region of the Galaxy (@immer2012). In Fig.\[fig:galactic\_mass\_radius\_distribution\] we present the 2-D Galactic distribution of ATLASGAL-MMB associations. In this figure we only include ATLASGAL-MMB associations that have masses and maser luminosities for which the sample is complete across the Galaxy (i.e., $M \ga$1,000[M$_\odot$]{} and $L_{\rm{MMB}}\ga$ 1,000Jykms$^{-1}$kpc$^2$). We find that the positions of the ATLASGAL-MMB associations are in reasonable agreement with the main structural features of the Galaxy, as determined by other tracers (taking into account the kinematic distance uncertainty due to peculiar motions is of order $\pm$1kpc; @reid2009). However, we also note that the degree of correlation is not uniform, with high source densities coincident with the near section of the Scutum-Centaurus arm and the southern end of the Galactic Bar. There is also a smaller number of sources that are distributed along the length of the Sagittarius arm. However, one of the most interesting and surprising features of Figs.\[fig:gal\_long\] and \[fig:galactic\_mass\_radius\_distribution\] is the lack of any clear enhancement in ATLASGAL sources or ATLASGAL-MMB associations corresponding to the Scutum-Centaurus arm tangent at $\ell\,\sim\,310\degr$-320. The low number of clumps with or without masers associated with the southern Scutum-Centaurus arm implies that this arm is not strongly forming stars of any type, and not just a lack of high-mass star formation. This is more surprising when we consider the distribution of CO emission, which is found to peak between $\ell =310-312$ (see Figs.6 and 7 presented in @bronfman1988). This would suggest that this arm is associated with significant amounts of molecular material, but that the clump forming efficiency is relatively low for some reason. Another interesting feature seen in Fig.\[fig:galactic\_mass\_radius\_distribution\] is the relatively low number of ATLASGAL-MMB asscociated clumps located at large Galactic radii. Since we have only plotted clumps with masses above 1,000[M$_\odot$]{} this low number of sources found outside the solar circle is not due to a lack of sensitivity. In Fig.\[galactocentric\_distribution\] we present the ATLASGAL-MMB source surface density as a function of Galactocentric radius. The three peaks in the Galactic radial distribution at $\sim$3, 5 and 10kpc correspond to the far 3-kpc spiral arm, a combination of the near section of the Scutum-Centaurus arm and the southern end of the Galactic Bar, and the Sagittarius spiral arm, respectively. There is a strong enhancement in the massive star-formation rate in the far 3-kpc arm with an otherwise relatively constant surface density between 3 and 7kpc, however, the surface density drops significantly at large radii. There is clearly a significant difference between the surface density inside and outside of the solar circle. The co-rotation radius is approximately the same as the solar circle (i.e, R$_{\rm{GC}}=$ 8.5kpc), which is where the spiral arms — in principle — are less important because the ISM is no longer being shocked as it runs into an arm. So one explanation could be that the spiral arms play an important role in creating the conditions required for massive star formation within the inner Galaxy (e.g., efficiently forming molecular clouds from the gas entering the spiral arm) compared to the outer Galaxy. The co-rotation radius is also where the metallicity has been found to drop sharply (e.g., @lepine2011) and therefore metallicity may also play a role. Since the ATLASGAL-MMB sample is primarily tracing massive star formation it is unclear whether this difference in the surface density of star formation between the inner and outer Galaxy is restricted to the massive SFR or if it is also found in the low- and intermediate-mass star formation; if it is a feature of the massive SFR then it could have implications for the IMF. However, it is also possible that the galactocentric massive star formation surface density is simply reflecting the underlying distribution of molecular and atomic gas in the Galaxy. This is something we will revisit when distances and masses are available for a larger fraction of the ATLASGAL compact source catalogue. The surface density distribution presented in Fig.\[galactocentric\_distribution\] has been determined using only the ATLASGAL-MMB associations that are above the clumps mass and maser luminosity completeness limit. By multiplying the surface density in each of the bins in this plot by the area of the bin annulas we estimate the total number of massive star forming clumps with masses $>$ 1000[M$_\odot$]{} in the Galaxy to be $\sim$560. The sample presented here therefore represents approximately 50percent of the whole Galactic population of massive star forming clumps that are associated with a methanol maser. We can take this analysis a step further and estimate the contribution these star forming clumps make to the Galactic SFR. If we take the median clump mass of $\sim$3000[M$_\odot$]{} and a SFE of 30percent then combined we estimate these clumps will produce clusters with a total stellar mass of $\sim5\times10^5$[M$_\odot$]{}. The formation time for the most massive stars in these cluster was calculated by @molinari2008 to be $\sim$1.5$\times10^5$yr and @davies2011 estimated the combined statistical lifetime of the MYSO and [UCH [ii]{}]{} region stages to be several $10^5$yr, and therefore a reasonable lower limit for the cluster formation time is likely to be $\sim$0.5Myr. However, most of the total cluster stellar mass comes from lower-mass stars, which take longer to form and will increase the cluster formation time to perhaps 1Myr. Using these two cluster formation times as an upper and lower limit we estimate that these methanol maser associated clumps have combined SFR of between 0.5-1[M$_\odot$]{}yr$^{-1}$, and therefore could be responsible for up to half the current Galactic SFR $\sim$2[M$_\odot$]{}yr$^{-1}$ (@davies2011). Luminosity-mass correlations ---------------------------- In Sect.\[sect:completeness\] we found a weak correlation between the clump mass and maser luminosity and speculated that this may be related to higher luminosities of the embedded young stars. In this section we will explore this possible link. In a recent paper @gallaway2013 reported the positional correlation of MMBs with other samples of young high mass stars such as the RMS (@urquhart2008) and with EGOs which are thought to trace shocked gas associated with the outflows of MYSOs (e.g., @cyganowski2008 [@cyganowski2009]). They identified 82 RMS sources ([UCH [ii]{}]{} regions and/or YSOs) within 2 of an MMB source and found that all of these have luminosities consistent with the presence of an embedded high-mass star. We have repeated the spatial correlation described by Gallaway et al., however, we relaxed the matching radius to 20 in order to determine the standard deviation of the offsets and set the association radius accordingly; this identifies 137 possible matches. In Fig.\[rms\_surface\_density\] we present a plot of the surface density as a function of angular separation between the matched RMS and MMB sources. The distribution is peaked at offsets between 0 and 1 and rapidly falls off to approximately zero for separations greater than 4. We consider ATLASGAL-MMB sources to be associated with an RMS source if the offset is less than 4; this identifies 102 reliable matches, which corresponds to approximately 20percent of the MMB sample in the $20\degr > \ell > 10\degr$ and $350 \degr > \ell >280\degr$ region (the RMS survey excluded sources within 10 of the Galactic centre due to the increased background confusion and larger distance uncertainties). Bolometric luminosities have been estimated for the RMS sources (@mottram2010 [@mottram2011b]) and thus allow us to investigate the luminosity distribution of these RMS-MMB associations. The fluxes used to estimate these luminosities are effectively clump-average values and are therefore a measure of the total cluster luminosity. However, if we assume that the luminosity is emitted from zero age main sequence (ZAMS) stars then the cluster luminosity is dominated by the most massive star in the cluster and so can be used to probe the correlation between it and the methanol maser. Since many of these masers are associated with [UCH [ii]{}]{} regions the ZAMS assumption is reasonable. The luminosities range from $\sim$100 to 10$^6$[L$_\odot$]{} with a peak in the distribution at 10$^{4.5-5.0}$[L$_\odot$]{} (see Fig.\[fig:rms\_luminosity\]). The RMS survey is complete for embedded young massive stars with luminosities a few times 10$^4$[L$_\odot$]{} (@davies2011) and so the fall off in the distribution for luminosities lower than this value is due to incompleteness. The minimum luminosity of a high-mass star is $\sim$1,000[L$_\odot$]{} (corresponding to a star of spectral type B3; @boehm1981 [@meynet2003]) and so the fact that most of the RMS-MMB sample ($\sim$90percent) have luminosities above this supports their association with high-mass star formation. While the majority of the MMB sources appear to be associated with massive stars we do find a handful of sources with luminosities towards the lower end of the distribution (i.e., 100–500[L$_\odot$]{}). Errors in the kinematic distance solution could possibly account for a few of these but it is unlikely to explain them all. It therefore seems likely that these methanol masers are associated with intermediate-mass stars with masses of several solar masses. This is consistent with the findings of @minier2003 who were able to put a lower limit of $\sim$3[M$_\odot$]{} for strong maser emission. It is possible that these are intermediate mass protostars that are still in the process of accreting mass on their way to becoming a high-mass protostar. This is consistent with the mass-size relationship discussed in the previous subsection, however, as we will show in the next few paragraphs, there is a strong correlation between clump mass and bolometric luminosity and therefore these low luminosity sources also tend to be associated with the lower mass clumps and would require a high SFE ($>$50percent) to form a massive star. In the upper and lower panels of Fig.\[rms\_luminosity\_mass\] we plot the bolometric luminosity as a function of clump mass and methanol maser luminosity for these ATLASGAL-MMB-RMS associated sources, respectively. In these plots we indicate the evolutionary type as classified by the RMS team, however, a KS test is unable to reject the null hypothesis that the two source types are drawn from the same populations and therefore we do not distinguish between them here. Again calculating the partial coefficients to remove the dependence of these parameters on distance we find a strong correlation between the bolometric luminosity and clump mass ($r$=0.78 and $p$-value $\ll 0.01$) and a weaker correlation between the bolometric and maser luminosities ($r$=0.42 and $p$-value $\ll 0.01$). The linear log-log fit to the bolometric luminosity and clump mass gives a slope of $0.94\pm0.04$ and so not only are these two parameters strongly correlated but their relationship is very close to linear. This value is slightly shallower than the slope of $\sim$1.3 reported by @molinari2008 from their analysis of a sample of 42 regions of massive star formation. However, we note that they use IRAS fluxes to determine source luminosities and this will tend to overestimate fluxes for the more distant sources, which are typically also the most massive sources. This would lead to a steepening of the luminosity-mass slope when compared to the RMS luminosities, which used the higher resolution 70[$\mu$m]{} MIPSGAL band to compute luminosities and therefore should be more reliable. \ Assuming that the clump mass remains relatively constant through the early embedded stages of massive star formation while the luminosity increases as both the accretion rate and the mass of the protostar increase, the source should move vertically upwards in the mass-luminosity plot. Once the [H [ii]{}]{} region has formed and starts to disrupt its host clump the source’s luminosity will be constant (luminosity of the ZAMS star ionizing the [H [ii]{}]{} region) and its mass will begin to decrease resulting in it moving horizontally to the left. Since there is no significant difference between the luminosity-mass distributions of MYSO and [UCH [ii]{}]{} region stages we can assume they have similar ages; at both of these stages core hydrogen burning has begun, however, the [UCH [ii]{}]{} regions are slightly more evolved and have started to ionise their surroundings. The fit to the data (solid red line in the upper panel of Fig.\[rms\_luminosity\_mass\]) is a crude approximation of the transition between protostellar and [H [ii]{}]{} region evolution. As Gallaway et al. point out, the MMB sources are likely to be associated with a broad range of evolutionary states from HMCs to [UCH [ii]{}]{} regions with their luminosity increasing through accretion as they develop. As previously mentioned only $\sim$20percent of ATLASGAL-MMB associated sources are associated with the later stages (i.e., YSO and [UCH [ii]{}]{}) and therefore the majority are younger and less evolved embedded objects whose luminosity is not yet sufficient to dominate that of its associated cluster. All of these sources will be located below the red line with more evolved [H [ii]{}]{} regions dominating the upper left part of the parameter space. We have found the mass and bolometric luminosity to be weakly correlated with the methanol maser luminosity. The mass and bolometric luminosity are strongly correlated with each other (i.e., $r=0.78$) and have very similar correlation values when compared to the maser luminosity (i.e., $r=0.42$ and 0.44, respectively), which might make the underlying cause of the correlation hard to distinguish. However, since the maser arises from a population inversion pumped by far-infrared and submillimetre photons emitted from the embedded YSO, it is the intrinsic luminosity of the YSO that is more likely the fundamental cause of these correlations. The weaker correlation between the bolometric and maser luminosities is probably related to the fact that the former is a measure of the total luminosity of the embedded proto-cluster, whereas the latter is more likely to be driven by a single cluster member, and no doubt maser variability also plays a role. A similar correlation between bolometric luminosity and water maser luminosity was reported by @urquhart2011b from a single dish survey of MYSOs and [UCH [ii]{}]{} regions identified by the RMS survey. Having found a correlation between the methanol and water maser luminosities for these two advanced stages we would conclude that the strength of the maser emission is dominated by the energy output of the central source and not driven by source evolution as suggested by Breen et al. (see discussion in Sect.\[lum\_vol\_correlation\]). One caveat to this is that both models of massive star formation (i.e., monolithic collapse and competitive accretion) predict that massive stars gain their mass through accretion from a circumstellar disk, and therefore their maser luminosity is likely to increase proportionally with their mass and luminosity. However, the large range of possible final masses of OB stars renders the maser luminosity insensitive to the current evolutionary stage of a particular stars in the same way that the bolometric luminosity is insensitive to evolutionary state of the embedded object. The nature of the unassociated MMB sources {#sect:non-atlas-mmb} ------------------------------------------ [c.....]{} & & & &&\ & & & &&\ MMB005.677$-$00.027$^\dagger$ & 0.79 & 0.23 & & &\ MMB006.881+00.093$^\dagger$ & 3.12 & 0.37 & 17.8 & 29.0 & 650.28\ MMB012.776+00.128$^\dagger$ & 0.84 & 0.21 & 13.0 & 29.0 & 197.67\ MMB013.696$-$00.156$^\dagger$ & 1.90 & 0.23 & 10.9 & -29.7 & 147.46\ MMB014.521+00.155$^\dagger$ & 1.40 & 0.20 & 5.5 & 14.9 & 33.52\ MMB015.607$-$00.255$^\dagger$ & 0.43 & 0.22 & 11.4 & -50.8 & 161.13\ MMB016.976$-$00.005$^\dagger$ & 0.64 & 0.20 & 15.5 & -1.4 & 270.72\ MMB018.440+00.045$^\dagger$ & 1.85 & 0.21 & 11.6 & 9.1 & 152.98\ MMB019.614+00.011$^\ddagger$ & 3.99 & 0.22 & 13.1 & 2.5 & 210.32\ MMB292.468+00.168 & 4.40 & 0.72 & 7.9 & 23.3 & 248.31\ MMB299.772$-$00.005$^\dagger$ & 15.65 & 0.47 & & &\ MMB303.507$-$00.721$^\dagger$ & 2.06 & 0.28 & 10.9 & -136.5 & 179.64\ MMB303.846$-$00.363$^\dagger$ & 7.40 & 0.27 & 11.9 & -75.3 & 210.62\ MMB303.869+00.194$^\ddagger$ & 0.90 & 0.27 & & &\ MMB307.132$-$00.476 & 1.20 & 0.34 & & &\ MMB307.133$-$00.477$^\dagger$ & 2.36 & 0.28 & & &\ MMB308.715$-$00.216$^\dagger$ & 1.04 & 0.26 & & &\ MMB311.551$-$00.055$^\dagger$ & 1.00 & 0.30 & 5.6 & -5.4 & 53.33\ MMB311.729$-$00.735$^\dagger$ & 0.46 & 0.31 & 14.3 & -183.6 & 346.68\ MMB312.501$-$00.084$^\dagger$ & 1.19 & 0.36 & 13.6 & -19.9 & 363.05\ MMB312.698+00.126$^\dagger$ & 1.65 & 0.30 & 14.4 & 31.6 & 338.24\ MMB312.702$-$00.087$^\dagger$ & 0.81 & 0.27 & 5.8 & -8.8 & 50.34\ MMB313.774$-$00.863$^\dagger$ & 14.30 & 2.16 & 3.3 & -49.4 & 128.08\ MMB316.484$-$00.310$^\dagger$ & 0.72 & 0.33 & & &\ MMB324.789$-$00.378$^\dagger$ & 1.15 & 0.24 & 15.2 & -100.0 & 301.03\ MMB325.659$-$00.022$^\dagger$ & 0.57 & 0.18 & 17.4 & -6.7 & 293.79\ MMB327.282$-$00.469$^\dagger$ & 5.40 & 0.26 & 14.5 & -118.4 & 296.61\ MMB327.863+00.098$^\ddagger$ & 1.58 & 0.25 & & &\ MMB328.385+00.131$^\dagger$ & 1.60 & 0.32 & 18.0 & 41.2 & 579.17\ MMB329.526+00.216$^\ddagger$ & 1.81 & 0.30 & & &\ MMB330.998+00.093$^\dagger$ & 0.70 & 0.20 & 12.8 & 20.7 & 177.95\ MMB331.900$-$01.186$^\star$ & 2.50 & 0.83 & 11.8 & -244.5 & 636.07\ MMB332.854+00.817$^\dagger$ & 1.10 & 0.25 & 11.8 & 167.5 & 192.35\ MMB332.960+00.135$^\dagger$ & 2.00 & 0.16 & 3.7 & 8.6 & 11.95\ MMB334.933$-$00.307$^\ddagger$ & 3.30 & 0.19 & 9.4 & -50.4 & 91.48\ MMB337.517$-$00.348$^\ddagger$ & 1.50 & 0.20 & 17.1 & -103.7 & 326.59\ MMB345.205+00.317$^\ddagger$ & 0.80 & 0.16 & 11.5 & 63.6 & 117.06\ MMB345.949$-$00.268$^\dagger$ & 1.53 & 0.29 & 13.9 & -65.1 & 306.05\ MMB348.723$-$00.078$^\dagger$ & 2.58 & 0.22 & 11.2 & -15.2 & 149.49\ MMB350.470+00.029$^\dagger$ & 1.44 & 0.16 & 1.2 & 0.6 & 1.34\ MMB350.776+00.138$^\dagger$ & 0.65 & 0.17 & 11.4 & 27.5 & 119.29\ MMB355.545$-$00.103$^\dagger$ & 1.22 & 0.55 & 11.5 & -20.7 & 404.14\ MMB356.054$-$00.095$^\ddagger$ & 0.52 & 0.19 & & &\ \ $^{\rm{a}}$ Sources with a superscript have been searched for mid-infrared emission by @gallaway2013: $\dagger$ and $\ddagger$ indicate infrared bright and infrared dark sources, respectively, and $\star$ identifies the sources they were unable to classify.\ In Sect.\[sect:unmatched\_mmb\] we identified 43 methanol masers that were not matched to an ATLASGAL source. In this section we will examine the available evidence to try and investigate the nature of these sources. We begin by inspecting the ATLASGAL maps at the locations of the [43]{} MMB sources. In many cases, the position of the maser is coincident with either low-surface brightness, diffuse submillimetre emission, which would have been filtered out by the background subtraction used in the source-extraction process, or weak compact emission that fell below the detection threshold used in the source extraction. We have measured the peak 870$\mu$m flux at the position of the MMB source and estimated the 3$\sigma$ noise from the standard deviation of nearby emission-free regions in the map, and have used the higher of these two values as the upper limit to the submillimetre flux. We have used these values with their assigned distances (as discussed in Sect.\[sect:distance\]) to estimate an upper limit for their masses using Eqn.1 and again assuming a dust temperature of 20K. These results are summarised in Table\[tbl:ATLASGAL\_dark\_sources\]. As previously mentioned, it is widely accepted that methanol masers are *almost* exclusively associated with high-mass star-forming regions. This is supported by many of the findings presented in this paper. One explanations is that these unassociated MMB sources are located at larger distances and that their submillimetre emission simply falls below the ATLASGAL detection sensitivity. Indeed, the median distance for the unassociated maser sample is $\sim$12kpc, which is much larger than the median value of $\sim$5kpc found for the ATLASGAL-MMB associations and the KS test showed the two distributions to be significantly different (see Fig.\[fig:atlas\_mmb\_distance\_hist\] for comparison of distance distributions and Sect.\[sect:distance\] for discussion). Looking at the estimated upper limits for the mass we find that they are all significantly lower than the 1,000[M$_\odot$]{} assumed to be the minimum required for massive star formation. However, as shown in Sect.\[mass-size-relationship\] it is at least feasible for less massive clumps to form massive stars, but in the majority of cases these sources would need to be significantly smaller than the beam. The only caveat is that for all of the other ATLASGAL-MMB associations we found a scale-free envelope, but these sources would need to be relatively discrete and isolated clumps that are not embedded in a larger structure. Alternatively, it is also possible that these methanol masers are associated with embedded sources that will go on to form intermediate-mass stars, in which case the low values obtained for the upper limits to the masses may not be all that important. Assuming this hypothesis is correct then, even though the dust emission is too weak to be detected, we might expect to see associated mid-infrared emission from diffuse nebulosity and evidence of extinction from dark lanes of dust often seen towards sites of star formation. [bf Approximately 80percent of these sources are associated with mid-infrared emission (these are indicated by superscripts given by the MMB name in Table\[tbl:ATLASGAL\_dark\_sources\]), which is similar to the proportion found by @gallaway2013 for the whole MMB catalogue (i.e., 83percent). We present a sample of these false-colour mid-infrared images in Fig.\[fig:irac\_images\_unassociated\_mmb\] again created by combining data extracted from the GLIMPSE archive.]{} The upper middle panel of this figure shows the mid-infrared image of the MMB source MMB303.507$-$00.721, which is located at a distance of $\sim$11kpc. This source is found to have extended 8$\mu$m emission commonly associated with star-forming regions and there is evidence of compact dust emission to the south-west of the mid-infrared emission. From a visual inspection of the mid-infrared emission we estimate that in approximately a third of cases the emission is consistent with this maser being associated with a more distant star-formation site, however, this is probably a lower limit. So for a significant fraction of these unassociated methanol masers it is likely that the lack of dust emission can be explained by them being more distant and their associated dust emission falling below the ATLASGAL detection threshold. However, there is also a small number of sources for which this explanation is not satisfactory. There are three sources that are located at relatively near distances (i.e., MMB292.468+00.168, MMB311.551$-$00.055 and MMB312.702$-$00.087) where we would expect to have detected their dust emission. There are two more that are located at the far distance, but where the far distance allocation puts them much farther from the Galactic mid-plane than expected for star forming regions (i.e., MMB311.729$-$00.735 and MMB331.900$-$01.186 that have $z$ distance of $-$183.6 and $-$244.5pc, respectively), which casts some doubt on the distance allocation if indeed these are star forming. However, there is another intriguing possibility which is that these masers may arise in the circumstellar shells associated with evolved stars (e.g., @walsh2003). A search of the SIMBAD database revealed that only ten of these methanol masers were previously known: one identified in the IRAS point-source catalogue; two detected in the BGPS, and so it is likely that their dust emission falls below our detection threshold but is detected by the BGPS due to their superior low surface-brightness sensitivity; and 7 are included in @robitaille2008 intrinsically red GLIMPSE source catalogue, six of which they classify as YSOs, and one (MMB328.385+00.131) they classified as a possible asymptotic giant branch star. In the lower right panel of Fig.\[fig:irac\_images\_unassociated\_mmb\], we present the mid-infrared image of the MMB source MMB328.385+00.131 that shows only a single point source coincident with the position of the methanol maser. This point source is isolated in the image and the lack of any extended 8$\mu$m emission or extinction feature, that are commonly associated with star forming regions, along with the classification made by @robitaille2008 from its mid-infrared colours, would suggest this maser might be associated with an evolved star. However, this could simply be due to a chance alignment along the same line of sight and so this association requires further investigation to test its reliability. Currently, there is not enough complementary data available for all of these sources to be able to properly evaluate these two explanations. The Hi-GAL survey of the inner Galactic plane at 70-500$\mu$m (@molinari2010a) will provide a way to definitively test these two possibilities (e.g., @anderson2012) and will be discussed in a future publication. Summary and conclusions ======================= The ATLASGAL survey (@schuller2009; $280\degr < \ell < 60\degr$) has identified the Galactic distribution of dust through its thermal emission at 870[$\mu$m]{} and is complete to all massive clumps above 1,000[M$_\odot$]{} to the far side of the inner Galaxy ($\sim$20kpc). In total the ATLASGAL survey has identified some 12,000 compact sources, many of which have the potential to form the next generation of massive stars. Methanol masers have been found to be associated with high mass star formation and we have therefore taken advantage of the availability of an unbiased catalogue of these objects compiled by the methanol multibeam (MMB; @green2009) survey team to identify a large sample of high-mass star forming clumps. Cross-matching these two surveys we have identified 577 ATLASGAL-MMB associated clumps within the overlapping region of both surveys (i.e, $280\degr < \ell < 20\degr$ and $|b| < 1.5\degr$) with two or more methanol masers being detected towards 44 clumps. We find $\sim$90percent of the matches are within 12 ($\sim$3$\sigma$) of the peak of the submillimetre emission revealing a strong correlation with column density and the location of a methanol maser within the clumps. We fail to identify any significant 870[$\mu$m]{} emission towards 43 MMB sources. Assuming a dust temperature of 20K and using distances provided by @green2011b, and derived here, we are able to estimate the clump masses and radii, column and volume densities, and methanol maser luminosities for almost 500 of the ATLASGAL-MMB associations. We find we are complete across the Galaxy to all dust clumps with masses larger than 1,000[M$_\odot$]{} hosting a methanol maser with luminosities $>$1,000Jykpc$^{2}$. We use these parameters to investigate the link between methanol masers and massive star forming clumps and as a probe of Galactic structure. Our main findings are as follows: 1. The clump radii cover a range from 0.1 to several parsecs with the larger clumps generally found at larger distances. With a median aspect ratio of 1.4 the ATLASGAL-MMB associations are fairly spherical centrally condensed structures, however, with a median $Y$-factor of $\sim$5 a significant amount of their mass is located outside the central region. The position of the methanol masers is strongly correlated with the peak column density at the centre of the clumps. We find no correlation between the aspect ratio and $Y$-factor with distance, which suggests that the envelope structures of these massive star forming clumps are scale-free. The apparently simple clump structure (with masers at the central col density peak and scale-free radial structure) suggests the formation of one central stellar cluster per clump. Formation of multiple clusters might be expected to be accompanied by more complex clump structure. 2. We are complete to all dust clumps harbouring a methanol maser with masses over 1,000[M$_\odot$]{} across the inner Galactic disk (i.e., $\sim$20kpc). The median clump dust mass ($\sim$3,000[M$_\odot$]{}) is significantly larger than the completeness level, which confirms that methanol masers are preferentially associated with massive clumps. Furthermore, assuming a Kroupa IMF and a star formation efficiency of 30percent we find that 72percent of these clumps (i.e., $M_{\rm{clump}} >$ 1,000[M$_\odot$]{}) are in the process of forming clusters hosting one or more 20[M$_\odot$]{} star(s). Although 28percent of the clumps have masses lower than 1,000[M$_\odot$]{} we find these are also more compact objects ($\simeq$0.3pc) that are likely to form either single stars or small multiple systems of bound stars that are also very likely to include a massive star. This is supported by the empirical mass-radius criterion for massive star formation (@kauffmann2010b) that shows that 97percent of ATLASGAL-MMB associations have masses and sizes that are consistent with all other known massive star forming clumps, including the lower mass clumps. We conclude that the vast majority of clumps associated with methanol masers are in the process of forming high-mass stars. 3. Inspecting the mass-radius relation for the ATLASGAL-MMB associations we find that a surface density of 0.05gcm$^{-2}$ provides a better estimate of the lower envelope of the distribution for masses and radii greater than 500[M$_\odot$]{}and 0.5pc, respectively, than the criterion given by @kauffmann2010b. This surface density threshold corresponds to a A$_K\sim2$mag or visual extinction, A$_V$, of $\simeq16$mag, which is approximately twice the required threshold determined by @lada2010 and @heiderman2010 for “efficient” low-mass star formation. This would suggest that there is a clear surface density threshold required for clumps before star formation can begin but a higher threshold is required to for more massive star formation. 4. Testing the evolutionary trend reported in the literature (i.e. @breen2011_methanol [@breen2012]) between the methanol maser luminosity and clump-averaged volume density we fail to find any correlation. Although we are able to reproduce the results of these previous studies we find that both parameters have a strong dependence on distance and that once this is removed the correlation between them drops to zero. However, we do find a the bolometric and methanol maser luminosities are correlated with each other. 5. We have identified seven clumps that have masses large enough to be classified as massive protocluster (MPC) candidates which are expected to form the next generation of young massive clusters (YMCs) such as the present day Archers and Quintuplet clusters. Using the Galactic plane coverage of this study and the number of MPC candidates detected we estimate the Galactic population to be $\le$20$\pm$6. This value is twice as many as previously estimated and similar to the number of currently known YMCs, which would suggest that only a few of these MPC candidates will successfully convert the 30percent of their mass into stars required to form a YMC. 6. The Galactic distribution reveals the Galactic centre region having a significantly lower star formation efficiency (SFE), than the rest of the Galaxy covered by this survey, which is broadly flat even towards the spiral arm tangents. The lower SFE is probably a reflection of the much more extreme environment found in the central region of the Milky Way. Interestingly we find no enhancement in either the ATLASGAL or ATLASGAL-MMB source counts in the direction of the Scutum-Centaurus arm tangent, from which we conclude that this arm is not actively forming stars of any type. 7. The galactocentric distribution reveals very significant differences between the surface density of the massive star formation rate between the inner and outer Galaxy. We briefly speculate on possible explanations, however, it is clear further work is required before the reasons behind this difference can be properly understood. Using the surface density distribution with the completeness levels applied we estimate the total Galactic population to be $\sim$560, which means that the sample presented here represents approximately 50percent of the whole population. We estimate the star formation associated with these methanol maser associated clumps may contribute up to 50percent of the Galactic star formation rate. 8. Bolometric luminosities are available from the literature for $\sim$100 clumps and these range between $\sim$100 to 10$^6$[L$_\odot$]{} with the distribution peaking at $\sim$10$^5$[L$_\odot$]{}. This confirms the association between methanol masers and massive young stars for 90percent of this sample of clumps, but also reveals that there are some masers associated with intermediate-mass stars. For lower luminosity clumps (i.e., $M_{\rm{clump}}<$1,000[L$_\odot$]{}) these may be intermediate-mass protostars that are still accreting mass and will eventually go on to form a high-mass star, however, these tend to be associated with the lower mass clumps and would therefore require a SFE $>$50percent. It therefore may be the case that a small number of these clumps are destined to only form intermediate-mass stars. 9. We investigated the available evidence for the 43 methanol masers towards which no 870[$\mu$m]{} emission has been detected and concluded that while perhaps 50percent may be more distant star forming regions where the dust emission has simply fallen below the ATLASGAL surveys sensitivity, the nature of the other masers is yet to be determined. This is the first of a series of three papers planned to use the ATLASGAL survey to conduct a detailed and comprehensive investigation of high-mass star formation. The main aim of these papers is to use the unbiased nature of the dust emission mapped by ATLASGAL over the inner Galactic plane to connect the results derived from different high-mass star formation tracers. In subsequent papers we will investigate the dust properties of an unbiased sample of ultra-compact HII regions identified from the CORNISH survey and a complete sample of massive YSOs identified by the RMS survey. Acknowledgments {#acknowledgments .unnumbered} =============== The ATLASGAL project is a collaboration between the Max-Planck-Gesellschaft, the European Southern Observatory (ESO) and the Universidad de Chile. This research has made use of the SIMBAD database operated at CDS, Strasbourg, France. This work was partially funded by the ERC Advanced Investigator Grant GLOSTAR (247078) and was partially carried out within the Collaborative Research Council 956, sub-project A6, funded by the Deutsche Forschungsgemeinschaft (DFG). L. B. acknowledges support from CONICYT project Basal PFB-06. JSU would like to dedicate this work to the memory of J.M.Urquhart. [^1]: E-mail: jurquhart@mpifr-bonn.mpg.de (MPIfR) [^2]: All of these observational programmes used either the Green Bank Telescope (FWHM $\sim$30;@urquhart2011b [@dunham2011b]) or the Effelsberg telescope (FWHM $\sim$40; @pandian2012 [@wienen2012]) and therefore have comparable resolution and sensitivity. [^3]: Beam-averaged volume densities would be significantly larger. However, we use clump-averaged values here in order to be consistent with the analysis of @breen2011_methanol. [^4]: Note that when deriving this relationship @kauffmann2010b reduced the dust opacities of @ossenkopf1994 by a factor of 1.5. This reduced value for the opacities has not been applied when determining the clump masses presented here and therefore we have rescaled the value given by @kauffmann2010b of 870 by this factor to the value of 580 given in Eqn.5 (cf. @dunham2011).

--- abstract: 'Existing face datasets often lack sufficient representation of occluding objects, which can hinder recognition, but also supply meaningful information to understand the visual context. In this work, we introduce Extended Labeled Faces in-the-Wild (ELFW)[^1], a dataset supplementing with additional face-related categories —and also additional faces— the originally released semantic labels in the vastly used Labeled Faces in-the-Wild (LFW) dataset. Additionally, two object-based data augmentation techniques are deployed to synthetically enrich under-represented categories which, in benchmarking experiments, reveal that not only segmenting the augmented categories improves, but also the remaining ones benefit.' author: - Rafael Redondo - Jaume Gibert bibliography: - 'eccv2020elfw.bib' title: | Extended Labeled Faces in-the-Wild (ELFW):\ Augmenting Classes for Face Segmentation --- Introduction ============ The existence of convenient datasets is of paramount importance in the present deep learning era and the domain of facial analysis is not exempt from this situation. Data should not only be accurate, but also large enough to describe all those underlying features fundamental for machine learning tasks. The first face datasets were acquired under controlled conditions [@phillips2005overview; @jesorsky2001robust; @phillips1998feret; @sim2001cmu], specially in relation to lightning, background, and facial expression. However, real-world applications in-the-wild operate under immeasurable conditions, which are often unrepeatable or at least an arduous task to be reproduced in the laboratory. In this respect, natural datasets acquired in-the-wild enable to cover a larger diversity, also with less effort. Labeled Faces in-the-Wild (LFW) [@LFWTech; @huang2008labeled], previous to the deep learning uprise, and a cornerstone to the present work, aimed at providing a large-scale face dataset to leverage applications in unconstrained environments, which are characterized by having extreme variations in image quality such as contrast, sharpness, or lighting, but also content variation such as head pose, expression, hairstyle, clothing, gender, age, race, or backgrounds. Additionally within LFW, semantic segmentation maps (labels) were released for a subset of faces with three different categories: *background*, *skin*, and *hair*. However, these maps lack contextual information given by either complementing or occluding objects and, particularly, they present several segmentation inconsistencies and inaccuracies. More specifically, beards and moustaches were inconsistently annotated either as skin or hair; any type of object on the head which obstacles identification is considered as part of the background, causing unnatural discontinuities in the facial semantic interpretation; common objects like sunglasses are simply ignored; and last but not least, not a few number of cases have irregular labeled boundaries due to the used super-pixel based annotation strategy. The main goal of this work is to create a set of renewed labeled faces by improving the semantic description of objects commonly fluttering around faces in pictures, and thus enabling a richer context understanding in facial analysis applications. To this end, we aim at extending LFW with more semantic categories and also more labeled faces, which partially solve the original LFW flaws and expand its range to a larger and more specialized real-life applications. In particular, as illustrated in Fig. \[fig:elfw\_examples\], we extended the LFW dataset in three different ways: (1) we updated the originally labeled semantic maps in LFW with new categories and refined contours, (2) we manually annotated additional faces not originally labeled in LFW, and (3) we automatically superimposed synthetic objects to augment under-represented categories in order to improve their learning. With such an extension, we additionally provide results for state of the art baseline segmentation models to be used for future reference and evaluate these data augmentation techniques. Related Works ============= The LFW dataset is contemporary to several other datasets with a similar goal, [*i.e.* ]{}providing vast and rich facial attributes in the form of images and labels. Examples are the Helen dataset [@le2012interactive], with $2,330$ dense landmark annotated images, the Caltech Occluded Faces in-the-Wild (COFW) [@burgos2013robust], comprising landmark annotations on $1,007$ images of occluded faces, and the Caltech 10000 Web Faces [@fink2007caltech], a larger dataset designed for face detection in-the-wild, which however lacks aligned faces. With time, datasets have grown larger and richer in attributes. One of the first widely-used public datasets for training deep models was the CASIA-Webface [@yi2014learning], with $500K$ images of $10K$ celebrities. Subsequently, the IARPA Janus Benchmark and successive upgrades (NIST IJB-A,B,C) [@klare2015pushing; @whitelam2017iarpa; @maze2018iarpa] released large datasets constructed upon still images and video frames, which were especially designed to have a more uniform geographic distribution. In such works, the authors claimed that the main limitation of previous databases, such as the Youtube Faces [@wolf2011face; @ferrari2018extended], the really mega in terms of quantity MegaFace [@Nech2017LevelPF], and the really wide in terms of variety and scale WiderFace [@yang2016wider], is that they were constructed with basic face detectors, and thus rejecting many valid faces due to far-reaching perspectives and facial expressions. It is also noted that benchmarks in datasets such as LFW are saturated, where the best face recognition performance exceeds a $99\%$ true positive rate, which suggests the need for an expansion of datasets. Another recent and also very large dataset is VggFace2 [@cao2018vggface2], which took advantage of internet image search tools to download a huge number of images to later manually screen false positives. Also recently, the large-scale celebrity faces attributes CelebA [@liu2018large] gathered over $200K$ aligned images of more than $10K$ celebrities, including $5$ landmarks and $40$ tagged attributes such as wearing hat, moustache, smiling, wavy hair, and others. Finally, the prominent MS-Celeb-1M [@guo2016ms] —recently shut down in response to journal investigations— got to collect over $10$ million images from more than $100K$ celebrities. Regarding components of heritage and individual identity reflected in faces, it is worth noting an outstanding study on diversity [@merler2019diversity], which provided 1 million annotated faces by means of coding schemes of intrinsic facial descriptions, mainly intended for face recognition. Until now, face datasets for semantic segmentation were mostly focused on facial parts such as lips, eyebrows, or nose [@warrell2009labelfaces; @liu2015multi]. At the time this work was carried out, a notorious CelebAMaskHQ [@lee2019maskgan] was released with $21$ categories over $30K$ high-quality images. Nonetheless, the labeling and detection of common occluding objects was out of their scope, but they can be certainly deemed as complementary to the present work. Considering that face recognition or face synthesis are not the final goal here, LFW was a good candidate which still offers a great variability to build upon. Moreover, it already provides pre-computed annotation segments and has been also heavily reported. The above mentioned limitations, though, should be considered when deploying applications for real environments. Labeled Faces in-the-Wild {#sec:lfw_relatedwork} ------------------------- Labeled Faces in-the-Wild (LFW) was originally created in the context of human face recognition, this is, the identification of particular individual faces. Although it is arguably an old dataset given the effervescent deep learning expansion, it has been regularly applied in numerous machine learning applications on computer vision. Examples are face verification [@taigman2014deepface; @schroff2015facenet; @parkhi2015deep; @wen2016discriminative; @cao2010face; @sun2015deepid3; @hu2014discriminative; @kumar2011describable; @amos2016openface; @liu2017sphereface], high level features for image recognition [@le2013building; @liu2015deep; @kumar2009attribute; @sun2014deep; @sun2015deeply; @li2013learning; @kumar2011describable; @huang2012learning; @bourdev2011describing; @wolf2010effective; @berg2013poof], large scale metrics learning [@koestinger2012large; @coates2013deep; @simonyan2013fisher; @nguyen2010cosine], face alignment [@cao2014face; @peng2012rasl; @saragih2009face], landmark and facial parts detection [@sun2013deep; @belhumeur2013localizing; @ranjan2017hyperface], generic image classification and similarity metrics [@chan2015pcanet; @larsen2015autoencoding; @mignon2012pcca], image retrieval [@wan2014deep; @kumar2011describable; @siddiquie2011image], age and gender classification [@levi2015age; @ranjan2017hyperface; @eidinger2014age], pose, gesture and gaze recognition [@dantone2012real; @ranjan2017hyperface; @rivera2012local; @zhu2015high; @zhang2015appearance], face frontalization [@hassner2015effective], and model warping [@saragih2011deformable], to name ‘a few’. Furthermore, other datasets have also been derived from it [@koestinger2011annotated]. The LFW dataset is made up of $13,233$ jpeg images of $250\times 250$ pixels from $5,749$ people, where $1,680$ people have two or more images[^2]. The authors advice that LFW has its own bias, as any other dataset. In particular, few faces present poor lightning exposure conditions and most pictures contain frontal portraits, because the Viola-Jones face detector [@viola2004robust], used for filtering –and cropping and resizing– fails on angular views, highly occluded faces, and distant individuals. LFW comes with different parallel datasets based on different alignment approaches: (1) the funneling method proposed by Huand et al. [@huang2007unsupervised], (2) the *LFW-a*[^3] commercial software, and (3) deep funneled [@Huang2012a]. Among these, the last two are claimed to provide superior results for most face verification algorithms. Additionally, all parallel versions have computed superpixel representations with the Mori’s online implementation[^4], an automatic local segmentation based on local color similarity [@mori2005guiding]. Finally, by means of these superpixels, LFW released $2,927$ face images originally labeled with 3 categories: *hair*, *skin*, and *background*. Extended Labeled Faces in-the-Wild ================================== ![ELFW insights. (Left) *Normalized appearance frequency* or the normalized number of class appearances per image in the whole dataset, where $1$ means that the class appears at every image. (Right) *Normalized area occupation* or the proportional area occupied by each class at every image where it does appear. Note that standard deviation (top-bar vertical brackets) relates to class variability, so that, as expected, *hair*, *beard-moustache*, and *head-wearable* are highly variable in size, while *background*, *skin*, and even *sunglasses* have in general small variations in relation to their normalized averaged size (top-bar numbers). Furthermore, both mean and standard deviation give an idea of the maximum and minimum areas throughout the ELFW dataset, concretely occurring for *background* and *beard-moustache*, respectively.[]{data-label="fig:elfw_stats"}](./assets/stats/elfw_stats.eps){width="0.7\linewidth"} The Extended Label Faces in-the-Wild dataset (ELFW) builds upon the LFW dataset by keeping its three original categories (*background*, *skin*, and *hair*), extending them by relabeling cases with three additional new ones (*beard-mustache*, *sunglasses*, and *head-wearable*), and synthetically adding facial-related objects (*sunglasses*, *hands*, and *mouth-mask*). The motivation under the construction of ELFW is the fact that most of the datasets with semantic annotations for face recognition do not explicitly consider objects commonly present next to faces in daily images, which can partially occlude the faces and, thus, hinder identification. As a matter of fact, LFW does not make any differences between hair and beard, sunglasses are confused by either hair or skin, hair is not properly segmented in the presence of a hat, or simply a very common object like hands occluding the face –even slightly– is not properly handled. For these reasons, in this section we (1) introduce ELFW, a new dataset especially constructed to deal with common facial elements and occluding objects, and (2) show means of augmenting samples with synthetic objects such as sunglasses, hands, or mouth-masks. Data collection --------------- Among the three released LFW datasets (see Sect. \[sec:lfw\_relatedwork\]), the one developed with the deep funneled approach was chosen for this work because the alignment method is publicly available and it has been reported to achieve superior face verification performance. From the $2,927$ images annotated with the original categories (*background*, *skin*, and *hair*), a group of $596$ was manually re-labeled from scratch because they contained at least one of the extending categories (*beard-moustache*, *sunglasses*, or *head-wearable*). Furthermore, from the remaining not labeled images with available superpixels —LFW was originally released with $5,749$ superpixels maps—, $827$ images having at least one of the extending categories were added up and labeled. In total, ELFW is made up of $3,754$ labeled faces, where $1,423$ have at least one of the new categories. Manual ground-truth annotation methodology {#sec:annotation} ------------------------------------------ The process of annotating images is never straightforward. Although difficulty varies with task, translating the simplest visual concept into a label has often multiple angles. For instance, how to deal with teeth, are they part of the skin face or they must be left out? Do regular glasses need to be treated as sunglasses even if the eyes’ contours can be seen through? To what extent does the skin along the neck need to be labeled? Are earrings a head-wearable as any regular headphones are? On the following, we summarize the guidance instructions elaborated to annotate the dataset to have a better understanding of its labels: - For simplicity, eyes, eyebrows, mouth, and teeth are equally labeled as *skin*. Neck and ears are also considered *skin*. - On the contrary, the skin of shoulders or hands are ignored, [*i.e.* ]{}labeled as *background*. - Helmets, caps, turbans, headsets, even glasses, and in general any object worn on the head —although partially occluding the face— lay under the same *head-wearable* category. - As an exception —and because they do not generally hamper identification—, regular glasses with no color shade whatsoever are labeled as *skin*. - Faces at the background which do not belong to the main face are ignored (*background*). - Likewise, occluding objects like microphones, flags or even hands are also considered as *background*. Following the same annotation strategy used by LFW, the workload was alleviated by initially labeling superpixels, preserving at the same time pixel-wise accurate contours. In order to improve productivity, a simple GUI tool was implemented to entirely annotate a superpixel with a single mouse click, having real-time visual feedback of the actual labels. In practice, scribbles were also allowed, which accelerated the annotation process. A second GUI tool allowed for manual correction of wrong segments derived from superpixels which, for instance, were outlining different categories at the same time. The whole (re)labeling process with the new extending categories was done in $4$ weeks by $4$ different people sharing the annotating criteria described above. The whole set of images was later manually supervised and corrected by one of the annotators. Some labeled and relabeled examples are shown in Fig. \[fig:elfw\_examples\]. See also Fig. \[fig:elfw\_stats\] to get deeper insights about the contributions of each labeled category to the ELFW dataset after annotation. Data augmentation {#sec:data_augmentation} ----------------- Due to the high dependence on large data for training deep models, it is increasingly frequent to enlarge relatively small datasets with synthetically generated images [@Gecer2018; @Shrivastava2017], which might fill the gap for real situations not depicted in the dataset and thus can help to better generalize to unseen cases, but also to balance under-represented categories. In this work, simple yet effective ways to automatically enlarge the proposed dataset have been used, from which ground-truth images can be trivially generated. Although these augmentations are released separately from the dataset, the code is open sourced, so that interested readers can use it at their will. The augmentation strategies are reported in the following sections. ### Category augmentation On the one hand, *sunglasses* is the worst balanced category throughout the collected data, see Fig. \[fig:elfw\_stats\]. On the other, the dataset does not present even a single case of an image with a common object typically present in faces, namely, *mouth masks*. When present, both types of objects usually occlude a large facial area and impede identification. In their turn, though, it is particularly easy to automatically add them to a given face. To this end, $40$ diverse types of sunglasses and $12$ diverse types of mouth masks were obtained from the Internet and manually retouched to guarantee an appropriate blending processing with a face. The whole collection of augmentation assets is depicted in Fig. \[fig:augmentation\_assets\]. By construction, all LFW faces should be detected in principle by the Viola-Jones algorithm [@viola2004robust], but interestingly not all of them were so detected by the used OpenCV implementation. Then, a total of $2,003$ faces were suitable for category augmentation. Before being attached, the sunglasses assets were resized proportionally to the interocular distance and made slightly transparent. Similarly, this distance was also used as a reference to estimate the face size to properly resize the masks. Note that no shading nor color correction was applied here, and although an artificial appearance is patent in some cases, the generated cases were effective enough to reinforce the category learning stage in a semantic segmentation scenario, see Sec. \[sec:experimental\_results\]. Examples of the described category augmentations can be seen in Fig. \[fig:elfw\_augmentation\]. ### Augmenting by occluding There are other common occluding objects in natural images but not attached to the face itself. As a matter of fact, LFW contains an important amount of them, such as a multiplicity of hand-held objects like microphones, rackets, sport balls, or even other faces. In fact, the variety of occlusions can be as broad as the variety of conditions for acquiring ‘clean’ faces. In particular, hands are, by nature, one of the most frequent elements among occluders [@mahmoud2011interpreting]. It is, however, an specially challenging *object* since the skin color shared with the face can entangle posterior face-hand discrimination. For these reasons, it appears reasonable to additionally use hands for data augmentation. Blending source hands into faces requires realistic color and pose matching with respect to the targeted face. To the extent of our knowledge, the most determined work on this regard is Hand2Face [@nojavanasghari2017hand2face], whose authors gave especial relevance to hands because their pose discloses relevant information about the person’s affective state. Other hand datasets made of images captured in first person view, such as Egohands [@bambach2015lending], GTEA [@li2015delving], or EgoYouTubeHands [@urooj2018analysis] are not suitable to be attached to faces in a natural way. The latter work, however, also unveiled a significant hand dataset in third person view, HandOverFace [@urooj2018analysis]. In the end, the hands used in this work were compiled from both Hand2Face and HandOverFace. To attach hands to faces we basically followed the approach described in [@nojavanasghari2017hand2face]. Firstly, the source head pose —originally with hands on it— is matched against all the target head poses in ELFW by using Dlib [@king2009dlib]. Two poses match if the distance between them —measured as the $L_2$ norm of the elevation, azimuth and rotation angles— is under a given solid angle threshold $\theta$. Secondly, the hands color is corrected according to each facial tone. For that, the averaged face color is measured inside a rectangular area containing the nose and both cheeks, which is largely uncovered for most of the current faces. The averaged color is transferred from the target face to the source hands by correcting the mean and standard deviation for each channel in the $l\alpha\beta$ color space [@reinhard2001color]. Before being attached, as with category augmentation, the hands are resized by using a scale factor relative to both origin and destination face sizes. Likewise, hands are also centered by considering their relative location from the source face center to the destination face center. Multiple examples of synthetic occluding hands are showed in Fig. \[fig:elfw\_occluding\] and the final hands usage distribution is illustrated in Fig. \[fig:elfw\_stats\_augmentation\]. ![Histograms illustrating the number of faces (vertical axis) which have been augmented with an specific number of different hands (horizontal axis). Such distributions are shaped by using $\theta=5^\circ$ head pose match between the source faces in HandOverFace or Hand2Face and the target faces in ELFW. It all adds up to $11,572$ new cases out of $3,153$ different faces and $177$ different hands.[]{data-label="fig:elfw_stats_augmentation"}](./assets/stats/elfw_stats_hand_augmentation.eps){width="1.0\linewidth"} Experiments: facial semantic segmentation ========================================= In this section we provide benchmarking results on the ELFW dataset for semantic segmentation, a natural evaluation framework for this released type of data. We chose two deep neural networks in the semantic segmentation state-of-the-art, namely the Fully Convolutional Network (FCN) originally proposed in the seminal work [@long2015fully], and a much recent architecture such as DeepLabV3 [@chen2017deeplab], which has been reported to perform remarkably well in several standard datasets. Baseline configuration {#sec:baseline_config} ---------------------- All experiments carried out here were configured alike in order to provide a common baseline for assessing the ELFW dataset. **Training hyper-parameters**: Both models, FCN and DeepLabV3, used a COCO train2017 pre-trained ResNet-101 backbone. Then, they were fine-tuned by minimizing a pixel-wise cross entropy loss under a $16$ batch sized SGD optimizer, and by scheduling a multi-step learning rate to provoke an initial fast learning with $10^{-3}$ at the very first epochs, latter lowered to $2^{-4}$ at epoch $35$, and finally $0.4^{-5}$ at epoch $90$, where the performance stabilized. Weight decay was set to $5^{-4}$ and momentum to $0.99$. In all experiments basic data augmentation such as random horizontal flips, affine shifts and image resize transformations were performed. An early-stop was set to 30 epochs without improving. **Augmentation factor**: In order to evaluate the proposed category augmentation strategies we defined $\sigma$ to be the factor denoting the number of object-based augmentation faces added to the main non-augmented training dataset. For instance, $\sigma=0.5$ means that an extra $50\%$ with respect to the base train set ($3,754$) of synthetically augmented images have been added for training, [*i.e.* ]{}$1,877$ augmented images. When more than one augmentation asset is used, this factor is uniformly distributed among the different augmentation types. **Validation sets**: To assess each of the augmented categories, separately and jointly, we used the same validation being careful with the nature of each data type. It was defined as $10\%$ of the base train set, [*i.e.* ]{}$376$ images in total, where $62$ out of $125$ images wore real sunglasses and $28$ out of $56$ faces were originally occluded by real hands. The remaining halves were reserved for training. The validation set was then randomly populated with other faces from the remaining set. **Hardware and software**: Each of the neural models was trained on an individual Nvidia GTX 1080Ti GPU. Regardless of the network architecture, the configuration with the smallest dataset ($\sigma=0$) required about $1$ training day, while the largest one ($\sigma=1$) took about a week. We employed the frameworks PyTorch $1.1.0$ and TorchVision $0.3.0$ versions, which officially released implementations of both FCN and DeeplabV3 models with the ResNet-101 pre-trained backbones[^5]. Validation tests {#sec:experimental_results} ---------------- ![Gain effect with different data augmentation types and ratios ($\sigma$) on global metrics for both FCN and DeepLabV3 architectures. The size of each training dataset (related to $\sigma$) is proportionally represented by each circular area.[]{data-label="fig:elfw_results_metrics"}](./assets/results/elfw_results_metrics.eps){width="0.82\linewidth"} ![Gain effect per class on Mean IU with different data augmentation types and ratios ($\sigma$) for both FCN and DeepLabV3 architectures. The size of each training dataset (related to $\sigma$) is proportionally represented by each circular area.[]{data-label="fig:elfw_results_classes"}](./assets/results/elfw_results_classes.eps){width="0.8\linewidth"} To report results, the same metrics in [@long2015fully] are considered, namely *Pixel Accuracy*, *Mean Accuracy*, *Mean IU* and *Frequency Weighted IU*. Pixel Accuracy is a class-independent global measure determining the ratio of correctly classified pixels. Mean Accuracy averages the corresponding true positive ratio across classes. Mean IU averages the intersection over union across classes. While Frequency Weighted IU weights the IU values by the corresponding normalized class appearances. In Tab. \[tab:global\_results\] both FCN and DeepLabV3 performances on the four global metrics are shown for sunglasses, hands, and both assets-based augmentation types and increasing factors. For each architecture and augmentation type, the reported values on all metrics correspond to the epoch —thus the same trained model— where the Mean IU reaches a maximum along the whole training stage. Mean IU is chosen over the others since it considers more accuracy factors and equally balanced classes. A visual representation of the performance gain is presented in Fig. \[fig:elfw\_results\_metrics\]. Overall, both augmentation techniques improve, although not always steady, the segmentation accuracy for both networks. Indeed, $\sigma=1$ tends to deliver the best results, [*i.e.* ]{}the more augmentation data, generally the better. DeepLabV3, performed on pair with FCN along the experiments. Although DeeplabV3 provides interesting multi-scale features, they probably do not make much of a difference in a dataset like this, since segments actually mostly preserve their size across images. Since all metrics are global, it is somehow hidden which of the classes are improving or deteriorating. However, Mean IU —which averages across all classes— showed a higher gain, revealing that some classes are certainly being boosted. In Tab. \[tab:classwise\_results\], the scores are dissected per class, in which Class IU is taken over Class Accuracy because the former supplements with a false positives factor. Reported values correspond to the exact same models shown in Tab. \[tab:global\_results\]. In Fig. \[fig:elfw\_results\_classes\] the gain per class versus augmentation data is illustrated. While FCN behaved irregularly, DeepLabV3 was able to take more profit from the larger augmented instances of the training set. Moreover, *sunglasses* experienced a higher gain when augmentation considered only sunglasses, *beard* was the one for hands augmentation —an expected outcome since hands tend to occlude beards—, and the exact same two categories underwent the highest boost when both augmentation categories were used. Field experiments ----------------- In this section, segmentation is qualitatively evaluated under different laboratory situations. In particular, we want to visualize the models’ generalization capacity on the *mouth-mask* category, which was purely synthetically added to the training set. The deployed model was an FCN trained with the three augmentation categories for $\sigma = 0.5$ at the best checkpoint for the global Mean IU across all epochs on the validation set described in Sec. \[sec:baseline\_config\]. A set of cases with target categories successfully identified is illustrated in Fig. \[fig:webcam\_success\]. Head-wearables such as caps, wool hats or helmets were properly categorized, sunglasses were identified, and hair was acceptably segmented in a wide range of situations. The last three frames show how the same sunglasses transited from *sunglasses* to *head-wearable* when moving from the eyes to the top of the head. The mouth-occluding objects were typically designated as *mouth-mask* as shown in Fig. \[fig:webcam\_mouthmasks\], which reveals the networks capability to generalize with purely synthetic data. In Fig. \[fig:webcam\_occludinghands\] a set of examples are depicted to show the ability of the segmentation network to cope with occluding hands. Note that those cases resembling the artificially augmented assets were properly segmented, even if the major part of the face gets occluded. Other explanatory failures and limitations are also depicted in Fig. \[fig:webcam\_severalfailures\]. Some simply happened spuriously, while others do have a direct link to the ELFW’s particularities. For instance, severe rotations lose the face track or beard, and can degenerate to a large hallo-effect. Hands were miss-classified as *head-wearable* if placed where wearables typically are expected —for instance, covering ears or hair—, because no *hand* data was augmented on such upper head locations. On its turn, head wearables may harden the classification of objects such as sunglasses. Finally, in contrast to Fig. \[fig:webcam\_occludinghands\], hands occluding the mouth were sometimes confused with mouth-masks, specially when the nose was occluded too, which again does not frequently occur in the actual data augmentation strategy. Conclusions =========== In this work, the ELFW dataset has been presented, an extension of the widely used LFW dataset for semantic segmentation. It expands the set of images for which semantic ground-truth was available by labeling new images, defining new categories and correcting existing label maps. The main goal was to provide a broader contextual set of classes that are usually present around faces and may particularly harden identification and facial understanding in general. Different category augmentation strategies were deployed, which yielded better segmentation results on benchmarking deep models for the targeted classes, preserving and sometimes improving the performance for the remaining ones. In particular, we have also observed that the segmentation models were able to generalize to classes that were only seen synthetically at the training stage. [^1]: ELFW dataset and code can be downloaded from <https://multimedia-eurecat.github.io/2020/06/22/extended-faces-in-the-wild.html>. [^2]: Varied erratas have been later published in consecutive amendments. For those meticulous readers, please visit the official website at <http://vis-www.cs.umass.edu/lfw/index.html>. [^3]: <https://talhassner.github.io/home/projects/lfwa/index.html> [^4]: <http://www.cs.sfu.ca/~mori/research/superpixels> [^5]: [ https://pytorch.org/docs/stable/torchvision/models.html]( https://pytorch.org/docs/stable/torchvision/models.html).

--- abstract: 'In this paper we show that it is possible to structure the longitudinal polarization component of light. We illustrate our approach by demonstrating linked and knotted longitudinal vortex lines acquired upon non-paraxially propagating a tightly focused sub-wavelength beam. Remaining degrees of freedom in the transverse polarization components can be exploited to generate customized topological vector beams.' author: - | F. Maucher$^{1,2}$, S. Skupin$^{3,4}$, S. A. Gardiner$^{1}$, I. G. Hughes$^{1}$\   bibliography: - 'bib.bib' title: Creating Complex Optical Longitudinal Polarization Structures --- The concept of light being a transverse wave represents an approximation that is suitable if the angular spectrum is sufficiently narrow [@Born:95]. However, many practical applications ranging from microscopy to data storage require tight focusing. Tight focusing implies a broad angular spectrum and the notion of light being transverse becomes inappropriate. Hence, the longitudinal polarization component can typically not be neglected [@Richards:PRS:1959; @Youngworth:OE:00]. To mention a few examples, a “needle beam” with particularly large longitudinal component was proposed in [@Chong:NatPhot:2008], and radial transverse polarization permits the significant decrease of the focal spot size [@Leuchs:OC:2000; @Leuchs:PRL:2003] while the generated longitudinal component may even dominate the interaction with matter [@Hnatovsky2011]. Last but not least, a M[ö]{}bius strip in the polarization of light was realized in [@Bauer:Science:2015]. In addition, there is current substantial interest in “structured light”, that is, generating customized light fields that suit specific needs in applications in a range of fields [@Allen:PRA:1992; @structured_light; @Franke-Arnold:17; @Maucher2:PRL:2016]. Since the proposal of the Gerchberg-Saxton algorithm [@Gerchberg:Optik:1972] 1972, advances in light shaping [@Grier:Nature:2003; @Whyte:NJP:2005; @Shanblatt:OE:11] now permit the realization of complex light patterns in the transverse polarization plane, including light distributions the optical vortex lines of which form knots [@Dennis:RoyProcA:2001; @Leach:NJP:2005; @Dennis:Nature:2010]. Knotted topological defect lines and their dynamics have been studied in diverse other settings, including for example classical fluid dynamics [@Moffatt:JFM:1969; @Moffatt:nature:1990; @Irvine:nature:2013], excitable media [@Paul:PRE:2003; @Maucher:PRL:2016; @Maucher:PRE:2017], and nematic colloids [@Tkalec:science:2011; @Martinez:NatMat:2014]. To date, the approach has typically been to determine the longitudinal polarization component of the electric field from given transverse components, and attempts to target complex structures in the longitudinal component have not yet been pursued. The reason for this is twofold. On the one hand, the longitudinal component is not directly accessible by beam shaping techniques. On the other hand, non-paraxial beam configurations are required, and topological light is usually studied in paraxial approximation. It is therefore not immediately evident that the whole range of three-dimensional light configurations known for transverse components can be realized in the longitudinal component as well. In this paper, we will show that complex light-shaping of the longitudinal polarization component is indeed possible. To this end, we firstly identify non-paraxial light patterns that give rise to vortex lines that form knots or links. Secondly, we invert the problem and derive how one must structure the transverse components of a tightly focused beam to give rise to a [*given*]{} complex pattern in the longitudinal component, and thus present the first example of non-transverse non-paraxial knots. Finally, we demonstrate that remaining degrees of freedom in the transverse polarization components allow for simultaneous transverse shaping, which could be interesting for applications, e.g., inscribing vortex lines into Bose-Einstein Condensates. ![The profiles $f^{\mathrm{ Hopf}}$ and $f^{\mathrm{ Trefoil}}$ of [Eqs. (\[eq:hopf\]) and (\[eq:trefoil\])]{} (a,e) at narrow widths contain evanescent waves. This is demonstrated in (c,g), where the profiles are shown in the transverse Fourier domain together with a circle of radius $k_0$. A spectral attenuation (d,h) according to [Eq. (\[eq:attenuation\])]{} removes the evanescent amplitudes as well as amplitudes close to $\bm k_\perp=0$ (see text for details), and alters $E^{\rm f}_z$ in the focal plane significantly (b,f).[]{data-label="fig:hopf_trefoil_evanescent"}](fig1.pdf){width="\columnwidth"} We begin with the equations describing a monochromatic light beam: $$\begin{aligned} \nabla^2{\bf E}({\bf r}_{\perp},z) + {k}^2_0{\bf E}({\bf r}_{\perp},z) &=0, \label{eq:nonparaxial}\\ \nabla\cdot{\bf E}({\bf r}_{\perp},z) =\nabla_{\perp}\cdot{\bf E}_{\perp}+\partial_z E_z&= 0.\label{eq:divE}\end{aligned}$$ Here, $k^2_0=\omega^2/c^2=(2\pi/\lambda)^2$, and we have introduced the transverse coordinates ${\bf r}_{\perp}=(x,y)$ and transverse electric field components ${\bf E}_{\perp}=(E_{x},E_{y})$ as we consider propagation in the positive $z$ direction. All three components of ${\bf E}$ in [Eq. (\[eq:nonparaxial\])]{} fulfil the same wave equation, and for a given field configuration ${\bf E^{\rm f}}({\bf r}_{\perp})$ at $z=0$ (e.g., at focus) the general solution for propagation in the positive $z$ direction reads $\hat{{\bf E}}({\bf k}_{\perp},z)=\hat{\bf {E}}^{\rm f}({\bf k}_{\perp})\exp(i k_z z)$, where $k_z({\bf k}_{\perp})=\sqrt{k_0^2-{\bf k}_{\perp}^2}$, ${\bf k}_{\perp} = (k_x,k_y)$ and the symbol $\hat{~}$ denotes the transverse Fourier domain. The prescribed field configuration ${ \hat{\bf E}^{\rm f}}$ must obey certain constraints. Firstly, in order to get a valid bulk solution, there must be no evanescent fields present, that is, ${ \hat{\bf E}^{\rm f}}=0$ for ${\bf k}_{\perp}^2\ge k_0^2$. Secondly, Eq. (\[eq:divE\]) implies for solutions propagating in the $z$ direction that ${\hat{E}_z^{\rm f}}({\bf k}_{\perp}=0)=0$. As preparation for what follows, we first investigate how to obtain a non-paraxial tightly focused knot or link in $E_z$, assuming that we can directly prescribe ${E}_z^{\rm f}$. For the transverse paraxial case, recipes to generate vortex lines in various shapes are known, and they usually involve linear combinations of Laguerre-Gaussian modes [@Dennis:Nature:2010; @Maucher:NJP:2016]. These recipes are not directly applicable to our problem, since there are evanescent fields, due to the nonparaxiality the wavelength cannot be scaled away and it would lead to Eq. (\[eq:divE\]) being violated. Nevertheless, we found that it is possible to adopt those recipes for the non-paraxial case by an educated guess. Starting from a given linear combination of Laguerre-Gaussian modes $f$, filtering in the transverse Fourier domain [@Goodman:16], $$H_{k_0}(\bm k_\perp)=e^{-\frac{1}{ 2\lambda^2\left(\sqrt{\bm k_\perp^2}-k_0\right)^2}}, H_{0}(\bm k_\perp)=1-e^{-(3\lambda{\bm k_\perp})^2},$$ chops off evanescent amplitudes as well as amplitudes close to $\bm k_\perp=0$, and the longitudinal polarization component at $z=0$ reads $$\hat E_z^{\rm f}(\bm k_\perp) = \begin{cases} \hat f(\bm k_\perp) H_{k_0}H_{0},& \text{for } \bm k_\perp^2<k_0^2 \\ 0 &\text{for } \bm k_\perp^2\geq k_0^2 \end{cases}. \label{eq:attenuation}$$ Since the higher-order Laguerre-Gaussian modes are broader in Fourier space and thus lose relative weight after attenuation, one must decrease the relative amplitudes of the lower-order modes to a certain extend. We have found that the following field structures produce a Hopf link or trefoil, respectively, $$\begin{aligned} f^{\mathrm{ Hopf}}&=4{\rm LG}_{00}^\sigma-5{\rm LG}_{01}^\sigma +11{\rm LG}_{02}^\sigma-8{\rm LG}_{20}^\sigma \label{eq:hopf}\\ \begin{split} f^{\mathrm{ Trefoil }}&=9{\rm LG}_{00}^\sigma- 20 {\rm LG}_{01}^\sigma + 40{\rm LG}_{02}^\sigma \\ & \quad- 18{\rm LG}_{03}^\sigma - 34 {\rm LG}_{30}^\sigma, \end{split}\label{eq:trefoil}\end{aligned}$$ for wavelength $\lambda=780~{\rm nm}$ and width $\sigma=370~{\rm nm}\approx \lambda/2$ of the usual Laguerre-Gaussian modes ${\rm LG}^{\sigma}_{ij}(\bm r_\perp)$. The resulting amplitudes before and after filtering for $f$ being either $f^{\mathrm{ Hopf}}$ or $f^{\mathrm{ Trefoil }}$ defined in [Eqs. (\[eq:hopf\]) and (\[eq:trefoil\])]{} are plotted in [Fig. \[fig:hopf\_trefoil\_evanescent\]]{} after normalization to unity. We note that individual mode amplitudes can be changed by about $10\%$ without altering topology, demonstrating a degree of robustness and hence experimental feasibility. Let us now verify that the presented patterns in the focal plane in fact give rise to vortex lines with the desired topology. It is straightforward to propagate the filtered component $E^{\rm f}_z$, as defined in [Eq. (\[eq:attenuation\])]{}, in the $z$-direction. The vortex lines throughout three-dimensional space are depicted by the black lines in [Figs. \[fig:hopf\_trefoil\_propagation\]]{}(a,b) together with a slice in the $z=0$ plane of the light profile phase. The obtained vortex lines are topologically equivalent to a Hopf link and a trefoil, as drawn in the insets. ![Propagation of the spectrally attenuated field shown in [Figs. \[fig:hopf\_trefoil\_evanescent\]]{}(b,f) gives rise to the vortex lines (black) in the forms of a Hopf link (a) and a trefoil (b). A phase slice is shown in the $xy$ plane at $z=0$. As comparison, idealised Hopf link and trefoil are shown as insets. []{data-label="fig:hopf_trefoil_propagation"}](fig2.pdf){width="\columnwidth"} We now address the main point of this paper, i.e. how to choose the transverse polarization components to obtain a given longitudinal polarization component. Because only the transverse components $E_x$ and $E_y$ are accessible to beam shaping, this point is also of great practical relevance. When inspecting [Eq. (\[eq:divE\])]{}, at a first glance the problem may seem to be ill-posed, given that only the longitudinal derivative of the longitudinal polarization enters, i.e., $\partial_zE_z$. However, in Fourier space it is easy to see from [Eq. (\[eq:divE\])]{} that a linearly polarized solution to this problem is given by $$E_x^{\rm f} =-i\!\int\limits_{-\infty}^x\! \mathcal{F}^{-1} \left[ k_z \hat{E}_z^{\rm f} \right]\!(x^\prime,y)dx^\prime, \quad E_y^{\rm f} = 0, \label{eq:xpol}$$ where $\mathcal{F}^{-1}[\hat{g}](x,y)=g(x,y)$ denotes the inverse transverse Fourier transformation. Obviously, an orthogonally polarized solution also exists, $$E_x^{\rm f} = 0, \quad E_y^{\rm f} = -i\!\int\limits_{-\infty}^y\! \mathcal{F}^{-1} \left[ k_z \hat{E}_z^{\rm f} \right]\!(x,y^\prime)dy^\prime. \label{eq:ypol}$$ Both $x$ and $y$ polarized solutions [Eqs. (\[eq:xpol\]) and (\[eq:ypol\])]{} evaluated for Hopf link and trefoil are depicted in Fig. \[fig:hopf\_trefoil\_focus\]. It is noteworthy that any superposition of real and imaginary parts of the solutions [Eqs. (\[eq:xpol\]) and (\[eq:ypol\])]{} is admissible, as long as the coefficients of this superposition add up to one. Furthermore, an arbitrary solenoidal field can be added without having an effect on the longitudinal component. We will discuss this later in more detail. ![ Amplitude and phase for the linearly polarized transverse components [Eqs. (\[eq:xpol\]) and (\[eq:ypol\])]{} that give rise to a longitudinal component forming the Hopf link [Fig. \[fig:hopf\_trefoil\_evanescent\]]{}(b), [Fig. \[fig:hopf\_trefoil\_propagation\]]{}(a) is shown in (a–d) and the trefoil [Fig. \[fig:hopf\_trefoil\_evanescent\]]{}(f), [Fig. \[fig:hopf\_trefoil\_propagation\]]{}(b) is shown in (e–h). The colormaps for each figure are on the right of the two rows of plots. []{data-label="fig:hopf_trefoil_focus"}](fig3.pdf){width="\columnwidth"} Unfortunately, the transverse polarization components computed from [Eqs. (\[eq:xpol\]) and (\[eq:ypol\])]{} are impractical, since, even though $E^{\rm f}_z$ has finite support, the components $E^{\rm f}_x$ or $E^{\rm f}_y$ are non-zero on a semi-infinite interval (see Fig. \[fig:hopf\_trefoil\_focus\]). However, simply attenuating these components by multiplying with, e.g., a sufficiently wide super Gaussian profile ${\rm SG}_N^{w}({\bm r}_{\perp})=\exp(-{\bm r}_{\perp}^{2N}/w^{2N})$ allows the resolution of the problem of semi-infinite light distributions without affecting propagation of the longitudinal component close to the optical axis. Evaluating $\nabla_{\perp}\cdot\left[{\rm SG}_N^{w}({\bm r}_{\perp}){\bf E}^{\rm f}_{\perp}({\bm r}_{\perp})\right]$ reveals that, where $\nabla_{\perp}{\rm SG}_N^{w}$ is large and points in the direction of ${\bf E}^{\rm f}_{\perp}$, additional satellite spots in the longitudinal component will appear. We have checked that using e.g. a super Gaussian with $N=5$ and $w=10 \lambda$ ensures that these additional spots are sufficiently far from the region of interest and both Hopf link and trefoil develop in the propagation of the modified $E_z$ component. So far, we have seen that the answer to the problem of how to choose ${\bf E}^{\rm f}_{\perp}(\bm r_\perp)$ for realizing a prescribed $E^{\rm f}_z$ is not unique, and there are certain degrees of freedom in the choice of ${\bf E}^{\rm f}_{\perp}$. The fundamental theorem of vector calculus (Helmholtz decomposition) allows us to decompose a (sufficiently well-behaved) vector field ${\bf F}$ into an irrotational (curl-free) and a solenoidal (divergence-free) vector field, and ${\bf F}$ can be written as ${\bf F}=-\nabla \phi+\nabla\times{\bf A}$. We wish to apply this theorem to the transverse plane, that is, we set ${\bf F}={\bf E}^{\rm f}_{\perp}(\bm r_\perp)$. In this case, the decomposition reduces to $${\bf E}^{\rm f}_{\perp}(\bm r_\perp) = -\nabla \phi(\bm r_\perp) + \nabla \times \left[A(\bm r_\perp){\bf e}_z\right], \label{eq:helmholtz}$$ with ${\bf e}_z$ the unit vector in $z$ direction. It is straightforward to verify that $$\hat \phi({\bf k}_{\perp}) = - i\frac{k_z({\bf k}_{\perp})\hat{ E}_z^{\rm f}({\bf k}_{\perp})}{{\bf k}_{\perp}^2} \label{eq:phi}$$ gives rise to a valid transverse polarization component ${\bf E}^{\rm f}_{\perp}$. The scalar function $A(\bm r_\perp)$ may be chosen arbitrarily, because the term ${\bf E}^{\rm sol}_{\perp} = \nabla \times \left[A(\bm r_\perp){\bf e}_z\right]$ produces the solenoidal part of ${\bf E}^{\rm f}_{\perp}$, which does not give rise to any longitudinal polarization component. The irrotational choice for ${\bf E}^{\rm f}_{\perp}$, that is, evaluating [Eqs. (\[eq:helmholtz\]) and (\[eq:phi\])]{} with $A(\bm r_\perp)=0$, for Hopf link and trefoil are shown in [Fig. \[fig:optimized\]]{}. ![Irrotational transverse amplitude and phase profiles producing a Hopf link (a–d) and a trefoil (e–h) in the longitudinal polarization component. []{data-label="fig:optimized"}](fig4.pdf){width="\columnwidth"} While the transverse polarization components shown in Figs. \[fig:hopf\_trefoil\_focus\] and \[fig:optimized\] produce exactly the same longitudinal field, they are completely different, in particular from a topological point of view. Unlike the light distributions in [Fig. \[fig:hopf\_trefoil\_focus\]]{}, which do not contain vortices in the transverse polarization components, the irrotational transverse polarization components in [Fig. \[fig:optimized\]]{} each feature phase singularities. Furthermore, note that the amplitudes required in the irrotational transverse polarizations are only roughly two to three times the peak amplitude in the longitudinal polarization. We have seen that all possible transverse polarization components producing a certain longitudinal component differ by a solenoidal field ${\bf E}^{\rm sol}_{\perp} = \nabla \times \left[A(\bm r_\perp){\bf e}_z\right]$, and the function $A(\bm r_\perp)$ represents the degrees of freedom one has when shaping the ${\bf E}^{\rm f}_{\perp}$. As our examples show, it is possible to control the topological structure of longitudinal and transverse electric field components simultaneously. Tightly focused beams containing vortex lines play a role in inscribing vortex lines with specific topology into Bose-Einstein condensates using two-photon Rabi-transitions [@Ruostekoski:PRA:2005; @Maucher:NJP:2016]. The demonstrated knotted or linked longitudinal vortex lines have an extent of roughly $1~\mu$m$^3$ and thus match the typical size of a Bose-Einstein Condensate. Being able to exploit the unique features of structured light in all three vector components of the electric field opens new avenues in controlling the interaction of light with matter. An important practical issue is to actually experimentally detect such a small structure in the longitudinal polarization component. Probing of the longitudinal field using molecules was achieved experimentally roughly 15 years ago [@Novotny:PRL:2001] and continues to be of interest for light-matter interactions [@Quinteiro:PRL:2017]. We propose using a tomographic method using a thermal Rubidium vapour cell that is very thin compared to the wavelength [@Sargsyan:OL:17] to experimentally access the longitudinal polarization component. Using an additional strong static magnetic field parallel to the optical axis and tuning the light field to resonantly drive a $\pi$-transition allows the selective coupling of the longitudinal polarization only. To separate the $\pi$-transition from the $\sigma^{\pm}$-transitions beyond Doppler broadening (roughly 0.5GHz at ) we need a sufficiently large magnetic field (roughly $B\sim 1T$). For such large magnetic fields, isolated pure $\pi$-transitions exist e.g. from $\ket{{\mathrm 5S_{1/2}m_jm_I}}=\ket{{\mathrm 5S_{1/2}\pm1/2\pm3/2}}$ to $\ket{{\mathrm 5P_{3/2}\pm1/2\pm3/2}}$. This method of light-matter coupling can however be extended to more general settings, where the angle of the magnetic field can be tuned and thus different components of vectorial topological light can superposed and inscribed into matter. In conclusion, we have presented a simple algorithm to realize an arbitrary (sufficiently well-behaved) field in the focal plane in the longitudinal polarization component, and elaborated on how to realize the transverse components for it. We have highlighted the importance of the occurrence of evanescent waves and discussed the important degrees of freedom in the choice of the transverse polarization components. Using this method has the potential to broaden the range of possible vectorial structured light fields extensively and lead to a range of applications in various fields in physics, including nonlinear optics and Bose-Einstein Condensates. This work is funded by the Leverhulme Trust Research Programme Grant RP2013-K-009, SPOCK: Scientific Properties Of Complex Knots. S.S. acknowledges support by the Qatar National Research Fund through the National Priorities Research Program (Grant No. NPRP 8-246-1-060).

--- abstract: 'Investors in stock market are usually greedy during bull markets and scared during bear markets. The greed or fear spreads across investors quickly. This is known as the herding effect, and often leads to a fast movement of stock prices. During such market regimes, stock prices change at a super-exponential rate and are normally followed by a trend reversal that corrects the previous over reaction. In this paper, we construct an indicator to measure the magnitude of the super-exponential growth of stock prices, by measuring the degree of the price network, generated from the price time series. Twelve major international stock indices have been investigated. Error diagram tests show that this new indicator has strong predictive power for financial extremes, both peaks and troughs. By varying the parameters used to construct the error diagram, we show the predictive power is very robust. The new indicator has a better performance than the LPPL pattern recognition indicator. JLS model, financial extremes, log-periodic power law, time series, network, error diagram.' author: - Wanfeng Yan - Edgar van Tuyll van Serooskerken title: 'Forecasting Financial Extremes: A Network Degree Measure of Super-exponential Growth' --- Introduction ============ Many models are designed to describe the dynamics of financial bubbles and crashes, among which the log-periodic power law (LPPL) (also known as the Johansen-Ledoit-Sornette (JLS)) model [@js; @jsl; @jls; @sornettecrash] stating that bubbles are generated by behaviors of investors and traders that create positive feedback in the valuation of assets. Therefore, bubbles are not characterized by an exponential increase of prices but rather by a faster-than-exponential growth of prices. This unsustainable growth pace often ends with a finite-time singularity at some future time $t_c$. After $t_c$, the old regime with a faster-than-exponential growth rate is broken, and often but not always, the new regime is a dramatic fall over a short period, a financial crash. Yan et. al [@rebound] extended the JLS model by introducing the concept of “negative bubbles” as the mirror image of standard financial bubbles. They found that positive feedback mechanisms also exist in negative bubbles, which are in general associated with large rebounds or rallies. Therefore, before financial extremes, like crashes and rebounds, a super-exponential growth in price is often observed. Measuring super-exponential growth in financial time series can be used to forecast financial extremes. Empirical observations of this super-exponential growth before crashes and rebounds in order to predict the turning points have been presented in many papers. For example, the 2006-2008 oil bubble [@oil], the Chinese index bubble in 2009 [@Jiangjebo], the real estate market in Las Vegas [@ZhouSorrealest08], the U.K. and U.S. real estate bubbles [@Zhou-Sornette-2003a-PA; @Zhou-Sornette-2006b-PA], the Nikkei index anti-bubble in 1990-1998 [@JohansenSorJapan99], the S&P 500 index anti-bubble in 2000-2003 [@SorZhoudeeper02], the South African stock market bubble [@ZhouSorrSA09] and the US repurchase agreements market [@repo]. In 2008, Lacasa et. al [@lacasa] presented an algorithm which converts a time series into a network. The constructed network inherits many properties of the time series in its structure. For example, fractal time series are converted into scale-free networks, which confirm that power law degree distributions are related to fractality. This method has been applied to the financial time series, [@QianJiangZhou10] as well as the energy dissipation process in turbulence [@LiuZhouYuan10]. In this paper, we combine the two powerful tools above and construct an indicator to measure the magnitude of the super-exponential growth of the stock price, by measuring the degree of the price network, generated from the price time series. By investigating twelve major international stock indices, we show that our indicator has strong power to forecast financial extremes. Methods ======= Lacasa et. al presented an algorithm which converts a time series into a network, called **visibility algorithm**. The detailed construction method is as follows: consider each pair of points in the time series, note these two points as $(t_i, y_i)$ and $(t_j, y_j)$, where $i<j$. Then if all the data points between them are below the line which connect these two points, make a link between these two points. In other words, there is a link between two points $(t_i, y_i)$ and $(t_j, y_j)$, if and only if: $$y_k < y_i + \frac{t_k-t_i}{t_j-t_i} (y_j-y_i), \forall i<k<j. \label{eq:linkvis}$$ The intuition of this algorithm is simple. Consider each point of the time series as a wall located at the x-axis value $t$ and with the height of the wall is equal to the y-axis value $y$. Assume someone standing on the top of the wall at $(t_i, y_i)$, if she can see the top of the wall at $(t_j, y_j)$, then we make a link between these two points. Of course, the height of this person is ignored here. In this paper, we extend the above construction to a new algorithm which can be called the **absolute invisibility algorithm**. This is just the opposite of the visibility algorithm. Consider two points $(t_i, y_i)$ and $(t_j, y_j)$, where $i<j$. Then, if all the data points between these two points are above the line which connect them, make a link between these two points: $$y_k > y_i + \frac{t_k-t_i}{t_j-t_i} (y_j-y_i), \forall i<k<j. \label{eq:linkabsnonvis}$$ To give an idea of what the networks look like in reality, Fig. \[fig:network\_examples\] shows examples of the constructed networks based on the S&P 500 Index daily close prices for April, 2014. The upper panel shows the network constructed by the visibility algorithm, while the lower panel shows the network built by the absolute invisibility algorithm. Note that the non-trading dates have been removed from the figures. First we start to build the connection between the LPPL model and the visibility/absolute invisibility algorithms. Then, we will explain how to use these two powerful tools to predict financial extremes. The LPPL model claims that during the bubble or negative bubble regime, the stock price grows or declines at a super-exponential rate, i.e. in a bubble regime, both the derivative and the second derivative of the **log-price** time series are positive; while in a negative bubble regime, both the derivative and the second derivative of the **log-price** time series are negative. The trick here is to use log-prices instead of normal prices. We know that an exponentially growing time series in a logarithmic-linear scale is represented as a straight line. Therefore, a super-exponentially increasing time series in a log-linear scale is represented as a **convex** line with a positive slope. In contrast, a super-exponentially decreasing time series in a log-linear scale is represented as a **concave** line with a negative slope. At this point, it is easy to find the connection between the LPPL model and the visibility/absolute invisibility algorithms. The networks, which are constructed by the visibility/absolute invisibility algorithms, contain the information about the super-exponential growth of a time series. In more detail, suppose that an observer stands on the point $(t_i, y_i)$ in a log-linear scaled wall array, if he or she has the visibility of another point $(t_j, y_j)$ from here, we can roughly conclude that the growth rate between these two points is super-exponential. The more points that $(t_i, y_i)$ can see, the higher the confidence that the time series grows in a super-exponential rate at $(t_i, y_i)$. The number of the points that $(t_i, y_i)$ can see is exactly the degree of the visibility network at point $(t_i, y_i)$! Based on the LPPL model, herding effect and imitation behavior generates super-exponential growth in stock market prices. The fast growth makes the financial system unstable and eventually leads to a sudden regime change, which is often, but not always, a crash in bubble regimes and rebound in negative bubble regimes. In consequence, the confidence of the super-exponential growth rate should be a good proxy to measure how close it is to the point of the regime shift — the financial extremes. We therefore use the degree of visibility/absolute invisibility network to predict the peaks and troughs in the financial market prices. However, in order to predict financial extremes, the degree of the visibility/absolute invisibility networks is not sufficient. We still need three more criteria: 1. To ensure the **prediction** condition, for any point in the time series, only the links to its left can be counted. The reason is obvious: at time $t_i$, we do not know the “future” information $\{t_j, y_j\}, j = i+1, i+2, \cdots$. 2. To ensure the **extreme** condition, the time series should be increasing before the peaks and decreasing before the troughs. Therefore, the link between $(t_i, y_i)$ and $(t_j, y_j)$ (suppose $t_i > t_j$) is made only if $y_i > y_j$ in a visibility network and $y_i < y_j$ in a absolute invisibility network. 3. To ensure the **normalization** condition, the same scope should be applied for all the points. There is no link between $(t_i, y_i)$ and $(t_j, y_j)$ if $|i - j| > S$, even though all the other conditions are satisfied. We build the financial extreme indicators based on the above discussion. A peak degree at time $t_i, i > S$, noted as $D_{peak}(i)$, is defined as the number of $j$s, where $j$ satisfies: $$\begin{aligned} &~&i-S \leq j \leq i-1, j \in \mathbb{N}, \\ &~&y_i > y_j, \\ &~&y_k < y_j + \frac{t_k-t_j}{t_i-t_j} (y_i-y_j), \forall j<k<i, \textrm{if} ~ i-j \geq 2.\end{aligned}$$ By definition, $D_{peak}(i) \in (0, S]$. The peak indicator is defined as $I_{peak}(i) = D_{peak}(i)/S \in (0, 1]$. Similarly, the trough indicator at time $t_i$ is defined as $I_{trough}(i) = D_{trough}(i)/S$, where $D_{trough}(i)$, is the number of $j$s, where $j$ satisfies: $$\begin{aligned} &~&i-S \leq j \leq i-1, j \in \mathbb{N}, \\ &~&y_i < y_j, \\ &~&y_k > y_j + \frac{t_k-t_j}{t_i-t_j} (y_i-y_j), \forall j<k<i, \textrm{if} ~ i-j \geq 2.\end{aligned}$$ Results ======= We choose twelve international stock indices over 21 years, between Jul. 7, 1993 and Jul. 7, 2014. They are: S&P 500 composite (S&PCOMP), Japan TOPIX (TOKYOSE), Hong Kong Hang Seng (HNGKNGI), Russell 1000 (FRUSS1L), FTSE 100 (FTSE100), STOXX Europe 600 (DJSTOXX), German DAX 40 (DAXINDX), France CAC 40 (FRCAC40), Brazil BOVESPA (BRBOVES), Turkey BIST National 100 (TRKISTB), Mexico Bolsa IPC (MXIPC35) and Thailand Bangkok S. E. T. (BNGKSET). During this period, there are 5479 trading days (note this as $T$). We construct the peak/trough indicators for each of these indices. In this paper, the look-back scope $S$ is set to be 262, since normally one calendar year has about 262 trading days. We take Hong Kong Hang Seng Index (HSI) as an example to give a general idea on how the indicators look like. Fig. \[fig:indicator\_examples\] shows the peak and trough indicators of HSI between Jul. 8, 1994 and Jul. 7, 2014 (due to the construction, as the look-back scope $S=262$, the indicators of the first 262 data points are not available). From the figure, one can easily find that when the peak indicator is high, the stock index is usually very close to a local peak; and when the trough indicator is high, the stock index is usually very close to a local trough. To better present this view, Fig. \[fig:indicator\_threshold\] is a summary of indicated peaks and troughs where the peak/trough indicators are greater than a threshold. By increasing the threshold, there are less and less indicated peaks and troughs. Most of them coincide with the real local peaks and troughs. It confirms that our new indicators have predictive power on real financial extremes. Fig. \[fig:indicator\_examples\] and Fig. \[fig:indicator\_threshold\] show the predictive power of the indicators qualitatively. We also present the predictability of the indicators quantitatively by introducing the error diagram method [@Molchan1; @Molchan2]. In order to implement the error diagram, we have to define peaks/troughs first. For one stock index, the peaks/troughs are defined as the points which have the maximum/minimum value in $b$ days before and $a$ days after. Here we use $b=131$ and $a=45$, which correspond to half a year and two months in trading days respectively. $$\begin{aligned} PK &=& \{(t_i,y_i)\}, \textrm{where} ~ y_i = \max(y_j), i-b \leq j \leq i+a, j \in \mathbb{N}, \label{eq:peakdefine}\\ TR &=& \{(t_i,y_i)\}, \textrm{where} ~ y_i = \min(y_j), i-b \leq j \leq i+a, j \in \mathbb{N}. \label{eq:troughdefine}\end{aligned}$$ To give a brief idea that how the peaks and troughs look like in reality, we show the four most important examples: S&P 500, FTSE 100, STOXX Europe 600, and Hong Kong Hang Seng. In Fig. \[fig:peak\_trough\_indicator\], we display the index values together with the peak/trough indicators. At the same time the peaks and troughs defined by (\[eq:peakdefine\]) and (\[eq:troughdefine\]) are marked by red circles and green crosses respectively. The time when peaks and troughs occurred are also indicated in vertical dash-dotted lines and dashed lines respectively. An error diagram for predicting peaks/troughs of a stock index is created as follows: 1. Count the number of peaks defined as expression (\[eq:peakdefine\]) or troughs defined as expression (\[eq:troughdefine\]). 2. Take the peak/trough indicator time series and sort the set of all indicator values in decreasing order. Consider that we have to use the first look-back scope days $S$ to generate the indicator and $a$ days in the end where we cannot define peaks/troughs (since we need $a$ days in the future to confirm whether today is a peak/trough or not). The indicator time series here we used is actually $I_{peak}(i)$ and $I_{trough}(i)$, where $S+1 \leq i \leq T-a$. 3. The largest value of this sorted series defines the first threshold. 4. Using this threshold, we declare that an alarm if the peak/trough indicator time series exceeds this threshold. Then, a prediction is deemed successful when a peak/trough falls inside the alarm period. 5. If there are no successful predictions at this threshold, move the threshold down to the next value in the sorted series of indicator. 6. Once a peak/trough is predicted with a new value of the threshold, count the ratio of unpredicted peaks/troughs (unpredicted peaks (troughs) / total peaks (troughs) in set) and the ratio of alarms used (duration of alarm period / prediction days, where prediction days equals to $T-a-S$). Mark this as a single point in the error diagram. 7. In this way, we continue until all the peaks/troughs are predicted. The aim of using such an error diagram in general is to show that a given prediction scheme performs better than random. A random prediction follows the line $y = 1 - x$ in the error diagram. This is simple because that if there is no correlation between the fraction of the predicted events should be proportional to the fraction of the alarms activated during the whole period. A set of points below this line indicates that the prediction is better than randomly choosing alarms. The prediction is seen to improve as more error diagram points are found near the origin $(0, 0)$. The advantage of error diagrams is to avoid discussing how different observers would rate the quality of predictions in terms of the relative importance of avoiding the occurrence of false positive alarms and of false negative missed peaks/troughs. By presenting the full error diagram, we thus sample all possible preferences and the unique criterion is that the error diagram curve be shown to be statistically significantly below the anti-diagonal $y = 1 - x$. In Fig. \[fig:error\_diagram\], we show error diagrams for different stock indices. The left panel shows the error diagram curves for peaks, while the right panel shows the error diagram curves for troughs. It is very clear that all these curve, both for peaks and troughs, are far below the random prediction line $y = 1-x$. This means our indicators are very powerful in predicting financial extremes. In general, troughs are better predicted than peaks. This might be due to the fact that stock moves are often wilder when investors panic. An error diagram curve always starts at the point $(0, 1)$ and ends at the point $(1, 0)$. In between, a random error diagram curve could be any monotone function in the unit square. Therefore, the p-values of our financial extreme indicators should be: $$p = A/A_{unit} = A, \label{eq:pvalue}$$ where $A$ is the area which is determined by the error diagram curve of the peak/trough indicator, the x-axis and the y-axis. Table. \[tb:pvalue\] shows the p-values of peak/trough predictions for all the tested stock indices. The average p-value for peaks and troughs are $0.0991$ and $0.0262$ respectively. This means that the predictability of our peak/trough indicators are significant in a significance level $10\%$ and $3\%$ respectively. One may doubt that the selection of the parameters $S$,$a$ and $b$ are arbitrary here. In fact, the selection above is only a reasonable practical decision as $S=262$, $b=131$ and $a=45$ represents the number of trading days of one year, six months and two months respectively. The prediction power of the indicator is generally very strong no matter how we choose the parameters. To prove this, for each stock index, we choose 10 values of $S$, $a$ and $b$: $$S = 200, 230, 260, \cdots, 470; a,b = 30,45,60, \cdots, 165. \label{eq:error_parm}$$ Therefore, we have $10^3=1000$ parameter combinations for each stock index. Fig. \[fig:boxplot\_pvalue\] shows the statistics of the p-values of the (a) peak and (b) trough predictions. Each box plot represents the 1000 p-values of the predictions: the lower and upper horizontal edges (blue lines) of box represent the first and third quartiles. The red line in the middle is the median. The lower and upper black lines are the 1.5 interquartile range away from quartiles. Points out of black lines are outliers. From the figure, it is clear that no matter how we choose the values of parameters $S$, $a$ and $b$, which market we are measuring, and which type of extremes (peaks or troughs) we are testing, the preditability of our indicator is constantly strong. The mean of the median p-value of peaks and troughs for all the stock indices are 0.0982, 0.0288 respectively. That means the predictability of our peak/trough indicators are significant in a significance level $10\%$ and $3\%$ respectively. Discussion ========== In this section, we will compare this new indicator with the LPPL pattern recognition indicator. The LPPL pattern recognition indicator is first presented by Sornette and Zhou [@SorZhouforecast06]. By introducing the pattern recognition method developed by Gelfand et. al [@gel], Sornette and Zhou transform the probability prediction of the financial critical time in the standard LPPL model into a quantitative indicator of financial bubbles and crashes. Yan el. al extended this indicator to financial “negative bubbles” and rebounds [@rebound], and made thorough tests on many major stock indices of the world [@YanRebWooSor]. Although the p-values of the predictability of this indicator are not clearly calculated in those papers, one can easily make an estimation from the error diagrams in these papers using (\[eq:pvalue\]). Fig. 5,8 of [@SorZhouforecast06] show the predictability of the peaks of Dow Jones Industrial Average Index and Hang Seng Index using the LPPL pattern recognition indicator. Similarly, Fig. 5,7 of [@rebound] show the preditability of the troughs of S&P 500 Index) and Fig.5-7 of [@YanRebWooSor] (Fig. 12-14 of the arXiv version) show the predictability of both the peaks and the troughs of Russell 2000 Index, Swiss Market Index and Nikkei 225 Index. For all these examples, the area $A$ determined by the error diagram curve, the x-axis and the y-axis are clearly larger than $0.1$. As $A_{unit} = 1$, the p-values in all these examples are greater than $0.1$. Given that the mean of the median p-value of peaks and troughs for all the stock indices are 0.0982, 0.0288 respectively for our new network indicator, we could conclude that on average the new indicator has a better predictive power than the LPPL pattern recognition indicator. In summary, we have constructed an indicator of financial extremes via the magnitude of the super-exponential growth of the stock price, by measuring the degree of the price network generated by visibility/absolute non-visibility algorithms with constrains. This new indicator has been applied to twelve major international stock indices. The peaks and troughs of the tested indices over the past 20 years can be effectively predicted by our indicator. The predictability of the indicator has been quantitatively proved by error diagrams. The performance of the indicator is robust to the parameters, and on average better than the LPPL pattern recognition indicator. [10]{} url \#1[`#1`]{}urlprefixhref \#1\#2[\#2]{} \#1[\#1]{} A. Johansen, D. Sornette, Critical crashes, Risk 12 (1) (1999) 91–94. A. Johansen, D. Sornette, O. Ledoit, Predicting financial crashes using discrete scale invariance, Journal of Risk 1 (4) (1999) 5–32. A. Johansen, O. Ledoit, D. Sornette, Crashes as critical points, International Journal of Theoretical and Applied Finance 3 (2000) 219–255. D. Sornette, Why Stock Markets Crash (Critical Events in Complex Financial Systems), Princeton University Press, 2003. W. Yan, R. Woodard, D. Sornette, Diagnosis and prediction of market rebounds in financial markets, Physica A 391 (2012) 1361–1380. D. Sornette, R. Woodard, W.-X. Zhou, The 2006-2008 oil bubble: [E]{}vidence of speculation and prediction, Physica A 388 (2009) 1571–1576. Z.-Q. Jiang, W.-X. Zhou, D. Sornette, R. Woodard, K. Bastiaensen, P. Cauwels, Bubble diagnosis and prediction of the 2005-2007 and 2008-2009 [Chinese]{} stock market bubbles, Journal of Economic Behavior and Organization 74 (2010) 149–162. W.-X. Zhou, D. Sornette, Analysis of the real estate market in [Las Vegas]{}: Bubble, seasonal patterns, and prediction of the [CSW]{} indexes, Physica A 387 (2008) 243–260. W.-X. Zhou, D. Sornette, [2000-2003 real estate bubble in the UK but not in the USA]{}, Physica A 329 (2003) 249–263. W.-X. Zhou, D. Sornette, [Is there a real estate bubble in the US?]{}, Physica A 361 (2006) 297–308. A. Johansen, D. Sornette, Financial “anti-bubbles”: log-periodicity in gold and [Nikkei]{} collapses, International Journal of Modern Physics C 10 (4) (1999) 563–575. D. Sornette, W.-X. Zhou, The [US]{} 2000-2002 market descent: How much longer and deeper?, Quantitative Finance 2 (6) (2002) 468–481. W.-X. Zhou, D. Sornette, A case study of speculative financial bubbles in the [South African]{} stock market 2003-2006, Physica A 388 (2009) 869–880. W. Yan, R. Woodard, D. Sornette, Leverage bubble, Physica A 391 (2012) 180–186. L. Lacasa, B. Luque, F. Ballesteros, J. Luque, J. C. [Nuño]{}, From time series to complex networks: The visibility graph, Proceedings of the National Academy of Sciences USA 105 (13) (2008) 4972 –4975. M.-C. Qian, Z.-Q. Jiang, W.-X. Zhou, Universal and nonuniversal allometric scaling behaviors in the visibility graphs of world stock market indices, Journal of Physics A: Mathematical and Theoretical 43 (33) (2010) 335002. C. Liu, W.-X. Zhou, W.-K. Yuan, Statistical properties of visibility graph of energy dissipation rates in three-dimensional fully developed turbulence, Physica A 389 (13) (2010) 2675–2681. G. M. Mochan, Earthquake prediction as a decision making problem, Pure and Applied Geophysics 149 (1997) 233–247. G. M. Mochan, Y. Y. Kagan, Earthquake prediction and its optimization, Journal of Geophysical Research 97 (1992) 4823–4838. D. Sornette, W.-X. Zhou, Predictability of large future changes in major financial indices, International Journal of Forecasting 22 (2006) 153–168. I. M. Gelfand, S. A. Guberman, V. I. Keilis-Borok, L. Knopoff, F. Press, E. Y. Ranzman, I. M. Rotwain, A. M. Sadovsky, Pattern recognition applied to earthquake epicenters in california, Physics of The Earth and Planetary Interiors 11 (3) (1976) 227–283. W. Yan, R. Rebib, R. Woodard, D. Sornette, Detection of crashes and rebounds in major equity markets, International Journal of Portfolio Analysis and Management 1 (2012) 59–79. Ticker Peak Trough --------- ---------- ---------- S&PCOMP 0.125485 0.023136 TOKYOSE 0.087145 0.053484 HNGKNGI 0.140173 0.013434 FRUSS1L 0.117058 0.021920 FTSE100 0.084874 0.019177 DJSTOXX 0.065708 0.016427 DAXINDX 0.127369 0.029723 FRCAC40 0.050226 0.033363 BRBOVES 0.074050 0.015495 TRKISTB 0.092147 0.016112 MXIPC35 0.101478 0.011147 BNGKSET 0.123534 0.060996 : \[tb:pvalue\]P-values of peak/trough predictions for all the tested stock indices. ![Examples of the constructed networks based on the S&P 500 Index daily close price in April, 2014. Left: the networks constructed by the visibility algorithm. Right: the network built by the absolute invisibility algorithm. Note that the non-trading dates have been removed from the figure.[]{data-label="fig:network_examples"}](fig/peakgraph2.eps "fig:"){width="45.00000%"} ![Examples of the constructed networks based on the S&P 500 Index daily close price in April, 2014. Left: the networks constructed by the visibility algorithm. Right: the network built by the absolute invisibility algorithm. Note that the non-trading dates have been removed from the figure.[]{data-label="fig:network_examples"}](fig/troughgraph2.eps "fig:"){width="45.00000%"} ![Example of the financial extreme indicators as well as the index value based on the Hong Kong Hang Seng Index daily close price for 20 years from Jul. 8, 1994 to Jul. 7, 2014. Upper: the peak indicator. Lower: the trough indicator.[]{data-label="fig:indicator_examples"}](fig/indicator.eps){width="90.00000%"} ![Financial extreme indicators with different thresholds. All the points which are marked by red circles are those whose peak indicators are greater than the threshold. All the points which are marked by green crosses are those whose trough indicators are greater than the threshold. As the threshold increases (from 0.1 to 0.2), less predicted peaks and troughs are marked. The sample data is the Hong Kong Hang Seng Index daily close price for 20 years from Jul. 8, 1994 to Jul. 7, 2014.[]{data-label="fig:indicator_threshold"}](fig/th.eps){width="95.00000%"} ![Peaks and troughs defined by (\[eq:peakdefine\]) and (\[eq:troughdefine\]) together with the peak/trough indicators. Four examples are S&P 500, FTSE 100, STOXX Europe 600, and Hong Kong Hang Seng. The peaks and troughs are marked by red circles and green crosses respectively. The time when peaks and troughs occurred are also indicated in vertical dash-dotted lines and dashed lines respectively. The sample data are the daily close price of these four indices for 20 years from Jul. 8, 1994 to Jul. 7, 2014.[]{data-label="fig:peak_trough_indicator"}](fig/ind_1spcomp.eps "fig:"){width="45.00000%"} ![Peaks and troughs defined by (\[eq:peakdefine\]) and (\[eq:troughdefine\]) together with the peak/trough indicators. Four examples are S&P 500, FTSE 100, STOXX Europe 600, and Hong Kong Hang Seng. The peaks and troughs are marked by red circles and green crosses respectively. The time when peaks and troughs occurred are also indicated in vertical dash-dotted lines and dashed lines respectively. The sample data are the daily close price of these four indices for 20 years from Jul. 8, 1994 to Jul. 7, 2014.[]{data-label="fig:peak_trough_indicator"}](fig/ind_5ftse100.eps "fig:"){width="45.00000%"} ![Peaks and troughs defined by (\[eq:peakdefine\]) and (\[eq:troughdefine\]) together with the peak/trough indicators. Four examples are S&P 500, FTSE 100, STOXX Europe 600, and Hong Kong Hang Seng. The peaks and troughs are marked by red circles and green crosses respectively. The time when peaks and troughs occurred are also indicated in vertical dash-dotted lines and dashed lines respectively. The sample data are the daily close price of these four indices for 20 years from Jul. 8, 1994 to Jul. 7, 2014.[]{data-label="fig:peak_trough_indicator"}](fig/ind_6djstoxx.eps "fig:"){width="45.00000%"} ![Peaks and troughs defined by (\[eq:peakdefine\]) and (\[eq:troughdefine\]) together with the peak/trough indicators. Four examples are S&P 500, FTSE 100, STOXX Europe 600, and Hong Kong Hang Seng. The peaks and troughs are marked by red circles and green crosses respectively. The time when peaks and troughs occurred are also indicated in vertical dash-dotted lines and dashed lines respectively. The sample data are the daily close price of these four indices for 20 years from Jul. 8, 1994 to Jul. 7, 2014.[]{data-label="fig:peak_trough_indicator"}](fig/ind_3hngkngi.eps "fig:"){width="45.00000%"} ![Error diagrams for twelve world major stock indices. Left: the error diagram curves for peaks. Right: the error diagram curves for troughs. The line $y=1-x$ represents the random prediction, the more the curves close to the origin $(0,0)$, the better the predictability.[]{data-label="fig:error_diagram"}](fig/ed_peak.eps "fig:"){width="45.00000%"} ![Error diagrams for twelve world major stock indices. Left: the error diagram curves for peaks. Right: the error diagram curves for troughs. The line $y=1-x$ represents the random prediction, the more the curves close to the origin $(0,0)$, the better the predictability.[]{data-label="fig:error_diagram"}](fig/ed_trough.eps "fig:"){width="45.00000%"} ![Statistics of the p-values of the peak (upper panal) trough (lower panal) predictions. Each box plot represents the p-values of the predictions for 1000 parameter selections of $S$, $a$ and $b$ defined in (\[eq:error\_parm\]). The lower and upper horizontal edges (blue lines) of box represent the first and third quartiles. The red line in the middle is the median. The lower and upper black lines are the 1.5 interquartile range away from quartiles. Points out of black lines are outliers. The lower the p-value is, the stronger the prediction power.[]{data-label="fig:boxplot_pvalue"}](fig/boxplot_peak.eps "fig:"){width="90.00000%"} ![Statistics of the p-values of the peak (upper panal) trough (lower panal) predictions. Each box plot represents the p-values of the predictions for 1000 parameter selections of $S$, $a$ and $b$ defined in (\[eq:error\_parm\]). The lower and upper horizontal edges (blue lines) of box represent the first and third quartiles. The red line in the middle is the median. The lower and upper black lines are the 1.5 interquartile range away from quartiles. Points out of black lines are outliers. The lower the p-value is, the stronger the prediction power.[]{data-label="fig:boxplot_pvalue"}](fig/boxplot_trough.eps "fig:"){width="90.00000%"}

--- abstract: 'We test the hypothesis that the classical and ultra-faint dwarf spheroidal satellites of the our Galaxy have been the building blocks of the Galactic halo by comparing their \[O/Fe\] and \[Ba/Fe\] vs. \[Fe/H\] patterns with the ones observed in Galactic halo stars. The \[O/Fe\] ratio deviates substantially from the observed abundance ratios in the Galactic halo stars for \[Fe/H\] $>$ -2 dex, while they overlap for lower metallicities. On the other hand, for the neutron capture elements, the discrepancy is extended at all the metallicities, suggesting that the majority of stars in the halo are likely to have been formed in situ. We present the results for a model considering the effects of an enriched gas stripped from dwarf satellites on the chemical evolution of the Galactic halo. We find that the resulting chemical abundances of the halo stars depend on the adopted infall time-scale, and the presence of a threshold in the gas for star formation.' --- Introduction ============ The $\Lambda$CDM paradigm predicts that a Milky Way-like galaxy must have formed by the assemblage of a large number of smaller systems. In particular, dwarf spheroidal galaxies (dSphs) were proposed in the past as the best candidate small progenitor objects, which merged through cosmic time to eventually give rise to the stellar Galactic halo (e.g. Grebel 2005). On the other hand, Fiorentino et al. (2015) using RR Lyrae stars as tracers of the Galactic halo (GH) ancient stellar component, showed that dSphs do not appear to be the major building-blocks of the GH, estimating an extreme upper limit of 50% to their contribution. ![\[O/Fe\] vs \[Fe/H\] (left panel) and \[Be/Fe\] vs \[Fe/H\] (right panel) ratios in the GH in the solar neighborhood for the reference model 2IM are drawn with the solid blue line. [*Models with the enriched infall from dSph*]{}: the magenta dashed dotted line and the red short dashed line represent the models 2IM+dSph and 2IM+dSph MIX, respectively. [*Models with the enriched infall from UfDs*]{}: the green dashed dotted line and the yellow short dashed line represent the models 2IM+UfD and 2IM+UfD MIX, respectively. Thinner lines indicate the ISM chemical evolution phases in which the SFR did not start yet in the GH, and during which stars are no created. [*Models of the dSph and UfD galaxies*]{}: The long dashed gray line represents the abundance ratios for the dSph galaxies, whereas long dashed black line for the UfD galaxies. [*Observational O data of the GH:*]{} Cayrel et al. (2004) (cyan circles), Akerman et al. (2004) (light green pentagons), Gratton et al. (2003) (dark green triangles). [*Observational Ba data of the GH:*]{} Frebel (2010). []{data-label="O1"}](OFE_ufd_dsph_2IM_ref_dati.eps "fig:") ![\[O/Fe\] vs \[Fe/H\] (left panel) and \[Be/Fe\] vs \[Fe/H\] (right panel) ratios in the GH in the solar neighborhood for the reference model 2IM are drawn with the solid blue line. [*Models with the enriched infall from dSph*]{}: the magenta dashed dotted line and the red short dashed line represent the models 2IM+dSph and 2IM+dSph MIX, respectively. [*Models with the enriched infall from UfDs*]{}: the green dashed dotted line and the yellow short dashed line represent the models 2IM+UfD and 2IM+UfD MIX, respectively. Thinner lines indicate the ISM chemical evolution phases in which the SFR did not start yet in the GH, and during which stars are no created. [*Models of the dSph and UfD galaxies*]{}: The long dashed gray line represents the abundance ratios for the dSph galaxies, whereas long dashed black line for the UfD galaxies. [*Observational O data of the GH:*]{} Cayrel et al. (2004) (cyan circles), Akerman et al. (2004) (light green pentagons), Gratton et al. (2003) (dark green triangles). [*Observational Ba data of the GH:*]{} Frebel (2010). []{data-label="O1"}](bafe_2im_dsp_ufd_fra_ref.eps "fig:") ![ As in Fig. 1 but for the 2IMW model. []{data-label="Ba1"}](OFE_wind_dsp_ufd_fra_ref_dati.eps "fig:") ![ As in Fig. 1 but for the 2IMW model. []{data-label="Ba1"}](BA_2IMW_dsp_ufd_fra_ref.eps "fig:") The SDSS (York et al. 2000) discovered a new class of objects characterized by extremely low luminosities, high dark matter content, and very old and iron-poor stellar populations: the ultra faint dwarf spheroidal galaxies (UfDs). In Spitoni et al. (2016) we test the hypothesis that dSph and UfD galaxies have been the building blocks of the GH, by assuming that the halo formed by accretion of stars belonging to these galaxies. Moreover, extending the results of Spitoni (2015) to detailed chemical evolution models in which the IRA is relaxed, we explored the scenario, in which the GH formed by accretion of chemically enriched gas originating from dSph and UfD galaxies. The chemical evolution models ============================= For the Milky Way we consider the following two reference chemical evolution models: 1. The classical two-infall model (2IM) presented by Brusadin et al. (2013). The Galaxy is assumed to have formed by means of two main infall episodes: the first formed the halo and the thick disk (with an infall time scale $\tau_H=0.8$ Gyr), the second one the thin disc ($\tau_D=7$ Gyr). 2. The two-infall model plus outflow of Brusadin et al. (2013; here we indicate it as the 2IMW model) with $\tau_H=0.2$ Gyr. In this model a gas outflow occurring during the GH with a rate proportional to the SFR through a free parameter is considered. In Tables 1, 2, and 3 of Spitoni et al. (2016) all the adopted parameters of the Milky Way, dSph and UfD models are reported. Here, we only underline that UfDs are characterized by a very small star formation efficiency (SFE) (0.01 Gyr$^{-1}$) and by an extremely short time scale of formation (0.001 Gyr). The time at which the galactic wind starts in dSphs is at 0.013 Gyr after the galactic formation, whereas for UfDs at 0.088 Gyr. As expected, the UfD galaxies develop a wind at later times because of the smaller adopted SFE. The nucleosynthesis prescriptions are the ones of Romano et al. (2010, model 15) and for Ba, the ones of Cescutti et al. (2006, model 1). Concerning the model for the GH with the enriched infall we assume that the gas infall law is the same as in the 2IM or 2IMW models and it is only considered a time dependent chemical composition of the infall gas mass. We tested two different models: - Model i): The infall of gas which forms the GH is considered primordial up to the time at which the galactic wind in dSphs (or UfDs) starts. After this moment, the infalling gas presents the chemical abundances of the wind. In Figs. 1 and 2 we refer to this model with the label "2IM(W)+dSph”or “2IM(W)+UfD”. - Model ii): we explore the case of a diluted infall of for the GH. In particular, after the galactic wind develops in the dSph (or UfD) galaxy, the infalling gas has a chemical composition which, by $50$ per cent, is contributed by the dSph (or UfD) outflows; the remaining $50$ per cent is contributed by primordial gas of a different extra-galactic origin. In all the successive figures and in the text, we refer to these models with the labels “2IM(W)+dSph(UfD) MIX”. The Results =========== The Results: the Galactic halo in the model 2IM ----------------------------------------------- In order to directly test the hypothesis that GH stars have been stripped from dSph or UfD systems, in the left panel of Fig. \[O1\], the predicted \[O/Fe\] vs. \[Fe/H\] abundance patterns for typical dSph and UfD galaxies are compared with the observed data in GH stars. The two models cannot explain the \[O/Fe\] plateau which GH stars exhibit for $\mathrm{[Fe/H]}\ge-2.0$ dex. Moreover, in left panel of Fig. 1 we also show the results with the enriched infall coming from dSph galaxies. We recall that a key ingredient of the 2IM model is the presence of a threshold in the gas density in the star formation (SF) fixed at 4 M$_{\odot}$ pc$^{-2}$ in the GH. Such a critical threshold is reached only at $t=0.356\,\mathrm{Gyr}$ from the Galaxy formation. During the first 0.356 Gyr in both “2IM+dSph” and “2IM+dSph MIX” models, no stars are created, and the chemical evolution is traced by the gas accretion. After the SF takes over, \[O/Fe\] values increase because of the pollution from massive stars on short time-scales. In the “2IM+dSph” model the first stars that are formed have \[Fe/H\] $>$ -2.4 dex. In this case, to explain data for stars with \[Fe/H\] $<$ -2.4 dex we need stars formed in dSph systems. Concerning the results with the enriched infall from UfD outflow abundances, model results for the GH still reproduce the data but with the same above mentioned caveat. In the right panel of Fig. 1, we show the results for the \[Ba/Fe\] vs. \[Fe/H\] abundance diagram. Chemical evolution models for dSphs and UfDs fail in reproducing the observed data, since they predict the \[Ba/Fe\] ratios to increase at much lower \[Fe/H\] abundances than the observed data. That is due to the very low SFEs assumed for dSphs and UfDs. The subsequent decrease of the \[Ba/Fe\] ratios is due to the large iron content deposited by Type Ia SNe in the ISM, which happens at still very low \[Fe/H\] abundances in dSphs and UfDs. In the right panel of Fig. 1, all our models involving an enriched infall from dSphs and UfDs deviate substantially from the observed trend of the \[Ba/Fe\] vs. \[Fe/H\] abundance pattern in GH stars. The Results: the Galactic halo in the model 2IMW ------------------------------------------------ In the reference model 2IMW the SFR starts at 0.05 Gyr. Comparing model “2IMW+dSph” in the left panel of Fig. 2 with model “2IM+dSph” in the left panel of Fig. 1, we can see that the former shows a shorter phase with the enriched infall of gas with SF not yet active than the latter. The model results for the model “2IMW+UfD” in the left panel of Fig. 2 overlap to the reference model 2IMW at almost all \[Fe/H\] abundances. Since in the UfD galactic model the wind starts at 0.088 Gyr and, at this instant, in the model 2IMW the SF is already active. Concerning the \[Ba/Fe\] vs \[Fe/H\] ratios (right panel in Fig. 2), we notice that the 2IMW model provides now a better agreement with the observed data than the 2IM model. By assuming an enriched infall from dSph or UfD galaxies, the predicted \[Ba/Fe\] ratios agree with the observed data also at $\mathrm{[Fe/H]}<-3\,\mathrm{dex}$. Conclusions =========== 1. The predicted \[O/Fe\] vs.\[Fe/H\] abundance ratios of UfD and dSph chemical evolution models deviate substantially from the observed data of the GH stars only for \[Fe/H\] $>$ -2 dex; we conclude that an evolution in situ in the GH is requested. On the other hand, we notice that for Ba the chemical evolution models of dSphs and UfDs fail to reproduce the observational observed data of the GH stars over the whole range of \[Fe/H\]. 2. The effects of the enriched infall on the \[O/Fe\] vs. \[Fe/H\] plots depend on the infall timescale of the GH and the presence of a gas threshold in the SF. The most evident effects are present for the model 2IM, characterized by the longest time scale of formation (0.8 Gyr), and the longest period without SF activity among all models presented here. 3. In the presence of an enriched infall of gas we need stars produced in dSphs or UfD s and accreted later to the GH, to explain the data at lowest \[Fe/H\]. 4. The optimal element to test different theories of halo formation is Ba which is easily measured in low-metallicity stars. In fact, we have shown that the predicted \[Ba/Fe\] vs. \[Fe/H\] relation in dSphs and UfDs is quite different than in the GH. Acknowledgments {#acknowledgments .unnumbered} =============== ES and FM acknowledge financial support from FRA2016 of the University of Trieste. [99]{} Akerman, C. J., Carigi, L., Nissen, P. E., Pettini, M., Asplund, M., 2004, A&A, 414, 93 Brusadin, G., Matteucci, F., Romano, D., 2013, A&A, 554, A135 Cayrel, R., Depagne, E., Spite, M., et al., 2004, A&A, 416, 1117 Cescutti, G., François, P., Matteucci, F., Cayrel, R., Spite, M., 2006, A&A, 448, 557 Fiorentino, G., Bono, G., Monelli, M., et al., 2015, ApJL, 798, L12 Frebel, A. 2010, Astron. Nachr., 331, 474 Gratton, R. G., Carretta, E., Desidera, S., Lucatello, et al., 2003, A&A, 406, 131 Grebel, E. K., 2005, IAU Colloq. 198: Near-fields cosmology with dwarf elliptical galaxies, 1 Romano, D., Karakas, A. I., Tosi, M., Matteucci F., 2010, A&A, 522, A32 Spitoni, E., 2015, MNRAS, 451, 1090 Spitoni, E., Vincenzo, F., Matteucci, F., Romano, D., 2016, MNRAS, 458, 2541 York, D.G., Adelman, J., Anderson, J.E., Jr., et al., 2000, AJ, 120, 1579

--- author: - 'Thiago Correa, Breno Gustavo, Lucas Lemos, Amber Settle' bibliography: - 'bibliography.bib' title: An Overview of Recent Solutions to and Lower Bounds for the Firing Synchronization Problem --- Introduction ============ Complex systems in a wide variety of areas such as biological modeling, image processing, and language recognition can be modeled using networks of very simple machines called finite automata. A finite automaton is an idealized machine that follows a fixed set of rules and changes its state to recognize a pattern, accepting or rejecting the input based on the state in which it finds itself at the conclusion of the computation. In general a subsystem can be modeled using a finite automaton if it can be described by a finite set of states and rules. Connecting subsystems modeled using finite automata into a network allows for more computational power. One such network, called a cellular automaton, consists of an $n$-dimensional array for $n > 1$ with a single finite automaton located at each point of the array. Each automaton reads the current state of its neighbors to determine how to modify its own state. The interest in studying cellular automata is not in the description of the local state transitions of each machine, but rather in studying the global behavior across the entire array. One of the oldest problems associated with cellular automata is the firing synchronization problem, originally proposed by John Myhill in 1957. In the firing synchronization problem each machine of the cellular automata is identical, except for a single designated machine called the general or initiator. The synchronization starts with every cell in a *quiescent* state, except the initiator which starts in the *general* state and is the one initiating the synchronization. The next state of each cell is determined by the cell’s state and the state of its neighbors, with a required rule being that a machine that is quiescent and has two neighbors that are quiescient must remain so in the next round of the computation. The goal of the problem is to synchronize the array of automata, which means that each machine enters a unique designated *firing* state for the first time and simultaneously at some point during the execution. As might be expected with a problem that has existed as long as the firing synchronization problem, there are a wide range of variants for the network and its initial configuration. In the original problem the initiator is placed at one end of the array, either on the left or the right. A problem that is more complex than the original problem is the generalized problem, defined by Moore and Langdon in 1968 [@MooreLangdon1968]. In this version the initiator can be placed anywhere in the array, although there is a restriction that there is a single initiator. The problem can be made more complex by requiring that the solution be symmetric, a requirement first introduced by Szwerinski in 1982 [@szwerinski1982time]. In this type of solution an automaton cannot distinguish its right and left neighbors, eliminating any directional information provided to the automaton and making the problem more difficult to solve. Another set of variants of the problem consider alterations to the underlying structure of the network. All of the problems described above operate on a one-dimensional array, which is the simplest configuration for a cellular automaton. In that configuration the end machines do not have neighbors, so that the end machines make state transitions based only on their own state and one existing neighbor. An alternate approach in one dimension is to connect the two end machines to each other, forming a ring rather than an array. The network can also be expanded to more than one dimension, producing a grid (two dimensions), a cube (three dimensions), or more generally a hybercube (four or more dimensions). Once a variant and configuration for the network has been established, it is important to measure the complexity of the solution to the problem. One measure of a solution is the time in which the solution causes the network to synchronize. A minimal-time solution is one that synchronizes the network as quickly as possible, and a non-minimal-time solution is one that synchronizes the network at some point but requires more steps/rounds than the minimal-time solution. For example, the minimal time required for the original problem on a one-dimensional array is $2n - 2$ steps for an array with $n$ machines. Any solution sychronizing in $2n - 2$ steps is minimal-time, while those requiring more than $2n - 2$ rounds are non-minimal-time solutions. Solutions can also be compared based on the number of states for each finite automaton, with solutions that require fewer states seen as more difficult and therefore better. Another way to compare solutions was introduced by Vollmar in 1982 [@vollmar1982some]. Vollmar’s metric is the state complexity of a solution, which is the combination of the number of states and number of transitions required by the solution. As with any long-standing problem, there are a large number of solutions to the firing synchronization problem. Our goal, and the contribution of this work, is to summarize recent solutions to the problem. We focus primarily on solutions to the original problem, that is, the problem where the network is a one-dimensional array and there is a single initiator located at one of the ends. We summarize both minimal-time and non-minimal-time solutions, with an emphasis on solutions that were published after 1998. We also focus on solutions that minimize the number of states required by the finite automata. In the process we also identify open problems that remain in terms of finding minimal-state solutions to the firing synchronization problem. Early results ============= While the focus of this paper is on recent solutions to and lower bounds for the firing synchronization problem (FSP), it is useful to summarize earlier results to provide context for more recent ones. In this section we summarize results providing solutions to the FSP using a (relatively) small number of states as well as known lower bounds on the number of states required to solve the FSP. This section focuses on solutions and lower bounds published prior to 1999. Minimal-time results -------------------- The first minimal-time solution to the original one-dimensional FSP was produced in 1962 when Goto gave a solution with over a thousand states [@Goto1962]. Once it became clear that a minimal-time solution was possible work quickly turned to minimizing the number of states required by the solution. In 1966 Waksman [@waksman1966optimum] gave a 16-state minimal-time solution, and Balzer [@Balzer1967] independently produced an 8-state solution using the same ideas. In 1987 Mazoyer produced a 6-state solution to a restricted version of the problem in which the initiator is always located at the left endpoint of the array [@Mazoyer1987]. In 1968 Moore and Langdon introduced the generalized problem in which the initiator can be located anywhere in the array and gave a 17-state minimal-time solution [@MooreLangdon1968]. An improvement on this solution was produced in 1970 with the publication of a 10-state solution [@VMP1970]. Further work was done by Szwerinski who produced a 10-state symmetric solution [@szwerinski1982time]. Szwerinski’s solution was subsequently improved to a 9-state solution [@settle1999new]. There were many fewer early solutions for the firing synchronization problem on a ring. In 1989 Culik modified Waksman’s solution to function on the one-dimensional ring [@Culik1989]. Like Waksman’s solution Culik’s ring solution uses 16 states. Non-minimal-time solutions -------------------------- An early non-minimal-time solution to the FSP was created by Fischer who gave a 15-state solution that synchronized an array of $n$ automata in 3$n$-4 steps[@Fischer:1965:GPO:321281.321290]. Later Mazoyer suggested that all solutions with few states must necessarily be minimal-time, a conjecture based on the idea that the simplest solution will naturally be the fastest. Yunés disproved the conjecture in 1994 by giving an implementation of a divide and conquer solution originally given by McCarthy and Minsky [@MM1972] that requires only 7 states and synchronizes in time $t(n) = 3n \pm 2 \theta_{n}logn+C$ where $0 \leq \theta_{n} < 1$ [@Yunes1994]. The solution is for the restricted version of the original problem where the initiator must be located at the left endpoint. A 6-state solution was produced in 1999 that allows the initiator to be location at either the left or right endpoint [@settle1999new]. The 6-state solution synchronizes $n$ automata in $2n - 1$ rounds if the initiator is located at the left endpoint and $3n+1$ rounds if the initiator is located at the right endpoint. The same authors produced a 7-state non-minimal-time solution to the generalized problem [@settle1999new]. The 7-state solution requires $2n-2+k$ rounds to synchronize an array of length $n$ when the initiator is located in position $k$ of the array. Also in that work is an 11-state non-minimal-time solution to the ring variant of the FSP [@settle1999new]. To synchronize a ring of length $n$ the solution requires $2n-3$ time steps if n is even and $2n-4$ time steps if n is odd. Lower bounds ------------ Early work on finding lower bounds for the number of states for solutions to the firing synchronization problem is limited. The earliest state lower bound for the FSP was claimed in 1967 by Balzer [@Balzer1967] and confirmed by Sanders in 1994 [@Sanders1994]. Sanders showed that there is no 4-state minimal-time solution to the restricted original problem where the initiator must be located at the right endpoint of the array. Sander’s (and Balzer’s) technique involved a modified exhaustive search in which a program examined all possible 4-state solutions to demonstrate that none of them correctly solved the problem. It is crucial for the programs that only minimal-time solutions are examined since it allows the program to discard any solution that has not achieved synchronization of $n$ automata by time step $2n-2$. The first analytical proofs of state lower bounds came in the late 1990s. One result showed that there is no minimal-time 3-state solution to the original FSP [@settle1999new]. A similar approach was able to show that there are also no 3-state non-minimal time solutions to the problem [@settle1999new]. Weaker results are produced for 4-state and 5-state solutions, showing that there are no solutions subject to a small number of constraints on the solution [@settle1999new]. A less elegant exhaustive case analysis showed that there is no 4-state minimal-time solution to the ring variant of the firing synchronization problem under certain restricted conditions [@settle1999new]. Summary of recent research ========================== The purpose of this paper is to document and classify recent results about the firing synchronization problem. Our main focus is on results published between 1998 and 2015. In this section we first describe the methodology we used to find the recent results. We then describe each of the results, categorizing them by type of problem and type of solution. Methodology ----------- In order to summarize current research on minimal-state solutions to the firing synchronization problem we conducted a literature review. We began by identifying pre-1998 authors who had published on the problem and from that the conferences and journals in which their work had been published. That resulted in the following list of conferences and journals: - Journals - Information and Computation - Information and Control - Information Processing Letters - International Journal of Unconventional Computing - Journal of Algorithms - Journal of Computer and System Sciences - Parallel Processing Letters - SIAM Journal on Computing - Theoretical Computer Science - Conferences - International Colloquium on Structural Information and Communication Complexity (SIROCCO) - International Conference on Cellular Automata for Research and Industry (ACRI) - International Conference on Membrane Computing - International Conference on Networking and Computing (ICNC) - international Workshop on Cellular Automata and Discrete Complex Systems - Symposium of Principles of Distributed Computing We then examined all of the journals and conferences looking for articles with relevant titles between 1998 and 2015. Among the journals only Information Processing Letters, Parallel Processing Letters, and Theoretical Computer Science yielded articles in the specified time period. In the specified time period only the International Colloquium on Structural Information and Communication Complexity (SIROCCO) and the International Conference on Cellular Automata for Research and Industry (ACRI) yielded relevant articles. We also searched for more recent articles written by pre-1998 authors. When we were aware of them, we included papers that had referenced pre-1998 articles. Any article found via any means also added to the list of authors included in the search. We also did searches for papers including relevant keywords, such as cellular automata, firing synchronization, and firing squad. In the remainder of the paper we summarize the articles containing results that minimize the number of states used to solve the firing synchronization problem. While our literature review may have missed some articles, either because they were published in venues of which we were unaware or by authors who were not in any way connected to previous results, we believe that the majority of articles containing state minimization results are represented. Background information ---------------------- In order to understand some of the results discussed later, some background information is necessary. We describe in this section Wolfram’s rules as well as discuss a solution technique common to many non-minimal-time results. ### Wolfram’s rules {#sec:wolfram} Stephen Wolfram is a scientist known for his work on cellular automaton and his contribution to theoretical physics. In 2002 he published *A New Kind of Science*[@Wolfram2002] in which he presents systems he calls *simple programs*. Generally these simple programs have a very simple abstract framework like simple cellular automata, Turing machines, and combinators. The focus of *A New Kind of Science* is to use these simple programs to study simple abstract rules, which are, essentially, elementary computer programs. In our survey, we found solutions based on some of Wolfram’s rules. Here we describe the basic structure of the rules and discuss two of them in particular. In Wolfram’s text elementary cellular automata have two possible values for each cell which are colored white (for 0) or black (for 1). The rules for a transition are based on triples, with the next color of a cell depending on its color and the colors of its two immediate neighbors. The configuration of the transition rules are always the same and are given in the following order: 1. three black squares (111) 2. two black squares followed by a white square (110) 3. a black square, a white square, and a black square (101) 4. a black square, and two white squares (100) 5. a white square and two black squares (011) 6. a white square, a black square, and a white square (010) 7. two white squares and a black square (001) 8. three white squares (000) Thus each set of transition rules can be thought of as an eight-digit binary number since each rule needs to transition either to white (0) or black (1). The rules are named based on the representation of the state produced in the automaton in binary. So rule 60 is the one that assigns the transitions the binary value 00111100. This means that it defines the transitions as 111 $\rightarrow$ 0, 110 $\rightarrow$ 0, 101 $\rightarrow$ 1, 100 $\rightarrow$ 1, 011 $\rightarrow$ 1, 010 $\rightarrow$ 1, 001 $\rightarrow$ 0, and 000 $\rightarrow$ 0. Another Wolfram’s rule mentioned in results we discuss later in this paper is rule 150 which specifies that the states above transition to the state encoded by the binary representation of the number 150 (binary value 10010110) ### 3n-step algorithm {#ssec:3nstep} The 3n-step algorithm is an interesting approach to synchronizing a two-dimensional cellular automaton due to its simplicity and straightforwardness. This type of approach is used when a minimal-time solution is not required. In the design of any 3n-step approach the most crucial step is to find the center cell of the array to be synchronized. The basic mechanism for doing this is to use two signals moving at different speeds. The first signal is called *a-signal* and it moves at the speed of one cell per unit step. The second signal is called *b-signal* and it moves at a speed of one cell per three unit steps. When the *a-signal* finds the end of the array it returns to meet the *b-signal* at the half of the array, but the reflected *a-signal* is called *r-signal*. If the length of the array is odd, the cell C$_{\lceil n/2 \rceil}$ becomes the general, if the length is even two generals are created and each general is responsible for synchronizing it is left and right halves of the cellular space. Recursion is an important part of this algorithm. After finding the center cell, the process has to be started over with the new general, or generals if it is an even array. And the simplest way to start over is to apply recursion on the left and right side of the now divided array. After many steps of recursion, which depends on the size of the array, the problem is reduced to a problem of size two which is the last step before firing. Overview of results ------------------- In the remainder of the paper we describe and categorize results for the FSP, with a focus on results published between 1998 and 2015. To provide an overview of the work we summarize the table below gives information about the results discussed. [|&gt;p[2.0cm]{}|c|&gt;p[2.0cm]{}|c|c|&gt;p[1.8cm]{}|c|c|]{} & & & & & & &\ Settle [@settle1999new] & 1999 & Szwerinski & 9 & - & $2n-1$ & arbitrary & symmetric\ Settle [@settle1999new] & 1999 & Mazoyer & 6 & 134 & $3n+1$ & left or right & -\ Settle [@settle1999new] & 1999 & Mazoyer + $3n$-step & 7 & 127 & $2n-2+k$ & arbitrary & -\ Noguchi [@Noguchi04] & 2004 & Balzer & 8 & 119 & $2n-2$ & left & -\ Umeo, Maeda and Hongyo [@umeo2006design] & 2006 & $3n$-step & 6 & 115 & max(k, n − k + 1) + 2n+ O(log n) & arbitrary & symmetric\ Umeo and Yanagihara [@umeo2009small] & 2009 & $3n$-step & 5 & 71 & $3n-3$ & left or right & $n=k^2$\ Umeo, Yunes and Kamikawa [@umeo20084] & 2008 & Wolfram’s rule 60 & 4 & - & $2n - 2$ & left & $N=2^n$\ Umeo, Yunès and Kamikawa [@umeo20084] & 2008 & Wolfram’s rule 150 & 4 & - & $2n-2$ & left & $N=2^n+1$\ Yunès [@yunes20084] & 2008 & Wolfram’s rule 60 & 4 & 32\* & $2^{n+1}-1$ & left & $N=2^n$\ Umeo, Yunès and Kamikawa [@umeo2009family] & 2009 & Computer-based investigation & 4 & - & $2n - 2$ & left & $n=k^2$, symmetric\ Umeo, Kamikawa and Nishioka [@umeo2010generalized] & 2010 & Computer-based investigation & 8 & 222 & $2n - 2$ & arbitrary & asymmetric\ In the table we list the author for the publication from which the results are drawn, the year in which the publication appears, the type of algorithm employed (described either by citing the author who first conceived it or the general approach taken), the number of states used in the solution, the number of rules (e.g. number of transitions) required by the solution, the time required to synchronize an array of $n$ automata, where the initiator is located when the synchronization is begun, and any important notes about the solution. Minimal-time solutions ---------------------- Recall that a minimal-time solution is one that takes $2n-2$ steps to synchronize an array of $n$ automata. Each section below summarizes a paper by a set of authors providing a minimal-time solution to the FSP. ### Settle and Simon: 2002 Settle and Simon [@settle2002smaller] provide a range of different solutions to the firing synchronization problem. The first solution is a minimal-time solution to the generalized problem, where the initiator can be located in any cell in the array. The 9-state minimal-time to the generalized problem is an improvement from the 10-states solution created by Szwerinski[@szwerinski1982time]. One common strategy to deal with problems that use the generalized problem is first make the generalized problem look like the original problem, and then solve the original problem. Szwerinski’s solution needed two states to transform the generalized problem into the original problem, and Settle and Simon and improved that to use only one more state. The second contribution of the paper is a 6-states non-minimal-time solution to the original problem, where the initiator must be on either the left or right side of the array. The 6-state automaton is based on Mazoyer’s 6-state solution, but the version by Settle and Simon the initiator may be on either the left and right side of the array. Mazoyer’s solution works by dividing the array in unequal parts of $2/3n$ and $1/3n$ to create subproblems that will also be divided recursively, and Settle and Simon’s solution works the same way. The third solution in the paper is a 7-state non-minimal-time solution to the generalized problem. The solution also based on the optimum time 6-state solution from Mazoyer. One state is added to the Mazoyer solution to transform it into a solution to the generalized problem. To do this Settle and Simon used the 3n-Step algorithm to allow the initiator to be put in any place of the array. The new state is used to find the middle of the array, from which point the work continues with the Mazoyer solution. ### Noguchi: 2004 The focus of Noguchi’s work is to provide a more straightforward solution with fewer rules to the 8-state minimal time problem on rings[@Noguchi04]. His 8-state solution uses a strategy inspired by Waksman[@waksman1966optimum] and Balzer[@Balzer1967] where waves are used to gather, store, and pass information about the system, typically by halving the array and placing initiators at the midpoints. This is not a solution to the generalized problem, so the general must be at the beginning of the line. The strategy used in the solution has a main wave that travels at full speed, and a reverse wave that goes back also at maximum speed. These waves are used to detect the middle point, quarter point, and so on of the line. Doing that reduces the problem to sub-problems of the original problem, and every sub-problem can also be reduced recursively. For every sub-problem, a new general and consequently a new main wave and reverse wave are created. The algorithm works sending multiple waves in different speeds. The primary wave travels a one cell per time, and after the primary wave, multiple middle waves are sent. This multiple waves will be responsible for the creation of the future middle points. The n-middle waves that are sent after the primary wave moves one machine in the time that the primary wave moves $2^n$ machines. This will ensure that when the primary wave reflects it will find the middle waves in the respective middle spots of the array, $1/2$, $1/4$, $1/16$, and so on. The primary wave also is responsible for sending backward waves. These backward waves are responsible for passing-through the middle waves. This contact serves as instruction for the middle waves to start the process again. ### Umeo, Yunès, and Kamikawa: 2008 In the work by Umeo, Yunès and Kamikawa[@umeo20084], some elements of a new family of time-optimal solutions to a less restrictive firing squad synchronization problem are presented. The solutions are based on elementary algebraic cellular automata. The authors present some 4-state solutions which synchronize every line whose length is $2^n$ or $2^n + 1$ based on Wolfram’s rules 60 and 150 described in Section \[sec:wolfram\]. Umeo, Yunès and Kamikawa were able to construct different 4-state solutions to the FSP which synchronizes every line of length $2^n$ or $2^n + 1$. Three of these solutions use only 33 transitions and one uses only 30 transitions. Rule 60 when run on a configuration of length a power of two where the left end cell is 1 (black) led to something that looked like a synchronization, although it should be noted that rule 60 is not a solution to the problem. So to create something able to synchronize every power of two the authors use a simple folding of the space-time diagram of rule 60 running on a line of length $2^{n+1}$, resulting in a space-time diagram of a 4-state automata running on a line of length $2^n$. Using this modification they could obtain the synchronization at the time $2n$ for a line of length $n$. Using the same concept, they also present a solution that synchronizes every line of length $N = 2^n + 1$ at time $2N - 1$ ($2N - 2$ steps), so that it is a strict optimum-time solution. They also noted that the number of non quiescent cells is not in the order of $n log(n)$ for a line of length $n$, neither it is in the order of $n^2$, but something in between. ### Umeo, Kamikawa and Yunès: 2009 In this paper Umeo, Kamikawa and Yunès[@umeo2009family] provide a partial solution to the FSP by presenting a family of 4-state solutions to synchronize any one-dimensional ring of length $n = 2^k$ where $k$ represents any positive integer. The authors consider only symmetrical 4-state solutions for the ring. In their approach they use a computer to search the transition rule set in order to find a FSP solution. They did this by generating a symmetrical 4-state transition table and computing the configuration of each transition table for a ring of length $n$. They assume that $Q$ is a quiescent state, $A$ and $G$ are auxiliary states and $F$ is the firing state. Their program starts from an initial configuration: $G\overbrace{Q,...Q}^{n-1}$ where $2 \le n \le 32$ and checks if each transition table yields a synchronized configuration: $\overbrace{FF,...F}^n$ during the time $t$ where $n \le t \le 4n$ and the state $F$ never appears before that. By doing this they found that there were 6412 successful synchronized configurations. Most of them included redundant entries, so they removed the redundancies and compared and classified the valid solutions into small groups. After that process they obtained seventeen solutions: four optimum-time solutions, four nearly-optimum time solutions, and nine non-optimum time solutions. All of these seventeen solutions can synchronize rings of length $n = 2^k$ for any positive integer $k$. They also converted the solutions into solutions for an open ring, that is, a conventional one-dimensional array. They found that the converted 4-state protocol can synchronize any one-dimensional array of length $n = 2^k + 1$ with the left-end general both in state $G$ and $A$ in optimum $2n -2$ steps, for any positive integer $k \ge 1$. ### Umeo, Kamikawa, Nishioka and Akiguchi: 2010 In this paper the authors Umeo, Kamikawa, Nishioka and Akiguchi [@umeo2010generalized] present a computer study on different solutions to the generalized firing synchronization problem (GFSP). Recall that the GFSP may be described as the original problem with the general on the far left or right of the array at time $t = -(k - 1)$, where $k$ is the number of cells between the general and the nearest end. This study reveals inaccuracies in previous solutions, such as redundant rules and unsuccessful synchronizations. The paper also introduces a new eight-state solution to the GFSP, which surpasses the previous best solution that had nine states. Their use of a computer-assisted approach helped the transition table of their new solution to not have flaws in redundancy or unsuccessful synchronizations. A six-state solution is also examined in the study. The study presented in this paper takes into account the state transition tables for each solution of the GFSP being analyzed. The first transition table studied is Moore and Langdon’s 17-state optimum-time solution [@MooreLangdon1968]. This solution had problems synchronizing a relatively large number of arrays with several positions of the initial general. The second transition table studied is Varshavsky, Marakhovsky and Peschansky’s 10-state optimum time solution [@VMP1970]. This solution also has unsuccessful synchronizations. The third table studied is Szwerinski’s optimum-time ten-state solution [@szwerinski1982time]. This solution had no errors in the table, so it didn’t present any unsuccessful synchronization; however, it had 21 redundant rules. The fourth table studied is Settle and Simon’s 9-state optimum time solution [@settle2002smaller]. This table had no errors, but it presented 16 redundant rules. The 8-state optimum-time solution is then presented. This solution has 222 rules none of them being redundant. The last table studied is Umeo, Maeda and Hongyo’s 6-state non-optimum-time solution [@umeo2006design]. This solution has 115 states none of them being redundant and is considered a 3n-step solution. Non-minimal-time solutions -------------------------- Recall that a non-minimal-time solution is one that takes more than $2n-2$ steps to synchronize an array of $n$ automata. Each section below summarizes a paper by a set of authors providing a non-minimal-time solution to the FSP. ### Umeo, Maeda and Hongyo: 2006 Umeo, Maeda and Hongyo [@umeo2006design] give a new 3n-step algorithm that improves the lower bound of the generalized firing synchronization to a 6-state symmetric solution. The paper has two parts. In the first part it provides a 6-state solution to the original problem with the initiator on the left side. In the second part it gives a 6-state solution to the generalized problem, where the initiator can be placed anywhere on the array. The main difference between the two solutions is that on the second one, more rules are used to transform the solution to the original problem on the solution to the generalized problem. This solution like other that uses the 3n - step algorithm starts with the propagation of the a and b signals, on this solution the propagation of the a- signal is represented by the state P. While P is going away on the right direction at speed of one cell per time, it takes a a state R and M alternatively at each step until either the b-signal or the r-signal arrives at the cell itself. The b-signal is represented by the propagation of a 1/3 speed signal where the cells take a state R, R and Z for each three steps. And finally, the R signal is represented as a 1/1 speed signal of the Z state. One of the key ideas used on this paper to improve the 3n-step algorithm is based on the use of the quiescent cells of the zone T. The zone T is the triangle area circled by a-, b-, and r-signals in the time-space diagram. In the implementations of Fischer and Yunès, all cells in zone T keep quiescent state and are always inactive during the computations. These authors use a strategy that depends on making all cells inside the zone T active. Using the quiescent cells from this zone the authors successfully reduced the number of states by reusing the Q states to help the r- signal (Z state) find the center of the array. Finally, note that the result presented in this paper is an improvement of Yunè’s 7-state solution. Further, the 6-state solution is the smallest one known at present in the class of non-trivial 3n-step synchronization algorithms. The authors also achieved a increase in the number of working cells from $O(n log n)$ to $O(n2)$, and with that a state-efficient synchronization algorithm. ### Umeo and Yanagihara: 2007 Here Umeo and Yanagihara [@umeo2007smallest] provide a partial 5-state solution to the firing synchronization problem. Using a 3n-step algorithm, this solution can synchronize any one-dimensional cellular array of length $n=2^k$ in 3n - 3 steps, for any positive integer k and with the initiator positioned on the right side. The first solution using this algorithm was from Fischer[@Fischer:1965:GPO:321281.321290] who gave a 15-state implementation. This 5-state solution is a small but partial solution to the original problem, since it has some limitations regarding the length of the array n. Other authors like Settle and Simon[@settle2002smaller] and Umeo[@umeo2006state] provided complete solutions to the generalized problem using the same kind of algorithm but with a higher number of states. Settle and Simon provided a solution using only 7 states[@settle2002smaller], and Umeo did using 6 states[@umeo2006state]. Unlike this one in both solutions the initiator can be placed in any part of the array. The first two states of the solution are used to create the ripple drivers that enable the propagation of the b- signal at 1/3 speed (state S). The a- signal is also realized using the first two states (state R). Every two steps the a- signal generates a 1/1 speed signal in state Q, this signal is transmitted in the reverse direction. Using the reverse State Q, a third state S is added that will be responsible for the b- signal. Each ripple driver can be used to drive the propagation of the b-signal to its right neighbor. The three-step separation of two consecutive ripples enables the b-signal to propagate at 1/3 speed. Finally, state L is used for a reflected R-signal. The return signal propagates left at 1/1 speed. Any cell where the return signal passes remains in a quiescent state. At time t=3n/2, the b-signal and the return signal meet at $Cn/2$ and $Cn/2+1$. At the next step, the cells $Cn/2$ and $Cn/2+1$ take a state L and R, and these states act like generals for the left and right half of the array, and the process starts over again recursively. ### Yunès: 2008 {#ssec:num1} In this paper Yunès[@yunes20084] presents a solution to the FSP based on Wolfram’s rule 60 discussed in Section \[sec:wolfram\]. He accomplishes his result using an algebraic approach instead of geometrical constructions. This solution solves the problem on an infinite number of lines but not all possible lines. Its state complexity is the lowest possible (4 states and 32 transitions). As mentioned before, running the rule 60 on a configuration where the left end cell is 1 (black) leads to something that looks like synchronization of lines of length which are powers of two, but it is not a solution to the problem. Yunès points that whatever the solution, if it synchronizes an infinite number of lines then for some $N$ the synchronization can only appear at time $2n -n$ for a line of length $n > N$. Then he designed the algorithm so that rule 60 ran sending a signal wave 1 from the left most cell. When the leading 1 reaches the right end cell, another symmetric rule 60 is launched. By doing that, a property of the global function is exploited so that the full interleaving of two basic configurations is reached and the synchronization appears naturally. This leads to the paper’s main theorem: There exists a 4-state solution to the firing squad synchronization problem which synchronizes all lines of length a power of 2 in $2^{n+1}-1$ steps. Furthermore the number of non-quiescent cells in the space-time diagram of a line of length $m = 2^n$ is $3 \cdot m^{\log{(3)}}$. They show that this theorem is optimal by proving that it is impossible to synchronize any line of length $n \ge 5$ with only 3 states. Using only 32 program instructions makes this solution very simple. Previous results most often used geometrical constructions, and this was the first time an algebraic approach produced a solution for the FSP. ### Umeo and Yanagihara: 2009 Umeo and Yanagihara present in this paper a partial solution to the firing synchronization problem with 5 states[@umeo2009small]. This solution only works with arrays of length $n = 2^k$, and it takes $3n - 3$ steps to synchronize, where $k$ is a positive integer. This solution uses a $3n$-step algorithm, which was explained in detail in Section \[ssec:3nstep\]. The advantage of the use of such algorithm is its simplicity, which makes the approach easily understandable. The internal set of 5 states for this solution is represented by $\mathcal{Q} = \{Q,L,R,S,F\}$. First the authors used a 2-state implementation for the wave, which implements the a-signal and enables the future propagation of the b-signal. The 2-states are $\{Q,R\}$. The states $\{Q,R,S\}$ implement the a- and b-signals which were explained in the subsection \[ssec:3nstep\]. The state $L$ was used to implement the search for the center cells. After all the center cells, for sections and subsections of the cellular space, were found they could achieve the firing state $F$. The solution described in this paper is proposed as not the smallest solution to the problem but interesting in its own way. The authors achieved a 5-state partial solution, since this solution only works for a certain set of arrays. Because of that we can consider it a small partial solution for the FSP. However, it is not the smallest, since a 4-state partial solution, presented in Section \[ssec:num1\] also exists. Surveys and generalized problems -------------------------------- In this section we summarize a survey article on the FSP and discuss an article that considers a solution for the FSP on a two-dimensional array. The two dimensional array is a cellular automaton composed of an array of m × n cells. The state of any cell is not only influenced by the state of the cell on the both sides, but also the cells at north and south. Several synchronization algorithms on two-dimensional arrays have been proposed, including Grasselli [@grasselli1975synchronization], Kobayashi [@kobayashi1977firing], Shinahr [@shinahr1974two] and Szwerinski [@szwerinski1982time]. ### Umeo: 2012 Umeo wrote a survey on solutions to the FSP that use a small number of states[@umeo2012recent]. The solutions discussed cover the FSP for one-dimensional arrays, two-dimensional arrays, multi-dimensional arrays and the generalized FSP (GFSP). The first optimum-time solution to the FSP, developed by Goto in 1962 [@Goto1962] had several thousands of states. After that Waksman in 1966 [@waksman1966optimum] presented a 16-state optimum-time solution. After that there was a 8-state solution by Balzer in 1967 [@Balzer1967], a 7-state solution by Gerken in 1987 [@gerken1987synchronisations] and finally a 6-state solution by Mazoyer in 1987 [@Mazoyer1987]. The GFSP has also been extensively studied, and the first optimum-time solution with a small number of states used 17. After that Varshavsky, Marakhovsky and Peschansky in 1970 [@VMP1970], Szwerinski in 1982 [@szwerinski1982time], Settle and Simon in 2002 [@settle2002smaller], Umeo, Hisaoka, Michisaka, Nishioka, and Maeda in 2002 [@umeo2003synchronization] presented solution with 10 states and 9 states. There is also a non-optimum-step GFSP solution by Umeo, Maeda, and Hongyo in 2006 [@umeo2006design] that works with 6 states. The paper presents theorems that point that one-dimensional arrays need at least $2n - 2$ steps to synchronize with the general in one of the ends. It also presents a theorem that says that there is no four-state full solution that can synchronize n cells, only four-state partial solution. Maeda et al: 2002 ----------------- The contribution of this paper [@maeda2002efficient] is a simple but efficient mapping scheme that enables the embedding any one-dimensional firing squad synchronization algorithm onto two dimensional arrays without introducing additional states. The paper gives a concise and small solution to this problem, where the rules from the 1D array can be easily converted to the 2D array. In the solution, the authors produce a series of conversion tables and rules to enable the embedding. First they split the m x n cells into groups g. This groups were formed by cells that were on the same line when the m x n board is turned 45 degrees. Because now we have two array lines, we need a state w to be the right and left end state. As there are more connections between the cells, to convert a cell from one dimension to two dimensions requires more rules. The authors provide a formula for the number of rules that a transformation (a, b, c) $\rightarrow$ d uses depending on the location on the grid. Open problems ============= Although the firing synchronization problem has been studied for decades, there are still several open problems. We discuss open problems for the original problem on the one-dimensional array, for the generalized problem on the one-dimensional array where the initiator can be located in any cell, and for the ring. Original problem ---------------- The most significant open problem in the area is whether there exists a complete 5-state solution to the FSP on a one-dimensional array. The smallest minimal-time solution so far for the original problem was created by Mazoyer which is a 6-state solution[@Mazoyer1987]. A 5-state solution would then optimal for the problem, since Balzer showed that there is no 4-state minimal-time full solution to the original problem[@Balzer1967], Which later was confirmed by Sanders [@Sanders1994]. Some partial 5-state solutions exist. Umeo gives a non-minimal-time partial solution using 5 states[@umeo2007smallest]. Yunès has produced solutions based on Wolfram’s rules, finding that there exists a 4-state solution to the FSP which synchronizes all lines of length a power of two[@yunes20084]. He also proved that there is no 3-state solutions able to synchronize a line of length $n \ge 5$. Various 4-state partial solutions as described by Yunès can be found in Table \[ovvtable\]. Since all the five-state solutions are partial solutions to the original problem, finding a complete solution would be a significant contribution to the area. It is still unknown whether it is easier to take an approach to the problem finding a minimal-time solution or a non-minimal-time solution. Evidence suggests that finding a non-minimal-time solution may lead to solutions with fewer states, as seen in Table \[ovvtable\]. The generalized problem ----------------------- Before 1998 the best solution for the minimal-time generalized problem was Mazoyer’s 10-state solution[@Mazoyer1987]. Since then his work has been improved in 1999 by Settle to a 9-state solution[@settle1999new]. In 2010 Umeo, Kamikawa and Nishioka [@umeo2010generalized] presented a 8-state minimal-time solution which is currently the smallest minimal-time solution to the generalized problem. Work has also been done on finding non-minimal-time solutions to the generalized problem. In 1999 Settle et al presented a 7-state non-minimal-time full solution to the generalized problem [@settle1999new], which improves the previous minimal-time solution presented by the same authors by two states. In 2006 Umeo, Maeda and Hongyo presented a 6-state non-minimal-time solution[@umeo2006design]. This evidence suggests that non-minimal-time solutions may require fewer states than minimal-time solutions, although there is no proof of that claim. Smaller solutions to the generalized problem are still open problems. Considering that the smallest solution has 6-states and that this number also applies to the original problem, both versions of the problem still do not have a 5-state solution. The 6-state solution to the generalized problem, unlike the 6-state solution to the original problem, is not a minimal-time solution. Because of that, finding a 6-state minimal-time solution to the generalized problem is also an open problem. The ring -------- There are some results on state lower bounds for the FSP on the ring. Berthiaume, Bittner, Perkovic, Settle and Simon showed that there is no 3-state full solution and no 4-state, symmetric, minimal-time full solution to the FSP for the ring[@berthiaume2004bounding]. Umeo, Kamikawa and Yunès proved that there is no 3-state partial solution to the firing synchronization problem for the ring [@umeo2009family]. Thus it is open whether or not there is an unrestricted 4-state minimal-time full solution to the FSP on the ring. Conclusion ========== Here we have summarized recent research on the firing synchronization problem, focusing primarily on results published between 1998 and 2015. We discussed results for the original problem on the one-dimensional array, the generalized problem where the initiator can be located in any cell of the array, the problem where the underlying network is a ring, and a paper which discusses how to modify one-dimensional solutions for multidimensional arrays. We also discuss the remaining open problems for the original, generalized, and ring problems.

--- abstract: 'Long-term evolution of a stellar orbit captured by a massive galactic center via successive interactions with an accretion disc has been examined. An analytical solution describing evolution of the stellar orbital parameters during the initial stage of the capture was found. Our results are applicable to thin Keplerian discs with an arbitrary radial distribution of density and rather general prescription for the star-disc interaction. Temporal evolution is given in the form of quadrature which can be carried out numerically.' author: - | D. Vokrouhlický$^{\,1,2\,}$[^1] and V. Karas$^{\,1,3\,\mbox{$\star$}}$\ $^{1}$Astronomical Institute, Charles University Prague, Švédská 8, CZ-15000 Praha, Czech Republic\ $^2$Observatoire de la Côte d’Azur, dept. CERGA, Av. N. Copernic, F-06130 Grasse, France\ $^3$Scuola Internazionale Superiore di Studi Avanzati, Via Beirut 2/4, I-34014 Trieste, Italy date: Received title: Stellar capture by an accretion disc --- = =msbm10 =msbm7 =msbm5 === \[firstpage\] accretion: accretion discs – celestial mechanics, stellar dynamics – stars: kinematics – galaxies: nuclei Introduction ============ Evolution of the orbit of a star under the influence of interactions with an accretion disc has been studied by numerous authors because this situation is relevant to inner regions of active galactic nuclei. The trajectory of an individual star is determined mainly by gravity of the central mass and surrounding stars while periodic transitions through the disc act as a tiny perturbation. The final goal is to understand the fate of a star, transfer of mass and angular momentum between the star and the disc, and also to determine how star-disc interactions influence the distribution of stellar orbits near a massive central object. An important and difficult task is to estimate the probability that a star gets captured from an originally unbound orbit, and to determine whether this probability is significant compared to other mechanisms of capture. Orbital evolution of a body crossing an accretion disc has been discussed with various approaches, first within the framework of Newtonian gravity, both in theory of the solar system (Pollack, Burns & Tauber 1979; McKinnon & Leith 1995) and for active galactic nuclei (Goldreich & Tremaine 1980; Syer, Clarke & Rees 1991; Artymowicz 1994; Podsiadlowski & Rees 1994). These studies have been generalized in order to account for the effects of general relativity [@VK93] and to model a dense star cluster in a galactic nucleus [@PL94; @R95]. It has been recognized that detailed physical description of the star-disc interaction is a difficult task (Zurek, Siemiginowska & Colgate 1994, 1996). In this letter a simplified analytical treatment of stellar orbital parameters is presented, describing the initial stage of star-disc collisions (when the star crosses the disc once per revolution). A great deal of our calculation is independent of microphysics of star-disc interaction. It is shown how our solution matches the corresponding Rauch’s solution which is valid in later stages, when eccentricity of the orbit becomes small enough. In the next section our approach to the problem is formulated and an analytical solution is given. Then, in Sec. \[details\], further details about the derivation of the results are presented, and finally a simple example of the orbital evolution is shown in Sec. \[example\]. Stellar capture by a disc ========================= Formulation and results {#formulation} ----------------------- Newtonian gravitational law is assumed throughout this paper. Our solution is based on the following assumptions: (i) : the disc is geometrically thin and its rotation is Keplerian; (ii) : at the event of crossing the plane of the disc, velocity of the star is changed by a tiny quantity. This impulse is colinear with the relative velocity of the star with respect to the material forming the disc; (iii) : the star crosses the disc once per revolution (the model is applicable to the initial phase of the stellar capture). The first item is a standard simplification in which the disc is treated in terms of vertically integrated quantities, while the second one can be expressed by the formula for an impulsive change of the star’s velocity: $$\delta\bmath{v}=\Sigma(\bmath{r},\bmath{v})\,\bmath{v}_{\rm rel}\,; \label{2one}$$ $\Sigma$ is an unconstrained function, given by a detailed model of the star-disc interaction, and $\bmath{v}_{\rm rel}$ is relative velocity of the star and the disc material. We stress, that our results are uniquely based on this assumption of colinearity, $\delta\bmath{v}\propto\bmath{v}_{\rm rel}$; the coupling factor $\Sigma$ is arbitrary and it can be as complex as necessary. In particular, $\Sigma$ contains information about the surface density $\akpa$ of the disc ($\akpa=0$ outside an outer edge $r=R_{\rm{d}}$ of the disc). The form of $\Sigma$ must be specified only for examination of temporal evolution of orbital parameters. We will assume, in analogy with Rauch (1995), a simplified formula for $$\begin{aligned} \Sigma(\bmath{r},\bmath{v}) &=& -{\pi R_\star^2\over m_\star}\, \akpa(r)\,{v_{\rm rel}\over v_\perp}\,\xi\,, \label{22two} \\ \xi &\approx& 1+\left({v_\star\over v}\right)^4\ln\Lambda \,,\label{sigma}\end{aligned}$$ when it is needed for purpose of an example. In eq. (\[22two\]) $R_\star$ denotes radius of the star, $m_\star$ is its mass, $v_\star$ escape velocity ($v^2_\star=2Gm_\star/R_\star$); $v_\perp$ is normal component of the star’s velocity to the disc plane, and $\ln\Lambda$ is a usual long-range interaction factor (Coulomb logarithm). The star’s orbit is traditionally characterized by the Keplerian osculating elements: semimajor axis $a$, eccentricity $e$, inclination $I$ to the accretion disc plane, and longitude of pericenter $\omega$ (measured from the ascending node). A derived set of parameters turns out to be better suited to our problem: $\alpha=1/a$, $\eta=\sqrt{1-e^2}$, $\mu=\cos{I}$, and $k=e\cos\omega$. We will show (Sec. \[details\]) that evolution of a stellar orbit following the capture by a disc can be written in a parametrical form: $$\begin{aligned} \alpha(\zeta) &=& \phi(\zeta)\left\{\alpha_0 \phi^{-1}(\zeta_0) + \sigma^2\left[\psi(\zeta)-\psi(\zeta_0)\right]\right\}\,, \label{one} \\ \eta^2(\zeta) &=& \zeta\phi(\zeta)\left\{\alpha_0 \phi^{-1}(\zeta_0) \sigma^{-2} + \left[\psi(\zeta)-\psi(\zeta_0)\right]\right\}\,, \label{two}\\ \mu(\zeta) &=& \sqrt{\zeta} + \theta(\zeta)\,, \label{three}\\ \mid\!k(\zeta)\!\mid{} &=& \zeta -1\,, \label{four}\end{aligned}$$ where the auxiliary functions $\phi(\zeta)$, $\theta(\zeta)$ and $\psi(\zeta)$ read $$\begin{aligned} \phi(\zeta) &\!=\!& {1\over C} \left(1\pm \sqrt{1-C+C\zeta} \right)^2\,, \label{five}\\ \theta(\zeta) &\!=\!& \mp{1\over C} \sqrt{{1-C+C\zeta\over \zeta}} \left(1\pm \sqrt{1-C+C\zeta}\right)\,, \label{six}\\ \psi(\zeta) &\!=\!& {1\over 1\pm\sqrt{1-C+C\zeta}}\left(2+{C\over 1\pm \sqrt{1-C+C\zeta}}\right). \label{seven}\end{aligned}$$ Formal parameter $\zeta$ of the solution decreases from its initial value $\zeta_0=1+e_0\mid\!\cos\omega_0\!\mid$ to the final value $\zeta_{\rm f}$, given by $$\zeta_{\rm f} = {2 R_{\rm{d}}\sigma^2\over 1+R_{\rm{d}}\sigma^2} \,. \label{fourteen}$$ At this instant, the orbit starts crossing the disc twice per revolution and our solution ceases to be applicable. Obviously, integration constants in (\[one\])–(\[seven\]) are determined in terms of the initial Keplerian orbital elements $(a_0,e_0,I_0,\omega_0)$ by $$\begin{aligned} \alpha_0 &=& {1\over a_0}\,, \label{eight}\\ \zeta_0 &=& 1 + e_0\mid\!\cos\omega_0\!\mid{}\,, \label{nine}\\ \sigma^{-2} &=& {a_0\eta_0^2\over \zeta_0}\,, \label{ten}\\ C &=& {z_0 (z_0 +2)\over (z_0+1)^2} {1\over 1-\zeta_0}\,, \label{eleven}\end{aligned}$$ where $$z_0 = -{1-\zeta_0\over \sqrt{\zeta_0}(\cos I_0 - \sqrt{\zeta_0})}\,.\label{twelve}$$ Upper signs in (\[five\])–(\[seven\]) are for the initial inclination $I_0$ greater than a critical value $I_\star$ given by $$\cos I_\star={1\over \sqrt{\zeta_0}}\,,\label{thirteen}$$ lower signs apply otherwise. Integration constant $C$ is singular ($C \rightarrow \infty$) for $I_0=I_\star$ ($z_0=-1$), and the solution can be simplified further. For instance, $\mu(\zeta)=1/\sqrt{\zeta}$ for all values of $\zeta$ down to $\zeta_{\rm f}$. Solution (\[one\])–(\[four\]) can be extended easily to the case of initially parabolic orbits by setting $\alpha_0=0$, $e_0=1$ and $\sigma^2=(\zeta_0/2 R_{\rm p})$. Here, $R_{\rm p}$ denotes pericenter distance of the initial parabolic orbit. It is worth mentioning that the parameter $\zeta$ does not determine the time-scale on which the evolution takes place. Indeed, eqs. (\[one\])–(\[four\]) do not provide temporal information because it depends on the precise form of $\Sigma$ in eq. (\[2one\]). On the other hand, the strength and the beauty of the parametric solution (\[one\])–(\[four\]) is in its independence on a particular model for $\Sigma$. We will also illustrate an example of temporal evolution later in the text, and only for this purpose the form of $\Sigma$ will be needed. Assuming relation (\[22two\])–(\[sigma\]) one obtains $$t-t_0=\frac{\pi}{\sigma^3\sqrt{GM}\Sigma_{\rm{c}}}\int_\zeta^{\zeta_0} \frac{\sigma^3{{\mbox{d}}}{z}}{z^{1/2}\alpha^{3/2}(z)\nu(z)\theta(z)} \label{time}$$ where $t_0$ is initial time, $M$ is the central mass, and $\Sigma_{\rm{c}}=(\pi{}R_\star^2/m_\star)\akpa(r_{\rm{c}})\xi$ with $r_{\rm{c}}=\sigma^{-2}$ being radial distance of the point of intersection with the disc. Function $\nu=v_{\rm{rel}}/v_\perp$ is determined by orbital parameters which themselves depend on $\zeta$ according to eqs. (\[one\])–(\[four\]). We note that Rauch conjectured that function $R=a\eta^2\cos^4(I/2)$ remains nearly conserved along the evolving stellar orbit and he used this function for estimates of the radius of the final, circularized orbit in the disc plane. In our notation, $$R(\zeta)={\zeta\over 4\sigma^2}\left[1+\sqrt{\zeta}+\theta \left(\zeta\right)\right]^2. \label{fifteen}$$ Hereinafter, we show that $R(\zeta)$ is a well-conserved quantity at later stages of the orbital evolution (when eccentricity is sufficiently small), but it fails to be conserved at the very beginning of the capture when the orbit is still nearly parabolic, close to an unbound trajectory. = Details of the solution {#details} ----------------------- In this section, we present more details of the derivation of the solution given above. Taking into account the fundamental assumption (\[2one\]), one can easily demonstrate that the Keplerian orbital elements are perturbed at each transition (due to interaction with the disc material) by quantities $$\begin{aligned} \delta (\sqrt{a}) &=& \Sigma\, \sqrt{a}\eta\, f_1 \,, \label{2two}\\ \delta (\sqrt{a}\eta) &=& \Sigma\, \sqrt{a}\eta\, f_2 \,, \label{2three}\\ \delta (\sqrt{a}\eta\mu) &=& \Sigma\, \sqrt{a}\eta\, f_3 \,, \label{2four}\\ \delta (k) &=& 2\Sigma\, (1+k)\, f_2\,. \label{2five}\end{aligned}$$ Here, we introduced auxiliary functions $$\begin{aligned} f_1 &=& \eta^{-3}\left\{\left(1+k\right)\left[1-\mu\sqrt{1+k} \right] + e^2 + k\right\}\,, \label{2six} \\ f_2 &=& 1-{\mu\over \sqrt{1+k}}\,, \label{2seven}\\ f_3 &=& \mu-{1\over \sqrt{1+k}}\,. \label{2eight}\end{aligned}$$ Combining eqs. (\[2three\]) and (\[2five\]) we find that $${\sqrt{1+k}\over \sqrt{a}\eta} = \sigma \label{2nine}$$ is conserved at the star-disc interaction. Hence, $\sigma$ is constant whatever the evolution of elements $a$, $e$ and $k$ is. In fact, condition (\[2nine\]) states that the initial Keplerian orbit has the same radius of intersection as the final orbit, after successive interactions with the disc. Longitude of the node is also conserved and can be set to zero without loss of generality. The above-given formulae (\[2two\])–(\[2nine\]) correspond to $k>0$ (i.e.$\mid\!\omega\!\mid{}<\pi/2$); for $k<0$ one should replace $k\rightarrow-k$. We note that all these relations can be easily reparametrized in terms of binding energy $E=GM/(2a)$, angular momentum $L=\sqrt{GMa}\eta$, and component of the angular momentum with respect to axis, $L_{\rm{z}}=\sqrt{GMa}\eta\mu$. Combining eqs. (\[2three\]) and (\[2four\]) with the help of (\[2nine\]), and introducing auxiliary variables $y=\sqrt{a}\eta\mu$ and $x=\sqrt{a}\eta$, we obtain differential equation $${{{\mbox{d}}}{y}\over{{\mbox{d}}}{x}}={x(\sigma{}y-1)\over\sigma{}x^2-y}\,. \label{2ten}$$ (We were allowed to change variations, $\delta$, to the differentials, ${{\mbox{d}}}$, assuming an infinitesimally small perturbation of the stellar orbit at each intersection with the disc.) The Abel-type differential equation (\[2ten\]) can be solved beautifully by standard methods of mathematical analysis (see, e.g., Kamke 1959). An appropriate change of variables gives directly a solution for the evolution of inclination, eq. (\[three\]). Similarly, considering (\[2two\]) and (\[2three\]) in terms of $\alpha=1/a$, we obtain, after brief algebraic transformations, $$-\sqrt{\zeta}\theta(\zeta)\, {{{\mbox{d}}}\alpha\over{{\mbox{d}}}\zeta} +\alpha (\zeta) = \sigma^2 \left[2-\zeta+\sqrt{\zeta}\theta\left(\zeta \right)\right] \label{2eleven}$$ with $\theta(\zeta)$ given by eq. (\[six\]). This is a linear differential equation, integration of which yields $\alpha(\zeta)$ and then, by eqs. (\[one\])–(\[two\]), also $\eta(\zeta)$. At this point, one can see that Rauch’s “quasi-integral” $R$ is changed at each passage across the disc according to $$\delta(\ln R)=2\Sigma\left(1-{1\over\sqrt{1+k}}\right)\,. \label{2twelve}$$ Realizing that $k\approx e$ we conclude that $\ln R$ indeed stays nearly constant at later stages of the orbit evolution, when eccentricity has decreased enough. On the other hand, at the very beginning of the capture process, when eccentricity is still high, $R$ fails to serve as a quasi-integral of the problem. Instead, its evolution is given by eq. (\[fifteen\]). For temporal evolution, eqs. (\[2two\])–(\[2nine\]) must be supplemented by additional relation, $$\delta(t)=\frac{2\pi}{\sqrt{GM}}\,a^{3/2}, \label{deltat}$$ which determines interval between successive intersections with the disc. Combining eq. (\[deltat\]) with (\[2nine\]) one obtains separated differential equation which yields formula (\[time\]). Recall that this last step requires assumption (\[22two\]) about the form of $\Sigma$. In the present case, $$\nu(\zeta)=\frac{1}{\zeta} \sqrt{\frac{2\zeta(1-\zeta-\sqrt{\zeta}\theta)+\zeta-\eta^2}{1- \zeta-2\sqrt{\zeta}\theta-\theta^2}}.$$ Relation for time is apparently too complicated to be integrated analytically but numerical evaluation is straightforward. = Example ------- We shall briefly demonstrate some properties of the analytical solution from Sec. \[formulation\]. We examine parabolic orbits ($\alpha_0=0$, $e_0=1$) with pericenter in the disc plane ($\omega_0=0$), and the pericenter distance $R_{\rm p}$ equal to the disc radius ($R_{\rm p}=R_{\rm d}=r_{\rm{c}}$). Initial inclination $I_0$ of the orbit to the disc plane is a free parameter in this example. Evolution of this set of orbits is split into two phases. First, we let the orbits evolve according to the solution of eqs. (\[one\])–(\[four\]) from the initial value $\zeta_0=2$ of the formal parameter $\zeta$ to its final value $\zeta_{\rm f}=1$. Figure \[fig1\] illustrates the evolution of the inclination $I(\zeta)$, measured in terms of the initial value $I_0$. Notice that the critical inclination $I_\star$ of eq. (\[thirteen\]) is $45\fdg$ Eccentricity of the orbits under consideration decreases according to a simple formula $e(\zeta)=\zeta-1$ (independently of $I_0$), leading eventually to circularized orbits at $\zeta=\zeta_{\rm f}$. We observe that orbits with $I_0<I_\star$ terminate at the final state $I_{\rm f}=0$, suggesting that the circularization time-scale is comparable to that necessary for grinding the orbit into the disc plane. On the other hand, when $I_0>I_\star$ the final circular orbits remain inclined significantly to the disc plane. ($I>90\degr$ corresponds to retrograde orbits.) Hence, for those orbits the circularization time-scale is shorter than the time necessary for tilting the orbit to the disc plane. Additional time to incline a circularized orbit is not much longer than circularization time, however.[^2] The difference is typically a factor of 10 for highly retrograde orbits. Secondly, we examine the evolution of circularized orbits which started with $I_0>I_\star$ and have settled to nonzero $I(\zeta=1)$. Because these orbits have zero eccentricity, there exists Rauch’s integral in the form $R_1=\sigma^2{a}\cos^4(I/2)=\zeta\cos^4(I/2)$. Here, we adopted a formal continuation of the $\zeta$ parameter to values smaller than unity (in this phase, $\zeta=a/R_{\rm d}$). For each orbit, we calculate the value of $R_1\equiv{}R(\zeta=1)$, so that the inclination is given by $$\mu(\zeta)=\sqrt{{4R_1\over \zeta}}-1\,. \label{3one}$$ Obviously, a given orbit terminates its evolution at $\zeta=4R_1$ when it is pushed completely into the disc plane. Dashed curves in Figure \[fig1\] correspond to constant values of $\zeta<1$. Figure \[fig2\] illustrates how function $R(\zeta)$ changes during the first circularization phase of the evolution. For each orbit we have chosen the same steps in $\zeta$ (0.2) in the range $1.8\geq\zeta\geq1.0$, as in Figure \[fig1\], and we computed the corresponding values of $R(\zeta)$ from eq. (\[fifteen\]). Our results agree with Rauch’s finding, namely that $R(\zeta)$ is conserved up to a factor of $\approx2$ for orbits with large eccentricity. During the second phase of the evolution the $R$-function is constant. Figure \[fig3\] shows time intervals $t_{\rm{e}}(\zeta)$ which elapse in the course of gradual circularization when eccentricity decreases from $e=\zeta-1$ to some terminal value (here, terminal eccentricity has been fixed to $e=10^{-3}$; notice that $t_{\rm{e}}$ goes to infinity for terminal eccentricity $e\rightarrow0$). We have verified the graph also by direct numerical integration of the corresponding orbits. Numerical factor standing in front of integral on the right-hand side of eq. (\[time\]) can be written in physical units in the form $$10^7\left(\frac{r_{\rm{c}}}{10^3\,R_{\rm{g}}}\right)^{9/4} \left(\frac{R_{\rm{g}}}{10^5R_\star}\right) \left(\frac{10^3R_{\rm{g}\star}}{R_\star}\right) \left(\frac{10^3}{\xi}\right)\,\mbox{yrs};$$ $R_{\rm{g}}=2GM/c^2$ and $R_{\rm{g}\star}=2Gm_\star/c^2$ are gravitational radii of the central mass and the star, respectively. A typical surface density profile of the disc has been assumed, as in eq. (1) of Rauch (1995). The value of $\xi\approx10^3$ corresponds to the estimate in addendum to Zurek, Siemiginowska & Colgate (1996). = Conclusion ========== We have found a solution describing the evolution of orbital parameters of a star orbiting around a massive central body in a galactic nucleus and interacting with a thin Keplerian disc. The solution is in a parametrical form valid for an [*arbitrary radial distribution of density and a very broad range of models of the star-disc interaction*]{}. Temporal evolution can be given in terms of quadrature provided the star-disc interaction is specified completely (in terms of function $\Sigma$). Our approach can be applied to other situations but the form of eq. (\[2ten\]) is linked to the assumption about interactions, eq. (\[2one\]). Also the situation when the orbit intersects the disc twice-per-revolution requires a specific form of $\Sigma$ to be given and, most likely, it does not allow a complete analytic solution. Our solution thus describes the initial phases of the stellar capture (large eccentricity) and it matches smoothly the low-eccentricity approximation. Apart from an interesting form of analytical expressions, our approach is useful as a part of more elaborate calculations. In an accompanying detailed paper, additional effects are taken into account (e.g., gravity of the disc) and distribution of a large number of stars is investigated (Vokrouhlický & Karas 1997, submitted to MNRAS). We thank the referee for comments concerning temporal evolution of the orbits and for other suggestions which helped us to improve our contribution. We acknowledge support from the grants GACR 205/97/1165 and GACR 202/96/0206 in the Czech Republic. Artymowicz P., 1994, ApJ, 423, 581 Goldreich P., Tremaine S., 1980, ApJ, 241, 425 Kamke E., 1959, Differentialgleichungen: Lösungsmethoden und Lösungen (Leipzig) McKinnon W. B., Leith A. C., 1995, Icarus, 118, 392 Pineault S., Landry S., 1994, MNRAS, 267, 557 Podsiadlowski P., Rees M. J., 1994, in: The Evolution of X-ray binaries, eds. S. S. Holt & C. S. Day, (AIP Press, New York), p. 71 Pollack J. B., Burns J. A., Tauber M. E., 1979, Icarus, 37, 587 Rauch K. P., 1995, MNRAS, 275, 628 Syer D., Clarke C. J., Rees M. J., 1991, MNRAS, 250, 505 Vokrouhlický D., Karas V., 1993, MNRAS, 265, 365 Zurek W. H., Siemiginowska A., Colgate S. A., 1994, ApJ, 434, 46; 1996, ibid, 470, 652 \[lastpage\] [^1]: Since November 1997: Astronomical Institute, Charles University Prague, V Holešovičkách 2, CZ-18000 Praha, Czech Republic. E-mail: vokrouhl@mbox.cesnet.cz; karas@mbox.cesnet.cz [^2]: We thank the referee for pointing out this fact, confirmed also by other estimates [@SCR91; @KL95].

--- abstract: 'The previous mean-field calculation \[Myaing Thi Win and K. Hagino, Phys. Rev. C[**78**]{}, 054311 (2008)\] has shown that the oblate deformation in $^{28,30,32}$Si disappears when a $\Lambda$ particle is added to these nuclei. We here investigate this phenomenon by taking into account the effects beyond the mean-field approximation. To this end, we employ the microscopic particle-rotor model based on the covariant density functional theory. We show that the deformation of $^{30}$Si does not completely disappear, even though it is somewhat reduced, after a $\Lambda$ particle is added if the beyond-mean-field effect is taken into account. We also discuss the impurity effect of $\Lambda$ particle on the electric quadrupole transition, and show that an addition of a $\Lambda$ particle leads to a reduction in the $B(E2)$ value, as a consequence of the reduction in the deformation parameter.' author: - 'H. Mei' - 'K. Hagino' - 'J.M. Yao' - 'T. Motoba' title: 'Disappearance of nuclear deformation in hypernuclei: a perspective from a beyond-mean-field study' --- Introduction ============ The nuclear deformation is one of the most important concepts in nuclear physics [@BM75; @RS80]. Whereas only those states with good angular momentum are realized in the laboratory, atomic nuclei can be deformed in the intrinsic frame, in which the rotational symmetry is spontaneously broken. This idea nicely explains the existence of rotational bands as well as enhanced electric transitions within the rotational bands in many nuclei. Theoretically, the nuclear deformation is intimately related to the mean-field approximation [@RS80; @BHR03], but there have also been recent attempts to describe the characteristics of deformed nuclei using symmetry preserved frameworks [@Dytrych13; @Maris15; @Stroberg16; @Jansen16; @GAB18]. In this paper, we shall discuss the nuclear deformation of single-$\Lambda$ hypernuclei [@Feshbach76; @Zofka80; @Zhou07; @MH08; @Schulze10; @Lu11; @Isaka11; @Myaing11; @Isaka12; @Isaka13; @Isaka14; @Lu14; @Cui15], where a $\Lambda$ particle is added to atomic nuclei. See Refs. [@HT06; @HY16; @GHM16] for reviews on hypernuclei. A characteristic feature of hypernuclei is that a $\Lambda$ particle does not suffer from the Pauli principle of nucleons, and thus its wave function can have a large probability at the center of hypernuclei. This may significantly affect the structure of atomic nuclei. In the history of hypernuclear studies, when the experimental data of strangeness-exchange $(K^-,\pi^-)$ reactions came out from CERN, Feshbach proposed the concept of “shape polarizabilty", that is, a possible change of nuclear radius and deformation induced by the hyperon participation [@Feshbach76]. Subsequently, Žofka carried out Hartee-Fock calculations for hypernuclei to analyze such effects on even-even nuclei with $Z=N$ and $A<40$ [@Zofka80]. He found that the relative change in quadrupole deformation should be maximum at $^9_{\Lambda}$Be and $^{29}_{~\Lambda}$Si in the $p$-shell and $sd$-shell, respectively, although the expected change was not so large (only of the order of 1-4% in $sd$-shell). See also Ref. [@Zhou07]. In modern light of nuclear structure studies, however, such response to the $\Lambda$ participation depends sensitively on the nuclear own properties such as softness and potential shape. As a matter of fact, based on the relativistic mean-field (RMF) theory, it was argued that the nuclear deformation may disappear in some nuclei, such as $^{12}$C and $^{28,30,32}$Si, when a $\Lambda$ particle is added to these nuclei [@MH08]. That is, those deformed nuclei turn to be spherical hypernuclei after a $\Lambda$ particle is put in them. See also Refs. [@Lu11; @Isaka11] for a similar conclusion. It has been shown that a softness of the potential energy surface in the deformation space is a primary cause of this phenomenon [@Schulze10]. In general, one expects a large fluctuation around the minimum when a potential surface is soft against deformation. This effect can actually be taken into account by going beyond the mean-field approximation with the generator coordinate method (GCM) [@RS80; @BHR03]. In addition, one can also apply the angular momentum and the particle number projections to a mean-field wave function, in which these symmetries are spontaneously broken. Such calculations have been performed recently not only for ordinary nuclei [@BH08; @Egido10; @Yao10; @Yao11; @Yao14; @Bally14; @Egido16] but also for hypernuclei [@MHY16; @Wu17; @Cui17]. We shall here apply the beyond-mean-field calculations to a typical soft hypernucleus, as the most appropriate theoretical treatment for the dynamical shape fluctuation. The aim of this paper is then to asses the effect beyond the mean-field approximation on the phenomenon of disappearance of nuclear deformation, which takes place in hypernuclei whose potential surface is soft. A similar work has been carried out with the anti-symmetrized molecular dynamics [@Isaka16]. Here, we instead employ the microscopic particle-rotor model based on the covariant density functional theory [@Mei14; @Xue15; @Mei15; @Mei16; @Mei17], in which the $\Lambda$ particle motion is coupled to the core wave functions described with the beyond-mean-field method. The paper is organized as follows. In Sec. II, we briefly summarize the microscopic particle-rotor model. In Sec. III, we apply this framework to the $^{31}_{~\Lambda}$Si hypernucleus, for which the disappearance of deformation has been found in the mean-field approximation, and discuss the impurity effect of $\Lambda$ particle on the structure of the soft nucleus, $^{30}$Si. We then summarize the paper in Sec. IV. Microsocpic Particle-Rotor Model ================================ We consider in this paper a single-$\Lambda$ hypernucleus. The Hamiltonian for this system reads, $$H=T_\Lambda+H_{\rm core}+\sum_{i=1}^{A_C}v_{N\Lambda}({\mbox{\boldmath $r$}}_\Lambda,{\mbox{\boldmath $r$}}_i),$$ where $T_\Lambda$ is the kinetic energy of the $\Lambda$ particle and $H_{\rm core}$ is the many-body Hamiltonian for the core nucleus, whose mass number is $A_C$. $v_{N\Lambda}({\mbox{\boldmath $r$}}_\Lambda,{\mbox{\boldmath $r$}}_i)$ is the nucleon-$\Lambda$ ($N\Lambda$) interaction, in which ${\mbox{\boldmath $r$}}_\Lambda$ and ${\mbox{\boldmath $r$}}_i$ denote the coordinates of the $\Lambda$ particle and of the nucleons, respectively. In the microscopic particle-rotor model, the total wave function for the system is described as $$\begin{aligned} \Psi_{JM_J}({\mbox{\boldmath $r$}}_\Lambda,\{{\mbox{\boldmath $r$}}_i\}) &=&\sum_{j,l}\sum_{n,I} {\cal R}_{jlnI}(r_\Lambda) \nonumber \\ &&\times [{\cal Y}_{jl}(\hat{{\mbox{\boldmath $r$}}}_\Lambda)\otimes\Phi_{nI}(\{{\mbox{\boldmath $r$}}_i\})]^{(JM_J)}, \label{wf}\end{aligned}$$ where $J$ is the angular momentum of the hypernucleus and $M_J$ is its $z$-component in the laboratory frame. ${\cal R}_{jlnI}(r_\Lambda)$ and ${\cal Y}_{jlm_j}(\hat{{\mbox{\boldmath $r$}}}_\Lambda)$ are the radial and the spin-angular wave functions for the $\Lambda$ particle, with $j$, $m_j$, and $l$ being the total single-particle momentum and its $z$-component, and the orbital angular momentum, respectively. In Eq. (\[wf\]), $\Phi_{nIM}(\{{\mbox{\boldmath $r$}}_i\})$ is a many-body wave function for the core nucleus, satisfying $H_{\rm core}|\Phi_{nIM}\rangle=\epsilon_{nI}|\Phi_{nIM}\rangle$, where $I$ and $M$ are the total angular momentum and its $z$-component in the laboratory frame for the core nucleus, and $n$ is the index to distinguish different states with the same $I$ and $M$. The radial wave function, ${\cal R}_{jlnI}(r_\Lambda)$, in Eq. (\[wf\]) is obtained by solving the coupled-channels equations given by, $$\begin{aligned} 0&=& \langle[{\cal Y}_{jl}(\hat{{\mbox{\boldmath $r$}}}_\Lambda)\otimes\Phi_{nI}(\{{\mbox{\boldmath $r$}}_i\})]^{(JM_J)} |H-E_J|\Psi_{JM_J}\rangle, \\ &=& \left[T_\Lambda(jl)+\epsilon_{nI}-E_J\right] {\cal R}_{jlnI}(r_\Lambda) \nonumber \\ &&~~~~~+\sum_{j',l'}\sum_{n',I'}V_{jlnI,j'l'n'I'}(r_\Lambda)\,{\cal R}_{j'l'n'I'}(r_\Lambda), \label{cc}\end{aligned}$$ with $$V_{jlnI,j'l'n'I'}(r_\Lambda)= \left\langle jlnI\left|\sum_{i=1}^{A_C}v_{N\Lambda}({\mbox{\boldmath $r$}}_\Lambda,{\mbox{\boldmath $r$}}_i) \right|j'l'n'I'\right\rangle,$$ where $|jlnI\rangle\equiv \left|[{\cal Y}_{jl}(\hat{{\mbox{\boldmath $r$}}}_\Lambda)\otimes\Phi_{nI}(\{{\mbox{\boldmath $r$}}_i\})]^{(JM_J)} \right\rangle$. In the microscopic particle-rotor model, the core wave functions, $\Phi_{nIM}$, are constructed with the generator coordinate method by superposing projected Slater determinants, $|\phi_{IM}(\beta)\rangle$, as, $$|\Phi_{nIM}\rangle = \int d\beta\,f_{nI}(\beta)|\phi_{IM}(\beta)\rangle, \label{proj_wf}$$ where $\beta$ is the quadrupole deformation parameter and $f_{nI}(\beta)$ is the weight function. In writing this equation, for simplicity, we have assumed that the core nucleus has an axially symmetric shape. Here, $|\phi_{IM}(\beta)\rangle$ is constructed as $$|\phi_{IM}(\beta)\rangle = \hat{P}^I_{M0}\hat{P}^N\hat{P}^Z|\beta\rangle, \label{mf}$$ where $|\beta\rangle$ is the wave function obtained with a constrained mean-field method at the deformation $\beta$, and $\hat{P}^I_{M0}$, $\hat{P}^N$, and $\hat{P}^Z$ are the operators for the angular momentum projection, the particle number projection for neutrons, and that for protons, respectively. Notice that the $K$-quantum number is zero in $\hat{P}^I_{M0}$ because of the axial symmetry of the wave function, $|\beta\rangle$. The weight function, $f_{nI}(\beta)$, in Eq. (\[proj\_wf\]) is determined with the variational principle, which leads to the Hill-Wheeler equation [@RS80], $$\begin{aligned} &&\int d\beta'\,\left[\langle\phi_{IM}(\beta)|H_{\rm core}|\phi_{IM}(\beta')\rangle \right. \nonumber \\ &&\left.~~~~~~~-\epsilon_{nI}\langle\phi_{IM}(\beta)|\phi_{IM}(\beta')\rangle \right]f_{nI}(\beta')=0.\end{aligned}$$ Notice that by setting $f_{nI}(\beta)=\delta(\beta-\beta_0)$ in Eq. (\[proj\_wf\]), one can also obtain the projected energy surface, $E_{J}(\beta_0)$, after solving the coupled-channels equations, Eq. (\[cc\]) [@Xue15]. (In this case, there is only one single state, $n=1$, in the core nucleus for each $I$.) See Refs. [@Mei14; @Xue15; @Mei15; @Mei16; @Mei17] for more details on the framework of the microscopic particle-rotor model. Deformation of the $^{31}_{~\Lambda}$S hypernucleus =================================================== We now apply the microscopic particle-rotor model to $^{31}_{~\Lambda}$Si as a typical example of hypernuclei which show the disappearance of nuclear deformation in the mean-field approximation. To this end, we employ the relativistic point-coupling model. For the core nucleus, $^{30}$Si, we use the PC-F1 [@PC-F1] parameter set, while we use PCY-S4 [@PCY-S4] for the $N\Lambda$ interaction. As we have shown in Ref. [@Mei16], the dependence of the results on the choice of the $N\Lambda$ interaction would not be large and the conclusion of the paper will remain the same, at least qualitatively, even if we use another set of the PCY-S interaction. The pairing correlation among the nucleons in the core nucleus is taken into account in the BCS approximation with a contact pairing interaction with a smooth energy cutoff, as described in Ref. [@PC-PK1]. We generate the reference states, $|\beta\rangle$, in Eq. (\[mf\]) by expanding the single-particle wave functions on a harmonic oscillator basis with 10 major shells. The coupled-channels calculations are also solved by expanding the radial wave functions, ${\cal R}_{jlnI}(r_\Lambda)$, on the spherical harmonic oscillator basis with 18 major shells. In the coupled-channels calculations, we include the core states up to $n_{\rm max}=2$ and $I_{\rm max}=6$. {width="9cm"} ![ The low-lying spectrum of the $^{30}$Si nucleus obtained with the GCM method with the covariant density functional with the PC-F1 set. The arrows indicate the electric quadrupole (E2) transition strengths, plotted in units of $e^2$fm$^4$. These are compared with the experimental data taken from Ref. [@Basunia10]. ](fig2r){width="9cm"} We first discuss the results for the core nucleus, $^{30}$Si. Figure 1 shows the potential energy curves for $^{30}$Si as a function of the deformation parameter, $\beta$. The energy curve in the mean-field approximation shows a shallow minimum at $\beta=-0.22$ (see the dotted line), which is similar to the energy curve for $^{28}$Si shown in Ref. [@MH08] obtained with the RMF theory with the meson-exchange NLSH parameter set [@NLSH]. For the projected energy curves, this calculation yields a well pronounced oblate minimum. For instance, for the $0^+$ configuration, the minimum appears at $\beta=-0.35$. The results of the GCM calculations for the spectrum as well as the $E2$ transition probabilities are shown in Fig. 2. The energy of each state is plotted also in Fig. 1, at the position of the mean deformation for each state. These calculations reproduce the experimental data reasonably well, even though the $B(E2)$ values for the intraband and the interband transitions are somewhat overestimated and underestimated, respectively. {width="8cm"} {width="8cm"} Let us now put a $\Lambda$ particle onto the $^{30}$Si nucleus and discuss the structure of the $^{31}_{~\Lambda}$Si hypernucleus. Fig. 3 shows the potential energy surface in the mean-field approximation, in which the curve for the hypernucleus (the solid line) is shifted in energy as indicated in the figure so that the energy of the absolute minima becomes the same as that for the core nucleus (the dotted line). One can see that the potential minimum is shifted from the oblate shape to the spherical shape by adding a $\Lambda$ particle to $^{30}$Si. As we have mentioned, the same phenomenon has been found also with another relativistic interaction, that is, the the meson-exchange NLSH interaction [@MH08]. Our interest in this paper is to investigate how this phenomenon is modified when the effect beyond the mean-field approximation is taken into account. Fig. 4 shows the projected energy curve, which includes the beyond mean-field effect. The solid line shows the energy for the 1/2$^+$ configuration of the $^{31}_{~\Lambda}$Si hypernucleus. One can notice that the energy at the spherical configuration is lowered when a $\Lambda$ particle is added, as has been indicated also in the previous mean-field calculations (see also Fig. 3) [@MH08; @Schulze10]. Moreover, the deformation at the energy minimum is shifted towards the spherical configuration, that is, from $\beta=-0.35$ to $\beta=-0.30$. Even though a care must be taken in interpreting the projected energy surface, which includes only the rotational correction to the mean-field approximation while the vibrational correction is left out [@Reinhard78], this may indicate that the collectivity is somewhat reduced in the hypernucleus. {width="8cm"} In order to gain a deeper insight into the effect of $\Lambda$ particle on the collectivity of the hypernucleus, Fig. 5 shows the spectrum of $^{31}_{~\Lambda}$Si for the positive parity states obtained with the microscopic particle-rotor model. One can observe that the spectrum resembles that in the core nucleus shown in Fig. 2. These positive parity states are in fact dominated by the $\Lambda$ hyperon in the $s$-orbit coupled to the positive parity states of the core nucleus. However, if one takes the ratio of the energy of the first 4$^+$ state to that of the first 2$^+$ state, $R_{4/2}=E(4^+)/E(2^+)$, the addition of a $\Lambda$ particle alters it from 3.083 to 2.829 with the PC-F1 parameter set. Here, the ratio for the hypernucleus is estimated as $E(9/2_1^+)/E(5/2_1^+)$. The $R_{4/2}$ ratio for the core nucleus is close to the value for a rigid rotor, that is, $R_{4/2}=3.33$. On the other hand, the $R_{4/2}$ ratio is significantly reduced in the hypernucleus. It is in between the rigid rotor limit and the vibrator limit, that is, $R_{4/2}=2.0$, even though the $R_{4/2}$ ratio is still somewhat closer to the rigid rotor value. This indicates a signature of disappearance of deformation found in the previous mean-field calculations [@MH08], even though the deformation does not seem to disappear completely and thus the spectrum still shows a rotational-like character. Of course, the weaker polarization effect of a $\Lambda$ particle, which has been found also in Ref. [@Isaka16], compared to that in the previous mean-field calculations is due to the beyond-mean-field effect, that is a combination of the effect of shape fluctuation and the angular momentum projection. In particular, the GCM calculations for the core nucleus indicate that the average deformation depends on the angular momentum (see Fig. 1). The impact of the $\Lambda$ particle may therefore be state-dependent as well. =1.5pt ----------------------- --------- -- ----------------------- --------- ---------- ------------- $I^\pi_i \to I^\pi_f$ $B(E2)$ $J^\pi_i \to J^\pi_f$ $B(E2)$ $cB(E2)$ $\Delta$(%) $2^+_1\to 0^+_1$ 63.60 $3/2^+_1\to 1/2^+_1$ 57.00 57.00 $-$10.38 $5/2^+_1\to 1/2^+_1$ 57.06 57.06 $-$10.28 $4^+_1\to 2^+_1$ 103.59 $7/2^+_1\to 3/2^+_1$ 92.14 102.38 $-$1.17 $7/2^+_1\to 5/2^+_1$ 10.22 102.24 $-$1.30 $9/2^+_1\to 5/2^+_1$ 102.36 102.36 $-$1.19 ----------------------- --------- -- ----------------------- --------- ---------- ------------- : The $E2$ transition strengths (in units of $e^2$ fm$^4$) for low-lying positive parity states of $^{30}$Si and $^{31}_{~\Lambda}$Si obtained with the PC-F1 parameter set for the $NN$ interaction. The c$B(E2)$ values denote the corresponding $B(E2)$ values for the core transition in the hypernucleus, defined by Eq. (\[cBE2\]). The changes in the $B(E2)$ is indicated with the quantity defined by $\Delta\equiv (cB(E2)-B(E2; {^{30}{\rm Si}}))/B(E2; {^{30}{\rm Si}})$. The calculated quadrupole transition strengths, $B(E2)$, are listed in TABLE I. Here we also show the $cB(E2)$ values, which are defined as [@Mei15], $$\begin{aligned} cB(E2: I_i\to I_f)&\equiv& \frac{1}{(2I_i+1)(2J_f+1)}\, \left\{ \begin{array}{ccc} I_f & J_f & j_\Lambda \\ J_i & I_i & 2 \end{array} \right\}^{-2} \nonumber \\ &&\times B(E2:J_i\to J_f), \label{cBE2}\end{aligned}$$ where $I_i$ and $I_f$ are the dominant angular momenta of the core nucleus in the initial and the final hypernuclear configurations, while $j_\Lambda$ is that for the $\Lambda$ particle. In the transitions shown in TABLE I, $j_\Lambda$ is 1/2. This equation is derived by relating $$\begin{aligned} &&B(E2:J_i\to J_f) \nonumber \\ &=& \frac{1}{2J_i+1}\, \left|\langle J_i||\hat{T}_{\rm E2}||J_f\rangle\right|^2, \\ &\sim& \frac{1}{2J_i+1}\, \left|\left\langle [j_\Lambda\otimes I_i]^{(J_i)} \left|\left|\hat{T}_{\rm E2}\right|\right| [j_\Lambda\otimes I_f]^{(J_f)}\right\rangle\right|^2, \end{aligned}$$ with $$B(E2:I_i\to I_f) = \frac{1}{2I_i+1}\, \left|\langle I_i||\hat{T}_{\rm E2}||I_f\rangle\right|^2,$$ where $\hat{T}_{\rm E2}$ is the $E2$ transition operator (which acts only on the core states). The table indicates that the $B(E2)$ transition strengths decrease by adding a $\Lambda$ particle into the core nucleus. This is consistent with the reduction in deformation in the hypernucleus as discussed in the previous paragraph. Summary ======= We have investigated the role of beyond-mean-field effects on the deformation of $^{31}_{~\Lambda}$Si. For this hypernucleus, the previous study based on the relativistic mean-field theory had shown that the deformation vanishes while the core nucleus, $^{30}$Si, is oblately deformed. Using the microscopic particle-rotor model, we have shown that the ratio of the energy of the first 4$^+$ state to that of the first 2$^+$ state is significantly reduced by adding a $\Lambda$ particle to $^{30}$Si, even though the spectrum of the hypernucleus $^{31}_{~\Lambda}$Si still shows a rotational-like structure. This implies that the addition of a $\Lambda$ particle to $^{30}$Si does not lead to a complete disappearance of nuclear deformation if the beyond-mean-field effect is taken into account, even though the deformation is indeed reduced to some extent. In accordance to this, the quardupole transition strengths have been found to be also reduced in the hypernucleus. Our study in this paper clearly shows that the beyond-mean-field effect plays an important role in the structure of hypernuclei. We emphasize that the microscopic particle-rotor model employed in this paper provides a convenient tool for that purpose, which is complementary to the generator coordinate method for the whole core+$\Lambda$-particle system [@MHY16]. We thank H. Tamura for useful discussions. This work was supported in part by JSPS KAKENHI Grant Number 2640263 and by the National Natural Science Foundation of China under Grant No. 11575148. [99]{} A. Bohr and B.R. Mottelson, [*Nuclear Structure Vol. II*]{} (Benjamin, Reading, MA, 1975). P. Ring and P. Schuck, [*The Nuclear Many Body Problem*]{} (Springer-Verlag, New York, 1980). M. Bender, P.-H. Heenen, and P.-G. Reinhard, Rev. Mod. Phys. [**75**]{}, 121 (2003). T. Dytrych, K.D. Launey, J.P. Draayer, P. Maris, J.P. Vary, E. Saule, U. Catalyurek, M. Sosonkina, D. Langr, and M.A. Caprio, Phys. Rev. Lett. [**111**]{}, 252501 (2013). P. Maris, M.A. Caprio, and J.P. Vary, Phys. Rev. C[**91**]{}, 014310 (2015). S.R. Stroberg, H. Hergert, J.D. Holt, S.K. Bogner, and A. Schwenk, Phys. Rev. C[**93**]{}, 051301(R) (2016). G.R. Jansen, M.D. Schuster, A. Signoracci, G. Hagen, and P. Navratil, Phys. Rev. C[**94**]{}, 011301 (2016). C.N. Gilbreth, Y. Alhassid, and G.F. Bertsch, Phys. Rev. C[**97**]{}, 014315 (2018). H. Feshbach, in [*Proc. of Summer Study Meeting on K-Physics and Facilities*]{}, edited by B. Palevsky, BNL report BNL-50579, p. 391 (1976). J. Žofka, Czech. J. Phys. [**B30**]{}, 95 (1980). X.R. Zhou, H.-J. Schulze, H. Sagawa, C.X. Wu, and E.-G. Zhao, Phys. Rev. C[**76**]{}, 034312 (2007). Myaing Thi Win and K. Hagino, Phys. Rev. C[**78**]{}, 054311 (2008). H.-J. Schulze, Myaing Thi Win, K. Hagino, and H. Sagawa, Prog. Theo. Phys. [**123**]{}, 569 (2010). B.-N. Lu, E.-G. Zhao, and S.-G. Zhou, Phys. Rev. C[**84**]{}, 014328 (2011). M. Isaka, M. Kimura, A. Dote, and A. Ohnishi, Phys. Rev. C[**83**]{}, 044323 (2011). Myaing Thi Win, K. Hagino, and T. Koike, Phys. Rev. C[**83**]{}, 014301 (2011). M. Isaka, H. Homma, M. Kimura, A. Dote, and A. Ohnishi, Phys. Rev. C[**85**]{}, 034303 (2012). M. Isaka, M. Kimura, A. Dote, and A. Ohnishi, Phys. Rev. C[**87**]{}, 021304 (2013). M. Isaka, K. Fukukawa, M. Kimura, E. Hiyama, H. Sagawa, and Y. Yamamoto, Phys. Rev. C[**89**]{}, 024310 (2014). B.-N. Lu, E. Hiyama, H. Sagawa, and S.-G. Zhou, Phys. Rev. C[**89**]{}, 044307 (2014). J.-W. Cui, X.-R. Zhou, and H.-J. Schulze, Phys. Rev. C[**91**]{}, 054306 (2015). O. Hashimoto and H. Tamura, Prog. Part. Nucl. Phys. [**57**]{}, 564 (2006). K. Hagino and J.M. Yao, in [*Relativistic Density Functional for Nuclear Structure*]{}, edited by J. Meng, Int. Rev. of Nucl. Phys. [**10**]{} (World Scientific, Singapore, 2016), p. 263. A. Gal, E.V. Hungerford, and D.J. Millener, Rev. Mod. Phys. [**88**]{}, 035004 (2016). M. Bender and P.-H. Heenen, Phys. Rev. C[**78**]{}, 024309 (2008). T.R. Rodriguez and J.L. Egido, Phys. Rev. C[**81**]{}, 064323 (2010). J.M. Yao, J. Meng, P. Ring, and D. Vretenar, Phys. Rev. C[**81**]{}, 044311 (2010). J.M. Yao, H. Mei, H. Chen, J. Meng, P. Ring, and D. Vretenar, Phys. Rev. C[**83**]{}, 014308 (2011). J.M. Yao, K. Hagino, Z.P. Li, J. Meng, and P. Ring, Phys. Rev. C[**89**]{}, 054306 (2014). B. Bally, B. Avez, M. Bender, and P.-H. Heenen, Phys. Rev. Lett. [**113**]{}, 162501 (2014). J.L. Egido, M. Borrajo, and T.R. Rodriguez, Phys. Rev. Lett. [**116**]{}, 052502 (2016). H. Mei, K. Hagino, and J.M. Yao, Phys. Rev. C[**93**]{}, 011301(R) (2016). X.Y. Wu, H. Mei, J.M. Yao, and X.-R. Zhou, Phys. Rev. C[**95**]{}, 034309 (2017). J.-W. Cui, X.-R. Zhou, L.-X. Guo, and H.-J. Schulze, Phys. Rev. C[**95**]{}, 024323 (2017). M. Isaka, Y. Yamamoto, and Th.A. Rijken, Phys. Rev. C[**94**]{}, 044310 (2016). H. Mei, K. Hagino, J.M. Yao, and T. Motoba, Phys. Rev. C[**90**]{}, 064302 (2014). W.X. Xue, J.M. Yao, K. Hagino, Z.P. Li, H. Mei, and Y. Tanimura, Phys. Rev. C[**91**]{}, 024327 (2015). H. Mei, K. Hagino, J.M. Yao, and T. Motoba, Phys. Rev. C[**91**]{}, 064305 (2015). H. Mei, K. Hagino, J.M. Yao, and T. Motoba, Phys. Rev. C[**93**]{}, 044307 (2016). H. Mei, K. Hagino, J.M. Yao, and T. Motoba, Phys. Rev. C[**96**]{}, 014308 (2017). T. Bürvenich, D. G. Madland, J. A. Maruhn, and P.-G. Reinhard, Phys. Rev. C [**65**]{}, 044308 (2002). Y. Tanimura and K. Hagino, Phys. Rev. C[**85**]{}, 014306 (2012). P.W. Zhao, Z.P. Li, J.M. Yao, and J. Meng, Phys. Rev. C[**82**]{}, 054319 (2010). M.M. Sharma, M.A. Nagarajan, and P. Ring, Phys. Lett. B[**312**]{}, 377 (1993). M.S. Basunia, Nucl. Data Sheets [**111**]{}, 2331 (2010); National Nuclear Data Center (NNDC) \[http://www.nndc.bnl.gov/\]. P.-G. Reinhard, Z. Phys. [**A285**]{}, 93 (1978).

--- abstract: 'We study a mathematical model for revenue management under competition with multiple sellers. The model combines the stochastic knapsack problem, a classic revenue management model, with a non coorperative game model that characterizes the sellers’ rational behavior. We are able to establish a dynamic recursive procedure that incorporate the value function with the utility function of the games. The formalization of the dynamic recursion allows us to establish some fundamental structural properties.' author: - Yingdong Lu bibliography: - 'Lu.bib' date: 'Received: date / Accepted: date' title: 'A Stochastic Knapsack Game: Revenue Management in Competitions' --- Introduction {#sec:intro} ============ A key model in revenue (yield) management is the following, a seller needs to sell a fixed amount of certain commodity before a fixed deadline to different buyers with individual price they are willing to pay, and the seller can dynamically adjust the selling price to maximize his/her overall revenue over time. Stochastic knapsack problem, also known as stochastic dynamic knapsack problem, a mathematical problem that captures the essences of this model, quantifies some of the most fundamental trade-offs in revenue management, and serves as an important building block for more complex and sophisticated models for real life applications. Consequently, the stochastic (dynamic) knapsack problem and its variations have been studied extensively, see, e.g. [@GallegovanRyzin1994], [@BitranMondschein95], [@FengGallego1995], [@PapastavrouRajagopalanKleywegt1996], [@FengGallego2000], [@FengXiao2000], [@VanSlykeYoung2000], [@ZhaoZheng2000], [@LinLuYao2008]. It is one of the fundamental models surveyed by Anould de Boer in [@DenBoer2015], please refer to that paper for more details, as well as references. It is natural to ask the question of what would happen if there are multiple sellers competing for the same demand stream from the buyers. In this paper, we generalize the classic stochastic knapsack problem, and formulate a mathematical model to capture the basic relations in this situation. An immediate goal is to formulate a dynamic recursion for calculating the optimal policies for sellers. In the single seller case, this is accomplished through the formulation of a dynamic program that computes the maximum expected revenue starting at any time with any amount of remaining inventory. However, in the case of multiple sellers, at each time period, the sellers’ decisions are inter-dependent. It is, therefore, not a trivial task to decide what will be the next best action even if every seller has the same forecast of the future demand arrivals. Another difficulty is that when multiple sellers are willing to sell the product, the buyer can have different ways to choose one of them to fulfill the demand, the difference in these selection rules has significant impact on the evolution of the system. To overcome these difficulties, we model the sellers as rational individual or institutions, and introduce a noncooperative game at each step of the dynamic recursion characterizing their behavior. Furthermore, we follow a static probabilistic selection rule, which will be described precisely later, that the buyer will use to select sellers. This selection rule, on one hand, reflects market power of the sellers, on the other hand, it allows the uncertainty that is natural in business reality. With this mechanism, the utility functions of the games are properly connected with the value functions of the dynamic recursion, thus help to identify pure strategy Nash equilibriums. Under the selection rule assumed, we are able to demonstrate that there is a unique Nash equilibrium of the game. In turn, assuming that the Nash equilibriums will be the strategy followed by all the sellers at each step, the dynamic recursion is able to proceed. Once establishing the dynamic recursion, we are able to extend the arguments that are effective for the single seller dynamic programming, and demonstrate that, in some cases, the value function exhibits remarkable rich monotonicity properties that provide insights to key trade-offs to the problem and can be helpful to dynamic pricing in practice. A related but different model is considered in [@doi:10.1287/mnsc.2013.1821], it is concluded that, under a differential game setting, the equilibrium structure enjoys simple structural properties. While the model studied here is quite different, but results are similar in spirit. The rest of the paper will be organized as follow. In Sec. \[sec:models\], we will introduce the basic mathematical models, and review preliminaries including some basic concepts in game theorety that will be needed for our analysis. In Sec. \[sec:dp\], we will discuss in details the dynamic recursion in which the game aspect of the problem is incorporated. In Sec. \[sec:monotonicity\], we establish some fundamental structural properties of the value functions of the dynamic recursion. Finally, we conclude the paper in Sec. \[sec:conclusions\] with a summary of our findings. Models and Preliminaries {#sec:models} ======================== Model Descriptions {#sec:model_descriptions} ------------------ Suppose that there are $N$ sellers, and each seller $n$, $n=1,2,\ldots, N$, has an initial inventory of $C_n$ units of product ( could be either goods or services) at the beginning of a common selling horizon. The selling horizon is discrete and of length $T<\infty$. At each time $t=1,2,\ldots, T$, demand for one unit of the product will emerge, and the buyer will post a price that he/she is willing to pay. To accommodate the event of no arrival, we can always include a class of demand with exceedingly low price. The sellers who have positive inventory need to decide whether they should accept or reject this demand. The buyer will then select one seller among all the sellers that accept the demand according to certain [*selection rule*]{}, and the selected seller will supply the product and collect the revenue. At the end of the selling horizon, all the remaining product will be savaged. The goal for each seller is to maximize his/her expected revenue. We assume that each seller does not have the information of the exact value of the initial inventory of other sellers, but has a distributional estimation of that quantity. We also assume that the distributional information of the future demand price is given to each seller, and no seller has any extra knowledge. In particular, we assume that the price of the demand realization at each period follow an independent and identically distributed discrete probability distribution $P$, with ${\mbox{\sf P}}[P=p_i]= \th_i$, $i=1,2,\ldots,I$. Suppose that, at each time $t$, when the demand is of class $i$, i,e. the price is $p_i$, a subset of sellers, denoted by $A_t(i)$ (which can be shortened to $A_t$ when there is no ambiguity), will accept the demand, decided based on the remaining time, demand type, remaining inventory and the selling history up to time $t$. The buyer will select only one seller among them, which means that there is a possibility that no seller is selected. There could be various selection rule models reflecting different market mechanisms, for example, a static rule ( the buyer chooses one product over the other overwhelmingly, which happens often in some local and monopoly market) and weighted rule (buyer assigns weights to the each product, then randomly, with probabilities determined by the weights, select ones that are available). In this paper, we will focus on a random allocation rule with static probabilities: each seller is associated with a probability $\pi_n$, $\sum_{n=1}^N \pi_n=1$. At each time, if a seller accepts, the probability of it being selected is always $\pi_n$, and with probability $1-\sum_{n \in A_t(i)}\pi_n$, no one is selected. At each time $t$, the phenomenon that the sellers are making independent decisions based on distributional information on the other sellers can be best modeled by a non coorperative strategic game, see, e.g. [@osborne1994course]. A Dynamic Recursion Formulation {#sec:dp} =============================== Our goal is to identify a strategy for a seller to achieve the best outcome, in terms of average revenue, under a reasonable assumption on other sellers’ behavior. Recall that each seller $n$, $n=1,2,\ldots, N$ with initial inventory $C_n$ is also given the distributional information of the inventory of all other sellers, either though statistical forecast or other business information inquiry, and any two sellers will be given the exact same distribution on the third seller. In addition, all the sellers do observe all the sells outcomes up to each decision time epoch, i.e. they know the amount each seller sold so far. It is our intention to derive a dynamic recursion for calculating the best outcome, hence the optimal strategy for each seller. Equivalently, given ${{\bf s}}=(s_1, s_2, \ldots, s_N)$ representing the amount of inventory has been sold so far by each seller, we seek to calculate $v_n(t, d_n, {{\bf s}})$, $n=1,2,\ldots, N$, the maximum expected revenue seller $n$ can collect starting from time $t$ and with remaining inventory $d_n$, for any time $t=1,2,\ldots, T$. We assume that the behavior of the sellers is modeled as a $N$-person game, and if sellers follow the Nash Equilibriums at each time period, a dynamic recursion can proceed. This will be argued inductively. At the last time period $T$, given a price realization, $p_i$, there are two strategies for each seller, accept or reject. The utility function of the game for seller $n$ will be the expected revenue collected by taking either action. If reject, of course, there is no revenue. It is clear that, if the random selection rule with static probabilities is followed, there exists a unique Nash equilibrium, that is, every seller will accept, as long as they have a positive inventory. In this case, the value function $v_n(T, {{\bf d}})$ has the following form, $v_n(T, {{\bf d}}) = \pi_n {{\bf\sf E}}[P],$ where $\pi_n$ are the probabilities in the section rule model. Now, suppose that we can calculate recursively all the value function $v_n(t+1, d_n, {{\bf s}})$ for any feasible ${{\bf s}}$, we demonstrate that there exists a unique pure strategy Nash equilibrium at time period $t$, and show how it is related to the calculation of the value function for time $t$, $v_n(t, d_n, {{\bf s}})$. There are two actions for each seller, accept or reject. The payoff function will be the expected revenue to be collected until time $T$. Therefore, the seller will consider the following [*balance inequality*]{}, whose left hand side (LHS) represents the price we get immediately, and right hand side (RHS) represents the future reward, $$\begin{aligned} \label{eqn:balance} p \ge {{\bf\sf E}}_{n, t}[ v_n (t+1, d_n, {{\bf s}})- v_n (t+1, d_n-1, {{\bf s}}+{{\bf e}}_n)],\end{aligned}$$ where ${{\bf\sf E}}_{n, t}$ is the expectation with respect to the information available at time $t$ for seller $n$, $p$ the generic price the class indicator is suppressed when there is no ambiguity). If holds, then the order will be accepted. Otherwise, if we have, $$\begin{aligned} \label{eqn:unbalance} p < {{\bf\sf E}}_{n, t}[ v_n (t+1, d_n, {{\bf s}})- v_n (t+1, d_n-1, {{\bf s}}+{{\bf e}}_n)],\end{aligned}$$ the order will be rejected. [**Remark**]{} The operator ${{\bf\sf E}}_{n,t}$ can be treated in a way as an conditional expectation, the information update each time is basically the confirmation that the random variable of each seller’s inventory is larger than the cumulative sales, which is updated at the end of each time period. The above defined strategy is a unique Nash Equilibrium. [**Proof** ]{}To prove that it is a Nash Equilibrium, let us discuss separately for those sellers depends upon their decisions. Suppose that for a particular seller $n$, the action is to accept, three events can happen, - sell $n$ is selected, with probability $\pi_n$; - some other seller $j$ in $A_t$ is selected, with probability $\pi_j$; - no seller is selected, with probability $1- \sum_{A_t} \pi_i$. Sum them up, the pay-off function has the following form, $$\begin{aligned} \pi_n{{\bf\sf E}}_{n, t}[p+v_{n} (t+1, d_n-1, {{\bf s}})]& + \sum_{j\neq n, j\in A_t} \pi_j {{\bf\sf E}}_{n, t}[v_n(t+1, d_n, {{\bf s}}+{{\bf e}}_j)]\\ &+ \left(1- \sum_{A_t} \pi_i\right) {{\bf\sf E}}_{n, t}[v_n(t+1, d_n, {{\bf s}})].\end{aligned}$$ If seller $n$ deviates from the strategy, i.e., rejects the demand, its payoff will be, $$\sum_{j\neq n, j\in A_t} \pi_j {{\bf\sf E}}_{n, t}[v_n(t+1, d_n, {{\bf s}}+{{\bf e}}_j)]+ \left(1- \sum_{A_t-\{n\}} \pi_i\right) {{\bf\sf E}}_{n, t}[v_n(t+1, d_n, {{\bf s}})].$$ From the , we know that seller $n$ could not be better off. In the case seller $n$ reject, the pay off is, $$\begin{aligned} \sum_{j\neq n, j\in A_t} \pi_j {{\bf\sf E}}_{n, t} [v_n(t+1, d_n, {{\bf s}}+{{\bf e}}_j)]+ \left(1- \sum_{A_t-\{n\}} \pi_i\right) {{\bf\sf E}}_{n, t}[v_n(t+1, d_n, {{\bf s}}+{{\bf e}}_j)].\end{aligned}$$ If the seller deviates from this strategy, the pay-off will become, $$\begin{aligned} \pi_n[p+ {{\bf\sf E}}_{n, t}[v_{n} (t+1, d_n-1, {{\bf s}})]] &+ \sum_{j\neq n, j\in A_t} \pi_j {{\bf\sf E}}_{n, t} [v_n(t+1, d_n, {{\bf s}}+{{\bf e}}_j)]\\ &+ \left(1- \sum_{A_t} \pi_i\right) {{\bf\sf E}}_{n, t} [v_n(t+1, d_n, {{\bf s}})].\end{aligned}$$ However, we know that $p+{{\bf\sf E}}_{n, t} [v_n(t+1, d_n-1, {{\bf s}})] <{{\bf\sf E}}_{n, t} [v_n(t+1, d_n, {{\bf s}})] $, therefore, the seller will be worse off. Suppose any other strategy that has a seller $n$, such that, $p+v_n(t+1, d_n-1, {{\bf s}}+{{\bf e}}_n) <v_n(t+1, d_n {{\bf s}}) $, but seller $n$ accepts the demand. We can see that deviation will lead to better pay-off. Meanwhile if there is a seller $n$ with $p+v_n(t+1, d_n-1, {{\bf s}}+{{\bf e}}_n) \ge v_n(t+1, d_n, {{\bf s}})$, but seller $n$ rejects, a deviation will lead to higher pay-off. $\Box$ The above arguments allow us to present the following dynamic recursion for the value function, $$\begin{aligned} &v_n(t, d_n, {{\bf s}}) = \sum_{i=1}^I \th_iw_n(t+1, d_n,{{\bf s}}, p_i), \label{eqn:main_recursion_1} \\ &w_n(t+1, d_n,{{\bf s}}, p_i) = \left(1-\sum_{m=1}^N \pi_m\right){{\bf\sf E}}_{n,t}[v_n(t+1,d_n, {{\bf s}})]\nonumber \\ &+ p_i \pi_n{\bf 1}\left\{{{\bf\sf E}}_{n,t}[v_n(t+1, d_n-1, {{\bf s}}+{{\bf e}}_n)+p_i] \ge {{\bf\sf E}}_{n,t}[v_n(t+1, d_n, {{\bf s}})\right]\} \nonumber \\ & +\sum_{m=1}^N \pi_m [v_n(t+1, d_n, {{\bf s}}+{{\bf e}}_m){\bf 1}\{{{\bf\sf E}}_{m,t}[v_m(t+1, d_m -1, {{\bf s}}+{{\bf e}}_m)+p_i] \ge {{\bf\sf E}}_{m, t}[v_m(t+1, d_m)]\} \nonumber \\ & +{{\bf\sf E}}_{n,t}[v_n(t+1, d_n+{{\bf s}}){\bf 1}\{{{\bf\sf E}}_{n,t}[v_n(t+1, d_n-1, {{\bf s}}+{{\bf e}}_n)+p_i < v_n(t+1, d_n, {{\bf s}})]\} ],\label{eqn:main_recursion_2} \\ & v_N(T, d_n, {{\bf s}}) = \pi_N {{\bf\sf E}}[p].\label{eqn:main_recursion_3}\end{aligned}$$ [**Remark** ]{} The information available at time $t$ is on the distribution on the initial capacity of all the other sellers, as well as the sales records in the past period. At time $t+1$, the sales records will be amended with what happened during time period $t$, the distribution inform hence is naturally updated, for example, if the original distributional estimation is $D$, and at time $t$, the total sales has been $s$, then that information should be updated to $D;D\ge s$. At time $t$, if there are sales by that seller, it should be updated to $D;D\ge s+1$, otherwise, it will stay at $D;D\ge s$. Monotonicity of the Value Functions and its Implications in Revenue Management {#sec:monotonicity} ============================================================================== In this section, we will establish monotonicity properties of the value function $v_n(t, d, {{\bf s}})$, based on the dynamic recursion formulated in Sec. \[sec:dp\]. The main result is stated in the following theorem, and its proof is presented in the Appendix. \[thm:monotone\] Under the random selection rule with static probabilities, the value function of the knapsack problem $v_n(t, d, {{\bf s}})$ for the $n$-th seller satisfies the following monotonicity properties. - Monotone in inventory ${{\bf d}}$, i.e. ${{\bf\sf E}}_{n, t-1}[v_n(t, d_n, {{\bf s}}) \ge {{\bf\sf E}}_{n, t-1}[v_n(t, d_n-1, {{\bf s}})]$; - Monotone in selling amount of competitors,$${{\bf\sf E}}_{n, t-1}[v_n(t, d_n, {{\bf s}})] \le {{\bf\sf E}}_{n, t-1}[v_n(t, d_n, {{\bf s}}+{{\bf e}}_i)] ;$$ - Monotone in time $t$, i.e. ${{\bf\sf E}}_{n, t-1}[v_n(t, d_n,{{\bf s}})] \ge {{\bf\sf E}}_{n, t}[v_n(t+1, d_n, {{\bf s}})]$ - “Concave” in $d_n$, i.e., $$\begin{aligned} &{{\bf\sf E}}_{n, t-1}[v_n(t,d, {{\bf s}})-v_n(t,d_n-1,{{\bf s}}+{{\bf e}}_n)]\nonumber \\ \ge & {{\bf\sf E}}_{n, t-1}[v_n(t,d_n+1,{{\bf s}})-v_n(t,d, {{\bf s}}+{{\bf e}}_n)];\label{eqn:concavity}\end{aligned}$$ - Submodular in $(t,{{\bf d}})$, i.e., $$\begin{aligned} \label{eqn:submodularity} &{{\bf\sf E}}_{n, t-1}[v_n(t,d, {{\bf s}})]-{{\bf\sf E}}_{n, t-1}[v_n(t,d, {{\bf s}}+{\bf e}_n)]\nonumber \\ & \ge {{\bf\sf E}}_{n, t}[v_n(t+1,d, {{\bf s}})]-{{\bf\sf E}}_{n, t}[v_n(t+1,d,{{\bf s}}+{\bf e}_n)].\end{aligned}$$ - Submodular in ${{\bf d}}$, i.e. $$\begin{aligned} \label{eqn:submodularity_more} &{{\bf\sf E}}_{n, t-1}[v_n(t,d,{{\bf s}})]-{{\bf\sf E}}_{n, t-1}[v_n(t,d, {{\bf s}}+{\bf e}_n)]\nonumber \\ & \ge {{\bf\sf E}}_{n, t-1}[v_n(t,d,{{\bf s}}-{\bf e}_m)]+{{\bf\sf E}}_{n, t-1}[v_n(t,d,{{\bf s}}+{\bf e}_n-{\bf e}_m)],\end{aligned}$$ with $m\neq n$. Recall that we raised several questions in the introduction, here, after the statement of the main monotonicity results, we need to use them to answer some of those questions. From (2) of Theorem \[thm:monotone\], we can immediately see that, For each individual seller, his/her total average revenue is a monotone decrease function of his/her competitors inventory surplus levels. [**Remark** ]{} It is apparent that the more the overall supply is, the less is the expected marginal gain for each individual unit. From (4), i.e. “concave in inventory” of Theorem \[thm:monotone\], we can conclude that If it is optimal to accept the at certain point, then it is also optimal to accept when your competitors have more inventory. [**Remark**]{} The intuition is that when there are more inventory in the hands of the competitors, they will be more aggressive, and it will then lower your expected marginal gain. Thus, you will be more likely to accept a lower price. The inequality in (6) tells us that A lower selling amount of his/her competitors will make a seller less likely to accept a fixed price; certainly, a higher selling amount will make the same seller more likely to accept the same price. Intuitively, observing more sells from ones competitors will make a seller more aggressive. Conclusions {#sec:conclusions} =========== In this paper, we extend the classic stochastic knapsack problem to model competitions between several sellers and effects on their dynamic pricing decisions. By utilizing dynamic programming techniques, together with a game theoretical model on the sellers’ behavior, we are able to identify a simple strategy, i.e. checking the balance inequality, for each seller, and a dynamic recursion for calculating the value functions required. Furthermore, we show that the value functions have several important first and second order monotonicity properties that are of important theoretical values and critical practical implications. Proof of Theorem \[thm:monotone\] {#sec:proof} ================================== [**Proof**]{} It is easy to see that (1) and (3) are trivial. We will prove the rest by backward induction on time $t$. First, it is trivial to check all of them at the end of selling season, time $T$. Next, suppose that at time period $t+1$ and later, the properties (2) and(4) through (6) hold. We want to extend all the result to time period $t$. Since selection is based on the static probabilities, to facilitate our discussion, denote $\Pi_0$ the event that no seller is selected, and $\Pi_i, i=1,\ldots, N$ the event that seller $i$ is selected. From our assumptions, it is clear that the probabilities of these events are $\pi_i, i=0,1,\ldots, N$, respectively. Furthermore, since all the demand random variables are i.i.d, it is suffice to focus on the event that the price of the demand is $p_i$, $i=1,\ldots, N$. We will use a generic notation $p$ to denote the price, for the ease of exposition. Validity of (2) {#validity-of-2 .unnumbered} --------------- Recall that, we need to establish ${{\bf\sf E}}_{i, t-1}[v_i(t, d_i, {{\bf s}})] \le {{\bf\sf E}}_{i, t-1}[v_i(t, d_i, {{\bf s}}+{{\bf e}}_j)]$, for $j \neq i$. Without loss of generality, it suffices to show, ${{\bf\sf E}}_{1, t-1}[v_1(t, d_1, {{\bf s}}) ]\le {{\bf\sf E}}_{1, t-1}[v_1(t, d_1, {{\bf s}}+{{\bf e}}_j)]$, for any $j>1$. We will argue that the inequality holds on each event $\Pi_i, i=0,1,\ldots, N$. On $\Pi_0$, since no seller is selected, it is easy to see that the inequality holds by induction, and the induction arguments also applies to $\Pi_i, i\neq 1$ and $i\neq j$. On $\Pi_1$, examine what happens at time $t$, the only case that is not straightforward is that seller one only accept given that the history is ${{\bf s}}$ but reject when it is ${{\bf s}}+{{\bf e}}_j$. In this case, the left hand side (LHS) of the inequality becomes ${{\bf\sf E}}_{1, t}[v_1(t+1, d_1-1, {{\bf s}}+{{\bf e}}_1)]+p$. By induction, it is less than or equal to ${{\bf\sf E}}_{1, t}[v_1(t+1, d_1-1, {{\bf s}}+{{\bf e}}_1+{{\bf e}}_j)]+p$. Meanwhile, ${{\bf\sf E}}_{1, t}[v_1(t+1, d_1-1, {{\bf s}}+{{\bf e}}_j+{{\bf e}}_1))]+p\le {{\bf\sf E}}_{1, t}[v_1(t+1, d_1, {{\bf s}}+{{\bf e}}_j)]$ due to the fact that this demand is not accepted when the history is ${{\bf s}}+{{\bf e}}_j$. Hence, the inequality follows. On $\Pi_j, j>1$, there are two cases need to be considered depending on whether seller $j$ accepts the demand. Case I, seller $j$ only accepts when the history is ${{\bf s}}$ not when it is ${{\bf s}}+{{\bf e}}_j$. In this case, we have both the LHS and the right hand side (RHS) equal to ${{\bf\sf E}}_{1, t}[v_1(t+1, d_1, {{\bf s}}+{{\bf e}}_j)]$. Case II, seller $j$ accepts in both cases. Then, the desired inequality is a consequence of ${{\bf\sf E}}_{1, t}[v_1(t+1, d_1, {{\bf s}}+{{\bf e}}_1)] \le {{\bf\sf E}}_{1, t}[v_1(t+1, d_1, {{\bf s}}+2{{\bf e}}_1)]$, which is the consequence of induction. Validity of (4) {#validity-of-4 .unnumbered} --------------- Without loss of generality, we only need to show, $$\begin{aligned} &{{\bf\sf E}}_{1, t-1}[v_1(t, d_1, {{\bf s}})] -{{\bf\sf E}}_{1, t-1}[v_1 (t, d_1 -1, {{\bf s}}+{{\bf e}}_1)]\\ \ge &{{\bf\sf E}}_{1, t-1}[v_1(t, d_1+1, {{\bf s}})] - {{\bf\sf E}}_{1, t-1}[v_1(t, d_1, {{\bf s}}+{{\bf e}}_1)]. \end{aligned}$$ Let us first consider case by case based on whether demand will be accepted by seller one. From the induction assumption for time $t+1$, we know that there are only the following cases, - the demand is only accepted when the inventory is at $d_1+1$ not when it is $d_1$; - the demand is accepted when the inventory levels are at both $d_1+1$ and $ d_1$; - the demand is rejected in either case. And we will discuss each case for events $\Pi_0, \Pi_1$ and $\Pi_j, j>1$. In Case I, on event $\Pi_0$, the inequality follows from induction, i.e. the concavity with respect to the inventory, at time $t+1$. On the event $\Pi_1$, the LHS becomes ${{\bf\sf E}}_{1, t}[v_1(t+1, d_1, {{\bf s}}) -v_1 (t+1, d_1-1, {{\bf s}}+{{\bf e}}_1)] $, and the RHS becomes $p$, then the inequality follows because the balance inequality is violated, which is exactly the reason the demand is not accepted when the inventory is at $(d_1, {{\bf s}})$. On $\Pi_j$, $j \ge 2$, since the decision of seller $j$ will not depend on the actual amount of inventory seller one has, but just the distribution, the RHS becomes, $ {{\bf\sf E}}_{1, t}[v_1(t+1, d_1+1, {{\bf s}}+{{\bf e}}_j ) - v_1(t+1, d_1, {{\bf s}}+{{\bf e}}_j+{{\bf e}}_1)]$. Hence, the inequality will follow from the concavity with respect to inventory from time $t+1$ due to induction assumption. In Case II, again, we only need to look at event $\Pi_1$, where the LHS becomes $p$ and the RHS becomes ${{\bf\sf E}}_{1, t}[v_1(t+1, d_1, {{\bf s}}) -v_1 (t+1, d_1-1, {{\bf s}}+{{\bf e}}_1)]$, and the inequality follows from the balance inequality. Finally, in Case III, the inequality follows from induction. Validity of (5) {#validity-of-5 .unnumbered} ---------------- Again, we need to show that, $$\begin{aligned} &{{\bf\sf E}}_{1, t-1}[v_1(t, d_1, {{\bf s}})] - {{\bf\sf E}}_{1, t-1}[v_1(t, d_1-1, {{\bf s}}+{{\bf e}}_1)] \\ \ge & {{\bf\sf E}}_{1, t-1}[v_1(t+1, d_1, {{\bf s}})] - {{\bf\sf E}}_{1, t-1}[v_1(t+1, d_1-1, {{\bf s}}+{{\bf e}}_1)].\end{aligned}$$ We will examine the inequality on each event $\Pi_i$, i=0,1, …, N. On $\Pi_0$, the inequality follows directly from the induction assumption. On $\Pi_1$, let us consider three subcases. First, it is again a straightforward conclusion from the induction assumption if the demand is not accepted for either inventory level. On the other hand if it is accepted for both inventory levels, then the inequality holds due to the induction assumption on the validity of (4) at time $t$ and $t+1$. If seller one only accepts when the inventory level is at $d_1$, but not when it is at $d_1-1$, the LHS will become $p$, then by the condition of accept, i.e. the balance inequality, it is larger than the RHS. On $\Pi_j, j \ge 2$, the inequality follows from the induction assumption on (6) if the demand is accepted for both inventory levels. By the distributional assumption, that is all that needs to be considered. Validity of (6) {#validity-of-6 .unnumbered} ---------------- It is our task to show that, for $j\ge 2$, $$\begin{aligned} &{{\bf\sf E}}_{1, t-1}[v_1(t, d_1, {{\bf s}})] - {{\bf\sf E}}_{1, t-1}[v_1(t, d_1 -1, {{\bf s}}+{{\bf e}}_1)] \\ &\ge {{\bf\sf E}}_{1, t-1}[v_1(t, d_1, {{\bf s}}-{{\bf e}}_j)] - {{\bf\sf E}}_{1, t-1}[v_1(t, d_1-1, {{\bf s}}-{{\bf e}}_j+{{\bf e}}_1)].\end{aligned}$$ On the event $\Pi_1$, we know that, by induction assumption, we only need to consider the case that the seller one accepts the demand when the inventory level is at $d_1$, but not when it is at $d_1-1$. In this case, the LHS becomes $p$. For the RHS, consider the two cases that seller one accepts in both cases and only accepts when the inventory is $d_1$ but not $d_1-1$. In the first case, it becomes $$\begin{aligned} {{\bf\sf E}}_{1, t}[v_1(t+1, d_1-1, {{\bf s}}-{{\bf e}}_j+{{\bf e}}_1)] - {{\bf\sf E}}_{1, t}[v_1(t+1, d_1-2, {{\bf s}}-{{\bf e}}_j+{{\bf e}}_1)].\end{aligned}$$ Then the inequality follows from the condition that the seller accepts when the inventory and history is $( d_1-1, {{\bf s}}-{{\bf e}}_j)$. In the second case, both the LHS and RHS become $p$. Now for the event $\Pi_j$, again, the one non-trivial case is similar. Hence, the LHS becomes, $$\begin{aligned} &{{\bf\sf E}}_{1, t}[ v_1(t+1, d_1, {{\bf s}}+{{\bf e}}_j)] - {{\bf\sf E}}_{1, t}[v_1(t+1, d_1-1, {{\bf s}}+ {{\bf e}}_j+{{\bf e}}_1)]\\ & \ge {{\bf\sf E}}_{1, t}[v_1(t+1, d_1, {{\bf s}})] - {{\bf\sf E}}_{1, t}[v_1(t+1, d_1-1, {{\bf s}}+{{\bf e}}_1 )],\end{aligned}$$ and the inequality thus follows by induction. This concludes the proof. $\Box$

--- abstract: 'Motivated by recent experiments, we investigate the excitation energy of a proximitized Rashba wire in the presence of a position dependent pairing. In particular, we focus on the spectroscopic pattern produced by the overlap between two Majorana bound states that appear for values of the Zeeman field smaller than the value necessary for reaching the bulk topological superconducting phase. The two Majorana bound states can arise because locally the wire is in the topological regime. We find three parameter ranges with different spectral properties: crossings, anticrossings and asymptotic reduction of the energy as a function of the applied Zeeman field. Interestingly, all these cases have already been observed experimentally. Moreover, since an increment of the magnetic field implies the increase of the distance between the Majorana bound states, the amplitude of the energy oscillations, when present, gets reduced. The existence of the different Majorana scenarios crucially relies on the fact that the two Majorana bound states have distinct $k$-space structures. We develop analytical models that clearly explain the microscopic origin of the predicted behavior.' author: - 'C. Fleckenstein' - 'F. Domínguez' - 'N. Traverso Ziani' - 'B. Trauzettel' title: Decaying spectral oscillations in a Majorana wire with finite coherence length --- Indroduction ============ Majorana fermions are fermionic particles which are their own antiparticles, i.e. $\gamma=\gamma^\dagger$.[@Majorana1937a] In condensed matter physics, these particles arise as quasiparticle excitations in topological superconductors.[@Volovik1999a; @Read2000a; @Kitaev2001a] Models for engineering topological superconductivity and its detection have been the matter of an extensive research over the last decade. The common ingredient in most of these models consists in proximitizing s-wave superconductivity into a system with strong spin-orbit interaction.[@Fu2008a; @Linder2010a; @Lutchyn2010a; @Oreg2010a; @Choy2011a] Their interest is not only fundamental but also practical because they exhibit non-abelian statistics[@Ivanov2001a; @Nayak2008a; @Alicea2011a] and therefore, can potentially be used in protocols for topological quantum computation. Signatures of Majorana bound states (MBSs) are predicted to appear in electrical conductance,[@Bolech2007a; @Law2009a; @Wimmer2011a; @Prada2012a], thermal conductance,[@Wimmer2010a; @Akhmerov2011a; @Sothmann2016a] ac-Josephson effect,[@Kwon2004a; @Jiang2011a; @Badiane2011a; @San-Jose2012a; @Dominguez2012a; @Houzet2013a; @Pikulin2011b; @Dominguez2017b; @Pico2017a] and studying the skweness of the $4\pi$-periodic supercurrent.[@Tkachov2013a] Indeed, experimental measurements confirm some of these predictions in the conductance,[@Mourik2012a; @Rodrigo2012a; @Deng2012a; @Das2012a; @Churchill2013a] Shapiro steps,[@Rokhinson2012a; @Wiedenmann2016a; @Bocquillon2016a], Josephson radiation[@Deacon2017a] and skweness of the supercurrent profile.[@Sochnikov2015a] In the last years, the quality of spin-orbit coupled quantum wires substantially increased.[@hg1; @hg2] Moreover, a new generation of proximitized Rashba wires were fabricated that exhibit a hard superconducting gap.[@Krogstrup2015a] Some of these devices showed robust zero bias conductance peaks,[@Zhang2016a; @Deng2017a] and others allowed to explore excitation energy oscillations produced by an external magnetic field.[@Albrecht2016a] In this article, we will focus on the study of conductance oscillations that arise in the Majorana-Rashba wire. It is well established,[@Cheng2009a; @Prada2012a; @Rainis2013a; @DasSarma2012a] that the origin of these oscillations resides in the spatial overlap between the MBSs typically located at the ends of the wire: The MBS wave functions exhibit an oscillatory exponential decay towards the center. In the limit of high magnetic fields, the finite energy resulting from the overlap between the modes is approximately given by[@Cheng2009a; @DasSarma2012a] $$\begin{aligned} \Delta E\approx \frac{\hbar^2k_\text{F,eff}}{m \chi}\cos\left(k_\text{F,eff} L\right) \exp\left(-\frac{2 L}{\chi}\right), \label{eq:Eoverlap}\end{aligned}$$ where $k_\text{F,eff}$, $L$, and $\chi$ are the effective Fermi wave vector, the length of the wire and the localization length of the MBS, respectively. Due to the fact that $k_\text{F,eff}$ and $\chi$ increase with the magnetic field, the resulting overlap, and hence the conductance, should exhibit an oscillatory pattern with an increasing amplitude. Recent experiments,[@Albrecht2016a] performed in Coulomb blockade Majorana islands[@Fu2010a; @Hutzen2012a; @Aguado2017a] show, however, clear deviations from this picture: For an increasing magnetic field, most samples experience a decaying amplitude of the oscillations, resulting into crossings and anticrossings. On top of that, some samples feature that oscillations remain pinned at zero energy for a wide range of magnetic field ($\sim40\,$mT). Furthermore, other samples manifest a vanishing conductance at high magnetic fields. ![$(a)$ Schematic of the system. The spin-orbit coupled wire is placed on top of a substrate and partially covered by a superconductor. We assume a space dependent proximity induced pairing amplitude $\Delta(x)$. $(b)$ Numerical tight-binding calculation of the lowest energy eigenvalues, as a function of the Zeeman energy $B$ for $N=200$, $L=2~\mathrm{\mu m}$, $\alpha=-20.2$ meVnm, $\Delta_0=1.26$ meV, $\mu=0$, $\xi=0.8~\mathrm{\mu m}$. $(c)$ Density of the corresponding eigenfunctions for $B=0.9~B_c$, and $(d)$ $B=1.1 B_c$ as a function of $x$. In $(c)$ and $(d)$ the unspecified parameters have the same value as in $(b)$.[]{data-label="Fig:profile"}](majorana2.pdf){width="1.\linewidth"} Motivated by these experimental results, some theoretical approaches introduced extra features into the original model:[@Lutchyn2010a; @Oreg2010a] Adding Coulomb interactions between the electrons in the wire and the dielectric environment leads to zero energy pinning.[@Dominguez2017a] Including leakage current effects, coming from the presence of a drain in the superconductor, this leads to a vanishing conductance.[@Danon2017a] Finally, the emergence of decaying oscillations can be obtained, taking into account orbital effects,[@Klinovaja2017a] or wires with multiple occupied subbands, high temperature, and simultaneous presence of Andreev bound states and MBSs.[@Chiu2017a; @Liu2017a] We, instead, study a simple scenario how topological decaying oscillations (see Fig. \[Fig:profile\](b)) can appear: We introduce a finite coherence length in the superconducting pairing, that is, $$\begin{aligned} \Delta(x)=\Delta_0 \tanh(x/\xi), \label{pairing}\end{aligned}$$ where $\xi$ is the coherence length of the superconductor [@Prada2012a; @Stanescu2012a; @Rainis2013a; @Osca2013a] (see Fig. \[Fig:profile\](a)). Such a model is appropriate in a wide range of experimentally relevant situations. Indeed, in genuinely one-dimensional problems, such as the one we aim to describe, the superconducting pairing potential varies on length scales comparable to the coherence length as the geometrical end of the superconductor is approached.[@Volkov1974a; @*Volkov1974b; @Likharev1979a; @Klapwijk2004a] The one-dimensional character of the physical setup is plausible, for instance, when superconductivity is induced by coating the nanowire with a thin film. Moreover, a smooth pairing potential is expected to be present if atoms of the coating are diffusing into the wire. In the latter case, however, the length $\xi$ is not directly related to the coherence length of the superconductor.[@Golubov2004a] The strength of the induced gap $\Delta_0$ does not only depend on the bare gap of the superconductor, but also on the contact between the wire and the Al film. To take all these options into account, we hence keep $\Delta_0$ and $\xi$ as independent variables. Under such a pairing potential, the critical field for observing MBSs reduces from $B_\text{c}=\sqrt{\Delta_0^2+\mu^2}$ to $B\approx \vert\mu\vert$ with $\mu$ the chemical potential.[@Prada2012a] When the wire is globally in the topological phase, the Majorana fermions are located close to the left and right ends of the wire (see Fig. \[Fig:profile\](d)). When, on the other hand, $\vert\mu\vert<B<B_{c}$, two MBS arise, placed close to the left end of the wire and the position $x_B$ satisfying $B=\sqrt{\Delta(x_B)^2+\mu^2}$ (see Fig. \[Fig:profile\](c)). It is interesting to note that in this case the magnetic field shifts the distance between MBSs, and thus, the maximum overlap between them decreases, which is reflected in their spectrum (see Fig. \[Fig:profile\](b)). Although some numerical results along these lines have been presented in Ref. , new experimental results motivate a more careful analysis and understanding of a position dependent pairing. Here, we study numerically and analytically the shape of MBS wave functions arising below the critical bulk field $\vert\mu\vert \lesssim B <B_\text{c}$ for an arbitrary coherence length $\xi$. In striking contrast to the constant pairing scenario, we find two different Majorana fermion solutions with different $k$ space structure. A decaying oscillatory wave function placed close to the left end of the wire and a gaussian-like wave function placed at $x_B$ characterize the system. The difference in the nature of the two Majorana fermions crucially influences their overlap, which, as $\alpha$, $\mu$ and $\xi$ are varied, can result in decaying oscillations, anticrossings or asymptotic decrease. All three scenarios have all been observed in experiments. As a further analysis of the properties of the model, we calculate the local linear conductance $G$ as a function of the applied magnetic field. We find that, in correspondence to the crossings and anticrossings in the lowest lying eigenvalues, $G$ develops non-quantized peaks. Interestingly, whenever $\xi$ is non-zero, the sharp transition between $G=0$ and $G=2e^2/h$, routinely associated to the topological phase transition, takes place for magnetic fields smaller than the value needed for the bulk topological phase transition. This behavior is in accordance with the above mentioned possibility of having Majorana bound states before the topological phase transition. The outline of the paper is as follows: In Sec. \[s2\] we present the Majorana-Rashba model.[@Lutchyn2010a; @Oreg2010a] Then, in Sec. \[s3\], we discuss qualitatively the main results. In Sec. \[sG\], we complement the qualitative analysis with quantitative calculations of the differential conductance. In Sec. \[s4\], we present analytical approaches to the problem and carefully characterize the oscillations as a function of the Rashba spin-orbit coupling strength and the chemical potential. Finally, we conclude in Sec. \[s5\] Model {#s2} ===== We study the Hamiltonian presented in Refs. \[, \] ${H}_c=\frac{1}{2}\int_0^{L} dx {\Psi}^{\dagger}(x)\mathcal{{H}}(x){\Psi}(x)$ with $$\begin{aligned} \mathcal{{H}}(x)&=&\left(\frac{-\partial_x^2}{2m^*}-\mu\right)\tau_z\otimes\sigma_0\nonumber\\ &-&i\alpha\partial_x\tau_z\otimes\sigma_z+B\tau_z\otimes\sigma_x+\Delta(x)\tau_x\otimes\sigma_z, \label{Eq:Hamilton}\end{aligned}$$ where $\Psi^{\dagger}(x)=[\psi_{\uparrow}^{\dagger}(x),\psi_{\downarrow}^{\dagger}(x),\psi_{\downarrow}(x),\psi_{\uparrow}(x)]$. The operators $\psi_{\uparrow,\downarrow}(x)$ annihilate a $\uparrow/\downarrow$ particle at position $x$ and the Pauli matrices $\sigma_i,~\tau_i$ with $i\in\{x,y,z\}$ act on spin- and particle-hole-space, respectively. In addition, $B=\frac{1}{2} g \mu_B B_x$ is the Zeeman energy, originating from a magnetic field applied in the $x$-direction $B_x$ ($B>0$ throughout the article), $m^*= 0.015 m_e$ if we choose the InSb effective mass, and $\mu$ is the chemical potential. The pairing potential $\Delta(x)$ is given by Eq. . Using standard finite difference methods, we discretize Eq.  yielding a $4N\times 4N$ matrix, henceforth called $\hat{H}_w$. Here, we use $N$ for the total number of sites, and thus, $L=a_0 N$ is the length of the wire with $a_0$ the lattice spacing. This Hamiltonian has $4N$ eigenstates, denoted as $\psi^\nu(x)= (u_\uparrow^\nu,u_\downarrow^\nu,v_\downarrow^\nu,v_\uparrow^\nu)$ with corresponding eigenvalues $\epsilon_\nu$. Conductance calculations are obtained by coupling the Majorana wire to normal contacts at each end. We account for the coupling to the leads by adding a constant and diagonal self-energy $\hat{\Sigma}_\text{L,R}^{r/a}=\pm i \hat{\Gamma}_\text{L,R}$, where $$\begin{aligned} &\hat{\Gamma}_\text{L}=\gamma_\text{L} \text{diag}(\hat{1},\hat{0},\hat{0},\cdots,\hat{0})_N,\label{gammaL}\\ &\hat{\Gamma}_\text{R}=\gamma_\text{R} \text{diag}(\hat{0},\hat{0},\hat{0},\cdots,\hat{1})_N.\label{gammaR}\end{aligned}$$ Here, $\hat{1}$ and $\hat{0}$, denote the identity and zero $4\times 4$ matrices. The subindex $N$ refers to the number of $4\times 4$ matrix entries, yielding a $4N\times 4N$ matrix $\hat{\Gamma}_\text{L,R}$. Here, $\gamma_l=\pi \rho_l t_l^2$ is the broadening of the level coupled to the normal lead, and $\rho_l$ is the density of states of the $l$-lead, and $t_l$ a coupling constant. Thus, we can construct the retarded and advanced Green’s function of the open system as $$\begin{aligned} \mathcal{G}^{r/a}(\omega)= \lim_{\eta\to \pm 0}[\omega-\hat{H}_w- \hat{\Sigma}_L^{r/a} - \hat{\Sigma}_R^{r/a}+ i\eta]^{-1},\end{aligned}$$ where $\mathcal{G}^{a}=(\mathcal{G}^{r})^\dagger$. Using Keldysh techniques,[@Cuevas1996a] and assuming a negligible quasiparticle contribution, one can express the zero bias conductance in the $l$-lead as $$\begin{aligned} \label{Eq:Cond} G_l= \frac{2e^2}{h} (T_{\text{LAR},l}+T_{\text{CAR},l}),\end{aligned}$$ where $$\begin{aligned} &T_{\text{LAR},l}= 4\text{Tr}\left[\hat{\Gamma}_{l}^e \mathcal{G}^r(0)\hat{\Gamma}_{l}^h \mathcal{G}^a(0)\right],\\ &T_{\text{CAR},l}=4\text{Tr}\left[\hat{\Gamma}_{l}^e \mathcal{G}^r(0)\hat{\Gamma}_{\overline{l}}^h \mathcal{G}^a(0)\right],\end{aligned}$$ are the local and crossed Andreev reflection at the $l$-lead, respectively. Here, $l=\text{L,R}$ and $\overline{l}=\text{R,L}$. Besides, $\hat{\Gamma}_{l}^{e/h}$ are $4N\times 4N$ matrices, keeping only the electron/hole-like contributions of Eqs.  and . Main results {#s3} ============ In this section, we characterize the effects that a finite coherence length introduce in the critical field and the oscillating pattern resulting from the hybridization of MBSs. To this aim, we diagonalize the discretized version of Eq.  and compare in Fig. \[fig:e\_B\_mu\] the lowest energy states in the parameter space $(B,\mu)$ for different coherence lengths: $\xi=10$nm, $\xi=200$nm, $\xi=400$nm and $\xi=1\mu$m. We can observe that for $B>\sqrt{\Delta_0^2+\mu^2}\equiv B_{c}$ (see blue curve in Fig. \[fig:e\_B\_mu\]) the qualitative topological properties of the Rashba-wire are still present, i.e. MBSs localized close to the two ends of the wire arise and oscillate with increasing amplitude for increasing magnetic fields. For $B<B_{c}$, MBSs arise for an increasing $\xi$, see Fig. \[fig:e\_B\_mu\](a)-(d). The reason for this appearance can be understood if we consider a slowly varying pairing potential. In this situation, the critical condition $B_\text{c}(x)=\sqrt{\Delta(x)^2+\mu^2}$ can be satisfied locally, and thus, two MBSs arise: One is placed close to the left end of the wire, $x_\nu\sim 0$, and another one at $x_B$, where the relation $B=\sqrt{\Delta(x_B)^2+\mu^2}$ is satisfied. Note that $x_B$, and thus, the distance between the MBSs, increases for an increasing magnetic field. Roughly speaking, the requirement for having Majorana fermions is hence no longer $B>B_{\text{c}}$, but becomes related to the existence of the point $x_B$, that is guaranteed to exist for $B>B_{\mu}\equiv |\mu|$. This behavior is indeed what we observe in Fig. \[fig:e\_B\_mu\]: For an increasing coherence length, zero energy states approach asymptotically to $B=\vert\mu\vert$ (see black curves in Fig. \[fig:e\_B\_mu\]). Interestingly, this means that MBSs are present whenever the system is in the quasi-helical regime of the spin-orbit coupled wire,[@Streda2003a; @Meng2013a; @Gambetta2015a; @Gambetta2014a] that is, whenever the system in the absence of superconductivity is effectively spinless. We observe deviations from this behavior for shorter coherence lengths (see Fig. \[fig:e\_B\_mu\] (b)), and zero-energy states can arise even for $B<\vert\mu\vert$. ![Numerical results for the lowest energy eigenvalues of the spin-orbit coupled wire as a function of Zeeman energy $B$ and chemical potential $\mu$ in meV. The calculations are done for $N=200$, $L=2~\mathrm{\mu m}$, $\alpha=-20.2$ meVnm, $\Delta_0=0.63$ meV and different values for the coherence length $\xi$: (a) $\xi=10$ nm, (b) $\xi=200$ nm, (c) $\xi=400$ nm and (d) $\xi=1~\mathrm{\mu m}$. We highlight the lines $B=\vert\mu\vert$ and $\mu=\pm \sqrt{B^2-\Delta^2}$ in black and blue, respectively.[]{data-label="fig:e_B_mu"}](drago.pdf){width="1\linewidth"} The existence of MBSs below $B_{\text{c}}$ is not the only interesting effect of a finite coherence length. The dependence that the lowest energy level has as a function of the applied magnetic field is also remarkable. Since the distance between the two Majorana fermions increases when the magnetic field is increased, the resulting overlap decreases, see Figs. \[Fig:profile\] (b) and \[Fig:MainResult1\] (a)-(b). This feature is often observed in experiments and is difficult to interpret. However, in the context of a finite coherence length, decaying oscillations for $B<B_{\text{c}}$ appear naturally. In this scenario, decaying oscillations are, interestingly, just one of the possible behaviors. It is worth to notice that decaying oscillations (Fig. \[Fig:MainResult1\] (a)) can evolve into anticrossings (Fig. \[Fig:MainResult1\] (c)) and finally into an monotonic decay to zero (Fig. \[Fig:MainResult1\] (d)) as the chemical potential or the spin-orbit coupling are increased. ![Numerical results of the low energy eigenvalues of a spin-orbit coupled wire of total length $N=200$, $L=2~\mathrm{\mu m}$ with $\alpha=-22.7$ meVnm, $\Delta_0=0.76$ meV, $\xi=0.2~\mathrm{\mu m}$ and different chemical potential: (a)-(b) $\mu=0.63$ meV, (c) $\mu=0.76$ meV, (d) $\mu=1.01$ meV. (a) and (b) have the same parameters with different scaling of the axes. The vertical dashed line in (b) represents $B=\vert\mu\vert$.[]{data-label="Fig:MainResult1"}](main.pdf) In order to understand the microscopic mechanisms that induce the different patterns, we analyze the physical properties of the lowest energy BdG wavefunctions. The Majorana fermion around $x_B$ is expected to be a non-oscillating function of $x$, in particular for $\mu=0$, a Gaussian [@Oreg2010a]. An oscillating hybridization energy can hence only emerge from an oscillating wavefunction of the Majorana fermion located at $x_{\nu}$. For a qualitative discussion of the wavefunction around $x_{\nu}$, we start by considering the case of a constant superconducting pairing. In this picture, MBSs mainly have dominant contributions from momenta $k$ around $k\equiv k_0=0$ and $k=\pm k_F$. Thus, the MBS wave function can be expressed as the linear combination $\psi\sim \psi_{k_0}+\psi_{k_\text{F}}+\psi_{-k_\text{F}}$.[@Klinovaja2012a] All these contributions have a spinor structure and decay exponentially, with a typical localization length related to the corresponding direct energy gap, that is, $$\begin{aligned} \psi_{j} \propto \exp(-\Delta_{j} x+ i k_j x),\end{aligned}$$ where the index $j$ resembles $k_0$ and $~\pm k_\text{F}$. Around $k_0$ the gap is given by $\Delta_{k_0}=\sqrt{B^2-\mu^2}-\Delta_0$, and does not depend on $\alpha$. In contrast, at $k=\pm k_F$, the gap is given by a fraction of the bare induced superconducting coupling $\Delta_{\pm k_F}=a_\Delta(\alpha,\mu,B)\Delta_0$. The behavior of $a_\Delta(\alpha,\mu,B)$ as a function of $\alpha$, for different values of $\mu$ is depicted in Fig. \[Fig:Relation\] (a)[@San-Jose2012a]. We can observe that as $\alpha$ or $\mu$ are decreased, $a(\alpha,\mu,B)$ and hence $\Delta_{\pm k_F}$ decrease. Note in passing that the contributions $\psi_{\pm k_\text{F}}$, oscillate with the wave vector $k_\text{F}$. Therefore, whenever $\Delta_{\pm k_F}/\Delta_{k_0}< 1$, the wave function will exhibit a spatial oscillatory pattern. In the limit of a slowly varying pairing potential, i. e. $k_F\gg 1/\xi$, it is possible to find similar expressions as in the constant pairing case (see Sec. \[s4\]). Due to the position dependent pairing, the exponents are then replaced by $$\begin{aligned} \Delta_{j} x\rightarrow \int_0^{x} \Delta_{j}(x') dx',\end{aligned}$$ where $\Delta_{j}(x)$ is the result of substituting $\Delta(x)$ into the direct gap expressions. An oscillatory pattern of the hybridization energy is observed when the oscillating contribution of the wavefunction located at $x_{\nu}$ is dominant around the location of the second Majorana ($x=x_B$). This condition is indeed fulfilled whenever $$\frac{ \int_0^{x_B} \Delta_{k_\text{F}}(x) dx}{\int_0^{x_B} \Delta_{k_0}(x) dx}< 1.$$ As shown in Fig. \[Fig:Relation\] (b) (see also Eq. for more details), the relation is satisfied for small values of $a_{\Delta}(\alpha,\mu,B)$, i.e. weak spin-orbit coupling and/or small $\mu$ close to the topological phase transition. For strong spin-orbit coupling and large $\mu$, however, the relation does not hold anymore and the oscillations disappear. A deeper analysis of the wavefunctions, going beyond simple scaling arguments, is presented in Sec. \[s4\]. Note that some physical behaviors described in this section could be transposed into the scenario where the superconducting pairing is roughly constant while the chemical potential acquires a spatial dependence due to, for instance, the presence of smooth confinement.[@Prada2012a; @Kells2012a; @CMoore2018a] ![$(a)$ $a_{\Delta}(\alpha,\mu,B)$ as a function of spin-orbit coupling $\alpha$ in meVnm with $B=0.35$ meV, $\Delta_0=0.63$ meV and $\mu=0.33$ meV (dotted), $\mu=-0.33$ meV (dashed), $\mu=0$ (solid). $(b)$ $R(\alpha,\mu,B)$ for 3 different values of $a$: $a_{\Delta}(\alpha,\mu,B)=1$ (dotted), $a_{\Delta}(\alpha,\mu,B)=0.8$ (dashed), $a_{\Delta}(\alpha,\mu,B)=0.6$ (solid).[]{data-label="Fig:Relation"}](figeq.pdf){width="1\linewidth"} Conductance {#sG} =========== We complement the analysis of the previous section with a quantitative calculation of the zero bias conductance $G_\text{L}:=G$ in the notation of Eq. (\[Eq:Cond\]). In the following, we will analyse two different scenarios: crossings/anticrossings and the asymptotic approach to zero. [*Crossings/anticrossings*]{}. — Anticrossings with an energy gap $\delta\epsilon$, give rise to $hG/(2e^2)\sim 2\gamma_L^2/(\delta\epsilon)^2\ll 1$, for $\delta \epsilon\gg \gamma_\text{L}$ (see Fig. \[Fig:cond\]). In turn, the crossing points exhibit conductance peaks with values between $4e^2/h<G<2e^2/h$, see Fig. \[Fig:cond\](b). Evidently, each MBS can contribute to the conductance since both MBS wave functions exhibit a finite weight at $x=0$[@Prada2017a]. In this situation, the left (right) MBS is (not) perfectly spin polarized.[@Sticlet2012a; @Prada2017a] This is known as a reason why it contributes to the conductance with $G\approx 2e^2/h$ ($G\lesssim 2e^2/h$), yielding a total subgap conductance of $G<4e^2/h$. This situation is similar to the crossings that can emerge when a non-proximitized part is added to the system before the leads.[@Cayao2015a] [*Asymptotic decay to zero*]{}. — The magnetic field shifts the MBS placed at $x_\text{B}$, yielding an exponential reduction of the energy. When the energy spectrum reaches zero, the conductance jumps abruptly to the quantized value $G=2e^2/h$, even for $B<B_\text{c}$. Interestingly, as $\xi$ is increased, the transition shifts towards smaller values of $B$, see Fig. \[Fig:cond\]$(d)$. The behavior discussed in this section is consistent with the qualitative discussion given in Sec. \[s3\] and with the analytical results given in the following section. ![(a)-(c) Lowest energy eigenvalue of the system (red, left vertical axis) and zero-energy conductance (blue, right vertical axis) as a function of $B/B_c$ with the parameters $N=180$, $L=1.8~\mathrm{\mu m}$, $\Delta_0=0.66$ meV, $\mu=0.2$ meV. Further $\xi=0$ in (a), while $\xi=0.5~\mathrm{\mu m}$ for (b)-(c), as well $\alpha=20.16~$meVnm in (a), (b) and (d), while $\alpha=25.2~$meVnm in (c). In (d), we illustrate the conductance calculation of (a) (yellow), (b) (blue) and a third one with the parameters of (c) but $\xi=0.25~\mathrm{\mu m}$. Furthermore, $\gamma_L=\gamma_R=2.52~$meV.[]{data-label="Fig:cond"}](CondEnergy.pdf){width="1\linewidth"} Analytical analysis {#s4} =================== In order to derive simple relations explaining the behavior observed by means of the numerical solution, we are inspired by Ref. and simplify the model in two regimes: strong and weak spin orbit coupling. Beyond Ref. , we will then solve the models in the presence of a non-uniform pairing potential. Effective model for strong spin-orbit coupling ---------------------------------------------- In the strong spin-orbit coupling regime, $m^*\alpha^2 \gg B,~ \Delta_0$, we complement the continuum model with the assumptions of a slowly varying superconducting pairing potential $m^*\alpha \gg 1/\xi$. Then, the continuum Hamiltonian can be further simplified to effective (linear) Hamiltonians around zero average momentum (i) and momentum $k_F\cong 2m^*\alpha$ (ii) (see also Fig. \[Fig:dispersion\] (a)-(b) for a schematic): $$\mathrm{(i)}~~\langle -i\partial_x\rangle\simeq 0, ~~~~~ \mathrm{(ii)} ~~\langle -i\partial_x\rangle\simeq \pm k_F.$$ For case (i), we are allowed to neglect the quadratic part of Eq. (\[Eq:Hamilton\]), resulting in the low energy Hamiltonian $$\begin{aligned} {H}_I&=&\int_0^{l} dx {\Psi}^{\dagger}(x)\bigg[-i\alpha\partial_x\tau_z\otimes\sigma_z\nonumber\\ &-&\mu\tau_z\otimes\sigma_0+B\tau_z\otimes\sigma_x+\Delta(x)\tau_x\otimes\sigma_z\bigg]{\Psi}(x). \label{Eq:ham3.1}\end{aligned}$$ In case (ii), we can perform a spin-dependent gauge transformation $${\Psi}(x)=e^{-2im^*\alpha x(\tau_0\otimes\sigma_z)}\tilde{\Psi}(x), \label{Eq:trafo}$$ where $\tilde{\Psi}(x)$ is a slowly varying function of $x$ with respect to $1/(m*\alpha)$ at low energy. After plugging Eq. (\[Eq:trafo\]) into Eq. (\[Eq:Hamilton\]) and linearizing, the transformed Hamiltonian becomes $$\begin{aligned} {H}_E&=&\int_0^{l} dx \tilde{\Psi}^{\dagger}(x)\bigg[i\alpha\partial_x\tau_z\otimes\sigma_z\nonumber\\ &-&\mu\tau_z\otimes\sigma_0+a_{\Delta}(\alpha,\mu,B)\Delta(x)\tau_x\otimes\sigma_z\bigg]\tilde{\Psi}(x), \label{Eq:ham3}\end{aligned}$$ where the fast oscillating terms are integrated out.\ Here, $a_{\Delta}(\alpha,\mu,B)\in \{0,1\}$ is determined by the dispersion relation of the lowest energy eigenvalue of Eq. (\[Eq:Hamilton\]) (assuming constant superconducting pairing $\Delta_0$). In case of strong spin orbit coupling, we obtain $a_{\Delta}(\alpha,\mu,B)\rightarrow 1$. The full behavior of $a_{\Delta}(\alpha,\mu,B)$ as a function of $\alpha$ is illustrated in Fig. (\[Fig:Relation\]) (a). For small $\alpha$, we find the analytical expression $$a_{\Delta}(\alpha,\mu,B)=\frac{\sqrt{\Delta_0^2+B^2}-B}{\Delta_0}+\frac{4m^{*2}(B+\mu)}{\Delta_0\sqrt{B^2+\Delta_0^2}}\alpha^2+\mathcal{O}(\alpha^3). \label{Eq:aDelta}$$ The rational behind the approximation scheme leading to Eqs. (\[Eq:ham3.1\]) and (\[Eq:ham3\]) is that, for strong spin-orbit coupling and for weak translational symmetry breaking by the applied superconducting pairing, we expect that the main effect of superconductivity is to renormalize the helical gap close to zero momentum and open a gap close to $k_F$, as schematically shown in Fig \[Fig:dispersion\]. Any low-energy eigenstate is then evaluated as linear combination of eigenstates of ${H}_I$ and ${H}_E$.\ ![Dispersion relation of the full continuum model with $E(k)$ in meV and $k$ in $\mathrm{nm}^{-1}$ in the strong spin-orbit regime ((a)-(b)) with $\alpha=-100$ meVnm, $\Delta_0=0.35$ meV, $\mu=0$ and (a) $B=0.1$ meV, (b) $B=0.33$ meV and weak spin-orbit regime ((c)-(d)) with $\alpha=-15$ meVnm, $\Delta_0=0.35$ meV, $\mu=0$ and (c) $B=0.7$ meV, (d) $B=0.33$ meV.[]{data-label="Fig:dispersion"}](dispersion.pdf) Effective model for weak spin-orbit coupling -------------------------------------------- In case of weak spin-orbit coupling, $m^*\alpha^2\ll B$, we additionally assume $\sqrt{m^*B}\gg 1/\xi$. To develop effective linear models for our purposes, in this case, we explicitly distinguish two regimes in parameter space: deep inside the topological phase and close to the phase transition. Far away from any boundary, the latter case is hence described within the linear Hamiltonian of Eq. (\[Eq:ham3.1\]) (see Fig. \[Fig:dispersion\] (d)), while, close to the boundaries, the contribution around $k=\pm k_F$ is still important. Deep inside the topological phase, the gap opened at $k=0$ is large compared to the gap opened at $k=\pm k_F$ given by $a_{\Delta}(\alpha,\mu,B)\Delta_0$ (see Fig. \[Fig:dispersion\] (c)). For weak spin-orbit coupling, $a_{\Delta}(\alpha,\mu,B)\ll 1$, the low energy physics is described around the points $k=\pm k_F$. An appropriate linear model has to take into account that spins are not (quasi-)helical in the weak spin-orbit regime but acquire a spin-tilting. To implement this feature in the linear model we, demand an artificial $\epsilon$, $\gamma$ and $\nu$, acting as magnetic field, chemical potential and Fermi velocity. The momentum-space Hamiltonian in the absence of superconductivity is given in the spin-resolved basis as $$\label{Eq:Hlin} \mathcal{H}^0_{\text{lin}}(k)=\begin{pmatrix} \nu k-\gamma & \epsilon\\ \epsilon & -\nu k-\gamma \end{pmatrix}.$$ Unlike former linear models of this paper, here we require $\gamma\geq \epsilon$ to make the model appropriate for our purpose. To connect those parameters to the physical parameters of the spin-orbit coupled wire, we demand three conditions to hold: (i) Fermi-surface, (ii) velocity at the Fermi points, and (iii) spin-tilting at the Fermi surface have to coincide in both models. The demands (i)-(iii) result in the following conditions $$\begin{aligned} k_{\text{F,lin}}&=&k_{\text{F,SOC}}\equiv k_{\text{F}},\nonumber\\ v_{\text{F,lin}}&=&v_{\text{F,SOC}}\equiv v_{\text{F}},\nonumber\\ \frac{\nu k_F+\sqrt{\nu ^2k_F^2+\epsilon^2}}{\epsilon}&=&\frac{k_F\alpha - \sqrt{B^2+ k_F^2\alpha^2}}{B}, \label{Eq:EquationSys}\end{aligned}$$ where the last equation originates from the eigenvectors of Eq. (\[Eq:Hamilton\]) with $\Delta(x)=0$ and Eq. (\[Eq:Hlin\]). The equation system (\[Eq:EquationSys\]) has the unique solution $$\begin{aligned} \label{Eq:Vel} \nu &=&\frac{(1+\kappa^2)v_{F}}{-1+\kappa^2},~~\epsilon=\frac{2\kappa(1+\kappa^2)v_{F}k_{F}}{(-1+\kappa^2)^2},\nonumber\\~~\gamma &=&\frac{(1+\kappa^2)^2v_Fk_F}{(-1+\kappa^2)^2}\end{aligned}$$ with the replacements $$\begin{aligned} \kappa &=&\frac{k_{F}\alpha-\sqrt{B^2+k_{F}^2\alpha^2}}{B},\nonumber\\ v_{F}&=& 4m^* k_{F}-\frac{k_{F}\alpha^2}{\sqrt{B^2+k_{F}^2\alpha^2}},\nonumber\\ k_{F}&=&\sqrt{2}\sqrt{m^{*2}\alpha^2+m^{*}\mu+m^{*}\sqrt{B^2+m^{*2}\alpha^4+2m^{*}\alpha^2\mu}}.\nonumber\\\end{aligned}$$ A feature that does not coincides in both models is the spin rotation length along the dispersion relation. However, this feature only plays a minor role for spectroscopic properties. The direction of the spin rotation along the dispersion, however, is the same in both models.\ Including superconducting pairing, the linearized model then becomes $$\begin{aligned} {H}_{\mathrm{lin}}&=&\int_0^{l} dx {\Psi}^{\dagger}(x)\bigg[i\alpha\partial_x\tau_z\otimes\sigma_z -\gamma\tau_z\otimes\sigma_0\nonumber\\ &+&\epsilon\tau_z\otimes\sigma_x+a_{\Delta}(\alpha,\mu,B)\Delta(x)\tau_x\otimes\sigma_z\bigg]{\Psi}(x). \label{Eq:ham3.2}\end{aligned}$$ Wave function at $x_B$ and new critical field {#Sec:subC} --------------------------------------------- The existence of zero energy MBS in the trivial phase can be fully understood within this analytical approach. We first focus on the large $\xi$ limit. In this regime, we can apply the linear approximations of Eqs. (\[Eq:ham3.1\]), (\[Eq:ham3\]) and (\[Eq:ham3.2\]). As a starting point, we concentrate on a system with no boundaries and demand the existence of one point in space, $x=x_B$, which satisfies the relation $\Delta^2(x_B)=B^2-\mu^2$. Around this point, the low-energy physics (for strong and weak spin-orbit coupling) is described within the linear approximation of Eq. (\[Eq:ham3.1\]) only, since it becomes gapless. We, therefore, search for zero-energy solutions $\mathcal{{H}}_I(x)\Phi(x)=0$, where we demand $\Phi(x)$ to be of the form $\Phi(x)=U\chi(x)$, with $$U=\frac{1}{\sqrt{2}}\left(\tau_0\otimes\sigma_0-i\tau_x\otimes\sigma_y\right).$$ This unitary transformation reorganizes the Hamiltonian in the Majorana basis. For $\chi(x)$ we further assume a solution of the form $$\chi(x)=(a,b,c,d)^T\exp[f(x)]. \label{Eq:sol1}$$ After plugging in this ansatz [@Timm2012a; @Fleckenstein2016a; @Traverso2017a; @Porta2018a], we obtain the solution $$\label{Eq:f(x)} f(x)=\pm \int dx \frac{\Delta(x)\pm \sqrt{B^2-\mu^2}}{\alpha}.$$ In the case of linear behavior of $\Delta(x)$ with respect to $x$ and $\mu=0$, we restore the groundstate solution of the displaced harmonic oscillator, solved in Ref. . With the solutions of Eq. (\[Eq:f(x)\]), we obtain the corresponding spinor structure of $\Phi(x)$ $$\begin{aligned} (u,v,\tilde{v},\tilde{u})^T=U(a,b,c,d)^T=\nonumber\\\frac{1}{\sqrt{2}}\left( \pm e^{\pm i\varphi}+i,ie^{\pm i \varphi}\pm 1,-ie^{\pm i \varphi} \pm 1,\pm e^{\pm i \varphi}-i\right)^T, \label{Eq:spinor}\end{aligned}$$ where $\varphi= \arccos(B/\mu)$. To fulfil the Majorana condition $\tilde{u}=u^*$ and $\tilde{v}=v^*$, we need $\exp[i\varphi] \in \mathbb{R}$, which is only true for $$B\geq |\mu|. \label{Eq:Bmu}$$ For a slowly varying $\Delta(x)$, the latter relation represents a bound for the formation of Majorana zero-modes. The hand-waving argument given in the previous section can hence be put on a formal basis.\ Compiling Eqs. (\[Eq:sol1\]), (\[Eq:f(x)\]) and (\[Eq:spinor\]) and imposing that the wavefunction is normalizable and centered around $x_B$, we explicitly obtain $$\Phi(x)=\frac{1}{\sqrt{N}}\begin{pmatrix} e^{i\varphi}+i\\ ie^{i\varphi}+1\\ -ie^{i\varphi}+1\\ e^{i\varphi}-i \end{pmatrix}e^{-\int_0^x dx' \frac{1}{\alpha}\left(\Delta(x')-\sqrt{B^2-\mu^2}\right)}, \label{Eq:Chi1}$$ with the normalization constant $N$. Wavefunction at $x_\nu$ in the strong spin-orbit coupling regime {#Sec:subD} ---------------------------------------------------------------- At the left end of the wire, where the proximity induced pairing decreases to zero, we have to distinguish between effective models of strong and weak spin-orbit coupling. For strong spin-orbit coupling, the low-energy physics is captured by a linear combination of eigenstates of Eqs. (\[Eq:ham3.1\]) and (\[Eq:ham3\]) $$\Psi(x)\simeq a_1\Psi_I(x)+b_1e^{-2i m^*\alpha x (\tau_0\otimes\sigma_z)}\Psi_E(x)$$ with the coefficients $a_1$ and $b_1$ to be derived by the boundary conditions, and $\Psi_l(x)$, $l\in I,E$, satisfying $$\mathcal{H}_l(x)\Psi_l(x)=0. \label{Eq:sol2}$$ The solution for $\Psi_I(x)$ is constructed by means of Eqs. (\[Eq:sol1\]), (\[Eq:f(x)\]) and (\[Eq:spinor\]). The solution for $\mathcal{H}_E(x)$ can be found in an analogous way after multiplying Eq. (\[Eq:sol2\]) (with $l=E$) from the left with $\tau_z\otimes\sigma_z$. Using the properties of Pauli matrices, especially $[\tau_y\otimes\sigma_0,\tau_0\otimes\sigma_z]=0$, where $[.,.]$ denotes the commutator, integration yields the solution $$\begin{aligned} \Psi_E(x)&=&\exp\bigg[\int_0^x dx'\frac{i}{\alpha}\bigg(\Delta(x')(\tau_y\otimes\sigma_0)\nonumber\\ &-&\mu(\tau_0\otimes\sigma_z)\bigg)\bigg]\Psi_0. \label{Eq:PsiE}\end{aligned}$$ Subsequently, the spinor $\Psi_0$ has to be chosen such that the wavefunction satisfies the Majorana condition $\Psi(x)=[u(x),v(x),v^*(x),u^*(x)]^T$, which results in four possible solutions. Furthermore, assuming a semi-infinite system ($x>0$), we have to satisfy the boundary condition $\Psi(0)=0$. Moreover, the solution has to decay away from $x=0$. The first condition implies that the spinors $\Psi_I(0)$ and $\Psi_E(0)$ are linearly dependent. Hence, from Eq. (\[Eq:PsiE\]), we select the solutions for $\Psi_E(x)$ which decay away from $x=0$ and combine them with their linearly dependent counterparts $\Psi_I(x)$. This leads to the only physical solution $$\begin{aligned} \Psi(x)&=&\frac{1}{\sqrt{N}} \begin{pmatrix} i-e^{i\varphi}\\ ie^{i\varphi}-1\\ -ie^{i\varphi}-1\\ -e^{i\varphi}-i \end{pmatrix}e^{\int_0^x dx'\frac{1}{\alpha}\left(\Delta(x')-\sqrt{B^2-\mu^2}\right)}\nonumber\\ &-&\frac{1}{\sqrt{N}}e^{-i(2m^*\alpha+\frac{\mu}{\alpha})x(\tau_0\otimes\sigma_z)} \begin{pmatrix} i-e^{i\varphi}\\ ie^{i\varphi}-1\\ -ie^{i\varphi}-1\\ -e^{i\varphi}-i \end{pmatrix} e^{-\int_0^x dx'\frac{\Delta(x')}{\alpha}}\nonumber\\ \label{Eq:MajoranaSol}\end{aligned}$$ with normalization constant $N$. Wavefunction at $x_\nu$ in the weak spin-orbit coupling regime {#Sec:subE} -------------------------------------------------------------- The wave function at $x_\nu$ for the case of weak spin-orbit coupling ($a_{\Delta}(\alpha,\mu,B)\ll 1$) is given by linear combination of eigenstates of Eq. (\[Eq:ham3.2\]), which indeed have the same form as the eigenstates of Eq. (\[Eq:ham3.1\]) with the replacements $B\rightarrow\epsilon$, $\mu\rightarrow\gamma$ and $\alpha\rightarrow-\nu$. A consequence of neglecting all other contributions in this linear approach is that we can only accomplish the boundary condition $\Psi(0)=0$ if we neglect the contribution of the spin orbit coupling in the spinors. If so, the only reasonable wave function is obtained by $$\begin{aligned} \Psi(x)=\frac{1}{\sqrt{N}}(-1+i,i-1,-i-1,-1-i)^T\nonumber\\ \sin\left(k_{F} x\right)\exp\left[-\int_0^x dx' \frac{a_{\Delta}(\alpha,\mu,B)\Delta(x')}{\nu}\right]. \label{Eq:sin}\end{aligned}$$ Overlap of wavefunctions ------------------------ The analysis of Secs. \[Sec:subC\]-\[Sec:subE\] is done for isolated Majorana fermions in (semi-)infinite space with a spatial variation of the superconducting pairing. However, since Majorana fermions always appear in pairs and our system is finite, there can be a finite hybridization energy between them. The hybridization energy is directly related to the overlap of the two wavefunctions. For $\Delta(x)$ defined in Eq. (\[pairing\]), in the regime where $B<B_{c}$, we can approximate the solution at the left end of the wire, where the proximity induced pairing decreases to zero, by the wavefunction of Eqs. (\[Eq:MajoranaSol\]), (\[Eq:sin\]), when spin-orbit coupling is strong/weak. On the other hand, around $x=x_B$ we have to take into account the wavefunction of Eq. (\[Eq:Chi1\]). For the strong spin orbit coupling regime, Eq. (\[Eq:MajoranaSol\]) has an oscillatory and a non-oscillatory part, while Eq. (\[Eq:Chi1\]) is non-oscillatory. Therefore, the hybridization energy will also be constituted by an oscillatory and a non-oscillatory part. Which of them is dominant is strongly dependent on the corresponding decay length and the relative position of the states. The hybridization energy is expected to show an oscillatory nature if the oscillatory part of Eq. (\[Eq:MajoranaSol\]) is dominant when $x=x_B$ is approached, This is the case if $$\int_0^{x_B} dx (1+a_\Delta (\alpha,\mu,B))\Delta(x)- \sqrt{B^2-\mu^2}< 0. \label{Eq:Relation4}$$ For $\Delta(x)$, following Eq. (\[pairing\]) with $B^2<\Delta_0^2+\mu^2$, $x_B$ is determined by $$x_B=\xi \mathrm{arctanh}\left(\frac{\sqrt{B^2-\mu^2}}{\Delta_0}\right). \label{Eq:xB}$$ ![Lowest energy eigenfunctions as a function of $x$ (left) with $B=1.134$ meV and eigenenergies (right) of the spin orbit coupled wire with proximity induced s-wave pairing with $\Delta_0=1.25$ meV, $N=250$, $L=2.5~\mathrm{\mu m}$, $\xi=0.8~\mathrm{\mu m}$, $\mu=0$ for three different values of the spin orbit coupling: (a)-(b) $\alpha=-15.12$ meVnm, (c)-(d) $\alpha=-37.8$ meVnm and (e)-(f) $\alpha=-75.6$ meVnm.[]{data-label="Fig:OscCombo"}](19_09.pdf){width="1\linewidth"} After performing the integration, we obtain $$\begin{aligned} R(\alpha,\mu,B)&\equiv &-\frac{1}{2}[1+a_\Delta(\alpha,\mu,B)]~\mathrm{ln}(1-\eta^2)\nonumber\\ &-&\eta~\mathrm{arctanh}(\eta)< 0, \label{Eq:Relation5}\end{aligned}$$ with $\eta=\sqrt{B^2-\mu^2}/\Delta_0$. $R(\alpha,\mu,B)$ is illustrated for different values of $a_{\Delta}(\alpha,\mu,B)$ in Fig. \[Fig:Relation\]. ![Numerical results for the lowest energy eigenvalue as a function of Zeeman energy $B$ and chemical potential $\mu$ in meV. The calculations are done for: $\Delta_0=0.63$ meV, $\xi=0.5~\mathrm{\mu m}$, $L=2~\mathrm{\mu m}$ and different spin-orbit coupling: (a) $\alpha=-10.1$ meVnm, (b) $\alpha=-20.2$ meVnm, (c) $\alpha=50.4$ meVnm and (d) $\alpha=-100.8$ meVnm.[]{data-label="Fig:AlphaDep"}](fig4.pdf){width="1.05\linewidth"} If $a_{\Delta}(\alpha,B,\mu)\rightarrow 1$, which is the case in the strong spin-orbit regime, Eq. (\[Eq:Relation5\]) can not be fulfilled. Hence, the long wave contribution will always dominate the behavior of the wavefunction at $x=x_B$ and the hybridization energy will show a non-oscillatory behavior, which is coherent with numerical results (Fig. \[Fig:OscCombo\] (e), (f)). For very large $\alpha$, similar arguments hold for the hybridization energy in the topological phase. Since the decay length of the wavefunction in the strong spin-orbit regime is proportional to $\alpha$, the overlap of the different wavefunctions is large, resulting in a suppression of zero-modes before the topological phase for large $\alpha$ (see Fig. \[Fig:AlphaDep\]).\ For $a_\Delta(\alpha,\mu,B)< 1$, on the other hand, the inequality Eq. (\[Eq:Relation5\]) can be fulfilled for some values of $B$ within $B^2<\Delta_0^2+\mu^2$ (see Fig. \[Fig:Relation\] (b)). The regime of dominant oscillatory behavior is amplified for small $a_\Delta(\alpha,\mu,B)$, i.e. weak spin-orbit coupling, a behavior explaining the numerical results (see Fig. \[Fig:OscCombo\] (a), (b)). In this regime, the wave function at $x=x_\nu$ provides strongly oscillatory character (Eq. (\[Eq:sin\])) leading to an oscillatory behavior of the hybridization energy (see Fig. \[Fig:OscCombo\] (a)-(b)).\ In the region between strong and weak spin-orbit coupling, instead, it is difficult to determine the analytical form of the wave function around $x_{\nu}$. It will be a mixture of Eqs. (\[Eq:MajoranaSol\]) and (\[Eq:sin\]). This is the regime, where we witness anticrossings in the hybridization energy, as the oscillatory and non-oscillatory contribution to the wave function at $x=x_B$ have similar decay lengths (Fig. \[Fig:OscCombo\] (c)-(d)).\ For smaller values of the gap parameter $\Delta_0$, all results remain qualitatively valid. However, the localization of the wavefunctions is reduced resulting in two major physical effects: Firstly, the overlap of the wavefunctions increases, and so does the hybridization energies. Secondly, the wavefunction centered around $x_B$ can significantly deviate from the Gaussian profile and can acquire oscillating contributions. This results in more complex hybridization energies close to the topological phase transition. The reason is that when the localization length of the wavefunction around $x_B$, which increases as $\Delta_0$ decreases, becomes comparable with the distance to the right end of the wire, then Friedel-like finite size oscillations become prominent [@Egger1995a; @Fabrizio1995a; @Eggert2009a; @Kyl2016a]. Since the two length scales involved are the localization length of the wavefunction and the distance to the boundaries of the wire, these effects become more pronounced in short samples. Role of the chemical potential ------------------------------ In the intermediate $\alpha$ regime, which ranges around $\alpha\sim 10-50$ meVnm, the chemical potential $\mu$ plays a crucial role since, especially as $\mu\rightarrow B$, $a_\Delta(\alpha,\mu,B)$ is a strongly asymmetric function with respect to $\mu\rightarrow -\mu$ (see Eq. (\[Eq:aDelta\]) and Fig. \[Fig:Relation\] (a)). As $a_\Delta(\alpha,\mu,B)$ controls the gap size at $k=\pm k_F$, oscillations are more pronounced in the negative $\mu$ regime, which is indeed consistent with numerical results (see Fig. \[fig:e\_B\_mu\] (d)). This allows us to witness low energy eigenvalues with oscillatory, anticrossing or monotonous convergence to zero also in dependence of the chemical potential $\mu$. For experimentally relevant values of $\alpha\sim 20$ meVnm, we indeed expect to be in the transition regime between strong and weak spin-orbit coupling. Interestingly, the tendency to an asymmetric behavior with respect to $\mu$ is maintained qualitatively even for experimantally relevant coherence length. With respect to our findings, the signature of a low energy conductance measurement could give an indication to the magnitude of the spin-orbit coupling as well as the chemical potential inside the wire. Conclusion {#s5} ========== Majorana fermions, i.e. zero-energy bound states, in a spin-orbit coupled quantum wire can exist even when the wire is not in the topological regime. The requirement is a finite coherence length $\xi$ of the proximity induced superconducting pairing amplitude. For slowly varying $\Delta(x)$ (large coherence length), we have given analytical and numerical demonstrations that the existence of zero-modes is possible in the whole $B\geq |\mu|$ region. Moreover, we have demonstrated that the momentum-space decomposition of the two Majorana fermions that can form before the topological phase transition are profoundly different. The one located at the end of the wire has an oscillating wavefunction, while the one located at the end of the locally topological region of the wire has a non-oscillating structure. This particular behavior implies a rich scenario for the hybridization energy. As a function of spin-orbit coupling and chemical potential, different behaviors can be obtained, ranging from a pattern of decaying oscillations, to anticrossings, and to a monotonous decay to zero energy. Oscillations are favored by weak spin-orbit coupling and tendentially small chemical potential, and their amplitude decays as a function of the magnetic field because the two Majorana fermions get separated in space. Stronger spin-orbit coupling and higher chemical potential favor, on the other hand, anticrossings and monotonous decay. We have interpreted the results by means of effective Dirac-like models, which allowed us to understand them as a consequence of different decay lengths characterizing the various momentum components of the Majorana fermion wavefunctions. We thank L. Chirolli for interesting discussions. Financial support by the DFG (SPP1666 and SFB1170 “ToCoTronics”), the Helmholtz Foundation (VITI), and the ENB Graduate school on “Topological Insulators” is acknowledged. [82]{}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 [****, ()](\doibase 10.1007/BF02961314) [****, ()](\doibase 10.1134/1.568223) [****,  ()](\doibase 10.1103/PhysRevB.61.10267) [****,  ()](http://stacks.iop.org/1063-7869/44/i=10S/a=S29) [****, ()](\doibase 10.1103/PhysRevLett.100.096407) [****,  ()](\doibase 10.1103/PhysRevLett.104.067001) [****,  ()](\doibase 10.1103/PhysRevLett.105.077001) [****,  ()](\doibase 10.1103/PhysRevLett.105.177002) [****,  ()](\doibase 10.1103/PhysRevB.84.195442) [****,  ()](\doibase 10.1103/PhysRevLett.86.268) [****,  ()](\doibase 10.1103/RevModPhys.80.1083) [****,  ()](\doibase 10.1038/nphys1915) [****,  ()](\doibase 10.1103/PhysRevLett.98.237002) [****,  ()](\doibase 10.1103/PhysRevLett.103.237001) [****,  ()](http://stacks.iop.org/1367-2630/13/i=5/a=053016) [****,  ()](\doibase 10.1103/PhysRevB.86.180503) [****,  ()](\doibase 10.1103/PhysRevLett.105.046803) [****,  ()](\doibase 10.1103/PhysRevLett.106.057001) [****,  ()](\doibase 10.1103/PhysRevB.94.081407) @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} [****,  ()](\doibase 10.1103/PhysRevLett.108.257001) @noop [****,  ()]{} [****,  ()](\doibase 10.1103/PhysRevLett.111.046401) @noop [****,  ()]{} [****,  ()](\doibase 10.1103/PhysRevB.95.195430) [****,  ()](\doibase 10.1103/PhysRevB.96.125438) [****,  ()](\doibase 10.1103/PhysRevB.88.075401) [****,  ()](\doibase 10.1126/science.1222360) [****,  ()](\doibase 10.1103/PhysRevLett.109.237003) [****,  ()](\doibase 10.1021/nl303758w) [****,  ()](\doibase 10.1038/nphys2479) [****,  ()](\doibase 10.1103/PhysRevB.87.241401) [****,  ()](\doibase 10.1038/nphys2429) [****,  ()](\doibase 10.1038/ncomms10303) [****,  ()](\doibase 10.1038/nnano.2016.159) [****,  ()](\doibase 10.1103/PhysRevX.7.021011) [****,  ()](\doibase 10.1103/PhysRevLett.114.066801) [****,  ()](\doibase 10.1038/nphys1626) [****,  ()](\doibase 10.1038/nphys4070) [****,  ()](\doibase 10.1038/nmat4176) @noop [ ()]{} [****,  ()](\doibase 10.1126/science.aaf3961) @noop [****, ()]{} [****,  ()](\doibase 10.1103/PhysRevLett.103.107001) [****,  ()](\doibase 10.1103/PhysRevB.87.024515) [****,  ()](\doibase 10.1103/PhysRevB.86.220506) @noop [****,  ()]{} [****,  ()](\doibase 10.1103/PhysRevLett.109.166403) [****,  ()](\doibase 10.1393/ncr/i2017-10141-9) [****,  ()](\doibase 10.1038/s41535-017-0012-0) [****,  ()](\doibase 10.1103/PhysRevB.96.125420) @noop [**** ()]{} [****,  ()](\doibase 10.1103/PhysRevB.96.054504) [****,  ()](\doibase 10.1103/PhysRevB.96.075161) [****,  ()](\doibase 10.1103/PhysRevLett.109.266402) [****,  ()](\doibase 10.1103/PhysRevB.88.144512) @noop [****,  ()]{} @noop [****,  ()]{} [****,  ()](\doibase 10.1103/RevModPhys.51.101) [****,  ()](\doibase 10.1007/s10948-004-0773-0) [****,  ()](\doibase 10.1103/RevModPhys.76.411) [****,  ()](\doibase 10.1103/PhysRevB.54.7366) [****, ()](\doibase 10.1103/PhysRevLett.90.256601) [****,  ()](\doibase 10.1103/PhysRevB.88.035437) [****,  ()](\doibase 10.1103/PhysRevB.91.235421) @noop [****,  ()]{} [****, ()](\doibase 10.1103/PhysRevB.86.085408) [****,  ()](\doibase 10.1103/PhysRevB.86.100503) [****,  ()](\doibase 10.1103/PhysRevB.97.165302) [****,  ()](\doibase 10.1103/PhysRevB.96.085418) [****,  ()](\doibase 10.1103/PhysRevLett.108.096802) [****,  ()](\doibase 10.1103/PhysRevB.91.024514) [****, ()](\doibase 10.1103/PhysRevB.86.155456) [****,  ()](\doibase 10.1103/PhysRevB.94.241406) [****,  ()](\doibase 10.1103/PhysRevB.95.205418) [****,  ()](\doibase 10.1103/PhysRevB.97.035433) [****,  ()](\doibase 10.1103/PhysRevLett.75.3505) [****,  ()](\doibase 10.1103/PhysRevB.51.17827) [****,  ()](\doibase 10.1103/PhysRevB.79.195114) [****,  ()](\doibase 10.1103/PhysRevB.94.115417)

--- abstract: 'Subgroup analysis is a frequently used tool for evaluating heterogeneity of treatment effect and heterogeneity in treatment harm across observed baseline patient characteristics. While treatment efficacy and adverse event measures are often reported separately for each subgroup, analyzing their within-subgroup joint distribution is critical for better informed patient decision-making. In this paper, we describe Bayesian models for performing a subgroup analysis to compare the joint occurrence of a primary endpoint and an adverse event between two treatment arms. Our approaches emphasize estimation of heterogeneity in this joint distribution across subgroups, and our approaches directly accommodate subgroups with small numbers of observed primary and adverse event combinations. In addition, we describe several ways in which our models may be used to generate interpretable summary measures of benefit-risk tradeoffs for each subgroup. The methods described here are illustrated throughout using a large cardiovascular trial ($N = 9,361$) investigating the efficacy of an intervention for reducing systolic blood pressure to a lower-than-usual target.' author: - 'Nicholas C. Henderson' - Ravi Varadhan bibliography: - 'bvsubgroup\_refs.bib' title: '**Bayesian Bivariate Subgroup Analysis for Risk-Benefit Evaluation**' --- \#1 1 [1]{} 0 [1]{} [**Doubly Compromised Estimation ...**]{} [*Keywords:*]{} Heterogeneity of treatment effect; patient-centered outcomes research; personalized medicine; benefits and harms trade-off Introduction ============ Both treatment effectiveness and treatment safety often vary according to patient sub-populations that are defined by observable baseline patient characteristics. Although the importance of evaluating the consistency of treatment effectiveness is often recognized when conducting subgroup analyses, heterogeneity in treatment safety is not examined as frequently. Moreover, when heterogeneity in treatment safety is addressed, such subgroup analyses are typically performed separately from the heterogeneity of treatment effect (HTE) analysis. While useful for addressing question of HTE and heterogeneity in adverse effects, such separate analyses ignore potentially important relationships between primary outcomes and safety outcomes. Dependencies between these two outcomes can substantially alter the risk-benefit considerations when compared with just analyzing their marginal distributions. In order to improve the relevance of subgroups analysis for assessing variation in patients’ risk-benefit profiles, it is critical to focus on examining differences in joint patient outcomes ([@Evans:2016]) within key patient sub-populations. Over the course of a clinical trial, patients experience a collection of outcomes of which some may be related to treatment benefit while others may be related to an adverse effect of treatment. From a patient-level perspective, a risk-benefit assessment involves examining the potential set of outcomes that would occur when taking a proposed treatment versus the potential set of outcomes that would occur under a control treatment. A treatment may be considered superior to the control if the likely set of outcomes under treatment is better than the likely set of outcomes under the control. In this article, we focus on the frequently occurring case where the recorded set of patient outcomes includes a time to some primary event and a binary indicator of whether or not a treatment-related adverse event occurred at some point during the period of patient follow-up. Similar ways of addressing both efficacy and safety have been proposed, for example, in the analysis of dose-finding studies ([@berry:2012]), but to our knowledge, little work has been done in the context of bivariate subgroup analysis. Methods based on Bayesian hierarchical models have a number of advantages when conducting subgroup analyses ([@jones:2011] or [@henderson:2016]), and many of these advantages should be more strongly felt when extending such methods to bivariate patient outcomes. It is well-recognized that subgroups defined by multivariate patient characteristics often have many subgroups with small within-subgroup samples sizes. This leads to very noisy estimates of subgroup-specific quantities - a phenomenon which is likely to be further exacerbated when examining the joint distribution of patient outcomes. A key advantage of a Bayesian approach coupled with hierarchical modeling is that subgroup-specific parameter estimates are partially driven by data in that particular subgroup and partially driven by patient outcomes in all the other subgroups. Allowing such “borrowing of information” across subgroups leads to shrunken estimates of subgroup parameters which frequently improves both the accuracy and stability of estimation ([@efron:1973]). In addition to simply generating more stable shrinkage estimates, Bayesian hierarchical models allow one to specify how subgroup parameters are related to one another. In this article, we focus on two approaches for modeling the relationships between the subgroup parameters (a saturated and an additive model), but many other alternatives could easily be incorporated into our framework for bivariate subgroup analysis. In this article, our main goal is to harness the advantages of Bayesian modeling to describe the joint distribution of a primary and an adverse event across patient subgroups. To this end, we outline a within-subgroup parametric model which assumes an exponential distribution for both adverse event-free and adverse-event-occurring survival. While such a parametric assumption may not be strictly correct, our assumption is that survival follows an exponential distribution within each subgroup rather than the entire population represented by the trial. We argue that this is a sensible choice given the potentially small number of patients within any particular subgroup. Moreover, our approach for direct modeling of patients’ joint distributions provides great flexibility to analyze patients’ risk-benefit tradeoffs from a variety of perspectives. For example, we demonstrate how our model may be used to assess heterogeneity with respect to either a composite outcome which weights adverse event-free and adverse event-occurring survival differently, and how our model may be used to evaluate heterogeneity with respect to a treatment effect based on the probability of achieving an improved outcome. This paper has the following organization. In Section 2, we begin by describing our motivating example - the SPRINT trial, and we describe both the subgroups to be analyzed and the primary and adverse events measured in this large clinical trial. In Section 3, we outline our parametric model for the joint distribution of the time-to-a-primary event and a binary adverse event, and we discuss its use in the context of bivariate subgroup analysis. Section 4 describes a general approach for specifying a prior distribution for the model parameters of Section 3, and we describe three particular ways of modeling the distribution of these parameters. In Section 5, we detail several interpretable measures of variation in patients’ risk-benefit profiles. Here, we describe how our Bayesian framework may be used to make inferences about such measures. Patient outcomes from the SPRINT trial are used to illustrate the use of these measures. Section 6 discusses the use of model diagnostics and model comparison, and Section 7 concludes with a brief discussion. The SPRINT Trial ================ The Systolic Blood Pressure Intervention (SPRINT) trial ([@nejm:2015]) was a large trial investigating the use of a lower-than-usual systolic blood pressure target among individuals deemed to be at increased cardiovascular risk. Specifically, the trial was designed to compare an intensive treatment (a systolic blood-pressure target of less than 120 mm Hg) versus the standard treatment (a systolic blood-pressure target of less than 140 mm Hg). In total, $9361$ individuals were enrolled in the trial, and each trial participant was classified as being at increased cardiovascular risk but without diabetes and had a systolic blood pressure greater than $130$ mm Hg. Of the $9361$ trial participants, $4678$ were randomized to the intensive treatment arm while $4683$ were randomized to the standard treatment arm. In the SPRINT trial, patient outcomes were recorded as a composite outcome where a primary event (PE) was said to occur if any one of the following five events took place: death from cardiovascular causes, stroke, myocardial infarction, acute coronary syndrome not resulting in myocardial infarction, or acute decompensated heart failure. At the conclusion of the trial (median follow up time of $3.26$ years), a total of $562$ PEs had occurred with $243$ PEs occurring in the intensive treatment arm and $319$ PEs occurring in the standard treatment arm. Based on these differences in the distribution of the PEs across treatment arms, the intervention was determined to have an overall benefit (an estimated log-hazard of $0.75$, confidence interval: $[0.64, 0.89]$). In addition to the primary outcome, participants were also monitored for the occurrence of serious adverse events (SAEs). The most common SAEs included acute kidney injury or acute renal failure, injurious fall, electrolyte abnormality, syncope, and hypotension. Each of the observed SAEs were classified according to whether or not they were possibly or definitely related to the intervention. Of the observed treatment-related SAEs, $220$ individuals in the intensive treatment group experienced a treatment-related SAE while only $118$ participants in the standard treatment group experienced a treatment-related SAE. While the marginal occurrence of SAEs in each treatment arm suggests that the standard treatment carries a lower risk of SAEs, the joint occurrence of primary events and SAEs may be more informative from a patient-centered perspective. That is, the risk-benefit tradeoffs involved in choosing a treatment strategy are more apparent when the likelihood of each PE-SAE combination can be examined. This can be particularly important in cases where the dependence pattern between the PE and SAE differs substantially across treatment arms. ----------------------- ----- -------- ----- -------- (r)[2-3]{} (r)[4-5]{} SAE No SAE SAE No SAE PE 18 301 30 213 No PE 100 4264 190 4245 ----------------------- ----- -------- ----- -------- : SPRINT trial: joint counts for primary events (PE) and serious adverse events (SAE) that were classified as possibly or definitely related to the intervention.[]{data-label="tab:sprint_joint"} Table \[tab:sprint\_joint\] shows the number of joint occurrences of PEs and treatment related SAEs in the SPRINT trial. As evidenced by the counts in Table \[tab:sprint\_joint\], most patients in both treatment arms remained free from both PEs and SAEs over the course of the trial. A notable difference between the two treatment arms is the difference in the proportion of PEs that were accompanied by an SAE. However, only examining the joint counts of PEs and SAEs, of course ignores the timings of the PEs. In the next section, we outline a subgroup-specific parametric model for describing the joint distribution of a time-to-event primary outcome and a binary safety outcome. Bivariate Subgroup Analysis =========================== Data Structure and Summary Statistics ------------------------------------- We assume that $n$ patients have been enrolled in a randomized clinical study where patients are monitored to determine whether they experience either a primary event (PE) or an adverse event (AE) (or both). For the $i^{th}$ individual in the study, $T_{i}$ denotes the time-to-failure for the PE of interest. We observe $Y_{i} = \min\{ T_{i}, C_{i} \}$ and an event indicator $\delta_{i} = I( T_{i} \leq C_{i})$ where $C_{i}$ denotes time-to-censoring and $I(\cdot)$ denotes the indicator function. In addition to observing the pair $(Y_{i},\delta_{i})$, we observe an indicator $W_{i}$ of whether or not patient $i$ experienced at least one AE during the study. That is, $W_{i} = 1$ if patient $i$ experienced at least one AE and $W_{i} = 0$ otherwise. We let $A_{i} = 1$ denote that patient $i$ was assigned to the treatment arm, and we let $A_{i} = 0$ denote that patient $i$ was assigned to the control arm. We assume that $G$ distinct patient subgroups have been defined according to multivariate patient characteristics and that $G_{i}$ is a variable which indicates membership in one of the $G$ subgroups. For example, suppose that patient subgroups have been formed on the basis of age (young/old) and sex (male/female) combinations. In this case, we would have $G = 4$ subgroups to represent each age/sex combination, and $G_{i}$ would be a variable indicating to which of the four categories the $i^{th}$ individual belongs. Subgroup analyses that only examine the primary outcome typically utilize a collection of summary statistics which measure treatment effectiveness in each subgroup considered. Such summary statistics are usually computed for marginally defined subgroups (i.e., subgroups defined by looking at one variable at-a-time) or subgroups which are defined according multivariate characteristics. In our analysis of bivariate patient outcomes, we require that one compute a pair of summary statistics $(D_{awg}, U_{awg})$ for each AE-treatment arm combination within each subgroup. The summary statistic $D_{awg}$ is the number of PEs that are observed to occur among the patients in subgroup $g$ $(g=1,\ldots,G)$, treatment arm $a$ $(a=0,1)$, and have AE status $w$ $(w=0,1)$. Similarly, $U_{awg}$ is the total follow-up time for the group of patients that are in subgroup $g$, treatment arm $a$, and have AE status $w$. In addition to $(D_{awg}, U_{awg})$, we require that one compute the summary statistics $V_{ag}$ for each treatment arm-subgroup combination. The summary statistic $V_{ag}$ is the total number of AEs observed for those patients in subgroup $g$ and treatment arm $a$. More formally, $(D_{awg}, U_{awg})$ and $V_{ag}$ are defined in terms of the individual-level responses as $$\begin{aligned} D_{awg} &=& \sum_{i=1}^{n} \delta_{i} I(A_{i}=a)I(W_{i}=w)I(G_{i}=g) \nonumber \\ U_{awg} &=& \sum_{i=1}^{n} Y_{i}I(A_{i}=a)I(W_{i}=w)I(G_{i}=g) \nonumber \\ V_{ag} &=& \sum_{i=1}^{n} W_{i}I(A_{i} = a)I(G_{i} = g) \nonumber\end{aligned}$$ Table \[tab:sprint\_summaries\] shows the summary statistics $(D_{awg}, U_{awg}, V_{ag})$ from the SPRINT trial using $G=8$ subgroups. These $8$ subgroups were formed by looking at all possible combinations of the following three baseline patient variables: chronic kidney disease (Yes/No), age ($\geq 75/< 75$), and sex (male/female). --------------------------- ----------- -------- ---------------------- ---------------------- ---------- ---------------------- ---------------------- ---------- --------- (r)[4-6]{} (r)[7-9]{} CKD Sex Age $(D_{01g}, U_{01g})$ $(D_{00g}, U_{00g})$ $V_{0g}$ $(D_{11g}, U_{11g})$ $(D_{10g}, U_{10g})$ $V_{1g}$ $N_{g}$ No $< 75$ Male (3, 92.00) (96, 5546.11) 29 (10, 174.05) (54, 5446.53) 61 3528 Yes $< 75$ Male (3, 48.41) (28, 1364.73) 16 (5, 103.15) (25, 1227.70) 32 858 No $\geq 75$ Male (1, 19.57) (46, 1277.47) 8 (1, 57.67) (26, 1265.94) 19 913 Yes $\geq 75$ Male (6, 40.00) (47, 977.71) 15 (7, 88.31) (38, 974.85) 31 730 No $< 75$ Female (0, 40.08) (31, 2641.97) 13 (0, 67.47) (25, 2705.38) 20 1706 Yes $< 75$ Female (2, 42.41) (12, 948.66) 14 (4, 50.23) (16, 1000.35) 16 617 No $\geq 75$ Female (0, 37.84) (16, 835.07) 12 (1, 60.98) (18, 778.16) 20 568 Yes $\geq 75$ Female (3, 30.81) (25, 612.09) 11 (2, 66.44) (11, 634.87) 21 441 --------------------------- ----------- -------- ---------------------- ---------------------- ---------- ---------------------- ---------------------- ---------- --------- : SPRINT trial: Summary statistics for the $8$ subgroups defined by the baseline variables: chronic kidney disease (CKD), age, and sex. The summary statistics $U_{awg}$ are computed using time measurements in years, but these are not provided here.[]{data-label="tab:sprint_summaries"} Distribution of Key Summary Statistics and Joint Distribution of Primary and Adverse Events ------------------------------------------------------------------------------------------- In our analysis, we assume the number of observed events $D_{awg}$ follows a Poisson distribution whose mean depends on the total follow-up time $U_{awg}$ and the hazard rate $\lambda_{awg}$ within the subset of patients with the $(a,w,g)$ combination of treatment arm $a$, AE status $w$, and subgroup $g$. Specifically, given $\lambda_{awg}$, we assume $D_{awg}$ is distributed as $$D_{awg}|\lambda_{awg} \sim \textrm{Poisson}(\lambda_{awg}U_{awg}). \nonumber$$ The assumed Poisson distribution for the summary statistics $D_{awg}$ is equivalent to assuming that, at the individual level, the time-to-failure $T_{i}$ of the primary event follows an exponential distribution with rate $\lambda_{awg}$. More specifically, the likelihood function that results from assuming $D_{awg}|\lambda_{awg} \sim \textrm{Poisson}(\lambda_{awg}U_{awg})$ is the same (ignoring constant terms) as the likelihood function that results from assuming that $$T_{i}|W_{i}=w, A_{i} = a, G_{i}=g \sim \textrm{Exponential}(\lambda_{awg}). \nonumber$$ To fully define the joint distribution of the time-to-primary event $T_{i}$ and the AE indicator $W_{i}$ given treatment arm and subgroup information, we now only need to specify the conditional distribution of $W_{i}$ given $A_{i}$ and $G_{i}$. Here, we assume that $W_{i}$ is a Bernoulli random variable with a success probability $p_{ag}$ that depends on the treatment arm and subgroup. This implies the distribution of $V_{ag}$ is given by $$V_{ag}|p_{ag} \sim \textrm{Binomial}(n_{ag}, p_{ag}), \nonumber$$ where $n_{ag}$ is the number of individuals in treatment arm $a$ and subgroup $g$. The parameters $(\lambda_{awg}, p_{ag})$ characterize the joint distribution of the primary and adverse event within the subgroup $g$ and treatment arm $a$. To see why this is the case, note that the joint probability $P(T_{i} > t, W_{i}=w|A_{i}=a, G_{i}=g)$ may be expressed as $$\begin{aligned} S_{ag}(t,w) &=& P(T_{i} > t, W_{i}=w|A_{i}=a, G_{i}=g) \nonumber \\ &=& P(T_{i} > t|W_{i}=w, A_{i}=a, G_{i}=g)P(W_{i}=w|A_{i} = a, G_{i}=g) \nonumber \\ &=& e^{-\lambda_{awg}t}p_{ag}^{w}(1 - p_{ag})^{1 - w}. \nonumber\end{aligned}$$ Comparing $S_{1g}(t, w)$ and $S_{0g}(t, w)$ for each subgroup enables one to compare the effect of the treatment on altering the risk-benefit profiles of patients in subgroup $g$. Modeling Subgroup Effects ========================= Regression Models for Subgroup Parameters ----------------------------------------- As described above, our model for the summary measures $(D_{awg}, U_{awg}, V_{ag})$ depends on the collection of hazard rate parameters $\lambda_{awg}$ and the AE probabilities $p_{ag}$. Because the number of parameters is potentially quite large, models which induce shrinkage are needed to produce stable estimates of these quantities. We describe a general regression models for $\lambda_{awg}$ and $p_{ag}$ which allow for shrinkage and allow for one to include information regarding how the subgroups parameters are related. In Sections 4.1.1-4.1.3, we describe three specific regression models, and in Section 4.2 we describe our prior specification for the parameters of two of these regression models. We consider a regression setting with design matrices $\mathbf{X}$ and $\mathbf{Z}$ for the hazard rate parameters $\lambda_{awg}$ and AE probabilities $p_{ag}$ respectively. The matrices $\mathbf{X}$ and $\mathbf{Z}$ are $G \times q_{x}$ and $G \times q_{z}$ respectively, and $\mathbf{x}_{g}^{T}$ and $\mathbf{z}_{g}^{T}$ denote the $g^{th}$ rows of $\mathbf{X}$ and $\mathbf{Z}$ respectively. The hazard rate parameters $\lambda_{awg}$ and the adverse-event probabilities are related to $\mathbf{x}_{g}$ and $\mathbf{z}_{g}$ via $$\log( \lambda_{awg} ) = \mathbf{x}_{g}^{T}{\mbox{\boldmath $\beta$}}_{aw} \qquad \textrm{and} \qquad \textrm{logit}( p_{ag} ) = \mathbf{z}_{g}^{T}{\mbox{\boldmath $\gamma$}}_{a}, \nonumber$$ where $\textrm{logit}(x) = \log\{x/(1-x)\}$. Here ${\mbox{\boldmath $\beta$}}_{aw}$ is the $q_{x} \times 1$ vector ${\mbox{\boldmath $\beta$}}_{aw} = (\beta_{aw,1}, \ldots, \beta_{aw,q_{x}})^{T}$, and ${\mbox{\boldmath $\gamma$}}_{aw}$ is the $q_{z} \times 1$ vector ${\mbox{\boldmath $\gamma$}}_{a} = (\gamma_{a,1}, \ldots, \gamma_{a,q_{z}})^{T}$. In total, this regression model involves $4q_{x} + 2q_{z}$ parameters. We describe a few possible ways of choosing $\mathbf{X}$ and $\mathbf{Z}$ below. ### Example 1: Saturated Model. {#sss:saturated} In this model, the number of regression parameters is equal to the number of summary statistics. The design matrices $\mathbf{X}$ and $\mathbf{Z}$ are assumed to be equal to the $G \times G$ identity matrix, and each ${\mbox{\boldmath $\beta$}}_{aw}$, ${\mbox{\boldmath $\gamma$}}_{a}$ are assumed to be $G \times 1$ vectors. This means that, for any combination of $(a,w,g)$, we have $$\log(\lambda_{awg}) = \beta_{aw,g} \qquad \textrm{and} \qquad \textrm{logit}(p_{ag}) = \gamma_{a,g}. \nonumber$$ In this model, there is no regression structure linking the parameters $\lambda_{awg}$ or the parameters $p_{ag}$ together in such a way that more strength is borrowed between subgroups which share a greater number of patient characteristics. Rather, the $\lambda_{awg}$ and $p_{ag}$ are treated separately with no additional information used to indicate the relationships among the subgroups. ### Example 2: Additive Model. {#sss:additive} In this model, the values of $\lambda_{awg}$ and $p_{ag}$ are determined additively from the variables comprising subgroup $g$. This approach is analogous to the Dixon and Simon model for subgroup analysis (see e.g., [@dixonsimon:1991], [@jones:2011], or [@henderson:2016]). The number of regression parameters in this model are determined by both the number of patient variables and the number of levels within each one of these variables. To define the additive model, we suppose that the $G$ subgroups are formed by $p$ patient variables, and the $j^{th}$ patient variable has levels $1,\ldots,p_{j}$. If we consider modeling $\lambda_{awg}$ for subgroup $g$, it is necessary to know the levels of each of the patient variables comprising subgroup $g$. If the level of the $j^{th}$ variable in subgroup $g$ is denoted by $g(j)$, then $$\begin{aligned} \log( \lambda_{awg} ) &=& \beta_{aw,1} + \sum_{j=1}^{p}\sum_{k=2}^{p_{j}} \mathbf{1}\{ g(j) = k \}\beta_{aw,Q_{j} + k} \nonumber \\ \textrm{logit}(p_{ag}) &=& \gamma_{a,1} + \sum_{j=1}^{p}\sum_{k=2}^{p_{j}} \mathbf{1}\{ g(j) = k \}\gamma_{a,Q_{j} + k}, \label{eq:anova_model}\end{aligned}$$ where $Q_{1} = 0$ and $Q_{j} = 1 - j + p_{1} + \ldots + p_{j-1}$ for $j > 1$. As an illustration of (\[eq:anova\_model\]), consider an example where four subgroups are formed from the combinations of the patient variables of age (young/old) and sex (male/female) and that male and young denote the first levels of the age and sex variables respectively. Suppose further that we label the four subgroups in the following manner: subgroup 1 denotes male/young (i.e., $g(1) = 1$, $g(2)=1$), subgroup 2 denotes male/old (i.e., $g(1)=1$, $g(2)=2$), subgroup 3 denotes female/young (i.e., $g(1)=2$, $g(2)=1$), and subgroup $4$ denotes female/old (i.e., $g(1) = 2$, $g(2)=2$). In this case, the $\lambda_{awg}$ would be expressed as $$\begin{aligned} \textrm{(Male/Young)} \qquad \log(\lambda_{aw1}) &=& \beta_{aw,1} \nonumber \\ \textrm{(Male/Old)} \qquad \log(\lambda_{aw2}) &=& \beta_{aw,1} + \beta_{aw,2} \nonumber \\ \textrm{(Female/Young)} \qquad \log(\lambda_{aw3}) &=& \beta_{aw,1} + \beta_{aw,3} \nonumber \\ \textrm{(Female/Old)} \qquad \log(\lambda_{aw4}) &=& \beta_{aw,1} + \beta_{aw,2} + \beta_{aw,3}, \nonumber \end{aligned}$$ and the relation between the $p_{ag}$ and $(\gamma_{a,1},\gamma_{a,2},\gamma_{a,3})$ would be expressed similarly. ### A Proportional Hazards Model An even more parsimonious approach than the regression models discussed above is to assume instead that the hazard ratios $\lambda_{a1g}/\lambda_{a0g}$ do not vary across subgroups. Under this assumption, the parameters $(\lambda_{a0g}, p_{ag})$ would be modeled as $$\log(\lambda_{a0g}) = \mathbf{x}_{g}^{T}{\mbox{\boldmath $\beta$}}_{a} \qquad \textrm{and} \qquad \textrm{logit}(p_{ag}) = \mathbf{z}_{g}^{T}{\mbox{\boldmath $\gamma$}}_{a}, \nonumber$$ and the hazard rates $\lambda_{a1g}$ would be obtained from the AE-free hazard rates $\lambda_{a0g}$ via $\lambda_{a1g} = \phi\lambda_{a0g}$ where $\phi$ is the constant hazard ratio. With this proportional hazards assumption, there are now $2q_{x} + 2q_{z} + 1$ regression parameters rather than the $4q_{x} + 2q_{z}$ parameters required for the regression models discussed in Sections \[sss:saturated\] and \[sss:additive\]. While this approach makes a quite restrictive assumption about how the AE-present and AE-free hazard rates are related, this could be a useful approach if the proportional hazards assumption $\lambda_{a1g} = \phi\lambda_{a0g}$ is justifiable and is seen to be reasonable in light of model diagnostics. Prior Specification ------------------- ### Prior for Saturated Model. For each value of $(a,w)$, we assume the parameters $\beta_{aw,g} = \log(\lambda_{awg})$ are drawn from a common normal distribution with mean $\mu_{aw}$ and variance $\tau_{aw}^{2}$ $$\beta_{aw,1}, \ldots, \beta_{aw,G}|\mu_{wa}, \tau_{aw} \sim \textrm{Normal}(\mu_{aw}, \tau_{aw}^{2}). \nonumber$$ The distribution of all the parameters $\beta_{aw,g}; w=0,1; a=0,1; g=1,\ldots G$ depends only on the following four vectors: ${\mbox{\boldmath $\mu$}}_{0} = (\mu_{00},\mu_{10})^{T}$, ${\mbox{\boldmath $\mu$}}_{1} = (\mu_{01}, \mu_{11})$, ${\mbox{\boldmath $\tau$}}_{0} = (\tau_{00}, \tau_{10})^{T}$, and ${\mbox{\boldmath $\tau$}}_{1} = (\tau_{01}, \tau_{11})^{T}$. The mean vector ${\mbox{\boldmath $\mu$}}_{a}$ for treatment arm $a$ is assumed to have the following bivariate normal distribution $${\mbox{\boldmath $\mu$}}_{a} \sim MVN_{2}\Bigg(\begin{bmatrix} 0 \\ 0 \end{bmatrix}, \begin{bmatrix} \sigma_{\mu,a}^{2} & \sigma_{\mu,a} \rho_{\mu,a}^{2} \\ \sigma_{\mu,a}^{2} \rho_{\mu,a} & \sigma_{\mu,a}^{2} \end{bmatrix} \Bigg), \nonumber$$ where $\mathbf{Z} \sim MVN_{2}({\mbox{\boldmath $\mu$}}, {\mbox{\boldmath $\Sigma$}})$ means that $\mathbf{Z}$ has a bivariate normal distribution with mean vector ${\mbox{\boldmath $\mu$}}$ and covariance matrix ${\mbox{\boldmath $\Sigma$}}$. Similarly, the distribution for ${\mbox{\boldmath $\tau$}}_{a}$ on the log scale is assumed to be given by $$\log({\mbox{\boldmath $\tau$}}_{a}) \sim MVN_{2}\Bigg( \begin{bmatrix} \log(1/2) \\ \log(1/2) \end{bmatrix}, \begin{bmatrix} \sigma_{\tau,a}^{2} & \sigma_{\tau,a}^{2} \rho_{\tau,a} \\ \sigma_{\tau,a}^{2} \rho_{\tau,a} & \sigma_{\tau,a}^{2} \end{bmatrix} \Bigg). \nonumber$$ For the correlation parameters $\rho_{\mu,a}$ and $\rho_{\tau,a}$, we assume that $\rho_{\mu,a} \sim \textrm{Uniform}(-1,1)$ and $\rho_{\tau,a} \sim \textrm{Uniform}(-1,1)$ for each $a$. Our prior distribution for the $\gamma_{a,g}$ is similar to the prior for the $\beta_{aw,g}$. Specifically, we start by assuming that $$\gamma_{a,1}, \ldots, \gamma_{a,G}|\mu_{a,\gamma}, \tau_{a} \sim \textrm{Normal}(\mu_{a,\gamma}, \tau_{a,\gamma}^{2}). \nonumber$$ We then assume that $\mu_{a,\gamma} \sim \textrm{Normal}(\log(1/2),\sigma_{\mu,a,\gamma}^{2})$ for $a=0,1$, and we assume that $\log(\tau_{a,\gamma}) \sim \textrm{Normal}(0,\sigma_{\tau,a,\gamma}^{2})$ for $a = 0,1$. In total, there are $8$ hyperparameters in this model. These are: $\sigma_{\mu,0}$, $\sigma_{\mu,1}$, $\sigma_{\tau,0}$, $\sigma_{\tau,1}$, $\sigma_{\mu,0,\gamma}^{2}$, $\sigma_{\mu,1,\gamma}$, $\sigma_{\tau,0,\gamma}$, and $\sigma_{\tau,1,\gamma}$. Our default choice is to set $\sigma_{\mu,0} = \sigma_{\mu,1} = \sigma_{\mu,0,\gamma} = \sigma_{\mu,1,\gamma} = 100$ and to set $\sigma_{\tau,0} = \sigma_{\tau,1} = \sigma_{\tau,0,\gamma} = \sigma_{\tau,1,\gamma} = 1$. Our justification for these default settings is discussed in more detail below. Our prior specification implies the $\log(\lambda_{awg}), g = 1,\ldots,G$ are drawn from a common normal distribution with mean $\mu_{aw}$ and variance $\tau_{aw}^{2}$. Because it is difficult to specify a range of reasonable values for the median of $\log(\lambda_{awg}), g = 1,\ldots, G$, we assign $\mu_{aw}$ a vague prior distribution by setting $\sigma_{\mu,a} = 100$, for $a = 0, 1$. A similar justification can be made for setting $\sigma_{\mu,a,\gamma} = 100$. Turning now to the variation in $\log(\lambda_{awg})$ we note that, for any pairs of subgroup indices $j,k$, the distribution of the hazard ratio (given $\tau_{aw}$) $\lambda_{awj}/\lambda_{awk}$ follows a log-normal distribution with parameters $0$ and $2\tau_{aw}^{2}$. Our prior distribution for $\tau_{aw}$ is based on the observation that the hazard ratio is most likely between $1/4$ and $4$. Because $P\{ \tfrac{1}{4} \leq \lambda_{awj}/\lambda_{awk} \leq 4|\tau_{aw} \} \approx 1 - 2\Phi(-1/\tau_{aw})$, this means that this approximate probability is greater than $0.68$ whenever $\tau_{aw} \leq 1$ and greater than $0.95$ whenever $\tau_{aw} \leq 1/2$. We want to choose a prior for $\tau_{aw}$ such that $P\{ \tfrac{1}{4} \leq \lambda_{awj}/\lambda_{awk} \leq 4|\tau_{aw} \}$ is fairly large for the most probable values of $\tau_{aw}$. This can be done by assuming that $\tau_{aw}$ follows a log-normal distribution with parameters $\log(1/2)$ and $1$ (i.e., setting $\sigma_{\tau,a} = 1)$. This means that $\tau_{aw}$ has a median of $1/2$ and that the probability that $\tau_{aw}$ is less than $2.6$ is approximately $0.95$. A similar justification can be made for setting $\sigma_{\tau,a,\gamma} = 1$. ### Prior for Additive Model. We assign vague priors to the intercept terms $\beta_{aw,1}$ and $\gamma_{a,1}$. Specifically, it is assumed that $$\beta_{aw,0} \sim \textrm{Normal}(0, \sigma_{0,\beta}^{2}) \qquad \textrm{and} \qquad \gamma_{a,0} \sim \textrm{Normal}(0, \sigma_{0,\gamma}^{2}), \nonumber$$ with $\sigma_{0,\beta}$ and $\sigma_{0,\gamma}$ set to default values of $\sigma_{0,\beta} = 100$ and $\sigma_{0, \gamma} = 100$. The hierarchical model for the remaining regression coefficients $\beta_{aw,j}, j = 1,\ldots, \sum p_{j} - (p-1)$ and $\gamma_{a,j}, j= 2,\ldots, \sum p_{j} - (p - 1)$ is the same as that used for the regression coefficients in the saturated model. That is, we assume that $$\begin{aligned} \beta_{aw,1},\ldots,\beta_{aw,q_{x}} &\sim& \textrm{Normal}(\mu_{aw}, \tau_{aw}^{2}) \nonumber \\ \gamma_{a,1}, \ldots, \gamma_{a, q_{z}} &\sim& \textrm{Normal}(\mu_{a,\gamma}, \tau_{a,\gamma}^{2}), \nonumber\end{aligned}$$ where $q_{x} = q_{z} = \sum_{j=1}^{p} p_{j} - (p-1)$. The prior distributions and default hyperparameters for $\mu_{a,\gamma}$, $\tau_{a,\gamma}$ and the four vectors ${\mbox{\boldmath $\mu$}}_{0} = (\mu_{00},\mu_{10})^{T}$, ${\mbox{\boldmath $\mu$}}_{1} = (\mu_{01}, \mu_{11})$, ${\mbox{\boldmath $\tau$}}_{0} = (\tau_{00}, \tau_{10})^{T}$, and ${\mbox{\boldmath $\tau$}}_{1} = (\tau_{01}, \tau_{11})^{T}$ are exactly the same as described for the saturated model. ### Posterior Computation The Bayesian bivariate subgroup models proposed here can be easily computed using STAN software. With STAN, we have implemented both the saturated and additive models. In our analysis of the SPRINT trial, we used $4,000$ posterior draws obtained from four chains each having $1500$ iterations with the first $500$ iterations treated as burn-in. The code for implementing our models can be obtained from our website: <http://hteguru.com/index.php/bbsga/> Measures of Risk-Benefit ======================== Heterogeneity in Joint Binary Outcomes -------------------------------------- A natural bivariate outcome of interest is survival past a specified time point $\kappa_{0}$ coupled with an indicator of whether or not an AE occurred during the follow-up period. For this bivariate outcome, there are four possible results: survival past $\kappa_{0}$/no AE occurs, survival past $\kappa_{0}$/AE occurs, failure before $\kappa_{0}$/no AE occurs, and failure before $\kappa_{0}$/AE occurs. The subgroup-specific posterior probabilities of each of these four outcomes may be obtained from the models detailed in Sections 3 and 4. The differences in these probabilities between treatment arms provide a measure by which to compare how treatment influences the probability of experiencing each of these four outcomes. If we define the function $F_{ag}(t,w)$ as $$\begin{aligned} F_{ag}(t,w) &=& P(T_{i} \leq t, W_{i}=w|A_{i}=a, G_{i}=g) \nonumber \\ &=& (1 - \exp(-\lambda_{awg} t))p_{ag}^{w}(1 - p_{ag})^{1-w}, \nonumber\end{aligned}$$ the differences in these four joint probabilities may be expressed as $$\begin{aligned} \theta_{g1} = S_{1g}(\kappa_{0}, 0) &-& S_{0g}(\kappa_{0},0) \nonumber \\ \theta_{g2} = S_{1g}(\kappa_{0}, 1) &-& S_{0g}(\kappa_{0}, 1) \nonumber \\ \theta_{g3} = F_{1g}(\kappa_{0}, 0) &-& F_{0g}(\kappa_{0},0) \nonumber \\ \theta_{g4} = F_{1g}(\kappa_{0}, 1) &-& F_{0g}(\kappa_{0}, 1) \nonumber \end{aligned}$$ If the PE is regarded as more undesirable than the adverse event, then the parameters $(\theta_{g1}, \theta_{g2}, \theta_{g3}, \theta_{g4})$ may be thought of as being ordered according to the desirability of the associated outcome. Specifically, $\theta_{g1}$ represents the difference in the probability of achieving the most desirable outcome, namely remaining both PE and AE free until time $\kappa_{0}$. Similarly, $\theta_{g2}$ represents the difference in the probability of achieving the second most desirable outcome, namely remaining free from the PE until time $\kappa_{0}$ while experiencing an AE at some time before $\kappa_{0}$. The parameter $\theta_{g3}$ represents the difference in probability of experiencing the PE while not experiencing the AE, and $\theta_{g4}$ represents the difference in probability of experiencing both the primary and adverse event before time $\kappa_{0}$. It is worth noting that $\sum_{j=1}^{4} \theta_{gj} = 0$ for each $g$. That is, if the treatment increases the probability of one outcome relative to control, it must decrease the probability of another outcome (or group of outcomes) by a corresponding amount. ![SPRINT trial with eight subgroups defined by chronic kidney disease (CKD), age, and sex. Posterior means and associated credible intervals using the saturated model are plotted for $\theta_{g1}$, $\theta_{g2}$, $\theta_{g3}$, $\theta_{g4}$. Note that the range of the x-axis can differ across $\theta_{g1}$, $\theta_{g2}$, $\theta_{g3}$, $\theta_{g4}$. For each $j$, the vertical line is placed at $\hat{\theta}_{j,ov}$, the average value of the posterior means $\hat{\theta}_{gj}$ across subgroups.[]{data-label="fig:four_forestplot"}](saturated_model_forest.eps){width="14cm" height="14cm"} Using the SPRINT trial data with the $8$ subgroups defined by chronic kidney disease status, age, and sex, Figure \[fig:four\_forestplot\] plots the posterior means of $(\theta_{g1}, \theta_{g2}, \theta_{g3}, \theta_{g4})$. The posterior quantities shown in Figure \[fig:four\_forestplot\] were computed using the saturated model. For each of the forest plots in Figure \[fig:four\_forestplot\], the solid vertical line represents the average value of the posterior means of $\theta_{gj}$ across subgroups, i.e., $\tfrac{1}{G}\sum_{g} \hat{\theta}_{gj}$ where $\hat{\theta}_{gj}$ is the posterior mean of $\theta_{gj}$. The quantity $\hat{\theta}_{j, ov} = \tfrac{1}{G}\sum_{g=1}^{G}\hat{\theta}_{gj}$ can be thought of as an estimate of the overall treatment effect for PE-AE category $j$. Looking at Figure \[fig:four\_forestplot\], the estimated overall treatment effect is negative for PE-AE categories 1 and 3 and is positive for categories 2 and 4 though the magnitude of the estimates for categories 3 and 4 is very small. Specifically, the values of $\hat{\theta}_{j,ov}$ were $\hat{\theta}_{1,ov} = -0.019$, $\hat{\theta}_{2,ov} = 0.025$, $\hat{\theta}_{3,ov} = -0.007$, and $\hat{\theta}_{4,ov} = 0.001$. These values suggest the intensive treatment increases the probability of 3-year PE-free survival while experiencing a treatment-related SAE, and it reduces the probability of experiencing a PE without an SAE. However, the intensive treatment also appears to reduce the probability of remaining both PE and SAE free for 3 years by an a similar amount to the increase in the probability of outcome $2$. Comparing the estimates $\hat{\theta}_{1,ov}$, $\hat{\theta}_{2,ov}$, and $\hat{\theta}_{3,ov}$ leads to a somewhat ambiguous picture about the effect of the intensive treatment on improving patient outcomes. A more complete evaluation of the risk/benefit effect of the intensive treatment would involve consideration of patients’ “baseline bivariate risk”. Specifically, the baseline bivariate risk refers to the probability of each PE-AE combination occurring under the control arm. Such baseline information would be useful in assessing the meaningfulness of any change $\theta_{gj}$ that the treatment induces on the probability of outcome $j$. Turning now to the variation in $\theta_{gj}$ across subgroups, two subgroups from Figure \[fig:four\_forestplot\] seem to stand out. The more prominent of these is the no-CKD/Age $< 75$/Female subgroup. In this subgroup, the $95\%$ credible interval for $\theta_{g2}$ does not cover the overall treatment effect estimate $\hat{\theta}_{2,ov}$, and the credible interval for $\theta_{g1}$ nearly does not cover $\hat{\theta}_{1,ov}$. This suggests the effect of the intensive treatment on altering a patients’ PE-AE outcome profile is somewhat different in the no-CKD/Age $<75$/Female subgroup than in the broader population represented by the trial. Specifically, compared to other groups in the trial, the intensive treatment for those in the no-CKD/Age $<75$/Female subgroup had almost no effect on any of the PE-AE probabilities as the estimates $\hat{\theta}_{gj}$ for this subgroup were much closer to zero when compared with their overall counterparts $\hat{\theta}_{j,ov}$. Another subgroup worth highlighting is the no-CKD/Age $< 75$/Male subgroup. Though not as extreme as the no-CKD/Age $< 75$/Female subgroup, the estimates of $\theta_{g1}$ and $\theta_{g2}$ are modestly closer to zero than the average values of these parameters, and while the credible intervals for these parameters still covered $\hat{\theta}_{1,ov}$ and $\hat{\theta}_{2,ov}$, most of the mass of these credible intervals were to the right and left of $\hat{\theta}_{1,ov}$ and $\hat{\theta}_{2,ov}$ respectively. ![SPRINT trial with eight subgroups defined by chronic kidney disease (CKD), age, and sex. Posterior means and associated credible intervals using the additive model are plotted for $\theta_{g1}$, $\theta_{g2}$, $\theta_{g3}$, $\theta_{g4}$. For each $j$, the vertical line is placed at $\hat{\theta}_{j,ov}$, the average value of the posterior means across subgroups.[]{data-label="fig:four_forestplot_additive"}](additive_model_forest.eps){width="14cm" height="14cm"} Figure \[fig:four\_forestplot\_additive\] displays posterior estimates of $(\theta_{g1}, \theta_{g2}, \theta_{g3}, \theta_{g4})$ from the additive model. One notable difference between the additive and the saturated model is in the overall estimates $\hat{\theta}_{1,ov}$ and $\hat{\theta}_{3,ov}$. Specifically, the estimate $\hat{\theta}_{1,ov}$ was much closer to zero in the additive model than in the saturated model. As with the saturated model, both the no-CKD/Age $<75$/Female and no-CKD/Age $< 75$/Male subgroups stand out in the additive model due to lower magnitudes of $\theta_{g2}$. The estimated values of $\theta_{g1}$ for the no-CKD/Age $<75$/Female subgroup is actually quite similar to those from the saturated model. However, due to differences between the saturated and additive models in the overall estimate $\hat{\theta}_{1,ov}$, the value of $\theta_{g1}$ for this subgroup does not appear different than the estimated overall value of this parameter. Comparing Utilities across Subgroups {#utility} ------------------------------------ While forest plots of $\theta_{g1},\ldots,\theta_{g4}$ can be useful for examining variability in the effect of treatment on altering patients’ risk-benefit profiles, interpreting such graphs can be somewhat challenging. Converting the bivariate outcomes into a composite score by weighting each outcome allows one to report heterogeneity with respect to a univariate score. Weighting methods for assessing risk-benefit have been described by a number of researchers including, for example, [@chuang:1994]. We consider here two approaches for weighting joint patient outcomes. A direct way of combining $\theta_{g1}, \ldots, \theta_{g4}$ to produce a univariate score would be to compute $$\label{ttg} \tilde{\theta}_{g} = b_{1}\theta_{g1} + b_{2}\theta_{g2} + b_{3}\theta_{g3} + b_{4}\theta_{g4}$$ for weights $b_{1}, \ldots, b_{4}$. Assuming that a patient receives a utility of $b_{j}$ when experiencing outcome $j$, $\tilde{\theta}_{g}$ represents the difference in expected utility between treatments for those in subgroup $g$. In other words, if one defines a composite outcome where a patient is assigned a composite of score of $b_{j}$ according to , then $\tilde{\theta}_{g}$ represents the subgroup-specific expected difference in this composite outcome. Rather than weight joint binary outcomes as is done in computing $\tilde{\theta}_{g}$, an alternative is to incorporate the survival time directly into the composite outcome. We define such a composite score $H_{i}$ which, for the $i^{th}$ patient, we define as $$H_{i} = b_{1}W_{i}\min\{ T_{i}, \tau \} + b_{2}(1 - W_{i})\min\{ T_{i}, \tau\},$$ where $\tau$ represents a truncation point that represents a follow-up period of interest. The composite measure $H_{i}$ may be viewed as an outcome whose expectation is a type of weighted restricted mean survival time (RMST) (see e.g., [@royston:2013]). We use $\min\{ T_{i}, \tau\}$ rather than $T_{i}$ to reduce the impact of model extrapolation beyond the follow-up period of interest $(0, \tau)$. With the measure $H_{i}$, a patient receives a weight of $b_{2}$ for each time unit of PE-free survival assuming he/she did not experience an AE over the time interval $(0, \min\{T_{i}, \tau\})$. Likewise, a patient receives a weight of $b_{1}$ for each unit of PE-free survival time assuming he/she did experienced an adverse event at some point in the time interval $(0, \min\{T_{i}, \tau\})$. The composite outcome $H_{i}$ bears some resemblance to the Q-TWist measure described in [@Gelber:1989] though $H_{i}$ does not explicitly distinguish between time with toxicity and time without toxicity. ![SPRINT trial with eight subgroups defined by chronic kidney disease (CKD), age, and sex. Utility analysis using the saturated model and composite outcome measure $H_{i}$. Posterior means and credible intervals for $\eta_{g}$ are shown for each subgroup. The solid vertical line is placed at the average value of the posterior means of $\eta_{g}$ across subgroups.[]{data-label="fig:utility_saturated"}](saturated_utility_forest.eps){width="14cm" height="14cm"} The expectation of the composite outcome $H_{i}$ conditional on subgroup and treatment arm information is given by $$\begin{aligned} E[H_{i}|A_{i}=a, G_{i}=g] &=& \frac{b_{1}p_{ag}(1 - e^{-\tau\lambda_{a1g}})}{\lambda_{a1g}} + \frac{b_{2}(1 - p_{ag})(1 - e^{-\tau\lambda_{a0g}})}{\lambda_{a0g}}. \nonumber\end{aligned}$$ Subgroup-specific treatment effects $\eta_{g}$ relative to this measure may then be defined as the difference in the expectation of $H_{i}$ in the treatment arm vs. the expectation of $H_{i}$ in the control arm $$\label{etag} \eta_{g} = E[H_{i}|A_{i}=1, G_{i}=g] - E[H_{i}|A_{i}=0, G_{i}=g].$$ Figure \[fig:utility\_saturated\] shows estimates of the treatment effects $\eta_{g}$ using the saturated model and two choices of weights $(b_{1}, b_{2})$. For each choice of weights, we set $b_{2} = 1$, and we set $b_{1}$ to $0.8$ and $0.5$ for the left-hand and right-hand plots respectively. When $b_{2} = 1$, $b_{2}$ may be interpreted as the proportion of value that one receives from a unit of extra life when an AE is known to occur as compared to the value of an extra unit of life when no AE occurs. Hence, a value of $b_{1}$ very close to one implies that AE-free survival and AE-occurring survival are valued similarly while values of $b_{1}$ much lower than one imply that AE-free survival is valued much more highly than AE-occurring survival. It is interesting to note that the younger subgroups (i.e., age $< 75$) tend to consistently receive less benefit treatment for both choices of weights. Indeed, all of the younger subgroups have posterior means less than the average value of the posterior means $\hat{\eta}_{g}$. In Figure \[fig:utility\_saturated\], two subgroups stand out in terms of being different than the average value of $\eta_{g}$ across subgroups. These are the no-CKD/Age $<75$/Female and yes-CKD/Age $< 75$/Female subgroups. Both of these subgroups do not appear to benefit when using either choice of weights to compute $H_{i}$. This is not that surprising for the no-CKD/Age $< 75$/Female subgroup as the four-outcome subgroup analysis from Figure \[fig:four\_forestplot\] suggested the intensive treatment had very little impact in changing the probabilities of any of the four outcomes. Similarly, the point estimates from Figure \[fig:four\_forestplot\] also suggested that the treatment had little effect on the yes-CKD/Age $< 75$/Female subgroup though the posterior uncertainty for these estimates was much greater than for the no-CKD/Age $< 75$/Female subgroup. Interestingly, the no-CKD/Age $<75$/Male subgroup did not show clear evidence of having a worse treatment effect $\eta_{g}$ than the average from the other subgroups. Heterogeneity in the Probability of Outcome Improvement ------------------------------------------------------- While using weights to create univariate composite measures can be useful, it is often difficult to justify a particular choice of weights in an analysis. An alternative to weighting is to order the outcomes according to their desirability and to use the probability of outcome improvement as the treatment effect to be measured. An advantage of approaches which rely on ordering outcomes (see e.g., [@claggett:2015] or [@follmann:2002]) is that they do not require that one assign numerical values to each outcome but only require that one be able to order outcomes according to their desirability. We consider here an ordering-based definition of treatment effect that compares the outcomes of two patients randomly drawn from each treatment arm. Specifically, we consider two treatment-discordant patients $i$ and $j$ (say $A_{i} = 1$ and $A_{j} = 0$) which are chosen randomly from subgroup $g$. If we could observe the failure times $T_{i}$ and $T_{j}$ along with $W_{i}$ and $W_{j}$, it would be possible to determine which patient had the superior outcome. That is, we could construct a function $O(\cdot, \cdot|\cdot,\cdot)$ such that $O(T_{i}, W_{i}|T_{j}, W_{j}) = 1$ when outcome $(T_{i}, W_{i})$ is superior to $(T_{j}, W_{j})$ and $O(T_{i}, W_{i}|T_{j}, W_{j}) = 0$ otherwise. While there are many ways of choosing $O(T_{i}, W_{i}|T_{j}, W_{j})$, Table \[tab:outcome\_ordering\] describes our approach for ordering pairs of bivariate outcomes. Here, we rely on an indifference parameter $\delta > 0$ which represents the additional gain in survival that would be needed to compensate for suffering from the AE. For example, suppose that $W_{i} = 1$ and $W_{j} = 0$ and that the indifference parameter is set to $\delta = 0.5$. In this case, patient $i$ would need an at least $50\%$ longer survival time than patient $j$ in order for the outcome of patient $i$ to be preferable to the outcome of patient $j$. -------------------------------- ------------------------- --------- $T_{i} > T_{j}(1 + \delta)$ $W_{i} = 1, W_{j} = 0$ $A = 1$ $T_{i} \leq T_{j}(1 + \delta)$ $W_{i} = 1, W_{j} = 0$ $A = 0$ $T_{j} > T_{i}(1 + \delta)$ $W_{i} = 0, W_{j} = 1$ $A = 0$ $T_{j} \leq T_{i}(1 + \delta)$ $W_{i} = 0, W_{j} = 1$ $A = 1$ $T_{i} > T_{j}$ $W_{i} = 1, W_{j} = 1$ $A = 1$ $T_{i} > T_{j}$ $W_{i} = 0, W_{j} = 0$ $A = 1$ $T_{i} \leq T_{j}$ $W_{i} = 1, W_{j} = 1$ $A = 0$ $T_{i} \leq T_{j}$ $W_{i} = 0, W_{j} = 0$ $A = 0$ -------------------------------- ------------------------- --------- : Our approach for comparing two joint outcomes $(T_{i}, W_{i})$ and $(T_{j}, W_{j})$ assuming that $A_{i} = 1$ and $A_{j} = 0$.[]{data-label="tab:outcome_ordering"} Using the outcome orderings described in Table \[tab:outcome\_ordering\], we define the subgroup-specific treatment effects $\phi_{g}$ as $$\label{phig} \phi_{g} = 2P\big\{ O(T_{i}, W_{i}|T_{j}, W_{j})= 1 \mid A_{i} = 1, A_{j} = 0, G_{i}=g, G_{j} = g \big\} - 1.$$ The parameter $\phi_{g}$ represents the probability that patient $i$ has a superior outcome to patient $j$ minus the probability that patient $j$ has a superior outcome to patient $i$ when assuming both patients $i$ and $j$ belong to subgroup $g$. Hence, $\phi_{g} = 1$ when patients in subgroup $g$ are sure to benefit from treatment, and $\phi_{g} = -1$ when patients in subgroup $g$ are certain to be harmed by treatment. Note that $\phi_{g}$ is a function of the subgroup-specific parameters $\lambda_{awg}$ and $p_{ag}$ and thus one may easily obtain draws from the posterior distribution of $\phi_{g}$ by transforming samples from the posterior distribution of $(\lambda_{awg}, p_{ag})$. More specifically, $\phi_{g}$ is related to the model parameters via $$\phi_{g} = 2\Big[ \frac{\lambda_{00g}p_{1g}(1 - p_{0g})}{\lambda_{11g}(1 + \delta) + \lambda_{00g}} + \frac{\lambda_{01g}(1 - p_{1g})p_{0g}}{\lambda_{10g}/(1 + \delta) + \lambda_{01g}} + \frac{\lambda_{00g}(1 -p_{1g})(1 - p_{0g})}{\lambda_{10g} + \lambda_{00g}} + \frac{\lambda_{01g}p_{1g}p_{0g}}{\lambda_{11g} + \lambda_{01g}} \Big] - 1. \nonumber$$ ![Sprint Trial: Posterior means of treatment effects $\phi_{g}$ with $\delta = 0.2$ (see Table \[tab:outcome\_ordering\]). Posterior means and credible intervals for $\phi_{g}$ are shown for both the saturated and additive models. The solid vertical lines are placed at the overall treatment effect estimates i.e, $\tfrac{1}{G}\sum_{g=1}^{G} \hat{\phi}_{g}$, where $\hat{\phi}_{g}$ is the posterior mean of $\phi_{g}$.[]{data-label="fig:ranking_forest"}](ranking_forest.eps){width="14cm" height="14cm"} Figure \[fig:ranking\_forest\] plots the posterior means of $\phi_{g}$ for each subgroup using both the saturated and additive models. For this analysis, we set $\delta$ to $\delta = 0.2$. This means that an individual receiving the intensive treatment would need at least $1.2$ times as much PE-free survival as under the standard treatment in order to be “compensated” for the fact that he/she would experience a treatment-related SAE under the intensive treatment while remaining SAE-free under the standard treatment arm. In Figure \[fig:ranking\_forest\], the “overall” treatment effects are represented by the solid vertical lines. These are computed as $\tfrac{1}{G}\sum_{g=1}^{G} \hat{\phi}_{g}$ where $\hat{\phi}_{g}$ is the posterior mean of $\phi_{g}$. The overall treatment effect estimates were $0.12$ and $0.13$ for the saturated and additive models respectively. As shown in Figure \[fig:ranking\_forest\], there is less clear variability across subgroup when using the treatment effect $\phi_{g}$. Specifically, all the credible intervals for both the saturated and additive models cover the average value of the posterior means $\hat{\phi}_{g}$. It is interesting to note that, on this scale, the estimated treatment effect for the no-CKD/Age $< 75$/Female subgroup is now very close to the estimated overall treatment effect which seems to stand in contrast to the results in Sections 5.1 and 5.3. While the width of the credible intervals in both the saturated and additive models prevents us from making any strong conclusions about this subgroup, it is worthwhile to note how the prominence of a treatment-covariate interaction can change when the treatment effect scale is changed. The differences between the saturated and additive models for the no-CKD/Age $\geq 75$/Female and yes-CKD/Age $\geq 75$/Female subgroups is also noteworthy. In addition to much greater shrinkage with the additive model, the difference between the estimates of $\phi_{g}$ in these $2$ subgroups changes sign between the saturated and additive models. This is due to the more structured borrowing of information among subgroups that is enabled by the additive model. Specifically, much information is shared among subgroups carrying the same CKD status with the no-CKD subgroups being consistently shrunken towards a larger value than the yes-CKD subgroups. Model Checking and Diagnostics ============================== Our approach assumes that, conditional on an AE indicator, survival follows an exponential distribution within each subgroup/treatment arm combination. We feel that such a simple parametric approach is needed due to the potentially large number of “cells” that arise from the number of subgroup/treatment arm/AE combinations where the sparse amount of within-cell data precludes more ambitious modeling. Despite this, our model treats patient survival as arising from a type of mixture-of-exponentials distribution and such a mixture can often provide a good fit to the observed overall survival patterns. ![SPRINT trial: Draws from the posterior predictive distribution for the saturated model. Kaplan-Meier estimates for the primary outcomes from the observed data are plotted in solid red. Kaplan-Meier estimates from fifty draws from the posterior predictive distribution are plotted in grey.[]{data-label="fig:pp_survival_curves"}](pp_draws.eps){width="18cm" height="12cm"} Posterior predictive checks ([@meng:1994]) are a useful tool for assessing the goodness-of-fit of a Bayesian model. Such checks are performed by first simulating outcomes from the posterior predictive distribution and computing a test statistic (or a collection of test statistics) of interest for each simulation replication. The distribution of the computed test statistics across posterior predictive draws is then compared with the observed value of this test statistic. An observed value of the test statistic that seems “typical” with respect to the posterior predictive distribution of this test statistic is an indication that the model provides a good fit or, at least, is not clearly deficient in some way. Figure \[fig:pp\_survival\_curves\] shows treatment-specific estimated survival curves from $50$ draws from the posterior predictive distribution. Here, the test statistics of interest are the treatment-specific Kaplan-Meier estimates. As shown in Figure \[fig:pp\_survival\_curves\], the Kaplan-Meier estimates for the observed data seem typical when compared with the collection of Kaplan-Meier estimates from the $50$ posterior predictive draws. Indeed, for both treatment arms, the observed Kaplan-Meier estimates are roughly centered among the posterior predictive survival curves, and the shapes of the posterior predictive survival curves largely resemble the linear shape of the observed Kaplan-Meier estimates. ![SPRINT trial: Posterior predictive checks. Samples from the posterior predictive distribution of the RMST (by treatment arm) using both the saturated and additive models. Solid vertical lines are placed at the observed values of the RMST.[]{data-label="fig:pp_rmst"}](pp_pval.eps){width="16cm" height="14cm"} Beyond graphical displays of survival, one can use univariate test statistics and posterior predictive p-values as a way of more formally assessing goodness-of-fit. Posterior predictive p-values represent the probability that hypothetical replications of the test statistic from the posterior predictive distribution is more extreme than the observed value of the test statistic. It is often recommended to try test statistics which are not directly modeled by the probability model used ([@gelman:2014]). Figure \[fig:pp\_rmst\] displays the posterior predictive distributions of the restricted mean survival time (RMST) using both the saturated and additive models. For the saturated model, the two-sided posterior predictive p-values are $0.30$ and $0.38$ for the standard and intensive treatment arms respectively. For the additive model, the two-sided posterior predictive p-values are $0.28$ and $0.40$ for the standard and intensive treatment arms respectively. These relatively large posterior predictive p-values do not indicate that either model has a serious flaw. --------------------------- ----------- ----------- (r)[2-2]{} (r)[3-3]{} DIC $12853.3$ $12863.4$ WAIC $12861.9$ $12865.2$ --------------------------- ----------- ----------- : Comparisons of the saturated and additive models using different information criteria. \[tab:IC\] Information criteria can be used to compare models such as the saturated and additive models that have differing number of parameters. The use of information criteria can be helpful whenever model checking procedures do not clearly show that one of the models is inadequate or when there is no strong a priori reason for favoring one model over the other. Two well-known information criteria that can be computed for Bayesian hierarchical models are the deviance information criterion (DIC) ([@spiegelhalter:2002]) and the widely-applicable information criterion (WAIC) ([@Watanabe:2010]). Both of these criterion are found by adding a penalty to negative two times the log-likelihood function evaluated at the posterior mean of the model parameters. The penalty is determined by the model degrees of freedom which are computed differently by DIC and WAIC. Lower values of both the DIC and WAIC imply better model performance. As shown in Table \[tab:IC\], the additive model had a DIC which was roughly $10$ less than the DIC of the saturated model. Differences in the DIC of less that $3-5$ are often not considered meaningful (see e.g., [@Louis:09]) so this difference of $10$ provides moderate support in favor of the additive model. Moreover, the additive model may be especially preferred if a parsimonious model is desired. In contrast to the DIC, the saturated and additive models are much closer when using WAIC as the comparison measure. Nevertheless, a difference of roughly $4$ in WAIC value still provides mild support in favor of the additive model. Thus, for the SPRINT data, there does not seem to be a good justification for using the saturated model as the log-likelihood (evaluated at the posterior mean of the parameters) for both models is roughly the same while the saturated model requires considerably fewer model parameters. Discussion ========== In this article, we have described a Bayesian approach for performing subgroup analysis with respect to both a primary and safety endpoint and have detailed its use in generating relevant within-subgroup risk-benefit measures. A key feature of our approach is that the benefit-safety endpoints are modeled jointly rather than used to perform separate subgroup analyses of benefit and safety. Summary assessment of risks and benefits, i.e. combining the multiple risks and benefit outcomes into a single utility measure, is a challenging problem. In section \[utility\] we have provided a few reasonable options, Eqs. \[ttg\], \[etag\], \[phig\]. Other summary measures are certainly possible. It should be remarked that both the primary and safety endpoints used in the SPRINT trial are themselves composite endpoints. Hence, it may be worth decomposing these composite measures further for a more thorough assessment of benefit and risk (e.g., death and other cardiac events are treated the same).

--- abstract: 'We calculate the massive Wilson coefficients for the heavy flavor contributions to the non-singlet charged current deep-inelastic scattering structure functions $F_L^{W^+}(x,Q^2)-F_L^{W^-}(x,Q^2)$ and $F_2^{W^+}(x,Q^2)-F_2^{W^-}(x,Q^2)$ in the asymptotic region $Q^2 \gg m^2$ to 3-loop order in Quantum Chromodynamics (QCD) at general values of the Mellin variable $N$ and the momentum fraction $x$. Besides the heavy quark pair production, also the single heavy flavor excitation $s \rightarrow c$ contributes. Numerical results are presented for the charm quark contributions and consequences on the unpolarized Bjorken sum rule and Adler sum rule are discussed.' --- DESY 16–148\ DO–TH 16/15\ September 2016\ [**The Asymptotic 3-Loop Heavy Flavor Corrections to the**]{} A. Behring$^a$, J. Blümlein$^a$, G. Falcioni$^a$, A. De Freitas$^a$, A. von Manteuffel$^b$, and C. Schneider$^c$ \ [*Platanenallee 6, D-15738 Zeuthen, Germany*]{} \ \ [*Johannes Kepler University, Altenbergerstraße 69, A-4040 Linz, Austria*]{} Introduction {#sec:1} ============ The flavor non-singlet charged current structure functions $F_{1,2}^{W^+ - W^-}(x,Q^2)$ can be measured in deep-inelastic neutrino(antineutrino)-nucleon scattering and in high energy charged lepton-nucleon scattering in $e p$ or $\mu p$ collisions. They are associated with the well-known unpolarized Bjorken sum rule [@Bjorken:1967px] and Adler sum rule [@Adler:1965ty] by their first moment, the former of which can be used for QCD tests measuring the strong coupling constant $a_s = \alpha_s/(4\pi) = g_s^2/(4\pi)^2$. These structure functions also allow for an associated determination of the valence quark distributions of the nucleon. The massless contributions to these combinations of structure functions have been calculated recently to 3-loop order [@Davies:2016ruz]. In the present paper we compute the asymptotic heavy flavor corrections to these flavor non-singlet structure functions in the region $Q^2 \gg m^2$ to the same order, with $m$ the heavy quark mass and $Q^2$ the virtuality of the process, and present numerical results in the case of charm quark contributions. The massless and massive QCD corrections at first order in the coupling constant have been computed in Refs. [@Bardeen:1978yd; @Furmanski:1981cw; @Gluck:1997sj; @Blumlein:2011zu][^1] and in Refs. [@Zijlstra:1992kj; @Moch:1999eb; @Buza:1997mg; @Blumlein:2014fqa; @Hasselhuhn:2013swa] to $O(a_s^2)$[^2]. The massive $O(a_s^2)$ corrections were calculated in the asymptotic representation [@Buza:1995ie], which is valid at high scales $Q^2$. To obtain an estimate of the range of validity, one may perform an $O(a_s)$ comparison with the complete result for the process of single heavy quark excitation [@Gluck:1997sj; @Blumlein:2011zu]. Likewise, a comparison is possible for the $O(a_s^2)$ corrections, which were given in complete form in Ref. [@Blumlein:2016xcy] for the Wilson coefficient with the gauge boson coupling to the massless fermions and assuming an approximation for the Cabibbo-suppressed flavor excitation term $s' \rightarrow c$, where the additional charm quark in the final state has been dealt with as being massless. The charged current scattering cross sections are given by [@Arbuzov:1995id; @Blumlein:2012bf] $$\begin{aligned} \frac{d \sigma^{\nu(\bar{\nu})}}{dx dy} &=& \frac{G_F^2 s}{4 \pi} \frac{M_W^4}{(M_W^2 + Q^2)^2} \\ && \times \Biggl\{ \left(1 + (1-y)^2\right) F_2^{W^\pm}(x,Q^2) - y^2 F_L^{W^\pm}(x,Q^2) \pm \left(1 - (1-y)^2\right) xF_3^{W^\pm}(x,Q^2)\Biggr\} \nonumber\\ \frac{d \sigma^{l(\bar{l})}}{dx dy} &=& \frac{G_F^2 s }{4 \pi} \frac{M_W^4}{(M_W^2 + Q^2)^2} \\ && \times \Biggl\{ \left(1 + (1-y)^2\right) F_2^{W^\mp}(x,Q^2) - y^2 F_L^{W^\mp}(x,Q^2) \pm \left(1 - (1-y)^2\right) xF_3^{W^\mp}(x,Q^2)\Biggr\}~, \nonumber\end{aligned}$$ where $x = Q^2/ys$ and $y = q.P/l.P$ denote the Bjorken variables, $l$ and $P$ are the incoming lepton and nucleon 4-momenta, and $s = (l+P)^2$. $G_F$ is the Fermi constant and $M_W$ the mass of the $W$-boson. $F_i^{W^\pm}(x,Q^2)$ are the structure functions, where the $+(-)$ signs refer to incoming neutrinos (antineutrinos) and charged antileptons (leptons), respectively. We will consider the combination of structure functions $$\begin{aligned} \label{eq:comb} F_{1,2}^{W^+ - W^-}(x,Q^2) = F_{1,2}^{W^+}(x,Q^2) - F_{1,2}^{W^-}(x,Q^2)\end{aligned}$$ in the following. The longitudinal structure function is obtained by $$\begin{aligned} \label{eq:FL} F_{L}(x,Q^2) = F_{2}(x,Q^2) - 2xF_{1}(x,Q^2).\end{aligned}$$ The combinations (\[eq:comb\]) can be measured projecting onto the kinematic factor $Y_+ = 1 + (1-y)^2$ in the case of $F_2$ for the differential cross sections at $x, Q^2 = \rm const.$ and by varying $s$ in addition, in the case of $F_L$. The main formalism to obtain the massive Wilson coefficients in the asymptotic range $Q^2 \gg m^2$, i.e. $L_{q,L,2}^{W^+ - W^-, \rm NS}$ and $H_{q,L,2}^{W^+ - W^-, \rm NS}$, has been outlined in Refs. [@Buza:1995ie; @Bierenbaum:2009mv; @Blumlein:2014fqa]. They are composed of the massive non-singlet operator matrix elements (OMEs) [@Ablinger:2014vwa] and the massless Wilson coefficients [@Davies:2016ruz] up to 3-loop order. The following representation of the structure functions is obtained $$\begin{aligned} F_{L,2}^{W^+ - W^-}(x,Q^2) &=& 2x \Biggl\{\Bigl[ |V_{du}|^2 (d - \overline{d}) + |V_{su}|^2 (s - \overline{s}) - V_{u} (u - \overline{u})\Bigr] \otimes \Bigl[ C_{q,L,2}^{W^+ - W^-, \rm NS} \nonumber\\ && +L_{q,L,2}^{W^+ - W^-, \rm NS} \Bigr] + \Bigl[ |V_{dc}|^2 (d - \overline{d}) + |V_{sc}|^2 (s - \overline{s}) \Bigr] \otimes H_{q,L,2}^{W^+ - W^-, \rm NS}\Biggr\},\end{aligned}$$ with one massless Wilson coefficient $C_{q,L,2}^{W^+ - W^-, \rm NS}$ and two massive Wilson coefficients $L_{q,L,2}^{W^+ - W^-, \rm NS}$, $H_{q,L,2}^{W^+ - W^-, \rm NS}$, see Sections \[sec:2\] and \[sec:3\]. The coefficients $V_{ij}$ are the Cabibbo-Kobayashi-Maskawa (CKM) [@Cabibbo:1963yz; @Kobayashi:1973fv] matrix elements, where $V_u = |V_{du}|^2 + |V_{su}|^2$, and the present numerical values are [@PDG] $$\begin{aligned} |V_{du}| &=& 0.97425,~~~ |V_{su}| = 0.2253,~~~ |V_{dc}| = 0.225,~~~ |V_{sc}| = 0.986~,\end{aligned}$$ with $$\begin{aligned} u - \overline{u} &=& u_v, \\ d - \overline{d} &=& d_v, \\ s - \overline{s} &\approx& 0~.\end{aligned}$$ In the following we will consider only the charm quark corrections with $m \equiv m_c$ the charm quark mass in the on-shell scheme. The transformation to the $\overline{\rm MS}$ scheme has been given in Ref. [@Ablinger:2014vwa]. We note that the 3-loop asymptotic charm quark corrections to the combination of structure functions $xF_3^{W^+ + W^-}(x,Q^2)$ have been calculated in Ref. [@Behring:2015roa] and related corrections to the twist-2 contributions of the polarized structure functions $g_{1,2}(x,Q^2)$ in Ref. [@Behring:2015zaa]. A series of asymptotic 3-loop heavy flavor Wilson coefficients have also been calculated for neutral current scattering along with the transition matrix elements in the variable flavor number scheme, see Refs. [@Ablinger:2016swq] for recent surveys. The Structure Function $F_L(x,Q^2)$ {#sec:2} =================================== The massive Wilson coefficients depend on the logarithms $$\begin{aligned} L_M = \ln\left(\frac{m^2}{\mu^2}\right),~~~~~~~~~L_Q = \ln\left(\frac{Q^2}{\mu^2}\right)~.\end{aligned}$$ Here $\mu$ denotes the factorization scale. For the Wilson coefficients in Mellin $N$ space we consider the following series in the strong coupling constant $$\begin{aligned} L_{q,2(L)}^{W^+ - W^-, \rm NS} = \delta_{2,0} + \sum_{k=1}^\infty a_s^k L_{q,2(L)}^{W^+ - W^-, {\rm NS},(k)}, \\ H_{q,2(L)}^{W^+ - W^-, \rm NS} = \delta_{2,0} + \sum_{k=1}^\infty a_s^k H_{q,2(L)}^{W^+ - W^-, {\rm NS},(k)}, \\ C_{q,2(L)}^{W^+ - W^-, \rm NS} = \delta_{2,0} + \sum_{k=1}^\infty a_s^k C_{q,2(L)}^{W^+ - W^-, {\rm NS},(k)}. \end{aligned}$$ In the following we drop the arguments of the nested harmonic sums [@HSUM] and harmonic polylogarithms [@Remiddi:1999ew] by defining $S_{\vec{a}}(N) \equiv S_{\vec{a}}$ and $H_{\vec{b}}(x) \equiv H_{\vec{b}}$. The 3-loop contributions to the Wilson coefficient $L_{q,L}^{W^+ - W^-, {\rm NS},(3)}$ in Mellin $N$ space are given by where $\hat{c}_{q,L}^{(3)}=c^{(3)}_{q,L}(N_F+1)-c^{(3)}_{q,L}(N_F)$ is the 3-loop massless contribution, cf. [@Davies:2016ruz]. The color factors in the case of QCD are $C_A= N_c = 3, C_F = (N_c^2-1)/(2N_c), T_F = 1/2$, $N_c = 3$ and $N_F$ denotes the number of massless flavors. Except for $\hat{c}^{(3)}_{q,L}(N_F)$, the Wilson coefficient is expressed by harmonic sums up to weight [w=3]{}. The polynomials $P_i$ above read By performing a Mellin inversion, the corresponding representation in $x$ space is obtained, which reads Here $\zeta_k, k \in \mathbb{N}, k \geq 2$, are the values of the Riemann $\zeta$ function at integer argument. Except for $\hat{c}^{(3)}_{q,L}(N_F)$, the Wilson coefficient is expressed by weighted harmonic polylogarithms of up to weight [w=3]{}. The contribution of the massive Wilson coefficient $H_{q,L}$ is found by combining the massless Wilson coefficient $C_{q,L}$ and $L_{q,L}$: $$H_{q,L}(x,Q^2) = C_{q,L}(N_F,x,Q^2) + L_{q,L}(x,Q^2)~.$$ Eq. (\[eq:FL\]) provides the relation to the Wilson coefficients of the structure function $F_1(x,Q^2)$. The Structure Function $F_2(x,Q^2)$ {#sec:3} =================================== The asymptotic massive 3-loop Wilson coefficient $L_{q,2}^{W^+ - W^-, {\rm NS}, (3)}$ in Mellin $N$ space reads where $\hat{c}^{(3)}_{q,2}=c^{(3)}_{q,2}(N_F+1)-c^{(3)}_{q,2}(N_F)$ is obtained from the 3-loop massless Wilson coefficient Ref. [@Davies:2016ruz]. Except for $\hat{c}^{(3)}_{q,2}(N_F)$, the Wilson coefficient is expressed by harmonic sums up to weight [w=5]{}. The polynomials in the equation above are defined as follows Here the constant $B_4$ is given by $$\begin{aligned} B_4 = - 4 \zeta_2 \ln^2(2) + \frac{2}{3} \ln^4(2) - \frac{13}{2} \zeta_4 + 16 {{\rm Li}}_4\left(\frac{1}{2}\right)~.\end{aligned}$$ By performing the Mellin inversion to $x$-space one obtains Here the $+$-prescription is defined by $$\int_0^1 dx g(x) [f(x)]_+ = \int_0^1 dx [g(x)-g(1)]f(x)~.$$ The contribution of the massive Wilson coefficient $H_{q,2}$ is found by combining the massless Wilson coefficient $C_{q,2}$ and $L_{q,2}$ by $$H_{q,2}(x,Q^2) = C_{q,2}(N_F,x,Q^2)+L_{q,2}(x,Q^2)~.$$ Except for $c^{(3)}_{q,2}(N_F)$, the Wilson coefficients are expressed by up to weight [w=4]{} harmonic polylogarithms. Note the emergence of a denominator $1/(1-x)^2$, cf. [@Ablinger:2014vwa], which is properly regularized by its numerator function in the limit $x \rightarrow 1$. We note that we have applied the shuffle algebra, cf. [@Blumlein:2003gb], which leads to a reduction of the number of harmonic polylogarithms compared to the linear representation, making the numerical evaluation faster. Numerical Results {#sec:5} ================= In the following we illustrate the asymptotic charm corrections up to 3-loop order to the charged current non-singlet combinations $F_{1,2}^{W^+-W^-}(x,Q^2)$ choosing the renormalization and factorization scales $\mu^2 = Q^2$. First we consider the behaviour of the corrections at small and large values of the Bjorken variable $x$. For those of the massless 3-loop Wilson coefficients see [@Davies:2016ruz]. The limiting behaviour for the two contributing functions $L_{q,i}^{W^+-W^-,\rm NS}(N_F+1) - \hat{C}_{q,i}^{W^+-W^-,\rm NS}(N_F)$ and $H_{q,i}^{W^+-W^-,\rm NS}(N_F+1) - {C}_{q,i}^{W^+-W^-,\rm NS}(N_F+1)$ are the same, see also [@Blumlein:2016xcy]. For the 3-loop contributions, yet for general values of $\mu^2$, at low values of $x$ one has $$\begin{aligned} L_{q,L}^{W^+-W^-\rm NS}(N_F+1)-\hat{C}_{q,L}^{W^+-W^-,\rm NS}(N_F) &\propto& a_s^3\,\frac{8}{3}C_F^2 T_F\, \ln^2(x) \\ L_{q,2}^{W^+-W^-,\rm NS}(N_F+1)-\hat{C}_{q,2}^{W^+-W^-,\rm NS}(N_F) &\propto& a_s^3 \Biggl\{ \frac{16}{27} C_A C_F T_F - \frac{5}{9} C_F^2 T_F \Biggr\} \ln^4(x)\end{aligned}$$ and at large $x$ $$\begin{aligned} L_{q,L}^{W^+-W^-,\rm NS}(N_F+1)-\hat{C}_{q,L}^{W^+-W^-,\rm NS}(N_F) &\propto& a_s^3 C_F T_F \Biggl\{\frac{128}{3} C_F L_Q \ln^2(1-x) + \Biggl[C_F \Biggl(\frac{896}{27} \nonumber\\ && + \frac{320}{9} L_M + \frac{32}{3} L_M^2 \Biggr) + 32 C_F L_Q^2 + \Biggl(- \frac{1088}{9} C_A \nonumber\\ && + \frac{80}{9} C_F + \frac{128}{9} T_F + \frac{256}{9} N_F T_F + \frac{128}{3} \zeta_2 C_A \nonumber\\ && - \frac{256}{3} \zeta_2 C_F \Biggr) L_Q \Biggr] \ln(1-x) \Bigg\} \\ L_{q,2}^{W^+-W^-,\rm NS}(N_F+1)-\hat{C}_{q,2}^{W^+-W^-,\rm NS}(N_F) &\propto& a_s^3 C_F T_F \Biggl\{ \frac{320}{9} C_F L_Q \left(\frac{\ln^3(1-x)}{1-x}\right)_+ \nonumber\\ && + \Biggl[C_F \Biggl(\frac{448}{9} + \frac{160}{3} L_M + 16 L_M^2\Biggr) + 48 C_F L_Q^2 \nonumber\\ && + \Biggl(-\frac{352}{9} C_A - \frac{512}{3} C_F + \frac{64}{9} T_F \nonumber\\ && + \frac{128}{9} N_F T_F\Biggr) L_Q \Biggr] \left(\frac{\ln^2(1-x)}{1-x}\right)_+\Biggr\}~.\end{aligned}$$ Below we plot the heavy flavor contribution to the structure function $F_1^{W^+-W^-}(x,Q^2)$ for the quark mass $m_c=1.59\, {\rm GeV}$ in the on-shell scheme [@Alekhin:2012vu] and the scales $Q^2=\mu^2=10,\,100,\,1000\,~{\rm GeV}^2$ for the complete structure function, including the massive and massless terms. ![The structure function $xF_1^{W^+-W^-}(x,Q^2)$, containing the 3-loop corrections including the asymptotic corrections for charm using $m_c^{\rm OMS} = 1.59~{\rm GeV}$ and the PDFs [@Alekhin:2013nda].[]{data-label="fig:1"}](F1rec_asy.pdf){width="0.7\linewidth"} ![The ratio of massive contributions to the structure function $xF_1^{W^+-W^-}(x,Q^2)$ over the complete structure function for $Q^2=10~{\rm GeV}^2$, containing the 3-loop corrections including the asymptotic corrections for charm using $m_c^{\rm OMS} = 1.59~{\rm GeV}$ and the PDFs [@Alekhin:2013nda]. For the dash-dotted line, asymptotic corrections at three loops and the complete heavy flavor contributions up to $O(a_s^2)$ [@Blumlein:2016xcy] are taken into account.[]{data-label="fig:2"}](F1ratios10asy.pdf){width="0.7\linewidth"} In Figure \[fig:1\] the scale evolution of the structure function $xF_1^{W^+-W^-}(x,Q^2)$ is shown in the range $Q^2 \in [10,1000]~{\rm GeV}^2$, including the asymptotic charm quark corrections to 3-loop order. ![The ratio of massive contributions to the structure function $xF_1^{W^+-W^-}(x,Q^2)$ over the complete structure function for $Q^2=100~{\rm GeV}^2$, containing the 3-loop corrections including the asymptotic corrections for charm using $m_c^{\rm OMS} = 1.59~{\rm GeV}$ and the PDFs [@Alekhin:2013nda]. For the dash-dotted line, asymptotic corrections at three loops and the complete heavy flavor contributions up to $O(a_s^2)$ [@Blumlein:2016xcy] are taken into account.[]{data-label="fig:3"}](F1ratios100asy.pdf){width="0.7\linewidth"} ![The ratio of massive contributions to the structure function $xF_1^{W^+-W^-}(x,Q^2)$ over the complete structure function for $Q^2=1000~{\rm GeV}^2$, containing the 3-loop corrections including the asymptotic corrections for charm using $m_c^{\rm OMS} = 1.59~{\rm GeV}$ and the PDFs [@Alekhin:2013nda]. For the dash-dotted line, asymptotic corrections at three loops and the complete heavy flavor contributions up to $O(a_s^2)$ [@Blumlein:2016xcy] are taken into account.[]{data-label="fig:4"}](F1ratios1000asy.pdf){width="0.7\linewidth"} Here and in the following we refer to the parton distribution functions [@Alekhin:2013nda]. As typical for non-singlet contributions, the profile is shifted from larger to smaller values of $x$ with growing values of $Q^2$. However, the effects are much smaller than in the singlet case. As it is well known, the validity of the asymptotic charm quark corrections in the case of $F_L(x,Q^2)$, and therefore for $F_2$ and in part for $xF_1$, is setting in at higher scales only due to the $F_L$ contribution, for details see [@Buza:1995ie]. We will discuss these aspects in the following figures for $xF_1$ and $F_2$. In Figure \[fig:2\] the corrections to $xF_1^{W^+-W^-}(x,Q^2)$ are illustrated for $Q^2 = 10~{\rm GeV}^2$ by adding the contributions from $O(a_s^0)$ to $O(a_s^3)$, showing an increasing degree of stabilization. We also present the exact heavy flavor corrections to $O(a_s^2)$ [@Blumlein:2016xcy], showing deviations in the range $x {\raisebox{-0.07cm } {$\, \stackrel{>}{{\scriptstyle\sim}}\, $}}10^{-2}$, while below there is exact agreement. The latter effect is due to the sufficiently large $W^2 = Q^2 (1-x)/x$ values through which the heavy quarks are made effectively massless for this structure function even at this low scale of $Q^2$. The charm quark corrections for $xF_1^{W^+-W^-}(x,Q^2)$ vary in a range of $-8 \%$ to $\sim 0\%$, depending on $x$, with a maximal relative contribution around $x \sim 3 \cdot 10^{-2}$. Figure \[fig:3\] shows that at $Q^2 = 100~{\rm GeV}^2$ the asymptotic corrections agree also in the case where we include the power corrections to larger values of $x \sim 0.3$ and for $Q^2 = 1000~{\rm GeV}^2$, Figure \[fig:4\], the agreement is obtained in the whole $x$ range. We turn now to the numerical illustration of the structure function $F_2^{W^+-W^-}(x,Q^2)$. In Figure \[fig:5\] we show the scaling violations of $F_2^{W^+-W^-}(x,Q^2)$ in the region $Q^2 \in [10,1000]~{\rm GeV}^2$, shifting the profile to lower values of $x$ with growing virtualities $Q^2$. Figure \[fig:6\] shows the contributions to $F_2^{W^+-W^-}(x,Q^2)$ at $Q^2 = 10~{\rm GeV}^2$ for growing order in the strong coupling constant stabilizing at 3-loop order, except of very large values of $x$. At $Q^2 = 10~{\rm GeV}^2$ comparing the results for $2xF_1$ and $F_2$ the effect of $F_L(x,Q^2)$ is clearly visible. The asymptotic expression is not yet valid in the charged current case, as the complete $O(a_s^2)$ charm quark corrections show. ![The structure function $F_2^{W^+-W^-}(x,Q^2)$, containing the 3-loop corrections including the asymptotic corrections for charm using $m_c^{\rm OMS} = 1.59~{\rm GeV}$ and the PDFs [@Alekhin:2013nda].[]{data-label="fig:5"}](F2rec_asy.pdf){width="0.7\linewidth"} Again the relative charm quark corrections vary in the range $[-8\%, \sim 0\%]$. As shown in Figure \[fig:7\], the asymptotic corrections agree with the case where the power corrections are included, except for a small range at very large $x$ values at $Q^2 = 100~{\rm GeV}^2$. Finally, this effect disappears for $Q^2 = 1000~{\rm GeV}^2$, see Figure \[fig:8\]. ![The ratio of massive contributions to the structure function $F_2^{W^+-W^-}(x,Q^2)$ over the complete structure function for $Q^2=10~{\rm GeV}^2$, containing the 3-loop corrections including the asymptotic corrections for charm using $m_c^{\rm OMS} = 1.59~{\rm GeV}$ and the PDFs [@Alekhin:2013nda]. For the dash-dotted line, asymptotic corrections at three loops and the complete heavy flavor contributions up to $O(a_s^2)$ [@Blumlein:2016xcy] are taken into account.[]{data-label="fig:6"}](F2ratios10asy.pdf){width="0.7\linewidth"} ![The ratio of massive contributions to the structure function $F_2^{W^+-W^-}(x,Q^2)$ over the complete structure function for $Q^2=100~{\rm GeV}^2$, containing the 3-loop corrections including the asymptotic corrections for charm using $m_c^{\rm OMS} = 1.59~{\rm GeV}$ and the PDFs [@Alekhin:2013nda]. For the dash-dotted line, asymptotic corrections at three loops and the complete heavy flavor contributions up to $O(a_s^2)$ [@Blumlein:2016xcy] are taken into account.[]{data-label="fig:7"}](F2ratios100asy.pdf){width="0.7\linewidth"} ![The ratio of massive contributions to the structure function $F_2^{W^+-W^-}(x,Q^2)$ over the complete structure function for $Q^2=1000~{\rm GeV}^2$, containing the 3-loop corrections including the asymptotic corrections for charm using $m_c^{\rm OMS} = 1.59~{\rm GeV}$ and the PDFs [@Alekhin:2013nda]. For the dash-dotted line, asymptotic corrections at three loops and the complete heavy flavor contributions up to $O(a_s^2)$ [@Blumlein:2016xcy] are taken into account.[]{data-label="fig:8"}](F2ratios1000asy.pdf){width="0.7\linewidth"} The Sum Rules {#sec:6} ============= For the combination of the charged current structure functions being considered here, there exist sum rules arising from the lowest Mellin moment. In the case of $F_2^{W^+-W^-}(x,Q^2)$, one obtains the Adler sum rule [@Adler:1965ty] and for $F_1^{W^+-W^-}(x,Q^2)$ the unpolarized Bjorken sum rule [@Bjorken:1967px], for which also the target mass corrections have to be considered, cf. [@Blumlein:2016xcy]. The Adler sum rule states $$\begin{aligned} \label{eq:ADLER} \int_0^1 \frac{dx}{x} \left[F_2^{\overline{\nu}p}(x,Q^2) - F_2^{{\nu}p}(x,Q^2)\right] = 2 [1 + \sin^2(\theta_c)]\end{aligned}$$ for three massless flavors. Here $\theta_c$ denotes the Cabibbo angle [@Cabibbo:1963yz]. The integral (\[eq:ADLER\]) neither receives QCD nor quark- or target mass corrections [@Blumlein:2012bf], cf. also [@Ravindran:2001dk; @Adler:2009dw]. Up to 2-loop order the vanishing of the heavy quark corrections has been shown in Ref. [@Blumlein:2016xcy]. Considering the limit of large scales $Q^2 \gg m^2$, this is confirmed at 3-loop order since the flavor non-singlet OMEs vanish for $N=1$ due to fermion number conservation [@Ablinger:2014vwa] and the first moment of the corresponding massless Wilson coefficient also vanishes [@Moch:2007gx]. The unpolarized Bjorken sum rule [@Bjorken:1967px] is given by $$\begin{aligned} \label{eq:UPBJ} \int_0^1 {dx} \left[F_1^{\bar{\nu} p}(x,Q^2) - F_1^{{\nu} p}(x,Q^2)\right] = C_{\rm uBJ}(\hat{a}_s),\end{aligned}$$ with $\hat{a}_s = \alpha_s/\pi$. The massless 1-loop [@Bardeen:1978yd; @Altarelli:1978id; @Humpert:1980uv; @Furmanski:1981cw], 2-loop [@Gorishnii:1985xm], 3-loop [@Larin:1990zw] and 4-loop [@Chetyrkin:14] QCD corrections have been calculated $$\begin{aligned} \label{eq:uBJSR} C_{\rm uBJ}(\hat{a}_s), &=& 1 - 0.66667 \hat{a}_s + \hat{a}_s^2 (-3.83333 + 0.29630 N_F) \nonumber\\ && + \hat{a}_s^3 (-36.1549 + 6.33125 N_F - 0.15947 N_F^2) \nonumber\\ && + \hat{a}_s^4 (-436.768 + 111.873 N_F - 7.11450 N_F^2 + 0.10174 N_F^3)~,\end{aligned}$$ setting $\mu^2 = Q^2$ for $SU(3)_c$. For $N_F = 3, 4$ the massless QCD corrections are given by $$\begin{aligned} \label{eq:uBJSRM0} C_{\rm uBJ}(\hat{a}_s, N_F=3) &=& 1 - 0.66667 \hat{a}_s - 2.94444 \hat{a}_s^2 - 18.5963 \hat{a}_s^3 - 162.436 \hat{a}_s^4 \\ C_{\rm uBJ}(\hat{a}_s, N_F=4) &=& 1 - 0.66667 \hat{a}_s - 2.64815 \hat{a}_s^2 - 13.3813 \hat{a}_s^3 - 96.6032 \hat{a}_s^4~. \end{aligned}$$ The massive corrections start at $O(a_s^0)$ with the $s'= (|V_{dc}|^2 d + |V_{sc}|^2 s) \rightarrow c$ transitions [@Gluck:1997sj; @Blumlein:2011zu] and have been given in complete form in Ref. [@Blumlein:2016xcy] to 2-loop order. The charm corrections at $O(\hat{a}_s^2)$ are of the same size as the massless $O(\hat{a}_s^4)$ corrections. Ref. [@Blumlein:2016xcy] also contains the target mass corrections. In the asymptotic case, the effect of the heavy flavor corrections reduces to a shift of $N_F \rightarrow N_F +1$ in the massless corrections since the massive OMEs vanish for $N=1$ due to fermion conservation, which holds to all orders in perturbation theory. Conclusions {#sec:7} =========== We have calculated the massive charm quark 3-loop corrections to the charged current Wilson coefficients for the structure functions $F_{1,2}^{W^+ - W^-}(x,Q^2)$ in the asymptotic region $Q^2 \gg m_c^2$ both in Mellin $N$ and $x$ space. The corresponding contributions are composed of two massive Wilson coefficients $L_{q}^{W^+-W^-,\rm NS}$ and $H_{q}^{W^+-W^-,\rm NS}$ for which the weak boson either couples to a massless ($L$) or a massive quark line ($H$), here in the $s' \rightarrow c$ transition. The massless 3-loop Wilson coefficients have been calculated in [@Davies:2016ruz] and the massive OMEs were presented before in [@Ablinger:2014vwa] as part of the present project to compute all massive 3-loop corrections to deep-inelastic scattering at high values of $Q^2$. The results have a representation in terms of nested harmonic sums and harmonic polylogarithms only. The charm quark corrections in case of both structure functions amount up to $\sim 8\%$, depending on $x$ and the 3-loop corrections stabilize lower order QCD results. At low values of $Q^2$, effects of power corrections are still visible, which we have illustrated using recent complete 2-loop results [@Blumlein:2016xcy], while for $Q^2 {\raisebox{-0.07cm } {$\, \stackrel{>}{{\scriptstyle\sim}}\, $}}100~{\rm GeV}^2$ the asymptotic representation is valid in a rather wide range of $x$. We also discussed potential contributions of the present corrections to the Adler and unpolarized Bjorken sum rules. In the former case, in accordance with the expectation, no corrections are obtained. For the Bjorken sum rule, the charm quark contributions lead to a shift of $N_F = 3$ by one unit in the massless result. There are no heavy quark contributions due to fermion number conservation, which is expressed by a vanishing first moment of the operator matrix element in the non-singlet cases. Therefore, only the massless terms contribute now with $N_F \rightarrow N_F + 1$. The 3-loop charm quark corrections to the structure functions $F_{1,2}^{W^+ - W^-}(x,Q^2)$ will improve the analysis of the HERA charged current data and are relevant for precision measurements in deep-inelastic scattering at planned facilities like the EIC [@EIC], LHeC [@LHEC] and neutrino factories [@NUFACT] in the future, which will reach a higher statistical and systematic precision than obtained in present experiments. [**Acknowledgment.**]{}  We would like to thank J. Ablinger, A. Hasselhuhn and A. Vogt for discussions, as well A. Vogt for providing us the effective numerical representations of the massless flavor non-singlet 3-loop Wilson coefficients of Ref. [@Davies:2016ruz]. This work was supported in part by the European Commission through contract PITN-GA-2012-316704 ([HIGGSTOOLS]{}) and the Austrian Science Fund (FWF) grant SFB F50 (F5009-N15). [99]{} J.D. Bjorken, Phys. Rev.  [**163**]{} (1967) 1767. S.L. Adler, Phys. Rev.  [**143**]{} (1966) 1144. J. Davies, A. Vogt, S. Moch and J.A.M. Vermaseren, arXiv:1606.08907 \[hep-ph\], and in preparation. W.A. Bardeen, A.J. Buras, D.W. Duke and T. Muta, Phys. Rev. D [**18**]{} (1978) 3998. W. Furmanski and R. Petronzio, Z. Phys. C [**11**]{} (1982) 293 and references therein. J. Blümlein, A. Hasselhuhn, P. Kovacikova and S. Moch, Phys. Lett. B [**700**]{} (2011) 294 \[arXiv:1104.3449 \[hep-ph\]\]. M. Glück, S. Kretzer and E. Reya, Phys. Lett. B [**398**]{} (1997) 381 \[Erratum: Phys. Lett. B [**405**]{} (1997) 392\] \[hep-ph/9701364\]. T. Gottschalk, Phys. Rev. D [**23**]{} (1981) 56. M. Buza and W.L. van Neerven, Nucl. Phys. B [**500**]{} (1997) 301 \[hep-ph/9702242\]. E.B. Zijlstra and W.L. van Neerven, Phys. Lett. B [**297**]{} (1992) 377. S. Moch and J.A.M. Vermaseren, Nucl. Phys. B [**573**]{} (2000) 853 \[hep-ph/9912355\]. J. Blümlein, A. Hasselhuhn and T. Pfoh, Nucl. Phys. B [**881**]{} (2014) 1 \[arXiv:1401.4352 \[hep-ph\]\]. A. Hasselhuhn, [3-Loop Contributions to Heavy Flavor Wilson Coefficients of Neutral and Charged Current DIS]{}, PhD Thesis TU Dortmund, 2013, DESY-THESIS-2013-050. M. Buza, Y. Matiounine, J. Smith, R. Migneron and W. L. van Neerven, Nucl. Phys. B [**472**]{} (1996) 611 \[hep-ph/9601302\]. J. Blümlein, G. Falcioni and A. De Freitas, Nucl. Phys. B [**910**]{} (2016) 568 \[arXiv:1605.05541 \[hep-ph\]\]. A. Arbuzov, D.Y. Bardin, J. Blümlein, L. Kalinovskaya and T. Riemann, Comput. Phys. Commun.  [**94**]{} (1996) 128 \[hep-ph/9511434\];\ J. Blümlein, M. Klein, T. Naumann and T. Riemann, PHE-88-01. J. Blümlein, Prog. Part. Nucl. Phys.  [**69**]{} (2013) 28 \[arXiv:1208.6087 \[hep-ph\]\]. I. Bierenbaum, J. Blümlein and S. Klein, Nucl. Phys. B [**820**]{} (2009) 417 \[arXiv:0904.3563 \[hep-ph\]\]. J. Ablinger, A. Behring, J. Blümlein, A. De Freitas, A. Hasselhuhn, A. von Manteuffel, M. Round, C. Schneider, and F. Wißbrock, Nucl. Phys. B [**886**]{} (2014) 733 \[arXiv:1406.4654 \[hep-ph\]\]. N. Cabibbo, Phys. Rev. Lett.  [**10**]{} (1963) 531. M. Kobayashi and T. Maskawa, Prog. Theor. Phys.  [**49**]{} (1973) 652. K.A. Olive et al. (Particle Data Group), Chin. Phys. C, 38, 090001 (2014). A. Behring, J. Blümlein, A. De Freitas, A. Hasselhuhn, A. von Manteuffel and C. Schneider,\ Phys. Rev. D [**92**]{} (2015) 11, 114005 \[arXiv:1508.01449 \[hep-ph\]\]. A. Behring, J. Blümlein, A. De Freitas, A. von Manteuffel and C. Schneider, Nucl. Phys. B [**897**]{} (2015) 612 \[arXiv:1504.08217 \[hep-ph\]\]. J. Ablinger, A. Behring, J. Blümlein, G. Falcioni, A. De Freitas, A. Hasselhuhn, A. von Manteuffel, M. Round, C. Schneider, F. Wißbrock, PoS (LL2016) 065 \[arXiv:1609.03397 \[hep-ph\]\];\ J. Blümlein, A. DeFreitas and C. Schneider, Nucl. Part. Phys. Proc.  [**261-262**]{} (2015) 185 \[arXiv:1411.5669 \[hep-ph\]\]. J.A.M. Vermaseren, Int. J. Mod. Phys. A [**14**]{} (1999) 2037 \[hep-ph/9806280\];\ J. Blümlein and S. Kurth, Phys. Rev. D [**60**]{} (1999) 014018 \[hep-ph/9810241\]. E. Remiddi and J.A.M. Vermaseren, Int. J. Mod. Phys. A [**15**]{} (2000) 725 \[hep-ph/9905237\]. J. Blümlein, Comput. Phys. Commun.  [**159**]{} (2004) 19 \[hep-ph/0311046\]. S. Alekhin, J. Blümlein, K. Daum, K. Lipka and S. Moch, Phys. Lett. B [**720**]{} (2013) 172 \[arXiv:1212.2355 \[hep-ph\]\]. S. Alekhin, J. Blümlein and S. Moch, Phys. Rev. D [**89**]{} (2014) 054028 \[arXiv:1310.3059 \[hep-ph\]\]. V. Ravindran and W.L. van Neerven, Nucl. Phys. B [**605**]{} (2001) 517 \[hep-ph/0102280\]. S.L. Adler, arXiv:0905.2923 \[hep-ph\]. S. Moch and M. Rogal, Nucl. Phys. B [**782**]{} (2007) 51 \[arXiv:0704.1740 \[hep-ph\]\]. G. Altarelli, R.K. Ellis and G. Martinelli, Nucl. Phys. B [**143**]{} (1978) 521, Erratum: Nucl. Phys. B [**146**]{} (1978) 544. B. Humpert and W.L. van Neerven, Nucl. Phys. B [**184**]{} (1981) 225. S.G. Gorishnii and S.A. Larin, Phys. Lett. B [**172**]{} (1986) 109. S.A. Larin, F.V. Tkachov and J.A.M. Vermaseren, Phys. Rev. Lett.  [**66**]{} (1991) 862. K.G. Chetyrkin, Talk at [Loops and Legs in Quantum Field Theory, Weimar, April 2014]{}, [https://indico.desy.de/conferenceOtherViews.py?view=standard&confId=8107]{} D. Boer, M. Diehl, R. Milner, R. Venugopalan, W. Vogelsang, D. Kaplan, H. Montgomery, S. Vigdor [*et al.*]{}, arXiv:1108.1713 \[nucl-th\];\ A. Accardi, J.L. Albacete, M. Anselmino, N. Armesto, E.C. Aschenauer, A. Bacchetta, D. Boer, W. Brooks [*et al.*]{}, arXiv:1212.1701 \[nucl-ex\]. J.L. Abelleira Fernandez [*et al.*]{} \[LHeC Study Group Collaboration\], J. Phys. G [**39**]{} (2012) 075001 \[arXiv:1206.2913 \[physics.acc-ph\]\]. D.M. Kaplan \[MAP and MICE Collaborations\], EPJ Web Conf.  [**95**]{} (2015) 03019 \[arXiv:1412.3487 \[physics.acc-ph\]\]. [^1]: The massive 1-loop corrections given in [@Gottschalk:1980rv] were corrected in [@Gluck:1997sj], see also [@Blumlein:2011zu]. [^2]: Some results given in [@Buza:1997mg] have been corrected in Ref. [@Blumlein:2014fqa].

--- abstract: 'We introduce a notion of probabilistic convexity and generalize some classical globalization theorems in Alexandrov geometry. A weighted Alexandrov’s lemma is developed as a basic tool.' address: 'Department of Mathematics, Penn State University, University Park, PA 16802' author: - Nan Li title: Globalization with probabilistic convexity --- [^1] Introduction {#introduction .unnumbered} ============ Recall that a length metric space $X$ is said to be an Alexandrov space with curvature bounded from below by $\kappa$, if for any quadruple $(p;x_1,x_2,x_3)$, the sum of comparison angles $$\begin{aligned} {\tilde\measuredangle_{\kappa}\left({p}\,_{x_2}^{x_1}\right)} +{\tilde\measuredangle_{\kappa}\left({p}\,_{x_3}^{x_2}\right)}+{\tilde\measuredangle_{\kappa}\left({p}\,_{x_1}^{x_3}\right)}\le2\pi, \label{comp.e1}\end{aligned}$$ or at least one of the model angles ${\tilde\measuredangle_{\kappa}\left({p}\,_{x_j}^{x_i}\right)}$ is not defined. We let ${\measuredangle\left[{q}\,_{s}^{p}\right]}$ denote the angle between two geodesics ${[\,qp\,]}$ and ${[\,qs\,]}$, which is defined by $\limsup{\tilde\measuredangle_{\kappa}\left({q}\,_{y}^{x}\right)}$, as $x\in{[\,qp\,]}$, $y\in{[\,qs\,]}$ and $x,y\to q$. The global comparison (\[comp.e1\]) is equivalent to the Toponogov comparison: for any geodesic ${[\,qs\,]}$ and any points $p\notin{[\,qs\,]}$, plus for $x\in{[\,qs\,]}\setminus\{q,s\}$, we have ${\measuredangle\left[{q}\,_{s}^{p}\right]}\ge{\tilde\measuredangle_{\kappa}\left({q}\,_{s}^{p}\right)}$ and ${\measuredangle\left[{x}\,_{q}^{p}\right]}+{\measuredangle\left[{x}\,_{s}^{p}\right]}=\pi$. In the Riemannian case, the sectional curvature is locally defined and the existence of its lower bound implies the corresponding global Toponogov comparison. It is interesting to consider the similar question in Alexandrov geometry: does local curvature bound imply global curvature bound? This property, if it holds, is so-called globalization property. For our convenience, we use the following definitions for a local Alexandrov space. An open domain $\Omega$ is called a $\kappa$-domain if for any geodesic ${[\,qs\,]}\subset\Omega$ and any points $p\in\Omega\setminus{[\,qs\,]}$ and $x\in{[\,qs\,]}\setminus\{q,s\}$, we have ${\measuredangle\left[{q}\,_{s}^{p}\right]}\ge{\tilde\measuredangle_{\kappa}\left({q}\,_{s}^{p}\right)}$ and ${\measuredangle\left[{x}\,_{q}^{p}\right]}+{\measuredangle\left[{x}\,_{s}^{p}\right]}=\pi$. Clearly, $X$ is an Alexandrov space if and only if it is a $\kappa$-domain. A length metric space $\mathcal U$ is said to be locally curvature bounded from below by $\kappa$ (local Alexandrov space), if for any $p\in \mathcal U$, there is a $\kappa$-domain $\Omega_p\ni p$. Let ${\text{Alex\,}}(\kappa)$ and ${\text{Alex\,}}_{loc}(\kappa)$ denote the collection of Alexandrov spaces and local Alexandrov spaces with curvature $\ge\kappa$, respectively. Let $\bar{\mathcal U}$ denote the metric completion of $\mathcal U$, that is, the completion of $\mathcal U$ with respect to its intrinsic metric. The Globalization theorem in [@BGP] states that if $\mathcal U\in{\text{Alex\,}}_{loc}(\kappa)$ is complete, that is, $\mathcal U=\bar{\mathcal U}$, then ${\mathcal U}\in{\text{Alex\,}}(\kappa)$. In general, $\mathcal U$, or $\bar{\mathcal U}$, may not be a global Alexandrov space if $\bar{\mathcal U}\neq \mathcal U$ (see Example \[example.1\]). This is partially because there are points in $\bar{\mathcal U}\setminus \mathcal U$ that are not contained in any $\kappa$-domain. However, this case is particularly important if one wants to prove $X\in{\text{Alex\,}}(\kappa)$, but the lower curvature bound can only be verified on a dense subset $\mathcal U\subset X$ (for instance, the manifold points), whose metric completion $\bar{\mathcal U}=X$ (see [@GW13] and [@HS13]). For a point $p\in\mathcal U$ and a subset $S\subset\mathcal U$, let $${{S}^{*p}}=\{q\in S: \text{there is a geodesic } {[\,pq\,]} \text{ connecting } p \text{ and } q \text{ in } \mathcal U\}.$$ We rephrase a few classical convexities using the above terminologies. - Convex – for every point $p\in\mathcal U$, ${{\mathcal U}^{*p}}=\mathcal U$. - A.e.-convex – for every point $p\in\mathcal U$, $\mathcal H^n(\mathcal U\setminus{{\mathcal U}^{*p}})=0$, where $n<\infty$ is the Hausdorff dimension of $\mathcal U$ and $\mathcal H^n$ is the $n$-dimensional Hausdorff measure. - Weakly a.e.-convex – for any $p\in\mathcal U$ and any $\epsilon>0$, there is $p_1\in B_\epsilon(p)$ so that $\mathcal H^n(\mathcal U\setminus{{\mathcal U}^{*p_1}})=0$. - Weakly convex – for any $p, q\in \mathcal U$ and any $\epsilon>0$, there is a point $p_1\in B_\epsilon(p)$ such that ${{B_\epsilon(q)}^{*p_1}}\neq\varnothing$. In [@Pet13], Petrunin shows that if $\mathcal U\in{\text{Alex\,}}_{loc}(\kappa)$ is convex, then $\bar{\mathcal U}\in{\text{Alex\,}}(\kappa)$. For example, $\mathcal U$ can be an open convex domain in $\mathbb R^n$. It is proved in [@GW13] that the above statement remains true if “convex" is replaced by “a.e.-convex". In this case, $\mathcal U$ can be an open convex domain in $\mathbb R^n$ with finitely many points removed. In this paper, we introduce a notion of probabilistic convexity and prove some globalization theorems related to it. Let $p\in\mathcal U$ and ${[\,qs\,]}$ be a geodesic in $\mathcal U$. Consider the probability that a point on ${[\,qs\,]}$ can be connected to $p$ by a geodesic in $\mathcal U$: $${\bf Pr}\left(p\prec{[\,qs\,]}\right)=\frac{\mathcal H^1\left({{{[\,qs\,]}}^{*p}}\right)} {\mathcal H^1\left({[\,qs\,]}\right)}.$$ Here $\mathcal H^1$ denotes the $1$-dimensional Hausdorff measure. We say that $\mathcal U$ is weakly $\mathfrak p_\lambda$-convex if for any $p,q,s\in\mathcal U$ and any $\epsilon>0$, there are points $p_1\in B_\epsilon(p)$, $q_1\in B_\epsilon(q)$, $s_1\in B_\epsilon(s)$ and a geodesic ${[\,q_1s_1\,]}\subset\bar{\mathcal U}$ so that ${\bf Pr}(p_1\prec{[\,q_1 s_1\,]})>\lambda-\epsilon$. By taking $s\in B_\epsilon(q)$, we see that if $\lambda>0$, then weak $\mathfrak p_\lambda$-convexity implies weak convexity. If $\mathcal U\in{\text{Alex\,}}_{loc}(\kappa)$, then weak a.e.-convexity implies weak $\mathfrak p_1$-convexity (see the proof of Corollary \[cor.ThmA\]). Our main results are stated as follows. If $\mathcal U\in{\text{Alex\,}}_{loc}(\kappa)$ is weakly $\mathfrak p_1$-convex, then its metric completion $\bar{\mathcal U}\in{\text{Alex\,}}(\kappa)$. The following theorem is about the optimal lower curvature bound for the metric completion. Suppose that $\mathcal U\in{\text{Alex\,}}_{loc}(\kappa)$ is weakly $\mathfrak p_\lambda$-convex for some $\lambda>0$. If $\bar{\mathcal U}\in{\text{Alex\,}}(\kappa_0)$ for some $\kappa_0\in\mathbb R$, then $\bar{\mathcal U}\in{\text{Alex\,}}(\kappa)$. With some extra argument, we prove the following corollary. \[cor.ThmA\] If $\mathcal U\in{\text{Alex\,}}_{loc}(\kappa)$ is weakly a.e.-convex then its metric completion $\bar{\mathcal U}\in{\text{Alex\,}}(\kappa)$. These results provide some connections between the global geometry and the local geometry via a probability of convexity, which may be used to attack a long-standing conjecture mentioned in M. Gromov¡¯s book [@Gromov-SaGMoC]: If $X\in{\text{Alex}^n(\kappa)}$ has no boundary, then a convex hypersurface in $X$ equipped with the intrinsic metric is an Alexandrov space with the same lower curvature bound. The answers to the following two questions remain unclear to the author. - [*Is Theorem A true or false if $\mathcal U$ is weakly $\mathfrak p_\lambda$-convex for some $\lambda\in(0,1)$?*]{} - [*Is Theorem B true or false if $\mathfrak p_\lambda$-convexity is replaced by weak convexity?*]{} A negative answer to the second question may provide some new examples for Alexandrov spaces. The main issue to adapt the proof from the complete case $\bar{\mathcal U}=\mathcal U$ to the incomplete case $\bar{\mathcal U}\neq\mathcal U$ is that there is not a priori uniform lower bound for the size of $\kappa$-domains in a bounded subset of $\mathcal U$. Our proof is divided into four steps. In Section 1 we establish the key tool “weighted Alexandrov’s lemma". In Section 2 we recall and prove some comparison results for thin triangles near a geodesic which can be covered by $\kappa$-domains. In Section 3, the probability condition and the weighted Alexandrov’s lemma are used to prove a global comparison through a combination of thin triangles. We complete the proof in Section 4 and give some examples in Section 5. The author would like to thank Dmitri Burago and Anton Petrunin for helpful discussions. [**Notation and conventions**]{} - $\mathbb M_\kappa^n$ – the $n$-dimensional space form with constant curvature $\kappa$. - ${{\textsf d}\left(p,q\right)}$ or $|pq|$ – the distance between points $p$ and $q$. - ${[\,pq\,]}$ – a minimal geodesic connecting points $p$ and $q$ if it exists. Once it appears, it will always mean the same geodesic in the same context. For simplicity, we let ${\,]pq\,]}={[\,pq\,]}\setminus\{p\}$, ${[\,pq[\,}={[\,pq\,]}\setminus\{q\}$ and ${\,]pq[\,}={[\,pq\,]}\setminus\{p,q\}$. - ${\tilde\measuredangle_{\kappa}\left({q}\,_{s}^{p}\right)}$ – the angle at $q$ of the model triangle $\tilde\triangle_\kappa pqs$, where $\tilde\triangle_\kappa pqs$ is a geodesic triangle in $\mathbb M_\kappa^2$ with the same lengths of sides as $\triangle pqs$. - ${\measuredangle\left[{q}\,_{s}^{p}\right]}$ – the angle at $q$ between geodesics ${[\,qp\,]}$ and ${[\,qs\,]}$. - $\psi(\epsilon\mid\delta)$ – a positive function in $\epsilon$ and $\delta$ which satisfies ${\displaystyle}\lim_{\epsilon\to 0^+}\psi(\epsilon\mid\delta)=0$. Weighted Alexandrov’s lemma =========================== Alexandrov’s lemma plays an important role in the classical globalization theorem. It is used to combine two small comparison triangles. In our case, we need to combine multiple triangles whose comparison curvatures are not necessarily the same. In this section, $X$ is a general length metric space. We assume all triangles are contained in a ball of radius $\frac{\pi}{10\sqrt\kappa}$ in the case $\kappa>0$. Recall the Alexandrov’s lemma. \[Alex.lem\] Let $p,q,s$ and $x\in{\,]qs[\,}$ be points in $X$. Then ${\tilde\measuredangle_{\kappa}\left({q}\,_{x}^{p}\right)}\ge{\tilde\measuredangle_{\kappa}\left({q}\,_{s}^{p}\right)}$ if and only if ${\tilde\measuredangle_{\kappa}\left({x}\,_{q}^{p}\right)}+{\tilde\measuredangle_{\kappa}\left({x}\,_{s}^{p}\right)}\le\pi$. By a direct computation, we have the following two weighted Alexandrov’s lemmas in implicit forms. \[tri.comp.2.2-im\] Let $p,q,s$ and $x\in{\,]qs[\,}$ be points in $X$. Suppose that $|qx|=b$ and $|xs|=d$. If there are $\kappa_1, \kappa_2\in\mathbb R$ such that $$\begin{aligned} {\tilde\measuredangle_{\kappa_1}\left({x}\,_{q}^{p}\right)}+{\tilde\measuredangle_{\kappa_2}\left({x}\,_{s}^{p}\right)}\le\pi, \label{tri.comp.2.2-e0} \end{aligned}$$ then there is $\lambda \in(0,1)$, such that for $\bar\kappa=(1-\lambda)\kappa_1+\lambda\kappa_2$, we have $$\begin{aligned} {\tilde\measuredangle_{\kappa_1}\left({q}\,_{x}^{p}\right)}\ge{\tilde\measuredangle_{\bar\kappa}\left({q}\,_{s}^{p}\right)}. \end{aligned}$$ \[triangle.comp.mult.im\] Let $p,q,s$ and $x_i, i=1,2,\dots,N$ be points in $X$, where $x_0=q$, $x_N=s$ and $x_i\in{\,]qs[\,}$ appear in the same order. If there are $\kappa_i\in\mathbb R$, such that $$\begin{aligned} {\tilde\measuredangle_{\kappa_i}\left({x_i}\,_{x_{i-1}}^{p}\right)}+{\tilde\measuredangle_{\kappa_{i+1}}\left({x_i}\,_{x_{i+1}}^{p}\right)}\le\pi, \end{aligned}$$ $i=1,2,\dots,N-1$, then there is $$\bar\kappa=\min_i\{\kappa_i\} +\psi\left(\max_{i,j}\{|k_i-k_j|\}\right.\left|\; \min_i\{|px_i|\},|qs|\right)$$ such that $$\begin{aligned} {\tilde\measuredangle_{\kappa_1}\left({q}\,_{x_1}^{p}\right)}\ge{\tilde\measuredangle_{\bar\kappa}\left({q}\,_{s}^{p}\right)}. \end{aligned}$$ When $|qs|$ is small enough, depending on $|pq|$ and $\kappa_i$, Taylor expansions can be used to get an effective weighted Alexandrov’s lemma. Let $${\textsf{sn}_{\kappa}}(t)=\left\{ \begin{array}{ll} \frac{\sin(\sqrt\kappa t)}{\sqrt\kappa}, & \hbox{$\kappa>0$,} \\ t, & \hbox{$\kappa=0$,} \\ \frac{\sinh(\sqrt{-\kappa} t)}{\sqrt{-\kappa}}, & \hbox{$\kappa<0$,} \end{array} \right. \quad \quad {\textsf{cs}}_\kappa(t)={\textsf{sn}_{\kappa}}'(t) =\left\{ \begin{array}{ll} \cos(\sqrt\kappa t), & \hbox{$\kappa>0$,} \\ 1, & \hbox{$\kappa=0$,} \\ \cosh(\sqrt{-\kappa} t), & \hbox{$\kappa<0$,} \end{array} \right.$$ and $${\textsf{md}}_\kappa(t)=\int_0^t{\textsf{sn}_{\kappa}}(a)\,da =\left\{ \begin{array}{ll} \frac{1-\cos(\sqrt\kappa t)}{\kappa}, & \hbox{$\kappa>0$,} \\ \frac{t^2}{2}, & \hbox{$\kappa=0$,} \\ \frac{1-\cosh(\sqrt{-\kappa} t)}{\kappa}, & \hbox{$\kappa<0$.} \end{array} \right.$$ For any $\kappa\in\mathbb R$ and $\triangle_\kappa ABC\subset\mathbb M_\kappa^2$, the $\kappa$-cosine law can be written in the following form $$\begin{aligned} {\textsf{md}}_\kappa(|BC|) &={\textsf{md}}_\kappa(|AB|)+{\textsf{md}}_\kappa(|AC|) -\kappa\cdot{\textsf{md}}_\kappa(|AB|){\textsf{md}}_\kappa(|AC|) \notag \\ &-{\textsf{sn}}_\kappa(|AB|){\textsf{sn}}_\kappa(|AC|)\cos{\measuredangle\left[{A}\,_{C}^{B}\right]}. \label{cos.law} \end{aligned}$$ Let $|AB|=c$, $|AC|=b$ and $|BC|=a$. The Taylor expansion of $|BC|$ at $b=0$ is $$\begin{aligned} a=c-b\cos({\measuredangle\left[{A}\,_{C}^{B}\right]})+\frac12\sin^2({\measuredangle\left[{A}\,_{C}^{B}\right]}) \frac{{\textsf{cs}}_\kappa(c)}{{\textsf{sn}_{\kappa}}(c)}\cdot b^2+O\left(b^3\right). \label{tri.comp.2.e1} \end{aligned}$$ The second order term in the expansion gives an indication of the curvature. The coefficient of the third term depends on both $c$ and $\kappa$. Let $f_c(\kappa)=\frac{{\textsf{cs}}_\kappa(c)}{{\textsf{sn}_{\kappa}}(c)}$ and view it as a function in $\kappa$. Sometimes we write $f_c(\kappa)$ as $f(\kappa)$ if $c$ is relatively fixed. Direct computation shows that $f(\kappa)$ is a $C^2$, concave and strictly decreasing function for $\kappa\in\left(-\infty,\; \left(\frac{2\pi}{c}\right)^2\right)$. \[tri.comp.2.2\] Let the assumption be the same as in Lemma \[tri.comp.2.2-im\]. Let $a=|pq|$ and $\bar\kappa\in\mathbb R$ such that $$\begin{aligned} f_a(\bar\kappa)=\frac{(b^2+2bd)f_a(\kappa_1)+d^2f_a(\kappa_2) }{(b+d)^2}. \label{tri.comp.2.2.e0} \end{aligned}$$ There is $\delta=\delta(a,\kappa_1,\kappa_2)>0$ so that if $|qs|<\delta$ then $$\begin{aligned} {\tilde\measuredangle_{\kappa_1}\left({q}\,_{x}^{p}\right)}\ge{\tilde\measuredangle_{\bar\kappa}\left({q}\,_{s}^{p}\right)}. \label{tri.comp.2.2.e00} \end{aligned}$$ By the monotonicity and the concavity of $f$, we see that $$\bar\kappa \ge\frac{(b^2+2bd)\kappa_1+d^2\kappa_2}{(b+d)^2} \ge\min\left\{\kappa_1, \; \frac{b^2\kappa_1+d^2\kappa_2}{b^2+d^2}\right\}.$$ Not losing generality, assume ${\measuredangle\left[{q}\,_{s}^{p}\right]}\neq 0,\pi$. Our goal is to combine $\triangle pqx$ and $\triangle pxs$ and get a globalized comparison. We start from finding a connection between ${\tilde\measuredangle_{\kappa_1}\left({q}\,_{x}^{p}\right)}$ and ${\tilde\measuredangle_{\kappa_2}\left({x}\,_{s}^{p}\right)}$, namely, (\[tri.comp.2.2.e8\]) and (\[tri.comp.2.2.e9\]). Let $|px|=c$. Applying (\[tri.comp.2.e1\]) on $\triangle_{\kappa_1} \tilde p\tilde q\tilde x\subset \mathbb M_{\kappa_1}^2$, we get $$\begin{aligned} c&= a-b\cos\left({\tilde\measuredangle_{\kappa_1}\left({q}\,_{x}^{p}\right)}\right) +\frac12\sin^2\left({\tilde\measuredangle_{\kappa_1}\left({q}\,_{x}^{p}\right)}\right) f_a(\kappa_1) \cdot b^2+O(b^3) \label{tri.comp.2.2.e2} \\ a&= c-b\cos\left({\tilde\measuredangle_{\kappa_1}\left({x}\,_{q}^{p}\right)}\right) +\frac12\sin^2\left({\tilde\measuredangle_{\kappa_1}\left({x}\,_{q}^{p}\right)}\right) f_c(\kappa_1) \cdot b^2+O(b^3) \label{tri.comp.2.2.e3}. \end{aligned}$$ Adding up (\[tri.comp.2.2.e2\]) and (\[tri.comp.2.2.e3\]) and taking in account $|a-c|<b$, we get $$\begin{aligned} &\cos\left({\tilde\measuredangle_{\kappa_1}\left({q}\,_{x}^{p}\right)}\right) +\cos\left({\tilde\measuredangle_{\kappa_1}\left({x}\,_{q}^{p}\right)}\right) \notag \\ &\qquad =\frac12\sin^2\left({\tilde\measuredangle_{\kappa_1}\left({q}\,_{x}^{p}\right)}\right) f_a(\kappa_1) \cdot b +\frac12\sin^2\left({\tilde\measuredangle_{\kappa_1}\left({x}\,_{q}^{p}\right)}\right) f_c(\kappa_1) \cdot b +O(b^2), \notag\\ &\qquad =\frac12\left(\sin^2\left({\tilde\measuredangle_{\kappa_1}\left({q}\,_{x}^{p}\right)}\right) +\sin^2\left({\tilde\measuredangle_{\kappa_1}\left({x}\,_{q}^{p}\right)}\right)\right) f_a(\kappa_1) \cdot b +O(b^2). \label{tri.comp.2.2.e4} \end{aligned}$$ Therefore, $$\begin{aligned} &\sin^2\left({\tilde\measuredangle_{\kappa_1}\left({x}\,_{q}^{p}\right)}\right) -\sin^2\left({\tilde\measuredangle_{\kappa_1}\left({q}\,_{x}^{p}\right)}\right) \notag\\ &\qquad=\left(\cos\left({\tilde\measuredangle_{\kappa_1}\left({q}\,_{x}^{p}\right)}\right) +\cos\left({\tilde\measuredangle_{\kappa_1}\left({x}\,_{q}^{p}\right)}\right)\right) \left(\cos\left({\tilde\measuredangle_{\kappa_1}\left({q}\,_{x}^{p}\right)}\right) -\cos\left({\tilde\measuredangle_{\kappa_1}\left({x}\,_{q}^{p}\right)}\right)\right) \notag\\ &\qquad\le 2f_a(\kappa_1) \cdot b +O(b^2). \label{tri.comp.2.2.e5} \end{aligned}$$ Plugging (\[tri.comp.2.2.e5\]) back into (\[tri.comp.2.2.e4\]), we get $$\begin{aligned} \cos\left({\tilde\measuredangle_{\kappa_1}\left({q}\,_{x}^{p}\right)}\right) +\cos\left({\tilde\measuredangle_{\kappa_1}\left({x}\,_{q}^{p}\right)}\right) \le\sin^2\left({\tilde\measuredangle_{\kappa_1}\left({q}\,_{x}^{p}\right)}\right) f_a(\kappa_1) \cdot b +O(b^2). \label{tri.comp.2.2.e6} \end{aligned}$$ The assumption $$\begin{aligned} {\tilde\measuredangle_{\kappa_1}\left({x}\,_{q}^{p}\right)}+{\tilde\measuredangle_{\kappa_2}\left({x}\,_{s}^{p}\right)}\le\pi, \label{tri.comp.2.2.e6.1} \end{aligned}$$ implies $$\begin{aligned} 0\le\cos\left({\tilde\measuredangle_{\kappa_1}\left({x}\,_{q}^{p}\right)}\right) +\cos\left({\tilde\measuredangle_{\kappa_2}\left({x}\,_{s}^{p}\right)}\right). \label{tri.comp.2.2.e7} \end{aligned}$$ Summing (\[tri.comp.2.2.e6\]) and (\[tri.comp.2.2.e7\]): $$\begin{aligned} -\cos\left({\tilde\measuredangle_{\kappa_2}\left({x}\,_{s}^{p}\right)}\right) \le -\cos\left({\tilde\measuredangle_{\kappa_1}\left({q}\,_{x}^{p}\right)}\right) +\sin^2\left({\tilde\measuredangle_{\kappa_1}\left({q}\,_{x}^{p}\right)}\right) f_a(\kappa_1) \cdot b +O(b^2). \label{tri.comp.2.2.e8} \end{aligned}$$ This also implies that $$\begin{aligned} &\sin^2\left({\tilde\measuredangle_{\kappa_2}\left({x}\,_{s}^{p}\right)}\right) -\sin^2\left({\tilde\measuredangle_{\kappa_1}\left({q}\,_{x}^{p}\right)}\right) \notag\\ &\qquad\le \left(\cos\left({\tilde\measuredangle_{\kappa_1}\left({q}\,_{x}^{p}\right)}\right) -\cos\left({\tilde\measuredangle_{\kappa_2}\left({x}\,_{s}^{p}\right)}\right)\right) \left(\cos\left({\tilde\measuredangle_{\kappa_1}\left({q}\,_{x}^{p}\right)}\right) +\cos\left({\tilde\measuredangle_{\kappa_2}\left({x}\,_{s}^{p}\right)}\right)\right) \notag\\ &\qquad\le 2f_a(\kappa_1) \cdot b +O(b^2). \label{tri.comp.2.2.e9} \end{aligned}$$ Apply (\[tri.comp.2.e1\]) again for $\triangle_{\kappa_2} \tilde p\tilde x\tilde s\subset \mathbb M_{\kappa_2}^2$ and by the property of $f$, $$\begin{aligned} |ps| &=c-d\cos\left({\tilde\measuredangle_{\kappa_2}\left({x}\,_{s}^{p}\right)}\right) +\frac12\sin^2\left({\tilde\measuredangle_{\kappa_2}\left({x}\,_{s}^{p}\right)}\right) f_c(\kappa_2)\cdot d^2+O(d^3) \notag \\ &\le c-d\cos\left({\tilde\measuredangle_{\kappa_2}\left({x}\,_{s}^{p}\right)}\right) +\frac12\sin^2\left({\tilde\measuredangle_{\kappa_2}\left({x}\,_{s}^{p}\right)}\right) f_a(\kappa_2)\cdot d^2+O(d^3). \label{tri.comp.2.2.e10} \end{aligned}$$ Plug (\[tri.comp.2.2.e8\]) into (\[tri.comp.2.2.e10\]), $$\begin{aligned} |ps| \le c&-d\cos\left({\tilde\measuredangle_{\kappa_1}\left({q}\,_{x}^{p}\right)}\right) +\sin^2\left({\tilde\measuredangle_{\kappa_1}\left({q}\,_{x}^{p}\right)}\right) f_a(\kappa_1) \cdot bd \notag\\ &+\frac12\sin^2\left({\tilde\measuredangle_{\kappa_1}\left({q}\,_{x}^{p}\right)}\right) f_a(\kappa_2) \cdot d^2+O(b^2d)+O(bd^2)+O(d^3). \label{tri.comp.2.2.e11} \end{aligned}$$ Substitute $c$ by (\[tri.comp.2.2.e2\]), $$\begin{aligned} |ps|\le a &-(b+d)\cos\left({\tilde\measuredangle_{\kappa_1}\left({q}\,_{x}^{p}\right)}\right) \notag\\ &+\frac12\sin^2\left({\tilde\measuredangle_{\kappa_1}\left({q}\,_{x}^{p}\right)}\right)\cdot \left(f_a(\kappa_1)\cdot(b^2+2bd)+f_a(\kappa_2)\cdot d^2\right) \notag\\ &+O(b^3)+O(b^2d)+O(bd^2)+O(d^3) \notag\\ \le a&-(b+d)\cos\left({\tilde\measuredangle_{\kappa_1}\left({q}\,_{x}^{p}\right)}\right) +\sin^2\left({\tilde\measuredangle_{\kappa_1}\left({q}\,_{x}^{p}\right)}\right) f_a(\bar\kappa)\cdot(b+d)^2+O((b+d)^3). \label{tri.comp.2.2.e12} \end{aligned}$$ where $\bar\kappa$ satisfies ${\displaystyle}f_a(\bar\kappa)=\frac{(b^2+2bd)f_a(\kappa_1)+d^2f_a(\kappa_2) }{(b+d)^2}$. Apply (\[tri.comp.2.e1\]) for $\triangle_{\bar\kappa} \tilde p\tilde x\tilde s\subset \mathbb M_{\bar\kappa}^2$: $$\begin{aligned} &|ps|=a-(b+d)\cos\left({\tilde\measuredangle_{\bar\kappa}\left({q}\,_{s}^{p}\right)}\right) +\sin^2\left({\tilde\measuredangle_{\bar\kappa}\left({q}\,_{s}^{p}\right)}\right) f_a(\bar\kappa)\cdot(b+d)^2 +O((b+d)^3). \label{tri.comp.2.2.e13} \end{aligned}$$ At last, we compare (\[tri.comp.2.2.e13\]) with (\[tri.comp.2.2.e12\]). Note that $$-\cos\left({\tilde\measuredangle_{\bar\kappa}\left({q}\,_{s}^{p}\right)}\right) +\cos\left({\tilde\measuredangle_{\kappa_1}\left({q}\,_{x}^{p}\right)}\right) =(b+d)\left({\tilde\measuredangle_{\bar\kappa}\left({q}\,_{s}^{p}\right)} -{\tilde\measuredangle_{\kappa_1}\left({q}\,_{x}^{p}\right)}\right)+O((b+d)^2)$$ and $$\sin\left({\tilde\measuredangle_{\bar\kappa}\left({q}\,_{s}^{p}\right)}\right) -\sin\left({\tilde\measuredangle_{\kappa_1}\left({q}\,_{x}^{p}\right)}\right) =\left({\tilde\measuredangle_{\bar\kappa}\left({q}\,_{s}^{p}\right)} -{\tilde\measuredangle_{\kappa_1}\left({q}\,_{x}^{p}\right)}\right)+O(b+d).$$ When $|qs|=b+d$ is small, we have $${\tilde\measuredangle_{\kappa_1}\left({q}\,_{x}^{p}\right)}\ge{\tilde\measuredangle_{\bar\kappa}\left({q}\,_{s}^{p}\right)}.$$ We now generalize Lemma \[tri.comp.2.2\] to the case of multiple triangles. \[triangle.comp.mult\] Let the assumption be the same as in Lemma \[triangle.comp.mult.im\]. Let $c_i=|x_ix_{i+1}|$ and $$\begin{aligned} \bar\kappa=\frac{{\displaystyle}\sum_{i=1}^N c_i^2\kappa_i +2\sum_{i=1}^{N-1} \left(c_{i+1}+\dots+c_N\right)c_i\kappa_i} {(c_1+c_2+\dots+c_N)^2}. \label{triangle.comp.mult.e1} \end{aligned}$$ There is $\delta=\delta(|pq|, \kappa_1\dots,\kappa_N)>0$ so that if $|qs|<\delta$ then $$\begin{aligned} {\tilde\measuredangle_{\kappa_1}\left({q}\,_{x_2}^{p}\right)}\ge{\tilde\measuredangle_{\bar\kappa}\left({q}\,_{s}^{p}\right)}. \label{triangle.comp.mult.e01} \end{aligned}$$ This is proved by a similar comuptation as in Lemma \[tri.comp.2.2\]. Summing up the cosine laws for the adjacent triangles, we get the following inequality, as a counterpart of (\[tri.comp.2.2.e8\]): $$\begin{aligned} -\cos&\left({\tilde\measuredangle_{\kappa_{i+1}}\left({x_i}\,_{x_{i+1}}^{p}\right)}\right) \le -\cos\left({\tilde\measuredangle_{\kappa_1}\left({x_1}\,_{x_2}^{p}\right)}\right) \\ &+\sin^2\left({\tilde\measuredangle_{\kappa_1}\left({x_1}\,_{x_2}^{p}\right)}\right) [f(\kappa_1)c_1+\dots+f(\kappa_i)c_i] +O((c_1^2+\dots+c_i^2)), \end{aligned}$$ A similar argument shows that when $|qs|$ is small, we have $$\begin{aligned} {\tilde\measuredangle_{\kappa_1}\left({q}\,_{x_2}^{p}\right)}\ge{\tilde\measuredangle_{\bar\kappa}\left({q}\,_{s}^{p}\right)}, \end{aligned}$$ where $\bar\kappa$ satisfies $$\begin{aligned} f(\bar\kappa)&= \frac{{\displaystyle}\sum_{i=1}^N c_i^2f(\kappa_i) +2\sum_{i=1}^{N-1}\left(f(\kappa_1)c_1+\dots+f(\kappa_i)c_i\right)c_{i+1}} {(c_1+c_2+\dots+c_N)^2}. \end{aligned}$$ By the convexity and monotonicity of $f$, $$\begin{aligned} \bar\kappa&\ge \frac{{\displaystyle}\sum_{i=1}^N c_i^2\kappa_i +2\sum_{i=1}^{N-1}\left(\kappa_1c_1+\dots+\kappa_ic_i\right)c_{i+1}} {(c_1+c_2+\dots+c_N)^2} \\ &=\frac{{\displaystyle}\sum_{i=1}^N c_i^2\kappa_i +2\sum_{i=1}^{N-1} \left(c_{i+1}+\dots+c_N\right)c_i\kappa_i} {(c_1+c_2+\dots+c_N)^2}. \end{aligned}$$ In the following we give a special case of Lemma \[triangle.comp.mult\], which will be used in our case. \[cor.mult.Alex\] Let the assumption be as in Lemma \[triangle.comp.mult\] for $i=1,\dots,2N$. Let $b_i=c_{2i-1}$ and $d_i=c_{2i}$. Assume $\kappa_{2i-1}=\kappa\ge\kappa_{2i}\ge\kappa^*$, $i=1,2,\dots,N$. Then (\[triangle.comp.mult.e01\]) holds for $$\begin{aligned} \bar\kappa =\frac{(b_1+\dots+b_N)^2(\kappa-\kappa^*)} {(b_1+\dots+b_N+d_1+\dots+d_N)^2}+\kappa^*. \label{cor.mult.Alex.e1} \end{aligned}$$ By the assumption and (\[triangle.comp.mult.e1\]), the $\bar\kappa$ in Lemma \[triangle.comp.mult\] satisfies $$\begin{aligned} \bar\kappa &\ge\frac{(b_1+\dots+b_N)^2\kappa +2(b_1+\dots+b_N)(d_1+\dots+d_N)\kappa^* +(d_1+\dots+d_N)^2\kappa^*} {(b_1+\dots+b_N+d_1+\dots+d_N)^2} \\ &=\frac{(b_1+\dots+b_N)^2(\kappa-\kappa^*)} {(b_1+\dots+b_N+d_1+\dots+d_N)^2}+\kappa^*. \end{aligned}$$ In the proof of Theorem A, we also need an estimate for the comparison curvature when a triangle extends without any curvature control. \[tri.comp.2.1\] Let $x,z,y_0,y\in X$ such that $y_0\in{\,]xy[\,}_{X}$. Suppose that $|xy_0|=r$ and $|xy|=a$. For any $\kappa\in\mathbb R$, there is $\kappa^*=\kappa^*(a,r,\kappa)\in\mathbb R$ such that ${\tilde\measuredangle_{\kappa}\left({x}\,_{z}^{y_0}\right)}\ge{\tilde\measuredangle_{\kappa^*}\left({x}\,_{z}^{y}\right)}$. Moreover, $\kappa^*$ is decreasing in $a$ and $\kappa^*\to\kappa$ as $a\to r$. In particular, when $|xz|<\delta(r,\kappa)$ is small enough, $\kappa^*$ can be chosen explicitly as $\kappa^*=f_a^{-1}(f_r(\kappa))$. Here $\kappa^*=f_a^{-1}(f_r(\kappa))\to-\infty$ as $r\to 0^+$ even if $\frac ar\to 0$ in a constant rate. The existence of $\kappa^*$ is obvious by direct computation. Suppose that $|xz|=b$ is small. By the Taylor expansion (\[tri.comp.2.e1\]) and the triangle inequality, $$\begin{aligned} |yz|&\le |y_0y|+|y_0z| \notag\\ &= |y_0y|+|xy_0|-b\cos\left({\tilde\measuredangle_{\kappa}\left({x}\,_{z}^{y_0}\right)}\right) +\sin^2\left({\tilde\measuredangle_{\kappa}\left({x}\,_{z}^{y_0}\right)}\right)f_r(\kappa) \cdot b^2+O(b^3) \notag\\ &=a-b\cos\left({\tilde\measuredangle_{\kappa}\left({x}\,_{z}^{y_0}\right)}\right) +\sin^2\left({\tilde\measuredangle_{\kappa}\left({x}\,_{z}^{y_0}\right)}\right)f_a(\kappa^*) \cdot b^2+O(b^3). \label{tri.comp.2.1.e1} \end{aligned}$$ On the other hand, $$\begin{aligned} |yz|= a-b\cos\left({\tilde\measuredangle_{\kappa^*}\left({x}\,_{z}^{y}\right)}\right) +\sin^2\left({\tilde\measuredangle_{\kappa^*}\left({x}\,_{z}^{y}\right)}\right)f_a(\kappa^*) \cdot b^2+O(b^3). \label{tri.comp.2.1.e2} \end{aligned}$$ Compare (\[tri.comp.2.1.e1\]) with (\[tri.comp.2.1.e2\]) in a way similar to Lemma \[tri.comp.2.2\]. We get that when $b$ is sufficiently small, $${\tilde\measuredangle_{\kappa^*}\left({x}\,_{z}^{y}\right)}\le{\tilde\measuredangle_{\kappa}\left({x}\,_{z}^{y_0}\right)}.$$ Comparisons near $\kappa$-geodesics =================================== In this section, we always assume that $\mathcal U\in{\text{Alex\,}}_{loc}(\kappa)$. By the standard globalization process, we may assume ${\text{diam}}(\mathcal U)\le\frac{\pi}{2\sqrt\kappa}$ if $\kappa>0$ and ${\text{diam}}(\mathcal U)\le 1$ if $\kappa\le0$. If the metric completion $\bar{\mathcal U}$ is locally compact, then for any $p,q\in\bar{\mathcal U}$, there exists a geodesic ${[\,pq\,]}$ connecting $p$ and $q$ in $\bar{\mathcal U}$. If $\bar{\mathcal U}$ is not locally compact, we consider its $\omega$-power $\bar{\mathcal U}^\omega$, where $\omega$ is a fixed non-principle ultrafilter on natural numbers (see [@AKP] for more details). $\bar{\mathcal U}$ can be viewed as a subspace of $\bar{\mathcal U}^\omega$. For any two points $p,q\in\bar{\mathcal U}$, there exists a geodesic ${[\,pq\,]}$ connecting $p$ and $q$ in $\bar{\mathcal U}^\omega$. In either case, geodesic ${[\,pq\,]}$ is well defined and ${[\,pq\,]}\subset\mathcal U$ means that $p$ and $q$ can be connected by a geodesic in $\mathcal U$. Let $\gamma$ be a geodesic. We call $\gamma$ a $\kappa$-geodesic if every point on $\gamma$ is contained in a $\kappa$-domain. By the definition, geodesic $\gamma_1$ is also a $\kappa$-geodesic with the same $\kappa$-domain covering, if $\gamma_1$ is close enough to $\gamma$. We start with recalling some results from [@Pet13]. \[Pet13-Lem2.3\] Let $\Omega_p$ and $\Omega_q$ be two $\kappa$- domains in $\bar{\mathcal U}$. Let $p\in\Omega_p$, $q\in\Omega_q$ and ${[\,pq\,]}\in\Omega_p\cup\Omega_q$. Then for any geodesic ${[\,qs\,]}\subset \Omega_q$, we have ${\measuredangle\left[{q}\,_{s}^{p}\right]}\ge{\tilde\measuredangle_{\kappa}\left({q}\,_{s}^{p}\right)}$ if $|qs|$ is sufficiently small. \[Pet13-Cor2.4\] Let $\Omega_1$ and $\Omega_2$ be two $\kappa$-domains in $\bar{\mathcal U}$. Assume $\Omega_3\subset\Omega_1\cup\Omega_2$ is an open set such that for any two points $x, y\in\mathcal U$, any geodesic ${[\,xy\,]}$ lies in $\Omega_1\cup\Omega_2$. Then $\Omega_3$ is a $\kappa$-domain. We observe that the proof of Lemma 2.5 in [@Pet13] also works for the case $x=p$. Thus the following stronger result holds. For completeness, we repeat Petrunin’s proof here. \[Pet13-Lem2.5.2\] Let ${[\,pq\,]}$ be a $\kappa$-geodesic and the points $x, y$ and $z$ appear on ${[\,pq[\,}$ in the same order. Assume that there are $\kappa$-domains $\Omega_1\supset{[\,xy\,]}$ and $\Omega_2\supset{[\,yz\,]}$. Then 1. geodesic ${[\,xz\,]}$ is unique; 2. for any $\epsilon>0$, there is $\delta>0$ such that ${[\,uv\,]}\subset B_{\epsilon}({[\,xz\,]})$ for any $u\in B_{\delta}({[\,xy\,]})$ and $v\in B_{\delta}({[\,yz\,]})$. In particular, there is an open set $\Omega_3\subset\bar{\mathcal U}$ which contains ${[\,xz\,]}$ and such that for any two points $u,v\in \Omega_3\cap\mathcal U$, any geodesic ${[\,uv\,]}$ lies in $\Omega_1\cup\Omega_2$. By Corollary \[Pet13-Cor2.4\], $\Omega_3$ is a $\kappa$-domain. \(1) follows from the fact that for any point $z$ in a $\kappa$-domain and any points $p,q,s\in\mathcal U$, ${\measuredangle\left[{z}\,_{q}^{p}\right]}+{\measuredangle\left[{z}\,_{s}^{p}\right]}+{\measuredangle\left[{z}\,_{s}^{q}\right]}\le2\pi$. This is proved exactly the same way as in [@BGP]. (2) is argued by contradiction. Assume that there exists $\epsilon>0$ and a sequence of geodesics ${[\,u_iv_i\,]}$, such that $u_i\to u$, $v_i\to v$ but ${[\,u_iv_i\,]}\not\subset B_\epsilon({[\,xz\,]})$. The ultralimit of ${[\,u_iv_i\,]}$ is a geodesic in $\bar{\mathcal U}^\omega$, connecting $u$ and $v$. Not losing generality, assume $v\neq u$. Then we obtain a bifurcated geodesic at $v\in{\,]pq[\,}$. For $r>0$ small enough, $v$ is contained in a $\kappa$-domain $B_r(v)\subset\bar{\mathcal U}$. It is straightforward to verify that $B_r(v)$ is also a $\kappa$-domain in $\bar{\mathcal U}^\omega$, a contradiction. \[2.k-domain\] Let ${[\,pq\,]}$ be a $\kappa$-geodesic. For any $y\in{\,]pq[\,}$, there exist $\kappa$-domains $\Omega_1, \Omega_2$ such that $\Omega_1\supset{[\,py\,]}$ and $\Omega_2\supset{[\,yq\,]}$. Let $\{V_i,\, i=1,2,\dots,N\}$ be a $\kappa$-domain covering of ${[\,pq\,]}$. Let $x_1=p$, $x_{N+1}= q$ and $x_i\in V_{i-1}\cap V_i$, $i=2,3,\dots,N$. We may assume that $x_i$ appear on ${[\,pq\,]}$ in the same order and $V_i\supset{[\,x_ix_{i+1}\,]}$. Not losing generality, assume $x_N=y$. By Lemma \[Pet13-Lem2.5.2\], there is a $\kappa$-domain $\Omega_1$ containing ${[\,x_1x_3\,]}$. Thus $\{\Omega_1,V_3,V_4,\dots,V_N\}$ forms a $\kappa$-domain covering of ${[\,pq\,]}$. Repeat applying Lemma \[Pet13-Lem2.5.2\] as the above, we will arrive at a position that ${[\,pq\,]}$ is covered by $\kappa$-domains $\Omega_{N-2}$ and $V_N$. In fact, we have ${[\,px_N\,]}\subset \Omega_{N-2}$ and ${[\,x_Nx_{N+1}\,]}\subset V_N$. Combining Lemma \[Pet13-Lem2.5.2\] and Corollary \[2.k-domain\], we get the following result immediately. \[geod.close\] Let ${[\,pq\,]}$ be a $\kappa$-geodesic and $x\in{[\,pq\,]}$, $z\in{\,]pq[\,}$. Then 1. geodesic ${[\,xz\,]}$ is unique; 2. for any $r>0$, there is $\epsilon>0$ such that for any $u,v\in B_\epsilon({[\,xz\,]})$, any geodesic ${[\,uv\,]}$ is a $\kappa$-geodesic contained in $B_r({[\,xz\,]})$. The following thin triangle comparison is contained in the proof of the main theorem in [@Pet13], which follows directly from Lemma \[Pet13-Lem2.3\] and Corollary \[2.k-domain\]. \[thin.comp\] Let ${[\,pq\,]}$ be a $\kappa$-geodesic. Then there exists $r>0$ such that ${\measuredangle\left[{q}\,_{s}^{p}\right]}\ge{\tilde\measuredangle_{\kappa}\left({q}\,_{s}^{p}\right)}$ for any $s\in B_r(q)\setminus\{q\}$. As an application of the above results, we get the following comparison. \[thin.comp.er\] Let ${[\,pq\,]}$ be a $\kappa$-geodesic. For any $p_0\in{\,]pq\,]}$, there exists $r>0$, depending only on $|pp_0|$ and the way that ${[\,pq\,]}$ sits in its $\kappa$-domain covering, such that for with any $u,v,w\in B_r({[\,p_0q\,]})$, we have ${\measuredangle\left[{u}\,_{v}^{w}\right]}\ge{\tilde\measuredangle_{\kappa}\left({u}\,_{v}^{w}\right)}$. By Lemma \[geod.close\], there is $\delta>0$ such that for any $x,y\in B_\delta({[\,p_0,q\,]})$, any geodesic ${[\,xy\,]}$ is a $\kappa$-geodesic. Apply Lemma \[geod.close\] once more. Take $r>0$ small so that for any $u,v\in B_r({[\,p_0q\,]})$, any geodesic ${[\,uv\,]}\subset B_\delta({[\,p_0q\,]})$. Thus for any $x\in{[\,uv\,]}$, geodesic ${[\,wx\,]}$ is a $\kappa$-geodesic. By Lemma \[thin.comp\], the function $g(x)=\textsf{md}_\kappa\circ {{\textsf d}\left(w,x\right)}$ is a $(1-\kappa g)$-concave function when restricted to ${[\,uv\,]}$. This implies that ${\measuredangle\left[{u}\,_{v}^{w}\right]}\ge{\tilde\measuredangle_{\kappa}\left({u}\,_{v}^{w}\right)}$. \[uniform.b1-1\] Let ${[\,pq\,]}$ be a $\kappa$-geodesic. For any $\kappa_1<\kappa$, there exists $r>0$ such that ${\measuredangle\left[{x}\,_{y}^{p}\right]}\ge{\tilde\measuredangle_{\kappa_1}\left({x}\,_{y}^{p}\right)}$ for any $x,y\in B_r(q)$. In particular, for any geodesic ${[\,qs\,]}\subset B_r(q)$, we have ${\measuredangle\left[{q}\,_{s}^{p}\right]}\ge{\tilde\measuredangle_{\kappa_1}\left({q}\,_{s}^{p}\right)}$ and ${\measuredangle\left[{s}\,_{q}^{p}\right]}\ge{\tilde\measuredangle_{\kappa_1}\left({s}\,_{q}^{p}\right)}$. For $\epsilon>0$ small, take $p_0\in{[\,pq\,]}$ such that $|pp_0|=\epsilon$. By Lemma \[thin.comp.er\], there is $r>0$ such that for any $x,y\in B_r(q)$, ${\measuredangle\left[{x}\,_{y}^{p_0}\right]}\ge{\tilde\measuredangle_{\kappa}\left({x}\,_{y}^{p_0}\right)}$. By Lemma \[tri.comp.2.1\], we get ${\tilde\measuredangle_{\kappa}\left({x}\,_{y}^{p_0}\right)}\ge {\tilde\measuredangle_{\kappa^*}\left({x}\,_{y}^{p}\right)}$, where $\kappa^*\to\kappa$ as $p_0\to p$. For any $\kappa_1<\kappa$, $\epsilon$ can be chosen small so that $\kappa^*\ge\kappa_1$. Then we have $$\begin{aligned} {\measuredangle\left[{x}\,_{y}^{p}\right]}={\measuredangle\left[{x}\,_{y}^{p_0}\right]}\ge{\tilde\measuredangle_{\kappa}\left({x}\,_{y}^{p_0}\right)} \ge {\tilde\measuredangle_{\kappa^*}\left({x}\,_{y}^{p}\right)} \ge{\tilde\measuredangle_{\kappa_1}\left({x}\,_{y}^{p}\right)}. \end{aligned}$$ \[exist.ks\] Let ${[\,qs\,]}$ be a $\kappa$-geodesic and point $p\notin{[\,qs\,]}$. There are $r_0>0$ and $\kappa^*=\kappa^*({[\,qs\,]},\, \sup\{{{\textsf d}\left(p,x\right)}:x\in{[\,qs\,]}\})\in\mathbb R$ such that for any $u\in B_{r_0}({[\,qs\,]})$ and $v\in B_{r_0}(u)$, we have ${\measuredangle\left[{u}\,_{v}^{p}\right]}\ge{\tilde\measuredangle_{\kappa^*}\left({u}\,_{v}^{p}\right)}$. By Corollary \[2.k-domain\], there exists $r_0>0$ so that for any $u\in B_{r_0}({[\,qs\,]})$, $B_{4r_0}(u)$ is contained a $\kappa$-domain. Let $w\in{[\,pu\,]}$ such that $|uw|=r_0$. Then we have ${\measuredangle\left[{u}\,_{v}^{w}\right]}\ge{\tilde\measuredangle_{\kappa}\left({u}\,_{v}^{w}\right)}$. Let $R=\sup\{{{\textsf d}\left(p,x\right)}: x\in{[\,qs\,]}\}+10r_0$. By Lemma \[tri.comp.2.1\], there is $\kappa^*=\kappa^*(R,r_0,\kappa)$ such that ${\tilde\measuredangle_{\kappa}\left({u}\,_{v}^{w}\right)}\ge{\tilde\measuredangle_{\kappa^*}\left({u}\,_{v}^{p}\right)}$. Therefore, ${\measuredangle\left[{u}\,_{v}^{p}\right]}={\measuredangle\left[{u}\,_{v}^{w}\right]} \ge{\tilde\measuredangle_{\kappa^*}\left({u}\,_{v}^{p}\right)}$. We would like to point out that when $|uv|<\delta(r,\kappa)$ is small, one can select $\kappa^*=f_{R}^{-1}(f_{r}(\kappa))$. Globalization with weak $\mathfrak p_\lambda$-convexity ======================================================= In this section, we always assume $\mathcal U\in{\text{Alex\,}}_{loc}(\kappa)$. We first need a better perturbation for weak $\mathfrak p_\lambda$-convexity. \[conv.geod.pert\] Suppose that $\mathcal U\in{\text{Alex\,}}_{loc}(\kappa)$ is weakly $\mathfrak p_\lambda$-convex. If $\lambda>0$, then for any $p,q,s\in\mathcal U$ and any $0<\epsilon<\lambda$, there is a geodesic triangle $\triangle\bar p\bar q\bar s$ such that 1. $\bar p\in B_\epsilon(p)$, $\bar q\in B_\epsilon(q)$ and $\bar s\in B_\epsilon(s)$; 2. ${[\,\bar p\bar q\,]}$ and ${[\,\bar p\bar s\,]}$ are $\kappa$-geodesics; 3. ${\bf Pr}\left(\bar p\prec{[\,\bar q\bar s\,]}\right)\ge\lambda-\epsilon$. Let $0<\epsilon_3\ll\epsilon_2\ll\epsilon_1<\epsilon/10$ be all small. We first select a $\kappa$-geodesic ${[\,p_1q_1\,]}$ that satisfies $p_1\in B_{\epsilon_1}(p)$ and $q_1\in B_{\epsilon_1}(q)$. Let $\bar p_1\in{[\,\bar p_1q_1\,]}$ such that $|p_1\bar p_1|=\epsilon_2$. There is a $\kappa$-geodesic ${[\,p_2s_2\,]}$ that satisfies $p_2\in B_{\epsilon_2}(\bar p_1)$ and $s_2\in B_{\epsilon_2}(s)$. Let $\bar p_2\in{[\,\bar p_2s_2\,]}$ such that $|p_2\bar p_2|=\epsilon_2$. By the definition of weak $\mathfrak p_\lambda$-convexity, there are points $\bar p\in B_{\epsilon_3}(\bar p_2)$, $\bar q\in B_{\epsilon_3}(q_1)$, $\bar s\in B_{\epsilon_3}(s_2)$ and a geodesic ${[\,\bar q\bar s\,]}$, such that ${\bf Pr}\left(\bar p\prec{[\,\bar q\bar s\,]}\right)>\lambda-\epsilon_3>\lambda-\epsilon$. By Lemma \[geod.close\], when $\epsilon_3\ll\epsilon_2\ll\epsilon_1$, ${[\,\bar p\bar q\,]}$ and ${[\,\bar p\bar s\,]}$ are both $\kappa$-geodesics. At last, we also have $$\begin{aligned} |q\bar q| &\le |qq_1|+|q_1\bar q|\le \epsilon_1+\epsilon_3<\epsilon, \end{aligned}$$ $$\begin{aligned} |s\bar s| &\le |ss_2|+|s_2\bar s|\le \epsilon_2+\epsilon_3<\epsilon \end{aligned}$$ and $$\begin{aligned} |p\bar p| &\le |pp_1|+|p_1\bar p_1|+|\bar p_1p_2|+|p_2\bar p_2|+|\bar p_2\bar p| \\ &\le 2\epsilon_1+2\epsilon_2+\epsilon_3<\epsilon. \end{aligned}$$ The size $r$ of the comparison in Lemma \[thin.comp\] depends on the way that ${[\,pq\,]}$ is contained in the $\kappa$-domains. When the $r$ is made larger, the comparison curvature may drop. The following two lemmas show that the defection of the lower curvature bound is controlled by the probability of points in ${[\,qs\,]}$ that can be connected to $p$ by a geodesic in $\mathcal U$. \[comp.X1\] Let $p\in\mathcal U$, $\kappa,\kappa^*\in\mathbb R$ and $l>0$. For any $\kappa_1<\kappa$, there is $\delta=\delta(|pq|,\kappa_1,\kappa^*)>0$ such that the following holds for any $\kappa$-geodesic ${[\,qs\,]}$ that satisfies $|qs|<\delta$ and $|pq|\ge l$. Suppose ${\measuredangle\left[{y_1}\,_{q}^{p}\right]}\ge{\tilde\measuredangle_{\kappa_1}\left({y_1}\,_{q}^{p}\right)}$ for some $y_1\in{\,]qs\,]}$ and ${\measuredangle\left[{x}\,_{y}^{p}\right]}\ge{\tilde\measuredangle_{\kappa^*}\left({x}\,_{y}^{p}\right)}$ for any $x,y\in{[\,qs\,]}$. Then $${\tilde\measuredangle_{\kappa_1}\left({q}\,_{y_1}^{p}\right)} \ge{\tilde\measuredangle_{\bar\kappa}\left({q}\,_{s}^{p}\right)},$$ for $\bar\kappa=\lambda^2\kappa_1+(1-\lambda^2)\kappa^*$, where $\lambda={\bf Pr}(p\prec{[\,qs\,]})$. In particular, if ${[\,pq\,]}$ is a $\kappa$-geodesic, then we have ${\measuredangle\left[{q}\,_{s}^{p}\right]} \ge{\tilde\measuredangle_{\bar\kappa}\left({q}\,_{s}^{p}\right)}$. Let ${{{[\,qs\,]}}^{*p}}$ be the set of points in ${[\,qs\,]}$ which can be connected to $p$ by a geodesic in $\mathcal U$. For every $x\in {{{[\,qs\,]}}^{*p}}$, $x\neq s$, by Corollary \[uniform.b1-1\], there exists $y\in{\,]xs\,]}$ such that ${\measuredangle\left[{x}\,_{y}^{p}\right]}\ge{\tilde\measuredangle_{\kappa_1}\left({x}\,_{y}^{p}\right)}$ and ${\measuredangle\left[{y}\,_{x}^{p}\right]}\ge{\tilde\measuredangle_{\kappa_1}\left({y}\,_{x}^{p}\right)}$. The collection of all such segments ${[\,xy\,]}$, together with ${[\,qy_1\,]}$, gives a covering of ${{{[\,qs[\,}}^{*p}}$. Let $x_1=q$. For any $\eta>0$ small, by Vitali covering theorem, there is a disjoint finite sub-collection ${\displaystyle}\{{[\,x_iy_i\,]}\}_{i=1}^N$ so that $$\begin{aligned} \sum_{i=1}^N|x_iy_i|\ge\mathcal H^1\left({{{[\,qs\,]}}^{*p}}\right)-\eta|qs| =(\lambda-\eta)|qs|. \label{comp.X.e4} \end{aligned}$$  Let $x_{N+1}=s$. By the construction, for all $i=2,3,\dots, N$, we have $$\begin{aligned} {\tilde\measuredangle_{\kappa_1}\left({x_i}\,_{y_i}^{p}\right)}\le{\measuredangle\left[{x_i}\,_{y_i}^{p}\right]} \text{\quad and \quad} {\tilde\measuredangle_{\kappa_1}\left({y_i}\,_{x_i}^{p}\right)}\le{\measuredangle\left[{y_i}\,_{x_i}^{p}\right]}. \label{comp.X.e5} \end{aligned}$$ By the assumptions, we have $$\begin{aligned} {\measuredangle\left[{x_i}\,_{y_{i-1}}^{p}\right]} \ge{\tilde\measuredangle_{\kappa^*}\left({x_i}\,_{y_{i-1}}^{p}\right)} \label{comp.X.e6} \end{aligned}$$ and $$\begin{aligned} {\measuredangle\left[{y_i}\,_{x_{i+1}}^{p}\right]} \ge{\tilde\measuredangle_{\kappa^*}\left({y_i}\,_{x_{i+1}}^{p}\right)}. \label{comp.X.e7} \end{aligned}$$ Since ${\tilde\measuredangle_{\kappa_1}\left({y_1}\,_{q}^{p}\right)}\le{\measuredangle\left[{y_1}\,_{q}^{p}\right]}$, by (\[comp.X.e5\]), (\[comp.X.e6\]) and (\[comp.X.e7\]), together with the fact that ${[\,qs\,]}$ is a $\kappa$-geodesic, we have $$\begin{aligned} {\tilde\measuredangle_{\kappa_1}\left({x_i}\,_{y_i}^{p}\right)}+{\tilde\measuredangle_{\kappa^*}\left({x_i}\,_{y_{i-1}}^{p}\right)} \le {\measuredangle\left[{x_i}\,_{y_i}^{p}\right]}+{\measuredangle\left[{x_i}\,_{y_{i-1}}^{p}\right]}\le\pi \end{aligned}$$ and $$\begin{aligned} {\tilde\measuredangle_{\kappa_1}\left({y_i}\,_{x_i}^{p}\right)}+{\tilde\measuredangle_{\kappa^*}\left({y_i}\,_{x_{i+1}}^{p}\right)} \le {\measuredangle\left[{y_i}\,_{x_i}^{p}\right]}+{\measuredangle\left[{y_i}\,_{x_{i+1}}^{p}\right]}\le\pi. \end{aligned}$$ Let $b_i=|x_iy_i|$ and $d_i=|y_ix_{i+1}|$, $i=1,2,\dots,N$. By (\[comp.X.e4\]), we have ${\displaystyle}\sum_{i=1}^Nb_i\ge(\lambda-\eta)\sum_{i=1}^N(b_i+d_i)$. Thus when $|qs|$ is small, by Corollary \[cor.mult.Alex\], we get $${\tilde\measuredangle_{\kappa_1}\left({q}\,_{y_1}^{p}\right)} \ge{\tilde\measuredangle_{\underline\kappa}\left({q}\,_{s}^{p}\right)}$$ for $$\begin{aligned} \underline\kappa &=\frac{(b_1+\dots+b_N)^2(\kappa_1-\kappa^*)} {(b_1+\dots+b_N+d_1+\dots+d_N)^2}+\kappa^* \\ &\ge (\lambda-\eta)^2(\kappa_1-\kappa^*)+\kappa^*. \end{aligned}$$ Let $\eta\to 0$. We get the desired result. \[comp.X\] Suppose that $\mathcal U$ is weakly $\mathfrak p_\lambda$-convex. Let both ${[\,pq\,]}$ and ${[\,qs\,]}$ be $\kappa$-geodesics. There is $\kappa^*=\kappa^*(p,{[\,qs\,]})\in\mathbb R$, so that for any $\underline\kappa<\lambda^2\kappa+(1-\lambda^2)\kappa^*$, $$\begin{aligned} {\measuredangle\left[{q}\,_{s}^{p}\right]} \ge{\tilde\measuredangle_{\underline\kappa}\left({q}\,_{s}^{p}\right)}. \label{comp.X.e0} \end{aligned}$$ If $\bar{\mathcal U}\in{\text{Alex\,}}(\kappa_0)$, then (\[comp.X.e0\]) holds for any $\underline\kappa<\lambda^2\kappa+(1-\lambda^2)\kappa_0$. Let $r_0>0$ and $\kappa^*=\kappa^*(p, {[\,qs\,]})$ be defined as in Lemma \[exist.ks\]. If $\bar{\mathcal U}\in{\text{Alex\,}}(\kappa_0)$, we take $\kappa^*=\kappa_0$. Fix $\epsilon_0>0$ small and take $\bar p_0\in{[\,pq\,]}$ such that $|p\bar p_0|=\epsilon_0$. Let $\delta =\delta({{\textsf d}\left(p,{[\,qs\,]}\right)} -2\epsilon_0,\kappa,\kappa^*)\in(0,r_0)$ be determined as in Lemma \[comp.X1\]. By Lemma \[thin.comp.er\], there exists $r\in(0,\delta/4)$ such that for any $p_1,x,y\in B_r({[\,\bar p_0q\,]})$, we have ${\measuredangle\left[{x}\,_{y}^{p_1}\right]}\ge{\tilde\measuredangle_{\kappa}\left({x}\,_{y}^{p_1}\right)}$. Let $q_i, i=1,\dots N+1$, be a partition of ${[\,qs\,]}$, such that $q_1=q$, $|q_iq_{i+1}|=\delta/2$ and $\delta/2\le|q_{N+1}s|\le\delta$. Let $v_0=q_1$ and $p_0=p$ and $0<\epsilon_i\ll\epsilon_{i-1}\ll r$ be small. By the definition of weak $\mathfrak p_\lambda$-convexity and Lemma \[conv.geod.pert\], we can recursively select $p_i, \bar p_i, u_i, v_i$ for $i=1,2,\dots,N$, such that 1. $p_i\in B_{\epsilon_i}(\bar p_{i-1})$, $u_i\in B_{\epsilon_i}(v_{i-1})$ and $v_i\in B_{\epsilon_i}(q_{i+1})$; 2. ${[\,p_iu_i\,]}$ and ${[\,p_iv_i\,]}$ are all $\kappa$-geodesics and ${\bf Pr}(p_i\prec{[\,u_iv_i\,]}) \ge \lambda-\epsilon_i$; 3. $\bar p_i\in{[\,p_iv_i\,]}$ and $|p_i\bar p_i|=\epsilon_i$;  Let $\epsilon=\sum_{i=1}^N\epsilon_i\ll\epsilon_0$. For a fixed $i$, when $\epsilon_i\to 0$, ${[\,p_iu_i\,]}$ converge to geodesic ${[\,\bar p_{i-1}v_{i-1}\,]}$. Thus when $\epsilon\to 0$, passing to a subsequence, ${[\,p_iu_i\,]}$ and ${[\,p_{i-1}v_{i-1}\,]}$ converge to the same limit geodesic ${[\,\bar p_0q_i\,]}$. Note that ${[\,u_iv_i\,]}$ converge to the geodesic ${[\,q_iq_{i+1}\,]}\subset{[\,qs\,]}$ (In the case that $\bar{\mathcal U}$ is not locally compact, we consider the $\omega$-power $\bar{\mathcal U}^\omega$ and the same argument applies). We have $$\begin{aligned} {\measuredangle\left[{v_i}\,_{u_i}^{p_i}\right]}\to{\measuredangle\left[{q_{i+1}}\,_{q_i}^{\bar p_0}\right]} \text{\quad and \quad} {\measuredangle\left[{u_{i+1}}\,_{v_{i+1}}^{p_i}\right]}\to{\measuredangle\left[{q_{i+1}}\,_{q_{i+2}}^{\bar p_0}\right]}, \end{aligned}$$ as $\epsilon\to 0$, $i=1,2,\dots,N-1$, since $q_i$, $i\ge 2$ are interior points of a $\kappa$-geodesic. Therefore, $$\begin{aligned} {\measuredangle\left[{v_i}\,_{u_i}^{p_i}\right]}+{\measuredangle\left[{u_{i+1}}\,_{v_{i+1}}^{p_i}\right]} \le{\measuredangle\left[{q_{i+1}}\,_{q_i}^{\bar p_0}\right]} +{\measuredangle\left[{q_{i+1}}\,_{q_{i+2}}^{\bar p_0}\right]}+\psi(\epsilon) =\pi+\psi(\epsilon). \label{comp.X.e3} \end{aligned}$$ Let $y_0\in{[\,q_1q_2\,]}$ such that $|q_1y_0|=r/2$. By Lemma \[geod.close\] and the fact that $|u_1v_1|\ge \delta/2-2\epsilon_1>r$, there exists $y_1\in{[\,u_1v_1\,]}$ such that $d(y_1,y_0)<\psi(\epsilon_1)$. Then $|q_1y_1|\le |q_1y_0|+|y_0y_1|\le\psi(\epsilon_1)+r/2<r$. Thus we have ${\measuredangle\left[{q_1}\,_{y_0}^{\bar p_0}\right]}\ge{\tilde\measuredangle_{\kappa}\left({q_1}\,_{y_0}^{\bar p_0}\right)}$ and ${\measuredangle\left[{y_1}\,_{u_1}^{p_1}\right]}\ge{\tilde\measuredangle_{\kappa}\left({y_1}\,_{u_1}^{p_1}\right)}$. Let $\kappa_1<\kappa$ such that $\underline\kappa=\lambda^2\kappa_1+(1-\lambda^2)\kappa^*$. The assumptions in Lemma \[comp.X1\] for $p_i$ and ${[\,u_iv_i\,]}$ are satisfied by the following construction: - $|u_iv_i|\le|q_iq_{i+1}|+4\epsilon_i <\delta$ and $|p_iu_i|\ge{{\textsf d}\left(p,{[\,qs\,]}\right)} -10\epsilon>{{\textsf d}\left(p,{[\,qs\,]}\right)} -\epsilon_0$. - ${[\,p_iu_i\,]}$ is a $\kappa$-geodesic. - $|u_iq_i|\le 2\epsilon_i<r_0$ and $|u_iv_i|<\delta<r_0$. Thus we have $$\begin{aligned} {\measuredangle\left[{u_i}\,_{v_i}^{p_i}\right]} \ge{\tilde\measuredangle_{\underline\kappa}\left({u_i}\,_{v_i}^{p_i}\right)}, \text{\quad \quad} {\measuredangle\left[{v_i}\,_{u_i}^{p_i}\right]} \ge{\tilde\measuredangle_{\underline\kappa}\left({v_i}\,_{u_i}^{p_i}\right)}, \label{comp.X.e1} \end{aligned}$$ and $$\begin{aligned} {\tilde\measuredangle_{\kappa_1}\left({u_1}\,_{y_1}^{p_1}\right)} \ge{\tilde\measuredangle_{\underline\kappa}\left({u_1}\,_{y_1}^{p_1}\right)}, \label{comp.X.e2} \end{aligned}$$ where $\underline\kappa=(\lambda-\epsilon)^2\kappa_1 +(1-(\lambda-\epsilon)^2)\kappa^*$. By (\[comp.X.e3\]) and (\[comp.X.e1\]), we have $$\begin{aligned} {\tilde\measuredangle_{\underline\kappa}\left({v_i}\,_{u_i}^{p_i}\right)} +{\tilde\measuredangle_{\underline\kappa}\left({u_{i+1}}\,_{v_{i+1}}^{p_i}\right)} \le {\measuredangle\left[{v_i}\,_{u_i}^{p_i}\right]}+{\measuredangle\left[{u_{i+1}}\,_{v_{i+1}}^{p_i}\right]} \le\pi+\psi(\epsilon). \end{aligned}$$ By (\[comp.X.e2\]) and Alexandrov’s lemma, we get $$\begin{aligned} {\tilde\measuredangle_{\kappa_1}\left({u_1}\,_{y_1}^{p_1}\right)} &\ge{\tilde\measuredangle_{\underline\kappa}\left({u_1}\,_{v_1}^{p_1}\right)} \notag \\ &\ge{\tilde\measuredangle_{\underline\kappa}\left({u_1}\,_{v_N}^{p_1}\right)}-N\psi(\epsilon) \ge{\tilde\measuredangle_{\underline\kappa}\left({u_1}\,_{v_N}^{p_1}\right)}-|qs|\frac{\psi(\epsilon)}{\delta}. \end{aligned}$$ We choose $\epsilon$ small so that $\epsilon, \psi(\epsilon)\ll r,\delta$. Then $$\begin{aligned} \left|{\tilde\measuredangle_{\kappa_1}\left({q}\,_{y_0}^{\bar p_0}\right)}-{\tilde\measuredangle_{\kappa_1}\left({u_1}\,_{y_1}^{p_1}\right)}\right|<\psi(\epsilon\mid r,\delta). \end{aligned}$$ Thus $$\begin{aligned} {\measuredangle\left[{q}\,_{s}^{p}\right]} &={\measuredangle\left[{q}\,_{y_0}^{\bar p_0}\right]}\ge{\tilde\measuredangle_{\kappa_1}\left({q}\,_{y_0}^{p_0}\right)} \ge{\tilde\measuredangle_{\kappa_1}\left({u_1}\,_{y_1}^{p_1}\right)} -\psi(\epsilon\mid r,\delta) \\ &\ge {\tilde\measuredangle_{\underline\kappa}\left({u_1}\,_{v_N}^{p_1}\right)}-\psi(\epsilon\mid r,\delta) \ge {\tilde\measuredangle_{\underline\kappa}\left({q}\,_{s}^{p}\right)}-\psi(\epsilon\mid r,\delta)-\psi(\epsilon_0)-\psi(\delta). \end{aligned}$$ At last, let $\epsilon\ll\delta\to 0$ and $\epsilon_0\to 0$. We get the desired comparison. \[comp.X.cor1\] Suppose that $\mathcal U$ is weakly $\mathfrak p_1$-convex. Let $(p;x_1,x_2,x_3)$ be a quadruple in $\mathcal U$. If ${[\,px_i\,]}$, $i=1,2,3$, are all $\kappa$-geodesics and the comparison angles ${\tilde\measuredangle_{\kappa}\left({p}\,_{x_j}^{x_i}\right)}$ are all defined, then $$\begin{aligned} {\tilde\measuredangle_{\underline\kappa}\left({p}\,_{x_2}^{x_1}\right)} +{\tilde\measuredangle_{\underline\kappa}\left({p}\,_{x_3}^{x_2}\right)} +{\tilde\measuredangle_{\underline\kappa}\left({p}\,_{x_1}^{x_3}\right)}\le2\pi. \label{comp.X.cor.e1} \end{aligned}$$ \[comp.X.cor2\] Suppose that $\mathcal U$ is weakly $\mathfrak p_\lambda$-convex and $\bar{\mathcal U}\in{\text{Alex\,}}(\kappa_0)$. Let $(p;x_1,x_2,x_3)$ be a quadruple in $\mathcal U$. If ${[\,px_i\,]}$, $i=1,2,3$, are all $\kappa$-geodesics and the comparison angles ${\tilde\measuredangle_{\kappa}\left({p}\,_{x_j}^{x_i}\right)}$ are all defined, then $$\begin{aligned} {\tilde\measuredangle_{\underline\kappa}\left({p}\,_{x_2}^{x_1}\right)} +{\tilde\measuredangle_{\underline\kappa}\left({p}\,_{x_3}^{x_2}\right)} +{\tilde\measuredangle_{\underline\kappa}\left({p}\,_{x_1}^{x_3}\right)}\le2\pi. \label{comp.X.cor.e1} \end{aligned}$$ Globalization in the metric completion ====================================== The following lemma, together with Corollary \[comp.X.cor1\] conclude the proof of Theorem A. \[pert.triangle\] Suppose that $\mathcal U$ is weakly convex and locally curvature bounded from below by $\kappa$. For any $p, x_i\in \bar{\mathcal U}$, $i=1,2,\dots,N$ and any $\epsilon>0$, there are points $\bar p\in B_\epsilon(p)$ and $\bar x_i\in B_\epsilon(x_i)$, $i=1,2,\dots, N$ so that geodesics ${[\,\bar p\bar x_i\,]}$ are all $\kappa$-geodesics. The idea has been used in the proof of Lemma \[conv.geod.pert\]. Let $\epsilon>\epsilon_1\gg\epsilon_2\gg\dots\gg\epsilon_N>0$ be small which will be determined later. First choose $p_1\in B_{\epsilon_1}(p)$ and $\bar x_1\in B_\epsilon(x_1)$ so that ${[\,p_1\bar x_1\,]}_{X}\subset \mathcal U$. Take $\bar p_1\in{[\,p_1\bar x_1\,]}$ with $|\bar p_1 p_1|=\epsilon_1$. Select $p_2\in B_{\epsilon_2}(\bar p_1)$ and $\bar x_2\in B_\epsilon(x_2)$ so that ${[\,p_2\bar x_2\,]}_{X}\subset \mathcal U$. By Lemma \[geod.close\], take $0<\epsilon_2\ll\epsilon_1$ so that ${[\,y\bar x_1\,]}$ are $\kappa$-geodesics for all $y\in B_{10\epsilon_2}(\bar p_1)$. In particular, ${[\,p_2\bar x_1\,]}_{X}$ is a $\kappa$-geodesic.  -0.2in Take $\bar p_2\in{[\,p_2\bar x_2\,]}$ with $|\bar p_2 p_2|=\epsilon_2$ and choose $p_3\in B_{\epsilon_3}(\bar p_2)$, $\bar x_3\in B_\epsilon(x_3)$ so that ${[\,p_3\bar x_3\,]}\subset \mathcal U$. Due to Lemma \[geod.close\], take $\epsilon_3\ll\epsilon_2$ small so that ${[\,y\bar x_2\,]}_{X}$ are $\kappa$-geodesics for all $y\in B_{10\epsilon_3}(\bar p_2)$. In particular, ${[\,p_3\bar x_2\,]}_{X}$ is a $\kappa$-geodesic. Take $\epsilon_3$ even smaller so that $\epsilon_3+2\epsilon_2<10\epsilon_2$. Thus $p_3\in B_{\epsilon_3}(\bar p_2)\subset B_{10\epsilon_2}(\bar p_1)$. Then we get ${[\,p_3\bar x_1\,]}$ is a $\kappa$-geodesic. Select $\bar p_3, p_4,\bar p_4, \dots,p_N$ recursively with $$|p_j\bar p_i| =\epsilon_j+2\epsilon_{j-1}+\dots+2\epsilon_{i+2}+2\epsilon_{i+1}<10\epsilon_i,$$ for any $j>i\ge 1$. Finally, we have that ${[\,p_N\bar x_i\,]}$ are $\kappa$-geodesics for all $0\le i\le N$. The point $\bar p=p_N$ is the desired point since $$|\bar pp| =|p_Np| \le |p_N\bar p_1|+2\epsilon_1 \le\epsilon_N+2\epsilon_{N-1}+\dots+2\epsilon_2+2\epsilon_1<\epsilon.$$ Assume $0<\lambda<1$. By Corollary \[comp.X.cor2\] and Lemma \[pert.triangle\] we get that $\bar{\mathcal U}\in{\text{Alex\,}}(\kappa_1)$, where $\kappa_1=\lambda^2\kappa+(1-\lambda^2)\kappa_0$. Repeat this argument recursively with $\kappa_{i-1}$ being replaced by $\kappa_i$ for $i=1,2,\dots$. We get $\bar{\mathcal U}\in{\text{Alex\,}}(\kappa_i)$, where $\kappa_{i+1}=\lambda^2\kappa+(1-\lambda^2)\kappa_i$. Clearly, $\kappa_i\to\kappa$ as $i\to \infty$. Thus $\bar{\mathcal U}\in{\text{Alex\,}}(\kappa)$. We will show that almost everywhere convexity implies weak $\mathfrak p_1$-convexity if $\mathcal U\in{\text{Alex\,}}_{loc}(\kappa)$ is finite dimensional. Let $p,q,s\in \mathcal U$. By the assumption, for any $r>0$, there is $p_1\in B_\epsilon(p)$, so that $\mathcal H^n\left(\mathcal U\setminus{{\mathcal U}^{*p_1}}\right)=0$. Choose $q_1\in B_\epsilon(q)$ and $\bar s\in B_{\epsilon/10}(s)$ so that ${[\,q_1\bar s\,]}\subset\mathcal U$. Take $s_0\in{[\,q_1\bar s\,]}$ such that $|s_0\bar s|=\epsilon/10$. Let ${\epsilon_1}>0$ be small so that $B_{\epsilon_1}(s_0)\subset B_\epsilon(s)$. Suppose that there is $\delta>0$ such that for every $s_1\in B_{\epsilon_1}(s_0)$ and every geodesic ${[\,q_1s_1\,]}$, $$\begin{aligned} \mathcal H^1\left({[\,\bar q_1s_1\,]}\setminus{{{[\,q_1s_1\,]}}^{*p_1}}\right)>0, \label{cor.ThmA.e1} \end{aligned}$$ where $\bar q_1\in{[\,q_1s_1\,]}$ such that $|q_1\bar q_1|=\delta$. The weak $\mathfrak p_1$-convexity follows if this is not true. By Lemma \[thin.comp.r\] and Alexandrov’s lemma, we can choose ${\epsilon_1}$ small so that for any $u\in B_{\epsilon_1}(s_0)$, $x\in{[\,q_1s_0\,]}$ and $y\in{[\,q_1u\,]}$, the comparison angle ${\tilde\measuredangle_{\kappa}\left({q_1}\,_{y}^{x}\right)}$ is decreasing in $|q_1x|$ and $|q_1y|$. Thus we have $$\begin{aligned} \frac{|xy|}{|us_0|}\ge c(\kappa)\cdot\frac{\min\{|q_1x|, |q_1y|\}}{|q_1s_0|}. \label{cor.ThmA.e2} \end{aligned}$$ Let $$T=\left(\bigcup_{s_1\in B_{\epsilon_1}(s_0)}\left({[\, q_1s_1\,]}\setminus{{{[\,q_1s_1\,]}}^{*p_1}}\right)\right) \setminus B_\delta(q_1)$$ and $S=\{v\in\mathcal U: |q_1v|=|q_1s_0|-{\epsilon_1}\}$. For any $x\in T$, $x\neq q_1$, there is $u_x\in B_{\epsilon_1}(s_0)$ such that $x\in T\cap{[\,q_1u_x\,]}$. Let $\bar x\in{[\,q_1u_x\,]}$ such that $|q_1\bar x|=|q_1s_0|-{\epsilon_1}$. Then $\bar x\in S$. Define a map $\phi\colon T\to S\times\mathbb R^+; x\to (\bar x, |q_1x|)$. Due to (\[cor.ThmA.e1\]), we have $\mathcal H^1\left(\phi(T\cap{[\,q_1u_x\,]})\right)>0$. Suppose that $\dim(\mathcal U)=n$. By Fubini’s theorem, $\mathcal H^n(\phi(T))>0$. Because ${{\textsf d}\left(q_1,T\right)}\ge\delta$, by (\[cor.ThmA.e2\]), we see that $\phi$ is $c(\kappa,|q_1s_0|,\delta)$-co-Lipschitz. Therefore, $\mathcal H^n(T)>0$. On the other hand, $T\subset\mathcal U\setminus{{\mathcal U}^{*p_1}}$. Thus $\mathcal H^n(T)\le\mathcal H^n(\mathcal U\setminus{{\mathcal U}^{*p_1}})=0$, a contradiction. Examples ======== \[example.1\] Let $\mathcal U$ be an open metric ball with radius $r$ in the unit sphere. If $r\le\pi/2$, both $\mathcal U$ and $\bar{\mathcal U}$ are convex. Petrunin’s Theorem [@Pet13] applies to this case. When $\pi/2<r<\pi$, neither $\mathcal U$ or $\bar{\mathcal U}$ is convex. When $r=\pi$, $\mathcal U$ is a.e. convex and Theorem A applies. \[eg.rational\] For any $\delta\in(0,1)$, we construct a locally flat, $2$-dimensional length space $\mathcal U$, whose metric completion $\bar{\mathcal U}\in{\text{Alex\,}}(0)$, but $\mathcal H^2(\mathcal U)\le\delta\cdot \mathcal H^2(\bar{\mathcal U})$. Let $X=[0,1]\times[0,1]$ be the unit square equipped with the induced Euclidean metric. Let $\{\gamma_i\}_{i=1}^\infty$ be the collection of all segments in $X$ whose coordinates of end points are pairs of rational numbers. Let $\mathcal U=\bigcup_{i=1}^\infty B_{r_i}(\gamma_i)$, where $\sum_{i=1}^\infty r_i=\delta/4$. Then $\mathcal H^2(\mathcal U)\le\delta< 1=\mathcal H^2(X)$. Now we show that $\bar{\mathcal U}=X$. For any two points $p, q\in X$ and any $\epsilon>0$, there are rational points $\bar p=(x_1,y_1)\in B_\epsilon(p)$ and $\bar q=(x_2,y_2)\in B_\epsilon(q)$. Thus for some $i$, ${[\,\bar p\bar q\,]}=\gamma_i\subset \mathcal U$. We have $$\textsf{d}_{\bar{\mathcal U}}(p, q) \ge\textsf{d}_X(p, q) \ge L(\gamma_i)-2\epsilon =\textsf{d}_{\mathcal U}(\bar p, \bar q)-2\epsilon \ge \textsf{d}_{\bar{\mathcal U}}(p, q)-4\epsilon.$$ Therefore, $\bar{\mathcal U}$ is isometric to $X$. [A]{} S. Alexander, V. Kapovitch, A. Petrunin, *Alexandrov geometry*, a draft avaliable at . Y. Burago, M. Gromov, G. Perel’man, *A.D. Alexandrov spaces with curvature bounded below*, Uspekhi Mat. Nauk, **47:2** (1992), 3-51; translation in Russian Math. Surveys, **47:2** (1992), 1-58. M. Gromov, *Sign and geometric meaning of curvature*, Rendiconti del Seminario Matematico e Fisico di Milano, **Vol 61**, Issue 1, (1991), 9-123. K. Grove, B. Wilking, *A knot characterization and 1-connected nonnegatively curved 4-manifolds with circle symmetry*, Preprint. J. Harvey, C. Searle, *Orientation and symmetries of alexandrov spaces with applications in positive curvature*, Preprint. A. Petrunin, *A globalization for non-complete but geodesic spaces*, Preprint. [^1]: The author was partially supported by the research funds managed by Penn State University.

--- author: - | P. Imkeller and I. Pavlyukevich\ Institut für Mathematik\ Humboldt-Universität zu Berlin\ Unter den Linden 6\ 10099 Berlin\ Germany date: 28 November 2000 title: 'Stochastic Resonance in Two-State Markov Chains' --- In this paper we introduce a model which provides a new approach to the phenomenon of stochastic resonance. It is based on the study of the properties of the stationary distribution of the underlying stochastic process. We derive the formula for the spectral power amplification coefficient, study its asymptotic properties and dependence on parameters. Introduction {#introduction .unnumbered} ============= The notion of *Stochastic Resonance* appeared about twenty years ago in the works of Benzi et al. [@Benzi83] and Nicolis [@Nicolis82] in the context of an attempt to explain the phenomenon of ice ages. The modern methods of acquiring and interpreting climate records indicate at least seven major climate changes in the last 700,000 years. These changes occurred with the periodicity of about $100,000$ years and are characterized by a substantial variation of the average Earth’s temperature of about $10 K$. The effect can be explained with the help of a simple energy balance model (for an extended review on the subject see [@Imkeller00]). The Earth is considered as a point in space, and its temporally and spatially averaged temperature $X(t)$ satisfies the equation $$\label{en_bal_determ} \dot X(t) = -U'(X(t)) - Q\sin{(\frac{2\,\pi t}{T})},$$ where $U(X)$ is a double-well potential with minima at $278.6K$ and $288.6K$ and saddle point at $283.3K$ and wells of equal depth. The second term in (\[en\_bal\_determ\]) corresponds to a small variation of the solar constant of about $0.1 \%$ with a period of $T=100,000$ years due to the periodic change of the eccentricity of the Earth’s orbit caused by Jupiter. The influence of this term reflects itself in small periodic changes of the depths of the potential wells. In this setting, the left well is deeper during the time intervals $(kT, (k+\frac{1}{2})T)$, whereas the right one is deeper during the intervals $((k+\frac{1}{2})T, (k+1)T)$, $k=0,1,2,\dots$ The trajectories of the deterministic equation (\[en\_bal\_determ\]) have two metastable states given by the minima of the wells. Due to the smallness of the solar constant $Q$ no transition between these states is possible. In order to obtain such transitions Benzi et al. [@Benzi83] and Nicolis [@Nicolis82] suggested to add noise to the system which results in considering the stochastic differential equation $$\label{en_bal_stoch} \dot X^{{\varepsilon},T}(t) = -U'(X^{{\varepsilon},T}(t)) - Q\sin{(\frac{2\,\pi t}{T})}+\sqrt{{\varepsilon}}\,\dot W_t,$$ ${\varepsilon}>0$, $\dot W$ a white noise. Now one can observe the following effect. Fix all parameters of the system except ${\varepsilon}$ and consider the typical behaviour of the solutions of (\[en\_bal\_stoch\]) for different values of ${\varepsilon}$. If the noise intensity is very small, the trajectory only occasionally can escape from the minimum of the well in which it is staying, and one can hardly detect any periodicity in this motion. If the intensity is very large, the trajectory jumps rapidly but randomly between the two wells and therefore also lacks periodicity properties. An interesting effect appears when the noise level takes a certain value ${\varepsilon}_0$: the trajectory always tends to be near the minimum of the deepest well and consequently follows the deterministic periodic jump function which describes the location of the deepest well’s minimum. It is very important to note that to produce this effect one needs all three of the following components to be present in the system (\[en\_bal\_stoch\]): the double-well potential for bi-stability, the noise to pass the potential barriers, and a small periodic perturbation to change the wells’ depths. The following are natural questions arising in the context of these qualitative considerations: how can one measure *periodicity* of the trajectories and, consequently, how does the quality of *tuning* of the noisy output to the periodic input be improved by adjusting the noise intensity ${\varepsilon}$? The formulation of the latter question suggests to consider the system (\[en\_bal\_stoch\]) as a random amplifier. The random system receives the harmonic signal of small amplitude $Q$ and usually large period $T$ as input. The stochastic process $X^{{\varepsilon},T}(t)$ is observed as the output. The input signal carries power $Q^2$ at frequency $1/T$. The random output has continuous spectrum and thus carries power at all frequencies. Benzi et al. [@Benzi83] considered the power spectrum of the output for different values of ${\varepsilon}$ and discovered a sharp peak at the input frequency for a certain optimal value of ${\varepsilon}_0$. This means that the random process $X^{{\varepsilon}_0,T}(t)$ has a big component of frequency $1/T$. The effect of amplification of the power carried by the harmonic considered as a response of the nonlinear system (\[en\_bal\_stoch\]) to optimally chosen noise was called *stochastic resonance*. In the past twenty years more than three hundred papers on this subject were published. An extensive description of the phenomenon from the physical point of view can be found in [@Grammaitoni98] and [@Anishchenko99]. The notion *stochastic resonance* is now used in a much broader sense. It describes a wide class of effects with the common underlying property: the presence of noise induces a qualitatively new behaviour of the system and improves some of its characteristics. Although stochastic resonance was observed and studied in many physical systems, only few mathematically rigorous results are known. The approach of M. Freidlin is briefly outlined in the next section of this paper. In sections \[st\_distr\], \[spa\] and \[extrema\] we introduce discrete-time Markov chains with transition probabilities chosen in such a way, that on a large temporal scale the *attractor hopping* behaviour of the underlying diffusion process is imitated in the limit ${\varepsilon}\to 0$. We investigate stochastic resonance for the Markov chains. The last section is devoted to generalizations and discussion. Large deviations approach {#freidlin} ========================= In this section we briefly survey rigorous mathematical results obtained by M. Freidlin in [@Freidlin00] using the theory of large deviations for randomly perturbed dynamical systems, developed in Freidlin and Wentzell (see [@WentzelFreidlin84]). Though the results of [@Freidlin00] are valid in a quite general framework, we confine our attention to a simple example of a diffusion with weak noise. Consider the SDE in $\mathbb R$ $$\label{sde} \dot X^{{\varepsilon},T}(t)= -U'(X^{{\varepsilon},T}(t), \frac{t}{T})+ \sqrt{{\varepsilon}}\dot W(t),$$ where $\dot W$ is a white noise and $U'(x,t)=\frac{\partial }{\partial x}U(x,t),$ with a time dependent potential just periodically switching between two symmetric double well states, i.e. $$U(x,t)=\sum\limits_{k\geq 0}U(x){\bf 1}_{[k,k+\frac{1}{2})}(t) + U(-x){\bf 1}_{[k+\frac{1}{2},k+1)}(t),$$ where $U(x)$ has local minima in $x=\pm 1$ and a saddle point in $x=0$, $\lim_{|x|\to\infty } U(x)=\infty $. We also fix the depths of the wells by two numbers $0<v<V,$ assuming that $U(-1)=-V/2$, $U(1)=-v/2$, and $U(0)=0$. Note, that $X^{{\varepsilon},T}$ is a Markov process which is not time homogeneous. In the following Theorem time scales are determined in which some form of periodicity is observed. Suppose $T=T({\varepsilon})$ is given such that $$\lim\limits_{{\varepsilon}\to 0}{\varepsilon}\ln{T({\varepsilon})}=\lambda >0.$$ a\) If $\lambda <v,$ then the Lebesgue measure of the set $$\{t \in [0,1] \,:\, |X^{{\varepsilon},T({\varepsilon})} (T({\varepsilon}) t)- \mbox{\rm sgn}{X_0}|>\delta \}$$ converges to 0 in $P_{X_0}$ probability as ${\varepsilon}\to 0$, for any $\delta >0$. b\) If $\lambda >v,$ then the Lebesgue measure of the set $$\{t \in [0,1] \,:\, |X^{{\varepsilon},T({\varepsilon})} (T({\varepsilon}) t)- \ophi(t)|>\delta \}$$ converges to 0 in $P_{X_0}$ probability as ${\varepsilon}\to 0$, for any $\delta >0$, where $$\ophi (t)=\sum\limits_{k\geq 0}- {\bf 1}_{[k,k+\frac{1}{2})}(t) + {\bf 1}_{[k+\frac{1}{2},k+1)}(t)$$ and $P_{X_0}$ denotes the law of the diffusion starting in $X_0$. $\square$ It is nessesary to explain why $\lambda = v$ is critical for the long time behaviour of the diffusion. At least intuitively, the answer follows from the asymptotics of the mean exit time from a potential well for the time-homogeneous diffusion. If the diffusion starts in the potential well with the depth $v/2$, its mean time ${\bf E}(\tau({\varepsilon})) $ needed to leave the well satisfies $${\varepsilon}\ln{{\bf E}(\tau({\varepsilon})) }\to v, \quad {\varepsilon}\to 0.$$ according to Freidlin and Wentzell [@WentzelFreidlin84]. This means, $X^{{\varepsilon},T({\varepsilon})}$ can leave neither the deep well with the depth $V/2$ nor the shallow one with the depth $v/2$ in time $T({\varepsilon})$ of order $e^{\lambda /{\varepsilon}}$ if $\lambda < v $. Therefore, $X^{{\varepsilon},T({\varepsilon})}$ stays in the $\delta $-neighbourhood of the minimum of the initial well. On the other hand, if $\lambda >v,$ $X^{{\varepsilon},T({\varepsilon})}$ has always enough time to reach the deepest well. In both cases, the Lebesgue measure of excursions leaving the $\delta $-tube of the deterministic periodic function $\ophi$ is exponentially negligible on the time scale $T({\varepsilon})$ as ${\varepsilon}\to 0$. The Theorem suggests the time scale which induces periodic and deterministic behaviour of the system (\[sde\]), and the Lebesgue measure as a measure of quality. In fact, it only gives a lower bound for the scale. In the next section, in the framework of discrete Markov chains approximating the diffusion processes just considered, we investigate different measures of quality which provide unique optimal tuning. Markov chains with time-periodic transition probabilities {#st_distr} ========================================================= For $m\in \mathbb N$, consider a Markov chain $X_m=(X_m(k))_{k\geq 0}$ on the state space $\mathcal S=\{-1,1\}$. Let $P_m(k)$ be the matrix of one-step transition probabilities at time $k$. If we denote $\pi^{-}_m(k)={\bf P}(X_m(k)=-1)$, $\pi^{+}_m(k)={\bf P}(X_m(k)=1)$, and write $P^*$ for the transposed matrix, we have $$\left( \begin{array}{c} \pi^{-}_m(k+1)\\ \pi^{+}_m(k+1) \end{array} \right) =P_m^*(k) \left( \begin{array}{c} \pi^{-}_m(k)\\ \pi^{+}_m(k) \end{array} \right).$$ In order to model the periodic switching of the double-well potential in our Markov chains, we define the transition matrix $P_m$ to be periodic in time with half-period $m$. More precisely, $$P_{m}(k)= \left\{ \begin{array}{ll} P_1, &\quad 0\leq k(\mbox{mod } 2m)\leq m-1, \\ P_2, &\quad m\leq k(\mbox{mod } 2m)\leq 2m-1, \end{array} \right.$$ with $$\label{P1P2} \begin{array}{c} P_1= \left( \begin{array}{cc} 1-{\varphi}& {\varphi}\\ \psi & 1-\psi \end{array} \right) = \left( \begin{array}{cc} 1-px^V& px^V\\ qx^v & 1-qx^v \end{array} \right), \\ P_2= \left( \begin{array}{cc} 1-\psi & \psi\\ {\varphi}& 1-{\varphi}\end{array} \right) = \left( \begin{array}{cc} 1-qx^v& qx^v\\ px^V & 1-px^V \end{array} \right). \end{array}$$ where ${\varphi}={\varphi}({\varepsilon},p,V)=pe^{-V/{\varepsilon}}$, $\psi=\psi({\varepsilon},q,v)=qe^{-v/{\varepsilon}}$, $x=e^{-1/{\varepsilon}}$, $0\leq p,q\leq 1$, $0<v<V<+\infty$, $0<{\varepsilon}<+\infty$. Sometimes, it will be convenient to consider $x\in [0,1]$. In these cases the ends of the interval will correspond to the limits ${\varepsilon}\to 0$ and ${\varepsilon}\to \infty $. In this setting, the numbers $V/2$ and $v/2$ clearly have to be associated with the depths of the potential wells, ${\varepsilon}$ with the level of noise. According to the Freidlin-Wentzel theory, the exponential factors in the one-step transition probabilities just correspond to the inverses of the expected transition times between the respective wells for the diffusion considered in the preceding section. This is what should be expected for a Markov chain *in equilibrium*, modulo the phenomenological *pre-factors* $p$ and $q$. They model the pre-factors appearing in large deviation statements, and add asymmetry to the picture. It is well known that for a time-homogeneous Markov chain on $S$ with transition matrix $P$ one can talk about *equilibrium*, given by the stationary distribution, to which the law of the chain converges exponentially fast. The stationary distribution can be found by solving the matrix equation $\pi =P^*\pi $ with normalizing condition $\pi^{-}+\pi^{+}=1$. For non time homogeneous Markov chains with time periodic transition matrix, the situation is quite similar. Enlarging the state space $S$ to $S_m = \{-1,1\}\times \{0,1,\dots , 2m-1\},$ we recover a time homogeneous chain by setting $$Y_m(k)=(X_m(k), k (\mbox{mod }2m)), \quad k\geq 0,$$ to which the previous remarks apply. For convenience of notation, we assume $S_m$ to be ordered in the following way:\ $\mathcal S_m=[(-1,0), (1,0), (-1,1),(1,1), \dots ,(-1,2m-1), (1, 2m-1)]$. Writing ${\rm A}_m$ for the matrix of one-step transition probabilities of $Y_m$, the stationary distribution $Q=(q(i,j))^*$ is obtained as a normalized solution of the matrix equation $({\rm A}_m^*-E)Q=0,$ $E$ being the unit matrix. We shall be dealing with the following variant of stationary measure, which is not normalized in time. Let $\pi _m(k)=(\pi^-_m(k), \pi^+_m(k))^*=2m(q(-1,k),q(1,k))^*$, $0 \leq k\leq 2m-1$. We call the set $\pi_m =(\pi _m(k))_{0\leq k\leq 2m-1}$ the stationary distribution of the Markov chain $X_m$. The matrix ${\rm A}_m$ of one-step transition probabilities of $Y_m$ is explicitly given by $${\rm A}_m=\left( \begin{array}{cccccccc} 0 & P_1 & 0 & 0 & \cdots &0 & 0 & 0\\ 0 & 0 & P_1 & 0 & \cdots &0 & 0 & 0\\ \vdots & & & & & & &\\ 0 & 0 & 0 & 0 & \cdots &0 & P_2 & 0\\ 0 & 0 & 0 & 0 & \cdots &0 & 0 & P_2 \\ P_2 & 0 & 0 & 0 & \cdots &0 & 0 & 0 \end{array} \right).$$ ${\rm A}_m$ has block structure. In this notation 0 means a $2\times 2$-matrix with all entries equal to zero, $P_1$, and $P_2$ are the 2-dimensional matrices defined in (\[P1P2\]). Applying some algebra we see that $({\rm A}_m^*-E) Q = 0$ is equivalent to $A_m'\, Q = 0,$ where $${\rm A}_m'=\left( \begin{array}{cccccccc} \widehat P-E & 0 & 0 &0 & \cdots &0 & 0 & 0\\ P_1^* & -E & 0 &0 & \cdots & 0 & 0 & 0\\ \vdots & & & & & & & \\ 0 & 0 & 0 & 0 & \cdots & -E & 0 & 0\\ 0 & 0 & 0 & 0 & \cdots & P_2^* & -E & 0\\ 0 & 0 & 0 & 0 & \cdots &0 & P_2^* & -E \end{array} \right)$$ and $\widehat P=P_2^*P_2^*\cdots P_1^*=(P_2^*)^m(P_1^*)^m$. But ${\rm A}_m'$ is a block-wise lower diagonal matrix, and so $A_m' Q = 0$ can be solved in the usual way to give For every $m\geq 1$, the stationary distribution $\pi _m$ of $X_m$ with matrices of one-step probabilities defined in (\[P1P2\]) is: $$\label{pi} \begin{array}{l} \left\{ \begin{array}{l} \pi_m^-(l)= \displaystyle\frac{\psi }{{\varphi}+\psi }+ \displaystyle\frac{{\varphi}-\psi }{{\varphi}+\psi}\, \frac{(1-{\varphi}-\psi )^l}{1+(1-{\varphi}-\psi )^m},\\ \pi_m^+(l)= \displaystyle\frac{{\varphi}}{{\varphi}+\psi }- \displaystyle\frac{{\varphi}-\psi}{{\varphi}+\psi}\, \frac{(1-{\varphi}-\psi )^l}{1+(1-{\varphi}-\psi )^m}; \end{array} \right. \\ \left\{ \begin{array}{l} \pi_m^-(l+m)=\pi _{m}^+(l),\\ \pi_{m}^+(l+m)=\pi_{m}^-(l),\qquad 0\leq l\leq m-1. \end{array} \right. \end{array}$$ **Proof:** $\pi _{m}(0)$ satisfies the matrix equation $((P_2^*)^m(P_1^*)^m-E)\pi_{m}(0)=0$ with additional condition $\pi^-_{m}(0)+\pi^+_{m}(0)=1$. To calculate $(P_2^*)^m (P_1^*)^m$, we use a formula for the $m$-th power of $2\times 2$-matrices, which results in $$\begin{array}{l} \left( \begin{array}{cc} p_{-1,-1}&p_{-1,1}\\ p_{1,-1}&p_{1,1} \end{array} \right)^m {}= \displaystyle\frac{1}{2-p_{-1,-1}-p_{1,1}} \left( \begin{array}{cc} 1-p_{1,1}&1-p_{-1,-1}\\ 1-p_{1,1}&1-p_{-1,-1} \end{array} \right)\\ {}+\displaystyle \frac{(p_{-1,-1}+p_{1,1}-1)^m}{2-p_{-1,-1}-p_{1,1}} \left( \begin{array}{cc} 1-p_{-1,-1}&-(1-p_{-1,-1})\\ -(1-p_{1,1})&1-p_{1,1} \end{array} \right) \end{array}$$ Using some more elementary algebra we find $$\begin{array}{ll} (P_2^*)^m(P_1^*)^m&=(P_1^mP_2^m)^*= \left( \begin{array}{cc} 1-\psi & \psi\\ {\varphi}& 1-{\varphi}\end{array} \right)^m \left( \begin{array}{cc} 1-{\varphi}& {\varphi}\\ \psi & 1-\psi \end{array} \right)^m \\ &=\displaystyle \frac{1}{{\varphi}+\psi} \left( \begin{array}{cc} {\varphi}& {\varphi}\\ \psi & \psi \end{array} \right) + (1-{\varphi}-\psi)^m\frac{{\varphi}-\psi}{{\varphi}+\psi} \left( \begin{array}{cc} -1 & -1\\ 1 & 1 \end{array} \right) \\ &+\displaystyle \frac{(1-{\varphi}-\psi)^{2m}}{{\varphi}+\psi} \left( \begin{array}{cc} {\varphi}& -\psi\\ -{\varphi}& \psi \end{array} \right), \end{array}$$ from which a straightforward calculation yields $$\left\{ \begin{array}{l} \pi_{m}^-(0)= \displaystyle\frac{{\varphi}+ \psi (1-{\varphi}-\psi )^m}{({\varphi}+\psi ) (1+(1-{\varphi}-\psi )^m)},\\ \pi_{m}^+(0)= \displaystyle\frac{\psi +{\varphi}(1-{\varphi}-\psi )^m}{({\varphi}+\psi ) (1+(1-{\varphi}-\psi )^m)}. \end{array} \right.$$ To compute the remaining entries, we use $\pi _m(l)=(P_1^*)^l\pi_m(0)$ for $0\leq l\leq m-1$, and $\pi _m(l)=(P_2^*)^l(P_1^*)^m\pi_m(0)$ for $m\leq l\leq 2m-1$ to obtain (\[pi\]). Note also the symmetry $\pi_{m}^-(l+m)=\pi_{m}^+(l)$ and $\pi_{m}^+(l+m)=\pi_{m}^-(l)$, $0\leq l\leq m-1$. $\square$ Spectral power amplification {#spa} ============================ The chain $X_{m}$ can be interpreted as amplifier of a signal. Our stochastic system may be seen to receive a deterministic periodic input signal which switches the double depths of the potential wells in (\[P1P2\]), i.e. $$I_{m}(l)=\left\{ \begin{array}{ll} V, &\quad 0\leq l(\mbox{mod }2m)\leq m-1,\\ v, &\quad m\leq l(\mbox{mod }2m)\leq 2m-1. \end{array} \right.$$ The output is a random process $X_{m}(k)$. The input signal $I_{m}$ admits a spectral representation $$I_{m}(k)=\frac{1}{2m}\sum_{a=0}^{2m-1}c_{m}(a) e^{-\frac{2\pi i k}{2m}a},$$ where $c_{m}(a)=(1/2m)\sum_{l=0}^{2m-1}I_{2}(l)e^{\frac{2\pi i a}{2m}l}$ is the Fourier coefficient of frequency $a/2m$. The quantity $|c_{2m}(a)|^2$ measures the power carried by this Fourier component. We are only interested in the component of the input frequency $1/2m$. Its power is given by $$\label{c2} |c_{m}(1)|^2= \frac{(V-v)^2}{4m^2}\csc^2{(\frac{\pi }{2m})}.$$ In the stationary regime, i.e. if the law of $X_m$ is given by the measure $\pi_m$, the power carried by the output at frequency $a/2m$ is a random variable $$\xi _{m}(a)=\frac{1}{2m}\sum_{l=0}^{2m-1}X_{m}(l)e^{\frac{2\pi i a}{2m}l}.$$ We define the *spectral power amplification* as the relative expected power carried by the component of the output with frequency $\frac{1}{2m}$. The *spectral power amplification coefficient* of the Markov chain $X_{m}$ with half period $m\geq 1$ is given by $$\eta_{m} = \frac{|{\bf E}_{\pi _{m}}(\xi _{m}(1))|^2} {|c_{m}(1)|^2}.$$ Here ${\bf E}_{\pi _{m}}$ denotes expectation w.r.t. the stationary distribution $\pi _{m}$. The explicit description of the invariant measure now readily yields the following formula for the spectral power amplification. Let $m\ge 1$. The spectral power amplification coefficient of the Markov chain $X_{m}$ with one-step transition probabilities  equals $$\eta_{m}= \frac{4}{(V-v)^2}\cdot \frac{({\varphi}-\psi)^2} {({\varphi}+\psi)^2 + 4(1-{\varphi}-\psi)\sin^2{(\frac{\pi}{2m})}}.$$ **Proof:** Using one immediately gets $$\begin{aligned} {\bf E}_{\pi_{m}}\xi _{m}(1) &=& \frac{1}{2m} \sum_{k=0}^{2m-1}{\bf E}_{\pi_{m}}X_{m}(k)e^{\frac{2\pi i}{2m}k}= \frac{1-e^{\pi i}}{2m} \sum_{k=0}^{m-1}(\pi^+_{m}(k)-\pi^-_{m}(k)) e^{\frac{2\pi i }{2m}k} \\ &=& \frac{2}{m}\frac{{\varphi}-\psi }{{\varphi}+ \psi } \left( \frac{1}{1-e^{\frac{\pi i}{m}}}- \frac{1}{1-(1-{\varphi}-\psi )e^{\frac{\pi i}{m}}} \right).\end{aligned}$$ Some algebra and an appeal to finish the proof. $\square$ Recall now that the one-step probabilities $P_1$ and $P_2$ depend on the parameters $0\leq p,q\leq 1$ and, what is especially important, on $0<{\varepsilon}<\infty $ which is interpreted as noise level. Our next goal is to *tune* the parameter ${\varepsilon}$ to a value which maximizes the amplification coefficient $\eta_m=\eta_m({\varepsilon}) $ as a function of ${\varepsilon}$. Extrema and zeros of $\eta_{m}({\varepsilon}) $. {#extrema} ================================================= In this section we study some features of the function $\eta_{m}({\varepsilon})$ and its dependence on $m\in \mathbb N$, $0<v<V<\infty$ and the pre-factors $0\leq p\leq 1$, $0\leq q\leq 1$. After substituting $e^{-1/{\varepsilon}}=x$ and writing $\eta _{m}({\varepsilon})=\eta _{m}(x),$ this function takes the form $$\begin{aligned} \label{e7} \eta_{m} (x)= \frac{4}{(V-v)^2} \frac{(px^V-qx^v)^2}{(px^V+qx^v)^2 + 4(1-px^V-qx^v)\sin^2{(\frac{\pi}{2m})}}\end{aligned}$$ In what follows, we assume $x\in [0,1]$. The boundaries $x=0$ and $x=1$ correspond to the limiting cases ${\varepsilon}= 0$ and ${\varepsilon}= \infty $. Denote $a_{m}=\csc{(\frac{\pi}{2m})}^2\geq 1$, $m\geq 1$. Our main result on optimal tuning is contained in the following theorem. a\) We have $\eta_{m} (x)\geq 0$, $\eta _{m}(0)=0$. b\) Let $0<\beta =\frac{v}{V}<1$ and $m\geq 1$ be fixed. There exists a continuous function $$p_-(q)=p_-(q; \beta ,m)=\displaystyle \frac{b(q; m,\beta )- \sqrt{b(q; m, \beta )^2-4a(q; m,\beta )(2-q)q}}{2a(q;m,\beta )},$$ where $a(q; m,\beta )= 1-a_{m}q(1-\beta )$, $b(q;m,\beta )=2-3(1-\beta )q +a_m(1-\beta )q^2$, with following properties: i\) $p_-(q)\geq 0$, $q\in [0,1]$$p_-(q)=0 \Leftrightarrow q=0$; ii\) $p_-(q)\leq q$, $q\in [0,1]$$p_-(q)=q \Leftrightarrow q=0$ or $q=1$, $m=1$; iii\) $\displaystyle \left. \frac{dp_-(q;m,\beta )}{dq}\right|_{q=0} =\beta $. Moreover for $m\geq 2$ 1\) If $(p,q)\in U_0=\{(p,q):0< q\leq 1, 0\leq p< p_-(q)\}$, $\eta_{m} (x)$ is strictly increasing on $[0,1]$. 2\) If $(p,q)\in U_1=\{(p,q):0< q\leq 1, p_-(q)< p\leq q\}$, $\eta_{m} (x)$ has a unique local maximum on $[0,1]$. 3\) If $(p,q)\in U_2=\{(p,q):0< q\leq 1, q\leq p\leq 1\}$, $\eta_{m} (x)$ has a unique local maximum on $[0,1]$ and a unique root on $(0,1]$. (See [Fig. ]{}) c)For any $\delta >0$ there exists $M_0=M_0(p,q,\beta ,\delta )$ such that for $m>M_0$ the coordinate of the local maximum $\widehat x_{m}\in[x_{m}(1-\delta ), x_{m}]$, where $$x_{m}=\left(\frac{\pi ^2}{2m^2pq}\frac{v}{V-v}\right)^{\frac{1}{V+v}}$$ **Proof:** Differentiate the explicit formula with respect to $x$ to determine the critical points and sets $U_0$, $ U_1$, $U_2$. The calculation of the resonance point in $U_1$, $U_2$ requires to find two points in some neighborhood such that the derivative is strictly monotone on the interval between them, and has different signs at the extremities. $\square$ **Remarks:**\ 1. The optimal tuning rule can be rewritten in the form $$m({\varepsilon})\cong \frac{\pi }{\sqrt{2pq}} \sqrt{\frac{v}{V-v}}e^{\frac{V+v}{2{\varepsilon}}}.$$ The maximal value of amplification is found as $$\lim\limits_{{\varepsilon}\to 0} \eta_{[m({\varepsilon})]}({\varepsilon}) = \frac{4}{(V-v)^2}.$$ 2\. We also see that the spectral power amplification as a measure of quality of stochastic resonance allows to distinguish a unique time scale, find its exponential rate ($\lambda =(V+v)/2$) together with the pre-exponential factor. [7]{} R. Benzi, G. Parisi, A. Sutera, A. Vulpiani, *A theory of stochastic resonance in climatic change*, SIAM J. Appl. Math., 43 (1983), 565–578. C. Nicolis, *Stochastic aspects of climatic transitions — responses to periodic forcing*, Tellus, 34 (1982), 1–9. P. Imkeller, *Energy balance models — viewed from stochastic dynamics*. P. Imkeller, J.-S. von Storch (eds.) *Stochastic Climate Models.* Birkhäuser: Basel, Boston 2001. L. Grammaitoni, P. Hänggi, P. Jung, F. Marchesoni, *Stochastic resonance*, Reviews of Modern Physics, 70 (1998), 223–287. V. S. Anishchenko, A. B. Neiman, F. Moss, L. Schimansky-Geier, *Stochastic resonance — noise induced order*, Physics – Uspekhi, 42 (1999), 7–36, (Uspekhi Fizicheskikh Nauk 169 (1999), 7–38, in Russian) M. Freidlin, *Quasi-deterministic approximation, metastability and stochastic resonance*, Physica D 137 (2000), 333–352 M. Freidlin, A. Wentzel, *Random perturbations of dynamical systems*, 2nd ed., Springer, Berlin,1998.

--- abstract: 'It is known that if a compact metric space $X$ admits a minimal expansive homeomorphism then $X$ is totally disconnected. In this note we give a short proof of this result and we analyze its extension to expansive flows.' author: - Alfonso Artigue title: Minimal expansive systems and spiral points --- Introduction ============ In [@Ma] Mañé proved that if a compact metric space $(X,\operatorname{dist})$ admits a minimal expansive homeomorphism then $X$ is *totally disconnected*, i.e., every connected subset of $X$ is a singleton. A homeomorphism $f\colon X\to X$ is *expansive* if there is ${\eta}>0$ such that if $\operatorname{dist}(f^n(x),f^n(y))<{\eta}$ for all $n\in{\mathbb Z}$ then $y=x$. We say that $f$ is *minimal* if the set $\{f^n(x):n\geq 0\}$ is dense in $X$ for all $x\in X$. In Theorem \[teoMaKaMin\] we give a short proof of Mañé’s result. Some definitions, results and proofs on discrete-time dynamical systems are trivially extended to flows, i.e., real actions $\phi\colon{\mathbb R}\times X\to X$. But, this is not the case for the problem considered in this note. In the context of flows we say that $\phi$ is *expansive* [@BW] if for all ${\varepsilon}>0$ there is ${\eta}>0$ such that if $\operatorname{dist}(\phi_t(x),\phi_{h(t)}(y))<{\eta}$ for some increasing homeomorphism $h\colon{\mathbb R}\to{\mathbb R}$, $h(0)=0$, then $y=\phi_s(x)$ for some $s\in (-{\varepsilon},{\varepsilon})$. In the case of expansive flows, Mañé’s proof was partially extended by Keynes and Sears [@KS]. They proved that if $X$ admits a minimal expansive flow *without spiral points* (see Section \[secExpFlow\]) then the topological dimension of $X$ is 1, i.e., local cross sections are totally disconnected. Our proof of Theorem \[teoMaKaMin\] avoids the problem of spiral points in the discrete-time case. However, its extension to flows is far from trivial, at least for the author. In the final section some remarks are given in relation with spiral points for homeomorphisms and flows. Minimal homeomorphisms ====================== Let $(X,\operatorname{dist})$ be a compact metric space and assume that $f\colon X\to X$ is a homeomorphism. Let us start proving a general [result not]{} involving expansivity. Recall that a *continuum* is a compact connected set. A continuum is non-trivial if it is not a singleton. We denote by $\dim(X)$ the topological dimension of $X$ [@HW]. For the next result we need to know that $\dim(X)>0$ if and only if $X$ has a non-trivial connected component. \[rmkSubCont\] [For a continuum $C$ it is known that the space of its subcontinua with the Hausdorff metric is arc connected, see [@IN]. Therefore, since the diameter function is continuous, given $0<\delta<\operatorname{diam}(C)$ there is a subcontinuum $C'\subset C$ with $\operatorname{diam}(C')=\delta$.]{} \[propContEst\] If $\dim(X)>0$ then for all $\delta>0$ there is a non-trivial continuum $C\subset X$ such that $\operatorname{diam}(f^n(C))\leq\delta$ for all $n\geq 0$ or for all $n\leq 0$. By contradiction assume that there are $\delta>0$, a sequence of continua $C_n$ and positive integers $k_n$ such that $\operatorname{diam}(C_n)\to 0$ and $\operatorname{diam}(f^{k_n}(C_n))>\delta$. Considering subcontinua and different times $k_n$, we can also assume that $\operatorname{diam}(f^{k_n}(C_n))=\delta$ [(Remark \[rmkSubCont\])]{} and $\operatorname{diam}(f^i(C_n))\leq\delta$ if $0\leq i\leq k_n$. Since $\operatorname{diam}(C_n)\to 0$ we have that $k_n\to+\infty$. Take a limit continuum $C$ of [$f^{k_n}(C_n)$]{} in the Hausdorff metric. By the continuity of $f$ we have that $\operatorname{diam}(f^i(C))\leq\delta$ for all $i\leq 0$. This proves the result. It can be the case that for every non-trivial continuum [$C\neq X$]{} it holds that: $\lim_{n\to+\infty}\operatorname{diam}(f^n(C))=\operatorname{diam}(X)$ and $\lim_{n\to-\infty}\operatorname{diam}(f^n(C))=0$. This is the case on an expanding attractor as for example the solenoid and the nonwandering set of a derived from Anosov. In [@Ka93] Kato introduced a generalization of expansivity that allowed him to extend several results of expansive homeomorphisms, including those obtained in [@Ma]. We recall that $f$ is *cw-expansive* [(*continuum-wise expansive*)]{}, if there is ${\eta}>0$ such that if $C\subset X$ is connected and $\operatorname{diam}(f^n(C))<{\eta}$ for all $n\in{\mathbb Z}$ then $C$ is a singleton. In this case we say that ${\eta}$ is a *cw-expansive constant* for $f$. We say that $C\subset X$ is ${\eta}$-*stable* if $\operatorname{diam}(f^i(C))\leq{\eta}$ for all $i\geq 0$. [[@Ma; @Ka93]]{} \[teoMaKaMin\] If $f\colon X\to X$ is a minimal homeomorphism of a compact metric space $(X,\operatorname{dist})$ and $\dim(X)>0$ then for all ${\eta}>0$ there is a non-trivial continuum $C\subset X$ such that $\operatorname{diam}(f^n(C))\leq{\eta}$ for all $n\in{\mathbb Z}$. Arguing by contradiction assume that $f$ is minimal, cw-expansive and $\dim(X)>0$. Consider a cw-expansive constant ${\eta}>0$. It is known [@Ka93] that in this case there is $m>0$ such that if $C$ is a ${\eta}$-stable continuum and $\operatorname{diam}(C)={\eta}/3$ then $\operatorname{diam}(f^{-m}(C))>{\eta}$. The proof is similar to the one of Proposition \[propContEst\]. Let $Y\subset X$ be a minimal set for $g=f^m$. In this paragraph we will show that $\dim(Y)=0$. Take an open set $U\subset Y$ such that $\operatorname{diam}(U)<{\eta}/3$. By contradiction assume that $\dim(Y)>0$. By Proposition \[propContEst\] there is a ${\eta}$-stable continuum $C_0\subset Y$ for $g$ (the case of $g^{-1}$ is analogous). Taking a negative iterate of $C_0$ we can assume that $\operatorname{diam}(C_0)\geq{\eta}/3$. Since $\operatorname{diam}(g^{-1}(C_0))>{\eta}$ and $\operatorname{diam}(U)<{\eta}/3$ there is a component $C_1$ of $g^{-1}(C_0)\setminus U$ with $\operatorname{diam}(C_1)\geq {\eta}/3$. In this way we construct a sequence of continua $C_k$ such that $\operatorname{diam}(C_k)\geq{\eta}/3$, $C_{k+1}\subset g^{-1}(C_k)$ and $C_k\cap U=\emptyset$ for all $k\geq 0$. Take $x\in \cap_{k\geq 0}g^{k}(C_k)$. Then $g^{-k}(x)\notin U$ for all $k\geq 0$. This contradicts that $Y$ is minimal for $g$ and proves that $\dim(Y)=0$. Now, since $f\colon X\to X$ is minimal, we have that $X=\cup_{i=1}^mf^i(Y)$ (disjoint union). Since $\dim(Y)=0$ we conclude that $\dim(X)=0$ and the proof ends. Let us illustrate the possible behavior of the diameter of the iterates of a continuum $C$ as in the previous theorem. If $f$ is an irrational rotation of a circle then $\operatorname{diam}(f^n(C))$ is constant (for a suitable metric) for all $n\in{\mathbb Z}$. In [@BKS] a minimal set on the two-dimensional torus is given containing a circle $C$ such that $\operatorname{diam}(f^n(C))\to 0$ as $n\to\pm\infty$. In [@F] Floyd gave an example of a compact subset $X\subset {\mathbb R}^2$ and a minimal homeomorphism $f\colon X\to X$. Each connected component of $X$ is an [interval, some]{} of them trivial, i.e., [singletons]{}. Therefore, $X$ is a nonhomogeneous space of positive topological dimension. This example also shows that on a minimal set there can be points not belonging to a non-trivial stable continuum (the trivial components). Assuming that $X$ is a Peano continuum and $f\colon X\to X$ is a minimal homeomorphism, is it true that every point belongs to a stable continuum? We do not know if the proof of Theorem \[teoMaKaMin\] can be *translated* to expansive flows. Another way to translate the result is solving the spiral point problem indicated in the next section. Spiral points {#secExpFlow} ============= In this section we wish to discuss the problem of *translating* Mañé’s proof for minimal expansive flows. As we said, it is related [to]{} spiral points. We start giving some remarks for spiral points of homeomorphisms, next we consider the corresponding concept for flows. Let $f\colon X\to X$ be a homeomorphism of a metric space $(X,\operatorname{dist})$. We say that $x\in X$ is a *spiral point* if there is $m\neq 0$ such that $$\lim_{n\to +\infty}\operatorname{dist}(f^n(x),f^{n+m}(x))=0.$$ The proofs by Mañé and Kato of Theorem \[teoMaKaMin\] use a simple lemma, [@Ma]\*[Lemma II]{} and [@Ka93]\*[Lemma 5.3]{}. This lemma [says]{} that if $x$ is a spiral point then $f$ has a periodic point. Therefore, non-trivial minimal sets have no spiral points. Let us give some details on this result. If a homeomorphism of a compact metric space $f\colon X\to X$ has a spiral point $x\in X$ then $\omega(x)$ is a finite union of continua $C_1,\dots,C_m$ such that $f(C_i)=C_{i+1}$ for $i=1,\dots,m-1$, $f(C_m)=C_1$ and $f^m\colon C_i\to C_i$ is the identity. Let $C_1$ be the $\omega$-limit set of $x$ by $f^m$. Since $x$ is a spiral point the continuity of $f$ implies that $f^m(y)=y$ for all $y\in C_1$. The sets $C_i$ have to be defined as $C_i=f^{i-1}(C_1)$. It only [remains]{} to prove that $C_1$ is connected. By contradiction assume that $C_1$ is a disjoint union of two non-empty compact sets $D,E\subset X$. Define $r=\inf\{\operatorname{dist}(p,q):p\in D, q\in E\}>0$ and $g=f^m$. Take $n_0$ such that for all $n\geq n_0$ it holds that $g^n(x)\in B_{r/3}(D)\cup B_{r/3}(E)$ and $\operatorname{dist}(g^n(x),g^{n+1}(x))<r/3$. If $g^{n_0}(x)\in B_{r/3}(D)$ we have that $g^{n}(x)\in B_{r/3}(D)$ for all $n\geq n_0$. In this case $E$ is empty. If $g^{n_0}(x)\in B_{r/3}(E)$ we conclude that $D$ is empty. This gives a contradiction that proves the result. \[rmkNontrivialSpiral\] Trivial examples of spiral points are points in the stable set of a hyperbolic periodic point. Let us give an example showing that the $\omega$-limit set of a spiral point may not be a finite set. Take a sequence $a_n\in {\mathbb R}$ such that $a_n\to+\infty$ and $|a_{n+1}-a_n|\to 0$ as $n\to+\infty$ (for example $a_n=\sqrt{n}$). Define a sequence of points in ${\mathbb R}^2$ by $x_n=(1/n,\sin(a_n))$ for $n\geq 1$. It is easy to prove that $\operatorname{dist}(x_{n+1},x_n)\to 0$ as $n\to+\infty$ (Euclidean metric in ${\mathbb R}^2$). Consider $X\subset{\mathbb R}^2$ a compact set containing $\{x_n:n\geq 1\}$ such that there is $f\colon X\to X$ satisfying $f(x_n)=x_{n+1}$ for all $n\geq 1$. By construction we have that the $\omega$-limit set by $f$ of $x_1\in X$ is $$\omega(x_1)=\{(0,y)\in {\mathbb R}^2:|y|\leq 1\}.$$ We have that $x_1$ is a spiral point and its $\omega$-limit set is formed by fixed points of $f$. We now consider the corresponding problem for flows. Consider $\phi\colon {\mathbb R}\times X\to X$ a continuous flow of the metric space $(X,\operatorname{dist})$. We say that $x\in X$ is a *kinematic spiral point* if there is $\tau\neq 0$ such that $\operatorname{dist}(\phi_t(x),\phi_{t+\tau}(x))\to 0$ as $t\to+\infty$. If a continuous flow of a compact metric space $\phi\colon {\mathbb R}\times X\to X$ has a kinematic spiral point $x\in X$ then $\phi_\tau(p)=p$ for all $p\in\omega(x)$. It is direct from the definitions. This implies that a non-trivial minimal set cannot have kinematic spiral points. We now remark that Theorem \[teoMaKaMin\] cannot be extended for kinematic expansive flows. A flow $\phi$ on a compact metric space $X$ is *kinematic expansive* if for all ${\varepsilon}>0$ there is ${\eta}>0$ such that if $\operatorname{dist}(\phi_t(x),{\phi}_t(y))<{\eta}$ for all $t\in{\mathbb R}$ then $y=\phi_s(x)$ for some $s\in(-{\varepsilon},{\varepsilon})$. This definition is weaker than Bowen-Walters expansivity. Some non-trivial minimal examples are known. In [@Mat] Matsumoto proves that the two-torus admits a $C^0$ minimal kinematic expansive flow. In [@ArK] it is shown that no $C^1$ minimal flow on the two-torus is kinematic expansive. On three-dimensional manifolds it is known that the horocycle flow of a compact surface of constant negative curvature is minimal. In [@Gura] Gura shows that this flow is *separating*, i.e., there is ${\eta}>0$ such that if $\operatorname{dist}(\phi_t(x),\phi_t(y))<{\eta}$ for all $t\in{\mathbb R}$ then $y=\phi_s(x)$ for some $s\in{\mathbb R}$. It [seems that]{} the horocycle flow is in fact kinematic expansive. Since the definition of expansive flow introduced by Bowen and Walters is stated using time reparameterizations the following definition is natural. We say that $x\in X$ is a *spiral point* if there is a continuous function $h\colon {\mathbb R}\to{\mathbb R}$ and $\tau>0$ such that $h(t)-t >\tau$ for all $t\geq 0$ and $\operatorname{dist}(\phi_t(x),\phi_{h(t)}(x))\to 0$ as $t\to+\infty$. In [@KS] the definition of spiral point is introduced using local cross sections of the flow but the definitions are equivalent. If there is $T>\tau$ such that $T>h(t)-t>\tau$ for all $t\geq 0$ (in the previous definition) then every orbit in the $\omega$-limit set of $x$ is [periodic or singular. This $\omega$-limit set may not be a single orbit. Consider, for example, the suspension of the homeomorphism in Remark \[rmkNontrivialSpiral\].]{} Spiral points appear in the Poincaré-Bendixon theory of two-dimensional flows. In fact, if $\phi$ is a continuous flow on the two-dimensional sphere then every point is spiral. [A *closed orbit* is a closed subset of $X$ of the form $\{\phi_t(x):t\in{\mathbb R}\}$. Since $X$ is compact, every closed orbit is a periodic orbit or a stationary point.]{} It can be the case that the $\omega$-limit set of a spiral point contains non-closed orbits. An example is illustrated in Figure \[figSpiral\]. In this case $h(t)-t\to+\infty$ as $t\to+\infty$. However, there is a fixed point (a special case of closed orbit) in the $\omega$-limit set of this spiral point. ![A spiral point in the plane.[]{data-label="figSpiral"}](figSpiral.pdf) Let us state the following question: if $\phi$ is a flow on a compact metric space $X$ and $x\in X$ is a spiral point, is there a closed orbit in $\omega(x)$? A positive answer, even assuming that $\phi$ is a minimal expansive flow, would prove (using [@KS]) that if a compact metric [space]{} admits a minimal expansive flow then $\dim(X)\leq 1$. Departamento de Matemática y Estadística del Litoral, Salto-Uruguay\ Universidad de la República\ E-mail: artigue@unorte.edu.uy

--- abstract: 'The isotopes $^{60}$Fe and $^{26}$Al originate from massive stars and their supernovae, reflecting ongoing nucleosynthesis in the Galaxy. We studied the gamma-ray emission from these isotopes at characteristic energies 1173, 1332, and 1809keV with over 15 years of SPI data, finding a line flux in combined lines of $(0.31\pm0.06) \times 10^{-3}\,{\mathrm{ph\,cm^{-2}\,s^{-1}}}$ and the line flux of $(16.8\pm 0.7) \times 10^{-4}\,{\mathrm{ph\,cm^{-2}\,s^{-1}}}$ above the background and continuum emission for the whole sky. Based on the exponential-disk grid maps, we characterise the emission extent of to find scale parameters $R_0 =7.0^{+1.5}_{-1.0}$ kpc and $z_0=0.8^{+0.3}_{-0.2}$ kpc, however the lines are too weak to spatially constrain the emission. Based on a point source model test across the Galactic plane, the emission would not be consistent with a single strong point source in the Galactic center or somewhere else, providing a hint for a diffuse nature. We carried out comparisons of emission morphology maps using different candidate-source tracers for both and emissions, and suggests that the emission is more likely to be concentrated towards the Galactic plane. We determine the /$\gamma$-ray flux ratio at $(18.4\pm4.2)\,\%$ , when using a parameterized spatial morphology model. Across the range of plausible morphologies, it appears possible that and are distributed differently in the Galaxy. Using the best fitting maps for each of the elements, we constrain flux ratios in the range 0.2–0.4. We discuss its implications for massive star models and their nucleosynthesis.' author: - 'W. Wang$^{1,2}$, T. Siegert$^{3,4}$, Z. G. Dai$^{1,5}$, R. Diehl$^{4,6}$, J. Greiner$^{4}$, A. Heger$^{7,8}$, M. Krause$^9$, M. Lang$^4$, M. M. M. Pleintinger$^4$, X.L. Zhang$^4$' title: 'Gamma-ray Emission of and Radioactivities in our Galaxy' --- Introduction ============ The radioactive isotope $^{60}$Fe is produced in suitable astrophysical environments through successive neutron captures on pre-existing Fe isotopes such as (stable) $^{54,56,57,58}$Fe in a neutron-rich environment. Candidate regions for production are the He and C burning shells inside massive stars, where neutrons are likely to be released from the $^{22}$Ne($\alpha$,n) reaction. production may occur any time during late evolution of massive stars towards core collapse supernovae (Woosley & Weaver 1995; Limongi & Chieffi 2003, 2006, 2013; Pignatari et al. 2016; Sukhbold et al. 2016; and references therein). There is also an explosive contribution to the yield by the supernova shock running through the carbon and helium shells (Rauscher et al. 2003). Electron-capture supernovae may be a most-significant producer of $^{60}$Fe in the Galaxy (Wanajo et al. 2013, 2018; Jones et al. 2016, 2019a). There are other possible astrophysical sources of $^{60}$Fe. From similar considerations, $^{60}$Fe can also be made and released in super-AGB stars (Lugaro et al. 2012). Furthermore, high-density type Ia supernova explosions that include a deflagration phase (Woosley 1997) can produce even larger amounts per event. Due to its long lifetime (radioactive half-life $T_{1/2} \simeq$2.6 Myr, Rugel et al. 2009; Wallner et al. 2015; Ostdiek et al. 2017), $^{60}$Fe survives to be detected in $\gamma$-rays after being ejected into the interstellar medium: $\beta$-decays to $^{60}$Co, which decays within 5.3 yr to $^{60}$Ni into an excited state that cascades into its ground state by $\gamma$-ray emission at $1173$ keV and 1332 keV. $^{26}$Al has a similarly-long radioactive lifetime of $\sim 10^{6}$ years, and had been the first live radioactive isotope detected in characteristic $\gamma$-rays at 1809 keV (Mahoney et al. 1978), thus proving currently ongoing nucleosynthesis in our Galaxy. Mapping the diffuse $^{26}$Al $\gamma$-ray emission suggested that it follows the overall Galactic massive star population (Diehl et al. 1995; Prantzos and Diehl 1996). If $^{60}$Fe and $^{26}$Al have similar astrophysical origins, with their similar radioactive life time, the ratio of $^{60}$Fe to $^{26}$Al $\gamma$-ray emissions in the Galaxy would be independent of the true specific distances and locations of the sources, and thus important for testing stellar evolution models with their nucleosynthesis and core-collapse supernova endings (Woosley and Heger 2007; Diehl 2013). The steady-state mass of these radioactive isotopes maintained in the Galaxy through such production counterbalanced by radioactive decay thus converts into a ratio for the $\gamma$-ray flux in each of the two lines through $$\frac{ I(^{60}Fe)}{ I(^{26}Al)} \sim 0.43 \cdot \frac{\dot{M}(^{60}Fe)}{\dot{M}(^{26}Al} \label{eq:fealfluxratio}$$ Timmes et al. (1995) carried the massive-star yields into an estimate of chemical evolution for $^{60}$Fe and $^{26}$Al in the Galaxy, predicting a $\gamma$-ray flux ratio of 0.16. Various revisions of models presented different ratio values ($\sim 0.5 - 1$, see Limongi & Chieffi 2003, 2006; Rauscher et al. 2002; Prantzos 2004; Woosley and Heger 2007). Different massive star regions, such as Scorpius-Centaurus or Cygnus, may show a different /ratio as the age of such associations dictates the expected fluxes (Voss et al. 2009). The average over the whole Milky Way, on the other hand, is determined by the number of massive stars and their explosions over the characteristic lifetimes of and , respectively, providing an independent measure of the core-collapse supernova rate in the Milky Way, and up to which masses stars actually explode. The yields of these two isotopes depend sensitively on not only the stellar evolution details such as shell burnings and convection, but also the nuclear reaction rates. Tur et al. (2010) found that the production of $^{60}$Fe and $^{26}$Al is sensitive to the 3$\alpha$ reaction rates during He burning, i.e. the variation of the reaction rate by a factor of two will make a factor of nearly ten change in the $^{60}$Fe/$^{26}$Al ratio. may be destroyed within its source by further neutron captures ($n,\gamma$). Since its closest parent, $^{59}$Fe is unstable, the $^{59}$Fe($n,\gamma$) production process competes with the $^{59}$Fe $\beta$ decay to produce an appreciable amount of . This reaction pair dominates the nuclear-reaction uncertainties in production, with $^{59}$Fe($n,\gamma$) being difficult to measure in nuclear laboratories due to its long lifetime and E1 and M1 reaction channels (Jones et al. 2019b). Using effective He-burning reaction rates can account for correlated behaviour of nuclear reactions and mitigate the overall nuclear uncertainties in these shell burning environments (Austin et al, 2017). Astronomical observations of the /ratio will help to constrain the nuclear-reaction aspects of massive stars, given the experimental difficulties to measure all reaction channels involved at the astrophysically-relevant energies. Such comparison and interpretation, however, relies on the assumption that and originate from the same sources (see, e.g., Timmes et al. 1995; Limongi & Chieffi 2006, 2013), and have the similar diffuse emission distributions, originating from nucleosynthesis in massive stars throughout the Galaxy. Therefore, the observational verification of the diffuse nature of emission is key to above interpretation of measurements of a /ratio in terms of massive-star models. Several detections of enriched material in various terrestrial as well as lunar samples (Knie et al. 2004; Wallner et al. 2015; Fimiani et al. 2016; Neuhäuser et al. 2019) confirm the evidence for a very nearby source for within several Myr. A signal from interstellar was first reported from the NaI spectrometer aboard the RHESSI spacecraft (2.6$\sigma$) which was aimed at solar science (Smith 2004). They also presented a first upper limit of $\sim 0.4$ (1$\sigma$) for the flux ratio of $^{60}$Fe/$^{26}$Al $\gamma$-ray emissions (Smith 2004). The first solid detection of Galactic emission was obtained from INTEGRAL/SPI measurements, detecting $\gamma$-rays with a significance of $4.9\sigma$ after combining the signal from both lines at 1173 and 1332 keV (Wang et al. 2007), constraining the flux ratio of $^{60}$Fe/$^{26}$Al $\gamma$-ray emissions to the range of $0.09- 0.21$ (Wang et al. 2007). Subsequent analysis of ten-year INTEGRAL/SPI data with a different analysis method similarly suggested a ratio in the range $\sim 0.08-0.22$ (Bouchet et al. 2015). In this paper, we use more than 15 years of SPI observations through the entire Galaxy, and carry out a broad-band spectral analysis in the $\gamma$-ray range 800 keV – 2000 keV. Rather than striving for high spectral resolution and line shape details, this wide-band $\gamma$-ray study aims at study of both and emission signals at 1173, 1332 and 1809keV simultaneously, i.e., using identical data and analysis methods, including data selection and background treatments. This paper is structured as follows. In §\[sec:dataobs\], we will describe the SPI observations and the data analysis steps. Our emission models to describe the 800–2000keV band are introduced in §\[sec:skymodels\]. We present our morphological as well as spectral findings in §\[sec:results\]. Implications and conclusion are shown in Section §\[sec:discussion\]. Observations and data analysis {#sec:dataobs} ============================== SPI observational data ---------------------- The [*INTEGRAL*]{} mission (Winkler et al. 2003) began with its rocket launch on October 17, 2002. The spectrometer SPI (Vedrenne et al. 2003) is one of INTEGRAL’s two main telescopes. It consists of 19 Ge detectors, which are encompassed into a BGO detector system used in anti-coincidence for background suppression. SPI has a tungsten coded mask in its aperture, which allows imaging with a $\sim 3^{\circ}$ resolution within a $16^{\circ} \times 16^{\circ}$ fully-coded field of view. The Ge detectors record $\gamma$-ray events from energies between 20 keV and 8 MeV. The performance of detectors, and the behaviour and variations of instrumental backgrounds, have been studied over the mission, and confirmed that scientific performance is maintained throughout the mission years (Diehl et al. 2018). The [*INTEGRAL*]{} satellite with its co-aligned instruments is pointed at predesignated target regions, with a fixed orientation for intervals of typically $\sim 2000$ s (referred to as [*pointings*]{}). The basic measurement of SPI consists of event messages per photon triggering the Ge detector camera. We distinguish events which trigger a single Ge detector element only (hereafter [*single event*]{}, SE), and events which trigger two Ge detector elements nearly simultaneously (hereafter [*multiple event*]{}, ME). The fast Pulse Shape Discriminator electronic unit (PSD), digitizes the shape of the current pulse, and allows suppression of background events, e.g. from localized $\beta$-decays within the Ge detectors (Roques et al. 2003), or from electronic noise. In this work, we use event data which hit only one detector (i.e., SE event data) and which carry the PSD flag for acceptable pulse shape (event type ‘PE’). We applied a selection filter to reject corrupted, invalid, or background-contaminated data. We apply selection windows to ‘science housekeeping’ parameters such as the count rates in several background-monitoring detector rates, proper instrument status codes, and orbit phase. In particular, we use the SPI plastic scintillator anti-coincidence counter (PSAC) mounted beneath the coded mask, and the rate of saturating events in SPI’s Ge detectors (i.e., events depositing $>$ 8 MeV in a single Ge detector; hereafter referred to as GeDSat rates) as background tracers. This selection leads to exclusions of strong solar-flare periods and other times of clearly increased / abnormal backgrounds. Additionally, regular background increases during and after passages through the Earth’s radiation belts are eliminated by a 0.1–0.9 window on orbital phase. As a final step after these primary selections, we perform a quality-of-fit selection using our best background model only (see §\[sec:dataanalysis\]), and exclude all further pointings which show deviations beyond $10\ \sigma$ above a fit of this background plus the expected signal contribution per pointing, thus eliminating pointings with abnormal background (note that SPI data are dominated by instrumental background counts, so that source counts from diffuse emission such as alone cannot deteriorate the fit to a pointing dataset significantly). These outliers are mainly due to missing ‘science housekeeping’ parameters which are interpolated later, or due to burst like events in the field of view, for example gamma-ray bursts, flares from X-ray binaries, or solar activity. The resulting dataset for our and study encompasses 99,864 instrument pointings across the entire sky, equivalent to a total (deadtime-corrected) exposure time of 213Ms. This includes data from INTEGRAL orbits 43 to 1950, or February 2003 to May 2018. The sensitivity (effective exposure time, effective area) of SPI observations is further reduced by the successive failure of four of the 19 detectors (December 2003, July 2004, 19 Feb 2009 and 27 May 2010). In Fig.\[fig:exposuremap\] we show the resulting exposure map from our cleaned data set. We bin detector events from the range between 800keV and 2000keV into seven energy bins: three of these address $\gamma$-ray line bands for and (1169 – 1176keV; 1329 – 1336keV; 1805 – 1813keV), and four continuum bands in between (800 – 1169keV; 1176 – 1329keV; 1336 – 1805keV; 1813 – 2000keV) are added to robustly determine the line flux above the diffuse $\gamma$-ray continuum. In the Appendix, we present an investigation of the impact of event selections using the PSD (see Fig.\[fig:sepsdcomparison\]). PSD selections succeed to suppress an apparent electronics noise that is visible in raw detector spectra in the energy range 1300 –1700 keV. We therefore use such PSD selections, although the impact on the resulting spectra from celestial sources is not clear except for this continuum band (see Appendix). ![Exposure sky map of fully-coded FOV in Galactic coordinates (the number at the color bar in units of sec) for the data selected from 15-year SPI observations for our and study (INTEGRAL orbits 43 – 1950).[]{data-label="fig:exposuremap"}](exposure){width="10cm"} Data analysis {#sec:dataanalysis} ------------- The raw data of SPI are dominated by the intense background (BG) radiation characteristic of platforms undergoing cosmic-ray (CR) bombardment. In SPI data analysis, we combine background models with a spatial model for sky emission to fit our data, allowing for adjustments of fit parameters for background and sky intensities. In general, the counts per energy bin, per detector, and per pointing are fitted by the background model described in Siegert et al. (2019), with details outlined shortly below, and the assumed sky map of celestial emission (e.g., the distribution obtained by COMPTEL or exponential disk models, see §\[sec:skymodels\]) as convolved into the domain of the SPI data space for the complete pointing sequence, by the instrument coded-mask response: D\_[e,d,p]{}=\_[m,n]{}\_[j=1]{}\^[k\_1]{} A\_[e,d,p]{}\^[j,m,n]{}\_s\^j I\_j\^[m,n]{} + \_t \_[i=1]{}\^[k\_2]{}\^i\_[b,t]{} B\^i\_[e,d,p]{} + \_[e,d,p]{}, \[eq:datafit\]where [*e,d,p*]{} are indices for data space dimensions: energy, detector, pointing; [*m,n*]{} indices for the sky dimensions (galactic longitude, latitude); $A$ is the instrument response matrix, $I$ is the intensity per pixel on the sky, $k_1$ is the number of independent sky intensity distribution maps; $k_2$ is the number of background components, $\delta$ is the count residue after the fitting. The coefficients $\beta_s$ for the sky map intensity are constant in time, while $\beta_{b,t}$ is allowed time dependent, see \[sec:bg\_char\] below. The sky brightness amplitudes $\beta_s$ comprise the resultant spectra of the signal from the sky. For this model fitting analysis, we use a maximum-likelihood method, implementing Poisson statistics which applies to such detector count analysis. Our software implementation is called [*spimodfit*]{} (Strong et al. 2005). The fitted model components are analysed for further consistency checks on possible systematics in residuals. We thus derive, per energy bin, best-fitted parameter values with uncertainties, and their covariance matrices. This provides flux estimates which are independent of spectral-shape expectations. ![SPI background tracer variations with time from saturated events in the Ge camera (GEDSAT). In the panel, the full SPI data base is shown in black, and the chosen data based on our selection criteria in red.[]{data-label="fig:gedsattracer"}](gesattot_tracer-new.ps){width="9cm"} Background characteristics {#sec:bg_char} -------------------------- Much of the radiation from instrumental background is promptly emitted directly from CR impacts. Background components arise from radioactive isotopes produced by the CR impacts. Local radioactivity in the spacecraft and instruments themselves thus will generate both broad continuum background emission and narrow gamma-ray lines from long and short-lived radio-isotopes. Varying with energy, background components may exhibit complex time variability due to their origins from more than one physical source (Weidenspointner et al. 2003). Background modelling by using the entire mission spectroscopy history has been established recently (Siegert et al. 2019). For the energy bands studied here, we follow this general approach, and model the background by fitting two components, one for the continuum and one for instrumental lines. From the mission spectral database (Diehl et al. 2018), we construct these background models at fine spectral precision. Then, we allow for adjustment of its overall normalisation in intensity, which accounts for the fact that the (small) celestial signal that had been part of the mission data used for this database now needs to be separated out. The instrument records several detector triggers which have sufficiently-high count rates, such as the one of onboard radiation monitors, the SPI anti-coincidence shield count rate, and the rate of saturating Ge detector events (events depositing $>$8 MeV in a Ge detector, GeDSat). These count rates reflect the current cosmic-ray flux which causes the prompt instrumental $\gamma$-ray background, while the long-term trends of radioactive build-up and decays is inherent to the spectral background data base. In this analysis, we use the GeDSat rates as a short-term background tracer and first-order description of the background variations with time scales of pointing-to-pointing, from 800 keV - 2000 keV. This tracing is sufficient in energy bins at or below the instrumental resolution, however in broader energy bins such as used in this work the superposition of the effects from different background lines blended together in a broader bin require re-scalings after suitable time-intervals. ![Expected significance (in $\sigma$ units, black points) above a background only description of the data in the 1169 to 1176 keV band, estimated from likelihood tests, BG vs. BG plus source, using different numbers of background parameters (bottom axis), i.e. varying the background on different time scales (top axis: re-scaling per number of INTEGRAL revolutions; or whenever an annealing was performed, a detector failed; or assuming a constant background). The red points show the Akaika Information Criterion ($\mathrm{AIC} = 2n_{par} - 2\ln(L)$, where $n_{par}$ is the total number of fitted parameters, right axis) which was used to find the required number of background parameters to describe the data in this bin sufficiently well. The optimum is found at one INTEGRAL orbit or 1,674 background fit parameters, respectively. The black points show the expected significance in the band with a total flux of $4 \times 10^{-4}\,{\mathrm{ph\,cm^{-2}\,s^{-1}}}$ as a function of fitted background parameters. See text for details.[]{data-label="fig:aicanalysisbgscalingfe1"}](expectations_fe60_1169-1176keV.ps){width="12cm"} The coefficients $\beta_{b,t}$ for background normalisations are therefore allowed time dependent, to cater for such effects, and for different background normalisations for each camera configuration of 19/18/17/16/15 functional detector elements, as well as for possible variations on time scales shorter than our background model was built. In Fig.\[fig:aicanalysisbgscalingfe1\], we show the optimal re-normalisation time scale for the energy bin including the 1173keV line. We find an optimum fit when re-normalising background at a time scale of one orbit, which corresponds to 1,674 background parameters per component. As a consistency check, we performed an estimate of the signal significance of the 1169–1176keV band (continuum plus line) as could be expected from earlier measurements (Wang et al. 2007). In this estimation we consider only the inner Galaxy, and assume the COMPTEL map as a tracer for the morphology of emission, and we adopt a total galaxy-wide flux of $4 \times 10^{-4}\,{\mathrm{ph\,cm^{-2}\,s^{-1}}}$. We then make use of the approach by Vianello (2018), who extended the Bayesian significance estimates of Li & Ma (1983), and assume herein that our background model parameters are normally distributed. This obtains the black line shown in Fig.\[fig:aicanalysisbgscalingfe1\], estimating a significance of $6\sigma$ for our optimal background model re-scaling. In case about half of the flux of $4 \times 10^{-4}\,{\mathrm{ph\,cm^{-2}\,s^{-1}}}$ would be degenerate and absorbed in the diffuse $\gamma$-ray continuum component, the line significance would be around $4\sigma$ in our estimate for a single line. The background rescaling investigations in the remaining energy bins show similar optimum time scales, except for the lowest energy bin (800– 1169 keV), which requires four parameters per orbit. The number of degrees of freedom is thus 1,591,831 for 1,595,180 data points in the , , and higher-energy continuum bins, taking into account all fitted parameters for the sky, detector failures and background variations. Analysing the resulting background variations in the line bands, we now investigate the candidate origins, based on our detailed spectral analysis of instrumental background with high spectral resolution. The most important background lines are the ones expected from the decay of [${\mathrm{^{60}Co}}$]{} in the instrument and the satellite, at the same line energies because this is the same cascade de-excitation in both cases. This $^{60}$Co background builds up in intensity with time, due to its 5.3yr radioactive lifetime, and thus will increasingly contribute to the total measured [${\mathrm{^{60}Co}}$]{} $\gamma$-ray line signal. In addition, there is a strong background line from activated Ge at 1337keV, which blends into the high-energy line at 1332keV. This Ge line, however, shows no radioactive build-up as the decay time is of the order of nano-seconds, and hence the count rate in this line closely follows the general activation of backgrounds. All these lines are superimposed onto an instrumental continuum background which is dominated by bremsstrahlung inside the satellite, and also includes Compton-scattered photons and a composite of weaker lines that escape identification in our deep spectral background analysis (Diehl et al. 2018). In Figs.\[fig:gedsattracer\] to \[fig:bgmodel\], we show characteristic background components as they vary with time, as determined from our data set. Here we focus on the total galactic emission, so that extrapolated estimates from the inner Galaxy towards the full Galaxy only may serve as a guidance, rather than a precise prediction. Nevertheless, we see that the steadily rising [${\mathrm{^{60}Co}}$]{} background line flux leads to a very shallow increase in the significance of celestial over time, shallower than count accumulation alone would suggest. While our previous result with only three years of data (Wang et al. 2007) showed a $4.9\sigma$ signal, in this case using both lines together and both SE and ME (single and multiple-trigger event types), which is consistent with our expectations, the increase of data by more than 200% in our current dataset would result in only $6\sigma$ significance for SE and the two lines combined. We now also understand the additional background time variation: because the background line rate increases almost linearly (Fig.\[fig:bgmodel\]), fitting the background requires more parameters than typical for SPI background that follows the solar cycle directly. Modelling in the Milky Way {#sec:skymodels} ========================== The signal is too weak to derive the spatial distribution or perform an imaging analysis. Therefore, we attempt to constrain the size of the emission region through fitting a parameterized geometrical model, an exponential disk profile, and we determine the scale radius and scale heights. Doing this for all energy bins between 800 and 2000keV, we obtain information on how this approach deals with known spatial distributions of $\gamma$-ray emission, thus helping to judge systematics limitations with the interpretation. The exponential-disk models in this analysis have been adopted in the following form: $$\rho(x,y,z) = A \exp\left(-R/R_0\right)\exp\left(-|z|/z_0\right){\mathrm{,~with~}}R=\sqrt{x^2 + y^2} \label{eq:expdisks}$$ In Eq.\[eq:expdisks\], $\rho(x,y,z)$ is the 3D-emissivity that is integrated along the line of sight $(l/b)$ to produce maps of sky brightness, with a pixel size of $1^{\circ} \times 1^{\circ}$. These maps are then folded through the coded-mask response to create expected count ratios for all selected pointings. The normalisation $A$, equivalent to $\beta_s^j$ in Eq.\[eq:datafit\], is determined (fitted) for a grid of scale radii $R_0$ and scale heights $z_0$ that we test. Here, we use a grid of 16 $R_0$-values, from 500pc to 8000pc in steps of 500pc, times 32 $z_0$-values between 10 and 460pc in steps of 30pc and between 500 and 2000pc in steps of 100pc. These models are independently fitted to all seven energy bins, to obtain a likelihood chart for all morphologies tested. From this, we can determine both the best-fitting flux as well as the best scale dimensions of the emission, plus their uncertainties. Doing this in all our energy bands, we can directly compare the emission size characteristics of versus , obtaining systematics information from the continuum bands in between (i.e. possible biases for scale dimensions, influenced by the continuum below the lines). All previous searches for (Wang et al. 2007; Harris et al. 2005; Bouchet et al. 2015), generally assumed that the diffuse emission follows the sky distribution of the line. In this study, we explore the morphology of by comparing with emission as well as continuum emission. We test different tracers of potential candidate sources for emission, and compare those tracer maps to that of emission as measured and deconvolved from $\gamma$-ray data. This provides an independent judgement of how similar and are (§\[sec:tracerresults\]), compared to tracers that may include some of the expected deviations from a strict correlation with . Similar to previous studies (Wang et al. 2009; Siegert 2017), we fit all-sky survey maps from a broad range of different wavelengths to our SPI data. From this we obtain a qualitative measure which maps are favoured at our chosen energies, respectively. We test a comprehensive set of maps to trace different emission mechanisms which may be related to our SPI data in the different energy bands. The list of tested maps (Tab.\[tab:tracermaps\]) includes also source tracers which may be weakly or not at all related to the candidate and sources. We use this list of tracers for all or energy bands, in order to reveal degeneracies and systematics, because differences between maps are hard to quantify through fit likelihoods in absolute terms. We also include a background-only fit for reference. In Tab.\[tab:tracermaps\] we comment on each map briefly to illustrate its main features. \[tab:tracermaps\] ![Spectral intensities (black) obtained from the fit to an exponential-disk model with $R_0 = 7$kpc and $z_0 = 0.8$kpc. The fitted total model, Eq. 4, is shown in red.[]{data-label="fig:specalfitnew"}](spec_al_fit_new.ps){width="0.7\linewidth"} Results {#sec:results} ======= From independently fitting spatial emission models to SPI data for each of our seven energy bands, we obtain intensity values for the celestial emission detected in each of these bands, for the same adopted spatial distribution model. In Tab.\[tab:chi2\], we presented the $\chi^2$ values for seven energy bands in the modelling fittings. So that the present fits are reliable, and the background model is acceptable. For further discussion and analysis, we fit each extracted set of sky-intensity values with $$\begin{aligned} F(E;C_0,\alpha,F_{60},F_{26}) & = & C_0 \times \left(\frac{E}{1000\,{\mathrm{keV}}}\right)^{\alpha} + \\\nonumber & + & F_{60} \times (T(1172.5\,{\mathrm{keV}},7\,{\mathrm{keV}}) + T(1332.5\,{\mathrm{keV}},7\,{\mathrm{keV}})) + \\\nonumber & + & F_{26} \times T(1809.0\,{\mathrm{keV}},8\,{\mathrm{keV}}){\mathrm{,}} \label{eq:specmodel}\end{aligned}$$ where $C_0$ is the continuum flux density nomalisation at 1000keV, $\alpha$ is the power-law index, and $F_{60}$ and $F_{26}$, respectively, are the integrated fluxes as derived from tophat functions, $T(E_0,\Delta E)$, centred at $E_0$ with bin width $\Delta E$. We link the parameters of the two lines as they are expected to reflect the same incident flux in intensity and width. The lines are expected to be somewhat broadened above instrumental line widths by 0.1–0.2keV due to large-scale galactic rotation of sources (Wang et al. 2009, Kretschmer et al. 2013), and the instrumental line width would be $\sim2.8$keV around the lines and $\sim3.2$keV around the line (Diehl et al. 2018). Within the 7keV bins for the lines and the 8keV for the line bands thus, $2.9\sigma$ (99.7%) of the expected line fluxes would be contained. \[tab:chi2\] Characterising the extents of and emission {#sec:expresults} ------------------------------------------- In Fig.\[fig:specalfitnew\], we show the 7-band spectral intensities as derived from an exponential disk with scale radius 7kpc and scale height 0.8kpc, as a typical example. We selected this as it reflects best-fit dimensions in the line band. In this example, the continuum is determined as $(2.4\pm0.2) \times (E/{\mathrm{1000\,keV}})^{(-1.3\pm0.2)} \times 10^{-5}\,{\mathrm{ph\,cm^{-2}\,s^{-1}\,keV^{-1}}}$. The and line fluxes are $(2.6 \pm 0.6) \times 10^{-4}\,{\mathrm{ph\,cm^{-2}\,s^{-1}}}$ and $(14.4 \pm 0.7) \times 10^{-4}\,{\mathrm{ph\,cm^{-2}\,s^{-1}}}$, respectively, which results in a /ratio of $(18.3\pm4.4)\,\%$. Our $16 \times 32 = 512$ scale size grid covers the full range of the Galactic plane, and therefore provides a measure of the emission extents for and as well as for the continuum emission expected from Bremsstrahlung and inverse-Compton interactions of cosmic-ray electrons. Each of the 512 exponential disk templates is treated individually in the first place, fitting its parameters without any priors or constraints. As a result, the absolute fluxes of continuum and lines vary with the emission dimensions. We find that with larger scale dimensions, the fluxes of continuum and lines increase. We find no strong variation of the line-to-continuum ratio for either or . This supports our assessment that the shape constraints that we derive are consistent and without major bias. The strong background line at 1337keV (see Section 2) could affect the 1332keV line result, which we therefore compare to the more-isolated 1173keV line; for our spatial results, we prefer to rely on the latter line for this reason, while the /flux ratio uses the data constraints from both Fe line bands combined. In our goal to determine the spatial extent of emission, we show next to each other for and in the 1173 and 1809keV lines the likelihood contour regions versus scale heights and scale radii (in Fig.\[fig:loglikprofiles\]). For the emission, we can obtain the characteristic scale radius of $R_0 =7.0^{+1.5}_{-1.0}$ kpc and scale height of $z_0=0.8^{+0.3}_{-0.2}$ kpc. However, for the emission lines, the constraints are very poor due to the weak signals, formally resulting in $R_0 = 3.5^{+2.0}_{-1.5}$ kpc, and $z_0= 0.3^{+2.0}_{-0.2}$ kpc. We will use the point source model test to exclude one point source model in the Galactic center (see Section \[sec:pointsource\]). In our goal to exploit maximum information for the /flux ratio while catering for the uncertainty of the spatial extent of emission, we include the quality of the fits to SPI data from this grid of exponential disk model fits in our assessment. We apply a weighting with the Aikaike Information Criterion (AIC, Akaike 1974), derived from the likelihood and the number of fitted parameters in each energy bin, thus taking the individual measurement and fitting uncertainties of each model map into account. This yields a line flux value of $(3.1\pm0.6) \times 10^{-4}\,{\mathrm{ph\,cm^{-2}\,s^{-1}}}$, and a line flux of $(16.8\pm 0.8) \times 10^{-4}\,{\mathrm{ph\,cm^{-2}\,s^{-1}}}$. The /ratio resulting from this emission-extent averaged analysis is $(18.4\pm4.2)\,\%$. The derived flux for the full Galaxy is also consistent with the previous map study with SPI data (Bouchet et al. 2015). In addition, a recent SPI analysis reported the flux values for both the inner Galaxy and the whole sky (Pleintinger et al. 2019): $\sim 0.29\times 10^{-3} {\mathrm{ph\,cm^{-2}\,s^{-1}}}$ for the inner region, and $\sim (1.7-2.1)\times 10^{-3} {\mathrm{ph\,cm^{-2}\,s^{-1}}}$ for the whole sky. We show the /ratio distribution from all 512 exponential-disk configurations in Fig.\[fig:fe60toal26rationew\], also indicating characteristic uncertainties. ![ /flux ratio for the grid of exponential disk models (blue, left axis). Including the uncertainties of the fluxes from each spectral fit, the total estimated /flux ratio from exponential disks is $(18.4 \pm 4.2)\,\%$. Alternative to exponential disks, we also show the flux ratios derived from a set of tracer maps (see section 4.2), as vertical lines according to their significance (right axis), together with their uncertainties as shaded bands. Clearly, these systematically show larger values compared to the exponential disk models. The IRIS (25 ${\mathrm{\mu m}}$) map shows the largest improvement above a background only description for both lines consistently (see Fig. 11), so that a flux ratio estimate from this map serves as a measure of the systematic uncertainty. We find the ratio of $0.24\pm 0.4$ based on the IRAS 25$\mu$m map, suggesting a systematic uncertainty of the order of $6\%$. []{data-label="fig:fe60toal26rationew"}](60fe26al_ratio_expdisks_selectmaps_new.ps){width="14cm"} Point source model test for the Galactic plane {#sec:pointsource} ---------------------------------------------- In this section, we aim to constrain the morphology of the weak in another way. We produced a catalogue of point source locations with $91\times 21 =1911$ entries between $l=-90^\circ - 90^\circ$ and $b=-20^\circ - 20^\circ$, in 2 degree steps. Then we used this catalogue to test a point source origin for both and emission lines in the Galactic plane. In this way, in Fig.\[fig:pointsourcescan\], we can check if and how extended the and emissions are. These morphology studies of and emission distributions suggested the $\gamma$-ray emissions are not attributed to one or several point sources in this region. In positive longitudes, the emission extended to $l\sim 35^\circ$ which may be contributed by Aquila, and at $l\sim 80^\circ$, the emission structure is Cygnus. The truncated structure in positive longitudes will be the influence of the exposure map partially (see Figure 1). In the negative longitudes, the emission extended to $l\sim -75^\circ$, probably Carina. However, these maps should not be interpreted as the real sky distribution map, but can imply that the emission is not point-like. The use of this simplified emission model at different galactic coordinates would yield large residuals in the raw SPI data space. Likewise, the emission morphology is unknown and could be particularly similar to . From these number of trials, we can estimate the influence of diffuse or point-like emission by taking into account that a background-only fit would result in a test-statistics defined as: $$TS = 2\left(\log (L_{BG}) - \log(L_{PS}(l/b))\right)\mathrm{,}$$ with $L_{BG}$ and $L_{PS}$ being the likelihoods of a background-only fit and a background plus point-source fit, respectively, for finding a point source by chance at a trial position $(l/b)$, would be distributed as $0.5 \chi^2$ with 3 degrees of freedom (2 position, 1 amplitude), we can compare how much the measured $TS$-distributions deviate. A single point source would have one (or more) high points that at high TS at a particular, sharply defined, $TS$-value. We can see that the case is deviating from the background-only case in more than a few points and consistently for $TS > 12$. This would be a signature of a diffuse signal for the emission. For comparison, the line case is shown as well. Therefore, we can exclude a single strong point source in the galactic center as well as somewhere else in this region as the origin of the detected emission. Investigations of different emission tracers {#sec:tracerresults} -------------------------------------------- In our effort to investigate the spatial distribution of emission, we fit the SPI data with a large set of maps representing different source tracers (see Tab.\[tab:tracermaps\]). These include the 408MHz map reflecting cosmic-ray electrons through their radio emission, cosmic-ray illuminated interstellar gas shining in GeV $\gamma$-rays, the COMPTEL and SPI maps reflecting radioactivity, different sets of infrared emission, and X-ray emission maps. The fit quality of these maps can be compared through their different likelihood ratios, where we normalize to a background-only reference (Fig.\[fig:tracerbars\]). For the line, as expected, we find again as best-fitting tracer map the SPI line map that had beed derived from a different data set and different analysis method (Bouchet et al. 2015), which supports consistency of our methods. For the line, the best fitting tracer map turns out to be the DIRBE 4.9$\mu$m map representing emission from small dust grains and star light from mostly M, K, and G-type stars. From our set of candidate-source tracers, Tab.\[tab:tracermaps\], we use the likelihood ratio of the combined continuum bins, the two lines, and the line, each normalized to a background only fit , as a measure for significance of a signal from the sky. We illustrate these signal significance levels and the resulting flux values in Fig.\[fig:tracerbars\]. For most of the tested maps, such as 408MHz, IRAS 25$\mu$m, COMPTEL emission, GeV $\gamma$-ray emission, and CO/dust/free-free emission maps, both the and radioactive-line bands show a significant detection of a signal from the sky in our SPI dataset, with significance levels of $>16\sigma$ for the line, and $>4\sigma$ for the lines. In general, for the cases for which we obtain a significant signal, such as the 408MHz map or the COMPTEL map, also shows a significant signal above the background. Somewhat surprisingly, however, the SPI map, which fits best at 1809keV, is particularly poor in detecting sky emission in the 1173keV line band of . This may be an indication that and may indeed have a different emission morphology. In Fig.\[fig:fe60toal26rationew\], we also presented the /ratio ranges derived from these tracer maps with the significant detections of both and emission lines. The all-sky emission maps observed by COMPTEL and SPI gave a ratio range of $0.15-0.24$, and 408MHz, IRAS 25$\mu$m and the Fermi $\gamma$-ray emission maps produced a ratio range of $\sim 0.2 -0.3$. The DIRBE $4.9\,{\mathrm{\mu m}}$ emission map can produce a highest detection significance level for lines, which gave a /ratio of 0.22 – 0.32. For some cases, such as the hard X-ray map from SWIFT/BAT, the soft X-ray (0.25 keV) map from ROSAT, the Planck CMB map, and the SZ-effect map, in both the and bands we obtain no or at most marginal detections of sky emission. The hard X-ray map (100-150 keV) is dominated by emission of point sources along the Galactic plane, such as X-ray binaries. Therefore, the non-detection of signals in and bands would be in line with both the and emission having a diffuse nature, rather than a strong contribution from such sources. The Planck SZ effect map follows the distribution of clusters of galaxies, which are mainly located at high Galactic latitudes. The ROSAT soft X-ray (0.25 keV) map is mostly bright also at high Galactic latitude regions due to the strong soft X-ray absorption by the Galactic plane. Non-detection of and emission signals with these two tracer maps therefore is consistent with our belief in origins of and signals in the plane of the Galaxy and its sources. To compare the acceptable fits for the lines and continuum from the set of tracer maps, we show the sample spectra in our energy bands in Fig.\[fig:spectrumphotontracerssample\] from six typical all-sky distribution models, including the diffuse emission maps from observations (408 MHz, $\gamma$-ray emission, infrared emission) and analytical formula (disk models), and point sources based on hard X-ray surveys. This provides an additional check against, or insight towards, possible systematics in our spectral fit results. ![Comparison of the spectra from $800$–$2000\,$keV for different candidate-source tracers in our continuum and line bands. We have included three $\gamma$-ray lines using six typical distribution models: a homogenous disk model (constant brightness along the Galactic plane with scale height 200 pc, see Wang et al. 2009), a COMPTEL maximum entropy emission map (Plüschke et al. 2001), an exponential disk model (scale radius $3.5\,$kpc, scale height $300\,$pc), the 408 MHz map (Remazeilles et al. 2015), the DIRBE infrared emission map at 2.5$\mu$m (Hauser et al. 1998), and a hard X-ray sky map derived by SWIFT/BAT surveys from $100\,$keV–$150\,$keV (bright point sources, Krimm et al. 2013).[]{data-label="fig:spectrumphotontracerssample"}](modelfit_spectra.eps){width="14cm"} The diffuse $\gamma$-ray continuum ---------------------------------- The hard X-ray to soft $\gamma$-ray Galactic diffuse emissions include the continuum and $\gamma$-ray lines such as positron annihilation line, emission lines, and line. The diffuse continuum emission would originate from several physical processes: inverse-Compton scattering of the interstellar radiation field, bremsstrahlung on the interstellar gas from cosmic-ray electrons and positrons; the neutral pion decays produced in interactions of the cosmic rays with the interstellar gas (see Strong et al. 2010 and references therein), and some unresolved hard X-ray/soft $\gamma$-ray sources. With the 7-year INTEGRAL/SPI observations, Bouchet et al. (2011) derived the hard X-ray spectrum from 20 keV to 2.4 MeV in the Galactic ridge region,with power-law index of $\Gamma\sim 1.4-1.5$ for the diffuse continuum, and a flux level of $\sim 10^{-5} \rm{ph}\ \rm{cm}^{-2} \rm{s}^{-1} \rm{keV}^{-1}$. Using the 15-year SPI data covering 800 keV to 2 MeV, we also determine the continuum spectra of the whole Galactic plane from the fittings, which have the average flux level of $\sim (2.0\pm0.4)\times 10^{-5} \rm{ph}\ \rm{cm}^{-2} \rm{s}^{-1} \rm{keV}^{-1}$ with a power-law index of $\Gamma\sim (1.3\pm 0.2)$. The continuum flux derived in this work is fitted from the whole Galactic plane, rather than only from the inner Galactic ridge (Bouchet et al. 2011). The spectral indices are consistent with each other. We conclude that our broad-band analysis also determines the Galactic continuum emission that underlies the targeted line emissions. Summary and discussion {#sec:discussion} ====================== With more than 15 years of INTEGRAL/SPI observations, we carried out a wide range of spatial model fits to SPI data from 800 – 2000 keV in seven energy bands, including the line bands for at 1773 and 1332 keV and for at 1809 keV, as well as wider continuum energy bands around these. We clearly detected the signals from both and emissions, as well as diffuse Galactic continuum emission. With only the SE data base, assuming the exponential-disk distribution model, we obtained a detection significance level of lines of $\sim 5.2\sigma$ with a combined line flux of $(3.1\pm0.6) \times 10^{-4}\,{\mathrm{ph\,cm^{-2}\,s^{-1}}}$, and the flux of $(16.8\pm 0.7) \times 10^{-4}\,{\mathrm{ph\,cm^{-2}\,s^{-1}}}$ for the whole Galaxy above the background continuum. From the consistent analysis approach, with identical data selections, response, and background treatments, we minimise biases or systematics, and obtain a result for the /flux ratio of $(18.4\pm4.2)\,\%$ based on the exponential disk grid maps. This large-scale galactic value is consistent with a local measurement from deposits of material on Earth (Feige et al. 2018). Since we do not know the real sky distribution of in the Galaxy, the derived /flux ratio will depend on the sky distribution tracers. In Fig.\[fig:tracerbars\], we compared the detection significance levels and flux values for and using different sky distribution tracers. The best fits for are the the SPI and the IRAS 25$\mu$m maps, while for , the best one are the DIRBE 4.9$\mu$m and IRAS 25$\mu$m maps. Thus, we can use these best fit maps to constrain uncertainties of the /flux ratio. If we use the same tracer for both and , i.e. the IRAS 25$\mu$m map, the /flux ratio is $0.20-0.28$; then using the different tracers, i.e., the SPI map for and the DIRBE map for , one finds the ratio range of $0.30 -0.44$. Using an astrophysically-unbiased geometrical description of a double-exponential disk, we explore a broad range of emission extents both along Galactic longitude and latitude for and . For emission we find $R_0 =7.0^{+1.5}_{-1.0}$ kpc and $z_0=0.8^{+0.3}_{-0.2}$ kpc. The $\gamma$-ray signal is weak and near the sensitivity limit of current $\gamma$-ray telescopes, so that imaging similar to what is obtained for $\gamma$-rays cannot be obtained at present. Formally, the scale radius and height are determined to be $R_0 = 3.5^{+2.0}_{-1.5}$ kpc, and $z_0= 0.3^{+2.0}_{-0.2}$ kpc. We carried out a point source model scan in the Galactic plane ($|l|<90^\circ; |b|<20^\circ$) for both and emission line cases. The morphology and test-statistics results suggest that the emission is not consistent with a strong single point source in the Galactic center or somewhere else in the Galactic plane. From our comparison with different sky maps, we provide the evidence for a diffuse nature of concentrated towards the Galactic plane, which is similar to that of . But it is possible that the and are distributed differently in the Galaxy. The ratio of $^{60}$Fe/ $^{26}$Al has been promoted as a useful test of stellar evolution and nucleosynthesis models, because the actual source number and their distances cancel out in such a ratio. A measurement therefore can help theoretical predictions and shed light on model uncertainties, which are a result of the complex massive star evolution at late phases and related nuclear reaction rate uncertainties. Timmes et al. (1995) published the first detailed theoretical prediction of this ratio of yields in and , giving a gamma-ray flux ratio $F(^{60}{\rm Fe})/F(^{26}{\rm Al})= 0.16\pm 0.12$. With different stellar wind models and nuclear cross sections for the nucleosynthesis parts of the models, different flux ratios $F(^{60}{\rm Fe})/F(^{26}{\rm Al})= 0.8\pm 0.4$ were presented (Prantzos 2004). Limongi & Chieffi (2006) combined their yields for stellar evolution of stars of different mass, using a standard stellar-mass distribution function, to produce an estimate of the overall galactic $^{60}$Fe/$^{26}$Al gamma-ray flux ratio around $0.185\pm 0.0625$. Woosley & Heger (2007) suggested that a major source of the large discrepancy was the uncertain nuclear cross sections around the creation and destruction reactions for the unstable isotopes and which cannot be measured in the laboratory adequately. A new generation model of massive stars with the solar composition and the same standard stellar mass distributes from 13 – 120 compared yields with and without effects of rotation (Limongi & Chieffi 2013). For the models including stellar rotation, they determined a flux ratio of $F(^{60}{\rm Fe})/F(^{26}{\rm Al})= 0.8\pm 0.3$. For the non-rotation models, they obtained a flux ratio of $F(^{60}{\rm Fe})/F(^{26}{\rm Al})= 0.2-0.6$; and if one only considers the production of stars from 13 – 40 , the predicted flux ratio reduces to $\sim 0.11\pm 0.04$. For stars more massive than 40 , the stellar wind and its mass loss effects on stellar structure and evolution contributes major uncertainty in ejecta production. But these stars may actually not explode as supernovae and rather collapse to black holes, so that their contributions may not be effective and could be ignored in a stellar-mass weighted galactic average. The measured values from gamma-rays suggest that the (generally) higher values from theoretical predictions may over-estimate and/or under-estimate production. This could be related to the explodability of massive stars for very massive stars beyond 35 or 40 . Acknowledgements {#acknowledgements .unnumbered} ================ We are grateful to the referee for the fruitful suggestions to improve the manuscript. W. Wang is supported by the National Program on Key Research and Development Project (Grants No. 2016YFA0400803) and the NSFC (11622326 and U1838103). Thomas Siegert is supported by the German Research Society (DFG-Forschungsstipendium SI 2502/1-1). The INTEGRAL/SPI project has been completed under the responsibility and leadership of CNES; we are grateful to ASI, CEA, CNES, DLR (No. 50OG 1101 and 1601), ESA, INTA, NASA and OSTC for support of this ESA space science mission. Appendix ======== In Fig.\[fig:sepsdcomparison\], we present the spectral examples from 800 keV - 2000 keV for the seven energy bands for both SE and PSD datasets. In the case of SE, the strong electronic noise cannot be suppressed in the band of 1336 - 1805 keV. While, this electronic noise does not affect the spectral counts for the PSD dataset. The reader can also refer to the supplementary information in Siegert et al. (2016), where we have done the test on the pule shape selections. Thus, in this work, we only refer to the PSD dataset. ![A comparison between the fitted broad spectra derived by the SE dataset and PSD dataset from 800 keV - 2000 keV. In the band of 1336 - 1805 keV, the strong electronic noise cannot be suppressed in the case of SE. []{data-label="fig:sepsdcomparison"}](fe60_analysis_SE_PSD_comparison_COMPTEL26Al.ps){width="15cm"} In the present work, we have tried to constrain the sky distributions of and lines in the Galaxy, so we determine the gamma-ray spectra of three gamma-ray lines (1173 keV, 1332 keV and 1809 keV) for the entire sky. In the previous work (Wang et al. 2007, 2009), we studied the spectra and fluxes of and using the maps only covering the inner Galaxy ($-30^\circ<l<30^\circ, -10^\circ<b< 10^\circ$). In the appendix here (see Fig.\[fig:specinnergal\]), we also show the spectral fitting of the broad spectrum with the same map as in Wang et al. (2007, 2009) for a comparison. For the inner Galaxy region, the flux is determined to be $(2.60\pm 0.13)\times 10^{-4} {\mathrm{ph\,cm^{-2}\,s^{-1}}}$, and the combined flux value is $(4.5\pm 0.8)\times 10^{-5} {\mathrm{ph\,cm^{-2}\,s^{-1}}}$. These flux value levels are consistent with the previous work (Diehl et al. 2006; Wang et al. 2007, 2009; Bouchet et al. 2015). Based on the used COMPTEL map in the inner Galaxy, the /flux ratio is $0.17\pm0.03$, consistent with the estimates from the full Galaxy, with smaller uncertainties because of the larger average exposure, and increased signal to noise ratio when more flux is actually expected in the analysed region. Of course, for the inner Galaxy, the diffuse gamma-ray continuum has a mean flux of $(4.3\pm 0.6)\times 10^{-6} \rm{ph}\ \rm{cm}^{-2} \rm{s}^{-1} \rm{keV}^{-1}$ at 1 MeV with a spectral index of $1.7\pm 0.3$. ![The fitted broad spectrum derived from the COMPTEL map only for the inner Galaxy $-30^\circ<l<30^\circ, -10^\circ<b< 10^\circ$. From this fitting, we derive the combined flux of $(4.5\pm 0.8)\times 10^{-5} {\mathrm{ph\,cm^{-2}\,s^{-1}}}$, and the flux of $(2.60\pm 0.13)\times 10^{-4} {\mathrm{ph\,cm^{-2}\,s^{-1}}}$. These values are consistent with the results in our previous work (Diehl et al. 2006; Wang et al. 2007, 2009).[]{data-label="fig:specinnergal"}](spectrum_InnerGalaxy_fit.ps){width="15cm"} Akaike, H. 1974, IEEE Trans. Autom. Control, 19, 716 Bouchet, L., Jourdain, E., & Roques, J.P. 2015, ApJ, 801, 142 Diehl, R. et al. 2006, Nature, 439, 45 Diehl, R. 2013, Reports on Progress in Physics, 76, 026301 Diehl, R. et al. 2018, A&A, 611, 12 Feige, J. et al. 2018, Phy. Rev. Lett., 121, 1103 Fimiani, L. et al. 2016, Phy. Rev. Lett., 116, 151104 Fujimoto, Y.; Krumholz, M. R.; Tachibana, S. 2018, MNRAS, 480, 4025 Haffner, L. et al. 2016, AAS Meeting No. 227, id.347.17 Jones, S. et al. 2016, A&A, 593, 72 Jones, S. et al. 2019, A&A, 622, 74 Jones, S. W. et al. 2019b, MNRAS, 485, 4287 Knie, K. et al. 2004, Phy. Rev. Lett., 93, 171103 Knödlseder, J. et al. 1999, A&A, 344, 68 Krause, Martin G. H. et al. 2015, A&A, 578, 113 Krimm, H. A. et al. 2013, ApJS, 209, 14 Kretschmer, K. et al. 2013, A&A, 559, 99 Kuffmeier, M. et al. 2016, ApJ, 826, 22 Limongi, M. & Chieffi, A. 2003, ApJ, 592, 404 Limongi, M. & Chieffi, A. 2006, ApJ, 647, 483 Limongi, M. & Chieffi, A. 2013, ApJ, 764, 21 Limongi, M. & Chieffi, A. 2018, ApJS, 237, 13 Lugaro, M. et al. 2012, Meteoritics & Planetary Science, 47, 1998 Mahoney, W.A., Ling, J.C., Jacobson, A.S., & Lingerfelter, R.E. 1982, ApJ, 262, 742 Mighell, K. J. 1999, ApJ, 518, 380 Neuhäuser, R., Giessler, F., Hambaryan, V. V. 2019, MNRAS in press (arXiv:1909.06850) Ostdiek, K. M. et al. 2017, Phys. Rev. C, 95, 055809 Pfrommer, C. et. al. 2017, ApJL, 847, L13 Pignatari, M. et al. 2016, ApJS, 225, 24 Pleintinger, Moritz M. M. et al. 2019, A&A, 632, 73 Plüschke, S. et al. 2001, ESA-SP, 459, 55 Prantzos, N. 2004, A&A, 420, 1033 Prantzos, N. & Diehl, R. 1996, Phys. Rep., 267, 1 Rauscher, T., Herger, A., Hoffman, R.D. & Woosley, S.E. 2002, ApJ, 576, 323 Rauscher, T., Herger, A., Hoffman, R.D. & Woosley, S.E. 2003, Nuclear Physics A, 718, 463 Roques, J.P. et al. 2003, A&A, 411, L91 Rugel, G. et al. 2009, Phy. Rev. Lett., 103, 072502 Schönfelder, V. et al. 1993, Advances in Space Research, 13, 647 Siegert, T. et al. 2016, Nature, 531, 341 Siegert, T. 2017, PhD Thesis, TU Munich Siegert, T. et al. 2019, A&A, 626, 73 Smith, D.M. 2004, ESA-SP, 552, 45 Strong, A. W. et al. 1994, A&A, 292, 82 Strong, A. W. et al. 2005, A&A, 444, 495 Sukhbold, T. et al. 2016, ApJ, 821, 38 Timmes, F.X. et al. 1995, ApJ, 449, 204 Timmes, F. X.; Woosley, S. E. 1997, ApJL, 481, L81 Vedrenne, G. et al. 2003, A&A, 411, L63 Vianello, G. 2018, ApJS, 236, 17 Voss, R. et al. 2009, A&A, 504, 531 Wallner, A. et al. 2015, Phy. Rev. Lett., 114, 041101 Wanajo, S., Janka, H.-T., Müller, B. 2013, ApJL, 774, L6 Wanajo, S. et al. 2018, ApJ, 852, 40 Wang, W. et al. 2007, A&A, 469, 1005 Wang, W. et al. 2009, A&A, 496, 713 Weidenspointner, G. et al. 2003, A&A, 411, L113 Winkler, C. et al. 2003, A&A, 411, L1 Woosley, S. E. & Weaver, Thomas A. 1995, ApJS, 101, 181 Woosley, S.E. 1997, ApJ, 476, 801 Woosley, S.E. & Heger, A. 2007, Physics Reports, 442, 269

harvmac tables [H]{} Ł[[L]{}]{} Ø[[O]{}]{} ¶[[P]{}]{} §[[S]{}]{} ø u v =cmmi10 scaled 2 |\#1 Ł[[L]{}]{} 0.022em \#1[0= 0-0 1.7320-0 0]{} \#1 \#1\#2\#3 = 0.5in \#2 \#3 .3in=0 Peter Cho[^1][Research supported in part by the National Science Foundation under Grant \#PHY-9218167.]{} Lyman Laboratory Harvard University Cambridge, MA 02138 0.3in **Abstract** We investigate the confining phase vacuum structure of supersymmetric $SO(11)$ gauge theories with one spinor matter field and $\Nf \le 6$ vectors. We describe several useful tricks and tools that facilitate the analysis of these chiral models and many other theories of similar type. The forms of the $\Nf=5$ and $\Nf=6$ quantum moduli spaces are deduced by requiring that they reproduce known results for $SU(5)$ SUSY QCD along the spinor flat direction. After adding mass terms for vector fields and integrating out heavy degrees of freedom, we also determine the dynamically generated superpotentials in the $\Nf \le 4$ quantum theories. We close with some remarks regarding magnetic duals to the $\Nf \ge 7$ electric $SO(11)$ theories. During the past few years, remarkable progress has been made in understanding nonperturbative aspects of $\CN=1$ supersymmetric gauge theories. Pioneering work in this area by Seiberg and collaborators has shed light upon such interesting strong interaction phenomena as phase transitions, confinement, and chiral symmetry breaking . Their studies have also opened up several new directions for model building which potentially have important phenomenological applications. Supersymmetric model investigations have thus yielded valuable insights into several basic issues in quantum field theory and particle physics. Many of the recent key advances were developed within the context of SUSY QCD which represents the prototype $\CN=1$ gauge theory . Unfortunately, it has often proven difficult to extend the new ideas beyond this relatively simple model to more complicated theories. For example, finding weakly coupled duals to strongly coupled models with no simplifying tree level superpotentials remains an outstanding challenge despite significant theoretical efforts to uncover patterns among known dual pairs. Confining phase analyses are generally more tractable than those which focus upon questions related to free magnetic and nonabelian Coulomb phases in various theories. But even addressing confinement issues in models with more complicated matter contents than those like SUSY QCD with fields in only fundamental representations frequently requires one to overcome nontrivial technical problems. In this note, we investigate the confining phase vacuum structure of a supersymmetric theory based upon an $SO(11)$ gauge group with one spinor field and $\Nf \le 6$ vectors. Our motivations for studying and presenting results on this particular model are threefold. Firstly, we wish to describe a number of useful tools that greatly facilitate the analysis of this nontrivial theory’s confining phase. These tricks can be applied to the study of many other supersymmetric theories’ low energy dynamics. While most of the simple methods which we employ have been known to nonperturbative SUSY model experts for some time , we believe it is worthwhile to discuss these previously undocumented techniques so as to make them accessible to a larger community. Secondly, the 32-dimensional spinor irrep of $SO(11)$ is pseudoreal. Since no mass term for it can be written down, our model is chiral. It may therefore have interesting applications for dynamical supersymmetry breaking. Finally and most importantly, understanding the confining phases of $\CN=1$ theories is invaluable in searching for duals. As we shall see, a dual to this $SO(11)$ model would act as a generator for several other magnetic descriptions of various electric theories. While we have not yet constructed such a dual, our present analysis restricts its possible form. Our article is organized as follows. In section 2, we discuss the low energy description of the microscopic $SO(11)$ model and identify gauge invariant operators which label its flat directions. We demonstrate that this theory confines when it contains $\Nf \le 6$ vector fields. We then proceed in section 3 to analyze the quantum moduli spaces in the $\Nf=5$ and $\Nf=6$ theories. After adding mass terms for vector fields and systematically integrating them out, we also deduce the dynamically generated superpotentials in the $\Nf \le 4$ quantum theories. Finally, we close in section 4 with some remarks and speculations regarding duals to the $SO(11)$ models with $7 \le \Nf \le 22$ vector fields. We begin our study of the $SO(11)$ model by listing its full symmetry group superfield matter content and one-loop Wilsonian beta function coefficient In the absence of any tree-level superpotential, the classical theory remains invariant under an arbitrary $G$ transformation. But in the quantum theory, each of the $U(1)$ factors in eqn.  is anomalous. The theta parameter in the $SO(11)$ Lagrangian undergoes a shift when an anomalous $U(1)$ rotation through angle $\a$ is performed. As a result, the spurion field acquires a $U(1)$ charge equal to the anomaly coefficient $C$. The charge assignments for $\Lambdab$ in  therefore simply equal the group theory coefficients of the $SO(11)^2 U(1)_\V$, $SO(11)^2 U(1)_\Q$ and $SO(11)^2 U(1)_\R$ anomalies . $G$ invariance restricts the possible form of any dynamically generated superpotential $\Wdyn$ which can arise within a low energy description of the $SO(11)$ model. For example, $\Wdyn$’s dependence upon $\Lambdab$ is completely fixed since it is the only field that carries nonvanishing R-charge. The net numbers of spinor and vector fields appearing within the nonperturbative superpotential are also determined by $U(1)_\V$ and $U(1)_\Q$ invariance. We thus easily find that $\Wdyn$ must assume the schematic forms listed in Table 1 as a function of $\Nf$. The results displayed in the table suggest that the $\Nf=5$ model is analogous to $\Nf=\Nc$ SUSY QCD inasmuch as the R-charge assignment for $\Lambdab$ vanishes in this case. As a result, no superpotential may be dynamically generated. But nonperturbative effects can alter the Kähler potential and quantum mechanically constrain the matter fields. The form such a constraint would have to take multiplied by a Lagrange multiplier field $X$ is shown in Table 1. The $\Nf=6$ $SO(11)$ model is similarly analogous to $\Nf=\Nc+1$ SUSY QCD. $\quad \Nf \quad $ $\quad R(\Lambdab) \quad $ $\quad \Wdyn \quad$ 0 10 $\bigl[\Lambda^{23} / Q^8 \bigr]^\fifth$ 1 8 $\bigl[\Lambda^{22} / V^2 Q^8 \bigr]^\quarter$ 2 6 $\bigl[\Lambda^{21} / V^4 Q^8 \bigr]^\third$ 3 4 $\bigl[\Lambda^{20} / V^6 Q^8 \bigr]^\half$ 4 2 $\Lambda^{19} / V^8 Q^8$ 5 0 $X \bigl[ V^{10} Q^8 - \Lambda^{18} \bigr]$ 6 -2 $V^{12} Q^8 / \Lambda^{17}$ Table 1: Schematic forms for dynamically generated superpotentials We next need to find $SO(11)$ invariant combinations of vectors and spinors that act as moduli space coordinates in the low energy effective theory. Equivalently, we need to determine the D-flat directions of the scalar potential in the microscopic theory. Identifying independent classical solutions to D-flatness conditions is generally a difficult task. However, we can avoid this complicated group theory exercise if we know instead the gauge symmetry breaking pattern realized at generic points in moduli space. The solution to this latter mathematical problem was worked out years ago in ref.  for a large class of theories including our particular $SO(11)$ model: Given this information, it is straightforward to count the number of $SO(11)$ singlet operators which enter into the low energy effective theory. In Table 2, we display the number of parton level matter degrees of freedom as well as the generic unbroken color subgroup as a function of $\Nf$. We also list the number of chiral superfields eaten by the superHiggs mechanism. The number of independent color-singlet hadrons in the low energy theory then simply equals the difference between the initial and eaten matter field degrees of freedom. $\quad \Nf \quad $ Parton DOF Unbroken Subgroup Eaten DOF Hadrons 0 32 $SU(5)$ $55-24=31$ 1 1 43 $SU(4)$ $55-15=40$ 3 2 54 $SU(3)$ $55-8=47$ 7 3 65 $SU(2)$ $55-3=52$ 13 4 76 1 55 21 5 87 1 55 32 6 98 1 55 43 Table 2: Number of independent hadron operators In order to figure out how to explicitly combine vector and spinor partons into gauge invariant hadrons, it is useful to recall some basic elements of $SO(11)$ group theory . The tensor product of two 32-dimensional spinor fields decomposes into irreducible $SO(11)$ representations as follows: Here $[n]$ denotes a tensor irrep with $n$ antisymmetric vector indices, and its “S” or “A” subscript indicates symmetry or antisymmetry under spinor field exchange. Since our model contains just one spinor flavor, all hadrons can only involve spinor products belonging to the symmetric ${11 \choose 1} = 11$, ${11 \choose 2}=55$ or ${11 \choose 5} = 462$ dimensional irreps. We contract vector fields into these spinor combinations using the $SO(11)$ Gamma matrices $$\Gamma_{11} = \sigthree \times \sigthree \times \sigthree \times \sigthree \times \sigthree$$ and charge conjugation matrix $C = \sigtwo \times \sigone \times \sigtwo \times \sigone \times \sigtwo$. We thus form the hadrons where Greek and Latin letters respectively denote color and flavor indices, square brackets indicate antisymmetrization and $\Vslash^i = V^i_\mu \Gamma^\mu$. Expectation values of these operators act as coordinates on the microscopic $SO(11)$ theory’s moduli space of independent flat directions. We can now check that the composite fields in eqn.  account for all independent hadronic degrees of freedom within the $SO(11)$ model’s confining phase. In Table 3, we list the number of nonvanishing hadrons as a function of flavor number. For $\Nf \le 4$, the $L$, $M$, $N$ and $O$ degrees of freedom sum up to the number of color-singlet composites entering into the low energy effective theory which we previously found in Table 2. This counting works in large part due to the antisymmetric flavor structure of the various hadrons in . When $\Nf=5$, the number of nonvanishing hadronic degrees of freedom exceeds the required number of composites by one. So a single constraint must exist among $L$, $M$, $N$, $O$, $P$ and $R$. This conclusion is consistent with our earlier finding that a relation among these fields is compatible with $R$-charge considerations in the $\Nf=5$ quantum theory. Similarly, a larger number of independent constraints must exist in the $\Nf=6$ case in order for the simple counting arguments to hold. By analogy with $\Nf=\Nc+1$ SUSY QCD, we expect these classical relations to persist in the quantum theory. $\quad \Nf \quad $ Hadrons $ \sp L \sp$ $\sp M \sp$ $\sp N \sp$ $\sp O \sp$ $\sp P \sp$ $\sp R \sp$ $\sp T \sp$ constraints 0 1 1 1 3 1 1 1 2 7 1 3 2 1 3 13 1 6 3 3 4 21 1 10 4 6 5 32 1 15 5 10 1 1 -1 6 43 1 21 6 15 6 6 1 -13 Table 3: Hadron degree of freedom count The simple tools which we have so far utilized to analyze the $SO(11)$ theory restrict the possible matter content of its low energy description. But since these heuristic methods are clearly not rigorous, we need further cross checks on our conclusions regarding the $SO(11)$ model’s vacuum structure. We therefore examine massless parton and hadron contributions to global ’t Hooft anomalies. To begin, we abandon the anomalous global symmetry in eqn.  and work instead with the nonanomalous group The generators of the new hypercharge and R-charge abelian factors are linear combinations of the three old $U(1)$ generators. After assigning the partonic matter fields the nonanomalous charges we can readily compute the quantum numbers under $G_{\rm new}$ of all the composite operators in eqn. . We next compare the $SU(\Nf)^3$, $SU(\Nf)^2 U(1)_Y$, $SU(\Nf)^2 U(1)_R$, $U(1)_Y$, $U(1)^3_Y$, $U(1)_R$, $U(1)^3_R$, $U(1)^2_Y U(1)_R$, and $U(1)^2_R U(1)_Y$ global anomalies at the parton and hadron levels as a function of $\Nf$. We find they precisely match when $\Nf=6$. This nontrivial agreement is consistent with our expectation that the $SO(11)$ model with six vectors and one spinor confines at the origin of moduli space like $\Nf=\Nc+1$ SUSY QCD. It strongly suggests that the low energy effective theory contains only the composite fields in  and no additional colored or colorless massless degrees of freedom. We further observe that all global anomalies match when $\Nf=5$ provided we include a field $X \sim \bigl( 1 ; 1 ; 0 , 2 \bigr)$ into the low energy spectrum. In the quantum theory, $X$ is naturally interpreted as a Lagrange multiplier which enforces a single constraint. Agreement between parton and hadron level anomalies occurs in other similar constrained $SO(\Nc)$ theories so long as Lagrange multiplier contributions are taken into account. It is also important to note that global anomalies do not match when $\Nf \ge 7$. The disagreement cannot be eliminated via inclusion of additional color-singlet composites into the low energy theory beyond those already listed in . So as in $\Nf = \Nc+2$ SUSY QCD , we interpret the anomaly mismatch as signaling the end of the $SO(11)$ model’s confining regime and the beginning of a new dual phase. Having established an overall picture of the $SO(11)$ theory’s confining phase, we are now ready to investigate its low energy structure in detail. We seek to determine how nonperturbative effects in the quantum theory modify the classical moduli space of degenerate vacua. One way to proceed is by starting with the $\Nf=0$ model and postulating that a superpotential consistent with the requirements of Table 1 is dynamically generated. We can then try to systematically “integrate in” vector flavors and construct superpotentials in the effective theories corresponding to larger values of $\Nf$ . This bottom-up approach unfortunately becomes intractable for $\Nf \ge 2$. Alternatively, we can follow a top-down procedure in which we first deduce the form of the nonperturbative superpotential for $\Nf=6$ flavors. Once $\Wdyn$ is known in this case, it is straightforward to methodically integrate out vector flavors and uncover the vacuum structure of the $SO(11)$ model for smaller values of $\Nf$. We will adopt this latter approach. Actually, it is technically easier to first determine the quantum constraint in the $\Nf=5$ theory. Recall that $SO(11)$ breaks down to $SU(5)$ when the spinor field develops a nonvanishing vacuum expectation value. Along the spinor flat direction, the $SO(11)$ constraint must reduce to the well-known $\Nf=\Nc=5$ relation $\det m - b \bar{b} = \Lambda_{SU(5)}^{10}$. This requirement fixes the exact form of the $SO(11)$ constraint. In order to embed $SU(5)$ inside $SO(11)$, we first choose a set of 24 fundamental irrep $SU(5)$ generators $t_a$ whose Cartan subalgebra members look like We next form annihilation and creation operators which satisfy the anticommutation relations $\{A_j, A_k \}=\{A_j^\dagger, A_k^\dagger \}=0$ and $\{A_j, A_k^\dagger\}=\d_{jk}$. The $32 \times 32$ matrices $T_a = A_j^\dagger (t_a)_{jk} A_k$ then generate the $SU(5)$ subgroup of $SO(11)$ in the spinor irrep . $SO(11)$ vectors and spinors break apart into $1+5+\bar{5}$ and $1+1+5+\bar{5}+10+\bar{10}$ under $SU(5)$. We can explicitly see how the vector decomposes by inverting the relationship in eqn.  between the operators $A_j$ and $A_j^\dagger$ which respectively transform as $5$ and $\bar{5}$ under $SU(5)$ and the Gamma matrices which transform as $11$ under $SO(11)$: Similarly, the $SU(5)$ irreps to which each of the spinor field’s 32 elements belong are readily identified by acting upon $Q$ with the four Cartan subalgebra generators: The names for the components of this row vector have been intentionally chosen to mimic familiar Standard Model and $SU(5)$ GUT nomenclature. Equal nonvanishing expectation values for the first and last entries in the spinor field break $SO(11) \to SU(5)$ while preserving D-flatness: This vev’s dependence upon a single parameter $a$ is consistent with the counting argument conclusion that the $SO(11)$ model has one independent spinor flat direction. Once the gauge symmetry is broken, we find that the $SO(11)$ hadrons decompose into the following combinations of $SU(5)$ mesons $m^{ij}$, baryons $b$ and $\bar{b}$, and singlets $\phi^i$: With this information in hand, we can assemble the hadrons into flavor singlet combinations and adjust their coefficients so as to recover the $SU(5)$ relation among mesons and baryons. After a lengthy computation, we thus deduce the quantum constraint in the $\Nf=5$ $SO(11)$ theory: Working in a similar fashion, we can determine the exact superpotential in the low energy $\Nf=6$ sigma model. Its form is tightly restricted by requiring that it be smooth everywhere on the moduli space and that its equations of motion yield valid classical relations among spinor and vector fields. Moreover, we must recover the $\Nf=5$ quantum constraint after giving mass to one of the vector flavors. The unique superpotential which satisfies these criteria is displayed below: We note that all combinations of $SO(11)$ hadrons consistent with symmetry and smoothness considerations enter into $W_{\Nf=6}$ with nonvanishing coefficients. Although this result may seem natural, other theories analogous to $\Nf=\Nc+1$ SUSY QCD are known to have zero coefficients for some [*a priori*]{} allowed terms in their superpotentials . So while it is relatively easy to figure out the basic polynomial form of the numerator in  as it is in many similar models , determining the numerical values for each term’s coefficient requires a detailed computation. Once the ground state structures of the $\Nf=5$ and $\Nf=6$ $SO(11)$ theories are known, it is straightforward to flow down to models with fewer vector fields. We simply add a tree level mass term $W_{\rm tree} = \mu_{ij} M^{ij}$ to eqn. , rotate the meson field $M^{ij}$ into diagonal form via a flavor transformation and then integrate out heavy vector flavors. Since the modified quantum theory with the meson mass term contains no sources which transform under the $SU(\Nf)$ flavor group like $N^i$ or $O^{[ij]}$, the expectation values of those fields containing heavy vectors must vanish. After systematically removing heavy vector flavors, we find the following tower of dynamically generated $SO(11)$ superpotentials: $$W_{(\Nf=4)} = {\Lambda_4^{19} \over L^2 M^4 - 4 L M^3 N^2 + 6 L M^2 O^2 - 12 M N^2 O^2 + O^4 }$$ $\downarrow$ $$W_{(\Nf=3)} = 2 \Bigl[ {\Lambda_3^{20} \over L^2 M^3 - 3 L M^2 N^2 + 3 L M O^2 - 3 N^2 O^2 } \Bigr]^{\half}$$ $$W_{(\Nf=2)} = 3 \Bigl[ {\Lambda_2^{21} \over L^2 M^2 - 2 L M N^2 + L O^2 } \Bigr]^{\third}$$ $\downarrow$ $$W_{(\Nf=1)} = 4 \Bigl[ {\Lambda_1^{22} \over L^2 M - L N^2} \Bigr]^{\quarter}$$ $\downarrow$ $$W_{(\Nf=0)} = 5 \Bigl[ {\Lambda_0^{23} \over L^2} \Bigr]^{1 \over 5}.$$ The strong interaction scales are related by requiring gauge coupling continuity across heavy vector thresholds: As a check, one can verify that these $\Nf < 5$ $SO(11)$ expressions properly reduce to their $SU(5)$ descendants along the spinor flat direction. The nonperturbative superpotentials in  lift the classical vacuum degeneracy and generate runaway scalar potentials. The $\Nf \le 4$ $SO(11)$ quantum theories can be stabilized by adding tree level superpotentials that increase along all D-flat directions. If $W_{\rm tree}$ preserves some global symmetry which is spontaneously broken in the true ground state, the chiral $SO(11)$ model should dynamically break supersymmetry . This condition on $W_{\rm tree}$ cannot be satisfied in the $\Nf=0$ theory, for all polynomial functions of the spinor field $Q$ break the global $U(1)_\R$ symmetry. Witten index arguments then suggest that SUSY remains unbroken in models with additional vector fields. Nevertheless, supersymmetry may be broken via other schemes such as coupling singlets to all the hadrons in the $\Nf=5$ quantum constraint . So whether SUSY can be dynamically broken in this $SO(11)$ theory remains an interesting open question. In this paper, we have investigated the low energy behavior of $SO(11)$ models containing $\Nf \le 6$ vectors and one spinor matter field. The tricks and tools which we used to analyze this particular theory can be profitably applied to the study of many other $\CN=1$ models. Knowing the pattern of gauge symmetry breaking at generic points in moduli space is especially valuable. Indeed, the confining phase structure of the $SO(11)$ theory is essentially fixed by $\Nc=5$ SUSY QCD along its spinor flat direction. Other $SO(\Nc)$ models of similar type are likewise restricted . It would be highly desirable to extend our understanding of the strongly interacting $SO(11)$ model into its dual phase. Our present confining phase results provide helpful clues in the search for a weakly coupled dual. In particular, the superpotential in eqn.  must be recovered from any magnetic dual to the $SO(11)$ electric theory when the number of vector flavors is reduced down to six. Our primary incentive for explicitly calculating $W_{\Nf=6}$ was in fact to determine which nonvanishing terms must be reproduced by a magnetic theory. The complex superpotential expression in  suggests the dual is not simple. Given that the spinor flat direction played a central role in our analysis of the $SO(11)$ theory’s confining phase, we naturally want to exploit it for studying the nonabelian Coulomb phase as well. Motivated by Seiberg’s dual to $\Nc=5$ SUSY QCD , we speculate that the magnetic gauge group $\tilde{G}$ contains $SU(\Nf-5)$ as a subgroup. Of course, other duals beside Seiberg’s could exist for $SU(5)$ gauge theory with $\Nf \ge 7$ quark flavors. So $\tilde{G}$ need not resemble $SU(\Nf-5)$ at all. In closing, we mention that a weakly coupled magnetic description of the electric $SO(11)$ theory would yield several other dual pairs as interesting by-products. For example after Higgsing the $SO(11)$ gauge group, one should find duals to $SO(6) \simeq SU(4)$ and $SO(5) \simeq Sp(4)$ models with $\Nf=4$ quark flavors and various numbers of antisymmetric fields. These special cases might shed light upon more general $SU(2\Nc)$ and $Sp(2\Nc)$ models with fundamental and antisymmetric matter. Finding weakly coupled duals to these theories with zero tree level superpotential remains an unsolved and challenging problem. [**Acknowledgments**]{} It is a pleasure to thank Howard Georgi and Matt Strassler for sharing their insights. I would also like to especially thank Per Kraus for collaborating at the beginning of this work and for numerous helpful discussions throughout this study. Finally, I am grateful to Csaba Csàki and Witold Skiba for their comments on the manuscript. [^1]: $^*$

--- author: - 'James F. Lutsko$^1$[^1] and [Jean Pierre Boon$^1$[^2]]{}' title: | Questioning the validity of non-extensive\ thermodynamics for classical Hamiltonian systems --- Introduction ============ The observation of natural phenomena and of laboratory experiments provides a wide spectrum of experimental results showing distributions of data that deviate from exponential decay as predicted for Boltzmann behavior [@davis]. It was the goal of non-extensive statistical mechanics developed originally by Tsallis [tsallis88]{} to offer a new approach to explain the non-Boltzmann behavior of non-equilibrium systems [^3]. More precisely the primary motivation for non-extensive thermodynamics is as a way to understand deformed exponential distributions (such as $q$-exponentials exhibiting long tails when $q>1$) found empirically in many areas of physics and other scientific disciplines [@swinney-tsallis]. The interest raised by this new approach has grown over the years and has produced an abundant literature [@tsallis09] reflecting new theoretical developments and a considerable number of applications to subjects as diverse as defect turbulence, energy distribution in cosmic rays, earthquake magnitude value distributions and velocity distributions in micro-organism populations or distributions of financial market data. Parallel to these developments, critical analyses were presented questioning the merits of the non-extensive theory [opponents]{} and thereby of its applications as well. These criticisms raised questions that often gave rise to ontological conflicts [@conflict]. Most of the criticisms are based on phenomenological analyses and thermodynamic arguments questioning the compatibility of the theory with classical statistical thermodynamics. More recently, the non-extensive theory was also re-examined on the basis of analyses demonstrating the necessary [[discreteness]{}]{} of systems to which the theory applies [@abe_10] and the limits of validity of the non-extensive formalism for a Hamiltonian system, the $q$-ideal gas, a model system of independent quasi-particles where the interactions are incorporated in the particles properties [[[abe\_99, boon\_lutsko\_11]{}]{}]{}. Here we merely adopt the viewpoint of analytical rigor to establish the validity limits of the non-extensive formalism for the general class of classical Hamiltonian systems with continuous variables and consequently of the class of physical systems to which the non-extensive interpretation applies. In the formalism, the $q$-exponential distributions arise as the result of a variational calculation maximizing the generalized entropy, the $q$-entropy, under the constraints of normalizability of the distribution function and of a prescribed average energy [@tsallis09]. The goal of the present work is to investigate the exact form of the distribution so derived for classical Hamiltonian systems from both the usual Tsallis entropy [tsallis88]{} and the homogeneous entropy [@lutsko_boon_grosfils]. The conclusion is found to be the same in both cases: the theory is only consistent in the thermodynamic limit for $0\leq q\leq1$. For finite systems of $N$ particles, the upper limit is $1+a/N$ for $a\sim{\mathcal{O}}(1)$. This means (i) that the non-extensive thermodynamics formalism cannot be used, at least in any straightforward way, to explain phenomena for which one observes that $q$ takes a value $q>1$ [[(corresponding to asymptotically power-law distributions)]{}]{} i.e. when the distributions exhibit extended exponential forms with long tails, and (ii) that, within the validity domain, the distribution has finite support, thus implying that configurations with e.g. energy above some limit have zero probability, a strange situation for systems with typical molecular potentials which are steeply repulsive at small distances. We treat successively the case according to the development based on the Tsallis entropy $S_{q}$ and the case based on the homogeneous entropy $S^{H}_{q}$. [[The reason for examining the two cases is the criterion of stability against perturbations of the probability distribution function, or [*[Lesche stability]{}*]{} [@lesche_abe]]{}]{} : the homogeneous entropy was shown to be Lesche stable while the Tsallis entropy is not [@lutsko_boon_grosfils]. We also present the results for the generalized ($q$-dependent) thermodynamic properties of Hamiltonian systems in both cases. Tsallis entropy =============== Non-extensive statistical mechanics is developed on the basis of three axioms: (i) the $q$-entropy for systems with continuous variables is given by [@tsallis09] $$S_{q}\,=\,k_{B}\,\frac{1-{K\int \rho ^{q}\left( \Gamma \right) d\Gamma }}{q-1}\,, \label{STq}$$where $\Gamma$ is the phase space variable and $K$ must be a quantity with the dimensions of $\left[ \Gamma \right] ^{q-1}$, i.e. $K=\hbar ^{DN\left( q-1\right) }$ ($D$ denotes the dimension of the system and $N$ the number of particles) and the classical Boltzmann-Gibbs entropy is retrieved in the limit $q\rightarrow 1$; (ii) the distribution function $\rho \left( \Gamma \right) $ is slaved to the normalization condition $$1\,=\,\int \rho \left( \Gamma \right) d\Gamma \,; \label{Norm}$$(iii) the internal energy is measured as $$U\,=\,{\int P_{q}\left( \Gamma \right) H\,d\Gamma }\,=\,\frac{\int \rho ^{q}\left( \Gamma \right) H\,d\Gamma }{\int \rho ^{q}\left( \Gamma \right) d\Gamma }\,, \label{Uesc}$$where $P_{q}\left( \Gamma \right) $ is the escort probability distribution [@beck_schlogl] which is the actual probability measure. We consider Hamiltonian systems with $H=T+V$ where $T$ is the classical kinetic energy. The distribution function $\rho \left( \Gamma \right) $ is obtained by maximizing the $q$-entropy subject to the normalization (\[Norm\]) and average energy (\[Uesc\]). Introducing the Lagrange multipliers $\bar{\alpha}$ and $\bar{\gamma}$, the variational method leads to $$0 = \frac{k_{B}Kq}{1-q}\rho^{q-1}(\Gamma)-\bar{\alpha} - q\bar{\gamma}\frac{(H-U)}{\int \rho ^{q}\left( \Gamma \right)d\Gamma }\rho^{q-1}(\Gamma) \,, \label{vareq}$$ which is solved to yield $$\rho \left( \Gamma \right) =\frac{\left( 1-\left( 1-q\right) \gamma \left( H-U\right) \right) ^{\frac{1}{1-q}}}{\displaystyle\int \left( 1-\left( 1-q\right) \gamma \left( H-U\right) \right) ^{\frac{1}{1-q}}d\Gamma }\,, \label{rhoT1}$$with $\gamma = \frac{\bar{\gamma}}{K\int \rho ^{q}\left( \Gamma \right)d\Gamma }$ and where the normalization condition (\[Norm\]) has been used to eliminate the multiplier $\bar{\alpha}$. The other, $\gamma $, is determined from the energy constraint (\[Uesc\]); using [(\[rhoT1\])]{} in (\[Uesc\]), we have $$0=\frac{{\displaystyle \int}\left( 1- \left( 1-q\right)\,{\gamma}\, \left( H-U\right)\right) ^{\frac{q}{1-q}}\left( H-U\right) d\Gamma }{{\displaystyle \int}\left( 1- \left( 1-q\right)\,{\gamma}\, \left( H-U\right)\right)^{\frac{q}{1-q}}d\Gamma } \,,$$which by multiplying the numerator by $(1-q)\gamma$ and adding and subtracting one gives $$\begin{aligned} {\int \left( 1-\left( 1-q\right) \gamma \,\left( H-U\right) \right) ^{\frac{q}{1-q}}d\Gamma } = {\int \left( 1-\left( 1-q\right) \gamma \,\left( H-U\right) \right) ^{\frac{1}{1-q}}d\Gamma }\,. \label{UT}\end{aligned}$$Note that the sign of the factor $f(\Gamma )\equiv 1-\left( 1-q\right) \gamma \left( H-U\right) $ is, so far, arbitrary so that we allow for the cancellation of a (possible imaginary) factor throughout these equations. However, this is only possible in the expression for the distribution [(\[rhoT1\])]{} if the sign of $f(\Gamma )$ is independent of $\Gamma $. Given this fact, the sign can be fixed by making the substitution $f(\Gamma )=s\left|f(\Gamma )\right|$ in the energy constraint which requires that $s^{\frac{q}{1-q}}=s^{\frac{1}{1-q}}$ so that $s=1$. Since the kinetic energy is in principle unbounded, the constant sign of the argument means that the distribution may have to be restricted to some subset of phase space so that it should be written as $$\rho \left( \Gamma \right) =\frac{\left( 1- \left( 1-q\right) \gamma \left( H-U\right) \right)^{\frac{1}{1-q}} \Theta\left( 1- \left( 1-q\right) \gamma \left( H-U\right) \right)}{{\displaystyle \int}\left( 1-\left( 1-q\right) \gamma\left( H-U\right)\right)^{\frac{1}{1-q}} \Theta\left( 1- \left( 1-q\right) \gamma \left( H-U\right) \right)d\Gamma}\,, \label{rhoT11}$$where $\Theta (x)$ is the step function which is one for $x>0$ and zero otherwise. We note that the introduction of the step function is in fact a redefinition of the variational problem in the sense that we have replaced $\rho(\Gamma)$ with $\bar{\rho}(\Gamma)\Theta(f(\Gamma))$ in Eq.(\[STq\]-\[Uesc\]) and then maximized with respect to $\bar{\rho}(\Gamma)$. If we do not do this, then there is simply no solution to the variational problem which is real and non-negative for all $\Gamma$. To see this, we note from the variational equation (\[vareq\]) that if $\rho(\Gamma)$ vanishes for some value of $\Gamma$ then it necessarily follows that $\bar{\alpha} = 0$, which is untenable since $\bar{\alpha}$ will generally assume a non-zero value due to the normalization condition. Therefore, we can only restrict the support of the distribution by redefining the variation problem. We now turn to the determination of the possible values for $q$. [*[Case:]{} $q<1$*]{}. The exponent occuring in the distribution is $\frac{1}{1-q}>0$ so that if $\gamma>0$, then the distribution must have finite support since for some sufficiently large value of $T$, the argument, $f(\Gamma)$, becomes zero, and from (\[rhoT1\]) we have $$\rho\left( \Gamma\right) =\frac{\exp_{q}\left( -\gamma\left( H-U\right) \right) }{\int\exp_{q}\left( -\gamma\left( H-U\right) \right) d\Gamma }\,, \label{rhoTq}$$ where [[$\exp_q(x) \equiv (1+(1-q)x)^{\frac{1}{1-q}}\Theta(1+(1-q)x)$ ]{}]{}is the $q$-exponential function. If $\gamma<0$, then $f(\Gamma)$ is always positive, and the distribution function never goes to zero so that there can be no finite support. This however leads to another problem since $f(\Gamma)$ is unbounded as the kinetic energy increases which, in turn, means that the integral of $f(\Gamma)$ over momenta will diverge so that the distribution cannot be normalized. We conclude that $\gamma<0$ is not allowed. We remark that it might be thought that one could introduce a limit on the kinetic energy, but this is not in keeping with the proposal that the nonextensive distribution be a generalization of the canonical distribution which describes [*[open systems]{}*]{} in contact with a reservoir. [*[Case:]{} $q>1$*]{}. The exponent being then $\frac{1}{1-q}<0$, we write the distribution function as$$\rho \left( \Gamma \right) =\frac{\left( 1+ \left\vert 1-q\right\vert \gamma \left( H-U\right) \right) ^{-\left\vert \frac{1}{1-q}\right\vert } \Theta \left(1+ \left\vert 1-q\right\vert \gamma\left( H-U\right) \right)}{\displaystyle \int \left( 1+ \left\vert 1-q\right\vert \gamma \left( H-U\right) \right) ^{-\left\vert \frac{1}{1-q}\right\vert } \Theta \left(1+ \left\vert 1-q\right\vert \gamma\left( H-U\right) \right) d\Gamma } \,.$$If $\gamma >0$, the distribution will be normalizable provided that $\left( \sum p_{i}^{2}\right) ^{-\left\vert \frac{1}{1-q}\right\vert }$ is integrable over $d^{ND}p$ (for large $p$) which is to say that $\left( P^{2}\right) ^{-\left\vert \frac{1}{1-q}\right\vert }P^{ND-1}dP$ be integrable for $P\rightarrow \infty $; this means we must have $-1>ND-1-2\left\vert \frac{1}{1-q}\right\vert $, or $1<q<1+\frac{2}{ND}$ which condition reduces to the classical Boltzmann result ($q=1$) in the thermodynamic limit [^4]. The other possibility is $\gamma <0$. We then write the distribution as$$\rho \left( \Gamma \right) =\frac{\left( 1-\left\vert \gamma \right\vert \left\vert 1-q\right\vert \left( H-U\right) \right)^{-\left\vert \frac{1}{1-q}\right\vert} \Theta \left( 1-\left\vert \gamma \right\vert \left\vert 1-q\right\vert \left( H-U\right) \right) }{\displaystyle \int \left( 1-\left\vert \gamma \right\vert \left\vert 1-q\right\vert \left( H-U\right) \right) ^{-\left\vert \frac{1}{1-q}\right\vert }\Theta \left( 1-\left\vert \gamma \right\vert \left\vert 1-q\right\vert \left( H-U\right) \right) d\Gamma }$$When integrated over momenta, this expression would show a singularity at $T=\frac{1}{\left\vert \gamma \right\vert \left\vert 1-q\right\vert }+U-V$, unless\ $\int {\left( X-\sum_{i}p_{i}^{2}\right) ^{-\left\vert \frac{1}{1-q}\right\vert }}dp^{3N} \sim \int_{0}^{\sqrt{X}}{\left( X-P^{2}\right)^{\left\vert \frac{-1}{1-q}\right\vert }}P^{3ND-1}dP \sim \int_{0}^{X}{\left(X-Y\right) ^{\left\vert \frac{-1}{1-q}\right\vert }}Y^{\frac{3ND-2}{2}}dY$ is finite; this requires that $1-\left\vert \frac{1}{1-q}\right\vert =1-\frac{1}{q-1}>0$, i.e. $q>2$ . But $\rho ^{q}\left( \Gamma \right) $ must also be integrable and in order that the singularity be integrable imposes $1-\frac{q}{q-1}>0$ which is incompatible with the condition $q>2$. Furthermore the maximization condition demands that the second derivative of the $q$-entropy [(\[STq\])]{} be $- k_B\,K\, q \,\rho ^{q - 2} < 0$, which is satisfied if $q > 0$. So in summary, the distribution function exists in the thermodynamic limit for $\gamma> 0$ and $0\leq q \leq 1$, and $\rho\left( \Gamma\right) $ has the form of a $q$-exponential with finite support. Homogeneous entropy =================== The homogeneous entropy was proposed as an alternative to the Tsallis entropy because it has various desirable properties that the Tsallis entropy does not share such as being Lesche stable and giving positive-definite heat capacities [@lutsko_boon_grosfils]. It is therefore interesting to ask whether it is subject to the same limitations as found for the Tsallis entropy. The analysis follows essentially the same lines as above except that in this case $\rho\left(\Gamma\right) $ is the physical probability [@lutsko_boon_grosfils] and so the energy is computed with the normal average. The homogeneous entropy reads [@lutsko_boon_grosfils] $$S_{q}^{H}=k_{B}\,\frac{1-\left( K\int\rho^{\frac{1}{q}}\left( \Gamma\right) d\Gamma\right) ^{q}}{1-q}\,, \label{SHq}$$ where $K$ is a quantity with the dimensions $\left[ \Gamma\right] ^{\frac{1-q}{q}}$, i.e. $K=\hbar^{DN\,\frac{1-q}{q}}$, and the normalization and energy constraints are $$1=\int\rho\left( \Gamma\right) d\Gamma\;\;\;\;;\;\;\;\;U={\int\rho\left( \Gamma\right) H\,d\Gamma}\,.$$ Following a similar analysis as given in the previous section, the conclusions are that the condition for normalizability is $0\leq q\leq1$ and $\gamma>0$ in which case the distribution function reads $$\rho\left( \Gamma\right) \,=\,\frac{\left( \exp_{q}\left( -\,\gamma \,\left( H-U\right) \right) \right) ^{q}}{\int\,d\Gamma\,\left( \exp _{q}\left( \,-\gamma\,\left( H-U\right) \right) \right) ^{q}}\;. \label{rhoHq}$$ Here $\gamma = \frac{\bar{\gamma}}{\left(K \int \rho ^{\frac{1}{q}} \left( \Gamma \right) d\Gamma \right)^q}$ where $\bar{\gamma}$ is the Lagrange multiplier used to fix the average energy and $\rho\left( \Gamma\right)$ has finite support. Thermodynamic properties ======================== Given that the formalism is constructed solely on the basis of the three axioms [(\[STq\])]{}, [(\[Norm\])]{} and [(\[Uesc\])]{}, the consistent way to define the thermodynamic temperature is through the thermodynamic definition ${\partial S}/{\partial U}= 1/T$. Considering $q<1$ and $\gamma >0$, we obtain from the Tsallis entropy [(\[STq\])]{} $$\begin{aligned} {T^T_q}=\frac{1}{ k_B\,\gamma}\left( K^{\frac{1}{1-q}} \int \exp_q\left(-\gamma\left(H-U\right)\right) d\Gamma \right)^{q-1} \,, \label{TqT}\end{aligned}$$and from the homogeneous entropy [(\[SHq\])]{} $$\begin{aligned} T^H_q \,=\,\frac{q}{k_B\,\gamma} \left( K^{\frac{q}{q-1}} \int \exp_q \left( - \frac{\gamma}{q}\left(H-U\,\right)\right)d\Gamma \right)^{1-q} \label{THq}\end{aligned}$$ With the explicit expressions of ${\gamma}$ (see [(\[rhoTq\])]{} and [(\[rhoHq\])]{}), [(\[THq\])]{} gives ${\bar{\gamma}}\,=\,{1}/{(k_B\,T^H_q)}$, the analog of the classical expression, whereas the equivalent relation for the Tsallis temperature is only obtained in the limit $q \rightarrow 1$. Correspondingly, the expressions for the specific heat $C_{V}=\left( \frac{\partial U}{\partial T_q}\right)$ are given by $$\begin{aligned} C^T_{V}= \frac{{\beta^T}/{\gamma}}{\left[ \frac{1}{q}\left(\frac{\beta^T}{\gamma}\right)^{4}\frac{1}{K^{2}\int\rho^{2q-1}\left( \Gamma\right) \left( \beta^T\left( H-U\right) \right)^{2}d\Gamma}-2\left( 1-q\right) \right]} $$ with the classical notation $\beta^T = 1/(k_BT^T_q)$, and by $$C^H_{V} = q^{\frac{1}{q}} \left( {\frac{\beta^H}{\gamma}} \right)^{\frac{1-q}{q}} \, {\frac{k_{B}}{K}}\int\rho^{\frac{2q-1}{q}}\left( \Gamma\right) \left( \beta^H\left(H-U\right) \right) ^{2}d\Gamma \label{CvH}$$ where $\beta^H = 1/(k_BT^H_q)$, or $$C^H_{V} = {\frac{1}{k_{B}\,(T_q^H)^2}}\int d\Gamma \rho \left( \Gamma\right) \left(H-U\right)^{2}\,\mathsf{C}_q(\Gamma) \label{CvH1}$$ with $\mathsf{C}_q(\Gamma) = q^{\frac{1}{q}}\left(\frac{\beta^H}{\gamma}\right)^{\frac{1-q}{q}}\frac{\rho^{\frac{q-1}{q}}\left( \Gamma\right)}{K}$. [(\[CvH1\])]{} is the generalization of the classical expression of the specific heat given in terms of the energy fluctuations: $C_{V} = {\langle (\Delta E)^2\rangle}/({k_{B}\,T^2})$. The thermodynamic temperatures are both positive while the generalized specific heat is only positive-definite in the case of the homogeneous entropy.[^5] Concluding comments =================== Non-exponential distributions are widely observed in nature. Non-extensive thermodynamics was motivated, in part, as a means of explaining the origin of such distributions which arise naturally as a result of maximizing the generalized entropy with the usual constraints of the normalization of the distribution and of fixed average energy. We have shown that when this procedure is applied to Hamiltonian systems, the resulting distributions only exist for the restricted range of $0<q<1+{\mathcal O}\left( \frac{1}{N}\right) $. Since the so-called “fat-tailed” distributions correspond to $q>1$, this means that generalized thermodynamics cannot be seen as an explanation of their occurrence for these systems. The problem with larger values of $q$ has to do with the existence of the integrals over momenta due to the unboundedness of the kinetic energy. One way around this would be to redefine the formalism so as to restrict the momenta a priori by making the ansatz ${\rho\left( \Gamma\right) =\Theta}\left( T_{\ast}-T\right) {\rho}_{\ast}{\left( \Gamma\right) }$, for some fixed positive number $T_{\ast}$, throughout the variational problem and maximizing to determine ${\rho}_{\ast}{\left( \Gamma\right) }$. However, this is obviously quite artificial and [*ad hoc*]{} since, for example, one could replace the step function by any function of $T$ that goes to zero sufficiently quickly as $T$ grows. This suggests the more straightforward conclusion that in the case of classical Hamiltonian systems, nonextensive thermodynamics does not provide a simple, natural explanation of distributions with fat tails. This work was [partly]{} supported by the European Space Agency under contract number ESA AO-2004-070. [99]{} A. Scott, ed., [*Encyclopedia of Nonlinear Science*]{} (Taylor and Francis, New York, 2005). C. Tsallis, *J. Stat. Phys.*, **52**, 479 (1988). C. Beck and E.G.D. Cohen, *Physica A*, **321**, 267 (2003). J. F. Lutsko and J. P. Boon, *Phys. Rev. E*, **77**, 051103 (2008). H.L. Swinney and C. Tsallis, eds., *Anomalous Distributions, Nonlinear Dynamics, and Nonextensivity*, *Physica D*, **193** (2004); J.P. Boon and C. Tsallis, eds., *Nonextensive statistical mechanics: new trends, new perspectives*, *Europhys. News*, **36/6**, [183-231]{} [(2005)]{}; Chap.7 in [@tsallis09]. , *Introduction to Nonextensive Statistical Mechanics* [(Springer, New York, 2009)]{}, see Bibliography. R. Luzzi, A.R. Vasconcelos and J. Galvao Ramos, *Science*, **298**, 1171 (2002); M. Nauenberg, *Phys. Rev. E*, **67**, 036114 (2003); D.H. Zanette and M.A. Montemurro, *Physics Lett. A*, **316**, 184 (2003); P. Grasberger, *Phys. Rev. Lett.*, **95**, 140601 (2005); D.H. Zanette and M.A. Montemurro, *Physics Lett. A*, **324**, 48 (2007). A. Cho, *Science*, **297**, 1269 (2002); A. Plastino, *Science*, **300**, 250 (2003); V. Latora, A. Rapisarda and A. Robledo, *Science*, **300**, 250 (2003); C. Tsallis, *Phys. Rev. E*, **69**, 038101 (2004); M. Nauenberg, *Phys. Rev. E*, **69**, 038102 (2004); R. Balian and M. Nauenberg, *Europhys. News*, **37**, [9]{} [(2006)]{}; F. Bouchet, T. Dauxois and S. Ruffo, *Europhys. News*, **37**, [9]{} [(2006)]{}; A. Rapisarda and A. Pluchino, *Europhys. News*, **37**, [10]{} [(2006)]{}; R. Luzzi, A.R. Vasconcelos and J. Galvao Ramos, *Europhys. News*, **37**, [11]{} [(2006)]{}; see also discussion in Chap.8 of reference [@tsallis09]. , *Europhys. Lett.*, **90**, [50004]{} [(2010)]{}. , *Physics Lett. A*, **375**, [329]{} [(2011)]{}. , *Europhys. Lett.*, **86**, [40005]{} [(2009)]{}. C. Beck and F. Schlögl, [*Thermodynamics of Chaotic Systems*]{}, (Cambridge University Press, London, 1993). [^1]: E-mail: [^2]: E-mail: [^3]: We note that other approaches exist such as super-statistics [@beck-cohen] and the fractional and nonlinear Fokker-Planck equations (see e.g. [@lutsko-boon] and references therein) [^4]: Note that $\rho^{q}\left( \Gamma\right) $ must also be integrable which imposes that $-1 > ND-1-\frac{2q}{q-1}$ or $q < 1+\frac {2}{ND-2}$, but this condition is weaker than the constraint $1<q<1+\frac{2}{ND}$. [^5]: [It was indeed shown that in the Tsallis formulation the specific heat can be negative [@abe_99].]{}

--- abstract: 'In this work, angular distribution measurements for the elastic channel were performed for the $^{9}$Be+$^{12}$C reaction at the energies E$_{Lab}$=13.0, 14.5, 17.3, 19.0 and 21.0 MeV, near the Coulomb barrier. The data have been analyzed in the framework of the double folding São Paulo potential. The experimental elastic scattering angular distributions were well described by the optical potential at forward angles for all measured energies. However, for the three highest energies, an enhancement was observed for intermediate and backward angles. This can be explained by the elastic transfer mechanism.' address: - 'Instituto de Física, Universidade de São Paulo, CEP 66318, 05315-970, São Paulo, SP, Brazil.' - 'National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, Michigan.' author: - 'R. A. N. Oliveira' - 'N. Carlin' - 'R. Liguori Neto' - 'M. M. de Moura' - 'M. G. Munhoz' - 'M. G. del Santo' - 'F. A. Souza' - 'E. M. Szanto' - 'A. Szanto de Toledo' - 'A. A. P. Suaide' title: 'Study of $^{9}$Be+$^{12}$C elastic scattering at energies near the Coulomb barrier' --- $^{9}$Be+$^{12}$C ,Elastic Scattering ,São Paulo Potential 24.10 Introduction ============ In the last few years, nuclear reactions involving weakly bound nuclei became a subject of interest due to the observation of flux enhancement for processes like nucleon transfer and breakup. Through the study of these processes [@1; @3; @I6], it is possible to obtain information about nuclear structure, such as single-particle states and nuclear cluster structure, as well as information about the influence of continuum states in the nuclear reaction dynamics [@4; @5; @6]. Additionally, the investigation on how these properties change from the stability line to regions far from the stability valley can also be addressed. In this context, the elastic scattering measurement and coupled channel analysis [@I9] are very important tools to investigate nucleon transfer and breakup, as they appear as competing mechanisms in the reproduction of the measured angular distributions. In this work, elastic scattering cross section measurements were performed for the ${^9}$Be+$^{12}$C reaction to study anomalies [@I9; @7; @8] in the extracted optical parameters values and the contribution of inelastic channels, transfer and compound nucleus formation in the elastic scattering process. The experimental data show an enhancement in the elastic cross sections at intermediate and backward angles. This behavior is typically observed in systems where projectile and target present the same core structure [@7; @8; @I7]. This effect can be understood in terms of a ${^3}$He transfer process, assuming that $^{12}$C has a ${^3}$He+$^{9}$Be cluster structure. The elastic scattering angular distributions were analyzed in a four steps procedure. Distinctively of previous works, our experimental elastic scattering data have been compared to optical model predictions using a empirical double folding potential [@9]. In this process, the normalization parameters for the real and imaginary potentials were adjusted to describe the data at forward angles ($\theta_{CM}$$\leq$80$^{\circ}$). The normalization of the real part of the potential shows a decrease as a function of the bombarding energy, and the normalization of the imaginary part of the potential is approximately constant. During the following two steps, we investigate the importance of the coupling to inelastic and transfer channels respectively, and finally in the fourth step we analyse the compound elastic contribution. Experimental Setup {#setup} ================== The experiment was performed at the University of São Paulo Physics Institute. The ${^9}$Be beam was delivered by the 8UD Pelletron accelerator with energies E$_{Lab}$=13.0, 14.5, 17.3, 19.0 and 21.0 MeV (E$_{CM}$=7.4, 8.3, 9.9, 10.8 and 12.0 MeV respectively) and hit a 40 $\mu$g/cm$^2$ thick $^{12}$C target. The charged particles produced in the ${^9}$Be+$^{12}$C reaction were detected by means of 13 triple telescopes [@10] separated by $\Delta\theta$ = 10$^{\circ}$ in the reaction plane, which covered the angular range from $\theta$ = 10$^{\circ}$ to $\theta$ = 140$^{\circ}$. The triple telescopes were composed of an ionization chamber, filled with 20 [*torr*]{} of isobutane gas, followed by a 150 $\mu$m silicon detector and a 40 mm CsI scintillator crystal. The identification of the elastic events was done by means of two-dimensional spectra like the one shown in Fig. \[biparametrico\]-a. The elastic yields were obtained by projecting the Z = 4 region on the E$_{Si}$ axis (energy measured by the Silicon detector) and identifying the elastic processes as shown in Fig. \[biparametrico\]-b. As ilustraded in this figure, we can see a considerable number of events due to target contamination. In order to subtract the contributions in the energy spectra from this target contamination, we have assumed Rutherford scattering for the involved cross sections. This is a good approximation in our experimental conditions. The uncertainties in the differential cross section were estimated considering the statistical uncertainty in the yield and the systematic uncertainty of 5% in the target thickness. Data Analysis and Discussion {#Analysis} ============================ Optical Model Calculations -------------------------- In the present work, the experimental data were analyzed using the FRESCO code [@I4] in the framework of the São Paulo potential (SPP) [@9] due to the recent success of this approach in describing nuclear reactions involving weakly bound nuclei [@I10; @I11]. Clearly, there are other options for the scattering potential, as shown in recent works by A. T. Rudchick [*et. al.*]{} [@8] and J. Carter [*et. al.*]{} [@7] who have used a Woods-Saxon shape and a double folding potential, respectively, and the effects of coupled channels to describe the ${^9}$Be+$^{12}$C scattering data. For the São Paulo potential, the radial dependence is a folding potential and the energy dependence takes non-local effects into account, in the form $$V(R,E)=V_F(R)\exp\left(-4\beta^2\right)\hspace{2mm},$$ where $\beta$=v/c, $v$ is the local relative velocity between the two nuclei and $V_F(R)$ is a folding potential obtained by using the matter distributions of the nuclei involved. In detail, the folding potential depends on the matter densities in the form $$V_F(R)=\int \rho(\overrightarrow{r}_1)\rho(\overrightarrow{r}_2)v_{nn}(\overrightarrow{R}-\overrightarrow{r}_1-\overrightarrow{r}_2)d^3r_1d^3{r_2}$$ where $v_{nn}(\overrightarrow{R}-\overrightarrow{r}_1-\overrightarrow{r}_2)$ is a physical nucleon-nucleon interaction given by $$v_{nn}(\overrightarrow{R}-\overrightarrow{r}_1-\overrightarrow{r}_2)=V_0\delta(\overrightarrow{R}-\overrightarrow{r}_1-\overrightarrow{r}_2)$$ with $V_0=-456$ MeVfm${^3}$ and the usage of a delta function corresponds to the zero range approach. Extensive systematics were performed in Ref. [@9], to provide a good description of matter and charge distribution. In the present work, we adopt the matter and charge diffuseness $a_m= 0.53$ fm and $a_c=0.56$ fm respectively, and the matter and charge distribution radius given by $R_M = 1.31A^{1/3}-0.84$ and $R_C = 1.76Z^{1/3}-0.96$ respectively. In this first step of the analysis, we considered the real and imaginary potential normalizations as free parameters that are adjusted in order to describe the forward angles of the angular distribution ($\theta_{CM}$ $\leq$ 80$^{\circ}$). In this case the nuclear potential is given by the equation $$V_{SPP}=(N_r+iN_i)V_F(R)\exp\left(-4\beta^2\right)\hspace{2mm}, \label{sppotential}$$ where the normalization coefficients $N_r$ and $N_i$ are reaction energy dependent. The results for the first step are shown as solid lines in Fig. \[elastico\]. We observe a good agreement for the forward angular region. However, a pronounced disagreement at backward angles for 17.3, 19.0 and 21.0 MeV, is an indication of the importance of other reaction mechanisms, not taken into account in the optical potential. For 13.0 and 14.5 MeV, the optical model description is reasonable. Finally, Fig. \[elastico\] also shows an angular distribution for E$_{Lab}$=19.0 MeV extracted from Ref. [@7]. These data are in good agreement with our data and with the optical model calculation at forward angles. The $N_r$ and $N_i$ energy dependence is depicted in Fig. \[nrni\]. The uncertainties are determined by a $\chi^2$ analysis. One can notice that N$_r$ increases in the vicinity of the Coulomb barrier energy (E$_{CM}$ $\approx$ 9 MeV [@8; @I5]). The values of N$_i$ are approximately constant for all energies, and they are in agreement with the results obtained in Ref. [@I1]. The increase of the N$_r$ parameter near the Coulomb barrier suggests the presence of a threshold anomaly [@I2; @I9]. However, no strong statement about this anomaly can be made due to the constant behavior of N$_i$. The results from the first step of the analysis were used to perform the coupling of the inelastic and transfer channels presented in the next sections. ![Angular distributions for the $^9$Be + $^{12}$C system at E$_{Lab}$ = 13.0, 14.5, 17.3, 19.0 and 21.0 MeV. The solid lines correspond to optical model fits, the dashed and dash dotted lines represent the same optical potential including the inelastic and elastic transfer mechanisms respectively (Color Online).[]{data-label="elastico"}](elasticoV3.eps) ![Best values for N$_r$ and N$_i$ as a function of the bombarding energy obtained from fits using the São Paulo potential (Equation \[sppotential\]) for the $^{9}$Be + $^{12}$C system. The lines are just to guide the eyes (Color Online).[]{data-label="nrni"}](NrNi.eps) Inelastic Channels ------------------ With the optical potentials previously obtained, we are able to take into account the effects of other channels. The second step of the analysis consisted in calculating the effects of inelastic channels in the theoretical elastic cross section, by considering 5/2${^-}$ and 7/2${^-}$ states of $^9$Be and 2${^+}$ of $^{12}$C. In our study the transitions to excited states of $^9$Be and $^{12}$C were calculated using the rotational model approach, where the coupling interaction V$_{\lambda}$([**r**]{}) of the multipole $\lambda$ is $$V_{\lambda}({\bf r})=-\delta_{\lambda}\frac{dV({\bf r})}{dr}$$ The Coulomb excitations are included by considering deformations on the charge distributions in the form $$V_{\lambda}^{C} = M(E\lambda)\frac{\sqrt{4\pi}e^2}{2\lambda+1} \left\{ \begin{array}{rcl} r^{\lambda}/r_c^{2\lambda+1}&r\leq r_c\\ 1/r^{\lambda}&r>r_c\\ \end{array} \right. \label{IN8}$$ where $$M(E\lambda)=\sqrt{(2J_{i}+1)B(E\lambda;J_{i}\to J_{f})}\hspace{0.2cm}. \label{IN11}$$ Therefore, to take into account the effects of inelastic process in the theoretical elastic cross section, we take from the literature [@7; @8; @11] the nuclear deformation parameters $\delta_{\lambda}$ and the reduced transition probabilities B(E$\lambda$:J$_{i}$$\to$J$_{f}$), in order to calculate the deformations on the symmetrical central potential. The parameters for each channel included in the calculations and the reduced transition probabilities are presented in table \[tabela1\]. [p[2cm]{} p[2cm]{} p[2 cm]{} p[2cm]{} p[2cm]{} ]{} Nucleus & Transition & $\lambda$ & B(E$\lambda$) & $\delta_{\lambda}$\ ${^9}$Be & $\frac{3}{2}^-$ $\to$ $\frac{5}{2}^-$ & 2 & 46.0$\pm$0.5 & 2.4\ \ ${^9}$Be & $\frac{3}{2}^-$ $\to$ $\frac{7}{2}^-$ & 2 & 33$\pm$ 1 & 2.4\ \ $^{12}$C & $0^+$ $\to$ $2^+$ & 2 & 42$\pm$ 1 & 1.52\ \ \ \[tabela1\] The results are shown as dashed lines in Fig. \[elastico\]. In general, we observed that the inclusion of these channels decreases the theoretical cross section when compared to the results obtained only with the optical potential. Comparing with the experimental data, we observed that the curves show a good description at forward angles. For intermediate and backward angles, the theoretical prediction underestimates the cross section. This demonstrates that the inclusion of these inelastic channels is not sufficient to explain the experimental results at intermediate and backward angles. $^{3}$He Cluster Transfer ------------------------- In order to improve the description of experimental elastic distributions at intermediate and backward angular region, a coupling to the $^{3}$He transfer was included in the third step of the analysis. [p[3cm]{} p[1.75cm]{} p[1.5cm]{} p[1.5cm]{} ]{} Systems & V$_{r}$(MeV) & r$_{r}$(fm) & a$_{r}$(fm)\ \ ${^9}$Be+$^{9}$Be$^{a}$ & 189.3 & 1.0 & 0.63\ \ ${^3}$He+${^9}$Be & 55.8$^{b}$ & 1.35 & 0.65\ \ \ [p[3cm]{} p[1.75cm]{} p[1.5cm]{} p[1.5cm]{}]{} A=C+v & J$^{\pi}$ & $nlj$ & A\ \ $^{12}$C=$^{9}$Be $\oplus$ $^{3}$He & 0$^{+}$ & 2p$_{3/2}$$^c$ & 1.224$^c$\ \ \[parametrospotencialoptico\] \ --------------------------------------------------------- $^{a}$ Ref. [@12; @13]. $^{b}$ Adjusted by FRESCO code to reproduce the cluster binding energy. $^{c}$ Ref. [@7; @14]. --------------------------------------------------------- : Optical parameters for DWBA calculations and spectroscopic factor for a A=C+v system.[]{data-label="tabela2"} \ The calculations for $^{12}$C($^9$Be,$^{12}$C)$^9$Be elastic transfer channel were done using the potential obtained in the optical model analysis. The $^{12}$C was considered as a cluster structure, composed of a $^9$Be core and a $^3$He valence particle in a single-particle state. The bound state wave functions were generated using a binding potential with a Woods-Saxon shape, with geometric parameters shown in table \[tabela2\]. The depth was adjusted to give the correct separation energy of the clusters. In the calculation, $^3$He is considered to be transferred from the 2p$_{3/2}$ single particle state in $^{12}$C(J$^{\pi}$=0${^+}$) to the same orbital on the $^9$Be(J$^{\pi}$=3/2${^-}$) projectile. The spectroscopic factor for the $^{12}$C$_{g.s}$ = $^9$Be$_{g.s}$ $\oplus$ $^3$He cluster structure was taken from the literature and is listed in table \[tabela2\]. The calculated angular distributions are shown in Fig. \[elastico\], as dash-dotted lines, and we can see the fair agreement with experimental data for 17.3, 19.0 and 21.0 MeV, showing the importance of the elastic transfer process at these energies. However, for 13.0 and 14.5 MeV, the importance of the inclusion of these channels is not clear. In the case of 19.0 MeV, one can notice that the theoretical prediction presents a good agreement when compared with the experimental data from Ref. [@7]. Compound Nucleus Formation -------------------------- The formation of compound nucleus is another mechanism that can also contribute to increase the cross sections at intermediate and forward angles [@15; @16]. Usually, in this process the nuclei fuse completely, forming an intermediate state ($^9$Be +$^{12}$C $\to$ $^{21}$Ne$^{*}$), which after a characteristic time, decays populating other open channels, including the entrance channel. For the latter case we have the compound elastic (CE), which for $^9$Be+$^{12}$C system is reflected in the process $^9$Be +$^{12}$C $\to$ $^{21}$Ne$^{*}$$\to$ $^9$Be +$^{12}$C. Residual Nucleus Level density parameter (MeV$^{-1}$)$^{a}$ Number of Discrete levels ------------------ -------------------------------------------- --------------------------- $^{20}$Ne 0.16 10 $^{20}$F 0.16 11 $^{13}$C 0.16 11 $^{18}$O 0.16 10 $^{17}$O 0.16 10 $^{12}$C 0.16 5 : Levels considered.[]{data-label="tabela3"} \ -------------------- $^{a}$ Ref. [@16]. -------------------- : Levels considered.[]{data-label="tabela3"} \ The calculation was performed using the Hauser-Feshbach STATIS code [@17]. The nuclear level density has been described by means of a level density expression given by Lang [@18], and the transmission coefficients were determined by an internal Fermi parametrization [@17]. The levels considered in this calculations are listed on table \[tabela3\] and are quite similar to those used in the Ref. [@16]. ![ Excitation function for Compound Nucleus Formation. The open circle points are to E$_{Lab}$ = 19.0 and 21.0 MeV. The calculated points, were normalized to agree with behavior of data from [@15] (Color Online).[]{data-label="secaototalcompound"}](secaototalcompound.eps) Finally the angular distributions are normalized in such a way that the total cross section agree with the results presented in the excitation function from Ref. [@15] (Figure \[secaototalcompound\]). When the obtained angular distributions are incoherently added to the $^{3}$He transfer results, we can see that the contribution of this reaction mechanisms is not relevant, even for our highest energies, as shown in Fig. \[total\] for the two highest energies E$_{Lab}$ = 19.0 and 21.0 MeV. ![Angular distributions for the $^9$Be + $^{12}$C system at E$_{Lab}$ = 19.0 and 21.0 MeV. The dot lines are the CE angular distributions from HF calculations. We can notice that the contribution of CE to differential cross section enhancement at backward angles is not significante (Color Online).[]{data-label="total"}](compound.eps) Conclusions =========== In this work we measured elastic scattering angular distributions for the $^9$Be+$^{12}$C light system at bombarding energies ranging from 13.0 MeV to 21.0 MeV. The double folding São Paulo potential was used in the analysis that was performed in four steps. In the first one we considered N$_{r}$ and N$_{i}$ as free parameters for the fits to the angular distributions at the forward angular region. The angular distributions calculated with the optical model have shown a good agreement with the 13.0 and 14.5 MeV experimental data. For 17.3, 19.0 and 21.0 MeV the agreement is reasonable at forward angles. However, the description at backward angles is not good, suggesting that the coupling to other mechanisms is important. In the second step of the analysis, no evidence of the coupling to inelastic channels was observed. Using the spectroscopic factors extracted from the literature for $^{12}$C=${^3}$He$\oplus$ ${^9}$Be, in the third step we took into account the ${^3}$He elastic transfer channel. For the ${^3}$He transfer we see a pronounced improvement in the description of the data for 17.3, 19.0 and 21.0 MeV, which suggests that this process is important at intermediate and backward angles. Finally in the fourth step, corresponding to the compound elastic calculation, the results suggest that this mechanism is not an important process at the energies studied in this work. The energy dependence of the N$_r$ parameter suggest the presence of the threshold anomaly. However, no strong conclusions could be made due to the constant value of N$_i$. Finally, to obtain information about the effects of the elastic transfer process and check the values of spectroscopic factors, it would be important to perform a more carefully analysis and measurements of the elastic scattering angular distributions at more backward angles. [99]{} M. Dasgupta, D. J. Hinde and K. Hagino, Nucl. Phys A 722, 196c (2003). F. A. Souza. et al., Nucl. Phys. A 821, 36 (2009). L.F. Canto, P.R.S. Gomes, R. Donangelo and M.S. Hussein, Phys. Rep. 424, 1 (2006). C. Beck, N. Keeley and A. Diaz-Torres, Phys. Rev. C 75, 054605 (2007). A. M. Moro and F. M. Nunes, Nucl. Phys A 767, 138 (2006). J. A. Tostevin, F. M. Nunes and I. J. Thompson, Phys. Rev. C 63, 024617 (2001). F. A. Souza, [*et. al.*]{}, Phys. Rev. C 75, 044601 (2007). J. Carter, [*et. al.*]{}, Nucl. Phys A 591, 349 (1995). A. T. Rudchick, [*et. al.*]{}, Nucl. Phys A 662, 44 (2000). W. von Oertzen and H. G. Bohlen, Phys. Rep. 19, 1 (1975). L.C. Chamon et al., Phys. Rev. C 66, 014610 (2002). M. M. de Moura [*et al*]{}., Nucl. Instr. and Meth. A 471, 368 (2001). I. J. Thompson, Comp. Phys. Rep. 7, 167 (1988). D. Pereira [*et al*]{}., Phys. Lett. B 670, 330 (2009). D. P. Souza [*et al*]{}., Nucl. Phys. A 836, 1 (2010). A. T. Rudchik [*et al*]{}., Nucl. Phys. A 660, 267 (1999). A. Barioni, [*et. al.*]{}, Phys. Rev. C 80, 034617 (2009). M. S. Hussein, P. R. S. Gomes, J. Lubian and L. C. Chamon, Phys. Rev. C 73, 044610 (2006). D. R. Tilley [*et al.*]{}, Nucl. Phys. A 745, 155 (2004). R. C. York and R. T. Carpenter, Nucl. Phys. A 282, 351 (1977). E. Ungricht [*et al*]{}., Nucl. Phys. A 313, 376 (1979). A. T. Rudchik [*et al*]{}., Nucl. Phys. A 667, 61 (2000). R. Cabezas, [*et al*]{}., Phys. Rev C 60, 067602 (1999). J. F. Mateja [*et al*]{}., Phys Rev C 20, 176 (1979). R. G. Stokstad, STATIS Program, Wright Nuclear Structure Laboratory, Yale University, INTERNAL REPORT 52 (1972). D. W. Lang, Nucl. Phys. 42, 353 (1963).

--- author: - | \ Institut für Theoretische Physik, Johann Wolfgang Goethe-Universität, Max-von-Laue-Str. 1, D-60438 Frankfurt am Main, Germany\ Frankfurt Institute for Advanced Studies, Ruth-Moufang-Str. 1, D-60438 Frankfurt am Main, Germany\ GSI Helmholtzzentrum für Schwerionenforschung GmbH, Planckstr. 1, 64291 Darmstadt, Germany\ E-mail: - | Jussi Auvinen\ Department of Physics, Duke University, POBOX 90305, Durham, NC 27707, USA\ E-mail: - | Jan Steinheimer\ Frankfurt Institute for Advanced Studies, Ruth-Moufang-Str. 1, D-60438 Frankfurt am Main, Germany\ E-mail: - | Marcus Bleicher\ Institut für Theoretische Physik, Johann Wolfgang Goethe-Universität, Max-von-Laue-Str. 1, D-60438 Frankfurt am Main, Germany\ Frankfurt Institute for Advanced Studies, Ruth-Moufang-Str. 1, D-60438 Frankfurt am Main, Germany\ E-mail: title: The beam energy dependence of collective flow in heavy ion collisions --- Introduction ============ In heavy ion collisions at ultra-relativistic energies nuclear matter can be studied under extreme conditions. At zero baryo-chemical potential and finite temperature, a cross-over between the hadron gas at low temperatures and the quark gluon plasma at high temperatures has been established by lattice QCD [@Borsanyi:2013bia; @Bazavov:2014pvz] and also been confirmed by experimental measurements at the Large Hadron Collider (LHC) and the Relativistic Heavy Ion Collider (RHIC) in combination with dynamical models [@Pratt:2015zsa]. At finite net baryon densities, current lattice calculations are not applicable and insights about the structure of the QCD phase diagram can only be achieved by experimental exploration in combination with detailed dynamical modeling and phenomenological approaches providing input on the equation of state in the whole temperature ($T$)-baryo-chemical potential ($\mu_B$) plane. The previous and currently running programs at CERN-SPS and the recent beam energy scan (BES) program at RHIC provide a comprehensive data set on general bulk observables and selected fluctuation/correlation measurements. In the future, the second generation of experiments with improved capabilities for rare probes will meticulously explore the exciting energy regime accessible at FAIR, NICA, and stage 2 of the BES@RHIC. The second goal, besides obtaining insight on the type of the phase transition of strongly interacting matter, is to understand the properties of the state of matter that is formed, the QGP. One of the most promising observables is the anisotropic flow, that is measured in terms of Fourier coefficients of the azimuthal distribution of produced particles. Directed flow $v_1$ and elliptic flow $v_2$ are analysed extensively to gain insights on the transport properties and the geometry of heavy ion collisions. Within the last 5 years the whole plethora of odd higher coefficients has been studied to investigate the initial state and its fluctuations in more detail. The flow coefficients - as indicated by the name - are a sign of collective behaviour and quantify the response of the system to spatial anisotropies, translated to momentum space by pressure gradients. To connect the final state particle distributions to the quantities of interest, e.g. the equation of state or the viscosity to entropy ratio of QCD matter, detailed dynamical models are necessary. The current state of the art is to use a combination of 3+1 dimensional (viscous) hydrodynamic evolution and hadronic transport for the non-equilibrium evolution in the late stages of the reaction. There are various ways to model/parametrize the initial state for these calculation ranging from Glauber/CGC type models, to classical Yang-Mills evolution, AdS/CFT or transport approaches. At lower beam energies the initial non-equilibrium evolution takes a non-negligible time and one needs to certainly pay attention to the type of initial dynamics that is required. In the present study, the calculations are performed within the Ultra-relativistic Quantum Molecular Dynamics (UrQMD) approach including a 3+1 dimensional ideal hydrodynamic evolution where appropriate. The main ingredients of the model are described in Section \[hybrid\_model\]. After that, Section \[flow\] contains a summary of the expectations for the beam energy dependence of anisotropic flow in heavy ion collisions and the three subsections \[elliptic\_flow\], \[tri\_flow\] and \[directed\_flow\] are devoted to $v_1$ to $v_3$ in more detail. The last Section \[summary\] summarizes the main conclusions. Hybrid Approach at non zero $\mu_B$ {#hybrid_model} =================================== Heavy ion collisions are processes that do not create a system at fixed temperature and density that can be studied over an extended period of time, but rather span a fluctuating finite region in the phase diagram evolving dynamically. Hybrid approaches, based on (viscous) hydrodynamics for the hot and dense stage of the evolution, and non-equilibrium transport for the dilute stages of the reaction are successfully applied to describe bulk observables at high RHIC and LHC energies. These approaches combine the advantages of hydrodynamics and transport and apply both approximations within their respective regions of validity. The equation of state and transport coefficients of hot and dense QCD matter are direct inputs to the hydrodynamic equations, which allows for a controlled modeling of the phase transition. Microscopic transport on the other hand describes all particles and their interactions and provides the full phase-space distribution of produced particles in the final state, which allows for an apples-to-apples comparison to experimental data. At higher beam energies the standard picture of the dynamical evolution of a heavy ion reaction consists of non-equilibrium initial state dynamics providing the initial state for the viscous hydrodynamics equations and a hadronic afterburner to account for the separation of chemical and kinetic freeze-out. At lower beam energies, it is not clear how close the produced system really gets to local equilibration and if it thermalizes in the whole phase-space. In general, it is expected that dissipative effects are larger for low beam energies and the hadronic interactions gain importance. In addition, the widely used assumption of boost invariance breaks down and a 3+1 dimensional calculation is inevitable. Also, the equation of state and the transport coefficients are required not only as a function of temperature, but in the whole $T-\mu_B$ - plane. The finite net baryon current needs to be conserved during the evolution. Going beyond the standard hybrid approach by exploring the non-equilibrium dynamics in the vicinity of a critical endpoint or a first order phase transition is the ultimate goal, where exploratory work has been performed so far [@Nahrgang:2011mg]. The dynamical approach that is used to calculate the results presented below is the Ultra-relativistic Quantum Molecular Dynamics (UrQMD) approach coupled to ideal relativistic fluid dynamics for the hot and dense stage (SHASTA) [@Bass:1998ca; @Bleicher:1999xi; @Petersen:2008dd]. This approach has recently been improved to include the finite shear viscosity during the hydrodynamic evolution as presented in another contribution to this conference and in [@Karpenko:2015xea]. The initial state is produced by generating nucleons according to Woods-Saxon distributions and computing dynamically the first binary interactions until the two nuclei have geometrically passed through each other. Event -by-event fluctuations of the positions of the nucleons and of the energy deposition per collision are naturally included. At $t_{\rm start} = \frac{2R}{\gamma v}$ the energy, momentum and net baryon density distributions are calculated by representing the individual particles with a three dimensional Gaussian distribution. Based on these initial conditions including fluctuating velocity profiles the ideal relativistic hydrodynamic equations are solved. There are different options for the equation of state, e.g. a bag model equation of state matched to a hadron gas at low temperatures to explore the sensitivity of observables to a strong first order phase transition (BM) or a more realistic equation of state provided by a chiral model that is fitted to lattice QCD at zero $\mu_B$, reproduces the nuclear ground state properties and incorporates constraints from neutron star properties. At a constant energy density a hypersurface is constructed and particles are sampled according to the Cooper-Frye formula. The hadronic rescattering and decays are treated by the hadronic transport approach. This approach incorporates most of the above mentioned ingredients that are necessary for a realistic dynamical description of heavy ion reactions at lower beam energies. The current strategy can be seen as a first step to take the well-established picture at high energies and explore how well it works at lower beam energies. Anisotropic Flow Observables {#flow} ============================ ![(Color online) Left: The calculated beam energy excitation function of elliptic flow of charged particles in Au+Au/Pb+Pb collisions in mid-central collisions (b=5-9 fm) with $|y|<0.1$(full line). This curve is compared to data from different experiments for mid-central collisions (see [@Petersen:2009vx] for refs). The dotted line in the low energy regime depicts UrQMD calculations with the mean field [@Li:2006ez]. Fig. from [@Petersen:2006vm]. Right: $v_2/\epsilon$ as a function of $(1/S)dN_{\rm ch}/dy$ for different energies and centralities for Pb+Pb/Au+Au collisions compared to data [@Alt:2003ab]. The results from mid-central collisions (b=5-9 fm) calculated within the hybrid model with different freeze-out transitions and different definitions of the eccentricity are shown by black lines. The green full lines correspond to the previously calculated hydrodynamic limits [@Kolb:2000sd]. Fig. taken from [@Petersen:2009vx].[]{data-label="figv2exc"}](v2exc_ch_withpot "fig:"){width="7cm"} ![(Color online) Left: The calculated beam energy excitation function of elliptic flow of charged particles in Au+Au/Pb+Pb collisions in mid-central collisions (b=5-9 fm) with $|y|<0.1$(full line). This curve is compared to data from different experiments for mid-central collisions (see [@Petersen:2009vx] for refs). The dotted line in the low energy regime depicts UrQMD calculations with the mean field [@Li:2006ez]. Fig. from [@Petersen:2006vm]. Right: $v_2/\epsilon$ as a function of $(1/S)dN_{\rm ch}/dy$ for different energies and centralities for Pb+Pb/Au+Au collisions compared to data [@Alt:2003ab]. The results from mid-central collisions (b=5-9 fm) calculated within the hybrid model with different freeze-out transitions and different definitions of the eccentricity are shown by black lines. The green full lines correspond to the previously calculated hydrodynamic limits [@Kolb:2000sd]. Fig. taken from [@Petersen:2009vx].[]{data-label="figv2exc"}](v2epsilon "fig:"){width="8cm"} Anisotropic flow is the collective response to geometrical structures in the initial state distribution of heavy ion collisions. If there are strong enough interactions spatial anisotropies are getting translated to momentum space throughout the evolution. Therefore, anisotropic flow is supposed to be very sensitive to the equation of state and to the transport properties of the produced matter. It is quantified by the Fourier coefficients of the azimuthal distribution of the produced particles $$v_n = \cos (n\phi -\Psi_n)$$ where $\Psi_n$ is the event plane and $n$ specifies the coefficient of interest. There are many different two-particle and many-particle methods to measure the anisotropic flow coefficients with different sensitivities to non-flow and fluctuations. In this article, either the event plane method (some $v_2$ results and $v_3$) or the theoretical definition with respect to the known reaction plane ($v_1 = \langle \frac{p_x}{p_x^2+p_y^2}\rangle$ and $v_2 = \langle \frac{p_x^2-p_y^2}{p_x^2+p_y^2}\rangle$) have been applied to calculate the flow observables. Elliptic flow is one of the main observables to support the claim that the quark gluon plasma behaves like a nearly perfect liquid. Therefore, the beam energy dependence of elliptic flow is supposed to be sensitive to the phase transition and the changing transport properties at lower collision energies. Triangular flow is the first odd flow component that is mainly sensitive to fluctuations and would be zero, if the initial state is taken as average over many events. Therefore, the hope is that measuring the beam energy dependence of triangular flow helps to disentangle the ’trivial’ initial state fluctuations from the interesting fluctuations that arise due to the critical dynamics. Last but not least directed flow is first of all a measure of the initial angular momentum in the heavy ion collision. The beam energy dependence was predicted to show a dip structure when a first order phase transition occurs during the evolution. In the following Sections, the beam energy dependence of each of these observables will be discussed in more detail. ![(Color online) Left: Fraction of elliptic flow generated during hydrodynamic evolution as a function of beam energy. Right: Magnitude of $v_2\{\textrm{EP}\}$ at the beginning of hydrodynamical evolution (squares), immediately after particlization (diamonds) and after the full simulation (circles) in midcentral collisions. Fig. from [@Auvinen:2013sba].[]{data-label="figv2phases"}](Gv2hydrofraction8294 "fig:"){width="6.5cm"} ![(Color online) Left: Fraction of elliptic flow generated during hydrodynamic evolution as a function of beam energy. Right: Magnitude of $v_2\{\textrm{EP}\}$ at the beginning of hydrodynamical evolution (squares), immediately after particlization (diamonds) and after the full simulation (circles) in midcentral collisions. Fig. from [@Auvinen:2013sba].[]{data-label="figv2phases"}](Bv2evob8294 "fig:"){width="6.5cm"} Elliptic flow - a measure of the perfect fluid {#elliptic_flow} ---------------------------------------------- ![(Color online) Left: Elliptic flow of pions in mid-central (b=5-9 fm) Pb+Pb collisions at $E_{\rm lab}=40A~$GeV and $E_{\rm lab}=160A~$GeV. The full and dashed black lines depict the hybrid model calculation, while the pure transport calculation is shown as the black dotted line. The colored symbols display experimental data obtained with different measurement methods by NA49 [@Alt:2003ab]. Fig. taken from [@Petersen:2009vx]. Right: Centrality dependence of elliptic flow of charged particles at midrapidity ($|y|<0.5$) for Pb+Pb collisions at $E_{\rm lab}=40A$ GeV (a) and $E_{\rm lab}=160A$ GeV (b). The horizontal lines indicate the results for averaged initial conditions using the hadron gas EoS (blue full line) and the bag model EoS (black dotted line) while the symbols (full circles for HG-EoS and open squares for BM-EoS) depict the results for the event-by-event calculation. Fig. taken from [@Petersen:2010md].[]{data-label="figv2_ptpi"}](v2_ptpi){width="7cm"} The second Fourier coefficient, the so called elliptic flow, is the response of the system to the initial almond shaped overlap region in non-central collisions. Due to fluctuations $v_2$ is also non-zero in central collisions, but it is mainly generated due to the overall shape deformation of the fireball. Only if the mean free path is small enough the initial spatial anisotropy is converted to the corresponding momentum space anisotropy. The beam energy dependence of elliptic flow is quite interesting. At very low energies, the spectator nuclei are blocking the interaction region and the so called ’squeeze-out’ leads to negative elliptic flow values with respect to the reaction plane. In this region the nuclear interactions are important and a hadron transport approach including mean fields can describe the elliptic flow rather well (see Fig. \[figv2exc\], left). At high beam energies the elliptic flow turns positive and grows as a function of beam energy. In the intermediate region around 10-40A GeV the hadron transport approach reaches the right “ball park” values while the underestimation at high beam energies is a sign for the importance of partonic interactions starting at around 160A GeV. Fig. \[figv2exc\] (right) shows the response function $v_2/\epsilon_2$ as a function of the charged particle density. The green lines indicate the expectations from ideal hydrodynamic calculations and that this line meets the experimental data at $\sqrt{s_{\rm NN}} = 200$ GeV was the basis for the claim that a perfect fluid has been created. The black lines show various UrQMD hybrid calculations to demonstrate that the generic behaviour of this curve can be reproduced once the non-equilibrium stages of the reaction are treated properly. To investigate how this energy dependence of elliptic flow is generated in the hybrid approach the contribution to the final integrated elliptic flow of charged particles has been calculated and the percentage is shown in Fig. \[figv2phases\] (left). At lower energies, the hydrodynamic evolution lasts only a few fm/c, therefore almost all the elliptic flow is generated by the hadronic transport approach. At higher energies the hydrodynamic evolution is necessary to reach the high elliptic flow values as discussed above. Fig. \[figv2phases\] (right) shows that the contribution from the late hadronic rescattering is constant at about 10 % throughout the whole energy regime investigated. Fig. \[figv2\_ptpi\] (left) shows the transverse momentum dependent elliptic flow of pions. NA49 measurements are compared to hybrid and pure hadronic transport calculations and again it is clearly visible that at 40 AGeV the hydrodynamic evolution does not play a role while at 160 AGeV the hydrodynamic evolution is crucial to reproduce the elliptic flow. The dependence on initial state fluctuations and the equation of state has been studied in Fig. \[figv2\_ptpi\] (right), where the centrality dependence of elliptic flow of charged particles is shown. There is no difference visible between the fluctuating events (symbols) and the averaged smooth initial conditions (horizontal lines) as expected for elliptic flow. The difference between the bag model equation of state with a strong first order phase transition and the hadron gas equation of state is not observable either. Triangular flow - a measure of fluctuations {#tri_flow} ------------------------------------------- Since 2010 event by event fluctuations in the initial state have been studied as the source of higher odd anisotropic flow coefficients [@Adare:2012kf; @Luzum:2013yya]. These odd coefficients average to zero without event by event fluctuations. To demonstrate that $v_3$ is really sensitive to fluctuations the beam energy dependence of triangular flow of charged particles is compared to the dynamical fluctuations of elliptic flow $\sigma_{v_2}$ in Fig. \[fig\_v3\] (left). In central collisions both have roughly the same value whereas in non-central collisions they have a similar magnitude only at high energies. At lower beam energies triangular flow decreases. ![(Color online) Left: $v_3\{\textrm{EP}\}$ compared with initial state fluctuations’ contribution to $v_2$, $\sigma_{v2}=\sqrt{\frac{1}{2}(v_2\{\textrm{EP}\}^2-v_2\{\textrm{RP}\}^2)}$ (squares). Right: Scaled flow coefficients $v_2/ \langle \epsilon_2 \rangle$ and $v_3/ \langle \epsilon_3 \rangle$ with respect to the average total hydro duration for impact parameter ranges $b = 0-3.4$ fm, $6.7-8.2$ fm and $8.2-9.4$ fm. Figs taken from [@Auvinen:2013sba].[]{data-label="fig_v3"}](Bv2fluctuations "fig:"){width="6.5cm"} ![(Color online) Left: $v_3\{\textrm{EP}\}$ compared with initial state fluctuations’ contribution to $v_2$, $\sigma_{v2}=\sqrt{\frac{1}{2}(v_2\{\textrm{EP}\}^2-v_2\{\textrm{RP}\}^2)}$ (squares). Right: Scaled flow coefficients $v_2/ \langle \epsilon_2 \rangle$ and $v_3/ \langle \epsilon_3 \rangle$ with respect to the average total hydro duration for impact parameter ranges $b = 0-3.4$ fm, $6.7-8.2$ fm and $8.2-9.4$ fm. Figs taken from [@Auvinen:2013sba].[]{data-label="fig_v3"}](Bscaledvnvshydrot "fig:"){width="6.5cm"} If non-zero triangular flow is measured at lower beam energies and it is generated by initial spatial fluctuations it serves as a nice probe to disentangle these ’trivial’ fluctuations from more interesting fluctuations due to the phase transition or critical endpoint. This option relies of course on the fact that a finite triangular flow needs to be present. As Fig. \[fig\_v3\] (right) shows, $v_3$ scales for different energies and centralities with the duration of the hydrodynamic evolution in the hybrid approach. The circles show that for elliptic flow the influence of the hadronic non-equilibrium evolution is much larger. Therefore, triangular flow is much more sensitive to a finite viscosity and might even vanish at lower beam energies. On the positive side, this disappearance of triangular flow can be seen as a more direct evidence that the perfect fluid QGP does not persist for a long enough time. Directed flow - a measure of the phase transition {#directed_flow} ------------------------------------------------- Last but not least, let us come to one more anisotropic flow observable. The directed flow measures the stream of particles within the reaction plane and has opposite sign in the forward and backward hemisphere. Nowadays, a rapidity-even $v_1$ dipole component has also been observed, but here we concentrate on the traditional rapidity odd measure. To plot the beam energy dependence one extracts the slope of directed flow at midrapidity. ![\[Color online\] Left: Slope of $v_1$ of protons and pions around mid rapidity extracted from the ideal 1-fluid calculations with a bag model and crossover EoS. For particle production we applied a Cooper-Frye prescription on a iso-chronous hypersurface. Right: Slope of $v_1$ of protons and pions around mid rapidity extracted from the ideal 1-fluid calculations with a bag model and crossover EoS. For particle production we applied a Cooper-Frye prescription on a iso-energy density hypersurface. Figs. taken from [@Steinheimer:2014pfa]. []{data-label="fig_fluid"}](dv1dy_hydro_IC "fig:"){width="6.5cm"} ![\[Color online\] Left: Slope of $v_1$ of protons and pions around mid rapidity extracted from the ideal 1-fluid calculations with a bag model and crossover EoS. For particle production we applied a Cooper-Frye prescription on a iso-chronous hypersurface. Right: Slope of $v_1$ of protons and pions around mid rapidity extracted from the ideal 1-fluid calculations with a bag model and crossover EoS. For particle production we applied a Cooper-Frye prescription on a iso-energy density hypersurface. Figs. taken from [@Steinheimer:2014pfa]. []{data-label="fig_fluid"}](dv1dy_hydro_IT "fig:"){width="6.5cm"} Based on the AGS and NA49 measurements there have been predicitions within hydrodynamic calculations that the slope of the directed flow of protons has a dip as a function of beam energy, signaling the phase transition to the quark gluon plasma [@Brachmann:1999xt; @Csernai:1999nf]. In light of the new STAR measurements from the RHIC beam energy scan program [@Adamczyk:2014ipa], this old observable has been recalculated with modern theoretical techniques. In Fig. \[fig\_fluid\] a one fluid calculation has been performed. This resembles the previous predictions enhanced by actual particle sampling and a $v_1$ calculation that can be compared to experimental measurements. On the left hand side, the rather unphysical scenario of a constant time freeze-out is shown, whereas the right hand side corresponds to a more realistic iso-energy density transition scenario. The two figures demonstrate, that the freeze-out dynamics has a significant effect on the structures in the excitation function of the slope of the directed flow of protons and pions. The large difference between the two different scenarios - with and without first order phase transition - disappears. ![\[Color online\] Left: Comparison of proton $v_1(y)$ for the different model applied. Right: Slope of $v_1$ of protons and anti-protons around mid rapidity extracted from the hybrid model calculations with a bag model and crossover EoS. We compare with standard UrQMD and experimental data.Figs. taken from [@Steinheimer:2014pfa]. []{data-label="fig_v1proton"}](protonv1 "fig:"){width="6.5cm"} ![\[Color online\] Left: Comparison of proton $v_1(y)$ for the different model applied. Right: Slope of $v_1$ of protons and anti-protons around mid rapidity extracted from the hybrid model calculations with a bag model and crossover EoS. We compare with standard UrQMD and experimental data.Figs. taken from [@Steinheimer:2014pfa]. []{data-label="fig_v1proton"}](dv1dy_hybrid_proton "fig:"){width="6.5cm"} Figs. \[fig\_v1proton\] and \[fig\_v1pion\] both show on the left hand side the directed flow as a function of rapidity at two different beam energies and on the right hand side the slope around mid rapidity that has been extracted to display the full beam energy dependence. Especially for the proton directed flow it is clear that the signal is very small around zero. Even though all the different calculations capture the positive slope it is obvious that pure one fluid calculations lead to too much directed flow, while surprisingly the pure hadron transport shows the best agreement with experimental data at this point. ![\[Color online\] Left: Comparison of pion $v_1(y)$ for the different model applied. Right: Slope of $v_1$ of negatively charged pions around mid rapidity extracted from the hybrid model calculations with a bag model and crossover EoS. We compare with standard UrQMD and experimental data.Figs. taken from [@Steinheimer:2014pfa]. []{data-label="fig_v1pion"}](pionv1 "fig:"){width="6.5cm"} ![\[Color online\] Left: Comparison of pion $v_1(y)$ for the different model applied. Right: Slope of $v_1$ of negatively charged pions around mid rapidity extracted from the hybrid model calculations with a bag model and crossover EoS. We compare with standard UrQMD and experimental data.Figs. taken from [@Steinheimer:2014pfa]. []{data-label="fig_v1pion"}](dv1dy_hybrid_pion "fig:"){width="6.5cm"} For pions the negative slope arises because of a shadowing effect. The newly produced particles are blocked by the spectators and therefore the slope of directed flow is negative at low beam energies. Again the fluid calculation overestimates the flow at low beam energies and for pions the hybrid UrQMD approach seems to describe the measurements rather well. More important than a quantitative comparison to experimental data is the conclusion that the hybrid calculations with different assumptions about the order of the phase transition do not show any sensitivity on these observables. Detailed measurements of the centrality dependence of directed flow are going to allow for better conclusions on the stopping mechanism, the nuclear mean fields and the equation of state that are all crucial to understand the dynamics of heavy ion collisions at low beam energies. Summary and Conclusions {#summary} ======================= Collective flow is one of the main observables to understand the properties of the quark gluon plasma and the dynamics of heavy ion reactions. Especially the beam energy dependence of anisotropic flow provides high potential for insights on the equation of state and the dependence of transport coefficients on the baryo-chemical potential. To connect the final state observables to the quantities of interest, hybrid approaches based on relativistic hydrodynamics and non-equilibrium transport provide a realistic description of the dynamics. The main conclusions are that the small change of elliptic flow as a function of the beam energy can be explained by the transport dynamics that gains importance at lower beam energies and replaces the diminished hydrodynamic evolution. Triangular flow is much more sensitive to the viscosity and cannot be generated by hadronic transport alone. The beam energy dependence of the slope of the directed flow around midrapidity has been studied and even though the old predictions are reproduced, there is no difference between a modern hybrid calculation with and without a strong first order phase transition. To reach final conclusions on this subject, more detailed studies including nuclear potentials and an improved treatment of the baryon stopping are necessary. Acknowledgements ================ H. P. acknowledges funding of a Helmholtz Young Investigator Group VH-NG-822 from the Helmholtz Association and GSI. This work was supported by the Helmholtz International Center for the Facility for Antiproton and Ion Research (HIC for FAIR) within the framework of the Landes-Offensive zur Entwicklung Wissenschaftlich-Oekonomischer Exzellenz (LOEWE) program launched by the State of Hesse. Computational resources have been provided by the Center for Scientific Computing (CSC) at the Goethe-University of Frankfurt. [99]{} S. Borsanyi, Z. Fodor, C. Hoelbling, S. D. Katz, S. Krieg and K. K. Szabo, Phys. Lett. B [**730**]{} (2014) 99. A. Bazavov [*et al.*]{} \[HotQCD Collaboration\], Phys. Rev. D [**90**]{} (2014) 9, 094503. S. Pratt, E. Sangaline, P. Sorensen and H. Wang, arXiv:1501.04042 \[nucl-th\]. M. Nahrgang, S. Leupold, C. Herold and M. Bleicher, Phys. Rev. C [**84**]{} (2011) 024912. S. A. Bass, M. Belkacem, M. Bleicher, M. Brandstetter, L. Bravina, C. Ernst, L. Gerland and M. Hofmann [*et al.*]{}, Prog. Part. Nucl. Phys.  [**41**]{} (1998) 255 \[Prog. Part. Nucl. Phys.  [**41**]{} (1998) 225\]. M. Bleicher, E. Zabrodin, C. Spieles, S. A. Bass, C. Ernst, S. Soff, L. Bravina and M. Belkacem [*et al.*]{}, J. Phys. G [**25**]{} (1999) 1859. H. Petersen, J. Steinheimer, G. Burau, M. Bleicher and H. Stocker, Phys. Rev. C [**78**]{} (2008) 044901. I. A. Karpenko, P. Huovinen, H. Petersen and M. Bleicher, arXiv:1502.01978 \[nucl-th\]. Q. f. Li, Z. x. Li, S. Soff, M. Bleicher and H. Stoecker, J. Phys. G [**32**]{} (2006) 407. H. Petersen, Q. Li, X. Zhu and M. Bleicher, Phys. Rev. C [**74**]{} (2006) 064908. P. F. Kolb, J. Sollfrank and U. W. Heinz, Phys. Rev. C [**62**]{} (2000) 054909. H. Petersen and M. Bleicher, Phys. Rev. C [**79**]{} (2009) 054904. J. Auvinen and H. Petersen, Phys. Rev. C [**88**]{} (2013) 6, 064908. C. Alt [*et al.*]{} \[NA49 Collaboration\], Phys. Rev. C [**68**]{} (2003) 034903. H. Petersen and M. Bleicher, Phys. Rev. C [**81**]{} (2010) 044906. A. Adare, M. Luzum and H. Petersen, Phys. Scripta [**87**]{} (2013) 048001 \[Phys. Scripta [**04**]{} (2013) 048001\]. M. Luzum and H. Petersen, J. Phys. G [**41**]{} (2014) 063102. J. Steinheimer, J. Auvinen, H. Petersen, M. Bleicher and H. Stöcker, Phys. Rev. C [**89**]{} (2014) 5, 054913. J. Brachmann, S. Soff, A. Dumitru, H. Stoecker, J. A. Maruhn, W. Greiner, L. V. Bravina and D. H. Rischke, Phys. Rev. C [**61**]{} (2000) 024909. L. P. Csernai and D. Rohrich, Phys. Lett. B [**458**]{} (1999) 454. L. Adamczyk [*et al.*]{} \[STAR Collaboration\], Phys. Rev. Lett.  [**112**]{} (2014) 16, 162301.